Closed meromorphic 1-forms

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 Kähler manifolds.