Some Applications of the Kohn{Rossi Extension Theorems

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SOME APPLICATIONS OF THE KOHNROSSI

EXTENSION THEOREMS

GEORGE MARINESCU

Abstract We prove extension results for meromorphic functions by

combining the KohnRossi extension theorems with Andreottis theory

on the algebraic and analytic dep endence of meromorphic functions on

pseudo concave manifolds Versions of KohnRossi theorems for pseudo

convex domains are included

Introduction

A naive idea of extending a meromorphic function b eyond its domain

of denition is to write as a quotient of two holomorphic functions and

then extend these functions At least for holomorphic functions dened

near the b oundary of complex manifolds this idea works as in the pro of

of the KohnRossi extension theorem see Theorem and the pro of of

subsequent Theorem But if is dened in a neighb ourho o d U of

the complement Y n M of a smo oth domain M whose b oundary Levi form

has at least one p ositive eigenvalue in a compact complex manifold Y we

cannot apply this metho d Indeed U is pseudo concave and any holomorphic

function on U is lo cally constant However it is rather simple to see that a

meromorphic function on any complex manifold can b e written as a quotient

of two holomorphic sections of an appropriate holomorphic line bundle We

can thus hop e to extend the line bundle and its sections The situation

in the ab ove example b ecomes particulary favourable in this resp ect if we

assume Y to b e Moishezon Then by Andreottis theory it can b e shown

that can b e written as a quotient of two holomorphic sections in some

tensor p ower E of a line bundle E Y with maximum Ko dairaIitaka

dimension Therefore we need to extend but the sections of E and this is

done using the theorems of KohnRossi The situation b eing global we have

to imp ose some supplementary hyp otheses namely one of the following see

Theorem

H M is q complete q n

H There exists one nonconstant holomorphic function in the neigh

b ourho o d of bM

Mathematics Subject Classication Primary D A Secondary F

Key words and phrases Extension of CR sections meromorphic function pseudo con

cave manifold Moishezon manifold Ko dairaIitaka dimension

The author was supp orted by a NATORoyal So ciety Fellowship at the University of

