Formulas y kerneles de integración explícitos en subvariedades regulares y singulares de Cn

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Centro de Investigación y de Estudios Avanzados

del Instituto Politécnico Nacional

Unidad Zacatenco

Departamento de Matemáticas

Fórmulas y kerneles de integración

explícitos en subvariedades regulares y

singulares de

C

n

.

Tesis que presenta

Luis Miguel Hernández Pérez

Para obtener el grado de

Doctor en ciencias

en la especialidad de

Matemáticas

Director de tesis:

Dr. Eduardo Santillan Zeron

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Center for Research and Advanced Studies

of the National Polytechnic Institute

Zacatenco Campus

Department of Mathematics

Explicit integration kernels and

formulae in regular and singular

subvarieties of

C

n

.

A dissertation presented by

Luis Miguel Hernández Pérez

To obtain the degree of

Doctor in science

in the speciality of

Mathematics

Thesis advisor:

Dr. Eduardo Santillan Zeron

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Introduction and objectives

The Bochner-Martinelli and Ramirez-Khenkin integration formulae are a pair of cornerstones of the field of several complex variables. LetΩbe a bounded domain in Cn with piecewise smooth boundary , and be a (0,q)-form whose

coeffi-cients are continuous functions on the closureΩ. If the differential (calculated

as a distribution) is also continuous in Ω, then the following identity holds in

for the Bochner-Martinelli kernels Bq(z,ξ),

ℵ =

Z

Ωℵ ∧Bq− Z

Ω(ℵ)∧Bq+

Z

Ωℵ ∧Bq−1

. (0.0.1)

It is impossible to enumerate all the applications of the integration formulae into complex analysis, geometry and other areas. We may mention for example their use for solving the Neumann -equation in strictly pseudoconvex domains ofCn. A natural problem is to produce integration formulae on general varieties. Let Ω be an open domain compactly contained in a smooth or singular complex

variety Σ. Ifhas a piecewise smooth boundary , the problem is to produce

integration formulae similar to (0.0.1) for differential forms ℵ such that ℵ and

are both continuous on Ω.

There is a vast literature in books and papers on integration formulae for smooth complex manifolds; see for example the references [9,10, 12, 15, 19]. An-dersson and Samuelsson also produced integration formulae for singular subvari-etiesΣofCn, but they delimited their work to analyse differentials formswhose restriction to the regular part of Σ extends onto a smooth form well defined on a

neighbourhood of Σ; see the references [1, 2, 3].

Thus, the main objective of this work is to propose a simple technique for producing explicit integration formulae in smooth and singular subvarieties of Cn.

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In the second chapter of this thesis, we use the work that professors Rupphen-tal y Zeron presented in [17], in order to produce explicit integration formulae on weighted homogeneous subvarieties. We firstly deduce those formulae on the particular case of the subvariety Σ = {z C3 : z

1z2 = zn3, n ∈ N, n ≥ 2},

and then we extend these formulae to arbitrary weighted homogeneous subvari-eties. We analyse this particular subvariety Σ, because it is simple enough so as to do the calculations explicitly and complicated enough so as to exemplify all the classical pathologies. We also obtained integral representations constructed around the Cauchy kernel or some of its variations by working on the particular weighted homogeneous subvariety Σ={z C3 : z

1z2 =zn3}. Thus, we wonder if

the Cauchy kernel is an intrinsic property of the weighted homogeneous subvari-eties for forms of degree zero or one. Amaizinly, the the answer is positive, as we will show at the end of chapter.

In the chapter three we include the work that professor Mats Andersson pre-sented in [1] and the work made by Peter Helgeson in his disssertation [11], in order to compare the proposed technique in these works with the technique pro-posed in this thesis.

We present in chapter four an alternative technique for producing integration formulae on smooth complex Stein submanifolds of Cn, based on the fact that every Stein submanifold inCn has a holomorphic retraction; see for example [21]. In general, it is quite difficult to find explicitly such a holomorphic retraction, so we exemplify the proposed technique with a practical example. We work on the smooth submanifold {z∈ Cn : n

j=1z2j =1}, also known as complex sphere. We must mention that some parts of this thesis were already published in the papper: "Integration Formulae and Kernels in Singular Subvarieties of Cn", CRM Proceedings and Lectures Notes. Volume (55), 2012.

The main result of this work is contained in chapter five. We propose there a simple technique for producing explicit integration formulae in subvarieties of Cn+1 generated as the zero locus of a polynomial smp(z) for s C and

z ∈ Cn. We consider polynomials of this kind, because the first entry s can be easily expressed as them-root ofp(z)and several of the main singular subvarieties presented in [4, 5, 7, 20] are the zero locus of such a polynomial. Nevertheless, the technique presented in this work can be applied to analyze other subvarieties of Cκ, with the conditions that some entries of z Σ can be easily expressed in

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vii

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Contents

Introduction and objectives v

1 Preliminaries 1

1.1 Theoperator . . . 1

1.2 Differential forms . . . 2

1.3 Currents . . . 10

1.4 Integration kernels . . . 10

2 Integration formulae on weighted homogeneous varieties 15 2.1 Introduction . . . 15

2.2 Practical example . . . 15

2.3 Integration formulae on weighted homogeneous subvarieties . . . . 21

3 Integral representation with weights 23 3.1 Weighted representation formulae . . . 23

3.2 Integral formulae on a Riemann surface inC2 . . . 26

3.3 Preliminaries . . . 26

3.4 Explicit integral representations onX . . . 30

3.5 Integral formulae on a Riemann surface inP2 . . . 33

3.6 A Cauchy-Green formula onX . . . 35

4 Bochner-Martinelli formulae on the complex sphere 43 4.1 Introduction . . . 43

4.2 Bochner-Martinelli formulae on the complex sphere . . . 43

5 Integration formulae and kernels in singular subvarieties of Cn 51 5.1 Introduction . . . 51

5.2 Basic properties . . . 55

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Chapter 1

Preliminaries

1.1

The

operator

The main operator in the field of several complex variables is indubitably the delta-bar differential operator. The importance of this operator lies at the base of the main definition in several complex variables: what is a holomorphic function?. Properly speaking, given a smooth function f defined from an open set U of the n-dimensional complex spaceCn into the complex plane C, this function f is holomorphic in U if only if f =0, where is defined as follows:

f =

n

k=1

f

kdz¯k

=

n

k=1

1 2

f

Rezk + i f

Imzk

dz¯k.

Notice that theoperator sends smooth functions into(0, 1)-differential forms. In a similar way, we may define a generaloperator which sends(p,q)-differential forms into (p,q+1)-differential forms, see for example [12].

