Deformations of Kähler manifolds with nonvanishing holomorphic vector fields

Jaume Amorós ; Mònica Manjarín ; Marcel Nicolau

Journal of the European Mathematical Society, Tome 14 (2012) no. 4, pp. 997-1040

Résumé

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 c1​(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×Cs fibering over a torus T=Cs/Λ. 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 dimX≤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.

Détail

DOI : 10.4171/jems/325

Classification : 32-XX, 14-XX, 37-XX, 00-XX
Keywords: Kähler manifold, deformation, vector field, Fujiki manifold