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# Complex Geometry, Analysis and Foliations

de au (Europe/Paris)
à International Center for Theoretical Physics ( Adriatico Building, Kastler Lecture Hall )
Strada Costiera, 11 I - 34151 Trieste Italy
 Description Our colleague Marco Brunella passed away suddenly in January 2012. Yet his profound, creative mathematics continues to have an impact on geometers and analysts. He would have been 50 years on September, 28th, 2014. This conference honours his memory. We have invited some of the best specialists in Complex Geometry, Analysis and Foliations to give talks about the latest developments of their field. They all have been influenced by Marco's work.  Documents: Contact Email: brunella.ictp@gmail.com
Go to day
• lundi 29 septembre 2014
• 08:30 - 09:30 Registration ( Adriatico Building, Office N°1 )
• 09:30 - 10:00 Opening
Short presentation by A. Verjovsky and then by C. Camacho.
• 10:00 - 11:00 Marco Brunella and the curvature of canonical line bundles 1h0'
The talk will present some striking results of Marco Brunella concerning the curvature of canonical and anticanonical line bundles of compact Kähler manifolds, and of foliations on such manifolds. These results all contain very deep ideas, and several ones are connected to important unsolved conjectures. We will try to give an overview of some of them.
 Intervenant: Jean-Pierre DEMAILLY (Grenoble)
• 11:00 - 11:30 Break
• 11:30 - 12:30 Index theorems and geodesic flow for meromorphic connections along foliations 1h0'
The study of meromorphic connections on Riemann surfaces is a classical topic, related for instance to the 21st Hilbert problem.
In this talk I shall introduce a novel point of view, with unexpected analytic, geometric and dynamical applications. More precisely, I shall show how to associate to holomorphic maps having a positive-dimensional fixed point set a foliation in Riemann surfaces with meromorphic connections along the leaves, and how to use this structure to prove several index theorems generalizing and extending both the classical holomorphic Lefschetz index theorem and the Camacho-Sad index theorems for foliations. Furthermore, I shall describe how to study with analytical and geometrical techniques the geodesic flow associated to a meromorphic connection, with the aim of describing the asymptotic behavior of the real geodesic defined by the connection. Finally, I shall describe a few applications of these results to the study of the dynamics of germs tangent to the identity, to the study of the flow of homogeneous vector fields, and to the study of meromorphic self-maps of the complex projective space.
 Intervenant: Marco ABATE (Pisa)
• 12:30 - 14:00 Lunch
• 14:00 - 15:00 Taxonomy of Class VII Surfaces 1h0'
In Kodaira's classification of compact complex surfaces Class VII hasn't been yet completely understood. An important part of Marco Brunella's mathematical work deals with dynamical properties of class VII surfaces. Part of this work was published posthumously.
In this talk we present new ways of subclassifying class VII surfaces, in which Marco's ideas and results play an important role. We also sketch the first steps of a tentative study of these surfaces based on the properties of their closed positive currents (work in progress together with Ionut Chiose).
 Intervenant: Matei TOMA (Nancy)
• 15:00 - 15:30 Coffee Break
• 15:30 - 16:30 Holomorphic foliations and invariant currents. 1h0'
In this talk, we will point out some numerical properties of codimension 1 foliations on projective manifolds which ensure the existence/inexistence of  holonomy invariant positive current.
 Intervenant: Frédéric TOUZET (Rennes)
• 16:30 - 17:00 Break
• 17:00 - 18:00 On deformations of elliptic fibrations, according to Cayley, Cremona, Halphen and Brunella. 1h0'
In the  first part of the talk I'll take the risk of doing  history of  mathematics, presenting results of Cayley, Cremona and Halphen on deformations of elliptic fibrations (without sections). After,  I'll  show some experiments  of degenerations of  the  configurations  treated by these  authors.  At last,  I'll  give  some ideas  of   Brunella's   general  result  on deformations of elliptic  fibrations and  singular holomorphic foliations,  which is not  widely diffused.

 Intervenant: Luis Gustavo MENDES (Porto Alegre)
• mardi 30 septembre 2014
• 10:00 - 11:00 Points singuliers de feuilletages holomorphes en dimension 3. 1h0'
IL s'agit d'un travail avec Alcides Lins neto et Marianna Vago où l'on donne une description des types de singularités modulo la connaissance de leur partie initiale.

