Closed meromorphic 1-forms

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, 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.

A residue formula for meromorphic connections and applications to stable sets of foliations

“…In this section, we first recall a residue formula for integrable meromorphic connections on line bundles based on recent survey by Pereira [19]. We will state it in a slightly more general setting, namely, for holomorphic connections defined away from reduced divisors, not assuming their singularities are polar.…”

“…Residue formulae computing the first Chern class of the line bundle in terms of the residues were shown for logarithmic connections on compact complex manifolds in [18] and for integrable meromorphic connections on projective manifolds in [8,Proposition 2.2]. Their extensions to arbitrary complex manifolds were provided recently by Pereira [19,Propositions 3.3 and 3.4], to which we refer the reader for the details.…”

“…[20, Proof of Proposition 7.1]). In Section 3, we offer their proofs (Lemma 3.2 and Theorem 3.5) following the recent survey by Pereira [19].…”

“…The residue formula stated above and Theorem 4.1 allow us to give a different proof of Brunella's conjecture in [2]. Also a residue formula recently given in [19] (see Theorem 3.1) simplifies the proof of the non-existence of real analytic Levi flat with transversely affine Levi foliation in [1]. We discuss these applications of residue formulae in Section 5.…”

“…Acknowledgements. The authors are grateful to Jorge Vitório Pereira for suggesting the extension argument for foliations over exceptional sets, which was crucially used in the proof of Theorem 5.2, and for pointing out his recent survey [19], which helps to improve some results in Section 3. We are also grateful to Stefan Nemirovski for his helpful comments, in particular for pointing out the reference [23].…”