Holography principle for twistor spaces

Misha Verbitsky111Partially supported by RFBR grants 12-01-00944- ,
10-01-93113-NCNIL-a, and AG Laboratory NRI-HSE, RF government grant, ag.
11.G34.31.0023.

<EMAIL_ADDRESS>
Abstract

Let $S$ be a smooth rational curve on a complex manifold $M$. It is called
ample if its normal bundle is positive: $NS=\bigoplus{\cal O}(i_{k})$,
$i_{k}>0$. We assume that $M$ is covered by smooth holomorphic deformations of
$S$. The basic example of such a manifold is a twistor space of a hyperkähler
or a 4-dimensional anti-selfdual Riemannian manifold $X$ (not necessarily
compact). We prove “a holography principle” for such a manifold: any
meromorphic function defined in a neighbourhood $U$ of $S$ can be extended to
$M$, and any section of a holomorphic line bundle can be extended from $U$ to
$M$. This is used to define the notion of a Moishezon twistor space: this is a
twistor space admitting a holomorphic embedding to a Moishezon variety
$M^{\prime}$. We show that this property is local on $X$, and the variety
$M^{\prime}$ is unique up to birational transform. We prove that the twistor
spaces of hyperkähler manifolds obtained by hyperkähler reduction of flat
quaternionic-Hermitian spaces by the action of reductive Lie groups (such as
Nakajima’s quiver varieties) are always Moishezon.

###### Contents

  1. 1 Introduction
    1. 1.1 Quasilines on complex manifolds
    2. 1.2 Holography principle
    3. 1.3 Moishezon twistor spaces
  2. 2 Twistor spaces and ample rational curves
    1. 2.1 Hyperkähler manifolds
    2. 2.2 Twistor spaces
    3. 2.3 Geometry of twistor spaces
    4. 2.4 Quasilines and ample curves in the twistor spaces
  3. 3 Holography principle for manifolds with ample rational curves
    1. 3.1 Holography principle for line bundles
    2. 3.2 The local holography principle
    3. 3.3 Holography principle for meromorphic functions
  4. 4 Moishezon twistor spaces
    1. 4.1 Vector bundles in a neighbourhood of an ample curve
    2. 4.2 Algebraic dimension of the field of meromorphic functions
    3. 4.3 Moishezon manifolds and ample rational curves
    4. 4.4 Moishezon twistor spaces
    5. 4.5 Twistor spaces and hyperkähler reduction
  5. 5 Appendix: Formal geometry and holography principle (by Dmitry Kaledin)

##  1 Introduction

###  1.1 Quasilines on complex manifolds

The present paper was written as an attempt to answer the following question.
Let $S\subset M$ be a smooth rational curve in a complex manifold, with normal
bundle $NS$ isomorphic to ${\cal O}(1)^{n}$.111Such rational curves are called
quasilines, see e.g. [BBI]. Is there a notion of a normal form for a tubular
neighbourhood of such a curve?

When the normal bundle is ${\cal O}(-1)^{n}$ instead of ${\cal O}(1)^{n}$, a
tubular neighbourhood of the curve has a normal form, obtained by blowing down
this curve to a point, and taking a sufficiently small Stein neighbourhood of
this point in the corresponding singular variety. When $NS={\cal O}(1)^{n}$,
no such normal form can be obtained. In fact, the birational type of the
manifold can be reconstructed from a complex analytic (and even formal)
neighbourhood of $S$. This was known already to Hartshorne ([Har, Theorem
6.7]).

However, one can associate with a quasiline an infinite-dimensional bundle
$\hat{\cal O}_{S}(M)$ over $S\cong{\mathbb{C}}P^{1}$, with

$\hat{\cal O}_{S}(M)=\lim_{\leftarrow}{\cal O}_{M}/I^{n}_{S}{\cal O}_{M},$

where $I_{S}$ is an ideal sheaf of $S$. This bundle is called a formal
completion of $M$ at $S$.

It is not hard to observe that the space of sections $H^{0}(S,\hat{\cal
O}_{S}(M))$ is finite-dimensional, and, moreover, $H^{0}(S,\hat{\cal
O}_{S}(M)\otimes_{{\cal O}_{S}}{\cal O}(i))$ is finite-dimensional for each
$i$ (4.1). For a long time, we expected that the algebra
$A_{S}:=\bigoplus_{i}H^{0}(S,\hat{\cal O}_{S}(M)\otimes_{{\cal O}_{S}}{\cal
O}(i))$ would provide a sort of an algebraic “normal form” of $S$ in $M$, in
such a way that the algebraic structure on $\hat{\cal O}_{S}(M)$ can be
reconstructed from this ring.

In this paper, we show that this approach works when $M$ is a Moishezon
manifold, and $M$ can be reconstructed from the ring $A_{S}$, up to birational
isomorphism (Subsection 4.3; this is not very surprising due to the above-
mentioned theorem of Hartshorne, [Har, Theorem 6.7]).

For non-Moishezon $M$, such as a twistor space of a simply connected, compact
hyperkähler manifold, this conjecture is spectacularly wrong. In this case,
$A_{S}=\bigoplus_{i}H^{0}({\mathbb{C}}P^{1},{\cal O}(i))$, and this ring has
no information about $M$ whatsoever (2.3).

###  1.2 Holography principle

The main technical tool of the present paper is the following theorem, called
the holography principle. Recall that an ample rational curve on a complex
variety $M$ is a smooth curve $S\cong{\mathbb{C}}P^{1}\subset M$ such that the
normal bundle $NS$ is decomposed as $NS=\bigoplus_{k}{\cal O}(i_{k})$, with
all $i_{k}>0$.

Theorem 1.1:  Let $S\subset M$ be an ample curve in a simply connected complex
manifold, and $U$ its connected neighbourhood. Suppose that $M$ is covered by
smooth complex-analytic deformations of $S$. Then

(i)

For any holomorphic vector bundle $B$ on $M$, the restriction map

$H^{0}(M,B){\>\longrightarrow\>}H^{0}(U,B)$

is an isomorphism.

(ii)

Let $\operatorname{Mer}(M)$, $\operatorname{Mer}(U)$ be the fields of
meromorphic functions on $M$ and $U$. Then the restriction map
$\operatorname{Mer}(M){\>\longrightarrow\>}\operatorname{Mer}(U)$ is an
isomorphism.

Proof: For line bundles, 1.2 (i) is implied by 3.1. For general vector
bundles, an elegant argument is given by D. Kaledin in the appendix to this
paper (Section 5). Holography principle for meromorphic functions is proven in
3.3.

The holography principle is not very surprising of one looks at the
neighbourhood of $S$ from the point of view of complex analysis. The normal
bundle to $S$ is obviously positive. Choose a Hermitian metric $h$ on a
neighbourhood of $S$ such that the Chern connection on $NS$ induced by $h$ has
positive curvature. Let $d_{S}:\;M{\>\longrightarrow\>}{\mathbb{R}}$ be the
Riemannian distance to $S$ in this Hermitian metric. Since $d_{S}$ around $S$
is close to the distance in $NS$, the form $dd^{c}d_{S}$ has $n-1$ positive
and 1 negative eigenvalue in a sufficiently small neighbourhood of $S$ ([D]).
This means that $S$ has a neighbourhood $U$ with a smooth boundary $\partial
U$ such that the Levi form on $\partial U$ has one negative and $\dim M-2$
positive eigenvalues. Then, a holomorphic function defined on an open subset
of $U$ and continuous on $\partial U$ can be extended outside of a boundary;
at least, this is the expectation one has from the solution of the Levi
problem.

###  1.3 Moishezon twistor spaces

Definition 1.2: Let $M$ be a compact complex manifold. Define the algebraic
dimension as $a(M):=\deg_{tr}\operatorname{Mer}(M)$, where
$\deg_{tr}\operatorname{Mer}(M)$ denotes the transcendence degree of the field
of global meromorphic functions on $M$.

Definition 1.3: A Moishezon variety is a compact complex variety satisfying
$a(M)=\dim M$.

The notion of a Moishezon manifold, as it is usually stated, makes no sense
for non-compact varieties. Indeed, $\deg_{tr}\operatorname{Mer}(M)=\infty$
even if $M$ is an open disk.

However, when $M$ contains an ample curve, the situation changes drastically.

Theorem 1.4:  Let $M$ be a complex manifold containing an ample rational
curve. Then $a(M)\leqslant\dim M$. Moreover, if $a(M)=\dim M$, there exists an
open embedding of $M$ to a Moishezon variety $M^{\prime}$ which satisfies
$\operatorname{Mer}(M^{\prime})=\operatorname{Mer}(M)$.

Proof: See 4.3.

Let now $M$ be a twistor space of $X$. Here, $X$ can be either a hypercomplex
(hyperkähler) manifold, a quaternionic, or quaternionic-Kähler manifold, or
Riemannian anti-selfdual manifold. We are not very specific, because the only
thing about $M$ which is used is existence of a large number of quasilines.
The twistor spaces are complex manifold covered by quasilines, usually non-
Kähler and non-quasiprojective (see 2.3 and 4.5).

Since $M=\operatorname{Tw}(X)$ is covered by quasilines which are by
definition ample, we can apply 4.2, and obtain that
$\deg_{tr}\operatorname{Mer}(M)\leqslant\dim M$. We call $M$ a Moishezon
twistor space if $\deg_{tr}\operatorname{Mer}(M)=\dim M$. This is equivalent
to an existence of an open embedding $M{\>\longrightarrow\>}M^{\prime}$ of $M$
to a Moishezon manifold (1.3).

Moishezon twistor spaces for compact 4-dimensional anti-selfdual manifolds
were discovered by Y.-S. Poon in [P], and much studied since then. Structure
theorems about such manifolds were obtained by F. Campana, [C], and partial
classification results by N. Honda (see e.g. [Ho]).

The definition given above extends the class of “Moishezon twistor manifolds”
significantly. Let, for instance, $M:=V{/\\!\\!/\\!\\!/}G$ be a hyperkähler
manifold which can be obtained using the hyperkähler reduction, where $V$ is a
flat hyperkähler manifold, and $G$ a compact Lie group (such as the Nakajima
quiver variety). Then the twistor space $\operatorname{Tw}(M)$ is always
Moishezon (4.5).

Acknowledgements: Many thanks to Hans-Joachim Hein, Claude LeBrun, Nobuhiro
Honda and Dima Kaledin for interesting discussions on the subject of this
article.

##  2 Twistor spaces and ample rational curves

###  2.1 Hyperkähler manifolds

Definition 2.1:  Let $M$ be a manifold, and $I,J,K\in\operatorname{End}(TM)$
endomorphisms of the tangent bundle satisfying the quaternionic relation

$I^{2}=J^{2}=K^{2}=IJK=-\operatorname{Id}_{TM}.$

The manifold $(M,I,J,K)$ is called hypercomplex if the almost complex
structures $I$, $J$, $K$ are integrable. If, in addition, $M$ is equipped with
a Riemannian metric $g$ which is Kähler with respect to $I,J,K$, the manifold
$(M,I,J,K,g)$ is called hyperkähler.

Consider the Kähler forms $\omega_{I},\omega_{J},\omega_{K}$ on $M$:

$\omega_{I}(\cdot,\cdot):=g(\cdot,I\cdot),\ \
\omega_{J}(\cdot,\cdot):=g(\cdot,J\cdot),\ \
\omega_{K}(\cdot,\cdot):=g(\cdot,K\cdot).$

An elementary linear-algebraic calculation implies that the 2-form
$\Omega:=\omega_{J}+\sqrt{-1}\>\omega_{K}$ is of Hodge type $(2,0)$ on
$(M,I)$. This form is clearly closed and non-degenerate, hence it is a
holomorphic symplectic form.

In algebraic geometry, the word “hyperkähler” is essentially synonymous with
“holomorphically symplectic”, due to the following theorem, which is implied
by Yau’s solution of Calabi conjecture.

Theorem 2.2:  Let $(M,I)$ be a compact, Kähler, holomorphically symplectic
manifold. Then there exists a unique hyperkähler metric on $(M,I)$ with the
same Kähler class.

Proof: See [Y], [Bes].

Remark 2.3: The hyperkähler metric is unique, but there could be several
hyperkähler structures compatible with a given hyperkähler metric on $(M,I)$,
if the holonomy of its Levi-Civita connection is strictly less than
$\operatorname{Sp}(n)$.

Definition 2.4:  Let $M$ be a hypercomplex manifold, and $L$ a quaternion
satisfying $L^{2}=-1$. Then $L=aI+bJ+cK$, $a^{2}+b^{2}+c^{2}=1$. The
corresponding complex structure on $M$ is called an induced complex structure.
The space $M$, considered as a complex manifold, is denoted by $(M,L)$. The
set of induced complex structures is naturally identified with $S^{2}$, which
we often consider as ${\mathbb{C}}P^{1}$ with the standard complex structure.

Definition 2.5:  ([V1]) Let $X\subset M$ be a closed subset of a hyperkähler
manifold $M$. Then $X$ is called trianalytic if $X$ is a complex analytic
subset of $(M,L)$ for every induced complex structure $L$.

Trianalytic subvarieties were a subject of a long study. Most importantly,
consider a generic induced complex structure $L$ on $M$. Then all closed
complex subvarieties of $(M,L)$ are trianalytic. Moreover, a trianalytic
subvariety can be canonically desingularized ([V2]), and this
desingularization is hyperkähler.

