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The dimension of harmonic currents on foliated complex surfaces

Bertrand Deroin, Christophe Dupont, Victor Kleptsyn

TL;DR

This work analyzes holomorphic foliations with hyperbolic singularities on complex surfaces and constructs a unique directed harmonic current $T$ with finite harmonic measure $\mu=T\wedge\mathrm{vol}_{g_P}$. It derives a transversal dimension formula, showing that the Hausdorff dimension of the restriction of $T$ to transversals equals the entropy-to-Lyapunov ratio $h_D/|\lambda|$, and proves exact dimensionality when holonomy is discrete. The paper extends Brunella’s inequality on $\mathbb{P}^2$, providing an explicit bound $\mathrm{dim}_H(T|_\Sigma)\le (d-1)/(d+2)$ for degree $d\ge 2$, and computes the Jouanolou foliation’s dimension as $1/4$. A key methodological core is a Limit Theorem for leafwise closed 1-forms, combined with a discretization of holonomy, and a detailed entropy analysis linking $h_D$ and $h_L$; in particular, $h_D=h_L$ when the holonomy pseudogroup is discrete on the pseudo-minimal set. The results reveal how dynamical quantities (Furstenberg entropy and Lyapunov exponents) govern the geometric regularity of harmonic currents, with implications for transverse measures and rigidity phenomena in complex foliations.

Abstract

Let $\mathcal{F}$ be a singular holomorphic foliation on an algebraic complex surface $S$, with hyperbolic singularities and no foliated cycle. We prove a formula for the transverse Hausdorff dimension of the unique harmonic current, involving the Furstenberg entropy and the Lyapunov exponent. In particular, we extend Brunella's inequality to every holomorphic foliation $\mathcal{F}$ on $\mathbb P^2$: if $\mathcal{F}$ has degree $d \geq 2$, then the Hausdorff dimension of its harmonic current is smaller than or equal to ${d-1 \over d+2}$, in particular the harmonic current is singular with respect to the Lebesgue measure. We also show that the Hausdorff dimension of the harmonic current of the Jouanolou foliation of degree $2$ is equal to $1/4$, and that the same property holds for topologically conjugate foliations on $\mathbb P^2$.

The dimension of harmonic currents on foliated complex surfaces

TL;DR

This work analyzes holomorphic foliations with hyperbolic singularities on complex surfaces and constructs a unique directed harmonic current with finite harmonic measure . It derives a transversal dimension formula, showing that the Hausdorff dimension of the restriction of to transversals equals the entropy-to-Lyapunov ratio , and proves exact dimensionality when holonomy is discrete. The paper extends Brunella’s inequality on , providing an explicit bound for degree , and computes the Jouanolou foliation’s dimension as . A key methodological core is a Limit Theorem for leafwise closed 1-forms, combined with a discretization of holonomy, and a detailed entropy analysis linking and ; in particular, when the holonomy pseudogroup is discrete on the pseudo-minimal set. The results reveal how dynamical quantities (Furstenberg entropy and Lyapunov exponents) govern the geometric regularity of harmonic currents, with implications for transverse measures and rigidity phenomena in complex foliations.

Abstract

Let be a singular holomorphic foliation on an algebraic complex surface , with hyperbolic singularities and no foliated cycle. We prove a formula for the transverse Hausdorff dimension of the unique harmonic current, involving the Furstenberg entropy and the Lyapunov exponent. In particular, we extend Brunella's inequality to every holomorphic foliation on : if has degree , then the Hausdorff dimension of its harmonic current is smaller than or equal to , in particular the harmonic current is singular with respect to the Lebesgue measure. We also show that the Hausdorff dimension of the harmonic current of the Jouanolou foliation of degree is equal to , and that the same property holds for topologically conjugate foliations on .

Paper Structure

This paper contains 53 sections, 37 theorems, 226 equations.

Table of Contents

  1. Overview, statements and notations
  2. Differential geometry and Harmonic currents
  3. Differential geometry for foliations
  4. Holomorphic foliations on surfaces
  5. Differential operators, curvature
  6. Hyperbolic singularities, angular domain
  7. The metric $m$ on the normal bundle $N_\mathcal{F}$
  8. The metrics $g_s$ and $g_P$ on the tangent bundle $T_\mathcal{F}$
  9. Smooth metrics
  10. Poincaré metric
  11. Relation between smooth metrics and the Poincaré metric
  12. Harmonic currents
  13. Directed harmonic currents
  14. The measure $\mu = T \wedge \text{vol}_{g_P}$
  15. Local expression of directed harmonic currents
  16. ...and 38 more sections

Key Result

Theorem 1

Let $\mathcal{F}$ be a foliation on an algebraic surface $S$ with hyperbolic singularities and no foliated cycle and $T$ be its harmonic current. For any holomorphic transversal $\Sigma$ intersecting the support of $T$, we have for the transversal measure $T_{\vert \Sigma}$ defined in Equation eq: l In particular, the Hausdorff dimensionThe Hausdorff dimension of a measure $\nu$ on the disc $\math

Theorems & Definitions (76)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • Definition 1.1
  • Lemma 1.2
  • proof
  • Remark 1.3
  • Definition 1.4
  • Lemma 1.5
  • ...and 66 more