Vol. 2, No. 3, 2020

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Monodromy and log geometry

Piotr Achinger and Arthur Ogus

Vol. 2 (2020), No. 3, 455–534
Abstract

A now classical construction due to Kato and Nakayama attaches a topological space (the “Betti realization”) to a log scheme over $ℂ$. 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.

Keywords
log geometry, monodromy, degeneration, fibration
Mathematical Subject Classification 2010
Primary: 14D05, 14D06, 14F25
Milestones
Received: 7 February 2018
Revised: 30 May 2019
Accepted: 18 June 2019
Published: 9 October 2019
Authors
 Piotr Achinger Instytut Matematyezny PAN Warsawa Poland Arthur Ogus Department of Mathematics University of California Berkeley United States