srv u I|BRAR IES

\llllllllllllll

\ meme»: STA

. » i» l l ll\lllllllllllll\lll

 

 

 

 

This is to certify that the

dissertation entitled
GENUS ONE ENUMERATIVE INVARIANTS IN pn
presented by

Eleny-Nicoleta Ionel

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Mathematics

76% 74/?pse

 

 

Major professor

DateMsfiMQ

MS U is an Affirmative Action/Equal Opportunity Institution 0- 12771

 

 

LIBRARY

Michigan State
University

 

 

 

PLACE ll RETURN BOX to romovo thin chookom from your rooord.
TO AVOID FINES roturn on or Moro doto duo.

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU In An Affirmative Action/Ema! Opportunity institution
Mona-9.1

 

GENUS ONE ENUMERATIVE INVARIANTS IN P"

By

Eleny-Nicoleta Ionel

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1995

ABSTRACT

GENUS ONE ENUMERATIVE INVARIANTS IN P”

By

Eleny—Nicoleta Ionel

In this thesis we prove recursive formulas for Tag, the number of degree d elliptic curves
with fixed j invariant in P". We use analysis to relate the classical invariant rd to
the genus one perturbed invariant RT” defined recently by Ruan and Tian (the later
invariant can be computed inductively).

Our approach is to start with RT La! and consider a sequence of perturbations
converging to zero. In the limit we get not only holomorphic tori (i.e. 74), but
also bubble trees with ghost base. We use Taubes Obstruction Bundle method to
compute the contribution of the ghost base stratum to the difference between the two

invariants.

ACKNOWLEDGMENTS

I would like to thank my advisor Prof. Thomas Parker for introducing me to the
subject and for the countless hours of discussions we had about it. Not only did
I learn a lot from these discussions, but they also helped me clarify my ideas and

provide rigorous mathematical approach.

I would also like to extend my thanks to Prof. Selman Akbulut, Prof. Ronald.
F intushel and to Prof. Jon Wolfson for their help.

iii

TABLE OF CONTENTS

List of Figures

0 Introduction.

1 Analysis
1.1 Setup ....................................
1.2 The Approximate gluing map ......................
1.3 The Obstruction Bundle .........................
1.4 The Gluing map ..............................
1.5 Completion of the construction .....................
1.6 The leading order term of the obstruction 7/): for t small ........
1.7 The enumerative invariant “rd .......................
1.8 The other contribution ..........................

2 Applications

2.1 Recursive formulas for c§(L")ev"( H j) ..................

2.2 Recursive formulas for CHE”) - ev'(Hj) .................

2.3 Applications to P", n S 3 . . .

3 Appendix

iv

14
18
21
27‘
33
37

39
4O

44
48

52

LIST OF FIGURES

0 Introduction.

A classical problem in enumerative algebraic geometry is to compute the number of

degree d, genus g holomorphic curves in P" that pass through a certain number of

constraints (points, lines, etc).

Let 0,; denote the number of degree d rational curves (9 = 0) through appropriate
constraints. For example 01 (pt, pt) = 1 (since 2 points determine a line). The first
nontrivial cases were computed around 1875 when Schubert, Halphen, Chasles et al.
found 02 for P2 and P3. Later, more low degree examples were computed in P2 and
P3, but the progress was slow. Then in 1993 Kontsevich [K] predicted, based on ideas
of Witten, that the number 04 of degree d rational curves in P2 through 3d — 1 points.

satisfies the following recursive relation:

_ 3d—1 ,, 3d—1 3

where (1,- ¢ 0, and 01 = 1. Ruan-Tian ( [RT], 1994) extended these formulas for ad in
any P".

When genus g = 1, the classical problem splits into two totally different prob-
lems: one can count (i) elliptic curves with a fixed complex structure, or (ii) elliptic
curves with unspecified complex structure (each satisfying the appropriate number of

constraints). This paper gives recursive formulas which completely solve the first of

these.

Thus our goal is to compute the number 7.; of degree d elliptic curves in P" with

fixed j invariant. This problem is considerable harder than the g = 0 case. The
simplest nontrivial case - 7'3 in P2 — was determined by Alluffi (1991) using classical
methods. Recently Pandharipande [Pan] has made more significant progress, using

the Kontsevich moduli space of stable curves to give recursive formulas for 7}; in P2.

We will approach this problem from a different direction, using analysis. This
approach is based on the ideas introduced by Gromov to study symplectic topology.

If (E, j) is a fixed Riemann surface, let
{f12 -> P" |5Jf = 0, [fl = d-le H2(P",Z) }/Aut(§3,j)

be the moduli space of degree d holomorphic maps f : E —> P", modulo the auto-
morphisms of (2,1) Each constraint, such as the requirement that the image of f
passes through a specified point, defines a subset of this moduli space.

Imposing enough constraints gives a 0-dimensional “cutdown moduli” space Md.
To see whether or not it consists of finitely many points, one looks at its bubble

tree compactification Ed [PW]. If the constraints are cut transversely, then all the

boundary strata of J—Vl-d are at least codimension 1, and thus empty. Unfortunately,
transversality fails at multiply-covered maps or at constant maps (called ghosts), so
791—4 is not a manifold.

This was a real problem until 1994, when Ruan and Tian considered the moduli

space M u of solutions of the perturbed equation:

51f = V($sf(x))

and used marked points instead of moding out by Aut(E, 3'). For a generic perturba-
tion 12 the moduli space M” is smooth and compact, so it consists of finitely many
points that, counted with Sign, give an invariant RT“ (independent of V).

In P”, the genus 0 perturbed invariant BTW is equal to the enumerative invariant
ad. The perturbed invariants satisfy a degeneration formula that gives not only
recursive formulas to compute the enumerative invariant ad in P", but also expresses
the higher genus perturbed invariants in terms of the genus zero invariants [RT]. For

convenience, these formulas are included in the Appendix.

Unfortunately, when 9 = 1, the perturbed invariant RT,” does not equal the

enumerative invariant Td. For example, for d = 2 curves in P2 the Ruan-Tian invariant

is RT” = 2 (cf. (A.2)), while 7'2 = 0 (there are no degree 2 elliptic curves in P2).

Thus while the Ruan-Tian invariants are readily computable, they differ from the
enumerative invariants rd. One should seek a formula for the difference between the
two invariants. For that, we take the obvious approach:

Start with the genus 1 perturbed invariant RTag,g and consider a sequence of generic
perturbations V -—> 0. A sequence of (J, V)-holomorphic maps converges either to a
holomorphic torus or to a bubble tree whose base is a constant map (ghost base).
Proposition 1.21 shows that the contribution of the (J,0)-holomorphic tori is a mul-
tiple of 74.

We show that the only other contribution comes from bubble trees with ghost
base such that the bubble point is equal to the marked point 2:1 6 T2. To compute
this contribution, we use the Taubes “Obstruction Bundle” method. Proposition 1.7
identifies the moduli space of (J, V)-holomorphic maps that are close to a bubble tree.
with the zero set of a specific section of the obstruction bundle. Studying the leading

order term of this section, we are able to compute the corresponding contribution

(Proposition 1.26). Adding both contributions, yields our main analytic result:

Theorem 0.1 Consider the genus 1 enumerative invariant Td(,31, . . . ,flk) in P“. Let
Lid be the n — 1 dimensional moduli space of I-marked rational curves of degree d in

P" passing through ,61, . . . ,flk. Let L —) L1,; be the relative tangent bundle, and denote
by Z —+ Lid its blow up defined by (1.17). Then:

_ _ 71-1 ”+1 n- i—l :-
Tl] Td(fila--°aflk) — RTd,l(,81 [IBZTH'MBk 1— _0 —(i+2)ev (H )6 1(L)

[where H‘ is a codimension i hyperplane in P", ev : Lid —> P" is the evaluation
map corresponding to the special marked point and n,- is the order of the group of

automorphisms of the complex structure j that fix a point.

Theorem 0.1 becomes completely explicit provided we can compute the top power

intersections ev"(H"""1)c'l(Z") We do this in the second part of the paper, in several

steps. For simplicity of notation, let

= c1(L')€ H2(ud,2), a = c1(Z*) e H2(L7d,Z), and y = ev'(H) (0.1)

where y E H 2(11.1,Z) or y E H 2(Zz~l¢,[,Z) depending on the context. In this notation,

Theorem 0.1 becomes:

' ' "-1 n+1 ~in—-—i ~
and(')= Z 0"“anan °)+Z(i+2)$y 1 'ludl (0'2)
i1+i2=n i=0

Proposition 2.2 gives recursive formulas relating E‘yj to x‘yj and Proposition 2.5
gives recursive formulas for x‘yj in terms of the enumerative invariant ad. Finally,
the recursive formulas for 0.1 are known (see [RT], [K]), so the right hand side of (0.2)

can be recursively computed.

In the end, we give applications of these formulas. We explicitly work out the
formulas expressing the number of degree d elliptic curves passing through generic .

constraints in P2 and P3 in terms of the rational enumerative invariant ad. For

example:

Proposition 0.2 Forj 7Q 0,1728, the number “rd = Td(pa,lb) of elliptic curves in P3
with fixed j invariant and passing through a points and b lines (such that 2a + b =
4d — 1) is given by:

‘rd(') = (d —1[d(d — 2)0'd(l, ) —

 

Z d2(2d1d2 — d)ad,(l, -)od,(-) (0.3)

d1+d2=d

Salt"

where ad(l, .) = 04(l,p“,lb) is the number of degree d rational curves in P3 passing
through some conditions as rd plus one more line. The sum above is over all decom-
positions into a degree d1 and a degree d2 component, d, aé 0, and all possible ways of

distributing the constraints p“, lb on the two components.
Using a computer program, one computes then specific examples, e.g.
73(1“) 2 6 - 25920 and 75(p,ll7) = 6- 15856790593536.

whenj 91$ 0,1728. To get rd forj = 0 orj = 1728 one simply divides the rd computed

for a generic j by 2 or 3 respectively.

1 Analysis

1.1 Setup

Let “rd be the genus one degree d enumerative invariant (with fixed j invariant) and
0,; be the genus zero degree d enumerative invariant in P". Using analytic methods,

we will compute 74 by relating it to the perturbed invariant RTM introduced by Ruan

and Tian [RT]. The later is defined as follows.

Let (2, j ) be a genus g Riemann surface with a fixed complex structure and l/
an inhomogenous term. A (J, u)-holomorphic map is a solution f : E —> P" of the

equation
was) = mm». (1.1)

For 29 +1 _>_ 3, let $1,...,x1 be fixed marked points on E, and 011,...,al, 61,...,flk

be various codimension submanifolds in P" such that
3

l 1:

index 51 = (n +1)d — n(g — 1) = 2(71 -|01.'[)+ 2(71 "1 - mil)

u

i=1 1:]
For a generic V, the invariant

RTd,g(ala° ' ' val I :61, ' ° ' 9/Bk)

counts the number of (J, V)-holomorphic degree d maps f : Z —> P" that pass through
fila-oqfik With f(x.-) 6 a,- for i = 1,...,l.

