Moduli of curves and moduli of sheaves

TL;DR

This work surveys and extends the Gromov–Witten/Donaldson–Thomas (GW/DT) correspondence to descendents in families of 3-folds, introducing a universal matrix that translates stable-map descendents into stable-pair descendents and proposing a robust conjectural framework for diagonal and general descendents. It establishes proven cases in toric geometries (torus-equivariant theories) and in relative settings, including the Calabi–Yau quintic and log Calabi–Yau pair degenerations, and further strengthens the theory through sign rules and generalized conjectures. The results connect curve-counting invariants across frameworks, enable Virasoro constraints to transfer to moduli of sheaves, and provide a machine-friendly structure (via the matrix \widetilde{\mathsf{K}}) for computing descendents in families. Collectively, the paper advances a unified, family-wise understanding of GW/DT correspondences with far-reaching implications for moduli problems in algebraic geometry and related areas.

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.