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Centro de Investigación y de Estudios Avanzadosdel Instituto Politécnico Nacional

Unidad Zacatenco

Departamento de Matemáticas

Fórmulas y kerneles de integraciónexplícitos en subvariedades regulares y

singulares de Cn.

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

México, D.F. Junio 2013

Center for Research and Advanced Studiesof the National Polytechnic Institute

Zacatenco Campus

Department of Mathematics

Explicit integration kernels andformulae in regular and singular

subvarieties of Cn.

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

México, D.F. June 2013

Introduction and objectives

The Bochner-Martinelli and Ramirez-Khenkin integration formulae are a pair of

cornerstones of the field of several complex variables. Let Ω be a bounded domain

in Cn with piecewise smooth boundary ∂Ω, and ℵ be a (0, q)-form whose coeffi-

cients are continuous functions on the closure Ω. 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 =∫

∂Ωℵ ∧ Bq −

∫Ω(∂ℵ) ∧ Bq + ∂

[ ∫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 Cn. 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 ∂ℵare 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-

eties Σ of Cn, 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

Cn.

In the first chapter of this thesis we briefly introduce some concepts that we

need for the development of this work.

vi Introduction and objectives

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 : z1z2 = 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 : z1z2 = 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 in Cn 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 : ∑nj=1 z2

j = 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 sm−p(z) for s ∈ C and

z ∈ Cn. We consider polynomials of this kind, because the first entry s can be

easily expressed as the m-root of p(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

terms

vii

Finally in chapter six we present the results obtained in this work and possible

suggestions for future research.

viii Introduction and objectives

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 C2 . . . . . . . . . . . . . 26

3.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4 Explicit integral representations on X . . . . . . . . . . . . . . . . . . 30

3.5 Integral formulae on a Riemann surface in P2 . . . . . . . . . . . . . 33

3.6 A Cauchy-Green formula on X . . . . . . . . . . . . . . . . . . . . . . 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

5.3 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

x CONTENTS

6 Results, conclusions and perspectives 65

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 Cn 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∂zk

dzk

=n

∑k=1

12

(∂ f

∂ Re zk+

i ∂ f∂ Im zk

)dzk.

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 like Lp or Hölder

regularity? We may go further on: Can we solve differential equations with this ∂

operator on varieties with singularities, and requesting that the solutions satisfy

an extra Lp or Hölder smooth condition at the singular points?.

2 Preliminaries

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 two n-dimensional complex coordinate systems (U, ϕU) and (V, ψV)

in X are compatible if either U ∩ V = ∅ or the map ϕU ϕ−1V is biholomorphic,

so that ϕU ϕ−1V 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 of n-dimensional complex atlases on X is called an n-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 an n-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 ϕ−1U is holomorphic.

Let X and Y 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)

with F(U) ⊂ Vsuch that ψV F ϕ−1U is a holomorphic map.

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︷︸︸︷

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 at x is denoted by Fr.

We have the following properties of Fr.

1. By convention, F0 = C. F1 = F(Tx) is the complexification of the 2n-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 on Tx

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 element ϕ ∈ Fr

uniquely in the form ϕ = Re(ϕ) + i Im(ϕ), where Re(ϕ) and Im(ϕ) are real-

valued r-forms at x. Then it follows that

dimR Fr =

(2nr

).

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)− i Im(ϕ).

(b) ϕ = ϕ.

(c) ϕ + ψ = ϕ + ψ.

(d) ϕ is real if and only if ϕ = ϕ.

4 Preliminaries

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

by

ϕ ∧ ψ(v1, ..., vr, vr+1, ..., vr+s)

:=1

r!s! ∑σ∈Sr+s

(sgn σ)ϕ(vσ(1), ..., vσ(r)) · ψ(vσ(r+1), ..., vσ(r+s)).

The sum is taken over all possible permutations σ of the set Sr+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)rsψ ∧ ϕ.

2. (ϕ ∧ ψ) ∧ω = ϕ ∧ (ψ ∧ω).

With the wedge product, the vector space

∧F :=

2n⊕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(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, then eit(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′.

From the definitions of wedge product and form of type (p, q), we can easily

deduced the following proposition.


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 that dzν 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 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 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 over C.

3. E is locally trivial.

6 Preliminaries

Local triviality means that there is an open covering Uii∈I of X together with

biholomorphic functions, called trivializations, such that Φi : π−1(Ui) → Ui ×Cr

and for each x ∈ 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 with π s = idU.

Let X be an n-dimensional complex manifold, we denote by F(X) the com-

plexified cotangent bundle T∗(X)⊗

C. It has the spaces F(Tx(X)) = Tx(X)∗⊗

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.

If E is a topological complex vector bundle of rank m over X, 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, ..., n form 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 x 7→ 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∂zν

dzν +n

∑ν=1

∂ f∂zν

dzν.


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 set U the differential d can be genreralized to the map d = dU :

Er(U) → Er+1(U) in the following way. Let ω = ∑1≤i1<...<ir≤2n ai1...ir(x)ωi1 ∧ ... ∧ωir is the basis representation in a coordinate neigbourhood U, 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. d is C-linear.

3. d d = 0.

4. If ϕ ∈ Er(U) and ψ ∈ Es(U), then d(ϕ ∧ ψ) = dϕ ∧ ψ + (−1)r ϕ ∧ dψ.

5. d is a real operator; that is dϕ = dϕ. In particular dϕ = d(Re ϕ) + i d(Im ϕ).

The differential dω is called the exterior derivative of the form ω.

We consider the decomposition of an r-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,...,jq dzi1 ∧ ...∧ dzip ∧ dzj1 ∧ ...∧ dzjq .

So a general r-form ω has the unique representation

ω = ∑p+q=r

∑|I|=p|J|=q

aI JdzI ∧ dzJ ,

8 Preliminaries

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∂zν

dzν and ∂ f =n

∑ν=1

∂ f∂zν

dzν,

here ∂ f has type (1, 0), ∂ f has type (0, 1), and d f is of type (1, 1).

Proposition 1.2.14. Let ϕ be a r-form of type (p, q). Then dϕ has a unique decomposition

dϕ = ∂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

∂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 general r-forms the derivatives with respect to a z and z are 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).

Notice that a smooth function f is holomorphic if and only if ∂ f = 0. Corre-

spondingly, it follows for a (p, 0)-form ϕ = ∑I=p aIdzI that ∂ϕ = 0 if and only if all

coefficients aI are holomorphic. These results motivate the following definition.


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

neigbourhood U ⊂ Y, and if hk are the components of h in these coordinates, then

in h−1(U) we define the pull-back h∗ϕ of ϕ with respect to h by:

(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 ϕ on U, the pull-back h∗ϕ is a (p, q)-form on h−1(U).

2. For every continous differential form varphi on U, 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 ∫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 ⊂ Cn, and g be a form of type (0, q− 1) on S as well. We say that λ = ∂g holds

in the distributional sense on S if and only if∫S

(λ ∧ σ + (−1)q−1g ∧ ∂σ

)= 0,

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

S.

10 Preliminaries

1.3 Currents

Let X be a complex manifold. A sequence ϕν = ∑I f IduI 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 : 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 if ψ is a differential form of degree 2n− r. Then ψ defines a current

TΨ of degree 2n− r by the formula

Tψ(ϕ) :=∫

Xψ ∧ ϕ, for all ϕ ∈ Er(X).

And if M ⊂ X is an r-dimensional differential submanifold. Then a current TM is

defined by TM(ϕ) :=∫

M ϕ. 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

variable y.

We fix z in G and let ε > 0, such that the ball Bε(z) = x ∈ Cn : ||x− z|| ≤ εis 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∫y∈Dz,ε

dy(Λ(y) ∧ F(y, z)) =∫

y∈∂Dz,ε

Λ(y) ∧ F(y, z). (1.4.1)

Notice that Λ ∧ F is a (n, n− 1)-form with respect to the variable y, and

∂Dz,ε = ∂G ∪ ∂Bε(z).

1.4 Integration kernels 11

Thus, if we expand (1.4.1), we have∫y∈Dz,ε

[∂yΛ(y)] ∧ F(y, z) +∫

y∈Dz,ε

Λ(y) ∧ (−1)q∂yF(y, z)

=∫

y∈∂GΛ(y) ∧ F(y, z)−

∫y∈∂Bε(z)

Λ(y) ∧ F(y, z). (1.4.2)

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

(i) limε→0

∫y∈∂Bε(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 =∫

∂GΛ ∧ F−

∫G(∂Λ) ∧ F− c∂

∫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 variable y, and that satisfies the conditions (i) and

(ii). Hence (1.4.3) is satisfied, for every (0, q)-differential form Λ 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 Cn, and

∂Λ = 0, then the form

λ(z) = −c∫

y∈GΛ(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 variety Σ

or on its regular part Σreg.

Let G ⊂ Σ be an open set with piecewise smooth boundary. We need to define

properly what the integral∫

G means.

12 Preliminaries

If the origin 0 does not lie on the closure G, the integral∫

G h is defined in the

usual sense for all (n− 1, n− 1)-forms h continuous on G.

If 0 ∈ G, we take ε > 0 such that the ball Bε(0) ∩ Σ ⊂ G, and define∫G

h := limε→0

∫G−Bε(0)

h.

Moreover, if h is not a (n− 1, n− 1)-form, we define∫

G h ≡ 0.

Let f be an (n− 1, n− 1)-form continuous on the closure G and 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∫G

d f =∫

∂Gf .

If 0 ∈ G, we can take ε > 0 such that Bε(0) ∩ Σ ⊂ G. Let Gε = G \ (Bε(0) ∩ Σ),

then: ∫Gε

d f =∫

∂Gε

f

=∫

∂Gf −

∫∂(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 order O(ε2n−3), the follow identity holds when

we take the limit when ε tends to zero,∫G

d f =∫

∂Gf .

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 on Greg. Let F(y, z) be a differential

form defined on G×G, continuous when y 6= 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

variable y 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 \ (Bε(0) ∪ Bε(z)). We apply Stokes’s theorem to the form

Λ(y) ∧ F(y, z) on the domain Dz,ε, and so we have∫y∈Dz,ε

dy(Λ(y) ∧ F(y, z)) =∫

y∈∂Dz,ε

Λ(y) ∧ F(y, z). (1.4.4)

Notice that Λ ∧ F is a (n− 1, n− 2)-form with respect to the variable y, and

∂Dz,ε = ∂G ∪ ∂(Bε(0) ∩ Σ) ∪ ∂(Bε(z) ∩ Σ)

1.4 Integration kernels 13

for ε small enough.

Thus, if we expand (1.4.4), we have∫y∈Dz,ε

[∂yΛ(y)] ∧ F(y, z) +∫

y∈Dz,ε

Λ(y) ∧ (−1)q∂yF(y, z)

=∫

y∈∂GΛ(y) ∧ F(y, z)−

∫y∈∂(Bε(0)∩Σ)

Λ(y) ∧ F(y, 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

z 6= 0):

i) limε→0

∫y∈∂(Bε(0)∩Σ)

Λ(y) ∧ F(y, z) = 0,

ii) limε→0

∫y∈∂(Bε(z)∩Σ)

Λ(y) ∧ F(y, z) = Λ(z),

iii) (−1)q∂yF = c∂zF, where c is a constant.

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

Λ =∫

∂GΛ ∧ F−

∫G

∂Λ ∧ F− c∂∫

GΛ ∧ 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 on G×G, continuous when

y 6= z, smooth of type (0, q) with respect to the variable z 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 form Λ

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

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∫

y∈GΛ(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

boundary, based on the fact that we may have an integral representation as in

(1.4.6).

14 Preliminaries

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 = zn3 , 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 ∈ Zn be a fixed integer vector with entries βk ≥ 1. A polynomial

Q(z) holomorphic on Cn 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) :=(sβ1z1, sβ2z2, ..., sβn zn

)(2.2.2)

and all points s ∈ C and z ∈ Cn. An algebraic subvariety Σ in Cn 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 ≥ 1 but all of them with

respect to the same fixed vector β.

16 Integration formulae on weighted homogeneous varieties

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 Σ if and only if

z lies in the regular part of Σ as well. Define de subvariety

Σ = z ∈ C3 : z1z2 = zn3 , n ∈N, n ≥ 2.

We analyze 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. The subvariety Σ 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 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

βk2πi

∫u∈C

fk(uβ ∗ z)(uβk zk)du ∧ du

u(u− 1)

is a solution to the ∂-equation ω = ∂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 kerneldu

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

∫u∈C

f1(un−1z1, uz2, uz3)un−2du ∧ du

u− 1

− z2

2πi

∫u∈C

f2(un−1z1, uz2, uz3)du ∧ duu− 1

− z3

2πi

∫u∈C

f3(un−1z1, uz2, uz3)du ∧ duu− 1

. (2.2.4)

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

cases in order to understand the formula (2.2.4)

2.2 Practical example 17

(a) Consider the case z3 6= 0. We may deduce that u = y3z3

, and so y =

(yn−13

z1zn−1

3, y3

z2z3

, 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−2 f1(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 variable y3 asdu ∧ duu− 1

=dy3 ∧ dy3

z3(y3 − z3).

Since y = (un−1z1, uz2, uz3), then u = y3/z3 and

ω(y) ∧ dy3

y3 − z3=

(n− 1)z1 y3n−2 f1(y)

z3n−1

dy3 ∧ dy3

y3 − z3

+z2 f2(y)

z3

dy3 ∧ dy3

y3 − z3

+ f3(y)dy3 ∧ dy3

y3 − z3

= (n− 1)z1 f1(un−1z1, uz2, uz3)un−2du ∧ du

u− 1

+ z2 f2(un−1z1, uz2, uz3)du ∧ duu− 1

+ z3 f3(un−1z1, uz2, uz3)du ∧ duu− 1

Hence, we can rewrite (2.2.4) as follows

g1(z) = −1

2πi

∫y3∈C

ω(y) ∧ dy3

y3 − z3.

(b) In case z2 6= 0, we may deduce u = y2z2

, and so y = (yn−12

z1zn−1

2, y2, y2

z3z2).

Moreover we can express ω(y) in (2.2.3) in terms of the coordinate y2:

ω(y) =(n− 1)z1

z2n−1 y2

n−2 f1(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 variable y2 asdu ∧ duu− 1

=dy2 ∧ dy2

z2(y2 − z2)


Since y = (un−1z1, uz2, uz3), then

ω(y) ∧ dy2

y2 − z2=

(n− 1)z1 y2n−2 f1(y)

z2n−1

dy2 ∧ dy2

y2 − z2

+ f2(y)dy2 ∧ dy2

y2 − z2

+z3 f3(y)

z2

dy2 ∧ dy2

y2 − z2.

