Kähler–Einstein metrics with prescribed singularities on Fano manifolds

Abstract

Given a Fano manifold ( X , ω ) {(X,\omega)} , we develop a variational approach to characterize analytically the existence of Kähler–Einstein metrics with prescribed singularities, assuming that these singularities can be approximated algebraically. Moreover, we define a function α ω {\alpha_{\omega}} on the set of prescribed singularities which generalizes Tian’s α-invariant, showing that its upper lever set { α ω ⁢ ( ⋅ ) > n n + 1 } {\{\alpha_{\omega}(\,\cdot\,)>\frac{n}{n+1}\}} produces a subset of the Kähler–Einstein locus, i.e. of the locus given by all prescribed singularities that admit Kähler–Einstein metrics. In particular, we prove that many K-stable manifolds admit all possible Kähler–Einstein metrics with prescribed singularities. Conversely, we show that enough positivity of the α-invariant function at nontrivial prescribed singularities (or other conditions) implies the existence of genuine Kähler–Einstein metrics. Finally, through a continuity method we also prove the strong continuity of Kähler–Einstein metrics on curves of totally ordered prescribed singularities when the relative automorphism groups are discrete.

Ding stability and Kähler-Einstein metrics on manifolds with big anticanonical class

“…explaining the non-appearance of multiplier ideal sheaves in the two definitions of volume of a line bundle. We now consider the case of a general singular θ-psh function ψ, no longer assumed to have analytic singularities, in which the theory is due to Trusiani [Tru22a] and Darvas-Xia [DX21]. In particular, the line bundle L is assumed to be pseudoeffective.…”

“…As mentioned, Trusiani has developed a very complete analytic theory of Kähler-Einstein metrics with prescribed singularities on Fano manifolds [Tru22a]. He proves also a converse to his analogue of Theorem 1.2, including uniqueness of Kähler-Einstein metrics with prescribed singularities up to automorphisms, by adapting Berndtsson's proof of strict convexity of the Ding functional [Ber15].…”

“…Thus a first step towards Theorem 1.1 is to develop the foundations of the theory of Kähler-Einstein metrics and the Ding functional in the big and klt setting. Our setting is morally related to recent work of Trusiani [Tru22a], who develops a theory of Kähler-Einstein metrics with prescribed singularities; Trusiani assumes −K X ample, but fixes the singularity type of the metrics of interest. By adapting his ideas, using foundational pluripotential-theoretic work of Darvas-di Nezza-Lu [DDNL18a,DDNL18c] and a key convexity result of Berndtsson-Pȃun [BP08], we prove the following: Theorem 1.2.…”

A relative Yau-Tian-Donaldson conjecture and stability thresholds

“…When a pair (Y, ∆) dominates a fixed Fano manifold X through a birational morphism Y → X, a log Kähler-Einstein for (Y, ∆) can be recovered from a Kähler-Einstein metric with prescribed singularities on X [54]. These metrics are given as weak solutions to the Einstein equation…”

“…ii) Exploit valuative criteria and analyze how these new algebro-geometric notions are related to the usual K-stability. An analytic analogue of the second goal has been addressed in [54] through the study of an extended Tian's α-invariant [51].…”