Laplacians in Odd Symplectic Geometry

We consider odd Laplace operators arising in odd symplectic geometry. Approach based on semidensities (densities of weight 1/2) is developed. The role of semidensities in the Batalin–Vilkovisky formalism is explained. In particular, we study the relations between semidensities on an odd symplectic supermanifold and differential forms on a purely even Lagrangian submanifold. We establish a criterion of ``normality’’ of a volume form on an odd symplectic supermanifold in terms of the canonical odd Laplacian acting on semidensities.