Hence, we can rewrite (2.2.4) as follows

g1(z) = −1

2πi

∫y2∈C

ω(y) ∧ dy2

y2 − z2.

(c) In case z1 6= 0, we may deduce that u = ( y1z1)

1n−1 ρk for k = 0, 1, ..., n − 2,

where ρ is the (n−1)-th root of unity, so that y = (y1, ( y1z1)

1n−1 ρkz2, ( y1

z1)

1n−1 ρkz3) is

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

ωk(y) = f1(y)dy1 +ρkz2

(n− 1)z1

(y1

z1

) 1n−1−1

f2(y)dy1

+ρkz3

(n− 1)z1

(y1

z1

) 1n−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:

duu− 1

=( y1

z1)

1n−1−1dy1

(n− 1)z1((y1z1)

1n−1 − ρk)

for k = 0, 1, ..., n− 2.

And

du ∧ duu− 1

=ρk( y1

z1)

1n−1−1( y1

z1)

1n−1−1

dy1 ∧ dy1

(n− 1)2|z1|2(( y1z1)

1n−1 − ρk)

for k = 0, 1, ..., n− 2.

2.2 Practical example 19

Since y = (un−1z1, uz2, uz3) and u = ( y1z1)

1n−1 ρk, then

ωk(y) ∧( y1

z1)

1n−1−1dy1

(n− 1)z1((y1z1)

1n−1 − ρk)

=(n− 1)z1(

y1z1)

1n−1−1 f1(y)dy1 ∧ dy1

(n− 1)2|z1|2(( y1z1)

1n−1 − ρk)

(2.2.5)

+z2 ρk( y1

z1)

1n−1−1( y1

z1)

1n−1−1

f2(y)dy1 ∧ dy1

(n− 1)2|z1|2(( y1z1)

1n−1 − ρk)

+z3 ρk( y1

z1)

1n−1−1( y1

z1)

1n−1−1

f3(y)dy1 ∧ dy1

(n− 1)2|z1|2(( y1z1)

1n−1 − ρk)

= (n− 1)z1 f1(un−1z1, uz2, uz3)un−2du ∧ du

u− 1

+ z2 f2(un−1z1, uz2, uz3)du ∧ duu− 1

+ z3 f3(un−1z1, uz2, uz3)du ∧ duu− 1

.

Notice that un−2 = ( y1z1)

1− 1n−1

ρ−k and that the term ( y1z1)

1n−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

variable y1 lies in the complex plane minus the negative real axis C, 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 = ( y1z1)

1n−1 ρk, 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 by n−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 C, and we add the result over all posible values of the index

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

g1(z) = −1

2πi

n−2

∑k=0

∫y1∈C

ωk(y) ∧( y1

z1)

1n−1−1dy1

(n− 1)z1((y1z1)

1n−1 − ρk)

,

where C is the complex plane minus the negative real axis; and we obviously use

the equation (2.2.5). We now procede to expand the above formula, using again


(2.2.5) and we have

g1(z) = − 12πi

n−2

∑k=0

∫y1∈C

f1(y)( y1

z1)

1n−1−1dy1 ∧ dy1

(n− 1)z1((y1z1)

1n−1 − ρk)

− z2

2πi

n−2

∑k=0

∫y1∈C

f2(y)ρk| y1

z1|

2n−1−2dy1 ∧ dy1

(n− 1)2|z1|2(( y1z1)

1n−1 − ρk)

− z3

2πi

n−2

∑k=0

∫y1∈C

f3(y)ρk| y1

z1|

2n−1−2y1 ∧ dy1

(n− 1)2|z1|2(( y1z1)

1n−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.dda

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

1a− ρk =

dda ∏k(a− ρk)

∏k(a− ρk)=

(n− 1)an−2

an−1 − 1, (2.2.7)

n−2

∑k=0

1ρka− 1

=n−2

∑k=0

(a

a− ρk − 1)=

n− 1an−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 =

( y1z1)

1n−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)

= − 12πi

∫y1∈C

f1(y)dy1 ∧ dy1

y1 − z1

− z2

2πi

∫y1∈C

f2(y)| y1

z1|

2n−1−2dy1 ∧ dy1

(n− 1)z1(y1 − z1)

− z3

2πi

∫y1∈C

f3(y)| y1

z1|

2n−1−2dy1 ∧ dy1

(n− 1)z1(y1 − z1).

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 kernel 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 Cn. The main idea behind their

work is to use a natural foliation induced by the mapping Hs(z) given below.

Hs(z) :=(sβ1z1, sβ2z2, ..., sβn zn

)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. Let Σ 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 z 6= 0 in Σ

f (z) = f (H1(z)) =−12πi

∫s∈C

[∂s f (Hs(z))

]∧ ds

s− 1(2.3.2)

under the assumption that the differential ∂s f (Hs(z)) exists and is continuous at every

s ∈ C.

Moreover let ℵ = ∑k fkdzk 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∗1ℵ = ∂z

(−12πi

∫s∈C

[H∗s ℵ] ∧ds

s− 1

). (2.3.3)

The deduction of formulae (2.3.2) and (2.3.3) is quite intuitive, but we must

proceed carefully. The integral in (2.3.3) is calculated according to the convention


used by Henkin and Leiterer in [12, p. 44]. It is easy to deduce that the pull-back

of ℵ with respect to Hs(z) is expressed in terms of the differentials dzk and ds, i.e.

H∗s ℵ =n

∑k=1

fk(Hs(z))sβk

(βk

zks

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 z in Σ, and the differential ∂s in (2.3.2) is calculated with respect to

the variable s 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 (Hs(z)) is continuous and has compact

support in the plane s ∈ C, so that we may apply Cauchy-Green formula on a

ball B with 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 point s ∈ 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

βk2πi

∫u∈C

fk(uβ ∗ z)(uβk zk)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 ∫s∈C

g(s, z) dzk ∧ ds = 0.

2

Lemma (2.3.1) is quite interesting; and it shows that indeed Cauchy kernel 1s−1

is an intrinsic property of the weighted homogeneous subvarieties for forms of

degree zero and one. However the main problem is that it does not seem to be a

natural way to generalize this result to forms with non compact support.

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)-form δH 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 call δH the contraction of

ω by H

24 Integral representation with weights

It is easy to see that the Cauchy kernel u(z, a) = dzz−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 set U ∈ Cn and, for any integer m, let Lm(U) =⊕nk=0 Ek,k+m(U). For instance, u ∈ Lm(U) can be written u = 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 Lmcurr(U) denote the corresponding space of currents. Fix a

point a ∈ Cn, let δz−a : Ep,q(U) → Ep−1,q(U) be contraction with the vector field

2πin

∑j=1

(zj − aj)∂

∂zj, and let ∇z−a = δz−a − ∂, so that ∇z−a : Lm(U) → Lm+1(U).

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

form we have an integral representation formula for holomorphic functions ϕ on

the closure D

ϕ(a) =∫

∂Dϕ[a] =

∫∂D

∂ϕ ∧ un + ϕ∂un =∫

∂Dd(ϕun) =

∫∂D

ϕun, a ∈ D.

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

called a weight with respect to a ∈ U.

If g is a weight with repect to z ∈ D, we can solve∇z−av = g in a neighborhood

U of the boundary ∂D, and we then get the weighted representation formula for

holomorphic functions ϕ on the closure D,

ϕ(a) =∫

∂Dϕu ∧ g +

∫D

ϕg, a ∈ D. (3.1.6)

3.1 Weighted representation formulae 25

If gj are weights and G(λ1, ..., λm) is holomorphic on the image of z 7→ (g10, ..., gm

0 ),

then one can form a new weight G(g). In particular, if m = 1 and g = 1 + (δz−a −∂)q, and q is 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 in Cn \ a. It will be called the Bochner-Martinelli

form. Replacing u by B(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 in Ω, D ⊂⊂ Ω, and ∇z−av = g in a neighbourhood

of the boundary ∂D, then

ϕ(a) =∫

∂Dϕ ∧ vn +

∫D

ϕgn, (3.1.8)

for holomorphic functions ϕ on the closure D.

If ∇z−a = g− [a] in Ω, then

ϕ(a) =∫

∂Dϕvn +

∫D

ϕgn −∫

D∂ϕ ∧ vn, (3.1.9)

for smooth functions ϕ on the closure D

Proof. See [1].

We work in Cn ×Cn instead of Cn. Let Ω ⊂ Cn a domain, η = z− ζ ∈ Ω×Ω

be fixed, and consider the subbundle E∗ = spandη1, ..., dηn 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

12πi

n

∑j=1

ηjej,

where ej is the dual basis to ηj.

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

bidegree (p, q) and let Lm =⊕

p Lp,p+m. If ∇η = δη − ∂, then we can solve

∇ηu = 1− [∆], with u ∈ L−1(Ω×Ω). (3.1.10)


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

fact, the Bochner-Martinelli section u = b∇ηb , where b = 1

2πi ∑ ∂||η||2||η||2 solves (3.1.10).

A form g ∈ L0(Ω×Ω) is a weight if ∇ηg = 0 and g0 ≡ 1 on ∆. As before if q

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 have ∂K = [∆]− P. Thus for D ⊂⊂ Ω we get

for smooth forms ϕ of type (0, q) on the closure D, the Koppelman’s formula

ϕ(z) =∫

∂Dϕ ∧ K +

∫D

∂ϕ ∧ K + ∂z

∫D

ϕ ∧ K +∫

Dϕ ∧ P. (3.1.11)

3.2 Integral formulae on a Riemann surface in C2

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 consider X = z ∈ C2 : f (z) =

0, where f is some holomorphic function such that d f 6= 0 on X.

Holomorphicity on X is defined locally in a natural way, since if ψii∈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 all i ∈ I. The set of holomorphic functions on X is 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 ∑nj=1(ζ j − zj)

∂∂ζ j

and that

∇ζ−z = δζ−z − ∂ζ .

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

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

δζ−zh = f (ζ)− f (z),

then h is called a Hefer form.

3.3 Preliminaries 27

The definition above does not provide any information of how to find h given

f . However, when working in convex domains it is always possible to derive an

appropriate Hefer form by the following lemma.

Lemma 3.3.2. Suppose that Ω is a convex domain in Cn and that f is holomorphic. Then

there are holomorphic functions hj(ζ, z) in Ω×Ω such thatn

∑j=1

(ζ j − zj)hj(ζ − z) = f (ζ)− f (z),

so hj(ζ − z) = 12πi ∑n

j=1 hj(ζ, z) is a Hefer form to f .

Proof. By a simple application of the complex chain rule we have

f (ζ)− f (z) =∫ 1

0

ddt

f (ζ − t(ζ − z))dt =∫ 1

0

ddt

f (ζ − t(ζ − z))dt =

=∫ 1

0

n

∑j=1

(ζ j − zj) f j(ζ − t(ζ − z))dt =n

∑j=1

(ζ j − zj)∫ 1

0f j(ζ − t(ζ − z))dt,

so we may choose hj(ζ, z) =∫ 1

0 f j(ζ − t(ζ − z))dt.

Note especially that if f is a polynomial, using this method, the functions hj

will be polynomials as well. Further, one should note that if f is r-homogeneous

then the hj’s will be homogeneous of degree (r− 1).

Consider a domain Ω ⊂ Cn and let f : Ω → Ω be a holomorphic function

such that d f 6= 0 on its zero set X = z ∈ Cn : f (z) = 0. Then, by the implicit

function theorem, we have that X is a smooth Riemann surface and also that the

reciprocal 1f is locally integrable, so that ∂ 1

f is well-defined in the current sense.

Thus, we may construct a form that apart from being smooth is a weight, as in

the following proposition.

Proposition 3.3.3. Let f : Ω → Ω be a holomorphic and assume d f 6= 0 on X. Then, if

h is a Hefer form to f , the form

g f = f (z)1f− h ∧ ∂

1f

is ∇ζ−z and g f0(z) = 1.

Proof. Obviously g f0(z) = 1 so we only need to check that g f is ∇ζ−z-closed,

which follows by the calculation

∇ζ−zg f = ∇ζ−z( f (z)1f)−∇ζ−z(h ∧ ∂

1f) = − f (z)∂

1f− (δζ−zh) ∧ ∂

1f=

= − f (z)∂1f+ ( f (z)− f (ζ))∂

1f= − f (ζ)∂

1f= −∂1 +

∂ ff

= 0.


Our aim is to derive an integral representing holomorphic functions, ϕ, de-

fined on X → C2. An idea of how to find such a formula would be to use known

results in the ambiet space, i.e., the Koppelman formula in C2, and then perform

a limit procedure to obtain a formula on X. First one should consider the problem

of finding a smooth extension of ϕ. However, by partition of unity it is easy to

construct such an extension and we will denote it by φ. Note that ∂-action on X

is then independent of the extension, since if ı : X → C2 we have that

ı∗∂φ = ∂ı∗φ = ϕ.

Now, let λ ∈ R with λ >> 0 and consider the form

gλ = f (z)| f |2λ 1f− h ∧ ∂| f |2λ 1

f+ 1− | f |2λ,

which is a weight that regularizes g f . Thus, by the Koppelman formula in Cn we

have the representation

φ(z) =∫

gλ ∧ g ∧ B ∧ ∂φ +∫

gλ ∧ gφ,

and, in particular, if we consider z ∈ X we have that

φ(z) =∫

h ∧ ∂| f |2λ 1f∧ g ∧ ∂φ ∧ B−

∫h ∧ ∂| f |2λ 1

f∧ gφ, (3.3.4)

where g is an arbitrary weight. We now consider the limit when λ tends to 0 since,

intuitively, this would shrink the support of the integrands down to X. Now, let

ξ be a smooth (1,1)-form and consider the integral∫∂| f |2λ 1

f∧ B ∧ ξ.

Since d f 6= 0, we may perform a change of variables so that f = 1ζ1

near every

point in X, and if β denotes the Bochner-Martinelli form in the new variables we

have ∫∂|ζ1|2λ dζ1

ζ1∧ β(ζ, z) ∧ ξ,

where ξ is a form of type (0, 1). Now, for simplicity, consider the case z = 0 so

that β(ζ) = O((|ζ1|+ |ζ2|)−1) and compute the above integral as∫∂|ζ1|2λ dζ1

ζ1∧ β(ζ) ∧ ξ =

∫ζ1

∂|ζ1|2λ dζ1

ζ1∧∫

ζ2

β(ζ) ∧ ξ =∫

ζ1

∂|ζ1|2λ dζ1

ζ1∧ ψ(ζ1) =

=∫

ζ1

λ|ζ1|2λ dζ1 ∧ dζ1

|ζ1|2∧ ψ(ζ1),

3.3 Preliminaries 29

where ψ is a continuous function. For our purpose it is sufficient to consider ψ to

be radial, and so we are interested in the limit of the integral∫ ∞

0λrλ−1ψ(r)dr.

