The weight-monodromy conjecture for proper smooth varieties over mixed-characteristic local fields

Deligne's weight-monodromy conjecture predicts that for a proper smooth variety $X$ over a mixed-characteristic local field $k$ (a finite extension of $\mathbb{Q}_p$), the $\ell$-adic Galois representation on each cohomology group $H^i(X_{\bar{k}},\bar{\mathbb{Q}}_\ell)$, $\ell\neq p$, is pure: the eigenvalues of a geometric Frobenius on the graded pieces of the monodromy filtration are Weil numbers of weight $i+j$. In the paper that introduced perfectoid spaces, Scholze proved the conjecture for varieties that are set-theoretic complete intersections in projective smooth toric varieties, by tilting to characteristic $p$ and reducing to Deligne's equicharacteristic theorem, while stating that in mixed characteristic the conjecture is wide open. The perfectoid method has since been extended to complete intersections in abelian varieties, and $p$-adic-coefficient, torsion, local, and fundamental-group analogues of the conjecture are known; the conjecture has also been reduced, via the cohomology of semistable Lefschetz pencils, to an arithmetic variant of a conjecture of Kashiwara for the projective line (Esnault–Kerz). No general proof is known: the conjecture for all proper smooth varieties over mixed-characteristic local fields (open core: dimension at least $3$, arbitrary bad-reduction geometry) remains open.

Background

Let $k$ be a mixed-characteristic local field: a field complete with respect to a discrete valuation, of characteristic $0$, whose residue field $\mathbb{F}_q$ is finite of characteristic $p$ (equivalently, a finite extension of $\mathbb{Q}_p$). Fix a separable closure $\bar{k}$ and write $G_k=\mathrm{Gal}(\bar{k}/k)$. For a prime $\ell\neq p$ and a proper smooth variety $X$ over $k$, the $\ell$-adic étale cohomology $V=H^i(X_{\bar{k}},\bar{\mathbb{Q}}_\ell)$ carries a continuous action of $G_k$. When $X$ has good reduction, the Weil conjectures (Deligne's Weil I) imply that eigenvalues of a geometric Frobenius element — a lift to $G_k$ of the map $x\mapsto x^{1/q}$ on the residue field — on $H^i$ are Weil numbers of weight $i$: algebraic numbers $\alpha$ with $|\alpha|=q^{i/2}$ for every complex absolute value.

For arbitrary (bad) reduction, Grothendieck's local monodromy theorem states that the inertia subgroup $I_k\subset G_k$ acts quasi-unipotently on $V$ — that is, some open subgroup of $I_k$ acts unipotently, as the exponential of a nilpotent operator; this produces a nilpotent monodromy operator $N:V\to V(-1)$, where $V(-1)$ is the Tate twist (tensoring with the dual of the $\ell$-adic cyclotomic character), satisfying $N\Phi=q\Phi N$ for any geometric Frobenius $\Phi$. The operator $N$ determines an increasing monodromy filtration $\mathrm{Fil}^N_j V$, characterized by $N(\mathrm{Fil}^N_j V)\subset \mathrm{Fil}^N_{j-2}V(-1)$ and $\mathrm{gr}^N_j V\simeq \mathrm{gr}^N_{-j}V(-j)$ for $j\geq 0$ via $N$.

Deligne's weight-monodromy conjecture (formulated in his Weil II) asserts that purity survives bad reduction: for all $i$ and $j$, the eigenvalues of $\Phi$ on $\mathrm{gr}^N_j H^i(X_{\bar{k}},\bar{\mathbb{Q}}_\ell)$ are Weil numbers of weight $i+j$. Equivalently, when $X$ admits a semistable model — a proper flat model over the ring of integers of $k$ whose special fiber is a divisor with normal crossings — the monodromy filtration on $H^i$ should coincide, up to shift, with the weight filtration. Deligne proved the conjecture for equal-characteristic local fields, originally under a simplifying assumption on the field of definition of the variety that was later removed (see Ito's equal-characteristic paper, arXiv:math/0308141); the mixed-characteristic case is the famous open part.

