Deformations of Kähler manifolds with nonvanishing holomorphic vector fields
type: Item
instance of: scholarly article
title: Deformations of Kähler manifolds with nonvanishing holomorphic vector fields
zbMATH Open document ID: 1266.32024
DOI: 10.4171/JEMS/325
author: Jaume Amorós
author: Mònica Manjarín
author: Marcel Nicolau
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2012-07-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/0909.4690
review text: Summary: We study compact Kähler manifolds \(X\) admitting nonvanishing holomorphic vector fields, extending the classical birational classification of projective varieties with tangent vector fields to a classification modulo deformation in the Kähler case, and biholomorphic in the projective case. We introduce and analyze a new class of tangential deformations, and show that they form a smooth subspace in the Kuranishi space of deformations of the complex structure of \(X\). We extend Calabi's theorem on the structure of compact Kähler manifolds \(X\) with \(c_1(X) =0\) to compact Kähler manifolds with nonvanishing tangent fields, proving that any such manifold \(X\) admits an arbitrarily small tangential deformation which is a suspension over a torus; that is, a quotient of \(F\times \mathbb C^s\) fibering over a torus \(T=\mathbb C^s/\Lambda\). We further show that either \(X\) is uniruled or, up to a finite Abelian covering, it is a small deformation of a product \(F\times T\) where \(F\) is a Kähler manifold without tangent vector fields and \(T\) is a torus. A complete classification when \(X\) is a projective manifold, in which case the deformations may be omitted, or when \(\dim X\leq s+2\) is also given. As an application, it is shown that the study of the dynamics of holomorphic tangent fields on compact Kähler manifolds reduces to the case of rational varieties.
Mathematics Subject Classification ID: 32J27
Mathematics Subject Classification ID: 32G05
Mathematics Subject Classification ID: 32M25
zbMATH DE Number: 6058741
zbMATH Keywords: Kähler manifold
zbMATH Keywords: deformation
zbMATH Keywords: vector field
zbMATH Keywords: Fujiki manifold
MaRDI profile type: Publication
Wikidata QID: Q61733161
arXiv ID: 0909.4690
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description: scientific article; zbMATH DE number 6058741