算术Okounkov体与adelic Cartier除数的正性

IF 0.9 1区 数学 Q2 MATHEMATICS Journal of Algebraic Geometry Pub Date : 2023-10-17 DOI:10.1090/jag/821
François Ballaÿ
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引用次数: 3

摘要

在代数几何中,k ronya定理和Lozovanu定理描述了一个卡地亚除数在一个投影变量上的丰富性和整洁性,这是根据其相关的Okounkov体的形状来描述的。我们在Arakelov几何的背景下证明了类似的结果,证明了线性R {\mathbb {R}} -Cartier因子D¯\overline {D}的算术丰度和整洁度是由Boucksom和Chen意义上的算术Okounkov体决定的。我们的主要结果推广到由Burgos Gil、Moriwaki、Philippon和Sombra建立的环测度R {\mathbb {R}} -因子在环上正性的任意射影变体准则。作为一个应用,我们证明了D¯\overline {D}的绝对极小值与Boucksom-Chen凹变换的极小值一致,并证明了在温和正假设下算术Hilbert-Samuel定理的一个逆。我们还建立了小点和子变种的一般网存在性的新判据。
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Arithmetic Okounkov bodies and positivity of adelic Cartier divisors
In algebraic geometry, theorems of Küronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov bodies. We prove the analogous result in the context of Arakelov geometry, showing that the arithmetic ampleness and nefness of an adelic R {\mathbb {R}} -Cartier divisor D ¯ \overline {D} are determined by arithmetic Okounkov bodies in the sense of Boucksom and Chen. Our main results generalize to arbitrary projective varieties criteria for the positivity of toric metrized R {\mathbb {R}} -divisors on toric varieties established by Burgos Gil, Moriwaki, Philippon and Sombra. As an application, we show that the absolute minimum of D ¯ \overline {D} coincides with the infimum of the Boucksom–Chen concave transform, and we prove a converse to the arithmetic Hilbert-Samuel theorem under mild positivity assumptions. We also establish new criteria for the existence of generic nets of small points and subvarieties.
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来源期刊
CiteScore
2.70
自引率
5.60%
发文量
23
审稿时长
>12 weeks
期刊介绍: The Journal of Algebraic Geometry is devoted to research articles in algebraic geometry, singularity theory, and related subjects such as number theory, commutative algebra, projective geometry, complex geometry, and geometric topology. This journal, published quarterly with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.
期刊最新文献
Moduli of ℚ-Gorenstein pairs and applications Splitting of Gromov–Witten invariants with toric gluing strata The higher Du Bois and higher rational properties for isolated singularities Arithmetic Okounkov bodies and positivity of adelic Cartier divisors Refined count of oriented real rational curves
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