[Paper Review] On the dynamics of codimension one holomorphic foliations with ample normal bundle

Marco Brunella|ArXiv.org|Jun 11, 2007

Advanced Differential Equations and Dynamical Systems6 references3 citations

TL;DR

This paper investigates codimension one holomorphic foliations on compact Kähler manifolds of dimension at least three with ample normal bundles, proving that such foliations cannot leave invariant a $C^{2,eta}$ Levi-flat hypersurface, and showing that the normal bundle cannot admit a hermitian metric with positive curvature on any neighborhood of such a hypersurface. The result supports a broader conjecture that all leaves accumulate to the singular set under these conditions.

ABSTRACT

We investigate the accumulation to singular points of leaves of codimension one foliations whose normal bundle is ample, with emphasis on the nonexistence of Levi-flat hypersurfaces.

Motivation & Objective

• To investigate whether leaves of codimension one holomorphic foliations with ample normal bundles accumulate to the singular set, extending Lins Neto's result from $\mathbb{C}P^n$ to general compact Kähler manifolds.
• To address the conjecture that every leaf accumulates to the singular set when the normal bundle is ample.
• To rule out the existence of $C^{2,eta}$ Levi-flat hypersurfaces invariant under such foliations.
• To analyze the curvature properties of the normal bundle in relation to the dynamics of the foliation.
• To explore the geometric and cohomological obstructions preventing the existence of such invariant hypersurfaces.

Proposed method

• Uses the theory of holomorphic foliations and their normal bundles, particularly the ampleness of $N_{\mathcal{F}}$, to analyze dynamical behavior.
• Applies the Baum–Bott formula to relate the second Chern class of $N_{\mathcal{F}}$ to residues at singular set components.
• Employs strong pseudoconvexity arguments on the complement of a compact invariant set $\mathcal{M}$ disjoint from $Sing(\mathcal{F})$, assuming such a set exists.
• Analyzes curvature properties of the Kähler form $\omega$ representing $N_{\mathcal{F}}$, showing $\omega \wedge \omega$ becomes $\partial\bar{\partial}$-exact after contraction of the singular set.
• Uses the $\partial\bar{\partial}$-lemma and contraction maps $\pi: X \to X_0$ to derive cohomological contradictions if such a hypersurface existed.
• Considers formal and convergent extension techniques for the Levi foliation on $M$, suggesting potential generalizations to non-invariant but smooth Levi-flat hypersurfaces.

Experimental results

Research questions

• RQ1 Can a codimension one holomorphic foliation with ample normal bundle leave invariant a $C^{2,eta}$ real hypersurface in a compact Kähler manifold of dimension $n \geq 3$?
• RQ2 Does the ampleness of the normal bundle $N_{\mathcal{F}}$ imply that every leaf accumulates to the singular set $Sing(\mathcal{F})$?
• RQ3 Can the normal bundle of a Levi-flat hypersurface admit a hermitian metric with leafwise positive curvature?
• RQ4 Is it possible to extend the Levi foliation and its normal bundle as a holomorphic object beyond a $C^{\infty}$-smooth Levi-flat hypersurface?
• RQ5 What cohomological obstructions arise if the complement of an invariant compact set $\mathcal{M}$ is strongly pseudoconvex and $N_{\mathcal{F}}$ is ample?

Key findings

• The normal bundle $N_{\mathcal{F}}$ of a codimension one holomorphic foliation on a compact Kähler manifold of dimension $n \geq 3$ cannot admit a hermitian metric with positive curvature on any neighborhood of a $C^{2,\alpha}$ real hypersurface $M$ that is invariant under the foliation.
• If a compact invariant set $\mathcal{M}$ disjoint from $Sing(\mathcal{F})$ exists, then $X \setminus \mathcal{M}$ is strongly pseudoconvex, and the Kähler form $\omega$ representing $N_{\mathcal{F}}$ satisfies $\omega \wedge \omega = \sum \mu_j \delta_{Z_j} + i\partial\bar{\partial}S$.
• After contracting the singular set $Z = Sing(\mathcal{F})$ to points via $\pi: X \to X_0$, the current $\pi_*(\omega \wedge \omega)$ becomes $\partial\bar{\partial}$-exact, which is considered highly unlikely for a strictly positive $(2,2)$-current arising from a Kähler form.
• The direct image $\pi_*(\omega^\wedge n)$ is a strictly positive measure and thus cannot be $\partial\bar{\partial}$-exact, indicating a cohomological inconsistency if such a $\mathcal{M}$ existed.
• The result implies that $\mathcal{M}$ cannot be a Levi-flat hypersurface of class $C^{2,\alpha}$, even if not globally invariant, under the assumption of leafwise positivity of the normal bundle.
• The paper suggests that the non-existence of such hypersurfaces is supported by cohomological obstructions and formal extension techniques, though a full generalization remains open.