The first part of this paper is devoted to the proof of Theorem 0.1.

Outline of the Proof of Theorem 0.1. The proof is done in several

steps. The basic idea is to start with the genus 1 perturbed invariant

RTd,1(Bi |fi2,.-.,Bz) (1-2)

and take a sequence of generic perturbations V -> 0. Denote by Mum, the moduli
space of (J,tV)-holomorphic maps satisfying the constraints in (1.2), and let

My = U Md,l,tu- (1.3)

:30

As t —> 0, a sequence of (J, tV)-holomorphic maps converges to a (J, 0)-holomorphic
torus or to a bubble tree ([PW]). Let 7W” denote the bubble tree compactification of

M " (for details on bubble tree compactifications, see [P]).

Proposition 1.21 shows that the number of (J,tV)-holomorphic maps converging

to a J-holomorphic torus is equal to

and(flla - - - ’51:)

where n,- = [Autx,( j )| is the order of the group of automorphisms of the complex

structure j that fix the point :rlnamely,

2 ifj9é0,1728
n,= 4 ifj=0 (1.4)
6 ifj=1728

These multiplicities occur because if f is a J-holomorphic map, then so is f o d for

any 45 6 Aut$,(j), but they get perturbed to distinct (J, tV)—holomorphic maps.

As t —-> 0, there are also a certain number of solutions converging to bubble trees.
Because the moduli space of (J,0)-holomorphic tori passing through ,61, . . . , fl], is 0
dimensional, the only bubble trees which occur have with a multiply-covered or a

ghost base (for these transversality fails, so dimensions jump up).

A careful dimension count shows that the multiply-covered base strata are still
codimension at least one for genus g = 1 maps in P". (This is not true for g 2 2.)
But at a ghost base bubble tree the dimension jumps up by n so these strata are n — 1
dimensional. There are actually 2 such pieces, corresponding to bubble tree where
(i) the bubble point is at the marked point 2:; and (ii) the bubble point is somewhere

else. To make this precise, a digression is necessary to set up some notation.

Let

M3 = { (f, 311,... ,yk) | f : S2 —+ P" degree d holomorphic, f(yj) 6 fly} (1.5)

be the moduli space of bubble maps, and Md = M3 / G be the corresponding moduli
space of curves, where G = PSL(2,C). Introduce one special marked point y E 5'2

and let

Ud={lf,y,y1,m,ykl | [faylaH-ayklEMd} (L6)

be the moduli space of I-marlced curves and

CVIUd-ipn, ev(lf9yaylr-°'aykl) =f(y) (1'7)

be the corresponding evaluation map. We will use f (y) to record the image of the I
ghost base

For generic constraints 51,...,fik the bubble tree compactification of U4 is a
smooth manifold that comes with a natural stratification, depending on the pos-
sible splittings into bubble trees and how the degree d and the constraints 61,. . . , fik
distribute on each bubble.

With this, the two “pieces” of the boundary of 17” are:
{x1} x 17.1 and T2 x ev'(,81) (1.8)

The first factor records the bubble point, while the image of the ghost base is encoded
in the second factor. For generic constraints each piece, as well as their intersection,

is a smooth manifold, again stratified.

To see which bubble trees with ghost base appear as a limit of perturbed tori,
we use the Taubes Obstruction Bundle. This construction must be performed on the
link of each strata. We do this first on the top statum of {1'1} x bid, which consists
of bubble trees with ghost base and a single bubble.

First we construct in Section 1.2 a set of approximate maps by gluing in the

bubble. The “gluing data” [ f , y, v] consists of a nonvanishing vector v tangent to the

bubble at the bubble point y. Proposition 1.4 shows that the obstruction bundle is
then diffeomorphic to ev“(TP").

In Section 1.4 we correct the approximate maps to make then (J, tV)-holomorphic
by pushing them in a direction normal to the kernel of the linearized equation. Those
approximate maps that can be corrected to solutions of the equation (1.1) are then
identified with the zero set of a section 1P: of the obstruction bundle. Proposition 1.7
shows that actually all the solutions of the equation (1.1) are obtained this way, i.e.
the end of the moduli space of (J, tV)-holomorphic maps is diffeomorphic to the zero

set of the section 21),.

To understand the zero set of wt it is enough to look at the leading order term of
its expansion as t —> 0. By Proposition 1.45 this has the form dfy(v) + it? where 17 is _

the projection of V on the obstruction bundle.

The construction described above extends naturally to all the other boundary
strata. Each bubble comes with “gluing data ” [f,-,y,-,v,-], consisting of a vector v,-
tangent to the bubble at the bubble point yg. But the leading order term of the

section wt depends only on the vectors tangent to the first level of nontrivial bubbles.

More precisely, let 2;, C 17d denote the collection of bubble trees for which the
image u = f (y) of the ghost base lies on h nontrivial bubbles. Geometrically, the
image of a bubble tree in 2;, has h components C1,...,Ch that meet at u. Let
le,I —> 3;, be the bundle whose fiber is TuCl EB - - - EB TuCh. The leading order term

of w, on 2;, is a section of W, equal to

a(f,y,vl) + wdé’ df1(y1)(v1) + . . . + dmmm) + w

where ([f,-, y;, v,-])[‘__.l is the gluing data corresponding to the bubbles C,, i = 1, . . . , h.

Unfortunately W -> 5.1 is not a vector bundle. But if we blow up each strata

2;, starting with the bottom one, then the total space of W is the same as the total
space of L, the blow-up of the relative tangent sheaf L —> bid. The leading order

term of wt descends as a map a + t1? : L —> ev“(TP"). Moreover, 17 doesn’t vanish on

Im(M) = ev.(fid) so it induces a splitting on the restriction

TP"/Im(M) = C(17) EB E.
Finally, we put all these pieces together in Proposition 1.26 to prove that the number
of (J, V)-holomorphic maps converging as V —> 0 to the boundary strata {2:1} x 17,, is
given by the Euler class cn_1(ev"(E) (8) L‘).

In Section 1.8 we show that the other boundary strata T2 x ev“(fil) gives trivial

contribution, concluding the proof of the Theorem 0.1.

1.2 The Approximate gluing map

Let Lid be the moduli space of 1-marked rational curves of degree d passing through

the conditions 61,. . . ,flk. In this section we construct a set of approximate maps

starting from {x1} x U0], the first boundary strata in (1.8). We will use a:

Cutoff function. In what follows, fix a smooth cutoff function 6 such that L3 (r) = 0

for r S 1 and fi(r) = 1 for r 2 2. Let fiA(r) = fi(r/\/X). Then 31 has the following

properties:

lfiAl S 1 a [dfixl S 2/\/X and dB) is supported in W S r _<_ 2%)?

The definition of the approximate gluing map on the top stratum. Let N
denote the top stratum of {1'1} x17 d. First we need to choose a canonical representative
of each bubble curve [ f, y] E N (recall that f (y) is the image of the ghost base). Using
the G = PSL(2, C) action, we can assume that y is the North pole and f is centered
on the vertical axis, which leaves a C" ’5 S1 x R+ indeterminancy. To break it off,

include as gluing data a unit vector tangent to the domain .5'2 of the bubble at the

bubble point y. The frame bundle

Fr={[f,y.ul|[f,yl€Uda “671,52, IUI=1} (1-9)

10

models the link of N. The notation [ f, y, u] means the equivalence class under the

action of G given by:

9- (Lyra) = (f 09—59(31), 9(a))

where the compact piece 5' 0(3) C G acts on the unit frame it by rotations and the
noncompact part acts trivially.

Fix a nonzero vector ul tangent to the torus at 3:1. This determines an identifica-
tion T1,,(T2) 1‘-:’ C such that u1 = 1, giving local coordinates on the torus at 3:1 = 0.

Similarly, let uo be a unit vector tangent to the sphere S2 at the north pole and

consider the identification
(T11T2,u1)'5(TN52,u0) (1.10)

that induces natural coordinates on the sphere via the stereographical projection
(such that N = 0, an = 1). These choices of local coordinates on the domain of the
bubble tree will be used for the rest of the paper. Fix also a metric on P” such that

we can use normal coordinates up to radius 1.

O In
Figure l. The domain of the bubble tree.

To glue, one needs to make sure that only a small part of the energy of f is
concentrated in a neighbourhood of y. The convention in [PW] is to rescale f until
50 of its energy is distributed in Hy, the hemisphere centered at y.

But since the constructions in the next couple of sections involve quite a few

estimates, we prefer to do a different rescaling, that will simplify the analysis. Choose

11

a representative of [f, y, U] such that
y = 0, u = 1, f centered on the vertical axis (1.11)

Since on the top strata [ f, y] cannot be a ghost, such representative is uniquely de-
termined up to a rescaling factor r E R+. We will choose this rescaling factor such

that moreover
max{ |V2f(2)l, lzl S 1} S 2 (1-12)
Note that if the degree of f is not 1, then imposing the extra condition

max{ IV’f(z)|, lzl :1}: 2 (1.13)

determines uniquely the representative. To see this, choose some representative f

~

as in (1.11) and look for a map f(2) = f(rz) satisfying also (1.13). The uniqueness

~

comes from the fact that the map s(r) = max{ [V2f(z)|, [2| S r} —2/r2 is decreasing.

If the degree of f is 1, (i.e. the image curve is a line), then we could replace (1.13)

by say |df(0)| = 1 and still have (1.12) satisfied.
Finally, the approximate gluing map

75 : Fr X (0,6) —> Maps(T2,X)

75( [fayauli A) = f) (114)

is constructed as follows: Choose the unique representative of [f , y, u] satisfying (1.11)

and (1.13). The approximate map f,\ is obtained by gluing to the constant map f (3;)
defined on T2 the bubble map f rescaled by a factor of A inside a disk D(0, x/X) C T2,

m2) = 3(1sz (3)

where the multiplication is done in normal coordinates at f (0)

12

rescale

 

Figure 2. The rescaled domain of the approximate map.
In what follows, we will denote by 01 = Fr X (0,5) the set of gluing data.

Weighted Norms. On the domain of f; we will use the rescaled metric g,\ =
dizdzdi, where
9M2) = (1 - mm )(b + b-IIZV) + 3A0?)

Define

l/p
”EH13“ = (/ [{IW;2 + IVéP’OK-z) for 6 vector field along f; and

1/p
“nup, = (flavor?) fornl-form along fA

The weighted norm of a vector field or 1-form on f A equals its usual norm off B (0, 2x/X)

and on B (0, \/X) it is equal with the norm of its pulled back on S 2 via a rescaling of
factor A. The usual Sobolev embeddings hold for this weighted norms with constants

independent of A.

