The stable cohomology of moduli spaces of sheaves on surfaces

Abstract

Let $X$ be a smooth, irreducible, complex projective surface, $H$ a polarization on $X$. Let $\gamma = (r, c, \Delta)$ be a Chern character. In this paper, we study the cohomology of moduli spaces of Gieseker semistable sheaves $M_{X,H} (\gamma)$. When the rank $r = 1$, the Betti numbers were computed by Göttsche. We conjecture that if we fix the rank $r \geq 1$ and the first Chern class $c$, then the Betti numbers (and more generally the Hodge numbers) of $M_{X,H} (r, c, \Delta)$ stabilize as the discriminant $\Delta$ 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 $K_X \cdot H \lt 0$, then the classes $[M_{X,H} (\gamma)]$ stabilize in an appropriate completion of the Grothendieck ring of varieties as $\Delta$ tends to $\infty$. Consequently, the virtual Poincaré and Hodge polynomials stabilize to the conjectural value. In particular, the conjecture holds when $X$ is a rational surface, $H \cdot K_X \lt 0$ and there are no strictly semistable objects in $M_{X,H} (\gamma)$.