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Title:
Generalized K\"ahler constant scalar curvature problem
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Abstract:
In this talk we consider the space of symplectic generalized K\"ahler (GK) structures on a given holomorphic Poisson manifold $(M,J,\pi)$. Fixing the cohomology class of the underlying symplectic form $[F]$ we denote this space by $GK_{[F]}(M,J,\pi)$. Recently Goto and Boulanger arrived at the moment map framework for the action of the group of hamiltonian diffeomorphisms $Ham(F)$ on the space of generalized almost K\"ahler structures, and used it to define the notion of the GK scalar curvature. Following the classical K\"ahler setup, we formulate the (generalized K\"ahler) Calabi Problem of finding a constant GK scalar curvature (cscGK) in $GK_{[F]}(M,J,\pi)$. We use the GIT interpretation of the moment map to develop the GK analogues of the notions of extremal metrics, Futaki invariant, Mabuchi energy and Mabuchi metric familiar from the K\"ahler setting. This allows us to recast Calabi-Lichnerowicz-Matsushima obstruction for the cscGK metrics in terms of the automorphism group of (J,\pi) and prove conditional uniqueness of cscGK metrics in $GK_{[F]}(M,J,\pi)$. In the special case when $(M,J,\pi)$ is a \emph{toric} Poisson manifold, we prove an existence result for the cscGK problem, constructing new examples of the cscGK metrics on P^2. (Based on a joint work with V.Apostolov and J.Streets)
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