# On the ADM mass of Kähler scalar flat ALE metrics

Claudio Arezzo111International Centre for Theoretical Physics ICTP,
<EMAIL_ADDRESS>and Karen Corrales 222Facultad de Matemáticas, Pontificia
Universidad Católica de Chile<EMAIL_ADDRESS>
(August 27, 2024)

###### Abstract

In this paper we study the behaviour of scalar flat Kähler ALE spaces and
their ADM mass under blow ups. In particular we prove that by blowing up
sufficiently many points at sufficiently big mutual distance one can produce
scalar flat metrics with arbitrarily large ADM mass. A general machinery for
producing scalar flat non Ricci flat ALE spaces of zero ADM mass is also
presented, using and integrating previous work by Rollin-Singer ([17]) and
Hein-LeBrun ([11]).

##  1 Introduction

The existence and geometric properties of Ricci and scalar flat metrics on ALE
Kähler spaces with a given structure group at infinity have been subject of
intense research in past few decades (see e.g. [8], [15], [14], [13], [7],
[10], [16]). In this paper we study to which extent the machinery developed in
[2] for constructing constant scalar curvature Kähler metrics on blow ups can
be used and extended in this setting. Given the fact that such spaces do not
carry holomorphic vector fields (which are the key obstruction for this
theory) it is not hard to get the existence of scalar-flat metrics on blow ups
of scalar flat ALE spaces (Theorem 3.1). While not the main focus of this
paper, we point out that the new ALE scalar flat spaces produced in Theorem
3.1 can in turn be used as new models for the gluing theory, hence getting,
even starting from compact cscK manifolds, infinitely may new families of cscK
manifolds of different topological types.

The new interesting problem studied in this paper is to estimate the maximal
size of an exceptional divisor that can be inserted in the original space. It
is known that in general this question can be very subtle on compact
manifolds, carrying implicitily the difficulties hidden in the notion of
$K$-stability and in the Tian-Yau-Donaldson Conjecture (see for example
Section $7.2$ point (2) in [9] and [19]). On top of its clear geometric
interest, the main motivation for this question comes from the recent and
beautiful work by Hein-LeBrun ([11]) connecting various geometric and
topological quantities of ALE spaces to their ADM mass, another natural
invariant arising from General Relativity recalled in Section 2.2 ([4]).

Hein-LeBrun’s topological formula for the ADM mass (Theorem 2.2) immediately
implies that the effect of blowing up one point on the ADM mass is to increase
it by an amount proportional to the volume of the exceptional divisor that one
can insert, while keeping the scalar curvature zero, hence pointing again to
the problem mentioned above.

The main result of this paper (Theorem 4.1) guarantees that this quantity
cannot decrease when performing a second blow up at a point sufficiently far
away from the first one. Iterating this scheme for multiple blow ups at
sufficiently many points at sufficiently big mutual distance, we get the main
applications in terms of ADM mass.

######  Theorem 1.1.

Let $(X,{\bf{\Gamma}},h,\omega)$ be an ALE scalar flat Kähler manifold with
structure group $\bf{\Gamma}$ and ADM mass $e\in\mathbb{R}$. Then for every
$\tilde{e}\geq e$ there exists an ALE scalar flat Kähler manifold with
structure group $\bf{\Gamma}$ with ADM mass $\tilde{e}$. Such manifold can be
obtained from $X$ by blowing up sufficiently many points at sufficiently big
mutual distance.

The above theorem indicates that, while $\Gamma$ does not determine the ADM
mass of an ALE scalar flat Kähler manifold with structure group $\bf{\Gamma}$
(something easy to observe even for trivial $\Gamma$), the interesting
quantity that can be algebraically determined by the structure group is

$inf\\{\mbox{ADM
mass}(X,{\bf{\Gamma}},h,\omega)\,|\,(X,{\bf{\Gamma}},h,\omega){\mbox{ ALE
scalar flat K\"{a}hler\ with structure group }}\,\bf{\Gamma}\\}.$

Theorem 1.1 gives also a general framework to produce scalar flat, non Ricci
flat, ALE spaces with vanishing ADM mass. Specific examples were known thanks
to Rollin-Singer ([17], Section 6.7) and Hein-LeBrun ([11], Theorem 4.7), but
our result gives a general machinery to produce them.

######  Corollary 1.2.

Let $(X,{\bf{\Gamma}},h,\omega)$ be an ALE scalar flat Kähler manifold with
structure group $\bf{\Gamma}$ and ADM mass $e<0$. Then there exists an ALE
scalar flat Kähler manifold with structure group $\bf{\Gamma}$ with vanishing
ADM mass. Such manifold can be obtained from $X$ by blowing up sufficiently
many points at sufficiently big mutual distance.

##  2 Preliminaries

###  2.1 The Kähler potential of a scalar flat ALE Kähler resolution.

We start by recalling the concept of Asymptotically Locally Euclidean (ALE for
short) Kähler manifold with structure group $\bf{\Gamma}$. Let
${\bf{\Gamma}}=(\Gamma_{j}),j=1...,k$ be a collection of finite subgroups of
the unitary group $U(m)$ each acting freely away from the origin. We say that
a complete noncompact Kähler manifold $(X,{\bf{\Gamma}},h,\omega)$ of complex
dimension $m$, where $h$ is the Kähler metric and $\omega$ is the Kähler form,
is an ALE Kähler manifold with structure group $\bf{\Gamma}$ if

  * •

there exist a compact subset $K$ such that the $j$-th connected component of
$X\setminus K$ is biholomorphic via a map $\pi_{j}$ to
$(\mathbb{C}^{m}\setminus B)/\Gamma_{j}$, where $B$ is a closed ball with
positive radius $R>0$, and

  * •

on each component in standard Euclidean coordinates the metric $\pi_{*}h$
satisfies

$\left|\frac{\partial^{\alpha}}{\partial
x^{\alpha}}\left(\big{(}\pi_{*}h)_{i\bar{j}}\,-\,\frac{1}{2}\,\delta_{i\bar{j}}\right)\right|\,\,=\,\,\mathcal{O}\left(|x|^{-\tau-|\alpha|}\right)\,,$
(2.1)

for some $\tau>0$ and every multindex $\alpha\in\mathbb{N}^{m}$.

Note that we are identifying the complex structure outside a compact subset
with the standard one. This puts a restriction (widely studied) on the class
of spaces allowed since in general its complex structure could be only
asymptotic to the standard one and in fact the complex structure could not
even admit holomorphic coordinates at infinity as shown for example by Honda
([12]) also in the scalar flat case.

