Centro de Investigación y de Estudios Avanzados
del Instituto Politécnico Nacional
Unidad Zacatenco
Departamento de Matemáticas
Fórmulas y kerneles de integración
explícitos en subvariedades regulares y
singulares de
C
n
.
Tesis que presenta
Luis Miguel Hernández Pérez
Para obtener el grado de
Doctor en ciencias
en la especialidad de
Matemáticas
Director de tesis:
Dr. Eduardo Santillan Zeron
Center for Research and Advanced Studies
of the National Polytechnic Institute
Zacatenco Campus
Department of Mathematics
Explicit integration kernels and
formulae in regular and singular
subvarieties of
C
n
.
A dissertation presented by
Luis Miguel Hernández Pérez
To obtain the degree of
Doctor in science
in the speciality of
Mathematics
Thesis advisor:
Dr. Eduardo Santillan Zeron
Introduction and objectives
The Bochner-Martinelli and Ramirez-Khenkin integration formulae are a pair of cornerstones of the field of several complex variables. LetΩbe a bounded domain in Cn with piecewise smooth boundary ∂Ω, and ℵ be a (0,q)-form whose
coeffi-cients are continuous functions on the closureΩ. If the differential∂ℵ (calculated
as a distribution) is also continuous in Ω, then the following identity holds in Ω
for the Bochner-Martinelli kernels Bq(z,ξ),
ℵ =
Z
∂Ωℵ ∧Bq− Z
Ω(∂ℵ)∧Bq+∂
Z
Ωℵ ∧Bq−1
. (0.0.1)
It is impossible to enumerate all the applications of the integration formulae into complex analysis, geometry and other areas. We may mention for example their use for solving the Neumann ∂-equation in strictly pseudoconvex domains ofCn. A natural problem is to produce integration formulae on general varieties. Let Ω be an open domain compactly contained in a smooth or singular complex
variety Σ. If Ω has a piecewise smooth boundary ∂Ω, the problem is to produce
integration formulae similar to (0.0.1) for differential forms ℵ such that ℵ and ∂ℵ
are both continuous on Ω.
There is a vast literature in books and papers on integration formulae for smooth complex manifolds; see for example the references [9,10, 12, 15, 19]. An-dersson and Samuelsson also produced integration formulae for singular subvari-etiesΣofCn, but they delimited their work to analyse differentials formsℵwhose 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 second chapter of this thesis, we use the work that professors Rupphen-tal y Zeron presented in [17], in order to produce explicit integration formulae on weighted homogeneous subvarieties. We firstly deduce those formulae on the particular case of the subvariety Σ = {z ∈ C3 : z
1z2 = zn3, n ∈ N, n ≥ 2},
and then we extend these formulae to arbitrary weighted homogeneous subvari-eties. We analyse this particular subvariety Σ, because it is simple enough so as to do the calculations explicitly and complicated enough so as to exemplify all the classical pathologies. We also obtained integral representations constructed around the Cauchy kernel or some of its variations by working on the particular weighted homogeneous subvariety Σ={z∈ C3 : z
1z2 =zn3}. Thus, we wonder if
the Cauchy kernel is an intrinsic property of the weighted homogeneous subvari-eties for forms of degree zero or one. Amaizinly, the the answer is positive, as we will show at the end of chapter.
In the chapter three we include the work that professor Mats Andersson pre-sented in [1] and the work made by Peter Helgeson in his disssertation [11], in order to compare the proposed technique in these works with the technique pro-posed in this thesis.
We present in chapter four an alternative technique for producing integration formulae on smooth complex Stein submanifolds of Cn, based on the fact that every Stein submanifold inCn has a holomorphic retraction; see for example [21]. In general, it is quite difficult to find explicitly such a holomorphic retraction, so we exemplify the proposed technique with a practical example. We work on the smooth submanifold {z∈ Cn : ∑n
j=1z2j =1}, also known as complex sphere. We must mention that some parts of this thesis were already published in the papper: "Integration Formulae and Kernels in Singular Subvarieties of Cn", CRM Proceedings and Lectures Notes. Volume (55), 2012.
The main result of this work is contained in chapter five. We propose there a simple technique for producing explicit integration formulae in subvarieties of Cn+1 generated as the zero locus of a polynomial 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 them-root ofp(z)and several of the main singular subvarieties presented in [4, 5, 7, 20] are the zero locus of such a polynomial. Nevertheless, the technique presented in this work can be applied to analyze other subvarieties of Cκ, with the conditions that some entries of z ∈ Σ can be easily expressed in
vii
Contents
Introduction and objectives v
1 Preliminaries 1
1.1 The∂operator . . . 1
1.2 Differential forms . . . 2
1.3 Currents . . . 10
1.4 Integration kernels . . . 10
2 Integration formulae on weighted homogeneous varieties 15 2.1 Introduction . . . 15
2.2 Practical example . . . 15
2.3 Integration formulae on weighted homogeneous subvarieties . . . . 21
3 Integral representation with weights 23 3.1 Weighted representation formulae . . . 23
3.2 Integral formulae on a Riemann surface inC2 . . . 26
3.3 Preliminaries . . . 26
3.4 Explicit integral representations onX . . . 30
3.5 Integral formulae on a Riemann surface inP2 . . . 33
3.6 A Cauchy-Green formula onX . . . 35
4 Bochner-Martinelli formulae on the complex sphere 43 4.1 Introduction . . . 43
4.2 Bochner-Martinelli formulae on the complex sphere . . . 43
5 Integration formulae and kernels in singular subvarieties of Cn 51 5.1 Introduction . . . 51
5.2 Basic properties . . . 55
Chapter 1
Preliminaries
1.1
The
∂
operator
The main operator in the field of several complex variables is indubitably the delta-bar differential operator∂. The importance of this operator lies at the base of the main definition in several complex variables: what is a holomorphic function?. Properly speaking, given a smooth function f defined from an open set U of the n-dimensional complex spaceCn into the complex plane C, this function f is holomorphic in U if only if ∂f =0, where∂ is defined as follows:
∂f =
n
∑
k=1∂f
∂z¯kdz¯k
=
n
∑
k=11 2
∂f
∂Rezk + i ∂f
∂Imzk
dz¯k.
Notice that the∂operator sends smooth functions into(0, 1)-differential forms. In a similar way, we may define a general∂operator which sends(p,q)-differential forms into (p,q+1)-differential forms, see for example [12].
