Deformations of Kähler manifolds with nonvanishing holomorphic vector fields

We study compact Kähler manifolds X admitting nonvanishing holomorphic vector fields, extending the classical birational classification of projective varieties with tangent vector fields to a classification modulo deformation in the Kähler case, and biholomorphic in the projective case. We introduce and analyze a new class of tangential deformations, and show that they form a smooth subspace in the Kuranishi space of deformations of the complex structure of X. We extend Calabi's theorem on the structure of compact Kähler manifolds X with c 1(X) = 0 to compact Kähler manifolds with nonvanishing tangent fields, proving that any such manifold X admits an arbitrarily small tangential deformation which is a suspension over a torus, that is, a quotient of F × ℂ s fibering over a torus T = ℂ s /A. We further show that either X is uniruled or, up to a finite Abelian covering, it is a small deformation of a product F × T where F is a Kähler manifold without tangent vector fields and T is a torus. A complete classification when X is a projective manifold, in which case the deformations may be omitted, or when dim 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 varieties. © European Mathematical Society 2012.