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

Abstract

In this paper, second installment in a series of three, we give a correspondence theorem to relate the count of genus g g curves in a fixed linear system of an abelian surface passing through g − 2 g-2 points to a tropical count. To do this, we relate the linear system defined by a complex curve to certain integrals of 1 1 -forms over cycles on 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-Göttsche type in abelian surfaces.