We introduce a notion of uniform Ding stability for a projective manifold with big anticanonical class, and prove that the existence of a unique Kähler-Einstein metric on such a manifold implies uniform Ding stability. The main new techniques are to develop a general theory of Deligne functionalsand corresponding slope formulas-for singular metrics, and hence 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. Contents 1. Introduction 1 2. Positive Deligne pairings and finite-energy spaces 7 3. Variational approach to Kähler-Einstein currents 15 4. Uniform Ding stability 20 5. Asymptotics of functionals and the main theorem 26 References 34
Given a polarized projective variety
(
X
,
L
)
{(X,L)}
over any non-Archimedean field, assuming continuity of envelopes, we define a metric on the space of finite-energy metrics on L, related to a construction of Darvas in the complex setting. We show that this makes finite-energy metrics on L into a geodesic metric space, where geodesics are given as maximal psh segments. Given two continuous psh metrics, we show that the maximal segment joining them is furthermore continuous. Our results hold in particular in all situations relevant to the study of degenerations and K-stability in complex geometry.
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