Edinburgh He thanks the Department of Mathematics Statistics for its hospitality

GEORGE MARINESCU

H The holomorphic line bundle E of maximal Ko dairaIitaka dimen

sion is trivial in the neighb ourho o d of M

Note that using the Thullentyp e theorem of Siu see also Ivashkovich

x and Shiman ab out the extension of a meromorphic map from

the Hartogs gure to a p olydisc one can prove that if is a meromorphic map

M with values in a Kahler manifold then has a meromorphic dened in Y n

continuation into bM Our result can b e viewed as global counterpart of the

previous theorem As for the b est global result let us mention the following

theorem of Ivashkovich every meromorphic map from a domain D in a

Stein manifold into a compact Kahler manifold K extends to a meromorphic

map from the envelop e of holomorphy D of D into K

The KohnRossi theorems are consequences of the global regularity of the

Neumann problem for domains M as ab ove Since the global regularity

holds also for some class of pseudo convex domains we prove in the

last section versions of those theorems for this class of domains

Aknow ledgements I wish to acknowledge my debt to Professor Guennadi

Henkin who is at the origin of the ideas used in this article He deserves a

large share of the credit for any originality the present pap er may p ossess

Meromorphic continuation

Let us give a brief account of the extension theory develop ed by Kohn and

Rossi Let M b e a smo oth domain in a complex manifold A smo oth function

on bM is said to b e CauchyRiemann CR if its dierential vanishes on the

subbundle C T bM T M A smo oth section over bM of a holomorphic

vector bundle is said to b e CR if it is represented in lo cal holomorphic

frames by vectors of CR functions In b oth cases we write f Of

course the restriction of a holomorphic section to bM is CR The crucial

lemma dep ends on the natural duality b etween the free b oundary conditions

imp osed by and is the formal adjoint and is the hilb ertian adjoint

of and the solution of the Neumann problem To simplify matters

let us take E V F merely a holomorphic section of the holomorphic

vector bundle F over a neighb ourho o d of bM We dene the Ho dge op erator

pq npnq

E F E F by d v ol where the exterior

pro duct is combined with the canonical pairing F F C Recall that

and on r forms Recall also that if a form

M vanishes on bM then clearly b elongs to Dom with C co ecients on

integration by parts and on such forms Consider a smo oth

extension of to M and assume that the Neumann problem is solvable

in bidegree n n for F which is the case if the Levi form of bM has

one p ositive eigenvalue

nn nn

M F E M F the Neumann op erator Denote by N E

Dene

N E M F

SOME APPLICATIONS OF THE KOHNROSSI EXTENSION THEOREMS

Consider now E M F which has compact supp ort in M

since has Clearly Dom and If is only CR then

of course hasnt compact supp ort but we can construct an extension

such that j alternatively Lemma of shows that in fact

is equivalent to Dom for any extension Thus

N N N

so that the Ho dge decomp osition given by the solution of the Neumann

N H where H is the N H reduces to N

orthogonal pro jector on the harmonic space H M F the space of

smo oth forms Dom such that and Assume that

H M F

Then if we denote N we have and Therefore

which implies On

M F In this extreme case the other hand we have Dom E

the b oundary conditions imp osed by the preceding relation are Dirichlet

conditions j so that on bM We easily infer that

in a neighb ourho o d of bM since real hyp ersurfaces are uniqueness

sets for holomorphic functions and sections in holomorphic vector bundles

Conversely if admits a holomorphic extension then and H van

ish so that condition is necessary and sucient for the holomorphic

extension Integrating by parts

for H M F we obtain

Lemma Let M be a relatively compact domain with smooth boundary

in a complex manifold Y such that the Levi form of the boundary bM has

at least one positive eigenvalue everywhere Let F be a holomorphic line

bund le over Y Then a given smooth CR section of F over bM has a

M which is holomorphic in M if and only if smooth extension over

for any H M F

On grounds of nite dimensionality of the harmonic space the Lemma

