Ding stability and K\"ahler-Einstein metrics on manifolds with big anticanonical class

Abstract

We introduce a notion of uniform Ding stability for a projective manifold with big anticanonical class, and prove that the existence of a unique K\"ahler-Einstein metric on such a manifold implies uniform Ding stability. The main new techniques are to develop a general theory of Deligne functionals - and corresponding slope formulas - for singular metrics, and to prove a slope formula for the Ding functional in the big setting. This extends work of Berman in the Fano situation, when the anticanonical class is actually ample, and proves one direction of the analogue of the Yau-Tian-Donaldson conjecture in this setting. We also speculate about the relevance of uniform Ding stability and K-stability to moduli in the big setting.