iii
PRiFACE These notes reproduce almost verbatim a course taught during the academic year 1962/63. The original notes, prepared by Joan Landman and Marion Weiner, were distributed to the class during the year. The present edition differs from the original only in that many mistakes have been corrected. I am indebted to Miss Weiner who prepared this edition and to several colleagues who supplied lists of errata. I intended the course as an introduction to the modern theory of several complex variables, for people with background mainly in classical analysis. The choice of material and the mode of presentation were determined by this aim. Limitations of time necessitated omitting several important topics. Every account of the theory of several complex variables is largely a report on the ideas of Oka. This one is no exception.
L.B.
Zurich, July 8, 1964.
iv
CONTENTS Preface, • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ii Chapter 1. Basic Facts about Holamorphic FUnctions ~1. Preliminaries ••••••••••••••••••••••••••••••••••••• · 1 82. An inequality...................................... 5 13. Proof of Hartogs' Theorem 1....................... 7 §4. Holomorphic mappings.: •••••••••••••••••••••••••••• 10 Chapter 2. Domains of Holomorphy 1. Examples and definitions •••••••••••••••••••••••••• 2. Convexity with respect to a family of functions... 3. Domains of convergence of power series ••••• ~ •••••• 4. Bergman domains ••••••••.•••••••••••••••••••••••••• 5. Analytic polyhedra •••••••.•••••••••••••••••.•••••• Chapter 3. Pseudoconvexity 1. P1urisubharmonic and pseudoconvex functions .•••••• 2. Pseudoconvex domains •••••••••••••••••••••••••••••• 3. Solution of the Levi Problem for tube domains •••••
1
12 14
18 22 24
26
30 35
Chapter 4.
1 1. 2.
3. 4.
~5.
Zeros of Holomorphic Functions. Meromorphic Functions. \'leierstrass Preparation Theor. em................... Rings of power series .•••••••••••••••••••••••••••• Meromorphic functions •••••••••••••••••••••••••••.• Removable singularities ••••••••.•.•••••••••••••••• Complex manifolds ••••••••••••••••••••••••• ·.. • • • • • • •
38
44 48
50 53
Chapter 5· The Additive Cousin Problem 1. The additive Problem formulated •••••••••••••• • •• • • ~ 19 2. Reformulation of the Cousin Problem ••••••••••••••• o 3. Reduction of the Cousin Problem to non-homogeneous Cauchy-Riemann equations •••••••••• 63
1
Chapter 6. Cohomology §1. Cohomology of a complex manifold with holomorphic functions as coefficients ••••••••••••• §2. Applications •••••••• ~ •••••••••• ~.................. §3. Other cohomologies ••••••••••••••••••••••••••••••••
69 74
76
Chapter 7. Differential Forms 1. Ring of differential forms in a domain •••••••.•••• 80 2. Differential forms on manifolds ••••••••••••••.•••• 84 3. Poincare Lemmas. • • • • • . • • • • • • • • • • • • • • • • • • • • • • • • • • • • 85
1
Chapter 8. Canonical Isomorphisms 1. De Rham 1 s Theorem. • • • • • • • • . • • • • • • • • • . • • • • . • • • • • • . • 89 2. Dolbeault 1 s Theorem •••••••••••••••••.••••••••••••• 93 3. Complex de Rham Theorem ••••••••••••••.•••••••••••• 96
1
v
Chapter 9. The Multiplicative Cousin Problem §1. The Multiplicative Problem, formulated •••••••••••• 98 ~2. The Multiplicative Cousin Problem is not always solvable ••••••••••••••••••••••• • ••••••••••• 100 §;. The solution of the Multiplicative Cousin Problem for polydiscs •••••••••••••••• -••••••••••••• 10'2: §4. Characteristic classes (From C.II to C.I) ••••••••• 106 ~
Chapter 10. Runge Regions 1. Preliminaries..................................... 110 2. Polynomial polyhedra •••••••••••••••••••••••••••••• 112 ;. Runge domains •••••••••••••••••••••••••••.••••••••• 113
1
Chapter 11. Cohomology of Domains of Holomorphy_ 1. Fundamental Lemma, stated ••••••••••••••• ~ ••••••••• 2. Applications of the Fundamental Lenuna ••••••••••••• .,. Preparation for the proof of the Fundamental Lemma ••••••••••••••••••••••••••••••••• §4. Proof of the Fundamental Lemma ••••••••••••••••••••
1
115 115
'%
117 120
Chapter 12. Some Consequ~nces of the Approximation Theorem 1. Relative convexitY···~···························· 128 2. Unbounded regions of holomorphy .•••••••••••••••••• 129 3. The Behnke... stein Theorem •.••••••••••.••.••••••.••• 130 4. Applications to the Levi Problem ••.•••••.•••••.•.• 132
1
Chapter 13. Solution of the Levi Problem 1. Reduction to a finiteness statement ..•..••.••.••.. 134 2. Reduction to an extension property .••••.•..••••••. 137 3. Proof of Propos! tion 2 •••••...••..••••.••.•.•••••• 140
1
Chapter 14. Sheaves 1. Exact sequences •.•..••••.•••••••.•.•.•..•..••••••• 2. Differential operators ..•••••••..••...•••••••••••• 3. Grcided groups .•... ....•.......•.............•..... 4. Sheaves and pre-sheaves ••••..•••••.•••.••••.•••••• Exact sequences of sheaves and cohomology •••.•••.• §~: Applications of the exact cohomology sequence theorem ••.•.••••..•.••.•.•.•..•.•.•••••.• §7. Proof of the exact cohomology sequence theorem .••• Chapter 15.
142 144 147
149 150
155 158
Coherent Analytic Sheaves
1. Definitions ....................................... 162
2. Oka's coherence theorem ...••••.••••••.•..••..•.•.• 163 3. Heierstrass Preparation Theorem, revisited •••••••• 165
4. The third step ••.••••••••.••••.•••.•.••••••••••••• 168 5. Consequences of Oka 1 s theorem ••.•.•.•••••••• , ••.•. 171 6. The sheaf of ideals of a variety ..••.•••.•.••••••• 173
vi
Chapter 16. FUndamental Theorems (semi-local form) Ill. Statement of the fundamental theorems for a box (semi-local form) .......................... .-.First step of the proof.. • • • • • • • • • • • • • • • • •.• • • • • • • • Reduction of (3) to Cartan 1 s theorem on holomorphic matrices ••••••••••••••••••••••••••• 4. Proof of Cartan's theorem on holomorphic matrices. 5· Ne\'l proof of the Oka-t~eil Approximation Theorem ••• 5. Fundamental theorems for regions of holomorphy (semi-local form) ••••••••••••••••••••••
1
Chapter 17. Coherent Sheaves in Regions of Holomorphy 1. Statement of the fundamental theorems • • • • • • • • • • . • 2. Prepara tiona for the proof •••••••••••••.•••••••••• 3. Proof of Theorem A•••••••••••••••••••••••••••••••• 4. Proof of Theorem· B••••••.•••••••••••••••••••••••••• 5· Applications of the fundamental theorems ••••••••••
175 175
•
17T 180 184 185 187 187 191 194 196
Chapter 18.
Stein Manifolds (Holomorphically Complete Manifolds) §1. Definition and examples ••••••••••••••••••••••••••• §2. An approximation theorem ••••••••••••••••••••••••.• §;. The fundamental theorems for Stein manifolds •••••• §4. Characterization of Stein manifolds .••••••••.•.•••
201 202
203 203
Appendix . •..........•..•.•..••.••.•.•....•••.....••.•... 205
1
Chapter 1.
Basic Facts about Holomorphic Functions § 1.
Preliminaries
We introduce the following notation:
a denotes the field of real numbers. ct denotes the field of complex numbers or the complex plane. ~n . "" denotes the space of n -tuples of complex numbers (z1, ...,zn)
= z.
0 then
(I f(z) I. E) is subharmonic. Proof. At a point where lrl < E, and thus in a neighborhood of this point, log max (I f(z) I, €) = log E = log (constant) is subharmonic. At a point where lrl > E, log max e/ 2, max log ( If I. E) is also subharmonic and log max =max log .. Theorem 3. Under the hypothesis of Theorem 2 and for
r. -- 1, z. -- p .eif.J J
log
J
~flz1.J ,znll~12~lJ ... .[1 _ l-p~ 0. Let g =log max n n
by the Cauchy integral representation on 1 variable and the existence of
Irl
the maximum of in D~ Since all the denominators are bounded away from zero as z1-z 0 , lr1-o
f at z 0• The arbitrariness of the choice of z 0 implies the continuity of f in the interior of n1• Now fix z 2, ...• zn; zi I < 1. Then
I
=
11
Q()
~
v=O
for
a (z 2••••• z )z 1
n
II
where
..!..( avr ~ 11
a (z·2, •••• z ) = 1 v n 11.
,
az1 / z 1=o
by the holomorphicity of f in z1• Near z1 = 0. (zl' ...• z n) lies in D1 and
£
here f is bounded and continuous. Thus a (z 2, ••• , z ) 11 n
=· _!_ 21T i
It
f{t, z 2••..• zn)
=p 1< P
..JI+l
d~
~
and therefore the a are holomorphic. By our induction hypothesis, the 11
a11 are continuous, and the proof will be complete once we show that the . power series converges normally for
lz11< 1.
Now, 1
lim sup I a (z 2, ...• z ) 111 < 1 11 n
v-ao
9
and Ia (z 2.... , z )I < _!_, II n II p
since the series converges for I z 11 < 1 + TJ1 for some TJ > 0: Set fZ
/\ = t zj = e
w.JI0 ~ ~ j < 211'
1
j
1
I
1\11 = ~rcz 2, ••• zn ) fa11 Cz 2, .•• 1
2~ ••• In
1:
1
zn )
1\II is open in 1\ ~ and • lim m( 1\.II ) = 0, v- co the measure of 1\ . For, if we let II Then
jv >· 1,.
} 1
(z 2, ••• , z n )~
where m ( 1\
II
1\].
) denotes
00
QJ. =
UA.
• II=J
II
and note that
and m(nQ.) = lim m(Q.) J
j-
00
J
then
() ~. = q, (the empty set). J
Thus m( () Q.) = 0 = lim m(c; .). J
J
Since Q. =:J 1\ ., m(/\ .)-0.
J i~ . Now let z. = r .e J,. r. < r J J J-
J
< -
< -
J
1 v(21r )0
1
1 11(21r )0 -
1
j = 2, ...• n.
By Theorem 3,
2 i& 2 i6n (1-r .) log (e •.•.• e II _ _ _ _2=---- dO 2" •• den _ __.,__ ___;_ j=2 l-2r. cos(cb . -6 .)+r. J J J J i8 i8n 2 2 n (1- r .) log a ( e •.•. , e ) _....J_ ___;_II_ _ _--=2-- d8 2" •. cWn. j=2 l-2r. cos (' . -e .)+r. J J J J
f Jnn
Ia
>I
J; Jn _
I
I
..• (\
-
< 1,
...
1\
v
10
Furthermore
1-2r. cos land Ia
II
I l
Definition 12. For any set KC D A0 (K) C.
= inf
zE K
open
:h
C(C
•
A0 (z) .
Note. Kc c D, read "K is relatively compact in 0 11 , is defined to
mean that the closure of K is compact and contained in D; hence A(cl K)> 0. Theorem 5. (Cartan-Thullen). If Dopenc (Cn , then the following conditions are equivalent: (i) D is a region of holomorphy. (ii) All boundary points of D are essential. " =A(K),
(iii) If KC CD, then A(K)
,.,
where K
"' '1:. =K.J and J is the family of
all functions holomorphic in D. (iv) D is holomorphically convex.
,..
i.e.
K·~-=
c: D implies Kc: ~=D.
Corollary 1. If n
=1,
then every open set is a region of holomorphy.
Corollary 2. Dis a region of holornorphy if and only if every compo• nent of D is a domain of holomorphy.
D1X
Cforollary 3. If o1c a;P and n 2c ([q are domains of holomorphy, then D2 c«:p+q is a domain of holomorphy. Corollary 4. If Da is a region of holomorphy for every a in some set
A, and if
nDa
is open, then
noa
is a region ofllolomorphy.
Proof. Assume Kc c ( f1 - Da ); then KC c Da for every a. Let Ka denote the hull of K in D with respect to all functions holomorphic in D . a a K is compact by assumption and Kc K . Thus K c K , and hence K a ,.. "' ,a a a is compact. Since K c KC ::-( :1 D ). a a
,.
no
I
" n
,.
16 ~·
In general, a union of regions of holomorphy is not a region
of holomorphy. However, if the regions are nested, i.e.
o1co2c ... ,
then their union is a region of holomorphy. This, however, we will prove much later. Cormlary 5. (Exercise) If D is geome.trically convex then Dis a region of holomorphy. Proof of Theorem 5. If D
D.
=
A
0, K is compact.
(iv) implies (i). Let D be any component of D. sequence of sets Dj' j = 1, 2, ... such that o1c c
D = ynj'
(e. g. Dj
=[z
zE
0,
liz
o 2c c
II < j, S 0(z) > j1}
Construct a
0 3... c
c D and
). Let (Pj} be dense
17
D. ([Pj}
D
in could be chosen ;_s the set of points o:_ whose coordinates in :m. 2n are rational). For each j we find a point z .t D such that z. ·p .6.0 (p.) 1 "' " J J J J and z.~ D.. ·we can find such a point since D. doesn't come arbitrarily Jl J J close to the boundary of D. Now, for each j, there exists a function r. boloJ morphic in D, such that < 1 in D. and lr.(zJI =A.> 1. Next choose JJ JJ J ~ integers N. > 0 such that J
II
.II
as z- z .. J J
So, the derivatives of g up to order j vanish at z .. On the other hand, if q is a boundary point of
D,
then there exists a
s~bsequence
fp.} of Jp.1 l 1 lv LJ j which converges to q. This means that the p. come arbitrarily close to lv
II -
the boundary, i.e . .6.(p. ) - ~. This implies that liz. -p. 0 and hence ~ ~ ~ that z. - q. Therefore if g is holomorphic at q, all the derivatives of g
lv
18 variish at q. This can only happen if g = 0, a contradiction. Hence g is not holomorphic at any boundary point. Note. Under a holomorphic mapping, !1 region of holomorphy is mapped into a region of holomorphy.
.-
§ 3.
A.
Domains of Convergence of Power Series
In this section we consider power series of the form ~ kn k (z 1-t 1) ..• (z -~ ) kn =0 1: · · n n n
co ao E ••• E
I) =0 for
~
=(~1' •.• , ~n) fixed.
~
(1)
V.le say that the series converges at a point
z =(zl' ... 'zn) if there is some arrangement of terms for which the series converges. Note. In the sequel, we take t = 0 ; i. e. we deal with series of the form co ~ k1 kn :L: • • • I: a. k zl . . . z k =0 k =0 Kl. · · n n
1
(1 I
)
n
Abel's Lemma. If the series (1') converges at some point
"z = (z" 1, z" 2, ..• , zn)' " then the series converges uniformly and absolutely in every compact subset of the polydisc z
n (r1R1 e , ... ,rnRn e )€Dforeacha, O
Ia
Therefore for 0
0 is given, there are
D such that A . -z? I< e: and f11. -z ~I < e: for all i. Hence 1
1
1
1
foralla, la.f =l.la.v. -a.z?l<e:and la.:\.-a.z?l< e:,av 1
a A e: D so that a z 0 e:
11
ao,
11
11
11
f.
Dand
as claimed. (2) is proved similarly by considering
the preimage in ct11 of a neighborhood of a boundary point of log D*. Since elK is compact, there is a ball Bin «:n, 0 such that dist (log K*, a(log B*nlog D*) ) > 6: for Bt'lo is a complete circular domain and KC
c
(BnD).
22
Let T = log B* () log D*. T is convex and c log ·D*
,..
V.l e will show
that log (K)*C T, by considering the convex hull T 1 ofT, i.e. the intersection of all closed half-spaces in .l.n containing T. Now, if~ ~ Tl' then there is a hyperplane P in lln separating ~ from T 1 and such that Pf'l(T~ ( ~ }>=~ Because Bfl Dis a complete circular domain,. if ~o E T, then Thus t(~ ~ m.nl~.1 o}. Suppose nn1r/ T1' if log I~ i I > log r for at least one i, say i = 1, then we may take P to E
m.nl
be the hyperplane : log Iz11 =log r + E, where e > o is such that logl~ 1 1 > log r +e. If logle.l T until it intersects log K* , P becomes a hyperplane to be considered in (the
"" intersection giving log (K)*), and we have translated P a distance> 6. A
~
Therefore log (K)* c cl T, and the distance of any point of log (K)* to such a hyperplane P is > 6 , s·o that the closed ball of radius 6 about any point of A
log (K)* c the closed half space::> cl T defined by every such hyperplane P and I'
thereforeccl T. By the convexity ofT, it follows that log(K)*C T, as claimt (iii) implies (i). (iii) implies that there exists a function f holomorphi in D and in no larger domain. At any point z e D, we may expand fin a power series which converges normally in a neighborhood of z. Thus there is a power series which converges in D and in no larger domain. § 4.
Bergman Domains
Definition 15. Let f(z 1, ... , z ,X) e «:, z. e «:, i = 1, ... , nand X e lB., n 1 such that f is continuous in all variables simultaneously and holomorphic in
A.
23 some domain DC «:0 for each fixed ). z
=(zl' ... , zn)
€
D, ~
E
E
I
I c J,. Then £z f(zl' ... , z n'). ) • 0.
rJ is called a Bergman Surface.
This is a surface of (real) dimension 2n-l. Definition 16. A Bergman domain is a domain bounded by a finite number of Bergman surfaces.
Example. Let n
= 2.
:
Let B
={ (zl' z 2l 11 z11< 1,
z 2 £ B('"l) }where
the B(z1) a~e domains parametrized by z1 and bounded by Jordan curves: z 2 = g(zl' e~ ), where g is continuous in all variables and holomorphic in z1 for Iz11 < 1 + € • Then B is a Bergman domain. For example, we might choose g(zl' e
iA
1 M ) = 2+zl e
For n arbitrary, D = {cz1, ... , zn)
Iz1l < 1,
z2
€
B(z 1),
z 3 £ B(zl' z 2), ... } . These domains will be called quasi-product domains. They reduce to product domains if B(z 1) =B1 independent of z 1, B(zl' z 2) = B 2, etc. We list the following properties, stated for any Bergman domain and proven for quasi -product domains in ct 2. Property 1. Every Bergman domain is a domain of holomorphy.
or
" ""z 2) is a boundary point then either Proof. If (zl' (a) 1~1 1 =1, and then ~... is singular at (~1 • i 2> and regular inside iA zl zl 1 ~ . s1ngular . ""' (b) "" z 2 = g (zz 1, e 0 ) and then 1s at (zl' z 2) and z 2 -g ( z1, e 0)
regular elsewhere. Property
2~
-
A Bergman domain has a distinguished boundary surface,
defined to be the set of those points of the boundary at which at least n Bergman surfaces intersect. (The distinguished boundary in our example is {< ei.P , g( ei4> , eiA )) fP , ~ E [ .0, 21r ,J ""'
J) .
24 Property 3. Iff is holomorphic in a neighborhood of a Bergman·
I
domain D then the maximum of If in D is achieved on the distinguished boundary. Proof. For each fixed zl' lf(d. z 2>1 has its maximum on the boundary
I
of B(z1); hence, consider f(zl' g(zl' e ) ) tion has its maximum on I z11 =1.
1.
But for each fixed A. this func-
Corollary. Iff is known on the distinguished boundary of a Bergman domain B, it is known throughout B. In fact, we have the following: Bergman Generalization of Cauchy' s Formula.
= (.l.) 2 211'.
211' 211' ('
r
if/> L\ _!& ( i~ iA )dqiUI\. . ~.,. aA e· , e
JJ
e e
00
§ 5.
Analytic Polyhedra
In this section we shall define analytic polyhedra. Vl e shall show that every holomorphy region is a limit of an increasing sequence of analytic polyhedra. A.
Definition 17. Let Dopenc ([n; f1, ... , fk holomorphic in D, and A = {z z E D and fj(z) < 1; j = 1, ...• kJ. If AC C D, A is called an
I
I
I
analytic polyhedron (of dimension n) . Corollary 1. Every analytic polyhedron is a region of holomorphy. Proof. Let B be an analytic polyhedron,
lr.(t)l =1 for J
some j, say j = 1. But then
~
e bndry B. Then
25
is holomorphic at every point of B and singular at
t.
Corollary 2. Every connected analytic polyhedron is a Bergman domain.
:
Proof. Let B be a connected analytic polyhedron, t
€
bndry B. Then
lr.l =1 for some j; i.e. every boundary point satisfies an equation of the J iA form f.(z1•... , z ) -e = 0 . J n B.
Theorem 7. Let D be a region of holomorphy. and K CC D. Choose
D such that 0
KC C Dopenc c D. 0
Then there exists an analytic polyhedron
A.
A in D such that KCC ACC D . 0
Proof. Lett
€
bndry D . There exists a holomorphic function 0
"'
gt in D such that jg~ l, and lg~(z)l < 1 inK. Hence. there exists a neighborhood Nt of each ~
E
I
I
bndry D0 such that g t (N t) > 1. But bndry D0 ,
being compact, is covered by a finite numrer of such neighborhoods. say Nt{" •.• Ntk .
Let A ={z Ig\ /zll< 1; j =1, ... , k}
Corollary. Let D be a region of holomorphy. Then there exists a sequence A .• j = l, 2, ... of analytic polyhedra in D such that J
A .C..C A "+lC CD, and D = J
J
co
U.J= l AJ..
uco
-
Proof. Choose D1~~C. c o ~~ i=l Di - D. 2 CC: ... c C D such that ~{~1 , Consider the sequence { Dj} ; there exists a subsequence. say DjJ • such that
Then by theorem
o" 1cc
rl.
there exist A. analytic polyhedra, such that
A 1Cc
1
" cc:. A cc ... CC D. o 2 2
26 Chapter 3. Pseudoconvexity § 1.
A,
Plurisubharmonic and pseudoconvex functions V>/ e have already introduced the notion of a continuous sub harmonic
function (I,§ 2). We now extend -this definition as follows: :
Definition 18. Let D be open in«:, 4>: D- B.. Then ,P is said to be subharmonic if: i) -
00
~ cP
< co
' 4>
f . . 00
ii) r,l is upper semi -continuous; i. e. lim sup 9 (p' ) < 9 (p) pr- p
-
-
iii) for any domain DCC D; if h is harmonic in D and continuous on 0
0
ii00 , then h > ~ on SD implies h > t/> in D , -
0
-
0
Property 1 (Mean Value Property, 1). Let
{I z-z0 I < r} C
D, ~ sub-
harmonic in D. Then 211'
f/1 (z ) < -21 0
-
'11
S (z + reiB) df/ • tP
0
0
Property 2 (Mean Value Property, II). Let
{lz-z0 l < r}c D, tP
sub-
harmonic in D. Then r 211'
t/>(z ) 0
1
< -2 7l'r
Jr Jr 0
~ (z 0 + re
ie ) r
drdf) .
0
Property 3 (Strong Maximum Principle). Let 4> be subharmonic in D. Let lVI = sup 0 9 . Then, in each component of D either 9 (z) < lVI or 9 is constant. _Property 4. If ; satisfies i), ii) and the integral condition of Property 1 or 2, then 4> is subharmonic in D. Property 5. If ; and r/1 are subharmonic, then max (9 , 1/J) is subharmonic. Proeerty 6. If ;
€
c 2,
; is subharmonic if and only if A 9 ~ 0 •
27 DC~
Definition 19. Let
n 6
;
:
D - B. • Then t; is said to be pluri-
subharmonic in D if: i) -oo
< ~ < o-J
•
ii) ; is upper semicontinuous iii) if (z16 .•• z ) £ D and a. £ ~ arbitrary6 i =16 ... n, then n 1 . t (t) =~ (z1 +tar •.. , ~n + ~a0) is subharmonic for small J t 6
I.
~
is said to be pseudoconvex if it is plurisubharmonic and continuous.
Remark. Statement iii) above is equivalent to the following: 4> o
T(zl' ... , zn) is subharmonic in each variable separately, for all
linear transformations T . Corollary 1. Let
DC~
n
• f:D- ct. Iff is holomorphic in D, then
log If I is plurisubharmonic. Furthermore, if f :f 0 on D, log
If I is pseudo-
convex. Corollary 2. Plurisubharmonic functions satisfy the strong maximum principle. Corollary 3. If Dc:«:n and cp, 1/J:D- It are plurisubharmonic, then so are
max(~, 1/J), 4>
+1/J, and AcP , A > 0.
Definition 20. Let q;
E
C 2. The the Hessian of 1/) is defined to be the
following matrix:
Note that His Hermitian, if ,P is real valued. Proposition 1. Let DC «:n.
f/J
:D- .IR,
fjJ €
c 2 . Then~
is pseudoconvex
if and only if the Hessian of 4> is positive semidefinite.
Proof. Consider i
.P(~)
= ,P(z 1 + ta1, ... , zn + ~an ) , a.1 £ «:
'
=1, ... , n . Now 0(~) is subharmonic if and only if At~ 0 . But
28
a2ri = 4 atat
6~
n~
a
=
.J=E1 az.J
4ar
n
a2! '\aj azka:Z. . J
4 E: j, k=l
=
a.J
Proposition 2. Let DC <en , D C C D , ~ :D - lB. such that fP is
--~-~---.
0
pseudoconvex in D. Then there exists a sequence [ ~ .) of pseudoconvex, L J. C«) functions in D such that ~ . - ~ uniformly in D . . 0 J 0 Proof. Define the "smoothing functions" K , £
K K
f
f
:«:n (~)
f
> 0--as follows:
-11.
>0
-
Support KfC
{IJ t II
f
Define:
4>
£(~)
=
l
Kf (z-t)
~ (~) dE 1 •••
di'Jn' where we
take~ =0
([!n
where it is undefined.