entails see also the pro of of Theorem

Theorem KohnRossi Let M as in Lemma and with connected

boundary Then every CR function on bM has a holomorphic extension to

al l of M

Note that if there exist a nonconstant CR functions connectedness of bM

is a sup eruous hyp othesis

Theorem KohnRossi Let Y be a complex manifold possesing at least

one nonconstant global holomorphic function Let M as in Lemma

GEORGE MARINESCU

Then for every holomorphic line bund le E on Y there exists a holomorphic

line bund le L suct that every smooth CR section of E L over the boundary

bM extends by a smooth section of E L over M holomorphic in M

Of course these three statements remain true for holomorphic sections de

ned near bM thanks to the uniqueness theorem Relying on these results

we shall prove the following

Theorem Let Y be a Moishezon manifold Y of dimension n

Assume that M is a smooth domain in Y subject to one of the fol lowing

hypothesis

H M is q complete q n

H The Levi form of bM has one positive eigenvalue everywhere and

there exists one nonconstant holomorphic function in the neighbour

hood of bM

H The Levi form of bM has one positive eigenvalue everywhere and

there exists a holomorphic line bund le E on Y of maximal Kodaira

Iitaka dimension which is trivial in the neighbourhood of M

Then any meromorphic function dened in a pseudoconcave neighbourhood

U of bM eg a connected neighbourhood of Y n M there exists a mero

morphic function on M U which agrees with it on U

Observe that in view of the theorem of Siu and Ivashkovich quoted in the

M Intro duction we may assume that is dened only on U Y n

Recall that a compact connected complex manifold is said to b e Moishezon

if the transcendence degree of its meromorphic function eld equals its com

plex dimension The KodairaIitaka dimension E of a holomorphic line

bundle on a compact complex manifold is the supremum of the generic rank

of the canonical rational maps dened by the nonzero sections in H E

for m if any and E otherwise By a wellknown result of

Siegel we have E deg tr K Y n dim Y where K Y is the

eld of meromorphic functions on Y Therefore if Y p ossess a line bundle

with maximum Ko dairaIitaka dimension then it is automatically Moishe

zon Conversely if Y is Moishezon then there exists a nonsingular pro jective

mo dication comp osition of blowups of nonsingular centres Y Y

e e

if we consider an ample divisor H on Y then the line bundle asso ciated to

H H has maximum Ko dairaIitaka dimension

Let us intro duce now the convexity notions used A real smo oth function

dened on a complex manifold X of dimension n is said to b e q convex

if its complex hessian i has at least n q p ositive eigenvalues at

every p oint of X is called exhaustion function if for every c R the set

f cg is relatively compact A complex manifold is said to b e q complete

if it is endowed with a q convex exhaustion function Let us consider now

a relatively compact smo oth domain M whose b oundary has a denition

function the Levi form of bM is the complex hessian of restricted to

the holomorphic tangent space T M C T bM

SOME APPLICATIONS OF THE KOHNROSSI EXTENSION THEOREMS

Following Andreotti a connected complex U manifold is called pseudo

concave if one can nd a nonempty op en subset N b U with psudo concave

b oundary bN ie for any p oint z bN given a holomorphic function

dened in the neighb ourho o d of z one can nd a neighb ourho o d V of z

such that maximum of jf j on V N is attained in a p oint of V N It

is wellknown that if Y is compact and M Y is a smo oth domain

whose b oundary Levi form has one p ositive eigenvalue everywhere then any

neighb ourho o d U of Y n M is pseudo concave Also any compact manifold

is pseudo concave take N to b e the complement of a twisted ball a stongly

pseudo convex set the Levi form of the b oundary is p ositive denite

Let us consider a connected pseudo concave manifold U of dimension n

and a holomorphic line bundle E over U We dene the graded ring

AE H X E

Since U is connected AE is an integral domain and we can consider the

quotient eld QE U b eing a manifold normal space QE is alge

braically closed in the meromorphic function eld K U

