vesentini

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Lectures On Levi Convexity

Of Complex Manifolds AndCohomology Vanishing Theorems

By

E. Vesentini

Tata Institute Of Fundamental Research, Bombay

1967

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Preface

These are notes of lectures which I gave at the Tata Institute of Funda-

mental Research in the Winter 1965.

Most of the material of these notes may be found in the papers [1],[2] and [16] listed in the Bibliography at the end of this book.

My thanks are due to M.S. Raghunathan for the preparation of these

notes and to R. Narasimhan and M.S. Narasimhan for many helpful sug-

gestions.

E. Vesentini

iii

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Contents

Preface iii

0 Prerequisites 1

1 Vanishing theorems for hermitian manifolds 7

1 The space L p,q . . . . . . . . . . . . . . . . . . . . . . 7

2 The space W p,q . . . . . . . . . . . . . . . . . . . . . . 15

3 W-ellipticity and a weak vanishing theorem . . . . . . . 25

4 Carleman inequalities . . . . . . . . . . . . . . . . . . . 30

2 W-ellipticity on Riemannian manifolds 39

5 W-ellipticity on Riemannian manifolds . . . . . . . . . . 39

6 A maximum principle . . . . . . . . . . . . . . . . . . . 44

7 Finite dimensionality of spaces of harmonic forms . . . . 508 Orthogonal decomposition in Lq . . . . . . . . . . . . . 57

3 Local expressions for and the main inequality 61

9 Metrics and connections . . . . . . . . . . . . . . . . . 61

10 Local expressions for ∂, ϑ and . . . . . . . . . . . . . 70

11 The main inequality . . . . . . . . . . . . . . . . . . . . 75

4 Vanishing Theorems 79

12 q-complete manifolds . . . . . . . . . . . . . . . . . . . 79

13 Holomorphic bundles over q-complete manifolds . . . . 85

14 Examples of q-complete manifolds . . . . . . . . . . . . 9215 A theorem on the supports of analytic functionals . . . . 95

v

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2 0. Prerequisites

is equivalent to the fact that every−∂ closed distribution is a holo-

morphic function1

(For a proof see for instance [5], Expose XVIII, [22])2

The sheaves A p,q being fine, the cohomology group H q( X ,Ω p) iscanonically isomorphic to the qth cohomology group of either oneof the complexes

0 / / H ( X , A p,o) / /

∂ / / H ( X , A p,1)

∂ / / . . .

∂ / / H ( X , A p,n) / / 0

0 / / H ( X , K p,o) / /

∂ / / H ( X , K p,1 )

∂ / / . . .

∂ / / H ( X , K p,n) / / 0.

2. We need the following theorem due to Leray.

LetG = U ii∈ I be a locally finite open covering of a paracompact

space X . Let F be a sheaf of abelian groups on X such that for

every q > 0, H q(U io∩ . . . ∩ U i p , F ) = 0 for every set (i0, . . . i p) of

p-elements in I. Then the canonical map

H q(G, F ) → H q( x, F )

is an isomorphic for all q ≥ o

For a proof see for instance [13] 209-210

3. The following result enables us to apply the above theorem tolocally free sheaves over homomorphic function on a complex

manifold.

Let X be a (para compact) complex manifold of complex dimen-

sion n. Given a vector bundle E an X and any covering G =

U iiǫ I there is a refinement V j jǫ J such that for j1, . . . , jk with

V j1 ∩ . . . ∩ V jk φ, for 0 ≤ p ≤ n, the sequence

1The idea of the proof is the following. Let α be any compactly supported function

of class C k with k > 0. Since−∂(T ∗ α) = 0, T ∗ α is a homomorphic function hence it is

C ∞. Then T itself is a C ∞ function [30, Vol. II, Theorem XXI, 50]. But −∂T = 0. ThenT is holomorphic.

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4 0. Prerequisites

Ω p( E ) (resp. A pq( E ), resp. K pq( E )) is the sheaf of germs of holo- 4

morphic E -valued p-forms (resp. diff erentiable ( p, q) forms withvalues in E , resp. ( p, q) currents with values in E ;

Γ( X , A pq( E )) = Global( p, q)C ∞ forms with values in E ;

Γ( X , K pq( E )) = Global(P, q) currents with values in E .

There is a one-one correspondence between global sections of

A pq( E ) and collections (ϕi)i∈ I where each ϕi is a vector valued

form,

ϕi =

ϕ′i

...ϕm

i

each ϕk

ibeing a scalar C ∞ ( p, q) forms on U i such that for x ∈ U i∩

U j we have ϕi( x) = ei j( x)ϕ j( x). One can set up a similar one-one

correspondence between sections of K pq( E ) and families (ϕi)i∈ I

of currents, each ϕk ii being defined on U i, satisfying ei jϕ j = ϕi

on U i ∩ U j.

We denote by ∂ the operators

1 ⊗ ∂ :E ⊗O

A pq −→ E ⊗ A p,q+1

and 1 ⊗ ∂ :E ⊗O

K pq −→ E ⊗ K p,q+1.

Then the sequences

0 → Ω p( E ) → A p,o( E )

∂→ . . .∂→ A p,n( E ) → 0

and 0 → Ω p( E ) → K p,o( E )

∂→ . . .∂→ K p,n( E ) → 0

are exact. Moreover the sheaves A pq( E ) and K pq( E ) are fine.

Hence if Φ denotes either the family of all closed subsets of X 5

or the family of all compact subsets of X , (more generally, anyparacompactifying family) and we set,

ΓΦ( X , A pq( E )) = σ|σ a section of A pq( E ) over X , Supp σ

∈Φ ,

then H q

Φ( X ,Ω p( E )) ≃ ϕ|ϕ ∈ ΓΦ( X , A p,q( E )); ∂ϕ = 0

| ∂ΓΦ( X , A p,q−1( E ))

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Chapter 1

Vanishing theorems for

hermitian manifolds

1 The space L p,q

All manifolds considered are assumed to be connected and paracompact. 6

Let π : E → X be a holomorphic vector bundle on a paracompact

complex manifold X of complex dimension n.

LetG =U i

iǫ I be a locally finite open coordinate covering of X and

ei j : U i ∩ U j → GL(mC)

i j∈ I × I transition functions defining E .

A hermitian metric along the fibres of E

is a collection hi:

U i →GL(m,C)iǫ I of C ∞ -maps such that for xǫ U i, hi( x) is a positive definite

hermitian matrix and for xǫ U i ∩ U j, h j( x) =t ei j( x)hi( x)ei j( x).

Lemma 1.1. Every holomorphic vector bundle on a complex manifold

admits a hermitian metric.

Proof. Let ρk

k ∈ I be a C ∞-partition of unity subordinate toU k

k ∈ I (=

G). Letho

i: U i → GL(m,C)

i∈ I be any family of C ∞-functions such

that for i ∈ I , x ∈ U i, hoi

( x) is a positive definite hermitian matrix. Then

the familyho

i

i∈ I defined by

hi( x) =k

ρk ( x)t eki( x)hok ( x)eki( x)

7

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8 1. Vanishing theorems for hermitian manifolds

is a metric along the fibres of E .

In particular, the holomorphic tangent bundle H → X admits a7

hermitian metric. Let G =U i

i∈ I be a covering of X by means of

coordinate open sets. Let Z 1i

, . . . , Z ni

be any system of coordinates in U i.

Then H → X is defined with respect to G by means of the transition

functions J i j : U i ∩ U j → GL(n,C) defined by

J i j( x) =

∂ Z α j

∂ Z ρl

β( x)

αβ

for x ∈ U i ∩ U j.

A hermitian metric on H is then a family of C ∞-hermitian -positive

definite matrix-valued functions gi : U i → GL(n,C

) such that

g =t J i jgi J i j

Identifying hermitian matrices with hermitian bilinear forms on the tan-

gent spaces through the basis∂

∂ Z 1. . .

∂

∂ Z nof the holomorphic tangent

space, we may write gi as αβ

giα ¯ βdZ αi d ¯ Z βα.

A hermitian metric on X defines on X (regarded as a diff erentiable

manifold), a Riemannian metric: in fact, if zαi = xαi + iyαi , then we have

ds2=

g

iαβ dZ αi · dI βi=

Re g

iαβ(dxαi dx

βi+ dyα

i dy βi

)

in the coordinate open set U i, in terms of the local real coordinates xi, yi.

Since X carries a complex structure, it has a canonical orientation de-

fined by this structure. Hence X has a canonical structure of an oriented

Riemannian manifold. We denote by ℓ the volume form on X with re-

spect to this oriented Riemannian structure.

Let C r ( X ), 0 ≤ r ≤ 2n (n= complex dimension of X ) denote the8

space of complex valued exterior diff erential forms of degree r . It is

a module over the algebra, F , (over the complex numbers) of complexvalued C ∞ function on X

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= τ(t i1) ∧ ... ∧ τ(t ir

) ∧ τ(t λ1) ∧ ... ∧ τ(t λm−r

)

and here the right side is zero unless i1,.., ir , λ1, . . ,λm−r =1, 2,..., m.

t 1, ..., t m are so chosen that τ(t 1) ∧ ... ∧ τ(t m) give the natural orienta-

tion,

∗(τ(t i1) ∧ ... ∧ τ(t ir

))(t λ1, ..., t λm−r

) = 0

if (i1, ..., ir , λ1,...,λm−r ) (1, 2, ..m) and

∗(τ(t i1) ∧ ... ∧ τ(t ir

))(t λ1,...t m−r )l =

= τ(t i1) ∧ τ(t ir

) ∧ τ(t λ1)...(t λm−r

).

Now, the right side is clearly ε · l where ε is the signature of the

permutation

(1, 2,..., n) (t i1,..., t ir

, t λ1,...t λm−r

).

It follows that

∗(τ(t i1) ∧ ... ∧ τ(t ir

)) = ετ(t µ1) ∧ ... ∧ τ(t µm−r

)

where 0 < µ1 < .. . < µm−r ≤ m are so chosen that10

t i1

, ..., t ir , t µ1

,.., t µm−r

= (1, 2,.., m).

Hence

(∗∗)(τ(t i1)∧), ∧ (t i1

, ..., t ir )

= ετ(t µ1) ∧ .. ∧ τ(t µr

) ∧ τ(t i1) ∧ ... ∧ τ(t ir

)

= εε′ · ℓ

where ε′ is the signature of the permutation

(1, 2, ..., m) (t µ1,.., t µr , t i1

,.., t ir ).

Hence εε′= (−1)r (m−r )

= (−1)r (m−1). It follows that

(∗∗)(τ(t i1)∧...∧τ(t i)(t i1

,.., t ir ) = (−1)r (m−1) τ(t i1

) ∧ ... ∧ τ(t ir )

(t i1, ..., t ir

).