Theorem 2.6:  ([V1], [V3]) Let $(M,I,J,K)$ be a hyperkähler manifold (not
necessarily compact). Then there exists a countable subset
$R\subset{\mathbb{C}}P^{1}$, such that for any induced complex structure
$L\notin R$, all compact complex subvarieties of $(M,L)$ are trianalytic.

Remark 2.7: For hypercomplex manifolds, 2.1 is (generally speaking) false,
though for manifolds with trivial canonical bundle a weaker form of this
result was obtained ([SV]).

###  2.2 Twistor spaces

Definition 2.8: Let $M$ be a Riemannian 4-manifold. Consider the action of the
Hodge $*$-operator: $*:\;\Lambda^{2}M{\>\longrightarrow\>}\Lambda^{2}M$. Since
$*^{2}=1$, the eigenvalues are $\pm 1$, and one has a decomposition
$\Lambda^{2}M=\Lambda^{+}M\oplus\Lambda^{-}M$ onto selfdual ($*\eta=\eta$) and
anti-selfdual ($*\eta=-\eta$) forms.

Remark 2.9: If one changes the orientation of $M$, leaving metric the same,
$\Lambda^{+}M$ and $\Lambda^{-}M$ are exchanged. Therefore, their dimensions
are equal, and $\dim\Lambda^{2}M=6$ implies $\dim\Lambda^{\pm}(M)=3$.

Remark 2.10: Using the isomorphism $\Lambda^{2}M=\mathfrak{so}(TM)$, we
interpret $\eta\in\Lambda^{2}_{m}M$ as an endomorphisms of $T_{m}M$. Then the
unit vectors $\eta\in\Lambda^{+}_{m}M$ correspond to oriented, orthogonal
complex structures on $T_{m}M$.

Definition 2.11: Let $\operatorname{Tw}(M):=S\Lambda^{+}M$ be the set of unit
vectors in $\Lambda^{+}M$. At each point $(m,s)\in\operatorname{Tw}(M)$,
consider the decomposition $T_{m,s}\operatorname{Tw}(M)=T_{m}M\oplus
T_{s}S\Lambda^{+}_{m}M$, induced by the Levi-Civita connection. Let $I_{s}$ be
the complex structure on $T_{m}M$ induced by $s$, $I_{S\Lambda^{+}_{m}M}$ the
complex structure on $S\Lambda^{+}_{m}M=S^{2}$ induced by the metric and
orientation, and

${\cal
I}:\;T_{m,s}\operatorname{Tw}(M){\>\longrightarrow\>}T_{m,s}\operatorname{Tw}(M)$

be equal to ${\cal I}_{s}\oplus I_{S\Lambda^{+}_{m}M}$. An almost complex
manifold $(\operatorname{Tw}(M),{\cal I})$ is called the twistor space of $M$.

Given a hyperkähler or hypercomplex manifold (2.1), one defines its twistor
space in a similar manner.

Definition 2.12: A twistor space $\operatorname{Tw}(M)$ of a hypercomplex
manifold $M$ is $S^{2}\times M$ equipped with a complex structure which is
defined as follows. Consider the complex structure $I_{m}:T_{m}M\to T_{m}M$ on
$M$ induced by $J\in S^{2}\subset{\mathbb{H}}$. Let $I_{J}$ denote the complex
structure on $S^{2}={\mathbb{C}}P^{1}$. The operator
$I_{\operatorname{Tw}}=I_{m}\oplus I_{J}:T_{x}\operatorname{Tw}(M)\to
T_{x}\operatorname{Tw}(M)$ satisfies
$I_{\operatorname{Tw}}^{2}=-\operatorname{Id}$. It defines an almost complex
structure on $\operatorname{Tw}(M)$.

The almost complex structure on the twistor space of a Riemannian 4-manifold
$X$ is integrable whenever $X$ is anti-selfdual ([AHS]) For a hypercomplex
manifold it is integrable as well ([K]). Twistor spaces are the main example
of the geometries we are working with.

###  2.3 Geometry of twistor spaces

Proposition 2.13:  Let $\operatorname{Tw}(M)$ be a twistor space of a compact
hyperkähler manifold. Then

(i)

$\operatorname{Tw}(M)$ is non-Kähler.

(ii)

The algebraic dimension of $\operatorname{Tw}(M)$ is 1.

Proof of (i): Let $\omega$ be the standard Hermitian form of
$\operatorname{Tw}(M)$. Then $dd^{c}\omega$ is a positive (2,2)-form ([KV,
(8.2)]). For any Kähler form $\omega_{0}$ on $\operatorname{Tw}(M)$, this
would imply

$\int_{\operatorname{Tw}(M)}d\left(\omega_{0}^{\dim_{\mathbb{C}}M-1}\wedge
d^{c}\omega\right)=\int_{\operatorname{Tw}(M)}\omega_{0}^{\dim_{\mathbb{C}}M-1}\wedge
dd^{c}\omega>0,$

which is impossible by Stokes’ theorem.

Proof of (ii): See 4.4.

###  2.4 Quasilines and ample curves in the twistor spaces

Definition 2.14: An ample rational curve on a complex manifold $M$ is a smooth
curve $S\cong{\mathbb{C}}P^{1}\subset M$ such that its normal bundle $N\\!S$
satisfies $N\\!S=\bigoplus_{k=1}^{n-1}{\cal O}(i_{k})$, with all $i_{k}>0$
(see [Ko]). It is called a quasiline if all $i_{k}=1$.

Claim 2.15: Let $M$ be a twistor space of a hyperkähler or 4-dimensional ASD
manifold, $m\in M$ a point, and $S_{m}$ the corresponding $S^{2}$ in
$\operatorname{Tw}(M)={\mathbb{C}}P^{1}\times M$. Then $S_{m}$ is a quasiline.

Proof: Since the claim is essentially infinitesimal, it suffices to check it
when $M$ is flat. Then $\operatorname{Tw}(M)=\operatorname{Tot}({\cal
O}(1)^{\oplus 2n})\cong{\mathbb{C}}P^{2n+1}\backslash{\mathbb{C}}P^{2n-1}$,
and $S_{m}$ is a section of ${\cal O}(1)^{\oplus 2n}$.

Existence of quasilines in twistor spaces is a very strong condition, and can
be used to obtain all kinds of geometric information; for example, see [C] and
[V4].

##  3 Holography principle for manifolds
with ample rational curves

###  3.1 Holography principle for line bundles

Throughout this paper, all neighbourhoods and manifolds are silently assumed
to be connected. One of the main results of the present paper is the following
theorem.

Theorem 3.1:  (holography principle for line bundles) Let $S\subset M$ be an
ample rational curve in a simply connected complex manifold, which is covered
by smooth, ample deformations of $S$, and $L$ a holomorphic line bundle on
$M$. Consider an open neighbourhood $U\supset S$. Then the restriction map
$H^{0}(M,L){\>\longrightarrow\>}H^{0}(U,L)$ is an isomorphism.

We deduce 3.1 from the following local result (3.1).

Remark 3.2:  Since $S$ is an ample curve, $S$ can be deformed in any normal
direction. Therefore, there exists an open neighbourhood $U\supset S$ which is
contained in a union of the set ${\mathfrak{S}}$ of all smooth, ample
deformations of $S$ intersecting $S$. Further on, we choose this neighbourhood
in such a manner that any $S_{1}\in{\mathfrak{S}}$ can be connected to $S$ by
a continuous family of deformations intersecting $S$.

Proposition 3.3:  Let $S\subset M$ be an ample rational curve in a connected
complex manifold, which is covered by deformations of $S$, and $L$ a
holomorphic line bundle on $M$. Consider a neighbourhood $U\supset S$ which is
is contained in a union of the set of all deformations of $S$ intersecting $S$
(3.1). Then for any smaller open neighbourhood $V\subset U$ of $S$, the
restriction map $H^{0}(U,L){\>\longrightarrow\>}H^{0}(V,L)$ is an isomorphism.

Proof of an implication “3.1 $\Rightarrow$ 3.1”.

Step 1: Choose a continuous, connected family $S_{b}$ of ample curves
parametrized by $B$ such that $\bigcup_{b\in B}S_{b}=M$, and choose a tubular
neighbourhood $U_{b}$ for each $S_{b}$, continuously depending on $b$. Then
the intersection $U_{b}\cap U_{b^{\prime}}$ for sufficiently close
$b,b^{\prime}$ always contains $S_{b}$ and $S_{b^{\prime}}$. By 3.1, $U_{b}$
can be chosen in such a way that $H^{0}(U_{b}\cap
U_{b^{\prime}},L)=H^{0}(U_{b},L)=H^{0}(U_{b^{\prime}},L)$.

Step 2: Since $B$ is connected, all the spaces $H^{0}(U_{b},L)$ are
isomorphic, and these isomorphisms are compatible with the restrictions to the
intersections $U_{b}\cap U_{b^{\prime}}$. Let now $f\in H^{0}(U_{b},L)$, and
let $\tilde{M}_{f}$ be the domain of holomorphy for $f$, that is, a maximal
domain (non-ramified over $M$) such that $f$ admits a holomorphic extension to
$\tilde{M}_{f}$. Since $\cup U_{b}=M$, and $f$ can be holomorphically extended
to any $U_{b}$, the domain $\tilde{M}_{f}$ is a covering of $M$. Now, 3.1
follows, because $M$ is simply connected.

###  3.2 The local holography principle

To prove 3.1 it remains to prove 3.1. If $\deg
L{\left|{}_{{\phantom{|}\\!\\!}_{S}}\right.}<0$, the statement of 3.1 is
vacuous. Therefore, we may always assume that $\deg
L{\left|{}_{{\phantom{|}\\!\\!}_{S}}\right.}\geqslant 0$, hence
$l{\left|{}_{{\phantom{|}\\!\\!}_{S}}\right.}$ is generated by global
sections. Let $S(M)$ be the space of deformations of the ample curve $S$ which
remain smooth, and $SS(M)$ the space of pairs $\\{(x,S_{1}):\ \ S_{1}\in
S(M),x\in S_{1}\\}$. Consider the natural forgetful maps
$\tau_{1}:\;SS(M){\>\longrightarrow\>}M$,
$\tau_{2}:\;SS(M){\>\longrightarrow\>}S(M)$, and let $E$ be the bundle
${\tau_{2}}_{*}\tau_{1}^{*}L$ on $S(M)$. Denote by $\deg_{S}L$ the degree of
the restriction of $L$ to $S$. Since $\dim H^{0}(S_{1},L)=\deg L+1$, $E$ is a
$(\deg L+1)$-dimensional vector bundle. Given a section $f$ of $L$ on $M$,
denote the corresponding section of $E$ by
$\tilde{f}:={\tau_{2}}_{*}\tau_{1}^{*}f$. When $\deg_{S}L=d$, the value of
$\tilde{f}$ at $S_{1}$ is uniquely determined by the restriction of $f$ to any
$d+1$ distinct points of $S_{1}$. Indeed,
$E{\left|{}_{{\phantom{|}\\!\\!}_{S_{1}}}\right.}$ is $d+1$-dimensional, and
any section $h\in H^{0}(S_{1},L)$ is uniquely determined by its values in
$d+1$ points. This gives a map

$L{\left|{}_{{\phantom{|}\\!\\!}_{z_{1}}}\right.}\times
L{\left|{}_{{\phantom{|}\\!\\!}_{z_{2}}}\right.}\times...\times
L{\left|{}_{{\phantom{|}\\!\\!}_{z_{d+1}}}\right.}\tilde{\>\longrightarrow\>}H^{0}(S_{1},L).$
(3.1)

Now, let $f$ be a section of $L$ on $V$. For any $S_{1}\in S(M)$ intersecting
$V$, we choose $d_{1}$ distinct points $z_{1},...,z_{d+1}\in S_{1}\cap V$, and
consider the section
$\varphi\left(f{\left|{}_{{\phantom{|}\\!\\!}_{z_{1}}}\right.},f{\left|{}_{{\phantom{|}\\!\\!}_{z_{2}}}\right.},...,f{\left|{}_{{\phantom{|}\\!\\!}_{z_{d+1}}}\right.}\right)\in
H^{0}(S_{1},L)$ defined using (3.1). When $S_{1}\subset V$, this section is
independent from the choice of $z_{1},...,z_{d+1}\in S_{1}$. Let $R_{V}$ be a
connected component of the set of all $S_{1}\in S(M)$ intersecting $V$ and
containing $S$. Since the map

$(z_{1},...,z_{d+1}){\>\longrightarrow\>}\varphi\left(f{\left|{}_{{\phantom{|}\\!\\!}_{z_{1}}}\right.},f{\left|{}_{{\phantom{|}\\!\\!}_{z_{2}}}\right.},...,f{\left|{}_{{\phantom{|}\\!\\!}_{z_{d+1}}}\right.}\right)$

is holomorphic and independent from the choice of $z_{1},...,z_{d+1}$ on an
open subset of $R_{V}$, it is independent of $z_{1},...,z_{d+1}$ everywhere on
$R_{V}$. This gives a section $\tilde{f}\in H^{0}(R_{V},E)$ extending the
section $\tilde{f}:={\tau_{2}}_{*}\tau_{1}^{*}f\in H^{0}(S(V),E)$.

By construction, $U$ is contained in a connected part $U_{1}$ of the union of
all deformations of $S$ intersecting $V$. To extend $f$ from $V$ to $U$, we
use $\tilde{f}$ to obtain an extension of $f$ to $U_{1}$, as follows.

