Realizations of Biextensions
Let k be a field of characteristic 0 and M(k) is the category of 1-motives over k. We prove that Biext^1(M_1,M_2;M_3)=Hom_{MR(k)}(T(M_1)\otimes T(M_2), T(M_3))
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arXiv subjects
Cristiana Bertolin
Publications and source records attributed to Cristiana Bertolin.
Some motivic pairings
In this note we prove the geometrical origin of pairings of abelian schemes. According to Deligne's philosophy of motives, this means that these pairings are motivic. We make also explicit the link between pairings and linear morphisms. We give a geometrical explanation of the main properties of the Weil pairings and of the Riemann forms of line bundles. We generalize our remarks to pairings of 1-motives.
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Deligne's conjecture on extensions of 1-motives
We introduce the notion of extension of 1-motives. Using the dictionary between strictly commutative Picard stacks and complexes of abelian sheaves concentrated in degrees -1 and 0, we check that an extension of 1-motives induces an extension of the corresponding strictly commutative Picard stacks. We compute the Hodge, the de Rham and the l-adic realizations of an extention of 1-motives. Using these results we can prove Deligne's conjecture on extensions of 1-motives.
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Extensions of Picard stacks and their homological interpretation
Let S be a site. We introduce the notion of extensions of strictly commutative Picard S-stacks. We define the pull-back, the push-down, and the sum of such extensions and we compute their homological interpretation: if P and Q are two strictly commutative Picard S-stacks, the equivalence classes of extensions of P by Q are parametrized by the cohomology group Ext^1([P],[Q]), where [P] and [Q] are the complex associated to P and Q respectively.
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Biextensions of 1-motives in Voevodsky's category of motives
Let k be a perfect field. In this paper we prove that biextensions of 1-motives define multilinear morphisms between 1-motives in Voevodsky's triangulated category of effective geometrical motives over k with rational coefficients.
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Multilinear morphisms between 1-motives
Let S be an arbitrary scheme. We define biextensions of 1-motives by 1-motives which we see as the geometrical origin of morphisms from the tensor product of two 1-motives to a third one. If S is the spectrum of a field of characteristic 0, we check that these biextensions define morphisms from the tensor product of the realizations of two 1-motives to the realization of a third 1-motive. Generalizing we obtain the geometrical notion of morphisms from a finite tensor product of 1-motives to another 1-motive.
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Biextensions of 1-motives by 1-motives
Let S be a scheme. In this paper, we define the notion of biextensions of 1-motives by 1-motives. If M(S) denotes the Tannakian category generated by 1-motives over S (in a geometrical sense), we define geometrically the morphisms of M(S) from the tensor product of two 1-motives M_1 and M_2 to another 1-motive M_3, to be the isomorphism classes of biextensions of (M_1,M_2) by M_3. Generalizing this definition we obtain, modulo isogeny, the geometrical notion of morphism of M(S) from a finite tensor product of 1-motives to another 1-motive.
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Motivic Galois theory for motives of niveau $\leq$ 1
Let T be a Tannakian category over a field k of characteristic 0 and π(T) its fundamental group. In this paper we prove that there is a bijection between the otimes-equivalence classes of Tannakian subcategories of T and the normal affine group sub-T-schemes of π(T). We apply this result to the Tannakian category T_1(k) generated by motives of niveau \leq 1 defined over k, whose fundamental group is called the motivic Galois group G_mot(T_1(k)) of motives of niveau \leq 1. We find four short exact sequences of affine group sub-T_1(k)-schemes of G_mot(T_1(k)), correlated one to each other by inclusions and projections. Moreover, given a 1-motive M, we compute explicitly the biggest Tannakian subcategory of the one generated by M, whose fundamental group is commutative.
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Le radical unipotent du groupe de Galois motivique d'un 1-motif
Let M be a 1-motive defined over a field of characteristic 0. To M we can associate its motivic Galois group, G_mot(M), which is the geometrical interpretation of the Munford-Tate group of M. We prove that the unipotent radical of the Lie algebra of G_mot(M) is the semi-abelian variety defined by the adjoint action of the semi-simplified of the Lie algebra of G_mot(M) on itself.
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