Now then, once we have any differential operator, a basic problem is to solve differential equations constructed with this operator. Hence, given a(0,q)-differential form ω, a natural question is to determine whether the differential equation

f = ω has a solution f?. We obviously require that ∂ω = 0. Moreover, does there exist a solution f which satisfies an extra smooth condition likeLpor Hölder regularity? We may go further on: Can we solve differential equations with this

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Solving the -equation is one of the main pillars of complex analysis, but it also has deep consequences on algebraic geometry, partial differential equations and other areas. For example, the classical Dolbeault theorem states that the -equation can be solved in all degrees on a Stein manifold; and its known that an open subset of Cn is Stein if and only if the -equation can be solved in all degrees on that set. Nevertheless, it is usually difficult to produce an explicit operator for solving the -equation on a given Stein manifold, even if we know that the equation can be solved.

1.2

Differential forms

Some of the material presented in this section was taken from the book [8, pp. 297-302].

Let X be a topological Hausdorff space.

Definition 1.2.1. An n-dimensional complex coordinate system(U,ϕU)in X consists of

an open set U ⊂X and a topological map ϕU from U onto an open set B ⊂Cn.

We say that twon-dimensional complex coordinate systems(U,ϕU)and(V,ψV) in X are compatible if either U∩V = ∅ or the map ϕU ϕ−1

V is biholomorphic, so that ϕU ϕV1 is bijective, holomorphic, and with holomorphic inverse.

A covering of X with a pairwise compatible n-dimensional complex coordi-nates systems is called an n-dimensional complex atlas on X. Two atlases are called equivalent if any two complex coordinates systems are compatible. An equivalence class ofn-dimensional complex atlases onXis called ann-dimensional complex structure on X.

Definition 1.2.2. An n-dimensional complex manifold is a topological Hausdorff space X with a countable basis, equipped with an n-dimensional complex structure.

Let X be ann-dimensional complex manifold, and B⊂X be an open set.

Definition 1.2.3. A complex function f : B→C is called holomorphic if for each x B

there is a coordinate system (U,ϕU) in X such that x∈ U and f ◦ϕU1 is holomorphic. Let X andY complex manifolds.

Definition 1.2.4. We say that the map F : X →Y is holomorphic if for each x∈ X there is a coordinate system (U,ϕU)in X at x and a coordinate system (V,ψV) in Y at F(x)

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1.2 Differential forms 3

Definition 1.2.5. Let A be an algebra over a ring R, and let D : A→ A be an R-linear map satisfying the Leibnitz rule: D(ab) = D(a)b+aD(b). Then D is called a derivation on A.

Let X be an n-dimensional complex manifold and x ∈ X be a point. The tangent space to X at x, denoted by Tx, is the vector space of all derivations of functions defined in a neighbourhood of x. We consider a complex-valued alternating multilinear forms on the tangent space Tx.

Definition 1.2.6. A complex r-form or a r-dimensional differential form at x is an alter-natingR-multilinear mapping

ϕ:

r−times

z }| {

Tx×...×Tx →C.

Alternating means that interchanging the values of two entries of ϕautomatically changes the sign of ϕ. The set of all complex r-forms atx is denoted by Fr.

We have the following properties ofFr.

1. By convention, F0 = C. F1 = F(Tx) is the complexification of the 2n-real dimensional vector space Tx∗, where Tx∗ is the dual space to Tx.

2. Since Tx is 2n-dimensional overR, every alternating multilinear form onTx with more than 2n arguments must be equal to zero. So that Fr = 0 for

r >2n.

3. In general, Fr is a complex vector space. We can represent an elementϕ∈ Fr

uniquely in the form ϕ=Re(ϕ) +iIm(ϕ), where Re(ϕ) and Im(ϕ)are real-valued r-forms atx. Then it follows that

dimRFr =

2n r

.

4. We associate with each element ϕ∈ Fr a complex-conjugate element ϕ∈ Fr

by setting ϕ(v1, ...,vr) := ϕ(v1, ...,vr). And so we have: (a) ϕ=Re(ϕ)−iIm(ϕ).

(b) ϕ= ϕ.

(c) ϕ+ψ= ϕ+ψ.

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Let ϕ ∈ Fr and ψ ∈ Fs be given. The wedge product ϕψ ∈ Fr+s is defined by

ϕψ(v1, ...,vr,vr+1, ...,vr+s) := 1

r!s!σ

S

r+s

(sgnσ)ϕ(vσ(1), ...,vσ(r)ψ(vσ(r+1), ...,vσ(r+s)).

The sum is taken over all possible permutations σof the setSr+s ={1, 2, ...,r+s}, and sgnσ is the sign of the permutation σ.

In particular we have (ϕψ)(v,w) = ϕ(v)·ψ(w)−ϕ(w)·ψ(v) for ϕ,ψ ∈ F1

and v,w∈ Tx. And in general

1. ϕψ= (−1)rsψϕ. 2. (ϕψ)∧ω = ϕ∧(ψω).

With the wedge product, the vector space

^ F:=

2n M

r=0

Fr

becomes a noncommutative graded associativeC-algebra with unit(1), it is called the exterior algebra at x.

Definition 1.2.7. Let p,q ∈ N∪ {0}such thatp+q =r. A r-form ϕis called a form of

type(p,q)if

ϕ(cv1, ...,cvr) = cp cq·ϕ(v1, ...,vr) for all c ∈C.

Proposition 1.2.8. Letϕbe a nonzero r-form of the type(p,q), then p and q are uniquely determined.

Proof. Suppose that ϕis of type(p,q)and of type (p′,q′), since ϕ6=0 there exist tangent vector v1, ...,vr such that ϕ(v1, ...,vr) 6=0. Then

ϕ(cv1, ...,cvr) = cp cq·ϕ(v1, ...,vr)

=cp′ cq′·ϕ(v1, ...,vr).

Therefore, cp cq =cp′ cq′ for each c ∈ C. If c = eit with t R, theneit(p−q) =

eit(p′−q′). This can hold only when pq = p′q′. Since p+q = p′+q′ = r, it follows that p = p′ and q=q′.

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1.2 Differential forms 5

Proposition 1.2.9. 1. If the form ϕis of type(p,q), then ϕis of type(q,p).

2. If ϕandψare both forms of type(p,q), then ϕ+ψandλ·ϕare also of type(p,q). 3. If ϕis a form of type(p,q) andψof type(p′,q′), then ϕψis of type

(p+p′,q+q′).

