Some refinements of the partial C0 estimate

Abstract

Relying on the recent work of Liu-Szekelyhidi we give a weak asymptotic estimate for the Bergman kernels of polarized Kahler manifolds with Ricci lower bound and Sobolev constant upper bound. We will also give a simple proof for the partial $C^0$ estimate along the (generalized) Kahler-Ricci flow on Fano manifolds.