  For a proper semistable curve $X$ over a DVR of mixed characteristics we
reprove the "invariant cycles theorem" with trivial coefficients (see
Chiarellotto, 1999) i.e. that the group of elements annihilated by the
monodromy operator on the first de Rham cohomology group of the generic fiber
of $X$ coincides with the first rigid cohomology group of its special fiber,
without the hypothesis that the residue field of $\cal V$ is finite. This is
done using the explicit description of the monodromy operator on the de Rham
cohomology of the generic fiber of $X$ with coefficients convergent
$F$-isocrystals given in Coleman and Iovita (2010). We apply these ideas to the
case where the coefficients are unipotent convergent $F$-isocrystals defined on
the special fiber (without log-structure): we show that the invariant cycles
theorem does not hold in general in this setting. Moreover we give a sufficient
condition for the non exactness.
