The Brauer group and the second Abel-Jacobi map for 0-cycles on algebraic varieties

Abstract

This paper focuses on the connection between the Brauer group and the $0$-cycles of an algebraic variety. We give an alternative construction of the second $l$-adic Abel-Jacobi map for such cycles, linked to the algebraic geometry of Severi-Brauer varieties on $X$. This allows us then to relate this Abel-Jacobi map to the standard pairing between $0$-cycles and Brauer groups (see [M], [L]), completing results from [M] in this direction. Second, for surfaces, it allows us to present this map according to the more geometrical approach devised by M. Green in the framework of (arithmetic) mixed Hodge structures (see [G]). Needless to say, this paper owes much to the work of U. Jannsen and, especially, to his recently published older letter [J4] to B. Gross.