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1. The space L p,q 13

Now, let E be a holomorphic vector bundle on X defined by (G =

U ii∈ I ; ei j : U i ∩ U j → GL(n,C)).Let h = (hi)i∈ I be a hermitian metric along the fibres. Let C p,q( X , E )

denote the space of C ∞ - forms of type ( p, q) on X with values in E

. The operator * of the associated Riemannian structure defines also a

canonical isomorphism

∗ : C p,q( X , E ) → C n− p,n− p( X , E )

as follows. If a form ϕ in C p,q( X , E ) is represented in U i by a column 13...ϕ1

i

ϕmi

where each ϕk i

is a scalar ( p, q) form, then ∗ϕ is represented by

...∗ϕ1

i∗ϕm

i in U i. Next, the hermitian metric h enables us to define an operator

# : C p,q( X , E ) → C q, p( X , E ∗),

where E ∗ is the dual-bundle to E .( E ∗ is defined by the transition func-

tions x t −1ei( x)

with respect to the covering G = U ii∈ I ). In fact if

we represent a ( p, q)-form ϕ in U i by a column

...ϕ1i

ϕmi

, where each ϕk i

is a scalar form of type ( p, q), then #ϕ in U i is given by the column of

( p, q)-forms

(#ϕ)i =hiϕ

1

ihiϕ

2i

hiϕmi

That (#ϕ)ii∈ I define forms with values in E ∗ follows from the fact

hiϕi = (t e jih je jiei jϕ j) = (t

e ji h jϕ j)

−

= et −1i j

h jϕ j

Remark . C p,q( X , E ) has a structure of an F -module where F is the 14

ring of global complex-valued C ∞-functions. For this structure of F -

modules on the C p,q( X , E )

∗ : C p,q( X , E ) → C n−q,n− p( X , E )

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is an F -linear isomorphism, while the operator

# : C p,q( X , E ) → C q, p( X , E ∗)

satisfies the condition

#( f .ϕ) = f (#ϕ) where f ∈ F , ϕ ∈ C p,q( X , E ).

It follows that the operators * and # define actually homomorphisms

∗ : A p,q( E ) → An−q,n− p( E ) (resp. A p,q( E )♯→ Aq, p( E ∗)

where Ars ( E )(resp. Ars ( E ∗))

is the vector-bundle of exterior diff erential forms of type (r , s) with val-

ues in E (resp. E ∗); here * is C-linear while # is anti linear.

In the special case when E is trivial bundle, the map # : C ( X , E ) →C ( X , E ∗) is simply the map

f − f .

Let α ∈ C p,q( X , E ) and β ∈ C rs( E ∗). Then (α ∧ β) is defined as a

scalar form of type ( p + r , q + s). For ϕ, ψ ∈ C pq( X , E ) we set

A E (ϕ, ψ)l = ϕ ∧ (# ∗ ψ). (We note that #∗ = ∗#.) If (supp ϕ ∩ supp ψ)

is compact, then

X |ϕ ∧ ∗#ψ| =

X | A E (ϕ, ψ)| < ∞

Let U be an open coordinate set in X and let ϕ, ψ be represented in

U by

ϕ =

1

p!q!ϕa

A ¯ Bdz A ∧ dz B

, ψ =

1

p!q!ψa

ABdz A ∧ dz B

,

where: a = 1, . . . , m = rankE , A = (α1, . . . , α p), B = ( β1, . . . , βq),15

1 ≤ αi, βi ≤ n, dz A= dzα1 ∧ . . . ∧ dzα p , dz β

= dz β1 ∧ . . . ∧ dz βq .

Then it follows from (1 II) that

A(ϕ, ψ) = 1 p! q!

hba

ϕa

ABφψbAB ,

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2. The space W p,q 15

(hba

)a,b=1,...,m being the local representation in U of the metric along the

fibres of E . We shall call A(ϕ, ϕ)1

2 the “length of the form ϕ.Sometimes it will be denoted also by |ϕ|.Let ϑ p,q( X , E ), be the space of C ∞ − ( p − q)− forms with compact

support. Then for ϕ,ψǫ D pq( X , E ) we can define a scalar product:

(ϕ, ψ) =

X

ϕ ∧ (∗#ψ) =

X

A E (ϕ, ψ).

Also, we denote (ϕ, ϕ) by ϕ2; is in fact a norm: the hermitian

bilinear form (ϕ, ψ) defines on D p,q( X , E ) the structure of a prehilbert

space over C. The completion of D p,q( X , E ) is denoted L p,q( X , E ).

This latter space is referred to in the sequel also as the space of squaresummable forms. Because of the Riesze-Fisher theorem L pq( X , E ) can

be identified with the space of square forms of type ( p, q)-i.e. the space

of ϕ such that X

A(ϕ, ϕ) < ∞.

2 The space W p,q

The operator ∂ : A p,q( E ) → A p,q+1( E ) defines homomorphisms (again

denoted by the same symbol)

∂ : C p,q( X , E )→

C p,q+1( X , E )

∂ : D p,q( X , E ) → D p,q+1( X , E ).

We can now define the adjoint of ∂ with respect to hermitian metric 16

on X and a hermitian metric along the fibres of the vector bundle π :

E → X , i.e. an operator

ϑ : C p,q( X , E ) → C p,q−1( X , E )

satisfying

(∂ϕ,ψ) = (ϕ,ϑψ) (1.1)

for ϕǫ D p,q

( X , E ), ψǫ D p,q+1

( X , E ). Such an operator, if it exists isimmediately seen to be unique. For the existence we have the

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( X , E ) the corresponding completion. Clearly, we have for ϕ,ψǫ D p,q

( X , E ) the inequality ϕ ≤ N (ϕ) , so that the identity

D p,q( X , E ) → D

p,q( X , E )

extends uniquely to a continuous linear map,

i : W p,q( X , E ) → L p,q( X , E ).

Proposition 1.1. i is injective.

Proof. Let ϕνǫ D p,q( X , E ) be a Cauchy sequence in N and assume that

ϕν

→0. Since ϕν is a Cauchy-sequence in N , ∂ϕν and ϑϕν are Cauchy

sequences in the norm . Hence ∂ϕν and ϑϕν tend to a limit inL p,q+1( X , E ) and L p,q−1( X , E ) respectively.

We denote these limits by u and v. By our identification of L p,q

( X , E ) as the space of square summable E -valued forms, u and v may be

regarded as forms on X with values on E .

Now for any ψ ∈ D p,q+1( X , E ) 18

(u, ψ) = limν→∞(∂ϕν, ψ) = lim

ν→∞(ϕν, ϑψ) = 0

since

ϕν

→0. Since ψ is arbitrary in D p,q+1( X , E ), U = 0. Similarly

v = 0. That is, ∂ϕν → 0 and ϑϕν → 0 in L p,q+1( X , E ) and L p,q−1( X , E )respectively. Thus N (ϕν) −→ 0 as ν −→ ∞, hence the proposition.

By continuity we obtain from (1.1) that, if ϕ ∈ W p,q( X , E ), ψ ∈W p,q+1( X , E ), then

(∂ϕ,ψ) = (ϕ,ϑψ)

Clearly, we may now regard W p,q( X , E ) also as a space of measur-

able E -valued forms on X . From the definitions, it is moreover clear that

if f ∈ W p,q( X , E ), ∂ f and ϑ f in the distribution sense are currents repre-

sentable by square summable ( p, q+1) and ( p, q−1) forms respectively.

In general W p,q( X , E ) is not the space of all square summable forms ω of

type ( p, q) whose ∂ and ϑ (in the sense of distributions) are again squaresummable. We have however, the

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Theorem 1.1. If the Riemannian metric (associated to the hermitian

metric) on X is complete, then

W p,q( X , E ) =ϕ | ϕ ∈ L p,∧q( X , E ); ∂ϕ ∈ L p,q+1( X , E ); ϑϕ ∈ L p,q−1( X , E )

Proof. We will first establish three lemmas which are needed for the

proof.

Lemma A. There exists C > 0 depending only on the dimension of X

such that for any scalar form u and any v ∈ C p,q( X , E ) ,

A E (u ∧ v, u ∧ v)( x) ≤ C | u |2 A E (v, v)( x)

where|

u|2 denotes the length of the scalar form a). (We recall that 19

A E (ϕ, ψ)ℓ = ϕ ∧ ∗#ψ). This is simply a lemma on finite dimensional

vector space with scalar products. We omit the proof.

Lemma B. Let p ∈ X be any point. The function ρ( x) = d ( p, x) =

distance from 0 of x, is locally Lipschitz continuous and wherever ρ has

partial derivatives, | d ρ |2≤ 2n.

Proof. We have by the triangle inequality

| ρ( x) − ρ( y) |≤ d ( x, y).

Now if U is a coordinate open set with coordinates ( X 1, . . . , X m) ( X

considered as a diff erentiable manifold of dimension m = 2n) and the

Riemannian metric is given on U by

gi jdxi x j then for any V ⊂ 0 ⊂ U ,

there exists λ and µ such that

λ

dxi2 ≤

gi jdxidx j ≤ µ

dxi2

so that if V is a ball about the origin in U , there exist constants C 1C 2 > 0

such that

C 1 | x − y |≤ d ( x, y) ≤ C 2 | x − y | for x, y ∈ V .

Thus | ρ( x) − ρ( y) |< C 2 | x − y | for x, y ∈ V . Hence the firstassertion.

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For the second assertion consider about a point p a neighbourhood U

in which we can introduce geodesic polar-coordinates. Let ( x1

, . . . , xm

)be a coordinate system such that xi( p) = 0 for 1 ≤ i ≤ m.

For ε > 0 we can choose U so small that | gi j |< δi j + ε, i.e.

d ( x, p)2 ≤ (1 + ε)

x2i

. In such a coordinate system, we have setting

ei = (0, . . . , 1 . . . , 0) (1 at the ith place)

∂∂ xi

( ρ) = Lt h→0

ρ(hei) −

(0)

h

On the other hand we have 20

|ρ(hei)

− ρ(0)

|≤d (0, h, ei)

≤(1 + ε)h

Since we are working with a coordinate system such that d ( x, 0) ≤(1+ε

x2i

12 . Hence ρ is Lipschitz continuous and, if the derivatives ex-

ist,∂ρ

∂ xi( p) ≤ 1. Now, by definition | d ρ |2= gi j

∂ρ

∂ X i∂ρ

∂ X jand in the coor-

dinate system introduced above, gi j(0) = δi j so that | ρ |2=m

1=1

∂ρ

∂ xi

2 ≤m. When X is a complex manifold of complex dimension n, we have,

| d ρ |2≤ 2n.

Lemma C. Let

D p,q

∂( X , E ) =

ϕ | ϕ ∈ L p,q( X , E ), ∂ϕ ∈ L p,q+1( X , E ), Supp ϕ ⊂⊂ X

D

p,qϑ

( X , E ) =ϕ | ϕ ∈ L p,q( X , E ),

ϑϕ ∈ L p,q( X , E ), ϑϕ ∈ L p,q−1( X , E ), supp ϕ ⊂ X

(where ∂ and ϑ are in the sense of distributions) and finally

D p,q( X , E ) = D

p,q

∂( X , E ) ∩ D p,q

ϑ ( X , E ).

Then

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(i) D p,q( X , E ) is dense in D p,q

∂( X , E ) w.r.t. the norm

P(ϕ) : P(ϕ)2= ϕ2

+ ∂ϕ2

and

(ii) D p,q( X , E ) is dense in D p,qϑ

( X , E ) with the norm

Q(ϕ) : Q(ϕ)2= ϕ2

+ ϑϕ2

(iii) D p,q( X , E ) is dense in D pq( X , E ) with the norm

N : N (ϕ)2= ϕ2

+ ∂ϕ2+ |ϑϕ2.

