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 Second Chern numbers of vector bundles and higher adeles Bull. Korean Math. Soc. 2017 Vol. 54, No. 5, 1699-1718 https://doi.org/10.4134/BKMS.b160687Published online July 3, 2017Printed September 30, 2017 Denis V. Osipov Steklov Mathematical Institute of Russsian Academy of Sciences Abstract : 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 $\Z$-torsor of a locally linearly compact $k$-vector space. Analogs of certain auxiliary results for the case of an arithmetic surface are also discussed. Keywords : vector bundles, Chern numbers, higher adeles, algebraic and arithmetic surfaces MSC numbers : 14J60, 14J20 Downloads: Full-text PDF