Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces
type: Item
instance of: scholarly article
title: Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces
zbMATH Open document ID: 0789.14002
DOI: 10.1007/BF01445104
author: Wolfgang Soergel
author: Lothar Göttsche
published in: Mathematische Annalen
publication date: 1994-06-20T00:00:00Z
full work available at URL: https://eudml.org/doc/165084
review text: For a smooth algebraic surface \(S\) over the complex numbers let \(S^{[n]}\) be the Hilbert scheme of zero-dimensional subschemes of length \(n\). The authors compute the singular cohomology \(H^*(S^{[n]}, \mathbb{Q})\). For the proof they use perverse sheaves. The result has been shown in special cases by \textit{G. Ellingsrud} and \textit{S. A. Strømme} in Invent. Math. 87, 343-352 (1987; Zbl 0625.14002) and for the case of a projective surface by \textit{L. Göttsche} [Math. Ann. 286, No. 1-3, 193- 207 (1990; Zbl 0686.14022)]. Using the mixed Hodge modules of Saito, the authors also describe the Hodge structure of \(H^*(S^{[n]}, \mathbb{Q})\), proving a conjecture of the first author. With the same methods they also compute the cohomology and its Hodge structure for higher order Kummer varieties, which were defined by \textit{A. Beauville} [J. Differ. Geom. 18, 755-782 (1983; Zbl 0537.53056)].
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 14F99
Mathematics Subject Classification ID: 14J99
zbMATH DE Number: 537296
zbMATH Keywords: Hilbert scheme
zbMATH Keywords: singular cohomology
zbMATH Keywords: perverse sheaves
zbMATH Keywords: Kummer varieties
MaRDI profile type: Publication
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OpenAlex ID: W1992553562
description: scientific article; zbMATH DE number 537296