Andreotti-Grauert theory by integral formulas (Progress in Mathematics)

0.13. Currents. Let X be an n-dimensional complex manifold. The space of all continuous linear maps

T:[C*o(X)]0 -* C

will be denoted by Cour (X). The elements of this space are called

currents on X. Recall some definitions from the theory of currents (for a presentation of this theory, see, for instance, [Griffiths/Harris 19781 and [6irka 1979]):

A current T on X is called of degree m (0<m<2n) if

Th = 0

for all hE[Cio(X)10 with 1 # 2n-m.

(0.9)

The degree of T will be denoted by deg T. The space of all currents of degree

on X will be denoted by Ceur(X).

A current T on X is called of bidegree (r,s) (or an (r,s)-ourrent), where 0
Th = 0

for all hf[Cicm(X)]0 with (1,m) # (n-r,n-s).

(0.10)

If f is a differential form with locally integrable coefficients on X, then by we denote the current on X which is defined by

18

h =

hE[C*°(X)]0.

fn h,

(0.11)

J

X

Notice that if f is of degree m reap. bidegree (r,s), then is also of degree m reap. bidegree (r,s), because, by definition,

f g = 0 X -

if g is a form of degree <2n.

If T is a current of degree m on X, then the currents dT,

)T and aT

are defined by

(dT)h = (-1)m+1T(dh), (aT)h = (-1)m+iT(ah), (5T)h = (-1)m+1T(5h),

hc[Coo(X)]0.

(0.12)

By Z*ur(X) we denote the space of all 5-closed currents on X, i.e. the space of all TEC*ur(X) with 5T=0. We net E*ur(X) = aC*ur(X), Zour(X) = Zcur(X)

s,r

*

n Cour(X) s,r

and

Ecur(X) = Eour(X) n Ccur(X)

s,r

s,r

*

(0<s,r
Now let E be a holomorphic vector bundle over X, and E* the dual bundle of E.

Then the space of all continuous linear maps

T:[C'(X,E*)] 0

C

will be denoted by C*ur(,E). The elements in C*ur(X,E) will be called E-valued currents on X. Notice that the E-valued currents on X may be viewed as "current sections" in E over X,

i.e. if (U3} is an open covering of X together with a collection of holomorphic trivializations

E I U

= XxCN

(N = rank E)

of B, and (g1J} is the corresponding cooycle of transition functions, then any E-valued current T over X can be identi_.fied with a collection of column vectors of "usual" currents Tk over U, such that

Ti = ai.iT.i

on UinUf.

19

If f is an E-valued differential form with locally integrable coefficients on X, then by we denote the E-valued current on X defined by

h =

I

fnh,

he(C°O(X,E*)]0.

(0.13)

If T is an E-valued current on X, then we say T is of class C<X (or T is a Co'-form) on X if T= for some f*C*'(X,E) (0
The notions of the degree and bidegree will be used (in the canonical sense) also for arbitrary E-valued currents. By Cmur(M,E) we denote the space of all E-valued currents of degree m on X, and by Csur(X,E) we denote the space of all E-valued (s,r)-currents on X. Since E is holomorphic, the 3-operator has an invariant meaning also for E*-valued forms. Therefore, this operator can be defined also for arbitrary E-valued currents: if T is an E-valued current of degree m on X, then we set

(3T)h = (-1)m+l(3h),

h5[Co*o(X,E*)]0.

(0.14)

Notice that if T is of bidegree (s,r), then 3T is of bidegree (s,r+1). Further notice that if f is an E-valued differential form on X such that 3f has locally integrable coefficients, then, by Stokes' theorem,

3 = <3f>.

(0.15)

By Zcur(X,E) we denote the space of all 3-closed E-valued currents i.e. the space of all TECour(X,E) with 3T=0. We set

on X,

Zcur(X,E)

s, r

= Zour(X/E) n Ccur(X,E), Ecur(X,E) = 3Ccur(X,E), and

s, r

Ea,r(X,E) = E*ur(,E) n cs,r(X,E) for all 0<s,r
0.14. If T is an E-valued current of degree s on X, then the support of T is, by definition, the smallest closed set McX such that Th=O for all h E [C00(X,E*)]0 with supp h c X\M. The support of a current T will be denoted by supp T.

For references we notice the following Proposition. Let T be an E-valued current on X. If supp T is compact then T is uniquely defined and continuous on the Frechet space C*0(X,E*), that is, then there exists a uniquely determined continuous linear map T:Co*o(X,E*)

20

C

such that Th=Th for all hc[C°*°(X,E*)]0.

Proof. Uniqueness: [C°°(X,E*)]0 is dense in C*°(X,E*) with respect to the topology of C0o(X,E*). Existence: Take an arbitrary

on X such that supp x - X and

1

Coo function

in a neighborhood of supp T. Set

Th = T(xh) for hE[C,°k°(XE*)]0_

Remark. Conversely, any continuous linear functional on C*°(X,E*) defines an S-valued current with compact support on X. Therefore, the space of E-valued currents with compact support on X can be identified with the dual space of the Frechet space

1.

The Martinelli-Bochner-Koppelman formula and the Kodaira firiteness

theorem

1.1. Almost C1 boundaries. Let 0 C Cn be a domain, X a subset of G, and let X be the

at of all x,X

such that, for some neighborhood U of x,

X A U is a closed CI submanifold of real codimension 1 in U.

We say X is a closed almost CI hypersurface in G if X is closed in G and the following two conditions are fulfilled:

(i) For any E>O, every compact set K C X\X, and each neighborhood U C Cn of K, there exists a neighborhood V C U of K with smooth boundary such that the (2n-l)-dimensional Euclidean volume of 6V

is < E. G, the (2n-l)-dimensional Euclidean volume

(ii) For each open U of U n X is finite.

Let X be a closed almost CI hypersurface in G. Then the points in X reap. X\X will be called the smooth reap. the non-smooth points of X. A function on X is said to be of class CI if it admits an extension into a neighborhood of X in Cn which is of class C1. A differential form (or a form) on X is, by definition, a differential form on X. Such a form f is called bounded reap. continuous on X if there exists a bounded reap. continuous differential form F in some neighborhood of X such that f=FIX. We say X is oriented if X is oriented.

If,

in this case, f is a form on

X. then we write

f:=J f, IX

provided the integral on the right hand side exists.

21

If D « C° is a domain, then we say the boundary of D is almost C1 if aD is a closed almost C1 hypersurface in Cn.

1.2. Orientation. He use the following orientation of Cn: If z1,...,zn are the canonical complex coordinates in Cn and t1....It2n are the real coordinates in Cn with zi = ti+itj+n (j=1,...,n), then the form dx1A...ndx2n is positive. If D Q Cn is a domain with almost C1 boundary, then we orient 3D (i.e., more precisely, the smooth part of 8D) by the orientation which is induced from D (so that Stokes' formula holds true).

1.3. The Martinelli-Bochner integrals BaD end BD. The form (cf. Sect. 0 for the notations) n-1

det(x-z,dx-da)4 .(x)

1

B = B(z,x)

1x-z1 °

(2ni)n

defined for all (z,x)ECnxQn with six is called the Martinelli-Bochner kernel. By Br=BI,(z,x) (r=O,...,n-1) we denote the sum of all monomials in B which are of bidegree (O,r) in z and hence of bidegree (n,n-r-1) in n-1 x. It is easy to see that Br, and B=

n-r-1

r

B (z,x)

n-l)det(x-z,dz,

l)'

(2ni)n

r

r

dx

Ix-zl

)A W(x)

n

for r=0,...,n-1.

Now let D.Cn be a domain with almost Cl boundary. Then, for any bounded differential form f on aD, we define by

(BoDf)(z) =

f(x)AB(z,x),

ztD,

(1.2)

J

xe

a Coo form BaD f in D. Further, since the singularity of B(z,x) at x=z is of order 2n-1 and

for any bounded differential form f

on D, we can define

f(x)AB(z,x),

(BDf)(z) = J

xtD

22

zID.

(1.3)

It is easy to see that so we obtain a continuous differential form BDf in D. Moreover, the following proposition holds true:

1.4. Proposition. (i) For any O<06<1, the Nartinelli-Boohner integral U. defines a bounded linear operator

BD: L'(D) 0 cot (ii) If fEL*o(D) is of class Ck for some kE{O,1,...,oo}, then

n Ck+oL(D).

BDf E

0<06<1

(iii) BD defines a continuous linear operator

BD:[Co*c(D)]O -) Co*o(D).

Proof. See, for instance, Lemma 1.8.5 and its proof in CH/L].

1.5. Proposition. Let D«Cn be a domain with almost C1 boundary. Then: (i) For any bounded (O,r)-form f on aD, BaDf is of bidegree (0,r), and

BaDf =

S

if 0
f(x>

(1.4)

x[aD and BaDf = 0 if r=n.

(ii) For any bounded (0,r)-form f on D, BDf is of bidegree (O,r-1), and

if 1
BDf =

(1.5)

J

xED

and BDf = 0 if r=0. Proof. By degree reasons, fABs=O if s
0

if s>r.

f X(aD

23

Since B =

n-1 Bs, this proves part (i). The proof of part (ii) is

similar. 0 1.6. Theorem (Martinelli-Bochner formula). Let D_Cn be a domain with almost C1 boundary. Then, for any continuous complex-valued function f on D such that of is also continuous on D, we have the representation

f = BaDf - BD3f

in D.

(1.6)

Proof. See, for instance, Sect. 1.9 in CH/L].

1.7. Theorem (Koppelman formula). Let DnC° be a domain with almost C1 boundary, and let f be a continuous (O,r)-form on D such that 5f is also continuous on D (1
(-1)rf = BaDf + 369E - BDSf

(1.7)

in D.

In particular, then 3BDf is continuous in D. Proof. See, for instance, Sect. 1.11 in CH/L].

Remark. For r=0, by Proposition 1.5 (ii), BDf=O and hence (1.7) coincides with (1.6).

1.8. The Martinelli-Bochner operator B' for forms and currents with compact support. For any continuous differential form f with compact support in Cn, the Martinelli-Bochner integral B f can be defined also D

with D=e: (B

f)(z):=

r

f(x)nB(z,x),

zECn.

J

If f is a continuous (O,r)-form with compact support in C° such that 3f is also continuous in C° (O
(1.8)

We want to generalize this to currents with compact support in C°. By p(z,x) we denote the form with B(z,x) =

i. e. 24

(-1)n+1 (3(z,x)A

(x),

(1.9)

n-1

det(x-a,dx da)

(-I)",

z,XECn.

(1.10)

Ix-zl2n

(27ri)n

Now, for any continuous differential form f with compact support in Cn, we define

(Af)(z) =

zECn.

(1.11)

J

x4 Cn

It follows from Proposition 1.4 (iii) that, in this way, a continuous linear map

A:CC,((Cn)]0 -* C*°(Cn) is defined. Hence, for any current T with compact support in Cn, the composition T-A is well-defined and continuous on [C°(Cn)]0 (cf. Sect. 0.14). Thus, for any current T with compact support in Cn, by setting

(B'T)h = T(ph) = T[

J x e4:n

for hE[C°°(Cn)]0' we obtain a current B'T on Cn.

1.9. Proposition. Let T be a current with compact support in Cn. Then: (i) If T is of bidegree (O,r), r>1, then B'T is of bidegree (O,r-1). (ii) If deg T = 0,

i.e. T is a distribution, then B'T=O.

(iii) If T is of bidegree (s,r) with s#0, then B'T=O.

Proof. If hE[C m(Cn)]0' then from (1.10) and (1.11) one obtains the following two assertions: a) 2h is of bidegree (n,m-1) if 1=n and 1<m
In particular, the bidegree of any monomial in bh is of the form (n,k) with k
To prove part M. we assume that T is of bidegree (O,r) with 1
25

assertion a), the bidegree of $h is $(n,n-r). Since T to of bidegree

(O,r). this implies that T(ph)s0. 1.10. Proposition. For any continuous (0, r)-fore f with compact support

I. Cn (0
B' = (-1)r.

00 Proof. Let he[Cn, n-r+1(On)]0. Then

(B')h = (Ah) = J f(a) Al h(x)4(z.x)ncJ (a) 94C°

(-I) r+1

x(Cn

J

j f(s)Ai (s,x)A w(a)Ah(x).

xa, ZECn Since

(a,x)A W(s) = (-1)n°(x,s)At.7($) = -B(x,s this implies

(B')h = (-1)rh.

1.11. Proposition. For any current T with compact support in C°, the i.e. there is a real-analy-

current B'T is real-enalytio in C° \ supp T,

tic form g on C° \ supp T such that

Ti(Cn \ suppT) _ .

Proof. It is sufficient to show that such a for. g exists over each open U cc C° \ supp T. Let such an open set U be fixed. Then we choose = 1 in a neighborhood of supp T and a Coo function x on Cn such that

%= 0 in some neighborhood of D. By Proposition I.D. we may assume that T is of bidegree (0,r) with 1
a

g(x)Apn-r(S.x)A W(s).

'ad =

(1.12)

J

Further, let bii

iR 1{s, x) be the (O,n-r)-forms in a depending on x

such that Sn-r(a'x)

b1

=

1.- i

1<11<...
r-1

(a.x)Adxi ..-ndxi 1

r-1

(1.13)

Set

bi

(M. X) _

...i

Z{s)(1-x (x))bi

r-1

1

...i

(a,x)A W($). r-1

Since x = 0 in a neigborhood of (1, then the assignment

Usx

)

bi

(1.14)

i

r-1

1

is a real-analytic map from U into C; n_r(Cn). Hence, the (O.r-1)-for.

dx1 A...Ad-xi

1<11<...
r-1

1

r-1

1

with

r-1 (x)'=

1

T( bi

..

1

i

(-.x)) r-i

is real-analytic in U. Now let hE(Cnoon-r+1(U)]O. Since

0 in a neighborhood of supp h and

= 1 in a neighborhood of eupp T. then we have the relation (B'T)h T(ph) = T[xp((1-x h)h)]. By (1.12) and (1.13), this implies that

(B'T)h = T

h(x)A x{.)(1-x>)pn-r(-'x)n

x6U

1

T

i

I h(x)Abi Jd

xEU

1

r-1

A...Adxi 1

r-1

r

Since the maps (1.14) are continuous as maps with values in

Cnon-r(Cn),

the integrals on the right hand side may be interchanged with T. So we obtain

(B'T)h = +

1 h(x)Agi

i

(x)dx1 .

A...Adx. 1

l

= +h,

r-1

xf.U

i.e. B'T=+. 1.12. Theorem (Koppelman formula for currents with compact support). For every current T with compact support in Cn, we have the representation

T = 3(B'T) + B'(3T).

(1.15)

Proof. By Proposition 1.9 we may assume that T is of bidegree (O,r) with 1
(-1) n-rh/L7 = 3BCn(h/r,) - Bn3(h/W) .

Further, since

t (x) _ (-1)n+1 B(z,x), we have

(Ph)(z) =

f

w(x)n W(z)

X,Cn

_ (-1)n+1BCnWo)(z)A W(z).

Together this implies that

(-1)r+lh = 3(ph) - P(3h).

Applying the current T to this equation and taking into account that B'T is of bidegree (O,r-1) (Proposition 1.9), we obtain

Th = (-1)r+l(T[a(ph)) - T[P(ah)] = (aT)(f3h) + (-1)r(B'T)(3h)

= [B'(3T)]h + (3(B'T)]h.

28

1.13. Theorem (Regularity of the 5-operator). Let E be a holomorphic vector bundle over a complex manifold X, and let T be an E-valued i.e. TECour(X,E). If

distribution on X,

5T = for some fEC0p1(X,E) with kE{O,l,...,oo}, then

for some

T =

n

gE

0+ '(X,9).

0
In particular, if 5T=O, then T= for some holomorphic section h of E over X.

i.e.

Z r(X,E) = <(9(X,E)>. Proof. Since the assertion is local, we may assume that X is a domain in C° and E is the trivial line bundle, i.e. TiC0cur (X). It is

sufficient to prove that, for every open U==X, there exists a form

gE

n

0<0(,
Ck+a'(U)

0, 1

with T= on U. To do this, we choose a C00 function x such that x = 1 in a neighborhood of U and supp x cc X. Then, by the Koppelman formula (1.16) and by Proposition 1.9 (ii), we have

T = xT = B' (8(f)) = B'(Tax) + B'<x f>

on U.

Since supp Tax cc X\U, from Proposition 1.11 we get a form g06C0 1(U) with

on U.

B' (Taz) =

Further, observe that, by Proposition 1.4 (ii),

Be,(x f)

I

I

Ck+«,(Q°)

O

and, by Proposition 1.10,

29

B'<xf> = -. Hence, the form

g:= 900 - BCn(xf) IU has the required property.

1.14. Dolbeault oohomology. Let E be a holomorphic vector bundle over an n-dimensional complex manifold X, and let 0<s,r
H9, r(X) = ZB r(X)/E9 r(X)

and

H9,r(M ,E) = Zs r(X,E)/Ee

and

Hs'r(X,E) = Zcur(X,E)/Ec f (X, 6). s,r cur

r(X,S)

(0
Hous'r(X) = Zour(X)/Ecur(X) our s,r s,r

The space Hs,r(,E) is called the Dolbeault cohomology group (or the a-cohomology group) of bidegree (s,r) of X with values in E. Further on, in the most cases, we restrict ourselves to the case a=0 (ef. the remark at the end of Sect. 0.1). The space HOO,r(X,E) will be called the

00

Dolbeault oohomology group of order r of X with values in E. Usually, the subscript oo is omitted, i.e.

HO'r(X,E):= HO,r(X,E). Below (Theorem 2.14) we shall see that, for all 0
HO,r(X,E) ---) HO,r.(,E) which is defined b y the assignment

C

-) E

(of. (0.13)) is an isomorphism. Therefore, HO'r(X,E) can be represented also by each of the groups HO, r(M,E) (0
viewpoint of the method of integral formulas (the principal tool used in the present monograph) it is especially advantageous to work with HD'r(X,K). Let us demonstrate this in the case of the Kodaira finiteness theorem:

30

1.15. Theorem (Kodaira finiteness theorem). If E is a holomorphic vector bundle over a compact n-dimensional complex manifold X, then

dim HO,r(X,,) < oo

for all O
To prove this theorem, we use the Koppelman formula (1.6) for forms .with compact support, the Holder estimates from Proposition 1.4 (i), and the fact that compact perturbations of the identity operator in a Benach space are Fredholm. The first step is the following

1.18. Lemma. Let UccD be a relatively compact open subset of a domain D=cC". Then there exist linear operators

Tr:CO r(D)

---) n C0,r-1(U)

(1
0
and

n

Kr:CO,r(D)

06

(1
CO,r(U)

O
5T !

f + Krf - Tr+13f

if 1
f

if r =n.

Proof. Choose a Coo function ;r on Cn with supp

x a

(1.16)

D and

1 in U.

Set

Trf = (-1)rBC" (1f) IF,

and

Krf = (-1)rBC" (SXAf) 10-

Then the assertions follow from Proposition 1.4 (i) and the Koppelman

formula (1.8). 0 By means of the local operators given by Lemma 1.16, now we prove

31

1.17. Theorem. Let E be a holomorphic vector bundle over a compact ndimensional complex manifold X. Then there exist linear operators

Tr:CO r(X,E) I

(1
CO,r-1(X,E)

I

0
Kr:C0 r(X,E) O,

--, n

CQ r(X,E)

(1
0
such that the following conditions are fulfilled: (i) For any O
the Banach space CO

r' E)

into the Banach space

COce-

,r-1(XE) reap.

CD r(X,E).

(ii) For every fECO r(,E) such that of is also continuous on X, we have the representation

Trf =

--

f + K f - Tr+1Sf r

if 1
f + Knf

if r=n

(1.17)

Proof. Since X is compact, we can find a finite open covering (Ui ) of

X such that each Ui has a neighborhood Ui which is biholomorphically equivalent to some domain in Cn and such that E is holomorphically trivial over U3. Then from Lemma 1.16 we get linear operators

nx

TrCO,r(,E) )' O

'

C0,r-1 (U. E)

0<01,<1

(1
and

06

Kr:CO r(X,E)

C0 r(Ui,E)

(1
O
which are bounded as operators acting from CO r(X,E) into C0 r_1(,E) reap. CO r(Uj,E) (for all 0
with &ECO,r+1(X,E), we have the representation

bTjf r

32

f + K=f - Tr+13f

if 1
f

if r=n.

Choose a Coo partition (Xj } of unity subordinated to {D3}. Setting

Tr =

XiT j, 3

and

-

4TJr +

x3K')r

if 1
Kr

if r--n,

3x ATn

we complete the proof. 0 1.18. Proof of Theorem 1.15. First consider the case r=0. Since X is compact, (.(X,E) is a Banaoh space with the norm 1-4O

X.

Since, by

!)ontel's theorem, any bounded set in (9(X,E) is relatively compact, this

space is finitely dimensional. It remains to observe that, by Theorem 1.13 (regularity of 3 ),

H0O'0(,E) = Zp,0('E) =f1(X,E).

Now let 1
1.17, we can find bounded linear operators

Tr:Z0,r(X,E)

Cp r_1(X,E)

Kr: Z0.r(X,S)

ZO r(X,E)

and

such that

3oTr = id + Kr

on

Z0,r(X,E),

(1.18)

where id is the identical operator. Since, oy Ascoli's theorem, the embedding

ZO r(, E)

0

Z0,r(X,E)

33

is compact, Kr is compact as an operator from Z0 r(X,E) into itself.

Hence, boTr is a Fredholm operator in Z0 r(X,E). Since the image of

this operator is contained in ED r(X,E), it follows that E0 r(,E) is finitely codimensional in Z00 r(,E).

F-1

2. Cauchy-Fantappie formulas, the Poincare 5-lemma the Dolbeault isomorphism and smoothing of the d-cohomology

2.1. Leray data

and Leray maps. Let D -

Cn be a domain with almost C1

boundary. A Cn-valued C1 map v=(vl,...,vn) defined on DxaD will be called a Leraydatum for D if

for all zdD and xebD.

# 0 If, moreover,

= 1

for all zcD and xEaD,

then v will be called a Leray map for D.

Remark. In CH/L] the notion of a Leray mag is used for arbitrary Leray data

.

Notice that, of course, any Leray datum w(z,x) leads to the

Leray map v(z,x):= w(z,x)/(x-z).

In Sect. 3.4 we shall consider genera-

lizations of both the notion of a Leray data and the notion of a Leray map. Then the difference between them becomes more substantial. 2.2. The Cauchy-Fantappie integral LV. Let D =- Cn be a domain with almost C1 boundary, and let v be

a Leray map for D.

Then we set n-1

Lv

Lv(z,x)

dot ( v ( z , x ) , a x))

(2xi)n

`

v z(v

z(

z , x

J(x)

(2.1)

for zeD and xEaD, and for any bounded differential form f on al), by

(Lvf)(z) =

I

f(x)ALv(z,x),

xeBD JJ

we define a continuous differential form Lvf in D.

34

zED,

(2.2)

By LI°=Lv(z,x) (O
which are of bidegree (O,r) in z and hence of bidegree (n,n-r-1) in x. It is easy to see that r _v L (z, x)

lnri)

r

(2iri)n

n-r 1

det(v(z,x),a v, ),3 v z x

(2.3)

for r=O,...,n-1.

2.3. Proposition. Let D = Cn be a domain with almost C1 boundary, and let v be a Leray map for D. Then, for any bounded (0,r)-form f on aD, Lvf is of bidegree (O,r), and we have

Lvf =

if O
!

(2.4)

xE3D

and Lvf = 0 if r=n. Proof. Repetition of the proof of Proposition 1.5.

2.4. The Cauchy-Fanta

ie integrals Rv. Let D - Cn be a domain with

boundary, and let v be a Leray map for D. Then, for all zED, xtaD and O
v 0 = v 0(z,x,t) = t X-z (x-zl

2

+ (1-t)v(z,x)

(2.5)

and n-1

iv = Rv(z,x,t) =

1

ndet(v0(z,x,t),(3z x+dt)v0(z,x,t),A W(x).

(2.8)

(tai)

Further, then, for any bounded differential form f on aD, by

{Rvf)(z) =

r

f(x)ARv(z,x,t),

zED,

(2.7)

(x,t)CbDx[0,1)

we define a continuous differential form Rvf in D. By Rv=Ri°(z,x,t) (r=0,...,n-i) we denote the sum of all monomials in r iv which are of bidegree (0,r) in z. Then

35

r

(n-2) M2)

Rv = r

(2Ti)n

n-r-l

det(v0 5 v0,(a x+d t)-0)A L (x) a

(2.6)

for r=O,...,n-1.

2.5. Proposition. Let D a Cn be a domain with almost Cl boundary, and let v be a Lerey map for D. Then, for any bounded (0,r)-form f on M, Rvf is of bidegree (O,r-1), and one has

Rvf =

if 1
(2.9)

.

(x,t)caDx[0, 1] and Rvf = 0 if r=0 or r=n.

Proof. Repetition of the proof of Proposition 1.5. Q

2.6. The operators Tv.

If D - Cn is a domain with almost Cl boundary and v is a Leray map for D, then, for any continuous differential form f on D, we net

T

v f

= BDf

+ Rvf.

2.7. Theorem (Cauohy-Fantappie formula for functions). Let D == Cn be a domain with almost C1 boundary, and let v be a Leray map for D. Then, for any continuous complex-valued function f on D such that 3f is also continuous on D, we have the representation

f=Lvf-Tv3f

on D.

(2.10)

Proof, In Sect. 3 we prove the more general Theorem 3.11. A direct proof of Theorem 2.7 is given, for instance, in Sect. 1.9 of [H/L]. Q Supplement to Theorem 2.7. The proof of Theorem 2.7 given in Sect. 1.9 of [H/L1 does not use any information about the kind of dependence of v(z,x) on s. So, actually, there the following more general statement is proved:

Let D Q Cn be a domain with almost C1 boundary, and let, for some point 20eD, a Cn-valued C1 map v on aD be given such that =1 for all xcaD. Then, for any continuous function f on D such that of is also continuous on D, we have the relation

36

n-1 r

f(s0) _ -(BDf)(zO) +

n

f(x)det(v(x),3v x( ))n w(x) J

xa aD n-1 J

1

f(x)det(v0(x,t),(

(2si)n

x

++d )v°(x t)), t

xa bD 0
where

v0t x-z

+ (1-t)v(x).

Ix-zl2

Remark. In (H/L], (2.10) is called the Lerey formula. Remark. If v(z,x)=(x-a)/Ix-z12, then Lv=BaD and Rv=O. Hence then the Cauohy-Fantappie formula (2.10) coincides with the Martinelli-Bochner formula (1.6).

2.8. Theorem (Cauchy-Fantappie formula for forms). Let DC=Cnbe a domain with almost C1 boundary, v a Leray map for D, and f a continuous (0,r)-form on b such that if is also continuous on S (0
f = L v f + 5Tvf - Tv3f.

(2.11)

In particular, then 3Tvf is continuous in D.

Remark. In [H/L], (2.11) is called the Koppelman-Leray formula. Remark. If v(z,x)=(x-a)/Ix-zI2, then Lv=BOD and Rv=O. Hence then (2.11) coincides with the Koppelman-formula (1.7).

2.9. Corollary. Let D -= Cn be a domain with almost C1 boundary, and let v be a Leray map for D such that, for some 0
(ii) v(z,x) depends holomorphically on z1"

.. 'zn-r+1'

(iii) v(z,x) depends holomorphically on Then, for each continuous (O,r)-form f on D such that 3f is also continuous on D, Lvf=O and hence the Cauchy-Fantappie formula (2.11) takes the form (-1)rf = 3Tvf - Tv5f

on D.

(2.12)

37

Proof. For r=n the assertion follows immediately from Proposition 2.3. If 0
r

v

(L f)(z) =

nrl (2ri)n

n-r-1

f(x)ndet(v(z,x),a v(z,x),k v(z,x))WW(x).

x

a

xcaD It remains to observe that, obviously, each of the conditions (ii) and (iii) implies the equation n-r-1

r

det(v(z,x),3zv(z,x),3xv(z,x)) = 0.

2.10. Remark. Let D -= Cn be a domain with C2 boundary, U a neighborhood of b, and 4:U- 4 IR a C2 function with D = {zeU: 4(z)
aq(z)

a4(z)

azl

azn

(2.13)

and

Q IRQ(z) _

ag(z)

i(z)

at1

at2n

(2.13')

ztU, where t=(t1,...,t2n):Cn--) 1R2n are the real coordinates in Cn with zj=ti(z)+itj+n(z), zcC'. Then, for all zeU and

,Cn,

Q(z),t(x)>,

2Re

(2.14)

where Re:= "real part of. Proof.

(z)

n / a9(z)

2Re = ReE

- i

j=1

n

8t1

n

ac(z)

1(t (x) + it

ate+n 1

38

atj

j

j+n

(x))

ag(z)

tj+n(x) _ .

t (x) + j=1

i

3=1 aj+n

2.11. Observation. Let D =. Cn be a convex domain with C2 boundary, U a neighborhood of D, and g:U- 4 iR a C2 function with D = {zEU: a(z)<0}.

Further, let t=(tl,...It2n):Cn--* 1R2n be the real coordinates in Cn with z3=ti (z)+itJ+n(z). Then by Remark 2.10, for any

the complex

plane

{zECn: = 0} is contained in the (real) tangent plane

{zECn: =0}

of aD at x. Since D is convex and hence all tangent planes of aD are contained in Cn\D, it follows that

# 0

for all xec3D and zeD,

i.e. QCQ is a Leray datum for D and the corresponding Leray map

VCQ(x) v(Q) = v(Q)(x.z):=

<0 Cg(x),x-z>

(2.15)

depends holomorphically on z. Therefore, by Corollary 2.9 (ii), for any continuous (0,r)-form f on D (l
(-1)rf = 8Tv(g)f - Tv(g) of

on D.

(2.16)

In particular, if f is a continuous 8-closed (O,r)-form on D (1
then the equation Du = f can be solved in D with

u:= (-1)rTv(Of. 2.12. Theorem (Poineare &-lemma for forms).

(2.17)

If D c Cn is a convex

domain with C2 boundary, then, for any continuous 3-closed (O,r)-form f on D, formula (2.17) gives a solution of the equation au = f in D. This solution is of class C°G in D for all O
39

Proof. This follows immediately from Observation 2.11 and Proposition 1.4 (ii).

2.13. Theorem (Poinoare 5-lemma for currents). Let Dc=Cn be a convex domain, and let T be a 5-closed current on D. Then, for any open UccD, there exists a current S on U such that

SS=T Proof. Choose a

Coo

function

on U.

such that

1 in neighborhood of U

and supp x =c D. Then, by Theorem 1.12,

xT = 5(B'xT) + B'(5;0T).

Thus it remains to solve the equation Su=B'(SxAT) on U. This can be done in view of Theorem 2.12, since, by Proposition 1.11, B'(3x4T) is of class Coo in a neighborhood of U, and, without loss of generality, U is

convex and with C°O boundary. [] In the following theorem and its proof we assume some familiarity with the general theory of sheaves (long exact cohomology sequenoes, the equation H('(9)=0 (q>l) for sheaves r which admit a partition of unity). For this materiel many references are available (see, for instance: [Hirsebruch 1966,

Wells 1973, Grauert/Remmert 1977, Griffiths/Harris

1976]). If E is a holomorphie vector bundle over a complex manifold X, then by (9E we denote the sheaf of germs of local holomorphic sections of

E, i.e. (9E is the map which attaches to any open U CX the space

(9(U,E): (9E(U):= NJ(U,E) for all open U C X. 2.14. Theorem (Dolbeault isomorphism and smoothing of the Dolbeault cohomology). Let E be a holomorphic vector bundle over an n-dimensional complex manifold X. Then

if

Hr(X,(9E) = 0

r>n+l,

(2.16)

and, for each rE{0,...,n}, there exist isomorphisms

Sr :H0 r(X,E) --> Hr(X,CJE)

(0<(Y- <00)

(2.19)

and

Sour'Hcr(X,E)

40

> Hr(X,l7E

(2.20)

such that, for all 0
HMr(X,E)

Hr(X,bE)

(2.21)

1<>

Hou=(X,K)

is commutative. In particular, for all O
HO,rM E )

<> a HO, rM E ) our

induced by the assignment Z r(X,E) a f

(

2 2 2) .

<-_> CZ r(X,E) (cf. Sect.

0.13) is an isomorphism.

Remark. The isomorphism (2.21) is called the Dolbeault isomorphism, and the inverse map of the isomorphism (2.22) is refered to as "smoothing of the Dolbeault cohomology".

Proof of Theorem 2.14. We consider the following sheaves: (Z r)E (Cour)E, and (ZD r)E (0
(ZO,r)E(U) = ZO,r(U,E)I (COur)E(U) = C0,r(U,8) (Zour)E(U) = Zc r(U,E)

for all open U C X. Further, if U C X is an open set, then by Cp r(U,E) (O
Sf is also of class Coin U. The corresponding sheaves will be denoted by

(CO, r}E(U) = C- r(U,E) for all open U C X;

Notice that if f is a section of (CO r)E and x is an arbitrary Coo function, then xf is again a section of (Cp r)E. Therefore, the sheaves (Cp r)B admit a partition of unity and so, for all O
41

Hq(X.(CO r) E) = 0

if

q>1.

(2.23)

if

q>1.

(2.23')

The same is true for the sheaves (COur)E:

Hq(X.(COur)E ) = 0

Now we observe that, by Theorem 1.13 (regularity of a), the map

,E) is an isomorphism, and, for all 0
=ZQO (X,E). Hence the assertion of the theorem holds true for r=0. Let r>l. Fix 0
0 - (Z0,s-1)E

..,n, the sequence of sheaves

)

(Cp s-1 )E

) (ZQ s)E --> 0

is exact. Let

s0 _O(I

0-*ZO

SO oa,s )H1(X,(Z°t 0,s-1 )E)

bq-1 oc,s

Hq(X,(Zoa O's-1 E)

a ZOG S(X,E)

oz's=,

l -)H1(,(C018-1 )E)

)H1(X,(Zo` )E)

)Hq(X,(zm 018-1 E)

oc,s

O's

Hq(X,(Zoc

O's

qoc,s

E )

)

be the corresponding long exact cohomology sequence. Then, by (2.23), it follows that, for s=1,...,n, the coboundary maps

Hq+1(X,(ZO,s-1)E) boa,e'Hq(X,(ZO<'s)E

(q>1)

(2.24)

are isomorphisms, and the coboundary map S s induces an isomorphism

42

SO

(X

s:H0' (,E) --i H1(X,(ZO a-1)E) 04.

If r>n+l, then,

(1<s
(2.25)

in view of the isomorphisms (2.24), we see that the

spaces Hr(X,(ZO 0)E) and Hr-n(X,(Z n)E) are isomorphic, Since (Zp n)E =(C0 n)E and, by Theorem 1.13 (regularity of 5),

0,0 ) E _ (7 E, (Zo.

this implies that, for r>n+l, Hr(X,oE) is isomorphic to

Hr-n(X,(CD n)E).

By (2.23), this yields (2.18).

For the remainder of this proof we may assume that 1
br o,.

= 6r-l o oG, 1

081

a0

oc, r-l° oc, r

Since the maps (2.24) and (2.25) are isomorphisms and (2.26) holds true, in this way we obtain an isomorphism

Sr ,O,r(X,E)

> Hr(X,O E

Now, analogously, we construct the isomorphism Scur First observe that, by Theorem 2.13 (Poincare 8-lemma for currents), for s=l,...,n, the sequence

our E 3 our E O -i (ZOcur s) E _.> (CO,s-1) -> (ZO's )

-> 0

is exact. Let

Sour,."Hq(X,(Zcus)E) - Hq+1(X,(ZOug-1)E

(q>0)

(2.27)

be the ooboundary homomorphisms from the corresponding long exact cohomology sequence. Then it follows from (2.23') that (2.27) is an isomorphism if q>1, and the maps Seur,s induce isomorphisms

O,s(X E) ^

cur"H cur

1 our E H (X'(ZO,s-1)

(1
(2.28)

)

Hence, the composition

43

Sr-1 0 ..051 050 our,1 cur,r-1 our,r is an isomorphism from HO,r,(X,X) onto Hr(X,(ZO 0)S). Since, by Theorem

1.13 (regularity of 3), also

8

(Zcur)B

<>

0,0

CJ

is an isomorphism, it follows that

Sour

<>

-1o Sr-1

our,lo..

O S1 o cur,r-10bcur,r

is an isomorphism from HO,r,(X,X) onto Hr(X,j9K). To complete the proof,

it remains to observe that, by definition of the ooboundary operators in the long exact oohomology sequences of short exact sheaf sequences, obviously, each of the diagrams

Sq

Hq(X,(Z' 0)X)

oa,s

<>

Hq(X,(Zour)E)

O,a

Hq+1(X,(ZO,s-1)$)

I <> 9 bour,a

(q>0,1<s
Hq+l(X,(ZO,sour-1 )X)

is commutative and hence (2.21) is commutative. 0 The preceding theorem contains, in particular, the following two statements: 1) If f is a Co- form (O can be solved with a current T. then we can solve 3u=f even with a CO'- form u (injectivity of (2.22)). 2) For any 3-closed current T, there exists a current S such that T-3S is of class Coo (surjeotivity of (2.22) for ow-,

=oo). Actually, we have the following more precise

2.15. Corollary. Let H be a holomorphic vector bundle over a complex manifold X. Then:

(i) For any X-valued 3-closed current T on X and every neighborhood U of supp T. there exists an X-valued current S on X such that T-3S is of class C°O on X and supp S C U. (ii) If f is an X-valued Ck form on X (k=O,1,...,oo) and T is an 8-valued current on X such that 8T = on X, then, for any neighborhood U of supp T, there exists a differential form

44

C*+° (X,&)

u G I

I

O
such that 5u = f on X and supp u C U.

Proof of part (i). Choose a neighborhood V of upp T such that V C U. Further, take C°O functions x0 and xl such that XO=1 in a neighborhood of X\V, xo=O in a neighborhood of supp T, X1=1 in a neighborhood of X\U, and

X1=0 in a neigborhood of V.

Since T is 5-closed and (2.22) is surjective, there exists an E-valued current S0 on X and a 5-closed Coo form f0 on X such that T-5S0= on X.

Then =-5s0 on X \ supp T. Since (2.22) is injective for oG=oo, this implies that there exists an E-valued C°0 form g0 on X \ supp T with f0=5g0 on X \ supp T. Over X\V, then

5(SO+<

g0>) = <-f0+5g0> = 0.

Since (2.22) is suriective, this implies that there is an E-valued current R over X\V and a 5-closed E-valued C°° form gl on X\V such that

S0+<0o>-5R= on X\V. Setting S=SO+<X0g0 x181>-5(X1R) we conclude the proof of part (i).

Proof of part (ii). Denote by 7 the sheaf on X which is defined as follows: If VCX is open, then T(V) is the space of all forms

C*+OG(V,E)

g f

0
such that also

5g E

n C*+O°(V,E).

0<0: <1 We remark that the sections of 3" admit multiplication by Coo functions.

Hence

H1(X,) = 0. Take open coverings {U

(2.29)

{Vi} Vi cc Wj, and {Wj} of X such that Uj each of the sets V. and Wj is biholomorphically equivalent to the unit ball in Cn. Then, by Theorem 2.12 (Poincare b-lemma for forms), we can find a family ,

J),

45

uj6

n C*+oo(VE)

0
On the other hand, since (2.22) is injective (for such that auj=f on V o,,=k) and 5T=, we can find an E-valued Ck-form g on X such that 5 dg=f on X. Set wi=uj-g on V3. Then wj(Zk(Vj,E) and, again by Theorem 2.12, we can solve the equations wi=avj with

vje I

Ck+ oL

(Uj,E).

1

O
Then vi-vjET(UinUj) and hence, by (2.29), we can find a family gi( -r(Ui)

with vi-vj=gi-gjon Uinuj. Setting h=ui-;gi on Ui, so we obtain a form

h(

n< 00l

Ck+OG(X,E) *

with ah=f on X. Then T- is 5-closed on X and since (2.22) is surjective for oc=oo, we can find an

E-valued current S on X as well as an E-

valued Coo form u00 on X such that T-+aS= on X \ supp T. Since h+uoo is of class Ck, repeating the arguments from above, we obtain a form

H (

f

Ck+oc(X \ supp T

,

E)

0
Coo

with &i=h+uoo on X \ supp T. Choose a

function x on X with x = 0 in

a neighborhood of supp T and x = I in a neighborhood of X\U. Then the form u:=h+uoo-a(x H) has the required properties.

3. Piecewise Cauchy-Fantappie formulas

3.0.Further on, for every collection K=(kl,...Ik1) of integers, we write IK):=l and K(s"):=(kl,...,ks-1'k8+1

"

-k

)

(1<s<1).

1

3.1. Domains with piecewise almost C1 boundary and frames for such

domains. Let D Q Cn be a domain. We say aD is piecewise almost C if there exist a neighborhood U of 3D and a finite number of closed almost C1 hypersurfaces Y1,...,YN in U (of. Sect. 1.1) such that the following conditions are fulfilled:

46

N

aD =

(i)

Yjn8D;

U

J=1

(ii) for all 1
(iii) for all 1
is 1

transversal (or empty).

If Dc=Cn is a domain with piecewise almost C1 boundary, then any collection Y=(Y1,...,YN) of closed almost C1 hypersurfaces in a neighborhood of 8D satisfying conditions (i)-(iii) will be called a frame for

D. Notice that, for each domain DcCn with pieowise almost C1 boundary, there exist different frames, in particular, aD is also a frame for D.

3.2. The manifolds SK. Let Dc=Cn be a domain with piecewise almost Cl boundary, and let Y=(Y1,...,YN) be a frame for D. Then we set Si = Sj(Y) = aD n Yi for j=1,...,N.

Further, for each ordered collection K=(k1,...,k1) of integers 1
Sk n...nsk SK = SK (Y)

=

1

if ki,...,kI are different in pairs 1

0

otherwise.

Then we have (as relations between sets) the equations N

aD =

Sj

(3.1)

j=1

and

(3.2)

where Kj:=(k1,...,k1,j). We orient the manifolds SK so that (3.1) and (3.2) hold true as relations between oriented manifolds.

3.3. The manifolds SKXAK and SKX&OK. Let D Q Cn be a domanin with piecewise almost C1 boundary, Y=(Y1,...III N) a frame for D, and SK=SK(Y).

By n we denote the simplex of all points (t0,...ItN)c RN+1 with t0,..,tN>O and 1. We orient n by dt1 ... dtN. N t

47

For every strictly increasing collection K=(k1,...,k1) of integers O
tk = 1}.

QK = s=1

We orient the manifolds QK so that

(-1) s+1 QK(s")

1

aQK =

s=1

(3.3)

Then, for any strictly increasing collection K=(k1,...,k1) of integers

1
defined by the form dtk A...ndtk

.

1

1

Q3 (j=O,...,N) are oriented by +1.

3.4. Leray maps and Leray data

.

Let DcCn be a domain with piecewise

almost C1 boundary, Y=(Y1,...,YN) a frame for D. and SK=SK(Y). A Leray map for (D,Y) (or for D) is, by definition, a map v which attaches to each strictly increasing collection K=(kl,...,k1) of integers 1
xISK,and tQQK such that, for any collection K=(k1,...,k1) of strictly increasing integers 1 = 1

(3.4)

and

vK(e..)(s,x,t) = vK(a,x,t)

if tcQK(s°) (1<s<1).

(3.5)

A Leray datum for (D,Y) (or for D) is, by definition, a family

W={WiI=1 of Cn-valued C3 maps

wj = Wj(S,X)

defined for all zcD and x653 such that

<wj(a;x),x-z)= 0

48

for all z 4D and xeSj (3=1,...,N).

(3.6)

If w={wi) is a Leray datum for (D,Y), then we define a Leray map for (D,Y) by setting

wj (a, X)

vK(z,x,t):=

(3.7) 3

<wi (z,x),x-z>

for any strictly increasing collection K=(kl,...,k1) of integers 1
3.5. The Cauchy-Fantappie integrals L. Let D - Cn be a domain with piecewise almost C1 boundary, Y=(Y1....,YN) a frame for D, and v a Leray map for (D,Y). Fix some strictly ino_reasing collections K=(k1,...,k1) of integers 1
We set n-i

LK = LK(z,x,t) =

1

(2ni)ndet(vK(z,x,t),((5z x+dt)vK(z,X, t))AW(X)

(3.8)

for zeD, xeSK,and t" 6K' For any bounded differential form f on SK, by

f(x)ALK(z,x,t),

(LKf)(z)

zcD,

(x, t )E SKxLK

we define a continuous differential form LKf in D.

v

By (IYC)r = (L)r(z'x,t) (O
r

v

(LK)r -

lnrl)

n-r-1

n det(vK,azvK,(c)x+dt)

Kv

(3.9)

(2ni)

for r=0,...,n-1.

3.6. Proposition. Let D - Cn be a domain with piecewise almost CI boundary, Y=(Yl,...,YN) a frame for D, v a Leray map for (D,Y), and K=(kl,...,k1) a strictly increasing collection of integers 1
49

LKf

=

if O
J

(3.10)

(x,t)eSe AK

and LKf = 0 if n-IKI.+1
Proof. Repetition of the proof of Proposition 1.5. O

3.7. The Cauchy-Fentappie integrals RK, Let D - Cn be a domain with piecewise almost C1 boundary, Y=(Y1,...,YN) a frame for D, v a Lerey map for (D,Y), and K=(k1,...,k1) a collection of strictly increasing integers 1
OK'

we set

vOK(z x t) = t0 x z 2 + (1-tO)vK(z,x,t'),

(3.11)

IX-21

where t':= (0,t1/(1-t0),...,tN/(1-t0)). We remark that if vK is of the form (3.7), then

w.(z,x)

x-z

(3.11')

vOK(z,x,t) = O1x-z12 + j ,K

n-1

RK =

(_l) RK(z,xt) _

IKI {21fi)R

det(vOK(z,x,t),(3z,x+d t)vOK(z,x,t))AJ(x)

for all zED, x&SK and tC n

(3.12)

OK'

Now, for any bounded differential form f on SK, by setting

f(x)ARK(z,x,t),

(RKf)(z) =

zcD,

(3.13)

J

(x, t) E SKxLOK

we define a continuous differential form RRf in D.

