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Making a survey of recent constructions of universal cohomologies we suggest a new framework for a theory of motives in algebraic geometry.

We construct a new Weil cohomology for smooth projective varieties over a field, universal among Weil cohomologies with values in rigid additive tensor categories. A similar universal problem for Weil cohomologies with values in rigid abelian tensor categories also has a solution. We give a variant for Weil cohomologies satisfying more axioms, like Weak and Hard Lefschetz. As a consequence, we get a different construction of Andr\'e's category of motives for motivated correspondences and show that it has a universal property. This theory extends over suitable bases., Comment: Appendix on gradings added

We prove that any rigid additive symmetric monoidal category can be mapped to a rigid abelian symmetric monoidal category in a universal way. This yields a novel approach to Grothendieck's standard conjecture D and Voevodsky's smash nilpotence conjecture.

Following Eilenberg-Steenrod axiomatic approach we construct the universal ordinary homology theory for any homological structure on a given category by representing ordinary theories with values in abelian categories. For a convenient category of spaces we then obtain a universal abelian category which can be actually described for CW-complexes as the category of hieratic modules.

We furnish any category of a universal (co)homology theory. Universal (co)homologies and universal relative (co)homologies are obtained by showing representability of certain functors and take values in $R$-linear abelian categories of motivic nature, where $R$ is any commutative unitary ring. Universal homology theory on the one point category yields "hieratic" $R$-modules, i.e. the indization of Freyd's free abelian category on $R$. Grothendieck $\partial$-functors and satellite functors are recovered as certain additive relative homologies on an abelian category for which we also show the existence of universal ones., Comment: References added

Following Nori's original idea we here provide certain motivic categories with a canonical tensor structure. These motivic categories are associated to a cohomological functor on a suitable base category and the tensor structure is induced by the cartesian tensor structure on the base category via a cohomological K\"unneth formula., Comment: Revised version to appear on JPAA

We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective. Showing that a suitable Betti--de Rham realization of 1-motives is fully faithful we can verify this period conjecture in several cases. The divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical Grothendieck period conjecture for algebraic cycles on smooth and proper schemes. These divisibility properties are treated in an appendix by B. Kahn (extending previous work of Bloch and Colliot-Th\'el\`ene--Raskind)., Comment: Appendix by B. Kahn added. Revised version to appear on Adv. Math

After introducing the Ogus realization of 1-motives we prove that it is a fully faithful functor. More precisely, following a framework introduced by Ogus, considering an enriched structure on the de Rham realization of 1-motives over a number field, we show that it yields a full functor by making use of an algebraicity theorem of Bost.

Making use of Freyd's free abelian category on a preadditive category we show that if $T:D\rightarrow \mathcal{A}$ is a representation of a quiver $D$ in an abelian category $\mathcal{A}$ then there is an abelian category $\mathcal{A} (T)$, a faithful exact functor $F_T: \mathcal{A} (T) \to \mathcal{A}$ and an induced representation $\tilde T: D \to \mathcal{A} (T)$ such that $F_T\tilde T= T$ universally. We then can show that $\mathbb{T}$-motives as well as Nori's motives are given by a certain category of functors on definable categories.

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]

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.

We introduce n-generated sheaves and n-motivic sheaves, describing completely for n = 0, 1 and proposing a conjecture for n > 1. We then obtain functors L\pi_0 and LAlb on DM_{eff}(k) deriving \pi_0 and Alb. The functor LAlb extends the one constructed (by the second author jointly with B.Kahn) to non-necessarily geometric motives. These functors are then used to define higher N\'eron-Severi groups and higher Albanese sheaves. The latter may be considered as an algebraic avatar of Deligne (co)homology., Comment: 54 pages, fully revised exposition with a new "structure theorem" for 1-motivic sheaves, see Theorem 1.3.10, including finitely presented (or constructible) 1-motivic sheaves

We introduce the "sharp" (universal) extension of a 1-motive (with additive factors and torsion) over a field of characteristic zero. We define the "sharp de Rham realization" by passing to the Lie-algebra. Over the complex numbers we then show a (sharp de Rham) comparison theorem in the category of formal Hodge structures. For a free 1-motive along with its Cartier dual we get a canonical connection on their sharp extensions yielding a perfect pairing on sharp realizations. We thus provide "one-dimensional sharp de Rham cohomology" of algebraic varieties., Comment: 30 pages

We prove a Lefschetz (1,1)-Theorem for proper seminormal varieties over the complex numbers. The proof is a non-trivial geometric argument applied to the isogeny class of the Lefschetz 1-motive associated to the mixed Hodge structure on H^2., Comment: 16 pages; Mathematische Zeitschrift (2008)

