Curve counting on abelian surfaces and threefolds - Bryan, Jim; Oberdieck, Georg; Pandharipande, Rahul; Yin, Qizheng (Algebraic geometry, 2018)
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 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).
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