Algebraic cones of LCK manifolds with potential
Liviu Ornea111Liviu Ornea is partially supported by Romanian Ministry of
Education and Research, Program PN-III, Project number PN-III-P4-ID-
PCE-2020-0025, Contract 30/04.02.2021, Misha Verbitsky222Misha Verbitsky is
partially supported by the HSE University Basic Research Program, FAPERJ
E-26/202.912/2018 and CNPq - Process 313608/2017-2.
Keywords: Projective orbifold, affine cone, normal variety, Hopf manifold, LCK
manifold 2020 Mathematics Subject Classification: 32Q40, 14N99, 32Q28, 53C55

Abstract

A complex manifold $X$, $\operatorname{\sf dim}X>2$, is called “an LCK
manifold with potential”, if it can be realized as a complex submanifold of a
Hopf manifold. Let $\tilde{X}$ be its ${\mathbb{Z}}$-covering, considered as a
complex submanifold in ${\mathbb{C}}^{n}\backslash 0$. We prove that
$\tilde{X}$ is algebraic. We call the manifolds obtained this way the
algebraic cones, and show that the affine algebraic structure on $\tilde{X}$
is independent from the choice of $X$. We give several intrinsic definitions
of an algebraic cone, and prove that these definitions are equivalent.

###### Contents

  1. 1 Introduction
    1. 1.1 LCK manifolds with potential
    2. 1.2 Algebraic cones
    3. 1.3 Projective normality
    4. 1.4 New results
  2. 2 Closed and open algebraic cones
    1. 2.1 Stein completions
    2. 2.2 Algebraic structures on algebraic cones
    3. 2.3 Algebraic cones and subvarieties in Hopf manifolds
    4. 2.4 Algebraic cones and holomorphic contractions
  3. 3 Locally conformally Kähler manifolds
  4. 4 The total space of an ample bundle, Remmert-Stein theorem and weak Stein completions
  5. 5 Montel theorem and Riesz-Schauder theorem
    1. 5.1 Montel theorem
    2. 5.2 The Banach space of holomorphic functions
      1. 5.2.1 Compact operators
      2. 5.2.2 Holomorphic contractions
      3. 5.2.3 The Riesz-Schauder theorem
  6. 6 Algebraic structures on weak Stein completions
    1. 6.1 Ideals of the embedding to a Hopf manifold
    2. 6.2 Algebraic structures on Stein completions: the existence
    3. 6.3 Algebraic structures on Stein completions: the uniqueness
  7. 7 Algebraic cones and LCK manifolds
    1. 7.1 Weighted projective spaces
    2. 7.2 Jordan-Chevalley decomposition
    3. 7.3 Algebraic cone obtained from an LCK manifold with potential
    4. 7.4 Submanifolds of Hopf manifolds obtained from algebraic cones
  8. 8 Closed algebraic cones and normal varieties

##  1 Introduction

###  1.1 LCK manifolds with potential

An LCK manifold with potential ([OV2]) can be defined in terms of a certain
Hermitian metric (3). For our purpose, another definition is more convenient.
In complex dimension $>2$, an LCK manifold with potential can be defined as a
compact complex manifold which admits a holomorphic embedding to a linear Hopf
manifold (2.3).333The equivalence of these two definitions is proven in
dimension $>2$ ([OV4]); for complex surfaces it holds up to Kato conjecture,
which is one of the greatest unresolved conjectures in the geometry of complex
surfaces. Note that the Kato conjecture is widely believed to be true.

Let $M$ be an LCK manifold with potential, embedded to a Hopf manifold
$\frac{{\mathbb{C}}^{n}\backslash 0}{\langle A\rangle}$. Then the
${\mathbb{Z}}$-cover $\tilde{M}$ of $M$ is a complex submanifold in
${\mathbb{C}}^{n}\backslash 0$.

The closure $\tilde{M}_{c}$ of $\tilde{M}\subset{\mathbb{C}}^{n}\backslash 0$
in ${\mathbb{C}}^{n}$ is a Stein variety with an isolated singularity, called
the (weak) Stein completion of $\tilde{M}$ (2.1). The nature of this
singularity was not explored so far; in the present paper, we relate its
geometry to the classical subject of projective normality, due to Zariski,
Muhly and Hodge-Pedoe ([Mu, Za, HP]).

The purpose of this paper is to study the algebraic geometry of the variety
$\tilde{M}_{c}$. We use the LCK geometry mostly as a motivation; almost all
the results are based on arguments of complex-analytic and algebraic geometric
nature which do not invoke the differential geometry of an LCK manifold.

###  1.2 Algebraic cones

Locally conformally Kähler manifolds with potential are in a sense very
“algebraic”. The difference between complex projective and Kähler geometry is
understood through the Kodaira embedding theorem: a Kähler manifold with
rational Kähler class admits a complex embedding into a projective space. In
this setup, one can say that the geometry of complex projective manifolds is
part of the Kähler geometry, and this part is certainly very algebraic.

The analogue of Kodaira embedding theorem is already known: an LCK manifold
with potential is precisely a complex manifold which admits a holomorphic
embedding to a Hopf manifold. This notion is already very close to algebraic
geometry. For example, there exists a $p$-adic version of the theory of Hopf
manifolds and their complex subvarieties, in the framework of the rigid
analytic spaces ([Sc]). The $p$-adic Hopf manifolds are known and well-studied
([Mus, Vo]).

The passage from LCK geometry to complex algebraic geometry is based on the
notion of the “algebraic cone”, introduced in [OV3]. The closed algebraic cone
of an LCK manifold with potential is its (weak) Stein completion
$\tilde{M}_{c}$ (2.1), equipped with a structure of an affine variety, and the
open algebraic cone is $\tilde{M}$, that is, $\tilde{M}_{c}$ without its
origin (apex) point. We give an intrinsic definition of the closed and open
algebraic cones in Subsection 2.2.

###  1.3 Projective normality

Let $X$ be a projective variety, and
$R:=\bigoplus_{i\in{\mathbb{Z}}^{\geqslant 0}}H^{0}(X,{\cal O}(i))$ the
corresponding graded ring. Since $R$ is finitely generated, its spectrum is an
affine variety, called the affine cone of $X$.

Recall that a complex variety $X$ is called normal if any bimeromorphic finite
map $X^{\prime}{\>\longrightarrow\>}X$ is an isomorphism.444We actually use an
equivalent definition (Subsection 8): a variety is normal if all locally
bounded meromorphic functions are holomorphic. We discuss the normality in
some detail in Section 8.

This is related to the notion of “projective normality” much studied in
classical algebraic geometry. A projective variety $X\subset{\mathbb{C}}P^{n}$
is called projectively normal if the ring
$R=\bigoplus_{i\in{\mathbb{Z}}^{\geqslant 0}}H^{0}(X,{\cal O}(i))$ of the
regular functions in its affine cone is integrally closed; this is equivalent
to the affine cone being normal. The projective normality is a very subtle
notion, because it greatly depends on the choice of projective embedding ([Ha,
Exercise I.3.18]). Before the advent of the Hodge theory, projective normality
was used to prove things such as the Riemann-Roch formula. O. Zariski and his
student H. T. Muhly, who were the first to study the projective normality in a
systematic manner, called it “arithmetic normality” ([Mu, Za]); however, the
“projective normality” stuck because it was used in the textbook by Hodge and
Pedoe, [HP].

###  1.4 New results

In this subsection, we summarize the new results of this paper.

From the Remmert-Stein theorem ([De, Chapter 2, §8.2]) it follows that the
closure $\tilde{M}_{c}$ of $\tilde{M}$ in ${\mathbb{C}}^{n}$ is a Stein
variety with a singular point in the origin (4). We call this closure a weak
Stein completion of $\tilde{M}$. Note that a weak Stein completion Ăâem a
prioriÎ depends on the embedding $M\hookrightarrow H$; however, for an
appropriate embedding, the completion $\tilde{M}_{c}$ is normal, and this
normal variety, called the Stein completion, is uniquely determined by the
complex geometry of $\tilde{M}$ (2.1).

We prove that the Stein completion $\tilde{M}_{c}$ homeomorphically and
holomorphically projects to all weak Stein completions (4).

We consider $\tilde{M}_{c}$ as a Stein variety with a unique isolated singular
point. In [OV2, OV3] we proved that this Stein variety is in fact an affine
cone over a projective variety (3). In this paper we describe this cone from
the algebraic-geometric point of view.

In 2.2, we prove that the (affine) algebraic structure of $\tilde{M}$ is
uniquely determined by its complex analytic structure, as long as the
${\mathbb{Z}}$-action is algebraic. Ipso facto this algebraic structure is
independent from the choice of the LCK manifold with potential
$M=\frac{\tilde{M}}{{\mathbb{Z}}}$ used to construct $\tilde{M}$.

We define a closed algebraic cone as an affine cone over a projective
orbifold, which is smooth outside of the origin. The variety $\tilde{M}_{c}$
is an example of a closed algebraic cone. In 2.3 we prove that this definition
is equivalent to the definition of the closed algebraic cone given in [OV3] in
terms of LCK manifolds with potential.

In this paper we study the set of closed algebraic cones which correspond to
the same open algebraic cone.

We use the Stein completion to prove that any open algebraic cone $\tilde{M}$
can be associated with a unique closed algebraic cone $\tilde{M}_{c}$, which
is normal (4). We also prove that the other closed algebraic cones associated
to $\tilde{M}$ are homeomorphic to $\tilde{M}_{c}$, because they are all
obtained by adding a point in the origin. We prove that the natural
homeomorphism map
$n:\;\tilde{M}_{c}{\>\longrightarrow\>}\tilde{M}_{c}^{\prime}$ to another
closed algebraic cone associated to $\tilde{M}$ is holomorphic. Moreover, $n$
is the normalization map.

##  2 Closed and open algebraic cones

In this section we give several definitions of closed and open algebraic
cones. In the main body of this paper we prove that all these definitions are
equivalent.

###  2.1 Stein completions

Let $K\subset{\mathbb{C}}^{n}$ be a compact subset. Hartogs theorem implies
that any holomorphic function on ${\mathbb{C}}^{n}\backslash K$ can be
extended to ${\mathbb{C}}^{n}$, if $n\geqslant 2$. Due to Rossi, the same
result is true for any Stein variety:

Theorem 2.1:  Let $X$ be a normal Stein variety, $\operatorname{\sf
dim}_{\mathbb{C}}X>1$, and $K\subset X$ a compact subset. Then every
holomorphic function on $X\backslash K$ can be extended to $X$.

Proof: [Ro, Theorem 6.6].

Remark 2.2:  Let $A$ be a commutative Fréchet algebra over ${\mathbb{C}}$ and
$\operatorname{{\sf Spec}}(A)$ the continuous spectrum of $A$, defined as the
set of all continuous ${\mathbb{C}}$-linear homomorphisms
$A{\>\longrightarrow\>}{\mathbb{C}}$. By [Fo1, Fo2], $\operatorname{{\sf
Spec}}(H^{0}({\cal O}_{X}))=X$ for any Stein variety $X$, where $H^{0}({\cal
O}_{X})$ is the algebra of holomorphic functions equipped with the topology of
uniform convergence on compacts. From [Fo2] it also follows that a Stein
manifold is determined uniquely from $H^{0}({\cal O}_{X})$ considered as a
ring with $C^{0}$-topology.

Definition 2.3:  Let $X$ be a normal Stein variety, and $K\subset X$ a compact
subset. By 2.1, the ring of functions on $X$ is identified with $H^{0}({\cal
O}_{X\backslash K})$; by 2.1, this ring with $C^{0}$ topology uniquely defines
$X$. Following [AS], we call $X$ the Stein completion of $X\backslash K$. A
weak Stein completion of $X\backslash K$ is any Stein variety $X^{\prime}$
containing a compact $K^{\prime}$ such that $X^{\prime}\backslash
K^{\prime}\cong X\backslash K$.

Remark 2.4:  Clearly, if $X\backslash K$ is normal, the normalization of a
weak Stein completion of $X\backslash K$ is the Stein completion of
$X\backslash K$.

