Gromov-Witten theory and Noether-Lefschetz theory for holomorphic-symplectic varieties

TL;DR

This paper explores the relationship between Gromov-Witten invariants and Noether-Lefschetz theory in holomorphic-symplectic varieties, proposing a new conjectural formula and extending known results for specific geometric classes.

Contribution

It introduces a new conjectural formula linking Gromov-Witten invariants to primitive classes in holomorphic-symplectic varieties, generalizing previous conjectures for K3 surfaces.

Findings

Evidence for the new conjectural formula for Gromov-Witten invariants.

Determination of generating series of Noether-Lefschetz numbers for Debarre-Voisin varieties.

Extension of results on HLS divisors on moduli spaces of Debarre-Voisin fourfolds.

Abstract

We use Noether-Lefschetz theory to study the reduced Gromov--Witten invariants of a holomorphic-symplectic variety of $K3^{[n]}$-type. This yields strong evidence for a new conjectural formula that expresses Gromov-Witten invariants of this geometry for arbitrary classes in terms of primitive classes. The formula generalizes an earlier conjecture by Pandharipande and the author for K3 surfaces. Using Gromov-Witten techniques we also determine the generating series of Noether-Lefschetz numbers of a general pencil of Debarre-Voisin varieties. This reproves and extends a result of Debarre, Han, O'Grady and Voisin on HLS divisors on the moduli space of Debarre-Voisin fourfolds.