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n-q-l-
'?£ [
o
< s < p
-163-
A. Andreotti Analogously oDe can find a similar fundamental seguence of ne15hborhoods
1u \ V~ IN:
such that
'\l
s> n-p-l [
Moreover, if
P
> 0,
2!
°
s < q < we can select the sequence {U,,}
in such a
Way that the restriction
is surjective,
u~.
1.e.
AnalogouslY, if
U11 is in the lIenvelope of holomorphyll of q), 0,
i. Uvs !A
we can select the sequence
Way that the restriction
such a
HO,O(U ) ~
1s surjective, i.e.
Uy
--'>
HO,o(U-) >!
is in the Ilenvelope of holomorphyll of U~.
According to the remarks made in the previo8S sections this theorem tells us when locally the Cauchy problem for cohomology classes 1s solvable. Two special cases will serve as an illustration
Case 1.
Assume that the Levi-form is non degenerate with
°'- p
< q
= n-l-p
Marking only the cohomology groups which are (possible) different
from zero, the situation is illustrated by the following picture
U+
1l0,o(u+)
S
HO,O(S)
HO,P(U+)
P
~s
-
U
HO,P(S)
Ho,q(S)
oOJf--~----c-po--,---------:q"'~,-j4-> HO,q(U-)
HO'O(U-)
Moreover one can show by application of Lemma did in the generalization
0
(7.2.2)
as we
f LeVi-problem to cohomology ~l.:;..SseB
-16S-·
A Andreotti
(point ~) in the proof) that the cohomology groups we have aarked are all infinite-dimensional. Case 2. and
Assume that the Levi-form is non-degenerate,
n
is odd
n-l = 2' then with the same conventions the situation is illustrated by the following picture
°< P =
U+
HO,o(U+)
q
HO,P(U+)
P
J,
HO,P(S) ~ HO,P(U+) • HO,P(U-)
HO,O(S)
S
,
)
° Again the groups marked in the picture are all infinite-dimensional In this case in dimension p the Cauchy problem is not solvable from either side, only the Riemann-Hilbert problem is solvable in that dimension. Remark. In both cases in dimensions p and q we are in the presence of systems of first order partial differential equations Lu = f on S which for infinitely many C'" functions f satisfying the integrability conditions have no solution u of class Cd'.
9.6. On
Non-validity of Poincare lemma for the complex 5
we can consider for any
the presheaf
s
the sheaf
as
QO,s
(5),
as •
defined by
fL _ QO,s(Jl.)
We thus get a complex of sheaves ;L* = 11 s:;,s operator
is:
:
QO,o VV'-
05
~
2J
QO,l
"'"
--4
QO,2 '1N"
with differential
~
...
It is natural to ask of this sequence of sheaves is exact. Indeed in that case~it woulj proviie a soft resolution of the sheaf tI(S) of germs of C"" functions f satisfying the restricted Cauchy-Riemann equations
is
f
= o.
-165-
A Andreoti.:
The answer to this question is in general negative as does show the folloWing Theorem (9.6.1). ~ S be a locally closed hYpersurface in a: n and let Zo € S be a point at which t he Levi form on the analytic tangent space of
S
~
Zo . ~
p
positive and
negative eigenvalues and is non-degenerate (so that
p+q
q
= n-l).
Then in the complex
the poincar& lemma is not valid in dimensions
p
and
q
~
holds in any other dimension.
f!221.
From theorem (9.5.1)
and the Mayer-Vietoris sequence we
deduce that there eXists a fundamental sequence of neighborhoods "'" of Zo in S. 'V = 1. 2, 3, •••• such that HO.s(w y ) = if s # P.q. Thus the Poincare lemma for is valid in
as
dimensions different from
p
and
q.
We have to ·show that this is no longer true in dimensions
and
q.
Let
w
°
be any neighborhood of
Zo
in
S.
p
First we
remark that
where of
(p)
= (n;l).
Indeed we can eelect a b.-is for the space
(0.1) forms in a neighborhood
of the form '11 ..... ~n-l.
Jl,
of
Zo
in
a:n • Jl n s
= w.
:i p •
Then
. ..
As such
QO,p(w)
C~(",)
S•
has a natural structure of a Frechet space.
-166_
A. Andr,eotti Given w
we can select a fundamental sequence of neighborhoods
wcro,,",
= 1,2,3••.. ,
of
Zo
we have remarked in section Set tor i.V I: (.oJ or LV)) Zp(W)
= Kerlds
BP(w)
= 1m
{;;S
in
S
such that
HPCw.)
#
°
as
9.5.
QO,p-l(w) -- QO'p(w)S
Since d s is a differential operator, it is continuous for the topology of Q*Cw). Therefore ZPCw) is a closed subspace of QO'p(w) and therefore a Frechet space. On its turn BPCw) = QO,p-l(w) / Zp-1Cw) inherits a quotient structure of a
Frechet space. Consider for every
v
the following Bet of continuous maps
r ~ ZP(w y )
ZPCw)
t i
"
BPCw) where Set
iv E>J
Then E v
is the natural injection and
= 1. CQ,~)
• Zp(uo),.. BPCVJi)
as a closed subspace of
ture of a Frechet space and we the
following
commut~tiJe
C·ffi
I
r
v
r),,)
the restriction map.
= i)M}
ZPC",) '- BPCw",)
•
has the struc-
complete t~e ~revious maps with
diagram
r." -_ _ ZPCw.) , 1'i
- 1 BPC",'y) rJ
l With j and d continuous. We have jCE,) = r: ivCBPCw v ))' Thus this spcice is a continuous im~Ge oy ~ linexr map of a Frechet space.
one of these two
By the BJllach open mapping theorem we mnst have ~roperties
-167-
A.. Andreotti
ii)
-1
or elee
ry
_tl
i~('" (wv)
is of first category in
Now as we have remarked in section
9.5-
we can construct for
each v an ele~ent ~\i" Zp(w) such that r" (~,/ 9'- i,-
on .Q. 2.
ls a domaln wl th a smooth
lf the complement of Jl
ls bounded.
Then a smooth functlon
can be extended to a smooth functlon h
ls holomorphlc ln al ••••• ~
f9:rall
ls connected and
11.
h
h
n
on
lf and only lf
on
h
b Jl.
such that
satlsfles (14)
that satlsfy (13).
Followlng Hormander
[1'1]
(theorem 2.3.2') we can
prove the above by flrst constructlng a smooth extenslon
H
such that (1.15") Slnce the supported
u
J> = 1
where
== 1, •..
j
a -problem for
0(
J E
can be solved wlth a compactly
Col (([ n) •
we set d. = d (j'fl)
wlth a compactly supported slon
h =
pH -
b J1
ln a small nelghborhood of
outslde a slightly larger neighborhood. ~
,0.
u
u
and vanlshes
Uslng ~ = ~ u
we obtaln the deslred exten-
as ln the previous theorem.
The functlon
H can be constructed by starting wlth any smooth extension
r
of
hand notlng that (14) implles that:
(1. 1/\)
near
rewrltlng this we have (1.17)
~(f-f.r)
b
Sl.
-187 -
J. J. Kohn
which implies that
and hence (1. 18}
Setting
H
(1.19'
2
f - fo r - f i r /2
s
we obtain (1.20)
as required. It should be mentioned that the tangential CauchyRiemann equations (14) have been studied extensively (see 30chner [2) Kohn
[15 ]
,Lewy [221
,
,Kohn and Rossi
[211
Andreotti and ~1l~ (-1]. etc.l.
In fact, the famous example of Lewy of an equation without solutions is one of these.
We will return to this
equation later. Our main concern will be to study (3) on a domain without any restriction on the support of c1..
Suppose, for
example, that the c( j E L 2 e0.) then.we wish to find L u~L2(rt) satisfying (), by this we mean that we want to find a sequence of smooth in
uy
defined on.1l
such that
(.D. )
L2 we have u = 11m u y and o(j = 11m u"z j' Suppose that there exists a point P E. b{l and a holo-
morphic function such that
f
f (p) = 0
defined· in a neighborhood and
f ~ 0
in
U
V
"JL - {p J.
of
P
Then
-188-
.r. J. Kohn we claim that If a solution of (3) eXists In
there eXists a holomorphlc function that
h
Is an open set containing
V
Ists no holomorphlc function h
p
=1
h
cannot be continued analytically over the point
That Is, If
g
L (SL) then 2 defined on Sl, suer
on
Sl.
V"
In a neighborhood of
P.
=J pf-· In lOin N
(1. 21)
F
and we chose
r
To see thiS let
f
then there ex-
P
defined on
g
C;(
P.
V such that U)
Now we define
such that
F by:
V"Sl~- U
F¢ L2 (SU.
N so large that
We set 0 =
and we have (1,0)
-d.(
E
a
of degree
)
r{-(d).
wlsh to prove (b)===>(d).
1:onal decomposition
().)O) and that the necessary an:! sufficient condition for the eXistance of a solution u S
= 0 and
..:l.}e,
satisfyin~
then u =l'*N",.
Tu = oc is that
It also follows that If
P: A--"7n.(T} is the ortho~onal projection onto7l(r) then (3.)))
Proof:
P = I
-l'*:n.
All that remains to be proven is ().))). that is.we
must show that the onerator P defined by ().))) is the orthogonal projection of A onto h(T). fP = 0
fin:l
J<J.? -
I) 1.
It suffices to show
71u),
-216-
J. J. Kohn
now TP slnce HT
=
T - TT*NT
= O.
~
=
T - LNT
-T*NT
Flnally P - I
T - T - HT
0
and hence
(k(P:- I) C f and ljI'i: a.
:z>
the
Then
p q • we obtatn. by tntegratton by parts:
(4.1 )
aofi) (;>.). =
('P,
f
(e
-).
jl
=
S(q(e-~),~}
dV - 5e-;\
bSt
JL (e
A
-ll
;:r(e'f),'I')
(>-)
where ~ ~ C'(fi), and
A
~
o.
\C , I
(4.2)
r
dV
f'
..:T
it
evaluated on dr.
has been normalized so that
= 1
on
blL
Before proceedtng we wl11 Hlustrate the formula (4.1) tn the case (4. J)
p = 0, q = 1, JlC{:n
(-Ifjz
.u) j
and
A = 0; tt reduces
+L(rf udS •
b-A
z
j
j
Stnce the boundary term tn (4.1) vantshes when ~ has comoact support 1n..{L we have
-219-
J. J. Kohn
(4.4 )
for all 'f'!'EQP,q-1
with compact support and since the set of
these is dense, we conclude that (4.5)
= e
('1',31/1)
Then, since
" j:( e
-)
~)
for
. = (T*'I"'/J) (A' ().)
for 1f t
;?;
and
we conclude that the boundary term in (4.1) must vanish for aU (4.6 )
If',-aP ,Q-1
and hence we obtain
S:) under the norm
ls dense ln 2
II T*'I'\\ ( ~)
2
+
2
\\S .)
+
h\\(:>.) .
Thls proposltlon enables us to prove estlmates for elements ln:O from whlch we can then deduce the cruc lal lnequal1 ty (J.15) of Lecture) for elem'ents ln~. n x n
Let
LL ,r '1 j
n
n
k
J = z: a L k21 lj k
L k:=1 lJ k
-b
lj
(c
)
be the (n-1)
lj
(4.1))
n
c
a lJ
for lj
k_
k
a
Let
be the
+'5"" b
k
then
k)
lJ
matrlces deflned by
(4.12)
(a
lj ~
(n-1)
matrlx deflned by
1 L. 1, j < n-1 ,
It then follows from the formula for the Levl-form «2.)1) of Lecture 2) that c
lJ
ls the Levl-form ln terms of our
basls. The followlng formula ls at the root of all the estlmates whlch we wl11 derlve here.
