A remark on semiglobal existence for <span class="mathjax-formula">$\bar{\partial }$</span>

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

A LBERTO S CALARI

A remark on semiglobal existence for ∂ ¯

Rendiconti del Seminario Matematico della Università di Padova, tome 97 (1997), p. 29-33

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ALBERTO

SCALARI (*)

ABSTRACT - It is

proved

ⁱⁿ [6] ^{that a domain Q of X = C’ is}

pseudoconvex

^{if and}

only

^if a-closed forms (of

positive degree)

^are

~-exact

in S~. We prove here

by elementary techniques

^that

pseudoconvexity

^{of Q is} still characterized

by solvability

of d-forms over

relatively

compact subsets of Q.

1. -

Semiglobal solvability

ⁱⁿ

C2.

Let X ⁼ C’ and let (9x be the sheaf of

holomorphic

^{functions on X.}

Let S~ be a domain of X with

C2-boundary

^{M =}

aS2,

and denote

by

z E ,S~ the Euclidian distance to M. Let z be a

point

^of

M;

^{one defines the}

Levi form of M at z

(from

the exterior side of

Q) by

where One denotes

by sM ( z ),

^and

the numbers of

respectively positive, negative,

^{and null}

eigenval-

ues of

L ( z).

^-

One denotes

by Ox), 0 j

^{n, the} space of a-closed

(0, j)-

forms on Q

(with Coo-coefficients)

modulo 9-exact forms.

We

begin

^our discussion in X ⁼

~2 .

The

proof

^{of the}

following

^state-

ment was

suggested

^word

by

^word

by professor

Alexander Tu-

manov.

(*) Indirizzo dell’A.:

Dipartimento

^di Matematica, Universita di Padova, ^Via

Belzoni 7, ³⁵¹³¹ Padova,

Italy.

30

PROPOSITION 1.1. Let Q’ be a domain

of

^{such that zo be-}

longs

^{to a}

compact component of c2 BQ’.

^Then

PROOF. Assume _zo ⁼ 0. Define:

It is easy to prove

that f is

^a

^(closed)

^{1-form in}

~2 B{zo}.

^Assume

^by

^ab-

surd that

Ox) ~ ~, _

^0. Then there exists _g in S2’:

ag = f in

Q’ whence

is

holomorphic

in S2’ since it is

holomorphic

^{in =}

0}

^{and bounded}

in S2’ far from 0.

By Hartogs’

theorem h extends to the

compact

compo- nents of

~2 BS2’

^{and in}

particular

^{to 0. But}

h ~ zl = o = -1 /z2

^does

not.

Q.E.D.

PROPOSITION 1.2. Let Q be a domain

of ^c2

^with

^C2 boundary

M ⁼ 8Q. For zo ^{E acssume}

LM ( zo )

^{0 . Then}

for

suitable S~’ cc S~ .

PROOF. Let _zo ⁼ 0.

(a)

^We prove that for convenient ,S~’ cc

Q,

^we

can find

T2’ D S~’ ,

zo ^e connected

along

^a

complex

^{curve L}

through

zo such that

any h E C9x (Q’)

^{extends to}

^{T2 ’.}

In fact in

complex

coordinates

z ⁼

(zl , z2),

^{z = x +}

iy,

we can assume:

Define

Let

Kct

⁼

I(Yl, Y2); I Yl, Y21 [ ct ~.

^{We have}

K~

^x

It ,

^{dc » 1 and for}

suitable t. Let B be a fixed

neighborhood

^of zo and Q’ ⁼

Q’t

^{cc S~ s.t.}

S~’ n B J

K~

^x

It . By [8,Theorem 5]

any

holomorphic

^{function on}

extends to

Kctl2 X ^{Jt .}

^In

particular (a)

follows with L

being

^{the z2}

axis.

(b)

Choose

complex

coordinates in

C2

s.t. the curve L of

(a)

^coin- cides with the _z2 axis. Define

f

^{as in the}

preceding Proposition,

^solve

a g

^{and set h =} zl g - ⁺

I Z2 12) (holomorphic

ⁱⁿ

^{S~’ ).}

Take S~’ such that

(a)

holds. Then h

(and

therefore also

h I L

⁼

- -1 /z2 )

^{extends at zo =} 0 which is a contradiction.

2. -

Semiglobal solvability

^{in C’.}

Let M=3Q a

C2

real submanifold of

_ ~ z

^{e X:}

d(z) &#x3E; 0~.

DEFINITION 2.1. Q is

pseudoconvex if

^and

only if

sm

(z)

⁼

0,

Vz e M.

It is classical that Sz is

pseudoconvex

^{if and}

only

^if

(9x)

⁼

0, Vj

&#x3E; 0. With the notation

Q~ = (z e Q; 3 &#x3E; E)

^{we can}

generalize

^the

above characterization as follows:

THEOREM 2.2. Q is

pseudoconvex if

^and

only if

(2.1 ) H~ ( ,~ E , (~x ) ~ s~ E, = 0 , ^{VE, VE’,}

^{with e’} &#x3E; E ;

0, Vj

&#x3E; 0 . PROOF. The

« Only

if» follows from the fact that

Q e

^is

pseudo-

convex

[6,

^Theorem

2.6.12].

