Holomorphic vector fields on super-Grassmannians

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

“…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]):…”

“…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).…”

“…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.…”

“…We have gr O E, where E = J /J 2 . Clearly, the action of G 0 on the sheaf O leaves invariant the filtration (8) and induced the action of this group on the locally free sheaf E, and hence on the corresponding vector bundle E, covering its standard action on M . Thus, E is a homogeneous vector bundle over M .…”

Vector fields on osp2m|2n(C)- and Vishnyakova1 2018Journal of Algebra3010View full textAdd to dashboardCite

“…Manin defined four series of complex homogeneous supermanifolds that correspond to four series of linear Lie superalgebras: gl m|n (C), osp m|2n (C), πsp n (C) and q n (C). The Lie superalgebras of holomorphic vector fields on these supermanifolds were studied in [BO,Bun,O1,O2,OS1,OS2,OS3,Ser] in the case of super-Grassmannians and in [V1, V2, V4, V6] in the case of other flag supermanifolds. We summarize the obtained results in the following table:…”