# Chern-Ricci flat balanced metrics on small resolutions of Calabi-Yau
threefolds

Federico Giusti  Department of Mathematics, Aarhus University, Ny Munkegade
118, 8000 Aarhus C, Denmark<EMAIL_ADDRESS>and  Cristiano Spotti
Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C,
Denmark<EMAIL_ADDRESS>
###### Abstract.

Given a (smoothable) projective nodal Kähler Calabi-Yau threefold, we show,
via a gluing construction, that all its - possibly non-Kähler - small
resolutions admit Chern-Ricci flat balanced metrics, which among other things
solve the dilatino equation appearing in the Hull-Strominger system.

###### Key words and phrases:

complex non-Kähler manifolds, balanced metrics, Chern-Ricci flat metrics,
Calabi-Yau manifolds, Hull-Strominger system

###### 2010 Mathematics Subject Classification:

53C55, 53C25,53C07

##  1\. Introduction

With the ultimate aim of geometrizing and classifying, one of the most studied
problems in complex geometry is the existence of hermitian metrics that can be
regarded as special. Through the years, the Kähler case is the one that has
been studied and understood the most, however, in the last decades the
interest towards the non-Kähler world has been increasing more and more,
leading to the search for special metrics also in this particular context.
While in the Kähler case special metrics arise naturally, the non-Kähler
scenario is too wild to guide us directly towards some central notion of
special metric, nevertheless, one can have indications on the path to follow
by watching the Kähler world. More specifically, given an $n$-dimensional
complex manifold $(M,J)$, if it is Kähler the obvious class of special (on a
first level) metrics is given exactly by Kähler metrics - which we recall
being hermitian metrics $h$ whose fundamental form $\omega:=h(J\\_,\\_)$ is
$d$-closed. In addition, this condition can also be combined with the notion
of Einstein metric (thanks to the properties of Kähler metrics) from the
general riemannian case, giving rise to the notion of Kähler-Einstein metrics,
which are universally regarded as the "most special" in the Kähler world.
Likewise, other notions of special Kähler metrics have been introduced and
studied (some of them are still central in the study of Kähler geometry), like
constant scalar curvature Kähler (cscK) metrics, or the more general class of
extremal Kähler metrics (introduced by Calabi in [C]), however they all share
the fact that they are giving a curvature condition on the metric, thus this
suggests that when searching for special metrics in the non-Kähler case we
shall ask for these metrics to be special under two aspects: the cohomological
one (satisfying a condition possibly generalizing the Kähler one) and the
curvature one.
Regarding the cohomological aspect, several conditions have been introduced
that generalize the Kähler one, and one of the most studied is given by
$d\omega^{n-1}=0$, identifying the class of balanced metrics (originally
introduced by Gauduchon in [G] as semi-Kähler metrics, and later on studied
further by Michelsohn in [M]), which is the class of metrics we are interested
in working with. Balanced metrics carry many interesting properties such as
the coincidence between the Hodge laplacian and the Dolbeault laplacian on
scalar functions (showed by Gauduchon in [G]), or the preservation of the
balanced condition for manifolds under holomorphic submersions proved in [M]
(showing a sort of duality between the Kähler condition and the balanced
condition). Also in [M], Michelsohn proved a characterization of balanced
metrics in terms of currents, which leads to the celebrated result from
Alessandrini and Bassanelli (see [AB]) showing that the class of compact
balanced manifolds is closed under proper modifications (condition not
satisfied by the class of Kähler manifolds). Moreover, balanced metrics ended
up being central in many interesting currently open problems, such as the
conjecture from Fino and Vezzoni (see [FV]) and the Gauduchon conjecture for
balanced metrics (see [STW], in which was solved in its original version for
Gauduchon metrics - identified by the condition
$\partial\overline{\partial}\omega^{n-1}=0$, which weakens the balanced
condition - posed by Gauduchon).
Moving instead on the curvature aspect, there are several known notions of
special metrics in the non-Kähler world such as Chern-Ricci flat metrics,
Bismut-Ricci flat metrics (which in the balanced case are equivalent to Chern-
Ricci flat metrics, see [AI]), Chern-Einstein metrics and many more.
The main goal of this paper is to construct Chern-Ricci flat balanced metrics
on the compact small resolutions of certain smoothable singular Kähler Calabi-
Yau threefolds. More specifically we wish to work on smoothable singular
threefolds $\tilde{M}$ whose singular set is made of ordinary double points
and are endowed with a Kähler Ricci-flat singular metric $\tilde{\omega}$.
Then, using the information on the asymptotics of this metric around the
singularity given by the results in [HS], together with what it is known on
the standard conifold (i.e. the local model of ordinary double points on
threefolds) and its small resolution to build, with a gluing construction
(inspired mostly by [BM], but also [AP] and [J]), Chern-Ricci flat balanced
metrics on the compact small resolutions of the singular threefold. The
strategy of the proof consists of two main steps: (1) a metric gluing between
the singular Calabi-Yau metric $\tilde{\omega}$ with the (rescaled) Candelas-
de la Ossa metrics $\omega_{co,a}$ (that are Kähler Calabi-Yau metrics on the
small resolution of the standard conifold, introduced in [CO]), and (2) an
Implicit Function Theorem deformation argument, where the deformation is a
balanced deformation (introduced in [FWW]) and all the analysis is performed
in suitable weighted Hölder spaces. Our main result is the following.

######  Theorem 1.1.

Let $(\tilde{M},\tilde{\omega})$ be a smoothable projective Kähler Calabi-Yau
nodal threefold (with $\tilde{\omega}$ a singular Calabi-Yau metric), and let
$M$ be a compact (not necessarily Kähler) small resolution of $\tilde{M}$.
Then $M$ admits a Chern-Ricci flat balanced metric $\hat{\omega}$ such that
$[\hat{\omega}^{2}]=[\tilde{\omega}^{2}]+\varepsilon^{4}[\mathbb{P}^{1}]$, and
$[\omega^{2}]$ converges to a nef class.

Here by "smoothable" we mean that $\tilde{M}$ admits a polarized flat
deformation to a smooth projective Calabi-Yau threefold (examples of such
manifolds are given by nodal quintic threefolds in $\mathbb{P}^{4}$), and the
reason why we require this condition to be satisfied is to be able to apply a
result from [HS].
Our interest towards Chern-Ricci flat balanced metrics comes actually from the
realm of Calabi-Yau geometry. Indeed, for a not necessarily Kähler Calabi-Yau
manifold (i.e. a complex manifold endowed with a holomorphic volume form) it
was introduced by Hull and Strominger (respectively in [Hu] and [S]) a system
of four equations coming from superstring theory known as the Hull-Strominger
system, whose solutions have proved to be extremely hard to construct (see
[GF] for a full presentation of the system and some known solutions, together
with several other references such as [AGF], [Fe], [FuY], [LY3], [P], [TY] and
the very recent [CPY2], [FeY] for the invariant case and [PPZ] for a flow
approach). The problem of solving This system, apart from its physical
meaning, carries great geometric interest, since it generalizes the Calabi-Yau
condition to the non-Kähler framework, and it holds a central role in the
geometrization conjecture for compact Calabi-Yau threefolds known as Reid’s
Fantasy (see [R]). This last conjecture, in particular, states that all
compact Kähler Calabi-Yau threefolds can be connected through a finite number
of conifold transitions (introduced by Clemens and Friedman, see [F]), i.e. a
procedure consisting of the contraction of a finite family of disjoint
$(-1,-1)$-curves in a compact Calabi-Yau threefold, followed by the smoothing
of the ordinary double points obtained from the previous step. These objects
thus show our interest towards singular threefolds with a finite number of
ordinary double points and our aim to find special metrics on their small
resolution; in particular the interest towards Chern-Ricci flat balanced
metrics is directly related to one of the equation of the Hull-Strominger
system, namely the conformally balanced equation, which on a compact Calabi-
Yau manifold $(X,\Omega)$ \- where $\Omega$ is the holomorphic volume form -
is an equation for hermitian metrics (actually their fundamental forms)
$\omega$ given by $d(||\Omega||_{\omega}\omega^{n-1})=0$ which is clearly
satisfied by balanced Chern-Ricci flat hermitian metrics. Moreover, Chern-
Ricci flat metrics correspond also to Hermite-Einstein metrics on the
holomorphic tangent bundle, thus this kind of metric appears as particularly
suited to be used as a starting point from which possibly build solutions for
the Hull-Strominger system (we will briefly discuss some ideas at the end of
the paper), and also portrays our construction (in some sense) as aiming
towards "reversing the arrow" of the construction done by Fu, Li and Yau in
[FLY] and Collins, Picard and Yau in [CPY1].
It is also interesting to notice that the deformation argument used in Section
4 proves also that on this class of manifolds the Gauduchon conjecture for
balanced metrics holds for certain classes nearby the boundary of the balanced
cone.
The paper is structured as follows. In Section 2 we recall some basic aspects
on ordinary double points and their resolutions, both regarding their geometry
and their topology, with focus on some fundamental results useful for our
construction. In Section 3 we present the first step of our work, consisting
of a gluing construction of a balanced metric made with the objects previously
introduced, together with the construction of a global "small" Chern-Ricci
potential for our new balanced metric. In the last section, i.e. Section 4, we
apply a deformation argument to obtain a Chern-Ricci flat balanced metric and
we discuss its associated balanced class, as well as its possible applications
to the search of solutions for the Hull-Strominger system.

Acknowledgements. Both the authors are supported by Villum Young Investigator
0019098.

The authors would like to thank Mario Garcia-Fernandez for useful
conversations and remarks.

##  2\. Ordinary double points on threefolds

In this preliminary section we shall recall some known facts about a certain
type of singularity on threefolds, regarding both their topology and their
geometry, and also fix some notation for the following sections.

The type of singularities we are interested in studying are called ordinary
double points (which are the most common kind of singularities in our
context), and are described by the model

$X:=\\{z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}=0\\}\subseteq\mathbb{C}^{4},$

which is known as the $3$-dimensional standard conifold, whose only singular
point is the origin. Then we have:

######  Definition 2.1.

A singular point $p$ in a singular threefold $Y$ is called ordinary double
point (ODP) if we can find a neighborhood $p\in U\subseteq M$ and a
neighborhood $0\in V\subseteq X$ such that $U$ and $V$ are biholomorphic
through a map that sends $p$ to $0$.

