COHOMOLOGICAL ARITHMETIC CHOW RINGS
COHOMOLOGICAL ARITHMETIC CHOW RINGS
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DOI：
10.1017/s1474748007000011
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发表时间：
2004-04
影响因子：
0.9
通讯作者：
J. B. Gil;Jürg Kramer;U. Kuehn
中科院分区：
文献类型：
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作者：
J. B. Gil;Jürg Kramer;U. Kuehn
我们开发了一种抽象算术 Chow 环的理论，其中无穷远纤维的作用是由计算合适的上同调理论的阿贝尔群复合体扮演的。作为这种形式主义的特例，我们恢复了 Gillet 和 Soulé 射影簇的原始算术交集理论。我们引入了算术 Chow 群的理论，它对于任意真态射是协变的，并且我们使用具有沿固定法线交叉除数的双对数奇点的微分形式的复形来开发算术 Chow 环的理论。最后一个理论适用于自守线束的研究。特别是，我们将关于对数奇异厄米线束的经典法尔廷斯高度推广到更高维度的循环。作为一个应用程序，我们计算模曲线乘积上赫克对应的法尔廷斯高度。
We develop a theory of abstract arithmetic Chow rings, where the role of the fibres at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. As particular cases of this formalism we recover the original arithmetic intersection theory of Gillet and Soulé for projective varieties. We introduce a theory of arithmetic Chow groups, which are covariant with respect to arbitrary proper morphisms, and we develop a theory of arithmetic Chow rings using a complex of differential forms with log-log singularities along a fixed normal crossing divisor. This last theory is suitable for the study of automorphic line bundles. In particular, we generalize the classical Faltings height with respect to logarithmically singular hermitian line bundles to higher dimensional cycles. As an application we compute the Faltings height of Hecke correspondences on a product of modular curves.