Deligne homology, Hodge-\({\mathcal D}\)-conjecture, and motives. (English) Zbl 0701.14019

Jannsen, Uwe

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

MSC:

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

Keywords:

Deligne-Beilinson homology and cohomology; Poincaré duality; mixed motives; analogue of mixed Hodge structures

Citations:

Zbl 0635.00005; Zbl 0588.14013