
Second SwissMAP Geometry&Topology conference  Diablerets 2015
Hôtel Les Sources, Les Diablerets, Switzerland, June 2226, 2015,
Speakers
In the morning, there will be two minicourses on dynamics in birational geometry, given by
Eric BEDFORD
Jeffrey DILLER

Indiana
Notre Dame

In the afternoon, we will have research talks of 1 hours. Speakers include
Sebastian BAADER
Ivan CHELTSOV
Julie DESERTI
Adrien DUBOULOZ
Vincent EMERY
Sergey GALKIN
Andriy REGETA
Susanna ZIMMERMANN

Bern
Edinburgh
Paris
Dijon
Bern
Moscow
Basel
Basel

Schedule
Participants are expected to arrive on Sunday for the dinner and go back on Friday after lunch. (Tell us if this is not the case).
Talks: In Les Diablerets, ( Hôtel Les Sources )

Monday
June 22

Tuesday
June 23

Wednesday
June 24 
Thursday
June 25 
Friday
June 26 
9h3010h30 
Minicourse
E. Bedford

Minicourse
E. Bedford 
Minicourse
J. Diller 
Minicourse
J. Diller 
Talk
J. Déserti 
11h0012h00 
Minicourse
J. Diller

Minicourse
J. Diller 
Minicourse
E. Bedford 
Minicourse
E. Bedford 
Talk
S. Galkin 
16h3017h30 
Talk
V. Emery

Talk
S. Baader 

Talk
A. Regeta 
17h4518h45 
Talk
S. Zimmermann

Talk
A. Dubouloz 

Talk
I. Cheltsov 
Minicourses  titles and abstracts
Eric BEDFORD  Dynamics of Birational Maps 

In this series of lectures we intend to outline some aspects of the dynamical theory of birational maps (in dim 2 and higher) which are at least close to being automorphisms. A general reference is [0] below. Specifically, we plan to work with the following topics:
1. The dynamics of the Hénon family of automorphisms of 2 (see [1], [2]). The Hénon family consists of (holomorphic) polynomial diffeomorphisms of 2, which have
birational extensions to 2. The purpose of this lecture(s) is to present the questions and
some methods that have been applied successfully to the case of biregular maps. This is intended to give a background which will illuminate the difficulties which arise in the
presence of indeterminate points.
2. A rather general discussion of dynamical degrees of rational maps (see [3]). These
are perhaps the most invariants of birational conjugacy. In (complex) dimension k, there
are dynamical degrees δ_{ℓ} in all codimensions 1≤ℓ≤k. The most basic of these is the first dynamical degree δ_{1}, but we will discuss the others, too.
3. Discussion of pseudoautomorphisms in dimension 3 (and higher) (see [4]). These are birational maps f for which neither f nor f 1 has an exceptional hypersurface. Under
iteration, these behave almost as well as automorphisms, but the indeterminacy locus can cause problems. We will illustrate this with examples.
[0]  E. Bedford, Invertible dynamics on blowups of k, arXiv:1411.0760 
[1]  E. Bedford and J. Smillie, Polynomial diffeomorphisms of 2: currents, equilibrium measure and hyperbolicity. Invent. Math. 103 (1991), no. 1, 6999. 
[2]  E. Bedford and J. Smillie, Polynomial diffeomorphisms of 2. II. Stable manifolds and recurrence. J. Amer. Math. Soc. 4 (1991), no. 4, 657679. 
[3]  E. Bedford, The dynamical degrees of a mapping. Proceedings of the Workshop Future Directions in Difference Equations, 313, Colecc. Congr., 69, Univ. Vigo, Serv. Publ., Vigo, 2011. 
[4]  E. Bedford and K. Kim, Dynamics of (pseudo) automorphisms of 3space: periodicity versus positive entropy. Publ. Mat. 58 (2014), no. 1, 65119. 
[5]  F. Perroni and DQ Zhang, Pseudoautomorphisms of positive entropy on the blowups of products of projective spaces. Math. Ann. 359 (2014), no. 12, 189209. 
[6]  E. Bedford, J. Diller, and K. Kim, Pseudoautomorphisms with invariant elliptic curves, arXiv:1401.2386 

