Poisson structures on Hilbert schemes of points of a surface and integrable systems
type: Item
instance of: scholarly article
title: Poisson structures on Hilbert schemes of points of a surface and integrable systems
zbMATH Open document ID: 0945.53049
published in: Manuscripta Mathematica
publication date: 2000-10-12T00:00:00Z
review text: The author proves that if \(S\) is a Poisson surface, i.e. a smooth projective surface with a Poisson structure, then the Hilbert scheme of points of \(S\) has a natural Poisson structure, induced by the one of \(S\). This generalizes previous results obtained by \textit{A. Beauville} [J. Differ. Geom. 18, 755-782 (1983; Zbl 0537.53056)] and \textit{S. Mukai} [Invent. Math. 77, 101-116 (1984; Zbl 0565.14002)] in the symplectic case, i.e. when \(S\) is an Abelian or K3 surface. Finally, the author applies his results to give some examples of integrable Hamiltonian systems naturally defined on these Hilbert schemes. In the case \(S= \mathbb{P}^2\) he obtains a large class of integrable systems, which includes the ones studied by \textit{P. Vanhaecke} [Prog. Math. 145, 187-212 (1997; Zbl 0873.58038)].
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14J25
zbMATH DE Number: 1239594
zbMATH Keywords: Poisson surface
zbMATH Keywords: Hilbert scheme
zbMATH Keywords: Poisson structure
zbMATH Keywords: integrable Hamiltonian systems
zbMATH Keywords: Hilbert schemes
author: Francesco Bottacin
reviewed by: Vincenzo di Gennaro
MaRDI profile type: Publication
full work available at URL: s002290050118
OpenAlex ID: W2095576397
DOI: 10.1007/S002290050118
description: scientific article; zbMATH DE number 1239594