On germs of holomorphic foliations admitting sectorial first integrals (Q2024074)

Let \(\mathcal{F}\) be a holomorphic foliation in a neighborhood \(U\subset \mathbb{C}^{2}\) of the origin \(0\). Assume that singular set of \(\mathcal{F}\) is \(0\) and that \(\mathcal{F}\) has at most a finite amount of separatrices through it (\(0\) is a non-dicritical singularity). Recall that a separatrix \(\Gamma\) through \(0\) is a germ of analytic irreducible curve which contains \(0\) and such that \(\Gamma\setminus\{0\}\) is contained in a leaf of \(\Gamma\). According to reduction of singularities, after a finite composition of blowing ups \(\pi\) over \(0\), the pull-back foliation \(\pi^{\ast}\mathcal{F}\) only has a finite amount of singularities, which are over the exceptional divisor \(E=\pi^{-1}(0)\). If all these simple singularities have linear part with two eigenvalues different from zero (non-degenerate singularities), \(\mathcal {F}\) is said to be a non-dicritical generalized curve. Here the authors mainly study a non-dicritical generalized curve \(\mathcal{F}\) when for a given separatrix \(\Gamma\) through \(0\) and any transverse disk \(\Sigma_{p}\subset U\) to \(\mathcal{F}\) with \(\Sigma_{p}\cap \Gamma=\{p\}:\) (1) There exists a sector \(S\subset \Sigma_{p}\) with vertex \(p\) and a holomorphic function \(\varphi:S\to\mathbb{C}\) constant on the sets \(L\cap S\) where \(L\) is a leaf of \(\mathcal{F}\) (\(\varphi\) is a sectorial first integral); (2) There exists a formal power series \(\tilde{\varphi}={\sum}_{i=0}^{\infty}a_{j}z^{j} \) such that for any proper subsector \(S'\subset S\) and \(k>0\), there exists a constant \(A_{k}\) verifying that \(|\varphi(z)-{\sum}_{j=0}^{k-1}a_{j}z^{j}|\leq A_{k}{|z|}^{k}\) in \(S'\) (\(\varphi\) admits \(\tilde{\varphi}\) as asymptotic expansion). They prove in the main result (Theorem A) that such a \(\mathcal{F}\) always admits a holomorphic first integral in a neighbourhood of \(0\in\mathbb{C}^{2}\). To prove it, they first consider the simplest case where \(\mathcal{F}\) has a non-degenerate simple singularity at \(0\) and show that \(\mathcal{F}\) has a holomorphic first integral of the form \(x^m y^n\), with \(m\) and \(n\) positive integers. It allows them to establish the existence of a sectorial first integral for any sector with a vertex on \(E\) and apply the results of [\textit{J. F. Mattei} and \textit{R. Moussu}, Ann. Sci. Éc. Norm. Supér. (4) 13, 469--523 (1980; Zbl 0458.32005)] to prove the existence of a first holomorphic first integral for \(\pi^{\ast}\mathcal{F}\) in a neighbourhood of \(E\) due to the finiteness of the holonomy groups of its components. The authors also study foliations with a saddle-node singularity at \(0\) (the other case of simple singularity) and establish the presence of a sectorial first integral. This work is a of interest for specialists in local theory of holomorphic foliations in \(\mathbb{C}^{2}\). It contains many nice examples and applications.