Geometric Analysis and Topology Seminar

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Geometric analysis on singular complex spaces

Speaker: Jian Song, Rutgers University

Location: Warren Weaver Hall 512

Date: Friday, February 2, 2024, 11 a.m.

Synopsis:

We establish a uniform Sobolev inequality and diameter bound for Kahler metrics, which only require an entropy bound and no lower bound on the Ricci curvature. We further extend our Sobolev inequality to singular Kahler metrics on Kahler spaces with normal singularities. This allows us to build a general theory of global geometric analysis on singular Kahler spaces including the spectral theorem, heat kernel estimates, eigenvalue estimates and diameter estimates. Such estimates were only known previously in very special cases such as Bergman metrics. As a consequence, we derive various geometric estimates, such as the diameter estimate and the Sobolev inequality, for Kahler-Einstein currents on projective varieties with definite or vanishing first Chern class.