Equivariant calculus on $μ$-character and $μ$K-stability of polarized schemes

Inoue, Eiji

doi:10.48550/arxiv.2004.06393

“…In this case, dµ 0 = e −ϕ 0 , and H g,Θ is reduced to H g . See [81,47] for related discussions. log( (dd c ϕ) n /n!…”

Section: Appendix: Non-archimedean Entropy and Generalized Mabuchi Fu...mentioning

confidence: 99%

“…Remark 5.11. During our completion of this paper, we noticed a preprint of Inoue [47] in which a more general framework of equvariant intersections is used to define an equivariant version of K-stability adapted to the usual Kähler-Ricci solitons.…”

mentioning

confidence: 99%

On the Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci soliton equations

Han¹,

Li²

2020

Preprint

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Let (X, D) be a log variety with an effective holomorphic torus action, and Θ be a closed positive (1, 1)-current. For any smooth positive function g defined on the moment polytope of the torus action, we study the Monge-Ampère equations that correspond to generalized and twisted Kähler-Ricci g-solitons. We prove a version of Yau-Tian-Donaldson (YTD) conjecture for these general equations, showing that the existence of solutions is always equivalent to an equivariantly uniform Θ-twisted g-Ding-stability. When Θ is a current associated to a torus invariant linear system, we further show that equivariant special test configurations suffice for testing the stability. Our results allow arbitrary klt singularities and generalize most of previous results on (uniform) YTD conjecture for (twisted) Kähler-Ricci/Mabuchi solitons or Kähler-Einstein metrics. Contents 1. Introduction and main results 1 2. Functionals and their dualities 5 3. Existence and properness 14 4. Generalized Mabuchi functional and its convexity 19 5. Non-Archimedean functionals and G-uniform stability 30 6. Yau-Tian-Donaldson conjecture for twisted Kähler-Ricci g-solitons 39 7. Stability via special test configurations 43 8. Examples 47 9. Appendix: Mabuchi functional as Kempf-Ness functionals 48 10. Appendix: Non-Archimedean Entropy and generalized Mabuchi functional 50 References 54

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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.

…”

mentioning

confidence: 96%

“…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].…”

mentioning

confidence: 97%

See 1 more Smart Citation

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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On the Yau‐Tian‐Donaldson Conjecture for Generalized Kähler‐Ricci Soliton Equations

Han

¹

,

Li

²

2022

Comm Pure Appl Math

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Letbe a polarized log variety with an effective holomorphic torus action, and be a closed positive torus invariant -current. For any smooth positive function defined on the moment polytope of the torus action, we study the Monge-Ampère equations that correspond to generalized and twisted Kähler-Ricci -solitons. We prove a version of the Yau-Tian-Donaldson (YTD) conjecture for these general equations, showing that the existence of solutions is always equivalent to an equivariantly uniform -twisted -Ding-stability. When is a current associated to a torus invariant linear system, we further show that equivariant special test configurations suffice for testing the stability. Our results allow arbitrary klt singularities and generalize most of previous results on (uniform) YTD conjecture for (twisted) Kähler-Ricci/Mabuchi solitons or Kähler-Einstein metrics.

show abstract

Equivariant calculus on $μ$-character and $μ$K-stability of polarized schemes

Cited by 5 publications

References 15 publications

On the Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci soliton equations

On the Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci soliton equations

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

On the Yau‐Tian‐Donaldson Conjecture for Generalized Kähler‐Ricci Soliton Equations

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