10th swiss-french workshop in Algebraic Geometry

Due to Covid restrictions, the workshop takes place 20 - 24 September 2021, in Charmey
(near Gruyères, Fribourg, Switzerland)

Mini-courses

Antoine Chambert-Loir
(Université Paris-Diderot)
Tame topology in number theory and geometry
  
Nicolas Perrin
(Université de Versailles Saint Quentin En Yvelines)
Geometry of spherical and G-varieties
  
Amos Turchet
(Università degli studi Roma Tre)
Diophantine geometry and hyperbolicity properties of algebraic varieties
  

Schedule

Monday  
September 20
Tuesday  
September 21
Wednesday  
September 22
Thursday  
September 23
Friday  
September 24
 






12h30 welcome
 7h30-8h30
  breakfast


8h45-9h45 
 mini-course 1

10h15-11h15 
 mini-course 2

11h45-12h45 
 mini-course 3

 7h30-8h30
  breakfast


8h45-9h45 
 mini-course 1

10h15-11h15 
 mini-course 2

11h45-12h45 
 mini-course 3

 7h30-8h30
  breakfast


8h45-9h45 
 mini-course 1

10h15-11h15 
 mini-course 2

11h45-12h45 
 mini-course 3

 7h30-8h30
  breakfast


8h45-9h45 
 mini-course 1

10h-11h 
 mini-course 2

11h15-12h15 
 mini-course 3

 lunch  lunch  lunch  lunch  bus at 12h42


14h30-15h30 
 mini-course 1

16h00-17h00 
 mini-course 2

17h30-18h30 
 mini-course 3

  19h dinner

time for discussion / enjoying the mountain side

  


17h20-18h10 
  Schefer

18h30-19h20 
  Benedetti

  19h30 dinner

time for discussion / enjoying the mountain side

  


 18h30 dinner

20-20h50 
  Posva

21h10-22h 
  Schmidt

time for discussion / enjoying the mountain side

  


17h20-18h10 
  Bot

18h30-19h20 
  Schneider

 19h30 dinner

Location

VIVA GRUYERE Charmey, Rte des Arses 4, 1637 Charmey

The journey to Charmey is 2h10 from Geneva, 2h30 from Basel/Zürich, 1h30 from Lausanne.
See timetables on www.cff.ch, the bus stop is "Charmey (Gruyère), Le Chêne". The place is very close to the bus stop.

Titles and abstracts

Mini-courses

Antoine Chambert-Loir - Tame topology in number theory and geometry
Summoned by Grothendieck in his Esquisse d'un programme (1985), tame topology is supposed to offer the flexibility of general topology without allowing its “pathological” constructions. Inspired by mathematical logic and real algebraic geometry, o-minimality is one solution to this program, proposed by van den Dries. The works of Peterzil and Starchenko showed that Serre's GAGA principle extends: if it is definable in an o-minimal structure, a complex analytic subset of C^n is necessarily algebraic.
In the last 10 years, these ideas have been made fruitful in number theory, where Zannier, Pila, then Tsimerman, Klingler, Ullmo and Yafaev proved the André-Oort conjecture concerning the geometry of subvarieties of Shimura varieties. An important tool is a counting theorem by Pila and Wilkie for points of R^n with rational coordinates with bounded numerator and denominator lying on a subset which is definable in an o-minimal structure.
Recently, Klingler, Bakker, Tsimerman, Brunebarbe used these ideas in Hodge theory, reproving for example a theorem of Cattani, Deligne and Kaplan regarding the algebraicity of the Hodge loci, or by proving a conjecture of Griffiths about the quasi-projectivity of the images of period maps.
The aim of the lectures is to present these notions of diverse origins and, as far as possible, to describe how they interact.

 
Nicolas Perrin - Geometry of spherical and G-varieties
For G a reductive group, spherical varieties are the simplest G-varieties from the G-equivariant point of view. They generalise both toric varieties (when G = T is a torus) and projective rational homogeneous spaces (eg. Grassmanians or quadrics). Spherical varieties admit a classification by colored fans extending the classification of toric varieties by fans. In these lectures, I will present basics on the action of a linear (reductive) group G on varieties and important invariants of G-varieties. I will explain how to use these invariants for the classification of spherical embeddings via colored fans and how the geometry of the fan controls the geometry of the variety.

 
Amos Turchet - Diophantine geometry and hyperbolicity properties of algebraic varieties
One of the fundamental problems in Diophantine Geometry is to describe the distribution of rational and integral points on varieties defined over a number field. While the situation is well understood for algebraic curves, in higher dimensions deep conjectures of Lang and Vojta predict that (in analogy with the one dimensional case) the distribution should be linked to geometric properties of the varieties. This mini-course will be a gentle introduction to this topic, with the goal of discussing modern techniques (due to Corvaja, Zannier, Levin, Autissier, Ru, Vojta…) and applications in new interesting cases. During the lectures we will also explain how these arithmetic properties are linked to various notions of hyperbolicity both in algebraic geometry and in complex-analysis.

