2023/08/04 by Dirk van Bree, van Bree, D., Amin Gholampour +5
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2308.02288
For a smooth projective surface X satisfying H1(X,ℤ) = 0 and w ∈ H2(X,μr), we study deformation invariants of the pair (X,w). Choosing a Brauer-Severi variety Y (or, equivalently, Azumaya algebra 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 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 h2,0(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 PGLr-SLr correspondence. As an application, we express SU(r) / μr Vafa-Witten invariants of X in terms of 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.