Example 2.5: Let $M\subset H$ be a submanifold in a Hopf manifold
$H=\frac{{\mathbb{C}}^{N}\backslash 0}{{\mathbb{Z}}}$, and
$\tilde{M}\subset{\mathbb{C}}^{N}\backslash 0$ its ${\mathbb{Z}}$-cover. By
the Remmert-Stein theorem (4), the closure of $\tilde{M}$ is Stein; however,
it is not necessarily normal (8). This is why we need the notion of a “weak
Stein completion”.

Example 2.6:  Let $P$ be a projective orbifold, and $L$ an ample line bundle
on $P$. Consider the total space $\operatorname{Tot}^{\circ}(L)$ of all non-
zero vectors in $L$. By 3 and 4 below, its (weak) Stein completion is obtained
by adding a single point, called the apex, or the origin. The weak Stein
completion of $\operatorname{Tot}^{\circ}(L)$ is not unique, as 8 shows;
however, all weak Stein completions of $\operatorname{Tot}^{\circ}(L)$ are
homeomorphic (4).

###  2.2 Algebraic structures on algebraic cones

The notion of an algebraic cone has two flavours: there are “closed algebraic
cones”, which can be obtained as weak Stein completions of “open algebraic
cones”. The latter can be defined in algebraic terms as follows.

Definition 2.7: Let $P$ be a projective orbifold, and $L$ an ample line bundle
on $P$. Assume that the total space $\operatorname{Tot}^{\circ}(L)$ of all
non-zero vectors in $L$ is smooth. An open algebraic cone is
$\operatorname{Tot}^{\circ}(L)$. The corresponding closed algebraic cone is
its weak Stein completion $Z$, such that the natural ${\mathbb{C}}^{*}$-action
can be extended to $Z$ holomorphically. By 4, the closed algebraic cone is
obtained by adding one point, called “the apex”, or “the origin”.

Further on, we shall rely on the notion of “holomorphic contraction”. To
simplify the terminology, we tacitly assume that the “contractions” are
invertible; however, most of the arguments remain valid without this
assumption.

Definition 2.8: A contraction of a topological space $X$ to a point $x\in X$
is a homeomorphism $\varphi:\;X{\>\longrightarrow\>}X$ such that for any
compact subset $K\subset X$ and any open set $U\ni x$, there exists $N>0$ such
that for all $n>N$, the map $\varphi^{n}$ maps $K$ to $U$.

An algebraic structure on a complex analytic variety $Z$ is a subsheaf of the
sheaf of holomorphic functions which can be realized as a sheaf of regular
functions for some biholomorphism between $Z$ and a quasi-projective variety.

One of the main results of this paper is the following theorem, proven in
Section 6 below. Note that any closed algebraic cone $X$ admits an algebraic
contraction $\gamma:\;X{\>\longrightarrow\>}X$ taking $v$ to $\lambda v$, for
$\lambda$ an invertible complex number, $|\lambda|<1$.

Theorem 2.9:  Let $X$ be a closed algebraic cone, and ${\cal A}_{1}$, ${\cal
A}_{2}$ two algebraic structures on $X$. Assume that the algebraic varieties
$(X,{\cal A}_{1})$ and $(X,{\cal A}_{2})$ are affine and admit (invertible)
algebraic contractions $\gamma_{1}$ and $\gamma_{2}$, that is, holomorphic
contractions compatible with the respective algebraic structures. Then ${\cal
A}_{1}={\cal A}_{2}$.

Proof: By 2.3 below, $X$ can be obtained in two different ways as a weak Stein
completion of ${\mathbb{Z}}$-coverings of LCK manifolds with potential
associated with the holomorphic contractions $\gamma_{1}$ and $\gamma_{2}$.
Then ${\cal A}_{1}={\cal A}_{2}$ by 6.3.

###  2.3 Algebraic cones and subvarieties in Hopf manifolds

The algebraic cones can be defined in terms of projective orbifolds as above;
there are two more definitions equivalent to this one. We could define open
algebraic cones as ${\mathbb{Z}}$-coverings of submanifolds of a linear Hopf
manifold (which is done in the present section) or define the closed algebraic
cones as Stein varieties with an isolated singularity admitting a holomorphic
contraction, as in Subsection 2.4 below.

Definition 2.10:  A linear Hopf manifold is a complex manifold
$H:=\frac{{\mathbb{C}}^{n}\backslash 0}{\langle A\rangle}$, where
$A\in{\sf{GL}}(n,{\mathbb{C}})$ is an invertible linear contraction. A
diagonal Hopf manifold is a linear Hopf manifold such that $A$ is
diagonalizable, and a classical Hopf manifold is a linear Hopf manifold such
that $A$ is a scalar matrix.

Theorem 2.11:  Let $M\subset H$ be a submanifold in a Hopf manifold, and
$\tilde{M}\subset{\mathbb{C}}^{n}\backslash 0$ its ${\mathbb{Z}}$-covering.
Then $\tilde{M}$ is an open algebraic cone. Moreover, any open algebraic cone
can be obtained this way.

Proof: Let $\tilde{M}$ be an open algebraic cone equipped with the standard
${\mathbb{C}}^{*}$-action, and $\gamma\in\operatorname{Aut}(\tilde{M})$ the
automorphism associated with $\lambda\in{\mathbb{C}}^{*}$, $|\lambda|<1$. By
7.4, $\frac{\tilde{M}}{\langle\gamma\rangle}$ admits an embedding to a linear
Hopf manifold. To obtain the converse assertion, we use the locally
conformally Kähler metrics with potential, defined below (3). As shown in
[OV3], any submanifold of a linear Hopf manifold is LCK with (proper)
potential. 7.3 implies that a Kähler ${\mathbb{Z}}$-cover of an LCK manifold
with proper potential is an open algebraic cone.

###  2.4 Algebraic cones and holomorphic contractions

Another equivalent definition of algebraic cones uses holomorphic
contractions. Note that this definition does not refer to algebraic
structures, however, an algebraic structure compatible with a contraction is
unique by 2.2.

Theorem 2.12: Let $X$ be a Stein variety with a unique isolated singularity,
admitting a holomorphic contraction. Then $X$ is biholomorphic to a closed
algebraic cone. Conversely, any closed algebraic cone can be obtained this
way.

Proof: By the same argument as used in the proof of 4, a closed algebraic cone
$X$ can be obtained as the closure of the preimage $\pi^{-1}(P)$, where
$P\subset{\mathbb{C}}P^{n}$ is a projective orbifold and
$\pi:\;{\mathbb{C}}^{n+1}\backslash 0{\>\longrightarrow\>}{\mathbb{C}}P^{n}$
the standard projection. Then $X$ admits a contraction
$v{\>\longrightarrow\>}\lambda v$, where $|\lambda|<1$.

Conversely, suppose that $X$ is a Stein variety with a unique isolated
singularity $x$ admitting a holomorphic contraction $\gamma$ to $x$. Then
$\frac{X\backslash\\{x\\}}{\langle\gamma\rangle}$ is a complex manifold
admitting a holomorphic embedding to a Hopf manifold (7.4), hence it is an LCK
manifold with proper potential (3). By 7.3, $X\backslash\\{x\\}$ is an open
algebraic cone.

##  3 Locally conformally Kähler manifolds

The main motivation for this paper comes from the theory of LCK (locally
conformally Kähler) manifolds, though we could state and prove everything in
it without using this important notion.

Definition 3.1: A complex manifold $(M,I)$ is called locally conformally
Kähler (LCK, for short) if it admits a covering $(\tilde{M},I)$ equipped with
a Kähler metric $\tilde{\omega}$ such that the deck group of the cover acts on
$(\tilde{M},\tilde{\omega})$ by holomorphic homotheties. An LCK metric on an
LCK manifold is an Hermitian metric on $(M,I)$ such that its pullback to
$\tilde{M}$ is conformal with $\tilde{\omega}$.

Remark 3.2: Equivalently, a Hermitian manifold $(M,I,g)$ is LCK if there
exists a closed 1-form $\theta$ (called the Lee form) such that the
fundamental form $\omega(x,y):=g(Ix,y)$ satisfies
$d\omega=\theta\wedge\omega$.

The special class of LCK manifolds we are concerned with is the following:

Definition 3.3:  An LCK manifold has a proper LCK potential if it admits a
Kähler ${\mathbb{Z}}$-covering on which the Kähler metric has a global and
positive potential function $\psi$ such that the deck group multiplies $\psi$
by a constant.555Such a function is called automorphic. In this case, $M$ is
called an LCK manifold with proper potential.

In the sequel, we will often tacitly omit the word “proper”.

Example 3.4:  The linear Hopf manifolds are LCK with potential, [OV6]. For a
classical Hopf manifold $H:=({\mathbb{C}}^{n}\setminus 0)/\langle A\rangle$,
$A=\lambda\operatorname{Id}$, $|\lambda|>1$, the flat Kähler metric
$\tilde{g}_{0}=\sum dz_{i}\otimes d\overline{z}_{i}$ on ${\mathbb{C}}^{n}$ is
multiplied by $\lambda^{2}$ by the deck group ${\mathbb{Z}}$. Also,
$\tilde{g}_{0}$ has the global automorphic potential $\psi:=\sum|z_{i}|^{2}$.

The main properties of LCK manifolds with (proper) potential are the
following.

Theorem 3.5:  ([OV2, OV3]) Let $M$ be a compact LCK manifold with proper
potential, and $\tilde{M}$ its Kähler ${\mathbb{Z}}$-cover. If
$\operatorname{\sf dim}_{\mathbb{C}}M\geqslant 3$, then the metric completion
$\tilde{M}_{c}$ is identified with the Stein completion of $\tilde{M}$, the
complement $\tilde{M}_{c}\setminus\tilde{M}$ is just one point. Moreover,
$\tilde{M}_{c}$ is an affine algebraic variety obtained as an affine cone over
a projective orbifold.

This is used to prove the following result (for an alternative, more
algebraic, proof, see 4).

Proposition 3.6:  Let $P$ be a projective orbifold, $\operatorname{\sf
dim}_{\mathbb{C}}P>1$, and $L$ an ample line bundle on $P$. Assume that the
total space $\operatorname{Tot}^{\circ}(L)$ of all non-zero vectors in $L$ is
smooth. Then a weak Stein completion of $\operatorname{Tot}^{\circ}(L)$ is
obtained by adding a point (called “the origin” or “the apex” elsewhere).

Proof. Step 1: Let $P$ be a principal ${\mathbb{C}}^{*}$-bundle over $X$, with
${\mathbb{C}}^{*}$-action denoted by $\rho_{P}(t):\;P{\>\longrightarrow\>}P$.
Consider the dual ${\mathbb{C}}^{*}$-bundle $P^{*}$, which has the same
underlying complex manifold but the group ${\mathbb{C}}^{*}$ acts on $P^{*}$
via $\rho_{P^{*}}(t)=\rho_{P}(t^{-1})$. There is a natural duality between $P$
and $P^{*}$ mapping $P\times_{X}P^{*}$ to a trivial ${\mathbb{C}}^{*}$-bundle
${\cal O}^{*}(M)$, taking a pair $(p,p)$ of sections of $P$ and $P^{*}$,
identified with $P$, to the unit section of ${\cal O}^{*}(X)$.

Let $\operatorname{Tot}^{\circ}(L)$ be the set of non-zero vectors in an ample
line bundle $L$ over a projective orbifold. Clearly, its dual bundle can be
identified with $\operatorname{Tot}^{\circ}(L^{*})$, hence the spaces
$\operatorname{Tot}^{\circ}(L)$ and $\operatorname{Tot}^{\circ}(L^{*})$ are
biholomorphic.

