# Errett Bishop theorems on Complex Analytic Sets: Chow’s Theorem Revisited
and Foliations with all leaves Compact on Kähler Manifolds

Carlos Martínez Aguilar  Unidad de Posgrado, UNAM, Mexico, Cto. de los
Posgrados, C.U., Coyoacán, 04510 Mexico City  Alberto Verjovsky  IMATE
Cuernavaca, UNAM, Mexico, Av Universidad, 62210 Cuernavaca, Morelos

###### Abstract

In this paper we present a series of seemingly unrelated results of Complex
Analysis which are in fact connected via a different approach to their proofs
using the results of Errett Bishop of volumes and limits of analytic
varieties. We start by proving Chow’s theorem by a technique suggested long
time ago in the beautiful book by Gabriel Stolzenberg. We think this approach
is very attractive and easier for students and newcomers to understand; also
the theory presented here is linked to areas of mathematics that are not
usually associated with Chow’s result. In addition, Bishop’s results imply
both Chow’s and Remmert-Stein’s theorems directly, meaning that this view is
simpler and just as profound as Remmert-Stein’s proof. After that, we give a
comparison table that explains how Bishop’s theorems generalize to several
complex variables classical results of one complex variable and prove Montel’s
compactness theorem using the techniques presented here. Finally we give an
alternative proof of a theorem of Edwards, Millet and Sullivan of foliations
with compact leaves for the case of complex foliations in Kähler manifolds.

###### keywords:

Chow’s Theorem, Bishop’s Theorems, Analytic Set, Algebraic Set

###### MSC:

32C25, 14-0X, 14-02

##  1 Introduction.

#### History and motivation

The purpose of this paper is to shed light on Errett Bishop’s results on
analytic varieties and their volumes, these statements listed as theorems 5, 4
and 7; these results were published in 1964 in an article called Conditions
for the Analycity of certain sets (see [1]). But the presentation of these
results we show here comes from the 1966 Gabriel Stolzenberg’s book Volumes,
Limits and Extensions of Analytic Varieties (see [12]) which also was the
catalyst for the exploration of Chow’s theorem via Bishop’s results.

In 1949 Wei-Liang Chow published the paper On Compact Complex Analytic
Varieties, which contains the proof of a long-conjectured result that gives a
deep link between analytic geometry and projective algebraic geometry. The
result, now known as _Chow’s theorem_ states that every closed analytic
subvariety of the complex projective space $\mathbb{CP}^{n}$ is algebraic. The
history regarding this result is related to the development of important
results of analytic sets of the XX century. The original proof of Chow was
done via analytic simplexes, later in 1953 Reinhold Remmert, and Karl Stein
proved that the closure of a purely $k$-dimensional analytic set is also an
analytic set under conditions which, in particular, imply that the complex
cone of a projective subvariety is also an analytic set. This is crucial for
the proof given in most textbooks on these topics that do not utilize the
techniques developed by Jean-Pierre Serre in 1956 his famous paper Géometrie
Algébrique et Géométrie Analytique colloquially known as GAGA. The proof
presented by us does not reference the algebraic properties of the ring of
germs of analytic functions nor nor does it make reference to quasi-coherent
sheaves, instead it is concerned with local properties of varieties and the
Hausdorff measure of said varieties respectively.
Inspired by this we show more results that link the theory of complex
algebraic sets and some well known theorems in complex analysis, see table
(2). Furthermore careful study of Bishop’s results pointed us to apply them to
foliations with compact leaves on Kähler manifolds in a manner similar to
Edwards, Millet and Sullivan in reference [4].

#### Analytic spaces

We begin with some basic definitions and terminology; we denote the sheaf of
holomorphic functions on $\mathbb{C}^{n}$ by $\mathcal{H}$ and for
$U\subset\mathbb{C}^{n}$ an open subset, the ring of holomorphic functions on
$U$ will be denoted by $\mathcal{H}(U),$ with the exception of
$\mathcal{H}_{n}:=\mathcal{H}(\mathbb{C}^{n})$. Given an open subset $U$ of
$\mathbb{C}^{n}$ and a finite subset
$\\{f_{1},\dots,f_{k}\\}\subset\mathcal{H}(U)$ we will denote the locus
consisting of points where all of these functions vanish by

$V_{U}(f_{1},\dots,f_{k}):=\big{\\{}z\in
U\,\big{|}\,f_{1}(z)=\dots=f_{k}(z)=0\big{\\}}.$

An analytic subset of $\mathbb{C}^{n}$ is a locally ringed subset
$X\subset\mathbb{C}^{n}$ such that for every $x\in X$ there are a
neighbourhood $U$ of $x$ and a finite subset
$\\{f_{1},\dots,f_{k}\\}\subset\mathcal{H}(U)$ with $X\cap
U=V_{U}(f_{1},\dots,f_{k})$, together with the local ring

$\mathcal{H}_{X}:=\mathcal{H}_{n}/\mathcal{I}(X)\hskip
2.84544pt,\text{where}\hskip
5.69046pt\mathcal{I}(X):=\big{\\{}f\in\mathcal{H}_{n}\,|\,f|_{U\cap
X}=0\big{\\}}.$

######  Definition.

An analytic space $(X,\mathcal{H}_{X})$ is a topological Hausdorff space $X$
together with a local ring structure $\mathcal{H}_{X}$ that is locally
isomorphic to an analytic subset of $\mathbb{C}^{n}$. We call a neighborhood
with its local isomorphism a chart.

