**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**

**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**

**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 **C**n _{with piecewise smooth boundary} * _{∂}*Ω

_{, and}

_{ℵ}

_{be a}

_{(}

_{0,}

_{q}

_{)}

_{-form whose }

coeffi-cients are continuous functions on the closureΩ_{. If the differential}_{∂}_{ℵ} _{(calculated}

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

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
of**C**n_{. 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_{. If} Ω _{has a piecewise smooth boundary} * _{∂}*Ω

_{, the problem is to produce}

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

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-etiesΣ_{of}**C**n_{, but they delimited their work to analyse differentials forms}_{ℵ}_{whose}
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
**C**n_{.}

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} _{∈} **C**3 _{:} _{z}

1z2 = zn_{3}, 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}_{∈} **C**3 _{:} _{z}

1z2 =zn_{3}}. 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 **C**n_{, based on the fact that}
every Stein submanifold in**C**n _{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∈ **C**n _{:} _{∑}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 **C**n_{", 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 **C**n+1 _{generated as the zero locus of a polynomial} _{s}m_{−}_{p}_{(}_{z}_{)} _{for} _{s} _{∈} **C** _{and}

z ∈ **C**n_{. 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 **C***κ*_{, with the conditions that some entries of} _{z} _{∈} Σ _{can be easily expressed in}

**vii**

**Contents**

**Introduction and objectives** **v**

**1** **Preliminaries** **1**

1.1 The*∂*operator . . . 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 in**C**2 _{. . . 26}

3.3 Preliminaries . . . 26

3.4 Explicit integral representations onX . . . 30

3.5 Integral formulae on a Riemann surface in**P**2 _{. . . 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** **C**n ** _{51}**
5.1 Introduction . . . 51

5.2 Basic properties . . . 55

**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 space**C**n _{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

*∂*z¯_{k}dz¯k

=

n

### ∑

k=11 2

*∂*f

*∂*Rez_{k} +
i *∂*f

*∂*Imz_{k}

dz¯k.

Notice that the*∂*operator sends smooth functions into(0, 1)-differential forms.
In a similar way, we may define a general*∂*operator 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*∂*

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 **C**n _{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,*ϕ*U)in X consists of

an open set U ⊂X and a topological map *ϕ*U from U onto an open set B ⊂**C**n.

We say that twon-dimensional complex coordinate systems(U,*ϕ*U)and(V,*ψ*V)
in X are compatible if either U∩V = ∅ _{or the map} _{ϕ}_{U} _{◦} * _{ϕ}*−1

V is biholomorphic,
so that *ϕ*_{U} _{◦}*ϕ*−_{V}1 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,*ϕ*U) in X such that x∈ U and f ◦*ϕ*−_{U}1 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,*ϕ*U)in X at x and a coordinate system (V,*ψ*V) in Y at F(x)

**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-nating**R**_{-multilinear mapping}

*ϕ*:

r−times

z }| {

Tx×...×Tx →**C**.

Alternating means that interchanging the values of two entries of *ϕ*automatically
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**_{.} _{F}1 _{=} _{F}_{(}_{T}_{x}_{)} _{is the complexification of the 2}_{n}_{-real}
dimensional vector space T_{x}∗, where T_{x}∗ is the dual space to Tx.

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

r >_{2}_{n}_{.}

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

dim**R**Fr =

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*.

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

*ϕ*∧*ψ*(v_{1}, ...,vr,vr+1, ...,vr+s)
:= 1

r!s!_{σ}_{∈}

### ∑

_{S}

r+s

(sgn*σ*)*ϕ*(v_{σ}_{(}_{1}_{)}, ...,v_{σ}_{(r)})·*ψ*(v_{σ}_{(r+}_{1}_{)}, ...,v_{σ}_{(r+s)}).

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

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

and v,w∈ Tx. And in general

1. *ϕ*∧*ψ*= (−1)rs_{ψ}_{∧}_{ϕ}_{.}
2. (*ϕ*∧*ψ*)∧*ω* = *ϕ*∧(*ψ*∧*ω*).

With the wedge product, the vector space

^ F:=

2n M

r=0

Fr

becomes a noncommutative graded associative**C**_{-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} _{ϕ}_{is called a form of}

type(p,q)if

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

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

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

*ϕ*(cv_{1}, ...,cvr) = cp cq·*ϕ*(v1, ...,vr)

=cp′ cq′·*ϕ*(v_{1}, ...,vr).