The vector space H X E is nite dimensional and if s s is a

basis we dene the canonical rational maps

X P x s x s x

We say that AE gives lo cal co ordinates at a p oint x if there exists m

such that is holomorphic at x and rank n If U is compact then

m x m

AE gives lo cal co ordinates at some p oint of U if and only if E n in

that case AE gives lo cal co ordinates on an op en dense set

Proof of Theorem We use the following basic result of Andreotti

which asserts that on a pseudo concave manifold U the algebraic dep en

dence of meromorphic functions is the same as the analytic dep endence A

corollary of this is that the elds K U and QF for some holomorphic

line bundle on U are isomorphic to simple algebraic extensions of elds of

rational functions with d and e variables resp ectively where e d n

Let us consider now the line bundle E given by the hyp othesis Since

E n AE gives lo cal co ordinates on an op en dense subset of Y

and a fortiori of U If s s give lo cal co ordinates at a p oint x

U and say s x then s s s s are analytically indep endent

meromorphic functions on U hence algebraically indep endent Therefore

the transcendence degree e of QE j equals n Consequently K U

QE j and for any given meromorphic function on U there exists sections

s t H U E for some m such that st

Assume that we have the hyp othesis H The manifold M b eing q

complete q n it is known that H M F for any holomorphic

vector bundle F on M By results of Hormander we have

n n

H M F H M F

GEORGE MARINESCU

so in particular

nn m n n m

H M E H M E

Therefore any CR section of E over bM has a holomorphic extension to

M by Lemma Applying this to the traces of s and t on bM together

with the uniqueness theorem we get sections s t H M E extending s

and t Then s t is a meromorphic function extending

Assume now H Theorem shows that there exists a nontrivial

holomorphic function in the neighb ourho o d of M Therefore for any holo

morphic vector bundle F Theorem gives a line bundle L such that any

holomorphic section of F L in the neighb ourho o d of bM extends holo

morphically in the interior of M An imp ortant feature of the line bundle

L is that it is trivial near bM from the divisor of a suitable holomorphic

function on Y we cut out all the comp onents intersecting the complement

of M Consequently any section in E near bM may b e viewed as a section

in E L and the same reasoning as ab ove concludes the pro of

Finally we prove the conclusion assuming H Since we do not have a

nonconstant holomorphic function anymore we have to prove a version of

k k

Theorem For k let us denote E E

Lemma Let Y be a Moishezon manifold Y of dimension n and M

a smooth domain satisfying H Then for every holomorphic line bund le

F on Y there exist k N and a holomorphic line bund le L such that every

smooth CR section of F L E over bM extends by a smooth section of

M holomorphic in M F L E over

Proof Consider the map

k nn nn k

H Y E H M F H M E F

s H s

where H is the pro jection on the space of harmonic forms and s is the

natural sesquilinear pairing It suces to show that the following claim is

true

Claim There exists k N and a section s H Y E such that for any

section H Y F we have H s

Indeed let b e any smo oth CR section of F over bM Then the following

relation holds for every H M F

Z Z Z

H s s s

bM bM bM

Since E is trivial in the neighb ourho o d of bM s is a section of F over

bM so by Lemma the section s has a smo oth extension to M which is

holomorphic in M Let us consider the divisor D consisting of the irreducible

comp onents counted with multiplicities of the zero set of s contained in M

The line bundle L is obtained as the line bundle asso ciated to D L D

SOME APPLICATIONS OF THE KOHNROSSI EXTENSION THEOREMS

Remark that L is trivial in the neighb ourho o d of bM Let us consider a

holomorphic section in L such that the divisor of is exactly D ie

Supp ose that C bM F L E is such that Since

L and E are trivial in the neighb ourho o d of bM induces a CR section

C bM F Then s C bM F has an extension S C M F

which is holomorphic in M We dene a meromorphic section of F L E

s S

Its p ole set in M is a union of branches of the zero set of s which intersect bM

