Signs in weight spectral sequences, monodromy-weight conjectures, log Hodge symmetry and degenerations of surfaces

Nakkajima, Yukiyoshi

Nakkajima, Yukiyoshi

Rendiconti del Seminario Matematico della Università di Padova, Tome 116 (2006), pp. 71-185.

EuDML MR Zbl | 1 citation dans Numdam

@article{RSMUP_2006__116__71_0,
     author = {Nakkajima, Yukiyoshi},
     title = {Signs in weight spectral sequences, monodromy-weight conjectures, log {Hodge} symmetry and degenerations of surfaces},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {71--185},
     publisher = {Seminario Matematico of the University of Padua},
     volume = {116},
     year = {2006},
     mrnumber = {2287344},
     zbl = {1167.14305},
     language = {en},
     url = {http://www.numdam.org/item/RSMUP_2006__116__71_0/}
}

TY  - JOUR
AU  - Nakkajima, Yukiyoshi
TI  - Signs in weight spectral sequences, monodromy-weight conjectures, log Hodge symmetry and degenerations of surfaces
JO  - Rendiconti del Seminario Matematico della Università di Padova
PY  - 2006
SP  - 71
EP  - 185
VL  - 116
PB  - Seminario Matematico of the University of Padua
UR  - http://www.numdam.org/item/RSMUP_2006__116__71_0/
LA  - en
ID  - RSMUP_2006__116__71_0
ER  -

%0 Journal Article
%A Nakkajima, Yukiyoshi
%T Signs in weight spectral sequences, monodromy-weight conjectures, log Hodge symmetry and degenerations of surfaces
%J Rendiconti del Seminario Matematico della Università di Padova
%D 2006
%P 71-185
%V 116
%I Seminario Matematico of the University of Padua
%U http://www.numdam.org/item/RSMUP_2006__116__71_0/
%G en
%F RSMUP_2006__116__71_0

Nakkajima, Yukiyoshi. Signs in weight spectral sequences, monodromy-weight conjectures, log Hodge symmetry and degenerations of surfaces. Rendiconti del Seminario Matematico della Università di Padova, Tome 116 (2006), pp. 71-185. http://www.numdam.org/item/RSMUP_2006__116__71_0/

Bibliographie
Cité par

[A] M. Artin, Algebraic approximation of structures over complete local rings. IHES Publ. Math., 36 (1969), pp. 23-58. | Numdam | MR | Zbl

[B] P. Berthelot, Cohomologie cristalline des schémas de caractéristique p > 0. Lecture Notes in Math. 407, Springer-Verlag, BerlinNew York (1974). | MR | Zbl

[BBM] P. Berthelot - L. Breen - W. Messing, Théorie de Dieudonné cristalline. II. Lecture Notes in Math. 930, Springer-Verlag (1982). | MR | Zbl

[BGR] S. Bosch - U. Güntzer - R. Remmert, Non-archimedean analysis. Grundlehren der Mathematischen Wissenschaften 261, SpringerVerlag (1984). | MR | Zbl

[BL] S. Bosch - W. Lütkebohmert, Formal and rigid geometry. I. Rigid spaces. Math. Ann. 295 (1993), pp. 291-317. | MR | Zbl

[BO1] P. Berthelot - A. Ogus, Notes on crystalline cohomology. Princeton University Press, University of Tokyo Press (1978). | MR | Zbl

[BO2] P. Berthelot - A. Ogus, F-isocrystals and de Rham cohomology. I. Invent. Math. 72 (1983), pp. 159-199. | MR | Zbl

[Ch] B. Chiarellotto. Rigid cohomology and invariant cycles for a semistable log scheme. Duke Math. J. 97 (1999), pp. 155-169. | MR | Zbl

[Cl] C. H. Clemens, Degeneration of Kähler manifolds. Duke Math. J. 44 (1977), pp. 215-290. | MR | Zbl

[Co] B. Conrad, Grothendieck duality and base change. Lecture Notes in Math. 1750, Springer-Verlag (2000). | MR | Zbl

[CI] R. Coleman- A. Iovita, The Frobenius and monodromy operators for curves and abelian varieties. Duke Math. J. 97 (1999), pp. 171-215. | MR | Zbl

[CL] B. Chiarellotto - B. Le Stum, Sur la pureté de la cohomologie cristalline. C. R. Acad. Sci. Paris, Série I, 326 (1998), pp. 961-963. | MR | Zbl

