VIRTUAL EVENT
RESEARCH SCHOOL  ECOLE DE RECHERCHE Geometry and Dynamics of Foliations (event 2251) Géométrie et dynamiques des feuilletages Dates: 1822 May 2020 & 2530 May 2020 This event is part of a series of three activities focusing on Foliation Theory and Complex Geometry at CIRMLuminy in 2020.
> see the Virtual workshop on Varieties with Trivial Canonical Class > register for the conference on Foliation Theory and Complex Geometry that will now take place on 2327 November 2020 
Due to the Covid19 epidemics, this workshop could not take place at CIRM as originally intended.
DESCRIPTION
Foliation Theory is a lively subject lying at a crossroad of many mathematical disciplines. The School “Geometry and Dynamics of Foliations” aims at presenting to young mathematicians some of the different techniques used by practitioners of the field. The topics of the minicourses give a panorama of recent developments in Foliation Theory, ranging from algebraicgeometric to dynamical contributions.
La théorie des feuilletages est un sujet très vivant qui se situe au carrefour de nombreuses disciplines mathématiques. L'école "Géométrie et dynamique des feuilletages" a pour but de présenter aux jeunes mathématicien.ne.s quelquesunes des différentes techniques utilisées par les praticiens du domaine. Les sujets des minicours donnent un panorama des développements récents de cette théorie, allant des contributions algébrogéométriques aux contributions dynamiques. 
SCIENTIFIC COMMITTEE
ORGANIZING COMMITTEE

MINI COURSES
Holomorphic Poisson Structures
Brent Pym (McGill University)
Lecture 1

Lecture 2

Lecture 3

Lecture 4

Abstract. 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 evendimensional 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.
Main references
Alexander Polishchuk. Algebraic geometry of Poisson brackets. J. Math. Sci. (New York) 84 (1997), no. 5, 1413–1444.
Mathscinet zbMATH Journal
Brent Pym. Constructions and classifications of projective Poisson varieties. Lett. Math. Phys. 108 (2018), no. 3, 573–632.
Mathscinet zbMATH Journal arXiv
Textbooks
JeanPaul Dufour and Nguyen Zung. Poisson structures and their normal forms. Progress in Mathematics, 242. Birkhäuser Verlag, Basel, 2005. xvi+321 pp.
Mathscinet zbMATH
Camile LaurentGengoux; Anne Pichereau; Pol Vanhaecke. Poisson structures. Grundlehren der Mathematischen Wissenschaften, 347. Springer, Heidelberg, 2013. xxiv+461 pp.
Mathscinet zbMATH
Problems
Poisson Structures.
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.
Main references
Alexander Polishchuk. Algebraic geometry of Poisson brackets. J. Math. Sci. (New York) 84 (1997), no. 5, 1413–1444.
Mathscinet zbMATH Journal
Brent Pym. Constructions and classifications of projective Poisson varieties. Lett. Math. Phys. 108 (2018), no. 3, 573–632.
Mathscinet zbMATH Journal arXiv
Textbooks
JeanPaul Dufour and Nguyen Zung. Poisson structures and their normal forms. Progress in Mathematics, 242. Birkhäuser Verlag, Basel, 2005. xvi+321 pp.
Mathscinet zbMATH
Camile LaurentGengoux; Anne Pichereau; Pol Vanhaecke. Poisson structures. Grundlehren der Mathematischen Wissenschaften, 347. Springer, Heidelberg, 2013. xxiv+461 pp.
Mathscinet zbMATH
Problems
Poisson Structures.
Lecture 1
Fano Foliations 0  Algebraicity of Smooth Formal Schemes and Applications to Foliations

