arXiv:0801.3153 Date: Mon, 21 Jan 2008 09:39:27 GMT (19kb)
Title: Remarks on 1-motivic sheaves Authors: Alessandra Bertapelle Categories: math.Algebraic Geometry Comments: 20 pages MSC-class: 14F42; 18E30 \ We generalize to perfect fields (without inverting the exponential characteristic) the construction of the category of 1-motives with torsion (introduced by L. Barbieri-Viale, A. Rosenschon and M. Saito in 2003) as well as the construction of the category of 1-motivic sheaves defined in a recent paper by L. Barbieri-Viale and B. Kahn. We extend a result of the latter paper showing that and have equivalent bounded derived categories. Over a field of characteristic zero, the previous constructions work also for Laumon 1-motives, i.e. allowing additive factors and formal -groups.
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AG (Algebraic geometry)
Mixed
Email from Thomas Riepe:
I found this very good for motivation when I wondered wha 1-motives shall be interesting: http://de.arxiv.org/abs/math/0102033 here a survey: http://de.arxiv.org/abs/math/0502476 Further, Cristiana Bertolin wrote very interesting looking texts and Frederic Paugam has an unpublished preprint on 1-motives and motivic monodromy on his website. Frederics remark on log motives made me curious about them - he thinks it has been the unpublished idea of Fontaine behind his modules (but then, in the 1980’s, motives were probably more SF than science?)
Barbieri-Viale: On the theory of 1-motives. Survey in Nagel-Peters.
Niranjan Ramachandran, From Jacobians to one-motives: exposition of a conjecture of Deligne (215–234) (have a paper copy)
Note to self: I have a pile of other references from the study group in Bordeaux.
1-motives were introduced by Deligne in his seminal Hodge III paper. The category of 1-motives is roughly the category of those mixed motives which arise from curves (not necessarily smooth or projective) and from abelian varieties.
See also Mixed motives, Laumon 1-motives
Nori motives, see reference below.
Various recent papers of Barbieri-Viale are available on his webpage
Deligne: Hodge III
arXiv:1206.5923 Nori 1-motives from arXiv Front: math.AG by J. Ayoub, L. Barbieri-Viale 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.
arXiv:1009.1900 On the derived category of 1-motives from arXiv Front: math.AG by Luca Barbieri-Viale, Bruno Kahn This is the final version of the 2007 preprint titled “On the derived category of 1-motives, I”. It has been substantially expanded to contain a motivic proof of (two thirds of) Deligne’s conjecture on 1-motives with rational coefficients, hence the new title. Compared to the 2007 preprint, the additions mainly concern an abstract theory of realisations with weight filtrations; Deligne’s conjecture is tackled though them by an adjunction game.
http://arxiv.org/abs/0801.3153: Remarks on 1-motivic sheaves, by Bertapelle.
arXiv:1205.6100 Generalized 1-motivic sheaves from arXiv Front: math.AG by Alessandra Bertapelle We extend the construction of the category of 1-motivic sheaves (introduced by Barbieri-Viale and Kahn) allowing quotients of connected algebraic k-groups by formal k-groups. We show that its bounded derived category is equivalent to the bounded derived category of the category of generalized 1-motives with torsion introduced in a previous paper by Barbieri-Viale and the author.
nLab page on 1-motives