Chapter V Stein Manifolds

Chapter V STEIN MANIFOLDS

Summary. Until this chapter we have studied function theory in open subsets of C" only. However, this is too special for many purposes. For example, the simultaneous analytic continuation of all functions analytic in an open subset of C" may lead to functions defined in a Riemann domain which is spread over C" with several sheets just as a Riemann surface may be in the case n = 1. We therefore introduce a class of complex analytic manifolds, the Stein manifolds, whose definition is modeled on the properties of domains of holomorphy in C". The definitions are given in section 5.1, and in section 5.2 we show that the existence and approximation theorems for the Cauchy-Riemann equations given in sections 4.2 and 4.3 can be extended to this more general case. In section 5.3 we prove that a Stein manifold can be represented concretely as a closed submanifold of a space CN of sufficiently high dimension. Conversely, such manifolds are always Stein manifolds. The maximal simultaneous analytic continuation of the analytic functions in a domain in c" is studied in section 5.4. We show that it leads to a Stein manifold spread over C" and that such manifolds can be characterized by a pseudoconvexity condition. (Sections 5.3 and 5.4 are not necessary for reading the later parts of this book.) At last the Cousin problems are stated in section 5.5 where we solve them for Stein manifolds. In section 5.6 we extend the results of section 5.2 to sections of analytic vector bundles over Stein manifolds. This is an essential preparation for the theory of coherent analytic sheaves to be given in Chapter VII. In section 5.7 finally we show that an analytic structure can also be defined by prescribing at each point of a manifold what differentials shall be of type (0,l) in such a way that a certain integrability condition is fulfilled. The result is important in the study of perturbations of complex structures, but the reader may bypass it here without any loss of continuity.

5.1. Definitions. A Hausdorff topological space R is called a manifold of dimension n if every point in R has a neighborhood which is homeomorphic to an open set in R". The concept of complex analytic manifolds is defined by means of a family of such homeomorphisms: 107

STEINMANIFOLDS

108

Definition 5.1.1. A manifold R (of dimension 2n) is called a complex analytic manifold of complex dimension n if there is given a family 9 of homeomorphisms K , called complex analytic coordinate systems, of open sets R, c R on open sets fiKc C" such that (i) I f K and K' E 9, then the mapping K'K-

: ~ ( 0n,R,,)

-+

K'(R, n a,,)

between open sets in C" is analytic. (Interchanging K and ,find that the inverse mapping is also analytic.)

(J R,

(ii)

=

K'

we

0.

K E S

(iii) If K,, is a homeomorphism of an open set Ro c R on an open set in C" and the mapping K K ~ -

: K J R ~n R,)

+

K ( Rn ~ R,)

as well as its inverse are analytic for every

K E

9, it follows that

K O E 9.

The condition (iii) is in a way superfluous. For if 9 satisfies (i) and (ii), we can extend 9 in one and only one way to a family 5' satisfying (i), (ii), (iii). In fact, the only such family 9' is the set of all mappings A complex analytic structure satisfying the condition (iii) relative to 8. can thus be defined by an arbitrary family 9 satisfying (i) and (ii), but if condition (iii) is dropped, there are many families defining the same structure. Such a family is called a complete set of complex analytic coordinate systems, and two such sets are called equivalent if they define the same structure. We shall say that n complex valued functions (z,, . . . ,zn) defined in a neighborhood of a point w in R are a local coordinate system at w if they define a mapping of a neighborhood of w into C" which is a coordinate system in the sense defined above. If f,,. . ,f, are analytic functions in a neighborhood of z(w) = (zl(w),. . . , z,(w)) in C", then (fl(z), . . . ,f,(z)) is another system of coordinates at w if and only if det(CA/lzj);,j=l # 0 at z(w). This follows from Theorem 2.1.2.

Definition 5.1.2. Let R, and R, be complex analytic manifolds. Then a mapping f : R , -+ R, is called analytic if ti, f K~ ~1 is analytic (where it is defined)for all coordinate systems K , in R l and ti2 in R2. It is of course sufficient to choose only coordinate systems in complete sets of coordinate systems in R, and R,. In particular, we have now 0

0

109

DEFINITIONS

defined the concept of analytic functions in a complex analytic manifold R ; the set of such functions with the topology of uniform convergence on compact subsets of R will be denoted by A@). It is obvious that R is a FrCchet space if R is countable at infinity, that is, if there exists a countable number of compact subsets K , , K,, . . . such that every compact subset of R is contained in some K j . In fact, the topology in A(R) is then defined by the seminorms 3 f + SUP K,

Ifl,

j = 1,2,

and the completeness is obvious. It is clear that every open subset of a complex manifold R has a structure of complex analytic manifold, so the concept of an analytic function (mapping) on an open subset is also well defined. Note that iffis analytic in d, c C", then f o K is analytic in R,. Hence by the definition of a complex analytic manifold, analytic functions do exist locally. We shall now define a class of manifolds where there is a good supply of globally defined analytic functions. As we shall see, complex function theory in such manifolds behaves essentially as in domains of holomorphy in C". Definition 5.1.3. A complex analytic manifold R of dimension n which is countable at infinity is said to be a Stein manifold i f (a) R

is holomorph-convex, that is, j? =

{ z ;z E R, If(z)l I sup lflfor eueryfE A @ ) ) K

is a compact subset of R for every compact subset K of R. diflererzt points in R, then f ( z l ) # f ( z Z )for some

(p) I f z1 and z 2 are f E A(Q).

(7) For every z E Q, one can ,find n functions fi, . . . ,f, a coordinate system at z.

E

A(R) which form

Example. By Theorem 2.5.5, every domain of holomorphy in C" is a Stein manifold. To give another example, we need a definition. Definition 5.1.4. A subset V of a complex analytic manifold R of dimension n is called an analytic submanifold of dimension m i f (i) V i s closed. (ii) In a neighborhood w of an arbitrary point u E V there exist local coordinates zl, . . . ,z, such that w n V = { w ;w E O, z,+ 1 ( ~ = ) .. . = Z"(W) = 0).

110

STEINMANIFOLDS

We can define a natural analytic structure on V by means of the coordinate systems (z,,. . . ,z,) when (zl,. . . ,z,) is a coordinate system for Q with the stated property. If f l , . . . ,f, is an arbitrary coordinate system for Q at U E r! then one can always find m among these functions which form a coordinate system for Vat v. In fact, since the Jacobian det($Jdzj) # 0

( i , j = 1,. . . ,n)

at z(v), one can choose i , , . . . , i, so that det(Z$Jaz,)

#0

(p,v = 1,. . . , m).

Hence the restrictions off;:,, . . . ,A, to Vform a local coordinate system at c.

Theorem 5.1.5. Every submanifold of a Stein manifold is a Stein manifold. Proof. (a), (p) are trivial since the restriction to a submanifold of a function which is analytic in the whole manifold is necessarily analytic. (y) follows from the remark just made. This theorem would have been false for every smaller class of manifolds containing all spaces C". In fact, we shall prove in section 5.3 that every Stein manifold of dimension n can be embedded as a submanifold of C2"+l

Finally, we prove that Theorem 2.6.11 can be extended to Stein manifolds.

Theorem 5.1.6. Let R be a Stein manifold, K a compact subset of 0, and o an open neighborhood of R. Then there exists a function cp E Cm(Q) such that (a) cp is strictly plurisubharmonic. (b) cp < 0 in K but cp > 0 in (c) (z ; z E Q, cp(z) < c> cc Qfor every c E R. Note that the notion of strict plurisubharmonicity is well defined for functions on a complex analytic manifold since it is invariant under analytic changes of variables. (See the proof of Theorem 2.6.4.)

10.

Proof of Theorem 5.1.6. By condition (a) in Definition 5.1.3 we can choose a sequence K , = K,K1,... of compact subsets of R such that f z j = K j , u K j = R, and K j is in the interior of K j + l for every j . Let ojbe an open set with K j c w j c K j + l and w1 c o. Since Rj = K j , we can for every j choose functions f j k E A(!2),k = 1, . . .,k j with absolute

L2 ESTIMATES AND EXISTENCE THEOREMS FOR

8 OPERATOR 111

THE

value < 1 in K j so that maxk Ifjk(z)I > 1, z E K j + 2 \ ~ j . By raising f j k to high powers we may even arrange that (5.1.1)

ki

(5.1.2) In view of condition ( y ) in Definition 5.1.3, we may also assume that among the functions f J k , k = 1, . . ., k j , it is possible to find n functions forming a system of local coordinates at any point in K j . Now form

The sum converges by (5.1.1), and cp(z) > j - 1 when cp is in fact in P ( R ) , for the series

1

Z E

C w j by (5.1.2).

~

&k(z)fjk(i)

j,k

converges uniformly on compact subsets of iz x R, so the sum is analytic in z and its complex conjugate is analytic in i.It is also clear that cp is plurisubharmonic, and cp is strictly plurisubharmonic since, if for some z n

1 w I tj&k(z)/dZ[

I= 1

=

0 for all j and k ,

hk

then w = 0 because there exist n functions forming a local system of coordinates at z . This completes the proof. Note that even log(2 + cp) is strictly plurisubharmonic by Corollary 1.6.8 and Theorem 1.6.12. Remark. In proving Theorem 5.1.6 we only used conditions (LY)and (7) in Definition 5.1.3. We shall see in section 5.2 that this leads to the conclusion that condition (8) is a consequence of the other conditions. 5.2. L2 estimates and existence theorems for the 8 operator. Let R be a complex analytic manifold of complex dimension n, which is countable at infinity. The decomposition of differential forms into forms of type @,q) and the definition of the 8 operator can immediately be extended to forms and functions on the manifold R, for all these concepts are invariant for analytic changes of coordinates (see section 2.1). In order to extend the Hilbert space techniques used in Chapter IV,

112

STEINMANIFOLDS

we must introduce hermitian norms on differential forms in 0. We therefore choose a hermitian metric on R, that is, a Riemannian metric which in any analytic coordinate system with coordinates zl, ’ . ., z , has the form n

where hjk is a positive definite hermitian matrix with C“ coefficients. The existence of such a hermitian structure is trivial locally, and is immediately proved in the large by means of a partition of unity. The invariant element of volume defined by the structure we denote by dV. (For definitions, see also Weil [ 11.) If f is a form of type (1,O) and f = X ; f j d z j in a local coordinate system, we set where (hJk)is the inverse .of defined, for

(hjk).