###  2.1. Topological aspects

These singularities arise naturally on threefolds when collapsing
$(-1,-1)$-curves, i.e. rational curves biholomorphic to $\mathbb{P}^{1}$ whose
normal bundle is isomorphic to $\mathcal{O}_{\mathbb{P}^{1}}(-1)^{\oplus 2}$,
and actually this procedure to obtain ODPs covers all the possibilities on
threefolds. Indeed, the standard conifold can be constructed in several ways,
one of which is the following: consider the rank $2$ bundle
$\mathcal{O}_{\mathbb{P}^{1}}(-1)^{\oplus 2}$ on $\mathbb{P}^{1}$ and notice
that the map

$([X_{1}:X_{2}],(w_{1},w_{2}))\mapsto(w_{1}X_{1},w_{1}X_{2},w_{2}X_{1},w_{2}X_{2})$

maps $\mathcal{O}_{\mathbb{P}^{1}}(-1)^{\oplus 2}$ onto $X$ \- since $X$
through a change of coordinates is biholomorphic to the set
$\\{W_{1}W_{2}-W_{3}W_{4}=0\\}$ \- sending the zero section onto the origin.
Moreover this map restricted to $\mathcal{O}_{\mathbb{P}^{1}}(-1)^{\oplus
2}\setminus\mathbb{P}^{1}$ (where $\mathbb{P}^{1}$ is meant as the zero
section) gives a biholomorphism with $X\setminus\\{0\\}$, proving our previous
statement. This shows us that these singularities always admit small
resolutions (with $\mathbb{P}^{1}$ as the exceptional curve) biholomorphic to
$\tilde{X}:=\mathcal{O}_{\mathbb{P}^{1}}(-1)^{\oplus 2}$, and it can be shown
that a singular threefold with $n$ ordinary double points admits exactly
$2^{n}$ small resolutions of this type (every singularity can be resolved with
a curve in two distinct bimeromorphic ways).
Since our following work aims to face the case of singular threefolds whith a
finite number of ODPs which admit a compact small resolution, in particular
those who do not admit Kähler metrics, before going further in exploring the
geometry of these singularities we will quickly show that this kind of
resolutions are actually a very common situation.

######  Remark 2.2.

Thanks to a result from Cheltsov (see [Ch]) we know that a hypersurface
$\tilde{M}$ in $\mathbb{P}^{4}$ of degree $d$ with only isolated ODPs is
factorial when $\tilde{M}$ has at most $(d-1)^{2}-1$ singularities, thus is in
particular $\mathbb{Q}$-factorial. We can then apply the work from Namikawa
and Steenbrink (see [NS]) to obtain that $\tilde{M}$ is smoothable, and hence,
thanks to the results from Friedman (see [F]) we have that any small
resolution $M$ of $\tilde{M}$ with exceptional curves $C_{1},...,C_{k}$,
$C_{i}\simeq\mathbb{P}^{1}$, satisfies necessarily a condition

$\sum_{i=1}^{k}\lambda_{i}[C_{i}]=0\quad\text{in}\>\>H_{2}(M,\mathbb{R}),\quad\text{where
each}\>\>\lambda_{i}\neq 0,$

which immediately implies that if $\tilde{M}$ has only one ODP, then $M$ can’t
be Kähler because it contains a homologically trivial curve (note that the
generic quintic threefold has exactly one node, hence satisfies this
situation).
Moreover, Werner proved in [W] that $M$ is projective if and only if all
$C_{i}$s are homologically non-trivial, and since $M$ is Moishezon,
projectivity is equivalent to Kählerness. Thus the class of examples above
lies in a larger one, since every small resolution with at least a
homologically trivial exceptional curve is non-Kähler.

###  2.2. Geometric aspects

The standard conifold $X$ is naturally endowed with a conical structure.
Indeed, we can introduce the function on $\mathbb{C}^{4}$

$r(z):=||z||^{\frac{2}{3}},$

which restricted to $X$ yields the conical distance to the singularity, and
can be used to define the metric

$\omega_{co,0}:=\frac{3}{2}i\partial\overline{\partial}r^{2},$

on the smooth part of $X$, which is clearly Kähler. Moreover, it can be seen
that $\omega_{co,0}$ is actually also Ricci flat, as well as a cone metric
over the link $L:=\\{r=1\\}\subseteq X$ which can be written as

$g_{co,0}=\frac{3}{2}(dr^{2}+r^{2}g_{L}),$

with $g_{L}$ a Sasaki-Einstein metric on the link $L$.
This metric structure of the standard conifold, with some further work, yields
also a Kähler Calabi-Yau structure on the small resolution. In fact, Candelas
and de la Ossa (see [CO]) constructed a family of metrics, depending on the
parameter $a>0$, of the form

$\omega_{co,a}:=i\partial\overline{\partial}f_{a}(r^{3})+4a^{2}\pi_{\mathbb{P}^{1}}^{*}\omega_{FS},$

where $\omega_{FS}$ is the Fubini-Study metric on $\mathbb{P}^{1}$, and
$f_{a}$ is smooth function satisfying the ODE

$(xf_{a}^{\prime}(x))^{3}+6a^{2}(xf_{a}^{\prime}(x))^{2}=x^{2},\qquad
f_{a}(x)\geq 0,$

on $[0,+\infty)$, which immediately gives $f_{a}(x)=a^{2}f_{1}(x/a^{3})$. Here
the function $r$ is simply the conical distance from the singularity re-read
on the resolution, hence portraying the conical distance from the exceptional
curve. Moreover, this family of metrics is such that as $a\rightarrow 0$ the
metrics $\omega_{co,a}$ converges, away from the exceptional curve, to the
standard cone metric $\omega_{co,0}$, and it is also asymptotic (at infinity)
to the cone metric $\omega_{co,0}$, and these facts can be seen explicitly
with the following expansion from [CPY1].

######  Lemma 2.3.

For $x\gg 1$, the function $f_{1}(x)$ has a convergent expansion

$f_{1}(x)=\frac{3}{2}x^{\frac{2}{3}}-2\log(x)+\sum_{n=0}^{+\infty}c_{n}x^{-\frac{2n}{3}}.$

This expansion is going to be crucial for our work, hence we will work again
with it in the next sections.

Before concluding this preliminary part, is extremely important for our
construction (and highly interesting just as a fact) to remark that the
standard cone metric $\omega_{co,0}$ is actually the (asymptotic) model near
the singularities for _all_ singular Kähler Calabi-Yau metrics around ODPs,
and this was shown by Hein and Sun in the following crucial theorem (Theorem
1.4 and Lemma A.1 from [HS]) which can be simplified with the following
statement (written only for threefolds):

######  Theorem 2.4 (Hein-Sun).

Let $\tilde{M}$ be a smoothable singular threefold whose singular set is a
finite family of ODPs endowed with a Kähler Calabi-Yau metric $\tilde{\omega}$
on its smooth part $M_{reg}$. Then for every singular point
$p\in\tilde{M}\setminus M_{reg}$ there exist a constant $\lambda_{0}>0$,
neighborhoods $p\in U_{p}\subseteq\tilde{M}$ and $0\in V_{p}\subseteq X$, and
a biholomorphism $P:V_{p}\setminus\\{0\\}\rightarrow U_{p}\setminus\\{p\\}$
such that

$P^{*}\tilde{\omega}-\omega_{co,0}=i\partial\overline{\partial}\varphi,\qquad\text{for
some}\>\varphi\in C_{2+\lambda_{0}}^{\infty},$

where $r$ is the conical distance from the singularities and
$C_{2+\lambda_{0}}^{\infty}$ is the space of smooth functions with decay rate
at zero of $2+\lambda_{0}$ (i.e. an $f\in C_{2+\lambda_{0}}^{\infty}$ is a
smooth function such that nearby zero it holds $|\nabla^{k}f|\leq
cr^{2+\lambda_{0}-k}$ for all $k\geq 0$).

##  3\. The gluing construction

We now start presenting the first part of our work, which consists of a gluing
construction of a family of balanced metrics on the compact small resolution
of a smoothable Kähler Calabi-Yau singular threefold. This type of gluing
constructions are not new in complex geometry, and have been used in the past
to construct significant examples, e.g. in [J], [AP] and [BM].
Before going ahead with the details of the construction it is significant to
make the following remark.

######  Remark 3.1.

Thanks to the results from Hironaka and Alessandrini-Bassanelli ([Hi] and
[AB]), we already know that compact small resolutions of smoothable Kähler
Calabi-Yau threefolds with a finite number of ODPs admit balanced metrics.
However, we are interested in finding special balanced metrics - in
particular, as we will see, with a small Chern-Ricci potential - hence we need
to perform an explicit construction in order to build metrics satisfying our
desired curvature properties.

In what follows we will lay out the initial data and see in details the
construction.
Let $\tilde{M}$ be a smoothable complex threefold obtained from the
contraction of a finite family of disjoint $(-1,-1)$-curves in a compact
complex threefold (thus the singular set of $\tilde{M}$ is made of a finite
number of Ordinary Double Points), let $M$ be a compact small resolution of
$\tilde{M}$ and suppose that the regular part $M_{reg}$ of $\tilde{M}$ is
equipped with $\tilde{\omega}$ a Kähler Calabi-Yau metric.
The idea of the gluing construction is to use the ingredients we introduced in
the previous sections, and notice that they are suitable to be glued together.
In particular, we know that around each one of the exceptional curves we have
the Candelas-de la Ossa metrics $\omega_{co,a}$ which are asymptotic, far from
the exceptional curve, to the cone metric; on the other hand we have that also
our background metric $\tilde{\omega}$ is asymptotic to the cone metric, but
this time when approaching the singularity. We then want to use the standard
conifold as a "bridge" to glue together the metrics $\tilde{\omega}$ and
$\omega_{co,a}$, while maintaining the balanced condition, and in order to be
able to do the analysis for this gluing construction is going to be crucial to
have a "concrete" construction of the small resolution $M$.
To do this, we will divide the process into three natural steps, and for
simplicity assume that $\tilde{M}$ has just one singularity (the process
obviously applies analogously to the case in which the singularities are any
finite number). In addition, since our resolution is known to be a crepant
resolution, we are also going to compute explicitly a holomorphic volume form
for $M$ (starting from the one on $\tilde{M}$), since such form is a crucial
ingredient for the deformation argument in the following section.

