Meromorphic first integrals: Some extension results (Q1607529)

Let \({\mathcal F}\) be a one-dimensional holomorphic foliation with possible isolated singularities on a complex surface \(M\). Such a foliation is given by a holomorphic vector field near each point of \(M\). By a theorem of \textit{A. Seidenberg} [J. Am. Math. 90, 248-269 (1968; Zbl 0159.33303)], each singularity of the foliation, after finitely many blow-ups, breaks into possible singularities that are either saddle-nodes, or simple (i.e. the ratio of its two eigenvalues at a singular point is in \(\mathbb{C}^*\setminus\mathbb{Q}^+\)). Let \(h\) be a meromorphic first integral of \({\mathcal F}\) on \(M\setminus S\), where \(S\) is a compact, smooth and connected holomorphic curve. The author shows that \(h\) extends to a meromorphic first integral on \(M\), when one of the following occurs: (A) \({\mathcal F}\) has finitely many separatrices through a singularity \(p\in S\), and all singularities in the Seidenberg desingularization at \(p\) are simple. (B) All singularities in the Seidenberg desingularization at each singularity of \({\mathcal F}\) in \(S\) are simple, and \(S\) has negative self-intersection number. (C) With the same first condition as in (B), \(S\) has self-intersection number \(n\geq 0\) and contains at least \(n+1\) the so-called ordinary dicritical singularities. The main ingredient is an extension result of the author: If \(T\) is a separatrix of the foliation \({\mathcal F}\) through a simple singular point \(p\) and \(h\) is a meromorphic first integral of \({\mathcal F}\) in a neighborhood of \(T\setminus\{p\}\) in \(M\), then \({\mathcal F}\) is holomorphically linearizable near \(p\) with resonant eigenvalues and \(h\) extends meromorphically to a neighborhood of \(p\). The author gives examples of non-extendable meromorphic first integrals for some singular foliations. The author also investigates the extendability of meromorphic first integrals for singular holomorphic foliations of codimension one in \(\mathbb{P}^n\) and for one-dimensional singular foliations in complex manifolds of higher dimension.