Holonomy and Integrability of $p$-Closed Foliations

Abstract: Motivated by the Ekedahl-Shepherd-Barron-Taylor conjecture, we study arithmetic criteria for the global and local integrability of foliations in characteristic zero. Our main result is a finite-holonomy theorem for the regular parts of invariant prime divisors of corank one foliations which are (p)-closed for almost all primes. As a consequence, we give a complete proof for corank one foliations of the statement proposed by Ekedahl-Shepherd-Barron-Taylor: a foliation on a smooth projective variety which is (p)-closed for almost all primes is algebraically integrable whenever it admits a compact leaf. We also introduce a local analytic notion of (p)-closedness for foliation germs on normal complex varieties and formulate local conjectures relating it to meromorphic and holomorphic first integrals. We establish holomorphic integrability for analytically (p)-closed foliations with canonical singularities on smooth surfaces and on klt surface germs. Combining these local results with the finite-holonomy theorem, we obtain holomorphic first integrals for certain analytically (p)-closed non-dicritical surface germs and for analytically (p)-closed simple singularities in arbitrary dimension. Finally, we formulate a connection-theoretic counterpart to local holomorphic integrability in terms of flat meromorphic extensions of the Bott partial connection, and verify it in dimension two.