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L ectures onCounterexamples in Several
Complex Variables
J ohn Erik Fornass Berit Stensones
AMS CHELSEA PUBLISHING American Mathematical Society * Providence, Rhode Island
Lectures onCounterexamples in Several
Complex Variables
John Erik Forn/ESS Berit Stensones
AMS CHELSEA PUBLISHINGAmerican Mathematical Society • Providence, Rhode Island
http://dx.doi.org/10.1090/chel/363.H
2000 Mathematics Subject Classification. Primary 32-01; Secondary 32D05, 32E05, 32A38, 32U05, 32V40, 32F45.
Library o f C ongress C ata log ing-in -P ub lication D ataFornaess, John Erik.
Lectures on counterexamples in several complex variables / John Erik Fornaess and Berit Stenspnes.
p. cm. — (AMS chelsea publishing)Originally published: Princeton, N.J. : Princeton University Press ; [Tokyo] : University of
Tokyo Press, 1987.Includes bibliographical references.ISBN 978-0-8218-4422-9 (alk. paper)1. Functions of several complex variables. I. Stens0nes, Berit, 1956- II. Title.
QA331.F67 2007515'. 94—dc22 2007026106
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10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07
TABLE OF CONTENTS
Introduction
L ec tu re 1: Some N o ta tio n s and D e f in i t io n s ............................................................ 2
2: Holomorphic F u n c tio n s ................................................................................... 6
3: Holomorphic C onvexity and Domains o f H o lo m o rp h y ....................... 11
U: S te in M a n i f o l d s .................................................................................................17
5: Subharmonic F u n c tio n s ................................................................................... 21
6: Subharmonic F u n c tio n s ( c o n t . ) ................................................................. 26
7: Subharmonic F u n c tio n s ( c o n t . ) ................................................................. 31
P lu risu b h arm o n ic F u n c tio n s ..................................................................... 3U
8; P lu risu b h arm o n ic F u n c tio n s ( c o n t . ) ................................................... 37
9: Pseudoconvex Domains . . ........................................................................... k2
10: Pseudoconvex Domains ( c o n t . ) ................................................................. k6
11: Pseudoconvex Domains ( c o n t . ) ................................................................ 50
12; I n v a r ia n t M e trics ............................................................................................ 55
13: B iholom orphic M a p s ........................................................................................60
Ik : C ounterexam ples to Sm oothing o f P lu risu b h arm o n icF u n c t i o n s ...............................................................................................................66
15: C ounterexam ples to Smoothing o f P lu risu b h arm o n icF u n c tio n s ( c o n t . ) ............................................................................................ 69
16 : Counterexam ples to Sm oothing o f P lu risu b h arm o n icF u n c tio n s ( c o n t . ) ............................................................................................ 73
17: Counterexam ples to Smoothing o f P lu risu b h arm o n ic
18 : Complex Monge Ampère E quation ................................................................. 83
19:
20:
2 1 :
22: CR-M anifolds ( c o n t . ) ................................................................................... 1°0
2k: S te in Neighborhood B a s is ........................................................................... 105
25: S te in N eighborhood B a s is ( c o n t . ) . . . . . .................................. 109
26 ; S te in N eighborhood B a s is ( c o n t . ) ......................................................... 113
27** Riemann Domains over £Cn ................................................................................ 115
28: The K ohn-N irenberg Example .................................................................... 119
29-* Peak P o i n t s ..................................................................................................... 123
30: Bloom’s E x a m p l e .................................................................................................. 126
31; D’A n g e lo 's E x a m p le ................................................... 129
32: I n t e g r a l M a n i f o l d s .............................................................................................. 133