Therefore, cp cq =cp′ cq′ for each c ∈ **C**_{. If} _{c} _{=} _{e}it _{with} _{t} _{∈} **R**_{, then}_{e}it(p−q) _{=}

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

**1.2 Differential forms** **5**

**Proposition 1.2.9.** 1. If the form *ϕ*is of type(p,q), then *ϕ*is of type(q,p).

2. If *ϕ*and*ψ*are both forms of type(p,q), then *ϕ*+*ψ*and*λ*·*ϕ*are also of type(p,q).
3. If *ϕ*is a form of type(p,q) and*ψ*of type(p′,q′), then *ϕ*∧*ψ*is of type

(p+p′,q+q′).

Notice thatdz*ν* is a form of type (0, 1), since dz*ν* =dz*ν*. Then dzi1 ∧...∧dzip ∧
dz_{j}_{1} ∧...∧dz_{j}_{q}, with 1 ≤ i1 < ... < ip ≤ n and 1 ≤ j1 < ... < jq ≤ n, is a form of
type (p,q).

**Theorem 1.2.10.** Any r-form *ϕ*has 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 dz_{i}_{1} ∧...∧dz_{i}_{p}∧dz_{j}_{1} ∧...∧dz_{j}_{q} 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)}(cv_{1}, ...,cvr) =

### ∑

p+q=rcp cq*ψ*_{(p}_{,}_{q)}(v_{1}, ...,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 over**C**_{.}

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

_{(}

_{U}

i) →Ui×**C**r
and for eachx ∈U_{i} the induced map pr◦Φ_{i} _{:} _{E}_{x} _{→} _{x}_{×}**C**r _{→}**C**r _{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 with*π*◦s=id_{U}.

Let X be an n-dimensional complex manifold, we denote by F(X) the
com-plexified cotangent bundle T∗(X)N**C**_{. It has the spaces} _{F}_{(}_{T}_{x}_{(}_{X}_{)) =} _{T}_{x}_{(}_{X}_{)}∗N**C**
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 (m_{r}) for each 0 ≤ r ≤ m, such that (Fr(E))x = Fr(Ex) for
every x ∈ X. If E is given by the transitions functions g_{ij}, then Fr(E) is naturally
given by the matrices g(r)_{ij} whose entries are the(r_{×}r)minors of g_{ij}.

**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 *ω*x at x. Notice
that if z_{1}, ...,zn are the local coordinates in a neighbourhood of x, then *ω*j := dzj
and *ω*n+j :=dzj, for j=1, ...,nform a basis of the 1-forms on this neighbourhood.
Moreover there is a representation

*ω*x =

### ∑

1≤i1<...<ir≤2n

ai1...ir(x)*ω*i1∧...∧*ω*ir,

where x_{7→} a_{i}_{1}_{...}_{i}_{r}(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

*∂*z*ν*dz*ν*+

n

### ∑

*ν*=1

*∂*f

**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* *∂*

*∂*z*ν* +

n

### ∑

*ν*=1

*ξν* *∂*

*∂*z*ν*,

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

*∂*z*ν* +

n

### ∑

*ν*=1

*ξν* *∂*f

*∂*z*ν*.

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* =∑_{1}_{≤}_{i}

1<...<ir≤2nai1...ir(x)*ω*i1∧...∧

*ω*_{i}_{r} is the basis representation in a coordinate neigbourhoodU, then

d_{U}(*ω*) :=

_{∑}

1≤i1<...<ir≤2n

da_{i}_{1}_{...}_{i}_{r}(x)∧*ω*_{i}_{1} ∧...∧*ω*_{i}_{r}.

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. dis**C**_{-linear.}

3. d◦d=0.

4. If *ϕ*∈ Er(U) and*ψ*∈ Es(U), then d(*ϕ*∧*ψ*) =d*ϕ*∧*ψ*+ (−1)r_{ϕ}_{∧}_{d}_{ψ}_{.}

5. dis a real operator; that isd*ϕ*=d*ϕ*. In particular d*ϕ*=d(Re *ϕ*) +i d(Im *ϕ*).
The differential d*ω* is called the exterior derivative of the form*ω*.

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

and use some notation, if I = (i_{1}, ...,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

a_{i}_{1}_{...}_{i}_{p}_{,}_{j}_{1}_{,...,}_{j}_{q}dz_{i}_{1}∧...∧dz_{i}_{p} ∧dz_{j}_{1}∧...∧dz_{j}_{q}.

