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0 on 8G) and '1/Ji 'Pi/'P· Then I:i 'l/Ji l on 8G. The system of the functions '1/Ji is called a partation of unity on 8G. The function g := 2::: �- I '1/Ji(!i is now a global defining function for G. We :=
leave it to the reader to check the details.
•
The Levi C ondition. For the remainder of this section let G c c en be a bounded domain with smooth boundary, and g : U = U(8G) -+ lR a global defining function. Then at any Zo E ac the real tangent space of the boundary is a
(2n -
I)-dimensional real subspace of Tz0 . The space
is called the complex (or holomorphic) tangent space of the boundary at zo. It is a (2n - 2)-dimensional real subspace of 7�0 , with a natural complex structure, so an ( n - 1 )-dimensional complex subspace3 . Definition. The domain G is said to satisfy the Levi condition (respec tively the strict Levi condition) at zo E 8G if Lev(g) is positive semidef init-e (respectively positive definite) on Hz 0(8G) . The domain G is called Levi convex (respectively strictly Levi convex) if G satisfies the Levi con dition (respectively the strict Levi condition) at every point z E 8G. Remark. The Levi conditions do not depend on the choice of the boundary function, and they are invariant under biholomorphic transformations. If
{! 1 = h . {!2,
with h > 0, then for
Z
E 80,
Lev(gi )(z , w ) = h (z) . Lev(g2) (z , w ) + 2 Re{ (8h)z ( w ) . (8g2)z (w ) } . So on Hz (8G) the Levi forms of g1 and g2 differ only by a positive constant. Aftme C onvexity. Recall some facts from real analysis:
A set M c JRn is convex if for every two points x , y E M , the closed line segment from x to y is contained in M . In that case, for each point xo E JRn - M there is a real hyperplane H c JRn with x0 E H and M n H = o . This property was already used in Section 1. 3
Hz (8G)
is often denoted by Tl •0 (8G).
4.
If a
E �n , U
=
Levi Convex Boundaries
67
U (a) is an open neighborhood and r.p U --+ � is at least �2 , :
then the quadratic form
is known as the Hessian of r.p at 4.3
a.
Proposition. Let G CC �n be a domain with smooth boundary, and {! a global defining function with ( dg) x -=!=- 0 fo r X E ac. Then G is convex if and only if Hess(g) is positive semidefinite on every tangent space Tx (aC) . Let G be convex, and Xo E ac an arbitrary point. Then T : = Tx0 (aC) is a real hyperplane with T n C = 0. For w E T and a(t) :=xo + tw we have (g o a) " ( O ) = Hess(g) (xo , w) .
PROOF:
Since g(xo) = 0 and g o a(t) > 0, it follows that g o a has a minimum at t = 0. Then (g o a)" (O) > 0, and Hess(g) is positive semidefinite on T. Now let the criterion be fulfilled, assume that 0 E G, and define g6 by
For small E and large N the set G, := {x : g6(x) < 0} is a domain. We have G, C C6 C G for E' < c:, and Uc;> O C6 = G. Therefore, it is sufficient to show that G, is convex. ,
The Hessian of {!c; is positive definite on Tx (ac) for every X E ac. Thus this also holds in a neighborhood U of aC. If E is small enough, then aC6 C U. We consider S := { (x, y) E C6
x
G, : txt ( 1 - t ) y E G, for O < t < 1 ) .
Then S is an open subset cf the connected set G, x G,. Suppose that S is not a closed subset. Then there exist points x0 , y0 E G, and a to E (0, 1) with toxo+ ( 1 - to)Yo E ace; . So the function t H {!c; 0 a(t) , with a (t) :=txo + ( 1 - t)yo , has amaximum at to. Then (g6 oa) " (t0) < 0 and Hess(g6) (a(t0 ) , x0 - y0) < 0. This is a contradiction. •
A domain G = {g < 0) is called strictly convex at xo E ac if Hess(g) is positive definite at x0. This property is independent of g and invariant under affine transformations. Now we return to Levi convexity. 4.4
Lemma.
Let U C en be open and r.p E � 2 (U; �) . Then
Lev( r.p) (z, w)
=
1
4
(Hess( r.p ) (z , w ) + Hess( r.p ) (z, iw ))
68
II.
Domains
oE
Holomorphy
This is a simple calculation!
PROOF:
•
4.5
Theor em. Let G c c en be a dom ain with smooth boundary. Then the following statements are equivalent: 1.
G is strictly T�evi convex. 2. The re is an open neighborhood U = U( EJG) and a strictly plurisubhar monic function g E 'l!f= (u; JR) such that U n G = {z E U : g(z) < 0) and (dg)z -=/:- 0 fo r z E U. 3. For eve ry z E EJG there is an open neighborhood W = W(z) C en , an open set V c en, and a biholomorphic map F : W --+ V such that F(WnG) is convex and even strictly convex at every point of F(W n EJG) . PROOF:
( 2) : We choose a global defining function g for G, and an open neighborhood U = U ( EJG) such that g is defined on U with (dg)z -=1- 0 for z E U. Let A > 0 be a real constant, and {!A : = eA e - 1. Then {!A is also a (1 )
===?
global defining function, and
Lev(gA)(z , w) The set K : = EJG
X
= Ae
A e (z ) [Lev(g) (z, w) + A l (8g)z (w) l 2 ]
•
S2n- l is compact, and
K0 : = {(z, w) E K : Lev(g) (z, w) < 0 } is a closed subset. Since Lev(g) is positive definite on Hz ( EJG) , we have ( ag)z (w) -=1- 0 for (z,w) E K0 . Therefore,
min Lev(g) (z , w) > - oo ,
M
K
min l (8g)z (w) l 2 > 0 .
C
Ko
We choose A so large that A ·
C +M
> 0. Then
Lev(gA )(z , w) = A · [Lev (g) (z, w) + A l (8g)z(w )l 2 ] > A · (M + AC) > 0 for (z, w)
E
K0 , and
Lev (gA)(z , w) > A 2 · 1 (8g)z(w)l 2 > 0 for (z, w) E K - Ko. So Lev(gA)(z, w) > 0 for every z E EJG and every w E en - {0}. By conti nuity, {!A is strictly plurisubharmonic in a neighborhood of EJG. (2 )
===?
( 3 ) : We consider
transformations:
a
point zo E EJG and make some simple coordinate
4.
Levi Convex Boundaries
69
By the translation z H w = z zo we replace zo by the origin, and a permutation of coordinates ensures that (!w1 (0) -=/=- 0. -
The linear transformation gives u1 = w . V' g(O) t, and therefore g( u )
-
2 Re (u . V'(g o w) (O) t ) + term sofdegree > 2 2 Re (u . Jw (O) t . V' g(O) t ) + terms of degree > 2
2 Re (w . V' g(O) t ) + terms of degree > 2 - 2 Re( u1) + terms of degree > 2. -
Finally, we write g( u) = 2Re(u1 + Q(u)) + Lev(g) (O, u) + . . . , where Q is a quadratic holomorphic polynomial, and make the biholomorphic transforma tion It
follows that
+ Lev(g) (O, v) + terms of order
g(v) = 2Re(v1)
> 3.
By the uniqueness of the Taylor expansion g(v)
=
Do(O) (v)
+ �Hess(g) (O, v) + terms of order
> 3,
and therefore Hess(g)(O, v) = 2 . Lev(g) (O, v) > 0 for v -=/=- 0 (in the new coordinates). Everything works in a neighborhood that may be chosen to be convex. (3)
===?
( 1 ) : This follows from Lemma 4.4: Hess(g) > 0 on Tz (8G)
===?
Lev(g) > 0 on Hz (8G) .
The latter property is invariant under biholomorphic transformations.
•
A Theorem of Levi. Let G cc en be a domain with smooth boundary. If G is strictly Levi convex, then it is easy to see that G is pseudoconvex.
We wish to demonstrate that even the weaker Levi convexity is equivalent to pseudoconvexity. For that purpose we extend the boundary distance to a function on en. dc(z)
:=
8c(z) 0 -8cn _c (z)
for z E G, for z E 8G, for z !f. G.
70
II.
Domains of Holomorphy
4.6 Lemma.
-de is
a
smooth defining function fo r G
PROOF: We use real coordinates x = ( x 1 , . . . , X N ) with N = 2n. It is clear that G = { x : -de( x) < 0} . Let Xo E ac be an arbitrary point and (! : U (xo) -+ JR. a local defining function. We may assume that Qx N (xo) -=/=- 0. Then by the implicit function theorem there is a product neighborhood U' x U" of xo in U and a smooth function h : U' -+ JR. such that { (x' xN ) E U' x U" : g(x' , xN ) = 0 } = { (x jl,( x')) : x' E U' } ,
'
It follows that O = "Vx' Q(x', h(x')) + gxN (x' , h (x')) . Vh(x ') . At the point (x' , h(x')) E 8G the gradient "Vg(x', h(x')) is normal to 8G and directed outward from G. Every point y in a small neighborhood of the boundary has a unique representation y = x + t . V g(x) , where t = -de( Y ) and X is the point where the perpendicular from y tO 80 meetS the boundary. Therefore, we define the smooth map F : U' x JR. -+ JR.N by y
F (x ' , t )
=
=
(x', h(x' )) + t . V g(x' , h (x')) .
Then there are smooth functions =
JJR,F (x', t) and therefore
A
and b such that
( VEhN(x') - I + t A(x') + t b(x') ·
.
E det ( N - I "V h (x')
)
h (x')) V h (x ') t ' (x gx , N det JJR,F(x' , 0) Qx N (x' , h(x')) "V h (x') t EN - 1 - (!x N (x' , h (x )) . de t O' 1 + II "V h (x' ) l l 2 - (!xN (x', h(x'))( l + 11Vh( x') l l 2 ) -=/=- 0. 1
·
(
)
It follows that there exists an c > 0 such that F maps U' x ( - s, s) diffeo morphically onto a neighborhood W = W (x0 ) , and U' x {0} onto ac n W . Moreover, since de (x + t V g(x) ) = -t for I t I < c and c small enough, it follows that de = ( t)J y - 1 is a smooth function near ac. If p ' is defined by p' (x', t ) := (x' ,0), then the projection ·
-
p = p'
o
y- 1 : X + t \l g(x) f-+ X, for X E 80, ·
is a smooth map, and de is given by de( Y ) = CJ . IIY - P( Y ) II , where CJ = 1 for y E G and CJ = - 1 elsewhere. For y # 8G we have
4.
Levi Convex Boundaries
71
N
- Pk ( y )) (8k v - (Pk )yv (Y)) ( · Yk : � II IIY (Y) p ( y ) j p (Y)) N J Y) IIY -:( y )ll · [Yv - Pv ( - (y _
Yv
-
and therefore
'
a . [y - p ( y) - Dp ( y ) ( y - p (y )) ] . IIY - P ( Y ) II Since g(p( y )) 0, it follows that Dp( y )("V g( p ( y ))) = 0. But y - p ( y ) is a multiple of \1 g( p ( y )). Together this gives . "Vg( p ( y)) y - p ( y) "V de(Y ) = a · II Y - (Y) II = ± II g( ( ))ll . P "V p y If y tends to 8G, we obtain that "Vde( Y ) of. 0. • "Vde(Y)
=
E.E. Levi showed that every domain of holomorphy with smooth boundary is Levi convex, and locally the boundary of a strictly Levi convex domain G is the "natural boundary" for some holomorphic function in G. Here we prove the following result, which is sometimes called "Levi's theorem". 4.7
Theorem. A domain G with smooth boundary is pseudoconvex if and only if it is Levi convex.
PROOF :
( 1 )Let G be pseudoconvex. The function -de is a smooth boundary function for G, and - log de = log 8e is plurisubharmonic on G, because of the pseudoconvexity. We calculate -
1
1
Lev( -de )(z , w) + I ( 8(de )) z (w) 1 2 · · de (z)2 de(z ) This is nonnegative in G. If z E G, w E T,, and ( 8 (de)) z (w) = 0, it follows that Lev(-de) (z , w) > 0. This remains true for z -+ 8G, so - de satisfies Lev( - log de )(z, w)
=
the Levi condition.
(2) Let G be Levi convex, and suppose that G is not pseudoconvex. Then in any neighborhood U of the boundary there exists a point z0 where the Levi form of - log 8e has a negative eigenvalue. This means that there is a vector wo such that
'P(((O) = Lev(log 8e) (z0, wo) > 0, for cp(() := log 8e (zo t (wo) . Consider the Taylor expansion
cp(()
) cp(O) + 2 Re(cp((O)( + 2 cp(((0 )( 2 ) + 'P(((O) I ( I 2 + - cp(O) t Re(A( + B(2 ) + >-1( 1 2 + . . . ,
72
II.
Domains of Holomorphy
with complex constants A , B and
a
real constant >. > 0.
We choose a point Po E 8G with 8e (zo) = IIPo - zo l l , and an arbitrary c > 0. Then an analytic disk 'ljJ : Dc:(O ) --+ en can be defined by
'lj;(() := Zo + ( wo + exp (A( + B(2) (Po - zo). We have 'lj; (O) = p0 , and we wish to show that 'ljJ (() E G, for 0 < ICI < c and E sufficiently small.
Since cp(( ) > cp(O) + R e(A( + B( 2 ) + (>. / 2 ) 1(1 2 near ( 8e (zo + (wo)
=
0, it follows th at
exp(cp( ( )) > exp ( cp (O)) I exp (A( + B( 2 ) I exp > 8e(zo) I exp (A( + B( 2 ) I - ll exp (A( + B( 2 ) (Po - zo) II , ·
·
·
C 1(1 2)
for ( small and -=f. 0 . This means that we can choose the E in such a way that 't/J(() E G, for 0 < ICI < c. The analytic disc is tangent to ac from the interior of G. Now f( () = de ( 't/J ( ( )) is a smooth function with a local minimum at ( Therefore ( 8de)p0 ('t/J'(O)) = (8! )0 (1) = 0 , and
f( ( )
=
Re (!((( 0 ) ( 2 ) + !((1(1 2 +terms of order >
Since Re ( f
=
0.
3.
f(((O) > 0 . This is a contradiction to the Levi condition at p0, because -de is a defining function for G.
=
•
Exer cises 1 . Prove Lemma 4.4. 2. Assume that G c c C2 has a smooth boundary that is Levi convex outside a point a that is not isolated in 8G. Show that G is pseudoconvex. 3. Assume that G C C2 is an arbitrary domain and that S c G is a smooth real surface with the following property: In every point of S the tangent to S is not a complex line. Prove that for every compact set K c G there are arbitrarily small pseudoconvex neighborhoods of S n K. 4. Assume that G c c C2 is a domain with smooth boundary. Then G is strictly Levi convex at a point Zo E ac if and only if the following condition is satisfied: There is a neighborhood U = U (zo), a holomorphic function cp : D --+ U with cp (O) = zo and cp'(O) -=f. 0, and a local defining function g on U such that (g o cp)(( ) > 0 on D - {0} and (g o cp) (((O ) > 0 .
5.
Holomorphic Convexity
73
5. Let G c c en be a domain with smooth boundary. If G satisfies the strict Levi condition at zo E 8G, then prove that the following hold: (a) There is no analytic disk r.p : D --+ en with
r.p (O) = z0
and
lim (-+ o
bc( r.p (()) llr.p ( ( ) - r.p (O) II 2
=
0
(b) There are a neighborhood U = U(z0) and a holomorphic function f in U with G n { z E U : f (z) = O} = {zo } . 6. A bounded domain G c en is called strongly pseudoconvex if there are a neighborhood u = u (ac) and a strictly plurisubharmonic function g E '6'2 (U) such that G n U = { z E U : g(z) < 0} . Notice that a strongly pseudoconvex domain does not necessarily have a smooth boundary ! Prove the following results about a strongly pseudoconvex bounded do main G: (a) G is pseudoconvex. (b) If G has a smooth boundary, then G is strictly Levi convex. (c) For every z E 8G there is a neighborhood U = U (z) such that U nG is a weak domain of holomorphy. 7. Let G c en be a pseudoconvex domain. Then prove that there is a family of domains G, C G such that the following hold: (a) G, CC Gv+l for every v. ( b) U:;" 1 Gv = G. (c) For every v there is a strictly plurisubharmonic function fv E '6'00 (Gv+I ) such that G, is a connected component of the set
{z
5.
E Cv+ I
:
fv (z) < 0} .
Holomorp h ic C on vexity
Affine C onvexityWe will investigate relationships between pseudocon vexity and affine convexity. Let us begin with some observations about convex domains in lRN . Let 2 be the set of affine linear functions f : JRN --+ lR with
f(x) = a 1x1 + . . . + aN X N t b,
a 1 , . . . , aN , b E lR.
If Jvf is a convex set and x0 a point not contained in M, then there exists a function f E 2 with f(x0) = 0 and J I M < 0. For any c E JR, the set {x E JRN : f (x) < c} is a convex half-space.
Definition .
Let
H( M)
M
c JRN be an arbitrary subset. Then the set
:= {x E JRN : f (x) < sup f, for all f E 2
is called the affine convex hull of M.
M
}
74
11.
Domains of Holomorphy
5. 1 Pr oposition. Let M, M1 , M2 c lRN be arbitrary subsets. Then 1.
M c H (M) . 2. H ( M) is closed and convex. 3.
H ( H ( M ))
4. If M1
c
=
H ( M) .
M2 , then H(M1 )
c
H(M2). 5. If M is closed and convex, then H(M) = M . 6. If M is bounded, then H ( M ) is also bounded.
PROOF:
( l ) is trivial.
5e
with f (xo) > supM f . By conti nuity, f(x) > sup, f in a neighborhood of xo. Therefore, H(M) is closed.
(2) If x0 rf_ H ( M ) , then there is an f E
If x0 , y0 are two points in H ( M ) , then they are contained in every convex half-space E = { x : f ( x) < sup M f } , and also the closed line segment from x0 to y0 is contained in each of these half-spaces. This shows that H(M) is convex.
( 3) We have to show that H (H( M )) C H(M) . If x
E H(H(M)) is an arbi
trary point and f an element of 2', then f (x) < supH ( M ) f < supM f , by the definition of H(M). (4) is trivial.
( 5) Let M be closed and convex. If xo rf_ M, then there is a point Yo
E M
such that d ist (xo , M) = dist(x0, Yo ) (because M is closed). Let zo be a point in the open line segment from x0 to yo. Then z0 rf_ M , and there is a function f E 5e with f(zo) = 0 and JI M < 0. Since t H f(txo + ( 1 - t)yo) is a monotone function, it follows that f(x0 ) > 0 and therefore x0 rf_ H ( M ). (6) If M is bounded, there is an R > 0 such that M is contained in the closed • convex set BR (O) . Thus H( M ) C BR (O). Remark.
H(M) is the smallest closed convex set that contains M .
5 . 2 Theor em . A domain that H(K) c c G.
G c JRN is convex if and only if K c c G implies
PROOF: Let G be a convex domain, and M c c G a subset. Then H(M) is closed and contained in the bounded set H( M ). Therefore, H(M) is compact, and it remains to show that H(M) C G. If there is a point xo E H(M) - G, then there is a function f E 5e with f (x0) = 0 and fi e < 0. It follows that sup M f < 0, and f(xo) > supM f . This is a contradiction to xo E H( M ).
5.
Holomorphic Convexity
75
On the other hand, let the criterion be fulfilled. If xo , Yo are two points of G, then K := { xo, Yo } is a relatively compact subset of G. It follows that H(K) is contained in G. Since H(K) is closed and convex, the closed line segment • from xo to Yo is also contained in G. Therefore G is convex.
H olomor phic C on vexity. Now we replace affine linear functions by holomorphic functions. Definition.
Let G C en be a domain and K c G a subset. The set
K = Kc :
{z E G
:
l f(z) l < s� I J I , for all f E
O(G) }
is called the holomorphically convex hull of K in G.
5 . 3 Pr oposition. Then
Let G c en be a dom ain, and K, K1 , K2 subsets of G.
1. K c K . 2. K is closed in G. 3. K = K . 4. If K1 C K2 , then K1 C K;. 5. If K is bounded, then K is also bounded. � �
�
�
PROOF:
( 1 )is trivial. �
(2) Let z0 be a point of G - K. Then there exists a holomorphic function f on G with l f(zo ) l > su pK i f l . By continuity, this inequality holds on an entire neighborhood U = U(zo) c G. So G - K is open. �
(3 ) SUPx l f l
tmPK i f l .
(4) is trivial.
(5) If K is bounded, it is contained in a closed polydisk pn (O, r ) . The coordi nate functions Zv are holomorphic in G. For z E K we have l zv l < supK i zv l I • r. Hence K is also bounded. �
Definition . A domain G c en is called holomorphically convex K cc G implies that K CC G. �
if
II.
76
Domains of Holomorphy
Example
In C every domain is holomorphically convex: Let K cc G be an arbitrary subset . Then K is bounded, and it remains to show that the closure of K is contained in G. If there is a point z0 E R - G, then Zo lies in aR n aG. We consider the holomorphic function f(z ) : = 1 /(z - z0) in G. If (z,_, ) is a sequence in K converging to zo, then l f ( z,_, ) l < supKifl < supKI ! I < oo . This is a contradiction. For n > 2, we will show that there are domains that are not holomorphically convex. But we have the following result. �
�
5.4
Pr op osition . If G morphically convex.
C
C" is an affine convex domain, then it is holo
PROOF: Let K be relatively compact in G. Then H ( K ) C C G. If zo is a point of G - H ( K ) , then there exists an affine linear function >. E £ with >.( zo) > supK >. . Replacing >. by >. - >. ( 0 ) we may assume that >. is a homo geneous linear function of the form >.(z)
=
2 Re(a 1 z1 +
·
·
·
+ a , z, ) .
Then f(z) := exp ( 2 . (a 1 z1 + . . . + a, z, )) is holomorphic in G, and
lf (-3 ) 1
=
exp ( >. ( z)) . Therefore, l f (zo )l > supKifl, and zo E G - k .This proves K c c • G. In general, holomorphic convexity is a much weaker property than affine convexity.
The C artan-Thullen Theor em . Let G c C" be a domain, and E > 0 a small real number. We define G, : =
{Z
E G : 8c (z) > s}.
Here are some properties of the set G,: 1 . If z E G, then there is an E > 0 such that 8c(z) > c . Therefore, G = U c: >O G,. 2. If c1 < c 2 , then G, => Gc: 2 • 3. G, is a closed subset of C" . In fact, if zo E C" - G, then 8c (zo) < c or z0 rf. G. In the latter case, the ball B, (zo) is contained in C" - Gc:. If z0 E G - Gc: and 8 := 8c(zo ) , then Bc:-o (zo) C C" - G,. So C" - G, is open.
f
a holomorphic function in G. If K C G, thenfo r any 8 with 0 < 8 < E there exists a constant C > 0 such that the following inequality holds:
5 . 5 Lemma. Let G c C" be a dom ain, K C G a compact subset, and
5.
77
Holomorphic Convexity
PROOF: For 0 < 8 < E, G := { z E G : dist(K, z) < 8} is open and relatively compact in G, and for any z E K the closed polydisk Pn (z , 6) is contained in G' c G. If T is the distinguished boundary of the polydisk and l f l < C on G', then the Cauchy inequalities yield a! . a! a f s < p < I D (Z ) I 8 1al � l f l 8 1al . C. •
5.6 Theor em (Car tan-Thullen). If G is a weak dom ain of holomorphy, then G is holomorphically convex. �
Let K CC G. We want to show that K c c G. Let dist(K, en - G) > dist (K, en - G) > 0. Clearly, K lies in G,. PROOF:
E
We assert that the holomorphically �onvex hull R lies even in G,. Suppose this is not so. Then there is a z0 E K - G,. Now let f be any holomorphic function in G. In a neighborhood U = U(z0 ) C G, f has a Taylor expansion
f(z)
= L a,_, (z
v>O
- zo) " , with
a,
=
1D" f(z0 ). v!
The function z H a,_, ( z) := ;, D " f ( z) is holomorphic in G. Therefore, l a,_, (zo) l < supK i a,_, (z) l . By the lemma, for any 6 with 0 < 8 < E there exists a C > 0 such that supK ia,_, (z) l < C/8 1 '-' l , and then
On any polydisk pn (z0 , 6) the Taylor series IS dominated by a geometric series. Therefore, it converges on P = pn ( z0 , E) to a holomorphic function f. We have f = f near z0 , and then on the connected component Q of zo in P n G. Since P meets G and en - G, it follows from Lemma 1..9 that there is a point Z l E p n 8Q n ac. Then f cannot be completely singular at z l . This is a contradiction, because f is an arbitrary holomorphic function in G, and G is a weak domain of holomorphy. • �
�
E xer ci se s
1 . Let G1 c G2 c en be domains. Assume that for every f E O(G1 ) there is a sequence of functions f,_, E O(G2 ) converging compactly on G1 to f. Show that for every compact set K c Gl it follows that Kc2 n cl = Kc l '
78
II.
Domains of Holomorphy
2. Let F : G 1 --+ G2 be a proper holomorphic map between domains in en ,
respectively em . Show that if G2 is holomorphically convex, then so is G1 . 3. Let G c en be a domain and S c G be a closed analytic disk with boundary bS. Show that S C (bS ) c· 4. Define the domain G C e 2 by�G := P 2 (0, 1 ) - P 2 (0, 1/2). Construct the holomorphically convex hull Kc for K : = { ( z1 , z2 ) : z1 = 0 and l z 2 l = 3/4}. Is Kc a relatively compact subset of G ? 5. Let F be a family of functions in the domain G. For a compact subset K c G we define -
{
K;: : = z E
}
G : lf(z ) l < s� l f l for all f E F . �
The domain G is called convex with respect to F, provided that K;: is relatively compact in G whenever K is. Prove: (a) Every bounded domain is convex with respect to the family 'l!f0 (G) of all continuous functions. (b) The unit ball B = B1 (0) is convex with respect to the family of holomorphic functions zi . z� with v, fJ, = 1 , . . . , n and k, l E No .
S in gu lar Fun ction s
6.
Nor m al Exh au stion s .Let G c en be a domain. lf G is holomorphically convex, we want to construct a holomorphic function in G that is completely singular at every boundary point. For that we use "normal exhaustions." Definition . A normal exhaustion of G is a sequence ( K, ) of compact subsets of G such that: 1 . K, C C ( Kv+ l t , for every v. 2. U::" 1 Kv = G .
6 . 1 Theor em. Any domain G in en admits a normal exhaustion. ff-G I S holomorphically convex, then there is a normal exhaustion (K, ) with K, K, fo r eve ry v. In the general case, K, := Pn (o, v) n G 1 ;v gives a normal exhaustion. 1f G is holomorphically convex, K" c c G for every v. We construct a new exhaustion by induction. PROOF:
Let Ki := K 1 . Suppose that compact sets Ki , . . . , K�-1 have been con structed, with Kj = Kj for j = 1, . . . , v - 1 , and Kj CC (JC;*+ 1 t . Then there exists a >.(v) E N such that K�_ 1 c ( K;.. ( v) t . Let K� : = K;.,(v) ·
6. Singular Functions �
It is clear that (K� ) is a normal exhaustion with K�
=
79
K� .
•
Unb ounded Holomor phic Function s . Again let G c en be a domain. �
6 . 2 Theor em. Let ( K, ) be a normal exhaustion of G with Kv = K, >. (J.L) a strictly monotonic increasing sequence of natural numbers, and (zJ.L) a sequence of points with zJ.L E K;.. ( J.L) + l - K;.. c,"l .
Then there exists a holomorphic function bounded.
f
in G such that
l f (z J.L ) I
is un
PROOF: The function f is constructed as the limit function of an infinite series f = 2:;:" 1 fw By induction we define holomorphic functions fJ.L in G such that:
2 - J.L for J.L > 1. J.L - 1 2 . I JJ.L(zJ.L) I > J.L + 1 + Ll fj (zJ.L) I for J.L > 2 . j=l 1.
I JJ.L I K>-<"l
<
:=
Let h 0. Now for J.L > 2 suppose that h , . . . , fJ.L - 1 have been constructed. Since zJ.L E K;..( J.L) + 1 - K;..( J.L) and K;.. ( J.L) = K;.. ( J.L) there exists a function g holomorphic in G such that l g(z J.L) I > q := supKA ( J.' ) lgl. By multiplication by a suitable constant we can make ,
If we set
and ( 2 ) .
fJ.L := g k
l g (zJ.L) I
> 1 > q.
with a sufficiently large k , then
fJ.L
has the properties ( 1 )
We assert that I: J.L JJ.L converges compactly in G . To prove this, first note that for K C G an arbitrary compact subset, there is a J.Lo E N such that J.L K C K;.. ( J.L ) · By construction supKifJ.L I < 2 - for J.L > J.Lo. Since the geo metric series I: 2 - J.L dominates I: JJ.L in K , the series of the JJ.L is normally convergent on K. This shows that r = 2: /L JJ.L is holomorphic in G. Moreover, o
l f(z /L ) I
>
I JJ.L(z /L ) I - L l fv(z !L ) I v # J.L
>
J.L + 1 -
(since z/L E K ( v )
(since L Tv = 1). v� 1 lf(zJ.L) I --+ oo for J.L --+ >
It follows that
L 2-v
>-.
for
v > J.L)
J.L
=·
•
1 1 . Domains of Holomorphy
80
The following is an important consequence: A
domain G is holomorphically convex if and only if fo r any infinite set D that is discrete in G there exists a function f holomorphic in G such that l f l is unbounded on D .
6 . 3 Theor em.
( l ) Let G be holomorphically convex, D C G infinite and discrete. Moreover, let ( Kv) be a normal exhaustion of G with Kv = K,. Then Kv n D is finite (or empty) for every v E N. We construct a sequence of points zJ.L E D by induction. PROOF:
�
Let z 1 E D - K1 be arbitrary, and >.(1) E N minimal with the property that z1 lies in K>- ( 1) + 1 · Now suppose the points z1 , . . . ,zJ.L- 1 and the numbers >. ( 1 ) , . . . , >. (J-L - 1 ) have been constructed such that z, E K.A ( v) +1 - K.A (v) • for v = 1 , .. . , /-L
-
1.
Then we choose zJ.L E D - K.A (J.L- 1)+1 and A(J-L) minimal with the property that zJ.L lies in K.A(J.L l + 1 . By the theorem above there is a holomorphic function f in G such that l f(zJ.L) I -+ oo for J-L -+ oo. Therefore, l f l is unbounded on D. (2) Now suppose that the criterion is satisfied, and K CC G. Then K C G, and we have to show that K is compact. Let (z,) be any sequence of points in K. Then �
�
sup{ l f(zv)l
:
v E N} <
sup l f l < oo , K
for every
f E O(G)
Therefore, { zv : v E N} cannot be discrete in G . Thus the sequence ( zv) has a cluster point z0 in G. Since K is closed, zo belongs to K. So G is • holomorphically convex. �
-
Sequences . For a domain G c en we wish to construct a sequence that
accumulates at every point of its boundary.
Let ( K, ) be a normal exhaustion of G . Then there exists a strictly monotonic increasing sequence A (J-L) of natural numbers and a se quence (zfl.) of points in G such that:
6 . 4 Theor em. 1.
zfl. E K.A(J.L) + 1 K.A(J.L) • for every f-L · 2 . If zo is a boundary point of G and U = U(z0) _
an open connected neigh borhood, then eve ry connected component of U n G contains infinitely many points of the sequence (z, ) .
PROOF: This is a purely topological result, since we make no assumption about G. The proof is carried out in several steps.
81
6 . Singular Functions
{ B" : v E N} be the countable system of balls with rational ( 1 ) Let B center and rational radius meeting 8G. Every intersection B, n G has at most countably many connected components. Thus we obtain a countable family =
C
= { CJ.L :
3
v E N such that CJ.L is a connected component of B, E B}.
(2) By induction, the sequences >-.(fl,) and (zJ.L) are constructed. Let z1 be arbitrary in CI - KI . Then there is a unique number >. (I) such that z i E K>-(1)+1 - K>-( I ) · Now suppose z i , . . . , zJ.L - I have been constructed such that Zj E cj n (KA(j)+l - K,A(j ) ), for j = 1 , . . . , f-L - 1.
We choose zJ.L E CJ.L - K>-(J.L - IJ+l and >-.(J-L) as usual. That is possible, since there is a point w E BV(J.L) n acfl. n ac if CJ.l. is a connected component of B v (J.L ) n G. Then en - KA(J.L-I ) +I is an open neighborhood of w and contains points of C,. (3) Now we show that property (2) of the theorem is satisfied, Let z0 be a point of ac, U = U(zo ) an open connected neighborhood, and Q a connected component of U n G. We assume that only finitely many zJ.L lie in Q , say ZI , . . . ,z m . Then U * := U - {z i , . . . , z m }
and
Q* : = Q - {z 1 , . . . , zm }
are open connected sets that contain no z,. Obviously, Q* is a connected component of G n U * . There is a point wo in U* n 8Q* n 8G, and a ball B, C U* with wo E B,. Then B, n G c U* n G. Moreover, B, n G must contain a point WI E Q*. The connected component C* of w1 in B, n G is a subset of the connected component of WI in U* n G. But C* is an element CJ.Lo of C. By construction it contains the point zJ.Lo . That is a contradiction. Infinitely many members of the sequence belong to Q. • 6.5 Theor em . If G is holomorphically convex, then it is a dom ain
morphy.
d
holo
�
PROOF: Let ( K, ) be a normal exhaustion of G with K = K, and choose " sequences A(J-L) E N and (z, ) in G such that z, E K>-(J.Ll+l - K>-(J.L) · We may assume that for every point z0 E ac, every open connected neighborhood U = U(z0) , and every connected component Q of U n G there are infinitely many z, , in Q.
Now let f be holomorphic in G and unbounded on D := {zJ.L clear that f is completely singular at every point Zo E 80.
:
f-L
E N} . It is •
82
II. Domains of Holomorphy
Remark. It is not necessary that a completely singular holomorphic func tion is unbounded. In 1978, D. Catlin showed in his dissertation that if G cc en is a holomorphically convex domain with smooth boundary, then there exists a function holomorphic in G and smooth in a neighborhood of G that is completely singular at every point of the boundary of G. Exer cises 1 . A domain G c c en is holomorphically convex if and only if for every z E 8G there is a neighborhood U(z) such that U n G is a domain of holomorphy. 2 . Let G1 c en and G2 c em be domains of holomorphy. Iff : G1 --+ e m is a holomorphic �pping then j - 1 ( G2) n G1 is a domain of holomorphy. 3 . Find a bounded holomorphic function on the unit disk D that is singular at every boundary point. ,
7.
E xamp les and Ap p lication s
Domains of Holomorphy 7 . 1 Pr op osition. holomorphy.
Every doma in in the complex plane e zs a domain of
PROOF: We have already shown that every domain in e is holomorphically • convex. Therefore, such a domain is also a domain of holomorphy.
7 . 2 Theor em. alent:
Th efollowing statements about domains G E en are equzv
1 . G is a weak doma in of holomorphy. 2. G is holomorphically convex. 3. For every infinite discrete subset D C G there exists a holomorphicfunc tion f in G such that I ! I is unbounded on D . 4. G is a doma in of holomorphy. The equivalences have all been proved in the preceding paragraphs. Fur thermore, we know that every domain of holomorphy is pseudoconvex. Still missing here is the proof of the Levi problem: Every pseudoconvex domain is holomorphically convex. We say more about this in Chapter V. Every affine convex open subset of en is a domain of holomorphy. The n-fold Cartesian product of plane domains is a further example. 7 . 3 Pr op osition. If G1 , . . . , Gn C e are arb itrary domains, then G G 1 x · · · x G is a domain d holomorphy. n
:=
7 . Examples and Applications
83
PROOF: Let D = {zJ.L = (zf , . . . , zf:: ) : f-L E N} be an infinite discrete subset of G. Then there is an i such that (zf) has no cluster point in Gi , and there is a holomorphic function f in Gi with limJ.L--> = I f ( zf) I = = · The function f in G, defined by [(z1 , . . . , z ) : = f (zi ) , is holomorphic in G and unbounded n on D. •
Remar k. The same proof shows that every Cartesian product of domains of holomorphy is again a domain of holomorphy.
C omplete Reinhar dt Domains . Let G c en be a complete Rein hardt domain (see Section 1 . 1 ) . We will give criteria for G to be a domain of holomorphy. For that purpose we define a map log from the absolute value space "f! to JRn by
Definition . A Reinhardt domain G is called logarithmically convex if log r(G n (C* )n) is an affine convex domain in JRn . Remark. For z = (z1 , . . . , zn ) E G we have lo g r (z ) = (loglz1 l , . . . , logiz i ) . n If z E (C* )n, then i zil > 0 for each i , and log r(z) is in fact an element of JRn . 7.4 Pr oposition. The doma in d convergence of a power series S(z) Ev > o av z " is logarithmically convex. PROOF: Let G be the domain of convergence of S(z) , and M : = log r(G n (C* )n) c JRn . We consider two points x , y E M and points z , w E G n (C* )n with log r (z) = x and log r(w) = y . If >. > 1 is small enough, >.z and :XW still belong to G n (C* )n. Since S(z) is convergent in >.z, Xw, there is a constant C > 0 such that l av l . >. 1 " 1 - l z" l < C
Thus
and
l av l . >. 1 " 1 . l w" l < C, for every v E N8
lav l . ;. l v l . l z" l t - l w" l l - t < C, for every v and 0 < t < 1 .
It follows from A bel's lemma that S(z) is convergent in a neighborhood of
This means that Zt E G and tx + ( 1 - t) y = log r (z t ) E M , for 0 < t < 1 . Therefore, M is convex. •
84
I I . Domains of Holomorphy
7 . 5 Pr op osition. Let G be a complete Reinhardt domain. If G is logarith mically convex, then it is holomorphically convex. PROOF: Let K be a relatively compact subset of G. Since G is a complete Reinhardt domain and K a compact subset of G, there are points z1 , . . . , z�,; E G n (C* )n such that
K c G'
:=
k
U P n ( O , qi )
i=l
C G,
where qi := T(z i ) ·
We consider the set Jlt = { m (z) = zv : v E N0} of monomials, which is a subset of O(G). For z E pn (o, Qi) and m E A we have Let Z
:=
{z 1 ,
•
•
,
•
l m(z) l
�
z �,; } . Then for z E K it follows that
<
sup lml K
<
sup lml G'
<
suplml, Z
for every m E A _ �
�
Suppose that K is not relatively compact in G. Then K has a cluster point Zo in aG, and it follows that l m(zo)l < SUP zlml , for every m E A _ Let h( z) : = log T ( z), for z E (C* ) n . Since G is logarithmically convex, the domain G0 := h( G n ( C* )n ) C JRn is affine convex. For the time being we assume that z0 E ( C* ) n . Then x0 := h(z0) E 8G0 , and there is a real linear function >.(x) = a1x1 + . . . + an X n such that >.(x) < >. (xo) for x E Go.
Let x = log T (z) be a point of G0, and u E JRn with Uj < Xj forj = 1 , . . . , n . Then eu1 < ex1 = l zj I , and therefore (since G is a complete Reinhardt do ,eun ) E G n (@*) " and u E Go . In particular, main) w = ( eu1 , •
•
•
>.(x) - naj = >.(x - nej ) < >.(x0), for every n E N.
Therefore, aj > 0 for j = 1 , .. . ,n. Now we choose rational numbers rj > aj and define >.(x) : = r 1 x1 + · . · + rn x n If we choose the r1 sufficiently close to aj , the inequality >. ( qi ) < >.(x0) holds for i = 1 , .. . , k, and it still holds after multiplying by the common denominator of the rj . Therefore, we may assume that the rj are natural numbers, and we can define a special monomial m0 by m0(z) := z? . . . z�n . Then l mo(zi ) l = e .A(q, ) < e >- (xo) = lmo(zo) l , for i = 1 , . . . k -
_
�
_
,
.
l mo ( zo) l > sup zlmo l , and this is a contradiction. 1f zo >t ( C * ) n , then after a permutation of the coordinates we may So
assume that ziol . . . zj0l =f. 0 and zj�1 = . . . = z�0l = 0. We can project on the space
7 . Examples and Applications
e1 and work with monomials in through as above.
variables
the
z
1
,
. . . , zz .
85
Then the proof goes
•
Now we get the following result: 7 . 6 Theorem. Let G c en be a complete Reinhardt domain. Then the following statements are equivalent: G 2. G 3. G 4. G 1.
is is is is
the dom ain of convergence of logarithmically convex. holomorphically convex. a dom ain of holomorphy.
a power se ries.
PROOF: We have only to show that if G is a complete Reinhardt domain and a domain of holomorphy, then it is the domain of convergence of a power series. By hypothesis, there is a function f that is holomorphic in G and completely singular at every boundary point. In Section 1 .5 we proved that for every holomorphic function in a proper Reinhardt domain there is a power series S' ( z ) that converges in G to f. By the identity theorem it does not • converge on any domain strictly larger than G.
An alytic Polyhed r a . Let G c en be a domain Definition . Let U C G, V1 , . . , Vk C e open subsets, and h , . . . , /k holomorphic functions in G. The set .