###  3.1. Step 1

We first glue together the metrics $\tilde{\omega}$ and $\omega_{co,0}$ nearby
the singularity. To do this, thanks to the nature of ODPs, we can introduce,
with an abuse of notation (that we will keep on using throughout the whole
paper - as it does not generate any confusion), a function $r$ on $\tilde{M}$
which coincides with the (pullback) conical distance from the singularity in a
neighborhood of said singularity (induced by the local biholomorphism $P$ from
Theorem 2.4), and is identically equal to $1$ far away from them (as a
consequence we will do the same with the standard cone metric
$\omega_{co,0}$). In order to perform this first step of the gluing, we will
need to work on the cone $X$ and pull back on $M_{reg}$ the object obtained.
Let $p>0$ and consider $\varepsilon>0$ sufficiently small, such that, thanks
to the result from Hein-Sun cited above, on the region $\\{0<r\leq
4\varepsilon^{p}\\}\subseteq X$ exists a constant $\lambda_{0}>0$ and is
defined a function $\varphi\in C_{2+\lambda_{0}}^{\infty}$ such that

$P^{*}\tilde{\omega}=\omega_{co,0}+i\partial\overline{\partial}\varphi.$

Hence, we introduce a cut-off function
$\chi_{\varepsilon}(x):=\chi_{1}(x/\varepsilon^{p})$ where $\chi_{1}$ is a
smooth function on $[0,+\infty)$ such that

$\chi_{1}(x):=\begin{cases}0\qquad&\text{if}\>x\leq 2,\\\ \text{Non
decreasing}\quad&\text{if}\>x\in(2,4),\\\ 1\qquad&\text{if}\>x\geq
4,\end{cases}$

and define the smooth real $(1,1)$-form

$\tilde{\omega}_{\varepsilon}:=\omega_{co,0}+i\partial\overline{\partial}(\chi_{\varepsilon}(r)\varphi),$

which for $\varepsilon$ sufficiently small is such that

$\omega_{\varepsilon}:=(P^{-1})^{*}\tilde{\omega}_{\varepsilon}$

defines a Kähler metric on $M_{reg}$ (extended to the whole $M_{reg}$ in the
obvious way). Indeed, we notice that on
$G_{\varepsilon}:=\\{2\varepsilon^{p}<r\leq 4\varepsilon^{p}\\}$ it holds

$|\omega_{\varepsilon}-\omega_{co,0}|_{\omega_{co,0}}=|i\partial\overline{\partial}(\chi_{\varepsilon}(r)\varphi)|_{\omega_{co,0}}\leq
c(r^{-2}|\varphi|+r^{-1}|\partial\varphi|_{\omega_{co,0}}+|i\partial\overline{\partial}\varphi|_{\omega_{co,0}})\leq
cr^{\lambda_{0}},$

from which on $G_{\varepsilon}$ we can write

$|\nabla_{\omega_{co,0}}^{k}(\omega_{\varepsilon}-\omega_{co,0})|_{\omega_{co,0}}\leq
Cr^{\lambda_{0}-k},\qquad\text{for all}\>\>k\geq 0.$

Moreover we notice that on $\\{0<r\leq 2\varepsilon^{p}\\}$ the metric
$\omega_{\varepsilon}$ is exactly conical, i.e. equal to $\omega_{co,0}$,
while on $\\{r>4\varepsilon^{p}\\}$ $\omega_{\varepsilon}$ coincides with the
background metric $\tilde{\omega}$, and on $G_{\varepsilon}$ is the only
region where $\omega_{\varepsilon}$ is not Ricci-flat.

###  3.2. Step 2

In this second step we instead perform the gluing between the Candelas-de la
Ossa metrics $\omega_{co,a}$ and the standard cone $\omega_{co,0}$, on the
small resolution $\hat{X}$ of the standard conifold $X$, preserving the
balanced condition. To do this we recall from Lemma 2.3 that the Candelas-de
la Ossa metrics are of the form

$\omega_{co,a}:=i\partial\overline{\partial}f_{a}(r^{3})+4a^{2}\pi^{*}\omega_{FS},$

which away from the exceptional curve have the expansion

$f_{a}(r^{3})=\frac{3}{2}r^{2}+a^{2}\psi_{a}(r),\qquad\text{where}\quad\psi_{a}(r)=3\log
r-3\log a+\sum_{n=0}^{+\infty}c_{n}a^{2n}r^{-2n}.$

This suggests introducing a large parameter $R\gg 1$ and a smooth cut-off
function $\chi_{R}(x):=\chi_{2}(x/R)$ on $[0,+\infty)$ such that

$\chi_{1}(x):=\begin{cases}1\qquad&\text{if}\>x\leq\frac{1}{2},\\\ \text{Non
increasing}\quad&\text{if}\>x\in\left(\frac{1}{2},1\right),\\\
0\qquad&\text{if}\>x\geq 1,\end{cases}$

from which we introduce the family of closed $(2,2)$-forms

$\omega_{a,R}^{2}=\left(i\partial\overline{\partial}\left(\frac{3}{2}r^{2}+a^{2}\chi_{R}(r)\psi_{a}(r)\right)\right)^{2}+2a^{2}i\partial\overline{\partial}\left(\chi_{R}(r)f_{a}(r^{3})\right)\wedge\pi^{*}\omega_{FS}.$

The reason why the notation $\omega_{a,R}^{2}$ makes sense is because the
results from Michelsohn ([M]) ensure us that every real positive closed
$(n-1,n-1)$-form admits a unique real positive $(1,1)$-form as a $(n-1)$th
root, and the forms we defined above happen to be positive for sufficiently
large $R$. Indeed, positivity is clear when $r\leq\frac{R}{2}$, where
$\omega_{a,R}^{2}=\omega_{co,a}^{2}$, and when $r\geq R$, where
$\omega_{a,R}^{2}=\omega_{co,0}^{2}$. In the gluing region $G_{R}:=\\{R/2\leq
r\leq R\\}$, instead, we need to do a few estimates to confirm the positivity.
In order to avoid having to impose limitations on the parameter $a$ we
substitute the functions $\psi_{a}$ with $\psi_{a}+3\log a$; this won’t alter
the metrics $\omega_{co,a}$ and any of its properties, thus it can be done
without problems. For simplicity, we will keep writing $\psi_{a}$, and in the
following the constant $c$ might vary from line to line. We have

$\displaystyle|\omega_{a,R}^{2}-\omega_{co,0}^{2}|_{\omega_{co,0}}\leq$
$\displaystyle 2a^{2}|\omega_{co,0}\wedge
i\partial\overline{\partial}(\chi_{R}(r)\psi_{a}(r))|_{\omega_{co,0}}+2a^{2}|i\partial\overline{\partial}(\chi_{R}(r)f_{a}(r^{3}))\wedge\pi^{*}\omega_{FS}|_{\omega_{co,0}}$
$\displaystyle+a^{4}|i\partial\overline{\partial}(\chi_{R}(r)\psi_{a}(r))|_{\omega_{co,0}}^{2}$
$\displaystyle\leq$ $\displaystyle
ca^{2}(|\omega_{co,0}|_{\omega_{co,0}}(r^{-2}|\psi_{a}|+r^{-1}|\partial\psi_{a}|_{\omega_{co,0}}+|i\partial\overline{\partial}\psi_{a}|_{\omega_{co,0}})$
$\displaystyle+|\pi^{*}\omega_{FS}|_{\omega_{co,0}}(r^{-2}|f_{a}(r^{3})|+r^{-1}|\partial
f_{a}(r^{3})|_{\omega_{co,0}}+|i\partial\overline{\partial}f_{a}(r^{3})|_{\omega_{co,0}})$
$\displaystyle+a^{2}(r^{-2}|\psi_{a}|+r^{-1}|\partial\psi_{a}|_{\omega_{co,0}}+|i\partial\overline{\partial}\psi_{a}|_{\omega_{co,0}})^{2})$
$\displaystyle\leq$ $\displaystyle
ca^{2}(|\omega_{co,0}|_{\omega_{co,0}}(r^{-2}|\psi_{a}|+r^{-1}|\partial\psi_{a}|_{\omega_{co,0}}+|i\partial\overline{\partial}\psi_{a}|_{\omega_{co,0}})$
$\displaystyle+|\pi^{*}\omega_{FS}|_{\omega_{co,0}}(1+|\omega_{co,0}|_{\omega_{co,0}}+r^{-2}|\psi_{a}|+r^{-1}|\partial\psi_{a}|_{\omega_{co,0}}+|i\partial\overline{\partial}\psi_{a}|_{\omega_{co,0}})$
$\displaystyle+a^{2}(r^{-2}|\psi_{a}|+r^{-1}|\partial\psi_{a}|_{\omega_{co,0}}+|i\partial\overline{\partial}\psi_{a}|_{\omega_{co,0}})^{2})\leq(\ast)$

To conclude the estimate let’s take a look at the single terms, keeping in
mind that we are taking the norms with respect to the standard cone metric. It
holds

  * •

$|\omega_{co,0}|_{\omega_{co,0}}=O(1)$;

  * •

$|\pi^{*}\omega_{FS}|_{\omega_{co,0}}=O(r^{-2})$;

  * •

$|\psi_{a}|\leq c(|\log r|+\sum_{n=0}^{+\infty}|c_{n}|a^{2n}r^{-2n})\leq
c|\log r|$;

  * •

$|\partial\psi_{a}|_{\omega_{co,0}}\leq
c(r^{-1}+\sum_{n=0}^{+\infty}|\tilde{c}_{n}|a^{2n}r^{-2n-1})\leq cr^{-1}$;

  * •

$|i\partial\overline{\partial}\psi_{a}|_{\omega_{co,0}}\leq
c(r^{-2}+\sum_{n=0}^{+\infty}|\tilde{\tilde{c}}_{n}|a^{2n}r^{-2n-2})\leq
cr^{-2}$;

from which we get

$(\ast)\leq ca^{2}(r^{-2}|\log r|+r^{-2}+r^{-2}(1+r^{-2}|\log
r|+r^{-2})+a^{2}(r^{-2}|\log r|+r^{-2})^{2})\leq ca^{2}r^{-2}|\log r|,$

and hence

$|\nabla_{\omega_{co,0}}^{k}(\omega_{a,R}^{2}-\omega_{co,0}^{2})|_{\omega_{co,0}}\leq
Ca^{2}r^{-2-k}|\log r|\qquad\text{for all}\>\>k\geq 0,$

which clearly implies the positivity also on $G_{R}$, as long as $R$ is chosen
to be sufficiently large.

###  3.3. Step 3

In this third and last step we want to glue together the metrics
$\omega_{\varepsilon}$ from step 1 with the metric $\omega_{a,R}$ from step 2
by matching isometrically the exactly conical regions. In order to do this we
are going to need to rescale by a constant $\lambda>0$ the metric on
$\hat{X}$, and we will now see that this constant is a geometric constant,
since it is dictated by the geometries of the two metrics we are gluing
together.
In what follows we will denote with $z$ the coordinates on $M_{reg}$ nearby
the singularity and with $\zeta$ the coordinates on $\hat{X}$, both given by
the identification with the standard conifold $X$. We then consider the
regions

$C_{R}:=\\{R/2\leq r(\zeta)\leq 4R\\}\subseteq\hat{X}\qquad\text{and}\qquad
C_{\varepsilon}:=\\{\varepsilon^{p}/2\leq r(z)\leq
4\varepsilon^{p}\\}\subseteq M_{reg}$

and define a biholomorphism between them by imposing

$\zeta=\left(\frac{R}{\varepsilon^{p}}\right)^{\frac{3}{2}}z.$

From this expression we have that on the identified region the following
identity holds

$r(\zeta)=r\left(\left(\frac{R}{\varepsilon^{p}}\right)^{\frac{3}{2}}z\right)=\frac{R}{\varepsilon^{p}}r(z)$

which yields
$\lambda=\lambda(\varepsilon,R):=\left(\frac{\varepsilon^{p}}{R}\right)^{2}$.
From this follows $\lambda r^{2}(\zeta)=r^{2}(z)$, and thus on the identified
conical regions $C_{R}^{\prime}:=\\{R\leq r(\zeta)\leq
2R\\}\simeq\\{\varepsilon^{p}\leq r(z)\leq
2\varepsilon^{p}\\}=:C_{\varepsilon}^{\prime}$ holds

$\lambda\omega_{co,0}(\zeta)=\omega_{co,0}(z),\qquad\text{and
consequently}\qquad\lambda\omega_{a,R}=\omega_{\varepsilon}.$

Hence, $\lambda$ is the needed rescaling factor, which allows us to define the
glued family of balanced metrics on the small resolution $M$ as

$\omega_{a,\varepsilon,R}:=\begin{cases}\lambda\omega_{a,R}\qquad&\text{on}\>r(\zeta)\leq
R,\\\ \omega_{co,0}\qquad&\text{on}\>\varepsilon^{p}\leq r(z)\leq
2\varepsilon^{p},\\\ \omega_{\varepsilon}\qquad&\text{on}\>r(z)\geq
2\varepsilon^{p}.\end{cases}$

Now we will move to the following part in which we will try to understand
better the geometry of this new family of metrics.