 

Research talks

Vladimiro Benedetti - Some divisors in the moduli space of Debarre-Voisin varieties
Projective cubic fourfolds provide a remarkable connection between different areas in algebraic geometry. The rationality question of such varieties has been conjectured to be controlled by properties coming from their Hodge structure and derived category. Debarre-Voisin (DV) hypersurfaces share some fundamental properties with cubic fourfolds, among which the fact that one can naturally associate to these Fano varieties some hyper-Kahler fourfolds. As in the case of cubics, the hyper-Kahler fourfold turns out to encode the rational Hodge structure of DV hypersurfaces. In this talk I will focus the attention on some divisors of the moduli space of DV varieties. By studying the geometry of such divisors and the associated hyper-Kahler, I will show how to deduce the integral Hodge conjecture for DV hypersurfaces. This is a joint work with Jieao Song.

 
Anna Bot - A smooth complex rational affine surface with uncountably many real forms
A real form of a complex algebraic variety X is a real algebraic variety whose complexification is isomorphic to X. Up until recently, it was known that many families of complex varieties have a finite number of nonisomorphic real forms. In 2019, Lesieutre constructed an example of a projective variety of dimension six with infinitely many, and now, Dinh, Oguiso and Yu found a projective rational surface with infinitely many as well. In this talk, I’ll present the first example of a rational affine surface having uncountably many nonisomorphic real forms. The first example with infinitely countably many real forms on an affine rational variety is due to Dubouloz, Freudenberg and Moser-Jauslin.

 
Quentin Posva - Gluing theory for stable varieties in positive characteristic
Stable varieties appear as degenerations of canonically polarized smooth varieties, and in characteristic zero they form a compact moduli space. I will report on the study of stable varieties in positive characteristic, in particular on the gluing techniques which allow to navigate between stable varieties, which might be non-normal and not suited for MMP techniques, and their normalizations.
 
Gerold Schefer - A p-adic equidistribution theorem
We will discuss some equidistribution theorems concerning the average value of a function on the Galois orbit of a root of unity. They are related to results of Baker, Ih and Rumely, and Chambert-Loir. Classically the test functions have to be continous, but we allow logarithmic singularities. To get an impression how the p-adic absolute value behaves, we will look at differences of roots of unity and compare it with the archimedian setting.

 
Harry Schmidt - Bounded hight and uniformness in rational dynamics
I will report on joint work with Myrto Mavraki on rational dynamical systems on the projective line. The focus will be particularly on polynomial dynamics.

 
Julia Schneider - Generating the plane Cremona group by involutions
Let k be a perfect field, and consider the birational transformations of the projective plane that are defined over k. I will discuss how to decompose such maps into involutions. In other words, I will show that the plane Cremona group over these fields is generated by involutions. This is joint work with Stéphane Lamy.

 

Participants

(the registration deadline is closed)

Ahmed Hashem Abdelhameed Abouelsaad (Basel)
Jefferson Baudin (EPFL)
Vladimiro Benedetti (Dijon)
Jérémy Blanc (Basel)
Anna Bot (Basel)
Jung Kyu Canci (Lucerne)
Antoine Chambert-Loir (Paris)
Gabriel Dill (Oxford)
Adrien Dubouloz (Dijon)
Marta Dujella (Basel)
Mani Esna Ashari (Basel)
Andrea Fanelli (Bordeaux)
Pascal Fong (Basel)
Pierre-Alexandre Gillard (Dijon)
Philipp Habegger (Basel)
Liana Heuberger (Angers)
Clémentine Lemarié-Rieusset (Dijon)
Irène Meunier (Toulouse)
Lucy Moser-Jausin (Dijon)
Stéphane Lamy (Toulouse)
Frédéric Mangolte (Angers)
Erik Paemurru (Basel)
Nicolas Perrin (Versailles)
Quentin Posva (EPFL)
Javier Alonso Carvajal-Rojas (EPFL)
Linus Rösler (EPFL)
Gerold Schefer (Basel)
Harry Schmidt (Basel)
Julia Schneider (Toulouse)
Roberto Svaldi (EPFL)
Ronan Terpereau (Dijon)
Amos Turchet (Rome)
Christian Urech (EPFL)
Immanuel van Santen (Basel)
Francesco Veneziano (Genova)
Robert Wilms (Basel)
Egor Yasinsky (Basel)
Sokratis Zikas (Basel)
Susanna Zimmermann (Angers)

Organisers

Philipp Habegger (University of Basel)
Ronan Terpereau (University of Burgundy)
Susanna Zimmermann (University of Angers)
Logistic support: Adrien Dubouloz (University of Burgundy)

The swiss-french workshops in Algebraic Geometry

Here are the previous ones:

1st, 2nd , 3rd, 4th, 5th, 6th, 7th, 8th 9th swiss-french workshop in Algebraic Geometry

Financial support

We gratefully acknowledge support from:
ANR FIBALGA
Institut de Mathématiques de Bourgogne
Swiss Academy of Sciences
Swiss mathematical society
University of Basel