# Deformations of Kähler manifolds with nonvanishing holomorphic vector fields

Jaume Amorós; Mònica Manjarín; Marcel Nicolau

Journal of the European Mathematical Society (2012)

- Volume: 014, Issue: 4, page 997-1040
- ISSN: 1435-9855

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topAmorós, Jaume, Manjarín, Mònica, and Nicolau, Marcel. "Deformations of Kähler manifolds with nonvanishing holomorphic vector fields." Journal of the European Mathematical Society 014.4 (2012): 997-1040. <http://eudml.org/doc/277305>.

@article{Amorós2012,

abstract = {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 $\textit \{tangential deformations\}$, 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$X\le s+2$ 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.},

author = {Amorós, Jaume, Manjarín, Mònica, Nicolau, Marcel},

journal = {Journal of the European Mathematical Society},

keywords = {Kähler manifold; deformation; vector field; Fujiki manifold; Kähler manifold; deformation; vector field; Fujiki manifold},

language = {eng},

number = {4},

pages = {997-1040},

publisher = {European Mathematical Society Publishing House},

title = {Deformations of Kähler manifolds with nonvanishing holomorphic vector fields},

url = {http://eudml.org/doc/277305},

volume = {014},

year = {2012},

}

TY - JOUR

AU - Amorós, Jaume

AU - Manjarín, Mònica

AU - Nicolau, Marcel

TI - Deformations of Kähler manifolds with nonvanishing holomorphic vector fields

JO - Journal of the European Mathematical Society

PY - 2012

PB - European Mathematical Society Publishing House

VL - 014

IS - 4

SP - 997

EP - 1040

AB - 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 $\textit {tangential deformations}$, 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$X\le s+2$ 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.

LA - eng

KW - Kähler manifold; deformation; vector field; Fujiki manifold; Kähler manifold; deformation; vector field; Fujiki manifold

UR - http://eudml.org/doc/277305

ER -

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