Conference in enumerative, real and birational geometry

6 - 10 June in Le Croisic

To see title and abstract click on name of speaker
To see title and abstract click on name of speaker

The ordinal of dynamical degrees of birational maps of the projective plane

For any birational map of the projective plane, one can consider its dynamical degree, which describes the dynamical behaviour of this map. Taking all dynamical degrees of birational maps together, we obtain a well-ordered subset of the real line, and we may ask: what is the ordinal of this set? In this talk I will give an introduction to the dynamical degree and ordinals, and prove that the set of dynamical degrees of all birational maps of the plane is of order type \omega^\omega, where \omega is the first infinite ordinal.

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Fewnomial bounds for real varieties and singularities

The topology of a real algebraic variety can be estimated by the topology of its complex part, which is roughly speaking determined by the degree, but finer estimates can be obtained in some cases by the complexity of the defining equations, typically, the the Newton polytopes or the number of monomials. In this talk, I will overview what is known on the subject and will also present works in progress.

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Algebraic models of the real affine plane and some foliated open surfaces

A fake real plane is a smooth geometrically integral real algebraic surface whose real locus is diffeomorphic to an open disc, whose complexification has the reduced rational homology type of a point, but is not algebraically isomorphic to the affine 2-space over the reals. I will review examples and some techniques to construct and study isomorphism classes of such fake planes up to biregular isomorphisms and birational diffeomorphisms which were developed in a series of joint papers with Frédéric Mangolte and Jérémy Blanc. Time permitting, I will also discuss some extensions of these methods towards the understanding of rational algebraic models of certain open differential surfaces foliated by closed real lines (work in no real progress with Frédéric Mangolte).

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Saturation, seminormalization and homeomorphisms of algebraic varieties

(joint work with François Bernard, Jean-Philippe Monnier and Ronan Quarez) We address the question under which conditions a bijective morphism between complex algebraic varieties is an isomorphism. Our two answers involve the seminormalization and saturation for morphisms between varieties, together with an interpretation in terms of continuous rational functions on the complex points of the variety. We propose also a version for algebraic varieties defined on an algebraically closed field of characteristics zero.

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About the linearity of finitely generated subgroups of the plane polynomial automorphism group

A group G is said to be linear if there exists a field K, an integer d, and an injective homomorphism from G to the general linear group GL(d,K). If L is a field, denote by Aut_L(A^2) the group of polynomial automorphisms of the affine plane over L. We address the following question raised by Cantat and Cornulier : Are finitely generated subgroups of Aut_L(A^2) necessarily linear? This is a work in progress with Rémi Boutonnet.

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Comparing bigraded and equivariant cohomology

Bigraded cohomology is defined as hypercohomology of tensor powers of the exponential morphism on the Klein quotient of a real algebraic variety. Vector bundles have Chern classes with values in these bigraded cohomology groups. There is a natural morphism from brigraded cohomology to Borel-Grothendieck equivariant cohomology of the complexification of the real algebraic variety. We show that the images of the aforementioned Chern classes of a vector bundle coincide with its equivariant Chern classes, as defined by Khan and Krasnov.

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Wall-crossing invariance of certain combinations of Welschinger numbers.

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Normal subgroups in the Cremona group, old and new

In the last decade many normal subgroups were found inside the plane Cremona group, following two distinct strategies: small cancellation theory or groupoid presentation using Sarkisov links.
In this talk I will mostly focus on the small cancellation side, and explain how the works of various people, including Lonjou, Shepherd-Barron and Urech, lead to a clean description of the loxodromic elements g in the Cremona group such that a high power of g generates a proper normal subgroup.

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Tropical homology over discretely valued fields.

The talk is about a work in progress with Emiliano Ambrosi. Ilia Itenberg, Ludmil Katzarkov, Grigory Mikhalkin and Ilia Zharkov proved in “Tropical homology” that for a smooth proper family of complex varieties over the punctured disk with smooth tropicalisation X the Hodge numbers of the general fiber coincide with the dimensions of the tropical homology groups of X. We explore the possibility of extending this result over more general discrete valued fields of arithmetic interest, such as R((t)) or Qp, the field of p-adique numbers. In the process of doing this, we get an action of the Galois group on the tropical homology groups and we compare this action, in certain cases, with the action defined by Tyler Foster in “Galois actions on analytifications and tropicalisation”.