Then~
f
(z)
f
C
co
• Furthermore, f f
in D , as follows: 0
4> (z)
=
l <en
~ f (z)=
5' fl:,n
Therefore
K f (t)
~ (z)d~ 1 •••
d 1}n
K(t)f(z-t)dE 1 ••• d1}. £ n
~
uniformly
29
I; (z)-; f(z) I ~
S
Kf(t)
a:n
I; (z)-; (z-t) I d~ 1 • • • cbJ0
< max
.-
l;(z)-; (z-t>l- 0 as
II t II < €, z
€
€-
0
D
0
since; is uniformly continuous on D • 0
Now ;
€
is plurisubharmonic since ;
€
(z)
=
S ctn
K (t) ; (z-0 d~ · ••• dTJ , € n
1
is essentially a linear combination of plurisubharmonic functions with positive coefficients. Proposition 3. Let A
domain
C
. ct, g:A - DC ct n; g holomorph1c. Then
; is pseudoconvex in D if and only if cb (g(t) ) is subharmonic and continuous for all such g. Proof. Assume; is pseudoconvex. By proposition 2, we may assume cb
€
00
C • Then 9g
and let
~(t)
€
C
00
• Now,
= fPg(t) .
Then
= as the Hessian of ; is positive definite. The converse is trivial, as the class of holomorphic g:A-D contains all linear transformations. Corollary. The image of a pseudoconvex function under a holomorphic mapping is pseudoconvex.
30 § 2. Pseudoconvex domains
~.
Definition 21. Let
{I t I < 1} C: C.
Then an analytic disc in D is a
mapping g: {It I < 1}- DC ftn, continuous on
{It I ~ 1} and holomorphic
in the interior.
The boundary of the analytic disc is the mapping g restricted to It J=1. z Set g(
{I t I ~ 1)) =:E. Abusing terminology by suppressing mention of g, we shall refer to
:E itself as the analytic disc and to a:E as the boundary. (Note that a:E in
general is not the set-theoretic boundary of the point set :E.) Theorem 8. Let DopenC: tr,n. Then the following are equivalent:
Jbe a sequence" of analytic discs in D. J
i) Let {:E.
then
~
.U :E .c. C D. ("Kontinuita•ssatz").
J=1
00
If
.v1 a :EJ.c c J-
D,
J
ii) -log A(z) is plurisubharmonic in D, where A(z) is taken in any norm. iii) For any analytic disc E in D, A(:E) = A(aE). iv) There exists a pseudoconvex function.; in D such that, for every N > 0 there exists a KC CD for
which~~ Non
D-K. (Informally, f =+co on
the boundary of DJ Exercises. a) In Euclidean space, i) has the following analog: Let Dopenc C Jln. Let { Ej} be a sequence of segments in D. If
Ua :E j c C p,
then U :E j r- c D. Show that this property holds if and only if
every component of D is convex. b) Find the analog of ii) in B. n. Definition 22. A region with any and hence all of the above properties is said to be pseudoconvex.
then
Corollary 1. Let o?penc cr::n, D. pseudoconvex. If n. 001D. is open, J J= J 00 J 0. is pseudoconvex. J=1 J
n.
Corollary 2. Let Dopenc a::n. Then D is pseudoconvex if and only if each component is pseudoconvex.
31 Corollary 3. The holomorphic image of a
pseud~onvex
region is
pseudoconvex. Corollary 4. Let
£A j} be a sequence of pseudoconvex domains such
that A .C A '+l' then UA . is pseudoconvex. J J J B. Proof of Theorem 8.
·
iii) implies i). U8I:.CC D and A(E.) = A(a:t.) imply cl( UE.)CD. J J J J That UI:. is bounded follows from the fact that g assumes its maximum J
over E on
II
{I t I =1}.
and Ua,; j is bounqed.
ii) implies iv). If Dis bounded, we may choose
If Dis unbounded, choose '(z) =max ~(z)
(-log~(z),
-log A (I:) i.e.
£:\(8:£)
A(E) .
But clearly,
iv) implies i). Let :E. be given by J
Consider the subharmonic functions -
~
(g.(O ) , j J
iv). Note that max
f/1 (g.(O)
It I
Z£1:.
J
(z) .
32 U a 1;.CC 0;
Now.
J
co
hence
sup ~ ( U j =1
But therefore
sup fjl ( U
a~ j) < 1\£ < co
co
E .) l
j=l
< M < co
• •
implying i) implies ii). (Proof due to Hartogs). Since i.\(z) is continuous, -log 6(z) is continuous. In fact -log b.(z) satisfies a Lipschitz· condition on compact subsets of D. To prove that -log
~(z)
is plurisubharmonic in D, it
is sufficient to show plurisubharmonicity at a point z
0
E
D.
Thus we must show
that, for the set of points z = z 0 + ~z 1 where z 0 and z1 are arbitrary points of D and ctn respectively, and t E ct is sufficiently small, -log 6(z 0 + t z 1) .=C/ltt) is subharmonic as a function of measure ~(z), z
E
D for
t.
II z111
For
It I< l.
small enough in the norm used to
Furthermore, 1/1(0 is subharmonic if it sat-
isfies the mean value property of subharmonic functions, and it is enough to show this for
~
= 0; i.e. to show that
s 211'
1/1(0)~ 2~
(! )
1/J(eiB)dB.
0
Let g(t) = h(~) + ih*(~). with h a real-valued function such that
d ~ 1.
h(eW) = f/;(eiB ), g holomorphic for
It I < 1,
claim that such a function exists.
Firstly, the continuity of 1/J implies the
and continuous for J
I
existence of a harmonic function h, in I~ < 1, equal to r/1 on I~
I = 1.
Vie Thus
his defined and is continuous on the closed unit disc. Secondly, take h* to be some conjugate function to h. Since g is now defined and holomorphic, and its
I I = l, its imaginary part satisfies a H~lder condition. Hence h* is continuous on f ~ I = 1. Next, let b be any vector in ctn with II b II = l, and let A satisfy 0 real part satisfies a Lipschitz condition on t
0 < "- < 1 . Consider the analytic disc in Cn 0
33
(1) E(O)C D. This is obvious. (2) If p £ E'(~) and~:- ~ .. then there exist p. £ Z(A.) such that p.-p.
(Namely, p. J (3)
3-g(to)
=z 0 U
O
35
1
either (i) c e em and ~ e sj = z I liz~ < J}or (ii) c- e OSJ and C i oD, or (iii) C c:_ dD n.nd C E dSj. For case (1), by hypothesis, there exists an NC about C such that NcnD is pseudoconvex. For case (ii), since Sj is convex, any ball about C, NC lying in D, satisfies NC~Dj is pseudoconvex. For case (iii), there exists a ball N ( C, r) such that N/) D is pseudo·convex. But Nl)s j is also pseudoconvex. Therefore N1\Dj is pseudoconvex. Therefore each Dj satisfies the hypothesis of this theorem and is bounded. life have already shotm that therefore D. is pseudoconvex, j = 1,2, •••• Since Djc Dj+l and \ D~j -> D, D is pseudoconvex (by Cor. 4 or Thm. 8). Establishing the converse of Theorem 9 is the Levi Problem.
§3.
Solution of the Levi Problem for tube domains
Definition 23. Let z. = xj + iYj• The set J n D = tJhere B is some open subset ofRn, is called a tube domain. B is called the base of the tube domain. Example. D = t {j)
I
= [ K, and we have obtained a convergent representation f
=
ev(fKew)
as claimed, unique by the above. Corollary 1. Let f be a convergent power series, and assume r vanishes at the origin. Then the set of zeroes of f in a neighborhood of the origin is of dimension n-1. Proof. By a linear change of variables, we may assume f is normalized with respect to z1 , K-1 ) f = h ( zK 1 + a1z1 + . + aK o o
For z2 , •.• ,zn small and fixed arbitrarily, a 1 , ••• ,aK are small, and z~ + a 1 z~-l + •.. + aK is a complex polynomial, with K roots (counting multiplicity). Furthermore, these zeroes are located 1n a neighborhood of the origin as they depend continuously on the ai. Definition 25. Let sclosed C. Dopen C G:n. Then S is said to be a (globally defined) analytic hyPersurface or analytic variety of codimension 1, if S is the set of zeroes of a function f ¢ 0, analytic 1n D.
44 Corollary 2. S is locally arcwise connected. Corollary 3. If n > 1 and f· is holomorphic in D d: en, then the zeroes of f are not isolated.
§ 2. Rings of power series A. As we have remarked previously, the set of convergent power series in z1 , ••• ,zn at a point forms a ring, as does the set of formal power series. We shall now state some algebraic results which will prove useful in the sequel. Refer to any standard algebra text, e.g. van der Waerden, Moderne Algebra, for proofs and details. Definition 26.- Let R be a commutative unitary ring. R is said to be an integral domain if: a E R, b E R, ab = 0 implies a = 0 or b = o. An element a e R is called a unit if a has an inverse in R. Elements a,b e R will be called equivalent, written a = b, if a = eb, where e is a unit. An element a e R is called reducible if a = be, where b and c are non-units; a is otherwise called irreoucible or prime. R is a unique factorization domain (U.F.D.) if every element may be written as a product of primes, unique up to order and equivalence. A subset I of R is an ideal if a, b E I implies a-b E I, and a E I, rt E R implies art E I. I is a proper ideal if I ~ R, and maximal if it is pr•oper and such that if ~ is any ideal satisfying I c. -.:P c.. R, · then I = 'I or ~ = R. A ring R is called a local ring if there exists a unique maximal ideal. R[t] denotes the ring of polynomials with coefficients in R and R[ t] the ring Ill power series in t with coefficients in &,· Let L ajtjE R[t], aj e R, t an indeterminate. L ajtj i~ called primitive if the coefficients aj of t have n8 common factor except units. (Note that r E iU t] implies r = ag with a e R and g e lt[ t] and primitive.} Two polynomials will be called relatively prime or coprime if they have no common polynomial factor. We use
the following results. Lemma a. (Gauss' Lemma) If R is UF.D, R(t] is UFD. Lemma b. Let p,q E R(t] (R[t]). Then p and q primitive implies pq primitive. Lemma c. Let P1 ,P 2 E R[t], R an integral domain. p1
=
p2 =
tK + al tK-1 + . . . +
~
tL + bl tL-1 + • . • + bL
•
Then there exists a polynomial r in.the coefficients a 1 ,bj called the resultant of P1 and P2, which is zero if and only if P1 and P2 have a common factor; i.e. if and only if there exist p,q,s E R[t], deg q > 0, such that P1 = pq, P2 = sq. Furthermore, there exist polynomials A and B such that AP 1 + BP 2 = r. Lemma d. No proper ideal I of a unitary ring R contains a unit. Lemma e. R is a local ring if the nonuni ts in R form an ideal. B. Definition 27. Let C9n denote the ring of formal power series at the origin in n complex variables. Property 1. c?n is an integral dom~in with unit. Property 2. The nonunits of ~n form an ideal; hence ~n is a local ring. Property 3. CJ n is UFD. P~oof. We use induction on n. For n = 1, units and elements of order 1 are irreducible. All elements of order > 1 are reducible, for if ord g = K > 1 , g(z)
=
zKg1 (z)
,
where g1 (z) 1s a unit, and the decomposition is unique. Hence, assume r9 n-l is UFD. We may assume f E (9 n is normalized at the origin. Then: K K-1 ) _ f = h ( z1 + a 1z1 + . • . + ~ = hp ,
46 where p is a Weierstrass polynomial; i.e. p E ~- 1 [z 1 ] and _is monic. Now f is prime in On if and only if p is prime in On-l [ z111 for: Assume p is reducible; i.e •. p = p1p2 • We may take p 1 ,p 2 to be Weierstrass polynomials. Hence f ~ (hp1 )p 2 . : Conversely, assume f is reducible: f =
fl f2
=
(hlpl )(h~2)
=
(hl h2){plp2) •
But p1p2 = p by uniqueness, and h1h2 = h. But (Qn_1[z 1] is UFO by assumption and Gauss' lemma. Now let f EOn· f
=
hp •
But
P = P1 ••• Pr where the pi are irreducible Weierstrass polynomials. f = hp 1 ••• Pr ,
Hence
and this decomposition is unique up to order and equivalence, for if f = tl ••• tt , ti = hi ri
,
by the Weierstrass Theorem and then rl ••• rt
=
P1 ••• Pr
=[
by uniqueness. Hence [r1 , •• • ,r 9 ~ p1 , ••• ,pr\ as C?n_1[z1 ] is UFD. Definition 28. Two holomorphic functions are said to be relatively prime or coprime at a point if their power series expansions at that point have no common irreducible factor other than a unit. Lemma f. Let f,g be holomorphic functions in D, 0 E D c ~n, n > 1 coprime at the origin, such that
47 f(O) = g(O) = 0. Then in any neighborhood of the origin, there exist points at which r vanishes and g does not, and points at which g vanishes and r does not. Proof. We may assume f and g are Weierstrass polynomials f
=
g
=
Suppose there exist no such points. resultant of f and g.
Let
r(a 1 ,bj)
be the
For each z 2, ••• ,zn in a neighborhood of the origin, there exist z1 such that f and g vanish simultaneously. But if f and g have a common zero viewed as polynomials in one variable, then f,g have a common factor. Hence r(ai,bj) is zero for each z 2 , ••. ,zn. But r is analytic, and hence r 0 near the origin, implies f,g have a common factor as polynomials in on. Lemma g. Let f,g be holomorphic functions coprime at the origin. They they are also coprime in some neighborhood of the origin. Proof. Let f = up g = vq ,
=
where p,q are Weierstrass polynomials, and let r
=
Ap + Bq
be the resultant of p and q. Let N be a neighborhood of the origin so small that u,p,v,q,A and B are convergent. Let a= (a 1 , ••• ,an) E N. To show f,g are coprime at a, it suffices to show p,q are coprime. The equation r
= Ap + Bq
persists where r,A,p,B and q are viewed as series about a, i.e. as series in ~i' where
48 ci p
Then
q
= zi - ai • = c~ + • • • = cr + ••• •
Now assume p and q have a common factor h "at a", h E ~-1 ( c1 ) • ~Ut p,q are primitive in t 1 j hence h is also.
where
hi
&n-1"
E
But
h divides r
A
E
k
E
~- 1 and 0 n-1 [ ~1] •
r.
Therefore
= hkA ,
a primitive power series in
k
k
=
ko
t 1,
+ kl ~1 + • • •
By comparing coefficients r
=
a relation in
h0 k0 A ,
Hence A divides
r,
(f)n- 1 •
and r
"X = hk.
But h and k are primitive, hence hk is primitive, hence h0 k0 must be a unit, i.e. hk is a unit, implying that h is a unit. Thus p and q are coprime at a.
§ 3.
Meromorphic . functions
Let x E Dopen C ~n, and consider functions which are each defined in some neighborhood of x in D. Call two such functions equivalent if they coincide on a neighborhood of x. This defines an equivalence relation, and the equivalence class of a function f at x, denoted by [f]x, is called the germ of f at x. If f is a holomorphic function, then [f]x amounts to a convergent power series. Germs at x form a ring with the obvious definition of
addition and multiplication. The ring ~x of germs of holomorphic functions is a commutative integral domain with identity and unique factorization. Topologize the · space. of germs 0 = &x by defining the following basis for the open se~~~ Let k E C), then k E (!)x and so k = [ f] x , where f is defined 0 0 1n an E-neighoorhood Nx of x0 in D. At each y E Mx ·, 0 take that class in c?Y 0 containing the direct analytic continuation of f, i.e. take [f] • Then define f]Y to be an open set and th~ collection of such JENxo sets to be the basis ot open sets. A holomorphic function f in D amounts to a continuous mapping, r : D -> which assigns to each point in D a holomorphic germ over that point. Now form the quotient field Mx of ~ for each x E D. Topologize M = U Mx as follows: Let l E M, then l E Mx , l=[~lx xe:Rnd is represented by [f1 lx /[f2lx 0 .... 0 0 0 where r 1 and f 2 a~ nolomorphic functions at x 0 , and because of unique factorization we may take f 1 and f 2 to be coprime at x0 • Let Nx 0 be a neighborhood of x0 in which f 1 and f 2 are defined and are still coprime. At each y E Nx , take that class in My represented by . [f1lyi[f 2ly· Thg union over Nx of these classes we define as an open set and the co~lection of all such sets we take as the basis for the topology. The elements of Mx are ~alled germs of meromorphic functions over x. Definition 29. A meromorphic function in D is a .continuous mapping which assigns to each point of D a meromorphic germ over that point. Meromorphic functions form a field. At a point z0 E D, a meromorphic function g is efined by the quotient of two functions f 1 ,f 2 coprime
U
U [
dJ
50
and holomorphic at z0 • a) If f 2 (z 0 ) ~ 0, i.e. f 2 is a unit, then f 1/f2 is holomorphic there and hence g is holomorphic at z0 and therefore in a neighborhood of z 0 • z0 i~ called a regular point of g. b) If f 2 (z 0 ) = 0 and f 1 (z 0 ) ~ 0, then g is said to have a pole at z0 • c) If f 2 (z 0 ) = 0 and f 1 (z 0 ) = 0, then z0 is called a point of indeterminacy of g. The set of such points has topological dimension 2n-4. Corollary 1. The set of regular points of g is open, and glregular pts is holomorphic. Corollary 2. If z0 is a pole of g, then there exists a neighborhood N of z 0 in which every point is a pole or a regular point. Furthermore, g has no isolated poles (n > 1), and, for each number M > 0 there exists a neighborhood NM of z0 in which jgj > M. Corollary 3. A point of indeterminacy is a limit point of zeroes and poles of g. Exercise. A point of indeterminacy of g is a limit point of zeroes of g - a, where a is any complex number. Poincar~'s Problem 1) Weak form: Given a domain D, is every function meromorphic in D a quotient of two functions holomorphic in D? 2) Strong. form: Given a domain D, g meromorphic in D, is g the quotient of two functions holomorphic in D and coprime at every point?
§4.
Removable singularities
In this section we shall state three theorems. The / first, Rado's theorem, facilitates the proof of the first theorem on removable singularities which is a direct generalization of the Riemann theorem 1n one complex variable.
51 A second theorem on removable singularities will be stated but not proven. Theorem 13 (Rad6) • Let f( zi, ••• , z .) be continuous in Dopen C. Cn and holomorphic in (D- {zlf(z) = 0""$). Then t is holomorphic in D. Proof (Heinz). A function of n variables is holomorphic if and only if it is holomorphic in each variable separately. It is sufficient to prove the theorem for fUnctions of one variable. If D 1 is any open disc whose closure is contained in D c: ~n, we must prove that f(z) is holomorphio in D1 • Without loss of generality, assume that D' is the unit disc (lzl, it is bounded there, and we may assume lf(z) I < 1 in (D' v ~ By hypothesis, f(z) is holomorphic in (D'-6). Construct a complex-valued harmonic function g(z) in D1 such that g(z) = f(z) on Then, consider the following functions for z E (D 1-6) and a > 0
r
r )
r-.
~l(z)
=
~2(z)
= Re [f(z)-g(z)] - a log I f(z) I
~7\(z)
=
Im [f(z)-g(z)] +a log If(z) I
=
Im [f(z)-g(z)] - a log If( z) I
Re [ f(z)-g(z)] +a log If(z) I
~
~4(z)
•
Note that for Z E (D 1 -6), a log .I f( z) I < 0. Now, as and points where z -> o(D'-6), which consists of r = o, either (i) Z -> Z 0 E (o(D 1-6) () and then [f(z)-g(z)]-> 0 or (ii} z -> z0 E (o(D 1 -6) ~ 6 ) and then [f(z)-g(z)] remains bounded and log lr(z)l ->- oo • In either event, ~ 1 (z} and ~ 3 (z) -> negative numbers while ~ 2 (z) and ~ 4 (z) -> positive numbers. But, since the ~ 1 are harmonic in (D 1 -6), they assume both their maximum and their minimum
r
r}
52 on
~(D 1 -6).
~ 2 (z)
Let
a
Hence, ~ 1 (z) and , 3 (z) and ~ 4 (z) are positive for all -> 0, then
are negative, and z E cl(Dt-6).
Re [ f ( z ) -g ( z ) ]
which implies Re [ ·f ( z) -g ( z) 1 ~ 0
~2(z) -> Re [f(z)-g(z)]
which implies Re [f(z)-g(z)]! 0
~ 3 (z) ->
Im [f(z)-g(z)]
which implies Im [f(z)-g(z)]
~4(z) ->
Im [f(z)-g(z)]
which implies
~l(z)
->
~
0 Im [f(z)-g(z)]! 0
for all z E cl(D'-6). Therefore f(z) = g(z) for all z E (cl(D'-6) Since d6 consists or points of and points of cl(D 1-6), f(z) = g(z) for z E ~6. In any component of the interior of A, f e 0 and thus g e 0. Therefore f(z) = g{z) in (D' U which means that f is harmonic in D'. Thus f has continuous first order partial derivatives. f satisfies the Cauchy-Riemann equations in D-6, hence by continuity, on o6 and on In the interior of 6, f = 0 and therefore satisfies the equations in 6. Hence f satisfies the Cauchy-Riemann equations in D1 and is therefore holomorphic in D1 • Theorem 14. Let Dopen C: ~n, and let g ~ 0 be holomorphic in D. Let f be holomorphio and bQunded in D - fzlg(z)= Then f is holomorphic in D. Proof. Consider the function if g .,. 0 gf
ur>.
r
r->,
r-.
oJ .
h
=
i
if g
0
=
0
Then h is continuous as f is bounded, and holomorphic where it is not zero. By Rad6's theorem, h is holomorphic. But g is holomorphic by assumption; hence f
is meromorphic. But f thus holomorphic.
=
~
is bounded,
thus without poles,
53 Theorem 15. Let Dopen C On; g,h holomorphic in D, not identically zero and relatively prime at every point or D. Let r be holamorphic in the set D- fzjg(z) = h(z) = Then f is holomorphic in D. Theorem 16. Let Ddoma1n c on and 1et g F 0 be holomorphic in D. Then (D - {zlg(z) = oj) is connected. Proof. Let S = (D zlg(z) = Oj). Suppose that S is not connected, then S = U VV where U and V are open, disjoint sets. Define a function h =l5 i~ ~ h is holomorphic where g ~ 0, and is bounded; therefore h is holomorphic in D. This is impossible since it implies that h is identically 1 in D.
oJ.
-l
.
§s.
Complex manifolds
Remark. From now on, "differentiable" means "C 00 11 • Definition 30. X is a (differentiable) manifold of real complex (com~lex ) dimension r, if the following conditions are satisfied: (1) X is a Hausdorff space. (2) Given an open set in X and a function defined in it, it is possible to say whether or not this function is f1fferentiable) holcmorphic • (3) There exist coordinates: every point in X has a neighborhood where r (real, differentiable) functions complex, holomorphic are defined such that they give a homeomorphism of this neighborhood onto a domain in ( ~~), and every function defined in this neighhorhood is (differentiable) if holomorphic and only if it is (differentiable as a function of x1 , •.• ,~) holomorphic in each variable of z1 , ••• ,zr • The coordinates are called local coordinates and such a neighborhood is called a coordinate patch. (4) There is a countable basis for the open sets of X, i.e. X is second countable. Remarks. On a complex manifold we may talk about holomorphic and meromorphic functions; on a differentiable
54
manifold, about differentiable functions. A connected 1-dimensional complex manifold is called a Riemann surface. Every c6mplex manifold is a differentiable manifold. Therefore it is natural to ask on which differentiable ~hifolds we can introduce the concept of holomorphic functions: that is, which differentiable manifolds can be given a complex structure. Necessary conditions are that the differentiable manifold be orientable and of even dimension, r = 2n. If n = 1, these conditions are also sufficient; however, if n > 1, they are not. In fact, in the latter case, necessary and sufficient conditions are not known. There are other differences between the cases n = 1 and n > 1: (i) When n = 1, axiom (4) in definition 30 is unnecessary as it follows from axioms (1), (2), and (3). When n > 1, axiom (4) is essential. (ii) If n = 1 and X is compact, there exist nonconstant meromorphic functions on X (i.e. on every closed Riemann surface there exist non-constant meromorphic functions). However, when n > 1, there are compact complex manifolds having no non-constant meromorphic functions. (iii) If n = 1 and X is not compact, there exist non-constant holomorphic functions (i.e. on every open Riemann surface), while if n > 1 this is not necessarily so. For example, let Y be a compact connected complex manifold of dimension n > 1, then X= (Y-fpj) is not compact and if there existed a non-constant holomorphic function on X, it would be holomorphic also at p (by Hartogs 1 Theorem), and it would be constant (by the maximum modulus theorem). Examples of Complex Manifolds. 1) vn 2) Any open subset of a complex manifold. 3) If X and Y are complex manifolds then X K Y is
55 a complex manifold, where a function r is holomorphic in · X~ Y if it is holomorphic in X and holomorphic in Y. 4) The complex projective space Pn(IJ)
=
t [(z
0
,z1 , ••• ,zn)] lzi are not all zeroJ·,
where [(z 0 , ••• ,z )] denotes the equivalence class of points (z 9, ••• ,z9l E tn+~, where two points (t0 , ••• ,Cnl and CC0 , ••• ,Cn) are equivale~t if and only if there is a t ~ o such that Cj = tt 3, j = O,l, ••• ,n, with local coordinates in a neighborhood of (z0 , ••• ,zn) where zi ~ 0 for some i, 0 ~ i ~ n, being z0/zi' ••• ,zi_1/zi, zi+l/zi, ••• , zn/zi, is a complex manifold. On this manifold there exist meromorphic functions; the ratio of two homogeneous polynomials of the same degree is such. 5) The special case of 4, P1 (t) =Riemann sphere=
[~e
U{oo3) •
6) Starting with a complex manifold of dimension n
1, omitting a single point, and imbedding P 1 (~), we will obtain a new complex manifold. This procedure is known as the cr-process. We do this for n = 2, starting with t 2 • First we define the space X to consist or two types of points: (z 1 ,z 2) ~ (O,o)j I = (Czl,z2) I (zl,z2) E c2 and (C 1 ,C 2 l -1 (O,O)J. II = { [( cl' C2)] I (cl' C2)E o: 2 and
X
>
I V II • Secondly, we make X into a Hausdorff space by defining the following basis of open sets: A neighborhood of a point p E I shall be a neighborhood in the ordinary topology_ of ~2 such that its closure does not contain (0,0). A neighborhood of a point p E II, p = [(C 1 ,C 2 )], and say cl ~ o, shall be the set of points; [(l,C)] E II satisfying IC - c~cll < E, and (zl,z2) E I satisfying =
z1 # O, lz 1 1 < c , lz 2 1 < c and lz~z 1 - ~~~ 1 1 < E, for some c > 0. Thirdly, we define local coordinates •. Near a point of I, take z1 and z 2 as coordinates. Near a point [(~ 1 ,~ 2 )] ~f II, where say C2 ~ 0 and hence [(~l'C2)] = [(~ 1/~ 2 ,1)] = [(~ 0 ,1)] say, take as local coordinates t and T I
(l+t)T
zl =
~O
z2 = T ~1
(l+t)~ 0
=
~2 = 1
•
On this new manifold X, the following holds: every holomorphic function on X is constant on II. For if f is holomorphic on X, it is holomorphic on X-P 1 (~) = c2-{o]. By Hartogs 1 Theorem, f is also holomorphic at the origin {0,0}. Therefore f approaches some complex number, a, as its argument goes to the origin. Hence near every point of P 1 (~), the value of f is close to a. Thus f =a on every point of P 1 (~), but then f must be identically a on II. 7) A globally presented, regularly imbedded analytic subvariety Y, of codimension r in an n-dimensional complex manifold X is defined as follows: Let f 1 , ••• ,fr be holomorphic functions defined on X, such that, for every x c X at which r1 (x) = ••• = fr(x}= 0, the rank of the Jacobian matrix J
=
ori "\ ( OZ"j)
is r, i.e. J is of maximal rank. The derivatives ofi/ozj are to be understood in the following way: Let N be a
57 coordinate patch containing x, define the local coordinates
and let
l
N -> ,n
Then Then Y,
as a subset or X,
is defined as:
Note that Y is closed in X, and that axioms 1, 2 and 4 are clearly satisfied. We now define the local coordinates in y. Let y E 1, with local coordinates j(y) =(z 1 (y), .•• ,zJy) defined in a neighborhood N of y, where N c.. X, Assume that, at y, det (ofi/ozj) = det A I o, J = l, ••• ,r by relabeling the zj if necessary. Define new coordinates ~1
==
fl(zl, ••• ,zn)
~r ==
fr(zl, ••• ,zn)
• • •
~n
zn • Then the transformation taking square rna trix; ==
z
to
A
is given by the
B
1
0 •
0 0
which is nonsingular as
~
det A I 0.
