Odd scalar curvature in anti-Poisson geometry

Recent works have revealed that the recipe for field-antifield quantization of Lagrangian gauge theories can be considerably relaxed when it comes to choosing a path integral measure ρ if a zero-order term νρ is added to the Δ operator. The effects of this odd scalar term νρ become relevant at two-loop order. We prove that νρ is essentially the odd scalar curvature of an arbitrary torsion-free connection that is compatible with both the anti-Poisson structure E and the density ρ. This extends a previous result for non-degenerate antisymplectic manifolds to degenerate anti-Poisson manifolds that admit a compatible two-form.