  We construct and study some vertex theoretic invariants associated to Poisson
varieties, specialising in the conformal weight $0$ case to the familiar
package of Poisson homology and cohomology. In order to do this conceptually we
sketch a version of the \emph{calculus}, in the sense of \cite{TT}, adapted to
the context of vertex algebras. We obtain the standard theorems of Poisson
(co)homology in this \emph{chiral} context. This is part of a larger project
related to promoting non-commutative geometric structures to chiral versions of
such.
