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A. L. Onishchik Yaroslavl State University Yaroslavl, Russia aonishch@aha.ru

On homogeneous sup ermanifolds asso ciated with irreducible compact Hermitian symmetric spaces
We consider the following problem: given an irreducible compact Hermitian symmetric space M = G/P , where G is a simply connected simple complex Lie group and P its parabolic subgroup, to describe, up to isomorphy, all homogeneous complex supermanifolds (M , O), whose reduction is M . An obvious example is the split supermanifold (M , ), where is the sheaf of holomorphic differential forms on M . It was proved in [1] that the only non-split homogeneous supermanifolds with this retract are the so-called -symmetric superGrassmannians Grn|n,k|k , constructed in [2]; in this case M = Grn,k , 0 < k < n, is the Grassmannian of k -subspaces in Cn . Several results concerning classification of non-split homogeneous supermanifolds (M , O) are known, where the retract is not fixed, but the odd part m of the dimension n|m = dim(M , O) does not exceed a given number, and also in the case M = CP1 . Here we consider the classification problem under certain restrictions on the representation of the subgroup P which determines the retract of (M , O). Another interpretation of is that it is dual to the "odd isotropy representation" of P acting in the odd tangent space to (M , O) at the point which is fixed under P . In the split case, we solve the problem for all completely reducible representations , while in the non-split case the very strong assumption that is irreducible is made. We prove that if this condition is satisfied, then in certain cases is dual to the usual isotropy representation, i.e., the retract is isomorphic to (M , ). These cases are as follows: M = Grn,2 , where n 5 is odd or n = 4, 6; M = Grn,k , 3 k n - k ; G = S p2n (C), n 2 (M is the symplectic isotropic Grassmannian); G = E6 ; G = E7 . Thus, for the Grassmannians listed above the only solution is Grn|n,k|k , while in the remaining cases no non-split homogeneous supermanifolds exist. For other irreducible compact Hermitian symmetric spaces M we obtain a list of possible irreducible representations , but we could not find any example of non-split homogeneous supermanifolds with reduction M and an irreducible odd isotropy representation except of Grn|n,k|k . The proofs are published in [3, 4]. References [1] A.L. Onishchik, Non-split supermanifolds associated with the cotangent bundle, Universitґ de Poitiers, Dґ e epart. Math., no. 109, Poitiers, 1997. [2] Yu.I. Manin, Gauge Field Theory and Complex Geometry, Springer-Verlag, Berlin, 1997. [3] A.L. Onishchik, Homogeneous supermanifolds over Grassmannians, J. Algebra, 313 (2007), 320-342. [4] A.L. Onishchik, On homogeneous supermanifolds over compact Hermitian symmetric spaces, SFB/TR-12, Schriftenreihe, no. 2, KЁ e.a., 2006. oln

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