The p-adic weight-monodromy conjecture for Hyodo–Kato cohomology

Let $Y$ be a proper smooth algebraic variety over $K$, and let $H^i_{\mathrm{HK}}(Y_{\bar{K}})$ denote its Hyodo–Kato cohomology, equipped with Frobenius, monodromy, and the monodromy filtration $M_\bullet$. For an integer $j$, write $\mathrm{gr}^M_j H^i_{\mathrm{HK}}(Y_{\bar{K}})$ for the $j$-th graded quotient. The p-adic weight-monodromy conjecture. The $i$-th Hyodo–Kato cohomology group $H^i_{\mathrm{HK}}(Y_{\bar{K}})$ is quasi-pure of weight $i$, meaning that $\mathrm{gr}^M_j H^i_{\mathrm{HK}}(Y_{\bar{K}})$ is Frobenius-pure of weight $i+j$. This is the $p$-adic analogue of Deligne's weight-monodromy conjecture for $\ell$-adic cohomology; the paper studies this prediction for varieties arising as complete intersections in toric varieties, while the general assertion is not established by the supplied text.