Any section $g\in H^{0}(R_{V},E)$ gives a function $\psi_{g}$ mapping a pair
$(x,S_{1})$, $x\in S_{1}\in R_{V}$ to
$g(S_{1}){\left|{}_{{\phantom{|}\\!\\!}_{x}}\right.}\in
L{\left|{}_{{\phantom{|}\\!\\!}_{x}}\right.}$. For the section $\tilde{f}$
constructed above, $\psi_{\tilde{f}}(x,S_{1})$ is independent from the choice
of $S_{1}$ whenever $S_{1}$ lies in $V$. The same analytic continuation
argument as above implies that $\psi_{\tilde{f}}(x,S_{1})$ is independent of
$S_{1}$ everywhere. For any $x\in U_{1}$, the set $f\in
L{\left|{}_{{\phantom{|}\\!\\!}_{x}}\right.}$ equal to
$\psi_{\tilde{f}}(x,S_{1})$, where $S_{1}\in R_{V}$ is an arbitrary curve
passing through $x$. This gives an extension of $f$ to $U_{1}$. 3.1 is proven.
We finished the proof of 3.1.

###  3.3 Holography principle for meromorphic functions

The following theorem is proven in the same way as 3.1. Given a complex
variety $M$, we denote the field of meromorphic functions on $M$ by
$\operatorname{Mer}(M)$.

Theorem 3.4:  Let $S\subset M$ be an ample curve in a simply connected complex
manifold, and $U\supset S$ a connected neighbourhood of $S$. Suppose that $M$
is covered by the union of all smooth, ample deformations of $S$. Then the
restriction map
$\operatorname{Mer}(M){\>\longrightarrow\>}\operatorname{Mer}(U)$ is an
isomorphism.

Proof: The same argument as used to deduce 3.1 from 3.1 can be used to reduce
3.3 to the following statement.

Proposition 3.5:  Let $S\subset M$ be an ample rational curve in a connected
complex manifold, which is covered by smooth, ample deformations of $S$.
Consider a neighbourhood $U\supset S$ which is contained in a union
${\mathfrak{S}}$ of all deformations of $S$ intersecting $S$ (3.1). Then for
any smaller open neighbourhood $V\subset U$ of $S$, the restriction map
$\operatorname{Mer}(U){\>\longrightarrow\>}\operatorname{Mer}(V)$ is an
isomorphism.

Proof: Define the degree $\deg_{S}(f)$ of a meromorphic function
$f\in\operatorname{Mer}(V)$ as the degree of the pole divisor of
$f{\left|{}_{{\phantom{|}\\!\\!}_{S_{1}}}\right.}$ for any deformation $S_{1}$
of $S$ transversal to the pole divisor of $f$. Denote by
$\operatorname{Mer}_{d}(V)$ the space meromorphic functions of degree
$\leqslant d$. To prove 3.3 it would suffice to show that the restriction map
$\operatorname{Mer}_{d}(U){\>\longrightarrow\>}\operatorname{Mer}_{d}(V)$ is
an isomorphism, for all $d$.

For each rational curve $S_{1}$, a degree $\leqslant d$ meromorphic function
is uniquely determined by its values in any $d+1$ distinct points on $S_{1}$.
Given a meromorphic function $f\in\operatorname{Mer}_{d}(V)$, and a
deformation $S_{1}$ of $S$ intersecting $V$, we can extend
$f{\left|{}_{{\phantom{|}\\!\\!}_{S_{1}\cap V}}\right.}$ to a degree $d$
meromorphic function $f_{1}$ on $S_{1}$ by computing its values at $d+1$
distinct points $z_{1},...,z_{d+1}$ of $S_{1}\cap V$. Whenever $S_{1}$ is in
$V$, this procedure gives $f{\left|{}_{{\phantom{|}\\!\\!}_{S_{1}}}\right.}$.
By analytic continuation, the values of $f_{1}(z)$ at any $z\in S_{1}$ are
independent from the choice of $z_{i}$ and $S_{1}$.

We have shown that $f_{1}$ is a well-defined meromorphic function on the union
$U_{1}$ of all deformations of $S$ intersecting $V$. By construction, on $V$
we have $f_{1}=f$. Since $U_{1}$ contains $U$, this implies that $f$ can be
extended from $V$ to $U$.

##  4 Moishezon twistor spaces

###  4.1 Vector bundles in a neighbourhood of an ample curve

Proposition 4.1:  Let $S\subset M$ be an ample rational curve, $U\supset S$
its neighbourhood, and $B$ a holomorphic bundle on $U$. Then $H^{0}(U,B)$ is
finite-dimensional.

Proof. Step 1: Let $I_{S}\subset{\cal O}_{U}$ be the ideal sheaf of $S$, and
$J^{r}_{S}(B):=B/I^{r+1}_{S}B$ the sheaf of $r$-jets of the sections of $B$.
Since $I^{r}_{S}/I^{r+1}_{S}=\operatorname{Sym}^{r}(N^{*}S)$, the bundle
$I^{r}_{S}/I^{r+1}_{S}$ is a direct sum of ${\cal O}(k_{i})$ with $k_{i}<-r$.
Therefore, $H^{0}(I^{r}_{S}B/I^{r+1}_{S}B)=0$ for $r$ sufficiently big.

Step 2: The sheaf $I^{r}_{S}B$ admits a filtration $I^{r}_{S}B\supset
I^{r+1}_{S}B\supset I^{r+2}_{S}B\supset...$ with associated graded sheaves
$I^{r}_{S}B/I^{r+1}_{S}B$ having no sections for $r\gg 0$. Therefore,
$H^{0}(I^{r}_{S}B)=0$ for sufficiently byg $r$.

Step 3: If $H^{0}(U,B)$ is infinite-dimensional, the map

$H^{0}(U,B){\>\longrightarrow\>}H^{0}(B/I^{r}_{S}(B))$

cannot be injective. Then, for each $r$, there exists a non-zero section with
vanishing $r$-jet: $f_{r}\in H^{0}(U,I^{r+1}_{S}B)$. This is impossible, as
shown in Step 2.

This proof is effective, and gives the following bound on the dimension of the
space of sections of $B$.

Corollary 4.2:  Let $S\subset M$ be an ample rational curve, $U\supset S$ its
neighbourhood, and $B$ a holomorphic bundle on $U$. Then

$\dim H^{0}(U,B)\leqslant\dim
H^{0}\left(S,\bigoplus_{d}\operatorname{Sym}^{d}(N^{*}S)\otimes_{{\cal
O}_{S}}B{\left|{}_{{\phantom{|}\\!\\!}_{S}}\right.}\right).$

The same argument can be applied to degree $d$ meromorphic functions. Recall
that the degree $\deg_{S}(f)$ of a meromorphic function
$f\in\operatorname{Mer}(V)$ is the degree of the pole divisor of
$f{\left|{}_{{\phantom{|}\\!\\!}_{S_{1}}}\right.}$ for any deformation of $S$
transversal to the pole divisor of $f$. We denote the space of meromorphic
functions of degree $\leqslant d$ on $U$ by $\operatorname{Mer}_{d}(U)$.

Corollary 4.3:  Let $S\subset M$ be an ample rational curve, and $U\supset S$
its neighbourhood. Then

$\dim\operatorname{Mer}_{d}(U)\leqslant\dim
H^{0}\left(S,\operatorname{Sym}^{\leqslant d}(N^{*}S)\otimes_{{\cal
O}_{S}}{\cal O}(d)\right).$ (4.1)

###  4.2 Algebraic dimension of the field of meromorphic functions

The conormal bundle $N^{*}\\!S$ is negative; clearly, the dimension (4.1) is
maximal when $S$ is a quasiline, and $N^{*}\\!S=\oplus{\cal O}(1)$. In this
case, the bound (4.1) is realized for a rational line in ${\mathbb{C}}P^{n}$.
Indeed, for a rational line $S$ in ${\mathbb{C}}P^{n}$, the sheaf of algebraic
functions in a neighbourhood of $S$ is isomorphic to of
$\bigoplus_{i}\operatorname{Sym}^{i}(N^{*}\\!S)$. This implies the following
simple numerical result.

Claim 4.4: Let $S\subset M$ be an ample rational curve, and $U\supset S$ its
neighbourhood. Then

$\dim\operatorname{Mer}_{d}(U)\leqslant\dim H^{0}({\mathbb{C}}P^{n},{\cal
O}(d)).$ (4.2)

where $n=\dim M$.

Corollary 4.5:  Let $M$ be a complex variety containing an ample rational
curve. Then the transcendence degree of $\operatorname{Mer}(M)$ satisfies
$\deg_{tr}\operatorname{Mer}(M)\leqslant\dim M$.

Proof: Consider the graded ring $\bigoplus_{d}\operatorname{Mer}_{d}(M)$.
Since $\dim\operatorname{Mer}_{d}(U)\leqslant\dim
H^{0}({\mathbb{C}}P^{n},{\cal O}(d))$, the Krull dimension of this ring is
$\leqslant n$. Therefore, the transcendence degree of its ring of fractions is
also bounded by $n$.

This observation is not new: it was known already to Hartshorne (in the
context of formal neighbourhoods). Applied to complex analytic spaces,
Hartshorne’s theorem can be stated as follows.

Theorem 4.6:  Let $S\subset M$ be a connected, positive-dimensional, smooth
subvariety in a complex manifold. Assume that the normal bundle of $S$ is
ample. Then the transcendence degree of the field $\operatorname{Mer}(M)$ of
meromorphic functions is no bigger than the dimension of $M$:

$\deg_{tr}\operatorname{Mer}(M)\leqslant\dim(M).$

Moreover, if equality is reached, $\operatorname{Mer}(M)$ is a finitely
generated extension of ${\mathbb{C}}$.

Proof: [Har, Theorem 6.7]; see also [KST].

###  4.3 Moishezon manifolds and ample rational curves

Hartshorne’s methods are already sufficient to prove the following general
result.

Proposition 4.7:  Let $S\subset M$ be an ample rational curve in a simply
connected complex manifold. Assume that
$\deg_{tr}\operatorname{Mer}(M)=\dim(M).$ Then there exists a meromorphic map
$\varphi:\;M{\>\longrightarrow\>}Z_{0}$ to an open subset of a projective
variety $M$, which is bijective onto its image outside of a complex analytic
subset of positive codimension.

Proof: By Hartshorne’s theorem (4.2), $\operatorname{Mer}(M)$ is a finitely
generated extension of ${\mathbb{C}}$. Let $\xi_{1},...,\xi_{N}$ be generators
of $\operatorname{Mer}(M)$, $D_{i}$ their pole divisors, and $L_{i}:={\cal
O}(D_{i})$ the corresponding line bundles. Then $\xi_{i}$ can be considered as
sections of $L_{i}$, and $\xi_{1},...,\xi_{N}$ – as sections of
$L:=\bigotimes_{i}L_{i}$. Consider now the subring of
$\bigoplus_{d}H^{0}(M,L^{d})$ generated by $\xi_{i}$, and let $Z$ be its
spectre. Clearly, $\dim M=\dim Z$, $Z$ is projective, and the natural rational
map $M\stackrel{{\scriptstyle\varphi}}{{{\>\longrightarrow\>}}}Z$ induces an
isomorphism
$\operatorname{Mer}(Z)\tilde{\>\longrightarrow\>}\operatorname{Mer}(M)$.

Let $\tilde{M}$ be a resolution of the base set of $\varphi$, such that
$\tilde{M}\stackrel{{\scriptstyle\tilde{\varphi}}}{{{\>\longrightarrow\>}}}Z$
is holomorphic. If $\tilde{\varphi}$ is ramified at some divisor $D$ in
$\varphi(\tilde{M})$, this divisor can be extended to $Z$ using 3.3 applied to
$U=\varphi(\tilde{M})\subset Z$. Taking the corresponding ramified covering
$\tilde{Z}$ of $Z$, we obtain a holomorphic map from $\tilde{M}$ to
$\tilde{Z}$, which is impossible, because $\operatorname{Mer}(\tilde{Z})$ is
strictly bigger than $\operatorname{Mer}(Z)$, and
$\operatorname{Mer}(\tilde{M})=\operatorname{Mer}(Z)$. Therefore,
$\tilde{\varphi}$ is bijective to its impage at its general point. Openness of
its image is a general property of bimeromorphic maps.

Theorem 4.8:  Let $S\subset M$ be an ample rational curve in a simply
connected complex manifold. Assume that
$\deg_{tr}\operatorname{Mer}(M)=\dim(M).$ Then there exists an open embedding
of $M$ to a Moishezon variety.

Proof: From 4.3 we obtain a line bundle $L$ on $M$ inducing a bimeromorphic
map $\varphi:\;M{\>\longrightarrow\>}Z_{0}$ to an open subset of a projective
variety $Z$. Resolving the base points of the inverse map if necessary, we may
assume that the inverse map $\psi:\;Z_{0}{\>\longrightarrow\>}M$ is
holomorphic. Then, $M$ is obtained from $Z_{0}$ by blowing down a certain
number of exceptional subvarieties $E_{i}$, obtained as common zero sets of a
certain number of meromorphic functions. Applying 3.3 to $Z_{0}\subset Z$, we
extend the meromorphic functions and the corresponding subvarieties $E_{i}$
and obtain closed exceptional subvarieties $E_{i}^{\prime}\subset Z$. Blowing
these down, we obtain a Moishezon variety which contains $M$ as an open
subset.

###  4.4 Moishezon twistor spaces

Definition 4.9:  Let $M=\operatorname{Tw}(X)$ be a twistor space of a simply
connected hyperkähler, hypercomplex, quaternionic or or 4-dimensional anti-
selfdual manifold, not necessarily compact. We say that $M$ is a Moishezon
twistor space if $\deg_{tr}\operatorname{Mer}(M)=\dim(M)$ (see 4.2).

