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.