2012/06/29 by E. G. Vishnyakova, Vishnyakova, E. G.
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG
paper · pdf · doi:10.48550/arxiv.1206.7017
21 pages
arxiv created 2014/07/08 · arxiv updated 2014/07/09
It is a classical result that any complex analytic Lie supergroup G is split \citekosz, that is its structure sheaf is isomorphic to the structure sheaf of a certain vector bundle. However, there do exist non-split complex analytic homogeneous supermanifolds. We study the question how to find out whether a complex analytic homogeneous supermanifold is split or non-split. Our main result is a description of left invariant gradings on a complex analytic homogeneous supermanifold G/H in the terms of H-invariants. 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.