P := { z E U :
fi (z) E v; ,
for i =
l , ..
. k} ,
is called an analytic polyhedron in G if P CC U. If, in addition, V1 = . . . = Vk = D , then one speaks of a special analytic polyhedron in G. Remark. An analytic polyhedron P need not be connected. The set U in the definition ensures that each union of connected components of P is also an analytic polyhedron if it has a positive distance from every other connected component of P.
7 . 7 Theorem. Eve ry connected analytic polyhedron P in G is a dom ain of holomorphy. PROOF: We have only to show that P is a weak domain of holomorphy. If zo E 8P, then there is an i such that fi(zo ) E 8v; . Therefore, f(z) • (fi ( z ) - /i(z o )) - 1 is holomorphic in P and completely singular at zo .
86
II. Domains of Holomorphy
E xa mple
Let
q<
1 be a positive real number, and 2 P := { Z = (zl , z2) E C : l z1 l < l, l z2 l < l and l zl . z2 1
< q} .
Then P (see Figure 1 1.7) is clearly an analytic polyhedron, but neither affine
q
Figu r e 1 1 .7.
An
analytic polyhedron
convex nor a Cartesian product of domains. So the analytic polyhedra enrich our stock of examples of domains of holomorphy. We will show that every domain of holomorphy is "almost" an analytic poly hedron. 7. 8
Theorem. If G c en is a domain of holomorphy, then there exists a sequence (Pv) of special analytic polyhedra in G with P" CC Pv+ i and u� l p" = G . PROOF: Let (Kv ) be a normal exhaustion of G with Kv = K"" If z E 8Kv+l is an arbitrary point, then z does not lie in K, C (Kv+I t , and therefore not in Kv · Hence there exists a function f holomorphic in G for which q == supKJ f l < IJ(z) l . By multiplication by a suitable constant we obtain q < 1 < I J (z) l , and then there is an entire neighborhood U = U(z) such that I f I > 1 on U. �
Since the boundary 8Kv+I is compact, we can find finitely many open neigh borhoods Uv,j of Zv,j E aKv+l ' j = 1 , . . . , kv ' and corresponding functions !v,j holomorphic in G such that l !v,j l > 1 on Uv ,j , and 8Kv+l c U�v 1 Uv ,j · We define P" == {z E (Kv +l t : lfv,j (z)l < l for j = l , .. . , kv } · Clearly, K, C Pv C (Kv+ l t · Furthermore, M :=Kv+l - (Uv,l U . . . U Uv,kJ is a compact set with P" C M C (Kv+l t · Consequently, P" CC Kv+l · Thus P" is a special analytic polyhedron in G. It follows trivially that the sequence • (Pv) exhausts the domain G.
8. Riemann Domains over en
87
In the theory of Stein manifolds one proves the converse of this theorem. Exercises 1. If R is a domain in the real number space JRn , then
TR
= R + ilRn := {z E en : (Re(zi ), . . . , Re(z )) E R }
n is called the tube doma in associated with R . Prove that the following properties are equivalent: (a) R is convex. (b) TR is (affine) convex. (c) TR is holomorphically convex. (d) TR is pseudoconvex. Hint: To show (d) ::::::::} ( a) choose x0 , yo E R. Then the function cp ( ( ) - In 8rR (x0 + ((Yo - x0 )) is subharmonic in D. Since 8rR ( x +iy ) = 8R(x) , one concludes that t � - ln 8R ( x0 + t (y0 - x0 )) assumes its maximum at t = 0 or t = 1 . 2 . Let G c en be a domain. A domain G c en is called the envelope of holomorphy of G if every holomorphic function f in G has a holomorphic extension to G. Prove: (a) If R c JRn is a domain and H(R) its affine convex hull, then G := H(R) + ilRn is the envelope of holomorphy of the tube domain G = R + i JRn. (b) If G c en is a Reinhardt domain and G the smallest logarithmically convex complete Reinhardt domain containing G, then G is the envelope of holomorphy of G . Hint: Use the convex hull of log T(G) . 3. Construct the envelope of holomorphy of the domain Gq P2 ( 0, ( 1 , q)) u P2 (0, (q, 1)) . 4. A domain G c en is called a Runge do main if for every holomorphic function f in G there is a sequence (P v ) of polynomials converging com pactly in G to f. �
�
�
�
:=
Prove that the Cartesian product of n simply connected subdomains of e is a Runge domain in en. 5. A domain G C en is called polynomially convex if it is convex with respect to the family of all polynomials (cf. Exercise 5.5). Prove that every polynomially convex domain is a holomorphically convex Runge domain.
8.
R iem ann Dom a in s
over
en
Riemann Dom ains. It turns out that for general domains in en the envelope of holomorphy (cf. Exercise 7 .2) cannot exist in en . Therefore, we have to consider domains covering en .
II.
88
Domains of Holomorphy
Definition . A (Rzemann) domain over e n is a pair (G, 1r) with the following properties: 1. G is a connected Hausdorff space. 4 2. 7r : G -+ e n is a local homeomorphism (that is, for each point X E G and its "base point" Z := 1r( X ) E e n there exist open neighborhoods U = U(x) c X and V = V(z) c en such that 1r : U -+ V is a homeomorphism). Remar ks
1. Let ( G, 1r) be a Riemann domain. Then G is pathwise connected, and
the map 1r : G -+ e n is continuous and open. The latter means that the images ci open sets are again open. 2. If (G, 1r") are domains over e n for v = 1 , .. . , l, and x, E G, are points over the same base point z0 , then there are open neighborhoods U" = Uv (xv) c G, and a connected open neighborhood V = V(zo ) C e n such that 1l"v luv : U" -+ V is a homeomorphism for v = l , .. . , l . Examples
1.
If G is a domain in en , then (G,idc) is a Riemann domain.
2. The Riemann surface of vz (without the branch point) is the set G := { (z, w) E C* x e : w 2 = z} .
If G is provided with the topology induced from e* x e , then it is a Hausdorff space. The mapping r.p : C* -+ G defined by ( � ( ( 2 , () is continuous and bijective. Therefore, G is connected. The mapping r.p is called a uniformization ci G.
Now let 1r : G -+ e be defined by 1r(z, w ) := z . Clearly, 1r is continuous. If (z0, w0) E G is an arbitrary point, then z0 -=/=- 0, and we can find a simply connected neighborhood V(z0) c @*. Then there exists a holomorphic function f in V with f 2 (z) - z and f(zo) = wo. We denote f (z) by vz. 1 The image W == f (V) is open, and the set 1r- (V) can be written as the union ci two disjoint open sets U± := {(z, ± f (z)) : z E V } = (V x ( ± W)) n G. Let f (z) == (z, f(z)). Then f : V -+ G is continuous, and_:rr o f(z) = z. The open set U == U+ i s a neighborhood ci (z0 , w0 ) , with f(V) = U and 1r(z, w) = (z, w) on U; that is, 1rlu : U -+ V is topological. Hence (G,1r) is a Riemann domain over C . �
4
�
general topological space X is said t o be connected if it is not the union of two disjoint nonempty open sets. A space X is called pathwise connected if each two points cf X can be joined by a continuous path in X . For open sets in en these two notions are equivalent. A
8 . Riemann Domains over en
89
The space G can be visualized in the following manner: We cover e with two additional copies of e , cut both these "sheets" along the positive real axis, and paste them crosswise to one another (this is not possible in !R 3 without self intersection, but in higher dimensions, it is). This is illu strated in Figure 1 1.8. 0
Figure
1 1 . 8 . The Riemann surface of
v'z
8. 1 Pr op osition (on the uniqueness of lifting). Let ( G 1r) be a domain over en and Y a connected topological space. Let y0 E Y be a point and 'l/.;1 , $2 : Y --+ G continuous mappings with 'l/J1 (y0) = 'l/J2 ( y0) and 1r o 'l/.;1 = 1r o 'l/.;2 . Then 'l/.;1 = 'l/.;2 . ,
Let M := {y E Y : 'lj.;1 (y) = 'lj.;2 (y) } . By assumption, y0 E M , so M -=1- 0. Since G is a Hausdorff space, it follows immediately that M is closed. Now let y E Y be chosen arbitrarily, and set x := 'lj.;1 (y) = 'l/.;2 (y) and z := 1r(x) . There are open neighborhoods U = U(x) C G and V = V(z) C en such that 7r : u --+ v is topological, and there is an open neigborhood W = W(y) with '1/J>- (W) c U for >. = 1 , 2. Then PROOF:
'1/Jd w = (7r I u) -
1
0 7r 0
'l/.;1 I w = (7r I u) -
1
0 7r 0
'l/J2 1 w = 'l/J2 1 w '
and therefore W C M. Hence M is open, and since Y is connected, it follows that M = Y. • Definition. Let Z o E en be fixed. A ( R iemann) domain over en with distinguished point is a triple Q = (G,1r, x0) for which: 1. (G,1r) is a domain over en. 2. x0 is a point of G with 1r(x0) :=z0.
90
II.
Domains of Holomorphy
Let Yj = (Gj , 1rj , Xj) be domains over e n with distin guished point. We say that YI is contained in Y2 (denoted by YI -< Y2 ) if there is a continuous map r.p GI --+ G2 with the following properties: 1. 1r2 o r.p 7ri (called "r.p preserves fibers" ). 2. r.p(x! ) x 2 .
Definition.
:
=
=
8.2 Proposition. with r.p( XI) = x2 is
If YI Y2 , then the fiber preserving map r.p : GI uniquely determined. -<
--+ G2
This follows immediately from the uniqueness of lifting.
The relation
8.3 Proposition.
1 . 9 -< Q . 2. YI -< Y2
and 92
-<
Q3
==;.
YI
"-< "
-<
is a weak ordering; that is:
93 .
The proof is trivial. Two domains YI ' Y2 over en with fundamental point are called isomorphic or equivalent (symbolically YI 92 ) if YI -< Y2 and Definition.
"'
92
-<
YI ·
Two domains Yj (Gj , 1rj , Xj ) , j 1, 2 , are isomorphic if and only if there exists a topological 5fiber preserving map r.p GI --+ G 2 with r.p(x! ) x 2 . =
8.4 Proposition. =
=
:
PROOF: If we have fiber preserving mappings 'PI : GI --+ G2 and 'P2 : G2 --+ GI , with 'PI (x i ) x2 and r.p2 (x 2) XI , it follows easily from the uniqueness of fiber preserving maps that r.p2 o 'PI = idc 1 and 'PI o 'P2 = idc2 • The other direction of the proof is trivial. • =
=
A domain g = ( G, 1r, x0) with 1r(x0) = z0 is called schlicht isomorphic to a domain 90 = ( G0 , idea , z0) with Go c e n .
Definition.
if it is
8.5 Proposition. with GI , G2 c en .
Let YJ = (Gj , ide; , Xj ) , j = 1, 2, be two schlicht domains Then Y I Y2 if and only if GI C G2 . -<
Example 5
Recall that a "topological map'' is a homeomorphism!
8.
Riemann Domains over en
91
z and z -1=- 0} and 7ri (z, w ) : = z. Then Let GI := {(z, w ) E C2 : w 2 (h = (GI , 7ri , (1, 1)) is the Riemann surface of y'z, with distinguished point (1, 1). The domain (h is contained in the schlicht domain (f2 (C, ide , 1), by cp(z, w ) := z . But the two domains are not isomorphic. =
=
Union of Riemann Domains.
We begin with the definition of the union of two Riemann domains. Let Qj = (Gj , 1rj , xj ) , j = 1, 2, be two Rie mann domains over (C n with distinguished point, and Zo := 7ri (xi ) = 1r2 (x2 ). We want to glue GI , G2 in such a way that XI and x2 will also be glued. To get a rough idea of the construction, assume that we already have a Riemann domain Q = ( G, 1r, x0) that is in some sense the union of QI and Q2 . Then there should exist continuous fiber preserving maps 'P I : GI --+ G with 'PI (xi ) x0 , and 'P2 : G2 --+ G with cp2 (x2 ) := x0 . If a : [0, 1] --+ GI and (3 : [0, 1] --+ G2 are two continuous paths with a(O) = XI , (3(0) = x2 and 7ri o a = 1r2 o (3, then /'I := 'PI o a and /'2 := cp2 o (3 are continuous paths in G with 7r O ')'I = 7r 0 ')'2 and /'I (O) = /'2 (0) = x0. Due to the uniqueness of lifting, it follows that /'I = ')'2 . This means that a(t) and (J(t) have to be glued for every t E [0, 1] . Unfortunately, this is an ambiguous rule. For example, we could say that x E GI and y E G2 have to be glued if 7ri (x) 1r2 (y) . Then the desired property is fulfilled, but it may be that there are no paths a from XI to x and (3 from x2 to y with 7ri o a = 1r2 o (3. =
=
Therefore, we proceed in the following way: Start with the disjoint union on this set with the GI U G2, and take the "finest" equivalence relation following property: .
rv
1. XI "' X2 . 2. If there are continuous paths a : [0 , 1] --+ GI and (3 : [0, 1] --+ G2 with a(O) = XI , (3(0)
=
x 2 , and 7ri .
o
a = 1r2 o (3, then a(1)
rv
(3(1) .
One can equip G := ( GI U G2)/ with the structure of a Riemann domain. This will now be carried out in a more general context. rv
Let X be an arbitrary set. An equivalence relation on X is given by a partition &: = {Xv : v E N} of X into subsets with:
1. 2.
UvE N Xv = X. X" n X�' = 0 for
v -/=- f-L. The sets X" are the equivalence classes. Now let a family (� )Lei of equivalence relations on X be given with � = {X�, : vL E NL } for L E I. We set N : = fl101 NL , and
92
II.
Domains of Holomorphy
Then &: = {Xv : v E N} is again an equivalence relation (simple exercise) , and it is finer than any � . This means that for every E I and every v E N , there is a VL E NL with X" c x�, . L
.
We apply this to the disjoint union X = u .AE L G,A , for a given family (�h).AEL of Riemann domains �h = (G>-, 1r>-, x>-) over en with distinguished point. An equivalence relation on X is said to have property (P) if the following hold: 1. X>- "' xll, for >., g E L. 2. If [0, 1] --+ G>- and (3 : [0 , 1] --+ Gil are continuous paths with a(O) rv (3(0) and 7r.A o a = 1rll o (3, then a(1 ) rv (3(1). We consider the family of all equivalence relations on X with property (P) . It is not empty, as seen above in the case of two domains. Therefore we can construct an equivalence relation (as above) that is finer than any equivalence relation with property (P) . We denote it by rv p . It is clear that 7r,A (X) = 7rll(y) if x E G.A , y E Gil, and x "' P y. The relation "'P also has property (P), and the elements of an equivalence class X" all lie over the same point z = z(Xv ) · We define G := Xj rvp and ?r(Xv ) := z(Xv ) · The equivalence class of all X,A will be denoted by x. z. If 8.6 Lemma. Let y E G.A and x E Gil be given with 1rll(x) = 7r>-(Y) a :
=:
we choose open neighborhoods U = U(y) C G.A, V = V(x) C Gil, and an open connected neighborhood W = W(z) such that 1r>-I : U --+ W and 1rll : V --+ W are topological mappings, then for r.p := (1rIl l v ) - o 7r.A U --+ V the following hold: 1 . r.p(y) = X. y, then r.p(y') y' for every y' E U. 2. If x :
"'P
"'P
PROOF: The first statement is trivial. Now let a [0, 1] --+ W be a con tinuous path with a(O) = z and a(1) = 7r>-(Y') for some y' E U. Then (3 := (7r>- lu ) - 1 o a and "( := r.p o (3 are continuous paths in U and V with (3(0) y rv p X = r.p(y) = "f(O) and 7r,A 0 (3 = 7rll 0 r.p 0 (3 = 7rll 0 "!· Therefore, y1 = {3 ( 1) rv p "( (1) = r.p(y1 ). • :
=
8. 7 Theorem.
There is a topology on G such that
is a Ri�mann domain over e n with distinguished point x, and all maps 'P>- : G>- --+ G with 'P>-(x) := equivalence class of x are continuous and fiber preserving.
8.
Riemann Domains over en
93
PROOF: (1) Sets of the form 'P>. (M) for M open in G>. together with G constitute a base of a topology for G. To see this it remains to show that the intersection of two such sets is again of this form. Let M C G>. and N C G12 be open subsets. Then 'P >. (M) n cp12 ( N)
=
cp12 ( N n cp 12 I ('P >. (M))).
But cp12 I ('P>. (M)) is open in G12 • In fact, let x E cp12 I ('P>.(M)) be given, and y E M be chosen such that 'P>. (Y) = cp12 (x) (and therefore y "'P x). Let z : = 7r>. ( Y) = 1r12 (x) . Then there exist open neighborhoods U = U(y) and V V(x) and an open connected neighborhood W = W(z) such that 7r>. : U --+ W and 1r12 : V --+ W are topological mappings. Let 'P : = ( 1r12 1 v ) -I o 7r>. : U --+ V. By the lemma, cp(y) x and cp(y') "' p y' for every y' E U. So V' := cp(MnU) is a neighborhood of x in G12, and since cp12 (cp(y')) = 'P>.( Y') for every y' E U, it follows that V' C 'P 12 I ( 'P>. (M) ) . =
=
Consequently, every 'P>. is a continuous map. (2 ) Remark: Since every y E G is an equivalence class 'P>.(x), we have M = U 'P>. ( 'P>. I ( M)) for any subset M C G. AEL (3) 7r : G --+ e n is continuous: Let V c e n be an arbitrary open set, and M : = :;r-I (V). Then 'P >. I (M) = 1r>. I (V) is open in G>. , and therefore M = u ).. E L 'P>. ('P ).. I (M )) is open in G. ( 4) G is a Hausdorff space: Let YI , Y2 E G with YI -=/=- Y2 , and Z I ir(y! ) , Z 2 : = ir (y2 ) . There are two cases. If z i -=/=- z2, then there are open neighborhoods VI (z i ) and V2 (z2) with VI n V2 = 0 . Then :;r- I (VI ) and :;r- I (V2) are disjoint open neighborhoods of YI and Y2 · If Z I = z2 , then we choose elements XI E G>. , x2 E G12 with 'P>. (x! ) = YI and cp12(x2 ) = y2 , and since X I and x2 are not equivalent , the above lemma implies that there are disjoint neigborhoods of YI and Y2 · (5) G is connected: Let y = 'P>. (x) be an arbitrary point of G. Then there is a continuous path a : [0 , 1] --+ G>. that connects the distinguished point X>. to x. Then 'P >. o a connects x to y. (6) 7r is locally topological: Let y = 'P>. (x) be a point of G, and z = ir(y) 7r>. (x) . Then there exist open neighborhoods U = U(x) C G>. and W = W(z) c e n such that 7r>. : U --+ W is a topological mapping. U := 'P>. (U) _ is an open neighborho od of y, with ir(U) 7r>. (U) = W. In addition, ir l u is injective, since 7r o 'P>. = 7r>. and 7r>. l u is injective. :=
=
=
94
II.
Domains of Holomorphy -
( 7) Clearly, the maps 'P>.. : G >.. --+ G are fiber preserving, and it was already • shown that they are continuous. Now g has the following properties:
1. 2.
9>-.
-<
Q , for every
X E L.
If Q* is a domain over en with 9>-.
PROOF:
_
Q* for every >., then g
-<
-<
Q*
(of the second statement)
If Q * is given, then there exist fiber preserving mappings r.p)_ : G>.. --+ G * . We
introduce a new equivalence relation N on the disjoint union X of the G>.. by x N x1
: -<==*
x E G>.. , x' E G12 and r.p)_ (x) = 'P� (x').
It follows from the uniqueness of lifting that N has the property (P).Now we define a map r.p : G --+ G* by r.p(r.p>.. (x))
:=
r.p)_ (x) .
Since "' p is the finest equivalence relation with property ( P ) ,r.p is well defined. • Also it is clear that r.p is continuous and fiber preserving. Therefore g is the smallest Riemann domain over en that contains all do mains Y>.. · -
Definition. The domain g constructed as above is called the union of the domains Y).. , and we write g = u ).. E L Y).. · Special cases:
1. 2.
From YI -< g and Y2 -< 9 it follows that YI U Y2 From YI -< Y2 it follows that YI U Y2 "' Y2 · 3. g u g "' g . 4 . YI u 92 "' 92 u YI . 5. YI u (92 u 93 ) "' ( QI u 92 ) u Y3 ·
-<
Q.
Example Let YI = ( GI , 7ri , xi ) be the Riemann surface of JZ with distinguished point XI = (1, 1) and Y2 = (G2 , id, x2 ) the schlicht domain
1
2
< l z l < 2)
with distinguished point x2 = 1 . Then YI u g2 = (G , ir, xo ), where G = (GI U G2 )/ "'P · I Let y E 1r;- (G2) C GI . Then we can connect y to the point XI by a path I a in 7r1 (G 2 ) , and 7ri (Y) to x2 by the path 7ri o a in G 2 . But XI "'P x2 , so
8. Riemann Domains over en
y
95
1r1 (y) as well. This shows that over each point of G2 there is exactly one equivalence class. "'p
Now let z E e - {0} be arbitrary. The line through z and 0 in e contains a segment a : [0, 1] --+ @* that connects z to a point z* E G2 . There are two paths a l , a 2 in Gl with 7ri O ctl = 7rl O ct2 = a Since al (1) rv p a2 ( 1 ) , it follows that a 1 (0) "' I' a2 (0) . .
Then it follows that (f! U �h Exercises
= (e - {0}, id, 1 ) .
define q:,t : en --+ en by (e i h z1 , . . . , e i tn Zn ) . '*" t ( Z1 , . . . , Zn ) A Riemann domain g = ( G, 1r, x0) is called a Re inhardt domain over en if 1r(x0) = 0 and for every t E 11 - (e* ) n there is an isomorphism 'Pt : g --+ g with 1r o 'Pt = q:,t o 1r. Prove: (a) If G c en is a proper Reinhardt domain, then g = ( G, id , 0) is a Reinhardt domain over en . 2 (b) Let G 1 , G2 c e be defined by
1 . For t = ( t 1 , . . . , t n ) E
11
•
rt.
.-
.=
P2
(0, 1)
-
{ (z , w) E
1 { (z , w) : l z l = and lwl < 2 } ' 2 P (0, 1 ) : lwl < �} · 1
2
Gluing G1 and G2 along { ( z , w) : � < l z l < 1 and lwl < � } one obtains a Riemann domain over e2 that is a Reinhardt domain over e2 , but not schlicht. Show that this domain can be obtained as the union of 0 1 = ci , id, ( £ , D and 02 = c2 , id, ( £ , � ) . 2. Let J = {0, 1 , 2, 3, . . . } C N0 be a finite or infinite sequence of natural numbers and Pi = pn(zi, ri) , i E ] , a sequence of polydisks in en. Assume that for every pair ( i, j) E .T x J an "incidence number" Eij E {0, 1 } is given such that the following hold: (a) Eij = Eji and Cii = 1 . (b) Cij = O if Pi n Pj = 0. (c) For every i > 0 in J there is a j < i with Eij = 1. (d) If Pi n Pj n Pk -=1=- 0 and Eij = 1 , then Cik =Ejk·
(
)
(
)
Points z E Pi and w E Pj are called equivalent (z "' w ) if z = w and Cij = 1 . Prove that G := u Pi/ rv carries in a natural way the structure of a Riemann domain over en . Let 1r : G --+ en be the canonical projection and suppose that there is a point Zo E n i EJ Pi. Is there a point Xo E G such that (G,7r, Xo ) can be written as the union of the Riemann domains ( Pj , id, z 0) ?
96
9.
1 1 . Domains of Holomorphy
T h e E nvelop e of Holomor p hy
H olomorphy on Riemann Domain s Definition . Let ( G, 1r) be a domain over en . A function f : G � e is called holomorphic at a point x E G if there are open neighborhoods U = U(x) c G and V = V(1r(x)) c en such that 1rlu : U � V is topological and f o ( 1rl u) - I : V � e is holomorphic. The function f is called holomorphic o n G if f is holomorphic at every point x E G. Remar k. A holomorphic function is always continuous. For schlicht domains in @" the new notion of holomorphy agrees with the old one. Definition . Let Qj = (Gj , 1rj , xj ) , j = 1, 2, be domains over en with distinguished point, and �h -< �h by virtue of a continuous mapping r.p : GI � G2 . For every function f on G2 we define fl c1 == f o r.p.
Pr op o sition. Iff : G2 � e is holomorphic and Q I holomorphic o n GI .
9.1
PROOF:
-<
92 , then f l c 1 zs
Trivial, since r.p is a local homeomorphism with 1r2 o r.p = 7ri -
•
Definition . 1. Let ( G ,7r) be a domain over en and X E G a point. If f is a holo morphic function defined near x , then the pair (!, x) is called a local holomorphic function at x. 2. Let (GI , 7rl ) , (G2, 7r2) be domains over en , and XI E GI , X2 E G2 points with 7ri (xi) = 1r2 (x2) == z . Two local holomorphic functions ( JI , x l ) , (h , x2) are called equivalent if there exist open neighbor hoods UI (xi) C GI , U2 (x2 ) C G2 , V(z), and topological mappings
I 7ri : UI � V , 1r2 : U2 � V with fi o (7ri i ul ) - = h 0 (7r2 1 u2 )- ·
3. The equivalence class of a local holomorphic function (!, x) is denoted
by fx ·
Remark. If ( h )x 1 = ( h)x 2 , then clearly, h (xi) = h (x2 ) · In particular, if GI = G2 , 7ri = 1r2 , and xi = x2 , then it follows that h and /2 coincide in an open neighborhood of xi = x2 .
9. The Envelope of Holomorphy
97
9 . 2 Pr op o sition. Let (G 1 , 1ri ) , (G2 , 1r2) be domains over en , a.x : [0, 1] --+ G>. continuous paths with 1r1 oa1 = 1r2 oa 2 . Additionally, let !>. be holomorphic on G.x , for >. = 1 , 2 . If (h)cq (O) = ( h ) a2 (o ) , then also ( !I ) a1 ( 1 ) = ( h ) a2 (1) · PROOF: Let M := { t E [0, 1] : (fi ) a 1 ( t ) = ( h ) a 2 (t) } · Then M of 0 , since 0 E M . It is easy to see that M is open and closed in [0, 1] , because of the identity theorem for holomorphic functions. So M = [0, 1] . • Pr opo sition. Let Qj = (Gj , 1rj , xj ) , j = 1 , 2, be domains ove r en with distinguished point, and 91 --< Y2 · Thenfo r every holomorphicfunction f on G1 there is at most one holomorphicfunction F on G2 with F / c 1 = f , i.e., a possible holomorphic extension o ff is uniquely determined.
9.3
PROOF: Let F1 , F2 be holomorphic extensions of f to G2 . We choose neigh borhoods U.x (x .x ) c G.x such that the given fiber-preserving map r.p : G1 + G2 maps U1 topologically onto U2 . We have FJ o r.p / u1 = f / u1 , for j = 1 , 2, and therefore F1 l u2 = F2 l u2 • It follows that (FI )x2 = (F2)x2 • Since each point of G2 can be joined to x2 , the equality F1 = F2 follows. •
Envelopes of H olomorphy Definition . Let g = ( G,1r, Xo) be a domain over e n with distinguished point and § a nonempty set of holomorphic functions on G. Let (Q.x)>.E L be the system of all domains over e n with the following properties: 1 . Q --< 9.x for every >. E L. 2 . For every f E § and every >. E L there is a holomorphic function F.x on G.x with F.x lc = f. Then H,y; (Q) == U .xE L Y>. is called the 9 - h u l l of Q. If § = 0( G) is the set of all holomorphic functions on G, then H(Q) Ho ( G ) (Q) is called the envelope of holomorphy of Q. If § = { f) for some holomorphic function f on G, then H1 (Q) = = Hu } (Q) is called the dom ain of existence of the function f .
Let g = ( G,1r, Xo) be a dom ain over e n ' § a nonempty set of holomorphic functions on G, and Hg; (Q) = (G, 1?, x0 ) the 9 - h u l l. Then the following hold:
9.4 Theorem. 1.
Q --< Hg; (Q) .
�
2. For each function f E § there exists exactly one holomorphic function F on G with Fie = f . �
II.
98
Domains of Holomorphy
IJC:h = (G 1 , 1ri , xi ) is a dom ain over en such that 9 --< YI and everyfunc tion f E .Jk can be holomorphically extended to G1 , then Y1 --< H,y; (Q) .
3.
PROOF: Hg; (Q) is the union of all Riemann domains 9:.. = (G;.. , 7r;.. , x;.. ) to which each function f E .Jk can be extended. We have fiber-preserving maps r.p;.. : G --+ G;.. and cp;.. : G;.. --+ G . �
Let "'p be the finest equivalence relation on X == U >. E L G;.. with property (P) . 6 Then 8 is the set of equivalence classes in X relative to "' P· We define a new equivalence relation c::: on X by
x
c:::
: -¢===?-
x'
Then
N
x E G;.. , x' E G, 1r;.. (x) = 1re (x '), and for each f E F and its holomorphic extensions F1 , F2 on G;.. , respectively G, we have (F:.. ) x = (Fe)x ' ·
has property (P):
(i)For any >. we can find open neighborhoods U = U(xo ) , V = V(x;.. ) , and W = W(1r(x0)) such that all mappings in the following commutative dia gram are homeomorphisms:
Then for f E .Jk and its holomorphic extension F;.. on G;.. we have that 1 F;.. o (7r;.. / v ) - 1 = F;.. o r.p;.. o (1r/ u ) - 1 = f o (1r/ u ) - is independent of >.. Therefore, all distinguished points x;.. are equivalent. (iqf a : [0, 1] --+ G;.. and (3 : [0, 1] --+ G, are continuous paths with a(O) "' (3(0) and 1r;.. o a = 1re o (3 , then (F:.. ) a(O) = (Fe)f3(0) · It follows that (F:.. ) a(l) = (Fe)f3( I ) as well, and therefore a(1) c::: (3(1). Since g --< 9:.. and 9:.. --< Hg; (Q) , it follows that Q --< HF (Q) . Furthermore, the fiber preserving map rp : = cp;.. o r.p;.. does not depend on >.. Now let a function f E .Jk be given. We construct a holomorphic extension F on G as follows: �
If y E 8 is an arbitrary point, then there is a >. E L and a point Y>. E G>. such that y = cp;.. (y;.. ), and we define
1f y = 'Pe(Ye) as well, then y;.. "'P Ye , and therefore y;.. "' Ye as well. It follows that F;.. (y;.. ) = Fe (Ye), and F is well defined. 6
For the definition of property (P) see page
92.
9 . The Envelope of Holomorphy
99
We have F o cp = F o VJ>-. o 'P>-. = F>-. o 'P>-. = on G. This shows that F is an extension of and from the equation F o $>-. = F>-. it follows that F is holomorphic (since $>-. is locally topological).
f
f,
•
The maximality of H,y; (Q) follows by construction.
The 9-hull Hg; (Q ) is therefore the largest domain into which all functions E § can ce holomorphically extended.
f
9. 5
Iden_!ity theorem. Let YJ = (GJ , 1rj , XJ ) , j = 1 , 2, be domains over e n , and Q = (G, ?r, x) the union of Q1 and G2. Let fJ : GJ --+ e be holomor phic functions and Q = ( G,1r, x) a domain with 9 --< Yj fo r j = 1 , 2-such that Then there is a holomorphic function on G with fJ , JfO f' J = 1 , 2 .
h i e. = hi e ·
f
f
h /e
f ie = J
h/ e ,
PROOF: Let and § == { ! } . Since Y1 --< Hg; ( Q ) and = == Y2 --< Hg; ( Q ) , it follows that 91 U Y2 --< Hg; ( Q ) . �
�
�
Let be a holomorphic extension o ff to G (where Hg; (Q) = G, 1?, x) ), and = = ]115. Then
1
f
(
(fleJ i e = 11e = (J10) l e = J1e = f. fl e1
Therefore, is a holomorphic extension o ff to Gj · Due to the uniqueness = fJ for j = 1 , 2. of holomorphic extension, •
J/ ej
P seu doconvexit y. Let P" c e n be the unit polydisk, (P n , H ) a Eu clidean Hartogs figure, and : p n --+ e n an injective holomorphic map ping. Then (
Then every holomorphic function G UP.
f o n G can be extended holomorphically to
The proposition follows immediately from the identity theorem. Definition. A domain ( G, 1r) over e n is called Hartogs convex if the fact that (P.'H) is a generalized Hartogs figure and x0 E G a point with 1i --< Q := (G,1r, xo) implies g u p "" r;; .
A domain Q = (G, 1r, xo) over en is called a dom ain of holomorphy if there exists a holomorphic function on G such that its domain ci existence is equal to Q.
f
1 00
II. Domains of Holomorphy
Remark. If G c e n is a Schlicht domain, then the new definition agrees with the old one.
9 . 7 Theor em. 1.
If Q = ( G,n, x0 ) is a dom ain over e n and § a nonempty set of holo morphic functions on G, then Hg; (Q) is Hartogs convex. 2. Every dom ain of holomorphy is Hartogs convex.
PROOF: Let (P, 'H) be a generalized Hartogs figure with 1i --< Hg; (Q). Then every function f E § has a holomorphic continuation to Hg; (Q) UP .There fore, Hg; (Q) U P + H.� ((]). On the other hand, we also have Hg; (Q) < • Hg; (Q) U P .s o Hg; (Q) U P "' Hg; (Q) . A Riemann domain (G, n) is called holomorphically convex if for every infinite discrete subset D C G there exists a holomorphic function f on G that is unbounded on D.
9.8 Theor em (Oka, 1 953). If a R iem ann dom ain ( G,n) is Hartogs pseu doconuex, it is holomorphically convex (and therefore a dom ain qf holornor phy) .
This is the solution of Levi's problem for Riemann domains over en . We cannot give the proof here. It seems possible to construct the holomorphic hull by adjoining Hartogs figures (cf. H . Langmaak, [La60] ) . It is conceivable that such a construction may be realized with the help of a computer, but until now (spring 2002) no successful attempt is known. We assume that parallel computer methods are necessary.
Boundary Point s . In the literature other notions of pseudoconvexity are used. We want to give a rough idea of these methods. Definition . Let X be a topological space. A filter (basis) on X IS a nonempty set R of subsets of X with the following properties: 1. 0 rf_ R. 2. The intersection of two elements of R contains again an element of the set R .
Example
1. If xo is a point of X , then every fundamental system of neighborhoods of xo in X is a filter, called a neighborhood filter of xo .
9 . The Envelope of Holomorphy
101
2 . Let (Xn ) be a sequence of points of X . If we define SN == { Xn n > N) , then R := {SN : N E N} is the so-called elementary filter of the sequence (x n ) . A filter is therefore the generalization of a sequence. Definition . A point x0 E X is called a cluster point of the filter R if x0 E A, for every A E R. The point x0 is called a limit of the filter R if every element of a fundamental system of neighborhoods of x0 contains an element of R. For sequences the new notions agree with the old ones.
If f : X --+ Y is a continuous map, then the image of any filter on X filter on Y , the so-called direct image.
IS
a
Definition . Let (G, 1r ) be a Riemann domain over e n . An accessible boundary point of ( G,n ) is a filter R on G with the following properties: 1 . R has no cluster point in G. 2. The direct image 1r ( R) has a limit zo E e n . 3. For every connected open neighborhood V = V(zo ) C e n there is exactly one connected component of 1r - 1 (V) that belongs to R. 4. For every element U E R there is a neighborhood V = V ( z0 ) such that U is a connected component of 1r - 1 ( V) .
Remark.
For a Hausdorff space X the following hold:
1 . A filter in X has at most one limit. 2. If a filter in X has the limit x0 , then filter .
x0
is the only cluster point of this
(for a proof see Bourbaki, [Bou66] , § 8 . 1 ) Therefore, the limit z0 in the definition above is uniquely determined. There is an equivalent description of accessible boundary points that avoids the filter concept. For this consider sequences (x , ) of points of G with the following properties: 1 . ( x , ) has nu cluster point in G. 2. The sequence of the images 1r ( x, ) has a limit zo E e n . 3. For every connected open neighborhood V = V(z0) C en there is an m E N such that for n, m > n0 the points x, and Xm can be joined by a continuous path a : [0, 1] --+ G with 1r o a([O, 1]) c V .
Two such sequences ( x, ) , (y,) are called equivalent if
7r(Yv) = Zo . 1 . limv--.oo 7r(Xv) = li mv 2. For every connected open neighborhood V = V ( z0 ) there is an n0 such that for n , m > m the points x, and Ym can be joined by a continuous path a : [0, 1] --+ G with a([O, 1]) c V. --; 00
n o
102
I I . Domains of Holomorphy
An accessible boundary point is an equivalence class of such sequences. Let 8 G be the set of all accessible boundary points of G . Even if G is schlicht, this set may be different from the topological boundary 8G. There may be points in ac that are not accessible, and it may be happen that an accessible boundary point is the limit of two inequivalent sequences. We define 6 == Gui!JG. If r0 = [xn] is an accessible boundary point, we define a neighborhood of r0 in G as follows: Take a connected open set U C G such that almost all Xn lie in U and 1r ( U) is contained in a neighborhood of z0 := limn---+ oo 1r(xn)· Then add all boundary points r = [yn] such that almost all Yn lie in U and limn-too 7r(Yn) is a cluster point of 1r ( U) . With this neighborhood definition G becomes a Hausdorff space, and * : G --+ e n with v
1r(x) :=
{
if x E G, 7r( x) nlim ---t oo 7r(xn) if x = [xn]
E i!JG,
is a continuous mapping. Definition . A boundary point r E 8G is called removable if there is a connected open neighborhood U = U(r) C G such that ( U, *) is a Schlicht Riemann domain over e n and 8 G n u is locally contained in a proper analytic subset of U. A subset M C 8G is called thin if for every ro E M there is an open neighborhood U = U(r0) C G and a nowhere identically vanishing holo morphic function f on U n G such that for every r E M n U there exists a sequence (xn) in U n G converging to such that limn-too f (x n ) = 0. v
r
Example Let G C e n be a (schlicht) domain and A C G a nowhere dense analytic subset. Then every point of A is a removable boundary point of G' := G - A. The points of the boundary of the hyperball Br (0) C e n are all not removable. Let B be a ball in the affine hyperplane H = { (zo , . . . zn ) E en+l : Zo = 1 } , and G c e n+ l - {0} the cone over B. Then every boundary point of G is not removable, since locally the boundary has real dimension 2 n + 1 . The set M := {0} is thin in the boundary, as is seen by choosing J (zo , . . . zn ) : = zo . ,
,
(G,1r) be a Riemann domain over e n . If r.p : D --+ G is a continuous mapping, 7r o r.p : D --+ en holomor phic , and (7r o r.p )' ( () -=/=- 0 for ( E D, then S := r.p(D) is called an analytic disk in G. The set bS := r.p(8D) is called its boundary.
Analytic Disk s . Let
v
9 . The Envelope of Holomorphy
l 03
Let I == [0, 1] be the unit interval. A family (St)tEI of analytic disks 'Pt (D) in G is called continuous if the mapping (z , t) H 'Pt (z ) is continuous. It is called distinguished if St c G for 0 < t < 1 and bSt c G for 0 < t < 1 . Definition . The domain G is called pseudoconvex if for every distin guished continuous family (St)tEI of analytic disks in G it follows that s1 c G.
The domain G is called pseudoconvex at r E 80 if there is a neighborhood U = U(r) C G and an c > 0 such that for every distinguished continuous family (St)tEI of analytic disks in G with 7r(St) C B10(7r(r) ) it follows that St n U c G for t E /. v
As in en one can show that a Riemann domain is pseudoconvex if and only if it is Hartogs pseudoconvex. 9.9 Theorem (Ok a). A R iemann dom ain (G, 1r) is pseudoconvex if and only if it is pseudoconvex at every point r E 80. 9. 10 Corollary. If (G, 1r) is a dom ain of holomorphy, then convex at every accessible boundary point.
0
is pseudo·
The converse theorem is Oka's solution of Levi's problem. Finally, we mention the following result: 9. 1 1 Theorem. Let (0,1r) be a Riemann dom ain over en , and
a thin set of non removable boundary points. point of 80 - M , then 0 is pseudoconvex. PROOF:
See [GrRe56] , §3, Satz 4.
If 0
M C 80 is pseudoconvex at every •
E xer cises 1 . Prove that a Reinhardt domain Q over e n must be Schlicht if it is a domain of holomorphy. 2. Prove that if ( G, 1r) is a Reinhardt domain, then for every f E 0( G) there is a power series S(z) at the origin such that f(x) = S(1r(x)) for E G. 3. Prove that the envelope of holomorphy of a Reinhardt domain is again a Reinhardt domain. 4. Prove that the Riemann surface of the function f(z) = log(z) has just one boundary point over 0 E C. 5 . Find a Schlicht Riemann domain in e 2 whose envelope of holomorphy is not schlicht . 6. Construct a Riemann domain g = (G, 1r , x 0) over e2 such that for all x,y E 1r-1 (1r(xo )) and every f E O (G) the equality f(x) = f(y) holds. X
104
II. Domains of Holomorphy �
7. Let (G, 1r ) be a Riemann domain and ( G, 7?) its envelope of holomorphy. If f is a holomorphic function on G and F its holomorphic extension to G, then f(G) F (G) 8. Consider =
G
.
