In one complex variable, a bounded domain in the complex plane carries two natural invariant Hermitian metrics: the Bergman metric and the canonical metric (a complete Hermitian metric with constant Gaussian curvature). When the domain has sufficiently smooth boundary so that the Bergman metric is complete, a consequence of a classical result of Qi-Keng Lu shows that these two metrics coincide if and only if the domain is the unit disk.

In higher dimensions, the Bergman metric can be defined analogously for bounded domains. However, the existence of a complete Kähler metric with constant holomorphic sectional curvature is generally too much to expect. Instead, Cheng–Yau and Mok–Yau proved the existence of a complete Kähler metric with constant Ricci curvature (the Kähler–Einstein metric) on such domains when they are pseudoconvex. The Bergman metric reflects the function theory and holomorphic geometry of the domain, while the Kähler–Einstein metric captures its pluripotential and complex geometric structure.

In this talk, I will discuss recent joint work with S. Y. Li (UC Irvine), M. Xiao (UCSD), and Hsiao (Taiwan)–Li (Wuhan), as well as related work by Ebenfelt–Treuner–Xiao and Savale–Xiao. We address longstanding questions concerning when the Bergman metric has constant holomorphic sectional curvature and when it is Einstein.

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.