## 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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