Foliations in complex geometry and dynamics

Speaker: Ekaterina Amerik (Moscow State University)
Title: Isotriviality of fibrations and the characteristic foliation on smooth hypersurfaces in holomorphic symplectic manifolds
Abstract: We prove some isotriviality criteria for fibrations arising from smooth algebraic foliations. An especially simple case is when the foliation in question is defined as the kernel of a nowhere vanishing global holomorphic $d-1$-form (where $d$ is the dimension of the ambient variety). Applying this to the characteristic foliation on a hypersurface in a holomorphic symplectic manifold, we obtain an extension of a result by Hwang and Viehweg from 2008, concerning its non-algebraicity.

Speaker: Tien Cuong Dinh (National University of Singapore)
Title: Unique ergodicity theorem for foliations
Abstract: In collaboration with Nguyen and Sibony, we obtained a quite general ergodic theorem for compact laminations or foliations by Riemann surfaces, possibly with singularities. This result says that with respect to any extremal directed harmonic current, ALMOST every leaf of the lamination/foliation is equidistributed. This is a version of the law of large numbers in the foliation theory.
For foliations in the projective complex plane which are generic in a geometric sense, Fornaess and Sibony proved that there is a unique directed harmonic current, up to a multiplicative constant. It follows that ALL leaves of the foliation are equidistributed with respect to this current. More recently, in collaboration with Sibony, we prove a similar result for generic foliations in an algebraic sense. It is known that such a foliation admits an invariant line. We show that the current of integration on this line is the unique directed harmonic current of the foliation and ALL leaves of the foliation are equidistributed with respect to this current.
The proof of the last result uses our recent theory of densities for positive harmonic currents. It allows us to obtain similar statements for foliations in compact Kaehler surfaces which are not necessarily projective. I will discuss this theory in the second part of my talk.

Speaker: George Marinescu (University of Kohln)
Title: Kodaira embedding theorem of Levi-flat CR manifolds
Abstract: Ohsawa and Sibony proved in 2000 the analogue of the Kodaira embedding theorem for compact Levi-flat CR manifolds. The purpose of this talk is to explain a proof using microlocal analysis of the Szego kernel, based on a joint work with Chin-Yu Hsiao.

Speaker: Takeo Ohsawa (Nagoya University)
Title: Levi flat hyper surfaces, results and questions around basic examples
Abstract: In the theory of several complex variables, Levi convexity is one of the most basic notions. From this viewpoint, strictly pseudoconvex hypersurfaces are generically important. Levi flat hypersurfaces belong to the other extreme. They are not so important in the basic theory, but quite interesting as distinguished objects. They arose first as counterexamples to the Levi problem on complex manifolds (H. Grauert, 1963). They cannot be approximated by strongly pseudoconvex hypersurfaces. In 1982, it was observed that a Levi flat hypersurface exists in Cˆ × (C/(Z ×√−1Z)) that bounds C∗ × {1 < |z| < 2}. Subsequently, boundaries of analytic disc bundles over compact Riemann surfaces have been studied. Results and questions around these examples will be discussed.

Speaker: Jorge Vitorio Pereira (IMPA)
Title: Birational geometry of foliations
Abstract: I will present the birational classification of foliated surfaces [after McQuillan, Brunella and Mendes] and digress about some of the difficulties to extend it for foliations of higher dimension/codimension. Time allowing, I will discuss the classification of (singular) Calabi-Yau foliations of codimension one and outline its proof.

Speaker: Julio Rebelo (University of Toulouse)
Title: Foliations on surfaces: some basic geometric aspects
Abstract: This talk is an introduction to the topic of singular holomorphic foliations. Consider a complex projective surface $M$ endowed with a (singular) holomorphic foliation $fol$. These foliations are closely related to meromorphic vector fields on $M$ and we will quickly review the geometric meaning of the definition along with the notions of leaf and of singular point. The holonomy representation associated with a leaf $L$ of $fol$ will then be introduced and applied, in particular, to describe the local structure of a foliation around a ``simple'' singular point. Some complementary remarks about the case of general singularities may also be made in connection with birational transformations.

The second part of the talk will be devoted to positive closed currents directed by a foliation. Let then $M$ and $fol$ be as above and assume that $T$ is a closed positive current on $M$ that, in addition, is {it directed}, by $fol$. The most basic example of these currents appears when $fol$ leaves invariant some algebraic curve $C subset M$: the corresponding current $T$ is nothing but the integration current over $C$. The condition for a current to be directed by a foliation will further be detailed and the geometric/dynamic consequences of the existence of a directed positive closed current for $fol$ will be clarified by means of a theorem due to Sullivan.

To close the discussion, we are going to seek a partial converse to the above mentioned example. Namely, we will look for sufficient conditions to ensure that every positive closed current directed by a certain foliation $fol$ must coincide with the integration current over an algebraic leaf (so that, in particular, the foliation must leave invariant some algebraic curve).