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Equivariant real structures of almost homogeneous three-dimensional SL(2)-varieties

In this talk, I will discuss a method of describing the real structures of almost homogeneous SL(2)-varieties which are compatible with each of the two real structures of the complex group SL(2). We start with the case of homogeneous varieties of the form SL(2) modulo a finite subgroup. Then, using the combinatorial Luna-Vust theory, we show how to treat the case of almost homogeneous threefolds. I will then discuss how this study relates to results of real structures on spherical varieties. This is joint work with Ronan Terpereau.

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Seminormalization and saturation for morphisms

Given a morphism between algebraic varieties, we introduce the associated notions of seminormalization and Lipman-Grothendieck saturation. We show that these two objects coincide when the morphism is finite but differ in general. These are tools used in the study of homeomorphisms between algebraic varieties (talk by G. Fichou). This is joint work with François Bernard, Goulwen Fichou and Ronan Quarez.

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Maximality of moduli spaces of vector bundles on curves

A basic topological property of real algebraic varieties says that the total mod 2 Betti number of the real locus is smaller than the total mod 2 Betti number of the complex locus, and varieties for which this inequality is an equality are called maximal. In some cases, maximality can be detected via a stronger property, involving the Hodge numbers of the variety. In joint work with Erwan Brugallé, we show that this is the case for moduli spaces of vector bundles on a non-singular real projective curve. This provides a new family of maximal varieties, with members of arbitrarily large dimension.

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Orthogonal groups in plane Cremona groups

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To see title and abstract click on name of speaker

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Topology and enumeration of real planar algebraic curves

The topology of a real planar algebraic curve is described in the neighbourhood of a singularity by a combinatorial invariant, namely a chord diagram. Most chord diagrams do not arise as such: we will characterize which ones do, and enumerate them.
Then we will introduce the concept of combinatorial curves, enriching that of combinatorial maps, in order to describe the global topology of connected singular algebraic curves in the real sphere.
From there, we shall count those topological types and deduce a bound on the number of connected singular algebraic curves of a given degree in the real projective plane.

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Typical topology of real polynomial singularities

In contrast with to the complex world, it makes sense to study the topology of real polynomials from a probabilistic point of view. I will present a general method to deal with the limit probability of differential geometric events (for instance: the probability of having a certain number of critical points; the probability that a level set is diffeomorphic to some fixed closed manifold; etc.), as the degree grows to infinity. Applying it to the case of Kostlan polynomials (a particularly nice model of random polynomials) one obtains an improvement of the results by Gayet and Welschinger about random hypersurfaces.

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Real forms of nilpotent orbits (and their closures) in complex semi-simple Lie algebras

Let G be a semi-simple complex algebraic group, which acts on its Lie algebra L(G) via the adjoint action, and let X be (the closure of) a nilpotent orbit in L(G). In this talk we will be interested in the real forms of X, i.e. the real algebraic varieties endowed with a real algebraic group action whose complexifications are isomorphic to X as G-varieties. This is a joint work with Michael Bulois and Lucy Moser-Jauslin (arXiv:2106.04444).

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Pinched handle decompositions of simplicial complexes

I will introduce a notion of pinched handle decompositions for finite simplicial complexes and prove their existence after finitely many stellar subdivisions at facets. This holds true in particular for closed triangulated manifolds regardless of being piecewise linear or not. I will then relate these decompositions to discrete Morse functions.

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Birational involutions of the projective plane

Birational involutions of the projective plane (or, equivalently, automorphisms of the field of rational functions in two variables of order 2) were studied already by the Italian school of algebraic geometry — Bertini, Castelnuovo, and Enriques. However, their explicit and complete description was obtained by Beauville and Bayle only in 2000 and only in the case of a complex projective plane. It turns out that for planes over algebraically non-closed fields the situation is much more complicated. I will talk about the joint work with I. Cheltsov, F. Mangolte and S. Zimmermann, in which we classified birational involutions of the real projective plane.



