Tropical curves in abelian surfaces II: enumeration of curves in linear systems

TL;DR

This paper establishes a correspondence between algebraic and tropical counts of genus g curves in abelian surfaces, introducing refined invariants and proving their invariance, advancing enumerative geometry in this setting.

Contribution

It provides a new correspondence theorem linking algebraic and tropical counts of curves in abelian surfaces and introduces refined invariants of Block-G"ottsche type.

Findings

Derived a correspondence theorem relating algebraic and tropical curve counts.

Expressed tropical multiplicities and proved invariance of refined multiplicities.

Introduced refined invariants for abelian surfaces.

Abstract

In this paper, second installment in a series of three, we give a correspondence theorem to relate the count of genus $g$ curves in a fixed linear system in an abelian surface to a tropical count. To do this, we relate the linear system defined by a complex curve to certain integrals of 1-forms over cycles in the curve. We then give an expression for the tropical multiplicity provided by the correspondence theorem, and prove the invariance for the associated refined multiplicity, thus introducing refined invariants of Block-G\"ottsche type in abelian surfaces.