ABSTRACT: In the 1990’s, Vafa-Witten tested S-duality of N=4 SUSY Yang-Mills theory on a complex algebraic surface X by studying modularity of a certain partition function. In 2017, Tanaka-Thomas defined Vafa-Witten invariants by constructing a symmetric perfect obstruction theory on the moduli space of Higgs pairs (E,\phi) on X. The instanton contribution (\phi=0) to these invariants is the virtual Euler number of moduli space of sheaves. I outline a method to calculate this contribution, when X is of general type, by reducing to descendent Donaldson invariants. For rank 2, this leads to verifications of a formula from Vafa-Witten. The method allows us to go beyond numbers: I propose refinements to virtual \chi_y genus, elliptic genus, and cobordism class, which involves weak Jacobi forms and Borcherds lifts thereof. I also give a new formula for rank 3 VW invariants on general type surfaces, correcting an error in the physics literature. Joint with Göttsche.