A combinatorial description of the monodromy of log curves

Abstract

Let \(k\) be an algebraically closed field of characteristic \(0\). For a log curve \(X/k^{\times }\) over the standard log point (Kato in Int J Math 11(2):215–232, 2000), 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 \(X\) (Du Bois in Bull Soc Math Fr 109(1):41–81, 1981). This can be seen as an analogue of the invariant cycles exact sequence for a semistable family (as in the complex, étale 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.