Poisson vertex cohomology and Tate Lie algebroids

We study sheaves on holomorphic spaces of loops (also referred to as arcs ) and apply this to the study of the complex, defined in [2], governing deformations of the 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 Tate ) sheaves of modules on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>ℒ</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> . In particular this allows us to easily compute the cohomology of the above in the case where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> is symplectic, we obtain de Rham cohomology of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> .