论文标题
Arakelov几何形状中的连续极小和渐近斜率
Successive minima and asymptotic slopes in Arakelov Geometry
论文作者
论文摘要
让$ x $成为全球字段$ k $的正常且几何积分的投影品种，让$ \ overline {d} $成为$ x $上的Adelic Cartier Divisor。我们证明了Chen的猜想，表明$ \ Overline {d} $的基本最小值$ζ_ {\ Mathrm {ess}}（\ overline {d}）$等于其渐近的最大斜率在中度假设下。作为一个应用程序，我们看到$ζ_ {\ mathrm {ess}}}（\ overline {d}）$可以通过boucksom- boucksom-the boucksom--倾斜变换来读取基础除数$ d $的Okounkov体。这给出了张关于连续的最小值的不平等现象的新解释，以及对任意投射品种的相等性标准，这是Burgos Gil，Philippon和Sombra的结果，涉及在圆环品种上的圆环计数分歧。当应用于投影空间$ x = \ mathbb {p} _k^d $时，我们的主要结果在研究Hermitian矢量空间的连续最小值研究中都有多个应用程序。我们获得了具有线性上限的绝对转移定理，回答了高德隆提出的问题。我们还在连续的斜坡和绝对的极小物之间进行了新的比较，从而扩展了高伦和雷蒙德的结果。
Let $X$ be a normal and geometrically integral projective variety over a global field $K$ and let $\overline{D}$ be an adelic Cartier divisor on $X$. We prove a conjecture of Chen, showing that the essential minimum $ζ_{\mathrm{ess}}(\overline{D})$ of $\overline{D}$ equals its asymptotic maximal slope under mild positivity assumptions. As an application, we see that $ζ_{\mathrm{ess}}(\overline{D})$ 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.