An analytic subspace of an analytic space $(X,\mathcal{H}_{X})$ consist of a
subset $Y\subset X$ such that for every $y\in Y$ there is a chart
$(U,\varphi)$ around $y$ with

$\varphi(Y\cap U)=V_{U}(\varphi(\eta_{1}),\dots,\varphi(\eta_{k})),$

and $\\{\eta_{1},\dots,\eta_{k}\\}\subset\mathcal{H}_{X}$, naturally we have a
local ring structure $(Y,\mathcal{H}_{Y})$ given by
$\mathcal{H}_{Y}:=\mathcal{H}_{n}/\mathcal{I}(Y)$, as before.

We denote complex projective space of dimension $n$ by $\mathbb{CP}^{n}$. It
is a complex manifold and therefore an analytic space with its usual affine
charts. Algebraic subsets on $\mathbb{C}^{n+1}$ are the vanishing points of a
finite number of polynomials in $\mathbb{C}[z_{0},\dots,z_{n}]$, note that
using projective coordinates, the set of vanishing points of homogeneous
polynomials are well defined on $\mathbb{CP}^{n}$. Analytic and algebraic
subsets of $\mathbb{CP}^{n}$ are going to be called projective analytic
subsets and projective algebraic subsets respectively.

######  Theorem 1 (Chow).

Every closed projective analytic subset is a projective algebraic subset.

In order to prove this statement, we proceed with some properties of volumes
of purely $k$-dimensional analytic subvarieties and their Hausdorff measures,
then we proceed to enunciate one of the key theorems for this proof, a result
that we call Bishop’s proper mapping theorem.

###  1.1 Volumes of analytic sets and Wirtinger’s inequality

#### Kähler manifolds

Volumes and metrics have a natural relationship that we are going to exploit
throughout this article. In the context of complex analytic spaces and
manifolds, Kähler geometry is special for its interplay between the metric and
the associated volumes both for the whole manifold and for its submanifolds.
Related to this is the result due to Wirtinger that among other things implies
that Kähler submanifols of a Kähler manifolds minimize the volumes of their
respective homological class. This will be of crucial importance when dealing
with the volume function of the leaves on a compact Kähler manifold in theorem
9.

######  Definition.

Let $M$ be a complex manifold with a complex structure $J:TM\rightarrow TM$,
an Hermitian metric on $M$ is a family real bilinear forms $\\{h_{p}\\}_{p}$
for all $p\in M$ where $h_{p}:T_{p}M\times T_{p}M\rightarrow\mathbb{C}$, such
that

$h_{p}(JX,JY)=h_{p}(X,Y)\hskip 8.5359pt\forall\\{X,Y\\}\subset T_{p}M\hskip
8.5359pt\forall p\in M.$ (1)

We will also denote $h(\cdot,\cdot)=\langle\cdot,\cdot\rangle$. We call a
manifold with a Hermitian metric a Hermitian manifold

We note that Riemannian manifolds of even dimension $(M,g)$ with a complex
structure $J$ have a natural Hermit1an metric given by

$h(X,Y):=g(X,Y)+g(JX,JY).$

All Hermitian manifolds $(M,h)$ have an associated 2-form $\omega$ given by
$\omega(X,Y)=\langle JX,Y\rangle$ which is clearly antisymmetric since
$J^{2}=-Id_{M}$.

######  Definition.

Let $(M,h)$ be a Hermitian manifold with Hermitian metric $h$ and associated
2-form $\omega$, $h$ is a Kähler metric if

$d\,\omega=0.$

We call a manifold with a Kähler metric a Kähler manifold or a Kählerian
manifold, the associated 2-form will be called a Kähler form or a Kähler
symplectic form.

Being a Kähler has some strong topological restrictions, for example powers of
the Kähler form $\omega^{k}$ are non trivial representatives of cohomology
classes on $H^{k}(M;\mathbb{R})$, meaning these groups are never trivial. More
important for our purposes is the fact that a complex submanifold of a Kähler
manifold is also Kähler.

######  Definition.

We say $N$ is a complex submanifold of $M$ if $N$ is a complex manifold such
that there exist $f:N\rightarrow M$ a holomorphic isometric immersion for some
Hermitian metrics $h_{N}$ and $h_{M}$ of $N$ and $M$.

By our definition of submanifold it follows that every complex submanifold of
a Kähler manifold is also Kähler since the clossedness of the 2-form follows
from the compatibility of $J$ and the derivative under the pullback $f^{*}$.
Other consequences are the following:

######  Theorem 2.

Any complex submanifold of a Kähler manifold is a minimal submanifold.

######  Theorem 3 (Wirtinger’s Inequality).

Let $M$ be a Kähler manifold with Kähler form $\omega_{M}$ and let
$f:N\rightarrow M$ be a closed oriented immersion of an oriented real manifold
of real dimension $2k$. Let $\omega=f^{*}\,\omega_{M}$, then

$\frac{\omega}{k!}\leq dVol_{N}\hskip 8.5359pt\text{where }dVol_{N}\text{is
the volume form of N,}$ (2)

and the equality holds if and only if $N$ is a complex submanifold of $M$.