This hermitian form is invariantly

By the Gram-Schmidt orthogonalization process every point in R has a neighborhood U where there are n forms wl, . . . ,anof type (1,O) with C“ coefficients such that at every point in U

(wj,wk) = d j k ,

j,k

=

l;..,n.

If we set f = x f , d , it follows that (f;,f) = C;Ifj12. More generally, a differential form f of type (p,q) can be written in a unique way as a sum

where h,Jare antisymmetric both in I and in J and 1’ means that summation is extended only over increasing multi-indices. We can define ( L f ) by

for this definition is independent of the choice of orthonormal basis ..., w”. (We leave the verification of this fact as an exercise.) As in Lemma 4.1.3, we choose once and for all a sequence v ] , , v ] * , . . . of 1 and qv = 1 on any compact functions in COm(R)such that 0 I q , I wl,

L~ ESTIMATES AND EXISTENCE THEOREMS FOR

THE

a OPERATOR113

subset of R when v is large. As a substitute for condition (4.1.6) we shall modify the hermitian metric so that

(5.2.1)

laqvl I 1 on R,

v = 1,2,

To see that this is possible we only have to note that given a hermitian metric we can choose a positive C" function M on R such that for all v.

laqvl I M

In fact, this means only a finite number of lower bounds for M on any compact subset of R. If we replace the metric by

M2

1hjk dzj d?k

the condition (5.2.1) will be fulfilled when the norm of qv is defined with respect to the new metric. From now on we keep the hermitian structure and the sequence q v j i x e d . Let cp be a function in C2(Q), and let L&,,(R,cp) be the space of all (equivalence classes of) forms of type (p,q), such that the coefficients are measurable in any local coordinate system, and

jlf12

~ ~ f = ~ ~ , z e - , d l / < m. The operator

3 defines linear, closed, densely defined operators T LtP.q)(Q'(P)

-+

qp,, + ,,(Qcp)

and S : LfP& + 1 ,(Q,cp)

+

L:p,q+ Z,(R.cp).

Lemma 5.2.1. D(p,4+,,(a) is dense in D,, n D, for the graph norm

f -

Ilfll,

+ IIT*fI/. + IlSfIl..

114

STEINMANIFOLDS

which gives

If".

IT*(vvf)- vvT*fI2

Hence q vf 4f i n the graph norm i f f € DT, n D,. It is therefore sufficient to approximate elements in DT* n D, which have compact support, and by means of a partition of unity we reduce the proof to the case when the support lies in a coordinate patch. In that case we can use the following classical lemma of Friedrichs. Lemma 5.2.2. Let cp E COm(RN)and Jcp dx = 1, let u E C(RN)have compact support and let a be a C' function in a neighborhood of the support of u. Put =

( J E W )

J v(x - &Y)qo(Y)

dY.

Then LIDkJEU - J , ( d k U ) -+ 0 in L2 when E -+ 0. Here aDkU is defined in the sense of distribution theory, Dk

=

ZjZxk.

Proof. The statement is obvious if u E C'. It is therefore enough to prove the inequality llaDkJev

- Je(aDku)/lLz

5

cIIuIILz,

for small E, assuming that a has bounded derivatives in the whole of RN. A simple computation gives we(x)

= aDkJ&u(x)

-

Je(aDkv)(x)

=

(. u(x - Ey)((a(x)

-

a(x - &Y))(Pk(Y)/&

+ (Dka)(x

-

&Y))dY) dy.

Here we have written qk = Dkcp. If C is a bound for lgradal, it follows that

Iw,(x)( 5 c /I.(

-

&Y)l(lYllcpk(Y)l

+ I d Y ) O dY,

so Minkowski's inequality for integrals gives

1 w I I L 2 5 c ~ ( l L ' l l ~ k ( Y ) l+ I d Y ) l ) dyllullL2. End of proof of Lemma 5.2.1. From Lemma 5.2.2 it follows irnmediately that if in a coordinate patch where f has compact support we apply the operator J , to (each component of).f, then

I/ T * ( J & 1f - J , T * f / l , hence in view of Lemma 4.1.4 IIT*(J,f) - T*f 1,

-+

+

0,

0 when

E

&

--f

+

0,

0, i f f € & .

L~ ESTIMATES AND EXISTENCE THEOREMS FOR

THE

3 OPERATOR 115

Similarly we prove that S ( J , f ) --t Sf if f~ Ds. (This is simpler since S has constant coefficients if the coordinates are analytic.) The proof of Lemma 5.2.1 is thus completed. Let U be a coordinate patch in 0 where there is a C" orthonormal . . . ,onfor forms of type (1,O). In terms of this basis we shall basis ol, now give expressions for the operators S and T* acting on forms f E D(p,4+ with support in U . This will prove that Lemma 4.2.1 remains essentially unchanged for such forms. If u E C ' ( U ) , we can write du =

2 & / ~ o ' w ' + 1Z U / Z ~ Q ' n

n

1

1

as a definition of the first-order linear differential operators a/do' and and if f = Z'fr,Ju' A aJ it = Cl follows that

?/c?d.Then we have &

where the dots indicate terms in which no h.Jis differentiated; they occur because 2oi and & j need not be 0. If the sum is denoted by Af, we obviously have - Afl I Clfl where C is independent off when the support off lies in a fixed compact subset of U . Now let u E D(p,4)have its support in U and form

where dots again indicate terms involving no differentiations. We shall integrate by parts here. First note that, with the notation

hjw Green's formula gives

jdu/;aW e-'+'dV = -

= e'+'2(we-q)/dd,

iUFe-'+'d~ +

rojO w

e-q

dv, u,

c;(u)

where oj E Cm(U ) . Integrating by parts in (5.2.2), we obtain

(5.2.3) T*f

= (-

1)"-

1' 1 I,K

hjfi,jg~'

A QR

+ ..

'

=

Bf

+ . . .,

j=1

where the dots indicate terms where no f I , j gis differentiated and which

116

STEINMANIFOLDS

do not involve cp. Hence IT*f - Bfl IC l f ( when f has its support in a fixed compact subset of U . With another constant independent o f f and of cp, we thus obtain (5.2.4)

lI4Il; +

IIBfIl; 5 2cllsfIl;

+ I/T*fjl.)2 + cllfll:.

The arguments which led to (4.2.3) still apply, so we get (5.2.5) llAfll;

+ IlBfll:

=

11'

1.3 j = 1

ldfI,J/6a-'12e - ' P d V +

Before repeating the integration by parts next performed in section 4.2, we must consider the commutators of the operators d / e d and 6, to obtain a substitute for (4.2.6). Thus, let w be a smooth function in U and form

81%

=

5

?w/&OkWk = k= 1

j,k= 1

?2w/?c?r'?wkdA wk +

i= 1

?W/(?wi8d.

Since &oi is a form of type ( l , ~ )we , may write (5.2.6)

zwi =

2 n

c;,a A gk,

j,k= 1

which gives &?w =

1(?2w/?c33'?wk+

j,k

c:, ?w/i?Wi)63' A wk. 1

If we replace w by W and take complex conjugates of all terms, we also obtain

a2w

=

1(32W/aoi?ak + 1zj,d w / a s i ) o i A ak. i

j .k

The identity 88w (5.2.7)

wkj

=

- 8 w therefore implies that

= d2w/zs?Wk

+ 1c;k aw/zwi = z 2 W / 2 W k ? & + 1z:j ?w/?(zi, 1

i

where the left-hand side is a definition. Note that with this notation we have ?& = E W j k d A sk. Hence w is plurisubharmonic precisely when the form positive definite.

c WjkAfk

iS

L z ESTIMATES AND EXISTENCE THEOREMS FOR

From (5.2.7) it follows that

( s k a w / a d - askw/aaj)= a 2 p / a d a W k W +

c

cik

THE

8 OPERATOR 117

aw/ami -

1cij awlad, i

i

or if we use the definition of hi and (5.2.7) again, with w replaced by 9, (5.2.8)

(skawiad - a s k w / a d )= 'pkjw + C cjkbiw - C 'ij awlad. I

i

Using Green's formula and (5.2.Q we now integrate by parts in (5.2.5). This gives, in view of (5.2.4),

where

In t , and in those terms in t 3 which involve the operators hi,another integration by parts leads to terms involving only the differential operators c?/dsi. If these are estimated by Cauchy-Schwarz' inequality, it follows from (5.2.9) that (5.2.10)

J C'(fr,j(2Re-'dV + 3 C' 1.J

(dh,j/'~?dI~ C' dV

I,J j = 1

2((/T*f/l;+

Ilsfll; + cllfll:)

for all f~ DfpSq+ with support in a fixed compact subset of U . Here C is a constant and II denotes the smallest eigenvalue of the hermitian symmetric form

1

(Pjktjtk. (5.2.11) (Note that II is independent of the choice of basis w', . . . ,w", for a change of basis means only a unitary transformation of the variables t , , . . ,t, .) The estimate (5.2.10) is of course mainly of interest when cp is plurisubharmonic.

118

STEINMANIFOLDS

We can now easily give a global version of (5.2.10).

Theorem 5.2.3. There exists a continuous function C on R such that

l2

+ 11% ;l),

(5.2.12) /(i - C)lf e P VdV I 4(/1T*f 1;

f E D(p.4+,,(R).

Here cp is an arbitrary function in C2(R) and 1, is the lowest eigenvalue of the form (5.2.1l), which is also a continuous function in R. The constant 4 here could be replaced by any number larger than 1.

Proof of Theorem 5.2.3. Let U j , j = 1,2,. . . be coordinate patches in R where (5.2.10) is applicable, chosen so that they form a locally finite covering of R. (That is, u U j = R and no compact set in R meets more ~ so that than a finite number of sets U j . ) Choose $ j Co"(Uj)

1t+bj2 = 1 in R. (If $ j ' ~Co"(Uj) and $' = C $i2 > 0 everywhere in R, we can take t+hj = $,'/$.) Now apply (5.2.10) to t+hj$ This gives J$j21f12Ae-'dVI 4(11$jT*f)l;+ II$,SfIl;)

+ CjJ u, (fI2

e-'dV.

Adding these estimates, we immediately obtain (5.2.12). Theorem 5.2.3 gives a perfect substitute for Lemma 4.2.1. Our discussion can therefore proceed along the lines of sections 4.2 and 4.3, so we shall make it brief.