So a generalr-form *ω* has the unique representation

*ω* =

### ∑

p+q=r_{|}

_{I}

### ∑

_{|}

_{=p}

|J|=q

and the differential of *ω* is given by

d*ω* =

### ∑

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

*∂*z*ν*dz*ν* and *∂*f =

n

### ∑

*ν*=1

*∂*f

*∂*z*ν*dz*ν*,

here*∂*f has type (1, 0), *∂*f has type(0, 1), andd f is of type (1, 1).

**Proposition 1.2.14.** Let*ϕ*be a r-form of type(p,q). Then d*ϕ*has a unique decomposition
d*ϕ*=*∂ϕ*+*∂ϕ*with*∂ϕ*a(p+1,q)-form and*∂ϕ*a (p,q+1)-form.

**Proof.** Let *ϕ*=∑_{I}_{,}_{J}aI JdzI∧dzJ. We define

*∂ϕ*:=

_{∑}

I,J

*∂*a_{I J}∧dz_{I} ∧dz_{J} and *∂ϕ*:=*∂*a_{I J}∧dz_{I} ∧dz_{J},

then d*ϕ* = *∂ϕ*+*∂ϕ* is the unique decomposition of the (p+q+1)-form d*ϕ* 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. *∂*and*∂*are**C**_{-linear operators with d}_{=}_{∂}_{+}_{∂}_{.}

2. *∂∂*=0,*∂∂* =0, and *∂∂*+*∂∂* =0.

3. *∂*,*∂*are not real. We have*∂ϕ* =*∂ϕ*and*∂ϕ*=*∂ϕ*.
4. If *ϕ*is a r-form and*ψ*is arbitrary, then

*∂*(*ϕ*∧*ψ*) = *∂ϕ*∧*ψ*+ (−1)r*ϕ*∧*∂ψ*,

*∂*(*ϕ*_{∧}*ψ*) = *∂ϕ*_{∧}*ψ*+ (−1)r*ϕ*_{∧}*∂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).

**1.2 Differential forms** **9**

**Definition 1.2.16.** Let *ϕ*be a p-form on the open set U ⊂X.
1. *ϕ*is called holomorphic if *ϕ*is of type(p, 0)and*∂ϕ*=0.
2. *ϕ*is called antiholomorphic if *ϕ*is 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 h_{k} are the components ofhin these coordinates, then
in h−1(U) we define the pull-backh∗*ϕ*of *ϕ*with respect to hby:

(h∗*ϕ*)(x) :=

_{∑}

p+q=r_{|}_{I}

### ∑

_{|}=p

|J|=q

(a_{I J}◦h)(x)dh_{i}_{1}(x)∧...∧dh_{i}_{p}(x)∧dh_{j}_{1}∧...∧dh_{j}_{q}(x).

And we have the following properties:

1. For every(p,q)-form *ϕ*onU, the pull-backh∗*ϕ*is 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 **C**n_{. The}
fact that *λ* is *∂*-closed (*∂λ* = 0) in the sense of distributions on S means that the
following integral vanishes _{Z}

S*λ*∧*∂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⊂**C**n_{, and} _{g}_{be a form of type} _{(}_{0,}_{q}_{−}_{1}_{)} _{on}_{S} _{as well. We say that}_{λ}_{=}_{∂}_{g}_{holds}
in the distributional sense onS if and only if

Z

S *λ*∧*σ*+ (−1)

q−1_{g}_{∧}_{∂σ}_{=}_{0,}

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

**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 *ϕν*tends 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 an**R**_{-linear map T} _{:} _{E}r_{(}_{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 if*ψ*is a differential form of degree 2n−r. Then*ψ*defines a current

TΨ of degree 2n−r by the formula

T*ψ*(*ϕ*) :=
Z

X*ψ*∧*ϕ*, for all *ϕ*∈ E
r_{(}_{X}_{)}_{.}

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

**1.4**

**Integration kernels**

Let G ⊂ **C**n _{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 let*ε* >_{0, such that the ball} _{B}_{ε}_{(}_{z}_{) =} _{{}_{x} _{∈} **C**n _{:} _{||}_{x}_{−}_{z}_{|| ≤} _{ε}_{}}
is contained in G. If Dz,*ε* = G\ B*ε*(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 thatΛ_{∧}_{F} _{is a}_{(}_{n}_{,}_{n}_{−}_{1}_{)}_{-form with respect to the variable}_{y}_{, and}