By ('i)r-(RK)r(z,x,t) (O
50

(n 1

r

n-r-1

r (2ffi)ndet(vOK,azvOK,(3x+dt)vOK)n(J(x)

(RK)r = (-1)11(1

(3.14)

for r=0,...,n-1.

3.8. Proposition. Let Dc-Cn be a domain with piecewise almost C1 boundary, Y=(Y1,....YN) a frame for D, v a Leray map for (D,Y), and K=(kl,...,kI) a strictly increasing collection of integers l
RKP =

f(x)n(RK)r-1( ,x,t)

1

if 1
(3.15)

(x,t)EJJSKxAOK

and RKf = 0 if r=0 or n-IKI+l
3.9. Proposition. Let D

c C° be a domain with piecewise almost C1

boundary, Y=(Y1,...,YN) a frame for D, v a Leray map for (D,Y), and K=(kl,...,kl) a strictly increasing collection of integers 1
(i) For all zeD, xcSK, and teQK, we have

(bz+dx t)Lv = 0

and hence

dx,t(LK)O(z,x,t)

dx,t(-K)r(z,x,t)

= 0,

-oz(LK)r_1(z,x,t)

if 1
z(LK)n-l(z,x,t) = 0.

(ii) For all zeD, xESK,and

we have

(8z+dx,t)Rv = 0 and hence

51

dx,t(RK)O(z,x,t) = 0, if 1
- -5z(RK)r-1(2 xt)

dx,t(RK)r(z'x't)

z(RK)n-1(z,x,t) = 0.

Proof. The proofs of parts (i) and (ii) are similar. We restrict ourselves to part (ii). By condition (3.4) in the definition of a Leray map and by (3.11), we have the relation n

Z-

1.

j=1

Therefore n

(xj-zj)(3z,x+dt) OK(z,x,t) = 0 j=1

and hence n-1

(az+dx t)RK = (oz x+dt)

1

(2)ri )

-

det(vOK,(a

n

[ A ( +d )v3 ] (2 i)n j=1 z,x t OK 1

x+dt)vOK)Atu(x)

W(x) = 0.

3.10. The operators Lv and Tv. Let D - Cn be a domain with piecewise almost C1 boundary,

a frame for D, and v a Leray map for

(D,Y). Then, for any continuous differential form f on D, we set

Lvf = ='LKf

Tvf = BDf +

and

RKf,

K

where the summations are over all strictly increasing collections K=(kl,...,k1) of integers 1
3.11. Remark. If f is a continuous differential form of bidegree (0,r) on D, then by Propositions 1.5, 3.6 and 3.8, Lvf is also of bidegree (O,r), T v f is of bidegree (O,r-1), and we have the following relations:

Lvf =

LKf 1
52

for all 0
L

vf

=0

if r=n,

RKf

TVF = BDf +

for all 0
l
vf

= BDf

T

vf

=0

where the summations are over all strictly increasing collections K=(kl,...,k1) of integers 1
3.12. Theorem (Piecewise Cauchy-Fantappie formula). Let D n Cn be a domain with piecewise almost C1 boundary, Y=(Y1,...,YN) a frame for D, and v a Leray map for (D,Y). Then, for any continuous (O,r)-form f on b such that bf is also continuous on D (0
(-1)rf = L v f + bTvf - Tvaf

in D.

(3.17)

In particular, then aTvf is continuous in D.

Remark. For r=n, formula (3.17) coincides with the Koppelman formula (1.7), because then, by Proposition 1.5 and Remark 3.11, v L f = BaDf = 0

and

T v f = BDf.

For r=0, i.e. if f is a function, in view of Remark 3.11, formula (3.17) takes the form

f =Lvf - Tvbf

in D.

(3.18)

Proof of Theorem 3.12. Let P be the set of all collections K=(k1,...AI) of strictly increasing integers 1
a(SKXQOK) _ (-1) IKI(SK)((K)

+

s=1

1)IKI+s+16KXAOK(s") + --SKjXOOK J=1

and, by Proposition 3.9, for all KFP and

dz,x,t[f(x)ARK(z,x,t)n v(z)l = bf(x)ARKv(z,x,t)nv(z)

53

+ (-1)r+lf(x)A

RK(z,x,t)Aav(z),

it follows from Stokes' formula that

j(NV.3f)AV + (_l)r+1J(RVf).5v = (-1)IKI K D

f

f(x)ARK(z,x,t)nv(z)

DxJSKxQK

D

IKI

+ (-1)IKI+s+i s=1

ff(x)ARR(zxt)nv(z) J

DxSKx Z\ OK(s - ) N

ff(x)ARR(z,x,t)Av(z)

+

j=1

for all ve(Coo nn_r(D)) 0 and REP. Since

(-l)IKIRK(z,x,t)

= LK(z,x,t)

if tepK

and since

J(RKf)AZ)V = (-1)rJ(& K)AV, D

D

this implies that

KUI

(RKbP - aRKf - LKP)(z)

(-1)IKI+s+1

If(x)ARK(z,x,t)

s=1

SKXns OK(s - ) N

Jf(x)A1 (z,x,t),

+

(3.19)

j=1

SKj xLOK

for all KeP and zED. Now we observe that, for all 1
54

n-1

I 1

(2ai)

ndet(

x

Ix-zl

n-i

x z 2l J(x) _

Ix-zl

1

det(x-z,dx-dz) A

(21i)°

J

Ix-z1

2n

LO(x)

= - B(z,x).

Therefore n

(`

I-

Jf(x)ARK(z,x,t) = -(BaDf)(z),

k=1

SkXAO and hence

ii

IKI

j`

Y-(-I)

Jf(x)nRK(z,x,t)

KeP s=1

SKxnj OK(s- )

(-1)IKI+e+1

-(BaDf)(a) + KeP

Jf(x)Ac(sx.t)

s=1

SKX'd

IKI>2

OK(.-)

for all zED. Thus, after summation over all KEP with 1
( L RKaf `

KeP I
-

a 6 RKf -

+ BaDf}(z)

L ;

cP 1
E

I
1

K (-1)IKI+s+1

e

2
SKX DOK(s")

N +

eP I
ff(x)R(z.x.t)

s=1

j=1

/

If(x)nRK(a,x,t)

(3.20)

J J SKj'A K

for all z D.

In view of Remark 3.11 and the Koppelman formula (1.7), the left hand side of (3.20) is equal to

55

[BaDf + SBDf - BD3f -LVf - STVf + TV3f](z)

_ [(-I) rf - LVf - 3TVf + Tv3fl(z).

Therefore, we have to prove that the right hand side of (3.20) vanishes.

First observe that

N

f(x)ARK(z,x,t) = 0, KeP

zeD,

(3.21)

j=1

IKI=n-r

SKjXAOK

because, for IKI=n-r, dim.

SKj = n+r-1, whereas all monomials in the

form f(x)nRK(z,x,t) are of degree >n+r in x. Further, we notice that, by condition (3.5) in the definition of a Leray map, for all KEP with IKI>2 and any 1<s
RK(z,x,t) = RK(s")(z,x,t)

if tCAOK(s")

(3.22)

In view of (3.21) and (3.22), the right hand side of (3.20) is equal to

KI

eP

(-1)IKI+s+1 ff(x)^__K(s")(z,x,t) 1

s=1

J

2
SKXL OK(s") N

Jf(x)AR(zxt).

+

K P j=1 1
(3.23)

SKjxIOK

In order to show that (3.23) vanishes, for any K=(k1,...,k1)EP, we set-

and

j(K,s) = ks (1<s
= (-1) IKI+s SK(s")j(K,s) for all 1<s<1, and, by definition (cf. K Sect. 3.2), we have SKj = 0 if j4c(K). Therefore, (3.23) is equal to

Then S

n-r f(x)^RR(s =2 56

KFP

IKI=l

-K(s")j(K,s)'(A0K(s")

)(z,x,t)

n-r +

1-

=

KEP

IKI=1-1 jEc(K)

{`

f(x)ARK(z,x,t) J

SKj'
To complete the proof it remains to observe that, for any 2<1
{(K,j): KEP,

IKI=1-1, jEc(K)}

and {(K(s"),j(K,s)): KEP,

IKI=1, 1<s<1}

are equal. U

3.13. Corollary. Let D == Cn be a domain with piecewise almost C1

boundary, Y=(Y1...,YN) a frame for D, v a Leray map for (D,Y), and O
IKI>n-r+1;

(ii) r>1 and vK(z,x,t) depends holomorphically on zl" .. '$n-r+l'

(iii) vK(z,x,t) depends holomorphically on x1,...,xr+IKI+1' Than, for each continuous (O,r)-form f on D,

LKf =0. In particular, if for each collection K=(kl,...,k1) of strictly increasing integers 1
(-1)rf = 5Tvf - Tvbf

in D.

(3.24)

Proof. By Proposition 3.6, LKf=0 if IKI>n-r+l. Now let IKI
57

f(x)A(LK)r(z,x,t),

LKf(z) =

J (x, t) f S In view of (3.9) and since dim this implies that

1R

SK = 2n - IKI and dim 9Z AK = IKI - 1,

fn-11 rn-r-11 LKf(z) = l r JIKI-1/

(27ri) n

f

f(x)AM(z,x,t)A W(x)

J

(x,t)ESKXQK

for all zeD, where

r

n-IKI-r

IKI-1

M(z,x,t) = det[vK(z,x,t),3zvK(z,x,t),axvK(z,x,t),dtvK(z,x,t)].

It remains to observe that, obviously, each of the conditions (ii) and (iii) implies that M(z,x,t)=O. Q

58

ca-CONVEX AN D

CHAPTER I I .

ca-CON CAVE MANIFOLDS

Summary. In this chapter we introduce the concepts of q-convex and q-concave manifolds and prove some elementary properties of them.

4. q-convex functions

4.1. The Levi form.

If M is a real C1 manifold and xiM, then by

TR x(M) we denote the (real) tangent space of M at x, and by TC,x(M) the complexified tangent space of M at x.

Now let X be an n-dimensional complex manifold. For xEX, by TX(X) we denote the subspace of TC x(X) which consists of all holomorphic tangent vectors: if zl,...,zn are holomorphic coordinates in a neighborhood of x, then TX(X) is spanned by a/azl,...,a/azn If Y is a real C1 submanifold of X and xEY, then we set TX(Y) = TX(X)nTj x(Y).

TX(Y) is called the holomorphic tangent space of Y at x. Let Q be a real-valued C2 function on X and xIX. Then we define an Hermitian form L (x) on TX(X) as follows: Choose holomorphic coordinates 211.1.1zn in a neighborhood of x and set

n

LQ(x)t ),

a2s(x)

tt =1 dzdazk i k

if t=

n =1

a

t .---(x). az j

Since

n

2

awravrs

i

(6-J) l

a24

ask

dwr/ az azk awa

i

59

if w1,..., n are other holomorphic coordinates, this definition is independent of the choice of holomorphic coordinates. The Hermitian form

L(x) is called the Levi form of 4 at X.

If Y is a real C1 submanifold of X and xEY, then by L1(x) we denote the restriction of L9(x) to TX(Y).

4.2. The Levi numbers p

of a function. Let H be a Hermitian form. If H

is the corresponding Hermitian matrix with respect to some linear coordinates, then by Sylvester's theorem on inertia the number p+(H) of positive eigenvalues of H and the number p_(i) of negative eigenvalues of H are independent of the choice of the coordinates. Put p (H)=p (H). Let X be a complex manifold and g a real-valued C2 function on X. Then, for all xEX, we set

pg(x) = P (LQ(x)).

If Y is a real C1 submanifold of X and

then we set

PQ}Y(x) = p (LQ)Y(x)).

4.3. Definition. Let X be an n-dimensional complex manifold, and 1
PQ(x) > q

for all xEX.

n-convex functions usually are called strictly plurisubharmonic (for such functions see, for instance, Section 1.4 in [H/L]). Remarks to this definition.

1. Clearly, each q-convex function is r-convex for all i
80

Then:

(i) Q is q-convex in a neighborhood of x if and only if there. exists

a q-dimensional complex submanifold Y of some neighborhood of x such that xcY and the restriction of q to Y is strictly plurisubharmonic. (ii) If de(x)t0 and a is q-convex in a neighborhood of x, then the submanifold Y in part (i) can be chosen so that, moreover, Proof. First suppose that a is q-oonvex in a neighborhood of x. Then we can choose a q-dimensional subspace T of TA(X) such that L9(x)IT is

positive definite and, moreover (after a small variation), dg(x)IT0, It is clear that then each complex submanifold Y with xEY and TX(Y)=T has the required properties. On the other hand, if Y is a q-dimensional complex submanifold such that xiY and 9IY is strictly plurisubharmonic, then po(x)>P+IY(x)=q. fl 4.5. Corollary. 1-convex functions do not have local maxima.

Proof. By Proposition 4.4 (i), this follows from the maximum prin-

ciple for strictly subharmonic functions of one complex variable. []

4.8. Theorem. Let X be an n-dimensional complex manifold and 9 a real-valued C2 function on X. Let 1
h(U)n{ZECn: 2q+1=...=2n=0}

is strictly convex (in the usual linear sense, i.e. the Hessian matrix with respect to the real and imaginary parts of z1,.,.,zq is. strictly positive definite).

Proof. In view of Proposition 4.4, we may assume that q=n. In this case the assertion follows from Theorem 1.4.14 and Proposition 1.4.8 (v)

in [H/L]. Cf 4.7. The orientation of complex manifolds. If X is a complex manifold, then we assume that X is oriented so that if zl,.... zn are local holo-

morphic coordinates on X and x1,...,x2n are the real coordinates with 0 1=x3+ixJ+n, then dx1A...ndx2n is positive.

4.8. Defining functions for oriented hypersurfaces. Let X be an n-dimensional complex manifold and M an oriented real C hypersurface (i_e. an oriented C1 submanifold of real dimension 2n-1). Then is a defining function for X if $ is a real-valued C1 function neighborhood U of M such that the following two conditions are

in X we say q in a fulfilled

61

(i) Q(x)=0 and dg(x)#0 for all xcM;

(ii) the orientation of M coincides with the orientation which is induced on M as a part of the boundary of the domain (xcU: Q(x)<0).

4.9. Lemma. Let X be an n-dimensional complex manifold and M an oriented real real C2hypersurfaoe in X. Suppose both e and (Q are defining functions for M. Then

peIM(x) = p 01M(x)

Moreover, then there exists a positive C1 function 4,

V of M such that (Q = u

(4.1)

for all xeM.

in a neighborhood

in V and

L ef)M(x) = q (x)LRIM(x)

for all xcM.

(4.2)

Proof. It is easy to see that there is a C1 function .j-> 0 in some

neighborhood V of M with

(Q = u(Q on V. Let x<M. Since lp(x)=4(x)=0 and

(P = i4-Q, then, for all tCTX(X), L(n(x)t = Ly (x) L(x) +

t(Lr )t7.

This implies (4.2), because t(4)=0 if ttTz(M). It is clear that (4.1)

follows from (4.2). Q 4.10. The Levi numbers p

of an oriented hypersurface. If M is an

oriented C2 hypersurface of a complex manifold, then we define

pM(x) = pgiM(x)

for all xeM,

where Q is an (arbitrary) defining C2 function for M. (By Lemma 4.9, this definition is correct.)

Convention. If D C X is a domain, then by aD we denote the oriented boundary of D. More precisely, then we assume that the C1 part (61)) 1

of

aD is oriented so that the following condition is fulfilled: if Q is a

defining function for (aD)1, then Q(x)<0 as Dax -,> (aD)1. Accordingly we use the notations paD(x) and po(x) if x is a C2 point of D.

4.11. Lemma. -Suppose that the following two conditions are fulfilled: (a) G C 1Rk is a domain and u is a real-valued C2 function on G which is convex;

62

(b) X is an n-dimensional complex manifold and

9 _ (91,...,Qk):X

) G

is a C2 map such that the functions

-V2,...,Qk are stictly plurisub-

harmonic on X and, for some 1
min 1
au x)

> 0

and

max l
axi

au (x)

> 0,

a"i

then uo9 is q-convex on X. (ii) If, for all xLG,

MAX au (x) 1
then

< 0

min au x l
and

< 0,

-uo(-g) is q-convex on X.

Proof. In view of Proposition 4.4 we may assume that q=n. In this case, assertion (i) of the lemma coincides with Theorem 1.4.12 (ii) in (H/L]. Assertion (ii) follows from (i).

4.12. Definition. For any t>0, we fix some positive Coo function

/p(x) _

on

^Y

1R such that, for all x(IIt,

(-x),

Ixl < xp(x) < Ixl +'8'

< 1,

dx

d'y

(x) dx

yo(x) = Ixl

if

(xl4/2.

and we set

maxp(ti,t2) =

y

2)

for tl,t2i fit. 63

4.13_ Lemma. For all

,>0, maxi is a convex C°O function on1R2 and,

moreover, for all (tilt 2)E 1R2, the following relations hold true:

a

(4.3)

maxx(tl,t2) > 0, min i=1,2 ati a

maxp(tl,t2) > 0,

(4.4)

maxp(tl,t2) = maxp(t2,t1),

(4.5)

max(ti,t2) < maxp(tl,t2) < max(tl,t2) + P.

(4.6)

maxp(t1,t2) = max(t1,t2)

(4.7)

.Max

i=1,2 ati

if It1-t21 > p.

Proof. Since the Hessian matrix of maxp is

1

t1-t2\

a

4 xp

2

1

-1

(_ 1

and since 1io >0, we see that maxp is convex. Since the first-order derivatives of max,. are

1

1 t2

2xp C---'and since

I

j

< 1, we obtain the estimates (4.3) and (4.4). Relations

(4.5)-(4.7) immediately follow from the correspoonding properties of the function

0

4.14. Corollary. Let X be an n-dimensional complex manifold, g a strictly plurisubharmonic Ck function on X (2
Proof. This follows from Lemmas 4.11 and 4.13. 0 Finally, let us notice the following obvious but important 4.15. Observation. Let X be an n-dimensional complex manifold, S> a q-convex function on X (lj0, the function Rj is q-convex on D.

64

5, q-convex manifolds

5.1. Definition. Let X be an n-dimensional complex manifold.

A C2 function Q:X

)iR is called an exhausting function for X if,

there is a compact set K C X such that, for all xEX\K, 9(x) ( sup q(z), zEX

and, for all real numbers eL

< :up sup (z),

the set {zEX: Q(z)<x.} is relatively compact in X.

An exhausting function for X will be called bounded or unbounded according as supX 9 < oo or supX 9 = oo.

If Q is an exhausting function for X. then we say 4 is g-convex at

infinity if there exists a compact set K - X such that Q is q-convex on X\K (1
We say X is completely q-convex if there exists an exhausting function for X which is (q+l)-convex everywhere on X (O
(n-l)-convex (resp. completely (n-1)-convex) complex manifolds will be called also pseudoconvex (resp. completely pseudoconvex). Remarks to this definition.

I. Recall that a complex manifold is completely pseudoconvex if and only if it is Stein (cf., for instance, Sect. 2.13 in [H/L]). II. It is clear that every compact complex manifold is pseudoconvex. On the other hand, no compact complex manifold is completely 0-convex,

because 1-convex functions do not have a maximum (Corollary 4.5). III. If Q is a bounded exhausting function for a complex manifold X, and c:= supX g, then := -ln(c-Q) is an unbounded exhausting function

for X. Therefore, it follows from Lemma 4.11 (i) that if X admits a bounded exhausting function which is q-convex at infinity (reap. everywhere on X), then X admits also an unbounded exhausting function which is q-convex at infinity (resp. everywhere on X). Notice that the opposite is not true. Example; The complex plane. IV. In the literature, it is also used to call a complex manifold q-convex if it is (n-q)-convex in the sense of Definition 5.1 (cf. Remark II following Definition 4.3).

65

5.2. Lemma (Morse's lemma). Every q-convex complex manifold X admits an unbounded exhausting Coo function without degenerate critical points which is (q+l)-convex at infinity. If X is even completely q-convex, then this function can be chosen to be (q+l)-convex everywhere on X. Proof. This follows from Observation 4.15, Proposition 0.5 in Appendix B. and Remark III following Definition 5.1. 0

5.3. Theorem. If X C M is a q-convex domain in a complex manifold M (0 < q < dime M - 1), then, for any pseudoconvex domain Y C M, the intersection XtY is q-convex; and for any completely pseudoconvex domain Y C M, the intersection XnY is completely q-convex.

In particular, every q-convex domain in a completely pseudoconvex manifold is completely q-convex.

Proof. As observed in Remark III following Definition 5.1, we can find an unbounded exhausting function (qX for X which is (q+l)-convex at infinity, and we can find an unbounded exhausting function Yy for Y which is strictly plurisubharmonic at infinity. Then, by Corollary 4.14, maxI((p , Yy) is an exhausting function for Xr1Y which is (q+l)-convex at infinity. If Y is completely pseudoconvex, then (py can be chosen to be strictly plurisubharmonic everywhere on Y, and, for any sufficiently large C
q-convex everywhere on XnY. Q

5.4. Definition. Let D - X be a non-empty domain in an n-dimensional complex manifold, and O
(ii) We may D is completely strictly q-convex if there exists a (q+1)-convex function Q in a neighborhood U$ of b such that

D = {zcU,5: q(z)
(iii) D will be called non-degenerate strictly q-convex (resp. non-degenerate completely strictly q-convex) if the function q in (i) (reap. (ii)) can be chosen without degenerate critical points (for the notions of a degenerate and a non-degenrate critical point, see Sect. 0 in Appendix B).

(iv) If q--n-1, then in (i)-(iii) we say also "pseudoconvex" instead of "q-convex".

88

Remarks to this definition.

1. The case aD = 0 is admitted. Thus, every compact complex manifold is non-degenerate strictly pseudoconvex in the sense of this definition. II.

It is clear that any strictly g-convex domain is q-convex (in the

sense of Definition 5.1). III. Strictly q-convex domains need not be smooth.

5.5. Proposition. Let X be a complex manifold and 9 a 1-convex function on X.

If the boundary of the domain D:= {zEX: P(z)<0} is of class C2,

then dp(y) # 0 for all yedD.

Proof. Repetition of the proof of Proposition 1.5.16 in [H/L]. 0

5.6. Corollary. Let X be an n-dimensional complex manifold, and D cc X a strictly q-convex domain (0
Proof. This follows immediately from Proposition 5.5. 0

5.7. Theorem. Let D == X be a strictly q-convex domain in an n-dimensional complex manifold X (0
(i) For each compact set K == D, there exists a strictly q-convex domain D' _= X with C°O boundary such that K == D' _= D. (ii) For every neighborhood U of D, there exists a strictly q-convex

domain D' _= X with C°O boundary such that b = D'

U.

Proof. (i) In view of Morse's Lemma 5.2, we can find an exhausting Coo function 4 for D without degenerate critical points and which is

(q+l)-convex at infinity. Since non-degenerate critical points are isolated (cf. Proposition 0.1 in Appendix B), there is a sufficiently large number o. < supD Y such that the set D':=(z4ED: e(z)
(ii) By part (i), it is enough to find a strictly q-convex domain D' U (the boundary of which need not be smooth) such that b - D'. To do this we take 4 and UaD as in Definition 5.4 (i) and we choose a Coo function x on X such that x = 0 on b and x > 0 on X\5. Then, by Observation 4.15, after shrinking UaD' for any sufficiently small £>0, the function _pE:= S + 6x is (q+1)-convex on U Further, it is clear that q,>0 on UaD\5. Hence, for each sufficiently small 5>0, the set 0.

D':=

9t(z)<5)

has the required property. Q

67

5.8. Lemma (cf. Sects. 4.8 and 4.10 for the notations). Let X be an n-dimensional complex manifold, M an oriented real C2 hypersurface in X, and Q a defining C2 function for M. Then, for each ycM, we can find a neighborhood Uy of y and a positive number Cy < on such that, for all

C>C, y the function eCB is (p+(y)+1)-convex on U y. Proof. Let q:=p+(y). Choose a q-dimensional subspace T of T,(M) such that L Q (y)IT is positive definite. Further, choose a (q+l)-dimensional subspace T of TY(X) with TnT,(M) = T, and set

K = {tET: Ytl1=1 and L9(y)t < 0), where 11.11

is some norm on T,(X). Then K is compact and t(q)#0 for all

ttK. Hence, we can find a number Cy
max IL (y)tj < C

tEK

>

min It(q)I2

trK

if C > C. y

Since Le Q(Y)t = 8CQ(CL9(y)t + C21t(Q)12),

then 11 eCQ(y)t >

if 0 $ tti and C > Cy_ Q

0

5.9. Theorem. Let X be an n-dimensional complex manifold, D - X a domain with C2 boundary, and 0
P D(x) > 4

for all xeaD.

Moreover, if this condition is fulfilled and e is a defining C2 function for 6D, then, for each sufficiently large positive number C,

eCQ

-1

is a defining function for BD which is (q+l)-convex in some neighborhood

of W. Proof. First suppose that D is strictly q-convex, i.e. there is a (q+l)-convex function p in a neighborhood U of aD such that DnU={x6U: (P(x)<0}. Since T'(aD) is of codimension 1 in TX(X), then

68

p+(L Y(aD(x)) > q

for all xEaD.

Since, by Corollary 5.6, d(p(x) # 0 for all xcaD and hence y is a defining function for aD, this means that p)D(x)>q for all xeaD. Now we suppose that paD(x)>q for all xEaD. If g is a defining C 2 function for 8D, then it is clear that, for every C>O, the function eC3-1 is also a defining function for al), and it follows from Lemma 5.8 that this function is (q+l)-convex in some neighborhood of aD if C is

sufficient1i

large. Cj

5.10. Corollary. Every relatively compact domain with C2 boundary in an arbitrary complex manifold is strictly 0-convex.

Proof. The assertion follows from Theorem 5.9, because the inequality

p+(x)>O is trivial. 0 5.11. Corollary. Let D cc X be a strictly q-convex domain in an n-dimensional complex manifold X (O
has the required properties. 0 5.12. Corollary. Let X be an n-dimensional complex manifold and D cc X a strictly q-convex domain (0
DuK C D' C DuU,

D\U C D",

D" C buU.

Moreover, if the boundary of D is of class Ck (2
Proof. Let 4 and UaD be as in Definition 5.4 (i), and let V - UaD be some smaller neighborhood of W. Choose a neighborhood W = U n V of K and a non-negative real-valued CO° function 1r on X such that > 0 on a h K and x = 0 on X\W. Since V is relatively compact in UaD, we can find f>0 such that p +F x and p -E x are (q+l)-convex in V (of. Observation 4.15). Then the domains

D':= Du{zeV: 'p(z) < EZ/ (z)) and

D":= (D\V) u{zeV: e(z) < - Ex (z)) 69

have the required properties.

If aD is of class Ck, then, by Corollary 5.11, we can assume that is Ck and 4(z)#0 for all ztUaD. Since V is relatively compact in UaD, ) then also d(Q + i # 0 on V if E is sufficiently small.

x

5.13. Theorem. Let D C X be a domain with Ck boundary (2
0 q.

(iii) There exist a neighborhood U of y and a (q+l)-dimensional closed complex submanifold Y of U such that yEY, the intersection YnaD is transversal, and p+ (Here aDnY is considered as a aDny (y) = q. real hypersurface of Y endowed with the orientation induced from DnY.) (iv) There exist a neighborhood U c X of y and a (q+l)-dimensional closed complex submanifold Y of some neighborhood of U such that the intersection Yn3D is transversal, and DnUAY is a strictly pseudoconvex domain with Ck boundary in Y. (v) There exist a neighborhood V c

X of y and a biholomorphic map h

onto a neighborhood h(U) of the origin in such that: i,(y)=0, the surface h(aDnU) intersects transversally the

from sc:,,e naighbor'!iood U of V C-'

suL ;pace Cq+1

{zcen: zq+2=...=zn=0},

and h(DnV)n 9+1 is a strictly convex set with Ck boundary in Cnn

(vi) There exist a neighborhood U of y and a q-dimensional closed complex submanifold Z of U such that Zn$ = {y} and

dist(x,y) = 0([dist(x,U)]1/2)

for Zex ---+ y,

(5.1)

where diet is the Euclidean distance with respect to some local holomorphic coordinates in a neighborhood of Y. Proof. It is easy to see that (v) (v)

(ii)

(iv)

(iii)

(ii) and

(vi), and it follows immediately from Lemma 5.8 that (i). So it remains to prove (i)

(v) and (vi)

Proof of (i) (v). Let U. p be as in condi'.ion (i). By Theorem 4.6, after shrinking U, we can find a biholomorphic map h from U onto a

neighborhood of the origin in Cn such that h(y)=0, the intersection of h(aDnU) with q+1 is transversal: and the restriction poh-1 ICnq+lnh(U)

70

is strictly convex. Now it is easy to find a neighborhood V == U of y such that h(VnD)fC5n +1 is strictly convex and with Ck boundary (for the

details, see, for instance, the proof of Lemma 1.5.23 (i) in [H/L}). Proof of (vi)

(ii). Without loss of generality we can assume

that X is a domain in Cn, Y=O, and Z = {zECn: zq+l=...=zn=°1.Let 4 be a defining C2 function for SD. Then it follows from condition (5.1) that

Izi2 = O(e(z))

as Z3z -* 0.

Since -9(0)=0, this implies that QIZ is strictly convex at 0. Since

strictly convex functions are strictly plurisubharmonie, it follows that p+IZ(0)=q. Since T6(Z) c TO(AD), we conclude that paD(0)>q. El

5.14. Theorem. Any strictly q-convex domain D == X in an n-dimensional completely pseudoconvex complex manifold X is completely strictly q-convex (0
Proof. Since D is strictly q-convex, we can find a real-valued C2 function If in some neighborhood UD of b with D={zEUD:(p(z)
if is (q+1)-convex in UD \ K.

If aD

is of class Ck, then, by Corollary 5.11, this function cQ can be chosen to be also of class Ck. Since X is completely pseudoconvex, we can find a function t}- which is of class Coo and strictly plurisubharmonic every-

where on X. Since D is compact, we can choose constants CO so that

max_ 6(4' (z) - C) < 0 ZED

and

min t(y' (z) - C) > max CQ(z). zED

zEK

Now it follows from Corollary 4.14 that, for sufficiently small >O, the function 9:= maxi{(y.-C),lp} is as in Definition 5.4 (ii).

5.15. Theorem. Let X be an n-dimensional complex manifold, and D =c X a domain such that, for some 0
71

real-valued C°° function x on X such that 0 < x < 1, supp

V and

x = 1 in some neighborhood of S. Set

Q = xcp +

(I - x)`{".

Then D={zcX: p(z)<0}, and 4 is (q+l)-convex in some neighborhood of S. It follows from Lemma 5,6 and Lemma 4.11 that, for a sufficiently large constant C>0, the function h:=eC2-1 is (q+l)-convex in some neighborhood U of

D. Clearly, DnU={zcU: h(z)<0}. O

5.16. Examples.

1. Let Fn be the n-dimensional complex projective space. For zcPn, by [z

0:

....zn ] we denote the homogeneous coordinates of z. Then, for all

0
D:= {(z0:...:zn]c Pn:

1z012+...+1zg12

<

Izq+112+...+1zn121

is a completely strictly q-convex domain with Coo boundary. In fact,

1Pq:= {[z0:...:zn]C1P

q+1=...zn=0}, :

if

(5.2)

then

12012+...+1zg12 1zq+11 +...+1zn12

is a (q+l)-convex function on

(5.3)

n\ Pq and D={zeIPn\ Pq: p(z)<1}.

II. If Pq is a q-dimensional projective subspace of the n-dimensional complex projective space Fn (0
(5.2) holds true, then (5.3) is an exhausting function for[Pn\1Pq which is (q+l)-convex everywhere in IPn\ Pq.

Remark. In [Barth 1970] it was proved that, for any q-dimensional \X is q-convex but not closed complex submanifold X of [Pn, the domain necessarily completely q-convex. Further, in [Barth 1970], examples are given which show that ff'n\X need not be q-convex if X is a q-dimensional

analytic set with singularities in

72

iPn.

8. q-concave functions and q-concave manifolds

B.1. Definition. Let X be an n-dimensional complex manifold.

A function Q will be called g-concave on X (l
If Q is an exhausting function for X (of. Definition 5.1), then will be called g-concave at infinity if there exists a compact set K ac X such that to is q-concave on X\K.

X will be called g-concave (0
I. Since 1-concave functions do not admit local minima (Corollary 4.5), there do not exist complex manifolds which admit exhausting functions which are 1-concave everywhere on X,

i.e. there do not exist "complete-

ly" 1-concave manifolds.

II. Every n-dimensional compact complex manifold is pseudoconoave.

III. Notice the following corollary of Lemma 4.11 (ii):

If u:(R --HER

is a concave and strictly increasing C2 function, then, for each q-concave function g, uog is q-concave." This implies that each q-concave complex manifold X admits a bounded exhausting function which is (q+l)concave at infinity.In fact, if Q is an unbounded exhausting function for 4:=-e_Q is an unbounded exhausting X which is q-convex at infinity, then function for X which is, by the corollary mentioned above, (q+l)-concave at infinity. Notice that, in general, the opposite is not true. Example: There do

not exist unbounded exhausting functions for the unit disc {zEC: IzIO, - is subharmonic in "1-£ 1; this is impossible. On the other hand, if g is a negative C2 function on the unit disc such that Q(z)=1-1/Iz12 for Iz1>1-1/2, then Q is a bounded exhausting function for the unit disc, which is 1-concave at infinity. IV. As observed at the end of Remark III, the unit disc in C is 0-concave. This implies (by the Riemann mapping theorem) that any finitely connected bounded domain in the complex plane is 0-concave. Also it is clear that the whole complex plane is 0-concave. In fact, each negative C2 function Q on C with Q(z)=-1/IzI2 for Izl>1 is an exhausting function for C, which is 1-concave at infinity. However (in distinction to the convex case - any domain in C is 0-convex), not every domain in C is 0-concave, because no complex manifold which consists of an infinite number of connected components is O-concave - this follows from the fact

73

that 1-concave functions do not admit local minima (of. Corollary 4.5). V.

In the literature, it is also used to call a complex manifold

g-concave if it is (n-g)-concave in the sense of Definition 6.1 (cf. Remark III following Definition 4.3 and Remark IV following Definition

5.1). 0 6.2. Lemma (Morse's lemma). Every q-concave complex manifold admits a bounded CO° exhausting function without degenerate critical points which is (q+l)-concave at infinity.

Proof. This follows from Observation 4.15, Proposition 0.5 in Appendix B, and Remark III following Definition 6.16.3. Theorem. Let M be an arbitrary n-dimensional complex manifold and let X,Y C M be two domains such that X is q-concave (0
Proof. As observed in Remark III following Definition 6.1, we can find exhausting functions eX and pY for X and Y, respectively, such that

QX is (q+l)-concave at infinity, gy is n-concave at infinity, and sup ex = sup QY = 0. Then, by Corollary 4.14, -maxI(- QX,- pY) is an exhausting function for XuY which is (q+l)-concave at infinity. 6.4. Definition. Let D cc X be an non-empty domain in an n-dimensional complex manifold X, and 0
(i) D will be called strictly g-concave if there exists a (q+l)-concave function a in a neighborhood U of BD such that D'U = (zcU: 9(z)<0}.

(Then, in particular, D is q-concave in the sense of Definition 6.1.) (ii) D is called non-degenerate strictly q-concave if the function Q in (i) can be chosen without degenerate critical points (for the notions of a degenerate and a non-degenerate critical point, see Sect. 0 in Appendix B).

(iii) Strictly (n-l)-concave domains will be called also strictly pseudoeoncave.

8.5. Proposition. Let D be a strictly q-concave domain in an n-dimensional complex manifold X (0
6.6. Theorem. Let X be an n-dimensional complex manifold, and D - X a strictly q-concave domain (0
74

(i) For each compact set K == D, there exists a strictly q-convex domain D'

D with Coo boundary such that K == D'.

(ii) For every neighborhood UD of D, there exists a strictly q-eon cave domain D' c UD with C°0 boundary such that D ¢ D'.

Proof. Repetition of the proof of Theorem 5.7. [l 6.7. Theorem. Let D == X be & domain with C2 boundary in an n-dimensional

complex manifold X, and 0
PdD(x) > q

for all x&dD.

Moreover, if this condition is fulfilled and y is an arbitrary defining C2 function for al), then, for any sufficiently large positive number C, the function

is a defining function for aD which is (q+l)-concave in some neighborhood of D.

Proof. Repetition of the proof of Theorem 5.9. El 6.8. Corollary. Every relatively compact domain with C2 boundary in a complex manifold is strictly 0-concave. Proof. This follows from Theorem 6.7, because the inequality paD(x)>0 is trivial. C] 6.9. Corollary. Let D cc X be a strictly q-concave domain in an n-dimensional complex manifold X (0
Proof. Repetition of the proof of Corollary 5.11, using Theorem 6.7

instead of Theorem 5.9. O 6.10. Corollary. Let D cc X be a strictly q-concave domain in an n-dimensional complex manifold X (0
DuK c D' C DuU,

D\U C D",

D" G DuU.

Moreover, if the boundary of D is of class Ck (0
75

domains D' and D" can be chosen also with Ck boundaries. Proof. Repetition of the proof of Corollary 5.12, using Corollary 6.9 instead of Corollary 5.11. Q

8.11. Theorem. Let D « X be a domain in an n-dimensional complex manifold such that, for some O
:

For each point xcaD, there exists a (q+l)-concave function

without degenerate critical points in some neighborhood U of x such that D n U = {zEU: Q(z)<0}. Then D is non-degenerate strictly q-concave. Proof. Repetition of the proof of Theorem 5.15. 6.12. Examples. I.

If D - X is a strictly q-convex domain in an n-dimensional

compact complex manifold, then X\D is strictly q-ooncave (0
Warning: If D is an arbitrary q-convex domain in a compact complex manifold X, then X\D need not be q-concave. Example: Let Y be an open subset of the Riemannian sphere IP1 which consists of infinitely many

connected components such that Y =1P1 \ (IP1 \ Y) Then D:=IP1 \ Y is 0-convex , but Y=JP 1\ D is not 0-concave (of. Remark IV following Definition 6.1).

II. With the notations from Example I in Sect. 5.18,

IPn\D =

{(z0:..

zn]EPn :

Iz0I2+...+Izq,2 > Izq+1(2+...+Izn(2}

is strictly q-concave, because D is strictly q-convex. III. Each q-dimensional projective subspaceiPg of Ipn (0
admits a basis of strictly q-concave neighborhoods, because q-ooncave (cf. Example II in Sect. 5.16).

78

jpn\pq

is

CHAPTER I I I_ THE CAU C H Y- R I EMANN

EQUATION ON q - CONVEX MANIFOLDS

Summary. In Sect. 7 a local Cauchy-Fantappie formula for non-degenerate strictly q-convex domains in n-dimensional complex manifolds is constructed, which yields local solutions of au = f0 r if r>n-q (Theorem 7.8). In Sect. 9 we prove 1/2-Holder estimates for these solutions. In Sect. 8, by means of the formula from Sect. 7, we prove a local uniform

approximation theorem for continuous a-closed (O,n-q-1)-forms given in a neighborhood of the closure of an n-dimensional non-degenerate strictly q-convex domain. Then, in Sect. 10, using the estimates from Sect. 9, we prove this result for forms given only on the closure of such a domain. In Sect.

11 we show that, via Fredholm operator theory in Banach spaces,

the local solutions with Holder estimates for the a-equation lead to the following version with uniform estimates of the Andreotti-Grauert finiteness theorem (Theorem 11.2): If D is a non-degenerate strictly q-convex domain in an n-dimensional complex manifold X, and E is a holomorphic vetor bundle over X, then dim

r>n-q, where HO42 >0(D,E):= ZO r(D,E)/Ep/2 'r

0(D,E) < oo for all )0(D,E).

1

In Sect.

12 we introduce the concept of a q-convex extension of a

complex manifold X, and prove that, with respect to such extensions, the Dolbeault cohomology classes of order r admit uniquely determined continuations if r>n-q (Theorem 12.14), where n = dim C X, and can be uniformly approximated if r=n-q-1 (Theorem 12.11 and Corollary 12.12).

Then, as a consequence, we obtain the classical Andreotti-Grauert finiteness theorem (Theorem 12.16): If E is a holomorphic vector bundle over an n-dimensional q-convex manifold X, then dim HO'r(X,E) < on for all r>n-q, where, in the completely q-convex case, even HO,r(X,E) = 0 for all r>n-q. Also in Sect. 12, we prove the following supplement to Theorem 11,2: If D is a non-degenerate completely q-oonvex domain in an n-dimensional complex manifold X. and E is a holomorphic vector bundle over X, then H1>Z > 0(D,E) = 0 for all r>n-q (Theorem 12.7).

77

7. Local solution of 3u=f0 r on strictly q-convex domains with r>n-q

Local Cauchy-Fantappie formulas with uniform estimates for non-degenerate strictly q-convex (reap, strictly q-concave) domains form an important tool in this monograph. In the construction of these formulas we use the fact that (q+l)-convex functions without degenerate critical points, with respect to appropriate local holomorphic coordinates, take an especially nice form - functions of this form will be called normalized. Before giving the definition let us recall the notion of the

7.1. Levi polynomial. Let U C Cn be an open set and Q:U -4 IR a C2 function. Then the function

n

F (z,x)

:= 2

Q

op(x)

n

ax ,i=1

a2e(x) (z - x )(z kxk)

j, k=1

j

ax

jaxk

defined for xcU and zeCn is called the Levi polynomial of Q. A direct computation shows (cf., for instance, Lemma 1.4.1 in [H/L]) that the real part of the Levi polynomial together with the Levi form (cf. Sect. 4,1) forms the second-order Taylor polynomial of Q,

n Q(z) _ g(x) - Re F(z,x) +

C

i.e.


--- (z-x)(z -x

J, k=1 axe axk

J

J

k

k

)

+ o(Ix z12)

(7.1)

for z - ) xEU (Re:= real part of). 7.2. Definition. Let U C Cn be an open set. A function Q:U -4 IR will be called normalized q-convex (10 such that

Re FQ (z,x)

> Q(x) - 9(z) +JB 1qJ ix.-z.12 - C Ix-zI2 J J J J=1

(7.2)

J=q41

for all

7.3. Lemma. Let X be an n-dimensional complex manifold, and IQ:X -i IR an arbitrary q-convex function (1
78

the canonical complex coordinates on Cn (h exists by Proposition 4.4 (i)). Then there is a neighborhood V C U of y such that 9oh-1 is normalized q-convex in h(V). 9oh-1

Proof. Since

is strictly plurisubharmonic with respect to

xl,...,xq, for every neighborhood V -r U of y, we can find oG>O with

29oh-1(x)t_

2

jtk > oG

j. k=1

axi-k

for all xEh(V) and t6Cq.

It j{

j=l

If V is sufficiently small, then, moreover, for some M
a29poh-1(x)

t jk t

axjaxk

max(j,k)>q+l

<

M

E_ It .12 It.,211/2 l iti2)l/2 + j=q+1 IJ=q+l i `J=l J

J

111

if xEh(V) and tEC°. Together this implies that

a2oh-1

(x)

tjtk > 2 L j,k=1

axj3xk

j=l

Itj2

?

+ M)

It, I2

j=q+l

for all xEh(V) and tECn. In view of (7.1), after shrinking V, the last estimate yields(7.2) with 9oh-1 and h(V) instead of _q and U .if we set

)6=o(/4 and C=4M2/oc+ 2M. E 7.4. Definition. [U,9,(p,D] will be called a g-convex configuration in Cn (O
function such that the domain D2:= relatively compact in U.

(ii) Y:U -+ R is a normalized (q+l)-convex function which has at most one critical point, and if such a critical point ysU exists, then (p(y)
79

If (U,Q,(p) fulfils conditions (i)-(iii) and the set D:={zcU: Q(z)
[U,q, y,D]. We distinguish 4 types of q-convex configurations [U,g, (p.D] in Cn: dg(z) # 0 for all zcU. 4 has a critical point ycU Type III: Q has a critical point ycU Type I:

Type II:

and 9(y)=0.

Type IV:

9 has a critical point y E U and p(y)>O.

and Q(y)
A q-convex configuration in Cr' will be called non-critical if it is

of type I, and critical otherwise. We say a q-convex configuration in Cn is smooth if it is of one of the types I,

III or IV, and non-smooth if

it is of type II.

Remark. If U, (p, D2 are as in condition (i), then the set {zcU:

dp(z)=0} consists precisely of all zeU with ',P(z) = minxU(V(x). There-

fore, then W2 = {zcU:p(z)=0} and d.p(z) # 0 for all zcaD2.

D 0

type I

type II

9=0

y

type III

type IV

FIGURE 1: q-convex configurations

7.5. Lemma. If Q:X >IR is an arbitrary (q+l)-convex function on an n-dimensional complex manifold X (O
Proof. By Lemma 7.3 we can find holomorphic coordinates h:V - Cn in some neighborhood V of y such that h(y)=O, h(V) is the open unit ball, and 9=h-1 is normalized (q+l)-convex in h(V). If dQ(y)40, then, after shrinking V. it can be assumed that moreover

80

d9(x) 4 0 if xeV and d9oh-1(z)Adlzl2 * 0 if zeh(V)\{0}. Therefore, then, for any 0
Now let dg(y)=0. Then y is a non-degenerate critical point which is not the point of a local minimum (by hypothesis) and not the point of a local maximum (by Corollary 4.5). So it follows from Proposition 0.6 in Appendix B that, after shrinking V, d9oh-1(z)Adlzl2 $ 0 for all

zeh(V)\{0} with _9.h(z) = 0. Since non-degenerate critical points are isolated (cf. Proposition 0.1 in Appendix B), after a further shrinking of V, dg(x) J 0 for all xEV\{y}. Thus, [h(V),goh-1,1212-r2] defines a q-convex configuration of type II if 0
7.6. The set Div(Q). Let Q:U -* IR be a normalized (q+l)-convex function defined on a domain U C Cn (0
Definition. By Div(Q) we denote the set of all n-tuples w=(wwn) of C1 functions wj:Cn'U

C which are obtained by the following

)

Construction: Let CO be constants such that inequality (7.2) is fulfilled. Choose C1-functions aj kU-K (j,k=1,...,n) such that

a29W ajk(x)

xl

(7.3) <

2n

for all xeU. Set as(x)

n if 1<j
_ajk(x)(xk-zk)

2

oxj

k=1

w3(z,x) =

219(x)

n - Iajk(x)(xk-zk) + (C+A)(xj-z j

2

G i

if q+2<j
k=l

Remark. If wEDiv(Q), then w(z,x) is holomorphic with respect to 21,...,zq+1, and it follows from (7.3) and (7.2) that

Re<w(z,x),x-z> > _q(x) - 6(z) +

Ix-zI2

for all x,z&U.