Theorem 5.2.4. Let R be a complex mani$old where there exists a strictly plurisubharmonic function cp such that { z ;z E R, cp(z) < c } cc R for every c E R. Then the equation 8u = f has (in the weak sense) a solution u E Lt,,,)(R,loc) f o r every f E Ltp,q+,,(R,loc) such that 8f = 0. Proof. Let us replace the function cp in (5.2.12) by ~ ( c p )where x is a convex increasing function. The lower bound 1for the form (5.2.11) can then be replaced by x'(cp)A (see the proof of Theorem 4.2.2), so we obtain from Theorem 5.2.3

j(xt(cp)1- c)ifi2e - x ( ~d~) If

,,w

I 4(11 T*~II:,,,+ isfii;(.,),

f E D(p.4+

x' is chosen so rapidly increasing that

x'(cp)I - c 2 4,

(5.2.13) it follows that (5.2.14)

/If

//;(q)

5

/IT*f

l / i ( q ) 4-

l sf

//i(q),

f

DT*

DS,

Lz ESTIMATES AND EXISTENCE THEOREMS FOR

THE

8 OPERATOR 119

if we apply Lemma 5.2.1. (The operator T* is of course the adjoint of Twith respect to the norms 1 l x ( a ) . ) If x satisfies (5.2.13), Lemma 4.1.1 (see also (4.1.5)) now shows that the equation au = f has a solution u E LfP,&x(cp)) for every f E LfP,¶+ l,(R,x(cp)) satisfying the equation 8f = 0, and u can be chosen so that (5.2.15)

l l ~ l l x w5

/If

l/x(s).

This proves the theorem. 0 I s I co, introduced after Theorem 4.2.2 are The spaces Wsp,q), obviously invariant under analytic changes of coordinates, so if we have a manifold R, the space Wsp,q)(R,loc)can be defined as the set of forms belonging to WsP,¶) in every coordinate patch. The proof of Theorem 4.2.5 applies with evident modifications since, for forms f E DT*n Ds with support in a coordinate patch, we obtain from (5.2.10) estimates of the derivatives df,,J/da and therefore of the derivatives dfI,,/dZj if z j are the local coordinates. Thus we leave as an exercise for the reader t o supply the details of the proof of the following theorem:

Theorem 5.2.5. Let Q be a complex manifold where there exists a strictly plurisubharmonic function cp E P ( Qsuch ) that { z ; zE R, cp(z) < c } c c Q for every c. Then the equation 8u = f has a solution UE W&$2,loc) for every f E WsP,¶+ ,,(Q,loc) such that 2f = 0. Every solution of the equation au = f has this property when q = 0. The last statement of course contains nothing new beyond Theorem 4.2.5. Corollary 5.2.6.

Under the hypotheses of Theorem 5.2.5, the equation such that E C;,¶+

8u = f has a solution u E C&)(R) for every f 2f = 0.

Since Theorem 2.7.10 can obviously be extended to manifolds, we obtain in view of Theorem 5.1.6

Theorem 5.2.7. I f R is a Stein manifold of dimension n, then H'(R,C) = 0 when r > n. We also obtain approximation theorems : Theorem 5.2.8. Let R be a complex manifold and cp a strictly plurisubharmonic function in Q such that K , = { z ; z E Q, cp(z) 5 c } cc R for every real number c. Every function which is analytic in a neighborhood of KO can then be approximated uniformly on K O by functions in A@). Proof. We can apply the proof of Lemma 4.3.1, using the estimate

120

STEINMANIFOLDS

(5.2.14) instead of (4.2.10). (The function cp in Theorem 5.2.8 is the, function p in Lemma 4.3.1.) The details will not be repeated. Combining Theorem 5.2.8 with Theorem 5.1.6, we obtain

Corollary 5.2.9. If R is a Stein manifold and K a compact subset of R with K = K , then every function which is analytic in a neighborhood of K can be approximated uniformly on K by functions in A@). It is now possible for us to prove a converse of Theorem 5.1.6. Theorem 5.2.10. A complex munifold R is a Stein manifold ifund only cp E P ( R ) such that R, = { z ;z E Q, cp(z) < c } cc R for every real number c. The sets fiCare then A(R) convex.

if there exists a strictly plurisubharmonic function

Proof. Theorem 5.1.6 states that functions cp with these properties exist in every Stein manifold. Conversely, suppose that such a function cp exists in the manifold R. We have to prove the three conditions (a), (p), (y) in Definition 5.1.3. First we give a lemma.

Lemma 5.2.11, For every Z ' E R there is a neighborhood w o and an analytic function uo E A(wo)such that uo(zo)= 0 and Reuo(z) < cp(z) - cp(zo) i f z o # z e w O . Proof. Let z l , . . . ,zn be local coordinates at zo, such that the coordinates of zo are all 0. The Taylor expansion of cp can be written n

where uo is a polynomial of degree I 2 with uo(0) = 0. Since the hermitian form is positive definite, it follows that

cp(4 > d o ) + Re uo(z), z # 0, in a neighborhood of the origin. This proves the lemma. End of proof of Theorem 5.2.10. First note that the hypotheses in the theorem are also satisfied by R,, with cp replaced by l/(c - cp). We therefore have an existence theorem for the 8 operator (Corollary 5.2.6) in Q, for every c. Let zo be an arbitrary point in SZ. We shall prove that there exist local coordinates at z" consisting of functions in A(R), that zo does not belong to the A(R)-hull of fiCfor any c < cp(zo),and that for any other point z 1 with cp(z') cp(zo) there is a function f E A(R) with f ( z o ) # f ( z ' ) . This will of course prove the theorem.

L2 ESTIMATES A N D EXISTENCE THEOREMS FOR

THE

a OPERATOR

121

Choose uo and wo according to Lemma 5.2.11 so that z 1 $ wo, and w0 is covered by one set of local coordinates. Then take other neighborhoods o1and w 2 of zo so that w1 cc

0 2

cc wo.

Let $ be a function in Com(02)which is equal to 1 in ol.We shall use I+$ as a standard cutoff function. Since the support of lies in aZ\ml, we can choose a > cp(zo) and E > 0 so that

a$

(5.2.16)

Re uo(z) <

if z E supp

--E

a$ and cp(z) < a.

From the proof of Theorem 5.2.4 for the 8 operator in Q,, it follows that there is a function cp, E Cm(Q,), bounded from below in Q,, such that the equation &I = f for every f~ L&lJQ,,cp,) with zj = 0 has a solution u E L2(Q,,cp,) with (5.2.17) IIUtlS, 5 llfllva. By Corollary 5.2.6, u is infinitely differentiable iff is infinitely differentiable. Now let u be an analytic function in wo and set with a large positive parameter t (5.2.18)

u, =

t+h eruo-

u,.

We want to choose u, so that u, E A(Q,) and u, is small. To do so we note that u, is analytic if (5.2.19)

av, = u eruo8$ =

where the last equality is a definition. By (5.2.16) we have the estimate ltftl/+,, = O(e-"), for fixed u, so application of (5.2.17) shows that (5.2.19) has a solution u, with (5.2.20)

~ ~ u i J=S eO(e-"),

t

-+

+00.

The equation (5.2.19) means in particular that u, is analytic in the complement of the support of 81) in Q,. By Theorem 2.2.3 we therefore obtain u,(zl) = u , ( z l ) -+ 0 and u,(zo) = u(zo) + u,(zo) -+ u(zo) when t + 00. Since u(zOycan be chosen equal to 1, it follows that u,(zl) # u,(zo) if t is large. By Theorem 5.2.8 we can approximate u, on Qv(zo) so closely by a function U E A(Q) that U ( z o )# U(z'). This proves condition (p). Furthermore, (5.2.20) implies that ~

1

a,

1u,12

dV-+ 0,

t

-+

a,

122

STEINMANIFOLDS

for every c < cp(zo). Hence Theorem 2.2.3 shows that u, + 0 uniformly on compact subsets of R,. If c' < c, it follows that < f on 0,. for large t while u,(zo) + 1. Approximating u, by functions in A(R), using Theorem 5.2.8, we conclude that zo is not in the A(R)-hull of fit,for any c' < cp(zo). Hence the A(Q)-hull of fi<.is equal to 0,. for every c'. This proves condition (a). Finally note that if the function U E A(wo)is chosen so that u(zo) = 0, the facts that du, = du - do, at zo, and that du,+ 0 at zo by Theorem 2.2.3, show that du, converges to du at zo when t -+ co. If u ' , -.., un are a coordinate system at zo, formed by functions vanishing there, the Jacobian of the corresponding functions u, . . ., urnE A(R,) with respect to u l , . . . , U" must therefore converge to 1 when t + a.Using Theorem 5.2.8 we can approximate these functions so closely by functions U ' , . . . , U" E A(R) that the Jacobian of U ' , . . . , U" with respect to u ' ; . . , u" is also # 0 at zo. Hence condition (7) in Definition 5.1.3 is also fulfilled.

Iu,~

',

Corollary 5.2.12. The condition (p) in Definition 5.1.3 is a consequence

of the other hypotheses.

Proof. This is an immediate consequence of Theorem 5.2.10 and the remark following Theorem 5.1.6, 5.3. Embedding of Stein manifolds. Let R be a complex analytic manifold of dimension n. Every f = (f,, . . . ,fN)e defines an analytic map

(5.3.1)

0 3z

+

( f l ( z ) , ' . . ,fN(z)) E CN.

Definition 5.3.1. The map (5.3.1) is called regular if it has runk n at every point in R, that is, iffor any point in Q there is a coordinate system formed by n of the functions f l , . . . , f N . If the inverse image of every compact subset of CN is a compact subset of R, the map is called proper. The condition of regularity can of course also be stated as follows: If (zl, . . . ,z,) are local coordinates, then the Jacobian matrix (dJ/&,), i = 1, .. ., N , j = 1.; . , n, has rank n at every point in the coordinate patch. It is clear that the range of a proper map (5.3.1) is closed, for every compact set is mapped on a compact set. If, in addition, the map is regular and one-to-one, the range is an analytic submanifold of CNwhich is isomorphic to R. Our aim is to construct such a map when R is a Stein manifold.

EMBEDDING OF

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123

MANIFOLDS

Lemma 5.3.2. If K is a compact subset of a complex manifold satisfying conditions (p) and ( y ) in Dejnition 5.1.3, then one can for some large NJind f E A(0)’ so that f is regular and one-to-one on K . Proof. By the Borel-Lebesgue lemma and condition (y) in Definition 5.1.3 we can choose ( f l , . - - fk)E , A@)‘ so that n among these functions form a coordinate system at any point in K . Then there is a neighborhood V of the diagonal in K x K such that (z’,z’‘)E V and fJ(z’)= fj(z”), j = 1,. . . , k, implies z’ = z”. Using (p), we can now choose a finite number of functions f k +1, . . . ,fN in A(R) so that f;(z’) = f;(z”), j = k + 1,. . ., N , and (z’,z ” ) K~ x K\ V implies z’ = z”. Altogether we then have a map with the required properties. Our next purpose is to show that N need not be chosen larger than 2n + 1. This follows by an argument of Whitney.

