1 November 2009 Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles
Luca Scala
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Duke Math. J. 150(2): 211-267 (1 November 2009). DOI: 10.1215/00127094-2009-050


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 Φ(L[n]) of the tautological bundle L[n] for the Bridgeland-King-Reid equivalence Φ:Db(X[n])DSnb(Xn) in terms of a complex CL of Sn-equivariant sheaves in DSnb(Xn) and we characterize the image Φ(L[n]⋅⋅⋅L[n]) in terms of the hyperderived spectral sequence E1p,q associated to the derived k-fold tensor power of the complex CL. The study of the Sn-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]L[n] and ΛkL[n]


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Luca Scala. "Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles." Duke Math. J. 150 (2) 211 - 267, 1 November 2009. https://doi.org/10.1215/00127094-2009-050


Published: 1 November 2009
First available in Project Euclid: 16 October 2009

zbMATH: 1211.14012
MathSciNet: MR2569613
Digital Object Identifier: 10.1215/00127094-2009-050

Primary: 14C05 , 14F05
Secondary: 18E30 , 20C30

Rights: Copyright © 2009 Duke University Press


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Vol.150 • No. 2 • 1 November 2009
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