Speakers: Adrien Dubouloz (Dijon), Mariusz Koras (Warsaw),
Frank Kutzschebauch (Bern), Karol Palka (Montreal)
Location: Banach Center IMPAN, Warsaw
Frank Kutzschebauch (Bern), Karol Palka (Montreal)
Location: Banach Center IMPAN, Warsaw
Workshop on Affine Algebraic Geometry
Warsaw, June 12 - 14, 2012
Warsaw, June 12 - 14, 2012
About the workshop:
Is it true that an endomorphism of Cn which has a constant nonzero jacobian is an automorphism? Is it true that X x Cn=Cm+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. The workshop aims to give some understanding of these tools. Some of the recent developments will be presented.
Registration:
People interested in participation should contact the organizers until April 30, 2012.
Series of lectures:
Log Minimal Model Program in dimension 2 and 3 (Adrien Dubouloz)
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 will give an overview of the genuine MMP for smooth projective surfaces, then I will discuss the log-MMP for surfaces, the singularities involved in the theory, main theorems and results. If time permits I will show some applications to log-unirationality questions, basics on the theory of deformation of log-rational curves or application to the study of automorphisms of rational quasi-projective surfaces (variations on the log-Sarkisov program).
References: http://arxiv.org/abs/0802.2441
Open algebraic surfaces (Mariusz Koras)
I will give 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 will be presented. I will explain the role of the theory of open surfaces in the proof of the Linearization Conjecture for C* actions on C3. I will also describe some recent results on singularities of Z-acyclic surfaces of general type.
References: http://arxiv.org/abs/1012.4120, Koras, M.; Russell, P. C*-actions on C3: the smooth locus of the quotient is not of hyperbolic type. J. Algebraic Geom. 8 (1999), no. 4, 603-694; Koras, M. On contractible plane curves. Affine algebraic geometry, 275-288, Osaka Univ. Press, Osaka, 2007
Applications of the Andersen-Lempert theory (Frank Kutzschebauch)
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 sketch the main ideas of the theory and show some applications.
References: http://arxiv.org/abs/1003.3434
Homologically trivial varieties and exotic structures on Cn (Karol Palka)
A smooth affine variety which is diffeomorphic, but non-isomorphic to the affine complex space Cn, is called an exotic Cn. I will discuss a classical result of Ramanujam on non-existence of an exotic affine plane C2. I will give 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 will also say what is known in higher dimensions mentioning in particular some results on Koras-Russell threefolds.
References: http://arxiv.org/abs/1003.2395, http://arxiv.org/abs/math/9801075.
Location and accomodation:
The meeting takes place at the Banach Center in the Institute of Mathematics of the Polish Academy of Sciences (IMPAN). Here is a description how to get there. The guest rooms of the Institute can accommodate most of the participants. Unfortunately, we cannot offer support of travel expenses. Expected arrival: June 11 afternoon or June 12 morning. Departure: June 14 or June 15.
Tentative schedule:
June 12, 13, 14
9:15-10:45 lecture 1
11:00-12:30 lecture 2
14:00-15:30 lecture 3
15:45-17:15 lecture 4
Other information for participants: On June 8 there is an opening football game, we have EURO 2012 in June.
Acknowledgements: ...
Is it true that an endomorphism of Cn which has a constant nonzero jacobian is an automorphism? Is it true that X x Cn=Cm+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. The workshop aims to give some understanding of these tools. Some of the recent developments will be presented.
Registration:
People interested in participation should contact the organizers until April 30, 2012.
Series of lectures:
Log Minimal Model Program in dimension 2 and 3 (Adrien Dubouloz)
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 will give an overview of the genuine MMP for smooth projective surfaces, then I will discuss the log-MMP for surfaces, the singularities involved in the theory, main theorems and results. If time permits I will show some applications to log-unirationality questions, basics on the theory of deformation of log-rational curves or application to the study of automorphisms of rational quasi-projective surfaces (variations on the log-Sarkisov program).
References: http://arxiv.org/abs/0802.2441
Open algebraic surfaces (Mariusz Koras)
I will give 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 will be presented. I will explain the role of the theory of open surfaces in the proof of the Linearization Conjecture for C* actions on C3. I will also describe some recent results on singularities of Z-acyclic surfaces of general type.
References: http://arxiv.org/abs/1012.4120, Koras, M.; Russell, P. C*-actions on C3: the smooth locus of the quotient is not of hyperbolic type. J. Algebraic Geom. 8 (1999), no. 4, 603-694; Koras, M. On contractible plane curves. Affine algebraic geometry, 275-288, Osaka Univ. Press, Osaka, 2007
Applications of the Andersen-Lempert theory (Frank Kutzschebauch)
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 sketch the main ideas of the theory and show some applications.
References: http://arxiv.org/abs/1003.3434
Homologically trivial varieties and exotic structures on Cn (Karol Palka)
A smooth affine variety which is diffeomorphic, but non-isomorphic to the affine complex space Cn, is called an exotic Cn. I will discuss a classical result of Ramanujam on non-existence of an exotic affine plane C2. I will give 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 will also say what is known in higher dimensions mentioning in particular some results on Koras-Russell threefolds.
References: http://arxiv.org/abs/1003.2395, http://arxiv.org/abs/math/9801075.
Location and accomodation:
The meeting takes place at the Banach Center in the Institute of Mathematics of the Polish Academy of Sciences (IMPAN). Here is a description how to get there. The guest rooms of the Institute can accommodate most of the participants. Unfortunately, we cannot offer support of travel expenses. Expected arrival: June 11 afternoon or June 12 morning. Departure: June 14 or June 15.
Tentative schedule:
June 12, 13, 14
9:15-10:45 lecture 1
11:00-12:30 lecture 2
14:00-15:30 lecture 3
15:45-17:15 lecture 4
Other information for participants: On June 8 there is an opening football game, we have EURO 2012 in June.
Acknowledgements: ...
