Let K be a complete non-archimedean field, let X be a normal projective variety over K, and let L be a line bundle on X. For every continuous metric varphi on L^an, define its plurisubharmonic envelope by P(varphi):=(suppsi∈operatornamePSH(X,L^an):psi≤varphi)^, where ^ denotes upper-semicontinuous regularization. The envelope conjecture asserts that P(varphi) is continuous, and hence is a global non-archimedean plurisubharmonic metric on L.

Non-Archimedean envelope conjecture

The conjecture asks whether a global envelope construction has the expected plurisubharmonicity on projective varieties. Earlier work established it in important special cases, while the general statement remained open.

Known results

Smooth varieties in characteristic zero or dimension at most two: envelope property proved (Boucksom–Jonsson, 2018 and 2022).
The conjecture is equivalent to comparison-map bijectivity for reduced, irreducible, unibranch projective varieties (Xia, 2023).
Continuous-envelope and Monge–Ampère results hold for curves and selected surfaces, with higher-dimensional extensions under resolution assumptions (Gubler, Jell, Künnemann, Martin, 2019).

September 2026 claimed proof

Lyuhui Wu’s preprint On the local theory of non-archimedean psh functions, reported September 23, 2026, develops a local theory and claims the envelope conjecture, subject to a residue-characteristic restriction. The preprint is unrefereed, so this is a claimed resolution rather than an established theorem.

Current status (as of September 2026): The conjecture is claimed solved by Wu’s unrefereed preprint under a residue-characteristic restriction, but independent verification is not recorded; the unrestricted statement remains unconfirmed.

Non-Archimedean envelope conjecture

Primary source

Additional references

Progress summary

Solutions 0