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Second Chern numbers of vector bundles and higher adeles. (English) Zbl 1397.14056

Summary: We give a construction of the second Chern number of a vector bundle over a smooth projective surface by means of adelic transition matrices for the vector bundle. The construction does not use an algebraic \(K\)-theory and depends on the canonical \(\mathbb Z\)-torsor of a locally linearly compact \(k\)-vector space. Analogs of certain auxiliary results for the case of an arithmetic surface are also discussed.

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
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