A virtual ${\rm PGL}_r$–${\rm SL}_r$ 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 $h^{2,0}(X) \gt 0$ , 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 ${\rm PGL}_r$ – ${\rm SL}_r$ correspondence. As an application, we express ${\rm SU}(r) / \mu _r$ Vafa–Witten invariants of $X$ in terms of ${\rm SU}(r)$ 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.