Curves on K3 surfaces and modular forms

Curves on surfaces and modular forms

Abstract.

We study the virtual geometry of the moduli spaces of curves and sheaves on surfaces in primitive classes. Equivalences relating the reduced Gromov-Witten invariants of surfaces to characteristic numbers of stable pairs moduli spaces are proven. As a consequence, we prove the Katz-Klemm-Vafa conjecture evaluating integrals (in all genera) in terms of explicit modular forms. Indeed, all invariants in primitive classes are shown to be governed by modular forms.

The method of proof is by degeneration to elliptically fibered rational surfaces. New formulas relating reduced virtual classes on surfaces to standard virtual classes after degeneration are needed for both maps and sheaves. We also prove a Gromov-Witten/Pairs correspondence for toric 3-folds.

Our approach uses a result of Kiem and Li to produce reduced classes. In Appendix LABEL:oldnew, we answer a number of questions about the relationship between the Kiem-Li approach, traditional virtual cycles, and symmetric obstruction theories.

The interplay between the boundary geometry of the moduli spaces of curves, surfaces, and modular forms is explored in Appendix LABEL:Pixton by A. Pixton.

Introduction

0.1. Stable maps and reduced classes

Let be a complex algebraic surface, and let be a nonzero effective curve class. The moduli space of stable maps from connected genus curves to representing has expected dimension

However, via the holomorphic symplectic form on , the standard obstruction theory for admits a trivial quotient. As a result,

The vanishing reflects the deformation invariance of Gromov-Witten theory: admits deformations for which is not of type and thus not represented by holomorphic curves.

A reduced obstruction theory, obtained by removing the trivial factor, yields a reduced virtual class1

of dimension . A rich Gromov-Witten theory is obtained by integrating codimension tautological classes on against . Such integrals are invariant with respect to deformations of for which the class remains of type .

The class is primitive if is not divisible.2 While the reduced Gromov-Witten theory of is defined for all , here we primarily study the primitive case.

0.2. Hodge classes

The rank Hodge bundle,

with fiber over the point is well defined for all . The Hodge bundle is pulled back from the moduli space of curves

when is at least . The top Chern class of is the most beautiful and well-behaved integrand in the reduced theory of . Define

(1)

Let be a polarized Calabi-Yau 3-fold which admits a -fibration,

Such a fibration determines a map of the base to the moduli space of polarized surfaces. The integrals precisely relate the Gromov-Witten invariants of to the intersection numbers of with Noether-Lefschetz divisors in the moduli of surfaces [gwnl].

0.3. Katz-Klemm-Vafa conjecture

Let be a primitive effective curve class. The Gromov-Witten partition function3 for is

The BPS counts are uniquely defined by

By deformation invariance4, both and depend only upon the norm . We write and for and respectively when .

The evaluation of in terms of modular forms was conjectured by S. Katz, A. Klemm, and C. Vafa [kkv]. The Fourier expansion of the discriminant modular form is

Define the series

where . Our first result is the proof of the Katz-Klemm-Vafa conjecture.

Theorem 1.

The invariants for primitive curve classes are determined by

By Theorem 1, the invariants are integers. The formula may also be directly written for the integrals . For , let be the Eisenstein series

where is the corresponding Bernoulli number.

Corollary 2.

For primitive curve classes,

Theorem 1 specializes in genus 0 to the rational curve counts on surfaces predicted by S.-T. Yau and E. Zaslow [yauz]. The Yau-Zaslow formula was proven for primitive classes in [beu, brl].

Of course, the integrals may also be considered in the non-primitive case. A complete conjecture is explained in [gwnl, clay] based on [kkv]. While the genus 0 integrals have been calculated for all classes in [gwyz], new methods appear to be required in higher genus.

0.4. Descendents

Let be a primitive effective curve class. The moduli space of stable maps from connected genus curves with ordered marked points comes with evaluation maps

Pulling back cohomology classes on via gives primary classes on . Descendent classes are obtained from the Chern classes of the cotangent lines

at the marked points.

