The p-adic weight-monodromy conjecture for proper smooth varieties over p-adic fields

Let $X$ be a proper smooth algebraic variety over $\mathbb{C}_p$ defined over $\bar{\mathbb{Q}}_p$. Its Hyodo–Kato cohomology $H^i_{\mathrm{HK}}(X^{\mathrm{an}})$ is a $(\varphi,N)$-module, and quasi-purity of weight $i$ means that each graded quotient of its monodromy filtration is Frobenius-pure of the corresponding shifted weight. The p-adic weight-monodromy conjecture. The $(\varphi,N)$-module $H^i_{\mathrm{HK}}(X^{\mathrm{an}})$ is quasi-pure of weight $i$. This is presented as an equivalent formulation of the weight-monodromy conjecture and is used in the paper to formulate the result for complete intersections in toric varieties; its general status is open in the supplied text.