Since L is trivial in the neighb ourho o d of bM in that neighb ourho o d

so that the b oundary value of is We infer that the p ole set of

do es not intersect bM so it is empty Therefore is a holomorphic

extension in M of the given

We turn now to the Claim Assume that the assertion of the claim would

b e false Then a simple algebraic argument shows that

k nn nn k

h Y E h M F h M E F

for the dimensions of the vector spaces in Set n dim Y Since F

has Ko daira dimension F n we have

k n

h Y E k for k

n k n

that is there are constants C C such that C k h Y E C k

for large k But the righthand memb er of is a constant due to the

fact that E is trivial on M Thus for large k cannot p ossibly hold

This nishes the pro of of the Lemma

Let b e a meromorphic function on U which can b e represented as a quo

tient st for s t H U E for suitable m We apply Lemma

to F E so that there exist k and L such that the conclusion of the

Lemma holds Since L and E are trivial near the b oundary we can view

s and t as sections in F L E near bM and we get sections s and t of

F L E over M extending them Then the meromorphic function s t

equals near bM and extents it over M

Remark A more transparent pro of of the extension of sections result

under hyp othesis H can b e given in the spirit of the Ehrenpreis pro of

of the Hartogs theorem One may use the the cohomology groups with

compact supp ort but in order to maintain ourselves in the spirit of the work

of Kohn and Rossi to o let us consider the Dirichlet cohomology groups

M j g f forms smo oth on

H M F

f for smo oth function on M j j g

bM bM

If the Levi form of bM has at least one p ositive eigenvalue then by the

KohnRossi version of the Serre duality

H M F H M F 0

GEORGE MARINESCU

and under H these groups vanish If is a CR section of F over bM

then there exists a smo oth extension to M such that j see

the pro of of We can then solve the equation with j

and will b e the desired holomorphic extension This argument can b e

carried out for forms to o yielding however a weak result for any p form

with values in F p n q one can nd a closed form in M having

the same tangential values on bM as For the extension of CR forms see

the work of C LaurentThiebaut and J Leiterer

Remark The hyp othesis H can b e weakened and the conclusion of

the Theorem still holds Namely one can assume that E Y is semi

p ositive with Ko dairaIitaka maximum dimension and E is trivial only in

the neighb ourho o d of the b oundary bM The pro of is the same the single

dirence is that the failure of results from asymptotic Morse inequali

ties that is

i k

n n nn k

ok cE dim H M E F

M n

for k In this formula icE is the curvature of E and M n is

the op en set of M where icE has exactly p ositive eigenvalue and n

negative ones Since E is supp osed to b e semip ositive on M and n we

infer that M n and the curvature integral in has to vanish

Therefore the righthand side of is ok so that this inequality cannot

b e true for large k Unfortunately the author has no example in which this

situation holds unless the bundle E is trivial in a whole neighb ourho o d of

M ie we have H For the latter see Example

Another application of Theorem is the extension of analytic hyp er

surfaces across strictly pseudo concave b oundaries It is wellknown that

analytic sets of dimension may b e extended across such b oundaries in

some small neighb ourho o d For analytic curves this is not always p ossible

see a counterexample in x subsection That is why we shall consider

the case when the ambient manifold has dimension Let M b e a strongly

pseudo convex domain ie the Levi form of bM is p ositive denite First

remark that M is convex in the sense that there exists an exhaustion

function of M which is convex strictly plurisubharmonic outside a

compact set K For almost all c sup j M fx M x cg is

K c

M has two comp onents one stongly pseudo convex The b oundary of M n

with p ositive denite Levi form and one with negative denite Levi form

Prop osition Let M a strongly pseudoconvex domain of dimension

Then for any analytic hypersurface H in M n M there exists an analytic

b b

hypersurface H in M such that H H

Proof Let E b e the holomorphic line bundle on M n M asso ciated to the

divisor H and let E b e the lo cally free sheaf of holomorphic sections in E

SOME APPLICATIONS OF THE KOHNROSSI EXTENSION THEOREMS

By the reduction principle of Grauert there exist a Stein complex space

f f f

M a prop er holomorphic surjection M M a nite subset A M

such that x is a connected nowhere dense analytic set of M for x A

and

M n A M n A

is biholomorphic In particular we see that there exist nonconstant holomor