If 'fI(:j5 and the
-222-
J. J. Kahn
support of 'f lies in (4.14)
KT*'f'U
2 (,,)
lrnu' + ns\L \\Ip\\) z
since the boundary term appears only when ~n
= 0
on
bll.
= nand
k
Observe that 2
',\J
°
and a
_228-
J. J. Kohn
(4.)))
2
+ \\sq>1I for all 'r ~ 0,
1\ =
where
and
2
(0)
rt,P.q
'f E-.u
+ 11'1'11
(\
('0)
-
COO (fl_'n lJ),
o
sr for some fixed large
e
s.
For the proof of this proposition see Hormander [11 J
To establish (4.))) in the case that pseudo-convex, choose ('I
s
SL
is strongly
sufficiently large so that
is Positive'definite as in (2.10) of section 2; it
.)
"ziZJ
will then be positive definite in a neighborhood A[jk]
Then from (4.16) we see that
~
of bM.
is positive definite
and then (4.))) follows from (4.14) by use of a partition of unity. Finally the same argument also yields the following result. Proposition.
If ?i,~c·"(ii)
in a neighborhood
VI
of
then (4.))) holds for all when
q
~
is strongly pluri-subharmonic bn
't:::
and i f 0
.n.
is pseudo-convex
and 'f'~:b,p,q
•
n ceo (fL (\ U) 0
1.
.-
In case there exists a i\·E'C (.11..)
which is strongly
pluri-subharmonic throu~hout il, then the integral in (4.))) can be all taken over 1"-
flO'-
thH we- ha ve
-229-
J. J. Kahn
(4.34) Henoe by the results of the prevlous leoture we obtaln the followlng result. Theorem.
If there exlsts a strongly plurl-subharmonl0 funoC~(rl)
tlon In
and If
11-
ls pseudo-oonvex wlth a smooth
boundary then the a-Neumann problem solutlon for
q
~
1
and
~
L p,q(.\l. 'LA)
sufflolently large.
The oondltlon that there eXlsts ),~ C"(Sl-)
2
for example.
= \z\
=
L
b,
~n,
plurl-subharmonio is satlsfied In
II
has a
a
jz ~
whloh is
taking
2
j
it
It Is also satisf1ed for
in a Steln
man1fold, by taklng N
2
/I =2:\h \ , 1
where
h 1 ••••• h n
poInts.
k
are holomorphl0 funotlons that separate
Thus we obtaln the operator
N,
for '[
> C.
and
for eaoh l ' given by
(4.35)
T*N«.
and L2 solutlon of the a-problem Now we have, If So( = 0: 2
1\ I'*No·
Il.
1s pseudo-convex and 1f 1n a ne1ghbor-
bJl
there 1s a strongly plur1-Bubharmon1c
If
W or
P = T*T + SS.
u = No(.
and
n - q
q
1;
~
or 11' the Lev1 form has
pos1t1ve e1genvalues or
q + 1
e1genvalues then for suffic1ently large 1:' problem has a solution on
LP,q(il,'l;)"),
the
e1~her
negative ~-Neumann
and the space
2
}(P,q are 1n
1s f1n1te d1mens10nal and cons1sts of elements which COO on
'J{p,q ,. 0 Proof I
Q.
for
Furthermore, i f
W =.iL
then
q ~ 1.
1be case
W..
..n.
follows from (4.34).
To establ1sh
the general case it suff1ces:to'show (by the results of Leoture 3) that 11'
l\lf"l\
1s bounded,
1f¥.L dtp,q
that 11'
then there 1s a subsequence
Buoh that
and
-245-
J. J. Kohn
ln L~,q. ~1
that
Jl -
=1
ln a nelghbrohood
Then, lf 'l:'
W.
V
C~(1L)
J 1e
To do thls, ohoose a functlon
of
bSl
such
and ls zero ln
ls sufflclently large, we obtaln the
followlng lneQuallty as a consequence of (4.33). 2
~ const·ql~,)(lrJ".!,'f)
115"1,,11
(6.6 )
~ const. (q(nl(P' q (\
2
tions the
H
J and
Ll"" ,Ln _ i ,L1 ,··· ,Ln _ 1
N = Ln
- Ln
it
therefore suffices to bound
II t\ -rL b
J< n
and for
u\\2
J
-r 2 .\lA b LU~ J
V,-r NU \\2.
and
The second of these is
immediate. (7. 34)
-r
II f\
Lull
2
0 0 0
(L u,P u) = (u,P Lu) + (uQ u)
z
J
J
J'
~ const.(~ulli\·ulL +
z
where
po, QO
2
I\ull )
are operators whose restrictions to
const. are pseudo-differential operators of order
r = O.
Finally: (7.35)
-t
111\
2
Nul\
0
= ((L -L )u, P u) n
n
in view ·of the above remark this is bounded by the right side of (7.33).
-267-
.r. J. Kohn The followlng theorem ls proven ln Kohn Theorem. of
PfbR
-rL
If
116].
ls pseudo-convex and lf ln a nelghborhood
each non-zero vector fleld of degree
and values ln
T 1 ,O(blL)
on
bSl
(1,0)
ls of finlte type at
P
and lf the Levl form ls dlagonallzable ln a nelghborhood of P
(1. e. ln C)
c
the
with
q
~
throughout a nelghborhood)
11 lJ
then there exlsts £ '> 0 on
S
c lJ
such that the 3-Neumann problem 0
ls
£-subeillptlc at
P.
-266-
J.J.Kohn
Lecture 8. The 1nduced Cauchy-Rlemann equat10ns. In thls lecture we w111 take up systemat1cally the euqat10ns on
b.Q wh1ch arlse 1n the extens10n problem
d1scussed 1n lecture 1.
We deflne'
(8.1) then 'f ~
q = 0
Observe that 1f
1'p,q 'f'
if = arj\
1
and LL,1.J
It is still an open question
whether on X which satisfies the conditions on the Levi-form for
q = i
ab u = 0
given in the above theorem the equation
has non-trivial local solutions.
To conclude these lectures we wish to point out an application of these results due to Kerzman (see [30J). Namely, if
H : L «(l)--> 2
H 0,0
denotes the orthogonal pro-
jection map onto the space of holomorphic functions, and if N is pseudo-local then
H is also pseudo-local.
can be expressed as (8.25)
Hu(z) = SK(Z.W)U(~') dV w' Il.
Then the pseudo-locality of
H
implies that
Now
H
-277-
J. J. Kohn
11]
A. ANSBEOTTI, and C.D. HILL, several artlcles to appear ln Ann. Scuola Norm. Sup. Plsa.
[2)
BOCHNER, S.
"Analytlc and meromorphlc contlnuatlon by
means of Green's formula,"
Ann. Math. (2) 44,
652~673
(1943). [31
EHRENPREIS, L. theorem,"
[4]
"A new proof and extenslon of Hartog's
Bull. Amer. Math. Soc. 67, 5007-509 (1961).
FOLLAND, G.B. and KOHN, J. J. the Cauchy-Rlemann complex,"
"The Neumann problem for Ann. of Math. Study Vol.
75, Prlnceton Unlv. Press, 1972. [5J
FOLLAND, G.B. and STEIN, E.M.
ab on
Mates for the
"Parametrlces and estl-
strongly pseudo-convex boundarles,"
Bull. Amer. Math. Soc. (to appear). [61
GAFFNEY, "r,P.,"Hllbert space methods ln the theory of harmonlc lntegrals," trans. Amer. Math. Soc. 78(1955) 426 - 444.
[7 J GRAUERT, H,- "Bemerkenswerte pseudo,)J th direct image R n* 1 of '} ~
n
is coherent on
S for
JI ~ 0
Grauert's proof uses the power series method.
The
idea is to expand a » -dimensional cohomology class in a power series in the variables of eral case to the case space).
~here
S
S
(after reducing the gen-
is an open subset of a number
The coefficients in the power series expansion may
not be cocycles, but, by using descending induction on
»
Grauert showed that they can be approximated by cocycles. Then he used induction on
dim S and applied the induction
hypothesis to the approximating cocycles to get the coherence of the
"th
direct image.
About ten years later Knorr (13) and Narasimhan (18) gave simplified presentations of Grauert's original proof. Recently Kiehl (10) used nuclear and homotopy operators and a form of Schwartz'! finiteness theorem to obtain a new proof for an important special case of Grauert's theorem. Then Forster-Knorr (3) and Kiehl-Verdier (12) succeeded in obtaining new proofs of Grauert's theorem along such lines. Their proofs make use of descending induction on JI , but does not
u~~
induction on
dim S.
This opens the way to
generalizing Grauert's theorem to relative-analytic spaces and such generalizations were carried out by Kiehl [11), Forster-Knorr (4) and Houze 1 (9).
Y-T.Siu
In
1962 Andreotti-Grauert [1] generalized the theorem
df Cartan-Serre in another direction.
They introduced the
concepts of strongly pseudo convexity and pseudo concavity and proved finiteness theorems for spaces which are strongly pseudo convex or pseudo concave. space x
~
is said to be strongly
I
X there exists an embedding
of x
U
«;N
on a complex
~
p-pseudoconvex if for every of an open neighborhood
T
onto a complex subspace of an open subset
r?-
and there exists a real-valued
such that
(
A function
'f
=
a21j1 - ) dZ i d Z j
Of·T
and the
Of
on
G
hermitian matrix
N - P + 1 . positive eigenvalues at
has at least
every point of
NXN
function
G of
G (where
zl' ••• , zN
are the coordinates
of (N).
(0.3)
Theorem (Andreotti-Grauert).
space and ~
'f:
['f < a#1 Suppose
.,
codh "l ~ r
{'f> b#!
such that
IT
and
is a coherent analytic and
p ~ " < r -q-n,
""1, R>i( nb )*, ~s an
a
with
codh'7 ~ r
R" n* '7 is coherent on
~somorp h'~sm ,
2B
f
or
- 287-
Y-T.Siu
This conjecture so far has not been completely proved. '!he special case vex case.
{1 < B.tI} = Z
{P > b# 1 = fO
The special case
pseudo concave case.
is called the pure pseudo conis called the pure
Partial results for these two pure Cases
were obtained by Knorr [14] and Siu (24,25].
Recently the
pure pseudo convex case was completely proved by Siegfried
[21] by using the methods of the new proofs of Grauert's theorem and the pure pseudo concave case was completely proved by Ramis-Ruget [19] by using the methods of the new proofs of Grauert's theorem together with duality.
Unfortunately these
methods cannot be applied to the mixed case, because any induction on the dimension of the direct image is impossible. A partial result on the mixed case was obtained by Siu [26]. In these lectures we will prove the following improved par-
tial result of the mixed case which is good enough for the known applications. (0.5)
Main Theorem.
Suppose
11: X --> S is a strongly
(p,q)-pseudoconvex-pseudoconcave holomorphic map (given with
If' and
~
sheaf on
< a# < b# < b*) .I:
such that Suppose
a* < a
t
< a < a# and
conclusions hold.
'1
dim S ~ n
codh'll .
and
is a coherent analytic and
'7?
I:IS,l1 (x) X
n
codh
7
for
x ~ X.
