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Explore the connections between algebraic geometry and metric geometry in this 55-minute differential geometry and physics seminar lecture. Learn about RCD (Riemannian Curvature-Dimension) structures on singular Kähler varieties, focusing on 3-dimensional projective varieties with klt (Kawamata log terminal) singularities. Discover how singular Kähler metrics with bounded Nash entropy and Ricci curvature bounded below induce unique compact RCD spaces homeomorphic to the underlying projective variety. Examine the specific case of singular Kähler-Einstein spaces of complex dimension 3 with bounded Nash entropy, understanding how they form compact RCD spaces that are both topologically and holomorphically equivalent to their underlying projective varieties. Gain insights into the interplay between algebraic, geometric, and analytic structures of klt singularities from birational geometry, and see how complex Monge-Ampère equations provide abundant examples of RCD spaces from algebraic geometry.