Lemma 5.3.3. If K is a compact subset of a complex manifold 0 of dimension n, and iff = (fi,. . ’ ,fN) E A(R)N,then f maps K on a compact set of Lebesgue measure 0 in C N if N > n. Proof. That,fmaps K on a compact set follows from the continuity off: In proving that the range has measure 0, we may assume that K is contained in one coordinate patch, with coordinates ( z l , . . . ,zn), for K can be covered by a finite number of coordinate patches. Since f ( z + [) = f ( z ) + O(l[l), the measure of the set in CN on which f maps a cube I in c“ with side E is therefore O ( E ~=~m(I)O(E2) ) because N > n. Now we can cover K by cubes with side E and total measure < m ( K ) $r 1, so it follows that m ( f ( K ) )< ( m ( K )+ I)O(E’). Hence f ( K ) is a null set.

’,

N 2 2n, is a regular map on the compact Lemma 5.3.4. I f f E subset K of R, then one can find ( a , , . . . , a N )E C N arbitrarily close to the origin so that +

(fi

- aifN+i,...,fN

-

aNfN+i)~A(n)~

is a regular map on K . In fact, this is true for all a E C N outside a set of measure 0.

Proof. We may assume that K is contained in one coordinate patch with coordinates ( z l , . . . ,zn). The vector a E C Nhas to be chosen so that, if

C i,,(i?f,/dz, 1

-

aj?fN+,/(?z,) = 0,

j = 1,.

.. N, 7

at some point in K and for some i € C N it follows that E.

=

0. With

STEINMANIFOLDS

124

a N + l = 1 and p = Ei A,afN+ Jazk, this condition can be rephrased as follows : The equations n

1;ika&/azk = p a j ,

j = 1, . . ., N + 1,

1

shall imply that 2 = 0. Since the matrix (afj/dzk)gE:;:::$has rank n, it is therefore sufficient to choose a so that ( a , l ) is not in the range of the

v, v = 1,2,. . . , it follows from Lemma If we first restrict 2 to a ball 1/11 I 5.3.3 that the range of the map (5.3.2) is of measure 0, for N + 1 > 2n, and since its intersections with the planes a N + l = constant are homothetic, they must all be of 2N-dimensional measure 0. Hence we have proved the lemma.

+

Lemma 5.3.5. I f f~ A(Q)”’, N 2 2n 1, is a regular one-to-one map on the compact subset K of 0, then one can find ( a l ,‘ . ’ ,a N )E CN arbitrarily close to the origin so that (fl

-

alfN+l,”’?fN

-

aNfN+l)EA(Q)N

is a regular one-to-one map on K . I n fact, this is true for all a E CNoutside a set of measure 0.

Proof. We know from Lemma 5.3.4 that the map is regular on K when a avoids a set of measure 0. We want to choose a so that for z’, z” E K the equations L(Z’)

-

ajfN+

1(Z’)

= fj(Z”) -

imply z’ = z”. With aN+ write them in the form

=

ajfN+

i = 1,

~(z”),

’

’

.> N ,

1 and /1 = f N + l(z’) - f N + l(z”),

h(z’) - fj(z”) = Laj,

j

=

1,.

+ ,. N

we can

+ 1.

Hence it is sufficient to show that a l r . . . , a N + can l be chosen with a N + = 1 so that these N 1 equations imply that /1 = 0, and therefore that z’ = z” since f is one-to-one. But the range of the map

+

C

X

K

x

K

3 (p,z’, z”) -+

is a null set since N

p(fI(2’) - fl(z”),. . . ,

+ 1 > 1 + 2n.

fN+l(z’)

- fN+l(z”))ECN+’

This proves the lemma.

EMBEDDING OF

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MANIFOLDS

125

Remark. Note that the geometrical meaning of the proof of Lemmas 5.3.4 and 5.3.5 is that we have projected C N + on the subspace CNdefined by z N + l = 0 along a direction which is not a tangent (respectively not even a chord) off ( K ) . It is now easy to prove the existence of regular one-to-one maps.

'

Theorem 5.3.6. Let R be a complex manifold which is countable at infinity and satisfies conditions (8) and ( y ) in Definition 5.1.3. Then (a) The set of all f E A(R)N which do not give a regular map of R into C N is of the j r s t category i f N 2 2n. (b) The set of all f E which do not give a regular one-to-one map of R into CN is of t h e j r s t category i f N 2 2n + 1. We recall that a set in a complete metrizable space is of the first category if it is contained in the union of countably many closed sets with no interior. A set of the first category has no interior, that is, its complement is dense. Proof of Theorem 5.3.6. We first prove (a). Since R is the union of countably many compact sets, it is sufficient to prove that for every which are not regular compact subset K of R, the set M of all f E on K is of the first category. Now M is obviously closed, for ifhe andf, + f w h e n j -+ co, then, i f f j is not regular at z j E K , it follows that f is not regular at any limit point of the sequence z j . It is therefore sufficient to prove that M has no interior point. To do so we choose according to Lemma 5.3.2 functions g , , . . . , g, E A(R) so that g = (gl,.. . , g,) is a regular map into C' on K . For every f E A(R)N we can now apply Lemma 5.3.4 repeatedly to the map ( L g ) into CN+'and conclude that r

6 =f,+ C a j , g k ,

j = I,..., N

1

is a regular map on K for suitable, arbitrarily small coefficients a j k . Thus f ' is not in M , so f is not an interior point of M . This completes the proof of (a). The proof of (b) is exactly parallel except for the fact that it depends on Lemma 5.3.5 instead of Lemma 5.3.4. It may therefore be left to the reader. It remains to discuss the existence of proper maps, which is much harder. First note that if we have a proper map f of R into CN,then { z ; z E Q Ifj(z)I < R, j = 1,. . . ,N } is relatively compact in R for every R and these analytic polyhedra, all defined by no more than N

126

STEINMANIFOLDS

inequalities, exhaust R. What we shall do first is therefore to discuss analytic polyhedra in a Stein manifold. We shall call an open relatively compact set P c R an analytic polyhedron of order N if for some fj E A@), j = 1, .. . ,N, the set P is a union of components of the open set

{ z ; z ~ RIfj(z)I , < 1, j = l;.., N } . Lemma 5.3.7. If R is a Stein manifold, K a compact subset of R with K = I?, and o a neighborhood of K , then there exists an analytic polyhedron P with K c P cc o. Proof. We may assume that o is relatively compact in 0. For every So, we can find f E A(R) so that If I < 1 in K but f ( z ) ( > 1. By the Borel-Lebesgue lemma we can therefore choose f l , . . . ,f N E A ( 0 ) so that

1

ZE

{z;z~Q lfj(z)I < 1, j = l , . . . ,N } contains K but does not meet So. Hence the intersection of this set with o is an analytic polyhedron P with the required properties. The next step is to decrease the order of the polyhedron with a device due to Bishop. Lemma 5.3.8. Let K be a compact set and P an analytic polyhedron of order N + 1 in R such that K c P. If N 2 2n, there exists an analytic polyhedron P' of order N such that K c P' c P. Proof. Let P be a union of components of the set

{z;zEQ~I~~(z)I < l,j=

l;..,N

+ 1).

Choose numbers co < c 1 < c z < c3 < 1 so that lfiz)I < co for + 1 when Z E K. We can choose f i t , . . . ,f with f +; 1 = f N + 1 so that

j = l , . . . ,N

( f i ' / f ~ + l , . . . , f N ' ~ N + l ) is

and

jj'

of rank n on ( z ; z E P , { f N f 1 ( Z )2I cz3,

is so close to j, that Ifi(z)I < co in K for j w = { z ; z E P Ifj'(z)l , <~

3 ,= j

l;.., N

=

1; .. ,N and

+ l} cc P.

In fact, this follows immediately from the proof of Theorem 5.3.6 since N 2 2n ; we can choose fi/f N + as fjlfN+ plus a linear combination with small coefficients of suitable functions in A(0). Now consider the open set

A y = {z;~~QIfj'(z)"fN+l(z)''I

< c l " , j = l,*..,N},

EMBEDDING OF

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127

MANIFOLDS

where v is a positive integer which will be chosen later. We shall prove that the union P,' of the components of A, which intersect K has the desired properties if u is large enough. First note that z E K implies \fi(z)v - f N +

l(z)YI

< 2cOv < c I v

if v is sufficiently large. Hence K c A, for large v. If we can prove that P,' c w when u is large, we will have an analytic polyhedron of order N with all the required properties. If P,' is not contained in o,then some point z E P,' must be on the boundary of o,for every component of P,' intersects K and therefore contains points in o. If IfN+,(z)I < c,, then Ifi(z)I' < c2'

+ c l v < c3'

when j 5 N if v is large,

which contradicts the assumption that z E do. Hence z is in the compact set

L

=

{ z ;Z E ao, tfN+l(z)I 2

CZ}.

Let L , be a compact subset of L contained in a coordinate patch with coordinates z l , . . z,. If z E L l n A, we have, with Fj = f , ' / f N + 1 , a ,

IFj(z)' - 11 < (cI/cz)y,

j

=

I , . . . ,N .

We shall prove that this implies that

(5.3.3)

max lfj'(z

j= 1

I

... N

+ i)'- f N + , ( z + i)"J > clV if Z E L , and

=

l/vz,

,

provided that v is sufficiently large. This will prove that no Z E L , can belong to a component of A, which intersects K , and so the proof will be completed when (5.3.3) is verified. To prove (5.3.3), we write

I f , ' ( . + 0" + [)"I = I4(z + c)' IfN+l ( z + ill 2 c,(l + ~ ( v - ' ) ) , we fN+l(z

Since have c,"(l + O(u-')) > c,"/2 if v is large. Now write

Fj(z

+ [)v

- 1 = Fj(Z)V((Fj(Z

+ [)Iv. I f N + 1(z + i)lv 2

- lI)fN+l(z

+ 5)/Fj(Z))' - 1) i-Fj(#

- 1,

and note that Taylor expansion gives

qz+ i ) / ~=~1 +( ~ ijco) + o ( ~ - ~ ) ,

where all the linear forms lj do not vanish simultaneously since the functions 5,j = 1 , . . ., N , have rank n on L,. Hence max llj([)l 2 clil 15jSN

128

STEINMANIFOLDS

for some c > 0. We have ( F j ( z + l)/Fj(z))" = 1 summing up the estimates above we obtain max

1sjSN

+')C

-fN+

l(z

+ [)'I

> c2"2-'(c/v

+ vl,(l) + O ( V - ~ )so,

+ o(v-')>> clY, Ill

if v is large enough. The estimates are uniform in z when completes the proof of the lemma. We can now prove the main result of this paragraph.

