1motives

General Introduction
1motives were introduced by Deligne in his seminal Hodge III paper. The category of 1motives 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 1motives
Nori motives, see reference below.
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Online References
Various recent papers of BarbieriViale are available on his webpage
Deligne: Hodge III
arXiv:1206.5923 Nori 1motives from arXiv Front: math.AG by J. Ayoub, L. BarbieriViale Let EHM be Nori's category of effective homological mixed motives. In this paper, we consider the thick abelian subcategory EHM1 generated by the ith relative homology of pairs of varieties for i = 0,1. We show that EHM1 is naturally equivalent to the abelian category M1 of Deligne 1motives 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 1motives. Thirdly, we verify a conjecture of Deligne on extensions of 1motives in the category of mixed realizations for those extensions that are effective in Nori's sense.
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Paper References
BarbieriViale: On the theory of 1motives. Survey in NagelPeters.
Niranjan Ramachandran, From Jacobians to onemotives: exposition of a conjecture of Deligne (215234) (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 1motives from arXiv Front: math.AG by Luca BarbieriViale, Bruno Kahn This is the final version of the 2007 preprint titled "On the derived category of 1motives, I". It has been substantially expanded to contain a motivic proof of (two thirds of) Deligne's conjecture on 1motives 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 1motivic sheaves, by Bertapelle.
arXiv:1205.6100 Generalized 1motivic sheaves from arXiv Front: math.AG by Alessandra Bertapelle We extend the construction of the category of 1motivic sheaves (introduced by BarbieriViale and Kahn) allowing quotients of connected algebraic kgroups by formal kgroups. We show that its bounded derived category is equivalent to the bounded derived category of the category of generalized 1motives with torsion introduced in a previous paper by BarbieriViale and the author.
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Other Information
Email from Thomas Riepe:
I found this very good for motivation when I wondered wha 1motives 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 1motives 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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