About derivations and vector-valued differential forms

Onishchik, A. L.

doi:10.1007/bf02433953

Cited by 5 publications

(2 citation statements)

References 6 publications

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“…. (M, T ) endowed with the bracket induced by the Lie bracket in T (for details, see [37]). First, it follows from the decomposition (3.26) that…”

Section: Propositionmentioning

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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“…. (M, T ) endowed with the bracket induced by the Lie bracket in T (for details, see [37]). First, it follows from the decomposition (3.26) that…”

Section: Propositionmentioning

confidence: 99%

Non-split supermanifolds associated with the cotangent bundle

Onishchik¹

2022

Preprint

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show abstract

“…. (M, T ) endowed with the bracket induced by the Lie bracket in T (for details, see [40]). First, it follows from the decomposition (3.26) that…”

Section: Proposition (What Ismentioning

confidence: 99%

Non-split supermanifolds associated with the cotangent bundle

Onishchik¹

2022

Communications in Mathematics

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Here, I study the problem of classification of non-split supermanifolds having as retract the split supermanifold $(M,\Omega)$, where $\Omega$ 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 $\omega$ on $M$ a supermanifold with retract $(M,\Omega)$ which is non-split whenever the Dolbeault class of $\omega$ is non-zero. In particular, this gives a non-empty family of non-split supermanifolds for any flag manifold $M\ne \mathbb{CP}^1$. In the case where $M$ is an irreducible compact Hermitian symmetric space, I get a complete classification of non-split supermanifolds with retract $(M,\Omega)$. For each of these supermanifolds, the 0- and 1-cohomology with values in the tangent sheaf are calculated. As an example, I study the $\Pi$-symmetric super-Grassmannians introduced by Yu. Manin.

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Arkadii L'vovich Onishchik (on his 70th birthday)

Ахиезер¹

2003

Russ. Math. Surv.

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About derivations and vector-valued differential forms

Cited by 5 publications

References 6 publications

Non-split supermanifolds associated with the cotangent bundle

Non-split supermanifolds associated with the cotangent bundle

Non-split supermanifolds associated with the cotangent bundle

Arkadii L'vovich Onishchik (on his 70th birthday)

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