Step 2: Replacing $L$ by $L^{*}$, we can assume that $L$ is anti-ample and it
is equipped with a metric with negative curvature. Then the function
$v{\>\longrightarrow\>}|v|^{2}$ is strictly plurisubharmonic on
$\operatorname{Tot}^{\circ}(L)$, by [Ca, (2.6)] or [Be, (15.19)], and defines
an LCK potential on the compact manifold
$M:=\frac{\operatorname{Tot}^{\circ}(L)}{\langle\gamma\rangle}$, where
$\gamma(x)=\lambda x$, $|\lambda|>1$. Applying 3 to
$\operatorname{Tot}^{\circ}(L)$, considered as a ${\mathbb{Z}}$-covering of
$M$, we obtain that the Stein completion of $\operatorname{Tot}^{\circ}(L)$ is
obtained by adding a point in the origin. By 2.1, $\tilde{M}_{c}$ is the
normalization of any weak Stein completion $Z$ of $\tilde{M}$. However, the
normalization map $\nu:\;\tilde{M}_{c}{\>\longrightarrow\>}Z$ is continous,
and identity outside of the origin, because $Z$ is smooth outside of the
origin. Therefore, $\nu$ is a diffeomorphism.

We give a more algebraic proof of 3, independent from the LCK geometry, in
Section 4. Note that the argument in Section 4 is stronger, because it works
even when $\operatorname{\sf dim}_{\mathbb{C}}P=1$.

3 was used to prove the following Kodaira type embedding theorem for LCK
manifolds with potential. Recall that a linear Hopf manifold is a manifold
$H:=\frac{{\mathbb{C}}^{n}\backslash 0}{\langle A\rangle}$, where
$A\in{\sf{GL}}({\mathbb{C}}^{n})$ is a linear contraction. In [OV3] it was
shown that all linear Hopf manifolds admit an LCK metric with potential.

Theorem 3.7: ([OV2, OV4]) Let $(M,I,\omega)$ be a compact LCK manifold with
potential, $\operatorname{\sf dim}_{\mathbb{C}}M\geqslant 3$. Then $(M,I)$
admits a holomorphic embedding to a linear Hopf manifold.

Remark 3.8: The converse assertion is also clearly true: a submanifold of a
linear Hopf manifold is LCK with potential ([OV3]). Therefore, in dimension
$\geqslant 3$, an LCK manifold with (proper) potential can be defined as a
manifold which admits a holomorphic embedding into a linear Hopf manifold. In
dimension 2, this is also true, if we assume the Kato conjecture (also known
as “the global spherical shell conjecture”); see [OV4, Section 5] for details.

##  4 The total space of an ample bundle, Remmert-Stein theorem and weak Stein
completions

In this section, we give an alternative proof of 3, independent from the LCK
geometry. We start by recalling the Remmert-Stein theorem.

Theorem 4.1:  (Remmert-Stein theorem)
Let $B$ be a complex analytic variety, and $C\subset B$ and $A\subset
B\backslash C$ complex analytic subvarieties. Assume that all irreducible
components $A_{i}$ of $A$ satisfy $\operatorname{\sf
dim}A_{i}>\operatorname{\sf dim}C$. Then the closure of $A$ is complex
analytic in $B$.

Proof: [De, §II.8.2].

Remark 4.2:  We will use 4 in one special case, which is also the case used in
the proof of Chow theorem ([De, Theorem II.8.10]). Suppose that
$A\subset{\mathbb{C}}^{n}\backslash 0$ is a complex analytic subset without
0-dimensional connected components; then its closure $\overline{A}$ in
${\mathbb{C}}^{n}$ is complex analytic. A posteriori, $\overline{A}$ is
isomorphic to a weak Stein completion of $A$, 2.1.

At this point we can give an alternative proof of 3.

Proposition 4.3:  Let $P$ be a projective orbifold, and $L$ an ample line
bundle on $P$. Assume that the total space $\operatorname{Tot}^{\circ}(L)$ of
all non-zero vectors in $L$ is smooth. Then a weak Stein completion of
$\operatorname{Tot}^{\circ}(L)$ is obtained by adding a point (called “the
origin” or “the apex” elsewhere). Moreover, the natural normalization map from
the Stein completion to a weak Stein completion is a homeomorphism.

Proof: The space $\operatorname{Tot}^{\circ}(L)$ can be interpreted
algebraically as follows. Consider the standard ${\mathbb{C}}^{*}$-action on
$\operatorname{Tot}^{\circ}(L)$ understood as a principal
${\mathbb{C}}^{*}$-bundle. A finite-dimensional
${\mathbb{C}}^{*}$-representation is a direct sum of 1-dimensional irreducible
representations, with ${\mathbb{C}}^{*}$ acting by $\rho(t)=t^{w}$; the number
$w$ is called the weight of the representation. Therefore, the category of
${\mathbb{C}}^{*}$-representations is equivalent to the category of graded
vector spaces. A ring with ${\mathbb{C}}^{*}$-action is the same as a graded
ring. In particular, the ring of fiberwise polynomial holomorphic functions on
$\operatorname{Tot}^{\circ}(L)$ is graded, with the functions of weight $w$
being polynomials of degree $w$ on each ${\mathbb{C}}^{*}$-orbit. We identify
the space $({\cal O}_{\operatorname{Tot}^{\circ}(L)})_{w}=({\cal
O}_{\operatorname{Tot}^{\circ}(L^{*})})_{w}$ of functions of degree $w$ with
the space of sections of $L^{\otimes w}$. The variety
$P\subset{\mathbb{P}}(H^{0}(L^{N})^{*})$ is identified with the graded
spectrum of the graded ring $\bigoplus_{w}H^{0}(X,L^{\otimes wN})$, whenever
$L^{N}$ is very ample. Then, the manifold $\operatorname{Tot}^{\circ}(L)$ is
the preimage of $P$ under the natural projection $H^{0}(L^{N})^{*}\backslash
0{\>\longrightarrow\>}{\mathbb{P}}(H^{0}(L^{N})^{*})$. The Stein completion of
$\operatorname{Tot}^{\circ}(L)$ is its closure in $H^{0}(L^{N})^{*}$, obtained
by adding the zero (called “the origin” or “the apex” elsewhere). It is
complex analytic by Remmert-Stein theorem (4), and normal by 8 below.

By 2.1, the Stein completion of $\operatorname{Tot}^{\circ}(L)$ is the
normalization of any weak Stein completion $Z$ of $\tilde{M}$. Since the
normalization map $\nu:\;\operatorname{Tot}^{\circ}(L){\>\longrightarrow\>}Z$
is continous, and identity outside of the origin, $\nu$ it is a homeomorphism.

##  5 Montel theorem and Riesz-Schauder theorem

In this section, we recall several basic notions of complex analysis and
functional analysis.

###  5.1 Montel theorem

Definition 5.1:  Let $M$ be a complex manifold, and ${\cal F}\subset
H^{0}({\cal O}_{M})$ a family of holomorphic functions. We call ${\cal F}$ a
normal family if for each compact $K\subset M$ there exists $C_{K}>0$ such
that for each $f\in{\cal F}$, ${\operatorname{{\sf sup}}}_{K}|f|\leqslant
C_{K}$.

Definition 5.2: The $C^{0}$-topology on the space of functions on $M$ is the
topology of uniform convergence on compacts.

Theorem 5.3:  (Montel)
Let ${\cal F}\subset H^{0}({\cal O}_{M})$ be a normal family of functions, and
$\overline{\cal F}\subset H^{0}({\cal O}_{M})$ its closure in the
$C^{0}$-topology. Then $\overline{\cal F}$ is compact in $C^{0}$-topology.

Proof: [Wu, Lemma 1.4].

###  5.2 The Banach space of holomorphic functions

We briefly recall some notions of functional analysis, which are standard.

Remark 5.4: Let $C^{0}(X)$ be the space of continuous functions on a compact
space, with the norm $\|f\|_{\operatorname{{\sf sup}}}:={\operatorname{{\sf
sup}}}_{x\in X}|f(x)|$. It is not hard to see that this space is Banach.
Similarly, if $X$ is a compact smooth manifold equipped with a connection
$\nabla$, the space $C^{k}(M)$ of $k$ times differentiable functions with the
norm $\|f\|_{C^{k}}:=\sum_{i=0}^{k}\|\nabla^{i}(f)\|_{\operatorname{{\sf
sup}}}$ is also Banach.

Theorem 5.5:  Let $M$ be a complex manifold, and let $H^{0}_{b}({\cal O}_{M})$
the space of all bounded holomorphic functions, equipped with the sup-norm
$|f|_{{\operatorname{{\sf sup}}}}:={\operatorname{{\sf sup}}}_{M}|f|$. Then
$H^{0}_{b}({\cal O}_{M})$ is a Banach space.

Proof: Let $\\{f_{i}\\}\in H^{0}_{b}({\cal O}_{M})$ be a Cauchy sequence in
the ${\operatorname{{\sf sup}}}$-norm. Then $\\{f_{i}\\}$ converges to a
continuous function $f$ in the ${\operatorname{{\sf sup}}}$-topology.

Since $\\{f_{i}\\}$ is a normal family (see 5.1), it has a subsequence which
converges in $C^{0}$-topology to $\tilde{f}\in H^{0}({\cal O}_{M})$, by
Montel’s 5.1. However, the $C^{0}$-topology is weaker than the
${\operatorname{{\sf sup}}}$-topology, hence $\tilde{f}=f$. Therefore, $f$ is
holomorphic.

####  5.2.1 Compact operators

Recall that a subset $X$ of a topological space $Y$ is called precompact, or
relatively compact in $Y$, if its closure is compact.

Definition 5.6:  A subset $K\subset V$ of a topological vector space is called
bounded if for any open set $U\ni 0$, there exists a number
$\lambda_{U}\in{\mathbb{R}}^{>0}$ such that $\lambda_{U}K\subset U$.

Definition 5.7:  Let $V,W$ be topological vector spaces. A continuous operator
$\varphi:\;V{\>\longrightarrow\>}W$ is called compact if the image of any
bounded set is precompact.

####  5.2.2 Holomorphic contractions

Theorem 5.8:  Let $X$ be a complex variety, and
$\gamma:\;X{\>\longrightarrow\>}X$ a holomorphic contraction to $x\in X$ such
that $\gamma(X)$ is precompact. Consider the Banach space $V=H^{0}_{b}({\cal
O}_{X})$ of bounded holomorphic functions with the sup-norm, and let
$V_{x}\subset V$ be the space of all $v\in V$ vanishing in $x$. Then
$\gamma^{*}:\;V{\>\longrightarrow\>}V$ is compact, and the eigenvalues of its
restriction to $V_{x}$ are strictly less than 1 in absolute value.666Since
$\gamma^{*}$ maps constants to constants identically, one cannot expect that
$\|\gamma^{*}\|<1$ on $V$. However, if we add a condition which excludes
constants, such as $v(x)=0$, we immediately obtain $\|\gamma^{*}\|<1$.

Proof. Step 1: For any $f\in H^{0}({\cal O}_{X})$ we have

$|\gamma^{*}f|_{{\operatorname{{\sf sup}}}}={\operatorname{{\sf
sup}}}_{x\in\overline{\gamma(X)}}|f(x)|.$

This implies that $\gamma^{*}(f)$ is bounded. Therefore, for any sequence
$\\{f_{i}\in H^{0}({\cal O}_{X})\\}$ converging in the $C^{0}$-topology, the
sequence $\\{\gamma^{*}f_{i}\\}$ converges in the ${\operatorname{{\sf
sup}}}$-topology. The set $B_{C}:=\\{v\in V\ \ |\ \ |v|_{{\operatorname{{\sf
sup}}}}\leqslant C\\}$ is precompact in the $C^{0}$-topology, because it is a
normal family. Then $\gamma^{*}B_{C}$ is precompact in the
${\operatorname{{\sf sup}}}$-topology.777The $C^{0}$-convergence for
holomorphic functions is strictly weaker that the ${\operatorname{{\sf
sup}}}$-convergence; for example, the sequence of bounded holomorphic
functions on an open disk, $f_{i}:=z^{i}$ converges in $C^{0}$-topology, but
does not converge in the ${\operatorname{{\sf sup}}}$-topology. This proves
that the operator $\gamma^{*}:\;V{\>\longrightarrow\>}V$ is compact.

It remains to show that its operator norm is $<1$ on $V_{x}$.