Let , and let

The insertion corresponds to the class on the moduli space of maps. Let

(2)

denote the reduced descendent Gromov-Witten invariants. By convention, the descendent vanishes if the degree of the integrand does not match the dimension of the reduced virtual class.

If only descendents of classes in and appear in (2), the bracket for primitive depends only upon the norm

by deformation invariance. Since the classes in are not monodromy invariant, the bracket (2) may depend upon if descendents of are present. When possible, we will replace the subscript of the descendent bracket by .

0.5. Point insertions

The evaluation of Theorem 1 extends naturally to the integrals

where is the Chern class of the Hodge bundle and is the point class.

Theorem 3.

For primitive classes on surfaces, we have

The last factor is related to the point insertions. In the case, when no points are inserted, Theorem 3 specializes to Theorem 1 by Corollary 2.

0.6. Quasimodular forms

The ring of quasimodular forms with possible poles at is the algebra generated by the Eisenstein series over the ring of modular forms with possible poles at . The ring of quasimodular forms is closed under . See [bghz] for a basic treatment.

By deformation invariance, the full descendent theory of algebraic surfaces is captured by elliptically fibered surfaces. Let be an elliptically fibered surface with section. Let

denote the section and fiber classes. A descendent potential function for the reduced theory of surfaces in primitive classes is defined by

for . For arbitrary insertions, we prove the following result.

Theorem 4.

is the Fourier expansion in of a quasimodular form with pole at of order at most .

The simplest of the series is the count of genus curves passing through points,

(3)

first calculated5 by J. Bryan and C. Leung [brl]. Formula (3) is also a specialization of Theorem 3.

In the non-primitive case, we conjecture the genus reduced descendent potential to be a quasimodular form of higher level. A precise statement is made in Section LABEL:ccllyy.

0.7. Stable pairs on surfaces

We will relate the reduced Gromov-Witten invariants of surfaces to integrals over the moduli spaces of sheaves on surfaces.

Let be a surface. A pair consists of a sheaf on supported in dimension 1 together with a section . A pair is stable if

  1. the sheaf is pure,

  2. the section has 0-dimensional cokernel.

Purity here simply means every nonzero subsheaf of has support of dimension 1. As a consequence, the scheme theoretic support of is a curve. The discrete invariants of a stable pair are the holomorphic Euler characteristic and the class6 .

Let be a nonzero effective curve class. Let be the moduli space of stable pairs satisfying

After appropriate choices [pt1], pair stability coincides with stability arising from geometric invariant theory in Le Potier’s study [LeP]. Hence, the moduli space is a projective scheme.

The class is irreducible if is not a sum of two nonzero effective curve classes.7 A basic result proven in [ky, pt3] is the following.

Proposition 5.

If is irreducible, is nonsingular of dimension .

When studying stable pairs, we will often assume is irreducible. In the irreducible case, depends, up to deformation equivalence, only upon the norm of . We will use the notation when .

0.8. Euler characteristic

Let be an irreducible effective curve class with norm .

Let be the cotangent bundle of the moduli space . Define the partition function

Here, denotes the topological Euler characteristic. We have written the stable pairs partition function in the variable instead of the traditional since the latter will be reserved for the Fourier expansions of modular forms.8 Since is empty if , we see is a Laurent series in .

The topological Euler characteristics of have been calculated by T. Kawai and K. Yoshioka. By Theorem 5.80 of [ky],

We require the signed Euler characteristics,

Therefore, equals

(4)

0.9. Correspondence

To prove Theorem 1, we formulate and prove a Gromov-Witten/Pairs correspondence in the setting of reduced classes.

Let be an irreducible effective curve class. We write the Gromov-Witten partition function as

Our Gromov-Witten/Pairs correspondence for the reduced theories of the 3-fold implies9

(5)

after the substitution . Together with the Euler characteristic calculation (4), the correspondence (5) immediately yields Theorem 1.