Lecture 2
Fano Foliations 1  Definition, Examples and First Properties

Lecture 3
Fano Foliations 2  Adjunction Formula and Applications

Lecture 4
Fano Foliations 3 Classification of Fano Foliations of Large Index

Abstract. 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 anticanonical 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 anticanonical class $K_F$.
This minicourse 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.
Main references
Carolina Araujo, Stéphane Druel. Characterization of generic projective space bundles and algebraicity of foliations. Comment. Math. Helv. 94, 4 (2019), p. 833853.
Mathscinet zbMATH Journal arXiv
Carolina Araujo, Stéphane Druel. On Fano foliations 2. Foliation theory in algebraic geometry, 1–20, Simons Symp., Springer, Cham, 2016.
Mathscinet zbMATH Book arXiv
Carolina Araujo, Stéphane Druel. On Fano foliations. Adv. Math. 238 (2013), p. 70118.
Mathscinet zbMATH Journal arXiv
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 anticanonical class $K_F$.
This minicourse 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.
Main references
Carolina Araujo, Stéphane Druel. Characterization of generic projective space bundles and algebraicity of foliations. Comment. Math. Helv. 94, 4 (2019), p. 833853.
Mathscinet zbMATH Journal arXiv
Carolina Araujo, Stéphane Druel. On Fano foliations 2. Foliation theory in algebraic geometry, 1–20, Simons Symp., Springer, Cham, 2016.
Mathscinet zbMATH Book arXiv
Carolina Araujo, Stéphane Druel. On Fano foliations. Adv. Math. 238 (2013), p. 70118.
Mathscinet zbMATH Journal arXiv
Problems
Fano Foliations.
Fano Foliations.
Lecture 1
An Overview of the Minimal Model Program

Lecture 2
MMP for corank1 Foliations (1)

Lecture 3
MMP for corank1 Foliations (2)

Lecture 4
MMP for corank1 Foliations (3)

Abstract
The goal of the Minimal Model Program (MMP) is to provide a framework in which the classification of varieties or foliations can take place. The basic strategy is to use surgery operations to decompose a variety or foliation into ``building block” type
objects (Fano, CalabiYau, or canonically polarized objects).
We first review the basic notions of the MMP in the case of varieties. We then explain work on realizing the MMP for foliations on threefolds (both in the case of codimension =1 and dimension =1 foliations). We explain and pay special attention to results such as the Cone and Contraction theorem, the Flip theorem and a version of the Basepoint free theorem.
References
Calum Spicer. Higherdimensional foliated Mori theory. Compos. Math. 156 (2020), no. 1, 1–38.
Mathscinet zbMATH Journal arXiv
Paolo Cascini, Calum Spicer. MMP for corank one on threefolds. Preprint 2018.
arXiv
Calum Spicer, Roberto Svaldi. Local and global applications of the Minimal Model Program for corank one foliations on threefolds. Preprint 2019.
arXiv
The goal of the Minimal Model Program (MMP) is to provide a framework in which the classification of varieties or foliations can take place. The basic strategy is to use surgery operations to decompose a variety or foliation into ``building block” type
objects (Fano, CalabiYau, or canonically polarized objects).
We first review the basic notions of the MMP in the case of varieties. We then explain work on realizing the MMP for foliations on threefolds (both in the case of codimension =1 and dimension =1 foliations). We explain and pay special attention to results such as the Cone and Contraction theorem, the Flip theorem and a version of the Basepoint free theorem.
References
Calum Spicer. Higherdimensional foliated Mori theory. Compos. Math. 156 (2020), no. 1, 1–38.
Mathscinet zbMATH Journal arXiv
Paolo Cascini, Calum Spicer. MMP for corank one on threefolds. Preprint 2018.
arXiv
Calum Spicer, Roberto Svaldi. Local and global applications of the Minimal Model Program for corank one foliations on threefolds. Preprint 2019.
arXiv
Complete Holomorphic Vector Fields and their Singular Points
Adolfo Guillot (UNAM Mexico)
Lecture 1
Complete Holomorphic Vector Fields and their Singular Points (1)

Lecture 2
Complete Holomorphic Vector Fields and their Singular Points (2)

Lecture 3
Complete Holomorphic Vector Fields and their Singular Points (3)

Lecture 4
Complete Holomorphic Vector Fields and their Singular Points (4)

Abstract. On a complex manifold, complex flows induce complete holomorphic vector fields. However, only very seldomly a vector field integrates into a flow. In general, it is difficult to say whether a holomorphic vector field on a noncompact manifold is complete or not (vector fields on compact manifolds are always complete). Some twentyfive years ago, Rebelo realized an exploited the fact that there are local (and not just global!) obstructions for a vector field to be complete. This opened the door for a local study of complete holomorphic vector fields on complex manifolds. In this series of talks we will explore some of these results.
What I will talk about is for the greater part contained or summarized in the articles in the bibliography. Their introductions might be useful as a first reading on the subject.
References
Adolfo Guillot, Julio Rebelo. Semicomplete meromorphic vector fields on complex surfaces. J. Reine Angew. Math. 667 (2012), 27–65.
Mathscinet zbMATH Journal
Adolfo Guillot. Vector fields, separatrices and Kato surfaces. Ann. Inst. Fourier (Grenoble) 64 (2014), no. 3, 1331–1361.
Mathscinet zbMATH Journal arXiv
What I will talk about is for the greater part contained or summarized in the articles in the bibliography. Their introductions might be useful as a first reading on the subject.
References
Adolfo Guillot, Julio Rebelo. Semicomplete meromorphic vector fields on complex surfaces. J. Reine Angew. Math. 667 (2012), 27–65.
Mathscinet zbMATH Journal
Adolfo Guillot. Vector fields, separatrices and Kato surfaces. Ann. Inst. Fourier (Grenoble) 64 (2014), no. 3, 1331–1361.
Mathscinet zbMATH Journal arXiv
Codimension one foliation with pseudoeffective conormal bundle
Frédéric Touzet (Université Rennes 1)
Lecture 1
Codimension One Foliations with PseudoEffective Conormal Bundle (1)

