Gromov-Witten theory of abelian varieties in families and modular forms

Abstract: This is the first paper in a series on the Gromov-Witten theory of the universal abelian variety over the moduli space of principally polarized abelian varieties of dimension $h$. We conjecture that the generating series of Gromov-Witten classes, when summed over the degree against the principal polarization, is a cycle-valued quasimodular form for $\mathrm{SL}_2(\mathbb{Z})$ and satisfy a holomorphic anomaly equation. These conjectures generalize the quasimodularity of the Gromov-Witten theory of elliptic curves to higher dimension and raise interesting questions regarding enumerative mirror symmetry for abelian varieties. In genus $1$ it specializes to a conjecture of Greer and Lian which was proven by Iribar Lopez after tautological projection. We also discuss a special family of abelian varieties with a conjectural relation to Siegel quasimodular forms of higher genus. The main result of the paper is a proof of the conjectures in genus $2$ after tautological projection. For that we introduce quotient Gromov-Witten invariants which are indexed by the characteristic polynomial of the curve class and are shown to determine all descendent Gromov-Witten invariants satisfying a degree conditions. We then give an explicit formula for all genus $2$ quotient invariants after tautological projection as the Shimura lift of the product of two Eisenstein series. The formula is based on a curious modular identity derived in a joint appendix with Brandon Williams.

• This paper refines Gromov-Witten theory for abelian varieties using family invariants over the moduli stack of principally polarized abelian varieties, $\mathcal{A}_h$.
• The study introduces quotient Gromov-Witten invariants indexed by characteristic polynomials and their reconstruction theorem which provides quasimodular and modular invariants.
• The main conjectures and results hinge on generating series representation, elliptic curve results generalization, and a Hodge-theoretic heuristic in mirror symmetry.

The family viewpoint

For a fixed abelian variety $X$ of dimension $h > 1$, the (non-reduced) Gromov-Witten invariants vanish in every non-trivial curve class: the presence of holomorphic 2-forms forces deformation to a complex torus without algebraic curves, and deformation invariance then kills all invariants. The reduced theory, developed over the past two decades for abelian surfaces and threefolds, circumvents this but lacks a uniform modular interpretation. This paper by Georg Oberdieck proposes a different framework: instead of fixing an abelian variety, one works with the universal family $\pi : \mathcal{X} \to \mathcal{A}_h$ over the moduli stack of principally polarized abelian varieties of dimension $h$, and pushes forward the standard virtual class of the relative moduli space $\mathfrak{M}_{g,n}(\pi,d)$ of stable maps of degree $d$ against the principal polarization. The resulting family invariants are cohomology classes on $\mathcal{A}_h$; they restrict to zero on each fiber yet are globally non-trivial, supported on Noether-Lefschetz loci. In dimension $h=1$ this recovers the classical quasimodular Gromov-Witten theory of a fixed elliptic curve.

The main conjectures

The central conjecture states that for any insertion $\Gamma \in H^*(\mathcal{X}^{\times n})$ and $2g-2+n>0$, the generating series

$h > 1$0

is a cycle-valued quasimodular form for $h > 1$1, i.e. lies in $h > 1$2. A companion holomorphic anomaly equation determines the dependence on the non-modular generator $h > 1$3 via gluing terms involving the Lefschetz dual correspondence $h > 1$4 constructed by K\"unnemann. A corollary of the conjectural HAE is a precise weight prediction: if $h > 1$5 is an eigenvector under multiplication-by-$h > 1$6 maps with multiplicity $h > 1$7, then $h > 1$8 has weight $h > 1$9.

Two motivations anchor these conjectures. First, they specialize at $\pi : \mathcal{X} \to \mathcal{A}_h$0 to the proven quasimodularity and HAE for elliptic curves. Second, in genus 1 with insertion $\pi : \mathcal{X} \to \mathcal{A}_h$1, the conjectures imply that $\pi : \mathcal{X} \to \mathcal{A}_h$2 is a modular form of weight $\pi : \mathcal{X} \to \mathcal{A}_h$3 valued in codimension $\pi : \mathcal{X} \to \mathcal{A}_h$4 cycles — precisely the Greer–Lian conjecture on Noether-Lefschetz cycles, proven by Iribar Lopez after tautological projection:

$\pi : \mathcal{X} \to \mathcal{A}_h$5

Quotient invariants and reconstruction

The paper introduces quotient Gromov-Witten invariants indexed not by degree but by the characteristic polynomial $\pi : \mathcal{X} \to \mathcal{A}_h$6 of a curve class $\pi : \mathcal{X} \to \mathcal{A}_h$7, defined via the self-adjoint endomorphism induced through the polarization. The moduli space decomposes into open-and-closed components $\pi : \mathcal{X} \to \mathcal{A}_h$8, and for $\pi : \mathcal{X} \to \mathcal{A}_h$9 the translation action of the abelian variety has finite stabilizers, so the quotient by translation is a proper DM stack carrying a virtual class defined via a $h$0-insertion at a rigidified marking.

The key structural result is a reconstruction theorem: for $h$1 and tautological insertions satisfying the expected degree constraint $h$2, the descendent invariant $h$3 equals a canonically computable polynomial $h$4 of weighted degree $h$5 times the quotient invariant $h$6. The polynomial is obtained from the intersection theory of $h$7-classes on $h$8 and symplectic invariant theory; concretely, for $h$9-cycles $\mathfrak{M}_{g,n}(\pi,d)$0 one gets power sums:

$\mathfrak{M}_{g,n}(\pi,d)$1

Consequently, quasimodularity of the descendent series implies quasimodularity of all quotient series $\mathfrak{M}_{g,n}(\pi,d)$2 of weight $\mathfrak{M}_{g,n}(\pi,d)$3, and conversely. The paper also derives, conditionally on the main conjectures, an explicit differential operator $\mathfrak{M}_{g,n}(\pi,d)$4 on symmetric functions governing the HAE for quotient invariants, including a boundary term $\mathfrak{M}_{g,n}(\pi,d)$5 when $\mathfrak{M}_{g,n}(\pi,d)$6.

