An Algebraic Description of the Monodromy of Log Curves

TL;DR

This paper defines an algebraic combinatorial monodromy operator on the log-de Rham cohomology of log curves over a standard log point, connecting it to classical monodromy and invariant cycles in degenerations.

Contribution

It introduces a new algebraic framework for describing monodromy of log curves, generalizing classical results and enabling explicit computations.

Findings

The invariant part of the monodromy action corresponds to Du Bois cohomology.

The construction recovers classical topological monodromy in complex degenerations.

Allows explicit calculation of monodromy operators in specific cases.

Abstract

Let $k$ be an algebraically closed field of characteristic $0$. For a log curve $X/k^{\times}$ over the standard log point, we define (algebraically) a combinatorial monodromy operator on its log-de Rham cohomology group. The invariant part of this action has a cohomological description, it is the Du Bois cohomology of $X$. This can be seen as an analogue of the invariant cycles exact sequence for a semistable family (as in the complex, \'etale and $p$-adic settings). In the specific case in which $k=\mathbb C$ and $X$ is the central fiber of a semistable degeneration over the complex disc, our construction recovers the topological monodromy and the classical local invariant cycles theorem. In particular, our description allows an explicit computation of the monodromy operator in this setting.