From 4.3, we immediately obtain the following corollary.

Corollary 4.10: Let $M$ be a Moishezon twistor space. Then $M$ admits an open
embedding to a Moishezon variety $M_{1}$. Moreover, $M_{1}$ is unique up to a
bimeromorphic equivalence.

Proof: The open embedding to a Moishezon variety follows from 4.3, and its
uniqueness is implied by an isomorphism
$\operatorname{Mer}(M)=\operatorname{Mer}(M_{1})$ (3.3).

It is easy to construct an example of a twistor manifold which does not belong
to this class. The twistor space of a K3 surface, and, more generally, any
compact hyperkähler manifold is never Moishezon.

Theorem 4.11:  Let $M$ be a compact hyperkähler manifold, and
$\operatorname{Tw}(M)$ its twistor space. Then
$\deg_{tr}\operatorname{Mer}(\operatorname{Tw}(M))=1$.

Proof: Let $Z\subset\operatorname{Tw}(M)$ be any divisor, and
$R\subset{\mathbb{C}}P^{1}$ a countable subset constructed in 2.1. For any
induced complex structure $L\notin R$, all complex subvarieties of $(M,L)$ are
even-dimensional. Therefore, $Z$ intersects the twistor fiber
$(M,L)=\pi^{-1}(L)\subset\operatorname{Tw}(M)$ non-transversally, or not at
all. By Thom’s transversality theorem, the intersection $Z$ with all fibers of
$\pi$ except a finite number is transversal. This means that $Z$ can intersect
only finitely many of the fibers of $\pi$. However, all these fibers are
irreducible divisors. Therefore, $D$ is a union of several fibers of $\pi$.
Since a meromorphic function is uniquely determined by its pole or zero
divisor, all meromorphic functions on $\operatorname{Tw}(M)$ are pull-backs of
meromorphic functions on ${\mathbb{C}}P^{1}$.

Remark 4.12: When $M$ is hypercomplex, no effective bounds on the
transcendence degree $\deg_{tr}\operatorname{Mer}(\operatorname{Tw}(M))$ are
known. We conjecture (based on empirical evidence) that the twistor space of a
compact hypercomplex manifold is not Moishezon, but this conjecture seems to
be difficult.

###  4.5 Twistor spaces and hyperkähler reduction

We recall the definition of hyperkähler reduction, following [HKLR] and [Nak].
This material is fairly standard.

We denote the Lie derivative along a vector field as
$\operatorname{Lie}_{x}:\;\Lambda^{i}M{\>\longrightarrow\>}\Lambda^{i}M$, and
contraction with a vector field by
$i_{x}:\;\Lambda^{i}M{\>\longrightarrow\>}\Lambda^{i-1}M$. Recall the Cartan’s
formula:

$d\circ i_{x}+i_{x}\circ d=\operatorname{Lie}_{x}$

Let $(M,\omega)$ be a symplectic manifold, $G$ a Lie group acting on $M$ by
symplectomorphisms, and $\mathfrak{g}$ its Lie algebra. For any
$g\in{\mathfrak{g}}$, denote by $\rho_{g}$ the corresponding vector field.
Cartan’s formula gives $\operatorname{Lie}_{\rho_{g}}\omega=0$, hence
$d(i_{\rho_{g}}(\omega))=0$. We obtain that $i_{\rho_{g}}(\omega)$ is closed,
for any $g\in{\mathfrak{g}}$.

Definition 4.13: A Hamiltonian of $g\in{\mathfrak{g}}$ is a function $h$ on
$M$ such that $dh=i_{\rho_{g}}(\omega)$.

Definition 4.14: $(M,\omega)$ be a symplectic manifold, $G$ a Lie group acting
on $M$ by symplectomorphisms. A moment map $\mu$ of this action is a linear
map ${\mathfrak{g}}{\>\longrightarrow\>}C^{\infty}M$ associating to each
$g\in{\mathfrak{g}}$ its Hamiltonian.

Remark 4.15: It is more convenient to consider $\mu$ as an element of
${\mathfrak{g}}^{*}\otimes_{\mathbb{R}}C^{\infty}M$, or, as it is usually
done, a function on $M$ with values in ${\mathfrak{g}}^{*}$.

Remark 4.16: Note that the moment map always exists, if $M$ is simply
connected.

Definition 4.17: A moment map $M{\>\longrightarrow\>}{\mathfrak{g}}^{*}$ is
called equivariant if it is equivariant with respect to the coadjoint action
of $G$ on ${\mathfrak{g}}^{*}$.

Remark 4.18:
$M\stackrel{{\scriptstyle\mu}}{{{\>\longrightarrow\>}}}{\mathfrak{g}}^{*}$ is
a moment map if and only if for all $g\in{\mathfrak{g}}$, $\langle
d\mu,g\rangle=i_{\rho_{g}}(\omega)$. Therefore, a moment map is defined up to
a constant ${\mathfrak{g}}^{*}$-valued function. An equivariant moment map is
is defined up to a constant ${\mathfrak{g}}^{*}$-valued function which is
$G$-invariant.

Definition 4.19: A $G$-invariant $c\in\mathfrak{g}^{*}$ is called central.

Claim 4.20: An equivariant moment map exists whenever
$H^{1}(G,{\mathfrak{g}}^{*})=0$. In particular, if $G$ is reductive and $M$ is
simply connected, an equivariant moment map is always possible to define.

Definition 4.21: Let $(M,\omega)$ be a symplectic manifold, $G$ a compact Lie
group acting on $M$ by symplectomorphisms,
$M\stackrel{{\scriptstyle\mu}}{{{\>\longrightarrow\>}}}{\mathfrak{g}}^{*}$ an
equivariant moment map, and $c\in{\mathfrak{g}}^{*}$ a central element. The
quotient $\mu^{-1}(c)/G$ is called the symplectic reduction of $M$, denoted by
$M{/\\!\\!/}G$.

Claim 4.22: The symplectic quotient $M{/\\!\\!/}G$ is a symplectic manifold of
dimension $\dim M-2\dim G$.

Theorem 4.23: Let $(M,I,\omega)$ be a Kähler manifold, $G_{\mathbb{C}}$ a
complex reductive Lie group acting on $M$ by holomorphic automorphisms, and
$G$ is a compact form of $G_{\mathbb{C}}$ acting isometrically. Then
$M{/\\!\\!/}G$ is a Kähler orbifold.

Remark 4.24: In such a situation, $M{/\\!\\!/}G$ is called the Kähler
quotient, or GIT quotient.

Remark 4.25: The points of $M{/\\!\\!/}G$ are in bijective correspondence with
the orbits of $G_{\mathbb{C}}$ which intersect $\mu^{-1}(c)$. Such orbits are
called polystable, and the intersection of a $G_{\mathbb{C}}$-orbit with
$\mu^{-1}(c)$ is a $G$-orbit.

Definition 4.26: Let $G$ be a compact Lie group, $\rho$ its action on a
hyperkähler manifold $M$ by hyperkähler isometries, and ${\mathfrak{g}}^{*}$ a
dual space to its Lie algebra. A hyperkähler moment map is a $G$-equivariant
smooth map $\mu:M\to{\mathfrak{g}}^{*}\otimes{\mathbb{R}}^{3}$ such that
$\langle\mu_{i}(v),g\rangle=\omega_{i}(v,d\rho(g))$, for every $v\in TM$,
$g\in{\mathfrak{g}}$ and $i=1,2,3$, where $\omega_{i}$ are three Kähler forms
associated with the hyperkähler structure.

Definition 4.27: Let $\xi_{1},\xi_{2},\xi_{3}$ be three $G$-invariant vectors
in ${\mathfrak{g}}^{*}$. The quotient manifold
$M{/\\!\\!/\\!\\!/}G:=\mu^{-1}(\xi_{1},\xi_{2},\xi_{3})/G$ is called the
hyperkähler quotient of $M$.

Theorem 4.28: ([HKLR]) The quotient $M{/\\!\\!/\\!\\!/}G$ is hyperkaehler.

Proof: We sketch the proof of Hitchin-Karlhede-Lindström-Roček theorem,
because we make use of it further on.

Let $\Omega:=\omega_{J}+\sqrt{-1}\>\omega_{K}$. This is a holomorphic
symplectic (2,0)-form on $(M,I)$. Let $\mu_{J},\mu_{K}$ be the moment map
associated with $\omega_{J},\omega_{K}$, and
$\mu_{\mathbb{C}}:=\mu_{J}+\sqrt{-1}\>\mu_{K}$. Then $\langle
d\mu_{\mathbb{C}},g\rangle=i_{\rho_{g}}(\Omega)$ Therefore,
$d\mu_{\mathbb{C}}\in\Lambda^{1,0}(M,I)\otimes{\mathfrak{g}}^{*}$. This
implies that the map $\mu_{\mathbb{C}}$ is holomorphic. It is called a
holomorphic moment map.

By definition, $M{/\\!\\!/\\!\\!/}G=\mu_{{\mathbb{C}}}^{-1}(c){/\\!\\!/}G$,
where $c\in{\mathfrak{g}}^{*}\otimes_{\mathbb{R}}{\mathbb{C}}$ is a central
element. This is a Kähler manifold, because it is a Kähler quotient of a
Kähler manifold.

We obtain 3 complex structures $I,J,K$ on the hyperkähler quotient
$M{/\\!\\!/\\!\\!/}G$. They are compatible in the usual way, as seen from a
simple local computation.

Theorem 4.29:  Let $V$ be a quaterionic Hermitian vector space, and
$G\subset\operatorname{Sp}(V)$ a compact Lie group acting on $V$ by
quaternionic isometries. Denote by $M$ the hyperkähler reduction of $V$. Then
$\operatorname{Tw}(M)$ is a Moishezon twistor space, in the sense of 4.4.

Proof: The holomorphic symplectic form on $(V,I)$ depends on $I$
holomorphically, giving a section

$\Omega_{tw}\in\Omega^{2}_{\pi}(\operatorname{Tw}(V))\otimes_{{\cal
O}_{\operatorname{Tw}(V)}}\pi^{*}{\cal O}(2).$

Here, $\pi:\;\operatorname{Tw}(V){\>\longrightarrow\>}{\mathbb{C}}P^{1}$ is
the twistor projection, and $\Omega^{2}_{\pi}(\operatorname{Tw}(V))$ the sheaf
of fiberwise holomorphic 2-forms. Consider the fiberwise holomorphic moment
map given by this form,
$\mu_{tw}:\;\operatorname{Tw}(V){\>\longrightarrow\>}\operatorname{Tot}({\mathfrak{g}}^{*}\otimes_{\mathbb{C}}{\cal
O}(2))$. Replacing the moment map by its translate, we can always assume that
$M{/\\!\\!/\\!\\!/}G=\mu_{{\mathbb{C}}}^{-1}(0){/\\!\\!/}G$ (that is, we
assume that the central vector $c$ used to define $M{/\\!\\!/\\!\\!/}G$
vanishes). Then $\operatorname{Tw}(M)$ is obtained as the space of polystable
$G_{\mathbb{C}}$-orbits in $\mu_{tw}^{-1}(0)\subset\operatorname{Tw}(V)$.
Here, “polystability” of an orbit is understood as the non-emptiness of the
intersection of this orbit with $\mu^{-1}(0)$; the set of such orbits is open.

The space
$\operatorname{Tw}(V)={\mathbb{C}}P^{2n+1}\backslash{\mathbb{C}}P^{2n-1}$ is
quasiprojective. Averaging the ring of rational functions with $G$, we obtain
that the field $G_{\mathbb{C}}$-invariant rational functions on
$\operatorname{Tw}(V)$ has dimension $\dim\mu_{tw}^{-1}(0)-\dim
G_{\mathbb{C}}=\dim\operatorname{Tw}(M)$, and hence $\operatorname{Tw}(M)$ is
Moishezon.

Corollary 4.30: Let $U$ be an open subset of a compact, simply connected
hyperkähler manifold, and $U^{\prime}$ an open subset of a hyperkähler
manifold obtained as $V{/\\!\\!/\\!\\!/}G$, where $V$ is flat and $G$
reductive. Then $U$ is not isomorphic to $U^{\prime}$ as hyperkähler manifold.

Proof:
$\dim_{tr}\operatorname{Tw}(U^{\prime})=\dim\operatorname{Tw}(U^{\prime})$ by
4.5, and $\dim_{tr}\operatorname{Tw}(U)=1$ by 4.4.

A twistor space of a manifold obtained by hyperkähler reduction is Moishezon,
and from the above argument it is easy to see that it is Zariski open in a
compact Moishezon variety. However, it is (almost) never quasiprojective.

Theorem 4.31:  Let $M$ be a hyperkähler manifold such that its twistor space
$\operatorname{Tw}(M)$ can be embedded to a projective manifold. Then, for
each induced complex structure $L$, the complex manifold $(M,L)$ has no
compact subvarieties of positive dimension.

Proof: Consider the anticomplex involution $\iota$ on $\operatorname{Tw}(M)$
mapping $(m,L)$ to $(m,-L)$. Suppose that $(M,L)\subset\operatorname{Tw}(M)$
has a compact subvariety. Since the $(M,L)$ is quasiprojective, this would
imply that $(M,L)$ contains a compact curve $S$. The curve $\iota(S)$ is also
holomorphic in $(M,-L)$.

Consider a Kähler form $\omega$ on $\operatorname{Tw}(M)$. Since $\iota$ is
antiholomorphic, $-\iota(\omega)$ is a closed, positive $(1,1)$-form.
Replacing $\omega$ by $\omega-\iota(\omega)$, we may assume that $\omega$
satisfies $\iota(\omega)=-\omega$.