Now then, once we have any differential operator, a basic problem is to solve differential equations constructed with this operator. Hence, given a(0,q)-differential form ω, a natural question is to determine whether the differential equation
∂f = ω has a solution f?. We obviously require that ∂ω = 0. Moreover, does there exist a solution f which satisfies an extra smooth condition likeLpor Hölder regularity? We may go further on: Can we solve differential equations with this∂
Solving the ∂-equation is one of the main pillars of complex analysis, but it also has deep consequences on algebraic geometry, partial differential equations and other areas. For example, the classical Dolbeault theorem states that the ∂ -equation can be solved in all degrees on a Stein manifold; and its known that an open subset of Cn is Stein if and only if the ∂-equation can be solved in all degrees on that set. Nevertheless, it is usually difficult to produce an explicit operator for solving the ∂-equation on a given Stein manifold, even if we know that the equation can be solved.
1.2
Differential forms
Some of the material presented in this section was taken from the book [8, pp. 297-302].
Let X be a topological Hausdorff space.
Definition 1.2.1. An n-dimensional complex coordinate system(U,ϕU)in X consists of
an open set U ⊂X and a topological map ϕU from U onto an open set B ⊂Cn.
We say that twon-dimensional complex coordinate systems(U,ϕU)and(V,ψV) in X are compatible if either U∩V = ∅ or the map ϕU ◦ ϕ−1
V is biholomorphic, so that ϕU ◦ϕ−V1 is bijective, holomorphic, and with holomorphic inverse.
A covering of X with a pairwise compatible n-dimensional complex coordi-nates systems is called an n-dimensional complex atlas on X. Two atlases are called equivalent if any two complex coordinates systems are compatible. An equivalence class ofn-dimensional complex atlases onXis called ann-dimensional complex structure on X.
Definition 1.2.2. An n-dimensional complex manifold is a topological Hausdorff space X with a countable basis, equipped with an n-dimensional complex structure.
Let X be ann-dimensional complex manifold, and B⊂X be an open set.
Definition 1.2.3. A complex function f : B→C is called holomorphic if for each x∈ B
there is a coordinate system (U,ϕU) in X such that x∈ U and f ◦ϕ−U1 is holomorphic. Let X andY complex manifolds.
Definition 1.2.4. We say that the map F : X →Y is holomorphic if for each x∈ X there is a coordinate system (U,ϕU)in X at x and a coordinate system (V,ψV) in Y at F(x)
1.2 Differential forms 3
Definition 1.2.5. Let A be an algebra over a ring R, and let D : A→ A be an R-linear map satisfying the Leibnitz rule: D(ab) = D(a)b+aD(b). Then D is called a derivation on A.
Let X be an n-dimensional complex manifold and x ∈ X be a point. The tangent space to X at x, denoted by Tx, is the vector space of all derivations of functions defined in a neighbourhood of x. We consider a complex-valued alternating multilinear forms on the tangent space Tx.
Definition 1.2.6. A complex r-form or a r-dimensional differential form at x is an alter-natingR-multilinear mapping
ϕ:
r−times
z }| {
Tx×...×Tx →C.
Alternating means that interchanging the values of two entries of ϕautomatically changes the sign of ϕ. The set of all complex r-forms atx is denoted by Fr.
We have the following properties ofFr.
1. By convention, F0 = C. F1 = F(Tx) is the complexification of the 2n-real dimensional vector space Tx∗, where Tx∗ is the dual space to Tx.
2. Since Tx is 2n-dimensional overR, every alternating multilinear form onTx with more than 2n arguments must be equal to zero. So that Fr = 0 for
r >2n.
3. In general, Fr is a complex vector space. We can represent an elementϕ∈ Fr
uniquely in the form ϕ=Re(ϕ) +iIm(ϕ), where Re(ϕ) and Im(ϕ)are real-valued r-forms atx. Then it follows that
dimRFr =
2n r
.
4. We associate with each element ϕ∈ Fr a complex-conjugate element ϕ∈ Fr
by setting ϕ(v1, ...,vr) := ϕ(v1, ...,vr). And so we have: (a) ϕ=Re(ϕ)−iIm(ϕ).
(b) ϕ= ϕ.
(c) ϕ+ψ= ϕ+ψ.
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
(sgnσ)ϕ(vσ(1), ...,vσ(r))·ψ(vσ(r+1), ...,vσ(r+s)).
The sum is taken over all possible permutations σof the setSr+s ={1, 2, ...,r+s}, and sgnσ is the sign of the permutation σ.
In particular we have (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 M
r=0
Fr
becomes a noncommutative graded associativeC-algebra with unit(1), it is called the exterior algebra at x.
Definition 1.2.7. Let p,q ∈ N∪ {0}such thatp+q =r. A r-form ϕis called a form of
type(p,q)if
ϕ(cv1, ...,cvr) = cp cq·ϕ(v1, ...,vr) for all c ∈C.
Proposition 1.2.8. Letϕbe a nonzero r-form of the type(p,q), then p and q are uniquely determined.
Proof. Suppose that ϕis of type(p,q)and of type (p′,q′), since ϕ6=0 there exist tangent vector v1, ...,vr such that ϕ(v1, ...,vr) 6=0. Then
ϕ(cv1, ...,cvr) = cp cq·ϕ(v1, ...,vr)
=cp′ cq′·ϕ(v1, ...,vr).
Therefore, cp cq =cp′ cq′ for each c ∈ C. If c = eit with t ∈ R, theneit(p−q) =
eit(p′−q′). This can hold only when p−q = p′−q′. Since p+q = p′+q′ = r, it follows that p = p′ and q=q′.
1.2 Differential forms 5
Proposition 1.2.9. 1. If the form ϕis of type(p,q), then ϕis of type(q,p).
2. If ϕandψare both forms of type(p,q), then ϕ+ψandλ·ϕare also of type(p,q). 3. If ϕis a form of type(p,q) andψof type(p′,q′), then ϕ∧ψis of type
(p+p′,q+q′).
Notice thatdzν is a form of type (0, 1), since dzν =dzν. Then dzi1 ∧...∧dzip ∧ dzj1 ∧...∧dzjq, with 1 ≤ i1 < ... < ip ≤ n and 1 ≤ j1 < ... < jq ≤ n, is a form of type (p,q).
Theorem 1.2.10. Any r-form ϕhas a uniquely determined representation
ϕ=
∑
p+q=r
ϕ(p,q),
where ϕ(p,q) are r-forms of type(p,q)
Proof. The existence of the above representation follows form the fact that the forms dzi1 ∧...∧dzip∧dzj1 ∧...∧dzjq constitute a basis of Fr. For the uniquenes assume that
ϕ=
∑
p+q=rϕ(p,q) =
∑
p+q=r˜
ϕ(p,q). Then
∑
p+q=rψ(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=rcp cqψ(p,q)(v1, ...,vr).