While the definition allows for an ALE space to have multiple components, a
remarkable Theorem by Hein-LeBrun (Propositions $1.5$ and $4.2$ in [11]) shows
that only the case $k=1$ can actually occur. We can then assume it thoughtout
the paper.

The behaviour of the Kähler potential of a scalar flat ALE Kähler metric is
quite well understood (see e.g.[2] Section $7$ and [1] for further details)

######  Proposition 2.1.

Let $(X,\Gamma,h,\omega)$ be a scalar flat ALE Kähler manifold with structure
group $\Gamma$. Then for $R>0$ large enough, we have that on
$X\setminus\pi^{-1}(B_{R})$ the Kähler form can be written as

  * •

if $dim(X)=m>2$, then, for some real constant $e_{X}$,

$\omega\,\,=\,\,i\partial\overline{\partial}\left(\,\frac{|x|^{2}}{2}\,+\,e_{X}\,|x|^{4-2m}+\,\psi_{\eta}\left(x\right)\right)\,,\qquad\mbox{with}\qquad\psi_{\eta}\,=\,\mathcal{O}(|x|^{3-2m})\,,$
(2.2)

  * •

if $dim(X)=2$, then for some real constant $e_{X}$,

$\omega\,\,=\,\,i\partial\overline{\partial}\left(\,\frac{|x|^{2}}{2}\,+\,e_{X}\,log|x|+\,\psi_{\eta}\left(x\right)\right)\,,\qquad\mbox{with}\qquad\psi_{\eta}\,=\,\mathcal{O}(|x|^{-1})\,,$
(2.3)

###  2.2 The ADM mass of a Kähler potential of an ALE Kähler manifold with
structure group $\Gamma$.

The key quantity studied in this paper is the celebrated ADM mass introduced
by Arnowitt-Deser-Misner ([4]), whose geometrical nature has been clarified by
Bartnik ([5]) and Chruściel ([8]).

In the context on ALE Kähler manifolds it has been the center of the beautiful
paper by Hein-LeBrun ([11]) which is the original motivation of our work.

Following Hein-LeBrun’s normalization conventions are chosen so that, at a
given end, the mass of an ALE manifold is given by

$m(X,h)=\lim\limits_{R\rightarrow\infty}\frac{{\bf{\Gamma}}(\frac{m}{2})}{4(m-1)\pi^{\frac{m}{2}}}\int_{S_{R}/\Gamma}[h_{kl,k}-h_{kk,l}]{\bf{n}}^{l}da_{E}$

where $S_{R}$ is the Euclidean coordinate sphere of radius $R$, $da_{E}$ is
the $(m-1)$-dimensional volume form induced on this sphere by the Euclidean
metric, and $\bf{n}$ is the outward-pointing Euclidean unit normal vector.

Let us also denote by $\clubsuit\colon H^{2}\rightarrow H^{2}_{c}(X)$ the
inverse of the isomorphism induced by the standard inclusion of compactly
supported forms into all differential forms.

Restricting our attention to the scalar flat case, their key result is the
following equivalence:

######  Theorem 2.2.

Let $(X,\Gamma,h,\omega)$ be a scalar flat ALE Kähler manifold with structure
group $\Gamma$. Then

$m(X,h)=-\frac{\langle\clubsuit(c_{1}(X),\omega^{m-1}\rangle}{(2m-1)\pi^{m-1}}=e_{X}\,\,.$

The first equality is a special case of Theorem C in [11], while the equality
$m(X,h)=e_{X}$ is a classical observation due to LeBrun ([15]) and Rollin-
Singer ([17], Section 6.8) in complex dimension $2$, whose proof extends
readily to any dimension.

##  3 Blowing up points

In order to fix notations needed in the sequel, for every point $p\in X$ we
denote by $(z_{1},\dots z_{m})$ normal coordinates centred at $p$, so that
near $p$ the metric $\omega$ is of the form

$\omega=\omega_{E}+i\partial\bar{\partial}\varphi_{1}(z),\quad\varphi_{1}(z)=\mathcal{O}(|z|^{4}),$
(3.1)

specifically, $\varphi_{1}(z)\leq c|z|^{4},$ where $c$ depends on the
curvature of $\omega$ at $p$.

It is also convenient to isolate, among all ALE scalar flat Kähler manifolds,
$(Bl_{0}\mathbb{C}^{m},\\{Id\\},h_{BS},\eta_{BS})$, where $h_{BS}$ denotes the
well-know Burns-Simanca metric, normalized so to have volume of the
exceptional divisor $E$ equal to $1$. Its asymptotic expansion at infinity can
be written as

$\displaystyle\eta_{BS}$
$\displaystyle=\omega_{E}+i\partial\bar{\partial}\varphi_{2}(w),\quad\varphi_{2}(w)=\mathcal{O}(|w|^{4-2m}),\quad\mbox{for}\;m>2$
(3.2) $\displaystyle\eta_{BS}$
$\displaystyle=\omega_{E}+i\partial\bar{\partial}\varphi_{2}(w),\quad\varphi_{2}(w)=\log(|w|^{2}),\quad\mbox{for}\;m=2$

in terms of the standard coordinates on $Bl_{0}\mathbb{C}^{m}\setminus
E\cong\mathbb{C}^{m}\setminus\\{0\\}$.

On the other hand, [2, Proposition 5.4] implies that there is no nonzero
holomorphic vector field vanishing somewhere on $X$ and then it is easy to
observe that the proof of [2, Theorem 1.1] readily extends to the ALE case.
More precisely we have

######  Theorem 3.1.

Let $(X,\Gamma,h,\omega)$ be a scalar flat ALE Kähler manifold with structure
group $\Gamma$. Given finitely many smooth points $p_{1},\dots p_{k}\in X$,
there exists $\varepsilon_{0}(X,p_{1},\dots,p_{k}):=\varepsilon_{0}>0$ such
that the blow-up of $X$ at $p_{1},\dots p_{k}$ carries zero scalar curvature
Kähler forms

$\omega_{\varepsilon}\in\pi^{*}[\omega]-\varepsilon^{2}(PD[E_{1}]+\dots+PD[E_{k}]),$

where the $PD[E_{j}]$ are the Poincaré duals of the $(2m-2)$-homology classes
of the exceptional divisors of the blow-up at $p_{j}$ and
$\varepsilon\in(0,\varepsilon_{0})$.

What is crucial for the scope of this paper is to make, as explicit as
possible, the dependence of $\varepsilon_{0}$ on all the geometric data built
in this theorem, such as the position of the points $p_{j}$ and the geometry
of $X$ and of $Bl_{0}\mathbb{C}^{m}$.

