Sobolev inequalities on Kähler spaces

Abstract

We establish a uniform Sobolev inequality for Kähler metrics, which only require an entropy bound and no lower bound on the Ricci curvature. We further extend our Sobolev inequality to singular Kähler metrics on Kähler spaces with normal singularities. This allows us to build a general theory of global geometric analysis on singular Kähler 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 Kähler-Einstein currents on projective varieties with definite or vanishing first Chern class.