On the constant scalar curvature K\"ahler metrics, general automorphism group

TL;DR

This paper establishes new estimates for scalar curvature equations and proves Donaldson's conjecture linking geodesic stability to the existence of constant scalar curvature Kähler metrics, especially when the automorphism group is nontrivial.

Contribution

It provides the first proof of Donaldson's conjecture in the case of nontrivial automorphism groups and connects properness of K-energy to cscK existence under these conditions.

Findings

Proved Donaldson's conjecture for nontrivial automorphism groups.

Derived estimates for scalar curvature equations with singular right hand sides.

Showed properness of K-energy implies cscK existence when automorphisms are nontrivial.

Abstract

In this paper, we derive estimates for scalar curvature type equations with more singular right hand side. As an application, we prove Donaldson's conjecture on the equivalence between geodesic stability and existence of cscK when $Aut_0(M,J)\neq0$. Moreover, we also show that when $Aut_0(M,J)\neq0$, the properness of $K$-energy with respect to a suitably defined distance implies the existence of cscK.