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 $\Phi (L^{[n]})$ of the tautological bundle $L^{[n]}$ for the Bridgeland-King-Reid equivalence $\Phi :D^b(X^{[n]})\to D_{{\mathfrak{S}}_n}^b(X^n)$ in terms of a complex $C_L^{•}$ of ${\mathfrak{S}}_n$-equivariant sheaves in $D_{{\mathfrak{S}}_n}^b(X^n)$ and we characterize the image $\Phi (L^{[n]}\otimes \cdot \cdot \cdot \otimes L^{[n]})$ in terms of the hyperderived spectral sequence $E_1^{p,q}$ associated to the derived $k$-fold tensor power of the complex $C_L^{•}$. 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 ${\Lambda }^k{L}^{[n]}$