######  Corollary 1.

Let $N$ be a complex compact submanifold with boundary of a Kähler manifold
$M$, then $N$ is a volume minimizing submanifold in its homology class

$H_{2k}(M,\partial N,\mathbb{Z}),$

meaning that for any real submanifold $X$ of real dimension $2k$ and boundary
$\partial N$ homologous to $N$ has

$Vol_{2k}(N)\leq Vol_{2k}(X)$

#### Volume forms

Let $\omega=-Im\langle\cdot,\cdot\rangle$ where $\langle\cdot,\cdot\rangle$ is
the standard Kähler metric in $\mathbb{C}^{n}$ and let $M$ be a complex
submanifold of $\mathbb{C}^{n}$ by Wirtinger’s inequality, if $Vol_{2k}(M)$ is
the volume of $M$ given by the Riemannian structure
$g=Re\langle\cdot,\cdot\rangle|_{M}$, then

$Vol_{2k}(M)=\frac{1}{k!}\int_{M}\omega^{k}.$ (3)

If $X$ is a purely $k$ dimensional analytic subset of $\mathbb{C}^{n}$ and
$\Sigma(X)$ is its singular locus, then $M=X\setminus\Sigma(X)$ is a complex
manifold and since $\Sigma(X)$ is an analytic subset of $X$ of lesser
dimension,

$Vol_{2k}(X)=Vol_{2k}(X\setminus\Sigma(X))=\frac{1}{k!}\int_{X\setminus\Sigma(X)}\omega^{k}=\int_{X}\omega^{k}.$

###  1.2 Hausdorff measure and Bishop’s proper mapping theorem

#### Hausdorff measure and metric

Besides the volumes of analytic sets, we will study the Hausdorff measure of
said sets in order to determine their dimension, we denote the
$\delta$-Hausdorff measure of a subset $S\subset\mathbb{C}^{n}$ by
$\mathbf{H}_{\delta}(S)$. Here we list some of the most important properties
of this measure for our purposes;

  1. 1.

If $\mathbf{H}_{\delta}(S)<\infty$, and $\delta<\kappa$, then
$\mathbf{H}_{\kappa}(S)=0$.

  2. 2.

If $f:X\rightarrow Y$ is a Lipschitz continuous function with Lipschitz
constant $\lambda$, then for any $\delta\in\mathbb{R}^{+}$ and $S\subset X$,
the following inequality holds

$\mathbf{H}_{\delta}(f(S))\leq\lambda^{\delta}\mathbf{H}_{\delta}(S).$

  3. 3.

If $X=\mathbb{R}^{n}$ and $S$ is a smooth submanifold of dimension
$k\in\mathbb{Z}^{+}$, then the volume of $S$ as a submanifold is related to
its Hausdorff measure by the formula

$Vol_{k}(M)=\alpha_{k}\mathbf{H}_{k}(S),\hskip
8.5359pt\alpha_{k}=\frac{\pi^{k}}{\Gamma(k/2+1)},$

where $\Gamma(z)$ is the Euler’s gamma function.

Related to the Hausdorff measure is the Hausdorff metric defined for $K_{1}$
and $K_{2}$ compact subsets of a metric space $(X,d)$ as

$d_{H}(K_{1},K_{2}):=\max_{x\in K_{1}}\left\\{d(x,K_{2})\right\\}+\max_{y\in
K_{2}}\left\\{d(y,K_{1})\right\\},$

the Hausdorff metric allows us to define a convenient notion of convergence of
closed subsets; let $\left\\{S_{n}\right\\}$ be any sequence of closed subsets
of $X$ then we say $S_{n}\rightarrow S$ for $S\subset X$ closed if for every
$K\subset X$ compact we have

$d_{H}(K\cap S_{n},K\cap S)\rightarrow 0.$

The first of Bishop’s results that is very useful for understanding some of
the analytical properties of purely $k$-dimensional subvarieties is the
following convergence theorem:

######  Theorem 4.

[Sequence theorem] Let $\\{V_{n}\\}$ be a sequence of purely $k$-dimensional
subvarieties of a domain $\Omega\subset\mathbb{C}^{n}$ such that
$V_{n}\rightarrow V$, with $V\subset\Omega$ a closed subset, if

$Vol_{2k}(V_{n})\leq M\hskip 5.69046pt\forall n\in\mathbb{N},$

then for the Hausdorff measure we have; $\mathbf{H}_{2k+1}(V)=0$, moreover $V$
is a purely $k$-dimensional analytic subvariety of $\Omega$.

As direct application of this result, one can show the following very useful
proposition

######  Theorem 5.

[Bishop’s proper mapping theorem] Let $\Omega\subset\mathbb{C}^{n}$ be a
domain that contains $0$ and let $S\subset\Omega$ be a closed subset, if
$\mathbf{H}_{2k+1}(S)=0$, then there is a suitable coordinate change of
$\mathbb{C}^{n}$, $(z_{1},\ldots,z_{n})$ and neighborhoods
$\Omega_{k}\subset\mathbb{C}^{k}\times\\{0\\}$ and
$\Omega_{n-k}\subset\\{0\\}\times\mathbb{C}^{n-k}$, such that
$0\in\Omega_{k}\times\Omega_{n-k}\subset\Omega$ and the projection

$\pi_{k}:S\cap\Omega_{k}\times\Omega_{n-k}\rightarrow\Omega_{k}\,,\hskip
8.5359pt\pi_{k}(z,w):=z,$

is a proper map.

######  Remark.

This applied to purely $k$-dimensional analytic subvarieties implies the
regular coordinates theorem, meaning the proper mapping is a finite sheeted
analytic covering.

##  2 Consequences of the proper mapping theorem

As mentioned, this result by Bishop can be used to prove many other important
results (see some of them in [12]), and one of the most significant is the
proof of Remmert-Stein’s theorem; this was generalized and proved by Bishop in
[1].

######  Theorem 6.

[Remmert-Stein] Let $\Omega\subset\mathbb{C}^{n}$ be an open subset and $Y$ an
analytic subset of $\Omega$ and let $X$ be an analytic subset of
$\Omega\setminus Y$. If $Y$ is of dimension at most $k-1$ and $X$ is of pure
dimension $k$, then the closure of $X$, $\overline{X}\cap\Omega$ is an
analytic subset of $\Omega$.

