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Quasimaps to moduli spaces of sheaves

Denis Nesterov

TL;DR

This work develops a comprehensive theory of quasimaps to moduli spaces of sheaves on a surface, proving that their moduli are proper and carry perfect obstruction theories, and establishes a canonical correspondence with moduli of sheaves on threefolds $S\times C$. By transporting quasimaps to a sheaf-theoretic framework, the authors derive wall-crossing formulas that relate the Gromov–Witten theory of $M(\mathbf{v})$ to Donaldson–Thomas theory of $S\times C$ with relative insertions, with explicit computations in the Hilbert-schemes case when $S$ is a del Pezzo surface. The paper further develops the determinant-line-bundle apparatus, stability notions (including $\epsilon$-stability), and an array of equivalences between quasimap moduli and relative moduli spaces of sheaves, including perverse-heart perspectives and stable-pair reformulations. A significant portion is devoted to Hilbert schemes, their ideal-sheaf interpretation, and the resulting $I$-function (vertex) framework, enabling concrete wall-crossing and computations that connect GW, DT, and PT theories in various geometries. Overall, the work builds a bridge between quasimap theory and DT/PT/GW correspondences, providing a robust toolkit for calculating invariants in surface- and threefold-derived settings and guiding future explorations on K3 and toric del Pezzo geometries.

Abstract

We develop a theory of quasimaps to a moduli space of sheaves $M$ on a surface $S$. Under some assumptions, we prove that moduli spaces of quasimaps are proper and carry a perfect obstruction theory. Moreover, they are naturally isomorphic to moduli spaces of sheaves on threefolds $S\times C$, where $C$ is a nodal curve. Using Zhou's theory of entangled tails, we establish a wall-crossing formula which therefore relates the Gromov-Witten theory of $M$ and the Donaldson-Thomas theory of $S\times C$ with relative insertions. We evaluate the wall-crossing formula for Hilbert schemes of points $S^{[n]}$, if $S$ is a del Pezzo surface.

Quasimaps to moduli spaces of sheaves

TL;DR

This work develops a comprehensive theory of quasimaps to moduli spaces of sheaves on a surface, proving that their moduli are proper and carry perfect obstruction theories, and establishes a canonical correspondence with moduli of sheaves on threefolds . By transporting quasimaps to a sheaf-theoretic framework, the authors derive wall-crossing formulas that relate the Gromov–Witten theory of to Donaldson–Thomas theory of with relative insertions, with explicit computations in the Hilbert-schemes case when is a del Pezzo surface. The paper further develops the determinant-line-bundle apparatus, stability notions (including -stability), and an array of equivalences between quasimap moduli and relative moduli spaces of sheaves, including perverse-heart perspectives and stable-pair reformulations. A significant portion is devoted to Hilbert schemes, their ideal-sheaf interpretation, and the resulting -function (vertex) framework, enabling concrete wall-crossing and computations that connect GW, DT, and PT theories in various geometries. Overall, the work builds a bridge between quasimap theory and DT/PT/GW correspondences, providing a robust toolkit for calculating invariants in surface- and threefold-derived settings and guiding future explorations on K3 and toric del Pezzo geometries.

Abstract

We develop a theory of quasimaps to a moduli space of sheaves on a surface . Under some assumptions, we prove that moduli spaces of quasimaps are proper and carry a perfect obstruction theory. Moreover, they are naturally isomorphic to moduli spaces of sheaves on threefolds , where is a nodal curve. Using Zhou's theory of entangled tails, we establish a wall-crossing formula which therefore relates the Gromov-Witten theory of and the Donaldson-Thomas theory of with relative insertions. We evaluate the wall-crossing formula for Hilbert schemes of points , if is a del Pezzo surface.
Paper Structure (47 sections, 49 theorems, 375 equations, 1 figure)

This paper contains 47 sections, 49 theorems, 375 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. Results
  4. Hilbert schemes of points
  5. More applications
  6. Methods
  7. Generalisations
  8. Acknowledgments
  9. Notation and conventions
  10. Stack of coherent sheaves
  11. Preliminaries
  12. Rigidification
  13. Determinant line bundles
  14. Quasimaps
  15. Preliminaries
  16. ...and 32 more sections

Key Result

Theorem A

Under our assumptions, there exists a moduli space $M_{\mathbf{v},\check{\beta}}^{\epsilon}(S\times C_{g,N})$ parametrizing sheaves on threefolds $S \times C$ for varying nodal curves $C$, and a natural isomorphism such that the degree of quasimaps $\beta$ together with the class $\mathbf{v}$ determine the Chern character of sheaves on $S\times C$.

Figures (1)

  • Figure 1: The Square

Theorems & Definitions (77)

  • Theorem A: Theorem \ref{['mapssheaves']}
  • Theorem B: Theorem \ref{['proper']}, \ref{['obsthe']}, \ref{['PT']}
  • Theorem C: Corollary \ref{['wallcrossingHilb2']}
  • Theorem D: Corollary \ref{['Delpezzowall']}
  • Theorem 2.1
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • Definition 3.4
  • Definition 3.5
  • ...and 67 more