Local meromorphic-integrability conjecture

Establish whether a foliation on a normal variety is analytically p-closed at every closed point if and only if it is meromorphically integrable at every closed point.

Background

This conjecture is the local analogue of the global arithmetic criterion. The paper introduces analytic p-closedness so that the arithmetic condition is invariant under analytic changes of coordinates, and conjectures that it characterizes local meromorphic first integrals.

The paper does not establish the equivalence in full generality; it proves several holomorphic-integrability results under stronger hypotheses, including canonical singularities on surfaces and selected higher-dimensional cases.

References

This leads to the following local conjectures. Let $X$ be a normal variety and let $\mathcal F$ be a foliation on $X$. For every closed point $x\in X$, the foliation $\mathcal F$ is analytically $p$-closed at $x$ if and only if it is meromorphically integrable at $x$.

— Holonomy and Integrability of $p$-Closed Foliations  (2608.26813 - Liu, 27 Aug 2026) in Introduction, Conjecture 2 (labelled \ref{conjecturemeromorphic})