Tropical Count of Curves on Abelian Varieties

TL;DR

This paper studies the enumeration of tropical genus g curves in g-dimensional tropical abelian varieties, confirming that the tropical counts align with complex counts for g=2,3, thus bridging tropical and complex enumerative geometry.

Contribution

It proves that tropical counts of genus 2 and 3 curves in g-dimensional tropical abelian varieties match complex counts, establishing a correspondence between tropical and complex enumerative geometry for these cases.

Findings

Tropical counts match complex counts for g=2 and 3.

Confirmed the correspondence between tropical and complex enumerative geometry.

Extended understanding of curve counting in tropical abelian varieties.

Abstract

We investigate the problem of counting tropical genus g curves in g-dimensional tropical abelian varieties. For g = 2, 3, we prove that the tropical count matches the count provided by G\"ottsche, Bryan-Leung, and Lange-Sernesi in the complex setting.