Let us then recall that in order to construct the scalar flat metric on
$Bl_{p_{1},\dots p_{k}}X$, the idea is to replace the metric $\omega$ on a
small neighbourhood of $p_{i}$ with a suitably scaled down copy of
$\eta_{BS}$·

Precisely, for a fixed $\varepsilon>0$ and
$r_{\varepsilon}=\varepsilon^{m-1/m}$, we will glue $\omega$ with
$\varepsilon^{2}\eta_{BS}$ on the annulus $B_{2r_{\varepsilon}}\setminus
B_{r_{\varepsilon}}$ around each $p_{i}$, under the change of variables
$z=\varepsilon w.$

Following (3.1) and (3.2), the metric on $X\setminus\\{p_{1}\dots,p_{k}\\}$,
which naturally extends to a metric on $Bl_{p_{1},\dots p_{k}}X$, is defined
by

$\tilde{\omega}_{\varepsilon}=\begin{cases}\omega\quad\mbox{on}\quad
X\setminus{\cup_{i=1}^{k}B_{2r_{\varepsilon}}(p_{i})}\\\
\omega_{E}+i\partial\bar{\partial}(\gamma_{1}(z)\varphi_{1}(z)+\varepsilon^{2}(1-\gamma_{1}(z))\varphi_{2}(\varepsilon^{-1}z))\quad\mbox{on}\quad
B_{2r_{\varepsilon}}(p_{i})\setminus B_{r_{\varepsilon}}(p_{i})\\\
\varepsilon^{2}\eta_{BS}\quad\mbox{on}\quad
B_{r_{\varepsilon}}(p_{i})\setminus\\{p_{i}\\}\end{cases}$ (3.3)

where $\gamma_{1}$ is a suitable cutoff function.

Once this pre-glued metric is built the problem is to find a function
$\varphi$ such that the metric
$\omega_{\varepsilon}=\tilde{\omega}_{\varepsilon}+i\partial\bar{\partial}\varphi$
has zero scalar curvature. In order to solve that equation, the most important
step is to show that the linear operator $L_{\tilde{\omega}_{\varepsilon}}$
associated to the scalar curvature is invertible and to obtain bounds on the
norm of the inverse in suitable Banach spaces: weighted Hölder spaces, which
now have to take into “weighted” account the behaviour of function near the
point $p_{j}$, with weight $\delta$, and at infinity, with weight
$\delta_{\infty}$.

The weighted spaces where to carry out the proof of Theorem 3.1 are the same
one as those used in [2] and [18] with the extra weight $\delta_{\infty}$.

If we denote $X^{*}:=X\setminus\\{p_{1}\dots,p_{k}\\}$,
$X_{r}:=X\setminus{\cup_{i=1}^{k}B_{r}(p_{i})}$ and $N_{p_{i}}$ by the
identification of $B_{r}(p_{i})\setminus\\{p_{i}\\}$ with
$Bl_{0}\mathbb{C}^{n}$, then we define

######  Definition 3.2.

Given $l\in\mathbb{N}$, $\alpha\in(0,1)$, $\delta,\delta_{\infty}\in(4-2m,0)$,
we define

  1. 1.

The weighted space $C^{l,\alpha}_{\delta,\delta_{\infty}}(X^{*})$ as the space
of functions $\varphi\in C^{l,\alpha}_{loc}(X^{*})$ for which the norm

$\|\varphi\|_{C^{l,\alpha}_{\delta,\delta_{\infty}}(X^{*})}:=\|\varphi\|_{C^{l,\alpha}_{\delta_{\infty}}(X_{r_{0}})}+\max_{i=1...k}\left\\{\sup_{r\in(0,2r_{0})}\left(r^{-\delta}\|\varphi(rz)\|_{C^{l,\alpha}(B_{2}(p_{i})\setminus
B_{1}(p_{i}))}\right)\right\\}$

is finite.

  2. 2.

The weighted space $C^{l,\alpha}_{\delta}(N_{p_{i}})$ is defined to be the
space of functions $\varphi\in C^{l,\alpha}_{loc}(N_{p_{i}})$ for which the
norm

$\|\varphi\|_{C^{l,\alpha}_{\delta}(N_{p_{i}})}:=\|\varphi\|_{C^{l,\alpha}(\\{|w|<2R_{0}\\})}+\sup_{R\in[R_{0},\infty)}\left(R^{-\delta}\|\varphi(Rw)\|_{C^{l,\alpha}(B_{2}\setminus
B_{1})}\right)$

is finite.

  3. 3.

The weighted space $C^{l,\alpha}_{\delta,\delta_{\infty}}(Bl_{p_{1},\dots
p_{k}}X)$ is defined for the gluing between the previous two, i.e. the space
of functions $\varphi\in C^{l,\alpha}_{loc}(Bl_{p_{1},\dots p_{k}}X)$ for
which the norm

$\displaystyle\|\varphi\|_{C^{l,\alpha}_{\delta,\delta_{\infty}}(Bl_{p_{1},\dots
p_{k}}X)}$
$\displaystyle=\|\varphi\|_{C^{l,\alpha}_{\delta_{\infty}}(X_{2r_{\varepsilon}})}+\|\varphi\|_{C^{l,\alpha}_{\delta}(N_{p_{i}})}$
(3.4)
$\displaystyle\;\;+\max_{i=1...k}\left\\{\sup_{r\in(r_{\varepsilon},2r_{\varepsilon})}\left(r^{-\delta}\|\varphi(rz)\|_{C^{l,\alpha}(B_{2}(p_{i})\setminus
B_{1}(p_{i}))}\right)\right\\}$

is finite.

Throughout this paper we will denote $X_{p_{1},\dots
p_{k}}:=X\setminus{\cup_{i=1}^{k}B_{2r_{\varepsilon}}(p_{i})}$ and
$A_{p_{i}}:=B_{2r_{\varepsilon}}(p_{i})\setminus B_{r_{\varepsilon}}(p_{i})$,
then

$\|\varphi\|_{C^{l,\alpha}_{\delta,\delta_{\infty}}(Bl_{p_{1},\dots
p_{k}}X)}=\|\varphi\|_{C^{l,\alpha}_{\delta_{\infty}}(X_{p_{1},\dots
p_{k}})}+\|\varphi\|_{C^{l,\alpha}_{\delta}(N_{p_{i}})}+\max_{i=1...k}\left\\{\|\varphi\|_{C^{l,\alpha}_{\delta}(A_{p_{i}})}\right\\}.$
(3.5)

######  Remark 3.3.

Note that the norm $\|\varphi\|_{C^{l,\alpha}_{\delta_{\infty}}(X_{p_{1},\dots
p_{k}})}$ uses to the metric $\omega$;
$\|\varphi\|_{C^{l,\alpha}_{\delta}(A_{p_{i}})}$ the pre-glued metric
$\tilde{\omega}_{\varepsilon}$ and
$\|\varphi\|_{C^{l,\alpha}_{\delta}(N_{p_{i}})}$ uses the metric
$\varepsilon^{2}\eta_{BS}$ for each $i$.

