Counting Elliptic Curves in K3 Surfaces

TL;DR

This paper computes genus one Gromov-Witten invariants for non-primitive classes on K3 surfaces, confirming a specific enumerative formula using advanced recursion and sum formulas.

Contribution

It introduces a novel approach combining genus two topological recursion and symplectic sum formulas to verify the Gottsche-Yau-Zaslow conjecture for non-primitive classes.

Findings

Verification of the Gottsche-Yau-Zaslow formula for index two classes

Development of methods relating genus two recursion to genus one invariants

Explicit calculations of Gromov-Witten invariants for non-primitive classes

Abstract

We compute the genus one family Gromov-Witten invariants of K3 surfaces for non-primitive classes. These calculations verify Gottsche-Yau-Zaslow formula for non-primitive classes with index two. Our approach is to use the genus two topological recursion formula and the symplectic sum formula to establish relationships among various generating functions.