Letting λ→ 0 and using the continuity of ψ yields the limit ψ(0), and so the first

integral in (3.3.4) is well-defined in the limit as well. Thus, for φ smooth we have

the representation formula

φ(z) =∫

h ∧ ∂1f∧ g ∧ B ∧ ∂φ +

∫h ∧ ∂

1f∧ gφ. (3.3.5)

At a first glance, introducing g f may seem a bit far-fetched, but the key moti-

vation lies in the factor ∂ 1f that will change the domain of integration from Cn to

X via the well-known Poincaré-Lelong formula.

Theorem 3.3.6. (Poincaré-Lelong). Let f : Ω → C be a holomorphic function such that

d f 6= 0 on X. Then1

2πi∂

1f∧ d f = [X].

Proof. Since d f 6= 0 on X we are free to make a change of coordinate sys-

tem having f as its last coordinate. Thus, it is sufficient to prove the equality1

2πi∂(

1z) ∧ dz = δ(0) in one complex variable. Pick a test function ϕ : C → C

supported in K with 0 ∈ K and note that ϕ∂(1z ) ∧ dz = ∂( ϕ

z ) ∧ dz− 1z ∂ϕ ∧ dz. Let

0 < ε < dist(0, K) and consider the set Kε = z ∈ K : |z| > ε over which 1z is

holomorphic, so that

∫Kε

∂ϕ

z∧ dz =

∫Kε

∂(ϕ

z) ∧ dz =

∫Kε

d(ϕ

zdz) =

∫∂K

ϕ

zdz−

∫ 2π

0ϕ(εeiθ)idθ

and letting ε→ 0 we get

∫K

∂ϕ

z∧ dz =

∫∂K

ϕ

zdz− 2πiϕ(0) = −2πiϕ(0),

by the continuity of ϕ. Thus, integrating ϕ∂ 1z ∧ dz over K yields

∫K

ϕ∂1z∧ dz = −

∫K

∂ϕ

z∧ dz = 2πiϕ(0),

and we are done.


3.4 Explicit integral representations on X

We are now ready to present our technique of finding integral representation

formulas on Riemann surfaces. First we introduce the concept of interior mul-

tiplication of a (p, q)-differential form with respect to a vector field. Let γ be a

vector field on X, then it is represented by a linear combination

γ =n

∑j=1

f j(z)∂

∂zj+ gj(z)

∂

∂zj,

where f j and gj are functions defined on X. Now for a differential for ω of type

(p, q), the interior multiplication of ω with respect to γ is the (p− 1, q− 1)-form

ıγ(ω) := γ¬ω, where ¬ is a linear operator that follows the rules

∂

∂zj¬dzj = δij

and

γ¬(α ∧ β) = (γ¬α) ∧ β + (−1)deg(α) ∧ (γ¬β).

Lemma 3.4.1. Let γ be any smooth (1, 0)-vector field such that γ¬d f = 2πi on X. Then,

if g is a compactly supported weight, we have for ϕ ∈ O(X)

ϕ(z) =∫

Xγ¬(h ∧ g)ϕ,

for all z such that g is a weight.

Proof. By the Poincaré-Lelong formula we know that ∂ 1f ∧ d f = 2πi[X] and

therefore

2πiγ¬[X] = γ¬(

∂1f∧ d f

)= 2πi∂

1f∧ (γ¬d f ) = −∂

1f

Substituting ∂ 1f in the integral representation (3.3.5) we get

φ(z) =∫

h ∧ (γ¬[X]) ∧ g ∧ B ∧ ∂φ +∫

h ∧ (γ¬[X]) ∧ gφ =

= −∫(γ¬[X]) ∧ h ∧ g ∧ B ∧ ∂φ−

∫(γ¬[X]) ∧ h ∧ gφ =

=∫[X] ∧ γ¬(h ∧ g ∧ B) ∧ ∂φ +

∫[X] ∧ γ¬(h ∧ g)φ =

=∫

Xγ¬(h ∧ g ∧ B) ∧ ∂φ +

∫X

γ¬(h ∧ g)φ

and since φ |X= ϕ and ∂ϕ = 0 we are done.

3.4 Explicit integral representations on X 31

Thus, in order to find a representation formula of ϕ on X all one needs is to

find a proper (1, 0)-field, γ, and derive the form γ¬(h∧ g). Though, even for sim-

ple choices of f , these calculations tend to become quite lengthy and for coming

calculations on Pn brute force will simply generate too complicated expressions.

It is therefore well worth the time considering the problem of expressing γ¬(h∧ g)

in a more clever way. Let B be the Bochner-Martinelli form, defined in C2 and

consider the form h ∧ g ∧ B. Since h is (1,0) and B is at least (1,0) we have that

h ∧ g ∧ B = h ∧ g0 ∧ B1, which is of full degree in the unbarred differentials dζ j.

Hence, there is a function C(z, ζ) such that

h ∧ g ∧ B = C(z, ζ)dζ1 ∧ dζ2,

and using the fact that δζ−zB = 1 and δζ−z = 0 on X we get

h ∧ g0 = δζ−z(h ∧ g ∧ B) = δζ−z(C(z, ζ)dζ1 ∧ dζ2)

= C(z, ζ)2πi((ζ1 − z1)dζ2 − (ζ2 − z2)dζ1), (3.4.2)

as a form on X. Thus, considering the part of X where ζ1 is a valid coordinate,

i.e., ∂ f∂ζ26= 0, interior multiplication in (3.4.2) by for instance ∂

∂ζ2yields

C(z, ζ) =1

2πig0

ζ1 − z1

∂

∂ζ2¬h

and so

h ∧ g ∧ B =1

2πi

(g0

ζ1 − z1

∂

∂ζ2¬h)

dζ1 ∧ dζ2.

Now, observe that the δζ−z operator is injective on forms of full degree in the

unbarred differentials dζ j. In fact, if we let α be such a form and δζ−zα = 0, then

δζ−z(B1 ∧ α) = α, but for degree reasons B1 ∧ α = 0 and therefore α = 0. For

now, our main interest is the form γ¬(h∧ g) integrated over the Riemann surface

X, and since γ is of type (1,0), and thus lowers the unbarred degree by one, the

part of h ∧ g contributing to the integral in Lemma (3.4.1) is the one of type (2, 1).

Choosing g = χ− ∂χ ∧ u, we have that g1 = −∂χ ∧ u1 and by the injectivity of

δζ−z we get

h ∧ g1 = ∂χ ∧ h ∧ u1 = ∂χ

(1

2πi1

ζ1 − z1

∂

∂ζ2¬h)∧ dζ1 ∧ dζ2.

The form γ¬(h ∧ g1) is then simply given by

γ¬(h ∧ g1) = −∂χ

(1

2πi1

ζ1 − z1

∂

∂ζ2¬h)∧ γ¬(dζ1 ∧ dζ2) (3.4.3)


end the problem of computting γ¬(h ∧ g) has been reduced to calculate ∂∂ζ2¬h

and γ¬(dζ1 ∧ dζ2). We formulate this result in the following proposition.

Proposition 3.4.4. Let D ⊂ C2 and f : D → C be a holomorphic function such that

d f 6= 0 on X. If X → C2 is the zero set of f and γ is any smooth (1,0)-vector field

such that γ¬d f = 2πi and h is a Hefer form to f , then for ϕ ∈ O(X) we have the

representation formula

ϕ(z) = − 12πi

∫∂D∩X

(ϕ

ζ1 − z1

∂

∂ζ2¬h)

γ¬(dζ1 ∧ dζ2)

on the part of X where ∂ f∂ζ26= 0.

We finalize this section with a practical example. Consider the function f :

C2 → C given by f (z) = z2 − z21 and its corresponding zero set X → C2 defined

by the parabola z2 = z21. Then d f = −2z1dz1 + dz2 6= 0 in C2 and we have a global

parametrization of X given by t 7→ (t, t2), with t ∈ C. Our aim now is to find a

Hefer form, h, and (1, 0)-field γ such that γ¬d f = 2πi. First, using Lemma (3.3.2),

we get

h1(ζ, z) =∫ 1

0−2(ζ1 − t(ζ1 − z1))dt = −2ζ1 + (ζ1 − z1)

∫ 1

02tdt = −ζ1 − z1,

h1(ζ, z) =∫ 1

0dt = 1,

and so h(ζ, z) = (−(ζ1+z1)dζ1+dζ2)2πi will work as a Hefer form to f . For the vector

field, a good choice would obviously be γ = 2πi ∂∂ζ2

; however notice that we

could equally well have chosen, e.g., πiζ1

∂∂ζ1

outside the origin. Now, plugging our

choices of h and γ into the formula (3.5.2) we get

γ¬(h ∧ g) = −∂χ

(1

2πi1

ζ1 − z1

∂

∂ζ2¬(−(ζ1 + z1)dζ1 + dζ2

2πi

))∧ 2πi

∂

∂dζ2¬(dζ1 ∧ dζ2)

=∂χ

2πi

(1

ζ1 − z1

)∧ dζ1 =

12πi

∂χ

ζ1 − z1∧ dζ1,

and using the parametrization (ζ1, ζ2) 7→ (τ, τ2) and (z1, z2) 7→ (t, t2) we end up

with the integral representation formula

ϕ(t) =1

2πi

∫∂D∩X

ϕ(τ)dτ

τ − t,

for ϕ holomorphic on X. As one probably would expect, having a global parametriza-

tion of X, the representation turns out to be nothing but the Cauchy integral

formula.

3.5 Integral formulae on a Riemann surface in P2 33

Using the assumption d f 6= 0 it is possible find an even more explicit local

form of (3.5.2), via the implicit function theorem. Let z ∈ X be fixed and assume

that t 7→ (t, p(t)), with t ∈ C, is a parametrization of X in some neighbourhood

U of z. Further, let χ be a cut-off function supported in this neighbourhood and

assume h(ζ, z) = h1(ζ,z)dζ1+h2(ζ,z)dζ22πi to be a Hefer form. Since ζ1 is a coordinate,

we may choose γ as

γ =2πi∂ f∂ζ2

∂

∂ζ2.

Thus

γ¬(dζ1 ∧ ζ2) = −2πidζ2

∂ f∂ζ2

,

and so, for all z ∈ U we have the representation formula

ϕ(t) =1

2πi

∫∂U∩X

h2∂ f∂ζ2

ϕ

τ − tdτ,

for ϕ holomorphic on X.

Notice that this formula is actually a weighted Cauchy-Green formula on X

since the function ψ(ζ, z) = h2∂ f

∂ζ2

clearly is holomorphic in z and ψ(z, z) ≡ 1 since

h2 = ∂ f∂ζ2

on the diagonal.

3.5 Integral formulae on a Riemann surface in P2

This section is devoted to find integral representation formulae of Cauchy-Green

type for smooth differential forms on some Riemann surfaces, X → P2. The

surfaces are defined as the zero set of a holomorphic, r-homogeneous polynomial

f : C3 → C such that d f 6= 0 on X.

The general idea is to start from the projective Koppelman formula and then

perform the same kind of calculations as in C2. However, before doing so, there

are some obstacles that need to be considered. Apart from the Koppelman repre-

sentation formula, the crucial ingredients that made the calculations in C2 possi-

ble were first the Poincaré-Lelong formula and second the notion of Hefer forms.

To begin with, when considering the Poincaré-Lelong formula we immediately

run into trouble since d f is not a projective form and thus undefined as a form

on P2. The reason for this lies in the fact that the ∂-operator is undefined acting

on sections taking values in non-trivial line bundles. We are therefore looking


for a substitute, D′, of ∂ making the operator D = D′ + ∂ well-defined on pro-

jective forms and preserving the Poincaré-Lelong formula with D instead of d.

For complex projective spaces, one such construction follows by the proposition

below.

Proposition 3.5.1. Let f : Cn+1 → C be an r-homogeneous function and define

D := d− r∂log|ζ|2.

Then the form D f is projective.

Proof. Since f is r-homogeneous, by assumption, we have that f (λζ) = λr f (ζ)

for any λ ∈ C. Differentiation with respect to λ then yields

rλr−1 f (ζ) =d

dλf (λζ) =

n

∑j=0

ζ j f j(λζ), ζ ∈ Cn+1,

and in particular, for λ = 1, we have the equality

r f (ζ) =n

∑j=0

ζ j∂ f∂ζ j

(ζ), ζ ∈ Cn+1. (3.5.2)

Then D f is projective, since

δzD f = δzd f − r f δzlog|ζ|2 = 2πi( n

∑j=0

ζ j∂ f∂ζ j− r f

)= 0.

Thus, if f : Cn+1 → C is holomorphic and r-homogeneous, we get a Poincaré-

Lelong formula that makes sense on complex projective spaces since

∂1f∧ D f = ∂

1f∧ (d f − r f ∂log|ζ|2) = ∂

1f∧ d f = 2πi[X]

on Cn+1 \ 0.For the notion of Hefer forms the question of finding corresponding objects on

Pn is a bit more complicated. Of course, having f given as an explicit polynomial

in Cn+1 there is no problem finding a (1,0)-form, h, such that δζ−zh = f (ζ)− f (z).

Though, once again the problem is that in general h is not a projective form. In

the following section we use an idea to get around this problem.

3.6 A Cauchy-Green formula on X 35

3.6 A Cauchy-Green formula on X

Let f : C3 → C be a holomorphic, r-homogeneous polynomial and let

h(z, w) = h0(z, w)dw0 + h1(z, w)dw1 + h2(z, w)dw2

be a Hefer form to f . Especially note that, since δw−zh = f (w)− f (z), using the

method from Lemma (3.3.2), one can always find h so that every hj is a polynomial

of homogeneity r − 1. Now, construct the form τ∗α h(z, ζ) from h(z, w) following

the scheme

zj 7→ zj

wj 7→ αζ j (3.6.1)

dwj 7→ γjα = dζ j −

ζ · dζ

|ζ|2 ζ j,

where α = ζ·z|ζ|2 −

∂(ζ·dζ)2πi|ζ|2 is the projective weight, and polynomials of differen-

tial forms are interpreted in the obvious way, changing scalar multiplication to

wedge products, which is well defined since α is of even degree. The form τ∗α h

is projective since all the γjα are so and it satisfies a Hefer like property by the

following lemma.

Lemma 3.6.2. Let f : C3 → C be a holomorphic and r-homogeneous polynomial. Then

∇zτ∗α h = f (z)− αr f (ζ).

Proof. We have that

∇zγjα = δz

(dζ j −

ζ · dζ

|ζ|2 ζ j

)+ ∂

z · dζ

|ζ|2 ζ j = 2πi(zj − αζ j),

and since ∇zα = 0,

∇zτ∗α h = ∇z

2

∑j=0

hj(z, αζ)γjα =

2

∑j=0

hj(z, αζ)∇zγjα = 2πi

2

∑j=0

hj(z, αζ)(zj − αζ j) =

= τ∗α (δz−wh(z, w)) = τ∗α ( f (z)− f (w)) = f (z)− f (αζ) = f (z)− αr f (ζ). (3.6.3)

Notice that the exact same arguments are valid even in Cn+1.