[D1] P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales. IHES Publ. Math. 35 (1968), pp. 107-126. | Numdam | MR | Zbl

[D2] P. Deligne, Théorie de Hodge, II. IHES Publ. Math. 40 (1971), pp. 5-57. | Numdam | MR | Zbl

[D3] P. Deligne, Théorie de Hodge, III. IHES Publ. Math. 44 (1974), pp. 5-77. | Numdam | MR | Zbl

[D4] P. Deligne, La conjecture de Weil, II. IHES Publ. Math. 52 (1980), pp. 137-252. | Numdam | MR | Zbl

[dJ] A. J. De Jong, Smoothness, semi-stability and alterations. IHES Publ. Math. 83 (1996), pp. 51-93. | Numdam | MR | Zbl

[DI1] P. Deligne - L. Illusie, Classes de Chern cristalline et intersections. Appendice de Relèvement des surfaces K3 en caractéristique nulle. In: Surfaces algébriques, Séminaire de Géométrie algébrique d'Orsay 1976-1978. Lecture Notes in Math. 868, Springer-Verlag (1981), pp. 58-79. | MR | Zbl

[DI2] P. Deligne - L. Illusie, Relèvements modulo p2 et décompositon du complexe de de Rham. Invent. Math. 89 (1987), pp. 247-270. | MR | Zbl

[DR] P. Deligne - M. Rapoport, Les schémas de modules de courbes elliptiques. In: Modular functions of one variables II, Lecture Notes in Math. 349, Springer-Verlag (1973), pp. 143-316. | MR | Zbl

[E] F. El Zein, Théorie de Hodge des cycles évanescents. Ann. Scient. Éc. Norm. Sup. 4e série 19 (1986), pp. 107-184. | Numdam | MR | Zbl

[EGA III-2] A. Grothendieck - J. Dieudonné, Éléments de géométrie algébrique. III-2. IHES Publ. Math. 17 (1963). | Numdam | MR | Zbl

[Fa] G. Faltings, F-isocrystals on open varieties: results and conjectures. The Grothendieck Festschrift II, Progr. Math. 87, Birkhäuser (1990), pp. 219-248. | MR | Zbl

[Fo1] J.-M. Fontaine, Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate. In: Journées de géométrie de algébrique de Rennes III, Astérisque 65 (1979), pp. 3-80. | Numdam | MR | Zbl

[Fo2] J.-M. Fontaine, Représentations p-adiques semi-stables. In: Périodes p-adiques, Astérsique 223 (1994), pp. 113-184. | Numdam | MR | Zbl

[Fr] R. Friedman, Global smoothings of varieties with normal crossings. Ann. of Math. 118 (1983), pp. 75-114. | MR | Zbl

[Fu1] K. Fujiwara, Etale topology and philosophy of log. The report of algebraic geometry symposium at Kinosaki (1990), pp. 116-123 (in Japanese).

[Fu2] K. Fujiwara, A proof of the absolute purity conjecture (after Gabber). Advanced Studies in Pure Math. 36 (2002), Azumino, pp. 153-183. | MR | Zbl

[FK] K. Fujiwara - K. Kato, Logarithmic etale topology theory. Preprint.

[FN] T. Fujisawa - C. Nakayama, Mixed Hodge structures on log deformations. Rend. Sem. Mat. Univ. Padova 110 (2003), pp. 221-268. | Numdam | MR | Zbl

[Gra] H. Grauert, Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen. IHES Publ. Math. 5 (1960), pp. 233-292. | Numdam | MR | Zbl

[Gro] M. Gros, Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique. Mém. Soc. Math. France (N.S.) 21 (1985), p. 87. | Numdam | MR | Zbl

[GH] P. Griffiths - J. Harris, Principles of algebraic geometry. John Wiley Sonc, Inc. (1978). | MR | Zbl

[GN] F. Guillén - V. Navarro Aznar, Sur le théorème local des cycles invariants. Duke Math. J. 61 (1990), pp. 133-155. | MR | Zbl

[Ha1] R. Hartshorne, Residues and duality. Lecture Notes in Math. 20, Springer-Verlag (1966). | MR | Zbl

[Ha2] R. Hartshorne Algebraic geometry. Graduate Texts in Math. 52, Springer-Verlag (1977). | MR | Zbl

[Hy] O. Hyodo, On the de Rham-Witt complex attached to a semi-stable family. Comp. Math. 78 (1991), pp. 241-260. | Numdam | MR | Zbl