Formal (mixed) Hodge structures FHS are introduced in such a way that the Hodge realization of Deligne's 1-motives extends to a realization from Laumon's 1-motives to formal Hodge structures of level 1, providing an equivalence of categories. For the sake of exposition we here confine our study to level 1 mixed Hodge structures. However, it is conceivable and suitable to consider formal mixed Hodge structures with arbitrary Hodge numbers: generalizing our definition herebelow it's not that difficult and we will treat such a matter nextly., Comment: 12 pages

This short note is an "elementary'' introduction to the conjectural theory of motives., Comment: 19 pages, to appear on Milan Journal of Math

This is an overview and a preview of the theory of "mixed motives of level 1" explaining some results, projects, ideas and indicating a bunch of problems., Comment: 45 pages; updated references & improved exposition

The goal of this paper is to introduce Hodge 1-motives of algebraic varieties and to state a corresponding cohomological Grothendieck-Hodge conjecture, generalizing the classical Hodge conjecture to arbitrarily singular proper schemes., Comment: 30 pages, LaTeX (using AMS symbols)

This paper has been superseded to a great extent by the following: Paper: math.AG/0511558 Title: The Neron-Severi group of a proper seminormal complex variety Authors: L. Barbieri-Viale, A. Rosenschon & V. Srinivas Paper: math.AG/0102150 Title: Deligne's Conjecture on 1-Motives Authors: Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito Journal-ref: Ann. of Math. (2), Vol. 158 (2003), no. 2, 593--633, Comment: This paper has been withdrawn

We describe algebraically defined cohomological and homological Albanese and Picard 1-motives (or mixed motives) of any algebraic variety in characteristic zero, generalizing the classical Albanese and Picard varieties. We compute Hodge, l-adic and De Rham realizations proving Deligne's conjecture for the concerned mixed Hodge structures. We investigate functoriality, universality, homotopical invariance and invariance under formation of projective bundles. We compare our cohomological and homological 1-motives for normal schemes. For proper schemes, we obtain an Abel-Jacobi map from the (Levine-Weibel) Chow group of zero cycles to our cohomological Albanese 1-motive which is the universal regular homomorphism to semi-abelian varieties. By using this universal property we get 'motivic' Gysin maps for projective local complete intersection morphisms. This paper is an extended version of our preliminary Comptes Rendus Note, Academie des Sciences, Paris, Vol. 326, 1998., Comment: 81 pages (428 K dvi file)

We define, in a purely algebraic way, 1-motives $Alb^{+}(X)$, $Alb^{-}(X)$, $Pic^{+}(X)$ and $Pic^{-}(X)$ associated with any algebraic scheme $X$ over an algebraically closed field of characteristic zero. For $X$ over $\C$ of dimension $n$ the Hodge realizations are, respectively, $H^{2n-1}(X)(n)$, $H_{1}(X)$, $H^{1}(X)(1)$ and $H_{2n-1}(X)(1-n)$., Comment: 5 pages, LaTeX, submitted as CR Note

After the work of Bloch and Srinivas on correspondences and algebraic cycles we begin the study of a birational class of algebraic varieties determined by the property that a multiple of the diagonal is rationally equivalent to a cycle supported on proper subschemes., Comment: 12 pages, LaTeX

Let $X$ be a complex projective surface with arbitrary singularities. We construct a generalized Abel--Jacobi map $A_0(X)\to J^2(X)$ and show that it is an isomorphism on torsion subgroups. Here $A_0(X)$ is the appropriate Chow group of smooth 0-cycles of degree 0 on $X$, and $J^2(X)$ is the intermediate Jacobian associated with the mixed Hodge structure on $H^3(X)$. Our result generalizes a theorem of Roitman for smooth surfaces: if $X$ is smooth then the torsion in the usual Chow group $A_0(X)$ is isomorphic to the torsion in the usual Albanese variety $J^2(X)\cong Alb(X)$ by the classical Abel-Jacobi map., Comment: 36 pages, LaTeX

Let $X$ be a non-singular projective complex surface. We can show that Bloch's conjecture (i.e., that if $p_g=0$ then the Albanese kernel vanishes) is equivalent to the following statement: If $p_g(X)=0$ then for any given Zariski open $U\subset X$ and $\omega\in H^2(U,{\bf C})$ there is a smaller Zariski open $V\subset U$ such that $$\omega\mid_V =\omega'+\zeta$$ where $\omega'\in F^2H^2(V,{\bf C})$ and $\zeta$ is integral., Comment: 4 pages, LaTeX