CUBIC FOURFOLDS AND THE KUGA-SATAKE CONSTRUCTION By Michael Annunziata A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mathematicsโ€”Doctor of Philosophy 2023 ABSTRACT Given a ๐พ3 surface ๐‘†, the Kuga-Satake construction associates to ๐‘† an abelian variety ๐พ๐‘†(๐‘†) known as the Kuga-Satake variety. Many similarities between cubic fourfolds ๐‘‹ and ๐พ3 surfaces ๐‘† have been studied, particularly via Hodge theory by Hassett and derived categories by Kuznetsov. We study how the Kuga-Satake construction fits into this theory by studying the Kuga-Satake varieties of cubic fourfolds and their associated ๐พ3 surfaces, endormorphism algebras of cubic fourfolds, and the derived category ๐ท ๐‘ (๐พ๐‘†(๐‘†)). Copyright by MICHAEL ANNUNZIATA 2023 To my parents. iv ACKNOWLEDGEMENTS I would first and foremost like to thank my advisor Rajesh Kulkarni for his guidance, mentorship, and friendship over the past six years. Without him, this dissertation would not have been possible. I will always be grateful for our weekly meetings during my time as a graduate student where we not only discussed mathematics but got to know each other as people. I would like to thank my committee members - Aaron Levin, Igor Rapinchuk, and Michael Shapiro. Thank you for sharing your time and insight with me since the time of my comprehensive exam. Thank you to my academic siblings - Nick Rekuski, Shitan Xu, Shen Yu, Yizhen Zhao. I will always remember our weekly seminars where you saw this work develop over the past few years. Thank you for your friendship and constant insight. Finally, I would like to thank my family and my partner. Being away from home for so long was not always easy but it was made possible by all of your support. Thank you all. The author was partially supported by the NSF grant DNS-2101761 during preparation of this dissertation. v TABLE OF CONTENTS LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER 2 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Hodge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Lattice Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 K3 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5 Cubic Fourfolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 CHAPTER 3 KUGA-SATAKE VARIETIES . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 The Kuga-Satake Construction . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 Kuga-Satake Varieties & Associated K3 Surfaces . . . . . . . . . . . . . . . . 33 3.4 Gushel-Mukai Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 CHAPTER 4 ENDOMORPHISM ALGEBRAS . . . . . . . . . . . . . . . . . . . . . 42 4.1 Endomorphism Algebras of K3 Surfaces . . . . . . . . . . . . . . . . . . . . . 43 4.2 Odd Unimodular Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Period Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.4 Endomorphism Algebras of Cubic Fourfolds . . . . . . . . . . . . . . . . . . . 59 4.5 Endomorphism Algebras and Associated K3 Surfaces . . . . . . . . . . . . . . 62 CHAPTER 5 DERIVED CATEGORIES AND FUTURE WORK . . . . . . . . . . . . 65 5.1 Background: Derived Categories . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.2 The Kuznetsov Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.3 Connecting Hodge Theory and Derived Categories . . . . . . . . . . . . . . . 74 5.4 Kuga-Satake Hodge Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.5 Derived Categories of Kuga-Satake Varieties . . . . . . . . . . . . . . . . . . . 79 5.6 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 vi LIST OF SYMBOLS ๐ดฮ› The discriminant group of the lattice ฮ› Br(๐‘‹) The Brauer group of ๐‘‹ C The complex numbers C The moduli space of smooth cubic fourfolds C๐‘‘ Cubic fourfolds admitting a labeling of discriminant ๐‘‘ c๐‘– (ยท) the ๐‘–-th Chern class ๐ถ๐‘™ (๐‘‰, ๐‘ž) The Clifford algebra associated to (๐‘‰, ๐‘ž) ๐ถ๐‘™ + (๐‘‰, ๐‘ž) The even Clifford algebra associated to (๐‘‰, ๐‘ž) ๐ถ๐‘œ๐‘Ÿ๐‘’๐‘  Corestriction of algebras CSpin(๐‘‰) The Clifford group on (๐‘‰, ๐‘ž) ๐ท ๐‘ (๐‘‹) The bounded derived category of coherent sheaves on a variety ๐‘‹ End(๐‘‰) The Hodge endomorphism algebra of the Hodge structure ๐‘‰ ร“๐‘› The n-th exterior algebra of ๐‘‰ ฮฆ๐พ The Fourier-Mukai transform with kernel ๐พ Gr(๐‘˜, ๐‘‰) The Grasmannian of ๐‘˜โˆ’dimensional subspaces of ๐‘‰ ๐ป ๐‘› (๐‘‹, ๐‘‰) Singular cohomology of a variety ๐‘‹ with coefficients in ๐‘‰ ๐ป ๐‘› (๐‘‹, ๐‘‰)0 Primitive cohomology of ๐‘‹ with respect to some ample divisor โ„Ž ๐‘,๐‘ž (๐‘‰) The ( ๐‘, ๐‘ž) Hodge number of ๐‘‰ ๐ป๐ป๐‘– (๐‘‹) The ๐‘–โˆ’th Hochschild homology of ๐ท ๐‘ (๐‘‹) โˆผ Isogeny of abelian varieties ๐œ… ๐‘† The Kuga-Satake class of ๐‘† ๐พ๐‘†(๐‘‰) The Kuga-Satake variety of a Hodge structure ๐‘‰ (ฮ›, ๐‘) A lattice ฮ› with bilinear form ๐‘ ๐‘™ (ฮ›) The number of generators of ๐ดฮ› N๐‘‘ The moduli space of polarized ๐พ3 surfaces of degree ๐‘‘ vii ๐‘ ๐‘†(๐‘†) The Nรฉron-Severi group of ๐‘† ๐‘‚ (๐‘‰) The orthogonal group on (๐‘‰, ๐‘ž) O ๐‘‹ The structure sheaf of a variety ๐‘‹ ๐‘ƒ๐บ ๐ฟ The projective general linear group ๐œŒ(๐‘†) The Picard rank of ๐‘† ๐œŒ๐‘‰ The algebraic representation defining the Hodge structure ๐‘‰ Q the rational numbers R the real numbers ๐‘† A ๐พ3 surface (๐‘†, ๐ฟ) A polarized ๐พ3 surface ๐‘† with polarization ๐ฟ ๐‘‡ (๐‘†) The transcendental lattice of ๐‘† ๐‘‡ (๐‘‰) The tensor algebra associated to ๐‘‰ T A triangulated category ๐‘‰ A free Zโˆ’module of finite rank or a finite-dimensional Qโˆ’vector space ๐‘‰ โˆจ The dual Hodge structure to ๐‘‰ (๐‘‰, ๐‘„) A polarized Hodge structure ๐‘‰ with polarization ๐‘„ ๐‘‰ (๐‘›) The ๐‘›โˆ’th Tate Twist of the Hodge structure ๐‘‰ ๐‘‹ A variety, typically a cubic fourfold unless otherwise noted ๐œ’(๐‘‹, O ๐‘‹ ) the arithmetic genus Z the integers viii CHAPTER 1 INTRODUCTION Associating abelian varieties to complex projective varieties has often been an effective strategy in algebraic geometry. Since many tools have been developed for studying abelian varieties, it is advantageous to be able to transfer geometric questions about projective varieties to questions about abelian varieties. A classic example of such an abelian variety is the Jacobian variety ๐ฝ (๐ถ) associated to a complex projective curve ๐ถ. In fact, the Torelli Theorem states that a nonsingular complex projective curve ๐ถ is completely determined by its Jacobian ๐ฝ (๐ถ) and the theta divisor. There have been several approaches to associating abelian varieties to higher dimensional varieties including Albanese varieties, intermediate Jacobian varieties, and Kuga-Satake varieties. Our study will focus on the Kuga-Satake construction, first developed in [KS67]. Kuga-Satake varieties have been traditionally used to study ๐พ3 surfaces. We say that a Hodge structure is of ๐พ3-type if it a weight two, polarized Hodge structure ๐‘‰ such that ๐‘‰ 2,0 = 1. Of course, given a ๐พ3 surface ๐‘†, its middle cohomology ๐ป 2 (๐‘†, Z) is an example of such a Hodge structure. We will review the Kuga-Satake construction in Chapter 3. The construction takes a weight two Hodge structure ๐‘‰ of ๐พ3-type and associates to it a Hodge structure of weight one. We will see that this corresponds geometrically to inputting a ๐พ3 surface ๐‘† into the construction and getting an abelian variety ๐พ๐‘†(๐‘†) out of it. It is known that the construction gives an inclusion of Hodge structures ๐‘‰ โ†ฉ! ๐ป 1 (๐พ๐‘†(๐‘‰), Q) โŠ— ๐ป 1 (๐พ๐‘†(๐‘‰), Q), so, in particular, it can be shown that the Hodge structure ๐‘‰ can be recovered from the Hodge structure on ๐พ๐‘† and so the Kuga-Satake construction is injective! This motivates the potential for transferring geometric questions on ๐‘† to questions on the abelian variety ๐พ๐‘†(๐‘†) instead. There is a Torelli theorem for ๐พ3 surfaces as well. That is, two ๐พ3 surfaces ๐‘†1 and ๐‘†2 are isomorphic if and only if there is a Hodge isometry ๐ป 2 (๐‘†1 , Z)  ๐ป 2 (๐‘†2 , Z) [PSS71]. In particular, the Kuga-Satake variety ๐พ๐‘†(๐‘†) determines the ๐พ3 surface ๐‘†. 1 Kuga-Satake varieties have proven useful in answering geometric questions about ๐พ3 surfaces. For example, it is an important ingredient in Deligneโ€™s proof of the Weil conjectures for ๐พ3 surfaces [Del71]. More recently, the construction has been used in proving cases of the Hodge conjecture for examples of self-products of ๐พ3 surfaces. We will discuss such an example of [Sch10] in Chapter 4. The Kuga-Satake construction has been generalized to varieties other than ๐พ3 surfaces. In [Mor85], Morrison studied the Kuga-Satake varieties associated to abelian surfaces. We detail this example in Example 3.2.9. Further, work has been done by Voisin in [Voi05] to provide an alternate viewpoint to the construction by putting a weight two Hodge structure on the exterior algebra โˆ— ๐‘‰. ร“ In our study, we aim to investigate the Kuga-Satake variety ๐พ๐‘†(๐‘‹) associated to a cubic fourfold ๐‘‹. It has been noted by many that the Hodge structure on the middle cohomology ๐ป 4 (๐‘‹, Z) of a cubic fourfold ๐‘‹ looks similar to ๐ป 2 (๐‘†, Z) of a ๐พ3 surface ๐‘†. In fact, its Tate Twist ๐ป 4 (๐‘‹, Z)(1), which we will define in Chapter 2, is actually a Hodge structure of ๐พ3-type. This observation was first put on solid footing by Hassett in [Has00] where the notion of an associated ๐พ3 surface was formalized. In particular, we say that a ๐พ3 surface ๐‘† is associated to a cubic fourfold ๐‘‹ if there is a Hodge isometry ๐ป 4 (๐‘‹, Z)(1) โŠƒ ๐พ โŠฅ ! ๐‘“ โŠฅ โŠ‚ ๐ป 2 (๐‘†, Z) (1) for appropriate sub-Hodge structures ๐พ โŠฅ and ๐‘“ โŠฅ . Associated ๐พ3 surfaces have been studied extensively by Hassett and others since then, and they play a role in rationality conjectures for cubic fourfolds. In particular, it is conjectured that a cubic fourfold is rational if and only if it possesses an associated ๐พ3 surface. Our goal is to study the Kuga-Satake construction for cubic fourfolds within the framework of associated ๐พ3 surfaces. In Chapter 3 we study the Kuga-Satake construction for cubic fourfolds. This leads to our first main result relating the Kuga-Satake construction to the theory of associated ๐พ3 surfaces. 2 Theorem 1.0.1. Suppose (๐‘‹, ๐พ) is a special cubic fourfold with associated ๐พ3 surface (๐‘†, ๐‘“ ) as in (1). Then ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐‘†) 2 , where โˆผ denotes isogeny of abelian varieties. We also show in Section 3.4 that these methods apply to other types of varieties. In particular there is a notion of associated ๐พ3 surfaces for Gushel-Mukai fourfolds. As a corollary to Theorem 1.0.1, we show: Corollary 1.0.2. Suppose ๐‘‹ is a Gushel-Mukai fourfold and ๐‘† is a ๐พ3 surface associated to ๐‘‹. Then ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐‘†) 4 . Next, we turn our attention to endomorphism algebras in Chapter 4. Given a ๐พ3 surface ๐‘†, one can consider the set of Hodge endomorphisms ๐‘‡ (๐‘†)Q ! ๐‘‡ (๐‘†)Q on its transcendental lattice, which we will denote by End(๐‘‡ (๐‘†)). In his foundational paper in [Zar83], Zarhin studied this endomorphism algebra and was able to show that End(๐‘‡ (๐‘†)) is in fact a field. Further, Zarhin shows that it is a number field that is either of totally real type or a CM-field. Our goal is to study End(๐‘‡ (๐‘‹)) for a cubic fourfold ๐‘‹. In doing so, we must take a couple of important detours. First, we study Nikulinโ€™s lattice theory in [Nik80]. Most of this lattice theory is developed for even, unimodular lattices. However, ๐ป 4 (๐‘‹, Z) is an odd, unimodular lattice, so we spend time developing the appropriate lattice theory for odd unimodular lattices. This results in: Proposition 1.0.3. Suppose ๐‘‡ is an even lattice of signature (21 โˆ’ ๐œŒ, 2). If 13 โ‰ค ๐œŒ โ‰ค 21, then there exists a primitive embedding ๐‘‡ โ†ฉ! ฮ›๐ถ4 such that ๐‘‡ โŠฅ  ๐‘ is odd. We also study the image of the period map for cubic fourfolds, which was developed by Laza and Looฤณenga in [Laz10] and [Loo09]. We prove the existence of a cubic fourfold ๐‘‹ with End(๐‘‡ (๐‘‹)) of totally real type. Further, we study how endomorphism algebras End(๐‘‡ (๐‘‹)) of cubic fourfolds ๐‘‹ are related to endomorphism algebras End(๐‘‡ (๐‘†)) of their associated ๐พ3 surfaces ๐‘†. In the final chapter, we take a derived category approach to associated ๐พ3 surfaces that was inspired by Kuznetsov in [Kuz16]. Kuznetsov studied the derived category ๐ท ๐‘ (๐‘‹) of a cubic 3 fourfold ๐‘‹ and noted that it contains a subcategory A ๐‘‹ := โŸจO ๐‘‹ , O ๐‘‹ (1), O ๐‘‹ (2)โŸฉ โŠฅ that looks quite similar to the derived category ๐ท ๐‘ (๐‘†) of a ๐พ3 surface ๐‘†. Kuznetsov conjectures that a cubic fourfold ๐‘‹ is rational if and only if A ๐‘‹  ๐ท ๐‘ (๐‘†) for a ๐พ3 surface ๐‘†. Addington and Thomas in [AT14] and others in [BLM+ 21] show that Kuznetsovโ€™s conjecture for the rationality of cubic fourfolds is actually equivalent to the conjecture inspired by Hassettโ€™s Hodge theoretic approach. This inspires us to study how the Kuga-Satake construction fits into the derived picture by investigating ๐ท ๐‘ (๐พ๐‘†(๐‘†)) and ๐ท ๐‘ (๐พ๐‘†(๐‘‹)). In doing so, we study the Kuga- Satake Hodge conjecture, which is a special case of the Hodge conjecture itself. We show the following: Proposition 1.0.4. There exist examples of associated ๐พ3 surfaces that satisfy the Kuga-Satake Hodge conjecture. Assuming the Kuga-Satake Hodge conjecture, which by the previous proposition is a fact in some cases, we are able to construct several functors involving the derived categories of Kuga-Satake varieties. We construct various Fourier-Mukai functors that result in the following commutative diagram in Chapter 5: ฮฆ O๐‘ ๐ท ๐‘ (๐พ๐‘†(๐‘†) 2 ) ๐‘† ๐ท ๐‘ (๐‘†) ฮฆ ฮฆ Oฮ“ ๐‘“ ๐ท ๐‘ (๐‘‹) ฮฆ O๐‘ ๐ท ๐‘ (๐พ๐‘†(๐‘‹) 2 ) ฮฆ Oฮ“ ๐ท ๐‘ (๐พ๐‘†(๐‘‹)) ๐‘‹ ฮ” and end by outlining possible next directions for research. 4 CHAPTER 2 BACKGROUND In this chapter we provide the main background theory and definitions required for our study. This chapter is outlined as follows: ยง2.1 We introduce basic Hodge theory and standard definitions, including the Hodge decom- position, Hodge diamond, and polarizations of Hodge structures. Important examples of Hodge-theoretical constructions are given, morphisms of Hodge structures are discussed, and we catalogue some basic facts that will be used later. ยง2.2 We briefly recall basic definitions and lattice theory that will be sufficient to use throughout. Discriminants of lattices and discriminant groups are discussed and the most important examples of lattices that we use are constructed. More advanced lattice theory is developed in Chapter 4 where it is primarily used in the context of endomorphism algebras. ยง2.3 We discuss facts about abelian varieties which are crucial for us in the context of the Kuga- Satake construction. In particular, we mainly approach complex tori and abelian varieties with their Hodge theory kept in mind. The Hodge diamond of an abelian variety is discussed in our context. The rest of the section is dedicated to showing a one-to-one correspondence between abelian varieties and polarized, integral Hodge structures of weight one. ยง2.4 In this section we define ๐พ3 surfaces and give basic examples. The Hodge diamond of a ๐พ3 surface is discussed. ยง2.5 We discuss cubic fourfolds, the main objects of our consideration throughout the rest of the dissertation. In particular, we point out important similarities between ๐พ3 surfaces and cubic fourfolds, and introduce the notion of associated ๐พ3 surfaces in the sense of Hassett. Motivating rationality questions are also discussed. 5 2.1 Hodge Theory In the following, suppose that ๐‘‰ is either a free Zโˆ’module of finite rank or a finite-dimensional Qโˆ’vector space. A reference for additional background in this section is [Huy16]. Definition 2.1.1. For ๐‘› โˆˆ Z, a weight-๐‘› Hodge structure on ๐‘‰ is a direct sum decomposition of the complexification ๐‘‰C B ๐‘‰ โŠ—Z C as รŠ ๐‘‰C = ๐‘‰ ๐‘,๐‘ž ๐‘+๐‘ž=๐‘› subject to the condition that ๐‘‰ ๐‘,๐‘ž = ๐‘‰ ๐‘ž,๐‘ where ๐‘ฃ โŠ— ๐‘ง B ๐‘ฃ โŠ— ๐‘ง for ๐‘ฃ โˆˆ ๐‘‰ and ๐‘ง โˆˆ C. We call the Hodge structure integral if ๐‘‰ is a free Zโˆ’module and we call it rational if ๐‘‰ is a Qโˆ’vector space. We say that a Hodge structure is of type (๐‘˜, ๐‘˜) if ๐‘‰ ๐‘,๐‘ž โ‰  0 only for ( ๐‘, ๐‘ž) = (๐‘˜, ๐‘˜). Example 2.1.2. The most important examples of Hodge structures that we consider are provided by cohomology of smooth projective varieties. Let ๐‘‹ be a smooth complex projective variety and let ๐‘‰ = ๐ป ๐‘› (๐‘‹, Q) denote singular cohomology. Then ๐‘‰C = ๐ป ๐‘› (๐‘‹, Q) โŠ— C = ๐ป ๐‘› (๐‘‹, C) by the universal coefficient theorem. We can decompose ๐‘‰C as รŠ ๐ป ๐‘› (๐‘‹, C) = ๐ป ๐‘,๐‘ž (๐‘‹) ๐‘+๐‘ž=๐‘› where ๐ป ๐‘,๐‘ž = ๐ป ๐‘ž (๐‘‹, ฮฉ ๐‘ ) and ฮฉ ๐‘ is the sheaf of differential ๐‘โˆ’forms on ๐‘‹. This defines a weight-๐‘› Hodge structure on ๐‘‰. Note that the same construction works when applied to the integral Hodge structure with ๐‘‰Z = ๐ป ๐‘› (๐‘‹, Z)/torsion. Definition 2.1.3. Let ๐‘‹ be a smooth complex projective variety. The Hodge numbers of ๐‘‹ are defined to be โ„Ž ๐‘,๐‘ž (๐‘‹) B dim(๐ป ๐‘,๐‘ž (๐‘‹)). The Hodge numbers of a variety are often kept track of in the Hodge diamond. For example, in dimension 2, the Hodge diamond of a surface looks like: 6 โ„Ž2,2 โ„Ž2,1 โ„Ž1,2 โ„Ž2,0 โ„Ž1,1 โ„Ž0,2 โ„Ž1,0 โ„Ž0,1 โ„Ž0,0 The Hodge diamond has several symmetries. In particular, โ„Ž ๐‘,๐‘ž = โ„Ž ๐‘ž,๐‘ since ๐ป ๐‘,๐‘ž = ๐ป ๐‘ž,๐‘ and โ„Ž ๐‘,๐‘ž = โ„Ž๐‘›โˆ’๐‘,๐‘›โˆ’๐‘ž due to Serre duality. These facts are often referred to as Hodge symmetry. There are several standard Hodge-theoretical constructions that we will use throughout. The following are all defined similarly for integral and rational Hodge structures: Example 2.1.4. 1. Suppose ๐‘‰ and ๐‘Š are both Hodge structures of weight ๐‘›. Then we can define the direct sum Hodge structure, which is again of weight ๐‘›, by (๐‘‰ โŠ• ๐‘Š) ๐‘,๐‘ž B ๐‘‰ ๐‘,๐‘ž โŠ• ๐‘Š ๐‘,๐‘ž . 2. Suppose ๐‘‰ is a Hodge structure of weight ๐‘› and ๐‘Š is a Hodge structure of weight ๐‘š. Then we can define the tensor product Hodge structure of weight ๐‘› + ๐‘š by รŠ (๐‘‰ โŠ— ๐‘Š) ๐‘,๐‘ž B (๐‘‰ ๐‘1 ,๐‘ž1 โŠ— ๐‘Š ๐‘2 ,๐‘ž2 ), ๐‘+๐‘ž=๐‘›+๐‘š where ๐‘ 1 + ๐‘ 2 = ๐‘ and ๐‘ž 1 + ๐‘ž 2 = ๐‘ž. 3. Suppose ๐‘‰ is a Hodge structure of weight ๐‘›. Then we define the dual Hodge Structure ๐‘‰ โˆจ of weight-(โˆ’๐‘›) by (๐‘‰ โˆจ ) โˆ’๐‘,โˆ’๐‘ž B HomC (๐‘‰ ๐‘,๐‘ž , C) ( ๐‘ + ๐‘ž = ๐‘›) 4. Define the Tate Hodge structure Z(๐‘›) as the weight โˆ’2๐‘› Hodge structure on Z such that Z(๐‘›) โˆ’๐‘›,โˆ’๐‘› is one-dimensional and Z(๐‘›) ๐‘,๐‘ž = 0 if ( ๐‘, ๐‘ž) โ‰  (โˆ’๐‘›, โˆ’๐‘›). This is a Hodge structure of type (โˆ’๐‘›, โˆ’๐‘›). If we just say the Tate structure, we are referring to Z(1). The rational Tate Hodge structures Q(๐‘›) are defined similarly by replacing Z with Q everywhere 7 5. Given an integral Hodge structure ๐‘‰ of weight ๐‘›, we can increase or decrease its weight by defining the Tate twist of ๐‘‰ by ๐‘‰ (๐‘˜) B ๐‘‰ โŠ— Z(๐‘˜). This is a Hodge structure of weight ๐‘› โˆ’ 2๐‘˜. The Tate twist is defined analogously for rational Hodge structures. Now that we have defined Hodge structures, we will need to be able to consider morphisms between them. Definition 2.1.5. Suppose ๐‘‰ and ๐‘Š are Hodge structures of the same weight ๐‘›. Then we define a morphism of Hodge structures as a linear map ๐‘“ : ๐‘‰ ! ๐‘Š such that ๐‘“C (๐‘‰ ๐‘,๐‘ž ) โŠ‚ ๐‘Š ๐‘,๐‘ž . If the weights of ๐‘‰ and ๐‘Š are different, the only morphism of Hodge structures is the zero map, called the trivial morphism. If the two Hodge structures do not have the same weight, then it is possible to define a morphism of Hodge structures between certain Tate twists. Suppose ๐‘‰ has weight ๐‘› and ๐‘Š has weight ๐‘› + 2๐‘˜ and we have a linear map ๐‘“ : ๐‘‰ ! ๐‘Š satisfying ๐‘“C (๐‘‰ ๐‘,๐‘ž ) โŠ‚ ๐‘Š ๐‘+๐‘˜,๐‘ž+๐‘˜ then we have a morphism of Hodge structures ๐‘“ : ๐‘‰ ! ๐‘Š (๐‘˜). We can also understand Hodge structures on ๐‘‰ through algebraic representations of Cโˆ— on ๐‘‰, i.e. morphisms of real algebraic groups ๐œŒ : Cโˆ— ! ๐บ ๐ฟ(๐‘‰R ). Proposition 2.1.6. Let ๐‘‰ be a finite-dimension Qโˆ’vector space. Then there is a bฤณection {Hodge structures of weight ๐‘› on ๐‘‰ } โ†” {real algebraic representations ๐œŒ : Cโˆ— ! ๐บ ๐ฟ(๐‘‰R )} such that ๐œŒ(๐‘Ÿ) = ๐‘Ÿ ๐‘› for all ๐‘Ÿ โˆˆ Rโˆ— . Proof. For a full proof see [VG00, Proposition 1.4]. We will sketch the idea of the proof here. Given a rational Hodge structure of weight ๐‘›, we can associate an algebraic representation by ๐œŒ : Cโˆ— ! ๐บ ๐ฟ(๐‘‰R ), ๐‘ง 7! (๐œŒ(๐‘ง) : ๐‘ฃ 7! (๐‘ง ๐‘ ๐‘งยฏ๐‘ž )๐‘ฃ) for ๐‘ฃ โˆˆ ๐‘‰ ๐‘,๐‘ž . In the other direction, an algebraic representation ๐œŒ defines a Hodge structure that is given by the decomposition ๐‘‰ ๐‘,๐‘ž = {๐‘ฃ โˆˆ ๐‘‰C | ๐œŒ(๐‘ง)๐‘ฃ = (๐‘ง ๐‘ ๐‘งยฏ๐‘ž )๐‘ฃ}. โ–ก 8 We now interpret our previous constructions of Hodge structures in terms of this correspondence. Example 2.1.7. 