WALL-CROSSING FOR TILT STABILITY By Nicholas Rekuski A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mathematics – Doctor of Philosophy 2022 ABSTRACT WALL-CROSSING FOR TILT STABILITY By Nicholas Rekuski Recent techniques from Bridgeland Stability have given new, interesting results about stable vector bundles on surfaces. However, in higher dimensions the theory is substantially harder, so there has been significantly less progress in this case. In this thesis, we develop theory for a related notion—tilt stability— then apply this theory to stable vector bundles. In the first part of this thesis, we recall and further develop the theory of tilt stability. This development culminates in a wall-crossing result for tilt stability. In the second part of this thesis, we apply our wall-crossing result to study stable vector bundles. Our first application is a criterion for when the restriction of a slope stable bundle to an integral subvariety is still slope stable. Our second application is to the theory of Lazarsfeld-Mukai bundles. Specifically, we show the Lazarsfeld-Mukai bundle associated to a Gieseker stable bundle is slope stable, and that slope stability and slope semistability are equivalent for Lazarsfeld-Mukai bundles associated to ample line bundles. This thesis is dedicated to the educators who have helped me get here. iii ACKNOWLEDGEMENTS I am thankful for everyone that has supported me throughout my graduate school career. I am thankful for my friends. To my graduate school friends Armstrong, Dan, Ioan- nis, Joshua, Michalis, and Mike; thank you for sharing this experience with me. To my pre-graduate school friends Davis, Owen, Sam, and Tim; thank you for sharing non-math graduate school experiences with me. I am thankful for my cohort. To my advisor Rajesh Kulkarni; thank you for both your mentorship and friendship. To my research group Mike, Yizhen, Shitan, and Shen; thank you for inspiring me. To my guidance committee Aaron Levin, George Pappas, Igor Rapinchuk, and Misha Shapiro; thank you for sharing your knowledge and time. I am thankful for my family. To my mom, thank you for listening to me complain about graduate school. I am thankful for my partner. To Kiegan, thank you for being my best friend and biggest cheerleader. You always brighten my day. The author was partially supported by the NSF grant DNS-2101761 during preparation of this thesis. iv TABLE OF CONTENTS LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi KEY TO SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Notation and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 CHAPTER 2 STABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 Stability for Coherent Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Stability Functions on an Abelian Category . . . . . . . . . . . . . . . . . . 22 2.3 Additional Properties of Very Weak Stability Functions . . . . . . . . . . . . 40 2.4 Stability Conditions on a Triangulated Category . . . . . . . . . . . . . . . . 62 CHAPTER 3 TILT STABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.1 Tilt Stability as a Very Weak Stability Function . . . . . . . . . . . . . . . . 71 3.2 Tilt Stability is a Very Weak Stability Condition for Rational β . . . . . . . 84 3.3 Tilt Stability is a Very Weak Stability Condition for Real Beta . . . . . . . . 101 3.4 Walls for Tilt Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.5 Bounding Actual Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.6 A Wall-Crossing Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 CHAPTER 4 APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.1 A Generalization of Bogomolov’s Restriction Theorem . . . . . . . . . . . . . 127 4.2 Stability of Lazarsfeld-Mukai Sheaves . . . . . . . . . . . . . . . . . . . . . . 130 CHAPTER 5 FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.1 Slope Stable Bundles on Rational Varieties . . . . . . . . . . . . . . . . . . . 139 5.2 Lazarsfeld-Mukai Bundles on Other Varieties . . . . . . . . . . . . . . . . . . 140 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 v LIST OF FIGURES Figure 1 Polyhedral on the Left: HN(A) is polyhedral on the left while HN(B) is not. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Figure 2 The Harder-Narasimhan Polygon: Position of Z(B) relative to Z(Ai ) and Z(Ai+1 ). Note the complex plane is rotated 90 degrees. In this orientation, the slope of the line between two points agrees with the slope of the quotient. In other words, with this orientation the slope of Z(B)Z(Ai ) is µσ (B/Ai ). . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Figure 3 Shapes and Orientation of Walls: The solid lines represent numerical walls for fixed E ∈ Db (X) satisfying rank(E) ̸= 0 and ∆H (E) ≥ 0. The dashed lines represent the α and β axes in R × R>0 . . . . . . . . . . . . 115 Figure 4 Bound on the Largest Actual Wall: The solid lines represent actual walls of E [1] when E is torsion-free. The horizontal dashed line represents our bound on the largest actual wall. The shaded region is the large volume limit. The actual walls of E is the same picture mirrored over β = µD H (E ) expect that weak σα,β -stability of E is equivalent to (H, D)- tilt twisted stability of E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 vi KEY TO SYMBOLS X A smooth projective variety H An ample divisor on X E , F , G Coherent sheaves on X Db (X) The bounded derived category of coherent sheaves. E, F, G Objects in the bounded derived category tilt σα,β The tilt stability associated to (β, α) ∈ R × R>0 CohD+βHH (X) The heart associated to tilt stability Y A integral subscheme of X µH (·) The slope ∆H (·) The discriminant associated to H chi (·) the i-th Chern character ci (·) the i-th Chern class chD i (·) the i-th D-twisted Chern character M A Lazarsfeld-Mukai sheaf H i Cohomology functors H i : Db (X) → Coh(X) A An abelian category A, B, C Objects in A D An R-divisor on X σ A very weak stability function Ak (X) The Chow ring of X GD H (·)(t) The reduced D-twisted Hilbert polynomial K0 (·) The Grothendieck group I(·) The imaginary part of a complex number R(·) The real part of a comple number Λ A finitely generated quotient of K0 (X) vii HN(·) The Harder-Narasimhan polygon. Q A quadratic form (T , F) A torsion pair WStab(X, Λ) The space of very weak stability conditions on X with respect to Λ W A numerical wall in the (H, D)-slice TX The tangent bundle on X viii CHAPTER 1 INTRODUCTION A natural way to study a variety is to consider the collection of all vector bundles on it. For example, the Jacobian—the collection of all line bundles of degree 0—is our most important tool for studying curves. However, in higher dimensions and higher ranks, the collection of all vector bundles is not well-behaved (i.e. there is never a coarse moduli space). For this reason, we consider the smaller class of stable bundles. Unlike vector bundles, there is a coarse moduli space of stable bundles. Furthermore, every vector bundle can be decomposed in an essentially unique way to stable bundles. For this reason, it suffices to study the moduli of stable bundles. While there exists a moduli of stable bundles, very basic properties about this moduli are unknown. For example, even for surfaces, it is unknown exactly when the moduli of stable bundles (with fixed topological invariants) is non-empty. That is to say, it is difficult to construct stable bundles with fixed topological invariants. To illustrate, it is completely open whether there exists a rank 2 stable bundle on P7 —projective space of dimension 7. In other words, for the most “basic” projective 7-fold, we do not know whether a certain moduli of stable bundles is non-empty. A much easier problem is to construct semistable bundles with fixed topological invari- ants. Semistable bundles are essentially bundles that are in the closure of the moduli of stable bundes. It is easier to construct semistable bundles because many categorical constructions with semistable bundles are still semistable (e.g. extensions, kernels, tensor products, exte- rior products): Lemma 1.0.1 (Lemma 2.2.15). Assume X is a smooth projective variety with ample class H. Consider a short exact sequence 0 → F → E → G → 0. 1 If F and G are slope semistable of the same slope then E is also slope semistable. In particular, if L is a line bundle on P7 then L ⊕ L is a semistable bundle of rank 2 on P7 . These categorical constructions almost never extend to stable bundles. To wit, if 0→F →E →G →0 is a short exact sequence with F and G stable of the same slope then E is never stable. On the other hand, in the setting of Bridgeland stability many such categorical con- structions do hold for stable objects. Bridgeland stability conditions are generalizations of slope stability from coherent sheaves to the bounded derived category, Db (X), due to Bridgeland [Bri07]. That is to say, Bridgeland stability conditions are notions of stability on certain abelian subcategories of the bounded derived category, called hearts, that satisfy a generalization of Harder-Narasimhan filtrations and Bogomolov’s inequality. While Bridgeland stability conditions do not have an associated Geometric Invariant Theory problem (in contrast to slope stability) they afford better deformation properties. Specifically, Bridgeland stability conditions are parameterized by a complex manifold. More- over, for fixed object E, this complex manifold has a wall and chamber structure such that Bridgeland stability of E only changes when crossing a wall. Each of these walls corresponds to a short exact sequence 0→F →E→G→0 in an abelian subcategory of Db (X). In practice, we do not work with this whole complex manifold; instead, we consider certain 2-dimensional slices parameterized by R × R>0 . In other words, for each (β, α) ∈ R × R>0 there exists a Bridgeland stability condition σα,βtilt on a heart CohD+βH H (X). Furthermore, for fixed object, the wall and chamber decomposition of these slice is very constrained—the walls must either be a unique vertical line or nested semicircles whose center lies on the horizontal axis. 2 In these slices, categorical constructions can often be deformed to give stable objects. One such construction is due to Bayer and Macrı̀ which we call a wall-crossing result. Theorem 1.0.2 ([BM11, Lemma 5.9]). Assume X is a smooth projective surface. Let F → E → G → F [1] tilt be a distinguished triangle such that F, G are σα,β -stable of the same slope for some (β, α) ∈ R × R>0 . If E ̸= F ⊕ G then there exists an infinitesimal deformation (β ′ , α′ ) of (β, α) such tilt that E is σα,β -stable. In other words, an extension of Bridgeland stable objects of the same slope is stable up to infinitesimal deformation. Many exciting results have used this construction to generalize previously known results about stable bundles. For example, [Bay18] generalizes Lazarsfeld’s Brill-Noether theorem on surfaces while [Kop20] generalizes Bogomolov’s restriction theorem. However, most such re- sults have been only surfaces. This is, in part, because it is very difficult to construct Bridge- land stable conditions on higher dimensional varieties. Furthermore, even when Bridgeland stability conditions exist in higher dimensions, the wall and chamber structure is much more complicated. To circumvent these problems in higher dimensions, we consider very weak stability conditions. Very weak stability conditions are weaker than Bridgeland stability conditions but are known to exist in higher dimensions. However, very weak stability conditions lose many of the properties we are accustomed to from Bridgeland stability (e.g. existence of Jordan-Hölder filtrations, uniqueness of Harder-Narasimhan filtrations, and Schur’s Lemma all fail). Like Bridgeland stability, there are 2-dimensional families of very weak stability conditions parameterized by R × R>0 . Furthermore, the wall and chamber structure is constrained like in the case of Bridgeland stability conditions. Since very weak conditions lose many of the properties of Bridgeland stability conditions, Bayer and Macrı̀’s wall-crossing argument does not hold. There have been some been a few 3 wall-crossing results for very weak stability, but they do not fully generalize their result. For example, [Sch20, Theorem 6.1.4] gives a wall-crossing result for P3 and uses it to describe the two components of the Hilbert scheme of twisted cubics in P3 in more detail. Similarly, Feyzbakhsh [Fey21, Proposition 4.2 and Corollary 4.3] gives a wall-crossing result for torsion objects and uses it to generalize Bogomolov’s restriction theorem to higher dimensions. These applications suggest that a general wall-crossing result will be a useful tool for moduli problems in higher dimensions. In this thesis we give such a result: Theorem 1.0.3 (Theorem 3.6.1). Assume X is a smooth projective variety. Let F → E → G⊕r → F [1] tilt be a distinguished triangle such that F and G are weakly σα,β -stable of the same slope. If G has good quotients and HomDb (X) (G, E) = 0 then there exists an infinitesimal deformation (β ′ , α′ ) of (β, α) such that E is weakly σαtilt ′ ,β ′ -stable. tilt Weak σα,β -stability is a slight variant of stability that only occurs for very weak stabil- ity conditions. Having good quotients is a notion that is vacuous for Bridgeland stability conditions and is equivalent to H −1 (G) = 0 in our case. We give some applications of our wall-crossing result to slope stable sheaves. The first is a generalization of Bogomolov’s restriction theorem to any integral subvariety. To explain, if E is slope stable, then E |H may not be slope stable. However, Mehta and Ramanathan showed that E |aH is slope stable for a ≫ 0 [HL10, Theorem 7.2.8]. In the case of surfaces in characteristic 0, Bogomolov gave an explicit lower bound on a [Bog93]. In a different thread, Kopper generalized Bogomolov’s theorem to restrictions of integral curves (rather than just hyperplanes) [Kop20, Theorem 3.3]. We generalize Kopper’s result to higher dimensions. Theorem 1.0.4 (Theorem 4.1.2). Let E be a reflexive, slope stable sheaf on X. Consider an integral subvariety ι : Y → X. If the following bounds are satisfied 4 ∆H (E ) 1 H n−2 · ch1 (E ) · Y H n−2 · Y 2 ˆ µH (E ) − − > − 2 2 rank(E )2 rank(E )H n−1 · Y 2H n−1 · Y ∆H (E (−Y )) 1 H n−2 · ch1 (E ) · Y H n−2 · Y 2 ˆ µH (E (−Y )) + + < − 2 2 rank(E )2 rank(E )H n−1 · Y 2H n−1 · Y then E |Y is slope stable. We also give two applications to Lazarsfeld-Mukai bundles. If we have a surjection E → F between sufficiently positive stable bundles then it is natural to wonder whether the kernel is stable. Lazarsfeld-Mukai bundles are the most basic instance of this scenario. If E is globally generated then there exists a natural surjection H 0 (E ) ⊗ OX → E → 0 with kernel M which is called the Lazarsfeld-Mukai bundle assocaited to E . On curves, stability of Lazarsfeld-Mukai bundles is well understood. In higher dimensions, much less is known. We list a some particularly noteworthy results: ˆ On Pn , the Lazarsfeld-Mukai bundle associated to OPn (d) for d ≥ 1 is slope semistable [Fle84, Corollary 2.2]. ˆ Assume X is a smooth curve of genus g. Let E be a slope semistable bundle on X with µH (E ) ≥ 2g + 1. The Lazarsfeld-Mukai bundle associated to E is slope stable [But94, Theorem 1.2]. ˆ Assume X is a K3 surface. Let L be a globally generated, ample line bundle on X The Lazarsfeld-Mukai bundle associated to L is slope stable [Cam12, Theorem 1]. If X is an abelian surface and L additionally satisfies L 2 ≥ 14 then the same conclusion holds [Cam12, Theorem 2]. ˆ Assume X be a smooth projective surface. Let L be a globally generated line bundle on X. If degH (L ) ≫ 0 then the Lazarsfeld-Mukai bundle associated to L is slope stable [ELM13, Theorem A]. 5 ˆ Assume X is a smooth projective variety of Picard rank 1. Let L be an ample line bundle on X. If degH (L ) ≫ 0 then the associated Lazarsfeld-Mukai bundle is slope stable [ELM13, Proposition C]. Our first result is for Del Pezzo surfaces: Theorem 1.0.5 (Theorem 4.2.4). Assume X is a smooth Del Pezzo surface over an alge- braically closed field of arbitrary characteristic. For ease of notation, let H = −KX which is ample by definition. Consider a globally generated, slope stable, torsion-free sheaf E with associated Lazarsfeld-Mukai sheaf M . If the following bounds are satisfied: ˆ 0 < degH (E ) ≤ KX 2 (h0 (E ) − rank(E )), ˆ ch2 (E ) > 0, ch2 (E ) 1 ˆ 2 + ≥ ∆H (E ) degH (E ) rank(E )2 then M is µH -stable. In fact, we show that this result holds with a weaker notion than slope stability. This is the first such result for higher rank sheaves on surfaces. Our second result shows slope stability and semistability are equivalent for Lazarsfeld- Mukai bundles associated to ample bundles: Theorem 1.0.6 (Theorem 4.2.8). Assume X is a smooth projectiive variety equipped with ample divisor H. Let M be the Lazarsfeld-Mukai bundles associated to OX (d) for d ≥ 1. M is slope stable if and only if M is slope semistable. 1.1 Notation and Assumptions Assume X is a regular projective variety over an algebraically closed field k. Unless stated otherwise, we assume dim(X) = n ≥ 2 and k is of characteristic 0. We denote an integral ample divisor on X by H. 6 We denote the bounded derived category of Coh(X) by Db (X) whose objects will be written as E, F, G. We write H i for the cohomology functors H i : Db (X) → Coh(X). Coherent sheaves will be written as E , F , G . Objects of a general abelian category A will be writen as A, B, C. If E ∈ Coh(X) then we will use the notation codim(E ) to denote codim(Supp(E ), X). If E ∈ Db (X), then we will write codim(E) to denote ! [ codim Supp(H i (E)), X . i∈Z 7 CHAPTER 2 STABILITY In this chapter, we provide background on stability for Coh(X), abelian categories, and triangulated categories. The chapter is as follows: §2.1 We introduce three notions of stability for coherent sheaves; µH -stability, (H, D)- Gieseker stability, and (H, D)-twisted stability. All the results and definitions of this section are standard. §2.2 We introduce very weak stability functions. A very weak stability function is essentially a generalization of µH -stability to any abelian category (not just Coh(X)). Most basic results about µH -stability have equivalent results for a very weak stability functions. Unfortunately, this means that some of the “failures” of µH -stability have correspond- ing “failures” for very weak stability functions. For example, Schur’s lemma is false and Harder-Narasimhan filtrations are not unique. To circumvent these failures, we introduce two classes of stable objects: σ-stable objects and weakly σ-stable objects. Weakly σ-stable objects are the natural generalization of µH -stable sheaves to a very weak stability function. σ-stability is a stronger notion of stability that is a generalization of simple objects in the full subcategory of Coh(X) generated by µH -semistable objects of fixed slope. Both of these notions are standard, but they are both called σ-stability in the literature. σ-stability is a partial fix for the failures mentioned above. Namely, Schur’s lemma holds as expected and Harder- Narasimhan filtrations are unique for σ-stable objects. We compare σ-stability and weak σ-stability in Lemma 2.2.11. We also define objects having good quotients and σ-pure objects. Having good quo- tients exactly determines when weakly σ-stability and σ-stability agree. σ-purity is a 8 generalization of a pure coherent sheaf. §2.3 We discuss additional properties of very weak stability functions including the Harder- Narasimhan property, the support property, and the existence/non-existence of Jordan- Hölder filtrations. For a very weak stability function, a σ-semistable object may not contain a σ-stable subobject. Therefore, σ-semistable objects may not have Jordan-Hölder filtrations. Fortunately, we can show that under relatively mild assumptions all objects of finite slope have what we call a “weak Jordan-Hölder filtration” which is a sufficient replace- ment for usual Jordan-Hölder filtrations. §2.4 We discuss very weak stability conditions on a triangulated category. The results and definitions of this subsection are standard. 2.1 Stability for Coherent Sheaves In this section we introduce three different notions of a stable coherent sheaf. Two of these definitions are well known - µH -stability (Definition 2.1.4) and (H, D)-Gieseker stability (Definition 2.1.6). The third notion is a newer notion due to Bridgeland called (H, D)- twisted stability (Definition 2.1.7). All three of these notions satisfy the seesaw inequality (Lemma 2.1.9). The strength of these three definitions is compared in Lemma 2.1.11. We end this section by stating the weak Bogomolov inequality (Lemma 2.1.15) and some cases where it holds in positive characteristic (Remark 2.1.16). Definition 2.1.1. Let E ∈ Coh(X), E ∈ Db (X), and D be a R-divisor. 1. We define the D-twisted Chern character chD (E ) = ch(E ) · exp(−D) where ch(E ) denotes the usual Chern character and exp(−D) is formally a power series in −D both viewed in the Chow ring with real coefficients A• (X) ⊗ R. 9 The first few terms of the D-twisted Chern character written in terms of Chern classes are chD (E ) = ch(E ) · exp(D)   1 2 = rank(E ) + c1 (E ) + (c1 (E ) − 2c2 (E )) + · · · 2 2 D3   D · 1−D+ − + ··· 2 6 = rank(E ) + (c1 (E ) − rank(E )D) D2   1 2 + (c1 (E ) − 2c2 (E )) − c1 (E ) · D + rank(E ) + ··· 2 2 where ci (E ) is the usual i-th Chern class of E . 2. We denote the k-th coefficient of chD (E ) by chD k (E ) which we view as an element of Ak (X) ⊗ R. 3. We define X −1 chD D k (E) = chk (· · · → E → E0 → E1 → · · · ) = (−1) chD i k (H (E)). i∈Z 4. If D = 0 we write chD k (E) = chk (E) which is the k-th coefficient of the usual Chern character ch(E). 5. For a nonnegative integer k and E ∈ Db (X), we set ⊕k H n−k · chD n D ≤k (E) = (H · rank (E), H n−1 · chD1 (E), . . . , H n−k · chD k (E)) ∈ R . 6. For ease of notation, we set degD H (E) = H n−1 · chD1 (E). Since X is an integral smooth variety over an algebraically closed field, by the Hirzebruch- Riemann-Roch theorem, degD H (E) and rank(E) agree with all usual definitions of the degree and rank. Furthermore, for the same reason, there is a group isomorphism An (X) = Z which we can make canonical by choosing the isomorphism where H n is positive. Therefore, we will identify objects in An (X) with the corresponding integer. 10 Example 2.1.2. If E ∈ Db (X) and ck (E) denotes the kth Chern class (which is defined on Db (X) inductively using the Chern character) then ˆ chD0 (E) = rank(E). ˆ chD1 (E) = ch1 (E) − D · rank(E) = c1 (E) − D rank(E). D2 ˆ chD2 (E) = ch2 (E) − D · ch1 (E) + 2 · rank(E) D2 = 21 (c1 (E)2 − 2c2 (E)) − D · c1 (E) + 2 rank(E). The vanishing of high codimension Chern characters is controlled by the support of the coherent sheaf. This lemma is well known, but the author could not find it written in the following level of generality. However, by a series of reductions, the following result can be reduced to a result about vector bundles whose support is smooth, irreducible, and quasi-projective in which case we can apply [Ful98, Example 18.2.1 with a similar argument argument as Example 15.3.1]. These reductions are technically difficult, so we give a different proof. Lemma 2.1.3. Assume E ∈ Coh(X). Let {Yi }m i=1 be the top dimensional irreducible com- ponents of Supp(E ). If codim(E ) = d then   0 :  0≤k ≤d−1 D chk (E ) = .  Pm  i=1 rank(E |Yi )[Yi ] : k = d  ≤d (E ) = (0, 0, . . . , +) if and only if E is supported in codimension In particular, H n−d · chD d. Proof. Define Y = Supp(E ) with the annihilator scheme structure (i.e. OY = K er(OX → E nd(E ))). For ease of notation, set U = X \ Y with induced subscheme structure and ι : U → X the associated open immersion. Since X is regular, there is a finite locally free resolution in Coh(X): 0 → Fn → Fn−1 → · · · → F1 → E → 0. 11 By construction, ι∗ E = 0, so we have an exact sequence of locally free sheaves in Coh(U ): 0 → ι∗ Fn → ι∗ Fn−1 → · · · → ι∗ F1 → 0. Since ι : U → X is an open immersion (a fortiori flat), ι∗ is exact, so, by additivity of the Chern character, X n Xn 0= (−1)k ch(ι∗ Fk ) = ι∗ (−1)k ch(Fk ) = −ι∗ ch(E ). k=1 k=1 In other words, ι∗ chk (E ) = 0 for all k. By excision, for all k we have the following exact sequence of groups: ι∗ Ak (Y ) ⊗ Q → Ak (X) ⊗ Q − → Ak (U ) ⊗ Q → 0. Since ι∗ chk (E ) = 0, by exactness, chk (E ) is in the image of the inclusion Ak (Y ) ⊗ Q → Ak (X) ⊗ Q. If 0 ≤ k ≤ d − 1, since codim(Y, X) = d, Ak (Y ) = 0, so chk (E ) = 0, as needed. If k = d then Ad (Y ) is the free abelian group generated by the top dimensional irreducible d Pm components, {Yi }m i=1 , of Y . Since chd (E ) is in the image of A (Y ) ⊗ Q, chd (E ) = i=1 ai [Yi ] for rationals ai . We claim ai = rank(E |Yi ). Let Vi0 be the largest dense open subscheme of Yi0 such that E |Vi0 is free (which exists by generic flatness). We have the following exact sequence Ad (Yi0 \ Vi0 ) → Ad (Yi0 ) → Ad (Vi0 ) → 0. Since dim(Yi0 \ Vi0 ) ≤ dim(Yi0 ), it follows that Ad (Yi0 \ Vi0 ) = 0. In other words, restriction gives an isomorphism Ad (Yi0 ) → Ad (Vi0 ). Thus, without loss of generality, we may assume E is free along Yi0 . Now, consider the closed immersion j : Yi0 → X which induces a morphism j ∗ : Ad (X) → A0 (Yi0 ). Since E is locally free along Yi0 , we find that Xm ∗ ∗ rank(E |Yi0 ) = rank(E |Yi0 ) = j chd (E ) = j ai [Yi ] = ai0 [Yi0 ]. i=1 12 It follows that Xm chd (E ) = rank(E |Yi )[Yi ], i=1 as desired. Last, since lower Chern characters vanish, by definition, chD k (E ) = chk (E ) for all 0 ≤ k ≤ d and R-divisors D. We recall the definition of µH -stability: Definition 2.1.4. Assume E ∈ Coh(X), E ∈ Db (X), and D is a R-divisor. 1. We define  D  degH (E) :   rank(E) rank(E) ̸= 0 µDH (E) =  +∞ :  rank(E) = 0 which we call the (H, D)-twisted slope of E. degH (E) 2. For ease of notation, we also set µH (E) = µ0H (E) = rank(E) which we call the usual slope of E. 3. We say that E ∈ Coh(X) is µH -(semi)stable if every proper nonzero subsheaf 0 → F → E satisfies µD D H (F ) (≤) µH (E /F ). The notation (≤) is short hand for “< (resp. ≤).” Formally, we say that +∞ = +∞ and any real number is strictly less than +∞. We maintain this convention throughout this write-up. Our definition of µH -stability differs slightly from the standard definition (e.g. [HL10, Definition 1.2.12]). If E is not torsion-free then Definition 2.1.4 is weaker than this standard notion. If E is torsion-free then our notion agrees with the standard notion (see Lemma 2.2.11 in view of Example 2.2.4). We use Definition 2.1.4 because it allows us to compare stability of all coherent sheaves using the same slope function - rather than using a different slope function for torsion sheaves. 