How to superize the notion of Kaehler manifold

TL;DR

This paper introduces a superized generalization of Kaehler manifolds, incorporating a continuous parameter linked to deformations of the Schouten bracket, and explores potential applications to hyper-Kaehler supermanifolds and infinite-dimensional supergeometry.

Contribution

It defines superized Kaehler and hyper-Kaehler manifolds with parameters on a singular supervariety, extending classical structures to the super setting and suggesting new avenues in supergeometry.

Findings

Superized Kaehler manifolds admit a continuous parameter.

Definitions of hyper-Kaehler supermanifolds depend on parameters on a supervariety.

Extension of Lie algebra actions to Lie superalgebras on differential forms.

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.