Weight-monodromy conjecture for certain threefolds in mixed characteristic

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The weight-monodromy conjecture claims the coincidence of the weight filtration and the monodromy filtration, up to shift, on the l-adic étale cohomology of a proper smooth variety over a complete discrete valuation field. Although it has been proved in some cases, the case of dimension ≥ 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.