We are now in a position to find Koppelman formulas on X since, by argu-

ments analogous to those in Proposition (3.3.3), the form given by

g f = f (z)1f− τ∗α h ∧ ∂

1f

(3.6.4)


is, apart from being smooth, a projective weight. Letting K = g f ∧ αM ∧ u and

P = g f ∧ αM in the Koppelman’s formula we will have the representation

ϕ(z) =∫

g f ∧ αM ∧ u ∧ ∂ϕ +∫

g f ∧ αM ∧ ϕ =

= −∫

τ∗α h ∧ ∂1f∧ αM ∧ u ∧ ∂ϕ−

∫τ∗α h ∧ ∂

1f∧ αM ∧ ϕ, (3.6.5)

for ϕ ∈ E0,0(X, LM+r−2). Note that, for the first integral, u is the type (1,0) at

least, all components of αM are of type (i, i), and we are integrating over P2 so

the only contributing term of τ∗α h is the (1, 0)-part. Since the γjα are (1, 0) this

practically means that hj(z, αζ) = hj(z, α0ζ). However, this does not hold in the

second integral since τ∗α h has a component of type (2,1). Let γ be a (1, 0)-vector

field such that γ¬d f = 2πi and assume that d f 6= 0 on X.

Then

2πiγ¬[X] = γ¬(

∂1f∧ D f

)= 2πi∂

1f

,

so that

ϕ(z) = −∫

τ∗α h ∧ ∂1f∧ αM ∧ u ∧ ∂ϕ−

∫τ∗α h ∧ ∂

1f∧ αM ∧ ϕ

=∫

τ∗α h ∧ (γ¬[X]) ∧ αM ∧ u ∧ ∂ϕ +∫

τ∗α h ∧ (γ¬[X]) ∧ αM ∧ ϕ

= −∫(γ¬[X]) ∧ τ∗α h ∧ αM ∧ u ∧ ∂ϕ−

∫(γ¬[X]) ∧ τ∗α h ∧ αM ∧ ϕ

=∫

Xγ¬(τ∗α h ∧ αM ∧ u) ∧ ∂ϕ +

∫X

γ¬(τ∗α h ∧ αM) ∧ ϕ

=∫

XKX ∧ ∂ϕ +

∫X

PX ∧ ϕ. (3.6.6)

To motivate why ϕ takes values in LM+r−2 we recall that the Koppelman for-

mula above is only well-defined as long as KX and PX both take values in the

trivial bundle L0 = C with respect to the ζ-variable, i.e., are 0-homogeneous in ζ.

It is sufficient to check the homogeneity of the form τ∗α h ∧ ∂ 1f ∧ αM ∧ u from the

first integral in (3.6.5). First,

τ∗α h(z, ζ) =2

∑j=0

hj(z, α0ζ)γjα,

and since γjα is 1-homogeneous in ζ and α0ζ is 0-homogeneous, the form τ∗α h is

1-homogeneous in ζ and (r− 1)-homogeneous in z. Further we have that

αM0,0 =

(z · ζ|ζ|2

)M

,


which is (−M)-homogeneous in ζ and M-homogeneous in z. Finally, the form ∂ 1f

is (−r)-homogeneous, and since u is 1-homogeneous in ζ and (−1)-homogeneous

in z, the form τ∗α h ∧ ∂ 1f ∧ αM is (2−M− r)-homogeneous in ζ and (M + r − 2)-

homogeneous in z. Consequently, in order to balance the homogeneities in the

integrands, the section ϕ must take values in LM+r−2.

We are now ready to formulate our main theorem.

Theorem 3.6.7. Let f : C3 \ 0 → C be an holomorphic and r-homogeneous polynomial

such that d f 6= 0 on the Riemann surface X = z ∈ C3 \ 0 : f (z) = 0 → P2. If

γ is any smooth (1,0)-vector field such that γ¬d f = 2πi on X, h is a Hefer form to f ,

and M ∈ N is fixed, we have for ϕ ∈ E0,0(X, LM+r−2) and ψ ∈ E0,1(X, LM+r−2) the

Koppelman formulas

ϕ(z) =∫

XKX ∧ ∂ϕ +

∫X

PX ∧ ϕ (3.6.8)

and

ψ(z) = ∂z

∫KX ∧ ψ, (3.6.9)

where KX = γ¬(τ∗α h ∧ αM ∧ u), PX = γ¬(τ∗α h ∧ αM).

Now, let us consider explicit expressions for KX and PX . Henceforth let K =

τ∗α h ∧ αM ∧ u and P = τ∗α h ∧ αM so that KX = γ¬K and PX = γ¬P. Consider the

form given by

ω′ = δζ(dζ0 ∧ dζ1 ∧ dζ2).

Then, by construction, ω′ is obviously a projective form of type (2,0) and thus the

pull-back of a form on P2 of full degree in the unbarred differentials. Therefore

every projective (2,0)-form in C3, σ, must be of the form

σ = C(z, ζ)ω′.

where C is a function that is homogeneous in both ζ and z. Having in mind that

the only component of K contributing to the integral is the one of type (2, 0), we

may write

K(z, ζ) = C(z, ζ)ω′ + Ω(z, ζ),

where Ω consists of all the remaining components of K. Thus, for (z, ζ) ∈ X× X,

we have by ∇z-action the equality

∇zK = −τ∗α h ∧ αM = C(z, ζ)δzω′ − ∂(C(z, ζ)ω′) +∇zΩ(z, ζ),


and so

−(

z · ζ|ζ|2

)M

τ∗α h = −τ∗α h ∧ αM0 = C(z, ζ)δzω′ =

= C(z, ζ)(2πi)2[(z2ζ1 − z1ζ2)dζ0 + (z0ζ2 − z2ζ0)dζ1 + (z1ζ0 − z0ζ1)dζ2].

Restricting ourselves to the parts of X where either ζ0 or ζ1 is non-zero and

the other one is a coordinate, we get an expression of C(z, ζ) by applying, for

instance, ∂∂ζ2¬ to the equality above as

C(z, ζ) = − 14π2

(z·ζ|ζ|2

)M

z0ζ1 − z1ζ0

∂

∂ζ2¬(τ∗α h).

The (2, 0)-component of K is then given by

(τ∗α h ∧ αM ∧ u)2,0 = C(z, ζ)ω′ = − 14π2

( (z·ζ|ζ|2

)M

z0ζ1 − z1ζ0

∂

∂ζ2¬(τ∗α h)

)ω′,

and thus

KX(z, ζ) = γ¬K = − 14π2

( (z·ζ|ζ|2

)M

z0ζ1 − z1ζ0

∂

∂ζ2¬(τ∗α h)

)(γ¬ω′).

Hence, to find an explicit form of KX it is sufficient to derive ∂∂ζ2¬(τ∗α h) and γ¬ω′

in accordance with the following proposition.

Proposition 3.6.10. Let X → P2 be the zero set of a holomorphic and r-homogeneous

polynomial f : C3 \ 0 → C. Then, if h is a Hefer form to f , we have on Ω = ζ ∈ X :

ζ0 6= 0, ∂ f∂ζ26= 0 ∪ ζ ∈ X : ζ1 6= 0, ∂ f

∂ζ26= 0 that KX in Theorem (3.6.7) is given by

KX(z, ζ) = − 14π2

( (z·ζ|ζ|2

)M

z0ζ1 − z1ζ0

∂

∂ζ2¬(τ∗α h)

)(γ¬ω′).

More explicit, by choosing γ = 2πi( ∂ f∂ζ2

)−1 ∂∂ζ2

, we have

KX =

(z · ζ|ζ|2

)M h2(z, α0ζ)− ζ2|ζ|2 ∑2

j=0 ζ jhj(z, α0ζ)

∂ f∂ζ2

(z0ζ1 − z1ζ0)(ζ1dζ0 − ζ0dζ1). (3.6.11)


In order to find PX one can make use of the results for KX since by the Kop-

pelman formula in Theorem (3.6.7) we have that PX = −∂ζKX outside of ∆.

Example 3.6.12. Consider P1 → P2 as the Riemann surface defined by the set z ∈ C3 :

z2 6= 0. A Hefer form to f (z) = z2 is then simply given by h(z, w) = dw22πi and we get

τ∗α h(z, ζ) =1

2πi

(dζ2 −

ζ · dζ

|ζ|2 ζ2

).

Since d f = dζ2, we have an obvious choice of vector field γ = 2πi ∂∂ζ2

valid globally on

P1 and thus,

KX(z, ζ) = − 14π2

( (z·ζ|ζ|2

)M

z0ζ1 − z1ζ0

∂

∂ζ2¬ 1

2πi

(dζ2 −

ζ · dζ

|ζ|2 ζ2

))2πi

∂

∂ζ2¬ω′ =

=1

2πi

( (z·ζ|ζ|2

)M

z0ζ1 − z1ζ0

|ζ0|+ |ζ1|2|ζ|2

)(ζ1dζ0 − ζ0dζ1) =

=1

2πi

(z·ζ|ζ|2

)M

z0ζ1 − z1ζ0(ζ1dζ0 − ζ0dζ1),

where we use that |ζ|2 = |ζ0|2 + |ζ1|2 on the surface. A parametrization of P1 is given

by the maps t 7→ [1, t, 0], with t ∈ C, on the part where z0 6= 0; and t 7→ [t, 1, 0], with

t ∈ C where z1 6= 0. Thus, we may express KX in local coordinates as

KX(t, τ) =1

2πi

(1 + τt

1 + |τ|2

)M dτ

τ − t,

and by a simple calculation of ∂KX(t, τ) we get an explicit expression of PX(t, τ) given

by

PX(t, τ) =M

2πi

(1 + τt

1 + |τ|2

)M−1 dτ ∧ dτ

(1 + |τ|2)2 .

Hence, we reach a Koppelman formula representing ϕ ∈ E0,0(X, LM−1) as

ϕ(t) =1

2πi

∫P1

(1 + τt

1 + |τ|2

)M dτ

τ − t∧ ∂ϕ +

M2πi

∫P1

(1 + τt

1 + |τ|2

)M−1 dτ ∧ dτ

(1 + |τ|2)2 ϕ.

Example 3.6.13. Consider the polynomial f (z) = z0z2 − z21 defined in C3 \ 0 and let

X → P2 be the corresponding zero set given by the equation z21 = z0z2. A complete

parametrization of X can then be given using two maps since on the set U0 = z0 6= 0


we have t 7→ [1, t, t2] with t ∈ C; and the remaining solutions are captured by s 7→[s2, s, 1], with s ∈ C, on the set U2 = z2 6= 0. A Hefer form corresponding to f is

given by

h(z, ζ) =1

2πi

(z2 + w2

2dw0 − (z1 + w1)dw1 +

w0 + z0

2dw2

),

and we get the form τ∗α as

2πiτ∗α h(z, ζ) =z2 + αζ2

2∧(

dζ0 −ζ · dζ

|ζ|2 ζ0

)− (z1 + αζ1) ∧

(dζ1 −

ζ · dζ

|ζ|2 ζ1

)+

+z0 + αζ0

2∧(

dζ2 −ζ · dζ

|ζ|2 ζ2

).

Since, for KX, we are only interested in the (0,0)-part of ∂∂ζ2¬(τ∗α h) we think of the α’s as

α0 and so

2πi(

∂

∂ζ2¬(τ∗α h)

)0,0

= −z2 + α0ζ2

2ζ2ζ0

|ζ|2 +(α0ζ1 + z1)ζ2ζ1

|ζ|2 +z0 + α0ζ0

2− z0 + α0ζ0

2ζ2ζ2

|ζ|2

=

(− z2ζ2ζ0

2|ζ|2 +z1ζ2ζ1

|ζ|2 +z0

2− z0ζ2ζ2

|ζ|2

)+

ζ · z|ζ|2

(− ζ2ζ2ζ0

2|ζ|2 +ζ1ζ2ζ1

|ζ|2 +ζ0

2− ζ0ζ2ζ2

|ζ|2

)=

ζ2

|ζ|2

(− z2ζ0

2+ z1ζ1 −

z0ζ2

2

)+

ζ · z|ζ|2

ζ2

|ζ|2 (ζ21 − ζ2ζ0) +

z0

2+

ζ · z|ζ|2

ζ0

2=

=ζ2

|ζ|2

(− z2ζ0

2+ z1ζ1 −

z0ζ2

2

)+

z0

2+

ζ · z|ζ|2

ζ0

2,

where we use that ζ21 − ζ2ζ0 = 0 on X in the last equality. For the vector field, γ, we

notice that d f = dζ0 − 2ζ1dζ1 + dζ2, and thus γ = 2πi ∂∂ζ2

is a valid choice. Then

γ¬ω′ = −4π2 ∂

∂ζ2¬(ζ0dζ1 ∧ ζ2 − ζ1dζ0 ∧ ζ2 + ζ2dζ0 ∧ ζ1) =

= −4π2(ζ1dζ0 − ζ0dζ1),

and so

KX =1

2πi

(ζ·z|ζ|2

)M

z1ζ0 − z0ζ1

[ζ2

|ζ|2

(− z2ζ0

2+ z1ζ1−

z0ζ2

2

)+

z0

2+

ζ · z|ζ|2

ζ0

2

](ζ1dζ0− ζ0dζ1).

Using the parametrization of X on the set U0, replacing ζ for τ and z for t, we get

KX =1

2πi

(1 + τt + τ2t2

1 + |τ|2 + |τ|4

)M(2 + |τ|2 + τt + 2τt|τ|22(1 + |τ|2 + |τ|4)

)dτ

τ − t, τ, t ∈ C.

(3.6.14)


In order to check whether the expression of KX is reasonable, we study the behaviour on

U0 ∩U1 where [1, t, t2] = [ 1t2 , 1

t , 1] = [s2, s, 1], and thus t = 1s . Substitution, with τ = 1

σ

and t = 1s , into (3.6.14) yields

KX =1

2πi

(σ2

s2

)M( 1 + σs + σ2s2

1 + |σ|2 + |σ|4)

)M(2 + |σ|2 + σs + 2σs|σ|22(1 + |σ|2 + |σ|4)

)dσ

σ− s,

and since f is 2-homogeneous we are representing sections taking values in LM, so, by

M-homogeneity of ϕ, we have

ϕ0 = 2πi∫

XKX(τ, t) ∧ ∂ϕ(τ) + 2πi

∫X

PX(τ, t) ∧ ϕ(τ) =

= 2πi∫

X

(σ2

s2

)M

KX(σ, s) ∧ ∂ϕ(1σ) + 2πi

∫X

(σ2

s2

)M

PX(σ, s) ∧ ϕ(1σ) =

= 2πi∫

X

(1s2

)M

KX(σ, s) ∧ ∂ϕ(σ) + 2πi∫

X

(1s2

)M

PX(σ, s) ∧ ϕ(σ) =

=

(1s2

)M

ϕ2

on U0 ∩U2.