13 Remark 2.1.5. We omit the divisor D from the notation of µH -stability because D does not affect stability. Specifically, if E is supported everywhere, we find H n−1 · chD1 (E ) µDH (E ) = rank(E ) n−1 H · ch1 (E ) − H n−1 · D rank(E ) = rank(E ) = µH (E ) − H n−1 · D Note that second summand is independent of the sheaf E . On the other hand, if E is torsion, then µDH (E ) = +∞ = µH (E ). D Therefore, we find that µD H (F ) (≤) µH (E ) if and only if µH (F ) (≤) µH (E ). µH -stability is also called slope stability, Mumford stability, and Mumford-Takemoto stability. Out of all notions of stability on Coh(X), µH -stability is the best behaved with respect to standard sheaf operations (e.g. twisting by line bundle, pullbacks, restriction to subvarieties, symmetric products, etc.). A weaker notion than µH -stabiliy is Gieseker stability. In fact, we discuss a slight gener- alization of Gieseker stability due to Matsuki and Wentworth [MW97, Definition 3.2] called (H, D)-Gieseker stability. Definition 2.1.6. Assume E is torsion-free and D is a R-divisor. 1. We define the reduced twisted Hilbert polynomial of E to be   χ(E ⊗OX (tH+D)) : rank(E ) ̸= 0   D rank(E ) GH (E )(t) =  +∞ :  rank(E ) = 0 where the Euler characteristic is defined formally as a polynomial in t via Riemann- Roch (see Remark 2.1.10 for the first few terms). Note that OX (tH + D) is an abuse of notation. Namely, since D is a R-divisor, OX (tH +D) may not be a line bundle. However, we can still define χ(E ⊗OX (tH +D)) formally via the Grothendieck-Riemann-Roch theorem. 14 2. We say E is (H, D)-Gieseker (semi)stable if every nonzero proper subsheaf F ⊆ E satisfies GD D H (F )(t) (≤) GH (E )(t) for all t ≫ 0. 3. We say E is H-Gieseker (semi)stable if it is (H, 0)-Gieseker (semi)stable. Unlike µH -stability, H-Gieseker stability is not invariant under twists by a line bundle. (H, D)-Gieseker stability partially accounts for this failure. Namely, for an integral divisor D, E (D) is H-Gieseker (semi)stable if and only if E is (H, D)-Gieseker (semi)stable. If Pic(X) = 1 then (H, D)-Gieseker (semi)stability and H-Gieseker stability are equivalent. Our last notion of stability on Coh(X) is a generalization of (H, D)-Gieseker stability due to Bridgeland ([Bri08, Definition 14.1]) which is called (H, D)-twisted stability. In short, twisted stability is weaker than µH -stability but stronger than Gieseker stability (Lemma 2.1.11). On surfaces, twisted (semi)stability and Gieseker (semi)stability are equivalent Lemma 2.1.11. Notation varies for (H, D)-twisted stability. Bridgeland originally called the notion twisted stable with respect to the pair (D, H). Some authors call this definition polyno- mial stability because it arises as a polynomial stability condition in the sense of [Bay09]. Others call it limit stability because it naturally arises in the large volume limit (see Lemma 3.2.3). Definition 2.1.7. Assume E ∈ Coh(X) and E ∈ Db (X). 1. Following [Bri08, Definition 14.1] we define  n−2 D  H ·ch2 (E) : rank(E) ̸= 0   D rank(E) νH (E) = .  +∞ :  rank(E) = 0 2. We say E ∈ Coh(X) is (H, D)-twisted (semi)stable if every proper nonzero subsheaf 0 → F → E satisfies either ˆ µD D H (F ) < µH (E /F ) or 15 ˆ µD D D D H (F ) = µH (E /F ) with νH (F ) (≤) νH (E /F ). The functions µD D D H , νH , and GH (t) all satisfy Rudakov’s seesaw inequality ([Rud97, Defi- nition 1.1]): In a short exact sequence of torsion-free coherent sheaves then the slopes must be monontically ordered with respect to the sequence. In the next subsection we work with a general abelian category, rather than just coherent sheaves, so we will prove the seesaw inequality in that generality. To this end, we recall the Grothendieck group of an abelian category. Definition 2.1.8. Assume A is an abelian category. We define the Grothendieck group of A, denoted K0 (A), to be the free abelian generated by objects in A with relations [A] = [B] + [C] whenever there is a short exact sequence 0 → B → A → C → 0. We will write K0 (X) = K0 (Coh(X)). One can define the Grothendieck group of a triangulated category similarly (Definition 2.4.3). We now state and prove Rudakov’s seesaw inequality following [Rud97, Lemma 3.2]. Lemma 2.1.9 (Seesaw Inequality). Assume A is an abelian category and d, r : K0 (X) → R are group homomorphisms. For ease of notation, for each A ∈ A set µ(A) = d(A)/r(A). If 0 → B → A → C → 0 is a short exact sequence in A such that r(A), r(B), r(C) ̸= 0 then one of the following inequalities must hold: 1. µ(B) < µ(A) < µ(C), 2. µ(B) > µ(A) > µ(C), or 3. µ(B) = µ(A) = µ(C). In particular, if 0 → F → E → G → 0 is a short exact sequence of coherent sheaves supported on X then one of the above inequalities must hold for µD D D H (·), νH (·), and GH (·)(t) for t ≫ 0. 16 Proof. Consider a short exact sequence 0 → B → A → C → 0 in A. Note that   1 r(B) d(B) µ(A) − µ(B) = det  , r(A)r(B) r(A) d(A) so µ(A) > µ(B) (respectively =, <) if and only if the determinant of the above matrix is positive (respectively zero, negative). Since r, d : K0 (A) → R are group homomorphisms (i.e. r and d are additive in short exact sequences), we can rewrite this determinant as     r(B) d(B) r(B) d(B) det   = det     r(A) d(A) r(B) + r(C) d(B) + d(C)   r(B) d(B) = det   r(C) d(C)   r(B) + r(C) d(B) + d(C) = det   r(C) d(C)   r(A) d(A) = det  . r(C) d(C) Therefore, we find that µ(A) > µ(B) (respectively =, <) if and only if µ(C) > µ(A) (respectively =, <), as needed. If we work in the generality of Lemma 2.1.9 and allow r(·) = 0 then we have no control over the ordering of the slopes. However, by restricting the possible values of r and d in a natural way (e.g. generalizing Lemma 2.1.3), than we can obtain a version of the seesaw inequality that holds for objects with r(A) = 0 (see Lemma 2.2.6). On a different note, the first few terms of reduced twisted Hilbert polynomial can be written solely in terms of µD H , νH , and invariants of X. This will allow us to compare µH -stability, (H, D)-Gieseker stability, and (H, D)-twisted stability. Remark 2.1.10. If E is a coherent sheaf that is supported everywhere then the first few D+KX /2 D+KX /2 terms of GD H (E ) can be written solely in terms of µH (E ), νH (E ) and invariants 17 of X. Specifically, by the Hirzebruch-Riemann-Roch theorem Z 1 GD H (E )(t) = ch(E (tH + D)) · td(TX ) rank(E ) X n ! n !  i c1 c21 + c2 c1 c2 Z  1 X D X (tH) = ch (E ) · · 1+ + + + ··· rank(E ) X i=0 i i=0 i! 2 12 24 rank(E ) · H n n (chD 1 (E ) + rank(E ) · c1 /2) · H n−1 = t + tn−1 rank(E )n! rank(E )(n − 1)! D (chD 2 2 (E ) + ch1 (E ) · c1 /2 + rank(E ) · (c1 + c2 )/12) · H n−2 + tn−2 + · · · rank(E )(n − 2)! H n tn D+KX /2 tn−1 = + µH (E ) n! (n − 1)! 2 ) · H n−2   n−2 D+KX /2 (c2 − 2KX t + νH (E ) + + ··· 12 (n − 2)! Using this calculation of the reduced twisted Hilbert polynomial we can obtain implica- tions among µH -stability, Gieseker stability, and twisted stability. Lemma 2.1.11. Assume E ∈ Coh(X) and D is a R-divisor. Each of the following state- ments implies the next a. E is µH -stable b. E is (H, D + KX 2 )-twisted stable. c. E is (H, D)-Gieseker stable. d. E is (H, D)-Gieseker semistable. e. E is (H, D + KX 2 )-twisted semistable. f. E is µH -semistable. Furthermore, if dim(X) = 2 then e ⇒ d and c ⇒ b. Proof. The implications a ⇒ b ⇒ c ⇒ d ⇒ e ⇒ f all follow from definition and Remark 2.1.10. 18 If dim(X) = 2 then H 2 t2 2 ) · H n−2   D+KX /2 D+KX /2 (c2 − 2KX GDH (t) = + µH (E )t + νH (E ) + 2 1 so the implications e ⇒ d and c ⇒ b immediately follow. We end this subsection by defining the twisted discriminant and noting a weak form of Bogomolov’s Inequality. Definition 2.1.12. Assume E ∈ Db (X) and D is a R-divisor. 1. We define the discriminant of E as ∆D (E) = chD 2 D 1 (E) − 2 rank(E ) ch2 (E). which is an element of A2 (X) ⊗ R. 2. We define the (H, D)-discriminant of E as D ∆H (E) = degD 2 H (E) − 2 rank(E )H n−2 · chD 2 (E). which is a real number. D 3. If E ∈ Coh(X) then we define ∆D (E ) and ∆H (E ) by viewing E as a chain complex supported in degree 0 in Db (X). D The bar notation in the (H, D)-discriminant is meant to signify that ∆H (E) is a real number. The following calculations allows us to compare these two variants of the discriminant. Lemma 2.1.13. Assume D is a R-divisor. 1. If E ∈ Db (X) then the disciminant ∆(E) is independent of the R-divisor D (as the notation suggests). D 2. If E ∈ Coh(X) then ∆H (E ) ≥ H n−2 · ∆(E ). 19 Proof. Assume D is a R-divisor. 1. By definition, D ∆(E) = chD 2 1 (E) − 2 rank(E) ch2 (E) D2 = (ch1 (E) − D rank(E))2 − 2 rank(E)(ch2 (E) − D · ch1 (E) + rank(E)) 2 = ch1 (E)2 − 2 rank(E) · ch2 (E), as claimed. 2. By part 1, it suffices to show degD 2 H (E) ≥ H n−2 · chD 2 1 (E) , but this inequality holds by the Hodge Index Theorem [Laz04, Corollary 1.6.3 (i)]. We note [Laz04, Corollary 1.6.3 (i)] requires that both H and chD 1 (E) are nef, but the argument from [Laz04, Corollary 1.6.3 (i)] holds if chD1 (E) is not nef and H is ample. D We note the analogous statment to Lemma 2.1.13.1 for ∆H is false. In other words, the (H, D)-discriminant truly depends on D: Example 2.1.14. Fix a very ample class H on X, set D = αH, and let E ∈ Db (X). By Example 2.1.2 we can rewrite D αH ∆H (E) = ∆H (E) = degH (E)2 − 2H n−2 · ch2 (E) rank(E) + 2 degH (E) rank(E)H n α + H n H n · rank(E)2 − rank(E)3 α2  D Therefore, we can always find an object E ∈ Db (X) such that ∆H depends on D, as claimed. Lemma 2.1.15 (Weak Bogomolov Inequality). Assume char(k) = 0. If E is torsion-free D and µH -semistable then ∆H (E ) ≥ 0. 20 Proof. By Lemma 2.1.13, it suffices to show H n−2 · ch1 (E )2 − 2 rank(E )H n−2 · ch2 (E )) ≥ 0. With this in mind, by Example 2.1.2, we can simplify the above expression as H n−2 · ch1 (E )2 − 2 rank(E )H n−2 · ch2 (E ) = H n−2 · c1 (E )2 − rank(E )(H n−2 · c1 (E )2 − 2H n−2 · c2 (E )) = 2 rank(E )H n−2 · c2 (E ) − (rank(E ) − 1)H n−2 · c1 (E )2 which is non-negative by the usual Bogomolov inequality [HL10, Theorem 7.3.1], as desired. Remark 2.1.16 (Stability in Positive Characteristic). All results in this subsection except the weak Bogomolov inequality (Lemma 2.1.15) remain valid over algebraically closed fields of positive characteristic. We provide a sketch a that the weak Bogomolov inequality fails over Raynaud surfaces. Using the same argument as [Gie79], one can show that if the weak Bogomolov inequality holds on X then Kodaira vanishing also holds. By [Ray78] for any algebraically closed field of positive characteristic, there is a smooth surface where Kodaira vanishing fails. Therefore, the weak Bogomolov inequality also fails for such surfaces. Surprisingly, the weak Bogomolov inequality holds for every surface that is neither quasi- elliptic nor of general type [Lan16, Theorem 7.1]. This result result can be extended to higher dimensions by Mehta-Ramanathan’s Restriction Theorem [HL10, Theorem 7.2.1] and Lemma 2.1.13 to give the following result valid in positive characteristic: Assume X is a smooth projective variety of dimension at least 2 (with very ample class H) over an algebraically closed field of positive characteristic p. Let H1 , H2 , . . . , Hn−2 be general hyperplanes in the linear system of H and set Y = H1 ∩ H2 ∩ · · · ∩ Hn−2 . Also, assume the Kodaiira dimension of Y is at 21 most 1 and Y is not quasi-elliptic. If E is torsion-free and µH -semistable then D ∆H (E ) ≥ 0. 2.2 Stability Functions on an Abelian Category In this section we introduce stability functions and very weak stability functions on any abelian category A (Definition 2.2.1). For a very weak stability function, we introduce two different notions of stable objects (Definition 2.2.3). For a stability function these two notions are equivalent (Lemma 2.2.11 in view of Lemma 2.2.8). The rest of this subsection consists of generalizing knnown results about stability func- tions to very weak stability functions. For example, we have variants of the seesaw inequality (Lemma 2.2.6), Schur’s Lemma (Lemma 2.2.13), and the extension of semistable objects of the same slope is semistable (Lemma 2.2.15). Broadly, stability functions and very weak stability functions are two generalizations of µH -stability to a general abelian category A. Stability functions have a theory similar to µH -stability on curves. Very weak stability functions have a theory similar to µH -stability on higher dimensional varieties. One goal of this subsection is to understand exactly which results hold for stability functions that do not hold for very weak stability functions. Stability functions are a variant of Rudakov’s stability structures [Rud97, Definition 1.1] due to Bridgeland [Bri07, Definition 2.1]. Very weak stability functions are a newer notion implicit in Bayer, Macrı̀, and Stellari [BMS16, Definition B.1] which are a variant of Toda’s weak stability functions [Tod10, Definition 2.7]. Definition 2.2.1. Assume A is an abelian category. Fix a finitely generated free abelian group Λ and a group homomorphism K0 (A) → Λ. 1. A stability function on A is a group homomorphism Z : Λ → C satisfying the positivity property: For every A ∈ A, IZ(A) ≥ 0 and if IZ(A) = 0 then RZ(A) < 0. 22 Note that Z(E) is an abuse of notation which denotes the following construction. If E ∈ A, there is a corresponding element in K0 (A) and so in Λ. The image of this element in Λ under Z is what we call Z(E). 2. A very weak stability function on A, denoted (Z : Λ → C, A) is a group homomorphism Z : Λ → C satisfying the weak positivity property: For every A ∈ A, IZ(A) ≥ 0 and if IZ(A) = 0 then RZ(A) ≤ 0. 3. We will denote a very weak stability function on A as a pair σ = (Z : Λ → C, A). If Λ is clear from context, we will just write σ = (Z, A). The positivity and weak positivity properties should be thought of as a generalization of Lemma 2.1.3. Even though it is not suggested by the notation (Z : Λ → C, A), a very weak stability function also depends on the group homomorphism K0 (X) → Λ. Since Z : Λ → C is a group homomorphism and there is a group homomorphism K0 (A) → Λ, Z is additive in short exact sequences of objects in A. Therefore, IZ and RZ are also additive in short exact sequences of objects in A where IZ(E) and RZ(E) denote the imaginary and real part of Z(E) respectively. Definition 2.2.2. Assume σ = (Z, A) is a very weak stability function and A ∈ A. We define the slope of A as  − RZ(A) : IZ(A) ̸= 0   IZ(A) µσ (A) =  +∞ :  IZ(A) = 0 where IZ(A) and RZ(A) denote the imaginary and real parts of Z(A) respectively. We have defined the slope of a very weak stability function, so we can essentially copy Definition 2.1.4 to obtain a notion of stability. We will call this notion weak σ-stability. However, there is also a stronger notion which we call σ-stability. 23 The definitions of σ-stability and weak σ-stability appear in [Bri07, Definition 2.2] and [Tod10, Definition 2.9] respectively. However, in both articles the definition is called σ-stable. For this reason, we have introduced the title of weak σ-stability. Definition 2.2.3. Assume σ = (Z, A) is a very weak stability function and A ∈ A is nonzero. 1. We say that A is σ-(semi)stable if every nonzero proper subobject 0 → B → A in A satisfies µσ (B) (≤) µσ (A). 2. We say that A is weakly σ-(semi)stable if every nonzero proper subobject 0 → B → A in A satisfies µσ (B) (≤) µσ (A/B). We consider some examples and non-examples of very weak stability functions. Example 2.2.4. Consider the function ZµDH : K0 (Coh(X)) → C defined by ZµDH (E ) = √ − degD H (E ) + −1 rank(E ). By Lemma 2.1.3 ZµDD is a very weak stability function when H dim(X) ≥ 1 and a stability function exactly when dim(X) = 1. Moreover, IZµDH (E ) = rank(E ) and −RZµDH (E ) = degD D H (E ), so µσµD = µH . For this H reason, if (Z, A) is a very weak stability function then IZ is called the generalized rank while −RZ is called the generalized degree. Weakly σ-(semi)stable objects are exactly µH - (semi)stable objects. In contrast, we will see in Example 2.2.9 that σ-stable objects are exactly skyscraper sheaves. We denote this very weak stability function as σµDH = (ZµDH , Coh(X)) and call it the very weak stability function associated to µH -stability. A slightly more general example arises from the generalization of µH -stability to coherent sheaves supported on a variety of codimension at least d [HL10, Definition and Corollary 1.6.9]: 24 Example 2.2.5. Consider the full abelian subcategory Cohd (X) of Coh(X) generated by coherent sheaves supported in codimension greater than or equal to d. We will define a very weak stability function on Cohd (X). We define the very weak stability function σd,µDH = (Zd,µDH : K0 (Cohd (X)) → C, Cohd (X)) where √ Zd,µDH (E ) = −H d+1 · chD d+1 (E ) + −1H d · chD d (E ). Note that σ0,µDH = σµDH from Example 2.2.4. Moreover, using a similar argument to that example, we find that σd,µDH is a very weak stability function whenever 0 ≤ d ≤ n − 1 = dim(X) − 1. For now, these are our only examples of very weak stability functions. In the next section, we will see a class of very weak stability functions called tilt stability. For technical reasons, (H, D)-Gieseker and (H, D)-twisted stability are not very weak stability functions. To explain, (H, D)-twisted stability involves three different topological invariants (rank, degDH , and H n−2 · chD 2 ) which cannot be written as a group homomorphism K0 (X) → C. Instead, the putative group homomorphism should be defined via K0 (X) → R⊕3 or K0 (X) → C[t]. Similarly, (H, D)-Gieseker stability involves dim(X) different topological invariants. In this case, the putative group homomorphism should be defined as a group homomorphism K0 (X) → R⊕ dim(X) or K0 (X) → C[t]. Allowing for group homomorphisms K0 (X) → R⊕k is essentially the definition of To- da’s weak stability function [Tod10, Definition 2.11]. Allowing for group homomorphisms K0 (X) → C[t] is essentially the definition of Bayer’s polynomial stability [Bay09, Definition 2.3.1]. νH is also a non-example of a stability function. For example, any sheaf supported in codimension 1 with negative second Chern character shows that νH does not satisfy the required positivity property. 25 The remainder of this subsection is about generalizing results about µH -stability to a very weak stability function or noting when the generalization fails. Our first such result is a variant of the seesaw inequality (Lemma 2.1.9) that holds for objects with generalized rank 0. This result can also be thought of as a refinement of a variant of [Tod10, Comments after Definition 2.7]. A weaker form of this inequality appears in [BM11, Comments before Remark 3.1.1]. Lemma 2.2.6 (Weak Seesaw Inequality). Let σ = (Z : Λ → C, A) be a very weak stability function. If 0 → B → A → C → 0 is exact in A then one of the following inequalities must hold: ˆ µσ (B) = µσ (A) = µσ (C) ˆ µσ (B) < µσ (A) < µσ (C) ˆ µσ (B) > µσ (A) > µσ (C) ˆ µσ (B) > µσ (A) = µσ (C) with Z(B) = 0 ˆ µσ (B) = µσ (A) < µσ (C) with Z(C) = 0 Proof. If IZ(B), IZ(A), IZ(C) ̸= 0 then this result follows by Lemma 2.1.9. Therefore, we may assume that at least one of IZ(B), IZ(A), IZ(C) is zero. If two of these values is zero, then by additivity of Z, the third must also be zero. In this case, we find that µσ (B) = µσ (A) = µσ (C) = +∞, as needed. Therefore, we may further assume that exactly one of IZ(B), IZ(A), IZ(C) is zero. If IZ(A) = 0, by additivity and nonnegativity of IZ, IZ(B) = IZ(C) = 0 which is the case above. If IZ(B) = 0, by definition of a very weak stability function, RZ(B) ≤ 0. Therefore, by additivity of Z, IZ(C) = IZ(A) and RZ(C) ≤ RZ(A) (with equality exactly when RZ(B) = 0). Since IZ ≥ 0, it follows that µσ (C) ≤ µσ (A) (with equality exactly when RZ(B) = 0). Therefore, we have shown that either µσ (B) = +∞ > µσ (A) > µσ (C) or 26 µσ (B) = +∞ > µσ (A) = µσ (C) with Z(A) = 0, as needed. Note that µσ (A) < +∞ for IZ(A) ̸= 0. If IZ(C) = 0 then a similar argument to the IZ(B) = 0 case holds. We now introduce necessary and sufficient conditions (see Proposition 2.2.12 and Lemma 2.2.11) where σ-stability and weak σ-stability agree. Definition 2.2.7. Assume σ = (Z, A) is a very weak stability function. We say that A has good quotients (with respect to σ) if every nonzero subobject 0 → B → A satisfies Z(A/B) ̸= 0. In particular, if A has good quotients then Z(A) ̸= 0. In fact, σ = (Z, A) is a stability function if and only if every object in A has good quotients. Similarly, if an object A does not have good quotients then it is not σ-stable. In particular, weakly σ-stable objects that are not σ-stable never have good quotients (Propo- sition 2.2.12). Lemma 2.2.8. Assume σ = (Z, A) is a very weak stability function. 1. σ is a stability function if and only if every object in A has good quotients. 2. Assume A ∈ A is σ-stable then A has good quotients. Proof. Assume σ = (Z, A) is a very weak stability function. 1. Assume σ is a stability function. Therefore, by definition, every nonzero object A ∈ A satisfies Z(A) ̸= 0. In particular, every proper subobject 0 → B → A satisfies Z(A/B) ̸= 0, as needed. Conversely, assume that σ is not a stability function. Therefore, we can find a nonzero object A ∈ A such that Z(A) = 0. Consider the short exact sequence 0 → A → A⊕2 → A → 0. Since 0 → A → A⊕2 is a proper nonzero subobject and Z(A⊕2 /A) = Z(A) = 0, 27 A⊕2 does not have good quotients. Thus, we have shown that there are objects of A that do not have good quotients, as needed. 2. Assume A ∈ A is σ-stable. Therefore, we know that every nonzero proper subobject 0 → B → A satisfies µσ (B) < µσ (A). By the generalized seesaw inequality, it follows that Z(A/B) ̸= 0, as desired. In general, for a very weak stability function we should not expect objects to have good quotients. For instance, with respect to µH -stability, the only objects with good quotients on X when dim(X) ≥ 2 are skyscraper sheaves. Moreover, this example shows that most objects in Coh(X) do not contain a nonzero σ-stable subobject. Example 2.2.9. Assume dim(X) ≥ 2 and consider the very weak stability condition σµDH associated to µD H . We will show σ-stable objects are exactly skyscraper sheaves. With this in mind, consider a nonzero coherent sheaf E on X, and a closed point x ∈ X with associated short exact sequence 0 → Ix → OX → ι∗ Ox → 0. We obtain a surjection E → E ⊗ ι∗ Ox → 0 which induces the following short exact sequence in Coh(X): 0 → C → E → E ⊗ ι∗ Ox → 0 By definition of the twisted Chern character, we find that H n−1 · chD≤1 (E ⊗ ι∗ Ox ) = (H · chD ≤1 (ι∗ Ox )) = (rank(E ) · 0, degH (E ) · 0) · chD n−1 n−1 D ≤1 (E ))(H for H n−1 · chD1 (ι∗ Ox ) = (0, 0) by Lemma 2.1.3. By additivity of ZµD H , it follows that ZµDH (E ⊗ ι∗ Ox ) = ZµDH (E ). In other words, if the surjection E → E ⊗ Ix is not an isomorphism then E does not have good quotients. 28 If E → E ⊗ ι∗ Ox is an isomorphism then E is supported in dimension 0. In other words, E ∼= Ox⊕k for some k ≥ 0. In this case, E has good quotients if and only if k = 0, 1. In all, we have shown that E ∈ Coh(X) has good quotients if and only if E is a skyscraper sheaf or 0. By Lemma 2.2.8, it follows that σ-stable objects are exactly skyscraper sheaves. Example 2.2.9 may suggest that if σ is a very weak stability function then objects with good quotients are exactly simple objects and 0—this is not the case! For example, in Lemma 3.1.13 we show that if E ∈ Coh(X) then E [1] has good quotients with respect to tilt stability. We also introduce a notion which is a generalization of a pure coherent sheaf which we call σ-purity. This is a necessary condition for the following lemma, but, more importantly, we will eventually see that σ-pure objects have Jordan-Hölder filtrations (Lemma 2.3.14). Definition 2.2.10. Assume σ = (Z, A) is a nonnegative stability condition. We say that a nonzero object E ∈ A is σ-pure if for every nonzero subobject 0 → F → E satisfies IZ(F ) ̸= 0. In particular, if A is σ-pure then IZ(A) ̸= 0. We now compare our two notions of stability plus a third that is similar to the usual definition of µH -stability (e.g. [HL10, Definition 1.2.12]). The author has not seen the following implications explicitly written together, but it is certain that most are well known to experts. For example, [BMT14, Remark 3.1.1] is essentially part 2 of the following lemma while [PT19, Remark 2.2] notes that the implication f ⇒ d is strict in general. Lemma 2.2.11. Assume σ = (Z, A) is a very weak stability function and let A ∈ A. Consider the following statements a. For every subobject 0 → B → A in A satisfying 0 < IZ(B) < IZ(A), we have µσ (B) ≤ µσ (A). b. For every subobject 0 → B → A in A satisfying 0 < IZ(B) < IZ(A), we have µσ (B) < µσ (A). 29 c. A is weakly σ-semistable. d. A is weakly σ-stable. e. A is σ-semistable. f. A is σ-stable. The following results hold: 1. There are implications a c e b d f 2. If A is σ-pure then b ⇒ d and a ⇒ c. 3. If A has good quotients with respect to σ then d ⇒ f . Proof. Assume σ = (Z, A) is a very weak stability function and A ∈ A. 1. f ⇒ e: This follows from definition. d ⇒ c: This follows from definition. b ⇒ a: This follows from definition. f ⇒ d: Assume A is σ-stable and assume 0 → B → A is a proper nonzero subobject. Since A is σ-stable, µσ (B) < µσ (A). By the generalized seesaw inequality, it follows that µσ (B) < µσ (A) < µσ (A/B), as needed. e ⇒ c: A similar argument to f ⇒ d holds. d ⇒ b: Assume A is weakly σ-stable assume 0 → B → A is a proper nonzero subobject satisfying 0 < IZ(B) < IZ(A). Since A is weakly σ-stable, µσ (B) < µσ (A/B). Moreover, since IZ(A/B) = IZ(A) − IZ(B) > 0, we know that Z(A/B) ̸= 0. Since µσ (B) < µσ (A/B) and Z(A/B) ̸= 0, by the seesaw inequality, µσ (B) < µσ (A), as needed. 30 c ⇒ a: A similar argument to d ⇒ b holds. c ⇒ e: Assume A is weakly σ-semistable and let 0 → B → A be a nonzero proper subobject. Since A is weakly σ-semistable, we know that µσ (B) ≤ µσ (A/B). By the generalized seesaw inequality, it follows that µσ (B) ≤ µσ (A), as needed. 2. Assume A is σ-pure. b ⇒ d: Assume that for every subobject 0 → B → A in A satisfying 0 < IZ(B) < IZ(A) we have µσ (B) < µσ (A). Consider a nonzero proper subobject 0 → B → A. By assumption, IZ(B) ̸= 0. If 0 < IZ(B) < IZ(A) then µσ (B) < µσ (A) by assumption. By the generalized seesaw inequality it follows that µσ (B) < µσ (A/B), as needed. If IZ(B) = IZ(A) then we find that IZ(A/B) = 0. Since IZ(B) = IZ(A) ̸= 0, µσ (B) < +∞ = µσ (A/B), as needed. a ⇒ c: A similar argument to b ⇒ d holds. 3. Assume A has good quotients with respect to σ. d ⇒ f : Assume A is weakly σ-stable. Let 0 → B → A be a nonzero proper subob- ject. Since A is weakly σ-stable, µσ (B) < µσ (A/B). Furthermore, since A has good quotients, Z(A/B) ̸= 0. Therefore, the generalized seesaw inequality tells us that µσ (B) < µσ (A), as needed. In view of Lemma 2.2.11.1 we no longer distinguish between weak σ-semistability and σ-semistability - we will just say an object is σ-semistable. Having good quotients is necessary to obtain the implication d ⇒ f above. Similarly, σ-purity is necessary to obtain the implications b ⇒ d and a ⇒ c: Proposition 2.2.12. Assume that σ = (Z, A) is a very weak stability function and A ∈ A. 31 1. If A is weakly σ-stable but not σ-stable then A does not have good quotients. 