1. The Tate Hodge structure Z(๐‘˜) corresponds to the algebraic representation given by ๐œŒ(๐‘ง)(๐‘ฃ) = (๐‘ง ๐‘งยฏ) โˆ’๐‘˜ ๐‘ฃ for each ๐‘ฃ โˆˆ ๐‘‰ (๐‘˜). 2. If ๐‘‰ and ๐‘Š are two Hodge structures corresponding to algebraic representations ๐œŒ๐‘‰ and ๐œŒ๐‘Š then ๐‘‰ โŠ— ๐‘Š corresponds to the algebraic representation ๐œŒ๐‘‰ โŠ— ๐œŒ๐‘Š . In particular, taking ๐‘Š = Z(1) to be the Tate Hodge structure, the Tate twist ๐‘‰ (1) corresponds to the algebraic representation ๐œŒ๐‘‰ (1) : ๐‘ง 7! (๐‘ง ๐‘งยฏ) โˆ’1 ๐œŒ๐‘‰ (๐‘ง). 3. A morphism of Hodge structures can be interpreted as a linear map ๐‘“ : ๐‘‰ ! ๐‘Š that satisfies ๐‘“ (๐œŒ๐‘‰ (๐‘ง)๐‘ฃ) = ๐œŒ๐‘Š (๐‘ง)( ๐‘“ (๐‘ฃ)) for ๐‘ฃ โˆˆ ๐‘‰. 4. If ๐‘‰ is a Hodge structure, then End(๐‘‰) also has a canonical Hodge structure. If ๐œŒ is the representation corresponding to the Hodge structure ๐‘‰, then the representation ๐œŒEnd (๐‘ง)( ๐ด) = ๐œŒ(๐‘ง)( ๐ด)(๐œŒ(๐‘ง)) โˆ’1 . corresponds to the Hodge structure on End(๐‘‰). For most of the following, we consider polarized Hodge structures. This is motivated by the fact that the cohomology of varieties we are working with also come equipped with a lattice structure given by the standard intersection form. Definition 2.1.8. Let ๐‘‰ be a rational Hodge structure of weight ๐‘› and suppose ๐œŒ๐‘‰ is its corresponding algebraic representation. Then a polarization of a Hodge structure is a morphism of Hodge structures ๐‘„ : ๐‘‰ โŠ— ๐‘‰ ! Q(โˆ’๐‘›) such that ๐‘„(๐‘ฃ, ๐œŒ๐‘‰ (๐‘–)๐‘ค) defines a positive definite symmetric form on ๐‘‰R . We call the pair (๐‘‰, ๐‘„) a polarized Hodge structure. Example 2.1.9. Let ๐‘‹ be a complex projective variety of dimension ๐‘›. Let ๐œ” โˆˆ ๐ป 2 (๐‘‹, Q) correspond to an ample divisor. Then for ๐‘˜ โ‰ค ๐‘› define a pairing by โˆซ ๐‘˜ (๐‘˜โˆ’1) ๐‘„(๐‘ฃ, ๐‘ค) B (โˆ’1) 2 ๐‘ฃ โˆง ๐‘ค โˆง ๐œ”๐‘›โˆ’๐‘˜ ๐‘‹ 9 where โˆง denotes the standard wedge product on ๐ป โˆ— (๐‘‹, Q). Then ๐‘„ defines a polarization on the primitive cohomology ๐ป ๐‘˜ (๐‘‹, Q)0 , which is defined as โŸจ๐œ”โŸฉ โŠฅ โŠ‚ ๐ป ๐‘˜ (๐‘‹, Q). This polarization ๐‘„ is sometimes called the Hodge-Riemann pairing. In particular, we are interested in the case where ๐‘› = ๐‘˜ = 2, i.e. middle cohomology of a surface. Example 2.1.10. If ๐‘† is a smooth projective surface then middle cohomology can be decomposed as ๐ป 2 (๐‘†, Q) = ๐ป 2 (๐‘†, Q)0 โŠ• Q ยท ๐œ”, where ๐œ” corresponds to O๐‘† (1). In this case, the Hodge-Riemann pairing gives โˆซ ๐‘„(๐‘ฃ, ๐‘ค) B โˆ’ ๐‘ฃ โˆง ๐‘ค. ๐‘‹ Of course, this is just the standard intersection form up to a sign change. This provides a polarization on ๐ป 2 (๐‘†, Q)0 . To provide a polarization on the full ๐ป 2 (๐‘†, Q), modify ๐‘„ to be positive on Q ยท ๐œ” (to satisfy the positive-definite requirement of the definition). We will often need to consider sub-Hodge structures or irreducible Hodge structures. Definition 2.1.11. Let ๐‘‰ be a Hodge structure of weight-๐‘›. โ€ข A sub-Hodge structure ๐‘Š of a Hodge structure ๐‘‰ is a sub-module or sub-vector space ๐‘Š โŠ‚ ๐‘‰ such that the Hodge structure on ๐‘Š is completely induced by the Hodge structure on ๐‘‰. In other words, ๐‘Š ๐‘,๐‘ž = ๐‘ŠC โˆฉ ๐‘‰ ๐‘,๐‘ž defines the Hodge structure on ๐‘Š. โ€ข In case of integral Hodge structures, a sub-Hodge structure ๐‘Š โŠ‚ ๐‘‰ is called primitive if ๐‘‰/๐‘Š is torsion-free. โ€ข A Hodge structure is called irreducible if it contains no non-trivial, proper, primitive sub- Hodge structures. We compile some facts concerning polarizations and sub-Hodge structures. Remark 2.1.12. Let (๐‘‰, ๐‘„) be a polarized Hodge structure of weight-๐‘›. โ€ข If the Hodge structure has even dimension ๐‘› = 2๐‘˜, then ๐‘„ defines a (โˆ’1) ๐‘˜โˆ’๐‘ โˆ’definite form on the subspace ๐‘‰R โˆฉ (๐‘‰ ๐‘,๐‘ž โŠ• ๐‘‰ ๐‘ž,๐‘ ). If ๐œŒ is the algebraic representation defining the Hodge 10 structure on ๐‘‰, then ๐œŒ(๐‘–) acts as ๐‘– ๐‘โˆ’๐‘ž on ๐‘‰ ๐‘,๐‘ž . Now, ๐‘– ๐‘โˆ’๐‘ž = ๐‘– 2๐‘โˆ’2๐‘˜ = (โˆ’1) ๐‘˜โˆ’๐‘ . Now since ๐‘„ is a polarization, we have that (โˆ’1) ๐‘˜โˆ’๐‘ ๐‘„(๐‘ฃ, ๐‘ฃ) = ๐‘„(๐‘ฃ, ๐œŒ(๐‘–)๐‘ฃ) > 0 by definition. โ€ข The restriction of the polarization ๐‘„ to any sub-Hodge structure ๐‘Š โŠ‚ ๐‘‰ defines a polarization. So, any sub-Hodge structure of a polarizable Hodge structure is also polarizable. We will use this fact when defining polarizations on transcendental and primitive sub-lattices of cohomology rather than full cohomology. โ€ข If ๐‘‰ is a polarized rational Hodge structure, then any sub-Hodge structure ๐‘Š โŠ‚ ๐‘‰ defines a direct sum decomposition ๐‘‰ = ๐‘Š โŠ• ๐‘Š โŠฅ , where the orthogonal complement is taken with respect to the polarization ๐‘„. This is because the polarization defines a symmetric, positive definite bilinear form on a finite dimensional vector space in the rational case. If ๐‘‰ is an integral Hodge structure, then ๐‘Š โŠ• ๐‘Š โŠฅ is in general only a finite index Hodge sub-structure. This distinction will be important for us to keep in mind in Chapter 3. 2.2 Lattice Theory We will make use of lattices throughout. The basic notions are defined here, with more advanced theory explored in Chapter 4. Definition 2.2.1. 1. A lattice is a pair (ฮ›, ๐‘) where ฮ› is a free Zโˆ’module of finite rank and ๐‘ :ฮ›ร—ฮ›!Z is a non-degenerate symmetric bilinear form. 2. We call the lattice even if ๐‘(๐‘ฅ, ๐‘ฅ) โˆˆ 2Z for all ๐‘ฅ โˆˆ ฮ› and odd otherwise. In particular, we need only one element ๐‘ฅ with ๐‘(๐‘ฅ, ๐‘ฅ) odd to be called an odd lattice. 3. The discriminant of a lattice is the determinant of the Gram matrix of ๐‘ with respect to any arbitrary basis of ฮ›, simply denoted by disc(ฮ›). Given any lattice, there is a natural injection into its dual lattice ฮ›โˆ— B Hom(ฮ›, Z) given by ๐‘–ฮ› : ฮ› โ†ฉ! ฮ›โˆ— ๐‘ฅ 7! ๐‘(๐‘ฅ, โ€“) The injectivity follows since the lattice is non-degenerate by definition. 11 Definition 2.2.2. The discriminant group of a lattice is defined to be ๐ดฮ› B ฮ›โˆ— /ฮ›. This is a finite group of order |disc(ฮ›)| by [Huy16]. The lattice ฮ› is called unimodular if ๐ดฮ› is trivial, which happens if and only if disc(ฮ›) = ยฑ1. The minimal number of generators of ๐ดฮ› is denoted by ๐‘™ (ฮ›). Definition 2.2.3. Given two lattices ฮ›1 and ฮ›2 , we can define the direct sum lattice ฮ›1 โŠ• ฮ›2 naturally by (๐‘ฅ 1 + ๐‘ฅ 2 , ๐‘ฆ 1 + ๐‘ฆ 2 )ฮ›1 โŠ•ฮ›2 B (๐‘ฅ 1 , ๐‘ฆ 1 )ฮ›1 + (๐‘ฅ 2 , ๐‘ฆ 2 )ฮ›2 where ๐‘ฅ 1 , ๐‘ฆ 1 โˆˆ ฮ›1 and ๐‘ฅ 2 , ๐‘ฆ 2 โˆˆ ฮ›2 . Note that the discriminant is multiplicative over direct sums, a fact that will be useful in later sections. The following are the most common examples of lattices of interest to us. Example 2.2.4. 1. ๐ผ denotes the lattice of rank one with Gram matrix simply (1). Its direct sum of ๐‘›-copies will be denoted by ๐ผ๐‘› . This is clearly a unimodular lattice of rank ๐‘›. 2. The hyperbolic lattice ๐‘ˆ is the rank two lattice with Gram matrix ยฉ0 1 ยช ยญ ยฎ. ยญ ยฎ 1 0 ยซ ยฌ This is a unimodular lattice of discriminant disc(๐‘ˆ) = โˆ’1. 3. The ๐ธ 8 โˆ’lattice is the unique positive definite, even, unimodular lattice of rank 8. It can be described by the set of points ๐ธ 8 B (๐‘ฅ๐‘– ) โˆˆ Z8 โˆช (Z + 12 ) 8 | ฮฃ๐‘ฅ๐‘– โˆˆ 2Z .  4. The lattice ๐ด2 is the rank 2 lattice with Gram matrix ยฉ 2 โˆ’1ยช ยญ ยฎ. ยญ ยฎ โˆ’1 2 ยซ ยฌ Note that this lattice is not unimodular. It has disc( ๐ด2 ) = 3 and discriminant group ๐ด ๐ด2  Z/3Z. One can more generally define lattices of type ๐ด๐‘› coming from root systems, but we will only make use of ๐ด2 . We will often consider sub-lattices. If ๐ฟ โ†ฉ! ฮ› is an injective morphism of free Z-modules, then one can use the natural inclusions ๐ฟ โ†ฉ! ฮ› โ†ฉ! ฮ›โˆ— โ†ฉ! ๐ฟ โˆ— 12 to show the relation disc(๐ฟ) = disc(ฮ›) ยท (ฮ› : ๐ฟ) 2 [Huy16, 14.0.2], where (ฮ› : ๐ฟ) denotes the index of the subgroup ๐ฟ in ฮ›. Definition 2.2.5. We call an embedding of lattices ๐ฟ โ†ฉ! ฮ› primitive if its cokernel is torsion free. As a word of caution, note that if ๐ฟ 1 โ†ฉ! ฮ› is a primitive embedding and ๐ฟ 2 B ๐ฟ 1โŠฅ , where the orthogonal complement is taken inside of ฮ›, then the inclusion ๐ฟ 1 โŠ• ๐ฟ 2 โ†ฉ! ฮ› is not necessarily primitive, though it is of finite index. We also consider the twists of lattices in the following sections. Definition 2.2.6. Given a lattice (ฮ›, ๐‘), its twist ฮ›(๐‘š) is defined by changing the form on ฮ› by multiplication by ๐‘š. In other words, ๐‘ ฮ›(๐‘š) B ๐‘š ยท ๐‘ ฮ› . The discriminant of a twisted lattice is related to the discriminant of the original lattice via disc(ฮ›(๐‘š)) = disc(ฮ›) ยท ๐‘š ๐‘Ÿ ๐‘˜ (ฮ›) . where rk(ฮ›) denotes the rank of ฮ›. 2.3 Abelian Varieties Abelian varieties will be important to us in the following, since the Kuga-Satake construction results in an abelian variety ๐พ๐‘†(๐‘‰). In this section, we collect basic facts about abelian varieties that we will use later. A reference for additional background is [Mil86] or [BL04]. A similar exposition of background is given in [Mac16]. Definition 2.3.1. Let ๐‘‰ be a complex vector space of dimension ๐‘› and let ฮ› โŠ‚ ๐‘‰ be a full lattice, i.e. ฮ› has rank 2๐‘› and ฮ› โŠ—Z R = ๐‘‰. Then a complex torus is defined to be ๐‘‹ := ๐‘‰/ฮ›. It is well known that if ๐‘‹ = ๐‘‰/ฮ› is a complex torus, then ๐œ‹1 (๐‘‹)  ๐ป1 (๐‘‹, Z)  ฮ›. Additionally, it is a fact from algebraic topology that for complex tori, ๐ป 1 (๐‘‹, Z) = Hom(๐ป1 (๐‘‹), Z). So for the case of a complex torus, ๐ป 1 (๐‘‹, Z) = Hom(ฮ›, Z). To compute cohomology of a complex torus, it is enough to make use of the Kรผnneth formula and note that as a real manifold, ๐‘‹  (R/Z) 2๐‘›  (๐‘† 1 ) 2๐‘› where ๐‘† 1 denotes the unit circle [Mil86, 13 Section 15]. This gives us that   ๐‘˜ 2๐‘› dim ๐ป (๐‘‹, Z) = ๐‘˜ and if the complex torus is algebraic, it has Hodge numbers given by     ๐‘,๐‘ž ๐‘› ๐‘› โ„Ž (๐‘‹) = ยท . ๐‘ ๐‘ž So, if we have an algebraic complex torus of dimension 2 (which we will call an abelian surface shortly), its Hodge diamond is as follows: 1 2 2 1 4 1 2 2 1 Now, we are working towards defining abelian varieties from complex tori. To do this, we first need to discuss Chern classes and polarizations of complex tori. Let X be a complex torus (or more generally, any complex manifold). Recall the exponential sheaf sequence: ๐‘’ 2 ๐œ‹๐‘– 0 โˆ’! 2๐œ‹๐‘–Z โˆ’! O ๐‘‹ โˆ’! O ๐‘‹โˆ— โˆ’! 0 This induces the usual long exact sequence in cohomology: ยท ยท ยท ! ๐ป 1 (๐‘‹, O ๐‘‹ ) ! ๐ป 1 (๐‘‹, O ๐‘‹โˆ— ) !1 ๐ป 2 (๐‘‹, Z) ! ยท ยท ยท . ๐‘ Definition 2.3.2. Identify ๐ป 1 (๐‘‹, O ๐‘‹โˆ— )  Pic(๐‘‹) and let a line bundle ๐ฟ โˆˆ Pic(๐‘‹). Define the first Chern class of the line bundle ๐ฟ to be the image of ๐ฟ under ๐‘ 1 as defined above, i.e. ๐‘ 1 (๐ฟ) โˆˆ ๐ป 2 (๐‘‹, Z). 14 When ๐‘‹ := ๐‘‰/ฮ› is a complex torus, by Kรผnneth formula, ๐ป 2 (๐‘‹, Z)  ๐ป 1 (๐‘‹, Z) ๐ป 1 (๐‘‹, Z)  ร“ ร“ Hom(ฮ›, Z) Hom(ฮ›, Z)  Hom(ฮ› โˆง ฮ›, Z), so the first Chern class ๐‘ 1 (๐ฟ) can be identified with a Zโˆ’valued alternating form on the lattice ฮ›. Further, we can identify exactly when an alternating form on the lattice ฮ› is induced from the first Chern class of a line bundle. Proposition 2.3.3. Let ๐ธ : ๐‘‰ ร— ๐‘‰ ! R be an alternating form. Then the following are equivalent: โ€ข ๐ธ represents the first Chern class of a line bundle ๐ฟ. โ€ข ๐ธ (ฮ›, ฮ›) โŠ‚ Z and ๐ธ (๐‘–๐‘ฃ, ๐‘–๐‘ค) = ๐ธ (๐‘ฃ, ๐‘ค) for all ๐‘ฃ, ๐‘ค โˆˆ ๐‘‰. Proof. See [BL04, Proposition 2.1.6]. โ–ก This motivates the definition of a polarization of a complex torus. We say that a line bundle ๐ฟ is positive definite if ๐ธ (๐‘ฃ, ๐‘–๐‘ฃ) > 0 for all ๐‘ฃ โˆˆ ๐‘‰ where ๐ธ is the alternating form associated to the line bundle ๐ฟ as in Proposition 2.3.3. Definition 2.3.4. A polarization of a complex torus ๐‘‹ is defined to be the first Chern class of a positive definite line bundle ๐ฟ. This means that a polarization is given by an alternating form ๐ธ : ฮ› ร— ฮ› ! Z under the above correspondence such that the Rโˆ’linear extension of ๐ธ satisfies ๐ธ (๐‘–๐‘ฃ, ๐‘–๐‘ค) = ๐ธ (๐‘ฃ, ๐‘ค) for all ๐‘ฃ, ๐‘ค โˆˆ ๐‘‰ and ๐ธ (๐‘ฃ, ๐‘–๐‘ฃ) > 0 for all ๐‘ฃ โˆˆ ๐‘‰ (positive definite). Compare this definition to the definition of a polarization of a Hodge structure given by 2.1.8. The following is well known, due to Lefschetz. Proposition 2.3.5. [Lef21] If ๐ฟ is a positive definite line bundle on a complex torus ๐‘‹, then ๐‘‹ is an algebraic variety with ๐ฟ 3 very ample. So, the map ๐œ‘ ๐ฟ 3 : ๐‘‹ ! P๐‘ is an embedding. We are finally ready to define an abelian variety. Definition 2.3.6. An abelian variety is defined to be a complex torus equipped with a positive definite line bundle. In particular, the above proposition implies that an abelian variety is a complex, projective variety. In fact, it is also an algebraic group. An important notion for us in Chapter 3 will be that of an isogeny of abelian varieties, as we will often consider Kuga-Satake varieties up to isogeny. Definition 2.3.7. A morphism of abelian varieties ๐‘“ : ๐ด ! ๐ต is called an isogeny if it is surjective 15 and has finite (zero-dimensional) kernel. The degree of the isogeny is the degree of the field extension [๐‘˜ ( ๐ด) : ๐‘“ โˆ— ๐‘˜ (๐ต)]. Isogeny is typically denoted by ๐ด โˆผ ๐ต. There are several equivalent notions of isogeny as seen in [Mil86]. Proposition 2.3.8. The following are equivalent: โ€ข ๐‘“ : ๐ด ! ๐ต is an isogeny. โ€ข dim( ๐ด) = ๐‘‘๐‘–๐‘š(๐ต) and ๐‘“ is surjective. โ€ข ๐‘‘๐‘–๐‘š( ๐ด) = ๐‘‘๐‘–๐‘š(๐ต) and ๐‘˜๐‘’๐‘Ÿ ( ๐‘“ ) is finite. โ€ข ๐‘“ is finite, flat, and surjective. It is often more appropriate to consider abelian varieties up to isogeny. For example, doing so makes direct sums work well. For example, given an abelian subvariety ๐ด1 โŠ‚ ๐ด, there always exists another abelian subvariety ๐ด2 โŠ‚ ๐ด such that ๐ด โˆผ ๐ด1 ร— ๐ด2 . Therefore, every abelian variety is isogenous to a direct sum of simple abelian varieties. Such statements do not hold when considering abelian varieties up to isomorphism instead of abelian varieties up to isogeny. Now that we have defined the necessary notions, we relate complex tori and abelian varieties back to Hodge structures. The following proposition will be our key consideration. Proposition 2.3.9. There is a one-to-one correspondence between the set of complex tori and the set of integral Hodge structures of weight one. Additionally, abelian varieties correspond to polarized, integral Hodge structures of weight one. Proof. Let ๐‘‹ = ๐‘‰/ฮ› be a complex torus. Then by definition we have ฮ› โŠ—Z R = ๐‘‰ is a complex vector space, so it comes equipped with a complex structure ๐ฝ, i.e. ๐ฝ 2 = โˆ’1. Set ๐ป 1,0 to be the ๐‘–โˆ’eigenspace under ๐ฝ. In other words, ๐ป 1,0 = {๐‘ฃ โˆˆ ๐‘‰ : ๐ฝ๐‘ฃ = ๐‘–๐‘ฃ}. Similarly, set ๐ป 0,1 to be the (โˆ’๐‘–)โˆ’eigenspace under ๐ฝ. Then clearly ๐ป = ๐ป 1,0 โŠ• ๐ป 0,1 defines a weight one Hodge structure. On the other hand, given a weight one integral Hodge structure on ๐ปZ , write ๐ปC = ๐ป 1,0 โŠ• ๐ป 0,1 . Then ๐ปZ โ†ฉ! ๐ป 1,0 and ๐ป 1,0 /๐ปZ is a complex torus. Now, the definition of a polarization on a complex torus was carefully constructed to exactly mirror that of a polarization of a Hodge structure, so abelian varieties correspond to polarized 16 Hodge structures. โ–ก Corollary 2.3.10. Further, there is a one-to-one correspondence between the set of abelian varieties up to isogeny and the set of polarized, rational Hodge structures of weight one. This consideration will be especially crucial for us when working with Kuga-Satake varieties that are constructed from rational Hodge structures. 2.4 K3 Surfaces In this section we recall relevant facts about ๐พ3 surfaces. Definition 2.4.1. A ๐พ3 surface ๐‘† is a smooth projective variety of dimension 2 that has trivial canonical bundle, i.e. ๐œ” ๐‘†  O๐‘† , and zero irregularity, i.e. ๐ป 1 (๐‘†, O๐‘† ) = 0. Example 2.4.2. We list a few basic examples of ๐พ3 surfaces. โ€ข A smooth quartic ๐‘† โŠ‚ P3C is a ๐พ3 surface. One of the more notable examples is that of the Fermat quartic given by ๐‘ (๐‘ฅ 4 + ๐‘ฆ 4 + ๐‘ง 4 + ๐‘ค 4 ) โŠ‚ P3C . It is easy to see that such a smooth quartic is indeed ๐พ3. From the standard short exact sequence 0 โˆ’! OP3 (โˆ’4) โˆ’! OP3 โˆ’! O๐‘† โˆ’! 0 we have that ๐ป 1 (๐‘†, O๐‘† ) = 0 since ๐ป 1 (P3 , OP3 ) = 0 = ๐ป 2 (P3 , OP3 (โˆ’4)). The canonical bundle is calculated directly from adjunction formula [Har13], giving ๐œ” ๐‘†  O๐‘† (โˆ’3 โˆ’ 1 + 4)  O๐‘† . โ€ข A smooth double cover of P2C branched along a smooth sextic curve is a ๐พ3 surface ๐‘† of degree 2. For example, one can use adjunction to see that ๐œ” ๐‘† is trivial. โ€ข Let ๐ด be an abelian surface over ๐‘˜ = C (any algebraically closed field ๐‘˜ where char(๐‘˜) โ‰  2 will also work). Then ๐ด always comes equipped with a natural involution ๐œ„(๐‘Ž) = โˆ’๐‘Ž for all ๐‘Ž โˆˆ ๐ด. This involution has 16 fixed points and so ๐ด/๐œ„ has 16 double point singularities. Its minimal resolution ๐‘† ! ๐ด/๐œ„ is a ๐พ3 surface that is called a Kummer surface. Proposition 2.4.3. The Hodge diamond of any ๐พ3 surface has the following form: 17 1 0 0 1 20 1 0 0 1 Proof. We have that ๐ป 0,0 = ๐ป 0 (๐‘†, O๐‘† ) = 1, so by Serre duality โ„Ž0,0 = โ„Ž2,2 = 1. By definition, โ„Ž0,1 = 0 since ๐ป 1 (๐‘†, O๐‘† ) = 0. By symmetry this gives โ„Ž1,0 = โ„Ž0,1 = โ„Ž2,1 = โ„Ž1,2 = 0, so all of the interesting Hodge numbers appear in the middle of the diamond. Of course, we also have that โ„Ž2,0 = 1 by definition since ๐ป 0 (๐‘†, ฮฉ2 )  ๐ป 0 (๐‘†, O๐‘† ) = 1. So, we only need to determine โ„Ž1,1 . The above gives us that the Euler characteristic ๐œ’(๐‘†, O๐‘† ) = 2. So the Noether formula ๐‘21 + ๐‘ 2 ๐œ’(๐‘†, O๐‘† ) = 12 tells us that ๐‘ 2 = 24, where ๐‘๐‘– is the ๐‘–โˆ’th Chern class of the tangent bundle. Note that ๐‘ 1 = 0 since ๐œ” ๐‘† is trivial. Now, ๐‘ 2 is equal to the topological Euler characteristic, and the above shows us that ๐‘ 1 = ๐‘ 4 = 1 and ๐‘ 2 = ๐‘ 3 = 0. So, we have 24 = 1 โˆ’ 0 + ๐‘ 3 โˆ’ 0 + 1. This implies ๐‘ 3 = 22, so we must have โ„Ž1,1 = 20. This completes the Hodge diamond. โ–ก Remark 2.4.4. Equipped with its intersection form, ๐ป 2 (๐‘†, Z) has the structure of an even unimod- ular lattice. This lattice is denoted by ฮ›๐พ3 , and it can be proved that for any ๐พ3 surface ๐‘† it is isomorphic to the lattice ๐ธ 8 (โˆ’1) โŠ•2 โŠ• ๐‘ˆ โŠ•3 , called the ๐พ3 lattice. Sub-lattices of ๐ป 2 (๐‘†, Z) will also be considered throughout. In particular, we will need the Nรฉron-Severi group and the transcendental lattice of ๐‘†. Definition 2.4.5. Given a ๐พ3 surface ๐‘†, the Nรฉron-Severi group is defined to be ๐‘ ๐‘†(๐‘†) B ๐ป 2 (๐‘†, Z) โˆฉ ๐ป 1,1 (๐‘†). 