~ r
.2.!! Let
Then the following
- 288-
Y.-T. Siu
s ii)
iii) iv)
~
S and
p $
v
B(U(D),.gl
D 1-->
CY(U(D),-§l
B (9)
,
J
r; (§)
on
S.
We derive the isomor-
phism statement by constructing a sheaf-homomorphism
- 2 9~v-l(7) on
U which is
a
right inverse of
6
This right inverse gives rise to a right inverse
e/I.ql ---> ~"-ll.g) 7!' ~V-1I§1 of 6 implies .r-(U, e'I-§I) which, together with
l)v I§)
the vanishing of th!! coherence of
R"n*-§,
JJ. _"-1 JJ. Rv+ 1 n*~ ••••• K n*~, yields the isomorphism
II "(1) ---> ~"-1('1l
1be construction of a right inverse
of
is based on the generalization of the following observation. For an open polydisc G in d: n • a continuous . .. -> n1:1 -> nIf) -> "1-> a
on an open neighborhood of Definition. (a)
a
Il.
D is an "] -privileged neighborhood if
the induced sequence p
> B(D) m _>
B( D)
P
a
is split exact, (b)
Coker a
---> "lois injective. t
When (a) is satisfied, one defines
B(D,"])
as Coker a •
This privilegedness is said to be in the sense of Cartan, Douady, or Grauert according as
B(D)
has the mean-
ing of i), ii), or iii). The definition of privilegedness and
B(D,;7)
is
independent of the choice of the resolution of ;7, because, by using Theorem B of Gartan-aka, we can
eas~ly
prove that
any two finite free resolution of ;7 on a Stein open neighborhood of
D- become isomorphic finite free resolutions
after we apply to each of them a finite number of modifications
[8, Def. VI.F.I], i.e. after we apply to each of
them a finite number of times the process of replacing it by its direct sum with some finite free resolution of the zero sheaf which has only two nonzero terms
( of. [8 • p.2a2.
-302-
Y_T.Siu (1.2)
For Banach spaces
EO' Fa
we denote by
L(EO' Fa)
the Banach space of all continuous linear maps from
EO
to
Fa • Suppose
S is an open subset of ~ nand
holomorphic Banach bundle on we denote by set
U of
S with fiber
E is a For
EO
s I;;: S
Es the fiber of E at s . For any open subS we denote by EI U the restriction of E to
U • Suppose fiber
Fa.
F
is a holomorphic Banach bundle on
A map '(: E --> F
S with
is called a bundle-homomor-
phism if for every open subset
U of
S for which there are
trivializations a: EI U
'"
there exists a holomorphic map
>
U x E
A(')
a
from
U to
L(EO.Fa )
such that
for
s I;;: U and
denote by
x I;;: EO
.
For any open subset
U
of
tlu the bundle-homomorphism Elu
induced by t
.
For
sl;;: S
-> Flu we denote by
(s
the map
S
we
Y-T. Siu
induced by '( • Suppose 0-;>
e
E(m) _;> E(m-l) _'_;> ••• _;> E(O)
is a complex of bundle-h9momorphisms of holomorphic Banach bundles on
S.
So
If for some
...
0-;>
is split exact, then there exists an open neighborhood
So
~
S
U of
such that the sequence
is split exact. To prove this, it suffices to prove the case where m· I
and
E(l), E(O)
the closed subspace of
are both trivial bundles.
Let
H be
which complements
Let
be the bundle-homomorphism induced by
e
and the inclusion
map Sx H
~s
o
e.....-..> S x (H Ell
is an isomorphism.
Im
eSo )
Since the invertible elements of
_304-
Y-T.Siu
L(E(l)$H, E(O)) So
£orm an open subset, there exists an open
So
neighborhood
U o£
So
(i.e. (a-/U)-l
phism
such that
IT
I U is a bundle-isomor-
is a bundle-homomorphism).
(1.3) Suppose S (respectively Q ) is an open subset o£ n 4:. (respectively iC N) and 7 is a coherent analytic shear· on S x .Q.
Fbr
we denote by 7(s)
the shear
where
are the coordinates of iC n • 7(s)
tl"'"
tn
n
be regarded in a natural way as a shear on Fbr
p
~
bundle on (n Let pose
1 , we denote by
whose f·iber is
,,: S x.Q.
s ~ S and
7
can
the trivial
B(.Q. 'n+J!5P)
B(Sl)p •
--> S be the natural projection. is
"-flat at
Is] x.Q
and
7
Sup-
admits a
£inite £ree resolution
on
S x.Q..
centered at z.
Suppose z
z
~
n
and
GC C
n
is an open polydisc
which is an 7(s)-privileged neighborhood of
Then there exists an open neighborhood
such that, for any open polydisc
U o£
DC U centered at
s
in s,
S
-305-
Y -T. Siu
Dx G is an I-privileged neighborhood of
(s,z)
To prove this, consider the following sequence of bundle-homomorphisms induced by (*):
Since
G is
"1
{s I x Q
at
I
0->
I(s)-priyileged and since by the
n-flatness of
the sequence
P
n+JD m(s)
p
n+JD lIs)
-> ... ->
->
tIs)
->
0
induced by (*) is exact, we conclude that the sequence (I), when restricted to the singleton
[s], is split exact.
(1.2), on some open neighborhood of
exact and Coker a
s
in
S. (I) is split
is a holomorphic Banach bundle.
that, for any bounded open polydisc
By
D centered at
Observe s ,
B(D, B(G'n+Nrd'il)
is naturally topologically isomorphic to
i B(D x G'n+Jl )
~
(0
i
~
m).
Hence. when
in a sufficiently small neighborhood
is split exact and to
B(D, Coker a).
B( D x G,
I)
U of
D is contained s
in
S,
is topologically isomorphic
To show the injectivity of
-·306.
Y·T. Siu
->
B(D xG, "])
7(S,Z) ,
it suffices to show the injectivity of
((Q( Coker
p: (where
V(Coker (1)
(1») s
->
7( s,z )
is the sheaf of germs of holomorphic sec-
tions of the bundle Coker (1) and. by induction on suffices to do the Case
n· 1.
c;;:
f Suppose
f
is nonzero.
integer
k
such that
n • it
Take
Ker P •
There exists a maximum nonnegative
with
By the
n-flatness of
'J. g c;;:
Since
G is an' F(s)-privileged neighborhood of
lows that
(1.4)
g(s)
Suppose
~
'J
and extend'
N
"] on ({; z
x
and if
Q.
If
z , it fol-
0 • contradicting the maximality of
k
is a coherent analytic sheaf on an open
Q of d: N •
subset ([N+l
Ker P •
"I
Identify4:,N
with the subset
0 x {N
of
trivially to a coherent analytic sheaf
Gee
Q is an open polydisc centered at
G is an 'i-privileged neighborhood of
z ,
~.
- 307-
YeT.Siu
t:or any bounded open disc
-
D C ([ .centered at
.!!!. 'I-privileged neighborhood ot:
0 , D x G is
(O,z).
Tb prove this, we can assume without loss ot: general-
ity that there exists an exact sequence
!!1m ->···->N d'l Ill' tJ'O m ->N" ->7->0
O->N on
.Q..
11
Dei.'ine
0->
by
(1
7" ->
°
is an exact sequence of coherent analytic sheaves on an open neighborhood of
D-.
If
privileged neighborhood of
D is an
' 1' -privileged
to ,then
and 1"-
D is an 1-privileged
-
~09-
Y-T. Siu
neighborhood of
to.
n-:
Tb prove this, we take finite free resolutions on
P
,
aI
m
0-> (l)m_> ... --> n 0->
(9 n
" a "m Pm
->
a
I
PI
I
I
I
o
-> nOPo _) 7' - "
n
"a." /API I
-> n'"
->
0
If
,,,PO nCI
'"1" ->iT
-> O.
We can construct the following commutative diagram
o
J, 0->
0->
",Pm
nv
t.,Pm n<J
t
a
m
->
!"
II
,,,Pm
0-> n'"
->
->
P1
_> nV
~
o
!Po II
-> nID
1
1o
!
->"7" iT --> 0
0
1 0
where i)
11)
" ( l j . 19 P j Ef) (!)pj and, except in the last column,the n n n vertical maps are the natural injections and projections. , a.. J
is of the form
:~ ) J
being a sheaf-homomorphism).
O'j: nVPj
-> nr!lj
- 310-
Y-T.
Siu
Let 1
S(D,
p.
nCO
J)
be a continuous linear map which, when composed with the map 1
;:;: B(D, n!f)Pj) -> induced by
B(D, n(lj-l)
1
a j , gives rise to a projection
Let
fl~: B(D, be a similar map.
• ncoPj-l)
->
Then a corresponding
can be defined by
Since clearly
,
"
0-> B(D,? l - > B(D,tl - > B(D,I l - > a is exact, the result follows. Before we state the principal theorem on privilegad polydiscs, we have to introduce a terminology. is a statement depending on for f'
f'
'f\
/7
lJ/Sy(P-l) By
induction hypothesis,
7(f,,)
converges to some element applying the open mapping
By
theorem to the map
induced by
r ' we Can find
such that
, f
g
"
->
Since it suffices to show that
r(y, n
(2.4)
k
<Xl
•
Suppcse
order of
Then
converges to
X is a complex space.
'f (i=l L aos o) ~ ~
Define the reduction
X as the smallest nonnegative integer
that. if
for some open subset
U of
X and
in
PO
such
i
-320-
Y-T.
for all
~
x
U , then
Siu
A complex space is reduced if
f · 0.
and only if its reduction order is zero.
(2.1) a rela-
By
tively compact open subset of a complex space has finite reduction order. Suppose the reduction order of
X is
~
P
(E,llllJ
map and
&'f ..,ag..r:..e:::;e::.;s~w:.:i:..:t:::h,-"th::.e~r:..:e:..:s:..:t:.:.r.::i",c",t=io=n
such that 0
"-,t ,F
~
ell!,11 .
stant independent of ~ , to, and
E!22f.
0
'f F•
where
e
is a con-
l!2,t
By considering the power series expansion in
t , we
observe that it suffices to prove the special case where n· 0
Take Stein open subsets
Gl C
C
•
.•
" Gee Gee G:! •
Take a finite Stein open covering 1t • lUi] (respectively t " VI."• n lUi Jl of G (respectively G) such that U(l) CC U· CC U" CC U(2) i i i i Take
t, l; Z~7.t2'
There exists
(9x)
? l; ck- l (?Ji. , (9X)
such that
S? • II?IIU ' where
e
~
is independent of E.,
~ Ill'
e 11e.l/m" and comes from applying the
-325_
y- T Siu
open mapping theorem to
-> By
().2) the restriction map
factors ,hrough a Hilbert space
7
jection of
H
Let
, 7
be the pro-
onto the orthogonal complement of the kernel
of the composite map
CC
Gz
Suppose
uP) C C uI2).
A'X-> A
(a)
X
be a finite Stein Let -
homomorphism such that
° l!!
is independent of the
Ll. for
0
Gz
Stein.
Let
pen covering of - G" (". 1,2)
a: ell A-X
Coker a
Q. E. D.
is a Stein complex space and
C C X are open subsets with
lk.. a lUI")} with
~(S)
Then
and satisfies the requirements.
Proposition.