1
Then the local coordinates
58
are CD Note. If X and Y are homeomorphic manifolds and the Cousin Problem is solvable in X then it is solvable in Y. Thus by the previous theorem, the Cousin Problem is solvable in any domain which is the product or simply connected domains in 1.
+
+
+j
Chapter 6. §1.
Cohomology
Cohomology of a complex·manifold with holomorphic fUnctions as coefficients
Let X be a complex manifold and U = {ui)' i e I, be a fixed open covering of X. (We always assume that ui ~ u 3 for i ~ J.) · · · A. Definition 32. An r-cochain ~ on U is a rule which assigns to every ordered intersection of (r+l) sets, Ui Ui , a holomorphiC fUnction fi i (z) o -r o ••• r defined 1n this intersection such that 1. (a) when the ij are distinct and nul ~ ~, fi 0 ••• ir(z) is a function holomorphic in lluiJ• j (b) when the iJ are either nondistinct or fluij = ~, fi o••• i r (z) = o. 2. f (odd permutation of (i0 .,.ir))=- ~(i 0 ••• ir) f (even permutation of (i0 ••• ir) )= f {i0 • • .ir) • Examples. 1. A 0-cochain is a rule f which assigns to every ui e U a holomorphic function f 1 (z) defined in ui. 2. A 1-cochain is a rule f which assigns to every ordered intersection ui ~u 3 , a holomorphic function rij{z) defined in ui A u 3 such that fij{z) = 0 if i = j or u1 f\ u j = ~, f ij = - f j i. Cochains of the same dimension form an Abelian group under addition, cr = Cr{x,u, 0 and any differentiable manifold X. In fact ~{X,U,C 00 ), r > 0 is already trivial, for every locally finite covering U. Proof. We define a homomorphism Q : cr(x,u,c 00 ) -> cr-l(x,u,c 00 ) r > o, so that f = Qof + oQf, and this is sufficient. Let i, wi} be a partition of unity subordinate to U (see Definition 31 and Proposition 3, 3, Chapter V). Define
= ~ mif(i1 0 ••• ir_1 ) , where this sum is understood as follows: wi f = 0 if f = 0 Q5(i 0 ••• ir-l)
or if wi = 0. Note that the local finiteness of U insures that almost all terms of this sum vanish at any point of X. Now
Note that this proof hinges upon the fact that Cr(X,U,C 00 ) is a module over globally defined C00 functions.
80 Chapter 7. Differential Forms §1. Rins of differential forms in a domain A. Definition 39. Let D Rn as follows: On zero forms, i.e., on functions defined in D, set
df
n
df = ~ axj dxj
.,
and on monomials: d(adxj 1
A. ••• Adxjr) =
( da) 1\ (dx j 1 1\ • • • 1\ dx j r) •
Extend d to RD linearly. Note that d is not an operator-homomorphism, i.e. d(fa) ~ fda , where f is a function on D and a E RD. Example (1). Let D c JR . . . , and observe that da on a zero form acts like a "gradient," on a pure 1-form like "curl," and on a pure 2-form like "divergence." Lemma 1. i) d2a = 0 for every a E RD ii) d (a 1\ ~ ) = da A B+ ( -1) r at\ df3, for every purer-dimensional form a. Proof of this lemma is left as an exercise. Hint: it suffices to consider only monomials Q 1 B. Definition 40. A form a E RD is said to be closed if da = 0. ~
82 A form a
f3
E
RD
is said to be exact if a
= df3 for some
RD. It is clear that ~he closed forms form a subgroup ~ of Rnl' and that the exact forms are a subgroup R~ of RD. Hence, we form the d-cohomology group R~ I R~ = closed forms I exact forma. Lemma 2. The closed forma form a subring of RD · in which the exact forms are a two-sided ideal. ( 1. e. 1) closed A closed = closed ii) closed A exact = exact iii) exact 1\ closed = exact) Proof. We may assume that a,f3 are monomials. 1) If da = df3 = 0, then d(a/\(3) = da/\(3 + (-l)degaaf\d~ E
ii)
If f3 = dy, da
= o.
= 0,
d(a Ar) = da A-y+ (-l)deg aaAdr = [ (-l)dega a] 1\ (3, i.e. a/\ f3 = d ( -1 ) dega a 1\ J' ) ,
and similarly for iii), ( f3 A a) = d(-y A a). Hence, the de Rham group is a ring, (the de Rham cohomology ring). C. NO\IJ assume D C. ron. vie identify an with lR 2n in the usual way, and observe that
= xj + iyj) j z j = xj - iyj zj
-
=
l, ••. ,n
are functions of xj and y j; hence we may apply "d", obtaining dzj = dxj + idyj j = 1, ••• ,n • dzJ = dxj - idyj
r
I
But the dxj,dyj generate the ring of differential forms on D; and the above equ:tions are solvable for dxJ,dyj in terms of dzj and dzj. Hence the dzj and dzj also generate the ring of forms, so that for any form a E R0 :
2n
a =
> L r;Q p+q==r
ai
i -j
-j dzi
1••• p 1•· • q
1
1\ • . •
11 < ••• Ap+l,q and Apq -> Ap,q+l as follows:
a:
()a
=
-oa
=
t j
t J
da dzj
dz
-da
d-z
ozJ
j
. J o, -o I
where a is a zero form, and extend to R0 as before. Now d = o + ~~ as can easily be verified. Hence d2 = 0 = (o+~) 2 = o2 + ~ 2 + (o~+~o). Note that (o+~) 2 Apq
= o2 Apq + ~ 2 Apq + (~~+~o) Apq
= Ap+2,q + Ap,q+2 + 2Ap+l,q+l -2
-
-
,.
so that o2 = o, o = 0 and oo = - oo. We nm.z define o-closed, o-closed, o-exa.ct, and o-exact forms, and form the associated cohomology groups:
-
84 0.-closed forms a-exact forms . and
--o-closed forms . a-exact forms
We may again verify that these groups are in fact cohomology rings. Observe that, if we restrict the coefficients to be holomorphic functions in D, becomes trivial and d = o. Hence, we can also form the cohomology ring:
o-
closed holomorRhic forms exact holomorphic forms where we define a holomorphic form as follows: Definition 42. The form L a1 i -j 7 dzi A ••• 1""' p l•••Jq 1 1\ dzip 1\ dz jl 1\ ••• A dz jq is said to be holomorphic if q = 0 and the
a
il ••• ip
are holomorphic functions.
E\2. Differential forms on ma!'lifolds Let D and 6 be domains in Rn and Rm respectively, and let f be a diffeomorphi3m from D onto 6 • Denote a point of D by x = (x1 , ... ,xn) and a point of 6 by ~ = ( ~ 1 , .. ., ~ m) • Then f ( x) = ~ , or ~ j = f j (x1 , •.. , xn), j = l, ••• ,m. There is an induced mapping f * associated with f which maps diffe~(.\ential for•ms in 6 into differential forms in D; if = :;:-- aj j ( ~ ) d~ j A . . . J\ d~ J' r , then ... _._ 1" • • r 1 f*a =
>
1. 2.
3. 1~.
aj 1 .•. j (f(x}) dfj A •.• Adfj r
1
r
and
f * preserves degrees f * ( a+f3) = f *a + f *f3 f * (a 1\f3) = f * aA f * f3 * 1( df a = f da •
Let D,6 be domains in ~n, and let f be a holomorphic mapping from D to 6. Let (z 1 , ••• ,zn) denote a point of D
85
:
and (~1, ..... ~n) denote a point or 6. The induced mapping f* (as above) has the properties 1. f* preserves bidegrees 2. f * (a + ~) = f * a + f.* t3 3· -r* (a A. t3) = f * a 1\ f *t3 4. ~r*a::;: f*~a and ()f*a= f *()a • For the case of holomorphic forms, since f is holomorphic and a holomorphic function of a holomorphic function is holomorphic, f * takes holomorphic forms into holomorphic forms. Let X be a differentiable manifold of dimension n. We define a differential form on X as follows: Definition 43. A differential form on X is a rule which defines a differential form in every ooordinate patch. Each coordinate patch ~a is diffeomorphic to a domain D in mn (by definition). A differential form on X associates with every coordinate patch Pa a differential form a in Da such that if Pa () Pa i ~ then the images of this 1 2 intersection in Da and ~ are diffeomorphic and the 1 2 induced map on forms takes a 1 into a 2 •
,
Poincare Lenunas The Poincare Lemmas state that in sufficiently "nice" domains any closed form not containing a 0-form is exact. More precisely, Theorem 23. ( Ta) Let D = x1 , ••• , xn) I Ixi I < Ri ~ oo j C Rn, and let a be a pure r-dimensional form, r > O, in D. - If da = 0 then a = dt3 for some t3 • (Tb} Let D = z1 , ••• , zn} I Iz j I < Rj ~ ooJC C:n, and let a be a pure r-dimensional holomorphic differential form, r > 0, in D. If da = 0 then a = dt3 for some holomorphic form t3 •
t(
1(
86 {Tc) Let D be the domain in (Tb), and let a be a (p,q)-form in D, q ~ 1. If oa. = 0 then a.= o~ for some (p,q-1)-form f3 • Proof. Lemma. Let i,j be fixed numbers, 1 ~ j, i,j = 1,2, ••• ,n. (La) Let D be the domain in {Ta). If ~ is a C00 • function in D, then there is a C00 function ~ in D such that ~ = o~/oxi and if o~/oxj = 0 then a~/oxj = o. xi Proof. Define 1[J(x1 , ••• ,~) = ~(x 1 , ••• ,xi-l, t ,xi+l, ••• ,xn) dt.
-
=
-
J 0
(Lb) Let D be the domain in (Tb). If ~ is a holomorphic function in D, then there is a holomorphic function 1[1 in D such that ~ = o1[1/ozi and if o~/ozj = 0 then o1[1/ozj = o. Proof. Since ~ is holomorphic in D, CD A. k ~ = ~ a 1c(z 1 , ••• ,z 1 , ••• ,zn)zi~~ Define k+l "ft;"" CD ak(z 1 , •.• ,zi, ••• ,zn)zi
1[J(zl,. • .,zn)
=~
k+l • k=O (Lc) Let D1 = ~(z 1 , ••• ,zn) I lzjl < rj < ro}C mn. If ~ is a C00 function in a neighborhood of D1 , then there is a C00 function t in perhaps a smaller neighborhood of 1 such that ~ = a,~/az 1 and if ~ is holomorphic in zj then so is 1[1. Proof. Define "' 1 f" ~(z1, ••• ,zi, ••• ,zn) 1[1( zl" ... , zn) = - ;r jj t-zi d~ d11 •
n
ICI k, (!3j-f3k) is a holomorphic form on Dk since c(Bj-13k) = 0 on Dk. . Therefore the coefficients of (~j-!3k) can be approximated as closely as desired by polynomials, and hence the form · (~j-13k) can be approximate~ by a form Pjk whose coefficients are these polynomials. Choose ej > 0 such that ~ ej < co, and define 113.-~~1 = ~ lb~ ~ - b~ 1' 1. Construct A J ~ •l••••:p ·1··· p 13 ~ as follO\'lS: 131 = 131 , ~ 2 = f:3 2 - 21 w~ere I f3 2-f31 1 = I ( 13 2 -~ 1 ) -P 21 1 < e1 on D1 , 13~ = !:'-,: - P7, 2 where ,.. "' "' 113 3-13 2 1 = I(!3 3-13 2 )-Q32 1 < c2 on D2 and Q32 = ~ + P32 , etc.
.!
/
-
_.._
-
"'\
/
/
"'
()pj ="'a!3j =a on Dj. Since lf3j-!3j+ll < ej on Dj, llm Bj = 13 exists uniformly on compact subsets of D, oo
j->oo
-
the
coefficients of ~ are C functions and of3 = a in D. (Cf. Part 2 of the proof of Theorem 19.) (ii) If a has bidegr~e (p,q), q > 2 then each Bj ~as bidegree (p,q-1). a(~j+l-~j) = 0 in a neighborhood of Dj. Therefore Bj+l:f3j = ayj for some form yj, in a Nj of Dj, by l._of (c). Let roj be a neighborhood real valued c00 function ~ 1 on Dj and 0 outside a ~ neighb~rhood Mj of Dj, Mj cc Nj. Set y j = rujy j:."'. Then yj ;s defined in Dj+l -~d 13j+l- f3j = ayj on Dj. Let P1 = ~ 1 , 13 2 = ~ 2 - dy 1 , ••• , and in general A. Bj = !3j- a(y 1 ~; 2 + ••• +yj_ 1 ). Then f3j+l-~j = (13j+l-f3j)-
=
1\
-
-
1\
"
.1\
"
-
-"'
'\
.
"
"
;::"'
d(r 1+••• +yj) + d(y 1+ ••• +yj_ 1 ) = (!3j+l-Bj) - orj -A
-
():] j = dP j = a desired f3.
on Dj.
Letting
j
->
oo
= o,
and
we obtain the
Chapter 8.
§ 1.
Canonical Isomorphisms De Rham 1 s Theorem
A. Definition 44a. Let X be a differentiable manifold with covering u. We say U is simple with respect to differential forms, or d-simple, if it is ?pen, locally finite, and, for the intersection of any finite number of sets of the cover u0 II • • • f\ ur, the Poincare lemma for 11 d" holds. Theorem 24a. Let X be a differentiable manifold. Then there exist arbitrarily fine d-simple coverings (i.e. every covering of X has ad-simple refinement). Proof. \'le shall first prove this theorem assuming X C IRn, X open. The case of an arbitrary manifold X is treated at the end of this section. Assume XC. iRn. Note that in any box { lx1-ail < ri, i = 1, ••• ,n ~ , Poincar~ 1 s lemma holds by Theorem 23. Furthermore, the intersection or any finite number of boxes is again a box; so it suffices to refine any covering U to a locally finite cover!~~ by boxes and this is easily done. Lemma la. If (X,U ,oP) = 0; r > 0, p! 0, \'lhere rf denotes the sheaf of germs of p-forms, and U is locally finite. Proof. The sections of 0° over U are C00 functions, so ~(X,U,o0 ) = Hr(X,U,C 00 ) = O, r > 0, by Theorem 22. Since any element of r.P, when multiplied by a C00 function, remains in r.P, we may establish this lemma by constructing a homomorphism Q : cr(x,u,oP) -> cr-l(x,u,rf), r > o, so that if f ~ cr(x,u,rf), then f = Qof + oQf precisely as before. Hence every cocycle is a coboundary. Corollary. Hr(x,oP) = 0; r > o, p! o, rf as above. Theorem 25a. Let X be a differentiable manifold, U a d-simple covering. Then the following groups are canonically isomorphic, where Cf denotes the sheaf of germs of closed p-forms in X:
90 (i) (ii)
( iii)
1( U ~ H X, ,cr ~
closed p+l-forms exact p+l-forms '
P!
0 •
r
>
o, p!
closed r-forms exact r-rorms-, r
>
o .•
Hr+lrx,u,oP) ~ Hr(x,u,QP+l)
r H (X,U,~) ~
I
0 •
Before proving this theorem (following A. Ueil) , we : introduce the notion of coelements. B. Definition 45. A coelement ~ of bidegree (r,p) is an r-cochain on a (fixed) covering with coefficients which are pure dimensional differential forms of degree p, i.e. if uj , ••• ,uj are distinct sets of the covering with nonempt~ inters~ction, then rrp(jo···jr) assigns to this intersection a pure difrerential form of degree p defined there. The coelements form a vector space over €. Define drrp = gr,p+l, v1here g assigns to each intersection "d" of the form which f assigns; d2f = 0. Define 5~P = hr+l,p in the usual way; 52 = 0. Clearly d5 = 5d (for 5 "adjusts" the domain, and d the range, of the coelements). Coelements f for \'lhich df = 0 are cochains with closed forms as coefficients, and if df = 5f = 0, the coelements are cocycles with closed forms as coefficients. If U is a d-simple covering, rrp a coelement, p > O, then a~P = 0 implies f = dg. If r > 0 and 5frp = o, then f = 5g by Lemma la. fr',p+l are l'le say that coelements rr+l,p and associated if there is a coelement grp such that f = 5g and f = dg. C. Proof of Theorem 25a. Note first that (1) and (ii) imply (iii), for closed (p+l)-forms Hl(X U =n) l+s( u ~n-s) exact (p+"l) -forms ~ , ,l r- ~ • • • ~ H X, ,1 r "' ••• ~ Hl+p(X,U,0°) , p ~0 1 -o and o = ~ •
91 Note also that r =n H (X, u ,( r) =
l rP I1df'P=or'P=O l1
~ohr-
.
,
r
> 0 •
,pldhr- ,p=O} (1) Ne associate a closed (p+l)-form on X (or, equivalently, a coelement fo,p+l or bidegree (O,p+l)) to each cocycle class in :H1 (x,u,oP) as follows: Let r1P be any cocycle; df = of = o. Using Lemma la, there exists a g 0 P such that og = r. Set fo,p+l = dg, and "' = 0; hence, we have assigned a closed note that df "' 5dg = dog= df = 0.) Denote by (p+l)-form. (Also, of= Sf~ the class in closed (p+l)-forms containing We ~ J exact (p+l)-forms make the following assertions: ~ "' ? a) ~ f) does not depend on the choice of g. f3) r{>= {r 2~ if (r1\ ,= tf~, i.e. if f 1 -r 2 = oh, dh=O. r) The class mapping tr1->if) is an isomorphism. "' Proof of a) Assume f ~ 5~ 1 = 5g8, and set 0"'r 1 = dg 1 ; r 2 = dg 2 • a) asserts that r 1-f2 = dh P whe~e "'h P is globally defined on X; i.e. 5h = 0. But r 1-r2 = d(g 1-g 2 ) and 5(g1 -g 2 ) = o. Proof of~) Suppose riP - f~P = oh0P, where 5f1 = 5f2 = df1 = df 2 = 0 and dh = o. Now f 2 = og 2, f 1 = og1 = og 2+5h = 5(g 2+h). Hence "'f 1 = dg1 = dg 2 + dh, "'r 2 = dg 2 , and "'r1 -r "' 2 = dh = o. Proof of r) It is clear that the association map fj -> ~ f is a homomor•phism. It is one-to-one for, assume fJ -> o; i.e • r = og, and dp 0P = 0. N0\'1 1 dg0P = 0 means g0P = dhO,p-l so that f = og = 5dh: hence Furthermore, it is onto, for, assume "'f is any closed (p+l)"' "' form. Then df = o, so f = dg by d-simplicity of u. Define f = 5g. Then df-= d5g = 5dg = 61 = 0, as "'f is globally defined, and 5f = 52g = 0. (ii) The proof in this case is essentially the same; let ~+l,p satisfy dtr+l,p = otr+l,p = o. Then there exists a grp such that 5g = f. Set "'r f ' p+l = dg, and
f.
i
i l
J
!rJ·= o.
.
92 'V
'V
observe df = 6f = o. With a similar notation, we prove a), ~), andy). a) Assume f = 0 = 6g ; then { dg~ = 0, for 6grp = o implies grp = 5hr-l,p, and dg = d6hr-l,V = 6dh. Now dh is closed, so:fdr} = f6(dhl).= o in Ffcx,u,oP+l). ~) r+l,p = 6hrp implies idhrp} = o, as dhrp =. o. y) t-le again have a homomorphism fr+l,pj ->trr,p+lj • It is one-to-one for, if f = 6grp and dgrp = o, then g = dhr,p-l, and r = 6g = 5dh , so if}= o. It is onto, 'V +1 'V 'V 'V for assume tr,P satisfies df = Bf = 0. Then f = dgrp by simplicity or u. Set r = 6grP. Then df = d6g = 6dg = 5f = 0 and 6f = o. Clearly {f) -> 't-f • D. Lemma 2a. Let X be a differentiable manifold; U,V d-simple coverings and V a refinement of u. Then the following diagrams commute, where r.P denotes the shear or germs of d-closed p-forms
l
i)
Hr+l(x,u,oP) - >
~
I-Ir+l(x,v,oP) - > ii)
E1 (X,U, (f ) 1
-
H (X,V,cf) iii)
Hr(x,u,~:;)
~ /"~'
~~
W(X,V,iC) ~
d-closed (p+l)-forms d-exacr-{P~fr:lorms
d-closed r-forms d-exact r-forms
where r > 0 and p > o. -Proof. For i), this lemma states that one obtains the same result by first mapping a coelement to any associate and then restricting the domain of definition; or by first restricting the domain and then associating it. The other commutativity claims are disposed or as easily.