This is an essential step towards the proof of Chow’s theorem if one is trying
to avoid using categorical methods of analytification and quasi-coherent
sheaves, this is because Remmert and Stein’s result imply that the $Cone(X)$
of an projective analytic subset of dimension $k$ $X\subset\mathbb{CP}^{n}$ is
an analytic subset of dimension $k+1$ in $\mathbb{C}^{n+1}$, where the cone is
defined by

$Cone(X):=\pi^{-1}[X]\cup\\{0\\},\hskip
8.5359pt\pi:\mathbb{C}^{n+1}\setminus\\{0\\}\rightarrow\mathbb{CP}^{n}.$ (4)

Here $\pi$ is the usual projection of $\mathbb{C}^{n+1}\setminus\\{0\\}$ onto
the projective space, so clearly $Cone(X)=\overline{\pi^{-1}[X]}$, and since
$\pi$ in an analytic projection, $\pi^{-1}[X]$ is an analytic subset. The
classical proof uses the fact that the cone is invariant under homothetic
transformations to show that the ideal of locally defined holomorphic
functions that vanish at the cone has a countable basis, then evoke Hilbert’s
famous basis theorem which proves that the ring of germs of holomophic
functions is Noetherian. This shows that $Cone(X)$ is in fact algebraic (see
this proof in [2]), but the same result can be proved without algebraic
methods with an equally simple proof using only the geometric and analytical
tools that we have presented thus far. We start the proof of Chow’s theorem by
citing another consequence of the proper mapping theorem and we give a sketch
of the proof.

######  Theorem 7.

[Bishop] Let $X$ be a purely $k$ dimensional subvariety of $\mathbb{C}^{n}$
and let $B(R,0)$ be the standard ball in $\mathbb{C}^{n}$ of radius $R$.
Suppose there is a constant $C\in\mathbb{R}^{+}$ such that

$Vol_{2k}(X\cap B(R,0))\leq CR^{2k}\hskip
5.69046pt\forall\,R\in\mathbb{R}^{+},$ (5)

then $X$ is algebraic.

Sketch of the proof: Let $\\{R_{n}\\}\subset\mathbb{R}^{+}$ be such that
$R_{n}\rightarrow\infty$ and define $V_{n}$ as the image of $V\cap B(0,R_{n})$
by the homothety $z\longmapsto z/R_{n}$, then $\\{V_{n}\\}$ is a sequence of
analytic sets of the unit ball with $Vol_{2k}(V_{n})<C$, and such that
$0\in\lim_{n\rightarrow\infty}V_{n}$. Then by the proper mapping theorem there
is a neighborhood of $0$, $\Delta=\Delta_{k}\times\Delta_{n-k}$ such that the
projection on the first factor $\pi_{k}$ is a $\sigma$-sheeted branched
covering for each $V_{n}\cap\Delta$, since $R_{n}\rightarrow\infty$ the balls
$B(0,R_{n})$ cover the whole of $\mathbb{C}^{n+1}$ and we deduce that the
projection $\pi_{k}$ restricted to $V$ is a $\sigma$-sheeted branched
covering. From this let us construct a set of canonically defining functions
for $V$; for each $z\in\mathbb{C}^{n}$ we denote

$\pi_{k}^{-1}[\pi_{k}(z)]\cap
V=\\{\alpha_{1}(z),\cdots,\alpha_{\sigma}(z)\\},$

and let
$P_{\alpha}:\mathbb{C}^{n}\times\mathbb{C}^{n}\rightarrow\mathbb{C}^{n}$ be

$P_{\alpha}(z,w):=\langle z-\alpha_{1}(z),w\rangle\cdots\langle
z-\alpha_{n}(z),w\rangle,$ (6)

we observe that if $z\in V$, $P_{\alpha}(z,w)=0$, so for $z$ outside of $V$ we
can chose a $w$ such that $\langle z-\alpha_{j}(z),w\rangle\neq 0$ for all
$j$. Now by expanding $P_{\alpha}$ in powers of $w$

$P_{\alpha}(z,w)=\sum_{|\mu|\leq\sigma}\eta_{\mu}(z)w^{\mu}$ (7)

then, because $\pi_{k}$ is a branched covering, the functions $\eta_{\mu}$ are
analytic. By applying a homothecy for a suitable $R_{n}$, one can prove using
Cauchy’s estimates that the functions $\eta_{\mu}$ are in fact polynomials of
degree at most $\sigma-|\mu|$ thereby proving the theorem.

###### Chow’s via Bishop.

By Theorem, 6 $Cone(X):=\overline{\pi^{-1}[X]}$ is an analytic set of
$\mathbb{C}^{n+1}$, and if $X\subset\mathbb{CP}^{n}$ is of dimension $k$ (real
dimension $2k$), then $Cone(X)$ has dimension $k+1$. Now the mapping
$\pi:\mathbb{S}^{2k+1}\rightarrow\mathbb{CP}^{n}$ known as Hopf’s Fibration,
fibres $\mathbb{CP}^{n}$ into circles, so by Fubini’s theorem;

$Vol_{2k+1}(Cone(X)\cap\mathbb{S}^{2n+1})=2\pi\,Vol^{p}_{2k}(X):=A,$

where $Vol^{p}$ is the projective volume, then $A$ is finite since $X$ is
compact. This means that the volume of the intersection $Cone(X)\cap B(R,0)$
grows at most polynomially as $R\rightarrow\infty$, since $Cone(X)$ is
homothetic-invariant, using polar coordinates

$Vol_{2k+2}(Cone(X)\cap B(R,0))=\int_{(Cone(X)\cap B(R,0)}dr\wedge\sigma_{r},$

where $\sigma_{r}$ is the $2k+1$ volume form of $S(0,r):=\\{\|z\|=r\\}$, we
can find a bound for the volume of $Cone(X)\cap B(R,0)$;

$0$
---
$C(X)$
---
$\mathbb{S}^{2n+1}$
---
Figure 1: Cone of $X$ having finite polynomially bounded volume.

