Deformations of Kähler manifolds with nonvanishing holomorphic vector fields

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 $tangentialdeformations$, 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.