Lecture 2
Codimension One Foliations with PseudoEffective Conormal Bundle (2)

Lecture 3
Codimension One Foliations with PseudoEffective Conormal Bundle (3)

Abstract. Let X be a projective manifold equipped with a codimension 1 (maybe singular) distribution whose conormal sheaf is assumed to be pseudoeffective. Basic examples of such distributions are provided by the kernel of a holomorphic one form, necessarily closed when the ambient is projective. More generally, due to a theorem of JeanPierre Demailly, a distribution with conormal sheaf pseudoeffective is actually integrable and thus defines a codimension 1 holomorphic foliation F. In this series of lectures, we would aim at describing the structure of such a foliation, especially in the non abundant case, i.e when F cannot be defined by a holomorphic one form (even passing through a finite cover). It turns out that \F is the pullback of one of the "canonical foliations" on a Hilbert modular variety. This result remains valid for 'logarithmic foliated pairs'.
Main references
Kevin Corlette, Carlos Simpson. On the classification of ranktwo representations of quasiprojective fundamental groups. Compos. Math. 144 (2008), no. 5, 1271–1331.
Mathscinet zbMATH Journal arXiv
JeanPierre Demailly. On the Frobenius integrability of certain holomorphic pforms. Complex geometry (Göttingen, 2000), 93–98, Springer, Berlin, 2002.
Mathscinet zbMATH Journal arXiv
Frédéric Touzet. On the structure of codimension 1 foliations with pseudoeffective conormal bundle. Foliation theory in algebraic geometry, 157–216, Simons Symp., Springer, Cham, 2016.
Mathscinet zbMATH Journal arXiv
Frédéric Touzet. Uniformisation de l'espace des feuilles de certains feuilletages de codimension un. Bull. Braz. Math. Soc. (N.S.) 44 (2013), no. 3, 351–391.
Mathscinet zbMATH Journal arXiv
Kevin Corlette, Carlos Simpson. On the classification of ranktwo representations of quasiprojective fundamental groups. Compos. Math. 144 (2008), no. 5, 1271–1331.
Mathscinet zbMATH Journal arXiv
JeanPierre Demailly. On the Frobenius integrability of certain holomorphic pforms. Complex geometry (Göttingen, 2000), 93–98, Springer, Berlin, 2002.
Mathscinet zbMATH Journal arXiv
Frédéric Touzet. On the structure of codimension 1 foliations with pseudoeffective conormal bundle. Foliation theory in algebraic geometry, 157–216, Simons Symp., Springer, Cham, 2016.
Mathscinet zbMATH Journal arXiv
Frédéric Touzet. Uniformisation de l'espace des feuilles de certains feuilletages de codimension un. Bull. Braz. Math. Soc. (N.S.) 44 (2013), no. 3, 351–391.
Mathscinet zbMATH Journal arXiv
Textbook
Marco Brunella. Birational geometry of foliations. IMPA Monographs, 1. Springer, Cham, 2015. xiv+130 pp.
Mathscinet zbMATH
Marco Brunella. Birational geometry of foliations. IMPA Monographs, 1. Springer, Cham, 2015. xiv+130 pp.
Mathscinet zbMATH
Value 
MAY 25 
MAY 26 
MAY 28 
14:00  14:40 
Brent Pym DISCUSSION SESSION 
Araujo  Druel DISCUSSION SESSION 

14:50  15:15 
Tiago Jardim da Fonseca 
Katsuhiko Okumura 
Roberto Svaldi (40 min talk) 
15:20  15:45 
 