For the fixed r-tuple (v1, ...,vr) we obtain a polynomial equation in the ring C[c,c]; and so all coefficients ψ
(p,q)(v1, ...,vr) must vanish. Since we can choose
v1, ...,vr arbitrarily, we have ϕ(p,q) = ϕ˜(p,q) for all p,q.
Definition 1.2.11. An holomorphic vector bundle E of rank r over an n-dimensional complex manifold X is a complex manifold satisfying the following conditions
1. There exists a holomorphic mapping π : E→ X.
2. For all x ∈ X the fiber of E, Ex = π−1(x) has the structure of an r-dimensional
vector space overC.
Local triviality means that there is an open covering{Ui}i∈I ofXtogether with biholomorphic functions, called trivializations, such that Φi : π−1(U
i) →Ui×Cr and for eachx ∈Ui the induced map pr◦Φi : Ex → x×Cr →Cr is a vector space isomorphism, where pr is the canonical projection.
Definition 1.2.12. Let E be an holomorphic vector bundle over X and U ⊂ X an open set. A continuous (differentiable, holomorphic) section in E over U is a continuous (dif-ferentiable, holomorphic) map s : U →E withπ◦s=idU.
Let X be an n-dimensional complex manifold, we denote by F(X) the com-plexified cotangent bundle T∗(X)NC. It has the spaces F(Tx(X)) = Tx(X)∗NC of complex covariant tangent vectors as fibers, so it is a topological complex vec-tor bundle of rank 2n. It even has a real-analytic structure, but not necessarily a complex analytic structure.
IfEis a topological complex vector bundle of rank moverX, we can construct a bundle Fr(E) of rank (mr) for each 0 ≤ r ≤ m, such that (Fr(E))x = Fr(Ex) for every x ∈ X. If E is given by the transitions functions gij, then Fr(E) is naturally given by the matrices g(r)ij whose entries are the(r×r)minors of gij.
Definition 1.2.13. An r-form or an r-dimensional differential form on an open set U ⊂X is a smooth section ω in the bundle Fr(U) := Fr(T(U)) for the tnagent vector bundle T(U).
We denote by Γ(U,Fr(X)) the vector space of holomorphic sections in Fr(X)
over U ⊂X.
So an r-form ω on U assigns to every point x ∈ U an r-form ωx at x. Notice that if z1, ...,zn are the local coordinates in a neighbourhood of x, then ωj := dzj and ωn+j :=dzj, for j=1, ...,nform a basis of the 1-forms on this neighbourhood. Moreover there is a representation
ωx =
∑
1≤i1<...<ir≤2n
ai1...ir(x)ωi1∧...∧ωir,
where x7→ ai1...ir(x) are smooth functions.
We denote by Er(U) the set of all smooth r-forms on U, and the subset of all smooth forms of type (p,q) by E(p,q)(U). Please do not confuse Er(U) with
Fr(U) = Fr(T(U)).
If f is a smooth function on U, then its differential d f ∈ E1(U) is given by
x 7→(d f)x, in local coordinates we have
d f =
n
∑
ν=1
∂f
∂zνdzν+
n
∑
ν=1
∂f
1.2 Differential forms 7
On the other hand, a smooth vector field is a smooth section of the tangent bundle T(X). So in local coordinates it can be written in the form
ξ =
n
∑
ν=1
ξν ∂
∂zν +
n
∑
ν=1
ξν ∂
∂zν,
where the coefficients ξν are smooth functions. Then we can apply d f to such a
vector field and we have
d f(ξ) =
n
∑
ν=1
ξν ∂f
∂zν +
n
∑
ν=1
ξν ∂f
∂zν.
For any open setU the differential d can be genreralized to the mapd = dU :
Er(U) → Er+1(U) in the following way. Let ω =∑1≤i
1<...<ir≤2nai1...ir(x)ωi1∧...∧
ωir is the basis representation in a coordinate neigbourhoodU, then
dU(ω) :=
∑
1≤i1<...<ir≤2n
dai1...ir(x)∧ωi1 ∧...∧ωir.
It is no difficult to show that this definition is independent of the choice of the local coordinates and that d has the following properties
1. If f is a smooth function,d f is the differential of f.
2. disC-linear.
3. d◦d=0.
4. If ϕ∈ Er(U) andψ∈ Es(U), then d(ϕ∧ψ) =dϕ∧ψ+ (−1)rϕ∧dψ.
5. dis a real operator; that isdϕ=dϕ. In particular dϕ=d(Re ϕ) +i d(Im ϕ). The differential dω is called the exterior derivative of the formω.
We consider the decomposition of anr-form into a sum of forms of type (p,q)
and use some notation, if I = (i1, ...,ip) and J = (j1, ...,iq) are multi-indices in increasing order, |I|= p and|J|=q are the lengths of I and J , we write
aI JdzI ∧dzJ
instead of
ai1...ip,j1,...,jqdzi1∧...∧dzip ∧dzj1∧...∧dzjq.
So a generalr-form ω has the unique representation
ω =
∑
p+q=r |I∑
|=p|J|=q
and the differential of ω is given by
dω =
∑
p+q=r∑
|I|=p
|J|=q
daI J∧dzI∧dzJ.
Notice that if f is a smooth function, then d f =∂f +∂f, where
∂f =
n
∑
ν=1
∂f
∂zνdzν and ∂f =
n
∑
ν=1
∂f
∂zνdzν,
here∂f has type (1, 0), ∂f has type(0, 1), andd f is of type (1, 1).
Proposition 1.2.14. Letϕbe a r-form of type(p,q). Then dϕhas a unique decomposition dϕ=∂ϕ+∂ϕwith∂ϕa(p+1,q)-form and∂ϕa (p,q+1)-form.
Proof. Let ϕ=∑I,JaI JdzI∧dzJ. We define
∂ϕ:=
∑
I,J
∂aI J∧dzI ∧dzJ and ∂ϕ:=∂aI J∧dzI ∧dzJ,
then dϕ = ∂ϕ+∂ϕ is the unique decomposition of the (p+q+1)-form dϕ into forms of pure type.
For generalr-forms the derivatives with respect to azand zare defined in the obvios way.
Theorem 1.2.15. 1. ∂and∂areC-linear operators with d=∂+∂.
2. ∂∂=0,∂∂ =0, and ∂∂+∂∂ =0.
3. ∂,∂are not real. We have∂ϕ =∂ϕand∂ϕ=∂ϕ. 4. If ϕis a r-form andψis arbitrary, then
∂(ϕ∧ψ) = ∂ϕ∧ψ+ (−1)rϕ∧∂ψ,
∂(ϕ∧ψ) = ∂ϕ∧ψ+ (−1)rϕ∧∂ψ.