2. If for every subobject 0 → B → A in A satisfying 0 < IZ(B) < IZ(A) we have µσ (B) (≤) µσ (A) but A is not weakly σ-(semi)stable then A is not σ-pure. Proof. Assume σ = (Z, A) is a very weak stability function and A ∈ A. 1. Assume A is weakly σ-stable but not σ-stable. Therefore, we can proper nonzero subobject 0 → B → A such that µσ (B) = µσ (A). By the generalized seesaw inequality either µσ (B) = µσ (A) = µσ (A/B) or Z(A/B) = 0. Since A is weakly σ-stable, we know that µσ (B) < µσ (A/B) so Z(A/B) = 0. In other words, A does not have good quotients, as desired. 2. Assume that for every subobject 0 → B → A in A satisfying 0 < IZ(B) < IZ(A) we have µσ (B) < µσ (A) but A is not weakly σ-stable. Since A is not weakly σ-stable, we can choose a proper nonzero subobject 0 → B → A such that µσ (B) ≥ µσ (A/B). By the generalized seesaw inequality we find that µσ (B) ≥ µσ (A). Therefore, by assumption, either IZ(B) = 0 or IZ(B) = IZ(A). If IZ(B) = 0 then we are done, so assume IZ(B) = IZ(A). It follows that IZ(A/B) = 0 so µσ (B) ≥ µσ (A) ≥ µσ (A/B) = +∞. In other words, IZ(B) = IZ(A) = 0. Thus, A is not σ-pure, as desired. A similar argument works for the statement involving σ-semistability. We now prove an analogue of Schur’s Lemma. Schur’s lemma holds as expected for σ- stable objects, but the results are weaker for weakly σ-stable objects. However, we truly need the weaker assumption of weakly σ-stable in order for our applications. This is because, as we saw in Example 2.2.9 objects of interest are generally not σ-stable—only weakly σ-stable. Lemma 2.2.13 (Schur’s Lemma). Let σ = (Z, A) be a very weak stability function. Assume f : A → B is a morphism of σ-semistable objects. 32 1. If µσ (A) > µσ (B) then f = 0. 2. Assume A is weakly σ-stable. If µσ (A) = µσ (B) then f is an injection or 0. 3. Assume B is σ-stable. If µσ (A) = µσ (B) then f is a surjection or 0. 4. Assume A and B are σ-stable. If µσ (A) = µσ (B) then f is an isomorphism or 0. Proof. Assume σ = (Z, A) is a very weak stability function and f : A → B is a morphism of σ-semistable objects. 1. Assume that f ̸= 0, so Im(f ) is a nonzero quotient of A. If Ker(f ) = 0 then A = Im(f ) so µσ (A) = µσ (Im(f )). If Ker(f ) ̸= 0, since A is σ-semistable, then µσ (Ker(f )) ≤ µσ (Im(f )). By the generalized seesaw inequality, we find that µσ (A) ≤ µσ (Im(f )). In either case, we find that µσ (A) ≤ µσ (Im(f )). On the other hand, if Im(f ) = B then we find that µσ (Im(f )) = µσ (B). If Im(f ) ̸= B, since B is σ-semistable, we know that µσ (Im(f )) ≤ µσ (B/ Im(f )) (for Im(f ) ̸= 0). By the generalized seesaw inequality, we find that µσ (Im(f )) ≤ µσ (B). In either case, we find that µσ (Im(f )) ≤ µσ (B). By combining the above two inequalities, we find that µσ (A) ≤ µσ (Im(f )) ≤ µσ (B), as desired. 2. Assume A is weakly σ-stable. Also assume that f ̸= 0 and f is not an injection. Therefore, we find that Ker(f ) is a nonzero proper subobject of A. Since A is weakly σ-stable, it follows that µσ (Ker(f )) < µσ (Im(f )). By the generalized seesaw inequality, it follows that µσ (A) < µσ (Im(f )). Since B is σ-semistable, by the same argument as part 1, µσ (Im(f )) ≤ µσ (B). Combining our two inequalities, we find that µσ (A) < µσ (Im(f )) ≤ µσ (B). Therefore, µσ (E) ̸= µσ (F ), as needed. 33 3. Assume A is σ-stable. Also assume that f is neither 0 nor a surjection. Therefore, we know that Im(f ) is a proper nonzero subobject of B. Since B is σ-stable, it follows that µσ (Im(f )) < µσ (B). Since A is σ-semistable, by the same argument as part 1, µσ (A) ≤ µσ (Im(f )). Combining these two inequalities, we find that µσ (A) ≤ µσ (Im(f )) < µσ (B). In particular, µσ (A) ̸= µσ (B), as desired. 4. Since A is σ-stable, by Lemma 2.2.11, A is weakly σ-stable. Therefore, by part 2, f is injective or 0. Since B is σ-stable, by part 3, f is surjective or 0. In all, this shows that f is either both surjective and injective or 0. Since A is abelian, f is both surjective and injective if and only if f is an isomorphism, as desired. The following result shows that σ-stability is necessary for Schur’s Lemma. Proposition 2.2.14. If B is weakly σ-stable and satisfies the following property: For every σ-semistable object A with µσ (A) = µσ (B) any morphism f : A → B is surjective or 0. then B is σ-stable. Proof. Consider a subobject 0 → A → B in A. Since B is weakly σ-stable, by Lemma 2.2.11, B is σ-semistable. Therefore, µσ (A) ≤ µσ (B). Assume that µσ (A) = µσ (B) Since B is σ-semistable, A is also σ-semistable. Therefore, by assumption, the inclusion 0 → A → B is surjective or 0. Thus, A → B is an isomorphism or A = 0. In either case, this shows that B is σ-stable, as desired. 34 We also describe the failure of Schur’s lemma for weakly σ-stable objects from a purely categorical view in Proposition 2.2.17. In analogy to µH -stability any extension of µσ -semistable objects of the same slope is also µσ -semistable. The argument for a very weak stability function is essentially the same as for µH -stability. Lemma 2.2.15. Assume σ = (Z, A) is a very weak stability function and 0 → B → A → C → 0 is a short exact sequence with both B and C σ-semistable. If µσ (B) = µσ (A) or µσ (A) = µσ (C) then A is σ-semistable. Proof. Consider a short exact sequence 0 → B ′ → A → C ′ → 0 in A. For ease of notation, define B ∩ B ′ → A to be kernel of the natural morphism A → (A/B) ⊕ (A/B ′ ). Similarly, define 0 → B + B ′ → A to be the image of the natural morphism B ⊕ B ′ → A. By the universal property of the kernel, there is an injection B ∩ B ′ → A. On the other hand, by the Second Isomorphism Theorem, B ′ /(B ∩ B ′ ) ∼ = (B + B ′ )/B and there is an injection (B + B ′ )/B → A/B ∼ = C. Since B and C are σ-semistable, we find that µσ (B ∩ B ′ ) ≤ µσ (B) and µσ (B ′ /(B ∩ B ′ ) ≤ µσ (C). Without loss of generality, assume µσ (B) = µσ (A). Therefore, by the Generalized Seesaw Inequality either Zσ (C) = 0 or µσ (C) = µσ (A). If µσ (A) = µσ (C) then, by the inequalities above, µσ (B ∩ B ′ ) ≤ µσ (B) = µσ (A) and µσ (B/(B ∩ B ′ )) ≤ µσ (C) = µσ (A). It follows by the Generalized Seesaw Inequality applied to 0 → B ∩ B ′ → B ′ → B/(B ∩ B ′ ) → 0 that µσ (B ′ ) ≤ µσ (A), as needed. If Z(C) = 0 then Z(B/(B ∩ B ′ )) = 0. It follows by the Generalized Seesaw Inequality that µσ (B ∩ B ′ ) = µσ (B ′ ), so µσ (B ′ ) = µσ (B ∩ B ′ ) ≤ µσ (B) = µσ (A). The naive generalization of Lemma 2.2.15. to σ-stable objects is false. Specifically, if B and C are σ-stable of the same slope, then A will never be σ-stable (for µσ (B) = µσ (A) = µσ (C)). The same issue happens with weakly σ-stable objects. However, this generalization is morally true up to deformation of the stability function within a parameter space of 35 stability conditions. Specifically, Theorem 3.6.1 essentially says that if B and C are weakly σ-stable and σ-stable respectively of the same slope and A ̸= B⊕C then A is weakly σ ′ -stable for a deformation σ ′ of σ. Lemma 2.2.15 essentially says the full subcategory of A generated by σ-semistable objects of a fixed slope is closed under extensions. This statement is not technically correct because the full subcategory of A generated by σ-semistable object of fixed slope is not abelian. We make this statement precise below and clarify the failure of Schur’s Lemma for weakly σ-stable objects. Definition 2.2.16. Let σ = (Z, A) be a very weak stability function and fix ϕ ∈ R ∪ {+∞}. 1. We define A(ϕ) to be the full subcategory of A generated by objects E ∈ A satisfying either E is σ-semistable with µσ (E) = ϕ or Z(E) = 0. 2. We also define A0 to be the full subcategory of A generated by objects E ∈ A satisfying Z(E) = 0. Proposition 2.2.17. Let σ = (Z, A) be a very weak stability function and fix ϕ ∈ R∪{+∞}. 1. A(ϕ) is a full extension closed abelian subcategory of A. 2. A0 is a full Serre subcategory of A (i.e. whenever 0 → B → A → C → 0 is exact in A, A ∈ A0 if and only if B, C ∈ A0 ). 3. A ∈ A(ϕ) is σ-stable if and only if A is simple in A(ϕ). Recall that an object in an abelian category is simple if there are no proper nonzero subobjects. 4. A ∈ A(ϕ) is weakly σ-stable if and only if π(A) is simple in A(ϕ)/A0 where π : A(ϕ) → A(ϕ)/A0 is the quotient functor. Proof. Assume σ = (Z, A) is a very weak stability function. 1. We will first show that A(ϕ) is closed under extensions. Therefore, assume we have a short exact sequence 0 → B → A → C → 0 in A with B, C ∈ A(ϕ). If Z(B) = 36 Z(C) = 0 then Z(A) = 0 so we are done. If Z(B) ̸= 0 with Z(C) = 0 then the generalized seesaw inequality tells us that µσ (A) = µσ (B). Similarly, if Z(C) ̸= 0 with Z(B) = 0 the same argument gives µσ (A) = µσ (C). In either case, by Lemma 2.2.15, E is σ-semistable. If Z(B) ̸= 0 and Z(C) ̸= 0 then the generalized seesaw inequality tells us that µσ (B) = µσ (A) = µσ (C) so by Lemma 2.2.15, E is σ-semistable. Thus, A(ϕ) is extension closed. Since Z(0) = 0 and A(ϕ) is an extension closed full subcategory of an abelian category (and so additive category) A, A(ϕ) is an additive category. Furthermore, since A is abelian, it remains to show that A(ϕ) has all kernels and cokernels. With this in mind, consider a morphism f : A → B in A(ϕ). We have the following short exact sequences in A: 0 → Ker(f ) → A → Im(f ) → 0 0 → Im(f ) → B → Coker(f ) → 0. First, assume Z(Ker(f )) = 0 then µσ (A) = µσ (Im(f )) ≤ µσ (B) = µσ (A), so µσ (Im(f )) = ϕ. By the seesaw inequality, it follows that Z(Coker(f )) = 0 or µσ (Coker(f )) = ϕ. If Z(Coker(f )) = 0 then we are done. If µσ (Coker(f )) = ϕ then Coker(f ) is a quotient of a σ-semistable object of the same slope, so Coker(f ) is σ- semistable. In all, we have shown that if Z(Ker(f )) = 0 then Ker(f ), Coker(f ) ∈ A(ϕ). A similar argument works in the case that Z(Coker(f )) = 0. Thus, we may assume that Z(Ker(f )) ̸= 0 and Z(Coker(f )) ̸= 0. Since A and B are σ-semistable, we find that ϕ = µσ (A) ≤ µσ (Im(f )) ≤ µσ (B) = ϕ, so µσ (Im(f )) = ϕ. Since Z(Ker(f )) ̸= 0 and Z(Coker(f )) ̸= 0, it follows by the seesaw inequality that µσ (Ker(f )) = ϕ and µσ (Coker(f )) = ϕ. Since Ker(f ) is a subobject of a σ-semistable object of the same slope, Ker(f ) is σ-semistable. Similarly, Coker(f ) is σ-semistable, as desired. 37 2. Assume 0 → B → A → C → 0 is exact in A(ϕ). If B, C ∈ A0 , by definition, Z(B) = Z(C) = 0. By additivity of Z, it follows that Z(A) = 0 so A ∈ A0 , as needed. Conversely, assume A ∈ A0 so Z(A) = 0. By additivity of Z, IZ(B) + IZ(C) = 0 and similarly for RZ. Since the image of Z lies in ∈ R × R+ >0 ∪ {0}, it follows that IZ(B) = 0 and IZ(C) = 0. Moreover, we know that RZ(B) ≤ 0 and RZ(C) ≤ 0 so we find that RZ(B) = Z(C) = 0 as well. In other words, B, C ∈ A0 , as needed. 3. First assume A ∈ A(ϕ) is σ-stable. Therefore, by definition, we know that any proper nonzero subobject 0 → B → A in A must satisfying µσ (B) < µσ (A) = ϕ. Since A(ϕ) is a full subcategory of A, it follows that A has no proper nonzero subobjects in A(ϕ). Second, assume that A ∈ A(ϕ) is simple. Consider a proper nonzero subobject 0 → B → A in A. Since A is simple in A(ϕ), we find that either µσ (B) ≤ µσ (A) or µσ (B) = µσ (A) with B not σ-semistable. However, since A is σ-semistable if µσ (B) = µσ (A), we find that B must be σ-semistable. Thus, we find that µσ (B) < µσ (A), so A is σ-stable, as desired. 4. Let π : A(ϕ) → A(ϕ)/A0 be the quotient functor. We know that π is essentially surjective, full, and exact. Furthermore, π(C) = 0 if and only if Z(C) = 0. First, assume that A ∈ A(ϕ) is weakly σ-stable. Consider a subobject 0 → B ′ → π(A) in A(ϕ)/A0 . Since π : A(ϕ) → A(ϕ)/A0 is full and essentially surjective, the injection π(f ) 0 → B ′ → π(A) can be written as 0 → π(B) −−→ π(A) where f : B → A is a 1 morphism in A(ϕ). Note that f is not necessarily injective. If Z(B) = 0 then π(B) = 0 and we are done, so assume Z(B) ̸= 0. By definition of A(ϕ), it follows that µσ (A) = µσ (B) = ϕ. Moreover, since π(f ) : π(B) → π(A) is injective and Z(B) ̸= 0, π(Im(f )) = Im(π(f )) ̸= 0. In other words, Z(Im(f )) ̸= 0 1 In fact, it is not difficult to show that if σ is not a stability function then π : A(ϕ) → A(ϕ)/A0 reflects injections (i.e. if π(f ) : π(B) → π(A) is injective then f : B → A is injective) if and only if ϕ ̸= +∞. 38 so by definition of A(ϕ), µσ (Im(f )) = µσ (A). On the other hand, since A is weakly σ-stable, µσ (Im(f )) < µσ (Coker(f )) or Im(f ) = A. Assume µσ (Im(f )) < µσ (Coker(f )) Since µσ (Im(f )) = µσ (A), it follows by the gen- eralized seesaw inequality that Z(Coker(f )) = 0. Equivalently, 0 = π(Coker(f )) = Coker(π(f )). Therefore, π(f ) : π(B) → π(A) is an isomorphism, as needed. For the other case, assume that Im(f ) = A. It follows that π(f ) : π(B) → π(A) is an isomorphism, as needed. Second, assume that π(A) is simple. Consider a proper nonzero subobject 0 → B → A → C → 0. Since π is exact, we obtain an exact sequence 0 → π(B) → π(A) → π(C) → 0 in A(ϕ)/A0 . Since π(A) is simple, it follows that π(B) = 0 or π(B) = π(A). If π(B) = π(A) then π(C) = 0, so Z(C) = 0. It follows by the generalized seesaw inequality that µσ (B) = µσ (A) < +∞ = µσ (C), as needed. If π(B) = 0 then Z(B) = 0, so µσ (B) = +∞ > µσ (A). Since A is σ-semistable, it follows that B = 0. In either case, we find that E is weakly σ-stable, as desired. The failure of Schur’s lemma for weakly σ-stable objects can be seen through the lens of the categories introduced above. Schur’s lemma for an abelian category states that a morphism between simple objects is either the zero morphism or an isomorphism. This result applied to A(ϕ) gives us a different proof of Lemma 2.2.13.4. On the other hand, since weakly σ-stable objects correspond to simple objects in A(ϕ)/A0 , and A(ϕ)/A0 is not a full subcategory of A. However, one can use Proposition 2.2.17.4 to obtain a different proof of 2 Lemma 2.2.13.2. 2 Here is a sketch of the proof. We may reduce to the case that µσ (A) ̸= +∞. As noted in the earlier footnote, π : A(ϕ) → A(ϕ)/A0 reflects injections if and only if ϕ ̸= +∞ or σ is a stability function. Therefore, by Schur’s lemma for an abelian category, we find that any morphism π(A) → π(B) must be an isomorphism or 0 Since π is full, essentially surjective and reflects injections; we find that any morphism A → B is an injection or 0. 39 2.3 Additional Properties of Very Weak Stability Functions In general, the notion of a stability function is too weak. For this reason, we consider stability functions that satisfy the Harder-Narasimhan and support properties. These can be thought of as generalizations of Harder-Narasimhan filtrations and Bogomolov’s Inequality for µH - stability respectively. In this subsection, we will define these notions. We will also discuss Jordan-Hölder filtrations. The new results in this subsection are about the existence/non-existence of Jordan-Hölder filtrations for very weak stability functions. If a stability function satisfies the Harder- Narasimhan and support properties then every σ-semistable object with good quotients has a unique Jordan-Hölder filtration. In contrast, there most very weak stability functions satisfying the Harder-Narasimhan and support properties have σ-semistable objects which do not have a Jordan-Hölder filtra- tion (Example 2.3.15) To this end, we introduce the notion of a weak Jordan-Hölder filtration which generalizes Jordan-Hölder filtrations. In contrast to Jordan-Hölder filtrations, for “real-life” very weak stability functions any σ-semistable object with finite slope has a weak Jordan-Hölder filtration (Lemma 2.3.14). Unfortunately, the trade-off for the existence is that weak Jordan-Hölder filtrations are not unique (Proposition 2.3.11). Definition 2.3.1. Assume σ = (Z, A) is a very weak stability function. 1. Let A ∈ A. We say that A has a Harder-Narasimhan filtration (with respect to σ) if there is a filtration of the form 0 = A0 → A1 → · · · → Am−1 → Am = A such that Ai /Ai−1 is σ-semistable for all i = 1, 2 . . . , m − 1, m and µσ (A1 ) > µσ (A2 /A1 ) > · · · > µσ (Am−1 /Am−2 ) > µσ (A/Am−1 ). 40 2. We say that σ satisfies the Harder-Narasimhan property if every object in A has a Harder-Narasimhan filtration with respect to σ. Definition 2.3.2. Assume that σ = (Z, A) is a very weak stability function and 0 = A0 → A1 → · · · → Am−1 → Am = A is a Harder-Narasimhan filtration of A. 1. We define µ+ σ (A) = µσ (A1 ). 2. We µ− σ (A) = µσ (A/Am−1 ). 3. We define the mass of A to be Xm mσ (A) = |Z(Ai /Ai−1 )|. i=1 If σ is a very weak stability function then a Harder-Narasimhan filtration is not necessarily unique. However, the length of the filtration and the slopes of the consecutive quotients are − well-defined. In particular, µ+ σ (A), µσ (A), and mσ (A) are well-defined. Furthermore, the objects A1 and A/Am−1 are a maximal destabilizing subobject and minimal destabilizing quotient respectively. Lemma 2.3.3. Assume σ = (Z, A) be a very weak stability function and A ∈ A has a Harder-Narasimhan filtration given by 0 = A0 → A1 → · · · → Am−1 → Am = A. 1. If 0 = B0 → B1 → · · · → Bn−1 → Bn = A is another Harder-Narasimhan filtration of A with respect to σ then a) m = n, 41 b) Z(Ai ) = Z(Bi ) for all i = 1, . . . , m. In particular, µσ (Ai /Ai+1 ) = µσ (Bi /Bi+1 ) for all i = 1, . . . , m − 1. c) There are injections Ai → Bi that make the following diagram commute: 0 A1 A2 ··· Am−1 Am = A . 0 B1 B2 ··· Bm−1 Bm = A Furthermore, if σ is a stability function then the injections Ai → Bi are isomor- phisms. 2. A1 is a maximal destabilizing subobject of A. In other words, every nonzero subobject 0 → B → A satisfies µσ (B) ≤ µσ (A1 ) and if equal then 0 → B → A factors through 0 → B → A1 . 3. A/Am−1 is a minimal destabilizing quotient of A. In other words, every nonzero quo- tient A → C → 0 satisfies µσ (C) ≥ µσ (A/Am−1 ) and if equal then A → A/Am−1 → 0 factors through A → C → 0. Proof. Assume σ is a very weak stability function and A ∈ A has a Harder-Narasimhan filtration. 1. Our proof essentially follows [HL10, Paragraphs before Theorem 1.3.7]. Consider two Harder-Narasimhan filtrations of E: 0 = A0 → A1 → · · · → Am−1 → Am = A 0 = B0 → B1 → · · · → Bn−1 → Bn = A. Without loss of generality, assume µσ (A1 ) ≤ µσ (B1 ). Let i be the smallest integer such that B1 → A factors through Bi → A (which exists because i = m satisfies this property). By minimality of i, the composition B1 → Ai → Ai /Ai−1 is nonzero. Thus, 42 by Schur’s Lemma (Lemma 2.2.13), µσ (B1 ) ≤ µσ (Bi /Bi−1 ). By combining the above inequality with the assumption µσ (A1 ) ≤ µσ (B1 ), we find that µσ (A1 ) ≤ µσ (B1 ) ≤ µσ (Ai /Ai−1 ) It follows by definition of a Harder-Narasimhan filtration that i = 1, so the injection B1 → E factors through an injection B1 → A1 , as needed. Since A1 is σ-semistable, it follows that µσ (B1 ) ≤ µσ (A1 ) so µσ (B1 ) = µσ (A1 ). Fur- thermore, since σ is a very weak stability function, IZ(B1 ) ≤ IZ(A1 ). By using the same argument as above with the roles of B1 and A1 switched (which we can do because µσ (F1 ) = µσ (E1 )), we find that IZ(A1 ) ≤ IZ(B1 ). This shows that IZ(A1 ) = IZ(B1 ). If IZ(A1 ) ̸= 0, since µσ (A1 ) = µσ (B1 ), it follows that RZ(A1 ) = RZ(B1 ) so Z(A1 ) = Z(B1 ). If IZ(A1 ) = 0 = IZ(B1 ), we know RZ(A1 ), RZ(B1 ), RZ(A1 /B1 ), RZ(B1 /A1 ) ≤ 0. Since RZ is additive in short exact sequences, it follows that RZ(A1 ) − RZ(B1 ) ≤ 0 and RZ(B1 ) − RZ(A1 ) ≤ 0. In other words, RZ(B1 ) = RZ(A1 ) so Z(B1 ) = Z(A1 ), as needed. Now, consider the following Harder-Narasimhan filtrations of A/A1 : 0 → A2 /A1 → · · · → Am−1 /A1 → Am /A1 = A/A1 0 → B2 /A1 → · · · → Bn−1 /A1 → Bn /A1 = A/A1 . Note that the above filtrations truly are Harder-Narasimhan filtrations by the third isomorphism theorem. By induction on m, we find that ˆ m − 1 = n − 1, ˆ Z(Ai+1 /A1 ) = Z(Bi+1 /B1 ) for all i = 1, . . . , m−1. Therefore, Z(Ai+1 ) = Z(Bi+1 ) for all i = 1, . . . , m − 1. 43 ˆ There are injections Ai /A1 → Bi /A1 that make the following diagram commute: 0 A2 /A1 ··· Am−1 /A1 Am /A1 = A/A1 . 0 B2 /A1 ··· Bm−1 /A1 Bm /A1 = A/A1 We already saw that Z(A1 ) = Z(B1 ), so we have proven parts a and b. It remains to show part (c). By the second isomorphism theorem, we can lift the injections Ai /A1 → Bi /A1 to injections Ai → Bi that makes the following diagram commute: 0 A2 ··· Am−1 Am = A . 0 B2 ··· Bm−1 Bm = A As we saw above, 0 → B1 → A factors through 0 → A1 → A, so we obtain the desired commutative diagram. Last, assume that σ is a stability function. Since Z(Ai ) = Z(Bi ), we find that Z(Bi /Ai ) = 0. Since σ is a stability function, it follows that Bi /Ai = 0 so the in- jections Ai → Bi are isomorphisms, as desired. 2. Assume 0 → A1 → A is a maximal destabilizing subobject of A. By definition of a Harder-Narasimhan filtration we know that B is σ-semistable. Now, assume that 0 → B → A is a proper nonzero subobject. By definition of a maximal destabilizing subobject, there is a Harder-Narasimhan filtration 0 → A1 → A2 → · · · → Am−1 → A We proceed by induction of the smallest integer i such that 0 → B → A factors through 0 → Ai → A. If i = 1 then 0 → B → A factors through 0 → A1 → A, as needed. Also, since A1 is σ-semistable µσ (B) ≤ µσ (A1 ). 44 Now, assume k is the smallest integer such that 0 → B → A factors through 0 → Ak → A. By minimality, there is a nonzero morphism f : B → Ak /Ak−1 . By definition of a Harder-Narasimhan filtration, Ak /Ak−1 is σ-semistable so µσ (Im(f )) ≤ µσ (Ak /Ak−1 ). In addition, there is an injection Ker(f ) → Ak−1 . Thus, by the inductive hypothesis we find that µσ (Ker(f )) ≤ µσ (A1 ) with equality exactly if the composition 0 → Ker(f ) → Ak−1 → A factors through 0 → A1 → A. Since µσ (Ker(f )) ≤ µσ (A1 ) and µσ (Im(f )) ≤ µσ (Ak /Ak−1 ) ≤ µσ (A1 ), by the gen- eralized seesaw inequality, µσ (B) ≤ µσ (A1 ). Now, assume that µσ (B) = µσ (A1 ). Therefore, by the generalized seesaw inequality, µσ (Ak /Ak−1 ) ≥ µσ (Im(f )) ≥ µσ (B) = µσ (A1 ). By definition of a Harder-Narasimhan filtration, it follows that k = 1. Thus, 0 → B → A factors through 0 → A1 → A, as desired. 3. The dual argument to part 2 holds. We can use the above result to obtain a generalization of Lemma 2.2.13.1. Alternatively, the following lemma can be thought of as a natural generalization of [HL10, Lemma 1.3.3] from µH -stability to any very weak stability function. Lemma 2.3.4. Assume that σ = (Z, A) is a very weak stability function and consider A, B ∈ A. Additionally assume A and B have Harder-Narasimhan filtrations with respect to σ. If µ− + σ (A) > µσ (B) then f = 0. Proof. For ease of notation, let 0 → A1 → · · · → Am−1 → A 0 → B1 → · · · → Bn−1 → B be Harder-Narasimhan filtrations of A and B. Also, assume that f : A → B is nonzero. Let i be the smallest nonzero integer such that the composition Ai → A → B is nonzero. 45 By minimality of i, we obtain a nonzero morphism Ai /Ai−1 → B. Let j be the smallest nonzero integer such that Ai /Ai−1 → B factors through Bj → B. By composition, we obtain a morphism Ai /Ai−1 → Bj → Bj /Bj−1 , and, by minimality of j, the morphism Ai /Ai−1 → Bj /Bj−1 is nonzero. By definition of a Harder-Narasimhan filtration, Ai /Ai−1 and Bj /Bj−1 are σ-semistable. Therefore, by Schur’s Lemma (Lemma 2.2.13.1), we find that µσ (Ai /Ai−1 ) ≤ µσ (Bj /Bj−1 ). By Lemma 2.3.3, it follows that µ− − σ (A) ≤ µσ (Ai /Ai−1 ) ≤ µσ (Bj /Bj−1 ) ≤ µσ (B) ), as desired. We also note that under natural assumptions, a very weak stability function satisfies the Harder-Narasimhan property. This result first appears in [BM11, Proposition B.2] for stability functions. The corresponding result for very weak stability functions first appears in [BMS16, Proposition B.2]. A simpler proof for stability functions is given in [Bay19, Section 3] which can be extended to very weak stability conditions by [MS17, Remark 4.14]. We reproduce this proof below. We need two technical definitions for proving the following lemma. The first definition is a natural generalization of [Sha77, Middle of Page 173 in view of Theorem 2] to very weak stability functions due to Bayer (Definition [Bay19, Definition 3.1]). The second definition first appears in [Bay19, Definition 3.2]. Definition 2.3.5. Assume σ = (Z, A) is a very weak stability function and A ∈ A. 1. We define the Harder-Narasimhan polygon of A (with respect to σ), written HN(A), to be the convex hull of the set {Z(B) | 0 → B → A is injective} ⊆ C. 2. We say that HN(A) is polyhedral on the left if the region in HN(A) to the left of the line connecting 0 to Z(E) is a polygon with finitely many vertices (see Figure 2.3). 46 Z(A) Z(B) HN(A) HN(B) 0 0 Figure 1 Polyhedral on the Left: HN(A) is polyhedral on the left while HN(B) is not. Note that if A has a Harder-Narasimhan filtration with respect to a very weak stability function σ = (Z, A) then the mass, mσ , is the length of the left boundary curve connecting 0 and Z(A) in HN(A). Lemma 2.3.6. Assume σ = (Z, A) is a very weak stability condition with A ∈ A. If HN (A) is polyhedral on the left and either of the following conditions holds ˆ σ = (Z, A) is a stability function or ˆ A is Noetherian (i.e. A satisfies the ascending chain condition) then A has a Harder-Narasimhan filtration with respect to σ. Proof. Assume HN (A) is polyhedral on the left and either of the above assumptions holds. Let 0 = z0 , z1 , z2 , . . . , zm−1 , zm = Z(A) be the vertices of the left boundary curve of HN (A) written with increasing y-coordinate. Since zi is a vertex of HN (A), zi cannot be written as a finite linear combination of elements in HN (A) \ {zi }. Therefore, by Carathéodory’s theorem there exists Ai ∈ A such that Z(Ai ) = zi . For fixed i, consider the set C = {B ∈ A | 0 → B → A is injective and Z(B) = zi } 47 with the ordering B ≤ B ′ if 0 → B → A factors through 0 → B ′ → A. If A is Noetherian then choose Ai to be a maximal element in the above poset. If σ is a stability function, choose arbitrary Ai . We first claim the injections 0 → Ai → A factor through 0 → Ai+1 → A for all i. There is a short exact sequence 0 → Ai ∩ Ai+1 → Ai ⊕ Ai+1 → Ai + Ai+1 → 0. By additivity of Z, we find Z(Ai ∩ Ai+1 ) + Z(Ai + Ai+1 ) Z(Ai ) + Z(Ai+1 ) = . 