Proof. We will prove (i) ; the proofs for the other two cases are similar.21

LetG = U ii∈ I be a locally finite covering of X such that E |U i is trivial.

Let m be the rank of E . Assume further that each U i is a coordinate open

set. Let V i | V i ⊂⊂ U ii∈ I be a shrinking of U i. Let ( ρi)i∈ I be a partition

unity subordinate to V ii∈ I . Let ϕ ∈ ˙mathscrD p,q( X , E )

σ . E |U i being

trivial, we may take ρiϕi to be a Cm-valued from on Cn with compact

support in V i. For any ε > 0, we can find a Cm valued C ∞ from ψi

with compact support in V i ⊂ Cn, such that || ∂ρiϕi − ∂ψi ||< ε and

ρiϕi −ψi < ε. This can be secured by the usual regularisation methods.

Since the support of ϕ is compact, we may take ψi = 0 except for a

finite number of i. Since E |U i is trivial and ρiϕi were regarded as scalarforms through suitable trivialisations, we may now revert the process

and consider ψi as E -valued C ∞-forms with support in V i. It follows

that ψ =

ψi is a C ∞-form with compact support and

ϕ − ψ =

ρiϕi −

ψi ≤

i

ρiϕi − ψi < M ε

and similarly ∂ϕ − ∂ψ < M ε where M is the number of indices of the

finite set

i | i ∈ I , U i ∩ support ϕ φ.

Since M is fixed for a given ϕ and ε is at our choice, the lemma isproved.

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Lemma C enables us to prove the following more general statement.

Theorem 1.1. If the Riemannian metric (associated to the hermitian22

metric) of X complete, then

i) D p,q( X , E ) is dense in the spaceϕ | ϕεL p,q( X , E ), ∂ϕ ∈ L p,q+1( X , E )

,

with respect to the norm P(ϕ);

ii) D p,q( X , E ) is dense in the spaceϕ | ϕ ∈ L p,q( X , E ), ϑϕ ∈ L p,q−1( X , E )

,

with respect to the norm Q(ϕ);

iii) D p,q( X , E ) is dense in the spaceϕ | ϕ ∈ L p,q( X , E ), ∂ϕ ∈ L p,q+1( X , E ), ϑϕ ∈ L p,q−1( X , E )

,

with respect to the norm N (ϕ).

Proof. We will prove i). The proofs of ii) and iii) are similar.

In view of Lemma C it is sufficient to prove that every distributive

form ϕ in L p,q( X , E ) such that ∂ϕ ∈ L p,q+1( X , E ) can be approximated

as closely as we want by forms ψ such that ψ,∂ψ are square summable

and supp ψ,⊂⊂

X .

Let µ :| R1 → [0, 1] be a C ∞ function such that

(i) µ(t ) = 1 if t < 1 and (ii) µ(t ) = 0 if t > 2.

Let M = Suppt

| d µdt

|. Let d ( x, y) be the distance function defined by

the complete Riemannian metric of X . Then we fix a point p ∈ X and 23

set ρ( x) = d ( x, p) for x ∈ X , the function

ων( x) = µ

ρ( x)

ν

ν > 0

is locally Lipschitz, and where the derivatives∂ρ∂ xi exist, then

| d ων |2≤| 1γ

∂µ∂t ρ( x)

ν d ρ |2≤ 2nM

2

ν2

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3. W-ellipticity and a weak vanishing theorem 25

Let f ν : X → R be a C ∞ function on X , satisfying the following27

conditions:

0 f ν 1,

f ν = 1 on Bν+1 − Bν,

Supp f ν ⊂ Bν+2 − Bν−1

Let d ( x, y) ( x, y ∈ X ) be the distance function determined by the

hermitian ds2, and let

εν = inf x∈ Bν

y∈∂ Bν+1

d ( x, y).

Then εν > 0. Consider the positive C ∞ function

F ( x) =

+∞ν=0

f ν( x)

εν

and the hermitian metric ds2= F ( x)ds2. Let d ( x, y) be the distance

function determined by ds2. We have

∽

d ( Bν, Bν+1)

x ∈ inf

Bν+1 − BνF ( x)d ( Bν, Bν+1)

1εν

εν = 1.

This implies that every Cauchy sequence for the distance d ( x, y) con-

verges. Hence ds2

is a complete hermitian metric.

3 W-ellipticity and a weak vanishing theorem28

We now introduce a seminorm on W p,q( X , E ) as follows: for ϕ,ψǫ W p,q

( X , E ), let b(ϕ, ψ) = (∂ϕ, ∂ψ)+(ϑϕ, ϑψ); then b is a positive semi definite

form and b(ϕϕ) defines a semi-norm on W p,q( X , E ). the map

j : W p,q( X , E ) → W p,qb

( X , E )

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3. W-ellipticity and a weak vanishing theorem 29

Finally (1.3) yields 32

ϑ x2 C f 2.

The uniqueness of x satisfying ∂ϑ x = f and ∂ x = 0 is easily

checked.

This completes the proof of the theorem.

It follows from the regularity theorem for elliptic systems that , if f

∈ C p,q ( X , E ) ∩ L p,q then G f (can be modified on a null set so that it)

will be of class C ∞ on X .

In particular we have the

Corollary. If f

∈ D p,q( X , E ) and ∂ f = 0 then there exists ψ

∈C p,q−1

( X , E ) such that ∂ψ = f

Remark. The corollary above clearly implies the following.

If E is W pq-elliptic with respect to a complete hermitian metric, then

the natural map

H q

k ( X ,Ω p( E )) → H q( X ,Ω p( E ))

where the left-side stands for the qth cohomology with compact supports

of X with values in Ω p( E ), is the trivial map α 0 for every α ∈

H q

k

( X ,Ω p( E )).

Remark. Let G =

U i

be a covering of X such that π : E → X is

defined with respect to G by holomorphic transitions ei j: U i ∩ U j →

GL(m,C)(m = rank of E ).

Let

hi

i∈ I

be a hermitian metric along the fibers of E ; then hi is a

C ∞ function on U i whose values are positive definite hermitian matrices.

We have on U i ∩ U j 33

hi =t e ji

h je ji,

and therefore t h−1i=

t (te−1

ji )t h−1 j

t e−1 ji

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4. Carleman inequalities 31

Conditions C 1, C 2, C 3 imply C ′1

, C ′2

, C ′3

below. This weaker set of

conditions are sufficient for the “vanishing theorems” we will now prove.

C ′1

. There is given a C ∞ function φ : X → R+ (this is the same as C 1).

C ′2

. Given C o > 0, there is a non-decreasing C ∞-function λ : R → R

that λ(t ) = 0 for t C o and λ(t ) > 0 for t > C o, such that, E is

W pq elliptic with respect to (ds2, eνλ(φ)h) for every positive integer

ν.

C ′3

. The W pq ellipticity constant is independent of ν that is, there is a 35

C > 0 such that for f ∈ D pq( X , E )

f 2

ν≤ C ∂ f 2

ν+ ϑν f

2

ν ,where ν stands for νλ and ϑν for ϑνλ.

Lemma 1.6. Assume given a constant C o > 0 and that the condition

C ′1

, C ′2

, C ′3

above are satisfied (C o in condition C ′2

is taken as the above

constant). For λ as in condition C ′2 , let L p,q

ν ( X , E ) denote the space

of E-valued forms on X which are square-summable with respect to

(ds2, eνλ(φ).h). Then for f ∈ ∩νL p,q

ν ( X , E )(q > 0) such that ∂ f = 0 , there

exist Ψν for every integer ν ≥ 0 such that Ψν ∈ L p,q−1ν ( X , E ) , ∂Ψν = f

and Ψν ν≤ C f ν (We assume that ds2 is complete).

Proof. This follows from Theorem 1.3.

Theorem 1.4. Assume that conditions C ′1

, C ′2

, C ′3

are satisfied. Then

for every square-summable form f of type ( p, q)(q > 0) having compact

support such that ∂ f = 0 , there exists Ψ ∈ L p,q−1( X , E ) such that, f =

∂Ψ , Ψ 2≤ C f 2 and (support of Ψ)

⊂ xφ( x) < (sup φ( y)) y ∈ support f ).

Proof. Let C o = sup φ( x), x ∈ support f . Then we have

f 2

ν= X e

νλ(φ)

A( f , f )dx=

Supp . f e

νλ(φ)

A( f , f )dX

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= Supp f

A( f , f )dx = X

A( f , f )dX =

f

2

Since νλ(ϕ( x)) = 0 on the support of f . It follows that f ǫ ∩ν

36

L p,qν ( X , E ) for every integer ν > 0. Hence by Lemma 1.6, we can find

ψνǫ L p,q−1ν2

( X , E ) for every νǫ Z+ such that−∂ψν = f and || ψ ||2ν≤ C || f ||2.

On the other hand, we have || ψν ||ν≥|| ψν || so that, || ψν ||2≤|| ψν ||2ν≤C || f ||2= C 1 say. It follows that we can assume (by passing to a subse-

quence if necessary) that ψν converges weakly in L p,q−1( X , E ) to a limit

ψ. We have || ψ ||< C || f ||. On the other hand, for every ε > 0,

φ( x)>c0+εe

νλ(φ)

A(ψν, ψν)dX ≤ C 1

and since λ is non-decreasing, we have,

eνλ(C 0+ε)

ϕ≥C 0+ε

A(ψν, ψν)dX ≤ C 1.

It follows that

ϕ≥C 0+ε

A(ψν, ψν)dX tends to zero and hence ψν → 0

almost everywhere in ϕ ≥ C 0 + ε. (The integrand is positive for all

ν.) Hence ψ = 0 on

x

|ϕ( x)

≥C 0 + ε

for every ε. Hence support

ψ ⊂ x | φ′( x) ≤ C 0. Finally for

u ∈ D n− p,n−q( X , E ∗), (−1) p+qψ,−∂u = (−1) p+q

ψ ∧

−∂u

= Lt (−1) p+q

ψν ∧

−∂u

=

f , u

.

Hence ∂ψ = f in the sense of distributions. This completes the proof

of the theorem.

Remark. The proof contains the following lemma (which has no con-37

nection with W p,q-ellipticity.

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4. Carleman inequalities 33

Lemma 1.7. Suppose given a C ∞ non-decreasing function λ : R → R

such that λ(t ) = 0 for t ≤ C 0, λ(t ) > 0 for t > C 0 , and a C ∞ functionϕ : X → R. Suppose further that there is given an f ∈ L pq( X , E ) such

that sup ϕ( x) on the support of f is C 0. If for every integer ν > 0 there

exists ψν such that ∂ψν = f and || ψν ||2ν≤ C || f ||2 for a constant C

independent of ν , then there exists ψ ∈ L p,q−1( X , E ) such that ∂ψ = f , ||ψ ||2≤ C || f ||2 and support ψ ⊂ x | ϕ( x) ≤ C 0. (Here by) || ψν ||2ν we

mean

eνλ(φ) A(ψν, ψν)dX ).

Suppose now that the function φ in C ′1

satisfies the following addi-

tional condition.