######  Remark 3.4.

It is important to remark that the proof of Theorem 3.1 depends on whether
$m=2$ or $m>2$. However, when $m=2$ we can still work weight spaces with
$\delta\in(-1,0)$, but we lose the uniform control of the inverse operator and
other problems appear. One way to overcome these problems is to construct a
better gluing metric than the one defined by (3.3). If we consider

$\hat{\omega}_{\varepsilon}=\begin{cases}\omega\quad\mbox{on}\quad
X\setminus{\cup_{i=1}^{k}B_{2r_{\varepsilon}}(p_{i})}\\\
\omega_{E}+i\partial\bar{\partial}(\gamma_{1}(z)\varphi_{1}(z)+\varepsilon^{2}(1-\gamma_{1}(z))\log|\varepsilon^{-1}z|)\quad\mbox{on}\quad
B_{2r_{\varepsilon}}(p_{i})\setminus B_{r_{\varepsilon}}(p_{i})\\\
\varepsilon^{2}\eta_{BS}\quad\mbox{on}\quad
B_{r_{\varepsilon}}(p_{i})\setminus\\{p_{i}\\}\end{cases}$

then we define
$\tilde{\omega}_{\varepsilon}=\hat{\omega}_{\varepsilon}+\varepsilon^{2}i\partial\bar{\partial}\left(\gamma_{1}(z)(\Gamma(z)-\log\varepsilon)\right)$,
where $\Gamma$ is a solution asymptotic to $\log|z|$ near $p_{i}$ of
$L_{\omega}(\Gamma)=c$, where $c$ is a constant and $L_{\omega}$ is the
linearized scalar curvature operato. For more details, see [18, Section 8.4].

Therefore, the new analysis carried out in this paper does depend on $m$ we
give the details only in the case $m>2$.

All proofs about the invertibility of the linearized scalar curvature,
$L_{\omega_{\varepsilon}}$, contained in [2] and [18], go through verbatim
also in this context so we do not reproduce them. However, we need to estimate
the dependence of norms of operators, both linear and non-linear, on the
geometric data of the construction. This is what we achieve in the next
section.

##  4 Dependence on the position of the points

The goal in this section is to study the behaviour of $\varepsilon_{0}$ on
Theorem 3.1, for the blow-up at one point $p$ with the one for the blow-up at
two points $p,q$, to prove that $\varepsilon_{0}(p,q)\geq\varepsilon_{0}(p)$
as $q$ goes to infinity.

Let $(X,\Gamma,h,\omega)$ be a scalar flat ALE Kähler manifold with structure
group $\Gamma$ and pick two points $p,q\in X$. In the sequel we will consider
$p\in X$ fixed and study the behaviour of all appearing quantities with $p$
fixed and moving $q$. Theorem 3.1 implies that

  * •

there exists $\varepsilon_{0}(p)>0$ such that for every
$\varepsilon\in(0,\varepsilon_{0}(p))$ the blow-up of $X$ at $p$ carries
scalar flat Kähler forms

$\omega_{\varepsilon}\in\pi^{*}[\omega]-\varepsilon^{2}aPD[E].$

  * •

there exists $\varepsilon_{0}(p,q)>0$ such that for every
$\varepsilon\in(0,\varepsilon_{0}(p,q))$ the blow-up of $X$ at $p,q$ carries
constant scalar curvature Kähler forms

$\omega_{\varepsilon}\in\pi^{*}[\omega]-\varepsilon^{2}a_{p}PD[E_{p}]-\varepsilon^{2}a_{q}PD[E_{q}].$

Note that with a slight abuse of notation we use $\omega_{\varepsilon}$ for
the pre-glued metrics $\tilde{\omega}_{\varepsilon}$ and the scalar flat
metrics $\omega_{\varepsilon}$ in both cases for $Bl_{p}X$ and $Bl_{p,q}X$
since there is no possibility of confusion.

Thus, our main estimate is

######  Theorem 4.1.

Let $(X,\Gamma,h,\omega)$ be a scalar flat ALE Kähler manifold with structure
group $\Gamma$ and $p,q\in X$. If $\varepsilon_{0}(p)$ and
$\varepsilon_{0}(p,q)$ are the constants where Theorem 3.1 holds then

$\varepsilon_{0}(p,q)\to\varepsilon_{0}\geq\varepsilon_{0}(p),\quad\mbox{as
}\;r(q)\to\infty$

where $r(q)$ is the distance between $q$ and a compact subset $K\subset X$
containing $p$.

In this case and following the metric defined in (3.3) we have

$\tilde{\omega}_{\varepsilon}=\begin{cases}\omega\quad\mbox{on}\quad
X_{p,q}\\\
\omega_{E}+i\partial\bar{\partial}\varphi_{p,\varepsilon}(z)\quad\mbox{on}\quad
A_{p}\\\
\omega_{E}+i\partial\bar{\partial}\varphi_{q,\varepsilon}(z)\quad\mbox{on}\quad
A_{q}\\\ \varepsilon^{2}\eta_{BS}\quad\mbox{on}\quad N_{p}\\\
\varepsilon^{2}\eta_{BS}\quad\mbox{on}\quad N_{q}\\\ \end{cases}$

where $\varphi_{p,\varepsilon}(z)$ and $\varphi_{q,\varepsilon}(z)$ are the
gluing potentials between $\varphi_{1}$ and $\varphi_{2}$, near $p$ and near
$q$, respectively.

