<i>p</i>-Adic Hodge theory for open varieties

Go Yamashita

doi:10.1016/j.crma.2011.10.016

Number Theory/Algebraic Geometry

p-Adic Hodge theory for open varieties

Presented by: Jean-Marc Fontaine

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

Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1127-1130.

Abstract
Résumé

This is an announcement of results whose proofs will be published elsewhere: We establish forms of the $C_{st}$ 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 $C_{pst}$ 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 $C_{st}$ 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 $C_{pst}$ 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: 2011-04-25
Accepted: 2011-10-19
Published online: 2011-11-01

DOI: 10.1016/j.crma.2011.10.016

Author's affiliations:

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

@article{CRMATH_2011__349_21-22_1127_0,
     author = {Go Yamashita},
     title = {\protect\emph{p}-Adic {Hodge} theory for open varieties},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1127--1130},
     publisher = {Elsevier},
     volume = {349},
     number = {21-22},
     year = {2011},
     doi = {10.1016/j.crma.2011.10.016},
     language = {en},
}

TY  - JOUR
AU  - Go Yamashita
TI  - p-Adic Hodge theory for open varieties
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 1127
EP  - 1130
VL  - 349
IS  - 21-22
PB  - Elsevier
DO  - 10.1016/j.crma.2011.10.016
LA  - en
ID  - CRMATH_2011__349_21-22_1127_0
ER  -

%0 Journal Article
%A Go Yamashita
%T p-Adic Hodge theory for open varieties
%J Comptes Rendus. Mathématique
%D 2011
%P 1127-1130
%V 349
%N 21-22
%I Elsevier
%R 10.1016/j.crma.2011.10.016
%G en
%F CRMATH_2011__349_21-22_1127_0

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/

References
Cited by

[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

Cited by Sources:

Comments - Policy