The weight-monodromy conjecture

Let $K$ be a discretely valued field with separable closure $K^{\operatorname{sep}}$, let $X$ be a variety over $K$ with semistable model, and let $E_r^{p,q}$ be its weight spectral sequence. Write $N$ for the monodromy operator on this spectral sequence and on $H^w_{\text{\rm \tiny \'et}}(X_{K^{\operatorname{sep}}},{\mathbb Q}_l)$, and write $W_\bullet$ for the weight filtration and $H^w_{\text{\rm \tiny \'et}}(X_{K^{\operatorname{sep}}},{\mathbb Q}_l)_\bullet$ for the monodromy filtration. Weight-monodromy conjecture. The top nonzero power of the monodromy operator

$$N^r: E_2^{-r,w+r} \rightarrow E_2^{r,w-r}$$

is an isomorphism for all $r,w$. In particular, the weight filtration on $H^i_{\text{\rm \tiny \'et}}(X_{K^{\operatorname{sep}}},{\mathbb Q}_l)$ coincides, up to a shift in degree, with the monodromy filtration; that is,

$$H^w_{\text{\rm \tiny \'et}}(X_{K^{\operatorname{sep}}},{\mathbb Q}_l)_{-r}/H^w_{\text{\rm \tiny \'et}}(X_{K^{\operatorname{sep}}},{\mathbb Q}_l)_{-r-1} \cong W_{w-r} H^w_{\text{\rm \tiny \'et}}(X_{K^{\operatorname{sep}}},{\mathbb Q}_l).$$

This conjecture asserts that the weights visible in the weight spectral sequence are governed by the nilpotent monodromy action on étale cohomology. Its status is not determined by the supplied source context.