The stable cohomology of moduli spaces of sheaves on surfaces

Let X be a smooth, irreducible, complex projective surface, H a polarization on X. Let γ = (r, c,∆) be a Chern character. In this paper, we study the cohomology of moduli spaces of Gieseker semistable sheaves MX,H(γ). When the rank r = 1, the Betti numbers were computed by Göttsche. We conjecture that if we fix the rank r ≥ 1 and the first Chern class c, then the Betti numbers (and more generally the Hodge numbers) of MX,H(r, c,∆) stabilize as the discriminant ∆ tends to infinity and that the stable Betti numbers are independent of r and c. In particular, the conjectural stable Betti numbers are determined by Göttsche’s calculation. We present evidence for the conjecture. We analyze the validity of the conjecture under blowup and wall-crossing. We prove that when X is a rational surface and KX · H < 0, then the classes [MX,H(γ)] stabilize in an appropriate completion of the Grothendieck ring of varieties as ∆ tends to ∞. Consequently, the virtual Poincaré and Hodge polynomials stabilize to the conjectural value. In particular, the conjecture holds when X is a rational surface, H · KX < 0 and there are no strictly semistable objects in MX,H(γ).