阿代尔向量丛的特征类及其应用
Characteristic Classes of Adelic Vector Bundles and Applications
AI总结：
本文为数域上射影簇的向量丛定义数值阿代尔特征类，构造重言数值相交代数，并证明受控的数值Bogomolov--Gieseker不等式，含曲线情形的等式与均匀间隙准则。
AI中文摘要：
我们为数域上射影簇上的向量丛定义了数值阿代尔特征类。度量数据由相伴射影丛上的重言商线丛承载。该构造使用了已建立的阿代尔线丛相交理论。高阶特征类是多线性相交泛函。它们对于同时受控且关于上确界范数为柯西列的模型序列是连续的。我们构造了由此产生的重言数值相交代数。我们还证明了受控的数值Bogomolov--Gieseker不等式。我们首先处理$K$上曲线的情形，其中数值不等式与Deligne配对及阿代尔Hodge指标定理相结合。然后我们转向更高维数，在取受控Zhang极限之前使用Moriwaki的模型级维数归纳。曲线定理包括等式和均匀间隙准则。
英文摘要：
We define numerical adelic characteristic classes for vector bundles on projective varieties over number fields. The metric data are carried by the tautological quotient line bundle on the associated projective bundle. The construction uses the established intersection theory of adelic line bundles. Higher characteristic classes are multilinear intersection functionals. They are continuous for simultaneously controlled model sequences that are Cauchy for the supremum norm. We construct the resulting tautological numerical intersection algebra. We also prove controlled numerical Bogomolov--Gieseker inequalities. We first treat the case of a curve over $K$, where the numerical inequality is combined with the Deligne pairing and the adelic Hodge index theorem. We then pass to higher dimension, using Moriwaki's model-level dimension induction before taking the controlled Zhang limit. The curve theorem includes equality and uniform-gap criteria.