C ′4

. For every C > 0, the set x | φ( x) < C ⊂⊂ X .

Remark. If C ′4 is satisfied in addition to C ′1, C ′2 and C ′3 then in Theorem1.4, we can assert that the support of ψ is compact.

Here we can state, as a corollary to Theorem 1.4 the following The-

orem 1.4′. If the hypothesis of Theorem 1.4 are fulfilled and if C ′4

holds,

then

H q

k ( X ,Ω p( E )) = 0.

Lemma 1.8. Let V ⊂ L p,q( X , E ) be the set V = V p,q( X , E ) = ϕ ∈L pq( X , E )

there exists ψ ∈ L p,q−1( X , E ) such that −∂ψ = ϕ and N = 38

N p,q( X , E ) = ϕ ∈ L p,q( X , E ) | ϑϕ = 0. Then N is the orthogonal

complement of V provided that the metric is complete.

Proof. Let ρ ∈ L p,q( X , E ). Then ρ ∈ orthogonal complement of

V if and only if ( ρ,−∂ψ) = 0 for every ψ ∈ L p,q−1( X , E ) with

−∂ψ ∈

L p,q−1( X , E ). Since the metric is complete, this is equivalent to ( ρ,−∂ψ) =

0 for every ψ ∈ D p,q−1( X , E ). Hence the lemma.

Theorem 1.5. Assume conditions C ′1

, C ′2

, C ′3

to hold for π : E → X. Let

f ∈ L p,q+1( X , E ) be such that sup ϕ( x) on the support of f is C 0. Suppose

further that ( f , g) = 0 for all g which belong to some L p,q+1ν , and are

such that ϑνg = 0(ν = 1, 2, ....)(1). Then there exists ψ ∈ L p,q( X , E ) such

that −∂ψ = f and support ψ

⊂ X

|ϕ( X )

≤C 0

.

1In view of the choice of λ, we have ( f , g) = ( f , g)ν.

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Theorem 1.8. Let π : E → K be a holomorphic vector bundle. Assume

that for a suitable complete hermitian metric ds2

on X and a suitablehermitian metric h along the fibres of E, conditions C ′

1 , C ′

2 , C ′

3and

C ′4

hold. Then the image ∂D pq( X , E ) of ∂ in D p,q+1( X , E ) is a closed

subspace for the usual topology on D pq( X , E ). Hence H q+1

k ( X ,Ω p( E ))

has a structure of a separated topological vector space.

Proof. The “usual” topology on D pq( H , E ) may be described as fol-

lows. Let K ν be an increasing sequence of compact sets such that

K ν ⊂o

K ν + 1 and U K ν = K . Further, let G = U ii∈ I be a locally finite

covering of X by open sets such that each U i is a relatively compact open

subset of a coordinate open set V i on X . Let Gν = U i|U i ∩ K ν φ. We

topologize each D pq(K ν, E ) = ϕ|ϕ ∈ D pq( X , E ), supp ϕ ⊂ν as follows:for U i ∈ Gν, ϕ|U i may be regarded as a C ∞ vector-valued function ϕi on

U i ; a fundamental system of neighbourhoods of zero in D pq(K ν, E ) is

given by, ϕ ∈ D pq(K ν, E )

|αϕi

< εα, for every U i ∈ Gν

where εα is an arbitrary family of positive reals.

D pq(K ν, E ) is a Frechet space. There is a natural injection

D pq(K ν, E ) → D

pq(K ν+1, E ).

The image of D pq(K ν, E ) is a closed subset of D pq(K ν+1, E ). The in-42

duced topology on the image coincides with the topology of D pq(K ν, E ).

This shows thatD pq( X , E ) is a strict inductive limit of the Frechet spaces

D pq(K ν, E ) [8] 66-67; [9] (225-227). The “usual” topology of D pq( X , E )

is the inductive limit topology.

A subset of D pq+1( X , E ) is closed if, and only if, it is sequentially

closed ([19], 228). Therefore it will be sufficient to prove that ∂D pq

( X , E ) is sequentially closed in D pq+1( X , E ).

Let ϕii∈N be a sequence of ( p, q)- forms of D pq( X , E ). Such that

the sequence ∂ϕii∈N ⊂ ∂D pq

( X , E ) converges to an element ϕ of D pq+1

( X , E ). The sequence ξϕi is a bounded set in D pq+1( X , E ).

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42 2. W-ellipticity on Riemannian manifolds

(ϕ is regarded as a 1-form with values in the tensor bundle). In the

local-coordinate system, a form ϕ ∈ C q

( X ) may be written in the form.

ϕ =

1

1

q!ϕ I dx I

where I runs over all q-tuples (i1, . . . , iq) of integers with 0 < i j n and

if I = (i1, . . . , iq) is a permutation (σ( ji), . . . , σ( jq)) of I ′ = ( j1, . . . , jq) ,

then

ϕ I = εσϕ I ′ .

(In particular, if I has repeated indices, ϕ I = 0). We have

A(ϕ, ψ) =

1

q! ϕ I ψ I

(ϕ, ψ ∈ C q

)

For ϕ ∈ C q and a q-tuple I = (i1, . . . , iq) ,

d ϕiI =∂ϕ I

∂ xi+

r

(−1)r ∂ϕi

∂ xir i1, . . . , ir , . . . iq

= iϕ I +

r

(−1)r ir ϕii1

, . . . ir . . . , iq (2.1)

as is seen by a direct computation. Similarly we have48

δϕ J =

−Σiϕ

i

J

(2.2)

where J is a (q − 1)-tuple and ϕi is the (q − 1)-form defined by

ϕi J =

gi jϕiJ

for any (q − 1)-tuple J = ( j1,... jq−l).

From (2.1) and (2.2), it follows that

δd ϕ I = −n

i=1

i(d ϕ)i I

= −n

i=1i

i

ϕ I −q

i=1

(−1)

r

i(ir ϕi

)ii...ˆir ...iq (2.3)

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6. A maximum principle 45

Proof. By Lemma 2.1 we have for ϕ ∈ Dq( X ),

ϕ 2+(κϕ,ϕ) = (d ϕ , d ϕ) + (δϕ, δϕ).

Since K is compact, there is C ≥ 0 such that (K ϕ, ϕ) > −CC |ϕ|−2

for every ϕǫ C q and each point of K . Hence,

ϕ 2+

X

A(k ϕ, ϕ) ≤ C ϕ 2k + d ϕ 2

+ δϕ 2 .

Since A(κϕ,ϕ) ≥ 0 on X − K we have for ϕ ∈ Dq,

ϕ 2≤ C ϕ 2

+ d ϕ 2+ δϕ 2

The lemma follows now from the fact that Dq is dense in W q. (Theorem

2.1.)

Remark. We have proved more: there is a C > 0 such that 51

ϕ2 C ϕ 2

K + d ϕ2+ δϕ2

Let X be a manifold as in the above lemma.

Let ϕ ∈ C q. Identity (2.8) shows that |ϕ|2 0 at each point of the

set Y where ϕ = 0 and A(K ϕ, ϕ) 0

Applying a classical lemma of E . Hopf ([36], 26-30) we see that

|ϕ|2 cannot have a relative maximum at any point of Y . Thus, if X iscompact, |ϕ|2 takes its, maximum in the set Supp f ∪ K .

If X is not compact we cannot draw the same conclusion. However

the following proposition will provide an estimate of |ϕ|2 on X in terms

of Sup |ϕ|2 on Supp f ∪ K .Proposition 2.3. Let X be a connected oriented and complete Rieman-

nian manifold. Assume give a compact set K on X such that for X

K , A(K ϕ, ϕ)( x) ≥ 0 for every ϕ ∈ C q. Suppose that ϕ ∈ C q ∩ Lq is such

that ϕ = f ∈ Lq. Then ϕ ∈ W q and sup x∈ X

|ϕ( x)|2 C 0 where

C 0 = Sup x∈supp f ∪K |ϕ( x)|2.

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For the proof of this proposition we utilise the following result

Lemma 2.4 (Gaff ney [12]). Let K be an oriented complete Riemannian

manifold and ω a C ∞ 1-form. Then if

X |ω|dX < ∞ and

X

|δω|dX < ∞ ,

we have X

δωdX = 0.

Proof. Let λ be a real value C ∞ function on such that λ(t ) = 1 for52

t 0 and λ(t ) = 0 for t 1. Let p0 ∈ X be a fixed point and ρ = ρ( x)

denote the distance function (from p0). Then ρ( x) is locally Lipchitz as

was remarked earlier (Lemma B of Chapter 1). Let 0 < r < R and let us

consider the form

λ ρ( x)

−r

R − r · ω

This form is locally Lipchitz and has compact support in the ball

x| ρ( x) R.

Hence by Stoke’s formula (which is applicable to Lipchitz continu-

ous form), X

d ∗ λ ρ( x) − r

R − r

· ω = 0;

We have,

X

1 R − r

λ ρ( x) − r −r · d ρΛ ∗ ω +

X

λ ρ( x) − r R − r

· d ∗ ω = 0.

Now set R = 2r . Then X

1

r λ′ ρ( x) − r

r

d ρΛ ∗ ω +

X

λ ρ( x) − r

r

· d ∗ ω = 0

Now |d ρ|2 (lemma B of Chapter 1).

Hence, since ∗ω is integrable, we have53 X

|d ρΛ ∗ ω|dX < ∞.

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6. A maximum principle 49

Since K is compact, there is a constant C 4 > 0 such that

| A(K u, u)| C 4 A(u, u)

for any u ∈ C q and at each point of K . Then we have X

| A(K ϕ, ϕ)|dX =

X −K

A(K ϕ, ϕ)dX +

K

| A(K ϕ, ϕ)|dX

X

A(K ϕ, ϕ)dX + 2

K

| A(K ϕ, ϕ)|dX

(K ϕ, ϕ) + 2C ||ϕ||2 2C ||ϕ||2 + ||d ϕ||2 + ||δϕ||2 < ∞.

Finally we have X

| A(ϕ, ϕ)|dX ≤ 1

2

X

A(ϕ, ϕ)dX +1

2

X

A(ϕ, ϕ)dX < ∞.

Then, it follows from (2.9) and from the fact that |λ′ | and |λ′′| are

bounded, that X

|λ(|ϕ|2)|dX < ∞.

Hence, by Lemma 2.4, X

λ(|ϕ|2)dX = 0, i.e. by (2.9)

X

λ′′(|ϕ|2)2(d |ϕ|2)2dX + 2

X

λ′(|ϕ|2). A(K ϕ, ϕ)dX + 2 X

λ′(|ϕ|2)|ϕ|2dX =

X

2λ′(|ϕ|2) A(ϕ, ϕ)dX . (2.11)

Let us consider now sup |ϕ| on Supp f ∪ k . If it is not finite there is 56

nothing to prove. If it is finite, we set C 0 = Sup |ϕ|2 on K

Support f .

