Lectures on Several Complex Variables*

Professor Ngaiming Mok

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 image005" -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