Deformations of Kähler manifolds with nonvanishing holomorphic vector fields

  • Jaume Amorós

    Universitat Politecnica de Catalunya, Barcelona, Spain
  • Mònica Manjarín

    Université de Rennes I, France
  • Marcel Nicolau

    Universidad Autonoma de Barcelona, Bellaterra, Spain


We study compact Kähler manifolds XX 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{tangential \ deformations}, and show that they form a smooth subspace in the Kuranishi space of deformations of the complex structure of XX. We extend Calabi's theorem on the structure of compact Kähler manifolds XX with c1(X)=0c_1(X) =0 to compact Kähler manifolds with nonvanishing tangent fields, proving that any such manifold XX admits an arbitrarily small tangential deformation which is a suspension over a torus; that is, a quotient of F×CsF\times \mathbb C^s fibering over a torus T=Cs/ΛT=\mathbb C^s/\Lambda. We further show that either XX is uniruled or, up to a finite Abelian covering, it is a small deformation of a product F×TF\times T where FF is a Kähler manifold without tangent vector fields and TT is a torus. A complete classification when XX is a projective manifold, in which case the deformations may be omitted, or when dimXs+2\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.

Cite this article

Jaume Amorós, Mònica Manjarín, Marcel Nicolau, Deformations of Kähler manifolds with nonvanishing holomorphic vector fields. J. Eur. Math. Soc. 14 (2012), no. 4, pp. 997–1040

DOI 10.4171/JEMS/325