How to superize the notion of Kaehler manifold

Abstract

The definition of Kaehler manifold is superized. In the super setting, it admits a continuous parameter, unlike their analogs on manifolds. This parameter runs the same singular supervariety of parameters that parameterize deformations of the Schouten bracket (a.k.a. Buttin bracket, a.k.a. anti-bracket) considered as deformations of the Lie superalgebra structure given by the bracket. The same idea yields definitions of several versions of hyper-Kaeahler supermanifolds depending on parameters that also run over a singular supervariety. Moreover, the same idea is potentially applicable to the Kaehler and hyper-Kaehler manifolds (or supermanifolds corresponding to the even tensors that define them); in these cases infinite-dimensional (super)manifolds should enter the picture. Strangely enough, "how to embody this idea for the case of only even tensors involved?" is an open problem. The actions of Lie algebras on the space of differential forms on symplectic, and hyper-Kaehler manifold (known already to A.Weil, and Verbitsky, respectively) are extended to actions of Lie superalgebras on the same spaces with values in a line bundle with a maximally non-integrable connections, see Leites D., Shchepochkina I., The Howe duality and Lie superalgebras. In: S.~Duplij and J.~Wess (eds.) "Noncommutative Structures in Mathematics and Physics", Proc. NATO Advanced Research Workshop, Kiev, 2000. Kluwer, 2001, 93--112; arXiv:math.RT/0202181.