Algebraic complete integrability of an integrable system of Beauville
type: Item
instance of: scholarly article
title: Algebraic complete integrability of an integrable system of Beauville
zbMATH Open document ID: 1144.14037
author: Jun-Muk Hwang
author: Yasunari Nagai
publication date: 2008-07-04T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10324
review text: The authors consider the following integrable system constructed by Beauville. Given a \(K3\) surface \(S\) of degree \(6\) in \({\mathbb P}^4\), let us consider the moduli space \(M^v_S\) of stable bundles with rank \(2\), \(c_1=0\) and \(c_2=4\) over \(S\). This space in endowed by the Mukai symplectic structure. Let a smooth cubic threefold \(X \subset {\mathbb P}^4\) contains \(S\) and \(M^v_X\) be the moduli space of stable bundles with rank \(2\), \(c_1=0\) and \(c_2=2\) over \(X\). The restriction of bundles over \(X\) to \(S\) is an embedding \(M^v_X \to M^v_S\) whose fibers are Lagrangian submanifolds. Varying \(X\) we obtain a fibration of an open dense subset of \(M^v_S\) by Lagrangian submanifolds which are invariant sets of some integrable Hamiltonian system, the Beauville system. The authors prove the following theorem: Let \(\bar{M}_S\) be the moduli space of semi-stable sheaves which compatifies \(M^v_S\) and \(\tilde{M}_S\) be the O'Grady's resolution of \(M_S\). Then there exists an open subset \(U\subset \tilde{M}_S\) such that the Beauville system extends to \(U\) and any fiber is the complement of a subvariety of codimension \(2\) in the intermediate Jacobian \(J(X)\). In particular, this implies that for any threefold \(X\) the Hamiltonian vector fields on \(M^v_X\) are extended to \(J(X)\), i.e. the system is algebraically completely integrable.
reviewed by: I. A. Taimanov
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 37J35
zbMATH DE Number: 5298312
zbMATH Keywords: moduli space of stable sheaves
zbMATH Keywords: \(K3\) surface
zbMATH Keywords: Lagrangian fibration
zbMATH Keywords: integrable system
MaRDI profile type: Publication
OpenAlex ID: W2520578099
cites work: Q4428936
cites work: Q4522263
cites work: Q3128810
cites work: Partial desingularisations of quotients of nonsingular varieties and their Betti numbers
cites work: La singularité de O’Grady
cites work: Symplectic structure of the moduli space of sheaves on an abelian or K 3 surface
cites work: Desingularized moduli spaces of sheaves on a K3
DOI: 10.5802/AIF.2360
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5298312