Semidensities on odd symplectic supermanifolds

We consider semidensities on a supermanifold E with an odd symplectic structure. We define a new Δ-operator action on semidensities as the proper framework for the Batalin-Vilkovisky (BV) formalism. We establish relations between semidensities on E and differential forms on Lagrangian surfaces. We apply these results to Batalin-Vilkovisky geometry. Another application is to (1.1)-codimensional surfaces in E. We construct a kind of ``pull-back'' of semidensities to such surfaces. This operation and the Δ-operator are used for obtaining integral invariants for (1.1)-codimensional surfaces