The space of finite-energy metrics over a degeneration of complex manifolds

Reboulet, Rémi

doi:10.48550/arxiv.2107.04841

Cited by 3 publications

(3 citation statements)

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“…) . Note that it has been extended to the case of general Kähler classes on nonprojective Kähler manifolds in [SD17, Definition 2.1], [SD18] (see also [DR17b, Section 3]) and to singular metrics on (relatively) ample line bundles in [Reb21].…”

Section: Mixed Volumes and Intersection Theory On The Riemann-zariski...mentioning

confidence: 99%

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

Dervan¹,

Reboulet²

2022

Preprint

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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

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Section: Mixed Volumes and Intersection Theory On The Riemann-zariski...mentioning

confidence: 99%

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

Dervan¹,

Reboulet²

2022

Preprint

Self Cite

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“…The result is an algebraic analogue of a theorem of Berndtsson on the convexity of the Ding-functional along geodesics in the space of Kähler potentials [Ber15]. In forthcoming work, Reboulet shows that Theorem 3.7 can in fact be deduced from Berndtsson's convexity result when X is smooth [Reb21]. The proof below is self-contained and purely algebraic.…”

Section: Convexitymentioning

confidence: 78%

The existence of the Kähler–Ricci soliton degeneration

Blum

Liu

et al. 2023

Forum of Mathematics, Pi

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“…for any u ∈ E 1 ψ with ψ-relative minimal singularities. It is also possible to extend the description of these functionals in terms of positive Deligne pairings to the whole space E 1 ψ (see also [53,Lemma 3.8] and [46]). 4.2.2.…”

Section: 1mentioning

confidence: 99%

A relative Yau-Tian-Donaldson conjecture and stability thresholds

Trusiani¹

2023

Preprint

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Generalizing Fujita-Odaka invariant, we define a function δ on a set of generalized b-divisors over a smooth Fano variety. This allows us to provide a new characterization of uniform K-stability. A key role is played by a new Riemann-Zariski formalism for K-stability. For any generalized b-divisor D, we introduce a (uniform) D-log K-stability notion. We prove that the existence of a unique Kähler-Einstein metric with prescribed singularities implies this new K-stability notion when the prescribed singularities are given by the generalized b-divisor D. We connect the existence of a unique Kähler-Einstein metric with prescribed singularities to a uniform D-log Ding-stability notion which we introduce. We show that these conditions are satisfied exactly when δ(D) > 1, extending to the D-log setting the δ-valuative criterion of Fujita-Odaka and Blum-Jonsson. Finally we prove the strong openness of the uniform D-log Ding stability as a consequence of the strong continuity of δ.

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The space of finite-energy metrics over a degeneration of complex manifolds

Cited by 3 publications

References 0 publications

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

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

The existence of the Kähler–Ricci soliton degeneration

A relative Yau-Tian-Donaldson conjecture and stability thresholds

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