In the last few decades, much progress has been made in birational algebraic geometry. The general viewpoint is that complex projective manifolds should be classified according to the behavior of their canonical class. As a result of the minimal model program (MMP), every complex projective manifold can be built up from 3 classes of (possibly singular) projective varieties, namely, varieties $X$ for which $K_X$ satisfies $K_X<0$, $K_X\equiv 0$ or $K_X>0$. Projective manifolds $X$ whose anti-canonical class $-K_X$ is ample are called Fano manifolds.

Techniques from the MMP have been successfully applied to the study of global properties of holomorphic foliations. This led, for instance, to Brunella's birational classification of foliations on surfaces, in which the canonical class of the foliation plays a key role. In recent years, much progress has been made in higher dimensions. In particular, there is a well developed theory of Fano foliations, i.e., holomorphic foliations $F$ on complex projective varieties with ample anti-canonical class $-K_F$.

This mini-course is devoted to reviewing some aspects of the theory of Fano Foliations, with a special emphasis on Fano foliations of large index. We start by proving a fundamental algebraicity property of Fano foliations, as an application of Bost's criterion of algebraicity for formal schemes. We then introduce and explore the concept of log leaves. These tools are then put together to address the problem of classifying Fano foliations of large index.[-]

In the last few decades, much progress has been made in birational algebraic geometry. The general viewpoint is that complex projective manifolds should be classified according to the behavior of their canonical class. As a result of the minimal model program (MMP), every complex projective manifold can be built up from 3 classes of (possibly singular) projective varieties, namely, varieties $X$ for which $K_X$ satisfies $K_X<0$, $K_X\equiv 0$ or $K_X>0$. Projective manifolds $X$ whose anti-canonical class $-K_X$ is ample are called Fano manifolds.

Techniques from the MMP have been successfully applied to the study of global properties of holomorphic foliations. This led, for instance, to Brunella's birational classification of foliations on surfaces, in which the canonical class of the foliation plays a key role. In recent years, much progress has been made in higher dimensions. In particular, there is a well developed theory of Fano foliations, i.e., holomorphic foliations $F$ on complex projective varieties with ample anti-canonical class $-K_F$.

This mini-course is devoted to reviewing some aspects of the theory of Fano Foliations, with a special emphasis on Fano foliations of large index. We start by proving a fundamental algebraicity property of Fano foliations, as an application of Bost's criterion of algebraicity for formal schemes. We then introduce and explore the concept of log leaves. These tools are then put together to address the problem of classifying Fano foliations of large index.[-]

The Minimal Model Program is a (partially conjectural) framework of the classification of algebraic varieties. In the early 2000s Brunella, Mendes and McQuillan observed that this framework could be adapted to the study of foliations on projective surfaces. In recent years this program of study has been developed for foliations on higher dimensional projective varieties. Our first lecture will review the Minimal Model Program for surface foliations. We will then survey the recent developments in the topic, focusing especially on three cases where the theory of minimal models of foliations is most developed, namely for rank one foliations, co-rank one foliations and algebraically integrable foliations. Time permitting we will explain a very recent development: adjoint foliated structures. These structures arise naturally as a way to address some of the unique challenges which arise when studying minimal model techniques in the setting of foliations.[-]

The Minimal Model Program is a (partially conjectural) framework of the classification of algebraic varieties. In the early 2000s Brunella, Mendes and McQuillan observed that this framework could be adapted to the study of foliations on projective surfaces. In recent years this program of study has been developed for foliations on higher dimensional projective varieties. Our first lecture will review the Minimal Model Program for surface foliations. We will then survey the recent developments in the topic, focusing especially on three cases where the theory of minimal models of foliations is most developed, namely for rank one foliations, co-rank one foliations and algebraically integrable foliations. Time permitting we will explain a very recent development: adjoint foliated structures. These structures arise naturally as a way to address some of the unique challenges which arise when studying minimal model techniques in the setting of foliations.[-]

The Minimal Model Program is a (partially conjectural) framework of the classification of algebraic varieties. In the early 2000s Brunella, Mendes and McQuillan observed that this framework could be adapted to the study of foliations on projective surfaces. In recent years this program of study has been developed for foliations on higher dimensional projective varieties. Our first lecture will review the Minimal Model Program for surface foliations. We will then survey the recent developments in the topic, focusing especially on three cases where the theory of minimal models of foliations is most developed, namely for rank one foliations, co-rank one foliations and algebraically integrable foliations. Time permitting we will explain a very recent development: adjoint foliated structures. These structures arise naturally as a way to address some of the unique challenges which arise when studying minimal model techniques in the setting of foliations.[-]

How to study the dynamics of a holomorphic polynomial vector field in $\mathbb{C}^{2}$? What is the replacement of invariant measure? I will survey some surprising rigidity results concerning the behavior of these dynamical system. It is helpful to consider the extension of this dynamical system to the projective plane.
Consider a foliation in the projective plane admitting a unique invariant algebraic curve. Assume that the foliation is generic in the sense that its singular points are hyperbolic. With T.-C. Dinh, we showed that there is a unique positive $dd^{c}$-closed (1, 1)-current of mass 1 which is directed by the foliation. This is the current of integration on the invariant curve. A unique ergodicity theorem for the distribution of leaves follows: for any leaf $L$, appropriate averages on $L$ converge to the current of integration on the invariant curve (although generically the leaves are dense). The result uses our theory of densities for currents. It extends to Foliations on Kähler surfaces.
I will describe a recent result, with T.-C. Dinh and V.-A. Nguyen, dealing with foliations on compact Kähler surfaces. If the foliation, has only hyperbolic singularities and does not admit a transverse measure, in particular no invariant compact curve, then there exists a unique positive $dd^{c}$-closed (1, 1)-current of mass 1 which is directed by the foliation( it's like uniqueness of invariant measure for discrete dynamical systems). This improves on previous results, with J.-E. Fornæss, for foliations (without invariant algebraic curves) on the projective plane. The proof uses a theory of densities for positive $dd^{c}$-closed currents (an intersection theory).[-]

The notion of a Poisson manifold originated in mathematical physics, where it is used to describe the equations of motion of classical mechanical systems, but it is nowadays connected with many different parts of mathematics. A key feature of any Poisson manifold is that it carries a canonical foliation by even-dimensional submanifolds, called its symplectic leaves. They correspond physically to regions in phase space where the motion of a particle is trapped.

I will give an introduction to Poisson manifolds in the context of complex analytic/algebraic geometry, with a particular focus on the geometry of the associated foliation. Starting from basic definitions and constructions, we will see many examples, leading to some discussion of recent progress towards the classification of Poisson brackets on Fano manifolds of small dimension, such as projective space.[-]

The notion of a Poisson manifold originated in mathematical physics, where it is used to describe the equations of motion of classical mechanical systems, but it is nowadays connected with many different parts of mathematics. A key feature of any Poisson manifold is that it carries a canonical foliation by even-dimensional submanifolds, called its symplectic leaves. They correspond physically to regions in phase space where the motion of a particle is trapped.

I will give an introduction to Poisson manifolds in the context of complex analytic/algebraic geometry, with a particular focus on the geometry of the associated foliation. Starting from basic definitions and constructions, we will see many examples, leading to some discussion of recent progress towards the classification of Poisson brackets on Fano manifolds of small dimension, such as projective space.[-]