Maps, sheaves, and K3 surfaces

TL;DR

This paper explores the relationship between curve counting theories on Calabi-Yau 3-folds with K3 fibrations, presents new results on descendent integrals on K3 surfaces, and surveys the proof of the Yau-Zaslow conjecture.

Contribution

It introduces new conjectures and results connecting curve and sheaf counting on K3 surfaces, advancing understanding in enumerative geometry.

Findings

Connection between curve and sheaf counting on K3 surfaces

New conjectures on descendent integrals on K3 surfaces

Survey of the proof of the Yau-Zaslow conjecture

Abstract

The conjectural equivalence of curve counting on Calabi-Yau 3-folds via stable maps and stable pairs is discussed. By considering Calabi-Yau 3-folds with K3 fibrations, the correspondence naturally connects curve and sheaf counting on K3 surfaces. New results and conjectures (with D. Maulik) about descendent integration on K3 surfaces are announced. The recent proof of the Yau-Zaslow conjecture is surveyed. The paper accompanies my lecture at the Clay research conference in Cambridge, MA in May 2008.