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Réduction semi-stable et décomposition de complexes de de Rham à coefficients. (Semi-stable reduction and decomposition of de Rham complexes with coefficients). (French) Zbl 0708.14014
The author uses the techniques from a paper by P.Deligne and J. Illusie [Invent. Math. 89, 247-270 (1987; Zbl 0632.14017)] to treat semi-stable reduction situations. As corollaries he gets algebraic proofs for degeneracy and vanishing results due to Saito, Steenbrink, Viehweg and Zucker.
Reviewer: A.Dimca

MSC:
14F40 de Rham cohomology and algebraic geometry
14F17 Vanishing theorems in algebraic geometry
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[1] P. Berthelot, Cohomologie cristalline des schémas de caractéristique \(p>0\) , SLN in Math., vol. 407, Springer-Verlag, Berlin, 1974. · Zbl 0298.14012
[2] J. A. Carlson and P. A. Griffiths, Infinitesimal variations of Hodge structure and the global Torelli problem , Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, pp. 51-76. · Zbl 0479.14007
[3] P. Deligne, Équations différentielles à points singuliers réguliers , Springer-Verlag, Berlin, 1970. · Zbl 0244.14004 · doi:10.1007/BFb0061194
[4] P. Deligne, Théorie de Hodge. II , Inst. Hautes Études Sci. Publ. Math. (1971), no. 40, 5-57. · Zbl 0219.14007 · doi:10.1007/BF02684692 · numdam:PMIHES_1971__40__5_0 · eudml:103914
[5] P. Deligne and L. Illusie, Cristaux ordinaires et coordonnées canoniques , Algebraic surfaces (Orsay, 1976-78), Lecture Notes in Math., vol. 868, Springer, Berlin, 1981, pp. 80-137. · Zbl 0537.14012
[6] P. Deligne and L. Illusie, Relèvements modulo \(p^ 2\) et décomposition du complexe de de Rham , Invent. Math. 89 (1987), no. 2, 247-270. · Zbl 0632.14017 · doi:10.1007/BF01389078 · eudml:143480
[7] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus , Inst. Hautes Études Sci. Publ. Math. (1969), no. 36, 75-109. · Zbl 0181.48803 · doi:10.1007/BF02684599 · numdam:PMIHES_1969__36__75_0 · eudml:103899
[8] J.-M. Fontaine and W. Messing, \(p\)-adic periods and \(p\)-adic étale cohomology , Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 179-207. · Zbl 0632.14016
[9] O. Hyodo, A note on \(p\)-adic étale cohomology in the semistable reduction case , Invent. Math. 91 (1988), no. 3, 543-557. · Zbl 0619.14013 · doi:10.1007/BF01388786 · eudml:143554
[10] O. Hyodo, On the de Rham-Witt complex attached to a semi-stable family , preprint, 1988. · Zbl 0742.14015 · numdam:CM_1991__78_3_241_0 · eudml:90090
[11] 1 L. Illusie, Complexe cotangent et déformations. I , Springer-Verlag, Berlin, 1971. · Zbl 0224.13014 · doi:10.1007/BFb0059052
[12] 2 L. Illusie, Complexe cotangent et déformations II , Springer-Verlag, Berlin, 1972. · Zbl 0238.13017 · doi:10.1007/BFb0059573
[13] K. Kato, On \(p\)-adic vanishing cycles (Applications of ideas of Fontaine-Messing) , preprint, 1985. · Zbl 0645.14009
[14] K. Kato, \(p\)-adic étale cohomology in the semi-stable reduction case , preprint, 1988.
