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

Xia, Mingchen

doi:10.48550/arxiv.2112.04290

Cited by 2 publications

(2 citation statements)

References 27 publications

(51 reference statements)

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“…Our slope formulas-which are the key to proving Theorem 1.1 from Theorem 1.2-begin by proving a slope formula for the Monge-Ampère energy for paths of singular metrics; for rays of metrics associated to test configurations, we show that the slope of the Monge-Ampère energy is the volume of L. In fact our results apply for a much more general class of energy functionals, which we call positive Deligne functionals, with our slope formulas then involving the positive intersection product of Boucksom-Favre-Jonsson when the subgeodesic has minimal singularities [BFJ09]. Allowing general singularity types, the algebro-geometric slope formula uses recent work of Darvas-Xia and Xia giving an algebraic interpretation of mixed Monge-Ampère masses [DX21,Xia21] (see also Trusiani [Tru22a, Section 3], who independently introduced the singularity classes relevant to these results). These sorts of slope formulae in the theory of Deligne pairings-which go back to work of Phong-Ross-Sturm in the ample setting [PRS08]-also play a crucial role in the general Yau-Tian-Donaldson conjecture addressing the existence of constant scalar curvature Kähler metrics in ample classes [BHJ19].…”

Section: Introductionmentioning

confidence: 93%

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: Introductionmentioning

confidence: 93%

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

Dervan¹,

Reboulet²

2022

Preprint

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“…Remark 1.4. Recently, M. Xia [Xia2] also constructed the Duistermaat-Heckman measure for ϕ ∈ E 1 NA (X, L). Though the construction is different from ours, both constructions give the same measure as these are continuous along decreasing nets ϕ i ϕ.…”

Section: −2πmentioning

confidence: 99%

Entropies in $μ$-framework of canonical metrics and K-stability, II -- Non-archimedean aspect: non-archimedean $μ$-entropy and $μ$K-semistability

Inoue¹

2022

Preprint

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This is the second in a series of two papers studying µ-cscK metrics and µK-stability from a new perspective, inspired by observations on µcharacter in [Ino3] and on Perelman's W -entropy in the first paper [Ino4].This second paper is devoted to studying a non-archimedean counterpart of Perelman's µ-entropy. The concept originally appeared as µ-character of polarized family in the previous research [Ino3], where we used it to introduce an analogue of CM line bundle adapted to µK-stability.We firstly show some differential of the characteristic µ-entropy μλ ch is the minus of µ λ -Futaki invariant, which connects µ λ K-semistability to the maximization of characteristic µ λ -entropy. It in particular provides us a criterion for µ λ K-semistability working without detecting the vector ξ involved in the µ λ ξ -Futaki invariant. We observe a family of filtrations {F ξ+τ (X ,L) } τ ∈[0,∞) associated to a test configuration (X , L) and a vector field ξ acting on (X, L) to consider the differential. We conceptualize such family of filtrations as polyhedral configuration and study its generalities. The concept implicitly appeared in many literatures involved in R-test configuration.In the latter part, we propose a non-archimedean pluripotential approach to the maximization problem. In order to adjust the characteristic µ-entropy μλ ch to Boucksom-Jonsson's non-archimedean framework, we introduce a natural modification μλ NA which we call non-archimedean µ-entropy. We extend the non-archimedean µ-entropy from the set of test configurations to a space E exp NA (X, L) of non-archimedean psh metrics on the Berkovich space X NA , which is endowed with a complete metric structure. We introduce moment measure χDϕ on Berkovich space for this sake, which can be considered as a hybrid of Monge-Ampère measure and Duistermaat-Heckman measure.We also compare our µ-framework with other frameworks: H-entropy framework in the context of Kähler-Ricci flow and Calabi energy framework in the context of Calabi flow. Some illustrations by toric examples are attached in Appendix.

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

Cited by 2 publications

References 27 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

Entropies in $μ$-framework of canonical metrics and K-stability, II -- Non-archimedean aspect: non-archimedean $μ$-entropy and $μ$K-semistability

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