On p-adic invariant cycles theorem

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.