15:50  16:15 

16:20  17:00 
Frédéric Touzet DISCUSSION SESSION 
Cascini  Spicer DISCUSSION SESSION 
ABSTRACTS
Bertrand Deroin  CNRS/IMPA, AGM
Dynamics and topology of the Jouanolou foliation I will report on some joint work with Aurélien Alvarez, which shows that the Jouanolou foliation in degree two is structurally stable, and that it has a nontrivial domain of discontinuity. This result is opposed to a series of results beginning in the sixties with the works of HudaiVerenov and Ilyashenko. Tiago Jardim da Fonseca  University of Oxford Higher Ramanujan Foliations I will describe a remarkable family of higher dimensional foliations generalizing the equations studied by Darboux, Halphen, Ramanujan, and many others, and discuss some related geometric problems motivated by number theory. Federico Lo Bianco  Institut Camille Jordan (Lyon) On symmetries of transversely projective foliations We consider a holomorphic (singular) foliation F on a projective manifold X and a group G of birational transformations of X which preserve F (i.e. it permutes the set of leaves). We say that the transverse action of G is finite if some finite index subgroup of G fixes each leaf of F. In a joint work with J. V. Pereira, E. Rousseau and F. Touzet, we show finiteness for the group of birational transformations of general type foliations with tame singularities and transverse finiteness for (nonvirtually euclidean) transversely projective foliations. In this talk I will focus on the latter result; time permitting, I will show how the presence of a transverse structure (projective, hyperbolic, spherical...) and the analysis of the resulting monodromy representation allow to reduce to the case of modular foliations on Shimura varieties and to conclude. Rémi Jaoui  University of Notre Dame A modeltheoretic analysis of geodesic equations in negative curvature. To any algebraic differential equation, one can associate a firstorder structure which encodes some of the properties of algebraic integrability and of algebraic independence of its solutions. To describe the structure associated to a given algebraic (nonlinear) differential equation (E), typical questions are:  Is it possible to express the general solutions of (E) from successive resolutions of linear differential equations?  Is it possible to express the general solutions of (E) from successive resolutions of algebraic differential equations of lower order than (E)?  Given distinct initial conditions for (E), under which conditions are the solutions associated to these initial conditions algebraically independent? In my talk, I will discuss in this setting one of the first examples of noncompletely integrable Hamiltonian systems: the geodesic motion on an algebraically presented compact Riemannian surface with negative curvature. I will explain a qualitative modeltheoretic description of the associated structure based on the global hyperbolic dynamical properties identified by Anosov in the 70’s for the geodesic motion in negative curvature. 
Katsuhiko Okumura – Waseda University
Topics on the Poisson varieties of dimension at least four It is thought that the classifications and constructions of holomorphic Poisson structures are worth studying. The classification when the Picard rank is 2 or higher is unknown. In this talk, we will introduce the classification of holomorphic Poisson structures with the reduced degeneracy divisor that have only simple normal crossing singularities, on the product of Fano variety of Picard rank 1. This claims that such a Poisson manifold must be a diagonal Poisson structure on the product of projective spaces, so this is a generalization of Lima and Pereira's study. The talk will also include various examples, classifications, and problems of highdimensional holomorphic Poisson structures. YenAn Chen  University of Utah Boundedness of Minimal Partial du Val Resolutions of Canonical Surface Foliations By the work of Brunella and McQuillan, it is known that smooth foliated surfaces of general type with only canonical singularities admit a unique canonical model. It is then natural to wonder if these canonical models have a good moduli theory and, in particular, if they admit a moduli functor. In this talk, I will show that the canonical models and their minimal partial du Val resolutions are bounded. João Paulo Figueredo  IMPA Regular foliations on rationally connected manifolds In this talk, we will consider the problem of classifying regular foliations on rationally connected manifolds over the complex numbers. Conjecturally, these foliations should have algebraic leaves. I will show this is true when the manifold has dimension three, and the foliation has codimension one and non pseudoeffective canonical bundle. Roberto Svaldi  EPFL Foliations and MMP: some applications I will explain how the recent developments in the theory of birational geometry of rank 2 foliations on 3folds have find quite a few applications in the study of the structure of such foliations and their singularities. In this talk that is a complement to the course of Cascini and Spicer, but which will be selfcontained, I shall briefly explain why the MMP terminates and then proceed to illustrate a few applications of these techniques, e.g., to the classification of canonical singularities, to the study of adjunction theory, and to the study of hyperbolicity properties of foliated 3folds. The work is in collaboration with Calum Spicer. Maxence Mayrand – University of Toronto Hyperkähler structures on symplectic realizations of holomorphic Poisson surfaces I will discuss the existence of hyperkähler structures on local symplectic groupoids integrating holomorphic Poisson manifolds, and show that they always exist when the base is a Poisson surface. The hyperkähler structure is obtained by constructing the twistor space by lifting specific deformations of the Poisson surface adapted from Hitchin's unobstructedness result. In the special case of the zero Poisson structure, we recover the FeixKaledin hyperkähler structure on the cotangent bundle of a Kähler manifold. 
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