Algebraic separatrices for non-dicritical foliations on projective spaces of dimension at least four

TL;DR

This paper proves that non-dicritical codimension one foliations on high-dimensional projective spaces always possess an invariant algebraic hypersurface, extending Rossi's algebraization results.

Contribution

It establishes the existence of invariant algebraic hypersurfaces for a broad class of foliations on high-dimensional projective spaces, using a strengthened algebraization theorem.

Findings

Non-dicritical foliations on projective spaces of dimension ≥4 have invariant algebraic hypersurfaces.

The proof enhances Rossi's algebraization theorem for analytic subvarieties.

The result applies to a wide class of foliations, broadening understanding of their geometric structure.

Abstract

Non-dicritical codimension one foliations on projective spaces of dimension four or higher always have an invariant algebraic hypersurface. The proof relies on a strengthening of a result by Rossi on the algebraization/continuation of analytic subvarieties of projective spaces.