The genus 2 theorem

The main result proves both conjectures in genus 2 after tautological projection, for all $\mathfrak{M}_{g,n}(\pi,d)$7. The proof rests on a Noether-Lefschetz decomposition: stable maps of genus $\mathfrak{M}_{g,n}(\pi,d)$8 factor through translates of abelian subvarieties of dimension at most $\mathfrak{M}_{g,n}(\pi,d)$9, so the rank-$d$0 part of the moduli space is expressed via finite \'etale covers as pushforwards from the universal families over $d$1, capped with Euler classes of $d$2.

The explicit formula for the genus 2 quotient invariant reads:

$d$3

where $d$4 is the weight-$d$5 Eisenstein series for the Weil representation on the lattice $d$6 — whose Fourier coefficients encode, by van der Geer's theorem, minus the degrees of Humbert divisors on $d$7 — and $d$8 denotes the Shimura lift. Hence $d$9 is modular of weight $\mathcal{A}_h$0 for $\mathcal{A}_h$1 and quasimodular for $\mathcal{A}_h$2, exactly matching the weight prediction $\mathcal{A}_h$3. More generally, all genus 2 quotient series satisfy

$\mathcal{A}_h$4

and satisfy the predicted HAE, e.g. $\mathcal{A}_h$5. Via the reconstruction theorem this yields quasimodularity of all genus 2 descendent invariants after tautological projection, lying in $\mathcal{A}_h$6.

The modular identity appendix

The formula hinges on a modular identity, derived jointly with Brandon Williams, equating an elaborate arithmetic expression built from degrees of Humbert divisors on moduli spaces $\mathcal{A}_h$7 with the Shimura lift $\mathcal{A}_h$8. The proof combines a Hecke-operator description of the Eisenstein coefficients $\mathcal{A}_h$9 for rescaled lattices $h=1$0 with two elementary multiplicative number-theoretic identities concerning sums involving M\"obius functions, divisor functions, and Euler products $h=1$1. It is worth noting candidly that the authors state the two number-theoretic identities were suggested by an AI system (ChatGPT 5.5 via the IMProofBench harness); the authors did not verify the AI document itself but wrote their own equivalent proof after understanding its strategy.

Mirror symmetry and Siegel modularity

The quasimodularity prediction admits a Hodge-theoretic mirror-symmetry heuristic following Golyshev–Lunts–Orlov: the family $h=1$2 should be mirror to the family of $h=1$3-fold powers of elliptic curves, with mirror symmetry interchanging the Mumford-Tate group action with the Looijenga-Lunts-Verbitsky Lie algebra action. Under this identification, the expected weight $h=1$4 matches the requirement that $h=1$5 be the asymptotic expansion of an almost-holomorphic section of $h=1$6. The fiberwise $h=1$7 action on Chow groups corresponds to monodromy of the mirror family, explaining the modularity. Extending this, the paper conjectures that for the family $h=1$8 (mirror to $h=1$9), the generating series indexed by positive-semidefinite half-integral matrices $\Gamma \in H^*(\mathcal{X}^{\times n})$0 are cycle-valued Siegel-quasimodular forms of genus $\Gamma \in H^*(\mathcal{X}^{\times n})$1 satisfying a HAE — a higher-genus analogue of a recent Siegel-modularity conjecture of Greer–Tayou in genus 1.

Limitations and open questions

Several qualifications bear directly on the strength of the results. All modularity statements in this paper are established only after tautological projection to the ring generated by Hodge bundle Chern classes; the full cycle-valued Conjectures remain open even in genus 2. The genus 2 computation relies on the reduced multiple cover formula for abelian surfaces of Bryan–Oberdieck–Pandharipande–Yin, which has been proven in most cases by Blomme–Carocci and, in full generality, only conditionally on a GW/PT correspondence. The announced extension to abelian surfaces in arbitrary genus (Theorem on $\Gamma \in H^*(\mathcal{X}^{\times n})$2) is likewise conditional on that multiple cover formula. Vanishing results (Propositions on vanishing and degree constraints) are expected but not proven without tautological projection. Open questions include: whether quasimodularity extends to log Gromov-Witten theory on toroidal compactifications (already difficult in genus 1); how to construct natural compactifications of the virtual cycles of $\Gamma \in H^*(\mathcal{X}^{\times n})$3 for $\Gamma \in H^*(\mathcal{X}^{\times n})$4, where the Torelli map has positive codimension and the invariants studied here vanish; and the role of the cycle part of $\Gamma \in H^*(\mathcal{X}^{\times n})$5 under mirror symmetry. For universal K3 or hyperkähler families, the relevant automorphic objects may need to be more general than quasimodular forms — possibly higher Green functions or Eichler integrals.

Conclusion

This paper reframes the Gromov-Witten theory of abelian varieties as a theory of family invariants over $\Gamma \in H^*(\mathcal{X}^{\times n})$6, conjecturally cycle-valued quasimodular forms satisfying a holomorphic anomaly equation, thereby generalizing Dijkgraaf's elliptic curve results to all dimensions and connecting them to Noether-Lefschetz special cycles, Shimura lifts, and enumerative mirror symmetry. The proof of the genus 2 case after tautological projection — via quotient invariants indexed by characteristic polynomials, the Noether-Lefschetz decomposition, and an explicit Shimura-lift identity — provides substantial evidence for the general framework, while leaving the unprojected statements, higher genera, and the Siegel-modular generalization as concrete open problems.