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.Comment: 15 page

T-motives

Considering a (co)homology theory $\mathbb{T}$ on a base category $\mathcal{C}$ as a fragment of a first-order logical theory we here construct an abelian category $\mathcal{A}[\mathbb{T}]$ which is universal with respect to models of $\mathbb{T}$ in abelian categories. Under mild conditions on the base category $\mathcal{C}$, e.g. for the category of algebraic schemes, we get a functor from $\mathcal{C}$ to ${\rm Ch}({\rm Ind}(\mathcal{A}[\mathbb{T}]))$ the category of chain complexes of ind-objects of $\mathcal{A}[\mathbb{T}]$. This functor lifts Nori's motivic functor for algebraic schemes defined over a subfield of the complex numbers. Furthermore, we construct a triangulated functor from $D({\rm Ind}(\mathcal{A}[\mathbb{T}]))$ to Voevodsky's motivic complexes.Comment: Added reference to arXiv:1604.00153 [math.AG

Voevodsky's mixed motives versus Kontsevich's noncommutative mixed motives

Following an insight of Kontsevich, we prove that the quotient of Voevodsky's category of geometric mixed motives DM by the endofunctor -Q(1)[2] embeds fully-faithfully into Kontsevich's category of noncommutative mixed motives KMM. We show also that this embedding is compatible with the one between pure motives. As an application, we obtain a precise relation between the Picard groups Pic(-), the Grothendieck groups, the Schur-finitenss, and the Kimura-finitenss of the categories DM and KMM. In particular, the quotient of Pic(DM) by the subgroup of Tate twists Q(i)[2i] injects into Pic(KMM). Along the way, we relate KMM with Morel-Voevodsky's stable A1-homotopy category, recover the twisted algebraic K-theory of Kahn-Levine from KMM, and extend Elmendorf-Mandell's foundational work on multicategories to a broader setting.Comment: This paper has been withdrawn for further analysi

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.Comment: new introduction

Nori 1-motives

Let EHM be Nori's category of effective homological mixed motives. In this paper, we consider the thick abelian subcategory EHM_1 generated by the i-th relative homology of pairs of varieties for i = 0,1. We show that EHM_1 is naturally equivalent to the abelian category M_1 of Deligne 1-motives with torsion; this is our main theorem. Along the way, we obtain several interesting results. Firstly, we realize M_1 as the universal abelian category obtained, using Nori's formalism, from the Betti representation of an explicit diagram of curves. Secondly, we obtain a conceptual proof of a theorem of Vologodsky on realizations of 1-motives. Thirdly, we verify a conjecture of Deligne on extensions of 1-motives in the category of mixed realizations for those extensions that are effective in Nori's sense

Motives from Diffraction

We look at geometrical and arithmetical patterns created from a finite subset of Z^n by diffracting waves and bipartite graphs. We hope that this can make a link between Motives and the Melting Crystals/Dimer models in String Theory.Comment: 19 pages, 1 ps-picture, latex 2e, to appear in Proceedings of EAGER conference, Leiden, september 200