Step 2: Since ${\operatorname{{\sf sup}}}_{X}|\gamma^{*}f|={\operatorname{{\sf
sup}}}_{\gamma(X)}|f|\leqslant{\operatorname{{\sf sup}}}_{X}|f|$, one has
$\|\gamma^{*}\|\leqslant 1$. If this inequality is not strict, for some
sequence $\\{f_{i}\\}$ of holomorphic functions $f_{i}\in V_{x}$ with
${\operatorname{{\sf sup}}}_{X}|f_{i}|\leqslant 1$ (that is, $f_{1}\in B_{1}$)
one has $\lim_{i}{\operatorname{{\sf sup}}}_{x\in\gamma(X)}|f_{i}(x)|=1$.
Since $B_{1}$ is a normal family, $f_{i}$ has a subsequence converging in
$C^{0}$-topology to $f$. Then $\\{\gamma(f_{i})\\}$ converges to $\gamma(f)$
in ${\operatorname{{\sf sup}}}$-topology, giving

$\lim_{i}{\operatorname{{\sf
sup}}}_{x\in\gamma(X)}|f_{i}(x)|={\operatorname{{\sf
sup}}}_{x\in\gamma(X)}|f(x)|=1.$

Since $f(x)=0$, $f$ is non-constant. By the maximum modulus principle, a non-
constant holomorphic function has no local maxima; this means that $|f(x)|>1$
somewhere on $X$. Then $f$ cannot be the $C^{0}$-limit of $\\{f_{i}\\}\subset
B_{1}$.

####  5.2.3 The Riesz-Schauder theorem

The following result is a Banach analogue of the usual spectral theorem for
compact operators on Hilbert spaces.

Theorem 5.9: (Riesz-Schauder, [Fr, Section 5.2])
Let $F:\;V{\>\longrightarrow\>}V$ be a compact operator on a Banach space.
Then for each non-zero $\mu\in{\mathbb{C}}$, there exists a sufficiently big
number $N\in{\mathbb{N}}$ such that for each $n>N$ one has
$V=\ker(F-\mu\operatorname{Id})^{n}\oplus\overline{\operatorname{im}(F-\mu\operatorname{Id})^{n}}$,
where $\overline{\operatorname{im}(F-\mu\operatorname{Id})^{n}}$ is the
closure of the image. Moreover, $\ker(F-\mu\operatorname{Id})^{n}$ is finite-
dimensional and independent on $n$.

Remark 5.10:  Define the root space of an operator
$F\in\operatorname{End}(V)$, associated with an eigenvalue $\mu$, as
$\bigcup_{n\in{\mathbb{Z}}}\ker(F-\mu\operatorname{Id})^{n}$. In the finite-
dimensional case, the root spaces coincide with the Jordan cells of the
corresponding matrix. Then 5.2.3 can be reformulated by saying that any
compact operator $F\in\operatorname{End}(V)$ admits a Jordan cell
decomposition, with $V$ identified with a completed direct sum of the root
spaces, which are all finite-dimensional; moreover, the eigenvalues of $F$
converge to zero.

We need the following corollary of the Riesz-Schauder theorem, which is
obtained using the same arguments as in the finite-dimensional case.

Theorem 5.11:  Let $F:\;V{\>\longrightarrow\>}V$ be a compact operator on a
Banach space. We say that $v$ is a root vector for $F$ if $v$ lies in a root
space of $F$, for some eigenvalue $\mu\in{\mathbb{C}}$. Then the space
generated by the root vectors is dense in $V$.

Definition 5.12:  Let $F\in\operatorname{End}(V)$ be an endomorphism of a
vector space. A vector $v\in V$ is called $F$-finite if the space generated by
$v,F(v),F(F(v)),...$ is finite-dimensional.

Remark 5.13: Clearly, a vector is $F$-finite if and only if it is obtained as
a combination of root vectors. Then 5.2.3 implies the following.

Corollary 5.14:  Let $F:\;V{\>\longrightarrow\>}V$ be a compact operator on a
Banach space, and $V_{0}\subset V$ the space of all $F$-finite vectors. Then
$V_{0}$ is dense in $V$.

We also need the following lemma, giving “a relative case” of Riesz-Schauder.

Lemma 5.15:  Consider the commutative square of continuous operators of Banach
spaces

$\begin{CD}W_{1}@>{K_{1}}>{}>W_{1}\\\ @V{R}V{}V@V{}V{R}V\\\
W_{2}@>{K_{2}}>{}>W_{2}\end{CD}$

with $R$ a surjective map, and both $K_{i}$, $i=1,2$, are compact. Restricting
$R$ to the space $W_{i}^{K_{i}}$ of $K_{i}$-finite vectors, we obtain a map
$R_{f}:\;W_{1}^{K_{1}}{\>\longrightarrow\>}W_{2}^{K_{2}}$. Then $R_{f}$ is
surjective.

Proof: Let $U_{2}\subset W_{2}$ be a finite-dimensional space of
$K_{2}$-finite vectors, and $U_{1}\subset W_{1}$ its preimage. Since $R$ is
surjective, the natural map $U_{1}\to U_{2}$ is surjective. Since
$U_{1}^{K_{1}}$ is dense in $U_{1}$, and $U_{2}$ is finite-dimensional, the
restriction of $R$ to $U_{1}^{K_{1}}$ is also surjective.

##  6 Algebraic structures on weak Stein completions

In this section, we put an algebraic structure on a weak Stein completion of a
Kähler ${\mathbb{Z}}$-covering of a submanifold in a Hopf manifold. We also
prove that this algebraic structure is unique. Later on, this algebraic
structure is used to show that this weak Stein completion is an algebraic
cone.

###  6.1 Ideals of the embedding to a Hopf manifold

Let $M\subset H$ be a subvariety of a linear Hopf manifold
$H=\frac{{\mathbb{C}}^{n}\backslash 0}{\langle A\rangle}$. In this subsection
we prove that the preimage of $M$ in ${\mathbb{C}}^{n}\backslash 0$ can be
defined by a set of polynomial equations.

We start with the following lemma.

Lemma 6.1:  Let $\gamma$ be an invertible linear contraction of
${\mathbb{C}}^{n}$. A holomorphic function on ${\mathbb{C}}^{n}$ is
$\gamma$-finite if and only if it is polynomial.

Proof: Clearly, a polynomial function is $\gamma$-finite. The operator
$\gamma^{*}$ acts on homogeneous polynomials of degree $d$ with eigenvalues
$\alpha_{i_{1}}\alpha_{i_{2}}...\alpha_{i_{d}}$, where $\alpha_{i_{j}}$ are
the eigenvalues of $\gamma$ on ${\mathbb{C}}^{n}$. Since $\gamma$ is a
contraction, all $\alpha_{i_{j}}$ satisfy $|\alpha_{i_{j}}|<1$. Therefore, any
sequence $\\{\alpha_{i_{1}}\alpha_{i_{2}}...\alpha_{i_{d}}\\}$ converges to 0
as $d$ goes to infinity. We obtain that every given number can be realized as
an eigenvalue of $\gamma^{*}$ on homogeneous polynomials of degree $d$ for
finitely many choices of $d$ only. Therefore, any root vector of $\gamma^{*}$
is a finite sum of homogeneous polynomials. This implies that the Taylor
decomposition of a $\gamma$-finite function $f$ can have only finitely many
components, otherwise the eigenspace decomposition of $f$ with respect to the
action of $\gamma^{*}$ is infinite.

This lemma can be used to prove the following theorem (see also 6.2 below).

Theorem 6.2:  ([OV3, Theorem 2.8], [OV5, Theorem 5.5])
Let $M\hookrightarrow H$ be a complex subvariety of a linear Hopf manifold,
and $\tilde{M}_{c}{\>\longrightarrow\>}{\mathbb{C}}^{N}$ the corresponding map
of weak Stein completions, with $\tilde{M}_{c}$ obtained as the closure of
$\tilde{M}\subset{\mathbb{C}}^{N}$ by adding the zero (4). Then
$\tilde{M}_{c}$ is an algebraic subvariety, that is, a set of common zeroes of
a system of polynomial equations.

###  6.2 Algebraic structures on Stein completions: the existence

Let $X\subset M\subset{\mathbb{C}}P^{n}$ be projective subvarieties of
${\mathbb{C}}P^{n}$. Then $Z=M\backslash X$ is called a quasi-projective
variety. The Zariski topology on $Z$ is the topology such that the closed
subsets of $Z$ are closed algebraic subvarieties. The regular functions on a
Zariski open subset $W\subset X$ are restrictions of the rational functions on
${\mathbb{C}}P^{n}$ which have no poles on $W$. This defines the sheaf of
regular functions on $Z$ as a subsheaf of the sheaf of holomorphic functions
in the Zariski topology on $Z$.

Recall that an algebraic structure on a complex analytic variety $Z$ is a
subsheaf of the sheaf of holomorphic functions which can be realized as a
sheaf of regular functions for some biholomorphism between $Z$ and a quasi-
projective variety.

Note that the algebraic structure is not unique, indeed, even the complex
manifold $({\mathbb{C}}^{*})^{2}$ has more than one algebraic structure
([Si]). In [Je], Z. Jelonek has constructed an uncountable set of pairwise
non-isomorphic algebraic structures on certain Stein manifolds.

It turns out that a Kähler ${\mathbb{Z}}$-cover of an LCK manifold with
potential is equipped with a distinguished (affine) algebraic structure, which
is uniquely determined by the complex structure (6.2, 6.3).

The main result of this section is the following theorem.

Theorem 6.3:  Let $M$ be an LCK manifold with proper potential, and
$\tilde{M}_{c}$ the Stein variety obtained as a weak Stein completion of its
Kähler ${\mathbb{Z}}$-cover $\tilde{M}$. Denote by
$\gamma\in\operatorname{Aut}(\tilde{M}_{c})$ the generator of the
${\mathbb{Z}}$-action on $\tilde{M}$ extended to $\tilde{M}_{c}$. This map is
an invertible holomorphic contraction centered in the apex $c$ of
$\tilde{M}_{c}$. Then $\tilde{M}_{c}$ admits an algebraic structure such that
the regular functions are precisely the $\gamma$-finite functions.

Proof. Step 1: 6.1 implies 6.2 when $\gamma$ is a linear contraction and
$\tilde{M}_{c}={\mathbb{C}}^{n}$.

Step 2: By definition, $M$ admits a holomorphic embedding to a linear Hopf
manifold. Consider the corresponding embedding
$\tilde{M}_{c}\hookrightarrow{\mathbb{C}}^{n}$. By 6.1, $\tilde{M}_{c}$ is an
algebraic subvariety of ${\mathbb{C}}^{n}$ given by a finite set of polynomial
equations.

To finish the proof of 6.2, it remains to show that a holomorphic function on
$\tilde{M}_{c}$ is polynomial if and only if it is $\gamma$-finite. It is
clear that any holomorphic function on $\tilde{M}_{c}$ obtained from a
polynomial function on ${\mathbb{C}}^{n}$ is $\gamma$-finite. It remains to
show the converse. Consider a $\gamma$-finite function on $\tilde{M}_{c}$. By
6.1, to prove that it is polynomial, it would suffice to prove that it can be
extended to a $\gamma$-finite function on ${\mathbb{C}}^{n}$.

Consider the exact sequence of sheaves of holomorphic functions on
${\mathbb{C}}^{n}$

$0{\>\longrightarrow\>}I_{\tilde{M}_{c}}{\>\longrightarrow\>}{\cal
O}_{{\mathbb{C}}^{n}}{\>\longrightarrow\>}{\cal
O}_{\tilde{M}_{c}}{\>\longrightarrow\>}0,$

where $I_{\tilde{M}_{c}}$ is the ideal sheaf of $\tilde{M}_{c}$. The
corresponding long exact sequence gives

$0{\>\longrightarrow\>}H^{0}({\mathbb{C}}^{n},I_{\tilde{M}_{c}}){\>\longrightarrow\>}H^{0}({\mathbb{C}}^{n},{\cal
O}_{{\mathbb{C}}^{n}}){\>\longrightarrow\>}H^{0}({\mathbb{C}}^{n},{\cal
O}_{\tilde{M}_{c}}){\>\longrightarrow\>}H^{1}({\mathbb{C}}^{n},I_{\tilde{M}_{c}}).$

Since ${\mathbb{C}}^{n}$ is Stein,
$H^{1}({\mathbb{C}}^{n},I_{\tilde{M}_{c}})=0$, hence the function $f$ is the
restriction of a holomorphic function $\tilde{f}\in
H^{0}({\mathbb{C}}^{n},{\cal O}_{{\mathbb{C}}^{n}})$. Finally, by 5.2.3,
$\tilde{f}$ can be chosen $\gamma$-finite.