**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}_{)}q_{∂}_{y}_{F}_{(}_{y}_{,}_{z}_{)}

=

Z

y∈*∂*G

Λ_{(}_{y}_{)}_{∧}_{F}_{(}_{y}_{,}_{z}_{)}_{−}

Z

y∈*∂*B*ε*(z)

Λ_{(}_{y}_{)}_{∧}_{F}_{(}_{y}_{,}_{z}_{)}_{.} _{(1.4.2)}

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

(i) lim

*ε*→0

Z

y∈*∂*B*ε*(z)

Λ_{(}_{y}_{)}_{∧}_{F}_{(}_{y}_{,}_{z}_{) =} Λ_{(}_{z}_{)}_{,}

(ii) (−1)q_{∂}

yF(y,z) = c*∂*zF(y,z), where c is a constant.
If we take the limit when*ε*tends to zero in (5.1.7),

Λ _{=}

Z

*∂*G

Λ_{∧}_{F}_{−}

Z

G(*∂*

Λ_{)}_{∧}_{F}_{−}_{c}_{∂}

Z

G

Λ_{∧}_{F}_{.} _{(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}_{, 0}_{≤}_{q} _{≤}_{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} **C**n_{, and}

*∂*Λ_{=}_{0, then the form}

*λ*(z) =−c
Z

y∈G

Λ(y)∧F(y,z)

satisfies the *∂*-equation

*∂*z*λ*(z) = Λ(z) on **C**n.

We are interesting in producing integration kernels on varieties with
singu-larities, so that we work with a practical example. Let Σ _{⊂}**C**n _{be a variety such}
that the origin of **C**n _{is an isolated singularity of} Σ _{and the regular part} Σ_{reg 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 Σ_{reg.}

LetG ⊂Σ_{be an open set with piecewise smooth boundary. We need to define}

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

If 0∈ G, we take*ε*>_{0 such that the ball} _{B}_{ε}_{(}_{0}_{)}_{∩}Σ_{⊂} _{G}_{, and define}

Z

Gh:=lim*ε*→0

Z

G−B*ε*(0)

h.

Moreover, if his not a (n_{−}1,n_{−}1)-form, we define R_{G}h_{≡}0.

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 take*ε* >_{0 such that} _{B}_{ε}_{(}_{0}_{)}_{∩}Σ _{⊂}_{G}_{. Let} _{G}_{ε}_{=}_{G}_{\}_{(}_{B}_{ε}_{(}_{0}_{)}_{∩}Σ_{)}_{,}

then:

Z

G*ε*

d f =

Z

*∂*G*ε*

f

=

Z

*∂*G f −
Z

*∂*(B*ε*(0)∩Σ)

f.

Recall that f is bounded because it is continuous on the compact set G. Since the
volumen of *∂*(B*ε*(0)∩Σ) is of the orderO(*ε*2n−3), the follow identity holds when

we take the limit when *ε*tends to zero,
Z

Gd f = Z

*∂*G f.

Let Λ_{be a smooth}_{(}_{0,}q)-differential form on the closure G, for 0≤ q _{≤}n_{−}1,
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 < _{2}* _{ε}* <

_{dist}

_{(}

_{y}

_{,}

_{z}

_{)}

_{, such that}

_{B}

_{ε}_{(}

_{0}

_{)}

_{∩}Σ

_{⊂}G and B

*ε*(z)∩

Σ _{⊂} _{G}_{. If} _{D}_{z}_{,}_{ε}_{=} _{G}_{\}_{(}_{Bε}_{(}_{0}_{)}_{∪}_{Bε}_{(}_{z}_{))}_{. We apply Stokes’s theorem to the form}
Λ_{(}_{y}_{)}_{∧}_{F}_{(}_{y}_{,}_{z}_{)}_{on the domain} _{D}_{z}_{,}_{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 thatΛ_{∧}_{F} _{is a}_{(}_{n}_{−}_{1,}_{n}_{−}_{2}_{)}_{-form with respect to the variable}_{y}_{, and}

**1.4 Integration kernels** **13**

for *ε*small 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}_{)}q_{∂}_{y}_{F}_{(}_{y}_{,}_{z}_{)}

=

Z

y∈*∂*G

Λ_{(}_{y}_{)}_{∧}_{F}_{(}_{y}_{,}_{z}_{)}_{−}

Z

y∈*∂*(B*ε*(0)∩Σ)

Λ_{(}_{y}_{)}_{∧}_{F}_{(}_{y}_{,}_{z}_{)}

−

Z

y∈*∂*(B*ε*(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∈*∂*(B*ε*(0)∩Σ)

Λ_{(}_{y}_{)}_{∧}_{F}_{(}_{y}_{,}_{z}_{) =}_{0,}

ii) lim

*ε*→0

Z

y∈*∂*(B*ε*(z)∩Σ)

Λ_{(}_{y}_{)}_{∧}_{F}_{(}_{y}_{,}_{z}_{) =}Λ_{(}_{z}_{)}_{,}

iii) (−1)q_{∂}

yF=c*∂*zF, where cis a constant.