ZE

=

w,

L,. This

Theorem 5.3.9. If R is a Stein manifold of dimension n, there exists an element f E A(R)2n+1which defines a one-to-one regular proper map of R into C2" +

Proof. According to Theorem 5.3.6 there exists a regular one-to-one map g into CZn+'. If we can construct f e A(R)'"+ such that

'

(5.3.4)

{z; z E Q, If(z)l 5 k

+ lg(z,I> == Q

for every k, the theorem will follow. (Here we have written If (z)l for maxi ljj(z)l and defined lg(z)l similarly.) For application of Lemmas 5.3.4 and 5.3.5 to the regular one-to-one mapping (Jg) then shows that there exist matrices ajk with constant, arbitrarily small coefficients, such that 2n+ 1

+ k1 = ~~

&(z) = jJz)

1

ajkgk,

j = 1,. . . , 2 n

+ 1,

defines a one-to-one regular mapping f ' into C'"'. If C, lajk[< 1, we have {z;zeR,If'(z)l Ik} = {z;zeR,If(z)l I k +lg(z)I} cc R,

sof' will also be proper. To constructfwe first note that, from (a) in Definition 5.1.3, it follows that there exists a sequence of compact subsets K j of R such that K j is in the interior of K j + for every j , Rj = K j and u p K j = R. By Lemmas 5.3.7 and 5.3.8 we can choose an analytic polyhedron Pi of order 2n such that K j c P j c Kj+l. Let ivj = SUP jgl. Pj

The condition (5.3.4) is then a consequence of the following: (5.3.5) 2 k Mk+l in P ~ + ~ for \ Pevery ~ k.

+ For (5.3.5) implies that I f 1

If1

: (Pj+l\Pj) = R\Pk. u

2k

+ lgl in P,,,\P,,

hence

If1

2 k

+ lgl in

EMBEDDING OF We first construct f l , . . . ,fin

(5.3.6)

max Ifiz)l > k

1
E

MANIFOLDS

STEIN

129

A(R) such that

+ Mk+

on apk for every k.

To d o so we note that by the definition of an analytic polyhedron of order 2n one can find ( h l k , . . . , h 5 , ) ~ A ( R ) 2 nso that maxj lhrl < 1 in __

= 1 on dP,. If we set hk= (a&:)"" with ak slightly larger than 1 and m, a large integer, we can successively choose a, and m, so that for every k

Pk- but maxj lhjkl

These conditions imply that the sums m

I

f, =

C hk, k= 1

j = 1, ... , 2 n ,

converge to functions in A(R), and the construction immediately gives (5.3.6). Now set G k = H k

max Ifj(z)I 5 k

{z;zEPk+l\Pk,

= { Z ; Z € Pk,

"ax

1SjS2n

l
5k

]f,(Z)I

+'k+l}r

+Mk+l).

From (5.3.6) it follows that these disjoint sets are compact. The A(R)-hull of G, u Hk is contained in K k + 2 and can obviously be written Gk u H , u H,' where H,' c p k +1 . (In fact, H ; is empty but this has no importance for us.) Using Theorem 5.2.8 to approximate by functions in A(R) a function which is 0 on H , u H,' and equal to some large constant on G,, we obtain successively functions h, E A(R) such that

C

Ihkl

< 2 - , in H,,

lhkl

2 1 -t k

+Mk+l +I 1 j
hjl

in G,,

Since G, c H k + c H k + 2 ..., the analytic functionf,,. fh+l=

satisfies the inequality if2, + l(z)l 2 k holds, which completes the proof.

k = 1,2;..

.

defined by

Chk 1

+ M,,

when z E G,. Hence (5.3.5)

130

STEINMANIFOLDS

5.4. Envelopes of holomorphy. In section 2.5 we proved that if R is a connected Reinhardt domain containing 0, or if R is a connected tube, there is a holomorphy domain d of the same type to which all functions which are analytic in R can be extended. We shall now give a general discussion of such results. It is not possible to restrict the study to open subsets of C“, so Stein manifolds naturally take the place of domains of holomorphy. Throughout this section we require all manifolds to be connected and countable at infinity without making this assumption explicitly in every statement. We shall say that a manifold d is a holomorphic extension of another manifold R if

(i) R is an open subset of d. (ii) The analytic structure of R is induced by that in d. (iii) For every u E A(R) one can find ii E A(d)so that u = ii in R. (The function ii is then uniquely determined by u, for d is connected.) We are interested in finding a Stein manifold d which is a holomorphic extension of a given manifold R. It is clear that R must then satisfy conditions (p) and ( y ) of Definition 5.1.3. Lemma 5.4.1. l f d is a holomorphic extension of R, one caiz for every compact subset K of d,find a compact subset K of R such that the A@)-hull of K contains K .

Proof. A(R) is a Frechet space with the topology defined by all seminorms of the form u E A(R), u + sup IUI, K

where K is a compact subset of R. In fact, the topology is defined by countably many seminorms since R is countable at infinity, and the completeness follows from Corollary 2.2.4. Similarly, A ( h ) is a FrCchet space. Now the fact that d is a holomorphic extension of R means that the restriction map A ( R ) -+ A(R)

is onto and, since it is continuous and one-to-one, the inverse is continuous by Banach’s theorem. For every compact set K c d, one can therefore find a compact set K c R and a constant C such that sup 161 5 t

c sup IiiI, K

ii E A(h).

Replacing ii by iik, taking kIh roots and letting k -+ “c, we conclude that C can be chosen equal to 1. This proves the lemma.

13 1

ENVELOPES OF HOLOMORPHY Theorem 5.4.2. If R is a Stein munifold, and extension of R, then R = d.

d

is a holoinorphic

Proof. If R # d, there must exist a boundary point z ~ ofd R, for would otherwise be open, hence d would not be connected. Let K be a compact neighborhood of z in fi. By Lemma 5.4.1, K n 0 is then in the A@)-hull of a compact subset of R, but this is impossible since R is a Stein manifold. Stein manifolds are maximal not only in the sense that they have no holomorphic extensions, but also in the sense that, if one can find a holomorphic extension which is a Stein manifold, it contains all natural holomorphic extensions :

d\R

Theorem 5.4.3. Let R , and R, bL' holornorphic extrnsions of R, and assuine thut R, is a Stein inanifold and that,functions in A(R,) gioe local coortlinutt)s rrrrywhert. in R, unrl srpurutc points in R,. Then there is an unnlytic isornorphisin cp qf 0, into 0, which is the identity on R ; if 0, is a Strin inanifold, it is un isomorphism onto. Hence there is apart from isomorphisms at most one holomorphic extension of R which is a Stein manifold. When such an extension exists, we call it the encelope cf hofornorphy of R. Proof of Theorem 5.4.3. I f U E A(R), we denote by E j u the analytic continuation of u to Rj.j = 1,2. When z 2 ER, and z1 E R , , we set ( p ( z 2 )= 2 , if (5.4.1)

for every

( E , u ) ( z , ) = (E,u)(z,)

U E

A(R).

Since analytic functions separate points in both R , and R,, this defines a one-to-one map of a subset of R, on a subset of 0,. The map cp is continuous and defined in a closed set. For let cp be defined at z2", v = 1,2,. . . and let z," -+ z 2 in 0,. Then these points form a compact set K , c R,, so by Lemma 5.4.1 there is a compact set K c R with A(R,)-envelope containing K , . Then ( p ( z , " ) ~k,, for every v by (5.4.1) and, since 52, is a Stein manifold, this is a compact set. Hence the sequence (p(z2")has a limit point z , , and

( E , u ) ( z , ) = lim ((E1u)((p(z2")) = lim ( E 2 u ) ( z 2 ' )= ( E 2 u ) ( z 2 ) , v- x "5

uE

A(W.

Thus z1 = ( ~ ( z , ) so , the limit point is unique and the sequence cp(z2") convergent. Next note that for any two points z1 E R , and z 2 ER,, one can find

132

STEINMANIFOLDS

',

f . . . ,f " E A(R) so that E jf ',. . . ,E j f n form a local system of coordinates at zl, j = 1,2. Indeed, given j = 1 or 2, we can choose f J ' , ' . ,. 'f E A(R) having this property at zj But if f k = alflk + a 2 f i k ,then E j ( f k ) ,k = l , . . . , n , is a coordinate system at z j except when (a,,a,) satisfies an algebraic equation (the vanishing of the Jacobian with respect to a coordinate system) and this equation is not identically fulfilled. Hence one can choose a, and a2 so that one obtains a coordinate system at both points. Now let M be the set of all z 2 in the domain of cp such that for every u E A(R) all derivatives of Eju at z j ( z , = cp(z2))with respect to E j f l , . . . , E j f " are the same for j = 1 and 2 if fl,. . . ,f " are chosen as above. It is obvious that M is closed. But M is also open. For the equations ( E l f k ) ( [ , )= ( E 2 f k ) ( l 2 )k, = 1 , . . . ,n, give an analytic isomorphism between connected neighborhoods of z 2 and z1 = (p(z2)if z 2 E M , and considering power series expansions in the local coordinates E j f k , k = l ; . . , n , we find that these equations imply (5.4.1) at and 1,. Hence M is equal to Q2 and cp is an analytic isomorphism of R, into R,. Since cp(R2) is a Stein manifold if R, is a Stein manifold, it follows from Theorem 5.4.2 that q(R2) must then be equal to R,. We shall now give a sufficient condition for the existence of an envelope of holomorphy.

Definition 5.4.4. A complex manifold R of dimension n is called a Riemann domain if analytic functions separate points in R and there is an analytic map cp:Q-+C" which is everywhere regular, that is, locally an isomorphism. One can of course imagine a Riemann domain as lying above c". Another way of stating the hypothesis is that there exist n functions (n = dimension) which form a local system of coordinates at every point. The main result of this section is the following theorem of Oka.

Theorem 5.4.5. Every Riemann domain R has an envelope of holomorphy

d, and d is also a Riemann domain.

