Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sheaves on surfaces and virtual invariants

L. Göttsche, M. Kool

TL;DR

The work develops a unifying framework for virtual invariants of moduli spaces of sheaves on smooth projective surfaces, grounded in Mochizuki's descendent formula and Seiberg-Witten data. It formulates and tests conjectures for virtual Euler characteristics, χ_y-genus refinements, elliptic genera, cobordism classes, Verlinde numbers, and Segre numbers across ranks, linking them to Vafa-Witten S-duality, K-theoretic refinements, and Higgs-pair theories. A central theme is a universal-function paradigm: invariants for any rank and surface are governed by a small set of universal series, computed via toric localization and expressed through lattice theta functions and SW-data. The results include conjectural formulas that match physics predictions (VW, LL) and rigorous partial verifications, together with a systematic computational approach to generate and test these universal structures. This framework provides a powerful bridge between geometric gauge theory, enumerative geometry, and modular/automorphic structures in the study of moduli spaces of sheaves on surfaces.

Abstract

Moduli spaces of stable sheaves on smooth projective surfaces are in general singular. Nonetheless, they carry a virtual class, which -- in analogy with the classical case of Hilbert schemes of points -- can be used to define intersection numbers, such as virtual Euler characteristics, Verlinde numbers, and Segre numbers. We survey a set of recent conjectures by the authors for these numbers with applications to Vafa-Witten theory, $K$-theoretic S-duality, a rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula, and a virtual Segre-Verlinde correspondence. A key role is played by Mochizuki's formula for descendent Donaldson invariants.

Sheaves on surfaces and virtual invariants

TL;DR

The work develops a unifying framework for virtual invariants of moduli spaces of sheaves on smooth projective surfaces, grounded in Mochizuki's descendent formula and Seiberg-Witten data. It formulates and tests conjectures for virtual Euler characteristics, χ_y-genus refinements, elliptic genera, cobordism classes, Verlinde numbers, and Segre numbers across ranks, linking them to Vafa-Witten S-duality, K-theoretic refinements, and Higgs-pair theories. A central theme is a universal-function paradigm: invariants for any rank and surface are governed by a small set of universal series, computed via toric localization and expressed through lattice theta functions and SW-data. The results include conjectural formulas that match physics predictions (VW, LL) and rigorous partial verifications, together with a systematic computational approach to generate and test these universal structures. This framework provides a powerful bridge between geometric gauge theory, enumerative geometry, and modular/automorphic structures in the study of moduli spaces of sheaves on surfaces.

Abstract

Moduli spaces of stable sheaves on smooth projective surfaces are in general singular. Nonetheless, they carry a virtual class, which -- in analogy with the classical case of Hilbert schemes of points -- can be used to define intersection numbers, such as virtual Euler characteristics, Verlinde numbers, and Segre numbers. We survey a set of recent conjectures by the authors for these numbers with applications to Vafa-Witten theory, -theoretic S-duality, a rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula, and a virtual Segre-Verlinde correspondence. A key role is played by Mochizuki's formula for descendent Donaldson invariants.

Paper Structure

This paper contains 26 sections, 9 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Virtual Euler characteristics
  3. Rank 2
  4. Application: Vafa-Witten invariants
  5. Rank 3
  6. Application: S-duality in rank 3
  7. Refinements
  8. Virtual $\chi_y$-genera
  9. Application: $K$-theoretic S-duality
  10. Virtual elliptic genera
  11. Virtual cobordism classes
  12. Virtual Verlinde numbers
  13. Hilbert schemes
  14. Rank 2
  15. Application: Verlinde formula for Higgs pairs
  16. ...and 11 more sections

Key Result

Theorem 2.5

Fix $\rho>1$. Then there exist $A,C_{ij} \in \mathbb Q(\!(q^{\frac{1}{2\rho}})\!)$, for all $1 \leq i \leq j \leq \rho-1$, and $B \in q^{\frac{\rho}{24}} \, \mathbb Q(\!(q^{\frac{1}{2\rho}})\!)$ with the following property.We suppress the dependence of these universal functions on $\rho$. Let $(S,H) where the sum is over all $(a_1, \ldots, a_{\rho-1}) \in H^2(S,\mathbb Z)^{\rho-1}$ satisfying

Theorems & Definitions (40)

  • Remark 1.1
  • Conjecture 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5: Laarakker
  • Remark 2.6
  • Conjecture 2.7
  • Remark 2.8
  • Remark 2.9
  • ...and 30 more