Toric degenerations, tropical curves and Gromov-Witten invariants of Fano manifolds

TL;DR

This paper introduces a tropical geometric approach to compute Gromov-Witten invariants of certain Fano manifolds that can degenerate into toric Fano varieties, expanding computational tools in algebraic geometry.

Contribution

It develops a novel tropical method for calculating Gromov-Witten invariants applicable to Fano manifolds with specific toric degenerations, including flag manifolds and moduli spaces.

Findings

Successfully computes invariants for flag manifolds of type A

Applies to moduli spaces of rank two bundles on genus two curves

Provides a new computational framework for Fano manifolds

Abstract

In this paper, we give a tropical method for computing Gromov-Witten type invariants of Fano manifolds of special type. This method applies to those Fano manifolds which admit toric degenerations to toric Fano varieties with singularities allowing small resolutions. Examples include (generalized) flag manifolds of type A, and some moduli space of rank two bundles on a genus two curve.