Immersivity of the contact line bundle of a complex-contact manifold and an application to the automorphism group

TL;DR

This paper studies complex-contact manifolds, showing that if the contact line bundle is immersive, the automorphism group aligns with a simple complex Lie algebra, extending previous results by relaxing key assumptions.

Contribution

It introduces the concept of immersivity of the contact line bundle and demonstrates its implications for the automorphism group of complex-contact manifolds.

Findings

Automorphism group is isomorphic to a simple complex Lie algebra for immersive contact line bundles.

Automorphism group is not connected if the Lie algebra type is A_{n}, D_{n+2} (n>1), or E_{6}.

Relaxed assumptions extend previous classifications of complex-contact manifolds.

Abstract

A connected Fano complex-contact manifold is isomorphic to the kaehlerian C-space of Boothby type with a natural complex-contact structure corresponding to a non-abelian simple complex Lie algebra if the contact line bundle is very ample. A. Beauville relaxed the provision to two assumptions that the contact line bundle is generically finite and that the automorphism group is reductive. We relax the provision to another one that that the contact line bundle is immersive, that is, the manifold admits a holomorphic immersion into some projective space associated with some holomorphic sections of the line bundle. As an application, we obtain that the automorphism group of a connected compact complex-contact manifold with immersive contact line bundle is isomorphic to the automorphism group of the corresponding simple complex Lie algebra of rank greater than one, which is not connected if and only if its type is A_{n}, D_{n+2} for n > 1 or E_{6}.