2 2 In other words, the midpoint of the line segment connecting Z(Ai ) = zi to Z(Ai+1 ) = zi+1 is equal to the midpoint of the line segment connecting Z(Ai ∩ Ai+1 ) to Z(Ai + Ai+1 ). By convexity of HN (A), it follows that Z(Ai ∩ Ai+1 ) lies on the line segment connecting Z(Ai ) to Z(Ai+1 ). This implies that IZ(Ai ) ≤ IZ(Ai ∩ Ai+1 ), IZ(Ai + Ai+1 ) ≤ IZ(Ai+1 ). There are natural injections Ai ∩ Ai+1 → Ai and Ai+1 → Ai + Ai+1 so we also find that IZ(Ai ) ≥ IZ(Ai ∩Ai+1 ) and IZ(Ai+1 ) ≤ IZ(Ai +Ai+1 ). This shows IZ(Ai ∩Ai+1 ) = IZ(Ai ) and IZ(Ai +Ai+1 ) = IZ(Ai+1 ). A similar argument gives that RZ(Ai ∩Ai+1 ) = RZ(Ai ) and RZ(Ai +Ai+1 ) = RZ(Ai+1 ). In all, we have shown Z(Ai ∩Ai+1 ) = Z(Ai ) and Z(Ai +Ai+1 ) = Z(Ai+1 ). If A is Noetherian, by maximality of Ai , we find that the natural injection Ai+1 → Ai + Ai+1 is an isomorphism. If σ is a stability condition, then Ai /(Ai ∩ Ai+1 ) = 0 so Ai ∩Ai+1 → Ai is an isomorphism. Thus, we we find either Ai ∩Ai+1 → Ai or Ai+1 → Ai +Ai+1 is an isomorphism. In other words, Ai → A factors through Ai+1 → A, as desired. We now show Ai+1 /Ai is σ-semistable. With this in mind, consider a subobject B/Ai induced by injections 0 → Ai → B → Ai+1 . It follows that IZ(Ai ) ≤ IZ(B) ≤ IZ(Ai+1 ). Furthermore, by convexity of HN (A), Z(B) must lie to the right of the line connecting Z(Ai ) 48 R Z(B) Z(Ai ) Z(Ai+1 ) I Figure 2 The Harder-Narasimhan Polygon: Position of Z(B) relative to Z(Ai ) and Z(Ai+1 ). Note the complex plane is rotated 90 degrees. In this orientation, the slope of the line between two points agrees with the slope of the quotient. In other words, with this orientation the slope of Z(B)Z(Ai ) is µσ (B/Ai ). to Z(Ai+1 ). Therefore, if m is the slope of Z(Ai )Z(Ai+1 ) and m′ is the slope of Z(Ai )Z(B) then −1/m′ ≤ −1/m. (see Figure 2.3). In other words RZ(B) − RZ(Ai ) RZ(Ai+1 ) − RZ(Ai ) µσ (B/Ai ) = − ≤− = µσ (Ai+1 /Ai ), IZ(B) − IZ(Ai ) IZ(Ai+1 ) − IZ(Ai ) as needed. Therefore, Ai+1 /Ai is σ-semistable, as claimed. Last, since HN (A) is convex and µσ (·) has the same ordering as arg(Z(·)), µσ (A2 /A1 ) > · · · > µσ (A/Am−1 ) If IZ : Λ → R is discrete then HN(A) will always be polyhedral on the left. A converse to Lemma 2.3.6 holds. Specifically, by the same argument as [Bay19, Proposi- tion 3.3], if A has a Harder-Narasimhan filtration with respect to σ then HN (A) is polyhedral on the left. In particular, we do not need additional assumptions on A nor σ. The next property we consider is the support property The support property first appears in [KS08, Section 2.1 Support Property]. By [BM11, Proposition B.4], in practice, the support property is equivalent to Bridgeland’s notion of locally finite [Bri07, Definition 5.7]. 49 The support property can be thought of as a generalized Bogomolov inequality. For our purposes, the most important consequence of the support property is that it induces a useful wall and chamber structure on the parameter spaces of stability conditions. From a different point of view, if σ is a stability function satisfying the support property then every σ-semistable object has a Jordan-Hölder filtration with unique factors up to permutation (see Proposition 2.3.11 and Proposition 2.3.10). Unfortunately, there is no similar result for very weak stability functions. Definition 2.3.7. We say that a very weak stability function (Z : Λ → C, A) satisfies the support property if either of the equivalent conditions are satisfied: 1. There exists a quadratic form Q on ΛR such that a) The Q is negative definite with respect to the kernel of Z (i.e. if Z(A) = 0 then Q(A) < 0), and b) If A ∈ A is semistable with respect to Z, then Q(A) ≥ 0 (where A is identified with the corresponding object in ΛR ). 2. For any norm || · || on ΛR ,   |Z(A)| inf : A ∈ A is σ − semistable >0 ||A|| To see that these two notions are equivalent consider the quadratic form Q(A) = |Z(A)|2 /C − ||A||2 where || · || is the standard norm on R⊕ rank(Λ) and C is a sufficiently positive real number. Since Λ is a finitely generated free abelian group, we know ΛR is a finite dimensional real vector space. Therefore, any two norms on ΛR are equivalent, so the second definition can be checked on a single norm (such as the Euclidean norm above). We note a linear algebra lemma about the quadratic form in short exact sequences 50 Lemma 2.3.8 ([Sch20, Lemma 2.7]). Consider a very weak stability function σ = (Z, A) satisfying the support property with respect to Q. If 0 → B → A → C → 0 is a short exact sequence in A satisfying the following: ˆ µσ (B) = µσ (A) or µσ (A) = µσ (C), ˆ Q(B), Q(C) ≥ 0 then Q(A) ≥ 0. We now define (weak) Jordan-Hölder filtrations and relate them to the support property. Definition 2.3.9. Assume σ = (Z, A) is a very weak stability function and A ∈ A is σ-semistable. 1. We say that A has a (weak) Jordan-Hölder filtration if there exists a filtration: 0 = A0 → A1 → A2 → · · · → Am = A such that Ai /Ai−1 is (weakly) σ-stable and µσ (Ai /Ai−1 ) = µσ (A) for all i = 1, . . . , m. 2. For a (weak) Jordan-Hölder filtrations, the consecutive quotients Ai /Ai−1 are called (weak) Jordan-Hölder factors. Proposition 2.3.10. Assume σ = (Z, A) is a stability function and fix ϕ ∈ R ∪ {+∞}. If σ satisfies the support property then the abelian category A(ϕ) is both Noetherian and Artinian (recall that A(ϕ) is defined in Definition 2.2.16). In particular, if σ satisfies the support property then every σ-semistable object in A has a Jordan-Hölder filtration. Proof. We will first show that A(ϕ) is Noetherian. Thus, fix an object A ∈ A(ϕ) and consider an ascending chain 0 = A0 → A1 → A2 → · · · → A 51 in A(ϕ). By additivity of Z, we find that Z(A) = Z(A1 /A0 ) + Z(A/A1 ) = Z(A2 /A1 ) + Z(A1 /A0 ) − Z(A2 /A1 ) + Z(A/A1 ) = Z(A2 /A1 ) + Z(A1 /A0 ) + Z(A/A2 ) = ··· Xk = Z(A/Ak ) + Z(Ai /Ai−1 ). i=1 for all integers k ≥ 1. Since Ai ∈ A(ϕ) for all i, we know that arg(Z(A/Ai ) = arg(Z(A)) = arg(Z(Ai )) for all i. In other words, Z(A/Ai ) and Z(A/Ai ) lie on the same ray in R×R+ >0 ∪{0} for all i. Thus, the above equality gives us X k X k Xk |Z(A)| = Z(A/Ak ) + Z(Ai /Ai−1 ) = |Z(A/Ak )| + |Z(Ai /Ai−1 )| ≥ |Z(Ai /Ai−1 )| i=1 i=1 i=1 for all k. nP o∞ k In short, we have shown that i=1 |Z(Ai /Ai−1 )| is bounded above. Therefore k=1 P∞ i=1 |Z(Ai /Ai−1 )| converges and so limi→∞ |Z(Ai /Ai−1 )| = 0. By definition of the support property, it follows that Z(Ai /Ai−1 ) = 0 for all i ≫ 0. Since σ is a stability function, it follows that Ai /Ai−1 = 0 for all i ≫ 0. In other words, the ascending chain eventually stabilizes, as needed. The argument that A(ϕ) is Artinian is essentially the same. Consider an object A ∈ A with µσ (A) = ϕ. Since A(ϕ) is both Noetherian and Artinian, any object in A(ϕ) has a Jordan-Hölder filtration. In other words, there is a filtration 0 → A1 → A2 → · · · Am−1 → A such that Ai /Ai−1 is simple in A(ϕ). By proposition 2.2.17.3, simple in A(ϕ) is equivalent to σ-stable. Therefore, we have shown that there is a filtration 0 → A1 → A2 → · · · → Am−1 → A in A (for A(ϕ) is a full subcategory) such that Ai /Ai−1 is σ-stable of slope ϕ = µσ (A), as desired. Unfortunately, Proposition 2.3.10 fails for very weak stability conditions. For example, we saw that in Example 2.2.9 that if σµDH is the very weak stability function associated to 52 µDH then the only objects with a σµD H -stable subobjects are direct sums of skyscraper sheaves. Consequently, the only σµDH -semistable objects with Jordan-Hölder filtrations are direct sums of skyscraper sheaves. To avoid this issue, we will work with weak Jordan-Hölder filtrations. However, in con- trast to Jordan-Hölder filtrations, weak Jordan-Hölder factors may not be unique. We only obtain the following: Proposition 2.3.11. Assume σ = (Z, A) is a very weak stability function and A ∈ A is σ-semistable. Given two weak Jordan-Hölder filtrations 0 = A0 → A1 → A2 → · · · → Al = A and 0 = B0 → B1 → B2 → · · · → Bm = A then l = m and there exists a permutation σ ∈ Sm such that Ai /Ai−1 is a subobject of Bσ(i) /Bσ(i−1) for all i. Moreover, if A• → A is additionally a Jordan-Hölder filtration then the injections Ai /Ai−1 → Bσ(i) /Bσ(i−1) are isomorphisms for all i. In other words, if A has a Jordan-Hölder filtration then every weak Jordan-Hölder is actually a Jordan-Hölder, and, in this case, the Jordan-Hölder factors are unique up to permutation. Proof. Choose the smallest i such that A1 is a subobject of Bi . By minimality of i, we obtain a nonzero morphism A1 → Bi /Bi − 1. By assumption, both A1 and Bi /Bi−1 are weakly σ- stable, so the nonzero morphism A1 → Bi /Bi−1 must be injective by Schur’s Lemma (Lemma 2.2.13.2). The result follows by induction on the weak Jordan-Hölder filtrations: 0 → A2 /A1 → A3 /A1 → · · · → Al /A1 = A/A1 and 0 → (B1 + A1 )/A1 → · · · → (Bi−1 + A1 )/A1 → Bi /A1 → Bi+1 /A1 → · · · → Bm /A1 = A/A1 . 53 If A• → A is a Jordan-Hölder filtration then the injections Ai /Ai−1 → Bσ(i) /Bσ(i−1) is an isomorphism by Schur’s lemma (2.2.13.4). In particular, Bσ(i) /Bσ(i−1) is σ-stable so B• → B is a Jordan-Hölder filtration. Even though weak Jordan-Hölder filtrations are not afforded the standard uniqueness properties, they exist in much more general scenarios. Specifically, we will show that under relatively weak assumptions, σ-semistable objects of finite slope have weak Jordan-Hölder filtrations (Lemma 2.3.14). We first need some basic facts about σ-pure objects. We also provide some examples to elucidate σ-purity. Recall that an object A ∈ A is σ-pure if every nonzero subobject 0 → B → A satisfies IZ(B) ̸= 0 (Definition 2.2.10). Example 2.3.12. Consider σµDH - the very weak stability condition corresponding to µD H. Assume E is σµDH -pure. Therefore, every nonzero subsheaf of E has positive rank. In other words, every nonzero subsheaf is supported everywhere, so E has no torsion subsheaves - i.e. E is torsion-free or 0. Conversely, if E is torsion-free or 0 then every nonzero subsheaf is supported everywhere so every nonzero subsheaf has positive rank. Thus, E is σµDH -pure. Recall that E is torsion-free if and only if E is pure and supported everywhere ([HL10, Definition 1.1.2]). In other words, we have shown that E is σµDH -pure if and only if E = 0 or E is pure and supported everywhere. More generally, consider σd,µDH from Example 2.2.5. Assume E ∈ Cohd (X) is σd,µdH -pure and consider a nonzero subsheaf 0 → F → E . By definition of Zd,µDH and σd,µDH -purity, H d · chDd (F ) ̸= 0, so F is supported in codimension at most d by Lemma 2.1.3. On the other hand, since F ∈ Cohd (X), F is supported in codimension at least d. Therefore, every nonzero subsheaf of E is supported in codimension d. In other words, E is pure of dimension d ([HL10, Definition 1.1.2]). By Lemma 2.1.3 we also find that if E is pure of codimension d then E is σd,µDH -pure. 54 Here are some easy facts about σ-purity. Each of these results have a corresponding result for µH -stability and torsion-free sheaves that is well known. Lemma 2.3.13. Assume σ = (Z, A) is a very weak stability function and A ∈ A. 1. If A is σ-pure then any nonzero subobject 0 → B → A is σ-pure. 2. Assume A has a Harder-Narsimhan filtration with maximal destabilizing subobject 0 → A1 → A. A is σ-pure if and only if IZ(A1 ) ̸= 0. In particular, a σ-semistable object A is σ-pure if and only if IZ(A) ̸= 0. 3. If 0 → B → A → C → 0 is a short exact sequence such that B and C are σ-pure then A is σ-pure. 4. If A is σ-pure and 0 → A1 → A is a maximal destabilizing subobject then A/A1 is 0 or σ-pure. Proof. Assume σ = (Z, A) is a very weak stability function and A ∈ A. 1. Assume A is σ-pure and 0 → B → A is a nonzero subobject. Consider a nonzero subobject 0 → C → B. Since C is a nonzero subobject of A, IZ(C) ̸= 0, as desired. 2. Assume A is σ-pure. In particular, IZ(A1 ) ̸= 0, as desired. Conversely, assume IZ(A1 ) ̸= 0 and consider a nonzero subobject 0 → B → A. By Lemma 2.3.3, µσ (B) ≤ µσ (A1 ). Since IZ(A1 ) ̸= 0, µσ (B) ≤ µσ (A1 ) < +∞. In particular, IZ(B) ̸= 0, as desired. 3. Consider a subobject 0 → B ′ → A such that IZ(B ′ ) = 0. Consider the composition f : B ′ → A → C. By additivity of IZ, IZ(Im(f )) = 0. Since C is σ-pure, it follows that Im(f ) = 0. Therefore, the injection 0 → B ′ → A factors through the injection 0 → B → A. However, since B is σ-pure and IZ(B ′ ) = 0, it follows that B = 0, as desired. 55 4. Assume A/A1 ̸= 0 and consider a subobject 0 → B → A/A1 satisfying IZ(B) = 0. We can find a subobject 0 → B ′ → A such that B = B ′ /A1 . Since IZ(B) = 0, by addi- tivity, IZ(B ′ ) = IZ(A1 ). If µσ (B ′ ) = µσ (A1 ), by definition of a maximal destabilizing subobject, B ′ = A1 and so B = 0. Therefore, we may assume µσ (B ′ ) < µσ (A1 ), so −RZ(B ′ ) < −RZ(A1 ). It follows that IZ(B) = 0 and RZ(B) = RZ(B ′ ) − RZ(A1 ) > 0 which contradicts the definition of a very weak stability function. Therefore, A1 = B ′ and B = 0, as claimed. We can now give an existence result for weak Jordan-Hölder filtrations for a very weak stability function: Lemma 2.3.14. Assume σ = (Z : Λ → R, A) is a very weak stability function and the image of IZ : Λ → R is discrete. If A ∈ A is σ-semistable, Noetherian, and satisfies IZ(A) ̸= 0 then A has a Jordan-Hölder filtration. Proof. Since IZ(A) ̸= 0 and A is σ-semistable, by Lemma 2.3.13.2, A is σ-pure. Consider the set C1 consisting of subobjects 0 → B → A such that µσ (B) = µσ (A). We define an order on C1 where 0 → B1 → A < 0 → B2 → A if IZ(B1 ) < IZ(B2 ). The identity 0 → A → A is in C, so C is nonempty. Furthermore, since IZ : Λ → R is discrete, any chain in C has a minimal object. Let D1 be the collection of all minimal objects in C1 . We define a partial order on D1 where 0 → B1 → A ≤ 0 → B2 → A if 0 → B1 → A factors through 0 → B2 → A. Since A is Noetherian, D1 has a maximal object. Let 0 → A1 → A be the maximal object of D1 . Since A is σ-pure, A1 is σ-pure by Lemma 2.3.13. We claim that A1 is weakly σ-stable. Since A1 is σ-pure, by Lemma 2.2.11, it suffices to show that for every subobject 0 → B → A1 in A satisfying 0 < IZ(B) < IZ(A1 ) we have µσ (B) < µσ (A1 ). However, by minimality of A1 in C1 , every subobject 0 < IZ(B) < IZ(A1 ) satisfies µσ (B) ̸= µσ (A1 ). Since A is σ-semistable, it follows that µσ (B) < µσ (A) = µσ (A1 ). Therefore, A1 is weakly σ-stable, as desired. 56 If IZ(A1 ) = IZ(A) then, by definition of C1 , A = A1 . In this case, we find that a Jordan- Hölder filtration of A is given by 0 → A. Thus, we may assume that IZ(A1 ) < IZ(A). Define C2 to be the set of all subobjects 0 → B → A satisfying the following ˆ 0 → A1 → A factors through 0 → B → A, ˆ µσ (B) = µσ (A), and ˆ IZ(B) > IZ(A1 ). We define an order on C2 where 0 → B1 → A < 0 → B2 → A if IZ(B1 ) < IZ(B2 ). Since IZ(A1 ) < IZ(A), the morphism 0 → A1 → A is contained in C2 , so C2 is non- empty. Furthermore, since IZ : Λ → R is discrete, any chain in C2 has a minimal object. Let D2 be the set of all minimal objects in C2 . We define a partial order on D2 where 0 → B1 → A ≤ 0 → B2 → A if 0 → B1 → A factors through 0 → B2 → A. Since A is Noetherian, D2 has a maximal object. Let 0 → A2 → A be a maximal object in D2 . Note that A2 /A1 is well-defined because 0 → A1 → A factors through 0 → A2 → A. We claim that A2 /A1 is weakly σ-stable and µσ (A2 /A1 ) = µσ (A). We first show µσ (A2 /A1 ) = µσ (A). By construction, µσ (A2 ) = µσ (A1 ), so, by the weak seesaw inequality (Lemma 2.2.6), µσ (A2 ) ≤ µσ (A2 /A1 ) with equality exactly when Z(A2 /A1 ) ̸= 0. Since IZ(A2 ) > IZ(A1 ), IZ(A2 /A1 ) > 0, so Z(A2 /A1 ) ̸= 0. Therefore, µσ (A) = µσ (A2 ) = µσ (A2 /A1 ), as needed. We now show A2 /A1 is weakly σ-stable. Assume 0 → B → A2 /A1 is a subobject. We can write B = B ′ /A1 where B ′ is a subobject of A2 , so there is a short exact sequence 0 → A1 → B ′ → B → 0 Since B ′ is a subobject of A2 and 0 → A1 → A factors through the composition 0 → B ′ → A2 → A, by minimality of A2 in C2 , one of the following must hold: ˆ B ′ = A2 , ˆ µσ (B ′ ) ̸= µσ (A), or 57 ˆ IZ(B ′ ) ≤ IZ(A1 ). First, if B ′ = A2 then B = A2 /A1 , so B is not a proper subobject of A2 /A1 . Second, if µσ (B ′ ) ̸= µσ (A), since A is σ-semistable, µσ (B ′ ) < µσ (A) = µσ (A1 ). By the weak seesaw inequality (Lemma 2.2.6), it follows that µσ (A2 /A1 ) = µσ (A1 ) > µσ (B ′ ) > µσ (B), as needed. Third, assume that IZ(B ′ ) ≤ IZ(A1 ). We may additionally assume that µσ (B ′ ) = µσ (A). By minimality of A1 in C1 , it follows that IZ(B ′ ) = IZ(A1 ). Therefore, B ′ is a minimal object of C1 . In other words, B ′ is an object in D1 . Since A1 is a maximal object in D1 and 0 → A1 → A factors through 0 → B ′ → A, we know that B ′ = A1 . It follows that B = B ′ /A1 = 0, so B is not a nonzero subobject of A2 /A1 . All cases considered, we find that A2 /A1 is weakly σ-stable, as desired. If IZ(A2 ) = IZ(A) then A = A2 by definition of C2 and the filtration 0 → A1 → A is a Jordan-Hölder filtration. If IZ(A2 ) < IZ(A) then we can continue this process inductively. Since IZ : Λ → R is discrete, this process will stop in finitely many steps to give us a Jordan-Hölder filtration. Later, we will use a deformation argument to extend this result to tilt stability—where IZ is not necessarily discrete and A is not necessarily Noetherian (Lemma 3.3.2). Lemma 2.3.14 does not extend to σ-semistable objects with IZ(A) = 0 as shown in the following example. In fact, in this example we completely classify coherent sheaves that have Jordan-Hölder filtration with respect to σµDH to illustrate the theory. Example 2.3.15. We first deduce some general facts. If IZ(A) = 0 then any short exact sequence 0 → B → A → C → 0 satisfies IZ(B), IZ(C) = 0. In this case, µσ (B) = µσ (A) = µσ (C) = +∞ Therefore, an object A satisfying IZ(A) = 0 is weakly σ-stable if and only if A contains no nonzero proper subobjects. In other words, and object satisfying IZ(A) = 0 is weakly σ-stable if and only if A is simple. 58 With this in mind, consider the very weak stability condition associated to µH -stability: σµDH . The only nonzero coherent sheaves with no nonzero proper subobjects are skyscraper sheaves. In other words, the weakly σ-stable objects with IZ(·) = 0 are skyscraper sheaves. As we saw in Example 2.2.9, skyscraper sheaves are in fact σ-stable. Assume E is σµDH -semistable. ˆ If dim(X) = 1 then σµDH is a stability function so E has a Jordan-Hölder filtratioon. ˆ Assume dim(X) ≥ 2 and E ∼ = ⊕m i=1 Oxi . Then E has a Jordan-Hölder filtration whose Jordan-Hölder factors are {Oxi }m i=1 . ˆ Assume dim(X) ≥ 2 and 0 < dim(E ) < dim(X). Every subsheaf and quotient of E satisfies IZµDH (·) = 0. Suppose, for contradiction, there is a weak Jordan-Hölder fiiltration of E : 0 = E0 → E1 → E2 → · · · → Em−1 → Em = E . By definition Ei /Ei−1 is weakly σµDH -stable for all i = 1, . . . , m. Therefore, Ei /Ei−1 is a skycraper sheaf, so dim(Ei /Ei−1 ) = 0 for all i. By induction, we find that 0 < dim(E ) = dim(E1 ). Thus, E1 is a weakly σµDH -stable object with IZµDH (E1 ) = 0 and dim(E1 ) > 0, a contradiction. ˆ If dim(X) ≥ 2 and dim(E ) = dim(X) (i.e. IZµDH (E ) = rank(E ) > 0) then E has a Jordan-Hölder filtration by Lemma 2.3.14. By definition of a Jordan-Hölder filtration each Jordan-Hölder factor must be supported everywhere. In particular, by Example 2.2.9, each Jordan-Hölder factor is weakly σµDH - stable but not σµDH -stable. Note that the very weak stability function associated to µD H satisfies both the Harder- Narasimhan and support properties. Example 2.3.16. The category Coh(X) is Noetherian and rank : K0 (Coh(X)) → R is discrete. Therefore, σµDH satisfies the Harder-Narasimhan property by Corollary 2.3.6. 59 By the weak Bogomolov inequality (Lemma 2.1.15), σµDH satisfies the support property D with respect to ∆H . We end this subsection by presenting an important consequence of the support property. Definition 2.3.17. Assume σ = (Z, A) is a very weak stability function satisfying the Harder-Narasimhan property. We say that a set of objects in S ⊆ A has bounded mass if sup{mσ (A) | A ∈ S} < +∞. Recall that mσ is the mass of A with respect to σ (Definition 2.3.2). It is clear that any finite collection of objects in Db (X) has bounded mass. The converse is false; if S has bounded mass then S may not be finite. However, if S has bounded mass and σ = (Z, A) is a very weak stability function satisfying the Harder-Narasimhan property and the support property (plus a technical assumption) then the image in Λ is finite. This result was first shown for K3 surfaces in [Bri08, Lemma 9.2]. Lemma 2.3.18. Assume σ = (Z : Λ → C, A) is a very weak stability function satisfying the Harder-Narasimhan property and the support property with respect to the quadratic form Q on ΛR . If Q has signature (2, rank(Λ) − 2) then every set of bounded mass has finite image in Λ. Proof. Assume that S ⊆ A has bounded mass. It suffices to show that {||A||2 | A ∈ S} is bounded in ΛR for some norm || · || on ΛR . Let M = sup{mσ (A) | A ∈ S}. Since Q has signature (2, dimR (ΛR ) − 2) we can find a basis {e1 , e2 , . . . , er } of ΛR such that ! X k Q(A) = Q ai e i = a21 + a22 − a23 − · · · − a2r . i=1 Since Q is negative definite with respect to the kernel of Z, by definition, Ker(Z) ⊗ R ⊆ span{e3 , e4 , . . . , er }. 60 On the other hand, by the fundamental theorem of linear algebra, dimR Ker(Z) ≥ rank(Λ) − dimR (C) = r − 2, so Ker(Z) ⊗ R = span{e3 , e4 , . . . , er } Thus, we can choose another basis {e′1 , e′2 , e3 , . . . , er } of ΛR such that Q (a1 e′1 + a2 e′2 + a3 e3 + · · · + ar er ) = |Z(a1 e′1 + a2 e′2 )| − a23 − · · · − a2r . where Z is defined on ΛR by linearity. Also, define a norm || · || on ΛR by ||a1 e′1 + a2 e′2 + a3 e3 + · · · + ar er || = |Z(a1 e′1 + a2 e′2 )| + a23 + · · · + a2r . By definition of mass, if Ai /Ai−1 is a σ-semistable factor of A ∈ S then mσ (Ai /Ai−1 ) < mσ (A). Furthermore, we know that A = m P i=1 Ai /Ai−1 in Λ. Therefore, if the set S ′ = {Ai /Ai−1 ∈ A | Ai /Ai−1 is a σ − semistable factor of some A ∈ S}. is finite in Λ then S is finite in Λ as well. Thus, without loss of generality, we may assume that every object A ∈ S is σ-semistable. Let A ∈ S and write A = a1 e′1 + a2 e′2 + a3 e3 + · · · + ar er in ΛR . Since A is σ-semistable, by the support propery, 0 ≤ Q(A) = Q (a1 e′1 + a2 e′2 + a3 e3 + · · · + ar er ) = |Z(A)| − a23 − · · · − a2r . In other words, a23 + a24 + · · · + a2r ≤ |Z(A)|. It follows that ||A|| = |Z(A)| + a23 + · · · + a2r ≤ 2|Z(A)| ≤ 2M. Hence, S is bounded in ΛR and so must be finite, as desired. While the assumption on the signature of Q in Lemma 2.3.18 may seem restrictive, in most cases we can reduce to this case using [Bay19, Lemma 7.2]. 61 2.4 Stability Conditions on a Triangulated Category In this section we recall very weak stability conditions on a triangulated category D. All the definitions and results of this subsection are standard. Broadly, a very weak stability condition on a triangulated category D is a very weak stability function satisfying the Harder-Narasimhan and support properties such that the associated abelian category embeds “nicely” into D. The prototypical example is µH -stability on Coh(X) with the natural embedding into Db (X). Surprisingly, the collection of stability conditions on D satisfy a deformation property - the collection is locally homeomorphic to Ck . In other words, there is a topology on the collection of stability condition on D such that each connected component is a complex manifold. For very weak stability conditions on D the deformation property is much weaker. Namely, there is a topology on the collection of very weak stability conditions on D that locally injects into Ck . We begin this subsection by introducing the heart of a bounded t-structure on D. These are abelian categories that embed “nicely” into D. Definition 2.4.1. Assume D is a triangulated category. We say that an full additive sub- category A of D is the heart of a bounded t-structure if both of the following conditions are satisfied. ˆ If k1 > k2 and A, B ∈ A then HomD (A[k1 ], B[k2 ]) = 0. ˆ For every nonzero object E ∈ D there exists integers k1 > k2 > · · · > km , objects A1 , A2 , . . . , Am ∈ A, and objects E1 , . . . , Em−1 ∈ D that fit into the following collection of distinguished triangles 0 E1 E2 ··· Em−1 E . A1 [k1 ] A2 [k2 ] ··· Am−1 [km−1 ] Am [km ] 62 where Ei Ei+1 Ai+1 denotes the distinguished triangle Ei → Ei+1 → Ai+1 → Ei [1] Traditionally, one defines t-structures then defines its heart. These notions first appear in [BBD82, Definition 1.3.1]. If we only consider bounded t-structures then there is a canonical bijection between t-structures in the sense of [BBD82, Definition 1.3.1] and Definition 2.4.1 by [Bri08, Lemma 3.1]. For this reason, we avoid introducing the theory of t-structures. The following lemma notes some standard facts about hearts of bounded t-structures. We will use these facts tacitly throughout. Lemma 2.4.2 ([BBD82]). Assume A is the heart of a bounded t-structure on D. 1. A is an abelian category. 2. Assume A, B, C ∈ A. Then 0 → B → A → C → 0 is exact in A if and only if B → A → C → B[1] is a distinguished triangle in D. 3. A is stable under extensions. In other words, if B, C ∈ A and B → A → C → B[1] is a distinguished triangle in D then A ∈ A. If A is an abelian category such that there is an equivalence of triangulated categories Db (A) = D then A is the heart of a bounded t-structure on D. However, the converse is false. Specifically, if A is the heart of a bounded t-structure on D then there may not be an equivalence of triangulated categories between Db (A) and D. A counter example is described in [MS17, Exercise 5.3] for Db (P1 ). However, if A is the heart of a bounded t-structure on D then there is a natural isomorphism of Grothendieck groups K0 (Db (A)) = K0 (A) = K0 (D). We defined the Grothendieck group of an abelian category in Definition 2.1.8. One can define the Grothendieck group of a triangulated category similarly. 63 Definition 2.4.3. We define the Grothendieck group of a triangulated category D, written K0 (D), to be the free abelian group generated by objects in D with relations E = F + G whenever there is a distinguished triangle F → E → G → F [1]. By definition of a triangulated category, for any object E ∈ D there is a distinguished triangle E → 0 → E[1] → E[1]. Therefore, by definition of K0 (D), we know that −[E] = [E[1]]. More generally, if k ∈ Z then (−1)k [E] = [E[k]]. Lemma 2.4.4. If A is the heart of a bounded t-structure on A then there is a natural isomorphism K0 (A) = K0 (D). Proof. The inclusion A → D induces a morphism ι : K0 (A) → K0 (D) which is well-defined by Lemma 2.4.2.3. We will construct an inverse ϕ : K0 (D) → K0 (A). Assume E ∈ D. By definition of the heart of a bounded t-structure we can find a collection of distinguished triangles: 0 E1 E2 ··· Em−1 E A1 [k1 ] A2 [k2 ] ··· Am−1 [km−1 ] Am [km ] appearing in the definition of the heart of a bounded t-structure. We define ϕ : K0 (D) → K0 (A) by ϕ([E]) = m km P i=1 [Ai ](−1) . We first claim that ϕ is well-defined. By uniqueness of this collection of distinguished triangles (Lemma 2.4.2.5), it suffices to show that this construction is additive in distin- guished triangles. Assume F → E → G → F [1] is a distinguished triangle. We proceed by induction on m appearing in Definition 2.4.2.b (i.e. the number of distinguished triangles in the collection for E). If m = 1 then E = A[k] for some A ∈ A and k ∈ Z. Therefore, we have a distinguished triangle F → A[k] → G → F [1]. Thus, if 0 G1 G2 ··· Gn−1 G C1 [l1 ] C2 [l2 ] ··· Cn−1 [ln−1 ] Cn [ln ] 64 is a collection of distinguished triangles appearing in Definition 2.4.2.b, then 0 G1 [−1] ··· Gn−1 [−1] G[−1] F C1 [l1 − 1] ··· Cn−1 [ln−1 − 1] Cn [ln − 1] A[k] is such a collection for F . Therefore n ! n ! X X ϕ(F ) + ϕ(G) = [Cn ](−1)li + [A](−1)k + [Cn ](−1)li −1 = [A](−1)k = ϕ(E), i=1 i=1 as desired. Consider the collection of distinguished triangles: 0 E1 E2 ··· Em−1 E . A1 [k1 ] A2 [k2 ] ··· Am−1 [km−1 ] Am [km ] The composition Em−1 → E → G gives us a distinguished triangle Em−1 → G → Cone(Em−1 → G). By applying the octohedral axiom to the distinguished triangles: Em−1 → E → Am [km ] Em−1 → G → Cone(Em−1 → G) E → G → F [1] we obtain a distinguished triangle Am [km ] → Cone(Em−1 → G) → F [1]. The inductive hypothesis applies to Am [km ] and Em−1 , so by considering the distinguished triangles Cone(Em−1 → G)[−1] → Em−1 → G F → Am [km ] → Cone(Em−1 → G). we find that ϕ(Em−1 ) = ϕ(G) − ϕ(Cone(Em−1 → G)) ϕ(Am [km ]) = ϕ(F ) + ϕ(Cone(Em−1 → G)). 