().4)
Gl
a
7
choice of
7 I~l· I
~(~)
Define
-> ([)~
be a sheaf-
A-X
,0 neighborhood n
is flat with respect
Then there exists an
0
pen
satisfying the following: Ll.(tO ,fl
yr:r(b.('O,P) x
C
there exists
.Q,
Gz,Im
linear over ([[ t] restriction map and
a)
->
r(A(tO,P) x Gl'
ck-l(A(tO,PlX U , 1m l
tt)
6 'f agrees with the
Ilf(E,) 1 12-l,t°, p:;;:
C is a constant independent of
Proof. disc.
there exists
such that
restriction map and where
k ~ 1
A(tO, f) C Q and
t"
°'f
Clll,ll,~
""2,t
to,
and ('
Consider first the special case where
X is a poly-
we can assume without loss of generality that there
exists an exact sequence P 0->(')1 Ax X
_>
Let
(a)m
(respectively
(b)
for the case L:f m
->c:>
(1
Pl
AxX
(blml denote
PO
->(9 (a)
AXX
(respectively
By using (1.6) and (1.3) to ob-
tain local solutions of (a) and by piecing together these local solutions by Cech cohomology, we conclude that (a)l implies (al m • clude that (a)m and (blm~l imply (b)m
holds and that (b)m_l
case follows by induction on
From (3.3l, we conHence the special
m
Fbr the general case, we prove (a) first.
We can
assume without loss of generality that i) ii)
I
is a complex subspace of an open polydisc
P, and
there exists a commutative diagram of sheaf-homomorphisms
-327-
Y T. Siu a
->
!9PO A' P
!quot. a
-> such that
N
Coker a
'" Po
([)
A,X
is isomorphic to
Coker a
under
the quotient map. Then (a) follows from ().l) and the special case. let
For (b)
m be a positive integer such that no more than
bers of
vs.
can intersect.
m mem-
We can assume without loss of
generality that we have an exact sequence p ([) m
_>
A'X rhen (b) follows from (a) and (3.). (3.5)
When the flatness condition on
Coker a
is dropped in
(3.4), the conclusions of (3.4) remain valid with the follow1ng modifications. i)
Y
and
l'
are linear over It , but may not be linear
over It[t). 11)
Fbr a fixed
to,
(' has to be sufficiently strictly
small. iii)
C may depend on
However, if
to
and
f .
is not a zero-divisor of any stalk of
- 328 -
Y-T. Siu
Coker
~
().6)
,then Suppose
fn
C can be chosen to be independent of
is a Stein complex space, 1 is a coherent p Ax X ., and 'f' : -->"] is a sheaf-
X
v
analytic sheaf on
AXX
epimorphism.
A(t O , p) C C A
Let
compact open subset of
I.
For
0 U, t
U be a relatively f ~ r(A(t O,f) xU, '1) de-
as the infimum of
fine the Grauert norm
/1111
and
where
'f
A different choice of
'f
would give an eqUivalent semi-norm.
When we use such a Grauert norm. we assume that a fixed
~
is chosen. Suppose
'It •. IU i I
pact open subsets of
X
is a collection of relatively comFbr
E, • define
When
().7)
to
~
0
,IIE-ll
0
tt, t 'F
Proposition
is simply denoted by
(Theorem B with Bounds).
11E.11?Jl,r Suppose
X. G"
- 329 -
Y-T. Siu
U.
( ». 1,2)
are as in (3.4).
'1
Suppose
is a coherent
analytic sheaf' on .t. x X flat with respect to
a:
and suppose
(OP
AXX
-> "]
n: Ax X
-> A ~
is a sheaf-epimorphism.
there exists an open neighborhood Q
of'
0
in
A
~
f'ying the following:
f£l:
(a)
O .t.(t .f)
en
linear over ([ [t] tion map and
(bl f£l:
.t.(t
'F) C S2.
linear over C [t]
a 1f equals the restric-
such that
/W( 7111
O
tion map and
there exists
0 ~ Gl't • f
c 111/1
•
~ t
0
•P
there exists
6 'f equals .the restric-
such that
IIr(~)11
Ul,t0 .f
~ CIl~1I
0
U2 ·t ,p
;
o
f .
to, {-linear
"f
C is a constant independent of !, , t • and
(~ '1 is not n-flat. for a f'ixed
'f
exist f'or
linear over t n - to n
p
sufficiently strictly small but maY not be . 0 [t] and C maY depend on t and f If
«
is not a zero-divisor of any stalk of .
be chosen to be independent of ~.
and
fn.)
Fbllows from (l.6). (1.3). and (3.4).
"l,
C can
- 330 -
Y-T.Siu
§4.
LeraY's Theorem with Bounds
(4.1)
First we examine the diagram-chasing proof of the
usual LeraY's theorem without bounds.
Suppose
herent analytic sheaf on a complex space
2t.
IUa la I;;: A
l(-
{Vilil;;: I
are Stein open coverings of with an index map": I
X such that
--> A
'1
is a co-
X and
l( refines
2t
LeraY's theorem states that
the restriction map L
->
Define
rf',>J (1X,
H (lX,'l)
is an isomorphism.
1
H (1(,7)
as the set of all
l()
such that
a,.
r(Ua(\···(\U
o
is skew-symmetric in a O' ••• , a,. 1 0 , •••
t
iJl.
De.fine
(IV. (\ ••• ~o
(w. ,7) ~
and skew-symmetric in·
- 331-
YcT.Siu
6 : r:!"" (1k, l{ ) 1
->
r:!'+l,"" (tt,
62 : r:!""(U,l{ )
->
r:!',"+l (tt, l{)
1\:
~
C (1(' ,F)
->
CO ,"
8 : r:!'(lt ,F) 2
->
r:!',O(Lt, l()
(It,
l( )
l()
as follows:
a
E. i O· •• 1" .IU(10 ('IV.10('I···IIV.1"
Consider the following commutative diagram:
- 332 -
Y -T. Siu
1
1
rex, 7 ) - >
0->
J
0->
°
°
°
CO (lk, 7 l 82 1
..L>
1)
2 (~, '1
~>
...
82 1
"I 9 6 6 1 CO ,a (lll, If) ...1.....> 2,0 (~, l() -1> ->
eO (If, '1)
'"
J
62 1 62 "I 91 6 61 0 - > 2(1(,7) - > CO, 1 (lX, J( ) - > 2,1 (!t, l( ) -1> 6J
62
6J
J
62
1
"
.
A sequence
is called a zigzag sequence if f*.J f
~
~
,£-v-l
J Z (J(,7)
~
c",L-"-l(1.t,l(')
The proof of Leray's theorem consists in showing that the
- 333 -
Y-T. Siu
correspondence (cohomol~~y
(where
class of f* •• )
f*,£
and
f£,*
~--->
(cohomology class of ft.*l
are the end-terms of a zigzag se-
defines an isomorphism between which agrees with the map defined by restriction.
HP(Ji',"!)
HJ(U.
'J) ->
and H1 (1(, "1)
This is shown in the following
three steps. al
Fbr every
f*,l c;;: Zl(l(,'1l
one can construct by the
Theorem B of Cartan-Oka a zigzag sequence with first term.
Likewise, for every
a zigzag sequence with bl
Ii'
fl
f*,t, f ~ .J-~-l
sequence, then
,* (0
.
•
as the last term.
~" J-l
0
Finally, since
it
follows that
f t,.
~
6(-gt_l.*)
• The proof of the "only
if" part is analogous. c)
The correspondence agrees with the restriction map.
because, if ~
ZJ. (11,7) ,
-
~~5
-
Y-T.Siu
the sequence
(f* .). ,I.
~O
f*
D,
•'"
f
•
v.(tO ,f)
C 52 and
from
(A(tO'flxvt' ,"J) lIl c1- l (A(tO'f)xl(,"J)
t.~ 67 A(t~ fl
~
1 l O C - (A(t 'f) x lJI, 7) linear over ([[t] 6(J(1,,7)
°
/lG(t.,?)]I,. ...,t,p
•
l,
2.!l
"J
is not
(J exist for
p
linear over ([ [t] tn -
on xl(}
such that
A(tO ,I') x VI.
~ c Max (111,11, ° ,117/1I(,t°, p), a,t,e
C is a constant independent of (When
of
,,-flat, for a fixed
l,,?,
°
f
t , and
to, ([ -linear
'f ' 'If,
sufficiently strictly small but maY not be and
C maY depend on
to
and
f.
If
t~ is not a zero-divisor of any stalk of "J , ~
can be chosen to be independent of ~.
fn .)
Fbllows the same line as in (4.1) except that (J.7l(bj
(Theorem B with bounds) is used instead of the Theorem B of Cartan-Oka.
- 33"7 -
Y-T, Siu
~.
Extension of Cohomology Classes
(5.1)
Andreotti-Grauert [1) proved that a
k-dimensional co-
homology class with coefficients in a coherent analytic sheaf
"J
can be extended across a strongly
k-pseudoconvex boundary
and across a strongly (r-k-l)-pseudoconcave boundary if codh "J ~ r So
We need the corresponding result with bounds.
we are going to examine one key point 0f Andreotti-
Grauert's result in such a way as can be carried over to the situation with bounds.
The precise statement of the situa-
tion with bounds is given in (5.2). Suppose
X is a complex space and
analytic sheaf on 6· X are open subsets. H
(t) I
Suppose, J.
~
"J
0 and
is a coherent Xl' DeC X
Assume the following.
l (AX (xln D),1)- 0 in case I ~ 1
Hl (4x D,'1) -> H (6 x(Xl '
f'\
D),
1)
is surjective in case 1. • 0
1+1 (4< D,'1) Let
X2 - Xl U D. (6 < Xl) U (6 x D)
is surjective for
->
1+ 1 (6 0, we can assume that are so chosen that,
H1'CH'CCP' 2 . • Q2:
contains
P (){ ~l
it 'i'i'
for some open polydisc
the Hartogs' f i~ure =
(p'
X (P" -
H" ,)
U (H2
xP",
> t} (cf, [1. pp.219-22a],.
show that the restriction map
It
suffices to
- 342-
Y-T. Siu
HN-P(Q2 • N(9)
1:
is injective. H" ,
P~ C lC •
j
J
U j
,
~
j-l"
~l
on
0
N-p TT P"j j~l
and
N-p n " TT P" j - HjP~~j+l
ltl
.
{ug 1 ) ,
1.%2
•
{ug2 ) ,
t. i ~
Ul '
fi
• .... t
Ul ' ..
HN- P (tt;.. N(9)
TT
" H. J
with
.... ,
(1 ~ j ~ N-p)
UN_pI UN_pJ
( i · 1,2)
is represented by a
ug i) (\ Ul n ••• n UN_ p ,
on
and
if and only if the holomorphic function
ug i )" Ul n ••• n UN_ p
function
gi
on
can be extended to a holomorphic
n ... n UN_ p
U l
where
GO
f.
1
~
L
f
Q,p+l' ••• ,aN·
-CI)
(i)
a +1 Z
zp+l' ••• , zN
of
4:
N
•
P
ap+l·· .a N p+l
is the Laurent series expansion in the last nates
N-p
H"
j~l
n
holomorphic function ~i •
~
HN-P(Ql' NIf))
Let
P XlTPp,X(P
An element
P"
Suppcse
->
Hence
L
N- P
coordi-
is injective.
