The University of Hong Kong
Domains in complex
Euclidean spaces of dimension n >
1 may exhibit properties fundamentally different from those of plane domains, starting
with the phenomenon discovered by Hartogs that for
certain domains holomorphic functions defined on them
can automatically be analytically continued to strictly bigger domains. We
will first of all use elementary analytic techniques such as the power series
method and the Cauchy integral formula to study the Hartogs
phenomenon and the related notions of domains of holomorphy
and holomorphic convexity. Generalizing subharmonic functions we have in Several Complex Variables
the notion of plurisubharmonic functions. Basic
properties of such functions and some relations between plurisubharmonicity
and the Hartogs phenomenon will be explained.
Domains of holomorphy are maximal domains for certain holomorphic functions. In terms of partial differential equation,
it is possible to characterize domains of holomorphy
by the solvability of Cauchy-Riemann equations. We will examine the case of a
domain with smooth boundary, which is a domain of holomorphy
if and only if defining functions of the boundary satisfy certain differential
inequalities, leading to the notion of (strictly) pseudoconvex
domains. We will explain in this context the solvability of the Cauchy-Riemann
equation for (0,1) forms with -estimates due to
Hörmander. Plurisubharmonic
weight functions will play an important role in the estimates.
The local study of holomorphic functions in Several Complex Variables
translates to the study of the algebra of germs of holomorphic
functions at a given base point in n-dimensional
complex Euclidean space, which is equivalently the algebra of convergent power
series in n complex variables. On the
geometric side (germs of) complex-analytic subvarieties
are defined by zeros of ideals of (germs of) holomorphic
functions, and the algebraic study of convergent power series lead to geometric
properties on complex-analytic subvarieties. In
this respect we will examine basic results such as the Weiestrass
Preparation and Division Theorems and derive some elementary properties on
germs of complex-analytic subvarities, e.g., in
relation to their decomposition into irreducible components.
A link between the
study of complex-analytic subvarieties and the
analytic theory lies in the interpretation of complex-analytic subvarieties as supports of closed positive currents. We
will introduce the notion of closed positive (p, p)-currents, define
the associated density numbers of Lelong, and give
the interpretation of complex-analytic subvarieties
in terms of closed positive currents. In this context we will also
examine plurisubharmonic functions and their
associated closed positive (1,1)-currents.
Lecture 12: |
May
17, 2005 (Tuesday) |
4:00
– 6:30pm |
Lecture 11: |
May 10,
2005 (Tuesday) |
4:00
– 6:30pm |
Lecture 10: |
April
26 2005 (Tuesday) |
4:00
– 6:30pm |
Lecture 9: |
April
19 2005 (Tuesday) |
4:00
– 6:30pm |
Lecture 8: |
April
12, 2005 (Tuesday) |
4:00
– 6:30pm |
Lecture 7: |
March
29, 2005 (Tuesday) |
4:00
– 6:30pm |
Lecture 6: |
March
22, 2005 (Tuesday) |
4:00
– 6:30pm |
Lecture 5: |
March
15, 2005 (Tuesday) |
4:00
– 6:30pm |
Lecture 4: |
March
8, 2005 (Tuesday) |
4:00
– 6:30pm |
Lecture 3: |
March
1, 2005 (Tuesday) |
4:00
– 6:30pm |
Lecture 2: |
February
23, 2005 (Wednesday) |
4:00
– 6:30pm |
Lecture 1: |
February
15, 2005 (Tuesday) |
4:00
– 6:30pm |
Lectures will be held in
Room 517, Meng Wah Complex,
HKU
*Lectures of a
graduate course MATH6203 Several Complex Variables of the joint
HKU-CUHK-HKUST Centre
for Advanced Study (Mathematics)
All are welcome