acts on the vector space of semidensities, and is a globally well-defined operator on the atlas of Darboux neighborhoods. Khudaverdian's Δ Δ π π operator depends only on the P-structure. It is manifestly nilpotent Δ Δ π π 2 = 0 , and of degree −1. Nevertheless, it is technically not a BV Δ operator as the vector space of semidensities has no multiplication. (The product of two semidensities is a density rather than a semidensity.) Given a fixed density ρ ρ , one may construct a nilpotent BV Δ operator as