Ding stability and Kähler-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 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

A relative Yau-Tian-Donaldson conjecture and stability thresholds

“…Dervan and Reboulet recently in [28], assuming −K X being big and klt, proved that the existence of a unique (weak) Kähler-Einstein metric implies a uniform Ding stability notion which is similar to the one studied in our setting (see Remark 4.15). We adapted to the prescribed singularities their use of Deligne pairings (Proposition 4.11).…”

“…Supposing −K X big, Xu in [60] proved that the K-stability condition (given in terms of the δ-invariant) of (X, −K X ) forced to have a klt anticanonical model, whose stability properties are essentially the same as that of (X, −K X ). Moreover, in [60] it is also showed that the uniform Ding stability introduced in [28] implies δ(X) > 1. This concludes the equivalence between the existence of weak Kähler-Einstein, the uniform Ding stability and δ(X) > 1 thanks to [22,28].…”

The birational geometry of GIT quotients

“…The space X, viewed naturally as a locally ringed space, then completely determines the birational geometry of X: birational varieties have isomorphic Zariski-Riemann spaces tautologically, while conversely the isomorphism type of X as a locally ringed space determines the birational equivalence class of X, as explained in Section 2.3. Zariski-Riemann spaces were originally introduced by Zariski [Zar39] in his study of resolution of singularities for surfaces; to this day, they see use in various areas of complex geometry, for example in the minimal model programme [Sho93], complex dynamics [DF21], K-stability [DR22,Tru23] and non-Archimedean pluripotential theory [BJ22].…”

K-stability for varieties with a big anticanonical class

“…There has been a tremendous progress in algebraic K-stability theory of log Fano pairs (see [Xu21] for a survey of the topic). In the recent works of [DZ22] and [DR22], the Kähler-Einstein problem is considered for Kähler manifold (X, ω) such that −K X is big. More precisely, in [DZ22] the authors prove a transcendental YTD theorem for twisted big Kähler-Einstein metrics.…”

“…Remark 1.3. In [DR22], Ding stability notions for a projective klt pair (X, ∆) with big −K X − ∆ are developed. If one assumes R = m∈r•N H 0 (−m(K X + ∆)) is finitely generated and denote by (Z, ∆ Z ) the anticanonical model, then one can show a similar statement to Theorem 1.2, i.e.…”