On the $p$-adic local invariant cycle theorem

TL;DR

This paper introduces a $p$-adic local invariant cycle theorem for proper schemes over a DVR, relating rigid and $p$-adic étale cohomology, and proves an isomorphism in the semistable case for certain slopes.

Contribution

It formulates a $p$-adic analogue of the local invariant cycle theorem and proves it for schemes with semistable reduction in a specific slope range.

Findings

Defines a specialization morphism linking rigid and $p$-adic étale cohomology.

Conjectures a $p$-adic local invariant cycle theorem for regular schemes.

Proves the specialization map induces an isomorphism on slope [0,1) part for semistable schemes.

Abstract

For a proper, flat, generically smooth scheme $X$ over a complete DVR with finite residue field of characteristic $p$, we define a specialization morphism from the rigid cohomology of the geometric special fibre to $D_{crys}$ 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 $l\neq p$. Our main result is that, if $X$ has semistable reduction, this specialization map induces an isomorphism on the slope $[0,1)$-part.