65 It follows that ϕ(E) = ϕ(Em−1 ) + ϕ(Am [km ]) = ϕ(G) + ϕ(F ), as claimed. Thus, we have shown that ϕ : K0 (D) → K0 (A) is well-defined. By construction, ϕ and ι are inverses. Remark 2.4.5. Assume that σ = (Z : Λ → C, A) is a very weak stability function and A is the heart of a bounded t-structure. If E ∈ D, we can identify E with its image in K0 (D) and so an element in K0 (A) by Lemma 2.4.4. Thus, we obtain an element in Λ associated with E. We will define Z(E) to be Z applied to this element in E. In particular, if E ∈ D then we can define the slope µσ (E). On the other hand, if we have a group homomorphism K0 (D) → Λ then we naturally obtain a group homomorphism K0 (A) = K0 (D) → Λ by Lemma 2.4.4. In short, fixing a group homomorphism K0 (D) → Λ is equivalent to fixing a group homomorphism K0 (A) → Λ. The definition of the heart of a bounded t-structure seems especially restrictive, but there are a few standard techniques to construct them. The first technique is to give an equivalence of triangulated categories Db (A) = D. We will eventually restrict our attention to Db (X) = Db (Coh(X)) so we have the “standard” heart Coh(X). Surprisingly, if X has a full strong exceptional collection and Q is the quiver associated to this collection then there is equivalence of triangulated categories Db (X) = Db (Repk (Q)) (see [Bon90, Theorem 6.2 and definitions within]). Therefore, Repk (Q) is the heart of a bounded t-structure on Db (X). However, since full strong exceptional collections are difficult to construct, this surprising fact is not sufficiently general for our applications. For example, a folklore conjecture is that a smooth projective surface admitting a full strong exceptional collection is rational. The second technique is to to find a semiorthogonal decomposition of D such that the semiorthogonal subcategories each admit the heart of a bounded t-structure. If each of the semiorthogonal subcategories is sufficiently “nice” (i.e. they satisfy a six functor formalism) 66 then we can glue the hearts of bounded t-structure of each subcategory to obtain a heart of a bounded t-structure on D ([BBD82, Theorem 1.4.10]). The third technique for constructing the heart of a bounded t-structure is by starting with the heart of a bounded t-structure and tilting with respect to a torsion pair. This method is due to [HRS96]. This is the only construction that we describe in more detail. Definition 2.4.6. Assume A is an abelian category. We say that a pair of full additive subcategories (T , F) of A is a torsion pair if the following two conditions hold: ˆ If T ∈ T and F ∈ F then HomA (T, F ) = 0, ˆ If A ∈ A then there exists a short exact sequence 0→T →A→F →0 with T ∈ T and F ∈ F. We call T a torsion class and F a torsion-free class. The short exact sequence appearing in the definition above is unique up to isomorphism of short exact sequences. The prototypical example of a torsion pair is given by the full additive subcategory of Coh(X) generated by torsion sheaves and the full additive subcategory generated by torsion- free sheaves (forming the torsion and torsion-free classes respectively). Torsion pairs first appear in [Dic66, Bottom of Page 224] where they are called a tor- sion theory. Dickson shows that torsion pairs are easy to construct on categories with all coproducts or products [Dic66, Theorem 2.3]. A similar argument gives a construction for Noetherian or Artinian categories. Using a stability function to construct a torsion pair was first used in [Bri08, Lemma 6.1]. The following is a natural generalization of that result to very weak stability functions. 67 Lemma 2.4.7. Assume that σ = (Z, A) is a very weak stability satisfying the Harder- Narasimhan property. The full additive subcategories of A generated by T = {A ∈ A | Any nonzero quotient A → C → 0 satisfies µσ (C) > 0} and F = {A ∈ A | Any nonzero subobject 0 → B → A satisfies µσ (B) ≤ 0} form a torsion pair. Proof. Let T ∈ T and F ∈ F. By definition of T and F, we know that µ− σ (T ) > 0 and µ+σ (F ) ≤ 0. It follows by Lemma 2.3.4 that HomA (T, F ) = 0, as needed. Consider A ∈ A with a Harder-Narasimhan filtration 0 → A1 → · · · → Am−1 → A. If µ− σ (A) > 0 then A ∈ T and a short exact sequence of the necessary form is given by 0 → A → A → 0 → 0, as needed. Therefore, we may assume that µ− σ (A) ≤ 0. If µ+σ (A) ≤ 0 then A ∈ F and a short exact sequence of the necessary form is given by 0 → 0 → A → A → 0, as needed. Therefore, we may also assume that µ+ σ (A) > 0. Choose the largest integer i such that µσ (Ai /Ai−1 ) > 0 (which exists because µ+ σ (A) > 0). Since Ai /Ai−1 is the minimal destabilizing quotient of Ai , by Lemma 2.3.3.3, Ai ∈ T . It remains to show that A/Ai ∈ F. Note that 0 → Ai+1 /Ai → Ai+2 /Ai → · · · → Am−1 /Ai → A/Ai is a Harder-Narasimhan filtration of A/Ai . In particular, Ai+1 /Ai is a maximal destabilizing suboject of A/Ai . Note that i + 1 ≤ m by the assumption µ−1 σ (A) ≤ 0, so Ai+1 is defined. By maximality of i, we find that 0 ≥ µσ (Ai+1 /Ai ) = µ+ σ (A/Ai ). On the other hand, since T ∈ T , we know that µ− σ (T ) > 0. It follows that A/Ai ∈ F, as desired. We give two more examples of torsion-pairs in Lemma 3.2.1 If we have a torsion pair, then the following result allows us to construct a heart of a bounded t-structure on the derived category. 68 Lemma 2.4.8 ([HRS96, Proposition 2.1]). Assume A is an abelian category. Let (T , F) be a torsion pair in A. 1. The full additive subcategory of Db (A) generated by A♯ = {E ∈ Db (A) | H0 (E) ∈ T , H−1 (X) ∈ F, Hi (X) = 0 otherwise} is the heart of a bounded t-structure on Db (A). In this case, we say that A♯ is obtained from A by tilting with respect to (T , F). 2. (F[1], T ) is a torsion pair in A♯ . Tilting the heart of a bounded t-structure generally does not preserve categorical prop- erties. For example, A can be Noetherian while A♯ is not (and vice versa). We now define stability conditions on a triangulated category D. Definition 2.4.9. Fix a group finite rank lattice Λ and a group homomorphism K0 (D) → Λ. 1. A very weak stability condition on D is a very weak stability function σ = (Z : Λ → C, A) such that ˆ A is the heart of a bounded t-structure on D and ˆ σ satisfies both the Harder-Narasimhan and support properties. 2. A Bridgeland stability condition (also called a stability condition) is a very weak stability condition σ = (Z : Λ → C, A) such that σ is a stability function (i.e. Z satisfies the positivity property of Definition 2.2.1). 3. Assume σ = (Z, A) is a very weak stability condition on D and A ∈ A is nonzero. We say that A is σ-(semi)stable (resp. weakly σ-(semi)stable) if A is σ-(semi)stable (resp. weakly σ-(semi)stable) in the sense Definition 2.2.3. In particular, if E ∈ D but E ̸∈ A then we will not discuss stability of E. 69 The surprising result of [Bri07] is that stability conditions on Db (X) with X a smooth variety actually form a topological space where each connected component is a complex manifold! Bridgeland’s argument can be generalized to include very weak stability conditions but we only obtain obtain a local embedding into Cn . Proposition 2.4.10 ([Bri07]). Fix a finite rank quotient K0 (X) → Λ and let WStab(X, Λ) be the set of all very weak stability conditions σ = (Z : Λ → C, A). We give WStab(X, Λ) the coarsest topology such that ˆ WStab(X, Λ) → HomZ (Λ, C) given by (Z, A) 7→ Z is continuous. ˆ WStab(X, Λ) → R ∪ {+∞} given by σ 7→ µ+ b σ (E) is continuous for all E ∈ D (X). ˆ WStab(X, Λ) → R ∪ {+∞} given by σ 7→ µ− b σ (E) is continuous for all E ∈ D (X). − Recall µ+σ and µσ are the slopes of the maximal destabilizing and minimal destabilizing ob- jects respectively- see Definition 2.3.2 The morphism WStab(X, Λ) → HomZ (Λ, C) given by (Z, A) 7→ Z is a continuous local injection. Furthermore, let Stab(X, Λ) is the subset of WStab(X, Λ) consisting of stability condi- tions with the subspace topology. The induced continuous function Stab(X, Λ) → HomZ (Λ, C) is a local homeomorphism. In other words, each connected component of Stab(X, Λ) is a complex manifold of dimension rank(Λ). 70 CHAPTER 3 TILT STABILITY In this chapter, we recall the construction of tilt stability. In addition, we describe the wall structure of the (H, D)-slice. This includes our new wall-crossing theorem. The chapter is structured as follows. §3.1 We recall the construction of tilt stability which is a family σα,β tilt of very weak stability functions parameterized by (β, α) ∈ R × R>0 , ample divisor H, and Q-divisor D (culumatively called an (H, D)-slice). We also completely characterize objects with tilt good quoitients and discuss a deformation property of σα,β -pure objects. §3.2 We show σα,βtilt is a very weak stability condition when β ∈ Q. In addition, we prove a tilt general form of the Large Volume Limit—which is a tool to compare σα,β -stable objects with µH -stable sheaves. §3.3 We show σα,β tilt is a very weak stability condition when β ∈ R. Using the theory of pure objects we developed, this argument is much simpler than previous arguments. tilt tilt In addition, we show that any σα,β -semistable object, E, with IZα,β (E) ̸= 0 has a weak Jordan-Hölder filtration. §3.4 We discuss the wall and chamber structure of the (H, D)-slice. §3.5 We give explicit bounds on the largest wall in the (H, D)-slice. §3.6 We prove the main technical theorem of this thesis: our wall-crossing result for tilt stability. 3.1 Tilt Stability as a Very Weak Stability Function In this subsection, we introduce tilt stability. For each β ∈ R, there is a heart of a bounded t-structure CohD+βH H (X) on Db (X). The general theory first appeared in [Bri08] for K3 71 surfaces, [AB13] for any surface, and [BMT14] for any variety. We also show that for each positive real number α > 0 there is a very weak stability tilt D+βH tilt function σα,β with heart CohH (X). We end this subsection by investigating σα,β -pure objects and objects with good quotients. We can completely classify objects in CohD+βH H (X) that have good quotients and, surprisingly, it contains more than just simple objects (Lemma tilt 3.1.13). We also show σα,β -pure objects have especially nice deformation properties (Lemma 3.1.10). Definition 3.1.1. Fix a Q-divisor D and an ample divisor H on X. For each β ∈ R we define ˆ THD+βH (X) to be the full subcategory of Coh(X) generated by {E ∈ Coh(X) | Any nonzero quotient E → G → 0 satisfies µD+βH H (G ) > 0} ˆ FHD+βH (X) to be the full subcategory of Coh(X) generated by {E ∈ Coh(X) | Any nonzero subsheaf 0 → F → E satisfies µD+βH H (F ) ≤ 0}. ˆ CohH D+βH (X) to be the full subcategory of Db (X) generated by {E ∈ Db (X) | H −1 (E) ∈ FHD+βH (X), H 0 (E) ∈ THD+βH (X), H i (E) = 0 if i ̸= −1, 0} Remark 3.1.2. Notice that if rank(E ) ̸= 0 then D+βH H n−1 · chD+βH 1 (E ) µH (E ) = rank(E ) n−1 H · ch1 (E ) − H n−1 · (D + βH) · rank(E ) = rank(E ) n−1 D H · ch1 (E ) − βH n · rank(E ) = rank(E ) = µDH (E ) − βH n = µDH (E ) − βH n 72 D+βH Therefore, µH (E ) > 0 if and only if µD n H (E ) > βH (and similarly for ≤). In short, THD+βH (X) can also be written as the full subcategory of Coh(X) generated by {E ∈ Coh(X) | Any nonzero quotient E → G satisfies µD n H (G ) > βH } while FHD+βH (X) can also be written as the full subcategory of Coh(X) generated by {E ∈ Coh(X) | Any nonzero subsheaf F → E satisfies µD H (F ) ≤ βH } n An injection of coherent sheaves does not induce an injection in CohD+βH H (X). However, a maximal µH -destabilizing subsheaf does induce an injection in CohD+βH H (X): Lemma 3.1.3. Let E ∈ Coh(X). 1. Assume E is not µH -semistable. If 0 → F → E is a maximal µH -destabilizing subobject then µ+ + H (E /F ) ≤ µH (F ). In particular, in this case, 0 → F [1] → E [1] is injective in CohD+βHH (X) for all β ≥ µD+ H (E ). 2. Assume E is µH -semistable. If 0 → F → E is a coherent subsheaf satisfying µH (F ) = µH (E ) = µH (E /F ) then 0 → F [1] → E [1] is injective in CohD+βH H (X) for all β ≥ µH (E ). Proof. Let E ∈ Coh(X). 1. We proceed by induction on the length of the Harder-Narasimhann filtration of E . If E has a Harder-Narasimhan filtration of length 2 then E /F is µH -stable with µH (E /F ) < µH (F ), as claimed. Consider a Harder-Narasimhan filtration 0 = E0 → E1 → · · · → Em−1 → Em = E . 73 By the inductive hypothesis, we know µ+ H (Em−1 /E1 ) < µH (E1 ). We have the following short exact sequence 0 → Em−1 /E1 → E /E1 → E /Em−1 → 0. Therefore µ+ + + H (E /E1 ) ≤ max{µH (E /Em−1 ), µH (Em−1 /E1 )}. However, µ+ H (E /Em−1 ) = µH (E /Em−1 ) < µH (E1 ) by definition of a Harder-Narasimhan filtration and µ+H (Em−1 /E1 ) ≤ µH (E1 ) by the inductive hypothesis, so we obtain the desired inequality. It follows that F [1], E [1], E /F [1] ∈ CohD+βH H (X) and so 0 → F [1] → E [1] → E /F [1] → 0 is exact in CohD+βHH (X), as claimed. 2. Since E is µH -semistable, F and E /F are also µH -semistable of slope µH (E ). In particular, F [1], E [1], E /F [1] ∈ CohD+βH H (X) and so 0 → F [1] → E [1] → E /F [1] → D+βH 0 is exact in CohH (X), as claimed. All coherent sheaves and their shift by [1] lie in CohD+βH H (X) for some β ∈ R depending on µD− D+ D− D+ H (E ) and µH (E ). Recall that µH (E ) (resp. µH (E )) is the slope of a minimal destabilizing quotient (resp. maximal destabilizing subobject) see Definition 2.3.2. Lemma 3.1.4. Assume E ∈ Coh(X) is nonzero. H (E ) if and only if E ∈ TH (X). In particular, E is a torsion sheaf if and D+βH 1. β < µD− only if E ∈ THD+βH (X) for all β ∈ R. H (E ) if and only if E ∈ FH (X) (and so E [1] ∈ CohD+βH D+βH 2. β ≥ µD+ H (X)). D+βH 3. E ∈ CohH (X) if and only if µD+ H (H −1 (E)) ≤ β < µD− 0 H (H (E)). Proof. Assume E ∈ Coh(X) is nonzero. 74 1. Assume β < µD− H (E ). Therefore, by Lemma 2.3.3.3, every quotient E → G → 0 H (E ) ≤ µH (G ) so E ∈ TH D+βH satisfies β < µD− D (X), as needed. Conversely, assume that E ∈ THD+βH (X). Therefore, every nonzero quotient E → G → H (G ) > β. In particular, if E → E /Em−1 → 0 is a minimal destabilizing 0 satisfies µD quotient then D− β < µDH (E /Em−1 ) = µH (E ), as desired. If E is a torsion sheaf then E is µH -semistable of slope µD− D H (E ) = µH (E ) = +∞. Therefore, E ∈ THD+βH (X) for all β ∈ R. Conversely, if E ∈ THD+βH (X) for all β ∈ R then β < µD− H (E ) for all β ∈ R. By Lemma H (E ) ≤ µH (E ), so µH (E ) = +∞. In other words, rank(E ) = 0 so E is 2.3.3.3, β < µD− D D a torsion sheaf, as desired. 2. Assume β > µD+ H (E ). By Lemma 2.3.3.2, every subsheaf 0 → F → E satisfies H (E ) ≥ µH (F ). Therefore, E ∈ FH D+βH β ≥ µD+ D (X), as needed. Conversely, assume E ∈ FHD+βH (X). By definition every nonzero subobject 0 → F → E satisfies µDH (F ) ≤ β. In particular, if 0 → E1 → E is a maximal destabilizing subobject then D+ β ≥ µDH (E1 ) = µH (E ), as desired. 3. This result follows from parts 1 and 2. The following lemma shows if E ∈ CohD+βH H (X) then E will still be in the heart for an infinitesimal deformations of β to the right. We need an additional assumption if we want to deform β to the left. We will see in Lemma 3.1.10 that this additional assumption is essentially purity with respect to tilt stability. 75 Corollary 3.1.5. Assume E ∈ CohD+β H 0H (X). 1. There exists ε > 0 such that E ∈ CohD+βH H (X) for all β ∈ [β0 , β0 + ε). 2. There exists ε > 0 such that E ∈ CohD+βH H (X) for all β ∈ (β0 − ε, β0 ] if and only if µD+ H (H −1 (E)) ̸= β0 or H −1 (E) = 0. Proof. Both of these results follow from Lemma 3.1.4.3. Deformations of the heart are generally well-behaved with respect to subobjects and quotients: Lemma 3.1.6. Assume E ∈ CohD+β H 0H (X) and E ∈ CohD+β H 1H (X) for real numbers β0 < β1 . 1. If 0 → F → E is a subobject in CohD+β H 1H (X) then F ∈ CohD+β H 0H (X). 2. If E → G → 0 is a quotient object in CohD+β H 0H (X) then G ∈ CohD+β H 1H (X). Proof. 1. Since β0 < β1 , we know that H 0 (F ) ∈ THD+β0 H (X). Therefore, it suffices to show H −1 (F ) ∈ FHD+β0 H (X). If F ̸∈ CohD+β H 0H (X) then we can find a subsheaf 0 → F → H −1 (F ) such that µD+β H 0H (F ) > 0. It follows that 0 → F → H −1 (F ) → H −1 (E) is a subsheaf satisfying µD+β H 0H (F ) > 0 so E ̸∈ CohH D+β0 H (X), as claimed. 2. The dual argument holds. For all positive real numbers α we define a very weak stability function on the heart CohD+βH H (X): Definition 3.1.7. Define ΛD H to be the image of H n−2 · ch≤2 : K0 (Db (X)) → Q⊕3 (the image lies in Q⊕3 because D is a Q-divisor). For all α ∈ R>0 and β ∈ R, we define the function σα,β tilt tilt = (Zα,β : Λ → C, CohD+βH H (X)) where α2 H n √ tilt Zα,β (E) = −H n−2 · chD+βH 2 (E) + rankD+βH (E) + −1H n−1 · chD+βH 1 (E). 2 76 We denote the associated slope by α2 −H n−2 · chD+βH 2 (E) + Hn · rank(E) µtilt α,β (E) =− 2 . H n−1 · chD+βH 1 (E) tilt To see that Zα,β is actually a group homomorphism ΛD H → C, notice that α2 n RZα,β tilt (E) = −H n−2 · ch2D+βH (E) + H · rankD+βH (E) 2  2 α2  n−2 D D β = −H · ch2 (E) + β degH (E) − − H n · rank(E) 2 2 and tilt IZα,β (E) = H n−1 · chD+βH 1 (E) = degD n H (E) − βH · rank(E). Remark 3.1.8. In fact, the calculation above shows that if rank(E) ̸= 0 then we can rewrite µtilt D D α,β in terms of µH , νH , α, and β: tilt RZα,β (E) µtilt α,β (E) = − tilt IZα,β (E)   β2 α2 H n−2 · chD D 2 (E) − β degH (E) + 2 − 2 H n · rank(E) = degD n H (E) − βH · rank(E)  2 2  β D νH (E) − βµD H (E) + 2 − α2 H n = . µDH (E) − βH n In particular, we find that 2 tilt 1 1 lim 2 µα,β (E) = − D n = − D+βH . α→∞ α µH (E) − βH µH (E) Lemma 3.1.9. Assume (β, α) ∈ R × R>0 and E ∈ CohD+βH H (X). tilt 1. IZα,β (E) ≥ 0. tilt 2. If IZα,β (E) = 0 then H n−2 · chD+βH ≤2 (H 0 (E)) = (0, 0, ≥ 0) 77 and H n−2 · chD+βH ≤2 (H −1 (E)) = (≥ 0, 0, ≤ 0). D+βH In particular, σα,β tilt = (Zα,β tilt : ΛD H → C, CohH (X)) is a very weak stability function (and a stability function when dim(X) = 2). tilt 3. IZα,β (E) = 0 if and only if codim(E) ≥ 2 or β = µD H (E). tilt 4. Zα,β (E) = 0 if and only if codim(E) ≥ 3. Proof. Assume (β, α) ∈ R × R>0 and E ∈ CohD+βH H (X). 1. There is a short exact sequence 0 → H −1 (E)[1] → E → H 0 (E) → 0 with H −1 (E) ∈ FHD+βH (X) and H 0 (E) ∈ THD+βH (X). By additivity, IZα,β tilt (E) = IZα,β tilt (H 0 (E)) − IZα,β tilt (H −1 (E)) = degD+βH H (H 0 (E)) − degD+βHH (H −1 (E)). Since H 0 (E) ∈ THD+βH (X), by definition, µD+βH H (H 0 (E)) > 0. If rank(H 0 (E)) = 0 then by Lemma 2.1.3, degD+βH H (H 0 (E)) ≥ 0. Otherwise, since µD+βH H (H 0 (E)) > 0, D+βH degH (H 0 (E)) > 0. In either case, degD+βH H (H 0 (E)) ≥ 0. On the other hand, since H −1 (E) ∈ FHD+βH (X), µD+βH H (H −1 (E)) ≤ 0 or H −1 (E) = 0. Either way, we D+βH find that degH (H −1 (E)) ≤ 0. In all, we have shown that tilt IZα,β (E) = degD+βH H (H 0 (E)) − degD+βH H (H −1 (E)) ≥ 0, as desired. D+βH 2. Assume IZα,β tilt (E) = 0 so degH (H 0 (E)) = degD+βH H (H −1 (E)). Therefore, by the same argument as part 1, 0 ≤ degD+βH H (H 0 (E)) = degD+βH H (H −1 (E)) ≥ 0. 78 Since degD+βH H (H 0 (E)) = 0 and H 0 (E) ∈ THD+βH (X), rank(H 0 (E)) = 0. Since H n−1 ·chD+βH ≤1 (H 0 (E)) = (0, 0), by Lemma 2.1.3, H n−1 ·chD+βH ≤2 (H 0 (E)) = (0, 0, ≥ 0). Since degD+βH H (H −1 (E)) = 0 and H −1 (E) ∈ THD+βH (X), H −1 (E) is µH -semistable (or 0). Therefore, by Bogomolov inequality, D+βH degD+βH H (H −1 (E))2 − 2 rank(H −1 (E))H n−2 · chD+βH 2 (H −1 (E)) = ∆H (H −1 (E)) ≥0 D+βH Since H n−1 · ch≤1 (≥ 0, 0), by rearranging the above inequality, H n−2 · chD+βH ≤2 (≥ 0, 0, ≤ 0), as claimed. By Lemma 2.4.7 and Lemma 2.4.8, we know that CohD+βH H (X) is the heart of a bounded t-structure on Db (X). Furthermore, by part 1, IZα,β tilt (E) ≥ 0 for all E ∈ Db (X). Moreover, if IZα,β tilt (E) = 0, by the argument above, tilt RZα,β (E) = RZα,β tilt (H 0 (E)) − RZα,β tilt (H −1 (E)) α2 = −H n−2 · chD+βH 2 (H 0 (E)) + rank(H 0 (E)) 2 α2 + H n−2 · chD+βH 2 (H −1 (E)) − rank(H −1 (E)) 2 ≤ 0, tilt as needed. Hence, σα,β is a very weak stability condition, as desired. 3. First assume IZα,β tilt (E) = 0. Therefore, degD H (E) − β rank(E) = 0. If rank(E) ̸= 0 then β = degD D H (E)/ rank(E) = µH (E), as needed. Therefore, assume rank(E) = 0. It follows by part 2 that H n−2 · chD+βH ≤2 (H 0 (E)) = (0, 0, ≥ 0) and H n−2 · chD+βH≤2 (H −1 (E)) = (0, 0, ≤ 0). Therefore, codim(E) ≥ 2 by Lemma 2.1.3, as needed. 79 Second, assume β = µD H (E) or codim(E) ≥ 2. In the first case, the result follows by direct computation. In the second case, the result follows by Lemma 2.1.3. tilt tilt 4. First assume Zα,β (E) = 0. In particular, RZα,β (E) = 0 so tilt 0 = RZα,β α2 = −H n−2 · chD+βH 2 (H 0 (E)) + rank(H 0 (E)) 2 α2 + H n−2 · chD+βH 2 (H −1 (E)) − rank(H −1 (E)) 2 tilt However, IZα,β (E) = 0 as well, by part 2, H n−2 · chD+βH ≤2 (H 0 (E)) = (0, 0, ≥ 0) and H n−2 · chD+βH ≤2 (H −1 (E)) = (≥ 0, 0, ≤ 0). tilt Using these inequalities and the equality involving RZα,β , it follows that H n−2 · chD+βH ≤2 (H 0 (E)) = (0, 0, 0) and H n−2 · chD+βH ≤2 (H −1 (E)) = (0, 0, 0). In other words, by Lemma 2.1.3, codim(E) ≥ 3, as desired. tilt We note σα,β -pure objects can be deformed to the left in CohD+βH H (X): Lemma 3.1.10. Assume E ∈ CohD+βH H (X) is nonzero. If E is σα,β tilt -pure for some α > 0 ′ then there exists ε > 0 such that E ∈ CohD+β H H (X) for all β ′ ∈ (β − ε, β]. Proof. It suffices to show β0 ̸= µD+ H (H −1 (E)). With this in mind, consider a maimxal µH - destabilizing subobject 0 → F → H −1 (E). By Lemma 3.1.3, 0 → F [1] → H −1 (E)[1] is in- jective in CohHD+βH (X). Furthermore, by definition, there is an injection 0 → H −1 (E)[1] → 80 E in CohD+βH H (X). Thus, F [1] is a subobject of E. Since E is σα,βtilt tilt -pure, IZα,β (F [1]) ̸= 0. In other words, −1 µD+ H (H (E)) = µD H (F ) ̸= β, as desired. We end this section by discussing reflexive sheaves and then classifying objects with good tilt quotients with respect to σα,β . Definition 3.1.11. A nonzero coherent sheaf E on X is said to be reflexive if any of the equivalent conditions are satisfied: ˆ The natural morphism E → E ∨∨ is an isomorphism. ˆ E is torsion-free and S2 (i.e. if x ∈ X satisfies dim(OX,x ) ≥ 2 then depthx Ex ≥ 2). ˆ E is torsion-free and satisfies the following property If 0 → E → E ′ → G → 0 is a short exact sequence with codim(Supp(G )) ≥ 2 then the short exact sequence splits. The equivalence of the first and second statement is [Har80, Proposition 1.3]. The equivalence of the second and third statements is a local cohomology calculation in view of [Har80, Proposition 1.6]. Furthermore, by definition, a reflexive sheaf is torsion-free, and every locally free sheaf is reflexive. Lemma 3.1.12. Assume E ∈ Coh(X). 1. Assume β ≥ µD+ H (E ). E is reflexive if and only if the following property holds If 0 → F → E [1] → G → 0 is a short exact sequence in CohD+βH H (X) with F ̸= 0 then codim(F ) ≤ 1. 81 2. Assume β < µD− H (E ). Every nonzero subsheaf of E is supported in codimension at most 1 (e.g. E is pure of codimension at most 1) if and only if the following property holds If 0 → F → E → G → 0 is a short exact sequence in CohD+βH H (X) with F ̸= 0 then codim(F ) ≤ 1. Proof. Assume E ∈ Coh(X). 1. First assume that E is reflexive. We have the induced exact sequence arising from cohomology: 0 → H −1 (F ) → E → H −1 (G) → H 0 (F ) → 0. We know that H −1 (F ) is torsion-free, so if H −1 (F ) ̸= 0 we are done. If H −1 (F ) = 0 then we have the following short exact sequence 0 → E → H −1 (G) → H 0 (F ) → 0. Conversely, consider the short exact sequence 0 → E → E ∨∨ → E ∨∨ /E → 0. E ∨∨ is torsion-free, so E ∨∨ ∈ FHD (X). Similarly, codim(E ∨∨ /E ) ≥ 2, so E ∨∨ /E ∈ THD (X). In all, this shows that this exact sequence is of the form given in the lemma statement, so codim(E ∨∨ /E ) ≤ 1. However, as noted above, codim(E ∨∨ /E ) ≥ 2 Therefore, we find that E ∨∨ /E = 0, so E is reflexive, as desired. 2. First assume every subsheaf of E is supported in codimension at most 1. Let 0 → F → E → G → 0 be a short exact sequence in CohD+βH H (X) with F ̸= 0. We have the following exact sequence from cohomology: f 0 → H −1 (G) → H 0 (F ) → − E → H 0 (G) → 0. 82 Therefore, we have the following short exact sequence 0 → H −1 (G) → H 0 (F ) → I m(f ) → 0 If H −1 (G) = 0 then H 0 (F ) = I m(f ), so 1 ≥ codim(H 0 (F )) = codim(F ) since every subsheaf of E is supported in codimension at most 1. If H −1 (G) ̸= 0 then H −1 (G) is torsion-free so H 0 (F ) is supported in codimension 0, as needed. The converse argument is similar to part 1. Lemma 3.1.13. Assume E ∈ CohD+βH H (X). E has good quotients with respect to σα,β tilt if and only if H 0 (E) = 0 or dim(X) = 2. Proof. Assume dim(X) ≥ 3 and H 0 (E) ̸= 0. Consider a closed point ι : x → X with surjection H 0 (E) → H 0 (E) ⊗ ι∗ Ox → 0. Since dim(X) ≥ 3, by Lemma 2.1.3, tilt Zα,β (H 0 (E) ⊗ ι∗ Ox ) = 0 Furthermore, by construction, there is a surjection E → H 0 (E)⊗ι∗ Ox → 0. Since H 0 (E) ̸= 0, H 0 (E) ⊗ ι∗ Ox is a nonzero quotient of E satisfying Zα,β tilt (H 0 (E) ⊗ ι∗ Ox ) = 0. In other tilt words, E does not have good quotients with respect to σα,β , as claimed. For the converse, assume dim(X) = 2 or H 0 (E) = 0. If dim(X) = 2 then σα,β tilt is a Bridgeland stability condition, so the result follows by Lemma 2.3.13.5. Thus, we may assume dim(X) ≥ 3 and H 0 (E) = 0. Therefore, consider a quotient E → G → 0 in CohD+βH H tilt (X) satisfying Zα,β (G) = 0. Since Zα,βtilt (G) = 0, by Lemma 3.1.9, codim(G) = 0. In particular, since H −1 (G) is torsion-free, H −1 (G) = 0. Furthermore, since H 0 (E) = 0, H 0 (G) = 0. Hence, G = 0 and so E has good quotients, as claimed. 83 3.2 Tilt Stability is a Very Weak Stability Condition for Rational β tilt We prove that if β ∈ Q then σα,β satisfies the Harder-Narasimhan property. Using a similar tilt argument, we also show that the collection of all σα,β -pure objects form a torsion-free class in CohD+βH H (X). In order to prove the support property, we also discuss the Large Volume Limit which is independently useful. Parts 1 and 2 of the following lemma were first shown in [BMT14, Lemma 3.2.4]. Parts 3 and 4 are new. Lemma 3.2.1. Assume D is a Q-divisor and H an ample divisor. 1. If 0 → E1 → E2 → · · · → E is an ascending chain in CohD+βH H (X) with IZα,β tilt (Ei ) = tilt IZα,β (Ei+1 ) for all i ≫ 0 then Ei = Ei+1 for all i ≫ 0. 2. If (β, α) ∈ Q × R>0 then CohD+βH H (X) is Noetherian and IZα,β tilt : ΛDH (X) → R is discrete. tilt In particular, if (β, α) ∈ Q × R>0 then σα,β satisfies the Harder-Narasimhan property. 