18 ๐‘ ๐‘†(๐‘†)Q can be defined similarly as ๐ป 2 (๐‘†, Q) โˆฉ ๐ป 1,1 (๐‘†). The transcendental lattice of ๐‘† is defined to be the orthogonal complement to ๐‘ ๐‘†(๐‘†) with respect to the intersection form as ๐‘‡ (๐‘†) B ๐‘ ๐‘†(๐‘†) โŠฅ โŠ‚ ๐ป 2 (๐‘†, Z). Note that when considered as rational Hodge structures, we have a direct sum decomposition ๐ป 2 (๐‘†, Q) = ๐‘ ๐‘†(๐‘†)Q โŠ• ๐‘‡ (๐‘†)Q , a fact that will be useful later. The rank of ๐‘ ๐‘†(๐‘†) is called the Picard rank of ๐‘†, denoted ๐œŒ(๐‘†). For a projective ๐พ3 surface ๐‘† โŠ‚ PC๐‘› , the Picard rank ranges between 1 โ‰ค ๐œŒ(๐‘†) โ‰ค 20. A ๐พ3 surface of maximal Picard rank 20 is sometimes called a singular ๐พ3 surface. We will often have to work with polarized ๐พ3 surfaces in the following. Definition 2.4.6. A polarized ๐พ3 surface (of degree 2๐‘‘) is a pair (๐‘†, ๐ฟ) where ๐‘† is a projective ๐พ3 surface and ๐ฟ โˆˆ Pic(๐‘†) is a primitive, ample line bundle with ๐ฟ 2 = 2๐‘‘. It is a well known fact that polarized ๐พ3 surfaces of degree 2๐‘‘ exist for arbitrary ๐‘‘ > 0, see for example [Bea11]. We will denote the moduli space of polarized ๐พ3 surfaces of degree ๐‘‘ by N๐‘‘ . 2.5 Cubic Fourfolds Throughout this section we work over ๐‘˜ = C. Definition 2.5.1. A cubic fourfold is a smooth hypersurface ๐‘‹ โŠ‚ P5 of degree 3. Remark 2.5.2. Cubic hypersurfaces (not necessarily smooth) in P5 are parametrized by P(C[๐‘ฅ, ๐‘ฆ, ๐‘ง, ๐‘ข, ๐‘ฃ, ๐‘ค] 3 ), since they are given by a choice of a degree 3 homogenous polynomial in 6 variables. This space has dimension     ๐‘›+๐‘‘ 8 โˆ’1= โˆ’ 1 = 55 ๐‘‘ 3 so P(C[๐‘ฅ, ๐‘ฆ, ๐‘ง, ๐‘ข, ๐‘ฃ, ๐‘ค] 3 )  P55 . The smooth cubic fourfolds correspond to a dense open set ๐‘ˆ โŠ‚ P55 . Therefore, the moduli space of cubic fourfolds is C = [๐‘ˆ/PGL6 (C)]. 19 Note that ๐‘ˆ has dimension 55 since it is dense in P55 and PGL6 (C) has dimension 62 โˆ’ 1 = 35 so that dim(C) = 55 โˆ’ 35 = 20. Cubic fourfolds are of great interest for rationality problems. That is, given a cubic fourfold ๐‘‹, does there exist a birational map ๐œ‘ : P๐‘› d ๐‘‹? Rationality of cubic surfaces is long known by classical methods. On the other hand, cubic threefolds are known to be irrational by methods of [CG72] using the intermediate Jacobian. Through work of Hassett, cubic fourfolds are expected to show more mixed behavior, though most cubic fourfolds are expected to be irrational. We will explore this idea through Hodge theory in this section. Before seeing more general theory, we briefly give a few examples of cubic fourfolds that are known to be rational. See [Has16, Section 1] for additional details. Example 2.5.3. 1. A cubic fourfold ๐‘‹ containing two disjoint planes ๐‘ƒ1 and ๐‘ƒ2 can be shown to be rational. The map is constructed in the expected way, by taking a points ๐‘ 1 โˆˆ ๐‘ƒ1 and ๐‘ 2 โˆˆ ๐‘ƒ2 and joining them with a line to generally get a third point on ๐‘‹. In this way one can construct a birational map ๐œ‘ : ๐‘ƒ1 ร— ๐‘ƒ2 d ๐‘‹. 2. Let ๐‘‹ be a cubic fourfold containing a plane ๐‘ƒ and another projective surface ๐‘Š such that deg(๐‘Š) โˆ’ โŸจ๐‘ƒ, ๐‘ŠโŸฉ is odd, where โŸจ , โŸฉ denotes the intersection pairing. Then ๐‘‹ can be shown to be rational by [Has99, Corollary 2.2]. We will revisit this example in Chapter 5. 3. If ๐‘€ is a skew-symmetric 2๐‘› ร— 2๐‘› matrix then the determinant of ๐‘€ can be written as det(๐‘€) = Pf(๐‘€) 2 where Pf (๐‘€) is a homogenous form of degree ๐‘› known as the Pfaffian of ๐‘€. A Pfaffian cubic fourfold is the zero locus of this form: ๐‘‹ = ๐‘ (Pf (๐‘€)) โŠ‚ P5 It was shown in [Tre84] that Pfaffian cubic fourfolds are rational. 20 Now we explore the Hodge theory of cubic fourfolds. Suppose ๐‘‹ โŠ‚ P5 is a cubic fourfold. The Hodge diamond of ๐‘‹ is of the form: 1 0 0 0 1 0 0 0 0 0 0 1 21 1 0 0 0 0 0 0 1 0 0 0 1 Remark 2.5.4. Let ๐‘‹ be a cubic fourfold. 1. Equipped with its intersection form, ๐ป 4 (๐‘‹, Z)  ๐ผ21 โŠ• ๐ผ2 (โˆ’1) by [Has00, Proposition 2.1.2]. Therefore, we can see that ๐ป 4 (๐‘‹, Z) is a unimodular lattice of signature (21, 2). It is an odd lattice since (โ„Ž2 ) 2 = 3 for a hyperplane class โ„Ž. 2. We also consider primitive cohomology ๐ป 4 (๐‘‹, Z)0 B {โ„Ž2 }โŠฅ . The signature of this lattice is (20, 2) but it is no longer unimodular since ๐ป 4 (๐‘‹, Z)0  ๐ด2 โŠ•๐‘ˆ 2 โŠ• ๐ธ 82 where each component is defined in 2.2.4. Since the discriminant is multiplicative, we have disc(๐ป 4 (๐‘‹, Z)0 ) = 3. 3. As was the case with ๐พ3 surfaces, we can define the Nรฉron-Severi group and transcendental lattice analogously. Since the Hodge conjecture holds for a cubic fourfold ๐‘‹ by [Zuc77], the Nรฉron-Severi group is given by ๐‘ ๐‘†(๐‘‹)  ๐ป 4 (๐‘‹, Z) โˆฉ ๐ป 2,2 (๐‘‹), and ๐‘‡ ๐‘†(๐‘‹) B ๐‘ ๐‘†(๐‘‹) โŠฅ โŠ‚ ๐ป 4 (๐‘‹, Z). The Picard rank ๐œŒ(๐‘‹) is the rank of ๐‘ ๐‘†(๐‘‹) and 1 โ‰ค ๐œŒ(๐‘‹) โ‰ค 21. This lattice structure looks very similar to that of a ๐พ3 surface, though the dimensions and signatures are not quite the same. That leads us to our next set of definitions. 21 Definition 2.5.5. 1. A cubic fourfold ๐‘‹ is called special if there exists an algebraic surface ๐‘‡ โŠ‚ ๐‘‹ that is not homologous to a complete intersection. 2. A labelling of a cubic fourfold is a choice of a rank two saturated sublattice ๐พ โŠ‚ ๐ป 2,2 (๐‘‹, Z) such that โ„Ž2 โˆˆ ๐พ. We call a sublattice ๐พ โŠ‚ ๐ฟ saturated if for every ๐‘™ โˆˆ ๐ฟ, if ๐‘›๐‘™ โˆˆ ๐พ for some ๐‘› โˆˆ Z, then ๐‘™ โˆˆ ๐พ. We denote a labelled cubic fourfold by (๐‘‹, ๐พ). Note that a cubic fourfold may have more than one labeling. 3. The discriminant ๐‘‘ of a labelled cubic fourfold is the determinant of the restriction of the intersection form on ๐ป 4 (๐‘‹, Z) to the sublattice ๐พ. We denote a cubic fourfold with a labeling ๐พ of discriminant ๐‘‘ by (๐‘‹, ๐พ ๐‘‘ ). Remark 2.5.6. Note that a very general cubic fourfold is not special. Indeed by [Voi86], for a very general cubic fourfold, ๐ป 2,2 (๐‘‹, Z)  Zโ„Ž2 . So for a very general cubic fourfold, any algebraic surface ๐‘‡ โŠ‚ ๐‘‹ is homologous to a complete intersection. Therefore, the notion of a special cubic fourfold really is special. Note that ๐พ ๐‘‘โŠฅ now has signature (19, 2) just as the primitive cohomology ๐‘“ โŠฅ B ๐ป 2 (๐‘†, Z)0 of a polarized ๐พ3 surface (๐‘†, ๐‘“ ) does. This motivates the following important definition. Definition 2.5.7. Let (๐‘‹, ๐พ) be a special cubic fourfold and (๐‘†, ๐‘“ ) a polarized ๐พ3 surface. We say that ๐‘† is associated to ๐‘‹ if there is a Hodge isometry ๐œ‘ ๐ป 4 (๐‘‹, Z)(1) โŠƒ ๐พ โŠฅ (1) โˆ’! ๐‘“ โŠฅ โŠ‚ ๐ป 2 (๐‘†, Z) i.e. ๐œ‘ is an isomorphism of weight 2 Hodge structures that respects lattice structures. We explore conditions for when a cubic fourfold to have an associated ๐พ3 surface. The following is given in [Has16, Proposition 20]. First, a lemma is needed. Lemma 2.5.8. Let ๐‘‘ > 0 be a positive integer such that ๐‘‘ โ‰ก 0 or 2 (mod 6). Then there exists an isomorphism of lattices ๐พ ๐‘‘โŠฅ โˆ’! ๐‘“ โŠฅ if and only if ๐‘‘ is not divisible by 4, 9 or any odd prime ๐‘ โ‰ก 2 (mod 3). 22 This motivates the following definition. Definition 2.5.9. We call an even, positive integer ๐‘‘ admissible if ๐‘‘ is not divisible by 4, 9 or any odd prime ๐‘ โ‰ก 2 (mod 3). The first few admissible values are ๐‘‘ = 14, 26, 38, 42. Now, we are ready to state Hassettโ€™s theorem as the following. Theorem 2.5.10. A labelled special cubic fourfold (๐‘‹, ๐พ ๐‘‘ ) admits an associated ๐พ3 surface if and only if ๐‘‘ is admissible. Proof. See [Has16, Proposition 20]. โ–ก We would like to understand how certain moduli spaces of cubic fourfolds relate to moduli spaces of ๐พ3 surfaces. First we fix some notation. As before, let C๐‘‘ โŠ‚ C denote special cubic fourfolds admitting a labeling of discriminant ๐‘‘. It can be shown that C๐‘‘ โŠ‚ C is an irreducible divisor, and it is non-empty if and only if ๐‘‘ โ‰ฅ 8 and ๐‘‘ โ‰ก 0, 2 (mod 8) [Has16, Theorem 13]. These divisors are often referred to as "Hassett divisors". โ€ฒ Further, let C๐‘‘ denote cubic fourfolds ๐‘‹ together with a choice of saturated embedding of ๐พ ๐‘‘ into ๐ป 4 (๐‘‹, Z). Let N๐‘‘ denote polarized ๐พ3 surfaces of degree ๐‘‘. Then Hassett proves the following results. โ€ฒ Proposition 2.5.11. Let ๐‘‘ be an admissible value. Then C๐‘‘ is irreducible and there exists an open โ€ฒ immersion of C๐‘‘ into N๐‘‘ . Further, we know when cubic fourfolds admit multiple associated ๐พ3 surfaces. We will use this fact later. Corollary 2.5.12. Let ๐‘‘ be an admissible value. If ๐‘‘ โ‰ก 2 (mod 6) then C๐‘‘ is birational to N๐‘‘ . Otherwise C๐‘‘ is birational to a quotient of N๐‘‘ . This possible ambiguity in having multiple associated ๐พ3 surfaces is addressed by Orlovโ€™s Theorem. We will discuss derived categories in much more detail in Chapter 5, but it is relevant to mention this result here. Theorem 2.5.13. [Orl97, Theorem 3.3] Let ๐‘†1 and ๐‘†2 be smooth projective ๐พ3 surfaces over C. 23 Then ๐‘†1 and ๐‘†2 are derived equivalent, i.e. ๐ท ๐‘ (๐‘†1 )  ๐ท ๐‘ (๐‘†2 ) if and only if there exists a Hodge isometry ๐‘‡ (๐‘†1 )  ๐‘‡ (๐‘†2 ) between transcendental lattices. Now, if ๐‘†1 and ๐‘†2 are both associated to a labelled cubic fourfold (๐‘‹, ๐พ), then we have Hodge isometries ๐ป 2 (๐‘†1 , Z)0  ๐พ โŠฅ  ๐ป 2 (๐‘†2 , Z)0 . Hence ๐‘‡ (๐‘†1 )  ๐‘‡ (๐‘†2 ) since the above are Hodge isometries. Therefore, by Orlovโ€™s Theorem, we conclude that if a cubic fourfold has multiple associated ๐พ3 surfaces then they are all derived equivalent. We end this section by giving a brief census of what is currently known about rationality of cubic fourfolds in terms of Hassett divisors C๐‘‘ : 1. Cubic fourfolds in C14 are known to be rational. Indeed, it can be shown that C14 is the closure of the Pfaffian locus that we saw earlier in this section. 2. In [RS18] it is shown by Russo and Staglianรฒ that all cubic fourfolds in C26 and C38 are rational. 3. Further, Russo and Staglianรฒ show in [RS19] that all cubic fourfolds in C42 are rational. Note that 14, 25, 38, 42 are in fact the first few admissible values of discriminants ๐‘‘. Since this is precisely when a cubic fourfold possesses an associated ๐พ3 surface ๐‘† by Theorem 2.5.10, this gives some evidence towards the following conjecture. Conjecture 2.5.14. A cubic fourfold ๐‘‹ is rational if and only if it possesses an associated ๐พ3 surface ๐‘†. Note, however, that there are no examples of irrational cubic fourfolds that are known to date, even though the conjecture implies that most cubic fourfolds should be irrational. We take this rationality conjecture as motivation to further exploring cubic fourfolds and their associated ๐พ3 surfaces in the following chapters. 24 CHAPTER 3 KUGA-SATAKE VARIETIES In this chapter we prove our first main result. Specifically, we aim to study the Kuga-Satake varieties of cubic fourfolds and their associated ๐พ3 surfaces in the interest of finding a relationship between them. The chapter is outlined as follows: ยง3.1 We review Clifford algebras of quadratic forms. A Clifford algebra is a unital associative algebra that is associated to a vector space or a Zโˆ’module equipped with a quadratic form ๐‘ž. These special algebras are one of the main ingredients in the Kuga-Satake construction, so we recall basic definitions and facts concerning them here, as well as briefly mentioning the Clifford group. ยง3.2 In this section, we recall the Kuga-Satake construction and show that it produces an abelian variety. We show that the construction works for cubic fourfolds and provide an example of a situation where the Kuga-Satake variety can be explicitly described in the case of an abelian surface. ยง3.3 We begin this section by proving an important lemma that is used to describe Kuga-Satake varieties associated to direct sums of Hodge structures. Lemma 3.3.1 allows us to describe additional types of Kuga-Satake varieties up to isogeny. We use this to prove one of our main results: If a cubic fourfold ๐‘‹ has an associated ๐พ3 surface ๐‘†, then ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐‘†) 2 , where โˆผ denotes isogeny of abelian varieties. We also prove a lemma that allows us to show a partial converse in the situation where we know more about the discriminants involved. This relates the Kuga-Satake construction to the theory of associated ๐พ3 surfaces. ยง3.4 In this section, we take an interesting interlude to show that our results can be applied to a different class of varieties that are known as Gushel-Mukai fourfolds. We quickly define these varieties and show that their Hodge theory is related to ๐พ3 surfaces in a similar way to cubic fourfolds. In particular, concepts such as "special", "admissible discriminant", and "associated ๐พ3 surface" are all generalized. We end with a couple of corollaries using our 25 main theorem in Section 3.3 to show similar results for Gushel-Mukai fourfolds. 3.1 Clifford Algebras One of the main ingredients in the Kuga-Satake construction is the Clifford algebra. This is a type of algebra that can be associated to any free Zโˆ’module of finite rank equipped with a quadratic form or finite dimensional vector space that is equipped with a quadratic form. Suppose ๐‘‰ is a free Zโˆ’module of finite rank attached with a quadratic form ๐‘ž. Recall the tensor algebra associated to ๐‘‰ is given by รŠ ๐‘‡ (๐‘‰) B ๐‘‰ โŠ—๐‘– . ๐‘–โ‰ฅ0 There is a natural Z/2Z-grading on ๐‘‡ (๐‘‰) given by decomposing it into even and odd components as ๐‘‡ (๐‘‰) = ๐‘‡ + (๐‘‰) โŠ• ๐‘‡ โˆ’ (๐‘‰) where ๐‘‡ + (๐‘‰) B โŠ—2๐‘– and ๐‘‡ โˆ’ (๐‘‰) B โŠ—(2๐‘–+1) . ร‰ ร‰ ๐‘–โ‰ฅ0 ๐‘‰ ๐‘–โ‰ฅ0 ๐‘‰ Definition 3.1.1. The Clifford algebra associated to (๐‘‰, ๐‘ž) is defined to be ๐ถ (๐‘‰, ๐‘ž) B ๐‘‡ (๐‘‰)/โŸจ๐‘ฃ โŠ— ๐‘ฃ โˆ’ ๐‘ž(๐‘ฃ) | ๐‘ฃ โˆˆ ๐‘‰โŸฉ . Also define ๐ถ๐พ (๐‘‰, ๐‘ž) B ๐ถ (๐‘‰, ๐‘ž) โŠ—Z ๐พ where we allow ๐พ โˆˆ {Q, R, C}. Note that the ideal defining ๐ถ (๐‘‰, ๐‘ž) is generated by even elements so the natural Z/2Z grading on ๐‘‡ (๐‘‰) descends to a grading on ๐ถ (๐‘‰, ๐‘ž) giving ๐ถ (๐‘‰, ๐‘ž) = ๐ถ + (๐‘‰, ๐‘ž) โŠ• ๐ถ โˆ’ (๐‘‰, ๐‘ž). The even Clifford algebra ๐ถ + (๐‘‰, ๐‘ž) will be of particular interest to us. Note that the even Clifford algebra is again a sub-algebra of the Clifford algebra. The tensor algebra ๐‘‡ (๐‘‰) comes with a natural anti-involution ๐œ„ : ๐‘‡ (๐‘‰) ! ๐‘‡ (๐‘‰), ๐‘ฃ 1 โŠ— . . . โŠ— ๐‘ฃ ๐‘› 7! ๐‘ฃ ๐‘› โŠ— . . . โŠ— ๐‘ฃ 1 . This descends to an anti-automorphism on ๐ถ (๐‘‰) since the ideal defining ๐ถ (๐‘‰) is invariant under ๐œ„, which we also denote by ๐œ„. 26 Proposition 3.1.2. The dimension of the Clifford algebra is given by dim๐พ (๐ถ๐พ (๐‘‰, ๐‘ž)) = 2๐‘› where ๐‘› = dim(๐‘‰). ร“โˆ— Proof. By [Huy16, Section 4.1.1], one can define an isomorphism ๐ถ๐พ (๐‘‰, ๐‘ž)  ๐‘‰๐พ between the Clifford algebra and exterior algebra of ๐‘‰. Under this isomorphism, a choice of an orthogonal basis ๐‘ฃ 1 , . . . , ๐‘ฃ ๐‘› of ๐‘‰ gives a basis of ๐ถ๐พ (๐‘‰, ๐‘ž) of the form ๐‘ฃ 1๐‘Ž1 ยท . . . ยท ๐‘ฃ ๐‘›๐‘Ž ๐‘› for ๐‘Ž๐‘– โˆˆ {0, 1}. There are 2๐‘› such basis elements. โ–ก Definition 3.1.3. The Clifford group associated to (๐‘‰, ๐‘ž) is defined to be CSpin(๐‘‰) B {๐‘” โˆˆ ๐ถ (๐‘‰, ๐‘ž) โˆ— | ๐‘”๐‘‰๐‘” โˆ’1 โŠ‚ ๐‘‰ }. To study the Clifford group in the next section, we use the fact that there is a natural orthogonal representation ๐œ : CSpin(๐‘‰) ! ๐‘‚ (๐‘‰) given by ๐‘” 7! (๐‘ฃ 7! ๐‘” ยท ๐‘ฃ ยท ๐‘” โˆ’1 ). By orthogonal representation we mean ๐œ(๐‘”) preserves ๐‘ž for any ๐‘”. Since ๐‘ž(๐‘ค) = ๐‘ค 2 for all ๐‘ค โˆˆ ๐‘‰, we see that ๐‘ž(๐‘”ยท๐‘ฃยท๐‘” โˆ’1 ) = (๐‘”ยท๐‘ฃยท๐‘” โˆ’1 )(๐‘”ยท๐‘ฃยท๐‘” โˆ’1 ) = ๐‘ž(๐‘ฃ)(๐‘”ยท๐‘” โˆ’1 ) = ๐‘ž(๐‘ฃ), so the representation is orthogonal. 3.2 The Kuga-Satake Construction The Kuga-Satake construction was first defined in [KS67]. The construction allows us to pass from certain Hodge structures of weight two to Hodge structures of weight one. Geometrically, this was first used in associating an abelian variety to a ๐พ3 surface. Since the original construction was developed, the Kuga-Satake construction has been generalized further. In particular, [Voi05] provides an alternative viewpoint to the construction, and [Mor85] applies the construction to abelian surfaces. We adapt the Kuga-Satake construction for cubic fourfolds ๐‘‹ and explore the relationship of ๐พ๐‘†(๐‘‹) with ๐พ๐‘†(๐‘†) for associated ๐พ3 surfaces ๐‘†. Definition 3.2.1. A Hodge structure ๐‘‰ is said to be of ๐พ3โˆ’type if it is a Hodge structure of weight two and ๐‘ฃ 2,0 = 1, where ๐‘ฃ ๐‘,๐‘ž B dim(๐‘‰ ๐‘,๐‘ž ). 27 Example 3.2.2. ๐พ3 surfaces and cubic fourfolds provide the most important examples of Hodge structures of ๐พ3โˆ’type for our purposes. 1. Let ๐‘† be a ๐พ3 surface. Then ๐ป 2 (๐‘†, Z) is of ๐พ3โˆ’type. We can also see that primitive coho- mology ๐ป 2 (๐‘†, Z)0 and the transcendental lattice ๐‘‡ (๐‘†) are sub-Hodge structures of ๐พ3โˆ’type. 2. Let ๐‘‹ be a cubic fourfold. Then ๐ป 4 (๐‘‹, Z) is of weight 4. Consider its Tate twist ๐‘‰ = ๐ป 4 (๐‘‹, Z)(1). This is now a Hodge structure of weight 2 with ๐‘ฃ 2,0 = 1, since โ„Ž3,1 (๐‘‹, Z) = 1. Now, let ๐‘‰ be a Hodge structure of ๐พ3-type and suppose that it has polarization โˆ’๐‘ž. The Kuga- Satake construction takes this weight 2 Hodge structure and gives a Hodge structure of weight one. The first step will be to endow ๐ถR (๐‘‰, โˆ’๐‘ž) with a complex structure. Let ๐‘‰ be one of the polarized Hodge structures in the example above. By the Hodge index theorem, if ๐‘‰ has rank ๐‘› then โˆ’๐‘ž has signature (๐‘› โˆ’ 2, 2). So, there is a choice of orthonormal basis of ๐‘‰R such that โˆ’๐‘ž can be written as โˆ’๐‘ž = โˆ’๐ธ 12 โˆ’ ๐ธ 22 + ๐ธ 32 + ยท ยท ยท + ๐ธ ๐‘›2 . Let ๐‘’๐‘– correspond to ๐ธ๐‘– in the Clifford algebra and set ๐ฝ B ๐‘’ 1 ยท ๐‘’ 2 โˆˆ ๐ถ๐‘™ R (๐‘‰, โˆ’๐‘ž). Since ๐‘’ 1 ๐‘’ 2 = โˆ’๐‘’ 2 ๐‘’ 1 we have that ๐ฝ 2 = (๐‘’ 1 ๐‘’ 2 ) (๐‘’ 1 ๐‘’ 2 ) = โˆ’(๐‘’ 1 ) 2 (๐‘’ 2 ) 2 = โˆ’1. Therefore multiplication by ๐ฝ defines a complex structure on ๐ถ๐‘™R (๐‘‰, โˆ’๐‘ž). Note that ๐ฝ is independent of the choice of orthonormal basis (up to a sign). Now, with the complex structure defined by ๐ฝ in hand, we define the weight one Hodge structure on ๐ถ๐‘™R (๐‘‰, โˆ’๐‘ž) by the algebraic representation: ๐œŒ : Cโˆ— โˆ’! GL(๐ถ๐‘™R (๐‘‰, โˆ’๐‘ž)) ๐‘ฅ + ๐‘–๐‘ฆ 7โˆ’! (๐‘ฃ 7! (๐‘ฅ + ๐ฝ ๐‘ฆ)(๐‘ฃ)) for any ๐‘ฃ โˆˆ ๐‘‰R . We perform the same construction to get a weight one Hodge structure on ๐ถ๐‘™R+ (๐‘‰, โˆ’๐‘ž) and ๐ถ๐‘™Rโˆ’ (๐‘‰, โˆ’๐‘ž) as well. Now, suppose ๐‘‰ is an integral Hodge structure. Then ๐ถ๐‘™ + (๐‘‰, โˆ’๐‘ž) โŠ‚ ๐ถ๐‘™R+ (๐‘‰, โˆ’๐‘ž) is a full lattice so the quotient ๐ถ๐‘™R+ (๐‘‰, โˆ’๐‘ž)/๐ถ๐‘™ + (๐‘‰, โˆ’๐‘ž) defines a complex torus. We show below that this is indeed an abelian variety. Definition 3.2.3. The Kuga-Satake variety associated to the integral Hodge structure ๐‘‰ of ๐พ3-type 28 is defined as ๐พ๐‘†(๐‘‰) B ๐ถ๐‘™ R+ (๐‘‰, โˆ’๐‘ž)/๐ถ๐‘™ + (๐‘‰, โˆ’๐‘ž). In order to show that this complex torus defines an abelian variety, it is sufficient to construct a polarization by our discussion in Proposition 2.3.9. We first choose two orthogonal vectors ๐‘ฃ 1 , ๐‘ฃ 2 โˆˆ ๐‘‰ such that ๐‘ž(๐‘ฃ ๐‘– ) > 0. This is possible since the signature of ๐‘ž is (2, ๐‘› โˆ’ 2). Now define a polarization by ๐‘„ : ๐ถ๐‘™ + (๐‘‰, ๐‘ž) ร— ๐ถ๐‘™ + (๐‘‰, ๐‘ž) โˆ’! ๐‘„(โˆ’1) (2) (๐‘ฅ, ๐‘ฆ) 7โˆ’! tr(๐‘ฃ 1 ยท ๐‘ฃ 2 ยท ๐œ„(๐‘ฅ) ยท ๐‘ฆ) where tr(๐ฟ) denotes the trace of an endormorphism ๐ฟ on ๐ถ๐‘™ + (๐‘‰, ๐‘ž) and ๐œ„ denotes the anti- automorphism of the Clifford algebra defined previously. Lemma 3.2.4. Q defines a polarization for the weight one Hodge structure on ๐ถ๐‘™ + (๐‘‰, ๐‘ž). Therefore, the Kuga-Satake variety ๐พ๐‘†(๐‘‰) is an abelian variety. Proof. [Huy16, Proposition 4.2.5] โ–ก Now that we have defined the Kuga-Satake variety ๐พ๐‘†(๐‘‰) associated to a Hodge structure ๐‘‰ of ๐พ3โˆ’type, and realized it as an abelian variety, we discuss basic properties of Kuga-Satake varieties. In the following, we may drop the polarization ๐‘ž in ๐ถ๐‘™ (๐‘‰, ๐‘ž) to simplify notation. Lemma 3.2.5. If dimC (๐‘‰) = ๐‘›, then dimC (๐พ๐‘†(๐‘‰)) = 2๐‘›โˆ’2 . Proof. We saw previously in Proposition 3.1.2 that dimC (๐ถ๐‘™ (๐‘‰)) = 2๐‘› . Therefore dimR (๐ถ๐‘™R+ (๐‘‰)) = 2๐‘›โˆ’1 . It follows that dimC (๐พ๐‘†(๐‘‰)) = 2๐‘›โˆ’2 โ–ก So in general the dimension of Kuga-Satake varieties will be quite high. This creates a difficulty in giving an explicit geometric description of ๐พ๐‘†(๐‘‰) in many cases. Definition 3.2.6. Let ๐‘‹ be a variety of dimension ๐‘› such that ๐ป ๐‘› (๐‘‹, Z)(๐‘˜) is a Hodge structure of ๐พ3โˆ’type for some Tate twist of degree ๐‘˜. Then we define the Kuga-Satake variety associated to ๐‘‹ to be the Kuga-Satake variety associated to the particular Hodge structure ๐‘‰ = ๐ป ๐‘› (๐‘‹, Z)(๐‘˜), i.e. ๐พ๐‘†(๐‘‹) B ๐พ๐‘†(๐ป ๐‘› (๐‘‹, Z)(๐‘˜)) with the polarization as defined in (2). 