Deformations of Kahler manifolds with non vanishing holomorphic vector fields

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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× Cˢ fibering over a torus T= Cˢ/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 c₁(X)=0 to general Kähler manifolds with non-vanishing vector fields. A complete classification when X is a projective manifold or when X≤ 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.