phic functions on M We may assume that A K so that the direct

f f f

M image E is a lo cally free analytic sheaf on M n M where M

c c c

Let us dene the rst absolute gapsheaf E which is the sheaf asso

ciated to the presheaf U lim inf U n A E where A runs over all

analytic subsets of U of dimension Because E is lo cally free and the

dimension of M is we see by the Riemann removable singularity theorem

that E E Since M contains no p ositivedimensional compact

analytic set Corollary of Andreotti and Siu shows that there exists a

F is E Then coherent analytic sheaf F on Y such that F j

f f

M nM

E By Prop osition a coherent analytic sheaf on M such that F j

M M n

of Andreotti and Siu we can take F to b e lo cally free that is

E can b e extended to a holomorphic line bundle F on M Let us consider

a section s of E whose divisor is H By Theorem applied to a domain

M with d c there exists a holomorphic line bundle L on M trivial on

bM such that every holomorphic section of F L in the neighb ourho o d of

bM can b e extended to M Since L is trivial on bM the section s can b e

d d d

viewed as a section in F L in the neighb ourho o d of bM There exists a

global section section S extending it and by considering the comp onents of

b b

the divisor of S we nd a hyp ersurface H of M such that H H

Remark It easily seen that the same reasoning shows that any mero

morphic function dened in the neighb ourho o d of bM of a strongly pseu

do convex domain M has a meromorphic extension to M We do not need

neither the domain to b e a subset of a Moishezon manifold nor the mero

morphic function to b e dened in a pseudo concave neighb ourho o d of bM

It suces only to remark that for any meromorphic function dened on

a complex manifold U there exist a holomorphic line bundle such that the

meromorphic function can b e written as the quotient of two global holo

morphic sections of that bundle To this end we cho ose a covering U of U

such that f h on U and the germs of the holomorphic functions f h

i i i i i

at every p oint of U are relatively prime Since f h f h on U U

i i i j j i j

we see that there exists a holomorphic function g on U U such that

ij i j

f g f and h g h Then fg g is the co cycle of the desired line

i ij j i ij j ij

bundle This line bundle at hand we can extend it as well as its sections as

in the previous pro of and then take the quotient of the extended sections

This statement can b e viewed as a particular case of the p owerful result of

Ivashkovich already quoted in the Intro duction

GEORGE MARINESCU

Let us give a few examples of domains for which our theorem applies We

b egin with an example of domain whose Levi form of the b oundary has at

least one p ositive eigenvalue and an example of pseudo concave manifold

Example Let Z b e a compact complex manifold and let Y b e an

analytic subset such that dim Y q dim Z Let X Z n Y Then

there exists a function X R such that

for any c R we have X fx X x cg b X that is is an

exhaustion function from b ellow

is q convex outside a compact set of X

For the pro of see Ohsawa Therefore for suciently small c the b ound

X if smo oth has a Levi form with at least one p ositive ary of the set X n

eigenvalue Consequently for small c any neighb ourho o d of the set X is

pseudo concave Note that in the terminology of X is a particular case

of q concave manifold

We give now an example of the situation in hyp othesis H

Example Let Y b e a Moishezon manifold carrying a line bundle E

with maximum Ko dairaIitaka dimension Let s H Y E D fs g

so that E is trivial on the complement of D Let Y Y the blow up of

1 1

center x Y n D Then D and x are compact disjoint sets The

pullback E E has maximum Ko dairaIitaka dimension and it is trivial

1 1

on the complement of D Let us consider an analytic set A Y n D

1 dim Y

of dimension q dim Y one can take A x P so that

X Y n A admits an exhaustion function from b ellow as in the preceding

e e e

example Therefore Y E and M Y n X for convenient very small c see

satisfy the hyp othesis H the Levi form of bM has n q p ositive

eigenvalues and E is trivial on M since X contains the compact set D

for c inf fx x D g

We end with an example of q complete manifold and of a very small pseu

do concave manifold

Example If Z is a compact complex manifold and E is a p ositive

holomorphic line bundle over Z and consider the global sections s s

q dim Z and Y fx Z s x s x g Then X Z n Y

is q complete Indeed if s are the lo cal representations of these sections as

holomorphic functions and if h are the lo cal representation of the p ositive

metric on E then log h js j is a q convex exhaustion function