Lemma 1.1 There exists 50 > 0 and constants C > 0 such that for any p _>_ 1 and
A S 50!

||dfxllm s C and 115mm 3 cw» (1.15)

Moreover on the annulus A: {x/X S [2] S 2x5} we have the following expansion:

BJfA = T? dfi ° df(y)(U) + 0W (1.16)

The estimates are uniform on 01 -> N.

13

Proof. Let B be the disk |z| S x/A. Note that df,\ vanishes for |z| 2 2\/A and by
the definition of the weighted norm on B,

Ildfxllpma = Ildfllp.B
But (1.12) implies that

1m6D<{|0lf(z)|, lzl S 1} S 2 (1-17)

In the same time, 51f; = 0 outside A. Hence we need only to consider what happens

in A. But on A

_ A l
lanAl S Cldfxl S C(ldflxl lf|+lfixl ldfl) W S Cfisgplfl'l'c S C

since sup lf(z)| S x/A sup [de S 2\/A in normal coordinates on P" at f(y). This
B B

concludes the first part of the proof. For the second part, notice that on A

EJfA=5JflA'f+fiA’3Jf='\/l—Xdfil—:T'f(é)

I
~

since f is holomorphic. But using (1.12) in normal coordinates on P" at f(y) and

y = 0, we get [f(z) — f(0) — df(0)(z)[ S 2|z|2 so

A,
b

A A
f (_) = -: -dfy(u) + 0(A) on A
Substituting this in the formula for EJfA we obtain (1.16). D

Extending the approximate gluing map. The approximate gluing map extends
naturally to the bubble tree compactification 17.1 of the moduli space of 1-marked
curves. For simplicity, let N denote some boundary stratum modeled on a bubble
tree B and corresponding to a certain distribution of the degree d 2 d1 + . . . + dm on

the bubbles. If [f,~, y,], i = 1, . . . , m are the bubble curves corresponding to the bubble

14

map f : B —-> P", then the gluing data Cl is a collection of unit vectors tangent to
each sphere in the domain at the corresponding bubble point together with gluing

parameters:
Gl={(lfiayiauila’\i)in=1luie T311523 luil 7e 07)“ S 5} (1'18)

Note that as long as f,- is not a constant map, then we can choose a unique
reresentative of [f,,y,-,u,-] as in ( 1.11), ( 1.13). Then Lemma 1.1 extends naturally to

N to give

Lemma 1.2 With the notations above, let f; be an approximate gluing map, and
A1,. ..Am be the corresponding annuli of radii A,- in which the cutoff functions are

supported. Then for 5 small enough, there exists a constant C such that:
”dele S C» ”ngAllpA S CAI/p

Moreover, .5ij = 0 except on the annuli A,- that correspond to nontrivial bubbles,

where

fir
[3|

 

5.113 = - dfi'dfi(yi)(ui) +000 (1-19)

The estimates above are uniform on 01 —> N.

We will see later that most of the important information is encoded in the first

level of nontrivial bubbles.

1.3 The Obstruction Bundle

In order to see which of the approximate maps can be corrected to solutions of the

equation 51f = V we need first to understand the behaviour of the linearization of

this equation over the space of approximate solutions.

15

Recall that transversality fails at a bubble tree with ghost base, so the linearization
at such bubble tree is not onto. The cause of that is the ghost base. Thus we start
by analysing the ghost maps:

Consider the moduli space of holomorphic maps f : T2 —-) P" representing 0 6

H2(P”). Obviously, the only such maps are the constant ones (ghosts). If Du is the
linearization of the section 5.1 : Maps(T2,P") —~> A0'1 at f : T2 —+ P", f(x) = u a
constant map, then
index Du = dim KerDu — dim CokerDu = c1(0) + n(l — 1) = 0
and
CokerDu = H1(T2,f"TP") ’-_‘-’ TuP" (canonically)

since f‘(TP") is a trivial bundle, so the elements on E H1(T2, f‘TP") are constant

on the torus, i.e. have the form w = X dz for some X 6 RP".

Now if f : B —> P" is a bubble tree map whose base is a ghost torus u = f (y) E P",

let D f be the linearization at f of the section a] : Maps(B,P”) —+ N“. Then
index D; = dim Keer — dim CokerD; = —1

To describe CokerD, we will use the following:

Definition 1.3 Iff : B —) P" is as above, let
B; C B consist of the domains of all the ghost bubbles with image f(y),
82 = B — B1 and
[3 c B denote the first level of bubbles that are not in 81.

Then CokerD I is n dimensional, consisting of l-forms a: such that

_ Xdz on Bl
w_ 0 01182

for some X 6 RP". In particular, there is a natural isomorphism

CokerD ’5 ev*(TP")

\. _ ./ (1.20)
ad

16

where ev : Ed -—> P" is the evaluation map. Since the moduli space of bubble trees
Z7: is compact, there exists a constant E > 0 such that D ID} has a zero eigenvalue

with multiplicity n, and all the other eigenvalues are greater than 2E.

When f A is an approximate map, let D; be the linearization of
5.; : Maps(T2,P”) —-> A0'1

at ,\ and D" its Lz-adjoint with respect to the metric g,\ on T2. Then D,\ is not
A

uniformly invertible. More precisely,

Lemma 1.4 For A > 0 small, the operator A; = DAD; has exactly n eigenvalues of

order x/A and all the others are greater than E. Moreover, over the set of gluing data ,

Gl, the span of low eigenvalues

A0’1(f,\*TP") <——> A0"

low low

l
GI

is a n-dimensional vector bundle (called the Taubes obstruction bundle ), naturally
isomorphic to the bundle
ev‘(TP") —> GI

where ev : Gl —+ P” is the evaluation map.

Proof. The proof is more or less the same as the one Taubes used for the similar
result in the context of Donaldson theory, [T1]. For each gluing data in GI, by cutting
and pasting eigenvectors we show that the eigenvalues of AA = D ,\ D; are 0(x/A) close
to those of Au 2 DuD;, where u is the point map in the base of the bubble tree.

Take for example the top stratum of ad, Choose {w,-, i = 1, n} a local orthonormal

base of CokerD E ev‘(TP") and define

 

wi<z> = fl (whiz) (1.21)

17

A straightforward computation shows that:

llDiwA||2.A S AIMIIwAHm (1-22)

Mai)“ = (st—+00) (1.23)

The Gramm-Schmidt orthonormalization procedure then provides n eigenvectors a7,

for A; with eigenvalues 0(\/A) such that
a", = a; + 0(,\)

The construction above extends naturally to the other substrata of bid. Note that for

example when Bl has other components besides T2 then D), is equal to no not only on
the ghost base, but on all BI and is extended with 0 starting from the first level of
nontrivial bubbles.

An adaptation of Taubes argument from [T1] shows that there are at most n low

eigenvalues of A). Therefore there is a well defined splitting

A0,l(f’\tTPn) : ADJ (onTPn) @ A%l(f’\auTPn)

low

The definition (1.21) combined with (1.20) provides the isomorphism AO’l E’ ev‘(TP"),

low

concluding the proof. CI

The partial right inverse of D,\. The restriction of D; D; to A? is invertible

(since all its eigenvalues are at least E). Define PA to be the composition of the

Lz-othogonal projection AO’1 —) A251 with the operator D§(D,\ D3)"l on A251. Then
PA : A0'1(fA”TP") —> A°(f,\"TP") (1.24)
is the partial right inverse of DA and satisfies the uniform estimate:
IIPAU||1.p.A _<_ ETIIUHM (1-25)

We will denote by 1d" : A0’1(f,\'TP") —) AO’l (fA'TPn) the projection onto the fiber

low

of the obstruction bundle.

18

1.4 The Gluing map

The next step is to correct the approximate gluing map to take values in the moduli

space M“, of solutions to the equation

are) = t - m, f(x)) (1.26)

where V is generic and fixed and t is a small parameter.
If f,\ is an approximate map, use the exponential map to write any nearby map

in the form f = exph(f), for some correction E E A0(f,\‘TP"). Let DA be the

linearization of the BJ-section at f,\ so
an = EJfA + DA“) + QA(€) (1.27) p
where Q; is quadratic in 5. Similarly,

l/(~'v,f(-’I=)) = 141?, fx($)) + dl/(E) + QM“)

so equation (1.26) can be rewritten as:
DAG) + NAN) = ”(17, fA($)) - 5fo (128)

where N),(€,t) = QA(§) — th(§) — tQAQ') is quadratic in ({,t).

The kernel of DA models the tangent directions to the space of approximate maps,
so is natural to look for a correction in the normal direction. More precisely, we will

consider the solutions of (1.28) of the form
f = exph(PA17) where 7r_(17) = 0 (1.29)
Since D).(P,\(n)) = 17 for such 17, then equation (1.28) becomes
'7 + NA,t(PA77) = W — ngA (1-30)

The existence of a solution of (1.30) is a standard aplication of the Banach fixed point

theorem combined with the estimates in the previous sections.

19

Lemma 1.5 There exists a constant 6 > 0 (independent of A, t) such that fort small
enough and for any a E A0'1(f,\‘TP") so that ”all“; < 6/2 the equation:

77 + NA,t(PA77) = a

has a unique small solution 77 6 A0’1(f,\'TP”) with ”77“,“ < 6. Moreover,

llnllm < 2llallw
and ifa is C°°, so in 7}.
Proof. Apply the contraction principle to the operator
T), : A0’1(fA‘TP”) —> A0’1(f,\'TP”)
Tm = a — NM(P,\17).

defined on a small ball centered at 0 in the Banach space A0’1(fA'TP") with the

weighted Sobolev norm Li. To prove that T is a contraction we note that:

”TAM - Tflhllm = ”NA,t(P2\771)_ NA.t(PA772)||p.A

and use some estimates of Floer. He proved in [F] that for the quadratic part Q of

(1.27), there exists a constant C depending only on ||df||p,A such that:

||Q!(€1)— Qf(€2)[[p,A S C( Héllllw + l|€2|l1.p.»\)||€1 — €2ll1.p.A (1-31)
[[Qf(€)llp.A S C IlElloo.A'||E||1.p.A- (1-32)

(Floer’s estimates are for the usual Sobolev norm, but the same proof goes through
for the weighted norms.) Since [Idfllpy is uniformly bounded by Lemma 1.2, the same
constant C works for all f,\ E I m(7,). Moreover, for t very small the same estimates

hold for the nonlinear part N A.t- Hence by (1.31):

HTWI — Tmillpa S 0( IIPA01||1.p.A + llPA772lll.p./\)“PA(771 - 772)l|1.p.A

S (7/15?2 ( llmllw + Ilmllm) - Hm - nzllp.»

