Riemann surface laminations with singularities. (English) Zbl 1159.32020
The paper under review is a well written survey about the authors’ theory of harmonic currents for laminations by Riemann surfaces.
Roughly speaking, a Riemann surface lamination of a complex manifold is a continuous “foliation” by complex curves with parameter space which is just a topological space. Such a lamination might have singularities, which is usually the case in nature. In the first section, the authors present the very basic definitions of laminations by Riemann surfaces, of reduced singularities and separatrices and provide many examples with remarks and observations. Then, they discuss linearization and normal forms (topological, formal and holomorphic), stating results and, in some cases, providing sketches of the proofs.
In the next section, they define directed positive closed currents for laminations contained in compact Kähler manifolds and they discuss existence, uniqueness, and other properties and show, almost always with proofs, how to use such objects to study laminations of \(\mathbb P^2\).
Next, the authors discuss of singular laminations without positive closed currents. They start proving a sort of “Brody reparametrization lemma” for laminations. Then, they show that generically for laminations in \(\mathbb P^2\) without singularities there are no positive directed closed currents, except integration along algebraic leaves. They continue by studying the Poincaré metric of the leaves and its regularity with respect to transversal variation, ending the paragraph discussing universal coverings maps of leaves.
The final section is the core of the survey, devoted to harmonic currents. These are, roughly speaking, the analogs of invariant measures in discrete dynamical systems. After proving existence and a way to construct such currents, they prove a series of properties of harmonic currents.
Finally, the authors discuss the (still open) problem of existence of minimal exceptional sets for foliations in \(\mathbb P^2\), showing, among other things, that if a foliation admits a positive closed directed current, then it has no minimal exceptional sets.
Roughly speaking, a Riemann surface lamination of a complex manifold is a continuous “foliation” by complex curves with parameter space which is just a topological space. Such a lamination might have singularities, which is usually the case in nature. In the first section, the authors present the very basic definitions of laminations by Riemann surfaces, of reduced singularities and separatrices and provide many examples with remarks and observations. Then, they discuss linearization and normal forms (topological, formal and holomorphic), stating results and, in some cases, providing sketches of the proofs.
In the next section, they define directed positive closed currents for laminations contained in compact Kähler manifolds and they discuss existence, uniqueness, and other properties and show, almost always with proofs, how to use such objects to study laminations of \(\mathbb P^2\).
Next, the authors discuss of singular laminations without positive closed currents. They start proving a sort of “Brody reparametrization lemma” for laminations. Then, they show that generically for laminations in \(\mathbb P^2\) without singularities there are no positive directed closed currents, except integration along algebraic leaves. They continue by studying the Poincaré metric of the leaves and its regularity with respect to transversal variation, ending the paragraph discussing universal coverings maps of leaves.
The final section is the core of the survey, devoted to harmonic currents. These are, roughly speaking, the analogs of invariant measures in discrete dynamical systems. After proving existence and a way to construct such currents, they prove a series of properties of harmonic currents.
Finally, the authors discuss the (still open) problem of existence of minimal exceptional sets for foliations in \(\mathbb P^2\), showing, among other things, that if a foliation admits a positive closed directed current, then it has no minimal exceptional sets.
Reviewer: Filippo Bracci (Roma)
MSC:
| 32S65 | Singularities of holomorphic vector fields and foliations |
| 32U40 | Currents |
| 30F15 | Harmonic functions on Riemann surfaces |
| 57R30 | Foliations in differential topology; geometric theory |
| 32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |
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