Jeffrey DILLER  Birational maps and dynamics on rational surfaces 

In this series of introductory lectures I will survey work done toward understanding dynamics of plane birational maps. My main goal will be to describe how the dynamics of such a map is determined by the linear pullback action it induces on the Picard group of an appropriately chosen rational surface.
Beginning with some generalities about rational maps and rational
surfaces, I will describe the important notions of algebraic stability and first dynamical degree. I will then describe the dynamical classication of birational surface maps by "degree growth" presented in the article [DF] by Favre and myself.
Next I will describe the way in which cohomological information
encapsulated by the pullback action induced by a birational selfmap
leads to an understanding of the ergodic theory of the map. A particular example [BD] studied by Bedford and myself will serve as useful
motivation for this. To deal with more general birational maps, I will introduce the notions of Green current, laminarity and geometric intersection of positive closed currents.
Any remaining time will be devoted to surveying more recent work
on dynamics of birational maps, perhaps including results about noninvertible rational maps, maps with invariant two forms, or the construction of automorphisms on blowups of the projective plane.
[BD]  Eric Bedford and Jeffrey Diller. Real and complex dynamics of a family of birational maps of the plane: the golden mean subshift. Amer. J. Math. 127 (2005), 595646. 
[DF] 
Jeffrey Diller and Charles Favre. Dynamics of bimeromorphic maps of surfaces.
Amer. J. Math. 123 (2001), 11351169. 
[Dil] 
Jeffrey Diller. Cremona transformations, surface automorphisms, and plane cubics.
Michigan Math. J. 60 (2011), 409440. With an appendix by Igor Dolgachev. 
[DDG] 
Jeffrey Diller, Romain Dujardin, and Vincent Guedj.
Dynamics of meromorphic maps with small topological degree I: from cohomology to currents. Indiana Univ. Math. J. 59 (2010), 521561. 
[DL] 
Jeffrey Diller and JanLi Lin. Rational surface maps with invariant meromorphic two forms. To apppear in Mathematische Annalen. Preprint available at arxiv.org. 

Talks  titles and abstracts
Sebastian BAADER  A partial order on plane curve singularities 

Links associated with plane curve singularities come with a natural fibre surface in the 3sphere. These are partially ordered by inclusion. Determining this order is interesting and difficult, possibly even equivalent to the deformation problem for plane curve singularities.

Ivan CHELTSOV  Cylinders in del Pezzo surfaces 

For a projective variety X and an ample divisor H on it, an Hpolar cylinder in X is an open ruled affine subset whose complement is a support of an effective divisor rationally equivalent to H.
This notion links together affine, birational and Kähler geometries.
I will show how to prove existence and nonexistence of Hpolar cylinders
in smooth and mildly singular del Pezzo surfaces (for different polarizations).
The obstructions comes from log canonical thresholds and Fujita numbers.
As an application, I will answer an old question of Zaidenberg and Flenner
about additive group actions on the cubic Fermat affine threefold cone.
This is a joint work with Jihun Park (POSTECH) and Joonyeong Won (KAIST).

Julie DESERTI  About a family of Jonquières maps 

In this talk I will consider the family of birational maps of the complex projective plane f_{α,β}=((α x+y)/(x+1),β y) with α, β in and α=β=1. I will give the degree growth of such maps, and prove that there are two domains of linearization with different behaviors. Finally I will adopt an other point of view and compute, using the results of Avila on SL(2,)cocycles, the Lyapunov exponant of the cocycle associated to f_{α,β}.

Adrien DUBOULOZ  Around the Cancellation Problem: affine spaces and algebraic tori 