######  Remark 4.2.

It is important to note that (3.1) and (3.2) imply that there exist constants
$c_{p},c_{q}$ and $c$ such that

$\begin{split}\|\varphi_{p,\varepsilon}(z)\|_{C^{4,\alpha}_{\delta}(A_{p})}&\leq\|\varphi_{1}(z)\|_{C^{4,\alpha}_{\delta}(A_{p})}+\varepsilon^{2}\|\varphi_{2}(\varepsilon^{-1}z)\|_{C^{4,\alpha}_{\delta}(A_{p})}\leq(c_{p}+c)r_{\varepsilon}^{4-\delta},\\\
\|\varphi_{q,\varepsilon}(z)\|_{C^{4,\alpha}_{\delta}(A_{q})}&\leq\|\varphi_{1}(z)\|_{C^{4,\alpha}_{\delta}(A_{q})}+\varepsilon^{2}\|\varphi_{2}(\varepsilon^{-1}z)\|_{C^{4,\alpha}_{\delta}(A_{q})}\leq(c_{q}+c)r_{\varepsilon}^{4-\delta},\end{split}$
(4.1)

where $c_{p}$ and $c_{q}$ depends on the curvature of $\omega$ around $p$ and
$q$, and $c$ does not depend on the point.

The simple but crucial observation, repeatedly used in the sequel, is that
since the metric $\omega$ is ALE then $c_{q}\leq c_{p}$ as $q$ goes to
infinity. Therefore,

$\|\varphi_{q,\varepsilon}(z)\|_{C^{4,\alpha}_{\delta}(A_{q})}\leq(c_{p}+c)r_{\varepsilon}^{4-\delta}.$
(4.2)

The existence of the constant $\varepsilon_{0}$ in the theory developed in [2]
and [3] comes from the estimates on the scalar curvature, its linear operator
and the quadratic part to use the Fixed point Theorem. Denoting by
$S(\omega_{\varepsilon})$ as the scalar curvature, $L_{\omega_{\varepsilon}}$
its linear part, $Q_{\omega_{\varepsilon}}$ its quadratic part and
$\mathcal{N}$ as the operator

$\mathcal{N}:C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X)\to
C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X),\quad\mathcal{N}(\varphi)=L_{\omega_{\varepsilon}}^{-1}(-S(\omega_{\varepsilon})-Q_{\omega_{\varepsilon}}(\varphi)),$

the key Proposition to prove the existence of cscK metrics is the following

######  Proposition 4.3.

see e.g. [18, Proposition 8.20] For some constant $c_{1}$, let

$\mathcal{U}:=\\{\varphi\in
C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X):\|\varphi\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X)}\leq
c_{1}r_{\varepsilon}^{2-\delta}\\}.$

Then, there exists $\varepsilon_{0}(p,q)$ such that for every
$\varepsilon<\varepsilon_{0}$, $\mathcal{N}$ is a contraction on $\mathcal{U}$
and $\mathcal{N}(\mathcal{U})\subset\mathcal{U}.$ In particular, $\mathcal{N}$
has a fixed point, which gives a cscK metric on $Bl_{p,q}X$ in the Kähler
class
$\pi^{*}[\omega]-\varepsilon^{2}a_{p}PD[E_{p}]-\varepsilon^{2}a_{q}PD[E_{q}]$.

The key estimates to prove the above Proposition are the following

(i)

there exists a constant $c_{1}$ such that if
$\|\varphi\|_{C^{4,\alpha}_{2,\delta_{\infty}}(Bl_{p,q}X)}\leq c_{1}$ then

$\|\mathcal{N}(\varphi)-\mathcal{N}(0)\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X)}\leq\frac{1}{2}\|\varphi\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X)}.$

(ii)

for the same constant $c_{1}$,
$\|\mathcal{N}(0)\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X)}\leq\frac{1}{2}c_{1}r_{\varepsilon}^{2-\delta}$.

######  Remark 4.4.

Note that

$\|\mathcal{N}(\varphi)-\mathcal{N}(0)\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X)}=\|L_{\omega_{\varepsilon}}^{-1}(Q_{\omega_{\varepsilon}}(0)-Q_{\omega_{\varepsilon}}(\varphi))\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}}\leq
K_{p,q}\|Q_{\omega_{\varepsilon}}(\varphi))\|_{C^{0,\alpha}_{\delta-4,\delta_{\infty}-4}}\leq
K_{p,q}c_{1}\|\varphi\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}}$
$\|\mathcal{N}(0)\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X)}=\|L_{\omega_{\varepsilon}}^{-1}(-S({\omega_{\varepsilon}}))\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X)}\leq
K_{p,q}\|S(\omega_{\varepsilon}))\|_{C^{0,\alpha}_{\delta-4}(Bl_{p,q}X)}\leq
K_{p,q}C_{p,q}r_{\varepsilon}^{4-\delta}.$

Therefore, (i) follows once $c_{1}$ is chosen less than $1/2K_{p,q}$ and
$\varepsilon_{0}(p,q)$ is chosen such that for every
$\varepsilon<\varepsilon_{0}$

$2K_{p,q}C_{p,q}r_{\varepsilon}^{2}<c_{1},\quad
r_{\varepsilon}=\varepsilon^{\frac{m-1}{m}}.$

Thus, the next two subsections will be dedicated to understand the definition
and construction of $C_{p,q}$ and $K_{p,q}$. Next, they will be compared with
the same constants $C_{p}$ and $K_{p}$ (in the case of $Bl_{p}X$) when $q$
goes to infinity.

###  4.1 Comparison between $C_{p,q}$ with $C_{p}$

In this part we will to compare the estimates on the scalar curvature for the
preglued metrics for $Bl_{p,q}X$ and $Bl_{p}X.$ It is known that there exist
$C_{p}$ and $C_{p,q}$ defined by

$\displaystyle C_{p,q}$ $\displaystyle:=$
$\displaystyle\inf\\{C>0\mid\|S(\omega_{\varepsilon})\|_{C^{0,\alpha}_{\delta-4}(Bl_{p,q}X)}\leq
C_{p,q}r_{\varepsilon}^{4-\delta}\\},$ $\displaystyle C_{p}$ $\displaystyle:=$
$\displaystyle\inf\\{C>0\mid\|S(\omega_{\varepsilon})\|_{C^{0,\alpha}_{\delta-4}(Bl_{p}X)}\leq
C_{p}r_{\varepsilon}^{4-\delta}\\}.$

######  Lemma 4.5.

There exists a compact subset $K$ which contains $p$ such that $C_{p,q}\leq
C_{p}$ for every $q\in X\setminus K$.

