Curve counting on abelian surfaces and threefolds

TL;DR

This paper advances the understanding of algebraic curve enumeration on abelian surfaces and threefolds, establishing new formulas, conjectures, and connections to modular forms and classical lattice counts.

Contribution

It provides complete genus results for primitive classes on abelian surfaces, proves hyperelliptic counts, and formulates conjectures for abelian threefolds using Jacobi forms.

Findings

Genus 2 counts are proven for all classes.

Counts match Euler characteristic calculations of moduli spaces.

Conjectures relate curve counts to Jacobi forms and classical lattice counts.

Abstract

We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov-Witten theory of K3 surfaces. We prove complete results in all genera for primitive classes. The generating series are quasimodular forms of pure weight. Conjectures for imprimitive classes are presented. In genus 2, the counts in all classes are proven. Special counts match the Euler characteristic calculations of the moduli spaces of stable pairs on abelian surfaces by G\"ottsche-Shende. A formula for hyperelliptic curve counting in terms of Jacobi forms is proven (modulo a transversality statement). For abelian threefolds, complete conjectures in terms of Jacobi forms for the generating series of curve counts in primitive classes are presented. The base cases make connections to classical lattice counts of Debarre, Goettsche, and Lange-Sernesi. Further evidence is provided by Donaldson-Thomas partition function computations for abelian threefolds. A multiple cover structure is presented. The abelian threefold conjectures open a new direction in the subject.