In view of our choice of the function λ all the terms on the left hand side

of (2.11) are non negative. We obtain

0 2 X

λ′(|ϕ|2)|ϕ|2dX X

2λ′(|ϕ|2) A(ϕ, ϕ)

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7. Finite dimensionality of spaces of harmonic forms 51

Theorem 2.3. Let X be a connected, oriented and complete Riemannian

manifold and assume that A(K ϕ, ϕ) C |ϕ|2

for all ϕ ∈ C q

for someC > 0 , outside a compact set K ⊂ X. Then for γ < C, M

qγ is finite

dimensional. In particular dim Hq < ∞.

Proof. Let ϕ ∈ Mqγ . Since ϕ ∈ W q, then by Lemma 2.3

ϕ2+ (K ϕ, ϕ) = d ϕ2

+ δϕ2

and by (2.12)

ϕ2+ (K ϕ, ϕ) = d ϕ2

+ ||δϕ2= γ ϕ2

Now K being compact, there is a constant C 2 > 0 such that (K u, u)

−C 2|u|2

for every u ∈ C q

and at each point of K . Since (K ϕ, ϕ) C 2|ϕ|2

on the complement X − K we have

ϕ2+C ϕ2

X −K C 2ϕ2K + γ ϕ2

or again,

ϕ2+C ϕ2

(C +C 2)ϕ2K + γ ϕ2

Hence, 58

(C − γ )ϕ2 (C +C 2)ϕ2

K (2.13)

Suppose now that Mqγ is infinite dimensional for γ < C , then the evalu-

ation map

ω ω( x) ( x ∈ K )(which associates to a form ω, the element of the qth exterior power of

the tangent space at x which is defined by ω) is a linear map into a finite

dimensional space. Hence the kernel of this map is of finite codimen-

sion. Since the finite intersection of subspaces of finite codimension

is non-zero (over an infinite field !) it follows that given any sequence

x1, . . . , xν, . . . of points of K , we can find forms ϕ∈ ∈ Mqγ such that

ϕν( xi) = 0 for i ν. We may further assume that ϕν = 1. It follows

then that we can choose a subsequence ψk = ϕνk of ϕν such that ψk

converges weakly to a limit ψ in Lq. Now this implies that ψ ∈ Mqγ ,

since for any ϕ

∈D q,

(ψ,∆ϕ) = Lt(ψk ,∆ϕ) = Lt(∆ψk , ϕ) = γ (ψ, ϕ).

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Proof. Since now 64

ϕ 2= d ϕ2

+ δϕ2= 0 for ϕǫ Hq

we have ϕ = 0 for ϕǫ Hq. A form invariant under parallel translation

is determined by its value at one point. Hence dim Hq

nq

. If ϕǫ Hq

is non-zero, then ϕ2 < ∞; on the other hand since ϕ = 0, | ϕ |2 is a

constant and the last assertion follows.

Lemma 2.7. Under the hypotheses of Theorems 2.3 and 2.4, for every

sǫ S q there exists a unique xǫ S q such that

s = x

in the sense of distributions. Moreover

x C ′ s ,

dx|2 + ||δ x2 ≤ C ′ s 2

C ′ being the constant which appears in Theorem 2.4. If s ǫ C q ∩ S q , then

(x can be modified on a null set in sch a way that) x ǫ C q ∩ S q and the

equation x = s holds in the ordinary sense.

Proof. S q as a (closed) subspace of the Hilbert space W q is a Hilbert

space. By theorem 2.4 the norm N is equivalent on S q to the norm

(||d ||2+ ||δ||

2

)

12

. f

(s, f ) is a continuous linear form on S

q

. Thus thereis a unique x ∈ S q such that

(s, f ) = (dx, d f ) + (δ x, δ f )for all f ǫ S q.

Let now ϕ be any element of W q and let h and ψ be its orthogonal

projections into Hq and S q. They are uniquely defined, and furthermore

ϕ = h + ψ

Since h is orthogonal to s and dh = 0, δh = 0, then65

(s, ϕ) = (s, ψ) = (dx, d ψ) + (δ x, δψ) = (dx, d ϕ) + (δ x, δϕ).

This proves the lemma.

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Moreover

x ≤ C ′ φ ,

δ x 2≤ C ′ φ 2

(C ′ being the constant introduced in Theorem 2.4)

Proof.

s (φ, s)

is a continuous form on S q . Then there exists a unique x ∈ S q such that

(φ, s) = (dx, ds) + (δ x, δs) for all s ∈ S q.

Moreover, by Theorem 2.4,

x 2≤ C ′( dx 2+ δ x 2) = C ′(φ, x) ≤ C ′ φ . x ,

dx 2+ δ x 2

= (φ, x) ≤ φ . x ≤ C ′ φ 2 .

Let now u ∈ Dq · u can be written

u = h + s, h ∈ Hq, s ∈ S q.

Since h⊥ d Dq−1 in Lq and dh = 0, δh = 0, then (ϕ, u) = (ϕ, s) =67

(dx, ds) + (δ x, δs) = (dx, du) + (δ x, δu) = ( x,

u) i.e.

ϕ = x

in the sense of distributions.

Let φ ∈ C q ∩ d Dq−1. By the regularity theorem, we can modify x on

a null set in such a way that x ∈ C q ∩ S q , and that the latter equation

holds in the ordinary sense. Thus, by proposition 2.2

δdx 2≤ 1

σ d φ 2

+σ dx 2= σ dx 2 for all σ > 0.

Henceδdx = 0, ϕ = d δ x.

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Theorem 2.6. Every current f ∈ Lq can be decomposed as the sum of

three currents f = h + d ϕ1 + δψ1

with h ∈ Hq, ϕ1 ∈ W q−1, ψ1 ∈ W q+1. Moreover

N (h)2= h 2≤ f 2, N (ϕ1)2 ≤ (C ′+1) ϕ 2, N (ψ1)2 ≤ (C ′+1) ψ 2

Proof. First of all any element f ∈ Lq can be expressed in a unique way

as the sum

f = h + ϕ + ψ

of three forms h ∈ Hq, ϕǫ d D q−1, ψǫδD q+169

|| f ||2 = ||h||2 + ||ϕ||2 + ||ψ||2.

Next we apply to ϕ and ψ Lemma 2.8 and 2.9 setting then

ϕ = δϕ1, ψ = d ψ1;

ϕ1 ∈ W q−1 ψ1 ∈ W q+1.

Moreover

N (ϕ1)2= ||ϕ1||2 + ||ϕ||2 ≤ (C ′ + 1)||ϕ||2 ≤ (C ′ + 1)|| f ||2

N (ψ1)2=

||ψ1

||2+

||ψ

||2

≤(C ′ + 1)

||ψ

||2

≤(C ′ + 1)

|| f

||2

Remark. If the hypotheses of Theorem 2.3 are satisfied, not only for the

forms of degree q, but also for those degree q − 1 and q + 1 then spaces

Hq−1, Hq+1, S q−1 and S q+1 can be introduced and one checks that

ϕ1 ⊥ Hq−1, i.e. ϕ1 ∈ S q−1

ϕ1 ⊥ Hq+1, i.e. ψ1 ∈ S q+1.

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Chapter 3

Local expressions for and

the main inequality

9 Metrics and connections

We now go back to the case of holomorphic vector bundles on a complex 70

manifold.

Let x be a complex manifold and π : E → X a holomorphic vector

bundle. Let U = U iiǫ I be a covering of X such that π : E → X is

defined with respect to U by transition functions ei j : U i ∩ U j → GL

(m, C) ( E is of rank m).

Let (hi)iǫ I be a hermitian metric along the fibres of E : then hi are C ∞

functions U i whose values are positive definite hermitian matrix such

that.

hi =t −e jih je ji =

t e−1i j h je

−1i j on U i ∩ U j.

Consider the 1-form li = h−1i

∂hi where ∂ is the exterior diff erentia-

tion with respect holomorphic coordinates. We have for this family of

forms the following.

Lemma 3.1. If li = h−1i

∂hi , then li are (1, 0) forms with values in M m(C)

and on U i ∩ U j we, have

li = e−1 ji l je ji + e−1

ji ∂e ji.

61

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64 3. Local expressions for and the main inequality

The bundle Θ∗ decomposes canonically as a direct sum

Θ∗⋍ Θ

∗ ⊕ Θ∗

where Θ (resp. Θ) is the holomorphic tangent bundle (resp. anti holo-74

morphic tangent bundle) of X . (The two bundles in the above decompo-

sition are regarded as diff erentiable bundles. As a diff erentiable bundle

Θ∗ (resp. (Θ

∗), is simply the bundle of C ∞− forms on X of type (1,0)

(resp. type (0,1)). The decomposition above gives a direct-sum repre-

sentations

A(Θ∗) ⋍ A(Θ∗) ⊕ A(Θ

∗),

and for any holomorphic vector bundle E on X ,

Γ( X , A( E ⊗ Θ∗)) ⋍ Γ( X , A( E ⊗ Θ∗)) ⊕ Γ( X , A( E ⊗ Θ∗

)).

Now if we are given a connection on E , it defines as we have re-

marked a covariant derivation

: Γ( X , A( E )) → Γ( X , A( E ⊗ Θ∗));

composing with the natural projections defined by the direct sum de-

composition above we obtain maps

′ : Γ( X , A( E )) → Γ( X , A( E ⊗ Θ∗

))

and ′′

: Γ( X ,A

( E ))→Γ( X ,

A( E

⊗Θ

∗

)).

Remarks. (1) As it has been remarked at the end of §3, the metric hion E induces a metric on the dual E ∗: this is simply given by the

family of positive definite matrix-valued functions

t h−1i

i∈ I

. The

corresponding connection is given locally on U i by the form −t li.75

It is also easy to see that the curvature form of this connection is

given by −t sii∈ I

(2) Let ¯ E denote the “anti holomorphic” bundle associated to E . ¯ E

is the bundle with transition functions ei j. Then t hi = hi define a

metric on ¯ E as well. In this case, the local forms

li = h−1i ∂ hi

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Proposition 3.1. h = 0

Proof. Let z1,.., zn be a coordinate system on U 1. In this proof we write

l for li and h for hi.

We have77

αh = (h)

∂

∂ zα

= ∂αhab −

c

lcbαhac

= ∂ xhab −

(h−1)cd ∂αhdb

hac

= ∂αhab −

δd a∂αh

db= ∂αhab − ∂αhab = 0.

Similarly one shows that

αh = h(∂

∂ zα ) = 0.

Hence the proposition.

(Here h is regarded as a function from U i into

C

m ⊗Cm∗

using the

given trivialisation of E | U i. A vector t ∈ Cm is an m-tuple denoted

(t a)1≤a≤m and t ǫ Cn

is again an m-tuple but is denoted (t a)1≤a≤m. Further

conventions are as follows: a vector in the dual of Cm is an m-tuple but

with subscripts instead of super scripts. In other words we write

t =

t aea for t ∈ Cm

t = t aea for t ∈ Cm

t =

t ae∗a for t ∈ Cm∗

t =

t ae∗a for t ∈ Cm∗

where ea (resp. ea, e∗a, e∗

a is the canonical (resp. conjugate of the

canonical, dual of the canonical, conjugate dual of the canonical) basis

of Cm).

Proposition 3.2. Let z1, . . . , zn be a coordinate system on U i. Then in78

U i , the curvature form can be written as

ai = (sab ¯ βα

)abd z β ∧ dzα,

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9. Metrics and connections 67

where

(sab ¯ βα)ab = ¯ βα − α ¯ β (3.1)

Proof. Let t = (t a) be a section of the bundle over U i.