«If»:

(a)

^{Let L be a}

complex

submanifold

of X,

^{and set w = L n}

Q,

We

= ( z

^{e OJ;}

^{d L ( z )}

&#x3E;

c} ^{where 6 L}

is the distance

along

^{L. Then}

(2.1 )

^im-

plies

that VE 3E’ such that

32

with E’ ^- 0 as E - 0. This follows from

adapting

^the

proof

^of

[6,

^Theo-

rem

4.2.9]

^{as we} show here.

Let f be

^{a 3-closed form}

Let

neighborhood of co and 0

^{= 0 in a}

neighborhood

e

Q;

7rz §t

a) I

^{X HL the}

orthogonal projection). By

^{a recour-}

sive

argument

^{it is not} restrictive to assume

codX L

^{= 1.} Choose coordi- nates so that L

= ~ z:

⁼

0}.

^Solve

thus

is 3-closed in

~,.

^{Solve 3G=F in}

~~,

E" &#x3E; E’ . Then

g : = j * G (~ :

^{solves ~,} E "’ &#x3E; E" with E "’ -~ O as E - 0. This proves

(2.2).

(b)

^{If n = 1} there is

nothing

to prove, and if n ⁼ 2 the theorem fol- lows from § ^1. Set

3,

^take zo ^e

3Q, vE

^and consider L ⁼ Cv (D

? ^Cu

(u

transversal to

M).

^Then

(2.2)

holds and this

implies

ⁱⁿ

particular

H’ (C2 B f Zo 1, O~)~’

⁼

^0,

^{cc to.} Thus with N ⁼

3co,

We can localize our statement:

DEFINITION 2.3. Q is

pseeudoconvex

at zo

if

^and

only if

sm

(z)

⁼

0,

Vz E M close _{to zo .}

Let B

(or B’)

range

through

^the

family

^of

neighborhoods

^{of the}

points

^{z close} to zo ^on M.

THEOREM 2.4. For _any B there exists B’ c B such that:

if

^and

only if

^{Q is}

pseudoconvex

^{at zo .}

PROOF. «If»:

by ^[1]

^we may find is

pseudoconvex, Q

^{c S~ n}

B, S~

n B’. Then

(2.3)

^follows.

« Only

^if»:

^(a)

^{X =}

^{~C2 .}

^{If Q is not}

pseudoconvex

^{at zo one} may find with

sM (zv) = 1.

^{Then the}

proof

^of

Proposition (1.2)

shows that

for _any B’

neighborhood

^of z, and for suitable Q’ = cc ,S~ . In

partic-

ular

(2.3)

^{is violated.}

(b)

^{Let X =}

~Cn , n ~ 3,

^{take L with}

codxL b 1,

^{define b = B n}

L,

cv ⁼ Q n L and

consider f

^{a-closed in o n b.}

By

^{the same}

argument

^{as in}

Theorem 2.2

f

^can be «lifted» to F 9-closed in S~’ n B’ for _any ,~2’ rela-

tively compact

^{in S~} and for suitable B’ . For _{any E} we take S~’

(depen-

dent on

E )

and B"

(independent

^of

^{E )}

^s.t.

(where n

^is the outward

normal)

and define

Solve

8 Gi

⁼

F¡

ⁱⁿ S~" f1 B"’

(independent

^of

E)

^then gl solves

8 gi = fi

ⁱⁿ

o" f1 b"’

(independent

^of

^{E )}

^{and thus}

3 g = f in -

^{nE +}

^(o"

n b"" where oj’ is _any

relatively compact

open subset of o and b"" a suit- able

neighborhood.

(c)

The end of the

proof

follows from

combining ^(a)

^and

^(b).

REFERENCES

[1] ^A. ANDREOTTI - H. GRAUERT, ^{Théorèmes de}

finitude pour la cohomologie

^des

éspaces complexes,

^{Bull. Soc. Math. France, 90 (1962),} pp. 193-259.

[2] A. ANDREOTTI - C. D. HILL, E. E. Levi

convexity

and the Hans

Lewy prob-

lem, ^{Part II:}

Vanishing

^{theorems, Ann. Sc. Norm.}

Sup.

^Pisa (1972), pp.

747-806.

[3] ^H. GRAUERT,

Kantenkohomologie, Compositio

^{Math., 44 (1981),} pp. 79- 101.

[4] ^{G. M.} HENKIN, ^H.

Lewy’s equation

^and

analysis

^on

pseudoconvex manifolds

(Russian), ^I.

Uspehi

^Mat. Nauk., ³² (3) (1977), pp. 57-118.

[5] ^G. M. HENKIN - J. LEITERER, Andreotti-Grauert

theory by integral formulas,

Birkhauser

Progress

ⁱⁿ Math., ⁷⁴ (1988).

[6] ^L.

HÖRMANDER,

^An Introduction to

Complex Analysis

^{in Several}

Complex

Variables, ^Van Nostrand, Princeton N.J. (1966).

[7] ^L.

HÖRMANDER,

L2 estimates and existence

theorems for

^{the ~} operator, ^Acta Math., ¹¹³ (1965), pp. 89-152.

[8] ^H. KOMATSU, ^A local version

of

Bochner’s tube theorem, ^{J. Fac.} Sci. Univ.

Tokyo,

^Sect. 1A, ¹⁹ (1972).

[9] ^G. ZAMPIERI, ^The Andreotti-Grauert

vanishing

^theorem

for

^dihedrons

of Cn,

J. Fac. Sci. Univ.

Tokyo,

to appear.

Manoscritto pervenuto ^{in redazione il 17}

maggio

^1995.