Proof. It suffices to prove this for forms of pure type, then the above formulas can be easily derived from the corresponding formulas for d and the uniqueness of the decomposition into forms of type(p,q).
1.2 Differential forms 9
Definition 1.2.16. Let ϕbe a p-form on the open set U ⊂X. 1. ϕis called holomorphic if ϕis of type(p, 0)and∂ϕ=0. 2. ϕis called antiholomorphic if ϕis of type(0,p)and∂ϕ=0.
Let h : X → Y be a smooth map between smooth complex manifolds. If
ϕ(x) = ∑p+q=r∑|I|=p
|J|=q
aI JdzI ∧dzJ, is a (p,q)-form in local coordinates z on a
neigbourhoodU ⊂Y, and if hk are the components ofhin these coordinates, then in h−1(U) we define the pull-backh∗ϕof ϕwith respect to hby:
(h∗ϕ)(x) :=
∑
p+q=r|I
∑
|=p|J|=q
(aI J◦h)(x)dhi1(x)∧...∧dhip(x)∧dhj1∧...∧dhjq(x).
And we have the following properties:
1. For every(p,q)-form ϕonU, the pull-backh∗ϕis a (p,q)-form on h−1(U). 2. For every continous differential form varphi onU, we have
d(h∗ϕ) = h∗(dh), ∂(h∗ϕ) = h∗(∂h), and ∂(h∗ϕ) =h∗(∂ϕ).
We may extend the notion of forms of type (p,q) with smooth coefficients to forms of type (p,q) whose coefficients are distributions. The action of the differential operator ∂ is understood in the sense of distributions. In particular, the action of the differential operator ∂ over those forms is described in the next paragraphs.
Let λ be a smooth form of type (0,q) defined on an open set S in Cn. The fact that λ is ∂-closed (∂λ = 0) in the sense of distributions on S means that the following integral vanishes Z
Sλ∧∂σ=0,
for every smooth form σ of type (n,n−q−1) defined in and with compact sup-port inside S.
Furthermore, let λ be a smooth form of type (0,q) defined on the open set
S⊂Cn, and gbe a form of type (0,q−1) onS as well. We say thatλ=∂gholds in the distributional sense onS if and only if
Z
S λ∧σ+ (−1)
q−1g∧∂σ=0,
for every smooth form σ of type (n,n−q) defined and with compact support in
1.3
Currents
Let X be a complex manifold. A sequence ϕν = ∑I fIduI of smooth r-forms on
X is said to be convergent to zero in Er(X) if ϕν together with all its derivatives
tends uniformly to zero. In particular, we say that the form ϕνtends uniformly to
zero if and only if all its coefficients tends uniformly to zero when ν→∞.
Definition 1.3.1. A current of degree 2n−r is anR-linear map T : Er(X) → C such
that if {ϕν} is a sequence of r-forms converging to zero, then T(ϕν) converges to zero in
C.
Notice that ifψis a differential form of degree 2n−r. Thenψdefines a current
TΨ of degree 2n−r by the formula
Tψ(ϕ) := Z
Xψ∧ϕ, for all ϕ∈ E r(X).
And if M ⊂X is anr-dimensional differential submanifold. Then a current TM is defined by TM(ϕ) :=RMϕ. Therefore, we say that a current of degree 2n−r has dimension r.
1.4
Integration kernels
Let G ⊂ Cn be a bounded open set with piecewise smooth boundary ∂G, and
Λ be a (0,q)-differential form on G, for 0 ≤ q ≤ n, such that its coefficients are
smooth functions on the closure G. Let F(y,z) be a differential form defined on a neighbourhood of G×G, continuous when y 6= z, smooth of type (0,q) with respect to the variable z, and smooth of type (n,n−q−1) with respect to the variabley.
We fix z inG and letε >0, such that the ball Bε(z) = {x ∈ Cn : ||x−z|| ≤ ε} is contained in G. If Dz,ε = G\ Bε(z), we apply Stokes’s theorem to the form
Λ(y)∧F(y,z)on the domain Dz,ε; and we obtain Z
y∈Dz,ε
dy(Λ(y)∧F(y,z)) = Z
y∈∂Dz,ε
Λ(y)∧F(y,z). (1.4.1)
Notice thatΛ∧F is a(n,n−1)-form with respect to the variabley, and
1.4 Integration kernels 11
Thus, if we expand (1.4.1), we have Z
y∈Dz,ε
[∂yΛ(y)]∧F(y,z) + Z
y∈Dz,ε
Λ(y)∧(−1)q∂yF(y,z)
=
Z
y∈∂G
Λ(y)∧F(y,z)−
Z
y∈∂Bε(z)
Λ(y)∧F(y,z). (1.4.2)
Now suppose thatF(y,z) satisfies the following conditions:
(i) lim
ε→0
Z
y∈∂Bε(z)
Λ(y)∧F(y,z) = Λ(z),
(ii) (−1)q∂
yF(y,z) = c∂zF(y,z), where c is a constant. If we take the limit whenεtends to zero in (5.1.7),
Λ =
Z
∂G
Λ∧F−
Z
G(∂
Λ)∧F−c∂
Z
G
Λ∧F. (1.4.3)
Thus, by a integration kernel we mean a differential F(y,z) form continuous when y 6= z, smooth of type (0,q) with respect to the variable z, smooth of type
(n,n−q−1)with respect to the variabley, and that satisfies the conditions (i) and (ii). Hence (1.4.3) is satisfied, for every(0,q)-differential formΛ on G, 0≤q ≤n,
whose its coefficients are smooth functions on the closure G.
We are interesting to produce integration kernels, because if we have an in-tegral representation as in (1.4.3), we automatically have a solution for the ∂ -problem as follows:
If 1≤ q ≤n−1, Λ is a smooth (0,q)-form, with compact support in Cn, and
∂Λ=0, then the form
λ(z) =−c Z
y∈G
Λ(y)∧F(y,z)
satisfies the ∂-equation
∂zλ(z) = Λ(z) on Cn.
We are interesting in producing integration kernels on varieties with singu-larities, so that we work with a practical example. Let Σ ⊂Cn be a variety such that the origin of Cn is an isolated singularity of Σ and the regular part Σreg of Σ is a smooth complex manifold of codimension 1. As we have seen, in order to
produce integration kernels we need that Stokes’s theorem holds on the varietyΣ
or on its regular part Σreg.
LetG ⊂Σbe an open set with piecewise smooth boundary. We need to define
If the origin 0 does not lie on the closureG, the integralRGhis defined in the usual sense for all (n−1,n−1)-formsh continuous on G.