(7.4)

7.7. Canonical Leray data and maps. Let [U,e, cp,D] be a q-convex configuration in Cn (0
{zEU: Q(z)=0},

D1:= {zEU: A(z)
Y2:= {zEU:(p(s)=O),

D2:= {zEU: p(z)
Y1

81

Then D=D1mD2 is a domain with piecewise almost C1 boundary and (Y1,Y2) is a frame for D (cf. Sect. 3.1).1j Now let w1cDiv(Q) (cf. Sect. 7.6). Then, by (7.4),

<w1(z,x),x-z> # 0

for all xEY1 and zeD1.

Further, we set w2(x)=V q(x) for xeU (cf. Remark 2.10), and recall that, by Observation 2.11, <w2(x),x-z> # 0

for all xaY2 and zdD2.

Thus, (w1,w2) is a Leray datum for (D; Y1,Y2) (cf. Sect. 3.4).

Definition. We say (w1,w2) is a canonical Leray datum for 1U,Q,(p,D]

(or for D) if w1eDiv(Q) and w,=0Y. We say v is a canonical Leray map for [U,Q, (P,D] (or for D) if v is the canonical combination (cf. Sect. 3.4) of some canonical Leray datum(w1,w2) for [U,4, (P,D]ii.e.

wj(z,x)

vj(z,x) =

(7.5)

<wj(z,x),x-z>

for ziD, xEYj (j=1,2), and

w (z,x)

v12(z,x,t) _ (1-t)

1

w (x) 2

+ t

<w1(z,x),x-z>

(7.6)

<w2(x),x-z>

for zeD, xeY1nY2 and 0
Remark. If v is a canonical Leray map for a q-convex configuration [U,',(P,D], then v1, v2, and v12 depend holomorphically on

zl" .. 'zq+l'

7.8. Theorem. Let [U,Q, (p,D] be a q-convex configuration in Cn (On-q, then Lvf=O and hence

(-1)r f = 3Tvf

- Tvaf

in D.

(7.7)

Proof. This follows from Corollary 3.13 (ii), because v1, v2, and v12 are holomorphic in z1,...,zq+l (cf. the remark at the end of Sect.

7.7). Q 1)

82

The fact that the volume of the smooth part of Y1 is finite follows, for instance, from Proposition 0.7 in Appendix S.

8. Local approximation of a-closed (0,n-q-l)-forms on strictly q-convex domains

In this section we prove the following approximation theorem: 8.1. Theorem. If [U,q, IQ,D] is a q-convex configuration in Cn (0
then any (O,n-q-l)-form which is continuous and 5-closed in a neighborhood of D can be approximated uniformly on D by (0,n-q-1)-forms which are continuous and 5-closed in tn. Remark.

A stronger result will be obtained in Sect.

10.

8.2. Proof of Theorem 8.1 for q=n-1. In this case we have to prove that every holomorphic function f in a neighborhood of D can be approximated uniformly on b by holomorphic functions in 6n. Of course, this is well known from the Oka-Cartan theory (see, for instance, Theorem 2.7.1 in (H/L]). However, to be independent, here we give a direct proof (using the same method as in the proof of Theorem 2.7.1 in CH/L]). First we prove the following D be a compact set and f a holomorphic function Statement 1. Let K in a neighborhood of D. Then f can be approximated uniformly on K by holomorphic functions in D2:={zEU:(P(z)
Proof of Statement 1. Let (w1,w2) be a canonical Leray data for D (cf. Sect 7.7). Then the corresponding Cauchy-Fantappie formula (3.18) takes the form n-1 (

2ni

n )

det(w1(z,x),d w1(z x)) f (z)

=

J

n w(x)

f( x)

<w(z,x),x-z>n p(x)=0 (f (x)<0

n-i +

f f(x)

det(w (x),dw ?) 2

2

AW(x)

<w2(x),x-z>n

1

Q(x)<0

Y(x)=0 n-1

f(x) det(v12(z,x,t),dx,tvl2(z,x,t))A J(x),

+

(8.1)

J

0
where v

is defined by (7.6). Since w1(z,x)is holomorphic in z, the 12 second two integrals in (8.1) define some holomorphic functions in D2.

83

Therefore, we only have to prove that the holomorphic function which is defined by the first integral in (8.1) can be approxmimated uniformly on K by holomorphic functions in D2. This is a consequence of the following Statement 2. For any fixed x0EU with 9(x0)=0, the function

1

<wI(z,x0),x0-z>

(zED)

can be approximated uniformly on K by holomorphic functions in D2.

Proof of Statement 2. Set (z,x)=<w1(z,x),x-z>. Since 9 does not have local minima and since, by (7.4), 4(z,x)#0 if zED and xcU\D, we can find points x1,...,xNED2\b such that (f(xN)=0,

g(xN) =

max XED2

(8.2)

Y(x),

and

4)(z,x

max 1-

)I

j 1

<

for j=1,...,N.

1

(z,xi)

zEK

Then

1

4(z,xj-1)

1

(z,xj)

00

k=1[

1

_

k

_(z,xj_1)

(z,xi)

I

for j=1,...,N, where the convergence is uniform on K. Therefore, for any

1<j
End of proof of Theorem 8.1 for g=n-1. Let f be a holomorphic function in a neighborhood W of_D. Take £>0 sufficiently small, set i(z)<&,y(z)<E}, and D2:= {zEU:(9(z)<E}. Then is an (n-l)-convex configuration, and D == D c= W. Therefore, it follows from Statement 1 that f can be approximated uniformly on b by holomorphic functions in D2. Since D2 is convex, this completes the proof of Theorem 8.1 for (Here we used the fact that any holomorphic function in a neighborhood of a convex compact set in Q;n can be approxi-

84

mated uniformly on this set by holomorphic functions in Cn. This can be proved similar as Statement 1.)

For O
f can be written

f = hdzq+2n...Adzn + ag,

(8.3)

where h is a Coo function on V which depends holomorphically on zl,...,zq+l, and g is a continuous (0,n-q-2)-form on V. Proof. Notice that, for all sufficiently small E>O, [U,g-E, (P-e]

defines also a q-convex configuration. Therefore it is sufficient to show that f can be written in the form (8.3), over D. Let v be a canonical Leray map for [U,9, (P,D] (cf. Sect. 7.7)). Then the piecewise

Cauchy-Fantappie formula (Theorem 3.12) yields the representation

f = (-1)n-q-1 Lvf + i!g

on D,

where g=(-1)n-q-4Tvf. Every canonical Leray map for [U,q,IP,D] is of class Coo, and holomorphic with respect to zl,...,zq+1' Since f is of

bidegree (O,n-q-1) and by (3.9) and (3.10), this implies that Lf is of the form hdzq+2 ... dzn where h is a C°° function on D which depends holomorphically on zi,...'zq+1' 8.4. Proof of Theorem 8.1 if q
5, then d(x g) is a continuous 3-closed (0,n-q-l)-form on Cn which is equal to 5g on 5. Therefore it is sufficient to prove that h can be approximated uniformly on 5 by Coo functions on Cn which depend holomorphically on zl,...,zq+l. To do this, first we fix some vector w=(wq+2,...,wn)ECn-q-1 and set

q+l

Cw

= (zeCn: zq+2=wq+2,...,zn=wn).

Then the set UnCN+l, the function are convex, and

£jUnCw+l,

and the set (aEU:tp(x)0)nCM+l

85

{ztU: cW0}nCO+l - UnCw+1

Further, then the restriction of

, to UnCW+1 is normalized (q+l)-convex.

Choose E>0 so small that

{zeU: p(z)<E, q(Z)
L:Cn-* H such that

D cc DE.,L'= {zEU: (p(z)«, (Q+L)(z)<E} oc V, and

(UnCw+1,s+L-t,(p-E,D, LnCW+l] is

a q-convex configuration in Cw+1

(or DnCw+l is empty). Since Theorem 8.1 is already proved for q=n-1, this implies that, for each >0 and all with w=(wq+2I...1wn)ECn-q-1

D,,CM+1 9 0 there exist a neighborhood W C Cn of DnCW+1 and a holomorphic function hw on Cn which is independent of zq+2,...,zn such that Ih-hwI
0
9. Uniform estimates for the local solutions of the 3-equation constructed in Sect.7

In Sect. 7, q-convex configurations were introduced, the notion of a canonical Leray map v for such configurations was defined, and it was observed (Theorem 7.8) that the corresponding Cauchy-Fantappie operator Tv (of. Sect. 3.10) solves the a-equation for forms of bidegree (O,r) with r>n-q. Here we prove the following important estimates for Tv:

9.1. Theorem. Let (U,4, y,D] be a q-convex configuration in Cn (00, and

DE:= {zeD:

Ip(z) < -E}.

Then there exists a constant C
II TvfH1/2,DS <

es

C140,1)

(9.1)

(for the definition of these norms, see Sect. 0.11). If (U,9,(p.D] is of type II and y¢U is the critical point of p, then, moreover,

YTvfIIO

D

< C(

E

sup

Iz-Yl) f110,D'

(9.2)

Z E supp f

and

sup z,xeDt Iz-yI
IITvf(z) - Tvf(x)II

Iz-xl /

< C(1 + Iln rl)r 1/2 pfq O,

D'

(9.3)

for each r>O.

In the proof of this theorem we use the following 9.2. Lemma. Let (U,g,(p,D] be a q-convex configuration in Cn, D2:=(zEU: and h(z,x):= Im<w(z,x),x-z> for z,xaU (Im:= imaginary

p(z)<0},

part of). Then the following assertions hold true:

Idxh(z,x)Ix=zI = Ids(z)l

(i)

(ii)

I

for all

dq(z)ndxh(z,x)Ix=z

for all z4U.

(iii) If tj=tj(x) are the real coordinates of xaCn with xj =tj(x)+ itj+n(x), then there is a constant K
(x) -

atj

(z)

<

Klx-zl

for all x,zED2 (j=l,..,2n).

atj

(iv) If [U,e,(Q,D] is of type II and y&U is the critical point of g, then there is a constant K
It(z,x)I < K(lx-yllx-zl + Ix-z12)

Proof. If ti=tj(x) are the real coordinates of +tj+n(x), then, by definition of Div(p),

for all z,xeD2.

with x=t(x)

87

n h(z,x) = J=1

I

(x-z)

(x)ti (x-z)f + O(Ix-z12),

J+n

J

J

and

(x)dti (x)I + O(Ix-zl). D

dxh(z,x) _ E Ct (x)dtj+n(x) J=1

J+n

J

ff

Proof of Theorem 9.1. Recall that

Tv = BD + Rl + R2 + R12.

It is well known (cf., for instance, Proposition 1 in Appendix 1 of [H/L1) that there exists a constant C
< C(1 + I1nIx-zll)Ix-sIIIfAO,D

for all z,xED. Hence, in particular, (9.1) and (9.3) hold true with BD instead of Tv. Further, it is easy to see (cf., for instance, Proposition 5 (i) in Appendix 1 of [H/L]) that (9.2) holds true with BD instead of Tv. Since S2d

= S12') D.

=0 (for the definitions of S2 and S12, see

Sect. 3.2), estimates (9.1)-(9.3) hold true also with R2 and R12 instead of Tv.

Therefore, it remains to prove that (9.l)-(9.3) hold true with Ri instead of Tv. Recall that (cf. Sect. 3.7)

Rvf(z) =

J

f(x)ARV(z,x,t),

z
S1xA01 where n-1

Ri(z,x,t):=

1

(2Ti)

n

det(v01(z,x.t).(az,x+d )v01(z,x,t))ACJ(x)

with

v01(z x t):= t02 + t1 Ix-zI

88

w(z,x)

<w(z,x),x-z>

,wn)EDiv(g). Further on (in this proof), we use the

for some w=(wl,

abbreviation 4 _ I(z,x) = <w(z,x),x-z>. Then

(az x+dt)v01 =

x-z

iw

aZ

Ix-zl2

+

t0L

xw -

dtl + tl

waz x 42

4

---L4(x-z)aZ xlx-zI2

dx-dz

Therefore, expanding the determinant in the definition of Ri(z,x,t) (using the rules collected in Proposition 0.6 and taking into account that only monomials of degree 1 in t contribute to the integral Rif) and integrating with respect to t, we obtain

Rif(z)

n-2 I-c

(' Y(x)A R5(z,x),

1

zED,

s-0 JS

1

where n-s-2 .

det(w,x-z,a Rs(z,x) _

z

r-s

w,dx-dz)

n-s-1 {x-zxl s+2

A j(x)

and cl,...,en-2 are complex numbers which are independent of f and z. This implies that the coefficients of Rif are linear combinations of integrals of the form

f

FG n-s-l l X -z I 2s+2 dx1 A....,...ndxnAL7(x),

(9.4)

Si

where 0<s
of

dQ(x)

wk(z,x)

+ O(Ix-z I)

and the product G contains at least one of the functions wk and at least

89

one of the functions xk-zk, there exists a constant C0
IG(z,x)I < C0(ild?(x)IIIx-zl +

Ix-zI2)

and

IIdzG(z,xI < CO(4dp(x)Il + Ix-zl) for all z,x D. Further, by (7.4), there is a constant oc>O with

I4(z,x)1 > oc;Ix-zI2

for all xeSI and zcD. Further, for some constant C1
C1(Idg(x)It + Ix-.1)

for all z,xED. Hence there is a constant C2
II Rif(z)II

r

C21(f1IO,D

I

I

S

II dgIIdS

I(x

zl 2n 3 +

l Slnsupp f

f J

dS

l

Ix zl 2n-2

(95)

Slnsupp f

where dSl is the Euclidean volume form of S1.

Taking into account the relation

G d

-

z f-s-1 lx-z12s+2 -

(n-s-i)GCV

d G z

n-s-l iX-zI2s+2

+

4n-slX-z12s++2

(s+l)GdzIx-zI2 n-s-lix-zl s+4'

we can find C3
11dg;JdSI

udRif(z)II < C31!fl!O,D

j

qI2I x=z12n 3 +

Slnsupp f

90

I

MIX-$i 2

Slnsupp f

dS1 111 x

-3

(9.6)

J

S1nsupp f

Set h(z,x) = Im 4(z,x). (Im

imaginary part of) Then, by (7.4),

there is a constant c)O such that

I(z,x)I > Ig()I + Ih(z,x)I + cix-zI2

(9.7)

for all xcSI aund zeD.

Now we first consider the case that the configuration (U,V, 1Q,D] is smooth. Then S1 is smooth and it is clear that

dS1

up

< oo.

FD I

Ix-.12n-2

Si

Moreover, then it follows from (9.7) and Proposition 1 (ii) in

Appen-

dix A (which can be applied in view of assertion (ii) in Lemma 9.2) that

sup zeD

2n-'3

< 00.

Si

Therefore and by (9.5), we can find a constant C4 < oo such that

yRif u0

C4hf IO,D

(9.8)

for all continuous differential forms f on D. Further, applying Proposition 1 (i) and (ii) in Appendix A to the integrals on the right hand

side of (9.6), we obtain C5
IdRif(z)M < C51e(z)I-1/2kfIO,D.

(9.9)

By Proposition 2 in Appendix A, it follows from (9.8) and (9.9) that, for some C6
91

IIRv1fIIl/2 D < C6IIfIIO D.

for any continuous differential form f on 5. This completes the proof of estimate (9.1) in the case that [U,Q, p,D] is smooth. Now we consider the case that [U,Q (Q,D) is not smooth, i.e. of type II. Then, without loss of generality, we can assume that the critical point of 9 is the origin.(So we have the situation considered in Propositions 1-3 in Appendix B.) In view of assertions (i)-(ii.) in Lemma 9.1 and estimate (9.7), we

can apply Proposition 3 in Appendix R to the integrals on the right hand side of (9.5), and we obtain a constant C7
IIR1flI0,D

C70f11O,D

(9.10)

and, moreover,

IIRLflIO,D < C7(

suP

z(supp f

Izl"fNO,D-

(9.11)

Applying Propositions 2 in Appendix B to the integrals on the right hand aide of (9.6), we obtain C8
C6(I1nIQ(z)II + 9dp(z)III9(z)1-1/2)IIf11o

IldRif(z)I

D.

(9.12)

for any continuous differential form f on b and all z(D. If &>0 and 0 <

x

1 is a continuous function on C° such that

<

x (z) = 0 for Izl<6/2 and x(z) = 1 for Izi>1, then it is clear that

lim Ri(X f)(z) for D3z ---+ 0 exists, and it follows from (9.11) (with (1 - Vf instead of f) that

I

v Since R f = Rv

I

Or f

I R1(1 - X) f II O, D

< oC7II f ljp, D'

+ Rl(1 - x)f and 9>0 can be chosen arbitrarily small,

so it follows that lin Rif(z) for D3z -+ 0 exists. Therefore, from (9.12) and Propositon 4 in Appendix B we obtain C9
92

IIRlf(z) - Rif(x)II sup z,xED Izl,ixl
1/2

Iln rl)r-112IIPU0

<

Iz-xl

D.

(9.13)

for all r>0 and any continuous differential form f on D.

It follows from (9.10) and (9.13) (with r=1) that (9.1) holds true with R1 instead of Tv,

i.e.

(9.1) is proved also for the case when the

configuration [U,Q, (1,D] in not smooth. Finally, we observe that esti-

mates (9.11) and (9.13) imply that (9.2) and (9.3) hold true with Ri instead of Tv.

10. Local uniform approximation of 3-closed (O,n-q-l)-forms on strictly q-convex domains

Here, by means of the estimates obtained in the preceding section, we prove the following strengthening of Theorem 8.1. ?0.1.

Theorem. Let [U,q, (p,D] be a q-convex configuration in Cn

(O0,and DE:= {zsD:

(f(z)<-t}. Then any continuous (O,r)-form

with n>r>n-q-1 or. D which is 3-closed in D can be approximated uniformly on DE by continuous 6-closed (O,r)-forms in Cn. Proof. Choose a Coo function Lr IR -) IR such that w (t)=l if t<1/2 and yr(t)=0 if t>1. We assume, without loss of generality, that 06U and :

if the configuration [U,p,t};,D] is not of type I, then 0 is the critical point of Q. For k=1,2,,.. and zECn, we set b0(z) _ tY (kIzl) if (U,9, (9,D] is not smooth, and bk(z) = 0 if [U,s, (p,D] is smooth. Then

1ldbkl'I0 en < k

max Idt(t)I

(10.1)

0
and

bk(z) = 0

if Izl

> k.

(10.2)

If (U,e, (,D] is smooth, then the surface {zf.U: 9(z)=0} is smooth. If FU,Q, (f,,Dj

is not smooth, i.e. of type II, then the surface

{zcU: 9(z)=0, z¢0} is smooth and b0(z)=1 for all z with Iz1<1/2k.

Therefore, in both cases, for all k=1,2,..., we can find a neighborhood Uk ca of DE as well as real-valued Coo functions bk on Cn and vectors akECn (j=1,...,N(k)
93

E

N(k)

bk = 1

(10.3)

on Uk,

j=o

and, for j=1,....N(k),

supp bk cc {zeU:pp(x)<0},

(10.4)

for all O
(D n supp bk) + tak C D

(10.5)

Now let f be some continuous (O,r)-form on D which is 3-closed in D, where n-q-1
Take a canonical Leray map v for [U,q,(p,D] and set

fk = bkf + T'(bbknf)

for k=1,2,... and j=0,1,...,N(k). Then, by Theorem 9.1,

fjcCO r(D\{zeU:(p(z)=0})

(10.6)

N(k) f == fj j=0

(10.7)

and, by (10.3), we have

on D7, k

The piecewise Cauchy-Fantappie formula (Theorem 3.12) yields the relation

= L",bkf) + 3Tv(bjf)

in D.

(10.8)

[bXfXAL.x.

(10.9)

By (10.4). Lv(bkf) =

J

R(x)=O

where the kernel Lr(z,x) is of the form

r

n-r-1

det(w(z,x),aaw(z,x),axw(z,x)) Lr(z,x) = c <w(z,x),x-z>n

94

for some wcDiv(Q) and some constant c (of. Sect. 3.5 and the proof of Proposition 0.9). In view of (7.4) and (10.4), we can find open sets Wk (j=0,...,N(k); k=1,2,... ) such that

D u({zcU: if(x)<0,9(z)=0} \ supp bk C Wk C U and

<w(z,x),x-z> # 0

if x.dD n supp bk and zeWi.

(10.10)

Therefore, the right hand side of (10.9) defines a Coo form gk on Wk with

91 = Lv(bkf)

on D.

Since w(z,x) depends holomorphically on zl,...,zq+1 and since r>n-q-1, it follows that

gk 6z0 r(Wk) (observe that

(10.12)

gk=0 if r>n-q). Moreover, by (10.10) and the definition

of Tv we obtain continuous (0,r-1)-forms hk on Wk which are C°O in Wk\(D n supp bk) such that

hk = Tv(bkf)

on D.

(10.13)

Since hk is Coo in Wk \ (DA supp bk) and by (10.8), one obtains that

(10.14)

By (10.8), (10.11) and (10.13), we have the relation fk=gk+ahk on D. Therefore, taking into account (10.12), (10.14), and (10.5), we can find neighborhoods Vk of D and numbers ck>0 (j=1,...,N(k); k=1,2,... ) such such that, by setting

Fk(z) = (gk+8hk)(z + akvk),

zeVk, forms Fk E ZO r(Vk) with

95

1

(10.15)

k

are defined.

By (10.1) and (10.2), it follows from estimate (9.2) in Theorem 9.1 that

lim

k---) oo

IIT'"(5b0 f)IIo

D

= 0.

(10.16)

Since f is continuous at 0, without loss of generality we can assume that, moreover, f(0)=0. Then it is clear that also lim

= 0 for

DE

k -* oo. Together with (10.16) this implies that

lim

k - moo Setting fk =

iIfk1IO,D6= 0.

(10.17)

N(O) Fk, now we obtain a sequence f of 3-closed

(0,r)-forms in some neighborhoods Vk of D.

In view of (10.7), it follows from (10.15) and (10.17) that lim Afk-fIIO DE = 0 for

k -p oo. Without loss of generality we may assume that & is so small that [U,9, (p-&,D I is also a q-convex configuration. If r=n-q-1, then the proof can be completed by applying Theorem 8.1 to (U,yo, (p-&,L) and the forms fk.

If r>n-q, then we proceed as follows: For all k, we

take a Coo function Xk on en with supp xk == Vk and ,/k = 1 on

.

By

Theorem 7.8, after shrinking Vk, we may assume that fk=&tk for some '"

,r-1(Vk)' Therefore, fk:=a(xkuk) is a continuous 3-closed (0,r)-form on Cn with fk=fk on 5

11. Finiteness of the

. O

Dolbeault cohomology of order r with uniform

estimates on strictly q-convex domains with r>n-q

11.1. Definition. Let D X be a domain in an n-dimensional complex manifold X. Suppose E is a holomorphic vector bundle over X or, more generally, E is a C°CC7 vector bundle over b (for the notion of a C°'12

vector bundle, see Sect. 0.12), where 0
We say the operator 8 is ou-regular at Zp r(,E) if there exists a linear operator

T:E° r 0(D,E) -3 C0,r-1(D,E)

96

1

such that the following conditions are fulfilled: (i) T is bounded as an operator acting from the normed space Ep = 0(D,E) endowed with norm 11.110 D into the Banach space CO,r-1(15,E)

endowed with the norm u Il D (for the notations, of. Sect.

0.11);

(ii) 3Tf = f for all ffiE r 0(D,E); (iii) T is compact as an operator with values in the Banach space CO,r-1(D,E)

(endowed with the norm 11.40,

D

Remarks to this definition.

I. For k6>0, condition (iii) follows from (i), because then the embedding of II.

C0,r-1(D,E)

into C0 r_1(D,E} is compact (Ascoli's theorem).

If 3 is o(,-regular at Zp r(D,E), then E ->O (D,E) is a closed

subspace of the Banach space ZO r(D,E) (with the norm 4.110 D). This

follows from the fact that 3 is closed as an operator between the Banach spaces

CO,r-1(13,E)

and Z r(D,E).

III. Simple arguments show (cf., for instance, the proof of Proposition 3 in Appendix 2 of (H/L]) that condition (ii) is equivalent to the following one:

(ii') There exists a compact linear operator K from

Eo6---->O

O,r

(5,E) into

C0 r(D,E) such that 3T - K is the identity operator on ED r (D,E). IV.

If 3 is oc-regular at Zg r(D,E) and, moreover,

dim[Z00 r(D,E)/ED r

(D,E)] < oo,

then 3 is also J3-regular at Z0 r(D,E) for all 0<0
D,E) C Ep r 0(5,E) C EO

Remark III and the relation Er0O

r

ME).

11.2. Theorem. Let D a= X be a non-degenerate strictly q-convex domain in an n-dimensional complex manifold X (0
dim

00,

where

HO r 'o( D,E):=

1/2

-

ZO 0,

r

(15,E)/E' /2

O, r

)0

(5,E).

97

Moreover, then there exist bounded linear operators

Tr:C0

C1/r-1(D.E),

r(15, E)

Kr:C r(D,E)

) CO,r(D,E)

(r=n-q,...,n) such that, for any continuous E-valued (O,r)-form f on D with n-q
8Trf + Tr+lDf = f + Krf

on D.

(11.2)

Sometimes the following generalization of this theorem to Co'b vector bundles is useful (see Sect. 0.12 for the definition of such bundles): 11.2'. Theorem. Let D cc X be a non degenerate strictly q-convex domain in an n -dimensional complex manifold X (0
bundle over D, where 0
(x-regular at 2g

O,r

ME), and

dim [2p r(D,E)/Eg r 0(,E)] < oo.

(11.3)

Moreover, then there exist bounded linear operators

Tr:Cp r ME)

Kr:CO r(,E)

0

CO,r-1(D,E),

CO r (D,E)

(r=n-q,...,n) which are even compact if oc<1/2 such that, for any continuous E-valued (O,r)-form f on D with n-q
3Trf + Tr+LDf = f + Krf

on D.

(11.4)

Proof of Theorems 11.2 and 11.2' (cf. the proof of the Kodaira finiteness Theorem 1.15). Since Theorem 11.2 is a special case of Theorem 11.2', we have to prove only Theorem 11.2'. (Notice that a direct proof of Theorem 11.2 would not give any simplifications.) So let the hypotheses of Theorem 11.2' be fulfilled.

In view of Lemma 7.5, Theorem 7.8 and Theorem 9.1, we can find open

98

sets Uj =- X (j=1,...,N
(O,r)-form f on 5 with n-q
aTrf + Tr+laf = f

on UjnD,

where Tn+1'-O.

By Lemma 1.16, we can find open sets Uj e= D (j=N+1,...,M
j' E)

compact linear operators

(r=1,...,n) and KJ r

from CO

O, r (D,E)

into CO/r(Uj,E) (r=l,...,n-1) such that, for any continuous E-valued

Mr)-form f on D with the property that of is also continuous on b,

aTrf + Tr+laf = f + Krf

on Uj,

where Kn:=O and Tn+1:=O.

Now we take a CO° partition of unity (xj}M=l subordinated to the covering (Uj}M=1, and define

j=l

xjTJ

and

Kr =

axfT3 + EI xjKJ j-1

for r=n-q,...,n_ It is clear that these operators have the required properties. It remains to prove that, for n-q
Mr = id + Kr

on ZO r(D,E)

(11.5)

(id:= identity operator). Since K is compact as an operator with values in C0 r(D,E), by Remark III following Definition 11.1, this implies that O is oc-regular at Z0 r(D,E). Further, this compactness of Kr and relation (11.5) yield that the restriction of doTr to ZO E) is a Fredholm operator. Since, obviously, the image of this operator is

r'

contained in E

)0

(D,E), it follows (11.3). Q

99

12. Global uniform approximation of 5--closed (O,n-q-l)-forms and

invariance of the Dolbeault cohomology of order >n-q with respect to q-convex extensions, and the Andreotti-Grauert finiteness theorem for the Dolbeault cohomology of order >n-q on q-convex manifolds

12.1. Definition. Let X be an n-dimensional complex manifold, and O
a) If D C X is a domain, then we say X is a g-convex extension of D if the following two conditions are fulfilled: (i) aD is compact.

(ii) There exists a (q+l)-convex function Q:U --) It in some open set U C X with X\D C U such that

D n U = (zEU: p(z)<0)

and, for any number 0 < oc < sup So(z),

zfU

the set {zeU: 0<9(z)
in this case, the function s in condition (ii) can be chosen without

degenerate critical points in U, then X will be called a non-degenerate q-convex extension of D.

b) If D C G C X are two domains, then we say G is a strictly q-convex extension of D in X if G is a q-convex extension of D such that, moreover, the set G\D is relatively compact in X and the following strengthening of condition (ii) is fulfilled:

(ii)' There exists a (q+l)-convex function Q:U - Qt in some neighborhood U c X of such that

D n U= (z6U:

If,

Q(z)
and

G r U= (zeU: 9(z)<1}.

in this case, the function 4 in condition (ii)' can be chosen

without degenerate critical points in U, then G will be called a non-degenerate strictly q-convex extension of D in X. c) Instead of "(n-l)-convex extension" we shall say also "pseudoconvex extension"

100

D

D

------ aX aDnx

------

aDn X

G\D

CA

X is a q-convex extension of D

a(inX

G is a strictly q-convex extension of D in X FIGURE 2

Remarks to this definition. In the following remarks, D C X is a domain in an n-dimensional complex manifold X such that X is a q-convex extension of D (O
I. For all numbers

0 < on < A < sup p(x), xtU the domain {z&U:

is a strictly q-convex extension in X of the

domain {z6U: s(z)
II. D and X need not be q-convex. For instance, if X={z4Cn: l<jz{2.

III. Since aD is compact, there exists only a finite number of connected components of X which have a non-empty intersection with both D and X\D. If X0 is some cod cted component of X such that X0 C X\D, then the restriction of 4 to X0 is a (q+l)-convex exhausting function for X0. Therefore, then X0 is completely q-convex.

IV. It is admitted that aD = 0 or D = 0 .

If D = {b , then X is

completely q-convex. If aD = 0 , then X\D is completely q-convex.

12.2. Definition. Let X be an n-dimensional complex manifold and 0
AI S A2.

(ii)

A2\A1 cc V

X.

101

(iii) There exist q-convex configurations [Uj,Qj, (pj,Dj] in Cn

(j=1,2) with U:=U1= U2 and (P:=IQl=(P2 such that, for some biholomorphic map h from U onto a neighborhood of V, V =

If,

and

VnA. = h(Di) for j=1,2.

in this case, the type of (U,qj, p,Dj] is Tj, then [A1,A2,V] will be

called of type (T1,T2).

A q-convex extension element [A1,A2,V] in X will be called smooth if it is of a type (T1,T2) with T1 # II and T2 # II, otherwise it will be called non-smooth.

If Al C A2 are domains in X, then we say A2 can be obtained from Al by means of a g-convex extension element in X if there exists a domain V C X such that CAl,A2'V] is a q-convex extension element in X. Instead of "(n-1)-convex extension element" we shall say also "pseudoconvex extension element".

mom

A2\A1

Al

/

.......-

ax

FIGURE 3: q-convex extension element [A1,A2,V] in X

Remark to this definition. If [U,s,p] defines a q-convex configuration in Cn (0O, [U,V, y-E] and M9, (P+0 also define qconvex configurations in Cn. This yields the following

Proposition. If CA1,A2'V] is a q-convex extension element in an n-dimensional complex manifold X (0
(i) For any neighborhood W c X of V, there exists a domain V+ with V cc V+ cc W such that [A1,A2,V+] is also a q-convex extension element

in X. 102

(ii) For any compact set K == V, there exists a domain V_ with K =c V_ _= V such that [A1,A2'V_] is also a q-convex extension element

in X. 12.3. Lemma. Let X be an n-dimensional complex manifold, let D C G C X be two domains such that G is a non-degenerate strictly q-convex extension of D in X (O
(ii)" There exists a (q+l)-convex function 4 without degenerate critical points in a neighborhood U of GG\D such that DnU={zeU:

Q(z)<0}, GnU={zeU: 9(z)<1}, and yo does not have local minima in U.

Then there exists a finite number of domains

D = A0 C A1 C. ... CAN =G such that, for all 1<j
Moreover, for any open covering {Ui}i6i of X, these domains Aj can be chosen so that, in addition, for each 1<j
Proof. It is sufficient to prove the following

Statement. Let Q:X -+ Qt be a (q+l)-convex function without degenerate critical points on an n-dimensional complex manifold X (0
Set Do,:= {zeX: Q(z)<m} for -oo0 with the following property: For all o&,j3 with -£
Dc" =AOcA1 c ...

C_ AN =Dp

such that, for each 1<j
[B1, B2'V] in X with B1=Aj_1, B2=Aj, and V = Ui for some iel. Proof of the statement. Since aD0 is compact, and non-degenerate critical points are isolated, there exists not more than a finite number of critical points of 9 which lie on 3D0. Denote these points by y1,...,yM (if exist - otherwise set M=0). By Lemma 7.5, for any l<j<M,

we can find holomorphic coordinates hj:Vj -4Cn in some neighborhood Vj of yj such that hj(V.) is the open unit ball in Cn, hj(yj)=0 and, for each 0
defines a q-convex configuration of type H. Choose these neighborhoods Vi so small that, moreover, Vj Ui for some i=i(j)cI, and

VjnVk = 0

if j -f- k.

(12.1) 103

We set

for j=1,...,M and a=1,2.

W = h-1({zECn: Izl
By Lemma 7.5, for any point ye D0\(1 U ... U W1), we can find holomor-

neighborhood Vy of y such that hy(Vy) is the

phic coordinates by in a

unit ball in Cn, hy(y)=0, and, for all 0
Vy a aD0\(Wi u ... U WM) and V

y

==.

U

i

for some i=i(y)EI. Set

N1y

= h

y

(12.2)

({zeCn: IzI<1/2}). Since

6D0\(WlU ...% U ) is compact, we can find points yM+1,

yN in

aD0\(Wiu ... u WM) (N
aD0\(WIu... uWI) C Wy

u...UWy

M+1

1

Set Wj = Wy1

j

.

N

for j=M+1,...,N. Then we can find L>0 with

N

DL \ D_E _c U Wj. j=1

+ ... + XN = 1 Choose Coo functions xj on X (j=1,...,N) such that xI on DE\D_E and, for any 1<j
(12.2),

Xj = 1 on W if 1<j<M.

Now let some numbers o&,fi with -E
Ak = {zc.X:

,(z) - o. < (ft-o,) 1

k

Xj(z)}

j=1

for k=0,...,N. Then it is clear that Do'= A0 C Al C ... C. AN = D)1, and (j=1,...,N) are q-convex extenit is easy to see that all sion elements in X if & is sufficiently small. Q

Remark to Lemma 12.3. Let us choose the number E in the proof of this Lemma so small that, moreover, do(z) $ 0 for all ze(DE \D_E)\{Y1" .. 'yM}

Then the q-convex extension elements [Aj-1' Aj,Wf] constructed in this

proof are of type (I,I) if j=M+1,...,N, and of type (IV,III), (II,III) 104

or (IV,II) according as oc
cular, if o.<0
12.4. Lemma. Let (A1,A2,V) be a q-convex extension element in an n-dimensional complex manifold X (0
(i) If n-q
and

fl = f2 on Al\U.

(ii) If Al is relatively compact in X, then for each n-q
(iii) Let n-q
is 1

relatively compact in X and a is 1/2-regular at Z00 n-q(A1,E). Then

E1/2--a0(A

O, r

O, r (A 2' E) o 2 .E) = ZO

(iv) Let n-q-1
E1/2--*0(A

O, r

1

,E).

is relatively compact in X 1

and (if r#n) 3 is 1/2-regular at Z00 r+l(A1,E). Then, for any 1/2 fEZO ,r(A1'E), there exists a sequence fkEZO/r(A2'E) with

lkm

f - fJ0, Al =

0.

We mention also the following generalization of this lemma (for the definition of C(`(9 vector bundles, see Sect. 0.12):

12.4'. Lemma. Let (A1,A2'V) be a q-convex extension element in an n-dimensional complex manifold X (0
(i) If n-q
f1 = f2 on Al\U.

(ii) If A is relatively compact in X, then for each n-q
105

is o.-regular also at Z r(A2,E). (iii) Let n-q
E 0(A2,E)

ZO r(A2,E)A E '0(AlIE).

=

(iv) Let n-q-1
r+1(A1,E).

Then, for any

fEZ r(A1,E), there exists a sequence fkkZ r(A2E) with

lm

f - fk10, Al =

0.

0

In the proof of Lemmas 12.4 and 12.4' we use the following corollary which follows immediately from Theorems 7.8, 9.1 and 10.1:

12.5. Corollary to Theorems 7. 8, 9.1 and 10.1. For any q-convex exten-

sion element [A1,A2,Vl in an n-dimensional complex manifold X (O
Tr: Z0 r(AnV) O,

i

CO/r 1(AjnV)

which is continuous (with respect to the Banach space topology of ) and the Fr4chet space topology of C1/2_1(AjnV) and such Zp r(Athat, for all fEZg r(Aj\V), 3Trf = f on AjiV.

(ii) If n-q-l
Zg r(V) ) Z00 r(AjnV)

is dense in the Frechet space Z r(AjnV). 0

Proof of Lemmas 12.4 and 12.4'.

It is sufficient to prove Lemma 12.4'.

Proof of part (i). Since x<1/2 and E is Co-(2 trivial over A2IV, it follows from Corollary 12.5 (i) that f1PA1nV = &u for some 1(Aln V,E). Choose a Coo function x on X such that x = 1 on

A2\A1 and supp x == U. Set f2 = fl - (xu)

106

Proof of part (ii). Since a is oc-regular at ZQ r(A1,E), we have a bounded linear operator Tl:Ec ---> 0(A1,E)

CO,r-1(Al,E)

0, r

which is compact as an operator with values in CO

0,r-1 (AVE)

and such

that aoT1 is the identity operator. Choose a neighborhood V' of AZ\AI with V' _= V. Then from Corollary 12.5 (i) we get a continuous linear operator

T2:Z0 r(A2nV,E)

) CO,r-1(A2n7E)

which is compact as an operator with values in CO,r-1(A2(%V',E) and such that 3oT2 is the restriction operator to A2nV'.

Denote by B the subspace of all fcZO r(A2,E) with

on X such that supp x =c V' and x = 1 in a T2 + (1-x)T1 a bounded linear Then TB is compact as an operator operator TB from B into Choose a Coo function

neighborhood of A2\A 1. Define by TB = CO,r-1(A2,E).

with values in

CO,r-1(A2,E),

and 3-TB - 3xA(T2+T1) is the identity

operator. Since 3XA(T2+T1) is compact in B and 3'TB(B) is contained in

E

C B, this implies that 3 is (36-regular at Z00 r(A2,E) (cf.

Remark III following Definition 11.1).

Proof of part (iii). Let fEZ0 r(A2,E) with fIA1EE 0(A1,E). Then fIA1 = 3u1 for some

ui6C0,r-1(Al,E).

Further, by Corollary 12.5 (i), we

can find u2EC0,r-1(A2nV,E) with fIA2nV = 3u2. Now we distinguish the cases n-q+l
First let n-q+l
relation

ul - 3(xv) = u2 + 8((1 - 'x)v),

and both sides of this equation together define a form ueC0,r-1(A2'E) with 3u=f. Now let r=n-q. Then, by hypothesis, A

1

is relatively compact in X and

is cc-regular at ZO n_q(A1,E). By pert (ii), this implies that 3 is o(,-regular also at Zg n q(A2,E). Therefore, it is sufficient to find a

107

sequence wk¢CO,n-q

such that the forms 3w1 are continuous on A2

1(AZ,E)

and

lie If - a'kIIO,A2 = 0. k

Choose a Coo function x on X with supp x == V and x = 1 in a neigh-

borhood of A. Further, by Corollary 12.5 (ii), we can find a sequence vkEZ0,n-q-1

(A2AV,E)

with

lk II

vk+u2-ul110,

Al n supp

X

= 0.

Then the sequence wk which is defined if we set wk = ul on Al and wk = (1-x)u1 +

Y(vk+u2) on A2nV has the required property.

Proof of part (iv). Let r(A1,E). Choose a Coo function x on X with supp x == V and x = 1 in a neighborhood of A2\A1. By Corollary 12.5 (ii), we can find a sequence vkcZo r(AZIV,E) such that

lie

IIf-vkIIO,Aln

supp

0

and hence

l

k

3XA(f-vk) IIO, AZ = 0.

II

Since 3XA(f-vk) = 3(x(f-vk)) on A

and since 3 is oC-regular at 1

ZO,r+l(A1,E),

belong to Em

it follows from part (iii) that the forms 3XA(f-vk) ---->O

i.e. SXA(f-vk) = awk for some

Since, by part (ii), 3 is oc-regular at ZO,r+l(A2'E), this sequence can be chosen so that, moreover, lie IIwkIIO,A2 = 0. Setting fk = f -

f-vk)

k + wk on Al and fk = vk + wk on A2\A1, now we obtain a sequence (G r(A2,E) fke Z0

with lie If-fkIO,A

k

2

= 0.

Q

12.6. Lemma. Let g:X -) R be a (q+1)-convex function without degenerate critical points on an n-dimensional complex manifold X (00 such that, for all r=l,...,n, the operator 3 is 1/2-regular at Z00 r(Dt). and

108

Zp

where D, := {z X: 4(z)
Proof. Since 4 has a local minimum at y and since all critical points of 4 are non-degenerate, we can find a neighborhood U of y such that, with respect to some holomorphic coordinates h:U -3 Cn, 4 is strictly convex in U. Choose £>O so small that Dt G U. Then DL is strictly convex with respect to these coordinates, and we could conclude the proof by the remark that the assertion of the lemma now follows from Theorem 2.2.2 in [H/L1, for instance. However, to be independent of [H/L], let us give also the following arguments: First observe that a is 1/2-regu-

lar at Zg r(DE), by Theorem 11.2. To prove that E

Zoo, r(D£),

we consider an arbitrary form fiZ0,r(DE). Then, by Theorem 2.12 (Poineare a--lemma), f = u on DL for some u4C /r_1(D£). In particular,

then the restriction of f to

D£/2:= {zEX: Q(z)
Since, by Theorem 11.2, 3 is 1/2-regular also at Z00 r(DE/2

and 4 does not have local minima in DE\DE/2, it follows from

Lemma 12.3 and assertions (ii) and (iii) in Lemma 12.4 that f belongs to El/2-->O

0

12.7. Theorem. let D cc X be a non-degenerate completely strictly q-convex domain in an n-dimensional complex manifold X (0
Z0 r(D,E)

In other words,

then there exist bounded linear operators Tr from Z00,r (5,E) into C11/r_1(D,E)

(r=n-q,...,n) with

3T f = f for all faZ0 r(D,E).

Notice also the following generalization of this theorem to C°" vector bundles (cf. Sect. 0.12):

12.7'. Theorem. let D

X be a non-degenerate completely strictly

q-convex domain in an n-dimensional complex manifold X (0
at Zg r(D,E), and E 0(D,E) = ZD r(D,E).

Proof of Theorems 12.7 and 12.7'. This follows from Lemma 12.6, Lemma 12.3 and assertions (ii) and (iii) in Lemma 12.4'. 0

12.8. Lemma. Let X be an n-dimensional complex manifold. Let D C G C X be domains in X such that G is a non-degenerate strictly q-convex exten-

109

sion of D in X (0
and fG = fD

on D\U.

(ii) Let n-q
E01/2---+0(G,E)

= Zg r(G,E)n

(iv) Let n-q-1
Z0,r+1(D,E).

Then the image of the restriction

Zl/r2,(G,E) --) Zi/r(D,E)

is dense in ZD/Y2,(D,E) with respect to the norm

D.2)

Notice also the following generalization of this lemma: 12.8'. Lemma. Let X be an n-dimensional complex manifold, and D L G C X domains in X such that G is a non-degenerate strictly q-convex extension of D in X (0
over G (0
(i) Let n-q
of aDnG, there exists fGEZ0 r(G,E) with fD-fGEE0

C

)'0$,E) and fG = fD

on D\U.

(ii) Let n-q
(iii) Let n-q
Eo -->0(G,E)

r

=

E

0, r 0(5,E).

(iv) Let n-q-1
regular at Z00 r+1(D,E). Then the image of the restriction map

1) For D = 0

we have Theorem 12.7.

2) Of course, this is not true with respect to the norm

110

1/2,D'

Zp r(G,E)

> Z0,r(D,E)

is dense in Zo r(D,E) with respect to the norm

O,D'

Proof of Lemmas 12.8 and 12.8'. This follows immediately from Lemmas 12. 6.

12.3 and 12.4'. O

12.9. Proposition.

1)

Let D cc X be a non-degenerate strictly q-convex

domain in an n-dimensional complex manifold X (Or>n-q,

E

ZO r(D,E)n EO

r(1), E).

Notice also the following generalization of this proposition:

oG

12.9'. Proposition. Let D - X be a non-degenerate strictly q-convex domain in an n-dimensional complex manifold X (Or>n-q,

Ep r 0(D,E) = ZO r0,E)0 EO r(D,E).

Proof of Propositions 12.9 and 12.9'.

