Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2017; 54(5): 1699-1718

Online first article July 3, 2017      Printed September 30, 2017

https://doi.org/10.4134/BKMS.b160687

Copyright © The Korean Mathematical Society.

Second Chern numbers of vector bundles and higher adeles

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

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