{ (z, w) � < l z l < l, lwl < 1} :
ei t , w) : : < r < l , t > O and lw
Determine the envelope of holomorphy of G. 9. Let G c e n be a domain and p G --+ JR a plurisubharmonic function. {z E G p ( z ) < c} c c G, 1f zo is an accessible boundary point of B then B is pseudoconvex at zo , in the sense of the last paragraph. :
==
:
C h ap t er III
Analyt ic S ets 1.
T h e Algeb r a of P ower Ser ies
T he Banach Algebr a Bt . In this chapter we shall deal more exten sively with power series in e n . Our objective is to find a division algorithm
for power series that will facilitate our investigation of analytic sets. We denote by C[z] the ring of formal power series I: v>o avz" about the origin. Let JR.+ be the set of n-tuples of positive real numbers. Definition . Let t = (t1 , . . . , tn ) E JR.+ and f = L: v> O avz" define the "number" I I J il t by
I I J ilt : =
{ Ev>ol av lt"
E
C [zl We
if this series converges,
otherwise.
Let Bt : = { ! E C [z� : II f l it < oo }. Remark.
One can introduce a weak ordering on JR.+ if one defines
For fixed f, the function t 1---t II J il t is monotone: If t < t * ,then II J i lt < I / f li t · · Definition . A set B is called a comp lex Banach algebra if the following conditions are satisfied: 1 . There are operations +
:
B B --+ B, · : C B --+ B x
x
and
o B :
x
B --+ B
such that (a) ( B , +, · ) is a complex vector space, (b) ( B ,+,o) is a commutative ring with 1 , (c) c . ( ! g) = ( c . f) g = f ( c . g) for all f, g E B and c E C. 2. To every f E B a real number 11!11 > 0 is assigned that has the properties ci a norm: o
( a) ( b)
o
o
li e · !II = lei . 1/!11 , for c E C and f E B , 1/ f + g /1 < 1/ f /1 + 1/ g ll , for f, g E B,
106
1 1 1 . Analytic Sets
IIIII = o � J = o . II! o g il < IIIII . ll g l l , for J, g E B. (c)
3.
is complete; i.e., every sequence in B that is Cauchy with respect to the norm has a limit in B.
4. B
Bt
1 . 1 Theor em. for any t E JR+. .
= {! E
C[z] II J ilt :
< oo} is a complex Banach algeb ra
Clearly, C[z] is a commutative C-algebra with 1. Straightforward calculations show that II· . . li t satisfies the properties ( 2a) , ( 2b) , (2c) and (3). It follows that Bt is closed under the algebraic operations, and all that remains to be shown is completeness. a >-J z" . Then for every Let ( f>.. ) be a Cauchy sequence in Bt with f>.. = L v>o - � c > 0 there is an n = n (c) E N such that for all A, f-L > n ,
PROOF:
Z::: l a �>-J - a �l l t" = ll f>.. - !M ilt v> O Since t" = t�1
•
•
•
t�n #
<E
0, it follows that E
for every v E Nn0 .
I ac>-J " ] < t" " - aC>
For fixed (a�>-J ) is therefore a Cauchy sequence in C which converges to a complex number a,. Let f = L: v> O avz". This is an element of C[z]. Given S > 0, it follows that there exists an n = n ( 8) such that Y,
=
:2::: I a�>-) - a�>-+ >
<
6
for >. > n and f-L E N
2
Let I C N0 be an arbitrary finite set. For any >. > n there exists a f-L N such that :Ev E I l a �" + >< l - a I t" < 8/2, and then "
:2::: I a�>- J v El
- a,
It"
< 8,
for >. > n .
In particular, ll f>.. - Jilt < 6. Thus f>.. - f (and then also and ( f>.. ) converges to f . Expan sion with Respect t o
additional notation: If v E N0 , t E JR+. , and E e n , write z
z1 .
= J-L( >.) E
f)
belongs to Bt,
•
For the following we require some
1 . The Algebr a of Power Series
107
(t1 , t' ) , and z = ( z1 , z1 ) . An element f = L v> O avz" E C[z] can be written in the form v = ( v1 , v1 ) ,
t
00
"""' f>. z A1 , f =�
=
with
I
f>. (z )
""
= �"'
I
I
a( A , v '){Z }" .
The series fA are formal power series in the variables z2 , . . . , zn. We call this representation of f the expansion off with respect to z1. Now the following assertions hold: 1.
2.
00
/>. E Bt ' for all A, and l: II!A IIt ' � < oo . ll zf . Jilt = t� . II J i l t , for s E No.
f E Bt
¢==;>
PROOF:
( l ) Since we are dealing with absolute convergence, it is clear that
ll!ll t
=
00
2: 1 1 />. ll t 't� . A=O
(2) We have z . f = I:r 0 fA z; + s . The right side is the unique expansion z . f with respect to z1 . Now the formula can be easily derived.
f
f
of •
Let B be a complex Banach algebra and (!>.) a sequence of elements of B. The series I::r 1 fA converges to an element f E B if the sequence Fn I:;�= 1 fA converges to f with respect to the given norm. 1 .2 Proposition. Eve ry f E B with Il l - !II < 1 is a unit in B with C onver gent Series in Banach Algebr a s . ==
f- 1
00
- """' (1
-6
-
f) A and ll f - 1 1 1
1 < - 1 - 11 1 -
111 ·
E :=
Ill - f 11 . Then 0 < c < 1 , and the convergent geometric series I::r 0 sA dominates the series I: r 0 ( 1 - ! ) A . As usual, it follows that this series converges to an element g E B. We have PROOF:
Let
n
n
( 1 - ( 1 - f)) . 2: ( 1 X=O
Jf'
t (l - !) A - I: (l - f f' A=O
As n tends to infinity we obtain
Xl
1 - ( 1 - J) n+ l f . g = 1 and 1/ g // <
I:: r 0 s A
= 1/(1
_ s) .
•
1 08
III. Analytic Sets
C onver gent Power Ser ies. A formal power series f = I:v> O avz" is called convergent if it is convergent in some polydisk P around the-origi n. In that case there exists a point t E P n JR+. , and since f converges absolutely
at t . it follows that f E Bt . On the other hand, if f E Bt , then by Abel's lemma f converges in P = pn ( o , t ) . 1 . 3 Theorem. :3 t E JR+. with II f li t < oo} is the set d conve rgent { f E C [z] 1. Hn :=
power serzes. 2. Hn is a C-algebra. 3. There is no zero divisor in H.: If f . g = 0 in H, then f = 0 or g = 0.
We have already proved the first part, and then the second part follows easily.
The last part is trivial, since C[z] contains no zero divisors. Remark. If f is convergent and f(O) = 0, then for every c > 0 there is a t E JR+. with II Jil t < c. In fact, since f (O) = 0, we have a representation f = Z1 h + · · · + zn fn . If 11 /ll t < oo, then also 11/i ll t < oo for i = 1 , _ . , n, and n
n
II Jilt = 2::.:: ti ll/ill t < max( t1 , · · · t n ) · 2::.:: 1 1/i ll t i=1 ,
This expression becomes arbitrarily small as t
--+
i=1
0.
When we go from Bt to Hn , we lose the norm and the Banach algebra structure, but we gain new algebraic properties: 1 . f E Hn is a unit ¢==;> f (O) -=1- 0. PROOF: One direction is trivial. For the other one suppose that f(O) -=1- 0. Then g == f . f(0) -1 1 is an element of Hn with g(O) = 0. So there exists a t with 1/ g//t < 1 , and f . f (0) 1 is a unit in Bt . Thus f is a unit • in Hn . 2 . The set m := {f E Hn f(O) = 0} of all nonunits in Hn is an ideal: -
-
:
(a) /I , h E m ===? h + h E m. (b) f E m and h E Hn ===? h f E m. ·
An ideal a in a ring R is called maximal if for every ideal b with a c b c R it follows that a = b or b = R. One can show that in any commutative ring with 1 -=/:- 0 there exists a maximal ideal. If a C R is maximal, then Rfa is a field. 1 .4 Theorem. The set m of nonunits is the unique maximal ideal in H, and Hn /m "'
C.
If a C Hn
is a proper ideal, then it cannot contain a unit. Therefore, it is contained in m . The homomorphism r.p Hn --+ C given by f H f( O ) is • surjective and has m as kernel. PROOF:
:
1 . The Algebra of Power Series
1 09
Distinguished Dir ection s . An element f E Hn is called zl- regular order k if there exists a power series fo ( z! ) in one variable such that:
of
1. f(z1 , 0, . . . , 0) = z f . fo (z1 ) . 2. fo (O) -1- 0. If f is zrregular of some order, f is called z1 -regular. Let f (z) = I:� 0 f>.. zr be the expansion of f with respect to z 1 . Then f is zrregular of order k if and only if fo(O' ) = . . . = fk_1 ( O') = 0 and fk (O') -1- 0. f is zl-regular if and only if f (z1 , 0, . . . , 0 ) � 0. We often need the following properties: 1. f is a unit in Hn -¢==? f is 21-regular of order 0. 2. If f>. is zl-regular of order k>. , for >. = 1 , 2 , then h . h is z1-regular of order k1 + k2 . There are elements f -1- 0 of Hn that f (0) = 0 which are not zrregular, even after exchanging the coordinates. Definition. map ac : en
Let c = ( c2 , . . . , en ) be an element of en - 1 . The linear --+ en with
is called a shear. The set I; of all shears is a subgroup of the group of linear automorphisms of en ' wilh ao = idcn . We can write ac (z) := z + z1 . (O , c ) . ln particular, we have ac (ei) = ( l ,c ) . Let f E Hn be a nonze ro e lement. Then there exists a shear a such that f o a is z1 -regular. 1 . 5 Theorem.
Assume that f converges in the polydisk P. If we had f(z1 , z') = 0 for every point (z1 , z') E P with z1 -1- 0, then by continuity we would have f = 0, which can be excluded. Therefore, there exists a point a = (a1 , a ' ) E P with a1 -1- 0 and f ( a) -1- 0. We define c := (a i ) - 1 . a ' and a : = ac . Now, PROOF:
So f o a(z1 , 01) � 0, and f o a is 21-regular.
•
Rem ark. If JI , . . . , /z are nonzero elements in Hn , then f == h . . fz -1- 0 converges on a polydisc P, and there exists a point a E P with f ( a ) -1- 0 and a1 -1- 0. As in the proof above we obtain a shear a such that h o a, . . . , fz o a are zl-regular. ·
110
1 1 1. Analytic Sets
Exercises
f(z) L: v>O avz" be a formal power series. (a) P r ve the "Cauchy estimates": f E Bt
1 . Let
=
===?
o
l av l <
l l ! l l t /t"
for
almost every v. (b) Prove that if there is a constant C with l av /s " < C , then f E Bt for t < s. (c) Let fn (z) = L: v>o av , nz" be a sequence of power series with //!n i l < C. If every sequence (a,,) converges in C to a number a, then show that (fn) is a Cauchy sequence in Bt converging to f(z) = L: v> O avz" , for every t < s . s
2. The Krull topology on Hn is defined as follows: A sequence Un) converges in Hn to f if for every k E N there is an no with f - fn E mk for n > no . What are the open sets in H,? Is Hn with the Krull topology a Hausdorff space? 3. Let B be a complex Banach algebra with 1. Show that for every f E B the series exp(f) = I::� o r fn! is convergent, and that exp(f) is a unit in B. 4. If f is a formal power series and f I:;� 0 P>. its expansion into homo geneous polynomials, then the order of f is defined to be the number =
ord(f) := min{s E No
: p8 -=f- 0).
Now let (fn) be a sequence of formal power series such that for every k E N there is an no with ord(fn) > k for n > no. Show that I:� 1 fn is a formal power series. Use this technique also for the following: If 91 , . . . , gm are elements of Hn with ord(gi) > 1 , then
:2::: afl. wfl. H :2::: afl. (g 1 (z) , . . . , 9m ( z))fl. J.L > O J.L > O defines a homomorphism rp Hm --+ Hn of complex algebras. :
2.
The Preparati o n The orem
Let a fixed element t E JR+. be chosen. When no confusion is possible we write B in place of Bt , B' in place of Bt' , and II ! I/ in place of II f 1 / t . The ring of polynomials in z 1 with coefficients in B' is denoted by B' [z1 ]. 2. 1 Weierstrass Formula in Bt . Let f and g = I:;� 0 g;.. z� be two elements of B. Assume that there exists an E No and a real number E with Division w ith Remainder in Bt .
0 < E < 1 such that
g8 is
a
s
unit in B' and ll z f
- g g-; 1 11 ·
< c t� . ·
111
2. The Preparation Theorem
Then there that
exists exactly one q E B f
=
and one r E
B'[z1 ]
with deg ( r ) <
s
such
q · g + r,
with
and
ll r ll < 1/ f l/ · 1 -1 c: Let us first try to explain the idea of the proof. If h E B, then there is a unique decomposition h qh · zf + rh , where rh E B' [z 1 ] and deg ( rh ) < s. If g is given, we define an operator T T9 : B --+ B by PROOF:
=
=
:
g ·
g; 1 . qh + rh . If T were an isomorphism, then f = T(T- 1 f) g · (g; 1 · qr-1 1 ) + rr-1 1 would T(h )
=
=
be the desired decomposition. One knows from Banach space theory that T is an isomorphism if idB - T is "small" in some sense. Since /l (id8 - T) (h) ll = ll zf - gg; 1 ll . llqh II , one can, in fact, conclude from the hypothesis of the theorem that id8 - T is "small." Now r- 1 2:r 0 (id8 - T) A . Since (id8 T ) 0 f = f and (id8 - T) 1 f (zf- g g_;- 1 )q1 , we obtain the following algorithm: =
=
Inductively we define sequences f>.. , q>.. , r>.. beginning with fo f zf q0 + ro . If f>.. zf q>.. + r>.. has been constructed for some >. > 0, then we define =
=
=
f>.. + I
:=
(z f - gg; I )q>.. ,
and obtain q>.. + I and r>.. + 1 by the unique decomposition
f>.. + 1 = zfq>.. + l + r>..+ I , r>.. + 1 E B'[z 1 ] with deg(r>.. + I ) < s. If we define q : I:: r 0 g_; 1 q>.. and r : 2:r 0 r>.. , then =
=
00
00
00
00
l::: (gg_; 1 q>.. + r>.. )
X=O
=
g · q + r.
112
1 1 1. Analytic Sets
When using this algorithm we do not need the abstract transformation T and the B anach theory of such transformations. However, it is necessary to prove the convergence of all of the series that were used. - ( zf - gg-;; 1 ) . Then ll h ll < c: . tf and gg-;; 1 = zf + h. From f>.. = zf q;.. + r;.. it follows that li rA II lll f>.. ll and 1/ q;.. /l . ll f>.. ll . Furthermore, from f>.. + I = - h · q;.. it follows that
For this let h
==
lt1 8
ll f>..+ I II
;..
.l h ll . ll q;.. ll < E .
II!AII
0 !A converges. Th us II /A II < E . II I II and Since 1 q < >-. s g 1 ll . II I II and li rA / / < c:;.. IIJII, g ll ; ;.. ll c: rt ll ; the series q = I: 0 g; 1 q;.. and r 0 r;.. also converge.
I;�
�
= I;�
The estimates for 1/ gsq/1 and IIl ii follow readily: < ll r ll
<
00
'L:0 II q;.. ll
<
l: l h ll
<
>-.=
00
>-.=0
f!S II J II ·
(X_
I >A = A=O
t1S IIJII ·
j'
1
1
-
!'"
II III · 1 - c: ·
It still remains to show uniqueness. Assuming that there are two expressions of the form it follows that
and ll (ql - q2)gs z f ll
I 1/(ql - q2)gs z f + (rr - r2 ) 1/
<
ll ( q 1 - q2 )gsh ll E · t� . II ( q1 - q2 )gs II E ll (q1 - q2 )gs zf ll· •
If
the assumptions of the theorem are satisfied and if in addition f E B'[z 1 ] , g E B' [z1] , and deg(g) = s, then q E B'[z1 ] with q = 0 or deg ( q) = deg( f) - s.
2.2 Corollary.
Let d := deg (!). For d < s we have the decomposition f Hence we have to consider only the case d > s. PROOF:
=
0 · g +f.
2. The Preparation Theorem
We assume that therefore
deg (JJ.L ) < d fo r f-L
=
1 13
0 , . . . , A . Then deg(q>-) < d - s , and
deg( f>- +d = deg (JA - r>- - gg; 1 q>-) lnax (d, s - l ,s + (d - s ) ) = d. Hence deg ( fA) < d and deg(q>-) < d - s for all A. It follows that deg( q) < d - s, and from f = g . q + r we can conclude that deg ( q) = d - s. •
The Weierstras s Condition . We use the notation from above. Definition. Let s E N0 . An element g = I:; 0 g>-z� E B satisfies the Weierstrass condition (or W-condition) at position s if 1 . gs is a unit in B'. 2. l l zf - gg; 1 ll < � tf .
�
Let R be an integral domain.' A polynomial f ( u) = fs u + fs - 1 u s- 1 + . . . + h u + fo E R[ u] is called monic or normalized if fs = 1 . A polynomial f E B'[zi] is normalized if and only if it is z 1 -regular of some order k < s. 8
lfg E B satisfies the W 2.3 Weierstrass preparation theorem in Bt · condition at position s, then there exists exactly one normalized polynomial w E B'[z 1 ] of degree s and one unit e E B such that g = e . w. PRooF: We apply the Weierstrass formula to f = zf . There are uniquely determined elements q E B and r E B'[zd with zf = q g + r and deg(r) < s (we choose an c < i such that ll zf - gg; 1 l l < stf ) . ·
But then zf - gg;1 = ( q - g;1 )g + r is a decomposition in the sense of the Weierstrass formula. Therefore, we have the estimate
llgs q - 1 11 < t� s ll zf - gg; 1 ll ·
1
� c < 1 � c < 1.
That means that g8 q and hence q is a unit in B . Let e := q - 1 and w == zf - r. Then w is a normalized polynomial of degree s, and e . w = q- 1 (zf - r) = g .
If there are two decompositions g = e 1 (zf - r!) = e2 (zf - r2 ) , then
z 1 = e1 1 . g + r1 = e2 I . g + r2 . 8
From the uniqueness condition in the Weierstrass formula it follows that e1 = e2 and r1 = r2 . • 2.4
z1 . 1
Corollary.
If g is
a
polynomial in
z1 ,
then e is also a polynomial in
An integral domain is a commutative nonzero ring in which the product of two nonzero elements is nonzero.
1 14
III. Analytic Sets
PROOF: We use the decomposition zf - gg; 1 = (q - g; 1 ) g + r . From the Weierstrass formula it follows that // r // < // z f - gg; 1 // · 1
Since w,
=
� E: < t� - 1 � c < tf .
1 , it is also true that // z f - ww; 1 // = // z f - w// = // r// < t� .
Therefore, g = e . w + 0 is a decomposition in the sense of the Weierstrass • formula, and the proposition follows from Corollary 2.2. The Weierstrass preparation theorem serves as a "preparation for the exami nation of the zeros of a holomorphic function." If the function is represented by a convergent power series g , and there exists a decomposition g = e· w with a unit e and a "pseudopolynomial" w (z1, z') = zf + A 1 ( z')z� - 1 + · · · + As (z') , then g and w have the same zeros. However, the examination of w is simpler than that of g .
Weier str ass P olyn omials. Now we turn to the proof of the Weierstrass formula and the preparation theorem for convergent power series. The ring Hn is an integral domain with l . If f E Hn and f (z) = I; 0 f>.. (z')z£ with h. = 0 for >. > s , then f is an element of the polynomial ring Hn - dz!] . If fs -=1- 0, then deg( f ) = s. lf f is normalized and f>.. (O') = 0 for >. < s, then f is zrregular exactly of order s, and f(z1 , O') = zf.
�
Definition. A normalized polynomial w E Hn - dz1 ] with and w ( z1 , O') = zf is called a Weierst rass polynomial.
deg(w)
=s
We have seen that a normalized polynomial w E Hn d z1 ] with deg(w) = s is a Weierstrass polynomial if and only if it is z1 -regular of order s . It follows easily that the product of two Weierstrass polynomials is again a Weierstrass polynomial. -
If g = e . w is the product of a unit and a Weierstrass polynomial of degree s,
then we also have that g is z 1 -regular of order s, since the unit e is z1-regular of order 0. We now show that conversely, every z 1 -regular convergent power series is the product of a unit and a Weierstrass polynomial.
2.5 Th eor em . Let g E Hn be z 1 -regular d orde r s . Then fo r every c > 0 and e v e y to E JR+. the re exists a t < to such that g lies in Bt, gs is a unit in Bt' , and //zf - gg_;-- 1 //t < c . tf .
PROOF: Let g = 0 gJ.. Z� be the expansion of g with respect to z . Then 1 g>. (O') = 0 for >. = 0 , 1 , . . . , s - 1 and g8(0') -=1- 0.
I;�
Since g is convergent, there exists a t1 < to with l / g / / t 1 < oo . Then g>. E Be1 for all >., and in particular,
2 . The Preparation Theorem
/(z ')
=
1 15
g8 ( 0') - 1 gs (z' ) - 1 E Bti .
Now, f(O') = 0. Thus there exists a t2 < t1 such that 1 1 ! / l t ' < 1 for all t < t2 . Therefore, g8 is a unit in Bt ' , and g an element of Bt . Let h := zf - gg; 1 . Then h E Bt for all t < t 2 , and we have an expansion h = 2::: � 0 h;.. z � with h. = 0 , h;.. = - g;.. g; 1 for >. -=/:- s, and h;.. (O') = 0 for >. = 0, 1 , . . . , s - 1 . 1f t1 > 0 is sufficiently small, then 00
E t1 . < t2 2' s
for all t = ( t 1 , t ' ) < t2. And since h;.. (O') = 0 for >. = 0 , . . . , s - 1 . for every small t 1 there exists a suitable t ' such that
Consequently, 1/ h//t < E . tf.
•
Remar k . In a similar manner one can show that if g 1 , . . . ,gN E C [z] are convergent power series and each gi is zrregular of order si , then for every E > 0 there is an arbitrary small t E JR+ for which gi E Bt , (g; )s, is a unit in Bt' and llzt' - gi (gi ) s/ 11 < E . t�'
Weier str ass Pr ep ar ation Theor em
2.6 Theorem (Weier str ass division for mula) . Let g E Hn be z 1 - regula r of order s. Thenfo r every f E Hn the re are uniquely dete rmined e lements q E Hn and r E Hn - dz1 ] with deg(r) < s such that f=
q · g + r.
Iff and g are polynomials in z1 with deg(g)
=
s, then q is also a polynomial.
PROOF: There exists a t E JR� and an E with 0 < E < 1 such that f and g lie in Bt., g8 is a unit in Bt ' , and ll z f - gg; 1 ll t < E . tf . It then follows from the division formula in Bt that there exist q and r with f = q . g + r .
Let two decompositions of f be given:
1 16
III. Analytic Sets
We can find a t E JR� such that f, q1 , q2 , r1 , r2 lie in Bt and g satisfies the vV-condition in Bt . From the Weierstrass formula in Bt it follows that q1 = q2 • and r 1 = r2 .
Let g E Hn be z1 regular of orde r s . Then the re exists a uniquely dete rmined unit e E Hn and a We ierst rass polynomial w E Hn - dzi] of degree s such that
2.7
Theorem (Weierstrass preparation theorem) .
g=e
·
w.
If g is a polynomial in z1 , then e is also a polynomial in z 1 . PRooF: There exists a t E JR� such that g satisfies the W-condition in Bt . The existence of the decomposition g = e . w with a unit e and a normalized polynomial w of degree s therefore follows directly from the preparation the orem in Bt . Since g is z1-regular of order s, the same is true for w. So w is a Weierstrass polynomial.
Now, w has the form w = zf - r, where r E Hn - 1 [z1] and deg ( r) < s. Thus, if there exist two representations g = e1 ( zf - r1 ) = e2 ( zf r2 ), it follows that 1 1 zf = e;- · g + r1 = e2 .g t r2 . The Weierstrass formula implies that e1 = e2 , • r1 = r2 and therefore w 1 = w2 -
.
Exercises
1 . Write a computer program to do the following: Given two polynomials f (w, x,y) (of degree n in w and degree m in x and y) and g(w, x, y) with g(w, 0, 0) = w 8 , the program uses the Weierstrass algorithm to determine y and r (up to order m in x and y) such that f = y . g + r. 2. Let f : p n- I x D --+ C be a holomorphic function andn 01< r < 1 be a real number such that ( H f(z' , () has no zero for z' E p - and r < I CI < 1 . 1 Then prove that there is a number k such that for every z' E p n the function ( H f(z' , () has exactly k zeros (with multiplicity) in D . Use this statement to give an alternative proof for the uniqueness in the Weierstrass preparation theorem. 3. Show that the implicit function theorem for a holomorphic function f : n e X (1: --+ (1: with f (O) = 0 and fzn (0) # 0 follows from the Weierstrass preparation theorem.
3.
Prim e Factorization
Unique Factorization. Let I be an arbitrary integral domain with 1 . Then I * : = I { 0} is a commutative monoid with respect to the ring multiplication, and the set JX of units of I is an abelian group . -
3.
Prime Factorization
1 17
Let a, b be elements of I*. We say that a divides b (symbolically a I b) if there exists c E I * with b = a . c. We can also allow the case b = 0. Then every element of I* divides 0, and a unit divides every element of I . Definition. Consider an element a E I * - Ix . 1 . a is called irreducible (or indecomposable) if from a = a1 . a2 (with a 1 , a2 E / * ) it follows that a 1 E / x or a2 E J x . 2. a is called p rime if a I a1a2 implies that a I a1 or a I a2 . Irreducible and prime elements can be defined in an arbitrary commutative monoid. In / * every prime element is irreducible, and in some rings (for example, in Z or in JR[X]) it is also the case that every irreducible element is prime. In Z[H] one can find irreducible elements that are not prime. Definition. I is called a unique facto rization domain (UFD) if every element a E J X can be written as a product of finitely many primes. One can show that the decomposition into primes is uniquely determined up to order and multiplication by units. In a UFD every irreducible element is prime and any two elements have a greatest common divisor (gcd). Every principal ideal domain2 is a UFD, and in this case the greatest common divisor of two elements a and b can be written as a linear combination of a and b. For example, Z and K[X] (with an arbitrary field K ) are principal ideal domains. So in particular, C [X] is a UFD .
Gaus s ' s Lemma. Let I be an integral domain. Two pairs ( a ,b), ( c ,d) E I x I * are called equivalent if ad = be. The equivalence class of a pair ( a,b) is called a fraction and is denoted by a fb. The set of all fractions has the structure of a field and is denoted by Q(I). We call it the quotient field of I. The set of polynomials f ( u) = a +a 1 u+ . + an un in u with coefficientsai E I 0 constitutes the polynomial ring J [ u] . The set J0[u] of monic polynomials in J[u] is a commutative monoid. Therefore, we can speak of factorization and irreducibility in J 0 [u] . 3 . 1 Gauss' s lemma) . Let I be a unique facto rization domain and Q = Q(I) . lf w 1 , w2 are ele ments of Q0 [u] with w1 w2 E J0 [u] , then W>, E JO[u] fo r >. = 1 , 2. PROOF: For >. = 1, 2, W>, = a>,,o + a;>,, I u + . . . + a>,,s,_ _ 1 us ,.. - l + u8" with ax,, E Q . Therefore, there exist elements d>- E I such that d>, . w>, E J [u] . We can choose d>, in such a way that the coefficients of d>, . W>, have no common divisor (such polynomials are called p rimitive). 2 A principal ideal domain is an integral domain in which every ideal is generated
by a single element.
1 1 1. Analytic Sets
1 18
We define d := d1d2 and assume that there is a prime element p with p I d. Then p doesn't divide all coefficients d>.a:>.,v of d>. . W>. . Let J-l>. be minimal such that p f d>.a>.,JL>- . Then JL ( d 1 wi) ( d2w2 ) = . . . + u �' ' + 2 ( da1 ,JL, a2,JL2 + something divisible by p ) + . . . .
Since I is a UFD, p doesn't divide (d1a1,JL, ) (d2a2,JL2 ) . So the coefficient of u�' ' + JL2 is not divisible by p. But since w1w2 has coefficients in I, every divi sor of d must divide every coefficient of d . w1 w2 = ( d1wi ) ( d2w2 ) . This is a contradiction ! 3 When d has no prime divisor, it must be a unit. But then d1 and d2 are also 1 units, and W>. = d>. (d>.w>.) belongs to I0 [u] . • 3 .2 Corollary. 1.
2.
3.
4.
Let I
re
a unique factorization domain.
If a E J0 [u] is prime in Q[u] , then it is also prime in I0 [u] . If a E J0 [u] is reducible in Q[u] , then it is reducible in J0 [u] . Every element of I0 [u] is a product offinitely many prime elements. If E J0 [u] is irreducible, it is also prime. a
PROOF: l . Let a E J0 [u] be a prime element in Q[u] . 1f a divides a product a' a" in I0[u] , then it does so in Q [u]. Therefore, it divides one of the factors in Q[u] . Assume that there is an element b E Q[u] with a' = ab. By Gauss's lemma b E J0 [u]. This shows that a is prime in J0 [u] .
E J0 [u] be a product of nonunits a 1 , a2 E Q[u) . If ci E Q is the 1 1 highest coefficient of ai , then c1 c2 = l ci 1 ai E Q0 [u) and a = ( c1 a1 ) ( c;- a2) . 1 By Gauss c; ai E I0 [u] , and these elements cannot be units there. So a is reducible in I0 [u] .
2. Let
a
,
3 . Every element a E J0 [u) is a finite product a = a 1 . . . az of prime elements of Q[u]. One can choose the ai monic, as in (2). Using Gauss's lemma several times one shows that the ai belong to I0 [u] . By (1) they arc also prime in JO [u].
4. Let a E J0 [u] be irreducible. Since it is a product of prime elements, it • must be prime itself. Remark. In the proof we didn't use that I is a unique factorization do main. We needed only the fact that Q[u] is a UFD (since Q is a field) and the statement of Gauss's lemma: If a 1 , a2 E Q0 [u) and a1 a2 E I0 [u] , then ai E I0 [u) for i = 1, 2. 3
The original version of Gauss's lemma states that the product of primitive poly nomials is again primitive. The reader may convince himself that this fact can be derived from our proof.
3. Prime Factorization
Factorization in Hn . I = H, .
1 19
Now the above results will be applied to the case
Definition. Let f E Hn, f = L� o P>. be the expansion of f as a series of homogeneous polynomials. One defines the order of f by the number
ord(f) : = min{>. E N0
:
P>. -=1- 0}
and
ord(O) := oo.
(See also Exercise 1 .4 in this chapter) Then the following hold:
ord(f) > 0 for every f E H,. 2. ord(f) = 0 -<==* f is a unit. 3. ord(h . h) = ord(h) t ord(f2). 1.
3.3
Theorem.
PROOF:
Hn
is
a
unique factorization domain.
We proceed by induction on n .
For n = 0, Hn = C is a field, and every nonzero element is a unit. In this case there is nothing to show. Now suppose that the theorem has been proved for n 1 . Let f E Hn be a no nu nit, f -=1- 0. Iff is decomposable and f = h · h is a proper decomposition, then ord(f) = ord(h ) + ord(/2), and the orders of the factors are strictly smaller than the order of f. Therefore, f can be decomposed into a finite number of irreducible factors. -
It remains to show that an irreducible f is prime. Assume that f I h /2, with f>. E (Hn )* for >. = 1 , 2. There exists a shear a such that h o a, h o a and f o a are zi-regular. If we can show that f o a divides one of the f>. o a , then the same is true for f and f>. . Therefore, we may assume that h, /2, and f are ZI-regular. By the preparation theorem there are units ei, e2, e and Weierstrass polyno mials WI , w2 , w such that h = e i -WI , h = e2 ·w2, and f = e · w. Then w divides wi w2. If wi w2 = q . w with q E Hn , then the division formula says that q is uniquely determined and a polynomial in zi . So w divides wiw2 in H�_ dzi] . Since w is irreducible in H,, it must also be irreducible in H�- I [zi] · By the induction hypothesis Hn- I is a UFD, and therefore w is prime in H�- I [zi] . It follows that w I WI or w I w2 in H�_dzi] and consequently in Hn. This means • that f I h or f I h in Hn ·
Hensel' s Lemma. Let w E Hn[u] be a monic polynomial of degree s. There is a polydisk P around 0 E en where all the coefficients of w converge to
1 20
III.
Analytic Sets
holomorphic functions. Therefore, we can look at w as a parametrized family of polynomials in one variable u . By the fundamental theorem of algebra w(O, u) splits into linear factors, and the same is true for every w(z, u) with z E P. We now show that such splittings are coherently induced by some splitting of w in H� [u] , at least in a neighborhood of 0 . 3.4
Let w(O, u) = TI�= I ( u - C>.)8" be the decomposition into linearfacto rs (with c, -=/=- cJL for v -=/=- f-L and si + · . . + s z = s). Then there are uniquely determined polynomials WI, . . . ,wz E H� [u] with the following prope rties: Hensel's lemma.
deg(w>.) = S>., fo r ).. = 2. W>. (O, u) = (u - C>.)8" . 3 . W = WI · · · · · Wz . 1.
1 , .. . , l .
PROOF: We proceed by induction on the number l. The case l We assume that the theorem has been proved for l - 1 .
=
1 is trivial.
First consider the case w(O, 0) = 0 . Without loss of generality we can assume that c1 = 0. Then w(O, u) = u8' h( u ) , where h is a polynomial over C with deg(h) = s - SI and h(O) # 0. So w is u-regular of order SI , and there exists a unit e E H� [u] and a Weierstrass polynomial wi with w = e . WI · Since wi (O, u) = u8' , it follows that •
l
e(O, u) = h(u) = IT (u - c>.)8" >.=2 By induction there are elements w2, . . . , wz E H�[u] with deg(w>.) = S>., W>. (O, u) = (u - C>.) 8" and e = w2 · · · w1 . Then w = wiw2 · · · wz is the de sired decomposition.
If w(O, O) -=/=- 0, then we replace w by w'(z, u) := w(z, u + ci ) and obtain a
decomposition w'
= wi . . . wf
as above. Define
This gives a decomposition w = w1 . . . wz in the sense of the theorem. The uniqueness statement also follows by induction. •
The Noetherian Property. Let R be a commutative ring with 1 . An R-module is an abelian group M (additively written) together with a composition R x M -+ M that satisfies the following rules:
r(xi + 2 2 ) = rxi + rx2 for r E R and xr, x2 E M . 2. (ri + r2)x = ri x + r2x for ri , r2 E R and x E M . 3 . ri (r2x) = (rir2)x for ri , r2 E R and x E M . 1.
3. Prime Factorization
4. 1
· x
121
= x for x E M.
These are the same rules as those for vector spaces (and the elements of a module are sometimes also called vectors). However, it may happen that rx = 0 even if r -=!=- 0 and x -=1=- 0. Therefore, in general, an R-module has no basis. So-called free modules have bases by definition. An example is the free module RQ := R x . . . x R (q times), with a basis of unit vectors. An example of a nonfree module is the Z-module M : = Z/6Z, where 2 · 3 = 2 · 3 = 0.
If M is an R-module, then a submodule of M is a subset following properties: 1. 2.
N
C M with the
x,
r
y E N =? x + y E N . E R and x E N =? r x E N .
A submodule of an R-module is itself an R-module. Example The ring R is also an R-module. The composition is the ordinary ring mul tiplication. In this case the submodules of R are exactly the ideals in R. An R-module M is called finite if there is a finite set { x 1 , . . . , xn } C M such that every x E M is a linear combination of the xi with coefficients in R. The free module Rq is obviously finite. But Z/6Z is also finite, being generated by the class I. Definition. An R-module M is called noethe rian if every submodule N C M is finite. A ring R is called noetherian if it is a noetherian R-module. This means that every ideal in R is finitely generated (in the sense of a module). 3.5
of
Proposition. ideals
becomes stationary,
Let R be a noetherian 1ing. Then any ascending chain Io
i.e.,
c
/1
c Jl c · · · c R
there is a ko such that h
=
ho fo r k > ko .
PROOF: The set J : = u� 0 h is obviously an ideal. Since R is noetherian, J is generated by finitely many elements h , . . . , fN . Each f" lies in an ideal h., . If ko = max ( k1 , . . . , kN ) , then all f" are elements of ho . So h = ho for
k > ko.
•
3.6 Theorem. PROOF:
IfR is
a
noethe rian ring, then Rq is a noetherian R-module.
We proceed by induction on q.
122
III.
Analytic Sets
The case q = 1 is trivial. Assume that q > 2 and the theorem has been proved for q - 1 . Let M c Rq be an R-submodule. Then I : = { r E R : 3 r' E Rq- I with (r, r' ) E M ) is an ideal in R and as such is finitely generated by elements ri , . . . , rz . For every r;.. there is an element r� E Rq I such that r;.. : = (r;.. , r� ) lies in M . The set M ' := M n ( {0} x Rq - I ) can be identified with an R-submodule of Rq - r , and by the induction assumption it is finite. Let r ;.. = ( 0, r�. ) , >. = I + 1 , .. . , p, be generators of M '. An arbitrary element x E M can be written in the form x = (xi , x' ) with J
XI E I. Then XI = I:>-. = l a;.. r;.. , a;.. E R, and
x-
l
2.:: a;.. r ;..
X=l
=
(0, x' -
l
2.:: a;.. r �)
X1
E M' .
That is, there are elements az + I , . . . , ap E R such that x
1
- 2.:: a;.. r ;.. = X1
p
2.::
>-.= 1 + 1
a;.. r;.. .
Hence { ri , . . . ,rp} is a system of generators for M . 3 .7 Riickert b a sis theorem.
noetherian.
•
The ring Hn of convergent power series is
PROOF: We proceed by induction on n. For n = 0, Hn = C, and the statement is trivial. We now assume that n > 1 and that the theorem has been proved for n - 1 . Let I C Hn be a nonzero ideal and g -=/=- 0 an element of I. Without loss of generality we can further assume that g is 2 1-regular of order s .
Let = <1>9 : Hn -+ ( Hn- I ) 8 be the Weierstrass homomorphism, which is defined in the following manner: For every f E Hn there are uniquely defined elements q E Hn and r = ro + r1zi + · · · + rs - I zf I E Hn- dzi] such that f = q · g + r. Let (f) : = (ro , . . . , rs- d ·
Now, is an Hn- rmodule homomorphism. By the induction hypothesis Hn- I is noetherian, and so (Hn- ! ) 8 is a noetherian Hn- rmodule. Since M : = @ ( Ijs an Hn-I -submodule, it is finitely generated. Let r;.. = (r6"l , . . . , r�?:) I ) , >. = 1 , .. . , l , be generators of M . If f E I is arbitrary, th en J = q · g + r with r = ro + ri zi + · · · + rs -I zf - I , and there are elements ai , · . . , az E Hn- I such that (ro, ri , · . . , rs - 1 ) = <1>9 ( ! ) L�= l a;.. r;.. . Hence we obtain the representation
4.
f
Branched Coverings
1 23
l
= q.
g + z:::a ).. . (rg'' ) + riA) Z1 + · + r�A\ zf- 1 ) , .>..= 1
The set {g, r<1l , . . . , rUl } with r< A) of generators of I .
·
=
·
rbA) + riA) z1 +
·
·
1 · + r�A) 1 zf - is a system •
Exercises 1 . Prove that O(q is not a UFD . 2 . Let M be a finite Hn-module, and m C Hn the maximal ideal. If M m . M , then M = 0 . 3. Let f n: pn - l x D --+ C be a holomorphic function such that for every z' E p - 1 there is a unique solution Zn = cp(z') E D of the equation f(z', zn ) = 0 . Use function theory of one variable to show that 'P is continuous, and use Hensel's lemma to show that 'P is holomorphic. 4. Let f E Hn be 2 1-regular, f = e · w with a unit e and a Weierstrass polynomial w E Hn - dz1 ] . Prove that f is irreducible in Hn if and only if w is irreducible in Hn - dz1] . 5. Show that f( z , w) : = z 2 - w 2 ( 1 - w) is irreducible in the polynomial ring
4.
