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Deligne’s conjecture on 1-motives. (English) Zbl 1124.14014

Let \(k\) be a field of characteristic \(0\) and let \(\overline k\) be its algebraic closure. An effective 1-motive \(M= [\Gamma @>f>> G]\) over \(k\) consists of a locally finite commutative group scheme \(\Gamma\) over \(k\) and a semi-abelian variety \(G/k\) together with a morphism of group schemes \(f: \Gamma\to G\) such that \(\Gamma(\overline k)\) is a finetely generated abelian group. Then \(\Gamma\) and \(G\) are identified, as 1-motives, respectively with \([\Gamma\to 0]\) and \([0\to G]\). An effective morphism of 1-motives consists of morphisms of \(k\)-group schemes \(u: \Gamma\to\Gamma'\) and \(v: G\to G'\) forming a commutative diagram with \(f\) and \(f'\). Morphisms of 1-motives are defined by inverting quasi-isomorphisms on the right, i.e they are represented by \(u\circ v^{-1}\) with \(v\) a quasi-isomorphism. \({\mathcal M}_1(k)\) is then the category of 1-motives over \(k\). Let \(\Gamma_{\text{tors}}\) be the torsion part of \(\Gamma\) and \(M_{\text{tors}}= \Gamma_{\text{tor}}\cap\text{Ker\,}f\) if \(M= [\Gamma@> f>> G]\). So one can define a torsionfree motive and similarly, a reduced motive. The free part of \(M\) is then the motive \([\Gamma/\Gamma_{\text{tor}}\to G/f(\Gamma_{\text{tor}})]\). If \(M\) is free then it is a 1-motive in the sense of Deligne. Moreover \(\operatorname{Hom}_{\text{eff}}(M, M')= \operatorname{Hom}(M,M')\), for \(M,M'\in{\mathcal M}_1(k)\), if \(M'\) is free. The category of Deligne 1-motives, denoted by \({\mathcal M}_1(k)_{\text{fr}}\) is a full subcategory of \({\mathcal M}_1(k)\).
Now let \(X\) be a complex algebraic variety and let \(H^j_{(1)}(X,\mathbb{Z})\) be the maximal mixed Hodge structure of type \(\{(0, 0), (0,1), (1, 0), (1,1)\}\) contained in \(H^j(X,\mathbb{Z})\). Let \(H^j_{(1)}(X,\mathbb{Z})_{\text{fr}}\) be the quotient of \(H^j_{(1)}(X,\mathbb{Z})\) by the torsion subgroup of \(H^j_{(1)}(X,\mathbb{Z})\). P. Deligne conjectured that the 1-motive corresponding to \(H^j_{(1)}(X,\mathbb{Z})_{\text{fr}}\) admits a purely algebraic description, i.e there exists a 1-motive \(M_j(X)_{\text{fr}}\) whose image \(r_{{\mathcal H}}(M_j(X)_{\text{fr}})\) under the Hodge realization functor is canonically isomorphic to \(H^j_{(1)}(X,\mathbb{Z})_{\text{fr}}(1)\). With the above notations the paper contains the following result
Theorem. Let \(Y\) be a closed subvariety of \(X\) and let \(W\) be a canonical integral weight filtration on the relative cohomology \(H^j(X,Y;\mathbb{Z})\). Then there exists a a canonical isomorphism of mixed Hodge structures \[ \phi_{\text{fr}}: r_{{\mathcal H}}(M_j(X, Y))_{fr}\to W_2 H^j_{(1)}(X, Y;\mathbb{Z})_{\text{fr}}, \] where \(M_j(X, Y)\) is a 1-motive asociated to \(X\) and \(Y\), such that \(M_j(X, Y)_{\text{fr}}\) is independent of the resolution choosen to define the weight filtration. Moreover in the above isomorphism the semi-abelian part and the torus part of \(M_j(X, Y)\) correspond respectively to \(W_1 H^j_{(1)}(X,Y;\mathbb{Z})_{\text{fr}}\) and \(W_0 H^j_{(1)}(X, Y;\mathbb{Z})_{\text{fr}}\). A similar result also holds for the \(\ell\)-adic and the de Rham realizations.
The result above implies Deligne’s conjecture up to isogeny. Also, as a corollary, Deligne’s conjecture without isogeny is reduced to show that \[ H^j_{(1)}(X,Y;\mathbb{Z})_{\text{fr}}= W_2 H^j_{(1)}(X,Y;\mathbb{Z})_{\text{fr}}. \] This is satisfied if \(\text{gr}^W_q(H^j(X, Y;\mathbb{Z}))\) are torsion free for \(q> 2\).

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14D07 Variation of Hodge structures (algebro-geometric aspects)
32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
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