20

Choosing 6 < E2/(4C) this implies
[ITATII — TAn2llp.A S 1/2 H771 - flzllpA

for any 771,772 6 B(0,5). Moreover, since ||T,\(0)||p,,\ S 6/2 then TA : B(0,6) —+
B(0, 5) is a contraction. Therefore TA has a unique fixed point 77 in the ball such that

moreover
ll'lllpA S ”TV? - TA(0)llp./\ + [ITA(0)llp.A S 1/2 ”77”va + [[TAmlllnA

so ||n||p,,\ S 2 IIT,\(0)||,,,,\ = 2||a||p,,\. Elliptic regularity implies that 17 is smooth when

ais. 0

Corollary 1.6 For t,A small enough, equation (1.30) has a unique small solution

”HUM S 5. Moreover,
L
”HUN 3 C(tlu|+Ar).

Proof. Follows immediately from Lemmas 1.2 and 1.4 and the estimate

1
llallpa = II it! - 5JfAllpA S thl + CAP- D

The gluing map. Let Cl be the set of gluing data. The gluing map is defined by
”'7, : 01—) Maps(T2,X)

776([fay9ul9 A) :- fA = CXPIA(P,\7])

where 17 = 17( f, y, u, A) is the unique solution to the equation (1.30) given by Corrolary
1.6.

By construction, 7y, is a local diffeomorphism onto its image. Moreover, if 7r{*( 1]) =

0 then f} is actually a solution of (1.26).

The obstruction to gluing. The section

at. : G! —> A°’1(fA*TP") given by

low

21

¢t(f,y,u, A) = rem) = new — 83ft) — Newman»

will be called the obstruction to gluing. Let Z, = z/J,’1(0) be the zero set of this section.

By applying the gluing construction to bubble trees in Z, we obtain a subset of the

moduli space M‘”.

1.5 Completion of the construction

We have seen in the previous section that applying the gluing construction to the
bubble trees in the zero set Z, we will get elements of the moduli space M (1.1”. It is
not clear yet why all the elements of this moduli space close enough to the boundary
stratum N can be obtained by the gluing procedure. The purpose of this section is l

to clarify this issue.

Recall the construction of the gluing map: Starting with a bubble tree we glue
in the bubble to obtain an approximate map f,\. Then we correct f; by pushing it
in a direction normal to the kernel of D,\ in order to get an element of the moduli
space M‘”. The key fact here is that the kernel of the linearization DA models the
tangent space to the approximate maps, and therefore, at least in the linear model,
it is enough to look for solutions only in a normal direction. For the construction to
be complete though, we need to show that the same thing is true for the nonlinear

problem.

More precisely, we will show that for t small, all the elements of the moduli space
Mm,” close to the boundary stratum N can be reached starting with an approximate
map and going out in a normal direction. The proof of the following Theorem is an
adaptation of the proof for the same kind of result in the context of Donaldson theory

[DK]. It is pretty technical and we include it just for continuity.

Theorem 1.7 The end of the moduli space Md.1,tu close to the boundary strata N
is difieomorphic to the zero set of the section 21),. More precisely, for 6 and t small

22

enough, there exists an isomorphism

MM,” 0 U5 ’5 {1(0) where (1.33)

U5 2 {f1T2 —) X [ 3f; S.t. f ': exph(§), [[6]]13), S 6 and [ngfll2.A S 63/2}(1.34)
and f,\ E Im'ye is some approximate map.

Proof. The proof consists of 2 steps. First, Lemma 1.8 shows that U5 is actually a
neighborhood of N in the bubble tree convergence topology. Second, recall that in
constructing the section if), we were looking for solutions of the equation (1.26) that

have the form

f=expfA(Pm) for some ||n||2,,\ S 6 (1.35)

To prove the Theorem it is enough to Show that for t small, all the solutions of the

equation (1.26) can be written in the form (1.35). This is a consequence of Proposition

1.9.

Lemma 1.8 U5 OW is a neighborhood of N in the bubble tree convergence topology.
More precisely, for any (J, tV)-holomorphic map fclose to the boundary strata/V there

exists an approximate map f,\ such that f can be written in the form

f=eXPf,(€) for some ”Ellms S 5

Proof. By contradiction, assume there exists a sequence fn of (J, th)-holomorphic
maps for tn —+ 0 such that fn do not have the required property. By the bubble
tree convergence Theorem ([PW]) there exists a bubble tree f such that fn —> f
uniform on compacts. Moreover, after rescaling the functions fn by some An, this
becomes a Luz-convergence. But this is equivalent to saying that fn is Ll'z')"1 close

to f. In particular, for A small enough, fn is Ll'm" close to f in, which contradicts

the assumption. D

Proposition 1.9 For small enough 6,t any map in U5 can be represented in the form

f = exph(P,\n) for some f,\ E Im'yc, [Inllm < 6 and 7r£*(17) = 0

23

Proof. We will use the continuation method. The key fact is that a neighborhood
of f; in Imry, is modeled by Afim and that P; spans the normal directions to Im'ye.

Let f 6 U5. By definition, there is f; 6 Imrye such that f = exp f5 5, where
”5”ng < 6. Consider the path f, = exph(s§). Let

S = {s 6 [0,1] I Elf), and ||17,||p,,\, < 6 such that f, = expfh(PA,77,)}. (1.36)

Note that by definition f = fi = exph(0) so 0 E S. We will show that S is both

open and closed and since it is nonempty, l E S.

S is closed. The only open condition in the definition of S is Hnsllpgs < 6. But since

31f... = 3d)».+DA.(PA.'7s)+NA(PA.Us) then

778 = ngs—EJfA, —NA9(P)\3773) SO
< 29' a C 2
['77:]le _ [I stllu + H JfA.[|2,A + Ellflsllm

S lngfsllz,,\ + CW/X‘f' Cllmllil (137)
We need to estimate [IBJf,[|2,A. Since

5st = ngA + SBA“) + NA(35) and Ein = 5Jf5 + DALE) + N505) then

5st = ngfl + (1 - $5fo + NA(8€) - SNAC)

The estimate (1.32) gives ||NA(§)||2,A S C ”(Him so

Hngsllm S [[EJfIHL’VX + ||51f5|l2,,\ + 2 C [[éllf’g'A < \/A + 63/2 + C 62

Therefore for A << 6,

”Estllm < 2 C 53/2 (138)

Using (1.38) in (1.37) we get

llmllu S 2 C 63/2 + CWT + Cllmllii

24

For small A S 63 , the constraint ||n,||2,,\ < 6 implies ||nsllm < 6/2 so it is a closed
condition too.

S is open. Assume that so 6 S, i.e. there exists an approximate map on such that

f,,, = exp ho (PAO(770))- We will show that s 6 S for s sufficiently close to so. For that

we need to find an approximate map fi, and an 17, 6 Ag; such that:
f: = exp;,(8€) = exp;,,(PA.m) (1-39)

It is enough to prove that the linearization of the equation (1.39) is onto at so. First

we prove that:

Lemma 1.10 A small neighborhood N5 of A in Im'y, is modelled by A0 More

1010'

precisely, there is a well defined map g : Aflow —> A251 such that any approximate map f

f 6 Inn. has the form f = exp,,(< + 1059(4)) for some 4 6 At... “cum 3 6.

Proof. The first statement is an immediate consequence of the way we constructed
the approximate maps. For the second part, notice that any f E Im'ye close to f,\ can

be written in the form f = exp f.\( x), with X small. Let X = C + Pm be the orthogonal

decom osition of X in A0 EB A0 , where 6 A0’1 recall that PX : Ao'1 —) A0 is an
P low ’7 E E E

isomorphism). Using the same techniques as in Section 1.4 we can prove that for any

C E A0 there exists a unique solution 17 = g(C) to the equation

low

77 + lepwl = 5Jf
which concludes the proof of Lemma. D

Since the notations are becoming cumbersome, we will illustrate for simplicity the

case so = 0. The general case follows similarly. Using Lemma 1.10 we can regard
the equation (1.39) as an equation in (Cm) 6 A90“, 69 A251. More precisely, for a

fixed 3 small, we need to find C 6 A2,,” and 77 6 A2; such that the approximate map
f = exp f,“ + PXg(C)) solves the equation:

€XPf(Pf’7) = eXPfJSE) (1-40)

25

The linearization of the equation (1.40) at (0,7)) is D : Afbw 69 A251 —> A0,
D(O.n)(z9 11) = Z + PAV9(Z) + PAD + 11(2, 77)
where II(z, n) is the derivative of PM? with respect to f;.

Our goal is to show that the operator D(o,,,) is an isomorphism in some appropriate

0.1
norms on A0 69 A E and A0.

low

Definition 1.11 On A0 EB A211 and A0 define the following norms:

low

H (z, n) Ila = IIZIILM + lln + V9(Z)l|2.x for any (2,11) 6 All... 69 11251

”5|le = ”1955”“ for anyt,t e A0

Consider the operator T : A0 EB A251 —> A0 given by T(z,n) = z + P),(n + Vg(z)).

low

Then T is continuous, since

llT(z,n)l|32 = HDAZ + n + V9(Z)||2.A < IIDAZIIM + “H + V9(Z)||2.A

S CA1/4llle1.2.A + lln + V9(Z)l|2.x S ||(z,n)||3,

for A small enough. Recall that the low eigenvalues of D ,\ are of order Al“, and thus

”DAZHM < ”MHZIIIJA 0“ A0

_ low'

Lemma 1.12 For A,6 small enough T is invertible, with the operator norm of the

inverse uniformly bounded [lT‘lll S CT (independent of A,6 ).

Proof. Leta = z + PX(n + Vg(z)). We need to estimate ||z||1,2,)( and “n + Vg(z)|[p,X
in terms of ”alle— Since DAG = DAz + n + Vg(z) then

26

MH + V9(Z)||2.A < ||O||82 + “DAZHZA S HOHBz + CAI/4HCH12A

S “OMB: + CAI/4 “01 — PM“ + Vg(z))lll,2,5

S Halls; + CAI/4 ||a||3, + CAI/4 H11 + V9(Z)||2.A

So for A small we get the uniform estimate ”n + Vg(z)||2,,\ < C1||a||B,. Then

llzllma = II0 - P501 + Vg(z))||1.2.A S Halls; + Cllnllza S C2llall32

thus ||(z,n)||1_r;1 S CT||T(z,n)||B,. This implies that T is injective. By construction,
index(T) = 0 so T is invertible, with [ITTIII S CT ( independent of A,6 ). D I
Lemma 1.13 Forz small, ||II(z,17)|[3, S C||n|[2,X||(z,0)||3,.