Br anched C over ings
Germs. Let B c en be an open set and Zo E B a fixed point. A local holomorphir: function at z0 is a pair (U, f) consisting of an arbitrary neigh borhood U = U(z0) C B and a holomorphic function f on U. Two such functions f : U --+ C and g : V --+ C are called equivalent if there is a neigh borhood W = W (z0) c U n V such that f i W = g i W . The equivalence class of a local holomorphic function (U, f) at z0 is called a g e rm (of holomorphic functions) and is denoted by fzo . The value f (z0) as well as all derivatives of f at z0 (and therefore the Taylor series of f at zo) are uniquely determined by the germ. On the other hand, if a convergent power series at z0 is given, then this series converges in an open neighborhood of z0 to a holomorphic function f , and the germ o ff determines the given power series. So the set Ozo of all germs of holomorphic functions at z0 can be identified with the @ algebra of all convergent power series of the form L:v > o ( z - z0 ) v . This algebra is isomorphic to the algebra Hn and has the same algebraic properties. Let fzo # 0 be any element of Ozo with f (z0) = 0. Then there are a neigh borhood U(z0) c B , a neighborhood W (O) C @+ a holomorphic function e on U, and holomorphic functions a1 , . . , a, on W such that after a suitable change of coordinates the following hold: .
1 24
III. Analytic Sets
e (z ) -!= for every z E U. 2. f(z) = e(z) . w (z - zo) for w (wi , w') = wi + ai (w')zf- I + . . . + as (w') .
1.
0
P seu d op olynomials. A pseudopolynomial of degree s over a domain G c en is a holomorphic function w in G X c that is given by an expression I w(u, z) = u s + h1 (z)u s - + · . . + hs (z) , with h i , . . . , hs E 0, where 0 = 0 (G) denotes the ring of holomorphic functions on G. The set of pseudopolynomials of any degree over G will be written as 0°[u] . We begin with several remarks on the algebraic structure.
4. 1 P r op o sition . If rmg 0
=
G zs a domarn, z. e., a connected open set, then the
0( G) zs an integral domarn.
PROOF: We need to show only that 0 has no zero divisors. Assume that JI , h are two holomorphic functions on G with both fi =/=- 0. Since G is a domain, their zero sets are both nowhere dense in G, and there is a point • z E G with h (z) . f2 (z) -!= 0. So h
. h =/=- 0.
It also follows that 0° [u] is free of zero divisors. We denote by Q the quo tient field of 0. Then the group Q[u] x of units in the integral domain Q[u] consists of the nonzero polynomials of degree 0. If 0* C 0 is the multiplica tive subgroup of not identically vanishing holomorphic functions on G, then Q [u] x n 0 = 0* .
4.2 P rop o sition .
If w� , w2 E Q0[u] are pseudopolynomials with WI . w2 E 0°[u] , then WI , W2 E 0° [u] .
PROOF: If w = U s + (JI/gi)us - I + . . . + Us /gs ) is an arbitrary element of Q0 [u] , then for all z E G the germs gi,z are not 0. For a moment we omit the i . If the quotient of fz and gz is holomorphic, i. e . , fz = h, . gz with h, E Oz , then h, is uniquely determined and there is a ball B around z in G such that hz comes from a holomorphic function h on B and the equation f = h . g is valid in B . If we take another point z' E B, the germ of h at this point is the quotient of the germs of f and g at this point. So z f--t hz (z) defines a global holomorphic function h on G. We write h = f j g. Thus, if w, := u s + ( (h )z/(g l )z)u s - I + · · · + (( Js )z/(gs )z) lies in O� [u] for every z E G, then w E 0° [u] . Now we apply Gauss's lemma in the unique factorization domain Oz Hn Let w := W I . w2 • Then (wi )z . ( w2)z = w, E O� [u] for every z E G. Conse quently, the coefficients of (wi )z are holomorphic, and by the remarks above • this means that w, E 0° [u] . "'
4 . Branched Coverings
1 25
The following is an immediate consequence of the above two propositions:
4.3 Theorem. Let G E en i. e., every element
is
be a domain. Then 0°[u] is a factorial monoid; a product cl finitely many primes.
Euclidean Domains Defin ition . An integral domain I is called a Euclidean domain if there is a function N : I * -+ N0 with the following property (division with remainder) : For all a, b E I , b i= 0, there exist q, r E I with 1 . a = q · b + r, 2. r = 0 or N (r) < N(b) . The function N is called the norm of the Euclidean domain. Examples
1. Z is a Euclidean domain with N(a) := Ia!.
2. lf k is a field, then k[x] is a Euclidean domain, by N( J ) : = deg ( J ) . Every Euclidean domain I is a principal ideal domain and thus factorial. lf a, b are elements of I, then the set of all linear combinations
r · a + s · b i= 0 ,
r, s E I ,
has an element d with N (d) minimal. The element d generates the ideal a = { ra + sb : r, s E I } and is a greatest common divisor of a and b. It is determined up to multiplication by a unit. Now assume again that G is a domain in en , 0 = 0( G) , and Q = Q( 0) . Then Q[u] is a Euclidean domain. 1f W I , W2 are pseudopolynomials in 0°[u] , there is a linear combination in w = r iwi + r2w2 =/=. 0 in Q[u] with minimal degree. It can be multiplied by the product of the denominators of the coefficients in r i and r2 . Then ri , r2 , and w are in O[u] , and w is a greatest common divisor of wi , w2.
The Algeb r aic Derivative. Let 0 and Q be as above. lf w E 0° [u] has positive degree, then it has a unique prime decomposition w = W I · . . w1 . The degree of each wi is positive. We say that w is (a pseudopolynomial) without multiple factors if all the w, are distinct. The (algebraic) derivative of a pseudopolynomial is defined as follows. lf w = l:�=O a v u v , then D(w) : = l:�= I v · a, . u v - I . Thus D (wi t w2) D (wi . w2)
_
D(wi) + D(w2 ) , D(w!) . w2 + W I . D (w2)
1 26
1 1 1. Analytic
Sets
4 . 4 Theorem. An element w E 0° [u] is without multiple factors if and o n ly zf a greatest common divisor of w and D (w) is a function h E 0* .
has the irreducible wi as a multiple factor, then D(w) is also divisible by wi · This is also true in Q[n] . So a greatest common divisor is certainly not a function h E 0*. Assume now that w = ni Wi has no multiple factor. Then PROOF:
If w
If the degree of the greatest common divisor 1 of w
and D(w) is positive, then 1 is a product of certain w, So at least one w, divides both w and D( w ) . Then wi divides w 1 . . . D (wi ) . . . w1 and hence D(wi ) · This is not possible, because D(wi ) has lower degree. So the degree of the greatest common divisor is 0, • and therefore it is a function h E 0*. Symmetr ic Polynomial s.
Definition . A polynomial p E Z[u 1, . . . ,us ] is called symmetric if for all i , j we have p (u1, . . . ,u i , . . . uj , . . . ,us) = p(u l , · . . , uj , . . . , u i , . . . ,us)·
There are the elementary !>ymmetric polynomials a1 , . . . , a, defined as follows: a 1 (u 1 , . . . , us ) - U1 + · · · + Us , a2 (u 1 , . . . , us ) - u l (u2 + . . . + us ) + u 2 (u3 + . . . + us) + . . · + us - 1Us ,
The following result is proved, e . g . , in the book [vdW66] 4.5 Theorem. lfp E Z[u1 , . . . , us] is symmetric, then there is exactly one polynomial Q(y 1 , . . . , Y s) E Z[y 1 , . . . , Ys ] such that p = Q( a 1 , . . . , a, ). Consider the special symmetric polynomial pv(u1 , . . . , us) = IJ (u i - Uj ) 2
The Discriminant.
i <j
(square of the Vandermonde determinant). Since it is symmetric, there is a uniquely determined polynomial Qv ( y1 , . . . , y,) with integral coefficients such that
4. Branched Coverings
1 27
Definition . If w u8 +h1 (z)u s - l + · . + hs(z) is a pseudopolynomial in 0° [u] , then A, = Q v (- h1 , h2 , . . . ( - 1 ) 8 hs ) is called the discriminant of w. It is a holomorphic function in G, and we denote its zero set by D,. .
=
,
It is well known from the theory of polynomials that where w 1 , . . . , w , are the zeros of the polynomial if and only if there is a pair i # j with w; = w1 .
u
f-t
w(u, z) . So �w (z)
=
0
Assume now that w is without multiple factors. Then there is a linear combi nation of w and D(w) that is a function h E 0* . We restrict to a point z E G with h(z) # 0. Then the greatest common divisor of w(u, z) and D(w) (u, z) is 1 . This means that w( u, z) has no multiple factors; i.e., the zeros of w( u, z) are all distinct. So �w (z) # 0, and D, is nowhere dense. Example Let G c en be a domain, a, b holomorphic functions in G, and u2 - a(z) . u + b(z) . In this case
w(u, z)
pv(u 1 , u 2 ) = IJ (u ; - u1 ) 2 = (u1 - u 2 ) 2 = (u1 + u2 ) 2 - 4 · u l u 2 . i <j
·
�w (z) = Q v ( a(z) , b(z)) = a(z) 2 - 4b(z). If z E G and �w (z) # 0, there are two different solutions of w(u, z) = 0. Hyper sur fa ces. We use the theory of pseudopolynomials to study ana lytic hypersurfaces. Such analytic sets are locally the zero set of one holomor phic function. Assume that f is a holomorphic function in a connected neigh borhood of the origin in en+ I that is not identically 0. Without loss of gener ality we may assume that A = N( f) contains the origin. Then a generic com plex line £ through 0 meets A in a neighborhood of 0 only at the origin. After a linear coordinate transformation, £ = { ( u ;z) : z = 0 } is the first coordinate axis. By the Weierstrass preparation theorem fco,o) = e (o,o) · wco,o) in the ring Hn+I , where e co, O ) is a unit in Hn+l , and W(o,o) E Hn [u] a Weierstrass poly nomial. We can represent the germs locally by holomorphic functions. Thus there is a domain G C e n containing 0 , and a disk D = { u E e : lui < r} such that in U : = D x G there are a holomorphic function e that does not vanish in U and a pseudopolynomial w over G with f = e . w in U. We may assume that A n (8 D x G) = 0 . Therefore, the zero set of f in U is that of W.
128
1 1 1. Analytic Sets
We can decompose w into prime factors. Using the fact that any power of a prime factor vanishes at the same points as the prime factor does, we may assume that w is without multiple factors. Then the discriminant A, is not identically zero in G. We set D, = {z E G : 6w (z) = 0 } .
4 . 6 Theorem (on branched cover ings). If z o E G - D, there are a neigborhood W = W(zo) C G - D, and holornorphic functions ft , . . . ,fs zn W with fi ( z ) -=/= J1 (z) fo r i -=/= j and z E W such that
w(n, z ) = (n - JI (z) ) · . . (n - fs (z)) in C W There are fewer than s points over any point zo E D, (see Figure III. l ) . X
·u
A
G
Figure 1 1 1. 1 . A branched covering over G
A point z E G above which there are fewer than s points is called a branch point. All points of the discriminant set Dw are branch points. Over all other points our set A is locally the union of disjoint graphs of holomorphic func tions, and is therefore regular. PROOF: For zo E G - D, the polynomial w(n, zo) has s distinct roots. We write w(n, zo) = (n - ci ) · · · (n - c8 ), where the ci all are distinct. If w(n, z ) = '1L3 + h1(z)n 8 - 1 + . . . + hs (z ) , then the germ
Wz0 := n 2 + (hi )z01L 8 - 1 + + (hs)z0 Hn- By Hensel's lemma it has is a polynomial over Oz0 w, , = w1 .zo . . . ws ,zo with the following properties: ·
"'
·
·
a
decomposition
4.
1 . w,,z0 (u, zo ) = U 2. deg (wi,zo ) = 1 .
- C;
Branched Coverings
1 29
for i = 1 , . . . , s .
We have w;,z0 u - ri , with ri E H,. There are a connected neighborhood W(zo) C G - Dw and holomorphic functions h , . . . , is in W such that the power series ri converge to j;. Since the germs of w and ( u - h) . . . ( u - is ) coincide at z0 , it follows from the identity theorem that =
wkxw
=
(u - h ) . . . ( u - is ) ,
and since W c G- D, it also follows that ii (z) -=!= i1 ( )for i -=!= j and z
z
E W.
•
Examples 1. Let G = e and w = zf - z2 . Then the discriminant is given by A, (z2) = 422, and Dw = {0} c C. If z2 E @*, there is a neighborhood W C e - D, where VZ2 is well defined. There we have w = (z1 - VZ2) . (z1 + VZ2) · This gives a surface above e that is a connected unbranched 2-sheeted covering over e- {0}. The point 0 is a branch point. This is the (branched) R iemann surface of y'Z. The unbranched part was already discussed in Section 1 1.8. 2. A completely different situation is obtained if we take w = zf - z� (z1 - z2 ) (z1 + z2 ) . The discriminant is 4z� in this case, and the discrim inant set D, is again the origin in C. The set A consists of two distinct sheets, which intersect above 0, and both are projected biholomorphically onto C. The set A - { 0 } , i.e., A without the branch point, is no longer connected. 3. In higher dimensions the situation is even more complicated. Let us con sider the analytic set A = N (f) , where f ( z1 , . . . , Zn ) = z�' + . ' . + z�n with si > 2 for i = 1 , . . . , n. This is a very simple holomorphic function. The derivatives are iz, = si . zf'- 1 , and their joint zero set consists only of 0 E e n . So all other points of A are regular. Every line £ through the origin lies completely in A, or i has a zero of order s with s > min( s 1 , . . . , s,) on £ at the origin. Therefore, there is no line that intersects A in 0 transversally. From this one can conclude that 0 is in fact a singular point of A (see, for example, Exercise 8.2 in Chapter I). Now we look on f : en -+ e as a fibration with general fiber ·
At = {z E e n : z�' + · · · + z�n = t} .
Then A = Ao has an isolated singularity, while all other sets At are regular everywhere. We call the family (At)tEIC a deformation of A.
130
III. Analytic Sets
The Unbranched Part . We assume that
w(u, z) a pseudopolynomial over A = { (u, z)
G
c en is a domain and
G of degree s. We set G' = G - Dw , E e x G : w(u, z) = 0} ,
A' = AIG' the part of A over G'. Then A' is an unbranched covering of G'. It is an n-dimensional submanifold of e x G', and we have the canonical projection 1r A --+ G. If ( u0 , z0 ) E A' is a point , there is a small neighbor hood and
:
B = B ( u0 , z0) c A' that is mapped by 1r holomorphically and topologically onto a ball around z0 in G'. We also call B a ball. The holomorphic map (7r i B ) - 1 : 7r(B) --+ en+l is a local parametrization of A'. A complex function f in B is called holomorphic if f o (7r i B ) - 1 is holomorphic. In particular, the components of 1r itself are holomorphic functions on B.
For holomorphic functions in B we have the same properties as for holomor phic functions in a domain of en . For example, the identity theorem remains valid, and we obtain the following results: 4. 7 Proposition. Assume that A1 is a connected component of A' and that M is an analytic subset of A1 . Then M = A1, or M is nowhere dense in A1 . 4.8 Proposition. If f is a holomorphic function on A', and A1 a connected component of A', then either f vanishes identically on A1 or its zero set is nowhere dense in A1 . 4.9 Proposition. Let A1 again be a connected component of A'. Assume that M is a nowhere dense analytic set in A1 and that f is a holomorphic function in A1 - M that is bounded along M . Then f has a unique holomor phic extension to A 1 . Decompositions. We consider the interaction between the decomposi
tion of a pseudopolynomial into irreducible factors and the decomposition of its zero set into "irreducible" components. 4.1 0 Proposition. Let G c en be a domain and w(u, z) a pseudopolyno mial over G without multiple factors. Then w is irreducible if and only if the intersection of its zero set A with e x ( G - E) is connected for every nowhere dense analytic subset E C G which contains the discriminant set Dw . Since locally over G' = G - Dw the set A' = AIG' looks like a domain in en , a nowhere dense analytic set does not disconnect A' , locally and globally. Therefore, we may assume that E = Dw·
PROOF:
If w is not irreducible, every factor wi defines an analytic set Ai over G - Dw . The intersection of different Ai is empty. So AI(G - Dw) is not connected.
4.
Branched Coverings
131
If, on the other hand, AI G1 has the connected components A;, i = 1 , . . . , s , with s > 0, then for any point z E G-Dw there is a ball B C G - Dw around z such that Ai lE splits into graphs of holomorphic functions f1 : j = 1 , . . . , si. In each case we form the pseudopolynomial Wi = ( u - h ) · · · (u - is , ) . The zero set of this wi is exactly Ai I B, and it determines w; and vice versa. So over the intersection of two different balls the pseudopolynomials must be the same, and thus we obtain global holomorphic functions wi in G - Dw . If z E Dw , then there is a neighborhood W of z such that Ai i (W - Dw ) C AIW is a bounded set. S o the coefficients of w; are bounded over this neighborhood and extend holomorphically to G. We also denote this extension by Wi , and • for reasons of continuity it follows that w = WI · · · Ws. If the w; are the irreducible factors of w, we call their zero sets A; the irre ducible components of A. The sets A� = A; IG' are the connected components of AIG'. 4.11 Proposition. Assume that w * , w are pseudopolynomials without mul tiple factors over a domain G and that A * = { w * = 0} C A = { w = 0}. Then w * is a factor of w. Let D denote the union of the discriminants of w * and w. It is a nowhere dense analytic set in G. Over G - D we decompose the two unbranched coverings into connected components. There we have A * = AI U · · · U As• and A = AI U · · · U As with s * < s. This yields pseudopolyno mials over G - D that extend to pseudopolynomials WI , . . . , Ws over G, with • w * = WI · · · Ws• and w = WI · · · Ws. This implies the result.
PROOF:
The following result is proved analogously. 4.12 Proposition. Assume that w is free of multiple factors and that A = {w = 0} is the disjoint union of two nonempty sets M', M" that are closed in C x G. Then there are pseudopolynomials w', w" over G with M' = {w' = 0}, M" = { w" = 0} , and w' · w" = w. The construction is first carried out outside Dw . \Ve set G' = G - Dw and use the fact that every nonempty open subset of A' = AIG' must be a union of connected components of A'. If w = WI · · · ws is the decomposition into irreducible factors, then we may assume that there is an s * with 0 < s * < s such that for w' = WI · · · Ws • and w" = Ws• + I · · · Ws we have M' IG' = { (u, z) E C x G' : w '( u , z) = 0} and M" IG' = {( u , z) E C x G' : w" ( u z) = 0} .
PROOF:
,
It is now essential that in a continuous family f ( u, z) of holomorphic functions of one variable u the zeros depend continuously on the family parameter z ( "continuity of roots" ) . We omit the proof here. If we apply this fact ( and the equations A = M' U M". w = w' · w" ) , we get that the sets M'IG' and
132
III.
Analytic Sets
M " I G' are not empty (i.e., 0 < s * < s) and that their closures in e x G are M', respectively M". • Project ions. In the next section we will investigate zero sets of several
holomorphic functions. Here we begin with the simplest case, the common zero set N of a pseudopolynomial w over G C en and an additional holomor phic function f in a neighborhood of A = { (u, z) E e x G : w(u, z) = 0}. Our method involves the projection of N to G.
4.13 Proposition. Assume that f = f(u, z) is a continuous function on A that is holomorphic outside of e x Dw and does not vanish identically in a neighborhood of any point of A. Then the projection of N = {! = w = 0} to G is an analytic set N' = {! = 0}, where f is a holomorphic function in G that does not vanish identically.
PROOF: If z E G - Dw , we have a ball B C G - Dw around it such that
over B our w has the form w(u, z) = (u - fi (z)) · · · (u - fs (z) ) . The function f does not vanish identically on any graph u = fi · Consequently,
f(z)
:=
f( JI (z) , z) · · · f( Js (z) , z)
does not vanish identically. In the usual way we obtain the holomorphic function f in the entire set G - Dw · It is bounded along Dw . So it extends to a holomorphic function in G. • Now consider the following situation: Assume that G is a domain in e n and that w is a pseudopolynomial over G without multiple factors. Let f be a holomorphic function in a neighborhood of A = {w(u, z) = 0} C e x G that does not vanish identically on any open subset of A and define
N
:=
{ (u, z) E e x G : w(u, z)
=
f(u, z)
=
0}.
Denote by N' the projection of N to G. We want to give a definition for "unbranched points" of N. The difficulty is that there may exist such unbranched points of N lying in the set of branch points of w.
4.14 Proposition. For any point z0 E N' there is an arbitrarily small linear coordinate change in z1 , . . . , Zn such that thereafter the line parallel to the z1 -axis through zo intersects N' in zo as an isolated point. In such coordinates there is a neighborhood U(zo) C G, a domain G' in the space en - l of the Variables Z1 = (z2 , . . . , Zn ) , and a pseudopolynomial w'(z1 , z') over G' such that { (z1 , z') E e x G' : w'(z1 , z') = 0} = N' n U. A "small" linear change of the coordinates z1 , . . . , Zn means here that the transformation is very near to the identity. Since f does not vanish
PROOF:
4.
Branched Coverings
133
identically, after a small generic coordinate transformation the line parallel to the zra.xis through zo intersects N' in z0 as an isolated point . And then • it is also clear that U, G' , and w ' with the desired properties exist. Let us now assume that we have chosen a point zo E N' and suitable coordi nates as above, and that U, G' , and w ' have also been chosen.
Definition. In the given situation, a point (u, z ) E N n (C x U) is called an unbranched point of N if z E N' - C x Dw' and there is a neighborhood V = V(z) C N' - C x Dw' with a holomorphic function g on V such that N n (C x V) is the graph { u = g(w) : w E V}. (Figure III.2 shows the situation. )
u
A
..... . :--...
o
examples of unbranched points �
Figure 111.2.
C
X
Dw'
Branched and unbranched p oints of N
4.15 Theorem. In the given situation, in every neighborhood of an arbi trary point ( u 1 z i ) of N n (C x U) there are unbranched points of N. ,
PROOF: We may assume that
z1
E N' - (C x Dw' ) . Then we take a small
neighborhood w = E X ul of (u l , z i ) such that the following hold: ul c u - (C X Dw' ) . 2 . ( u � , zi ) is the only point of A above
1.
z1
in W.
134
3.
III. Analytic Sets
The set A n W is defined by a pseudopolynomial w * over
Setting N� c N' n projection 1r : W --+
U� to be the image set of W n N under U1 , take Ut so small that N� = N' n U1 .
U1 . the canonical
We restrict w * to N� and replace possible multiple factors by one factor at a time. So we obtain a new pseudopolynomial w1 without multiple factors over N� such that
{ (u, z) E E x N� : w1 (u, z) = 0 } , E a suitable disk. near to (u1 , z1) we can find points (u z , z2) E N n W lying over
NnW=
Arbitrarily N{ - Dw, . All of these points are unbranched points of N, since we can find neighborhoods W2 (u 2 , z2 ) c W and U2 (z 2 ) C U1 with the same properties as W and U1 • Now choose U2 so small that it contains no point of Dw, and that Wt i N' n u2 decomposes into linear factors. Then every sheet of A I (N' n U2) • with the property that it contains points of N is a graph over N' n U2 . Exercises 1 . Prove that every symmetric polynomial in u1, . . . , Us can be written as a polynomial in the power SUmS Sk : = U � + . . . + u�. 2 . Calculate the discriminant of a cubic polynomial. 3. Let D : = Dr (O) C C and f be a holomorphic function in an open neigh borhood of D without zeros in 8D. 1f f has in D the zeros c1, . . . ,c8 (some of them may be equal), then
_ 1 r 21r i len
f u, z1 ,
:
4. Let
D
Zn -
("'' ! '(( ) d( = t em . f(( ) j=l 1
x P" --+ C be a holomorphic function in the variables
Assume that f is u-regular of order s and that for every z E P" the function u f--t J(u, z) has exactly s zeros u1 (z) , . . . , u 8 (z) (with multiplicity) in D. Show that the coefficients of the "pseudopoly nomial" w(u, z) : = n;=l (u - Uj (z)) are holomorphic. 5. Prove the "continuity of roots": Let f(u, z) be u-regular at the origin. Show that there is an r > 0 such that if g(z) is a function defined in a neighborhood of 0 with lg(z) I < r and f(g(z), z ) = 0 ,then g is continuous at 0. 6. A complex function f on an analytic set A in a domain G C en is called holomorphic if it is locally the restriction of a holomorphic function in the ambient space. (a) Let A : = {(w, z) E C 2 : w2 - z3 } be the Neil parabola. There is a bijective holomorphic parametrization of A given by w = t3 and z = t2 Describe the holomorphic functions on A as functions of the parameter t. Is there a meromorphic function on A that has a pole at oo with main part tm ? •
•
•
,
•
5.
135
Irreducible Components
(b) Show that the analytic set A = {(w, z 1 , z2 ) E C 3 : w2 = z1 z2 } is not regular at the origin. Consider the holomorphic map (t 1 , t2 ) �--+ (t 1 t2 , ti, t§). It is a "two-to-one" map. Describe the local holomorphic functions on A in t1 , t2 • 7. Prove that there is a topological holomorphic map from @* onto A := { (w;) E C2 : W 2 = Z1 z2 }. 8. Prove that there is no topological holomorphic map from @* onto the "elliptic surface" A := { (� ) E C 2 : w 2 = (z 2 - 1)(z 2 - 4)}. 9. Define the pseudopolynomial w E O(C 2 )[u] by w(u, z 1 , z2 ) := u 2 - u . z 1 and determine the discriminant set and the irreducible components of A := {w = 0}. Let f on A be defined by f(u, z 1 , z2 ) := z2 . (u 1 ) . Consider N := {f = w = 0} c C 3 and determine the projection N' c C 2 and the set of unbranched points of N. -
5.
I r r edu cib le C omp on en t s
Embedded -Analytic S et s . We wish to study general analytic sets. Since it is easier to work with pseudopolynomials than with arbitrary holo morphic functions, we introduce the notion of "embedded-analytic sets." These are subsets of the common zero set of finitely many pseudopolyno mials, and they are not a priori analytic by definition. But later on, it will turn out that they are indeed analytic. Assume that G c e n- d = { z ' = (zd+ I , . . . , zn )} is a domain and that wi (zi; z ' ) ,i = 1 , .. . , d, are pseudopolynomials over G without multiple fac tors. The zero sets of the single Wi intersect transversally4 in cd X G. We denote by D c G the union of the d discriminant sets belonging to the wi. We call it the union discriminant set. We put
A := {(z1 , . . . , zd , z ' ) : wi (zi; z') = O, for i =
1 , .. . , d and
z' E G}.
�
Over any ball B C G - D the set A lB consists of finitely many disjoint holomorphic graphs. Every graph is contained in a connected component Z of AI ( G - D ) . We call the closure of Z in A an irreducible embedded-analytic component of A. �
�
�
�
Definition . If A is defined as above, any union of finitely many irreducible embedded-analytic components of A is called an embeddedanalytic set of dimension n - d. Two submanifolds M, N C en intersect transversally at a point z E M n N if the entire space is spanned by vectors that are tangent to M or N at z. fu our case the common zero set of the pseudopolynomials w; contains enough vectors to span en. Therefore, � say that these zero sets intersect transversally. �
4
136
1 1 1. Analytic Sets
By definition any embedded-analytic set A can be decomposed into finitely many irreducible components.
The surrounding set A :J A is not uniquely determined. Sometimes we can make A smaller by throwing away those irreducible factors of w; that do not vanish identically on A . Then the wi are uniquely determined by A . �
Assume that A is an embedded-analytic set in A c r,cd xG and that f is a holomorphic function in a neighborhood of A that does not vanish identically on any open subset of A. If N = {Z E A : f(z) = 0 } , then fo r any point z0 E N there is an arbitrarily small linear change cl the coordinates z' such that the affine space parallel to the (z1 , . . . , zd+l)- "axis " through z0 intersects N in a n isolated po int. If z1 E N is any point near z0, then there are unbranched po ints qf N arbitrarily near z1 . A t all of these points N is a submanifold of dimension n - d - 1 . 5 . 1 P r opo sition .
PROOF:
We proceed as in the proof for the last theorem of the previous section. The procedure to find unbranched points will be denoted by ( * ) .
First we construct the projection f of j, which is holomorphic in G. For this observe that if D is the union discriminant set of the polynomials w 1 , . . . ,wd and z' E G - D, we always have the same number of points z�, . . . ,zs in A over z'. We set f(z') = f(z1) . . . f(zs) and obtain j, which is holomorphic on G - D. Since it is bounded along D , we can extend it holomorphically to G. Therefore, the projection set is N' = {J-(z') = 0}. Assume now that z� E N' . Then, after an arbitrarily small linear coordinate change in the variables z', the line L parallel to the Zd+ 1-axis through z� intersects N' in an isolated point. Then by Weierstrass's theorem we can find a neighborhood U = U(z�) C G, a domain G', and a pseudopolynomial w'(zd+ l , z") over G' without multiple factors such that U n N ' is equal to the set {z' = (zd+ l • z") E C x G' : w'(zd+ I , z") = 0}, with z" = (zd+2 • . . . , zn ) ·
Since the space r,c d x L intersects N at z0 in an isolated point, it remains to prove the existence of unbranched points. We apply ( * ) to N n (Cd x U) and prove in the same way as before that for points z1 E N n (C d x U) there are unbranched points of N arbitrarily near to z1 . Of course, at these points N • is a submanifold of dimension n - d - 1 . For the following we use the same notation and hypotheses as above. 5 . 2 Theorem.
Let N' C G be the projection cl N . For every point zo x G, after a suitable linear change of the coordinates z' (z0, z�) E N there is a neigborhood U U(z�) C G, a domain G' in the space of the variables Zd+2 , . . . , Zn , and a pseudopolynomial w' over G' without multiple factors such that:
C r,c d
I. N'
n
U=
{w' = 0} .
=
5.
2. N n (e d
x
Irreducible Components
U) is an embedded-analytic set of dimension
n -
d
-
137
1.
PROOF: We use the notation and results from the proof above. Thus the first statement is clear. We set �+ I :=w'. Restricting the Wi to ed X (N' n U) and projecting them down to e d X G' we get pseudopolynomials w i ( Zi; z") ' over G'. Then N n (e d x U) is in the joint zero set of w 1 , . . . , w d+I · Let A be the union of those irreducible components of this set that contain the unbranched points of N. Since N is the closure of unbranched points, it +I follows that N n (e d x G) C A . By the mapping theorem that we prove in the next paragraph, every irreducible component of A is in N. So we have the desired equality. •
Images of Embedded -Analytic Set s . Assume that G = e d X G' c en is a domain and A c G an irr educible embedded-analytic set over G'.
Let G1 = e dl X Gi c en, be a domain, A I c G 1 an embedded-analytic set over Gi , F a holomorphic mapping from a neigh borhood of A C G into GI such that - C A I for some nonempty open subset U c A. Then F(A) c 4 5 . 3 M apping theo r em .
We denote by D C G the union of the discriminant sets of the wi (zi, z' ) that define the surrounding set A for A . It is sufficient to prove that F (A n (ed x (G ' - D ) ) ) C A I . Since we can connect two points of A n (ed x (G' - D ) )by a chain of arbitrarily small balls, it is enough to give the proof for such a ball. So we may replace A by a ball in en- d and may assume that F is defined in a neighborhood of B . PROOF:
�
Let A I be an embedded-analytic set in
� (WI , W' ) = · · · = W � d 1 ( Wd 1 , W ) = 0 } A 1 -- { WI 0 . So - c 4 Then w; o Flu 0, and by the identity theorem wi o FJB �
I
_
For an arbitrary point v E G� we choose a small transformation of the coordinates in ed , such that AI is also embedded in a set Ay = { wY = . . . = wd, = 0} . The transformation can be made arbitrarily small, and we can do it so that
A1 n .Ar n (ed,
x
{v}) = A I n (ed ,
x
{v}) .
Then AI is given by the infinite set of holomorphic equations wi = 0, w'[ = 0, v E G� . If F maps a nonempty open part of B into A 1 , then by the identity theorem wi o F = w; o II 0, and consequently - c A I · This completes the proof. • _
Remark. Assume that A is an analytic set in a domain G c en and that z0 is a point of A . If zo is a regular point of dimension n - d, then there is a neighborhood U = U(zo) c G such that A n U is an embedded-analytic set.
138
llL Analytic Sets
In fact, there is a neighborhood U with holomorphic functions JI , . . . , jd such that N(fi , . . . , fd ) = A n u and the rank of the Jacobian is d everywhere. We may assume that i_
1, . . . , d J - 1, . . . , d
) # 0.
Then the transformation F ( z 1 , . . . , zn ) = (h (z) , . . . !d ( z) , zd+ l , . . . , zn ) maps a neighborhood of z0 biholomorphically onto a neighborhood of the image point. If the inverse is given by ,
then A is given by the equations
Zd
=
gd ( O, . . . , 0, Zd+ l , . . . , Zn ) ·
So A n U is an embedded-analytic set.
Local Decomposition. We use embedded-analytic sets to show that an arbitrary intersection of analytic sets is again an analytic set. First we consider the following situation. Let G c e n be a domain and Zo E G a point. Assume that at z0 a set 9of local analytic functions f is given such that 1 . For every f E 9 there is a connected open neighborhood U ( z0) C G with f E O(U) and f =/=- 0. 2. f(zo) = 0. We want to construct a "maximal" analytic set S* in a neighborhood U* ( zo ) c G such that for each zero set N of finitely many elements f E :7 there is a neighborhood V = V(z0 ) c G with S* n V c N n V. Then S* is uniquely determined near z0 and can be considered as the common zero set of the functions f E 9 Lt may be nontrivial even if the domains of definition of the functions f tend to the point z0 . For example, if :7 is the set of the func tions fn (z) : = z!/ (1 - nzl ) , defined on Un := {z E C n : Re(z l ) < 1/n}, then S* is the analytic set { z1 = 0} in an arbitrary neighborhood of 0 , whereas the intersection of the Un does not contain any neighborhood of the origin. We employ the results from the beginning of this section several times and carry out an induction on the codimension of the embedded-analytic sets obtained from the functions f E Y' .
5.
Irreducible Components
139
(A) We begin with one arbitrarily chosen function f E Y'. The equation f 0 gives an analytic set5 S of codimension 1 . We decompose S into irreducible components si in a neighborhood U(zo) (given by pseudopolynomials in c X G'), and we choose the neighborhood U so small that the Si stay irreducible in the whole neighborhood. =
(B) Next we try to obtain codimension 2. If every function f
E .'/'
vanishes identically near z0 on Si, we leave Si unchanged (and have it as a codimension ! component for our S* ) . Otherwise, there is an f' E :7 that does not vanish identically in any small neighborhood of z0 on Si. We apply Theorem 5.2: After an arbitarily small linear change of the coordinates z' the set Si n { f' 0} is a finite union of irreducible embedded-analytic sets Sij of codimension two, which stay irreducible if we pass to some smaller neighborhood of z0 . The Sij are embedded in the zero set of two pseudopolynomials wi1 ( z1 ; z3 , . . . , z,) and w�(z2 ; z3 , . . . , zn )· =
(C) Now codimension 3 follows. For this we need consider only the Sij . Leave Sij unchanged if every f vanishes on Sij (and get codimension 2 components for S* ). Otherwise, find an f" E :7 not vanishing identically on Sij , and (after an arbitrarily small coordinate change of the variables z" (z3 , . . . , z,)) the set Sij n { ! " 0} is the union of a finite set of irreducible embedded-analytic sets Sij k , given in the zero set of three pseu dopolynomials w� k (z;.. ; z4 , . . . , zn ), >. 1 , 2 , 3. =
=
=
(D) Continuing, it is possible to obtain components of codimension 1 , 2, . . . , n 1 , and finally one reaches dimension 0. If there is a ! -dimensional component S Si, .. . in - l such that not every f E :7 vanishes on S , we have to replace S by the one-point set {z0 } . Then the procedure stops. Only finitely mar,.y steps were necessary. -
=
We obtained a finite system :70 of local holomorphic functions j, f ', J " , . . . and a finite system A of irreducible embedded-analytic sets Si, Si1, Sij k , . . . and may assume that they all are defined in one neighborhood U(z0) C G, that every s E A of dimension d is embedded in a set e n- d X Gd, and that the union discriminant sets Ds C Gd C <e d belong to the embedding of S. The necessary linear coordinate change in z' can be made at the beginning of the procedure, i.e., once for all steps of the procedure. If S E A is an irreducible embedded-analytic set that has an open part in the union of the other sets of A , then it also has an open part in an irreducible S' E A, S' i= S . It follows by the mapping theorem that it is completely contained in S '. Then we simply throw it away and denote the new system again by A . After finitely many steps we have that the intersection of every 5
An exact definition of dimension and codimension of analytic sets will be given later. Here we use embedded-analytic sets, for which the dimension has already been defined. An analytic hypersurface is obviously embedded-analytic.
140
111.
Analytic Sets
S with the union of the rest of A is nowhere dense in S . Moreover, the points of S over Ds are nowhere dense in S . We denote by S* Usu t S the union of all remaining components. Then S* is given by the finitely many holomorphic functions f E Y0 . Therefore, it is an analytic set. 1f Y c Y is an arbitrary finite subset, then in a small neighborhood of z0 every f E Y vanishes at every z E S * .So S* c N(Y') . Obviously, S* is uniquely determined by this property. =
A
A
Finally, we want to show that the decomposition into irreducible embedded analytic sets is unique. For that we use the notion of regularity for points of embedded-analytic sets just as in the analytic case. Clearly, the intersection of two different S, E A contains no regular point. So the points of every S E A are regular if they are not in such an intersection and not over DA . We denote the set of regular points of S* by S* and set S S n S* for S E A . Then for S E A the sets S are the connected components of S * . Since the set S* is uniquely determined in a neighborhood of zo , this is also true for its connected components. And since the closure of S is S, the irreducible embedded-analytic components S are also uniquely determined near zo . .
.
.
=
.
.
5 .4 Theor em.
The intersection of (even infinitely m any) analytic sets is an analytic set and is locally afinite union of components that are irreducible embedded-analytic sets. This decomposition is locally uniquely determined.
Let {A ' : L E I } be a family of analytic sets in a domain G c e n , and Zo E A == n,E/ A ' an arbitrary point. We consider the system y of all local holomorphic functions f such that:
PROOF:
1 . f is defined in an open neighborhood U of z0 (depending on f). 2. f =j:. 0 near zo. 3. There is an L E I such that f vanishes near z0 on A , . As above, z0 is contained in an analytic set S* that is the union of irreducible embedded-analytic sets S and that is given by a finite subsystem Yo c Y.
If z is a point of A that is sufficiently near z0 , then every f E Yo is defined at z and vanishes on some A , and consequently at z . This shows that A c S * in a neighborhood of z0 . On the other hand, let z be a point in the intersection of S* with a small neighborhood of z0. Any analytic set A , is given by finitely many holomorphic functions fL . . . , J'N that belong to the system Y. Then by construction every ft vanishes on every embedded-analytic component S of S * ,in particular at z . Therefore, S* C A , for all L. Thus S * is contained S* near zo . in the intersection A of the A , and we have the equality A Since S* is an analytic set that has a unique decomposition into irreducible s embedded-analytic sets, this completes the proof. =
5.
Irreducible Components
141
Analyticit y . Now we are able to prove the following result, which we announced at the beginning of the section: 5.5 P ropo sition . Every embedded-analytic set A in a domain red X G c e n is an analytic set. PROOF: As in the last part of the proof of the mapping theorem, it follows that the embedded-analytic set A is given as the joint zero set of infinitely many holomorphic functions. Theorem 5 .4 shows that A is an analytic set. • Consequently, every analytic set has locally a unique decomposition into ir reducible analytic components.
The Zariski T op ology. We prove that the system of all analytic sets has the properties of the system of closed sets of a topology. 5 . 6 Theor em.
The system A the following properties:
ci
all analytic sets in a domain G
c en
and the empty set belong to A . 2. If A 1 , . . . , At E A, then also A = Ut= l A i E A . 3. If I is an index set and {A , : L E I } a collection cl analytic sets in then A = n, E J A, is also an analytic set in G .
has
1. G
G,
PROOF:
( 1 ) G is defined by the zero function, and 0 by the constant function 1 .
(2) Let z E A = A 1 U · · · U At . Then in a neighborhood U(z) there are holomorphic functions fi,J : i = 1 , . . . , l j = 1 , .. . , d, such that for all i we have U n Ai = N ( fi , l , . . . , !i, dJ · It follows that U n A = N ( h ,1, . . . ft,J1 :ji = 1 , . . . , di) ·
3) This is Theorem 5 .4.
8
So the analytic sets are the closed sets of a topology in G. We call this topology the (analytic) Zariski topology of G. It plays an important rule in complex algebraic geometry.
Global Decomp ositions . Assume that G c e n is a domain and A c G . an analytic subset. We call A irreducible if the set of regular points A c A is connected. It follows that A has the same dimension d at all regular points. This number d is called the dimension of A and is denoted by dim( A ). Every irreducible embedded-analytic set is also an irreducible analytic set. 5 . 7 Theor em.
Every analytic set A has a unique decomposition into count ably m any irreducible analytic subsets A, . The covering .9/ = { A i : i = 1 , 2, 3, . . . } is locally finite.
