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Completely Integrable Foliations: Singular Locus, Invariant Curves and Topological Counterparts

Javier Ribón

TL;DR

This work advances the theory of completely integrable holomorphic foliations with codimension $q\ge 2$ by revealing a topological backbone to CI phenomena: either the singular set is sizable or there are infinitely many separatrices, and such behavior is captured by topological constructs like the total holonomy group. By introducing topological complete integrability (TCI) and developing a robust neighborhood/transverse-section framework, the authors show how to construct holomorphic first integrals in the isolated-separatrix case and prove the existence of dicritical codimension-1 invariant varieties under CI. They also provide a sharp characterization for TCI vector fields, tying CI to diagonal linear models with integer eigenvalues and a finite total holonomy group, thereby linking geometric, topological, and algebraic facets of CI foliations. Overall, the paper connects topological invariants to analytic integrability, yielding new tools to detect and construct first integrals in higher codimension foliations with isolated singularities.

Abstract

We study codimension $q \geq 2$ holomorphic foliations defined in a neighborhood of a point $P$ of a complex manifold that are completely integrable, i.e. with $q$ independent meromorphic first integrals. We show that either $P$ is a regular point, a non-isolated singularity or there are infinitely many invariant analytic varieties through $P$ of the same dimension as the foliation, the so called separatrices. Moreover, we see that this phenomenon is of topological nature. Indeed, we introduce topological counterparts of completely integrable local holomorphic foliations and tools, specially the concept of total holonomy group, to build holomorphic first integrals if they have isolated separatrices. As a result, we provide a topological characterization of completely integrable non-degenerated elementary isolated singularities of vector fields with an isolated separatrix.

Completely Integrable Foliations: Singular Locus, Invariant Curves and Topological Counterparts

TL;DR

This work advances the theory of completely integrable holomorphic foliations with codimension by revealing a topological backbone to CI phenomena: either the singular set is sizable or there are infinitely many separatrices, and such behavior is captured by topological constructs like the total holonomy group. By introducing topological complete integrability (TCI) and developing a robust neighborhood/transverse-section framework, the authors show how to construct holomorphic first integrals in the isolated-separatrix case and prove the existence of dicritical codimension-1 invariant varieties under CI. They also provide a sharp characterization for TCI vector fields, tying CI to diagonal linear models with integer eigenvalues and a finite total holonomy group, thereby linking geometric, topological, and algebraic facets of CI foliations. Overall, the paper connects topological invariants to analytic integrability, yielding new tools to detect and construct first integrals in higher codimension foliations with isolated singularities.

Abstract

We study codimension holomorphic foliations defined in a neighborhood of a point of a complex manifold that are completely integrable, i.e. with independent meromorphic first integrals. We show that either is a regular point, a non-isolated singularity or there are infinitely many invariant analytic varieties through of the same dimension as the foliation, the so called separatrices. Moreover, we see that this phenomenon is of topological nature. Indeed, we introduce topological counterparts of completely integrable local holomorphic foliations and tools, specially the concept of total holonomy group, to build holomorphic first integrals if they have isolated separatrices. As a result, we provide a topological characterization of completely integrable non-degenerated elementary isolated singularities of vector fields with an isolated separatrix.
Paper Structure (21 sections, 26 theorems, 88 equations)

This paper contains 21 sections, 26 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Completely integrable foliations
  3. Closedness of Leaves
  4. Stability of leaves
  5. The singular set and the set of separatrices
  6. Existence of a separatrix
  7. The neighborhood of the set of separatrices
  8. Construction of holomorphic first integrals of ${\mathcal{F}}$
  9. End of the proof of Theorem \ref{['teo:infinite']}
  10. The completely integrable case
  11. Existence of a dicritical hypersurface
  12. The neighborhood of the set of separatrices
  13. Construction of holomorphic first integrals
  14. End of the proof of Theorem \ref{['teo:exist_cod1']}
  15. TCI germs of vector fields
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\mathcal{F}$ be a completely integrable germ of holomorphic foliation of codimension $q \geq 2$ in $\hbox{\rm{(}${\mathbb C}^{n},{\bf 0}$\rm{)}}$ with an isolated singularity at the origin. Then, there are infinitely many invariant analytic varieties of dimension $\dim (\mathcal{F})$ through th

Theorems & Definitions (73)

  • Definition 1
  • Theorem 1
  • Corollary 1
  • Definition 2
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 1
  • proof
  • Corollary 2
  • ...and 63 more