If 0∈ G, we takeε>0 such that the ball Bε(0)∩Σ⊂ G, and define
Z
Gh:=limε→0
Z
G−Bε(0)
h.
Moreover, if his not a (n−1,n−1)-form, we define RGh≡0.
Let f be an(n−1,n−1)-form continuous on the closure Gand smooth on the regular part Greg = G\ {0}. If 0 does not lie on the closure G, we automatically
have that the Stokes’s theorem holds; that is Z
Gd f = Z
∂G f.
If 0∈ G, we can takeε >0 such that Bε(0)∩Σ ⊂G. Let Gε =G\(Bε(0)∩Σ),
then:
Z
Gε
d f =
Z
∂Gε
f
=
Z
∂G f − Z
∂(Bε(0)∩Σ)
f.
Recall that f is bounded because it is continuous on the compact set G. Since the volumen of ∂(Bε(0)∩Σ) is of the orderO(ε2n−3), the follow identity holds when
we take the limit when εtends to zero, Z
Gd f = Z
∂G f.
Let Λbe a smooth(0,q)-differential form on the closure G, for 0≤ q ≤n−1, and such that its coefficients are differentiable onGreg. LetF(y,z)be a differential form defined onG×G, continuous wheny6=z, smooth of type(0,q)with respect to the variable z 6= 0, and smooth of type (n−1,n−q−2) with respect to the variabley 6=0.
Fix z 6= 0 in G. Let 0 < 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
Z
y∈Dz,ε
dy(Λ(y)∧F(y,z)) = Z
y∈∂Dz,ε
Λ(y)∧F(y,z). (1.4.4)
Notice thatΛ∧F is a(n−1,n−2)-form with respect to the variabley, and
1.4 Integration kernels 13
for εsmall enough.
Thus, if we expand (1.4.4), we have Z
y∈Dz,ε
[∂yΛ(y)]∧F(y,z) + Z
y∈Dz,ε
Λ(y)∧(−1)q∂yF(y,z)
=
Z
y∈∂G
Λ(y)∧F(y,z)−
Z
y∈∂(Bε(0)∩Σ)
Λ(y)∧F(y,z)
−
Z
y∈∂(Bε(z)∩Σ)
Λ(y)∧F(y,z). (1.4.5)
Now suppose that F(y,z) satisfies the following conditions (remember that
z6=0):
i) lim
ε→0
Z
y∈∂(Bε(0)∩Σ)
Λ(y)∧F(y,z) =0,
ii) lim
ε→0
Z
y∈∂(Bε(z)∩Σ)
Λ(y)∧F(y,z) =Λ(z),
iii) (−1)q∂
yF=c∂zF, where cis a constant.
If we take the limit whenε→0 in (1.4.5), we get
Λ=
Z
∂G
Λ∧F− Z
G∂
Λ∧F−c∂
Z
G
Λ∧F. (1.4.6)
Thus, by a integration kernel on an open set G ⊂ Σ with piecewise smooth
boundary, we mean a differential form F(y,z)defined onG×G, continuous when
y6=z, smooth of type(0,q)with respect to the variablez 6=0, smooth of type(n−
1,n−q−2) with respect to the variable y 6= 0, and that satisfies the conditions (i), (ii) and (iii) above. Hence (1.4.6) is satisfied for every(0,q)-differential formΛ
on G, 0 ≤q ≤n−1, whose coefficients are continuous functions on the closureG
and differentiable on the regular part Greg.
Like in Cn, if Λ is a continuous form with compact support in G (the origin
0 may be contain in the support of Λ) such that it is smooth and ∂-closed on
Greg=G\ {0}. Then, the following form
λ(z) =−c Z
y∈G
Λ(y)∧F(y,z)
satisfies the ∂-equation
∂zλ(z) =Λ(z).
Thus, we can solve the ∂-equation on open sets of Σ with smooth piecewise
Chapter 2
Integration formulae on weighted
homogeneous varieties
2.1
Introduction
The main objective of this chapter is to analyze the work that professors Rup-phental and Zeron presented in [17], in order to produce integration formulae and integral kernels for the representation of measurable functions well defined and with compact support in a weighted homogeneous subvariety. We firstly deduce explicit integration formulae and integral kernels on the particular sub-variety Σ = {z ∈ C3 : z1z2 = zn
3, n ∈ N, n ≥ 2}; and then we extend
these formulae to arbitrary weighted homogeneous subvarieties. We analyse this particular subvariety Σ because it is simple enough so as to do the calculations
explicitly and complicated enough so as to exemplify all the classical pathologies.
2.2
Practical example
Definition 2.2.1. Letβ∈ Znbe a fixed integer vector with entries βk ≥1. A polynomial
Q(z)holomorphic onCn is said to be weighted homogeneous of degree d ≥1with respect
to the vector βif the image Q(Hs(z))is equal to sdQ(z) for the mapping
Hs(z) := sβ1z1,sβ2z2, ...,sβnzn (2.2.2)
and all points s ∈ Cand z ∈ Cn. An algebraic subvariety ΣinCn is said to be weighted
homogeneous with respect to β if it is the zero locus of a finite number of weighted ho-mogeneous polynomials Qj(z) of (possibly different) degrees dj ≥ 1but all of them with
Notice that H1(z) in (2.2.2) is the identity mapping and Hs(z) is always an automorphism on the weighted homogeneous subvariety Σ for s 6= 0 fixed. In
particular, when s 6=0, the image Hs(z) lies in the regular part of Σif and only if
zlies in the regular part of Σas well. Define de subvariety
Σ={z∈ C3 : z1z2=zn
3, n∈ N, n≥2}.
We analyze this particular subvarietyΣbecause it is simple enough so as to do the
calculations explicitly and complicated enough so as to exemplify all the classical pathologies. The subvarietyΣis weighted homogeneous of degree nwith respect
to the vector β= (n−α,α, 1) for any integer 1≤ α <n. The calculations become
simpler if we take β= (n−1, 1, 1). Fix the general(0, 1)-measurable form
ω(y) = f1(y)dy1+f2(y)dy2+ f3(y)dy3, (2.2.3)
whose coefficients fk are all Borel-measurable functions well defined on Σ. We also suppose that each fk is essentially bounded and has compact support in Σ
and that y1,y2, and y3 are the cartesian coordinates of C3. We know from the
work of Rupphental and Zeron [17] that:
g1(z) =
3
∑
k=1βk
2πi Z
u∈C fk(u
β∗z)(uβkzk)du∧du
u(u−1)
is a solution to the ∂-equationω =∂g1 on the regular part Σ\ {0}.