If we take the limit when*ε*→0 in (1.4.5), we get

Λ_{=}

Z

*∂*G

Λ_{∧}F_{−}
Z

G*∂*

Λ_{∧}F_{−}c*∂*

Z

G

Λ_{∧}F. (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 G_{reg.}

Like in **C**n_{, 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}

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

*λ*(z) =−c
Z

y∈G

Λ_{(}_{y}_{)}_{∧}_{F}_{(}_{y}_{,}_{z}_{)}

satisfies the *∂*-equation

*∂*z*λ*(z) =Λ(z).

Thus, we can solve the *∂*-equation on open sets of Σ _{with smooth piecewise}

**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} _{∈} **C**3 _{:} _{z}_{1}_{z}_{2} _{=} _{z}n

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.** Let*β*∈ **Z**n_{be a fixed integer vector with entries} _{β}_{k} _{≥}_{1}_{. A polynomial}

Q(z)holomorphic on**C**n _{is said to be weighted homogeneous of degree d} _{≥}_{1}_{with respect}

to the vector *β*if the image Q(Hs(z))is equal to sdQ(z) for the mapping

Hs(z) := s*β*1z1,s*β*2z2, ...,s*β*nzn (2.2.2)

and all points s _{∈} **C**_{and z} _{∈} **C**n_{. An algebraic subvariety} Σin**C**n _{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

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 Σ_{as well. Define de subvariety}

Σ_{=}_{{}_{z}_{∈} **C**3 _{:} _{z}_{1}_{z}_{2}_{=}_{z}n

3, n∈ **N**, n≥2}.

We analyze this particular subvarietyΣ_{because it is simple enough so as to do the}

calculations explicitly and complicated enough so as to exemplify all the classical
pathologies. The subvarietyΣ_{is weighted homogeneous of degree} _{n}_{with 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 f_{k} is essentially bounded and has compact support in Σ

and that y1,y2, and y3 are the cartesian coordinates of **C**3. We know from the

work of Rupphental and Zeron [17] that:

g_{1}(z) =

3

### ∑

k=1*β*_{k}

2*π*i
Z

u∈**C** fk(u

*β*_{∗}_{z}_{)}(u*β*kzk)du∧du

u(u_{−}1)

is a solution to the *∂*-equation*ω* =*∂*g_{1} on the regular part Σ_{\ {}0}.

We rewrite previous expression for g_{1}(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 **C**3 and expand the expression for

g1(z),

g1(z) = (1−n)z1

2*π*i
Z

u∈**C** f1(u

n−1_{z}

1,uz2,uz3)u

n−2_{du}_{∧}_{du}

u−1

− z2

2*π*i
Z

u∈**C** f2(u

n−1_{z}

1,uz2,uz3)du∧du

u−1

− z3

2*π*i
Z

u∈**C** f3(u

n−1_{z}

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

**2.2 Practical example** **17**

(a) Consider the case z3 6= 0. We may deduce that u = y_{z}3_{3}, and so y =

(yn_{3}−1 z1

zn_{3}−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

z_{3}n−1 y3

n−2_{f}

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

du_{∧}du
u−1 =

dy_{3}_{∧}dy_{3}
z3(y3−z3).

Sincey= (un−1z_{1},uz_{2},uz_{3}), thenu =y_{3}/z_{3} and

*ω*(y)∧ dy3

y3−z3 =

(n−1)z1 y3n−2f1(y)

z3n−1

dy3∧dy3

y3−z3

+ z2f2(y)

z3

dy_{3}∧dy_{3}
y3−z3

+ f3(y)dy3∧dy3

y_{3}−z_{3}

= (n_{−}1)z_{1}f_{1}(un−1z_{1},uz_{2},uz_{3})u

n−2_{du}_{∧}_{du}

u−1

+ z_{2}f_{2}(un−1z_{1},uz_{2},uz_{3})du∧du

u−1

+ z_{3}f_{3}(un−1z_{1},uz_{2},uz_{3})du∧du

u−1

Hence, we can rewrite (2.2.4) as follows

g1(z) =− 1

2*π*i
Z

y3∈**C**

*ω*(y)∧ dy3

y3−z3.

