Deligne's weight-monodromy conjecture

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. Deligne's weight-monodromy conjecture

   Let $K$ be a $p$-adic local field with residue field of cardinality $q$, let $\bar K$ be an algebraic closure, and let $X$ be a smooth proper $K$-scheme. Its geometric étale cohomology groups $\rm{H}^i(X_{\bar K},\Q_\ell)$ carry the monodromy filtration $\mathrm{Fil}_{\mathrm{M}}^\bullet$. A $q$-Weil number of weight $w$ is an algebraic number whose complex absolute values are $q^{w/2}$. Weight-monodromy conjecture. The eigenvalues of any geometric Frobenius lift on

   $$\operatorname{gr}^j_{\mathrm{M}} \rm{H}^i(X_{\bar K},\Q_\ell)$$

   are $q$-Weil numbers of weight $i+j$ for every pair of integers $i,j$. This is Deligne's conjecture, motivated by the analogy with limit mixed Hodge structures; it is a central prediction about the interaction between Frobenius weights and monodromy in the degeneration of smooth proper varieties. The source discusses important cases proved in equal characteristic and by Scholze under a complete-intersection hypothesis, while the general statement remains open.

   source: David Hansen and Bogdan Zavyalov, “Arithmetic Properties Of -adic Étale Cohomology and Nearby Cycles of Rigid-Analytic Spaces”, arXiv:2301.01800 (2025).