  We consider the possibility of adding a Grassmann-odd function \nu to the odd
Laplacian. Requiring the total \Delta operator to be nilpotent leads to a
differential condition for \nu, which is integrable. It turns out that the odd
function \nu is not an independent geometric object, but is instead completely
specified by the antisymplectic structure E and the density \rho. The main
impact of introducing the \nu term is that it makes compatibility relations
between E and \rho obsolete. We give a geometric interpretation of \nu as
(minus 1/8 times) the odd scalar curvature of an arbitrary antisymplectic,
torsion-free and \rho-compatible connection. We show that the total \Delta
operator is a \rho-dressed version of Khudaverdian's \Delta_E operator, which
takes semidensities to semidensities. We also show that the construction
generalizes to the situation where \rho is replaced by a non-flat line bundle
connection F. This generalization is implemented by breaking the nilpotency of
\Delta with an arbitrary Grassmann-even second-order operator source.