It is sufficient to prove Proposi-

tion 12.9'. Let r(D,E)/1E0 r(,E). Take a non-degenerate strictly q-convex domain G cc D such that D is a non-degenerate strictly q-convex extension of G in X. Then it follows from Theorem 1.13 (regularity of 3) that f1GEEc T 0(0,E). Since, by Theorem 11.2, 5 is oc-regular at ZO,r(G,E), now we may apply Lemma 12.8' (iii) and obtain that f belongs

to ED r 0(5,E). D 12.10. Proposition. Let X be an n-dimensional complex manifold, let D =c G =c X be rlon-degenerate strictly q-convex domains such that G is

a non-degenerate strictly q-convex extension of D in X (O
If n-q
3D, there exists a form f0eZO,r(G,E) with fD = fG on D\U. 1)

and

Below we prove the more complete Theorem 12.15.

2) The assertions of this proposition will be repeated also in the more complete theorems given below: part (i) follows from Theorems 12.13 and 12.15 (i), part (ii) follows from Theorems 12.14 and 12.15 (ii), part (iii) is contained in Theorem 12.11.

ill

(ii)

for all n-q
ZQ

(iii) If n-q-l
ZO,r(G'E) ) ZO,r(D,E) is dense in Zg r(D,E) (with respect of the norm Notice also the following generalization of this proposition to Co"? vector bundles (see Sect. 0.12):

12.10'. Proposition. Let X be an n-dimensional complex manifold, let D c= G c

X be non-degenerate strictly q-oonvex domains such that G is

a non-degenerate strictly q-convex extension of D in X (O
let E be a CO 0 vector bundle over G. Then the following assertions hold true:

(i) If n-q
aD, there exists a form f0Zg r(G,E) with

fl)-fGEE r 0(D,E) and

fD = fG on D\U.

(ii) Ep r

0

(G,E) = Z0 r(G'E)n E0 r(,E) for all n-q
(iii) If n-q-l
ZO,r(G,E)

Zg r($,E)

is dense in Zp r(D,E) (with respect of the norm 1'10,D)`

Proof of Propositions 12.10 and 12.10'. It is sufficient to prove Proposition 12.10'.

Assertion (1) follows from Lemma 12.8' (i) and Proposition 12.9.

To prove part (ii), we consider a form fEZ0 r(6,E)flE0r(,&). By Proposition 12.9,

Since, by Theorem 11.2, 5 is

1/2-regular at Zo n_q(D,E), we may apply Lemma 12.8' (iii) and obtain

that f belongs to E = 0 Assertion (iii) follows from Lemma 12.8' (iv), because, by Theorem 11.2, & is 0-regular at ZO,r+1('E) for n-q-l
12.11. Theorem. Let E be a holomorphic vector bundle over an n-dimensioX a non-degenerate strictly q-convex nal complex manifold X, and D domain such that X is a non-degenerate q-oonvex extension of D

112

(O
ZQ,r(X,E) -----a ZO r(D,E) is dense in ZO r(D,E) (with respect to the norm Proof. Since X is a non-degenerate q-convex extension of D, there is a sequence of non-degenerate strictly q-convex domains Dk == X (k=0,1,...

)

such that DO = D, each Dk+1 is a non-degenerate strictly

q-convex extension of Dk in X, and

00

X = U Dk. k=0

Now let a form fgEZg r(D0,E) and a number Z>0 be given. Then, by Theorem 12.10 (iii), we can find a sequence fkeZg r(Dk,E) with Hence F:= lie fk exists uniformly ,12k for k=1,2,... .

ifk fk-1IIo,Dk-1 <

on the compact subsets of X, belongs to

estimate II F-f0I10, D

and satisfies the

E' 0

12.12. Corollary. Let E be a holomorphic vector bundle over an n-dimensional complex manifold X, and D = X an arbitrary (not necessarily non-degenerate) strictly q-convex domain such that X is a q-convex extension of D. If n-q-1
(0,r)-farm in a neighborhood of D can be approximated uniformly on D by E-valued continuous 5-closed (O,r)-forms on X. Proof. This follows from Theorem 12.11, for D has a basis of strictly q-convex neighborhoods with C°O boundary (of. Theorem 5.7 (ii)). 12.13, Theorem. Let E be a holomorphic vector bundle over an n-dimensional complex manifold X, and D.== X a non-degenerate strictly q-convex domain such that X is a non-degenerate q-convex extension of D (0
(1) For any fDaZO,r(D,E) and each neighborhood U of 3D, there exists a form fXEZ r(X,E) such that fD fx = fD on D\U.

(ii) E0,r(,E) = Z r(X,E)n Eg r(,E),

1)

Proof. Since X is a non-degenerate q-convex extension of D. there is a sequence of non-degenerate strictly q-convex domains Dk c X (k=0,1,...

) such that DO=D, each Dk+l is a non-degenerate strictly

1} Part (ii) will be generalized in Theorem 12.14 (injectivity of the restriction map (12.3)).

113

,I-convex extension of Dk it.

X,

and 00

X = J Dk. k=0

Now let a form feZO rM E) and a neighborhood U of aD be given. Then from Proposition 12.10 (i) we obtain a sequence fkeZO r(Dc,E) (k=0,1,... )such that fk+1

f1=f0 on D\U, and fk+l=fk on Dk-1 if k>1. Then fX:= lie fk is a form with the properties required in assertion (i).

To prove (ii), let a form feZ0 r(,E) with fID a E00 r(,E) be given. Then from Proposition 12.10 (ii) and (iii) we obtain a sequence O ukECO,r-1 (D E) such that 3u = f on D and 4 -u < 1/2 k. Then k' k II11u

k+1 k6 ,Dk

k

u:= lim uk exists uniformly on the compact subsets of X, and solves the equation

Cu=f on X.

12.14. Theorem. Let E be a holomorphic vector bundle over an n-dimensional complex manifold X, and D == X a strictly q-convex domain such that X is a q-convex extension of D (0
HO'r(X,E)

>

HO,r(D,E)

(12.3)

is an isomorphism.

Proof. In view of the Dolbeault isomorphism (Theorem 2.14) it is sufficient to prove that the restriction map

HO'r(X,E)

is an isomorphism

HO'r(D,E)

(12.3')

.

In view of a lemma of M.Morse (of. Proposition 0.5 in Appendix B) and Observation 4.15, we can find a strictly q-convex domain G -= D with Coo boundary such that both X and D are non-degenerate q-convex extensions of G. Then it follows from Theroem 12.13 (ii) that the restriction map HOO'r(,E) - H00'r(G,E) in injective. Hence (12.3') is injective.

In order to prove that (12.3') is also surjective, we consider a form then we obtain a form E). Applying Theorem 12.13 (i) to

0

PEZO

r'

fX e Z00,r ME) such that f-fX is 3-exact in G, and, by part (ii) of Theorem 12.13, we may conclude that f-fX is 3-exact even on D. U

114

12. 15.-Theorem. Let D cc X be a not,-degenerate. striet.)y c ofly x dut:I-. in an n-dimensional complex manifold (0
a form

such that fD-fD E EO r(, E) and fD = fD on D'\U.

Z

(ii) The restriction map 0 r H1

/2--a0(D, E) --* HO'

r (, E)

(12.4)

is an isomorphism, where

Z0

We mention also the following generalization of this theorem to C°r(% vector bundles (cf. Sect. 0.12):

12.15'. Theorem. Let D == X be a non-degenerate strictly q-convex domain in an n-dimensional complex manifold (0
bundle over b (0
r(D.E) and each neighborhood U of OD, there exists

(i) For any

a form fD E ZO r(D,E) such that fD - fD E EO r(D,E) and fD = fD on D\U.

(ii) The restriction map

ZO r(D,E)/Eo 0(D,E)

HO'r(D,E)

is an isomorphism.

Proof of Theorems 12.15 and 12.15'.

It is sufficient to prove Theorem

12.15'.

First we prove assertion (i). Let fDEZ0 r(,E), and let U be a neighborhood of M. Choose a non-degenerate strictly q-convex domain G c= D such that D\U am G and D is a non-degenerate strictly q-convex extension of G in X. Then f 16 belongs to Z0 O(G,E), and, by Proposition D 12.10 (i), we get a form fD E ZO r(D,E) with f0-fD EE' = 0(G,E) and fD = fD on D\U. By Theorem 12.13 (ii), fD -fDbelongs even to E0 r(D,E). Hence part (i) is proved. Assertion (ii) follows from (i) (surjectivity of (12.4)) and Proposition 12.9' (injectivity of (12.4)). 0

1)5

12.16 Theorem (Andreotti-Grauert finiteness theorem). Let E be a holomorphic vector bundle over an n-dimensional complex manifold X. If X is q-convex (0
dim HO,r(X,E) < 00

for all n-q
(12.5)

If X is completely q-convex (O
for all n-q
HO,r(X,E) = 0

(12.6)

Proof. If X is q-convex, then, by 'a lemma of M.Morse (cf. Proposition

0.5 in Appendix B) and Observation 4.15, we can find a strictly q-convex

domain D - X with Coo boundary such that X is a non-degenerate q-convex extension of D.

It follows from Theorems 12.14 and 12.15 (ii) that

dim HO,r(X,E) = dim HO,r(D,E) = dim HO 0

0

12-

D,E).

By Theorem 11.2, this implies (12.5). If X is completely q-convex, then this domain D can be chosen to be completely strictly q-convex, and, by Theorem 12.7, we obtain (12.6).

116

CHAPTER I 'V.

THE CAU C H Y- R I EMAN N

EQUATION ON ca-CONCAVE MANIFOLDS

Summary. Sectiont13-15 are organized analogously as Chapter II. In Sect.

13 a local Cauchy-Fantappie formula for non-degenerate strictly

q-concave domains is constructed, which yields local extension of holomorphic functions (local Hartogs extension phenomenon), as well as local solutions of au = f0 r for all 1
14, first we prove

1/2-Holder estimates for these local solutions, repeating word for word the arguments from Sect.

9.

Then, using the same arguments as in Sect.

11, from these estimates we deduce the following version with uniform estimates of the Andreotti-Grauert finiteness theorem: If D is a non-degenerate strictly q-convex domain in a complex manifold X and E is a holomorphic vector bundle over X, then dim H1

1/20

O
(D,E):= Z O

O, r

(D,E)/E1/2--->0(D,E).

O, r

no for all In Sect.

15 we

introduce the concept of a q-concave extension of a complex manifold, and prove that the Dolbeault cohomology classes of order O
Sections 16-19 are devoted to the Dolbeault cohomology of order q of q-concave manifolds. In Sect. 16 we prove that the extensions of such cohomology classes along a q-concave extension are uniquely determined (if exist). Sect. 17 is devoted to the special case of (linearly) concave domains X C Cn. Here we establish the Martineau isomorphism between Hn,n-1(X) and the space of holomorphic functions on the dual domain of X, where the emphasis is on the boundary behavior of this isomorphism. In Sect. 18 we prove the Andreotti-Norguet theorem. The main assertion of this theorem can be formulated as follows: If D is an n-dimensional complex domain which is both strictly q-concave and strictly (n-q-1)convex, then dim HO'q(D,E) = oo. Further, a version with uniform estimates of this theorem is obtained. In Seat. 19 we prove the following

117

version with uniform estimates of the Andreotti-Vesentini separation theorem: For any non-degenerate strictly q-concave domain D in a compact complex manifold X and all holomorphic vector bundles E over X, the

space EO,q(b,E) is closed with respect to the max-norm. Then, as a consequence, we obtain the classical Andreotti-Vesentini separation theorem.

13. Local solution of du=f0

r

on strictly q-concave domains with

1
13.1. Definition. [U,g,(Q,H,D] will be called a g-concave configuration in Cn (1
(a) H = H(z), zCn, is a function of the form

H(z) = H'(z) + M

lzj12,

(13.1)

.1=q+2

where H' is a holomorphic polynomial in z¢Cn and M is a positive number.

(b)

rp(z)

< 0

for all zeU with Re H(z) = 9(z) = 0.

(c)

D = {zEU: 94z)<0, (Q(z)<0, Re H(z) < 01 1 0 .

(d)

d Re H(z) # 0

for all zEU with Re H(z) = 0,

d Re H(z) A d (p(z) # 0 d Re H(z) A d9(z) # 0

for all zeU with Re H(z) = (P(z) = 0, for all zeU with Re H(z) _ 9(z) = 0.

If (U,g, (P,H] is so that [U,-4,(p] defines a q-convex configuration,

conditions (a),(b), and (d) are fulfilled, and the domain D:= (zeU: 9(z)<0, (,(z)<0, Re H(z) < 01

is non-empty, then we say [U,Q, ,H] defines the q-concave configuration

[U,4, (p,H,D]. A q-concave configuration (U,g,cp,H,D] will be called of type I, III or IV according as [U,-g, (P] defines a q-convex configuration of type I,

118

II, III of IV.

II,

=0 D

y

0

type I

type II

type III

----- (P = 0

""""' Re H = 0

type IV

FIGURE 4: q-concave cofigurations

13.2. Lemma. Let -P:X -) IR be a (q+l)-concave function without degenerate critical points on an n-dimensional complex manifold (0
[h(V), Poh-1, Izl2-r 2, H]

defines a q-concave configuration in Cn.

This configuration is of type 1

if d4(y) & 0, and of type II if da(y) = 0.

Proof. By Lemma 7.5, there exist holomorphic coordinates h:V -i Cn in a neighborhood V of y such that h(V) is the unit ball, h(y) = 0, [h(V),-goh-l,Iz12-r2]

and, for any 0
defines a q-convex configura-

tion, which is of type I if dg(y)¢& 0, and of type II if da(y) = 0. Fix 0
-goh-1

is normalized (q+1)-convex (of. Def. 7.2), there are constants C0 with +1 n Izj)2 -Re F 1(z,0) > Oh-1(z)) + p1HIzjI2 - C Qoh 3=1 j=q+2

for all zeh(V). Setting

n H(z)

(z,0) + (C+A) y Iz.12

F

Qoh

j=q+2

we obtain a polynomial of the form (13.1) such that Re j(.) > AIz12 for

119

all zth(V) with 9(hW) > 0. Hence, (h(V),goh-llzl2-r2,H-?r2/2] fulfils all conditions in order to define a q-concave configuration,

except for (possibly) condition W. By Lemma 0.3 in Appendix B, for almost all complex-linear maps L:Cn- )C (any real-linear map is the real part of some complex-linear map), the function

r2

/

Re

f

+ L(z)

H(z)

11

2

does not have degenerate critical points. By Proposition 0.4 in Appendix B, the same is true for the restrictions of this function to the surface "(p=0" as well as to the smooth part of "9=0". This implies that, for

almost all complex-linear maps L:Cn+C and almost all real numbers the function

fir2 H(z):= H(z) - - + L(z) + b,

ztCn,

2

fulfils condition (d) in Definition 13.1.

If, moreover, L and e are

chosen sufficiently small, then H fulfils also the other conditions

in this definition. Q

13.3. The set Div(H). Let H be of the form (13.1),

i.e.

n

H(z) = H'(z) + M E IzjlztCr', j=q+2

where H' is a holomorphic polynomial and M is a positive number. Definition. By Div(H) we denote the set of all n-tuples

V = (V1__ vn) of complex-valuedC1 functions vj:Cn,,n-+C which are obtained by the following Construction. Take holomorphic polynomials v'. = v'(z,x) (j=1,...,n) in (z,x)ECn)Cn such that

n

H'(x) - H'(z) _

v 120

v

for j=4+2,...,n.

Remark. For any vtDiv(H), we have the relation

= H'(x) - H'(z) + M : (Ix12 - (z12 + zjxj - xjz j=q+2

and hence Re = Re H(x) - Re H(z).

13.4. Canonical Leray data

(13.2)

and maps. Let [U,9, (Q,H,D] be a q-concave

configuration in Cn (0
trl = 9, xr2 =(Q, 14

Yj = {z(U: %rj(z)=0),

= Re H, and

Dj = {zEU: yrj(z)<0}

for j=1,2,3. Then D = D1nD2,D3 is a domain with piecewise almost C1 boundary, and (Y1,Y2'Y3) is a frame for D (cf. Sect. 3.1). Proposition. Let (wl,w2'w3) so that

w1 = w1(z,x) =

w*(x,z)

for some w E Div(-q)

(13.3)

w2 = C(P w3EDiv(H)

(cf. Sect. 7.6 for the definition of Div(-9), and Remark 2.10 for the 7(4). Then there is a constant ct>0 such that

definition of

Re<w1(z,x),x-z> > 9(x) - 9(z) + oL(x-zI2

(13.4)

for all z,xEU. Moreover, then <wj(z,x),x-z>

0

(13.5)

for all x4Yj, ztD3, and j=1,2,3. Hence (wl,w2'w3) is a Leray datumfor (D; Y1,Y2,Y3). Proof.

(13.4) follows from (7.4). (13.5) follows from (13.4) if j=1,

and from (13.2) if j=3. If j=2, then (13.5) follows from the fact that D2 is convex and, for any the set {zeCn: <w2(z,x),x-z>=0} is the complex tangent plane of 8DZ=Y2 at the point x. 0

Definition. We say (wl,w2,w3) is a canonical Leraydatumfor [U,9,(Q,H,D] (or for D) if (13.3) holds true, and we say v is a canonical Leray map for [U,9,(P,H,D] (or for D) if v is the canonical combina-

121

tion (cf. Sect. 3.4) of some canonical Leray datumfor D.

13.5. Lemma. Let [U,V, lp,H,D] be a q-concave configuration in Cn,

D1,D2,D3 be as in Sect. 13.4, and let

K:=

22\D1

)).

let

Then the follow

ing assertions hold true: (i) The domains D3 and D3nD2 are pseudoconvex.

(ii) For any neighborhood W c U of K, there exists a strictly q-convex domain G with K cc G cc W such that U is a q-convex extension of G (cf. Def.

12.1).

(iii) If n-q-l
Proof of part (i). This follows from the fact that U and D2 are convex, and Re H is a continuous plurisubharmonic function (cf., for instance, Sect. 1.4 and 1.5 in [H/L]).

Proof of part (ii). Since K = {zEU:p(z)
< 01,

there exists E>0 with

K =c

(P(z)<E,-g(z)<&,- Re H(z) < F,1 cc W,

and we can take ey->O so small that, for the function

iy (z):= max (

Re H(z) - L + cx lz12

ztU,

we have K cc (zEU: j (z)<0) cc W. Since the functions p(z)-£, and - Re H(z) - 6 + (xiz12 are strictly plurisubharmonic with respect to 21,...,zq+1, it follows from Corollary 4.14 that, for each )B>0, the function

t

(z):= maxp(p(z)-E, -R(z)-E,

- Re H(z) - £ + oclzl2),

z6U,

is also strictly plurisubharmonic with respect to z1,...,zq+l. In particular, y% is (q+l)-convex. Without loss of generality, we can assume that(p is an unbounded exhausting function for U.

Then also y is an

unbounded exhausting function for U. Thus, for any sufficiently small p>O, the domain G:= {zEU: L$(z)
12.12. 13.6. Lemma. Let [U,4, if,H,D] be a q-concave configuration in Cn (1
Leray map for D. Then, for any continuous (O,r)-form f on D, the follow-

122

ing

assertions hold true (for the definition of the operators Lv,LZ,

L3,Li3, see Sect. 3.5): v (i) If O
(ii) If 1
If r=0,

i.e.

f is a function, then LZf

is holomorphic in D and admits a holomorphic extension into D2.

L13f(iii) =

If 0
If r=q-1 and of = 0, then also

0.

Proof of part (i). Since w1(z,x) and w3(z,x) depend holomorphically on xi....,xq+1' this follows from Corollary 3.13 (iii). Proof of part (ii). Since w2 is independent of z, the map v2(z,x) = w2(x)/<w2(x),x-z> depends holomorphically on zED, and, by (13.5), it admits a holomorphic extension into D2.

If r>1, this implies that

LZf = 0, by Corollary 3.13 (ii). Now let r=0. Then, by Proposition 3.6, n-1

LZf(z) =

1

'o (2ni)

zED,

f f(x)det(v2(z,x),o 2((z, ))A w(x), S2

and we see that LZf is holomorphic in D and admits an holomorphic extension into D2.

Proof of part (iii). If 0
Now let r=q-1 and of=0. Recall that, by Proposition 3.6,

L13f(z) =

f(x)A(L13)q-1(z,x,t),

J

S13)
n-q-1

(L13)q-1 = A det(v13'dtvl3'av13, xv13)A

W(x).

(13.6)

Set Wz = (zEU: 4(x)>Q(z),Re H(x)>Re H(z)) for all zaD. Then any Wz is a neighborhood of D2\(D1uD2), and it follows from (13.2) and (13.4) that v13(z,x,t) is defined and continuously differentiable for

all zeD, xeWz, and teA13. Hence the right hand side of (13.6) defines a continuous form F(z,x,t) for zED, xEWz and t(L 13. Since v13(z,x,t) depends holomorphically on x1....,xq+1, we have relation

123

bxF(z,x,t) = 0

(13.7)

for all zED, xEWz, and teL13. Let Pq_1'be the set of all strictly increasing collection I=(il,...,iq_1) of integers 1
F(z,x,t) = Y- FI(z,x,t)ACJ(x)Adt1 dzl, IePq-1

where dz1:= dzi A...Adzi

if I=(i1,...,iq_1). Then q-1

1

L13f(z) _

-_

IePq-1

[ f dti J f(x)AFI(z,x,t)nc.1(x) Adzl

013

S13

for all zeD. Therefore, it is sufficient to prove that, for all

(z,t)EDxQ13 and IEP 9-1'

J f(x)AFI(z,x,t)AW(x) = 0.

(13.8)

S13

To do this, we fix (z,t)ED* A13 and Ir-Pq_1. By (13.7), the form

is 3-closed on Wz. Since Wz is a neighborhood of D2\(D1uD3 and

is of bidegree n-q-1, from Lemma 13.5 (iii) we get a

sequence FI of 3-closed continuous (O,n-q-1)-forms on U such that

FI -0

uniformly on D2\(D1uD3). Since S13 <. D2\(D1uD3), it

follows that

r

fAF1(z,x,t)A w(x) = lim f f(x)AFI(x)A w(x).

(13.9)

k

J

xES13

xeS13

Since the forms f and FI are 3-closed, the forms f(xjAFi(x)ACJ(x) are closed on S1. Since S13 is the boundary of S11 this implies, by Stokes' theorem,

f f(x)AFI(x)A LO (X) = 0 S13

124

for all k. Therefore, (13.6) follows from (13.9). Q 13.7. Lemma. Let [U,9, t,H,D] be a q-concave configuration in Cn (1
Mr:ZD r(D) -* Z0 r(D2fD3)

which is continuous with respect to the Banach space topology of ZOO, r (D) and the Frechet space topology of Z00 r(D2nD3) such that

v

= Mrf D L23f

for all fEZ0 r(D) (for the definition of L23, see Sect. 3.5).

Proof. By (13.5), the map v23(z,x,t) is defined and continuously differentiable for all zED2nD3, xaS23 and ten23. Therefore, by setting

r

(n-1) Mr =

n-r 1

r

,(x)

(27ri)n

we obtain a continuous differential form Mr = Mr(z,x,t) defined for zED2nD3, xcS2J, and

23. Put

M f =

I

r

(x, t)ES23'A23 for all continuous differential forms f on D. Since S23n(D2nD3) = S , it is clear that in this way continuous linear operators

Mr:C r(D)

) C r (D2nD3)

are defined. By Proposition 3.6, MrfID = L2 3f for all fECU r(D). Fix f6Z0 r(D). It remains to prove tnat aMrf = 0.

In the same way as

i) We do not consider the case n-l
125

in the case of Proposition 3.9, we see that azMr = xMr+1' where Mr+1:= 0 if r=n-2. Since Mr+1 contains the factor dtln w(x) and hence f"axMr+1 =

dx,t(f"Mr+1),

aMrf(z) _

this implies that

dx t[f(x)"Mr+1(z,x,t)]

- J

(x,t)ES23"L23 for all z(D2nD3. Hence, by Stokes' formula (aS23 = 0 ')'

8Mrf(z) _

+

I

f(x)AMr+l(z,x,t)

S23xaL23 for all

Since f(x)IMr+i(z,x,t) is of degree 1 in t whereas

aA23 is of dimension zero, it follows that 5Mrf = 0.

13.8. Theorem (Hartogs extension). Let [U,9, cp,H,D] be a 1-concave configuration in Cn,

D:= {zeU: ((z) < 0 and Re H(z) < 0}.

Then any holomorphic function in D admits a holomorphic extension into D.

IIIII

D

FM ReH=0

Proof of Theorem 13.8. Since, for any sufficiently small £>0, [U,g+E, t+s,H+E] also defines a 1-concave configuration, it is sufficient to prove that the theorem holds true for functions which are holomorphic in some neighborhood of D. Let f be such a function, and let v be a canonical Leray map for [U,Q, (Q,H,D]. Then, by Lemma 13.6, Lvf = L3f = L1 3f = 0. Hence the piecewise Cauchy-Fantappie formula

126

(3.18) then takes the form f = L2f + L?3f on D. We complete the proof by the remark that,

in view of Lemma 13.6 (ii) and Lemma 13.7, both L2f and

1,23f admit holomorphic extensions into D.

13.9. Corollary. Let 9:X -* R be a 2-concave function without degenerate critical points on an n-dimensional complex manifold X (n>2), and Then, for any point yEaD, there exists a

let D:=

neighborhood U of y such that any holomorphic function in UnD admits a holomorphic extension into U.

Proof. This follows from Theorem 13.8 and Lemma 13.2. 0 13.10. Theorem. Let [U,9, y,H,D) be a q-concave configuration in Cn (1
(i) For any 8-closed continuous (O,r)-form f in D, there exists a continuous (0,r-1)-form u in D with au = f. (ii) Set H(z)<0}.

D:=

Let v be a canonical Leray map for D, Tv the corresponding Cauchy-Fantappie operator introduced in Sect. 3.10, and

Mr:Zg r(D) -- ZQ r(D) the continuous linear operator from Lemma 13.7. Then, for any fcZ0 r(D), we have the representation

f =dTvf + Mr f

in D

.

(13.10)

Moreover, then there exists a continuous (O,r-1)-form g on D with Mrf = dg on D. Hence u:= Tvf + g solves the equation au = f in D. r(D). Then, by Lemma 13.6, Lif = L2f Proof of part (ii). Let fEZ0 O Lif = Li3f = 0, and hence the piecewise Cauchy-Fantappie formula (see Theorem 3.12) takes the form f = 3Tvf + L23f in D. Since L2 3f = Mrf

on D (cf. Lemma 13.7), this implies (13.10). Since, by Lemma 13.5 (i) and Theorem 5.3, D is completely pseudoconvex, it follows from Theorem 12.16 that M f = ag for some continuous (O,r-l)-form g on D. r

If f assertion (i) is contained this special case by means the proof of Theorem 12.13 Proof of part (i).

admits a continuous extension onto D, then in (ii). The general case follows from of a modification of the arguments given in (ii) (the approximation theorem for holomor-

127

phic functions which we need in the case r=1 follows from the extension

Theorem 13.8 and the fact that D is pseudoconvex). []

13.11. Corollary. Let 9:X -> IR be a q-concave function without degenerate critical points on an n-dimensionsal complex manifold (1
D:= {zsX:

neighborhood U of y such that, for any 5-closed continuous (O,r)-form f on DnU, there exists a continuous (0,r-1)-form u on DAU with au = f.

Proof. This follows from Lemma 13.2 and Theorem 13.10 M.

14. Uniform estimates for the local solutions of the 5-equation obtained in Sect. 13, and finiteness of the Dolbeault cohomology of order r with

uniform estimates on strictly q-concave domains with 1
In Sect. 13 q-concave configurations were introduced, the notion of a canonical Leray map v for such configurations was defined, and we saw (Theorem 13.10) that the corresponding Cauchy-Fantappie operator Tv (of. Sect. 3.10) together with the extension operator Mr from Lemma 13.7 solves the 5-equation for forms of bidegree (O,r) with 1
14.1. Theorem. Let [U,q, (p,H,D] be a q-concave configuration in Cn

(10, and

Re H(z)<-t).

DE:=

Then there exists a constant C
MTvf%l,2,DE < CIIfU0,D'

(14.1)

(for the definition of the norms, see Sect. 0.11). If [U,y, (,H,D] is of type II and yEU is the critical point of j), then, moreover,

II T°f 11

126

0DI.


sup

L sesupp f

I z-y IJ II fIl O, D'

(14.2)

sup

z, xeDF

IITvf(z) - iV2(x)II

< C(i +

in

rl)r1/2NfM0

(14.3)

D

Iz-xl

Iz-yI
for each r;-U.

DE

D\DE

----e == 0, 0 .

R

H

O,

(pRe=H-

FIGURE 5: The domains D and DE in Theorems 14.1 and 14.2

Proof of Theorem 14.1. This is a word for word repetition of the proof of Theorem 9.1, with the following two exceptions: 1) The operator TV now is of the-form

T°=BD+R1 +RZ+R3+R12 +R13' and we have to add the remark that since S3nDE = S13n&E =0

,

estimates

(14.1)-(14.3) hold true also with R3 and R13 instead of TV.

2) The map w(z,x) now does not belong to Div(Q), but it is of the

form w(z,x) = -w*(x,z)

with w EDiv(-Q),

(14.4)

and, instead of Lemma 9.2, we have to use the following Lemma. If w(z,x) is of the form (14.4) and h(z,x) = Im<w(z,x),x-z>, then assertions (i)-(iv) in Lemma 9.2 hold true.

128

-E

Proof of the latter lemma. If tj are the real coordinates on Cn with xj=tj(x)+itj+n(x), xeCn, then immediately from the definition of Div(-9) and the relation h(z,x) = Im<w*(x,z),z-x> one obtains that

-Q(a)

rag(z)

h(z,x) = 1

ti(x z)

tj+n(x-z) -

Ill

+ 0(Ix-Z1 2

atj+n

atj

and

a89(z)

l

d h(z,x) = 0_?_(_-_)dtj+n(x-z) - -dt (x-z)I + O(ix-zI) x ate+n , ati J .

It is easy to see that the last two relations yield assertions (i)-(iv) in Lemma 9.2.

14.2. Theorem (of. Figure 5). Let [U,Q, (P,H,D] be a q-concave configura-

tion in Cn (l0, and

DL:= {zc.U: (Q(z)<-&,Re H(z)< -E}.

Then, for all 1
R:ZO r(D)

+ p"c-1(Dt)

such that

aRf=fI

(14.5)

for all feZg r(D).

Proof. Let D,v,Tv, and Mr be as in Theorem 13.10. Since D is pseudoconvex (Lemma 13.5 (i)), we can find a strictly pseudoconvex domain G with Coo boundary such that DL == G cc D (cf. Theorem 5.7 (i)). Since any strictly pseudoconvex domain in Cn is corm letely strictly pseudocon-

vex (cf. Theorem 5.3), then, by Theorem 12.7, there exists a continuous

linear operator R :ZOO r(G) - C1/r_1(G) such that 8%f = f for all

130

fiZ0 r(G). Set R = Tv + RGMr. Since Mr acts continuously from Z0 into Z0 r(D), RG acts continuously from Zp r(G) into C1/r-1(G), and G C = D, we see that RGMr is bounded as an operator from Z0,r(D) into

CO,r_1(D.). Together with estimate (14.1) in Theorem 14.1 this implies that RE is bounded as an operator from Z r(D) into CO,r_1(DE). Relation (14.5) follows from (13.10).

14.3. Definition. Let D == X be a domain in an n-dimensional complex manifold X. and 1
concave function s:U -3IR in some neighborhood U of aD such that

D n U =

If, in this case, the function So can be chosen without degenerate critical points in U, than we say the boundary of D is non-degenerate

strictly q-concave with respect to X.

MWM

D

X\D aDnX ax

FIGURE 6: The boundary of D is strictly q-concave with respect to X

Remarks to this definition.

I. Of course, the boundary of any strictly q-concave domain in a complex manifold is q-concave with respect to this manifold, because, by definition, q-concave domains are relatively compact. In distinction to this, a domain with q-concave boundary with respect to some manifold

131

need not be relatively compact in that manifold. II. In Sect. 12 we did not introduce the notion of a (not necessarily relatively compact) "domain with strictly q-convex boundary", because the theorems which we cm, prove for such domains are not nice enough, in our opinion.

III. Let X be a complex manifold, and D c X a domain whose boundary is non-degenerate strictly 1-concave with respect to X. Then any Car) vector bundle over 15 (for the definition of such bundles, see Sect.

0.12) admits a holomorphic extension into a neighborhood of D. This follows from the fact that, in view of Corollary 13.9, the transition functions of such bundles admit holomorphic extensions. In view of this circumstance, in distinction to Sections 11 and 12, in

Sections

14 and 15 we do not formulate results for C°'(1 vector

bundles.

Theorem 14.4 (Hartogs extension). Let X be an n-dimensional complex manifold with n>2, and D C X a domain whose boundary is strictly 1-ooncave with respect to X. Further, let E be a holomorphio vector bundle over X. Then there exists a neighborhood U of aD such that the restriction map

O(DuU,E)

(?(D\U,E)

(14.6)

is an isomorphism.

Proof (of. Fig. 7). By Lemma 13.2, for any point ycaD, we can find a neighborhood Gy of y, a 1-concave configuration [Uy,s>y, (Qy,Hy,Wy] in Cn,

and a biholomorphic map h

from a neighborhood of Uy onto a neighborhood y of Gy such that D Gy = hy(Wy) and

= hy({zaUy: Ty(z)
(14.7)

For every yflaD, we choose a number by>0 sufficiently small, and set

WY:= {zEWy: Qy(z)<- E}. Then

[Uy,Qy+sy, ,py,Hy,Wy] is also a 1-concave

configuration. If the neighborhoods Gy are chosen so small that E is

trivial over each 0, then it follows from Theorem 13.8 that the restriction maps

132

(14.8)

(7(Gy,E)

are surjective. Since aD is compact, we can find a finite number of points y1,...,YNE6D with aD C Gy1 u...uGyN. Since the sets hy(W') are

relatively compact in D, we can find a neighborhood U of aD such that

h

yl

(W'

y1

) u ... u hyN ( W' ) C D\U

and

DuU c (D\U) L) G

yl

yN

u ... U G yN

Now it follows from the surjectivity of the restriction maps (14.8) that (14.6) is surjective. Further, we can choose U so that the intersection of D\U with each connected component of DuU is non-empty. Then (14.6) is also injective.

D\Gy

IIIIIIIIIIIIIillllllIi,I

hy(Wy) by (Wy\W3, )

Gy\hy(Wy)

FIGURE 7: To the proof of Theorem 14.4

14.5. Theorem. Let D - X be a non-degenerate strictly q-concave domain in an n-dimensional complex manifold X (1
dim

oo,

where

133

N12--.

0(D,E):= Z r(D,fi)/EO,r 0($,S)

Proof. By Lemma 13.2 and Theorem 14.2, we can find open sets Uj (j=1....,N
ZQ r(S,E) into C1/r-1(UjeD,E) such that aT3f = f on U Z for all r(D,E).

Further, taking into account Lemma 1.16, we can find open

sets Uj - D (j=N+1,...,M
aT3f = f + Of on U

0/2(0.,E)

such that, for all f&Z0,r(b'E),

.

Choose a Coo partition of unity {xj}M subordinated to {Uj}M, and set

T

j)

XjT3

and

K =

axjeT3 + j=N 1

3

Then T is a bounded linear operator from ZO r(D,fi) into

K is a bounded linear operator from Z r(D,E) into

CO/r_1(D,E),

CO/r(D,E), and

STf = f + Kf for all fEZ0 r(D,E)_ This implies the assertion of the theorem (of.

Remark III following Definition 11.1). El

134

15. Invariance of the Dolbeault cohomology of order O
15.1. Definition. Let X be an n-dimensional complex manifold, and O
a) If D c X is a domain, then we say X is a g-concave extension of D if the following conditions are fulfilled: (i) aD is compact.

(ii) Every connected component of X has a non-empty intersection with D.

(iii) There exists a (q+l)-concave function j);U --4 R in a neighborhood U C X of X\D such that DnU = {zeU: 9(z)
0 < o. < sup

(z),

the set {zeU: O<e(z)
in this case, the function q in condition (iii) can be chosen

without degenerate critical points in U, then X will be called a non-degenerate q-concave extension of D.

b) If D C G are two domains in X, then we say G is a strictly q-concave extension of D in X if G is a q-concave extension of D such that, moreover, the set G\D is relatively compact in X and the following strengthening of condition (iii) is fulfilled:

(iii)' There exists a (q+l)-concave function g:U - R in some neighborhood U of G\D such that

DnU = {zEU: Q(z)
If,

and

GnU = {zEU: Q(z)
in this case, the function p in condition (iii)' can be

chosen without degenerate critical points in U, the G will be called a non-degenerate strictly q-concave extension of D in X.

Remarks to this definition. I.

If X is an n-dimensional complex manifold and D C G are two

135

domains in X. then G is a strictly q-ooncave extension of D in X if and only if X\b is a strictly q-convex extension of X\d in X. II.

If Q: U -> I is as in condition (iii), then, for e11

0
the domain Du{zcU: Q(z)
domain Du{zcU: P(z)<%1-

D

D

X\D

G\D dX

G is a strictly q-concave extension of D with respect to X

X is a q-concave extension of D

FIGURE 8

15.2. Theorem (Hartogs extension). Let E be a holomorphic vector bundle over a complex manifold X, and let D C X be a domain such that X is a 1-concave extension of D. Then the restriction map

U(X,E)

' (i(D,E)

is an isomorphism.

Proof. In view of a lemma of Morse (Proposition 0.5 in Appendix B) ) of domains

and Observation 4.15, we can find a sequence Dk (k=0,1,...

136

in X such that D0 c D, each Dk+1 is a non-degenerate strictly 1-concave extension on Dk, and

00

X=UI

Dk.

k=0 If follows from Theorem 14.4 that all restriction maps

((Dk+1'E)

) (9(Dk,E)

are isomorphisms. Since D0 c D, this implies that the restriction map

> (9(D,E)

(') (X, E)

is an isomorphism. D

15.3. Corollary. For any 1-concave complex manifold X and all holomorphic vector bundles E over X, we have

dim (((X, E) < on.

Proof. Since X is 1-concave, we can find domains D -= G -= X such that X is a 1-concave extension of G, and G is a 1-concave extension of D. Then, by Theorem 15.2, the restriction maps

UM E)

)

(9(G, E)

and

(9(G, E)

) (90, E)

are isomorphisms. Since D and G are relatively in X, this implies that all sections in O(G,E) and (9(D,E) are bounded. Hence (9(G,E) and (9(D,E) form Banach spaces with respect to the sup-norm. Since D is relatively compact in G, the restriction map

(9(G, E)

+ (2(D, E)

is compact. Since this map is also an isomorphism and compact isomor-

137

phisms between Banach spaces exist only in the finite dimensional case,

this implies the assertion. D

FIGURE 9: q-concave extension element (A1,A2,V] in X

15.4. Definition. let X be an n-dimensional complex manifold, and l
[A1,A2'V] will be called a g-concave extension element in X if A1,A2, and V are domains in X such that the following conditions are fulfilled (of. Def.

13.1 for the definition of a q-concave configuration):

(i)

Al C A2,

(ii)

A2\Al cc V cc X.

(iii) There exist q-concave configurations [Uj,yoj,(pj,Hj,Dj] in Cn (j=1,2) with U.=U1=U2, 'p:= pl ='P2, and H:=H1=H2

such that, for some

biholomorphic map h from U onto some neighborhood of V.

V=

Ip(z)
and

ViA. = h(Di)

If,

138

for j=1,2.

in this case, [U,Qj, p,H,Dj] is of type Tj, then (A1,A2,V]

will be called of type (T1,T2).

If Al C A2 are domains in X, then we say A2 can be obtained from Al by means of a g-concave extension element in X if there exists a domain V C X such that [A1,A2,V] is a q-concave extension element in X.

15.5. Lemma. let X be an n-dimensional complex manifold, and let D C G c X be domains such that G is a non-degenerate strictly q-convex extension of D in X (1
D=AOCAIC ... CAN =G such that, for all 1<j
Moreover, for any open covering {U0}1 1 of X, these domains Ai can be chosen so that, in addition, for each 1<j
and a domain V == Ui such that (Aj-1,AjV] is a q-concave extension element in X.

Proof. Since (q+l)-concave functions do not admit local minima, this is a straightforward modification of the proof of Lemma 12.3, using

Lemma 13.2 instead of Lemma 7.5. []

15.6. Lemma. Let X be an n-dimensional complex manifold, [A1,A2'V] a q-concave extension element in X (2
exists a form f2EZO,r(A2'E) such that over, f2=f1 on Al\U.

(ii) If A

is relatively compact in X, and

is 1/2-regular at

1

Z0 O,r(A2,E), then d is 1/2-regular also at ZO r(A2,E).

1/2-*0 (A2,E)

(iii) EO r

0

1/2--+0 (AV

= ZO r(A2,E) n 'O

,r

E).

Proof. The proofs of parts (i) and (ii) are repetitions of the proofs of the corresponding assertions in Lemma 12.4, where instead of Corollary 12.5 (i) we have to use Theorem 14.2. The proof of part (iii) is a repetition of the proof of Lemma 12.4 (iii) in the case n-q+l
139

15.7. Proposition 1). Let X be an n-dimensional complex manifold, let D S X be a domain whose boundary is non-degenerate strictly q-concave with respect to X 2)(2
E1/i2,--0(5,E)

O,

= Z00 r(5,E)n E0 r(D,E) 0

for all 1
Proof. Since the boundary of D is non-degenerate strictly q-convex with respect to X, we can find a domain G C D such that D is a non-degenerate strictly q-concave extension of G in X. Then, by Lemma 15.5 and Lemma 15.6 (iii),

El/2-400,E)

r(5,E)0

Since G C D and, by Theorem 1.13 (regularity of a),

E0 rM E) =

this implies the assertion. 0

15.8. Proposition 3). Let X be an n-dimensional complex manifold, let D C G C X be domains such that G is a non-degenerate strictly q-concave extension of D in X (2
and F = f on U\D.

(ii) If D is relatively compact in X and a is 1/2-regular at Z0 r(D,E), then 5 is 1/2-regular also at Z0 r(d,E). (iii)

E1/2--).0(G,E)

= Z0 r(0,E)n E0 r(D,E).

1) This proposition is contained also in Theorem 15.12.

2) of. Def. 14.3 Assertion (i) of this proposition is contained also in Theorem 15.9. and part (iii) follows from Theorems 15.11 and 15.12.

140

Proof. Part (ii) follows immediately from Lemma 15.5 and part (ii) of Lemma 15.6. Taking into account Proposition 15.8, assertions (i) and (iii) follow also from Lemma 15.5 and the corresponding assertions

in Lemma 15.6. C]

15.9. Theorem. Let E be a holomorphic vector bundle over an n-dimensional complex manifold X, and let D c X be a domain such that X is a non-degenerate q-concave extension of D (2
form FcZ0,r(X,E) such that f-F E

and F=f on D\U.

Proof. Since X is a non-degenerate strictly q-concave extension of D, ) of domains Dk c X such that

we can find a sequence Dk (k=0,1,...

DO = D, each Dk+1 is a non-degenerate strictly q-concave extension of

Dk, and 00

X= I

I

Dk.

k=0 Let feZ0 r(D,E). Then, by Proposition 15.8 (i), we can find a sequence fkEZp r(Dk,E) (k=1,2,...

fk fk_lEE0,r(Dk,E)

)

such that

and

f1=f on D\U,

and

fk = fk-1 on Dk-2 if k>2.

Now F:= lim fk has the required properties. Q

15.10. Lemma 1). Let E be a holomorphic vector bundle over an n-dimensional complex manifold X, and let D be a domain in X such that X is a non-degenerate q-concave extension of D (2
E0 r(X,E) = Z00 r(X,E)n E00 r(D,E)

Proof. Let feZ0 r(,E) with

for all 1
f'D E EO r(D,E) be given, and let Dk be

as in the proof of Theorem 15.9.

It is sufficient to find a sequence ukECO,r-1(Dk,E) (k=0,1,... ) such that, for all k=0,1,... ,

The assertion of this lemma is contained also in Theorem 15.11.

141

(15.1)

and

Uk = uk-i

on 15k-

(15.2)

l'

where D_1:= 0 . By Proposition 18.7, we can find

u0EC0,r-1(DO,E)

with au0 = f on D0.

Assume, for some m>0, a collection ukeCk- r_1(Dk,E) (k=0,...,m) satis-

fying (15.1) and (15.2) for k=O,...,m is already constructed. Then, by

0 Proposition 15.8 (iii), we can find geC0 r-1(Dm+1'E) with 3g = f on Dm+l' If r=1, then g - um is a holomorphic section on Dm, and, by Theorem 15.2, there exists a holomorphic section h:X

) E with h = g - um on

Dm. Hence, in this case, um+1.- g - h is a section in C0,0(Dm+l'E) with Dum+1 = f on Dm+l and um+l = um on Dm.

Now let r)2. Then g - um

Z0,r-1(D'mE)

and, by Proposition 16.8 (i),

we can find a form vaZO,r-1(Dm+1'E) such that g - um - v = 3w for some

weC0 r_2(DmE). Choose a Coo function x on X with supp x == Dm. Then with dum+1 = f on 5

I// = 1 in Dm-, and

g - v - 5(x w) is a form in C0,r-1(Dm+1'E)

and um+1 = um on Dm-l. 0 m+1

15.11. Theorem. Let D C X be a domain in an n-dimensional complex manifold X such that X is a q-concave extension of D (1
HO'r(X,E)

` HO'r(D,E)

is an isomorphism (for the definition of HO,r, see Sect. 1.14). Proof. In view of the Dolbeault isomorphism (Theorem 2.14) it is sufficient to prove that the restriction map

H00 r(M, E)

) H0, r(D, E)

(15.3)

is an isomorphism. In view of a lemma of Morse (Proposition 0.5 in Appendix B) and Observation 4.15, we can find a domain G c D such that

142

both X and D are non-degenerate q-concave extensions of G. Since the assertion is already proved for r=0 (Theorem 15.2), we can assume that 1
HO'r(,E) - HO'r(G,E) and hence (15.3) is injective. In order to prove the surjectivity, now let fEZ0 r(S,E). Then we can apply Theorem

15.9 to fld and obtain FEZO r(X,E) with f-FEED r(,E). By Lemma 15.10, it follows that f-FEE 0

(D'E).

