A Guided Tour to Normalized Volume

Li, Chi; Liu, Yuchen; Xu, Chenyang

doi:10.1007/978-3-030-34953-0_10

Cited by 36 publications

(31 citation statements)

References 64 publications

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“…See [LLX18] for a survey on the recent study of the normalized volume function, especially the guiding question, the Stable Degeneration Conjecture (see [Li18,7.1] and [LLX18, 4.1]).…”

Section: Filtrations Induced By Divisorsmentioning

confidence: 99%

Uniqueness of K-polystable degenerations of Fano varieties

Blum

2019

Ann. of Math. (2)

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We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with recently proven boundedness and openness statements, the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable Q-Fano variety is finite.

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“…See [LLX18] for a survey on the recent study of the normalized volume function, especially the guiding question, the Stable Degeneration Conjecture (see [Li18,7.1] and [LLX18, 4.1]).…”

Section: Filtrations Induced By Divisorsmentioning

confidence: 99%

Uniqueness of K-polystable degenerations of Fano varieties

Blum

2019

Ann. of Math. (2)

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“…The first part of our main theorem is a characteristic p > 0 analog of a result on normalized volume by Y. Liu and C. Xu [LX17, Corollary 2.11]. We thank both Y. Liu and C. Xu for inspiring discussions about the relation between F -signature and normalized volume, also see [LLX18,Theorem 6.14] and [Liu18].…”

Section: Introductionmentioning

confidence: 79%

-Signature Under Birational Morphisms

Polstra

Schwede

et al. 2019

Forum of Mathematics, Sigma

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“…Here vol (W,(1−β)∆),p is the normalised volume at the singularity p ∈ W , defined as the infimum of local volumes of valuations centred at p normalised by their log discrepancies, as in [LLX18] 2 , the surface W must be itself defined as cubic surface in P 3 whose singular locus (if non-empty) consists of A 1 or A 2 singularities and the limit divisor ∆ is a hyperplane section.…”

Section: Proofs Of the Main Theoremsmentioning

confidence: 99%

“…Here

{\hat{normalvol}}_{false(W, false(1 - β false) normalΔ false), p}

is the normalised volume at the singularity

p \in W

, defined as the infimum of local volumes of valuations centred at

p

normalised by their log discrepancies, as in [35]. It follows from the definition that the normalised volume satisfies

{\overset{vol}{true}}_{(W, (1 - β) Δ), p} ⩽ {\overset{vol}{true}}_{W, p}, for any β \in false(1 - lct (W, Δ), 1 false),

where

lct (W, Δ)

denotes the log canonical threshold of the pair

(W, Δ)

.…”

Section: Proofs Of the Main Theoremsmentioning

confidence: 99%

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Applications of the moduli continuity method to log K‐stable pairs

Gallardo

Martinez-Garcia

Spotti

2020

J. London Math. Soc.

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The 'moduli continuity method' permits an explicit algebraisation of the Gromov-Hausdorff compactification of Kähler-Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the 'log setting' to describe explicitly some compact moduli spaces of K-polystable log Fano pairs. We focus on situations when the angle of singularities is perturbed in an interval sufficiently close to one, by considering constructions arising from Geometric Invariant Theory. More precisely, we discuss the cases of pairs given by cubic surfaces with anticanonical sections, and of projective space with non-Fano hypersurfaces, and we show ampleness of the CM line bundle on their good moduli space (in the sense of Alper). Finally, we introduce a conjecture relating K-stability (and degenerations) of log pairs formed by a fixed Fano variety and pluri-anticanonical sections to certain natural GIT quotients. GIT d of degree d hypersurfaces in P n. The first statement of the above result is a special case of the following natural expectation, at least when the automorphism group has no non-trivial characters. Conjecture 1.3. Let X be a K-polystable Fano variety. Then for any sufficiently large and divisible l ∈ N, there exists β 0 = β 0 (l, X) such that for all D ∈ | − lK X | and β ∈ (β 0 , 1), the pair (X, (1 − β)D) is log Kpolystable if and only if [D] ∈ P(H 0 (−lK X)) is GIT-polystable for the natural representation of Aut(X) on H 0 (−lK X). Moreover, the Gromov-Hausdorff compactification M K X,l,β of the above pairs (X, (1 − β)D) is homeomorphic to the GIT quotient P(H 0 (−lK X)) ss //Aut(X). Note that the automorphism group Aut(X) of a K-polystable Fano is always reductive. If X is smooth or a Gromov-Hausdorff degeneration, this follows from by Matsushima's obstruction [Mat57] and Chen-Donaldson-Sun [CDS15]. Otherwise, this has been very recently established in [ABHLX20] via purely algebro-geometric techniques Once its different steps are established, the application of the moduli continuity method (the proof of theorems 1.1 and 1.2) is just a couple of paragraphs. However, in order to keep the reader focused on the main goal of the paper, we recall here what the different steps of the moduli continuity method are, noting that each of them requires involved technical proofs to accomplish them. Suppose that we have a Gromov-Hausdorff

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A Guided Tour to Normalized Volume

Cited by 36 publications

References 64 publications

Uniqueness of K-polystable degenerations of Fano varieties

Uniqueness of K-polystable degenerations of Fano varieties

-Signature Under Birational Morphisms

Applications of the moduli continuity method to log K‐stable pairs

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