Notice thatdzν is a form of type (0, 1), since dzν =dzν. Then dzi1 ∧...∧dzip ∧ dzj1 ∧...∧dzjq, with 1 ≤ i1 < ... < ip ≤ n and 1 ≤ j1 < ... < jq ≤ n, is a form of type (p,q).

Theorem 1.2.10. Any r-form ϕhas a uniquely determined representation

ϕ=

p+q=r

ϕ(p,q),

where ϕ(p,q) are r-forms of type(p,q)

Proof. The existence of the above representation follows form the fact that the forms dzi1 ∧...∧dzip∧dzj1 ∧...∧dzjq constitute a basis of Fr. For the uniquenes assume that

ϕ=

p+q=r

ϕ(p,q) =

p+q=r

˜

ϕ(p,q). Then

p+q=r

ψ(p,q) =0 for ψ(p,q) = ϕ(p,q)ϕ˜(p,q), and so we have

0=

p+q=r

ψ(p,q)(cv1, ...,cvr) =

p+q=r

cp cqψ(p,q)(v1, ...,vr).

For the fixed r-tuple (v1, ...,vr) we obtain a polynomial equation in the ring C[c,c]; and so all coefficients ψ

(p,q)(v1, ...,vr) must vanish. Since we can choose

v1, ...,vr arbitrarily, we have ϕ(p,q) = ϕ˜(p,q) for all p,q.

Definition 1.2.11. An holomorphic vector bundle E of rank r over an n-dimensional complex manifold X is a complex manifold satisfying the following conditions

1. There exists a holomorphic mapping π : E X.

2. For all x ∈ X the fiber of E, Ex = π−1(x) has the structure of an r-dimensional

vector space overC.

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Local triviality means that there is an open covering{Ui}i∈I ofXtogether with biholomorphic functions, called trivializations, such that Φi : π−1(U

i) →Ui×Cr and for eachx ∈Ui the induced map pr◦Φi : Ex x×Cr Cr is a vector space isomorphism, where pr is the canonical projection.

Definition 1.2.12. Let E be an holomorphic vector bundle over X and U ⊂ X an open set. A continuous (differentiable, holomorphic) section in E over U is a continuous (dif-ferentiable, holomorphic) map s : U →E withπ◦s=idU.

Let X be an n-dimensional complex manifold, we denote by F(X) the com-plexified cotangent bundle T∗(X)NC. It has the spaces F(Tx(X)) = Tx(X)∗NC of complex covariant tangent vectors as fibers, so it is a topological complex vec-tor bundle of rank 2n. It even has a real-analytic structure, but not necessarily a complex analytic structure.

IfEis a topological complex vector bundle of rank moverX, we can construct a bundle Fr(E) of rank (mr) for each 0 ≤ r ≤ m, such that (Fr(E))x = Fr(Ex) for every x ∈ X. If E is given by the transitions functions gij, then Fr(E) is naturally given by the matrices g(r)ij whose entries are the(r×r)minors of gij.

Definition 1.2.13. An r-form or an r-dimensional differential form on an open set U ⊂X is a smooth section ω in the bundle Fr(U) := Fr(T(U)) for the tnagent vector bundle T(U).

We denote by Γ(U,Fr(X)) the vector space of holomorphic sections in Fr(X)

over U ⊂X.

So an r-form ω on U assigns to every point x ∈ U an r-form ωx at x. Notice that if z1, ...,zn are the local coordinates in a neighbourhood of x, then ωj := dzj and ωn+j :=dzj, for j=1, ...,nform a basis of the 1-forms on this neighbourhood. Moreover there is a representation

ωx =

1≤i1<...<ir≤2n

ai1...ir(x)ωi1∧...∧ωir,

where x7→ ai1...ir(x) are smooth functions.

We denote by Er(U) the set of all smooth r-forms on U, and the subset of all smooth forms of type (p,q) by E(p,q)(U). Please do not confuse Er(U) with

Fr(U) = Fr(T(U)).

If f is a smooth function on U, then its differential d f ∈ E1(U) is given by

x 7→(d f)x, in local coordinates we have

d f =

n

ν=1

f

zνdzν+

n

ν=1

f

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1.2 Differential forms 7

On the other hand, a smooth vector field is a smooth section of the tangent bundle T(X). So in local coordinates it can be written in the form

ξ =

n

ν=1

ξν

zν +

n

ν=1

ξν

zν,

where the coefficients ξν are smooth functions. Then we can apply d f to such a

vector field and we have

d f(ξ) =

n

ν=1

ξν f

zν +

n

ν=1

ξν f

zν.

For any open setU the differential d can be genreralized to the mapd = dU :

Er(U) → Er+1(U) in the following way. Let ω =∑1i

1<...<ir≤2nai1...ir(x)ωi1∧...∧

ωir is the basis representation in a coordinate neigbourhoodU, then

dU(ω) :=

1≤i1<...<ir≤2n

dai1...ir(x)∧ωi1 ∧...∧ωir.

It is no difficult to show that this definition is independent of the choice of the local coordinates and that d has the following properties

1. If f is a smooth function,d f is the differential of f.

2. disC-linear.

3. d◦d=0.

4. If ϕ∈ Er(U) andψ∈ Es(U), then d(ϕψ) =dϕψ+ (−1)rϕdψ.

5. dis a real operator; that isdϕ=dϕ. In particular dϕ=d(Re ϕ) +i d(Im ϕ). The differential dω is called the exterior derivative of the formω.

We consider the decomposition of anr-form into a sum of forms of type (p,q)

and use some notation, if I = (i1, ...,ip) and J = (j1, ...,iq) are multi-indices in increasing order, |I|= p and|J|=q are the lengths of I and J , we write

aI JdzI ∧dzJ

instead of

ai1...ip,j1,...,jqdzi1∧...∧dzip ∧dzj1∧...∧dzjq.

So a generalr-form ω has the unique representation

ω =

p+q=r |I

|=p

|J|=q

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and the differential of ω is given by

dω =

p+q=r

|I|=p

|J|=q

daI J∧dzI∧dzJ.

Notice that if f is a smooth function, then d f =f +f, where

f =

n

ν=1

f

zνdzν and f =

n

ν=1

f

zνdzν,

heref has type (1, 0), f has type(0, 1), andd f is of type (1, 1).

Proposition 1.2.14. Letϕbe a r-form of type(p,q). Then dϕhas a unique decomposition dϕ=∂ϕ+∂ϕwith∂ϕa(p+1,q)-form and∂ϕa (p,q+1)-form.

Proof. Let ϕ=∑I,JaI JdzI∧dzJ. We define

∂ϕ:=

I,J

aI J∧dzI ∧dzJ and ∂ϕ:=aI J∧dzI ∧dzJ,

then dϕ = ∂ϕ+∂ϕ is the unique decomposition of the (p+q+1)-form dϕ into forms of pure type.

