On K-stability and the volume functions of ℚ-Fano varieties: Table 1.

Fujita, Kento

doi:10.1112/plms/pdw037

Cited by 48 publications

(60 citation statements)

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“…As a spin-off of our proof, we get the an equivalent characterization of K-semistability using divisorial valuations in Theorem 3.7. This generalizes a result about Fujita's divisorial stability in [Fuj15a] (see also [Li15a]).…”

Section: Introductionsupporting

confidence: 86%

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K-semistability is equivariant volume minimization

Li¹

2017

Duke Math. J.

154

186

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This is a continuation to the paper [Li15a] in which a problem of minimizing normalized volumes over Q-Gorenstein klt singularities was proposed. Here we consider its relation with K-semistability, which is an important concept in the study of Kähler-Einstein metrics on Fano varieties. In particular, we prove that for a Q-Fano variety V , the K-semistability of (V, −K V ) is equivalent to the condition that the normalized volume is minimized at the canonical valuation ord V among all C * -invariant valuations on the cone associated to any positive Cartier multiple of −K V . In this case, it's shown that ord V is the unique minimizer among all C * -invariant quasi-monomial valuations. These results allow us to give characterizations of K-semistability by using equivariant volume minimization, and also by using inequalities involving divisorial valuations over V .Definition 2.5 ([Ber15, Fuj15b]).1. Let (V, L)/C 1 be a normal semi-test configuration of (V, r −1 K −1 V ) and (V,L)/P 1 be its natural compactification. Let D (V,L) be the Qdivisor on V satisfying the following conditions: L) is a Z-divisor corresponding to the divisorial sheafL(r −1 KV /P 1 ). The Ding invariantDing invariant Ding(V, L) of (V, L)/C is defined as:3. V is called Ding semistable if Ding(V, L) ≥ 0 for any normal test configuration (V, L)/C of (V, −r −1 K V ).

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Section: Introductionsupporting

confidence: 86%

“…2. If F is a prime divisor on V , then A V (F ) = 1 and Θ V (F ) is nothing but (a multiple of ) Fujita's invariant η(F ) defined in [Fuj15a]. In particular, the following theorem can be seen as a generalization of result about Fujita's divisorial stability in [Fuj15a] (see also [Li15a]).…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

K-semistability is equivariant volume minimization

Li¹

2017

Duke Math. J.

154

186

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“…β(F ) ≥ 0) for all prime divisors F on X leads to be the (weaker) notion of divisorial stability (resp. divisorial semistability) studied in [Fuj15b]. Note that every prime divisor on X is dreamy by [BCHM10, Corollary 1.3.2].…”

Section: Introductionmentioning

confidence: 99%

A valuative criterion for uniform K-stability of ℚ-Fano varieties

Fujita

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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171

232

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“…In this section, we carry out calculations to show that there is a close relation between minimization of vol with Fujita's divisorial stability ( [Fuj15a]) which is a consequence of K-stability ([Tia97], [Don02], see also [Ber12]). The calculations will essentially show that the derivative of vol at the canonial valuation on the cone along some directions of C * -invariant valuations is given by Fujita's invariant on the base.…”

mentioning

confidence: 99%

Minimizing normalized volumes of valuations

2017

Math. Z.

134

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For any Q-Gorenstein klt singularity (X, o), we introduce a normalized volume function vol that is defined on the space of real valuations centered at o and consider the problem of minimizing vol. We prove that the normalized volume has a uniform positive lower bound by proving an Izumi type estimate for any Q-Gorenstein klt singularity. Furthermore, by proving a properness estimate, we show that the set of real valuations with uniformly bounded normalized volumes is compact, and hence reduce the existence of minimizers for the normalized volume functional vol to a conjectural lower semicontinuity property. We calculate candidate minimizers in several examples to show that this is an interesting and nontrivial problem. In particular, by using an inequality of de-Fernex-Ein-Mustaţȃ, we show that the divisorial valuation associated to the exceptional divisor of the standard blow up is a minimizer of vol for a smooth point. Finally the relation to Fujita's work on divisorial stability is also pointed out.

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On K-stability and the volume functions of ℚ-Fano varieties: Table 1.

Cited by 48 publications

References 48 publications

K-semistability is equivariant volume minimization

K-semistability is equivariant volume minimization

A valuative criterion for uniform K-stability of ℚ-Fano varieties

Minimizing normalized volumes of valuations

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