Abstract
Absolute Hodge cohomology is presented as a Poincaré duality theory that generalizes Deligne-Beilinson cohomology in the sense that it includes the weight filtration. In this way it applies to general schemes over the complex numbers. The relation with motivic cohomology is again given by a regulator map that is conjectured to have dense image, at least for smooth schemes that can be defined over a number field. This conjectured property induces the classical Hodge Conjecture for smooth, projective varieties.
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Notes
For definitions and properties of (rigid) tensor categories, cf. [DMOS] or [Sa].
We assume the choice of an ‘orientation’ to dispose of twists.
This means that C has an internal Horn with the properties stated in the section on motives and that every object X of C is reflexive, i.e. isomorphic to its bidual Xvv.
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© 1992 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden
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Hulsbergen, W.W.J. (1992). Absolute Hodge cohomology, Hodge and Tate conjectures and Abel-Jacobi maps. In: Conjectures in Arithmetic Algebraic Geometry. Aspects of Mathematics, vol 18. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-322-85466-7_9
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DOI: https://doi.org/10.1007/978-3-322-85466-7_9
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
Print ISBN: 978-3-528-06433-4
Online ISBN: 978-3-322-85466-7
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