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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 Feix-Kaledin hyperkähler structure on the cotangent bundle of a Kähler manifold.
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 Fe...

53D17 ; 53C26 ; 53C28 ; 32G05

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I will explain how the recent developments in the theory of birational geometry of rank 2 foliations on 3-folds 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 self-contained, 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 3-folds.
The work is in collaboration with Calum Spicer.
I will explain how the recent developments in the theory of birational geometry of rank 2 foliations on 3-folds 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 self-contained, I shall briefly explain why the MMP terminates and then proceed to illustrate a few applications of these techniques, e.g., ...

14E30 ; 37F75 ; 32S65

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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 pseudo-effective canonical bundle.

14E30 ; 37F75 ; 14J30

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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.

14C20 ; 14E99 ; 32M25

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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 high-dimensional holomorphic Poisson structures.
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 ...

53D17 ; 14J45 ; 14C17

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To any algebraic differential equation, one can associate a first-order 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 non-completely integrable Hamiltonian systems: the geodesic motion on an algebraically presented compact Riemannian surface with negative curvature. I will explain a qualitative model-theoretic 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.
To any algebraic differential equation, one can associate a first-order 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 ...

12H05 ; 37D40 ; 53D25 ; 53C22

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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 (non-virtually 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.
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 ...

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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.

14D23 ; 14K99 ; 37F75 ; 11J81 ; 11G18

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Virtualconference  Dynamics of Jouanolou foliation
Deroin, Bertrand (Auteur de la Conférence) | CIRM (Editeur )

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 non-trivial domain of discontinuity. This result is opposed to a series of results beginning in the sixties with the works of Hudai-Verenov and Ilyashenko.

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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 non-compact manifold is complete or not (vector fields on compact manifolds are always complete). Some twenty-five 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.
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 non-compact manifold is complete or not (vector fields on compact manifolds are always complete). Some twenty-five years ago, Rebelo realized an exploited the fact that there are local (and not just global!) obstructions for a ...

34M05 ; 57S20 ; 32C99

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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 non-compact manifold is complete or not (vector fields on compact manifolds are always complete). Some twenty-five 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.
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 non-compact manifold is complete or not (vector fields on compact manifolds are always complete). Some twenty-five years ago, Rebelo realized an exploited the fact that there are local (and not just global!) obstructions for a ...

34M05 ; 57S20 ; 32C99

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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 non-compact manifold is complete or not (vector fields on compact manifolds are always complete). Some twenty-five 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.
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 non-compact manifold is complete or not (vector fields on compact manifolds are always complete). Some twenty-five years ago, Rebelo realized an exploited the fact that there are local (and not just global!) obstructions for a ...

34M05 ; 57S20 ; 32C99

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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 non-compact manifold is complete or not (vector fields on compact manifolds are always complete). Some twenty-five 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.
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 non-compact manifold is complete or not (vector fields on compact manifolds are always complete). Some twenty-five years ago, Rebelo realized an exploited the fact that there are local (and not just global!) obstructions for a ...

34M05 ; 57S20 ; 32C99

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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 $K_X0$. Projective manifo...

14E30 ; 37F75 ; 14M22

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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 $K_X0$. Projective manifo...

14E30 ; 37F75 ; 14M22

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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, Calabi-Yau, 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.
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, Calabi-Yau, 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 ...

14E30 ; 37F75

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

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, Calabi-Yau, 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.
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, Calabi-Yau, 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 ...

14E30 ; 37F75

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

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, Calabi-Yau, 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.
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, Calabi-Yau, 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 ...

14E05 ; 14E30 ; 37F75

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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, Calabi-Yau, 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.
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, Calabi-Yau, 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 ...

14E30 ; 37F75 ; 14E05

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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 Jean-Pierre 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 pull-back of one of the "canonical foliations" on a Hilbert modular variety. This result remains valid for "logarithmic foliated pairs''.
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 Jean-Pierre Demailly, a distribution with conormal sheaf pseudoeffective is actually integrable and thus defines a c...

37F75

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