###  3.4. Description of the volume

We are mainly interested in the Chern-Ricci tensor - and in particular on its
potential - hence, in order to understand it, we are going to need to obtain
information on the volume of our metrics
$\omega=\omega_{\varepsilon,R}:=\omega_{1,\varepsilon,R}$. To do this, we need
to estimate the distance between our glued metrics and the standard cone
metric on the gluing region, since in the other parts of the manifold we
already know the geometry from the initial data. Moreover, inside the gluing
region there is also an exactly conical region - whose geometry is also
understood - which separates the two gluing regions from the first two steps,
hence we can just estimate the said distance separately on the two regions and
then take the maximum.
Clearly, the metric is unaltered on the gluing region from step 1, thus we
still have on $G_{\varepsilon}$ that

$|\nabla_{\omega_{co,0}}^{k}(\omega_{\varepsilon}-\omega_{co,0})|_{\omega_{co,o}}\leq
cr^{\lambda_{0}-k},$

for all $k\geq 0$.
On the other hand, since in step 3 we had to rescale the metric nearby the
exceptional curve, we are going to need to repeat the estimate to understand
how the rescaling affected the distance from the standard cone metric. At some
point we are going to link the parameters $\varepsilon$ and $R$. Again the
constant $c$ might be varying from line to line. We have

$\displaystyle(\omega_{\varepsilon,R}^{2}-\omega_{co,0}^{2})(\zeta)=$
$\displaystyle\lambda^{2}(2\omega_{co,0}\wedge
i\partial\overline{\partial}(\chi_{R}(r)\psi_{1}(r))+2i\partial\overline{\partial}(\chi_{R}(r)f_{1}(r^{3}))\wedge\pi^{*}\omega_{FS}$
$\displaystyle+i\partial\overline{\partial}(\chi_{R}(r)\psi_{1}(r))^{2})(\zeta)=(\ast)$

Let’s now change coordinates from $\zeta$ to $z$, recalling that $\lambda
r^{2}(\zeta)=r^{2}(z)$ and noticing that, by construction,
$\pi^{*}\omega_{FS}$ is invariant under rescalings.

(1) $\displaystyle(\ast)=$
$\displaystyle\lambda^{2}\left(\lambda^{-1}\omega_{co,0}\wedge
i\partial\overline{\partial}\left(\chi_{R}\left(\lambda^{-\frac{1}{2}}r\right)\left(3\log
r-\frac{3}{2}\log\lambda+\sum_{n\geq
0}c_{n}\lambda^{n}r^{-2n}\right)\right)\right.$
$\displaystyle+\pi^{*}\omega_{FS}\wedge
i\partial\overline{\partial}\left(\chi_{R}\left(\lambda^{-\frac{1}{2}}r\right)\left(\lambda^{-1}\frac{3}{2}r^{2}+3\log
r-\frac{3}{2}\log\lambda+\sum_{n\geq 0}c_{n}\lambda^{n}r^{-2n}\right)\right)$
$\displaystyle\left.+\left(i\partial\overline{\partial}\left(\chi_{R}\left(\lambda^{-\frac{1}{2}}r\right)\left(3\log
r-\frac{3}{2}\log\lambda+\sum_{n\geq
0}c_{n}\lambda^{n}r^{-2n}\right)\right)\right)^{2}\right)(z).$

What we want to do now is estimate the norm (with respect to the cone metric)
of this quantity, thus we start computing the derivatives that appear in the
above expression. We have

$\displaystyle\partial\overline{\partial}\left(\chi_{R}\left(\lambda^{-\frac{1}{2}}r\right)\left(3\log
r-\frac{3}{2}\log\lambda+\sum_{n\geq 0}c_{n}\lambda^{n}r^{-2n}\right)\right)$
$\displaystyle=$
$\displaystyle\frac{\lambda^{-1}}{R^{2}}\chi_{R}^{\prime\prime}\left(\lambda^{-1/2}r\right)\left(3\log
r-\frac{3}{2}\log\lambda+\sum_{n\in\mathbb{N}}c_{n}\lambda^{n}r^{-2n}\right)\partial
r\wedge\overline{\partial}r$
$\displaystyle+2\frac{\lambda^{-1/2}}{R}\chi_{R}^{\prime}\left(\lambda^{-1/2}r\right)\left(3r^{-1}+\sum_{n\in\mathbb{N}}\tilde{c}_{n}\lambda^{n}r^{-2n-1}\right)\partial
r\wedge\overline{\partial}r$
$\displaystyle+\frac{\lambda^{-1/2}}{R}\chi_{R}^{\prime}\left(\lambda^{-1/2}r\right)\left(3\log
r-\frac{3}{2}\log\lambda+\sum_{n\geq
0}c_{n}\lambda^{n}r^{-2n}\right)\partial\overline{\partial}r$
$\displaystyle+\chi_{R}\left(\lambda^{-1/2}r\right)\left(-3r^{-2}+\sum_{n\in\mathbb{N}}\tilde{\tilde{c}}_{n}\lambda^{n}r^{-2n-2}\right)\partial
r\wedge\overline{\partial}r$
$\displaystyle+\chi_{R}\left(\lambda^{-1/2}r\right)\left(3r^{-1}+\sum_{n\in\mathbb{N}}\tilde{c}_{n}\lambda^{n}r^{-2n-1}\right)\partial\overline{\partial}r,$

and

$\displaystyle\partial\overline{\partial}\left(\chi_{R}\left(\lambda^{-\frac{1}{2}}r\right)\left(\lambda^{-1}\frac{3}{2}r^{2}+3\log
r-\frac{3}{2}\log\lambda+\sum_{n\geq 0}c_{n}\lambda^{n}r^{-2n}\right)\right)$
$\displaystyle=$
$\displaystyle\left(\frac{\lambda^{-1}}{R^{2}}\chi_{R}^{\prime\prime}\left(\lambda^{-1/2}r\right)r^{2}+4\frac{\lambda^{-1/2}}{R}\chi_{R}^{\prime}\left(\lambda^{-1/2}r\right)r+2\chi_{R}\left(\lambda^{-1/2}r\right)\right)\partial
r\wedge\overline{\partial}r$
$\displaystyle+\left(\frac{\lambda^{-1/2}}{R}\chi_{R}^{\prime}\left(\lambda^{-1/2}r\right)r^{2}+2\chi_{R}\left(\lambda^{-1/2}r\right)r\right)\partial\overline{\partial}r$
$\displaystyle+\partial\overline{\partial}\left(\chi_{R}\left(\lambda^{-\frac{1}{2}}r\right)\left(3\log
r-\frac{3}{2}\log\lambda+\sum_{n\geq 0}c_{n}\lambda^{n}r^{-2n}\right)\right)$

From here we can obtain the estimates

$\displaystyle\left|\partial\overline{\partial}\left(\chi_{R}\left(\lambda^{-\frac{1}{2}}r\right)\left(3\log
r-\frac{3}{2}\log\lambda+\sum_{n\geq
0}c_{n}\lambda^{n}r^{-2n}\right)\right)\right|_{\omega_{co,0}}$
$\displaystyle\leq$ $\displaystyle c\frac{\lambda^{-1}}{R^{2}}\left(|\log
r|+|\log\lambda|+\sum_{n\in\mathbb{N}}|c_{n}|\lambda^{n}r^{-2n}\right)$
$\displaystyle+c\frac{\lambda^{-1/2}}{R}r^{-1}\left(1+\sum_{n\in\mathbb{N}}|\tilde{c}_{n}|\lambda^{n}r^{-2n}+|\log
r|+|\log\lambda|+\sum_{n\geq 0}|c_{n}|\lambda^{n}r^{-2n}\right)$
$\displaystyle+cr^{-2}\left(1+\sum_{n\in\mathbb{N}}|\tilde{\tilde{c}}_{n}|\lambda^{n}r^{-2n}+\sum_{n\in\mathbb{N}}|\tilde{c}_{n}|\lambda^{n}r^{-2n}\right)$

and

$\displaystyle\left|\partial\overline{\partial}\left(\chi_{R}\left(\lambda^{-\frac{1}{2}}r\right)\left(\lambda^{-1}\frac{3}{2}r^{2}+3\log
r-\frac{3}{2}\log\lambda+\sum_{n\geq
0}c_{n}\lambda^{n}r^{-2n}\right)\right)\right|_{\omega_{co,0}}$
$\displaystyle\leq$ $\displaystyle
c\left(\frac{\lambda^{-1}}{R^{2}}r^{2}+\frac{\lambda^{-1/2}}{R}(r+1)\right)+\left|\partial\overline{\partial}\left(\chi_{R}\left(\lambda^{-\frac{1}{2}}r\right)\left(3\log
r-\frac{3}{2}\log\lambda+\sum_{n\geq
0}c_{n}\lambda^{n}r^{-2n}\right)\right)\right|_{\omega_{co,0}}$

In order to conclude the estimate, we impose $R:=\varepsilon^{-q}$, with $q>0$
to be chosen, from which follows $\lambda=\varepsilon^{2(p+q)}$. Also we are
now going to use that we are in the region $\\{2\varepsilon^{p}\leq r\leq
4\varepsilon^{p}\\}$, which among the others implies the estimate

$|\log r|\leq|\log 4|+p|\log\varepsilon|\leq c(1+|\log\varepsilon|).$

We thus get

$\displaystyle\left|\partial\overline{\partial}\left(\chi_{R}\left(\lambda^{-\frac{1}{2}}r\right)\left(3\log
r-\frac{3}{2}\log\lambda+\sum_{n\geq
0}c_{n}\lambda^{n}r^{-2n}\right)\right)\right|_{\omega_{co,0}}$
$\displaystyle\leq$ $\displaystyle
c\varepsilon^{-2p}\left(1+|\log\varepsilon|+\sum_{n\in\mathbb{N}}|c_{n}|(1/2)^{2n}\right)$
$\displaystyle+c\varepsilon^{-2p}\left(1+\sum_{n\in\mathbb{N}}|\tilde{c}_{n}|(1/2)^{2n}+1+|\log\varepsilon|+\sum_{n\geq
0}|c_{n}|(1/2)^{2n}\right)$
$\displaystyle+c\varepsilon^{-2p}\left(1+\sum_{n\in\mathbb{N}}|\tilde{\tilde{c}}_{n}|(1/2)^{2n}+\sum_{n\in\mathbb{N}}|\tilde{c}_{n}|(1/2)^{2n}\right)\leq
c\varepsilon^{-2p}|\log\varepsilon|$

and

$\displaystyle\left|\partial\overline{\partial}\left(\chi_{R}\left(\lambda^{-\frac{1}{2}}r\right)\left(\lambda^{-1}\frac{3}{2}r^{2}+3\log
r-\frac{3}{2}\log\lambda+\sum_{n\geq
0}c_{n}\lambda^{n}r^{-2n}\right)\right)\right|_{\omega_{co,0}}$
$\displaystyle\leq$ $\displaystyle
c(1+\varepsilon^{-p}(\varepsilon+1))+\left|\partial\overline{\partial}\left(\chi_{R}\left(\lambda^{-\frac{1}{2}}r\right)\left(3\log
r-\frac{3}{2}\log\lambda+\sum_{n\geq
0}c_{n}\lambda^{n}r^{-2n}\right)\right)\right|_{\omega_{co,0}}$
$\displaystyle\leq$ $\displaystyle
c(\varepsilon^{-p}+\varepsilon^{-2p}|\log\varepsilon|)\leq
c\varepsilon^{-2p}|\log\varepsilon|$

