Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles

Abstract

Let $X^{[n]}$ be the Hilbert scheme of $n$ points on the smooth quasi-projective surface $X$, and let $L^{[n]}$ be the tautological bundle on $X^{[n]}$ naturally associated to the line bundle $L$ on $X$. As a corollary of Haiman's results, we express the image ${\mathbf {\Phi}}(L^{[n]})$ of the tautological bundle $L^{[n]}$ for the Bridgeland-King-Reid equivalence ${\mathbf{\Phi}}:{\mathbf D}^b(X^{[n]}) \rightarrow \mathbf{D}^{b}_{\mathfrak{S}_n}(X^n)$ in terms of a complex ${\mathcal C}^{\bullet}_L$ of $\mathfrak{S}_n$-equivariant sheaves in ${\mathbf D}^b_{\mathfrak{S}_n}(X^n)$ and we characterize the image ${\mathbf{\Phi}}(L^{[n]} \otimes \cdots \otimes L^{[n]})$ in terms of the hyperderived spectral sequence $E^{p,q}_1$ associated to the derived $k$-fold tensor power of the complex ${\mathcal C}^{\bullet}_L$. The study of the ${\mathfrak S}_n$-invariants of this spectral sequence allows us to get the derived direct images of the double tensor power and of the general $k$-fold exterior power of the tautological bundle for the Hilbert-Chow morphism, providing Danila-Brion-type formulas in these two cases. This easily yields the computation of the cohomology of $X^{[n]}$ with values in $L^{[n]} \otimes L^{[n]}$ and $\Lambda^k L^{[n]}$