We shall first extend R as far as possible by the classical method of forming power series expansions of the functions in A(R). We shall then use the results of section 5.2 to prove that the manifold so obtained is a Stein manifold. Let z E R. The restriction of the map cp to a suitable neighborhood of z is by hypothesis an analytic isomorphism onto a neighborhood of

ENVELOPES OF HOLOMORPHY

133

cp(z). Let n be the inverse. We can then define the derivatives dUu(z) when u E A(R) by a u U ( z ) = a'u(.n(i))< = &)'

It is clear that 3% E A@). For a fixed zo we now form the power series expansion

c (i dz,))"

(5.4.2)

a

-

W Z d b!

When i = cp(z) for some z sufficiently close to zo, the sum converges and is equal to u(z). Let r,, be the supremum of all I such that the power series for every u E A(R) converges in { cp(zo)} ID,where

+

D={z;lzjl
and put D,, = {cp(z,)} + r,,D. Then the series (5.4.2) defines an analytic function u,, in D,, for every function u which is analytic in R, and we have for every CI

(W,, = a.(%,).

(5.4.3)

We shall now make a Riemann domain R,, out of the disjoint union R u D,,. To do so we identify z E R and [ E D,, if u(z) = u,,([) for every u E A(R). When u = cpj, a coordinate of cp, it follows that cpj(z) = C j , that is, cp(z) = i. Hence the map cp extends to R,, if we define it as the identity on D,,. Further, if Z E R is identified with (ED,,, we obtain u(w) = u,,(cp(w)) for all w close to z,for u(w) = =

c (cp(w) c tcp(w)

-

cp(4)"a"u(z)la!

-

YP 3'u,,(iKa!

= U,,(cp(W))?

where the second equality follows from (5.4.3) and the definition of the equivalence relation. It is now clear that R,, is a Hausdorff space with the strongest topology for which the natural maps of R and of D,, into R,, are continuous, and the extension of the function cp to R,, makes it a Riemann domain. Note that the natural maps R -+ R,, and D,, -+ R,, are analytic isomorphisms. We identify R with its image, and R,, is then a holomorphic extension of a. Now take a dense countable subset zo, z l , . . . of R and form successively i2,,,(R,,),,, . . . . These manifolds increase to a Riemann domain R' which is a holomorphic extension of !2. To R' we apply the same method to obtain a Riemann domain Q" = (Q')' and so on. Let 6 be the limit of these Riemann domains. It is again a Riemann domain which extends

STEINMANIFOLDS

134

R holomorphically, but now we have the power series (5.4.4)

6' = h.

In fact, if for some 2 E 6

c (i @(i))" L?"u"(Z)/Cr!, z

-

where u" is the analytic continuation of u E A@), converges in the polydisc D, = {$j(Z)} + rD for every U E A(R), then there exists an open neighborhood D, of i which is mapped homeomorphically by @ on D, In fact, if Z E R'"), we can by construction find 6, c R("+'I. For every 5 ~ we h denote by d(5) the boundary distance of Z in fi, that is, the supremum of all r such that there is a neighborhood d of Z which is mapped homeomorphically on {@(Z)) + rD by @. By the arguments just given, r is the radius of the largest polydisc where (5.4.4) converges for every u E A(R). The proofs of Theorems 2.5.4 and 2.6.5 can therefore be repeated in order to prove that -log d is plurisubharmonic in fi. (d is finite and continuous everywhere unless h = C".) Theorem 5.4.5 is thus a consequence of the following

Theorem 5.4.6. Let R be a Riemann domain such that the boundary distance d isjinite and - log d is plurisubharmonic in R. Then R is a Stein manifold. Note the close relation to Theorem 5.2.10. However, -log d is not necessarily smooth and { z ; -log d(z) < c) need not be compact. Combination of Theorem 5.1.6 with the proof of Theorem 2.6.7 also shows that -log d must be plurisubharmonic if R is a Stein manifold. Proof of Theorem 5.4.6. Let R, = { z ;z E R, d(z) > c}. We claim that the sets R, are Stein manifolds when c > 0 and have the Runge property with respect to one another. (Cf. Theorem 4.3.3 which obviously extends to Stein manifolds in view of Corollary 5.2.9.) To do so, we have to show that, if 0 < b < c and K is a compact subset of R,, then I?,, is a compact subset of R,, and therefore by Theorem 4.3.3 independent of b. When 0 < E < c0, the set K lies in one component of R, which we denote by R,'. Fix a point zo E R,' and let p(z) be the distance from zo t o z in R,' with respect to the element of arc length ldq(z)l. The set { z ;z E RE',p(z) < C} is relatively compact for every C. In fact, if it is relatively compact for one value of C, it follows by application of the Borel-Lebesgue lemma that it is also relatively compact when C is replaced by C + 4 2 . Choosing a function x E Com(D) with x 2 0 and JxdA = 1, we set P,(4 = Jp(n(cp(z) - m

x ( 0d45),

ENVELOPES OF HOL~MORPHY

135

where n is’the analytic inverse of cp which is defined in {cp(z)).+ d(z)D, and maps cp(z) on z. It is clear that pa E Cm(R;,) if 6 < E. Writing p n = J which is a Lipschitz continuous function with Lipschitz constant 1, we have for w in a neighborhood of ~ ( z ) 0

p,(nw) =

1 f(w - 6i)xK)d W .

If D is a first-order derivative, we obtain by differentiating under the integral sign and changing variables D p , ( ~ w )=

J”(Df)(S)x((w - O/S)d2(O/S2”.

Since / O f (I 1, it follows that IDp,(~w)l 5 1 and that each derivative of Dp, is bounded by a constant depending only on 6. Hence it is possible to choose a constant Cd so that

n

P,’(z) = P,(z)

+ c d C lipj(z)l’ 1

is strictly plurisubharmonic in R;&, and since p - pa I 6, it is still true that ( z ; z E a;,, p,’(z) < C) is relatively compact in R for every C. NOWlet pa be the regularization of -log d , defined similarly in Rd. From Theorem 2.6.3 it follows that pa is plurisubharmonic. We have -logd 5 p , 5 -log@ - 6). Set

0 = { 2 ; 2 E R,, p,(z) < - log(24).

Then 0 contains R,, since 0 < 6 < E, and 0 is contained in R2,. Let 38 < E~ and 3~ < b, and let 0’ be the component of 0 which contains K . Set in 0‘ q(2) = -(P,(Z) + 6 + log c)/(Pa(4 + log(24). If 6 is small enough, we have p,(z) 5 -6 - logc in K since K c 9,. Thus q 0 in K ; but q(2)+ co if z approaches a boundary point of 0‘. Since q is a convex increasing function of pa, it is plurisubharmonic. Now consider the function

$(z) = p,’(z)

+ E.q(z),

where i. is a positive parameter. If C is the maximum of p,’(z) in K , then $ ( z ) 5 C in K , and { z ; z ~ O ’ , ( C / (< z ) y) c c 0’ for every y. Indeed, pa’ < y 1 in this set, so it is relatively compact in R, and since q < y / i the closure contains no boundary point of 0‘. Hence it follows from Theorem 5.2.10 that 0‘ is a Stein manifold and that $ ( z ) 5 C in K O , .

+

136

STEINMANIFOLDS

Since this is true for every 2 > 0, it follows that q 0 in I?,,, hence' < -1ogc in I?,.. This implies that -logd(z) < -log c in I?,., so that I?,. is a compact subset of R,, and since 0' contains the components of R, which intersect K , it follows that R,, is a compact subset of R,. Hence all R, are Stein manifolds and they have the relative Runge property. Now let K,, K,, . . . be an increasing sequence of compact subsets of R such that every compact subset of R is contained in one of them, let Kj c R,,, and assume that K j is A@,,)-convex, where bj is a sequence of positive numbers -+ 0. Any function which is analytic in a neighborhood of K j can then be uniformly approximated on K j by a function This function can in turn be approximated which is analytic in Rb,+,. arbitrarily closely on K j + by functions which are analytic in Rbj+ and so on. Hence all functions which are analytic in a neighborhood of K j can be uniformly approximated on K j by functions which are analytic in R. It follows that the A(R)-hull of each K j is equal to K j ; hence R is a Stein manifold.

p,(z)

5.5. The Cousin problems on a Stein manifold. The results of section 5.2 make it easy to extend the Mittag-Leffler and Weierstrass theorems to Stein manifolds. We prefer to give the results in a form parallel to Theorem 1.4.5. Theorem 5.5.1. Let R be a Stein manifold and Rj open subsets of R such that f2 = u? R j . If g j k E A(Qj n a,), j , k = 1,2,. . . and (5.5.1)

gjk

=

-gk,

;gij + gj,

+ gki = 0

in Ri n Rj n

for all i, j , k,

then one can find functions g j E A(Rj) such that

(5.5.2)

gjk

=gk -

g j in Rj n Rkfor all j and k ;

in other words, theJirst Cousin problem has a solution. Proof. We just have to repeat the proof of Theorem 1.4.5. Thus we choose a partition of unity and define functions h, as there. Then hk E crn(nk) and hk - hj = gjl, in Rj n Qk. This implies that Jh, = Jhj in Rj n R,, so there is a form II/ E C&)(R) such that II/ = ah, in i2, for every k. By Corollary 5.2.6, the equation au = - II/ has a solution u E crn(n), and the functions g k = h, + u then have all the required properties.

THECOUSINPROBLEMS ON

A

STEINMANIFOLD

137

Next we consider the second Cousin problem, the analogue of the Weierstrass theorem. In doing so, we denote by A * @ ) the set of functions in A(R) which are everywhere different from 0, so that their reciprocals are also in A@). Theorem 5.5.2. Let R be a Stein manifold and Q j open subsets of R such that Q = u;"Rj. If gjkEA*(Q,nQ,),j,k = 1,2;.., and i f

(5.5.3)

gjkgkj

= 1; g i j g j k g k i = 1

in Qi n 0, n Qk for a[/ i, j , k,

then one can find g j E A*(Qj)so that

(5.5.4)

gjk

in Q j n f l k for all j and k,

= gkg,-'

provided that there exist nonvanishing functions gk E C(Q,) satisfying (5.5.4). Proof. Let c j be nonvanishing functions in C(Rj) such that gjk

=

ckc,- 1 .

First assume that all the sets Rj are simply connected. Then we can write cj = ebJ, where bjE C(Rj). If we set hj, = b, - b j , we obtain g j k = exphjkr so that hj, is a unique continuous definition of the logarithm of gjk and therefore analytic in Rj n 51,. We have (5.5.5)

hij = - h j i ;hi, + hjk + hki = 0 in Ri n Q j n R k .