For generalr-forms the derivatives with respect to azand zare defined in the obvios way.

Theorem 1.2.15. 1. andareC-linear operators with d=+.

2. ∂∂=0,∂∂ =0, and ∂∂+∂∂ =0.

3. ,are not real. We have∂ϕ =∂ϕand∂ϕ=∂ϕ. 4. If ϕis a r-form andψis arbitrary, then

(ϕψ) = ∂ϕψ+ (−1)rϕ∂ψ,

(ϕψ) = ∂ϕψ+ (−1)rϕ∂ψ.

Proof. It suffices to prove this for forms of pure type, then the above formulas can be easily derived from the corresponding formulas for d and the uniqueness of the decomposition into forms of type(p,q).

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1.2 Differential forms 9

Definition 1.2.16. Let ϕbe a p-form on the open set U ⊂X. 1. ϕis called holomorphic if ϕis of type(p, 0)and∂ϕ=0. 2. ϕis called antiholomorphic if ϕis of type(0,p)and∂ϕ=0.

Let h : X → Y be a smooth map between smooth complex manifolds. If

ϕ(x) = ∑p+q=r|I|=p

|J|=q

aI JdzI ∧dzJ, is a (p,q)-form in local coordinates z on a

neigbourhoodU ⊂Y, and if hk are the components ofhin these coordinates, then in h−1(U) we define the pull-backh∗ϕof ϕwith respect to hby:

(h∗ϕ)(x) :=

p+q=r|I

|=p

|J|=q

(aI J◦h)(x)dhi1(x)∧...∧dhip(x)∧dhj1∧...∧dhjq(x).

And we have the following properties:

1. For every(p,q)-form ϕonU, the pull-backh∗ϕis a (p,q)-form on h−1(U). 2. For every continous differential form varphi onU, we have

d(h∗ϕ) = h∗(dh), (h∗ϕ) = h∗(h), and (h∗ϕ) =h∗(∂ϕ).

We may extend the notion of forms of type (p,q) with smooth coefficients to forms of type (p,q) whose coefficients are distributions. The action of the differential operator is understood in the sense of distributions. In particular, the action of the differential operator over those forms is described in the next paragraphs.

Let λ be a smooth form of type (0,q) defined on an open set S in Cn. The fact that λ is -closed (∂λ = 0) in the sense of distributions on S means that the following integral vanishes Z

Sλ∂σ=0,

for every smooth form σ of type (n,n−q−1) defined in and with compact sup-port inside S.

Furthermore, let λ be a smooth form of type (0,q) defined on the open set

S⊂Cn, and gbe a form of type (0,q1) onS as well. We say thatλ=gholds in the distributional sense onS if and only if

Z

S λσ+ (−1)

q−1g∂σ=0,

for every smooth form σ of type (n,n−q) defined and with compact support in

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1.3

Currents

Let X be a complex manifold. A sequence ϕν = ∑I fIduI of smooth r-forms on

X is said to be convergent to zero in Er(X) if ϕν together with all its derivatives

tends uniformly to zero. In particular, we say that the form ϕνtends uniformly to

zero if and only if all its coefficients tends uniformly to zero when ν→∞.

Definition 1.3.1. A current of degree 2n−r is anR-linear map T : Er(X) C such

that if {ϕν} is a sequence of r-forms converging to zero, then T(ϕν) converges to zero in

C.

Notice that ifψis a differential form of degree 2n−r. Thenψdefines a current

TΨ of degree 2n−r by the formula

Tψ(ϕ) := Z

Xψϕ, for all ϕ∈ E r(X).

And if M ⊂X is anr-dimensional differential submanifold. Then a current TM is defined by TM(ϕ) :=RMϕ. Therefore, we say that a current of degree 2n−r has dimension r.

1.4

Integration kernels

Let G ⊂ Cn be a bounded open set with piecewise smooth boundary G, and

Λ be a (0,q)-differential form on G, for 0 q n, such that its coefficients are

smooth functions on the closure G. Let F(y,z) be a differential form defined on a neighbourhood of G×G, continuous when y 6= z, smooth of type (0,q) with respect to the variable z, and smooth of type (n,n−q−1) with respect to the variabley.

We fix z inG and letε >0, such that the ball Bε(z) = {x Cn : ||xz|| ≤ ε} is contained in G. If Dz,ε = G\ Bε(z), we apply Stokes’s theorem to the form

Λ(y)∧F(y,z)on the domain Dz,ε; and we obtain Z

y∈Dz,ε

dy(Λ(y)∧F(y,z)) = Z

y∈Dz,ε

Λ(y)F(y,z). (1.4.1)

Notice thatΛF is a(n,n1)-form with respect to the variabley, and

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1.4 Integration kernels 11

Thus, if we expand (1.4.1), we have Z

y∈Dz,ε

[yΛ(y)]∧F(y,z) + Z

y∈Dz,ε

Λ(y)(1)qyF(y,z)

=

Z

y∈G

Λ(y)F(y,z)

Z

y∈Bε(z)

Λ(y)F(y,z). (1.4.2)

Now suppose thatF(y,z) satisfies the following conditions:

(i) lim

ε→0

Z

y∈Bε(z)

Λ(y)F(y,z) = Λ(z),

(ii) (−1)q

yF(y,z) = czF(y,z), where c is a constant. If we take the limit whenεtends to zero in (5.1.7),

Λ =

Z

G

ΛF

Z

G(

Λ)Fc

Z

G

ΛF. (1.4.3)

Thus, by a integration kernel we mean a differential F(y,z) form continuous when y 6= z, smooth of type (0,q) with respect to the variable z, smooth of type

(n,n−q−1)with respect to the variabley, and that satisfies the conditions (i) and (ii). Hence (1.4.3) is satisfied, for every(0,q)-differential formΛ on G, 0q n,

whose its coefficients are smooth functions on the closure G.

We are interesting to produce integration kernels, because if we have an in-tegral representation as in (1.4.3), we automatically have a solution for the -problem as follows:

If 1≤ q ≤n−1, Λ is a smooth (0,q)-form, with compact support in Cn, and

Λ=0, then the form

λ(z) =−c Z

y∈G

Λ(y)∧F(y,z)

satisfies the -equation

zλ(z) = Λ(z) on Cn.

We are interesting in producing integration kernels on varieties with singu-larities, so that we work with a practical example. Let Σ Cn be a variety such that the origin of Cn is an isolated singularity of Σ and the regular part Σreg of Σ is a smooth complex manifold of codimension 1. As we have seen, in order to

produce integration kernels we need that Stokes’s theorem holds on the varietyΣ

or on its regular part Σreg.