93 Theorem 26a. Let X be a differentiable manifold for which there exist arbitrarily fine d-simple covering~. Then the follott1ing groups are canonically isomorphic: ( I) (de Rham): Hr(X ~) ~ d-closed r-rorms r > 0 ' - d-exact r-forms ' Hr(x,u,e) ~ Hr(t,e) , r > o , (II) (Leray):
::
for any d-simple covering u. Proof. Hr(X,I) is a direct limit of groups ~(X,U,~); U any covering of X, where the class of all coverings is directed by 11 is a refinement of." By .Theorem 24a, the d-simple coverings are cofinal, hence it suffices to consider only d-simple coverings in the direct limit. By (iii) of Theorem 25a Hr(X U e) ~ d-closed r-forms for ~ny d-simple ' ' ' - d-exact r-forms U; hence I and II. We now complete Theorem 24a. Note that Theorem 25a does not require the existence of arbitrarily fine d-simple coverings. Now let U be any covering of X. Let V be a locally finite refinement of U such that each v e V lies entirely in a coordinate patch and the intersection of any finite number of v•s is contractible to a point. (Such a covering can be constructed using l4hitney•s imbedding theorem.) Since every finite intersection of sets of V is diffeomorphic to an open set in mn, the de Rham theorem applies. If the r th cohomology group of such an open set in IRn with complex coefficients is trivial, then every clo.sed r-form is exact: that this cohomology is trivial is a lmown result. §2. Dolbeault's Theorem This section is Section 1 applied to complex manifolds and the operator ~. A. Definition 44b. Given a complex manifold X and covering U, we say that U is simple with respect to (p,q) differential forms, q > 1, or • a-simple if it is open,
-
94
-
locally finite, and, for the intersection of any finite number , of sets of the covering, the Poincare lemma for o holds. Theorem 24b. Let X be a complex manifold. Then there exist arbitrarily fine ~-simple coverings. Proof. As before, assume X c. ~n, open. Once again / use Theorem 23, which establishes the Poincare lemma for o, for coverings by polydiacs {lzj- ajl < RjJ • The completion of this theorem for a manifold is remarked on at the end of this section. Lemma lb. Hr(x,u,r.P) = o, r > 0; where r.P now denotes the sheaf of germs or forms of type (O,p), and U is locally finite. Proof. We may again use a partition of unity argument as in Lemma la. Theorem 25b. Let X be a complex manifold, U a ~-simple covering. Then, if f.P denotes the sheaf of germs of ~-closed forms of type (0 1 p), there exist canonical isomorphisms between the following groups: i) Hl(x,u,r.P) ~ a~closed forms or type (O,p+l) I - ~-exact forms of type (O,p+l) ii) Hr+l(x,u,oP) ~ Hr(x,u,oP+1 )
-
~
-
-
~-closed
forms of type (O,r) ~-exact forms of type (O,r)
where r > 0, p ~ 0. Proof. 1) and ii) proceed precisely as in Theorem 25a. iii) is implied by i} and ii), also as before, when one notes that a a-closed form of type (0,0) is a holomorphic fUnction, and conversely; i.e. r.P = (!). Lemma 2b. Let X be a complex manifold; U, V a-simple coverings where V is a refinement of u. Then the following diagrams commute, where f.P denotes the sheaf or germs of a-closed forms of type (O,p), and r > 0, p! 0:
-
- -·----
95 i)
-
ii)
~-closed
forms of type (O,p+l) ~-exact forms of type (O,p+l)
iii)
~ ~-closed forms of type (O,r) ~ o-exact forms of type (O,r)
-
Theorem 26b. Let X be a complex manifold for which there exist arbitrarily fine a-simple coverings. Then -there exist canonical isomorphisms between I. (Dolbeault) Hr(X,(0) "' ~-closed forms of type (O,r) , · - ~-exact forms of type (O,r) r > 0 •
II. (Leray) Hr{x,u,(Q) ~ Hr{X, 0) , r > 0 • Corollary. If D (_ (In, D a polydisc, then W(D, U) = 0 for all r > 0. We shall eventually use this corollary to establish the result for any region of holomorphy. Remark. The general Dolbeault theorem reads as follows: ~r(X,sheaf of germs of holomorphic forms of degree s) ~ ~-closed forms of type (s,r) • ~-exact forms of type (s,r) However, we shall require only the restricted result of 26b, I. B. In order to complete Theorem 24b, we require the result that in domains of holomorphy all the above cohomology groups are trivial (proof in Chap.ll). Assuming this result, we have the Poincare lemma with respect to ~ for holomorphy domains, so that any locally finite covering by domains of holomorphy is o-simple. Hence it~suffices to establish that arbitrarily fine coverings by domains of holomorphy exist. /
-
96
§3. Complex de Rham theorem Once again, we attempt to repeat 1 for complex manifolds X, holomorphic forms, and the operator d. Definition 44c. Given a complex manifold X ·and covering u, we say that U is a-simple or simple with respect to holomorphic forms, if it is open, locally finite, and, for the intersection of any finite number of sets of the covering, / the Poincare lemma for d holds. (Recall that o = d on holomorphic forms.) Theorem 24c. Let X be a complex manifold. Then there exist arbitrarily fine d-simple coverings. Proof. For XC ~n, open, the proof is again immediate and proceeds as before. At this point, hO\Ilever, we find that no Lemma lc exists. Hence, we must modify Theorem 25c as follm'ls: Theorem 2~c. Let X be a complex manifold, U a a-simple covering, for which Hr(x,u,rP) = o, r > 0, where r.P denotes the sheaf of germs of holomorphic p-forms. Let oP denote the shear of germs of closed holomorphic p-forms. TPen the following groups are canonically isomorphic: closed holomorphic (p+l)-forms ) Fl(X u Cf) i ~ exact holomorPEic (p+l)-forms 1
'
'
ii)Rr+lrx,u,OP)~ ~(x,u,r.P+1 ) Hr(X u r,) closed holomorphic r-forms , ' ' ~ exact· holomorphic r-forms where p ~ 0, r > 0. . _ Proof. As before, under the remark that 0° = e. Lemma 2c. We state here merely that the analogous commutativity lemma is valid, assuming the missing Lemma lc for all manifolds and coverings used. Theorem 26c. (Complex de Rham). Let X be a complex manifold for which there exist arbitrarily fine o-simple coverings and such that Hr(x,u,rru) = 0, 'r > o, for all d-simple coverings U. Then, there exists a canonical iii)
97 isomorohism between: Hr[X ~) ~ ~osed holomoQEhic r-forms r>O · · ' - exact holom_f)rp 1c r-forms ' • (We shall not need the complex Leray theorem.) Let us assume that we have already proven that in a domain of holomorphy the cohomolozy with holomorphic coefficients is trivial (not closed forms). Then the hypothesized Lemma lc holds, so that the theorems of this section hold for domains of holomorphy. closed holomorphic r-forms We remark that the group eXac£ holomorphic r-forms is clearly trivial for r > n, where n is the dimension of the manifold. Hence Theorem 27. Let X be a complex manifold of dimension n, such that ~~,u,nP) = o, r > 0 for all o-simple Qoverings u. Then ~(X,C) = O, r > n. This theorem gives a topologically necessary condition for a differentiable manifold X of real dimension 2n to be a complex manifold.
98 Chapter 9· §1.
The Multiplicative Cousin Problem
The Multiplicative Problem, formulated
A. This second Cousin problem is a generalization-of the Weierstrass problem in one complex variable: . Given a domain D ~ 0, a discrete set or points, av, bv and positive integers nv, mv, find a function r, meromorphic in D, with zeroes at av of order nv, and poles at bv or order mv. We.now formulate the multiplicative problem, referred to as C.II in the sequel: Multiplicative Cousin Problem. Let X be a complex manifold U =lui$' 1 e I, an open covering, and let functions Fi be defined and meromorphic in ui' such that Fi/F j is holomorphic in ui /) u 3_. Does there exist a function F; o, defined and meromorphic in X, such that F/Fi is holomorphic in u1? ~· C.II is precisely C.I, written multiplicatively. As in the Weierstrass problem, where it is sufficient to find a fUnction with given zeroes of given order, we shall find we need only consider holomorphic functions Fi. We shall also show that c.II is not always solvable. As before, we shall formulate C.II using sheaves and cohomology groups. B. Definition 46. Let X be a complex manifold, and let ;?t.denote the sheaf, over X, or germs or meromorphic functions unde~ ~1 tiplica tion, where fll = ~EX ~~ and ~ consists or the germs or meromorphic functions at x, and is a multiplicative group. We topologize as \'le did (!) I by defining a subbasis for the topology utilizing the topology or X, as follows: Let m E ·~n i then m E and g E m is defined in a X ~ neighborhood N of x. For each y e N, let ~ g ~ e m. Y be the equivalent class or meromorphic functions in //1. Y .... containing g. Then the sets Ng = {.[ g ~ e MY I y e N for
m
m
J
99 each choice or g e m, and ror each m e ,J?t. , form the subbasis of lfl • Let denote the subsheaf of invertible holomorph1c elements of )ll, ; such that f c. }11 and 1-. C. /rl ror each x E X. Form the quotient groups }ll an~ set ''7'1 -%. X X ~~~ = xeX /'~~x with the quotient topology. The sheaf of germs of divisors of X is the quotient shear A/.:/- . Note that elements of 71!. / j are equivalence classes of germs of meromorphic functions, where germs represented by two functions F1 , F2 are equivalent (at x} if F1/F2 is a local unit, i.e. if F1/F2 is holomorphic and nonvanishing in a neighborhood of x. We note also that a set of Cousin data associated with the covering U may be regarded as a section of l!L;..jover X. Definition 47. A divisor on X is a section over X of AI -f. 1 i.e. is an equivalence class of sets of Cousin data, in the sense that two sets of Cousin data are equivalent if their "mesh" is a set of Cousin data. A divisor on X is integral (positive) if all germs are germs of holomorphic functions. Definition 48. A divisor a on X is principal if there ex1.sts a meromorphic function F defined in X such thatthe divisor it defines (F)= a. Hence, we may state C.II in the following equivalent way: given a divtsor on X, is it princ:ipal? Lemr;a 1. If every integral divisor of the complex manifold X is principal, then every C.II is solvable. Proof. It is enough to show that every divisor is a quotient of integral divisors. Let p E X, and let FP be a meromorphic function defined in a neighborhood NP of p such that (FP) is the restriction of a to NP. FP = ~p/tp' where $P and tp are holomorphic functions defined in Np
j-- ·
-
u
;3':,,
lCC
and coprime at p, and hence in a neighborhood of p, say Np. Let q c NP, and Fq = ~ql'q the corresponding function at q, defined in Nq. He shall be done if ~p "' ~q and ?/lp "' ?/1q' where defined, \'/here ''"'" means equivale~ce modulo 2J-. Not'l Fp"' Fq, hence ~p/'P"' ~q/'q' so ~p?/lq"' ?f!p~q' at q. But ~p?/lq and ~p~q are holomorphic, and are equivalent in a neighborhood of q. We may choose this neighborhood so that ~q' ?/1 are coprime. But ~p divides ?/lp~q; hence ~P divides ~q• Similarly, ~q divides ~p' as ~p' ?/IP were coprime in Np. Therefore ~P/~q is a unit, i.e. ~p "' ~q' and similarly 1frq• Thus (~p I p E X) and (?/IP I p E X) are integral divisors. C. Theorem 28. Let X be a complex manifold such that C.II is allr:ays solvable. Then so is the strong Poincare problem, i.e. every globally defined meromorphic function is the ratio of two holomorphic functions, coprime at every point of X. Proof. Let F be the global meromorphic ~~ction. Then (F) = a/~ , where a and ~ are principal integral divisors; for as was shown in the proof of Lemma 1, on any complex manifold X every divisor is a quotient of integral divisors, and since C.II is solvable every integral divisor is principal; hence F/f/g = rn1/rn 2 is a local unit. Therefore F/f/g is holomorphic and equivalent to 1 at each point of X; so F/f/g is a global unit, say G. Thus F = fG/g, where fG and g are coprime at each point of X. Theorem 29. Let X be a complex manifold such that C.II is always solvable, and Y a regularly imbedded analytic hypersurface~ Then Y may be globally presented. Proof. Exercise for the reader.
'P "'
§2.
The Multiplicative Cousin Problem is not always solvable The following example is due to Oka. Let XC ~ 2 be defined as follO\'lS:
101 X= [Cz 1,z 2 ) I 3/4 < lzjl < 5/4, j = 1,2}. Note that X, as a product domain, is a domain of holomorphy. This shows, incidentally, that ''C.I implies C.II" is false. Set ! = f z1-z 2-1 = I') X, and note that Y is a closed subset of X consisting of two distinct components, for (z1 ,z 2) = (x1 +iy1 , x2 +iy2 ) e Y implies 0 < x1 < 1, where x1 - 1 = x2 • Hence y1 = y2 -# 0; but Y is nonempty and (z 1,z 2 ) e Y implies 1 2 ) e Y. Hence, we see also that the components of Y lie in (Im z1 > O, Im z 2 > 0) and (Im z1 < 0, Im z 2 < 0). Now define the divisor 1 of X as follows: . r(z 1-z 2-l), for (Im z1 > 0, Im z 2 > 0)
OJ
(z ,z
1
-
t
1
outside the upper component of Y
This clearly defines a set of Cousin data, for which, we claim, the Cousin problem has no solution. For, assume there exists a solution F(z 1 ,z 2 )~ and consider its restriction to
[1z1 1 =
1,
lz 2 1 =
If
g(a,p)
=
F( eia, eit3) •
Now g(a,p) is a continuous, periodic fUnction or both a and t3 • Furthermore, g has precisely one zero, for the upper component of y n(lzll=l,lz21=1J ={<ei1T/3,ei 21T/3)J, and g "' 1 elsewhere. Nou consider the edge curve l in the a,B plane as indicated in the figure, and the edge curve r-2 about (v/3,21T/3) within rl and oriented B similarly. 21T · - ( ---: ( 21T 1 21T) Since g is periodic in a and t3 it is clear that arg g(a,p) can be defined as a single-valued function along 1 • Furthermore, by connecting the tHo curves as a in the second figure, we obtain the following:
r
i
r-
102
-~~· _J
rv
J
d log g
=
r2
rl
III
J d log g ;
arg g(a,~) is also single-valued along r- 2 • Now F(z1 ,z 2 ) = h(z 1,z 2 )•(z1-z 2-l), where h(z 1,z 2 ) is a unit. In the region enclosed by 2 , we may define a single-valued branch of the log; so i.e.
r
J d log g = J d log [ ... ei~ + eia r2
\Je
-1] •
r2
may calculate this latter integral explicitly.
S'f3
=
f3'
j'
r2
+-¥
la a +); d log (-eif3 + e = J d log =
obtain
Set
1
1 a -1)
r'2
r2 I
where now encircles the origin in the (a• ,f31 )plane, and -eif3'+2vi/3 + eia•+vi/3_1 has a zero at (0,0). Set -eifj '+2'1fi/3 + eia r +'ITi/3 - 1 = u + iv • Now
= - sin (f3•-a•+w/3) Hence, if ,-.2 was chosen 1 small enough, the region enclosed by 2 • So
J
r-
d log (u+iv)
r;
=
J
I o(u,v)
d(at,pt)
d log (a'+
r;
and this contradition establishes our claim.
•
'< 0
in
i~t) f. 0 ,
103 ~3.
The solution of the Multiplicative Cousin Problem for polydiscs
Theorem 30 (Cousin). Let DC 0 be given, and let r be any fUnction holomorphic in x3• Choose l e 1~ with L ei = e. Let tKi) i_?: j be a sequence of compact sets K1 cc Ki+l' Ki C Xi+l' U Ki = X, and K1 is the given set Kj for i = j. By hypothesis, there is a function gl holomorphic in xj+l with lf-gll<el compact on Kj. Similarly, for Kj+l C: Xj+l' since Xj+l has the Runge property with respect to x3+2 ' there is a g 2 holomorphic 1n x3+2 with lg 2-g1 1 < e2 on Kj+l' etc. Since 2: ei < oo, lim gk = g exists uniformly on k~oo . compact subsets K or X, because every KcK1 for some l. g is therefore holomorphic in X and satisfies Ir-g I ~ I r-g.t I + Ig .e -gt+ 1 1 + • • • < e.e +a l +l + • • • ~ e on Kj • (2) By Dolbeault 1s theorem, Wl(X, 0 if and only if ~ closed (O,J) fo~~ = 0 tor q > 01 i e. ~ exact (o,q ror.ms ' • if and only if for a differential form a in X or type (O,q) with ~a = 0 there is a f3 or type (O,q-1) such that a = af3 •
-
112
(a) Let q = 1, and let a, defined in X, be a (O,l) form with oa = O. By Dolbeault 1 s. theorem for Xj, there are differentiable functions_ ~1'~2···· defined in xl,x2, ••• reapectively,~uch that o~j =a in x3• Consider ~he sum: ~1 + (~2-~1) + (~3-~2) + ••• • In Xl and X2' o(~2-~3) = 0 and hence ~ 2 -~ 3 b~ing holomorphic can be approximated on · any compact s~bset o.t ~ 2 by a holomorphic function P23 on X, Set ~ 2 = ~2 , ~ 3 = ~ 3 - P23 , etc., using the Runge trick as before (cr. proof of Theorem 23, p. 88). (b) Let q > 1, and let a , defined in X, be of type (O,q)_ with ~a= o. Again, there are ~j in x3 such t~t opj =a in x3, where ~j ~~s type (O,q~l). ~ince o(~ 3 ~ 2 ) = 0 in x2, (~ 3 -~2 ) = ay 2 in x2 • Let ~3 = f33 2' etc. (of. p. 88).
-.
or
§2. Polynomial Polyhedra Definition 51. Let p1, ••• ,pr be polynomials in I IPj(z)l < 1 for j = l, ••• ,ri. 1 , ... ,zn. Let A= A is an open set. If Acc en, then A is called a polynomial polyhedra~ (of dimension n). Note that a polynomial polyhedron is a region of ho1omorphy. Theorem 32. (Oka-Weil). Let X be a polynomial poly~ hadron of dimension n, then (1) Hq(x,c?> = o for all q > o (2) If f(z) is holomorphic -·in X then f = ~ qj where the qj are polynomials in z1 , ••• ,zn and the pk used to define X, and the sum converges normally in X. Proof. (1) X is a bounded set and therefore lies in a polydisc; assume X lies in the unit po1ydisc, i.e. XC: (lzjl < 1, j = l, ••• ,n). Consider cn+r where r is the number of polynomials defining X. In cn+r consider L: = (z 1, ••• ,zn, t 1, ••• ,Cr) I lz 3 1.! 1, J=l,, •• ,n, ci = Pi
113 polydisc. Consider the analytic hypersurraces 0 = fi(z 1 , ••• 1 zn~C 11 ••• ,Cr) = Ci- pi(z1 , ••• 1zn). The fi are defined everywhere and are clearly holomorphic fUnctions in cn+r. With (lzjl < 1 1 ICil < 1) as the complex manifold X in lemma a and noting that the Jacobian or the fi has maximal rank everywhere because ari/~Ck = 5ik~= the hypothesis of lemma a is satisfied and since Z = ~ 0 , Hq( 2: 0 ~ D> = 0 1 tor all q > 0 and every holomorphic function on ~ 0 is a restriction of a holomorphic fUnction in the open polydisc (lzjl < 1, ICil < 1). But a holomorphic function in the open polydisc can be written as a power series. Hence every holomorphic function on ~0 is the restriction of a series, 1 ~ J1 Jn 11 r ~ aj ••• j i ••• 1 zl ••• zn cl ···Cr I converging normally inn 1~ 0 : We claim that ~ 0 is holomorphically equivalent to X. For, define the mapping (z 1 , .•• ,zn) -> (z 1 , ••• ,zn, p1 (z), ••• ,pn(z)). It is of rank n and one-to-one. The preimage of Lo is X. Hence Hq(X, f9) = Hq( ~ 0 , g) = 0 for all q > 0. (2) We have already obtained that every holomorphic function on ~ 0 is a restriction of a series jl i 2: aj 1 ••• ir z1 ••• err converging normally in ~ 0 • But~ ~ 0 , Ck = pk(z) and thus the above series is a series in only the zj, converging normally in x. §3.
Runge domains
Definition 52. Let K C.c r,n. The polynomial hull of K, = 0 I for every polynomial p with lp(z)l ~-1 on K, IP cz0 >I ~ l} . Definition 53. Xopenc: en is polynomially convex if K c ~X implies that K* c::: c. X. Definition 54. A Runge region is a region of holomorphy in which every holomorphic function can be expanded in a
* K
,-tz
114
normally convergent series of polynomials. Theorem 33. Let Xopen C: en. The following statements are equivalent: (1) X is polynom1ally convex. (2) X = L)Xj, where the Xj are polynomial polyhedra, xj c c xj+l. (3) X is a Runge region. Proof. (1) implies (2). SL~ce X is polynomially convex, X is holomorphically convex, By the Cartan-Thullen theorem, X is a region of holomorphy. Continue the proof by adapting the proof for analytic polyhedra, Theorem 7 and its corollary, (p. 25).
(2) implies (3). By the Oka-Weil Theorem, Theorem 32, each polynomial polyhedron is Runge in the next one, and Hq(Xj,£9) = 0 for all q < o. Apply lemma~. Hq(X,(?) = 0 for q > 0. Hence by Theorem 21 (p. 75), X is a region of holomorphy. Let f be holomorphic in X and Kcompac\: x. Then Kc Xj, for some j, and by the Oka-Weil Theorem, f can be approximated as closely as desired by a polynomial in K. Hence X is a Runge region. (3) implies (1). Since X is a Runge region it is a region of holomorphy and therefore is holamorphically convex, A A i.e. if Kccx, then KccX where K is the hull of K with respect to holomorphic functions on X. We claim that "K = K* and hence K* ~c X. Indeed, K "' c: K* since the family of all holomorphic functions on X is larger than the A family of polynomials. It r~ains to show that K* c: K. Let z0 e K* and let f be a holomorphic function in X such that jf(z)l ~ 1 for z e K. For every e > O, there is a polynomial p(z) satisfying jp(z}-f(z)l ~ £ on K V f z0 ~ , since X is a Runge region. But then Ip (z ) I ~ 1 + e on K, and since z0 e K* , Ip ( z0 ) I ~ 1 + e • Hence lf(z 0 ) I~ 1 + 2e, "and because e is arbitrary, If(z 0 ) I ~ 1, i.e. z0 e K.
115 Chapter 11. §1.
Cohomology of Domains of Ho1omorphy Fundamental Lemma, stated
( 1) Let Kcompact c G:n. Let x* denote the polyn·omial hull or K; x* = {z I IP(z)l <max IP(C)I for every polynomial 7 * K PJ. Note_that K is bounded and compact and that K*
=llf D
K c Dj
I D is a polynomial polyhedron,
(Proof easy) • ( 2) Recall that an analytic polyhedron De ~n was . defined as follows: there exists an open set G c mn and functions rl' ••• I r holomorphio in G such that D c. c. G v ' and D-c~z I z e G, lrj(z)l < 1, j = l, ••• ,v.f. ~ince 7 Dec G, hence bounded, He shall assume that G c. II z II < lJ, the unit polydisc. (3) Let D be an analytic polyhedron. Then ~' the Oka image of the closure of D, is defined as follows
i
L- = [Cz,C) I lz 1 1, ICJI i
L
~
1, z
E
= l, ••• ,n;
G, CJ=fJ(z 1 , ••• ,zn); j
l = 1, ••• 1 VJ•
c a:n+v, and is closed. We now stcte the fundamental lemma. Fundamental Len11a ( Oka) • Let ~ be as above.
---·-
--.;;.
>=~·· L.--
'--
Then
"'---
•
We shall prove the lemma in this chapter; the proof of a more general form of the lemma will come later (p. 196).
§2. !l2Plications of the Fundamental Lemma Let D ,_._ c Gopen (: en be an analytic polyhedron. ~ Observe that the Oka mapping of c1 D -> 2._ c: 111~ n+v 1
A.
(zl, ••• lzn)
->
(zl, ••• ,znlcl~···,Cv>
I
given by: zi = zi, cj = fj(zl, ••• ,zn); is defined on all of G: call the image of G under this mapping 11 2:: " . Note 1
116
c.z:=
that 2:: is closed and 1 • Hence there exists an e > 0 such that if dist [(z1 , ••• ,zn,C 1 , •.• ,cv>' L J < e:, then (z1 , ••• ,zn) e: G; where the distance of a point p e: ¢m from a set S c. a:m is defined as infse:siiP-sll = infse:s·lmax IP1-s 1 1} ir=l, •••,m Let > e: denote th~ e:-neighborhood of)! 1 defined by E = { w I dis t ( w, 0, R > 1 such that D(O,l-e) ~ D(O,l) D(O,R) CD. Then + ~ A in D(O,R).
+
+ =-
=
7
Define u(z)
=
log lzl -log A
log R -log
11-e(
11-e:
•
Then u is harmonic in D(O,R) - D(O,l-e), + < u and u = 0 for lzl = 1- e. Hence, +Cl) < A log ~/log~ ; ~-E ~-E letting e ~ O, we obtain +Cl) = 0. Proposition 2. If + ~ 0 and log + is subharmonic, then + is subharmonic. Proof. (i) and (ii) follow from the monotonicity of "log", and (iii) from the inequality:
Proposition 3. Let +Cz) ~ 0 be defined and upper is not sub harmonic semicontinuous in Dc. a such that log at z0 e D, and +(z) dz
•
lzl=r Using upper semicontinuity, let +n be continuously differentiable in D, +n .J. cf>. Let hn(z) be the harmo~ic function for which log +n(z) = hn(z) on lzl=r. Let hn be the conj~gate harmonic function, and set -h -ih 7/1
n
=e n
n.
Now on tlzl=rJ, log ~17/lnl = (log 4>> - hn =log+ • log +n < o, i.e. +11fln1 < 1. If 4>(0) 11/Jn(o) I = K;: 1, then 4> 11/lnJ ~ ~11/lnJ, so 11/ln(z)l is the required function. If +(0) + ln lfn{O)I < o, i.e. ln +Co) < hn(O) = 2~ hn(z) dz = 2.~} ln 4>n(z) dz, Iz I=r Iz I=r therefore ln 4>(0) ~ n!~ l/21rr. ln cf>n(z) dz, so lzl=r ln ~(0) ~ 2~ ln +(z) dz, contradicting choice of disc. lzl=r
i
i
J
f
J
§4.
Proof of the fundamental lemma
Statement. Let Gopencc ,n and assum~ (for the sake of s1mplici1ty) that G cc(lzj(1 ~ 1 for 1 = 1, ... ,v}: l5 is compact. Let L: =fz then R1 (z) = o. If z E S and zj -> z, then for large j, either zj is and then limR 1 (zj) = 0 ~ R1 (z), or zj E S zJ•>z and then R1 (zj) =max lc 1-r1 1 = lci-f1 (zJ,zJ)I for some (zJ,zJ,Cj) e 2:*· By the compactness of L*, there is a subsequence [(zJ,zJ,Cj)7 converging to a point (z,Z,C)e 2:* j A j 1\ and z !zR1 ( z ) = IC1- f 1 ( z, Z) I ~ max IC1- f 1 ( z, Z) i = R1 (z) •
J
Hence R1 is upper semicontinuous. Now, assume that log R1 is not subharmonic at some z0 Eo, then there is a closed disc in n, [lz-z0 1 ~ p}, and a function t(z) holomorphic in the open disc and
122
continuous on its closure, such that R1{z) lt(z)l < 1 on lz-z0 1 = p, ,and therefore < 1 - 2E for some 1/4 > E > o, and R1 (z0 ) lt(z0 )1 = 1. Define F(z,Z,~,t) = (~ 1 -f1 (z,Z))t(z) - (l+t)eia for all ~~ (z,Z) E G, lz-z0 f < p, ~nd t E T = [t E e I lt-t0 1 < E and t 0 E [O,al] , wher~ a and a are fixed real numbers to be determined later. We claim that this F satisfies the hypothesis of the proposition with K = L c en+v , K* = 2:* , T = T and G in the propos! tion, call it GP, = G f\(lz-z 0 1 < p) and all~. Indeed: a) F :l.s holomorphic in Gp X T and continuous in cl GP J( cl T. b) If r- = ~z I z E bdry {G A(lz-z0 l" = ~. Proof: Bdry (G ll (I z-z0 I
L ~ ~.