By Theorem 5.5.1 there exist functions hk E A(Rk) such that hjk = h, - hj in Rj n Rk.

Writing g k = exp hk E A*@,), we have then solved the second Cousin problem (5.5.4). We now drop the assumption that the sets Qj are simply connected. be another covering of R with open simply connected sets Let {R,'}:= R,' such that for every v there is a positive integer i, for which R,' c Riv. (That is, the covering {Rv') is a refinement of the covering {R,}.) Set g,;

=

giviu in R,,' n R,'.

Then g:, satisfies (5.5.3). I f (5.5.4) holds for some continuous nonwith c,' = ci,. Hence the first vanishing c j , we obtain 81, = C,,'C,'-' part of the proof shows that one can find g', E A*(R,') so that for all v, p g,:

= g,'gv'-

in 0,' n R,'.

138

STEINMANIFOLDS

In particular, this implies that in Ri n R,’ n R,,’ c Ri n Rivn Rip we have g,,’g,’-’g. [&.I .g.. I,”

=

1.

Here we have used (5.5.3). But this means that gp’giwi= gy‘giyi in R,’ n R,,’ n Ri, so there is a uniquely defined element gi E A*(Ri) such that g , = g,’giVi in Ri n R,’ for every v. Now we obtain g,gJ

’

= ’%.‘gi”,(gv’gi”j)- = g.i d g--” ji =

gJk-

in R,’ n R j n R, for all v, j, k . This proves (5.5.4). Theorem 5.5.2 is a case of the O k a principle: on a Stein manifold it is “usually” possible to d o analytically what one can d o continuously. In section 7.4 we shall make the topological restriction in Theorem 5.6.2 explicit by interprcting i t as the vanishing of a certain cohomology class. In particular, this condition is always fulfilled if H 2 ( 0 , Z )= 0 (and only then). An example of a Cousin problem which cannot be solved is therefore obtained by choosing a Stein manifold for which this group is not zero. Such a n example was given after Theorem 4.2.7. F o r a direct discussion of a n unsolvable Cousin problem, see O k a [ 31. It is natural to ask if the topological difficulties which we encountered in Theorem 5.5.2 would disappear if the second Cousin problem is stated as follows (cf. Theorem 1.4.3’):

Gicen an open covering in,) of R und functions f; E A(Ri) such that f,/& E A*(Rj n R,) for all j and k; ,find f E A(R) so that ,f/&E A*(Rj) for euery j . If we set g j k = f,/&,in Rj n R, and g j = f / & , this reduces to the Cousin problem as studied in Theorem 5.5.2. (See also the proof of Theorem 1.4.3’ by means of Theorem 1.4.5.) However, it is not obvious a priori that for arbitrary g j k satisfying (5.5.3) one can find A(Rj) so that

f,/&.

Thus the Cousin problem stated above might be more = special than that considered in Theorem 5.5.2. The two statements a r e equivalent though, for we have the following theorem :

gjk

Theorem 5.5.3. Let R,, j = 1,2, . . . , be an open covering of a Stein manifold R, let gjk E A*(Rj n 9,) satisfy (5.5.3). Then one can ,find functions f j E A(Rj) not identically 0 so that (5.5.6)

f,

= g,,

f k in R j n R, for all j and k.

Note that we d o not assert that f;. E A*(Qj), so the theorem does not

SECTIONS OF

A

VECTORBUNDLE

139

solve the second Cousin problem but merely asserts the equivalence of two different ways of posing it. The proof of Theorem 5.5.3 will be given in the next section in a somewhat more general context.

5.6. Existence and approximation theorems for sections of an analytic vector bundle. Let R be a complex manifold. Then an analytic vector bundle B over R with N-dimensional fiber is an analytic manifold B together with (i) an analytic map p : B + R called the projection ; (ii) a vector space structure in each fiber B, = p - ’ ( z ) ; such that B is locally isomorphic to the product of an open subset of R and C”. This means that for each z g R there exists an open neighborhood w and an analytic mapping cp of p - *(Q) onto Q x C N such that cp- is analytic, and for every z E o,cp maps B, onto ( z ) x C N and the composed map B, 5 ( z } x CN+ C N is linear, hence a linear isomorphism. Let .(Ri]ie,be an open covering of R such that for each i there is an analytic map cpi of p-’(Ri) onto Ri x C” with the properties listed above. Then g IJ. . = c p . c pJ. - I 1

can be regarded as an analytic map of Ri n Rj into the group GL(N,C) of invertible N x N matrices with complex coefficients, and we have (5.6.1)

gijgji = identity in Ri n Rj, g i j g j k g k i= identity in Ri n Rj n R,.

A system of such N x N matrices g i j with coefficients analytic in Ri n R j is called a system of transition matrices. We recall that the data in the second Cousin problem are precisely a system of transition functions. The bundle B can be recovered from the system of transition matrices just defined. In fact, let B’ be the set of all (i, z , w) E I x R x C N such that z € R i . We say that two elements (i, z , w) and (i’, z’, w’) in B’ are equivalent if z = z’ and w ’ = gi.i(z)w. That this is an equivalence relation follows from (5.6.1). It is easy to verify that the space of equivalence classes, with the projection induced by the map B ‘ 3 (i, z , w) + z , is an analytic vector bundle for an arbitrary system of transition matrices.

140

STEINMANIFOLDS

If {gij} is defined by means of a given analytic vector bundle, the construction gives back an isomorphic bundle. If o is an open subset of R, a C" (or analytic) section of B over o is by definition a C" (or analytic) map o 3 z -+ u(z)E B, such that p(u(z))= z.

If we have a covering {R,} as above, this means that cpi u (or analytic) map of w n Ri into CNsuch that 0

(5.6.2)

=

u i is a C"

ui = gijuj in o n Ri n Rj.

Conversely, any system of C" (or analytic) maps u i of o n Ri into C N with these properties corresponds to precisely one C" (or analytic) section of B over o.To prove Theorem 5.5.3 therefore means to construct a nontrivial analytic section of any analytic line bundle over a Stein manifold. We can also define the space of locally square integrable sections of B over o to be a system of N-tuples u i of locally square integrable functions in o n Ri satisfying (5.6.2). Similarly we can define distribution sections. We shall write A(w,B), C"(w,B), L2(o,B,loc), W'(o,B,loc), . . . for the spaces of sections thus defined. Note that these definitions are independent of the choice of covering. In order to carry over the L2 methods we also have to define corresponding spaces of forms, for example C;,,,(o,B). This we do by means of a covering (R,} as above. To define an element u E C;",,,,(o,B) thus means to give for every i an N-tuple ui of forms in C;",,,(o n Ri) such that ui = gijuj in o n R, n Rj.

(A different covering gives an isomorphic space.) Since g i j is analytic, it follows that

-

du, = g i j a u j ,

hence the N-tuples auj of forms of type (p,q + 1) define an element in C&+ lJo,B). Similarly, the operator is defined on 9 ; p , q , ( ~ , B ) . Next we have to define hermitian norms on B-valued forms of type (p,q). To do so we first choose, as in section 5.2, a C" hermitian metric in R so that (5.2.1) is valid. Then we choose a C" hermitian metric in B, that is, a C" function in B which restricted to each fiber B, is a positive definite hermitian symmetric form. This can be done by means of a partition of unity. In a neighborhood U of any point in Q we can by

SECTIONS OF

A

141

VECTORBUNDLE

the Gram-Schmidt orthogonalization procedure construct C" sections b,, . . . , bN of B such that bl(z),. . ., bN(z)form an orthonormal basis for B, for every Z E U . Every B-valued form u of type (p,q) in U can then be written in one and only one way as a sum N

u = Cu'b, 1

where uv is a scalar form of type @,q) in U . We set

c N

lu12

=

[uv12.

1

This definition is of course independent of the choice of the basis b,. The spaces L&q)(R,B, cp) are now defined as in section 5.2, and so are B, cp) to L:p,4+l)(R,B, cp) and from the operators T and S from L~p,q,(R, Lfp.4+ ,)(Q B, cp) to L:P.4+2)(Q,B, cp), respectively. Since

(~(v,u) - v ~ & (=~ ( a ~ ,A u(' I la^/,(^ 1

~ 1 ~

by the definitions above, the proof of Lemma 5.2.1 is applicable with only formal changes to prove that D(p,q+l,(R, B, cp) is dense in D, n D,, with respect to the graph norm. Furthermore, N

au

=

1&"bV + . . . 1

where dots indicate terms involving no differentiation of u. If f E C&+,)(U,B,cp) and we write T*f = X:(T*f)'b,, it follows that (T*f )' apart from terms involving no derivatives is given by (5.2.3) with f replaced by f ' . Applying (5.2.10) to f " and adding for v = 1, N, we conclude that (5.2.10) is valid for f E D(p,4+l)(U,B). Repetition of the proof of Theorem 5.2.3 then again gives (5.2.12), and existence and approximation theorems follow as before : * * a ,

Theorem 5.6.1. Let R be a Stein manifold and B an analytic vector bundle over R. Then the equation du = f has a solution U E W&@,B) for every f E W&.q+,,(R,B) such that df = 0. Every solution of the equation 8u = f has this property when q = 0. Theorem 5.6.2. Let R be a Stein manifold and cp a strictly plurisubharmonic function in R such that K , = ( z ;z E a, ~ ( z ) c> is compact for every real number c. Let B be an analytic vector bundle over R. Every analytic section of B over a neighborhood of K O can then be uniformly

142

STEINMANIFOLDS

approximated (in the sense of the hermitian metric on B ) by sections belonging to A(R,B). The proofs need not be repeated. Since every point in a Stein manifold forms a holomorph-convex set, it follows from Theorems 5.1.6 and 5.6.2 that for every Z,ER and bo E B,, one can find sections u E A(R,B) with u(z,) arbitrarily close to 6,. Hence one can find N analytic sections M I , . . . ,uN of B over R such that u'(z,), . . . , uN(zo)are linearly independent, which implies Corollary 5.6.3. Let B be a vector bundle over a Stein manifold 0. For every zo E R and eoery 6, E B,,, one can jind an analytic section u of B over R such that u(zo) = b,. In particular, this shows that the bundle defined by the transition functions of a second Cousin problem has a nontrivial section, that is, we have proved Theorem 5.5.3.