3. Define T and F to be the full additive subcategories of CohD+βH H (X) generated by T = {E ∈ CohD+βH H (X) | IZα,β (E) = 0} F = {E ∈ CohD+βH H (X) | E is σα,β tilt − pure} respectively. If (β, α) ∈ R × R>0 then (T , F) is a torsion pair in CohD+βH H (X). 4. Define T and F to be the full additive subcategories of CohD+βH H (X) generated by D+βH tilt T = {E ∈ CohH (X) | E has good quotients with respect to σα,β } F = {E ∈ CohD+βH H (X) | Zα,βtilt (E) = 0} respectively. If (β, α) ∈ R × R>0 then (T , F) is a torsion pair in CohD+βH H (X). Proof. Assume D is a Q-divisor and H an ample divisor. 84 1. Assume 0 → E1 → E2 → · · · → E is an ascending chain in CohD+βH H (X) satisfying tilt tilt IZα,β (Ei ) = IZα,β (Ei+1 ) for all i ≫ 0. Without loss of generality, we may assume that this equality holds for all i > 0. tilt tilt tilt Since IZα,β (Ei ) = IZα,β (Ei+1 ), so by additivity, IZα,β (Ei /Ei+1 ) = 0. By Lemma 3.1.9.2, rank(Ei /Ei+1 ) ≥ 0 for all i ≫ 0, so by additivity rank(Ei ) ≥ rank(Ei+1 ) for all i ≫ 0. Since Ei is a subobject of E, there is an injection 0 → H −1 (Ei ) → H −1 (E) and so rank(Ei ) = rank(H 0 (Ei )) − rank(H −1 (Ei )) ≥ 0 − rank(H −1 (E)). Thus, rank(Ei ) is discrete, bounded below by − rank(H −1 (E)), and decreasing; so rank(Ei ) = rank(Ei+1 ) for all i ≫ 0. By additivity, rank(Ei /Ei+1 ) = 0 for all i ≫ 0 and so codim(Ei /Ei+1 ) ≥ 2 by Lemma 3.1.9.3. In particular, H −1 (Ei /Ei+1 ) = 0 for all i ≫ 0. Without loss of generality, we may assume H −1 (Ei /Ei+1 ) = 0 for all i > 0. On the other hand, there is a chain of surjections E → E/E1 → E/E2 → · · · D+βH in CohH (X) which induces a chain of surjections in Coh(X): H 0 (E) → H 0 (E/E1 ) → H 0 (E/E2 ) → · · · . Since Coh(X) is Noetherian, the above chain of surjections eventually stabilizes, so H 0 (E/Ei ) = H 0 (E/Ei+1 ) for all i ≫ 0. Without loss of generality, we may assume that this isomorphism holds for i > 0. There is a short exact sequence 0 → Ei+1 /Ei → E/Ei → E/Ei+1 → 0 D+βH in CohH (X) which induces the following exact sequence in Coh(X): 0 H −1 (Ei+1 /Ei ) H −1 (E/Ei ) H −1 (E/Ei+1 ) . H 0 (Ei+1 /Ei ) H 0 (E/Ei ) H 0 (E/Ei+1 ) 0 85 Using the vanishing of H −1 (Ei /Ei+1 ) and the isomorphisms above, we have the fol- lowing short exact sequence in Coh(X): 0 → H −1 (E/Ei ) → H −1 (E/Ei+1 ) → H 0 (Ei+1 /Ei ) → 0 Taking the dual, we obtain the following exact sequence in Coh(X): 0 H 0 (Ei+1 /Ei )∨ H −1 (E/Ei+1 )∨ H −1 (E/Ei )∨ E xt1 (H 0 (Ei+1 /E1 ), OX ) E xt1 (H −1 (E/Ei+1 ), OX ) ··· As stated above, codim(H 0 (Ei /Ei+1 )) ≥ 2 so H 0 (Ei /Ei+1 )∨ = 0 and E xt1 (H 0 (Ei+1 /E1 ), OX ) = 0. Therefore, we have an isomorphism H −1 (E/Ei )∨ = H −1 (E/Ei+1 )∨ for all i > 0. Since H −1 (E/Ei ) is torsion-free or 0, there is a natural injection H −1 (E/Ei ) → H −1 (E/Ei )∨∨ = H −1 (E/E1 )∨∨ . for all j > 0. Therefore, we have the following ascending chain in Coh(X): 0 → H −1 (E/E1 ) → H −1 (E/E2 ) → · · · → H −1 (E/E1 )∨∨ . Since Coh(X) is Noetherian, H −1 (E/Ei ) = H −1 (E/Ei+1 ) for all i ≫ 0. Hence, we have shown H j (E/Ei ) = H j (E/Ei+1 ) via the natural morphisms for all i ≫ 0 and j ∈ Z. In other words, E/Ei = E/Ei+1 for all i ≫ 0 and so Ei = Ei+1 for all i ≫ 0, as desired. 2. Fix β ∈ Q. Note that tilt IZα,β (E) = degH (E) − H n−1 · (D + βH) · rank(E) is discrete because D is a Q-divisor and β ∈ Q. Consider an ascending chain 0 → E1 → E2 → · · · → E. 86 D+βH tilt in CohH (X). Since IZα,β is additive in short exact sequences and non-negative, tilt tilt tilt tilt IZα,β (Ei ) ≤ IZα,β (Ei+1 ) ≤ IZα,β (E). Therefore, since IZα,β (Ei ) is discrete, increas- tilt tilt tilt ing, and bounded above by IZα,β (E), IZα,β (Ei ) = IZα,β (Ei+1 ) for all i ≫ 0. By applying part 1, we find that CohD+βH H (X) is Noetherian, as claimed. Since CohD+βH H (X) is Noetherian and IZα,β tilt : ΛD H → R is discrete, by Lemma 2.3.6, tilt σα,β satisfies the Harder-Narasimhan property. tilt 3. Assume T ∈ T and F ∈ F. Consider a morphism f : T → F . By additivity of IZα,β , tilt tilt IZα,β (Im(f )) = 0. Since F is σα,β -pure, Im(f ) = 0, as needed. D+βH Now, let A ∈ CohH (X). Consider the collection C = {T ∈ T | there is an injection 0 → T → A}. which we give the natural poset structure. We know C = ̸ ∅ because 0 ∈ C. Consider a chain 0 → T1 → T2 → · · · → A tilt tilt in C. Since IZα,β (Ti ) = 0 = IZα,β (Ti+1 ), by part 1, Ti = Ti+1 for all i ≫ 0. Therefore, by Zorn’s lemma, C has a maximal element. Let T be a maximal element of C. We claim that A/T ∈ F. Assume that 0 → B → A/T is a subobject satisfying IZα,β tilt (B) = 0. Therefore, we can write B = B ′ /T and by additivity of IZα,β tilt tilt , IZα,β (B ′ ) = 0. However, by maximality of T , it follows that B ′ = T and so B = 0, as desired. tilt Therefore, A/T is σα,β -pure and so A/T ∈ F, as desired. 4. The “dual” argument to part 3 holds. tilt We need some more theory before proving the support property for σα,β . In particular, we tilt need a variant of the large volume limit for σα,β -stability Loosely, the large volume limit gives tilt an equivalence between σα,β -stable objects for α ≫ 0 with (H, D)-twisted stable objects. 87 The following lemma is a preliminary step to the large volume limit. Lemma 3.2.2. Assume E ∈ Coh(X) is torsion-free and nonzero 1. Let 0 → F → E → G → 0 be exact in CohD H (X) with F ̸= 0. a) If E is µH -semistable then µD D H (F ) ≤ µH (E ). b) If E is (H, D)-twisted semistable then either ˆ µD D H (F ) < µH (E ) or ˆ µD D D D H (F ) = µH (E ) with νH (F ) ≤ νH (G) c) If E is (H, D)-twisted stable and F ̸= E then either ˆ µD D H (F ) < µH (E ) or ˆ µD D D D H (F ) = µH (E ) with νH (F ) < νH (G) 2. Let 0 → F → E [1] → G → 0 be exact in CohD H (X) with G ̸= 0. a) If E is µH -semistable then µD D H (E [1]) ≤ µH (G). b) If E is µH -stable, codim(F ) ≤ 1, and G ̸= E [1] then µD D H (E [1]) < µH (G). Proof. Assume E ∈ Coh(X) is nonzero and torsion-free. 1. Let 0 → F → E → G → 0 be exact in CohD H (X) with F ̸= 0. The induced long exact sequence in Coh(X) is f 0 → H −1 (F ) → 0 → H −1 (G) → H 0 (F ) → − E → H 0 (G) → 0. Since H −1 (F ) = 0, we find that µD 0 D D H (H (F )) = µH (F ) and similarly for νH . The above exact sequence induces two short exact sequences in Coh(X): 0 → H −1 (G) → H 0 (F ) → I m(f ) → 0 0 → I m(f ) → E → H 0 (G) → 0. 88 Since H −1 (G) ∈ FHD (X), H −1 (G) is torsion-free or 0. Furthermore since E is torsion- free, I m(f ) is 0 or torsion-free, It follows that H 0 (F ) is also torsion-free or 0. By assumption, 0 ̸= F = H 0 (F ) so H 0 (F ) is torsion-free. Since H −1 (G) ∈ FHD+βH (X) and H 0 (G) ∈ THD+βH (X), H −1 (G) ∼ = H 0 (F ) if and only H 0 (F ) = 0. However, F = H 0 (F ) is nonzero, so H −1 (G) ̸= H 0 (F ) and so I m(f ) ̸= 0. We have two cases, either H −1 (G) = 0 or H −1 (G) ̸= 0. If H −1 (G) = 0 then H 0 (F ) = I m(f ), so µD H (H (F )) = µH (I m(f )). If H 0 D −1 (G) ̸= 0, by definition of FHD+βH (X) and THD+βH (X), we find that µD H (H −1 (G)) < 0 ≤ µD 0 H (H (F )). It follows 0 D by the seesaw inequality that µD H (H (F )) < µH (I m(f )) (we obtain a strict inequality because I m(f ) is torsion-free). All cases considered, we find that µD 0 H (H (F )) ≤ H (I m(f )) with equality exactly if H −1 µD (G) = 0. a) Assume E is µH -semistable. Since I m(f ) ̸= 0, we know that µD H (I m(f )) ≤ D 0 D µDH (E ). As we saw above, µH (H (F )) ≤ µH (I m(f )), so µD D 0 D H (F ) = µH (H (F )) ≤ µH (E ), as needed. b) Assume E is (H, D)-twisted semistable. By Lemma 2.1.11 and part 1a, µD H (F ) ≤ D D µD D H (E ). Therefore, it suffices to show that if µH (F ) = µH (E ) then νH (F ) ≤ D νH (G). Assume µD H (F ) = µH (E ). Since E is µH -semistable, µH (I m(f )) ≤ µH (E ). It D D D H (F ) = µH (I m(f )), so we find that H −1 follows that µD D (G) = 0. It follows that we have the following sequence is exact in Coh(X): 0 → H 0 (F ) → E → H 0 (G) → 0. If codim(H 0 (G), X) ≤ 1 then H n−2 · chD 0 ≤1 (H (G)) ̸= (0, 0) by Lemma 3.1.9 so µD 0 D D 0 H (H (F )) = µH (E ) = µH (H (G)) by the generalized seesaw inequal- 89 D ity. By definition of (H, D)-twisted semistability, it follows that νH (H 0 (F )) ≤ νH (H 0 (G)) as needed. If codim(H 0 (G), X) ≥ 2 then we find that D D νH (F ) = νH (H 0 (F )) < +∞ = νH D (H 0 (G)) = νH D (G), as needed. c) The same argument as part 1b holds. 2. Assume 0 → F → E [1] → G → 0 is exact in CohD H (X) with F ̸= 0. The induced long exact sequence in Coh(X) is: g 0 → H −1 (F ) → E → − H −1 (G) → H 0 (F ) → 0 → H 0 (G) → 0. Since H 0 (G) = 0, we find that µD 0 D D H (H (G)) = µH (G) and similarly for νH . The above exact sequence induces the following two short exact sequences in Coh(X): 0 → H −1 (F ) → E → I m(g) → 0 0 → I m(g) → H −1 (G) → H 0 (F ) → 0. Since I m(g) is a subsheaf of torsion-free H −1 (G), I m(g) is torsion-free or 0. Since H −1 (G) ∈ FHD+βH (X) and H 0 (F ) ∈ THD+βH (X), H −1 (G) = H 0 (F ) if and only H −1 (G) = 0. However, 0 ̸= G = H −1 (G) by assumption so I m(g) ̸= 0, as claimed. Since H −1 (G) ∈ FHD+βH (X), H −1 (G) ̸= 0, and H 0 (F ) ∈ FHD+βH (X), −1 µD H (G) = µH (H D (G)) ≤ 0 < µD 0 H (H (F )). Therefore, by the generalized seesaw inequality, µD D H (I m(g)) ≤ µH (G) with equality exactly if codim(H 0 (F )) ≥ 2. a) Assume E is µH -semistable. If H −1 (F ) = 0 then E = I m(g) and so µD D D D H (E [1]) = µH (E ) = µH (I m(g)) ≤ µH (G) 90 by the inequality above, as needed. If H −1 (F ) ̸= 0, since E is µH -semistable and I m(g) ̸= 0, we find that µD H (E ) ≤ −1 µD D H (I m(g)). Moreover, as noted above, µH (I m(g)) ≤ µH (H D (G)) = µD H (G). Thus, we find that D D D µD H (E [1]) = µH (E ) ≤ µH (I m(g)) ≤ µH (G). In either case, µD D H (E [1]) ≤ µH (G), as desired. b) Assume E is µH -stable, E [1] ̸= G, and codim(F, X) ≤ 1. Since codim(F, X) ≤ 1, H −1 (F ) ̸= 0 or H −1 (F ) = 0 with codim(H 0 (F )) ≤ 1. First, assume that H −1 (F ) ̸= 0. Since I m(g) ̸= 0, we know that H −1 (F ) ̸= E . More- over, since H −1 (F ) ̸= 0 and H −1 (F ) ̸= E , since E is µH -stable, we find that −1 µDH (H (F )) < µD D H (I m(g)). By the generalized seesaw inequality µH (E ) < µD D D H (I m(g)). Using the inequality µH (I m(g)) ≤ µH (G) from above, we find that µD D D D H (E [1]) = µH (E ) < µH (I m(g)) ≤ µH (G), as needed. Second, assume that H −1 (F ) = 0 with codim(H 0 (F ), X) ≤ 1. As stated above, H (I m(g)) < µH (G). Therefore, since E = I m(g), D in this case, µD µD D D D H (E [1]) = µH (E ) = µH (I m(g)) < µH (G), as desired. Recall by Lemma 3.1.12.1 that the codimension assumption above is satisfied for every subobject 0 → F → E [1] exactly if E is reflexive. Furthermore, note that the inequalities appearing in Lemma 3.2.2 are not the naive generalization of the inequalities in the definition of µH -(semi)stability nor (H, D)-twisted (semi)stability. 91 We now prove the Large Volume Limit. The terminology comes from physics, but the tilt idea is that σα,β -stability for α ≫ 0 is equivalent to notions of stability on Coh(X). The Large Volume Limit is one of only a few techniques to detect µH -stability in the (H, D)-slice. The other techniques are to show σα,β tilt -stability when β = µD H (E ) (see Proposition 3.2.6), or tilt to show that a torsion object is σα,β -stable (see Lemma 4.1.1). This lemma originally appeared in [Bri08, Section 14] for stability functions on a K3 surface. Part 2 is most comonly written with the weaker conclusion that E is µH -semistable. Lemma 3.2.4 is the technical tool that allows us to strengthen this conclusion. Lemma 3.2.3 (Large Volume Limit). Assume E ∈ Coh(X) is torsion-free. 1. E is (H, D)-twisted stable if and only if E is weakly σα,β tilt -stable for all α ≫ 0 and all β < µD H (E ). 2. If E [1] is weakly σα,β tilt -stable (equivalently σα,β tilt -stable by Lemma 3.1.13) for all β > µDH (E ) then E is µH -stable. 3. If E is reflexive and µH -stable then E [1] is weakly σα,β tilt -stable ((equivalently σα,β tilt -stable by Lemma 3.1.13) for all α ≫ 0 and all β > µD H (E ). Proof. Assume E is torsion-free. 1. First, assume that E is (H, D)-twisted stable. Let β < µD H (E ), so by Lemma 3.1.4, E ∈ CohD+βH H (X). Consider a short exact sequence 0 → F → E → G → 0 in CohH D+βH (X) with F ̸= 0 and F ̸= E . By writing out the long exact sequence in cohomology we find that H −1 (F ) = 0. Therefore, µD H (F ) = µH (H D −1 (F )) > β where the inequality holds by Lemma 3.1.4. Since E is (H, D)-twisted stable, by Lemma 3.2.2.1c, either ˆ µD D H (F ) < µH (E ) or ˆ µD D D H (F ) = µH (E ) with νH (F ) < νH (G). D 92 We will consider each case. Assume µD D D D H (F ) < µH (E ). Since β < µH (F ) < µH (E ), we know that −1/(µD D H (F ) − β) < −1/(µH (E) − β). Furthermore, µD H (F ) < µH (E ) and E is torsion-free and nonzero, so rank(F ) > 0 and D rank(E ) > 0. Therefore Remark 3.1.8 applies and we find that 2 tilt 1 1 2 lim µα,β (F ) = − D <− D = lim 2 µtilt (E ) α→∞ α 2 µH (F ) − β µH (E) − β α→∞ α α,β In other words, µtilt tilt α,β (F ) < µα,β (E ) for all α ≫ 0. By the generalized seesaw inequality, it follows that µtilt tilt α,β (F ) < µα,β (G) for all α ≫ 0, as desired. D D D Second, assume that µD H (F ) = µH (E) with νH (F ) < νH (G). We shall consider two subcases: rank(G) = 0 or rank(G) ̸= 0. If rank(G) = 0, since µD D H (F ) = µH (E ), we find that degD D H (F ) = degH (E ) so degH (G) = 0. By Lemma 2.1.3, it follows that H n−2 ·chD2 (G) ≥ 0. Since H n−2 ·chD 2 (G) ≥ 0 and rank(F ) = rank(E ) > 0, by additivity of the Chern characters, νH D (F ) ≤ νH D (E ) with equality exactly if H n−2 · chD 2 (G) = 0. By Remark 3.1.8, we find that   β2 α2 νHD (F ) − βµD H (F ) + 2 − 2 µtilt α,β (F ) = µD H (F ) − β  2  α2 νH (F ) − βµH (E ) + β2 − D D 2 = µD H (E ) − β  2  D β α2 νH (E ) − βµD H (E ) + 2 − 2 ≤ µD H (E ) − β µtilt α,β (E ) with equality exactly if H n−2 · chD 2 (G) = 0. If H n−2 · chD tilt 2 (G) = 0 then µα,β (G) = D+βH +∞ > µtilt tilt α,β (E ) where IZα,β (E ) = degH (E ) is nonzero because rank(E ) ̸= 0 and D+βH µH (E ) > β. If H n−2 · chD tilt tilt tilt 2 (G) ̸= 0, µα,β (F ) < µα,β (E ) < µα,β (G) by the generalized seesaw inequality. In either scenario, we find that µtilt tilt tilt α,β (F ) ≤ µα,β (E ) < µα,β (G) for all α > 0, as needed. 93 If rank(G) ̸= 0 then by the seesaw inequality, we find that µD D D H (F ) = µH (E ) = µH (G). Thus, µD D D D H (F ) = µH (G) and νH (F ) < νH (G), so by Remark 3.1.8,  2  D β α2 νH (F ) − βµD H (F ) + 2 − 2 µtilt α,β (F ) = D µH (F ) − β  2  D β α2 νH (F ) − βµD H (G) + 2 − 2 = D µH (G) − β  2  D D β α2 νH (G) − βµH (G) + 2 − 2 < µDH (G) − β µtilt α,β (G) for all α > 0. In every case, we have shown µtilt α,β (F ) < µα,β (G) for all α ≫ 0, so E is weakly σα,β - tilt tilt stable for all α ≫ 0, as desired. For the converse, suppose that E is weakly σα,β tilt -stable for all α ≫ 0 and β < µD H (E ). In particular, E ∈ THD+βH (X) for all β < µD H (E ). Taking the limit as β approaches µDH (E ) from the left, we find that nonzero every quotient of E in Coh(X) has slope at least µD H (E ). In other words, E is µH -semistable. Consider a short exact sequence 0 → F → E → G → 0 in Coh(X). We already know that E is µH -semistable, so assume µD D H (F ) = µH (G ). We must show that νHD (F ) < νH D (G ). Since E ∈ CohD+βH H (X) and G is a quotient of E , we find that G ∈ CohH D+βH (X) for all β < µD H (E ). Thus, we have a short exact sequence in CohH D+βH (X): 0 → F → E → G → 0. Since E is weakly σα,β tilt -stable, by definition, µtilt tilt α,β (F ) < µα,β (G ) for all α ≫ 0. By the generalized seesaw inequality, we find that 94 µtilt tilt α,β (E ) < µα,β (G ) for all α ≫ 0. Therefore, by Remark 3.1.8,   D β2 α2 νH (E ) − βµD H (E ) + 2 − 2 = µtilt α,β (E ) µD H (E ) − β < µtilt α,β (G )   D β2 α2 νH (G ) − βµD H (G ) + 2 − 2 = µD H (G ) − β  2  D β α2 νH (G ) − βµDH (E ) + 2 − 2 = µD H (E ) − β D D D for all α ≫ 0. Solving for νH (G ) on the right, we find that νH (E ) < νH (G ). By the D D seesaw inequality we find that νH (F ) < νH (G ), as desired. 2. Assume E [1] is weakly σα,β tilt -stable for all α ≫ 0 and β > µD H (E ). In particular, E ∈ FHD+βH (X) for all β > µD H (E ). If we take the limit as β approaches µH (E ) from D the right, we find that every subsheaf of E (in Coh(X)) has slope at most µD H (E ). In other words, E is µH -semistable. Consider a short exact sequence 0 → F → E → G → 0 in Coh(X) with F ̸= 0, E . We already know that E is µH -semistable, so assume µD D H (F ) = µH (G ). We must show that D D D νHD (F ) > νH D (E ). Since µD H (F ) = µH (G ), by the seesaw inequality, µH (F ) = µH (E ). Furthermore, since E is µH -semistable, F [1] ∈ CohD+βH H (X) for all β > µD H (E ). Thus, we have a short exact sequence 0 → F [1] → E [1] → G → 0 in CohD+βH H (X). Since E [1] is weakly σα,βtilt -stable and, has good quotients (by Lemma 3.1.13), by Lemma 2.2.11 we find that E is σα,β tilt -stable. Since F is a proper nonzero subobject of E , we find that µtilt tilt tilt tilt α,β (F ) = µα,β (F [1]) < µα,β (E [1]) = µα,β (E ). It follows by remark 3.1.8 that for 95 α ≫ 0 and β > µD H (E ) that   D β2 α2 νH (F ) − βµD H (F ) + 2 − 2 = µtilt α,β (F ) µDH (F ) − β < µtilt α,β (E )   D β2 α2 νH (E ) − βµD H (E ) + 2 − 2 = µD H (E ) − β   D β2 α2 νH (E ) − βµD H (F ) + 2 − 2 = µD H (F ) − β Since µDH (F ) < β, we can simplify the above inequality to find νH (F ) > νH (E ). D D By assumption, F is nonzero, torsion-free (because it is a subsheaf of E which is torsion-free), and µD H (F ) = µH (E ) = µH (G ). Therefore, F , E , and G all have D D √ nonzero rank, so the seesaw inequality (with respect to −H n−2 · chD 2 (·) + −1 rank(·)) D D D D holds. Since νH (F ) > νH (E ), it follows that νH (F ) > νH (G ). In all, we have shown that every proper nonzero subsheaf 0 → F → E satisfies either ˆ µD D H (F ) < µH (E /F ) or ˆ µD D D H (F ) = µH (E /F ) with νH (F ) > νH (E /F ). D By Lemma 3.2.4, it follows that E is µH -stable, as desired. 3. Assume E is reflexive and µH -stable. Therefore, E [1] ∈ CohD+βH H (X) for all β > H (E ). Consider a short exact sequence 0 → F → E [1] → G → 0 in CohH D+βH µD (X) with F ̸= 0, E . By Lemma 3.2.2 in view of Lemma 3.1.12, we find that µD H (E [1]) < µD H (G). By considering the long exact sequence on cohomology, we find that µH (G) = D µD H (H −1 (G)) ≤ β because H −1 (G) ∈ FHD+βH (X). We already saw that µD H (E ) < β, so by Remark 3.1.8, it follows that 2 tilt 1 1 2 tilt limα→∞ µ α,β (E [1]) = − < − = lim α→∞ µ (G). α2 µDH (E [1]) − β µD H (G) − β α2 α,β Note that this inequality still holds in the case that β = µD H (G) if we formally say −1/(µD tilt tilt H (G) − β) = +∞. It follows that µα,β (E [1]) < µα,β (G) for all α ≫ 0 β > µH (E ). D 96 Since µtilt tilt tilt tilt α,β (E [1]) < µα,β , by the seesaw inequality, µα,β (F ) < µα,β (G) for all α ≫ 0 and β > µD H (E ). In other words, E [1] is weakly σα,β -stable for all α ≫ 0 and all β > µH (E ). tilt D By Lemma 3.1.13, E [1] has good quotients, so by Lemma 2.2.11, we find that E [1] is tilt σα,β -stable for all α ≫ 0 and all β > µD H (E ), as desired. Lemma 3.2.4. Assume E is a torsion-free sheaf. If E satisfies the following property for every proper nonzero subsheaf 0 → F → E one of the following holds: ˆ µD D H (F ) < µH (E /F ) or ˆ µD D D D H (F ) = µH (E /F ) with νH (F ) ≥ νH (E /F ) then E is µH -stable. Proof. We will prove the contrapositive, so assume that E is not µH -stable. We will first show that E ⊕2 does not satisfy the property in the lemma statement then show that E does not satisfy the property. If E is not µH -semistable then we are done, so assume E is µH -semistable. Therefore, by definition, we can choose a proper nonzero subsheaf 0 → F → E such that µD H (F ) = µDH (E /F ). On the other hand, let Y be a closed subvariety of X of codimension 2 satisfying H n−2 · chD 2 (F ) rank(E ) − H n−2 · chD 2 (E ) rank(F ) H n−2 · Y ≥ 2 . (3.1) rank(E ) We have a surjection OX → ι∗ OY → 0 and so we have a short exact sequence 0 → K → E → E ⊗ ι∗ OY → 0. Since Y is supported in codimension 2, H n−2 · chD2 (E ⊗ ι∗ OY ) = rank(E )H n−2 · Y. It follows that H n−2 · chD 2 (K ) = H n−2 · chD 2 (E ) − rank(E )H n−2 · Y. 97 In all, we have constructed a short exact sequence 0 → F ⊕ K → E ⊕2 → (E /F ) ⊕ (E ⊗ ι∗ OY ) → 0. Therefore, by additivity of the Chern character, H n−2 · chD 2 (F ) + H n−2 · chD 2 (E ) − H n−2 · Y rank(E ) D νH (F ⊕ K ) = . rank(F ) + rank(E ) Note that rank(E ) = rank(K ) because E ⊗ ι∗ OY is supported in codimension 2. On the other hand, H n−2 · chD2 (E ) D νH (E ⊕2 ) = . rank(E ) By using Equation 3.1, we find that νH D (F ⊕ K ) ≤ νH D (E ⊕2 ). By the seesaw inequality, we find that νHD (F ⊕ K ) ≤ νH D ((E /F ) ⊕ (E ⊗ ι∗ OY )). Also, by construction, µDH (F ⊕ K ) = µH (E ) = µH ((E /F ) ⊕ (E ⊗ ι∗ OY )). D D In all, E ⊕2 does not satisfy the property in the lemma statement. We will now show that E does not satisfy the given property. Consider the short exact sequence 0 → E ∩ (F ⊕ K ) → F ⊕ K → (F ⊕ K )/(E ∩ (F ⊕ K )) → 0 where the intersection is with respect to the inclusion 0 → E → E ⊕2 into the first component. Since rank(F ⊕ K ) > rank(E ), we know that E ∩ (F ⊕ K ) ̸= 0, E . It follows that (F ⊕ K )/(E ∩ (F ⊕ K )) ̸= 0, E as well. Using a similar argument to Lemma 2.2.15, E ∩ (F ⊕ K ) and (F ⊕ K )/(E ∩ (F ⊕ K )) are subsheaves of E . Moreover, that same argument tells us that µD D H (E ∩(F ⊕K )) = µH (E ) and similarly for (F ⊕K )/(E ∩(F ⊕K )). Furthermore, as we saw above, νH D (F ⊕ K ) ≤ νH D (E ⊕2 ) = νHD (E ). Therefore, by the D D D D seesaw inequality, either νH (E ∩(F ⊕K )) ≤ νH (E ) or νH ((F ⊕K )/(E ∩(F ⊕K )) ≤ νH (E ). In other words, E does not satisfy the proper in the lemma statement, as desired. 98 There is also a variant of the Large Volume Limit for σ-semistability which is essentially as one would expect. Interestingly, in the semistable case, we do not need the reflexive assumption. There is also a variant of the Large Volume Limit for chain complexes. tilt Lemma 3.2.5 ([Bri08, Proposition 14.2]). If E is σα,β -semistable for all α ≫ 0 then one of the following holds: ˆ H −1 (E) = 0 and H 0 (E) is µH -semistable. ˆ H −1 (E) is µH -semistable and codim(H 0 (E)) ≥ 2. The following cultural aside notes that µH -stability can also be detected along the vertical wall in the (H, D)-slice (we will discuss walls in the next subsection). Note that we do not need the reflexive assumption in this case. This proposition can be proven using the same techniques as the Large Volume Limit. Proposition 3.2.6. Assume E is torsion-free and for ease of notation set β0 = µD H (E ). E is µH -stable if and only if E [1] is σα,β0 -stable for all α > 0. tilt D We will eventually see that σα,β satisfies the support property with respect to ∆H . For now, we will only prove this result when β ∈ Q. We use a similar argument to [BMT14, Section 7.3]. tilt Lemma 3.2.7. If (β, α) ∈ Q × R>0 then σα,β satisfies the support property with respect to D tilt ∆H . In particular, if (β, α) ∈ Q × R>0 then σα,β is a very weak stability condition. tilt tilt Proof. Since β ∈ Q, IZα,β : Λ → R is discrete so we can proceed by induction on IZα,β . First, assume that IZα,β tilt (E) = 0. By definition IZα,β tilt (E) = degD+βH H (E) which is inde- pendent of α, so µtilt tilt α,β (E) = +∞ for all α > 0. In other words, E is σα,β -semistable for all α ≫ 0. By Lemma 3.2.5, one of the following must hold: ˆ H −1 (E) = 0 and H 0 (E) is µH -semistable. 99 ˆ codim(H 0 (E)) ≥ 2 and H −1 (E) is µH -semistable. In the first case, by Bogomolov’s Inequality (Lemma 2.1.15) D D ∆H (E, E) = ∆H (H 0 (E), H 0 (E)) ≥ 0. In the second case, by Lemma 2.1.3, −1 H n−2 · chD ≤2 (E) = −H n−2 · chD ≤2 (H (E)) + (0, 0, d) with d ≥ 0. It follows by the calculation above and Bogomolov’s Inequality (Lemma 2.1.15) that D ∆H (E, E) = degD 2 H (E) − 2 rank(E)(H n−2 · chD 2 (E)) −1 = degD H (H (E))2 + 2 rank(H −1 (E))(d′ − H n−2 · chD 2 (H −1 (E))) −1 ≥ degD H (H (E))2 − 2 rank(H −1 (E))(H n−2 · chD 2 (H −1 (E))) ≥0 This completes the base case. D Second, assume that IZα,β (E) > 0. If E is σα,β -semistable for all α ≫ 0 then ∆H (E, E) ≥ 0 by the same argument as above. Therefore, we may assume that E is not σαtilt ′ ,β -semistable for some α′ > 0. Since µtilt α,β is continuous in α, we can find α1/2 ∈ [α0 , α1 ] such that E has a nonzero proper subobject 0 → F → E satisfying µα1/2 ,β0 (F ) = µα1/2 ,β0 (E). tilt Since E is not σα,β 0 -semistable for all α > 0 we know IZ tilt (E) ̸= 0. Since E is not weakly σαtilt 1/2 ,β0 -semistable and IZα1/2 ,β0 (E) ̸= 0, by the same argument as Lemma 2.3.14, we can find a σαtilt 1/2 ,β0 -stable subobject 0 → Ê → E satisfying µtilt tilt α1/2 ,β0 (Ê) = µα1/2 ,β0 (E) and IZαtilt 1/2 ,β0 (Ê) < IZαtilt 1/2 ,β0 (E). In short, we have a short exact sequence 0 → Ê → E → E/Ê → 0 all of the same slope at (β0 , α1/2 ) with Ê and E/Ê both σα1/2 ,β0 -semistable. Since IZαtilt 1/2 ,β0 (Ê) < IZαtilt 1/2 ,β0 (E), we know that IZαtilt 1/2 ,β0 (E/Ê) < IZαtilt 1/2 ,β0 (E), so by D D the inductive hypothesis ∆H (Ê) ≥ 0 and ∆H (E/Ê) ≥ 0. Therefore, by Lemma 2.3.8, we D find ∆H (E) ≥ 0, as needed. 100 D By Lemma 3.1.9, Ker(Zα0 ,β0 )tilt = {(0, 0, 0)} and so ∆H is vacuously negative definite. Last, we find σαtilt 0 ,β0 satisfies the Harder-Narasimhan property by Lemma 2.3.6. 3.3 Tilt Stability is a Very Weak Stability Condition for Real Beta tilt In this subsection we show that σα,β is a very weak stability condition for all (β, α) ∈ R × R>0 . Furthermore, we show that the induced morphism R × R>0 → WStab(X) is a homeomorphism onto its image. For this reason, we will say that tilt stability is a continuous family. There are currently two techniques to extend the results from the previous section to all β ∈ R. tilt The first technique, from [BMS16, Appendix B], is to show that σα,β for β ∈ R \ Q tilt must correspond to a point of WStab(X) that is close to σα,β 0 for β0 ∈ Q. This method is useful because it automatically shows tilt stability forms a continuous family. However, this method is technically difficult because it requires a detailed understanding of WStab(X) (which is done by essentially reducing to arguments about Stab(X)). tilt The second technique, from [Fey18, Section 2.3], is to show σα,β satisfies the Harder- Narasimhan and support properties directly then show that tilt stability forms a continuous family. This is method is useful because it avoids an analysis of WStab(X). However, it is technically difficult because the argument requires a detailed understanding µtilt+ α,β (E) as a function of β. tilt We introduce a third technique which is to extend our results to β ∈ R for σα,β -pure objects then to extend to all objects using that pure objects form a torsion-free class. Since tilt σα,β -pure have nice deformation properties, the existence of Harder-Narasimhan filtrations and the support property are easy to show. Furthermore, this new technique allows us to construct weak Jordan-Hölder filtrations (which were not previously known to exist). 