Generalized Kähler Constant Scalar Curvature Problem

Explore the generalized Kähler constant scalar curvature problem in this comprehensive lecture by Y. Ustinovskii from Leigh University. Delve into the space of symplectic generalized Kähler (GK) structures on holomorphic Poisson manifolds and examine the moment map framework for the action of hamiltonian diffeomorphisms. Learn about the formulation of the Calabi Problem in the GK context and discover the development of GK analogues to key concepts in Kähler geometry, including the Futaki invariant, Mabuchi energy, and Mabuchi metric. Investigate the Calabi-Lichnerowicz-Matsushima obstruction for cscGK metrics and understand the conditional uniqueness of these metrics. Conclude with an exploration of existence results for the cscGK problem in toric Poisson manifolds, including new examples of cscGK metrics on P^2.