###  6.3 Algebraic structures on Stein completions: the uniqueness

In this subsection, we prove that the algebraic structure constructed on
$\tilde{M}_{c}$ in 6.2 is uniquely determined by the complex geometry of the
Stein variety $\tilde{M}_{c}$.

We start with the following lemma, showing that a function is $\gamma$-finite
(that is, polynomial) if and only if it admits a certain growth condition.
This is similar to a well-known result of complex analysis which states that
an entire holomorphic function on ${\mathbb{C}}^{n}$ is polynomial if and only
if it has polynomial growth.

Lemma 6.4:  Let $M$ be an LCK manifold with proper potential, and
$\tilde{M}_{c}$ the Stein variety, obtained as a weak Stein completion of its
Kähler ${\mathbb{Z}}$-cover $\tilde{M}$. Denote by
$\gamma\in\operatorname{Aut}(\tilde{M}_{c})$ the holomorphic contraction
generating the ${\mathbb{Z}}$-action on $\tilde{M}_{c}$. Choose a compact set
$K\subset\tilde{M}_{c}$ containing an open neighbourhood of the apex. Denote
by ${\cal B}\subset H^{0}({\cal O}_{\tilde{M}_{c}})$ the following ring of
functions on $\tilde{M}_{c}$:

${\cal B}:=\\{f\in H^{0}({\cal O}_{\tilde{M}_{c}})\ |\ \exists\;C>0\text{\
such that\ }\forall i\ \ {\operatorname{{\sf
sup}}}_{K}|(\gamma^{*})^{-i}|f|<C^{i}\\}.$ (6.1)

Then ${\cal B}$ coincides with the space of $\gamma$-finite functions.888We
call the function satisfying (6.1) the function of polynomial growth. This
terminology is justified because for $\gamma$ a linear contraction of
${\mathbb{C}}^{n}$, (6.1) is equivalent to having polynomial growth.

Proof. Step 1: Let $f$ be a $\gamma$-finite function, and $W$ the space
generated by $\\{f,(\gamma^{*})f,(\gamma^{*})^{2}f,...\\}$. Let
$\|\cdot\|_{K}$ be the norm on $W$ defined by $\|f\|:={\operatorname{{\sf
sup}}}_{K}|f|$, and let $C:={\operatorname{{\sf
sup}}}_{\|f\|=1}\|(\gamma^{*})^{-1}f\|$ be the operator norm of the operator
$(\gamma^{*})^{-1}\in\operatorname{End}(W)$ in this norm. Then
${\operatorname{{\sf sup}}}_{K}|(\gamma^{*})^{-i}f|\leqslant
C^{i}{\operatorname{{\sf sup}}}_{K}|f|$, and $f$ has polynomial growth, in the
sense of (6.1).

Step 2: Suppose that $f$ has polynomial growth, in the sense of (6.1), and let
$W$ be the space of all functions generated by
$\\{f,(\gamma^{*})f,(\gamma^{*})^{2}f,...\\}$. Then all elements of $W$ have
the same growth as $f$, with the same bound $C$, hence the closure
$\overline{W}$ of $W$ in the norm $\|f\|:={\operatorname{{\sf sup}}}_{K}|f|$
consists of functions with polynomial growth.

The condition (6.1) holds for all $f\in W$ if and only if $(\gamma^{*})^{-1}$
has finite norm on $W$. Therefore,
$\gamma^{*}{\left|{}_{{\phantom{|}\\!\\!}_{\overline{W}}}\right.}$ is
invertible. To finish the proof of 6.3, we need to prove that the norm of
$(\gamma^{*})^{-1}$ is infinite on $\overline{W}$ if $\overline{W}$ is
infinite-dimensional.

The operator $\gamma^{*}$ on $\overline{W}$ is compact; by the Riesz-Schauder
theorem, it has the Jordan cell decomposition with eigenvalues converging to
0, unless $\overline{W}$ is finite-dimensional. The norm of a linear operator
$A$ with eigenvalues $\alpha_{i}$ satisfies $\|A\|\geqslant{\operatorname{{\sf
sup}}}|\alpha_{i}|$. Therefore, a compact operator cannot be invertible on an
infinitely-dimensional Banach space: the inverse operator would have infinite
norm.

Theorem 6.5:  Let $\tilde{M}_{c}$ be a weak Stein completion of a Kähler
${\mathbb{Z}}$-covering of an LCK manifold with potential. Suppose that
$\tilde{M}_{c}$ can be obtained from two different LCK manifolds, and let
$\gamma_{1},\gamma_{2}\in\operatorname{Aut}(\tilde{M}_{c})$ be the
corresponding holomorphic contractions. Consider the affine algebraic
structures ${\cal A}_{1}$ and ${\cal A}_{2}$ associated with the LCK
structures on $\frac{\tilde{M}}{\langle\gamma_{1}\rangle}$ and
$\frac{\tilde{M}}{\langle\gamma_{2}\rangle}$ as in 6.2. Then ${\cal
A}_{1}={\cal A}_{2}$.

Proof. Step 1: Let $\gamma_{1},\gamma_{2}\in\operatorname{Aut}(\tilde{M}_{c})$
be holomorphic contractions centered in the apex $c$, associated with two LCK
manifolds with proper potential such that their Kähler ${\mathbb{Z}}$-covers
have the same weak Stein completion $\tilde{M}_{c}$. We prove 6.3 using 6.3.
To show that the algebraic structures induced by $\gamma_{1}$ and $\gamma_{2}$
are the same, it would suffice to show that the spaces of $\gamma_{1}$\- and
$\gamma_{2}$-finite functions on $\tilde{M}_{c}$ coincide. By 6.3, this would
follow if the growth estimates (6.1) for $\gamma_{1}$ and $\gamma_{2}$ are
equivalent.

We have reduced 6.3 to the following statement. Let $K\subset\tilde{M}_{c}$ be
a compact subset which contains an open neighbourhood of the apex
$c\in\tilde{M}_{c}$. Then a function $f\in H^{0}({\cal O}_{\tilde{M}_{c}})$
which satisfies ${\operatorname{{\sf
sup}}}_{K}(\gamma_{1})^{-n}|f|<C_{1}^{n}$, also satisfies ${\operatorname{{\sf
sup}}}_{K}(\gamma_{2})^{-n}|f|<C_{2}^{n}$ for some other constant $C_{2}>0$
and all $n>0$.

Step 2: Suppose that there exists an integer $d>0$ such that
$\gamma_{1}^{d}(K)\subset\gamma_{2}(K)$. Then the polynomial growth estimate
(6.1) for $\gamma_{2}$ follows from the growth estimate for $\gamma_{1}^{d}$,
which is equivalent to the growth estimate for $\gamma_{1}$. Therefore, 6.3
would follow if we prove that the integer $d$ always exists.

Recall that a continuous map $f:\;M{\>\longrightarrow\>}M$ fixing $x\in M$ is
called a contraction centered in $x\in M$ if for each compact subset $K\subset
M$, and each open neighbourhood of $x$, a sufficiently big iteration of $f$
gives $f^{N}(K)\subset U$. The maps $\gamma_{1},\gamma_{2}$ are contractions
centered in the apex $x$, hence for some $N>0$, the set $\gamma_{1}^{N}(K)$
lies in the interior of $\gamma_{2}(K)$ which is a neighbourhood of $x$. 6.3
is proven.

##  7 Algebraic cones and LCK manifolds

In Section 2, we gave three equivalent definitions of an algebraic cone: an
open algebraic cone is the total space of an ample ${\mathbb{C}}^{*}$-bundle,
or a Kähler ${\mathbb{Z}}$-covering of a submanifold in a Hopf manifold, or a
Stein variety admitting a holomorphic contraction, with the apex removed. In
this section, we prove this equivalence. We start by recalling some basic
facts about the weighted projective spaces.

###  7.1 Weighted projective spaces

For an introduction and background material about the weighted projective
spaces, see [BG, §4.5].

Recall that any representation $V$ of ${\mathbb{C}}^{*}$ is a direct sum of
1-dimensional representations isomorphic to $\rho_{w}$, with
${\mathbb{C}}^{*}$ acting by $\rho_{w}(t)(z)=t^{w}z$. Such a representation is
called representation of weight $w$, and the decomposition of $V$ into
subrepresentation of constant weight – the weight decomposition.

The following trivial claim is left as an exercise to the reader.

Claim 7.1:  Let $\rho$ be ${\mathbb{C}}^{*}$ acting on ${\mathbb{C}}^{n}$.
Assume that $\rho$ contains a contraction. Then all weights of $\rho$ are
positive or negative.

Claim 7.2:  Let $\rho$ be ${\mathbb{C}}^{*}$ acting on ${\mathbb{C}}^{n}$ with
weights $w_{1},...,w_{n}\in{\mathbb{Z}}^{>0}$. Then its orbit space
${\mathbb{C}}P^{n-1}(w_{1},...,w_{n})$ is equipped with a structure of a
projective orbifold, and ${\mathbb{C}}^{n}\backslash 0$ can be identified with
the total space of an ample ${\mathbb{C}}^{*}$-bundle over
${\mathbb{C}}P^{n-1}(w_{1},...,w_{n})$.

Proof: [BG, Proposition 4.5.3].

Definition 7.3: The orbifold ${\mathbb{C}}P^{n-1}(w_{1},...,w_{n})$ is called
the weighted projective space.

Claim 7.4:  Let $\rho$ be ${\mathbb{C}}^{*}$ acting on ${\mathbb{C}}^{n}$ with
weights $w_{1},...,w_{n}\in{\mathbb{Z}}^{>0}$, and
$Z\subset{\mathbb{C}}^{n}\backslash 0$ be a $\rho$-invariant submanifold. Then
the orbit space $Z/{\mathbb{C}}^{*}$ is a projective orbifold in the
corresponding weighted projective space
${\mathbb{C}}P^{n-1}(w_{1},...,w_{n})$.

Proof: [BG, Proposition 4.6.2].

Corollary 7.5:  Let $\rho$ be ${\mathbb{C}}^{*}$ acting on ${\mathbb{C}}^{n}$
with weights $w_{1},...,w_{n}\in{\mathbb{Z}}^{>0}$, and
$Z\subset{\mathbb{C}}^{n}\backslash 0$ a $\rho$-invariant submanifold. Then
$Z$ is an open algebraic cone.

###  7.2 Jordan-Chevalley decomposition

We recall a few definitions and results from the algebraic group theory,
generalizing the Jordan normal form to arbitrary algebraic group.

Definition 7.6: An element of an algebraic group $G$ is called semisimple if
its image is semisimple for some exact algebraic representation of $G$, and is
called unipotent if its image is unipotent (that is, exponential of a
nilpotent) for some exact algebraic representation of $G$.

Remark 7.7: For any algebraic representation of an algebraic group $G$, the
image of any semisimple element is a semisimple operator, and the image of any
unipotent element is a unipotent operator ([Hu, §15.3]).

Theorem 7.8: (Jordan-Chevalley decomposition, [Hu, §15.3])
Let $G$ be an algebraic group, and $A\in G$. Then there exists a unique
decomposition $A=SU$ of $A$ in a product of commuting elements $S$ and $U$,
where $U$ is unipotent and $S$ semisimple.

###  7.3 Algebraic cone obtained from an LCK manifold with potential

Theorem 7.9:  Let $M$ be an LCK manifold with proper potential, and
$\tilde{M}$ its Kähler ${\mathbb{Z}}$-cover. Then $\tilde{M}$ is an open
algebraic cone.

