Mabuchi geometry of big cohomology classes with prescribed singularities

Xia, Mingchen

doi:10.48550/arxiv.1907.07234

Cited by 4 publications

(6 citation statements)

References 39 publications

(42 reference statements)

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“…We briefly note that we will occasionally work over normal (or even unibranch, in the sense of [Gro64, Chap. 0, 23.2.1]) analytic spaces rather than complex manifolds, in which case a general reference is [Xia19b]. In particular, the integration by parts formula still holds by virtue of [Xia19b, Theorem 3.33].…”

Section: Positive Deligne Pairings and Finite-energy Spacesmentioning

confidence: 99%

“…Although out of the scope of the present article, we note that a slight modification of our definition can accommodate for the case of metrics with prescribed singularities (originally introduced in [RN17] and studied e.g. in [Tru22b,Tru20] in the ample case, and in [Xia19b] in the big case).…”

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

confidence: 99%

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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: Positive Deligne Pairings and Finite-energy Spacesmentioning

confidence: 99%

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

View full text Add to dashboard Cite

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“…We have assumed X to be smooth only for simplicity. All of our constructions work equally well when X is normal, based on the pluripotential theory in these settings developed in [Xia21].…”

Section: Construction Of Partial Okounkov Bodiesmentioning

confidence: 87%

“…For ℓ, ℓ ′ ∈ R 1 (X, ω), define ℓ ∧ ℓ ′ as the greatest geodesic in R 1 (X, ω) that lies below both ℓ and ℓ ′ . It is shown in [Xia21] that ∧ is well-defined and (R 1 (X, ω), d 1 , ∧) is a complete rooftop metric space.…”

Section: A Generalization Of Boucksom-chen Theoremmentioning

confidence: 99%

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Partial Okounkov bodies and Duistermaat--Heckman measures of non-Archimedean metrics

Xia¹

2021

Preprint

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Let X be a smooth projective variety. We construct partial Okounkov bodies associated to Hermitian pseudo-effective line bundles (L, φ) on X. We show that partial Okounkov bodies are universal invariants of the singularity of φ. As an application, we generalize the theorem of Boucksom-Chen and construct Duistermaat-Heckman measures associated to finite energy metrics on the Berkovich analytification of an ample line bundle.

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The closures of test configurations and algebraic singularity types

Darvas¹,

Xia²

2020

Preprint

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Given a Kähler manifold X with an ample line bundle L, we consider the metric space of L 1 geodesic rays associated to the first Chern class c 1 (L). We characterize rays that can be approximated by ample test configurations. At the same time, we also characterize the closure of algebraic singularity types among all singularity types of quasi-plurisubharmonic functions, pointing out the very close relationship between these two seemingly unrelated problems. By Bonavero's holomorphic Morse inequalities, the arithmetic and non-pluripolar volumes of algebraic singularity types coincide. We show that in general the arithmetic volume dominates the non-pluripolar one, and equality holds exactly on the closure of algebraic singularity types. Analogously, we give an estimate for the Monge-Ampère energy of a general L 1 ray in terms of the arithmetic volumes along its Legendre transform. Equality holds exactly for rays approximable by test configurations.Various other cohomological and potential theoretic characterizations are given in both settings. As applications, we give a concrete formula for the non-Archimedean Monge-Ampère energy in terms of asymptotic expansion, and show the continuity of the projection map from L 1 rays to non-Archimedean rays. We also show that the closure of ample test configurations and filtrations gives the same set of rays.

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Mabuchi geometry of big cohomology classes with prescribed singularities

Cited by 4 publications

References 39 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

Partial Okounkov bodies and Duistermaat--Heckman measures of non-Archimedean metrics

The closures of test configurations and algebraic singularity types

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