Tunisian Journal of Mathematics

Monodromy and log geometry

Piotr Achinger and Arthur Ogus

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/tunis.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


A now classical construction due to Kato and Nakayama attaches a topological space (the “Betti realization”) to a log scheme over . We show that in the case of a log smooth degeneration over the standard log disc, this construction allows one to recover the topology of the germ of the family from the log special fiber alone. We go on to give combinatorial formulas for the monodromy and the d 2 differentials acting on the nearby cycle complex in terms of the log structures. We also provide variants of these results for the Kummer étale topology. In the case of curves, these data are essentially equivalent to those encoded by the dual graph of a semistable degeneration, including the monodromy pairing and the Picard–Lefschetz formula.

Article information

Tunisian J. Math., Volume 2, Number 3 (2020), 455-534.

Received: 7 February 2018
Revised: 30 May 2019
Accepted: 18 June 2019
First available in Project Euclid: 13 December 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.) 14D06: Fibrations, degenerations 14F25: Classical real and complex (co)homology

log geometry monodromy degeneration fibration


Achinger, Piotr; Ogus, Arthur. Monodromy and log geometry. Tunisian J. Math. 2 (2020), no. 3, 455--534. doi:10.2140/tunis.2020.2.455. https://projecteuclid.org/euclid.tunis/1576206288

Export citation


  • P. Berthelot, L. Breen, and W. Messing, Théorie de Dieudonné cristalline, II, Lecture Notes in Mathematics 930, Springer, 1982.
  • A. A. Beĭlinson, J. Bernstein, and P. Deligne, “Faisceaux pervers”, pp. 5–171 in Analysis and topology on singular spaces, I (Luminy, France, 1981), Astérisque 100, Soc. Math. France, Paris, 1982.
  • P. Deligne, “Théorie de Hodge, II”, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57.
  • A. Grothendieck, “Eléments de géométrie algébrique, III: Étude cohomologique des faisceaux cohérents, II”, Inst. Hautes Études Sci. Publ. Math. 17 (1963), 5–91.
  • D. Husemoller, Fibre bundles, 3rd ed., Graduate Texts in Mathematics 20, Springer, 1994.
  • L. Illusie, Complexe cotangent et déformations, I, Lecture Notes in Mathematics 239, Springer, 1971.
  • L. Illusie, “Autour du théorème de monodromie locale”, pp. 9–57 in Périodes $p$-adiques (Bures-sur-Yvette, France, 1988), Astérisque 223, 1994.
  • L. Illusie, “An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology”, pp. 271–322 in Cohomologies $p$-adiques et applications arithmétiques, II, edited by P. Berthelot et al., Astérisque 279, Société Mathématique de France, Paris, 2002.
  • L. Illusie, K. Kato, and C. Nakayama, “Quasi-unipotent logarithmic Riemann–Hilbert correspondences”, J. Math. Sci. Univ. Tokyo 12:1 (2005), 1–66.
  • M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren der Math. Wissenschaften 292, Springer, 1990.
  • K. Kato, “Logarithmic structures of Fontaine–Illusie”, pp. 191–224 in Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989.
  • F. Kato, “Log smooth deformation and moduli of log smooth curves”, Internat. J. Math. 11:2 (2000), 215–232.
  • K. Kato and C. Nakayama, “Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over ${\bf C}$”, Kodai Math. J. 22:2 (1999), 161–186.
  • K. Kato and T. Saito, “On the conductor formula of Bloch”, Publ. Math. Inst. Hautes Études Sci. 100 (2004), 5–151.
  • C. Nakayama, “Logarithmic étale cohomology”, Math. Ann. 308:3 (1997), 365–404.
  • C. Nakayama and A. Ogus, “Relative rounding in toric and logarithmic geometry”, Geom. Topol. 14:4 (2010), 2189–2241.
  • W. Nizioł, “Toric singularities: log-blow-ups and global resolutions”, J. Algebraic Geom. 15:1 (2006), 1–29.
  • A. Ogus, “On the logarithmic Riemann–Hilbert correspondence”, Doc. Math. Extra Vol. (2003), 655–724. Kazuya Kato's fiftieth birthday.
  • A. Ogus, Lectures on logarithmic algebraic geometry, Cambridge Studies in Advanced Mathematics 178, Cambridge University Press, 2018.
  • M. Rapoport and T. Zink, “Über die lokale Zetafunktion von Shimuravarietäten: Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik”, Invent. Math. 68:1 (1982), 21–101.
  • T. Saito, “Weight spectral sequences and independence of $\ell$”, J. Inst. Math. Jussieu 2:4 (2003), 583–634.
  • P. Deligne, Cohomologie étale (Séminaire de Géométrie Algébrique du Bois Marie), Lecture Notes in Math. 569, Springer, 1977.
  • A. Grothendieck, Groupes de monodromie en géométrie algébrique, I: Exposés I–II, VI–IX (Séminaire de Géométrie Algébrique du Bois Marie 1967–1969), Lecture Notes in Math. 288, Springer, 1972.
  • P. Deligne and N. Katz, Groupes de monodromie en géométrie algébrique, II: Exposés X–XXII (Séminaire de Géométrie Algébrique du Bois Marie 1967–1969), Lecture Notes in Math. 340, Springer, 1973.
  • J. Steenbrink, “Limits of Hodge structures”, Invent. Math. 31:3 (1975/76), 229–257.
  • J. H. M. Steenbrink, “Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures”, Math. Ann. 301:1 (1995), 105–118.
  • T. Tsuji, “Saturated morphisms of logarithmic schemes”, Tunis. J. Math. 1:2 (2019), 185–220.