We rewrite previous expression for g1(z), in order to obtain an integral for-mula that depends on the Cauchy kernel instead of the non-holomorphic kernel
du
¯
u(u−1). Thus, fix the point z = (z1,z2,z3) in C3 and expand the expression for
g1(z),
g1(z) = (1−n)z1
2πi Z
u∈C f1(u
n−1z
1,uz2,uz3)u
n−2du∧du
u−1
− z2
2πi Z
u∈C f2(u
n−1z
1,uz2,uz3)du∧du
u−1
− z3
2πi Z
u∈C f3(u
n−1z
1,uz2,uz3)
du∧du
u−1 . (2.2.4)
Define y = (y1,y2,y3) := (un−1z1,uz2,uz3), we need to analyze the following
2.2 Practical example 17
(a) Consider the case z3 6= 0. We may deduce that u = yz33, and so y =
(yn3−1 z1
zn3−1,y3 z2
z3,y3). Moreover we can express ω(y) in (2.2.3) in terms of the
coor-dinate y3:
ω(y) = (n−1)z1
z3n−1 y3
n−2f
1(y)dy3+z2
z3f2(y)dy3+f3(y)dy3;
and we also express the common kernel in the three integrals of (2.2.4) in terms of the variabley3as
du∧du u−1 =
dy3∧dy3 z3(y3−z3).
Sincey= (un−1z1,uz2,uz3), thenu =y3/z3 and
ω(y)∧ dy3
y3−z3 =
(n−1)z1 y3n−2f1(y)
z3n−1
dy3∧dy3
y3−z3
+ z2f2(y)
z3
dy3∧dy3 y3−z3
+ f3(y)dy3∧dy3
y3−z3
= (n−1)z1f1(un−1z1,uz2,uz3)u
n−2du∧du
u−1
+ z2f2(un−1z1,uz2,uz3)du∧du
u−1
+ z3f3(un−1z1,uz2,uz3)du∧du
u−1
Hence, we can rewrite (2.2.4) as follows
g1(z) =− 1
2πi Z
y3∈C
ω(y)∧ dy3
y3−z3.
(b) In case z2 6= 0, we may deduce u = y2
z2, and so y = (y
n−1 2
z1
zn2−1,y2,y2 z3
z2).
Moreover we can express ω(y) in (2.2.3) in terms of the coordinatey2:
ω(y) = (n−1)z1
z2n−1 y2
n−2f
1(y)dy2+ f2(y)dy2+z3
z2f3(y)dy2;
and we also express the common kernel in the three integrals of (2.2.4) in terms of the variabley2as
du∧du u−1 =
Sincey= (un−1z1,uz2,uz3), then
ω(y)∧ dy2
y2−z2 =
(n−1)z1 y2n−2f1(y)
z2n−1
dy2∧dy2
y2−z2
+ f2(y)dy2∧dy2
y2−z2
+ z3f3(y)
z2
dy2∧dy2
y2−z2 .
Hence, we can rewrite (2.2.4) as follows
g1(z) =− 1
2πi Z
y2∈C
ω(y)∧ dy2
y2−z2.
(c) In case z1 6= 0, we may deduce that u = (y1
z1) 1
n−1ρk for k = 0, 1, ...,n−2,
where ρ is the (n−1)-th root of unity, so that y = (y1,(yz11)
1
n−1ρkz2,(y1
z1) 1
n−1ρkz3) is
a multivalued function. Moreover, we also have the multivalued form,
ωk(y) = f1(y)dy1+ ρ
kz
2
(n−1)z1
y1
z1
1 n−1−1
f2(y)dy1
+ ρ
kz
3
(n−1)z1
y1
z1
1 n−1−1
f3(y)dy1.
Where ωk is the expansion of (2.2.3) in the k-th branch of the (n−1)-th root function. We also express the common kernel in the three integrals of (2.2.4) in terms of the variable y1:
du u−1 =
(y1
z1) 1
n−1−1dy1
(n−1)z1((yz11)
1
n−1 −ρk)
for k=0, 1, ...,n−2.
And
du∧du u−1 =
ρk(y1
z1) 1 n−1−1(y1
z1) 1 n−1−1
dy1∧dy1
(n−1)2|z
1|2((yz11) 1
n−1 −ρk)
2.2 Practical example 19
Sincey= (un−1z1,uz2,uz3) and u= (yz11)
1
n−1ρk, then
ωk(y)∧ (
y1
z1) 1
n−1−1dy1
(n−1)z1((yz11)
1
n−1 −ρk)
= (n−1)z1(
y1
z1) 1
n−1−1f1(y)dy1∧dy1
(n−1)2|z
1|2((yz11)
1
n−1 −ρk)
(2.2.5)
+ z2 ρ
k(y1
z1) 1 n−1−1(y1
z1) 1 n−1−1
f2(y)dy1∧dy1
(n−1)2|z
1|2((yz11)
1
n−1 −ρk)
+ z3 ρ
k(y1
z1) 1 n−1−1(y1
z1) 1 n−1−1
f3(y)dy1∧dy1
(n−1)2|z
1|2((yz11)
1
n−1 −ρk)
= (n−1)z1f1(un−1z1,uz2,uz3)u
n−2du∧du
u−1
+ z2f2(un−1z1,uz2,uz3)du∧du
u−1
+ z3f3(un−1z1,uz2,uz3)
du∧du u−1 .
Notice that un−2 = (y1
z1)
1− 1 n−1
ρ−k and that the term (y1
z1) 1
n−1−1 appears in the
first three lines of the above expansion, because we work in the k-th branch of the (n−1)-th root function, so that we may assume for practical reasons that the variabley1lies in the complex plane minus the negative real axisCb, recall that the real line has zero Lebesgue measure in C2. Moreover, in the last three lines of the above expansion we recover the variable u that lies in the complex plane.