Proof. By differentiating the relation Dfan = n with respect to f at f,\ we get

301(Pxn)(2) + Dr(11(zn7)) = 0 8°

N DA(I1(an)) “2.5 = II 301(PM7)(Z) ”2.5.

Using the expansion of

0,6 = (v: + M) 0 V: on + éwatm

NIH

(cf. [MS]) then
||3D1(PA77)(Z) H2» S C'llleoo.5llP5nl|1,2,,\

uniformly in a neighborhood of f X. Therefore

||11(Z,n)||32 = IID;(11(z,n))||2.A S CIIleoo.,\lle||1.2,5

S Clllexmllnllm=C||2||82||77||2.A- D

27

If we choose 6 small enough then for “17“” < 6, ||I1(z,n)||3, S CT/2 ||(z,n)||1_r32
where CT is the constant in Lemma 1.12 so D(o,,,)(z,n) = T(z,n) + II(z,n) is still

invertible. This concludes the proof of Proposition 1.9. D

1.6 The leading order term of the obstruction wt for t small

Next step is to identify the leading order term of the section if), as t —) 0. Let N

denote some stratum of 175 and G1 —> N denote the gluing data as in (1.18). For
the sake of the gluing construction, the gluing data has to be defined on the domain

of the bubble tree. But we will see in a moment that the important information is
encoded in the image curves. Introduce first some notation: If u,- 6 Ty,5'2 is a unit ‘

frame and A,- is the gluing parameter, let
v, = A,- - u,- E TMSQ, (v,- # 0) denote the gluing data.

Definition 1.14 For any [f,y,v] 6 01, such that f : B —> P" is an element of N,
let ([f,-,y.-,v,-])}"_:_l be the bubble maps together with the gluing data and let u be the

image of the ghost base (so u = f,(y,-) for allj E B). Set

am, .21. v1) = i8 «n(ijvj). X.- >w.- (1.41)
'=1je§
17(x) = if!“ V(z,u) , w,(z) )w, (1.42)

where {a}, = ngz, i = 1,n} is an orthonormal base of H1(T2,u*TP"), X,- 6 Top"
and B is as in (1.3).

Note that a depends only on the gluing data on the first level B of essential
bubbles, and 17 depends only on the image of the ghost base. Then

28

Lemma 1.15 Using the notation above, let f) be an approximate gluing map. Then
fort and [AI = «A? + . . . A? small enough,
my) —_- 17(u) + O(|A|) (1.43)
was) = a([f,y,vl) + 0(lAl3/2)- (1.44)
and the section 16, has the form

MU, it v]) = Mu) + a([f.y, v]) + 0(IAI5/4 + t IAI + t2). (1.45)

The estimates above are uniform on N.

Proof. For the first 2 relations, it is enough to check them on components. As-

sume for simplicity that B consists of a single bubble [f,y,v]. If (.0 = X dz is an -

element of the base for H 0", let (J; be the element of the local orthonormal frame for

A?,;,lu(f,\'TP") provided by Lemma 1.4. Then

Klaus-55H S llVlloonA—wxllm S CA so

(MBA) = (V,WA>+0(A)

On the other hand, using the definition of w;

(M) = / <u(z,f(y)),w>= /<V(z,f(y)),w)+0(A) so
Izl>\/X T2

(mu—15> = /<V(z,f(y)),w)+0(x\)

T2
which gives (1.43). Similarly,
[<5Jf6’wA-EAH S [ngfAII2,A[|wA—E,\||2,A g CAI/2), S CA3”

and using the estimate (1.16) and the definition of w; we get

\/X

6'

figlzlgm/X

dfi(df(y)(U),X> + 0W) = A(df(y)(U)a X) + 0W)

 

(ng/MWA) =

29

Combine the previous 2 relations we get
(51f5,51)=(df(y)(/\U),X)+ 003/2) = (df(y)(v),X) + 003/2)

which implies (1.44).
The general case when B has more bubbles follows in a similar maner using the

relation (1.19) and the fact that w is 0 pass the first level of nontrivial bubbles.

Finally, the relation (1.45) is a consequence of (1.15) provided we have an estimate

of the the quadratic part. For that use (1.32) to get

( NA(PA77) , «75) S ||N5(P517) “4/33 ”5AM“ S Cllnllm Hulk/3,5 IIWI|4

s OHM”? + t) 0(IAI3" + t).

Thus the quadratic part is 0(IAI5/4 +t [Al + t2). [:1

The definition of L —> L75 . From this point on, since we are going to look at
the leading order term, it will become easier if we forget part of the gluing data. We

have already observed that the map a depends only on the gluing data on the first

level B of essential bubbles. Moreover, if we denote by

w = Z df,(y,~)(v,-) 6 RP” (1-46)

.765

then the map a and the linear part 1,6, of wt become respectively

«M = i<w,Xi>wi (1.47)

i=1

than) = t17(u)+a(w) (1.48)

Introduce a space W together with a projection 7r : W -> 175 such that the fiber of
7r at a 1-marked curve (possibly with more components) is the span of the tangent

planes to all the image bubbles that meet at the marked point. By definition w E W

30

so (1.46) defines a projection p : Cl —-> W. Note though that 71' : W —-> 175 is not a
vector bundle, and that W it is equal to the relative tangent bundle L —> 115 on the

top strata of 35.
Here is a more precise description of W. Stratify Ed by letting 3;, be the union

of all boundary strata such that the image of the marked point is on h nontrivial

bubbles, i.e.
Z), ={sz—1P" | Bhashelements} (1.49)

Note that 72 D 773 D . . . and each 7;, is a smooth variety with normal crossings. For

transversality arguments we need to use the moduli space 2;, obtained from Z), by
collapsing all the ghost bubbles up to the first level of essential bubbles. The natural
projection

q : Z}, —) 2),
has fiber (10,}, = Mo,h+1, the moduli space of h + 1 marked points on the sphere.

Moreover,
dimZ), = n — h —1 and diméh = n — 2h +1 (1.50)
In particular, 2;. ¢ (6 only for h S [13,31].

Let L,- be the pullback of the relative tangent bundle to the i’th factor of 2),. When

the constraints B1, . . . , B), are in generic position, the fibers of L1, . . . , L), over a point
in 2), are linearly independent subspaces of P". This is because linear dependence

imposes n + 1 — h conditions, and 2'), is only n — 2h + 1 dimensional. So on 2;,
leh =q‘(L1 EB..-€BLh) (1.51)

Remark 1.16 Since not all the gluing parameters can be zero, a dimension count
argument similar to the one above shows that w defined by (1.46) is an element of

W - {0}, the space nonzero vectors in W, thus p : C1 -—> W — {0}.

Note that leh is nothing but the normal bundle of Z), in H5, for any 2 S h S

[133-1]. This observation allows us to get a line bundle out of W as follows:

31

Definition 1.17 Let N = [1311]. Blow up 175 along ZN (the bottom strata), then

blow up the proper transform of ZN_1 and so on, all the way up to blowing up the

proper transform of Z; and denote by
p : L75 —> L15

the resulting manifold. Similarly, after the first blow up, extend L over the excep-
tional divisor 2N as the universal line bundle over P(N2N), the projectivization of

the normal bundle of ZN, and so on. Let L —> L75 denote the blow up ofL constructed

above.

By definition, the total space of L —) L75 is the same as p“(W). From now on, we will

make this identification.

Note that both the map a and the linear part 1;, of w, pull back to L — {0} as

a(w) = :(w,X,-)X, (1.52)
1,6,(w) = t17(1r(w))+a(w) (1.53)

where 7r : L —) P" is the composition L —> L75 3+ P". For simplicity of notation, we
have also denoted by ev : L75 —) P" the composition L75 —p)Ll5 fi+ P". Note that by

definition, a is a linear map but {6, is not, and we have the following diagramm:

L—{O} “—1”; ev“(TP") TP"
1 1 110
L1,, = V. 2+ P"

Proposition 1.18 As t -) 0 the zero set of the section if), is homotopic to the zero

set of its leading order term

223. : Z — {0} —> ev"(TP")

32

Proof. In generic conditions and for t small enough the zero sets of both sections
w, : Cl —) ev’(TP") and p‘p.(wt) : L — {0} —> ev‘(TP”)

consist of points lying on the top stratum of L15 and L75 respectively. But on the top
stratum, the projection pr : Cl —> L — {0} is an isomorphism, thus the two zero sets
are diffeomorphic for t small. Note that (1.45) gives

p.(t/)t(w)) = “7(a) + a(w) + 0(lwI5/4 + t le + t?)

Finally, Lemma 1.19 gives that w = 0(t) on the zero set of 16,, so

p.(¢t(w)) = Mu) + a(w) + 0(t5")

giving the desired homotopy as t —> 0. Cl

Lemma 1.19 The linear map a : L— {0} —) ev"(TP") defined in (1.52) has no zeros
when the constraints B1, . . . ,5) are in a generic position, thus there exists C > 0 such

that
[a(w)] Z Clwl (1-54)
Moreover, there exists a uniform constant C on L — {0} such that the zero set of if),

is contained in [w] S Ct.

Proof. First part is a standard transversality argument and dimension count. Note

that a induces a map
a (8) id: L (8) L” —> ev"(TP”) 8) L” i.e.
a®idzfi5 x C ——> ev*(TP”)®L‘

Because of the C'-equivariance of a, the zero set of a : L — {0} —> ev“(TP") is the

same as the zero set of the section
5 12,, —> ev‘(TP") a Z:

5(x) = (a®id)(x,1)

33

If the constraints Bl, . . . “6;. are in generic position, then it is transverse to the zero

set of ev"(TP"). But the base L75 is only 12 — 1 dimensional, while the fiber is n
dimensional, so generically ii and thus a has no zeros.

For the second part, note that on the zero set of p.(1,b,)
0 = my.) = a(w) + tV(u) + 0(lwl5/4 + lel/2 t + t2) so
a(w) = —t17(u) — 0(|w|5/4 + lel/z t + t?)
which combined with (1.54) gives

C|w|S|a(w)| 5 t|17(u)|+C(|w|5/4+|w|1/2t+t2) i.e.

(tune—61mm) 3 Ct

For t and w small, the left hand side is positive, completing the proof. D

1.7 The enumerative invariant 75

Next step is to find the zero set of the leading order term of IPt- As a warm-up we
will discuss first the limit case t = 0. The constructions described in the previous

sections apply equally in this case, giving:

Proposition 1.20 Let N be a ghost base boundary stratum of H5. Then the moduli
space of J -holomorphic tori close to N is isomorphic to the zero set of a section in

the obstruction bundle over the space of gluing data

Wlfuyuvz‘lili) = a([f:,y.',v.']]?‘___1)+ GUNS/4)

where a is defined by (1.41). Moreover, for generic constraints 61,. . .,B1, the number

of J -holomorphic tori that define the enumerative invariant

7.1%,. . . fix)

is finite, and the moduli space of these holomorphic tori is at a positive distance from

the ghost base boundary strata of the bubble tree compactification.

