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Unique ergodicity for singular holomorphic foliations of $\mathbb{P}^3(\mathbb{C})$ with an invariant plane

Félix Lequen

TL;DR

This work proves a unique ergodicity result for singular holomorphic foliations of $\mathbb{P}^3(\mathbb{C})$ with hyperbolic singularities and an invariant plane, under no foliation cycles. The authors adapt the Deroin–Kleptsyn framework to the singular context by building a Brownian-motion-based transverse dynamics on leaves and deriving negative Lyapunov exponents via Nguy\^en’s integrability, followed by contraction and a similarity principle to propagate ergodic behavior across nearby leaves. The culmination is a unique positive harmonic current directed by the foliation, given (up to scale) by the extension by zero of the Fornæss–Sibony current on the invariant plane, thus generalizing the 2D result of Dinh–Sibony to a 3D projective setting. The results provide a rigorous link between harmonic currents, transverse diffusion, and geometric ergodicity in a high-dimensional holomorphic foliation context, with implications for the statistical behavior of leaves and their Brownian motion.

Abstract

We prove a unique ergodicity theorem for singular holomorphic foliations of $\mathbb{P}^3(\mathbb{C})$ with hyperbolic singularities and with an invariant plane with no foliation cycle, in analogy with a result of Dinh-Sibony concerning unique ergodicity for foliations of $\mathbb{P}^2(\mathbb{C})$ with an invariant line. The proof is dynamical in nature and adapts the work of Deroin-Kleptsyn to a singular context, using the fundamental integrability estimate of Nguyên.

Unique ergodicity for singular holomorphic foliations of $\mathbb{P}^3(\mathbb{C})$ with an invariant plane

TL;DR

This work proves a unique ergodicity result for singular holomorphic foliations of with hyperbolic singularities and an invariant plane, under no foliation cycles. The authors adapt the Deroin–Kleptsyn framework to the singular context by building a Brownian-motion-based transverse dynamics on leaves and deriving negative Lyapunov exponents via Nguy\^en’s integrability, followed by contraction and a similarity principle to propagate ergodic behavior across nearby leaves. The culmination is a unique positive harmonic current directed by the foliation, given (up to scale) by the extension by zero of the Fornæss–Sibony current on the invariant plane, thus generalizing the 2D result of Dinh–Sibony to a 3D projective setting. The results provide a rigorous link between harmonic currents, transverse diffusion, and geometric ergodicity in a high-dimensional holomorphic foliation context, with implications for the statistical behavior of leaves and their Brownian motion.

Abstract

We prove a unique ergodicity theorem for singular holomorphic foliations of with hyperbolic singularities and with an invariant plane with no foliation cycle, in analogy with a result of Dinh-Sibony concerning unique ergodicity for foliations of with an invariant line. The proof is dynamical in nature and adapts the work of Deroin-Kleptsyn to a singular context, using the fundamental integrability estimate of Nguyên.
Paper Structure (28 sections, 53 theorems, 174 equations)

This paper contains 28 sections, 53 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Context and motivation.
  3. Statement and related results
  4. Outline of proof
  5. Organisation of the proof
  6. Background
  7. Notations
  8. Holomorphic foliations by curves
  9. Holonomy
  10. Local model of hyperbolic singularities
  11. Poincaré metric on the leaves
  12. Harmonic currents and foliation cycles
  13. Harmonic measures
  14. Brownian motion on the leaves
  15. Geometry of the foliation
  16. ...and 13 more sections

Key Result

Theorem 1

Let $\mathscr{F}$ be a singular holomorphic foliation by curves on $\mathbb{P}^2(\mathbb{C})$, seen as a compactification of $\mathbb{C}^2$. Assume that all the singularities are hyperbolic and that the only invariant algebraic curve is the line at infinity $L_\infty$. Then there is a unique positiv

Theorems & Definitions (93)

  • Theorem 1: Dinh-Sibony DinhSibony
  • Theorem 2: Fornæss-Sibony
  • Theorem 3: Dinh-Nguyên-Sibony DNSKahler
  • Theorem 4
  • Remark
  • Theorem 5: Geometric Birkhoff theorem
  • Definition 1
  • Theorem 6: Canille Martins-Lins Neto CMLN, Bacher Bacher
  • Theorem 7: Bacher Bacher
  • Definition 2
  • ...and 83 more