The Partial $C^{0}$-estimate along a general continuity path and applications

TL;DR

This paper develops a new partial $C^{0}$-estimate along a continuity path with conic singularities on Fano manifolds, leading to progress on the Yau-Tian-Donaldson conjecture for certain cases.

Contribution

It introduces a novel partial $C^{0}$-estimate along a generalized continuity path with singularities, advancing the understanding of Kähler-Einstein metrics on Fano manifolds.

Findings

Establishment of a new partial $C^{0}$-estimate with conic singularities

Proof of reductivity of automorphism groups of limit spaces

Progress on the Yau-Tian-Donaldson conjecture for specific holomorphic vector fields

Abstract

We establish a new partial $C^{0}$-estimate along a continuity path mixed with conic singularities along a simple normal crossing divisor and a positive twisted $(1,1)$-form on Fano manifolds. As an application, this estimate enables us to show the reductivity of the automorphism group of the limit space, which leads to a proof of Yau-Tian-Donaldson Conjecture admitting some types of holomorphic vector fields.