We can now rewrite the formula (2.2.4), where g1 is defined, in terms of the form (2.2.5). Since u lies in the complex plane and we use the change of vari-able u = (y1
z1) 1
n−1ρk, we need to calculate the integrals in (2.2.4) over the Riemann
surface where the(n−1)-th root function is well defined; and this surface is com-posed byn−1 branches glued together. For practical reasons, integrating over this Riemann surface is equivalent to integrating over each of the (n−1)branches and adding all the integrals together. Hence, we integrate with respect to the variable
y1, when it lies in Cb, and we add the result over all posible values of the index
k=0, ...,n−2. We then rewrite (2.2.4) as follows:
g1(z) = − 1
2πi
n−2
∑
k=0Z
y1∈Cb
ωk(y)∧ (
y1
z1) 1
n−1−1dy1
(n−1)z1((yz11)
1
n−1 −ρk)
,
(2.2.5) and we have
g1(z) = − 1
2πi
n−2
∑
k=0Z
y1∈Cb
f1(y) (
y1
z1) 1
n−1−1dy1∧dy1
(n−1)z1((y1
z1) 1
n−1 −ρk)
− z2
2πi
n−2
∑
k=0Z
y1∈Cb f2(y)
ρk|y1
z1| 2
n−1−2dy1∧dy1
(n−1)2|z
1|2((yz11)
1
n−1 −ρk)
− z3
2πi
n−2
∑
k=0Z
y1∈Cb f3(y)
ρk|y1
z1| 2
n−1−2y1∧dy1
(n−1)2|z
1|2((yz11)
1
n−1 −ρk)
. (2.2.6)
In order to simplify the above formula, we need the following properties of the (n−1)-th root of the unity ρ, that can be easily deduced for every a∈ C,
1. n−2
∑
k=0ρk =0, 2.
n−2
∏
k=0(a−ρk) = an−1−1, 3. d
da
n−2
∏
k=0(a−ρk) = (n−1)an−2.
The identities below are easily infered, from the three above equations.
n−2
∑
k=01
a−ρk =
d
da∏k(a−ρk)
∏k(a−ρk) =
(n−1)an−2
an−1−1 , (2.2.7)
n−2
∑
k=01
ρka−1 = n−2
∑
k=0
a
a−ρk −1
= n−1
an−1−1. (2.2.8)
We use (2.2.7) in the first integral of the right hand side of (2.2.6) with a = (y1
z1) 1
n−1, and (2.2.8) in the second and third integrals of the right hand side of
(2.2.6), and we have (in case z1 6=0)
g1(z) = −
n−2
∑
k=0hk(z)
= − 1
2πi Z
y1∈Cb f1(y)
dy1∧dy1 y1−z1
− z2
2πi Z
y1∈Cb
f2(y)|
y1
z1| 2
n−1−2dy1∧dy1
(n−1)z1(y1−z1)
− z3
2πi Z
y1∈Cb f3(y)
|y1
z1| 2
n−1−2dy1∧dy1
2.3 Integration formulae on weighted homogeneous subvarieties 21
Working on the particular weighted homogeneous subvariety Σ = {z ∈ C3 :
z1z2 = zn3} and with the techniques presented in [17], we have obtained in the
cases (a), (b) and (c) integral representations constructed around the Cauchy ker-nel or some of its variations. Thus, we wonder if the Cauchy kerker-nel is an intrinsic property of the weighted homogeneous subvarieties for forms of degree zero or one. Amaizinly the the answer is positive as we will show below.
2.3
Integration formulae on weighted homogeneous
subvarieties
Ruppenthal and Zeron have indirectly deduced in [17, 18] some integration for-mulae on weighted homogeneous subvarieties ofCn. The main idea behind their work is to use a natural foliation induced by the mapping Hs(z)given below.
Hs(z) := sβ1z1,sβ2z2, ...,sβnzn for s∈ C and z∈ Cn.
The following pair of integration formulae are deduced from the Cauchy-Green formulae; see for example [12, p. 9]. The differentials are all defined and calcu-lated in the sense of distributions.
Lemma 2.3.1. LetΣbe a weighted homogeneous subvariety given as in definition (2.2.1)
and f be a continuous function defined on Σ and with compact support. The following
identity holds for every z6=0inΣ
f(z) = f(H1(z)) = −1
2πi Z
s∈C
h
∂sf(Hs(z))
i
∧ ds
s−1 (2.3.2)
under the assumption that the differential ∂sf(Hs(z)) exists and is continuous at every
s∈ C.
Moreover letℵ =∑k fkdz¯kbe a continuous(0, 1)-form well defined and with compact support on Σ. Assume that ℵ is also ∂-closed in the regular part of Σ. The following identity holds on the regular part as well
ℵ = H1∗ℵ=∂z
−1 2πi
Z
s∈C[H
∗
sℵ]∧
ds s−1
. (2.3.3)
used by Henkin and Leiterer in [12, p. 44]. It is easy to deduce that the pull-back ofℵ with respect to Hs(z) is expressed in terms of the differentialsdz¯k andds¯, i.e.
Hs∗ℵ =
n
∑
k=1fk(Hs(z))sβk
βkzk
s ds+dzk
. (2.3.4)
The integral in (2.3.3) is then calculated only over those monomials that contain the volume form ds¯∧ds, i.e. over the monomial that are of degree 2 with respect to the variable s. However the differential ∂z in (2.3.3) is calculated with respect to the variable zinΣ, and the differential∂s in (2.3.2) is calculated with respect to
the variables inC.
Proof. All the differentials are calculated in the sense of distributions. Let f
be a continuous function defined on Σ and with compact support. Take a fixed
point z 6= 0 in Σ. The function s 7→ f(Hs(z)) is continuous and has compact
support in the plane s ∈ C, so that we may apply Cauchy-Green formula on a ball Bwith centre at the origin ofCand radius large enough; see for example [12, p. 44]. Identity (2.3.2) is the resulting formula. We only need to assume that the differential ¯∂f(Hs(z))exists and is continuous at every points ∈C.
Identity (2.3.3) is directly deduce from the main formula presented by Rup-penthal and Zeron in [17, p. 443].
g1(z) =
3
∑
k=1βk
2πi Z
u∈C fk(u
β∗z)(uβkzk)du∧du
u(u−1) .
We just need to calculate the integral in (2.3.3) according to the convention used by Henkin and Leiterer in [12, p. 44], so that the integral is calculated only over those monomials of the pull-back (2.3.4) that contain the differential ds¯. In other words, given any measurable function g(s,z), one always assume that the follow-ing integral vanishes Z
s∈Cg(s,z)dz¯k∧ds =0.
✷
Chapter 3
Integral representation with weights
In this chapter we include the work that professor Mats Andersson presented in [1] and the work made by Peter Helgeson in his disssertation [11]. In [1] the professor Andersson describes a new approach to representation formulae for holomorphic functions and provide a general method to generate weighted inte-gral formulas. In [11] the main idea is to start with weighted Koppelman formulas on the projective space P2, and by a limit procedure to obtain Cauchy-Green for-mulae on an embedded Riemann surface. We include these works in order to compare the proposed technique in these works with the technique proposed in this thesis, specially with the techniques presented in the chapter 5, that is the main chapter in this work.