33: Peak S e ts f o r A ( D ) ..............................................................................................138
3k: Peak S e ts . S tep 1 ................................................................................................ lU l
35: Peak S e ts . S tep 2 ................................................................................................ 1^5
36: Peak S e ts . S tep 3 ................................................................................................ 1^8
37: Peak S e ts . S tep h. .............................................................................. . 159
38: Sup-Norm E s tim a te s f o r th e 3 - E q u a t i o n ................................................... 165
39: S ib o n y ’s 3-Example ......................................................................................... l68
1+0: H y p o e l l ip t ic i ty f o r 3 ............................................... 17^
1+1; In n e r F u n c t i o n s ................................................... 178
k2: In n e r F un ctio n s ( c o n t » ) - ........................ l81+
U3: Large Maximum Modulus S e t s ............................................... 189
kk: Zero S e ts ................................................................. 19^
1+5; N o n ta n g e n tia l Boundary L im its o f F u n c tio n s in H (Bn ) . . . 202
1+6: W erm er's E x a m p le .......................................................................... 212
1+7; The Union P r o b l e m .............................................................................................. 2ll+
1+8: Riemann D o m a i n s ................................................................................................... 218
1+9; Runge E x h a u s t io n ...................................................................................................222
Lecture 23; Pseudoconvex Domains without Pseudoconvex Exhaustion . . . 102
51; Peak S e ts in Weakly Pseudoconvex B oundaries . , ............229
52; Peak S e ts in Weakly Pseudoconvex B oundaries ( c o n t . ) . . . 23h
53; The K obayashi M e t r i c .......................... . . . . . . . . . . . . 236
B i b l i o g r a p h y .................................................................................................................................2̂ +2
Lecture 50; Runge Exhaustion (cont.) . , . ............................... 227
INTRODUCTION
These n o te s a r e from a g ra d u a te co u rse in P r in c e to n d u rin g 82 /83 and th e
f a l l o f 83.
The pu rpose was to c o l l e c t some o f th e coun terexam ples in th e s e v e r a l
complex v a r ia b le th e o ry w hich w ere s c a t t e r e d th ro u g h o u t th e l i t e r a t u r e . T h is
c o l l e c t io n i s by no means com ple te .
D uring th e f i r s t few w eeks, th e co u rse c o n s is te d o f an in t r o d u c t io n to
some o f th e b a s ic co n c ep ts o f th e th e o ry , in p a r t i c u l a r th o se needed l a t e r
f o r th e exam ples.
T h is m inim ized th e r e s u l t s needed to be ta k e n f o r g ra n te d and made i t
p o s s ib le to s t a t e th e s e sim ply w ith a p p ro p r ia te r e f e r e n c e s .
My th a n k s go to E. Low, A. N o e ll , P. Sm ith and B. S ten s^ n es who gave
some o f th e l e c t u r e s , w h ile th e n o te s were w r i t t e n by B. S te n s^ n e s .
My th a n k s a ls o go to Jay B elanger who p ro o fre a d th e m a n u sc r ip t.
The l e c tu r e s a re in th e same o rd e r a s th e y were g iven and hence th e r e
a re jumps back and f o r th between v a r io u s to p ic s b u t th e v a r io u s exam ples can
be re a d in d e p e n d e n tly .
John E rik F o rn æ ss
2k2
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2hj
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ISBN 978-0-8218-4422-9
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9780821844229
About this book
C ounterexam ples are rem arkably effective for u n d erstand ing th e m eaning, and th e lim itations, o f m athem atical results. Fornæss and Stensones look a t som e o f th e m ajor ideas o f several com plex variables by considering counterexam ples to w hat m ig h t seem like reasonable variations o r g en eralizations. T he first p a r t o f th e b ook reviews som e o f th e basics o f the theory , in a self-contained in tro d u c tio n to several com plex variables. T he counterexam ples cover a variety o f im p o rtan t topics: th e Levi p rob lem , p lurisubharm onic functions, M onge-A m père equations, C R geom etry, function theory , and th e d equation .
T he book w ou ld be an excellent supp lem ent to a g raduate course on several com plex variables.
9780821844229
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