Hypersurfaces Invariant by Pfaff Equations

TL;DR

This paper investigates conditions for the existence of meromorphic first integrals in Pfaff equations of any codimension on complex manifolds and counts invariant hypersurfaces for certain holomorphic foliations.

Contribution

It extends previous results by providing new conditions for integrability and enumerative formulas for invariant hypersurfaces in complex foliations.

Findings

Conditions for meromorphic first integrals in Pfaff equations

Enumeration of invariant hypersurfaces in projective holomorphic foliations

Generalization of prior integrability criteria

Abstract

We present results expressing conditions for the existence of meromorphic first integrals for Pfaff equations of arbitrary codimension, integrable or not, on complex manifolds. These results are in the same vein as previous ones by J-P. Jouanolou and E. Ghys. We also prove an enumerative result counting the number of hypersurfaces invariant by a projective holomorphic foliation with split tangent sheaf.