15.12. Theorem. Let X be an n-dimensional complex manifold, and let D C X be a domain whose boundary is non-degenerate strictly q-concave with respect to X (of. Def. 14.3). Further, let E be a holomorphic vector bundle over X.

Then, for all l
HO,rME)

is an isomorphism, where

0,r 1/2>0 (D,E):= ZO0,r (5,E)/El/2

HO r

>O(S,E).

The assertion on the surjeotivity of this restriction map admits the following strengthening: For any

r' E)

and each neighborhood U of

of aD, there exists FEZ0 r($,E) with f-FEED r(,E) such that, moreover, FI(D\U) = f.

Proof. By Proposition 15.7, (15.5) is injective.

To complete the proof, let

r(D,E) be given. Take a domain G C D

such that D\U = G\U and D is a non-degenerate strictly q-concave extension of G in X. Then f1d belongs to Z00 ,r(6,E) and, by Proposition 15.8

(i), we can find FEZ0 r(5,E) with f-FEED r(G,E) and F = f on D\U. By

Lemma 15.10, it follows that f-F belongs even to EO r(D,E). El

15.13. Theorem (Andreotti-Grauert finiteness theorem). Let E be a holomorphie vector bundle over a q-concave n-dimensional complex manifold X (1
dim HO'r(X,E) < on

for all 0
143

Proof.

In view of a lemma of Morse (Proposition 0.5 in Appendix B)

and Observation 4.15, we can find a non-degenerate strictly q-concave domain D cc X such that X is a non-degenerate q-concave extension of D. Since the assertion is already proved for r=0 (Corollary 15.3), we may assume that 1
dim HO'r(X,E) = dim HO,r(D,E) = dim H0 r(D,E).

In view of Theorems 15.12 and 14.5 (ii), this implies that

dim HO'r(,E) = dim

oo

.

16. A uniqueness theorem for the Dolbeault cohomology of order q with respect to q-concave extensions

Let X be an n-dimensional complex manifold, and let D c X be a domain such that X is a q-concave extension of D (O
) H0,r(D)

H0,r(X)

(18.1)

is an isomorphism. This is not true for r=q. For instance, consider the case r=q=0. Then (16.1) takes the form

) ?(D),

(9(X)

(16.2)

and it is clear that (16.2) need not be surjective (for instance, if X is the complex plane, and D is the unit disc). However, since, by definition, the intersection of D with any connected component of X is non-empty, (16.2) is injective. In the present section we show that this uniqueness property of holomorphic functions admits the following generalization to cohomology classes:

16.1. Theorem. Let E be a holomorphic vector bundle over an n-dimensional complex manifold X, and let D C X be a domain such that X is a Then the restriction map q-concave extension of D (0
144

HO'q(X,E)

> H0'q(D,E)

(16.3)

is injective. If q>1, then even the following more precise statement

q_1(,E)

holds true: If feZ0 q(,E) and ueC0

with du = f on D, then, for

any neighborhood U of aD, there exists a from vEC0 q_1(X,E) such that 5v = f on X and v = u on D\U. The main step in the proof of this theorem is the following

16.2. Lemma. Let CA1,A2'V] be a q-concave extension element in the n-dimensional complex manifold X (1
(X,E) and u1cc0 q_1(AI,E) with au1 = f on

A1. Then, for any neighborhood U of A2\A1, there exists u2EC0 q-1(A2,E) such that au2 = f on A2 and u2 = u1 on Al\U.

Proof. Since, by Lemma 13.5 (1), V is pseudoconvex and hence (cf. Theorem 5.3) completely pseudoconvex, it follows from Theorem 12.16 (or

from Theorem 2.8.1 in [H/L]) that the equation v = f can be solved on V with Now we distinguish the cases q=1 and q>2. If q=1, then ul - v is a holomorphic section of E over AInV, and it follows from Theorem 13.8 that there is a holomorphic section h:V -p E such that h = ul - v on A1nV. Hence, setting u2 = u1 on Al and u2 = v+h

on V, we obtain a continuous section u2:A1uV -> E with Au2 = f on A1U V and u2 = ul on A1. Since A2 C_ A1uV this completes the proof for for q=1.

Now let q>2..We may assume that U == V. Then, by Theorem 14.2, u1 - v = aw on A1nU for some weC0 q_2(A1n U,E). Choose a CO° function x on X such that supp x cc U and

1 on AZ 1. Then, on Aln U, we have

ul - NX w) = v

+ 5((1-X )w).

The two sides of the last equation together define a form u25C0 q_i(A1uU,E) with 3u2 = f on A1uU and u2 = ul on Al\U. Since A2 S A1UU. this completes the proof.

Proof of Theorem 16.1. For q=0 the assertion of the theorem follows from the uniqueness theorem for holomorphic functions. So we can assume that q>1. In view of a lemma of Morse (of. Proposition 0.5 in Appendix B) and Observation 4.15, for every neighborhood U of aD, we can find a domain G C D such that X is a non-degenerate q-concave extension of G, and G\U = D\U. Now the assertion of the theorem follows from Lemma 15.5

and Lemma 16.2. Q

145

17, The Martineau theorem on representation of the Dolbeault cohomology of a concave domain by the space of holomorphic functions on the dual domain

17.1. Definition.

Let X be a domain of the form

X = Cn \ D, where D cc Cn is a convex (with respect to the real-linear structure of n C ) domain such that 06D. Then the domain

X*:= {yaCn: $ 1

for all

(17.1)

will be called the dual domain of X.

17.2. Propositon. If D, X, and Xf are as in the preceding definiton,

then X* is a bounded domain of holomorphy which contains the origin. Moreover, for any point yEaX , there exists a sequence Pk (k=1,2,...

) of (n-1)-dimensional complex planes in Cn which converges

(in the obvious sense) to some (n-l)-dimensional complex plane P in Cn 0 such that

Pkn X*

_

for all k>1,

but yeP0.

If the boundary of D is of class C1, then these planes can be chosen so that, moreover,

PO n X* _ {Y}

Proof.

It is clear that OsX*. That X* is bounded follows from the

fact that OED. To complete the proof, we fix yeaX*. Then, by (17.1), we can find z0eaD with =1. Take a sequence zkeD (k=1,2,... ) which converges to z0, and set

Pk = {xECn: =1}

Then Pk converges to P

0

for k=0,1,...

.

and YEP0. Further, by (17.1), Pkn)(*

for k>1.

It remains to prove that P0nX* _ {y} if aD is of class C1. To do

146

this, we consider an arbitrary point weP0nX*. Then =1 and, since z0eaD, then it follows from (17.1) that wcbR*. Since w4* X*, the plane

{zfiCn: =1}

does not intersect D. Since =1, this implies that this plane is the complex tangent plane of aD at z0 if aD is of class C1. Since the complex tangent plane is uniquely determined, it follows that PO^X* consists only of one point.

Remark. Let D, X. XyeaX*, and Pk be as in Proposition 17.2, and set

00

U:= Cn

I

I

k

0

Pk.

If aD is of class C1, then X*\{y} C U, and there exists a function hECO(J(X*UU) (cf. Sect. 0.11 for the definition of CO) which is singular at y (cf. Definition 17,5 (i) below).

In fact, take holomorphic functions hk on Cn\Pk which have a pole on Pk (k=1,2,...

),

and set

-

00

h(z) _

&khk(z)

(zEU)

k=1

for sufficiently small Ek

0.

17.3. The Fentappie-Martineau map MX. Let X be a domain of the form X = Cn\D, where D c= Cn is a convex domain with OsD. Then, for any feZn,n-1(X), we define a holomorphic function Mxf on X* as follows: Let ysX

Since D is compact, we can find a convex domain Gy a Cn with Coo

boundary such that D c= Gy and

for all

y

(17.2)

Define

MXf(y) =

f(x)

(n-l)n

(21ri)

(1-<x,Y>)

n

(17.3)

bay

147

By Stokes' theorem, this definition is independent of the choice of Gy. The continuous linear map

MX:Zn,n-1(X) -- O(X obtained in this way will be called the Fantappie-Martineau map for X.

17.4. Proposition 1). Let X and MX be as in Definition 17.3. Then

Ker MX - En,n-1(X),

where Ker MX:= (fEZn0 n-1(X): MXf = 01.

Proof. Let fsEn,n-1(X). Then f=5u for some ueC° n-2(X), and, for any fixed yeX#, the form

f(x)

= d

r

u(x)

1

x l (1-<x,y>)n J

(1-<x,y>)n

is exact on aGy, where Gy is as in Definition 17.3. Therefore, it follows from Stokes' theorem and (17.3) that MXf(y) = 0. Now let f E Ker MX.

In order to prove that fEEn,n-1(X), we take a

number R
Iz1
Consider also the larger ball BR+1:= {zECn: Izl
wl(z,x) = (z11 ...1zn)

for xs BR and zEBR+l\BR, and

w2(z,x) = (X1,..

for xEaBR+1 and ecBR+l\BR. Then (wl,w2) is a Leray data for (D;-aBR,aBR+1) (cf. Sect. 3.4), and if F is the (O,n-1)-form on X with f(z) = F(z)Adz1A...P.dzn, then the corresponding piecewise

Cauchy-Fantappie formula (Theorem 3.12) yields the representation 1) The assertion of this proposition is contained also in Theorem 17.9.

148

n-1 1

I

F(x)ndet(z, dz )Adxln... A dxn

F(z) = g(z) -

(27ri)n

n

I

aBR

for all zcBR+1\BR' where g is some form which belongs to E0 O n n-1(BR+1\B R) (the integral over 8BR+1 vanishes, because w2 is independent of z). ,

Since F(x)ndx1n...ndxn = f(x), therefore it is sufficient to prove that the integral n-1 f(x)

j

det(z, dz

)

n

xEaBR

vanishes for all zeBR+l\-R. By Proposition 0.9, this integral is equal to n-1 f(x)

1
,x>-11

n

detl z l Izl2

,

d z Iz(2

xeaBR

Since = 1 for all xeB

and zcBR+1\BR, it follows that the

last integral is equal to (-l)n(2ri)n/(n-1)!MXf(z/Izl2) = 0. O

17.5. Definition. Let D C X be a domain in a complex manifold X, and yeaD.

(i) If h is a holomorphic function on D, then we say, as usually, h is singular at y (or y is singular for h) if, for each neighborhood U of y, the following condition is fulfilled: If W is a connected component of UnD with yeW, then there do not exist a neighborhood V of y and a holomorphic function v on V with v=h on VnW. (ii) If f is a 8-closed (s,r)-form with r>l on D, then we say f is singular at y if there do not exist a neighborhood U of y and a continuous (s,r-l)-form u on UnD such that 8u=f on UnD.

17.6. Definition. Let X be a domain of the form X = Cn\D, where D . Cn is a strictly convex domain with C2 boundary such that OED. Then, clearly, for any yEaX* there exists a uniquely determined point zeaX

149

with =1. Conversely, for each zcaX, there exists a uniquely determined ycaX* with =1. Thus, by means of the equation = 1

a bi,ective map X: ax

)

-ax*

is defined. Obviously, this map is continuous 1).

17.7. Theorem. Let X be a set of the form X = Cn\D, where D - Cn is a strictly convex domain with C2 boundary such that 0ED. Then there exists a continuous linear map

N: b(X*)

> Zn.n-1(X)

such that the following assertions hold true: (i) If MX is the Fantappie-Mertineau map for X, then MxNh = h for

all h(e,(X*). (ii) Let Y be a subset of aX

If he 12(X*) admits a continuous

extension onto X*uY, then Nh admits a continuous extension onto

X1(Y). In particular,

NC0(2 (X*) C Zn,n-1(X) (for the definition of COCA , see Sect. 0.11). Moreover, if heO(X*)

admits a holomorphic extension into some neighborhood of X*uY, then Nh admits a continuous and a-closed extension into some neighborhood of

Xu4Xl(Y). (iii) A function h e(r)(X*) is singular at a point yEaX* if and only

if the form Nh is singular at the point

¢X1(y).

dim [Zn-1(X)/En,n-1(X)] < cc.

(iv)

Proof. Take a strictly convex C2 function 9:Cn -+IR with the follow-

ing properties

:

In the proof of Theorem 17.7 below we shall see that $Xis even a C1 diffeomorphism.

150

(a)

D = {zECn: 9(z)<0};

(b)

BO(O) = min

g(z);

zccn (c) there exists RR or IzI
domain, and Tz:= {xECn: =0} is the complex tangent plane of

aDz at z (cf. Remark 2.10 for the definition of Q ). Hence

= 0

(17.4)

for all z,xCCn with fi(x)
= 0

(17.5)

for all zeCn\{0}. Therefore, setting

V CQ(z)

e



for zECn\{0}, we obtain a C1 map 0Q from &\{0} into itself. It is easy to show that S is even an orientation preserving C1 diffeomorphism from Cn\{0} onto itself. Further, it is clear that

(17.6)

1

for all zcCn\{0}. In particular (cf. Def. 17.6),

Q(z) _ §X(z).

(17.7)

Moreover, since D = {9<0}, it follows from (17.4) that <x,Q9(z)> # 1 if zeX and xeD. Hence 4)(X).C X*. Since, on the other hand, by (17.6), 4

(D\{0}) C Cn\X* and 4) is a bijection of Cn\{0}, we see that

49(X) = X*

and

(

(D\{0}) = Cn\X*.

(17.6)

Using the diffeomorphism 9'Q, now we construct the map N. Let he(9(X*) be given. Since h is holomorphic and, by Proposition 0.10,tJ'(z)n w(y) is closed on M:= {(z,y)E (Cn \{0}) X X*: =l},

151

it follows that also the form h(y) W'(z)A W (Y) is closed for all (z,y) in M. Since, by (17.6) and (17.8), the map Xaz -> is a C1 map from X into M,

it follows that the form

Nh(z):= h(

ZEX,

+Q(z)),,J'(z)A W(4 (z)),

(17.9)

is closed on X. So a continuous linear map

Zn n-1(X)

N:(9(X

is defined. It remains to prove assertions (i)-(iv), First we prove (i). Let hE C?(X*) and yEX*\{0}. Further, let Gy be as in Definition 17.3. Then

MNh(y) _

-

h(0p(x))W'(x)Aw(4Q(x))

- (n-1)! (27ri)n

(1 - <x,z>)n

xE8Gy

In view of (17.6), this yields the relation

MXNh(Y)

LJ(z)

(n-1)!

<4 '(z),z-y>n

(2)ti)

(17.10)

zcYaGy) Set W* _ +S(Cn\dy)u{0}. Since Cn\dy is relatively compact in X, and 4Q is an orientation preserving diffeomorphism, it follows from (17.8) that

W* cc X* and

60 _ 49(aGY).

Since, by (17.6),

(17.12)

whereas, by (17.2), <x,y>#1 for all

XE(iy, we see that 4y1(y) 4t Gy, i.e.

yE W*. Further, by (17.2) and (17.6), 152

(17.13)

1 - <+-1(a),y> = 0

if zeBW*

(17.14)

From (17.10)-(17.14), by the Cauchy-Fantappie formula (cf. the supplement following Theorem 2.7), now we obtain that MXNh(y) = h(y). This proves assertion (i).

Assertion (ii) follows immediately from (17.9). Next, we prove assertion (iii). Let hE d(X*) and ys X*. If h is not singular at y,

i.e. if h admits a holomorphic extension into some neigh-

borhood of y,then, by (17.9), Nh is not singular at

(y).

Conversely, assume that Nh is not singular at o(y), i.e. there exists a neighborhood U of 4 1(y) such that Nh = 5u on UnX for some ueC0

n-2(UnX).

Take a Coo function

a neighborhood V c= U of

on Cn with supp

U and

1 in

'(y). Since D is strictly convex, we can find

a convex set Dy c D with Dyn(Cn\V) = Dn(Cn\V) and

(Y) E Cn\Dy. Set

Xy = Cn\Dy. Since yEdX* and hence, by (17.7), 091(y) _

1(y) is the X only point zED with =1, so we see that yeXY. Now we set f = Nh - 3( r u). Then feZn,n-1(Xy) and it follows from Definition 17.3 that

MX(fiX) = MX (f)lx..

Since, by Proposition 17.4, MO(x u) = 0 and, by assertion (i), MxNh = h, this yields the relation

h = (MX f)IX*.

Hence h admits a holomorphic extension into XY 3 y, and so h is not singular at y. This proves assertion (iii).

Assertion (iv) follows immediately from (ii) and (iii).

17.8. Corollary. Let X be an n-dimensional complex manifold, Q:X -) I2 an (n-1)-concave function, and D:= {xeX: q(z)<0}. Then, for any ycaD with dp(y) = 0, there exist arbitrarily small pseudoconvex neighborhoods U of y such that

dim [Zp,n-1(nD)/EU,n-1(UAD)] = on.

More precisely, for any yeaD with de(y) # 0, there exist arbitrarily small pseudoconvex neighborhoods U of y such that the following asser-

153

tion holds true: for each x0ndD, we can find an open set Vx C X with

(UnD)\{x} a Vx

as well as a form

x et Vx,

which is singular at x.

Proof. Since dg(y) # 0 and -9 is strictly plurisubharmonic, by an appropriate choice of local holomorphic coordinates in a neighborhood of

Y. we can achieve that - is strictly convex (cf., for instance, Theorem 1.4.14 in CH/L}). Hence, we may assume that X is an open subset of Cn and -g is a strictly convex function on x. Then it is easy to find a

strictly convex C2 domain G = X such that GC D and, for some neighborhood U of y, ( nU = DnU. Without loss of generality, we may assume that OEG. Set Y = Cn\d.

Then, by the remark following the proof of Proposition 17.2, for each point xedDnU = oGnU, we can find a holomorphic function h in a neighborhood of ?*\{4)(x)} which is singular at 4y(x). Now the assertion follows

from Theorem 17.7. 0 17.9. Theorem (Martineau theorem). Let X be a domain of the form

X = Cn\b, where D - Cn is a convex 1) domain with OED. Then the the Martineau map MX induces an isomorphism

X:H0,n-1(X)

' b(X*)

(for the definition of H0n'n-1, see Sect. 1.14).

Proof. That MX is well-defined and injective we know from Proposition 17.4. In order to prove that MX is surjective, we consider a function hc,((X*). We have to find fEZn0 ,n-1(X) with M X f = h. Since D is convex, we can find a sequence of strictly convex open sets Dk Q Cn (k=1,2,.. ) with C2 boundaries such that

Dk » Dk+1

(k=1,2,...

and

00

D=

n

Dk.

k=1

1) We do not assume that D is strictly convex.

154

)

(17.15)

Set Xk = Cn\Dk. Then Xk cc Xk+1' Xk

in Xk+1

(k=1,2,...

), and

00

00

X=U

X* =

Xk,

X* I

k=1

I

k=1

From Theorem 17.7 (1) we obtain a sequence fk5Zn,n-1(Xk) with

MXkfk=hlXk

(k=1,2,...

).

Now we want to modify this sequence so that we obtain a sequence FkEZO'n-1(Xk) such that again

MXkFk

=

for

k=1,2,...

for

k=3,4,...

(17.16)

hi * and, moreover,

F kl

Xk-2

= Pk-i

.

(17.17)

lXk-2

Set F1=f1 and assume that, for some m>1, a collection FkEZn,n-1(Xk) (k=1,...,m) is already costructed such that (17.16) and (17.17) hold true for k=1,...,m. In view of Proposition 17.4, then fm+1-Fm = ou on X for some ueC° n-2(X m). Take a Coo function x on @° with supp x cc Xm and 1 on Xm_i, and set Fm+l = fm+l

-

3(; Cu). Then it is clear that

(17,17) holds true for k=m+1, and it follows from Proposition 17.4 that also (17.16) is valid for k=m+1. Setting f = lim Fk we conclude the

proof. 0 Remark. Under the hypotheses of Theorem 17.9 there exists a continuous linear operator

N: (9(X*) -- Zn, n-1(X) such that MXNh = h for all hE (9(X*).

In fact, take a strictly convex C2 function q:X -)IR such that

inf i(z) = 0, zeX

sup :up Q(z) = on,

and, for any O
155

compact in Cn, and the set Doc,u(Cn\X) is open. Then a modification of the arguments given in the proof of Theorem 17.7 (i) shows that the

required map N may be defined by setting Nh(z) =

1'(z)n 400 9(z))

for all zcX and he b(X*), where 02(z):= VCRo(z)/.

17.10. Corollary. Let X be a domain of the form X = Cn\D, where D == Cn is a convex domain with 0¢D. Then

dim Proof. Notice that

HO,n-1(X) =

00.

HO,n-1(X) = Hn,n-I(X)

(smoothing of the Dolbeault cohomology)

and, by Theorem 2.14 Hn,n-1(X) = There-

fore, the assertion follows from Theorem 17.9 and the fact that, by

Proposition 17.2, dim (O(X*) = oo. 0 Remark. In distinction to the strictly concave case, where we have the local result given by Corollary 17.8, for Corollary 17.10 there does not exist a local version. In fact, under the hypotheses of Corollary 17.10 it is possible that there exist points ycaX such that, for any sufficiently small pseudoconvex neighborhood, UnX is also pseudoconvex HO,n-1(U'X) and hence = 0.

18.Solution of the E. Levi problem for the Dolbeault cohomology: the Andreotti-Norguet theorem on infiniteness of the Dolbeault cohomology of order q on q-concave-(n-q-1)-convex manifolds

If f is a holomorphic function on a domain D c Cn such that, for some

sequence YkeD with yk ->YceD, sup fk(yk) = oo, then f is singular at y. This admits the following generalization to a-closed (0,r)-forms:

18.1. Lemma. Let D c Cn be a domain, f a continuous a-closed (0,r)-form on D (0
hk in a neighborhood of Y such that Y={h=0}, Yk={hk=0}, dh(z)t0 for all zeY, and hk tends to h together with the first-order derivatives, uniformly on the compact sets.

156

Y = {y}u(Y0D), and the following condition is fulfilled: for any neighborhood V - U of y, there exists a continuous 5-closed (r,0)-form g on VnD with

= 00.

sup k

Then f is singular at y (of. Def. 17.5).

Proof. Assume f is not singular at y and r>1 (for r=0 the assertion is trivial). Then there exist a neighborhood V of y and a continuous (0,r-l)-form u on VnD such that au = f on VnD. Choose an open ball B c= VnU centered at y such that the intersection YnaB is transversalThen, for any sufficiently large k, the intersection YknaB is also transversal, and it follows from Stokes' theorem that, for any continuous 5-closed (r,0)-form g on VnD,

I1

sup k

1

IY

J fng

UAg

sup k

< 00.

J YnaB

nB

k

CI

The main result of this paragraph is the following 18.2. Theorem (Andreotti-Norguet theorem, local version). Let D C Cn be a domain with C2 boundary, let O
(y) = n-q-1

and

paD(y) = q

(for the definition of pI (y), see Sect. 4.10). Then there exists a neighborhood U of y such that, for all neighborhoods V c U of y, dim HO'q(DnV) = oo.

(18.1)

More precisely, then even the following statement holds true: There exists a neighborhood U of y such that, for all xeaDnU, we can find a q(Ux) which is singular neighborhood Ux of (5nU)\{x} and a form at x (cf. Def.

17.5).

Proof. We may assume that q>1; for q=0 the assertion is trivial, because then, with respect to appropriate local holomorphic coordinates,

157

aD is strictly convex in a neighborhood of y (of., for instance, Theorem 1.4.14 in [H/L]). Since paD(y)=q, using assertion (vi) in Theorem 5.13, we can find a

neighborhood U of y such that, for any xEaDnU, there is a q-dimensional closed complex submanifold Zx of U which depends continuously on x such that Zx = {x} u{Zx r 1D1. Since P D(y)=n-q-1, after shrinking U, by Lemma

5.8, there exists an (n-q)-convex function in U with UrD = {xcU:g)(z)
Since Zx depends continuously on x, after a further shrinking of U, for any xEaDAU, we can find holomorphic coordinates zx,...,zn in U which depend continuously on x such that

Zx = {weU: zx(w)=...=z°-q(w)=0}.

By Proposition 4.4 (i), these coordinates can be chosen so that, moreover, Q is strictly plurisubharmonic with respect to zx,...,zn_q, and, by Lemma 7.3, we may assume that yo is even normalized (n-q)-convex with

respect to these coordinates. By Proposition 5.5, dQ(y) # 0. Now it is clear that the assertion of the theorem is a consequence of the following more special Lemma. Let Q be a normalized (n-q)-convex function in a neighborhood U C Cn of the origin such that 9(0) = 0 and dg(0) $ 0. Let D:= {zEU_ q(z)<0} and Z:= {zECn: z1=..=zn_q=0}. If ZnU = {0}u(ZnD), then, for any

sufficiently large number Woo, the (O,q)-form

dzn-g+1 ... dzn f(z)--

(18.2)

[FR(z,0) + M(Izn-q-1

is

+ ...

+

Iznl2)]n-q-1

8-closed in a neighborhood of 5\{0}, but singular at the point 0.

Proof of this lemma. Since Z is contained in the complex tangent plane of aD at 0, the restriction of (of. Sect. 7.1 for the definition of the Levi polynomial 9) to Z is a quadratic homogeneous polynomial. Therefore, we can find N
IF2(z,0)I < N1z12

for all zEZ.

(18.3)

be as in estimate (7.2) (in Definition 7.2) with n-q instead of q, and let Woo be so large that

Let C.

M > C +p, and

158

(18.4)

M + N - ,in 4q.

(18.5)

Set

+ )znl2).

h(z) = Fe(z,0) + M(Izn_q+112 + ...

Than it follows from (18.4) and (7.2) that

Re h(z) > -Q(z) + PIzl2

for all zEU.

(18.6)

In particular, h(z) # 0 for all z in some neighborhood

of (D\{O})nU. Therefore equation (18.2) defines a continuous (O,q)-form f in this neighborhood. Since h is holomorphie with respect to zl,...,zn_q, this form is 8-closed.

It remains to prove that f is singular at 0. We want to use Lemma 18.1.

Re h(z)

(18.3) and (18.5) we get the estimates 1h(z)1 > (M-N)Iz12 >

Re

< (M+N)1z12 and

Ih(z)I sin 7r/4q. Hence

ov 1

for all zcZ,

>

(h(z)) q

(18.7)

Izj2q

where o.:= (M+N)-1sin 71/4. Let t1,...It2n be the real coordinates in

n

with zj =ti+itJ+n. Set g(z) = dzn_q+1n...Adzn/(2i)q and

k(

= - Z

(0)

...

a (0)) n

for k=1,2,...

.

Then the manifolds Zk converge to Z, and

dxn-g+lA... Adxnndx2n-q+ lA...Adx2n

fAg =

hq

Therefore, by (18.7), for each neighborhood V =a U of 0,

sup Re r

k

fAg = 00.

J

ZknV

So it follows from Lemma 18.1 that f is singular at 0. Q Theorem 18.2 admits several globalizations. For instance, we have

159

18.3. Theorem (Andreotti-Norguet theorem, a global version). Let E be a holomorphic vector bundle over an n-dimensional complex manifold X, and let D cc X be a strictly (n-q-l)-convex domain (O
dim H0,q(D,E) = oo.

More precisely, then there exists a neighborhood U of y such that, for any xcaDnU, we can find a neighborhood Ux of D\(x} and a form

fxEZ q(UxE) which is singular at x. Proof. By Theorem 18.2, we have a neighborhood U of y such that, for

we can find a neighborhood Wx of (DU)\{x} and a form

any

FXEZD q(Wx,E) which is singular at x.

Fix xcaD and a neighborhood U. C= U of x. Since D is strictly (n-q-l)-convex, there is an (n-q)-convex function Q in some neighborhood G of 8D such that DIG = (zeG: 9(z)<0). Take

&>O sufficiently small.

Then DE:= Du{zeG: Q(z)-E}, and we have the relation DLn(U\Ux) Wx. Take a CO° function X on X such that supp cc U\D_ and 1 in some neighborhood of Ux\D. Then, after extending by zero, 3(x Fx) is a continuous a-closed (O,q+l)-form on D

which vanishes on D_

.

Hence, by

Theorem 12.14, we can solve the equation $u = 8(% F ) with IAECO q(DE,E). Setting fx:= Fx - u we conclude the proof. L]

x

Notice also the following globalization of Theorem 18.2: 18.4. Theorem (Andreotti-Norguet theroem, a second global version). Let E be a holomorphic vector bundle over an n-dimensional complex manifold X, and let D cc X be a strictly (n-q-l)-convex domain such that at least one of the following two conditions is fulfilled: (i) X is a q-concave extension of D;

(ii) X is an (n-q-1)-convex extension of D, and there exists at least one point yEaD such that aD is of class C2 in a neighborhood of y and paD(Y) = q. Then

dim HO'q(X,E) = oo.

Proof. By Theorem 18.3, this follows from Theorem 16.1 (if condition (i) is fulfilled) and Theorem 12.14 (if condition (ii) is fulfilled).

[-;

The form fx in Theorem 18.2 which is constructed in the proof of this theorem is unbounded at x. Actually, this form can be chosen to be continuous in x:

160

18.5. Theorem (Andreotti-Norguet theorem, local version with uniform estimates). Let D C Cn be a domain with C2 boundary, let O
dim [Zp,9(DnV)/Zp,9(DnV)nE0 9(DnV)] = oo.l)

More precisely, then there exists a neighborhood U of y such that, for each xeaDnU, there is a neighborhood Ux of (DtU)\{x} and a form fxEZO,q(f]x) which is singular at x.

Proof. We may assume that q>1 (for q=O the assertion is trivial, because then, with respect to appropriate local holomorphic coordinates, aD is strictly convex in a neighborhood of y - of., for instance, Theorem 1.4.14 in (H/L]).

First we prove the following Statement A. Let [U,9, p,D] be an (n-q-l)-convex configuration of type I in Cn, and ycaD with (p(y)
exist a neighborhood Wy of the set {zaD: z#y and p(z)
Proof of Statement A. By Theorem 18.2, we can find a neighborhood V of y, a neighborhood Vy of (DnV)\(y) and a form fyEZO q(Vy) which is

singular at y. Fix some neighborhood V' = V of y, and choose L>O sufficiently small. Then [U,9-E, p,D&] with Df:= {zeU:(P(z)
function x on Cn with supp x « B and x = 1 on V'. Then, after extending by zero, a(% fy) is a continuous and 5-closed (O,q+1)-form on DE. Hence, by Theorem 7.8, we can solve the equation au = a(x fy) with ucC00 q(Df).

Setting Wy = (D.\V')uVy and Fy = x fy - u, we conclude the proof of Statement A.

Next we prove the following Statement B. Let [U,9, (p.D] be an (n-q-1)-convex configuration of

type I in Cn (O
,p(y)
any neighborhoood V of y, there exist a neighborhood Y of (zeb: y(z)
Proof of Statement B. Fix a neighborhood V1 s V of y. Then it is easy to find an (n-q-l)-convex configuration such that the following conditions (a)-(e) are fulfilled:

1) For q=0, this means dim CO(20oV) = oo (cf. Sect. 0.11 for the definition of CO(7).

161

(a) U' = U, (b) Q'

(p'

= p.

< 4 and hence D c D'.

(c) p'(y) < 0 and hence yED'.

(d) i' _ 9 on U1\V

.

(e) There exists a point xEaD'\D with p-.,(x) = q such that,

for each neighborhood Y of the set {zLD':(p(z)
)

such that: DO = V'nD; each Dk+l can be

obtained from Dk by means of a q-concave extension element in Y (of. Def.

15.4);

00

D'\(D\V') =

Dk;

k=O and, for all k, there is an integer N(k) with Dk0(DN(k)\DN(k)-1)

=0. By statement A, we can find a neighborhood Y of {zED': z#x,(P(z)

aDnV, fI(V'n

D)

obtain the contradiction that f belongs to EO q(D'\(D\V')). So statement B is proved.

Now we go to prove the assertion of the theorem. By Lemma 7.5, we may assume that D belongs to some (n-q-1)concave cofiguration [W,g, y,D] of type I in n, where tp(y)<0. Moreover we may assume that paD(z) = q for all zeaD with(p(z)
In order to

find the required form fx, first we take an (n-q-1)-convex configuration (UnD)\{x} C D' C U\{x}, and [W',g', (P',D'] such that W' = W, P6D'(z) = q for all

U. It is sufficient to construct a form fE

Z00 q(D') which is singular at x.

Take a sequence xkedD'\{x} which converges to x such that xk

xl if

k -/--l. Taking into account Observation 4.15, we can find neighborhoods

Vk of xk such that VknVl = QS if k O 1 and, for all k, the domain

00

Dk:= D'LI

J

V.

j=1

jIL k belongs to some (n-q-1)-convex configuratiion [Wk' k' pk,Dk] with Wk = W and (pk =(Q, where, moreover, PoDk (z) = q for all zEaDknU. Further, we

162

balls Bk =c Vk centered at xk. Then, by statement B, we can

fix opa

0

find 19 sequence of forms fkEZO q(Dk) such that, for each k, fk is

"s ngular at aDnBk". Finally, we choose a sequence of numbers fk 0_0 which are so small that the series f:=

O° Lkfk converges uniformly on 1

the compact subsets of D'. So we obtain a form feZ0 q(D'). It remains to show that f is singular at x:

Each of the forms f-fk admits a continuous and a-closed extension into the ball Bk. Since the 3-equation is solvable on balls, it follows that, for all k, the form f-fk is not "singular at 6D'i1Bk". Since, by

construction, each fk is "singular at aD'!Bk", this implies that, for any k,

f is "singular at aD'n Bk". Therefore, f is singular at x. 0

Theorem 18.5 admits the following globalization:

18.6. Theorem (Andreotti-Norguet theorem, global version

with uniform

estimates )_ Let D == X be a strictly (n-q-l)-convex domain in an n-dimensional complex manifold X (O
If there exists at least one point yEaD with

PaD(y) = q, then

dim [Z0,q(D,E)/Zp q(D,E)n E0 q(D,E)) = oo.

More precisely, then there exists a neighborhood U of y such that, for any xEaDnU, we can find a form fxEZ0 q(D,E) which is singular at x, and which admits a continuous and 3-closed extension into some neighborhood of D\{x}.

Proof. This is a repetition of the proof of Theorem 18.3, where

instead of Theorem 18.2 we have to use Theoreml8.5. [] 10.7. Remark (Supplement to Theorems 18.2 and 16.5). In the next section we shall prove the Andreotti-Vesentini theorem: For any non-degenerate strictly q-concave domain D in a compact complex manifold, the space ED q(D) is closed with respect to the uniform convergence on the compact subsets of D, and the space E0

is closed with respect to uniform

convergence on D.

This theorem expresses a global property of the domain D, i.e. the corresponding local statement is not true.

In fact, let D,q,y, and U be as in Theorem 18.2. Since paD(y)=n-q-1, then, by Lemma 7.5 and Theorem 10.1, there exist arbitrarily small pseudoconvex neighborhoods V of y such that the image of the restriction map

EO,q(V) = Z0,4(V)

ZO,q(DnV)

183

is dense in the Frechet space ZO q(DnV) (one can prove that this is true V of Y). If, even for all sufficiently small moreover, V c U. then, by (18.1) this implies that the space EO q(DnV) is

not closed in the Freohet space ZO q(DnV).

Under the hypotheses of Theorem 18.8, in the same way we see that the spaoe Z0, q(bnV) n EO q(DnV) is not closed in ZO q(DnV).

19. The Andreotti-Vesentini separation theorem for the Dolbeault cohomology of order q on q-concave manifolds

Let Y be an n-dimensional complex manifold, and X

Y a domain which

is both non-degenerate strictly q-concave and non-degenerate strictly (n-q-1)-convex (1
HO,q(X) f

is of infinite dimension. So the question arises whether the natural topology of this space, the factor topology of the Freohet space Z00 q(X)

with respect to the subspace EO q(X), is separated? In other words: Is EO.q(X) closed with respect to uniform convergence on the compact subsets of X? 1) Under certain additional hypotheses the answer is affirmative. For instance, if Y is compact, then one has the following 19.1. Theorem (Andreotti-Vesentini theorem). Let E be a holomorphic vector bundle over an n-dimensional compact complex manifold Y, and let X C Y be a q-concave domain (1
() For all O
hence (of. Remark II following Def.

respect to uniform convergence on D.

1) For dim HO'q(X) < oo this is always the case, by Banach's open mapping theorem.

164

(ii) Denote by Z0 q(D,E) the space of all f&ZO q(D,E) such that

Ding=0 8D

for any geZ0,n-q-1

(X\D,E*).

Then there exist a bounded linear operator T

acting from the Banach space ZO,q(D,E) (endowed with the norm II IO,D) into the Banach space CO/q_1(D,E) (with the norm 11-11 ) and a compact 1/2,D

linear operator K from ZO q(D,E) into itself such that

daT = id + K

on ZO,q(D,E)

(id:= identity operator). Moreover, then

E0 q(D,E)n Z0,9(6,E) C ZO,q(D,E)

Hence each of the spaces E

--*O

(D,E) (0<x<1/2) and E0

4(D,E)nZ0

(D,E)

is a closed and finitely codimensional subspace of Z0 q(D,E).

Theorems 19.1 and 19.2 admit the following generalizations:

19.1'.Theorem (Andreotti-Vesentini theorem). Let E be a holomorphic vector bundle over an n-dimensional complex manifold Y which is (n-q)-convex (1
Y, X is a q-concave extension of Y\d.

Then E0 q(,E) is closed with respect to uniform convergence on the compact subsets of X.

19.2'. Theorem (Andreotti-Vesentini theorem with uniform estimates). Let E be a holomorphic vector bundle over an n-dimensional complex manifold

X, and 1
(5,E), and

hence E --+O (D,E) is closed with riect to uniform convergence on (ii) Denote by Z0 q(D,E) the space of all

q(6,E) such that

J fag = 0 aG

165

for any

,n-q-1(G,E*).

from ZD q(D,E) into

Then there exist a bounded linear operator T

CO/q_1(D,E) and a compact linear operator from

Z0 q(D,E) into itself such that

$oT = id + K

(19.1)

on ZD q(D,E).

Moreover, then

E0,q(D,E)nZ0,q(D,E) C Zo.q(D,E).

(19.2)

Hence each of the spaces E 4 0(D,E) (0
I. The authors believe that, under the hypotheses of Theorem 19.2',

E0 q(D,E)n Z0 q(,E). (By Theorem 19.2' we know

even

only that the difference between these spaces is finitely dimensional.) II.

In Theorems 19.2 and 19.2', assertion (i) follows from assertion

(ii) (cf. Remark II following Definition 11.1).

III. The authors believe that, under the hypotheses of Theorem 19.2, the space is not topologically complemented in the Banach space

Z00

q(D,E) (cf. point 6 in Problems at the end of this monograph).

The remainder of the present Section 19 is devoted to the proof of Theorems 19.1' and 19.2' (Theorems 19.1 and 19.2 are special cases of them). By means of Theorems 12.14 and 16.1, Theorem 19.1' can be easily obtained from Theorem 19.2' - the corresponding arguments will be given at the end, in Sect. 19.14. In the following Seats. 19.3-19.13 we prove Theorem 19.2'. In all these sections we assume that the hyptheses of Theorem 19.2' are fulfilled and use the notations from this theorem.

19.3. Construction of the operators N0 "' W2. Let e;e -* IR be a strictly (q+l)-convex function without degenerate critical points in a neighborhood a of aG such that G AO = {See: p(z)
find domains Al = A2 a A3 = $, q-convex configurations [Uj,4j, tpj,DI in Cn, and biholomorphic maps h from A3 onto U (3=1,...,N
Pi:E*®AIA

(3=1,...,N)

be fixed, where r = rank E, and A is the bundle of holomorphic

166

(19.4)

(n,0)-forms on X. We fix also some canonical Leray maps vj for [Uj,gj, IQj,Dj] (of. Sect. 7.7). Then, by Theorems 9.1 and 7.8, the v.

corresponding Cauchy-Fantappie operators T j definebounded linear operators

Tvj:CO (D O,r J

C1/2 (h (GnAi)) 0,r-1 j

(1
(19.5)

such that v.

v.

(-1)n_of

= aTn-gf + Tn-q+ldf

on hj(GnA

(19.6)

for all DECO n_q(Dj) with the property that of is also continuous on Dj. Let

WJ.C0 (GnA2 E n,r j r

C1/2 n,r-1

'

'

(1
be the operators which are defined, via the biholomorphic maps hj and the isomorphisms (19.4), by the operators (19.5) (here we identify E*-valued (n,r)-forms with E*OA-valued (O,r)-forms). Then (19.6) takes the form

(-1)n-qf

= aWn-qf + Wn-q+laf

on GAA

(j=1,...N)

(19.7)

such that of is also continuous on

for all f6C0

Further, we choose some open sets Al cc A _= G (j=N+1,...,M
C A1U...UA1NUAN+1u...u AM,

(19.8)

and, for any N+1<j<M, A is a ball with respect to appropriate holomorphic coordinates in a neighborhood of A. From Observation 2.11 and Proposition 1.4 (i) then we obtain bounded linear operators

Wj:CO r n(A?,E*) ,

r i

1/2 -- C n,n -1 (A1j ' E

(1
-2

0 such that, for all fECn,n-q(A3'E*) with the property that of is also

continuous on A

,

167

(-1)°-qf = a-n_gf + Wn_q+laf

on A

By (19.5) we can find Coo functions

(j=N+1,...,M).

(19.9)

on X such that

supp Iij cc Ai and (19.10)

in some neighborhood of G. Now, setting (-1)°-gxiWn-q)xjf),

Wff =

(-1)n-gaxj^Wn-q(xjf),

(-1)n-g+1Xjwn-q+1(,jk.Af)

W2f =

for fEC0 n-q(d,E*), we obtain bounded linear operators

WO:C° n-g(G,B*)

--> Cn/n-q-1(G,E

and

Cn,n-q(G'E*)'

Cn,n-q(G,E*)

Wj'Wj

From (19.7) and (19.9) it follows that

aW0f = f + Wif + W2f

for all

n,n-q (G,E*) and j=1,.:.,M.

19.4. Lemma. There exist a bounded linear operator

S:C0

n-q

MECn n-q-1M E

and a topologically closed and finitely codimensional subspace Z0 n_q(G,E*) of Z0 n_q(G,E*) such that (id:= identity operator)

M

aa) j=1

168

id

(M,E*). on Z0 n,n-q

(19.12)

19.5. Proof of Lemma 19.4. Set

W =

Wj j

V =

and

J=1

=1

(W1+W2 J

i

Then, by (18.11) and (19.10),

5oW = id + V

on Z0,n_q(G,E*)

(19.13)

Zn,n-q(d,E*).

(19.14)

and hence

V(Z0,n-q((1,E*)) S

Since the embedding operator C1/n_q(G,E*) -+ C0,n_q(0,E*) can be approximated with respect to the uniform operator norm by finitely dimensional linear operators, the operator V considered as an operator with values in C0,n_q(G,E*) has the same property. Therefore, we can find a finitely dimensional linear operator F1 from C0 n_q(G,E*) into Cn,n_q(G,E*) such that the uniform operator norm of V-F1 is < 1/2. Then id+V-F1 is invertible in C0 n_g1G,E*), and, setting F2 = F1(id+V-F1 we obtain a finitely dimensional linear operator F2 such that, by (19.12),

3W(id+V-F1)-1f = f+F2f

(19.15)

for all f E (id+V-F1)(Zn,n-q((i,E*)). We set

Zn, n_q(G, E*} = Ker F1 n Ker F2 /1(id+V) (Zn n_q(0, E*) n Ker F1), where Ker F is the apace of all f6C n_q(d,E*) with Fif=O. Further, we not

S = -

Since V restricted to ZO itseln

Zo n_q(d,E*) into

f

, ,

W(id+V-F1)-1

n-q (G,$*) is a compact linear operator from and since Ker F1 and Ker F2 are closed and

finitely codimensional subspaces of Cn n_q(G,fi*), then Zn n_q(G,fi

is a closed and finitely codimensional subspace of Z0 (G,$*). n,n-q Further, since

169

W+SV = W[id-(id+V-F1)-1V] = W(id+V-F1)-1(id-F1),

it follows from (19.15) that 3(W+SV)f = f for all feZn

n-q(G,E*),

i.e.

(19.12) holds true.

19.6. Lemma. Denote by

Fn,n-q-1(6,E*)

the space of all fECn,n-q-1((I,E*)

such that of is continuous on d. Set

M

Af = f = 7 j=1

for feF0,n-q-1(0,E*)' where W0,W1,W are the operators from Construction 19.3, and S is the operator from Lemma 19.4. Then Zn,n-q-1(d,E*) is a finitely codimensional subspace of Im A := A(Fn,n-q-1

19.7. Proof of Lemma 19.6. Let Fn,n-q-1(G,E*) be the space of all where Z0 n-q(d,E*) is the finite(G,E*) with 3f Zn n-q(d,E*), fEF0,n-q-1 ly codimensional subspace of Zn n_q(d,E*) from Lemma 19.4. Then it is clear that Fn,n-q-1 (d,E*) is of finite codimension in

Fn,n-q-10,E*).

Heno A(Fn,n-q-1(G,E*)) is of finite codimension in Im A. This completes the proof, because, by (19.12) in Lemma 19.4,

Min

n q 1(G,E*)) C

0,n-q-1(0,E*)

C Im A.