LetG ⊂Σbe an open set with piecewise smooth boundary. We need to define

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If the origin 0 does not lie on the closureG, the integralRGhis defined in the usual sense for all (n−1,n−1)-formsh continuous on G.

If 0∈ G, we takeε>0 such that the ball Bε(0)Σ G, and define

Z

Gh:=limε→0

Z

G−Bε(0)

h.

Moreover, if his not a (n1,n1)-form, we define RGh0.

Let f be an(n−1,n−1)-form continuous on the closure Gand smooth on the regular part Greg = G\ {0}. If 0 does not lie on the closure G, we automatically

have that the Stokes’s theorem holds; that is Z

Gd f = Z

G f.

If 0∈ G, we can takeε >0 such that Bε(0)Σ G. Let Gε =G\(Bε(0)Σ),

then:

Z

Gε

d f =

Z

Gε

f

=

Z

G f − Z

(Bε(0)∩Σ)

f.

Recall that f is bounded because it is continuous on the compact set G. Since the volumen of (Bε(0)∩Σ) is of the orderO(ε2n−3), the follow identity holds when

we take the limit when εtends to zero, Z

Gd f = Z

G f.

Let Λbe a smooth(0,q)-differential form on the closure G, for 0≤ q n1, and such that its coefficients are differentiable onGreg. LetF(y,z)be a differential form defined onG×G, continuous wheny6=z, smooth of type(0,q)with respect to the variable z 6= 0, and smooth of type (n−1,n−q−2) with respect to the variabley 6=0.

Fix z 6= 0 in G. Let 0 < 2ε < dist(y,z), such that Bε(0)Σ G and Bε(z)∩

Σ G. If Dz,ε = G\((0)(z)). We apply Stokes’s theorem to the form Λ(y)F(y,z)on the domain Dz,ε, and so we have

Z

y∈Dz,ε

dy(Λ(y)∧F(y,z)) = Z

y∈Dz,ε

Λ(y)F(y,z). (1.4.4)

Notice thatΛF is a(n1,n2)-form with respect to the variabley, and

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1.4 Integration kernels 13

for εsmall enough.

Thus, if we expand (1.4.4), we have Z

y∈Dz,ε

[yΛ(y)]∧F(y,z) + Z

y∈Dz,ε

Λ(y)(1)qyF(y,z)

=

Z

y∈G

Λ(y)F(y,z)

Z

y∈(Bε(0)∩Σ)

Λ(y)F(y,z)

Z

y∈(Bε(z)∩Σ)

Λ(y)∧F(y,z). (1.4.5)

Now suppose that F(y,z) satisfies the following conditions (remember that

z6=0):

i) lim

ε→0

Z

y∈(Bε(0)∩Σ)

Λ(y)F(y,z) =0,

ii) lim

ε→0

Z

y∈(Bε(z)∩Σ)

Λ(y)F(y,z) =Λ(z),

iii) (−1)q

yF=czF, where cis a constant.

If we take the limit whenε→0 in (1.4.5), we get

Λ=

Z

G

ΛF Z

G

ΛFc

Z

G

ΛF. (1.4.6)

Thus, by a integration kernel on an open set G ⊂ Σ with piecewise smooth

boundary, we mean a differential form F(y,z)defined onG×G, continuous when

y6=z, smooth of type(0,q)with respect to the variablez 6=0, smooth of type(n−

1,n−q−2) with respect to the variable y 6= 0, and that satisfies the conditions (i), (ii) and (iii) above. Hence (1.4.6) is satisfied for every(0,q)-differential formΛ

on G, 0 ≤q ≤n−1, whose coefficients are continuous functions on the closureG

and differentiable on the regular part Greg.

Like in Cn, if Λ is a continuous form with compact support in G (the origin

0 may be contain in the support of Λ) such that it is smooth and -closed on

Greg=G\ {0}. Then, the following form

λ(z) =−c Z

y∈G

Λ(y)F(y,z)

satisfies the -equation

zλ(z) =Λ(z).

Thus, we can solve the -equation on open sets of Σ with smooth piecewise

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Chapter 2

Integration formulae on weighted

homogeneous varieties

2.1

Introduction

The main objective of this chapter is to analyze the work that professors Rup-phental and Zeron presented in [17], in order to produce integration formulae and integral kernels for the representation of measurable functions well defined and with compact support in a weighted homogeneous subvariety. We firstly deduce explicit integration formulae and integral kernels on the particular sub-variety Σ = {z C3 : z1z2 = zn

3, n ∈ N, n ≥ 2}; and then we extend

these formulae to arbitrary weighted homogeneous subvarieties. We analyse this particular subvariety Σ because it is simple enough so as to do the calculations

explicitly and complicated enough so as to exemplify all the classical pathologies.

2.2

Practical example

Definition 2.2.1. LetβZnbe a fixed integer vector with entries βk 1. A polynomial

Q(z)holomorphic onCn is said to be weighted homogeneous of degree d 1with respect

to the vector βif the image Q(Hs(z))is equal to sdQ(z) for the mapping

Hs(z) := sβ1z1,sβ2z2, ...,sβnzn (2.2.2)

and all points s Cand z Cn. An algebraic subvariety ΣinCn is said to be weighted

homogeneous with respect to β if it is the zero locus of a finite number of weighted ho-mogeneous polynomials Qj(z) of (possibly different) degrees dj ≥ 1but all of them with

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Notice that H1(z) in (2.2.2) is the identity mapping and Hs(z) is always an automorphism on the weighted homogeneous subvariety Σ for s 6= 0 fixed. In

particular, when s 6=0, the image Hs(z) lies in the regular part of Σif and only if

zlies in the regular part of Σas well. Define de subvariety

Σ={z C3 : z1z2=zn

3, n∈ N, n≥2}.

We analyze this particular subvarietyΣbecause it is simple enough so as to do the

calculations explicitly and complicated enough so as to exemplify all the classical pathologies. The subvarietyΣis weighted homogeneous of degree nwith respect

to the vector β= (n−α,α, 1) for any integer 1≤ α <n. The calculations become

simpler if we take β= (n−1, 1, 1). Fix the general(0, 1)-measurable form

ω(y) = f1(y)dy1+f2(y)dy2+ f3(y)dy3, (2.2.3)

whose coefficients fk are all Borel-measurable functions well defined on Σ. We also suppose that each fk is essentially bounded and has compact support in Σ

and that y1,y2, and y3 are the cartesian coordinates of C3. We know from the

work of Rupphental and Zeron [17] that:

g1(z) =

3

k=1

βk

2πi Z

u∈C fk(u

βz)(uβkzk)du∧du

u(u1)

is a solution to the -equationω =g1 on the regular part Σ\ {0}.

