Lecture 5
Notes about Exercise 1
Lemma. Let U and V be as in Theorem 1 above. 0,q (U ), = 0 then there exists 0,q1 (U )
such that = on V .
Proof. Choose a polydisk W so that V W , W U . Choose C0 (W ) with 1 on a neighborhood
0,q1
(W ) so that 0 = on W .
Lecture 4
Applying Hartogs Theorem
Let X Cn be an algebraic variety, codC X = 2. And suppose f O(Cn X ). Then f extends holomorphically to f O(Cn ).
Sketch of Proof : Cut X by a complex plane (P = C2 ) transversally. Then f |P O(P cfw_p) so by
hartog, f |
Lecture 3
Generalizations of the Cauchy Integral Formula
There are many, many ways to generalize this, but we will start with the most obvious
Theorem. Let D Cn be the polydisk D = D1 Dn where Di : |zi | < Ri and let f O(D) C (D )
then for any point a = (
Lecture 2
Cauchy integral formula again. U an open bounded set in C, U smooth, f C (U ), z U
1
1
f ( )
1
f
f (z ) =
d +
( )
d d
2i U z
2i U
z
the second term becomes 0 when f is holomorphic, i.e. the area integral vanishes, and we get
f ( )
1
d
f (z ) =
Chapter 1
Several Complex Variables
Lecture 1
Lectures with Victor Guillemin,Texts:
Hormander: Complex Analysis in Several Variables
Griths: Principles in Algebraic Geometry
Notes on Elliptic Operators
No exams, 5 or 6 HWs.
Syllabus (5 segments to course,