142
III. Analytic Sets
We decompose A into connected components. Let A' be such a component. It has dimension d in all of its points. PROOF:
We consider a point z 0 E A that lies in A' . In a neighborhood U = U(zo) C G we have a decomposition of A into finitely many irreducible embedded A analytic components 1 ' . . . , A,. By Af we denote the set of points of Ai that are not over the union discriminant set. Some d-dimensional Af meet A '. Their union A* is contained in A' and dense in A' n U . Hence, the closure of A* in U is equal to A' n U. But A * is an analytic set. From this it follows that A' is an analy!ic set, that only finitely many A' locally). Since the intersect U and that the union of all A ' is A (as it istopology of G is countable, it follows that the set of the A' is countable. • '
If A is irreducible and A = A1 arbitrary analytic sets, then A = A 1 or A = A2.
5 . 8 Corollary.
Sometimes this condition is used
as
U
A2 , where A1 , A2 are
the definition of irreducibility.
5 . 9 Proposition. Let A, B C G be irreducible analytic sets. If there is an open set U C G such that A n U -=/= 0 and A n U C B n U, then A C B .
P ROOF:
This is an immediate consequence of the mapping theorem.
•
Another corollary is the following: 5 . 10 Identity theorem (for analytic set s ) . Let A, B C G be irreducible analytic sets. If there is a point zo E A n B and an open neighborhood U = U(zo) C G with A n U = B n U, then A = B . 5 . 11 Prop osition. Let A , B c G be analytic subsets with A C B. If A is irreducible, then A is contained in some irreducible component of B . U .x. EA B.x. be the unique decomposition into irreducible Let B components. We can choose an open set U C G and a finite set {>..1 , . . . , >- t } C A such that U n A i= 0 is irreducible and U n B = (U n B.x., ) U · · · U (U n B.x.1 ) Then U n A = U n A n B = (U n A n u . . . U ( U n A n B.x.1 ) . Thus there is an index j such that U n A = U n A n Aj , so U n A c U n B.x.j . It follows • that A is contained in B.x.j .
PROOF:
=
B.x.tJ
Now we can generalize the notion of the dimension to arbitrary analytic sets. Definition . If A C G is an analytic set with irreducible components Ai, then dim( A) := supi dim( A i) is called the (complex) dimension of A.
6. Regular
and
Singular Points
143
In general, the dimension of an analytic set can be oo. But if C1 cc G is a rel atively compact subdomain, then only finitely many irreducible components intersect C1 . So the dimension of A n C 1 is finite. An analytic set is called pure-dimensional of dimension d if all its irreducible components have the same dimension d. Exercises 1. Let A be an analytic set near the origin in e n . Assume that every ir reducible component of A has dimension > 1. Show that there exists a neighborhood U = U(O) such that A n U is the union of irreducible one-dimensional analytic sets containing 0. 2 . Consider A := {(z1 , z2 ) E e 2 : z� = zr + zD. Show that A is irreducible, but A has a nontrivial decomposition into irreducible components in a small neighborhood of the origin. 3. Let A1 1 A2 c e n be analytic sets. Prove that A1--=-A2· is analytic. 4. Let {A; : i E N} be a locally finite family of irreducible analytic sets in a domain G c en . Suppose that Ai � A . for i # j and prove that the Ai are the irreducible components of theif union. Regu lar and Sin gu lar P oin t s
6.
Comp a ct Analytic Set s .
Our goal is to prove the following simple
proposition. 6 . 1 Prop osition. If G c e n is a domain and A c G a n irreducible compact analytic set, then A consists of a single point. We first prove a lemma. 6 . 2 Lemma. Iff is a holomorphic function in a neighborhood of A, then
f !A
is constant.
PROOF: We assume that the dimension of A is n - d. Since A is compact, there is a point z0 E A where Iii takes its maximum. After a linear coordinate change there is a neighborhood U = U(z0) c G and a domain G' c e n-d such that A n U is an embedded-analytic set over G'. Denote by D c G' the union discriminant set. Over every z" E G' - D our A n U has points. The point z0 lies over some z0 E C' and we may assume that it is the only point of A n U over z0. It remains to construct the elementary symmetric functions associated with f !Anu over G' - D. For this, if z1, . . . ,z 8 are mapped onto z", we define fi(z") = ai(J(zi ) , . . . ,f(zs)). These functions are holomorphic on G' - D and bounded along D. So they extend to holomorphic functions in G'. The s
:
144
III.
Analytic Sets
absolute value of every extension takes on its maximum at z�. By the max imum principle each such is constant in G'. Since the values of f can be reconstructed from the values of the f; , it follows that f is constant over G' , in particular in some open subset of A. Since A is connected, it follows that • f is constant on A and then by continuity also on A. PROOF of the proposition: All coordinate functions z; must be constant on • A . So A is a single point. A consequence is that every compact analytic subset A C G consists of only finitely many points.
Embeddin g of Analytic S et s . Assume that A c G is an analytic set in a domain G c e n , that 0 E A, and that the plane P = {zd+l = · · · = zn = 0} intersects A in an isolated point.
In a neighborhood U(O) the set A is an analytic subset d a n embedded-analytic set of dimension n - d that is defined over a domain G' in the space d variables z' = (zd+l , . . . , zn ) . If the set d ( n- d)- dimensional regular po ints is dense in A , then A is itself a n embedded-analytic set.
6 . 3 Theorem .
Remark.
No coordinate transformation is necessary for this statement !
PROOF: By definition, A is the zero set N(JI , . . . , fN ) of finitely many holomorphic functions in a neighborhood of 0. Since A n P = { 0} is isolated, there is an i such that j; does not vanish identically in any neighborhood of 0 on the z1-axis. Consequently, f; is z1-regular, and we can apply the Weierstrass preparation theorem, which implies that A is locally contained in the zero set of a pseudopolynomial w(z1 ; zz , . . . , zn ) · Now we proceed by induction on d. In the case d = 1 there is nothing t o prove. 1f d > 1 we consider the projection 7r : e n -+ en -l with Z f--t Z1 = (zz , · . . , Zn ) · To h , . . . , fN there are associated as usual holomorphic functions f 2 , . . . , f N of z' such that
1r({w = h = . . · = fN = 0} ) = A := {z' : j 2 (z') = . . . = J N (z') = 0) in some neighborhood of 0 . The intersection of P' = {z' : Zd+ I = . . . = zn = 0} with A contains 0 as an isolated point. So we have for A the same situation, but with one dimension fewer. By the induction hypothesis it follows that there are pseudopolynomi als wz , . . . , wd such that A locally is contained in the set {wz (zz ; z ") = . . . = wd (zd ; z") = 0} . By exchanging z1 with Zz, we obtain a pseudopolynomial w1 (z1 ; z") such that A is contained in {w 1 = Wz = . . . = Wd = 0} .
6 . Regular and Singular Points
1 45
- d are dense in A , we can take the union A* of those irreducible components of the embedded analytic set that contain such a regular point. Then A * and A are identical. This completes the proof. If the regular points of dimension
n
•
Remark. This result is also known as the "embeddingtheorem of Remmert and S Lein."
Again we consider a domain G c en, an analytic set A c G, and a domain G' c en- d such that 7r(G) c G' (where 1f : z M z' (zd+ I , . . . , z )). Suppose n that there exists a domain G* C G such that for every z' E G' there exists a neighborhood U U(z') c c G' with (e d x U) n G * c c (e d x U) n G and (ed X U) n A c G * . Then the following holds. =
=
If the set d regular (n - d) - dimensional po ints of A is dense in A , then A is a n embedded-analytic set over G'.
6 .4 Pr op osition .
PROOF: We take an arbitrary point zb E G'. It follows from the hypotheses that the set (ed x {z6}) n A is compact and analytic. Therefore, it consists of finitely many points. Each of these points has a neighborhood such that the restriction of the set A to this neighborhood is an embedded-analytic set over a neighborhood of zb . By multiplying the pseudopolynomials by the same distinguished variable belonging to our various points we obtain d uniquely determined pseudopolynomials over a neighborhood U'(zb) C G' such that A n (e d x U') is just their joint zero set. But these pseudopolynomials glue together to form global pseudopolynomials over G' . •
Regular P oints of an Analytic Set . Assume again that G c en is a domain, and A C G an analytic set. 6.5 Theorem .
For any z0 E A there is a fixed neighborhood U(z0 ) c G with finitely m a ny holomorphic functions JI , . . . , JN whose joint zero set is A n U such that fo r all d at every regular point z E A n U of dimension n - d the rank of their Jacobian at z is equal to d. It is remarkable that this statement is also true for a singular point z0 of A . PROOF: After applying a linear coordinate transformation in en we can find a neighborhood U U(zo) C G such that A n U is a finite union of irreducible embedded-analytic components. To give these in the canonical form a further coordinate transformation is not necessary. We denote by A' the union of all ( n - d)-dimensional irreducible components of A n U and choose pseudopolynomials w1 , . . . , wd of minimal degree such that A' is contained in the common zero set of the wi . =
�
Let D c A' be the set of points that lie over the union discriminant set DA' . Its dimension is equal to n - d - 1 . Now we carry out the proof in
146
III.
Analytic Sets
several steps and construct sets D t , D2 ,.... . . . , D n- d+ I = 0 with D i + I c Di and dim (Di ) = n - 4- i . We begin with DI :=D. The Jacobian of w1 , . . . , Wrt has rank d in A' - D1 (implicit function theorem). �
�
Next we decompose D1 into irreducible components C>.. and choose, if possi ble, in each C>.. a point z ;-, where A' is regular. We can make U so small that only finitely many C>.. occur in U. Then we apply another linear transforma tion in en that is near the identity such that in a small neighborhood of any Z>.. the set A' can be written as an ( n - d)-dimensional holomorphic graph {z
:
Zi
=
J>.. , i (Zd+l , . . . , zn )
for i = 1, . . . , d} .
Finally, we apply a linear transformation that is very near the identity to the variables ZI , . . . , Zrt such that for every >. and every point z in A' above (z�+ l ' . . . ,z�) the first d coordinates of z are distinct. Now we use Proposition 6.4. In the new coordinates (and in a slightly smaller neighborhood, which we again denote by U) the set A' is again an embedded-analytic set contained in the common zero set of pseudopolynomi als WI , . . . ,wd. We choose wi with minimal degree. We can assume that the components of D1 are still irreducible in U, and that the points z ;-, are still in U. In a neighborhood of z;-. we have a decomposition �
with w;_ i (z>.. ) -=/= 0. So the Jacobian determinant of WI , . . . ,wd with respect to the �ariables .z t ' . . . , z,t does not vanish at any Z>.. - We denote the zero set of this Jacobian in DI n U by D2 . It-has dimension n - d - 2, and WI ' . . . , Wrt , WI ' . . . ,wd have rank d on A' - D2 · �
�
Now apply the same procedure to D2 and ob.t ain an ( n- d D3 and continue in this way until reaching D n- d+ I = 0. �
- 3)-dimensional
By putting all of the pseudopolynomials together, in a small neighborhood U of zo we get holomorphic functions Jr , . . . , jd, fd+ I , . . . . fN (with N = ( n - d + 1) . d ) whose rank is d in every regular point of A' n U. Since the pseudopolynomials always were chosen with minimal degree, it follows that A' = N(h , . . . , JN ) near zo .
Now set A + = A n U - A' . It is the union of the remaining irreducible com ponents of A n U. We may assume that U is so small that A + is the common zero set of finitely many holomorphic functions gi , . . . , gs in U. Multiplying the fi by the g1 yields finitely many holomorphic functions in U that de scribe the set A n U. No point of A' n A + is a regular point of A. For every z E A' - A + there is a gj that does not vanish there. So the rank of the Jacobian of the fi . gj is equal to d at every nonsingular point of A' - A + .
6.
Regular and Singular Points
147
The same procedure can be used for every d and the corresponding A', and consequently, in finitely many steps we obtain a neighborhood of z0 and holomorphic functions in this neighborhood with the desired properties. •
The Singular L o cus. From the preceding theorem we conclude the following: 6 . 6 Theorem.
Th e set Sing(A) again an analytic set.
singular points
d
an analytic set A is
The intersection of two irreducible components of A belongs to Sing( A) . The union S of all these intersections is an analytic set.
PROOF:
Assume that z0 is a point of an irreducible component A' of A and dim(A') = n - d. Then there is a neighborhood U = U (z0 ) C G with holomorphic functions h , . . . N vanishing exactly on A' n U such that their Jacobian has rank d in each of the regular points. Let S* be the analytic set of all points of A' n U where all d x d minors of the Jacobian vanish. Clearly, S* is contained in Sing( A') n U. On the other hand, at any point of Sing( A') n U the Jacobian of h , . . . N cannot have rank d. So Sing(A') n U = S *, and Sing( A') is analytic in G.
,f
,f
The union of S and the sets Sing(A') for all irreducible components set Sing(A) . It is analytic, since the union is locally finite.
A'
is the •
The set Sing(A) is called the singular locus of A.
Extending Analytic Set s . Let G c
cc n
be a domain.
6 . 7 Lemma. Let Zo = (ziO} ' . . . ' z�0)) c e n be a n arbitrary point and E = { z : Zi = zi0l for i = 1 , .. . , d} a n affine plane of codimension d containing zo . IfA C G is a n irreducible analytic set of positive dimension that is n o t a
subset
of
E, then there is a n open dense subset C C ed such that
fc( zl , . . . , zn ) : = c1 ( z1
- zi0> ) + . . . + cd (zd - zA0> )
does not vanish identically o n Afo r every c = (c1 , . . . ,cd ) E C . In particular, fo r any hyperplane Ho C en containing E there is a hyperplane H arbitrarily close t o H0 and also containing E such that dim(Ai ) < dim( A) - l fo r every irreducible component A, of A n H . We define cp :
:= fc·
: !ciA
fc(z)
fc
=
148
llL Analytic Sets
Our main tool for extending analytic sets is the following.
If E = {z E en : Z l = . . . = Zd = 0} is a n ( n - d) dimensional plane and A an analytic set in G - E , whose irreducible compo nents all have d imens ion n - d + l with 0 < 1 < d , then the closure A of A in G is an analytic set in G . 6. 8 P r op osition .
The proposition is of a local nature. We may assume that A is irreducible and 0 E E n G. It is enough to construct a continuation of A into a neighborhood of 0 . PROOF:
Let c = (O, . . . , O, ed+l , . . . , c,) be an arbitrary point of E n G. We consider the following family of ( d - 1 )-dimensional planes through c : For
i = 1, . . . , n - d + l j = I, . . . , d - l we have the linear map LA : e d - l -+ e n - d+ l and define P (c, A) := c t { (w', LA (w' )) : w' E So P(c, A) consists of vectors w
ed - 1}.
= (WI , · . . , wd - t , Wd - l +l • · . . , wn ) with
d-l for i = 1 , . . . , 1, :2::: aij . wj d- l ed -l + i + Z::: aij . w, for i = l + 1 , . . . , n - d + l . ,=1 Then every P( c, A ) meets E exactly in c = ( 0 , . . . ,O,ed+l , . . . ,en )· If 0 is the zero matrix, then P( c) == P( c, 0) is the plane ed - l { ( 0 , . . . , 0, ed+l ' . . . ,en )} ' �1
X
In the next chapter we will introduce Grassmannian manifolds and a topology on the set of linear subspaces (with fixed dimension) of a given vector space. In our case it follows that a neighborhood of c + Po is given by the set of all planes P = c + P with P ffi (Ox e n - d+ l ) = e n . This shows that every ( d - l ) dimensional plane through c that is near P(c, O)is of the form P ( c, A ) . We choose real numbers 0 < r1 < r2 and r > 0 so small that the "shell" S = {z = (
'
z ,z
"
) E ed - l e n- d+ l . r1 x
" ' < li z II < r2 and i z l < r }
is a relatively compact open subset of G. Only finitely many irreducible com ponents A , of A enter S . We can find a hyperplane H1 containing E that intersects the A, not at all or in codimension 1. Let A , be the finitely many irreducible components of H1 n A , that enter S. We can find another hyper plane H2 containing E that intersects all A , at most in codimension 2. We continue this procedure running through the irreducible components of the A, n H2 . After finitely many steps we have n - d + l hyperplanes such that their intersection P is a plane P( c, A ) that meets A n S in at most finitely
6. Regular and Singular Points
149
many points. By the above lemma we can choose this plane arbitrarily near
Po := P(c).
Now apply a linear coordinate transformation very near to the identity that leaves E invariant and maps our plane P (c, A) onto Po and replace the transformed shell S by a new shell S' (in the new coordinates) that is a little bit smaller such that S' is contained in the old (tran sformed) S. Thi s can be done so that 8S' n P0 n A 0. =
Earlier we proved that A is an embedded-analytic set in a neighborhood of the points of the intersection A n P0 n S' over a domain G' in the space of variables Zd- l + I , . . . , Zn . So there is a small closed ball B c cn- d+ l around the origin such that (cd - l x B) n 8S' n A remains empty, every irreducible component of (<e d- l x B) n S' n A enters P0 n S', and every plane through a point of B and parallel to P0 intersects A n S' in at most finitely many points.
Zd+ I , . . .
,
Z
n
S'
E
c
I I . . . .
0
A n Bo
Figure
11 1.3. Intersecting A
with B0 and Po
Zl , . . . , Zd-l
Now, the set B0 {z' E <e d- l : ll z' ll < r2} x B is a neighborhood of the origin in e n , and each of our parallel planes through points of B E meets A n B0 in a compact analytic set and therefore in at most finitely many points (cf. Figure 111.3). At all of these points the set A is locally an embedded-analytic set over B. By multiplying the pseudopolynomials by the same distinguished variable that we obtained for the different intersection points over the same base point in B - E, we have that An ( cd-l X (B-E) ) is -
150
Ill.
Analytic Sets
an embedded-analytic set over B - E . The coefficients of the corresponding pseudopolynomials over B - E are bounded along E . Hence, they can be analytically extended to B. This means that A n (B0 - E ) has a unique analytic continuation to B0 • • The proposition just proved is also true if A c G - E is an analytic set, whose irreducible components all have dimension greater than n - d, since we can write A as a finite union of pure-dimensional analytic sets. As a consequence we have the following theorem.
Assume that G c en is a domain, K c G an ( n - d)- dimensional analytic subset, and A an analytic subset of G - K all components of which have d imension > n - d. Th en the closure A of A in G is a n analytic set in G. 6 . 9 Theorem of Remmert-Stein .
PROOF : If zo E K is a regular point, then K can be transformed in a neighborhood of z o to a plane E. So A is analytic in a neighborhood of zo and therefore in all regular points of K. We can replace K by the set K1 of singular points of K, which is analytic again and has lower dimension. By the same argument we show that A is analytic at all regular points of K1 . • Continuing in this way we prove that A is analytic in G.
This theorem first was proved by
R.
Remmert and
K.
Stein; see [ReSt53] .
The Local Dimen sion . We show how our results are linked with the classical dimension theory of analytic sets.
Let G c en he a domain, A c G an analytic set, and Zo E A a point. There is an open neighborhood U = U(z0) c G such that U n A is a finite union of irreducible analytic components A 1 , . . . , At . If we choose U small enough, then the A, are uniquely determined. Definition .
In the given situation the uniquely determined number dimz0 (A) := max dim(A.>-) .\ = 1 ,
.
.. , 1
is called the (local) dimension of A at z 0. The set A has dimension 0 at z0 E A if and only if there is an open neigh borhood U = U(z0) c G such that U n A = {z0}.
Let k : = dimz0 (A) be positive. Th en k is the smallest number with the property that there are holomo rphic functions . . . , ik in a small neighborhood U of z0 such that z0 is isolated in A n N(JI , . . . , !k ) ·
6 . 1 0 Pr oposition .
h,
PROOF: If dimz0 (A) = k, then there must be at least one k-dimensional irreducible component Pt of A at z0 .
6 . Regular and Singular Points
151
If f is any holomorphic function near z0 , then either !l A' 0 (and therefore A n N ( f) still k-dimensional) or A' n N(J) is ( k - I)-dimensional. So at least k functions are required. _
On the other hand, by Lemma 6 . 7 we can find a holomorphic function h near z0 that does not vanish identically on any irreducible component A' of dimension k at z0• It follows that A' n N(h ) has dimension k - 1 for all such components A '. We can repeat this process, and after k steps we reach dimension zero, so that z0 is isolated in A n N ( h , . . . , fk ) . • Definition . If A has dimension k at z0 , then any system {h , . . . , fk } of holomorphic functions with AnN( h , . . . , fk ) = { z0} is called a parameter system for A in z0 .
6. 1 1 Ritt' s lemma. Let B C A be closed analytic sets in a domain G C en . Th en B is nowhere dense in A zj and only if dimz (B) < dimz (A) for every z E B . PROOF:
Let the criterion be fulfilled, and z0 be an arbitrary point of B. Then there exists an open neighborhood U of z0 in G and a parameter system { it , . . . , fk } on U for B at z0• Since dimz0 (A) > k, it is not possible that z0 is isolated in AnN( h , . . . , fk ) · This means that (A-B) nN(h , . . . , fk )nW -=1= 0 for every neighborhood W = W ( z0) . So B is nowhere dense in A. On the other hand, let B be nowhere dense in A, and z0 a point of B . In a small neighborhood U of z0 we have unique decompositions into irreducible components: BnU
=
B1
U
...
U Bm
and
A = A1
u
· · .
u
A1 .
Every component B; is contained in a component Aj(i ) , and for any open neighborhood W = W(zo ) we have (Aj (i) B; ) n W -=1= 0, because otherwis e, there would exist points z E W n B; where B is dense in A. So dim( B;) < dim(Aj(i ) ) for all i . It follows that dimz0 (B) < dimz0 (A) . • _
Let G c en be a domain, A c G an analytic set, and Z o E A a point. 1f dimz0 (A) = k , then the number n - k is called the codimension of A at z0. 6 . 1 2 Second Riemann exten sion theorem . Suppose that n > 2 and that the analytic set A C G has everywhere at least codimension 2 . Th en any holomo rphic function f on G - A has a holomorphic extension to G . PROOF:
We may assume that A is irreducible of codimension d > 2. If z0 is a regular point of A, then there is a neighborhood U of z0 such that U n A
152
III.
Analytic Sets
is biholomorphic ally equivalent to an open subset of a linear subspace E of codimension d. By the theorem on removable singularities (see Section II. l) f can be holomorphically extended to zo . We repeat this procedure. Beginning with the set Sing(A), which has codi mension d + 1 , after finitely many steps only a set of isolated points remains. Since f can also be extended to these points, we obtain the desired result. • Exercises 1 . Let A 1 , A2 c en be analytic sets. Show that dim(A 1 n A2 ) > dim( A! ) +
dim(A2 ) - n. 2. Assume that G c en is a domain and j a nonconstant holomorphic function on G. Prove that there is an at most countably infinite set Z c e such that A, := {z E G : f(z) = c ) is regular for c E e - Z. 3. Let G c en be a domain and A c G an analytic set. Prove that for any k > 0 the closure of the set A k : {z E A : dimz (A) = k } is either empty or is a pure k-dimensional analytic subset of G. 4. Let G c en be a domain and fl , . . . , fm holomorphic functions on G. Denote by N the common zero set N(fi , . . . , fm ) · Show that if the rank of the Jacobian of h , . . . , fm at some point zo E N is equal to r, then there is a neigborhood U U(z0) c G and a closed submanifold M c U of dimension less than or equal to n - r such that U n A c M . 5 . Let A be an analytic set in a domain G c en. For every point Zo E A the set lz0 (A) := { (J)z0 E Oz0 : f \ A 0} is an ideal in Oza · Show that there are a neighborhood U of z0 and holomorphic functions h , . . . , fk on U such that: (a) A n U = N ( h , . . . , fk ) · (b) lz0 (A) is generated by the germs (JI )z0 , • • • • (Jk )z0 • Show that the vector space n Tzo (A) = w E en l::: w vfzv (zo ) = 0 for every f E lz0 (A) =
=
{
v=l
has dimension n - rkz ( h ' . . . , fk ) . It is called the Zariski tangential space of A at zo. Let A be defined as in Exercise 5, and suppose that z0 is a regular point of A. Show that Tz0 (A) is the set of tangent vectors a(O) (see Section I. 7) , where a : I -+ en is differentiable, a(I) c A, and a ( O ) Zo . Let A be an analytic set in a domain G c en ' and let Zo E A be an arbitrary point. The embeddinf? dimension of A at zo is the smallest integer e such that there is an open neighborhood U = U(zo) and a closed submanifold M c U of dimension e with A n U C M . It is denoted by embdimz0 (A). Prove that z f--t embdimz (A) is upper semicontinuous on A, and that embdimz0 (A) dimc (Tzo (A)). Consider the analytic set A { w exp ( l /z ) } c {( w , z ) E e 2 : z -=!= 0}. Determine the closure of A in e 2 . 0
6.
}
=
7.
a.
=
=
=
C h ap ter IV
C omp lex M an ifold s 1.
T h e C omp lex Str u ctu r e
C omplex C oor din ates . Let X be a Hausdorff space, i. e . a topological space satisfying the Hausdorff separation axiom. Sometimes such a space is also called a separated space or a 2'2-space. Hausdorff spaces are the most common in topology (for example, every metric space is a Hausdorff space), but non-Hausdorff spaces do arise, in particular in algebraic geometry. The space en with the Zariski topology is not Hausdorff. ,
We think that a space X is too big if there exists a discrete subset with the cardinality of the continuum. Therefore, we demand that the topology of X have a countable base. In this case X is said to satisfy the second axiom
Vv C Ur(v) for every v E N. The refinement map is not uniquely determined, but we can fix it once and for all. A covering Y {Vv : v E N } is called locally finite if each x E X has a neighborhood U U(x) such that U meets only finitely many Vv. =
=
Definition . A Hausdorff space X is called paracompact if every open covering f/ of X has a locally finite open refinement W. Every compact space is paracompact. Furthermore, every locally compact space with countable basis is paracompact.
154
IV. Complex Manifolds
For the moment we only assume that X is a Hausdorff space. Definition. An n-dimensional complex coordinate system ( U,cp) in X consists of an open set U C X and a topological map cp from U onto an open set B c en . If p E X is a point, then every coordinate system ( U, cp) in X with p E U is called a coordinate system at p . The entries z; in z = cp(p) are called the complex coordinates of p (with respect to ( U, cp) ) .
If j is a complex function in U, we can consider it as a function of the complex coordinates zt , . . . , Zn , by Two (n-dimensional) complex coordinate systems ( U,cp) and (V,1/;) in X are called (holomorphica lly) compatible if either U n V = 0 or the map
cp o 1/J - 1 : 1/J ( U n V) --+ cp(U n V) is biholomorphic (see Figure IV . I ) .
u
Figure IV. l .
v
Change of coordinates
The sets B,p : = 1/J ( U n V ) and B'P := cp(U n V ) are open subsets of e n . 1f z; (respectively Wj ) are the complex coordinates with respect to cp (respectively '!(;) , then compatibility of the coordinate systems means that the functions z; = z;(W t , . . , w,) and wj = Wj (zt , . . . , zn ) are holomorphic. .
1 . The
Complex Structure
155
A covering of X with pairwise compatible n-dimensional complex coordinate systems is called an n-dimensional complex atlas on X . Two such atlases A 1 and A2 are called equivalent if any two coordinate systems ( U,cp) E A 1 and (V, 1/;) E A2 are compatible. An equivalence class of en-dimensional) complex atlases on X is called an n-dimensional complex structure on X. It contains a maximal atlas that is the union of all atlases in the equivalence class. Definition. An n-dimensional complex manifold is a Hausdorff space X with countable basis, equipped with an n-dimensional complex struc ture. Every complex manifold is locally compact and paracompact. Examples 1 . The complex n-space en is an n-dimensional complex manifold. The
complex structure is given by the coordinate system (en , id) . 2. If X is an arbitrary n-dimensional complex manifold, then any nonempty open subset B c X is again an n-dimensional complex manifold. For p E B there is a coordinate system ( U,cp ) in X at p . Then (Un B, cp l uns ) is a coordinate system in B at p. All of these coordinate systems are compatible. 3. Let G c en be a domain and X c G a k-dimensional complex sub manifold. Of course, X is a Hausdorff space (in the relative topol ogy) with countable basis. For z0 E X there are open neighborhoods W W(z0) c G and B B(O) c en and a biholomorphic map F : W --+ B such that =
=
F(W n X)
=
{(w 1 , . . . , wn ) E B : wk + I
=
·
· · =
W n
=
0} .
Let pr' : en --+ e k be the projection (w l , . . . ' W ) M (w l , . . . ' wk ) · We n define U : W n X and cp := pr' o F : U --+ e k . Then ( U,cp) is a k dimensional complex coordinate system in X at z0. If ( V,'lj; ) is another coordinate system, with 1/J pr' o E , then =
=
is holomorphic. So we get a complex structure on X . 4. Finally, let ( G,1r) be a Riemann domain (over C " ). Then G is a connected Hausdorff space, and for every p E G there is an open neighborhood U = U(p ) such that B == 1r (U) is open and cp : 1r l u : U --+ B is topological. Then ( U, 'P) is a complex coordinate system. If 1j; = 1r l v is another coordinate system, then for x E U n V we have cp(x ) 1/; (x) = 1r(x) : z and =
=
=
156
IV. Complex Manifolds
Therefore, the coordinate systems are compatible. We get a complex structure on G. One can prove that G has a countable basis (cf. [Gr55] , §2] ). So every Riemann domain over en is an n-dimensional complex manifold.
H olomorp hic Fun ctions. Let X be an n-dimensional complex mani fold. Definition . A complex function f on an open subset B c X is called holomorphic if for each p E B there is a coordinate system ( U, cp) at p such that f o cp- 1 : cp(U n B ) --+ e is holomorphic. If z1 , . . . , zn are the complex coordinates with respect to ( U,cp) , then 1 (z1 , . . . , zn ) f--t f o cp - (z1 , . . · , Zn )
is a holomorphic function in the ordinary sense. If zv = zv ( w1 , . . . , Wn ), where w1 , . . . , w, are the complex coordinates with respect to a coordinate system ( V, 7/!), then f 0 7/•- 1 ( W1 , . . . , Wn ) = f 0 \0- 1 ( Z1 ( Wt , . . . , Wn ) , . . . , Z n ( W1 , . . . , Wn ) ) is also holomorphic. So the definition of holomorphy is independent of the coordinate system. We denote the set of holomorphic functions on B by O(B). It is a @-algebra with unit element. Examp le Let G c en be a domain and X c G a k-dimensional complex submanifold. We consider a complex coordinate system ( U,cp) in X , where U is the inter section of X with an open set W c G and cp pr' o F , with a biholomorphic 0}. map F : W --+ B C en such that F(U) {w E B : Wk +1 = . . . w, If f is a holomorphic function on G, then =
=
=
fix o cp - 1 (w1 , · . . , wk )
=
.f o F- 1(w1 , . . . , wk , O, .
=
. . , 0)
is holomorphic. Therefore, fix is a holomorphic function on the complex manifold X . 1 . 1 Identity theorem.
Let X be connected. If f, g are two holomo rphic functions o n X that coincide in a nonempty open subset U C X , then f = g. PROOF: Let W {x E X : f(x) g (x)}. Then wo -=/= 0 , since U C W. Assume that there exists a boundary point xo of W" in X and let ( U,cp) be a coordinate system at x0 with cp(x0 ) = 0. Then all derivatives of f o cp - 1 and g o cp - 1 must coincide at 0. It follows that the power series of these functions around the origin are equal. But then f g in a whole neighborhood of x0 • and this is a contradiction. =
=
=
1. The Complex Structure
157
If there were a point x E M : = X - W" that was not an interior point of M , then x would be a boundary point of W " . This shows that M must be open. Since X is connected, M has to be empty. • 1 . 2 M aximum pr inciple.
Let X be connected, f E O(X ) , and x0 E X a point such that I i i has a local m aximum at x0 . Th en f is constant. The functions f and g := f(x0) are both holomorphic on X . If (U, rp) is a coordinate system at x0 and B : = rp(U) , then fo := f o rp -1 is holomorphic on B, and l fo l has a local maximum at z0 := rp(x0 ). Thus there is an open neighborhood B' = B'(z0) c B such that fo is constant on B' and f is constant on U' := rp-1 (B'). So f l u = g l u , and by the identity theorem • f = g; i . e . , f is constant.
PROOF:
1 . 3 Cor ollary.
If X is compact and connected, then every holomo rphic
function o n X is constant. The conlinuous funclion I J I Lakes ils maximum al some poinl of • X . Now the corollary follows from the maximum principle. PROOF:
1 . 4 Cor ollary. mension in e n .
There zs no compact complex submanifold of positive di
Let X c e n be a compact connected submanifold. Then the stan dard coordinate functions zv lx must be constant, for v = 1 , .. . , n . This means that X is a single point. If X is not connected, it is a finite set of • points. PROOF:
Remark.
Another proof is given in Section 1 1 1 .6.
Riemann Sur fa ces. An (abstract) R iemann surface is by definition a ! -dimensional connected complex manifold. Example The complex plane e and every domain in e are Riemann surfaces. Recall the Riemann surface of .,fZ, X
= { ( w, z ) E e 2 : w2 = z, z -/= 0} .
Since X is a Riemann domain over e, it is a !-dimensional connected complex manifold. From the projection 1r := pr 2 1 x : X -+ e we get complex coordinate systems (U, rp) with rp : = 7r l u and sufficiently small U .
158
IV. Complex Manifolds
The function f : X -+ C with j ( w z) with f 2 = z. ,
:=
w is a global holomorphic function
Example
The Riemann sphere C C U { oo} is a compact connected Hausdorff space. We have two coordinate systems (C, rp) and (C - { 0 }, 1/;) with rp(z) = z and 1/J (z) = ljz. On @* = C n (C - { 0 }) we have rp o 1j;- 1 (z) = 1/z, and this is holomorphic. So C is a compact Riemann surface. Every global holomorphic function on C is constant, but there are nontrivial meromorphic functions, for example f(z) = z (with one pole at oo ) . Here a me romorphic fun ction on X is a function f that is holomorphic outside a discrete subset P c X and satisfies JiQJ, I f (x) l 00 for every p E P. =
=
The points of P are called the poles of f
H olomor phic M appings . Let F : X -+ Y be a continuous map be tween complex manifolds. Definition. The map F defined above is called holomorphic if for any p E X there is a coordinate system ( U, rp) in X at p and a coordinate system ( V, 1/; ) in Y at F(p) with F(U) c V such that 1/J o F o rp- 1 : rp(U) -+ 1/;(V)
is a holomorphic map .
The map F : X -+ Y is holomorphic if and only if for any open subset V C Y and any f E O(V) it foll ows that f o F E O(F- 1 ( U) ) . 1 .5 Proposition .
The proof is an easy exercise. The category of complex manifolds consists of a class of objects, the complex manifolds, and a class of sets such that to any pair ( X , Y ) of objects there is assigned a set O(X, Y ) (which may be empty) , the set of holomorphic maps between X and Y. In a general category this set would be called the set of morphisms from X to Y . For ( G F ) E O(Y, Z) x O(X, Y ) we always have the composition G o F E O(X, Z) such that the following axioms hold: ,
1 . If H o G and G o F are defined, then ( H o G) o F = H o ( G o F ) . 2. For any manifold X we have the identity map idx E O(X , X) such that idy o F F and F o idx F , if the compositions are defined. =
=
l . The Complex Structure
1 59
Another example for a category is given by the topological spaces and contin uous mappings. If we replace in our definitions above the field e by lR and the word "holomorphic" by "differentiable", we get the category of differentiable manifolds and differentiable mappings. From every n-dimensional complex manifold we obtain a 2n-dimensional differentiable manifold by "forgetting the complex structure." A holomorphic function f : X --+ e is obviously a holomorphic mapping. More generally, in the case of a Riemann surface a meromorphic function f on X may be viewed as a holomorphic mapping f : X --+ C. Definition. A biholomorphic m ap F : X --+ Y is a topological map such that F and F- 1 are holomorphic. If there exists a biholomorphic map between X and Y , then the manifolds are called isomorphic or biholomorphically equivalent, and we write X Y . "'
Remar k. 1f X is a complex manifold and ( U, cp ) a complex coordinate system with cp ( U) = B c en , then cp : U --+ B is a biholomorphic map.
C ar tesian P r o d u ct s . Assume that X1 , . . . , X , are complex manifolds of dimension n 1 , , n,. Then the set X = X1 x . . . x Xm carries a natural topology generated by the sets U = U1 x · · · x Um , U; c X; open for i = 1 , . , m. One sees immediately that X is a Hausdorff space with countable basis. .
.
.
.
.
Given complex coordinate systems (U; , cp; ) in X; , for i = 1 , a coordinate system ( U, cp ) for X by
.
.
.
, m,
one defines
It is clear that two such coordinate systems are compatible. So we obtain an n-dimensional complex atlas and a complex structure on X . The projections p; : X --+ X; are holomorphic maps for i = 1 , . . , m. .
A simple example is en
=
e
X
•
.
•
X
c.
n times The Cartesian product of two complex manifolds X 1 , X2 satisfies the follow ing unive rsal property:
Given any complex manifold Y and any two holomorphic maps F : Y --+ X1 and G : Y --+ X2 , there exists exactly one holomorphic map H : Y --+ X1 x X2 with F = p1 o H and G = P2 o H . Although trivial in our case (we set H important in more general categories.
-
(F, G) ) , this property becomes
1 60
IV. Complex Manifolds
Analytic Sub sets . Let X be an n-dimensional complex manifold. Definition . A subset A c X is called analytic if for each point p E X there are a (connected) open neighborhood U = U(p) and finitely many holomorphic functions !I , . . . , fm on U such that
UnA
= { q E U : fi (q) = 0 for i =
1 , . . . , m}.
We call A an analytic hypersurface if we can always take m = 1 . From the definition it follows that A is a closed subset of X . Locally, an analytic set in X is the same as an analytic set in an open set B c e n . So most properties of analytic sets in e n can be transferred. 1 . 6 Propositio n . If X is connected and A C X analytic, then either A = X or A is nowhere dense and X - A is connected.
Assume that A f= X . If A is somewhere dense in X , then A contains interior points (because it is closed in X ) . Since X is connected, the interior of A has a boundary point p E X - A (same argument as in the proof of the identity theorem). We take a connected neighborhood U = U(p) such that A n U = {q E U : fi (q) = · · · - fm (q) = 0}. Then U contains an open subset V (consisting of interior points of A ) where !I , . . . , fm vanish identically. By the identity theorem they vanish on the whole set U, and p cannot be a boundary point of the interior of A . This is a contradiction, and it follows that A is nowhere dense. PROOF:
If X - A is not connected, it can be decomposed into two nonempty open subsets U1 , U2 . The function f : X - A -t e with f(x) = O on U1 and j(x) = 1 on U2 is holomorphic and bounded. By Riemann's extension theorem (which can be applied locally) there exists a holomorphic function f on X that coincides with f outside A . Since f can take only the values 0 and 1 , it is locally constant. But on the connected manifold X every locally constant • function is constant. This is a contradiction.
-
-
Let JI , . . . , fm be holomorphic functions that are defined on an open subset U c X . Let p E U be a point and (V, 1/J) a complex coordinate system in X at p . The mapping f (!I , . . . , fm) : U -+ e m is holomorphic, and we define =
i
J _
-
_
This is something like a Jacobian matrix of f at coordinate system 1/J. Since
1, . . . , m 1, . . . , n
p,
).
but it depends on the
1.
161
The Complex S tructure
we have This shows that
is independent of the chosen coordinate system. Definition . An analytic set A c X is called regular (of codimension d) at a point p E A if there are an open neighborhood U = U (p) c X and holomorphic functions h, . . , fd on U such that: 1 . A n u = { q E u : fi ( q ) = . . . = fd ( q ) = o } . .
2.
rkp (fi , . . . , fd )
=
d.
The number n - d is called the dimension of A at p .
If A is regular at every point, A is called a complex submanifold of X 1 .7
Propositio n . An analytic set A is regular of co dimension d at p E A if and only if there is a complex coordinate system ( U, cp) in X at p with cp(U) = B C en and cp(U n A ) = {W E B : Wn- d+ l = . . = w, 0} .
=
.
If A is a complex sub manifold cl X , then A itself is a complex manifold.
PROOF: Let (U, 1j;) be an arbitrary coordinate system at p and W := 1/; ( U) . Then A == 1/; (A n U) is an analytic subset of W that is regular of codimension d at z0 := 1/J(p), and there exists a biholomorphic map f!rom W onto an open { w : Wn- d+ l 0 and f ( A ) neighborhood B B( O ) c en with f(zo ) . . . w. 0). We take cp : f 1/;. _
=
=
=
=
=
=
o
If A c X is a submanifold, then A inherits a natural complex structure from • X . This can be demonstrated in the same way as in the case X en . =
Example Let F : X -+ Y be a holomorphic map from an n-dimensional manifold into an m-dimensional manifold. Then GF
is called the graph of F.
:=
{(x , y) E X
x
Y : y
=
F(x) )
162
IV. Complex Manifolds
Let Po E X be a point and qo := F(p0) E Y. We choose coordinate systems ( U,cp ) in X at p0 and ( V,?,b) in Y at q0 , with F(U) C V. Then ( U x V,cp x ?,b) is a coordinate system in X x Y at (p0 , q0 ) E G F · Writing 1,b o F = (!I , . . . , fm ) we get 1 GF n (U x V ) = { ( cp x 1,b) - ( z , w ) : j;
a
1 cp - ( z ) - w; = 0 for i = 1 , . . . , m}.
