Variétés kähleriennes dont la première classe de Chern est nulle
type: Item
instance of: scholarly article
title: Variétés kähleriennes dont la première classe de Chern est nulle
zbMATH Open document ID: 0537.53056
author: Arnaud Beauville
published in: Journal of Differential Geometry
publication date: 1983-01-01T00:00:00Z
review text: The first part of this paper recalls the decomposition theorem for compact Kähler manifolds with \(c_ 1=0\). Up to a finite covering, such a manifold splits as a product of a complex torus and compact Kähler manifolds with holonomy group SU(m) or Sp(r). Compact manifolds with holonomy SU(m) (\(m\geq 3)\) are projective, with trivial canonical bundle and no holomorphic p-forms for \(p<m=\dim(X)\). Compact manifolds with holonomy Sp(r) are characterized by the existence of a unique (up to a scalar) holomorphic 2-form which is everywhere non-degenerate: they are called Kähler symplectic manifolds. The second part of the paper gives examples of Kähler symplectic manifolds in any (even) dimension. They are constructed from symmetric products of K3 or abelian surfaces. Then the period map for symplectic manifolds is studied. The situation is analogous to the case of K3 surfaces: the space \(H^ 2(X,{\mathbb{C}})\) has a natural quadratic form, defined over \({\mathbb{Z}}\), and the period map is a local isomorphism from the moduli space into the quadric in \({\mathbb{P}}(H^ 2(X,{\mathbb{C}}))\) defined by this form.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 3853948
zbMATH Keywords: first Chern class
zbMATH Keywords: Kähler manifolds
zbMATH Keywords: Kähler symplectic manifolds
zbMATH Keywords: abelian surfaces
zbMATH Keywords: K3 surfaces
zbMATH Keywords: moduli space
Wikidata QID: Q115187906
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214438181
OpenAlex ID: W1942165802
DOI: 10.4310/JDG/1214438181
description: scientific article; zbMATH DE number 3853948