We rewrite previous expression for g1(z), in order to obtain an integral for-mula that depends on the Cauchy kernel instead of the non-holomorphic kernel

du

¯

u(u−1). Thus, fix the point z = (z1,z2,z3) in C3 and expand the expression for

g1(z),

g1(z) = (1−n)z1

2πi Z

u∈C f1(u

n−1z

1,uz2,uz3)u

n−2dudu

u−1

− z2

2πi Z

u∈C f2(u

n−1z

1,uz2,uz3)du∧du

u−1

− z3

2πi Z

u∈C f3(u

n−1z

1,uz2,uz3)

du∧du

u−1 . (2.2.4)

Define y = (y1,y2,y3) := (un−1z1,uz2,uz3), we need to analyze the following

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2.2 Practical example 17

(a) Consider the case z3 6= 0. We may deduce that u = yz33, and so y =

(yn3−1 z1

zn3−1,y3 z2

z3,y3). Moreover we can express ω(y) in (2.2.3) in terms of the

coor-dinate y3:

ω(y) = (n−1)z1

z3n−1 y3

n−2f

1(y)dy3+z2

z3f2(y)dy3+f3(y)dy3;

and we also express the common kernel in the three integrals of (2.2.4) in terms of the variabley3as

dudu u−1 =

dy3dy3 z3(y3−z3).

Sincey= (un−1z1,uz2,uz3), thenu =y3/z3 and

ω(y)∧ dy3

y3−z3 =

(n−1)z1 y3n−2f1(y)

z3n−1

dy3∧dy3

y3−z3

+ z2f2(y)

z3

dy3∧dy3 y3−z3

+ f3(y)dy3∧dy3

y3−z3

= (n1)z1f1(un−1z1,uz2,uz3)u

n−2dudu

u−1

+ z2f2(un−1z1,uz2,uz3)du∧du

u−1

+ z3f3(un−1z1,uz2,uz3)du∧du

u−1

Hence, we can rewrite (2.2.4) as follows

g1(z) =− 1

2πi Z

y3∈C

ω(y)∧ dy3

y3−z3.

(b) In case z2 6= 0, we may deduce u = y2

z2, and so y = (y

n−1 2

z1

zn2−1,y2,y2 z3

z2).

Moreover we can express ω(y) in (2.2.3) in terms of the coordinatey2:

ω(y) = (n−1)z1

z2n−1 y2

n−2f

1(y)dy2+ f2(y)dy2+z3

z2f3(y)dy2;

and we also express the common kernel in the three integrals of (2.2.4) in terms of the variabley2as

dudu u−1 =

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Sincey= (un−1z1,uz2,uz3), then

ω(y)∧ dy2

y2−z2 =

(n−1)z1 y2n−2f1(y)

z2n−1

dy2∧dy2

y2−z2

+ f2(y)dy2∧dy2

y2−z2

+ z3f3(y)

z2

dy2∧dy2

y2−z2 .

Hence, we can rewrite (2.2.4) as follows

g1(z) =− 1

2πi Z

y2∈C

ω(y)∧ dy2

y2−z2.

(c) In case z1 6= 0, we may deduce that u = (y1

z1) 1

n−1ρk for k = 0, 1, ...,n−2,

where ρ is the (n−1)-th root of unity, so that y = (y1,(yz11)

1

n−1ρkz2,(y1

z1) 1

n−1ρkz3) is

a multivalued function. Moreover, we also have the multivalued form,

ωk(y) = f1(y)dy1+ ρ

kz

2

(n−1)z1

y1

z1

1 n−1−1

f2(y)dy1

+ ρ

kz

3

(n−1)z1

y1

z1

1 n−1−1

f3(y)dy1.

Where ωk is the expansion of (2.2.3) in the k-th branch of the (n−1)-th root function. We also express the common kernel in the three integrals of (2.2.4) in terms of the variable y1:

du u−1 =

(y1

z1) 1

n−1−1dy1

(n−1)z1((yz11)

1

n−1 −ρk)

for k=0, 1, ...,n−2.

And

du∧du u−1 =

ρk(y1

z1) 1 n−1−1(y1

z1) 1 n−1−1

dy1∧dy1

(n−1)2|z

1|2((yz11) 1

n−1 −ρk)

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2.2 Practical example 19

Sincey= (un−1z1,uz2,uz3) and u= (yz11)

1

n−1ρk, then

ωk(y)∧ (

y1

z1) 1

n−1−1dy1

(n−1)z1((yz11)

1

n−1 −ρk)

= (n−1)z1(

y1

z1) 1

n−1−1f1(y)dy1dy1

(n−1)2|z

1|2((yz11)

1

n−1 −ρk)

(2.2.5)

+ z2 ρ

k(y1

z1) 1 n−1−1(y1

z1) 1 n−1−1

f2(y)dy1∧dy1

(n−1)2|z

1|2((yz11)

1

n−1 −ρk)

+ z3 ρ

k(y1

z1) 1 n−1−1(y1

z1) 1 n−1−1

f3(y)dy1∧dy1

(n1)2|z

1|2((yz11)

1

n−1 −ρk)

= (n−1)z1f1(un−1z1,uz2,uz3)u

n−2dudu

u−1

+ z2f2(un−1z1,uz2,uz3)du∧du

u−1

+ z3f3(un−1z1,uz2,uz3)

dudu u−1 .

Notice that un−2 = (y1

z1)

1− 1 n−1

ρ−k and that the term (y1

z1) 1

n−1−1 appears in the

first three lines of the above expansion, because we work in the k-th branch of the (n−1)-th root function, so that we may assume for practical reasons that the variabley1lies in the complex plane minus the negative real axisCb, recall that the real line has zero Lebesgue measure in C2. Moreover, in the last three lines of the above expansion we recover the variable u that lies in the complex plane.

