A workshop on "The Topology of Algebraic Varieties" will be held in Montreal from September 24 to 28, 2012 preceded by three related mini-courses from September 21 to 23, 2012. Additional information can be found at the following website: www.crm.umontreal.ca/2012/Topology12/index_e.php This is a workshop on connections between simple topological invariants of a normal algebraic variety, and other natural invariants, especially those coming from birational and affine algebraic geometry. Major themes will include the simple connectedness of rationally connected varieties, the related development of the Shafarevich map via cycle space theory, and the resolution of the Shafarevich conjecture on the universal cover of a smooth variety (with linear fundamental group) via Hodge theory in various guises. Among other central subjects there will be recent advances in the theory of plumbing for surface singularities that bridged many classical problems on algebraic varieties with knots and the theory of 3-manifolds, and the key role of the Bogomolov-Miyaoka-Yau inequality on (orbifold) Chern numbers of singular varieties in several new break-through results in affine algebraic geometry.

The conference will bring together experts in algebraic/analytic geometry as well as relevant topologists working in low dimensional complex singularity theory for fruitful exchange of information and interaction on an international scale. It will introduce modern topological and analytical tools used in this area to a new generation of researchers.

Project Report

The workshop organized by Rajendra Gurjar (TATA institute of Mathematics), Shulim Kaliman (University of Miami, Mathematics Department), Steven Lu (UQAM Mathematics Department), and Pe- ter Russell (McGill University) took place from September 21, 2012 till September 28, 2012 at the Centre de Recherches Math ?ematiques in Montreal. The workshop attracted 55 senior mathematicians from North America, Europe, and Asia and a considerable number of young researchers. The workshop started with three mini-courses of four lectures each: the first one was on topological methods in the study of singularities, (Anne Pichon and Walter Neumann); the second one was on the geometry of non-compact Kaehler manifolds (Terrence Napier and Mohan Ramachandran), and the third one was on the generalization of the Bogomolov- Miyaoka-Yau inequality to singular quasiprojective varieties and the modern approach to Nevanlinna theory vai the BMY inequality (Adrian Langer and R.Kobayashi). Among the workshop presentations there were solutions of old problems. Here are some examples. M. Koras presented an idea of solution of the Coolidge-Nagata conjecture that states that any plane rational curve with locally irreducible singularities can be rectified by birational automorphisms of the plane. F. Kutzschebauch gave a sketch of the solution of the Gromov- Vaserstein problem which implies that any invertible matrix with entries being holomorphic functions on a Stein variety, that is in the same connected component as the identity matrix (in the space of invertible matrices), is a product of elementary matrices. J. Keum’s lecture on the algebraic version of Montgommery-Yang problem (suggested by J. Kollar) was very clear and nice. His joint work with D. Hwang came very close to a full solution. In L. Katzarkov’s lecture fantoms of derived categories were used as a new invariant that allows to show non-rationallity of a variety even in the case when the intermediate Jacobian does not work. D. Arapura discussed a cyclic cover Y of prime degree over a projective space whose branched locus is a hyperplane arrangement D with normal crossings, and the fact that the generalized Hodge conjecture holds for any toroidal resolution of singularities of Y . M. Miyanishi lectured on A1 and A1∗-fibrations of affine threefolds. He introduced several nontrivial results, including a global analogue of a theorem by Mumford. It states that a normal contractible affine surface with a simply connected smooth part is smooth. F. Canatese explained new results on moduli space of curves whose automorphism groups contain a given finite group, and gave a description of irreducible components of such a moduli space in the case of some specific groups. H. Flenner, reporting on joint work with S. Kaliman and M. Zaidenberg, presented a definitive result on deformations of affine surfaces with an A1-fibration. B. Purnaprajna reported on joint work with R. Gurjar on the fundamental groups of surfaces with finite automorphism groups. It leads to a solution of the Shafarevich conjecture for certain interesting classes of surfaces. B. Oliveira reported on his ongoing investigation with Bogomolov on closed symmetric differentials (of degree two) and their implications on the fundamental group while R. Kobayashi reported on his recent foray into Ka ?hler Ricci solitons via his Hamiltonian version of the Kahler Ricci flow. F. Campana explained his joint work with J. Winkelmann on Oka varieties. J. Winkelmann gave an extremely enlightening and self contained exposition of the usefulness of the Kobayashi metric in resolving problems in complex geometry, a particular example being his solution of a conjecture of Kollar concerning the holomorphic convexity of the universal cover of a singular variety. M. McQuillan gave an intensive lecture that well explained his solution of Lang’s conjecture concerning holomorphic curves in surfaces of general type that improved upon the Chern inequality bound of Bogomolov. In so doing, he has explained the recent solutions of resolution of singularities for one dimensional foliations in higher dimensional spaces. S. S. Abhyankar reported on new results on local fundamental groups and Galois theory. His talk, in retrospect, had a special poignancy since sadly Professor Abhyankar passed away not long after the conference. Nice expository lectures were delivered by M. Nori on monodromy in algebraic geometry, by F. Eyssidieux, who used his very well-written survey on the Shafarevich Conjecture, and by A. Langer, who talked on vanishing theorems for cohomology of line bundles of varieties defined over fields of positive characteristic.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Tie Luo
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University of Miami
Coral Gables
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