34

Proof. For the second part, note that w and A'lz/J have the same zero set, so as
A —> 0 the limit of the end of the moduli space of J—holomorphic tori is modeled by
the zero set of the section a. But we have seen that generically a has no zeros, and
thus there are no J -holomorphic tori in the neighborhood of that boundary stratum.

C]

Now we are finally ready to evaluate the contribution from the interior.

Proposition 1.21 Fort small, the number of (J,tV)-holomorphic maps that satisfy
the constraints in the definition of RT5,1(B1 I B2, . . . H61) and are close to some (J, 0)-

holomorphic torus is equal to

and(flla ° ' ° Hal)

where n,- = [Aut5,(j)| is the order of the group of automorphisms of the complex .

structure j that fix the point x1.

Proof. Recall that RT5,1(BI | S2,...,B1) counts the number of solutions of the
equation

CJf($) = V($af($))

such that f(xl) 6 fll and f passes through 32, . . . ,6).
A generic path of perturbations converging to 0 provides a cobordism M " to the

solutions of the equation
5Jf($) = 0

such that f(xl) 6 31 and f passes through B2,...,fl1. A (J,0)-holomorphic torus
f : T2 —-> P" is a smooth point of this cobordism, i.e. all the intersections are
transversal and the cokernel H0'1(T2, f”(TP”)) vanishes (since f’(TP") is positive

for the standard complex structure).

But the invariant r5(,81,...,31) counts the number of such solutions mod the
automorphism group of j. Imposing the condition f (x1) 6 51 reduces the stabilizer

to just Aut5,(j). Cl

35

Remark 1.22 Note that the pertubed invariant counts the number of (J, V)- holomor—

phic maps with Sign. This sign is determined by the spectral flow of the linearization
D, to 55. In the limit, when V = 0, we have D; = 5,; thus all (J,0)-tori have a

positive sign. This agrees with the way they were counted classically to obtain 75.

Lemma 1.23 For generic V the section 17 : ev.(fi5) —+ TP” defined by (1.42) has no

ZCT‘OS.

Proof. For generic V, the section V is transverse to the zero section. But the fiber of
T P” is n dimensional, and the base Im(M) = ev.(L_(5) is only it — 1 dimensional, so

17 has no zeros generically. 0

Remark 1.24 The zeros u E P" of 17 give the location of the point maps u that can be
perturbed away to get genus one (J, V)-holomorphic maps representing 0 E H2(P"). .
Since index=0 then generically 17 has finitely many zeros. But Im(M) is a codimension

1 subvariety in P" that doesn’t depend on V. Then we can choose V generic so that

its zeros do not lie in Im(M), and thus V(f(y)) 75 0 for any [f, y] 6 H5.

Moreover, Lemma 1.18 showed that as t —> 0 the zero set Z, of 4’: is homotopic

to the zero set Zo of the map
wo : L — {0} —> ev"(TP”)
211000) = 17(7r(w))+ a(w)
where a, 17 are defined in (1.52), (1.42) and it : L —) P" is the composition L —+

L75 9), P". We have also made a change of variables w —> w/ t.

We are ready now to identify the zero set Zo. Since 17(u) 75 0 on lm(M) then it

induces a splitting of the obstruction bundle:
TP"|Im(M) = C < 17 > 63 E (1.55)
where E is an n — 1 dimensional bundle, so

ev"(TP”) = C < 17 > EB ev'E (1.56)

36

Lemma 1.25 The number of zeros (counted with multiplicity) of wo is equal to

~

Cn—1(€V'(E) <3 L')
Proof. Using (1.56) map ibo : L — {0} -—> ev*(TP”) splits as

w1(w)= 17(7r(w)) + a1(w) (1.57)
ib2(w) = a2(w) (1.58)

where a,- denotes the corresponding projection of a(w). The map a; : L — {0} —>

ev"(E) is C‘-equivariant, so tensored with the identity on L“ induces a C"'-equivariant
map

(12:21.1 x C' —> ev'(E) (8) L"

that has the same zero set as a2. Let
52 :Ll5 ——) ev“(E) (8 L” given by 52(x) = ("12(x, 1)

Then the zero set of a2 is equal to Z (52) x C‘. To find the zero set of wo, for any
(x, v) E Z(52) x C“ solve the equation

0 = 1/21(x,v)= 17(x) + a1(x,v) = 17(x) + v - a1(x,1)

Note that a1 3:5 0 on Z(a2) since a has no zeros so for any x E Z(Zig) there exists a
unique v E C“ such that

—V(x) = v - a1(x, 1)

This implies that there exists an isomorphism between the zero set of zbo and the zero

set of 5;. To complete the proof, note that for generic V the section 52 is transversal to

the zero section of ev"( E) (8?, so its zero set is given by the Euler class of ev*(E)® LI.

[3

Finally, we can compute the boundary contribution

37

Proposition 1.26 Fort small, the number of (J,tV)-holomorphic maps that satisfy
the constraints in the definition of RT5,1(51 I 32,. . .,B1) and are close to the boundary

strata {x1} x H5 is equal to

2 2+2

i=0

)ev'(H"_l"') - c'l(L‘)

where L is the blow up of the relative tangent bundle L defined in (1.17).

Proof. As we have seen previously, the moduli space of (J,tV)-holomorphic maps
that satisfy the constraints in the definition of RT5,1(BI | ,82, . . . , B)) and are close to

the boundary strata {x1} x H is diffeomorphic to the zero set of the section wo. Using

Lemma 1.25, the later is equal to

~

c._1(ev*(E) a Z‘) = Z ev‘(cn-.--1(E))-ci(L‘)

But by definition c,(E) = c,-(TP”) = (”T1)Hi, completing the proof. D

1.8 The other contribution

In the previous sections we have described in great length the gluing construction
corresponding to the strata {x1} x L(5, that consists of a ghost base and a bubble
at the marked point x1. Finally, it is the time to sketch the gluing construction
corresponding to other boundary stratum T2 x ev'wi) and to explain why it does

not give any contribution.

Proposition 1.27 Fort small, the number of (J,tV)-holomorphic maps that satisfy
the constraints in the definition of RT5,1(fi1 | ,82, . . . ,BI) and are close to the boundary

strata T2 x ev‘(fl1) is equal to 0.

38

Proof. Construct first the space of approximate maps. The only difference from
the gluing construction decribed in Section 1.2 is that we need to allow the bubble
point x E T2 to vary. Since the tangent bundle of the torus is trivial, choose an
isomorphism

TT2 2’ T2 X C
which gives an identification T5T2 E C for all x E T2 (providing local coordinates on

T2). The set of gluing data will then be modeled on:
T2 X Fr X (0,5)
where
Fr = { [f,ywl l [f,y] e ev‘wl), u e Ty52|u|=1}
is the restriction of the frame bundle over L15 defined by (1.9).

To glue, use the unit frame it E TyS2 to identify T5,T2 E Ty52 which will induce

natural coordinates on the sphere via the stereographic projection.

Then all the constructions decribed in Sections 1.2-1.7 extend to this case. Since

the holomorphic l-form w E H0’1(T2, C) is constant along the torus, then the isomor-

phism between the obstruction bundle and ev"(TP") is independent of the bubble

point, so
H‘"1 E p‘ev*(TP") ev‘(TP")
\« I/ l
T2 X ev“(fi1) 41-) ev‘(fi1)

Moreover, the linear part of the section w, that models the end of the moduli space
is also independent of the bubble point. But a dimension count shows that the zero

set of a T2-equivariant section in the obstruction bundle must be empty generically.

Cl

2 Applications

In this second part of the paper we explain how one can compute the top power inter-
sections c’[(L‘)ev“(H "‘14) involved in Theorem 0.1. The programm is simple: first
we find recursive formulas for the top intersections c‘[(L")ev"'(H ”'14) (see Proposi-

tion 2.2), where L is the relative tangent bundle of L15, and object well known to the

algebraic geometers. Next we exploit the fact that L is a blow up of L to compute
its coresponding top intersections recursively in Proposition 2.5.

Unfortunately, the notation becomes quickly pretty complicated if we insist on
keeping track of all the information, so we chose to indicate at each step only the new

changes, leaving out the data that stays the same.
Notations. If Bo, . . . , B), are various codimension constraints let

llama; 31,-..,flk)= eVmwo) [ud( ifila-°-aIBk)]

denote the moduli space of 1-marked cuves in P" passing through Bo, . . . , Bk, such

that the special marked point is on Bo and let

Md(/6031819 - - ' afik)

denote the corresponding moduli space of curves (in which we forget the special

marked point).

In particular, let L15 = L15( ; (31, . . . ,flk) be the moduli space of 1—marked curves

that appears in Theorem 0.1. If i, j Z 0 are such that i + j: dim L15 then let
am I 31.....31) = cm ev*(HJ') [at] (2.1)

denote the top intersection. Moreover, if L75 is the blow-up L15 as in (1.17), let
x = c1(L“) e H2(L(5,Z), a = c1(L')€ H2(L75,Z) and y = ev*(H) (2.2)

39

40

where y 6 H2(L15, Z) or y E H2(L15, Z) depending on the context. Note that
$11103]. | ' ) = a«"iyj [L15]: 15" [u5(HJ'; )1 (2-3)

Remark 2.1 Using the notation above and the degeneration formula (A.2), Theorem

0.1 becomes:

"ml ' ) = Z 05(H‘1,H‘2, ')+ E: (n +1) rift/"‘1’i ' [L75] (13-4)

f1+i2=n i=0 z + 2

2.1 Recursive formulas for c‘i(L*)ev*(Hj)

Let L15 be some r-dimensional moduli space of 1-marked curves of degree d through ,
some constraints ,8“. . .,,8k (not necessarily the same as in Theorem 0.1) and let

L —-) L15 be its relative tangent sheaf. In this section we give recursive formulas for
am I -) = ciiL')ev‘(H")[udJ
where the constaints 31,. . . , )8), are dropped from the notation.

Proposition 2.2 For every r-dimensional moduli space L15 of any degree d 2 1, there

are the following recursive relations:

¢d(0aj I ') = 03(Hj, ') (2-5)
. . 2 . . l . .
¢d(l +13] l ) : _'d_¢d(zi.7 +1|°)+ E¢d(za.7 l H29 ') (26)
d2 ' ° fl i2
+ d§=dfi¢danJlH a ')'0d2(H 9 )
1 i+l 2 d3 ' 1 ' Hi; 2 7
+ (— ) dl+d2=d gash“ _ ,21[°)'0'52( a ') ( ' l

for any i 2 0, where the sums above are over all possible distributions of the con-
straints 31,. ..,Bk on the two factors and d1,d2 7i 0. When i = 0, the last term in
(2.6) is missing.