Example 3.6.15. Consider the polynomial f (z) = z30 + z3

1 + z32 in C3 \ 0 and its zero

set X → P2. Using the Riemann-Hurwitz formula it is readily checked that X is a

Riemann surface of genus 1. A Hefer form to f is given by

h(z, w) =1

2πi

2

∑j=0

(z2j + zjwj + w2

j )dwj,

and thus

τ∗α h(z, ζ) =1

2πi

2

∑j=0

(z2j + zjαζ j + (αζ j)

2) ∧(

dζ j −ζ · dζ

|ζ|2 ζ j

).

Further we have that

2πi(

∂

∂ζ2¬(τ∗α h)

)0,0

=∂

∂ζ2¬( 2

∑j=0

(z2j + α0zjζ j + α2

0ζ2j )

)(dζ j −

ζ · dζ

|ζ|2 ζ j

)

= (z20 + α0z2ζ2 + α2

0ζ22)−

ζ2

|ζ|2 [z20ζ2 + z2

1ζ1 + z22ζ2 +

α0(z0ζ20 + z1ζ2

1 + z2ζ22) + α2

0(ζ30 + ζ3

1 + ζ32)] =

= (z20 + α0z2ζ2 + α2

0ζ22)−

ζ2

|ζ|2 [z20ζ2 + z2

1ζ1 + z22ζ2 + α0(z0ζ2

0 + z1ζ21 + z2ζ2

2)].


where we use that ζ30 + ζ3

1 + ζ32 = 0 on X in the last equality. Choosing γ to be the vector

field

γ =2πi∂ f∂ζ0

∂

∂ζ0=

2πi3ζ2

0

∂

∂ζ0,

on the part of X where ζ0 6= 0 we end up with KX given by

KX(z, ζ) =1

2πi

(ζ·z|ζ|2

)M

z0ζ1 − z1ζ0

[z2

0 +

(ζ · z|ζ|2

)z2ζ2 +

(ζ · z|ζ|2

)2

ζ22 −

− ζ2|ζ|2

(z2

0ζ2 + z21ζ1 + z2

2ζ2 +

(ζ · z|ζ|2

)(z0ζ2

0 + z1ζ21 + z2ζ2

2)

)]ζ2dζ1 − ζ1dζ2

3ζ20

.

The examples presented above illustrate that although explicit calculations of

KX and PX may be lengthy and tedious, they are doable. We are now going to

take a closer look on the properties of the form KX. As seen in Example (3.6.12)

and (3.6.13), KX(τ, t) is of the form

KX(τ, t) = ψX(τ, t)dτ

τ − t,

where ψX(τ, t) is holomorphic in t and ψX(τ, τ) ≡ 1, which is to be expected since

(3.6.8) and (3.6.9) should be weighted Cauchy-Green formulas on X. This can

actually be seen in a general setting from the expression in Proposition (3.6.10).

From (3.6.11) it is clear that ψX(ζ, z) is always holomorphic in z and therefore

also in local coordinates t. It is, however, not obvious that ψX(z, z) ≡ 1 and we

consider this problem a bit more carefully. To begin with, we note that2

∑j=0

hj(ζ, ζ)ζ j = 0,

since on the diagonal we have that hj =∂ f∂ζ j

, and the result follows from (3.5.2).

Further, on the part Ω ⊂ X considered in Proposition (3.6.10), we have that ∂ f∂ζ26=

0, and γ may be chosen as

γ =2πi∂ f∂ζ2

∂

∂ζ2.

Therefore

ψX(ζ, ζ) =

(h2(ζ, ζ)− ζ2

|ζ|22

∑j=0

ζ jhj(ζ, ζ)

)1∂ f∂ζ2

=

=h2(ζ, ζ)

∂ f∂ζ2

= 1,

which shows that (3.6.8) and (3.6.9) really are weighted Cauchy-Green formulas

on X.

Chapter 4

Bochner-Martinelli formulae on the

complex sphere

4.1 Introduction

In this chapter we propose an alternative technique for producing integration

formulae on complex Stein submanifolds, based on the fact that every Stein sub-

manifold in Cn has a holomorphic retraction, see [21]. In the practice it is difficult

to find explicitly such a holomorphic retraction, so we exemplify the technical

proposal with a practical example.

4.2 Bochner-Martinelli formulae on the complex sphere

The following result presented in [21] is the base of this section.

Proposition 4.2.1. Let X be a Stein closed submanifold in Cn. There exist an open Stein

neighborhood W of X and a holomorphic retraction π : W → X, so that π(x) = x for

every x ∈ X.

Proof. See [21].

We begin defining the positive volume form in the space Cn,

dvol(z) := dz1 ∧ dz2 ∧ ...∧ dzn ∧ dz1 ∧ dz2 ∧ ...∧ dzn

= (−1)n(n−1)

2 (2i)nn∧

j=1

[dxj ∧ dyj], (4.2.2)

for zj = xj + iyj in C. Notice that

(−1)ndu∧ du∧ dvol(z) = du∧ dz1∧ dz2∧ ...∧ dzn∧ du∧ dz1∧ dz2∧ ...∧ dzn = dvol(u, z).

44 Bochner-Martinelli formulae on the complex sphere

Then Fubini’s theorem automatically yields the following identity for every L1

function h defined on the space Cn+1,

(−1)n∫(u,z)

h(u, z)du ∧ du ∧ dvol(z) =∫(u,z)

h(u, z)dvol(u, z)

=∫

z

[ ∫u

h(u, z)du ∧ du]

dvol(z), (4.2.3)

where (u, z) lies in C×Cn; see for example [6, p. 334]. The combination of Fubini

and Stoke’s theorems yields an interesting result. Let Uz be a collection of open

sets in C parametrized by the points z ∈ Cn, such that Uz is empty for |z| large

enough. Assume that h(u, z) is differentiable with respect to the variable u and

that each boundary ∂Uz is piecewise smooth and depends smoothly on z, then∫z∈Cn,u∈∂Uz

h(u, z)du ∧ dvol(z) =∫

z∈Cn,u∈Uz

∂h∂u

du ∧ du ∧ dvol(z) =

= (−1)n∫

z

[ ∫u∈Uz

∂h∂u

du ∧ du]

dvol(z) = (−1)n∫

z

[ ∫u∈∂Uz

h(u, z)du]

dvol(z).

The volume form dvol is also used for integrating functions on the complex

Stein submanifold defined in the following paragraph.

Define p(z) :=n

∑j=1

z2j and Σ := z ∈ Cn|p(z) = 1 be the complex submanifold

of the complex manifold Cn, also known as the complex sphere. Notice that Σ has

complex dimension n− 1, therefore it has 2n− 2 real dimension as a real manifold.

We choose this submanifold because it is easy to find an open neighborhood and

the holomorphic retraction described in the Proposition 4.2.1.

Let π : U → Σ be the projection given by:

π(z) =z

+√

p(z).

Where U = z ∈ Cn : Re(p(z)) > 0 is an open neighborhood of Σ and +√

p(z)

is the principal branch of the square root in Re(p(z)) > 0. We can easily verify

that π is well defined and holomorphic on U and π(Σ) = Σ ⊂ U. Moreover

π(z) = z for each z ∈ Σ.

Let Ω be an open set of Σ with piecewise smooth boundary. If ϕ is a smooth

function defined on the closure Ω, we have that the following classical Bochner-

Martinelli formula holds, see for example [12],

(ϕ π)(a) =∫

∂G(ϕ π)(z)B(z, a)−

∫G

∂z(ϕ π)(z) ∧ B(z, a), (4.2.4)

4.2 Bochner-Martinelli formulae on the complex sphere 45

for a in the open set G := π−1(Ω) ∩ |p(z) − 1| < 12. Where B(z, a) is the

Bochner-Martinelli kernel defined by

B(z, a) =(n− 1)!(−1)

n(n−1)2 ω(z, a)

(2πi)n||z− a||2n , (4.2.5)

and

ω(z, a) :=n

∑j=1

(−1)j−1(zj−aj) [dzj] ∧ dz.

Where

dz = dz1 ∧ dz2 ∧ ...∧ dzn,

and

[dzj] = dz1 ∧ ...∧ dzj−1 ∧ dzj+1 ∧ ...∧ dzn.

We introduce some aditional notation, [dzj,p] means that in the wedge product

dz1 ∧ dz2 ∧ ...∧ dzn do not appear the differentials dzj and dzp.

The main idea is to apply the Bochner-Martinelli integration formulae on the

open set G and the function ϕ π, and then to rewrite it in terms of integrals over

the closure Ω ⊂ Σ.

Notice that ∂G = |p(z)− 1| = 12 , π(z) ∈ Ω ∪ |p(z)− 1| ≤ 1

2 , π(z) ∈ ∂Ω.Define z = x

√1 + w for a vector x in π(U) ⊂ Σ and a point |w| ≤ 1

2 in C. We

easily have that p(z) = 1 + w, π(z) = x, and more important:∫∂G(ϕ π)(z)B(z, a) =

∫x∈Ω

∫|w|= 1

2

ϕ(x)B(x√

1 + w, a)

+∫

x∈∂Ω

∫|w|≤ 1

2

ϕ(x)B(x√

1 + w, a). (4.2.6)

The first integral on the right hand side of (4.2.6) is calculated over those

monomials in B(x√

1 + w, a) that contain either dw or dw, because the variable

w runs along a circumference in C; while the integral in the second integral on

the right hand side of (4.2.6) is calculated calculated over those monomials in

B(x√

1 + w, a) that contain the volume form dw∧ dw, because the variable w runs

along an open disc in C.

If we expand B(x√

1 + w, a) in (4.2.5), we have

B(x√

1 + w, a) = B1 + B2 + B3 + B4.

Where B1 is the term that does not contains any term dw or dw, that is:

B1 =(n− 1)!(−1)

n(n−1)2

(2πi)n

n

∑j=1

(−1)j−1(xj√

1 + w−aj)√

1 + w|1 + w|n−1

||x√

1 + w− a||2n[dxj] ∧ dx.


The term B2 agrupates all the monomials that contain dw but not dw, that is:

B2 =(n− 1)!(−1)

n(n−1)2

(2πi)n

[n

∑j=1

∑p>j

(−1)p+j−1(xj√

1 + w−aj)(√

1 + w)3|1 + w|n−3xp

2||x√

1 + w− a||2ndw ∧ [dxj,p] ∧ dx

+n

∑j=1

∑p<j

(−1)p+j(xj√

1 + w−aj)(√

1 + w)3|1 + w|n−3xp

2||x√

1 + w− a||2ndw ∧ [dxj,p] ∧ dx

].

Then

B2 =(n− 1)!(−1)

n(n−1)2

(2πi)n

n

∑j=1

∑p 6=j

hpj(xj√

1 + w−aj)(√

1 + w)3|1 + w|n−3xp

2||x√

1 + w− a||2ndw∧ [dxj,p]∧ dx.

Where

hpj =

(−1)p+j−1 if p > j

(−1)p+j if p < j.

The term B3 agrupates all the monomials that contain dw but not dw, that is:

B3 =(n− 1)!(−1)

n(n−1)2

(2πi)n

n

∑j,p=1

(−1)p+n+j−1(xj√

1 + w−aj)|1 + w|n−1xp

2√

1 + w||x√

1 + w− a||2ndw∧ [dxj]∧ [dxp].

The term B4 agrupates all the monomials that contain dw ∧ dw, that is:

B4 =(n− 1)!(−1)

n(n−1)2

(2πi)n

[n

∑j,r=1

∑p>j

(−1)p+r+n+j(xj√

1 + w− aj)|1 + w|n−2xpxr

4√

1 + w||x√

1 + w− a||2ndw ∧ dw ∧ [dxj,p] ∧ [dxr]

+n

∑j,r=1

∑p<j

(−1)p+r+n+j−1(xj√

1 + w− aj)|1 + w|n−2xpxr

4√

1 + w||x√

1 + w− a||2ndw ∧ dw ∧ [dxj,p] ∧ [dxr]

].

Then

B4 =(n− 1)!(−1)

n(n−1)2

(2πi)n

n

∑j,r=1

∑p 6=j

gjpr(xj√

1 + w− aj)|1 + w|n−2xpxr

4√

1 + w||x√

1 + w− a||2ndw ∧ dw ∧ [dxj,p] ∧ [dxr].

Where

gjpr =

(−1)p+r+n+j if p > j

(−1)p+r+n+j−1 if p < j.

Notice that B1 and B2 are both equal to zero on Ω, because dx = dx1 ∧ ...∧ dxn

is an n-form and Ω has complex dimension n− 1. Therefore, in order to calculate


the first integral on the righthand side of (4.2.6), we only integrate over those

monomials that contain the differential dw wedge the volume form on Ω, because

the variable w runs along a circumference in C and Ω has real dimension 2n− 2.

That is, we only integrate over those monomials of the form dw ∧ [dxj] ∧ [dxp].

And we have ∫x∈Ω

∫|w|= 1

2

ϕ(x)B(x√

1 + w, a) =∫

x∈Ω

∫|w|= 1

2

ϕ(x)B3

=(n− 1)!(−1)

n(n−1)2

2(2πi)n

n

∑j,p=1

∫x∈Ω

∫|w|= 1

2

(−1)p+n+j−1ϕ(x)(xj√

1 + w− aj)|1 + w|n−1xp√1 + w||x

√1 + w− a||2n

dw ∧ [dxj] ∧ [dxp](4.2.7)

In the same form the terms in the second integral of (4.2.6) are only integrated

over those monomial that contain the volume form dw ∧ dw wedge the volume

form of the boundary of Ω, because the variable w runs along an open disc in C

and the boundary of Ω has real dimension 2n− 3. That is, we only integrate over

those monomials of the form dw ∧ dw ∧ [dxj,p] ∧ [dxr]. And we have∫x∈Ω

∫|w|= 1

2

ϕ(x)B(x√

1 + w, a) =∫

x∈Ω

∫|w|= 1

2

ϕ(x)B4

=(n− 1)!(−1)

n(n−1)2

4(2πi)n

n

∑j,r=1

∑p 6=j

∫x∈∂Ω

∫|w|≤ 1

2

gjpr ϕ(x)(xj√

1 + w− aj)|1 + w|n−2xpxr√

1 + w||x√

1 + w− a||2ndw ∧ dw ∧ [dxj,p] ∧ [dxr] (4.2.8)

Before calculating the integral∫

G ∂z(ϕ π)(z) ∧ B(z, a) in (4.2.4), we need to

calculate the differential ∂z(ϕ π)(z) as follows

∂z(ϕ π)(z) =n

∑p=1

∂((ϕ π)(z))∂zp

dzp

=n

∑p=1

n

∑k=1

(∂ϕ

∂zk π(z)

)· ∂

∂zp

(zk√p(z)

)dzp

where∂

∂zp

(zk√p(z)

)dzp =

1

p(z)32

p(z)− zp

2 if p = k

−zkzp if p 6= k.


