Deformations of Kähler manifolds with nonvanishing holomorphic vector fields

Amorós, 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 -