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Deligne homology, Hodge-\({\mathcal D}\)-conjecture, and motives. (English) Zbl 0701.14019
Beilinson’s conjectures on special values of L-functions, Meet. Oberwolfach/FRG 1986, Perspect. Math. 4, 305-372 (1988).
[For the entire collection see Zbl 0635.00005.]
After an outline of Deligne-Beilinson homology and cohomology - at first of complex spaces, then for schemes X and simplicial schemes \((X_ n)\)- and their Poincaré duality, follows a treatment of the Hodge theory for Borel-Moore homology. Then the Riemann-Roch theorem for X, involving Chern maps with values in the Deligne-Beilinson homology and cohomology groups, is presented, as well as Beilinson’s Hodge-\({\mathcal D}\)-conjecture [A. A. Bejlinson, J. Sov. Math. 30, 2036-2070 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Math. 24, 181-238 (1984; Zbl 0588.14013)]. Finally, a reformulation by Deligne of Beilinson’s conjecture for motives with coefficients in a number field, in terms of mixed motives, which are a sought-for analogue of mixed Hodge structures, is discussed.
Reviewer: J.H.de Boer

14F99 (Co)homology theory in algebraic geometry
14A20 Generalizations (algebraic spaces, stacks)
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32C38 Sheaves of differential operators and their modules, \(D\)-modules