Title: Formal geometry of cotangent stacks
Abstract: Derived Poisson geometry finds a number of applications in the geometry of smooth schemes and topological field theory. The treatment in the literature begins with a solution, due to Calaque–Pantev–Toën–Vaquié–Vezzosi, to the fundamental problem of formulating the correct definition of a Poisson derived stack. Their solution is a careful application of formal geometry to descend the usual definition from affine schemes using certain kinds of crystals; the output is a Lie algebra called Pol(X). We attempt to answer the question: what is a deformation-theoretic interpretation of the resulting object? To do so, we perform an alternate construction of Pol(X) by combining the symplectic geometry of the cotangent stack with a formal geometry approach to Cartan calculus. Our main theorem states that the existing definition of Pol(X) and the alternate construction give equivalent theories of Poisson structures. In particular, since Pol(X) recovers ordinary Poisson structures on smooth schemes, we gain an alternative perspective on usual Poisson deformation theory.
Alex Karapetyan
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