Unique ergodicity for foliations on compact Kähler surfaces. (English) Zbl 1518.37061
Let \(\mathcal{F}\) be a (possibly singular) holomorphic foliation by Riemann surfaces on a compact Kähler surface \(X\).
In order to study the dynamical properties of foliations, the authors develop the formalism of directed \(dd^c\)-closed currents. The pioneering work of L. Garnett [J. Funct. Anal. 51, 285–311 (1983; Zbl 0524.58026)] and D. Sullivan [Invent. Math., 36, 225–255 (1976; Zbl 0335.57015)] developed the ideas of harmonic measures and foliated cycles for nonsingular foliations.
In the present paper, given a singular holomorphic foliaton \(\mathcal{F}\), a positive \(dd^c\)-closed \((1,1)\)-current \(T\) is said to be directed by \(\mathcal{F}\) if \(T \wedge \Omega=0\), for every local holomorphic 1-form \(\Omega\) defining \(\mathcal{F}\) on \(X\).
This novel technique allows the interplay between cohomological intersection and geometric intersection. The present article studies the cohomological properties of tangent currents.
Assume that \(\mathcal{F}\) is generic in the sense that all the singularities are hyperbolic, and moreover it admits no directed positive closed \((1,1)\)-current. Then there exists a unique (up to a multiplicative constant) positive \(dd^c\)-closed \((1,1)\)-current directed by \(\mathcal{F}\). This is a very strong ergodic property of \(\mathcal{F}\) showing that all leaves of \(\mathcal{F}\) have the same asymptotic behavior.
The proof uses an extension of the theory of densities to a class of non-\(dd^c\)-closed currents. This is independent of foliation theory and represents a new tool in pluripotential theory. A complete description of the cone of directed positive \(dd^c\)-closed currents is also given when \(\mathcal{F}\) admits directed positive closed currents.
This work is a new step in the study of Kähler surfaces, already initiated by the authors in previous publications. For previous results about foliations on the projective plane \(X=\mathbb{P}^2\) with an invariant line see the papers of the authors [Invent. Math. 211, No. 1, 1–38 (2018; Zbl 1395.37030); J. Algebr. Geom. 27, No. 3, 497–551 (2018; Zbl 1393.32017)].
Let us remark that the whole paper is based on the technical mastery and deep ideas of N. Sibony [Duke Math. J. 52, 157–197 (1985; Zbl 0578.32023)].
In order to study the dynamical properties of foliations, the authors develop the formalism of directed \(dd^c\)-closed currents. The pioneering work of L. Garnett [J. Funct. Anal. 51, 285–311 (1983; Zbl 0524.58026)] and D. Sullivan [Invent. Math., 36, 225–255 (1976; Zbl 0335.57015)] developed the ideas of harmonic measures and foliated cycles for nonsingular foliations.
In the present paper, given a singular holomorphic foliaton \(\mathcal{F}\), a positive \(dd^c\)-closed \((1,1)\)-current \(T\) is said to be directed by \(\mathcal{F}\) if \(T \wedge \Omega=0\), for every local holomorphic 1-form \(\Omega\) defining \(\mathcal{F}\) on \(X\).
This novel technique allows the interplay between cohomological intersection and geometric intersection. The present article studies the cohomological properties of tangent currents.
Assume that \(\mathcal{F}\) is generic in the sense that all the singularities are hyperbolic, and moreover it admits no directed positive closed \((1,1)\)-current. Then there exists a unique (up to a multiplicative constant) positive \(dd^c\)-closed \((1,1)\)-current directed by \(\mathcal{F}\). This is a very strong ergodic property of \(\mathcal{F}\) showing that all leaves of \(\mathcal{F}\) have the same asymptotic behavior.
The proof uses an extension of the theory of densities to a class of non-\(dd^c\)-closed currents. This is independent of foliation theory and represents a new tool in pluripotential theory. A complete description of the cone of directed positive \(dd^c\)-closed currents is also given when \(\mathcal{F}\) admits directed positive closed currents.
This work is a new step in the study of Kähler surfaces, already initiated by the authors in previous publications. For previous results about foliations on the projective plane \(X=\mathbb{P}^2\) with an invariant line see the papers of the authors [Invent. Math. 211, No. 1, 1–38 (2018; Zbl 1395.37030); J. Algebr. Geom. 27, No. 3, 497–551 (2018; Zbl 1393.32017)].
Let us remark that the whole paper is based on the technical mastery and deep ideas of N. Sibony [Duke Math. J. 52, 157–197 (1985; Zbl 0578.32023)].
Reviewer: Jesus Muciño Raymundo (Morelia)
MSC:
| 37F75 | Dynamical aspects of holomorphic foliations and vector fields |
| 37C86 | Foliations generated by dynamical systems |
| 32M25 | Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions |
| 32S65 | Singularities of holomorphic vector fields and foliations |
| 32U15 | General pluripotential theory |
| 32U40 | Currents |
Keywords:
currents; density of currents; hyperbolic singularity; singular holomorphic foliation; tangent currentReferences:
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