101 D+βH tilt Lemma 3.3.1. Assume E ∈ CohH (X) satisfies IZα,β (E) ̸= 0. Define C to be the collection: D+βH tilt {F ∈ CohH (X) | 0 → F → E is a σα,β tilt − semistable and µtilt α,β (F ) ≥ µα,β (E)}. The collection {H n−1 · chD ≤1 (F )}F ∈C is finite. Proof. We first claim {rank(F )}F ∈C is finite. Since 0 → F → E is a subobject, rank(F ) = rank(H 0 (F )) − rank(H −1 (F )) ≥ − rank(H −1 (E)). In other words, {rank(F )}F ∈C is bounded below. If rank(F ) ≤ 0 then we are done, so we may assume rank(F ) > 0. By direct computation, tilt D tilt IZα,β (F )2 ∆H (F ) α2 RZα,β (F ) =− + + rank(F ). 2 rank(F ) 2 rank(F ) 2 Therefore, by definition of C, |µtilt tilt tilt α,β (E)|IZα,β (E) ≥ −µα,β (E)IZα,β (F ) tilt tilt ≥ RZα,β (F ) tilt IZα,β (E)2 α2 ≥− +0+ rank(F ) 2 rank(F ) 2 tilt IZα,β (E)2 α2 ≥− + rank(F ). 2 2 In other words, ! tilt 2 tilt IZα,β (E)2 |RZα,β (E)| + ≥ rank(F ) α2 2 so {rank(F )}F ∈C is bounded above. Therefore, the set {rank(F )}F ∈C is finite. By definition of C, tilt tilt 0 ≤ IZα,β (F ) ≤ IZα,β (E). In other words, 0 ≤ degD H (F ) − β rank(F ) ≤ IZα,β (E). tilt Since {rank(F )}F ∈C is finite, {H n−1 · chD ≤1 (F )}F ∈C is finite as well, as claimed. 102 Lemma 3.3.2. Assume E ∈ CohD+βH H (X) is σα,β tilt -pure. 1. There exists a Harder-Narasimhan filtration 0 → E1 → E2 → · · · → Em−1 → E tilt with respect to σα,β . tilt 2. If E is σα,β -semistable then there exists is a weak Jordan-Hölder filtration 0 → E1 → E2 → · · · → Em−1 → E. tilt with respect to σα,β . tilt Proof. 1. For ease of notation, let C be the collection of all σα,β -destabilizing subobjects tilt of E. By Lemma 3.3.1, {IZα,β (F )}F ∈C is finite. Therefore, by Lemma 3.2.1.1 any ascending chain in C must stabilize. Therefore, let E1 be a maximal object of an ascending chain. If E1 = E we are done, so assume E ̸= E1 . Furthermore, if E/E1 is tilt σα,β -semistable then µtilt tilt tilt α,β (E1 ) > µα,β (E) > µα,β (E/E1 ) tilt tilt by the seesaw inequality (note that Zα,β (E1 ) since E is σα,β -pure). In this case, 0− > tilt E1 − > E is a Harder-Narasimhan filtration with respect to σα,β . Threfore, we may tilt tilt assume E/E1 is not σα,β -semistable. By Lemma 2.3.13.4, either E/E1 is either σα,β - tilt tilt pure or 0. Since E ̸= E1 , E/E1 ̸= 0 so E/E1 is σα,β -pure. Since E/E1 is σα,β -pure, by tilt the same argument as above, we can choose a maximal σα,β -subobject 0− > E2 − > E1 . We can continue the argument above to obtain a filtration 0 → E1 → E2 → · · · → E such that Ei+1 /Ei is σ-semistable for all i and µσ (E1 ) > µσ (E2 /E1 ) > µσ (E3 /E2 ) > · · · . 103 Furthermore, by induction, µtilt tilt α,β (Ei ) > µα,β (E) for all i. Therefore, by Lemma 3.3.1, tilt {IZα,β (Ei )}i≥1 is finite. Thus, by Lemma3.2.1, Ei = Ei+1 for all i ≫ 0. Hence, we tilt have constructed a Harder-Narasimhan filtration of E with respect to σα,β , as desired. tilt 2. By the same argument as part 1, any ascending chain of σα,β -destabilizing subobjects tilt must stabilize and IZα,β takes only finitely many values over all destabilizing subob- jects. Therefore, Lemma 2.3.14 gives us a Jordan-Hölder filtration. Remark 3.3.3. Note that the same argument as Lemma 3.3.1 and Lemma 3.3.2 allow for tilt infinitesimal deformations of (β, α). Specifically, by Lemma 3.1.10 if E is σα,β -pure then ′ there exists ε > 0 such that E ∈ CohD+β H H (X) for all β ′ ∈ (β − ε, β]. The rest of the arguments go through as expected. tilt In particular, if E is σα,β -pure then there exists ε > 0 and a Harder-Narasimhan filtration 0 → E1 → E2 → · · · → Em−1 → E tilt ′ with respect to σα,β ′ for all β ∈ (β − ε, β + ε). Lemma 3.3.4. Assume (β, α) ∈ R × R>0 . tilt 1. σα,β satisfies the Harder-Narasimhan property tilt D 2. σα,β satisfies the support property with respect to ∆H . tilt In other words, for all (β, α) ∈ R × R>0 , σα,β is a very weak stability condition. D+βH Proof. Assume E ∈ CohH (X). By Lemma 3.2.1.3 there is a short exact sequence 0 → T → E → E/T → 0 tilt tilt where E/T is σα,β -pure and IZα,β (T ) = 0. By Lemma 3.3.2.2, there is a Harder-Narasimhan filtration 0 → E1 /T → E2 /T → · · · → Em−1 /T → E/T 104 of E/T . We claim that 0 → T → E1 → · · · → Em−1 → E tilt is a Harder-Narasimhan filtration of E. It remains to show that T is σα,β -semistable and µtilt tilt tilt tilt α,β (T ) > µα,β (E1 /T ). However, since IZα,β (T ) = 0, µα,β (T ) = +∞ so T is σα,β -semistable. tilt tilt tilt Moreover, since E/T is σα,β -pure, IZα,β (E1 /T ) ̸= 0 so µtilt tilt α,β (T ) > µα,β (E1 /T ), as desired. D By Lemma 3.1.9, ∆H is vacuously negative definite. Therefore, assume E ∈ CohD+βH H (X) tilt is σα,β -semistable. If IZα,β tilt (E) = 0, by Lemma 3.1.9, either β = µD H (E) or codim(E) ≥ 2. In the first case, D D since D is a Q-divisor, β ∈ Q, ∆H (E) ≥ 0 by Lemma 3.2.7. In the second, case, ∆H (E) = 0 by direct calculation. tilt tilt Thus, we may assume IZα,β (E) ̸= 0. In particular, by Lemma 2.3.13, E is σα,β -pure. By Remark 3.3.3, we can choose β ′ < β such that β ′ ∈ Q and E is σα,β tilt ′ -semistable. The result follows by Lemma 3.2.7. tilt We now show that if we vary (β, α) continuously then σα,β varies continuously with respect to the topology on WStab(X, Λ). Lemma 3.3.5. There is a continuous injection R × R>0 → WStab(X) (where R × R>0 has tilt the usual Euclidean topology) given by (β, α) 7→ σα,β . Proof. By Lemma 3.3.2.2 there is a short exact sequence 0 → T → E → E/T → 0 tilt tilt tilt where E/T is σα,β -pure and IZα,β (T ) = 0. If E is σα,β -pure (i.e. T = 0) then the result follows by Remark 3.3.3. Suppose E is not σα,β tilt -pure, so µtilt+α,β (E) = +∞ and T is a maximal tilt σα,β -destabilizing subobject of E. By Lemma 3.1.9 either β = µD H (T ) or codim(T ) ≥ 2. In ′ the first case, there exists ε > 0 such that 0 → T → E ois a subobject in CohD+β H H (X) for tilt+ all β ′ ∈ (β − ε, β] or β ′ ∈ [β, β + ε). Therefore, limβ→µDH (T ) µtilt α,β (T ) = +∞, so µα,β (E) is continuous. If codim(T ) ≥ 2 then µtilt+ α,β (E) = +∞ for all (β, α) ∈ R × R>0 , as needed. 105 The argument for continuity of µtilt− α,β (E) is nearly identical. Definition 3.3.6. We define the (H, D)-slice to be the image of R × R>0 in WStab(X) under the continuous injection above. 3.4 Walls for Tilt Stability In this subsection we will see that the (H, D)-slice has a locally finite collection of real codimension submanifolds (called walls) such that stability only changes when crossing over a wall. Furthermore, the possible walls are either nested semicircles with center along the line α = 0 or a unique vertical wall. It is well-known that there is a largest semicircular wall, but we provide an explicit bound on the radius of this wall. We first describe “numerical” walls. Numerical walls are class of real codimenson 1 submanifolds that include all “actual” walls. Even though most numerical walls are not actual walls, they are relatively easy to describe and restrict the possible actual walls. Definition 3.4.1. A numerical wall associated E is the set of points (β, α) in the (H, D)-slice of the form: W (E, F ) = {(β, α) | µtilt tilt α,β (E) = µα,β (F )} ⊆ R × R>0 for some object F ∈ Db (X) (not necessarily in the heart CohD+βH H (X)). Remark 3.4.2. A numerical wall W (E, F ) can equivalently be defined by W ′ = {(β, α) | RZα,β tilt (E)IZα,β tilt (F ) = RZα,β tilt (F )IZα,β tilt (E)}. To see this, first note that W (E, F ) ⊆ W ′ . Now, assume that (β, α) ∈ W ′ . If IZα,β tilt (F ) ̸= 0 and IZα,β tilt (E) ̸= 0 then it is clear that tilt tilt (β, α) ∈ W (E, F ). Thus, assume that IZα,β (F ) = 0 or IZα,β (E) = 0. Without loss of tilt tilt tilt generality, we may assume that IZα,β (F ) = 0. It follows that 0 = RZα,β (F )IZα,β (E) so tilt tilt tilt RZα,β (F ) = 0 or IZα,β (E) = 0. We claim that RZα,β (F ) ̸= 0. 106 tilt tilt D Since IZα,β (F ) = 0, F is σα,β -semistable so, by Lemma 3.3.4, ∆H (F ) ≥ 0. We also find that β = µD tilt H (F ). If RZα,β (F ) = 0, we can solve for α to find q D −∆H (F ) α=± . rank(F ) D tilt tilt Since ∆H (F ) ≥ α ̸∈ R>0 , so RZα,β (F ) ̸= 0. Therefore, IZα,β (E) = 0 as needed. Definition 3.4.3. We say a numerical wall W is an actual wall associated to E if for some (β, α) ∈ W there exists a short exact sequence 0→F →E→G→0 D+βH in CohH (X) such that ˆ W = W (E, G) or W = W (E, F ), ˆ F and G are σα,β tilt -semistable (in particular, F, G ∈ CohD+βH H (X)), ˆ µtilt α,β (F ) = µα,β (F ), ˆ W ̸= R × R>0 (i.e. W is 1-dimensional). We call 0 → F → E → G → 0 a destabilizing sequence associated to W at (β, α). tilt By definition and Lemma 3.3.5, actual walls control σα,β -(semi)stability. Local finiteness of actual walls follows by [Bri08, Section 9] for surfaces and [BMS16, Proposition B.5] in general. Lemma 3.4.4. Assume E ∈ Db (X) is nonzero. 1. Let W be the collection of all actual walls associated to E in the (H, D)-slice. If E is weakly σαtilt 0 ,β0 -stable for some (β0 , α0 ) in the complement of W then E is weakly σα,β tilt - stable for all (β, α) in the connected component in the complement of W containing (β0 , α0 ). In other words, weak σ-stability is constant in chambers. 107 2. There exists a locally finite collection of walls, W, in the (H, D)-slice such that if 0 → E1 → E2 → · · · → Em−1 → E is a Harder-Narasimhan filtration of E with respect to σαtilt 0 ,β0 for some (β0 , α0 ) then 0 → E1 → E2 → · · · → Em−1 → E is a Harder-Narasimhan filtration of E with respect to σα,β for all (β, α) in the con- nected component in the complement of W containing (β0 , α0 ). In other words, Harder- Narasimhan filtrations are constant in chambers. 3. There exists a locally finite collection of walls, W, in the (H, D)-slice such that Jordan- Hölder filtrations of each semistable factor of E are constant. We will first introduce some constraints on numerical walls (and thus on actual walls). The following standard result, first appearing in [Mac14] for surfaces, gives initial restrictions on the class of numerical walls. Lemma 3.4.5. Fix objects E, F ∈ Db (X). 1. W (E, F ) is given by the equation: xα2 + xβ 2 − 2yβ + 2z = 0 where x = degD D H (E) rank(F ) − degH (F ) rank(E) y = (H n−2 · chD 2 (E)) rank(F ) − (H n−2 · chD2 (F )) rank(E) z = (H n−2 · chD D 2 (E)) degH (F ) − (H n−2 · chD D 2 (F )) degH (E) In particular, ˆ If x ̸= 0 then W (E, F ) is a semicircle centered at (y/x, 0) with radius squared y 2 /x2 − 2z/x, so W (E, F ) = ∅ if y 2 /x2 − 2z/x ≤ 0. 108 ˆ If x = 0 and y ̸= 0 then W (E, F ) is a vertical line given by the equation β = − yz . ˆ If x, y = 0, and z ̸= 0 then W (E, F ) = 0. ˆ If x, y, z = 0 then W (E, F ) = R × R>0 . 2. If rank(E) ̸= 0 and x ̸= 0 then W (E, F ) is a semicircle with radius squared D ∆H (E) ρ2 = (µDH (E) − c) − 2 rank(E)2 where (c, 0) is the center. If, in addition, rank(F ) ̸= 0 then D D νH (E) − νH (F ) c= D D . µH (E) − µH (F ) 3. If rank(E) ̸= 0 then there is a unique vertical numerical wall given by the equation β = µD H (E). The shapes of the numerical walls are shown in Figure 3.4. Proof. 1. Consider the wall W (E, F ) = {(β, α) ∈ R × R>0 | µtilt tilt α,β (E) = µα,β (F )}. Equivalently, by Remark 3.4.2, W (E, F ) is all points (β, α) such that tilt tilt tilt tilt RZα,β (E)IZα,β (F ) = RZα,β (F )IZα,β (E) which simplifies to α2  D D  0= degH (E) rank(F ) − degH (F ) rank(E) 2 β2   + degD H (E) rank(F ) − deg D H (F ) rank(E) 2   − β (H n−2 · chD 2 (E)) rank(F ) − (H n−2 · ch D 2 (F )) rank(E) + (H n−2 · chD D 2 (E)) degH (F ) − (H n−2 · chD D 2 (F )) degH (E). 109 If we multiply each side of this equation by 2 and set D x = degDH (E) rank(F ) − degH (F ) rank(E) y = (H n−2 · chD2 (E)) rank(F ) − (H n−2 · chD 2 (F )) rank(E) D z = (H n−2 · chD D 2 (E)) degH (F ) − (H n−2 · chD 2 (F )) degH (E), we obtain the desired result. Assume that x ̸= 0, so we can divide each side of the above equation by x then complete the square to find that y y2 z α2 + (β − )2 = 2 − 2 . x x x y y2 − 2 xz ,  In other words, W (E, F ) is a semicircle centered at x ,0 with radius squared x2 as needed. Assume that x = 0 and y ̸= 0, so the above equation simplifies to β = − yz . Assume that x, y = 0 then W (E, F ) is given by the equation 2z = 0. Thus, if z = 0 then W (E, F ) = R × R>0 , and if z ̸= 0 then W (E, F ) = ∅. 2. Assume rank(E) ̸= 0 and x ̸= 0 We will consider two cases. Either rank(F ) = 0 or rank(F ) ̸= 0. If rank(F ) = 0 then it we find that x = − degD H (F ) rank(E) and y = −(H n−2 · chD2 (F )) rank(E). It follows that y/x = (H n−2 · chD D 2 (F ))/ degH (F ). Similarly, we find that z (H n−2 · chD D 2 (E)) degH (F ) − (H n−2 · chD D 2 (F )) degH (E) = x − degD H (F ) rank(E) D (H n−2 · chD 2 (F )) = −νH (E) + µD H (E) D . degH (F ) 110 In all, we find that D y 2 ∆H (E) y y2 (µDH (E) − ) − =µ D H (E) 2 − 2µ D H (E) + x rank(E)2 x x2 degD 2 H (E) − 2 rank(E)(H n−2 · chD2 (E)) − 2 rank(E) y 2 H n−2 · chD2 (F ) D = 2 − 2µD H (E) D + 2νH (E) x degH (F ) y2 z = 2 −2 x x =ρ2 , as needed. If rank(F ) ̸= 0 then, by Remark 3.1.8, W (E, F ) is given by the solutions (β, α) of β 2 α2    D νH (E) − βµD H (E) + − (µD H (F ) − β) 2 2  2 α2  D   D D β = νH (F ) − βµH (F ) + − (µH (E) − β). 2 2 Which simplifies to α2 D β2 D 0= (µH (E) − µD H (F )) + (µ (E) − µD H (F )) 2 2 H D D D + β(νH (F ) − νH (E)) + νH (E)µD D D H (F ) − νH (F )µH (E) D Since rank(E), rank(F ) ̸= 0 and x ̸= 0, we find that µD H (E) ̸= µH (F ). Therefore, we can multiply the above equation by 2, divide each side by µD D H (E) − µH (F ), and complete the square to obtain  D D 2  D D 2 2 νH (E) − νH (F ) νH (E) − νH (F ) α + β− D = µH (E) − µD H (F ) µD D H (E) − µH (F ) ν D (E)µD D D H (F ) − νH (F )µH (E) −2 H . µD D H (E) − µH (F ) In other words, W (E, F ) is a semicircle centered at (c, 0) where D D νH (F ) − νH (E) c= D D . µH (E) − µH (F ) 111 It remains to show that D 2 ∆H (E) ρ = (µD H (E) − c) −2 . rank(E)2 However, D 2 ∆H (E) 2 D 2 D (µDH (E) − c) − = c2 − 2µD D H (E)c + µH (E) − µH (E) + 2νH (E) rank(E)2 D = c2 − 2µD H (E)c + 2νH (E). and D D D µD H (E)νH (E) − µH (E)νH (F ) −2µD D H (E)c + 2νH (E) = −2µH (E) D µD D H (E) − µH (F ) D D ν D (E)µD H (E) − νH (E)µH (F ) + 2νH D (E) H µD D H (E) − µH (F ) ν D (E)µD D H (F ) − νH (F )µH (E) D = −2 H D . µDH (E) − µH (F ) In all, we have shown that D D 2 D D D D (E)µD  ∆H (E) νH (E) − νH (F ) νH H (F ) − νH (F )µH (E) (c−µD H (E)) − 2 = D D −2 = ρ2 , rank(E)2 µH (E) − µH (F ) µD H (E) − µ D H (F ) as desired. 3. Assume rank(E) ̸= 0. By part 1, it suffices to show −z/y = µD H (E) for all F with x = 0 and y ̸= 0. Since x = 0, we know degD D H (F ) rank(E) = degH (E) rank(F ). It follows that z (H n−2 · chD D 2 (E)) degH (F ) − (H n−2 · chD D 2 (F )) degH (E) − =− y (H n−2 · chD2 (F )) rank(E) − (H n−2 · chD (E)) rank(F ) 2 −1 (H n−2 · chD D 2 (E)) degH (F ) rank(E) − (H n−2 · chD D 2 (F )) degH (E) rank(E) = rank(E) (H n−2 · chD 2 (F )) rank(E) − (H n−2 · chD (E)) rank(F ) 2 −1 (H n−2 · chD D 2 (E)) degH (E) rank(F ) − (H n−2 · chD D 2 (F )) degH (E) rank(E) = rank(E) (H n−2 · chD 2 (F )) rank(E) − (H n−2 · chD (E)) rank(F ) 2 (H n−2 · chD2 (E)) rank(F ) − (H n−2 · chD2 (F )) rank(E) = −µD H (E) (H n−2 · ch2 (F )) rank(E) − (H n−2 · chD D 2 (E)) rank(F ) = µD H (E), 112 as needed. We refine Lemma 3.4.5.4 to show that if degD H (E) ≥ 0 and rank(E) ̸= 0 then semicircular walls are nested. Note that these assumptions are not at all restrictive. Specifically, by the D tilt support property, if ∆H (E) < 0 then for all (β, α) ∈ R×R>0 , E is not σα,β -semistable. In par- ticular, there are no actual walls associated to E. Also, if rank(E) = 0 then by Lemma 3.4.5, D all semicircular numerical walls associated to E have center (−(H n−2 · chD 2 (E))/ degH (E), 0) and so must be nested. D Lemma 3.4.6. Assume rank(E) ̸= 0 and ∆H (E) ≥ 0. For all c ∈ R satisfying |c−µD H (E)| > q D ∆H (E)/ rank(E)2 define Bc to be the semicircle in R × R>0 with center c and and radius squared D ∆H (E) ρ2c = (µD H (E) 2 − c) − . rank(E)2 1. Bc does not intersect the vertical line β = µD H (E). 2. If Bc1 and Bc2 are on the same side of the vertical line β = µD H (E) (i.e. both c1 , c2 < D µDH (E) or c1 , c2 > µH (E)) then one of the following must hold: ˆ Bc1 = Bc2 , ˆ Bc1 is nested within Bc2 (i.e. Bc1 is a subset of the region strictly between Bc2 and the real axis), or ˆ Bc2 is nested within Bc1 . 3. Numerical walls associated to E in the (H, D)-slice are all disjoint. On each side of the unique vertical wall, numerical walls are nested semicircles. The relation between numerical walls is represented in Figure 3.4. D Proof. Assume rank(E) ̸= 0 and ∆H (E) ≥ 0. Define Bc as in the lemma statement. 113 D 1. Without loss of generality, assume c < µD H (E). Since ∆H (E) ≥ 0, by definition of Bc , D ∆H (E) ρ2c = (µD H (E) − c) −2 ≤ (µD 2 H (E) − c) . rank(E)2 Since c < µD D H (E) and ρc > 0, it follows that ρc < µH (E) − c so ρc + c < µH (E). In D other words, the right endpoint of Bc lies to the left of the vertical line β = µD H , as claimed. 2. Without loss of generality, assume that c1 , c2 < µD H (E). If c1 = c2 then Bc1 = Bc2 , so assume, without loss of generality, c1 < c2 . We will show that c1 + ρc1 > c2 + ρc2 and c1 − ρc1 < c2 − ρc2 . Since c1 < c2 < µD H (E), D D ∆H (E) ∆H (E) ρ2c2 = (µD H (E) 2 − c2 ) − 2 ≤ (µD 2 H (E) − c1 ) − 2 = ρ2c1 rank(E) rank(E) and so ρc2 < ρc1 . Since c1 < c2 , it follows that c1 − ρc1 < c2 − ρc1 < c2 − ρc2 , as needed. D By part 1, neither Bc1 nor Bc2 intersect the line β = µD H (E), so ρc1 ≤ µH (E) − c1 and ρc2 ≤ µDH (E) − c2 . It follows that ρc1 + ρc2 ≤ (µD D H (E) − c1 ) + (µH (E) − c2 ). Moreover, since c1 < c2 , we find that (c2 − c1 )(ρc1 + ρc2 ) ≤ (c2 − c1 )((µD D H (E) − c1 ) + (µH (E) − c2 )) 2 = (µD 2 D H (E) − c1 ) − (µH (E) − c2 ) = ρ2c1 − ρ2c2 = (ρc1 − ρc2 )(ρc1 + ρc2 ). Since ρc1 + ρc2 > 0, we can divide each side of the above inequality by ρc1 + ρc2 to find that c2 − c1 ≤ ρc1 − ρc2 . Equivalently, we find that c2 + ρc2 ≤ c1 + ρc1 , as needed. 114 α β = µD H (E) β Figure 3 Shapes and Orientation of Walls: The solid lines represent numerical walls for fixed E ∈ Db (X) satisfying rank(E) ̸= 0 and ∆H (E) ≥ 0. The dashed lines represent the α and β axes in R × R>0 . 3. By Lemma 3.4.5, all numerical walls associated to E are either the unique vertical wall or semicircles of the form D 2 ∆H (E) ρ = (µDH (E) 2 − c) − . rank(E)2 where ρ is the radius and c is the center. The result the follows by part 1 and part 2. Remark 3.4.7. Assume that Bc is a semicircle appearing in Lemma 3.4.6. If s is an intersection point of Bc with the line α = 0 (i.e. s = c − ρ or s = c + ρ) then D µD (E) + s ∆H (E) c= H − . 2 2 rank(E)2 (µDH (E) − s) It follows by Lemma 3.4.5.2 that we can write the radius squared in terms of the endpoints of Bc as D 2 (∆H (E) − rank(E)2 (µD 2 2 H (E) − s) ) ρ = 4 rank(E)2 (µD H (E) − s) 2 The following is a consequence of the fact that numerical walls do not intersect. We follow the same proof as [ABCH13, Lemma 6.3]. Lemma 3.4.8. If 0 → F → E → G → 0 is a destabilizing sequence for W (E, F ) at (β0 , α0 ) then F, E, G ∈ CohD+βH H (X) for all (β, α) ∈ W (E, F ). 115 Proof. Assume 0 → F → E → G → 0 is a destabilizing sequence in CohD+β H 0H (X). We first claim that F, E, G ∈ CohD+βH H (X) for all (β, α) ∈ W (E, F ). We will just show that D+βH E ∈ CohH (X) for all (β, α) ∈ W (E, F ). The same argument holds for F and G. If W (E, F ) is a vertical wall, then the result follows because CohD+βH H (X) is independent of α. Therefore, assume that W (E, F ) is a semicircular wall. If β > β0 then H −1 (E) ∈ FHD+βH (X), so we just need to show that H 0 (E) ∈ THD+βH (X). Assume for contradiction that H 0 (E) ̸∈ THD+βH (X) for some (β, α) ∈ W (E, F ). Choose a smallest such β which we call β1 such that H 0 (E) ̸∈ THD+β1 H (X) and (β1 , α1 ) ∈ W (E, F ). Thus, we have a quotient H 0 (E) → G → 0 in Coh(X) such that µD H (G ) = β1 . By minimality of β1 , G is µH -semistable and for all β ∈ [β0 , β1 ) we have a short exact sequence 0 → K → E → G → 0 in CohD+βH H (X). Since G is µH -semistable, by Bogomolov’s inequality, we know 2 D that νH D − µD H (G ) /2 = −∆H (G ) ≤ 0, so   D β2 α21 νH (G ) − βµD H (G ) + 2 − 2 lim− µtilt α1 ,β (G ) = lim− β→β1 β→β1 µDH (G ) − β  D 2  D µH (G ) α21 νH (G ) − µDH (G ) 2 + 2 − 2 = lim− β→β1 µD H (G ) − β D α2 −∆H (G ) − 21 = lim− β→β1 µD H (G ) − β = −∞ because α1 > 0. By Lemma 3.4.6, no two numerical walls can interesect, so we know that β = µD H (E) does not intersect W (E, F ). Therefore, we know that µtilt α,β > −∞ for all (β, α) ∈ W (F, G) and since limβ→β1− µtiltα,β (G ) = −∞, we know that there is a region on W (F, G) such that µtilt tilt α,β (E) > µα,β (G ). However, by Lemma 3.4.6 no two numerical walls can intersect, so µtilt tilt tilt α,β (E) > µα,β (G )) for all (β, α) ∈ W (E, F ). In particular, µα0 ,β0 (E) > µα0 ,β0 (G ). By tilt the seesaw inequality, we find that µtilt tilt α0 ,β0 (K) > µα0 ,β0 (E) which tells us that K = 0 or E is not σ-semistable. If E is not σαtilt 0 ,β0 -semistable, then 0 → F → E → G → 0 is not a 116 destabilizing sequence at (β0 , α0 ), a contradiction. If K = 0 then E = G which contradicts α0 ,β0 (E) > µα0 ,β0 (G ). Hence, we find that H (E) for all (β, α) ∈ W (E, F ). that µtilt tilt 0 Now, assume that β < β0 then H −1 (E) ∈ THD+βH (X), so we just need to show that H −1 (E) ∈ FHD+βH (X). Assume for contradiction that H 0 (E) ̸∈ THD+βH (X) for some (β, α) ∈ W (E, F ). It is clear that if H −1 (E) ̸∈ THD+βH (X) for some (β1 , α1 ) ∈ W (E, F ) then H −1 (E) ̸∈ THD+βH (X) for all β ≤ β1 . Therefore, we can choose the largest β, which we call β1 , such that H −1 (E) ∈ THD+βH (X) and (β1 , α1 ) ∈ W (E, F ). Therefore, we have a nonzero subobject 0 → F → H −1 (E) in Coh(X) such that µD H (F ) = β1 . By maximality of β1 , we know that H 0 (E) is µH -semistable and we have a short exact sequence F → E → C → 0 in CohD+βH H (X) for all β ∈ [β1 , β0 ]. Since F is µH -semistable, we know that D D νH (F ) − µD 2 H (F ) /2 = −∆H (F ) ≤ 0, so   β2 α21 νHD (F ) − βµDH (F ) + 2 − 2 lim+ µtilt α1 ,β (F ) = lim+ β→β1 β→β1 µDH (F ) − β  D 2  2 µH (F ) α21 νHD (F ) − µD H (F ) + 2 − 2 = lim+ β→β1 µD H (F ) − β D α2 −∆H (F ) − 21 = lim+ β→β1 µDH (F ) − β = +∞ because α1 > 0. By Lemma 3.4.6, no two numerical walls can interesect, so we know that β = µD H (E) does not intersect W (E, F ). Therefore, we know that µtilt α,β (E) < +∞ for all (β, α) ∈ W (F, G) and since limβ→β1+ µtilt α,β (F ) = +∞, we know that there is a region on W (F, E) such that µtilt tilt α,β (F ) > µα,β (E). However, by Lemma 3.4.6 no two numerical walls can intersect, so µtilt tilt tilt tilt α,β (F ) > µα,β (E)) for all (β, α) ∈ W (E, F ). In particular, µα0 ,β0 (F ) > µα0 ,β0 (E). In other words, either F = 0 or E is not σαtilt 0 ,β0 -semistable. If E is not σαtilt 0 ,β0 -semistable, then 0 → F → E → G → 0 is not a destabilizing sequence at (β0 , α0 ), a contradiction. Also, F ̸= 0 117 by assumption, a contradiction. Hence H −1 (E) ∈ FHD+βH (X) for all (β, α) ∈ W (E, F ), as needed. All cases considered, we find that E ∈ CohD+βH H (X) for all (β, α) ∈ W (E, F ), as desired. The same argument shows that F, G ∈ CohD+βH H (X) for all (β, α) ∈ W (E, F ) as well. By a direct calculation, it immediately follows that destabilizing sequences do not depend on the given point: Lemma 3.4.9. Assume E is a torsion-free sheaf. 1. If 0 → F → E → G → 0 is a destabilizing sequence for W (E, F ) at (β0 , α0 ) then 0 → F → E → G → 0 is a destabilizing sequence at all (β, α) ∈ W (E, F ). 2. If 0 → F → E [1] → G → 0 is a destabilizing sequence for W (E, F ) at (β0 , α0 ) then 0 → F → E [1] → G → 0 is a destabilizing sequence at all (β, α) ∈ W (E, F ). 3.5 Bounding Actual Walls In this subsection, we will give some explicit bounds on the radius of actual walls associated to torsion-free sheaves in the (H, D)-slice. These result cans be thought of as effective versions of the large volume limit. The following lemma first appeared in [ABCH13, Lemma 6.4] for surfaces. Lemma 3.5.1. Fix E ∈ Db (X) and assume W is an actual semicircular wall associated to E with radius ρ and center (c, 0). If 0 → F → E → G → 0 is a destabilizing sequence associated to W then ˆ µDH (H −1 (F )), µD H (H −1 (E)), µD H (H −1 (G)) ≤ c − ρ and ˆ µD 0 D 0 D 0 H (H (F )), µH (H (E)), µH (H (G)) ≥ c + ρ. 118 Proof. Since W is an actual wall, we know that 0 → F → E → G → 0 is a stabilizing sequence at (β0 , α0 ). By Lemma 3.4.9; F, E, G ∈ CohD+βH H (X) for c − ρ < β < c + ρ. Therefore, if we let β approach c − ρ, we obtain the first result. If we let β approach c + ρ we obtain the second result. If E ∈ Coh(X) then Lemma 3.5.1 shows that any actual wall of E must lie entirely to the left of the unique vertical wall β = µD H (E ). Similarly, any actual wall of E [1] must lie entirely to the right of the unique vertical wall β = µD H (E ). This is provided by the following lemma. Part 1 uses essentially the same argument as [CH16, Proposition 8.3]. Part 2 uses a similar argument to [Kop20, Theorem 3.3]. D Lemma 3.5.2. Let E be a torsion-free sheaf satisfying ∆H (E ) ≥ 0. Assume W is an actual semicircular wall associated to E (resp. E [1]) with associated destabilizing sequence 0 → F → E → G → 0 (resp. 0 → F → E [1] → G → 0). Let ρ be the radius of W . 1. If H −1 (G) ̸= 0 (resp. rank(H 0 (F )) ̸= 0) then s D ∆H (E ) ρ≤ . 4(rank(E ) + 1) D In particular, W is empty if ∆H (E ) = 0. 2. If H −1 (G) = 0 (resp. rank(H 0 (F )) = 0) then   1 D 1 ρ≤ ∆H (E ) − . 2 rank(E )2 D D In particular, in this case, W is empty if either ∆H (E ) = 0 or ∆H (E ) = 1 with rank(E ) = 1. Proof. We will only consider the case of E . The arguments for E [1] are essentially the same. Furthermore, if ∆H (E ) = 0 then W is empty by [BMS16, Proposition A.8]. Thus, we may D assume ∆H (E ) > 0. Taking cohomology of the destabilizing sequence gives us the following exact sequence of coherent sheaves: 0 → H −1 (G) → H 0 (F ) → E → H 0 (G) → 0. 119 1. Assume H −1 (G) ̸= 0, so H 0 (F ) is torsion-free; in particular, H 0 (F ) has non-zero rank. By Lemma 3.5.1, c + ρ ≤ µD H (H (F )). Since H (F ) has nonzero rank, by 0 0 additivity, (c + ρ) rank(H 0 (F )) ≤ degD 0 H (H (F )) −1 D = degD H (H (G)) + degD H (E ) − degH (H (G)) 0 −1 = µDH (H (G)) rank(H −1 (G)) + degD D 0 H (E ) − degH (H (G)) Using the bounds from Lemma 3.5.1, it follows that −1 (c + ρ) rank(H 0 (F )) ≤ µDH (H (G)) rank(H −1 (G)) + degD D 0 H (E ) − degH (H (G)) ≤ (c − ρ) rank(H −1 (G)) + degD D 0 H (E ) − degH (H (G)). Either H 0 (G) is torsion or has positive rank. If H 0 (G) is torsion, by Lemma 2.1.3, then degD 0 H (H (G)) ≥ 0. Therefore, ρ(rank(H 0 (F )) + rank(H −1 (G))) ≤ c(rank(H −1 (G)) − rank(H −1 (F ))) + degD H (E ) and so ρ(rank(H 0 (F )) + rank(H −1 (G))) ≤ −c rank(E ) + degD H (E ) = (µD H (E ) − c) rank(E ) On the other hand, by additivity, rank(E ) + 2 ≤ rank(E ) + 2 rank(H −1 (G)) = rank(H 0 (F )) + rank(H −1 (G)), so ρ(rank(E ) + 2) ≤ (µD H (E ) − c) rank(E ). If rank(H 0 (G)(̸= 0, by Lemma 3.5.1, then (c + ρ)(rank(H 0 (F )) + rank(H 0 (G))) ≤ (c + ρ) rank(H 0 (F )) + µD 0 H (H (G)) rank(H (G)) 0 ≤ (c − ρ) rank(H −1 (G)) + degD H (E ) 120 In other words, by additivity, ρ(rank(H 0 (G)) + rank(H 0 (F )) + rank(H −1 (G))) ≤ c(rank(H −1 (G)) − rank(H 0 (F )) − rank(H 0 (G))) + degD H (E ) = −c rank(E ) + µD H (E ) rank(E ) = (µD H (E ) − c) rank(E ). On the other hand, by additivity, rank(H 0 (G) + rank(H 0 (F )) + rank(H −1 (G)) = rank(E ) + 2 rank(H 0 (G) ≥ rank(E ) + 2. Thus, we have found that ρ(rank(E ) + 2) ≤ (µD H (E ) − c) rank(E ) in both cases. By Lemma 3.5.2, both sides of the above inequality are positive, so ρ2 (rank(E ) + 2)2 ≤ (µD 2 2 H (E ) − c) rank(E ) . By Lemma 3.4.5, D 2 ∆H (E ) ρ = (µD H (E ) 2 − c) − , rank(E )2 so we find that D ! ∆H (E ) D ρ2 (rank(E ) + 2)2 ≤ ρ2 + rank(E )2 = ρ2 rank(E )2 + ∆H (E ) rank(E )2 Equivalently, D ρ2 (4 rank(E ) + 4) ≤ ∆H (E ) so D 2 ∆H (E ) ρ ≤ 4(rank(E ) + 1) 121 2. Since H −1 (G) = 0, we have the following exact sequence in Coh(X): 0 → H 0 (F ) → E → H 0 (G) → 0. By weak the seesaw inequality (Lemma 2.2.6), one of the following inequalities must hold: ˆ µD 0 D H (H (F )) ≤ µH (E ) ˆ µD 0 D H (H (G))µH (E ). First suppose µD 0 D H (H (F )) ≤ µH (E ). Since W is a semicircular wall, by Lemma 3.4.5.1, µD 0 D D 0 D H (H (F )) ̸= µH (E ). In other words, µH (H (F )) < µH (E ) so 1 µD 0 H (H (F )) + 2 ≤ µD H (E ). rank(E ) Therefore, by Lemma 3.5.1, 1 c + ρ ≤ µD 0 D H (H (F )) ≤ µH (E ) − rank(E )2 We obtain the same inequality in the case of µD 0 D H (H (G)) ≤ µH (E ). 