41

Proof. The first relation follows by definition, and provides the initial step of the
recursion. The second one requires more work. In what follows, we will identify a

cohomology class like c1( L) with a divisor representing it. Then:
Lemma 2.3 On L15, we have the following relation:

1 2 l
c1(L‘) - —7-t - —ev'(H) + — Z d§M5,,5, (2.8)

— d2 d d2d1+d2=d

where ’H denotes the extra condition that the curve passes through H2, and M5,,52
denotes the boundary stratum corresponding to the splittings in a degree d1 I-marked
curve and a degree d; curve, for d,- 75 0 (for all possible distributions of the constraints

B1,. . . ,flk on the two components).

Proof. Fix 2 hyperplanes in generic position in P". Each curve in L15 intersects a

hyperplane in (1 points. Then the moduli space Y = ev;+1 (H) flev2+2( H) of l-marked

curves passing through 61,. . . , m, H, H is a d2 fold cover of L15:

"zy—‘TUd, [fayla°°°ayl¢9a7b;y]—)[f7y1$"'3yk;yl
Define the section

(a - b)dy
y - a)(y - b)

 

s(lfiy1,'--,yk,a,b;y]) =(

Then 3 is a section in the relative cotangent bundle L‘, and it extends to the compact-
ification L75. As a and b are getting closer together, the section 3 converges to 0. Thus
its zero set is the sum of the divisors {a = b} and M(y ; a,b), where M(y ; a,b) is
the sum of all boundary strata corresponding to splittings into a degree d1 l-marked
bubble and a degree d2 bubble containing a, b for d = d; + d2. Note that d,- 75 0. The
infinity divisor is {y = a} + {y = b}. Thus

7r"(cl(L")) = {a = b} + M(y; a,b) - {y = a} - {y = b}

Note that
52.555 = «newt?»

42

When projecting down to L15, the divisor {a = 6} becomes ’H, and the divisors
{y = a}, {y = b} become each d - ev‘(H). The rest amounts to summing over all
codimension 1 boundary strata. The boundary strata ./\/i5,,52 appears with coefficient
d3 in 7r.(M(y,a ; b)). Combining all the pieces together completes the proof of

Lemma. Cl

Remark 2.4 We could have chosen any 2 marked points out of the already existent
ones, and then express c1(L) in terms of them. But then this expression would not
look independent of choice. Nevertheless, with some work, one can actually see that
all these divisors are homotopic. We have chosen to introduce 2 new marked points

to avoid this issue.

Relation (2.8) provides the basic relation for proving (2.6):

2

, 2. 1 . d .
61+‘(L') = "61(0) - ev“(11) + —C’1(L‘)-‘H + E: -’- 61(L") - Mm.

d d2 d1+d2=d d2
so taking a cup product with ev“(Hj) we get:
. . 2 . . 1 . . 2
¢d(z +13] l l : _Eqbdhaj +1l)+ §¢d09J [H 'I ) (2'9)
d; i L:- I1 Hj M
+ Z 8-2- Cl( )ev ( )' d1.42
di+dz=d

Next, we need to understand the restriction of L to the boundary stratum M5,,52.
Let
P : M41312 _> udl

be the projection on the first component (the one that contains the special marked

point y). If A, B are the 2 special points of M5,,5, (where the 2 components meet),
and

evA X evB :L15l X .6452 —-> P" X P"

be the corresponding evaluation map. Then by definition

M5152 = (evA X eVB)‘([A]) (2.10)

43

where A is the diagonal of P" X P". Moreover, it is known that as divisors,
61(Lfl/Mdiidz :ptc1(L:t)+{y = A} (211)
where L A = Llu d1 is the relative tangent bundle of L15,. Next step is to find
' o i Z I: i— I:
CI(L )/Md1,d2 = 2(1) p Cl ((LA) . ({y : AI)! (212)
1:0

For the self intersection of the divisor {y = A} note that its normal bundle N inside

M5,,5, is nothing but p‘(LA)/{y = A}, so for l > 0,

({y = Al)’ = CAN)“ = (-1)"1P'C'1"1(L'A)' [{y = All

Substituting in (2.12) and after some algebraic manipulations we get:
61(13): [Mdhdal = P‘Cl(L31)+(-1)"‘P‘Cf"(L2) ' [{y = AH (213)

Compute the intersections above inside L15, X M5,. Then (2.10) combined with

the relation [A] = )3 H"1 X H ‘9 gives

1°1+12=n

eVI(Hj) ' [Mdmbl : Z udI(Hji Hi1, ' ) X Md2(H£29 ')

ii+i2=n

ev"(H") . [{y= ad] 2 ud,(H"1 ; -)x M5,(H‘2, .)

11+12=n+j

where we sum over all possible distributions of the constraints on the two components.

The relations above imply

ev‘IHJ‘) -p*c:(L:.) - [MW = Z (ciao-1115111; H“, -))>< Mam”, -)
i1+i2=n
= 23 and I H“, .).a.,(H12, -) (214)
i1+i2=n
eV‘(Hj) 'P'Ci—1(L'A) ' [{y = yAll = Z (Ci-1(LA) 'UdIULIi1 ; '))X M5,(H‘2, ')
ii+i2=n+j

= Z ¢d1(i_19il l ' )' ad2(Hi21 ') (215)

i1+i2 =n+j

44

Substituting these relations in (2.9) using (2.13) we get (2.6), which concludes the

proof of Proposition 2.2. D

2.2 Recursive formulas for c’i(f.*) -ev*(Hj)

Let 11.1 = Ud( ; ,81, . . . ,flk) the some r—dimensional moduli space of 1-marked curves
and L7,; be its repeated blow-up along 293:1], . . . , Z; as in (1.17). Next we give recur-

sive formulas for computing the top intersections iiyj:

Proposition 2.5 If ad is some r-dimensional cutdown moduli space of 1-marked

curves then for i +j = r, the top intersection

 

~. .~ .. (51+52_1)i] . .~ ~

x‘JU = x'JU - ~ ~ “”UIXU2 2.16

yld] yld] .2.l<1-xl)<1—x.)._,yly2[d m )
J1+J2=fl+1

where i,- = c1(Z‘|ad ) and y,- = ev“(H) on 114., with d, ¢ 0 fort = 1,2.

Proof. Recall the construction of ILL: starting with Ltd, we first blow up along ZlgétL],

then we blow up the proper transform of Zl"—§’—‘] and so on, up to blowing up the

proper transform of 22. Since E extends as the blow up of L then

~ [ii-‘1 ~
c.(L) = 61(L) — z: z. (2.17)
h=2

where 2;, is the exceptional divisor corresponding to Zh.

For simplicity, let N = [11%;] and E”) be the partial blow up of L along ZN, . . . , 2;,
only. Denote by
52(h) = —c1(Z(h)) (2.18)

Recall that an element of 2;, has h components having a point in common, the

image of the ghost base. Then:

45

Lemma 2.6 Using the notations above,

 

(El-1»...+Ez':,,—1)i
if:

Why-[am = E‘(h+1)yj[filh+l)l— (1-5,)....-(1- h) .

7r‘yj [1042119)
h

where i,- and y.- are c1(Z“) and ev"(H) corresponding to the i’th component of each

element ofprZh, the proper transform of Zh.

Proof. Let 1r : L25") —-> 1.75““) be the blow up of “15"“) along prZh, and 2;, be that

exceptional divisor. Then
claim) = 1r’c1(L(h+1))—[Zh] i.e. :1:(h) = 7r“:c(h + 1) + [21,] so
5‘(h)yj[1«7§h)l = 7r‘( m + 1),.- )+ 2 (DH Mk +1)yj )[Z'hJ’ (2.20)
(:1
Thus we need to understand terms like
7r"(a)[2h]l for a = Ei'1(h +1)yj, [Z 1

But this kind of intersections were computed by Fulton [Ful]. Here is a brief sketch
of the argument: Let N be the normal bundle of the proper transform of Z}, and
E = ON(1) be the hyperplane class in P(N). Recall that Z}, is codimension h.

Combining the relation
21.: c‘.“‘(0~(—1))
with the definition of the Segree class of the normal bundle ([Ful] p 47)
3,-(N) '0 = "-(Ci'l+'(01v(1)) ' 7r‘(a))
one gets

7T.(7T'(0) ° [Zhlll = (4)1.la ' Si—hUV)

46
In our case we look at top intersections (i.e. |a| +1 = r) so for I Z 1
7r'(a) - [23,]! = (—1)'_loz - s,-_h(N)[prZh] (2.21)

as integers. To complete the proof, we need to find the Segree class of N.
Note that 2;, has components indexed by the different distributions of the degree
on the h bubbles:

where d.- 74 0 for i = 1,. . . ,n. When we blow up, the components of prZh become

disjoint, and the component per, ,,,,, d». can be viewed as a subset of

(M0.h+l X IL, X . . . X (Lg/Sh (2.22) .

where Mo'h+1 is the moduli space of h+1 marked points on a sphere, and the symetric
group 5;, acts freely on the h bubbles by permuting them (giving the same bubble

tree). Consider the projections:

~

~ ~ 1". ~
M0,}..H X 21,11 X . . . X 21¢, ——+ ad.
Jr 7T J, 71'

p.
M0,h+1 X Lid, X . . . X Udh ——-) Lid,

be the projections on the i’th factor, and L,- be the relative tangent bundle of the i’th

factor. The normal bundle of Z}, is isomorphic to
p'i'Ll EB . . . 6917214.
so the normal bundle of the proper transform of 2;, inside 215"“) is
N =flZ1®...®p‘,§Zh,

where Z,- is the full blow up of L,- as in (1.17). Thus:

 

s(N) = _ (2.23)

47

where 55.- is the pull back via p,- of the first Chern class of the relative cotangent bundle

Z" of Lid... Putting together (2.20), (2.21) and (2.23) we get:

Emmi") = 5‘(h+1)yi[fi.$"“’1

 

‘ i 1 ~. . ..
- I" 7r" 1"", h— J rZh
+ 53W 1) [(1—51)-...-(1—a)l._. ( ( ”y )[p ]

l=h

(7r‘5(h +1) —1)i
(1—51)-...-(1—§5h)

 

= 5‘(h+1)yj[57.ih+l)l‘[ l "'yjlprzhl

But
5(h+1)[przh]=51+...+§fh

which gives (2.19). D

Next, note that all the cohomology classes involved in the last term of (2.19) are
pulled back from 2h, the result of collapsing the ghost bubbles in prZh to a point.
But note that dim 2;. < dim 3;, unless h = 2 (the difference is dim Mo,h+1) so a
top intersection on 2;, of classes pulled back from 2;, vanishes unless h = 2. This,

combined with (2.19) and the fact that ZU‘) = L when h = N + 1, implies:

(51 + 52 —1)i
(1 — 551)(1— Eh

 

mad] = x‘yjlu.1—[ )l «‘yiiprza (224)

But prZ; = 22 = ev5(A) for A is the diagonal of P" so

1 ~ ' ~ ' i .‘2 ~ ~
[Przzl = - Z [Ud.(H”;') Xud2(H";-)l= Z yl‘yz lad. xudzl (2.25)
2 d1+d2=d d1+d2=d
31+32=n c1+u2=n

The factor of 1/2 in front of the sum comes from action of the symmetric group
S; on the 2 bubbles (yielding the same bubble tree). After distributing y on the 2

components (2.24) becomes (2.16). D

48

2.3 Applications to P", n g 3

In this section we apply the inductive algorithms described in the previous section to

compute the elliptic enumerative invariant rd in P2 and P3.