5.7. Almost complex manifolds. Let R be a C" manifold of dimension 2n. We shall say that R has an almost complex structure if we are given two mappings Po,l and P,,, of the space CG,(R) of complex valued first-order differential forms into itself such that (i) Po,l and P,,, are linear over Cm(R). (ii) Po,l and P,,, are complementary projections in the sense that Po,l + P1,, = identity, Po,lPl,o = 0. (iii) Po,land PI,, are complex conjugate in the sense that ___

fECiW. p0,J = PI,Of> Condition (i) implies that Po,l and Pl,o induce linear mappings on the complexified cotangent space at every point. Thus these operators can be defined for differential forms over open subsets of R only. From and that conditions (ii) it follows that Pi,l = Po,l, that Pf,, = project the Pl,oPo,l= 0. In view of (iii), the operators Po.l and complexified cotangent space at each point on two complex conjugate n-dimensional subspaces. Any complex structure in R defines in a natural way an almost complex structure where P,.,f and Po.]f are the parts of f spanned by differentials of analytic functions and their complex conjugates, respectively. The Cauchy-Riemann equations can then be written Po,ldu = 0, so the space of (local) analytic functions is determined by Po,l. Hence there is at most one complex structure defining a given almost complex structure.

ALMOSTCOMPLEX MANIFOLDS

143

For any almost complex structure we shall write 8u = Po,l du if u is a differentiable function. The almost complex structure is then defined by an analytic structure if and only if for every point X E Q there exist C" functions u ' , ..., U" in a neighborhood of x such that auj = 0 for every j and du', . . . ,du" are linearly independent at x. The problem to decide when an almost complex structure is defined by an analytic structure is thus purely local, so we may assume that Q c R'". Moreover, of degree 1 with we may assume that there are n forms c o ' ; . . , c o " P0,,d' = 0 (we say that 0' is of type (1,O) as in the analytic case) which are linearly independent at each point in 0 . Indeed, we can choose formsf', . . ., f" such that Pl,ofl,. . ., P,,,f" are linearly independent at = P l , J j have the desired any given point in Q, and then the forms d. property in some neighborhood. As in section 5.2, we can therefore write (5.7.1)

du =

C 2ul2d'Oi + 1i 3 u / W d

as a definition of the linear first-order differential operators i/o"wj and d / a D . The equation au = 0 means that ?u/&Y = 0 for all j . From (5.7.1) it follows that all first-order differential operators are linear combinations of the operators 2/&d and C-;/C;d. Using this fact we can extend Lemma 4.2.4. We write D" = (?/SX,)"' . . . (?/Sx,,Y"'

Lemma 5.7.1. Let Q be an open set in R'", let u E L'(Q) have compact support in R, and assume that ?u/&jE L 2 ( 0 ) ,j = 1,. .., n. Then it follows that u E W '(Q). If K is u conzpuct subset of 0,then (5.7.2)

C

la1 5 1

IID'uIIL~ I C(Ilul/Lz +

2 ~ ~ ? ~ / 2 ~ 3~ '~ ~S ~U ~PuP2c) K . n

1

Proof. We first prove (5.7.2) when u E Co"(K). Denote the adjoint of

i3/i3GJ by S j . Then we have ( ' l ~ u / C ' d (d' x =

[ ( 2 u / 2 d 1 2dx + J((hjZ/&ijj

-

8ji3/2d)u)Udx.

Since hj + Z / 2 d is of order 0, the differential operator in the last integral is of order 1, so we obtain

if we recall that all first-order operators are linear combinations of the operators 2 / 2 0 . and 2 / Z d . The last term in this estimate can be replaced

144

STEINMANIFOLDS

by

(n

+ 1)C211uII& + 1

la1 i 1

IID"u1/&/2.

This gives (5.7.2) when u E Com(K).Now if u only satisfies the hypotheses in the lemma, it follows from Lemma 5.2.2 with the notation used there that a J , u / d d - J,du/a6J -+ 0 in L2 when E -+ 0. Hence an application of (5.7.2) to J,u - J,u shows that D"J,u is L2 convergent when E + 0, which proves that D"u E L2 when lc11 5 1. Lemma 5.7.2. Let R be an open set in R2" and let u E L2(R,loc). If & / d d E W'((R,loc),j = 1, . . . ,n, for some positive integer s, it follows that u E W" +'(R,loc).If K is a compact set in R and Q' c R is a neighborhood of K , rhrn where C does not depend on u. The proof follows from Lemma 5.7.1 by the arguments used in part (a) of the proof of Theorem 4.2.5, so we leave it as an exercise. For later reference we note that the Sobolev lemma shows (see the proof of Corollary 4.2.6) that u has continuous derivatives of order s + 1 - 2n. Taking s = 2n in (5.7.3), we obtain the estimate

We shall now determine when an almost complex structure is defined by an analytic structure. Recall that forms f of degree 1 with P,,,f= 0 (resp. P,,,f = 0) are said to be of type (1,O) (resp. (0,l)). Since any form of degree 1 can be written in a unique way as a sum of a form of type (1,O) and a form of type (O,l), it follows that a form of degree p can be written in one and only one way as a sum of forms of type (a,b),a 2 0, b 2 0, a b = p , where by a form of type (a,b) we mean a linear combination of exterior products of a forms of type (1,O) and b forms of type (0,l). In general, the differential dfof a formfof type (p,q) can have components of type (a,b) for all a and b with a + b = p + q + 1. However, if the almost complex structure is defined by an analytic structure, we know that dfis the sum of a form of type 0, + 1,q) and another of type (p,q + 1). The condition in the following definition is therefore necessary for an almost complex structure to be defined by an analytic structure.

+

ALMOST COMPLEX MANIFOLDS

145

Definition 5.7.3. An almost complex structure is called integrable if dfhas no component o f t y p e (2,O) w h e n f i s o f t y p e (0,l). Iff is of type (l,O), it follows that df has no component of type (2,0), that is, df has no component of type (0,2). Hence it follows that iff is of type (p,q), then dfis a sum of a form dfof type (p 1,q) and another 8fof type (p,4 + 1). In fact, it suffices to verify this for a form

+

f

=

g , A . . . A g, A h l A ‘ . . A h,

where g , ; . . , g , are of type (1,O) and h,, . . . , h , are of type (O,l), and then the statement follows from Definition 5.7.3 and the remark following it. Having now defined the operators d and in general, we conclude that (2.1.4) is valid, that is,

a

32

=

88 + ad

=

22

=

0.

For an integrable almost complex structure, we thus have the same formalism as for an analytic structure, and indeed we shall prove

Theorem 5.7.4. Every integrable almost complex structure is defined b y a unique analytic structure. Proof. The uniqueness is obvious, as was pointed out above. What we have to prove is therefore that for any x E 52 there exist n C“ functions u l , . . . , U” in a neighborhood of x such that au = 0 and the dlfferentials du’, . . . , dun are linearly independent at x. We may assume in the proof . . . , onas discussed that R c R2”,that x = 0, and that there are forms ol, above which are of type (1,O) and linearly independent at each point in 52. Now the reader will recall that in section 5.2 we worked with such forms corresponding to an analytic structure. However, the fact that the forms W J were obtained from an analytic structure was used only in proving (5.2.7) which was a consequence of the identity 8 w + 8dw = 0. Since this is true in the integrable almost complex case too, the proofs of Theorems 5.2.3 and 5.2.4 remain valid then. To take advantage of this fact we have to find a function cp which is “plurisubharmonic with respect to the almost complex structure.” To do so, we note that if $ ( x ) = IxI2, we have at 0

This is > 0 when t # 0, since the operators

are linearly independent.

146

STEINMANIFOLDS

For a suitable 6 > 0, the ball R' = {x;IxI < 6) is thus in R and Ct,bjktjtk is uniformly positive definite there, so that l/(S2 - t,b) satisfies the hypotheses of Theorem 5.2.4 in R'. Hence we can choose cp equal to a convex increasing function of l/(S2 - $) so that, for everyfof type (0,l) in R' with $ = 0 and l / f l l q < co, there is a function u such that au = f and (5.7.5)

To use this we shall choose u so that f = au is nearly 0, then choose 8u = f and u is as small as f: Then a(u - u) = 0, and u - u should not be 0 iff is much smaller than u. Now we can only obtain good approximate solutions of the equation very close to the origin, so we proceed as follows. Let u l , . . . , u" be linear forms with du' = 0' at 0. Denote the mapping x + E X , E > 0, by n, and consider the almost complex structure defined by the forms X , * W ' , . . . , n,*wn in R'. All that we have proved above is true uniformly in E when 0 < E < 1 ; and, since du' - n,*w'/E is O ( E ) together with all its derivatives when E + 0, we conclude that u so that

a

D" 2,uj=

O(E) for all a,

if & is the 2 operator with respect to the almost complex structure defined by the forms n,*uk. Hence it foll3ws from (5.7.5) that we can find u,J so that zEv,J = 3,uJ in R' and /lz~;llp = O(E).In view of (5.7.4) i t follows that the derivatives of u,J at 0 are O(E). Hence the differentials of the functions U' = u' - u,' are linearly independent at 0 if E is sufficiently small, and since 8,U-l = 0 it follows that the functions U ' ( X / E ) are solutions of the original 3 equation. Notes. The class of manifolds which are now called Stein manifolds was first introduced by Stein [I]. In the seminars of Cartan [l], the theory of coherent analytic sheaves on Stein manifolds was developed (see Chapter VII). It contains the results of sections 5.2, 5.5, and 5.6 apart from the solution of the Levi problem (Theorem 5.2.10) which is due to Grauert [l]. For the origin of the methods we have used, see the notes to Chapter IV. They are not restricted to Stein manifolds; in fact, in Hormander [ l ] they were used to prove more general results due to Andreotti and Grauert [l]. The embedding theorem in section 5.3 is due to Bishop [l] and Narasimhan 111. We have mainly followed the proof of Bishop, who even proved that there is a proper map into C + ' .The results of section 5.4 are due to Oka [5]. The construction of the envelope of holomorphy can be done somewhat more elegantly by means of the notion of sheaf which we shall introduce in Chapter

ALMOSTCOMPLEX MANIFOLDS

147

VII. (See Cartan [ l ] and Malgrange [I].) However, the classical construction we give is basically the same apart from the fact that it may be too explicit. We refer to Bishop [2] for another proof that the envelope of holomorphy is a Stein manifold which does not rely on a solution of the Levi problem. The results on integrable almost complex structures proved in section 5.7 are due to Newlander and Nirenberg [ I ] but our proof is essentially that of Kohn [I]. For the applications to the study of perturbations of analytic structures we refer to a recent survey article by Nirenberg [ 11.