When
- 343-
Y-T.Siu ~i
0 , the function
3
ZN-p(Ui , NO)
fi
when regarded as an element of
is the co boundary of CN-p-l(lJ., 1
0)
N
where
(i) hl ••• (N_p) h
(i)
gi
L
1\
01 •• • v ... (N-p)
ap",:l f: 0 ap+;_l ~ 0 ap+>l ;$ -1
It follows that, if
fl
is the restriction of
f2
and
fl
is the co boundary of some {s.
j
JO··· N-p-l
)
(a.,b.) C (-"',lD)
is a
if
function,
a. < a#< b# < b. ' p,q ~ 1 , r ~ 0 , and analytic sheaf on L!> x X
11)
'I' is strongly strongly
iii)
iv)
p-pseudoconvex on
q -pseudo convex on
< b < b. respect
to
the
pro-
1t: L!> x X -> L!>
Suppose
- 345-
Y-T. Siu
Si c C b.
is an open neighborhood of
are finite collections of Stein open subsets of
X , each of
which is relatively compact in some (but in general not the same) Stein open subset of
X.
a* < a l < a#
Then, for
and
b# < b l < b* ' there exist a* < a2 < al and b l < b 2 < b* (independent of which satisfy the following.
ii)
(a) (Surjectivity). bl
~
b
W~,
ii)
(T(i O )' .... L(i,,))
We can regard
case, we identify
Lemma.
•
-r;*-§
is
is a coherent analytic sheaf on
u:
as a refinement of Q .g Iu.'.
Fix a multi-index
struct a free sheaf system (a)
and a morphism
e
Jt/
0
R/
0.
(0. 0 )
0 •
-
(1(0. 0 ) }II
a
e
0
crPo.)
on
(0. ))
' crpo. 0
on
tt
e
$J 0.
0
There exists a sheaf-epimorphism Define
,
the identity map of
o
for
0.
0
C/. a
•
The construction is complete.
Vk1up
,
(0. ) 0 such that tto (0. ) 0 and obtain
->§IU.
(a )
from
J'pa)
We construct them
(a),
0
(.,(0.'
(10.'
It suffices to con-
is surjective, because'we can set ~ -
e
JI! -
there exist a free sheaf system
-§ -
for
0.
0
Cae p
- 3.50-
Y-T.Siu
(6·3)
Suppose
tn· (UiJ~.l
of a complex space on
zt.
X
-§
and
~ (§a' 'Y'~a) is a sheaf system
Introduce the cocha1n
• (where
is a collection of open subsets
At
is as in (6.11l
~roup
11
r(U.
-§ )
a '._ l;;: C (U'. l
c2(zt)
E, or ~
R ).
is given by
by
(~1+V.- ):~o with
Denne
Cl : c2(tt) - >
Let
cL+ 1 (lJI)
-352-
Y-T.Siu where
Define
by
(6.4)
Proposition.
With the notations of (6·3), if each
l«:. (~,7))
H~C' (ltl) -> H is an isomorphism for all 1.
is Stein, then the map duced by f)
9*:
Ui
in-
(This proposition can easily be proved by a spectral sequence argument.
However we prefer to present a more elementary
proof. because it can be carried over to the case with bounds,) ~.
(a) (Surjectivity).
Take
~
l, (;
z1 (V!,'~)
•
By
Theorem
B of Cartan-Oka. one can construct, by induction on 11 1H ~ (Vl,-") "'1+11 II ' "'-C I(,
•
,
(O~1I
d.
~
~
copies of
For
F' •
with power series expansion
L
f
~ (;;: N3 ~ ~i
6" 0.
~-l_>
Et:
1
Ir~+l,(l
""
Let
-> "+1 _>
1
"+ 1 "-1 -> Eb.+l -> F1.+1 ..• --> Eb.+l ->
359_
Y-T. Siu
be a commutative diagram of trivial Banach bundles and bunY dIe-homomorphisms on f::> with 6~ 6 - l = where a1 $
~
CL
'" '" "'I' "'2;
for some integers
(12
sequence of complexes.
°,
that is, we have a
Some statements concerning these com-
plexes which we will consider later are true only for "'I + c
~
~ "'2 -
'"
ing only on
n
c
where
c
is a positive integer depend-
(and other given numbers).
Such a restric-
tion will be clear from the proofs and will not be explicitly stated.
We will be interested in the behavior of these com-
plexes in a neighborhood of
°(,0)
So we will sometimes re-
by some suitable A(
place A
to (;;: A
For
and
d (;;:Pi:I~
°
~ t ,d] let ~"'[
denote the
vth cohomology sheaf of the complex
°
°
°
~ "'->(!)(Fh~-l )(t ,d) -> (D(Fh){t ,d) -> (D(E,.~+l )(t ,d) -> ••. When
to
when
d =
=
0,
¥~ [to ,d]
~
is simply denoted by 1f",[d] , and,
(CD, ••• , C1l) , it is simply denoted by
For
'"
~
FI", •
< P let
•
•
r p ,P-l r p _l ,p_2
For
When
to (;;: A
and
d (;;:N~
d. (00, ... , 00)
let
,HI Cl,t0 ,d'f
is simply denoted by
- ,6.0-
Y-T.Siu ~
Consider the following conditions
~
~
(E) • (M) • (F)
•
(Br n
(Quasi·- epimorphism with Ihunds).
(Et
stant
There exists a con-
C with the following property.
f, ~ ~(tO.d'f) ~
E,:
-
with
'f
0
a..P.t
.d.r
6;t,.
a. < P •
there exist
0
G;:
(f,)
Fbr
v 0 Ett(t.d.r)
with
~
6a.E,-O
such that 1· )
~ r p+ 1.P
f, -
H) Hi)
~
(M)
stant
'f
0 a..P,t .d.p
1f
0 a..P.t .d.p
(Quasi-monomorphism with Ihunds). C with the following property.
" 0 ,d.p) t,G;: I\.(t ~.
and
and
~-l
rpa.... - 6p
and? G;:
?
F,i"-1 (t 0 .d.f)
are linear over
There exists a conFbr
a. < P •
satisfying
there exists
'fa..P.t 0.d.f «..,. ? l"''- E'"-l( b.+l t 0 .d." ) such that
i)
«: [t]
" 6a."0
·- 3,6.1- .
Y-T. Siu
'f'o:,~,t0 ,d,p is bilinear over ([ t] •
iii)
(d and
(Finite-dimensionality along the Fibers). Fbr to c,; A d ~-l(tO,d'f) ->
~
1 ...->
F~-l
O:,t0 ,d,p
"-l( 0
~+l t
->
1 ->
F~
O:,t 0 ,d,p
~+l
~
0 (t ,d,p) ->....
1
->
1
1 ...->
0
~(t ,4,,0)
~+l
->... F 0 O:,t ,d,p
1
~ 0 "+1 0 ,d 'f) -> ~+l(t ,d,p) -> ~+l (t ,d,p) ->...
where i) ii)
the composite vertical maps are
~ 0
O:,t ,d,p
---> ~
~+l,o:
(to,d,p) factors through a Hilbert
0:+1
space iii) iv)
v)
the middle row is a complex of Frechet spaces ~
.
dim H (F 0 ) < '" 4: O:,t ,d,/, H"(F' 0 ) --> H"(F' 0 ) O:,t ,d,p o:+l,t ,d,p
is bijective
-362-
Y-T.Siu
(Bl: (Finite Generation with fuunds). a)
~
Im(~
° t E: A. °
--->~
«,t
for
(1 ~ i ~ k)
~
/(\
is finitely generated over n~tO
0)
a,+l,t
and let
A be the
~
N o ' . Then, J:or f a.+ 1, t
there exists a constant Ii:
S c;;:~ (to 'f)
~in'>.l < ,....
a.+l,tO
7 c;;: such that
satisJ:ying the J:ollowing.
C
f
bl e ongs to
a (1), ... , a (k)
of
sufJ:iciently strictly small,
6~E,.
with
D O-submodule
n t
°
such that the image oJ:
A , then there exist
c;;: r(A(tO ,f), /J)
(Bl~
M:>reover, by the condition of factorization
through a Hilbert space given in
"
(F)
,in the statement of
-378 -
Y-T. Siu
(B)~
we can choose
a (1)
and
?
so that the map
is linear over ([ (t] • By induction on )l
(E) , (M) HI, (F{ ~ )1+1 ( E) (M) ,and
,
on
n
n , it follows from (704) that
= } (B)'> •
n
Under the assumption of
~
(F) , we are going to show, by induction
that the natural map
e
.f!:.c?!!!
to
is an isomorphism. that
to. 0
)l
(F)
•
n ~ 0
m~
trarily
e
follows
immedia~ely
follows from
»
(E)
•
from Suppose
for some
,t,*
(Ef
-~ ~+l[d(l)])
~~[d(L)]
by ~
which by (7.3) is independent of the choice of
. The above condition
By
induction on
n , condition
ural maps from W'[d(L)]o for
I < £
~
ii l
n
is satisfied i f ii)
~
A = H(t+ I
is satisfied if the nat-
to W~[d(L-Il]o are surjective
(A by-product of this surjectivity condition
is that all the constants in the proof of (7.2) are independent of
p , because of the last sentence of (7.2)).
From
the exact sequence
(where
0-
is defined "by multiplication by
tt l, we obtain
the exact sequence ->Fi:(d(tl l _>~:(d(Ll) _>At:(d(l-ll) _>flrl(d(Ll
j
_> •••
Hence the surjectivity condition just mentioned is satisfied if
9f6 • 0
for
)I
< ,. < v + n.
Of course, in general, this
- 381-
Y-T. Siu
last condition is not satisfied. (Ej#. (M)#, (F)#
for certain
Under the assumption of
#'s, we are going to modify
the complexes so that the new complexes satisfy this last condition.
.•• __> L"-l __>
-/ > L"+l __> L' __
...
is a complex in which the maps of (holomorphic) bundle-homomorphisms.
Let
__> L"-l __>
L~
~
__>
L~+l
__>
~
... --> ~-l _> ~ _> E(+l _> be a commutative diagram of bundle-homomorphisms such that for Note that, to have define
a
by any open polydisc re-
latively compact in A)_ al
The complexes comp1 exes
bj
The complexes
~ cl
'""Wo-
satisfy
The complexes
~
satisfy
Fh
satisfy
~
(Et
and
(E/
-
satisfy
(M)'
~ the complexes
satisfy
(Ft
===9
(M)'
~
the complexes
(F)~
Statement cl is clear-
and
satisfy
Let us prove bj:
Suppose
_ 383~
Y-T. Siu •
for some
,
t, Ell f
t
ni tion of
...
~
.....
G;:
Eil- 1 (t0 ,d,p)
N~
~-l
6tt ,6~
and
tt
1+1
By applying property
equation, (since
~
"
~ 6 p (-lll, • Ej
of the complexes
6~:i~:if ~ LP _> LP+l_> .. , _> L8 ->'0 of trivial vector bundles of finite rank on ~(fO)
pO G: iR~
(where
and ~(fo) C A) in which the maps are holomorphic
bundle-homomorphisms and there exists a commutative diagram
_ 386 -
Y-T. Siu
I:
I\
a-~
Il-
o-d+ l ,'J
~ -> ~+l
r~"lta.
of complex-homomorphisms on
.
-of rr-
Proof.
A P
P
~
s
+
1 , there exist a complex
ill' LP
_>
.
