On holomorphic functions on a compact complex homogeneous supermanifold

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.Comment: 29 page

Superdistributions, analytic and algebraic super Harish-Chandra pairs

“…We realize that in both cases the theory is very similar to the differential one, however given some crucial differences between the smooth category and the analytic and algebraic one, we believe the present work is justified, given the importance of this theory for practical purposes together with the lack of an appropriate and complete available reference, though we are aware that a good step towards a complete clarification of these issues, for the analytic setting only, appears in the papers [21], [22]. This paper was put on the web on June 2011.…”

Graded covering of a supermanifold I. The case of a Lie supergroup

“…Since any morphism F of supermanifolds preserves the filtration (3), the morphism gr(F ) is defined. Summing up, the functor gr is a functor from the category of supermanifolds to the category of split supermanifolds, see for example [Vi2,Section 3.1] for details. We can apply the functor gr to a Lie supergroup G and we will get a split Lie supergroup gr(G).…”

Vector fields on glm|n(C)-flag supermanifolds

“…Let us give an explicite description of a flag supermanifold in terms of charts and local coordinates (see also [Man,V3]). Let us take two nonnegative integers m, n ∈ Z and two sets of non-negative integers k = (k 1 , .…”