Hence, putting together this estimates we can finally obtain

$\displaystyle|\omega_{\varepsilon,R}^{2}-\omega_{co,0}^{2}|_{\omega_{co,0}}$
$\displaystyle\leq$ $\displaystyle
c\varepsilon^{4(p+q)}\left(\varepsilon^{-2(p+q)}\left|i\partial\overline{\partial}\left(\chi_{R}\left(\lambda^{-\frac{1}{2}}r\right)\left(3\log
r-\frac{3}{2}\log\lambda+\sum_{n\geq
0}c_{n}\lambda^{n}r^{-2n}\right)\right)\right|_{\omega_{co,0}}\right.$
$\displaystyle+\varepsilon^{-2p}\left|i\partial\overline{\partial}\left(\chi_{R}\left(\lambda^{-\frac{1}{2}}r\right)\left(\lambda^{-1}\frac{3}{2}r^{2}+3\log
r-\frac{3}{2}\log\lambda+\sum_{n\geq
0}c_{n}\lambda^{n}r^{-2n}\right)\right)\right|_{\omega_{co,0}}$
$\displaystyle\left.+\left|i\partial\overline{\partial}\left(\chi_{R}\left(\lambda^{-\frac{1}{2}}r\right)\left(3\log
r-\frac{3}{2}\log\lambda+\sum_{n\geq
0}c_{n}\lambda^{n}r^{-2n}\right)\right)\right|_{\omega_{co,0}}^{2}\right)$
$\displaystyle\leq$ $\displaystyle
c\varepsilon^{4(p+q)}(\varepsilon^{-4p-2q}|\log\varepsilon|+\varepsilon^{-4p}|\log\varepsilon|^{2})\leq
c\varepsilon^{2q}|\log\varepsilon|$

which implies, on the whole gluing region, that for all $k\geq 0$ holds

$|\nabla_{\omega_{co,0}}^{k}(\omega_{q,\varepsilon}^{2}-\omega_{co,0}^{2})|_{\omega_{co,0}}\leq
cr^{m-k},$

where $m=m(\lambda_{0},p,q)$.
In order to obtain geometric information from this estimate, we are going to
need to recall a result from Michelsohn ([M]). This result gives an explicit
isomorphism between the cone of strictly positive $(1,1)$-forms and the cone
of strictly positive $(n-1,n-1)$-forms. In particular, if
$\Psi\in\bigwedge^{n-1,n-1}$ and $\psi\in\bigwedge^{1,1}$ are strictly
positive forms such that $\psi^{n-1}=\Psi$, we can always find a base
$\\{e_{i}\\}$ of $(1,0)$-forms that "diagonalizes" simultaneously $\Psi$ and
$\psi$, giving the expressions

$\Psi=\sum_{j=1}^{n}\Lambda_{j}\widehat{e_{j}\wedge
Je_{j}}\qquad\text{and}\qquad\psi=\sum_{j=1}^{n}\lambda_{j}e_{j}\wedge
Je_{j},$

where $J$ is the complex structure and $\widehat{e_{j}\wedge Je_{j}}$ means
the wedge product of all the terms of the form $e_{k}\wedge Je_{k}$ except the
one corresponding to the index $j$. Moreover, Michelsohn’s theorem gives us
also a formula relating the coefficients of $\psi$ and $\Psi$ in this basis,
that is

$\lambda_{j}=\frac{\left(\Lambda_{1}\cdot...\cdot\Lambda_{n}\right)^{\frac{1}{n-1}}}{\Lambda_{j}}\qquad\text{for
all}\>\>j=1,...,n.$

We will then use this to obtain an expansion for $\omega$. In order to have a
more compact notation we will introduce the notation $O(r^{l})$ to denote the
decay of a function (or of the norm of a tensor) and all its derivatives; to
be clearer, saying that some tensor $\theta$ is such that $\theta=O(r^{l})$
means that for all $k\geq 0$ holds $|\nabla_{\omega_{co,0}}^{k}\theta|\leq
cr^{l-k}$.

######  Remark 3.2.

Notice that we can choose a basis $\\{e_{j}\\}$ of $(1,0)$-forms diagonalizing
simultaneously $\omega_{co,0}$ (we can actually assume it to be the identity)
and $\omega$; this means that also $\omega_{co,0}^{2}$ and $\omega^{2}$ are
diagonal (in the sense of $(n-1,n-1)$-forms, implying that also the term
$O(r^{\lambda_{0}})$ is necessarily diagonal with respect to this basis. Thus
we can write

$\omega^{2}=\sum_{j=1}^{n}(\alpha_{j}+O(r^{m}))\widehat{e_{j}\wedge Je_{j}}$

and applying Michelson’s theorem with $\Lambda_{j}=\alpha_{j}+O(r^{m})$, we
obtain $\omega=\sum_{j=1}^{n}\lambda_{j}e_{j}\wedge Je_{j}$, with

$\lambda_{j}=\frac{((\alpha_{1}+O(r^{m}))(\alpha_{2}+O(r^{m}))(\alpha_{3}+O(r^{m})))^{\frac{1}{2}}}{\alpha_{j}+O(r^{m})}=\frac{(\alpha_{1}\alpha_{2}\alpha_{3})^{\frac{1}{2}}}{\alpha_{j}}+O(r^{m}),$

which implies, again thanks to Michelson’s theorem

$\omega=\sum_{j=1}^{n}\left(\frac{(\alpha_{1}\alpha_{2}\alpha_{3})^{\frac{1}{2}}}{\alpha_{j}}+O(r^{m})\right)e_{j}\wedge
Je_{j}=\omega_{co,0}+O(r^{m}).$

Hence we can write the volume form as $\omega^{3}=\omega_{co,0}^{3}+O(r^{m})$.

###  3.5. The Chern-Ricci potential

In order to use this description of the volume to estimate the Chern-Ricci
potential on the gluing region we are also going to need to understand how the
holomorphic volume form of the resolution is related to the holomorphic volume
of our background Calabi-Yau singular manifold.
Before doing it we start by fixing some notation. Denote

  * •

with $\Omega$ the holomorphic volume of $M_{reg}$ such that

$\tilde{\omega}^{3}=i\Omega\wedge\overline{\Omega};$

  * •

with $\Omega_{0}$ the holomorphic volume of the cone $X$ such that

$\omega_{co,0}^{3}=i\Omega_{0}\wedge\overline{\Omega_{0}};$

  * •

with $\chi$ the holomorphic volume of the resolution $\hat{X}$ such that

$(\lambda\omega_{co,1})^{3}=i\chi\wedge\overline{\chi}.$

In order to explain clearly how these forms are related and how we can use
them to construct a holomorphic volume on $M$ we shall take a step back and
watch closer at the topological gluing between $M_{reg}$ and $\hat{X}$.
Call $P$ the biholomorphism given from Theorem 2.4, and we can suppose
$\varepsilon$ to be sufficiently small so that we can assume that $P$ is
defined (eventually after a restriction) on the set $\\{0<r(z)\leq
4\varepsilon^{p}\\}$. Moreover, if we call $Q$ the biholomorphism introduced
at the beginning of Step 3 of the gluing construction, we can assume it to be
defined not only on the region written before, but actually on the whole
$\\{0<r(\zeta)\leq 4R\\}$. This allows us to give a topologically concrete
description of $M$, that is

$M=\frac{M_{reg}\>\overset{\circ}{\bigcup}\>\hat{X}}{\sim},$

where the equivalence relation $\sim$ is defined as

$y\in M_{reg},\>x\in\hat{X},\>y\sim x\>\>\text{if and only
if}\>\>P^{-1}(y)=Q^{-1}(x).$

Thanks to this explicit description we can observe that on the gluing region
holds

$Q^{*}\chi=\Omega_{0}$

and there exists a holomorphic function $h$ on the gluing region such

$P^{*}\Omega=fQ^{*}\chi=h\Omega_{0},$

and $h$ can actually be extended to the whole $\hat{X}$. We can however obtain
even more information on $h$ recalling again Theorem 2.4; indeed said theorem
guarantees us that

$P^{*}\Omega\wedge\overline{P^{*}\Omega}=\Omega_{0}\wedge\overline{\Omega_{0}}+i\partial\overline{\partial}\Phi$

where $\Phi$ is a smooth $(2,2)$-form that behaves as $=O(r^{2+\lambda_{0}})$
nearby the singularity. From this we get that

$(|h|^{2}-1)\Omega_{0}\wedge\overline{\Omega_{0}}=i\partial\overline{\partial}\Phi,$

and thus

$|h|^{2}=1+O(r^{\lambda_{0}})\>\>\text{on (a neighborhood of $\mathbb{P}^{1}$
in)}\>\>\hat{X},$

from which, by continuity, we have that $|h|^{2}\equiv 1$ on the exceptional
curve. Thus we can define the holomorphic volume $\hat{\Omega}$ of $M$ by
gluing together $h\chi$ and $\Omega$, and thus define a global Chern-Ricci
potential as

$f=f_{a,q,\varepsilon}:=\log\left(\frac{i\hat{\Omega}\wedge\overline{\hat{\Omega}}}{\omega^{3}}\right).$

We now conclude this section by describing the behaviour of $f$ in all the
regions of $M$, to show that it is suitable to apply a deformation argument
similar to the one done in [BM]. We have

  * •

on $\\{r(z)>4\varepsilon^{p}\\}$ hold $\omega=\tilde{\omega}$ and
$\hat{\Omega}=\Omega$, thus $f\equiv 0$;

  * •

on $\\{2\varepsilon^{p}\leq r(z)\leq 4\varepsilon^{p}\\}$ hold
$\omega=\omega_{co,0}+O(r^{m})$ and
$\hat{\Omega}\wedge\overline{\hat{\Omega}}=\Omega_{0}\wedge\overline{\Omega_{0}}+O(r^{\lambda_{0}})$,
from which we have

$f=\log\left(\frac{\omega_{co,0}^{3}+O(r^{m})}{\Omega_{0}\wedge\overline{\Omega_{0}}+O(r^{\lambda_{0}})}\right)=\log(1+O(r^{\tilde{m}}))=O(r^{\tilde{m}}),$

where $\tilde{m}=\min\\{\lambda_{0},m\\}$;

  * •

on$\\{\varepsilon^{p}\leq r(z)\leq 2\varepsilon^{p}\\}$ hold
$\omega=\omega_{co,0}$ and
$\hat{\Omega}\wedge\overline{\hat{\Omega}}=i(1+O(r^{\lambda_{0}}))\Omega_{0}\wedge\overline{\Omega_{0}}$,
from which follows $f=O(r^{\lambda_{0}})$;

  * •

on $\\{\varepsilon^{p}/2\leq r(z)\leq\varepsilon^{p}\\}$ hold
$\omega=\omega_{co,0}+O(r^{m})$ and
$\hat{\Omega}\wedge\overline{\hat{\Omega}}=\Omega_{0}\wedge\overline{\Omega_{0}}+O(r^{\lambda_{0}})$,
implying again $f=O(r^{\tilde{m}})$;

  * •

on $\\{r(z)<\varepsilon^{p}/2\\}$ hold $\omega=\lambda\omega_{co,1}$ and
$\hat{\Omega}\wedge\overline{\hat{\Omega}}=i(1+O(r^{\lambda_{0}}))\Omega_{0}\wedge\overline{\Omega_{0}}$,
giving once again $f=O(r^{\tilde{m}})$.

