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8/4/2025 (Aula B): "Four open problems on the Bergman metric", Andrea Loi, Università degli Studi di Cagliari (seminari del gruppo di Geometria Differenziale e Topologia)

Abstract: The Bergman metric is a key object in complex differential geometry and several complex variables, offering a canonical Kähler metric intrinsically tied to the geometry of complex domains and manifolds. Its structure connects complex analysis, differential geometry, and algebraic geometry in profound ways. 

This seminar consists in three parts. We will begin with an overview of local Kähler geometry to recall the necessary background. The second part will focus onto the Bergman metric itself, presenting its fundamental properties and geometric meaning. In the final part, we will explore four open problems that revolve around the geometry of the Bergman metric. These questions concern fundamental aspects such as curvature, covariant derivatives, Einstein-type conditions, and the relationship between the Bergman metric and metrics with constant holomorphic sectional curvature.