Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A virtual $\mathrm{PGL}_r$-$\mathrm{SL}_r$ correspondence for projective surfaces

D. van Bree, A. Gholampour, Y. Jiang, M. Kool

Abstract

For a smooth projective surface $X$ satisfying $H_1(X,\mathbb{Z}) = 0$ and $w \in H^2(X,μ_r)$, we study deformation invariants of the pair $(X,w)$. Choosing a Brauer-Severi variety $Y$ (or, equivalently, Azumaya algebra $\mathcal{A}$) over $X$ with Stiefel-Whitney class $w$, the invariants are defined as virtual intersection numbers on suitable moduli spaces of stable twisted sheaves on $Y$ constructed by Yoshioka (or, equivalently, moduli spaces of $\mathcal{A}$-modules of Hoffmann-Stuhler). We show that the invariants do not depend on the choice of $Y$. Using a result of de Jong, we observe that they are deformation invariants of the pair $(X,w)$. For surfaces with $h^{2,0}(X) > 0$, we show that the invariants can often be expressed as virtual intersection numbers on Gieseker-Maruyama-Simpson moduli spaces of stable sheaves on $X$. This can be seen as a $\mathrm{PGL}_r$-$\mathrm{SL}_r$ correspondence. As an application, we express $\mathrm{SU}(r) / μ_r$ Vafa-Witten invariants of $X$ in terms of $\mathrm{SU}(r)$ Vafa-Witten invariants of $X$. We also show how formulae from Donaldson theory can be used to obtain upper bounds for the minimal second Chern class of Azumaya algebras on $X$ with given division algebra at the generic point.

A virtual $\mathrm{PGL}_r$-$\mathrm{SL}_r$ correspondence for projective surfaces

Abstract

For a smooth projective surface satisfying and , we study deformation invariants of the pair . Choosing a Brauer-Severi variety (or, equivalently, Azumaya algebra ) over with Stiefel-Whitney class , the invariants are defined as virtual intersection numbers on suitable moduli spaces of stable twisted sheaves on constructed by Yoshioka (or, equivalently, moduli spaces of -modules of Hoffmann-Stuhler). We show that the invariants do not depend on the choice of . Using a result of de Jong, we observe that they are deformation invariants of the pair . For surfaces with , we show that the invariants can often be expressed as virtual intersection numbers on Gieseker-Maruyama-Simpson moduli spaces of stable sheaves on . This can be seen as a - correspondence. As an application, we express Vafa-Witten invariants of in terms of Vafa-Witten invariants of . We also show how formulae from Donaldson theory can be used to obtain upper bounds for the minimal second Chern class of Azumaya algebras on with given division algebra at the generic point.
Paper Structure (22 sections, 34 theorems, 144 equations)

This paper contains 22 sections, 34 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. SLr invariants
  3. PGLr invariants
  4. Main results
  5. Consequences
  6. Application to Vafa-Witten theory
  7. Application to $c_2^{\mathrm{min}}$ of Azumaya algebras
  8. Models of twisted sheaves
  9. Brauer-Severi varieties, Azumaya algebras, and gerbes
  10. Twisted sheaves
  11. Brauer group
  12. Moduli of twisted sheaves
  13. Chern characters
  14. Moduli spaces
  15. Invariants of moduli of twisted sheaves
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\widetilde{w} \in H^0(B,R^2 f_* \mu_r)$ be a section. Suppose $r$ is prime and $\mathrm{gcd}(r,\widetilde{w}_b \mathcal{H}_b)=1$ for some (and hence all) closed points $b \in B$. Then $\mathsf{Z}_{(\mathcal{X}_{b},\mathcal{H}_{b}),\widetilde{w}_{b}}^{\mathrm{PGL}_r, \mathsf{P}}(q)$ is independe

Theorems & Definitions (83)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Definition 2.1
  • Theorem 2.2: Skolem-Noether
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • Proposition 2.5
  • ...and 73 more