Deformations of Kahler manifolds with non vanishing holomorphic vector fields

Details Deformations of Kahler manifolds with non vanishing holomorphic vector fields Deformations of Kahler manifolds with non vanishing holomorphic vector fields

2009-09-25

arxiv.org/abs/0909.4690v4

Mathematics

Algebraic Geometry

38 pages, change in the order of the sections, to appear in JEMS

Scientific paper

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 $F\times \mbb C^s$ 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 $\dim X\leq s+2$ 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.

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