In the 1950’s, E. Calabi  initiated a far-reaching program of finding, on a given compact Kähler manifold $X$, a canonical representative of the space of Kähler metrics that belong to a fixed de Rham class. He proposed as a candidate of such representative the notion of constant scalar curvature Kähler metric, including the Kähler–Einstein metrics  as a special case.  Calabi’s program was one of the most active areas of research in Kähler geometry during the last half-century. The central conjecture in the field, which is still open in full generality, is the Yau–Tian–Donaldson (YTD) conjecture. It states, broadly speaking, that the full obstruction for the existence of a constant scalar curvature Kahler metric can be expressed in terms of a complex-algebraic notion of K-polystability of  $X$. This correspondence, if established, will have further deep implications for the definition of well-behaved moduli spaces of Kähler manifolds. 

In the 1990’s, an extension of Kähler geometry emerged from studies in $(2,2)$ supersymmetric quantum field theory in physics. These geometric structures were later rediscovered, and given the name of generalized Kähler (GK) structures, in the context of Hitchin’s generalized geometry program. In the ensuing decades it has become clear that GK geometry is a deeply structured extension of Kähler geometry with novel implications for complex, symplectic and Poisson geometry.

In this talk I will explain how,  guided by an infinite dimensional momentum map picture,  one can extend Calabi’s notion of constant scalar curvature Kahler metric to the  generalized Kahler context. This setup will naturally lead us to an algebro-geometric notion of Poisson K-polystability of a polarized complex Poisson manifold,  and to a Yau-Tian-Donaldson type conjecture on such manifolds. I will discuss a resolution of this conjecture on the complex projective space.  

This talk is based on joint works with Jeffrey Streets, Yury Ustinovskiy and Brent Pym.