Proof: Since E e o0 C s, either ~ e int S or ~ £ bdry s. a) If ~ e 1nt s, then there is a sequence of points ~ j -> ~ , ~ j e iS"'\). Since ~ j e S, there are points (~3,z.1,cJ> e L, and ·by 3), c~3~z3,c3> e L:• By the compactness of ,.-1 there is a convergent subsequence j j j jj .. j (~ , Z 1 C ) , ~ -> ~ 1 Z -> Z, C -> C and (~ 1 Z, C) E L • b) If ~ e bdry S and L 0 h 1\ such that if ~~-~I < E then L: 0, a number b, 0 < b < 1, and N polynomials PJ(Z,C) such that a) if lz-~1 < e and for all j, IPj(Z,C)I < 1, then (z,Z) e G and b) if lz-~ I < e and (z,Z,~) e L: then for all j, IP j(Z,C) I < b. Proof. a) If (~ 1 Z,C) e 2: then (~ 1 Z) e G, and there is a point (~ 1 Z,C) e ~ by 4). Since G is open, there is a neighborhood of (~,Z,C) such that every point in this neighborhood has its first n-coordinates in G; i.e. there is a 6 > 0 such that if the distance from (z,Z,C) to ~(~} is less than 6, then (z,Z) e G. But since
125 = Z: ~, and Zk, ~lc such that (~k,zk,Ck) E ~ and max IPj(~k,zk,~k)l > 1 for each k. Since ;;- i~ompact ~here is a subsequ~nce (~k,zk,Ck) ."'\ /\ which converges to a point (~ ,z, C) E > • renee in the limit we obtain a point E 2: IP 1 1 ltl so that F(z,Z,C,a) # 0 * "' for "' 0, independent of q, such that - Mllz-q 11 3 < R < Ml~-q113 • Hence (z-q)*H (z-q) + R > (a-MIIz-qlllllz-qll2: - E = a/M; then for II z-qll < E, (z-q) - *H(z-q)-1-R > 0 Take j-
+
+
except at q where it vanishes. But in D" B ~ < 0; therefore Q f 0. Lemma 3. Let Dope~c. _,n be a strictly pseudoconvex region. There is an E > 0 such that if q E bd~y D and B is the polydisc of radius~ E about q, then B tj D is : a region or holomorphy. Proof: We will show that every boundary point is essential. Take E = 1/2 E2 , the E of lemma 2. Bdry (B flD) = ( (bdry B)() D) V(B /)bdry D) U(bdry Bnbdry D). If y E bdry B, then since B is a domain of holomorphy, there is a function holomorphic in B and singular at y. If y E (bdry D)A B, then the polynomial Q, of lemma 2, corresonding to y, is holomorphic, Q(y) = 0, and Q(z) F 0 in D A B. Hence 1/Q is holomorphic in DAB and singular at y. B. Assuming proposition 1, we will show that every boundary point or a strictly pseudoconvex D is essential. Let q E bdry D. Let B be the polydisc of lemma 2 and Q(z) the quadratic polynomial. Let ru > 0 be a c00 function whose support lies in the interior of B and ru(q) > o. Consider ~ - tn, where t > 0 is small. By lemma 1, if t is small enough, n1 = [~ - tm < 0] is strictly pseudoconvex. DC Dl cc. F be continuous linear maps from E into F. If A is onto and B is compact then the range of A + B has finite codimension, i.e. dim F(A+B)E 0 here. Hence DCcDN. Also,
l=l
141 D1 satisfies 2' with respect to D and D2 satisfies 2 1 with respect to D2 ~ 1 , 2 = 2, .•. ,~. Hence every closed (0,1) form in D is cohomologous in D to a closed (0,1) form in D1 which in turn is cohomologous in n1 (and hence D) to a closed (0,1) form in D2, etc. up to DN. Take for D0 , in 2, the set DN.
142
Chapter 14.
§1.
Sheaves
Exact sequences
In the following, all groups are Abelian and ·all maps are homomorphisms. : A. Definition 57. A sequence is a collection of groups AJ and maps ~J : A;--> AJ+l' written [Aj,~J) or: •••
.
>
j-1
Aj-l
>
~j
Aj
Aj+l
>
> • •• •
The sequence is said to be exact at Aj if 1m ~j"l=ker ~j' where 1m ~j-l = [ala E Aj , there exists b E Aj-l such that ~j-lb = aJ ker ~ j
= [ al a
E
Aj
1
~j a =
0} .
The sequence { Aj~~;I is called Aj' for every j. Definition 58. A collection of to form a commutative diagram if all leading from a group A to a group the same result: e.g. the diagram A
<j>
exact if it is exact at maps and groups is said compositions of maps B in the collection give
> B
~w c commutes if t~(a) = Q(a) for every a E A. Remarks. 1) Clearly, 0 -> A -> 0 is exact if and only if A = 0. J. 2) 0--> ~> B is exact if and only if ~ is one-to-one. In this case, we may regard A as a subgroup of B, for A1 = ~(A)C: B, and ~ : A -> A1 is an isomorphism. Hence the diagram ~ 0->A->B
t~
tid
0 ->Ali> B
143 is commutative (1 will always denote the inclusion map and id the identity map). 3) A !_> B -> 0 is exact if and only if 4> . is onto. Here,.the homomorphism theorem of group theory implies B"' A/ker denote!3 "is isomorphic to"). Hence, we may "factor" ~ as follows:
+ ("'
-
A _j_> B j " ' -/
A/ker
++1
This diagram commutes, where here j (as always) denotes the canonical projection and is onto; and 1 , the map induced is an isomorphism. by 4) Combining 2) and 3), 0 -> A i.> B L> C -> 0 is called a short exact sequence if and only if ' is 1-1, t is onto and 1m = ker ~. Remark. Utilizing the above remarks, the following · diagram commutes and both horizontal sequences are exact: ~
+
+,
+
+
0-> A-> B->
tid
~t
Note that ' : A -> A1 , id : B -> B and ~l-1 : C -> B/A1 are all isomorphisms. Isomorphic groups may be identified; hence short exact sequences should be thought of as being in the rorrri: 0
->
A
..!....>
B
..J.....>
B/A - > 0 •
Proposition 1. Let I=[ a,l3,-y, •••j be directed by " > "; and let there be given sequences {Aj,+j~ exact for each a, such that, for a < 13 the following diagram exists and is commutative:
B.
,
144
:
where the a~ a
< f3 < 1
satisfy the following compatibility condition: implies
a
r
=
aj1 ar
•
Then the limit sequence lAj = 11m Aj , ~j = 11m ~~ exact and the following diagram commutes:
a,!J> Aaj+l ->
• • • -> Aj
t1 ti
J
is
•••
• • • -> A • ~> A -> • • • J 'flj j+l
•
Remark. The Aj are defined as follows: Set Sj =lJ Aaj; define an equivalence relation """" on aer a a2 Sj as follows: s 1 E Aj 1 , s 2 E Aj are equivalent, s 1 ~ s 2 , if there exists a f3 such that ~ < ~~ a 2 < f3 and alf3 a2f3 aj (s~= aj (s 2 ). Then Aj = Sj/~, and the group structure is canonical. Alternatively, define a thread to be a set of elements tga) such that for every a E I, g 0 E Aj; and for every pair a,p E I such that a A satisfying d2 = 0 is called a
145 differential operator. The sequence A ~> A~> A is not necessarily exact, but ker d ~1m d as d2 = 0. Hence \'1e may define: H(A)
3
ker d/im d,
the der}ved group of A. H(A) is a measure of the deviation from exactness or the above sequence, in the sense that H(A) = 0 if and only if the sequence is exact. We say x E A is closed if dx = 0 ; x E A is exact if there exists a y E A such that X = dy. Denote the homology class in H(A) of an element x E A by [x]. Definition 61. If A,B are groups with differential opera tors, l'le say that f : A -> B is an allot'lable homomorphism if fd 1 = d2 f; i.e. if the following diagram commutes: dl A
B
> A
d
2
> B
Examples. The group of cochains on a space, with boundary operator; continuous maps are allowable. The additive group of differential forms, d the differential; differential maps are allowable. Chains on a simplex, boundary operator; simplicial maps are allowable. Proposition 2. An allowable map f : A --> B induces a homomorphism f*
such that if g
H(A) - > H(B) ,
B -> C allowable, then: (gf) * = g* f *
and
146
Proof. Define f * [x] = [ fx] mapping of H(A) -> H(B} I for implies dx = 0 and X = dy implies
1
X
r* is
Then
e A.
d(fx)
=
rx =
f(dy)
f(dx)
=
a
= od(fy) •
B. Proposition 3. Let A,B,C be groups with differential operators, and let the following be a short exact sequence of allowable maps: O->A!_>B£_>C->O. Then there exists a canonical homomorphism D such that the follO\dng diagram is exact (viewed as an infinite, repeated sequence) H{A) *
D/ 'J
H(C) ~ HlB) g*
Proof. (Recall the \'Tell proof of de Rham 1 s theorem!} Exactness at H(B): {Does not need D) i) Let a e: H(A); \\le must show g*f * (a) = o. But, let x e: a ; g* f * (a) = [gf{x)] = 0 as gf = 0. 1i) Let y e: B, dy = o, such that g* [y] = 0. We wish to find x e: A, dx = 0, such that f * [x] = [y]. Now g* [y] = [gy] = 0, implies gy = dz, z e: c. But g is onto; hence there exists a y1 e: B such that gy1 = z, which implies gy = dz = dgy1 = g(dy1 ). Hence y-dy1 e: ker g, so there exists an x e: A ·such that fx = y-dy1 , and x is closed for dfx = f{dx) = dy = 0, and f is one-to-one. But f * [x] = [y-dy1 ] = [y], as required. Construction of D: Let z e: C, dz = o. Now g is onto, so there exists a y e: B such that z = gy. But 0 = dz = dgy = g( dy), so dy e: ker g implies that there exists an x e: A such that fx = dy, and x is closed as before. Note that x is unique once y has been chosen. Set D[z] = [x]; D is well defined if [x) is independent of y. Bence, let y e: B such that O=gy.
147 Then y -e ker g, so there exists a unique x e A such that y = fx. Therefore dy = f dx, so D[O] = [dx] = O. D is clearly homomorphic. Exactness at H(C): i) Let y e B, dy = o. Then ng*[y] = D[gy] = [x), where rx = dy. But f is one-to-one, so d7 = 0 implies X=
0.
ii) Let z e C, dz = o such that D[z] = 0; i.e. there exists a y e B such that z = gy; dy = f.x and D[z] = [x] = 0. Hence, x = dx1 • Set y1 = y-rx1 • Then y1 is closed, !or dy1 = dy- dfx1 = fx- f(dx1 ) = f(x-dx1 )=0. Furthermore, g [y1 ] = [gy-gfx1 ] = [gy] = [z], as gf = o. Exactness at H(A): i) Let z e C, dz = 0. Then f *D[z] = [fx), where z = gy and dy = fx. But then f *D[z] = [dy] = O. ii) Let x e A, dx = o such that r* [x] = 0; i.e., fx = dy. Set z = gy. Then z is closed, for dz = dgy = gdy = gfx = 0; and D[z] = [x].
§3. Graded groups Definition 62. A group A is called graded if, for every integral J, there exists a subgroup AJ such that X E A implies X= Xjl+ ••• + Xj ; Xji E Aji' k < m and the representation is unique.k Note that this uniqueness implies Aj nAk = O, j ~ k. An element xJ E Aj is called pure (j-)dimensional. Examples. Chains and cochains on a simplicial complex are graded by their dimension. Differential forms are graded by their degrees. In both these cases, Aj = 0 for j < 0. Definition 63. A differential operator d on a graded group A is said to respect the grading if there exists an integer r, called the shift of d, such that d AjC: Aj+r for every j. (In practice, r is almost always i 1.)
148 A map f : A -> B of graded groups with differential operators is called allowable if the differential operators have the same shift and f preserves dimension. Corollary. If A is a graded group with differential operator which respects the grading, then th! derived group H(A) is graded; and [x e A.o dx;::~ j H (A)
t em j-r j
=
Corollary. An allowable map groups induces homomorphisms
·
f : A -> B of graded
f* : Hj(A) -> Hj(B) •
Proposition 4. Let 0--> A_£> B_s> G--> 0 be a short exact sequence of graded groups and allowable homomorphisms. Then there exist maps d such that the following sequence is exact:
* Hj(B) lL> * Hj(C) .2_> Hj+r(A) -> ••. -> Hj(A) .f._> ••• where r is the shift of the differential operators. Note: There are lrl distinct sequences. Proof. Utilizing proposition 3, there exists a D: H(C) -> H(A). As a map of graded groups, D : Hj(C)->Hj+r(A),. for: suppose z e C is pure j-dimensional, dz = o. Then there exists a y such that gy = z, and dim y = j. There exists an x such that dy = fx and dim (dy) = j+r = dim x. But D[z) = [x], so shift D = r =shift d; renameD "d"; then the exactness result of proposition 3 and the above corollaries conclude the proof. Example. Let X be a topological space; A c. X a subspace. Let C(Z) denote the graded group of chains over Z, with standard boundary operator, a. Then 0 -> C(A)
1 ->
C(X}
j ->
.
C(X)/C(A) :: C(X,A}
->
0
is an allO\-mble short exact sequence of graded groups, and proposition 4 implies the exactness of the sequence ••• -> Hj(A}
2>
Hj(X)
~>
Hj(X,A)
~>
Hj-l(A)
- > •••
149
§4.
Sheaves and pre-sheaves
A. Recall the defini~ion of "sheaf''· (Chapter 6, §;, Defn. 36); we rephrase it as follol·Js: Definition 64. A sheaf of Abelian groups, ~~ is defined . as follous: Let X, the base space, be a paracompact Hausdor~f spa~e. For every x E X, let Sx be an associated Abelian group called the stalk of the sheaf over x; and set S = ~EX Sx, whose topology is smallest such that i) the projection map p : S -> X, defined by p(s) = x if s E Sx' is continuous and a local homeomorphism. ii) the group operations in the stalks are continuous; i.e. s -> -s is a continuous map of S into S; and (s 1 ,s 2 )->s 1+s 2 , defined on the set R of pairs (s1 ,s 2 ) such that s 1 ,s 2 belong to the same stalk, is a continuous map of the subset R of S x S into S. vie remark that the stalks are discrete. Let Y c. X; then S (Y) = UxEY Sx, with the induced topology, is called the induced sheaf of S ~ Y. A section over X is a map t : X -> S, continuous, such that p • t = idx. A section over Y c X is a section of S(Y) over Y. Remarks. Every sheaf has at least one section, the zero section, given by t : x -> 0 E Sx. If two sections coincide at a point, they coincide in a neighborhood of this point. Corollary. Let Yopel~ X. Then t(Y) is open in S. B. Definition 65. Let X. be a paracompact Hausdorff space. Let U = { u1 \ be an open covering of X such that u1 , u2 E U implies u1 ~ u2 E U. Let r-(ui) be an Abelian group associated to each ui E U, such that, if ui c. uj there exists a homomorphism '~'ij : r(uj) -> r(ui) satisfying the compatibility condition: ui c uj c uk
implies
~'ji
?'kj
= Yki •
The collection (x,r-(ui),rij) is called a prehseaf. Proposition 5. To each presheaf there may be associated a sheaf, called the sheaf defined by the presheaf.
150
Proof. Take X as the base space. For each x e X the collection tui I ul e U, x e ui} is directed; set Sx equal to the direct limit of the groups r(ui). Set S = SX, with topology defined as follows: s e S implies . s e Sx. Now x e Ux e U; then [sy I sy e SY; y e is an open set and the collection of all such sets is a basis for the topology of S • Note that the 1-(ui) form sections of the sheaf S defined by the presheaf. Proposition 6. Every sheaf is defined by same presheaf. Proof. Take = I uopen in Let r-(u) be the sections over u; and define the rij by restriction. c. A subset T of a sheaf S is itself a sheaf if and only if T is open and Tx = T llsx is a subgroup of sx. Then T is called a subsheaf of s. Example. S is the sheaf of germs of continuous functions and T is that subset of ~ consisting of all the germs of ceo functions.
4ex
ux'f
u lu
xJ .
§5. Exact sequences of sheaves and cohomology Unless otherwise stated, all sheaves have the same fixed base space X. Definition 66. Let s 1 and s 2 be two sheaves. A continuous map of s1 into .§ 2 such that ~(s 1 ,x> C...8 2,x and ~ I 8l,x is a group homomorphism, is called a homomorphism of the sheaf s 1 . into the sheaf ~ 2 • The subset of ~l mapped into the neutral elements of 8 2, Sxt' is called the kernel of ~; denoted ker ~. The ker ~ is an open set, for the set of neutral elements of ~ 2 is the image of the null section of 82 over X and this is open in .§ 2 (cr. corollary of §4, Chap. 14), and since ~ is continuous, the preimage of the set of neutral elements of 82 is open in 81 . The ker ~ f'\8 1 ,x is the kernel of the group homomorphism ' I 8l,x and thus
foe
151 is a subgroup or s1 ,x. Therefore ker' is a subshear or ~ 1 • The image of ,, ~{~l) c~ 2 ; denoted 1m ,, is a subshear of ~ 2 : that is open follows from the· continuit7 of ~~ the commutivity of ' and the projection map, i.e. p2' = p1 , and the fact that p1 and p2 are local homeomorphisms, and, as before, 1m ' f)s 2,x is a subgroup of s2,x, so that 1m ' is a ·subsheaf of ~ 2 • Hence we can form the quotient sheaves s2 11m ~ and ~ 1 1 ker ~' the cokernel of ' and coimage of ,, respectively. Definition 67 The se~uence or sheaves and shear homomorphisms I!j h..> I!j+l :.1±1> .§j+2 is called~ exact when, for each x e X, the sequence sj,x ~~--~j+l,x .l+l> Sj+2,x is exact. Example. Let T be a subsheaf of S, and let 0 denote the null sheaf, i.e. the sheaf whose stalks are the trivial i groups over each point. The sequence 0 -> T -> S -> S/:£ -> .Q is exact by definition. We have already defined the cohomology groups, Hq{X,S), q > 0, of a paracompact space X with coP.fficients in a sheaf S • For convenience, define Hq(X,~) = 0 for q < 0. Note that H0 (X,S) is the group of global sections of the sheaf. Let S and T be two aheaves and let ' be a homomorphism of ~ into T. We claim that for each q, ' induces a homomorphism ~*of Hq(X,S) into Hq(X,T). Consider any open covering of X, U =lui)-. The group of cochains C{X,U,S) is a graded group with differential operator (the coboundary) which respects grading (the shift is +1). Hence the derived group H(C(X,U,S)) is graded and its pure dimensional parts are the cohomology groups of the covering with coefficients in S. Similarly we have C(X,U,T) and H(C(X,U,T)). Now, an element of C(X,U,S) is an assignment of sections of ~~ and ~ maps S continuously into T, thus ~ maps C(X,U,S) into C(X,U,T). ~ is in fact an allowable homomorphism and hence induces a homomorphism of the derived
im'
152
groups H(C(X,U,S)) and H(C(X,U,T)). Taking the direct * . limit, we obtain the desired homomorphism Theorem 41. (Exact cohomology sequence). Let 0 ->·A!> B i> C -> 0 be a short exact sequence of sheaves.
+•
-
-
-
-
-
~> H2(X,!)
--> •••
Assume the theorem for now. (It is proved in §7, p. 158.) Definition 68. A sheaf S is fine if and only if, for any locally finite open co~ering~X, U = fui!' i E I, there exist homomorphisms ~ 1 of S into S such that 1. ~i(Sx) = 0 for xi ui and 2.
fu
~1 =
identity.
(The sum is finite at each point because U is locally finite and ~ satisfies 1.) Example. Let X be a paracompact differentiable manifold, and let § be the sheaf of germs of differential forms of degree p. Let U be a locally finite covering of X and let wi) be a partition of unity subordinate to U. Defin~ ~i to be multipli'cation by wi' Then 11~· are ·homomorphisms of § into § satisfying 1. and 2. above., so that S is a fine sheaf. Theorem 42. If S is a fine sheaf, then Hq(X,§) = 0 for all q > 0. Proof. The proof is the exact analogue of the C00 case: theorem 22, p. 78. Let q > 0 be fixed and let U = fui~ be a locally finite covering of X. Define the homomorphism Q: Cq{X,U,S) -> Cq-l{X,U,S) by Qf(i ••• i 1 ) = ~~if(ii 0 ••• iq 1 ) •
i
-
f
-
0
q-
ne:
,
-
Verify that . S- = Q5 )- + 5Q j~ exactly as before. Hence if ot =0, then f = 5QJ, Thus Hq{X,U,S) = 0 and then the direct limit Hq(X,S) = O.
153 Definition 69. A resol¥tion of~a shea€ ~ is an exact -1=1 0 1 sequence of sheaves .Q -> ..§ -> ~ - > A1 - > • • • such that Hq(X,._!j) = 0 for all j ~ 0 and q > o. A resolution is called a fine resolution when all the !j are fine sheaves.
=
~amples.
Let X be a connected differentiable manifold and let ! = ~ (Sx = ~ and the topology is the discrete one). Let Aj be the sheaf of germs of .differential forms of i d d degree j. The sequence .Q -> tl! -> !a -> _!1 -> • • • is a fine resolution of e. Proof. Note that if X were not connected we would have to take for ~ the sheaf of germs of functions which are constant on each component of X in order that the sequence 0 -> ~ -> !o -> .!h -> • • • be exact at A0 • We have already established that Hq(X,A j) ;: 0 for all j ~ 0 and q > 0 (cr. corollary p. 89) and that the Aj are fine sheaves. The exactness of the sequence at ~ and at A0 is immediate, and exactness at Aj, j > O, follows from the Poincare lemmas. 2. Let X be a complex manifold and let S be C9 and Aj be the sheaf of germs o! differential forms of type (O,j). The sequence .Q -> (!) .!> !o ...2> A1 d> • • • is a fine resolution of () • . Proof. As in example 1., the Aj are fine sheaves, and the exactness of the sequence at Aj, j > O, follows from the Poincare lemmas. Th~orem 43.~ (Abstract de Rham). Let -1=1 o/Q ~1 0 -> .§. > !Q -> ,!1 -> • , • be a resolution of a sheaf-~· Consider the inguced cohomology sequence * 1.
0
0
-> H
(X,S)
~ ~0 -=s: H (X,A0 }
1m ~;_ 1 C ker ~;
1
0
0
-> H
(X,A1 )
~1
-> , • , •
Then
and HP(x,S) ~ ker ~;/1m ~;-l for all p > o. canon. isom.
154
Applying this theorem to the above examples or resolutions or sheaves, we obtain for 1. HP(x ~) closed p-forms P > 0 1 ' ~ exact P•forms 1 i.e. the de Rham Theorem (Theorem 26a, p. 93) 1 and for 2. = HP(x 1 t/) ~ ~-closed (O,p) forms 1 P > 0 , - ~-exact (0 1 p) forms i.e. the Dolbeault Theorem (Theorem 26b 1 p. 95). Proof of Theorem 43. For j = 0 1 1 1 2 1 • • • 1 set BJ = ker = 1m 'J-l since the resolution sequ~nce is exact. For each j 1 the sequence 0 -> ker ~j !_> .!j ~> 1m ~j - > .Q is exact by cons¥ruction; rewrite it as .Q -> ~j .!_> AJ .!J> ~j+l -> 0. By the exactness theorem (Theorem 41) 1 the sequence ~.
'j
Hq(X,~j)
Hq(X,~j+l)
Hq+1 (X 1 Bj) -> Hq+l(X,Aj) is exact for q ~ 0 and j ;: 0. By hypothesis, ~(X,~j) = 0 for q > o and j ! 0. Hence Ifi(X,~j+l) ~ Hq+ (X,BJ) for q > 0 and j > 0. Then HP(X,S) = HP(X,Bn) ~ HP-l(X,B1 ) 2 -1 --v"' HP- (X,B 2 ) ~ ••• ~ H (X,Bp_1 ). Now BP = ker 'P is a sub sheaf of Ap. t1e claim that X0 (X,BP) ~ ker ~;. Indeed, the ker ~~ is the set of those global sections of AP that ~~ maps into the null section of ~+1, but, by the definition of ~;, this set is precisely the set of global sections of ~p· Consider, next, the exact sequence ->
0
->
0
.
0
0 -> H (X,~j) - > H (X;~·j) -> H (X,~j+l)
E...> H 1 (x,~j)
->
H 1 (x,~J)
-> • • •
-
for j = p-1, p
>
i.e. _
J..* onto H1 (X,BP_ ), Since 5 is a homomorphism from ker ~P 1 1 ) "' ker ~P* I ker 5 1m ~*p- 1 = ker 5 C ker ~*p and H (X,B -p- 1 J..* J..* = ker o/p I 1m ~p-l' by exa;tness. * Hence HP(x,s) ~ ker ~P I 1m ~p-l.
o,
155
66.
Applications of the exact cohomology sequence theorem
I.
Let X be a complex manifolq.
Let
~:
sheaf of germs of homomorphic functions
?It:
sheaf of germs of meromorphic functions
We may view be the inclusion. Al
JtL ;
-dJ
as a subsheaf of let i \'le form the exact sequence i
~
j
.,
: & ->h,.
.()
0->~->I!J:->~1~-> 0.
Recall that a section of .t!:J: I Q. over X is an equivalence class of sets of data for a C.I problem. Using the exact cohomology sequence theorem, there exists an exact sequence o ~ i* o lAo j* 0 .),. /tl 1 AI 0->H (X, C1) - > -R (X, rrl) -> H (X,nt/ V) - > H (X,
Hq(X,
!2_)
Hq+l(X,~y) Note that Hq(x,Q
Hq(X, Q/Qy)
-> ->
Hq+l(X, ~)
I 2,y) ~ Hq(Y,
->
->
•••
Q) .
Ue nO\'l claim that: q
->
Theorem II. Hq(X, (9) = 0 = Hq+l(X, {9) for fixed o implies Hq(Y, (0)= o. (cr. Chapt-; 6, §2, Theorem 20.) For, using exactness, tze obtain immediately:
-
q
t)
0.