(b) In case z_{2} 6= 0, we may deduce u = y2

z2, and so y = (y

n−1 2

z1

zn_{2}−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

z_{2}n−1 y2

n−2_{f}

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

z_{2}f3(y)dy2;

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

du_{∧}du
u−1 =

Sincey= (un−1z1,uz2,uz3), then

*ω*(y)∧ dy2

y_{2}−z_{2} =

(n−1)z1 y2n−2f1(y)

z_{2}n−1

dy2∧dy2

y_{2}−z_{2}

+ f2(y)dy2∧dy2

y2−z2

+ z3f3(y)

z_{2}

dy2∧dy2

y_{2}−z_{2} .

Hence, we can rewrite (2.2.4) as follows

g1(z) =− 1

2*π*i
Z

y2∈**C**

*ω*(y)∧ dy2

y_{2}−z_{2}.

(c) In case z_{1} 6= 0, we may deduce that u = (y1

z1) 1

n−1*ρ*k for k = 0, 1, ...,n−2,

where *ρ* is the (n−1)-th root of unity, so that y = (y1,(y_{z}_{1}1)

1

n−1*ρ*kz_{2},(y1

z1) 1

n−1*ρ*kz_{3}) is

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

*ω*_{k}(y) = f_{1}(y)dy_{1}+ *ρ*

k_{z}

2

(n−1)z_{1}

y1

z_{1}

1 n−1−1

f2(y)dy1

+ *ρ*

k_{z}

3

(n−1)z_{1}

y1

z_{1}

1 n−1−1

f3(y)dy1.

Where *ω*_{k} 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 y_{1}:

du u−1 =

(y1

z1) 1

n−1−1_{dy}_{1}

(n−1)z1((y_{z}_{1}1)

1

n−1 −*ρ*k)

for k=0, 1, ...,n−2.

And

du∧du u−1 =

*ρ*k(y1

z1) 1 n−1−1(y1

z1) 1 n−1−1

dy1∧dy1

(n−1)2_{|}_{z}

1|2((yz11) 1

n−1 −*ρ*k)

**2.2 Practical example** **19**

Sincey= (un−1z1,uz2,uz3) and u= (y_{z}1_{1})

1

n−1* _{ρ}*k

_{, then}

*ω*_{k}(y)∧ (

y1

z1) 1

n−1−1_{dy}_{1}

(n−1)z1((y_{z}_{1}1)

1

n−1 −*ρ*k)

= (n−1)z1(

y1

z1) 1

n−1−1_{f}_{1}(_{y})_{dy}_{1}∧_{dy}_{1}

(n−1)2_{|}_{z}

1|2((y_{z}1_{1})

1

n−1 −*ρ*k)

(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((y_{z}_{1}1)

1

n−1 −* _{ρ}*k)

+ z3 *ρ*

k_{(}y1

z1) 1 n−1−1(y1

z1) 1 n−1−1

f_{3}(y)dy_{1}∧dy_{1}

(n_{−}1)2_{|}_{z}

1|2((y_{z}_{1}1)

1

n−1 −*ρ*k)

= (n−1)z1f1(un−1z1,uz2,uz3)u

n−2_{du}_{∧}_{du}

u−1

+ z2f2(un−1z1,uz2,uz3)du∧du

u−1

+ z3f3(un−1z1,uz2,uz3)

du_{∧}du
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
variabley_{1}lies in the complex plane minus the negative real axis**C**b_{, recall that the}
real line has zero Lebesgue measure in **C**2_{. 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 g_{1} 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−1*ρ*k_{, 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

y_{1}, when it lies in **C**b_{, and we add the result over all posible values of the index}

k=0, ...,n_{−}2. We then rewrite (2.2.4) as follows:

g_{1}(z) = − 1

2*π*i

n−2

### ∑

k=0Z

y1∈**C**b

*ω*_{k}(y)∧ (

y1

z1) 1

n−1−1_{dy}_{1}

(n−1)z1((y_{z}1_{1})

1

n−1 −* _{ρ}*k)

,

(2.2.5) and we have

g_{1}(z) = − 1

2*π*i

n−2

### ∑

k=0Z

y1∈**C**b

f_{1}(y) (

y1

z1) 1

n−1−1dy_{1}∧dy_{1}

(n−1)z_{1}((y1

z1) 1

n−1 −*ρ*k)