Known mixed-characteristic cases at the time of the source paper: all varieties of dimension at most $2$ (the semistable case via the Rapoport–Zink weight spectral sequence — a spectral sequence from the cohomology of the strata of the special fiber of a semistable model converging to $H^*(X_{\bar{k}})$ — combined with de Jong's alterations, which replace $X$ by a generically finite cover admitting semistable reduction); certain threefolds whose special-fiber components have maximal Picard number (Ito); varieties $p$-adically uniformized by Drinfeld upper half spaces — rigid-analytic analogues of complex upper half spaces whose quotient by a suitable arithmetic lattice is the variety (Ito); and several classes of Shimura varieties — moduli spaces of arithmetic origin whose cohomology is accessible through automorphic forms — handled by automorphic methods, as recounted in Scholze's introduction. The source paper added the class of geometrically connected proper smooth varieties that are set-theoretic complete intersections in a projective smooth toric variety — that is, varieties whose underlying set is cut out in the toric variety (a normal variety with a dense algebraic torus action, with geometry controlled by combinatorial data) by exactly as many equations as their codimension.

Scholze's method introduced perfectoid spaces — a class of analytic adic spaces that are, informally, perfect modulo $p$: the $p$-th power map gives an automorphism of the structure sheaf after reduction modulo $p$ — and the tilting construction, which functorially exchanges mixed-characteristic and characteristic-$p$ perfectoid spaces while preserving the étale topos, and hence realizes the Fontaine–Wintenberger isomorphism of absolute Galois groups. Passing to the inverse limit of projective space under the $p$-th power map on coordinates produces a perfectoid projective space together with a Galois-equivariant projection from its tilt; the conjecture then reduces to Deligne's equicharacteristic theorem, provided the (highly transcendental, 'fractal') preimage of $X$ can be approximated by an algebraic variety with the same $\ell$-adic cohomology. Such an approximation is available for toric ambient spaces, and later for abelian ones, but not for arbitrary $X$; this is precisely where the method stops and the general conjecture remains open.

Problem Statement

Prove Deligne's weight-monodromy conjecture in the following generality, with no restrictions on dimension, reduction type, or geometry beyond properness and smoothness. Let $k$ be a mixed-characteristic local field (a finite extension of $\mathbb{Q}_p$) with residue field $\mathbb{F}_q$ of characteristic $p$; let $X$ be a proper smooth variety over $k$; fix a prime $\ell\neq p$ and set $V=H^i(X_{\bar{k}},\bar{\mathbb{Q}}_\ell)$ with its continuous $G_k=\mathrm{Gal}(\bar{k}/k)$-action, monodromy operator $N$, and monodromy filtration $\mathrm{Fil}^N_\bullet V$. The claim to be established is: for every $i$, every $j$, and every geometric Frobenius $\Phi\in G_k$, each eigenvalue of $\Phi$ acting on $\mathrm{gr}^N_j V$ is a Weil number of weight $i+j$, i.e. an algebraic number $\alpha$ with $|\alpha|=q^{(i+j)/2}$ for every complex absolute value. All currently known classes — varieties of dimension at most $2$; set-theoretic complete intersections in projective smooth toric varieties or in abelian varieties; $p$-adically uniformized varieties; certain threefolds with maximal Picard number; and the automorphic cases — fail to cover the general case, so the open core is the conjecture for arbitrary proper smooth varieties of dimension at least $3$ over finite extensions of $\mathbb{Q}_p$.