('.I hT (X, ;.;;; __/~y)
~
pq+l (X I ;.:_y ,/) ) l
,
so that it is enough to show Hq+l (X, :!) ) = 0. -Y Let a be a cochain in Hq+l (X, 0 ) • Then r • a is a cochain with L~y coefficients, a~ f is holomorphic in x. Hance, multiplication by f induces a homomorphism f*: Hq+l (X, {_?) -> Hq+l (X, {9 y). Clearly f * is onto and one-to-one; therefore an isomorphism. III. As a last application, we obtain another old result: Let X be a complex manifold; and let ~* denote the sheaf of germs of invertible holomorphic functions under multiplication. Note that the sections are the nowhere
157 vanishing globally defined holomorphic functions. Let ~* denote the sheaf of germs of meromorphic functions under multiplication. Then {!) * C /It, and \lle obtain the exact sequence: , * . I * * o -> {) *. -i > [1 L> iJJ. I Q. - > .Q • The sections of fit* I~* are divisors in ~' i.e. equivalence classes of sets of data for the C.II problem. Using the exact cohomology sequence theorem, we obtain the exact sequence: * 0 .-}1 * j 0 k * ., * 5 1 ;() * ••• - > H (X, ttl ) - > H (X,~ I C.t ) - > H (X,£ ) -> •••
Here j * takes a meromorphic function into the C.II problem it solves: therefore any C.II problem a can be solved if 5a = 0.
This, however, is not particularly illuminating for we know little about H1 {x,C?*); so, we imbed this group in another exact sequence involving "simpler" coefficient groups. We have an exact sequence: 0 ->
i /-f'\ exn (11 ~-> ~ ~> .;!__
* ->
0 ,
where here ~ is viewed as a sub sheaf of CJ, giving exactness at 2.; 11 exp 11 iS the map: exp (s)-: 21fiS and rll (1 21ris the exactness at ~ is clear since ker exp =[ s , e = and exactness at 0* follovJS from the fact that every nonvanishing holomorphic function is locally an exponential. Hence, \•le obtain th~ exact sequence: 1 . exp 1 If * d 2 .
e
••. - > H
(x,O)
- > B (X,£ ) - > H (X,~) - >
... ,
but this gives rise to a (canonical) map: C
=
d • 5 : H0
- *I (X,ii~
0*) ->
2
H {X, H 2 (X,~ is one-to-one. Thus C(D) = 0 implies D is principal.
158 -··
So, we have re-established the Oka-Serre Theorem: Theorem III. There exists a map C : H 0 (X,~*/c?*)->H 2 (x,ZJ such that 1) D principal implies C(D) = 0 2) H1 (x,i2) = o and C(D) = o implies n· is principal. • (Cf. Chapter 9, §4, Theorem 31.)
--
.
~7.
Proof of the exact cohomology sequence theorem
nottl restate and then prove the theorem: Theorem 44. Let 0 -> A !> B ~> C -> 0 be an exact sequence of sheaves. Then there exists an exact sequence: \~e
*
f • • • - > Hj ( X, A ) -> Hj ( X,B )
-
Proof. Let U = consider the sequence:
}
-
L*> Hj ( X, C) -> 5 Hj+l (X , A ) -> , •• -
--
-
ui~ be a covering of X;
and
0 -> Cq(U,~) !_> Cq(U,~) 1L> Cq(U,C) where Cq(U,A) denotes the group of q-cochains on the covering Us and f and g denote the induced mappings. We claim this sequence is exact. At Cq(U,A) we must show ker f = 0; therefore assume a e Cq(U,A); f(a) = 0. Now f(a) = 0 means f • ~ 0 ••• iq(x) = 0 e Bx for all x e ui 0 n ... /)uiq. By exactness, f is one-to-one so ai i (x) = 0 e Ax for 0 ••• q every x E ui ll ... /)ui ; i. e. a = 0. At ·_cq{U~B), \rle m8st show im f = ker g. Let a e Cq(U ,A). Then g • f{a)i i (x) = gfai i (x) = 0, since gf = 0, -o• • • q
o· • • q
for every x e ui (l ••• (}ui ; hence g • f( al 0 q im f c. ker g. Now let ~ e ker g; i.e. g~~i 0 ••• 1q (x)
=
= 0 for every
x e ui fl .•. (l u1 • For each x e ui () ••• (', ui 0
q
0
0 so
q
there exists
an aio· •• iq(x) e Ax such that
ra1 ••• i (x) = f3 i ••• 1 (x). 0 q 0 q We claim that the assignment a defined by the ai 1 (x) is a cochain; i.e. that ai i is a section over 0 ·~~ ~ ••• 1• o· · • q
o
nu
q
159 ai ••• i (x) e Ax so p • ai i = i~ ·i 1) ••• () ui • o q o• • • q ""U 0 q To show continuity, let L. e S = ai i (ui ll ... ()ui ) ·1 --u a o• • • q o q = f f3i ••• i (ui () ••• I) ui ) , an open set for Now
-
q
0
ui /) ••• ,1 ui o
continuous.
q
0
is open,
f3i
q
.
i
o• • • q
is a section and
Let Nx= be a neighborhood of XO
f
is
in Sa ; then
0
a-il •• i [Nx] =a-il i • r·l. f(Nx) = (f• ai i )-lf(Nx) o • q o o· • • q o o• • • q o = pi-l i f(Nx ) which is open for f is an open mapping. o· • • q o Hence, 1m f = ker g. We cannot complete this sequence to a short exact sequence, for g may not be onto [e.g. take sequence 0 ...(") 0 l0 .),.; c (X,z) -> c (X,~) -> c (X,d/.Q)] • So, define C~(U,C) = g[Cq(U,B)], a subgroup of Cq(U,_Q) comprised of "liftable" cochains. Hence, we now have the follo't'ling short exact sequence of groups and allol'lable maps: 0 -> Cq(U,A) _1> Cq(U,B) ~> Cq(U,C) -> 0 • -
-
a
-
Hence, we obtain the exact cohomology sequence: ••• -> Hq(U,A) -> Hq(U,~) -> Hi(U,C) -> Hq+l(U,A)->,. For refinements of U, we have the desired commutativity, so that we may appeal to the proposition of Chapter 14, §1, to obtain the exactness of the limit sequence:
*
*
5
••• -> Hq(X,A)f-> Hq(X,~)g-> Hi(X,C) -> Hq+l(X,!) -> 1-'le
• ••
shall therefore be done if we can shot'l Hi(X,C) ~ Hq(X,C)
(canonically!), and this shall be proven by showing that for each cochain ~ in -Cq(U,C) for a locally finite covering U, there is a refinement V in which ~ is liftable; then the limit groups are isomorphic, since we have an injective {one-to-one) map i : ci(U,C) --> Cq(U,_Q) which commutes with the boundary operator and the"refinement" maps of the direct limit procedure.
160
(Since X is paracompact, we may restrict ourselves to the locally finite coverings U.) Let U = [ui~ be a gi~en locally finite covering. Let W= !wii be a new locally finite covering, refining. U and such that wi C c.. ui for each i, possible since paracompactness implies normality •. Let ~! Cq(U,C). To each x eX we assign an open vxc X such that
-
i)
X E VX
ii) x E wi iii) x E uj iv) x i uj v)
implies implies implies
X E Ui () ••• {)ui 0
q
Observe that once we establish the existence of the theorem is proved. Let x E X; using the local finiteness of U, vi there exist integers r,s < oo such that x E wj , ••• ,wj; r
i
uk , • •., uk 1
s
and no other wi' uk.
v~
wj {\ ••• 1l wj t\ uk f) ••• /)uk ; clearly v! is 1 r 1 s an open neighborhood of x and satisfies i), 11), and iii). Note that any smaller neighborhood of x will also satisfy these. For each k F k1 , ••• ,ks; xi uk' which implies Set
=
X t Wk' SO there_exiS~S an Open 2neigibor[tJ~d Vk,X Of~X such that vk,x(\ wk = ..,. Set vx = vx (\ k,x , k
1
k 1 , ••• , k 8
an open neighborhood of x l'Jhich now satisfies i), ••• , iv). To satisfy v), observe that X E u1 /1 ••• /)ui can only 0 2 q happen if ui E { uk 1 • • • , uk 1 and that vx C ui f\ ••• () ui • j 1 s 0 q 2 2 Now ~ 1· 0 ••• i q I vx is a section of C over vx • Hence i (v 2 ) is an open. ~i ... i (v~) is open in C, so g- 1 ~
I
0
q
non-empty subset of B {g is onto).
o/i 0 •·• q
X
There exists a
'
161 bx
E
g- 1
~i•••i (v~) such that p ~bx = x, and neighborhood q B
0
of bX in - such that1 p : NX -> p(NX ) is a homeo. 2 morphism. Let MX = Nx()g- ~i •• • i (vX ), and set 0 v = p o g(M ) C v2 • q X X X 1 Now g I Mx is one-to-one; and g- ~i i : v -> M cr•q X X: is a section of B over vx , mapped by g onto the section
N
x.
~io••iq : vx
->
g(Mx) of C.
162
Chapter 15.
Coherent Analytic Sheaves §1.
Definitions
Definition 71. An analytic sheaf ~ is a ,sheaf whose base space X is a complex manifold (or subspace of one); and such that each element of the sheaf can be multiplied by the germ of a holomorphic function; more precisely, each stalk ~x is an c?x-module, and this multiplication is continuous. vie have the notions of subsheaf, sheaf homomorphism, induced sheaf, factor sheaf, etc., as before. r Examples. t9 , the sheaf of germs of (r-dimensional) vector-valued holomorphic functions is an analytic sheaf. The sheaf of germs of continuous functions is an analytic sheaf. The sheaves Q_ * and ftt~ are not analytic, for there is no distributive 1a-v1 for multiplication by germs of holomorphic functions {recall that the operation in the stalks of these sheaves is multiplication). Definition 72. An analytic sheaf ~ is ~lobally finitely generated if there exist a finite number of global sections ·"' x' s 1 ,s 2, ,sk Sl.,tch that for every x E X, t E ;::r t = ~l ( s 1 )x + . • • + ~k ( sk) x where ~ j E CJ x. Examples. The sheaf 0; section "1". The sheaf CJ r ; r s~tions (0, ••• ,0,1,0,. .,0). Definition 73. An analytic sheaf is locally finitely generated if every point x E X has a neighborhood Nx such that the induced sheaf d- {N ) over Nx is globally X finitely generated. Let be an analytic sheaf, and s 1 , ••• , sk sections of ~(U), uopenc X. Let x E U. If there exists a tuple (~ 1 ,~ .. ,~k) E {9 ~ such that 1 ••
1
;1:
~l(sl)x + ··~ + ~k(sk)x = 0 ' the tuple (~ 1 ,.~.,~k) is called a relation between the sections s 1 , •.• ,sk at x.
+--
Note that the collection of all such relations forms an analytic "sheaf of relations" (between s 1 , ••• ,sk) contained 1n t!) k (over the space U). . Definition 74. The analytic sheaf ~ is called a coherent analytic sheaf if: 1) It is locally finitely generated • . 2) For every open U C X, the sheaf of relations of any finite number of sections over U is also locally finitely generated. Note that the definition is local. Remark. For convenience, we will call a coherent analytic sheaf, a coherent sheaf.
-
§2.
Oka's coherence theorem
The aim of this section is the statement and two steps of the proof of a three-step theorem due to Oka. The last section of the proof will be postponed until two theorems are established. Theorem 45. (Oka) The sheaf of germs of vector-valued holomorphic functions is coherent; i.e. (noting the local character of coherence) let Dopenc..:: ¢n, and let { aij ( z) i=l, ••• sq, j=l, ••• ,p be holomorphic fUnctions defined in D. Let xED; then there exists an open D1 C: D, x E D1 , with the following property: For any ~ E D1 the holomorphic solutions (~ 1 , ••• ,~p) of
J,
p
* :
1=~ aij(z) ~j(z)
defined in some neighborhood of
= 0 ~"
,
i
= 1, ••. ,q
may be written as:
L
~j(z) = (=l ~v(z) ~~(z) ,
j = 1, ..• ,p
where the ~j; j=l, ••• ,p, v=l, ••• ,L 1. III. If the theorem is true for same n and all q, then it is true for n+l and q = 1. It is clear that these steps complete the theorem; and that I holds, since a holomorphic function of no variables is a constant. We may assume x = 0 e D, with no loss of generality. II. Let us first introduce the following abbreviations: . p
(a)
tj; a1 j (z)
-
-
p
~j(z)
=
0,
?
{f3 ) -
'i J~ 1 ai j ( z )
~j ( z)
=
0,
(y)
-
~
~j(z)
=
0
(~j)
=
tl
(~l,
aqJ(z)
t
i = 1, ••• ,q
i = 1 , ••• , q
-15
J
... ,~p) .
,,
By hypothesis, (y) has a finite pseudobasis (~j), rt=l, ••• ,K 1. Note that constant term. N Let s < N, an integer. Set zj = ~j' j = l, ••• ,n-1; and define: G(~l'"""'~n-l'z)
=
Then G is a holomorphic function of its variables in a
166 neighborhood of the origin and of order s, so that Theorem 12 applies~ Hence: G((l, ••• ,(n-l'z) = H((l, ••• ,(n-l'z)[zs+Bl(Cl, ••• ,(n-l)zs-1+ 0
••• + Bs(Cl~ ••• ,cn-1>1 and the B1 vanish at the origin. It is now enough to show that the Ci occur only as C~n. Take Q to be a primitive N th root of unity; ~hen: G(C1, ••• ,Q(i, ••• ,cn-1'z) = H(C1, ••• ,9(i ••• ,cn-1z[zs + Bl(Cl, ••• ,Q(i, ••• ,cn-1)zs-1~ •• ] =
G(Cl, ••• ,ci, ••• ,tn-1'z)
But the expansion of the ~leierstrass Theorem 12 is unique. Corollary. Let P(Z,z) = zs + a1 (z)zs-l + ••• + as(Z), where the aj are holomorphic in a neighborhood D of the origin. Let (C,c) e ~n, C c D. Then P(Z,z) = PI (Z,z) PII (Z,z) where
P·t(Z,z)
=
PII( Z, z )
zr + a1 (Z)zr-l + ••• + ar(Z)
= zt + 13 1 ( Z ) zt-1 + ••• + f:3t ( Z )
with Bi' a. holomorphic in a neighborhood of C, that pi"(c .. z)J= (z-c)r and PII(C,c) f. o. Proof. Changing variables, set:
z' = z - c,
z'
=
such
z - c
I
and define: Q(Z ,z•)· = P(Z 1 +C,z'+c). Then Q(Z 1 ,z') = QI(z 1 ,z')QII(z 1 ,z'); where QII is a unit and QI(z',z•) = z'r + a1 (Z 1 )z 1r-l + ••• + ar(Z 1 ). Set: PI{z,z) = QI(z-c,z-c) pii(z,z)
=
QII(z-c,z-c) •
Then PII(C,c) I 0 and PI(C,z) = QI(O,z-c) = (z-c)r. It is clear that pi is a monic polynomial.
167 B. Theorem· 47. (Division Theorem) _ I.et there be given a polynomial P = zs + a 1 (Z)zs-l + •.• + as(Z) with the aj holomorphic in a-neighborhood of the origin, and aj(O) = 0. Let f(Z,z) be holomorphic near the origin. Then: f(Z,z}
=
q(Z,z) P(Z,z) + R(Z,z)
where R is a polynomial in z of degree < s, with coefficients holomorphic in a neighborhood of the origin, and q is a holomorphic function in a neighborhood of the origin. Furthermore, this representation is unique. Proof. We first establish uniqueness. Suppose 0 =
qP + R 00
=
Let
v ~ 1;
( ~ qj(Z)zj)P + R •
we equate the coefficients of zs+v, 0
=
qv + qv+lal + ••• + qv+sas
obtaining •
But ai(O) = O, hence q (0) = 0. But then ~q./~zj m_o, v M ml v n-1 j = l, ••• ,n-1. Similarly, one finds ~ qv/~ z1 ••• ~ zn_ 1=0, hence q 0. But then R = 0. We now assume f = ~ fj(Z)zJ. This is no loss of generality, since the te -s of order < s in z may be included in R. Assume aj(Z) =NO (llzllj); this may be achieved by replacing zi by ~i' N > j, i = l, ••• ,n-1 in f and P, since aj(O) = o.·. The transformation back to the zi is achieved as in Theorem 46 since we have already established uniqueness.
=
NOl'l
zs = AsP + asl (Z)za-1 + • •• +ass (Z) where A = 1 and a . = -aj' Hence, multiplying successively S SJ by zm, m = 1,2, ••• and substituting appropriately gives: z s+m = As+m ( Z,z }P + as+m,l ( Z) za-1 + .•• +as+m,s(Z)
168 where in fact:
As+m+l =
zAs+m +.Asas+m,l
as+m+l,p = as+m,l asp + as~,p+l;
0
(ci -c i)
p
= -
1 ( r~ prr =
Using the division theorem, p-1
-"'
Cp =
q
'\
0 i Pi)
=
(Pi ci)
where q, R are polynomials and p II
JJ.i P Pi
1
P cP + :n
-t=t
~ 1=!
I
P = P cP + prr
I
p.L
Pi)
•
I q P
+R ,
deg R < deg PI.
+ ..!1,
)
Hence
•
But PII ~ is holomorphic, so R vanishes of order I p (deg P ) > deg R; hence R o. Thus:
=
PII C = q p
'
i = l, •.. ,p,
i = l, •.. ,p;
But so
Hence
~
i
Pi
c1 = 0
if and only if
171
~ .J'- gij(Z)
j+.:~=S f;r
1TB(Z)
=
0 ;
s = 0, ••• ,3a ,
whel"e the 1ri.t are knol'm functions; and this sys tern of equations involves n variables. Hence the solutions have a finite pseudobasis by the induction bypothesis, thus completing the proof.
.-
§s.
~sequences of Oka's theorem A. Remarks on coherent sheaves. 1. Coherence is a local property. A sheaf is coherent if and only if every x E X has a neighborhood ia uhich the induced sheaf is coherent. 2. A subsheaf Q of a coherent sheaf vf , is coherent if and only if it is locally finitely generated. For, any section of G is a section of ·d, since Gx is a subgroup of ~ • Hence, the sheaf of relations R of any finite X number of sections of G is the sheaf of relations between these sections, conside;ed as sections of .~. Since ~?L is coherent, R is locally finitely generated. B. Corollary. If~4 is coherent and s 1 , ••• ,sk are sections of 2:, then the sheaf of relations R(s 1 , ••• ,sk) is coherent. ,..
J
173 Theorem 49. Suppose A and ~ are coherent sheaves and f : A -> B is a homomorphism of ~ into ~. Then 1m f, ker f, coker f = B /im f, and coim f = A I ker f are coherent sheaves. Proof. It is sufficient to prove that 1m f and ker f are coherent, as the coherenc~of coker f and coim f is then given by Theorem 48. To establish the coherence of 1m f = f(~) and ker f, we need only show that they are locally finitely generated since we already know that 1m f is a sub sheaf of B and l{er f is a sub sheaf of A ( cf. p. 151). 1. Since A is locally finitely generated, every x E X has a neighborhood Nx in \'lhich a fin! te number of sections s 1 , ••• ,sk of A(Nx) over Nx generate (A(Nx))Y, y E Nx. Their images under f generate ((1m f)(Nx))y' y E Nx. Thus 1m f is locally finitely generated. 2. For kerf, take x, Nx and s 1 , ••• ,sk as above. Since f(s 1 ), ••• ,f(sk) are sections of B(Nx) over Nx' R(f(s 1 ), ••• ,f(sk)) is a coherent sheaf. Consider the following mapping g : Ry -> Ay of Ry into Ay' y E Nx
01
k
->
~ ~i(si)y •
Since f( ~ ~ 1 (si)y) = ~ ~ 1 f(s 1 )Y = 0, g is a homomorphism of R into (ker f) I N • But every element X of {{ker f) I Nx)y is of the form L ~i ( s 1 )Y with ~ 1 E CJx and ~ ~ 1 f(si)y = 0. Hence g is onto. Therefore ker f I Nx is the image under a homomorphism of a coherent sheaf and thus is coherent, by part 1.
§6.
The sheaf of ideals of a variety
Definition 75. Let X be a complex manifold. An analytic set V in X is a closed subset of X such that
174 for every v e: V there is a neighborhood Nv of v and functions ~ 1 , ••• , ~k holomorphic in Nv such that V/) Nv = fx e: X I ~ 1 (x) = ~ 2 (x) = ••• = ~k(x) = ~· Definition 76. For x e: X and V an analytic set in X, let e/~ be the subset of C?x of all the germs of holomorphic functions vanishing on V. (If xi V then &x = (? ~ ; if V E X then 0~ is trivial.) C)~ is a subgroup of l9x and an ideal. ~, (9 ~ \'lith /'("'\ XEA induced topology is a subsheaf of ~ called the sheaf of germs of the ideals of the analytic set V, denoted ~V(X). Theorem 50. (Cartan) If V is an analytic set in a complex manifold X then ~v(X) is coherent. He \'lill only prove a wealcer form of this theorem. Theorem 51. If V is a regularly imbedded, analytic subvariety of codimension lc in an n-dimensional complex manifold X, then ~v(X) is coherent. Proof. ~ v(X) is a subsheaf of a coherent sheaf. Hence it suffices to ShOH that .;) V(X) is locally finitely generated. Let x e: X. If x i V then there is a neighborhood Nx of x \·lith Nx 11 V = ~. In Nx, .J V(X) = Q, and l'Ie are done. If x e: V, then by the definition of V we can introduce local coordinates z1 , ••• ,z such that n ~ V = (Cz 1 , ••• ,zn) EX I z1 = z 2 = ••• = zk = Oj, in some neighborhood Nx of X in X. Now, if f e (d V(:NX) )X, then f is the germ of u functi~n holomorphic in Nx and vanishing on V, so that f = O.)i zi, where the -w1 are holomorphic in Nx. Since z1 , ••• , zk ~ sections of v-iv(Nx) over Nx and the mi e &x, the proof is complete. Remark:. CJ (X)/~Jv(X) is a coherent sheaf, by Theorem 48. Its stalk over every point off of V is trivial. On V, since (cJ v(V) )x is trivial, its stalk is the stalk of 0 (V). Hence, on V this sheaf can be ictentitied vJith the sheaf C) (V) of germs of holomorphic functions on V. c0 (X) /,J~(X) is the trivial extension of i2(v) to X.
c?,
f=I
175 Chapter 16.
§I.
Fundamental Theorems (semi-local form)
Statement of the fundamental theorems for a box {semi-local form)
Notation. By an open box in 0n, we mean a [(z1 , ••• ,zn) ea.n I ai<xi 2n. It is enough to consider Hq(x,u,Jt) for U a locally finite covering of X. Since if X c E2n we can refine U to a covering in which more than 2n+l sets always have an empty intersection, we are done. That (1) is true follovJs from the definition of a coherent sheaf as being locally finitely generated. To prove (2), assume Theorem 52Aj for j ~ ~· Let K be .··/ the given degenerate box of dimension r and ~ the coherent sheaf over X. Then there is a box x0 with K c c. x0 c. c X and jl(X0 ) globally finitely generated; call the generating sections s 1 , ••• ,sm. This means that there exists a homomorp~ism f 1 of1 .9ml(x0 } onto :.£.(x0 }. For, define
.J:
f
1
:
(9 ·~1 -> 1:.X .1\.
1
X E
X0 by(f !1: )-> >
~i ( si) X.
~mr generate ( :f(x0 ) )x at every
It is
onto because the s 1 x E x0 • Therefore, denoting ker r1 by a1 , we have the following i ,/'lml f, .-1 exact sequence 0 -> a1 -> £. -""> it:_ -> 0 • By Theorem 49, a1 is coherent. Apply Theorem 52Ar to a1 over x0 . We get a new x0 , call it X~ and G 1 (X~) globally finitely generated. Then there exist m2 generating sections •••• Denoting ker r 2 by G2, we get the exact sequence 11m2 -> G -> 0. Again, G is coherent and .Q -> G2 -> L./ 2 1 we can continue this process, as far as we want, up to 4n,
177
§3. Reduction of (3) to Cartan's theorem on holomorphic matrices Lemma 1. Let K be an r+l-dimensional degenerate ,.. closed box as in Theorem 52 A +l' given by: K = l a:l:<x-t < ai, "' 7 r .~... - J..B1~Y 1~ f3 1 5 • Assume, e.g. that a 1 < a 1 (or similarly, .,.-
/'
that
f3i
Kj+l of closed, nest~d boxes such that in no neighborhood or any box is the sheaf induced by_~i globally finitely generated. But the Kj intersect in a point, and a point surely has such ~ neighborhood; and this contradiction establishes the claim. . . A )\ Lemma 2. Let K1 , K2 be two closed boxes, given as follO\'JS: ( "' "" 1\ . Pl. ~ xl ~ ~' 131 ~ Y1 ~ f31
~1
=
K2 =
~j ~ j ~ 7xj (;;:1-E ~ X1 ~ ; (
X
~ Yj ~ ~ j (11 ~ Y1 ~ ;1
f3 j
1
1
1
where a1 ~ ~1 -E~ Le~ ~ be an analytic sheaf over a neighborhood or K1 U K2 • Let there be givxn sections a 1 , ••• ,ar of ~over a neighborhood of K1 generating the stalks of ~(K 1 ) at every point; and, similarly, sections b1 , ••• , b s of d- over a neighborhood o~ 2;._ Furthermore, assume that in some neighborhood of K1 f'. K2 ,
J:..
K
a 1 = ~ ~ij bj and bj = ~ tjk ak ; ~ij' tjk functions holomorphic in this neighborhood. Then there exist sections c1 , ••• ,cN over a neighborhood A A of K1 U K2 such that, in ~ neighborhood of K1, ai= L:~ijcj; and, in a neighborhood of K2 , bi = ~ tijcj. Claim. Lemma 2 implies Lenuna 1. Let K1 , K2 be as in Lemma 1, where we tal 0 so small that Kl f1 K2 ()X0 • By the induction hypothesis, there exists a neighborhood X0 of K1 K2 such that Theorems 52Ar and Br apply; now ~
>
n
179 consider the map rJ._s -> i_(x0 ) given by . (~ 1 , ••• ,~s) -> ~ ~ibi. This is a sheaf homomorphism, and the bi generate the stalks of ~(X0 ) at every point, so that the sequence d) s _.., ( ) ~ -> ~- x0 -> ~ is exact. We comple~e to a short exact sequence:
,.
and G is coherent.