Then

¯ βt = ∂ ¯ βt = (∂ ¯ βt a)

Hence

(α ¯ βt )a= ∂α∂ t a

β+

b

labα∂ ¯ βt b

where l =

lab β

dzα is the connection form.

αt = ∂αt + (labα)(t ) = ∂αt a +b la

bα t b

so that

¯ βαt = ∂ ¯ β∂αt a +

b

∂ ¯ βlabαt b +

b

labα∂ ¯ βt b

= ∂ ¯ β∂αt a + sab ¯ βα

t b +

b

labα∂ ¯ βt b.

Hence −(α ¯ βt − ¯ βαt )a=b

sab ¯ βα

t b.

This proves the proposition.

The equation (3.1) is known as the Bianchi Identity.

(We remark that s is a form of type (1,1).).

The following lemma is easy to prove. 79

Lemma 3.3. # = #.

Next we specialize the preceding considerations to the case E = Θo

the homomorphic tangent bundle. Let z1, . . . , zn be a local coordinate

system in an open set U ⊂ X. Let

g =

gαβdz−αdz β

be the expression for the hermitian metric in this coordinate system. The

associated ∂-connection is then given in U by the (1,0)- form

C α βγ dzγ

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9. Metrics and connections 69

γ

Γγ

βγ = −r

S γ

βγ

Since Θ ⋍ Θ⊕Θ as a diff erentiable vector bundle, we have a direct

sum decomposition

Γ( X , A(Θ ⊗ Θ∗)) ⋍ Γ( X , A(Θ ⊗ Θ∗)) ⊗ Γ( X , A(Θ ⊗ Θ∗

))

and hence a natural projection 81

Γ( X , A(Θ ⊗ Θ∗)) → Γ( X , A(Θ ⊗ Θ∗));

composing this projection with D and , we obtain two linear maps

′′ : Γ( X , A(Θo)) → Γ( X , A(Θo ⊗ Θ∗o))

and D′′ : Γ( X , A(Θo)) → Γ( X , A(Θo ⊗ Θ∗o)).

If the metric is Kahler, these two maps coincide. In a similar way,

composing the natural projection

Γ( X , A(Θ ⊗ Θ∗)) → Γ( X , A(Θ ⊗Θ∗o))

with and D we define two linear maps,

′ : Γ( X , AΘo)) → Γ( X , A(Θo ⊗ Θo

∗))

D′

: Γ( X ,AΘ

o))

→Γ( X ,

A(Θ

o ⊗Θ

∗o))

which coincide if the hermitian metric of X is a Kahler metric.

Suppose now that we are given a vector bundle E on X and hermitian

metrics on X and along the fibres of E . We then have canonical ∂ -

connections on the bundle E and Θo and a ∂- connection on Θo. These

connections extend canonically to connections on each of the bundles

p

ΛΘ∗o ⊗

q

ΛΘ∗o ⊗ E

We denote by the covariant derivation in any of these bundles: 82

: Γ( X , A( p∧Θ∗

o ⊗ q∧Θ∗o ⊗ E )) → Γ( X , A(

p∧Θ∗o ⊗ q∧Θ∗ ⊗ E ⊗ Θ∗))

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and

T = − ∗ #−1

S ∗ # : C pq

( X , E ) → C p,q

−1

( X , E ) (3.8)

so that

ϑ = θ + T . (3.9)

Finally let

= ∂ϑ + ϑ∂

If the hermitian metric on X is a Kahler metric, then86

∂ =∼

∂, ϑ =

∼

ϑ, =∼

.

For ϕ ∈ C pq( X , E )

ϑϕa AB′ = (−1) p−1∇αϕa

Aα B′ , (3.10)

so that, exactly as in the case of the Laplacian in Chapter 2, we have

∼ϕa AB

= −∇α∇αϕa

AB+

qr =1

(−1)r −1(∇α∇ βr − ∇ βr

∇α)ϕa Aα

B′r

(3.11)

where

∇α= gαβ∇ β,

and A = (α1, ........α p), B = ( β1, ......βq), B′r = ( β1, ......., ˆ βr ,....,βq).

In view of the Ricci identity, the summand of (3.11) can be ex-

pressed by

qr =1

(−1)r −1(∇α∇ βr − ∇ βr

∇α)ϕa Aα

B′r = (

∼

K ϕ)a AB (3.12)

where∼

K is a mapping

∼

K : C pq( X , E ) → C pq( X , E ),

which is linear over C ∞ functions, whose local expression involves lin-early (with integral coefficients) only the coefficients of the curvature87

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Hence, by (3.14),

∼

− ∗−1∼∗ =

∼

K− ∗−1∼

K ∗ . (3.15)

If the metric on X is Kahler, this identity becomes

− ∗−1∗ =

∼

K − ∗−1∼

K ∗ .

A direct computation shows that, in this case,∼

K − ∗−1∼

K∗ does not

depend on the curvature form of the Kahler metric. We have in fact (see

[2], 103)

((

∼

K − ∗−1

∼

K∗))a AB =

q

i=1

(−1)i sab βi βϕb β

AB′i+

p

j=1

(−1) j sabαα j ϕbα A′ j

¯ B + sab β βϕb

AB

where A′ j= (α1, . . . , α j,....,α p).

Starting from this formula, it is easy to check that

∼

K − ∗−1∼

K∗ = e(s)Λ − Λe(s),

where89

e(s) : C pq( X , E ) → C p+1,q+1( X , E )

is the linear mapping locally defined by

(e(s)ϕ)i =√ −1sa

ib ∧ ϕb,

and Λ is the classical operator of the Kahler geometry (see e.g. [35],

42). Thus we have, in the Kahler case,

− ∗−1∗ = e(s)Λ − Λe(s)

which, by (1.7), can also be written

E − #−1 E ∗ # = e(s)Λ − Λe(s).

This formula, which was first obtained in [4], 483, yields W -ellipti-city conditions on Kahler manifolds (see [2], ).

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11. The main inequality 75

If the metric on X is not Kahler, then it follows from Lemma 3.4 and

from (3.15) that

− ∗−1∗ =

∼

K − ∗−1∼

K ∗ +G1ϕ + G2′ϕ +G3∇′′ϕ

where

G1 : C pq( X , E ) → C pq( X , E ),

G2 : C pq( X , E ⊗ Θ∗) → C pq( X , E ),

G3 : C pq( X , E ⊗ Θ∗) → C pq( X , E ),

are linear over C ∞ functions. Their local expressions involve the torsion

tensor and its first covariant derivatives. If the hermitian metric is a 90

Kahler metric, then G1 ≡ 0, G2 ≡ 0, G3 ≡ 0.

11 The main inequality

We shall now establish an integral inequality which, under convenient

hypotheses, yields a sufficient condition for the W pq- ellipticity of E .

Let ϕ ∈ C pq( X , E )(q > 0). Let ξ and η be two tangent vector fields

to X , defined by

ξ = ( ξ β = hbaγ ϕa A β ¯ B′ϕb ¯ Aγ B′

, ξ ¯ β= 0)

η = (ηγ = 0, ηγ = hba βϕa A β ¯ B′ .ϕb ¯ Aγ B′)

We have (η being the complex dimension of X )

div ξ =

2ni=1

Di ξ i =

n β=1

β ξ β −α,β

S βα βξ α,

where

β ξ β = hba βγ ϕa A β ¯ B′ .ϕbaγ B′

+ hbaγ ϕa A β ¯ B′ . ¯ Bϕb ¯ Aγ B′

= hbaγ βϕa A β ¯ B′ .ϕbaγ B′

+ hbaγ ϕa A β ¯ B′ . ¯ Bϕb ¯ Aγ B′

+ hba( βγ − γ β)ϕa A β ¯ B′ .ϕb ¯ Aγ B′

.

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11. The main inequality 77

= 2∂ϕ2+ +c(| S |2 ϕ, ϕ),

∼ϑϕ2 ≤ 2(ϑϕ2+ T ϕ2) ≤ 2ϑϕ2

+ c X

| S |2 A(ϕ, ϕ)dX

= 2ϑϕ2+ c(| S |2 ϕ, ϕ),

c being a universal positive constant (depending only on the dimension

of X ).

Furthermore, again by the Schwartz inequality, we have 93

1

p!(q − 1)!|

S α β β ξ β | ≤ 1

p!(q − 1)!| S ξ |≤ q | S | A(′′ϕ,′′ϕ)

12 A(ϕ, ϕ)

12

≤ qε

2| S |2 A(ϕ, ϕ) +

q

2ε A(′′ϕ,′′ϕ),

1

p!(q − 1)!| S βγ β

ηγ | ≤ 1

p!(q − 1)!| S η |≤ q | S | A(

∼

ϑϕ,∼

ϑϕ)12 A(ϕ, ϕ)

12

≤ qε

2| S |2 A(ϕ, ϕ) +

q

2ε A(

∼

ϑϕ,∼

ϑϕ),

for any ε > 0.

Substituting these estimates in (3.16) we obtain for any ε > 0

1 − q

ε

′′ ϕ2

+

∼

K ϕ −

2c + 2qε +qc

ε

| S |2 ϕ, ϕ

≤2

∂ϕ

2+ +2

1 +

q

ε ϑϕ

2.

Setting, for instance ε = 2q , and

K =∼

K −

5

2c + 4n2

| S |2 . Id : C pq( X , E ) → C pq( X , E ) (3.17)

we obtain the following

Proposition 3.3. For any ϕǫ D pq( X , E ) (q > 0) we have

12′′ϕ2

+ (K ϕ, ϕ) ≤ 3 ∂ϕ2+ ϑϕ2 .

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If the hermitian metric on X is a K¨ ahler metric, the above proposition

can be considerably sharpened. In fact, in this case, S ≡ 0 , ∂ = 0 , 94∼

ϑ = ϑ. It follows from (3.16 ) that for every ϕǫ D pq( X , E ) (q > 0) satisfies

the identity

′′ϕ2+ (K ϕ, ϕ) = ∂ϕ2

+ ϑϕ2 (K¨ ahler) (3.18)

where we have set K =∼

K . Also in the hermitian case the choice of the

numerical constants can be improved.

As a corollary to proposition 3.1 we have the following

Theorem 3.1. If there exists a positive constant C 2 such that

A(K ϕ, ϕ) ≥ C 2 A(ϕ, ϕ)

for every ϕǫ C pq( X , E ) (q > 0) and at each point of X, then E is W p,q-

elliptic.

Remark. Eff ect on K of a change of the hermitian metric on E .

Let ϕ : X R be any C ∞- function. We wish to examine how the

operator K above changes when we replace the hermitian metric h on E

by h′= eϕh. (The hermitian metric on the base X remains undisturbed).

First, the curvature form s′ of h′ is given by

s′= s +α,β

∂2ϕ∂ zα∂ z β

dzα ∧ dz β. I m,

where s, as above, is the curvature form of h; ( zα)1≤α≤n is a local coordi-

nate system in an open set U ⊂ X on which a local trivialization of E is

assumed given; the symbol I m stands for the (m × m) identity matrix. It

is immediate from (3.13) that

(K ′ϕ)a

AB= (K ϕa) AB +

i

(−1)i ∂2ϕ

∂ zα∂ z βiϕaα

AB′i

.

where K ′ is the analogue of K , defined now with respect to the new95hermitian metric h′.