P
LP+ 1
;;1'+1
_> ... _>
PL
S
->
°
of trivial vector bundles of finite rank on some L\{po) a commutative diagram
such that the mapping cones
for
I' ~ )I ~
s •
P
p~ of
0"""
P ..
satisfy
and
- 387
-
Y-T. Siu
The case
#.
s + 1
is trivial, because one Can set
L• • To go from the step # to the step # - 1 , we ob# serve that, by (7.5 J, #Fh has properties (E)#-~(M)#, and (F)#-I. has property (8)#-1. By (7.2), n
°
ih
is finitely generated over shrinking t,(I) ,
n®O'
One can find (after
pO)
... , A generate
whose images in #-1
L#-I
A over nOO'
• ~(DO)X((k I
#-1 L~
#L~
,/
#-1
Let
be defined by
t, (1),
••• , .;(k J •
u-"-I
#-1 a
~
Define
Define
-
388 -
Y-T. Siu
Then the complex
/L-l L·
requirement. Q.E.D.
and the map
IT·
/L-l
0.
satisfy the
-
389 _
Y -T. Siu
§8.
Right Inverses of Coboundary Maps
As in § 7. suppose ••• _ >
~-l
~
_>
v
~+ 1
6
_>
1 _> S
a+l,a.~
for some sufficiently strictly
p ,we aSsume that 'l!~ and ~~+l are defined for
( 1, ••• , 1)
"P and, from now on, x (l
maps for that particular
(8.2)
=
",p+ 1 and '" (l
deno te the
f·
We are going to define
such that (* ) on
lO(~-l) fur
~(tO, pl C f::> and
f, G: ~(tO 'f)
with power series
expansion
define
where
~A
is regarded as an element of
~(l, ••• , 1).
Be-
-
291-
Y-T. Siu
cause of the norm estimates for
:r.P
;.p+l
~~+l' ~~
• we have
'f~E,c;;. Eii~}(tO,pJ To verify that this definition can give us a sheafhomomorphism. we have to prove its compatibility with re. , f 0 Let strictions. Suppose A(t 'I' ) CC A(t '1') p"
t
E, c;;. Fb:(t '1')
be the restriction of
~.
Then we have the
power series expansion I
E,
Hence
It follows from the Il:-linearity of ~~+l' !Ii~+l. &~ norm estimates that
equals the restriction of 'Y~ l.,
I
,
to oC:>(t • I' )
It remains to verify the identity (*). O
A(t '1') C CA. take
For
and their
_ 392 -
Y-T.Siu Let E~
r At
Or _ (t - t)
P
Then
E(E A t;i. and I by definition of
Since
) (t A A-r.t ~
oA - t ) f
'£ ~ •
~~+l' ~rl both are linear over ~ll+l)
to
and
Fix two integers II
9lI~+1)
is an isomorphism.
Let (p ~ >! ~ s) •
Assume that 9>fp.
... ," a-I
""P+ 1 •
'1
are coherent on
A.
Let
8~-1
Im(I!l(Et:- l ) ~>
(0«))
Ker(O(~) 6~ (O(Eh+ll) and. for any open polydisc Q CA. let
H~ (n) Suppose that there
exists. for every
a • a sheaf-homomor-
phism
such that
We are going to prove the following two statements for any open polydisc
n CA.
_ 394 _
Y-T. Siu
Let
Consider the following two statements.
!f(n, S:) -> !f(n, S:+V_P+2)
1)"
k
~
.
1 •
~
!fIn, 't~) -> !f(n. ~~+"-P+2)
2)"
k
~
has zero image for
has zero image for
1 •
===? • exact rows
First, let us show that ibe commutative diagram with
1)
2)
p
for
...
~
v <s
N
S·
0->
~
1
(.)
"
~
~"~
->
~~
J
- > W->O
~
II J
o - > S~+"_P+2 ->~~+"-P+2 - > '#->0
yields the commutative diagram with exact rows
!f(n,e:)
1 •
!f (n, S ~+. -p+2) Since
9f
~
->
!fen, ~:)
!f(n,9f")
1
II
~~
- > Hk(Q. ~ ~+"-P+2) - >
is coherent. !f(Q,~')
i ~~a+~-pt-3 -> (!)(~+~-P+3) ->Ea+~_p+3 - > 0 "+1
yields the commutative diagram with exact rows
Jf(n,(!)«(»)
Jf (Q,B~+1)
->
1
1
':/
Jf "+1 Jf(Q,(9«(+"_P+3 ») - > (Q,lla+~ -p+3) - >
e
•
0
(k
~ 1)
and
factors through the map
Now we are ready to prove
o~ .,
Jf+l (Q, 1.~a+~ _P+3 l
Jf(n, (9(~+"_pt-3)) ~
The result follows from the fact that rr
9:'a) Ia-
Jf+l(Q •
->
i)
together with the vanishing of
for
k
~
1 , implies that
for
p
~
y < s
and
2)
for
p ;;i y ;;i s
and
2 )..
..
l)p
==> for
holds. 1)..
and
ii) •
The existence
Jf(n,C9( (+2)) Since
for any
p~"<s.
II~
~
~
, we have
0 2)
~
1)
"The diagram (* )" s
yields the commutative diagram with exact rows
-
396 -
Y-T.Siu
r(Q, ~~)
~
(0, By
l)s.
->
r(S"l. ~S)
1
1
II
'f~+s-pt2) -> r(Q, 'N ) -> ~(Q,e~+s_p+2) T
S
T is surjective and
i)
>
p.
tence of
The case
e~+l'
S· P
Suppose
To prove
follo:ws.
we have to distinguish between the case s
W-(S"l, 13 ~)
->
s
~
p
ii),
and the case
follows immediately from the exiss
>
p.
(t)~;i
The diagram
yields the commutatiYe diagram with exact rows
->
Suppose an element r(Q. ?fs).
Since
1-
(n.
't~)
is mapped to
Then
factors through the map
it follows from
,uch that
E, of r
2)s_1
Hl (Q
that there exists
0
s-l) 7a+l
6) t
in
- 391 -
Y-T.Siu
Hence
ii)
is proved.
(9·2)
Suppose
E;;
is a sequence of complexes of trivial
§7.
Banach bundles as in that the complexes p ~ " ~ Max(s,p+n)
( By
Fix two integers s;? p satisfy (E)• , (M) '+1 ,and
(7.6) (after replacing
A ( pO)) there exists a complex of finite rank on A
f:!,
.
Assume .
( F)
for
by some
L' of trivial vector bundles
and there exists a commutative diagram
O-~+l
\ ~
(--> r~+ 1 lei
of complex-homomorphisms on f:!,
R'
I:L
for
0
f
a-a.•
such that the -mapping cones
satisfy
p ~ y ~ s.
by some
(+1
By the results of §g,
(after replacing A
A(PO») there exists a sheaf-homomorphism
such that
on
V(~-l) , where
are as in (7.5).
-
398 -
Y-T.Siu
fur any object derived from
E~"
we put a ~
on top
of its symbol to denote the. corresponding object derived from
E;; d ~I\I~
let
For
~.6 and
to
e~(tO ,d)
~ 1lI~ , 'f~ induces a sheaf-homomorphism
d
from
B~(tO,d): to
to I;: A and
fur any open polydisc Q C A and for
-> (9(F&)
Im((Q(Erl)(tO,d)
C9(F&;~) (to ,d)
(to,dl)
such that
°
~p-lep 60.+2 o.(t ,d)
°
~ on 5~(t ,d).
By applying (9.1) to the complexes of bundles
C9(~)(tO,d) , we obtain the following.
associated to
°
-
°
~ t ,d], " ' , Ws - 1 [t ,d] ~p[ to ~ A and all
i) 11)
°
~y IIa. (.Q.,t ,d)
d ~ f::I~
are coherent on with
d n ';
~o t -> r(n, ,H
Ker(~(n,tO,dl ->
°
<Xl
,d] l
,
A
If
for all
then, for
p ~ y ~ s ,
is surjective
r(n, 1fo[tO,d]))
°
-yO) C Ker (~~ ~ (Q,t .,d) -> 1Ia.+»-p+3 (n,t ,d)
n
for any open polydisc d
Since
n
.;
<Xl
C A and for all
to ~ A and all
•
(9(~)(tO,d)
is the mapping cone of
-
.399 -
Y -T. Siu
and rest, i9(~) (to ,d))
is the mapping cone of
we have the following two long exact sequences:
•°
°
->~(l(t ,d]->~(l(t ~~ ,d]
_>PIi"+l(la(L) (to,d)) ->9J:+ l (tO,d]->
->
°,d) - > N'/b.(S2,t°,d)
~(Q,t ~
- > "+1 (f2, to ,d) _ > ••.
- > H"+l(r(Q, (9(L) (to ,dl))
From the first long exact sequence and
(M)~+l
(p ~ ~ < s) ,
it follows that 9J. P (O(L)(tO,dl) ->PI\P(tO,d] ->~P(tO,d]-> a/P+l(ID(L)(tO,dl) _ > .•• _>PI\s-l(tO,d]·_> ,#S-l(tO ,d] - > i\l-s (OlLllt O ,d)) - > Ws ( to ,d]
is exact.
From the sharp form of the Five-Lemma, we conclude
1£ ~P(tO,d],
the following. 4 for all iJ
for
to '- A
and all
" ' , ~s(tO,d] d
are coherent on
'-N~ with d n i
°
Q)
°
,
~
~ ~ ,d]) p;;:'Ji ~ s , l\.(.Q,t ,dJ ->r(Q,9J.(t
is
surjective ii)
for
p
r(S2, .J(tO ,d]l)
C Ker(I{ (U,tO ,d) - > ~+~-P+3 (S2,tO 'dl)
for any open
-
400 -
Y-T.Siu
polydisc Q d
n
f
Ol
•
C
A
and for all
to (;;:
A
and all
d (;;: N~
with
-
401 _
Y-T.Siu §10.
Proof of Coherence
(10.1)
~
SUppose
is a sequence of complexes of trivial
Banach bundles "as in §7. s;?p+n.
Fix two integers
Assume that the complexes
(M)Hl, (Ft
p, s
.. R'
r;;: ~ ,
We use the notations of ·§9.
for
aj
d r;;:N~
and for any open polydisc
a
•
Ker ( ~11+1 (n,t a ,d)
C
we
p:;: 11 ;'i s,
ct
~+l(n,tO,d))
• (n,t0,d)) -> ~+l
1m (~ (n,t ,d) b)
( E), •
satisfy
We assume the following two condi tions for to
such 'that
-> HI
Ker ( Ib.
HI
~
a
(n,t-a ,d) )
(n,t ,d)
->
are going to prove by induction on
ent on
~
Assume
n
~[tO,d]
p:;:. < s .
for ~
1.
The Case
n
•
that
n =
a
901-
is coher-
is trivial.
The induction hypothesis states that
is coherent for
p;'i 11 < s , to
r;;: L>,
and
d r;;:N~
Since coherence is a local property, by (9.2) w.thout loss of generality we can assume the following for any open polydisc Q d
r;;: N~ i)
with For
d
n
'I
<Xl
C L>
to
r;;: ~
and all
•
p~)I<s,
surjective.
and for all
is
_ j02 _
Y-T.Siu ii)
For
p < ~ < s , Ker(~(Q,tO,dl ->r(.R,91[tO,d)J)
°
j ~+'-P+3 (U,t
°,d))
As in (7.3J we use the following notation.
Fbr
C Ker ( 'hj( Sl,t ,d) - >
(10.2)
Lemma.
divisor for
Suppose
p:;:
)/+1 tm ...!f n 1+0
11
< s
and
tn
m ~ 1 , let
is not a zero-
R n+ and p is suffif < P' I in
ciently strictly small, then for
to ~A(fJ
and
is contained in
!!22!.
Consider the following commutative diagram
"+l(A(p))
i H:(A( p' I)
which comes from the commutative diagram
"~No
-
403-
Y-T.