Thus we can write globally (on $M$) that

$|f|\leq cr^{\tilde{m}}.$

##  4\. The deformation argument

In this last section we will see that what was built in the previous section
are exactly the ingredients we need to introduce a deformation argument in the
same fashion as [BM], in order to obtain a balanced Chern-Ricci flat metric on
our small resolution $M$. We will also analyze the cohomology class of the
metric obtained and see why said metric is interested in the framework of the
Hull-Strominger system.

###  4.1. The strategy

We will now set up the problem for this section. First of all we need to
introduce a deformation of the metric that preserves the balanced condition,
and this can be done using the one introduced in [FWW]

$\omega_{\psi}^{2}:=\omega^{2}+i\partial\overline{\partial}(\psi\omega),\quad\psi\in
C^{\infty}(M,\mathbb{R})\>\>\text{such that}\>\>\omega_{\psi}^{2}>0.$

Notice that again, thanks to the results of Michelsohn in [M], writing
immediately $\omega_{\psi}^{2}$ makes sense.
Thus the problem we are interested in solving, following what was done in
[BM], is the equation

$\omega_{\psi}^{3}=e^{f}\omega^{3}$

for $\psi\in C^{\infty}(M,\mathbb{R})$ such that $\omega_{\psi}^{2}>0$.

######  Remark 4.1.

The equation introduced above makes sense, because, as we’ve seen,
$f=O(r^{\tilde{m}})$, thus $e^{f}=1+O(r^{\tilde{m}})$, meaning that
$e^{f}\omega^{3}$ is nearby $\omega^{3}$ itself, hence it makes sense to try
to obtain it as a small deformation of $\omega$.

For practicality is useful to reformulate our equation as an operator on the
space of smooth functions, thus we introduce
$F:C^{\infty}(M,\mathbb{R})\rightarrow C^{\infty}(M,\mathbb{R})$ as

$F(\psi)=F_{\varepsilon}(\psi):=\frac{\omega_{\psi}^{3}}{\omega^{3}}-e^{f}.$

Our aim is to solve this equation through a fixed point argument, i.e. turning
the problem in a new one that can be solved by applying Banach’s Lemma. In
order to achieve this, the first step to take is to start studying the
linearization at $0$ of the operator $F$. To do this we shall introduce the
notation $\omega_{0}^{\prime}:=\frac{d}{dt}_{|_{t=0}}\omega_{tu}$, where
$\omega_{tu}$ is the curve corresponding to the tangent vector $u\in
C^{\infty}(M,\mathbb{R})$, and compute the derivative at zero of
$\omega_{tu}^{3}$ in two different ways:

  * $\frac{d}{dt}_{|_{t=0}}\omega_{tu}^{3}=3\omega^{2}\wedge\omega_{0}^{\prime};$

  * $\frac{d}{dt}_{|_{t=0}}\omega_{tu}^{3}=i\partial\overline{\partial}(u\omega)\wedge\omega+\omega^{2}\wedge\omega_{0}^{\prime}.$

Even though none of these two expressions are explicit, we can put them
together to obtain an explicit one, that is

$d_{0}F(u)=Lu=L_{\varepsilon}u:=\frac{3}{2}\frac{i\partial\overline{\partial}u\wedge\omega^{2}+ui\overline{\partial}\omega\wedge\partial\omega}{\omega^{3}}.$

######  Remark 4.2.

The estimates done in the previous sections ensure us that, on the gluing
region from Step 2 of the gluing construction, hold

$\partial\omega=O(r^{2(q/p)-1}|\log
r|)\quad\text{and}\quad\partial\overline{\partial}\omega=O(r^{2(q/p)-2}|\log
r|),$

and $\partial\omega=0$ and $i\partial\overline{\partial}\omega=0$ everywhere
else. Thus if $q>p$, then
$\partial\omega,i\partial\overline{\partial}\omega\rightarrow 0$ as
$\varepsilon\rightarrow 0$, showing that $L$ is a bounded operator, and
implying that $\tilde{m}=m=\lambda_{0}$ (assuming $\lambda_{0}\leq 2$).

Before starting the analytical part, we shall establish once for all a value
for $p$ and $q$, and in light of Remark 4.2, we have that a good choice is
given by

$p=\frac{2}{5}\quad\text{and}\quad q=\frac{3}{5},$

from which we get $\lambda=\varepsilon^{2}$ and $m=\frac{2}{5}\lambda_{0}$.

###  4.2. Weighted analysis

Our aim is now to study the invertibility of this linear operator, and we wish
to do this in suitable weighted function spaces. In order to introduce said
spaces we shall start by introducing a weight function useful in our
situation, and for simplicity we may assume that the biholomorphism $P$ is
defined on the region $\\{r(z)\leq 1\\}$ (this is true up to a rescaling).
Define then

$\rho=\rho_{\varepsilon}(z):=\begin{cases}\varepsilon&\text{on}\>r(z)\leq\varepsilon,\\\
\text{non decreasing}&\text{on}\>\varepsilon\leq r(z)\leq 2\varepsilon,\\\
r(z)&\text{on}\>2\varepsilon\leq r(z)\leq 1/2,\\\ \text{non
decreasing}&\text{on}\>1/2\leq r(z)\leq 1,\\\ 1&\text{on}\>r(z)\geq
1,\end{cases}$

where we recall that $z$ are the "small" coordinates around the singularity in
$\tilde{M}$. Using this weight function we can introduce the weighted Hölder
norm and its corresponding weighted Hölder spaces
$C_{\varepsilon,b}^{k,\alpha}(M)$, where $k\geq 0$, $\alpha\in(0,1)$ is the
Hölder constant, $b\in\mathbb{R}$ is the weight and $\varepsilon$ indicates
the dependence on the metric $\omega$ obtained by the gluing construction done
above. We define

$\displaystyle||u||_{C_{\varepsilon,b}^{k,\alpha}(M)}:=$
$\displaystyle\sum_{i=0}^{k}\sup_{M}|\rho^{b+i}\nabla_{\varepsilon}^{i}u|_{\omega}$
$\displaystyle+\underset{d_{\varepsilon}(x,y)<inj_{\varepsilon}}{\sup}\left|\min\left(\rho^{b+k+\alpha}(x),\rho^{b+k+\alpha}(y)\right)\frac{\nabla_{\varepsilon}^{k}u(x)-\nabla_{\varepsilon}^{k}u(y)}{d_{\varepsilon}(x,y)^{\alpha}}\right|_{\omega},$

where $inj_{\varepsilon}$ is the injectivity radius of the metric $\omega$,
and thus interpret $F$ (and $L$) as operators defined as
$F:C_{\varepsilon,b}^{2,\alpha}(M)\rightarrow
C_{\varepsilon,b+2}^{0,\alpha}(M)$.
Following now a blended strategy between the one from Székelyhidi (as in
Chapter 8 in [Sz]) and the one from Tosatti and Weinkove (see [TW]), we
consider a variation of our operator by fixing a point $x\in M$, where $x$ is
away from the gluing region (for example we can assume $x$ to be outside of
the open set on which the biholomorphism $P$ is defined, and in the region
where the weight is identically equal to $1$), and defining

$\tilde{F}(\psi):=\frac{\omega_{\psi}^{3}}{\omega^{3}}-e^{f+ev_{x}(\psi)}$

(where $ev_{x}$ is the evaluation operator at $x$), with consequent
linearization at zero

$\tilde{L}=L+e^{f}ev_{x}.$

Thus the new problem to solve is

(2) $\tilde{F}(\psi)=0,$

whose solution clearly still gives a balanced Chern-Ricci flat metric.

We then wish to prove the following lemma.

######  Lemma 4.3.

With the same notations as above, there exist $b\in\mathbb{R}$ and $c>0$ such
that for sufficiently small $\varepsilon$ it holds

$||u||_{C_{\varepsilon,b}^{2,\alpha}}\leq
c||\tilde{L}u||_{C_{\varepsilon,b+2}^{0,\alpha}},$

for all $u\in C_{\varepsilon,b}^{2,\alpha}$.

###### Proof.

Suppose by contradiction that the above inequality does not hold. This means
that for all $n\in\mathbb{N}$ we can find $\varepsilon_{n}>0$ and $u_{n}\in
C_{\varepsilon_{n},b}^{2,\alpha}$ such that $\varepsilon_{n}\rightarrow 0$ as
$n\rightarrow 0$, $||u_{n}||_{C_{\varepsilon_{n},b}^{2,\alpha}}=1$ and

(3) $||\tilde{L}u_{n}||_{C_{\varepsilon_{n},b+2}^{0,\alpha}}<\frac{1}{n}.$

In the first place we analyze what happens on $M\setminus\mathbb{P}^{1}\simeq
M_{reg}$, i.e. away from the exceptional curve. The properties of the sequence
$\\{u_{n}\\}_{n\in\mathbb{N}}$ guarantee us that up to subsequences we may
assume $u_{n}\rightarrow u_{\infty}$ uniformly on compact subsets of $M_{reg}$
in the sense of $C_{b}^{0,\alpha}$, with respect to the background Calabi-Yau
metric $\tilde{\omega}$. Moreover, since for any compact set $K\subseteq
M_{reg}$ there exists $n_{K}\in\mathbb{N}$ such that for all $n\geq n_{K}$, on
$K$ it holds $\omega=\tilde{\omega}$ and hence
$\nabla_{\omega}=\nabla_{\tilde{\omega}}$, we actually have
$C_{b}^{2,\alpha}$-convergence (again uniformly on compact subsets of
$M_{reg}$). Also, for the same reason, on any compact subset of $M_{reg}$, for
$n$ sufficiently large the Chern-Ricci potential is identically zero, thus we
get $\tilde{L}=L+ev_{x}$; and still choosing $n$ sufficiently large, we end up
with $\omega=\tilde{\omega}$ on $K$, resulting in

$Lu_{n}=\frac{3}{2}\frac{i\partial\overline{\partial}u_{n}\wedge\tilde{\omega}^{2}}{\tilde{\omega}^{3}},$

i.e. a scalar multiple of the laplacian of $u_{n}$ with respect to the Kähler
metric $\tilde{\omega}$. Thus taking the limit for $n\rightarrow+\infty$ of
$\tilde{L}u_{n}$ yields

