Deformations of Kahler manifolds with non vanishing holomorphic vector fields

In this article we study compact K"ahler 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 K"ahler manifold $X$ admits an arbitrarily small deformation of a particular type which is a suspension over a torus; that is, a quotient of $F\times \text{mbb}{C}^s$ fibering over a torus $T=\mbb C^s/\Lambda$. 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 K"ahler manifolds with $c_1(X)=0$ to general K"ahler 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 K"ahler manifolds reduces to the case of rational manifolds.