The existence of the Kähler-Ricci soliton degeneration

Blum, Harold F.; Liu, Yuchen; Xu, Chenyang; Zhuang, Ziquan

doi:10.48550/arxiv.2103.15278

Cited by 3 publications

(4 citation statements)

References 40 publications

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“…This functional is a variant of invariants that appeared in previous works of Tian-Zhang-Zhang-Zhu, Dervan-Székelyhidi and Hisamoto (see [74] for more details). The results from [38,54,14] together prove the following algebraic version of Hamilton-Tian conjecture:…”

Section: Stability Of Fano Varietiesmentioning

confidence: 78%

Canonical Kähler metrics and stability of algebraic varieties

Li¹

2022

Preprint

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Section: Stability Of Fano Varietiesmentioning

confidence: 78%

Canonical Kähler metrics and stability of algebraic varieties

Li¹

2022

Preprint

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“…This is still a far-off dream, but seems plausible as indeed such picture for Kähler-Ricci flow on Fano manifold, the Hamilton-Tian conjecture, is completely realized in H-entropy formalism (cf. [He,CSW,DS,HL2,BLXZ]). We also refer to analytic result for Calabi flow [Xia1].…”

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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“…The question is then reduced to a finite generation problem for the quasi-monomial valuation v * . This has been resolved in [16] based the important work [73].…”

Section: Two Test Configurations (Xmentioning

confidence: 99%

“…We will revisit this transformation by understanding how the corresponding transversal complex Monge-Ampère equations are transformed under the deformation of Reeb vector fields. Finally we explain how this PDE/pluripotential point of view allows one to combine the version of YTD conjecture for g-solitons on log Fano pairs proved in [48], the algebraic results from [16,73] and the discovery in [1, 2] to prove the YTD conjecture for general Fano cones.…”

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confidence: 99%

Notes on weighted Kähler-Ricci solitons and application to Ricci-flat Kähler cone metrics

2021

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We first survey the author's joint work with Jiyuan Han on the Yau-Tian-Donaldson (YTD) conjecture for g-weighted Kähler-Ricci solitons (or g-solitons). We then review recent works of Apostolov-Canderbank-Jubert-Lahdili which establish a connection between a particular g-soliton equation with Ricci-flat Kähler cone metrics (or equivalently Sasaki-Einstein metrics). The main interest in this connection is the transformation of a possibly irregular Sasaki-Einstein metric to a particular g-soliton equation on any quasi-regular quotient. We will revisit this transformation by understanding how the corresponding transversal complex Monge-Ampère equations are transformed under the deformation of Reeb vector fields. Finally we explain how this PDE/pluripotential point of view allows one to combine the version of YTD conjecture for g-solitons on log Fano pairs proved in [48], the algebraic results from [16,73] and the discovery in [1, 2] to prove the YTD conjecture for general Fano cones. Contents1. Yau-Tian-Donaldson conjecture for g-soliton equations 1.1. g-soliton equations 1.2. Generalized Futaki invariant and Matsushima type result 1.3. Energy functionals 1.4. Analytic criterion 1.5. Stability of (X, T) 1.6. Yau-Tian-Donaldson conjecture 1.7. Log Fano case 2. Application of g-soliton equations to Kähler Ricci-flat cone metrics 2.1. Kähler Ricci-flat cone metric 2.2. Deformation of Reeb vector fields 2.3. Transveral Kähler deformation 2.4. Reduction to g-soliton equation on orbifolds 2.5. Stability of Fano cones vs. g-weighted stability of Fano orbifolds Appendix A. Chern curvature of associated holomorphic line bundle Appendix B. Moment map associated to torus actions Appendix C. Filtrations

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The existence of the Kähler-Ricci soliton degeneration

Abstract: We prove an algebraic version of the Hamilton-Tian Conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a reduced uniformly Ding stable triple, which admits a Kähler-Ricci soliton when the ground field = C.

Cited by 3 publications

References 40 publications

Canonical Kähler metrics and stability of algebraic varieties

Canonical Kähler metrics and stability of algebraic varieties

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

Notes on weighted Kähler-Ricci solitons and application to Ricci-flat Kähler cone metrics

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