Total Cohomology of Hilbert schemes of Orbisurfaces I: Reduction to Stacky Loci

Abstract

We generalize Nakajima's theory of Hilbert schemes of points on surfaces to Hilbert schemes of points on \textit{orbisurfaces}, and obtain initial factorization based on stratification by stabilizers. While the usual formula is retained for the trivial stabilizer stratum and for those strata of codimension $1$, genuine difficulties arise with stacky points strata. In the local picture of an isolated stacky point, the theory reduce to the study of $\Quot^\bfv[\CC^2/G]$ for a \textit{small} finite subgroup of $GL_2(\CC)$. The generating series for their total cohomology is a geometric correction of the generating series for total cohomology of ordinary smooth surface schemes. In addition, we obtain an action of the Borel-Moore homology of the punctual fibre stacks on the total Hilbert cohomology by virtual correspondences, in analogy with the Fock space description in Nakajima's theory. The punctual fibre stacks are genuinely different from the symmetric stack, and has natural realizations via motivic integration.