29 One of the more important facts that comes from the construction is that we can recover the Hodge structure on ๐‘‰ from the Hodge structure on ๐พ๐‘†(๐‘‰). We provide a proof of this fact in the case of a cubic fourfold. Proposition 3.2.7. Suppose ๐‘‹ is a smooth cubic fourfold and ๐‘‰ = ๐ป 4 (๐‘‹, Z). Then there is an inclusion of Hodge structures of weight four ๐‘‰ โ†ฉโˆ’! ๐ถ๐‘™ + (๐‘‰ (1)) โŠ— ๐ถ๐‘™ + (๐‘‰ (1)) โŠ— Q(โˆ’1). Furthermore, the Hodge structure of weight four on ๐‘‰ can be recovered from the Hodge structure of weight one on ๐พ๐‘†(๐‘‹). Proof. The proof is similar to that of [Huy16, Proposition 4.2.6], adapted to our situation. Choose an element ๐‘ฃ 0 โˆˆ ๐‘‰ (1) with ๐‘ž(๐‘ฃ 0 ) โ‰  0. Consider the map ๐œ‘ : ๐‘‰ (2) โ†ฉโˆ’! End(๐ถ๐‘™ + (๐‘‰ (1))) ๐‘ฃ 7โˆ’! (๐œ‘(๐‘ฃ) := ๐‘“๐‘ฃ : ๐‘ค 7! ๐‘ฃ ยท ๐‘ค ยท ๐‘ฃ 0 ). This is an embedding since ๐‘“๐‘ฃ (๐‘ฃ 1 ยท ๐‘ฃ 0 ) = ๐‘ž(๐‘ฃ 0 )๐‘ฃ ยท ๐‘ฃ 1 for all ๐‘ฃ 1 โˆˆ ๐‘‰. We must now show that ๐œ‘ is a morphism of Hodge structures of weight zero. Recall that a Hodge structure on ๐‘‰ corresponds to a representation ๐œŒ๐‘‰ of Cโˆ— on ๐‘‰. Let the induced representations ๐œŒ๐‘‰ , ๐œŒ๐‘‰ (1) , ๐œŒ๐‘‰ (2) , ๐œŒ๐ถ๐‘™ , and ๐œŒ ๐ธ๐‘›๐‘‘ correspond to the Hodge structures on ๐‘‰, ๐‘‰ (1), ๐‘‰ (2), ๐ถ๐‘™ + (๐‘‰ (1)), and End(๐ถ๐‘™ + (๐‘‰ (1))) respectively. Then, to show that ๐œ‘ is a morphism of Hodge structures we must show ๐‘“ ๐œŒ๐‘‰ (2) (๐‘ง) (๐‘ฃ) = ๐œŒ ๐ธ๐‘›๐‘‘ (๐‘ง) ยท ๐‘“๐‘ฃ (3) for all ๐‘ง โˆˆ Cโˆ— and ๐‘ฃ โˆˆ ๐‘‰ (2) by Example 2.1.7(3). By definition, the evaluation of the left hand side for any ๐‘ค โˆˆ ๐ถ๐‘™ + (๐‘‰ (1)) is given by ๐‘“ ๐œŒ๐‘‰ (2) (๐‘ง) (๐‘ฃ) (๐‘ค) = (๐œŒ๐‘‰ (2) (๐‘ง)(๐‘ฃ)) ยท ๐‘ค ยท ๐‘ฃ 0 . The evaluation of the right hand side of (3) for any ๐‘ค โˆˆ ๐ถ๐‘™ + (๐‘‰ (1)) is given by (๐œŒ ๐ธ๐‘›๐‘‘ (๐‘ง) ยท ๐‘“๐‘ฃ )(๐‘ค) = ๐œŒ๐ถ๐‘™ (๐‘ง) ๐‘“๐‘ฃ (๐œŒ๐ถ๐‘™ (๐‘ง) โˆ’1 (๐‘ค)) = ๐œŒ๐ถ๐‘™ (๐‘ง)(๐‘ฃ ยท ๐œŒ๐ถ๐‘™ (๐‘ง) โˆ’1 ๐‘ค ยท ๐‘ฃ 0 ) 30 by Example 2.1.7(4). So, it suffices to show the following identity holds: ๐œŒ๐‘‰ (2) (๐‘ง)(๐‘ฃ) = ๐œŒ๐ถ๐‘™ (๐‘ง) ยท ๐‘ฃ ยท ๐œŒ๐ถ๐‘™ (๐‘ง) โˆ’1 . This computation is checked in [Huy16, Proposition 2.6]. Note that by definition, ๐œŒ๐ถ๐‘™ (๐‘ง) acts via left multiplication by an element in ๐ถ๐‘™ + (๐‘‰R ). This follows since for ๐‘ง = ๐‘ฅ + ๐‘–๐‘ฆ โˆˆ C, ๐‘ฅ, ๐‘ฆ, ๐ฝ โˆˆ ๐ถ๐‘™ + (๐‘‰R ), where ๐ฝ = ๐‘’ 1 ยท ๐‘’ 2 as before. This identity shows that this element, which we identify with ๐œŒ๐ถ๐‘™ (๐‘ง), is contained in the Clifford group CSpin(๐‘‰ (2)). Now, use the fact that ๐ถ๐‘™ + (๐‘‰ (1)) โˆ—  ๐ถ๐‘™ + (๐‘‰ (1)) โŠ— Q(1) to get End(๐ถ๐‘™ + (๐‘‰ (1)))  ๐ถ๐‘™ + (๐‘‰ (1)) โŠ— ๐ถ๐‘™ + (๐‘‰ (1)) โˆ—  ๐ถ๐‘™ + (๐‘‰ (1)) โŠ— ๐ถ๐‘™ + (๐‘‰ (1)) โŠ— Q(1). Combined with the above result, this gives ๐‘‰ (2) โ†ฉ! ๐ถ๐‘™ + (๐‘‰ (1)) โŠ— ๐ถ๐‘™ + (๐‘‰ (1)) โŠ— Q(1) and tensoring with Q(โˆ’2) gives the desired embedding of Hodge structures ๐‘‰ โ†ฉ! ๐ถ๐‘™ + (๐‘‰ (1)) โŠ— ๐ถ๐‘™ + (๐‘‰ (1)) โŠ— Q(โˆ’1). Next suppose we have a Hodge structure of weight four on ๐‘‰ as above. Then we get a weight one Hodge structure on ๐ถ๐‘™ + (๐‘‰ (1)). We claim that this Hodge structure can be used to recover the Hodge structure on ๐‘‰. Recall that we have constructed the map ๐œ‘ above of Hodge structures of weight zero. Since ๐œŒ๐ถ๐‘™ (๐‘ง) โˆˆ CSpin(๐‘‰ (2)), so we apply the orthogonal representation ๐œ : CSpin(๐‘‰ (2)) ! ๐‘‚ (๐‘‰ (2)) to get ๐œ(๐œŒ๐ถ๐‘™ (๐‘ง)) = ๐œŒ๐‘‰ (2) (๐‘ง). In this way we recover the Hodge structure of weight zero on ๐‘‰ (2). Now, by definition, ๐œŒ๐‘‰ (2) (๐‘ง) = (๐‘ง ๐‘งยฏ) โˆ’2 ๐œŒ๐‘‰ (๐‘ง). This allows us recover the Hodge structure on ๐‘‰ by ๐œŒ๐‘‰ (๐‘ง) = (๐‘ง ๐‘งยฏ) 2 ๐œŒ๐‘‰ (2) (๐‘ง). โ–ก Corollary 3.2.8. The Kuga-Satake Construction is injective: ๏ฃฑ ๏ฃผ ๏ฃฑ ๏ฃผ ๏ฃฒ polarized Hodge structures of ๏ฃด ๏ฃด ๏ฃด ๏ฃฝ ๏ฃด ๏ฃฒ polarized Hodge structures of ๏ฃด ๏ฃด ๏ฃด ๏ฃฝ ๏ฃด ๐พ๐‘† : โ†ฉโˆ’! . ๏ฃด ๐พ3 โˆ’ type ๏ฃด ๏ฃด ๏ฃด ๏ฃด weight one ๏ฃด ๏ฃด ๏ฃด ๏ฃณ ๏ฃพ ๏ฃณ ๏ฃพ 31 Proof. By the previous proposition, we can recover the Hodge structure on ๐‘‰ directly from the Hodge structure on ๐พ๐‘†(๐‘‰) via the orthogonal representation ๐œ : CSpin(๐‘‰ (๐‘˜)) ! ๐‘‚ (๐‘‰ (๐‘˜)) for the appropriate Tate-Twist of degree ๐‘˜. โ–ก We now study an example of a Kuga-Satake variety that we can explicitly describe. The following is due to Morrison in [Mor85], who first studied Kuga-Satake varieties associated to abelian surfaces. Example 3.2.9. Let ๐ด be an abelian surface. Then ๐พ๐‘†( ๐ด) โˆผ ๐ด8 . We will outline a sketch of the main ideas of the proof. We saw previously that the Hodge diamond of an abelian surface looks like: 1 2 2 1 4 1 2 2 1 This implies that the Kuga-Satake variety ๐พ๐‘†( ๐ด) has dimension 26โˆ’2 = 16, so ๐ด8 is at least a reasonable candidate. We also know from our discussion in Chapter 2 that an abelian variety is completely determined by its weight one Hodge structure. Therefore, this problem reduces to comparing ๐ป 1 ( ๐ด, Z) and ๐ป 1 (๐พ๐‘†( ๐ด), Z). ร“2 Recall from Chapter 2 that for an abelian surface, ๐ป 2 ( ๐ด, Z)  ๐ป 1 ( ๐ด, Z) as Hodge structures. With this in mind, identify ๐ป 2 ( ๐ด, Z) as the subspace of Hom(๐ป 1 ( ๐ด, Z) โˆ— , ๐ป 1 ( ๐ด, Z)) given by alter- nating morphisms. Similarly, identify ๐ป 2 ( ๐ด, Z) โˆ— with a subspace of Hom(๐ป 1 ( ๐ด, Z), ๐ป 1 ( ๐ด, Z) โˆ— ). 32 With these identifications, one can define a map: โˆง2 ๐ป 1 ( ๐ด, Z) โˆ’! End(๐ป 1 ( ๐ด, Z) โŠ• ๐ป 1 ( ๐ด, Z) โˆ— ) ยฉ 0 ๐‘ฃยช ๐‘ฃ 7โˆ’! ยญ ยฎ. โˆ’๐‘ฃ โˆ— 0 ยซ ยฌ Now, this morphism induces a bฤณective map on the Clifford algebra as ๐ถ๐‘™ + (โˆง2 ๐ป 1 ( ๐ด, Z)) ! End(๐ป 1 ( ๐ด, Z)) โŠ• End(๐ป 1 ( ๐ด, Z) โˆ— ). It is possible to show by direct computation that End(๐ป 1 ( ๐ด, Z)) โŠ• End(๐ป 1 ( ๐ด, Z) โˆ— ) is isomorphic as a Hodge structure to (๐ป 1 ( ๐ด, Z) โŠ• ๐ป 1 ( ๐ด, Z) โˆ— ) 4 , for example [Huy16, Proposition 4.3.1]. Since an abelian variety is always isogenous to its dual [Mil86, Chapter 7], it follows that for abelian surfaces we have ๐พ๐‘†( ๐ด) โˆผ ( ๐ด ร— ๐ด) ห† 4 โˆผ ๐ด8 . In the next section, we develop more theory and look at additional examples of Kuga-Satake varieties. 3.3 Kuga-Satake Varieties & Associated K3 Surfaces Before proving our main result, we need a lemma that allows us to analyze the Kuga-Satake variety of a direct sum of Hodge structures. Lemma 3.3.1. Suppose ๐‘‰ is a Hodge structure of ๐พ3 type and ๐‘‰ can be written as a direct sum of Hodge structures ๐‘‰ = ๐‘‰1 โŠ• ๐‘‰2 . Without loss of generality, suppose ๐‘‰2 is of type (1, 1) and ๐‘‰1 is of ๐พ3โˆ’type. Then ๐ถ๐‘™ + (๐‘‰) is isomorphic to 2dim ๐‘‰2 โˆ’1 copies of ๐ถ๐‘™ + (๐‘‰1 ) ร— ๐ถ๐‘™ โˆ’ (๐‘‰1 ). Thus dim ๐‘‰2 ๐พ๐‘†(๐‘‰)  ๐พ๐‘†(๐‘‰1 โŠ• ๐‘‰2 ) โˆผ ๐พ๐‘†(๐‘‰1 ) 2 . Proof. The even Clifford algebra decomposes as ๐ถ๐‘™ + (๐‘‰) = ๐ถ๐‘™ + (๐‘‰1 โŠ• ๐‘‰2 )  (๐ถ๐‘™ + (๐‘‰1 ) โŠ— ๐ถ๐‘™ + (๐‘‰2 )) โŠ• (๐ถ๐‘™ โˆ’ (๐‘‰1 ) โŠ— ๐ถ๐‘™ โˆ’ (๐‘‰2 )). Since ๐‘‰ is of ๐พ3โˆ’type, we must have that either ๐‘‰1 or ๐‘‰2 are also of ๐พ3โˆ’type and so the other must be pure of type (1, 1). Since ๐‘‰2 is of type (1, 1), this means that the representation defined by ๐‘‰2 is 33 dim(๐‘‰2 ) โˆ’1 trivial, so by [Poz13, Remark 3.2.6] we have that ๐ถ๐‘™ + (๐‘‰1 ) โŠ— ๐ถ๐‘™ + (๐‘‰2 )  ๐ถ๐‘™ + (๐‘‰1 ) 2 as Hodge dim(๐‘‰2 ) โˆ’1 structures and similarly ๐ถ๐‘™ โˆ’ (๐‘‰1 ) โŠ— ๐ถ๐‘™ โˆ’ (๐‘‰2 )  ๐ถ๐‘™ โˆ’ (๐‘‰1 ) 2 as Hodge structures. So we have an isomorphism of Hodge structures: dim(๐‘‰2 ) โˆ’1 ๐ถ๐‘™ + (๐‘‰)  (๐ถ๐‘™ + (๐‘‰1 ) โŠ• ๐ถ๐‘™ โˆ’ (๐‘‰1 )) 2 . Now, since the Kuga-Satake variety defined by ๐ถ๐‘™ + (๐‘‰) and the Kuga-Satake variety defined by dim(๐‘‰2 ) ๐ถ๐‘™ โˆ’ (๐‘‰) are isogenous by [Huy16, Remark 4.2.3], we have that ๐พ๐‘†(๐‘‰) โˆผ ๐พ๐‘†(๐‘‰1 ) 2 . โ–ก Remark 3.3.2. The above lemma is very useful for switching between different Hodge structures coming from the cohomology of a variety used to define Kuga-Satake varieties. โ€ข Let ๐‘‹ be a variety of dimension ๐‘› and let ๐ป ๐‘› (๐‘‹, Z)(๐‘˜) be of ๐พ3โˆ’type for some Tate twist. Then the transcendental lattice ๐‘‡ (๐‘‹) is also of ๐พ3โˆ’type so we can define the Kuga-Satake variety associated to ๐‘‡ (๐‘‹) as ๐พ๐‘†(๐‘‡ (๐‘‹)) (up to appropriate Tate Twist if necessary). This is now an abelian variety of dimension 2dim(๐‘‡ (๐‘‹))โˆ’2 = 2dim(๐ป ๐‘› (๐‘‹,Z))โˆ’๐œŒ(๐‘‹)โˆ’2 where ๐œŒ(๐‘‹) is the Picard rank of ๐‘‹. For a ๐พ3 surface ๐‘†, for example, the dimension of ๐พ๐‘†(๐‘‡ (๐‘†)) is 220โˆ’๐œŒ(๐‘†) . It is natural to ask if there is a relationship between ๐พ๐‘†(๐‘‹) B ๐พ๐‘†(๐ป ๐‘› (๐‘‹, Z)(๐‘˜)) and ๐œŒ(๐‘‹) ๐พ๐‘†(๐‘‡ (๐‘‹)). Lemma 3.3.1 implies that ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐‘‡ (๐‘‹)) 2 , a rather useful fact to keep on hand. An interesting example occurs when the Picard rank is maximal. For example, if a ๐พ3 surface has Picard rank ๐œŒ(๐‘†) = 20, then dim(๐พ๐‘†(๐‘‡ (๐‘†))) = 1, i.e. it is an elliptic curve. This shows that in the case of maximal Picard rank, ๐พ๐‘†(๐‘†) is isogenous to a product of elliptic curves. โ€ข If ๐‘‹ is a projective variety and โ„Ž is an ample class, then one can consider primitive coho- mology โ„ŽโŠฅ B ๐ป ๐‘› (๐‘‹, Z)0 (๐‘˜) with respect to the ample class. Then ๐ป ๐‘› (๐‘‹, Z)0 (๐‘˜) is again a Hodge structure of ๐พ3โˆ’type, so we can define a Kuga-Satake variety by ๐พ๐‘†(๐‘‰, โ„Ž) B ๐พ๐‘†(๐ป ๐‘› (๐‘‹, Z)0 (๐‘˜)). By again using Lemma 3.3.1 we have that ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐‘‹, โ„Ž) 2 . โ€ข Putting the above together, we have the following relations between Kuga-Satake varieties up 34 to isogeny: ๐œŒ ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐‘‹, โ„Ž) 2 โˆผ ๐พ๐‘†(๐‘‡ (๐‘‹)) 2 . So we can easily translate between different Kuga-Satake varieties that are natural in our context. With these new facts we can explore some new important examples of Kuga-Satake varieties. We will use the example of a Kummer surface again in Chapter 5. Example 3.3.3. Recall the construction of a Kummer surface outlined in Example 2.4.2. Given an abelian surface ๐ด, one can associate to it a Kummer surface ๐‘† which is always a ๐พ3 surface. Since we know ๐พ๐‘†( ๐ด) โˆผ ๐ด8 for an abelian surface by Example 3.2.9, a Kummer surface is a natural next choice to understand the Kuga-Satake construction. The Hodge structure on ๐ป 2 (๐‘†, Q) of a Kummer surface ๐‘† is isomorphic to ๐ป 2 ( ๐ด, Q) โŠ• Q(โˆ’1) 16 by [Huy16, Example 3.2.5]. Essentially, the cohomology of the Kummer surface is given by the cohomology of the abelian surface plus cohomology coming from the 16 exceptional divisors coming from the blowup. In particular, this also implies that ๐œŒ(๐‘†) = ๐œŒ( ๐ด) + 16. By the same argument, it is also known that ๐‘‡ (๐‘†)Q  ๐‘‡ ( ๐ด)Q is an isomorphism of Hodge structures. Therefore, we have that ๐พ๐‘†(๐‘‡ (๐‘†)) โˆผ ๐พ๐‘†(๐‘‡ ( ๐ด)). Now, using the remarks following Lemma 3.3.1, we have a sequence of isogenies ๐œŒ(๐‘†) ๐œŒ(๐‘†) 16 ๐พ๐‘†(๐‘†) โˆผ ๐พ๐‘†(๐‘‡ (๐‘‹)) 2 โˆผ ๐พ๐‘†(๐‘‡ ( ๐ด)) 2 โˆผ ๐พ๐‘†( ๐ด) 2 since ๐œŒ(๐‘†) = ๐œŒ( ๐ด) + 16. Since ๐พ๐‘†( ๐ด) โˆผ ๐ด8 , we can conclude 19 ๐พ๐‘†(๐‘†) โˆผ ๐ด2 . We also mention the following example of Paranjape in [Par88, Main Theorem] since we will refer to it in the following chapters. The proof is quite involved, so we only state the result here. 35 Example 3.3.4. Let ๐‘† ! P2 be a double cover that is ramified over 6 lines in general position. We saw in Example 2.4.2 that such an ๐‘† must be a ๐พ3 surface. Paranjape computes that 18 ๐พ๐‘†(๐‘‡ (๐‘†)) โˆผ ๐‘ƒ2 where ๐‘ƒ is a certain abelian fourfold that is known as a Prym variety. It is constructed geometrically as a cover of a certain genus 5 curve. See [Par88, Section 2] for more background and explicit details. Combined with Lemma 3.3.1, we have that for ๐พ3 surfaces ๐‘† which are realized as double covers of P2 ramified over six lines 18 19 ๐พ๐‘†(๐‘†) โˆผ ๐พ๐‘†(๐‘‡ (๐‘†)) 2 โˆผ (๐‘ƒ2 ) 2 โˆผ ๐‘ƒ2 , so the Kuga-Satake variety in this case can be explicitly described as a product of specially constructed abelian fourfolds up to isogeny. We now come to our first main result in which we relate the Kuga-Satake variety of a cubic fourfold ๐‘‹ and its associated ๐พ3 surface ๐‘†. Theorem 3.3.5. Suppose (๐‘‹, ๐พ) is a special cubic fourfold with associated ๐พ3 surface (๐‘†, ๐‘“ ). Then ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐‘†) 2 . Proof. Since ๐‘† is associated to ๐‘‹, ๐พ โŠฅ  ๐‘“ โŠฅ . Let ๐‘‰ = ๐ป 4 (๐‘‹, Z). Then ๐‘‰Q  ๐พQโŠฅ โŠ• ๐พQ and ๐‘‰ (1)Q is a weight two rational Hodge structure of ๐พ3 type. Additionally, ๐‘‰ (1)Q  ๐‘“QโŠฅ โŠ• ๐พ (1)Q where ๐‘“QโŠฅ is of ๐พ3 type and ๐พ (1)Q is of type (1, 1). Apply Lemma 3.3.1 with ๐‘‰1 = ๐‘“QโŠฅ and dim ๐‘‰2 ๐‘‰2 = ๐พ (1)Q . Then dim ๐‘‰2 = 2 and ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†( ๐‘“ โŠฅ ) 2 = ๐พ๐‘†( ๐‘“ โŠฅ ) 4 . Finally, by Remark 3.3.2, there is an isogeny ๐พ๐‘†(๐‘†) โˆผ ๐พ๐‘†( ๐‘“ โŠฅ ) 2 . Combined with the above, we arrive at the desired result ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†( ๐‘“ โŠฅ ) 4 โˆผ ๐พ๐‘†(๐‘†) 2 . โ–ก We can also prove a partial converse to the above theorem. To do that, we need the following lemma. 36 Lemma 3.3.6. Suppose ๐ด, ๐ต are Hodge structures of ๐พ3โˆ’type of the same rank and ๐ถ is a Hodge structure that is pure of type (1, 1). Suppose further that ๐ด โŠ• ๐ถ  ๐ต โŠ• ๐ถ. Then ๐ด  ๐ต as Hodge structures. Proof. First observe that ๐ด1,1  ๐ต1,1 since both of these are trivial Hodge structures of the same rank. Further, we have that ( ๐ด โŠ• ๐ถ) 2,0 = ๐ด2,0 and (๐ต โŠ• ๐ถ) 2,0 = ๐ต2,0 since ๐ถ is pure of type (1, 1). Let ๐œ‘ : ๐ด โŠ• ๐ถ ! ๐ต โŠ• ๐ถ be the Hodge isomorphism in the statement. Then we must have ๐œ‘C (( ๐ด2,0 ) = ๐œ‘C (( ๐ด โŠ• ๐ถ) 2,0 ) = (๐ต โŠ• ๐ถ) 2,0 = ๐ต2,0 . Since ๐œ‘ is a Hodge morphism, ๐œ‘C ( ๐ด0,2 ) = ๐ต0,2 as well. It follows that ๐ด  ๐ต. โ–ก Theorem 3.3.7. Suppose ๐‘‘ is an admissible value, (๐‘‹, ๐พ ๐‘‘ ) is a labelled cubic fourfold of discrim- inant ๐‘‘ and (๐‘†, ๐‘“ ) is a polarized ๐พ3 surface of degree ๐‘‘. Suppose ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐‘†) 2 . Then ๐‘† is associated to ๐‘‹. Proof. Since ๐‘‘ is an admissible value, we have that ๐‘“ โŠฅ  ๐พ ๐‘‘โŠฅ as lattices by Lemma 2.5.8. To show that ๐‘† is associated to ๐‘‹, it suffices to show that ๐‘“ โŠฅ  ๐พ ๐‘‘โŠฅ as Hodge structures. Note that ๐พ๐‘†(๐พ ๐‘‘โŠฅ โŠ• ๐พ ๐‘‘ ) โˆผ ๐พ๐‘†(๐‘‹) and ๐พ๐‘†( ๐‘“ โŠฅ โŠ• ๐พ ๐‘‘ ) โˆผ ๐พ๐‘†(๐‘†) 2 since ๐พ ๐‘‘ is pure of type (1, 1) and ๐พ๐‘†(๐‘†) 2 โˆผ ๐พ๐‘†( ๐‘“ โŠฅ ) 4 . Since ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐‘†) 2 , injectivity of the Kuga-Satake construction implies ๐พ ๐‘‘โŠฅ โŠ• ๐พ ๐‘‘  ๐‘“ โŠฅ โŠ• ๐พ ๐‘‘ as Hodge structures. Lemma 3.3.6 implies that ๐พ ๐‘‘โŠฅ  ๐‘“ โŠฅ as Hodge structures. โ–ก The additional assumption that we begin with an admissible value of ๐‘‘ is necessary. Consider the following where ๐‘‘ = 2 is not admissible. Example 3.3.8. Consider Example 19 of [Has16]. A cubic fourfold ๐‘‹ containing a plane ๐‘ƒ gives a ๐พ3 surface (๐‘†, ๐‘“ ) of degree 2. However, if ๐พ is the rank two labelling of ๐‘‹ then [VG05, 9.7] shows that ๐พ โŠฅ โŠ‚ ๐‘“ โŠฅ as an index 2 sublattice. In particular, there is no lattice isomorphism ๐พ โŠฅ  ๐‘“ โŠฅ and ๐‘† is not associated to ๐‘‹. However, we still have an embedding ๐พ โŠฅ โ†ฉ! ๐‘“ โŠฅ of lattices of rank 21. Therefore rk( ๐‘“ โŠฅ /๐พ โŠฅ ) = 0, so by [Huy16, Remark 2.5] there is an isogeny ๐พ๐‘†(๐พ โŠฅ ) โˆผ ๐พ๐‘†( ๐‘“ โŠฅ ). So, ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐พ โŠฅ ) 4 โˆผ 37 ๐พ๐‘†( ๐‘“ โŠฅ ) 4 โˆผ ๐พ๐‘†(๐‘†) 2 . In particular, ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐‘†) 2 , but ๐‘† is not associated to ๐‘‹ under this choice of labeling and polarization. However, it is unclear if ๐‘‹ may be associated to ๐‘† under a different choice of polarization (of admissible degree). 3.4 Gushel-Mukai Varieties Our result can be applied to other types of varieties. We provide an aside on one such example here, that of Gushel-Mukai varieties. These varieties are studied in [DIM15] and [Deb20] where basic definitions and properties are given. We will focus on Gushel-Mukai fourfolds as our previous results are applicable to them. Definition 3.4.1. A Gushel-Mukai fourfold ๐‘‹ is a smooth complex Fano variety of dimension 4 and Picard rank 1, so that Pic(๐‘‹) = Z๐ป, such that ๐‘‹ is of index 2 and degree 10. That is, โˆ’๐พ ๐‘‹ โ‰ก 2๐ป and ๐ป 4 = 10. The following proposition gives further geometric attributes of Gushel-Mukai fourfolds, these are sometimes taken as a definition. Proposition 3.4.2. [Deb20, Theorem 1.1] A smooth Gushel-Mukai fourfold ๐‘‹ as given by the above definition can be described as ร“2 ๐‘‹ = Gr(2, ๐‘‰5 ) โˆฉ P(๐‘Š9 ) โˆฉ ๐‘„ โŠ‚ P( ๐‘‰5 ) ร“2 where ๐‘„ is a quadric and ๐‘Š9 โŠ‚ ๐‘‰5 is a vector subspace of dimension 9 and ๐‘‰5 is a vector space of dimension 5. These varieties are studied because they have interesting properties similar to those of cubic fourfolds. In particular, questions about their rationality are open and they have interesting period maps and derived categories (as studied in [KP18] in particular). The Hodge diamond of a Gushel-Mukai fourfold looks similar to that of a cubic fourfold. A full computation is provided by [IM11, Lemma 4.1] and the Hodge diamond looks like the following: 38 1 0 0 0 1 0 0 0 0 0 0 1 22 1 0 0 0 0 0 0 1 0 0 0 1 As was the case with cubic fourfolds, all of the interesting structure comes from the middle cohomology. We will refer to middle cohomology ฮ› ๐‘‹ B ๐ป 4 (๐‘‹, Z) as the Gushel-Mukai lattice. Equipped with the standard intersection form, it is a unimodular lattice of signature (22, 2) [DIM15, Proposition 5.1], i.e. ฮ› ๐‘‹  (1) 22 โŠ• (โˆ’1) 2 . By the classification of unimodular lattices, 22 โˆ’ 2 = 20 is not divisible by 8. So, in particular, ฮ› ๐‘‹ is an odd, unimodular lattice as in the case of the cubic fourfold lattice. We would like to establish the notion of a "special" Gushel-Mukai fourfold as was done with cubic fourfolds. Using the description in 3.4.2, set ๐บ B Gr(2, ๐‘‰5 ). Then ๐ป 4 (๐บ, Z) embeds into ๐ป 4 (๐‘‹, Z) as a rank 2 sub-lattice. Its orthogonal complement is sometimes referred to as the "vanishing cohomology" of X, i.e. ๐ป 4 (๐‘‹, Z)van B ๐ป 4 (๐บ, Z) โŠฅ โŠ‚ ๐ป 4 (๐‘‹, Z). The vanishing cohomology is an even lattice of signature (20, 2) and it is isomorphic to ๐ป 4 (๐‘‹, Z)van  2๐ธ 8 โŠ• 2๐‘ˆ โŠ• 2๐ด2 , where all of the components are as defined in Chapter 2. ๐ป 4 (๐‘‹, Z)van always contains a special rank 2 sublattice ฮ›2 as defined in [DIM15, Proposition 5.1]. Definition 3.4.3. We call a Gushel-Mukai fourfold ๐‘‹ special if it contains a surface ๐‘‡ that does not come from Gr(2, ๐‘‰5 ). This is equivalent to the condition that the rank of ๐ป 2,2 (๐‘‹) โˆฉ ๐ป 4 (๐‘‹, Z) is at 39 least 3. Note that a very general Gushel-Mukai fourfold is not special by [DIM15, Corollary 4.6]. Recall that for special cubic fourfolds, we had to choose a rank 2 sublattice ๐พ. Since the dimension of ๐ป 4 (๐‘‹, Z) for Gushel-Mukai fourfolds is one higher, we must choose a special rank 3 sublattice. A labeling of a Gushel-Mukai fourfold is a choice of a primitive, positive-definite rank 3 sublattice ๐พ โŠ‚ ๐ป 4 (๐‘‹, Z) that contains ฮ›2 . We define the discriminant of (๐‘‹, ๐พ) to be the determinant of the intersection form on ๐พ. Proposition 3.4.4. [Deb20, Proposition 4.11] Let (๐‘‹, ๐พ) be a special, labelled Gushel-Mukai fourfold of discriminant ๐‘‘. Then ๐พ โŠฅ is isomorphic to ๐ป 2 (๐‘†, Z)0 for a polarized ๐พ3 surface (๐‘†, ๐‘“ ) of degree ๐‘‘ if and only if ๐‘‘ โ‰ก 2, 4 (mod 8) and the only primes that divide ๐‘‘ are ๐‘ โ‰ก 1 (mod 4). As with cubic fourfolds, the proposition motivates the following definitions. Definition 3.4.5. 