Degenerations and limit Frobenius structures in rigid cohomology

TL;DR

This paper introduces a new limiting Frobenius structure for degenerations of projective varieties over finite fields, which can be computed effectively and may connect to the cohomology of semistable limits.

Contribution

It defines a novel limiting Frobenius structure in rigid cohomology for degenerations satisfying a p-adic lifting condition, with computational methods and conjectural links to semistable cohomology.

Findings

Constructed explicit examples supporting the conjecture.

Demonstrated effective computability of the limiting Frobenius structure.

Proposed a conjectural relation to the cohomology of semistable limits.

Abstract

We introduce a "limiting Frobenius structure" attached to any degeneration of projective varieties over a finite field of characteristic p which satisfies a p-adic lifting assumption. Our limiting Frobenius structure is shown to be effectively computable in an appropriate sense for a degeneration of projective hypersurfaces. We conjecture that the limiting Frobenius structure relates to the rigid cohomology of a semistable limit of the degeneration through an analogue of the Clemens-Schmidt exact sequence. Our construction is illustrated, and conjecture supported, by a selection of explicit examples.