for X i i log h i log js j the rst term of the sum is

p ositive denite b eing the curvature of E and the second vanishes on the set

fs s s s g say s which has dimension at least n q

so that i has at least n q p ositive eigenvalues Thus Z n X is

pseudo concave for large c Since Z n X c R forms a fundamental system

of neighb ourho o ds of Y every neighb ourho o d of Y is pseudo concave

SOME APPLICATIONS OF THE KOHNROSSI EXTENSION THEOREMS

Using the last example we obtain the following striking result Let b e a

meromorphic function dened in a neighb ourho o d of the set Y ab ove then

extends as a meromorphic function to Z and Z b eing pro jective it admits

a p ositive line bundle is in fact rational by the wellknown theorem of

Weierstrass Hurwitz and Chow But this is only a particular case of a more

general result of Dingoyan asserting that a meromorphic function on a

pseudo concave pro jective manifold is in fact rational

Extension of CR sections from weakly pseudoconvex

boundaries

In this section we consider weakly pseudo convex domains in complex

manifolds A smo oth domain M in a complex manifold X is said to b e

pseudo convex if the Levi form of its b oundary is p ositive semidenite every

where The extension pro cedure used so far involves the the global regularity

of the Neumann problem That is why we consider the supplementary

condition that the dual of the bundle where the section to b e extended lives

is p ositive near bM This condition was intro duced by Kohn in the

following form there exists a smo oth nonnegative function X R

which is strictly plurisubharmonic in the neighb ourho o d of bM This hy

p othesis implies that that M is convex Theorem and any line

bundle is p ositive near bM Let us intro duce the following hermitian metrics

on the trivial line bundle X C

g expt t

This gives L inner pro ducts on M Ho dge op erators formal ad

of The L spaces ob joints and hilb ertian adjoints

t t

M under the norms k k are denoted L M tained by completing E

top ologically all are the same Consider the LaplaceBeltrami op erator

dened by

t t t

Dom fu Dom Dom u Dom u Dom g

t t t t

and the kernel of

H M fu Dom Dom u u g

t t

together with the orthogonal pro jector H on this subspace For s N

M the space of p q forms with s times continuously dif denote by E

ferentiable co ecients on M The result ab out the global regularity is the

following

Theorem Kohn Let M be a smooth pseudoconvex domain in the

complex manifold X assume that there exists a smooth nonnegative real

function on X which is strictly plurisubharmonic in the neighbourhood of

bM Then for any s N there exists a real T such that for t T and

s s

q we have that

i has closed range and H M is a nite dimensional subspace of

M E s

GEORGE MARINESCU

pq pq

There exists a bounded operator N L M L M such that

t t

and we have the strong Hodge decomposition ii Range N Dom

N N H L M

t t t

t t

on their domains iii N commutes with

t t

iv N and H map E M into E M

t t

Therefore we have the following analogue of Lemma

Lemma Let s and t T like in Theorem Then for any

smooth CR function on bM there exists E M such that j

and in M if and only if

for al l H M

Proof Let b e an arbitrary smo oth extension of to M Then we can

achieve that j by replacing by

L r b

where r is a dening function for M and L is a vector eld which

satisfy L r near bM As b efore we consider

M E

t t

and by It is clear that Since j we have Dom

t t

iii of Theorem this implies that N Then ii shows that

N if and only if H Since id on functions the last

t t

condition is equivalent to by integration by parts Consider the form

N By iv applied to s we infer that E M Moreover

M so that j b ecause s Dene now E Dom

E M by It is now clear that is the desired

extension

We can state now an analogue of Theorem

Theorem Let M be a smooth pseudoconvex domain with connected

boundary assume that there exists a smooth nonnegative real function

on X which is strictly plurisubharmonic in the neighbourhood of bM Then

for any smooth CR function on bM and any s there exists an s times

M holomorphic in M continuously dierentiable extension on

The pro of is essentially the same as of Theorem from For the

convenience of the reader we sketch it here

Proof Remark rst that the set of CR functions forms a ring If the only

CR functions on bM are the constants then the conclusion follows Assume

now that there exist at least one nonconstant CR function f it necessarily

takes an innity of values The key observation is that we can nd a monic

p olynomial P with complex co ecients such that P f has an extension in

M holomorphic in M Indeed take a basis in H M E

k t

SOME APPLICATIONS OF THE KOHNROSSI EXTENSION THEOREMS