3.1
Weighted representation formulae
Let X be a smooth complex manifold, and recall that a smooth vector field is a smooth section of the tangent bundle T(X). So in local coordinates it can be written in the form
ξ =
n
∑
ν=1
ξν
∂ ∂zν +
n
∑
ν=1
ξν
∂ ∂zν,
where the coefficients ξν are smooth functions.
Definition 3.1.1. Let H be a vector field and let ω be an r-form on a smooth complex manifold X, r ≥1. We can define an(r−1)-formδH by the formula
(δHω)(x)(v2, ...,vr) = ω(H(x),v2, ...,vr), for v2, ...,vr ∈ Tx,
where Tx denotes the tangent vector space of X at x ∈ X. We callδH the contraction of
It is easy to see that the Cauchy kernel u(z,a) = zdz−a satisfies the following equations in the sense of distributions.
(z−a)u(z) = dz and ∂u = [a], (3.1.2) where [a] denotes integration (evaluation) at a considered as an (n,n)-current. In order to generalize Cauchy’s formula to higher dimensions it might seem to be most natural to look for forms u that satisfy the second equation in (3.1.2), since each such solution gives rise to a representation formula by Stokes’theorem. We have to introduce some notation. Let Ep,q(U) denote the space of smooth forms of type (p,q) in the open setU ∈ Cn and, for any integer m, let Lm(U) =
Ln
k=0Ek,k+m(U). For instance, u ∈ Lm(U) can be writtenu =u1+...+un, where the index denotes the degree in dz, so that uk is a form of type (k,k−1). In the same way we let Lm
curr(U) denote the corresponding space of currents. Fix a point a ∈ Cn, let δz−a : Ep,q(U) → Ep−1,q(U) be contraction with the vector field 2πi
n
∑
j=1(zj−aj) ∂
∂zj, and let ∇z−a = δz−a−∂, so that ∇z−a : L
m(U) → Lm+1(U).
Notice that when n =1, the Cauchy kernel is the unique solution in L−1
curr(U)to
∇z−au(z) = 1−[a]. (3.1.3)
Recall equation (3.1.2). If n>1, (3.1.3) means that
δz−au1=1, δz−auk+1−∂uk =0, ∂un = [a], (3.1.4) and any such u will be called a Cauchy form (with respect to a). If uis a Cauchy form we have an integral representation formula for holomorphic functions ϕon the closure D
ϕ(a) =
Z
∂D ϕ[a] = Z
∂D∂ϕ∧un +ϕ∂un = Z
∂Dd(ϕun) = Z
∂D ϕun, a∈ D.
Definition 3.1.5. A smooth form g ∈ L0(U) such that ∇z−ag = 0 and g0(a) = 1 is
called a weight with respect to a ∈U.
Ifgis a weight with repect toz ∈ D, we can solve∇z−av= gin a neighborhood
U of the boundary ∂D, and we then get the weighted representation formula for holomorphic functions ϕon the closureD,
ϕ(a) =
Z
∂Dϕu∧g+ Z
3.1 Weighted representation formulae 25
If gj are weights andG(λ1, ...,λm)is holomorphic on the image ofz7→ (g10, ...,g0m), then one can form a new weight G(g). In particular, ifm=1 and g=1+ (δz−a−
∂)q, and qis a form of type (1, 0), then
G(g) =
n
∑
k=0G(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=1b∧(∂b)k,
is smooth and solves∇z−a =1 inCn\ {a}. It will be called the Bochner-Martinelli form. ReplacingubyB(z,a)in (3.1.6) we have the integral representation formula, known as the Bochner-Martinelli formula.
Proposition 3.1.7. If g is a weight inΩ, D ⊂⊂Ω, and∇z−av =g in a neighbourhood
of the boundary ∂D, then
ϕ(a) =
Z
∂Dϕ∧vn+ Z
Dϕgn, (3.1.8)
for holomorphic functions ϕon the closure D. If∇z−a =g−[a]inΩ, then
ϕ(a) =
Z
∂D ϕvn+ Z
D ϕgn− Z
D∂ϕ∧vn, (3.1.9)
for smooth functions ϕon the closureD Proof. See [1].
We work inCn×Cn instead ofCn. Let Ω⊂Cn a domain, η =z−ζ ∈ Ω×Ω
be fixed, and consider the subbundle E∗ = span{dη1, ...,dηn} of the cotangent bundle T1,0∗ (the space of forms of type (1, 0)) over Ω×Ω. Let E be its dual
bundle and let δη be the contraction with the section
1 2πi
n
∑
j=1ηjej, whereej 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 letLm =LpLp,p+m. If ∇η =δη−∂, then we can solve
where[∆]denotes the(n,n)-current integration over the diagonal∆ inΩ×Ω. In
fact, the Bochner-Martinelli section u= ∇b
ηb, whereb=
1 2πi∑
∂||η||2
||η||2 solves (3.1.10).
A form g ∈L0(Ω×Ω)is a weight if ∇ηg =0 andg0 ≡1 on∆. As before ifq
is any smooth form in L−1 and G(0) = 1, then g = G(∇ηq) is a weight. If g is a
weight we can solve
∇ηv=g−[∆],
and if K = vn and P = gn, then we have∂K = [∆]−P. Thus for D ⊂⊂Ω we get for smooth forms ϕof type (0,q) on the closureD, the Koppelman’s formula
ϕ(z) =
Z
∂D ϕ∧K+ Z
D∂ϕ∧K+∂z Z
D ϕ∧K+ Z
Dϕ∧P. (3.1.11)
3.2
Integral formulae on a Riemann surface in
C
2By means of the theory of integral representation formulas in Cn, we are now going to find explicit formulas representing holomorphic functions on Riemann surfaces, X, embedded in C2. In particular we will considerX ={z∈ C2 : f(z) = 0}, where f is some holomorphic function such thatd f 6=0 on X.
Holomorphicity on X is defined locally in a natural way, since if {ψi}i∈I is a collection of charts we say that ϕ : X → C is holomorphic on X if and only if
∂(ϕ◦ψ−1) = 0 for alli ∈ I. The set of holomorphic functions on Xis denoted by
O(X).
3.3
Preliminaries
Before presenting the general idea of how to find integral formulas for holo-morphic functions on Riemann surfaces, we need to introduce some concepts and results that will be of fundamental importance for the rest of this text. Re-call that δζ−z is the contraction by the vector field 2πi∑j=n 1(ζj−zj)∂ζ∂
j and that
∇ζ−z =δζ−z−∂ζ.
Definition 3.3.1. Let U ⊂Cn be open and f : U → Cbe a holomorphic mapping. If h
is a holomorphic(1, 0)-form such that
δζ−zh = f(ζ)− f(z),
