Ekedahl–Shepherd-Barron–Taylor/Bost conjecture on algebraicity of leaves from p-curvature

Establish that for a foliation F ⊂ T_{M/R} on a smooth R-scheme M, where R is a finitely generated subalgebra of C, the complex-analytic leaves are analytifications of algebraic subvarieties of M_C if and only if the reduction F mod p is closed under p-th powers for almost all primes p, i.e., F has vanishing p-curvature.

Background

This conjecture extends the p-curvature philosophy from flat connections to algebraic foliations. It posits an equivalence between an arithmetic condition (p-closedness for almost all primes) and a geometric/algebro-analytic conclusion (algebraicity of complex leaves).

The authors prove this conjecture in a non-abelian setting for isomonodromy foliations under additional hypotheses, providing evidence and applications.

References

Conjecture [Ekedahl--Shepherd-Barron--Taylor {Conjecture F, Bost }] Let R\subset \mathbb{C} be a finitely-generated \mathbb{Z}-algebra, and M a smooth R-scheme. Let \mathscr{F}\subset T_{M/R} be a foliation, i.e.~a sub-bundle of Lie algebras. Then the complex-analytic leaves of \mathscr{F} are analytifications of algebraic subvarieties of M_{\mathbb{C}} if and only if \mathscr{F}\bmod p is closed under p-th powers mod p for almost all primes p, i.e.~it has vanishing p-curvature.

— p-Curvature and Non-Abelian Cohomology  (2601.07933 - Lam et al., 12 Jan 2026) in Introduction, Subsection “Main results (applications)”, Conjecture [Ekedahl–Shepherd-Barron–Taylor; Bost]