Now, since the cohomology classes of $S$ and $\iota(S)$ are equal, we have

$\int_{S}\omega=\int_{\iota(S)}\omega=-\int_{\iota(S)}\iota(\omega)$

On the other hand, $\int_{S}\omega=\int_{\iota(S)}\iota(\omega)$ by functorial
properties of integral. This implies $\int_{S}\omega=0$, giving a
contradiction.

Remark 4.32: As shown in [V3], a general fiber of the map
$\operatorname{Tw}(M){\>\longrightarrow\>}{\mathbb{C}}P^{1}$ has only even-
dimensional complex subvarieties.

##  5 Appendix: Formal geometry and holography principle (by Dmitry Kaledin)

In this Appendix, we will try to explain the constructions of the paper in a
slightly more general context of $\mathcal{D}$-modules and “formal geometry”
of Gelfand and Kazhdan [GK]. To save space, we only sketch the proofs, and we
work in the algebraic setting (generalization to complex-analytic varieties is
immediate, exactly the same arguments work).

Assume given a smooth algebraic variety $X$ over a field $k$ of characteristic
$0$. We will work with coherent $\mathcal{D}$-modules over $X$, that is, with
sheaves of left modules over the algebra $\mathcal{D}_{X}$ of differential
operators on $X$ which are finitely generated over $\mathcal{D}$. Any coherent
$\mathcal{D}$-module $\mathcal{E}$ is also a quasicoherent sheaf of
$\mathcal{O}_{X}$-modules; recall that if $\mathcal{E}$ is coherent over
$\mathcal{O}_{X}$, then it comes from a vector bundle on $X$ equipped with a
flat connection. In particular, the structure sheaf $\mathcal{O}_{X}$ is a
$\mathcal{D}$-module; it corresponds to the trivial line bundle. For any
coherent sheaf $\mathcal{E}$ on $X$, we can consider the induced
$\mathcal{D}$-module $\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}$;
this is coherent over $\mathcal{D}_{X}$ but not over $\mathcal{O}_{X}$.
Coherent $\mathcal{D}$-modules form an abelian category, and we can consider
its derived category.

Assume given an open subvariety $U\subset X$ with the embedding map $j:U\to
X$, and let $Z=X\setminus U\subset X$ be the complement $U$. Then the complex
of quascoherent sheaves
$R^{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}j_{*}\mathcal{O}_{U}$ on $X$ has
a natural structure of a complex of $\mathcal{D}$-modules, and we have an
exact triangle

$\begin{CD}\delta_{Z}@>{}>{}>\mathcal{O}_{X}@>{}>{}>R^{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}j_{*}\mathcal{O}_{X}@>{}>{}>\end{CD}$
(5.1)

of complexes of $\mathcal{D}$-modules on $X$, where $\delta_{Z}$ is supported
at $Z$. We will need the following standard result.

Lemma 5.1:  In the notation above, for any coherent sheaf $\mathcal{E}$ on
$X$, we have a natural identification

$\operatorname{Hom}_{\mathcal{O}_{X}}(\delta_{Z},\mathcal{E})\cong\Gamma(X,\mathcal{E}\otimes\widehat{\mathcal{O}}_{X,Z}),$

where $\widehat{\mathcal{O}}_{X,Z}$ is the formal completion of structure
sheaf $\mathcal{O}_{X}$ at the closed subscheme $Z\subset X$.

Sketch of a proof. Since $\delta_{Z}$ is supported at $Z$, the natural map

$\operatorname{Hom}_{\mathcal{O}_{X}}(\delta_{Z},\mathcal{E})\to\operatorname{Hom}_{\mathcal{O}_{X}}(\delta_{Z},\mathcal{E}\otimes\widehat{\mathcal{O}}_{X,Z})$

is an isomorphism. On the other hand, we obviously have

$\operatorname{Hom}_{\mathcal{O}_{X}}(R^{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}j_{*}\mathcal{O}_{U},E\otimes\widehat{\mathcal{O}}_{X,Z})=0,$

and the claim then immediately follows from the long exact sequence associated
to the exact triangle (5.1). $\square$

For any map $f:X\to Y$ of smooth algebraic varieties, the pullback functor
$f^{*}$ extends to a functor between $\mathcal{D}$-modules, and its derived
functor extends to a functor between the derived categories of
$\mathcal{D}$-modules; we will denote this last functor by $f^{?}$. We have

$f^{?}=f^{*}[\dim X-\dim Y],$

where $f^{*}$ is the standard pullback functor for $\mathcal{D}$-modules, and
$[-]$ stands for cohomological shift. If $f$ is a smooth map, or a closed
embedding, or a composition of the two, then $f^{?}$ has a left-adjoint
functor $f_{?}$ given by

$f_{?}=f_{!}[\dim Y-\dim X],$

where $f_{!}$ is the standard functor of direct image with compact supports.
Then 5 has the following corollary.

Lemma 5.2:  Assume given a map $f:Y\to X$ of smooth algebraic varieties, and
assume that $f$ factors as

$\begin{CD}Y@>{\pi}>{}>Z@>{\iota}>{}>X,\end{CD}$

where $Z$ is smooth, $\iota$ is a closed embedding, and $\pi$ is a smooth map
with contractible fibers. Then for any vector bundle $\mathcal{E}$ on $X$, we
have a natural identification

$\operatorname{Hom}_{\mathcal{D}_{Y}}(f^{?}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}^{*}),\mathcal{O}_{Y})\cong\Gamma(X,\mathcal{E}\otimes\widehat{\mathcal{O}}_{X,Z}),$
(5.2)

where $\mathcal{E}^{*}$ is the dual vector bundle, and
$\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}^{*}$ is the corresponding
induced $\mathcal{D}$-module.

Proof. For any $\mathcal{D}$-module $\mathcal{F}$ and $\mathcal{O}_{X}$-module
$\mathcal{E}$, we have

$\operatorname{Hom}_{\mathcal{D}_{X}}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E},\mathcal{F})\cong\operatorname{Hom}_{\mathcal{O}_{X}}(\mathcal{E},F),$

so that 5 provides an identification

$\operatorname{Hom}_{\mathcal{D}_{X}}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}^{*}\otimes_{\mathcal{O}_{X}}\delta_{Z},\mathcal{O}_{X})\cong\Gamma(X,\mathcal{E}\otimes\widehat{\mathcal{O}}_{X,Z}),$

and by adjunction, it suffices to construct an isomorphism

$f_{?}f^{?}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}^{*})\cong\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}^{*}\otimes_{\mathcal{O}_{X}}\delta_{Z}.$

But under our assumptions on $f$, we have
$f_{?}\mathcal{O}_{X}\cong\delta_{Z}$, and we are done by the projection
formula. $\square$

We note that the left-hand side of (5.2) admits a slightly different
interpretation. Recall that for any coherent sheaf $\mathcal{E}$ on $X$, the
jet bundle $J^{\infty}\mathcal{E}$ of $\mathcal{E}$ is a (pro)coherent sheaf
on $X$ given by

$J^{\infty}\mathcal{E}=\pi_{2*}\pi_{1}^{*}\mathcal{E},$

where $\pi_{1},\pi_{2}:\widehat{X}\to X$ are the two natural projections of
the completion $\widehat{X}$ of the product $X\times X$ near the diagonal
$X\subset X\times X$. In terms of $\mathcal{D}$-modules, we have

$J^{\infty}\mathcal{E}\cong\operatorname{\mathcal{H}om}_{\mathcal{O}_{X}}(\mathcal{D}_{X},\mathcal{E}).$

The jet bundle $J^{\infty}\mathcal{E}$ carries a canonical flat connection,
and in the assumptions of 5, we have

$\displaystyle\operatorname{Hom}_{\mathcal{D}_{Y}}(f^{?}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}^{*}),\mathcal{O}_{Y})$
$\displaystyle\cong\operatorname{Hom}_{\mathcal{D}_{Y}}(\mathcal{O}_{Y},\operatorname{\mathcal{H}om}_{\mathcal{O}_{Y}}(f^{?}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}^{*}),\mathcal{O}_{Y}))$
$\displaystyle\cong\operatorname{Hom}_{\mathcal{D}_{Y}}(\mathcal{O}_{Y},f^{?}(\operatorname{\mathcal{H}om}_{\mathcal{O}_{X}}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}^{*},\mathcal{O}_{X})))$
$\displaystyle\cong\Gamma^{\nabla}(Y,f^{*}J^{\infty}\mathcal{E}),$

where $\Gamma^{\nabla}(-)$ stands for the space of flat global sections. Then
(5.2) reads as

$\Gamma^{\nabla}(Y,f^{*}J^{\infty}\mathcal{E})\cong\Gamma(X,\mathcal{E}\otimes\widehat{\mathcal{O}}_{X,Z}).$
(5.3)

Remark 5.3: The assumptions of 5 are in fact too strong. Firstly, it is
clearly enough to require that the fibers of $\pi$ are non-empty and
connected, so that their top degree cohomology with compact supports is one-
dimensional – and under the assumptions as stated, we not only obtain an
isomorphism of $\operatorname{Hom}$’s but also of the $\operatorname{RHom}$’s,
so that (5.3) extends to an isomorphism

$H^{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}_{DR}(Y,f^{*}J^{\infty}\mathcal{E})\cong
H^{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}(X,\mathcal{E}\otimes\widehat{\mathcal{O}}_{X,Z})$

of cohomology groups. Secondly, one probably does not have to require that
$\pi$ and $Z$ are smooth – some assumptions are needed, but they can
considerably relaxed. However, since even the stronger assumptions work for
us, we did not pursue this.

Assume now that we are given smooth proper algebraic varieties $Y$, $X$, and a
family of closed embedding from $Y$ to $X$ parametrized by a smooth algebraic
variety $T$ – that is, we have a map

$f:Y\times T\to X$

such that for any $t\in T$, the corresponding map $f_{t}:Y=Y\times t\to X$ is
a closed embedding. Moreover, assume given a coherent sheaf $\mathcal{E}$ on
$X$, denote by $\rho:Y\times T\to T$ the projection, and denote

$\Phi_{f}\mathcal{E}=\rho_{*}^{\nabla}f^{*}J^{\infty}\mathcal{E},$

where $f^{*}J^{\infty}\mathcal{E}$ is the pullback of the jet bundle
$J^{\infty}\mathcal{E}$ equipped with its natural flat connection $\nabla$,
and $\rho_{*}^{\nabla}$ stands for sheaf of relative flat sections. Then
$\Phi_{a}\mathcal{E}$ is a sheaf on $T$, the base of the family, and by (5.3),
the fiber $(\Phi_{f}\mathcal{E})_{t}$ at a point $t\in T$ is given by

$(\Phi_{f}\mathcal{E})_{t}=\Gamma^{\nabla}(Y,f_{t}^{*}J^{\infty}\mathcal{E})\cong\Gamma(X,\mathcal{E}\otimes\widehat{\mathcal{O}}_{X,Y_{t}}),$

where $Y_{t}\subset X$ is the image of the closed embedding $f_{t}:Y\to X$. On
the other hand, $\Phi_{f}\mathcal{E}$ carries a natural flat connection, and
if we assume that $f$ is smooth with contractible fibers, we have

$\Gamma^{\nabla}(T,\Phi_{f}\mathcal{E})=\Gamma^{\nabla}(T\times
Y,f^{*}J^{\infty}\mathcal{E})\cong\Gamma(X,\mathcal{E}),$

again by (5.3).

We now note that this is exactly the situation that we have in the paper.
Namely, we take $X$ to be the twistor space of a hyperkähler manifold $M$, we
take $T=M_{\mathbb{C}}$ to be the complexification of the real-analytic
manifold underlying $M$, we take $Y=\mathbb{C}P^{1}$, and we let

$f:T\times Y\to X$

be the standard family of twistor lines (real points in this family correspond
to horizontal sections of the twistor fibration $X\to\mathbb{C}P^{1}$
parametrized by points of $M$). Then since the normal bundle to any line in
our family is a sum of several copies of $\mathcal{O}(1)$, the family is
unobstructed, and moreover, it remains unobstructed even if we fix a point at
a twistor line, so that the map $f$ is smooth. Its fibers are small polydiscs,
thus contractible, and all the assumptions of 5 are therefore satisfied.

## References

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Misha Verbitsky
Laboratory of Algebraic Geometry,
Faculty of Mathematics, National Research University HSE,
7 Vavilova Str. Moscow, Russia


# The convergence of circle packings on tori

Yanwen Luo  Department of Mathematics, Rutgers University, Piscataway, NJ,
08854<EMAIL_ADDRESS>and  Xu Xu  School of Mathematics and
Statistics, Wuhan University, Wuhan 430072, P.R. China<EMAIL_ADDRESS>
###### Abstract.

The notion of circle packings has been explored intensively since Thurston
[17] proposed it as an approximation to conformal maps. One of the fundamental
questions about circle packings is whether it converges to the Riemann mapping
if the radius of the circles tends to zero. In this paper, we prove the
convergence of circle packings to the maps induced by the uniformization
theorem in a different sense proposed by Gu-Luo-Wu [3] for surfaces of genus
one. This is the first result of the convergence of circle packings on
surfaces with nontrivial topology.