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Chapter 4

Vanishing Theorems

12 q-complete manifolds

Let X be a complex manifold and φ : X → R a C ∞-function. The Levi 96

form L (φ) of φ is the hermitian quadratic diff erential form on X defined

as follows: let z1,..., zn be a coordinate system on an open set U ⊂ X ;

then

L (φ)

∂

∂ zα,

∂

∂ z β

=

∂2ϕ

∂ zα∂ z β.

(L (ϕ) is thus a hermitian form on the holomorphic tangent space).

Definition 4.1. A C ∞ function φ : X → R is strongly q-pseudo-convex

if the Levi form L (φ) has at least (n − q) positive eigen-values at every

point of X.

Definition 4.2. A complex manifold X is q-complete if there exists a C ∞

function φ : X → R such that

(i) φ is strongly q-pseudo-convex

(ii) for c ∈ R , the set x | x ∈ X , φ( x) < c

is relatively compact in X.

Lemma 4.1 (E. Calabi). Let H be a hermitian quadratic di ff erential form on X and G a hermitian metric on X . Assume that H has at least p

79

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80 4. Vanishing Theorems

positive eigen values. Let ε1( x),...,εn( x) be the eigen values of H (w.r.t.

G) at x in decreasing order: εr ( x) ≥ εr +1( x)Then given c1, c2 > 0 , G can be so chosen that 97

l H ( x) = c1ε p( x) + c2 Inf(0, εn( x)) > 0 for all x ∈ X .

Proof. Let G be any complete hermitian metric whatever. Let σ1( x), . . . ,

σn( x) be the eigen-values of H with reference to G arranged in de-

creasing order. We construct now a metric G on X whose eigen val-

ues are functions of σi( x)

1≤i≤n as follows: let λ : X → R be a C ∞−function (we will impose conditions on λ letter); let U be a coordi-

nate open set in X with holomorphic coordinates ( z1,..., zn); in U , we

have G = GU αβdzα

dz β

so that (GU αβ)αβ is a function whose valuesare positive definite hermitian matrices; then the matrix valued functionGU = (GU αβ)αβ where G =

GU αβdzαdz β in U is defined by

G−1U = G−1

U

∞=

r =0

λ( x)r

(r + 1)!( H U G

−1U )r

where H U is the matrix valued function ( H U αβ) defined by

H =

H U αβdzαd z β

in U .

We now assert that GU define a global hermitian deff erential form on

X and under a suitable choice of λ, it is positive definite. To see that GU

defines a global hermitian diff erential form on X , we need only prove

the following. Let V be another coordinate open set with coordinates

complex (w1,..., wn). Let J =∂( z1, ..., zn)

∂(w1, ..., wn)be the Jacobian matrix. As98

before let GV = (GV αβ) be defined by G =

GV αβdwαd w β in V . Then

if

GV is defined starting from GV as

GU from GU , we have

J Gt U J = GV .

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13. Holomorphic bundles over q-complete manifolds 85

Lemma 4.5. Let X be a hermitian manifold and φ a strongly q-pseudo

convex proper function such that at each x ∈ X

lL (φ)( x) = εn−q( X ) + n Inf(0, εn( x)) > 0,

where ε p are the eigen values of L (ϕ) with respect to the metric on

X arranged in decreasing order. Then given any continuous function

g : X → R there exists a sequence aν1≤ν<∞ of real constants such that

for any C ∞ function µ : R → R satisfying (i) µ′(t ) > 0 (ii) µ′′(t ) ≥ 0 and

(iii) µ′(t ) ≥ aν for ν ≤ t < ν + 1 , we have

lL ( µ(φ))( x) > g( x).

Proof. Since the sets K ν = x|ν ≤ φ( x) ≤ ν + 1 are compact andlL (φ)( x) > 0 for all x ∈ X , we can find aν such that

aνlL (φ)( x) > g( x) for x ∈ K ν

the lemma now follows from Lemma 4.4

13 Holomorphic bundles over q-complete

manifolds

Lemma 4.6. Let E be a holomorphic vector bundle on a q-complete

complex manifold. Let ϕ : X → R be a strongly q-pseudo-convex C ∞

function on X such that x|φ( x) < c ⊂⊂ X for every c ∈ R. Let ds2 be a

hermitian metric on X such that lL (φ)( x) > 0 with respect to this metric.

Let h be any hermitian metric on X. Then there is a sequence (aν)1≤γ<∞of real constants such that the following property holds.

Let µ be any C ∞ functions on R such that µ′(t ) ≥ aν for ν ≤ t ≤ ν+1 104

and µ′(t ) > 0 , µ′′(t ) ≥ 0 for all t. Let K and (resp. K − µ′) be the opera-

tors (linear over C ∞ functions) from C rs ( X , E ) −→ C rs ( X , E ) defined in

chapter 3. (See (3.12) and (3.17 )) with respect to the hermitian metrics

ds2 and h (resp. h = e− µ(φ)h. Then for ψ ∈ C r ,s( X , E ) with s > q, we

have

A− µ(K − µψ, ψ) ≥ e− µ(φ) A(ψ, ψ).

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Proof. From the Remark at the end of Chapter 3,

K − µψ = K ψ −(−1)i ∂2 µ(φ)

∂ β z ∂ z βi

ϕa A

β

B′i

.

It follows that

A− µ(K − µψ, ψ) = e− µ(φ)L ( µ(φ))(ψ, ψ) + e− µ(φ) A(K ψ, ψ)

≥ e− µ(φ)lL ( µ(φ) + f A(ψ, ψ),

Where f : X −→ is a continuous function depending on K . By

Lemma 4.4. (aν)1≤ν<∞ can be so chosen that for any convex µ satisfying

(i) µ′(t ) > 0, (ii) µ′′(t ) ≥ 0 (iii) µ′(t ) > aν for ν ≤ t < ν+1,

we have

c1lL ( µ(ϕ) + f ≥ 1.

Hence

A− µ(ϕ)(K − µψ, ψ) ≥ e− µ(ϕ) A(ψ, ψ).

Theorem 4.1. Let X be a q-complete manifold so that there exists ϕ :105

X −→ satisfying : (i) ϕ is strongly q-pseudoconvex and (ii), for c ∈, x | ϕ( x) ≤ c is compact X. Let E be a holomorphic vector bundle

on X. Then there exists a complete hermitian metric ds2 on X and a

hermitian metric h along the fibres of E and real constants aν1≤ν<∞and c > 0 such that for every C ∞ function µ :

−→ satisfying

(i) µ′(t ) > 0 , (ii) µ′′(t ) ≥ 0 and (iii) µ′(t ) ≥ aν for v ≤ t < ν+1

we have

3(∂ψ− µ + ϑ− µψ2− µ) ≥ ψ2

− µ for every ψ ∈ D r ,s( X , E ), s > q.

Hence E is W rs -elliptic for s > q, with respect to the metric e− µ(ϕ)h.

Proof. From proposition 3.1., we have

(K − µψ, ψ) ≤ 3∂ψ2

− µ + ϑ− µψ2

.

(The subscript − µ means that the operators scalar product etc. are

defined with reference to the metric ds2 on the base and e− µ(ϕ)h along

the fibres of E ). The theorems then follow from Lemma 4.2, Lemma 4.6and the above inequality.

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Theorem 4.3. The cohomology groups H sk

( X ,Ωr ( E )) (cohomology with 107

compact supports) vanish for s ≤ n − q − 1 for any vector bundle E onthe q-complete complex manifold X . Moreover, H

p−q

k ( X ,Ωr ( E )) has a

structure of a separated topological vector space.

Proof. The theorem can be proved by applying Serre’s duality to Theo-

rem 4.2. It can also be given the following direct proof, which is based

on the results of §4.

Let us choose a metric h′ along the fibres of the dual bundle E ∗ in

such a way that conditions 1), 2), 3), 4) of the Remark after Theorem

4.1. be satisfied (by E ∗). Corresponding to the metric e−λ(φ)h′ on E ∗,

we choose the “dual” metric t (e−λ(φ)h′)−1= eλ(φ)t

h′−1 on E . Let ϕ ∈C rs ( X , E ), with s

≤n

−q

−1. Then

∗#ϕ

∈C n

−rn

−s( X , E

∗). Since

n − s ≥ q + 1, ∗#ϕ satisfies the inequality

A E ∗,−λ(K E ∗ ,−λ ∗ #ϕ, ∗#ϕ) ≥ A E ∗ ,−λ(∗#ϕ, ∗#ϕ),

i.e. A E ∗ ,−λ(K E ∗ ,−λ ∗ #ϕ, ∗#ϕ) ≥ A E ,λ(ϕ, ϕ).

Hence, for any ϕ ∈ D rs ( X , E ), with s ≤ n − q − 1 we have

ϕ2 E ,λ ≤ 3(∂ ∗ #ϕ2

E ∗ ,−λ + ϑ E ∗ ,−λ ∗ #ϕ2 E ∗ ,−λ)

i.e. by (1.4), (1.5), (1.6),

ϕ2 E ,λ ≤ 3(∂ϕ2

E ,λ + ϑ E ,λϕ2)

This inequality holds for any ϕ ∈ D rs ( X , E ), with s ≤ n − q − 1, and108

for any C ∞, nondecreasing, convex function λ : → .

Our theorem follows from Theorem 1.4. of §4 and from Theorem

1.8.

Let now X be a q-complete manifold and φ : X → R+ a strongly

q-pseudo convex function on X such that for

c ∈ , Bc = x| x ∈ X , φ( x) < c ⊂⊂ X . We set Bc = Y for some c ∈ .

Lemma 4.7. Bc is q-complete.

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13. Holomorphic bundles over q-complete manifolds 89

Proof. Let µ be a C ∞, nondecreasing convex function on (−∞, c) such

that t | t ∈ , µ(t ) < t is relatively compact in (−∞, c) for everyt ∈ . Then the two conditions of Definitions 4.2 are satisfied by the

function µ(φ). This proves the lemma

We adopt the following notation

F rs( X , E ) =ψ ∈ L rs

loc( X , E ); ∂ψ = 0

.

Qrs( X , E ) =ψ ∈ L rs

−νφ( X , E ) for positive integer ν, ∂ψ = 0

.

Finally ρ : F rs( X , E ) → F rs (Y , E ) is the restriction map. Clearly

Qrs( X , E ) ⊂ CF rs ( X , E ) and we have

Theorem 4.4. If s ≥ q, ρ(Qrs ( X , E )) is dense in F rs (Y , E ). (The metricon X and that along the fibres are chosen as ds2 and h respectively in

the Remark after Theorem 4.1.)

Proof. We denote Qrs( X , E ) simply by Q. Let µ be a continuous linear

functional on F rs (Y , E ) which vanishes on r (Q). It is sufficient to prove

that µ vanishes on all of F rs (Y , E ).