− z2

2*π*i

n−2

### ∑

k=0Z

y1∈**C**b
f2(y)

*ρ*k|y1

z1| 2

n−1−2dy_{1}∧dy_{1}

(n−1)2_{|}_{z}

1|2((y_{z}1_{1})

1

n−1 −*ρ*k)

− z3

2*π*i

n−2

### ∑

k=0Z

y1∈**C**b
f3(y)

*ρ*k|y1

z1| 2

n−1−2y_{1}∧dy_{1}

(n−1)2_{|}_{z}

1|2((y_{z}1_{1})

1

n−1 −*ρ*k)

. (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*ρ*k =0,
2.

n−2

### ∏

k=0(a−*ρ*k) = an−1−1,
3. d

da

n−2

### ∏

k=0(a−*ρ*k) = (n−1)an−2.

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

n−2

### ∑

k=01

a−*ρ*k =

d

da∏k(a−*ρ*k)

∏_{k}(a−*ρ*k) =

(n−1)an−2

an−1_{−}_{1} , (2.2.7)

n−2

### ∑

k=01

*ρ*k_{a}_{−}_{1} =
n−2

### ∑

k=0

a

a−*ρ*k −1

= n−1

an−1_{−}_{1}. (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 z_{1} 6=0)

g1(z) = −

n−2

### ∑

k=0h_{k}(z)

= − 1

2*π*i
Z

y1∈**C**b
f1(y)

dy_{1}_{∧}dy_{1}
y1−z1

− z2

2*π*i
Z

y1∈**C**b

f_{2}(y)|

y1

z1| 2

n−1−2dy_{1}∧dy_{1}

(n−1)z1(y1−z1)

− z3

2*π*i
Z

y1∈**C**b
f3(y)

|y1

z1| 2

n−1−2dy_{1}∧dy_{1}

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

Working on the particular weighted homogeneous subvariety Σ _{=} _{{}_{z} _{∈} **C**3 _{:}

z1z2 = zn_{3}} 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 of**C**n_{. The main idea behind their}
work is to use a natural foliation induced by the mapping Hs(z)given below.

Hs(z) := s*β*1z1,s*β*2z2, ...,s*βn*zn for s∈ **C** and z∈ **C**n.

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.** LetΣ_{be 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

2*π*i
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} f_{k}dz¯_{k}be 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 = H_{1}∗ℵ=*∂*z

−1
2*π*i

Z

s∈**C**[H

∗

sℵ]∧

ds s−1

. (2.3.3)

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.

H_{s}∗ℵ =

n

### ∑

k=1f_{k}(Hs(z))s*β*k

*β*_{k}zk

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 differential}_{∂}_{s} _{in (2.3.2) is calculated with respect to}

the variables in**C**_{.}

**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}_{(}_{H}_{s}_{(}_{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 of**C**_{and 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].

g_{1}(z) =

3

### ∑

k=1*β*_{k}

2*π*i
Z

u∈**C** fk(u

*β*_{∗}_{z}_{)}(u*β*kzk)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∈**C**g(s,z)dz¯k∧ds =0.

✷

**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 **P**2_{, 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*

*∂*
*∂*z*ν* +

n

### ∑

*ν*=1

*ξν*

*∂*
*∂*z*ν*,

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)-form*δ*H by the formula

(*δ*H*ω*)(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 call*δ*H the contraction of

It is easy to see that the Cauchy kernel u(z,a) = _{z}dz_{−}_{a} 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 ∈ **C**n _{and, for any integer} _{m}_{, let} Lm_{(}_{U}_{) =}

L_{n}

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 u_{k} 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 _{∈} **C**n_{, let} _{δ}_{z}_{−}_{a} _{:} _{E}p,q_{(}_{U}_{)} _{→} _{E}p−1,q_{(}_{U}_{)} _{be contraction with the vector field}
2*π*i

n

### ∑

j=1(zj−aj) *∂*

*∂*z_{j}, and let ∇z−a = *δ*z−a−*∂*, so that ∇z−a : L

m_{(}_{U}_{)} _{→} _{L}m+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}

*δ*z−au1=1, *δ*z−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 *ϕ*on
the closure D

*ϕ*(a) =

Z

*∂*D *ϕ*[a] =
Z

*∂*D*∂ϕ*∧un +*ϕ∂*un =
Z

*∂*Dd(*ϕ*un) =
Z

*∂*D *ϕ*un, a∈ D.

