On the geometry of supermanifolds with even and odd Kählerian structures

Even and odd Kählerian structures are constructed on supermanifolds associated with the tangent bundles of Kählerian manifolds. Mechanics that are bi-Hamiltonian with respect to the corresponding Poisson brackets are found; they determine Killing vectors of the Kählerian structures. An analog of the operator $\Delta$ in the Batalin–Vilkovisky quantization method is constructed; it corresponds to the divergence operator of the base manifold.