###### Key words and phrases:

conformal maps, circle packings, uniformization theorem, convergence

###### 2000 Mathematics Subject Classification:

30C35, 53A55, 30C20, 30C62

##  1\. Introduction

In 1985, Thurston [18] proposed “the Finite Riemann Mapping Theorem”, claiming
that the maps induced from circle packings converge to the Riemann mapping.
This conjecture initiated a flurry of research about the convergence of maps
constructed from circle packings to the corresponding conformal maps. Rodin-
Sullivan [10] firstly confirmed the uniform convergence of the piecewise
linear maps on a simply connected domain in the plane induced from the
standard hexagonal circle packings to the Riemann mapping. A series of results
including He [6], Doyle-He-Rodin [2] and He-Schramm [7] proved the convergence
of the first and second derivatives of the induced maps from the circle
packings extended by Möbius transformations to their smooth counterparts.
Eventually, He-Schramm [8] showed the $C^{\infty}$ convergence of hexagonal
circle packings to the Riemann mapping. It also leads to the study of discrete
analogs of concepts and theorems in complex analysis and Riemann surfaces. See
[11] for a comprehensive survey on this topic.

In contrast to most of the results above concerning about planar domains in
the plane, little is known about the convergence of circle packings to
conformal diffeomorphisms provided by the classical uniformization theorem for
Riemannian surfaces with non-trivial topology. In this case, it is difficult
to define a canonical map from the circle packings on general surfaces to
surfaces with constant curvature. Meanwhile, computing such a map on general
surfaces has a wide range of applications in surface parameterizations [19].
Therefore, we need to justify the convergence of the discrete conformal maps
induced from circle packings to the smooth conformal maps on general surfaces.

The goal of this paper is to demonstrate the convergence of circle packings to
uniformization maps in an alternative framework proposed in Gu-Luo-Wu [3] and
Wu-Zhu [13] when the surface is a smooth torus. This scheme of convergence is
proposed for a new theory of discrete conformal geometry called vertex
scaling. The fundamental results about this new theory include a discrete
uniformization theorem for polyhedral surfaces [4, 5, 15], an infinite
rigidity theorem similar to the hexagonal circle packings in the plane [14],
and the convergence of discrete conformal maps to their smooth
counterparts[9]. Effective algorithms are also available to compute conformal
maps [12].

We will apply this convergence scheme and prove that the uniformization factor
given by the uniformization theorem can be approximated by the discrete
uniformization factor produced by the deformation of radii of circle packings.
The main idea comes from the work by Wu-Zhu [13].

###  1.1. Set Up.

Assume that $M$ is a closed orientable smooth surface of genus one, and $T$ is
a triangulation of $M$. Denote the set of vertices, edges, and faces of $T$ as
$V=V(T),E=E(T),F=F(T)$ respectively. Suppose $M$ is equipped with a Riemannian
metric $g$, $T$ is a _geodesic triangulation_ if any edge in $T$ is an
embedded geodesic segment in $(M,g)$. Therefore, each face in $F$ of a
geodesic triangulation is a geodesic triangle in $(M,g)$. It induces an edge
length function $l\in\mathbb{R}^{E}_{>0}$. We denote the geodesic
triangulation with an edge length function on $(M,g)$ as $(T,l)$. The area of
$(M,g)$ is denoted as $Area(M,g)$.

Conversely, let $l$ be a positive function on $E$ satisfying triangle
inequalities for each face of $T$. Then we can determine a _Euclidean
triangular mesh_ $(T,l)_{E}$ by gluing Euclidean triangles with prescribed
edge lengths $l$ by isometries. Note that a Euclidean triangular mesh produces
a piecewise flat metric on $M$ with all the singular points in $V(T)$. The
area of $(T,l)_{E}$, which is the sum of areas of Euclidean triangles in the
mesh, is denoted as $Area((T,u*l)_{E})$.

Given $(T,l)_{E}$, let $\theta^{i}_{jk}$ denote the inner angle at $i$ in the
Euclidean triangle $\triangle ijk\in F$, and _discrete curvature_ $K_{i}$ at a
vertex $i\in V$ be the angle defect

$K_{i}=2\pi-\sum_{jk:\triangle ijk\in F}\theta^{i}_{jk}.$

Note that $(T,l)_{E}$ is flat if and only if $K_{i}=0$ for any $i\in V$.

Thurston [17] introduced circle packings on surfaces with non-positive Euler
characteristic. We summarize relevant concepts as follows.

######  Definition 1.1.

The edge length function of a Euclidean triangular mesh $(T,l)_{E}$ is a
circle packing metric if for any $ij\in E(T)$

(1)
$l^{2}_{ij}=e^{2\rho_{i}}+e^{2\rho_{j}}+2e^{\rho_{i}+\rho_{j}}\cos\Theta_{ij},$

for a function $\Theta:E\to[0,\pi/2]$ and a function $\rho:V\to\mathbb{R}$.
The function $\Theta$ is called a conformal weight, and the function $\rho$ is
called a logarithmic radius. A circle packing metric is uniform if $\rho$ is a
constant function. Two circle packing metrics $(T,l)_{E}$ and
$(T,l^{\prime})_{E}$ are discrete conformal if their conformal weights are
identical.

Note that $r_{i}=e^{\rho_{i}}$ is the radius of the circle centered at vertex
$i$. If we draw a triangle $\triangle ijk$ with lengths given by
$l_{ij},l_{ik},l_{jk}$ and three circles of radius $r_{i},r_{j},r_{k}$ at
vertices $i,j,k$ respectively in the plane, then $\Theta_{ij}$ will be the
intersection angle of the two circles centered at $i$ and $j$.

Any two discrete conformal circle packing metrics $l$ and $l^{\prime}$ are
differed only by their logarithmic radii $u_{i}=\rho^{\prime}_{i}-\rho_{i}$.
Such a vector $u\in\mathbb{R}^{V}$ is called a _discrete conformal factor_ ,
and we denote this relation as $l^{\prime}=u*l$. If we start with a circle
packing metric $(T,l)_{E}$ with a conformal weight $\Theta$ and logarithmic
radius $\rho$, the discrete conformal factor $u$ acts on the initial
logarithmic radius by $u\to\rho+u$, which is equivalent to $l^{\prime}=u*l$.
We can produce all the other circle packing metrics discrete conformal to the
initial metric $l$ through this action.

Let $\theta^{i}_{jk}(u)$ and $K_{i}(u)$ denote the inner angle and the
discrete curvature at the vertex $i$ in $(T,u*l)_{E}$ respectively. Set
$K(u)=[K_{i}(u)]_{i\in V}\in\mathbb{R}^{V}$. Then $u\in\mathbb{R}^{V}$ is
called a _discrete uniformization factor_ if $K(u)=0$.

###  1.2. Main results.

By the uniformization theorem, for any $(M,g)$ there exists a unique function
$\bar{u}$ on $M$ such that $(M,e^{2\bar{u}}g)$ is a flat torus with unit area.
We will prove that the discrete uniformization factor $u\in\mathbb{R}^{V}$
induced by a sufficiently regular and dense geodesic triangulation $(T,l)$ on
$(M,g)$ approximates the smooth uniformization factor $\bar{u}\in
C^{\infty}(M)$. To characterize the regularity of a geodesic triangulation, we
use the following definition.

######  Definition 1.2.

A Euclidean triangular mesh $(T,l)_{E}$ is called _$\epsilon$ -regular_ if

  1. (a)

The discrete metric $l$ is a circle packing metric with some conformal weight
$\Theta$ so that $\cos\Theta_{ij}\geq\epsilon$ for all $ij\in E$;

  2. (b)

Any inner angle $\theta_{jk}^{i}$ in $(T,l)_{E}$ satisfies
$\theta^{i}_{jk}\geq\epsilon$.

For a geodesic triangulation $(T,l)$ on a Riemannian surface, it is
$\epsilon$-regular if the corresponding Euclidean mesh $(T,l)_{E}$ is
$\epsilon$-regular.

######  Remark 1.3.

Note that an $\epsilon$-regular geodesic triangulation $(T,l)$ has a uniform
upper bound on the degrees of vertices by $\lceil 2\pi/\epsilon\rceil$.

Let $|x|=\max_{i\in A}|x_{i}|$ denote the maximal norm of a vector
$x\in\mathbb{R}^{A}$ when $A$ is finite. Define the _size_ of a geodesic
triangulation $(T,l)$ to be $|l|$. The main result of this paper is

######  Theorem 1.4.

Suppose $(M,g)$ is a closed orientable Riemannian surface of genus one, and
$\bar{u}=\bar{u}_{M,g}\in C^{\infty}(M)$ is the unique function on $M$ such
that $e^{2\bar{u}}g$ is the metric of constant curvature with unit area.
Assume $T$ is a geodesic triangulation in $(M,g)$ with edge length
$l\in\mathbb{R}^{E(T)}_{>0}$. Then for any $\epsilon>0$, there exists a
constant $\delta=\delta(M,g,\epsilon)>0$ such that if $(T,l)_{E}$ is
$\epsilon$-regular, uniform, and $|l|<\delta$, then

  1. (a)

there exists a unique discrete conformal factor $u\in\mathbb{R}^{V}$, such
that $(T,u*l)_{E}$ is globally flat and $Area((T,u*l)_{E})=1$, and

  2. (b)

$\big{|}u-\bar{u}|_{V(T)}\big{|}\leq C|l|$ for some constant
$C=C(M,g,\epsilon)$.

Theorem 1.4 implies that the discrete conformal factor $u$ defined on the set
of vertices approaches the smooth conformal factor $\bar{u}$ restricted to the
set of vertices as the size of geodesic triangulations tends to zero.

######  Remark 1.5.

The uniqueness of the discrete uniformization factor was proved by Thurston
[17] in terms of the rigidity of circle packing metrics with respect to the
discrete curvature. The existence of arbitrarily dense, $\epsilon$-regular,
and uniform geodesic triangulations on any $(M,g)$ can be deduced from the
work by Colin de Verdiere [1]. It provides a family of geodesic triangulations
on any Riemannian torus whose angles of triangles are arbitrarily closed to
$\pi/3$ and sizes tend to zero.

Acknowledgements
The authors thank Dr. Tianqi Wu and Dr. Xiaoping Zhu for valuable
communications. The research of the second author is supported by the
Fundamental Research Funds for the Central Universities under Grant no.
2042020kf0199.

Data Availability
Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

##  2\. Preliminaries and Preparations

We summarize the main tools in Wu-Zhu [13] to prove Theorem 1.4 in this
section. It consists of four subsections, containing basic facts about
calculus on graphs, properties of geodesic triangles and geodesic
triangulations, and the curvature deformation with respect to discrete
conformal factors. In addition, the last subsection provides a proof of a
cubic estimation for controlling the deviation between smooth conformal
deformation of metrics and discrete conformal deformation of metrics.

###  2.1. Discrete calculus of graphs.

Let $G=(V,E)$ be a graph with the set of vertices $V$ and the set of edges
$E$. Define three vector spaces $\mathbb{R}^{V}$, $\mathbb{R}^{E}$ and
$\mathbb{R}^{E}_{A}$, where

  1. (a)

the vector $x\in\mathbb{R}^{E}$ is symmetric, i.e., $x_{ij}=x_{ji}$;

  2. (b)

the vector $x\in\mathbb{R}^{E}_{A}$ is anti-symmetric, i.e., $x_{ij}=-x_{ji}$.

A vector $x\in\mathbb{R}^{E}_{A}$ is called a _flow_ on $G$. A positive vector
in $\mathbb{R}^{E}$ is called an _edge weight_ $\eta$ on $G$. Given an edge
weight $\eta$ and a vector $x\in\mathbb{R}^{V}$, the _gradient_ $\nabla
x=\nabla_{\eta}x$ is a flow defined by

$(\nabla x)_{ij}=\eta_{ij}(x_{j}-x_{i}).$

The _divergence_ $div(x)$ of a flow $x\in\mathbb{R}^{E}_{A}$ is a vector in
$\mathbb{R}^{V}$ defined by

$div(x)_{i}=\sum_{j\sim i}x_{ij}.$

Given an edge weight $\eta$, the _Laplacian_
$\Delta=\Delta_{\eta}:\mathbb{R}^{V}\to\mathbb{R}^{V}$ is defined as $\Delta
x=\Delta_{\eta}x=div(\nabla_{\eta}x)$. Specifically,

$(\Delta x)_{i}=\sum_{j\sim i}(\nabla x)_{ij}=\sum_{j\sim
i}\eta_{ij}(x_{j}-x_{i}).$

The Laplacian is a linear operator on $\mathbb{R}^{V}$. It could be
represented as a $|V|\times|V|$ symmetric matrix. It is well-known that if $G$
is connected, $ker(\Delta)$ is an $1$-dimensional space defined by

$\mathbb{1}=\\{x\in\mathbb{R}^{V}:x=(a,...,a)\text{ for some
}a\in\mathbb{R}\\}.$

Note that $\Delta_{\eta}$ is an invertible operator on $\mathbb{1}^{\perp}$
for any edge weight $\eta\in\mathbb{R}^{E}_{>0}$. Let $\Delta^{-1}_{\eta}$
denote this inverse of $\Delta_{\eta}$, and we have

######  Lemma 2.1 (Lemma 2.2 in [13]).

The map $(\eta,y)\longmapsto\Delta_{\eta}^{-1}y$ is smooth from
$\mathbb{R}^{E}_{>0}\times\mathbb{1}^{\perp}$ to $\mathbb{1}^{\perp}$.