To complete the proof of Theorem 1, we must establish the reduced Gromov-Witten/Pairs correspondence for . There are two main ideas in the argument:

  1. Let be the rational elliptic surface obtained by blowing-up the base locus of a pencil of cubics in . Let be a nonsingular member of the pencil. Using special degenerations of elliptically fibered surfaces to unions of rational elliptic surfaces , we prove a new formula relating the reduced virtual classes of to the standard virtual classes of . We prove the formula separately for stable maps and stable pairs.

    Figure 1. A degeneration of surfaces
  2. Since is isomorphic to blown-up at points, is deformation equivalent to a toric 3-fold. We prove a Gromov-Witten/Pairs correspondence for toric 3-folds following [moop].

Together (i) and (ii) yield the correspondence (5) and complete the proof of Theorem 1.

We have no direct approach to the integrals on . The moduli space of stable maps has contracted components and subtle virtual contributions. The nonsingularity of the corresponding moduli spaces of stable pairs is remarkable. Theorem 1 provides a model use of the Gromov-Witten/Pairs correspondence.

Part (i) constitutes the technical heart of the paper. The primitivity of is crucial. In Section LABEL:npd, we state a degeneration formula in the non-primitive case which leads to much more subtle invariants of . Unfortunately, the toric correspondence (ii) is not sufficient to conclude a Gromov-Witten/Pairs correspondence for non-primitive classes . The non-primitive degeneration formula will be pursued in a sequel [mpt2].

0.10. Point insertions for stable pairs

Let be an irreducible effective curve class of norm .

The linear system of curves of class is -dimensional. Let

(6)

be the canonical morphism obtained by sending to the support of . A point incidence condition for stable pairs corresponds to the pull-back of a hyperplane . The integral for stable pairs associated to point conditions is defined by

By Bertini, the subvariety

is nonsingular of dimension for generic hyperplanes. Using Gauss-Bonnet, the Euler characteristics of the spaces are expressible in terms of the integrals by the formula

(7)

In fact, equation (7) may be easily inverted to express in terms of the Euler characteristics.

Theorem 6.

The point conditions for irreducible classes on surfaces are evaluated by

Point conditions in the reduced Gromov-Witten theory of are evaluated by Theorem 3. We derive Theorem 3 from Theorem 1 using degeneration and exact Gromov-Witten calculations for Hodge integrals. Theorem 3 then implies Theorem 6 by the equivariant Gromov-Witten/Pairs correspondence for .

We do not know a direct approach along the lines of [ky] for determining the integrals or the Euler characteristics of .

0.11. Plan of the paper

We start, in Section 1, with a precise statement of the Gromov-Witten/Pairs correspondence for the reduced theory of with primary insertions, leaving many of the proofs for later Sections. Elliptically fibered surfaces are reviewed in Section 2. The degeneration formulas in terms of the standard virtual classes of the rational elliptic surface are proven in Section 3 for stable pairs and in Section LABEL:gwd for Gromov-Witten theory. We give full details for stable pairs and a briefer account for the more standard Gromov-Witten theory.

The Gromov-Witten/Pairs correspondence for toric 3-folds is established in Section LABEL:gwptor, completing the proof of Theorem 1. Theorems 3 and 6 are proven in Section LABEL:pint. The quasimodularity of Theorem 4 is obtained in Section LABEL:qmod from a boundary induction in the tautological ring of the moduli space of curves using the strong form of Getzler-Ionel vanishing proven in [fpm].

Our approach uses a result of Kiem-Li [KL] to construct reduced classes10 . In Appendix LABEL:oldnew, we compare the Kiem-Li method to standard virtual cycle techniques. The inquiry leads naturally to a counterexample to a question of Behrend and Fantechi concerning symmetric obstruction theories that is explained in Section LABEL:oldnewl.

In Appendix LABEL:Pixton, by A. Pixton [pix], the interplay between Theorem 1 and boundary expressions for in low genus are explored.

