Moduli of curves and moduli of sheaves

TL;DR

This paper explores the deep relationships between moduli spaces of curves and sheaves on 3-folds, focusing on the Gromov-Witten/Donaldson-Thomas correspondence and its extensions in algebraic geometry.

Contribution

It presents a comprehensive framework connecting moduli of curves and sheaves, including recent developments in correspondences for families of 3-folds.

Findings

Proof of the correspondence for the Calabi-Yau quintic 3-fold

Development of descendent and relative correspondences with A. Pixton

Framework for families of 3-folds in moduli problems

Abstract

Relationships between moduli spaces of curves and sheaves on 3-folds are presented starting with the Gromov-Witten/Donaldson-Thomas correspondence proposed more than 20 years ago with D. Maulik, N. Nekrasov, and A. Okounkov. The descendent and relative correspondences as developed with A. Pixton in the context of stable pairs led to the proof of the correspondence for the Calabi-Yau quintic 3-fold. More recently, the study of correspondences in families has played an important role in connection with other basic moduli problems in algebraic geometry. The full conjectural framework is presented here in the context of families of 3-folds. This article accompanies my lecture at the ICBS in July 2024.