Let $(G,l)$ be a metric graph with the length defined by a positive vector
$l\in\mathbb{R}^{E}_{>0}$. A notion of the $C$-isoperimetric condition on it
was introduced in [13] as a discrete version of the isoperimetric inequality
on Riemannian surface. Set the boundary for any $U\subset V$ as

$\partial U=\\{ij\in E:i\in U,\text{ and }j\notin U\\},$

and define the l-perimeter of $U$ and the l-area of $U$ respectively as

$|\partial U|_{l}=\sum_{ij\in\partial
U}l_{ij}\quad\text{and}\quad|U|_{l}=\sum_{ij\in E(U)}l_{ij}^{2}.$

A metric graph $(G,l)$ satisfies the $C$-_isoperimetric condition_ for some
constant $C>0$, if for any $U\subset V$,

$\min\\{|U|_{l},|V|_{l}-|U|_{l}\\}\leq C|\partial U|_{l}^{2}.$

This notion of $C$-isoperimetric metric graphs is an essential tool to prove
the main theorem due to the following discrete elliptic estimation.

######  Lemma 2.2 ((Lemma 2.3 in [13]).

Assume $(G,l)$ is $C_{1}$-isoperimetric, and
$x\in\mathbb{R}^{E}_{A},\eta\in\mathbb{R}^{E}_{>0},C_{2}>0,C_{3}>0$ are such
that

  1. (i)

$|x_{ij}|\leq C_{2}l_{ij}^{2}$ for any $ij\in E$;

  2. (ii)

$\eta_{ij}\geq C_{3}$ for any $ij\in E$.

Then

$|\Delta^{-1}_{\eta}\circ
div(x))|\leq\frac{4C_{2}\sqrt{C_{1}+1}}{C_{3}}|l|\cdot|V|_{l}^{1/2}.$

###  2.2. Lemmas about geodesic triangles and geodesic triangulations.

We will constantly replace small geodesic triangles on Riemannian surfaces by
Euclidean triangles in this paper. Hence, it is necessary to estimate the
difference of angles, areas, and curvatures between them. The following two
lemmas provide a bound on this difference.

Given a geodesic triangle $\triangle ABC$ on a Riemannian surface, we will
denote its inner angles as $A,B,C$, and its lengths of edges opposite to
$A,B,C$ as $a,b,c$. The area of $\triangle ABC$ will be denoted as $|\triangle
ABC|$.

######  Lemma 2.3 (Lemma 3.5 in [13]).

Given a Euclidean triangle $\triangle ABC$ with edge lengths $a$, $b$, and
$c$, if all the angles in $\triangle ABC$ are at least $\epsilon>0$,
$\delta<\epsilon^{2}/48$, and

$|a^{\prime}-a|\leq\delta a,\quad|b^{\prime}-b|\leq\delta
b,\quad|c^{\prime}-c|\leq\delta c,$

then

  1. (1)

$a^{\prime},b^{\prime},c^{\prime}$ form a Euclidean triangle with opposite
inner angles $A^{\prime},B^{\prime},C^{\prime}$ respectively;

  2. (2)

$|A^{\prime}-A|\leq\frac{24}{\epsilon}\delta;$

  3. (3)

$\bigg{|}|\triangle A^{\prime}B^{\prime}C^{\prime}|-|\triangle
ABC|\bigg{|}\leq\frac{576}{\epsilon^{2}}\delta\cdot|\triangle ABC|,$

where $|\triangle ABC|$ is the area of $\triangle ABC$.

The next lemma on geodesic triangles provides a weak equivalence between edge
lengths and areas in a Euclidean triangle with lower bounds on angles.

######  Lemma 2.4 (Lemma 4.1 [13]).

Suppose $\triangle_{E}ABC$ is a Euclidean triangle with the edge lengths
$a,b,c$. If all the inner angles in $\triangle_{E}ABC$ are at least
$\epsilon>0$, then

$\frac{\epsilon}{8}a^{2}\leq|\triangle_{E}ABC|\leq\frac{1}{\epsilon}a^{2}.$

Starting from a geodesic triangulation $(T,l)$ on a Riemannian surface
$(M,g)$, we will deform the Riemannian metric $g$ in the smooth conformal
class to the metric $e^{2u}g$. The following lemma ensures that when the size
of the geodesic triangulation $(T,l)$ is sufficiently small, there exists a
geodesic triangulation $(T^{\prime},\bar{l})$ isotopic to $(T,l)$ relative to
$V$ with respect to the metric $e^{2u}g$ such that the angle deviation is
small. Moreover, the $1$-skeleton of $(T^{\prime},\bar{l})$ , when regarded as
a metric graph, is $C$-isoperimetric.

######  Lemma 2.5 (Lemma 4.4 in [13]).

Suppose $(M,g)$ is a closed Riemannian surface, and $T$ is a geodesic
triangulation of $(M,g)$, and $l\in\mathbb{R}_{>0}^{E(T)}$ denotes the
geodesic lengths of the edges.

  1. (a)

Given a conformal factor $u\in C^{\infty}(M)$, there exists a constant
$\delta=\delta(M,g,u,\epsilon)>0$ such that if $|l|<\delta$ then there exists
a geodesic triangulation $T^{\prime}$ with edge length $\bar{l}$ in
$(M,e^{2u}g)$ where $V(T^{\prime})=V(T)$, and $T^{\prime}$ is isotopic to $T$
relative to $V(T)$. Furthermore,

$|\bar{\theta}^{i}_{jk}-\theta^{i}_{jk}|\leq\frac{\epsilon}{2},$

where $\theta^{i}_{jk}$ is an angle in $(T,l)_{E}$, and
$\bar{\theta}^{i}_{jk}$ is the corresponding angle in
$(T^{\prime},\bar{l})_{E}$.

  2. (b)

If $(T,l)_{E}$ is $\epsilon$-regular, there exists a constant
$\delta=\delta(M,g,\epsilon)$ such that if $|l|<\delta$, $(T,l)$, as a metric
graph on the $1$-skeleton of $T$, is $C$-isoperimetric for some constant
$C=C(M,g,\epsilon)$.

###  2.3. Differential of the discrete curvature.

The explicit formula for the derivative of angles in Euclidean triangles with
respect to discrete conformal factors is given in [20, 16].

######  Lemma 2.6 ([20], Lemma 2.5 in [16]).

Let $\theta^{i}_{jk}(u)$ be the inner angle at $i$ in triangle $\triangle ijk$
in $(T,u*l)_{E}$, and $K(u)\in\mathbb{R}^{V}$ is the discrete curvature. Then

$\displaystyle\frac{\partial\theta_{jk}^{i}}{\partial
u_{j}}=\frac{\partial\theta_{ik}^{j}}{\partial u_{i}}$ $\displaystyle=$
$\displaystyle\frac{r_{i}^{2}r_{j}^{2}\sin^{2}\Theta_{ij}+r_{i}^{2}r_{j}r_{k}(\cos\Theta_{jk}+\cos\Theta_{ij}\cos\Theta_{ik})+r_{i}r_{j}^{2}r_{k}(\cos\Theta_{ik}+\cos\Theta_{ij}\cos\Theta_{jk})}{l_{ik}l_{ij}^{3}\sin\theta_{jk}^{i}}.$

Define an edge weight $\eta\in\mathbb{R}^{E}$ as

$\eta_{ij}(u)=\frac{\partial\theta_{jk}^{i}}{\partial
u_{j}}+\frac{\partial\theta_{jm}^{i}}{\partial u_{j}},$

where $\triangle ijk$ and $\triangle ijm$ are triangles in $T$. Then we have

(2) $\frac{\partial K}{\partial u}=-\Delta_{\eta(u)}.$

This connection between the differential of the discrete curvatures and the
Laplacian operator will be crucial to construct a flow to locate the discrete
uniformization factor.

###  2.4. Cubic estimation.

The following lemma illustrates the deviation between distances induced by
discrete conformal factors and smooth conformal factors. It is an analogue to
Proposition 5.2 in [13] in the setting of circle packing metrics.

######  Lemma 2.7.

Suppose $(M,g)$ is a closed Riemannian surface, and $u\in C^{\infty}(M)$ is a
conformal factor. Assume that $x,y\in M$ such that

$d_{g}(x,y)=l=\sqrt{e^{2\rho}+e^{2\rho}+2e^{2\rho}I}=e^{\rho}\sqrt{2+2I},$

where $I\geq 0$. Then there exists $C=C(M,g,u,I)>0$ such that

$|d_{e^{2u}g}(x,y)-\sqrt{e^{2(\rho+u(x))}+e^{2(\rho+u(y))}+2e^{2\rho+u(x)+u(y)}I}|\leq
Cd_{g}(x,y)^{3}.$

###### Proof.

By replacing $g$ with $e^{2u}g$ and consider $g$ as a conformal change of
$e^{2u}g$ by the conformal factor $-2u$, it suffices to prove that

$d_{e^{2u}g}(x,y)-\sqrt{e^{2(\rho+u(x))}+e^{2(\rho+u(y))}+2e^{2\rho+u(x)+u(y)}I}\leq
Cl^{3}$

for some constant $C=C(M,g,u)>0$.

Let $\gamma:[0,l]\rightarrow(M,g)$ be a shortest geodesic connecting $x,y$.
Let $h(t)=u(\gamma(t))$ and $s_{e^{2u}g}(\gamma)$ be the length of the curve
$\gamma$ in the new metric $e^{2u}g$. Then

$\displaystyle
d_{e^{2u}g}(x,y)-\sqrt{e^{2(\rho+h(0))}+e^{2(\rho+h(l))}+2e^{2\rho+h(0)+h(l)}I}$
$\displaystyle\leq$ $\displaystyle
s_{e^{2u}g}(\gamma)-\sqrt{e^{2(\rho+h(0))}+e^{2(\rho+h(l))}+2e^{2\rho+h(0)+h(l)}I}$
(3) $\displaystyle=$ $\displaystyle\left[\int_{0}^{l}e^{h(t)}dt-l\cdot
e^{h(l/2)}\right]+\left[l\cdot
e^{h(l/2)}-\sqrt{e^{2(\rho+h(0))}+e^{2(\rho+h(l))}+2e^{2\rho+h(0)+h(l)}I}\right].$

The first term in Equation (2.4) is estimated by the mid-point rule
approximation of definite integral of a function $f(t)\in C^{2}[0,l]$ given by

$\left|\int_{0}^{l}f(x)dx-l\cdot
f(\frac{l}{2})\right|\leq\frac{l^{3}}{24}\max_{0\leq t\leq
l}|f^{\prime\prime}(t)|.$

Note that $h(t)$ and its first two derivatives are determined by $u$,
$\gamma$, and their derivatives up to the second order. Since $M$ is compact,
and all the functions involved are smooth functions, then there exists a
constant $C_{1}=C_{1}(M,g,u)>0$ such that the maximum of $h$ and its first and
second derivatives are bounded by $C_{1}$. Therefore,

$\left|\int_{0}^{l}e^{h(t)}dt-l\cdot e^{h(l/2)}\right|\leq C_{1}l^{3}.$

The second term in Equation (2.4) is

$\displaystyle l\cdot
e^{h(l/2)}-\sqrt{e^{2(\rho+h(0))}+e^{2(\rho+h(l))}+2e^{2\rho+h(0)+h(l)}I}$
$\displaystyle=$ $\displaystyle
e^{\rho}e^{h(l/2)}\sqrt{2+2I}-e^{\rho}\sqrt{e^{2h(0)}+e^{2h(l)}+2e^{h(0)+h(l)}I}$
$\displaystyle=$ $\displaystyle
e^{\rho}\frac{(2e^{2h(l/2)}-e^{2h(0)}-e^{2h(l)})+2I(e^{2h(l/2)}-e^{h(0)+h(l)})}{e^{h(l/2)}\sqrt{2+2I}+\sqrt{e^{2h(0)}+e^{2h(l)}+2e^{h(0)+h(l)}I}}.$

It can be bounded with help of the following inequality for a function in
$f\in C^{2}[0,l]$ given by

$\left|f(\frac{l}{2})-\frac{1}{2}[f(0)+f(l)]\right|\leq\frac{l^{2}}{4}\max_{0\leq
t\leq l}|f^{\prime\prime}(t)|.$

There exist $C_{2}=C_{2}(M,g,u)>0$, and $C_{3}=C_{3}(M,g,u)>0$ such that

$\left|2e^{2h(l/2)}-e^{2h(0)}-e^{2h(l)}\right|\leq C_{2}l^{2}$

and

$\left|e^{2h(l/2)}-e^{h(0)+h(l)}\right|\leq
e^{\xi}\cdot\left|2h(l/2)-h(0)-h(l)\right|\leq C_{3}l^{2},$

where $\xi$ is between $2h(l/2)$ and $h(0)+h(1)$.

Set $\mu$ be the minimum of $u$ in $M$, then

$l\cdot
e^{h(l/2)}-\sqrt{e^{2(\rho+h(0))}+e^{2(\rho+h(l))}+2e^{2\rho+h(0)+h(l)}I}\leq\frac{e^{\rho}l^{2}(C_{2}+C_{3})}{2e^{\mu}\sqrt{2+2I}}\leq
C_{4}l^{3},$

for some $C_{4}=C_{4}(M,g,u,I)>0$.

Hence,

$|d_{e^{2u}g}(x,y)-\sqrt{e^{2(\rho+u(x))}+e^{2(\rho+u(y))}+2e^{2\rho+u(x)+u(y)}I}|\leq
Cd_{g}(x,y)^{3},$

for some $C=C(M,g,u,I)>0$. ∎

######  Remark 2.8.

Lemma 2.7 shows that if $x$ and $y$ are two vertices in a geodesic
triangulation determining one edge, the length of this edge in the new
conformal metric $e^{2u}g$ is approximated by the length of this edge in the
discrete conformal metric, which is obtained from the discrete conformal
change by $u$.