1. An integer satisfying the conditions of Proposition 3.4.4 is called an ad- missible discriminant. 2. We say that the ๐พ3 surface (๐‘†, ๐‘“ ) satisfying the condition of Proposition 3.4.4 is associated to the Gushel-Mukai fourfold ๐‘‹. The first admissible values of ๐‘‘ are 10, 20, 26. There is a conjecture relating the existence of associated ๐พ3 surfaces to rationality similar to the conjecture for cubic fourfolds. Conjecture 3.4.6. A Gushel-Mukai fourfold ๐‘‹ is rational if and only if there is an associated ๐พ3 surface. Now that we have set up the theory of Gushel-Mukai fourfolds similarly to that of cubic fourfolds, we can apply the proof of Theorem 3.3.5 to get: Corollary 3.4.7. Suppose ๐‘‹ is a Gushel-Mukai fourfold and ๐‘† is a ๐พ3 surface associated to ๐‘‹. Then ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐‘†) 4 . It is natural to ask if these concepts for Gushel-Mukai fourfolds and cubic fourfolds are inter- twined. There is a notion of an "associated cubic fourfold" for Gushel-Mukai fourfolds. This is addressed in the following proposition. 40 Proposition 3.4.8. [DIM15, Proposition 6.6] Let (๐‘‹, ๐พ ๐‘‹ ) be a labelled Gushel-Mukai fourfold of discriminant d. Then ๐พ ๐‘‹โŠฅ is isomorphic to ๐พ๐‘ŒโŠฅ for a special, labelled cubic fourfold (๐‘Œ , ๐พ๐‘Œ ) (also of discriminant ๐‘‘) if and only if either: 1. ๐‘‘ โ‰ก 2, 20 (mod 24) and the only odd primes that divide ๐‘‘ are ๐‘ โ‰ก ยฑ1 (mod 12) or 2. ๐‘‘ โ‰ก 12, 66 (mod 72) and the only primes ๐‘ โ‰ฅ 5 that divide ๐‘‘ are ๐‘ โ‰ก ยฑ1 (mod 12). In particular, the first values of such a ๐‘‘ are 2, 12, 26, 44. Again, we can apply 3.3.5 to get: Corollary 3.4.9. Let ๐‘Œ be a cubic fourfold associated to a Gushel-Mukai fourfold ๐‘‹. Then ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐‘Œ ) 2 . In particular, note that the values of ๐‘‘ for which Gushel-Mukai fourfolds and cubic fourfolds possess an associated ๐พ3 surface ๐‘† are not disjoint. Consider the following intriguing example. Example 3.4.10. For ๐‘‘ = 26, a labelled cubic fourfold (๐‘Œ , ๐พ๐‘Œ ) of discriminant 26 possesses a unique associated ๐พ3 surface ๐‘† by Corollary 2.5.12 since 26 โ‰ก 2 (mod 4). Combining our above results, we get that (๐‘Œ , ๐พ๐‘Œ ) is associated to a Gushel-Mukai fourfold (๐‘‹, ๐พ ๐‘‹ ) also of discriminant 26 and so we have ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐‘Œ ) 2 โˆผ ๐พ๐‘†(๐‘†) 4 . 41 CHAPTER 4 ENDOMORPHISM ALGEBRAS In this chapter, we study endomorphism algebras of cubic fourfolds. Endomorphism algebras of ๐พ3 surfaces have been extensively studied and used to answer Hodge conjecture type questions. This chapter is outlined as follows: ยง4.1 Given a rational Hodge structure ๐‘‰, one can study its endormorphism algebra EndHod (๐‘‰) given by all homomorphisms of Hodge structures ๐‘‰ ! ๐‘‰. This was studied in detail by Zarhin in [Zar83] which we outline. In particular, Zarhin showed that for an irreducible Hodge structure ๐‘‰ of ๐พ3-type, EndHod (๐‘‰) is a field and furthermore, it is a number field that is either totally real or a CM field. We explore how these results are useful for answering Hodge conjecture type questions. Next, we discuss known results for the transcendental lattice ๐‘‡ (๐‘†) of ๐พ3 surfaces in both cases, and finish by outlining Van Geemenโ€™s proof in [VG08] that there exists ๐พ3 surfaces ๐‘† with transcendental lattice ๐‘‡ (๐‘†) such that EndHod (๐‘‡ (๐‘†)) = ๐น for any given totally real number field ๐น. ยง4.2 In this section, we outline Nikulinโ€™s lattice theory for even, unimodular lattices in [Nik80]. We modify this theory for odd, unimodular lattices with applications to ฮ›๐ถ4 for a cubic fourfold ๐‘‹. ยง4.3 We study the period map for cubic fourfolds through work of Laza and Looฤณenga. In particular, we work towards understanding the image of the period map. ยง4.4 In this section we prove our result for the existence of cubic fourfolds with endomorphism algebras of totally real type. ยง4.5 We explore some applications of endomorphism algebras in the case of ๐พ3 surfaces associated to cubic fourfolds in the sense of Section 2.5. We prove a few results as corollaries to the previous sections, and in the vein of [Sch10], we explore connections between Kuga-Satake varieties and endomorphism algebras of totally real type in the associated ๐พ3 framework. 42 4.1 Endomorphism Algebras of K3 Surfaces We begin this chapter by examining known results about endomorphism algebras for ๐พ3 sur- faces. These results inspire our approach for studying endomorphism algebras for cubic fourfolds. Most of the following results are found in [Zar83] and [VG08]. Let ๐‘† denote a ๐พ3 surface. In this section, we consider the Hodge structure for the transcendental lattice ๐‘‡ (๐‘†) instead of the full middle cohomology ๐ป 2 (๐‘†, Z). Recall the definition as follows. Definition 4.1.1. Let ๐‘‰ be a Hodge structure of ๐พ3โˆ’type. The transcendental lattice ๐‘‡ of ๐‘‰ is the minimal primitive sub-Hodge structure ๐‘‡ โŠ‚ ๐‘‰ with ๐‘‰ 2,0 = ๐‘‡ 2,0 โŠ‚ ๐‘‡C . Note that the transcendental lattice is clearly still of ๐พ3โˆ’type since โ„Ž2,0 (๐‘‡) = โ„Ž2,0 (๐‘‰) = 1. In the case of a ๐พ3 surface, ๐‘‡ (๐‘†) will denote the transcendental lattice of ๐ป 2 (๐‘†, Z). It is important to note that the transcendental lattice is an irreducible Hodge structure. The following result is well-known, we include a proof for the sake of completeness. Proposition 4.1.2. The transcendental lattice ๐‘‡ (๐‘‰) of a Hodge structure ๐‘‰ of ๐พ3โˆ’type is an irreducible Hodge structure of ๐พ3โˆ’type. Proof. Let ๐‘† โŠ‚ ๐‘‡ be a nontrivial sub-Hodge structure of the transcendental lattice. If ๐‘† is pure of type (1, 1) then (๐‘† โŠฅ ) (2,0) โŠ‚ ๐‘‰ 2,0 โŠ‚ ๐‘†CโŠฅ where the โŠฅ is taken inside ๐‘‡. So ๐‘† โŠฅ โŠ‚ ๐‘‡ is a nontrivial Hodge structure of ๐พ3โˆ’type. This contradicts minimality of ๐‘‡. If ๐‘† is not pure of type (1, 1), then ๐‘‰ 2,0 โŠ‚ ๐‘†C so minimality and primitivity of ๐‘‡ again implies that ๐‘† = ๐‘‡. So, ๐‘‡ contains no nontrivial sub-Hodge structures and by the discussion above, ๐‘‡ is a Hodge structure of ๐พ3โˆ’type. Therefore the transcendental lattice is an irreducible Hodge structure of ๐พ3โˆ’type. โ–ก Recall the definition of a homomorphism of Hodge structures from Definition 2.1.5. We use the following notation for the main object of study in this chapter. Remark 4.1.3. Let ๐‘‰, ๐‘Š be (rational) Hodge structures. Let Hom(๐‘‰, ๐‘Š) denote the Qโˆ’vector space of homomorphisms of Hodge structures from ๐‘‰ to ๐‘Š. Then the endomorphism algebra of ๐‘‰ is defined to be End(๐‘‰) B Hom(๐‘‰, ๐‘‰). Note that End(๐‘‰) naturally has the structure of a Qโˆ’algebra with the product given by composition of endomorphisms. 43 Lemma 4.1.4. If ๐‘‰ is an irreducible Hodge structure then End(๐‘‰) is a division algebra. Proof. Note that kernels and images of homomorphisms of Hodge structures are sub-Hodge struc- tures. Therefore, if ๐‘‰ is irreducible, then for any nonzero ๐‘“ โˆˆ End(๐‘‰), its kernel must be zero and the image must be ๐‘‰ by Proposition 4.1.2. This means it must be an isomorphism. In particular, ๐‘“ has an inverse and so End(๐‘‰) is a division algebra. โ–ก In particular, the above implies that End(๐‘‡ (๐‘†)) is a division algebra for the transcendental lattice of a ๐พ3 surface ๐‘†. Zarhin shows that much more holds in this case: Theorem 4.1.5. [Zar83, Theorem 1.6] Let ๐‘† be a ๐พ3 surface. Then End(๐‘‡ (๐‘†)) is a commutative field. Proof. There is a natural homomorphism ๐‘– : End(๐‘‡ (๐‘†)) ! ๐ป๐‘œ๐‘š C ๐ป 2,0 (๐‘†) = C. Since End(๐‘‡ (๐‘†)) is a division algebra by Lemma 4.1.4, ๐‘– : End(๐‘‡ (๐‘†)) โ†ฉ! C must be an embedding and so it is a commutative field. โ–ก The same proof works for End(๐‘‡ (๐‘‹)) of a cubic fourfold. Note that since ๐‘‡ (๐‘†) is finite dimensional, End(๐‘‡ (๐‘†)) is necessarily a number field. We will consider two main types of number fields in the following theory, totally real number fields and CM fields. Definition 4.1.6. Let ๐พ be a number field. 1. A number field ๐พ is a totally real number field if every embedding ๐‘– : ๐พ โ†ฉ! C satisfies ๐‘–(๐พ) โŠ‚ R. 2. A number field ๐พ is a totally imaginary field if there is no embedding ๐‘– : ๐พ โ†ฉ! R. 3. ๐พ is a CM-field if it is totally imaginary and if it is a quadratic extension ๐พ/๐น of a totally real number field ๐น. โˆš The most basic examples are provided by quadratic fields ๐พ = Q( ๐‘‘) for ๐‘‘ โˆˆ Z square-free. If ๐‘‘ > 0, the field is totally real. If ๐‘‘ < 0, the field is totally imaginary and it is a CM-field with ๐น = Q. This leads us to the following important result of Zarhin. We will use it frequently in the remainder of the chapter. 44 Theorem 4.1.7. Let ๐‘‰ be a Hodge structure of ๐พ3โˆ’type and suppose End(๐‘‰) is a commutative field (for example, consider End(๐‘‡ (๐‘†)) for a ๐พ3 surface ๐‘†). Then either 1. End(๐‘‰) is a totally real number field or 2. End(๐‘‰) is a CM-field. Proof. See [Zar83, Theorem 1.5.1]. โ–ก Endomorphism algebras have proven to be useful tools for studying the Hodge conjecture for products. Given a smooth projective variety ๐‘‹, recall that the Hodge conjecture states that the space of Hodge classes Hod ๐‘˜ (๐‘‹) of degree 2๐‘˜ given by Hod ๐‘˜ (๐‘‹) B ๐ป 2๐‘˜ (๐‘‹, Q) โˆฉ ๐ป ๐‘˜,๐‘˜ (๐‘‹) is spanned by algebraic ๐‘˜-cycles. The conjecture is of course still open for ๐‘˜ โ‰  0, 1, dim(๐‘‹) โˆ’ 1, dim(๐‘‹). Now, we explore how endomorphism algebras naturally appear in the Hodge conjecture for self-products. For smooth projective varieties ๐‘‹, ๐‘Œ , the Kรผnneth formula and linear algebra give รŠ ๐ป ๐‘  (๐‘‹ ร— ๐‘Œ , Q)  ๐ป ๐‘™ (๐‘‹, Q) โŠ— ๐ป ๐‘š (๐‘Œ , Q) ๐‘™+๐‘š=๐‘  รŠ  Hom(๐ป 2 dim(๐‘‹)โˆ’๐‘™ (๐‘‹, Q), ๐ป ๐‘š (๐‘Œ , Q)). ๐‘™+๐‘š=๐‘  Note that each summand is a Hodge substructure of the left side. It is easy to see that under this correspondence, Hodge cycles of degree 2๐‘˜ correspond to homomorphisms of Hodge structures. Now, apply the above to ๐‘‹ = ๐‘Œ = ๐‘† for a ๐พ3 surface ๐‘†. Note that the Hodge conjecture is not known in general for the self-product ๐‘† ร— ๐‘†. The only nontrivial case in the Hodge conjecture in this case is that of ๐ป 4 (๐‘† ร— ๐‘†, Q). We know that ๐ป 1 (๐‘†, Q) = 0 and ๐ป 3 (๐‘†, Z) = 0 by Proposition 2.4.3. We can also see that ๐ป 4 (๐‘‹, Q) โŠ— ๐ป 0 (๐‘Œ , Q)  Hom(๐ป 0 (๐‘†, Q), ๐ป 0 (๐‘†, Q)) and ๐ป 0 (๐‘‹, Q) โŠ— ๐ป 4 (๐‘Œ , Q)  Hom(๐ป 4 (๐‘†, Q), ๐ป 4 (๐‘†, Q)) 45 are spanned by the classes {๐‘} ร— ๐‘† and ๐‘† ร— {๐‘} respectively for a point ๐‘, so they are algebraic. The only remaining case is HomHod (๐ป 2 (๐‘†, Q), ๐ป 2 (๐‘†, Q)). ๐ป 2 (๐‘†, Q) splits as a direct sum of Hodge structures as ๐ป 2 (๐‘†, Q) = ๐‘ ๐‘†(๐‘†)Q โŠ• ๐‘‡ (๐‘†)Q . Now, since ๐‘ ๐‘†(๐‘†)Q is pure of type (1, 1) and ๐‘‡ (๐‘†)Q is irreducible by Proposition 4.1.2, we have that HomHod (๐‘ ๐‘†(๐‘†)Q , ๐‘‡ (๐‘†)Q ) = 0 and HomHod (๐‘‡ (๐‘†)Q , ๐‘ ๐‘†(๐‘†)Q ) = 0. In the first case, observe that the image of such a Hodge morphism would define a Hodge sub-structure of ๐‘‡ (๐‘†)Q . In the second case, the fact that ๐‘ ๐‘†(๐‘†)Q2,0 = 0 implies that the kernel of such a Hodge morphism would define a Hodge sub-structure of ๐‘‡ (๐‘†)Q . The Neron-Severi part ๐‘ ๐‘†(๐‘†)Q is algebraic by definition and HomHod (๐‘ ๐‘†(๐‘†)Q , ๐‘ ๐‘†(๐‘†)Q ) is spanned by classes of curves ๐ถ1 ร— ๐ถ2 โŠ‚ ๐‘† ร— ๐‘†. Therefore, we have reduced the problem of the Hodge conjecture for self-products of ๐พ3 surfaces ๐‘† ร— ๐‘† to considering HomHod (๐‘‡ (๐‘†)Q , ๐‘‡ (๐‘†)Q ) and this is of course just End(๐‘‡ (๐‘†)Q ) by definition. So, the Hodge conjecture for ๐‘† ร— ๐‘† is equivalent to the condition that End(๐‘‡ (๐‘†)Q ) be generated by algebraic classes. This is the main motivation for studying endomorphism algebras of Hodge structures. By Theorem 4.1.7, studying End(๐‘‡ (๐‘†)) reduces to two cases. The CM case is well understood due to a result of Mukai. Theorem 4.1.8. If ๐‘“ โˆˆ End(๐‘‡ (๐‘†)) is a Hodge isometry, then ๐‘“ corresponds to the class of an algebraic cycle in ๐ป 4 (๐‘† ร— ๐‘†, Q). Proof. See [Muk03, Theorem 2]. โ–ก However, in [RM08, Theorem 5.4], the author uses Mukaiโ€™s theorem and shows the following: Theorem 4.1.9. If End(๐‘‡ (๐‘†)) is a CM field, then it is spanned by Hodge isometries. In particular, the Hodge conjecture holds for self-products of ๐พ3 surfaces ๐‘† ร— ๐‘† if End(๐‘‡ (๐‘†)) is of CM type. 46 On the other hand, if End(๐‘‡ (๐‘†)) is totally real, then the problem of the Hodge conjecture for self products is much more open. In the totally real case, it is easy to show that the only Hodge isometries in End(๐‘‡ (๐‘†)) are ยฑid. So Mukaiโ€™s result no longer helps. We will note some examples and results about the Hodge conjecture in the totally real case in Section 4.5. For the remainder of this section, we will outline Van Geemenโ€™s proof for the existence of ๐พ3 surfaces ๐‘† with End(๐‘‡ (๐‘†)) totally real. We will point out where details of Van Geemenโ€™s proof will no longer work in the case of cubic fourfolds, which will lead to our focus in the next two sections. First, a Hodge-theoretic lemma is needed. Lemma 4.1.10. Let ๐น be a totally real field and let ๐‘š โˆˆ Z, ๐‘š โ‰ฅ 3. Then there exists an irreducible, polarized Hodge structure (๐‘‰, ๐œŒ, ๐œ“) of ๐พ3โˆ’type with End(๐‘‰) = ๐น and dim๐น (๐‘‰) = ๐‘š, where ๐œŒ denotes the algebraic representation defining the Hodge structure and ๐œ“ denotes the polarization. Proof. See [VG08, Lemma 3.2]. โ–ก Now, Van Geemen shows that such a Hodge structure can be realized as the transcendental lattice of a ๐พ3 surface. We reproduce the proof here directly to illustrate the changes needed in our study of cubic fourfolds. Proposition 4.1.11. [VG08, Proposition 3.3] Given a totally real number field ๐น and integer ๐‘š โ‰ฅ 3 such that ๐‘š [๐น : Q] โ‰ค 10, there exists an (๐‘š โˆ’ 2)โˆ’dimensional family of ๐พ3 surfaces such that End(๐‘‡ (๐‘†)) = ๐น for a general member of that family. Proof. Let (๐‘‰, ๐œŒ, ๐œ“) be a polarized, irreducible Hodge structure of ๐พ3โˆ’type with End(๐‘‰) = ๐น and period ๐œ” by Lemma 4.1.10. Choose a free Zโˆ’module ๐‘‡ โŠ‚ ๐‘‰ of rank ๐‘‘ = dimQ (๐‘‰) such that ๐œ“ is integer valued on ๐‘‡ ร— ๐‘‡. Then by [Nik80, Theorem 1.10.1], there is a primitive embedding of ๐‘‡ into the ๐พ3 lattice ฮ›๐พ3 , ๐‘‡ โ†ฉ! ฮ›๐พ3 . The surjectivity of the period map for ๐พ3 surfaces implies the existence of a polarized ๐พ3 surface ๐‘† with End(๐‘‡ (๐‘†))  ๐‘‡ corresponding to the period ๐œ”. The proof of [VG08, Lemma 3.2] shows that there are (๐‘š โˆ’ 2) moduli. โ–ก So, if we want to work towards a similar result for cubic fourfolds, there are two main difficulties in following a similar process. 47 First, the lattice theory used by Van Geemen was developed by Nikulin in [Nik80] for even, unimodular lattices. While the ๐พ3 lattice ฮ›๐พ3 is an even, unimodular lattice, the cubic fourfold lattice ฮ›๐ถ4 is odd and unimodular, and the lattice ฮ›๐ถ4,0 is even and non-unimodular. Odd, unimodular lattices are easier to understand than even, non-unimodular lattices. We study odd, unimodular lattices in Section 4.2 and apply the results to ฮ›๐ถ4 instead of ฮ›๐ถ4,0 . Second, the proof uses surjectivity of the period map for ๐พ3 surfaces. It is well known that the period map for cubic fourfolds is not surjective. However, by work of Laza and Looฤณenga, we understand the image of the period map for cubic fourfolds explicitly. We dedicate Section 4.3 to the study of this period map. 4.2 Odd Unimodular Lattices In [Nik80, Theorem 1.12.2], Nikulin proves an important result about the existence of embed- dings of sub-lattices into even, unimodular lattices. The goal of this section is to work through Nikulinโ€™s lattice theory to prove a similar result for odd unimodular lattices. Recall that a lattice is a pair (๐ฟ, ๐‘) where ๐ฟ is a free Zโˆ’module of finite rank and ๐‘ : ๐ฟ ร— ๐ฟ ! Z is a non-degenerate, integral, symmetric, bilinear form on ๐ฟ. We saw the following examples in Chapter 2, but we compile them here as they will be the most important examples for this chapter. Example 4.2.1. Given an algebraic variety ๐‘‰ of dimension ๐‘›, the intersection form on the variety endows its middle cohomology ๐ป ๐‘› (๐‘‰, Z) with the structure of a lattice. 1. For a ๐พ3 surface ๐‘†, ๐ป 2 (๐‘†, Z) is isomorphic to the lattice ฮ›๐พ3 = ๐ธ 8 (โˆ’1) โŠ•2 โŠ• ๐‘ˆ โŠ•3 . This lattice is even and unimodular. 2. For a cubic fourfold ๐‘‹, ๐ป 4 (๐‘‹, Z) is isomorphic to the lattice ฮ›๐ถ4 = โŸจ1โŸฉ 21 โŠ• โŸจโˆ’1โŸฉ 2 [Has16]. This is an odd, unimodular lattice since ๐‘(โ„Ž2 , โ„Ž2 )๐ถ4 = 3 โˆ‰ 2Z for a hyperplane section โ„Ž, and disc(ฮ›๐ถ4 ) = 1. 3. Denote the orthogonal complement of โ„Ž2 in ฮ›๐ถ4 by ฮ›๐ถ4,0 = {โ„Ž2 }โŠฅ . This is an even lattice and ฮ›๐ถ4,0 โŠ‚ ฮ›๐ถ4 . However, ฮ›๐ถ4,0 โŠ‚ ฮ›๐ถ4 is not unimodular. ฮ›๐ถ4,0  ๐ด2 โŠ• ๐‘ˆ โŠ•2 โŠ• ๐ธ 8โŠ•2 by 48 [Has16] where ยฉ2 1 ยช ๐ด2 = ยญยญ ยฎ. ยฎ 1 2 ยซ ยฌ Hence disc( ๐ด2 ) = 3. Since the discriminant is multiplicative across direct sums, ฮ›๐ถ4,0 is not unimodular. We need several numerical facts in this section. Most of the following are found in [Huy16, Chapter 14] or [Zar83]. We provide proofs when it was not available in literature. For a lattice ๐ฟ, we denote its dual free lattice HomZ (๐ฟ, Z) by ๐ฟ โˆ— . For any integral lattice ๐ฟ, there is a canonical finite index inclusion ๐‘– ๐ฟ : ๐ฟ โ†ฉ! ๐ฟ โˆ— of order |disc(๐ฟ)|, given by ๐‘™ 7! ๐‘(๐‘™, โ€“). This is well-defined since ๐ฟ is an integral lattice. Lemma 4.2.2. If an embedding of lattices ๐‘† โ†ฉ! ๐ฟ has finite index, then disc(๐‘†) = disc(๐ฟ) ยท (๐ฟ : ๐‘†) 2 . Proof. We have the chain of inclusions ๐‘† โ†ฉ! ๐ฟ โ†ฉ! ๐ฟ โˆ— โ†ฉ! ๐‘† โˆ— so that |๐‘† โˆ— /๐‘†| = |๐ฟ/๐‘†| ยท |๐ฟ โˆ— /๐ฟ| ยท |๐‘† โˆ— /๐ฟ โˆ— |. This gives disc(๐‘†) = disc(๐ฟ) ยท (๐ฟ : ๐‘†) ยท |๐‘† โˆ— /๐ฟ โˆ— |. Now, it is easy to show that ๐‘† โˆ— /๐ฟ โˆ— and ๐ฟ/๐‘† are isomorphic, so |๐‘† โˆ— /๐ฟ โˆ— | = |๐ฟ/๐‘†| = (๐ฟ : ๐‘†), giving the result. โ–ก Recall the following definition. Definition 4.2.3. Define the discriminant group of ๐‘† to be ๐ด๐‘† = ๐‘† โˆ— /๐‘†, which is a finite abelian group of order |disc(๐‘†)|. 49 We extend the bilinear form on ๐‘† naturally to a bilinear form on ๐‘† โˆ— as follows: Since ๐‘† โˆ— /๐‘† is a finite group, given ๐‘ 1 , ๐‘ 2 โˆˆ ๐‘† โˆ— we have ๐‘› ยท ๐‘ 1 , ๐‘š ยท ๐‘ 2 โˆˆ ๐‘† for some ๐‘›, ๐‘š โˆˆ Z. Define 1 ๐‘(๐‘ 1 , ๐‘ 2 ) ๐‘†โˆ— = ๐‘(๐‘› ยท ๐‘ 1 , ๐‘š ยท ๐‘ 2 ) ๐‘† . ๐‘›๐‘š It is easy to see that this is well-defined. Note that the form on ๐‘† โˆ— restricts to the original form on ๐‘†, and the form on ๐‘† โˆ— is now Q-valued. The following description of ๐‘( , ) ๐‘†โˆ— will be useful. Lemma 4.2.4. If ๐‘ โˆ— โˆˆ ๐‘† โˆ— and ๐‘™ โˆˆ ๐‘† then ๐‘(๐‘ โˆ— , ๐‘™) ๐‘†โˆ— = ๐‘ โˆ— (๐‘™). Proof. Since ๐‘† has finite index in ๐‘† โˆ— , ๐‘‘๐‘ โˆ— โˆˆ ๐‘† for some ๐‘‘. That is, some multiple of ๐‘ โˆ— can be described as ๐‘‘๐‘ โˆ— = ๐‘(๐‘ฅ, ) ๐‘† for some ๐‘ฅ โˆˆ ๐‘†, ๐‘‘ โˆˆ Z. Therefore 1 1 1 ๐‘(๐‘ โˆ— , ๐‘™) ๐‘†โˆ— = ๐‘(๐‘ฅ, ๐‘™) ๐‘† = ๐‘(๐‘ฅ, โˆ’) ๐‘† (๐‘™) = ยท ๐‘‘๐‘ โˆ— (๐‘™) = ๐‘ โˆ— (๐‘™). ๐‘‘ ๐‘‘ ๐‘‘ โ–ก Definition 4.2.5. Given an abelian group ๐ด, we will use ๐‘™ ( ๐ด) throughout to denote the minimal number of generators of ๐ด. This may be referred to as the length of ๐ด. Example 4.2.6. By [Huy16, Example 14.0.3], for ฮ› = ฮ›๐ถ4,0 , ๐ดฮ› = Z/3Z . Since ๐ดฮ› is a cyclic group, ๐‘™ ( ๐ด) = 1. Remark 4.2.7. If ๐ด is a finite abelian group then we can decompose ๐ด into its ๐‘โˆ’components ๐ด ๐‘ . ร‰ If ๐‘ is a bilinear form on ๐ด then its restriction to ๐ด ๐‘ will be denoted ๐‘ ๐‘ . Note that ๐‘ = ๐‘ ๐‘๐‘. Definition 4.2.8. Define the discriminant bilinear form ๐‘ ๐‘† : ๐ด๐‘† ร— ๐ด๐‘† ! Q/Z to be ๐‘ ๐‘† (๐‘ 1 + ๐‘†, ๐‘ 2 + ๐‘†) = โŸจ๐‘ 1 , ๐‘ 2 โŸฉ๐‘†โˆ— + Z, where ๐‘ 1 , ๐‘ 2 โˆˆ ๐‘† โˆ— . We will be primarily concerned with odd lattices. However, if a lattice ๐ฟ is even then one can further define the discriminant quadratic form similarly by ๐‘ž ๐ฟ : ๐ด ๐ฟ ! Q/2Z. It will be useful to decompose the discriminant form as follows: Lemma 4.2.9. If ๐‘† is an integral lattice then its discriminant bilinear form decomposes as (๐‘ ๐‘† ) ๐‘ = ๐‘ ๐‘†โŠ—Z ๐‘ . 