Duke Mathematical Journal

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

Luca Scala

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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 $\Lambda^k L^{[n]}$

Article information

Source
Duke Math. J. Volume 150, Number 2 (2009), 211-267.

Dates
First available in Project Euclid: 16 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1255699340

Digital Object Identifier
doi:10.1215/00127094-2009-050

Mathematical Reviews number (MathSciNet)
MR2569613

Zentralblatt MATH identifier
1211.14012

Subjects
Primary: 14C05: Parametrization (Chow and Hilbert schemes) 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]
Secondary: 18E30: Derived categories, triangulated categories 20C30: Representations of finite symmetric groups

Citation

Scala, Luca. Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles. Duke Math. J. 150 (2009), no. 2, 211--267. doi:10.1215/00127094-2009-050. https://projecteuclid.org/euclid.dmj/1255699340.


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