Monodromy and Log Geometry

TL;DR

This paper explores how log geometry and monodromy can be used to recover topological information of degenerations, providing combinatorial formulas and extending results to Kummer etale topology, especially for curves.

Contribution

It introduces methods to recover topology from log structures in degenerations and derives combinatorial formulas for monodromy and differentials, extending classical results.

Findings

Recovered topology from log special fiber in log smooth degenerations.

Derived combinatorial formulas for monodromy and $d^2$ differentials.

Extended results to Kummer etale topology and curves, linking to dual graphs.

Abstract

A now classical construction due to Kato and Nakayama attaches a topological space (the "Betti realization") to a log scheme over $\mathbf{C}$. 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 etale 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.