Some refinements of the partial $C^0$ estimate

TL;DR

This paper improves understanding of Bergman kernels and provides a simplified proof for the partial $C^0$ estimate in the context of K"ahler manifolds with Ricci bounds, relevant to complex geometry and Ricci flow.

Contribution

It offers a weak asymptotic estimate for Bergman kernels under certain bounds and simplifies the proof of the partial $C^0$ estimate along the K"ahler-Ricci flow.

Findings

Established a weak asymptotic estimate for Bergman kernels.

Provided a simplified proof for the partial $C^0$ estimate.

Applied results to the (generalized) K"ahler-Ricci flow on Fano manifolds.

Abstract

Relying on the recent work of Liu-Sz\'ekelyhidi we give a weak asymptotic estimate for the Bergman kernels of polarized K\"ahler 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) K\"ahler-Ricci flow on Fano manifolds.