Neighborhoods of curves with a prescribed number of foliations

Abstract

Given a connected projective curve $C \subset \mathbb{P}^n$, $n \geq 2$, and an integer $0 \leq \ell \leq n$, we construct an n-dimensional (non compact) complex manifold, obtained as a neighborhood of an embedded copy of $C$, which carries exactly $\ell$ codimension one holomorphic foliations; moreover, every codimension one distribution on it is one of these foliations. We also determine the field of meromorphic functions of these manifolds: it can be prescribed to be $\mathbb{C}$ or a purely transcendental extension of transcendence degree one, and no larger field is possible as soon as the number of foliations is finite. This extends to arbitrary dimension, and refines, previous constructions of neighborhoods of curves in surfaces without foliations or without non-constant meromorphic functions.