Dice Question Streamline Icon: https://streamlinehq.com

Algebraicity vs. integrality for leaves of algebraic foliations

Establish, for a smooth R-scheme X over a finitely generated Z-algebra R⊂C equipped with a foliation F⊂T_X and a point x∈X(C), the equivalence between (i) algebraicity of the complex leaf of F through x; (ii) integrality of the formal leaf (descent to a finitely generated R-algebra); and (iii) ω(p)-integrality of the formal leaf.

Information Square Streamline Icon: https://streamlinehq.com

Background

This conjecture formulates a global criterion for when an individual (possibly sporadic) leaf of an algebraic foliation is algebraic, extending the analogy of p-curvature from flat bundles to foliations.

It is intended to capture arithmetic signatures of algebraic leaves, including those arising from non-linear isomonodromy flows such as Painlevé VI and Schlesinger systems.

References

Conjecture Let R\subset \mathbb{C} be a finitely-generated \mathbb{Z}-algebra and X a smooth R-scheme equipped with a foliation \mathscr{F}\subset T_X. Let x\in X(\mathbb{C}) be a point. Then the following are equivalent: (1) (algebraicity) The leaf of \mathscr{F}{\mathbb{C} through x is algebraic; (2) (integrality) The formal leaf of \mathscr{F}{\mathbb{C} through x descends to a finitely-generated R-algebra; (3) (\omega(p)-integrality) The formal leaf \mathscr{F}_{\mathbb{C} through x is \omega(p)-integral.

— Algebraicity and integrality of solutions to differential equations (2501.13175 - Lam et al., 22 Jan 2025) in Conjecture 1.3, Section 1.2 (Generalization to foliations)