On the Tropical Discs Counting on Elliptic K3 Surfaces with General Singular Fibres

TL;DR

This paper explores the tropical geometry of elliptic K3 surfaces with various singular fibres, extending the correspondence between open Gromov-Witten invariants and tropical disc counting using Lagrangian Floer theory.

Contribution

It provides local models for all Kodaira singular fibre types and generalizes the tropical disc counting correspondence to these cases.

Findings

Established local models for singular fibres I_n, II, III, IV

Extended the Gromov-Witten/tropical disc correspondence

Enhanced understanding of tropical geometry in elliptic K3 surfaces

Abstract

Using Lagrangian Floer theory, we study the tropical geometry of K3 surfaces with general singular fibres. In particular, we give the local models for the type $I_n$, $II$, $III$ and $IV$ singular fibres in the Kodaira's classification and generalize the correspondence theorem between open Gromov-Witten invariants/tropical discs counting to these cases.