M. Barbieri Viale: Sharp cohomologies and formal Hodge structures.

Abstract. The concept of formal Hodge structure is pointing to that of "sharp" singular cohomology of a complex algebraic variety (also "sharp" de Rham over a field of zero characteristic and "sharp" crystalline in positive characteristics). Sharp singular cohomology is a formal Hodge structure containing, in the underlying algebraic structure, a formal group which is an extension of ordinary singular cohomology mixed Hodge structure. Following Grothendieck strategy to construct a cohomology we take care of the 1-motivic cohomology via Laumon 1-motives and "sharp" realizations. Also "sharp" mixed motives are expected.