
Journal of Lie Theory 25 (2015), No. 2, 459476 Copyright Heldermann Verlag 2015 The Splitting Problem for Complex Homogeneous Supermanifolds Elizaveta Vishnyakova Université du Luxembourg, 6 rue Richard CoudenhoveKalergi, 1359 Luxembourg vishnyakovae@googlemail.com It is a classical result that any complex analytic Lie supergroup G is split (see J.L. Koszul, Graded manifolds and graded Lie algebras, Proceeding of the International Meeting on Geometry and Physics (Bologna), Pitagora, 7184 (1982)), that is, its structure sheaf is isomorphic to the structure sheaf of a certain vector bundle. However, there do exist nonsplit complex analytic homogeneous supermanifolds. We study the question how to find out whether a complex analytic homogeneous supermanifold is split or nonsplit. Our main result is a description of left invariant gradings on a complex analytic homogeneous supermanifold G/H in the terms of Hinvariants. As a corollary to our investigations we get some simple sufficient conditions for a complex analytic homogeneous supermanifold to be split in terms of Lie algebras. Keywords: Lie supergroup, complex homogeneous supermanifold. MSC: 51P05, 53Z05, 32M10 [ Fulltextpdf (343 KB)] for subscribers only. 