2009/09/25 by Jaume Amorós, Amoros, Jaume, Mònica Manjarín +3 · 1 citation
Mathematics · #Geometry and complex manifolds #Algebraic Geometry and Number Theory #Advanced Differential Equations and Dynamical Systems
paper · pdf · doi:10.48550/arxiv.0909.4690
In this article we study compact K "ahler manifolds X admitting\nnon-singular holomorphic vector fields with the aim of extending to this\nsetting the classical birational classification of projective varieties with\ntangent vector fields. We prove that any such a K "ahler manifold X admits an\narbitrarily small deformation of a particular type which is a suspension over a\ntorus; that is, a quotient of F\× mbb Cs fibering over a torus T= mbb\nCs/\Λ. We derive some results dealing with the structure of such\nmanifolds. In particular, we prove an extension of Calabi's theorem describing\nthe structure of compact K "ahler manifolds with c1(X)=0 to general K "ahler\nmanifolds with non-vanishing vector fields. A complete classification when X\nis a projective manifold or when \dim X\≤ s+2 is also given. As an\napplication, it is shown that the study of the dynamics of holomorphic tangent\nfields on compact K "ahler manifolds reduces to the case of rational manifolds.\n