Acknowledgements

Much of the work presented here was completed at MSRI in 2009 during a program on modern moduli in algebraic geometry. We thank the organizers for creating a stimulating environment. We thank J. Bryan, B. Conrad, C. Faber, H. Flenner, D. Huybrechts, B. Bakker, D. Joyce, A. Klemm, A. Marian, D. Oprea, E. Scheidegger and S. Yang for may related conversations. Correspondence with A. Boocher and D. van Straaten lead to the example in Appendix LABEL:oldnew. We are particularly grateful to Jun Li for discussions and an advanced copy of [LiWu].

D.M. was partially supported by a Clay research fellowship. R.P. was partially supported by DMS-0500187 and the Clay Institute. R.T. was partially supported by an EPSRC programme grant.

1. Reduced Gromov-Witten/Pairs correspondence

1.1. Stable maps

Let be a complex projective surface, and let be a primitive effective curve class. Consider the noncompact Calabi-Yau 3-fold

equipped with the -action defined by scaling the second factor. Let

denote the inclusion given by the identification .

Let be the moduli space of connected genus stable maps to representing the class . Since is a Calabi-Yau -fold, the moduli space has expected dimension with respect to the standard obstruction theory. Since has a holomorphic symplectic form, admits a reduced obstruction theory and reduced virtual class,

The construction of the reduced theory exactly follows Section 2.2 of [gwnl]. Although is not compact, the -fixed locus

is compact, so we can consider the reduced residue invariants11

Here, is the first Chern class of the standard representation of and the generator of , the -equivariant cohomology of a point. The relationship between the residue invariants of and the invariants (1) of is the following.

Lemma 7.

Proof.

The result is a direct consequence of the virtual localization formula of [GP],

The first equality is by localization. The denominator on the right is the equivariant Euler class of the virtual normal bundle. Over a stable map , the virtual normal bundle has fiber

where is the normal bundle to in . Since , we have

from which the above formula follows. ∎

If is irreducible, then and the reduced virtual class is pulled back from the projection to the first factor. In the irreducible case, Lemma 7 is immediate. An alternative proof of Lemma 7 for primitive is obtained by deforming to the irreducible case.

1.2. Stable pairs

Let the moduli space of stable pairs on with

We will construct in Section 3.3 a reduced virtual class in dimension ,

Again, we consider the reduced residue invariants

By deformation invariance of the reduced theory, the invariant can be computed when is irreducible.12 By standard arguments13

in the irreducible case. By Proposition 5, is nonsingular of dimension . The obstruction bundle of the standard deformation theory [HT, pt1] of has fiber

over the moduli point of the pair

Here, is the canonical bundle and the isomorphism is by Serre duality. Since is the tangent space to , the moduli of stable pairs on , and is trivial with the standard representation, the obstruction bundle is

The reduced class is obtained by removing the trivial factor , as we show in Section LABEL:spsym.

Lemma 8.

.

Proof.

We calculate the residue of the top Chern class of the reduced obstruction bundle,

The second equality comes from localisation. We have omitted all of the terms in which do not contribute. ∎

1.3. Point insertions

For both theories of , we can define reduced residue invariants with point insertions. For Gromov-Witten theory, define14

where the evaluation maps are taken to

and is the point class.

For stable pairs, the product is equipped with a universal sheaf . Define operations

by the slant product

where and are the projections of to the first factor and to (via the second factor). Notice that is the pull-back via the map of (6) of the universal curve in . Define the residue invariants

following Section 6.1 of [pt2].

The calculations of Lemmas 7 and 8 immediately extend to yield the following formulas,

1.4. Correspondence

The reduced Gromov-Witten/Pairs correspondence is stated in terms of the generating series

The stable pairs series is a Laurent function in since is empty for sufficiently negative . The above partition functions specialize to the partition function and of Sections 0.8 and 0.9 when and .

Theorem 9.

For primitive ,

  1. is a rational function of .