$\int_{(Cone(X)\cap
B(R,0))}dr\wedge\sigma_{r}=\int^{R}_{0}r^{2k+1}dr\,\int_{Cone(X)\cap\mathbb{S}^{2n+1}}\sigma_{1}=\frac{A}{2k+1}R^{2k+2},$

so setting $C=\frac{A}{2k+1}$ by Theorem 7 this means $Cone(X)$ is algebraic
and therefore $X$ is also algebraic. ∎

######  Remark.

An immediate consequence of this is the following: a purely $k$-dimensional
analytic subset $X\subset\mathbb{C}^{n}$ is algebraic if and only if it is
contained after some unitary change of coordinates in the union of a ball
$B(R,0)$ and a cone

$K:=\\{(z_{1},z_{2})\in\mathbb{C}^{n}\,|\,\|z_{2}\|<C\|z_{1}\|\hskip
5.69046pt\textrm{with}\hskip
5.69046pt0<C\,,\,z_{1}\in\mathbb{C}^{k}\,,z_{2}\in\mathbb{C}^{n-k}\\}.$

#### Similarities with complex analysis

The following table shows the similarities between the results shown here and
others in the classical theory of holomorphic functions of one variable. The
last theorem in this table (Montel’s theorem) will be proved for holomorphic
functions of several complex variables using bishop’s results.

  * One complex variable |  Several complex variables
---|---
• Liouville’s theorem |  • Bishop’s theorem
$|f(z)|\leq C\,R^{k}$ for $|z|\leq R$ for all $R\in\mathbb{R}^{+}$ with $f$ entire and $k$ a positive integer, then $f$ is a polynomial |  $Vol_{2k}(X\cap B(R,0))\leq CR^{2k}$ for all $R\in\mathbb{R}^{+}$ with $X$ analytic subvariety, implies $X$ is algebraic
• Riemann’s extension theorem |  • Bishop’s generalization of Remmert-Stein’s theorem
If $f:(\Omega\setminus E)\subset\mathbb{C}\rightarrow\mathbb{C}$ is a holomorphic function and $E$ is a compact subset of capacity $0$ then $f$ is extendible to a holomorphic function on the whole region $\Omega$. |  Let $U\subset\mathbb{C}^{n}$ be a bounded open subset and let $B\subset U$ be closed with $X\subset U\setminus B$ a purely k dimensional analytic subset such that $B\subset\overline{X}$. If $B$ has capacity $0$ relative to the algebra of analytic functions on $X$ that are continuously extendible to $\overline{X}$, and if it exists $f:U\rightarrow\mathbb{C}^{k}$ proper on $B$ with $f(B)$ not an open connected subset of $\mathbb{C}^{k}$, then $\overline{X}\cap U$ is an analytic subset of $U$ (see [1][theorem 4]).
• Montel’s compactness theorem |  • Bishop’s sequences of analytic sets with bounded theorem
Let $\\{\Gamma_{i}\\}$ be a sequence of graphs of uniformly bounded holomophic functions $f_{i}:\Delta\rightarrow\mathbb{C}$ such that $\Gamma_{i}\overset{d_{H}}{\longrightarrow}\Gamma$, with $\Gamma\subset\mathbb{C}^{2}$ a closed subset and $\Delta$ the unit disk in $\mathbb{C}$, then $\Gamma$ is the graph of a holomorphic function. |  Let $\\{V_{i}\\}$ be a sequence of analytic suvariety with uniformly bounded volume such that $V_{i}\overset{d_{H}}{\longrightarrow}V$, then $V$ is an analytic subvariety ([12][pp. 30]).

## Proof of Montel’s Theorem via Bishop

#### Normal families

We now proceed to prove a generalization of Montel’s result of compact subsets
of holomorphic functions on the unit disk. First definitions, let $X$ be a
metric space, we denote by $C(X)$ the space of continuous functions from $X$
to $\mathbb{C}$, we say a subset $F\subset C(X)$ is normal if for any sequence
$\\{f_{n}\\}_{n\in\mathbb{N}}\subset F$ there is a subsequence
$\\{f_{n_{k}}\\}_{k\in\mathbb{N}}\subset F$ that is uniformly convergent. The
open unit ball on $\mathbb{C}^{n}$ will be denoted $B$ with its closure on
$\mathbb{C}^{n}$ denoted by $\overline{B}$. Let

$\mathcal{A}(B):=C(\overline{B})\cap\mathcal{H}(B),$

be the Banach algebra of holomorphic functions in $B$ that are continuous on
$S:=\partial B$, then we will prove the following result:

######  Theorem 8 (Montel’s Theorem).

Let $F\subset\mathcal{A}(B)$ a set of locally bounded functions, i.e. for
every $z\in B$ there exist an open neighborhood $z\in\Omega\subset B$ such
that

$\|f|_{\Omega}\|_{\infty}=\sup\\{|f|\,|\,z\in\Omega\\}\leq C_{\Omega},\hskip
5.69046pt\,\text{for some }C\in\mathbb{R}^{+},$

then $F$ is a normal set.

