On Odd Parameters in Geometry

1) In 1976, looking at the list of simple finite-dimensional complex Lie superalgebras, J. Bernstein and I, and independently M. Duflo, observed that some divergence-free vectorial Lie superalgebras have deformations with odd parameters and conjectured that no other simple Lie superalgebras have such deformations.Here, I prove this conjecture and overview the known classification of simple finite-dimensional complex Lie superalgebras, their presentations, realizations, and relations with simple Lie (super)algebras over fields of positive characteristic.(2) Any ringed space of the form (a manifold M , the sheaf of sections of the exterior algebra of a vector bundle over M ) is called split supermanifold.Gaw , edzki (1977) and Batchelor (1979) proved that every smooth supermanifold is split.In 1982, P. Green and Palamodov showed that a complexanalytic supermanifold can be non-split.So far, researchers considered only even obstructions to splitness.This lead them to the conclusion that any supermanifold of superdimension m|1 is split.I'll show there are non-split supermanifolds of superdimension m|1 ; e.g., certain superstrings, the obstructions to their splitness depend on odd parameters.