On the p-adic local invariant cycle theorem

For a proper, flat, generically smooth scheme X over a complete discrete valuation ring with finite residue field of characteristic p, we construct a specialization morphism from the rigid cohomology of the geometric special fibre to D-cris of the p-adic etale cohomology of the geometric generic fibre, and we make a conjecture ("p-adic local invariant cycle theorem") that describes the behavior of this map for regular X, analogous to the situation in l-adic etale cohomology for l not equal p. Our main result is that, if X has semistable reduction, this specialization map induces an isomorphism on the slope [0, 1)-part