A generalization of Higgs bundles to higher dimensional varieties
type: Item
instance of: scholarly article
title: A generalization of Higgs bundles to higher dimensional varieties
zbMATH Open document ID: 1027.14021
DOI: 10.1007/PL00004797
author: Francesco Bottacin
published in: Mathematische Zeitschrift
publication date: 2004-01-11T00:00:00Z
review text: In the paper under review, the author studies a generalization of Higgs bundles to higher dimensional varieties. Properties of the moduli space parametrizing these generalized Higgs bundles are investigated intensively. In particular, a Poisson structure on this moduli space and its relation with parabolic structure over divisors are found. The main tool employed in this paper is deformation theory. In section one, for a fixed line bundle \(L\) on an \(n\)-dimensional smooth variety \(X\), the author introduces the notion of \((n-1)\)-cohomology \(L\)-twisted \textit{Higgs pairs} \((E, \overline{\phi})\) where \(E\) is a locally free sheaf on \(X\) and \(\overline{\phi} \in H^{n-1}(X, {\mathcal E}nd(E) \otimes L)\), and the notion of \textit{stability} for these pairs \((E, \overline{\phi})\). The moduli space of these pairs is denoted by \({\mathcal P}(L)\). The existence of a univeral Higgs pair on \({\mathcal P}(L)\) is discussed. In section two, the author uses infinitesimal deformation theory to study the tangent and cotangent bundles of \({\mathcal P}(L)\), and proves that they are related to the cohomology groups of certain complexes. A homomorphism \(B_s: T^*{\mathcal P}(L) \to T{\mathcal P}(L)\) is constructed, where \(s\) is a section of \(\omega_X^{-1} \otimes L\) and \(\omega_X\) denotes the canonical line bundle of \(X\). The map \(B_s\) is further investigated in section three. Section four is devoted to tangent vector fields on \({\mathcal P}(L)\). After reviewing some general results about symplectic and Poisson structures in section five, the author proves in section six that the antisymmetric contravariant \(2\)-tensor \(\theta_s\) coming from \(B_s\) defines a Poisson structure on \({\mathcal P}(L)\). This Poisson structure coincides with the canonical symplectic structure when \(L = \omega_X\) and \(s\) is the identity section. In section seven, relations between the common Higgs bundles and the generalized ones are analyzed. In the final section…
reviewed by: Zhenbo Qin
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14B10
zbMATH DE Number: 1419356
zbMATH Keywords: instanton bundle
zbMATH Keywords: Higgs bundle
zbMATH Keywords: Poisson structure
zbMATH Keywords: deformation theory
zbMATH Keywords: moduli space
zbMATH Keywords: parabolic vector bundles
MaRDI profile type: Publication
full work available at URL: pl00004797
OpenAlex ID: W2087547979
description: scientific article; zbMATH DE number 1419356