论文标题
算术Okounkov的身体和Adelic Cartier除数的积极性
Arithmetic Okounkov bodies and positivity of adelic Cartier divisors
论文作者
论文摘要
在代数的几何形状中，库罗尼亚和洛佐瓦努的定理表征了卡地亚分裂的增强性和变形，从其相关的okounkov尸体的形状来看，对投射品种的变化。我们证明了在Arakelov几何形状的背景下的类似结果，表明Adelic $ \ Mathbb {r} $ - Cartier Divisor $ \ operline {d} $的算术增大和细胞是由Arithmetic Okounkov Bodies在Boucksom and Chen and Chen的意义上确定的。我们的主要结果推广到任意投射品种的标准，该标准是感谢物，由Burgos Gil，Moriwaki，Philippon和Sombra建立的感谢您的福利$ \ mathbb {r} $的阳性。作为一个应用，我们表明，绝对最低$ \叠加{d} $与boucksom的最低含量相吻合 - 倾斜的凹形变换，我们证明在轻度积极假设下与算术希尔伯特 - 塞缪尔定理进行了交谈。我们还为存在小点和亚地区的通用网络建立了新的标准。
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 $\mathbb{R}$-Cartier divisor $\overline{D}$ 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 $\mathbb{R}$-divisors on toric varieties established by Burgos Gil, Moriwaki, Philippon and Sombra. As an application, we show that the absolute minimum of $\overline{D}$ 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.
