On holomorphic functions on a compact complex homogeneous supermanifold

TL;DR

This paper investigates conditions under which non-constant holomorphic functions exist on compact homogeneous complex supermanifolds, extending classical results and computing holomorphic function algebras on flag supermanifolds.

Contribution

It establishes criteria for the non-existence of non-constant holomorphic functions on such supermanifolds and describes vector bundles on split homogeneous supermanifolds, including explicit calculations for flag supermanifolds.

Findings

Non-constant holomorphic functions can exist on compact supermanifolds with compact reduction.

Criteria are provided for when such functions do not exist.

The algebra of holomorphic functions on classical flag supermanifolds is explicitly computed.

Abstract

It is well-known that non-constant holomorphic functions do not exist on a compact complex manifold. This statement is false for a supermanifold with a compact reduction. In this paper we study the question under what conditions non-constant holomorphic functions do not exist on a compact homogeneous complex supermanifold. We describe also the vector bundles determined by split homogeneous complex supermanifolds. As an application, we compute the algebra of holomorphic functions on the classical flag supermanifolds which were introduced by Yu.I. Manin.