We are going to prove this by means of Bishop’s results. First, we note that
the graph of a holomorphic function $f:B\rightarrow\mathbb{C}$ is an analytic
subvariety of pure dimension $n$ on $B\times\mathbb{C}$ since
$\Gamma_{f}=V(w-f(z))$, where $\Gamma_{f}$ denotes the graph of $f$, and $z\in
B,\,z=(z_{1},\dots,z_{n})$, $w\in\mathbb{C}$, also if $\\{f|_{\Omega}\\}$ are
bounded for all $f\in F$, then $Vol_{2n}(\Gamma_{f|_{\Omega}})$ have uniformly
bounded volume as we shall see;

###### Proof.

Let $\epsilon\in(0,1)$ and we take
$B_{\epsilon}=\overline{B(1-\epsilon,0)}\subset B$, since $B_{\epsilon}$ is a
compact subset of $B$ we have that $F$ is uniformly bounded in $B_{\epsilon}$,
so let $C\in\mathbb{R}^{+}$ be a bound for $|f(z)|$ for all $z\in
B_{\epsilon}$ and $f\in F$, then by Cauchy’s integral formula for the open
ball (see [13] [Ch. 3]), if $S_{\epsilon}:=\partial B_{\epsilon}$, then

$f(z)=\int_{S_{\epsilon}}\frac{f(\zeta)}{(1-\langle
z,\zeta\rangle)^{n}}\,d\sigma(\zeta)\hskip 8.5359pt\,\forall z\in
B_{\epsilon}\setminus S_{\epsilon},$ (8)

where $\sigma$ is the usual Lebesgue measure in $S_{\epsilon}$. Taking the
derivative with respect to $z_{j}$ we obtain

$\dfrac{\partial f}{\partial
z_{j}}(z)=\int_{S_{\epsilon}}\frac{f(\zeta)n\zeta_{j}}{(1-\langle
z,\zeta\rangle)^{n+1}}\,d\sigma(\zeta)\hskip 8.5359pt\,\forall z\in
B_{\epsilon}\setminus S_{\epsilon},$ (9)

therefore we have the following bound for the partial derivatives of all $f\in
F$ and $z\in B_{\epsilon}\setminus S_{\epsilon}$,

$|\dfrac{\partial f}{\partial z_{j}}(z)|\leq
C\frac{nVol_{2n-1}(S_{\epsilon})(1-\epsilon)}{(1+(1-\epsilon)^{2})^{n+1}}.$
(10)

Now to use the sequence theorem 4, we use this constant bound of the
derivatives to show that for a sequence of graphs of functions
$\\{f_{n}\\}_{n\in\mathbb{N}}$, the $2n$ dimensional volumes of their graphs
are uniformly bounded in $B_{\epsilon}$ by a constant $M\in\mathbb{R}^{+}$
this is clear since the volume of $\Gamma_{f}$ is given by

$Vol_{2n}(\Gamma_{f})=\int_{B_{\epsilon}}|\lambda(z)|\,dx_{1}\cdots
dx_{n}dy_{1}\cdots dy_{n},$ (11)

where $z_{j}=x_{j}+iy_{j}$ and $\lambda:B_{\epsilon}\rightarrow\mathbb{R}$ is
given by the pullback of the parametrization function $\varphi(z)=(z,f(z))$

$\varphi^{*}\omega=\lambda(z)(dx_{1}\wedge
dy_{1})\wedge\cdots\wedge(dx_{i}\wedge dy_{n}),$

with $\omega$ the volume form of $\Gamma_{f}\subset B\times\mathbb{C}$, a
straight forward calculation shows

$\lambda(z)=1-2Re\big{(}\nabla f\big{)}+\|\nabla f\|^{2},\hskip 5.69046pt,$
(12)

with

$\nabla f=\Big{(}\dfrac{\partial f}{\partial z_{1}},\dots,\dfrac{\partial
f}{\partial z_{n}}\Big{)},\hskip 8.5359ptRe\big{(}\nabla
f\big{)}:=\sum_{j=1}^{n}Re\Big{(}\dfrac{\partial f}{\partial z_{j}}\Big{)}.$

Therefore the volumes of all the graphs are uniformly bounded by (10) and
(11), now the only thing left to show is the convergence with respect to the
Hausdorff metric of a sub-sequence of $\\{\Gamma_{f_{n}}\\}$ in
$B_{\epsilon}$. This follows immediately from the fact that $|f(z)|$ is
uniformly bounded in $B_{\epsilon}$ for all $f\in F$, so the image of all
$f_{n}$ are inside a compact set, therefore for $z\in S_{\epsilon}$ there
exist a convergent subsequence $\\{f_{n_{k}}(z)\\}$ and by Cauchy’s integral
formula $\\{f_{n_{k}}(z)\\}$ converges uniformly on $B_{\epsilon}\setminus
S_{\epsilon}$, which means $\Gamma_{f_{n_{k}}}\rightarrow\Gamma_{f}$ with
$\Gamma_{f}\subset B_{\epsilon}\times\mathbb{C}$ and by theorem 4 we know
$\Gamma_{f}|_{B_{\epsilon}}$ is an analytic subvariety of pure complex
dimension $n$. Moreover it is clearly the graph of a holomorphic function
$f:B_{\epsilon}\setminus S_{\epsilon}\rightarrow\mathbb{C}$ since at each
fibre $\\{z_{0}\\}\times\mathbb{C}$, the intersection of each graph
$\Gamma_{f_{n_{k}}}$ gives a unique point $(z_{0},f_{n_{k}}(z_{0}))$ and if it
was the case that there were two distinct points $(z_{0},w_{1})$ and
$(z_{0},w_{2})$ at the intersection of
$(\\{z_{0}\\}\times\mathbb{C})\cap\Gamma_{f}$, then taking a compact set $K$
containing $\\{f_{n_{k}}(z_{0})\\}\cup\\{w_{1},w_{2}\\}$ Hausdorff convergence
implies that the limit of $\\{f_{n_{k}}(z_{0})\\}$ would not be unique. Thus
we have proved that $\Gamma_{f}$ is the the graph of a holomorphic function in
$B_{\epsilon}$ for all $\epsilon\in(0,1)$, by analytic continuation we thus
have $f$ is defined in $B$ and it is the limit of a subsequence. ∎

## Foliations on Kähler manifolds with all leaves compact

In this final section we apply Bishop’s sequence theorem (Theorem 4) to prove
a version of Theorem 1 in [4], for the particular case of complex foliations
on Kähler manifolds with all leaves compact. Finally we end with an example of
such a foliation and calculate the volume function for its leaves.