Hence, we have an exact cohomology sequence:
Ho(Xo,.Q. s) - > Ho(Xo,i·(xo)) - > Hl(xo,g) •
If x0 is small enough, Theorem 52Br applies, so is onto; but H1 (x0 ,G) = o. Hence H0 (X0 ,f2_s) -> E0 (x0 this means that every section of over x0 is a linear combination of the bi ; in particular, the aj are. In a similar manner, the bi are a linear combination of the aj. Proof of Lemma 2. Under the hypothesis of Lemma 2,
,J::)
.1:
ai = ~ ~ijbj and bj = ~ ~jka~ wh~re holomorphic in a neighborhood of K1 /! K2 • vle adopt the following notation: . a
=
~
=
and b = (
{~ij)
and
~
= 1\
l: );
~ij'
tjk are
column vectors
{tij);
matrices holo-
..,
morphic in a neighborhood of K1 /.) K2 • Then the hypothesis takes the form: ~b
= a
~a
=
b.
Now consider the {r+s) )l..(r+s) matrices defined below, which satisfy the follO\'Jing relations: . ·' 0 t a ' \ I a.l I • 1
o\ -r ~
\f ~
cr
\'
li or
1
8
'
I
/U J
I ;1~ ; ( -Ir = { ar / I
\
~1 J bS/
~
I
I
I
\ 0
I \
~rb
.•
:
~: j
=
0
~1 •
•
bs
,.
180
where Ij
denotes the
j
j
;w.
Define
unit matrix.
,/
'
!
(_-r_r.---· ( Ir M
=
\
:
\
0
0
\ "''.
~
Then M( ~) = (~), and M is a" nons~lar matrix holomorphic in a neighborhood of K1 K2 • -1M , where M is a holomorphic If we can write M = M1 1 2 A nonsingular matrix defined in a neighborhood of K2 , and M2 a holomorphic no)nsingular matrix defined in a neighborhood of Kl. Then M2\~ = Ml ( Set
n
n
/' cl c
\
= (I :•
)
\.. cr+s Then the cj are global sections generating Sf in a /1 ""\ ; ) neighborhood of K1 t) K2 , since the rows of M ( ~ are sections generating .J- in a n;._ighborhood ~~, ~ , M1 ( ~ )
1
j'
sections in a neighborhood of K2 ; and M2 { /\
"'
\0
=
M1 ( 0 )
b
in a neighborhood of K1 tl K2; and both M1 , M2 are invertible. Hence, the proof of Theorems A and B is reduced to: Lerruna 3. (Cartan 1 s Theorem on Holomorphic Matrices) Let M be a holomorphic nonsingular matrix defined in a neighborhood " 1\ of K 1 ~ K2 • Then there exist hol~morphic nonsingular matrices A, defined in a ~eighborhood 'of K1 , and B, defined in a neighborhood of K2 , such that M = ~A.
§4. A.
Proof of Cartan•s Theorem on Holomorphic Matrices
Let A be an
II z II
= max
Iz j j=l, •• ,n N )< N rna trix; set
Recall that
II AII = Note that
I aij I ~ II AII ~
sup
I
where
i ~x II
N max I a 1 j
I •
.
z = ( z1 , ••• , zn)
E
en.
181
Let A( z) denote an N >< N matrix whose entries are fUnctions holomorphic in a domain De ¢n. Set:
Ill A Ill
max
=
Z E
and
Ill Alila where 0 < a < 1 and
Ia1 j(z)-aij(z)l J\
~
H A. ct.
Hlz-zl
=
D
II A( z) n
Ill AIll + H
,
is the smallest constant such that
•
Note that if D1 c c D, domains, then IliA lila ,D ~ kii(A lin· 1 It is lmown that Ill Ill is a norm; and if n a DC ct , the N)(N matrices whose entries are fUnctions holomorphic in D form a Banach space under this norm. Furthermore-: IIIABIII
~
IIIAIII ·IIIBIII
Ill AB lila
~
c
I
Ill A II Ia nIB lila •
(The proof of the above statements is left as an exercise.) Note that, for any matrix A, eA e ~ ~~ is dominated by
)
111~111n.
A3/3- A4/4
If
IliA Ill
< 1,
then
log
(I~A) = A -
A2/2 +
+ ••• is dominated by JIIAIII + IIIAIII 2/2 + ••••
Furthermore, eA is always nonsingular, and eloe(I+A) = I+A. B. The following propositions establish Lemma ;: Proposition 1. Let D be a polydisc, D1 c c D and M a holomorphic nonsingular matrix defined in D. Given e > 0, there exists a nonsingular·_ entire matrix P such that M = PM1 in n1 and II II -M1 ~ ID < e • Proposition 2. There exists an 1 e· ·> 0 such that, if Ill I-r-i lla < E then Lemma 3 holds for M. Claim. Propositions 1 and 2 imply Lemma 3. Proof. By Propos! tion 1, M = PM1 , Ill I-r-i1 lll D < e • 1 Then in a smaller domain, IIII-M1 111a < e • Eence, M1 = BA by Proposition 2. But P is holomorphic everywhere, so M = (PB)A.
182
of Proposition 1.
~roQf
o.
about
M(z)
Assume that D is a polydisc
~·lrite:
=
M(O)lM(0)-1M(.fz)JtM-1 (tz) M(.fz>} •••
[M-1(¥z> L
an integer.
M(z~ ,
:
Now
llli-M- 1 (~z)M(K~lz) Ill ~ IIIM- 1 (~z) Ill IIIM(!z)-M(¥!z> Ill for L sufficiently large,
0 < K < L.
z) K+l M-1 (K L M(L z)
as the log series converges. M(z)
=
< e
Hence: =
NK(z) e
,
So:
N0 (z)
M(O) e
••• e
NL_1 (z)
•
h
By going to a smaller polydisc, D, if necessary, for each NK(z) there exists a polynomial sequence PKj(z) such that Poj(z) PlJ(z) PL :j,f) IIPKj(z)-NK(z) 11 0 -> o. Set Mj(z) = M(O)e e ••• e - •.•
o,
and det M I 0 for j sufficien~ly large. DHence M(z) = Mj(z~ [M~ 1 (z) M(z)I , -1 J and Mj (z) M(z) converges uniformly to I with j; and M. is a matrix of entire functions, for all j. J "" .. "' Proof of Proposition 2. Let K1 , K2 be given as follows: t 1\ ·" .-.. ) 0 1 ~ xl ~ al ' 131 ~ Y1 ~ t31 Then
UIM.(z)-r~(z)
Kl
=
l\
1{2
=
111.'\
->
1
1\
(laj~xj~aj, J
1\
cal-£"" xl
.-.
A
13j~Yj~f3j;
j=2, ••• ,n
A
~ al_. tll ~ yl ~ tll
a 1 < d1-E, as in lemmas 2 and 3. "\ A. M is defined in a neighborhood N of K1 /) K2; we write M = M(z), where z ~ z1 and the dependence on z 2, ••• ,zn is suppressed. Now M = I + X, and lllxllla < E where g is to be chosen later. In the z1-plane, let r be
where
""'
1\
a smooth analytic Jordan curve· containing K1 () K2 and .lying in N (more precisely, their projections on the z1-plane). Let 1 1 and r 2 d~~ote disjoint closed arcs, segments of 1, such that 1 I) K1 C.. int 1 1 . and 1 I) K2 < 1nt 1 2 , where "int 1i" denotes the segment 1i without its endpoints, as indicated in the diagram. Let G denote the bounded component of the complement of 1 in the z1-plane, and G its closure. Take o- to be a real-valued C0 function defined on 1, 0 < ~< 1, -t-~-H-+---H-+11-----+---> such that cr= 0 on 1 1 and xl cr· = 1 on 1 2 • For any matrix Y holomorphic in G and continuous inn, define:
-
T (Y) 1
=
1
2~i
f
Y( ~) CT ( ~) d'"
t-z
~
1 r
T2(Y)
=
y -
T (Y) 1
,
= __!_ fY(~>l_l-cJ(C)j dC 2~i.
~-z
1 I\
Then T1 (Y) is holomorphic in a neighborhood of Ki; and the linear operator Y -> T.(Y) is bounded in G; ~ IIIT1 (Y) !Ia ~ cii!YIIIa· tie ltdsh to solve the equation: .
I
+X
+ T1 (Y))
(I
+ T2 (Y))
=
(I
=
(I + T1 (Y)) (I + Y - T1 (Y)) ,
for some matrix Y holomorphic in G and continuous in for Ill XIll a sufficiently small; i.e. we wish to solve: X
=
Y- T1 (Y) T1 (Y) + Y T1 (Y)
=
Y +
F(Y) • Therefore, define T(Y) =X- F{Y), for Y E S = tY I Y holomorphic in G and continuous in G, IIIY-XIIIa < E, lllxlla < Note that T(S) c. s. l'le claim that, if
EJ·
E
G;
184 is small enough, T is contracting; so that the contracting mapping_ principle applies and there exists a unique Y0 c: S such that Y0 =X- F(Y0 ), as desired. IIIT1 (Y) Ill a~ c lilY Ill a lc( c,C) depending on
Now constant
implies that for some c and C (of p. ·181),
IIIT(Y)-XIII a= IIIF(Y) Ill a-~ kii!YIII~~ k[ IIlY-XIII a+ IIIXIII al 2 < 4ke 2 ;: so we require E < l/4k. Furthermore:
Ill T(Y) -T ( Z) I!.Ia
=
Ill F (Y) - F {Z) II Ia
=
Ill (Y-Z)Tl (Y)+ZTl (Y-Z)+Tl (Z)Tl (Z-Y)
+ T1 (Y) ~
T1 (Z-Y)111a
IIIY-Zllla K(K+l) (IIIYII6 + lllzllla)
IIIa
for some constant K(c,C). Now IIlY-X < e:, henc~ 0 ~ IIIYIIIa ~ 2c:, so IIIT(Y)-T(Z) Ill a~ 4c: K(IC+l) IIIY-ZIIIa, and we require also E < l/[4K(K+l)]. §5.
New proof of the Oka-Weil Approximation Theorem
Theorem 53. Let X be an analytic polyhedron, xc:::_Gopen c ~n, X= c: G I lrj(z) 1 o. By Theorem 52B, q v, H (D 1 ,~} = 0 for all q > o.
Chapter 17.
Uoherent Sheaves in Regions of Holomorphy
§1. Statement of the Fundamental Theorems Theorem 55A. Let X be a region of holomorphy and da coherent sheaf over X. Then global sections of .::J generate ~x at every x E X. Theorem 55B. Under the same hypothesis as in Theorem 55A,, Hq(X,~) = 0 for all q > 0. These theorems have numerous applications which will be given later on. .. A
:
~2.
Preparations for the proof
Notation. (z 1 , ••• ,zn_1 ) = Z, zn = z so that (z 1 , ••• ,zn) = (Z,z). Theorem 56. (Cartan) Let P(Z,z) be a Ueierstrass polynomial or degree s, P(Z,z) = z8 + a1 (Z)zs-l + ••• + as(Z); the aj are holomorphic in a neighborhood of the origin and aj(O) = o. Let r 1 , ••• ,rn-l'rn = r be> 0 and such that each aj is holomorphic for lzjl ~ rj and let P{~,z) 1 0 for L(Z,z) I lzjl~ rj, j = 1, ••• ,n~l and lzl = r; • Let f(Z,z) be holomorphic in D = L{Z,z) I lzjl~ rj for all ~J; and let I rl ~ 1 on D. Then (1) f = QP + R where Q is holomorphic in 1nt D and R is a polynomial of degree s-1 in z with holomorphic S-1 j coefficients in int D, R = bj(Z)z , and
f-o
IQ(Z,z) I ~ K and bj(Z) I ~ K \'lhere IC does not depend on r. Proof. (This proof is independent of the Division Theorem, Theorem 47, and hence gives a new proof of 1t.) · 1 r {z, C) dC Let Q(Z,z) = 2-iii. ICI=r P(Z,~)(~-z) ; Q is (2)
j'
holomorphic in int D.
Then
188
r
(
~ P(Z,~)-P(Z,z) ] d'" • But PTZ;0 lc l=r s-j s-j ( ) ( ) s-1 P Z,~ -P Z,z = ~ a (Z) C - z , a = 1. Carrying t-z j=O j ~ - z _1 0 out the division gives R(Z,z) ~ 0 bj(Z) zj, where
R(Z z) = 1
211-I J
I
e-z
';,
jt
bJ(Z)
=
2-;Ij
~~~~~~ (Cs-j-1 + alcs-J-2 + a2cs.-J-3+ •••
ICI=r + as-j-1) dC. Hence R is of the required form. Now to estimate Q and the coefficients of R. For ICI = r and lzjl ~ rj, j = l, ••• ,n-1, lrl ~ 1 and !PI has a lO't'ler bound b > 0 and laj I < c, a constant, for all j. Hence lbj{Z)I ~ K1, a constant independent of r. To estimate Q write Q =(f-R)/P. Then l+K 1 (l+r+ ••• +rs-l) IQI ~ = K2, a constant independent of r. Take K =max (K1 ,K 2 ). /()q. Theorem 57. Let Mj be a submodule of L/ 0 J (vectorvalued holomorphic functions in ¢nnear the origin) of dimensi~n q., j = l, ••• ,L; i.e. each Mj is a set of columns
rh\ ~ J ;( : )C ~i l\ ~qji)
holomorphic near zero, such that this set is
closed under addition, and under multiplication by holomorphic functions near the origin. Then (1) Each M: has a finite basis B., i.e. for every J J Mj there exist a finite number Nj of elements of Mj such that any other element of r~j is a linear combination of these with holomorphic coefficients. (2) After a linear change of variables, \~e can find a SEquence of polydiscs D about the origin D1 =' D2 ·:;) ••• , l v 1\ D = 1OJ , and the finite bases Bj are defined v -> in D1, such that if ~ E ~1 . and is holomorphicNin some Dv, and Iff (z) II~ s~p l~i(z~l < 1 then r~ ?~1 j1{1£ ~ I ZEDV
189 ->
->
where (~ 1 , ••• ,+N) = B; and the tg are holomorphi~ in Dv and II 'If nil < J K v ~a constant depending on v). Proof. We use a double induction. First we show that if for a fixed n this theorem holds for ideals (1-dimensional modules), it holds in general. Then we use induction on n. ;J
-
Consider Mj, · j
Let ( tl )
fixed.
£
Mj, Consider the
9qJ/ ideal I 1 of the ideal r2 for 1 = 0; = ~q _1 = O.
+
all those ~ which can occur as a ~ 1 ; then of all those 9 which ·can occur as a ~ 2 etc. till Iqj of +q 1 s for 91 = 2 = ••• By hypothesis each Ik j has a finite basis
+
" jfk k .. ~k = j , •. • ,jr ~.
Then the basis Bj
1
(1~ \ I o \ k I o . I 0 1) I 12 ., (: I I i2 i 0 . \ Jj '\ ~ j ''"'\ j~J 0
I
for -~J
is
t
l? ,
13
= l, ... ,r3)
\
qj Part (2) follows similarly. Now, for n = 0 the l-1j are the finite (qj) dimensional vector spaces of all qj-tuples of constants, and the statements are obvious • Assume the theorem for n-1 and every finite number of modules. We must prove it for n and any finite number of ideals. Assume that none of the ideals Ij is identically zero, so that \'le may pick a non-ideritic~lly zero element from each. Make a linear transformation such that these elements are normalized with respect to the variable zn. Consider any one of the ideals I with element 9 ~ 0. Then 9 = ~ 0p; where 0 is a unit and p is a lleierstrass polynomial, say of degree s, p = ~ s + a 1 {Z}zs-1 + .•. + a 8 {Z), a.j(O} = 0. Assume, without loss of generality, that 0 doesn't appear, then p e I. Let t be any element of I. t is holomorphic in some closed neighborhood N of the \
I
\
+j
+
+
190 origin. In N, 1~1 ~ c, a constant. Hence ~/c = JC e I, is holomorphic 1n N, and IJI ~ 1 . there. In perhaps a smaller closed neighborhood D or the origin we may apply Theorem 56. Then J( = qp + r; q is bolomorphic in int D, lql < K, and r = b0 (z) + b1 (Z)z + ••• + b 8 _ 1 (z)zs-l 1 where·tb; bj are holomorphic in 1nt D and lb j I ~ K. Since and qp e I so does r. Consider all a-tuples
.J.
(b 0 (z)
)
/ : such that b0 + b1z + ••• + b 8 1zs-l e I. They \ bs-l(Z) form a module in the n-1 variables z. By hypothesis, this . ~~(Z) ( •v
module has a finite basis
)
v = 1 1 • • • ,m;
1
Bs-l(Z)
so that j
m bj(Z) = ~ av(Z) Bj(Z);(*) av holomorphic in int D, v~ ~ j ~-1 _E!.,_ v.
= 0,1, ••• ,s-1. Then r
=
>
j=O
bj(Z)z = /
3=0
(
> _av(Z)B]Z)) z j
V=I
and therefore
~
Hence
_.
iP
=
.r· and
=
qp
+L
=
qp
+ "v
>s-1
j;n
v j
Bjz
,
(2: avBj)zj "'v(
~ B~zj
) '
(**)
.
v = l, ••• ,mf is a finite basis
\
I. Do this for all the ideals.·. Ij. f ~g(z) \ Now for each module L. consisting of all s-tuplesl :j ) \ b s-1 J (Z 1 such that > b~ zk e I., l·le find polydiscs DvCO:n- \ s-l for
1{;0
J
with the required property {2). Take a sequence or polydiscs I n I D in ~ such that every hyperplane z = constant /)Dv is V I I .,1. ( contained in DV 1 DV +l C DV , /I , DV = { 0J 1 and the bases for the Ij are defined in D1 • Consider any ~ e Ij, say given by(**}. Then by our induction hypothesis, in(*) the II aj II~ Kv and we already have Iql ~ K. Therefore q and aj,
191 the coefficients 1n the basis expansion of t, do have norms bounded by a constant depending on v. -> Theorem 58. Let f.i be a submodule of Let ~vi be a sequence of elements of M which are all ho!~morphic in a fixed neighborhood ~~ of the ~~igin. If the ~v converge uniformly in N to ~ , then ~ e M. Proof. The ~: converge componentwise; ~~ ->·~j uniformly 1n N, j = l, ••• ,q. In any compact subset of N the ~~ are uniformly bounded; assume ~~~~ ~ 1. By Theorem 57, perhaps a smaller polydisc D, ~~ = t~
t
06 .
i:t
:e
iJe i
}j
where the are the basis vectors and the 'if/~ are holomorphic in D with ll¢,e II~ KD. H~nce) for j fixed and for each fixed .e we have a sequence (1/lp,'s of uniformly bounded holomorphic functions in D. Thus ~ 1/J~ ~ contains a subsequence which converges normally, say to '¥_e1, . Therefore for each j there isNa subs:quence of .
~j
a = ~=l~vsjv' The
( .1 \ v
=\ : \
\ form a Banach space Aj of vector-valued
~Lj I
holomorpl:.ic functions under the norm,
192
Aj is complete by Theorem 5~; Let Aj denote the subspace of Aj of relations, i.e. + e Aj if and only if s = ~ 'v sjv = 0 • Aj is a closed linear subspace of Aj, by The~rem 58. Hence we may form -~J/A~ = BJ.-> r+J • Denote an element of Bj, represen~~d by ~I· by Bj is a Banach space with norm Jl[+llj =-~nf_> max l~vl •
+ e [ cV]
v ,xj
Proposition. (Oka-Weil Theorem for sections). Let be a s·:;tion of .£.: over x3 such that fS= 2:v ~v sjv
~
and [~] e: Bj. Let e: > 0 be given. Then there exists a se~tion -r -~f over X such that II rr- -r 113 < e:, i.e. lfcr--rllj = II[}J IIJ < e , where (o-·- -r) I x3 = >--v lv sjv'
.1:
Proof. sjk:
We claim that every section
~;l~~k)
sj+l,l ,
where the JC1k) are holomorphic in
a neighborhood of K.; k = l, ••• ,L .• For, consider the J ( ,f\Lj+l J(Nj))x -> (jt(Nj))x, ma~pin~ f: ~; x e Nj, given by ( f;l
\
Ij \
Lj+l
)'->
f=IJ._e(sj+l,.e>x.
Lj+l
~Lj+l(Nj)
into ..
generate
.,
~(Nj)
r isahomomorphismof
and is onto because the sj+l,.t
( .;T (N j)) x at every x
E
Nj.
Hence
where Q : ker r, is an exact sequence of coherent sheaves. By Theorem 54B there I\ A is an an~lytic p_olyhcdron Nj such that Kj C. N 3 ~c Nj L and Hq(N3,a) = 0 for all q > o. Therefore HO(Nj, J2. j+l)->
0 -> G -> QLj+l(Nj) -> _i(Nj) -> O,
0
1\
./
H (Nj 1~) -> 0 is an exact sequence, implying that the mapping j_ ,.. is onto. Hence every section of ·L~j) is a LJ:.esj+l,.t where the Ap, are holomorphic in Nj; therefore so is each sjk' h Nou, ()= ~ sj and sjv = ~·r(v) ~~·! sj+l,.e' w ere ~v and Jl~v) are h~lom~rphic in a neighborhood of Kj. Thus
>
193 ; ?/I.e = 2:v ~v .X.lv) is ho1omorphic in a neighborhood of Kj. On Xj, the ?/I.e can be approximated by functions ?/J~l) ho1omorphic 1n a neighborhood of Kj+1 ' rf'=
2:~ 1fr.esj+l,£
I?/J2-?/11 1 ) I
max X I~ I for [ ~1 v' J v y- ~v sJ.V = s and II s llj+l = inf -> maxv X ltv I. for [1/1] , j+1
1,,, ( 2 ) '~',t
~ ?'' (1 ) ·;r ( .e ) I
- L--
T"
£
J-
Take any representation of the in terms of the
-~ J-(v)
s j -1:1, v - 2-I.L
->
->
over all representatives
'if!
I.L
E
[1/1 ] ,
s =
s jJ..L •
sj+l,v
Then
2: ~(!.,sj+l,v'
194 Since v depends only on
j,
say v
II s I j ~
= l, ••• ,Lj+l; X j, IJ.(~ ) I
Lj +1 inf max I~v II X.J v ) I • On !J.,v,Xj bounded by a constant bj depending only on
j.
is
~herefore
lslj ~ bj Lj+l
1~! max I~vI ~ b jL j+lll s llj+l. This [~] :V,Xj proves the proposition. To complete the proof or Theorem A, we must show that -·l every stalk ~x' x e X, can be generated by global sections. So, let ~ e X. Then x lies in some Xj. The sections sj 1 , ••• ,sjL or Xj generate the stalk at every point of Xj. By t~e proposition, there are global sections tj 1 , ••• ,tjL such that on Xj j L t jk =
( *)
f=~
(
Bkt
+ 1/tkt ) s jt
'
where the tke are holomorphic in Xj and l~ktl < e there. For e sufficiently small, the transformation (*) is nonsingular, so that we may solve for the sjt in terms of the tjk" Hence the tjk generate ~·
§4.
Proof of Theorem B
First we define the tensor product sheaf -j- ® 0°' q over X, where 0°'q is the sheaf of germs of differential forms of type (O,q). This sheaf is called the sheaf of germs of differential forms of type (O,q) with values in the sheaf j-. Let x e X. Both .J:X and 0°'q. are r0x modules. X · Consider al~ finite sum~ (~jfj~tjllj), for ~j' 1/Jj e c.9x and f j E j-x , ID j E Ox' q. Define addition of t\110 sums in the natural way, -
>
:;= aj + N
M
I
~ aj =
I
f
a 1 + ••• +aN +a 1 + ••• + aM.
AllOt'/
interchanges of the order of terms of a sum and drop any term w:tth ~j' fj, "if/j or IDj equal to zero. Identify the terms (~jf,fJ\ljjiDj) and (fj®~i/JJIDj). Then these finite
195 sums modulo the identification form an Abelian group Gx = J'$-o~,q· Under the follO\'ling topology, we obtain X ..., 0 q the sheaf :r~ 0 ' : Take a representative of any element _g e Gx , ~(~Jfj~V!Jcoj). In a sufficiently small neighborhood Nx of x, the ~j and V!j are holomorphic functions, the f j are sections of .;i: over Nx, the co j are differential forms, and the projection maps of the sheaves ::1- and 0°'q are homeomorphisms. Then, for each y E Nx, .
a~sign tha~ class in
(~
l.t
I~
I'))
Gy" ~ ('f1Jfjtt9V!fj 1 for which +j and ~J are the 1direct a~alytic continuation~ or +j and "J' and the fj and coj are sections of ~ and Oo,q through rj and coj, respectively. We define the collection of all these classes to be an open set; and these open sets are to form a basis for the topology. _
N~l" ~efine ~ ( 2:_ (~Jf l~~ Jcoj)) = 2: (+Jr J®~ imJ). Then ~: 4 ® 0° 1 q -> .:; ® oo, q+1 is a homomorphism of the
sheaves; ~d ~ 2 = 0. Since Hq(x,Q) = 0 for all q > O, (cf. Theorem 36, p. 117), by Dolbeault's theorem (Theorem 26B, / p. 95) we have the Poincare lemma with respect to in X. i ·tr_ Ao "d · ~J Cl o 1 o ·-:1..r.:\ o 2 j Hence the sequence .2 ->_->~iS 0 - > L 0' 0 ' ->.£_r,b0 ' -> • • • is exact. For, at J-@ 0°' 0 ker ~ is :Jr~J CO, and else\'lhere exactness follo~IS from the Poincare lermna. This sequence is a resolution of ' (~jfj~ 1/lrj)) = ~ (~jfj~) V'j(aiDj)),; and then proceed as in the example on p. 152. - Then by the Abstract de Rham Theorem (p. 153), Hq(X~J.:!: (~ closed (O,q) forms with values in:/-)/(~ exact (O,q) forms with values in :f). Now exhaust X by analytic polyhedra Xj. Tben for every j, Hq(X ., )-) = 0 for all q > 0 by Theorem 54B. As in the proof J -that Hq(X, 0) = 0 for all q > 0 (Theorem ;6), we obtain a lemma ~ for-;he sheaf ~~ and hence Hq(X,~) = 0 for all q>O.
