RIGIDITY OF FLAG SUPERMANIFOLDS
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Springer Science+Business Media New York (2020)
RIGIDITY OF FLAG SUPERMANIFOLDS E. G. VISHNYAKOVA∗ Departamento de Matem´atica Universidade Federal de Minas Gerais, Av. Antˆonio Carlos, 6627 CEP: 31270-901, Belo Horizonte Minas Gerais, Brazil and Laboratory of Theoretical and Mathematical Physics Tomsk State University Tomsk 634050, Russia [email protected]
Abstract. We prove that under certain assumptions a supermanifold of flags is rigid, that is, its complex structure does not admit any non-trivial small deformation. Moreover under the same assumptions we show that a supermanifold of flags is a unique non-split supermanifold with given retract.
Introduction It is a classical result that any flag manifold is rigid, see [Bott]. In other words its complex structure does not possess any non-trivial small deformation. In general this statement is false for a flag supermanifold, see [BO], [Va]. For instance the projective superspace CP1|m , where m ≥ 4, see [Va], and the super-grassmannian Gr2|2,1|1 , see [BO], are not rigid. In [O4] it was proved that the super-grassmannian Grm|n,k|l is rigid if m, n, k, l satisfy the following conditions 0 < k < m,
0 < l < n,
(k, l) 6= (1, n − 1), (m − 1, 1), (1, n − 2), (m − 2, 1), (2, n − 1), (m − 1, 2). The idea of the paper [O4] is to compute the 1st cohomology with values in the tangent sheaf showing their triviality. This implies the rigidity of the supergrassmannian in this case. DOI: 10.1007/S00031-020-09629-6 Supported by FAPEMIG, grant APQ-01999-18, by Tomsk State University, Competitiveness Improvement Program and by CAPES/HUMBOLDT Foundation, Research Fellowship. Received October 3, 2019. Accepted September 14, 2020. Corresponding Author: E. G. Vishnyakova, e-mail: [email protected] ∗
E. G. VISHNYAKOVA
In this paper we compute the 1st cohomology with values in the tangent sheaf T of a flag supermanifold showing their triviality under the following conditions 0 < kr < · · · < k1 < k0 = m
and 0 < lr < · · · < l1 < l0 = n
and (kr , lr ) 6= (1, lr−1 − 1), (kr−1 − 1, 1), (1, lr−1 − 2), (kr−1 − 2, 1), (2, lr−1 − 1), (kr−1 − 1, 2). Therefore [Va] the result shows that under these conditions any flag supermanifold is rigid. We use the results of [O4] and the fact that a supermanifold of flags of length n is a superbundle with base space a super-grassmannian and fiber a flag supermanifold of length n − 1. The paper is organized as follows. In Section 1 we give necessary definitions and discuss properties of supermanifolds, split supermanifolds, superbundles and split superbundles. Section 2 is devoted to the study of split superbundles. In Sections 3 m|n and 4 we recall and prove some facts about a supermanifold of flags Fk|l considered as a superbundle. We compute the Lie superalgebra of vector fields on the retract of m|n Fk|l in Section 5. In Section 6 we prove the K¨ unneth Formula for supermanifolds in some cases. Finally we obtain our main result, Theorem 36, in the last section. 1. Main definitions 1.1. Supermanifolds We use the word “supermanifold” in
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