1-motives
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General Introduction
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.
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Search results
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Online References
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 EHM1 generated by the i-th relative homology of pairs of varieties for i = 0,1. We show that EHM1 is naturally equivalent to the abelian category M1 of Deligne 1-motives with torsion; this is our main theorem. Along the way, we obtain several interesting results. Firstly, we realize M1 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.
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Paper References
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.
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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
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.
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Other Information
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?)
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