Holomorphic plane fields with many invariant hypersurfaces

TL;DR

This paper proves that a holomorphic plane field on a compact complex manifold with infinitely many invariant hypersurfaces must be integrable and admit a meromorphic first integral, extending understanding of such geometric structures.

Contribution

It establishes a new criterion linking the abundance of invariant hypersurfaces to the integrability of holomorphic plane fields.

Findings

Infinite invariant hypersurfaces imply the existence of a meromorphic first integral.

Holomorphic plane fields with many invariants are necessarily integrable.

The result applies to non-integrable fields, broadening classical integrability conditions.

Abstract

In this paper we show that a (non necessarily integrable) holomorphic plane field on a compact complex manfold $M$ having an infinite number of invariant hypersurfaces must admit a meromorphic first integral $F:M\longrightarrow \mathbb{C}^{p}$. In particular, it is integrable.