Siu
->0
where i)
t-
is the natural map
iiI
a
is defined by multiplication by
iii)
b
is defined by multiplication by
iv)
c
is defined by multiplication by
° A+m
(t n - t n )
m (t n - to) n A (t n _ to) n
Let
be defined by multiplication by
be induced by
"+1
By
r a + l,a
(t n - t~)
( B)~+l
m
Let
(applied to
A = 0),
Since t n f sufficiently strictly small, Ker J C Ker g m9f"+ 1 , it follows that is not a zero-divisor of n t
Ker'f C
(* )
for
°
When
t~
=
Ker
° , both
y
1
phisms and, hence, (*) trivially holds.
and
'If
are isomor-
for
_ 404 _
Y-T. Siu
One has
a
because
h = gf.
I
gf1;
a =9
It follows that
-rB
~
The following lemma is in codimension
~
1.
a
=
I
a ,
h"t
a and
1m
eC
rmo- .
strengthened form of (10.2)
Its proof is similar to that of (10.2).
Its consequence (10.5) will be needed only for the proof of the coherence of 9t p •
(10.3)
Lemma.
d.e+l' ••• , d n ~ to
~
n.
12.!:
s;;- p+-2 , 1 ;$ £,
Suppose
N,
nee A
A ~N*
~+3 (Q,tO ,e A)) is contained in
Im(~+3(n,tO,e"') -> ~+3(Q,tO,eA)) Proof.
Since ¥p+-l[t O,e"']
is coherent on Do, by consider-
ing the increasing sequence of subsheaves consisting of the kernels of the sheaf-homomorphisms ¥p+-l[ to, e"'] defined by multiplication by conclude that there exists not a zero-divisor of
(t
m~
l
~O
- tJ)m
as
such that
(t - t~·lm a/-p+l[ to ,e"']x L
_> ~p+l[ to ,e"'] m varies, we t
l
- tJ
for
is
x ~ 52 •
-
4c05 -
Y-T. Siu Consider the following commutative diagram p 0 ClO 0P ,,0 A T p+l 0 H..+ (>i ...
By (10.1) i) and ii), the image of
• ( " ),t 0 ,d A+m ) !\.+o-P+J A(P
in
belongs to the image of
r(A(P " ),
A0) HoJ ( A(p" ) ,t 0 ,d X+m)
1H;
t
in
~ (" 0 A+m) !\t+~-P+J A(p ), t ,ct
By (10.2), for
ciently strictly small, the image of ~
('
0
A
!\.+~-P+J A(p ),t ,ct )
~
!\t+O_P+J(A( f
k
E b.~.
'0
•
in
1. 1.
By
(B)~, for f
i=- 1
b.S. 1.
1.
in
j, - \ ) 1(+1(A(
1.
in
1.
P')}
t
A
,ct ) •
It follows that the image of
I
%+"_P+J(A( p))
i~l
k E
), t
i=l
suffi-
belongs to the image of
i(A( 1" in
k E b.~.
I
p
belongs to
sufficiently strictly small, the image of equals the image of
k E
i=l
c.l.. 1
1
in
-
413 -
Y-T. Siu
(1 ~ i ~ k) •
Since
and
it follows that
Since
A is arbitrary,
Hence
R
(III)
is coherent on ~(P) ~ ~
The case
p.
The only difference between this
case and the previous Case is that, when
v = p , (lO.lj ii)
no longer implies that, for open polydisc 9. C ~, to G;: ~ and
d G;:~~
with
d
Ker(th (n,tO ,d) (. . ) {
When
n
n
1 , (**)
can assume that
OJ
-> C
=
f
,
r(n, W~[tO ,d])) ~
Ker(~ (n,t
0
,d)
~
0
-> ~+~_p+3(Q,t
is clearly satisfied for >I n ~ 2 Since s ~ p+n , '>.,p+l ~
=
P
,dl)
So we
is coherent
- 414 _
Y -T.
on .6
by (II).
Siu
Now we modify the argument of (II) to avoid
the use of (**) when
~ ~
p.
We pick up the argument of
(II) at the point where k
-
"h
~ r(A( F
1: b. E. i=l l'i
p"
By (10.5), for h
•
depending on i)
h
,
->
t
•
sufficiently strictly small, there exists such that
h
as
CD
), "" 0 Wp )
_>
h
CD
belongs to
ii)
By
,
(B)~, for
p
sufficiently strictly small, the image of
~
k . 1: bi';i
in
1~1
•
!\.+3n_2(A(f l)
~+3n_2(6(f'))
equals the image of
k 1: c. ~ .
i=l
1
1
for some (l5- i 5- k )
,As in (II), we conclude that
Since
h
• ->
trariness of
CD
h
as
h
->
CD
,
it follows from the arbi-
that k '-0 1O or(6(f ),,,-) n t
.
in
-
415 _
Y-T.Siu
Hence
R. is coherent on
(10.7)
A(
f ) .
Proof of Main Theorem.
So (:;:
Fbr every
S
one can find a proper holomorphic
map rr with finite fibers from an open neighborhood
So
into an open subset
G of en
We have
It is easy to see that an analytic sheaf;1
R°o-*~ is coherent on
ent if and only if M is an
IDs ,s-module
for some
codh
where
roS,S M
~
codh (!)
on
U
is coher-
G. Moreover, if
s (:;: U , then codh (!)
M
n ..(s)
M is regarded naturally as an
particular, if
U of
M;;- n , then
S,S
n([).. ( s ) -module.
M is a flat
module (see (A.8)-(A.12) of the Appendix).
In
nlD..(s)-
Hence for the
proof of the Main Theorem we can assume without loss of generality that
S ~ A
and
"J
is
n-flat.
Moreover, in the
course of the proof, any replacement of A
by A(P) (with
P(:;:1R~l does not result in any loss of generality. In (6.5) we have constructed a sequence of complexes
Et:
of trivial Banach bundles on
A.
By (5.2) and the re-
sults of Andreotti-Grauert [1] these complexes (El~ ,(M) >'+1 , ( F ') , and (10.1) a), b)
By (10.6),
, 11
1s coherent on A for
for p ~ "
~
P~ ~ 0(
"
ijx
"i
"Ix i
o.
linear function
fur any f(t,z)
(i - 1,2)
(i - 1,2)
00, (t ,z j ~ such that
~}
Xz -
.
Xl ' there exists a
o 0 f(t,z ) = 0
and
f(t,z)
- 419 -
Y-T. Siu
is nowhere zero on
Xl.
fined by multiplication by Xl • Supp Ker'8
and
"if -> "] on Xz be de-
Let 8: f
Since
Supp Coker
e
f
is nowhere zero on
are subvarieties of the
Stein space
and hence are Stein.
From the cohomology sequence of the
short exact sequence
e
0-> Ker a -> "J-> Ima -> 0 • we conclude that }>.
.,.
e
induces an isomorphism I 2.,
(R n ••) 0 t
->
( RI n. 2 (Im e) )
t
o·
Consider the following exact sequence
o2
(R
n.7)
t
0
E,
-> I 2.,
->
(R n •
.Tl
t
0
coming from the short exact sequence
o -> Im e C-> "J -> We are going to prove that ~ show that diagram
Ker
7•
O.
Coker a
-> 0
is surjective.
It suffices to
Consider the following commutative
- 420 _
Y -T. Siu
1 2
".1)
(R
a2
0 t
->
1 2.,.
(R ".,) 0 g
I
t
" where map.
62
al'
are induced by
a2 •
Since
,81
Xl
Hence
,,: X
1
on
Ker
-->
and
8
= 0
is the restriction
is an isomorphism on
By applying the Main Theorem to
together with the function '1"
D
X, we conclude that
El 2
g
is an isomorphism it suffices Since
is an isomorphism.
the map sheaf
C,
and
Ker $2 • 0
show that
to
? c,
e
and .;
g
and the
is an isomorphism.
is surjective.
Since
Supp CokerE
is Stein, the image of the natural map
0 0 over n+ N({) 0 0 • Since E, (t,z) (t,z) is surjective, by Nakayama's Lemma, the image of the natural generates
(Coker a)
map
•
o 2.,
..,
(R ".,) 0 t
generates
'} 0
neighborhood phism cr: n+
0
(t ,z )
P
U'
•
of
--> "}
- > • (t 0 ,z 0 )
By letting
to on
in
zO
vary, for some open
D we can find a sheaf-epimor-
-
42} -
Y-T. Siu
By applying the same argument to
some open neighborhood
U"
of
Ker to
in
7- ,
instead of
(T
U'
for
we can find a
sheaf-epimorpnism n+
T :
rtf!
->
Ker S is said to be stronglY
A holomarphic map
I-pseudoconvex ~:
Siu
cf
if there exist a
X ---> (~,c.)
C
(-00,00)
function
and a real number
< c. .uch
c
that i) ii)
"I
{'f~ c}
is proper for
< c.
If is strongly I-pseudoconvex on
(When the additional condition c < c.
c
{f:;: c}
~
[ 'f> c#}
{'/< c}- for
is added, this definition agrees with a special Case
of strongly Fbr
(p,q)-pseudoconvex-pseudoconcave maps. ) f c;;: r(X,lOx)
image of the germ of
COx ,x
f
xc;;: X let
and at
x
fIx)
denote the
under the natural map
-> (Ox ,xl"4..x ,x
11: •
We are going to prove the following result concerning blowing down.
Ii'
n: X ---> S is strongly
is Stein, then
c;;: r(X,®X)
S
X is holomorphically convex (that is, for
every discrete sequence f
I-pseudoconvex and
such that
{x.}
in
X there exists
f(x.) - > 00
have the holomorphic convexity of
as
v ->
00).
Once we
X, we can blow down
X
by the Reduction Theorem of Remmert (whose generalization to the unreduced case can be proved in a way analogous to the reduced case [30)).
- 424 _
Y -T.
Siu
The result on blowing down will be proved by using the finite generation of
(Rln;~) s
for
s G;: Sand
c# < c < c*
Fbr
such a finite generation, it suffices to consider the Case where
S= A
and
s
=
O.
Strictly speaking this finite
generation does not follow from the Main Theorem, because in general is not
n-flat.
Ox
However, this can be obtained
from the argument used in proving the Main Theorem.
In the
proof of the Main Theorem, the flatness is needed to get a sequence of complexes for for
p
~ ~
< r-q-n.
gettin~ ri~ht
(
satisfying
~
(E)
,(M)
~+l
~
, ( F)
Such a sequence of complexes is needed
inverses of coboundary maps (§8) and
global isomorphisms (§9) which. in turn. are essential for proving the coherence of the direct image sheaves under consideration. HOwever, when only the finite generation of the stalks of the direct image sheaves under consideration is needed, it Can be proved directly by the arguments of §7 without using the sequence of complexes, 'provided that n-flat on
['I' S is a strongly
(Rl,,;COx) s •
(12.2)
Lemma.
holomorphic map and
S is a single point, then
l-pseudoconvex X is holo-
morphicallY convex. Proof. in
Take
{'f > c}
{x,,} '.
c# < c < c. Let.j
and take a discrete sequence
be the ideal-sheaf of the subvariety
The exact sequence
yields the commutative diagram with exact rows r(X,COx)
{X.,}
7 rex, HI (Xc , D be the natural projection. Al c:;: IR and a function
~:
IR --> IR
~
~
is strongly plurisubharmonic on some open neighborhood of A ~ Al
when Proof.
and
For a xC:;:Q
and for
rf
B;;-~(A)
function
on an open subset Q
h
a c:;: ([m , let
and
m
denote
1J
J
1
d(h;x,a)
of er m
a2h) (x)a.a. _
(
E
L(h;x,a)
i,j=ldz.dZ. and let
K
denote
m
Clh (x)a . • i=ldZ 1 E
i
Let in
'i = 'i' f
<e n+ N
a c:;: S
•
and
y=
y. f
and let
S
be the unit sphere
It suffices to show that for fixed
there exist
,
A (x, a I c:;:
IR
and
,
xC:;: K and
B (. ,x, a) c:;:
IR
such that,
- 432 _
Y-T. Siu I
A;;- A (x,a)
if'
and
I
B;;- B (A,x,a) • then
Direct computation shows that
+ [
I:
i 0 i
1
on
has an open neighborhood
~~N ~
on
n(Kj • 0
which
by a ( nonproperj holomorphic map n+N n
--> a:
with the natural projection 0 •
K
_ 434 _
Y-T.
