Chern classes of tautological sheaves on Hilbert schemes of points on surfaces
type: Item
instance of: scholarly article
title: Chern classes of tautological sheaves on Hilbert schemes of points on surfaces
zbMATH Open document ID: 0919.14001
author: Manfred Lehn
published in: Inventiones Mathematicae
publication date: 1999-08-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9803091
review text: Let \(X\) be a smooth projective surface and \(X^{[n]}\) the Hilbert scheme of generalized \(n\)-tuples on \(X\). The paper deals with the Chern classes of tautological bundles of the form \(p_*({\mathcal O}_\Xi\otimes q^*E)\), where \(E\) is a vector bundle on \(X\) and \((p,q):\Xi\to X^{[n]}\times X\) is the universal family. It gives an algorithmic description of the action of these Chern classes on the cohomology of the Hilbert schemes within the framework of Nakajima's oscillator algebra [\textit{H. Nakajima}, Ann. Math., II. Ser. 145, No. 2, 379-388 (1997; Zbl 0915.14001)]. This method leads to an identification of the cohomology ring of \((\mathbb{A}^2)^n\) with a ring of explicitly given differential operators on a Fock space. The paper ends with the computation of the top Segre classes of tautological bundles associated to line bundles on \(X^{[n]}\) up to \(n=7\), extending computations of Severi, LeBarz, Tikhomirov and Troshina, and gives a conjecture for the generating series.
reviewed by: Manfred Lehn
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 1295384
zbMATH Keywords: Hilbert scheme
zbMATH Keywords: Chern classes of tautological bundles
zbMATH Keywords: differential operators on a Fock space
MaRDI profile type: Publication
OpenAlex ID: W3104878773
arXiv ID: math/9803091
DOI: 10.1007/S002220050307
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description: scientific article