Deformations of Kähler manifolds with nonvanishing holomorphic vector fields (Q437809)

Summary: 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 \(F\times \mathbb C^s\) fibering over a torus \(T=\mathbb C^s/\Lambda\). We further show that either \(X\) is uniruled or, up to a finite Abelian covering, it is a small deformation of a product \(F\times T\) 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 \(\dim X\leq s+2\) 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.