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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