Degenerations and limit Frobenius structures in rigid cohomology

Abstract

AbstractWe introduce a ‘limiting Frobenius structure’ attached to any degeneration of projective varieties over a finite field of characteristicpwhich satisfies ap-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.