Deformations of Kahler manifolds with non vanishing holomorphic vector fields

Abstract

In this article we study compact Kähler manifolds $X$ admitting non-singular holomorphic vector fields with the aim of extending to this setting the classical birational classification of projective varieties with tangent vector fields. We prove that any such a Kähler manifold $X$ admits an arbitrarily small deformation of a particular type which is a suspension over a torus; that is, a quotient of $F\times \mbb C^s$ fibering over a torus $T=\mbb C^s/Λ$. We derive some results dealing with the structure of such manifolds. In particular, we prove an extension of Calabi's theorem describing the structure of compact Kähler manifolds with $c_1(X)=0$ to general Kähler manifolds with non-vanishing vector fields. A complete classification when $X$ is a projective manifold 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 manifolds.

arXiv comment38 pages, change in the order of the sections, to appear in JEMS