Cover: - Pure - Ab. mixed - Triang. mixed - CTs: generalized/ordinary, geom/absolute. - Various constructions of motives, including realizations. - Explain first the dream, and then what is known. For example, explain why we cannot prove nice properties of pure motives, but we can define them. - Try to clarify whether we really need abelian mixed motives.
Scholl-Deninger section 2.9 explains how Deligne cohomology in a certain index range is isomorphic to a certain Ext^1 group in the cat of R-mixed Hodge structure with the action of a real Frobenius. (From this one would guess that Bette-Deligne is a geometric-absolute pair. Have working defs for MMQ and MMZ in terms of realizations, ref to Deligne: La groupe fondamental…, Scholl: Remarks on special values of L-functions, and LNM1400. For such motives get maps to Ext groups also for motivic cohomology, these are conjecturally isos. Some more details about this picture.
Expected dimension of cats of motives: This I believe is 1 (or 2??) for number fields, which maybe explains the form of the order of vanishing conjecture according to Kim, i.e. it’s an “Euler char” but only 2 terms can possibly contribute.
Is it possible to say that a geometric-absolute pair is associated to every completion of a number field? Is this correct? The mystery is really de Rham-Betti-Deligne-absolute Hodge…
Relate the idea of oriented theories to Gillet’s construction, for which the requirements are products, projective bundle thm, and weak Gysin.
Note: Motivic cohomology should be in the bounded derived cat of mixed motives, according to Jannsen. Here R is the functor from correspondences to the derived cat of mixed motives.
Jannsen end of p 285: Expectation that the cohomological dimension of mixed motives over a field is the Kronecker dimension of the field, so 1 in the global field case. and 0 for finite fields.
There is some discussion of the “integral part” of motivic cohomology in Bondarko’s weights paper: http://front.math.ucdavis.edu/1007.4543
nLab page on Beilinson conjectures MOTIVES AND CT NOTIONS