
Journal of Lie Theory 33 (2023), No. 4, 9651004 Copyright Heldermann Verlag 2023 On Odd Parameters in Geometry Dimitry Leites Department of Mathematics, University of Stockholm, Sweden dimleites@gmail.com (1) In 1976, looking at the list of simple finitedimensional complex Lie superalgebras, J. Bernstein and I, and independently M. Duflo, observed that some divergencefree 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 finitedimensional 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. K. Gawedzki (1977) and M. Batchelor (1979) proved that every smooth supermanifold is split. In 1982, P. Green and V. P. Palamodov showed that a complexanalytic supermanifold can be nonsplit. So far, researchers considered only even obstructions to splitness. This lead them to the conclusion that any supermanifold of superdimension m1 is split. I'll show there are nonsplit supermanifolds of superdimension m1; e.g., certain superstrings, the obstructions to their splitness depend on odd parameters. Keywords: Simple Lie superalgebra, deformation, nonsplit supermanifold. MSC: 58A50, 17B60. [ Fulltextpdf (288 KB)] for subscribers only. 