Jérémy Blanc - Universität Basel - Mathematik
Second SwissMAP Geometry&Topology conference - Diablerets 2015
Hôtel Les Sources, Les Diablerets, Switzerland, June 22-26, 2015,
In the morning, there will be two mini-courses on dynamics in birational geometry, given by
Jeffrey DILLER
Notre Dame

In the afternoon, we will have research talks of 1 hours. Speakers include
Sebastian BAADER
Vincent EMERY

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 )
June 22
June 23
June 24
June 25
June 26
9h30-10h30 Mini-course
 E. Bedford
 E. Bedford
 J. Diller
 J. Diller
 J. Déserti
11h00-12h00 Mini-course
 J. Diller
 J. Diller
 E. Bedford
 E. Bedford
 S. Galkin
16h30-17h30 Talk
 V. Emery
 S. Baader
 A. Regeta
17h45-18h45 Talk
 S. Zimmermann
 A. Dubouloz
 I. Cheltsov

Mini-courses - 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 in all codimensions 1≤ℓ≤k. The most basic of these is the first dynamical degree δ1, but we will discuss the others, too.
   3. Discussion of pseudo-automorphisms 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 blow-ups 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, 69-99.
[2] E. Bedford and J. Smillie, Polynomial diffeomorphisms of 2. II. Stable manifolds and recurrence. J. Amer. Math. Soc. 4 (1991), no. 4, 657-679.
[3] E. Bedford, The dynamical degrees of a mapping. Proceedings of the Workshop Future Directions in Difference Equations, 3-13, Colecc. Congr., 69, Univ. Vigo, Serv. Publ., Vigo, 2011.
[4] E. Bedford and K. Kim, Dynamics of (pseudo) automorphisms of 3-space: periodicity versus positive entropy. Publ. Mat. 58 (2014), no. 1, 65-119.
[5] F. Perroni and D-Q Zhang, Pseudo-automorphisms of positive entropy on the blowups of products of projective spaces. Math. Ann. 359 (2014), no. 1-2, 189-209.
[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 self-map 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 non-invertible 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), 595-646.
[DF] Jeffrey Diller and Charles Favre. Dynamics of bimeromorphic maps of surfaces. Amer. J. Math. 123 (2001), 1135-1169.
[Dil] Jeffrey Diller. Cremona transformations, surface automorphisms, and plane cubics. Michigan Math. J. 60 (2011), 409-440. 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), 521-561.
[DL] Jeffrey Diller and Jan-Li 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 3-sphere. 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 H-polar 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 non-existence of H-polar 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 fα,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 fα,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 counter-examples 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 counter-examples 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) Gauss-Bonnet theorem gives a lot of information about the volume spectrum of hyperbolic n-manifolds. 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 K-theory counterparts to Grothendieck's ring of varieties, what is the explicit description of K1(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 K0(Vars) by endomorphisms.
Definition of higher Ki(Vars) and an explicit construction of K1(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 n-space n determines n up to isomorphism. In this talk we present a similar result about the quotient nd, where μd=<ξ∈*d=1> is a cyclic group of order d: if X is an irreducible affine normal variety such that Aut(X)= Aut(nd) as ind-groups, then X =nd as varieties. If X is not necessarily normal then we classify all such X with an isomorphism Aut(X)= Aut(nd) of ind-groups.
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 non-trivial 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.

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)
Jean-Philippe Furter (La Rochelle)
Sergey Galkin (Moscow)
Isac Héden (Basel)
Johannes Josi (Geneva/Paris)
Nikita Kalinin (Geneva)
Grigory Mikhalkin (Geneva)
Pierre-Marie Poloni (Basel)
Andriy Regeta (Basel)
Maria Fernanda Robayo (Basel)
Immanuel Stampfli (Bremen)
Christian Urech (Basel)
Francesco Veneziano (Basel)
Susanna Zimmermann (Basel)

Financial support
We gratefully acknowledge support from the SwissMAP/FNS project http://www.nccr-swissmap.ch/