The partial C ⁰-estimate along a general continuity path and applications

We establish a new partial C⁰-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 new proof of the Yau-Tian-Donaldson conjecture.