On Albanese and Picard 1-motives with \({\mathbb{G}_a}\)-factors
The author constructs Laumon-1-motives \({{\text{Pic}}^+_a(X), {\text{Alb}}^-_a(X), {\text{Pic}}^-_a(X)}\), and \({{\text{Alb}}^+_a(X)}\) associated to an algebraic variety \(X\) with complete singular locus, whose associated étale (Deligne-)1-motives coincide with Picard and Albanese motives constructed by Barbieri-Viale and Srinivas. The motivation to construct such Laumon-1-motives comes from Barbieri Viale's program whose aim is to find ``sharp'' versions of singular, de Rham, and crystalline (co)homology that conserve more properties of the underlying variety than the classical cohomology theories. The idea is that the ``sharp'' singular (co)homology groups associated to an algebraic variety should be formal Hodge structures whose underlying formal groups are extensions of the ordinary singular (co)homology groups. Using the equivalences of category between formal Hodge structures of level smaller or equal to 1 and the category of Laumon-1-motives, the ``sharp'' singular (co)homology groups of level smaller or equal to 1 could be visualized by Laumon-1-motives.
scientific article; zbMATH DE number 5640856
scientific article