Remarks on Grassmannian supermanifolds

Author:
Oscar Adolfo Sánchez-Valenzuela

Journal:
Trans. Amer. Math. Soc. **307** (1988), 597-614

MSC:
Primary 58A50; Secondary 14M15

DOI:
https://doi.org/10.1090/S0002-9947-1988-0940219-3

MathSciNet review:
940219

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Abstract: This paper studies some aspects of a particular class of examples of supermanifolds; the *supergrassmannians*, introduced in [**Manin**]. Their definition, in terms of local data and glueing isomorphisms, is reviewed. Explicit formulas in local coordinates are given for the Lie group action they come equipped with. It is proved that, for those supergrassmannians whose underlying manifold is an ordinary grassmannian, their structural sheaf can be realized as the sheaf of sections of the exterior algebra bundle of some canonical vector bundle. This realization holds true equivariantly for the Lie group action in question, thus making natural in these cases the identification given in [**Batchelor**]. The proof depends on the computation of the transition functions of the *supercotangent bundle* as defined in a previous work [**OASV 2**]. Finally, it is shown that there is a natural *supergroup action* involved (in the sense of [**OASV 3**]) and hence, the supergrassmannians may be regarded as examples of *superhomogeneous spaces*--a notion first introduced in [**Kostant**]. The corresponding *Lie superalgebra* action can be realized as superderivations of the structural sheaf; explicit formulas are included for those supergrassmannians identifiable with exterior algebra vector bundles.

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DOI:
https://doi.org/10.1090/S0002-9947-1988-0940219-3

Article copyright:
© Copyright 1988
American Mathematical Society