  2. After the variable change ,

Theorem 9 is not a specialization of the Gromov-Witten/Pairs correspondence for 3-folds conjectured in [pt1]. The main difference is the occurrence of the reduced class. Since the reduced class suppresses contributions from stable maps with disconnected domains, the correspondence here may be viewed here as concerning only connected curves. Theorem 9 will be proven in Sections 2-LABEL:gwptor.

2. Elliptically fibered surfaces

2.1. Elliptic fibrations

We fix here some notation which will be used throughout the paper. Let be an elliptically fibered surface

(8)

with a section. We assume is smooth except for 24 nodal rational fibers. Let

denote the classes of the section and the elliptic fiber. The intersection pairings are

By deformation invariance, the reduced Gromov-Witten and stable pairs theories for primitive effective classes depend only on the norm . By deformation invariance, we can fully capture the both theories for primitive classes on all algebraic surfaces by studying

on elliptically fibered surfaces .

2.2. Rational elliptic surface

A rational elliptic surface is obtained by blowing-up the 9 points of the base locus of a generic pencil of cubics. The pencil determines a map

with nonsingular elliptic fibers (except for 12 nodal rational fibers). Let be one of the 9 sections of , and let be a fixed elliptic fiber with distinguished point

(9)

Let and be two copies of a rational elliptic surface . Let , , and be identical choices of the auxiliary data. A reducible surface

is obtained by attaching and along the respective fibers (with the corresponding distinguished points identified). The singular surface is elliptically fibered over a broken rational curve,

(10)

The fibration (10) has a distinguished section .

2.3. Degeneration

The fibration (10) is a degeneration of (8). More precisely, there exists a family of fibrations

(11)

over a pointed curve with the following properties:

  1. is a nonsingular 3-fold, and is a nonsingular surface.

  2. has a section.

  3. When specialized to nonzero , we obtain a nonsingular elliptically fibered surface of the form (8).

  4. When specialized to , we obtain (10).

  5. The relative canonical bundle is trivial.

Denote by the degenerating family of surfaces obtained from composing (11),

Since the section and fiber classes are globally defined by (ii), the sub-lattice of spanned by and is fixed by the monodromy of around .

The degenerating family (11) will play an essential role in our proof of Theorem 9.

3. Reduced stable pairs

3.1. Definitions

Let be a complex algebraic surface, and let be an effective curve class. Let

We include as the fiber over ,

Let be the quasi-projective moduli space of stable pairs on with holomorphic Euler characteristic and class

Strictly speaking, to construct , we apply Le Potier’s results [LeP] to the projective 3-fold

to obtain a projective moduli space containing as an open subscheme.

We can also consider stable pairs on families of surfaces. Let

be a smooth15 family of surfaces, and let

be the corresponding family of 3-folds. We consider as a subvariety via the inclusion

Let be a section of the local system with fiber over .

By making smaller if necessary, we can choose a holomorphic 2-form which is symplectic on every fiber of ,

(12)

In particular, is trivial. By [LeP], there is a family of moduli spaces

representing the functor taking -schemes to the set of flat families of stable pairs in the class on the fibers of

In addition, there is a universal sheaf on , flat over , with a global section , such that the restriction of

(13)

to the fiber over any closed point over is the corresponding stable pair .

3.2. Standard obstruction theory

As in [pt1], given a stable pair on , we let denote the complex of sheaves

in degrees and . When the section is onto, is quasi-isomorphic to the kernel , the ideal sheaf of the Cohen-Macaulay curve which is the scheme theoretical support of . Similarly we let

denote the universal complex.

From the perspective of [pt1], the trace-free Ext groups,

provide deformation and obstruction spaces for the stable pair . More generally, let denote the derived dual of the truncated relative cotangent complex of , and consider the map

(14)

given by the image of the relative Atiyah class of under the projection

Here and are the projections from to and respectively.

Proposition 10.

The map (14) is a perfect theory for the morphism .

Proof.

The result is proved in [pt1, Section 2.3] and [HT, Theorem 4.1] for projective morphisms . Since the fibers of our are noncompact, we need a small modification to check that the complexes

(15)