On the rigidity of super-Grassmannians

Onishchik, A. L.

doi:10.1007/bf00773551

Cited by 8 publications

(6 citation statements)

References 2 publications

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“…By Theorem 24 and Theorem 16 we obtain that H 1 (π −1 (U 0 ), T h pq ) = 0. Further by Lemma 23 and Theorem 16 we have Let us recall some results of Bott [Bott], see also [O4,Section 3], that we will essentially use to prove the next theorem. Let E be a homogeneous bundle over B 0 and x 0 ∈ B 0 be the origin of Z I 1 ⊂ B 0 , see (16).…”

Section: St Cohomology Group With Values In the Sheaf T 22mentioning

confidence: 99%

Rigidity of flag supermanifolds

Vishnyakova¹

2019

Preprint

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Section: St Cohomology Group With Values In the Sheaf T 22mentioning

confidence: 99%

Rigidity of flag supermanifolds

Vishnyakova¹

2019

Preprint

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“…Our final aim is to calculate the cohomology of the tangent sheaf T = Der O for the considered isotropic super-Grassmannian in degrees 0 and 1. As it was done in [5,8,9], we start by the similar calculation for the Z-graded sheafT = Der gr O. We use the following exact sequence of sheaves (see [4,7]):…”

Section: Tangent Sheaf Cohomology Of the Retractmentioning

confidence: 99%

“…The mappings α and β are homomorphisms of sheaves of Lie algebras. As in [5,8,9], we calculate first the cohomology of A p and B p , using the theorem of Bott about homogeneous vector bundles (see [1], Theorem IV ). We need some notation concerning weights and roots of the group G 0 = GL n (C).…”

Section: Tangent Sheaf Cohomology Of the Retractmentioning

confidence: 99%

“…In [10,11] their cohomology with values in certain classes of locally free sheaves were studied, but the tangent sheaf does not lie in these classes. On the other hand, the papers [5,8] are devoted especially to the calculation of the 0-and 1-cohomology of the tangent sheaf of super-Grassmannians (case (1)), and in [9] we do the same things for the isotropic super-Grassmannians of maximal type, associated with an even symmetric bilinear form (case (2)). In the present paper, the same problem for the isotropic super-Grassmannians of maximal type, corresponding to the case (3), is considered.…”

Section: Introductionmentioning

confidence: 99%

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On isotropic super-Grassmannians of maximal type associated with an odd bilinear form

Onishchik¹,

Serov²

2005

American Mathematical Society Translations: Series 2

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We study the super-Grassmannian IGr n|n,s|t (C) formed by those graded subspaces of dimension s|t, s + t = n, of the vector superspace C n|n that are totally isotropic with respect to an odd skew-symmetric bilinear form b, defined in this superspace. We prove that IGr n|n,s|t (C) is a non-split supermanifold whenever t ≥ 1, s ≥ 2, and a rigid one whenever t ≥ 3, s ≥ 4. If t ≥ 2, s ≥ 3, then the Lie superalgebra of vector fields on this super-Grassmannian coincides with the simple classical Lie superalgebra πsp n|n (C) that annihilates the form b.

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“…Note that this special question initiated the study exposed in this paper. For the three other series of super-Grassmannians this calculation was performed in [34], [40], [41].…”

Section: Introductionmentioning

confidence: 99%

Non-split supermanifolds associated with the cotangent bundle

Onishchik¹

2022

Preprint

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Here, I study the problem of classification of non-split supermanifolds having as retract the split supermanifold (M, Ω), where Ω is the sheaf of holomorphic forms on a given complex manifold M of dimension > 1. I propose a general construction associating with any d-closed (1, 1)-form ω on M a supermanifold with retract (M, Ω) which is non-split whenever the Dolbeault class of ω is non-zero. In particular, this gives a non-empty family of non-split supermanifolds for any flag manifold M = CP 1 . In the case where M is an irreducible compact Hermitian symmetric space, I get a complete classification of non-split supermanifolds with retract (M, Ω). For each of these supermanifolds, the 0-and 1-cohomology with values in the tangent sheaf are calculated. As an example, I study the Π-symmetric super-Grassmannians introduced by Yu. Manin.

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On the rigidity of super-Grassmannians

Cited by 8 publications

References 2 publications

Rigidity of flag supermanifolds

Rigidity of flag supermanifolds

On isotropic super-Grassmannians of maximal type associated with an odd bilinear form

Non-split supermanifolds associated with the cotangent bundle

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