  We consider semidensities on a supermanifold E with an odd symplectic
structure. We define a new $\Delta$-operator action on semidensities as the
proper framework for Batalin-Vilkovisky 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 $\Delta$-operator are
used for obtaining integral invariants for (1.1)-codimensional surfaces.
