- Work in Progress
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 should be 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 one should take care of the 1-motivic cohomology via Laumon 1-motives and "sharp" realizations. Also "sharp" mixed motives are expected.
The first talk I gave on this subject was on June 16, 2005 (Current Geometry -
The International Conference on problems and trends of contemporary geometry, Naples) However, this project was presented very roughly in a letter to M. Saito on July 1, 2001. I drafted a general theory in a covering letter to A. Beilinson (November 28, 2005, when submitting the paper "Formal Hodge Theory") which is here
A more recent text of a talk (Paris 7, January 10, 2007 & Paris 13, May 18, 2007) on "sharp" theories along with a list of references is
here
Slides for the Brixen Summer School (September 12-17, 2011) on "Cohomology theories" are here
On the relations between Deligne 1-motives and Voevodsky motives we (jointly with B. Kahn) have a long paper titled "On the derived category of 1-motives" The (almost) final version is the arXiv:1009.1900v1 [math.AG] linked here
An abstract theory of realisations with weight filtrations and the proof of Deligne's conjectures (with rational coefficients) via the motivic Albanese functor LAlb is included.
This project started jointly with B. Kahn on February 2002. We consider the category of Deligne1-motives over a perfect field k of exponential characteristic p and its derived category for a suitable exact structure after inverting p. Providing a fully faithful embedding into an étale version of Voevodsky's triangulated category of geometric motives we show that there is "almost" a left adjoint, the motivic Albanese functor LAlb. It is a left adjoint rationally. Composing with motivic Cartier duality, we then obtain a related motivic Picard functor RPic. Applied to the motive of a variety we thus get bounded complexes of 1-motives that provide the 1-motives predicted by Deligne. To prove Deligne's conjectures one has just to make use of weight filtrations on a given realization.
We planned a paper to be split (at least) in two Parts titled "On the derived category of 1-motives: I, II" A preliminary version of Part I was published on line on the
K-Theory archive as a preprint N. 800 (October 5, 2006) & updated to N. 851 (June 11, 2007). In Part I we constructed the functor LAlb and we computed it completely for smooth and to a large extent for
general schemes (in characteristic 0). One application is a new proof of Roitman's torsion theorem and its generalisations: we provide a new motivic version which extends this theorem to any algebraic scheme (in characteristic 0). The proof of Deligne's conjectures via realisations applied to LAlb is an achievement of Part II of our project. Actually, the final version of the 2007 preprint is now titled "On the derived category of 1-motives" & it is Part I substantially expanded to contain Part II. Part I is also available in the math arxiv.org & as an IHÉS prepublication (including an index of notations already) Prépublication Mathématique de l'IHÉS (M/07/22), June 2007, 144 pages.
- Pre-E-Printed
- Printed (from 2001)
Printed & e-printed versions available from journal's homepage.
L. Barbieri-Viale & A. Bertapelle: Sharp
de Rham realization Advances in Mathematics
Vol. 222, Issue 4 (2009) 1308-1338 (DOI http://dx.doi.org/10.1016/j.aim.2009.06.003)
L. Barbieri-Viale:
Formal Hodge Theory Mathematical Research Letters (International Press) Vol. 14 Issue 3 (2007) 385-394.
L. Barbieri-Viale: On
the theory of 1-motives in Algebraic Cycles and Motives London Mathematical Society Lecture Note Series, Vol. 343, Cambridge University Press, London, 2007. (Proceedings of the workshop on the occasion of the 75-th birthday of J.P. Murre - 2004, Lorentz Center, Leiden).