So GF is locally defined by the functions gi (p, q ) := j; (p) - w; o 1,/J( q ) , for i = 1 , . . . , m. Since rk (po ,qo) (g1 , . . . ,gm ) = m, we see that G F is an n-dimensional submanifold. The diagonal �x c X x X is a special case, which is given as the graph of the identity: �x = { (x , x') E X x X : x = X'}. Example
Let A = { (w, z1 , z2) E C3 : w2 = Z1 Z2 } · The projection p : (w , z1 , z2 ) M 2 (z1 , z2 ) realizes A as a branched covering over C that is the zero set of the pseudopolynomial w( w ; z1 , z2 ) = w2 - z 1 z2 . Outside the discriminant set Dw = {(z1 , z2 ) : z1 z 2 = 0} it always has two regular leaves over C2 . So A is everywhere 2-dimensional and regular outside Dw . It is even regular outside the origin, since V'w(w, z1 , z2 ) vanishes only at ( 0 , 0 , 0 ). One can show that 0 is, in fact, a singular point; e . g . , by using Exercise 8.2 in Chapter I. 2 The map cp : C -+ A with cp(t1 , t2 ) : = ( t1 t2, ti , t§ ) is surjective. We call it a unifo rmization. The Jacobian
vanishes exactly at (0, 0) . The image point (0, 0 , 0) = cp(O, 0 ) is then called a nonuniformizable point. Analogously to the situation in en one proves that the set S ing ( A ) of singular points of an analytic set A is a nowhere dense analytic subset. The set A is called irreducible if A - Sing (A) is connected. To every analytic set A C X there is a uniquely determined locally finite system of irreducible analytic sets (A>.) >.cA such that A is the union of all these irreducible components A>. .
Differ entiab le Function s. Let X be an n-dimensional complex man ifold, B c X an open set. A function f : B + C is called differentiable (respectively smo oth), if for every complex coordinate system ( U,cp) with U n B -!= 0 the function j o cp - 1 is differentiable (respectively infinitely dif ferentiable) on cp(U n B ) . We denote the IR-algebra of real-valued smooth
1 . The Complex Structure
163
functions on B by g(B ) and the «:>algebra of complex-valued smooth fuc tions by g ( B, q.
1 . 8 Pr op osition. In every complex manifold X there is a sequence of com
pact subsets (K, ) with Kn C (Kn + 1 t and U:' 1 Kn = X .
PROOF: Since X is locally compact with countable basis, we can find a countable basis ( Bv ) v E N of the topology of X such that each Bv is compact. We take K1 := B 1 . lf n 1 is the minimal number such that K1 C B 1 U · · · UB nu then k 1 > 2, and we take K2 := B 1 U . . . U Bn 1 , and so on. •
We call ( Kn ) n EN a compact exhaustion of X
1 .9 Pr oposition. Let an open covering fl = { U, : L E I } if X be given, and two real numbers r, r' with 0 < r' < r . Th en there is a locally finite open
refinement Y = { V, : A E L } of fl such that the following hold:
For each A E L there exists a complex coordinate system (V.>- , 'P>- ) in X with 'P .>- (V.>-) = Br ( O ) . 2. Th e open sets 'P.>- 1 (Br' ( 0 ) ) also cover X . 1.
PROOF: We use a compact exhaustion ( K, ) and define M1 := K1 and Mn == Kn (Kn - 1 t for n > 2. Then (Mn ) is a covering of X by compact sets. -
We consider a fixed M = Mn . For each x E M there is an index L = L (x) E I and an opcn ncighborhood V = V(x) c U, ll ((Kn + l t - Kn - 2 ) · Wc can makc V so small that there is a complex coordinate system rp : V --+ Br (O ) with rp(x ) = 0, and we define V' : = rp- 1 ( Br' ( 0) ) . The set M is covered by finitely many neighborhoods V� 1 , , V� m like our V'. Then . n .
•
.
.
Y := { Vn i : n E N, i = ,
l , ..
.
, mn } 8
is the desired covering.
Definition . A (smo oth) partition of unity on X is a family (rp, ) ,EI of smooth real-valued functions such that: 1 . rp, > 0 everywhere. 2. The system of the sets supp( rp, ) is locally finite. 3. E, Ef rp, = 1.
1 . 10 Theor em. For any open cowering fl = { U,
pa rtition
d
unity ( rp, ) with supp(rp , ) C U, .
:
L
E I } of X there is a
164
IV. Complex Manifolds
PROOF : We have a locally finite refinement Y = { VA : A E L } of f/ and complex coordinates cpA : VA -+ Br (O) as in Proposition 1 .9. If 11, : e n -+ lR is a smooth function with 0 < 1/;(z) < 1 , 1/; (z) E 1 on Br' (O) and 1/;(z) = 0 on e n - B, ( 0 ) ,we define a smooth function 1/;A on X by 1/;A = 1/J 0 cpA on VA and 1/;A (x ) = 0 otherwise.
Let T : L -+ I be a refinement map (with VA c U (A ) ) · Then 1f/ = { W, : L E I } with W, := U AET -1 (, VA is an open refinement of f/ with W, C U, . ) In addition it is locally finite, because for 17 E X there is a neighborhood P = P(x) such that P n VA -1= a only for A E La, La c L finite. But then P n W, -1= 0 only for L = T(A ) , A E La. 7
We define cp, : = 'l: AET - 1 ( ,) 1/JA . The sum is finite at every point. So cp, is c A 1 ( Br' (0)) , smooth and has its support in W, . Every 11 E X lies in a set p where 1/;A is positive. Therefore, cp == '2: , cp, is well defined and everywhere • positive. Now we can define the partition of unity by cp, == cp, j cp. 1 . 1 1 Cor ollar y. Let U C X be an open set and V CC U a n open subset. Then there exists a function f E G"(X) with f lv = 0 and f l cx-u) = 1 . PROOF: The system { U, X - V} is an open covering of X . Let { cp1, cp2 } be a partition of unity for this covering. Then supp(cpl ) c U, supp(cp2 ) c X - V, • and cp1 + cp2 = 1 . We take f == 'P2 ·
Tangent Vector s . Let X be an n-dimensional manifold and a E X an arbitrary point. Defin ition . such that
A derivation on X at a is an R-linear map
v[f · g]
=
v[f] · g (a) + f(a) · v[g]
v
:
G"(X) -+ lR
for f, g E G"(X)
If c is constant, then v[c] = 0 for every derivation v .
1 . 1 2 Pr op osition . Iff E G"(X) and f l u = 0 fo r some open neighborhood U = U(a) C X , then v[f] = 0 fo r every derivation v at a. PROOF: We choose a function g E G"(X) such that g l v = 0 for some open neighborhood V = V (a) CC U and g l cx - u ) = 1 . Then g . f = f, and from the derivation rule it follows that
v [ f] = v [g . f] = v [g] . f(a) + g (a) . v[f] = 0 . •
1.
The Complex S tructure
165
1 . 1 3 C or ollar y. Iff,g are twofunctions of g(X) with f l u = g l u fo r some open neighborhood U = U( a ) C X, then v[f] = v[g] fo r every derivation v at the point a . It follows from the corollary that we can restrict derivations to locally defined functiOnS and WOrk With COOrdinateS. If cp = ( Z 1 , . - - , Zn ) : U -+ en iS a coordinate system at a and zv = xv + i y, then we define partial derivatives at a by
( a[}y, ) a [!] : = (f
0 cp- 1 )y, (cp(a) ) ,
for i = 1 , . . . , n. The partial derivatives depend on the chosen coordinate system, but once we have made our choice, every derivation v at a has a unique representation'
v=
b
t a ( a� ) a + t bi ( a8y, ) a ' i= 1 i= 1 ;
'
with ai = v[xi] and i = v[yi] for i = 1 , . . , n. .
In en the space of derivations is isomorphic in a natural way to the space of tangent vectors. But what is a tangent vector on a complex manifold X ? We start with a differentiable path a : I -+ X , where I c lR is an interval with 0 E I, and a(O) = a . Let (U, cp) be a coordinate system in X at a. Then we a,) and get the tangent vector can write cp o a = ( a 1 , .
.
.
•
( cp o a)' ( 0 ) = (a � ( 0 ), . . . , a� (0)).
Unfortunately, this vector depends on the coordinate system. But a tangent vector at a should somehow be completely determined by a pair (p ,c ), where cp is a coordinate system at a and c = ( c1 , . . . , c,) E en an arbitrary vector. In this sense the tangent vector to a is given by the pair (cp, (cp o a)'(O ) ) . If we take another coordinate system 1/;, then (1/; o a) ' (O ) = (cp o a) ' (O) . J'I/J o
Therefore, we call two pairs ( cp, c ) and ( 1/J, c' ) equivalent if the Jacobian of 1/J o cp- 1 at cp( a) transforms c into c' , i.e., if
c = c' . Jcpo'I/J ' ( 1/; (a )) t . An equivalence class is called a tangent vector at a . The set Ta (X) of all tangent vectors at a is called the tangent space. It carries the structure of a complex vector space, which can be defined on representatives: 1 For the proof use the fact that every smooth function f on a domain G c en has near Zo E G a unique representation f(z) = L�= l gv (z) (xv - xe) + L �= l hv (z) (Yv ye) with gv(zo ) = fxv (zo ) and hv (zo ) = fYv ( zo ) -
IV. Complex Manifolds
1 66
( rp, c i ) + ( rp, c2 ) .A · (rp, c ) .
-
(rp, c1 + c2 ) , (rp, .\ - c ) .
The C omp lex Str u cture on the Sp ace of Der ivation s. If f = g + ih is a complex-valued smooth function on the open set B c X and v a derivation at a E B, then we defme v[f] : =v[g ]
+ i v[h]
.
1 . 1 4 Pr op osition . For every c E en and every coordinate system rp at a there is a unique derivation v at a such that v[f] = = c . V(j
o
rp- 1 ) (rp(a)) t fo r every holomorphicfunction f
The derivation v depends only o n the equivalence class of ( rp, c ) . P ROOF: If a coordinate system rp = (z 1 , . . . , zn ) with zv = x v vector c = a + ib are given, then v can be defmed by
+ i yv and a
t (�) t ( )
8 av a + bv a v a v= Yv a v= 1 1 If f is a holomorphic function, then fYv = ifxv and fxv = fzv · Consequently, v
:=
n
v[j] = :l) a v + i bv)(J o rp- 1 ) xv (rp(a)) = C · '\! (! o rp- 1 ) (rp(a)) t . v 1 o=
The uniqueness follows from the equations v[x v ]
+ i v [y v]
= v[zv] = Cv
If the pair ( rp, c) is equivalent to ( ?,b, c '), then c'
. '\1
( f o 1,b - 1 ) ( 1,b (a) ) t
c ' · Jcpo'I/J (1,b (a) ) t · '\1 (J o rp - 1 ) ( rp( a)) t . V ( fo rp - 1 ) ( rp( a)) t . -1
-
c
Therefore, v is determined by the equivalence class of ( rp, c ) .
•
The assignment ( rp, c ) f--t v induces a real vector space isomorphism between the tangent space Ta (X) and the space of derivations at a. It follows that the tangent space has complex dimension n . The pair ( rp, ev ) is mapped onto the derivation (8/ox v )a , and (rp, ie,) onto (o jo yv ) a 1 . 1 5 Pr op osition . A complex structure o n the space cl derivations at a is
given by
J(v ) [j] = i . v [f], fo r every holomorphicfunctio n f .
1 . The Complex Structure Obviously, J is lR-linear, and J o J(v) = -v.
P ROOF:
167
•
If v corresponds to the tangent vector given by ( cp, c ) , then J ( v) corresponds to ( cp, ic) . One must distinguish carefully between the real number J (v) [ !] and the complex number i . v [ !] if f is a real-valued smooth function. The differential operators ( a I azv ) and ( aI az v ) are not real-valued deriva tions, and therefore they do not correspond to tangent vectors. But they are nevertheless useful. If v is a derivation, then a
a
n
n
u=l
v=l
2::= av u o cp - 1 )x v + � b v ( f o cp- 1 )y v
v [ Jl
n
L av . ( (f 0 cp- 1 ) zv + ( J 0 cp- 1 )zv )
V= [ n
v= 1 n
n
u=l
if Cv := av
v
=1
+ ib, for v = 1 , . . . , n.
Therefore, every (real-valued) derivation can be written in the form
v=
I:C v (a� ) + t cv ( � ) v Zv v=l v=l a
a
·
The Indu ced M apping. Let F : X --+ Y be a holomorphic map be tween complex manifolds. Let x E X be an arbitrary point, y := F( x) € Y. Defin ition . defmed by
The tangential map F* = (F* )x : Tx (X) --+ Ty (Y) 1s
( F* v) [g ] = = v [go F] , for derivations v and functions g E G"(Y) .
The map F* is linear, acting on tangent vectors as follows:
if cp is a coordinate system at x, and 1/J is a coordinate system at
y.
Now we have an assignment between the category of complex manifolds (with a distinguished point) and the category of vector spaces. To any manifold
168
IV.
Complex Manifolds
X and any point a E X there is associated the tangent space Ta (X) . To any holomorphic map F : X -+ Y with F(a) = there is associated the homomorphism F* : Ta ( X ) -+ Tb (Y ) . This assignment has the following
b
properties:
(idx ) * (G o F) ,
-
i drx (X) ' G * o F (if G *
:
Y -+ Z is another
holomorphic map).
Such an assignment is called a covariant functo r. If it interchanged the order of the maps, it would be called a contravariant functor. Remark. For historical reasons the elements of the tangent space are called contravariant vectors and the elements of its dual space covariant vectors. But the tangent functor behaves covariantly on the tangent vectors and con travariantly on the covariant tangent vectors in Ta (X ) '· One should keep this in mind.
Immer sion s and Sub mer sion s. We are particularly interested in the case where the (local) Jacobian of a holomorphic map F : X -+ Y has maximal rank. If n = dim (X) and m = dim(Y) , then the rank is bounded by min(n, m ) . Only two cases are possible: Definition . The holomorphic map F is called an immers ion at x if rk(F* ) = n < m, and F is called a s ubmersio n at x if rk(F* ) = m > n. In the first case ( F* ) is injective; in the second case it is surjective. x
We call F an immersion (respectively submersion) if it is an immersion (re spectively submersion) at every point x g X. Remark. If F : X -+ Y is an inj ective immersion, then for every x E X there are neighborhoods U(x) c X and V(F(x)) c Y such that F(U) is a submanifold of V . In addition, if X is compact, then F(X) is a submanifold of Y. We omit the proof here. 1 . 1 6 Theorem .
Let tio ns are equivalent:
xo E X
be a point,
y0
== F( xo) . The following condi
1 . F is a submersio n at xa. V(yo) C Y with 2. There are neighborhoods U U(x0) C X and V F(U) c V , a manifo ld Z, and a holomorphic map G : U -+ Z such that x f--t (F(x) , G(x)) defines a biholomorphic map fro m U to an open subset d V X Z. 3 . There is an open neigborhood V = V( Yo) C Y and a holomorphic map s : V -+ X with s (y0) = x0 and F o s = idv . ( Then s is called a local section for F. ) =
=
1.
The Complex Structure
1 69
PROOF: ( 1 ) ===? ( 2 ) : We can restrict ourselves to a local situation and assume that U = U(O) c en and V = V(O) c em are open neighborhoods, and F : U -+ V a holomorphic map with F(O) = 0 and rk(Jp (O)) = m.
We write Jp (O) = (J;, ( o ) , J� (O) ) , with J;, (O) E Mm,m (C) and J� (O) E Mm , n-m (
JF (O) =
( J;, (O) 0
J� (O)
En-m
),
for z' E
e m , z" E en-m ,
and therefore det JF (O) -=/= 0.
By the inverse function theorem there are neighborhoods U(O) c U and W(O) c en such that F : U -+ W is biholomorphic.
Z := e n-m
We observe that is a complex manifold, and G : = pr, : U -+ Z with (z' , z" ) f--t z" is a holomorphic map such that ( F,G) = F is biholomor phic near 0 , �
( 2 ) ===? (3) : If U, V , Z, and G are given such that F(U) c V and (F, G) U -+ W c V x Z is biholomorphic, then s : V -+ X can be defined by
s(y) : = (F, G) - 1 (y, G(x0)). Then (F, G) (s(y0 )) = (y0, G(x0)) = (F, G)(x0 ) , and therefore s(y0) = x0 . Furthermore, (F,G) o s(y) = (F,g) o (FG)- 1 (y, G(x0 )) = (y, G(x0)) . Thus F o s(y) = Y · (3) ===? ( l ) : If s is a local section for F, with s(y0 ) = x0, then F* o s * (v) = V for every v E Ty 0 (Y) , Thus it follows immediately that F* is surjective. • L
If F : X -+ Y is a submersio n, then fo r each y E Y the fiber F- 1 (y) is empty or an ( n - m)-dimensiona l submanifold if X , In the latter case Tx (F - 1 (y )) = Ker((F* )x) for all x E F - 1 (y) . 1 7 Cor ollary,
PROOF : We consider a point x0 E X , Let M := p - 1 (yo ) be the fiber over y0 : = F(x0). Then we can find neighborhoods U = U(x0) C X , V = V(y0 ) c Y , a manifold Z, and a holomorphic map G : U -+ Z such that ( F,G) : U -+ W C Vx Z is biholomorphic. It follows that M n U = (F, G)- 1 ( {y0} x Z) n U is a manifold of dimension n m,
-
Since FIM is constant, we have F* ITx0 (M) = 0. This means that Txo (M) c Ker(F* ). Since these spaces have the same dimension, they must be equaL •
1 70
IV. Complex Manifolds
G luing. Assume that X is a set that is the union of a countable collection (Xv)v EN of subsets such that the following hold: 1 . For every Y
E N there is a bijection cp" : X" -+ Mv , where M" is an
n-dimensional complex manifold. 2. For every pair ( YJL) E N x N the subset cpv (Xv n X, ) is open in Mv , and the map
is biholomorphic. 3. For every pair of points a E X , and b E X , with a -=/= b there are open neighborhoods U(cpv (a)) C Mv and V(cpiL (b)) C MIL with cp" 1 (U) n 1 cpiL (V) = o _
1 . 1 8 Pr op osition .
Under the above conditions there is a unique complex structure on X such that the X" are open in X and the cp" : X" -+ Mv are biholomorphic. PROOF: We give only a sketch of the proof and leave the details as an exercise for the reader.
A subset U C X is called open if cpv (U n Xv) is open in M" for every Y. Then the collection of open sets has the properties of a topology on X . In addition, for every open set W c M" the set cp" 1 (W) is open in X . Consequently, the maps cp" : X" -+ Mv are homeomorphisms. From the last hypothesis it follows that the topology on X is Hausdorff, and since the collection of the X" is countable, it has a countable basis.
If U c A1" is open and 1/J : U -+ e n a coordinate system, then 1/J := 1/J o cp" : cp" 1 (U) -+ en is a coordinate system for X . One checks easily that two such coordinate systems are biholomorphically compatible. So we obtain a • complex structure on X . One says that X is obtained by gluing the manifolds M" . Another way to describe this process is the following. Let there be given a collection of complex manifolds Mv , open subsets M"IL c Mv , and biholomorphic maps cpv11. : MIL" -+ M"IL (including 'Pw = idMJ · Consider pairs (x, v) with x E M" . Then (x, Y ) is called equivalent to (y,f-1) if X
E M"IL '
y E MIL" and x = cp"IL (y).
The set X of equivalence classes is the result of the gluing process. one has to add a condition that ensures the Hausdorff property.
Of
course,
Exer cises 1 . Let M be a compact connected complex manifold with dim(M) > 2 and N c M a closed submanifold of codimension greater or equal to 2. Show that every holomorphic function f : M - N -+ e is constant.
2.
2. Let
Complex Fiber Bundles
171
X be a Riemann surface.
(a) If f is a meromorphic function on X , and P the set of poles, show that f(x) for x E X - P, i(x) := oo for x E P
{
3.
4.
5.
6.
defines a holomorphic map j : X -+ C. (b) Prove that every holomorphic map f : X -+ C. that is not identically oo defines a meromorphic function on X . Let f : X -+ Y be a nonconstant holomorphic map between Riemann surfaces, x0 E X , and y0 : = f(x0) E Y . Prove that there is a k > 1 such that there are complex coordinates cp : U ( x0 ) -+ C. and 1/.J : V (y0) -+ C. with: (a) cp (x0 ) = 0, 1/.J(yo) = 0. (b) f (U) c V. (c) 1/.J o f o cp - 1 ( z ) = z k . The general linear group GL n (
X xz Y
:=
{ ( x , y) E X
x
Y
:
f(x) = f(y)}
is a complex submanifold of X x Y .
2.
C omp lex Fib er B u n d le s
Lie Gr oup s and Tr an sformation Gr oup s. Assume that G is a set that has the structure of a group and at the same time that of an n dimensional complex manifold. The inverse of g E G will be denoted by 9 - 1 , the identity element by e, and the composition of two elements 9 1 , g2 E G by 9192 · Defin ition . erties hold:
We call G a complex Lze group if the following two
prop
IV. Complex Manifolds
172
1 . The mapping g f--t g - I (from G to G) is holomorphic. 2. The mapping (gi , g2 ) f--t gig2 (from G x G to G) is holomorphic. There are many examples of complex Lie groups. The simplest one is the space en , where the composition is vector addition. Another example is the group e* with respect to ordinary multiplication of complex numbers. The most important example is the general linear group GL n (C) : = {A E Mn (
Its complex structure is obtained by considering it as an open subset of e n . The multiplication of matrices is bilinear, and the determinants appearing in the calculation of the inverse of a matrix A are polynomials in the coefficients of A . Every matrix A E GL n (q defines a linear and therefore holomorphic map
Then
·
=
=
·
·
=
=
=
We want t o generalize this situation. Let X be a complex manifold and G a complex Lie group. Definition. We say that G acts analytically on X (or is a complex Lie transformation group on X ) if there is a holomorphic mapping
The holomorphic map x f--t
Often we write gx instead of
r
:=
Zwi + . . . t zw2n
is a subgroup of the (additive) group en generated by WI ' . . . , W2n · The group r acts on en by
2.
Complex Fiber Bundles
173
Fiber Bundles . Let X and F be complex manifolds and G a complex Lie group acting analytically and faithfully on F . Definition. A topological (respectively holomorphic)fiber bundle over X with structure group G and typical fiber F is given by a topological space (respectively a complex manifold) P and a continuous (respectively holomorphic) map 1r : P -+ X , together with 1 . an open covering � { U< : L E I } of X , 2. for any L E I a topological (respectively biholomorphic) map =
'P<
: 1f -
l (U<) -+ UL X F
with pr, o 'P< 1r, 3. for any pair of indices 1 L , K) E I x I a continuous (respectively holo morphic) map g<,. : U< n U,. -+ G with =
for x E U<,.
:=
U< n U,.
and p E F .
The maps 'P< are called local trivializations and the maps g,,_ a system qf transition functions. Since G is acting faithfully, we get the following compatibility condition:
Now let a system of transition functions (g<,.) be given such that the com patibility condition is satisfied. Using the gluing techniques mentioned at the end of Section 1 , a suitable bundle space P, a projection 1r : P -+ X , and local trivializations can be constructed as follows: Identifying (x, p) and (x, g<,_(x)p), we can glue together the Cartesian prod ucts U,. x F and U< x F over U<><· Due to the compatibility condition this works in a unique way over U<><>-. . The obvious projection from P to X is continuous. Therefore, P is a fiber bundle over X with structure group G and typical fiber F . If the transition functions g <>< are holomorphic, then P carries the structure of a complex manifold, and the projection and the local trivializations are holomorphic. So P becomes an analytic fiber bundle in that case. Examp l e
Let X be an n-dimensional complex manifold. There is an open covering � = { U< : L E I } ,together with complex coordinate systems
IV. Complex Manifolds
1 74
We take C as typical fiber and @* as structure group, acting on C by multi plication. Then we can define transition functions g<,. : U<,. -+ C* by
The compatibility conditions are satisfied because of the chain rule and the determinant product theorem. So by the gluing procedure described above we get a holomorphic fiber bundle over X that is called the canonical bundle and is denoted by Kx .
Eq uivalence. Let 7rp : P --+ X and 7rQ : Q --+ X be two topological or holomorphic fiber bundles over the same manifold X , with the same fiber F and the same structure group G. We assume that there is an open covering {U< : L E I } of X such that there are trivializations 'P< : 1r/ (U<) --+ f/ U< x F and 1/;< : 1rQ 1 (UJ --+ U< x F . =
Defin ition . A fiber bundle isomorphism between P and Q is a topo logical (respectively biholomorphic) map h : P --+ Q with 7rQ o h 7rp such that for any L E I there is a continuous (respectively holomorphic) map h, : U< --+ G with =
The bundles are called equivalent in this case. We give a description of bundle equivalence in the context of transition func tions. Let (g: ,. ) be the system of transition functions for P with respect to f/ , and (g:�) the corresponding system for Q. Then we have 1/J< o h o cp � 1 (x, g ;,_ (x)p) 1/J< o h o cp,_ 1 (x, p) 1/J< o 1/;;; 1 (x, h,. (x)p)
(x, h< (x)g;" (x )p)
_
(x , g;� ( x)h ,. (x)p).
Since G is acting faithfully, it follows that ( C)
Two systems of transition functions (g;,_) and (g:�) with respect to the same covering are called topologically (respectively analytically) equivalent or cohomologous if there are continuous (respectively holomorphic) maps h< : U< --+ G satisfying condition ( C ) . Equivalent fiber bundles have equivalent systems of transition functions. On the other hand, it is easy to see that fiber bundles constructed from equiv alent systems of transition functions are themselves equivalent. Now we will demonstrate that the latter remains valid in passing to finer coverings.
2.
Complex Fiber Bundles
1 75
Let us assume that there are given systems of transition functions g' = (g:,.J and g" = (g:� ) with respect to a covering fl = { U< : L E I} . We call them equivalent as well if there is a refinement Y = { V, : v E N } of fl (with refinement map T : N -+ I ) and a collection of maps h, : Vv -+ G such that
To show that the systems are equivalent in the old sense, we define h< : U< -+ G by In fact, h, is well defined, since on U< n V,, we have
an d therefore g :� (v ) hv (g ;T (v ) ) - l get
The bundles are equivalent! If two bundles are given with respect to two different coverings, then they are
called equivalent if they are equivalent with respect to a common refinement, for example, the intersection of the coverings. Then everything works as above.
C omplex Vector Bundles . manifold.
Let X be an n-dimensional complex
Defin ition . A complex topological (respectively holomorphic) vector bundle of rank r over X is a topological (respectively holomorphic) fiber bundle V over X with
1 76
IV. Complex Manifolds
section (or simply section) in V over U is a continuous (differentiable, holomorphic) map s : U -+ V with 1r o s = idu . We denote by r(U, V) (or O(U, V)) the vector space of holomorphic cross sections in V over U , by tf:(U, V) the space of differentiable sections, and by �0 (U, V) the space of continuous sections. The vector bundle V is called globally generated if the canonical map r(X, V) -+ Vx with s r-+ s(x) is surjective for every x E X . Let (UL , <1\)<El be a collection of vector bundle charts
0
(X ' z . gL t< (X ) t )
for
( X ' z ) E ULt< X e r .
If s is a holomorphic section in V, then defines a system of holomorphic maps SL : UL -+ er ' and we obtain the compatibility condition s< (x) = s ,. (x) , g<,_(x) t
on
U<>< ·
On the other hand, any such system (s< ) defines a global section s . Example We can define vector bundles by giving a system of transition functions. The construction of the bundle space is carried out with the same gluing technique as for general fiber bundles. If X is an arbitrary n-dimensional complex manifold, and (U< , 'P< )<El a com plex atlas for X , then { Un � E I} . defines a system of transition functions with respect to � The corresponding vector bundle T(X ) is called the tangent bundle of X . It results from gluing (x, c) E u,. X e n with (x, c . gL,_(x) t ) E UL X e n , for x E U<,. . Therefore, we can identify the fiber (T(X) ) with the tangent space Tx (X). The local trivializations
x
For a E U< the trivialization
A holomorphic section in T (X ) is also called a holomorphic vector field.
2.
Complex Fiber Bundles
1 77
Defin ition . Let V, W be two holomorphic vector bundles over X . A vector bundle homomorphism between V and W is a fiber preserving holomorphic map TJ : V -+ W such that for any x E X a linear map TJx : Vx -+ Wx is induced. The map TJ is called a vector bundle isomorphism if TJ is bijective and TJ, 1)- 1 both are vector bundle homomorphisms.
A holomorphic vector bundle V of rank r over X is called trivial if it is isomorphic to the bundle X X c r . This is equivalent to the existence of a frame { 6 , . . . , �r } of holomorphic sections �i E f( X, V ) such that for every x E X the elements 6 ( x ) , . . . , �r (x) E Vx are linearly independent. In this case V is globally generated. But there are also nontrivial bundles that are globally generated.
A holomorphic map TJ : V -+ W is a vector bun dle homomorphism if and only if for each pair of vector bundle charts
W- l 0 1) 0
=
( 2 ,Z .
h(x) t ) .
We omit the elementary proof.
Standar d C on str u ction s . Let X be an n-dimensional complex man ifold. We can think of a vector bundle over X as a parametrized family of vector spaces. Therefore, numerous constructions from linear algebra carry over to the theory of vector bundles. 1 . The dir ect sum : If V, W are two vector bundles over X , then the direct sum, or Whitney sum, V EB W : = V x x W carries a vector bundle structure that is defined as follows:
Let "1£ = { U< : � E I } be an open covering of X such that there are vector bundle charts
can be defined by
If g<,. , respectively h,, are transition functions for V , respectively W , then the matrices
178
IV. Complex Manifolds
are transition functions for V EB W. The fiber of the Whitney sum is given by (V EB W)x = Vx x Wx · 2. The dual bun dle: Let 1r : V -+ X be a holomorphic vector bundle of rank r with tri vializations
We will show that for every x E X there is a natural isomorphism ( V' ) x -+ (Vx ) ' = HomiC ( Vx , C). Given elements x
E
U< , v
E
Vx , and A
E
' (V )x , we define
using the vector space isomorphisms (
cr .
For
X
E UL n u" we obtain (
I
X
(
This shows that the definition of A(v) is independent of the trivializations. 3 . Ten sor power s of a line bundle: Let 1r : F -+ X be a line bundle with transition functions g<" : U<" -+ @ * . Defin ition . For k E N, the tensor power p k is the line bundle defined by the transition functions l". We give an interpretation of p k using the dual bundle 7r1 : F' -+ X . As sume that there is an open subset U c X and a holomorphic function f : (1r' )- 1 (U) -+ C. If 1/;< : (F' ) I u, -+ U< x C are trivializations (with ¢< o 1/J" 1 ( x, z ) = (x, z · g<�< (x ) - 1 ) ) , then we have a power series expansion J o 1j;-; 1 (x, z ) = on
00
l:::a v ,< (x) z v
v =O
1/JL(F'I u, n u ) with holomorphic functions av,L on u n UL . Over UL n u" n u the following holds:
2.
Complex Fiber Bundles
1 79
00
f o 1/J" 1 ( x, z) = J o 1/;;- 1 (x, z · g< �< (x ) - 1 ) = :2::: (av, < (x)g<�< ( x ) - v ) z v , v =O
and therefore av,< = av, �< . g�" . This means that av = ( av,J is a cross section in p k over U. So every holomorphic function ! on F ' that is homogeneous of degree k on the fibers is a section in p k . In particular, F/: can be identified with the space L k ( F� , q of k-linear functions f : F� x . . . x F� -+ C. For e l, .
.
.
, e k E Fx the tensor product e 1 ® . . . ® e k E ( F k )x is defined by
The tensor power e ® · . . ® e of an element e E Fx is denoted by ek for short. Finally, we define p- k : = ( F') k "' ( F k )'. 4 . The ten sor pr oduct: Let p : V -+ X be a vector bundle of rank r with transition functions G, : U<" -+ GL r (q , and 1r : F -+ X a line bundle with transition functions g<" (with respect to the covering f/ = { U< : L E I} ) . Defin ition . The tensor product V ® F is the vector bundle of rank given by the transition functions
r
Let
and on the other hand,
It follows that Consequently, 17 = (17<) is an isomorphism
a
cross section in V ® F, and we obtain for every x
(V ® F)x "' HomiC ( F� , V,).
For v E Vx and e E Fx the tensor product
v
®e
E ( V ® F) x is defined by
(v ® e) (.\) := .\ (e) . v.
180
IV.
Complex Manifolds
Lifting of Bundles. Let f : X -+ Y be a holomorphic map between complex manifolds and p : V -+ Y a vector bundle of rank T. Definition .
The
j* V : = X
lifted
bundle. or pullback. f*V over X is defined by
x y V :=
{(x , v ) E X
x V :
J (x) = p (v)}.
The bundle projection p : f* V -+ X is given by p(x, v ) : = x. The fiber of j * V over x E X is given by (f* V)x = Vf(x) · Therefore, the lifted bundle is trivial over the preimage sets f- 1 (y). One has the following commutative diagram: v
j*V -1-
p
X
�
-1-
y
p
If f/ = { U, E I } is an open covering of Y such that V is trivial over U, , then f/ := { U, = rl (U, ) : E I } is an open covering of X such that !* v �
: L �
L
�
is trivial over U, . If
iP,(x, v )
:=
(x, (
1f G, are transition functions for V with
i.e., f* V is given by the transition functions G, o f . If � E r(U, V ) is a holomofJZ_hic se�tion over some open subset U C Y, then � can be lifted to a section � E r(U, j*V) given by
-
�(x) : = (x, �( J (x) ) ) .
Subbundles and Quotien t s .
Defin ition . Let 1r : V -+ X be a vector bundle of rank T. A subset W c V is called a sub bundle (of rank p ) of V if there is a pdimensional linear subspace E c cr ' and for any X E X an open neighborhood u = U(x) and a trivialization
One sees immediately that:
Complex Fiber Bundles
2.
1. 2. 3.
181
Wx W n Vx is always a pdimensional linear subspace of Vx. W is a submanifold of V . W is itself a vector bundle. =
Example
Let V be a vector bundle of rank r on X and Y c X a submanifold. If we denote the natural injection Y '---+ X by j, then Vly := j *V is a vector bundle of rank r on Y. We apply this to the tangent bundle T( X). Choose an open covering %' = (U,),EI such that there are complex coordinate systems cp, = (zL . . . ,z�) for X in U, with the following properties: 1 . U, n Y = {z:t + 1 = . . . = z�. = 0}. 2. zJ: , . . . , z:t are complex coordinates for Y A trivialization
If?,
: j *T (X ) I u, nY -+ (U, n
(y, [cp" c] )
r+
Y )x
en is given by
(y, c) .
A tangent vector v belongs to Ty (Y) c Ty (X) if and only if there is a differ entiable path a : I -+ Ywith a( O) = x and (cp, o a) ' ( O) = c . This is equivalent to the statement that c = ( c1 , . . . , cd , 0, . . . , 0) . Therefore �P;- 1 (U, X {c E en :
cd+ l
= . . . =
Cn
=
0} )
= T(Y ) I u , ,
and T(Y) is a subbundle of j * T(X) . If G, := then
(oztjoz; I v, p, = 1 , .. . , n) are the transition functions for T(X ) , GU <
. ( ' ) ( gu, (z', 0)
O) Z
-
_
n
gz" ( z' , 0)
g,K = ((8zt/oz;)IY I p, = 1 , .. . , d) , are the transition functions for and gZ" ( z' , O) = ((oztjoz;)( z' , O ) I v, p, = d + 1 , .. . , n) . Let V be a vector bundle of rank If W c V is a subbundle, then we define the quotient bundle VjW by (VjW)x := Vx/Wx. We have to show that there are trivializations for VjW. If E c er is a subspace such that there are trivializations If? : V l u -+ u X e r with Wlu = �J?- 1 (U X E ) , then we can choose a subspace F c e with E ffi F = e and define If? : ( VjW) I u -+ u X F where T(Y),
v,
r.
by
lf?( v mod W ) : = prF (IJ? (v) ) ,
where pr, : E ffi F -+ F is the canonical projection. Using trivializations as above one obtains transition functions
1 82
IV. Complex Manifolds
for V such that g,,. are transition functions for W , and h, are transition functions for VjW. Example
The quotient bundle Nx (Y) := j* T(X) jT(Y) is called the normal bundle of Y in X . Exercises
1. Let G be a complex Lie group that acts analytically on a complex man ifold X . Then for every x E X the stab ilizer G, := { g E G : gx = x} is a closed Lie subgroup of G, i.e. , a subgroup and a (closed) complex submanifold. Prove that there is a unique complex structure on G fGx such that the canonical projection 1r : G -+ G /Gx is a holomorphic submersion. 2 . Let 1r : V -+ X be a complex vector bundle of rank r . For x E X let !JlJx be the set of bases of Vx. Prove that the disjoint union of the !JlJx , x E X , carries the structure of a fiber bundle over X with structure group and typical fiber equal to GLr (C). 3. Let cp : V -+ W be a vector bundle homomorphism over X . Suppose that rk(cpx ) is independent of x E X . Prove that
Ker ( cp) :=
U
x EX
Ker(cpx )
and
Im(cp) :=
U
x EX
Im(cpx )
are subbundles of V , respectively W. Show that Im(cp) "' V/ Ker(cp). 4. Let X = C be the Riemann sphere. Determine the transition functions for T(X) for the canonical bundle Kx and for the normal bundle of { oo} in X . 5 . Let f : X -+ Y be a holomorphic map. Prove that there is a uniquely determined vector bundle homomorphism f' : T(X) -+ f*T(Y) with (f' )x = (f. )x for X E X. 6. Let p : V -+ X be a holomorphic vector bundle. Show that there are vector bundle homomorphisms h : p* V -+ T(V) and k : T(V) -+ p*T(X) over V with Im(h) = Ker(k).
3.
Cohomology
C oh omology Gr oup s . Let X be an n-dimensional complex manifold and 1r : V -+ X a complex vector bundle of rank T. Assume that there is an open covering f/ = { U, : L E I } of X . We consider sections in V on the sets U, and on intersections U,,. := U, n U,. , respectively U,,.>- := U, n U,. n U>- .
3.
Cohomology
1 83
A 0-dimensional cochain with values in V (with respect to � ) is a function s that assigns to every L E I a section s, E r(U" V ) . The set of 0-dimensional cochains is denoted by C0 ( � , V ) . Defin ition .
A !-dimensional cochain (with values in V )is a function � that assigns to every pair 1 L , K) E I x I a section �"' E r(Uu, , V ) .The set of !-dimensional cochains is denoted by C 1 (0/o, V ) . Finally, a 2-dimensional cochain (with values in V ) is a function A that assigns to every triple 1 L, K, v) E I x I x I a section A,, E r(U uw , V ) , and the set of all these 2 -dimensional cochains is denoted by C2 (� ' v ) . Assume that a 0-dimensional cochain s is given. One may ask whether the sections s, E r ( U, , V ) can be glued together to a global section s E r(X, V ) . For that it is necessary and sufficient that s, = s" on U," . This can be expressed in another way: If we assign to every 0-cochain s a 1-cochain 6s by ( Os ) , " := s" - s,, then s defines a global section if and only if 6s = 0. Defin ition .
The coboundary operators
are defined by · .
(on u,l< ) '
A cochain s E C0 (� , V ) (respectively � E C 1 ( � , V ) ) is called a cocycle if 6s = 0 (respectively o� = 0). The sets of cocycles are denoted by Z0 (� , V) (respectively Z 1 (� , V) ) . Remark s
The sets of cochains and the sets of cocycles are all complex vector spaces. 2. We can identify Z0 (�, V ) with r( X , V ) . 3. A 1-cochain � is a cocycle if and only if the following compatibility con dition holds: �w = �'" + ��
4. Sometimes we need cocycles of degree 2. We call an element A E C2 (� , V ) a cocycle if the following compatibility condition holds:
The set of all these cocycles is denoted by Z2 (�, V ) .
184
IV. Complex Manifolds
A co boundary is an element of the image of the co boundary operator. The sets of coboundaries are again vector spaces, denoted by B 1 (Uf/ , V) = OC0 (Uf/ , V) and B2 (Uf/ , V) : = OC 1 ( Uf/ , V). For completeness we define B0 (Uf/, V) : = 0 . 3 . 1 Pr o p o sition .
PROOF:
Bi (Uf/, V) c Zi (Uf/ , V ) to r i = 0, 1 , 2 .
The case i
=
0 is trivial. For �
=
O.s E B 1 (Of/ , V) we have
( TJvl" - 1],1" + TJw ) - ( TJr<J" 1],1" + 1],1< ) + ( TJr< v - TJw + TJ,r< ) T}vl" - TJr<J" + TJr< v = Ar< VJ" -
•
•
Hi (Uf/ , V) : = zi (Uf/ , V ) /B' (Uf/ , V) is called the ith coho mology group of V with respect to 0/o.