We can now rewrite the formula (2.2.4), where g1 is defined, in terms of the form (2.2.5). Since u lies in the complex plane and we use the change of vari-able u = (y1

z1) 1

n−1ρk, we need to calculate the integrals in (2.2.4) over the Riemann

surface where the(n−1)-th root function is well defined; and this surface is com-posed byn−1 branches glued together. For practical reasons, integrating over this Riemann surface is equivalent to integrating over each of the (n−1)branches and adding all the integrals together. Hence, we integrate with respect to the variable

y1, when it lies in Cb, and we add the result over all posible values of the index

k=0, ...,n2. We then rewrite (2.2.4) as follows:

g1(z) = − 1

2πi

n−2

k=0

Z

y1∈Cb

ωk(y)∧ (

y1

z1) 1

n−1−1dy1

(n−1)z1((yz11)

1

n−1 −ρk)

,

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(2.2.5) and we have

g1(z) = − 1

2πi

n−2

k=0

Z

y1∈Cb

f1(y) (

y1

z1) 1

n−1−1dy1∧dy1

(n−1)z1((y1

z1) 1

n−1 −ρk)

− z2

2πi

n−2

k=0

Z

y1∈Cb f2(y)

ρk|y1

z1| 2

n−1−2dy1∧dy1

(n−1)2|z

1|2((yz11)

1

n−1 −ρk)

− z3

2πi

n−2

k=0

Z

y1∈Cb f3(y)

ρk|y1

z1| 2

n−1−2y1∧dy1

(n−1)2|z

1|2((yz11)

1

n−1 −ρk)

. (2.2.6)

In order to simplify the above formula, we need the following properties of the (n−1)-th root of the unity ρ, that can be easily deduced for every a∈ C,

1. n−2

k=0

ρk =0, 2.

n−2

k=0

(a−ρk) = an−1−1, 3. d

da

n−2

k=0

(a−ρk) = (n−1)an−2.

The identities below are easily infered, from the three above equations.

n−2

k=0

1

a−ρk =

d

da∏k(a−ρk)

k(a−ρk) =

(n−1)an−2

an−11 , (2.2.7)

n−2

k=0

1

ρka1 = n−2

k=0

a

a−ρk −1

= n−1

an−11. (2.2.8)

We use (2.2.7) in the first integral of the right hand side of (2.2.6) with a = (y1

z1) 1

n−1, and (2.2.8) in the second and third integrals of the right hand side of

(2.2.6), and we have (in case z1 6=0)

g1(z) = −

n−2

k=0

hk(z)

= − 1

2πi Z

y1∈Cb f1(y)

dy1dy1 y1−z1

− z2

2πi Z

y1∈Cb

f2(y)|

y1

z1| 2

n−1−2dy1∧dy1

(n−1)z1(y1−z1)

− z3

2πi Z

y1∈Cb f3(y)

|y1

z1| 2

n−1−2dy1∧dy1

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2.3 Integration formulae on weighted homogeneous subvarieties 21

Working on the particular weighted homogeneous subvariety Σ = {z C3 :

z1z2 = zn3} and with the techniques presented in [17], we have obtained in the

cases (a), (b) and (c) integral representations constructed around the Cauchy ker-nel or some of its variations. Thus, we wonder if the Cauchy kerker-nel is an intrinsic property of the weighted homogeneous subvarieties for forms of degree zero or one. Amaizinly the the answer is positive as we will show below.

2.3

Integration formulae on weighted homogeneous

subvarieties

Ruppenthal and Zeron have indirectly deduced in [17, 18] some integration for-mulae on weighted homogeneous subvarieties ofCn. The main idea behind their work is to use a natural foliation induced by the mapping Hs(z)given below.

Hs(z) := sβ1z1,sβ2z2, ...,sβnzn for s∈ C and z∈ Cn.

The following pair of integration formulae are deduced from the Cauchy-Green formulae; see for example [12, p. 9]. The differentials are all defined and calcu-lated in the sense of distributions.

Lemma 2.3.1. LetΣbe a weighted homogeneous subvariety given as in definition (2.2.1)

and f be a continuous function defined on Σ and with compact support. The following

identity holds for every z6=0inΣ

f(z) = f(H1(z)) = −1

2πi Z

s∈C

h

sf(Hs(z))

i

∧ ds

s−1 (2.3.2)

under the assumption that the differential sf(Hs(z)) exists and is continuous at every

s∈ C.

Moreover letℵ =∑k fkdz¯kbe a continuous(0, 1)-form well defined and with compact support on Σ. Assume that ℵ is also -closed in the regular part of Σ. The following identity holds on the regular part as well

ℵ = H1∗ℵ=z

−1 2πi

Z

s∈C[H

sℵ]∧

ds s−1

. (2.3.3)

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used by Henkin and Leiterer in [12, p. 44]. It is easy to deduce that the pull-back ofℵ with respect to Hs(z) is expressed in terms of the differentialsdz¯k andds¯, i.e.

Hs∗ℵ =

n

k=1

fk(Hs(z))sβk

βkzk

s ds+dzk

. (2.3.4)

The integral in (2.3.3) is then calculated only over those monomials that contain the volume form ds¯∧ds, i.e. over the monomial that are of degree 2 with respect to the variable s. However the differential z in (2.3.3) is calculated with respect to the variable zinΣ, and the differentials in (2.3.2) is calculated with respect to

the variables inC.

Proof. All the differentials are calculated in the sense of distributions. Let f

be a continuous function defined on Σ and with compact support. Take a fixed

point z 6= 0 in Σ. The function s 7→ f(Hs(z)) is continuous and has compact

support in the plane s ∈ C, so that we may apply Cauchy-Green formula on a ball Bwith centre at the origin ofCand radius large enough; see for example [12, p. 44]. Identity (2.3.2) is the resulting formula. We only need to assume that the differential ¯f(Hs(z))exists and is continuous at every points ∈C.

Identity (2.3.3) is directly deduce from the main formula presented by Rup-penthal and Zeron in [17, p. 443].

g1(z) =

3

k=1

βk

2πi Z

u∈C fk(u

βz)(uβkzk)du∧du

u(u−1) .

We just need to calculate the integral in (2.3.3) according to the convention used by Henkin and Leiterer in [12, p. 44], so that the integral is calculated only over those monomials of the pull-back (2.3.4) that contain the differential ds¯. In other words, given any measurable function g(s,z), one always assume that the follow-ing integral vanishes Z

s∈Cg(s,z)dz¯k∧ds =0.

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Chapter 3

Integral representation with weights

In this chapter we include the work that professor Mats Andersson presented in [1] and the work made by Peter Helgeson in his disssertation [11]. In [1] the professor Andersson describes a new approach to representation formulae for holomorphic functions and provide a general method to generate weighted inte-gral formulas. In [11] the main idea is to start with weighted Koppelman formulas on the projective space P2, and by a limit procedure to obtain Cauchy-Green for-mulae on an embedded Riemann surface. We include these works in order to compare the proposed technique in these works with the technique proposed in this thesis, specially with the techniques presented in the chapter 5, that is the main chapter in this work.