 

Registration open. By email to susanna.zimmermann-AT-XunivX-angers.fr (delete the letter X).

Schedule and location

The conference will be hosted at the Domaine port aux rocs in Le Croisic (France).
Schedule to be announced.

Monday  
6 June
Tuesday  
7 June
Wednesday  
8 June
Thursday  
9 June
Friday  
10 June


 

 



 9h-9h50 
  Bihan

 10h10-11h 
  Moser-Jauslin

 11h20-12h10 
  Furter



 9h-9h50 
  Kharlamov

 10h10-11h 
  Dubouloz

 11h20-12h10 
  Manzaroli



 9h-9h50 
  Fichou

 10h10-11h 
  Bot

 11h20-12h10 
  Simon



 9h50-11h 
  Terpereau

 11h15-12h05 
  Schneider



 15h40-16h30 
  Huisman

 17h-17h50 
  Lamy




 14h30-15h20 
  Shaw

 15h50-16h40 
  Schaffhauser




 14h30-15h20 
  Stecconi

 15h50-16h40 
  Monnier




 14h30-15h20 
  Yasinsky

 15h50-16h40 
  Welschinger




 

 


Participants

We are fully booked out.

Eduardo Alves da Silva (IMPA)
Michele Ancona (Strasbourg)
Turgay Akyar (METU)
François Bernard (Angers)
Aurore Boitrel (Angers)
Thomas Blomme (Geneva)
Anna Bot (Basel)
Frédéric Bihan (Savoie Mont Blanc)
Erwan Brugallé (Nantes)
Thibaut Chailleux (Angers)
Jules Chenal (Lyon)
Jean-Baptiste Campesato (Angers)
Alex Degtyarev (Bilkent)
Adrien Dubouloz (Dijon)
Nicolas Dutertre (Angers)
Goulwen Fichou (Rennes)
Pascal Fong (Basel)
Jean-Philippe Furter (Bordeaux)
Maria Rosario Gonzalez-Dorrego (Madrid)
Ilia Itenberg (Jussieu)
Andrés Jaramillo Puentes (Essen)
Yann Le Dreau (Rennes)
Frédéric Mangolte (Angers/Marseille)
Léo Mathis (Trieste)
Irène Meunier (Basel/Toulouse)
Jean-Philippe Monnier (Angers)
Johannes Huisman (Brest)
Stéphane Lamy (Toulouse)
Cédric Le Texier (Toulouse)
Kostya Loginov (HSE Moscow)
Anh Nguyen (Nantes)
Matilde Manzaroli (Thübingen)
Lucy Moser-Jauslin (Dijon)
Javier Orts (Lisbon)
Renata Picciotto (Angers)
Quentin Posva (Lausanne)
Marie-Françoise Roy (Rennes)
Johannes Rau (Univ. de los Andres)
Sumit Roy (Pohang)
Julia Schneider (Toulouse)
Kris Shaw (Oslo)
Christopher-Lloyd Simon (ENS Lyon)
Michele Stecconi (Nantes)
Ronan Terpereau (Dijon)
Christian Urech (EPFL Lausanne)
Jean-Yves Welschinger (CNRS, Uni Lyon 1)
Egor Yasinski (Polytechnique)
Susanna Zimmermann (Angers)

Organisation

Organisors / scientific committee

Erwan Brugallé (Nantes)
Frédéric Mangolte (Angers/Marseille)
Susanna Zimmermann (Angers)
 
 
 
 
 
 
 
 
 
 
 

  

Financial support:

ANR FIBALGA
ANR ENUMGEOM
Fédération de recherche Mathématiques des Pays de Loire
GDR GAGC
GDR Singularité et Applications
LAREMA
Région Pays de la Loire (Étoiles Montantes)
SFR MathSTIC
Centre Henri Lebesgue