Proof: Let $\gamma$ be the generator of the ${\mathbb{Z}}$-action on
$\tilde{M}$. Using 3, we can embed $M$ to a linear Hopf manifold. This gives a
holomorphic map $\tilde{M}{\>\longrightarrow\>}{\mathbb{C}}^{n}\backslash 0$
taking $\gamma$ to a linear contraction
$A\in\operatorname{End}({\mathbb{C}}^{n})$. Let $\tilde{M}_{c}$ be the closure
of $\tilde{M}$ in ${\mathbb{C}}^{n}$. By 6.1, $\tilde{M}_{c}$ is equipped with
a natural algebraic structure. To prove 7.3 it remains only to show that this
variety admits a holomorphic ${\mathbb{C}}^{*}$-action, free outside of the
origin $c$, and acting with eigenvalues $|\alpha_{i}|<1$ on the Zariski
tangent space $T_{c}\tilde{M}_{c}$.

Let ${\cal G}_{A}$ be the Zariski closure of $\langle A\rangle$ in
${\sf{GL}}({\mathbb{C}}^{n})$. This is a commutative algebraic group, acting
on the variety $\tilde{M}_{c}\subset{\mathbb{C}}^{n}$. To prove that
$\tilde{M}_{c}$ is an algebraic cone, we find a ${\mathbb{C}}^{*}$-action
contracting $\tilde{M}_{c}$ to the origin and apply 7.1.

Let $A=SU$ be the Jordan-Chevalley decomposition for $A$, with $S,U\in{\cal
G}_{A}$. Since ${\cal G}_{A}$ preserves $\tilde{M}_{c}$, the endomorphisms $S$
and $U\in\operatorname{End}({\mathbb{C}}^{n})$ also act on $\tilde{M}_{c}$.
Since the eigenvalues of $S$ are the same as the eigenvalues of $A$, the map
$S$ is a linear contraction. Let $G_{S}\subset{\cal G}_{A}$,
$G_{S}=e^{{\mathbb{C}}\log S}$ be a one-parametric subgroup containing $S$. We
prove that $G_{S}$ can be approximated by subgroups of ${\cal G}_{A}$
isomorphic to ${\mathbb{C}}^{*}$; then these subgroups also contain a
contraction, and we can apply 7.1.

Consider the map taking any $A_{1}\in{\cal G}_{A}$ to its unipotent component
$U_{1}$. Since ${\cal G}_{A}$ is commutative, this map is a group
homomorphism. Therefore, its kernel ${\cal G}_{s}$ (that is, the set of all
semisimple elements in ${\cal G}_{A}$) is an algebraic subgroup of ${\cal
G}_{A}$. A connected commutative algebraic subgroup of
${\sf{GL}}({\mathbb{C}}^{N})$ consisting of semisimple elements is always
isomorphic to $({\mathbb{C}}^{*})^{k}$ ([BT, Proposition 1.5]). The one-
parametric subgroups ${\mathbb{C}}^{*}\subset({\mathbb{C}}^{*})^{k}$ are dense
in $({\mathbb{C}}^{*})^{k}$ because one-parametric complex subgroups
${\mathbb{C}}^{*}\subset({\mathbb{C}}^{*})^{k}$ can be obtained as
complexification of subgroups
$S^{1}\subset{\sf{U}}(1)^{k}\subset({\mathbb{C}}^{*})^{k}$, and those are
dense in ${\sf{U}}(1)^{k}$. Therefore, the contraction $S\in{\cal
G}_{s}=({\mathbb{C}}^{*})^{k}$ can be approximated by an element of
${\mathbb{C}}^{*}$ acting on $\tilde{M}_{c}$.

We have obtained a ${\mathbb{C}}^{*}$-action $\rho$ on $\tilde{M}_{c}$
containing a contraction. By 7.1, all weights of this
${\mathbb{C}}^{*}$-action are positive (or negative, in which case we replace
$\rho$ by its opposite). Then, 7.1 implies that $\tilde{M}$ is an open
algebraic cone.

###  7.4 Submanifolds of Hopf manifolds obtained from algebraic cones

Lemma 7.10:  Let $\gamma$ act on a complex variety $\tilde{M}_{c}$ by
contractions, contracting $\tilde{M}_{c}$ to a point $c$. Then the
corresponding ${\mathbb{Z}}$-action on
$\tilde{M}:=\tilde{M}_{c}\backslash\\{c\\}$ is properly discontinuous, hence
$\tilde{M}/{\mathbb{Z}}$ is Hausdorff; it is a manifold when $\tilde{M}$ is a
manifold.

Proof: By definition, the ${\mathbb{Z}}$-action is properly discontinuous if
every point has a neighbourhood $U$ such that the set $\\{g\in{\mathbb{Z}}\ \
|\ \ g(U)\cap U\neq\emptyset\\}$ is finite. Let $x\in\tilde{M}$ and $K$ be the
compact closure of an open neighbourhood of $U\subset\tilde{M}$ containing
$x$. Since $\tilde{M}_{c}$ is Hausdorff, there exists a neighbourhood $W\ni c$
such that its closure does not intersect $K$. By definition of contractions,
there exists $N>0$ such that $\gamma^{n}(K)\subset W$ for all $n\geqslant N$.
This implies that $\gamma^{n}(K)\cap K=\emptyset$. This also implies that
$K\cap\gamma^{-n}(K)=\emptyset$. We have shown that $\gamma^{n}(K)\cap
K=\emptyset$ for all $n\notin[-N,N]$.

The following theorem, applied to a weak Stein completion of an open algebraic
cone, allows us to obtain a holomorphic embedding of its
${\mathbb{Z}}$-quotient to a Hopf manifold.

Theorem 7.11:  Let $\tilde{M}_{c}$ be a Stein variety equipped with a
holomorphic contraction $\gamma$, contracting it to the point $c$. Assume that
the complement $\tilde{M}:=\tilde{M}_{c}\backslash c$ is smooth. By 7.4,
$\tilde{M}/\langle\gamma\rangle$ is a complex manifold. Then there exists a
holomorphic embedding $j:\;\tilde{M}/\langle\gamma\rangle\hookrightarrow H$ to
a Hopf manifold.

Proof: Let $R$ be the ring of $\gamma$-finite holomorphic functions on
$\tilde{M}_{c}$, $I$ the maximal ideal of $c$, and and $V\subset I$ a finite-
dimensional $\gamma$-invariant space generating $I$. By 5.2.2, the action of
$\gamma^{*}$ is compact on $I$ and has all eigenvalues $<1$. By Riesz-Schauder
theorem, $R$ is dense in ${\cal O}_{\tilde{M}_{c}}$ (5.2.3). Therefore, the
functions in $V$ separate the points in $\tilde{M}$, for $V$ sufficiently big.

By Montel’s theorem, $R$ is dense in $C^{1}$-topology whenever it is dense in
$C^{0}$-topology; therefore, the differentials of the functions $f\in R$
generate $T^{*}_{x}\tilde{M}$ for all $x\in\tilde{M}$. This implies that the
tautological map $\tilde{M}{\>\longrightarrow\>}V^{*}$ is a holomorphic
embedding. This map is by construction $\gamma$-equivariant, hence it induces
a holomorphic embedding $\tilde{M}/\langle\gamma\rangle\hookrightarrow
H=V^{*}/\langle\gamma\rangle$.

##  8 Closed algebraic cones and normal varieties

Definition 8.1:  Recall that a complex variety $X$ is called normal if any
locally bounded meromorphic function on an open subset $U\subset X$ is
holomorphic ([De, Definition II.7.4]). In algebraic geometry, a variety is
normal if all its local rings are integrally closed; these two notions are
equivalent for a complex variety obtained from an algebraic one ([De, Theorem
II.7.3], [Ku, Satz 4, p. 122]).999We are grateful to Francesco Polizzi for
this reference; please see the excellent Mathoverflow thread
https://mathoverflow.net/questions/303406/algebraic-vs-analytic-normality/ for
more details about the complex analytic and complex algebraic normality.

Remark 8.2: Each complex variety $A$ admits the normalization, that is, a
normal variety $B$ equipped with a finite holomorphic map
$n:\;B{\>\longrightarrow\>}A$. The normalization is unique up to an
isomorphism. In the sequel, we will usually consider the normalization maps
$n:\;B{\>\longrightarrow\>}A$ which are homeomorphisms. The concept might seem
alien, but in fact it is very natural. For example, consider the plane curve
$C$ defined by the equation $x^{2}=y^{3}$. This curve is clearly singular in
zero. It can be parametrized by a variable $t$ with $t^{2}=y$ and $t^{3}=x$;
this parametrization defines a map $n:\;{\mathbb{C}}{\>\longrightarrow\>}C$
which is finite, hence $n$ is the normalization of $C$. However, $n$ is
clearly a homeomorphism.

Claim 8.3:  Let $G$ be a finite group holomorphically acting on a normal
complex variety $Y$, and $X=Y/G$ the quotient variety. Then $X$ is normal.

Proof: Let $f$ be a locally bounded meromorphic function on $X$. Its pullback
$\tau^{*}f$ to $Y$ is clearly meromorphic and locally bounded, hence
holomorphic. Since $\tau^{*}f$ is $G$-invariant, it is holomorphic on $Y$.

We are interested in the normality of a closed algebraic cone.

Let now $X$ be a projective variety and $L$ an ample line bundle on $X$. The
homogeneous coordinate ring of $X$ is the ring
$\bigoplus_{i}H^{0}(X,L^{\otimes i})$. This ring is finitely generated
whenever $L$ is ample ([La, Theorem 2.3.15]). Indeed, for an ample line bundle
$L$ and $i$ sufficiently big, the cohomology $H^{>0}(X,L^{\otimes i})$
vanishes, hence the number $\operatorname{\sf dim}H^{0}(X,L^{\otimes i})$ is
equal to the holomorphic Euler characteristic $\chi(L^{\otimes i})$. The
latter is polynomial in $i$ by Riemann-Roch-Hirzebruch theorem, which implies
that finitely many generators are enough to generate this ring.

A projective variety $(X,L)$ for which the ring
$\bigoplus_{i}H^{0}(X,L^{\otimes i})$ is integrally closed is called
projectively normal. By definition, the affine variety associated with this
ring is the affine cone of $X$, which is a closed algebraic cone in our
parlance.

The closed algebraic cone, obtained from an open algebraic cone by taking the
Stein completion, is normal by definition of Stein completion. However, the
projective normality is a very tricky condition, depending on the choice of
the bundle $L$.

This was the subject of Exercise I.3.18 from Hartshorne’s “Algebraic Geometry”
([Ha]), where Hartshorne considers the same smooth complex curve with two
different projective embeddings; the affine cone of the first is normal, and
the affine cone of the second is not.

In Exercise II.5.14, [Ha], Hartshorne gives a criterion for projective
normality:

Proposition 8.4:  A projective variety $X\subset{\mathbb{C}}P^{n}$ is
projectively normal if and only if $X$ is normal, and the restriction map

$H^{0}({\mathbb{C}}P^{n},{\cal O}(i)){\>\longrightarrow\>}H^{0}\left(X,{\cal
O}(i){\left|{}_{{\phantom{|}\\!\\!}_{X}}\right.}\right)$

is surjective for all $i\geqslant 0$.

Remark 8.5: This proposition can be applied to algebraic cones, because the
quotient singularities are normal (8), and the orbifolds have only quotient
singularities.

