On p-adic invariant cycles theorem

Abstract

For a proper semistable curve X over a DVR of mixed characteristics we reprove the ”invariant cycles theorem” with trivial coefficients (see [Ch99]) 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 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 [CoIo10]. 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.