###### Proof.

By definition of the weighted norm, we have

$\begin{split}\|S(\omega_{\varepsilon})\|_{C^{0,\alpha}_{\delta-4}(Bl_{p}X)}&=\|S(\omega_{E}+i\partial\bar{\partial}\varphi_{p,\varepsilon})\|_{C^{0,\alpha}_{\delta-4}(A_{p})}\\\
\|S(\omega_{\varepsilon})\|_{C^{0,\alpha}_{\delta-4}(Bl_{p,q}X)}&=\max\left\\{\|S(\omega_{E}+i\partial\bar{\partial}\varphi_{p,\varepsilon})\|_{C^{0,\alpha}_{\delta-4}(A_{p})},\|S(\omega_{E}+i\partial\bar{\partial}\varphi_{q,\varepsilon})\|_{C^{0,\alpha}_{\delta-4}(A_{q})}\right\\}\\\
\end{split}$ (4.3)

Moreover, given that
$S(\omega)=(g_{\omega})^{\alpha\bar{\beta}}(R_{\omega})_{\alpha\bar{\beta}}$,
then we compute

$\displaystyle(g_{p,\varepsilon})_{j\bar{k}}$ $\displaystyle=$
$\displaystyle\delta_{jk}+\frac{\partial^{2}\varphi_{p,\varepsilon}}{\partial
z_{j}\partial\bar{z}_{k}}(z)$ $\displaystyle(g_{q,\varepsilon})_{j\bar{k}}$
$\displaystyle=$
$\displaystyle\delta_{jk}+\frac{\partial^{2}\varphi_{q,\varepsilon}}{\partial
z_{j}\partial\bar{z}_{k}}(z)$
$\displaystyle(R_{p,\varepsilon})_{\alpha\bar{\beta}}$ $\displaystyle=$
$\displaystyle
2(g_{p,\varepsilon})^{j\bar{k}}\left[\left(\frac{\partial^{3}\varphi_{p,\varepsilon}}{\partial
z_{\alpha}\partial
z_{j}\partial\bar{z}_{k}}\right)\left(\frac{\partial^{3}\varphi_{p,\varepsilon}}{\partial\bar{z}_{\beta}\partial
z_{j}\partial\bar{z}_{k}}\right)-\left(\frac{\partial^{4}\varphi_{p,\varepsilon}}{\partial
z_{\alpha}\partial\bar{z}_{\beta}\partial
z_{j}\partial\bar{z}_{k}}\right)\right]$
$\displaystyle(R_{q,\varepsilon})_{\alpha\bar{\beta}}$ $\displaystyle=$
$\displaystyle
2(g_{q,\varepsilon})^{j\bar{k}}\left[\left(\frac{\partial^{3}\varphi_{q,\varepsilon}}{\partial
z_{\alpha}\partial
z_{j}\partial\bar{z}_{k}}\right)\left(\frac{\partial^{3}\varphi_{q,\varepsilon}}{\partial\bar{z}_{\beta}\partial
z_{j}\partial\bar{z}_{k}}\right)-\left(\frac{\partial^{4}\varphi_{q,\varepsilon}}{\partial
z_{\alpha}\partial\bar{z}_{\beta}\partial
z_{j}\partial\bar{z}_{k}}\right)\right]$

It is immediate to check that there exists a constant $c_{1}$ such that

$\|\nabla^{i}\varphi_{p,\varepsilon}\|_{C^{4-i,\alpha}_{\delta-i}}\leq
c_{1}\|\varphi_{p,\varepsilon}\|_{C^{4,\alpha}_{\delta}},\quad\|\nabla^{i}\varphi_{q,\varepsilon}\|_{C^{4-i,\alpha}_{\delta-i}}\leq
c_{1}\|\varphi_{q,\varepsilon}\|_{C^{4,\alpha}_{\delta}},$

where the constant $c_{1}$ is independent of the function and on the points.

Thus, there exist constants $C_{1}$, independent of $\varphi_{p,\varepsilon}$
and $\varphi_{q,\varepsilon}$, respectively, such that

$\|S(\omega_{E}+i\partial\bar{\partial}\varphi_{p,\varepsilon})\|_{C^{0,\alpha}_{\delta-4}(A_{p})}\leq
C_{1}\|\varphi_{p,\varepsilon}\|_{C^{4,\alpha}_{\delta}(A_{p})}$
$\|S(\omega_{E}+i\partial\bar{\partial}\varphi_{q,\varepsilon})\|_{C^{0,\alpha}_{\delta-4}(A_{q})}\leq
C_{1}\|\varphi_{q,\varepsilon}\|_{C^{4,\alpha}_{\delta}(A_{q})}$

Using Remark 4.2 taking a sufficiently big compact subset $K$ containing $p$,
we obtain

$\begin{split}\|S(\omega_{\varepsilon})\|_{C^{0,\alpha}_{\delta-4}(Bl_{p,q}X)}&=\max\left\\{\|S(\omega_{E}+i\partial\bar{\partial}\varphi_{p,\varepsilon})\|_{C^{0,\alpha}_{\delta-4}(A_{p})},\|S(\omega_{E}+i\partial\bar{\partial}\varphi_{q,\varepsilon})\|_{C^{0,\alpha}_{\delta-4}(A_{q})}\right\\}\\\
&\leq\max\left\\{C_{1}\|\varphi_{p,\varepsilon}\|_{C^{4,\alpha}_{\delta}(A_{p})},C_{1}\|\varphi_{q,\varepsilon}\|_{C^{4,\alpha}_{\delta}(A_{q})}\right\\}\\\
&\leq C_{1}(c_{p}+c)r_{\varepsilon}^{4-\delta}.\end{split}$ (4.4)

for $q\in X\setminus K$.

Therefore, $C_{p,q}\leq C_{1}(c_{p}+c)=:C_{p}.$ ∎

###  4.2 Comparison between $K_{p,q}$ with $K_{p}$

In this part we will to compare the bounds on the norm of the inverse linear
operator associated to the scalar curvature in the construction of metrics on
$Bl_{p,q}X$ and on $Bl_{p}X.$

It is known that for every $\varphi\in
C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X)$ and $\psi\in
C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p}X)$ there exist $K_{p,q}$ and
$K_{p}$ defined by

$\displaystyle K_{p,q}$ $\displaystyle:=$
$\displaystyle\inf\\{K>0\mid\|\varphi\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X)}\leq
K\|L_{\omega_{\varepsilon}}(\varphi)\|_{C^{0,\alpha}_{\delta-4,\delta_{\infty}-4}(Bl_{p,q}X)}\\},$
$\displaystyle K_{p}$ $\displaystyle:=$
$\displaystyle\inf\\{K>0\mid\|\psi\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p}X)}\leq
K\|L_{\omega_{\varepsilon}}(\psi)\|_{C^{0,\alpha}_{\delta-4,\delta_{\infty}-4}(Bl_{p}X)}\\}.$

######  Lemma 4.6.

$K_{p,q}$ goes to $K_{0}\leq K_{p}$ as $q$ goes to infinity.