19.6. Construction of the operators T,K, and L. Let 6 and IP be as at the beginning of Construction. 19.3. Then, by Lemma 13.3, we can find open Bk sets Bik == Bk == e, q-concave configurations [Vk,6k,yrk,Hk,Gk] in

c, and biholomorphic maps gk:B4->Vk (k=1,...,M'
{zeVk: 'rk(z)
G'(DnBk) = Gk; E is holomorphically trivial over Bk. k

Further on, let some holomorphie vector bundle isomorphisms

Qk:EIB

) Bk ® Cr

(1
be fixed. Moreover, we fix some canonical Leroy maps uk for (of. Sect. 13.4). [Vk'$k'rk'Hk'Gk]

170

(19.16)

Let

Tr:C0,r(DnBk,E)

C0,r-1(DnBk,E)

and

Lr:CO r(DnBk,E) -- C r(DnBk,E)

be the operators which are induced, via the isomorphisms (19.16) and the

u

u

biholomorphic maps gk, by the operators (-1)rT k and respectively (for the definition of

Luk

(-1)rL k,

and Tuk, see Sects. 3.b and

3.7). Choose Coo function .1,...,, on X such that supp

>`k == Bk and

(19.17)

in some neighborhood of 6G. Then, by the piecewise Cauchy-Fantappie formula (Theorem 3.12),

akf = .5Tq Ukf) + q+l(aXknf) + LQ(akf)

for all

on D

(19.16)

q(D,E) and 1
By (19.17) we can find a Coo function )0 on X with supp a0

a0 +)i

X\G and

(19.19)

1

in some neighborhood of D. Since G' is non-degenerate strictly

(n-q)-convex, by Theorem 11.2, for n-q
O, r-1 (G',E) and KOr from COO, r (G''E) O, r (G',E) Into C1/2

1/2

into C 0

(d',E) such that

Of =

3T0()%f) + q+1 TO (bxf) q 0 0

-

K0(a0f) q

on D

(19.20)

for all feZ0 q(D,E).

Now, for any feZO q(D,K), we define

171

if =k q(Tkf),

if =

M'

k

\0KgU0f),

M'

Lf =

(19.21)

Since Tr and Kr are bounded linear operators from C0 r(D',E) into C /r_1((i',E) reap. C

and since, by Theorem 14.1, the operators

Tr (1
T is a bounded linear operator from ZO q(D,E) into Cp q_1(b,E), and K is a bounded linear operator from

(19.22)

ZO.q(D,E) into C

a(Tf) = f + Kf + Lf for all

on b\aG

(19.23)

q(D,E).

By (19.22), K is compact as an operator from Z0 q(D,E) into CD q(D,E). This is not true for the operator L. However, we have the following lemma, which is the key to the proof of Theorem 19.2':

19.9. Lemma. Let A and Is A be as in Lemma 19.6, and let (Im A) space of all fez0 q(b,E) with

f

fng = 0

L

be the

for any g Im A.

BG

Then, for each fe(Im A)

L ,

the form Lf (which is defined by (19.21)

immediately only on D\aG) admits a continuous extetSsion onto D. There-

fore, a linear operator L:(Im A)1

) C0 q(D,E) is well-defined. This

operator is compact (here both (I A)

and C0 q(D,E) are considered as

Banach spaces endowed with the norm 11-11

172

0,D).

We divide the proof of this lemma into three parts. The first part is a reduction to some estimates for certain local integrals. Then these estimations will be proved in parts 2 and 3. In the following proof we meet many constants. Therefore, all "large" constants will be denoted by C, i.e. if we write "f(y)
19.10. Part 1 of the proof of Lemma 19.9: reduction to local estimates. By definition of L, it is sufficient to prove that, for all 1
) xkLq(Tkf).

f

Therefore, let us fix 1
the canonical Lerey data for [Vk,c,y ,Hk,Gk] such that uk is the canonical combination of (vilw 2'w3). Set

Xk- 9kl

A.

%(t)
Since supp

u

all fECg q(Gk). Therefore, by definition of

((-1)qALUkof))(t)

=

Llk,

r f(e)AL(t,s)nw(s)

(19.24)

gk(aGnBk)

for all feC q(Gk) and teGk, where

n-q-1

qtl n-1 (-1) (

r--,

)

q X(t)X(s)

L(t,s): (2ni)°

q i

det(w Is w at w 1 ) 1 s 1'

(19.25)

<w1(s,t),s-t>n

Let pz,px:XxX -3 X be the projections defined by pz(z,x)=z and px(z,x)=x, and let p*E and p*E be the pull-backs of E onto XXX with respect to pz and px. Then, for any Hom(p)E,p$E)-valued continuous differential form Y(z,x) defined for (s,x)fDxoG, and for each E-valued continuous differential form f on aG, the integral

173

f f(x)AY(s,x),

zED,

xcaG is well-defined, and the result of this integration is an E-valued differential form on D. Recall that, by (19.4),

Re <w1(t,s),s-t> > k(s) - 5k(t) +cls-t12

(19.28)

for all s,tEVk. Therefore the form L(t,s) is defined and continuous for all t,sEVk with 5k(s)>d'k(t) and %#t. Since g = -50 gk, it follows that

the pull-back of L(t,s) with respect to gkxgk is defined and continuous for all z,xEBk with 9(z)?9(x) and x$z. We denote this pull-back by Lx(z,x). Since supp >k == BI and hence Lx(z,x) = 0 if (z,x) d: BkxBk,

after extending by zero, Lx(z,x) is defined and continuous for all (s,x)EDxd with x$z.

By Lx(z,x) we denote the continuous Hom(p*E,p*E)-valued differential form which is defined for all (z,x)cDxd with x#z by the following two conditions: 1) Lx(z,x) = 0 if (z, x)

Bkx

2) with respect to the isomorphism (19.16), Lx(z,x) is represented by the matrix Lx(z,x)Ir, where Ir is the rxr unit matrix.

Let gk,...,gk be the components of the biholomorphic map gk:Bk--4Vk and Wk = dgkA...Adgk. Then, by definition of the operator Lk and by (19.24), one has

f(x)ALX(z,x)n Ok(x)

(),kLQ(kf)](z} =

(19.27)

J

xEaG

for all fsZg q(D,E) and zeD\aG For fixed zsD\aG, the form Lx(z,x)A wk(x) may be interpreted as a vector of continuous E5-valued differential forms on G, and it follows from (19.25) that these forms belong to the space Fn,n-q-1(G,Et) defined in Lemma 19.6. Therefore

(A(Lx(z,-)A 0k)I(x)

(zeD\aG,

is a well-defined Hom(pxE,pzE)-valued differential form on (D\dG)xG, which is of bidegree (O,q) in z, and of bidegree (n,n-q-1) in x. If

174

1

ff(Im A)

,

then (19.27) may be written

[XkLQ(Xkf)](z) =

Jf(x)A

111lLX(z,x)ALk(x)

(19.28)

-

xE aG

zeD\aG. By definition of A, one has M

LX(z,x)Awk(x) - [A(LX(z, Mwk)](x)=7Kj(z,x), J-

where Kj(z,x) are the Hom(pXE,paE)-valued differential forms defined by

Kj(z,x) _

](x),

zed\aG, xEd (1<j<M). In view of (19.28), this implies that

[>`kLq()kf)](z)

M

r

= = I f(x)AKj(z,x)

(19.29)

j=1

xEaG

for all fe(Im A)1 and zeD\aG. Therefore it is sufficient to show that, for all 1<j<M, the map

f

)

(19.30)

I

X E aG

defines a compact operator from CD q(G,E) into itself. So, let us fix also 1<j<M (for the rest of the proof of Lemma 19.9).

Denote by a metric on X which is locally defined as the Euclidean metric with respect to appropriate Coo coordinates. For any E>0, we choose a Coo function d,.:XxX -9 [0,1] such that d11 (z,x)=0 if dist(z,x)>E, and d&(z,x)=l if dist(z,x)
K£(z,x) _

Wk)](.)

for zeD\aG and xsd. Since 1-dt(z,x) = 0 if dist(z,x)<E/2 and aLX(z,x)n wk(x) is continuous for all (z,x)EDxd with x#z, we see that, for any £>0, the assignment

175

538

(1-dt(z,-))3LX(0,-)A Wk

may be interpreted, locally, as a vector of continuous maps with values in the Banaoh space C0,n_q(G,S*). Since the operator

is

continuous with respect to 1-0O,G, this implies that, for each t>0, the form Kj(z,x)-K£(z,x) is continuous for all (z,x)E6Xd. Hence, for any E>0, the map

f f(x)n[Kj(-,x)-K (-,x)],

f

xic

defines a compact linear operator from C0O.q ($,E) into itself. In order to prove that (19.30) defines a compact operator from C00 9(L,2) into itself, therefore it is sufficient to show that

lira sup f Q[(z,x)Qd6 = 0, E---)0 seD J xfiaG where d6 is a measure on aG which is locally defined to be the (2n-l)-dimensional Euclidean volume with respect to appropriate C°° coordinates in X. Since the operator S is bounded with respect to 11.110

G'

therefore, it remains to show (in order to prove Lemma 19.9)

that

Wk)](x)IId6 = 0

lim

E-40 sup J

for 1=1,2,3. If N+l<j<M, this is trivial, because then supp

(19.31)

hj

a G (of.

Construction 19.3) and hence, by definition of WI,

wk)(x) = 0

if xEaG.

So we may assume that 1<j
Let (p1,p2) be the canonical Leray data for [Uj,gj,(Qj,Dj] such that vj is the canonical combination of (pl,p2). Set

Z = xj'hj 176

Since supp x « {teUj:Lpj(t)
x Tvj(x f) = X BD3 (x f) + x R1,(/I, f). v.

v.

axnT (xf) = axnBD (xf) + 3xnR1J(xf), J v.

v.

XT (axAY)

= xBDi(3xnP) + xRl3(aXAY).

We set q-1 n-q-1

BO(t,s)

= x (t)x (s)

det(s-t,inddt Is-tl

Aw(s)1

(19.33) 0

q-1 n-q-1

B1(t,s) = ax (t) X (s) det(s-t, din dt AW(s))

(19.33)1

Is-tl

q-1 n-q

B2(t,s) =

det(s-t, 2n , x (t)3x (s) 10-ti

dt)ALO

(s)

(19.33)2

Further, we introduce the abbreviation

P1(t,s)

s-t

r = r(t.s,y) = y

+

{s-t12

(19.34)

(1 -y)



and set

q-1 n-q-1

R0(t,s,Y) = x (t)x (s) det(r,dyr,3sr, atr )A W(s),

n rq-1 n-q-1

R1(t,s,y) = 3x(t)Ax(s) det(r.dyr,6ar, atr )AW(s), q-2 n-q r-+ r,

R2(t,,Y) = x (t)ax (s)Adet(r,dyr,asr,3tr)AW(s)_ Then, by definition of BD

and R13 (of. Sects. 1.3 and 3.7), for any J

n_q(Dj) and all tfDJ, we have

177

(xBD 3 (xf)J(t)

3/C'BD

= b0 J f(x)ABO(t,s), aeDi

(xf)](t) = b1 3

(19.38)0

f(x)%B1(t,s),

J

(19.36) 1

seD.J

[x BD (3xAf)](t) = b2

f(x)AB2(t,s s Ds

(19.36)2

1

[x RIi(x f)](t) = s0 J f(s)ARO(t,s,y),

(19.37) 0

sEh 0
v

f(s)ARl(t,s,Y),

[axAR1J(xf)](t) = al

(19.37) 1

0
C X R 1J(3x Af)](t) = a2

i

f(s)AR2(t,s,y),

(19.37) 2

0
where b1 and al are some constants which depend only on n and q (similar as in (19.25)). ^1 Now, for all x,u6A with xfu, by BX(x,u) we denote the pull-back of B (t,s) with respect to hxh (1=0,1,2). Since xi = 0 outside A than

BX(x,u)=0 if (x,u) e,

Therefore, after extending by zero, we may

assume that the forms BX(x,u) are defined and continuous for all x,u&X with x#u. Analogously, by RX(x,u,y) (xEG, uE)G, 0
1) BX(x,u) and RX(x,u,y) vanish if (x,u) $ 2) with repeet_to the isomorphism (19.4), BX(x,u) is represented by the matrix B1(x,u)Ir, and RX(x,u,y) is represented by the where Ir is the rxr unit matrix. matrix RX(x,u,y)I

178

Then it follows from (19.32)11 (19.36)1, and the definition of the operators W (of. Construction 19.3) that

II[W(df(z, )aLx(z,

C

J dE(z,u)3uLx(z,u)ABX(w,u) u&G

.

d£(z,u)auLx(z,u)ARX(w,u,Y)

uEOG O
for all z6D, 00, wEG, and 1=0,1,2. Thus, for the proof of (19.31), it is sufficient to show that

lim

sup

fd(z u)uLx(z,u)AB(w,u))

1im

E-0 zED

d5 = 0

(19.38)

uE G

and

lim sup E-30 zED

lim

JdE($,u)3uLX(z,u)ARX(w,u,y)

G3w-.x xfaG

d6 = 0

(19.39)

uEG O
for 1=0,1,2. Now we want'to express (19.38) and (19.39) in the coordinates hJ. First notice that, by definition of B1, R1, L, and d., for all sufficiently small

.>0, we have

dE(z,u)auLx(z,u)ABX(w,u) = 0

(19.40)

dE(z,u)auLx(z,u)ARX(w,U,Y) = 0

(19.41)

and

if u 4 B,

i or w 4 Ai or z 4= BknA (1=0,1,2). Further, by definition

of L ,B1, and R1, there is a holosorphic map H from Bk*A3 into the group of invertible oosplex rxr matrices such that

179

BULX(z,u)ABX(w,u) = H(z,u)auLX(s,u)ABX(w,u)

for all zEDnBknAj,

(1=0,1,2), and

-1

ii

N

for all zeDn 3nA3

3,

(19.42)

= H(a,u)SuLX(z,u)ARX(w.u,Y)

wEGnA3 , ui3an

3 nA3,

(19.43)

O
We introduce also the following abbreviations:

Y1:= h(BknAI)

for 1=0,1,2; on Y3;

:=gkohi

a(t,s):=

for s,tEY3;

A(t,s):=

for s,ttY3;

D-:= Di nY2 = {tEY2: 9j(t)<0};

D+:= {tEY2: gj(t)>0}; S:= {tEY2: ej(t)=O};

for e,tEY3.

G& (t,s):=

Then qj = -5koA on Y3 and hence, by (19.26), Re A(t,s) > pi(t) - QC(s) + cjs-t(2

(19.44)

for all s,tEY2. Therefore, setting b=),kohi 1 and

n-q-1

q+l n-1l (-1)

det(a, dsa

(

P(t,s) =

(2li )n

q

)

b(t)b(s)

q ta) ,

(19.45)

An

we obtain a continuous differential form defined for all s,tEY2 with Qj(t)>Qj(s). By (19.25), P(t,s) is the pull back of LX(z,u) with respect to

h-1. Therefore, it follows from (19.40)-(19.43) that

J dE.(z,u)5uLX(z,u)ABX(w,u) ucG 160

< C

J aE(hj(z),s)asP(hi (z),s)ABl(hi (w),s)

and

d(z,u)buLX(z,u)Akl(w,uY) J

ucbG o
< C

J

Ge(hi(z),s)38P(hi(z),s)AR1(hi(w),s,y)

sES 0
for all zcDnBknA and (1=0,1,2). Taking into account again (19.40) and (19.41), so we see: in order to prove (19.38) and (19.39) it is sufficient to show that, for 1=0,1,2,

lim

sup

I(B1,E,t) = 0

E--)0 tED+ and

lim

sup

I(R1,£,t) = 0,

E--)0 tED+

where (dS:= (2n-1)-dimensional Euclidean volume form of S)

I(B1,E,t):=

- lim

I

f GE(t,s)asP(t,s)nB'(z,s)

z-4x

sED

lim D 3z--->x

S CS

D

dS

XES and

I(R1,E,t):=

I GE(t,s)8sP(t,s)4Rl(z,s,y)

dS.

0
XES

181

This will be done in Seats. 19,11 and 19.12. Actually, we shall obtain even the following more precise estimates: If 5>0 is an arbitrarily small (fixed) number, then

I(B1,L,t) <

Cal/2-6

(19.46)

and

I(R1,L,t) <

for all

CE1/2-6

(19.47)

E>O and 1=0,1,2.

19.11. Part 2 of the proof of Lemma 19.9: proof of estimate (19.46). From the definition of B1 (of. (19.33)1) it is clear that in the definition of I(B1,L,z) the limes may be interchanged with the integration over S. Hence

I(B1,L,Z) < C

JdS

111 GE(a,u)3uP($,u)Ilp B1(x,u)IIdG

xES

for all zED+, E>0, and 1=0,1,2, where d6 is the 2n-dimensional Euclidean volume form in Cn. Since is C1 in a neighborhood of D , we can find

K0 with Iz-uI>KL. Therefore, it follows that

I(B1,E,a) < C J dS xeS

I I5uP(z,u)Iu)IId6

(19.48)

ueD Iu-zI
for all zeD+,E>0, and 1=0,1,2.

Recall that, by the definition of canonical Leray maps for q-concave configurations (cf. Soot. 13.4), w is of the form w1(a,u) = -w(u,a) 1 for some wEDiv(-6k). Since -6k is normalized (q+1)-convex, this implies that wl(z,u) depends holomorphically on ul,...,uq+1 (cf. Sect. 7.8). Taking into account that a(z,u)=w1(P(z),P(u)) and )B is biholomorphio, so

we see that

n-q-1 q

r _ _ _ _ 1 r1 det(a, 8 ua .3a a) - 0 3u An

182

and hence, by (19.45), n-q-1

q

q+1 n-1 det(a,r---, 5 a ( , (-1) u q aP(z,u) = b(z)ab(U)A (2 i)n An

a

a)

Since 6k = -ejo,B-1, moreover, it follows from (19.49) that Ia(z,u)I < C(Ildgf(u)II + lu-at) for all z,uEY2. Hence

for all ztD+ and ucD

I

I

(19.50)

l

IA($,u)In

Together with (19.48) this yields the estimate

.

(u) pB'(x,u){I

---d5

I(B1,E,$) < C dS

r

+ CJ

dS

B1(x,u)

C

IA(z,u)Inlu-sl-1d6

IA(z,u)°

IId

J XES

u-z

+

IIdqj

P(z,u)11 < C

II au

u(D Iu-zI
xfS

ucD Iu-sl
for all zED+, C>0, and 1=0,1,2. SincellBl(x,u)lj
and (19.44)), this implies that

IA(z,u)I>olu-zI2 (of. (19.33)

1

I(B , E, z) < C

IIdgj(u)IldS IA(z,u)I

n

dS lu-x)

d6

n-1 + CIz-uI - 2n1

ueD

lu-xl

(19.51)

2n-1

x S

lu-zI
for all zeD

dS

)u-zI
F>0,and 1=0,1,2.

By Proposition 3 (1) in Appendix B, we have

d5

1

Iu- xIdS

1

<

[diet-

(

u, S] 1 /4

t

dS lu-x

C

1-1/4 I

- [dist(u,S)]

1/4

By Proposition 5 (i) in Appendix B, this implies

I

d6

J Is ueD l u-s l
dS

i

< C

1/2

(19.52)

lu

xeS

103

for all zeD+ and E>O. Since sod=0 on S, we have IQj(u)I < CIu-xl for all ucD

Therefore

and

dS < C x12n 1

r

1u

for all ueD

.

dS

`

lu -xl2)1u

xI2n-3

By Proposition 2 (i) in Appendix B, this implies that

dS lu-x1

J

2n-1 <

C(1 + Ilnlgj(u)))

XES

for all uED

.

Hence

rlldy.(u)Ild6 J

fd9(u)j(l + lnl9.(u)I{)

dS lu-xI2n-i

IA(z,u)I°

xES

uED

(19.53)

IA(z,u)In

uED lu-zl
Iu-zl
for all zED+ and 1>0. By (19.44), for all

d6

< C

J

4Oj(u)1+1u-z12)2lu-z12n-4

IA(z,u)ln >

Since oj=0 on S, we can find a constant K'
and

such that IQ,(u)I
with dist(u,D+)
integral on the right hand side of (19.53) can be estimated by

C

[

I!dgj(u)II(1 + 11n?p3(u)II)

d6.

(IQj(u)I+lu-zl2)21u-z12n-4

uED I9j(u)I
In view of Proposition 5 (ii) in Appendix B, this implies that, for every fixed &0, the right hand side of (19.53) is bounded by C6; l/2-d Together with (19.51) and (19.52) this implies (19.46).

19.12. Part 3 (and end) of the proof of Lemma 19.9: proof of estimate

(19.47). Set 0- = 4-(w,u) = - 29.(u)

for w,uiUi, where p1EDiv(i3) is as in Sect. 19.10. Then, by (7.4),

184

Re 4_(w,u) > o(-Rj(u) - P3(W) + lu-wl2)

(19.64)

for all w,uEUY In particular, 4_(w,u)#0 for all w,u E SuD , the case u=weS. Therefore, the map

U-w v_:= v-(w,u,t):= t IU-wi

except for

pl(w,u) 2

+ (1-t)

4_(w,u)

is defined and of class C1 for all O
with w#u.

Therefore

r-- r-R0(w,u,t):= x (w)x (u) det(v_,dtv_,(l

n-q-1

q-1

uv-,wv_ )A w(u), n-q-1

q-1

R1(w,u,t):= ox (w)AX (u) det(v_,dtv_, 3uvwv

L) (u),

q-2 n-q

R2(w,u,t):= x (w)ax (u)Adet(vdty ,duv_,wv )na1(u) are continuous differential forms for all 0
with w'u.

We have (of. (19.35) 1)

Rl(w,u,t) = R1(w,u,t)

if u4S (1=0,1,2).

(19.55)

Since

det(u-w,u-w,a11...,an-2) = det(pi,pl,al,...,an-2) =.0

(19.56)

for any collection al,...,an-2 of vectors of differential forms, it follows that

R0 R_(w,u,t):=

r- rI q-1

x (w)x (u)(2t-l)dt '_(w,u)iu-w(

n-q-1

det(v_,dtv_,3uv_,awv_ )A w(u),

2

ax (W)AX (u) (2t-1)dt l R(w,u,t):= 2

r-z r-q-1

n-q-1

det(v _,dtv _,duv_,awv_ )A w(u),

_(w,u)lu-wl

2 R(w,u,t):=

x (w)ax (u)A(2t-1)dt 2

r-, q-2

n-q

-

det(v_,dtv_,3uv_,wv_)AWJ(u),

185

for all O
a p

du

a v_ = t-

+ (1-t)

(1-t)p a

t3 lu-wl2

1

- u a-(u-w) - - -p1 1u-w!

lu-wl

and

-dw

3 P

%I"- = t2 + (1-t)

w 1

to lu-wl2

(1-t)p o

-(u-w) - - ~ pi lu-wl

lu-wl

+_

this implies (taking into account again (19.56)) that 1 n-

R0(w,u,t) = x (w)x (u)

/

1

r==

R1(w,u,t) = ax (w)AZ (u)

1

r=0

PredtARrs'

--1 L s-

dt.A

Rrs,.

R2(w,u,t) = x (w)3x(u)A) 2_ pYe tARrs r--j s=0 for all O
q-1-r s n-q-1-s r-2 r-- ri r---I det(p1,u-w,du, dupl.dw, r

awpl

Rre

(r+s+)

^ w(u)

and

q-2-r s

r

det(p1,u-w,du,

R

-r-slu_wl

n-q-s i1Vl,dw,

awpl) n w(u) r+s+

Since 4_(w,u)#O if wcD

end uESuD (of. (19.54)), it follows from (19.57)1 that the forms R1(w,u,t) have a singularity of order <2n-3 at

u=WED,aedthe forms 8 R'(w,u,t) are continuous for all 0
u 4Y1, we obtain that

188

J G(a,u)3P(a,u)ARl(w,u,t) = juG&(z,u)luP(z,u)AR1(w,u,t) ueD 0
usS O
+ (-1)n JG(z.u)AuP(z,u)AuRl(w,u,t)

(19.58)

ueD 0
for all eeD+, w*D , E>0, and 1=0,1,2.

Recall that, by (19.54), Iu-w12
.

Further,

it is clear that

ml

m2

m3

m4

det(pi,u-v,du,&uPl,dw,3wP1)I < Clu-w11P1(w,u)I

and

ml

m2

m3

m4

I3u[det(p1,u-v, du,3uPl,drr,3wpl)3 < C(lu-wl + Ip1(w,u)I) for all if ml,...,m4>0 is an arbitrary collection of integers with l+...+m4=n-2. Therefore, it follows from (19.57)1 that

lu-wl + IP (w,u)I n

IP (w,u)IB3 + (w,u)u lu-;I n-3

lt_(w,u)Ilu-wl

lPl(w,u)lu3ul u-wlIJ +

(19.59) 14_(w,u)Iiuwl2n-

for all 0
and 1=0,1,2. Since pE Div(pj) (of. Sects. 7.6

and 7.7), one has the estimate

1p1(w,u)I < Odqj(w) + 1u-w1)

(19.60)

for all w,uESUD . In view of the definition of 4_(w,u), this implies that also uau¢_(w,u)u < C(lldoi(w)II + lu-wl) for all w,uESUD . Therefore, it follows from (19.59) that

187

11d9j(w)II

1

-1 u

1+-(w,u)IIu-wl2n-3

l

d43 (w){2

(19.61) I

and 1=0,1,2. Further, by (19.57) 1 and (19.60),

for all 0
we have

Rl(w,u,t)II

C

1

4

II

for all 0
+

IIdej(w)I I4-(w,u)I1u-w1 n 3

(19.62)

and 1=0,1,2.

we set S. = {wED : Qj(w)=-1/m}. Since soj does Now, for m=1,2,... not have degenerate critical points, for all sufficiently large m, Sm is ,

smooth and, for any continuous function f on SOD

f dS =

lim

m >oo

,

we have

f dSm,

(19.63)

nY

Sny

where dSm is the Euclidean volume form of Sm; this can be proved by similar arguments as in point 6 in Appendix B. Recall that, for any vi

continuous differential form g on D1, the form R1f.admits a continuous extension onto SOD

(of. the beginning of the proof of Theorem 9.1). In is continuous on Dj and vani-

view of (19.37) 1 and since GF(z,

shes on

for all fixed £>0 and zED+, this

implies that, for all fixed E>0,

zED+,

fl,

E

z(w):=

and 1=0,1,2, the function

JGL(z,u)uP(z,u)AR1(w,ut)I

,

wED ,

II

uES 0
admits a continuous extension onto SOD which will be denoted by f z. Since, for all sufficiently small &>O, supp f1 Z =C Y1, therefore it follows from (19.63) that 188

I(RI,E,z) _

PL zdS =

I

lie

S

fI zdSm

I

(19.84)

S

m

for all zeD+ and all sufficiently small 6>0 (1=0,1,2). Take a number KK& (cf. the beginning of Sect. 19.11). Then it follows from (19.58) that

f1 z(w) < CJiIaUP(z,u)IIIIRl(w,u,t)Ild5 + CJ II auP(z,u)IIIIauRl(w,u,t)`Id5 ufD

uaD lu-zl
lu-zI
for all zeD+, wED ,£>0, and 1=0,1,2, where d6 is the Euclidean volume form in Cn. In view of (19.64), this implies that

I(R1,E,z) < C

sump

1 dSm wiSm

J

I`aUP(z,u)1IIIRl(w,u,t)IId6

uED lu-zI
o
C sup J dSm III

J uuD

m wES

m

lu-zl
o
for all z6Dall sufficiently small g>0, and 1=0,1,2. In view of (19.61) and (19.62), this implies that

I(R1,e,z) < C amp f IIauP(z,u)I(I1(u)+I2(u)+I®(u))dK uED lu-zl
for all zED+, all sufficiently small £>O, and 1=0,1,2, where

I1(u):=

1 I

dS ,

weS m

189

Idg (M) Im(u)'-

dSm,

10_(w,u)Ilu-wl n I

WtS

m dQ, (W) Il 2 dSm. u)121u-w1 2 33

Wes m

Taking into account (19.50), so we see that

(Iu-z1+id93(u)4)

1

I(R ,E,z) < C sup 1$_(z,u)In

m

2

1

3

(Im (u)+Im(u)+Im(u))ds(19.65)

U ED

lu-zI
for all zED+, all sufficiently small E>0, and 1=0,1,2. Now we go to estimate the integrals I:(u). By (19.54), we have

dSm I1(u) -< C Is

(I9J(u)I+lu-wl2)Iu-WI2n-3'

weSm

IId9 (w)llds°

I2(u) < C

'

(Ipj(u)I+Iu-wl2)lu-wl2n-

°

WES5

jj d4,) (W) u 2dSm

1.3 (u) < CJ

(Ipj(u)i+lt(w,u)I+lu-wl

for all

)

1u-W1

and , where t(w,u):= Im 4_(w,u) = Is . There-

fore and by Lemma 9.2, we can apply Proposition 2 in Appendix B to the integrals I:(u), and obtain the estimate

U

Ie(u)+I2(u)+I3(u) < CI1 + I1nl j u)11 + `l

for all uED

and m=1,2,...

.

Since IId9,(w)I < C(I1d9 (u)II + lu-wl), we see that

190

i

Ip3(u)I

) 1 11

(19.86)

d3

I2(u) < Cjde(u)u m 3

d8

m

CIQ_(w,u)Ilu-wl m

+

_

IQ_(w,u)Ilu-wl ° wESm

n-

wESm

and

dS

I3(u) < CJjd2i(u)II

dS

+C

e 14_(w,u)I Iu-wl n-4

lu-wl2n-

10_(w,u)I

wESm

wESe

dSa + CIIdgj(u)02 IQ_(w,u)121u-wl2n-3

We s

for all uED

and m=1,2,...

Taking into account that, by (19.54),

.

I+-(w,u)I>clu-w12 for all u,w&D

,

this implies that

d6 I1(u)+I2(u)+I3(u) < Cjdej(u)4 m

m

t

dS

m nI4_(w,u)Ilu-wl

+ C

m 2n IQ-(w,u)Ilu-W1 WESm

wESm

+ Cjjdgj(u)112

dS

I WESm

2

IQ_(w,u)1 lu-wl

for all uED

and m=1,2,...

(19.67)

n_

.

Now let some 6>0 be given. We want to prove (19.47) for this Without lose of generality we may assume that 5<1/2. In view of (19.54), we have

clgj(u)15/21u_w12n-1-S

IQ-(w,u)Ilu-w12n-3

>

I0(w,u)Ilu-w12n-2

Iej(U)11/2+S/21u-wi2n-1-S

>

clgf(u)l1+5/21u-w12n-1-5

IQ_(w,u)121u-w12n-3

>

for all uED , w6Sm (m=1,2,...

). Further, by Proposition 3 (i) in

Appendix B, we have

191

dSm < oo.

sup

lu-wl2n-1-6

m S

m

Therefore it follows from (19.67) that II

11(u)+I2(u)+l3(u) <

NdQj(u)Il

1

(19.68)

Iqj(u)I1+S/2J

for all uED

and m=1,2,...

.

Since Qj(z)>O and cj(u)<0 if z&D+ and uED ,

there is a constant K'
such that {uED

for all zED

lu-zl
:

:

(19.69)

gj(u)
Recall that, by (19.44), IA(z,u)l

for all z6D+ and uED abbreviations

> c(lgj(u)1 + IIm A(z,u)l + Iu-z12)

.

For 2eD+ and E,>0, we introduce the following

J' r IQ

-j 1(&,z) =

I

j

lu-zi

2n-1

d6'

uED

lu-sI
Ildgj(u)Q Ip.(u) I-1/2-J/2

J (E, z) =

d6,

(I9j(u)1+1u-z12)21u-zi n-

2

uED Igj(u)I
J3(E,z) =

11dej(u)1121g.(u)I-1-a/2

f

J (Igj(u)1+IIm A(z,u)I+lu-zl

2

)

ueD

I_qj(u)I
Ifdg.(u)il(1 + I1njgj(u)Ij)

4(E,z) _

(IQj(u)I+lu-zl 2

uED

I-Pj(u)I
192

(19.70)

)2

iu-zl

2n-4 d5 l

iu-zl

2n-5 d6,

(u)1-1/2

udQ (u)jj21Q

J5( ., a) =

J

i 2 (1ej(u)I+IIm A(e,u)I+lu-al

)

lu-al

n- d6.

ueD 14j (u)I
Taking into account (19.69) and (19.70), then it follows from (19.85), (19.66), and (19.68) that

I(R1,E,a) < i=1 1

for all aeD+, all sufficiently small L>0, and 1=0,1,2. So it is sufficient to prove that, for i=1,..,5,

Ji(E,s) <

C&1/2-8

(19.71)

for all seD+ and all E>O.

Since qj does not have degenerate critical points, it is easy to show (for instance, by means of Proposition 0.7 in Appendix B) that dist(u,S) < Clgj(u)I1/2 and hence

j (E,a) < C r [dist(u,S)7-a d6 lu-al2n-1

1

J

uED lu-sI
for all s.D+ and E>0. Therefore, if i=1, then (19.71) follows from Proposition 5 (1) in Appendix B (with oc=a and 6=1/2-8). If i=2, then

(19.71) follows from Proposition 5 (ii) in Appendix B (with m=1/2+8/2, ,8=1/2-6, and d=2), and if i=4, then, in view of the estimate

1 + jlnlQ (u)II <

Clgj(u)I-6/2

(ueD

(19.71) follows also from Proposition 5 (ii) (with ox.=3/2, )3=1/2-8, and d=1).

It remains to consider the cases i=3 and i=5. Since

A(s,u) = <w1(p(s)"8(u)),p(u)-)5(a)> = -4wi(A(u),)8(a)),$(u)-p(a)) for some wiEDiv(-6k), where 6and ,8 is biholomorphic, it

193

follows from the lemma stated in the proof of Theorem 14.1 that the following statements (i)-(iv) hold true: (i)

cJIdQj(z)H
IIdQJ(a)II

for all z,uEY2.

II d2 (z)Adulm A(z,u)lu=z

II

>

c11dgj(zA2

for all z,u¢Y2. (iii)

a Im

a Im A(z,.) (u) -

atm

(z)

< C)u-zl

atm

for all z,ueY2 and m=l,...,2n, where tl,...,t2n are the real coordinates

in Cn. (iv) If dpj(u0)=O for some uOeS (this is possible only when [uj,pj,{.p.Dj] is of type II), then

IIm A(z,u)l < C(lu-u011u-zI + Iu-z12)

for all z,ucY2.

Hence, for i=3, estimate (19.71) follows from Proposition 5 (iii) in Appendix B (with

and d=2), and, for i=5, (19.71)

follows from the same proposition (with o<,=l/2,,8=1/2-8, and d=l). Thus, estimate (19.47) and hence Lemma 19.9 is proved.

19.13. Proof of Theorem 19.2'. Since, by Lemma 19.6, Z n-q-1(6,E*) is a 1 is a finitely codimen(Im A)

finitely codimensional subspace of Im A, sionel subspace of ZQ q(D,E).

Let T,K,L be the operators from construction 19.8. Since K is bounded 1/2 (D,E) (cf. assertion (19.22)), as an operator from Z00 q(D,E) into then it follows from Lemma 19.9 that the restriction of K+L to ZO q(D,E) is compact as an operator with values in C0 q(D,E). We denote this

restriction by K. Further, let T be the restriction of T to ZD q(D,E). 0 Then (19.23) implies (19.1) and, except for relation (19.2), assertion

194

(ii) of Theorem 19.2' is proved. Since, by Stokes' theorem Ep 4 0(B,E) is contained in ZO q(D,E) for all O
It remains to prove (19.2). Let feZO q(D,E)n E0 q(D,E) and geZ° n_q-1(G,E*). We have to show that

(19.72)

fng = 0. J

aG

Let 9:6 -+ LR be as at the beginnig of construction 19.3. Take 1>0 so small that Gu{zEO: &)(z)
is a non-degenerate strictly q-convex extension of G. By Proposition which converges to

12.10 (iii), we can find a sequence g uniformly on G. Then

fng = Jim k aG

I

}

fngk.

aG

Since fEZg q(D,E) and, therefore, the forms fngk are closed on GE\G, this implies, by Stokes' formula, that

f fAg = Jim f fngk. k JJJ

aG

W&

Since f is 3-exact on D and hence the forms fngk are exact on M. C D, so we see (again by Stokes' theorem) that the integrals on the right hand side of the last relation vanish for all k. Hence (19.72) holds true.

19.14. Proof of Theorem 19.1'. Since G is relatively compact in Y, and Y

is (n-q)-convex, we can find a relatively compact domain G' - Y such that G - G', Y is an (n-q)-convex extension of G' and hence Y\G is an (n-q)-convex extension of 0'\G. Set D=G'\G. Then, by Theorem 12.14, the restriction map

HO' qmd, E)

g0, q(D, E)

is an isomorphism. On the other hand, X is a q-concave extension of Y\a

195

and thus, by Theorem 16.1, the map

HI.NX,E)

Ho' q(y\(I, 11)

is injeetive. Together this implies that the restriction map HO,q(X,9)

HO,q(D,B)

(19.73)

is injeetive. Further, by Proposition 0.5 in Appendix B and Observation 4.15, we may assume that a(3 and aG' are smooth. Then, by Theorem 19.2', E1/2--+0 (,E) is closed with respect to the norm p'uo D. Since 0 4 (19.73) is injective, this implies that 900 q(X,E) is closed with respect

the space

to uniform convergence on the compact subsets of X. Theorem 19.1' is proved.

196

CHAPTER V. ROME APPLICATION.

20. Solvability criterions for a=f and duality between the Dolbeault cohomology with compact support and the usual Dolbeault cohomology

Let X be an n-dimensional complex manifold, and 1
fAg = 0

for all gi[Znon-q(X)]0.

(20.1)

1

X

It arises the question whether, conversely, (20.1) implies solvability of $u=f? Of course, a necessary condition for an affirmative answer is

that E q(X) is closed with respect to uniform convergence on the compact subsets of X. We shall see that this is also sufficient (Proposition 20.1).

Perhaps it should be recalled here that we know already the following

three conditions on X, each of which guarantees that EO

(X) is

closed:

1) X is (n-q)-convex (cf. the Andreotti-Grauert Theorem 12.16).

2) X is (q+1)-concave (cf. the Andreotti-Grauert Theorem 15.13). 3) X is a q-concave domain in a compact complex manifold (cf. the Andreotti-Vesentini Theorem 19.1) or, more generally, X is as in Theorem 19.1'.

In particular, if X is compact, then E0 q(X) is closed.

Now let f be a 3-closed continuous (O,q)-form with compact support in X. Then, again by Stokes' formula, for the solvability of c5u=f with uE[C0 q_1(X)l0 it is necessary that

fAg = 0

for all gEZnoon_q(X).

(20.2)

J

X 197

We shall see that condition (20.2) is also sufficient for the solvabiliif and only if the space 3[C 00 (X)] ty of &u=f with uE[C0 (X)) is O,q-1 0 O,q-1 0 closed with respect to the topology of [C0 q(X)JO (cf.

(0.8)). Further,

we shall see that this is the case if and only if En,n-q+l(X) is closed with respect to uniform convergence on the compact subsets of X (Propositions 20.2 and 20.3).

It is the aim of the present section to prove these solvability criterions, and then, by means of them, to establish duality relations between the Dolbeault cohomology with compact support and the usual Dolbeault cohomology. As an application, we prove the Hartogs' extension theorem of Kohn-Rossi.

20.1. Proposition. Let E be a holomorphic vector bundle over an n-dimensional complex manifold, and 1
tion uECO

q-1

ME) if and only if

j gAf = 0

for all g([Znoon q(X,E*)J0.

(20.3)

X Proof. That condition (20.3) is necessary follows from Stokes' theorem. To prove the converse, we assume that f does not belong

to EO q(,E). Since E0(, qE) is closed, then, by the Hahn-Banaeh theorem, we can find a continuous linear functional F on Cp q(,E), i.e. an E *-valued (n,n-q)-current with compact support (cf. Sect. 0.14), such that F(f) # 0 and F(E0 q(X,E)) _ {0}, i.e. 5F=O. By, Corollary 2.15 (i),

we can find a form g6[Znon q(,E*))0 and an E*-valued (n,n-q-l)-current S with compact support such that F = + 3S. If we interpret aS as a

continuous linear functional on C0 q(,E) (cf. Sect. 0.14), then W vanishes on (Z q(X,E)) = {0). Hence (f) = F(f) # 0, i.e. condition

(20.3) is violated. U 20.2. Propositon. Let E be a holomorphic vector bundle over the n-dimensional complex manifold X, and 1
X. Let PE[Z0 q(X,E)]O. Then the equation au=f can be solved with Uc[CO q_1(X,E)]0 if and only if

j gAf = 0

for all geZooon_q(X,E*).

x Proof. That condition (20.4) is necessary follows from Stokes' theorem. To prove the converse, we assume that condition (20.4) is

196

(20.4)

fulfilled.

Then we define a linear functional F on EO

n,n-q+1(X,E*)

as follows: If

hEEn,n-q+l(X,E*), then we take veCn n_q(X,E*) with &=h, and set

F(h) =

f fAv.

X This definition is correct. In fact, let v,wECn n_q(M,E*) with 3v=3w=h.

Then by Theorem 2.14 (smoothing of the Dolbeault cohomology), we can find gcZnoon_q(X,E*) such that v-w-g = 3p for some piCn,n-q-1(X,E*).

Hence, by Stokes' formula,

J fn(v-w-g) = 0,

X i.e., by condition (20.4),

J fnv =

1 fnw.

X

X

Since, by hypothesis, the space En

,n-q+1(,E*)

is closed, it fol-

lows from Banach's open mapping theorem that F is continuous. By the Hahn-Banach theorem, there exists a continuous linear extension F of F If we interpret this extension as a current with onto CO ,n-q+l(M,E*).

compact support (cf. Sect. 0.14), then

,

for all hE[Cnon-q(X,E)]0, we

have

(-1)q-1(3F)(h) = F(8h) =

r fAh = (h), JX

i.e. = 3((-1)q-1 F]. By Corollary 2.15 (ii), this implies that f=3u

for some uc[C0,q-1(X,E)]0. 0 If E is a holomorphic vector bundle over an n-dimensional complex manifold X and is closed, then it follows from ProposiEn,n-q+l(X,E*)

tion 20.2 that the space 3[Cp q_1(X1E)]0 is closed with respect to the topology of (Co q(X,E)]0 (cf. (0.8)). The converse is also true:

20.3. Proposition. Let E be a holomorphic vector bundle over an n-dimensional complex manifold X, and let 1
199

a[Co q_1(X,8)]0 is closed with respect to the topology of (Co q(X,S)]0 if and only if

Ee,n-q+1(,b*)

is closed with respect to uniform conver-

gence on the compact subsets of X.

Proof. As already observed, it follows from Proposition 20.2 that this condition is sufficient. We prove the converse. Assume that a[C0,4-1(X,E)]0 is closed. By Stokes' theorem, En,n-q+l(X,E*) is

contained in the closed subspace of all

f

for all gc[Zp q_1(X,E)]0.

fng = 0

(20.5)

X

We shall prove that this subspace is even equal to

En,n-q+l(M,E*).

Fix feZ0 n_q+l(X,E*) such that (20.5) is fulfilled. Setting

F(h)

fAh, J

X

we define a continuous linear functional on [C01q-1(X'E)]0.

By (20.5) there exists a uniquely determined linear functional H on 8(C0O0 q_1(X,E)l0 such that the diagram

is commutative. Since, by hypothesis, the space 3[C0 q_1(,E)]0 is closed with respect to the topology of [CO q(X,E)l0 and since the operator 8 is continuous with respect to this topology, it follows from Banach's open mapping theorem that H is continuous. By the Hahn-Banach theorem, H admits a continuous extension to some E -valued (n,n-q)-current H on X. Since (20.6) is commutative, we have

aH (h) _ (-1)'H(ah)

fAh = (h),

(-l)'F(h)

X

200

for all hc[C0 q_1(X,E)J0, i.e. c)H = . By Theorem 2.14 (smoothing of 0 the Dolbeault cohomology), this implies that f6E0 n-q+1 (X,E).

Let E be a holomorphic vector bundle over an n-dimensional complex manifold X, and let O
Recall the smoothing isomorphism (Theorem 2.14)

HO,q(X,E) = Ho'q(X,E) = Zg q(,E)/ED,q(,E).

Further on,we assume that HO'q(X,E) is endowed with the factor topology of the Frechet space ZO q(X,E) with respect to the subspace EO q(X,E).

If E0 q(,E) is closed.in Z0 q(,E), then (and only then) H0'q(M,E) is also a Frechet space. By (HO'q(X,E))* we denote the space of continuous linear functionals on H0'q(X,E).

For any fixed he[Zooon_q(,E*)]0, the map

'

Z00 q(,E) 3 u

J

uAh

X

is continuous and linear on ZD q(X,E). By Stokes' theorem, this map vanishes on E00 q(,E) and hence it defines a continuous linear functional on H0'q(X,E). Thus, a homomorphism

[Zoon-q(,E*)]0

(HO,q(X,E))*

§

is defined. Again by Stokes' theorem, this homomorphism vanishes on

a[Cnn-q-1

(X,E*)]0 (if q
[P,,,-q( X,E*)]0

)

(HO,q(X,E))*

(20.7)

is induced, where

[H"'-9(X,B*)]

0

(M,E*)10/arc °O (Zoo n,n-q-1 (M,E*)] 0 n,n-q

.

(20.8)

The homomorphism (20.7) will be called the canonical homomorphism from [H"'-q(X,E*)]0 into (HO,q(X,E))*.

201

20.4. Theorem. Let E be a holomorphic vector bundle over an n-dimensional complex manifold X.

Then, for all O
phism (20.7) is surjective. Moreover, the following assertions hold true: (i) The canonical homomorphism

[(Hn,O(X,E*)]0

(HO'n(X,E))*

(20.9)

is an isomorphism. If E0 0 , n(X,E) is closed with respect to uniform convergence on the compact subsets of X, then, moreover,

dim [Hn,O(X,E*)]0 = dim HO'n(X,E).

(20.10)

(ii) The canonical homomorphism

[(Hn'n(X,E*)]0

(6(X,E))*

(20.11)

is an isomorphism if and only if ED 1(,E) is closed with respect to uniform convergence on the compact subsets of X. In this case

dim [Hn,n(X,E*)]0 = dim 6(X,E).

(20.12)

(iii) Let l
dim [Hn'n-q(X,E*)]0 = dim HO'q(X,E).

(20.13)

Proof of: (20.7) is always surjective. Let FE(HO'q(X,E))* be given.

