On the volume of graded linear series and Monge–Ampère mass

Hisamoto, Tomoyuki

doi:10.1007/s00209-012-1133-6

Cited by 7 publications

(7 citation statements)

References 22 publications

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“…The main result of [His12] gives an analytic description of the volume. The analytic counterpart of the volume is the following generalized equilibrium metric, which is originated from [Berm07].…”

Section: Analytic Representation Of Volumementioning

confidence: 99%

“…Theorem 2.5 (The main theorem of [His12]). Let W be a graded linear series of a line bundle L, such that the natural map X PW * k is birational to the image for any sufficiently large k. Then for any fixed smooth Hermitian metric h = e −ϕ we have vol(W ) = X MA(P W ϕ).…”

Section: Analytic Representation Of Volumementioning

confidence: 99%

“…Then it can be proved algebraically that the limit of spectral measures is given by the Lebesgue-Stieltjes measure of the volume function vol(W λ ) in λ. The main theorem of [His12] interpret each volume into the Monge-Ampère measure of associated equilibrium metric P W λ ϕ. The Legendre transformation of this family of equilibrium metrics is nothing but the weak geodesic ray ϕ t so that we may complete the proof by the recent developed techniques of pluripotential theory.…”

Section: Introductionmentioning

confidence: 97%

“…Let us briefly explain the outline of our proof of Theorem 1.1. The proof is based on the analytic study for graded linear series, which was exploited in [His12]. We apply it to [WN10]'s family of graded subalgebras…”

Section: Introductionmentioning

confidence: 99%

“…Acknowledgments.The author would like to express his gratitude to his advisor Professor Shigeharu Takayama for his warm encouragements, suggestions and reading the drafts. The author also would like to thank Professor Sébastien Boucksom for his indicating the relation between the authors preprint [His12] and the paper of Julius Ross and David Witt-Nyström. The author wishes to thank Doctor David Witt-Nyström, Professor Robert Berman and Professor Bo Berndtsson for stimulating discussion and helpful comments during the author's stay in Gothenburg.…”

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confidence: 99%

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On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold

Hisamoto

2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We apply the result of [His12] to the family of graded linear series constructed from any test configuration. This solves the conjecture raised by [WN10] so that the sequence of spectral measures for the induced C * -action on the central fiber converges to the canonical Duistermatt-Heckman measure defined by the associated weak geodesic ray. As a consequence, we show that the algebraic p-norm of the test configuration equals to the L p -norm of tangent vectors. Using this result, We may give a natural energy theoretic explanation for the lower bound estimate on the Calabi functional by [Don05] and prove the analogous result for the Kähler-Einstein metric.2000 Mathematics Subject Classification. Primary 32A25, Secondary 32L10, 32W20.

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Section: Analytic Representation Of Volumementioning

confidence: 99%

Section: Analytic Representation Of Volumementioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold

Hisamoto

2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Thresholds, valuations, and K-stability

Blum

Jönsson

2020

Advances in Mathematics

120

181

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Let X be a normal complex projective variety with at worst klt singularities, and L a big line bundle on X. We use valuations to study the log canonical threshold of L, as well as another invariant, the stability threshold. The latter generalizes a notion by Fujita and Odaka, and can be used to characterize when a Q-Fano variety is K-semistable or uniformly K-stable. It can also be used to generalize volume bounds due to Fujita and Liu. The two thresholds can be written as infima of certain functionals on the space of valuations on X. When L is ample, we prove that these infima are attained. In the toric case, toric valuations acheive these infima, and we obtain simple expressions for the two thresholds in terms of the moment polytope of L.arXiv:1706.04548v1 [math.AG] 14 Jun 2017for all positive integers p, q. This, together with the Noetherianity of X, implies that {J (X, (c/p) · a p )} p∈N has a unique maximal element that is called the c-th asymptotic multiplier ideal and denoted by J (X, c · a • ). Note that J (X, c · a • ) = J (X, (c/p) · a p ) for all p divisible enough.Asymptotic multiplier ideals also satisfy a subadditivity property. See [Laz04, Theorem 11.2.3] for the case when X is smooth.Corollary 5.6. Let a • be a graded sequence of ideals on X. If m ∈ N * and c ∈ Q * + , thenNext we give a containment relation for the multiplier ideal associated to the graded sequence of valuation ideals. The result appears in [ELS03] in the case when v is divisorial.Proof. It is an immediate consequence of the valuative criterion for membership in the multiplier ideal [BdFFU15, Theorem1.2] that J (X, c · a • (v)) ⊂ a cv(a•(v))−A(v) (v).Since v(a • (v)) = 1 (see [Blu16b, Lemma 3.5]), the proof is complete.

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Canonical growth conditions associated to ample line bundles

Nyström¹

2018

Duke Math. J.

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On the volume of graded linear series and Monge–Ampère mass

Cited by 7 publications

References 22 publications

On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold

On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold

Thresholds, valuations, and K-stability

Canonical growth conditions associated to ample line bundles

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