 Intervenant: Dominique CERVEAU (Rennes)
• 11:00 - 11:30 Break
• 11:30 - 12:30 Circle actions on the 7-sphere with unbounded periods and non-linearizable multicentres 1h0'
We give an example of a free circle action on the 7-dimensional sphere whose orbits have unbounded lenghts (equivalently: unbounded periods).As an application we construct a smooth vector field X in a neighbourhood U of the origin in the 8-dimensional real space such that : U - (0) is foliated by closed integrale curves, the differential DX(0) generate  a 1-parametri group of rotations, but X is not orbitally equivalent to its linearization at the origin, hence proving that Poincare' Centre Theorem, true for planar non-degenerate centers is not generalizable in 8 dimensioni.
 Intervenant: Massimo VILLARINI (Modena)
• 12:30 - 14:00 Lunch
• 14:00 - 15:00 From umbilical foliations to the plurisubharmonic variation of the Poincaré metric 1h0'  Intervenant: Jorge Vitorio PEREIRA (IMPA)
• 15:00 - 15:30 Coffee Break
• 15:30 - 16:30 From character varieties to isoperiodic foliations: a transfer principle 1h0'
Schiffer variations are surgery operations that takes an abelian differential on a curve to another one with the same periods. Viewed in the moduli space of abelian differentials of a fixed genus g>=2, they draw a complex algebraic foliation of dimension 2g-3, called the isoperiodic foliation. Its transverse structure is modelled on an open set contained in the group of complex periods, on which the mapping class group acts via the symplectic group. We will see that the (rich) dynamical properties of this latter are also satisfied by the isoperiodic foliation: this phenomenon is what we call the transfer principle. The fact that it holds relies on the connectivity of certain moduli spaces of abelian differentials on curves with prescribed periods.
This is a work in collaboration with Gabriel Calsamiglia and Stefano Francaviglia.
 Intervenant: Bertrand DEROIN (Paris)
• 16:30 - 17:00 Break
• 17:00 - 18:00 A generalization of Malmquist's theorem 1h0'
 The centennial theorem of Malmquist states that a non-autonomous algebraic ordinary differential equation of the first order having an entire solution is in fact a Riccati equation. We will speak about related results concerning algebraic differential equations having at least one single-valued solution.

• mercredi 1 octobre 2014
• 09:20 - 09:50 Brunella's Local Alternative 30'
It is a local version of a conjecture of Brunella which says that a codimension 1 foliation in the projective three-dimensional space P^3 either has an invariant algebraic surface or each leaf is sub-foliated by a one-dimensional foliation. In this local take, we have the following "local conjecture": a germ of holomorphic codimension 1 foliation in C^3,0 either possesses a germ of analytic invariant surface, or there exists a neighborhood of the origin wherein each leaf contains a germ of analytic curve. We give a positive answer to this local conjecture for certain types of foliations.