Defin ition .
We have H0 (Uf/ , V)
=
r(X, V ) , independently of the covering, and we have
The canonical map from Z1 (Of/ , V) to H 1 (Of/, V) will be denoted by q. We do not want to elaborate on H2 , because we need cases.
Refinement s . Let 'f/ = {V, there is a refinement map T : N To : C0 (Uf/ , V) -+ C0('f/, V) and T1 (To .S )n := ( 8T(n) ) l vn
it
only in very special
: n E N} be a refinement of Of/ . Then -+ I with Vn C U7 (n) . It induces maps : C1 (Uf/ , V) -+ C 1 ( 'f/ , V) by
and
h� ) nm := (�T(n)T(m) ) I Vnm
Then O(To.s) = T1 (6.s), and if 6� = 0, then also O(T1 � ) = 0. Therefore, T induces a map T* : H1(Uf/, V ) -+ H 1 ('f/, V) by T* (q(� ) )
:=
q(T1 �).
By the remarks above it is clear that T* is well defined, and it is a vector space homomorphism.
3.
The map 7* is independent
3 . 2 P r op osition .
is injective.
of
Cohomology
185
the refinement map 7 and
PROOF: Let a : N -+ I be another refinement map and � a cocycle with respect to f/ . We have to show that 71 � - a � is a coboundary: 1
�T(n)T(m) I Vnm - �a(n)a(m) I Vnm (�T(n)a(n) + �a (n)T(m) ) - (�a (n)T(m) + �T(m)a(m)) �T(n)a (n) lvnm - �T(m)a(m) l vnm •
We define TJ E C° C'f/, V) by
Then 71� - a 1 � = OTJ. Now we consider a cocycle � with respect to f/ such that 71 � = O s for some s E C0 (''f/ , V ) . We want to see that � itself is a coboundary. On \;
n U, we have
Since �T(n)T(m) I Vnm = (s m - sn ) l vnm ' it follows that :=
Therefore, we can define h, on U, by h, I u, nvn �
h, - hi< = (�
=
0 ( - h ) . So 7* is injective.
•
We have seen that if Y is a refinement of f/ , then H 1 ( f/ , V ) can be identified with a subspace of H1 (Y, V ) . Therefore, we form the union of the spaces H1 ( f/ , V ) over all coverings f/ and denote this union by H 1 (X, V ) . We call it the absolute, or Cech, cohomology group of X with values in V . Acyclic Coverings. Let
vector bundle over X .
X be a complex manifold and p
:
v
-+
X
a
The covering f/ is called acyclic, or a Leray covering for V , if H1 (U" V) = 0 for every L E I . If Y = {V, : n E N } is another covering, then U, n Y := { U, n Vn : n E N } is a covering of U" and since H 1 ( U, n Y, V) is a subspace of H1 (U" V ) , we have also H1 (U, n Y, V ) 0. =
1 86
IV. Complex Manifolds
3 .3 Theorem. If ut/ = {U, : L E I } is acyclic and Y = {V, : n E N } is a refinement, with refinement map T : N -+ I , then T* : H1 (Of/ , V) -+ H1 ( Y, V) is bijective. PRooF:
We start with an TJ E Z1 (1/, V) and define TJ ( ' ) E Z1 (U, n Y, V) by 1)�,),
: = T}n m l u,nVnm , for all n, m with U, n "
f 0.
Since Of/ is acyclic, there is an element g ( <) E C0 (U, n "f/ , V) with TJ ( < ) = il ' ) . ( ) ) ) ) Then TJnm + g�' = g� on U, n V, m , and therefore gr,::( - g� = g}/: - gr, on U," n Vn m · Now we define � E Z1 ( Uf/ , V) by � " i u,K n Vn : = gk ) - g�" ) , and h E C0 ( Y , V)
' n r )) (on " = v, n ur ( n ) ) · by h, : = g� ( Then on Vnm we have
h � - TJ) nm
�r ( n ) r (m ) - TJnm g� ( n) ) g{r (m )) g �r ( n )) g� (m ) ) h, - hm . _
_
(gF( n ))
_
g �r ( n )) )
_
So TJ = T1 � + Sh, and T* is surjective.
3 . 4 Corollary.
•
Ij 0l/ is an acyclic covering of X , then H 1 (X, V ) = H 1 (Of/ , V).
PROOF: We have H 1 (UZ£ , V) c H1 (X, V). 1f a is an element of H1 (X, V), then there is a covering Y with a E H 1 ( Y , V) . Now we can find a common refinement 1f/ of Of/ and Y. Then H 1 (Y, V) c H 1 (1f/, V) = H1 (Uf/ , V ) , and • therefore a E H 1 (Of/ , V ) .
Gener alization s. The simplest case of a vector bundle over X is the trivial line bundle Ox : = X x C. Therefore, the associated Cech cohomology group H 1 (X, Ox ) plays an important role for the function theory on X . The trivial fiber bundle X x @* is not a vector bundle, but it is not so far from that. If 1r : P -+ X is a general analytic fiber bundle, then a section in P over an open set U c X is a holomorphic map s : U -+ P with 1r o s = id u . If the typical fiber of P is an abelian complex Lie group, then the set r ( U, P ) of all sections in P over U carries in a natural way the structure of an abelian group. In the case of the bundle Ox : = X x C* we have a canonical isomorphism r ( U, Ox ) "' O* (U) : = { f E O(U)
:
f (x ) i= 0 for every x E U}.
3.
Cohomology
187
This is a multiplicative abelian group . If f/ = { U, : L E I } is an open covering of X , then we can form cochains � : ( L , K ) f--t �"' E O* (U," ) . The set C1 (f£ , 0x ) of all these cochains forms a (multiplicative) abelian group, and we can define the subgroups Z 1 (f/ , O:X ) and B 1 (f/ , O:X ) of cocycles and coboundaries : A cochain � is called a cocycle if �w = �'"�"v on U," v , and � is called a coboundary if there are functions s, E O* (U,) such that �'" = s"s;:- 1 on U," .
Since all the groups are abelian, we have the quotient group
which we call the first cohomology group with values in O:X (with respect to the covering 0.?/ ). Just as in the case of vector bundles we can pass to finer coverings and finally form the Cech cohomology group H 1 (X, O:X ) . There is a nice interpretation for the elements of H 1 (X, O:X ). Every cocycle with values in O:X defines a line bundle over X , and this bundle is independent of the covering. Two cocycles e and � II define equivalent line bundles if and only if there are functions h, with ��" = ��� h"h;:- 1 , i.e., if and only if the cohomology classes of e and e' are equal. Therefore, H1 (X , O x ) is the set of isomorphy classes of line bundles over X . This group is also called the Picard group of X and is denoted by Pic(X). The group structure is induced by the tensor product. The identity element corresponds to the trivial bundle Ox and the inverse to the dual bundle. In the same way as above we can form cohomology groups of any fiber bundle P whose typical fiber is an abelian group. If the fiber of P is a nonabelian group, things become a little bit more complicated. We can define cocycles with values in P , but they do not form a group. In the set Z 1 ( f/ , P) of cocycles we can introduce an equivalence relation by (I
<,
(II
"-' <,
:
<¢==:?-
(I = h - 1 g'" II h" . 3 h, W1th "''" , •
The set H1(f/ , P ) of all equivalence classes is called the cohomology set with values in P (with respect to f/ ) . Usually it is not a group, but there is a distinguished element, represented by the cocycle � with �'" = 1 for all L , K . Passing to finer coverings and forming the cohomology set H 1 (X, P ) causes no problems. The most important nonabelian case is the cohomology set
whose elements correspond to the isomorphy classes of vector bundles of rank r over X . The distinguished element is represented by the trivial bundle X X
1 88
IV.
Complex Manifolds
abelian group for every open set U c X , and that we can restrict sections over U to open subsets V c U. Having this in mind we can define another sort of cohomology group . Let K c C be a subgroup with respect to the addition, for example K = Z, IR, or C. For an open set U c X we define the (abelian) group K(U) by K(U)
:=
{f : U -+ K
:
f is locally constant } .
Then K(U) is an abelian group . 1f V c U is an open subset and f an element of K(U) , then flv E K(V). If U is connected, then K(U) "' K . We can define groups of cochains C0( %' , K ), C 1 ('32, K ) , and C2 ( %' , K ) just as we did it for vector bundles. If � is an element of C 1 ( %' , K ) , then �"' E K(Uu, ) · Cocycles, coboundaries, and cohomology groups are defined in the usual way. For example, we have
The Singular C oh omology. We want to give a short overview of co homology groups of topological spaces and their relation to Cech cohomology as defined above. For proofs see [Gre67] . For
q
E No the set
A,
:=
{x
=
( xo , . . . , xn ) E JR q + l
n
: L x; i=O
=
l , x; >
0}
is called the q-dimensional standard simplex. The 0-dimensional simplex is a point, � 1 a line segment, � 2 a triangle, and so on. Let X be a topological space. We assume that all spaces here are connected and locally connected. A singular q-simplex in X is a continuous map a : A, -+ X . If X is a complex manifold and if there is an open neighborhood U = U(� q ) and a smooth map a : U -+ X with &l�q = a , then a is called a differentiable q-simplex. A singular q-chain in X is a (formal) linear combination n 1 a1 + . . . + n k ak of singular q-simplices with n i E Z. The set Sq (X) of all singular q-chains in X is the free abelian group generated by the singular q-simplices. For a q-simplex defined by It
a
and i
=
is called the ith face of a.
0, . . . , q the
(q
- I)-simplex
ai
:
� q- 1 -+ X is
3.
Cohomology
1 89
The (q - 1 )-chain aa := '2:.I=o ( - 1 ) i ai is called the boundary of a. The bound ary operator a induces a homomorphism
and it follows easily that a 0 a = 0 . Definition . The group Hq (X ) := { c E Sq (X) : de = O } jaSq + 1 (X) is called the qth singular homology group of X . In general, H0 (X ) is isomorphic to the free abelian group generated by the connected components of X . Since we assume all spaces to be connected, we have H0 (X) "' Z . If X is an n-dimensional complex manifold, then Hq (X ) = 0 for q > 2n. Now for
q
> 0 we define the group of singular q-cochains to be
Then the coboundary operator o : Sq (X) --+ Sq + 1 (X) is defined by
Jj [c] := j[ac] , for f E Sq (X) and c E Sq + 1 (X) . Obviously, we have 6 o 6 = 0. We can define cocycles (elements f of Sq (X ) with o j = 0) and coboundaries (elements of the form og with g E sq-1 (X ) ) . Defin ition . The group Hq (X) : = { ! E Sq (X) : oj = O} joSq - 1 (X ) is called the qth singular cohomology group of X . From above it is clear that H0 (X) "' Z.
A topological space X is called contractible if there is a point x0 E X and a continuous map F : [0 ,1 ] x X --+ X with F(O, x) = x and F(1, x ) = x0 • In that case H1 (X ) = {0} . The space X is called locally conlraclible if every poinl of X has arbitrarily small contractible neighborhoods. Among the connected topological spaces there is a big class of spaces (including the so-called CW complexes) that are locally contractible and have the following properties:
1 . Hq (X) "' Hq (X, Z) for q = 0, 1 , 2, . . ..
2 . Every open covering of X has a refinement tt/ = { U, acyclic in the sense that H1 (U,, Z) = 0 for every L E J .
:
L
E I } that is
We call such spaces good topological spaces. For example, every (connected) complex manifold is a good topological space, and also every irreducible an alytic set. Finitely generated abelian groups are classified as follows:
190
IV. Complex Manifolds
If G is a finitely generated abelian group, then there are uniquely deter mined numbers r E No and n 1 , . . . , ns E N with ni > 2 and nilni+l for i 1 , . . . , s 1 such that =
-
The number r is called the rank of G, and zr the free part of G. The sum of the ZjniZ is called the torsion part of G . In many cases cohomology can be computed from homology:
If X is a compact good topological space (for example, a compact connected complex manifold), then Hq (X) is finitely generated for every q, and for q > 1 there is an isomorphism
3.5 Theorem.
H q (X) '"" (free part of Hq (X)) EB (torsion part of Hq _ 1 (X) ) .
The rank of H 1 (X) is called the first b1 (X).
Betti number of X , and is denoted by
Now we give an application of Cech cohomology methods to singular coho mology. Let X and Y be good topological spaces. 3.6 Theorem (Kiinneth formula) .
We choose open coverings 02/ = {U, : L E I} of X and )/ = {Vn n E N} of Y such that all the U,, Vv and all their pairwise intersections are connected. We write Wm := U, X Vn and Wm,l<m : = Wm n Wl<m = u,l< X Vnm · A cocycle '1/! E Z 1 (%' X 1/ , Z) is given by constant maps '1/Jm , l<m : Wm,l<m --+ z.
P ROOF:
We identify any cocycle � E Z 1 (%' , Z) with a cocycle [ E Z 1 (%' x 1/, Z) by [n ,�<m = �'" and also any rJ E Z 1 (1/, Z) with an fj E Z 1 (%' x 1/, Z). This ' induces natural injections
with Im(j1 ) n Im(j 2 ) =
{0}, and therefore an injective map
j : H 1 (%' , Z)
x
H 1 (1/, Z) --+ H 1 (%'
x
1/, Z)
by j ( a , b) := J 1 (a) + jz (b). Given a cocycle '1/J E Z 1 (%' x 1/, Z), for any L E I we define '1/J, E Z 1 (1/, Z) by ('1/J ,) n m : = '1/J m,,m · The cohomology class of '1/J, in H 1 (1/, Z) is independent of the i n dex We see this as follows. L
3.
Cohomology
191
(a) If Uu, f. 0 , then by the co cycle property we have 1/JLn,K,n + 1/JKn,Km + 1/JK,m,Lm - 1/'K,n ,Km 1/JLn,Kn - 1/Jim,Km ·
Setting ('Pu,)n := '1/!m .>
For each there is an 'rJL E C0 ( "f/ , Z) with '1/Jo = '1/!L - 61/ ( TJL ) , and we define 'f) E C0 (6!/ X Y', Z) by TJm : = (TJL )n . Then
1
-
: = '1/J - '1/Jo - OrJ E Z1(6!/
x
"f/, Z) and
IL = '1/!L - ('1/!L - 01/ (TJL)) - 01/ (TJL ) =
Now we construct a (} E Z1 ( 6!1 , Z) with For
n
E N define Qn E Z1 (6!1 , Z) by
Since lm,Lm
=
0 for all
n, m,
and
L,
Q=
0 for every
L.
T
in the case Vnm f. 0 we have
and therefore (}m = (}n . If Vnm = 0, we can argue in the same way as above in (b) , because Y is connected. So we have a (} E Z1 ( 6!1 , Z) with Qn = (} for every (}LI< = lm,>
Therefore,
Q=
1
-
=
lm,><m
n
and
(for arbitrary
)
n, m .
and '1/! = Q + '1/!o + OrJ. This shows that j is surjective.
•
Exercises :
1 . Let 6!1 = { U1 , Uz , U3 } be the covering of X : = { z E C 1 < l z l < 2} given by U1 : = { x + i y : y < x} , Uz : = {x + i y : y < -x } , and u3 : = {X + iy : y > 0}. Calculate H1 ( 6!1 ' Z). 2. Prove that dime H1 ((C2 - { 0}, 0) = oo , where 0 denotes the trivial line bundle over C2 . Hint: Use Laurent series.
192
IV. Complex Manifolds
3 . Show that H 1 (e, O) 0. 4. Show that %' = {Ui : i 1 , . . ' n} with ui := {z : Zi f. 0} is an acyclic covering of en - { 0} for 0. 5. Let X be an n-dimensional complex manifold such that Hq (X) = Z for q even, 0 < q < 2n, and Hq (X) 0 otherwise. Calculate the singular cohomology of X. =
=
.
=
4.
Meromorphic Functions and Divisors
Let X be an n-dimensional complex manifold and x E X a point. Two holomorphic functions f, g defined near x are called equivalent at x if there exists a neighborhood U U(x) with f l u = g\u- The equivalence class of f at x is called the germ of f at x. We denote the germ by fx and the set of all germs by Ox· Having fixed a complex coordinate system r.p U -+ B C en at x, we may identify the set Ox with the ring Hn of convergent power series by The Ring of Germs.
=
:
fx H
Taylor series of f o r.p - 1 at r.p ( x).
So Ox has the structure of a local e-algebra. 2 An element fx E Ox is a unit if and only if f(x) f. 0. Of course, Ox is also noetherian and a unique factorization domain.
Let f, g be holomorphic functions near x0 E X . If the germs fxo , 9xo are relatively prime in Ox0 , then there is an open neigborhood U U(xo ) C X such that fx, 9x are relatively prime in Ox for x E U. PROOF: We can work in en and assume that x0 0 and that fo and go are Weierstrass polynomials in z1 . Since fo and g0 are relatively prime in Hn ,
4.1 Proposition. =
=
they are also relatively prime in H� _ 1 [z! ] . If Q is the quotient field of Hn _ 1 , it follows from Gauss ' s lemma that fo and g0 are relatively prime in Q[ z1 ] .
We can find a linear combination
where a, b E Hn _ 1 [ z1 ] , and h E (Hn _ I )* is the greatest common divisor of fo and 90· If u U' X U" c e X en - 1 is a sufficiently small neighborhood of the origin, the power series a, b converge to pseudopolynomials over U" and h to a holomorphic function on U'' that does not vanish identically. =
2
A commutative «:::-algebra A with unity is called a local «:::-algebra if the set m of nonunits forms an ideal in A and the composition of canonical homomorphisms C '-t A __,. A/m is surjective.
4.
Meromorphic Functions and Divisors
1 93
Let w = ( w1 , w' ) be a point in U. If i.pw is a common factor of fw and gw , then it divides hw , which has degree 0 as a polynomial in z1 . So r.pw does not depend on z1 and does not vanish identically in zz , . . . , Zn · Therefore, i.pw is z1 -regular of order 0, and by the preparation theorem it is, up to a unit, a Weierstrass polynomial of degree 0; i.e., r.pw is already a unit. •
This shows that fw and gw are relatively prime.
Let X be an n-dimensional complex man ifold. We consider analytic hypersurfaces A C X. Then locally A is given as the zero set of one holomorphic function f. We always assume that f does not vanish identically and therefore A is nowhere dense in X. If locally A = { ( z1 , . . . , zn ) : Zn = 0}, then every holomorphic function g vanishing on A is of the form g(z1 , . . . , zn ) = Zn · g(z1 , . . . , zn ) · We will generalize this result to the arbitrary case. Analyt ic Hypersurfaces .
Every hypersurface A lytic set of dimension n - 1 .
CX
4.2 Proposition.
is a pure-dimensional ana
After choosing appropriate coordinates, we may assume that A is contained in an open set u c en and f is a Weierstrass polynomial w in Zl without multiple factors such that P ROOF:
A = N(w) = {z E U
:
w ( z)
= 0}.
Since N(w) is a branched covering over some domain G C en -I , every irre • ducible component of N (w) has dimension n - 1 . 4.3 Theorem (Nullstellensatz for hypersurfaces) .
Let A C X be an
analytic hypersurface and xo an arbitrary point of A. 1 . There exists an open neighborhood U = U(x0) C X and a holomorphic function f on U such that: (a) U n A = {x E U : f(x) = 0}. (b) If h is a holomorphic function on a neighborhood V = V(x0) C X with hi = 0, then there is a neighborhood W = W(x0) C U n V and a holomorphic function q on W such that hl w = q · (fl w ) . � 2. If f is any holomorphic function defining A in U and h is again a holo morphic function on a neighborhood V = V(x0) vanishing on A, then there exists a k E N and a holomorphic function q on a neighborhood W = W(x0) c U n V such that hk l w = q · U l w ). A
Again we work in en and assume that x0 0. Let f be an ar bitrary defining function for A near the origin. After choosing appropriate
PROOF:
=
194
IV. Complex Manifolds
coordinates we find a unit e and a Weierstrass polynomial w in z1 such that f = e . w. We choose a neighborhood U = U' x U" of the origin such that:
1 . A n U = {(z1 , z') E U
0�.
w(z1 , z' ) = 2 . There is a prime factorization w = w1 1 :
•
•
•
w;1 on U.
We define w : = w1 · . . wz . This is a pseudopolynomial without multiple factors that also defines A in U.
A, which we may assume to be defined on U, then by the division formula there is a holomorphic function q near the origin
1f h is a function vanishing on
and a pseudopolynomial r with deg(r)
<
deg(w) such that near 0 ,
h =q · w+r. Since w has no multiple factors, the greatest common divisor of w and ow j 8z1 is a not identically vanishing holomorphic function g of zz , . . . , zn , and we can find pseudopolynomials q 1 , qz with
We may assume that everything is defined on U. Suppose that there is a z� E U" such that w((, z!J ) E C[(] has a multiple zero (0 . Then w((o , z�) = 8wj8z1 ((o , z�) = 0, and therefore g(z�) = 0. Hence, if g(z') f. 0, then w((, z') has exactly s :=deg(w) distinct zeros. Since h\ N (w) = 0, h((, z') has at least these s distinct zeros. Using this fact and the division formula, it follows that r(z') = 0 for z' E U" N(g). Therefore, by the identity theorem r = 0 and h = q · w. Taking f = w yields the first part of the theorem. -
Let k := max(k 1 , . . . , kz ) . Then
This proves the second part of the theorem.
•
Every local holoniorphic function f that satisfies the conditions of the first part of the Nullstellensatz will be called a minimal defining function for A. Now let A be an irreducible analytic hypersurface in X , and h a holomorphic function on some open subset U c X with h\(AnU) = 0. For xo E U n A there exists a neighborhood V = V(x0 ) c U and a minimal defining function f for A on V . Then ordA,x0 (h) : = max{m E N
:
:l q with h = fm . q near xo }.
It follows from the Nullstellensatz that ordA,x0 ( h ) > 1 , and from the unique factorization into primes that it is finite. Furthermore, it is independent of j, because if h , fz are two minimal defining functions, then we have equations
4.
h
=
q1 q2)
qz fi . It follows that h So q1 and qz must be units.
q1 !z and
= 0.
Meromorphic Functions and Divisors
fz =
=
195
q1qzh and therefore h ( l
-
For every point xo E A there is a neighborhood U = U( xo) such that ordA,x (h) > ordA,x0 (h) for x E U n A.
4 .4
Pr op osition . If A is irreducible, h holonwrphic in a neighborlwod cl A , and h i A = 0, then the number ordA,x (h) is independent ofx E A. PROOF: Let x0 E A be an arbitrary point. In a neighborhood U of x0 there exist a decomposition A n U = A 1 U . . . U A1 into irreducible components and minimal defining functions f>.. for A.\. Since h vanishes on every A.\, there exist k1 , . . . , kz E N and a holomorphic function q on a neighborhood V = V ( x0 ) C U such that =
h
j�l . . . j1kt q, and (f>.. ) x0 f qx0 for A •
= 1 , . . . , l.
Since (!I )x 0 , (fz )x0 are irreducible, it follows that ( f.\ )xo and qxo are rel atively prime for A = 1 , . . . , l . But then ( f>.. ) x and y, remain relatively prime for x sufficiently close to x0 , say in a neighborhood W ( xo ) c V. •
Let
•
•
,
( ) : ordA,x (h) . It is necessary to consider two cases:
n x
=
( a) 1f A is irreducible at x0 , then l = 1 , and it is clear that n ( x ) = n ( x0 ) for x E W n A. Consequently, x H n ( x ) is a locally constant integer-valued function on the set A of regular points of A. Since A is globally irreducible, A is connected and n ( x ) globally constant on A. Let n * E N be the value of this function. •
(b) If 1 > 1 , then f == h . . . fz is a minimal defining function for A at x0 . With m := min(k 1 , . . . , kz ) we have h
=
f1k1 . . . Jlkz . q
=
Jm . (},
where (} is a holomorphic function near x0 . Therefore, n ( x0 ) >
m.
We assume that m = k.\. In every small neighborhood of x0 there are regular points x E A.\ that do not belong to A, for p, f- A . Then n ( x ) = n *, and f>.. is a minimal defining function for A at x. Since h = J�>- . q , with a holomorphic function q, and (h )x f qx it follows that n ( x ) = k.\. So m < n (xo ) < n ( x ) = n * = k.\ = m, and therefore n ( xo ) = n *. ,
•
Now we define ordA (h)
:
=
One easily sees that
{ �onstant value of ordA,x (h)
if h i A = 0, otherwise.
1 96
IV.
Complex Manifolds
Mer omorphic Function s. Let X be an n-dimensional complex mani fold. We consider holomorphic functions that are defined outside of an ana lytic hypersurface. In the !-dimensional case these are holomorphic functions with isolated singularities. Definition . Let A c X be an analytic hypersurface. A complex-valued function m on X - A is called a meromo rphic function on X if for any point x E X there are holomorphic functions g, h on an open neighbor hood U = U(x) c X such that N (h) c A n U and m = g j h on U - A. Obviously, m is holomorphic on X - A. In particular, every holomorphic function f on X is also meromorphic on X . Different meromorphic functions may be given outside of different analytic hypersurfaces. If m>-. : X - A>-. --+ C are meromorphic functions on X , then m1 ± m2 and m1 . m2 are meromorphic functions on X , given as holomorphic functions on X - (A1 U Az). m
:
X - A --+ C is a meromorphic function, for p E A we have two possibilities: If
(a) There is a neighborhood U = U(p) c X such that m is bounded on U - A . Then there is a holomorphic function m on u with ml u-A = m l u - A , and p is called a removable singularity for m. (b) For any neighborhood V = V(p) c X and any n E N there is a point x E V - A with lm(x)l > n. If m = g/h near p , then h must vanish at p , because otherwise we would be in situation (a) . Now there are again two possibilities: (i) If g(p) f- 0, then limx -+ p l m(x) l = + oo, and we have a pole at p. (ii) The other possibility is g(p) = 0. This cannot occur in the case n = 1, since it may be assumed that the germs gp and hp are relatively prime, but it is possible for n > 1 . The behavior of m is extremely irregular in that case: We take any c E C. Then gP - c · hp and h, are relatively prime, and therefore there exists a sequence (xv) of points in N(g - ch) - N( h) with limv -+oo Xv = p . This means that m(xv) = c for every v. We call p a point cl indeterminacy in this case. In the case n = 1 a meromorphic function is a function that is holomorphic except for a discrete set of poles. For n > 1 we have the polar set
P( m)
:=
{p E X
: m
is unbounded in any neighborhood of p } .
The polar set consists of poles and points of indeterminacy. We show that P( m) is an analytic hypersurface. Let p E X be an arbitrary point and U = U(p) c X a connected neighbor hood, where m is the quotient of g and h and N (h) c A . We may assume that p E A and gP, h, are relatively prime. Then
4.
m bounded near p
So P(m) n U
=
{x E U
{::=} {::=} {::=} {::=} :
h (x)
Meromorphic Functions and Divisors
3 rp (holomorphic near p ) with g hp l gp hp is a unit h(p ) # 0. =
=
rp
1 97
.h
o}.
If Z c X is an irreducible hypersurface, then we define ordz ( m) as follows: If m g/h near x , then ordz,x ( m) == ordz,x (g) - ordz,x (h). If we choose gx , hx relatively prime, this definition is independent of g and h. Now it follows exactly as above that ordz,x ( m) is constant on Z. =
4.5 Identity theorem for mer om orphic fun ctions. Let X be connected, m : X - A --+ C a meromorphic fun ction, and U C X a nonempty open set such that ml u - A 0. Th en P(m) 0 and m 0. =
=
=
PROOF: The set X - P(m) is connected, m is holomorphic there, and U (AU P(m) ) is a nonempty open subset of X - P(m). By the identity theorem for holomorphic functions it follows that m 0 on X - P(m). But then m is globally bounded and P(m) 0 . • =
=
The set Al(X) of meromorphic functions on X has the structure of a ring with the function m 0 as zero element. We set =
Al(X) - {0}
{ m E At (X) : m vanishes nowhere identically } If m E Al(X ) * has a local representation m gjh , the zero set N(g) is independent of this representation. Therefore, we can define the global zero set N (m ) , which is an analytic hypersurface in X . Outside of P(m) UN(m) , m is holomorphic and without zeros. Therefore, 11m is also holomorphic there and has local representations l jm hjg. So l jm is also meromorphic, and consequently Al(X) is a field. For this it is essential that X is connected! =
=
4 . 6 Levi' s exten sion theorem. Let A C X be an analytic set that has at least codimension 2 , and let m be a meromorphic fun ction on X - A. Th en there exists a meromorphicfun ction m on X with mix-A m. =
PROOF: Since the statement is true for holomorphic functions, we may assume that P( m) # 0 . So it is an analytic hypersurface in X - A. By the theorem of Remmert-Stein, Q : P(m) is an analytic set in X . By Riemann's second extension theorem the holomorphic function m on ( X - Q) - A has a holomorphic extension m to X - Q. =
Let p E A n Q be a point. We have to show that m is locally meromorphic at p. We choose an open neighborhood U U(p) c X and a function g E O (U) such that : =
198
IV.
Complex Manifolds
1 . Q n U c N (g) . 2 . N(g) = N1 U . . . U Nk is a decomposition into irreducible components.
Since dim ( A) < dim ( Ni ) , there are points a , E Ni - A and neighborhoods Vi = Vi ( a i) c U - Asuch that m = P;/qi on v; - Q and N(qi) c Qnv; c N(g); i.e., g\N(qi ) = 0. From the Nullstellensatz it follows that there is a number si E N and a holomorphic function ri such that g8 i = r, . qi . Then m = Pirig - Si near a,. This means that there exists an s E N such that g5 m is holomorphic near a 1 , . . . , ak •
Thus N : = (U - A) n P(g8m) is empty or an analytic hypersurface that is contained in N (g) -A. In the latter case it is a union of irreducible components of N (g) - A, and this is impossible, since every such component contains a point a,. So N must be empty, and g5m is holomorphic in U - A. By Riemann's second extension theorem there is a holomorphic extension h of • g5m on U. Then g - s h is meromorphic on U with (g- sh) \ u-A = m.
Divisors . Let X be a connected complex manifold, m E .4(X) * , and Z c X an irreducible analytic hypersurface. If Z c P( m), then ordz ( m) is a negative integer, and if Z c N (m) , then ordz (m) E N. In all other cases we have ordz (m) = 0. If P(m) = U, EI P, and N (m) = U.\EL N.\ are the decompositions of the polar set and the zero set into irreducible components, then the formal sums (m),
:=
2 )- ordpJm)) . P,
and
LEI
(m ) o
:=
L ordN"' (m)
.\EL
. N:>,
are called respectively the divisor cl poles and the divisor if zeros of m. Finally, div(m ) : = (m ) 0 - ( m ) , is called the divisor of m. From the remarks above it is clear that
div ( m) =
L
z cx
ordz (m)
·
Z,
where the sum is over all irreducible hypersurfaces
Z in X
Definition . Let (Z,), EI be a locally finite system of irreducible ana lytic hypersurfaces Z, E X . If for every L E I a number n, E Z is given, then the formal linear combination
is called a divisor on X . The divisor is called positive or
effective if n,
> 0 for every L E I.
4.
Meromorphic Functions and Divisors
1 99
Divisors can be added or multiplied with integer constants in an obvious way. Therefore, the set 5?(X) of all divisors on X has the structure of an abelian group. As we just have seen, there is a map div : At'(X )* --+ 5?(X). Since we have div(m 1 m2 ) = div (m 1 ) + div(m 2 ) , div is a group homomorphism. Sometimes it is useful to generalize the notion of a divisor a little bit. Let (A) , E r be a locally finite system of (arbitrary) analytic hypersurfaces in X , and ( n J ,EI a system of integers. Then for every L E I we have a decomposition
into irreducible components. The system {At : L E I, -\, E L,} is agam locally finite, and we define 2:,EI n, . A, : = 2: ,EI 2:: .\ , E L, n, . A � , . With this notation it is possible to restrict divisors to open subsets: If U c X is open and D = 2:,EJ n, . Z, a divisor on X , then
D u := 4 . 7 Pr oposition .
If A c X is an analytic hypersurface and f a m inimal definingfunctionfor A in an open set U with U n A f- 0, then
The proof is more or less straightforward. Now let an arbitrary divisor D on X be given. Then for any point p E X there is an open neighborhood U = U (p) c X , a finite system { Zi : i = 1 .. . , N } of irreducible hypersurfaces Zi c U, and a system of numbers ni E Z such that ,
N
D iu = I >i Zi . ·
i=l In addition, there is a neighborhood V = V(p) c U such that there exist minimal defining holomorphic functions fi for Zi in V. Then
di v
N
N
=l
i =l
(iIJ it'') = L ni . zi n
v
= Dlv
In this way every divisor is locally the divisor of a meromorphic function.
200
IV.
Complex Manifolds
Associated Line B undle s . Let X be a connected n-dimensional com plex manifold. If Z c X is an analytic hypersurface, then there is an open covering %' = { U, : L E I } of X with the following property: If U, n Z -=/= 0 , then there is a minimal defining function f, E O(U,) for Z. Setting f, == 1 if U, n Z = 0 , in Uu, we get the two relations
with suitable holomorphic functions
gu,
and g,"'" Then
Since f, does not vanish identically, it follows that g,"g'" 1 on U," . This shows that gu, E 0* (Uu-,) and g"' = g,"1 . Furthermore, on U,">-. we have the compatibility condition =
The system of the nowhere vanishing functions g," = f,j f" defines a holo morphic line bundle on X, which we denote by [Z}.It is easy to show that this definition does not depend on the covering and the functions k 4.8
Proposition.
1. Th ere is a section sz E f(X, [ Z] }vith Z = {x 2. [Z]is trivial over X Z .
EX
: sz (x) = 0} .
-
PROOF: The system of holomorphic functions f, defines a global section sz with {x E U, : sz (x) = 0) = {x E U, : f, (x) = 0) = U, n z . Then it is clear • that [ 41 - z is trivial.
We can generalize the concept of associated line bundles to the case of divi sors. If D is a divisor on X , then there is an open covering %' = { U, : L E I } of X , and meromorphic functions m, on U, with Di u, = div ( m, ) . It follows that the functions are nowhere vanishing holomorphic functions on U," . They define a line bun dle, which we denote by [D]Jf D = k . Z, then [D) [Z] k . If D = D1 + D2, then [ D ]= [D1] Q9 [D2] . Thu s the map 6 : 51( X) --+ Pic( X ) , is a homomorphism of groups. 4.9
Theorem.
Th e sequence v¢!(X ) *
o(D) d
==
isomorphy class of [D] ,
group homomorphisms
� 51(X) � Pic(X)
4.
Meromorphic Functions and Divisors
201
is exact. PROOF: ( l ) Let m f- 0 be a meromorphic function. Then [div(m)] is given by only one transition function mjm = 1 . Therefore, o o div(m) = 1 .
( 2 ) Let D be a divisor on X with J(D) = 1 . We assume that D i u, = div(m,) and [D] is represented by g,"' = m,jm"' . Since [ D ]ls trivial, there are nowhere vanishing holomorphic functions h, with h, . g,"' = 1 . h, on U,"' . Then h, . m, = h,
·
m,
Therefore, a meromorphic function m on X can be defined by ml u, : = h, . m, . Obviously, div(m) l u, = div(m,) = D i u, , and therefore div (m) = D . •
Meromorphic Sections . Let X be a connected n-dimensional complex manifold. Any analytic hypersurface Z c X leads to a line bundle [ Z jq, gether with a global holomorphic section sz that vanishes exactly on Z. The construction of sz fails in the case of an arbitrary divisor D and its associ ated line bundle [ D] Therefore, we introduce the notion of a meromorphic section. Definition. Let 1r : L --+ X be an analytic line bundle and A c X an analytic hypersurface. A holomorphic section s E f(X A L ) is called a meromorphic section over X in L if for every point x E X there is an open neighborhood U = U(x) c X , a function h E O(U), and a holomorphic section t E f(U, L ) such that: 1. h · s = t over U - A . 2. N(h) c A n U. -
,
If we have a system of trivializations rp, : 1r - 1 (U,) --+ U, x C and transition functions g,"' , then we have the following description of s .
For x E U, - A we define s,(x) by rp, o s(x) = (x, s, (x) ). Then s, = g,"' · s"' on U,"' - A. If we choose the U, small enough, there are holomorphic functions h, t, on U, such that h, . s, = t, over U, - A and N ( h, ) c A n U, . This means that s, is a meromorphic function on U, . We could have as well said that a meromorphic section is a system (s,) of meromorphic functions with s, = g"" . s"' . If Z is an irreducible hypersurface, then ordz(s,) = ordz (s"') , and we denote this number by ordz(s). The sum
div(s)
:= ZC X irreducible hypersurface
ordz (s) . Z
is called the divisor of the meromorphic section s.
202
IV. Complex Manifolds
Now we have the solution for our problem: Let D be the divisor on X given by D \ u div(m, ) . Then [D] is described by the transition functions 9u< = m,m"' 1 , and the system of the m, defines a global meromorphic section sv of [D]with div(sv) = D. ,
=
So far our definitions seem to be purely tautological. But for example, they allow us to determine the space of holomorphic sections of [D ]in terms of meromorphic functions on X . For that we need the following notation: If D�, D2 are two divisors on X , then D 1 > Dz if and only if D1 - Dz is a positive divisor. 4 . 1 0 Theo r em. Let D = 2: z n z . Z be a divisor on X . Th en there is a natural is omorphism {m E v¢!(X ) * : div(m) > - D) ----+ r(X, [D]) . ""
PROOF: Let sv be the global meromorphic section of [D]with div( sv) D . Then for any meromorphic function m on X also t : = m · s v is a meromorphic section of [ D}. =
If
m
is a meromorphic function with div(m) > - D, then div(t) = div(m) + div(sv) = div(m) + D > -D + D = 0.
This means that t is a holomorphic section. The map m H m · s v is obviously injective. Let t E r(X, [D] ) be given. If D \ u
,
=
div(mJ , then
Hence t,m-; 1 = t "' m"' 1 on U,"' ' and there exists a meromorphic function m on X with m\u, = t,m-; 1 . Therefore, div(m) \ u, = div(tJ - div(m,) > - D \ u, • So the map is an isomorphism. ·
Example
Let X
=
C be the Riemannian sphere and D
r(x, [D] )
=
=
n · 00 · Then
{ m E v¢!(X)* : ord00 (m) > - n and ordp(m) > 0 otherwise } .
The holomorphic sections in [D] are just the meromorphic functions on X that have a pole of order at most n at oo . Exer ci ses
1 . Let x be a point in a complex manifold X and let fx , 9x be nonunits in Ox . Prove that fx and 9x are relatively prime if and only if dim(N (f, g)) < n -
2.
5.
Quotients and Submanifolds
203
2 . Let G c en be a domain, A c G an analytic hypersurface, U c G an open subset with U n A -=!= 0 , and f : U --+ C a minimal defining function for A. Prove that Sing(A) n U = {z E U : f(z) = 0 and \7 f(z) = 0}. 3. Consider the meromorphic function m(z 1 , z2) := 2 212 1 on C2 . Show that the closure X of the graph { (z1 , zz , w) : z1 -=/= 0 and w = m(z 1 , z2 ) } in C 2 x C is an analytic hypersurface. Determine a minimal defining function for X at (0, 1 , oo ) . 4. Classify the singularities of m(z1 , zz) :=sin(z1 )1 sin(z1 z2 ) . 5. Let L --+ X be a holomorphic line bundle. Prove that L = [D ]for some divisor D on M if and only if L has a global meromorphic section s -=!= 0. 6. Let X be a compact Riemann surface. Show that for every nonconstant meromorphic function f on X the numbers of zeros and poles are equal (counted with multiplicity) .
5.
Qu otien t s a n d Sub manifold s
T op ological Qu otients. Let X be an n-dimensional complex manifold and '"" an equivalence relation on X . If x,y E X are equivalent, we write x '"" y or R( x, y ) . For x E X let X (x) := { y E X : y rv x} = { y E X : R(y , x ) }
be the equivalence class of x in X . These classes give a decomposition of X into pairwise disjoint sets. The set X1R of all equivalence classes is called the topological quotient of X modulo R. Let 1r : X --+ XI R be the canonical projection given by 1r : x H X (x) . Then XI R will be endowed with the finest topology such that 1r is continuous. This means that U c XI R is open if and only if 1r� 1 (U) c X is open. We call this topology the quotient topology.
A set A c X is called saturated with respect to the relation R if
5 . 1 P r op o si tio n . 1.
2. 3.
A saturated -{==} A = UxEA X ( x ) . If U C XIR is open, the-n <+= l (fff') is open and saturated. If W C X is open and saturated, then 1r(W) C XIR is open.
The proof is trivial.
204
IV. Complex Manifolds
5 .2 Pr oposition. Let Z be an arbitrary topological space. A map f X/ R --+ Z is continuous if and only if f o 1r : X --+ Z is contin uous. This statement is also trivial, since ( f o 1r) - 1 ( U ) = Jr - 1 ( f - 1 ( U )) .