###### Proof.

Let $\varphi\in C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X)$. We define
the functions $\chi\varphi$ on $Bl_{p}(X)$ and $\bar{\chi}\varphi$ on
$Bl_{q}X$ as follows

$\chi\varphi=\begin{cases}\varphi(z)\quad\mbox{on }X_{p,q}\\\
\varphi(z)\quad\mbox{on }A_{p}\\\ \varphi(z)\quad\mbox{on }N_{p}\\\
\gamma(\frac{|z|}{r_{\varepsilon}})\varphi(z)\;\mbox{on }A_{q}\\\
0\quad\mbox{on }N_{q}\end{cases}$
$\bar{\chi}\varphi=\begin{cases}0\quad\mbox{on }X_{p,q}\\\ 0\quad\mbox{on
}A_{p}\\\ 0\quad\mbox{on }N_{p}\\\
\bar{\gamma}(\frac{|z|}{r_{\varepsilon}})\varphi(z)\;\mbox{on }A_{2q}\\\
\varphi(z)\quad\mbox{on }A_{q}\\\ \varphi(z)\quad\mbox{on }N_{q}\end{cases}$

where $\gamma,\bar{\gamma}:\mathbb{R}\to[0,1]$ are smooth functions such that

$\gamma(x)=\begin{cases}1\quad\mbox{if}\quad x\geq 2\\\ 0\quad\mbox{if}\quad
x\leq 1\end{cases}$ and $\bar{\gamma}(x)=\begin{cases}0\quad\mbox{if}\quad
x\geq 3\\\ 1\quad\mbox{if}\quad x\leq 2\end{cases}$

Note that it is possible to identify $\bar{\chi}\varphi$ around $p$, i.e. to
compute the norm of the same function but with the metric near $p$. In this
case, there exists a constant $K_{1}$, depending on $p$, such that

$\begin{split}\|\bar{\chi}\varphi\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{q}X)}&=\|\bar{\chi}\varphi\|_{C^{4,\alpha}_{\delta_{\infty}}(X_{q})}+\|\bar{\chi}\varphi\|_{C^{4,\alpha}_{\delta}(A_{q})}+\|\bar{\chi}\varphi\|_{C^{4,\alpha}_{\delta}(N_{q})}\quad\mbox{($\bar{\chi}\varphi$
has compact support)}\\\ &\leq
K_{1}(\|\bar{\chi}\varphi\|_{C^{4,\alpha}(X_{p})}+\|\bar{\chi}\varphi\|_{C^{4,\alpha}_{\delta}(A_{p})})+\|\bar{\chi}\varphi\|_{C^{4,\alpha}_{\delta}(N_{p})}.\end{split}$

Since

$\|\varphi\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X)}=\|\varphi\|_{C^{4,\alpha}_{\delta_{\infty}}(X_{p,q})}+\max\left\\{\|\varphi\|_{C^{4,\alpha}_{\delta}(A_{p})},\|\varphi\|_{C^{4,\alpha}_{\delta}(A_{q})}\right\\}+\|\varphi\|_{C^{4,\alpha}_{\delta}(N_{p})}+\|\varphi\|_{C^{4,\alpha}_{\delta}(N_{q})},$

then we have

$\displaystyle\|\varphi\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X)}\leq\|\chi\varphi\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p}X)}+\|\bar{\chi}\varphi\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{q}X)}$
(4.5) $\displaystyle\quad\leq
K_{p}\|L_{\omega_{\varepsilon}}(\chi\varphi)\|_{C^{0,\alpha}_{\delta-4,\delta_{\infty}-4}(Bl_{p}X)}+K_{1}(\|\bar{\chi}\varphi\|_{C^{4,\alpha}(X_{p})}+\|\bar{\chi}\varphi\|_{C^{4,\alpha}_{\delta}(A_{p})})+\|\bar{\chi}\varphi\|_{C^{4,\alpha}_{\delta}(N_{p})}$
(4.7) $\displaystyle\quad\leq
K_{p}\left(\|L_{\omega}(\chi\varphi)\|_{C^{0,\alpha}_{\delta_{\infty}-4}(X_{p,q})}+\|L_{\omega}(\chi\varphi)\|_{C^{0,\alpha}_{\delta_{\infty}-4}(A_{q})}+\|L_{\omega_{\varepsilon}}(\chi\varphi)\|_{C^{0,\alpha}_{\delta-4}(A_{p})}+\|L_{\varepsilon^{2}\eta_{BS}}(\chi\varphi)\|_{C^{0,\alpha}_{\delta-4}(N_{p})}\right.$
(4.9)
$\displaystyle\quad\quad\quad\quad\left.+K_{1}\|L_{\omega}(\bar{\chi}\varphi)\|_{C^{0,\alpha}(X_{p})}+K_{1}\|L_{\omega_{\varepsilon}}(\bar{\chi}\varphi)\|_{C^{0,\alpha}_{\delta-4}(A_{p})}+\|L_{\varepsilon^{2}\eta_{BS}}(\bar{\chi}\varphi)\|_{C^{0,\alpha}_{\delta-4}(N_{p})}\right)$
(4.10) $\displaystyle\quad\leq
K_{p}\|L_{\omega_{\varepsilon}}(\varphi)\|_{C^{0,\alpha}_{\delta-4,\delta_{\infty}-4}(Bl_{p,q}X)}+K_{p}\\!\left(\|L_{\omega}(\chi\varphi)\|_{C^{0,\alpha}_{\delta_{\infty}-4}(A_{q})}\\!+\\!K_{1}(\|L_{\omega}(\bar{\chi}\varphi)\|_{C^{0,\alpha}(X_{p})}\\!+\\!\|L_{\omega_{\varepsilon}}(\bar{\chi}\varphi)\|_{C^{0,\alpha}_{\delta-4}(A_{p})})\right).$
(4.12)

Moreover, as $L$ is a linear operator at each metric, we get that there exists
a constant $K_{2}$, depending on $p$, such that

$\begin{split}\|L_{\omega}(\chi\varphi)\|_{C^{0,\alpha}_{\delta_{\infty}-4}(A_{q})}&\leq
K_{2}\|\chi\varphi\|_{C^{4,\alpha}_{\delta_{\infty}}(A_{q})}\\\
\|L_{\omega}(\bar{\chi}\varphi)\|_{C^{0,\alpha}(X_{p})}&\leq
K_{2}\|\bar{\chi}\varphi\|_{C^{4,\alpha}(X_{p})}\\\
\|L_{\omega_{\varepsilon}}(\bar{\chi}\varphi)\|_{C^{0,\alpha}_{\delta-4}(A_{p})}&\leq
K_{2}\|\bar{\chi}\varphi\|_{C^{4,\alpha}_{\delta}(A_{p})}\end{split}$