 Intervenant: Marianna RAVARA VAGO (Rennes)
• 09:50 - 10:00 Break
• 10:00 - 11:00 Kähler threefolds without subvarieties 1h0'
Let $M$ be a compact Kahler manifold without non-trivial complex subvarieties. Using Brunella's alternative for holomorphic foliations, Nadel's vanishing theorem and Demailly's regularization of positive currents, we prove that $M$ is a compact torus. This is a joint work with F. Campana and J.-P. Demailly.
 Intervenant: Misha VERBITSKY (Moscow)
• 11:00 - 11:30 Break
• 11:30 - 12:30 Foliations with a compact leaf 1h0'
We investigate foliations on projective surfaces having a compact leaf.
This is a joint work in progress with Benoît Claudon, Jorge Vitorio Pereira and Frédéric Touzet.
 Intervenant: Frank LORAY (Rennes)
• 12:30 - 14:00 Lunch
• 14:00 - 15:00 On periodic orbits in complex billiards 1h0'
A conjecture of Victor Ivrii (1980) says that in every billiard with smooth boundary the set of periodic orbits has measure zero. This conjecture is closely related to  spectral theory. Its particular case  for triangular orbits was proved  by M. Rychlik (1989),  Ya. Vorobets (1994) and other mathematicians, and for quadrilateral orbits in our joint work with Yu. Kudryashov (2012).
We present a new approach to planar Ivrii's conjecture for billiards with piecewise-analytic boundary: to study its complexified version with reflections from holomorphic curves. The direct complexification of Ivrii's conjecture is false in general.
It would be interesting for real applications to classify the counterexamples: complex billiards with open sets of periodic orbits of a given period.  We will show that the only "nontrivial" counterexamples with four reflections are formed by couples of confocal conics. We will discuss a small result concerning odd number of reflections. We provide applications of these results to real billiards, including Plakhov's Invisibility Conjecture and Tabachnikov's commuting billiard problem.
 Intervenant: Alexey GLUTSYUK (Lyon and Moscow)
• 15:00 - 15:30 Coffee Break
• jeudi 2 octobre 2014
• 09:20 - 09:50 Complete vector fields on affine surfaces 30'
see joint pdf.
 Intervenant: Matthias LEUENBERGER (Bern) Documents:
• 09:50 - 10:00 Break
• 10:00 - 11:00 Marco and the theory of Anosov flows in dimension 3 1h0'
At the beginning of his career, Marco Brunella published five papers related to Anosov flows in dimension 3.
These papers had a great influence on the subsequent development of the theory.
I would like to review these papers and present the present status of the question.
 Intervenant: Etienne GHYS (Lyon)
• 11:00 - 11:30 Break
• 11:30 - 12:30 Foliations and webs inducing Galois coverings 1h0'
Motivated by previous work of Cerveau and Déserti, we introduce the notion of Galois holomorphic foliation
on the complex projective space as those whose Gauss map is a Galois covering when restricted to an appropriate Zariski open subset.
We characterize Galois foliations on $\mathbb P^2$ belonging to certain classes, which include homogeneous
foliations and we give a geometric characterization of Galois foliations in terms of their inflection divisor and their singularities.

 Intervenant: Marcel NICOLAU (Barcelona)
• 12:30 - 14:00 Lunch
• 14:00 - 15:00 The bi-disc theorem 1h0'
 The identification of surfaces with negative Kodaira dimension which are not fibred in rational curves with the natural foliations on bi-disc quotients can reasonably be considered the centre piece of the classification of foliated surfaces. It was very much a collaborative effort with Marco, and, curiously, I have never given a talk specifically devoted to this theorem.
 Intervenant: Michael McQUILLAN (Roma)
• 15:00 - 15:30 Coffee Break
• 15:30 - 16:30 On cohomological invariants of complex manifolds 1h0'
We will focus on algebraic aspects of the $\delta_1\delta_2$-Lemma for bounded double complexes, characterizing it in terms of special cohomologies.
We will apply such a result to complex and symplectic manifolds.
We will also report on some results on Dolbeault-Massey triple products.
• 16:30 - 17:00 Break
• 17:00 - 18:00 Germs of singular holomorphic two dimensional foliations 1h0'
see joint pdf.
 Intervenant: Alcides LINS NETO (Rio de Janeiro) Documents:
• vendredi 3 octobre 2014
• 09:20 - 09:50 Smooth foliations on compact homogeneous kähler varieties 30'
see joint pdf.
 Intervenant: Federico LO BIANCO (Rennes) Documents:
• 09:50 - 10:00 Break
• 10:00 - 11:00 The Green-Griffiths locus of quotients of bounded symmetric domains 1h0'
The Green-Griffiths locus is a closed subset of a compact projective manifold which contains the image of all entire curves contained in the manifold. In this talk we shall describe this locus for compact quotients of bounded symmetric domains. It turns out that the following dichotomy holds : either the uniformizing bounded symmetric domain is the ball and the Green-Griffiths locus is empty, or the Green-Griffiths locus is the whole manifold. This is a joint work with E. Rousseau.


 Intervenant: Simone DIVERIO (Paris)
• 11:00 - 11:30 Break
• 11:30 - 12:30 On bihermitian structures on Kato surfaces 1h0'
There has been much works recently on bihermitian structures on compact complex surfaces, especially in the Kähler case in relation with generalized Kähler geometry. On the other hand , for non-Kähler surfaces we have so far still rather few examples. Recently, however, Apostolov, Bailey and Dloussky have obtained a new nice sufficient condition for their existence. In this talk I will explain how their resut can produce, together with the result of Brunella on locally conformally Kähler metrics on Kato surfaces, new examples of bihermitian structures on some parabolic Inoue surfaces and intermediate Kato surfaces.
This is a joint work with M. Pontecorvo.
 Intervenant: Akira FUJIKI (Osaka)
• 12:30 - 14:00 Lunch