Tunisian Journal of Mathematics

Monodromy and log geometry

Piotr Achinger and Arthur Ogus

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Abstract

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

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

Dates
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
https://projecteuclid.org/euclid.tunis/1576206288

Digital Object Identifier
doi:10.2140/tunis.2020.2.455

Mathematical Reviews number (MathSciNet)
MR4041282

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

Keywords
log geometry monodromy degeneration fibration

Citation

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


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