**Definition 3.1.5.** A smooth form g ∈ L0_{(}_{U}_{)} _{such that} _{∇}_{z}_{−}_{a}_{g} _{=} _{0} _{and g}_{0}_{(}_{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 *ϕ*on the closureD,

*ϕ*(a) =

Z

*∂*D*ϕ*u∧g+
Z

**3.1 Weighted representation formulae** **25**

If gj are weights andG(*λ*1, ...,*λ*m)is holomorphic on the image ofz7→ (g1_{0}, ...,g_{0}m),
then one can form a new weight G(g). In particular, ifm=1 and g=1+ (*δ*z−a−

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

G(g) =

n

### ∑

k=0G(k)(*δ*z−aq)(*∂*q)k−1

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

Notice that, if

b(z,a) = 1

2*π*i

*∂*||z−a||2

||z−a||2 ,

then the form

B(z,a) = 1

∇z−a =
1
1−*∂*u =

n

### ∑

k=1b∧(*∂*b)k,

is smooth and solves∇z−a =1 in**C**n\ {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, and_{∇}z−av =g in a neighbourhood

of the boundary *∂*D, then

*ϕ*(a) =

Z

*∂*D*ϕ*∧vn+
Z

D*ϕ*gn, (3.1.8)

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

*ϕ*(a) =

Z

*∂*D *ϕ*vn+
Z

D *ϕ*gn−
Z

D*∂ϕ*∧vn, (3.1.9)

for smooth functions *ϕ*on the closureD
**Proof.** See [1].

We work in**C**n_{×}**C**n _{instead of}**C**n_{. Let} Ω_{⊂}**C**n _{a domain,} _{η}_{=}_{z}_{−}_{ζ}_{∈} Ω_{×}Ω

be fixed, and consider the subbundle E∗ = span{d*η*_{1}, ...,d*η*n} of the cotangent
bundle T_{1,0}∗ (the space of forms of type (1, 0)) over Ω_{×}Ω_{. Let} _{E} _{be its dual}

bundle and let *δη* be the contraction with the section

1
2*π*i

n

### ∑

j=1*η*_{j}e_{j},
wheree_{j} is the dual basis to *η*_{j}.

Let L_{p}_{,}_{q} _{denote the space of sections to the exterior algebra over} _{E}∗_{⊕}_{T}∗
0,1 of

bidegree (p,q) and letLm _{=}L_{p}L_{p}_{,}_{p+m}_{. If} _{∇}_{η}_{=}_{δ}_{η}_{−}_{∂}_{, then we can solve}

where[∆_{]}_{denotes the}_{(}_{n}_{,}_{n}_{)}_{-current integration over the diagonal}∆ _{in}Ω_{×}Ω_{. In}

fact, the Bochner-Martinelli section u= _{∇}b

*η*b, whereb=

1
2*π*i∑

*∂*||*η*||2

||*η*||2 solves (3.1.10).

A form g ∈L0_{(}Ω_{×}Ω)is a weight if ∇*η*g =0 andg0 ≡1 on∆. As before ifq

is any smooth form in L−1 _{and} _{G}_{(}_{0}_{) =} _{1, then} _{g} _{=} _{G}_{(}_{∇}_{η}_{q}_{)} _{is a weight. If} _{g} _{is a}

weight we can solve

∇*η*v=g−[∆],

and if K = vn and P = gn, then we have*∂*K = [∆]−P. Thus for D ⊂⊂Ω we get
for smooth forms *ϕ*of 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

D*ϕ*∧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 **C**n_{, we are now}
going to find explicit formulas representing holomorphic functions on Riemann
surfaces, X, embedded in **C**2_{. In particular we will consider}_{X} _{=}_{{}_{z}_{∈} **C**2 _{:} _{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 {*ψ*_{i}_{}}_{i}_{∈}_{I} is a
collection of charts we say that *ϕ* : X → **C** _{is holomorphic on} _{X} _{if and only if}

*∂*(*ϕ*◦*ψ*−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 2*π*i∑j=n 1(*ζ*j−zj)_{∂ζ}∂

j and that

∇*ζ*−z =*δζ*−z−*∂ζ*.

**Definition 3.3.1.** Let U ⊂**C**n _{be open and f} _{:} _{U} _{→} **C**_{be a holomorphic mapping. If h}

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

*δ _{ζ}*

_{−}

_{z}h = f(

*ζ*)− f(z),