First, we extend µ to a continuous linear form on L rsloc

(Y , E ) (Hahn- 109

Banach extension theorem). By the representation theorem it follows

that there is a ψ ∈ L rs (Y , E ) with compact support in Y such that µ(u) =

(ψ, u) for every u ∈ L rsloc

(Y , E ).

Let c = supϕ( x) | x ∈. Support ψ. Since Support ψ is compact inY = Bc, then

c < c.

Now, let λ : → be a real C ∞ nondecreasing, convex function

such that

λ(t ) = 0 for t ≤< cλ(t ) = 0 for t < c,

0 ≤ λ′ (t) ≤ 1.

Then, for t > c,λ(t ) ≤ t − c ≤ t .

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Theorem 4.5. Let X be a Stein manifold and ϕ : X → R a strongly

0-pseudo convex C ∞ function on X such that for c ∈ R,

X | φ(X) < c ⊂⊂ X. Then the pair ( X , Bc) , c ∈ R is

a Runge pair. In other words, the set of holomorphic functions on Bc

which are restrictions of global holomorphic functions on X is dense in

the space of all holomorphic functions on Bc in the topology of uniform

convergence on compact subsets (of Bc).

14 Examples of q-complete manifolds

Let X be an m-complete manifold and f 1, . . . , f q+1 be any q+1 holomor-

phic functions on V . Let Z = x| f i( x)

=0 for 1

i

q+

1.Assertion Y = V − Z is (m + q)-complete.

Proof. Let ϕ : V → R be a strongly m-pseudo-convex function V such

that for every c ∈ R,

Bc = x | x ∈ V , ϕ( x) < c

⊂⊂ X .

Because of the second condition, we can choose a C ∞ function113

λ : R→R such that λ

′(t ) > 0, λ

′′(t )≥

0,

λ(t ) > 0 for t ∈ R, λ(t ) → ∞ as t → ∞ and further

λ(ϕ) >

q+1i=1

f i f i on Y .

Set ψ = λ(ϕ) − logq+1i=1

f i ¯ f i. Since L (logq+1i=1

f i ¯ f i) is positive semi-

definite with at most q positive eigen-values, ψ strongly (m + q)-pseudo

convex. It remains to prove that

B′c = X | x ∈ Y , ψ( x) < c ⊂⊂ y

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Theorem 4.6 ([32]). Let X be q-complete of complex dimension n. Let

Ω

n

denote the sheaf of germs of holomorphic n-forms on X. The naturalmap

H q( X ,Ωn) → H n+q( X ,C)

is surjective.

Proof. For any complex manifold, we have spectral sequence ( E st r )o ≤115

r ≤ ∞ such that E ∞ is the associated graded group of H ∗( X ,C), and

E st 1= H t ( X ,Ωs) [11]. In the case when X is q-complete, H t ( X ,Ωs) = 0

for t > q. In particular for t +s = n+q, E st t = 0 for all (s, t ) diff erent from

(n, q). Hence E s,t r = 0 for (s, t ) (n, q), s + t = n + q. Moreover, all of

E n,q1

are cycles for d 1 since Ωn+1= 0. Hence E

n,q1

→ E n,q2

is surjective.

For r ≥ 2, E n−r ,q+

r −1r = 0 since q + r − 1 > q and E n+r ,q+r +1 = 0. Hence E

n,q2∽ E

n,q∞ . Hence H n+q( X ,C) ≃ E

n,q∞ = E

n,q2

. (We note that E s,t ∞ = 0 for

s + t = n + q, (s, t ) (n, q)). This proves the theorem.

We consider now the subsets of C2n

Z 1 = z ∈ C2n

zi = 0 for i > n, X 1 = C

2n − Z 1,

Z 2 = z ∈ C2n

zi = 0 for i ≤ n, X 2 = C

2n − Z 2,

Z = Z 1 ∪ Z 2, Y = C2n − Z = X 1 ∩ X 2.

The analytic set Z ⊂ C2n has complex codimension n at each point.

We shall prove that Y is not (n − 1)-complete, when n ≥ 2.Since the point 0 is a complete intersection of codimension 2n in

C2n, then U = C2n − 0 is (2n − 1)-complete. Let Ω2n be the sheaf of

group of holomorphic 2n-forms. Then by Theorem 4.6.

H 2n−1(U ,Ω2n) → H 2n−1(U ,C)

is surjective. On the other hand, since U is contractible on the unit116

sphere of C2n, then H 2n−1(U ,C) ≈ C. Hence

H 2n−1(U ,Ω2n) 0.

The exact cohomology sequence of Mayer-Victoris yields

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≥ 2e−ψ(1 + | z|2)−2 A(u, u) = 2(1 + | z|2)−2 AA−ψ(u, u),

since L (ψ)u, u ≥ 2L (log(1+ | z|2))u, u ≥ 2(q+ | z|2)−2 A(u, u). Hence

K −ψ is positive definite with eigenvalues ≥ 2(1 + | z|2)−2 at the point z;

K −1−ψ is also positive definite, and its eigenvalues are ≤ 1

2(1 + | z|2)2. By

consequence we have

| A−ψ(u, v)| ≤ A−ψ(K −1−ψu, u) A−ψ(K −ψv, v)

≤ 1

2 A−ψ((1 + | z|2)2u, u) A−ψ(K −ψv, v).

Hence, for forms u, v of type ( p, q), we have

|(u, v)−ψ |2 ≥ 12(1 + | z|2)u2

−ψ(K −ψv, v)−ψ, (4.2)

whenever the forms u, v are such that the norms on the right are finite.

On D p,q(Cn), we introduce the norm M by118

M (u)2= (K −ψu, u)−ψ + ∂u2

−ψ + ϑ−ψu2−ψ,

and let V p,q be the completing of D p,q with respect to this norm. Obvi-

ously V pq

⊂L

pq

loc. For u

∈D p,q, we have, by (3.18),

(K −ψu, u) ≥ ∂u2−ψ + ϑ−ψu2

−ψ.

Thus the norm M on D p,q is equivalent to that defined by

M ′(u)2= ∂u2

−ψ + ϑ−ψu2−ψ.

Let ω ∈ L p,qloc

, and let ω−ϕ < ∞. Then, for u ∈ D p,q, we have, by

(4.2) (dropping the factor 12

as we may)

|(u, ω)−ψ

|2

≤ (1 +

| z

|2)ω

2

−ψ M ′(u)2

= ω2−φ M ′(u)2 (4.3)

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at each point of X and for every ϕ ∈ C pq( X , E ). Let ε( x) be the least

eigenvalue of the (positive definite) hermitian from A(K ϕ, ϕ) x( x ∈ X )acting on the space of the ( p, q)-form with values in E. Then ε( x) > 0.

If ϕ ∈ C pq( X , E ) , ∂ϕ = 0 is such that X

1

ε( x) A(ϕ, ϕ)dX < ∞,

there exists a from ψ ∈ C p,q−1( X , E ) such that

∂ψ = ϕ, X

A(ψ, ψ)dX ≤ 3

X

1

ε( x) A(ϕ, ϕ)dX .

Proposition 4.3. Let φ be a plurisubharmonic function on Cn such that

there exists a constant C > 0 for which |φ( z) − φ( z′)| ≤ C whenever

| z − z′| ≤ 1. Let V be a (complex) subspace of Cn of codimension k,

and f a holomorphic function on V such that

| f |2e−ϕd V v < ∞ (d V v =

Lebesgue measure on V ). Then, there exits a holomorphic function F on

Cn with F |V = f and

Cn

|F |2e−ϕ(1 + | z|2)3k dv M

V

| f |2e−ϕd V v,

where M is a constant independent of f ; here dv = d Cn v is the Lebesgue121

measure in Cn

Proof. We may clearly suppose that V has codimension 1 in Cn. We

assume that V is the subspace zn= 0.

Then f is a holomorphic function in Cn, depending only on z1, . . . ,

zn−1. Let λ : R → R be a C ∞ function on R, such that

0 λ(t ) 1,

λ(t ) = 1 for t 1

4

λ(t ) = 0 for t 1.

Let C 1 = Sup d λdt .

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is a C ∞ plurisubharmonic function on Cn, such that

φε( z) → φ( z) as ε → 0.

Moreover123

| φε( z) − φ( z) | Cn

α( z′) | φ( z − ε z′) − φ( z) | d ( z′) C ,

and therefore | φε( z) − φε( z′) | 3C for | z − z′ | .

Hence there exists a C ∞ function µε : Cn → C, such that

∂µε = ω εn

e−φ

(1+ | z |2)2| µε |2 dv ec

Cn

e−φε

(1+ | z |2)2| µε |2 dv C 3

V

e−φ | f |2 dvV .(C 3 = C 21πe5c)

Letting ε → 0, we can find a distribution µ such that

∂µ = ω1

Cn

e−φ

(1+ | z |

2

)2

|µ

|2 dv C 3 V

e−φ

|f

|2 dvV .

Consider now the distribution

F = λ(| zn |2) f − zn µ.

We have

∂F = ∂λ · f − zn∂µ = 0.

Hence F is a holomorphic function Cn.Furthermore124

Cn

e−φ

(1+ | z |2)3 − | F |2 dv

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15. A theorem on the supports of analytic functionals 103

Proof. Suppose that µ is supported by K . Then, by definition

| ˜ µ(ζ )| = | µ(eζ )| ≤ C ε sup z∈K ε

|eζ ( z)| = C εesup

K ε

z,ζ ≤ C εe H K (ζ )+ε|ζ |

To prove the converse, we proceed as follows. H is a closed sub-

space of the space C of continuous function on Cn; hence (Hahn - Ba-

nach) µ extends to a linear functional µ : C → C, hence defines a

measure. It is clearly sufficient to prove that any ε > 0, there exists

distribution ν with support in K ε such that

µ( f ) = ν for all f ∈ H .

Let ξ ′ be the space of distributions with compact support in Cn

. For 127ν ∈ ξ ′, let ν denote the Fourier transform of ν considered as a function

of 2n real variables u1, . . . , u2n:

ν(u1, . . . , u2n) = ν(e′u),

where e′u( z) = exp (−i(u1 Re z1 + u2 Im z1 + . . . + u2n Im zn)).

Clearlyν has an extension to a holomorphic function on C2n; further

since linear combination of the function eζ are dense in H , we have

ν(g) = µ(g) for all g

∈H if and only if µ(eζ ) = ν(eζ ) for all ζ , i.e. if

and only if ∽

µ(ζ ) =ν(iζ 1, −ζ i,..., iζ n, −ζ n). (4.4)

Therefore, because of the Paley-Wiener theorem, it is sufficient to

construct an entire function ν on C2n satisfying (4.4), for which, we

have further,

|ν(u) | C ε(1+ | u | |) N φ(u). (4.5)

for some N > 0; here φ(u) = sup x∈K ε

( x1 Im u1 + . . . + x2n Im u2n) and z j =

x2 j−1 + ix2 j.

Consider the subspace V of C2n consisting of points (iζ i, −ζ 1, ..., i

ζ n, −ζ n), where ζ =

(ζ 1,...,ζ n) ∈ C

n

. If u=

(iζ 1, ...,−ζ n), we haveφ(u) = H K ε (ζ ) = H K (ζ ) + ε|ζ |.

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106 BIBLIOGRAPHY

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