Counting Curves in Elliptic Surfaces by Symplectic Methods

TL;DR

This paper computes Gromov-Witten invariants of elliptic surfaces using symplectic techniques, confirming the Yau-Zaslow Conjecture for primitive classes in K3 surfaces.

Contribution

It introduces explicit formulas and methods for calculating family GW invariants of elliptic surfaces, including TRR and symplectic sum formulas.

Findings

Confirmed Yau-Zaslow Conjecture for primitive classes in K3 surfaces

Established TRR and symplectic sum formulas for elliptic surfaces

Computed explicit family GW invariants for primitive classes

Abstract

We explicitly compute family GW invariants of elliptic surfaces for primitive classes. That involves establishing a TRR formula and a symplectic sum formula for elliptic surfaces and then determining the GW invariants using an argument from \cite{ip3}. In particular, as in \cite{bl1}, these calculations also confirm the well-known Yau-Zaslow Conjecture \cite{yz} for primitive classes in $K3$ surfaces.