50 Proof. See [Nik80, Proposition 1.7.1]. โ–ก The discriminant form is an important tool in studying the classification of lattices. Definition 4.2.10. We say two lattices ๐‘† and ๐ฟ have the same genus if ๐ฟ โŠ— Z ๐‘  ๐‘† โŠ— Z ๐‘ for all primes ๐‘ and ๐ฟ โŠ— R  ๐‘† โŠ— R. Note that all equivalent forms have the same genus, but not all forms of the same genus are equivalent. Consider the following result of Nikulin. Theorem 4.2.11. The genus of an even lattice ๐ฟ is determined by its signature (๐‘™+ , ๐‘™โˆ’ ) and discrim- inant form ๐‘ž ๐ฟ . Proof. [Nik80, Corollary 1.9.4] โ–ก Now we will begin to explore embeddings of lattices in more detail. Definition 4.2.12. Fix a lattice ๐‘†. A lattice ๐ฟ such that ๐‘† โ†ฉ! ๐ฟ is an embedding and ๐ฟ/๐‘† is a finite abelian group will be called an overlattice of ๐‘†. Given an overlattice ๐ฟ of ๐‘†, we have a series of embeddings ๐‘† โ†ฉ! ๐ฟ โ†ฉ! ๐ฟ โˆ— โ†ฉ! ๐‘† โˆ— where each embedding is of finite index. Define ๐ป ๐ฟ := ๐ฟ/๐‘†. Taking quotients by S, we have a chain ๐ป ๐ฟ := ๐ฟ/๐‘† โŠ‚ ๐ฟ โˆ— /๐‘† โŠ‚ ๐‘† โˆ— /๐‘† = ๐ด๐‘† where ๐ด๐‘† is the discriminant group. We would like to understand subgroups of the discriminant group ๐ด๐‘† . To do that, we need a few lemmas. Definition 4.2.13. Let ๐‘ ๐‘† be the discriminant bilinear form on ๐ด๐‘† . We say a subgroup ๐ป โŠ‚ ๐ด๐‘† is ๐‘ ๐‘† โˆ’isotropic if ๐‘ ๐‘† | ๐ปร—๐ป = 0. The following lemma was presented in [Nik80, Proposition 1.4.1] for even overlattices. We adapt the lemma for all overlattices. In doing so, we consider ๐‘ ๐‘† โˆ’isotropic subgroups instead of ๐‘ž ๐‘† โˆ’isotropic subgroups as Nikulin does, since discriminant quadratic forms are only defined for even lattices. 51 Lemma 4.2.14. Fix a lattice ๐‘†. There is a bฤณective correspondence between finite index overlattices ๐‘† โŠ‚ ๐ฟ and ๐‘ ๐‘† โˆ’isotropic subgroups ๐ป โŠ‚ ๐ด๐‘† . Proof. Let ๐ฟ be a finite index overlattice of ๐‘†. Then the corresponding ๐‘ ๐‘† โˆ’isotropic subgroup of ๐ด๐‘† is given by ๐ป ๐ฟ = ๐ฟ/๐‘†. This is clearly a subgroup of ๐ด๐‘† . To see why it is isotropic, choose ๐‘™1 , ๐‘™2 โˆˆ ๐ฟ. Then ๐‘ ๐‘† ( ๐‘™ยฏ1 , ๐‘™ยฏ2 ) = โŸจ๐‘™1 , ๐‘™2 โŸฉ ๐ฟ + Z = 0ฬ„ + Z, where โŸจ๐‘™1 , ๐‘™2 โŸฉ ๐ฟ โˆˆ Z since ๐ฟ is an integral lattice. In the other direction, let ๐ป โŠ‚ ๐ด๐‘† = ๐‘† โˆ— /๐‘† be a ๐‘ ๐‘† โˆ’isotropic subgroup. Consider the quotient map ๐‘ž : ๐‘† โˆ— ! ๐ด๐‘† . and the preimage ๐ฟ := ๐‘ž โˆ’1 (๐ป) of ๐ป. ๐ฟ contains ๐‘† since ๐ป โŠ‚ ๐‘† โˆ— /๐‘† and ๐‘† โ†ฉ! ๐ฟ is of finite index since ๐‘† โŠ‚ ๐ฟ โŠ‚ ๐‘† โˆ— and ๐‘† โ†ฉ! ๐‘† โˆ— is of finite index. ๐ฟ is integral since ๐ป being ๐‘ ๐‘† โˆ’isotropic implies โŸจ , โŸฉ๐‘†โˆ— restricts to an integral form on ๐ฟ. These constructions are clearly inverses of each other which gives the bฤณection in the statement. โ–ก The following technical lemma is necessary for studying subgroups of ๐ด๐‘† . Lemma 4.2.15. Suppose ๐ฟ is a finite index overlattice of ๐‘†. Then the following all hold. 1. ๐ฟ โˆ— /๐‘† = ๐ป ๐ฟโŠฅ in ๐ด๐‘† 2. ๐ป ๐ฟโŠฅ /๐ป ๐ฟ  ๐ฟ โˆ— /๐ฟ = ๐ด ๐ฟ 3. (๐‘ ๐‘† |๐ป ๐ฟโŠฅ )/๐ป ๐ฟ = ๐‘ ๐ฟ Proof. 1. Let ๐‘™ โˆ— โˆˆ ๐ฟ โˆ— and ๐‘š โˆˆ ๐ฟ. Then ๐‘ ๐‘† ( ๐‘™ยฏโˆ— , ๐‘š) ยฏ = โŸจ๐‘™ โˆ— , ๐‘šโŸฉ๐‘† + Z = ๐‘™ โˆ— (๐‘š) + Z = 0ฬ„ + Z since ๐‘™ โˆ— โˆˆ ๐ฟ โˆ— = ๐ป๐‘œ๐‘š(๐ฟ, Z). So ๐ฟ โˆ— /๐‘† โŠ‚ ๐ป ๐ฟโŠฅ . ยฏ = 0 for all ๐‘™ โˆˆ ๐ฟ. Let ๐‘ ยฏโˆ— โˆˆ ๐ป ๐ฟโŠฅ . Then ๐‘ ๐‘† ( ๐‘ ยฏ, ๐‘™) =โ‡’ โŸจ๐‘ โˆ— , ๐‘™โŸฉ๐‘†โˆ— โˆˆ Z, for all ๐‘™ โˆˆ ๐ฟ =โ‡’ ๐‘ โˆ— (๐‘™) โˆˆ ๐‘ for all ๐‘™ โˆˆ ๐ฟ by Lemma 4.2.4 =โ‡’ ๐‘ โˆ— โˆˆ ๐ฟ โˆ— =โ‡’ ๐‘ ยฏโˆ— โˆˆ ๐ฟ โˆ— /๐‘†. Therefore ๐ฟ โˆ— /๐‘† = ๐ป ๐ฟโŠฅ . 2. Since ๐ฟ โˆ— /๐‘† = ๐ป ๐ฟโŠฅ and ๐ป ๐ฟ = ๐ฟ/๐‘†, we can see that ๐ด ๐ฟ := ๐ฟ โˆ— /๐ฟ  (๐ฟ โˆ— /๐‘†)/(๐ฟ/๐‘†) = ๐ป ๐ฟโŠฅ /๐ป ๐ฟ as claimed. 52 3. Note that ๐ป ๐ฟ is ๐‘ ๐‘† โˆ’isotropic. Also โŸจ , โŸฉ๐‘†โˆ— | ๐ฟ โˆ— = โŸจ , โŸฉ ๐ฟ โˆ— since ๐ฟ โˆ— โ†ฉ! ๐‘† โˆ— . The discriminant- bilinear form ๐‘ ๐ฟ : ๐ด ๐ฟ ร— ๐ด ๐ฟ ! Q/Z is given by โŸจ , โŸฉ ๐ฟ โˆ— +Z. By the previous part, ๐ด ๐ฟ = ๐ป ๐ฟโŠฅ /๐ป ๐ฟ . So, ๐‘ ๐‘† induces a bilinear form ๐ป ๐ฟโŠฅ /๐ป ๐ฟ ร— ๐ป ๐ฟโŠฅ /๐ป ๐ฟ ! Q/Z by โŸจ , โŸฉ๐‘†โˆ— + Z. In other words, (๐‘ ๐‘† |๐ป ๐ฟโŠฅ )/๐ป ๐ฟ = ๐‘ ๐ฟ . This completes the proof. โ–ก Definition 4.2.16. We say two lattices ๐‘† and ๐พ are orthogonal, denoted ๐‘† โŠฅ ๐พ, if there is a primitive embedding of ๐‘† into a unimodular lattice ๐ฟ such that (๐‘†) โŠฅ ๐ฟ  ๐พ. The following proposition relates the discriminant forms of orthogonal lattices. Our argument is similar to [Nik80, Proposition 1.5.1], but we do not require ๐ฟ to be an even lattice and we consider discriminant-bilinear forms instead of discriminant-quadratic forms. Note that ๐‘† โŠ• ๐พ โ†ฉ! ๐ฟ is of finite index. We will apply the lemmas above for this embedding. Proposition 4.2.17. Two lattices ๐‘† and ๐พ are orthogonal if and only if ( ๐ด๐‘† , ๐‘ ๐‘† )  ( ๐ด๐พ , โˆ’๐‘ ๐พ ). Proof. We first prove the forward direction. So, let ๐‘† and ๐พ be two lattices that are primitively embedded in a unimodular lattice ๐ฟ. In particular, we have a primitive embedding ๐‘† โ†ฉ! ๐ฟ where ๐ฟ is unimodular with ๐‘† โŠฅ = ๐พ and ๐‘† โŠ• ๐พ โ†ฉ! ๐ฟ is of finite index. So, we have the following chain of inclusions ๐‘† โŠ• ๐พ โ†ฉ! ๐ฟ โ†ฉ! ๐ฟ โˆ— โ†ฉ! (๐‘† โŠ• ๐พ) โˆ— . Taking quotients, this gives a map ๐œ™ as a composition: ๐œ™ : ๐ป ๐ฟ := ๐ฟ/(๐‘† โŠ• ๐พ) ! ๐ฟ โˆ— /(๐‘† โŠ• ๐พ) ! (๐‘† โŠ• ๐พ) โˆ— /(๐‘† โŠ• ๐พ)  ๐ด๐‘† โŠ• ๐ด๐พ . Composing with either projection to ๐ด๐‘† or ๐ด๐พ , this composition gives two maps ๐œ™ ๐‘† : ๐ป ๐ฟ ! ๐ด๐‘† and ๐œ™ ๐พ : ๐ป ๐ฟ ! ๐ด๐พ . We claim that both maps are isomorphisms which would show that ๐ด๐‘†  ๐ด๐พ as groups. First, note that since ๐‘† is primitive in ๐ฟ, we have that ๐ฟ  ๐‘† โŠ• (๐ฟ/๐‘†). Now ๐ฟ โˆ—  ๐‘† โˆ— โŠ• (๐ฟ/๐‘†) โˆ— and so projection gives us a surjective map ๐‘“ โˆ— : ๐ฟ โˆ— ! ๐‘† โˆ— . Now, the key point to note is that ๐ฟ is unimodular, so ๐ฟ  ๐ฟ โˆ— . This gives us a surjection ๐‘“ โˆ— : ๐ฟ  ๐ฟ โˆ— ! ๐‘† โˆ— . Given ๐‘ โˆ— โˆˆ ๐‘† โˆ— , let ๐‘™ ๐‘  โˆˆ ๐ฟ be 53 an element such that ๐‘“ โˆ— (๐‘™ ๐‘  ) = ๐‘ โˆ— . Since ๐พ is also primitive in ๐ฟ, we can apply the same argument to ๐พ โˆ— . So we have elements ๐‘™ ๐‘  โˆˆ ๐ฟ, ๐‘™ ๐พ โˆˆ ๐พ such that ๐œ™ ๐‘† (๐‘™ ๐‘  ) = ๐‘ ยฏโˆ— and ๐œ™ ๐พ (๐‘™ ๐‘˜ ) = ๐‘˜ยฏโˆ— . This shows that each map is surjective. Next, we see that that ๐œ™ ๐‘† and ๐œ™ ๐พ are embeddings since ๐‘† and ๐พ are primitive in ๐ฟ: Suppose ยฏ = 0 for some ๐‘™ โˆˆ ๐ฟ. Write ๐‘™ = ๐‘ โˆ— + ๐‘˜ โˆ— where ๐‘ โˆ— โˆˆ ๐‘† โˆ— and ๐‘˜ โˆ— โˆˆ ๐พ โˆ— since ๐ฟ โŠ‚ ๐‘† โˆ— โŠ• ๐พ โˆ— . Then ๐œ™ ๐‘† ( ๐‘™) ๐‘™ ๐‘™ ๐‘™ ๐‘™ ยฏ = ๐‘ ยฏโˆ— = 0, so ๐‘™ = ๐‘ ๐‘™ + ๐‘˜ โˆ— where ๐‘ ๐‘™ โˆˆ ๐‘†. Now, ๐‘† โŠ• ๐พ โŠ‚ ๐ฟ is of finite index, so ๐‘ ๐‘™โˆ— โˆˆ ๐‘† since ๐œ™ ๐‘† ( ๐‘™) ๐‘™ ๐‘™ ๐‘› ยท ๐‘™ = ๐‘› ยท ๐‘ ๐‘™ + ๐‘› ยท ๐‘˜ ๐‘™โˆ— โˆˆ ๐‘† โŠ• ๐พ for some ๐‘› โˆˆ Z. This implies that ๐‘› ยท ๐‘˜ ๐‘™โˆ— โˆˆ ๐พ. But ๐ฟ/๐พ is torsion-free since ๐พ is primitive in ๐ฟ. Therefore ๐‘˜ ๐‘™โˆ— โˆˆ ๐พ. So ๐‘™ โˆˆ ๐‘† โŠ• ๐พ, which means ๐‘™ยฏ = 0. Injectivity of ๐œ™ ๐พ can be seen similarly. We now have isomorphisms ๐ด๐‘†  ๐ด๐พ . To see why ๐‘ ๐‘†  โˆ’๐‘ ๐พ under this isomorphism, apply Lemma 4.2.15 to ๐‘ ๐‘†โŠ•๐พ = ๐‘ ๐‘† โŠ• ๐‘ ๐พ and ๐‘ ๐ฟ . Setting ๐ฟ/(๐‘† โŠ• ๐พ) = ๐ป ๐ฟ , by proof of Lemma 4.2.14, we see that ๐ป ๐ฟ is ๐‘ ๐‘†โŠ•๐พ โˆ’isotropic, ๐ป ๐ฟโŠฅ = ๐ฟ โˆ— /(๐‘† โŠ• ๐พ), and ๐‘ ๐ฟ = ((๐‘ ๐‘† โŠ• ๐‘ ๐พ )|๐ป ๐ฟโŠฅ /๐ป ๐ฟ . But, ๐ฟ  ๐ฟ โˆ— , so ๐ป ๐ฟโŠฅ = ๐ป ๐ฟ . This means ๐ป ๐ฟโŠฅ is ๐‘ ๐‘†โŠ•๐พ โˆ’isotropic and so ๐‘ ๐ฟ = ((๐‘ ๐‘† โŠ• ๐‘ ๐พ )|๐ป ๐ฟโŠฅ /๐ป ๐ฟ = 0, giving ๐‘ ๐‘† = โˆ’๐‘ ๐พ . So, we have ( ๐ด๐‘† , ๐‘ ๐‘† )  ( ๐ด๐พ , โˆ’๐‘ ๐พ ). This completes the proof of the forward direction. Now we suppose ๐‘† and ๐พ are two lattices, and there exists an isomorphism ( ๐ด๐‘† , ๐‘ ๐‘† )  ( ๐ด๐พ , โˆ’๐‘ ๐พ ). Since ๐ด๐‘†โŠ•๐พ  ๐ด๐‘† โŠ• ๐ด๐พ , we consider the graph of this isomorphism ฮ“ in ๐ด๐‘†โŠ•๐พ . Observe that ฮ“ is ๐‘ ๐‘†โŠ•๐พ -isotropic since ๐‘ ๐‘† = โˆ’๐‘ ๐พ . So, by Lemma 4.2.14, ฮ“ corresponds to an overlattice ๐ฟ. The unimodularity follows from the last part of Lemma 4.2.15. This proves the backward direction. โ–ก The following lemma is adapted from [Nik80, Proposition 1.12.1], again by replacing even lattices with arbitrary lattices, and ๐‘ž ๐‘† with ๐‘ ๐‘† . Lemma 4.2.18. A lattice with signature (๐‘ + , ๐‘ โˆ’ ) and bilinear-discriminant form ๐‘ ๐‘† exists if and only if a lattice with signature (๐‘ โˆ’ , ๐‘ + ) and bilinear-discriminant form โˆ’๐‘ ๐‘† exists. Proof. If a lattice ๐‘† has signature (๐‘ + , ๐‘ โˆ’ ) and bilinear form โŸจ , โŸฉ๐‘† , then by multiplying the form by โˆ’1 we get a lattice ๐‘‡ with signature (๐‘ โˆ’ , ๐‘ + ) and bilinear form โŸจ , โŸฉ๐‘‡ = โˆ’โŸจ , โŸฉ๐‘† . Extending โŸจ , โŸฉ๐‘‡ to ๐‘‡ โˆ— we have โŸจ , โŸฉ๐‘‡ โˆ— = โˆ’โŸจ , โŸฉ๐‘†โˆ— . Therefore, ๐‘๐‘‡ = โŸจ , โŸฉ๐‘‡ โˆ— + Z = โˆ’โŸจ , โŸฉ๐‘†โˆ— + Z = โˆ’๐‘ ๐‘† , so ๐‘‡ has signature 54 (๐‘ โˆ’ , ๐‘ + ) and bilinear-discriminant form โˆ’๐‘ ๐‘† as required. The other direction is implied as well. โ–ก Before discussing the embedding theorem for odd unimodular lattices, we need a result of Nikulin. In particular, we need the corollary to [Nik80, Theorem 1.16.5]. In the following, by an invariant (๐‘ก+ , ๐‘กโˆ’ , ๐‘ ๐‘† ) of a lattice ๐ฟ, we mean a lattice ๐ฟ with signature (๐‘ก+ , ๐‘กโˆ’ ) and discriminant bilinear form ๐‘ ๐‘† . Theorem 4.2.19. An odd lattice ๐‘† with invariants (๐‘ก+ , ๐‘กโˆ’ , ๐‘ ๐‘† ) exists if the following conditions are satisfied simultaneously: 1. ๐‘ก+ โ‰ฅ 0 2. ๐‘กโˆ’ โ‰ฅ 0 3. ๐‘ก+ + ๐‘กโˆ’ > ๐‘™ ( ๐ด๐‘† ) 4. ๐‘ก+ + ๐‘กโˆ’ > ๐‘™ ( ๐ด๐‘†2 ) + 2. Proof. See [Nik80, Corollary 1.16.6]. โ–ก Now, we can put everything together to state the embedding theorem for odd unimodular lattices. Applying Theorem 4.2.19 we have the following useful statement: Corollary 4.2.20. There exists a primitive embedding of an even lattice ๐‘† with invariants (๐‘ก+ , ๐‘กโˆ’ , ๐‘ ๐‘† ) into an odd unimodular lattice ๐ฟ of signature (๐‘™+ , ๐‘™โˆ’ ) with odd orthogonal complement ๐พ if the following conditions are satisfied: 1. ๐‘™+ โˆ’ ๐‘ก+ โ‰ฅ 0 2. ๐‘™โˆ’ โˆ’ ๐‘กโˆ’ โ‰ฅ 0 3. ๐‘™+ + ๐‘™ โˆ’ โˆ’ ๐‘ก+ โˆ’ ๐‘กโˆ’ โ‰ฅ ๐‘™ ( ๐ด๐‘† ) 4. ๐‘™+ + ๐‘™ โˆ’ โˆ’ ๐‘ก+ โˆ’ ๐‘กโˆ’ > ๐‘™ ( ๐ด๐‘† ) + 2. Proof. Assuming the conditions above, Theorem 4.2.19 implies the existence of an odd lattice ๐พ with invariants (๐‘™+ โˆ’ ๐‘ก+ , ๐‘™โˆ’ โˆ’ ๐‘กโˆ’ , โˆ’๐‘ ๐‘† ). In particular, note that ๐‘† and ๐พ are orthogonal by Proposition 4.2.17. In particular, it implies that there exists a primitive embedding of an even lattice ๐‘† with invariants (๐‘ก+ , ๐‘กโˆ’ , ๐‘ ๐‘† ) into some odd unimodular lattice ๐ฟ of signature (๐‘™ + , ๐‘™โˆ’ ). โ–ก We will explore how the above embedding results apply in the case of a cubic fourfold. Recall 55 that for a cubic fourfold ๐‘‹, ๐ป 4 (๐‘‹, Z)  ฮ›๐ถ4 is an odd unimodular lattice of signature (21, 2). Proposition 4.2.21. Suppose ๐‘‡ is an even lattice of signature (21 โˆ’ ๐œŒ, 2). If 13 โ‰ค ๐œŒ โ‰ค 21, then there exists a primitive embedding ๐‘‡ โ†ฉ! ฮ›๐ถ4 such that ๐‘‡ โŠฅ  ๐‘ is odd. Proof. Conditions 1 and 2 of Corollary 4.2.20 are automatically satisfied since ๐œŒ โ‰ฅ 0. Now, denoting (๐‘™+ , ๐‘™โˆ’ ) = (21, 2) and (๐‘ก+ , ๐‘กโˆ’ ) = (21 โˆ’ ๐œŒ, 2) we have ๐‘™+ + ๐‘™โˆ’ โˆ’ ๐‘ก+ โˆ’ ๐‘กโˆ’ = 21 + 2 โˆ’ (21 โˆ’ ๐œŒ) โˆ’ 2 = ๐œŒ. Next the conditions 3 and 4 of Corollary 4.2.20 are satisfied if ๐œŒ โ‰ฅ ๐‘™ ( ๐ด๐‘‡ ) and ๐œŒ > ๐‘™ ( ๐ด๐‘‡2 ) + 2. Note that ๐‘‡ is a free Zโˆ’module so ๐‘‡  Z23โˆ’๐œŒ . So รŠ ๐‘‡ โˆ— = Hom(๐‘‡, Z)  Hom(Z23โˆ’๐œŒ , Z)  Hom(Z, Z)  Z23โˆ’๐œŒ . 23โˆ’๐œŒ So ๐‘™ (๐‘‡ โˆ— ) = 23 โˆ’ ๐œŒ. Hence the quotient ๐ด ๐ฟ = ๐ฟ โˆ— /๐ฟ satisfies ๐‘™ ( ๐ด ๐ฟ ) โ‰ค 23 โˆ’ ๐œŒ since the image of the generators of ๐ฟ โˆ— under the quotient map form a (not necessarily minimal) generating set of ๐ด ๐ฟ . Now, ๐œŒ โ‰ฅ 23 โˆ’ ๐œŒ โ‰ฅ ๐‘™ ( ๐ด๐‘‡ ) for ๐œŒ > 12, so condition 3 is satisfied for ๐œŒ > 12. For condition 4, note that ๐‘™ ( ๐ด๐‘‡ ) = max ๐‘ ๐‘™ ( ๐ด๐‘‡๐‘ ). So, ๐œŒ > 25 โˆ’ ๐œŒ โ‰ฅ ๐‘™ ( ๐ด๐‘‡ ) + 2 โ‰ฅ ๐‘™ ( ๐ด๐‘‡2 ) + 2 for ๐œŒ โ‰ฅ 13. Therefore, all conditions of 4.2.20 are satisfied for 13 โ‰ค ๐œŒ โ‰ค 21 and the required embedding ๐‘‡ โ†ฉ! ฮ›๐ถ4 exists. โ–ก We use this result in Section 4.4. 4.3 Period Maps In this section, we will first recall the period map for ๐พ3 surfaces. In particular, it is well known that the period map is surjective. A similar result for cubic fourfolds is false. In particular the works of Voisin, Laza, and Looฤณenga shows that, unlike the situation for ๐พ3 surfaces, the period map for cubic fourfolds is not surjective. However, we know the exact image of the map. Definition 4.3.1. Define the moduli space of marked ๐พ3 surfaces to be ๐‘ B {(๐‘†, ๐œ‘)}/โˆผ where ๐‘† is a ๐พ3 surface and ๐œ‘ is a marking of ๐‘†, i.e. an isomorphism of lattices ๐œ‘ : ๐ป 2 (๐‘†, Z) ! ฮ›๐พ3 . 56 Two marked ๐พ3 surfaces (๐‘†1 , ๐œ‘1 ) and (๐‘†2 , ๐œ‘2 ) are considered to be equivalent under โˆผ if there is an isomorphism ๐‘“ : ๐‘†1 ! ๐‘†2 preserving the marking. Definition 4.3.2. Consider the Cโˆ’linear extension of the bilinear form โŸจ , โŸฉ on the complex vector space ฮ›๐พ3,C B ฮ›๐พ3 โŠ—Z C. The period domain associated with ฮ›๐พ3 is defined to be ๐ท B {๐‘ฅ โˆˆ P(ฮ›๐พ3,C ) | โŸจ๐‘ฅ, ๐‘ฅโŸฉ = 0, โŸจ๐‘ฅ, ๐‘ฅโŸฉ ยฏ > 0} The period domain ๐ท is naturally related to Hodge structures of ๐พ3โˆ’type by the following observation. Proposition 4.3.3. There exists a bฤณection between ๐ท and the set of Hodge structures of ๐พ3โˆ’type on ฮ›๐พ3 such that for all non-zero (2, 0)โˆ’classes ๐œŽ: 1. โŸจ๐œŽ, ๐œŽโŸฉ = 0 2. โŸจ๐œŽ, ๐œŽโŸฉ ยฏ >0 3. ๐œŽ is perpendicular to ฮ›1,1๐พ3 . Proof. See [Huy16, Proposition 6.1.2]. โ–ก Definition 4.3.4. The period map for ๐พ3 surfaces is the map ๐‘ ! ๐ท defined by sending a marked ๐œ‘ ๐พ3 surface (๐‘†, ๐œ‘) to the complex line ๐ป 0 (๐‘†, ฮฉ2 ) โŠ‚ ๐ป 2 (๐‘†, C)  ฮ›๐พ3,C . By the above proposition, this line determines a point in the period domain ๐ท. Now, we state the Torelli theorem for ๐พ3 surfaces, which implies that a ๐พ3 surface is determined by its Hodge structure. Theorem 4.3.5. The period map for ๐พ3 surfaces ๐‘ ! ๐ท is surjective. Proof. See [Tod80, Theorem 1]. โ–ก With the period map for ๐พ3 surfaces as motivation, we also consider the period map for cubic fourfolds. We have seen previously that the Hodge structure of cohomology of cubic fourfolds is similar to that of ๐พ3 surfaces, so it is reasonable to expect that some similar results may hold. Definition 4.3.6. Define the period domain for cubic fourfolds as D = {๐œ” โˆˆ P(ฮ›๐ถ4,0 โŠ—Z C) | โŸจ๐œ”, ๐œ”โŸฉ = 0, โŸจ๐œ”, ๐œ”โŸฉ ยฏ < 0}0 57 where the final subscript denotes a choice of a connected component. Let M0 denote the space of marked, smooth cubic fourfolds where a marking means an isomorphism ๐ป 4 (๐‘‹, Z)  ฮ›๐ถ4 . Then the period map is given by M0 ! D/ฮ“ by sending ๐‘‹ to ๐ป 1 (๐‘‹, ฮฉ3 ). Note that here ฮ“ represents the monodromy group. Proposition 4.3.7. The period map for cubic fourfolds is injective. Proof. See [Voi86]. โ–ก The map is not surjective, we now discuss the image of the period map for cubic fourfolds. We have the following definitions from [Laz10]. Definition 4.3.8. 1. Let ฮ›๐ถ4 denote the cubic fourfold lattice and fix โ„Ž โˆˆ ฮ›๐ถ4 such that โ„Ž2 = 3. For any rank two sublattice ๐‘€ โ†ฉ! ฮ›๐ถ4 primitively embedded in ฮ›๐ถ4 with โ„Ž โˆˆ ๐‘€, define a hyperplane D ๐‘€ to be D ๐‘€ = {๐œ” โˆˆ D | ๐œ” โŠฅ ๐‘€ }. Note that D ๐‘€ is the restriction of a hyperplane ๐ป ๐‘€ โŠ‚ P(ฮ›๐ถ4,0 โŠ—Z C) to D. 2. We say D ๐‘€ is a hyperplane of determinant ๐‘‘ = det(๐‘€). 3. The hyperplanes of a given determinant form an arrangement of hyperplanes in the period domain D and the arrangements of determinant 2 and 6 are denoted by H2 and H6 . Remark 4.3.9. The motivation for considering H2 and H6 comes from [Has00], where Hassett notes that the period map for cubic fourfolds misses this hyperplane arrangement. Laza and Looฤณenga showed that in fact, the period map is surjective onto the complement of this arrangement. The image of this arrangement in D/ฮ“ coincides with the image of the union of two hypersurfaces H2 โˆช H6 by [Has00, Proposition 3.2.2, Proposition 3.2.4]. Theorem 4.3.10. The image of the period map for cubic fourfolds is the complement of the hyperplane arrangements H2 โˆช H6 . Proof. See [Laz10, Theorem 1.1]. โ–ก We use this result in the next section. We now discuss endomorphism algebras of Hodge structures of cubic fourfolds. 