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### Verlinde formulas on surfaces

## Abstract

Let $S$ be a smooth projective surface with $p_g>0$ and $H^1(S,{\mathbb Z})=0$.

We consider the moduli spaces $M=M_S^H(r,c_1,c_2)$ of $H$-semistable sheaves on $S$ of rank $r$ and

with Chern classes $c_1,c_2$. Associated a suitable class $v$ the Grothendieck group of vector bundles

on $S$ there is a deteminant line bundle $\lambda(v)\in Pic(M)$, and also a tautological sheaf $\tau(v)$ on $M$.

In this talk we derive a conjectural generating function for the virtual Verlinde numbers, i.e. the virtual holomorphic

Euler characteristics of all determinant bundles $\lambda(v)$ on M, and for Segre invariants associated to $\tau(v)$ .

The argument is based on conjectural blowup formulas and a virtual version of Le Potier's strange duality.

Time permitting we also sketch a common refinement of these two conjectures, and their proof for Hilbert schemes of points.