On p-adic invariant cycles theorem

International audienceFor a proper semistable curve X over a DVR of mixed characteristics and perfect residue field we prove the " invariant cycles theorem " with trivial coefficients, i.e. that the group of elements of the first de Rham cohomology group of the generic fiber of X annihilated by the monodromy operator coincides with the first rigid cohomology group of its special fiber. This is done using an explicit description of the monodromy operator on the de Rham cohomology of the generic fiber of X with coefficients a convergent F-isocrystal (see [11]) and the proof exhibits an interesting interplay between this cohomology group and the com-binatorics of the graph of the reduction of X. The result was proved in a different way in the case the DVR has finite residue field in [4]. We also study the case where the coefficients are unipotent convergent F-isocrystals on the special fiber of X (without log-structure): we show that the invariant cycles theorem does not hold in general in this setting and give a sufficient condition for non-exactness