58 4.4 Endomorphism Algebras of Cubic Fourfolds In this section, we prove the result for the existence of cubic fourfolds ๐‘‹ with End(๐‘‡ (๐‘‹)) of totally real type. Lemma 4.4.1. Let (๐‘‰, โ„Ž, ๐œ“) be a polarized weight 2 Hodge structure with ๐‘ฃ 2,0 = 1 and denote ๐‘‰ 2,0 = C๐œ” for an element ๐œ” โˆˆ ๐‘‰C . Then (๐‘‰, โ„Ž, ๐œ“) is an irreducible Hodge structure of ๐พ3-type if and only if ๐œ“C (๐œ”, ๐‘ฃ) = 0 for ๐‘ฃ โˆˆ ๐‘‰ implies ๐‘ฃ = 0. Proof. See [VG08, Lemma 1.7]. โ–ก In Proposition 4.2.21, we showed that there exists a primitive embedding of an even lattice ๐‘‡ into ฮ›๐ถ4 under certain conditions. We interpret this in geometric situations for cubic fourfolds. In the following statement, let D๐‘‡ denote the period domain associated to ๐‘‡C , i.e. D๐‘‡ = {๐œ” โˆˆ P(๐‘‡ โŠ—Z C) | โŸจ๐œ”, ๐œ”โŸฉ = 0, โŸจ๐œ”, ๐œ”โŸฉ ยฏ < 0} โŠ‚ P(๐‘‡C ). Additionally, let H2 , H6 โŠ‚ Dฮ›๐ถ4,0 โŠ‚ P(ฮ›๐ถ4,0,C ) be the (open subsets of) hyperplanes as in Definition 4.3.8. Note that if ๐‘‡C โŠ‚ ฮ›๐ถ4,0,C , then D๐‘‡ = Dฮ›๐ถ4,0 โˆฉ P(๐‘‡C ). Theorem 4.4.2. Let ๐‘‡ be an even lattice of signature (21 โˆ’ ๐œŒ, 2) with 13 โ‰ค ๐œŒ โ‰ค 21. Then there exists a primitive embedding ๐‘– : ๐‘‡ โ†ฉ! ฮ›๐ถ4 . Further, assume that D๐‘‡ is not contained in (H2 โˆฉ P(๐‘‡C )) โˆช (H6 โˆฉ P(๐‘‡C )). Then there exists cubic fourfold ๐‘‹ such that under this embedding ๐‘‡ (๐‘‹)  ๐‘‡. Proof. By Proposition 4.2.21, there exists a primitive embedding ๐‘‡ โ†ฉ! ฮ›๐ถ4 . Let ๐‘ = ๐‘‡ โŠฅ . To define a Hodge structure on ๐‘‡, we need to choose a period ๐œ” โˆˆ ๐‘‡C so that the Hodge structure on ๐‘‡ is of ๐พ3-type and such that ๐œ” is in the image of the period map for cubic fourfolds. This will then show that ๐‘‡ can be realized as the transcendental lattice of a cubic fourfold ๐‘‹. Since ๐‘‡ is a lattice, it comes equipped with a bilinear form ๐œ“. Our goal is to a find period in D๐‘‡ so that (๐‘‡, ๐œ”, ๐œ“) is a polarized Hodge structure of weight 2. Moreover, by Lemma 4.4.1, the Hodge structure corresponding to a period ๐œ” โˆˆ D๐‘‡ is an irreducible Hodge structure if ๐œ“C (๐œ”, ๐‘ก) โ‰  0 for 59 all nonzero ๐‘ก โˆˆ ๐‘‡. For any nonzero ๐‘ก โˆˆ ๐‘‡, define a hyperplane ๐ป๐‘ก := {[๐œ”] | ๐œ“C (๐œ”, ๐‘ก) = 0} โŠ‚ P(๐‘‡C ). Now since the period domain D๐‘‡ associated to ๐‘‡C is a nonempty open subset in a quadric in P(๐‘‡C ), we have that ๐ป๐‘ก โˆฉ D๐‘‡ is of codimension โ‰ฅ 1 in D๐‘‡ by [VG08, Lemma 3.2]. This ร implies that D๐‘‡ โ‰  ๐‘ก (๐ป๐‘ก โˆฉ D๐‘‡ ). This together with our hypothesis on D๐‘‡ implies that D๐‘‡ โ‰  ร ๐‘ก (๐ป๐‘ก โˆฉ D๐‘‡ ) โˆช (H2 โˆฉ D๐‘‡ ) โˆช (H6 โˆฉ D๐‘‡ ). Choosing ๐œ” corresponding to a point in the complement gives ๐œ” โˆˆ ๐‘‡C โŠ‚ ฮ›๐ถ4,0,C . Since ๐œ” is in the image of the period map, this defines a cubic fourfold ๐‘‹ with ๐ป 4 (๐‘‹, Z)  ฮ›๐ถ4 . Now the transcendental lattice ๐‘‡ (๐‘‹) and the lattice ๐‘‡ are both irreducible Hodge substructures of ๐พ3 type in ๐ป 4 (๐‘‹, Z)(1) with nontrivial intersection. This follows by Proposition 4.1.2 and by our hypothesis on ๐‘‡. By definition of irreducibility of Hodge structures, this implies that ๐‘‡ = ๐‘‡ (๐‘‹). This finishes the proof. โ–ก Remark 4.4.3. A similar embedding result to the above is shown in [May11, Corollary 5.2] in terms of roots of lattices. We use the following lemma in what comes next: Lemma 4.4.4. Let ๐น be a totally real field. Then for any integer ๐‘š โ‰ฅ 3 there exist rational Hodge structures of ๐พ3-type (๐‘‰, ๐œ”, ๐œ“) with End๐ป๐‘œ๐‘‘ (๐‘‰) = ๐น and ๐‘‘๐‘–๐‘š ๐น ๐‘‰ = ๐‘š. Proof. See [VG08, Lemma 3.2]. โ–ก Remark 4.4.5. The signature of ๐œ“ that is constructed in the proof of [VG08, Lemma 3.2] is given by (๐‘š [๐น : Q] โˆ’ 2, 2). We have the following proposition towards proving the existence of cubic fourfolds ๐‘‹ with End(๐‘‡ (๐‘‹)) a totally real field: Proposition 4.4.6. Let ๐น be a totally real field, ๐‘š โ‰ฅ 3 an integer, and (๐‘‰, ๐œ”, ๐œ“) a rational ๐พ3- type Hodge structure of signature (๐‘š [๐น : Q] โˆ’ 2, 2) with End๐ป๐‘œ๐‘‘ (๐‘‰) = ๐น and ๐‘‘๐‘–๐‘š ๐น ๐‘‰ = ๐‘š. If 3 โ‰ค ๐‘š [๐น : Q] โ‰ค 10, then ๐‘‰C โŠ‚ ฮ›๐ถ4,0,C . Further, if ๐œ” โˆ‰ (H2 โˆฉ P(๐‘‰C )) โˆช (H6 โˆฉ P(๐‘‰C )), then there exists a cubic fourfold ๐‘‹ such that 60 End(๐‘‡ (๐‘‹)) = ๐น and dim๐น (๐‘‡ (๐‘‹)) = ๐‘š. Proof. Choose a free Z-module ๐‘‡ โŠ‚ ๐‘‰ of rank dimQ ๐‘‰ such that ๐œ“ is integer valued on ๐‘‡ ร—๐‘‡. This is possible: Otherwise, we can choose a new ๐‘‡ by multiplying the basis elements by appropriate scalars to clear denominators. Now, apply Proposition 4.2.21 to get a primitive embedding ๐‘‡ โ†ฉ! ฮ›๐ถ4,0 . This gives ๐‘‡C = ๐‘‰C โ†ฉ! ฮ›๐ถ4,0,C . The hypothesis of Proposition 4.2.21 are satisfied since the signature of ๐œ“ is given by (๐‘š [๐น : Q] โˆ’ 2, 2). Since the corresponding period ๐œ” โˆˆ ๐‘‡C is in the image of the period map, apply Theorem 4.4.2 to find a cubic fourfold ๐‘‹ such that ๐‘‡ (๐‘‹)  ๐‘‡. Such an ๐‘‹ gives the result, i.e. End(๐‘‡ (๐‘‹)) = ๐น. โ–ก The above proposition illustrates the difficulty in proving an analogue of Van Geemenโ€™s Propo- sition 4.1.11 for cubic fourfolds. In particular, ๐œ” may be contained in (H2 โˆฉ P(๐‘‰C )) โˆช (H6 โˆฉ P(๐‘‰C )). If this happens, then ๐œ” is not in the image of the period map. However, we show below that there exist examples of cubic fourfolds ๐‘‹ with End(๐‘‡ (๐‘‹)) of totally real type, though we do not have any control on ๐น. Lemma 4.4.7. Let ๐‘‡ be an irreducible, rational Hodge structure of ๐พ3โˆ’type. If dimQ (๐‘‡) is odd then ๐น = End(๐‘‡) is totally real. Proof. The proof is taken from [Huy16, Remark 3.3.14]. We have that dim๐‘„ (๐‘‡) = dim๐น (๐‘‡) ยท [๐น : Q]. So, [๐น : Q] divides dim๐‘„ (๐‘‡). But [๐น : Q] is even for any CM-field. Therefore, if dimQ (๐‘‡) is odd, then ๐น is totally real. โ–ก Proposition 4.4.8. For a cubic fourfold ๐‘‹ with even Picard rank ๐œŒ(๐‘‹), End(๐‘‡ (๐‘‹)) is of totally real type. Proof. The proof is immediate: Since dimQ (๐‘‡ (๐‘‹)) = 23 โˆ’ ๐œŒ(๐‘‹), we can see that End(๐‘‡ (๐‘‹)) is totally real by Lemma 4.4.7. โ–ก Corollary 4.4.9. There exist cubic fourfolds for any even Picard rank 2 โ‰ค ๐œŒ โ‰ค 20 with End(๐‘‡ (๐‘‹)) a totally real field. 61 Proof. By [Laz21, Section 1], there exists cubic fourfolds of any Picard rank 1 โ‰ค ๐œŒ(๐‘‹) โ‰ค 21. Apply Proposition 4.4.8 to complete the proof. โ–ก 4.5 Endomorphism Algebras and Associated K3 Surfaces Now that we have shown that cubic fourfolds of real type exist, we relate endomorphism algebras of cubic fourfolds to those of associated ๐พ3 surfaces when this makes sense and to the Kuga-Satake construction. We now know that there exist ๐พ3 surfaces and cubic fourfolds that are of real type. However, it is a priori unclear if there exists associated ๐พ3 surfaces such that the ๐พ3 surface and the cubic fourfold are both of real type. We prove that such examples do exist. Lemma 4.5.1. If a ๐พ3 surface ๐‘† is associated to a cubic fourfold ๐‘‹, then either both End(๐‘‡ (๐‘†)) and End(๐‘‡ (๐‘‹)) are CM fields or both are totally real fields. Additionally, the Picard ranks are related by ๐œŒ(๐‘‹) = ๐œŒ(๐‘†) + 1. Proof. By definition, ๐‘† being associated to ๐‘‹ means that there is a Hodge isometry ๐‘” : ๐พ โŠฅ (1) ! ๐ป 2 (๐‘†, Z)0 for some labeling ๐พ (1) โŠ‚ ๐ป 4 (๐‘‹, ๐‘)(1). In particular, note that ๐‘‡ (๐‘‹) โŠ‚ ๐พ โŠฅ and ๐‘‡ (๐‘†) โŠ‚ ๐ป 2 (๐‘†, Z)0 . Since the map ๐‘” is a Hodge isometry, it must respect both the Hodge structures and the lattice structures. So, we have that ๐‘‡ (๐‘‹) 2,0 = ๐พ โŠฅ (1) 2,0  ๐ป 2 (๐‘†, Z)02,0 = ๐‘‡ (๐‘†) 2,0 . So, ๐‘‡ (๐‘‹) โŠ‚ ๐พ โŠฅ (1)  ๐ป 2 (๐‘†, Z)0 with ๐‘‡ (๐‘‹) 2,0 = ๐‘‡ (๐‘†) 2,0 . By irreducibility of ๐‘‡ (๐‘†) and ๐‘‡ (๐‘‹) from Proposition 4.1.2, we have that ๐‘‡ (๐‘‹)  ๐‘‡ (๐‘†). This gives the first part of the statement. This also implies that ๐œŒ(๐‘‹) = ๐œŒ(๐‘†) + 1 โ–ก We need an additional statement in order to prove the main result in the subsection. Theorem 4.5.2. There exists associated ๐พ3 surfaces of any Picard rank 1 โ‰ค ๐œŒ(๐‘†) โ‰ค 20. Proof. See [ABP20, Theorem 1.1]. The authors use intersections of Hassett divisors to prove their result. โ–ก Now, we prove the following proposition. Proposition 4.5.3. There exists ๐พ3 surfaces ๐‘† associated to cubic fourfolds ๐‘‹ such that End(๐‘‡ (๐‘†)) and End(๐‘‡ (๐‘‹)) are both totally real. 62 Proof. Choose a cubic fourfold ๐‘‹ and associated ๐พ3 surface ๐‘† such that the Picard rank ๐œŒ(๐‘‹) of ๐‘‹ is even. This is possible by Theorem 4.5.2. Now, by Corollary 4.4.9 we have that End(๐‘‡ (๐‘‹)) is totally real. By Lemma 4.5.1, End(๐‘‡ (๐‘†)) must be totally real as well. This concludes the proof. โ–ก The remainder of this section is motivated by [Sch10]. In that paper, the author proves the following result for ๐พ3 surfaces. Theorem 4.5.4. Let ๐‘† be a ๐พ3 surface and let End(๐‘‡ (๐‘†)) be totally real of degree ๐‘‘ over Q. Let ๐พ๐‘†(๐‘†) denote the Kuga-Satake variety of ๐‘†. Then there exists another abelian variety ๐ต such that ๐‘‘โˆ’1 ๐พ๐‘†(๐‘‡ (๐‘†)) โˆผ ๐ต2 and the endomorphism algebra of ๐ต can be described by EndQ (๐ต) = ๐ถ๐‘œ๐‘Ÿ๐‘’๐‘  ๐ธ/Q๐ถ + (ฮฆ) where ฮฆ : ๐‘‡ (๐‘†) ร— ๐‘‡ (๐‘†) ! End(๐‘‡ (๐‘†)) is the symmetric bilinear form from [Zar83], ๐ถ + is the associated even Clifford algebra, and ๐ถ๐‘œ๐‘Ÿ๐‘’๐‘  is the corestriction of algebras. Proof. See [Sch10, Theorem 1]. โ–ก Schlickewei uses this theorem to prove an interesting result, namely that the Hodge conjecture holds for ๐‘† ร— ๐‘† where ๐‘† is a ๐พ3 surface that is realized as a double cover of P2 ramified over six lines. Since the proof of Theorem 4.5.4 only requires that ๐‘‡ is an irreducible Hodge structure of ๐พ3โˆ’type and End(๐‘‡ (๐‘†)) is of totally real type (see Section 3.3 of [Sch10]), the same result holds for the Kuga-Satake varieties of cubic fourfolds. Using our result in Theorem 3.3.5, we have the following corollary. Corollary 4.5.5. Let ๐‘† be a ๐พ3 surface associated to a cubic fourfold ๐‘‹ such that End(๐‘‡ (๐‘†)) and End(๐‘‡ (๐‘‹)) are totally real of degree ๐‘‘ over Q. Then there exists another abelian variety ๐ต such ๐œŒ(๐‘‹)+๐‘‘โˆ’1 that ๐พ๐‘†(๐‘‹) โˆผ ๐ต2 and the endomorphism algebra of ๐ต can be described by EndQ (๐ต) = ๐ถ๐‘œ๐‘Ÿ๐‘’๐‘  ๐ธ/Q๐ถ + (ฮฆ) as in Theorem 4.5.4. 63 Proof. By Theorem 3.3.5, we have ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐‘†) 2 . We also saw in Chapter 3 that ๐พ๐‘†(๐‘†) โˆผ ๐œŒ(๐‘†) ๐พ๐‘†(๐‘‡ (๐‘†)) 2 . In Lemma 4.5.1, we showed that ๐œŒ(๐‘‹) = ๐œŒ(๐‘†) + 1. Putting this together, we have a string of isogenies: ๐œŒ(๐‘†)+1 ๐‘‘โˆ’1 ๐œŒ(๐‘‹) ๐œŒ(๐‘‹)+๐‘‘โˆ’1 ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐‘†) 2 โˆผ (๐พ๐‘†(๐‘‡ (๐‘†))) 2 โˆผ (๐ต2 )2 โˆผ ๐ต2 . The description of the endomorphism algebra of ๐ต is direct from Theorem 4.5.4. โ–ก 64 CHAPTER 5 DERIVED CATEGORIES AND FUTURE WORK In this chapter, we explore Kuznetsovโ€™s derived categorical approach to studying cubic fourfolds. In particular, we work towards trying to understand the relationship between ๐ท ๐‘ (๐พ๐‘†(๐‘‹)) and ๐ท ๐‘ (๐พ๐‘†(๐‘†)) when ๐‘† is a ๐พ3 surface associated to ๐‘‹. The goal is to examine how the derived category of Kuga-Satake varieties may fit into the general framework. The chapter is outlined as follows: ยง5.1 We discuss the background material on derived categories with a particular focus on excep- tional objects, semiorthogonal decompositions, Fourier-Mukai transforms, and Hochschild homology. The section ends with an explicit calculation for ๐ท ๐‘ (๐‘†) of a ๐พ3 surface. ยง5.2 In this section, we outline Kuznetsovโ€™s approach to studying the derived categories of cubic fourfolds. In particular we define the Kuznetsov component A ๐‘‹ , prove facts about the derived category ๐ท ๐‘ (๐‘‹) of a cubic fourfold ๐‘‹ such as its Hochschild homology and semiorthogonal decompositions, and note similarities between Kuznetsovโ€™s approach using derived categories and Hassettโ€™s approach using Hodge theory. We end the section with a few important examples, including Pfaffian cubic fourfolds and cubic fourfolds containing a plane. ยง5.3 We discuss recent works which show that the derived categorical and Hodge theoretic ap- proaches to studying cubic fourfolds and associated ๐พ3 surfaces are the same. In particular, we mention the important results of [AT14] and [BLM+ 21]. These results, along with our result in Theorem 3.3.5 motivate our attempt to relate derived categories of Kuga-Satake varieties into the general framework. ยง5.4 In this section, we study the Kuga-Satake Hodge Conjecture, which is a special case of the standard Hodge conjecture. We use this conjecture to obtain an algebraic cycle used in the construction of Fourier-Mukai transforms. We prove that the Kuga-Satake Conjecture holds for at least some associated ๐พ3 surfaces. ยง5.5 We use the Kuga-Satake Hodge conjecture result of the previous section to construct Fourier- 65 Mukai transforms involving the derived category of the Kuga-Satake variety. ยง5.6 In our final section, we sketch potential future directions of research using the Fourier-Mukai functors. 5.1 Background: Derived Categories Throughout this section, T will denote a triangulated category. We assume basic familiarity with abelian and triangulated categories. For a reference, see [Huy06, Chapter 1]. Definition 5.1.1. Suppose T1 , . . . , T๐‘› is a sequence of full triangulated subcategories of T such that for all ๐‘– < ๐‘—, HomT (T๐‘— , T๐‘– ) = 0 and T1 , . . . , T๐‘› generate T . (That is, T is the smallest full triangulated subcategory containing all of T1 , . . . , T๐‘› .) Then we say T1 , . . . , T๐‘› is a semiorthogonal decomposition of T and we write T = โŸจT1 , . . . , T๐‘› โŸฉ. Much of our focus will be on semiorthogonal decompositions. A standard method for con- structing semiorthogonal decompositions is using exceptional collections. Definition 5.1.2. An object ๐ธ โˆˆ T is called exceptional if Hom(๐ธ, ๐ธ) = ๐‘˜ and Ext๐‘š (๐ธ, ๐ธ) = 0 for all ๐‘š โ‰  0. In addition, we call a collection of exceptional objects {๐ธ 1 , . . . , ๐ธ ๐‘› } an exceptional collection if Ext๐‘š (๐ธ๐‘– , ๐ธ ๐‘— ) = 0 for all ๐‘– > ๐‘—. Recall that ๐ท ๐‘ (๐‘‹) B ๐ท ๐‘ (Coh(๐‘‹)) denotes the bounded derived category of coherent sheaves on a variety ๐‘‹. The next proposition allows us to use exceptional collections to construct semiorthogonal decompositions of ๐ท ๐‘ (๐‘‹). Proposition 5.1.3. [BO95, Theorem 3.5] Let ๐‘‹ be a smooth projective variety and suppose {๐ธ 1 , . . . , ๐ธ ๐‘› } is an exceptional collection in ๐ท ๐‘ (๐‘‹). Define its orthogonal complement to be A B โŸจ๐ธ 1 , . . . , ๐ธ ๐‘› โŸฉ โŠฅ B ๐น โˆˆ ๐ท ๐‘ (๐‘‹) : Extโ€ข (๐ธ๐‘– , ๐น) = 0 โˆ€๐‘– = 1, . . . , ๐‘› .  Then there is a semiorthogonal decomposition: ๐ท ๐‘ (๐‘‹) = โŸจA, ๐ธ 1 , . . . , ๐ธ ๐‘› โŸฉ. We also consider maps between derived categories. The most natural way to do so is using Fourier-Mukai transforms, defined as follows. 66 Definition 5.1.4. Let ๐‘‹, ๐‘Œ be smooth projective varieties and let ๐พ โˆˆ ๐ท ๐‘ (๐‘‹ ร— ๐‘Œ ). Denote the two natural projections by ๐‘ 1 : ๐‘‹ ร— ๐‘Œ ! ๐‘‹ and ๐‘ 2 : ๐‘‹ ร— ๐‘Œ ! ๐‘Œ . Then the Fourier-Mukai transform with kernel ๐พ is defined to be ฮฆ๐พ : ๐ท ๐‘ (๐‘‹) โˆ’! ๐ท ๐‘ (๐‘Œ ) F 7โˆ’! ๐‘… ๐‘ 2 โˆ— (๐ฟ ๐‘ โˆ—1 F โŠ— ๐ฟ ๐พ) where ๐‘…โ€“โˆ— denotes the derived direct image, ๐ฟโ€“โˆ— denotes the derived pullback, and โŠ— ๐ฟ denotes the derived tensor product. In the following, we only consider Fourier-Mukai functors between derived categories. Example 5.1.5. We list a few examples of Fourier-Mukai kernels: โ€ข Consider the structure sheaf of the diagonal Oฮ” โˆˆ ๐ท ๐‘ (๐‘‹ ร— ๐‘‹). Then the Fourier-Mukai transform with kernel ๐พ = Oฮ” gives the identity on ๐ท ๐‘ (๐‘‹), since ๐‘… ๐‘ 2 โˆ— ( ๐‘ โˆ—1 F โŠ— ๐ฟ Oฮ” ) = F . So ฮฆOฮ”  id. โ€ข If ๐‘“ : ๐‘‹ ! ๐‘Œ is a morphism of varieties, consider the structure sheaf of the graph Oฮ“ ๐‘“ of ๐‘“ . Then the Fourier-Mukai transform with kernel ๐พ = Oฮ“ ๐‘“ is isomorphic to the derived pushforward ฮฆOฮ“ ๐‘“ : ๐ท ๐‘ (๐‘‹) ! ๐ท ๐‘ (๐‘Œ ). โ€ข If ๐ด is an abelian variety and ๐ดห† is its dual, then ๐ด ร— ๐ดห† comes equipped with a special line bundle called the Poincarรฉ bundle P, see [Mil86] for reference. The Fourier-Mukai functor ฮฆP : ๐ท ๐‘ ( ๐ด) ! ๐ท ๐‘ ( ๐ด) ห† gives an equivalence of categories. This is an important example since ๐ด and ๐ดห† are in general not isomorphic varieties, so it illustrates that non-isomorphic varieties may be derived equivalent. An important invariant of a triangulated category T is its Hochschild homology ๐ป๐ปโ€ข (T ). The following facts and additional background can be found in [Kuz09]. We are primarily interested in Hochschild homology of derived categories and their admissible subcategories. In this context, we use ๐ป๐ปโ€ข (๐‘‹) to denote ๐ป๐ปโ€ข (๐ท ๐‘ (Coh(๐‘‹)). It is an invariant in the sense that a derived equivalence ๐ท ๐‘ (๐‘‹)  ๐ท ๐‘ (๐‘Œ ) induces an isomorphism ๐ป๐ปโ€ข (๐‘‹)  ๐ป๐ปโ€ข (๐‘Œ ). Hochschild homology is directly related to Hodge theory through the Hochschild-Kostant- 67 Rosenberg Theorem, which is useful for explicitly describing ๐ป๐ปโ€ข (๐‘‹): Theorem 5.1.6 (HKR Isomorphism Theorem). Let ๐‘‹ be a smooth projective variety and let ๐ป ๐‘,๐‘ž (๐‘‹) denote its standard Hodge decomposition as in Chapter 2. Then there is an isomorphism of graded ๐‘˜โˆ’vector spaces รŠ ๐ป๐ป๐‘– (๐‘‹)  ๐ป ๐‘,๐‘ž (๐‘‹). ๐‘žโˆ’๐‘=๐‘– With this theorem we can compute Hochschild homology of many varieties of interest since we know their Hodge diamonds. For example, this applies to ๐พ3 surfaces, cubic fourfolds, abelian varieties, and Gushel-Mukai varieties. Hochschild homology also works nicely with semiorthogonal decompositions. Proposition 5.1.7. [Kuz09, Theorem 7.3] Let ๐‘‹ be a smooth projective variety and suppose there is a semiorthogonal decomposition of its derived category as ๐ท ๐‘ (๐‘‹) = โŸจA1 , . . . , A๐‘› โŸฉ. Then its Hochschild homology decomposes as a direct sum: รŠ ๐‘› ๐ป๐ปโ€ข (๐‘‹)  ๐ป๐ปโ€ข (A๐‘– ). ๐‘–=1 The following is also useful in combination with the above proposition: Lemma 5.1.8. [Kuz10, Lemma 2.5] If ๐ธ is an exceptional object in ๐ท ๐‘ (๐‘‹), then the sub- triangulated category โŸจ๐ธโŸฉ is equivalent to the derived category of a vector spaces ๐ท ๐‘ (๐‘˜). In particular, its Hochschild homology is ๐‘˜ [0]. Proof. The functor ๐ท ๐‘ (๐‘˜) ! ๐ท ๐‘ (๐‘‹), ๐‘‰ 7! ๐‘‰ โŠ— ๐ธ is fully faithful since ๐ธ is an exceptional object, so ๐ท ๐‘ (๐‘˜)  โŸจ๐ธโŸฉ. โ–ก We briefly mention yet another useful tool, called the Serre functor. Definition 5.1.9. Let ๐‘˜ be a field and T be a ๐‘˜-linear triangulated category. A Serre functor on T is an exact functor ST : ๐‘‡ ! ๐‘‡ such that Hom(F , ST (G))  Hom(G.F ) โˆง for all F , G โˆˆ T and the dual is taken in the category of ๐‘˜-vector spaces. 68 For a smooth projective variety ๐‘‹, there exists a Serre Functor given by S๐ท ๐‘ (๐‘‹) (F ) = F โŠ— ๐œ” ๐‘‹ [๐‘‘๐‘–๐‘š(๐‘‹)], where ๐œ” ๐‘‹ is the canonical sheaf of ๐‘‹. By the Serre Functor of ๐‘‹, we mean this functor. We apply these tools to a ๐พ3 surface now. Example 5.1.10. Let ๐‘† be a ๐พ3 surface. Recall that its Hodge diamond has the form: 1 0 0 1 20 1 0 0 1 We use the HKR Theorem to compute the Hochschild homology of ๐ท ๐‘ (๐‘†) by reading off the columns of the Hodge diamond as ๐ป๐ปโ€ข (๐‘†)  ๐‘˜ [2] โŠ• ๐‘˜ 22 [0] โŠ• ๐‘˜ [โˆ’2]. The canonical sheaf ๐œ” ๐‘† of a ๐พ3 surface is trivial, so its Serre functor is the shift by 2 functor, S๐ท ๐‘ (๐‘†)  [2]. 5.2 The Kuznetsov Viewpoint In Chapter 2, we saw a conjecture motivated by the framework of associated ๐พ3 surfaces via a Hodge theoretical approach: A cubic fourfold is rational if and only if it possesses an associated ๐พ3 surface. In [Kuz10], Kuznetsov provides an alternative approach to establishing a criterion for rationality via derived categories. We summarize the main ideas here. In the following, let ๐‘‹ denote a cubic fourfold and ๐ท ๐‘ (๐‘‹) its derived category of coherent sheaves. Consider the line bundles O ๐‘‹ , O ๐‘‹ (โˆ’1), O ๐‘‹ (โˆ’2) on ๐‘‹. Computing their cohomology is a direct application of the Kodaira Vanishing Theorem. Recall that the Kodaira Vanishing Theorem says that if L is an ample invertible sheaf on ๐‘‹, then ๐ป ๐‘– (๐‘‹, L โˆ’1 ) = 0 for ๐‘– โ‰ค 4. We have that 69 dim(๐ป 0 (๐‘‹, O ๐‘‹ )) = 1 and dim(๐ป ๐‘› (๐‘‹, O ๐‘‹ (๐‘˜))) = 0 for โˆ’ 2 โ‰ค ๐‘˜ โ‰ค 0, (๐‘›, ๐‘˜) โ‰  (0, 0). Now, the basic properties of Ext imply that {O ๐‘‹ , O ๐‘‹ (1), O ๐‘‹ (2)} form an exceptional collection in ๐ท ๐‘ (๐‘‹). Definition 5.2.1. Let ๐‘‹ be a cubic fourfold and consider the exceptional collection {O ๐‘‹ , O ๐‘‹ (1), O ๐‘‹ (2)} in ๐ท ๐‘ (๐‘‹) as shown above. Then the Kuznetsov component A ๐‘‹ of ๐‘‹ is the orthogonal complement of this exceptional collection: A ๐‘‹ := โŸจO ๐‘‹ , O ๐‘‹ (1), O ๐‘‹ (2)โŸฉ โŠฅ . The Kuznetsov component appears in the semiorthogonal decomposition: Lemma 5.2.2. Let ๐‘‹ be a cubic fourfold and let A ๐‘‹ be the Kuznetsov component as defined above. Then there is a semiorthogonal decomposition of ๐ท ๐‘ (๐‘‹) as ๐ท ๐‘ (๐‘‹) = โŸจA ๐‘‹ , O ๐‘‹ , O ๐‘‹ (1), O ๐‘‹ (2)โŸฉ . Proof. Apply Proposition 5.1.3. โ–ก The sub-category A ๐‘‹ has interesting properties. We compute its Hochschild homology as follows. Example 5.2.3. We use the HKR Theorem and the Hodge diamond of the cubic fourfold ๐‘‹, reproduced here for reference: 70 1 0 0 0 1 0 0 0 0 0 0 1 21 1 0 0 0 0 0 0 1 0 0 0 1 Recall by Lemma 5.1.7 that รŠ 2 รŠ ๐ป๐ปโ€ข (๐‘‹)  ( ๐ป๐ปโ€ข (O ๐‘‹ (๐‘–))) ๐ป๐ปโ€ข (A ๐‘‹ ). ๐‘–=0 By Lemma 5.1.8, {O ๐‘‹ , O ๐‘‹ (1), O ๐‘‹ (2)} form an exceptional collection, so each of them has Hochschild homology  ๐‘˜ [0]. Therefore, using the above decomposition and tallying the middle column of the Hodge diamond via the HKR Theorem, we have that ๐ป๐ป0 (๐‘‹)  ๐‘˜ 25 , and therefore ๐ป๐ป0 (A ๐‘‹ )  ๐‘˜ 22 . The exceptional objects do not contribute to Hochschild homology outside of the degree 0 part, so we have that ๐ป๐ปโ€ข (A ๐‘‹ )  ๐‘˜ [2] โŠ• ๐‘˜ 22 [0] โŠ• ๐‘˜ [โˆ’2]. It is also possible to prove that the Serre functor on A ๐‘‹ is shifting by 2, i.e. SA ๐‘‹  [2]. Comparing with Example 5.1.10, the resemblance between the triangulated category A ๐‘‹ and the derived category ๐ท ๐‘ (๐‘†) for a ๐พ3 surface ๐‘† is striking! This observation leads to Kuznetsovโ€™s conjecture in [Kuz10, Conjecture 1.1]. 