The Cancellation Problem, usually attributed to Zariski, asks whether an algebraic variety whose product with an affine space is an affine space is an affine space itself. The answer to this question is positive in dimensions 1 and 2 and was recently shown to be negative in positive characteristic in dimension 3. In characteristic zero, it remains widely open starting from dimension 3, illustrating in particular the lack of effective characterization of affine spaces among algebraic varieties. One can ask more generally whether two varieties whose products with an affine space are isomorphic are isomorphic themselves; in the affine case, this amounts to asking whether the coefficient ring of a polynomial ring is uniquely determined by its structure. Many counterexamples to this generalized problem are known starting from dimension 2, all arising as locally trivial affine bundles over uniruled varieties.
A natural variant from the algebraic point of view is to replace polynomial rings by rings of Laurent polynomials, the question becoming whether the coefficient ring of such a ring is uniquely determined by its structure. In geometric terms this corresponds to asking whether two varieties whose products with an algebraic torus are isomorphic are isomorphic themselves. The answer turn out to be positive in the case where one of the varieties is itself an algebraic torus, but the general case is more intricate. After giving a short survey of classical and more recent results and counterexamples to the classical Zariski Cancellation Problem for affine spaces, I will focus on the variant for algebraic tori and explain some of the tools involved in the construction of varieties which fail cancellation.

Vincent EMERY  Volumes of hyperbolic manifolds 

For n even, the (generalized) GaussBonnet theorem gives a lot of
information about the volume spectrum of hyperbolic nmanifolds. In
contrast, the situation for n odd appears to be much more complicated
(but also much more interesting). I will discuss some aspects, in
particular the question of commensurability of volumes in dimension n=3.

Sergey GALKIN  Calculus of algebraic dynamics 

First I recall how one can do calculus of algebraic geometry with values
in Grothendieck's ring of varieties, with examples of formulas,
and relation to birational geometry.
Then I explain what are the higher Ktheory counterparts to Grothendieck's ring of varieties, what is the explicit description of K_{1}(Vars),
and what could be a calculus of automorphisms there.
Finally, I explain a tentative construction of the Hochschild homology counterpart, HH(Vars). I hope one should be able to do calculus of endomorphisms in this group, and I will explain how to twist Chern characters of known formulas in K_{0}(Vars) by endomorphisms.
Definition of higher K_{i}(Vars) and an explicit construction of K_{1}(Vars) was explained to me by Evgeny Shinder three years ago. It was independently found by Inna Zakharevich about the same time, and it appears in her recent preprints 1506.06197,
1506.06200

Andriy REGETA  Characterization of some affine toric varieties 

Recently Hanspeter Kraft proved that the automorphism group of the affine nspace n determines n up to isomorphism. In this talk we present a similar result about the quotient n/μ_{d}, where μ_{d}=<ξ∈*ξd=1> is a cyclic group of order d: if X is
an irreducible affine normal variety such that Aut(X)=
Aut(n/μ_{d}) as indgroups,
then X =n/μ_{d}
as varieties. If X is not necessarily normal then we classify all
such X with an isomorphism Aut(X)=
Aut(n/μ_{d}) of indgroups.

Susanna ZIMMERMANN  Abelian quotients of the real Cremona groups 

The Cremona group of the complex plane contains many normal subgroups, all of which are of uncountable infinite index. There is no proper normal subgroup containing a nontrivial element of degree 1, 2, 3, or 4.
What about the Cremona group of the real plane?
I will present an abelian quotient of it, which implies the existence of normal subgroups of index equal to any given power of 2, all of them containing every map of degree 1, 2, 3, and 4.

How to come The journey to Les Diablerets is around 2 hours from Geneva, 3h from Basel/Zurich. See timetables on www.cff.ch.
The train station of Les Diablerets is at 15 minutes by foot from the Hotel.
Participants
Sebastian Baader (Bern)
Eric Bedford (Indiana)
Jung Kyu Canci (Basel)
Cinzia Bisi (Ferrara)
Jérémy Blanc (Basel)
Ivan Cheltsov (Edinburgh)
Jeffrey Diller (Notre Dame)
Julie Déserti (Paris)
Adrien Dubouloz (Dijon)
Vincent Emery (Bern)
JeanPhilippe Furter (La Rochelle)
Sergey Galkin (Moscow)
Isac Héden (Basel)
Johannes Josi (Geneva/Paris)
Nikita Kalinin (Geneva)
Grigory Mikhalkin (Geneva)
PierreMarie Poloni (Basel)
Andriy Regeta (Basel)
Maria Fernanda Robayo (Basel)
Immanuel Stampfli (Bremen)
Christian Urech (Basel)
Francesco Veneziano (Basel)
Susanna Zimmermann (Basel)