(4) $L_{\infty}u_{\infty}+u_{\infty}(x)=0.$

We shall now prove that $u_{\infty}$ is necessarily identically zero on the
whole $M_{reg}$. Indeed, take $\delta>0$ and $B_{\delta}$ a ball of radius
$\delta$ around the singularity, and notice that

$0=\int_{M_{\delta}:=M_{reg}\setminus
B_{\delta}}L_{\infty}u_{\infty}+u_{\infty}(x)dV_{\tilde{\omega}}=-\int_{\partial
B_{\delta}}i\overline{\partial}u_{\infty}\wedge\tilde{\omega}^{2}+\text{Vol}(M_{\delta})u_{\infty}(x).$

Writing then with $d\hat{V}$ the volume induced on $\partial B_{\delta}$
induced by $\tilde{\omega}$, we notice that

$\left|\int_{\partial
B_{\delta}}\overline{\partial}u_{\infty}\wedge\tilde{\omega}^{2}\right|\leq\int_{\partial
B_{\delta}}|\overline{\partial}u_{\infty}|_{\omega}d\hat{V}\leq
c\delta^{4-b},$

thus assuming $b<4$ and taking the limit for $\delta\rightarrow 0$ in (4) we
get $u_{\infty}(x)=0$, and, consequently, that $u_{\infty}$ is harmonic on the
whole $M_{reg}$. Notice now that, since $L_{\infty}u_{\infty}=0$, we get

(5)
$0=\int_{M_{\delta}}u_{\infty}L_{\infty}u_{\infty}\tilde{\omega}^{3}=\int_{M_{\delta}}u_{\infty}i\partial\overline{\partial}u_{\infty}\wedge\tilde{\omega}^{2}$

and since $\tilde{\omega}$ is balanced (actually Kähler) it holds

$d(i\overline{\partial}u_{\infty}\wedge(u_{\infty}\tilde{\omega}^{2}))=u_{\infty}i\partial\overline{\partial}u_{\infty}\wedge\tilde{\omega}^{2}+i\partial
u_{\infty}\wedge\overline{\partial}u_{\infty}\wedge\tilde{\omega}^{2},$

which integrated on $M_{\delta}$ and combined with (5) gives

(6) $\displaystyle 0$ $\displaystyle=\int_{\partial
B_{\delta}}u_{\infty}i\overline{\partial}u_{\infty}\wedge\tilde{\omega}^{2}=\int_{M_{\delta}}i\partial
u_{\infty}\wedge\overline{\partial}u_{\infty}\wedge\tilde{\omega}^{2}$
$\displaystyle=2\int_{M_{\delta}}i\partial
u_{\infty}\wedge\overline{\partial}u_{\infty}\wedge\ast\tilde{\omega}=2\int_{M_{\delta}}\langle
i\partial
u_{\infty}\wedge\overline{\partial}u_{\infty},\tilde{\omega}\rangle\tilde{\omega}^{3}=2\int_{M_{\delta}}|\partial
u_{\infty}|^{2}\tilde{\omega}^{3}.$

But since

$\left|\int_{\partial
B_{\delta}}u_{\infty}i\overline{\partial}u_{\infty}\wedge\tilde{\omega}^{2}\right|\leq\int_{\partial
B_{\delta}}|u_{\infty}|_{\omega}|\partial u_{\infty}|_{\omega}d\hat{V}\leq
c\delta^{4-2b},$

thus choosing $b<2$ and taking the limit for $\delta\rightarrow 0$ in (6) we
get $\int_{M_{reg}}|\partial u_{\infty}|_{\omega}^{2}\tilde{\omega}^{3}=0$,
that is $u_{\infty}\equiv const.$, and since $u_{\infty}(x)=0$, necessarily
$u_{\infty}\equiv 0$ on the whole $M_{reg}$.
Let now $M_{c}:=\\{r(z)\geq 1/2\\}\subseteq M_{reg}$ be the compact set on
which $u_{n}\rightarrow 0$ uniformly in $C_{b}^{2,\alpha}$. To obtain a
contradiction we want to prove that $\\{u_{n}\\}_{n\in\mathbb{N}}$ admits a
subsequence uniformly convergent to zero in $C_{b}^{2,\alpha}$ also on
$A:=\\{r(z)<1/2\\}$.
In order to work in this region, it is simpler to shift to the "large"
coordinates $\zeta$, i.e. the coordinates on $\hat{X}$ away from the
exceptional divisor. It is then useful to recall the relations

$\zeta=\varepsilon^{-3/2}z\quad\text{and}\quad r(z)=\varepsilon r(\zeta),$

from which we can write down the explicit identification

$\left\\{r(z)<\frac{1}{2}\right\\}=A\simeq\tilde{A}=\tilde{A}_{\varepsilon}=\left\\{r(\zeta)<\frac{1}{2}\varepsilon^{-1}\right\\};$

this last set $\tilde{A}$ is the one we will be working on.
The first thing to do is rewrite the weight function in terms of this
coordinates on $\tilde{A}$, resulting in

$\rho=\begin{cases}\varepsilon&\text{on}\>r(\zeta)\leq 1,\\\ \text{non
decreasing}&\text{on}\>1\leq r(\zeta)\leq 2,\\\ \varepsilon
r(\zeta)&\text{on}\>2\leq r(\zeta)\leq 1/2\varepsilon^{-1}.\end{cases}$

Notice that the entire gluing region of the metric (from the previous step) is
entirely contained inside the third region, i.e. $\\{2\leq r(\zeta)\leq
1/2\varepsilon^{-1}\\}$.
We now go back to our sequence $\\{u_{n}\\}_{n\in\mathbb{N}}$. Since
$||u_{n}||_{C_{\varepsilon_{n},b}^{2,\alpha}}=1$ for all $n\in\mathbb{N}$, we
have in particular that on all $\tilde{A}_{n}:=\tilde{A}_{\varepsilon_{n}}$
holds

$|\rho^{b}u_{n}|\leq c.$

Introducing then the new sequence

$U_{n}:=\varepsilon_{n}^{b}u_{n},$

the above weighted estimates for $u_{n}$ imply the following ones for this new
sequence:

$\begin{cases}|U_{n}|\leq c&\text{on}\>r(\zeta)\leq 1,\\\ |U_{n}|\leq
c&\text{on}\>1\leq r(\zeta)\leq 2,\\\ |U_{n}|\leq
cr^{-b}(\zeta)&\text{on}\>2\leq r(\zeta)\leq
1/2\varepsilon_{n}^{-1}.\end{cases}$

This estimates for $U_{n}$ suggest us to introduce a new weight function
$\tilde{\rho}=\tilde{\rho}_{n}$ on $\tilde{A}_{n}$ given by

$\tilde{\rho}(\zeta)=\begin{cases}1&\text{on}\>r(\zeta)\leq 1,\\\ \text{non
decreasing}&\text{on}\>1\leq r(\zeta)\leq 2,\\\ r(\zeta)&\text{on}\>2\leq
r(\zeta)\leq 1/2\varepsilon_{n}^{-1},\end{cases},$

with which we get that

$|\tilde{\rho}^{b}U_{n}|\leq c,$

and analogous weighted estimates also for $\nabla U_{n}$ and
$\nabla^{2}U_{n}$, hence again by Ascoli-Arzelà theorem we have that
$U_{n}\rightarrow U_{\infty}$ uniformly on compact sets of $\hat{X}$ (since
$\tilde{A}_{n}\rightarrow\hat{X}$) in the sense of
$\tilde{C}_{b}^{2,\alpha}=C_{b}^{2,\alpha}(\tilde{\rho})$, where this last
space is the weighted Hölder space on $\hat{X}$ identified by the weight
$\tilde{\rho}$ and the Candelas-de la Ossa metric $\omega_{co,1}$.
On the other hand it holds

$\rho^{b+2}Lu_{n}=\tilde{\rho}^{b+2}\Delta_{\omega_{co,1}}U_{n},$

hence

(7)
$||Lu_{n}||_{C_{\varepsilon_{n},b+2}^{0,\alpha}}=||\Delta_{\omega_{co,1}}U_{n}||_{\tilde{C}_{b+2}^{0,\alpha}},$

and since
$\frac{1}{n}>||\tilde{L}u_{n}||_{C_{\varepsilon_{n},b+2}^{0,\alpha}}$ and
$u_{n}(x)\rightarrow 0$ hold, taking the limit in (7) we obtain that
$U_{\infty}$ is harmonic. Moreover, if we assume $b>0$, then $U_{\infty}$ is a
harmonic function decaying at infinity, thus necessarily $U_{\infty}\equiv 0$,
and thus $U_{n}\overset{\tilde{C}_{b}^{2,\alpha}}{\rightarrow}0$ uniformly on
compact sets of $\hat{X}$.
If we are now able to prove that $U_{n}$ admits a subsequence converging
uniformly to zero on the whole $\hat{X}$ in the sense $\tilde{C}_{b}^{0}$ we
get our contradiction, and we are done. Indeed, if
$U_{n}\overset{\tilde{C}_{b}^{0}}{\rightarrow}0$ uniformly (up to
subsequences) on $\hat{X}$, then scaled Schauder estimates imply that also
$U_{n}\overset{\tilde{C}_{b}^{2,\alpha}}{\rightarrow}0$ uniformly, which is
the same as saying
$u_{n}\overset{\tilde{C}_{\varepsilon_{n},b}^{2,\alpha}}{\rightarrow}0$
uniformly on $\\{r(z)<1/2\\}$. Thus $\\{u_{n}\\}_{n\in\mathbb{N}}$ up to
subsequences is uniformly convergent to zero on the whole manifold $M$, which
is a contradiction with the fact that
$||u_{n}||_{C_{\varepsilon_{n},b}^{2,\alpha}}=1$ for all $n\in\mathbb{N}$.
Now we will prove that the said uniformly convergent subsequence exists. If by
contradiction this was not the case, since we have the uniform convergence on
compact sets, we would be able to find $\gamma>0$ and
$\\{x_{n}\\}_{n\in\mathbb{N}}\subseteq\hat{X}$, $x_{n}\in\tilde{A}_{n}$, such
that $R_{n}:=r(\zeta(x_{n}))\rightarrow+\infty$ and
$R_{n}^{b}U_{n}(x_{n})\geq\gamma$ for all $n\in\mathbb{N}$. But
$\\{x_{n}\\}_{n\in\mathbb{N}}$ is a bounded sequence (since it lies in
$\\{r(z)<1/2\\}$), thus if we name $r_{n}:=r(z(x_{n}))$ and recall that
$r_{n}=\varepsilon_{n}R_{n}$, up to subsequences we can end up into two cases:

  * (i)

if $x_{n}\rightarrow x_{\infty}$ such that $r(x_{\infty})>0$, then, since
$u_{n}$ is uniformly convergent on compact sets on $M_{reg}$, we get that
$u_{n}(x_{n})$ is bounded, giving