Denote by p the canonical projection from Z0 q(,E) onto HO'q(X,E). Set H=Fop. By definition of the topology of HO'q(X,E) given above, p is continuous. Hence H is continuous. By the Hehn-Banach theorem, H admits continuous linear extension onto C0 q(X,E), i.e. H admits an extension to some E*-valued (n,n-q)-current H with compact support on X (cf. Sect. 0.14). Since H vanishes on Ep q(X,E), we have &H=O. By Corollary 2.15 (i), we can find ge[Zn0 n_q(X,E*)]0 with

H - = 3S

202

(20.14)

for some E -valued (n,n-q-1)-current S with compact support on X.

In

order to prc.ve that (20.7) is surjective, now it is sufficient to show that

H(h) =

f gnh x

for all hEZ0 q(,E). Let a form h Z

q(,E) be given. By Theorem 2.14 (smoothing of the Dolbeault cohomology), we can find h00EZO q(,E) with h-h00EE0 q(,E)- Since H vanishes on Eg q(,E), then H(h)=H(h00), and, by Stokes' theorem,

f gAh = X

f gnh00. X

Thus, it remains to prove that

H(h00)

(20.15)

gAhO.O. J

X

Take a Coo function x on X such that supp X- X and

1 in some

neighborhood of supp H u supp g U supp S. Then

H(X hoo) = H(hoo),

gnx ho =

J

gnh00,

and oS(Xhoo) = 0.

Hence (20.15) follows from (20.14).

Proof of assertion (i), Since Zp n(X,E) = C0+n(,E), it is trivial that (20.9) is injective. Since (20.7) is always surjective, so it is proved that (20.9) is an isomorphism. If Eg n(X,E) is closed, then HO'n(X,E) is.a Freshet space, and, by the Hahn-Banach theorem, dim HO'n(X,E) = dim (HO'n(X,E))*. Since (20.9) is an isomorphism, this implies (20.10). Proof of assertions (ii) and (iii). First we prove that, for all 0
q+l(X,E)

is closed.

Assume (A)q is true. Then, in particular, (20.7) is injective,

203

acc, {gE[Z0on_q(X,E*)]0:

gnh = 0

for all hEZ0 q(X,E) ).

(20.16)

J

X

Therefore (A)q implies that 5[Cn°On_q_1(M,E*)]0 is closed with respect to

the topology of [Cn°O0_q(,E*)]0. In view of Proposition 20.3 (applied to the bundle E*®K, where K is the bundle of holomorphic (n,0) forms on X), this means that (A) q implies (B)

q"

Assume that (B) q is true. Then, by Proposition 20.2 (again applied to

E*®K), the space a(C0.n-q-1(,E*)]0

M:= [Zn n-q(,E*)]0 n is equal to the right hand side of (20.16). Since by Corollary 2.15 (ii), M is always equal to the left hand side of (20.16), this means that (20.16) holds true, i.e. (20.7) is injective. Since (20.7) is

always surjective, so it is proved that (B)q S (A) q*

Now we prove assertion (ii). The equivalence of (A)0 and (B)0 means that (20.11) is an isomorphism if and only if Eg 1(X,E) is closed. This implies

(20.12), because dim (1(X,E) = dim((9(X,E))* by the Hahn-Banach

theorem.

Finally, we prove assertion (ii). The first part of this assertion means that (A)q (B)q. Assume E00 q+1(, E) is closed, i.e. (B) q holds true. Then also (A) is fulfilled, i.e. (20.7) is an isomorphism. 9 If, moreover, E q(, E) is closed, then HO'q(X,E) is a Frechet space, and (20.13) follows from the Hahn-Banach theorem and the fact that

(20.7) is an isomorphism. 0 20.5. Corollary. If X is an n-dimensional complex manifold without compact connected components, then, for any holomorphic vector bundle E

over X, (HO,n(X,E))* =

i.e.

E00

0,

n(,E) is dense in Z0 n(X,E).

Proof. This follows immediately from part (i) of Theorem 20.4, because, by uniqueness of holomorphic functions. [Hn,O(X,E*)10 = 0.

0

20.6. Theorem. If X is a 0-convex n-dimensional complex manifold without compact connected components, then, for any holomorphic vector bundle E

204

over X,

HO'n(X,E) = 0.

Proof. By the Andreotti-Grauert Theorem 12.16, E0 n(,E) is closed.

Thus, by Corollary 20.5, E00 n(,E) = ZD n(,E). 0

Remark to Theorem 20.6. By Corollary 5.10, each domain X - Y with C2 boundary in an arbitrary complex manifold Y is 0-convex. It is easy to show that, moreover, a complex manifold X is 0-convex if it admits an exhausting function which has only a finite number of critical points. Question. Does there exist complex manifolds which are not 0-convex? 20.7. Theorem. If E is a holomorphic vector bundle over a q-convex n-dimensional complex manifold X (0
dim [HO'r(X,E)]O = dim

HO,n-r(X,E ®K)

< o0

(20.17)

for all 0
If q=n-1, then, moreover,

dim [HO,n(X,E)]0 = dim U(X,E*®K).

(20.18)

Proof. Let 0
dim HO,n-r(X,E*®K) < oo

and

dim

HO,n-r+l(X,E*®K)

< oo.

Hence, by Theorem 20.4 (iii),

dim [Hn,r(X,E®K*)]0 = dim

HO,n-r(X,E®K) < oo.

Since E ®K*-valued (n,r)-forms may be viewed as (O,r)-forms with values in (E ® K*)®K = E, this coincides with (20.17). Let q=n-1. Then, by the Andreotti-Grauert Theorem 12.16, dim HO'1(X,E*OK)

< oo and it follows from Theorem 20.4 (ii) that

dim [Hn,n(X,E®K*)]0 = dim (2(X,E*®K). This coincides with (20.18), identifying E

K*-valued (n,n)-forms and

E-valued (O,n)-forms. 0

205

20.8 Theorem. If E is a holomorphic vector bundle over a q-concave n-dimensional complex manifold X (2
dim [HO'r(X,E)]0 = dim HO,n-r(X,E* K) < oo

(20.19)

for all n-q+2
on X. Proof. Let n-q+2
dim

HO,n-r(X,E*®K)

< on

and

dim

HO,n-r+l(X,E*®K)

< oo.

Hence, by Theorem 20.4 (iii) and (ii),

dim [Hn,r(X,E®K*)]0 = dim

HO,n-r(X,E*®K) < oo.

K*-valued (n,r)-forms and E-valued (0,r)-forms, this implies (20.19). El Identifying E

If the manifold X in Theorem 20,8 is a domain in some compact complex manifold Y and E is a holomorphic vector bundle over Y, then the conclusion of Theorem 20.8 can be strengthened:

20.9. Theorem. Let X C Y be a q-concave domain in an n-dimensional compact complex manifold Y (1
dim

HO, n-r(X, E *OK) < oo,

and, by the Andreotti-Vesentini theorem 19.1, the space EOn c+1(X,E*®K) is closed. If q=1 and hence n-r=0, then this implies, by Theorem 20.4 (ii), that

dim [Hn,n(X,E (9 K*)]O = dim HO'0(X,E*®K) < oo.

(20.20)

If 2
dim [,n,r(X,E ® K*)]0 = dim

206

HO,n-r(X,E*®K)

< oo.

(20.21)

Identifying E (VK*-valued (n,r)-forms and E-valued (0,r)-forms, (20.20)

and (20.21) imply that (20.19) holds true also for r=n-q+l. El 20.10. Theorem. Let E be a holomorphic vector bundle over an n-dimen-

sional compact complex manifold Y, and let X - Y be a domain which is both q-concave and (n-q-l)-convex (1
dim[HO'r(X,E)]0 = dim

HO,n-r(X,E*®

K)

for all 0
(20.22)

If r#n-q, then, moreover,

dim [H0,r(X,E)]0 < oo.

(20.23)

If X is a q-concave extension of some strictly (n-q-1)-convex domain

D - X, then dim [HO'n-q(X,E)]0 = 00.

Proof.

(20.24)

If 0
Theorem 20.7. If n-q+1
dim HO,q+l(X,E* K) < oo,

and, by the Andreotti-Vesentini Theorem 19.1, E0 q(X,E*®K) is closed. Thus, we can apply part (iii) of Theorem 20.4 and obtain that

dim [H°'°-q(X,E K)]0 = dim H°'4(X,E ®K).

Since E ®K*-valued (n,n-q)-forms may be interpreted as E-valued (0,n-q)-forms, this implies (20.22) for r=n-q. If X is a q-concave extension of a strictly (n-q-1)-convex domain, then (20.24) follows from (20.22) and the Andreotti-Norguet Theorem 18.4.

As an interesting application of Theorem 20.7, now we obtain the following version of the Hartogs' extension phenomenon:

20.11. Theorem (Kohn-Rossi theorem). Let X be a connected non-compact 1-convex complex manifold such that there exists at least one non-constant holomorphic function on X. 1) Then, for any compact set M -- X such 1)

There exist connected non-compact pseudoconvex manifolds without non-constant global holomorphic functions (see (Grauert 1963])

207

that X\M is connected, the restriction map (.?(X)

O(X\M)

>

is an isomorphism.

Proof. The injectivity is clear, because X is connected. We prove the surjectivity. Let fE(9(X\M) be given. Since X is 1-convex, by Theorem

20.7, (20.25)

N:= dim [HO'1(X)]0 < cc.

Choose a Coo function x on X such that supp x == X and x = 1 in a neighborhood of M. Without loss of generality we may assume that f is not constant. Then the forms fk8xe(Z0,1(X)]0 (k=0,1,... ) are linearly independent. Therefore, by (20.25), we can find a vector 0

(a0,...,a1)

in CN+1 such that

N

-L-akfk3X = du k=0 for some uE[C00(X)10. Setting

N

F=

x)

(1

k=00

akfk - u,

we obtain a holomorphic function F on X with N

F = Eakfk

on X\[(supp

U (supp u)]

k=0

By hypothesis, there is a non-constant holomorphic function h on X. Then are linearly independent. also the forms hkfdx E[Z° 1(X)JO (k=0,1,... )

Again by (20.25), we can find a vector 0 = (b0,...,bN)ECN+1 with

N

bkhkfa k

=

')v

for some vE[C°O(X)]0. Setting

G

obkhk k

208

and

H= (1- X )Gf - v

we obtain holomorphic functions G - 0 and H _& 0 on X such that H = Gf

on X\[(suppx ) U (supp V) 3,

(20.26)

and

IakHGN-k k=0

N

= GNY akfk = GNF k=0

on X\[(supp

U(supp u) U(supp v)J.

Since X is connected, it follows that N

kHkGN-k = GNF

on X.

k=0

Therefore

ak(G) = F

on X\iz6X: G(z)=0},

k=0 Since G * 0 and (a 0'-'-'&N)

0, this implies that H/G is a locally

bounded meromorphic function on X,

i.e. H/G is holomorphic on X. Since,

by (20.26), H/G=f in some open part of X\M, and X\M is connected, it follows that H/G=f on X\M. 20.12. Remark to Theorem 20.11. If X mm Y is a non-degenerate strictly pseudoconvex domain with dX # 0 in some complex manifold Y, then there exist non-constant holomorphic functions on X. In this case, even dim ((X) = oo. This well-known fact can be proved, for instance, in the same way as statement (A) in the proof of Theorem 23.3 below. On the other hand, this is a special case of Theorem 18.4.

20.13, Theorem. Let E be a holomorphic vector bundle over an n-dimensional complex manifold Y, and X am Y a domain with C1 boundary. Suppose, for some 0
(ii) X is (n-q+i)-concave;

(iii) Y is compact and X is (n-q)-concave.

Let U be a neighborhood of 3X, and fEZ0 q(U,E). Then there exists a form FEZ q(X,E) with F=f in some neighborhood of dX if and only if

J fAg = 0

for all g6Zn,n-q-1(X,E*).

(20.27)

209

Proof. By Stokes' theorem it is clear that condition (20.27) is necessary. Assume (20.27) is fulfilled. Take a Coo function x on Y such

that x = 1 in a neighborhood of aX and supp x e U. Then 5 Af E(ZO,q+l(X,E)JO and, by (20.27),

gA(axAf) = J gAf = 0

(20.28)

ax

X

for all gEZ0,n-q-1(X,E*). Since at least one of the conditions (i),

(ii), (iii) is fulfilled, it follows from at least of one of the Theorems 12.16, 15.13, and 19.1 that En,n_q(X,E*) is closed with repect to uniform convergence on the compact subsets of X. By this and by (20.28),

it follows from Propoe4tion 20.2 that the equation o% = %A f can be solved with ue[C0 q(,E)]O. Set F = x f - U.

21. The domain Iz012+...+Izg12< Izq+112+...+Izn12

Let Pn be the n-dimensional complex projective space, and let E be a

holomorphic vector bundle over Fn. For zEIPn, by [z0:..:zn] we denote the homogeneous coordinates of z. Let 0
D:= {ze pn_

2012+...+Izg12

<

1zq+112+...+1zn12),

As observed in Sects. 5.16 and 8.12, D is completely strictly q-convex and strictly (n-q-l)-concave. Hence the theory developed in the preceding

sections admits several applications to this domain.

Let us notice

some of them.

21.1. Theorem. (i) For all integers 1
dim HO'r(D,E) < on

for all 0
(iii)

HO'r(D,E) = 0

if n>r>n-q.

(iv)

dim HO,n-q-1(D,E) = oo.

.(ii)

Proof. (i) follows from the Andreotti-Grauert Theorems 12.16 and 15.13, and the Andreotti-Vesentini Theorem 19.1. (ii) follows from the Andreotti-Grauert Theorems 12.16 and 15.13. (iii) follows from the Andreotti-Grauert Theorem 12.16. (iv) follows from the Andreotti-Norguet Theorem 18.4.

210

21.2. Theorem. (i) For all integers 1
at ZO (ii)

r(15, E).

dim H.0(D,E) < oo 0

(iii) (iv)

if 0
dim HO

if n>r>n-q.

n-q-1

1/2-40 (D,E)

Proof. (i) follows from Theorems 11.2, 14.5 and 19.2, (ii) follows from Theorems 11.2 and 14.5. (iii) follows from Theorem 12.7. (iv) follows from Theorem 16.6.

21.3. Theorem. For all integers 0
(HO,n-r(D,E*)}*

[Hn'r(D,E)]0

is an isomorphism and

dim [Hn'r(D,E)]0 = dim

(HO,n-r(D,E*})*

(21.1)

Hence, by assertions (ii)-(iv) in Theorem 21.1,

dim [HO'r(D,E)]0 < 00

if 0
dim [HO,r(D,E)]0 = 0

if O
dim [HO,q+1(D,E)]0 = 00. Proof. This follows from Theorem 21.1 (i) and Theorem 20.4.

22. The condition Z(r)

It is clear that the results on q-convex and q-concave manifolds presented above admit several generalizations to manifolds which satisfy appropriate combinations of convexity and concavity conditions. An example we met already in Sect. 19: The boundary of the domain D in Theorem 19.2' consists of two pieces, aG' and -6G, where aG' is strictly (n-q)-convex and -aG is strictly q-concave. 1)

H10/2-_'O(D,E):= Zoo

,

E)/E6, r

)0(D,E)

211

More generally, one can give the following

22.1. Definition. Let D - X be a domain in an n-dimensional complex manifold X. We say D is a non-degenerate strictly Z(r) domain (1
Example. Let 9:X - IR be a strictly plurisubharmonic function without degenerate critical points on an n-dimensional complex manifold X such that the surface {zEX: p(z)=0} is compact (for instance, X=Cn and 9=1212-1). Then, for any 1O, the set {zEX: -E
22.2. Theorem. Let D cc X be a non-degenerate strictly Z(r) domain in an n-dimensional complex manifold X (?
dim

0(D,E) < oo.

Proof. This is a straightforward combination of the proofs of Theorems 11.2 and 14.5.

For references, we notice also the following obvious 22.3.Corollary to Theorem 22.2 and Lemma 12.6 (iii) and (iv). Let D -c X

be a non-degenerate strictly Z(r) domain in an n-dimensional complex manifold X (1
(i)

E1/2-40(G,E)

=

ED/2-->0(D,E).

Z0 r(d,E)n

In particular, for any fEZ r(G,E) with f=0 on D, the equation &u=f can be solved with ufCO/r-1(G,E). (ii) The image of the restriction map

ZO,r-1(G,E) -- ZO,r-1(D,E)

is dense in ZO,r-1(D,E) (with respect to uniform convergence on D). 1)

)0

HO1/2 ,0(D,E):= Zoo r(5,E)/E1/2

212

(5,E)

23.The Rossi theorem on attaching complex manifolds to a complex manifold along a strictly pseudoconcave boundary of real dimension >5

Throughout this section, X is an n-dimensional complex manifold, and

g:X -9 IR is a strictly plurisubharmonic function without degenerate critical points such that the set D:= {zEX: 0<_>(z)<1} is relatively compact in X.

23.1. Theorem (see [Rossi 1965]). If n>3, then D is biholomorphically equivalent to a neighborhood of the boundary of some relatively compact

domain Y - Z in a complex manifold Z (i.e. one can "fill in" the pseudoconcave "holes" of D).

In the present section we prove the following tesult which is the first step in the proof of Theorem 23.1:

23.2. Theorem. If n>3, then we can find an integer N such that D is N

biholomorphically equivalent to some complex submanifold of C

(of

course, this submanifold is not closed in CN).

We do not present here the arguments which lead from Theorem 23.2 to Theorem 23.1. Instead, following (Rossi 1965], we refer to Theorem XII D 6 in [Gunning/Rossi 1965], which gives a version of Theorem 23.1 with isolated singularities in Z. We remark also that, modulo isolated Theorem 23.1" is a singularities, the implication "Theorem 23.2 special case of the more recent (and more powerful) Harvey-Lawson theorem (see [Harvey/Lawson 1975] and [Harvey 1977]).

In (Rossi 1965) a counterexample is given (which will be reproduced in the next section) which shows that the conclusion of Theorem 23.1 need not be true if n=2. Since the arguments which lead from Theorem 23.2 to Theorem 23.1 work also for n=2, so also the conclusion of Theorem 23.2 need not be true if n=2. We shall prove (Theorem 23.3 below) that, for n=2, the conclusion of Theorem 23.2 (and therefore the conclusion of Theorem 23.1) holds true if and only if the operator a is 0-regular at Z0 l(Dt) for all 0
Thus, Rossi's counterexample shows also that the separation Theorem 19.2' is not true without the hypothesis that the q-concave "holes" in D

can be "filled in" by the relatively compact complex maniNd G.

23.3. Theorem. Let n=2. Then there exists an integer N such that D is biholomorphically equivalent to a complex submanifold of CN if and only if, for any 0
213

Proof of Theorems 23.2 and 23.3. If n=2 and D is biholomorphically equivalent to a complex submanifold of CN, then, by the arguments mentioned above (for instance by the Harvey-Lawson theorem), it follows that D is contained as an open subset in some larger complex manifold Y, such that, for any O
non-degenerate strictly.pseudoconvex domain Gt _ Y.

In view of the

separation Theorem 19.2', this implies that if n=2 then the 3-operator is 0-regular at Z0 l(Dt) for all 03 then, obviously, the O

domains Dt are non-degenerate strictly Z(l) domains. Hence it follows from Theorem 22.2 that the operator 3 is 0-regular at ZO,l(Dt)"

Therefore, it remains to prove that, for any n>2, the following statement holds true: If the operator 5 is 0-regular at Z00 l(Dt) for all

0
Assume the 3-operator is 0-regular at Z01(Dt) for all 0
(A) If x,y are different points in D, then there exists a sequence fk (k=1,2,...

)

of holomorphic functions on D such that lim fk(y) = 0 and

lim fk(x) = 1.

(B) If xED and vcTX(D) is a holomorphic tangent vector (of. Sect. 4.1), then there exists a sequence Fk (k=1,2,...

)

of holomorphic

functions on D such that lim aFk(x) = v. We shall prove (A) and (B) simultaneously. Let x,yeD such that x,4y and, without loss of generality, Q(y)
Take a neighborhood U cc D of x such that there exist holomorphic coordinates in a neighborhood U of x. Then we can find a holomorphic function on U with h(x)=0 and dh(x)=v, and, using the Levi polynomial (cf. Sect. 7.1), we can find a holomorphic function g on U such that g(x)=0 and

Re g(z) > p(x) - g(z) + Ix-z12

(23.1)

r

for all zEU. Take a Coo function x on X with supp x cc U and X = 1 in a neighborhood of x. Setting sk =

e-kg;x

sequences sk,S' E ZD l(D) (k=1,2,...

and Sk=he-kg ax we obtain two

). Since 3X = 0 in a neighborhood of

x and outside supp x it follows from (23.1) that there is a number Q(x)
l im II SkilO, Db =

I im 1 1' 0 0 ,D

0'

(23.2)

Since sk and Sk vanish outside supp x and since supp x cc D, we can find

0<&
and Db is a

non-degenerate strictly (n-l)-convex extension of DE, now it follows from Lemma 12.8 (ii) and (iii) that 5 is 0-regular also at Z0

0

1(F)b),

and

sk,SkEE0,1(Db). Taking into accont.(23.2), so we obtain sequences wk,Wk of continuous functions on 6b with awk=sk' 3Wk=Sk and

lk e-kg

Setting fk

INklIO, Db =

l im 11 WkIl0, Db

- wk and Fk = x he-

= 0.

- Wk, we obtain sequences of

continuous functions on bb which are holomorphic in Db and such that

limn fk k

xe II0,Db

xhe-kg1

= lim 11 Fk k

10' Db

(23.3)

= 0.

Since D is a non-degenerate strictly pseudoconvex extension of Db and 3 is 0-regular at Z0 1(Db), from part (iv) of Lemma 12.8 we obtain sequences fkand Fk of continuous functions on D which are holomorphic in D and such that

Ilfk - fkIIO,Db

k

and

11 Fk

-

k'

Fk1IO,Db

.

By (23.3), then

lim fl fk k

- x e-

IIFk u0 D b = lim k

- xhe-k9O

D

b

= 0.

(23.4)

Since p(y)0, it follows from (23.1) that lim

(y)e-kg(y)

= 0 for k -9 oo. Moreover, x

(x)e-kg(x)

= I for all k. Therefore, it follows from (23.4) that the sequence fk has the properties as in statement (A).

x

Since h(x)=g(x)=O and x = 1 in a neighborhood of x, it follows that a(x he-kg )(x) = ah(x) =v for all k. Therefore, it follows from (23.4) that the sequence Fk has the properties as in statement (B). El

215

24. Rossi's example of a real 3-dimensional strictly pseudoconcave boundary which cannot be embedded into CN

Set

6 = {z6C2: IzI<3 and Iz11 > 1281'

Choose E>0 so small that E<1/128 and the functions

a+,e-:C2

Q2

defined by

9+(z) = IZzl

z2 2I(1

±

+

1+1z21

12),

zf C2

1-2

are strictly plurisubharmonic on 6. This is possible by Observation 4.15 and since 21211(1+Iz2I2) is strictly plurisubharmonic for all zEC2 with zl#0.

Set

U+ = {zEC2: zl t 0}

and

U_ _ {zEC2: z2 # 0}.

We introduce a new complex structure on C2\{0} - U+uU- by saying that the functions

2

z2

zl

u+:=

and

Eu +

v+:=

zl

1+fu+I

2

are holomorphic coordinates on U+, and the functions

z2

z

u := 1

and

v

2

fu -

z2

are holomorphic coordinates on U_. This is correct, because

u+

and

v+ = v u2 + Eu-

on U+nU_.

(24.1)

The complex manifold formed by C2\{0} with this new complex sructure will be denoted by M.

216

24.1. Lemma. On G:= {zeC2: l/2
U+ = {z&U+: 21z11>1z21}

and.

U_ =

21z21>Iz1l1.

Then lu+I<2 and 1/128
u+

v+ +

2I (1+Iu+12)

l+lu+i

on U+,

i.e.

Iz12 = P+' h+ on U+, and since the functions

are strictly

plurisubharmonic with respect to the standard structure of Cn, this implies that 1z12 is strictly plurisubharmmonic with respect to the complex structure of M.

24.2. Theorem. There does not exist a complex manifold M with the following

properties:

(i) K is an open subset of M;

(ii) there exists a relatively compact domain D aD = {zECn:

M such that

IzI=1}.

Proof. By (2.41), v+u+=v_u_-C on U+nU_. Therefore, the two sides of this equation together define a holomorphic function on M, which will be denoted by v0. Moreover it follows from (24.1) that the functions v+ and v_ are holomorphic on M. Consider the holomorphic map 4:M -e C3 which is defined by

(z) _ (vO(z),v+(z),v_(z)).

First we prove that rank do = 2 everywhere on M. Set N+

= {zEM: v+(z)=O} and notice that N + GU_. Since Eu+ 14 0 on

U+n U_ (? N+nN_), it follows from (24.1) that N+nN_ _ 0. Since v+ and u+

217

are holomorphic coordinates on U+, and v_ and u_ are holomorphic coordi-

nates on U, we see that v+ and v0=v+u+ form holomorphic coordinates on U+\N+, and v- and v0= v_u_-E form holomorphic coordinates on U_\N_. Since N+ C U

and N+ AN- _ 0, this implies that, everywhere on M, at

least oneof the pairs (v+,v0) or (v_,v0) is a system of local holomorphic coordinates. Hence rank d4 = 2 everywhere on M.

Now we assume that there exists a complex manifold M with the properties (i) and (ii). Without loss of generality, we can assume that M=MuD. Then, by Lemma 24.1, D is strictly pseudoconvex, and, by Theorem 20.11

(see also Remark 20.12), the restriction map 0(M) -) (2(M) is surjective.Therefore, 4) admits a holomorphic extension 4) onto M. Denote by Crit 4) the set of all zEM with rank d4)(z) < 2.

Since Crit 4) C M\M, and

M\M is compact, we see that Crit 0 (being an analytic set)consists of a finite number of connected compact sets. Hence v0(Crit 4))

Take an arbitrary complex number c, and set Lc

is finite.

{zEM: v0(z)=ev+(z)},

Lc is an 1-dimensional closed analytic subset of M.

If zEM\U+, i.e. z1=0 and z2'0, then u_(z) = 0 and therefore it follows from (24.1) that

ov+(z) = 0. On the other hand, v0(z) = v-(z)u2(z) -£ # 0 for zEM\U+. Hence LcnM C. U+. Since v0=v+z2/z1 and v+=0 on U+, we have

La M = {zET2\{0}: z2=cz0.

(24.2)

In particular, LcnM has the topological type of a punctured plane. Since v0-u+v+ on U+, we have

V0 =

zlz2 2

z2

z2/zl

zi l+Iz2/z11

2

U+

Therefore, v0 is not constant on Lc, and, for any point x which belongs to the boundary of LcnM\M) in L0, we obtain

v (x) = 0

lim

Lc3z--4x

v (z) = 0

c 2 1+1c12

(24.3)

Hence v0 is a non-constant holomorphic function on L. which is constant on the boundary of the compact set Lcn(M\M). Since L. is of complex dimension 1, this is possible only if LcnM\M) is finite. Since, topologically, LcnM is a punctured plane, and since Lc is closed in M, it follows that Len(M\M) consists of precisely one point xc. Since 4)(z)(-z) for all zEM, and since, by (24.2), -zELo\{xc} for all zcLc\{xc}, we see that xccCrit 4). By (24.3), this implies

218

-

c

2

E v0(Crit 4).

1+Icf

This is impossible, because c is an arbitrary complex number, whereas

v0(Crit 4) is finite. O

219

NoTBS

To Sections 1 and 2. For some historical remarks concerning the Martinelli-Bochner-Koppelman and the Cauchy-Fantappie formulas we refer to the Notes at the end of Chapter 1 in [H/L] (in [H/L1 these formulas are called the Leray resp. the Koppelman-Leray formula). The Kodaira theorem on finiteness of the cohomology of compact complex manifolds with coefficients in holomorphic vector bundles (which is equivalent to Theorem 1.15, via Dolbeault isomorphism) was obtained in [Kodaira 1953] by means of the theory of harmonic integrals. The elementary proof by means of the Martinelli-Bochner formula given here is due to [Toledo/Tong 1976] (see also [Henkin 1977]).

In [Cartan/Serre 1953] this theorem was gene-

ralized to cohomology with coefficients in coherent analytic sheaves. The Poincare 8-lemma for smooth forms (Theorems 2.12) is due to Grothendieck and Dolbeault (see [Cartan 1953/54, Dolbeault 1953,1956]). The regularity of the 5-operator (Theorem 1.13), the Poincare 5-lemma for currents (Theorem 2.13), and the Dolbeault isomorphism Hsr(X) ti Hr(sheaf of holomorphic (s,0)-forms on X) = Hs'r(X) 00 cur (Theorem 2.14) were obtained in [Dolbeault 1953,1956]. With respect to these results we refer also to [de Rham 1955, Schwartz 1955, Griffiths/Harris 1978, Cirka 1979]. To Section 3. For historical remarks concerning the piecewise Cauchy-

-Fantappie formula (Theroem 3.12 and Corollary 3.13) corresponding to the canonical combination of a Leray data (cf. Sect. 3.4), we refer to the Notes at the end of Chapter 4 in [H/L] (in (H/L] these formulas are called the Leray-Norguet resp. the Koppelman-Leray-Norguet formula). For the case of a general Leray map (in the sense of Sect. 3.4), the piecewise Cauchy-Fantappie formula was established and applied to the tangent Cauchy-Riemann equation in [Ajrapetjan/Henkin 1984]. We remark that, in the present monograph, we use only piecewise Cauchy-Fantappie formulas corresponding to the canonical combination of a Leray data. The general formula will be used only in the subsequent parts of the pending book mentioned in the Preface. To Sections 4-6. The concepts of q-convexity and q-concavity were developed in the works [Rothstein 1955] and [Andreotti/Grauert 1962]. The

220

fact that q-convex domains in Stein manifolds are completely q-convex (Theorem 5.3) was observed in (Coen 1969]. For more recent developments of these concepts we refer to [Grauert 1981, Diederieh/Fornaess 1985, 1986, Peternell 1985].

To Sections 7 and 9. Local solutions with uniform estimates of the equation au=f0 r on strictly q-convex domains with r>n-q (cf.Theorems 7.8 and 9.1), for the first time, were obtained in the case of C2 domains in [Fischer/Lieb 1974] as a generalization of the corresponding results proved in [Grauert/Lieb 1970, Henkin 1970, Lieb 1970, Henkin/ Romanov 1971] for strictly pseudoconvex domains (for such estimates on pseudoconvex domains, cf. also the Notes at the ends of Chapters 2,

3,

and 4 in [H/L1).

To Sections 11 and 12. Theorem 11.2, for the first time, was proved for domains with C2 boundary in [Fischer/Lieb 1974]. The Andreotti-Grauert Theorem 12.16 was otained in [Andreotti/Grauert 1962]. Moreover,

in that

work, a more general result for cohomology with coefficients in arbitrary coherent analytic sheaves is proved. In the more special situation of (s,r)-forms (with values in the trivial bundle), another proof of this theorem was given in [Hormander 1965], which leads to a sharpened result, with weighted L2 estimates.

Notice that the proof of Theorem 11.2 obtained in [Fischer/Lieb 1974] makes use of the Andreotti-Grauert Theorem 12.16, whereas the proof given here is independent of Theorem 12.16; moreover, we prove Theorem 12.16 by means of Thacrem 11,3. Our proof of Theorem 12.16 consists in an inductive procedure with respeet to the levels of an exhausting function which is strictly (q+l)-convex at infinity. Theorem 12.14 on the extension of the Dolbeault cohomology classes of order >n-q along q-convex levels, obtained by this procedure, may be considered as a natural supplement to the classical Hartogs extension theorem for holomorphic functions. Observe that this inductive procedure was also used in Sect. 2.12 of [H/L] in solving the 5-equation on completely pseudoconvex manifolds. Sections 11 and 12 of the presept work contain all results of Sections 2.11 and 2.12 in [H/L] (as the special cane q=n-1).

Notice also that, in distinction to [Fischer/Lieb 1974], we do not assume that the boundary of the domain D in Theorem 11.2 is smooth.

However, in order to limit the technical difficulties, we restrict ourselves to the non-degenerate case, The extra work which we have to do (in order to obtain all necessary estimates also in the case of non-degenerate critical points on the boundary) pays off. For instance, in the inductive procedure mentioned above, we need not worry about the levels with critical points. In [H/L1 extra arguments were necessary in order to "jump over" such critical levels. In our opinion, there is no doubt that Theorem 11.2' holds true also

221

for arbitrary strictly q-convex domains. For q=n-1, i.e. for strictly pseudoconvex domains, this is proved (see Theorem 3.2.1 in CH/L] if CX=O, and [Bruna/Burgues 1986] if oc=l/2).

Also, the authors are sure that the following theorem holds true: If, under the hypotheses of Theorem 12.7, the boundary of D is of class C°O, then, for all n-q
Tr(ZD r(D,E) C CD+r/2(D,E)

for all k=1,2,...

,

where Tr is the operator from Theorem 12.7. For q=n-1, i.e. for strictly pseudoconv`x domains D, this is proved

(see [Siu 1974, Greiner/Stein 1977, Lieb/Range 1986]). Notice also the following fundamental consequence of this theorem, which was obained, for the first time, for q--n-1,

in [Kohn 1965]:

If, under the hypotheses of Theorem 12.7, the boundary of D is of class C°O, then, for any n-q
Tr:Z r(D,E) >

CO,r-1(D,E)

and a sequence of constants Ck< oo such

that, for all fEZp r(D,E), 3Trf = f and

k r(D,E) C Ck+r/i (D,E) Tr(ZD

for all k=1,2,..

To Sections 13-15. Local solutions with uniform estimates of the equation du=f0 r on strictly q-concave domains with 1
Notice also that, in distinction to [Lieb 1979], we do not assume that the boundary of the domain D in Theorem 14.5 is smooth. As in the

222

cast of Theorem 11.2 we admit non-degenerate critical points. In our opinion, there is no doubt that Theorem 14.5 holds true also for arbitary strictly q-concave domains. Finally, let us notice that, in distinction to [Lieb 19793, we construct a formula which solves the a-equation locally "without shrinking of the neighborhood" (Theorem 13.10 (i)). Moreover, the following theorem holds true:

If [U,4,(Q,H,D3 is a q-concave configuration in Cn (1
there exists a solution u of Su=f which is also continuous on D. Moreover, this solution can be given by a bounded (with respect to the supnorm) linear operator the norm of which tends to zero if the diameter of D tends to zero.

To prove this theorem it is necessary to strengthen the estimates given in Theorem 14.1. This can be done by means of a construction from [Henkin 1977].

Corollary 13.9 and Theorem 15.2 on the extension of holomorphic functions along 1-concave levels generalize the classical Hartogs extension theorem. A further natural generalization of this fact is given by Theorems 13.10 and 15.11 on the extension of the Dolbeault cohomology classes of order r along q-concave levels with 0
(9(K) of all

holomorphic functions in some neighborhoods of K. A proof of the latter result was obtained independently also in [Ajzenberg 1966].

To Section 18. The Andreotti-Norguet theorem (Theorems 18.2 and 18.3) was proved in [Andreotti/Norguet 1966]. The proof given here and Lemma 18.1 are taken from [Grauert 1981]. The corresponding fact with uniform estimates (Theorems 18.5 and 18.6), for the first time, was obtained in [Andreotti/Hill 1972] in the case when D is of class Coo (then the form fx in Theorems 18.5 and 18.6 can be chosen to be also of class Coo).

To Section 19. The Andreotti-Vesentini separation theorem (Theorem 19.1') was obtained in (Andreotti/Vesentini 1965]. The corresponding result with uniform estimates (Theorem 19.2'), in the case when D is strictly paeudoconcave (i.e. q=n-1) and of class C2, was proved in (Henkin 1977]. In its general form, Theorem 19.2' seems to be new. We remark that, in our opinion, there is no doubt that Theorem 19.2' holds true also for arbitrary strictly q-concave domains D (i.e. the

223

assumption that the domain D in Theorem 19.2' is non-degenerate can be omitted).

Moreover, the authors are sure that also the following theorem holds true:

If, under the hypotheses of Theorem 19.2', D is of class C°°, then,

moreover

T(ZO, O

q(D,E)n Zk

O

,q

(D,E)) C Ck+1/2(D,E)

-

O, q-1

for all k=1,2,.

where ZO q(D,E) and T are as in part (ii) of Theorem 19.2'.

To Section 20. The general facts concerning the concept of duality were obtained in [Serre 1955]. Theorems 20.6-20.10 follow from these general facts combined with the Andreotti-Grauert Theorems 12.16, 15.13, the Andreotti-Vesentini Theorem 19.1', and the Andreotti-Norguet Theorem 18.4.

The Hartogs extension Theorem 20.11 was proved in [Kohn/Rossi 1965].

Theorem 20.13 on the extension of 3-closed forms, in the case of Coo forms, follows from the results of [Andreotti/Grauert 1962] and [Andreotti/Vesentini 1965]. Let us mention also the following two statements A and B, which complete Theorem 20.13:

A. Let D - X be a completely strictly q-convex domain with Coo boundary in an n-dimensional complex manifold X (0
U a neighborhood of aD, and U+:=(X\D)nU. Then, for any 0
S(ZD r(U+,E)) C Z° r/2 (DuU,E)

for all 2 < od
This statement follows from the results of [Kohn/Rossi 1965] (if oe,=oo), [Folland/Stein 1974], and [Rothschild/Stein 1976).

B. Let D '= X be a CO° domain in an n-dimensional compact complex manifold X. Suppose D is both strictly q'convex and strictly (n-q-1)-concave (0
(i) There exists an E-valued a-closed (O,r)-form F of class C°° on DuU with FIU = f.

224

1/2

fng = 0

(ii)

for all V Zn°On-r-1(D,E*).

J

aD

For q=n-1 and D - Cn, this result was obtained in [Dautov 19721 (see also [Ajzenberg/Dautov 1983]). Then, in [Henkin 1977], it was proved for the case when(also)q=n-1 but D is not necessarily oontained in Cn.

225

P RO BLR MS

1. Linearly g-concave domains. A domain D C Cn (resp. a domain D contained in the n-dimensional complex projective space

IPn)

is said to be

linearly q-convex if, for any zcCn \D (resp. z 61Pn\D), there exists a

complex q-dimensional plane Cz C Cn(resp. a q-dimensional complex Cn\D (resp. z Pz C_ Pn\D). If, projective plane P C ) such that ztCq z S Z_ moreover, this plane can be chosen continuously depending on z, then

we say D is continuously linearly q-convex.

Example. The domains Iz012+.--+Izg12 < Izq+1 I2+

+Iznl2

considered in Sect. 21, are continuously linearly q-convex. Problems.

(i) Is any linearly q-convex domain q-convex? (ii) Is any

continuously linearly q-convex domain completely q-convex?

The authors believe that, at least under the hypothesis of the second question, the answer to the first one is affirmative, i.e.: any continuously linearly q-convex domain is q-convex. It seems that in this case an exhausting function which is (q+l)-convex at infinity can be constructed patching together, by means of procedure

maxi (of.

Def. 4.12), appropriate local (q+l)-convex functions with joint (q+l)-dimensional subspaces of TZ(D) on whioh their Levi forms are positive definite.

2. Embedding of q-convex manifolds into linearly (q+k)-convex domains. Let D cc X be a strictly q-convex domain in an n- dimensional complex

manifold X. Problem. Does there exist an integer k and a linearly (q+k)-convex D Q Cn+k such that D is biholomorphically equivalent to a closed submanifold of D? Remark. If q=n-1, the answer is affirmative. This was proved by Henkin and Fornaess (see [ (irka/Henkin 1975, Fornaess 1976)). 3. Extension of 81--closed forms from complex submanifolds into q-convex domains. Let E be a holomorphic vector bundle over an n-dimensional

complex manifold X, D -- X a completely strictly q-convex domain (O
226

Problem. Prove that, for all n-q-1
Zp

Zg r(DAM,E)

r(', E)

is surjective.

Remarks. For r=0 and q=n-1 this is proved (cf. Sect. 4.11 in [H/L)). If M is a smooth complete intersection in X, aD is of class C2, and

the intersection MA W is transversal, then the problem is also solved (together with P.L.Poljakov, unpublished). Let us outline the idea in the following more special situation: Assume X is a domain in 6n,

E is

the trivial line bundle, k=1, and D - X is a strictly q-convex domain with C2 boundary which has, moreover, the following property: There exist C1 functions F1(x,z),...,Fn_q(x,z) defined for xcaD and z cc n which are complex linear with respect to z such that, for any

point xeaD, the complex plane Px:= {zEX: F1(x,x-z)=...=Fn_q(x,x-z)=0)

satisfies the following three conditions:

(i) Px is complex q-dimensional and contained in the complex tangent plane of aD at x;

(ii) Pxn D = 0 ; (iii) Iz-xI = O([dist(z,aD))1/2) for Px3z -p X.

Set A =

IF112+,,.+1Fn_g12.

Let w(z,x) _

((wl(z,x),...,wn(z,x)) be the

vector of functions such that

n

A(z,x) _ Y wj(z,x)(xj-z.). j=1

Then w(z,x) is a Leray data for D. Set

w(z,x)

X-Z

v0(z,x,t) =

(1-t)

t

IX-z12

A(z,x)

Now let M be a smooth complete intersection of codimension 1 in X such that the intersection Mn8D is transversal, i.e. M = {zeX: g(z)=0), where g is a holomorphic function on X with dg(z) # 0 for all ze M, and dglaD(z) # O for zcaDnM. Further, let f"00 r( Set

227

n-r-1 f(x)gdet('-ff, rdx

Hf(a) = c1

n

r

df)A ldx dg(x)

Ix-zI

f xe MnD

- . r-r

+ e

n-r-2

f(x)Adet(v d v0 3 z v0, 3v° )Ate(x) x dg(x) t

2

zED\M 1

xeMnaD O
where c1, c2 are appropriate constants. In this way we obtain a continuous (O,r)-form Hf on D\M, and, estimating the integrals in the definition of Hf, one can prove that gHf admits a continuous extension onto b such that, after the correct choice of the constants Cl and c2,

gHf,M = f.

4. Uniform approximation. Let K

c X be a compact set in the n-dimensio-

nal complex manifold X such that, for some (q+l)-convex function

Q:X -) R (O
K = {zEX: 9(z)<-0).

(Here it is permitted that 9 has degenerate critical points.) Problem. Prove that any (O,n-q-1)-form f of class Ck in a neighborhood of K which is 3-closed in the interior of K can be approximated uniformly on K, together with all derivatives of order < k, by 3-closed Coo forms in a neighborhood of K (k=0,1,2,...

).

For q=n-1, this is proved (see Chapter 3 in [H/L1 for k=0, and [Bruna/Burgues 1986] for the general case). 5

The a-equation on analytic sets. Let X be a k-dimensional closed

analytic subset of the unit ball B in Cn which is a local complete intersection, i.e. for any zEX there are helomorphic functions f1'...Ifn kin a neighborhood U C B of z such that

XnU = {xEU: fi(x)=...=fn-k(x)=0).

Let S(X) be the singular locus of X and

d:= dim X - dim S(X).

228

It is well-known that then any a-closed (O,r)-form on X\S(X) with r
0 < r
HO,r(Y) is bijeotive.

The following problem is related to the first Riemann extension theorem:

Problem. Suppose f is a 5-closed (O,r)-form on X\S(X) with r=d-1 which admits a CO° extension into B. Does it follow that f is 3-exact on X\S(X)?

If X is of codimens ion 1, i.e. X = {z(B: g(z)=0}

and

S(X) = {z6X: dg(z)=0}

for some holomorphic function g on B, then the answer is affirmative. Let us outline the proof of this fact (obtained together with P.L. Poljakov): In view of the cohomological generalization of the second Riemann extension theorem mentioned above it is sufficient to prove that, for any b>0, f admits a (5-closed Coo extension into BE\S(X), where Bi:={z6Cn: IzI<1-£}. To obtain this extension, first we define a Colo form Hf on Bt\X as follows:

n r-r

Hf(z) =

lim

--'0

f(x)n

c1

n-r-l

det(x-z,dz, dx

2n-

to x

dg(x)

Ix-zl xs XnB6

Ildg(x)I >S r

+ lim

S-10

n-r-2

f(x)n det(v0 ,dtv0,azv°,axv° )A fitj

c2

()

(x)

x(XnaB

U dg(x)II >S 0
229

for zEBE\M, where

v0 = v0(z,x,t) =

t

x-z IX-ZI

2

x

+ (1-t)

<X,X-z>

and c1, 0 are appropriate constants. The limits for 5 -> 0 exist in 2 view of a theorem of Coleff and Herrera (see [Coleff/Herrera 1978]). By this theorem, moreover:

1)0

I

f(x)n h(x))

= l )0 "A

xExnBc

I

f(x)n g(x

X BL

Ig(x) 1=a

11dg(x)I`>S

for any smooth form h with compact support in B. Using the latter relation and the fact that f is

5-closed on X\S(X) one can prove that

Hf is 5-closed in B&\X. Further, estimating the integrals in the definition of Hf, it can be shown that the form F:=gHf admits a continuous extension into B&\S(X) such that, after the correct choice of c1 and c2,

FIX\S(X) = f-

6. Let D - X be a non-degenerate strictly q-concave domain in a compact complex manifold X. Then we know from the Andreotti-Vesentini Theorem 19.2 that E0 q(D) is a closed subspace of the Banach space Z0,q(b).

Problem. Prove that E0 q(D) is not complemented in ZO q(D), i.e. prove that there does not exist a closed linear subspace F of ZO q(D) such that F n E0 q(6) _ {0} and F + E00 q(6) = ZO q.(D).

7. Let D - X be a domain in an n-dimensional complex manifold X of the form

D = {zcX: e1(z)<0, Q2(z)>0}

where 41 and Q2 are strictly plurisubharmonic functions with

dQl A d92 # 0 Problems.

if 91 = 92 = 0.

(i) Prove that, for n>4, the space E0 0 , 1(D) is a closed sub-

space of the FrEchet space -ZO 1(D), and EO 1(D) is a closed subspace of

the Banach space ZO 1(5).

230

(ii) Find an example which shows that this is not true for n=3.

Remark. If X = C3 and if there exists a pluriharmonic function g0 such that

{9o = 0) n D = (g1= Q2= o)fl D,

then the spaces ED 1(D) and E00 1(D) are closed (cf. Sect.