3.1

Weighted representation formulae

Let X be a smooth complex manifold, and recall that a smooth vector field is a smooth section of the tangent bundle T(X). So in local coordinates it can be written in the form

ξ =

n

ν=1

ξν

zν +

n

ν=1

ξν

zν,

where the coefficients ξν are smooth functions.

Definition 3.1.1. Let H be a vector field and let ω be an r-form on a smooth complex manifold X, r ≥1. We can define an(r−1)-formδH by the formula

(δHω)(x)(v2, ...,vr) = ω(H(x),v2, ...,vr), for v2, ...,vr ∈ Tx,

where Tx denotes the tangent vector space of X at x ∈ X. We callδH the contraction of

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It is easy to see that the Cauchy kernel u(z,a) = zdza satisfies the following equations in the sense of distributions.

(z−a)u(z) = dz and u = [a], (3.1.2) where [a] denotes integration (evaluation) at a considered as an (n,n)-current. In order to generalize Cauchy’s formula to higher dimensions it might seem to be most natural to look for forms u that satisfy the second equation in (3.1.2), since each such solution gives rise to a representation formula by Stokes’theorem. We have to introduce some notation. Let Ep,q(U) denote the space of smooth forms of type (p,q) in the open setU ∈ Cn and, for any integer m, let Lm(U) =

Ln

k=0Ek,k+m(U). For instance, u ∈ Lm(U) can be writtenu =u1+...+un, where the index denotes the degree in dz, so that uk is a form of type (k,k−1). In the same way we let Lm

curr(U) denote the corresponding space of currents. Fix a point a Cn, let δza : Ep,q(U) Ep−1,q(U) be contraction with the vector field 2πi

n

j=1

(zj−aj)

zj, and let ∇z−a = δz−a−, so that ∇z−a : L

m(U) Lm+1(U).

Notice that when n =1, the Cauchy kernel is the unique solution in L−1

curr(U)to

∇z−au(z) = 1−[a]. (3.1.3)

Recall equation (3.1.2). If n>1, (3.1.3) means that

δz−au1=1, δz−auk+1−uk =0, un = [a], (3.1.4) and any such u will be called a Cauchy form (with respect to a). If uis a Cauchy form we have an integral representation formula for holomorphic functions ϕon the closure D

ϕ(a) =

Z

D ϕ[a] = Z

D∂ϕ∧un +ϕ∂un = Z

Dd(ϕun) = Z

D ϕun, a∈ D.

Definition 3.1.5. A smooth form g ∈ L0(U) such that zag = 0 and g0(a) = 1 is

called a weight with respect to a ∈U.

Ifgis a weight with repect toz ∈ D, we can solve∇z−av= gin a neighborhood

U of the boundary D, and we then get the weighted representation formula for holomorphic functions ϕon the closureD,

ϕ(a) =

Z

Dϕu∧g+ Z

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3.1 Weighted representation formulae 25

If gj are weights andG(λ1, ...,λm)is holomorphic on the image ofz7→ (g10, ...,g0m), then one can form a new weight G(g). In particular, ifm=1 and g=1+ (δz−a−

)q, and qis a form of type (1, 0), then

G(g) =

n

k=0

G(k)(δz−aq)(q)k−1

k! , is a weight if G(0) = 1.

Notice that, if

b(z,a) = 1

2πi

||z−a||2

||z−a||2 ,

then the form

B(z,a) = 1

∇z−a = 1 1−u =

n

k=1

b∧(b)k,

is smooth and solves∇z−a =1 inCn\ {a}. It will be called the Bochner-Martinelli form. ReplacingubyB(z,a)in (3.1.6) we have the integral representation formula, known as the Bochner-Martinelli formula.

Proposition 3.1.7. If g is a weight inΩ, D ⊂⊂Ω, andz−av =g in a neighbourhood

of the boundary D, then

ϕ(a) =

Z

Dϕ∧vn+ Z

Dϕgn, (3.1.8)

for holomorphic functions ϕon the closure D. If∇z−a =g−[a]inΩ, then

ϕ(a) =

Z

D ϕvn+ Z

D ϕgn− Z

D∂ϕ∧vn, (3.1.9)

for smooth functions ϕon the closureD Proof. See [1].

We work inCn×Cn instead ofCn. LetCn a domain, η =zζ ×

be fixed, and consider the subbundle E∗ = span{dη1, ...,dηn} of the cotangent bundle T1,0∗ (the space of forms of type (1, 0)) over Ω×. Let E be its dual

bundle and let δη be the contraction with the section

1 2πi

n

j=1

ηjej, whereej is the dual basis to ηj.

Let Lp,q denote the space of sections to the exterior algebra over ET∗ 0,1 of

bidegree (p,q) and letLm =LpLp,p+m. If η =δη, then we can solve

(36)

where[∆]denotes the(n,n)-current integration over the diagonalin×. In

fact, the Bochner-Martinelli section u= b

ηb, whereb=

1 2πi∑

||η||2

||η||2 solves (3.1.10).

A form g ∈L0(×Ω)is a weight if ∇ηg =0 andg0 ≡1 on∆. As before ifq

is any smooth form in L−1 and G(0) = 1, then g = G(ηq) is a weight. If g is a

weight we can solve

ηv=g−[∆],

and if K = vn and P = gn, then we haveK = [∆]−P. Thus for D ⊂⊂Ω we get for smooth forms ϕof type (0,q) on the closureD, the Koppelman’s formula

ϕ(z) =

Z

D ϕ∧K+ Z

D∂ϕ∧K+z Z

D ϕ∧K+ Z

Dϕ∧P. (3.1.11)

3.2

Integral formulae on a Riemann surface in

C

2

By means of the theory of integral representation formulas in Cn, we are now going to find explicit formulas representing holomorphic functions on Riemann surfaces, X, embedded in C2. In particular we will considerX ={z C2 : f(z) = 0}, where f is some holomorphic function such thatd f 6=0 on X.

Holomorphicity on X is defined locally in a natural way, since if {ψi}iI is a collection of charts we say that ϕ : X → C is holomorphic on X if and only if

(ϕψ−1) = 0 for alli ∈ I. The set of holomorphic functions on Xis denoted by

O(X).

3.3

Preliminaries

Before presenting the general idea of how to find integral formulas for holo-morphic functions on Riemann surfaces, we need to introduce some concepts and results that will be of fundamental importance for the rest of this text. Re-call that δζ−z is the contraction by the vector field 2πi∑j=n 1(ζj−zj)∂ζ

j and that

ζ−z =δζ−z−∂ζ.

Definition 3.3.1. Let U ⊂Cn be open and f : U Cbe a holomorphic mapping. If h

is a holomorphic(1, 0)-form such that

δζzh = f(ζ)− f(z),

Figure

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