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Characteristic classes for algebraic vector bundles with Hermitian metric. I. (English) Zbl 0715.14018
The authors define characteristic classes for arbitrary hermitian vector bundles on arithmetic varieties (extending Arakelov theory). Let X be a regular, flat, quasiprojective variety over \({\mathbb{Z}}\). Let E be an algebraic vector bundle on X and h an hermitian metric on the corresponding holomorphic vector bundle \(E_{\infty}\) on the set \(X_{\infty}\) of complex points of X. One defines a Chern character ch(E,h) in the graded group \(CH(X)_{{\mathbb{Q}}}=\oplus_{p\geq 0}CH^ p(X)\otimes {\mathbb{Q}} \) (where \(CH^ p(X)\) is the arithmetic Chow group of codimension p of X), which is characterized by the properties: functoriality, additivity, multiplicativity, normalization and commpatibility with Chern forms. One proves that ch induces an isomorphism from \(K_ 0(X)\otimes_{{\mathbb{Z}}}{\mathbb{Q}}\) to \(CH(X)_{{\mathbb{Q}}}.\)
This part I of this paper has four sections: In section 1, one studies Bott-Chern secondary characteristic classes and in section 2, one studies the first Chern class of an hermitian line bundle in the language of this paper. In section 3, a splitting principle for CH of Grassmannians is proved. Using this principle, in section 4, one defines characteristic classes and one proves their properties.
[For part II of this paper see ibid., No.2, 205-238 (1990; Zbl 0715.14006)].

MSC:
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
57R10 Smoothing in differential topology
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