Poisson Vertex Cohomology and Tate Lie Algebroids

We study sheaves on holomorphic spaces of loops and apply this to the study of the complex, defined in \cite{BdSHK}, governing deformations of the \emph{Poisson vertex algebra} structure on the space of holomorphic loops into a Poisson variety. We describe this complex in terms of the (continuous) de Rham-Lie cohomology of an associated Lie algebroid object in locally linearly compact topological (alias \emph{Tate}) sheaves of modules on $\mathcal{L}^{+}M$. In particular this allows us to easily compute the cohomology of the above in the case where $\pi$ is symplectic - we obtain de Rham cohomology of $M$.