Affine Algebraic Geometry

Workshop:
Affine Algebraic Geometry

12 - 14 July, 2012, Warsaw

The workshop concentrated around fundamental questions concerning noncomplete algebraic varieties:
-Is it true that an endomorphism of Cn which has a constant nonzero jacobian is an automorphism?
-Is it true that X x Cn=C(m+n) imples X=Cm? Is a nonsingular contractible complex variety necessarily rational?
-Can we embed Cn into Cm in a nonstandard way?
-How to describe exotic algebraic structures on Cn viewed as a differential manifold?
We know only partial solutions to these problems. Affine algebraic geometry gives tools to study them and the aim of the workshop was to give some understanding of these tools and to present some of the recent developments.

During the workshop twelve 1-hour lectures divided into four series were given:

  1. Frank Kutzschebauch: Applications of the Andersen-Lempert theory
    The group of holomorphic automorphisms of Cn for n>2 is infinite dimensional and in general not well understood. An intensive research on this group was initiated in the end of the 80's by the ground breaking work of E. Andersen and L. Lempert. The theory describes complex manifolds such that the local phase flows on their holomorphically convex compact subsets can be approximated by global holomorphic automorphisms. This leads to a construction of holomorphic automorphisms with prescribed local behavior. We sketched the main ideas of the theory and showed some applications. References: http://arxiv.org/abs/1003.3434
  2. Adrien Dubouloz: Log Minimal Model Program in dimensions 2 and 3
    The Minimal Model Program is considered to be the right framework to study birational geometry of projective varieties in higher dimension. The study of quasiprojective varieties requires a relative version of the program.
    I gave an overview of the genuine MMP for smooth projective surfaces, then I discussed the log-MMP for surfaces, the singularities involved in the theory, main theorems and results. I showed some applications to log-unirationality questions, basics on the theory of deformation of log-rational curves and application to the study of automorphisms of rational quasi-projective surfaces (variations on the log-Sarkisov program).
    http://alpha.science.unitn.it/~andreatt/scuoladott1.pdf
    http://www.math.univ-toulouse.fr/~slamy/stock/notesDiablerets.pdf
    http://www.math.univ-toulouse.fr/~slamy/stock/jung_translation.pdf
    http://www.math.univ-toulouse.fr/~slamy/stock/dubouloz_lamy_Feb2012.pdf http://www.math.ens.fr/~debarre/NotesGAEL.pdf
    http://arxiv.org/abs/alg-geom/9707016
  3. Karol Palka: Homologically trivial varieties and exotic structures on Cn
    A smooth affine variety which is diffeomorphic, but non-isomorphic to the affine complex space Cn, is called an exotic Cn. I discussed a classical result of Ramanujam on non-existence of an exotic affine plane C2.
    I gave a review of recent results on a class of Q-homology planes, i.e. algebraic surfaces which have the same rational homology as the plane. Cylinders over some of these surfaces give examples of exotic C3's. I said what is known in higher dimensions mentioning in particular some results on Koras-Russell threefolds.
  4. Mariusz Koras: Open algebraic surfaces
    I gave an outline of the results in the theory of open surfaces obtained since 70's mostly by Japanese mathematicians. Several classical problems have been solved using these methods.
    New proofs of classical Abhyankar-Moh-Suzuki and Lin-Zaidenberg theorems and applications to embeddings of C* into C2 have been presented. I have explained the role of the theory of open surfaces in the proof of the Linearization Conjecture for C* actions on C3. I also described some recent results on singularities of Z-acyclic surfaces of general type.

Organizing Committee

  • Karol Palka
  • Mariusz Koras

Speakers

Participants

  • Bodnár József
  • Buczyńska Weronika
  • Buczyński Jarosław
  • Danielewski W.
  • Dubouloz Adrien
  • Farnik Michal
  • Finston David
  • Heden Isac
  • Jelonek Zbigniew
  • Koras Mariusz
  • Kutzchebauch Frank
  • Langer Adrian
  • Lason Michal
  • Leire Gregorio
  • Mondal Pinaki
  • Palka Karol
  • Pintér Gergő
  • Pragacz Piotr
  • Tirabassi Sofia
  • Włodarczyk J.