1 In other words, W is contained in the semicircle with right endpoint µD H (E ) − rank(E )2 whose center and radius satisfy the equation of Lemma 3.4.5.2. In other words, 1 D 1 ρ≤ ∆H (E ) − , 2 rank(E )2 as claimed. The results of this section and the previous section are illustrated in Figure 3.5. 3.6 A Wall-Crossing Theorem tilt If W is an actual semicircular wall associated to E and E is weakly σα,β -stable on one side tilt of the semicircular wall then E is σα,β -unstable on the other side of the semicircular wall. 122 tilt σα,β -stability of E [1] is equivalent to µH - stability of E α   D β = µD H (E) α= 1 ∆H (E ) − 1 2 rank(E )2 β Figure 4 Bound on the Largest Actual Wall: The solid lines represent actual walls of E [1] when E is torsion-free. The horizontal dashed line represents our bound on the largest actual wall. The shaded region is the large volume limit. The actual walls of E is the same picture mirrored over β = µD H (E ) expect that weak σα,β -stability of E is equivalent to tilt (H, D)-twisted stability of E . tilt The converse is false, E may be σα,β -unstable on both sides of an actual semicircular wall. When dim(X) = 2, [BM11, Lemma 5.9] gives a partial converse. There have been a few related results to higher dimensions, but they do not fully general- ize Bayer and Macrı̀’s result. For example, in the case of X = P3 , [Sch20, Theorem 6.1.4] gives a partial converse. Similarly, in the case of destabilizing sequences 0 → F → E → G → 0 satisfying H −1 (E) = 0, codim(H 0 (E)) = 1, and H 0 (G) = 0; [Fey21, Proposition 4.2 and Corollary 4.3] gives a partial converse. Using the theory developed in the previous sections, we can give a higher dimensional generalization. Theorem 3.6.1. Assume F, G ∈ Db (X), G has good quotients, and there exists (β0 , α0 ) ∈ W (F, G) such that F and G are weakly σαtilt 0 ,β0 -stable. Let 0 → F → E → G⊕r → 0 be a short exact sequence in CohD+βH 0H (X) for some positive integer r. If the following conditions are satisfied ˆ HomDb (X) (G, E) = 0, 123 ˆ there exists ε > 0 such that µtilt tilt ⊕r α,β0 (F ) < µα,β (G ) for all α ∈ (α0 , α0 + ε). tilt then there exists δ > 0 such that E is weakly σα,β 0 -stable for all α ∈ (α0 , α0 + δ). Before giving the proof, we remark on the assumptions. The requirement that G has good quotients allows us to use the full strength of Schur’s Lemma (Lemma 2.2.13). The assumption HomDb (X) (G, E) = 0 is essentially avoiding the case E = F ⊕ G. In fact, if r = 1 then it suffices to assume E ̸= F ⊕ G. Last, since W is an actual wall, there exists ε > 0 such that either µtilt tilt α,β0 (F ) < µα,β0 (G) for all α ∈ (α0 , α0 + ε) or α ∈ (α0 , α0 − ε). The first case is considered in Theorem 3.6.1. The same proof will give a similar result in the second case. By Lemma 2.3.13 and Lemma 3.1.9, the good quotients assumption is vacuous when dim(X) = 2. Therefore, Theorem 3.6.1 truly generalizes [BM11, Lemma 5.9] in the case of tilt stability. Proof. Choose ε > 0 as in the theorem statement. By Lemma 3.4.4, and decreasing ε if tilt necessary, we may assume that F and G are weakly σα,β 0 -stable for all α ∈ [α0 , α0 + ε), Harder-Narasimhan filtrations of E are constant in the region {β0 } × (α0 , α0 + ε), and Jordan-Hölder filtrations of semistable factors of E are also constant in that region. For some α ∈ (α0 , α0 + ε) consider a Harder-Narasimhan filtration of E: 0 → E1 → E2 → · · · → Em−1 → E. tilt with respect to σα,β 0 . By choice of ε above, this filtration is a Harder-Narasimhan filtration tilt of E with respect to σα,β 0 for all α ∈ (α0 , α0 + ε). By Lemma 2.3.3, E → E/Em−1 → 0 is a tilt minimal destabilizing quotient with respect to σα,β 0 . Since E/Em−1 is a minimal destabilizing quotient of E and IZα0 ,β0 (E) ̸= 0, we find IZαtilt 0 ,β0 (E/Em−1 ) ̸= 0 as well. In particular, by Lemma 3.3.2 and choice of ε, E/Em−1 has a weak Jordan-Hölder filtration 0 → Ê1 → Ê2 → · · · → Êl−1 → E/Em−1 124 tilt with respect to σα,β 0 for all α ∈ (α0 , α0 + ε). In particular, E → (E/Em−1 ) → (E/Em−1 )/El−1 → 0 tilt is a weakly σα,β 0 -stable quotient of minimal slope for all α ∈ (α0 , α0 +ε). For ease of notation, rewrite this quotient as E → Ê → 0. Since E is σαtilt 0 ,β0 -semistable, it follows that µtilt tilt tilt α0 ,β0 (Ê) = µα0 ,β0 (E) and so Ê is σα0 ,β0 - semistable. Therefore, by Schur’s Lemma (Lemma 2.2.13), the composition F → E → Ê must be an injection or 0. First assume the composition F → E → Ê is 0. Therefore, by exactness of the short exact sequence 0 → F → E → G⊕r → 0, E → G⊕r → 0 factors through E → Ê → 0. In particular, there is a surjection G⊕r → Ê → 0. Since Ê and G are tilt weakly σα,β 0 -stable for all α ∈ (α0 , α0 + ε), by Schur’s Lemma (Lemma 2.2.13), µtilt α,β0 (Ê) ≥ ⊕r µtilt tilt tilt α,β0 (G ). Moreover, by construction, µα,β0 (Ê) ≤ µα,β0 (E) for all α ∈ (α0 , α0 + ε). By ⊕r assumption, µtilt tilt α,β0 (F ) < µα,β0 (G ) for all α ∈ (α0 , α0 + ε) so, by the weak seesaw inequality, ⊕r µtilt tilt α,β0 (E) < µα,β0 (G ) (we obtain strict inequality because W (F, G) is semicircular and so tilt Zα,β 0 (G) ̸= 0). Combining these inequalities, we find ⊕r ⊕r µtilt tilt tilt α,β0 (G ) ≤ µα,β0 (Ê) ≤ µα,β0 (E) < µα,β0 (G ), tilt for all α ∈ (α0 , α0 + ε) which is not possible. Therefore, the composition F → E → Ê is injective. Since F → E → Ê is injective, we have a short exact sequence 0 → F → Ê → Q → 0 with a surjection G⊕r → Q → 0. We claim Q = G⊕s for 0 ≤ s ≤ r. Since G has good quotients, by Lemma 2.3.13, G⊕r also has good quotients. Therefore, if Zαtilt 0 ,β0 (Q) = 0 then Q = 0 or Q = G⊕r , as claimed. If Zαtilt 0 ,β0 (Q) ̸= 0 then, by the weak seesaw inequality (Lemma 2.2.6) µtilt tilt tilt tilt α0 ,β0 (F ) = µα0 ,β0 (Ê) = µα0 ,β0 (Q). In particular, Q is σα0 ,β0 -semistable. Thus, by Schur’s Lemma (Lemma 2.2.13), the surjection G⊕r → Q → 0 induces an isomorphism G⊕s = Q for some 0 ≤ s ≤ r. 125 Thus, we have shown we have a short exact sequence 0 → F → Ê → G⊕s → 0. By the nine lemma, the kernel of the surjection E → Ê is G⊕r−s . If s ̸= r then we obtain a nonzero morphism HomDb (X) (G, E) which is a contradiction. Therefore, we must have s = r tilt which means that E → Ê is an isomorphism. In particular, E is weakly σα,β 0 -stable for all α ∈ (α0 , α0 + ε), as desired. There is also a dual version of Theorem 3.6.1 corresponding to walls of the form 0 → F ⊕r → E → G → 0. Remark 3.6.2. If X satisfies Bogomolov’s Inequality then the results of this chapter hold over an algebraically closed field of characteristic p > 0. In particular, the results hold for Del Pezzo surfaces (see Remark 2.1.16). 126 CHAPTER 4 APPLICATIONS In this chapter, we apply our wall-crossing result in two scenarios. The first scenario is a restriction theorem for µH -stable sheaves. The second scenario is µH -stability of Lazarsfeld- Mukai sheaves. 4.1 A Generalization of Bogomolov’s Restriction Theorem Bogomolov’s Restriction theorem states that on a smooth projective surface, µH -stable bun- dles remain stable when restricted a general divisor D ∈ |aH| for a ≫ 0 [Bog93]. In fact, Bogomolov is even able to give an effective bound on a. John Kopper gives a generalization of Flenner’s theorem for surfaces [Kop20, Theorem 3.3]. Specifically, Kopper proved that µH -stability on a smooth projective surface is preserved when restricting to any divisor of sufficiently high degree (with an effective bound on the degree) - not just a divisor in some multiple of the very ample class. We generalize this result to higher dimensional varieties. A preliminary lemma shows a codimension 1 sheaf is µH -stable on its support iff E is tilt weakly σα,β -stable on the ambient space. This result is now standard. Lemma 4.1.1. Consider an integral subvariety ι : Y → X. Let E be a torsion-free sheaf on Y . If E is σα,β tilt -(semi)stable for some (β, α) ∈ R × R>0 in the (H, D)-slice then E is µH|Y -(semi)stable. Proof. Since codim(ι∗ E ) = 1, rank(ι∗ E ) = 0 so H n−2 · chD+βH (ι∗ E ) α,β (ι∗ E ) 2 µtilt = degH D+βH (ι∗ E ) H n−2 · ch2 (ι∗ E ) − H n−2 · (D + βH) · ch1 (ι∗ E ) = degH (ι∗ E ) By a similar argument to Lemma 2.1.3, we find rank(E )H n−2 · Y H n−2 · ch2 (ι∗ E ) = H n−2 · ch1 (E ) − . 2 127 Therefore, since ch1 (ι∗ E ) = rank(E ), degH (E ) H n−2 · Y α,β (ι∗ E ) = µtilt − rank(E ) 2 where degH (E ) is the degree of E on Y with respect to the ample divisor induced by H. Now, consider a proper, nonzero subsheaf 0 → F → E in Coh(Y ). In particular, we obtain an injection 0 → ι∗ F → ι∗ E in CohD+βH H (X) for all β ∈ R. In particular, since ι∗ E tilt is σα,β -(semi)stable, H n−2 · Y H n−2 · Y µH|Y (F ) − = µtilt (ι α,β ∗ F ) (≤ ) µ tilt α,β ∗(ι E ) = µ H|Y (E ) − . 2 2 Therefore, E is µH|Y -(semi)stable, as desired. Theorem 4.1.2. Consider a reflexive, µH -stable sheaf E on X. Consider an integral sub- variety ι : Y → X. If the following bounds are satisfied ∆H (E ) 1 H n−2 · ch1 (E ) · Y H n−2 · Y 2 1. µH (E ) − − > − 2 2 rank(E )2 rank(E )H n−1 · Y 2H n−1 · Y ∆H (E (−Y )) 1 H n−2 · ch1 (E ) · Y H n−2 · Y 2 2. µH (E (−Y )) + + < − 2 2 rank(E )2 rank(E )H n−1 · Y 2H n−1 · Y then E |Y is µH|Y -stable. Proof. Since E is reflexive, we have a short exact sequence 0 → E (−Y ) → E → E |Y → 0. This induces a distinguished triangle E → E |Y → E (−Y )[1] → E [1]. We claim that this distinguished triangle lies in CohβH H (X) for some β. Since E is µH -stable, E (−Y ) is µH -stable as well. Since E and E (−Y ) are µH -stable, by the Large Volume Limit (Lemma 3.2.3), E is weakly σα,β tilt -stable and E (−Y )[1] is σα,βtilt -stable for all α ≫ 0 and −µD H (E ) + degH (−Y ) < β < µH (E ). D 128 On the other hand, by additivity Y2   ch≤2 (OX (−Y )) = 1, −Y, . 2 Therefore, by multiplicativity of the Chern character, degH (E (−Y )) = degH (E ) − rank(E )H n−1 · Y H n−2 · Y 2 H n−2 · ch2 (E (−Y )) = H n−2 · ch2 (E ) − H n−2 · ch1 (E ) · Y + rank(E ) . 2 Therefore, by Lemma 3.4.5, the center of W (E , E (−Y )[1]) is H n−2 · ch1 (E ) · Y H n−2 · Y 2 − . rank(E )H n−1 · Y 2H n−1 · Y The assumed inequalities guarentee W (E , E (−Y )[1]) lies between µH (E (−Y )[1]) and µH (E ). Furthermore, by Lemma 3.5.2 and the assumed inequalities, W (E , E (−Y )[1]) is larger than every actual wall of E and E (−Y )[1]. In particular, E is weakly σα,β tilt -stable and E (−Y )[1] is σα,β tilt -stable along W (E , E (−Y )[1]). Note HomDb (X) (E (−Y )[1], E |Y ) = Ext−1 Db (X) (E (−Y ), E |Y ) = 0. Furthermore, since H ·Y ̸= 0, H n−2 ·ch≤2 (E ) and H n−2 ·ch≤2 (E (−Y )) are not scalar multiples. tilt In particular, µtilt α,β (E (−Y )) < µα,β (E ) in either the chamber directly above or directly below W (E , E (−Y )[1]). In either case, by Theorem 3.6.1, E |Y is weakly σα,β tilt -stable in either the chamber directly above or below the actual wall W (E , E (−Y )[1]). The desired result follows by Lemma 4.1.1. In the case that Y is a multiple of the ample divisor H then the inequalities in Theorem 4.1.2 simplifies to: Corollary 4.1.3. Assume E is a reflexive µH -stable sheaf and H is ample divisor with H n = 1. If 1 a> + ∆H (E ) rank(E )2 then E |aH is µH -stable. 129 This result first appeared (via a similar argument) in [Fey21, Theorem 1]. The asymptotic form of this corollary (i.e. α ≫ 0 rather than an explicit bound) is the Mehta-Ramanathan theorem [HL10, Theorem 7.2.8]. A slightly more general form of this corollary first appeared in [Lan04] 4.2 Stability of Lazarsfeld-Mukai Sheaves In this section, we apply our wall-crossing theorem to study Lazarsfeld-Mukai bundles asso- ciated to sufficiently positive, stable bundles. Definition 4.2.1. Assume E is a globally generated, torsion-free sheaf. In other words, there is a short exact sequence 0 → ME → H 0 (E ) ⊗ OX → E → 0. (4.1) We call ME the Lazarsfeld-Mukai sheaf associated to E . If E is clear from context, we just write M instead of ME . The Lazarsfeld-Mukai sheaf is also called the kernel or syzygy sheaf. ⊕h0 (E ) As short hand, we will write OX instead of H 0 (E ) ⊗ OX . In this section we show the Lazarsfeld-Mukai bundle associated to a sufficiently positive stable bundle is also stable. Broadly, by shifting Equation 4.1 we obtain the following distinguished triangle: ⊕h0 (E ) 0 → E → M [1] → OX [1] → 0. If E is stable and sufficiently positive then M [1] is σα,β tilt -stable by Theorem 3.6.1. This stability enforces severe constraints on the first few Chern characters of a maximal µH - destabilizing subsheaf of M . In many cases, we can use these constraints to show that M is stable. This general method of proof has been used in many recent results (e.g. [Bay18],[Kop20], and [Fey21]). Our case differs from these because we are working with torsion-free sheaves— tilt while each of these recent works only considers torsion sheaves. For a torsion sheaf, σα,β - 130 stability is equivalent to µH -stability along its support ([Kop20, Lemma 2.6]). For torsion-free sheaves this result is false, so more work is needed to show µH -stability. Lemma 4.2.2. Let E be a globally generated, torsion-free, H-twisted stable sheaf on X with associated Lazarsfeld-Mukai sheaf M . If the following bounds are satisfied: ˆ degH (E ) > 0, ˆ H n−2 · ch2 (E ) > 0, and n−2 ˆ 2 H deg ·ch(E2 (E ) ) + 1 rank(E )2 > ∆H (E ) H ⊕h0 (E ) then W (OX [1], E ) is an actual wall associated to M [1] in the (H, 0)-slice, and M [1] is tilt σα,β -stable in the chamber directly above this wall. Proof. There is an exact sequence in Coh(X): ⊕h0 (E ) 0 → M → OX →E →0 which induces the following exact sequence in CohβH H (X) for all β ∈ [0, µH (E )): ⊕h0 (E ) 0 → E → M [1] → OX [1] → 0. Since OX is µH -stable and E is H-twisted stable, by the Large Volume Limit (Lemma 3.2.3), OX [1] and E are weakly σα,β tilt -stable for all β ∈ [0, µH (E )) and α ≫ 0. In fact, since ∆H (OX ) = 0, OX [1] is weakly σα,β tilt -stable for all β ≥ 0 and all α > 0. Furthermore, since n−2 2 H deg ·ch(E2 (E ) ) + 1 rank(E )2 > ∆H (E ), H s   ∆H (E ) 1 1  H n−2 · ch2 (E ) max , ∆H (E ) − < .  4(rank(E ) + 1) 2 rank(E )2  degH (E ) ⊕h0 (E ) On the other hand, by direct calculation and Lemma 3.4.5.1, the radius of W (OX [1], E ) is H n−2 · ch2 (E )/ degH (E ). Therefore, E is weakly σα,β tilt -stable for all (β, α) lying above the ⊕h0 (E ) ⊕h0 (E ) wall W (OX [1], E ). This shows that W (OX [1], E ) is an actual wall associated to M [1] in the (H, 0)-slice. 131 It remains to show M [1] is σα,β tilt -stable for all (β, α) in the chamber directly above ⊕h0 (E ) ⊕h0 (E ) W (OX [1], E ). By Lemma 3.1.13, OX [1] has good quotients. Furthermore, by defi- nition of a Lazarsfeld-Mukai sheaf, HomDb (X) (OX [1], M [1]) = H 0 (M ) = 0. For ease of notation, set  n−2 · ch2 (E ) H n−2 · ch2 (E )  H ⊕h0 (E ) (β0 , α0 ) = , ∈ W (OX [1], E ). degH (E ) degH (E ) By direct computation, ⊕h0 (E ) degH (E )2 ε(2H n−2 · ch2 (E ) + degH (E )ε) µtilt α0 +ε,β0 (OX [1]) − µtilt α0 +ε,β0 (E ) = 2H n−2 · ch2 (E )(degH (E )2 − rank(E )H n−2 · ch2 (E )) which is positive as long as degH (E )2 − rank(E )H n−2 · ch2 (E ) > 0. However, since E is H-twisted stable (a fortiori µH -semistable by Lemma 2.1.11), this in- equality follows from Bogomolov’s Inequality. Hence, by Theorem 3.6.1, M [1] is weakly σα,β tilt -stable for (β, α) in the chamber directly above W . In fact, by Lemma 3.1.13 and Lemma 2.2.11, M [1] is σα,β tilt -stable for all such (β, α), as desired. Lemma 4.2.3. Let M be the Lazarsfeld-Mukai sheaf associated E satisfying the assumptions of Lemma 4.2.2. 1. If M is not µH -semistable and 0 → N → M is a maximal µH -destabilizing subsheaf then H n−2 · ch2 (N ) H n−2 · ch2 (M ) ≤ . degH (N ) degH (M ) 2. If M is µH -semistable and 0 → N → M is a subsheaf satisfying µH (N ) = µH (M ) and rank(N ) < rank(M ) then H n−2 · ch2 (N ) H n−2 · ch2 (M ) < degH (N ) degH (M ) 132 Proof. Assume M is the Lazarsfeld-Mukai sheaf associated to E . ⊕h0 (E ) 1. Since W (E , OX [1]) has endpoints (0, 0) and (2H n−2 ·ch2 (E )/ degH (E ), 0) and walls are locally finite, by Lemma 4.2.2, there exists α0 > 0 such that M [1] is σα,0 tilt -stable for all α ∈ (0, α0 ). By Lemma 3.1.3, 0 → N [1] → M [1] is a subobject in Coh0H H (X) and so 2 H n−2 · ch2 (N [1]) − α2 rank(N [1]) = µtilt α,0 (N [1]) degH (N [1]) < µtilt α,0 (M [1]) 2 H n−2 · ch2 (M [1]) − α2 rank(M [1]) = degH (M [1]) for all α ∈ (0, α0 ). Taking the limit as α approaches 0 gives H n−2 · ch2 (N ) H n−2 · ch2 (M ) ≤ , degH (N ) degH (M ) as claimed. 2. By the same argument as part 1, we find H n−2 · ch2 (N ) H n−2 · ch2 (M ) ≤ . degH (N ) degH (N ) If we have equality, by Remark 3.1.8, µtilt tilt α,β (N [1]) = µα,β (M [1]) for all (β, α) ∈ R×R>0 . In particular, M [1] is not σα,β tilt -stable for β > µH (M [1]). This contradicts Lemma 4.2.2, so we must have H n−2 · ch2 (N ) H n−2 · ch2 (M ) < degH (N ) degH (M ) as claimed. Theorem 4.2.4. Assume X is a smooth Del Pezzo surface over an algebraically closed field of arbitrary characteristic. For ease of notation, let H = −KX which is ample by definition. Let E be a globally generated, torsion-free, (H, H2 )-Gieseker stable sheaf on X with associated Lazarsfeld-Mukai bundle M . If the following bounds are satisfied: 133 ˆ 0 < degH (E ) ≤ KX 2 (h0 (E ) − rank(E )), ˆ ch2 (E ) > 0, H n−2 · ch2 (E ) 1 ˆ 2 + ≥ ∆H (E ) degH (E ) rank(E )2 then M is µH -stable. Proof. Consider a maximal µH -destabilizing subsheaf 0 → N → M . Since H = −KX , by the Hirzebruch-Riemann-Roch theorem, χ(N ) ch2 (N ) 1 rank(N ) = − + . degH (N ) degH (N ) 2 degH (N ) Since 0 → N → M is a maximal µH -destabilizing subsheaf, by definition and Lemma 4.2.3, χ(N ) ch2 (N ) 1 rank(N ) ch2 (M ) 1 rank(M ) χ(M ) = − + < − + = . degH (N ) degH (N ) 2 degH (N ) degH (M ) 2 degH (M ) degH (M ) Since X is a Del Pezzo surface, χ(N ) χ(M ) h1 (E ) < = ≤ 0. degH (N ) degH (M ) degH (M ) Since M is a Lazarsfeld-Mukai bundle, h0 (M ) = 0 and so h0 (N ) = 0. We claim h2 (N ) = 0 as well. Since N is torsion-free, there is a natural injection 0 → N → N ∨∨ whose cokernel is supported in dimension 2. Furthermore, N ∨ is reflexive and so locally free [Har80, Corollary 1.4]. Thus, by Serre duality, ∨∨ ∨ h2 (N ) = h2 (N ) = h0 (N ⊗ ωX ). However, since N is µH -semistable, ∨ µ+ H (N ⊗ ωX ) = −µH (N ) + degH (ωX ) < −µH (M ) + degH (ωX ) < 0, 2 where the last inequality follows from assumed bound: degH (E ) ≤ KX (h0 (E ) − rank(E )). Since µ+H (N ∨ ⊗ ωX ) < 0, ∨ h2 (N ) = h0 (N ⊗ ωX ) = 0, 134 as claimed. Therefore, we have shown h1 (N ) χ(N ) 0≤− = < 0, degH (N ) degH (N ) a contradiction. Hence, M must be µH -semistable. We now claim M is µH -stable, so consider a subsheaf 0 → N → M with µH (N ) = µH (M ). Furthermore, by Lemma 2.2.11, we may assume rank(N ) < rank(M ). By Lemma 4.2.3 and the Hirzebruch-Riemann-Roch theorem, χ(N ) ch2 (N ) 1 rank(N ) ch2 (M ) 1 rank(M ) χ(M ) = − + < − + = . degH (N ) degH (N ) 2 degH (N ) degH (M ) 2 degH (M ) degH (M ) By the same argument as above, χ(N ) < 0 and χ(M ) > 0 so we find χ(N ) χ(M ) 0= < degH (N ) degH (M ) which is a contradiction. Thus, M is µH -stable, as claimed. Last, by remark 3.6.2, this argument holds in arbitrary characteristic. In the case of smooth Del Pezzo surfaces and H = −KX , Theorem 4.2.4 completely generalizes the best previously known result [TLZ21, Theorem 3.7]. Example 4.2.5. We note Theorem 4.2.4 cannot be weakened to just the inequalities 0 < degH (E ) and 0 < ch2 (E ). Consider the tangent bundle TP2 on P2 . There is a short exact sequence 0 → ΩP2 (1)⊕3 → H 0 (TP2 ) ⊗ OP2 → TP2 → 0. In particular, the Lazarsfeld-Mukai bundle associated to TP2 is not µH -stable even though TP2 is µH -stable ([OSS80, Chapter 2 Theorem 1.3.2]) with degH (TP2 ), ch2 (TP2 ) > 0. We note that ch2 (E ) 1 1 2 + 2 = 1 + < 3 = ∆H (TP2 ), degH (E ) rank(E ) 4 so TP2 does not satisfy the positivity properties of Theorem 4.2.4. Interestingly, TP2 (2) does satisfy the required positivity properties, so we find that the Lazarsfeld-Mukai bundle associated to TP2 (2) is µH -stable. 135 We note that any sufficiently high twist of a torsion-free µH -stable sheaf satisfies the assumptions of Theorem 4.2.4. Corollary 4.2.6. Assume X is a smooth Del Pezzo surface. For ease of notation, let H = −KX and reg(E ) the Castelnuovo-Mumford regularity of E [Laz04, Definition 1.8.1]. Assume E is a µH -stable, torsion-free sheaf. If s   4∆ (E ) 1  H d ≥ max + 1 − µH (E ) + , ∆H (E ) − µH (E ), reg(E ), 0  rank(E )2 2  then the Lazarsfeld-Mukai bundle associated to E (d) is µH -stable. Proof. By Mumford’s theorem ([Laz04, Theorem 1.8.3]), E (d) is globally-generated, so M is well-defined. By definition of the Castelnuovo-Mumford regularity, h1 (E (d)), h2 (E (d)) = 0. There- fore, by the Hirzebruch-Riemann-Roch theorem and direct computation, E (d) satisfies the assumptions of Theorem 4.2.4. The result follows. In particular, the Lazarsfeld-Mukai associated to OX (−dKX ) is µH -stable for d ≥ 1. Remark 4.2.7. We expect Theorem 4.2.4 to extend to smooth, projective surfaces such that h1 (OX ), h2 (OX ) = 0 and either KX is ample or numerically trivial. Specifically, the argument from Theorem 4.2.4 goes through in this setting (at least in characteristic 0) except, possibly, the vanishing of h2 (N ). We end by showing µH -semistability and µH -stability are equivalent for Lazarsfeld-Mukai bundles associated to ample line bundles. Theorem 4.2.8. Assume X is a smooth projective variety equipped with ample divisor H. Let M be the Lazarsfeld-Mukai bundle associated to OX (d) for d ≥ 1. The following are equivalent 1. M is µH -stable. 136 2. M is µH -semistable. 3. Every actual wall to the right of µH (M ) associated to M [1] in the (H, 0)-slice is nexted ⊕h0 (OX (d)) within the wall W (OX [1], OX (d)). Proof. 1 ⇒ 2: This follows from definition. ⊕h0 (OX (d)) 2 ⇒ 3: Suppose W (OX [1], OX (d)) is not the largest wall associated to M [1] in the (H, 0)-slice. Therefore, we can choose a larger actual semicircular wall W with destabilizing sequence 0 → F → M [1] → G → 0. Since M is the Lazarsfeld-Mukai bundle associated to OX (d), s ∆H (M ) d H n−2 · ch2 (M ) = = . 4(rank(M ) + 1) 2 degH (M ) Therefore, by Lemma 3.5.2, rank(H 0 (F )) = 0. Furthermore, by Lemma 3.4.5.1, W inter- sects the line β = 0 at −(H n−2 · ch2 (M [1]) degH (F ) − H n−2 · ch2 (F ) degH (M [1])) 0<α=2 . degH (M [1]) rank(F ) − degH (F ) rank(M [1]) By the same argument as Lemma 4.2.3, the numerator of this expression is negative. There- fore, degH (M [1]) rank(F ) − degH (F ) rank(M [1]) < 0 or, since rank(H 0 (F )) = 0, µH (M [1]) < µH (F ). Furthermore, since rank(H 0 (F )) = 0, by Lemma 2.1.3, degH (H 0 (F )) ≥ 0. It follows that degH (H −1 (F )) µH (H −1 (F )) = rank(H −1 (F )) degH (H −1 (F )) − degH (H 0 (F ) > rank(H −1 (F )) = µH (F ) > µH (M [1]) = µH (M ). 137 In particular, taking cohomology of 0 → F → M [1] → G → 0 gives us an injection 0 → H −1 (F ) → M with µH (H −1 (F )) > µH (M ). In other words, M is not µH -semistable, as claimed. 3 ⇒ 1: This follows from Lemma 4.2.2 and the Large Volume Limit (Lemma 3.2.3). We note that Theorem 5.8 and Lemma 4.2.2 gives us an explicit description of the largest actual wall of M [1] in infinitely many cases. In practice, such a description seems difficult to find unless degH (M [1]) ≤ 0 or rank(M [1]) = 0. In fact, as far as the author knows, the only other example outside of these cases is spherical bundles on K3 surfaces. 138 CHAPTER 5 FUTURE WORK In this chapter, we discuss more possible applications of our wall-crossing result and further investigations into the applications above. 5.1 Slope Stable Bundles on Rational Varieties As stated in the introduction, it is unknown whether there exists a slope stable bundle of rank 2 on P7 . In fact, in characteristic 0, it is unknown whether there exists a slope stable bundle of rank 2 on Pn for any n ≥ 5. On P4 over C, all known examples come from the Horrocks-Mumford bundle construction. This bundle is constructed by considering certain representations of Z/5Z. This construction suggests that low rank indecomposable vector bundles on Pn for n sufficiently large tend to rise via representation theoretic considerations rather than geometric ones. On the other hand, if X is equipped with an exceptional collection (e.g. many rational varieties), then there exists a quiver Q and a triangulated equivalence Db (X) = Db (Rep(Q)) where Rep(Q) is the representations of Q [Bon90]. Therefore, we obtain a heart Rep(Q) in Db (X). Furthermore, there is a notion of stability on Rep(Q) due to [Kin94], and this induces a very weak stability condition on the heart Rep(Q). We can often find stable objects in the (H, D)-slice that correspond to stable representations of Q. This has already been done for P2 , P1 × P1 , and P3 in [AB13], [AM17], and [Sch20] respectively. Moreover, with sufficient understanding of the walls in the (H, D)-slice, we hope to use Theorem 3.6.1 to relate slope stable vector bundles to stable representations of Q. Further, it is very easy to calculate whether there exists a stable representation with given fixed invariants, so we hope to use this putative correspondence to construct slope stable bundles. 139 5.2 Lazarsfeld-Mukai Bundles on Other Varieties Theorem 4.2.4 shows that Lazarsfeld-Mukai bundles associated to sufficiently positive stable bundles on Del Pezzo surfaces are slope stable. As noted in Remark 4.2.7, the same argument would work on any surface with h1 (OX ), h2 (OX ) = 0 granted h2 (N ) = 0. Therefore, in future work, we hope to investigate this vanishing more thoroughly. Moreover, in future work we aim to show Theorem 4.2.4 extends to higher dimensional ⊕h0 (E ) Fano varieties. One possible argument would be to show W (OX [1], E ) is the largest actual wall associated to ME . This has been done when E is an ample line bundle and X is a K3 suface. However, we believe this method is too hopeful in general. 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