Example 8.6:  Let $X\subset{\mathbb{C}}P^{n}$ be a projective manifold, which
is not contained in a projective subspace
${\mathbb{C}}P^{k}\subset{\mathbb{C}}P^{n}$ of smaller dimension. Consider a
linear projection
$p:\;{\mathbb{C}}P^{n}{\>\longrightarrow\>}{\mathbb{C}}P^{n-1}$ centered in a
point $z\notin X$. Then the restriction
$p{\left|{}_{{\phantom{|}\\!\\!}_{X}}\right.}$ is holomorphic. Assume that
$p:\;X{\>\longrightarrow\>}{\mathbb{C}}P^{n-1}$ is also injective (this can be
always achieved if $2\operatorname{\sf dim}X<n-1$ for an appropriately general
choice of $z$; indeed, $p$ is not injective if and only if the center $z$ does
not belong to the secant variety101010A secant variety of
$X\subset{\mathbb{C}}P^{n}$ is the Zariski closure of the union of all lines
${\mathbb{C}}P^{n}\subset{\mathbb{C}}P^{n}$ intersecting $X$ in at least two
points. of $X$, which has dimension $\leqslant 2\operatorname{\sf dim}X+1$).
Consider the natural map of affine cones
$u:\;C(X){\>\longrightarrow\>}C(p(X))$. Outside of the origin $c$, this map is
biholomorphic, hence it is birational and finite. By definition, a variety $Z$
is normal if any bimeromorphic finite map $Z_{1}{\>\longrightarrow\>}Z$ is an
isomorphism. Were $C(p(X))$ normal, this would imply that $u$ is an
isomorphism. However, the Zariski tangent space $T_{c}C(X)$ is
$n+1$-dimensional, because $C(X)\subset C({\mathbb{C}}P^{n})$ generates the
vector space $C({\mathbb{C}}P^{n})={\mathbb{C}}^{n+1}$ (otherwise $X$ would
have been contained in a smaller dimensional projective subspace). On the
other hand, $\operatorname{\sf dim}T_{c}C(p(X))\leqslant n$, hence
$u:\;C(X){\>\longrightarrow\>}C(p(X))$ is not an isomorphism.111111We are
grateful to Yu. Prokhorov for this example.

This example makes sense, if one considers the following statement.

Claim 8.7:  Let $\tilde{M}$ be an open algebraic cone, and ${\mathfrak{S}}$
the set of all closed algebraic cones $\tilde{M}_{c}$ obtained by adding the
origin to $\tilde{M}$. Then there exists only one closed algebraic cone
$Z\in{\mathfrak{S}}$ which is normal, and for any other
$Z^{\prime}\in{\mathfrak{S}}$, its normalization is $Z$. Moreover, the
normalization map $Z{\>\longrightarrow\>}Z^{\prime}$ is a homeomorphism.

Proof: Let $Z$ be the Stein completion of $\tilde{M}$; by definition, it is
normal. Since all holomorphic functions on $\tilde{M}$ can be extended to $Z$,
this variety is equipped with a finite, bijective, bimeromorphic map to any
closed algbraic cone $Z^{\prime}$ associated with $\tilde{M}$. Indeed, the
holomorphic functions on $Z^{\prime}$ are holomorphic on $\tilde{M}$, hence
they can be extended to its Stein completion. This implies that $Z$ is the
normalization of all other $Z^{\prime}\in{\mathfrak{S}}$.

Acknowledgements: We are grateful to Yuri Prokhorov for 8 and an important
discussion about the projective normality, and to Andrey Soldatenkov for his
insightful comments about the algebraic structures.

## References

  * [AS] A. Andreotti, Y. T. Siu, Projective embeddings of pseudoconcave spaces, Ann. Scuola Norm. Sup. Pisa 24, 231-278 (1970).
  * [Be] A. Besse, Einstein manifolds, Springer, 1987.
  * [BT] A. Borel, J. Tits, Groupes réductives, Publications Mathématiques de l’IHÉS, 27 (1965), 55-151.
  * [BG] C. P. Boyer, K. Galicki, Sasakian geometry, Oxford Mathematical Monographs. Oxford University Press, Oxford, 2008.
  * [Ca] E. Calabi, Métriques kählériennes et fibrés holomorphes, Ann. Ecole Norm. Sup. 12 (1979), 269-294.
  * [De] J.-P. Demailly, Complex analytic and differential geometry, https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf
  * [Fo1] O. Forster, Uniqueness of Topology in Stein Algebras, In Function Algebras, F. Birtel ed., pp. 157-163, Scott Foresman, Chicago, 1966.
  * [Fo2] O. Forster, Zur Theorie der Steinschen Algebren und Moduln, Math. Z. 97 (1967), 376-405.
  * [Fr] A. Friedman, Foundations of modern analysis, Dover, 2010.
  * [Ha] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
  * [HP] W. V. D. Hodge, D. Pedoe, Methods of algebraic geometry. Vol. III. Book V: Birational geometry. Cambridge, at the University Press, 1954\.
  * [Hu] J. E. Humphreys, Linear algebraic groups, GTM 21, 4th ed., Springer, 1998.
  * [Je] Z. Jelonek, Simple examples of affine manifolds with infinitely many exotic models, Adv. Math. 284 (2015), 112-121.
  * [Ku] N. Kuhlmann, Die Normalisierung komplexer Räume, Math. Ann. 144 (1961), 110-125.
  * [La] R. Lazarsfeld, Positivity in Algebraic Geometry I, Springer, 2004.
  * [Mu] H. T. Muhly, A remark on normal varieties. Ann. of Math. (2) 42 (1941), 921-925.
  * [Mus] G. Mustafin, p-adic Hopf varieties, Functional Anal. Appl. 11 (1977), no. 3, 234–235 (1978).
  * [OV1] L. Ornea, M. Verbitsky, An immersion theorem for compact Vaisman manifolds, Math. Ann. 332 (2005), 121-143.
  * [OV2] L. Ornea, M. Verbitsky, Locally conformal Kähler manifolds with potential, Math. Ann. 348 (2010), 25-33.
  * [OV3] L. Ornea, M. Verbitsky, Locally conformally Kähler metrics obtained from pseudoconvex shells, Proc. Amer. Math. Soc. 144 (2016), 325-335.
  * [OV4] L. Ornea, M. Verbitsky, Embedding of LCK manifolds with potential into Hopf manifolds using Riesz-Schauder theorem, “Complex and Symplectic Geometry ”, Springer INdAM series, 2017, 137-148.
  * [OV5] L. Ornea, M. Verbitsky, Lee classes on LCK manifolds with potential, arXiv:2112.03363.
  * [OV6] L. Ornea, M. Verbitsky, Non linear Hopf manifolds are locally conformally Kähler, arXiv:2202.12398.
  * [Sc] P. Scholze, p-adic geometry, Proceedings of the International Congress of Mathematicians - Rio de Janeiro 2018. Vol. I. Plenary lectures, 899-933, World Sci. Publ., Hackensack, NJ, 2018\.
  * [Ro] Rossi, H. Vector fields on analytic spaces, Ann. Math. 78(3), 455-467 (1963).
  * [Si] C. Simpson, Subspaces of moduli spaces of rank one local systems, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 3, 361-401.
  * [Vo] H. Voskuil, Non-Archimedean Hopf surfaces Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 2, 405-466.
  * [Wu] H. Wu, Normal families of holomorphic mappings, Acta Math. 119 (1967), 193-233.
  * [Za] O. Zariski, Complete linear systems on normal varieties and a generalization of a lemma of Enriques-Severi, Ann. of Math. (2) 55 (1952), 552-592.

Liviu Ornea
University of Bucharest, Faculty of Mathematics and Informatics,
14 Academiei str., 70109 Bucharest, Romania, and:
Institute of Mathematics “Simion Stoilow” of the Romanian Academy,
21, Calea Grivitei Str. 010702-Bucharest, Romania
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Misha Verbitsky
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Estrada Dona Castorina, 110
Jardim Botânico, CEP 22460-320
Rio de Janeiro, RJ - Brasil, also:
Laboratory of Algebraic Geometry,
Faculty of Mathematics, National Research University Higher School of
Economics, 6 Usacheva Str. Moscow, Russia
<EMAIL_ADDRESS><EMAIL_ADDRESS>


# CipherSniffer: Classifying Cipher Types

Brendan Artley, Greg Mehdiyev
Simon Fraser University
Burnaby, BC, Canada
<EMAIL_ADDRESS>

###### Abstract

Ciphers are a powerful tool for encrypting communication. There are many
different cipher types, which makes it computationally expensive to solve a
cipher using brute force. In this paper, we frame the decryption task as a
classification problem. We first create a dataset of transpositions,
substitutions, text reversals, word reversals, sentence shifts, and
unencrypted text. Then, we evaluate the performance of various tokenizer-model
combinations on this task.

##  1 Introduction

Identifying ciphers is a challenging problem. Human code-breakers typically do
this by looking for patterns in the encrypted text. Common strategies include
analyzing common letter pairings, counting character frequencies, and trying
to solve the small words first. These strategies work well for humans, but we
want to see if language models can pick up on these strategies as well.

We evaluated different tokenization methods and model types on this problem.
We were interested in comparing word-level tokenizers and subword-level
tokenizers on this task. This problem is challenging because some cipher
methods shuffle the order of characters arbitrarily, resulting in a large
amount of out-of-bounds vocabulary. It was interesting to see how the models
adapted to pick up on cipher hints rather than the semantic meaning from the
text.

##  2 Related Work

Deciphering classic ciphers with language models is an area with existing
research. In Berg-Kirkpatrick and Klein (2013) the authors implement a trigram
HMM model using beam search to solve substitution ciphers. They show how using
HMMs with millions of random restarts results in significant increases in
decipherment. This research was enhanced by Kambhatla et al. (2018), where the
authors showed how to reduce the search space of the beam search algorithm by
scoring the entire plaintext at each decipherment step. These papers focus on
models that directly solve ciphers, but could be improved by the
classification models in our paper.

We also acknowledge the work done by Krishna et al. (2019) in classifying
ciphers using SVMs, HMMs, and CNNs. They find that an HMM architecture is the
most accurate classifier. Also, the author shows how models get more accurate
as the length of text increases. They experiment with cipher lengths between
10 and 10000 characters whereas we experiment with cipher lengths between 7
and 443 characters.

Finally, we drew inspiration for the cipher types from Rayner et al. (2006)
and Sakaguchi et al. (2016). In Rayner et al. (2006), the authors show that
people find it difficult to identify words when the start and end characters
are jumbled rather than when the internal characters are jumbled. Sakaguchi et
al. (2016) takes this further by implementing a semi-character level RNN that
is more robust on a spelling correction task. They do this by embedding the
starting character, ending character, and internal character counts in three
separate vectors. These two papers motivated us to create the word reversal,
text reversal, and character shift ciphers.

##  3 Approach

We framed our task as a 6-label classification problem, with 5 cipher classes
and 1 unencrypted text class. Then, we experimented with various model and
tokenizer combinations and evaluated each model based on its performance on a
hidden test set.

###  3.1 Ciphers

The five ciphers that we used were substitution cipher, transposition cipher,
word reversal, text reversal, and character shift. Except for text reversal,
each encipherment method preserves the position of the whitespace in the text.
We chose these ciphers as they each modify the text differently.

When we applied each cipher to the dataset, we iteratively looped through the
cipher types. This ensured that the dataset had an equal class distribution
and made training and evaluation easier. The data is saved in a txt file
format, with the first character of each line corresponding to its class
label. The label encoding that we used is shown in table 1.

Label | Encoded Label
---|---
`Substitution` | 0
`Transposition` | 1
`Text Reversal` | 2
`Character Shift` | 3
`Word Reversal` | 4
`Unencrypted` | 5
Table 1: Label mapping

#### Substitution Cipher:

The substitution cipher replaces each letter in the alphabet with a new
letter. We also added a derangement condition, which means that letters can
not be replaced with themselves. This ensures that the generated substitution
cipher is distinct from the unencrypted text. In Figure 1, we show a possible
letter mapping of the substitution cipher. See figure 1.

Figure 1: Substitution Cipher

#### Transposition Cipher:

The transposition cipher changes the position of each letter in the sentence.
We add the condition to keep the whitespace characters in their original
space. This method often results in letters mapping to themselves, but we add
a condition to redo the transposition if the encrypted text matches the
unencrypted text. See figure 2.

Figure 2: Transposition Cipher

#### Word Reversal:

Word reversal is when every word in a sentence is reversed. The position of
each whitespace character remains unchanged. See figure 3.

Figure 3: Word Reversal

#### Text Reversal:

Text reversal is when every character in a sentence is reversed. This includes
whitespace characters. See figure 4.

