Comptes Rendus
Number Theory/Algebraic Geometry
p-Adic Hodge theory for open varieties
Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1127-1130.

This is an announcement of results whose proofs will be published elsewhere: We establish forms of the Cst conjecture of Fontaine–Jannsen for proper semistable pairs over a complete discrete valuation ring R of mixed characteristic (0,p) with perfect residue field, and partially properly supported cohomology. We derive the conjecture Cpst for separated K-schemes of finite type, where K is the fraction field of R. The proof is based on the method of syntomic complexes and p-adic vanishing cycles. A new ingredient is the use of hollow log schemes à la Ogus to provide tubular neighborhoods of intersections of components of divisors with normal crossings.

Cette Note annonce des résultats dont les démonstrations seront publiées ailleurs. Ils concernent des formes de la conjecture Cst de Fontaine–Jannsen pour les paires semistables propres sur un anneau de valuation discrète complet R de caractéristique mixte (0,p) à corps résiduel parfait et des groupes de cohomologie partiellement à support propre. On en déduit la conjecture Cpst pour les K-schémas séparés de type fini, où K est le corps des fractions de R. La méthode de démonstration est celle des complexes syntomiques et des cycles évanescents p-adiques. Un nouvel ingrédient est lʼutilisation de log schémas creux à la Ogus, qui fournissent des voisinages tubulaires dʼintersections de composantes de diviseurs à croisements normaux.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2011.10.016
Go Yamashita 1

1 Toyota Central R&D Labs., Inc., 41-1, Aza Yokomichi, Oaza Nagakute, Nagakute-cho, Aichi-gun, Aichi-ken, 480-1192, Japan
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Go Yamashita. p-Adic Hodge theory for open varieties. Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1127-1130. doi : 10.1016/j.crma.2011.10.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.10.016/

[1] L. Berger Représentations p-adiques et équations différentielles, Invent. Math., Volume 148 (2002), pp. 219-286

[2] A.J. de Jong Smoothness, semistability and alterations, Inst. Hautes Etudes Sci. Publ. Math., Volume 83 (1996), pp. 51-93

[3] R. Hartshorne On the De Rham cohomology of algebraic varieties, Inst. Hautes Etudes Sci. Publ. Math., Volume 45 (1975), pp. 5-99

[4] R. Hartshorne Algebraic de Rham cohomology, Manuscripta Math., Volume 7 (1972), pp. 125-140

[5] O. Hyodo; K. Kato Semi-stable reduction and crystalline cohomology with logarithmic poles, Astérisque, Volume 223 (1994), pp. 221-268

[6] K. Kato Logarithmic structures of Fontaine–Illusie, Algebraic Analysis, Geometry, and Number Theory, Johns Hopkins University Press, Baltimore, 1989, pp. 191-224

[7] A. Ogus F-crystals on schemes with constant log structure, Special issue in honor of F. Oort, Comp. Math., Volume 97 (1995), pp. 187-225

[8] T. Tsuji p-Adic étale cohomology and crystalline cohomology in the semistable reduction case, Invent. Math., Volume 137 (1999), pp. 233-411

[9] T. Tsuji Poincaré duality for logarithmic crystalline cohomology, Compositio Math., Volume 118 (1999) no. 1, pp. 11-41

[10] T. Tsuji Semi-stable conjecture of Fontaine–Jannsen: a survey, Astérisque, Volume 279 (2002), pp. 323-370

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