On odd parameters in geometry

Abstract

1) In 1976, looking at simple finite-dimensional complex Lie superalgebras, J.~Bernstein and I, and independently M.~Duflo, observed that certain divergence-free vectorial Lie superalgebras have deformations with odd parameters and conjectured that other simple Lie superalgebras have no such deformations (unpublished). Here, I prove this conjecture and overview the known classification of simple finite-dimensional complex Lie superalgebras, their presentations, realizations, and (very sketchily) relations with simple Lie (super)algebras over fields of positive characteristic. 2) Any supermanifold which is a 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. Gawȩdzki (1977) and Batchelor (1979) proved that every smooth supermanifolds is split. In 1982--83, P. Green and Palamodov showed that a~complex-analytic supermanifold can be non-split. So far, researchers considered only even obstructions to splitness, and hence concluded that any supermanifold of superdimension $m|1$ is split. I'll show there are non-split supermanifolds of superdimension $m|1$ \textbf{over a~base with odd parameters}, although every $m|1$-dimensional supermanifold over $\text{Spec}\mathbb{C}$ is split; e.g., certain superstrings, the obstructions to their splitness depend on odd parameters.

40 pages; the strange words in Theorem 5.1 are struck out, references updated; otherwise coincides with the published version, but an incomplete proof is completed

Cited by in corpus (1)

• Deformations of Symmetric Simple Modular Lie (Super)Algebras