A YTD correspondence for constant scalar curvature metrics

Abstract

Given a compact K\"ahler manifold, to better understand Mabuchi's $K$ energy we introduce a family of $K^\beta$ energies, whose favorable properties are similar to those of the Ding energy from the Fano case. The construction uses Berman's transcendental quantization, and we show that the slope of the $K^\beta$ energies along test configurations can be computed using intersection theory. With these ingredients in place we provide a uniform Yau-Tian-Donaldson correspondence that characterizes the existence of a unique constant scalar curvature K\"ahler metric using test configurations. Combining our techniques with the non-Archimedean approach to $K$-stability pioneered by Boucksom--Jonsson, we show that the properness of the classical $K$ energy can be tested by checking its slope along a distinguished subclass of Chi Li-type models, called log discrepancy models, thus yielding another $G$-uniform Yau--Tian--Donaldson correspondence.