Motives and algebraic de rham cohomology

Details Motives and algebraic de rham cohomology Motives and algebraic de rham cohomology

1999-08-18

arxiv.org/abs/math/9908093v1

Mathematics

Algebraic Geometry

Latex2e, to appear

Scientific paper

In this paper, we define a certain Hodge-theoretic structure for an arbitrary variety X over the complex number field by using the theory of mixed Hodge module due to Morihiko Saito. We call it an arithmetic Hodge structure of X. It is shown that extension groups of arithmetic Hodge structure do not vanish even for degree $\geq2$. Moreover, we define higher Abel-Jacobi maps from Bloch's higher Chow groups of X to these extension groups. These maps essentially involve the classical Abel-Jacobi maps by Weil and Griffiths, and Mumford's infinitesimal invariants of 0-cycles on surfaces.

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