[HK] O. Hyodo - K. Kato, Semi-stable reduction and crystalline cohomology with logarithmic poles. Périodes p-adiques (Bures-sur-Yvette, 1988). Astérisque 223 (1994), pp. 221-268. | Numdam | MR | Zbl

[IKN] L. Illusie - K. Kato - C. Nakayama, Quasi-unipotent logarithmic Riemann-Hilbert correspondences. J. Math. Sci. Univ. Tokyo 12 (2005), pp. 1-66. | MR | Zbl

[Il1] L. Illusie, Complexe de de Rham-Witt et cohomologie cristalline. Ann. Scient. Éc. Norm. Sup. 4e série 12 (1979), pp. 501-661. | Numdam | MR | Zbl

[Il2] L. Illusie, Autour du théorème de monodromie locale Périodes p-adiques (Bures-sur-Yvette, 1988). Astérisque 223 (1994), pp. 9-57. | Numdam | MR | Zbl

[Il3] L. Illusie, Logarithmic spacse (according to K. Kato). In: Barsotti symposium in algebraic geometry, V. Cristante and W. Messing (eds), Perspective in Math., Academic Press (1994), pp. 183-203. | MR | Zbl

[Il4] L. Illusie, An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology. In: Cohomologies p-adiques et applications arithmétiques (II), Astérisque 279 (2002), pp. 271-322. | Numdam | MR | Zbl

[It1] T. Ito, Weight-monodromy conjecture for p-adically uniformized varieties. Invent. Math. 159 (2005), pp. 607-656. | MR | Zbl

[It2] T. Ito, Weight-monodromy conjecture over equal characteristic local fields. Amer. J. Math. 127 (2005), pp. 647-658. | MR | Zbl

[Ka] T. Kajiwara, The monodromy-weight conjecture and the hard Lefschetz theorem for H1 of projective log semistable varieties. Preprint.

[Kf] F. Kato, Log smooth deformation theory. T^ohoku Math. J. 48 (1996), pp. 317-354. | MR | Zbl

[Kk1] K. Kato, Logarithmic structures of Fontaine-Illusie. In: Algebraic analysis, geometry, and number theory, Johns Hopkins Univ. Press (1989), pp. 191-224. | MR | Zbl

[Kk2] K. Kato, Logarithmic degeneration and Dieudonné theory. Preprint.

[Kk3] K. Kato, A letter to Y. Nakkajima (in Japanese) (1997).

[Kü] K. Künnemann, Projective regular models for abelian varieties, semistable reduction, and the height paring. Duke Math. J. 95 (1998), pp. 161-212. | MR | Zbl

[Kz] N. Katz, Slope filtration of F-crystals. Astérisque 63, Soc. Math. de France (1979), pp. 113-163. | Numdam | MR | Zbl

[KN] K. Kato - C. Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over C. Kodai Math. J. 22 (1999), pp. 161-186. | MR | Zbl

[KS] M. Kashiwara - P. Schapira, Sheaves on manifolds. Grundlehren der Mathematischen Wissenschaften 292, Springer-Verlag, Berlin (1994). | MR | Zbl

[KwN] Y. Kawamata - Y. Namikawa, Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties. Invent. Math. 118 (1994), pp. 395-409. | MR | Zbl

[Mi] J. S. Milne, Étale cohomology. Princeton University Press (1980). | MR | Zbl

[Mo] A. Mokrane, La suite spectrale des poids en cohomologie de HyodoKato. Duke Math. J. 72 (1993), pp. 301-337. | MR | Zbl

[Nak1] C. Nakayama, Logarithmic étale cohomology. Math. Ann. 308 (1997), pp. 365-404. | MR | Zbl

[Nak2] C. Nakayama, Nearby cycles for log smooth families. Compositio Math. 112 (1998), pp. 45-75. | MR | Zbl

[Nak3] C. Nakayama, Degeneration of l-adic weight spectral sequences. Amer. J. Math. 122 (2000), pp. 721-733. | MR | Zbl

[Nak4] C. Nakayama, A projection formula for log smooth varieties in log étale cohomology. Preprint. | MR | Zbl

[Nakk1] Y. Nakkajima, Liftings of log K3 surfaces and classical log Enriques surfaces in mixed characteristics. J. of Algebraic Geometry 9 (2000), pp. 355-393. | MR | Zbl

[Nakk2] Y. Nakkajima, Monodromies and weight filtrations, and the types of simple normal crossing log surfaces with torsion canonical sheaves. Preprint.

