Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties

Han, Jiyuan; Li, Chi

doi:10.48550/arxiv.2009.01010

Cited by 11 publications

(49 citation statements)

References 64 publications

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“…The following gives an extension of the above theorem. Analogous results for other frameworks are known by [Der2] and [HL2]: the central fibre of an 'optimal degeneration' is 'semistable' in a suitable sense depending on the framework.…”

Section: Resultssupporting

confidence: 54%

“…It is shown by [HL2] that for a Q-Fano variety which is modified K-semistable with respect to a proper vector ξ, the maximum of ȞNA is achieved by ϕ ξ , so that we conclude the following.…”

Section: −2πmentioning

confidence: 55%

“…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%

“…It is called R-degeneration in [DS] and R-test configuration in [BJ4]. A gemstone of polyhedral configuration is appeared in [HL2], where polyhedral configuration is recognized as a geometric realization of a single finitely generated filtration. Here we strengthen polyhedral configuration is useful because it realizes an intuitive treatment of family of filtrations rather than a single filtration.…”

Section: −2πmentioning

confidence: 99%

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

Inoue¹

2022

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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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Section: Resultssupporting

confidence: 54%

Section: −2πmentioning

confidence: 55%

Section: −2πmentioning

confidence: 99%

Section: −2πmentioning

confidence: 99%

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

Inoue¹

2022

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“…However, for the nontrivial soliton case, it seems that the implication remains unknown. Fortunately, we still know that the K-semistabily is equivalent to the existence of K-polystable degeneration [8]. Thus this problem can be reduced to researching whether the existence of polystable degeneration implies the lower boundedness of the modified K-energy.…”

Section: Introductionmentioning

confidence: 99%