SCV Seminar

# Several Complex Variables Seminar

This semester (Spring 2008) the seminar meets Thursdays 10:00am-11:00am, in Milner 317.

Jan. 24:
Emil Straube, Texas A&M University
Plurisubharmonic Defining Functions (After Fornaess and Herbig) I
Abstract: Let D be a smoothly bounded domain in complex space of dimension at least two. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. Then the Diederich-Fornaess exponent can be chosen arbitrarily close to 1, and the closure of D admits a Stein neighborhood basis.

Jan. 31:

Emil Straube, Texas A&M University
Plurisubharmonic Defining Functions (After Fornaess and Herbig) II
Abstract: Let D be a smoothly bounded domain in complex space of dimension at least two. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. Then the Diederich-Fornaess exponent can be chosen arbitrarily close to 1, and the closure of D admits a Stein neighborhood basis.

Feb. 7 <

Emil Straube, Texas A&M University
Plurisubharmonic Defining Functions (After Fornaess and Herbig) II
Abstract: Let D be a smoothly bounded domain in complex space of dimension at least two. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. Then the Diederich-Fornaess exponent can be chosen arbitrarily close to 1, and the closure of D admits a Stein neighborhood basis.

Feb. 14

Zach Teitler, Texas A&M University
Introduction to multiplier ideals: from analysis to algebraic geometry
Abstract: I will give a completely elementary introduction to the theory of multiplier ideals. These are defined in analysis in terms of local integrability and in algebraic geometry in terms of resolution of singularities. I will explain both definitions without assuming any background knowledge. My primary goal is to explain the relationship between the two definitions. Future talks, including an upcoming talk in the algebraic geometry seminar, will explore some of the main questions in the area and some of the successful applications of multiplier ideals to answer open questions.

Feb. 21

Zach Teitler, Texas A&M University
Introduction to multiplier ideals: from analysis to algebraic geometry II
Abstract: I will finish the development of part 1 of the talk and compute an example in some detail. I will briefly mention a few notable results obtained using the theory of multiplier ideals, with the goal of giving fuller explanations in future talks. This talk (part 2) will be very heavily dependent on material from part 1.

Feb. 28

Zach Teitler, Texas A&M University
Title: Properties and applications of multiplier ideals, I
Abstract: I will present the fundamental cohomology vanishing property of multiplier ideals, with an algebraic-geometry approach. I will present Nadel's groundbreaking use of multiplier ideals to show the existence of Einstein-K\"ahler metrics on certain projective manifolds. Here there will be some effort to relate the algebraic geometry and analysis viewpoints. Time permitting other applications may be described. I will recall all relevant definitions. In particular, if you missed my last two talks, we won't really need that material. (Notes from those talks have been typed up and are available upon request.) Properties and applications II will go over Siu's famous proof of the deformation invariance of plurigenera, after spring break.

Mar. 6

Zach Teitler, Texas A&M University
Title: Properties and applications of multiplier ideals, II
Abstract: TBA

### Note Special Day and Time: Friday, April 18

Apr. 18
Siqi Fu, Rutgers University-Camden
Title: Spectral theory of the d-bar-Neumann Laplacian and applications
Abstract: I will talk about interplays between spectral behavior of the d-bar-Neumann Laplacian and geometric properties of the underlined space, and applications of the spectral theory to problems in complex geometry. Part of the talk is based on joint work with Howard Jacobowtiz.

Apr. 24

Al Boggess, Texas A&M University
Title: A Global CR Approximation Theorem for hypersurface graphs in Several Complex Variables
Abstract: This seminar is a continuation of another one given at the end of last fall in which I presented an overview of the theory of CR functions. However this seminar should be self-contained. I will state and outline the proof of the following result, which is joint work with Roman Dwilewicz and Dan Jupiter: if $\omega$, is an open subset of a real hypersurface in C^n, that is given as a graph over a convex subset in R^{2n-1}, then $\omega$ is CR-Runge in the sense that continuous CR functions on $\omega$ can be approximated by entire functions on C^n in the compact open topology of $\omega$. As presented in the seminar last fall, there are counter examples to show that this result does not hold for graphed CR submanifolds in higher codimension.