On p-adic invariant cycles theorem

TL;DR

This paper revisits the p-adic invariant cycles theorem for proper semistable curves, extending previous results to cases with non-finite residue fields and exploring the theorem's limitations with unipotent coefficients.

Contribution

It provides a new proof of the invariant cycles theorem without the finite residue field assumption and investigates its failure with unipotent convergent F-isocrystals.

Findings

The invariant cycles theorem holds without the finite residue field hypothesis.

The theorem does not generally hold for unipotent convergent F-isocrystals.

A sufficient condition for the theorem's non-exactness is provided.

Abstract

For a proper semistable curve $X$ over a DVR of mixed characteristics we reprove the "invariant cycles theorem" with trivial coefficients (see Chiarellotto, 1999) 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 $\cal V$ 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 Coleman and Iovita (2010). 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.