Also, we may assume that there exist constants $K_{3}$ and $K_{4}$ depending
only on $p$ as $q$ goes to infinity, such that

$\|\bar{\chi}\varphi\|_{C^{4,\alpha}(X_{p})}\leq
K_{3}\|\bar{\chi}\varphi\|_{C^{4,\alpha}(X_{q})},\quad\|\bar{\chi}\varphi\|_{C^{4,\alpha}_{\delta}(A_{p})}\leq
K_{4}\|\bar{\chi}\varphi\|_{C^{4,\alpha}_{\delta}(A_{q})}.$

Thus, there exists a constant $K$ which depends on $p$ as $q$ goes to infinity
such that

$\|\varphi\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X)}\leq
K_{p}\|L_{\omega_{\varepsilon}}(\varphi)\|_{C^{0,\alpha}_{\delta-4,\delta_{\infty}-4}(Bl_{p,q}X)}+K_{p}K\left(\|\chi\varphi\|_{C^{4,\alpha}_{\delta_{\infty}}(A_{q})}+\|\bar{\chi}\varphi\|_{C^{4,\alpha}(X_{q})}+\|\bar{\chi}\varphi\|_{C^{4,\alpha}_{\delta}(A_{q})}\right).$

Moreover, properties on the weighted norms imply

$\|\chi\varphi\|_{C^{4,\alpha}_{\delta_{\infty}}(A_{q})}\leq
r(q)^{\delta_{\infty}}\|\chi\varphi\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{q}(X))},$
$\|\bar{\chi}\varphi\|_{C^{4,\alpha}(X_{q})}+\|\bar{\chi}\varphi\|_{C^{4,\alpha}_{\delta}(A_{q})}\leq
r(q)^{\delta_{\infty}}\|\bar{\chi}\varphi\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{q}(X))},$

where $r(q)$ is the distance between $q$ and a compact subset $K\subset X$
containing $p$.

Since $r(q)$ goes to infinity as $q$ goes to infinity and $\delta_{\infty}<0$,
we get

$\|\varphi\|_{C^{4,\alpha}_{\delta,\delta_{\infty}}(Bl_{p,q}X)}\leq
K_{p}\|L_{\omega_{\varepsilon}}(\varphi)\|_{C^{0,\alpha}_{\delta-4,\delta_{\infty}-4}(Bl_{p,q}X)},\quad\mbox{as}\;q\;\mbox{goes
to infinity}.$

∎

###### Proof of Theorem 4.1.

Following Remark 4.4 and (i), (ii), if we choose $c_{1}=1/2K_{p,q}$ then

$\varepsilon_{0}(p,q):=\left(\frac{1}{2K_{p,q}\sqrt{C_{p,q}}}\right)^{\frac{m}{m-1}}$

and for every $\varepsilon<\varepsilon_{0}(p,q)$ we have

$K_{p,q}C_{p,q}r_{\varepsilon}^{2}<\frac{1}{2}c_{1}.$

Using Lemma 4.5 and Lemma 4.6, we get, for $q$ sufficintely far away from $p$

$\varepsilon_{0}(p,q):=\left(\frac{1}{2K_{p,q}\sqrt{C_{p,q}}}\right)^{\frac{m}{m-1}}\geq\left(\frac{1}{2K_{p,q}\sqrt{C_{p}}}\right)^{\frac{m}{m-1}}\longrightarrow\left(\frac{1}{2K_{p}\sqrt{C_{p}}}\right)^{\frac{m}{m-1}}.$

Therefore, $\varepsilon_{0}(p,q)$ goes to
$\varepsilon_{0}\geq\varepsilon_{0}(p).$ ∎

##  5 ADM mass under blow ups, proof of the main results

We now analyze the effects of the previous results on the ADM mass of a scalar
flat ALE space. Thanks to the formula due to Hein-LeBrun ([11]) (Theorem 2.2),
this is easily achieved by the following

######  Proposition 5.1.

Let $m(X,h)$ and $m(Bl_{p}X,h_{\varepsilon})$ be the ADM mass of
$(X,\Gamma,h,\omega)$ and $Bl_{p}X$ respectively, where $h_{\varepsilon}$ is
the scalar flat metric produced out of $h$ by Theorem 3.1. Then,

$m(Bl_{p}X,h_{\varepsilon})=m(X,h)+\frac{(m-1)\varepsilon^{2(m-1)}}{(2m-1)\pi^{m-1}}.$

###### Proof.

Since $\omega_{\varepsilon}\in\pi^{*}[\omega]-\varepsilon^{2}PD[E_{p}]$ and we
have normalized the Burns-Simanca metric in a way that
$[\eta_{BS}]=-PD[E_{p}]$ then

$[\omega_{\varepsilon}]=[\omega]+\varepsilon^{2}[\eta],\quad
c_{1}(Bl_{p}X)=\pi^{*}(c_{1}(X))-(m-1)PD[E_{p}].$

By Theorem 2.2 we have

$\displaystyle m(Bl_{p}X,h_{\varepsilon})$ $\displaystyle=$
$\displaystyle-\frac{\langle\clubsuit(c_{1}(Bl_{p}X)),[\omega_{\varepsilon}]^{m-1}\rangle}{(2m-1)\pi^{m-1}}$
$\displaystyle=$
$\displaystyle-\frac{\langle\clubsuit(c_{1}(X)-(m-1)PD[E_{p}]),([\omega]-\varepsilon^{2}PD[E_{p}])\rangle}{(2m-1)\pi^{m-1}}$
$\displaystyle=$ $\displaystyle
m(X,h)+\frac{(m-1)\varepsilon^{2(m-1)}}{(2m-1)\pi^{m-1}}.$

∎

###### Proof of Theorem 1.1:.

Proposition 5.1 shows that each time we blow up a point $p$ we can increase
the ADM mass of any amount up to

$\frac{(m-1)\varepsilon_{0}(p)^{2(m-1)}}{(2m-1)\pi^{m-1}}.$

Theorem 4.1 shows that iterating the process taking a second point
sufficiently far away from the first one, we can increase again the ADM mass
by the same amount (at least). Of course, blowing up points does not change
the structure groups at infinity of the original ALE space. Given that the
process can be iterated infinitely many times, the theorem is proved. ∎

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