Closed meromorphic 1-forms

TL;DR

This paper reviews the properties of closed meromorphic 1-forms and their associated foliations, connecting classical foliation results with the theory of meromorphic forms and exploring applications in algebraic and Kähler geometry.

Contribution

It provides a comprehensive overview linking foliation theory with meromorphic 1-forms, including classical results, and applies these insights to algebraic separatrices and Kähler geometry.

Findings

Classical foliation results are explained via meromorphic 1-forms.

Application to algebraic separatrices in semi-global settings.

Insights into the geometry of hypersurfaces with trivial normal bundle.

Abstract

We review properties of closed meromorphic $1$-forms and of the foliations defined by them. We present and explain classical results from foliation theory, like index theorems, the existence of separatrices, and resolution of singularities under the lenses of the theory of closed meromorphic $1$-forms and flat meromorphic connections. We apply the theory to investigate the algebraicity separatrices in a semi-global setting (neighborhood of a compact curve contained in the singular set of the foliation), and the geometry of smooth hypersurfaces with numerically trivial normal bundle on compact K\"ahler manifolds.