Figure 4: Text Reversal

#### Character shift:

Character shift is when all non whitespace characters are shifted in the
sentence. See figure 5.

Figure 5: Character shift

###  3.2 Models/Tokenizers

The models tested were GRUs, LSTMs, and BERT. The GRU and LSTM models were
implemented in Keras (Chollet et al., 2015), and the BERT (Devlin et al.,
2018) model was implemented in Pytorch (Paszke et al., 2019). The embedding
dimension for each model was 300, except for BERT which had an embedding
dimension of 768. At the end of model training, we would restore the model
weights that achieved the highest accuracy on the validation dataset.

The tokenization strategies tested were Byte-Pair encoding, WordPiece,
character-level, and word-level tokenization. These were built using Python
(Van Rossum and Drake Jr, 1995) and the tokenizers module from HuggingFace
(Wolf et al., 2020).

We also tested pre-trained GloVe embeddings (Pennington et al., 2014) as a
baseline model. We did this because GloVe is a state-of-the-art static word
embedding architecture used across many NLP tasks. The pre-trained GloVe
embeddings were trained on the Wikipedia 2014 dataset and Gigaword corpora
(Graff et al., 2003). We also trained custom glove embedding weights using our
corpora by modifying the code from the GloVe GitHub repository.

For the most part, the code for this project was implemented from scratch. To
implement the frozen GloVe embeddings and the BERT training script, we used
existing code. In these cases, we provide references to source code directly
in the CipherSniffer code. Finally, all the code for this paper is open source
and accessible in the CipherSniffer repository. We provide an example training
notebook for those who want to reproduce the results from the paper. We also
provide all the trained models and tokenizers for those who do not want to
train them from scratch.

###  3.3 General Architecture

Each model takes an input of raw text and predicts an output class. Moreover,
each model outputs a 6-dimensional vector scaled by the softmax function. Each
scalar value is the predicted probability of the input text belonging to each
class. We added a feed-forward network on top of each model to improve the
accuracy and reduce the likelihood of overfitting. The feed-forward network
structure is identical for each model. This network consists of two
512-dimensional linear layers with the ReLU activation function and 50%
dropout. See figure 6 for the LSTM model architecture.

Figure 6: LSTM Architecture

##  4 Experimental setup

###  4.1 Dataset

The dataset that we used for this task was the Gigaword dataset (Graff et al.,
2003). This open-source dataset contains article summaries and headlines for
3,993,608 news articles. In our experiments, we only used the summaries from
this dataset. We cleaned the data by lowercasing the text and removing words
that contained non-ASCII-lowercase characters. This made the average sentence
length 177 characters and reduced the character vocabulary from 128 to 27.

Next, we created a subset of 55,000 data points for model training. We further
split this set into 50,000 for training, 2,500 for validation, and 2,500 for
an unseen test set. The remaining 3,943,608 article summaries were used as a
corpus for the tokenization models. Due to the computational demand of
training non-parallelizable RNN and LSTM models, we did not make the
classifier training, validation, and test set any larger. See table 2 for more
information on the dataset.

Data | Rows | Size (MB)
---|---|---
Tokenizer | 3,943,608 | 660.8
Train | 50,000 | 8.6
Valid | 2,500 | 0.4
Test | 2,500 | 0.4
Table 2: CipherData

###  4.2 Metrics

During training, we optimized model parameters using the cross-entropy loss.
The cross-entropy loss is the negative sum of each label denoted as “ti”
multiplied by the log of the predicted probability of each class. Each “ti”
value in the summation is zero, except for the true label which has a value of
one. This is because there is only one true label for each data point. The
equation for the cross-entropy loss is shown below.

CE Loss $=-\sum_{c=1}^{M}y_{o,c}\log(p_{o,c})$

During model training, we monitored the training and validation accuracy at
the end of each epoch. If the validation accuracy of the model was better than
all previous epochs, we saved the model weights. This was done to ensure that
we were not overfitting to the training data and that we had a highly accurate
model. After the model finished training, we restored the saved model weights
and performed a final evaluation on a hidden test set.

Accuracy $=\frac{TP+TN}{TP+TN+FP+FN}$

##  5 Results

In table 3 we show the accuracy of each model. We trained each model for 10
epochs, with a learning rate of 0.001, a batch size of 1024, and an ADAM
optimizer. Each model was trained on a single NVIDIA Tesla P100 GPU. Due to
the number of model-tokenizer combinations, we did not perform any extensive
hyperparameter tuning.

The BERT model with a wordpiece tokenizer achieved the highest overall test
accuracy of 99.00%. It correctly classified all data points in the
substitution, transposition, character shift, and unencrypted text classes. It
scored 99.04% on the word reversal class and 90.65% on the text reversal
class.

The second best model was a GRU with GloVe embeddings trained on our corpora.
This model achieved an overall test accuracy of 97.00%. It achieved the
highest accuracy on the text reversal class with an accuracy of 95.20%, and it
also matched BERT on word reversals with a score of 99.04%.

###  5.1 Tokenizer Accuracy

On average, the best tokenization strategy was the subword-level tokenizer.
Models that used subword tokenizers scored an average overall accuracy of
96.13%. We believe that this was due to their ability to capture common words
and handle out-of-bounds vocabulary at the same time.

The word-level tokenizers were more unpredictable, scoring an average accuracy
of 92.31%. On the low end, the models that used frozen GloVe embeddings
performed very poorly. They struggled to classify the substitution and
transposition ciphers, scoring roughly 70-80% accuracy on each class. This was
likely due to the high number of out-of-bounds vocabulary created from
applying ciphers to the text. On the high hand, the trained GloVe embeddings
on our corpora of cipher text performed on par with the subword-level
tokenizers.

The character-level tokenizers scored an average accuracy of 89.40% and the
results were quite interesting. They struggled to classify word reversals and
text reversals with accuracy scores in the range of 58-69% and 79-86%
respectively. Further, all the character-level tokenization models scored
higher than the word-level tokenization models on the transposition and
substitution classes. The results from the character-level tokenizer were more
accurate than we expected, as there were only 28 possible embeddings in this
model.

###  5.2 Other

All the models we tested had no problem classifying original text and
character shifts. The classes that were difficult for all the models were the
word reversal and text reversal classes. This was likely due to the similar
character order of the ciphered text, but it is not entirely evident why the
models struggled here. For more comprehensive model results, see table 3.

We also recognize the difference in tunable parameters for each model.
Moreover, the BERT uncased model has 109 million tunable parameters compared
with the GRU and LSTM models which have 9.2 and 9.3 million tunable
parameters. These models have only 8% of the tunable parameters of the BERT
model, which could explain the difference in performance.

##  6 Future Work

It would be interesting to identify the root cause of some notable results in
this paper. This includes the difficulty of classifying word reversals and
character reversals and the high accuracy of a GRU model with GloVe
Embeddings.

Second, human codebreakers typically use whitespace characters as cipher
hints. This leads us to believe that a sentencepiece tokenizer (Kudo and
Richardson, 2018) may outperform the BPE and wordpiece models. This tokenizer
treats the input text as a sequence of Unicode characters rather than
splitting on whitespace, and may provide more cipher hints to the model.

Model | Level | Subs | Trans | T-Rev | Chr-S | W-Rev | Original | Acc
---|---|---|---|---|---|---|---|---
BERT | S | 1 | 1 | 0.9065 | 1 | 0.9904 | 1 | 0.9900
GRU GLOVE | W | 0.9543 | 0.9448 | 0.9520 | 0.9784 | 0.9904 | 1 | 0.9700
LSTM BPE | S | 1 | 0.9880 | 0.9041 | 0.9880 | 0.8681 | 0.9976 | 0.9576
LSTM WP (pre) | S | 1 | 1 | 0.8609 | 0.9952 | 0.8897 | 0.9952 | 0.9568
GRU WP (pre) | S | 0.9976 | 1 | 0.8129 | 0.9928 | 0.9400 | 0.9976 | 0.9568
GRU WP | S | 1 | 0.9976 | 0.8609 | 0.9976 | 0.8825 | 1 | 0.9564
GRU BPE | S | 1 | 1 | 0.8801 | 0.9880 | 0.8705 | 0.9976 | 0.9560
LSTM WP | S | 1 | 0.9832 | 0.8321 | 0.9952 | 0.9257 | 0.9976 | 0.9556
LSTM GLOVE | W | 0.9255 | 0.9544 | 0.9257 | 0.9832 | 0.9353 | 0.9976 | 0.9536
LSTM Character Level | C | 0.9952 | 0.9976 | 0.7938 | 0.9544 | 0.6930 | 0.9952 | 0.9048
GRU GLOVE (pre) | W | 0.7548 | 0.8106 | 0.8849 | 0.9065 | 0.9664 | 1 | 0.8872
GRU Character Level | C | 0.9880 | 0.9736 | 0.8609 | 0.9329 | 0.5803 | 0.9639 | 0.8832
LSTM GLOVE (pre) | W | 0.8005 | 0.7146 | 0.9353 | 0.9233 | 0.9185 | 0.9976 | 0.8816
Avg Accuracy |  | 0.9551 | 0.9511 | 0.8777 | 0.9720 | 0.8808 | 0.9954 |

Table 3: Model Accuracy

Table Abbreviations: S(Subword), W(Word), C(Character), Subs(Substitution),
Trans(Transposition), T-Rev(Text Reversal), Chr-S(Character Shift), W-Rev(Word
Reversal)

## References

  * Berg-Kirkpatrick and Klein (2013) Taylor Berg-Kirkpatrick and Dan Klein. 2013.  Decipherment with a million random restarts.  In _Proceedings of the 2013 Conference on Empirical Methods in Natural Language Processing_ , pages 874–878, Seattle, Washington, USA. Association for Computational Linguistics.
  * Chollet et al. (2015) Francois Chollet et al. 2015.  Keras.
  * Devlin et al. (2018) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018.  BERT: pre-training of deep bidirectional transformers for language understanding.  _CoRR_ , abs/1810.04805.
  * Graff et al. (2003) David Graff, Junbo Kong, Ke Chen, and Kazuaki Maeda. 2003.  English gigaword.  _Linguistic Data Consortium, Philadelphia_ , 4(1):34.
  * Kambhatla et al. (2018) Nishant Kambhatla, Anahita Mansouri Bigvand, and Anoop Sarkar. 2018.  Decipherment of substitution ciphers with neural language models.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 869–874, Brussels, Belgium. Association for Computational Linguistics.
  * Krishna et al. (2019) Krishna, Stamp, Austin, and Troia. 2019.  Classifying classic ciphers using machine learning.
  * Kudo and Richardson (2018) Taku Kudo and John Richardson. 2018.  Sentencepiece: A simple and language independent subword tokenizer and detokenizer for neural text processing.  _CoRR_ , abs/1808.06226.
  * Paszke et al. (2019) Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. 2019.  Pytorch: An imperative style, high-performance deep learning library.  In _Advances in Neural Information Processing Systems 32_ , pages 8024–8035. Curran Associates, Inc.
  * Pennington et al. (2014) Jeffrey Pennington, Richard Socher, and Christopher Manning. 2014.  GloVe: Global vectors for word representation.  In _Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 1532–1543, Doha, Qatar. Association for Computational Linguistics.
  * Rayner et al. (2006) Keith Rayner, Sarah White, Rebecca Johnson, and Simon Liversedge. 2006.  Raeding wrods with jubmled lettres: There is a cost.  _Psychological science_ , 17:192–3.
  * Sakaguchi et al. (2016) Keisuke Sakaguchi, Kevin Duh, Matt Post, and Benjamin Van Durme. 2016.  Robsut wrod reocginiton via semi-character recurrent neural network.
  * Van Rossum and Drake Jr (1995) Guido Van Rossum and Fred L Drake Jr. 1995.  _Python reference manual_.  Centrum voor Wiskunde en Informatica Amsterdam.
  * Wolf et al. (2020) Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Remi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander Rush. 2020.  Transformers: State-of-the-art natural language processing.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ , pages 38–45, Online. Association for Computational Linguistics.