######  Theorem 9.

Let $M$ be a compact connected Kähler manifold of complex dimension $n$ and
$\mathfrak{F}$ a holomorphic foliation with leaves of complex dimension $d<n$
and with all leaves compact, then:

  1. 1

The $2d$-dimensional volume (with respect to the Kähler metric) of the leaves
is uniformly bounded.

  2. 2

The quotient space $M/\mathfrak{F}$ is a complex orbifold, with singularities
corresponding to leaves with nontrivial holonomy (which by the first
proposition is a finite group).

###### Proof.

First, we prove the continuity of the volume function
$\nu:M\rightarrow\mathbb{R}^{+}$ given by $z\mapsto
Vol_{2d}(\mathcal{L}_{z})$, for the set of generic leaves:

$H_{0}=\\{x\in M\,|\,\mathcal{L}_{x}\text{ has zero holonomy}\\},$

where $\mathcal{L}_{x}$ denotes the leaf through $X$. We note here that the
set $H_{0}$ is a dense open set of $M$ (see [5]). Let $\\{z_{n}\\}\subset
H_{0}$ be a sequence such that each $z_{i}$ is on a different leaf
$\mathcal{L}_{z_{i}}$ of $\mathfrak{F}$ and such that $z_{n}\rightarrow z\in
H_{0}$, since all the leaves are compact we have that
$\mathcal{L}_{z_{i}}\rightarrow\mathcal{L}$ for the Hausdorff metric, where
$\mathcal{L}\subset M$ is a non-empty closed set. Now let $\mathcal{L}_{z}$ be
the leaf containing $z$, since $\mathcal{L}_{z}$ has zero holonomy, by Reeb’s
stability theorem, there exists a tubular neighborhood of $\mathcal{L}_{z}$,
say $U$ which is biholomorphic to $\mathcal{L}_{z}\times D$ where
$D\subset\mathbb{C}^{n-d}$ is an open disk (ball) and such that $U$ is a
saturated open subset of $M$ with every leaf of $U$ mapped biholomorphically
to the sets $\mathcal{L}_{z}\times\\{x\\}$, therefore every leaf in $U$ is
homologous to $\mathcal{L}_{z}$ and by Corollary 1, all leaves in $U$ have the
same volume, since $z\in U$ and $z_{n}\rightarrow z$ there is a large enough
positive integer $N$ such that all leaves $\mathcal{L}_{z_{k}}$ have the same
volume for $k>N$. Therefore by Theorem 4, $\mathcal{L}$ is an analytic
subvariety of $U$ of complex dimension $d$ with it’s volume equal to
$\lim_{n\rightarrow\infty}Vol_{2d}(\mathcal{L}_{z_{n}})$. Since tangency to
$\mathfrak{F}$ is defined locally by the null space of $d$ holomorphic
1-forms, by Hausdorff convergence this tangency is preserved on the limit so
$\mathcal{L}$ is tangent to $\mathfrak{F}$ and therefore
$\mathcal{L}=\mathcal{L}_{z}$. Now $\nu$ is not in general continuous but
lower semicontinuous, the semicontinuity can be proved by showing that the
leaf space $M/\mathfrak{F}$ is Hausdorff [4][p. 20], which we will prove, but
more than that, the volume function $\nu$ is in fact discretely lower-
semicontinuous, meaning that for any $n\in\mathbb{Z}^{+}$, $z\in M$ and
$\epsilon\in\mathbb{R}^{+}$, there is a small enough neighborhood of $z$ such
that

$\nu(y)>n\,\nu(z)\hskip 5.69046pt\text{or}\hskip
5.69046pt|\nu(y)-k\,\nu(z)|<\epsilon\hskip 5.69046pt\text{for some
}\,k\in\\{1,\cdots,n\\}.$

We prove this fact locally. Given a tubular neighborhood of $\mathcal{L}_{z}$,
$W$ there is a bundle retraction $\rho:W\rightarrow\mathcal{L}_{z}$ with
$\rho^{-1}(x)$ homeomorphic to a disk, for every leaf $\mathcal{L}_{y}$, the
restriction
$\rho|_{W\cap\mathcal{L}_{y}}:(W\cap\mathcal{L}_{y})\rightarrow\mathcal{L}_{z}$
is a codimension zero submersion, so if $y$ is sufficiently close to $z$, then
$\mathcal{L}_{y}\subset W$ and also is image under $\rho$ covers all of
$\mathcal{L}_{z}$ and is therefore by compactness a finitely sheeted covering
space of $\mathcal{L}_{z}$ with covering transformation
$\rho|_{\mathcal{L}_{y}}$ which proves the discrete lower-semicontinuity of
$\nu$ locally and therefore $\nu$ is locally bounded. We also note that the
previous proof is also true for $X\subset M$ a compact saturated set. With
this we proceed to show that the set where $\nu$ is not bounded, also known as
the “bad set”:

$B:=\\{x\in M\,|\,\nu\,\text{is not bounded in a neighborhood of }x\\},$

is a saturated compact set of codimension greater or equal to 2 (see [6] ).
For each $x\in B$ there exist a chart around $x$, $U\subset M$ such that
$\nu|_{U}$ is bounded, therefore given a convergent sequence in $U$,
$x_{n}\rightarrow x$, we have that their corresponding leaves in,
$\\{U\cap\mathcal{L}_{n}\\}$ (which we can suppose to be generic) converge to
$U\cap B$ by the fact that $B$ is a compact saturated set. Therefore since the
volume of $\\{U\cap\mathcal{L}_{n}\\}$ is uniformly bounded, by Theorem 4, he
have that $U\cap B$ is an analytic subvariety of complex dimension $d$, this
means that $B$ is analytic set of $M$ with real codimension at least 2, which
means that $M\setminus B$ is a connected open subset since $M$ is connected,
then by discrete lower semicontiuity the volume is therefore bounded.

