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In this article we study compact K\"ahler manifolds $X$ admittingnon-singular holomorphic vector fields with the aim of extending to thissetting the classical birational classification of projective varieties withtangent vector fields. We prove that any such a K\"ahler manifold $X$ admits anarbitrarily small deformation of a particular type which is a suspension over atorus; that is, a quotient of $F\times \mbb C^s$ fibering over a torus $T=\mbbC^s/\Lambda$. We derive some results dealing with the structure of suchmanifolds. In particular, we prove an extension of Calabi's theorem describingthe structure of compact K\"ahler manifolds with $c_1(X)=0$ to general K\"ahlermanifolds with non-vanishing vector fields. A complete classification when $X$is a projective manifold or when $\dim X\leq s+2$ is also given. As anapplication, it is shown that the study of the dynamics of holomorphic tangentfields on compact K\"ahler manifolds reduces to the case of rational manifolds.

Deformations of Kahler manifolds with non vanishing holomorphic vector fields