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            1. Workshop on Several Complex Variables and Algebraic Geometry(new)

              2007-05-16 來源:數學科學研究中心








              Time & Place

              Stephen Yau

              (University of Illinois at Chicago)

              Holomorphic De Rham cohomology and its application to the complex Plateau problem.



              John Erik Fornaess

              (University of Michigan)


              Plurisubharmonic defining functions



              Anatoly Libgober

              (University of Illinois at Chicago)

              Higher genera in algebraic geometry


              Edward Bierstone

              (University of Toronto)


              Functoriality in resolution of singularities




























              Abstracts of Lectures


              Stephen Yau (UIC)

              Title: Holomorphic De Rham cohomology and its application to the complex Plateau problem.


              Let X be a compact connected strongly pseudoconvex CR Manifold of real dimension 2n-1 in {C}^{n+1}. Tanaka introduced a spectral sequence E_r^{(p,q)}(X) with E_1^{(p,q)}(X) being the Kohn-Rossi cohomology group and E_2^{(k,0)}(X) being the holomorphic De Rham cohomology denoted by H_h^k(X). We study the holomorphic De Rham cohomology in terms of the s-invariant of the isolated singularities of the variety V bounded by X. We give a characterization of the singularities with vanishing s-invariants. For n > 2, Yau used the Kohn-Rossi cohomology groups to solve the classical complex Plateau problem in 1981. For n=2, the problem remains unsolved for over a quarter of a century. In the joint work with H.S. Luk, we make progress in this direction by putting some conditions on X so that V will have very mild singularities. Specifically, we prove that if dim X=3 and H_h^2(X)=0, then X is a boundary of complex variety V with only isolated quasi-homogeneous singularities such that the dual graphs of the exceptional sets in the resolution are star shaped and all curves are rational.



              John Erik Fornaess (University of Michigan)

              Title: Plurisubharmonic defining functions.


              I will talk about recent joint work with Herbig on the existence of Stein neighborhoods of some pseudoconvex domains. The Stein neighborhoods are constructed as sublevel sets of plurisubharmonic functions.



              Anatoly Libgober (UIC)

              Title: Higher genera in algebraic geometry


              I will describe birational invariance of higher arithmetic genus, its generalizations and applicaiton of higher genus to non multiplicativity in locally trivial fibrations with nontrivial monodromy.



              Edward Bierstone (University of Toronto)

              Title: Functoriality in resolution of singularities


              Algorithms for resolution of singularities in characteristic zero are based on Hironaka's idea of reducing the problem to a simpler question of desingularization of an "idealistic exponent" or "marked ideal"). How can we determine whether two marked ideals are equisingular in the sense that they can be resolved by the same blowing-up sequences? From the point of view of this question of equisingularity or functoriality, I will compare recent algorithms and discuss results and open problems.