  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.