Proposition 2.7 The number Td(p3d’1) of degree d elliptic curves in P2 with fixedj

invariant and passing though 3d — 1 points is

”(pm—1) : 3((1 - l)0d(p3d-l) (2.26)

nJ- 2

where 04 is the number of rational curves through 3d points. and n,- is the order of

the group of automorphisms of the complex structure j fixing a point.

Proof. For n = 2, relation (0.2) gives:
njrd(p3d“) = 0.,(1, 1, p3d-1) — 3ev‘(H) — c1(i') (2.27)

where L —> 114 is the relative tangent bundle over the moduli space of l-marked
rational curves of degree d passing through 3d — 1 generic points and I: is its blow up
as in (1.17).

The moduli space Md of unmarked curves is n — 2 = 0 dimensional, consisting
of od(p3d'1) curves. In particular, we do not need to blow up to do the intersection

theory, i.e. Z = L. Using (2.2) (or easier by inspection)

2
61(L') = -g0d(l,P3d-1) = -20d(P3d—l)

e“v“(H) = 02(l.p3d"‘) = d0d(P3d-l) and 0d(l.l.p3d“) = dzad(p3d‘l)

So pluging them back in (2.27) we obtain

1
Td(P3d'l) = —(d2 - 34 + 2) 0d

"J”

49

which gives (2.7). C]
In particular,

(d;‘)ad if j ¢ 0,1728

3.1—1) (cl-1),” ifj = 0 (2.28)

Td(P
(d-‘)ad ifj = 1728
This formula was recently obtained by Panharipande [Pan] using different methods.

Similarly, we can prove that:

Proposition 2.8 The number rd 2 rd(p“,lb) of elliptic curves in P3 with fixed j
invariant and passing through a points and b lines (such that 2a + b = 4d -- 1) is given

 

by:
2 d — 1 d — 2 2
Ta = ( )( )od(l) — —- Z d2(2d1d2 — d)oagl (1)ch2 (2.29)
dn] dnJ d1+d2=d
where od(l) = od(p“,lb, l) is the number of degree rational curves in P3 passing

through same conditions as rd plus one more line. By the term 0311(1)on we understand
the sum over all decompositions into a degree d1 and a degree d2 bubble such that the

constraints are distributed in all possible ways on the bubble, and d,- 75 0.

Proof. When n = 3, Theorem 0.1 becomes:

nJ-rd(p°,lb) = Z od(Hi‘, Hi9,p°,lb) — 6ev'(H2) — 4ev'(H)c1(Z') — cf(Z"‘)(2.30)
t1+t2=3

The moduli space Md of degree d unmarked curves passing through a points and b

lines is n — 2 = 1 dimensional, with a finite number of bubble trees in the boundary.

Then Proposition 2.8 is a consequence of (2.30) and the following Lemma after some

simple algebraic manipulations:

50

Lemma 2.9 We have the following relations

2 od(Hi‘, Hi’,p“, 1”) = 2d - ad(p°,lb+1) (2.31)
f1+t2=3
ev'(H2) = Ud(p“,lb+l) (2.32)
an ~71: :- t 1 1 2
ev (H) - c1(L )= ev (H) - c1(L )= —dad(l) + 2 Z dldzod,(l)od,(2.33)
d1+d2=d
c1(L")2 = — Z dgad,(l)od, (2.34)
d1+d2=d
c1(Z')2 = —2 Z d2ad,(z)ad, (2.35)
d1+d2=d

Proof. Relations (2.31) and (2.32) follow immediately by definition. The remaining

relations involve more work. To prove (2.33), use (2.8) to get

1

eV*(H)°Cl(L‘) = fiev‘ui )[Ud( ;H2)l—-0d(H2)+ +E Z ‘12 0011(H Hil)0d2(H.2)
51:33.3.
1 1
= 703mg.- 2 d§(adl(H,1)ad2(H)+ad,(H,H)ad,(l))
d1+d2=d
1 1 2 2
= —d0d(l)+—d_5 Z d2(d1d20dn(l)0d2+d10dlad2(l))

d1+d2 =d

which gives (2.33) after switching the indices in the second sum.
For simplicity we have recorded only the new constraints, whereas the old ones

get distributed in all possible ways. The equality between ev'(H) - c1(L") and ev’( H)

c1(Z") is a consequence of (2.16).

Next, we need to evaluate cf(L"). We use (2.8)

afar) = -§CI(L*)-ev*(H>-[ud1+ Z gamma-[um;H‘*,-)1-a..(H‘a-)
1 ,_ di 1' i
+ ECI()L)[uth2,')]_d;_da-2—[ud1(Hl;')]'Ud2(H2,’) (2°36)

I1+32=3

51

The first term is given by (2.33). Using (2.8) again, we get:

01(3) ' [Ud( :Hza') l = —203(l)

. 2 . .
61(L‘) ° lud1( iHilv')] : —d_10'd1(H,H”,-) : ‘20d1(H”9')

Plugging these relations in (2.36), expanding the sums and then combining the terms

together, we get (2.34).

Finally, to get (2.35), use (2.16)

 

~ 1 (51+:‘52 — 1)’ 1 - '
2 = 3L" [ " ~ udl(H";-) Xud2(H'2;°)
2 61%;” (1‘ $1)“ — 2:2) 0[ ]
i1+i2=3
1 . .
= x2—'2' Z 081(H”,')'0d2(H'2a') =$2‘ Z d10d1(lv’)°0d2(')
1113:: “HF"

after expanding the sum and then combining terms together. D

If we distribute the constraints in Proposition 2.8 in all possible ways, formula (2.29)

becomes:

7301“,”) = 2(d _ijgd — 2)0.z(p“.l"+1) (2.37)

 

2 d ° b
+ _ Z Z d2(2d1d2 - d) 0d1(pal’lbl+l) . ad2(pa2,lb2) (a)( )
njd d1=l 01:0 bl

01

Example. Using a computer program based on (2.37) and the recursive formulas
(A.3) for 0.1, one recovers for example that in P3 all the degree 2 elliptic invariants

are 0 (fact known for a very long time) but also one gets new examples, like:
7'3”“) 7507,11?)
j # 0,1728 6 - 25920 6 - 15856790593536

j = 0 3 - 25920 3 - 15856790593536

 

j = 1728 2 - 25920 2 - 15856790593536

APPENDIX

3 Appendix

If we let ad(j1,j2, . . . ,jk) = 0,1(Hjl, Hi1, . . , Hj“) be the genus 0 enumerative invari-
ant in P", then Ruan-Tian proved that the genus zero perturbed invariant and the

genus zero enumerative invariant are equal in P", i.e.
0,1(1‘13'l , H”, . . . , Hj") = RTMUPl , H”, H"3 lHj‘, . . . , H“) (A.1)
Consequences of Ruan-Tian degeneration formula are:

RTd,l(fil '1827": 7/61) = Z Ud(Hi1aHizvfilw°'afil) (A2)

£1+i2=n
and that ad in P" satisfies the following recursive formula: for jl Z jg Z 2 jk 2 2,

0d(jlaj27j3) = —0d(jlvj2 +1aj3 — 1) + d0d(jl +j3 —1aj2)— d0d(jl +j27j3 — 1)
d-l n
+ 2 D ad,(]1,32,i)ad,(j3 - 1,n — 2') — 0.1.01.5 —1,z')ad.(j2,n - z')) (A-3)
d1=l i=0
where ad(jl,j2,j3) = od(j1,j2,j3,j4, . . . ,jk) and the conditions Hj‘, . . . , Hj" are dis—
tributed in the right hand side in all possible ways. Note that 01 (pt, pt) = 1 gives the

initial step of the recursion.

52

BIBLIOGRAPHY

BIBLIOGRAPHY

[AM]

[K]

[D]

[DK]

[F]

[Ful]
[CH]

[MS]

[Pan]

[PW]

[P]

[RT]

[T1]

[T2]

P.S. Aspinwall, D.R. Morrison, Topological field theory and rational curves,
Comm. Math. Phys. 151 (1993), 245-262.

M. Kontsevich, Y. Manin, Gromov- Witten classes, quantuum cohomology and
enumerative geometry, preprint.

S.K. Donaldson, The orientation of Yang-Mills moduli spaces and 4-manifold
topology, J.Diff.Geom. 26(1987), 397-428.

S.K. Donaldson, P. Kronheimer, ”The geometry of Four Manifolds”, Oxford
Math. Monographs, Oxford Univ. Press, Oxford, 1990.

A. Floer, The Unregularized gradient flow of the symplectic action, Comm.
Pure. Appl. Math. XLI(1988), 775-813.

W. Fulton, ”Intersection Theory”, Springer-Verlag, Berlin, 1984.

P. Griffiths, J. Harris, ”Principles of Algebraic Geometry”, Wiley, New York
(1978)

D. McDuff, D. Salamon, ”J-holomorphic Curves and Quantuum Cohomol-
ogy”, Univ.Lect.Series, AMS, Providence, Rhode Island, 1994.

R. Panharipande, A note on elliptic plane curves with fixedj invariant,
preprint, May 1995.

TH. Parker, J. Wolfson, Holomorphic maps and bubble trees, J. Geom. Anal-
ysis,3(l993), 63-97.

T.H.Parker, Compactified moduli spaces of pseudo-holomorphic curves,
preprint.

Y.Ruan, G.Tian, A mathematical theory of quantuum cohomology, J. Diff.
Geom. 42( 1995), 259-367.

C.H.Taubes, Self-dual connections on non-self dual 4-manifolds, J. Diff.
Geom. 17(1982), 139-170.

C.H.Taubes, Self-dual connections on J-manifolds with indefinite intersection

form, J.Diff.Geom. 19(1984), 517-560.

53

HICHIGRN STRTE UNIV. LIBRRRIES
llllllllllllllllllllllllllllllHlllllllllllllllll
31293015559374