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

Abstract

For a smooth projective surface $X$ satisfying $H_1(X,\mathbb{Z}) = 0$ and $w \in H^2(X,\mu_r)$, 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 $\mathrm{SU}(r) / \mu_r$ 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.