  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 $M$.
