Curve counting on abelian surfaces and threefolds

Abstract

We study the enumerative geometry of algebraic curves on abelian surfaces and three- folds. 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 quasi-modular 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 gen- erating series of curve counts in primitive classes are presented. The base cases make connections to classical lattice counts of Debarre, G ̈ottsche, 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.