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 Dcris\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{cris}$$\end{document} of the p-adic étale 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\ell }$$\end{document}-adic étale cohomology for ℓ≠p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\ell }\ne p$$\end{document}. Our main result is that, if X has semistable reduction, this specialization map induces an isomorphism on the slope [0, 1)-part.