TAILIEUCHUNG - Đề tài " Deligne’s conjecture on 1-motives "

We reformulate a conjecture of Deligne on 1-motives by using the integral weight filtration of Gillet and Soul´ on cohomology, and prove it. This implies e the original conjecture up to isogeny. If the degree of cohomology is at most two, we can prove the conjecture for the Hodge realization without isogeny, and even for 1-motives with torsion. j Let X be a complex algebraic variety. We denote by H(1) (X, Z) the maximal mixed Hodge structure of type {(0, 0), (0, 1), (1, 0), (1, 1)} contained in j j H j (X, Z). Let H(1) (X, Z)fr. | Annals of Mathematics Deligne s conjecture on 1-motives By L. Barbieri-Viale A. Rosenschon and M. Saito Annals of Mathematics 158 2003 593 633 Deligne s conjecture on 1-motives By L. Barbieri-Viale a. Rosenschon and M. Saito Abstract We reformulate a conjecture of Deligne on 1-motives by using the integral weight filtration of Gillet and Soule on cohomology and prove it. This implies the original conjecture up to isogeny. If the degree of cohomology is at most two we can prove the conjecture for the Hodge realization without isogeny and even for 1-motives with torsion. Let X be a complex algebraic variety. We denote by Hj1 XffL the maximal mixed Hodge structure of type 0 0 0 1 1 0 1 1 contained in Hj XffL . Let Hj1 X 2 fr be the quotient of Hj1 X 2 by the torsion subgroup. P. Deligne 10 conjectured that the 1-motive corresponding to Hj1 XffL fr admits a purely algebraic description that is there should exist a 1-motive Mj X fr which is defined without using the associated analytic space and whose image rn Mj X fr under the Hodge realization functor rH see loc. cit. and below is canonically isomorphic to H X z fr 1 and similarly for the Z-adic and de Rham realizations . This conjecture has been proved for curves 10 for the second cohomology of projective surfaces 9 and for the first cohomology of any varieties 2 see also 25 . In general a careful analysis of the weight spectral sequence in Hodge theory leads us to a candidate for Mj X fr up to isogeny see also 26 . However since the torsion part cannot be handled by Hodge theory it is a rather difficult problem to solve the conjecture without isogeny. In this paper we introduce the notion of an effective 1-motive which admits torsion. By modifying morphisms we can get an abelian category of 1-motives which admit torsion and prove that this is equivalent to the category of graded-polarizable mixed Z-Hodge structures of the above type. However our construction gives in general nonreduced effective 1-motives

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