71 Conjecture 5.2.4. A cubic fourfold ๐‘‹ is rational if and only if the triangulated subcategory A ๐‘‹ โŠ‚ ๐ท ๐‘ (๐‘‹) is equivalent to the derived category of a ๐พ3 surface, i.e. A ๐‘‹  ๐ท ๐‘ (๐‘†) for some ๐พ3 surface ๐‘†. There are several known examples of rational cubic fourfolds ๐‘‹ such that A ๐‘‹  ๐ท ๐‘ (๐‘†) for a ๐พ3 surface ๐‘†. We mention two such well known examples here (and compare to the examples seen in Chapter 2). Pfaffian cubics provide the most explicit examples where the Kuznetsov component of ๐‘‹ is realized as the derived category of a ๐พ3 surface. The following is proved in [Kuz06]: Theorem 5.2.5. Let ๐‘‹ be a Pfaffian cubic fourfold. Then there exists a smooth ๐พ3 surface ๐‘† such that A ๐‘‹  ๐ท ๐‘ (๐‘†). Let ๐‘Š be a 6โˆ’dimensional vector space and consider P(โˆง2๐‘Š โˆ— ), the space of skew forms on ๐‘Š. Define the Pfaffian space as Pf(๐‘Š) B ๐œ” โˆˆ P(โˆง2๐‘Š โˆ— ) | ๐œ” is degenerate .  Recall that a Pfaffian cubic fourfold is given by ๐‘‹ = P(๐‘‰) โˆฉ Pf(๐‘Š โˆ— ) for some 6โˆ’dimensional vector space ๐‘Š and 6โˆ’dimensional vector subspace ๐‘‰ โŠ‚ โˆง2๐‘Š โˆ— . In [Kuz10, Section 3], Kuznetsov constructs the ๐พ3 surface explicitly as the zero locus of a global section on ๐‘‰ โˆ— โŠ— OGr(2,๐‘Š) (1) on the Grassmanian Gr(2, ๐‘Š). Here OGr(2,๐‘Š) (1) is the very ample class corresponding to the Plรผcker embedding. In other words, ๐‘† = ๐‘ (๐‘ ) โŠ‚ Gr(2, ๐‘Š) for a global section ๐‘ . Before discussing cubic fourfolds that contain a plane, we take a brief detour into twisted ๐พ3 surfaces. We recall the basic terminology here and refer the reader to [Huy16, Chapter 18] for more detailed background. Definition 5.2.6. Let ๐‘‹ be any scheme. 1. An Azumaya algebra A over ๐‘‹ is defined to be an O ๐‘‹ โˆ’algebra that is coherent as an O ๐‘‹ โˆ’module and รฉtale locally isomorphic to ๐‘€๐‘› (O ๐‘‹ ). 2. An Azumaya algebra A is called trivial if A  End(๐ธ) for a locally free sheaf ๐ธ. 72 3. Two Azumaya algebras are defined to be equivalent, i.e. A1 โˆผ A2 if there exists locally free sheaves E1 , E2 such that A1 โŠ— End(E1 )  A2 โŠ— End(E2 ). 4. The Brauer group Br(๐‘‹) is defined to be the set of equivalence classes of Azumaya algebras on ๐‘‹ where the equivalence relation is given above and the group operation is tensor product โŠ—: Br(๐‘‹) B {Azumaya algebras A}/โˆผ . With the definition of a Brauer class, we can define a twisted ๐พ3 surface as follows. Definition 5.2.7. A twisted ๐พ3 surface is a pair (๐‘†, ๐›ผ) where ๐›ผ โˆˆ Br(๐‘†). Now, let ๐‘‹ be a cubic fourfold containing a plane ๐‘ƒ. Then Kuznetsov relates the theory of such cubic fourfolds to twisted ๐พ3 surfaces. Theorem 5.2.8. [Kuz10, Theorem 4.3] If ๐‘‹ is a cubic fourfold containing a plane, then its Kuznetsov component can be realized as the derived category of a twisted ๐พ3 surface. That is, there exists a ๐พ3 surface ๐‘† and a Brauer class ๐›ฝ โˆˆ Br(๐‘†) such that A ๐‘‹  ๐ท ๐‘ (๐‘†, ๐›ฝ). In particular, to prove that the Kuznetsov component of a cubic fourfold containing a plane is equivalent to the derived category of a ๐พ3 surface, it is sufficient to prove that the Brauer class ๐›ฝ that appears in the above theorem is trivial. Example 5.2.9. Let ๐‘‹ be a cubic fourfold containing a plane ๐‘ƒ and a projective surface ๐‘‡ not homologous to ๐‘ƒ. Consider the intersection index given by ๐›ฟ(๐‘‡) B deg(๐‘‡) โˆ’ โŸจ๐‘ƒ, ๐‘‡โŸฉ. We saw in Chapter 2 that such a cubic fourfold is rational if ๐›ฟ(๐‘‡) is odd by a result of Hassett. Furthermore, such examples exist by [Has99]. Kuznetsov shows in [Kuz10, Proposition 4.7] that such a cubic fourfold with odd intersection index has trivial Azumaya class ๐›ฝ. Therefore, this provides an example of a rational cubic fourfold such that A ๐‘‹  ๐ท ๐‘ (๐‘†). The above examples illustrate why one may expect there to be a direct relationship between Kuznetsovโ€™s conjecture for the rationality of cubic fourfolds and the conjecture using Hassettโ€™s 73 Hodge theoretical approach which we will explore in the next section. It is worth mentioning that there is an analogous derived categorical approach for Gushel-Mukai fourfolds as well. As we explored in Section 3.4, there is a notion of associated ๐พ3 surface for Gushel-Mukai fourfolds ๐‘Œ . As in the case of cubic fourfolds, we can define a subcategory A๐‘Œ โŠ‚ ๐ท ๐‘ (๐‘Œ ) and it is conjectured that a Gushel-Mukai fourfold is rational if and only if A๐‘Œ  ๐ท ๐‘ (๐‘†) for a ๐พ3 surface ๐‘†. See [KP18] for more details. 5.3 Connecting Hodge Theory and Derived Categories In [AT14], Addington and Thomas showed that Kuznetsovโ€™s derived category criterion for ratio- nality coincides with Hassetโ€™s associated ๐พ3 notion. This is summarized in [AT14, Theorem 1.1]. Theorem 5.3.1. If A ๐‘‹  ๐ท ๐‘ (๐‘†) for some ๐พ3 surface ๐‘†, then ๐‘‹ โˆˆ C๐‘‘ for some admissible value of ๐‘‘. Conversely, for each admissible value of ๐‘‘, the set of cubics ๐‘‹ โˆˆ C๐‘‘ satisfying A ๐‘‹  ๐ท ๐‘ (๐‘†) for some ๐พ3 surface ๐‘† is a Zariski open dense subset of C๐‘‘ . In light of the above theorem, Kuznetsovโ€™s conjecture would therefore imply that generic ๐‘‹ โˆˆ C๐‘‘ for admissible ๐‘‘ is rational. The proof is involved, but we mention some ingredients of their strategy here since the approach uses theory relevant to our study. The authors begin by defining a weight two Hodge structure on the Kuznetsov component A ๐‘‹ . They show that there is a Hodge isometry with the usual Mukai lattice ๐ป โˆ— (๐‘†, Z) when A ๐‘‹  ๐ท ๐‘ (๐‘†), and interpret the relationship between this lattice and ๐ป 4 (๐‘‹, Z). Next, they consider the intersection C๐‘‘ โˆฉ C8 for admissible ๐‘‘, which they show to be a non-empty intersection. The motivation for considering C8 comes from Theorem 5.2.8: Since every cubic fourfold in C8 contains a plane by [Voi86], we already know that A ๐‘‹  ๐ท ๐‘ (๐‘†, ๐›ผ) for some twisted ๐พ3 surface for every ๐‘‹ โˆˆ C8 . They then use lattice theory to show that there must be cubic fourfolds ๐‘‹ โˆˆ C๐‘‘ โˆฉ C8 where ๐›ผ is trivial, hence A ๐‘‹  ๐ท ๐‘ (๐‘†) for a ๐พ3 surface ๐‘† for these cubic fourfolds. Finally they use Deformation theory to show that A ๐‘‹  ๐ท ๐‘ (๐‘†) for a Zariski open subset of C๐‘‘ . The authors mention in [AT14, Section 7.4] that it may be possible to use stability conditions of some kind to pass from a Zariski open subset of C๐‘‘ to all of C๐‘‘ . Indeed, using this approach the 74 authors in [BLM+ 21, Part VI] show that "generic" can be replaced with "every". In other words, the two notions of associated ๐พ3 surfaces entirely coincide. Theorem 5.3.2. [BLM+ 21, Corollary 29.7] Let ๐‘‹ be a cubic fourfold. Then ๐‘‹ has an associated ๐พ3 surface in the sense of Hassett if and only if there exists a smooth projective ๐พ3 surface ๐‘† such that A ๐‘‹  ๐ท ๐‘ (๐‘†) in the sense of Kuznetsov. Addington and Thomas proved another interesting result about how the Hodge isometry that realizes a ๐พ3 surface as an associated ๐พ3 surface is induced by a certain algebraic cycle. Theorem 5.3.3. [AT14, Theorem 1.2] Let ๐‘‹ โˆˆ C๐‘‘ be a special cubic fourfold of admissible discriminant. Then there exists a polarized ๐พ3 surface ๐‘† of degree ๐‘‘ and an algebraic cycle ๐‘ โˆˆ ๐ด3 (๐‘‹ ร— ๐‘†)Q that realizes the Hodge isometry ๐พ โŠฅ (1)Q ! ๐ป 2 (๐‘†, Q)0 . In other words, the associated condition can be realized by an algebraic cycle on ๐‘‹ ร— ๐‘†. In the following, we proceed in a similar fashion for the Kuga-Satake construction. This motivates the next section. 5.4 Kuga-Satake Hodge Conjecture Our initial goal is to construct a functor ๐ท ๐‘ (๐‘†) ! ๐ท ๐‘ (๐พ (๐‘†) 2 ). Of course, the natural approach is to construct a Fourier-Mukai transform using a kernel ๐พ โˆˆ ๐ท ๐‘ (๐‘† ร— ๐พ๐‘†(๐‘†) 2 ). Our idea is to use an algebraic cycle ๐‘ โŠ‚ ๐‘† ร— ๐พ๐‘†(๐‘†) 2 in the kernel ๐พ. The Fourier-Mukai transform is most meaningful when the class ๐‘ depends on the Kuga-Satake construction itself. This brings us to the Kuga-Satake Hodge Conjecture. By the Kuga-Satake construction, there is an inclusion ๐ป 2 (๐‘†, Q) โŠ‚ ๐ป 1 (๐พ๐‘†(๐‘†), Q) โŠ— ๐ป 1 (๐พ๐‘†(๐‘†), Q). By application of Kรผnneth formula on ๐ป 2 (๐พ๐‘†(๐‘†) ร— ๐พ๐‘†(๐‘†), Q) we have that ๐ป 1 (๐พ๐‘†(๐‘†), Q) โŠ— ๐ป 1 (๐พ๐‘†(๐‘†), Q) โŠ‚ ๐ป 2 (๐พ๐‘†(๐‘†) ร— ๐พ๐‘†(๐‘†), Q). Composing these inclusions gives us an injective map ๐‘– : ๐ป 2 (๐‘†, Q) โ†ฉ! ๐ป 2 (๐พ๐‘†(๐‘†) 2 , Q)). 75 Recall that by the Kรผnneth formula, there is a chain of isomorphisms รŠ ๐ป 4 (๐‘† ร— ๐พ๐‘†(๐‘†) 2 , Q)  ๐ป ๐‘™ (๐‘†, Q) โŠ— ๐ป ๐‘š (๐พ๐‘†(๐‘†) 2 , Q) ๐‘™+๐‘š=4 รŠ  Hom(๐ป 4โˆ’๐‘™ (๐‘†, Q), ๐ป ๐‘š (๐พ๐‘†(๐‘†) 2 , Q)). ๐‘™+๐‘š=4 Therefore, the injection ๐‘– corresponds to an element ๐œ… ๐‘† โˆˆ ๐ป 4 (๐‘† ร— ๐พ๐‘†(๐‘†) 2 , Q) that is known as the Kuga-Satake class. This leads us to a special case of the standard Hodge Conjecture. Conjecture 5.4.1. (Kuga-Satake Hodge Conjecture) The Kuga-Satake class ๐œ… ๐‘† constructed above is algebraic. The Kuga-Satake Hodge Conjecture was developed for surfaces, but we can make a similar conjecture for cubic fourfolds with some quick alterations. If ๐‘‹ is a cubic fourfold, then our formulation of Proposition 3.2.7 gives an inclusion ๐ป 4 (๐‘‹, Q)(1) โŠ‚ ๐ป 1 (๐พ๐‘†(๐‘‹), Q) โŠ— ๐ป 1 (๐พ๐‘†(๐‘‹), Q). We can ignore the Tate twist if we are only concerned with cohomology. Now, by applying the Kรผnneth formula in the same way we get an injective map ๐‘– : ๐ป 4 (๐‘‹, Q) โ†ฉ! ๐ป 2 (๐พ๐‘†(๐‘‹) 2 , Q)). We now get a chain of isomorphisms รŠ ๐ป 6 (๐‘‹ ร— ๐พ๐‘†(๐‘‹) 2 , Q)  ๐ป ๐‘™ (๐‘‹, Q) โŠ— ๐ป ๐‘š (๐พ๐‘†(๐‘‹) 2 , Q) ๐‘™+๐‘š=6 รŠ  Hom(๐ป 6โˆ’๐‘™ (๐‘‹, Q), ๐ป ๐‘š (๐พ๐‘†(๐‘‹) 2 , Q)). ๐‘™+๐‘š=6 So the inclusion corresponds to an element ๐œ… ๐‘‹ โˆˆ ๐ป 6 (๐‘‹ ร— ๐พ๐‘†(๐‘‹) 2 , Q). Conjecture 5.4.2. (Kuga-Satake Hodge Conjecture for Cubic Fourfolds) The Kuga-Satake class ๐œ… ๐‘‹ constructed above is algebraic. 76 The conjecture is still open even for K3 surfaces and a few cases are known. In the next section, we assume that the Kuga-Satake Hodge conjecture holds in order to construct the Fourier-Mukai functors. Since this is a significant assumption, we provide an example of an associated ๐พ3 surface for which the conjecture holds. We briefly survey currently known results for the Kuga-Satake Hodge Conjecture. As mentioned earlier, the conjecture is still open for all but a few special varieties. Recall in Example 3.3.3 it was shown that for a Kummer surface ๐‘† associated to an abelian 19 surface ๐‘†, ๐พ๐‘†(๐‘†) can be explicitly described as ๐พ๐‘†(๐‘†) โˆผ ๐ด2 . It is known that the Kuga-Satake Hodge conjecture holds for Kummer surfaces. To show this, first consider the following result of [MZ99, Theorem 0.1] and [RM08, Theorem 3.15]. Theorem 5.4.3. Let ๐ด be an abelian surface. Then the Hodge conjecture holds for products ๐ด๐‘› for arbitrary ๐‘› โˆˆ Z+ . Now, one can show: Proposition 5.4.4. The Kuga-Satake Hodge conjecture holds for Kummer surfaces. Proof. See [Huy16, Example 4.3.4] or [VV22, Remark 2.15]. The following facts give the proof: 1) 19 If ๐‘† is a Kummer surface associated to an abelian surface ๐ด, then we have shown that ๐พ๐‘†(๐‘†) โˆผ ๐ด2 19 in Example 3.3.3. 2) The Hodge conjecture holds for ๐ด2 by Theorem 5.4.3. 3) The correspondence ๐‘‡ (๐‘†)Q  ๐‘‡ ( ๐ด)Q is algebraic (given by the graph of the rational map ๐ด d ๐‘†). โ–ก Before proceeding with our main construction, we need one more result: Lemma 5.4.5. The set of complex projective Kummer surfaces are dense in the image of the period map for ๐พ3 surfaces. In other words, if (๐‘†, ๐œ‘) denotes a marked ๐พ3 surface, then P : {(๐‘†, ๐œ‘) | ๐‘† is Kummer} ! D has dense image. Furthermore, this also holds for polarized ๐พ3 surfaces: For any ๐‘‘ > 0, Kummer surfaces are dense in N๐‘‘ , the moduli space of polarized ๐พ3 surfaces with polarization of degree ๐‘‘. Proof. See [Huy16, Remark 3.24]. โ–ก 77 Proposition 5.4.6. There exist examples of associated ๐พ3 surfaces that satisfy the Kuga-Satake Hodge conjecture. Proof. By Lemma 5.4.5, Kummer surfaces are dense in N๐‘‘ for all admissible discriminants ๐‘‘. Choose an admissible ๐‘‘ satisfying ๐‘‘ โ‰ก 2 (mod 6), for example, ๐‘‘ = 14. By Corollary 2.5.12, a dense open subset of N๐‘‘ for a degree ๐‘‘ are associated ๐พ3 surfaces. So, by density of Kummer surfaces in N๐‘‘ , there must exist Kummer surfaces that are associated ๐พ3 surfaces in N๐‘‘ for such a ๐‘‘. Now, applying Proposition 5.4.4 we get examples of associated ๐พ3 surfaces for which the Kuga-Satake Hodge Conjecture holds. โ–ก Corollary 5.4.7. Let ๐‘‹ be a cubic fourfold with an associated ๐พ3 Kummer surface ๐‘†. Then ๐‘‹ satisfies the Kuga-Satake Hodge Conjecture for cubic fourfolds. 19 20 Proof. By Theorem 3.3.5, we have that ๐พ๐‘†(๐‘‹) โˆผ ๐พ๐‘†(๐‘†) 2 โˆผ ( ๐ด2 ) 2 โˆผ ๐ด2 . We also have that ๐‘‡ (๐‘‹)Q (1)  ๐‘‡ (๐‘†)Q is a Hodge isometry by the proof of Lemma 4.5.1. So, ๐‘‡ (๐‘‹)Q (1)  ๐‘‡ (๐‘†)Q  ๐‘‡ ( ๐ด)Q where the second isometry is algebraic. Applying Theorem 5.3.3, we see that ๐‘‡ (๐‘‹)Q (1)  ๐‘‡ (๐‘†)Q is induced by an algebraic cycle. Therefore, the correspondence ๐‘‡ (๐‘‹)Q (1)  ๐‘‡ ( ๐ด)Q is algebraic as well. The result follows from by Theorem 5.4.3. โ–ก The Kuga-Satake Hodge Conjecture is known in a few other cases as well. We mention these results and their applications: Example 5.4.8. 1. The Kuga-Satake Hodge Conjecture holds for abelian surfaces by Theorem 5.4.3 together with Example 3.2.9. 2. In [Par88], Paranjape uses his description of the transcendental lattice of ๐พ3 surfaces that are realized as double covers of P2 ramified over 6 lines to show that the Kuga-Satake Hodge Conjecture holds for such ๐พ3 surfaces. 3. In [Sch10, Theorem 2], the author uses Paranjapeโ€™s result and a study of the endomorphism algebra of ๐พ3 surfaces with real multiplication to show that the full Hodge conjecture holds for self-products ๐‘† ร— ๐‘† of ๐พ3 surfaces ๐‘† that are double covers of P2 ramified over 6 lines. 4. Inspired by results of [Sch10], the author of [Var22] shows that if the Kuga-Satake Hodge 78 Conjecture holds for a family of ๐พ3 surfaces of generic Picard rank 16, then the Hodge conjecture holds for all powers of ๐พ3 surfaces in that family. In particular, this generalizes the results of [Sch10]. 5.5 Derived Categories of Kuga-Satake Varieties In this section, we assume that the Kuga-Satake Hodge conjecture for both ๐พ3 surfaces ๐‘† and cubic fourfolds ๐‘‹ holds. We want to relate ๐ท ๐‘ (๐พ๐‘†(๐‘†)) and ๐ท ๐‘ (๐พ๐‘†(๐‘‹)) when ๐‘† is an associated ๐พ3 surface to ๐‘‹ in an effort to examine how they fit into Kuznetsovโ€™s derived categorical viewpoint. We focus on ๐ท ๐‘ (๐พ๐‘†(๐‘†)) since the approaches are similar. First, recall that ๐พ๐‘†(๐‘†) is an abelian variety, so we have some basic facts that hold for ๐ท ๐‘ ( ๐ด) for any abelian variety ๐ด. Example 5.5.1. Let ๐ด be an abelian variety and ๐ดห† denote its dual abelian variety. 1. Recall that the Poincarรฉ bundle P on ๐ด ร— ๐ดห† induces an equivalence of categories by the Fourier-Mukai transform with kernel P, ๐ท ๐‘ ( ๐ด)  ๐ท ๐‘ ( ๐ด). ห† 2. We can use Theorem 5.1.6 to compute Hochschild homology for an abelian variety since รŠ ๐ป๐ป ๐‘˜ ( ๐ด) = ๐ป ๐‘,๐‘ž ( ๐ด) ๐‘โˆ’๐‘ž=๐‘˜ and we know from Chapter 2 that the Hodge numbers of any abelian variety of dimension ๐‘› are given by     ๐‘,๐‘ž ๐‘› ๐‘› โ„Ž ( ๐ด) = ยท . ๐‘ ๐‘ž For example, if ๐ด is an abelian surface, then we have ๐ป๐ป0  ๐‘˜ 6 . 3. If ๐ด is an abelian variety, then its canonical bundle ๐œ” ๐ด is trivial. This is easily seen as the tangent bundle of any group variety is trivial, so the cotangent bundle is trivial and so the canonical bundle is trivial. It is a fact that for any variety ๐‘‹ with trivial canonical bundle ๐œ” ๐‘‹ , its derived category ๐ท ๐‘ (๐‘‹) has no nontrivial semiorthogonal decompositions [KO15, Theorem 1.4]. 79 We want to construct Fourier-Mukai functors ๐ท ๐‘ (๐‘†) ! ๐ท ๐‘ (๐พ๐‘†(๐‘†)) (or to ๐ท ๐‘ (๐พ๐‘†(๐‘†) 2 ). Unfortunately, there is in general no geometric description of ๐พ๐‘†(๐‘†). However, the Kuga-Satake Hodge Conjecture provides us with a potential algebraic cycle on ๐‘† ร— ๐พ๐‘†(๐‘†) 2 . Assuming the Kuga-Satake Hodge conjecture, let ๐œ… ๐‘† โŠ‚ ๐‘† ร— ๐พ๐‘†(๐‘†) 2 be the codimension 2 ๐‘– algebraic Kuga-Satake class. Taking Poincarรฉ duals we get an algebraic surface ๐‘ ๐‘† โŠ‚ ๐‘† ร— ๐พ๐‘†(๐‘†) 2 . In [VG00, 10.2], Van Geemen shows that the Kuga-Satake Hodge conjecture is equivalent to the following condition: ๐‘ ๐‘† induces an isomorphism ๐‘‡ (๐‘†)  ๐‘‡ (๐‘†) via ๐œ‹โˆ— ๐œ™โˆ— ๐‘‡ (๐‘†) โ†ฉโˆ’! ๐ป 2 (๐‘†, Q) โ†ฉโˆ’! ๐ป 2 (๐พ๐‘†(๐‘†) 2 , Q) โˆ’! ๐ป 2 (๐‘†, Q) where ๐œ‹, ๐œ™ are defined by the following diagram: ๐œ™ ๐‘๐‘† ๐พ๐‘†(๐‘†) 2 ๐œ‹ ๐‘† Now, consider the sheaf ๐‘–โˆ— O๐‘๐‘† , we refer to it as simply O๐‘๐‘† to simplify the notation. We construct the Fourier-Mukai transform with kernel ๐พ = O๐‘๐‘† to get a functor ฮฆO๐‘๐‘† : ๐ท ๐‘ (๐‘†) ! ๐ท ๐‘ (๐พ๐‘†(๐‘†) 2 ). We outline a possible approach to understanding properties of this functor in the next section. Now, recall the basic examples of Fourier-Mukai functors given in Example 5.1.5. Let ๐‘† be associated to a cubic fourfold ๐‘‹. Then by our result in Theorem 3.3.5, we have an isogeny ๐‘“ : ๐พ๐‘†(๐‘†) 2 ! ๐พ๐‘†(๐‘‹). The Fourier-Mukai functor with kernel given by the structure sheaf of the graph of ๐‘“ gives a pushforward functor: ฮฆOฮ“ ๐‘“ : ๐ท ๐‘ (๐พ๐‘†(๐‘†) 2 ) ! ๐ท ๐‘ (๐พ๐‘†(๐‘‹)). So this far we have constructed functors 80 ฮฆ O๐‘ ๐ท ๐‘ (๐พ๐‘†(๐‘†) 2 ) ๐‘† ๐ท ๐‘ (๐‘†) ฮฆ Oฮ“ ๐‘“ ๐ท ๐‘ (๐พ๐‘†(๐‘‹)) Now, we make similar constructions for ๐ท ๐‘ (๐‘‹). We construct a functor ๐ท ๐‘ (๐พ๐‘†(๐‘‹) 2 ) ! ๐ท ๐‘ (๐‘‹) in the same way as for the ๐พ3 surface. Let ๐‘ ๐‘‹ โŠ‚ ๐พ๐‘†(๐‘‹) 2 ร— ๐‘‹ be the Kuga-Satake Hodge cycle and consider O๐‘ ๐‘‹ โˆˆ ๐ท ๐‘ (๐พ๐‘†(๐‘‹) 2 ) ร— ๐‘‹). This gives a functor ฮฆO๐‘๐‘‹ : ๐ท ๐‘ (๐พ๐‘†(๐‘‹) 2 ) ! ๐ท ๐‘ (๐‘‹). Finally we construct a functor ๐ท ๐‘ (๐พ๐‘†(๐‘‹)) ! ๐ท ๐‘ (๐พ๐‘†(๐‘‹) 2 ) using the kernel given by the graph of the diagonal morphism ฮ” : ๐พ๐‘†(๐‘‹) ! ๐พ๐‘†(๐‘‹) 2 Considering Oฮ“ฮ” โˆˆ ๐ท ๐‘ (๐พ๐‘†(๐‘‹) ร— ๐พ๐‘†(๐‘‹) 2 ) we get a functor: ฮฆOฮ“ฮ” : ๐ท ๐‘ (๐พ๐‘†(๐‘‹)) ! ๐ท ๐‘ (๐พ๐‘†(๐‘‹) 2 ). Putting this all together, we get the following diagram: ฮฆ O๐‘ ๐ท ๐‘ (๐พ๐‘†(๐‘†) 2 ) ๐‘† ๐ท ๐‘ (๐‘†) ฮฆ ฮฆ Oฮ“ ๐‘“ ๐ท ๐‘ (๐‘‹) ฮฆ O๐‘ ๐ท ๐‘ (๐พ๐‘†(๐‘‹) 2 ) ฮฆ Oฮ“ ๐ท ๐‘ (๐พ๐‘†(๐‘‹)) ๐‘‹ ฮ” where ฮฆ : ๐ท ๐‘ (๐‘†) ! ๐ท ๐‘ (๐‘‹) is simply the composition. Note that this functor ฮฆ is Fourier-Mukai since it is the composition of Fourier-Mukai functors. So, for a ๐พ3 surface ๐‘† associated to a cubic fourfold ๐‘‹, we have constructed a functor ๐ท ๐‘ (๐‘†) ! ๐ท ๐‘ (๐‘‹) using the Kuga-Satake construction. 5.6 Future Work The construction above raises some questions: What are the properties of the functor ฮฆ : ๐ท ๐‘ (๐‘†) ! ๐ท ๐‘ (๐‘‹)? What is the image of ฮฆ? What is the relationship between ๐ท ๐‘ (๐‘†) and A ๐‘‹ ? 81 A key obstacle in answering these questions is a lack of understanding of the functors ฮฆO๐‘๐‘† : ๐ท ๐‘ (๐‘†) ! ๐ท ๐‘ (๐พ๐‘†(๐‘†) 2 ). A possible approach to prove that ฮฆO๐‘๐‘† is fully faithful is to consider its right adjoint, which we will denote by ฮฆ ๐‘… . It is well known that the functor ฮฆO๐‘๐‘† fully faithful if and only if ฮฆ ๐‘… โ—ฆ ฮฆO๐‘๐‘†  Id. The right adjoint has a description as follows: Proposition 5.6.1. Let ๐‘‹, ๐‘Œ be smooth projective varieties. Let ฮฆ๐พ : ๐ท ๐‘ (๐‘‹) ! ๐ท ๐‘ (๐‘Œ ) be a Fourier-Mukai functor with kernel ๐พ โˆˆ ๐ท ๐‘ (๐‘‹ ร— ๐‘Œ ). Then ฮฆ๐พ has a right adjoint given by ฮฆ๐พ ๐‘… , where the kernel ๐พ ๐‘… is given by ๐พ ๐‘… B ๐พ โˆจ โŠ— ๐ฟ ๐ฟ๐‘ž โˆ— ๐œ” ๐‘‹ [dim(๐‘‹)] where ๐‘ž : ๐‘‹ ร— ๐‘Œ ! ๐‘‹ is projection. Proof. See [Muk81, Theorem 2.2]. โ–ก Note that this simplifies in our case since ๐œ” ๐‘† is trivial and dim(๐‘†) = 2. So, we focus on the dual (๐‘–โˆ— O๐‘๐‘† ) โˆจ . In the case that ๐‘ ๐‘† โŠ‚ ๐‘† ร— ๐พ๐‘†(๐‘†) 2 is smooth, we have the following: Proposition 5.6.2. If ๐‘– : ๐‘‹ โ†ฉ! ๐‘Œ is a smooth closed subvariety of codimension ๐‘ then (๐‘–โˆ— O ๐‘‹ ) โˆจ  ๐‘…๐‘– โˆ— ๐œ” ๐‘‹ โŠ— ๐ฟ ๐œ”๐‘Œโˆ— [โˆ’๐‘]. Proof. See [Huy06, Corollary 3.40]. โ–ก If the algebraic cycle ๐‘ ๐‘† is smooth, the above proposition gives (๐‘– โˆ— O๐‘๐‘† ) โˆจ  ๐‘–โˆ— ๐œ” ๐‘๐‘† [โˆ’2]. 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