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Notes on absolute Hodge cohomology. (English) Zbl 0621.14011
Applications of algebraic K-theory to algebraic geometry and number theory, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Boulder/Colo. 1983, Part I, Contemp. Math. 55, 35-68 (1986).
[For the entire collection see Zbl 0588.00014.]
In a previous paper [J. Sov. Math. 30, 2036-2070 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24, 181-238 (1984; Zbl 0588.14013)] the author has developed the absolute Hodge cohomology theory for algebraic schemes over $${\mathbb{C}}$$. In the paper under review, he constructs for every such scheme X a complex $$R\Gamma(X,{\mathbb{Z}})$$ of mixed Hodge structures. The cohomology groups $$H^ i(R\Gamma(X,{\mathbb{Z}}))$$ give back the usual mixed Hodge structure on $$H^ i(X,{\mathbb{Z}})$$ as defined by Deligne. The derived category $$D({\mathcal H})$$ of mixed Hodge structure comes into play and one defines the absolute Hodge complex $$R\Gamma_{{\mathcal H}}(X,{\mathbb{Z}}(i))$$ of X as $$Hom_{D({\mathcal H})}({\mathbb{Z}}(-i), R\Gamma(X,{\mathbb{Z}}))$$. For $$A={\mathbb{Q}}$$ or $${\mathbb{R}}$$ one has an analogous complex $$R\Gamma_{{\mathcal H}}(X,A(i)).$$
The absolute Hodge cohomology groups of X are defined by $$H^ j_{{\mathcal H}}(X,A(i))=H^ j(R\Gamma_{{\mathcal H}}(X,A(i)))$$. For $$A={\mathbb{Q}}$$, these groups form a twisted Poincaré duality theory in the sense of Bloch and Ogus. They contain much information about algebraic cycles. $$H^ j_{{\mathcal H}}(X,{\mathbb{Q}}(i))$$ is an extension of a ”Hodge-theoretic part” (which is discrete) by an ”intermediate Jacobian part” (which is continuous) and thereby carries a natural topology. For example, if X is smooth and projective, $$H_{{\mathcal H}}^{2i}(X,{\mathbb{Z}}(i))$$ is an extension of the group of integral (i,i)-classes on X by the Griffiths intermediate Jacobian $$J^ i(X)$$. One disposes in this case of a cycle map from $$CH^ i(X)$$, the group of algebraic cycles on X of codimension i modulo rational equivalence, to $$H_{{\mathcal H}}^{2i}(X,{\mathbb{Z}}(i))$$, which induces the Abel-Jacobi mapping on the subgroup of cycles which are homologous to zero. More generally, for X smooth let $$K_ i(X)$$ be Quillen’s K-groups. The Adams operations define a filtration $$\gamma^.$$ on $$K_ i(X)\otimes {\mathbb{Q}}$$. Beilinson defines the ”motivic cohomology groups” of X by $$H^ j_{{\mathcal M}}(X,{\mathbb{Q}}(i))=gr^ i_{\gamma}K_{2i-j}(X)\otimes {\mathbb{Q}}$$. These fulfill the requirements of a universal twisted Poincaré duality theory, hence one has Chern character maps, called regulators $$r_{{\mathcal H}}: H^ j_{{\mathcal M}}(X,{\mathbb{Q}}(i))\to H^ j_{{\mathcal H}}(X,{\mathbb{Q}}(i)).$$
The author’s first conjecture is that $$r_{{\mathcal H}}$$ has dense image. In particular it should be surjective to the ”Hodge-theoretic part”. For X smooth projective and $$j=2i$$ this is the Hodge (i,i) conjecture.
For schemes X over $${\mathbb{R}}$$, there exists a real version $$H_{{\mathcal H}/{\mathbb{R}}}$$ of absolute Hodge cohomology. If 2 is invertible in A, $$H^ j_{{\mathcal H}/{\mathbb{R}}}(X,A(i))$$ is the part of $$H^ j_{{\mathcal H}}(X\otimes {\mathbb{C}},A(i))$$ which is invariant under complex conjugation. - If X is smooth and proper over $${\mathbb{Q}}$$ and $$j<2i$$, the $${\mathbb{Q}}$$-de Rham cohomology of X determines a $${\mathbb{Q}}$$-structure on the 1-dimensional $${\mathbb{R}}$$-vector space $$\det (H^ j_{{\mathcal H}/{\mathbb{R}}}(X,{\mathbb{R}}(i)))$$. On the other hand, the restriction of the regulator $$r_{{\mathcal H}}$$ to a certain subspace $$H^ j_{{\mathcal M}}(X,{\mathbb{Q}}(i))_{{\mathbb{Z}}}$$ of $$H^ j_{{\mathcal M}}(X,{\mathbb{Q}}(i))$$ defines another $${\mathbb{Q}}$$-structure of this space, if certain conjectures hold.
The relation between these two $${\mathbb{Q}}$$-stuctures gives an element $$C\in {\mathbb{R}}^*/{\mathbb{Q}}^*$$. It is conjectured that his C is related to the L-function of $$H^{j-1}(X)$$ in the following way. This L-function should have a zero of order $$\dim_{{\mathbb{Q}}}H^ j_{{\mathcal M}}(X,{\mathbb{Q}}(i))_{{\mathbb{Z}}}$$ at $$s=j-i$$ and the leading coefficient j-i in its Laurent expansion should map to C. The author indicates how to formulate this conjecture for arbitrary motives over a number field.
One final remark: one may regret the numerous misprints in this article of such an outstanding content.
Reviewer: J.H.M.Steenbrink

##### MSC:
 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14F40 de Rham cohomology and algebraic geometry