On the Concavity of the Arithmetic Volumes

Ikoma, Hideaki

doi:10.1093/imrn/rnu156

Cited by 10 publications

(17 citation statements)

References 18 publications

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“…The following definition of ampleness is the one used by Ikoma in [Iko15b]. Definition 3.14.…”

Section: Arithmetic Cartier Divisorsmentioning

confidence: 99%

“…When is a number field, we use arithmetic intersection theory to adapt the above argument. We first reduce the problem to a fixed model and work with an arithmetic variety on , and then apply an arithmetic analogue of Boucksom, Demailly, Păun and Peternell's theorem due to Ikoma [Iko15b]. To do so we also use an arithmetic Bertini-type theorem due to Moriwaki [Mor95].…”

Section: Introductionmentioning

confidence: 99%

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Successive minima and asymptotic slopes in Arakelov geometry

Ballaÿ¹

2021

Compositio Math.

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Let $X$ be a normal and geometrically integral projective variety over a global field $K$ and let $\bar {D}$ be an adelic ${\mathbb {R}}$ -Cartier divisor on $X$ . We prove a conjecture of Chen, showing that the essential minimum $\zeta _{\mathrm {ess}}(\bar {D})$ of $\bar {D}$ equals its asymptotic maximal slope under mild positivity assumptions. As an application, we see that $\zeta _{\mathrm {ess}}(\bar {D})$ can be read on the Okounkov body of the underlying divisor $D$ via the Boucksom–Chen concave transform. This gives a new interpretation of Zhang's inequalities on successive minima and a criterion for equality generalizing to arbitrary projective varieties a result of Burgos Gil, Philippon and Sombra concerning toric metrized divisors on toric varieties. When applied to a projective space $X = {\mathbb {P}}_K^{d}$ , our main result has several applications to the study of successive minima of hermitian vector spaces. We obtain an absolute transference theorem with a linear upper bound, answering a question raised by Gaudron. We also give new comparisons between successive slopes and absolute minima, extending results of Gaudron and Rémond.

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“…The following definition of ampleness is the one used by Ikoma in [Iko15b]. Definition 3.14.…”

Section: Arithmetic Cartier Divisorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Successive minima and asymptotic slopes in Arakelov geometry

Ballaÿ¹

2021

Compositio Math.

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“…This result has firstly been proved in the case where D and E are Cartier divisors (cf. [13]), and then be extended to the general case of adelic R-Cartier divisors by Ikoma [22] (the normality hypothesis on the arithmetic variety in the differentiability theorem in loc. cit.…”

Section: 28mentioning

confidence: 99%

“…The adelic R-Cartier divisor D is said to be nef if it is relatively nef and if the height function h D is non-negative (see[29, §4.4]). If D is nef, one has (see[22, Proposition 3.11])(14) D d · D = (D (d+1) ) = vol(D).…”

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confidence: 99%

On isoperimetric inequality in Arakelov geometry

Chen¹

2018

Bul. Soc. Math. France

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“…(3): If L i are associated to continuous Hermitian line bundles on an O K -model of X, then the assertion follows from the projection formula [18, Proposition 2.4.1] (see also [16,Lemma 2.3]). In general, we can assume that L 0 , .…”

Section: (Nef )mentioning

confidence: 99%