Then

∂z(ϕ π)(z) =1√p(z)

n

∑p=1

(∂ϕ

∂zp π(z)

)dzp −

1

p(z)32

n

∑p=1

n

∑k=1

(∂ϕ

∂zk π(z)

)zpzkdzp

We set z = x√

1 + w in the above equation, with x a vector in π(U) ⊂ Σ and a

point |w| ≤ 12 in C, and we have

∂z(ϕ π)(z) = ∂x ϕ(x)−( n

∑p=1

xpdxp

)( n

∑k=1

∂ϕ

∂xk(x))

dw

− 1

2(1 + w)

n

∑p=1

∑k 6=p

xp∂ϕ

∂xk(x)dw,

sincen

∑p=1

xp2 = 1 then

n

∑p=1

xpdxp = 0, and

∂z(ϕ π)(z) = ∂x ϕ(x)− 1

2(1 + w)

n

∑p=1

∑k 6=p

xp∂ϕ

∂xk(x)dw.

Now if z = x√

1 + w with x a vector in π(U) ⊂ Σ and a point |w| ≤ 12 in C in

the second integral on the right hand side of (4.2.4), we have∫G

∂z(ϕ π)(z) ∧ B(z, a) =∫

x∈Ω

∫|w|≤ 1

2(∂x ϕ(x)− 1

2(1 + w)

n

∑p=1

∑k 6=p

xp∂ϕ

∂xk(x)dw

)∧ B(x

√1 + w, a) (4.2.9)

Notice that B1 and B2 in the expansion of B(x√

1 + w, a) are both equal to zero

on Ω, because dx = dx1 ∧ ... ∧ dxn is an n-form and Ω has complex dimension

n − 1.Therefore, in order to calculate the integral (4.2.9), we only integrate over

those monomials that contain the volume form of real dimension two wedge the

volume form on Ω, since the variable w runs along an open disc in C and Ω has

real dimension 2n− 2.That is, we only integrate over those monomials of the form

dw ∧ dw ∧ [dxp] ∧ [dxr] which come from the wedge product:

∂ϕ(x) ∧ B4 −(

1

2(1 + w)

n

∑p=1

∑k 6=p

xp∂ϕ

∂xk(x)dw

)∧ B3.

Notice that(1

2(1 + w)

n

∑p=1

∑k 6=p

xp∂ϕ

∂xk(x)dw

)∧ B3 =

( n

∑r=1

xrdxr

)(∑k 6=r

∂ϕ

∂xk(x))∧

∧[(n− 1)!(−1)

n(n−1)2

(2πi)n

n

∑j=1

∑j 6=r

cpjr(xj√

1 + w−aj)|1 + w|n−2xp

4√

1 + w||x√

1 + w− a||2ndw ∧ dw ∧ [dxr,j] ∧ [dxp]

]= 0,


becausen

∑p=1

xpdxp = 0, sincen

∑p=1

xp2 = 1.

Then if z = x√

1 + w with x a vector in π(U) ⊂ Σ and a point |w| ≤ 12 in C we

have ∫G

∂z(ϕ π)(z) ∧ B(z, a) =∫

x∈Ω

∫|w|≤ 1

2

∂x ϕ(x) ∧ B4. (4.2.10)

The previous work can be rewritten into the main result of this chapter. Hence,

applying Fubini’s theorem in (4.2.7), we define the function

F1,j(x, a) :=∫|w|= 1

2

(xj√

1 + w− aj)|1 + w|n−1√

1 + w||x√

1 + w− a||2ndw,

and the integral kernel

K1(x, a) :=n

∑j=1

(−1)j−1F1,j(x, a) ∧( n

∑p=1

(−1)pxp ∧ [dxj] ∧ [dxp]

).

Applying again Fubini’s theorem in (4.2.8) and (4.2.10) to define the function

F2,j(x, a) :=∫|w|≤ 1

2

(xj√

1 + w− aj)|1 + w|n−2

√1 + w||x

√1 + w− a||2n

dw ∧ dw,

and the integral kernel

K2(x, a) :=n

∑j=1

F2,j(x, a) ∧( n

∑r=1

∑p 6=j

gjprxp xr[dxj,p] ∧ [dxr]

),

where

gjpr =

(−1)p+r if p > j

(−1)p+r+j−1 if p < j.

Notice that we are introducing the factor (−1)n in the definition of the kernels

K1 and K2, that comes from the fact that we are applying Fubini’s theorem see

equation (5.2.2).

Now we can present our main result of this chapter.

Theorem 4.2.11. (Main) Let Σ be the complex sphere defined as the zero locus of the

polynomial p(z) =n

∑j=1

z2j , and π : U → Σ be the projection given by:

π(z) =z√p(z)

,


where U = z ∈ Cn : Re(p(z)) > 0 is an open neighborhood of Σ and the square root√p(z) is well defined. Given an open domain Ω in Σ with piecewise smooth boundary,

considere a smooth function ϕ on the closure Ω. Then we have the following identity

ϕ =(n− 1)!(−1)

n(n−1)2

4(2πı)n

(B∂Ω(ϕ)− BΩ(∂ϕ)

)on Ω.

Where

BΩ(∂ϕ) =∫

x∈Ω∂ϕ ∧ K2(x, a),

B∂Ω(ϕ) = 2∫

x∈∂Ωϕ(x) ∧ K2(x, a)

+ 2∫

x∈Ωϕ(x) ∧ K1(x, a).

Chapter 5

Integration formulae and kernels in

singular subvarieties of Cn

5.1 Introduction

The material presented in this chapter was already published in the volume 55 of

the CRM Proceedings and Lecture Notes; see [13].

Let Ω be a bounded domain in Cn with piecewise smooth boundary ∂Ω, and

ℵ be a (0, q)-form whose coefficients are continuous functions on the closure Ω.

If the differential ∂ℵ (calculated as a distribution) is also continuous on the clo-

sure Ω, then the following identity holds in Ω for the Bochner-Martinelli kernels

Bq(z, ξ),

ℵ(z) =∫

∂Ωℵ ∧ Bq −

∫Ω(∂ℵ) ∧ Bq + ∂

[ ∫Ωℵ ∧ Bq−1

]; (5.1.1)

see [12, 14].

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 (5.1.1) for differential forms ℵ such that ℵ and ∂ℵare both continuous on the closure Ω. One must be careful when we define ℵ and

∂ℵ on the singular part of Σ, but the details are clarified after Definition 5.1.5.

We propose in this chapter a simple technique for producing explicit integra-

tion formulae in a family of singular subvarieties of Cn, and for general contin-

uous differential forms ℵ. The main idea is to push down the integration kernels

from a neighbourhood of the subvariety into the subvariety itself.

We restrict our analysis to those subvarieties Σ in Cn+1 generated as the zero

52 Integration formulae and kernels in singular subvarieties of Cn

locus of a polynomial sm−p(z) for s ∈ C and z ∈ Cn, because the first entry s

can be easily expressed as the m-root of p(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 Σ of Cκ such that some entries of z ∈ Σ can be easily expressed in

terms of the other ones. We may work for example on the subvariety with a

simple-elliptic singularity D5 given in [7, p. 63] as the zero locus of the system of

polynomials

x2 + y2 − λzw and xy− z2 − w2,

for we have that x± y =√

λzw± 2(z2 + w2). Fix from now on Σ as the zero locus

of the polynomial sm−p(z) for (s, z) in C×Cn, m ≥ 1, and p 6≡ 0. The m-root of

p(z) is naturally defined in Σ, but we need to extend this property to a set that

contains Σ and has non empty interior in Cn+1. Thus define the closed set

Π =(s+ν, z) ∈ C×Cn : (s, z) ∈ Σ, |ν| ≤ |s|m

, (5.1.2)

A really important fact is that for every point (u, z) in Π, there is a unique

solution s ∈ C to the system

sm = p(z) and |u− s| ≤ |s|/m. (5.1.3)

It is obvious that there is only one solution s when m = 1 or p(z) vanishes.

Thus assume that p(z) does not vanish, m ≥ 2, and there are two different solu-

tions s and ρs to the system (5.1.3). It is easy to see that ρ 6= 1 is an m-root of the

unity and

|1− ρ| ≤ |1− u/s|+ |ρ− u/s| ≤ 2/m,

but this is impossible. Hence the following natural projection is well defined

π : Π→ Σ with π(u, z) = (m√

p(z), z) (5.1.4)

and m√

p(z) the unique solution s to (5.1.3). The projection π is only differentiable

at those points (u, z) of Π where p(z) is different from zero. On the other hand

we will work with general differential forms.

Definition 5.1.5. Let Σ ⊂ Cn+1 be the zero locus of τm−p(ξ) for τ ∈ C and ξ ∈ Cn.

Given any open or closed set X in Σ, a general differential form ℵ of bidegree (0, q) in X

is any linear combination

ℵ = ∑|I|=q

f I(τ, ξ)dξ l + ∑|J|=q−1

gJ(τ, ξ)dτ ∧ dξ J , (5.1.6)

where the coefficients f I and gJ are all arbitrary functions defined on X.

5.1 Introduction 53

The representation (5.1.6) is by no means unique. For example the differential

form dsm is obviously equal to dp(z) in Σ. We are specially interested in the case

when X is the closure of a domain in Σ and the coefficients f I and gJ in (5.1.6)

are all continuous functions. In this case the coefficients of the pull-back π∗ℵ are

not necessarily continuous, because the differential of the m-root of p(z) in (5.1.4)

is involved. However the coefficients of p(z)π∗ℵ are indeed continuous. We may

now introduce the integration kernels deduced from the Bochner-Martinelli ones.

Definition 5.1.7. Consider the functions

F(s, x) :=∫|w|<1/m

dw ∧ dw(x + |w−s|2)n+1 , (5.1.8)

F∗(s, x) :=∫|w|=1/m

(w−s) dw(x + |w− s|2)n+1 , (5.1.9)

defined for s ∈ C, the real numbers x ≥ 0, and the integer numbers n and m greater than

or equal to one. These functions are well defined and continuous at those points (s, x)

with x > 0 or |s| > 1m . Define as well the following integration kernels for the complex

vectors (τ, ξ) and (s, z) in C×Cn,

K1(τ, ξ; s, z) := F(

s−τ

τ,‖ξ−z‖2

|τ|2

)ω(ξ; z)

2πi|τ|2n , (5.1.10)

K2(τ, ξ; s, z) := F∗(

s−τ

τ,‖ξ−z‖2

|τ|2

)dω(ξ; z)

2nπi|τ|2n , (5.1.11)

where the total differential dω stands for (dξ+dz)ω and

ω(ξ; z) :=n

∑j=1

(ξ j−zj) dξ j

(2πi)n ∧∧k 6=j

[(dξk−dzk) ∧ dξk

].

Finally recall that an open set G in Σ is said to have piecewise-smooth bound-

ary ∂G if the closure G is compact and ∂G is the union of a finite number of

compact smooth submanifolds (with boundary); so that the (2n−1)-real dimen-

sional volume of ∂G is always finite. The following theorem is the main result of

this work.

Theorem 5.1.12. (Main) Let Σ be a subvariety in Cn+1 given as the zero locus of the

holomorphic polynomial sm−p(z) for (s, z) in C×Cn, m ≥ 1, and p 6≡ 0. The singular

part of Σ is contained in H = 0×Cn. Given an open domain G in Σ with piece-

wise smooth boundary ∂G, consider any (0, q)-form ℵ whose coefficients f I and gJ are

continuous functions on the closure G according to the expansion (5.1.6). Assume that


the differential ∂ℵ exists in the sense of distributions and is continuous in the difference

G\H; moreover it has a continuous extension into G, so that we may say that ∂ℵ exists

and is continuous in G. Then the following identity holds for (s, z) in G\H and every

fixed integer κ ≥ 2,

pκ(z)ℵ(−1)qn!

= B∂G(pκπ∗ℵ)− BG(pκπ∗∂ℵ) + ∂zBG(pκπ∗ℵ)

where

BG(pκπ∗ℵ) =∫(τ,ξ)∈G

[pκ(ξ)π∗ℵ

]∧K1(τ, ξ; s, z), (5.1.13)

BG(pκπ∗∂ℵ) =∫(τ,ξ)∈G

[pκ(ξ)π∗∂ℵ

]∧K1(τ, ξ; s, z), (5.1.14)

B∂G(pκπ∗ℵ) =∫(τ,ξ)∈∂G

[pκ(ξ)π∗ℵ

]∧K1(τ, ξ; s, z) + (5.1.15)

+ (−1)q∫(τ,ξ)∈G

[pκ(ξ)π∗ℵ

]∧K2(τ, ξ; s, z).

We must point out that the main difference between the previous theorem and

Koppelman formula in [12, p. 57] is the extra integral term in the operator B∂G.

However there is a simple technique for removing this extra term: Let (s, z) be

any fixed point in Σ with the first entry s 6=0, The kernel K2 in (5.1.11) is smooth at

every point (τ, ξ) in some neighbourhood of Σ, because of the following identity

F∗(

s−τ

τ,‖ξ−z‖2

|τ|2

)=∫|u−τ|= |τ|m

|τ|2n (u−s) du(‖ξ−z‖2 + |u−s|2)n+1 .

Suppose there is a form K3 smooth on a neighbourhood of G×G such that

(∂ξ + ∂z)K3(τ, ξ; s, z) = K2(τ, ξ; s, z). (5.1.16)

The kernel K1 in (5.1.10) satisfies a similar equation according to (5.2.17), but

K1 is unfortunately singular when ξ=z. We can easily calculate from (5.1.16) that

the second integral term in (5.1.15) can be rewritten in terms of K3, i.e.

(−1)q∫

G

[pκπ∗ℵ

]∧(K2−∂zK3) +

∫G

[pκπ∗∂ℵ

]∧K3 =

∫∂G

[pκπ∗ℵ

]∧K3.

Therefore K1 + K3 is the integration kernel that we are looking for, because the

main result in the main Theorem 5.1.12 can be rewritten as follows :

pκ(z)ℵ(−1)qn!

=∫

ξ∈∂G

[pκπ∗ℵ

]∧(K1+K3)−

∫ξ∈G

[pκπ∗∂ℵ

]∧(K1+K3)

+ ∂z

∫ξ∈G

[pκπ∗ℵ

]∧(K1+K3).