Analytic Decomposition s. If X is an n-dimensional complex mani fold and R an equivalence relation on X , one can ask whether Xj R carries the structure of a complex manifold such that 1r is a holomorphic map. As sume that such a structure exists. Then Xj R must be a Hausdorff space. If rp : U --+ e k is a complex coordinate system for X/ R, then U = = 1r- 1 (U) is an open saturated set in X , and f = = rp o 1r : U --+ ek a holomorphic map with r- 1 (f(x)) = 7r - 1 (1r(x)) = X(x). So the fibers o ff are equivalence classes, and the equivalence classes must be analytic sets. If additionally 1r is a submersion, then rkx (f) = k for every x E U, and the fibers are ( n - k) dimensional manifolds. We now show that these conditions are also sufficient for the existence of a complex structure. �
Let X be an n-dimensional complex manifold and 2 = {Z, : L E I } a decom position of X into d-dimensional analytic sets. For x E X let t(x) E I be the uniquely determined index with x E Z,(x) . Then there is an equivalence rela tion R on X such that the equivalence class X (x) is exactly the analytic set Zt(x) . We consider the topological quotient X I R and the canonical projection 1r : X --+ XjR and assume that the following conditions are fulfilled: 1 . Xj R is a Hausdorff space. 2. For any x0 E X there exists a saturated open neighborhood U of X(xo) in X and a holomorphic map f : Q --+ en- d such that (a) r-1 (f(x)) = X(x) for all X E U. (b) rkx (f ) = n - d for x E U. �
�
5 . 3 Theorem. Under the conditions above, Xj R carries a unique structure of an (n - d)-dimensional complex manifold such that 7f : X --+ XjR is a
holomorphic submersion.
�
PROOF: Let x0 E X be �given. Jhen there is an open n�ghborhood U of X(x0) in X with Jr-1 (1r (U) ) = U, and a submersion f : U --+ en - d whose fibers are equivalence classes X(x). If z0 : = f(x0 ), then there is an open � n neighborhood W = W(z0) c e - d and a holomorphic section s : W --+ U (with s(zo) = xo and f o s = idw ) . For z E W we have f- 1 (z) = X (s (z)), and therefore 7r - 1 (1r(s (W)))
=
U
zEW
X(s (z))
=
U
ZEW
r - 1 (z)
=
f-1 (W)
This is an open set, so 1r(s(W)) C XjR is open as well. We define a complex coordinate system rp : 1r( s(W)) --+ en - d by
5.
cp(7r(s(z)) )
Quotients and Submanifolds
205
:=z.
Then cp (7r (x) ) = f(x). This shows that if is well defined and continuous. It is also bijective, with cp - 1 (z) = 7r(s(z) ) , and therefore a homeomorphism. Now let '1/J be another coordinate system given by '1/J(K(t(z) )) a local section for some suitable submersion g . Then
:=z,
where t is
The coordinate transformations are holomorphic.
•
Pr operly Discontinu ou sly Acting Gr oup s. Let G be a complex Lie group acting analytically on an n-dimensional complex manifold X . Then R (x, y )
: -{=}
:l g E G with y = gx
defines an equivalence relation on X . The equivalence class X ( x) = { y E X :J g E G with y = g x} is called the orbit of x under the group action and is also denoted by Gx. The topological quotient Xj R is called the orbit space and is also denoted by XjG. We consider a very special case. Definition. The group G acts properly discontinuously if for all x , y E X there are open neighborhoods U = U(x) and V = V (y) such that
{g E G : gU n V # 0} is empty or a finite set. Here the orbits Gx are discrete subsets of X and are therefore 0-dimensional analytic subsets. If the action is free, we want to show that all conditions are fulfilled for X I G to be a complex manifold and 1l" : X --+ XjG a holomorphic submersion (which means in this case that 1l" is an unbranched covering). 5 . 4 L emma . Let G act freely and properly discontinuously o n X and let xo , y0 E X be given.
1 . If there is a go E G with Yo = gox0 , then there are neighborhoods U = U(xo) and V = V(yo ) such that gU n V = 0 fo r g # g0 . In the case Yo xo and go e one can choose V U. 2. If gxo # Yo fo r every g E G, then there are neighborhoods U = U(x0 ) and V = V(yo ) such that gU n V = 0 fo r every g E G. =
PROOF:
=
=
At first we cho ose neighborhoods Uo (x0 ) and V0 (y0) such that M
:=
{g E G : gUo n Vo # 0}
206
IV. Complex Manifolds
is finite or empty. There is nothing to prove if M = {g0 } in the first case or M = 0 in the second case. Therefore we assume that there are elements gi , · . . , gN , N > 1 , with M = {go , gi , . . . , gN } in the first case and M = {g i , . . . , gN } in the second case. Then we define Y>.. := g>.. xo , for A = 1 , .. . , N . Since G acts freely, Y>.. -=/= yo for A = 1 , , N . _
.
We choose neighborhoods W>.. = W>.. (Y>.. ) and V = V(yo) c Vo such that W>.. n V = 0, and we choose a neighborhood U = U(xo) c U0 such that g>.. U c W>.. , for A = 1 , .. . ,N. Then gU n V = 0 for g -=/= go in the first case, H and g E G in the second case. 5.5
Let G act freely and properly dicontinuously on X . Th en X/G has the unique structure if an n-dimensional complex manifold, so that 1r : X --+ X/G is an unbranched holomorphic covering. PROOF: Let U C X be an open set. Then 7f - I (1r(U) ) = U E gU is an Theorem.
g G
open set, and therefore 1r(U) is also open. For xo E X we can choose an open neigborhood U = U(x0) such that gU n U = 0 for g -=/= e. Then 1r : U --+ 1r(U) is bijective.
( 1 ) We have to show that X/G is a Hausdorff space. Let X I , x2 E X be given, with 1r(x l ) -=!= 1r(x2 )- Then gxi -=/= x2 for every g E G. There are open neighborhoods U = U(x l ) and V = V(x2 ) with gU n V -=/= 0 for every g E G. Then 1r(U) and 1r(V) are disjoint open neighborhoods of 1r(x l ) and 1r (x2 ) . (2) We verify the other conditions. Let x0 E X be given and choose a small open neigborhood U = U(x0 ) c X such that 1r : U --+ 1r(U) is a homeomor phism and such that there exists a complex coordinate system rp : u --+ en . Then f : fJ :=7r - I (7r(U)) --+ e n can be defined by f(gx) : = rp(x) , for a: E U and g E G. It is clear from above that f is well defined. The fibers of f are the G-orbits, and on gU we have f(y) = .rp(g - I (y) ) . This shows that f is holomorphic, and rky (f) = n for every y E U.
1f U is small enough, then 1r - I (1r(U)) = U gEG gU, with pairwise disjoint sets gU that are topologically equivalent to 1r (U) . So 1r is an unbranched covering. •
c omplex Tori. Let {WI , . . . ,W2n } be a real basis of e n . Then the discrete group r := Zwi + . . + Zw2n acts freely on e n by translation. The set .
Aw is the orbit of w .
:=
f + w = { w + w : w E r}
The group r acts properly discontinuously On e n : Let zo , Wo E e n be given. If w0 w0 + z0 for some wo E r , choose =
5. E
<
-::
Quotients and Submanifolds
207
. inf { II w II : w E r - J 01 1.
Then (w+ Bc (zo)) n Bc (wo) = 0, unless w = Wo. If wo - Zo tf. r and
E
<
-l
. dist(wo , f t Zo) ,
2
then (w+ B6 (zo)) n B6 (w0) = 0 for every w . The n-dimensional complex manifold rn = TF == e n ;r is called a complex torus, and r is called the lattice of the torus. The set P == { z = t i WI + · · · + t2 n w2n : 0 < ti < 1) contains a complete system of representatives for the equivalence classes. Therefore, Tn = 1r( P) is a compact space. The map
h WI + . . . + t2 n W2 n rl ( e 27r it 1 , induces a homeomorphism T" --+ S I X . . . X s i .
•
5. 6
Pr op osition .
H I (S I , Z) = Z.
•
•
27r it2n ) e ,
2 n times
27r it : PR OO F: Let U. I ·. = { e2 1r it . _ l4 < t < 4l} U2 .· 3n e 0 < t < 2 r t and U3 := { e 1 ' : � < t < 1 ) . Then %' == { UI , U2 , 3} is an acyclic o)Je'rl covering for S I with UI 23 = 0. Therefore, every triple � = ( a,b, c ) E Z3 is a cocycle in z i ( %' , Z) . It is a coboundary if and only if there is a triple (u , v, w) E Z3 with ·
a = v - u'
b = w - u'
J
'
and
c=
w-v
This is the case if and only if a + c = b. Since every cocycle has the form (a , b, c) = (0, b - a - c, 0 ) + (a, a + c, c) = ( b - a c ! . (0, 1, 0) + 8 (0, a , a + c ) , it follows that H I (%', Z) is generated by the class of (0, 1, 0). • _
From the Kunneth formula it follows that HI (Tn , .Z) the first Betti number of T" is equal to 2n.
'""
.z2n and therefore '
H opf Manifold s. Let (} > 1 be a fixed real number and n > 1. Then the (multiplicative) group r := {Qk : k E Z} acts freely on e n - { 0 } by z rl (}k . z. The action is properly discontinuous. To see this, we define the sets Ur := {z E e n : r < llzii < Qr}, for r > O.
Then the sets (}k Ur are pairwise disjoint. If two points are given, one can find an r > 0 and a k E Z such that z2 E V : = QkUr . The case k = 0 is allowed. Now (}8 U n V
Z I , Z2 E en - { 0 } z i E U = = Ur and = 0 , unless s = k.
208
IV. Complex Manifolds
So H = Hr : = (e n - {0})/f is an n-dimensional complex manifold, and the canonical projection 1r : en - {0} --+ H is an unbranched holomorphic covering. H is called a Hopf manifold. The map
zH
exp
( (
)
In z II z 2n .l II , In (! W
)
induces a diffeomorphism H --+ S 1 X S2n - l . Here S2n - l is the (2n - 1 ) dimensional sphere in JR2 n = e n with 2n - 1 > 3.
k 5 . 7 Proposition . For k > 2, H1 ( S , Z) PROOF:
{U1 , U2 }
We have Sk with
=
{x
= 0.
= (xi , . . . , x k+l )
{ XE
Sk k { XE S
: :
llxll - 1 ) . Then 'P£ -
- & < X k+ l <1!11 Jl -1 < X k+l < -t
is an open covering of Sk with contractible sets, and ll'ii
=
{x
E sk
:
- &
< Xk +l < -t
= Z 1 ( 'P£ , Z) = Z and C0 ( 'P£ , Z) = Z2 . The !!!!! iii Z) is given by J(a , b) :=b - a. Then • = 0. It follows that H 1 ( H, Z ) = -- x S2 k - l , Z) = Z, and br ( H) = 1 .
is connected. Therefore, C1 ( 'P£ , Z) coboundary map 5 : C0 (f£ , 3111 --+ obviously, B 1 (f£ , Z) = Z and 1111
•••
The C omplex Pr oj ective Sp ace. In X : = e n +l the equivalence relation
R(z , w )
: -{=}
3
AE
C*
with w
- {0} we consider
= AZ.
The equivalence class L, of z is the set L, = ez - { 0 } , the complex line through z and 0 without the origin. So we have a decomposition of X into ! -dimensional analytic sets. We can also look at these sets as the orbits of the canonical action of e• on X by scalar multiplication. Definition . The topological quotient ]pm : = X/ R = is called the n-dimensional complex projective space.
(e n+l - {0}) /e*
Let n : X = e n +l - {0} --+ Jpm be the canonical projection, with n(z) : = L, and let two points z = (zo , . . . , zn ) , w = (wo, . . . , wn ) be given. We have
n(z) = n(w)
3 A E e• with wi = AZi for i = 0 , . . . , n zi wi = for all i , j where the fractions are defined.
wJ·
zJ·
5.
Quotients and Submanifolds
209
So 1r (z) does not determine the entries zj , but the ratios zi : Zj · Therefore, we denote the point x = 1r(z0 , . . . , zn ) by (zo : . . . : zn ) and call zo , . . . , Zn the homog eneous coordinates of x. If z0 , . . . , zn are homogeneous coordinates of x , then so are Azo , . . . , A Z n for every A E @*.
If W c X is an open set, then 1r-1 (1r(W)) = U .\ E IC * A · W is a saturated open set in X , and therefore 1r(W) is open in P". For example, this is true for
fji := {z = (zo , . . . , zn) R C'..,.. The sets Ui
:=
-
{ 0 � : zi f. O )
e X,
i = O, . . . , n .
1r( Ui) form an open covering of Jpm
We show that ]pm is a Hausdorff space: Let z , w E X be given, with L, f. 4 Then w z z * ·. = and w := II z II = 1 }. Therefore, are distinct points of S2n+l = {X E lR2n +2 = e n + 1 : we can find an E > 0 such that B6 (z * ) n B6 (w* ) = 0. Then U == 1r ( B6 (z * )) and V := 1r ( B6 (w* )) are disjoint open neighborhoods of 1r(z) , respectively 7r(w). *
l j;lj
l l xl
Now let a point z0 = ( z6o) , . . . , z�o) ) E X be given. Then there exists an index i with zt) f. 0, and Zo lies in Ui . We define fi : ui --+ e n by �
fi ( zo , · · · , Zn )
: ==
�
Zo Zi- 1 Zi+ 1 (,··., ,
Zn \
,... , - I.
Then
If a point u = ( u0 , . . . , u, ) E U; is given, we define a holomorphic section s : e n --+ � by s(z1, . . . , zn) := ( Ui Zl , . . . , uizi, ui , Ui Zi + 1 , . . . , uizn ) · Then
(
)
o Un s U ' . . . ' Ui - 1 ' Ui + 1 ' . . . ' . = ( Uo ' . . . ' Un ) ' "u,' Ut Ui Ui
and
fi s(z� , . . . , Zn ) = (z1 , . . . , zn ) · Therefore, fi is a submersion, and rkz(fi) = n for every z. o
Altogether this shows that Jpm is an n-dimensional complex manifold and 1r : e n +I - { 0} --+ Jpm a holomorphic submersion. Since every equivalence class Lz has a representative in the sphere s2 n+l , Jpm = 1r( S2 n +l ) is compact.
210
IV.
Complex Manifolds
Local coordinates are given by the maps lf!i : Ui
means that
'Pi ( zo : . . . : Zn ) =
---+
e n with 'Pi o 1r = fi ·
This
\ -v ' . . . ' o ' . . . .. ) . z, z,., -
.
7.r:
' .
The set is biholomorphically equivalent to en . We call it an affine part of Jpm. If we remove Uo from lPm , we get the so-called (projective) hyperplane at infinity
H0
{ ( Zo : · · · : Zn ) E lP'n : zo - {(0 : t 1 : . . . : tn) : (h , .
= .
0}
. , tn ) E e n -
(0) ) .
It has the structure of an (n - I )-dimensional complex projective space. If we continue this process we get
lP'n lP'n - 1
en en -1
u
U
lP'n - 21 ' lP'n - '
It remains to study lP' 1 = {(zo : zi ) : (zo , zi ) E e 2 - ( 0 ) ) . But this is the union of e = (( 1 : t ) : t E e} and oo : = (0 : 1 ) , with t = zi/z0 . In a neighborhood of oo we have the complex coordinate zo / z1 = 1 jt. So we see that lP' 1 = e = e U { oo} is the well-known Riemann sphere. The hyperplane H0 is a regular analytic hypersurface, given by
H0 n ui
=
� ( zo : . . . : Zn ) E ui : �0
=
0 �.
Therefore, U0 is dense in P". It should be remarked that there is no reason to distinguish between Uo and the other sets Ui . Everything above could have been done as well with the affine p art ui and the hyperplane Hi := {7r ( z) E lP'n : Zi = 0 } .
Meromorphic Functions. On a compact complex manifold every global holomorphic function is constant. But we know already from the ex ample of the Riemann sphere that there may exist nonconstant meromorphic functions. In this regard we consider the compact manifolds defined above, beginning with the complex projective space lP'n .
A nonconstant polynomial p(t) = l::�v i =O avt v is a holomorphic function on the set Uo = {(1 : t 1 : . . . : tn ) I t = (t 1 , . . . , tn ) E e n }. In fact, it defines a meromorphic function on lP'n with polar set H0. We see this as follows: 3
The hat signalizes that the i th term is to be left out.
5.
Quotients and Submanifo1ds
211
The functions tM = zM /zo, p, > 1 , are holomorphic coordinates on U0 , and w.\ := z.\/ Zi , A f. i likewise on Ui . Therefore, on ui - Ho = ui n Uo we have
( :o ) .. .
I.e. , p = gj h on ui - Ho , where g(w)
h(w) := w�
L av ( :� )
:=
"
'
...
( :n
v 2:: �vi=O av w� -l l w�' . . . w�n an d
are holomorphic functions on ui with
N(h) = (W E Ui : Wo = 0} = Ui D Ho . So there are numerous global meromorphic functions on projective space. Now let T = e n /f be an n-dimensional complex torus, and rr : e n ---+ T the canonical covering. If m is a meromorphic function on T, then m o rr is a meromorphic function on en ' which is periodic with respect to the generators w 1 , . . . ,w2 n of the lattice I?. In the case n = 1 such meromorphic functions always exist; they are the !?-elliptic functions. We shall later see that for n > 2 the existence of !?-periodic functions depends on the lattice I?. In fact there are complex tori with no nonconstant meromorphic functions. Finally, consider the Hopf manifold H = ( e n - { 0 } ) /f with r = {Qk : k E Z} and n > 1. Let m be a meromorphic function on H. Since the canonical projection rr : e n - ( 0 ) ---+ H is a covering, m := m o rr is meromorphic on en - ( 0 ) . Since n > 1 , it follows from Levi ' s extension theorem that m can be extended to a meromorphic function on e n . On any line L through the origin in en , m must have isolated poles or be identically oo . But since m comes from H, poles on L must have a cluster point at the origin, which is impossible unless m is constant on L. The same argument works for any other value of 6 . A meromorphic function on e n - (0) that is constant on every line through the origin comes from a meromorphic function on the projective space Jpm - I _ This means that if h : H ---+ ]pm- l is the canonical map, then a bijection J!t (lP'n- l ) ---+ J!t (H) is defined by m H m o h . On the n-dimensional Hopf manifold there are not "more" meromorphic functions than on ( n - I)-dimensional projective space.
Gras smannian M anifolds . The set of 1-dimensional complex subvec tor spaces of en +l can be identified with the n-dimensional projective space, and we have given it a complex structure. Now we do the same for the set Gk n of k-dimensional sub spaces of en . The idea is the following: If V0 c e n is a fixed element of Gk ,n, then we choose an ( n - k)-dimensional subspace ,
212
IV. Complex Manifolds
Wo c en such that Vo EB Wo = e n . We are looking for a topology on Gk , n such that the set of all k-dimensional subspaces V with V EB Wo = e n is a neighborhood of Vo in Gk , n·
But how to get complex coordinates? In the case G 1 , n + 1 = ]pm we consider, for example, V0 = Ceo with e0 = ( 1 , 0, . . . , 0 ) and Wo = { ( zo , . . . , zn ) : zo = 0). Then Vo EB Wo = c n + I ' and a vector z = ( zo , . . . ' Zn ) cj 0 generates a 1dimensional space v with v EB Wo = en + I if and only if Zo cj 0. Multiplication by a nonzero complex scalar does not change the space V . Therefore, V is uniquely determined by
) z0- 1 . z = z0- 1 . ( z0, z �
The map
f
:V
c-t
=
1
( ,.z0- 1 . z ) �
With Z = ( Z 1 , · . . , Zn ) .
z0 1 . z E C" gives the familiar local coordinates.
When we try to transfer this procedure to higher k, we use another viewpoint. Every V with V EB Wo = C n +l has the form Graph( tp v ) of a linear map 'PV : c --+ e n given by f ( V) = 'PV ( l ) .lf Vo c e n is a k-dimensional subspace and V0 EB W0 = en , then every other k-dimensional subspace V c e n with V EB Wo = en has the form Graph(cpv) for 'PV E Homc ( Vo , Wo ) · Fixing bases of Vo and Wo , the matrix of 'PV with respect to these bases gives local coordinates in Mk, n - k (C) '"" C k (n - k) . Now we will do this job in detail. An ordered k-tuple of linearly independent vectors a1 , . . . , ak E e n can be combined in a matrix
with rk(A) = k. The set
St (k, n) := {A E Mk , n (C)
:
rk(A) = k}
is called the complex Stiefel manifold of type (k, n ) . Since its complement in Mk, n (C) '"" C k n is an analytic set given by the vanishing of all (k x k) minors of A, St (k, n ) is an open set in Mk , n (C) and therefore a complex manifold. The group GL k (C) acts on St(k, n ) by multiplication from the left, and every orbit of this group action represents exactly one k-dimensional subspace of e n . The topological quotient
Gk , n = St(k, n ) / GL k (C) is called the c omplex Grassmannzan of type ( k,n ) . If, for example, Wo = rw = ( w l , · . . , w n ) : w 1 = · · · = w k = o } , then a matrix A E St ( k, n ) represents a basis of a k-dimension al space V with V EB Wo = e n if and only if A = (Ao [A) with Ao E GLk (C) and A E Mk , n - k (C) .
5.
Quotients and Submanifolds
213
In this case v has the form Grap h ( lf!V) for a linear map lpv : e k -t e n - k . Of course, V is uniquely represented by the matrix A0 1 . A = ( E k IA0 1 . A) , and A0 1 · A is the matrix of rpv with respect to the standard bases. Now we consider the set of multi-indices
yrk , n := { I = (i l , · · · , ik ) E Nk : l < i 1 < . . . < i k < n } . For any A E St(k, n ) there exists an I = (i 1 , . . . , i k ) E yrk , n such that
Ar Then there
IS
(Ar iAr) .
:=
a permutation matrix P r E GL n (C) such that A . P r -
For fixed I we define
Vr := {A E St(k, n ) : det Ar f. 0} . .---....._..-
�
We remark that ( G . A) r = G . Ar and ( G . A) , = G . Ar for G E GL k (q . Therefore, Vr is invariant under the action of GL k (q . 5. 8 Lemma.
Let 7rk , n : St( k , n )
-t
Gk , n be the canonical projection. Th en
Let A E 7r;; � (7rk, n (Vr)) be given. Then there is an A* E Vr with 7rk, n (A) = 7rk , n (A* ) . This means that there is a matrix G E GL k (q with A = G . A * . Since Vr is invariant under the action of GL k (C) , A lies in Vr . The converse inclusion is trivial. • PROOF:
So Vr is a saturated open subset of St( k, n ) , and Ur := Kk , n (Vr) is open in Gk, n · We leave it to the reader to show that Gk, n is a Hausdorff space.
If Er C e n is generated by ei, , . . . , eik , and Fr C en by the remaining ej , then Er EB Fr = e n , and every k-dimensional subspace V c e n with V EB Fr = e n is represented by a matrix A E Vr. The uniquely determined matrix A[ 1 . Ar E Mk, n - k (C) describes the linear map rpv : Er -t Fr . Therefore, we define the holomorphic map fr : Vr -t Jl.fk, n - k (C) � e k( n - k )
by
fr (A) := A r 1 . Ar.
It is clear that fr is holomorphic, and since
fr (G · A) = (G . Ar) - 1 . (G . Ar)
=
fr (A) ,
2 14
IV.
Complex Manifolds
f1 respects the fibers of 1fk,n · It remains to show that fr has maximal rank. For that we construct sections:
Let A E Vr be given. Then we define s : Mk,n - k (q --+ Vr by s (B)
:=
Ar . (Ek !B) · P / .
This is a holomorphic map with s(fr (A) ) = (Ar!Ar) . P/ 1 = A and fr o s (B)
=
f1 (A1 . (Ek ! B) . P / )
=
Ar 1 . (Ar . B ) = B .
So fr is a submersion and Gk,n a complex manifold of dimension k ( n - k). Comp lex coordinates are given by rpr : Ur --+ Mk,n - k (q with rpr (1fk,n(A))
:=
A[ 1 . Ar.
-t
Let Sk,n = = {A E St (k, n ) : A · A = Ek } be the set of orthonormal systems of k vectors in e n . Then Sk ,n is a compact set, and 1fk,n : Sk , n --+ Gk,n is surjective. So Gk , n is compact.
Submanifolds and Nor m al B u n dles. Let X be an n-dimensional complex manifold. A holomorphic map f : Y --+ X is called an embedding if there is a submanifold Z c X such that f induces a biholomorphic map from Y onto Z. Every embedding is an immersion, but in general not every (injective) immersion is an embedding. The following proposition has been already mentioned in Section 1 . 5.9
Pr op osit ion . Let f : Y --+ X be a holomo rphic map between com plex manifolds. If Y is compact and f an injective immers ion, then f is a n embedding. PROOF: Every immersion defines a local embedding. To see this, we may consider a holomorphic map f from a neighborhood V = V (O) c em into a neighborhood u = U(O) c em X e n - m with f(O) = (0, 0) and rk Jr (O) = m. We write f = (f1 , f2 ) and assume that already rk Jr1 (0) = m. Then for the map F : v X e n -m --+ en with
we have det JF (O, 0) = det Jr1 (0) f. 0. So there exist neighborhoods V* V* (O) c V and W = W(O, 0) c U such that F : V* x V* --+ W is biholomor phic. Since F(z, 0) = f(z), the image f(V*) = F(V * x { 0 } ) is a submanifold of W . We have proved that for every point xo E Y there are neighborhoods V = V(xo) c Y and W = W(f( xo)) C X such that J (V) is a closed submanifold of W. We have to show that there is a small open neighborhood U = U(f(x0)) c W such that f(Y) n U = f(V) n U. Suppose that there is
5. Quotients and Submanifolds
215
a sequence Xn E Y - V with f(x n ) --+ f (x0 ) . Since Y is compact, we can assume that (xn ) converges to an element x* E Y - V .Since f is injective, • f(x* ) f. f(x0 ) . But f (x n ) must converge to f(x* ) . This is impossible. If Y is a regular hypersurface in a complex manifold X , then there are two line bundles associated to Y ,namely the normal bundle Nx (Y) , which is defined on Y, and the bundle [ Y]on X . We will show that these bundles coincide on Y.
Choose an open covering %' = (Uc) cEI of Y in X such that there are complex coordinates z� , . . . , z� for X in U" with the following properties: 1 . Y n Uc = { X E U, : Z� ( X) = 0}. 2 . z L . . . , z �, _ 1 are complex coordinates for Y.
We have already seen in Section 2 that the normal bundle Nx (Y) is given with respect to u" n y by the transition functions
On the other hand, the line bundle [ Y)s defined by the transition functions f""' : = z� / z�. But z� ( z� , . . . , z� ) = f""' ( z� , . . . , z� ) · z� implies
az " (z�, . . . ' z� - 1 , 0) = !cl< (z�, . . . ' z� 1 , 0) . a"'�� n This yields the so-called first adjunction formula. 5 . 10
Fir st adj un ction formula. [Y] h,· = Nx (Y) fo r every regular hypersurface Y C X.
The normal bundle Nx (Y) is naturally related to the canonical bundles Kx and Ky . If g""' ' respectively G, are the transition functions for T(Y) , re spectively T(X) , then
Therefore, det G, o j = det g""' . h, and det g" "'1 = (det G" "'1 o j ) is equivalent to the second adjunction formula: 1
5 . 11
. h, 1
which
S econd adjun ction formula. Ky = j* Kx ® Nx (Y) fa r every regular hypersurface Y c X .
This formula can be generalized to higher codimensions; see, for example,
[GriHa78] .
216
IV. Complex Manifolds
Pr oj ective Algebr aic M anifolds. canonical projection and x E Jpm , we define
If
- {0}
1r : c n + l
---+
Jpm is the
This is a complex line through the origin in cn +l , and we have C(1r(z)) = Cz for z E cn +l {0}.
-
A set X c c n + l is called a conical set or a cone if it is the union of a family of complex lines through the origin. This means that �
zEX
�
AZ E X for A E C.
===?
If X is an arbitrary subset of ]pm , then
X :=
U
xEX
U
f(x) = 7f - 1 (X)
(0)
is a conical set.
Let X c c n + l be a conical set, f a holomorphic function near the origin, and f = 2:� 0 Pv its expansion into homogeneous polynomi als. If there is an E > 0 such that f i Bc ( D) nX - 0, then Pv l x = 0 fa r ev ery
5. 1 2 Lemma.
l/ .
PROOF:
Let
z
f. 0 be an arbitrary point of B6 (0) n X. Then 00
u=o
vanishes identically for I A I < I . So Pv ( z) = 0 for every v , and since X 1s • conical, P v I x 0 for every v . Now let F'r , . . . ,Fk be homogeneous polynomials in the variables zo , Then the analytic set �
X
:=
{z
�
E
en +
is a cone. If we set X ' := X set
X In U;
by
1
:
F1 (z)
= . . . = Fk (z) = 0 }
- {0 ) , then the image X
:= 1r (X ' ) c ]pm is the
= { (ZQ : . . . : Zn ) : F ( zo , . . . , Zn ) = = Fk (ZQ ,
= {(zo
1
.. . . . : z,)
=
. . . , z,.
·
•
·
· ··
,
Zn )
= 0}
•
z; f. 0}, we can define holomorphic functions hv zo Zn . , . . . Fv = ' ) J; ,v (zo : . . . : Zn : z, z, {x E U; : fi , 1 (x) = . . . = J; , k (x) = 0}, and consequently X
Then X n U; is an analytic set.
:
(
)
5. Quotients and Submanifolds
217
Defin it ion . An analytic set X c Jpm that is the zero set of finitely many homogeneous polynomials is called a (p rojective) algebraic set. The subsets X n Ui are called (affine) algebraic.
E
A complex manifold X is called projective algebraic if there is an N N and a holomorphic embedding j : X --+ lP'N such that j (X) is a regular algebraic set.
Every analytic set X in projective space is the zero set of finitely m any homogeneous polynomials F1 , . . . ,F8 such that if x X is a regular point of codimension d, then rkz ( F1 , . . . , F, ) = d fa r every z 1r - 1 (x) . 5 . 1 3 Theorem of Chow.
E E
PROOF: If X c lP'n is a nonempty analytic set, then X' = 1r - 1 (X) is also analytic. Since dimz (X') > J for all z c n +l _ { 0}' by the extension theorem of Remmert-Stein its closure X = X' u { 0} is analytic in cn + I .
E
By Theorem 6.5 in Chapter III we can find an open neighborhood U = U(O) c C n + I and finitely many holomorphic functions ft , . . . ,fm on U with N(ft , . . . , f ) = U n and rkz ( !I , . . . , fm) = d at any regular point z of m dimension n+1 -d in U n . Now we expand fi into homogeneous polynomials Pi ,v· Then Pi ,v l x = 0 for all i , v.
X
X
Let h c (')0 Hn + I be the ideal that is generated by all i = 1 , . . . , m . Since 11 C I, C . . . C 00 '""
Pi, v,
v < k,
is an ascending chain of ideals in a noetherian ring, it must become station ary. Thus there are homogeneous polynomials F1 , . . . , F8 such that every Pi v is a finite linear combination of the Fa . But then every fi is also a linear combination of the F,: ,
a=l It is clear that N ( ft , . . . , fm ) = N (F1 , . . . , F8) near the origin. But since X is a cone, even X = N ( F1 , . . . , F, ) . Setting �
A (z) = ( ai, a(z) we have
i = 1, . . . , m CY = 1 , . . . , s
)'
A (z) . J(F, , . . . , Fs ) (z) for z E X near 0. Therefore, d = rk, (!I , . . . ,Jm ) < rk, (F1 , . . . , Fs) . If X is regular of codimen sion d at x, then X is also regular of codimension d at every z E 1r- 1 (x) , J(f, , ... ,Jm) (z) =
218
IV. Complex Manifolds
because 1r is a submersion. It follows that rkz (F1 , . . . , Fs) cannot be greater than d at these points. Since the rank is constant along 1r - 1 ( x) , it must be • equal to d at every z E 1r - 1 (x) . Examples
1 . Let L C e n+ I be a complex linear subspace of codimension q . Then there are linear forms y l , . . . , yq On en + l SUCh that L
=
{z E e n+ I
:
tp 1 (z)
=
...
=
tpq (z)
=
0}.
Since the linear forms are homogeneous polynomials of degree 1 ,
IP' ( L)
:=
{ ( zo : . . . : z,) E IP'n
:
tpf.t ( zo , . . . , Zn )
=
0 for f..L
=
L
. . . , q}
is a regular algebraic set. We call !P'( L) a (projective) linea r subspace. It is isomorphic to IP'n - q . 2. We now show that the Grassmannian manifolds are projective �lgebraic. Let 0 < k < n be given, and N : = G) - 1 . We identify 1\ e n with eN+ l and define the Plucker embedding pl : G k . n --+ IP'N as follows: 1f a subspace v c e n has the basis { al , . . . , ak } , then
It is clear that this is a well-defined injective map . To see that it is a holomorphic immersion, we choose another descripton. As above, when we introduced the Grassmannian, we use the set of multi-indices
To any I E Y'k, n there corresponds a permutation matrix P r E GL n (e) such that for A E St(k, n) we get A . P I = (Ar iAr) . We define p : St(k, n) --+ e N + l - { 0} by p(A)
:=
(det Ar I I E Y'k , n ) ·
Then p(G . A) = det G . p(A) , so p induces a map p : G k, n --+ IP'N such that the following diagram commutes: p
-
St(k, n)
cl'k, n
p
5.
Quotients and Submanifolds
L
a 1 /\ · · · /\ ak =
I E..Yfk ,n
219
det AI - ei ,
and therefore p = pl. Let 'P I : UI --+ Mk , n - k (C) be a complex coordinate system for Gk , n · Then rp i 1 (B) = 7rk,n ((Ek iB) - P[ 1 ) and 1 1 p o rp [ ( B ) = Jr o p ((Ek i B ) · P[ ) = 1r (det ((Ek i B) · P[ 1 ) J :
J E
Y'k , n) ·
Obviously, p is a holomorphic map. Every matrix
E
B E Mk ,n - k (q can be written in the form B = (b� 1 , . . . , b�) +
with b, C k for p, = k + 1 , .. . , n . We have det ((Ek iB) . P/) I = 1 , and if J = (I- {iv }) U {p,} for some 11 { 1 , . . . , k } and p, {k + 1 , .. . , n } , then
E
E
and therefore det ((En iB) . P[ 1 ) J = bvw So p o rp / (B) contains the components 1 , b, for all 11, p, and some other components. It follows that pl is an immersion. Since Gk,n is compact, pl is an embedding.
Pr oj ective H yper sur fa ces . The simplest example of an analytic hy persurface in Jpm is the hyperplane H0 { z0 0). =
=
If S E Mn+I (q is asymmetric matrix, then qs (z) := z·S·z t is a homogeneous
polynomial of degree 2. The hypersurface Q s : = { ( zo : . . . : z,)
: qg ( zo , . . . , Zn) = 0}
is also called a hyperquadric. It follows easily from the classification of sym metric matrices that Q 8 "" QT (biholomorphic) if and only if rk(S) = rk(T) . In particular, every quadric of rank n + 1 is biholomorphically equivalent to the standard hyperquadric Qn -1 = { (zo : . . . : z, ) : z5 + · · . + z� = 0). Since Qn - 1 n Uo = (( 1 : t 1 : . . . : tn) : ti t . . . + t � = - 1 }, Qn - 1 has no singularity in Uo . The same works in every Ui , so Qn _ 1 is a projective algebraic manifold. Now consider the Grassmannian G 2 ,4 '-+ lP'5 . Since dim (G 2 4) = 4, it is a , hypersurface. From multilinear algebra one knows that a 2-vector w E f\ 2C 4 is decomposable (i.e., of the form w = a A b ) if and only if w 1\ w = 0. So G2 ,4 = {1r(w) : w12W34 - w13 W24 + W14 W2 3 = 0}
220
IV.
Complex Manifolds
is a hyperquadric that is isomorphic to Q4 . This means that there is a 1 to 1 correspondence between the set of proj ective lines in lP'3 (which is the same as the set G2 ,4 of planes in C4) and the set of points of a 4-dimensional quadric.
Every analytic hypersurface Z C lP'n single homogeneous polynomial. 5 . 14 Theor em .
a�
the zero set of a
�
PROOF: If Z c lP'n is an analytic hypersurface, then Z = 1r -1 (Z) U ( 0 ) c c n+ 1 is also an analytic hypersurface. Therefore, there exists an open neigh borhood U = U(O) c c n + 1 , a point z E U n Z ,and a holomorphic function f : U -+ C with \7 f(z) f. 0 and Z n U c N(f). Making U smaller if necessary, we can find a holomorphic function g dividing f such that U n Z = N(g) . Without loss of generality we may assume that u U' X U" c c X e n and g (z 1 , z') = Z � + ak- 1 (z')z� - 1 + · . . + ao (z') �
�
=
is a Weierstrass polynomial. If a�<,v is the homogeneous part of a, of degree v, then
Pk (z1 , z')
:= z� + qk -1 , 1 (z')z� - 1 + · · · + qo,k (z')
is the homogeneous part of g of degree k. Since Z is conical, it follows that Pk l z 0.
=
Now there is a dense open subset V c U" such that { t E U ' : g(t, z') = 0} consists of exactly k points, for every z' E V. Since N(g) n U c N(p k ) n U and deg(pk ) = k, we have g Pk over V and then everywhere in U, by the • identity theorem. So Z = N(p k ) · �
=
We can choose a polynomial p with minimal degree such that
Z
=
{ (zo : . . . : Zn ) E lP'n : p(zo , . . . , Zn ) = 0} .
Then by the degree of Z we understand the number deg(p). For example, deg(H) = l for any hyperplane, and deg(Q) = 2 for any hyperquadric. Now, let Z c lP'n be an arbitrary hypersurface of degree k, defined by some homogeneous polynomial p of degree k. Then
Z n Ui
= { (zo : . . . : Zn )
E
Ui : zi k .p(zo , . . . , Zn )
=
0},
( ZJ I zi) k . and the line bundle [ Z !b given by the transition functions 9i; In particular, for every hyperplane H we have the same line bundle [H] with transition functions ZJ I zi · =
Defin ition . If H c lP'n is a hyperplane, then the line bundle 0 ( 1 ) [H] is called the hyperplane bundle. The kth tensor power of the hyperplane bundle is denoted by O(k).
5.
Quotients and
Submanifolds
221
If Z c ]pm is a hypersurface of degree k, then [ Z,j: O(k).
A homogeneous polynomial F of degree k induces a global section sF f(lP'n , O(k)) by
(sF ) i ( zo : . . . : zn ) : = zj k F(zo, . . . , zn ) In fact, (sF) i is a holomorphic function on ui ' with
Obviously,
{X
E JP'n
:
SF (X ) =
0}
=
E
for zi f. 0.
{ ( Zo : . . . : Zn ) : F ( Zo ,
.
·
. , Zn ) = 0}.
On the other hand, let s be an arbitrary global holomorphic section of O (k) . Then s can be represented by holomorphic functions S i : ui --+ c with
: Zn ) On Ui n Uj , Z� · Si ( Zo : and we obtain a holomorphic function f : c n + l - ( 0 ) --+ c with ZJ . S j ( Zo : . . . : Zn ) f( z )
=
.
z� si (1r(z))
=
·
·
·
on 7r -1 (Ui) ·
There is a holomorphic continuation F of f on c n+ l with F(>..z) = A • • F ( z ) . This means that F is a homogeneous polynomial of degree k, with sF = s . Consequently, we have the following result.
EN
For k the vector space f(lP'n , O(k) ) is isomorphic to the space of homogeneous polynomials of degree k in the variables z0 , . . . , zn . Analytic hypersurfaces in lP'n are exactly the zero sets of global holomorphic sections cf O(k) . there is a homogeneous polynomial For any Zo c n + l - ( 0 ) and any k F of degree k with F(z0) f. 0. Therefore O(k) is generated by global sections. 5 . 1 5 Pr oposition .
EN
E
The bundle 0(1) can be described in purely geometric terms. For this we note that if -1 Po := (0 : . . . : 0 : 1 ) lP'n 1 , then a projection 1r0 : JP'n + 1 - {Po} --+ lP'n is given by
E
7ro (zo : . : zn : Zn + l ) : = ( zo : . . . : zn ) We obtain local trivializations Yi : Jr() 1 (Ui) --+ ui X c by ·
·
(
·
)
lf!i ( Zo : . . . : Zn : Zn+ 1 ) : = ( Zo : . . . : Zn ) , Zn+ l Zi with lf!i 1 ( (zo : . . . : zn ) , c) = (zo : . . . : Zn : czi) Obviously this gives the 1 + !ran_sition functions Zj zi _If F . is a linear form on e n ! ' then the section sF IS g1ven at x = 1r ( z ) = �zo . . . . . z ) by n SF (x) = 'Pi 1 (x, z;- 1 F(z) ) = (zo : . . . : Zn : F(z) ) . ·
.
,
222
IV. Complex Manifolds
The Euler Sequence. Here we discuss the tangent bundle of the pro jective space in more detail. Let 1r : c n+ l - {0} ---+ ]pm be the