Weight-monodromy conjecture for certain threefolds in mixed characteristic

TL;DR

This paper proves the weight-monodromy conjecture for certain threefolds in mixed characteristic, using spectral sequences and the Hodge index theorem, advancing understanding in algebraic geometry.

Contribution

It provides a proof of the conjecture for threefolds with specific semistable models and Picard number conditions, extending previous results in mixed characteristic.

Findings

Proved the weight-monodromy conjecture for specific threefolds in mixed characteristic.

Established a $p$-adic analogue using Mokrane's spectral sequence.

Analyzed the weight spectral sequence via the Hodge index theorem.

Abstract

The weight-monodromy conjecture claims the coincidence of the weight filtration and the monodromy filtration, up to shift, on the $l$-adic \'etale cohomology of a proper smooth variety over a complete discrete valuation field. Although it has been proved in some cases, the case of dimension $\geq 3$ in mixed characteristic is still open so far. The aim of this paper is to give a proof of the weight-monodromy conjecture for a threefold which has a projective strictly semistable model such that, for each irreducible component of the special fiber, the Picard number is equal to the second $l$-adic Betti number. Our proof is based on a careful analysis of the weight spectral sequence of Rapoport-Zink by the Hodge index theorem for surfaces. We also prove a $p$-adic analogue by using the weight spectral sequence of Mokrane.