5.2 Basic properties 55

For example, if Σ ∼= Cn is the hyperspace given as the zero locus of the trivial

polynomial s−1, we automatically have

F(0, x) =∫|w|<1

dw ∧ dw(x + |w|2)n+1 =

2πin

[1xn −

1(x + 1)n

],

F∗(0, x) =∫|w|=1

w dw(x + |w|2)n+1 =

2πi(x + 1)n+1 .

The following identity can be easily calculated using (5.2.16),

(∂ξ + ∂z)ω(ξ; z)

(1 + ‖ξ−z‖2)n =dω(ξ; z)

(1 + ‖ξ−z‖2)n+1 = nK2(1, ξ; 1, z).

Recall the definition (5.1.10)–(5.1.11) for the kernels K1 and K2. The form that

we are looking for is the classical Bochner-Martinelli kernel for Σ ∼= Cn,

K1(1, ξ, 1, z) +ω(ξ; z)

n(1 + ‖ξ−z‖2)n =ω(ξ; z)

n‖ξ−z‖2n .

Considering again the general result presented in the main Theorem 5.1.12,

one of the remaining problems is to find a simple smooth solution K3 to equation

(5.1.16), in order to eliminate the extra integral term of the operator B∂Ω in (5.1.14).

The next section of this work is devoted to analyze the basic properties of the

functions and kernels introduced in Definition 5.1.7.

5.2 Basic properties

We begin defining the positive volume form in the space Cn,

dvol(ξ) :=n∧

j=1

[dξ j ∧ dξ j

2πi

]=

1πn

n∧j=1

[dxj ∧ dyj], (5.2.1)

for ξ = x + iy in Cn. Fubini’s theorem automatically yields the following identity

for every L1 function h defined on the space Cn+1,

∫(u,ξ)

h(u, ξ)du ∧ du ∧ dvol(ξ) =∫

ξ

[ ∫u

h(u, ξ)du ∧ du]

dvol(ξ), (5.2.2)

where (u, ξ) lies in C×Cn; see for example [6, p. 334]. The combination of Fubini

and Stoke’s theorems yields an interesting result. Let Uξ be a collection of open

sets in C parametrized by the points ξ ∈ Cn, such that Uξ is empty for |ξ| large


enough. Assume that h(u, ξ) is differentiable with respect to the variable u and

that each boundary ∂Uξ is piecewise smooth and depends smoothly on ξ, then∫ξ∈Cn,u∈∂Uξ

h(u, ξ)du ∧ dvol(ξ) =∫

ξ∈Cn,u∈Uξ

∂h∂u

du ∧ du ∧ dvol(ξ) = (5.2.3)

=∫

ξ

[ ∫u∈Uξ

∂h∂u

du ∧ du]

dvol(ξ) =∫

ξ

[ ∫u∈∂Uξ

h(u, ξ)du]

dvol(ξ).

The volume form dvol is also used for integrating functions on the subvariety

Σ. Recall that Σ is the zero locus of τm−p(ξ) for (τ, ξ) in C×Cn, m ≥ 1, and

p 6≡ 0. Consider an open set G in Σ with compact closure G and a continuous

function h defined on G. It is easy to see that∣∣∣∣ ∫(τ,ξ)∈G

h(τ, ξ) dvol(ξ)∣∣∣∣ ≤ m‖h‖G

∫ξ∈η(G)

dvol(ξ)

for the projection η(τ, ξ) = ξ and the positive volume form dvol in (5.2.1). Notice

that η is an m-branched covering from Σ onto Cn. The ramification locus of η

coincides with the zero locus of p, and so it has codimension one. In particular, if

we introduce the open set

Gβ = (τ, ξ) ∈ G : |τ| < β ⊂ Σ for β > 0, (5.2.4)

its image η(Gβ) is equal to ξ ∈ η(G) : |p(ξ)| < βm, and so the 2n-real dimen-

sional volume of this image converges to zero when β→ 0. Whence

limβ→0

∫(τ,ξ)∈Gβ

h(τ, ξ) dvol(ξ) = 0. (5.2.5)

Assume now that the open set G ⊂ Σ has piecewise-smooth boundary ∂G. It

is easy to see that the set Gβ given in (5.2.4) has piecewise-smooth boundary as

well. In particular

∂Gβ = (τ, ξ) ∈ ∂G : |τ| < β ∪ (τ, ξ) ∈ G : |τ| = β, (5.2.6)

and so the (2n−1)-real dimensional volume of every ∂Gβ is finite. Notice that we

may use the positive m-root of |p(ξ)| instead of |τ| in the equation above because

every point (τ, ξ) lies in Σ. Consider the differential form ω defined a pair of lines

below (5.1.11). There are exactly n monomials in ω that contain no differential dzk;

they are indeed

ω0j (ξ; z) =

(ξ j−zj) dξ j

(2πi)n ∧∧k 6=j

(dξk ∧ dξk) (5.2.7)

5.2 Basic properties 57

for 1 ≤ j ≤ n. Each differential dξω0j is equal to dvol(ξ) according to (5.2.1). Let h

be any continuous function defined on G. It is easy to prove that the integral of

hω0j over the boundary ∂Gβ in (5.2.6) satisfies the inequality∣∣∣∣ ∫

(τ,ξ)∈∂Gβ

h(τ, ξ)ω0j (ξ; z)

∣∣∣∣ ≤ C‖h‖GβVolume[2n−1](∂G)+

+mC‖h‖Gβ

∫ξ∈η(G),|p(ξ)|=βm

d|ξ j| ∧∧k 6=j

∣∣∣∣dξk ∧ dξk2i

∣∣∣∣,where C > 0 is a real constant that only depends on the form ω and the size and

shape of G. Since G is compact, the image η(G) is contained in some compact

ball. Whence there exists a real constant M2 > 0 such that∣∣∣∣ ∫(τ,ξ)∈∂Gβ

h(τ, ξ)ω0j (ξ; z)

∣∣∣∣ ≤ M2‖h‖Gβ(5.2.8)

for all constants β > 0, indexes 1 ≤ j ≤ n, and continuous functions h defined

on G. We also need to analyze the functions and kernels introduced in Defini-

tion 5.1.7, so we introduce the functions

F(s, x) =∫|w|<1/m

dw ∧ dw(x + |w−s|2)n+1 , (5.2.9)

F∗(s, x) =∫|w|=1/m

(w−s) dw(x + |w−s|2)n+1 , (5.2.10)

for s ∈ C, the real number x ≥ 0, and the integer n ≥ 1. These functions are well

defined and continuous at those points (s, x) with x > 0 or |s| > 1m . We can easily

deduce that [n + x

∂

∂x

]F(s, x) =

∫|w|< 1

m

n|w−s|2 − x(x + |w−s|2)n+2 dw ∧ dw.

Let dw be the total differential calculated with respect to the variable w. The

identity below easily holds for s 6= w or x > 0,

dw

[(w−s) dw

(x + |w−s|2)n+1

]=

x− n|w−s|2(x + |w−s|2)n+2 dw ∧ dw.

The application of Stokes’ theorem to (5.2.10) yields

− F∗(s, x) = nF(s, x) + xFx(s, x), (5.2.11)

whenever x > 0 or |s| > 1m . Nevertheless one can deduce that F∗(s, x) is well

defined and continuous at those points (s, x) with x ≥ 0 and |s| 6= 1m . We analyze


as well the integration kernel introduced in (5.1.10) for the complex vectors (τ, ξ)

and (s, z) in C×Cn,

K1(τ, ξ; s, z) = F(

s−τ

τ,‖ξ−z‖2

|τ|2

)ω(ξ; z)

2πi|τ|2n , (5.2.12)

where

ω(ξ; z) =n

∑j=1

(ξ j−zj) dξ j

(2πi)n ∧∧k 6=j


]. (5.2.13)

We shall prove that the anti-holomorphic differential of the kernel K1 in (5.2.12)

is equal to minus the integration kernel

K2(τ, ξ; s, z) = F∗(

s−τ

τ,‖ξ−z‖2

|τ|2

)dω(ξ; z)

2nπi|τ|2n , (5.2.14)

where F∗ is given in (5.2.10) and the total differential dω = (dξ+dz)ω satisfies

dω(ξ; z)n

=(∂ξ+∂z)ω(ξ; z)

n=

n∧k=1


2πi

]. (5.2.15)

The first equality above holds because ω(ξ; z) is of bidegree (n, n−1) with

respect to the vector (ξ, z). We also have that[(∂ξ + ∂z)‖ξ−z‖2

]∧ω(ξ; z) = ‖ξ−z‖2 dω(ξ; z)/n. (5.2.16)

The previous identity and equations (5.2.11)–(5.2.15) can be used for calculat-

ing the anti-holomorphic differential of the kernel K1 in (5.2.12):

K2(τ, ξ; s, z) = −[

nF +‖ξ−z‖2

|τ|2 Fx

]dω(ξ; z)

2nπi|τ|2n (5.2.17)

= −(∂ξ + ∂z)K1(τ, ξ; s, z),

for τ 6= 0 and z 6= ξ, or for τ 6= 0 and |s−τ| > |τ|m . Recall that the notation Fx in

(5.2.11) and (5.2.17) stands for the partial derivative of F(s, x) with respect to the

second variable x. Finally we need to express the Bochner-Martinelli-Koppelman

formula according to the positive volume form dvol introduced in (5.2.1). Let Ω

be an open set in Cn with piecewise smooth boundary, and ℵ be a continuous

(0, q)-form defined on Ω. Assume that the differential ∂ℵ is continuous on Ω

and has a continuous extension onto the closure Ω, then the following integral

formula holds in Ω,

(−1)qℵ(z)(n−1)!

=∫

∂Ω

ℵ ∧ω

‖ξ−z‖2n −∫

Ω

(∂ξℵ) ∧ω

‖ξ−z‖2n + ∂z

∫Ω

ℵ ∧ω

‖ξ−z‖2n , (5.2.18)

5.3 Main theorem 59

where ω = ω(ξ; z) is given in (5.2.13). The integrals in (5.2.18) are all calculated

according to the convention introduced in [12, p. 44]. For example the last two in-

tegrals in (5.2.18) are calculated only over those monomials whose degree is equal

to 2n with respect to the variable ξ, so that they contain the form dvol(ξ). Equa-

tion (5.2.18) is easily deduced by rewriting the Bochner-Martinelli-Koppelman for-

mula presented in [12] or [14]. This equation can be directly deduced from Stoke’s

theorem as well, we only need to observe that the following identity holds,

(∂ξ + ∂z)

(ω(ξ; z)‖ξ−z‖2n

)= 0

according to (5.2.16). The problem is then reduced to prove (5.2.18) when ℵ is

equal to the smooth form h(ξ)dξ J with |J| = q and Ω is the ball with centre at z

and radius ε > 0 small enough. One may calculate in particular that

limε→0

[ ∫‖ξ−z‖=ε

h(ξ)dξ J∧ω

ε2n +∫‖ξ−z‖<ε

∂zh(ξ)dξ J∧ω

‖ξ−z‖2n

]= h(z) lim

ε→0

∫‖ξ−z‖<ε

[(−1)q dξ J∧(∂ξω)

ε2n + ∂zdξ J∧ω

‖ξ−z‖2n

]=

(−1)qh(z)dzJ

(n−1)!.

We may now prove the main result of this chapter.

5.3 Main theorem

Let Hβ = |s| ≤ β ×Cn be a closed set in Cn+1 for β ≥ 0. Assume from now on

that Σ is a singular subvariety in Cn+1 given as the zero locus of the holomorphic

polynomial τm−p(ξ) for (τ, ξ) in C×Cn, m ≥ 1, and p 6≡ 0. The singular part of

Σ is obviously contained in H0. Recall the closed set Π of Cn+1 and the natural

projection π from Π onto Σ defined in the formulae (5.1.2)–(5.1.4). Given any

bounded domain G in Σ with piecewise smooth boundary ∂G, we define the

bounded open set Ω in Cn+1 as the interior of the inverse image π−1(G\Hβ), i.e.

Ω =(τ+ν, ξ) ∈ C×Cn : (τ, ξ) ∈ G\Hβ, |ν| < |τ|

m

. (5.3.1)

The difference G\Hβ is obviously contained in Ω. Let ℵ be a differential form

of bidegree (0, q) and with continuous coefficients in the closure G; i.e. ℵ is a

linear combination

ℵ = ∑|I|=q

f I(τ, ξ)dξ I + ∑|J|=q−1

gJ(τ, ξ)dτ ∧ dξ J , (5.3.2)


where the coefficients f I and gJ are all continuous functions on G. We assume that

∂ℵ exists in the sense of distributions and is continuous on Σ\H0. Recall that the

singular part of Σ is contained in H0. Moreover the given hypotheses imply that

the differential ∂ℵ has a continuous extension onto G, so that we may say that ∂ℵexists and is continuous in G. The pull-back of ℵ with respect to the projection π

given in (5.1.4) can be easily calculated in a simple formula,

π∗ℵ = ∑I

f I(σ(ξ), ξ)dξ I + ∑J

gJ(σ(ξ), ξ)[σ(ξ)∂p(ξ)

m p(ξ)

]∧dξ J , (5.3.3)

where π(u, ξ) = (σ(ξ), ξ) and the function σ(ξ) = m√

p(ξ) is the unique solution to

(5.1.3). It is easy to see that the (0, q)-forms ℵ and π∗ℵ are equal in G\H0. One of

the main problems is that the coefficients of the pull-back π∗ℵ are not necessarily

continuous in the closure G, because we have to deal with the differential of

σ(ξ). The same can be said about the coefficients of π∗∂ℵ. Nevertheless this

problem can be easily solved by multiplying the pull-backs π∗ℵ and π∗∂ℵ times

the holomorphic polynomial pκ(ξ) for any integer exponent κ ≥ 2.

The main idea is to apply the Bochner-Martinelli-Koppelman integration for-

mulae on the open set Ω and the form pκ(ξ)π∗ℵ, and then to rewrite it in terms

of integrals over the set G in Σ. We suppose that the domain G has piecewise

smooth boundary, i.e. ∂G is the union of a finite number of smooth real mani-

folds. It is then easy to deduce that Ω in (5.3.1) has piecewise boundary for any

β > 0. In particular

∂Ω =(τ+ν, ξ) ∈ C×Cn : (τ, ξ) ∈ G\Hβ, |ν| = |τ|

m∪ (5.3.4)

∪(τ+ν, ξ) ∈ C×Cn : (τ, ξ) ∈ ∂(G\Hβ), |ν| ≤ |τ|m

.