[15] N. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin , Inst. Hautes Études Sci. Publ. Math. (1970), no. 39, 175-232. · Zbl 0221.14007 · doi:10.1007/BF02684688 · numdam:PMIHES_1970__39__175_0 · eudml:103909
[16] N. Katz, The regularity theorem in algebraic geometry , Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, Paris, 1971, pp. 437-443. · Zbl 0235.14006
[17] N. Katz, Algebraic solutions of differential equations (\(p\)-curvature and the Hodge filtration) , Invent. Math. 18 (1972), 1-118. · Zbl 0278.14004 · doi:10.1007/BF01389714 · eudml:142177
[18] N. Katz, Travaux de Dwork , Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 409, Springer, Berlin, 1973, 167-200. Lecture Notes in Math., Vol. 317. · Zbl 0259.14007 · numdam:SB_1971-1972__14__167_0 · eudml:109810
[19] N. Katz and T. Oda, On the differentiation of de Rham cohomology classes with respect to parameters , J. Math. Kyoto Univ. 8 (1968), 199-213. · Zbl 0165.54802
[20] P. May, Simplicial Objects in Algebraic Topology , Van Nostrand Mathematical Studies, No. 11, Van Nostrand, 1967. · Zbl 0165.26004
[21] D. Mumford, Semi-stable reduction , Toroidal Embeddings I, SLN in Math., vol. 339, 1973, pp. 53-108.
[22] A. Ogus, \(F\)-crystals and Griffiths transversality , Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, Tokyo, 1978, pp. 15-44. · Zbl 0427.14007
[23] M. Saito, Modules de Hodge Polarisables , vol. 553, Pub. RIMS, 1986. · Zbl 0691.14007
[24] M. Saito, Mixed Hodge Modules , vol. 585, Pub. RIMS, 1987. · Zbl 0727.14004
[25] M. Saito, Introduction to Mixed Hodge Modules , vol. 605, Pub. RIMS, 1987. · Zbl 0753.32004
[26] J. Steenbrink, Limits of Hodge structures , Invent. Math. 31 (1975/76), no. 3, 229-257. · Zbl 0303.14002 · doi:10.1007/BF01403146 · eudml:142362
[27] J. Steenbrink, Mixed Hodge structure on the vanishing cohomology , Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 525-563. · Zbl 0373.14007
[28] S. Zucker, Hodge theory with degenerating coefficients. \(L_2\) cohomology in the Poincaré metric , Ann. of Math. (2) 109 (1979), no. 3, 415-476. JSTOR: · Zbl 0446.14002 · doi:10.2307/1971221 · links.jstor.org
[29] H. Esnault and E. Viehweg, Logarithmic de Rham complexes and vanishing theorems , Invent. Math. 86 (1986), no. 1, 161-194. · Zbl 0603.32006 · doi:10.1007/BF01391499 · eudml:143391
[30] G. Faltings, Crystalline cohomology and \(p\)-adic Galois representations , preprint, Princeton Univ., 1988. · Zbl 0805.14008
[31] G. Faltings, \(F\)-isocrystals on open varieties, results and conjectures , preprint, Princeton Univ., 1988. · Zbl 0736.14004
[32] K. Kato, Logarithmic structures of Fontaine-Illusie , 1988, notes. · Zbl 0776.14004
[33] 1 A. Grothendieck, Éléments de Géométrie Algébrique. IV. Étude locale des schémas et des morphismes de schémas. I , Inst. Hautes Études Sci. Publ. Math. (1964), no. 20, 259, rédigés avec la collaboration de J. Dieudonné. · Zbl 0136.15901 · doi:10.1007/BF02684747 · numdam:PMIHES_1964__20__5_0
[34] 2 A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II , Inst. Hautes Études Sci. Publ. Math. (1965), no. 24, 231, rédigés avec la collaboration de J. Dieudonné. · Zbl 0135.39701 · numdam:PMIHES_1965__24__5_0 · eudml:103851
[35] 3 A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III , Inst. Hautes Études Sci. Publ. Math. (1966), no. 28, 255, rédigés avec la collaboration de J. Dieudonné. · Zbl 0144.19904 · doi:10.1007/BF02684343 · numdam:PMIHES_1966__28__5_0 · eudml:103860
[36] 4 A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV , Inst. Hautes Études Sci. Publ. Math. (1967), no. 32, 361, rédigés avec la collaboration de J. Dieudonné. · Zbl 0153.22301 · numdam:PMIHES_1967__32__5_0 · eudml:103873
[37] P. Berthelot, A. Grothendieck, and L. Illusie, Théorie des intersections et théorème de Riemann-Roch , Séminaire de Géométrie Algégrique du Bois-Marie 1966/67, SLN in Math., vol. 225, Springer-Verlag, Berlin, 1971, xii+700. · Zbl 0218.14001
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