Let X be a smooth projective complex variety and let F be a foliation on X. Assume that, for almost all primes p, the reduction of F modulo p is p-closed, meaning that its tangent sheaf is closed under the restricted p-th power operation on vector fields. If F admits a compact leaf, then F is algebraically integrable.

Ekedahl–Shepherd-Barron–Taylor conjecture

The conjecture links arithmetic $p$ -closedness of foliations with algebraic integrability. No proposer or original date is identified in the retrieved sources.

Known results

July 2025: $p$ -integrability claims equivalence between the conjecture and the Grothendieck–Katz $p$ -curvature conjecture for foliations from integrable connections.
January 2026: p-curvature and non-abelian cohomology proves a non-abelian Katz-type theorem and obtains many new cases, under an additional arithmetic hypothesis not known in general.
April 2026: Non-abelian $p$ -curvature and a non-abelian Katz’s formula announces further partial cases, not a full proof.

August 2026 corank-one result

In August 2026, Holonomy and Integrability of $p$ -Closed Foliations claimed the conjecture for corank-one foliations באמצעות a finite-holonomy theorem, relating arithmetic $p$ -closedness to integrability. This is a claimed advance, not a verified resolution of the broader conjecture.

Current status (as of August 2026): The corank-one case is claimed solved but remains unverified; the broader Ekedahl–Shepherd-Barron–Taylor conjecture remains open.

Ekedahl–Shepherd-Barron–Taylor conjecture

Primary source

Additional references

Progress summary

Solutions 0