(12.8)
Theorem.
If
n: X ---> S
convex holomorphic map and
S is
is a stronglY
Siu
I-pseudo-
X is holo_
Stein, then
morphically convex. Proof.
By (12.4), every point
neighborhood
s
G
S admits an open
U such that
is holomorphically convex for some
?:
of
c# < c < c..
---> R be the Remmert quotient of
i) ii) iii)
iv )
Let
(that is,
G
R is Stein
7
is a proper surjective holomorphic map
R07.(QG = (DR
?-l(x)
is connected for every
x r~ R) •
Since no compact irreducible positive-dimensional subvariety of
G Can intersect
It> c#J
by virtue of the maximum
principle for strongly plurisubharmonic functions, ? maps G n ['f> c#J piece
biholomorphically onto its image.
n-l(Ul
n
['1' > c#J
and
R through?
Hence we Can and obtain a
~
c~mplex
space
R
holomorphic map
n R
induces a strongly
---> U
Since there exists a strongly
plurisubharmonic function on
piece together all the Remmert quotients form a complex space convex map
,
,
n : X
X
---> S.
R is
R, by (12.4),
the uniqueness of Remmert quotients, as
,
I-pseudoconvex
n
s
Stein.
By
varies, we Can
R together and
induces a strongly Since, for every
s
~
I-pseudoS ,
-
435
.
Y-T.Siu
'1 ,., (n )- (U) = R
, X
is Stein, by (12.7/
there is a proper holomorphic map rr
X
follows that
Theorem.
(12·9)
i)
X.
from
X
to
Suppose
," X
-->
Then for
>!
> =
S
,
X
,it
?
is a strongly is a coherent analytic
1
~
R n* '] is coherent on
S
R~ n* 1--> R~n~ '1 is an isomorphicm for
ii)
Since
is holomorphically convex.
l-pseudoconvex holomorphic map and sheaf on
is Stein.
c#
< c < c*
H~(n-l(U),1) -> r(U,R~n"l) is an isomorphism for
iii)
any Stein open subset ~.
Let
,
,
n , X ,~
U of
S.
be as in the proof of (12.8).
a- is proper, by Grauert's direct image per case), for
JL;;
° , RJL~ "}
theo~em
is coherent on
Since
(for the pro-
X'
and
W of
X
, is an isomorphism for any Stein open subset Since Ir for
~ ~
maps
{'f > c#}
biholomorphically onto its image,
1
By applying Grauert's direct image theorem (for the proper
case) to
I
i
n' Supp R cr* 7, it follows that, for
Y RO(n' )*(R a-*'1)
is coherent and
>I;;
1 ,
-
436 -
Y -T. Siu
for every Stein open subset
U of
S.
It follows that
-
'i~7
-
Y -T. Siu
§l).
Relative Exceptional Sets
(1).1)
Suppose
variety
A of
above
S
n: X ---> S is a holomorphic map.
is said to be proper nowhere discrete
X
nlA
if
A sub-
is proper and every fiber of
itive-dimensional at any of its points.
A subvariety
X which is proper nowhere discrete above exceptional relative to
nlA
is pos-
A of
S is said to be
if there exists a commutative
S
diagram of holomorphic maps
!Ii
-> Y
X
S
such that
f
i)
is proper
every fiber of ~lf(A)
ii)
~
iii)
X-A
maps
o .
R ~*lDx
iv)
has dimension
biholomorphically onto
~ 0
y- (A)
COy •
3
The following result on relative exceptional sets is a consequence of (12.8).
(1) .2)
and
Theorem.
Suppose
A is a subvariety of
crete above
S
.
Then
and only i f for every
A
n: X-> S is a holomorphic map X which is proper nowhere dis-
is exceptional relative to
s
L
T
equals
Let Sl
Stein open neighborhood of points
x,y
of
k o ->""'X2 ,x.... "" I.
X.
ish on Sl
and the natural projec-
~
be a relatively compact in
S.
Take two distinct
Consider the two exact sequences
... k ,;2 k ->W.X ,x.. [, - > (""'X " """X x )@.c->O,
where w.,X ,x,y = IoMX,x f\ ly large, beth
into
"
-
= IO(L* I .
~
T of
~ of ,,-l(T)
there exists a holomorphic embedding
IF'N" T
is
roM
X,y •
By
Rl"*(-x'X,y~.ck)
(14.2), for and
k
suff'icient-
k Rl,,*(-i,[email protected] )
Van-
It follows that the two maps
are both surjective on Q.
Since
Q
is Stein, it follows
Y-T.Siu
that there exist enough holomorphic sections of
n-1(Q)
to construct an embedding ~.
L*
over
- 4012 -
Y-T.Siu §15.
Extension of Complex Spaces
(15.1 ) Theorem. Suppose X is a complex space with 19[n+l] Ox and S is a Stein space of dimension ~ n X Suppose n: x-> S is a holomorphic map and ~: X ---> (a*,b*)
C
function such that a* < a < b < b*.
(~~,~) n
II a $
iv)
'f $ b}
n:
X--->
n =
-X
the restriction of
n
X
S such that
X which intersects every
';Ix
Proof (sketch).
is proper for
-
X is an open subset of branch of
iii)
is a strongly Dlurisubharmonic
Then there exist uniquelY a Stein space
and a holomorphic map
ii)
.
to
-
X-
!'f'>aJ
First we prove the case
is proper for
n = O.
It follows
from CO[l] = (!) ~o X X that, outside a subvariety of dimension , in X, codh 19X ~ ) . We can choose a* < a < b < b < b* such that let
codh 19x ~)
X~. Ic < 'f' < d}.
ideal-sheaf on
X
on Let
such that
the following commutative diagram
Fbr arbitrary coherent Consider on Xab'
-
443 _
Y-T Siu It- (X~ ,.1)
.. I
VI
It-(X~' ,.1) coming from
0->..5 ->
Ox ->
f[VS -> 0
By the results of Andreotti-Grauert [1],
Since
J
that
a
cr is an isomorphism. b' = I()X on Xa , T is an isomorphism. It follows is an epimorphism. From the arbitrariness of ..5 ,
we conclude that there exist a holomorphic map
such that, for some
a < a i)
,
R is an epi-
codhRM when
M is
R is regular of dimension
codh M agrees with the maximum of
n-l
n ,
such that there
exists an exact sequence
O->R
Pi
Suppose space
X.
- > ... ->R
7
codh
Sk(7l
"J
~ k
->R
Po
->M->O.
is a coherent analytic sheaf on a complex
We define
codh"J x
Let
PI
as the function
-> codh'n
"1.
'VX,x x
denote the set of points of
X where
-
447 -
Y-T. Siu
(A.2j
«:
n
o
Lemma (Frenkelj.
,
D
and ~
A is a subvariety of
X and
.
7
is a coherent
Then ~~ "1 = 0 for A H (X-A,?) is bijective
~ r -J
and injective for
~ =
r-d-l •
We can assume without loss of generality that an open subset of «:n.
When
'1
=
lJ
and
X is
A is regular, i t
is a direct consequence of Frenkel's lemma. and, when the singular set
,
A
of
A is nonempty, it follows from the
long exact sequence
When
"1 f
n([) , we use a local finite free resolution of
? .
Now we define relative gap-sheaves with respect to a subvariety.
Suppose
A is a subvariety of a complex space
- 418 _
Y-T,Siu
7 C1
X and
the sheaf
are coherent analytic sheaves on
7 [A]1 U
X.
Define
by the pre sheaf
1-->
1l I(sIU-A)
Is {s
~
r(u,11/ (sIU-A)
~
A of
(A.6j
Proposition.
coherent and ~.
'1[dl~='1lSd(~/,]}J-9
Suppose '1 C
on a complex space
1
Q. E. D.
Supp.fj/1 ~
P.
JkfjC "l
are coherent analytic sheaves
whose radical d
and
P
7
is a prix is of dimension d
(7[d_lJ§)X = 'x
For some open neighborhood
exists a coherent ideal sheaf is
dl •
Hence 1[dJ~ is
x ~ X such that
X and
mary submodule of ~x
Proof.
~
dim SuPP(1[dJ§/7l ~ d •
Lemma.
dimx
for some subvariety
U of dimension
Follows from (A.)) and (A.5j.
(A. 7 )
Then
r(U-A,:/)
J
U of
x
in
there
U whose stalk at x k There exists k ~ ~ such that p x C Hence on some open neighborhood and
on
9 Ix'
Let
Y = SuPP('7[d_lJ§/:f1
Y.
Since
and let
f
dim Y < d • there exists
be the ideal sheaf of f ~
Jx-P.
Nullstellensatz,
for some
X
i
~
f\l
Since
f
Q
P • it follows that
C • E. D.
By the
- 450
-
Y -T. Siu
(A.a)
Proposition.
Suppose
is not a zero-divisor of
1x
n
dimx V(f) Supp O[k] 7< k V(f) ~ Supp (!)X/f~ Suppose
is a coherent analytic sheaf
X, x ~ X ,and
on a complex space
Proof.
?
f ~ r(X,(!)X)
Then
f
x
if and only if for all
k ~N*
'
where
0 for is a zero-divisor. Then fxs x x , where U is an open neighsome s~r(U,7) with Sx f borhood of x • Let k = dim Supp s Then x Supp s C V(f) () Supp O[k]1 .
f
°
Suppose
f
.
is not a zero-divisor.
x
the kernel of the sheaf-homomorphism multiplication by borhood
U of
y~
If
U •
By considering
'1 --->1
defined by
f , we conclude that, for some open neigh-
x , fy
dimxV(f)
for some open subset
is not a zero-divisor of
n Supp
0lk]7~ k
W of
U
1y
for some
k
7y
(A.9)
Proposition.
on a complex space
Suppose X.
Supp O[d]'] equals the Equivalently, for if
dimx Sd (7) < d
for
,
then,
,
which, because of the Nullstellensatz, contradicts a non-zero-divisor of
for
fy
being
y ~ U •
7
Then the
is a coherent analytic sheaf d-dimensional component of
d-dimensional component of
x ~ X , dimx Supp O[d]7< d
Sd (7) •
i f and only
-451_
Y-T. Siu
f!:22!.
We prove the equivalent statement.
The "if" part
follows from
dimx Supp OEd] 7 < d There exists an open subset U of
Fbr the "only if" part. we assume that and dimx Sd(7) X - Supp OEd] 7 ing
=
d.
such that
f ~ r(U,~)
unSd (7l
and
branch of
After replac-.
Sd ("]) n u
V(f)
does not contain any
un Supp O[k]
(A.IO)
=
for any
7y
?
k
"lx