The second assertion follows easily from Reeb’s stability theorem, since for
every leaf with null holonomy $\mathcal{L}$ we have an open laminated set $U$
biholomorphic to $\mathcal{L}_{z}\times D$ so locally $M/\mathfrak{F}$ is
homeomorphic to $D$, also $M/\mathfrak{F}$ is Hausdorff since every leaf is
compact so if $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are two distinct leaves,
then there are $\\{\epsilon_{1},\epsilon_{2}\\}\subset\mathbb{R}^{+}$ such
that the sets

$D_{i}:=\\{z\in M\,|\,d_{H}(\mathcal{L}_{i},z)<\epsilon_{i}\\},\hskip
5.69046pti\in\\{1,2\\},$

are disjoint, so intersecting with a laminated tubular neighborhood of
$\mathcal{L}_{i}$ we have that $M/\mathfrak{F}$ is Hausdorff. Finally if
$\mathcal{L}$ has holonomy, then by the boundedness of the volume function,
the holonomy group $H(\mathcal{L})$ is finite (see [4][p. 20]) and
$M/\mathfrak{F}$ is locally homeomorphic to $D/H(\mathcal{L})$ where
$\mathcal{L}$ has a tubular neighborhood homeomorphic to $\mathcal{L}\times
D$. ∎

Now for an example of this phenomenon let $\Sigma$ be the complex Klein curve
i.e. $\Sigma$ is a compact Riemann surface of genus $3$ with a hyperbolic
structure given by the action from the discrete group of Möbius
transformations

$\Gamma:=\Big{\\{}T\in
PSL(2,\mathbb{Z})\,\,\Big{|}\,\,T=\begin{pmatrix}a&&b\\\
c&&d\end{pmatrix}\equiv\begin{pmatrix}1&&0\\\ 0&&1\end{pmatrix}\mod
7\Big{\\}},$

which means $\Gamma/\mathbb{H}^{2}=\Sigma$, this group has as fundamental
domain of a compact hyperbolic 14-sided polygon (see [7]). Therefore
$\pi_{1}(\Sigma)\cong\Gamma$ is a compact hyperbolic Riemann surface, so there
exist a monomorphism

$\varphi_{7}:\pi_{1}(\Sigma)\rightarrow\Gamma\subset
PSL(2,\mathbb{R})=Isom(\mathbb{H}^{2}).$

Given this monomorphism we construct the suspension bundle
$E(\varphi_{7})\rightarrow\Sigma$ (compare [10] [pp. 24]) and its associated
foliation $\mathfrak{F}(\varphi_{7})$ as the bundle on $\Sigma$ given by the
discrete action of $\Gamma$ on $\mathbb{H}^{2}\times\mathbb{H}^{2}$ namely

$T(z,w):=(T\,z,T\,^{-1}w)\hskip 5.69046pt\forall T\in\Gamma.$ (13)

The quotient of $\mathbb{H}^{2}\times\mathbb{H}^{2}$ by this action is clearly
a compact complex manifold by the fact that $\Gamma$ has a compact fundamental
domain. $E(\varphi_{7})$ is a foliated space with leaves $\mathcal{L}_{z}$
given by the images of
$\\{\mathbb{H}^{2}\times\\{z\\}\,|\,z\in\mathbb{H}^{2}\\}$ by the quotient
map. The leaves of the foliation are transverse to the sections of the disk
bundle, moreover all of them are compact since all leaves are covered by
$\Sigma$. The tangent space of $E(\varphi_{7})$ on a point $\overline{(z,w)}$
is isomorphic to $T_{z}(\mathcal{L}_{z})\bigoplus T_{w}(\mathbb{H}^{2})$, also
since $\Gamma$ acts via isometries, then the quotient is a hyperbolic manifold
with a Kähler metric given by $h_{\mathcal{L}_{z}}\oplus h_{\mathbb{H}^{2}}$
where $h_{\mathcal{L}_{z}}$ is the Kähler metric of the leaf through $z$.

The leaves isomorphic to $\Sigma$ have hyperbolic area equal to the area of
the 14-sided polygon fundamental domain, which is comprised of fourteen
triangles with angle sum of $4/7\pi$ , therefore the total area of the surface
is $8\pi$ this also can be calculated by the fact that $\Sigma$ is a genus $3$
surface, meaning that its Euler characteristic is $\chi(\Sigma)=-4$, so by
Gauss-Bonnet the area is as indicated. Now the group of (orientation-
preserving) isometries of $\Sigma$ is of order 168 we show that there is a
characterization of all its subgroups up to its conjugacy classes, in
particular this characterization shows that there aren’t quotients of $\Sigma$
by any of its subgroups with characteristic $2$, this implies that all leaves
of $\mathfrak{F}(\varphi_{7})$ are either $\Sigma$, a torus or the complex
projective curve (sphere) see [7]. The following is the list of all conjugacy
classes for the subgroups of orientation-preserving isometries of $\Sigma$;
this list is obtained by carefully studying the platonic tessellations of
$\Sigma$.

  * Order 1

The identity class.

  * Order 2

There are 21 classes of involutions.

  * Order 3

The class of order 3 has 56 elements.

  * Order 7

There are two classes of order 7 rotations each with 24 elements.

  * Order 4

There are 42 classes of order 4 translations.

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