References

1. Peter Scholze, Perfectoid spaces, arXiv:1111.4914; DOI 10.1007/s10240-012-0042-x (Publications Mathématiques de l'IHÉS 116 (2012) 245–313), https://arxiv.org/abs/1111.4914
2. Pierre Deligne, La conjecture de Weil : II, DOI 10.1007/bf02684780 (Publications Mathématiques de l'IHÉS 52 (1980) 137–252), https://doi.org/10.1007/bf02684780
3. A. J. de Jong, Smoothness, semi-stability and alterations, DOI 10.1007/bf02698644 (Publications Mathématiques de l'IHÉS 83 (1996) 51–93), https://doi.org/10.1007/bf02698644
4. Michael Rapoport, Thomas Zink, Period Spaces for "p"-divisible Groups (AM-141), Annals of Mathematics Studies 141, Princeton University Press, 1996, DOI 10.1515/9781400882601, https://doi.org/10.1515/9781400882601
5. Tetsushi Ito, Weight-monodromy conjecture for certain threefolds in mixed characteristic, arXiv:math/0212109, https://arxiv.org/abs/math/0212109
6. Tetsushi Ito, Weight-monodromy conjecture for p-adically uniformized varieties, DOI 10.1007/s00222-004-0395-y (Inventiones mathematicae 159 (2004) 607–656), arXiv:math/0301201, https://arxiv.org/abs/math/0301201
7. Tetsushi Ito, Weight-monodromy conjecture over equal characteristic local fields, arXiv:math/0308141, https://arxiv.org/abs/math/0308141
8. Yoichi Mieda, Note on weight-monodromy conjecture for p-adically uniformized varieties, arXiv:1411.5959, https://arxiv.org/abs/1411.5959
9. Peter Scholze, p-adic Hodge theory for rigid-analytic varieties, arXiv:1205.3463; DOI 10.1017/fmp.2013.1 (Forum of Mathematics, Pi 1 (2013) e1), https://arxiv.org/abs/1205.3463
10. Bhargav Bhatt, Matthew Morrow, Peter Scholze, Integral p-adic Hodge theory, arXiv:1602.03148; DOI 10.1007/s10240-019-00102-z (Publications Mathématiques de l'IHÉS 128 (2018) 219–397), https://arxiv.org/abs/1602.03148
11. Bhargav Bhatt, Peter Scholze, Prisms and prismatic cohomology, arXiv:1905.08229; DOI 10.4007/annals.2022.196.3.5 (Annals of Mathematics 196 (2022), no. 3), https://arxiv.org/abs/1905.08229
12. L. Alexander Betts, Daniel Litt, Semisimplicity and weight-monodromy for fundamental groups, arXiv:1912.02167, https://arxiv.org/abs/1912.02167
13. Kazuhiro Ito, On a torsion analogue of the weight-monodromy conjecture, arXiv:2008.11905, https://arxiv.org/abs/2008.11905
14. Federico Binda, Hiroki Kato, Alberto Vezzani, On the p-adic weight-monodromy conjecture for complete intersections in toric varieties, arXiv:2207.00369, https://arxiv.org/abs/2207.00369
15. Peter Wear, Perfectoid covers of abelian varieties and the weight-monodromy conjecture, arXiv:2303.05610, https://arxiv.org/abs/2303.05610
16. Hélène Esnault, Moritz Kerz, Semistable Lefschetz pencils, arXiv:2311.15886; DOI 10.1515/crelle-2025-0084 (Journal für die reine und angewandte Mathematik, 2025), https://arxiv.org/abs/2311.15886
17. David Hansen, Bogdan Zavyalov, Arithmetic properties of ℓ-adic étale cohomology and nearby cycles of rigid-analytic spaces, arXiv:2301.01800, https://arxiv.org/abs/2301.01800
18. Kazuya Kato, Chikara Nakayama, Sampei Usui, Logarithmic Tate conjectures over finite fields, arXiv:2502.16974, https://arxiv.org/abs/2502.16974
19. Kazuya Kato, Chikara Nakayama, Sampei Usui, Logarithmic geometry and Frobenius, II, arXiv:2511.17019, https://arxiv.org/abs/2511.17019
20. Federico Binda, Alberto Vezzani, A motivic approach to rational p-adic cohomologies, arXiv:2508.16196, https://arxiv.org/abs/2508.16196
21. Alexander Petrov, Bogdan Zavyalov, Formality for rigid-analytic spaces satisfying the weight-monodromy conjecture, arXiv:2607.14517, https://arxiv.org/abs/2607.14517
22. Yoshiaki Yamamura, Weights in étale cohomology over mixed-characteristic local fields and applications to anabelian geometry, arXiv:2608.01089, https://arxiv.org/abs/2608.01089
23. Takeshi Saito, Perfectoid spaces and the weight-monodromy conjecture, after Peter Scholze, DOI 10.1007/s40306-013-0044-x (Acta Mathematica Vietnamica 39 (2014) 55–68), https://doi.org/10.1007/s40306-013-0044-x