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

1-motivic sheaves and the Albanese functor

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 sheave

Albanese and Picard 1-motives

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 Not

Tensor product of motives via K\"unneth formula

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 JPA

The Neron-Severi group of a proper seminormal complex variety

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