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SGA 7, tome 1, exp. IX, contains in its introduction and in section 13.4 remarks about ideas and conjectures of Deligne on a “théorie de Néron pour motifs de poids quelconque”. Would someone please give an epitome of that theory or references? (This is Thomas Riepe's request at the secret blogging seminar, mildly edited.)

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up vote 4 down vote accepted

Deligne commented last year:

""Neron model" is perhaps misleading. It is only the case of unipotent (rather than quasi-unipotent) local monodromy I want to consider. The questions I had in mind were :

-What can one say about a motive about the field of fraction of a discrete valuation ring ; what objects over the residue field can one get (in some/all cohomology theory, or perhaps even motivically ?).

-model : for polarized variations of Hodge or admissible variations of mixed Hodge structures on the punctured disk, nilpotent orbits theorems. This is also the story of tangential base points in my fundamental group of P^1 minus 3 points. One should get motives over the punctured Zariski tangent space. There is work of Ayoub to get motives in a sense better than "system of realizations". How the more precise SL(2) orbit theorem fits in is unclear.

-model (function fields,l-adic) : the relation between monodromy weight filtration and weights (in my Weil II).

-model : Raynaud analytic description of degenerating abelian varieties as, in a rigid analytic category, a quotient by a lattice of an extension of an abelian variety by a torus. The resulting abelian variety over the (complete) field of fractions is an analytic, not algebraic, construct. Modulo any power of the maximal ideal of the discrete valuation ring, it however makes algebraic sense. What to hope for for more general motives is unclear. I wonder about it in [71] of my list of publications.

The classical story of Neron models is about points and extending them to some model. Points make motivic sense (as extensions in a mixed motives category), but I did not have them in mind."

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Thank you! Do you have a digital copy of Deligne's [71]?, i.e., MR1416353 (98a:14015) Deligne, P.(1-IASP) Local behavior of Hodge structures at infinity. (English summary) Mirror symmetry, II, 683--699, AMS/IP Stud. Adv. Math., 1, Amer. Math. Soc., Providence, RI, 1997. – Thanos D. Papaïoannou Oct 27 '09 at 17:49

A new article on generalized Neron models in the arxiv.

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