Contact Kaehler Manifolds: Symmetries and Deformations

TL;DR

This paper classifies complex compact Kaehler manifolds with contact structures that are almost homogeneous and have second Betti number at least 2, showing they are mostly projectivised tangent bundles with specific exceptions.

Contribution

It extends the classification of contact Kaehler manifolds by identifying conditions under which they are projectivised tangent bundles and describes their deformation behavior.

Findings

Such manifolds are projectivised tangent bundles when almost homogeneous with b2 ≥ 2.

Global projective deformations typically preserve the bundle type.

Exceptions include certain unstable bundles over projective spaces.

Abstract

We study complex compact Kaehler manifolds $X$ carrying a contact structure. If $X$ is almost homogeneous and $b_2(X) \geq 2$, then $X$ is a projectivised tangent bundle (this was known in the projective case even without assumption on the existence of vector fields). We further show that a global projective deformation of the projectivised tangent bundle over a projective space is again of this type unless it is the projectivisation of a special unstable bundle over a projective space. Examples for these bundles are given in any dimension.