Holmstrom Deligne homology

Deligne homology

Brief memo notes from Jannsen

Deligne homology was defined by Beilinson: Notes on absolute Hodge cohomology, and by Gillet (ref). For a complex variety $X$ , can define groups $H^D_a(X, A(b) )$ , where $A$ is $\mathbb{Z}$ or $\mathbb{R}$ or some other coefficient ring. It is the extension of two groups described in terms of Borel-Moore homology and de Rham homology, and its Hodge filtration. Bloch-Ogus properties. No pairing, but a duality isomorphism

H^i_D(X, A(j) ) \to H^D_{2d-i}(X, A(d-j) )

for smooth $X$ of dim $d$ . This suffices to define the Gysin morphisms needed for algehraic correspondences. Also easy def of cycle class map, leading to Abel-Jacobi map.

To define Deligne homology, use standard simplicial methods of Deligne.

In section 2, we treat the Hodge theory of Borel-Moore homology This leads to Beilinson’s absolute Hodge cohomology.

Beilinson has defined Chern maps and Chern characters

c, ch: K_{2j-i}(X) \to H^i_D(X, \mathbb{Q}(j) )

and also homological counterparts

\tau: K'_{a-2b}(X) \to H_a^D(X, \mathbb{Q}(b) )

which together form a Riemann-Roch theorem (ref to Gillet). Many constructions in K-theory are defined via K’-theory (Gysin maps, Quillen spectral sequence), so this is useful even of one is primarily interested in the regulator maps

r = ch: K_{2j-i}(X)^{(j)} \to H^i_D(X, \mathbb{R}(j) )

for a smooth and proper variety $X$ .

Beilinson’s conjecture on the surjectivity of $r \otimes \mathbb{R}$ and results of Suslin and Soule on the Adams eigenspaces lead to a conjecture on the coniveau filtration on $H^i_D(X, \mathbb{R}(j) )$ . This is the Hodge-D-conjecture, reviewed in section 3.

Section 4: Beilinson’s conjecture for motives with coefficients. Relations to mixed motives and special values.

category: [Private] Notes

Deligne homology

Defined for simplicial schemes, see Jannsen.

Jannsen Thm 1.19: We get a Bloch-Ogus theory on the category of all schemes which are separated and of finite type over $F$ , where $F$ is either $\mathbb{C}$ or $\mathbb{R}$ .