[Henkin 1977] and

13 of the present work). It seems that the solution of the first

problem can be reduced to this case by means of the Rossi theorem (cf. Sect.23). An example which solves the second problem, probably, can be found by means of Rossi's example (cf. Sect 24).

A positive solution of the first problem would give a new approach to the deep theorem of M.Kuranishi (see [Kuranishi 1982]) and T.Akahori (see [Akahori 1987)) on local embedding into CR of strongly pseudoconvex CR-structures M with dirt M > 7; a counterexample for dirt M = 5 could be obtained, probably, by means of an example as in the second problem.

231

APPENDIX A. ESTIMATION Q W SOME I N TEQRAL S (THE SMOOTH CASE)

In this Appendix A and the following Appendix B we prove estimates for some integrals over hypersurfaces reap. domains in IRn which are used in the basic text of the present monograph. Appendix A is devoted to the case of smooth hypersurfaces rasp. domains with smooth boundary. In Appendix B we consider the more general case of surfaces with non-degenerate singularities. In both appendixes the notations introduced in Sects. 0.1-0.4 of the basic text will be used. 0. Preliminaries. We use the following notations:

m>2 is an integer, and x1,...,x0 are the canonical coordinates in Rm.

R is a positive number and BR is the open ball with radius R centered at the origin in

IRe.

Q is a real-valued C2 function in some neighborhood of 8R such that with p(x)=0.

d9(x)#0 for all xe

D:? {xEBR: yo(x)<0}.

S:= {xSBR: yo(x)=0}, and we assume that S # ( .

dS is the Euclidean volume form of S, i.e. dS is the C1 form of degree m-1 on S which is defined by the following condition: If v is a differential form of degree m-1 in a neighborhood of S such that d9(x)AV(x) =

dQ(x)Adx1A..-.Adam

(1)

for all xIS, then v'S=dS. We orient S by dS.

In this Appendix A we prove the following three propositions: 1. Proposition. (i) There exists C0, C + Clln bl

if 1=3

dS < J (E+lx-y1 )Ix-Y12CE 1/2 J1

xES

232

(2)

if 1=2.

(ii) Suppose h(y,x) is a C1-function which is defined for x,y in some neighborhood of $R such that the following two conditions are fulfilled:

h(y,y) = 0

for all V dg

dg(Y)ndxh(y,x) IX--y

if YES.

0

(3)

Then there exists a constant C0,

I

C

if 1=1

Cb1/2

if 1=2.

dS

(4) Ix-YIm-3

(E,+Ih(y,x)I+Ix-YI2)

xcs

2. Proposition. There exists a constant C
udf(x)u < IQ(x)I-1/2

for all

then

If(x)-f(y)I < CIx-YI1/2

for all x,yED.

3. Proposition. (i) There exists a constant C
dx1n...ndxm < C.

(19(x)1+Ix-yl

(5)

)21x-YIm-4

XEBR

(ii) Let h(y,x) be as in Proposition 1 (ii), and let d>0 and o.,, >0 be numbers with

Then there exists a constant C0,

233

r

I

R(x) 1-o4 dx n...ndx

J

°0

-d

)2lx-Ylm-

(Ig(x)I+lh(Y,x)I+Ix-YI

< Cry.

(6)

xeBR IQ(x)I
4. Proof of Proposition 1 (i). It is sufficient to prove that for each zE9 there exist a neighborhood U of z and a constant CO, the integral

dS

t

(7) (F:+ix-yl2)Ix-ylm-1

J

xc Sn U

can be estimated by the right hand side of (2).

Fix ze8. Since dy(z)#0, we can find a neighborhood U of z and an index 1<j<m such that

dS = Fdx1n... _...ndxmis

on SnU,

where F is a real-valued function which is bounded by some constant K
dt1n...ndtm_1 K J (E+Itl2)Itlm-

tEIRm-1

ItI<2R Therefore it is sufficient to prove the following statement: For any number bO,

dtIA...ndtm-1

J

r L + LIlnEl

if 1=3

L C 1/2

if 1=2.

<{l

(E+Itl2)Itl,-1

(8)

tERm-1

Itl
d x 1-2 sm

J 0

234

dx.

.

E+x2

Since

0

and

1/2

b

( d-

<

J E+X2

00

f dx )

x

1/2

0

0

d Z _ 2 E 1/2

J

F

this implies (8).

It is sufficient to prove that for each

5. Proof of Proposition 1 (ii).

zES there exist a neighborhood U of z and a constant C0, the integral dS

(9) (F,+Ih(Y,x)I+Ix-Y12)1Ix-Ylm-3

I

xcSnU can be estimated by the right hand side of (4). Fix zcS. Since h and p are C1 functions and in view of (3), we can find a neighborhood U of z and integers 1<j
dS < Kldxh(Y,x)Rdxla......ndx,ISI

j, k

on UnS, where K
all X,yEU, Ih(Y,x)12 + E

s#j, k

Ixs-ysl2 < Llx-y12.

Therefore, the integral (9) can be estimated by

dt In . . . ndtm

K(1+L)1L J

tE

(F.+Itll+It12)11tlm-3'

Rm-1

Itl
where

235

b =

max

I1h(Y,x)12+

s

x,YE rn%

1

Ixs-ye12 11/2

k

J

So it remains to find a constant Woo (which depends only on b and m) such that, for all E>0,

M

dt1A...Adtm-1

J

if 1=1

<

(E+lti+Itl2)

(10)

1/2

Itlm-3

if 1=2,

M F-

t E Mm-1 Itl
If m=2 and 1=1, then (10) holds true with M=b.

If =2 and 1=2, then the integral in (10) can be estimated by b 2

r J

< ME 1/2

wdx 2 +x

= ln(l+- )

max

E1/2 ln(l+ b2) < on.

e

0

where 2

M =

0
E

Now we consider the case when m>3 and 1=2. Then the integral in (10) can be estimated by b

ds A...Ads 2 J

sE 181
J

d2 2 m(E+x+Isl ) 181

-

0

Qim-2

4

J

s fi Isl
1

(F,+Isl

2

)Isl

3 m-3'

Qtm-2

Therefore, in this case, (10) follows from (8).

Finally, let m>3 and 1=1. Then (10) follows from the more precise estimate

dt A...Adt 1

1

m-1

< 8s (Itll+It12)ItIm-3

m-3

rl/2

for all r>0,

(11)

tE IRm Itll
where sm-3 is the Euclidean volume of the (m-3)-dimensional unit sphere in

236

!Re-2. To prove the latter estimate we observe that the integral on

the left hand side of (11) is bounded by r

r

dsln...Adsm-2

r

r

2

J

0

(x+lsl

2m3

00

Jdx ( x+u2,

)ISIm-3

1

0

SE IRm-2

0

where

xlr00

00

du

dug J

X+U

J U2 f

x1/2

0

0

d2 = 2x 1/2.

+

x

J

11

6. Proof of Proposition 2. After passing to appropriate local coordinates, we may assume the

pdf(x)II

Then D={xcBR: xl<0} and

q=x1.

<

If x,yED and min(Ix11,1Y11) >

for all xED.

Ix11-1/2

Ix-YI, then this implies

1

1

=

l

(12)

Jf(tx+(1_t)Y)dt

l

<_

(

J

0

Ix-y dt 1/ Itx1+(1-t)yll

<

Ix-Y1

0

If x,YED with xj=yj for all 2<j<m, then from (12) we obtain

If(x)-f(Y)I = l f

rtf(txl+(1-t)Yl,x2,...,xm)dtl

0 1

1x1-ylldt

1/2 1/2 -IY11 J Itxl+(1-t)yll /2 = 211x11

I

< 21x-YI

1/2

Together this implies that, for all x,y D, 1f(x)-f(Y)I < If(x)-f(-Ix-YI,x2,...,xm)I

+If(-Ix-yl,x2,...,x +If(-Ix-YI,y2,..-,Ym) - f(Y)I < 51x-y11/2

237

7. Proof of Proposition 3 (i). Since de(x)40 if p(x)=0, we can find 5>0 It is enough to prove that

> 6 if 1q(x)1 < 3 (xaL ).

such that 1tdq(x)U

dxlA...Adxm

sup

< oo.

(13)

(Ig(x)I+Ix-yl2)21x-y1m-4

Y-BR

J

XEBR I-P(xI
< 5, we can find a constant K
Since Udg(x)jj > 5 if IQ(x)I

finite number of open sets U1....,UN c BR such that

{xEBR:Ip(x)I<&} C UIU...UUN

and, for each 1<1
form a system of C1 coordinates on U, with

on U1,

dx1A...ndxm = hIld9Adx1A... ^ ....Adxml

where h1 is a positive continuous function < K on U1. Therefore, the integral in (13) is bounded by

N

K

sup Y6 "R

IdQndx 1 A...

(

... ndxm' '

j(1)

I

1=1

(Igl+lx-yl2)21x-y1m-4

xEU1

< NK J

ds 4...nds m-1 1

dt J

(Itl+Isl-)

Islet-4

tE IR1

se IRM-1 IsI<2R

dsin...ndsm-1 < 2NK

J

Be

Islm-2

0

&im-1

Isl<2R 8. Proof of Proposition 3 (ii). For any Woo, we can find C00,

238

de A...nds 1

m-2 J

Rm-2

-r

se Isl<M

dx

IxI -o'dt

J

(Ixl+ltl+Isl2) Islet-3-d

-oo

r

Ix1-o6dx

ds1A...Adsm-2

= 2

(IxI+IsI2)ISIm-3-d

J IRm-2

se

-r

Isl<M r

dslA...Adsm-2 <

2 J se Isl<M

JB-1

Islm-2-(d+l-2oa-2

dx < Corp.

Ixl

)

(14)

-r

Hm-2

For a>O and yEBR we set

Fa(y) _ {xCBR: max(f?(x),Ix-YI)
Then, by (3), we can find constants 0<&<1 and C1
xm

xl,

k, l form a system of C1 coordinates on F6(y) with

dx1A. ndxm < C1Idg(x)ndxh(y,x)Adxln...

on F0(y).

IQ(x)1-oodx1A...A dxm

r J

...ndxml

(IQ(x)I+Ih(Y,x)I+Ix-yl

2 2 >

Ix-YI

m-3- d

-

C2

(15)

xEF5(y) Ig(x)I
Since d9(x)=0 for all xEBR with Q(x)=0, it is clear that there is a constant C3O,

239

J

I9(x)I-oedx1A...Adxm < C3r1-off

C3rk

XEBR

19(x)I
Hance, for all 0
C3

Ig(x)I-o'dx1A...Adx® t

(Ig(x)I+Ih(y,x)I+Ix-yl xeBR\Fa(y) IQ(x)I
Together with (15) this implies (6).

240

Ix-YIm-4

) 2

am

pp

APPENDIX n. ESTIMATION OF SOME INTEGRALS (THE NON -SMOOTH CASE)

Here we generalize the estimates from Appendix A to the case of a hypersurface defined by a function which may have non-degenerate critical points. We use again the notations introduced in Sects. 0.1-0.4 of the basic text of the present monograph.

First we recall some facts about non-degenerate critical points (for more information about such points, see, for instance, (Milnor 1963, 1965a, 1965b, Hirsch 1976, Wallace 1968]).

0. The notion of a non-degenerate critical point. If A:X-3Y is a C1 map between two C1 manifolds X and Y, then z4X is called a critical point of A if the rank of the differential of A at z is not maximal. Now let f:X- R1 be a C2 function on a C2 manifold X. and let zeX be a critical point of f, i.e. df(z)=0. Then z is called a non-degenerate critical point of f if, for any system of C2 coordinates x1,...,xn in a neighborhood of z,

(a2f(z) In

detlL

(1)

axiaxJ

i,J=1

=0.

Otherwise, z is called a degenerate critical point of f. Remark. A direct computation shows that, under the hypothethis df(z)=0, condition (1) is independent of the choice of the coordinates , , n.

x1,

0.1. Proposition. Non-degenerate critical points are isolated, i.e. if

f:X> R1 is a C2 function on a C2 manifold X and sEX is a non-degenerate critical point of f, then there exists a neighborhood U of z such that df(x)#0 for all xEU\(z}.

Proof. Condition (1) means that the differential of the map X3p

>

(af (P),...,af (P)/rETRn 1 axn

is of maximal rank at z. Hence, by the implicit function theorem, this

241

map is 1-1 in some neigborhood U of z, and, in particular, z is the only point in U which is mapped onto zero.

From Proposition 0.1 one obtains immediately the following

0.2. Corollary. Let f:X ->IR be a C2 function without degenerate critical points on a C2 manifold X, and K - X a compact set. If g:X -+ JR is a C2 function such that the first- and second-order derivatives of g are sufficiently small, uniformly on K, then f+g

does not have degenerate critical points in K. 0 The following important lemma is due to M. Morse (cf. Lemma A in §2 of (Milnor 1965a]):

0.3. Lemma. If D C IRk is a domain and f:D-+ R is a C2 function, then, for almost all real-linear maps L:

Rk--*

have degenerate critical points in D.

JR, the function f+L does not

(Here "almost all" means that the

Lebesgue measure of the set of all real-linear maps L; Rk-4 iR such that f+L has degenerate critical points is zero.)

Proof. Let A:D -+Rk be the C1 map defined by A(x) =

( of x)

- ` axl

af(x)

axk

Denote by Crit(A) the set of all critical points of A in D, and by Ly, yEIRk, the linear map Ly:

Since

,

Rk)

IR defined by Ly(x) = x1y1 + ...

+ xnyn'

by Sard's theorem (cf., for instance, Proposition 1.4.7 in

(Marasimhan 1968]), the Lebesgue measure of A(Crit(A)) is zero, it is sufficient to prove that, for all y Rk\A(Crit(A)), f+Ly has only non-degenerate critical points. To do this, we consider a point y Rk such that f+Ly has at least one degenerate critical point in D. If x is such a point, then y = A(x) and dA(x)=0, i.e. y A(Crit(A)).

We need also the following generalization of this lemma: 0.4. Proposition. Let X C Rn be a C2 submanifold of

JRn and f:X--> R.

a C2 function. Then, for almost all real-linear maps L: 1Rn3 JR, the function f+LIX does not have degenerate critical points in X. Proof. Fix zEX and choose linear coordinates xl,...Ixn in

,Rn

such

that, for some neighborhood U of z, x1,...1xk are C2 coordinates in UnX. It is sufficient to prove the following

Statement. For almost all linear maps L:1Rn-+ R, (f+L)JUnX does not have degenerate critical points in UnX.

Proof of the statement. Let xi,...,xR: IRn -) R1 be the linear maps with xt(x)=xi, E' the space of all linear maps on 1Rn spanned by

xi,...,xk, and E" the space of all linear maps on Mn spanned by

242

xk+1"

.,xn.

Since h:=(x1,...,xk) is a C2 diffoomorphism from on UnX

onto D:= h(UnX)

1R

,

then for any fixed L"tE", it follows from Lemma

0.3 applied to (f+L")oh-1 that, for almost all linear maps L: ik_

.

1R,

the function (f+L")+Loh does not have degenerate critical points in UAX. Since E' coincides with the space of all maps of the form Loh, so the following statement is obtained: If, for L"£ E", M(L") denotes the set of all L'EE' such that (f+L"+L')IUnX has at least one degenerate critical point in UnX, then the Lebesgue measure of M(L") is zero for

all L"eE". In view of Fubini's theorem, this completes the proof. 0 0.5. Proposition. Let X be a Coo manifold which is countable at infi-

nity, and f:X --)IR a C°0 function. Then there exists a sequence of CO° functions fj:X-4 IR (j=1,2,... ) without degenerate critical points which converges to f together with the first- and second-order derivatives, uniformly on the compact subsets of X. Moreover, if there is a compact set K cc X such that f does not have degenerate critical points on K, then this sequence fj can be chosen so that fj = f on K.

Proof. Since X is countable at infinity, we may assume that X is a Coo submanifold of some)R°. By Proposition 0.1 we can find a neighborhood U == X of K such that f does not have degenerate critical points

in U. Choose a Coo function x on X such that x = 0 on K and x= 1 in some neighboorhood of X\U. By Poposition 0.4 we can find a sequence of linear maps Li: IRna IR which tends to zero and such that all f+LilX do not have degenerate critical points in X. By Proposition 0.2, for each j>1, we can choose an index ij so large that the function

does not have degenerate critical points in X too. 0 O.S. Proposition. Let f be a real-valued C2 function in a neighborhood U of the origin in IRn such that 0 is a non-degenerate critical point of

f which is neither the point of a local minimum nor the point of a local maximum of f. Then there exists £>0 such that

df(x)A dlxl # 0

for all x with 0
Proof. Let q be the number of second-order derivatives of f at point of f, then this matrix has appropriate change of the linear

positive eigenvalues of the matrix of 0. Since 0 is a non-degenerate critical n-q negative eigenvalues and, after an coordinates in ]Rn, Taylor's formula

gives

243

f = f(0) +

x2 + o(lx12)

x2 j=1

for x-)0.

(2)

j=q+l

Hence n + o(IxI)

for x>0,

(3)

dxjndxk + o(1x1)

for x >0.

(4)

df = 2 11 x dx. - 2

x dx

j

J=1

j=q+1

j

J

and so

df(x)A dlxl = 8 j=1 k=q+l lxl

Since the origin is neither the point of a local minimum nor the point

of a local maximum of f and hence 1
0.7, Proposition. Let X be an n-dimensional real C2 manifold, f:X-> R a C function, z6X a non-degenerate critical point of f, and q the number of positive eigenvalues of the matrix of second-order derivatives of f at z. Then there exist C1 coordinates x=(xl,.,.,xn):U - IRn in some neighborhood U of z such that x(z)=0 and

f=

x2J

j=q+1

x2 J

on U.

References for the proof. For the case of a CO° function, this result is due to M. Morse and the proof can be found, for instance, in [Milnor 1983]. For the general case, see Lemma 1.1 and Exercise **1 in Section

6.1 of (Hirsch 1976]. Q Remark. All estimates given below can be proved also without Proposition 0.7 (using only relation (2)). However, we shall use Proposition 0.7, in view of the technical simplifications obtained in this way. 1. Notations. Further on, in this appendix, we use the following notations:

m>2 is an integer.

R is a positive integer and BR is the open ball with radius R centered at the origin in

Qtm.

4 is a real-valued C2 function in some neighborhood of BR such that the following conditions are fulfilled: (i) d9(x);60 for all xEBR with x#0;

(ii) g(0)=0, dp(0)=0 and the Hessian matrix

244

Ca2.e"" lm axPxk

;,k=l

has q positive and m-q negative eigenvalues where 1
Sb:= {xEBR: q(x)=b} for any real number b * 0. S:=S0:= {xEBR\{0}: 9(x)=0}, dSb is the Euclidean volume form of Sb,

i.e. the form of degree m on

Sb which is defined by the following condition:

If v is a differential form of degree m-1 in a neighborborhood of Sb such that dq(x)nv(x) = d.(x)Adx1n...n dxm

(5)

for all xESb, then VISb = dSb.

We orient the surfaces Sb by dSb. In this appendix we prove the following four propositions.

2. Proposition. (i) There exists a constant C
b4 IR, yeBR and £>0,

r

dSb <

C(1+IIn£I)

(6)

<

C(l+I In £, I+Ildg(y)II E 1/2 ]

(7)

J ( F,+ I x-y 12) x-Y 10°-3 XESb

and

(ds(x)IdSb

1 (b+ix-yl2)Ix-ylm-

-

X Sb

(ii) Suppose h(x,y) is a C1 function defined for x,y in a neighborhood of $R such that, for some constant K
((N

de(Y)u < II dxh(Y, x) Ix=Y11 < Kldq(Y)J

I-1CIIdq(Y)j2
(yaBR),

(8)

(YEFIR) I

(9)

245

an(y,x)

ah(Y.x)

<

x=y

axe

Ih(y,x)I

Klx-y1

(x,ycBR,

(x,yELR)

K(Ixllx-yl + Ix-y12)

<

1<,j<m),

(10)

axe

(11)

Then there exists C0,

UdQu2d

(

_ < C[1+I In E I+Rdq(y)R6 112)

J (f+Ih(Y,X)I+Ix-YI XESb

)

(12)

Ix-YIm-

3. Proposition. (i) For each oa>0, there exists a constant Co`
Ix-

J

xESb

M-1-0c,

Coy raz

-

(13)

I

Ix-zI
(ii) If h(y,x) is as in Proposition 2 (i), then there exists a constant C
<

(14)

(Ih(Y,X)I+Ix-YI2)IX-yim-3

XE S

(xl
4. Proposition. There exists a constant C
IIdpCy)u IIdf(y)u

<

l1nlg(Y)Il

+

(YCD)

1g( Y)l 1/2

(15)

and

f(0):=

his

f(y)

(16)

Day--+0

exists (for 2
If(x)-f(y)I

246

< C(i+Iln rI)r1/2Ix-YI1/2.

where r=max(Ix1,ly1)

(17)

5. Proposition. (i) For all oc,$>0 with c,+,B<1, there exists a constant C
dxlA...Adxo

r

J

Cry,

[dist(x,S)] W

(18)

Ix-yIm-1

XEBR Ix-yI
where dist(x,S) = inf{Ix-sI: seS).

(ii) Let h(y,x) be as in Proposition 2 (ii), and let d>0 and o6,p > 0 be numbers with

oc+'y<2. Then there exists a constant CO,

pdQ(x)II Ig(x) I-OGdx1A...f dxm

l

c

<

(19)

(x)I+Ix-yI2)2Ix-YIm-3-CA.

(I Q

xtBR IQ(x)I
(iii) Let h(y,x) be as in Proposition 2 (ii), and let d>0 and cc-J5 > 0 be numbers with

oG + b <

d+l 2

Then there exists a constant C0,

IIdQ(x)1121Q(x)I-oydx1A..Adxm

J

(12(x)1+Ih(y,x)I+lx-YI

)

p -

Ix-Ylm

X EBR

12(x)I
8. Preparations for the proofs of Propositions 2 - 5. 0.7 and by Propositions 1 and 2 in Appendix A, tions 2 - 5, we may assume that

x j=1

n

By Proposition

in the proofs of Proposi-

xs

(21)

j-q+l

and therefore

247

n

xdx - 2 7- x dx . j j

dQ = 2 J=1

(22)

j-q+l

j

J

For j=l,...,m, we set

U _ {xEBR: xj > m }

Uj = {xEBR: xj < - m }.

and

..,xm form

Then, by (22), for all 1<j<m and bcR, the functions x1,... J

as well as on

a system of coordinates on

From (5) and (22)

it follows that

dSb = I x1n... ...AdxmlSb

for all btR.

on

I

(23)

J

Further, notice that

Sb C Uiu...UUmuU1u...UUm

for all b11R.

(24)

Since, by (22),

7. Proof of Proposition 2 (1).

IIdq(x)d < pdg(y)8 + 21x-YI, estimate (7) is a consequence of (6) and the following estimate:

dSb

I

(E+Ix-yl

<

2

CE 1/2.

(25)

)lx-ylm-

XESb

In view of (23) and (24), the integrals in (6) and (25) are bounded by

dx1n........ndxm

m J=1

Ixl J

Ix

(s+lx-yl )Ix-ylm-

I

j

.,xm are coordinates on UnSb as well

with 1=3 rasp. 1=2. Since x1, ... J

and since lxl/lxjl<m on as on grals in (6) and (25) are bounded by

248

so it follows that the inte-

+I' 2_dtm-I

f

2m

dt,

(atl

)It1m

tE ItI<2R

IRm-1

with 1=3 resp.

1=2. Therefore (6) and (25) follow from estimate (8) in

Appendix A. 0 8. Proof of Proposition 3 (1). By (23) and (24), the integral in (13) is bounded by

IxI

m

dx1A...,,...Adxm

I

Ix-Ylm-1-a

3=1

Ix:1

(U+VU )n Sb

Ix-al
this implies that the integral in (13) can be

since IxI/Ixil<m on

estimated by

dt1n...A dtm-1

r

2m2

= 2m2s Itlm-l-oc

J

tE

m-2

0-1

ItI
where sm-2 is the Euclidean volume of the (m-2)-dimensional unit

sphere. 9. Proof of Proposition 4. For x6D and 0
v(x,t) = ((1-t)x1,...,(1-t)xq(1+t)xq+1,...,(1+t)xm).

Then

So(v(x,t)) = (1+t2)g(x) - 2tIxI2

and hence s(v(x,t)) < -2tIxI2

(xED, 0
(26)

Further,

249

Iv(x,t)I < 21x1

(xED, 0
(27)

Iv(x,t)-xl

(xED, 0
(28)

and < tlxl

By (26) and (27), 1nlg(v(x,t))I < K(l+llnlxll+lln tI)

for some constant K
Ildq(v(x, t)) 11 Kt-1/2 IQ(v(x,t))l1/2

Therefore it follows from (15) that, for some constant C1
ddf(v(x,t))q < C1(1+llnlxll+t-1/2)

(xED, 0
(29)

Since lbv(x,t)/atl
t( If(v(x,t))-f(x)I =

I

J L f(v(x,s))dsl < 4C1Ixl(l+llnlxll)t1/2

(30)

0

for all xeD and 0
1

If(v(x,l))-f(0)I =

ff(tv(x1))dtI < 4C1Ixl(1+llnlxll)

(31)

0

for all xED.

Now let x,yED be fixed, and r=max(Ixl,lyl). In order to prove (17), we distinguish the cases lxl<41x-yl and Ixl>41x-yl. Let lxl<41x-yI. Then IYI<5Ix-yI and it follows from (30) and (31) that

If(x)-f(y)I < If(x)-f(v(x,1))I + If(v(x,l))-f(0)I 4 If(0)-f(v(Y,1))l + If(v(y,1))-f(y)1

< 20 C1 (1x11/2(1+11nixll) + Iyl1/2(1+I1nIY11))Ix-Yll12

250

This implies (17), because IXI,IyI
is

O
increasing for

Now let Ixl>41x-yI. Then we can find a C1 map w:(0,l] - )D such that Iw(t)I
w(0)=x, w(1)=y,

u

<

Iw(t)I < C21xI

(O
(32)

(O
(33)

2 -

and dw(tt) dI <

C21x-yI

We define

g(t) = v(w(t), 1-

J)

(O
Then it follows from (26) and (32) that

g(g(t)) < - 2C2Ix-yIIxI

Since

dy(y) <21y1, (27) and (32) imply that

(0
(34)

dg(g(t)) <4C2;x1. Hence

II dp(g(t))d <

IQ(g(t))l1/2

4C 2

2

1x1 1/2 1x-YI -1/2 '

and it follows from (15) and (34) that, for some constant C3
Ildf(g(t))Il

Since, by (33),

<

Ixl1/21x-YI-1/2).

C3(1 + Ilnlx-yllxll +

Idg(t)/dtl < 2C2Ix-yl and since (xl,Ix-yi
inequality implies that, for some constant C4
1

If(g(1)) - f(g(0))I =

I

I

dtf(g(t))dt

C4(1+Iln rI)ri/21x-Y11/2

0

By (30), we have

If(x) - f(g(0))I < 4C1(1+I1nlxI))lxl1/21x-YI1/2

251

and (since IyI<21xI)

8C1(1+Ilnlyll)lx11/21x-yI1/2.

If(g(1)) - f(Y)I <

Taking into account that Ix1,lyl,lx-yl < r, we see that the last three estimates imply (17).

10. Further preparations for the proofs of Propositions 2 (ii), 3 (ii), and 5. Set

., 2

5,

and

8K m

where K is as in (8) - (11). We may assume that & < 1. For each yeBR, we define

F(y) = {xcBR:

Ix-YI<),Iyl}

and

E(Y) _ {xEBR:

Ix-yl<6IYI}.

10.1. Lemma. For any yeBR\{0}, we can find indices 1
coordinates on F(y), and

dx1A...Adxm

1

<

>.IYI

for xeF(y).

Idgrdx1A...,,...Adxml

(35)

B

(ii) The functions ,xm

form a system of C1coordinates on E(y), and hence, for all b.1R, the functions ...,xm

form a system of C1 coordinates on SbnE(Y). Moreover,

dx1A...Adxm

and, for all be IR, 252

1 Idg-1X(y,x)-dxl.... IZ-1 9IYI

..Adxml

on E(y)

(36)

dSb

1

-11YI

Id h(y,x)A dx n...

x

1

k "' Adx

m

on S

b R(y).

I

(37)

and set

Proof. Fix

v() ii = 2x ah(Y,x) -

2x.ah(y,x).

i axj

J

axi

Since 1dg(y)11=2IyI and

dg(Y)Adxh(Y,x)lx=y

= = vij(y)dxi(Y)ndxj(y), 1
it follows from (9) that, for some 1
(38)

KIY12

IVkl(Y)I >

m

Choose sE{k,l} such that -1

ah(y,x)I Iy81

> !IVkl(Y)I

axs

x=y

Then, by (38) and (8). and taking again into account that Ildg(y)II

= 2IyI, we obtain that

IYSI > 42 2 = 4alyl 2K m and hence

Ixsl

This implies (35).

> 3)Iyl # 0

if xeF(y).

(39)

In particular, deAdx1A....... Adxm # O on F(y), i.e.

(F(y) is convex!) the functions Q,xl,... ...xm form a system of C1 coordinates on F(y).

In view of (8) and (10), it follows from (38) that

Ivk,i(x)I > -Iyl2

if xeF(y).

(40)

2b3

Since dgAdxh(y,x)Adx1A...C'-1...Adxm = vk ldxln..ndxm on F(y),

this implies (36). In particular, dgndxh(y,x)Adx1A...

..ndxm # 0

on F(y).

(40a)

Set

IQ(x)-Q(Y)I2

rr

e = jXeBR:

X IYl

!tt

EIixi-YiI2

ids

<

a6IYI2}.

Since udyl < 31y1 on F(y) and by (35), then it is easy to show that E(y) -c 9

F(y).

In view of (40a) and since 0 is convex with respect to the coordinates it follows that the functions

s

k, l

...,xm

form a system of C1 coordinates on E(y).

It remains to prove (37). By (5), we have

dS

b

= jdsl`

Ivk,II

Idx h(y,x)Adx1 A.......ndx mISb k,l

on SbnE(y).

I

Taking into account that QdQn<31YI on E(y), together with (40) this

implies (37). 0

11. Proof of Proposition 2 (ii). Since

h dQ(x)l12 = 41x14 < 41Y12 + 81ylix-yi + 41x-y12

and since (6) and (25) are already proved, it remains to find a constant C
IYIdSb <

f

CF 1/2

(£ + Ih(y,x)I + Ix-YI2)2Ix-y1m-

X Sb

for all bER, E>0 and yED. Taking into account again (25) (and the definition of E(y)), we see that there is a constant C
254

IYIdSb

(

(E + Ih(Y,x)I + Ix-YI

1

IX-YIm-3

)

xcSb\E(Y)

1

$

dSb

r

<

(E + Ix-YI2)2

J

-1/2 C 4.-

Ix-y1m-2

XESb

for all bE [R, &>0 and yED. Therefore it is sufficient to find a constant C
IYId J

xESbnE(y)

< CE 1/2

( E. + Ih(y,X) I + Ix-YI2)

(41)

Ix-:YIm-3

Since h is a C1 function in a neighborhood of BLEB and h(z,z)=0 for all zEB12, there is a constant c>0 such that Iz-vI > c(Iz-vI + Ih(z,v)1)

(42)

for all z,vc 9. Therefore and by Lemma 10.1, the left hand side of (41) is bounded by

dtIA...ndtm_1

1

so0°+i

r J

(F- +

tE &m-1

I ti I+

It

)

1 t 1 m-

Iti
where

b:=

max

(Ih(z,v)12 + I2-v1211/2

Now (42) follows from estimate (10) in Appendix A.

12. Proof of Proposition 3 (ii). Since

dq(x) =21x1<21y1+21x-yl and

since (13) is already proved, it is sufficient to find a constant C
1

}

IYIdS

(Ih(y,x)I+Ix-yl

<

Cr.

)Ix-YIm-$

xES IxI
255

Notice that, for all O
1

IdS

(lh(Y,x)l+lx-yl

J

dS

1

Ix YI0°

)Ix-ylm-3

xeS\ E(y)

XE S

Ixl
lxl
By (13) it is therefore sufficient to find a constant C
IyldS2

r

(Ih(y,x)I+lx-yl )Ix-yl

J

M-5

(43)

Cr.

<

xESnE(y) Ixl
If c is as in (42), then it follows from Lemma 10.1 that the left hand side of (43) is bounded by

dt1A...ndtm-1 1 Scm-1

J

( I t

tE

1

I + I t I

)Itlm

IRm-1

1t11
where k(r):=

sup Ih(y,x)I. xeE(y),Ixl
Since Ix-yl<&Iyl<2Sr if xEE(y) and Ixl
in Appendix A. Q 13. Proof of Proposition 5 (i). Clearly,

dx 1A .

J

.

. ndx to

m- I

xEBR Ix-yl
dx1A...Adxm <

rb

I xEBR Ix-yl
Therefore, it is sufficient to prove that

256

[dist(x,S)]oclx-Ylm-141

1A...Adxm sup YEBR

00.

[dist(x,S)]0.lX_Ylm 1+

J

<

XaBR

To do this, we fix some 8>O such that oc+A+E
sup YeBR

.

. nd>M

Ix-YIN_ +

(dist(x,S)J

XEBR dist(x,S)>Ix-yj

dx1n...Adx m

p <

<

00,

Ix-Ylm-l+oc+Q

YE$R

xeBR

then it is enough to prove that

dx1A...Adxm

r

9 p

YEBR

XBR

(dist(x,S)]oc +p+EIx-YI m-1=t

<

00.

Let U and U be the sets introduced in point 6 of this appendix. Fix and set

1<j<m,

Wk = {xEU6: 2-k-1
for k=0,1,...

.

Then, by (22) and the definition of U6J,

dx1A...Adxm < m2kIdQAdxIA...,,...AdxmI

on Wk.

Since 4=0 on S, OES and {dp(x)U=21x1, we have

l

(x)I

< 21-kldist(x,S)I

for xeWk.

The last two estimates imply that, for all ytBR,

dxl A A. .

I

. ndxm

+) 5

+F-

Ix-YIm-1-b

XCWk

257

< m2 k+(1 k)(oa+)6+f,)

IdI(x)Adx1A .

r

IQ(x)lo-+

J

.

.

j

...( Adxm

+E Ix-Yl0-1-G

xEWk

dk dtiA ... A dtm

ds

r

2m2k(1-oc

<

Isl°G+£+p

-dk

to

1

Itim-1-F ,

l IRm-1

Iti<2R

where

2

dk:= max Iq(x) I < Max

xEWk

I x 1

xeWk

< 2-2k -

Since

dtIA...ndtm-1 < Itlm-l-c

1

00

tiiRm ItI<2R and

2-2k

ds

2(2

Isl°C+t+

J

2k

-2-2k

it follows that, for some constant C
C(201)2k.

sup YEBR

xeWk

Since oc+Jj+i
<1, this implies that, for some C
00

d x1A...Adxm sup YEBR

f xEBR

258

[dist(x,S)°c,

Ylm-1-E

(2OC+j3+t-12k

< oo.

13. Proof of Proposition 5 (ii) and (iii). Let U and Ui be the sets introduced in point 6 of this appendix. Then, by (21) and (22), the form a system of coordinates on U4 as well as functions i

on Uj, and

dx A...Adx 1

m

m

Id Adx 1A ........Adxm

on U+ uU

I

3

.i

.

J

(44)

First we proof part (ii). Since

(IQI+lx-yl

2 2 Ix-YI m 3-d >IgI 1-oG-$ Ix-YI m 1+2cx )

it is sufficient to find a constant C1O,

lIdq(x)1!dx1A... Adxm 1

J

(45)

Ix-ylm-l+2o+2p-d

,b

19(x) 1

x 6 BR

q(x)
Since the functions .,xl,...,....,xm are coordinates on U+ and on Uj, J

and (44) holds true, the left hand side of (45) is bounded by r

2m

ds1A...A dsm-1 ISIm-1+2o.+2p-

J

;m-1

se Is 1<2R

1

Itl

dt.

-r

Since m-1+2oo+2,-d<m-l and rr I

Itl ldt < 2rp,

(46)

-r this implies (45).

Now we prove part (iii). We distinguish the integrals over E(y) and over BR\E(y). Since Ildg(x)Q < 21yI+21x-yI < (2/(3+2)Ix-y1 for x 4 E(y) (cf. point 6 of this appendix), one obtains that

259

jdq(x)u21g(x)I-W dx1n...Adxm

r

Ix-ylm-3-d

(1-9(x)1+Ih(y,x)I+lx-YI2)

J

xEBR\E(y) Ixl
<

f2

+ 2)

J

Ix-YIm-4-

(19(x)I+Ix-YI2)

XEBR I9(x)I
for all YE$R and r>O. Since

Ig(x)-oc (19(x)1+Ix-yl

1

2 2

,BIx-YIm-

Ix-ylm-

IQ(x)I1

)

+ oc+

this implies that, for all yEBR and r>O,

ud9(x)12I9(x)I-°Gdx1A...Adxm JJ

(Ip(x)I+Ih(Y,x)I+lx-YI2)

Ix-YIm-

-d

xEBR\E(y) 19(x)I
<

(

/2 +2]

k d9(x)JIdx1A...Adxm

J

Il

lg(x)l

1-

Ix-YI

m-2+2oc+2 -

(47)

XEBR 19(x)I
Since the functions S°,xl,..._ ..,xm are coordinates on U and on J

and (44) holds true, the right hand side of (47) is bounded by

r

+ 2)2m 1

de A...Ads 1 pm-2 lelm-2+2oc+ -d

( J

se

Qgm-1

-

f ItIA ldt.

-r

Isl<2R

Since m-2+2o.+2,8-d<m-1 and by (46), this implies that, for some constant C2
260

(

Ild9(x)h214(x)I-°Odx1A...Adxm

J

< (Ig(x)I+lh(y,x)I+lx-yl2)2Ix-ylm-3-d

C2 r

(48)

XEBR\E(y) IQ(x)I
for all YEBR and r>O. Since Ildg(x)I1<31Y1 for xEE(y), it follows from

Lemma 10.1 (ii) that

r

xEE(y) st

IIdQ(x)II21g(x) I-CCdx1A...Adxm (IQ(x)I+Ih(y,x)I+Ix YI

Ix-Ylm-

-d

)

IQ(x)I
00

dsln..Adsm2 Idt sE Mm-2

-r

2)2le Islm3-d.

I (1tl+lul+I

sl

-00

Isl<2R Together with (14), Appendix A, and (48), this implies (20).

Z81

Ajrapetjan, R.A., and G.M. Henkin 1984: Integral representations of differential forms on Cauchy-Riemann manifolds and the theory of CR-functions(Russian).Uspehi Mat. Nauk 39 (1984), no. 3, 39-106. Ajzenberg, L.A. 1966: The general form of continuous linear functionals on the space of

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1987:

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747-806.

Andreotti, A., and F. Norguet 1966: Problhme de Levi et convexite holomorphe pour les classes de cohomologie. Ann. Scuola norm. sup. Pisa 20 (1966), 197-241. Andreotti, A., and E. Vesenteni 1965: Carleman estimates for the Laplace-Beltrami equation on comple manifolds. Publications Mathematiques de 1'Institut des Haute Etudes Scientifiques, 25, 1965. Barth, W.

1970:

Der Abstand von einer algebraischen Mannigfaltigkeit is komplex-projektiven Raum. Math. Ann. 187 (1970), 150-162. Bruna, J., and J.M. Burgues 1966: Holomorphic approximation and estimates for the a-equation on strictly pseudoconvex non smooth domains. Universitat Autonoma de Barcelona, Seccio de Matematiques, Preprint 1988. Cartan, H. 1953/54:

Seminaire E.N.S., 1953-54, Ecole Normale Superieure, Paris.

262

Cartan H., and J.-P. Serre 1953: Un theorAme de finitude concernant les varietes analytiques oompactes. C.R. Acad. Sci. Paris 237 (1953), 128-130. Chern, S.S. 1956: Complex manifolds. The University of Chikago, Autumn 1955 - Winter 1956.

Cirka, E.M., and G.M. Henkin 1975: Boundary properties of holomorphic functions of several complex variables (Russian). Current problems in mathematics 4, Moscow 1975, 12-142.

Coon, S.

1969:

Sullomobogia degli a perti q-completi di uno spazio di Stein. Ann. Scuola Norm. Super. Pisa 23 (1969), no. 2, 289-303. Coleff, N.R., and M. Herrera 1978: Les courants residues associes A une forme meromorphe . Lect. Notes Math. 633 (1978).

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Dautov, Sh.A. 1972: On forms which are orthogonal to holomorphic functions with respect to integrating over the boundary of a strictly pseudoconvex domain (Russian). Deklady Akad. Nauk SSSR 203 (1972), no. 1, 16-18. ,

de Rham, G. 1955: Varietds differentiables, Hermann, Paris 1955.

Diederich, K., and J.E. Fornaess 1985: Smoothing q-convex functions and vanishing theorems. Invent, math. 82 (1985), 291-305.

- 1986: Smoothing q-convex functions in the singular case. Math Ann. 273 (1988), 665-671.

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-- 1981: Kantenkohomologie. Compositio Math. 44 (1981), 79-101. Grauert, H., and I. Lieb 1970: Des Ramirezsche Integral and die LSsung der Gleichung 5f=Y, im Bereich der beschr8nkten Formen. Rice Univ. Studies 56 (1970), no. 2, 29-50.

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I.

Ann. Math. 102

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ibid

1979:

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1963:

Morse Theory. Princeton University Press, Princeton, New Jersey, 1963.

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1955:

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Siu, Y.-T. 1974: The 3-problem with uniform bounds on derivatives. Math. Ann. 207, 163-176.

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287

LIST O Y SYMBOLS

real, complex numbers

real, complex Euclidean space of dimension n topological closure of the set X for x =

x1yl+...+xnyn

<X, Y>

Y C= X deg f IfI

for x,Y E Cn

(1x112+...+1x112)1/2

I x l

Cn

for x E Cn

means that Y is contained in a compact subset of X degree of the differential form f absolute value of a differential form f of maximal degree; 11

IIf(x)II

norm of a differential form f at the point x; 12

supp f

support of f,

det

of matrices of differential forms; 13

w(x), W, (x)

15

II-IIDM

sup-norm; 16

11

IIa

d

a -Holder norm; 16

exterior differential operator holomorphic component of d; 10 antiholomorphic component of d; 10



the current defined by the form f; 19,20

QC, V

complex, real gradient; 38

BD' BaD

B'

Martinelli-Bochner integral for currents with compact support; 25

Lv

Cauchy-Fantappie integral,; 35, 52

Rv

Cauchy-Fantappie integral; 35

Tv LLi

Ilk 268

sum of Cauchy-Fantappie integrals; 36, 52 .Cauchy-Fantappie integral; 49

Cauchy-Fantappie integral; 51

SK

piece of a boundary; 47

K

simplex; 47 1 if K=(kl,...,kl); 46

{KI

(k1,....... ,k1) if K=(kl,...,k1); 46

K(s")

e

T$ x(M)

tangent space of M at x; 59

Tc X(M)

complexified tangent space of M at x; 59

TX(X)

space of holomorphic tangent vectors in TC x(X); 59

TX(Y)

TX(X) T x(Y) if Y is a submanifold of X; 59

LR(x)

Levi form of o at x; 60

L91Y(x)

restriction of L0(x) to TX(Y); 60

po(x), pp(x)

number of positive, negative eigenvalues of Lo(x); 80

P;IY(x),

PPIY(x)

number of positive, negative eigenvalues of L°)Y(x); 60 = PRIM(M), where 9 is a defining function for M; 62

PM(x), PM(x)

dual domain of X; 146

X*

Notations which contain the symbols L°O, C°', Z°C, (0<0G,,B
Zour,

Eour, and (`I,

Eo6, C0

as for instance C*°(M,E), Cm (M,E),

C0°Or(M,E), E°°r(,E) and [Csor(M,E)]0 are introduced in Sections s,

s,

0.11-0.13.

OE

sheaf of germs of holomorphic sections of E; 40

Hq(rf)

6ech cohomology group with coefficients in

Hs,r(X)

Heur(X) Heur(X,E)

[Ha,r(X,E)10

= Za r(X)/ES'r(X); 30 ZS r(X,E)/Ecur(X,E); 30 = Zs r(X.E)/Esur(X,E); 30 [Z°°r(X,E)l0/[ES°r(X,E))0; 201

269

SUBJECT INDEX

almost C1 boundary

normalized q-convex function

21

- C1 hypersurface (closed) Cauchy-Fantappie formula - integral

- Cauchy-Fantappie formula

- q-convex manifold

pseudoconvex manifold

65

66

- extension

66

18

Dolbeault cohomology

manifold

19

- element 19

exhausting function

60

FantapTie-I-ILrtineau map frame 47 Kol.pclman formula 24

14d,

Levi form

65

- pseudoconvex domain

-

extension

- - extension 100

76

oc, -regular 96

24

22, 25

D -cohomology

22

- - q-convex domain critical point

66

241

q-concave extension q-convex extension

135

100

strictly pseudoconcave domain - pseudoconvex domain

- - q-concave domain 74 - extension 135 q-convex domain

- extension

100

66

66

74

74-

135

q-convex domain

59

non-degenerate ccml:lately strictly 66 pseudoconvex domain

270

28

-

Martinelli-9ochner formula

- kernel

manifold

- q-concave domain

34, 48

- integral

60

- pseudoconcave domain

34, 48

- polynomial

101

function

strictly plurisubharmonic function 60

- - for currents with compact support Leray rata

100

extension

146

E-valued current

- map

138

73

q-convex at infinity exhausting function 65

30

domain with piecewise almost C1 boundary 46 dual domain

73 65

135

- - element

- with values in a vector bundle

46

53

q-concave at infinity exhausting function 73

65

- strictly pseudoconvex domain - - q-convex domain

pseudoconcave manifold

21

manifold

current

piecewise almost C1 boundary

36, 37, 53

34, 35, 49, 50

closed almost C1 hypersurface

78

order of a Dolbeault cohomology group 30

21

30

66

74

b6