for some xed t T and let c f i k j k

s ij j

Cho ose complex numb ers a i k such that a c for all j

i i ij

i i

and a Put P z a z Since f we have P f

i b b

Also P f for every H M Therefore by Lemma

there exist h E M such that h P f on bM and h in M Since

P f hits an innity of values h is nonconstant

Now the idea is to write f as a quotient of two functions in E M

holomorphic in M Using the same reasoning as ab ove but replacing f

with h f i k we nd a p olynomial Q such that Qhf has an

M holomorphic in M Set G Qh Of course extension F E

f F G on bM and F G is holomorphic in the complement of the zero

set of G If we show that F G has a holomorphic extension to all of M

we are done For this we use the Riemann removable singularity theorem

Note that h b eing nonconstant so is G Qh to the eect that fG g

is thin So we have to show that F G is lo cally b ounded in M For this

purp ose we return to the p olynomial P First consider the holomorphic

k k

functions G P F G and G h they have the same b oundary values so

they are identical Consequently P F G h on M n fG g Since h is

lo cally b ounded P F G is lo cally b ounded But P b eing monic this entails

that F G is lo cally b ounded to o

It is known that in general there exists no strictly plurisubharmonic

function near the b oundary bM However in Grauerts example there exists

a p ositive line bundle near bM This case was studied by K Takegoshi

who extended the regularity theorem as follows

Theorem Let M be a smooth domain in a complex manifold X

and let E X be a holomorphic line bund le which is positive near bM

Then for every s N there exists a rank ms such that for m ms and

q we have that

pq m

the LaplaceBeltrami operator acting on L M E has closed i

pq m

range and its kernel H M E is a nite dimensional subspace of

M E E

pq m pq m

There exists a bounded operator N L M E L M E such

that

ii Range N Dom and we have the strong Hodge decomposition

pq m

N H L M E N

m m m

t t

iii N commutes with on their domains

m m

pq m

iv N and H the orthogonal projection on H M E

m m

pq m m

map E M E into E M E

Thus we have the following version of Lemma

Lemma Let s and m ms like in Theorem Then for

m m

any smooth CR section C bM E there exists E M E

GEORGE MARINESCU

such that j and in M if and only if

nn m

for al l H M E

The only thing we must prove is that there exists a smo oth extension

such that j This is easily seen b ecause formula carries over

the bundle case Indeed the transition functions are holomorphic and the

vector eld L of typ e so that L is still a smo oth section

We shall now restrict attention to a particular case of pseudo convex do

mains Let X b e a weakly complete manifold ie there exists a plurisub

harmonic smo oth exhaustion function X R If c is a regular value

of then M X f cg is a pseudo convex domain and also a a weakly

complete manifold

Theorem Let X be a weakly complete Kahler manifold and c a

regular value of the exhaustion function Assume that E X is a semi

positive holomorphic line bund le which is positive near bX Then for s

m m

X E any CR section C bX E extends to a section E

c c

which is holomorphic in X

Proof Let m ms such that Lemma holds By the representation

theorem of

nn m nn m n n m

H X E H X E H X E

c c c

where n dim X But now the hyp otheses we made on X and E are the

same with the hyp otheses of the vanishing theorem of Takegoshi to the

eect that these groups vanish Therefore relation is trivially satised

and the conclusion follows by Lemma

We obtain thus an asimptotically vanishing theorem for the b oundaries of

some sp ecial pseudo convex domains

Corollary Assume that E X is a positive line bund le on the weakly

complete manifold X and the exhaustion function satises

rank exp dim X

If c is a regular value of then for suciently large m any CR section

C bX E vanishes identical ly

Proof We have only to remark that by Theorem of Ohsawa p

a seminegative line bundle F of typ e q seminegative with at least

dim X q negative eigenvalues on a weakly complete Kahler manifold

satises H X F for p dim X q r where rank exp r In

m m

our case F E q and r dim X so that H X E

and the extension theorem ab ove entails that is the b oundary value of the

zero section

SOME APPLICATIONS OF THE KOHNROSSI EXTENSION THEOREMS

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eds Vieweg Braunschweig Germany

Institute of Mathematics of the Romanian Academy PO Box RO

Bucharest Romania

Current address Department of Mathematics Statistics The University of Edin

burgh James Clerk Maxwell Building Kings Buildings Mayeld Road Edinburgh EH

JZ UK

Email address georgemathsedacuk