The same estimate holds if the condition on $\rho$ is relaxed to the
assumption that there exists a constant $C=C(M,g,u)$ such that
$|\rho_{i}-\rho_{j}|\leq Cl$ for all $i,j\in V$ and the condition on $I$ is
relaxed to $I>-1$. We omit the proof here.

##  3\. Proof of the main theorem

In this section, we will prove the main theorem following the framework in
[13].

Recall that the main theorem is

######  Theorem 1.4.

Suppose $(M,g)$ is a closed orientable Riemannian surface of genus one, and
$\bar{u}=\bar{u}_{M,g}\in C^{\infty}(M)$ is the unique function on $M$ such
that $e^{2\bar{u}}g$ is the metric of constant curvature with unit area.
Assume $T$ is a geodesic triangulation in $(M,g)$ with edge length
$l\in\mathbb{R}^{E(T)}_{>0}$. Then for any $\epsilon>0$, there exists a
constant $\delta=\delta(M,g,\epsilon)>0$ such that if $(T,l)_{E}$ is
$\epsilon$-regular, uniform, and $|l|<\delta$, then

  1. (a)

there exists a unique discrete conformal factor $u\in\mathbb{R}^{V}$, such
that $(T,u*l)_{E}$ is globally flat and $Area((T,u*l)_{E})=1$, and

  2. (b)

$\big{|}u-\bar{u}|_{V(T)}\big{|}\leq C|l|$ for some constant
$C=C(M,g,\epsilon)>0$.

In the rest of this paper, the notion $a=O(b)$ represents that if
$\delta=\delta(M,g,\epsilon)$ is sufficiently small, then $|a|\leq C\cdot b$
for some constant $C=C(M,g,\epsilon)$. Moreover, we will replace
$\bar{u}|_{V(T)}$ by $\bar{u}$ for simplicity when there is no confusion.

The proof consists of the following three steps.

  1. (1)

Prove that $(T,\bar{u}*l)_{E}$ is close to a flat Euclidean triangular mesh
$(T,\bar{l})_{E}$, which means that

$K(\bar{u})=div(x)=O(|l|^{2})$

for some flow $x\in\mathbb{R}^{E}_{A}$ with $x_{ij}=O(l_{ij}^{2}).$

  2. (2)

Construct a smooth deformation of the discrete conformal factor
$u(t):[0,1]\rightarrow\mathbb{R}^{V}$ starting at $u(0)=\bar{u}$ such that

$K(u(t))=(1-t)K(\bar{u}).$

Note that at the end of this deformation, the curvature $K(u(1))\equiv 0$. As
a result, we locate a discrete uniformization factor. Furthermore, this
deformation is bounded by the size of the regular geodesic triangulations, in
the sense that

$|u(1)-\bar{u}|=O(|l|).$

  3. (3)

Deform $u(1)$ by a small constant vector to $\tilde{u}$ so that the resulting
flat torus has unit area, then

$|u(1)-\tilde{u}|=O(|l|).$

$(T,l)$ on $(M,g)$ $(T,l)_{E}$ as a Euclidean mesh $(T,\bar{l})$ on
$(M,e^{2\bar{u}}g)$ $(T,\bar{u}*l)_{E}$ is close to flat
$\bar{u}$$\bar{u}|_{V(T)}$ Figure 1. The framework of the proof.

###### Proof.

We assume that $(T,l)$ is $\epsilon$-regular, uniform, and $|l|\leq\delta$
where $\delta=\delta(M,g,\epsilon)$ is a sufficiently small constant to be
determined. Then $l$ is a circle packing metric given by a constant
logarithmic radii $\rho$ and a conformal weight $\Theta$.

If $\delta$ is sufficiently small, then by Lemma 2.5, there exists a geodesic
triangulation $T^{\prime}$ of $(M,e^{2\bar{u}}g)$ homotopic to $T$ relative to
$V(T)=V(T^{\prime})$. Let
$\bar{l}\in\mathbb{R}^{E(T)}\cong\mathbb{R}^{E(T^{\prime})}$ denote the
geodesic lengths of edges of $T^{\prime}$ in $(M,e^{2\bar{u}}g)$. Since
$T^{\prime}$ and $T$ are combinatorial equivalent and isotopic, we will denote
$(T^{\prime},\bar{l})$ as $(T,\bar{l})$. Clearly,
$(T,\bar{l})_{E}=(T,\bar{l})$ is isometric to $(M,e^{2\bar{u}}g)$ and flat
since $(M,e^{2\bar{u}}g)$ is a flat torus.

###  3.1. Step 1

By Lemma 2.7,

(4) $\bar{l}_{ij}-(\bar{u}*l)_{ij}=O(|l|^{3}).$

Then by Lemma 2.3, the angle deviation satisfies

(5)
$\alpha^{i}_{jk}:=\bar{\theta}^{i}_{jk}-\theta^{i}_{jk}(\bar{u})=O(|l|^{2}),$

where $\bar{\theta}^{i}_{jk}$ denotes the inner angle in $(T,\bar{l})_{E}$,
and $\theta^{i}_{jk}(\bar{u})$ denotes the inner angle in $(T,\bar{u}*l)_{E}$.
Define a flow $x\in\mathbb{R}^{E}_{A}$ as

$x_{ij}:=\frac{\alpha^{i}_{jk}-\alpha^{j}_{ik}}{3}+\frac{\alpha^{i}_{jk^{\prime}}-\alpha^{j}_{ik^{\prime}}}{3}=O(l_{ij}^{2}),$

where $\triangle ijk$ and $\triangle ijk^{\prime}$ are adjacent triangles.
Since $\alpha^{i}_{jk}+\alpha^{j}_{ik}+\alpha^{k}_{ij}=0$, then

$div(x)_{i}=\sum_{j:j\sim i}x_{ij}=\sum_{jk:\triangle ijk\in
F(T)}\left(\frac{\alpha^{i}_{jk}-\alpha^{j}_{ik}}{3}+\frac{\alpha^{i}_{jk}-\alpha^{k}_{ij}}{3}\right)=\sum_{jk:\triangle
ijk\in F(T)}\alpha^{i}_{ij}=K_{i}(\bar{u}).$

Since degrees of vertices of $\epsilon$-regular triangulations are uniformly
bounded, we have

$K_{i}(\bar{u})=\sum_{j:j\sim i}x_{ij}=O(|l|^{2}).$

It implies that the curvature of the metric $(T,\bar{u}*l)$ is closed to zero
when $\delta$ is sufficiently small. By Lemma 2.5, angles in $(T,\bar{l})_{E}$
have a lower bound by $\epsilon/2$, although $(T,\bar{l})_{E}$ might not be a
circle packing metric. By equation (5), $(T,\bar{u}*l)_{E}$ is
$\epsilon/3$-regular for sufficiently small $\delta>0$.

###  3.2. Step 2

Define

$\tilde{\Omega}=\\{u\in\mathbb{1}^{\perp}:(T,u*l)_{E}\text{ is
}\frac{\epsilon}{5}\text{-regular}\\},$

and

$\Omega=\\{u\in\tilde{\Omega}:|u-\bar{u}|\leq 1\text{, $(T,u*l)_{E}$ is
$\frac{\epsilon}{4}$-regular}\\}.$

Then $\bar{u}$ is in the interior of $\Omega$. Consider the system of
differential equations on $int(\tilde{\Omega})$,

(6)
$\left\\{\begin{array}[]{lcl}u^{\prime}(t)=\Delta_{\eta(u)}^{-1}K(\bar{u})=\Delta_{\eta(u)}^{-1}\circ
div(x),\\\ u(0)=\bar{u},\end{array}\right.$

where the edge weight $\eta(u)$ is given by Theorem 2.6. By Lemma 2.1, the
right-hand side of (6) is a smooth function of $u$, so the equation (6) has a
unique solution $u(t)$ with the maximum existing open interval in $\Omega$
given by $[0,T_{0})$ for $T_{0}>0$. Moreover, by Theorem 2.6,

$K(u(t))=(1-t)K(\bar{u}).$

Therefore, we only need to show that $T_{0}\geq 1$ for sufficiently small
$\delta>0$. Then $u(1)$ is a discrete uniformization factor.

By Lemma 2.5, $(T,l)$ is $C$-isoperimetric for some constant
$C=C(M,g,\epsilon)$. For any $u\in\Omega$, $(T,u*l)$ is
$(e^{4|\bar{u}|+1}C)$-isoperimetric since $|u|\leq|\bar{u}|+1$. To apply Lemma
2.2, we need to bound related quantities $\eta$ and $|V|_{l}$ as follows.

Define $r_{i}(u)=e^{\rho+u_{i}}$ when $u\in\Omega$. There exists a constant
$R=R(M,g)$ such that for any $i,j\in V$,

$r_{i}/r_{j}=e^{u_{i}-u_{j}}\leq e^{|\bar{u}_{i}-\bar{u}_{j}|+2}\leq R.$

Then

$\displaystyle\frac{\partial\theta_{jk}^{i}}{\partial u_{j}}$
$\displaystyle=\frac{r_{i}^{2}r_{j}^{2}\sin^{2}\Theta_{ij}+r_{i}^{2}r_{j}r_{k}(\cos\Theta_{jk}+\cos\Theta_{ij}\cos\Theta_{ik})+r_{i}r_{j}^{2}r_{k}(\cos\Theta_{ik}+\cos\Theta_{ij}\cos\Theta_{jk})}{l_{ik}l_{ij}^{3}\sin\theta_{jk}^{i}}$
$\displaystyle\geq\frac{\epsilon
r_{i}r_{j}r_{k}(r_{i}+r_{j})}{l_{ik}l_{ij}^{3}}\geq\frac{\epsilon
r_{i}r_{j}r_{k}}{l_{ik}l_{ij}^{2}}\geq\frac{\epsilon
r_{i}r_{j}r_{k}}{(r_{i}+r_{k})(r_{i}+r_{j})^{2}}\geq\frac{\epsilon}{8R^{3}}.$

Hence,

$\eta_{ij}(u)=\frac{\partial\theta_{jk}^{i}}{\partial
u_{j}}+\frac{\partial\theta_{jm}^{i}}{\partial
u_{j}}\geq\frac{\epsilon}{4R^{3}}.$

In addition, by Lemma 2.4 and the fact that angles of $(T,\bar{l})_{E}$ has a
lower bounded by $\epsilon/2$,

$\displaystyle|V|_{l}=\sum_{ij\in E}l_{ij}^{2}=O(\sum_{ij\in
E}\bar{l}_{ij}^{~{}2})=O(\sum_{ijk\in
F}(\bar{l}_{ij}^{~{}2}+\bar{l}_{jk}^{~{}2}+\bar{l}_{ik}^{~{}2}))$
$\displaystyle=$ $\displaystyle O(\sum_{ijk\in F}|(\triangle
ijk,\bar{l})_{E}|)=O(|(T,\bar{l})_{E}|)=O(1).$

Here $|(T,\bar{l})_{E}|$ denotes the area of the piecewise flat metric on $M$.
Combining these two bounds with the fact that $x_{ij}=O(|l|^{2})$ in Step $1$,
we can apply Lemma 2.2 to conclude that

$|u^{\prime}(t)|=O(|l|),$

for any $t\in[0,T_{0})$. If $T_{0}<1$,

$|u(T_{0})-\bar{u}|=O(|l|).$

Therefore, $u(T_{0})\in int(\Omega)$ if $\delta$ is sufficiently small, which
contradicts with the maximal property of $T_{0}$. Thus, $T_{0}\geq 1$ and
$(T,u(1))_{E}$ is globally flat and

(7) $|u(1)-\bar{u}|=O(|l|).$

###  3.3. Step 3

Let $\tilde{u}$ be the discrete uniformization factor with
$Area(T,\tilde{u}*l)_{E}=1$ and

$\tilde{u}-u(1)\in\mathbb{1}=\\{x\in\mathbb{R}^{V}:x=(a,...,a)\text{ for some
}a\in\mathbb{R}\\}.$

To estimate their difference, notice first that

$e^{-|u(1)-\bar{u}|}(\bar{u}*l)_{ij}\leq(u(1)*l)_{ij}\leq
e^{|u(1)-\bar{u}|}(\bar{u}*l)_{ij}.$

By equation (4),

$\displaystyle|(u(1)*l)_{ij}-\bar{l}_{ij}|=|(u(1)*l)_{ij}-(\bar{u}*l)_{ij}|+|(\bar{u}*l)_{ij}-\bar{l}_{ij}|$
$\displaystyle\leq$
$\displaystyle(e^{2|u(1)-\bar{u}|}-1)(\bar{u}*l)_{ij}+O(l_{ij}^{3})=O(|l|\cdot\bar{l}_{ij}).$

Applying Lemma 2.3, we deduce that

$\Big{|}|(\triangle ijk,u(1)*l)_{E}|-|(\triangle
ijk,\bar{l})_{E}|\Big{|}=O(|l|)|(\triangle ijk,\bar{l})_{E}|.$

Since $Area((T,\bar{l})_{E})=Area((T,\tilde{u}*l)_{E})=1$, we take the sum
over all triangles in $F$, then

$|Area(T,u(1)*l)_{E}-Area(T,\tilde{u}*l)_{E}|=O(|l|).$

This implies that

$|\tilde{u}-u(1)|=O(|l|).$

Therefore, we have

$|\tilde{u}-\bar{u}|\leq|\tilde{u}-u(1)|+|\bar{u}-u(1)|=O(|l|).$

Then $\tilde{u}$ is the desired conformal factor, and the proof of Theorem 1.4
is completed.

∎

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