Commentarii Mathematici Helvetici

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Volume 87, Issue 3, 2012, pp. 521–587
DOI: 10.4171/CMH/262

Higher arithmetic Chow groups

José Ignacio Burgos Gil[1] and Elisenda Feliu[2]

(1) Instituto de Ciencias Matemáticas (CSIC, UAM, UCM, UC3), C/ Nicolás Cabrera, 15, 28049, Madrid, Spain
(2) Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

We give a new construction of higher arithmetic Chow groups for quasi-projective arithmetic varieties over a field. Our definition agrees with the higher arithmetic Chow groups defined by Goncharov for projective arithmetic varieties over a field. These groups are the analogue, in the Arakelov context, of the higher algebraic Chow groups defined by Bloch. For projective varieties the degree zero group agrees with the arithmetic Chow groups defined by Gillet and Soulé, and in general with the arithmetic Chow groups of Burgos. Our new construction is shown to be a contravariant functor and is endowed with a product structure, which is commutative and associative.

Keywords: Arakelov geometry, higher Chow groups, Beilinson regulator, intersection theory, Deligne cohomology

Burgos Gil José Ignacio, Feliu Elisenda: Higher arithmetic Chow groups. Comment. Math. Helv. 87 (2012), 521-587. doi: 10.4171/CMH/262