[Nakk3] Y. Nakkajima, p-adic weight spectral sequences of log varieties. To appear in J. Math. Sci. Univ. Tokyo. | MR | Zbl

[Nakk4] Y. Nakkajima, Weight filtrations on log crystalline cohomologies of families of open simple normal crossing log varieties in characteristic p > 0. In preparation.

[Ny] N. O. Nygaard, Closedness of regular 1-forms of algebraic surfaces. Ann. Scient. Éc. Norm. Sup. 4e serie 12 (1979), pp. 33-45. | Numdam | MR | Zbl

[NS] Y. Nakkajima - A. Shiho, Weight filtrations on log crystalline cohomologies of families of open smooth varieties in characteristic p > 0. Preprint. | MR

[O] A. Ogus, A. F-crystals on schemes with constant log structure. Compositio Math. 97 (1995), pp. 187-225. | Numdam | MR | Zbl

[R1] M. Raynaud, Géométrie analytique rigide d'après Tate, Kiehl,... Table ronde d'analyse non archimedienne. Bull. Soc. Math. France, Mémoire 39/40 (1974), pp. 319-327. | Numdam | MR | Zbl

[R2] M. Raynaud, 1-motifs et monodromie géométrique. In: Périodes p-adiques, Seminaire de Bures, 1988. Astérisque 223, Soc. Math. de France (1994), pp. 295-319. | Numdam | MR | Zbl

[RZ] M. Rapoport - T. Zink, Über die lokale Zetafunktion von Shimuravarietäten, Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik. Invent. Math. 68 (1982), pp. 21-101. | MR | Zbl

[SaM] M. Saito, Modules de Hodge polarisables. Publ. Res. Inst. Math. Sci. 24 (1988), pp. 849-995. | MR | Zbl

[SaT] T. Saito, Weight spectral sequences and independence of l. J. of the Inst. of Math. Jussieu. 2 (2003), pp. 1-52. | MR | Zbl

[SGA 1] A. Grothendieck - M. Mme Raynaud, Revêtements étales et groupe fondamental. Lecture Notes in Math. 224, Springer-Verlag (1971). [SGA 41 2] P. DELIGNE, Cohomologie étale. Lecture Notes in Math. 569. Springer-Verlag (1977). | MR

[SGA 4-3] A. Grothendieck - ET AL., Théorie des topos et cohomologie étale des schémas. Lecture Notes in Math. Vol. 305. Springer-Verlag (1973). | MR

[SGA 7-I] A. Grothendieck, Groupes de monodromie en géométrie algébrique. Lecture Notes Math. 288 (1972), Springer-Verlag. | MR

[St1] J. H. M. Steenbrink, Limits of Hodge structures. Invent. Math. 31 (1976), pp. 229-257. | MR | Zbl

[St2] J. H. M. Steenbrink, Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures. Math. Ann. 301 (1995), pp. 105-118. | MR | Zbl

[ST] J.-P. Serre - J. Tate, Good reduction of abelian varieties. Ann. of Math. 88 (1968), pp. 492-517. | MR | Zbl

[SZ] J. H. M. Steenbrink - S. Zucker, Variation of mixed Hodge structure. I. Invent. Math. 80 (1985), pp. 489-542. | MR | Zbl

[Ta] J. Tate, Rigid analytic spaces. Invent. Math. 12 (1971), pp. 257-289. | MR | Zbl

[Ts1] T. Tsuji, p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case. Invent. Math. 137 (1999), pp. 233-411. | MR | Zbl

[Ts2] T, TSUJI, Poincaré duality for logarithmic crystalline cohomology. Compositio Math. 118 (1999), pp. 11-41. | MR | Zbl

[Ue] K. Ueno, Compact rigid analytic spaces with special regard to surfaces. Algebraic geometry, Advanced Studies in Pure Math. 10 (1987), Sendai 1985, 765-794. | MR | Zbl

[Us] S. Usui, Recovery of vanishing cycles by log geometry. Tohoku Math. J. 53 (2001), pp. 1-36. | MR | Zbl

[V] M. Van Der Put, Serre duality for rigid analytic spaces. Indag. Math. (N.S.) 3 (1992), pp. 219-235. | MR | Zbl

[Z] S. Zucker, Variation of mixed Hodge structure. II. Invent. Math. 80 (1985), pp. 543-565. | MR | Zbl