-
196 §5.
Applications of the Fundamental Theorems
The following results are all obtained relatively easily from the fundamental theorems A and B. Some of these results have been obtained previously, with more effort. Note. In the following, X is a region of Molomorphy and ~ a coherent analytic sheaf over X. A. Theorem 59a. Let V be an analytic set in X; i.e. V C X, and every .point p e V has a neighborhood Np such that X ~NP is the set of common zeroes of a finite number of functions defined and holomorphic in Np. Then V = fx eX I fi(x) = 0 for every i E where the· fi are fUnctions holomorphic in X and I is some index set. We cannot prove this theorem, since it relies on the fact that the sheaf ~V(X) is coherent (Theorem 50). However, we can establish: Theorem 59. If V is a regularly imbedded analytic subvariety in X, then there are ~nctions fi' i E I, holomorphic in X, such that V = 1 x EX I fi(x) = 0 for 7 every i E IJ. Proof. ~V(X) is coherent by Theorem 51. Hence, by Theorem 55A, the global sections of ~V(X) generate the stalk at every point. For any point p E X - V, the stalk (;j V(X) )P contains the germ "1" j hence "1" is a linear combination of functions holomorphic on X and vanishing on V (with appropriate coefficients). Hence at least one function does not va~ish at pj hence there is a function holomorphic in X, = 0 on V and ~ 0 at p. Theorem 60. Oka 1 s Fundamental Lemma; a general form. Let V be a regularly imbedded subvariety in X; P the closure of a polynomial polyhedron in X. Then cl (V(\ P) = (cl (VAP)) * , its polynomial hull. Pro~f. cl (V f) P) C P which is defined by polynomial inequalities. Hence there exists a polynomial polyhedron P' such that PC(. P •c:·c X and JV(P 1 ) is globally finitely
rJ
197
generated (applying Theorem A to the coherent sheaf ~v{X)). At any point p e P - V, {Jv
= o. But C) is coherent, so Theorem B applies. Theorem 62. In X, every ~-closed form is ~-exact. Proof. Appeal to the Dolbeault isomorphism theorem and Theorem B. Theorem 6;. Suppose there exist finitely many local -1 sections s 1 , ••• ,sr generating all the ~~x' x eX. Then every global section s is of the form s = ~ ~jsj where the are holomorphic in X. Proof. Consider the sheaf homomorphism (Q r(X) -> ~
'j
defined by: hypothesis;
.+1 (:
~r x
-
->
2::
~i ( si) x
-
• This map is onto by .
hence we may form the exact sequence:
o
->
Q
->
~r
->
.J-
->
.Q
•
G is also coherent; hence we obtain the exact cohomology sequence: H0 (x, H0 (x,E-) -> H1 (X,G) • Now H1 (X,G) = 0 by Theorem B; and since H0 (X,~r), H0 (x,j_) -;:re the global sections in ;J_r, respectively; Theorem 63 is complete. Corollary 1. Let Ucc X, U open. Then there exist finitely many global sections s 1 , ••• , sr of such that every section s of' l_(u) is of the form s = L ~jsj, where the ~j are holomorphic in u.
J:.
,-1
Proof. By Theorem 6:;, it is enough to shou that there exist a finite number of global sections of ~ generating the stalks of ~(U) at. every point. ~t this is Theorem A. Corollary 2. Let D c· ~n, D open. Then the following are equivalent: i) D is a region of holomorphy ii) lfuenever ~l, ••• , ~r are holomorphic functions in D without common zeroes, there exist holomorphic functions t 1 , •• ,'/fr in D such that 2: ~ j'/1 j ;; 1. Proof. i) implies ii). View the ~i as global sections of the sheaf ~(D). It is thus enough to show that they generate tbe stalks {J.... at every point x E D, as "1" is a global section and Theorem 63 applies. But this is just the hypothesis of ii). ii) implies i). If D has no boundary points, D = ~n and so is a region of holomorphy. Hence, assume D has boundary points; we shall show that every such point is essential. Let "a" E bdry D; a = (a1 , ••• ,an). Consider the n holomorphic functions ~. = zj-a .• They have a common zero at the point a, J . J only; hence they have no common zero in D. Hence there exist '/fj, holomorphic in D, such that 2: V!j(z) (zj-a .) 1 in D. If the ?{lj are all holomorphic in a neighborhood Y.a", ( 2:: V!j(z)(zj-aj))a = 1. But this is clearly a contradiction, so at least one of the ?frj is singular at "a". B. Recall that, in a region of holomorphy, vJe have an extension theorem for functions defined on regularly imbedded, globally presented hypersurfaces. This theorem extends as follows (X still denotes a region of holomorphy): Theorem 64. Let Yc X be a regularly imbedded subvariety. Then every function holomorphic on Y is the restriction of a function holomorphic on X. Proof. Consider the sheaf Jy(X). lrJe may form the exact sequence: A
=
->
~{X)
->
SJ - ;J.y
->
0 '
199
J
where Q; y = Q (Y) ( cf. Remark p. 174). have the exact cohomology sequence:
H0 (X,~) and
->
H0 (Y,i2)
->
t:le therefore
H 1 (X,~y)
H1 (x,~yl = 0 by Theorem B. Theorem 65. Let points z j e
-, x, f z j) discrete, be given together with numbers aj. Then there exists a function ~~ holomorphic ~n X, such that ~(zj) = aj. Proof. z j ~ is a regularly imbedded subvariety, of dimension zero. Theorem 66. lvith the zj as above, let polynomials Pj(z), or degree Nj, be given. Then there exists a function ~~ holomorphic in X, such that in some neighborhood N.+l or zj, ~(z) = Pj(z) + O(llz~ J ); i.e. has any given Taylor expansion up to any given order. Proof. Consider the sheaf ~ 1 defined by its stalks as follows. If for x e X, xI= zj set = (2 X • If ::1_ r X for x e X, x = z j set :;-x = germs of functions l'lhose Taylor expansions about zj have no terms of order< N.; . J i~e. which vanish at zj of order at least Nj+lj( • .:}- is an open subsheaf of (X). For coherence, l'le must ---~ show that ~ is locally finitely generated. But, for points x I= zj, this is clear; and at zj the stalks are generated by the polynomials in z-zj of degree Nj+l. Hence, we may form the exact sequence:
!
+
J
i
CJ-
-
o
->
J-
->
(9
->
C) I
f._
->
o
and therefore the exact cohomology sequence: H0 (x,t..l)
by Theorem B.
->
H0 (X,
;9/.f.)
->
But (o
o
(fJ/j)x = J l germs of polynomials of degree oo; contradicting ( zn) e "K. Condition 2 is trivial, as is Condition 3 once it is observed that every point of X has local coordinates such that zn+l = • •• = zN = 0 describe X. iii) .If X is a Stein manifold, and r is holomorphic on X, then the set x I f{x) -1 0~ is also a Stein manifold. iv) The product of two Stein manifolds is also one. He leave it to the reader to verify that iii) and iv) are Stein manifolds, while stating the following theorems (without pl''Oof). ~
t
202
Theorem 68. (Stein) Every universal covering space or a Stein manifold is again a Stein manifold. Theorem 69. (Behnke-Stein) Every open Riemann surface is a Stein manifold. vie remark that there exist manifolds which are not Stein man1folds; that conditions 2 and 3 can be replaced by a ''K-completeness" condition. (A complex manifold X is K··complete if for every x E X there exist finitely many functions f 1 , ••• ,rK holom~rphic on X such that x is a~ isolated point of the set (y EX I f 1 y = f 1x, ••• ,fKy = fKxj.), and that: Theorem 70. (Grauert) Conditions 1, 2, and 3 imply condition 0.
§2.
An approximation theorem
Definition 78. An analxt~c polyhedron Y in a complex manifold X is defined as follows: Y (f ~z), ••• ,fN(z)) is one to one, of maximal ranlc, of Y into ll~jl < 1, j = 1, ••• ,~IJ.. The image of Y in the disc is a regularly imbedded analytic subvariety of the disc. Therefore by Theorem 64, g can be extended to a function G holomorpbic in the disc. Hence in the disc G = ci i ~ 1 1 ••• ~NN and this series l' • • N converges normally. But, setting ~i = fi(z 1 , ••• ,zn), vle
1
2::
203
obtain the desired normally convergent expansion.
§;.
The fundamental theorems· for Stein manifolds
Theorem 72. Theorems A and B hold for Stein manifolds. Corollary. All consequences of these theorems, except Corollary 2, hold also. In particular, we have the complex de Rham theorem: I
~) closed holomorphic q-forms • ,. ~ exact fiolomorphic q-rorms Note that this result shows also that the cohomology of differential forms on any Stein manifold is trivial. These statements need no proof!
Hq(X
§4.
Characterization of Stein manifolds
Theorem 73. Let X be a manifold satisfying condition 0. Then the follo\·ling are equivalent: i) X is Stein. ii) H1 (x, :f:.) = 0 for every coherent sheaf of ideals ~i i.e. for every coherent subsheaf of (9 . Proof. i) implies ii): Theorem B. ii.) implies i) : Recall the corollaries of theorems A and B: Given a discrete sequence of points, there exists a function taking prescribed values. This implies holomorphic convexity and separation of points. At every point there exists a function with a prescribed expansion in terms of local coordinates. This ii~lplies the cxio"Cencc of' local coordin:J:~c::.: ~1l1ich
are holomorphic functions. Now recall that the proof of these corollaries required only Theorem Bin the form of 11). Theorem 74. (Grauert-Harasimhan) Let X be a complex manifold satisfying condition 0. Then the follol'ring are equivalent:
204
i) X is Stein. ii) There exists a strongly plurisubharmonic realvalued function ' on X such that {' < aJccx, for every a • Proof of this theorem is essentially that of the solution • to the Levi problem, and will not be given here. Theorem 75· (Bishop; Narasimhan) Let X be a complex manifold of dimension n. Then the following are equivalent: i) X is holomorphically equivalent to a regularly imbedded subvariety or e2n+l. ii) X is Stein. Note that this gives an imbedding theorem for regions or hol--omorphy. Proof. i) implies ii). Clear by the examples. ii) llnplies i) will not be proved here. One must find 2n+l functions such that the mapping defined by them is one to one, of maximal rank, and "proper'' in that the inverse image of a compact set is compact. Ue do not establish this, but make the following remarks: This mapping is not unique. However, in the space of all holomorphic maps X -> c2n+l , under the topology of normal convergence, the fUnctions of 1) are dense.
205
appendix This appendix is concerned with proving the theorem of L. Schwartz
ap~earing
in chapter 13 on
139. We actually
pa~e
prove a weaker theorem than is stated there, but one which nonetheless suffices for our purposes. We assume that E and F are vector spaces, each having a nested sequence
!"I '
!'In ' .
of norms defined on it;
!!
II II n +1
, n = 1, 2, • • • •
Furthermore, E and F are metric spaces with metric
• Under the topology induced by each metric, we assume that E and F are complete and separable, property that fer each n, {x f E bounded (relatively compact) The theorem He are Theorem (L.
l.Ji
~oing
Scbt-rart.~).
alld
that E has the r.dded
r II X lln+l ~
th respect
1} is totally
to the norm
II II n.
to prove is the follovdng: If A and B are continuous linear
mappine-s of E into F and A is onto and B is compact, then
F/(A+B)E is finite dimensional.
206
Before proceeding with the proof,
= C0 (D·, U @')~Z 1 (D0 , U", {f)
E
1,
and F
note that if
~e
= z1 (D, U1 , {j), then
E and F
have the properties assumeJ. above, for: The nestej sequence of norms on each space is defined as follows. Kij1C~
First, in each
1 c Ki,j+l /~ u1 1 .
u 1 1 ~U 1
take a sequence of subsets
I n eac h uj, ""1 uk " tr
It
¢ or s e ts o.~.r-- u" take a
1 F
...,
sequence of subsets Kjkl c c KJk,J+l / (u/'{)uk"); and in each u. 1 nu. 1 /:¢of sets of U1 ta:.e a sequence of subsets J
l
I
I
KiJkc C.Kij,k+l
/'
"\
(ui 1 / !UJ 1 ) .
, Then for J C: E; i.e. .f = g + h,gE; C0
aSSiO.:nS the hOlOmOri)hiC fUnCtion g.l tO U.l ~
I
and hE_ z1 assigns
the holomorphic function hjk to uj"/\uk" /: ¢; define
,•
=max i·zG K ' in
Clearly these are norms and
II
lin ~
II II
The separability of E and F is obvious.
n+l for all n
= l,
2, ••••
Finally, the totally boundedness of {xE::E in the norm
II II n
I II xll n+l ~ lJ
follows from the fact that a uniformly
bounde1 sequence o ~. . holomorphic functions contains a normally . convergent subsequenqe. I.
Preliminaries. Hencerorth x and y shall 1enote elements of E and F,
respectively.
(e and f shall 1enote elements of the dual spaces).
Let sn dS n
so n
= LX t: E = {xE E = ~X €: E
Note.
'-
llxlln
I II xlln I II x lin
~ 1},
Ln
= ·[yt:
F ,. 11 Yltn
~ ~~
I II Ylln I II Ylln
= ~},
.,
= lJ, )
-12 for k 'f j. The proof is based on the followini. J
Lemma.
p -
If E is a Banach space and G is a closed, linear,
proper subset of a linear set DeE, then there is an x 0 ED with
II x 0 II
-
II x 0
= 1 and
1
GII ~ 2•
I
Proof. I
II
there is a y E: G with d ~
and let Di 2 =
..
~
1_a1 , a 2~
•
= diatance
I
to G. Then I I I I X -y x - y II ~ 2d. Set X • 0 tlx 1 -y' II the linear space spanned by ~~
Take x ' D - G and let d
of x
Since every finite dimensional linear
subspace of a Banach space is closed, we may apply the above len,ma to Gl and Dl2 as subsets of A-:~ (N) with norm exists e 1 ~ D12 with
II e1 11; = 1
Next, apply the lemma to G12 get e 2 ( D123 with
II e 2 11;
and
II e 1
= {a1 , a2}
= 1 and
II e 2
u;.
II
~~for all e E G1 •
-ell;
= {~, a 2 , a3f
and n123 - e·H;
"
spanned by
and
II ek
~
- e II~
,::
1
and
~ ~ for all e f. a12 •
Continue this process, obtaining a sequence {ek·~
II ek 11; = 1
Hence, there
t
A~~'(N) with
..
2 for all e belonging to the space
~, ••• , ~; but then II ek - e j ~~~- ~ ~ for k 'f
On the other hand, since -[ek~~A-:t-(N), ek = AJ.~ofk' fk E: N.
j.
212
By Proposition A, there are
{eiJ
mp and CP such that
Corollary 2, (B*fk) contains a subsequence
II fk II"~ ~ CP. Then by which ~ is Cauchy in the and
consta~ts
norm
II
*
*
However, B*fk = .~A fk = -ek,
liP •
can have no Cauchy subsequence in the norm
II I~
by
construction. Theorem 2.
Given n, there exist mn and Kn < oo such that if
e '"(A~~+B.;~)F.;~ and and II f
II: ~ n
Proof.
II e II~
Kn II e
< oo, then there is an f€ F* ·t-rith e = (A:~+B*)r
II~ •
Let p be as in Proposition B, and let mn be given by
Proposition A.
It suffices to consider only n
have established the theorem for n
=p
of {A.;~ +B.;:-), as before.
ll.;~+l n
< ·II -
ll.;t- • n
p;
since once we
we have it at once for all
n < p by choosing· for such n, the constants mn and recalling that II
~
=mp
and Kn
= Kp ,
Let N denote the null space
~ie claim that it is sufficient to prove ~
,,
.,
.,
that there is a Kn < oosuch that for all fE F"'' 1rlith II(A"'"'+B'.. )fll~ <en, * Kn .. .. .~. .. .. It f - Nlllrb ~ 2ll (A"''"+B-''")fll~. Indeed, if (A"'"'+B.,..)f = e, then there will exist a ~t-- f - N satisfying II~~~~!- < K II (A:~+B:~)fll{:- = K llell* m - n n n n ... ,,. " •t. n and (A·.r+B''" )~ = (A·.r+B"'' )f = e.
213 By Proposition B, a subsequence of the Vi' call them again
o,
satisfies IIB*(wi-fj>ll; i,,
and therefore
with .,
II ' i
"'u: r
-
v
=
((A'~+a·,;-)'IJ )x
II 'ti o,
1
I:t o.
O, implying
11: r,1 o. Henge there is a 'It~ F* II"' - nNII* ! ~· Then t t N, lvhile
and
~
~
t 1 ( (A+B)x) =::
,, ... lim ( (A''"+s''" )\lfi )x = 0
1-+ro N, a contradietion.
i~oo
III.
1j'
"l
- vj
n t( (A+B)x) = lim
for all x E E means that "'
,·
IIB*('lti-wj)ll~
Thenbythe triangle inequality, IIA.~~(\Ifi'•tj)ll~ by Proposition A that
iVi1
Main Results. Let YoE (A+B)E, and let M ={r
fF~~
I f(yo)
=
o}.
Call
(A~t-+B* )M = L.
Lemma 1.
Suppose e 1 € L,
and sup:pose ei (x)
y+ e(x) for all
x
~ C for some n and C
EE.
< oo,
Then e €- L.
By Theorem 2, we can find f 1 E F~~ with e 1 = (A-:~ +B"'.t-) f i
Proof. and
II ei II~
II f.ll~~ l. m
< K. n _
~le claim that fit H. ~
for l-lhich e 1 = (A-""+B'
)t 1 •
Set
w1 =
Indeed, there are ~-£ N
.. " fi - ~i' then (A''"+B'~)\jti
l.
= O.
Since y o E (A+B)E, there is a sequence 4x ~ ~ E such that .. a (A+B)x ~ y • Hence ti (y ) =lim vi( (A+B)x ~ =lim (L\~~+B-::}~.. ~=0, a a o o ~ 00 a a-;.oo J. implying that \It i E lJf, but then f i = \If i + 'if M. Now, because the
tf i}
is uniformly b0unded in the norm
it follot-rs that a subsequence of the
{r~,
call them again
II II~ ,
{ri~,
converges at every point y ofF, fi (y)--;:. f(y), f =:-F~~ and f(y 0
)
so that f € 11. But then, ( (A-:~ +B·::- )f )x = f ( (A+B )x} = lim f i ( (A +B )x) .. ., i~oo lim ( (A';-+B-·")fi )x lim ei (x) = e (x) for all xE E. Thus i~oo.. .. i-+oo e = (A"+B"';-)f! L.
=
=
n
=0
214
Before proceeding, He recall some notions from functional The Hahn-Banach Theorem states that if S is a linear
analysis.
subspace of T, a Hausdorff locally convex topological vector space, and if x{ (j, an open convex subset of T such thatcf.!ls = ~~ then there is a closed hyperplane H:JS with HncJ= ~.
= H +' ~x~,
theorem it follows that since T
(,.
~
From this
i.e. H + the linear
space spanned by x, if we define fer t f T, ~( t) = 'A where t = h + 'Ax, then ~ is a continuous linear functional on T whose nullspace is H. Hence there exists a continuous linear functional ~ on T satisfying ~(x)
=1
and ~(S) = 0.
Theorem
=1
that e(x ) 0
Proof. If n
= 1,
3.
If e E- E.:~ but e
t L,
= 0 for
while g(x ) 0
then there is an x 0
,
every g tL.
(1) Since e E E.::·, there is an n for Hhich
Therefore !!ell~:.< oo for some n > 2. we will assume
-
II e II~
II e II~
< oo.
II II~~ II l!~,llell~
i.e. llell{!- < oo, then, since n
E E such
< ro.
In order to simplify notation
< oo : the proof in the general case is
(essentially) the same. (2) then
There is an
r\
>
0 such that if gE::L and llg- ell~
tt_. such~
Proof. ·Assume that no (gj )fL satisfying llgj-
every xE E, Jgj(x) - e(x)l llgj II~ ~ 1 +
II e II~
exists.
Then there is a sequence·
ell~~"~ land llgj- ell~ j
j
O. But
II
gj -ell; ~
O, i.e. for 1 implies
< oo so that by Lemma l, e f: L; a contradiction.
215 Take t1_ smaller so that 2 Yl_ ~
(3)
II g- ell~ ~ 2rt•·
llg-
1, then if g{ L and
ell~ >'1.·
as1 •
(4) Let {x1 (l)} be a dense sequence in exists since
as1 C
E a separable metric space.
II g
an. integer N1 > 0 such that if gEL and
Such a sequence
Then there exists
- e II~ ~ 2 yt the follow•
\
ing inequalities cannot hold simultaneously
Proof.
Assume the contradiction, then there is a sequence
(gN)t L trJith 1
= 1,
II~- ell~~ 2yt
~rLfor
ell~~ 2f1. implies that II gNII~ ~ 2rt_+ llell; < oo
llgN-
••• , N.
and lgN(x 1 (l))- e(xi(l))l
so that there is a subsequence, call it again (p.N), such that at every xE E, gH(x) 1r> g(x), g t L by Lemma 1, and lg(xi (l)) - e(xi (l))
I·~ 'l for
i
= 1,
2, ••••
llg-
e II~ ~ 2 rt.'
Hence }g(x) -e(x)j
~ tl,
for xt ~s 1 , which implies that llg- e II~ ~rv contradicting (3). (5)
.
Let {xi
(2)7
J
be a dense sequence in
exists an N2 such that if g' L and llg- e
os2 •
11; ~ 3y
Then there
the following
inequalities cannot hold simultaneously
i = 1 i
Proof:
= 1,
=1,
Assume the contradiction, then there is a sequence
(gN) E L with
for i
,
II gN
••• , N1
- e II; j
~ 3rL
and lgN(x1 (l)) - e(x1 ( 1 )) I
lgN(x 1 ( 2 )) - e(x 1 ( 2 )) I
~ 2fl
for 1
~ rL
= 1,
••• , N2 •
216
II gl'J - e II;~
3rt_ implies that llgN
11; ~ 3f'L + lie II;~ 3yt + lie II; < oo
so that there is a subsequence of the (gN) conver.ging at every point of E to a gtL satisfying lg(x.(l))- e(x_(l))l I~
ytfor i
= 1,
g€ L,
. (4) ••• , N1 ,• contradict1ng •
* ~k Continue this process: hence if g ~ L and llg - e Ilk
rt
then the following inequalities cannot hold simultaneously
Ig {x.1 (s) )
- e(x. (s) ) I < s 1
-
rt,,
i
.
= 1,
•. . , Ns , s = 1, ... , k- 1.
x(l) x(l) { (7) Let {an} denote the sequence ~ , ~ , x(l) (2) (2) x( 2 ) (3) .., Nl xl x2 N2 xl ] 27l' 2V(_' ••• , 2rt' 3yt' .•. EE. Since
. x(k)
\I ·
and
II
lin~
II
lln+l' for any fixed k,
II
anllk
7
Jl
1
~'l,ik = k~
... , tl'
0, i.e. an ~0
in E. (8)
II
* +l ~ Ilk
* IfgE:Lthenforsomek, llg- ellk 1 by ( 8 ) • 1 gEP i -: there is a continuous linear functional on 'f._ 0 \-Tbich is 0
subspace of Th~refore
,. 0 where vE: )L , v --
II P
I! 0 L •
on P and 1 at q.
This means that there is a sequence of complex
numbers ak l'lith [:lakl < oo such-that if gf: L then [::akg('1c) = 0
= 1.
and Cake(~)
From (7) and Clakl < oo, we see that the sequence of
(10)
partial sums an x 0
EE.
of r=:ak'1c is Cauchy in E and therefore converges to
Then e (x 0
)
= 1, lvhile g(x 0 )
=0
for every gt Lj
completing the proof. Lemma 2.
(A+B)E is closed. Let y 0 f:. (A+B)E.
Proof.
so we may as well assume y 0 f (y ) o o
= 1.,
so that f
f
M.
o·
If y 0
F 0.
=0
then we know y 0 E (A+B)E,
.
Then there is an f 0 f p'H' such that
Set e
= (A*+B-:~)f o'
o
e
J L:
o.,..
for if
= {A~:-+B~~ )' 0 , ~ 0 E· M, so that if w0 = f 0 - ~ 0 , then = o, but wo (y o ) = lim w ( {A+B)x ) = lim ( (A;:-+B*)w )x = o a o a o a
e0 E L. then e , 0
(A~:-+B.;~)w
o
~~
implies ''''~'o f L and hence f o E L. -
there exists an x f E such that e (x ) 0
gEL.
0
Let fE F~:-, then f - f(y 0 )f 0
then e - f(y )e 0
0
E L,
0
0
= (A+B)x
0
E
t
M.
=1
=0
and g{x ) 0
Therefcre. if e 0
&
e{x ) = f(y ), i.e. { (A"+3" )f)x Hence y
0
for every
= (A.;;.+B.;~)f,
which means t~at e(x ) - f(y )e (x ) = 0, or &
0
¢.
a
Since e o t E" but e o - L, by '11heorern 3
(A+B)E.
0
0
0
= f( (A+B)x 0 ) = f{y 0 )
0
A
fer all fE: F". .
218
Theorem
4.
Proof.
Let K = (A+B)E. By Lemma 2, K is a closed subspace of F,
F/(A+B)E is finite dimensional.
If F/K is infinite dimensional, then there ~s a sequence of linearly independent y i ~ F, so that K0
=K, K1 =K0 + {y1~, K2 =K1
+ {12}, •• , are
closed subspaces ofF satisfying K1 properly contained in Ki+l' For each i, then, there is a pi such that IIKi- Yi+liiPi > 0. Hence we can find continuous linear ftL."'lctionals fiEF{~ vlith fi (Ki) =0 and fi(yi+l) = 1. i, f i (K)
= o.
The fi a~e linearly independent and for every
Therefore for every x EE, ( (A-:~o +B-:1- )f i )x =r1 {{A+B)x)
so that f.c N, and N therefore is infinite dimensional. I
l
contradicts Theorem 1 and completes the proof.
This
=0
1