Recall that Hβ is equal to |s| ≤ β ×Cn, so that |τ| > β in (5.3.4); moreover

the singular part of Σ is contained in H0. We also need the Bochner-Martinelli

kernel written in terms of the vectors (u, ξ) and (s, z) in C×Cn; see equations

(5.2.13) and (5.2.18), or references [12, 14] :

B(u, ξ; s, z) =(u−s) du ∧ dω(ξ;z)

n + (du−ds)∧ du ∧ω(ξ; z)2πi(|u−s|2 + ‖ξ−z‖2)n+1 , (5.3.5)

where ω(ξ; z) and dω(ξ; z) are respectively defined in formulae (5.2.13) and (5.2.15).

Whence, since the form pκπ∗ℵ in (5.3.3) and its ∂-differential are both continuous

in the closure Ω for any fixed integer κ ≥ 2, the Bochner-Martinelli-Koppelman

formula (5.2.18) yields the following identity for (s, z) in Ω,

pκπ∗ℵ(−1)qn!

= B∂Ω(pκπ∗ℵ)− BΩ(pκπ∗∂ℵ) + (∂s+∂z)BΩ(pκπ∗ℵ), (5.3.6)

5.3 Main theorem 61

The open set Ω has piecewise smooth boundary and the operators BΩ and B∂Ω

are naturally given by the formula

BM(pκπ∗ℵ) =∫(u,ξ)∈M

[pκ(ξ)π∗ℵ

]∧B(u, ξ; s, z), (5.3.7)

where M is equal to Ω or its boundary∂Ω. The integral above is calculated ac-

cording to the convention introduced in [12, p. 44], so that only those terms whose

degree is equal to the real dimension of M are integrated. We have that the forms

π∗ℵ and ℵ are equal in G\H0, because of (5.3.2)–(5.3.3) and since G\Hβ is natu-

rally contained in Ω. Hence we only need to rewrite the operators BΩ and B∂Ω as

integrals over the open set G or its boundary ∂G, in order to deduce the desired

integration formula for the singular subvariety Σ. Recall the functions F and F∗

given in (5.2.9)–(5.2.10), and rewrite them using the change of variable w equal tou−τ

τ ,

F(

s−τ

τ,‖ξ−z‖2

|τ|2

)=

∫|u−τ|< |τ|m

|τ|2n du ∧ du(‖ξ−z‖2 + |u−s|2)n+1 , (5.3.8)

F∗(

s−τ

τ,‖ξ−z‖2

|τ|2

)=

∫|u−τ|= |τ|m

|τ|2n (u−s) du(‖ξ−z‖2 + |u−s|2)n+1 . (5.3.9)

Now we may analyze the operator BΩ defined according to (5.3.5)–(5.3.7).

Since the real dimension of Ω is equal to 2n+2, the integrals on Ω must be cal-

culated only over those monomials that contain all the differentials du, du, dξ,

and dξ. The coefficients of π∗ℵ in (5.3.3) are locally constant with respect to the

variable u; moreover the form π∗ℵ contains neither the differential du nor du.

Hence

BΩ(pκπ∗ℵ) =∫(u,ξ)∈Ω

[pκπ∗ℵ] ∧ du ∧ du ∧ω(ξ; z)2πi(|u−s|2 + ‖ξ−z‖2)n+1

=∫(τ,ξ)∈G\Hβ

∫|u−τ|< |τ|m

du ∧ du ∧ [pκπ∗ℵ] ∧ω(ξ; z)2πi(|u−s|2 + ‖ξ−z‖2)n+1 (5.3.10)

=∫(τ,ξ)∈G\Hβ

F(

s−τ

τ,‖ξ−z‖2

|τ|2

)[pκ(ξ)π∗ℵ

]∧ ω(ξ; z)

2πi|τ|2n .

The last two equalities above follows from Fubini’s theorem (5.2.2) and the

definition of Ω in (5.3.1). The integral with respect to the variable u yields in

particular the function F in (5.3.8). Moreover, since the variable u in (5.3.10) runs

along an open disc in C, we can easily understand why only those monomials in

(5.3.5) that contain the volume form du∧du can be integrated in (5.3.10). We can


simplify the final identity in (5.3.10) by using the kernel K1 defined in (5.2.12),

BΩ(pκπ∗ℵ) =∫(τ,ξ)∈G\Hβ

[pκ(ξ)π∗ℵ

]∧K1(τ, ξ; s, z). (5.3.11)

A similar formula can be deduced for BΩ(pκπ∗∂ℵ) given in (5.3.6)–(5.3.7).

Nevertheless the operator B∂Ω(pκπ∗ℵ) in (5.3.6) must be calculated carefully, be-

cause the boundary ∂Ω the union of two sets; see equation (5.3.4). Whence

B∂Ω(pκπ∗ℵ) =∫(τ,ξ)∈∂(G\Hβ)

∫|u−τ|< |τ|m

du∧du∧[pκπ∗ℵ]∧ω(ξ; z)2πi(|u−s|2+‖ξ−z‖2)n+1

+(−1)q∫(τ,ξ)∈G\Hβ

∫|u−τ|= |τ|m

(u−s)du∧[pκπ∗ℵ]∧dω(ξ; z)2nπi(|u−s|2+‖ξ−z‖2)n+1 (5.3.12)

−∫(τ,ξ)∈G\Hβ

∫|u−τ|= |τ|m

ds∧du∧[pκπ∗ℵ]∧ω(ξ; z)2πi(|u−s|2+‖ξ−z‖2)n+1 .

The integrals in the first line of (5.3.12) are calculated over those monomials

in (5.3.5) that contain the volume form du∧du, because the variable u runs along

an open disc in C; while the integrals in the second and third lines are calculated

over those monomials in (5.3.5) that contain either du or du, because the variable

u runs along a circumference in C. A direct application of Fubini’s theorems

(5.2.2)–(5.2.3) with the kernels K` and functions given in (5.2.12), (5.2.14), (5.3.8),

and (5.3.9) yields the following formula for (5.3.12) minus the term in third line,

B∗δΩ(pκπ∗ℵ) :=∫(τ,ξ)∈δ(G\Hβ)

[pκπ∗ℵ

]∧K1(τ, ξ; s, z) (5.3.13)

+(−1)q∫(τ,ξ)∈G\Hβ

[pκπ∗ℵ

]∧K2(τ, ξ; s, z).

We do not include the term in the third line of (5.3.12) because it is cancelled

by the antiholomorphic differential ∂sBΩ(pκπ∗ℵ) in (5.3.6), i.e.∫(τ,ξ)∈G\Hβ

∫|u−τ|= |τ|m

ds∧du∧[pκπ∗ℵ]∧ω(ξ; z)2πi(|u−s|2+‖ξ−z‖2)n+1

= (n+1)∫(τ,ξ)∈G\Hβ

ξ 6=z

∫|u−τ|< |τ|m

(u−s)ds∧du∧du∧[pκπ∗ℵ]∧ω

2πi(|u−s|2+‖ξ−z‖2)n+2

= ∂sBΩ(pκπ∗ℵ).

Recall the expression of BΩ(pκπ∗ℵ) given in the second line of (5.3.10). Bochner-

Martinelli-Koppelman formula (5.3.6) produces then the desired result on G\Hβ

for any β > 0, recall that the form ℵ and its pull-back π∗ℵ coincide on G\H0

because of the system of equations (5.3.2)–(5.3.3),

pκ(z)ℵ(−1)qn!

= B∗∂Ω(pκπ∗ℵ)− BΩ(pκπ∗∂ℵ) + ∂zBΩ(pκπ∗ℵ). (5.3.14)

5.3 Main theorem 63

Finally we need to calculate the limit of the previous equation as β converges

to zero. Let (s, z) be any fixed point in G\H0, so that neither s nor p(z) vanishes.

Functions F and F∗ in (5.3.8)–(5.3.9) are obviously continuous with respect to

(τ, ξ) for ξ 6= z; moreover

limτ→0

1|τ|2n F

(s−τ

τ,‖ξ−z‖2

|τ|2

)= 0 if ξ 6= z. (5.3.15)

The same equality holds for F∗ instead of F. On the other hand the coefficients

of the forms ω in (5.2.13) and pκπ∗ℵ in (5.3.3) are all continuous on the closure G,

because we have chosen the exponent κ ≥ 2. Therefore equations (5.3.10)–(5.3.11)

can be rewritten as follows

BΩ(pκπ∗ℵ) = ∑|J|=q−1

dzJ

∫(τ,ξ)∈G\Hβ

hJ(τ, ξ; s, z)dvol(ξ) (5.3.16)

for some coefficients hJ that are continuous at each (τ, ξ) in G with ξ 6= z. The

term dzJ appears in the formula above because π∗ℵ is a (0, q)-form and the inte-

gral in the third line of (5.3.10) is calculated over an open set in the regular part

of the 2n-real dimensional subvariety Σ, so that we integrate only those monomi-

als that contain the volume form dvol(ξ) defined in (5.2.1). Consider the set Gβ

defined in equation (5.2.4), so that[G ∩ Hβ/2

]⊂ Gβ ⊂

[G ∩ Hβ

]. (5.3.17)

Recall that Hβ = |s| ≤ β × Cn and that sm = p(z) for every (s, z) in the

compact set G. The polynomial p(z) does not vanish because (s, z) lies in G\H0.

Hence there exists a real number θ > 0 small enough such that p(ξ) and p(z)

are different for every point (τ, ξ) in Gθ, and so each coefficient hJ in (5.3.16) is

continuous on Gθ. Equations (5.2.5) and (5.3.17) yields a simple way to calculate

the limit as β goes to zero of the operator BΩ(pκπ∗ℵ) expressed in (5.3.11) or

(5.3.16). We can define in particular

BG(pκπ∗ℵ) :=∫(τ,ξ)∈G

[pκ(ξ)π∗ℵ

]∧K1(τ, ξ; s, z) (5.3.18)

= limβ→0

∫(τ,ξ)∈G\Hβ

[pκ(ξ)π∗ℵ

]∧K1(τ, ξ; s, z).

The operator BG(pκπ∗∂ℵ) can be defined in a similar way,

BG(pκπ∗∂ℵ) :=∫(τ,ξ)∈G

[pκ(ξ)π∗∂ℵ

]∧K1(τ, ξ; s, z). (5.3.19)


Let (s, z) be again a fixed point in G\H0, so that neither s nor p(z) vanishes.

The operator B∗∂Ω(pκπ∗ℵ) in (5.3.13) can be expanded as follows

B∗∂Ω(pκπ∗ℵ) = ∑|I|=q

dzI

∫(τ,ξ)∈G\Hβ

h∗I (τ, ξ; s, z)dvol(ξ) (5.3.20)

+ ∑|I|=q

dzI

n

∑j=1

∫(τ,ξ)∈∂(G\Hβ)

hj,I(τ, ξ; s, z)ω0j (ξ; z).

for some coefficients h∗I and hj,I that are continuous at (τ, ξ) in G with ξ 6= z. The

form ω0j was defined in (5.2.7) and appears in connection to ω in (5.2.13) because

the real dimension of the boundary of G\Hβ is equal to 2n−1. The coefficients h∗Iand hj,I are obviously determined by the form pκπ∗ℵ in (5.3.3) and the functions

F and F∗ given in (5.3.8)–(5.3.9); moreover they are all continuous on the compact

set Gθ for the parameter θ > 0 deduced some paragraphs above and the set

Gβ defined in (5.2.4). Finally observe that all the coefficients of pκπ∗ℵ in (5.3.3)

contain the factor p(ξ) because of the exponent κ ≥ 2; moreover F is continuous

on Gθ and satisfies equation (5.3.15). Whence there exists a real constants M3 > 0

such that

|hj,I(τ, ξ; s, z)| ≤ M3|p(ξ)| and ∂(Gβ) = ∂(G ∩ Hβ) (5.3.21)

for all points (τ, ξ) in Gθ, indexes j, I, and real numbers β > 0. Since |p(ξ)| is

less than or equal to βm for every point (τ, ξ) in Hβ, we easily have that equations

(5.2.8) and (5.3.21) yields that

limβ→0

∣∣∣∣ ∫(τ,ξ)∈∂(G∩Hβ)

hj,I(τ, ξ; s, z)ω0j (ξ; z)

∣∣∣∣ = 0.

Thus the operator B∂G(pκπ∗ℵ) may be defined as the limit when β goes to

zero of the operator B∗∂Ω(pκπ∗ℵ) expressed in (5.3.13) and (5.3.20),

B∂G(pκπ∗ℵ) :=∫(τ,ξ)∈∂G

[pκπ∗ℵ

]∧K1(τ, ξ; s, z) (5.3.22)

+(−1)q∫(τ,ξ)∈G

[pκπ∗ℵ

]∧K2(τ, ξ; s, z).

The desired result on G\H0 is obtained by applying the limit β → 0 to the

Bochner-Martinelli-Koppelman formula (5.3.14), recall that the form ℵ and its

pull-back π∗ℵ coincide on G\H0 because of (5.3.2)–(5.3.3),

pκ(z)ℵ(−1)qn!

= B∂G(pκπ∗ℵ)− BG(pκπ∗∂ℵ) + ∂zBG(pκπ∗ℵ).

Chapter 6

Results, conclusions and perspectives

When we worked on the particular weighted homogeneous subvariety Σ = z ∈C3 : z1z2 = zn

3, we obtained integral representations constructed around the

Cauchy kernel or some of its variations, and so we wondered wether the Cauchy

kernel is an intrinsic property of the weighted homogeneous subvarieties for

forms of degree zero or one well defined and with compact support. Amaizinly,

the answer was positive, as it was shown in chapter one. The proof of these fact

was an immediate consequence from the Cauchy-Green formulae. However, one

of the main problems founded is that it does not seem to be a natural way to gen-

eralize this result to forms with non compact support. We can obviously analyze

this problem in a future work.

In Chapter number four we presented a technique for producing Bochner-

Martinelli formulae on smooth complex Stein submanifolds of Cn, the main idea

is to use the natural explicit holomorphic retraction that exists from a neighbour-

hood of the submanifold into the submanifold itself; see [21].

The main result of this thesis was to present in chapter number five a sim-

ple technique for producing explicit integration formulae in subvarieties of Cn+1

generated as the zero locus of a polynomial sm−p(z) for s ∈ C and z ∈ Cn. Nev-

ertheless the technique presented in this work can be applied to analyze other

subvarieties such that some entries can be easily expressed in terms of the other

ones. The main difference between the integration formula obtained and Kop-

pelman formula in [12, p. 57] is the extra integral term in the operator B∂G, see

theorem 5.1.12.

Considering again the general result presented in the main Theorem 5.1.12, the

remaining problems is to find a simple smooth solution K3 to equation (5.1.16),

66 Results, conclusions and perspectives

in order to eliminate the extra integral term of the operator B∂Ω in (5.1.14), and

caculate the Hölder and Lp estimates.

References

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