$0<\gamma\leq
R_{n}^{b}U_{n}(x_{n})=(R_{n}\varepsilon_{n})^{b}u_{n}(x_{n})=r_{n}^{b}u_{n}(x_{n})\underset{n\rightarrow\infty}{\longrightarrow}0,$

which is a contradiction;

  * (ii)

if $r_{n}\rightarrow 0$, we consider the biholomorphisms
$\sigma_{n}:B_{n}\rightarrow A\setminus\\{0\\}$, given by

$\sigma_{n}(z^{\prime}):=r_{n}^{3/2}z^{\prime},$

where $B_{n}:=\\{0<r(z^{\prime})<\frac{r_{n}^{-1}}{2}\\}\subseteq X$. Then, if
we endow $B_{n}$ with the metric

$\eta_{n}:=r_{n}^{-2}\sigma_{n}^{*}\omega,$

using (1) it is easy to notice that the couple $(B_{n},\eta_{n})$ converges to
$(X,\omega_{co,0})$, i.e. the standard Calabi-Yau cone. If we then introduce
the functions

$w_{n}:=r_{n}^{b}\sigma_{n}^{*}u_{n}$

on $B_{n}$, we notice that the pullback of the weight function $\rho$ gives

$\rho^{\prime}(z^{\prime})=\sigma_{n}^{*}\rho(z^{\prime})=\begin{cases}\varepsilon_{n}&\text{on}\>r(z^{\prime})<R_{n}^{-1},\\\
\text{non decreasing}&\text{on}\>R_{n}^{-1}\leq r(z)\leq 2R_{n}^{-1},\\\
r_{n}r(z^{\prime})&\text{on}\>2R_{n}^{-1}\leq
r(z)<\frac{r_{n}^{-1}}{2},\end{cases}$

from which we get (pulling back the inequality $\rho^{b}u_{n}\leq 1$)

(8) $r^{b}(z^{\prime})w_{n}(z^{\prime})\leq 1$

on each $z^{\prime}\in X$ (assuming $n$ to be sufficiently large). Hence, this
shows that for any compact $K\subseteq X$, we can chose $n\in\mathbb{N}$ to be
sufficiently large in order to have $K\subseteq B_{n}$ and
$\rho^{\prime}(z^{\prime})=r_{n}r(z^{\prime})$ on the whole $K$, and get that
$w_{n}$ is uniformly bounded on $K$; and since this works for any compact
$K\subseteq X$, we obtain that - up to subsequences -
$\\{w_{n}\\}_{n\in\mathbb{N}}$ converges uniformly on compact sets of $X$ to a
function $w_{\infty}$, and from (8) we get that $w_{\infty}$ is decaying at
infinity. Moreover, recalling that $R_{n}^{b}U_{n}(x_{n})\geq\gamma$ for all
$n\in\mathbb{N}$, if we introduce the sequence
$y_{n}:=\sigma_{n}^{-1}(x_{n})$, it is straighforward to notice that from its
definition follows that $w_{n}(y_{n})\\\ geq\gamma$ and
$||y_{n}||_{\eta_{n}}=1$ for all $n\in\mathbb{N}$, thus implying that - up to
subsequences - $y_{n}\rightarrow y_{\infty}\in X$, and hence

(9) $w_{\infty}(y_{\infty})>0.$

Now, if we recall the definition of the operator $\tilde{L}$ and take the
pullback with respect to $\sigma_{n}$ of $\rho^{b+2}\tilde{L}u_{n}$, it is
immediate to see that on every compact $K\subseteq X$ we get

(10)
$\sigma_{n}^{*}\left(\rho^{b+2}\tilde{L}u_{n}\right)=\frac{3}{2}r^{b+2}(z^{\prime})\frac{i\partial\overline{\partial}w_{n}\wedge\eta_{n}^{2}}{\eta_{n}^{3}}+\frac{3}{2}\frac{i\overline{\partial}\eta_{n}\wedge\partial\eta_{n}}{\eta_{n}^{3}}+r_{n}^{b+2}r^{b+2}(z^{\prime})u_{n}(x).$

Analyzing then the single terms, we can notice that as $n\rightarrow+\infty$

    * •

$\sigma_{n}^{*}\left(\rho^{b+2}\tilde{L}u_{n}\right)\rightarrow 0$ from the
hypothesis (3);

    * •

$\frac{i\partial\overline{\partial}w_{n}\wedge\eta_{n}^{2}}{\eta_{n}^{3}}\rightarrow\Delta_{\omega_{co,0}}w_{\infty}$

from the above observations;

    * •

$r_{n}^{b+2}r^{b+2}(z^{\prime})u_{n}(x)\rightarrow 0$ trivially;

    * •

$\frac{i\overline{\partial}\eta_{n}\wedge\partial\eta_{n}}{\eta_{n}^{3}}\rightarrow
0,$

since from Remark 4.2 we can pullback with $\sigma_{n}$ the inequality

$|\partial\omega|_{\omega}\leq cr^{2}|\log r|,$

and obtain

$|\partial\eta_{n}|_{\eta_{n}}\leq cr_{n}^{3}r^{2}(z^{\prime})(|\log
r_{n}|+|\log r(z^{\prime})|)\rightarrow 0.$

Hence, taking the limit as $n\rightarrow+\infty$ in (10) we obtain

$\Delta_{\omega_{co,0}}w_{\infty}\equiv 0,$

and since it holds on every compact $K\ subseteqX$, we get that $w_{\infty}$
is harmonic on the whole cone $X$. But now, since it holds (8), we can apply
Lemma 6.9 from [CPY1] (actually just its proof applied to $\pm w_{\infty}$,
without the hypothesis $w_{\infty}\geq 0$), and obtain that necessarily
$w_{\infty}\equiv 0$ on $X$, which is a contradiction, as it holds (9).

Thus the proof is complete. ∎

As a direct consequence we get

######  Lemma 4.4.

The operator $\tilde{L}:C_{\varepsilon,b}^{2,\alpha}(M)\rightarrow
C_{\varepsilon,b+2}^{0,\alpha}(M)$ defined above is an isomorphism.

###### Proof.

Notice that $L$ is elliptic and $ev_{x}$ is a compact operator, thus $L$ and
$\tilde{L}$ share their index, which is zero. Moreover, by Lemma 4.3 we have
that $\tilde{L}$ is injective, thus we automatically get that $\tilde{L}$ is
also surjective and has bounded inverse, thus $\tilde{L}$ is an isomorphism. ∎

With this result we can now show how to reformulate the original equation as a
fixed point problem.
In order to do this we shall introduce the operators
$\hat{F},E,G:C_{\varepsilon,b}^{2,\alpha}(M)\rightarrow
C_{\varepsilon,b+2}^{0,\alpha}(M)$ defined as

$\hat{F}(\psi):=\frac{\omega_{\psi}^{3}}{\omega^{3}},\quad
E(\psi):=e^{f+ev_{x}(\psi)}\quad\text{and}\quad G(\psi)=e^{f}ev_{x}(\psi),$

from which we can write

$\tilde{F}=\hat{F}-E.$

Now, we can consider the expansion

$\hat{F}(\psi)=\hat{F}(0)+L(\psi)+\hat{Q}(\psi),$

and thus rewrite Equation (2) as

$\hat{F}(0)+L(\psi)+\hat{Q}(\psi)-E(\psi)=0.$

Here, we notice that $\tilde{L}=L-G$, thus we can rewrite Equation (2) once
more and get

$\hat{F}(0)+\tilde{L}(\psi)+\hat{Q}(\psi)+G(\psi)-E(\psi)=0,$

and using now Lemma 4.4, we get that Equation (2) is therefore equivalent to

(11) $\psi=\tilde{L}^{-1}(E(\psi)-G(\psi)-\hat{F}(0)-\hat{Q}(\psi))=:N(\psi),$

i.e. the search for a fixed point for the operator
$N:C_{\varepsilon,b}^{2,\alpha}(M)\rightarrow
C_{\varepsilon,b}^{2,\alpha}(M)$. To do this, we will have to identify the
open set on which we wish to apply Banach’s Lemma, and show that on said open
set $N$ can be restricted and gives rise to a contraction.
The first thing to do, is the following remark.

######  Remark 4.5.

If $C,\tau>0$, and $\varphi$ is a function on $M$ such that
$||\varphi||_{C_{\varepsilon,-2}^{2,\alpha}}\leq C\varepsilon^{\tau}$, thanks
to Remark 4.2 it is straightforward to see that

$||i\partial\overline{\partial}(\varphi\omega)||_{C_{\varepsilon,0}^{0,\alpha}}\leq||\varphi||_{C_{\varepsilon,-2}^{2,\alpha}}\leq
C\varepsilon^{\tau},$

thus we’re guaranteed that, choosing $\varepsilon$ to be sufficiently small,
$\omega_{\varphi}^{2}>0$, and thus it’s square root $\omega_{\varphi}$ exists
and is a balanced metric. Moreover, we can apply again the argument used in
Remark 3.2, and obtain that if
$||\varphi||_{C_{\varepsilon,-2}^{2,\alpha}}\leq C\varepsilon^{\tau}$, then

$|\omega_{\varphi}-\omega|_{\omega}\leq
c||\varphi||_{C_{\varepsilon,-2}^{2,\alpha}}\leq c\varepsilon^{\tau},$

which also implies that $\omega_{\varphi}\rightarrow\omega$, as
$\varepsilon\rightarrow 0$.

Thanks to this remark, we have a suggestion on how to choose the open set on
which apply Banach’s Lemma, hence we introduce

$U:=\\{\varphi\in
C_{\varepsilon,b}^{2,\alpha}\>|\>||\varphi||_{C_{\varepsilon,b}^{2,\alpha}}<\tilde{c}\varepsilon^{b+2+\tau}\\}\subseteq
C_{\varepsilon,b}^{2,\alpha},$

and we notice that for every $\varphi\in U$ it holds
$||\varphi||_{C_{\varepsilon,-2}^{2,\alpha}}\leq C\varepsilon^{\tau}$, with
$C$ independent of $\varphi$ and $\varepsilon$.
We will now prove that on $U$, the operator $N$ is a contraction. In
particular, given $\varphi_{1},\varphi_{2}\in U$, we want to estimate

$N(\varphi_{1})-N(\varphi_{2})=\tilde{L}^{-1}(E(\varphi_{1})-G(\varphi_{1})-(E(\varphi_{2})-G(\varphi_{2}))-(\hat{Q}(\varphi_{1})-\hat{Q}(\varphi_{2}))).$

Starting by analyzing the $\hat{Q}$-term, we notice that by the Mean Value
Theorem we can find $t\in[0,1]$ such that

$\hat{Q}(\varphi_{1})-\hat{Q}(\varphi_{2})=d\hat{Q}_{\eta}(\varphi_{1}-\varphi_{2})=(L_{\eta}-L)(\varphi_{1}-\varphi_{2}),$

where $\eta=t\varphi_{1}+(1-t)\varphi_{2}\in U$, and $L_{\eta}$ is the
linearization of $\hat{F}$ at $\eta$. With the same strategy used to compute
$L$ we can easily obtain an expression for $L_{\eta}$, and thus get

$(L_{\eta}-L)(\varphi_{1}-\varphi_{2})=\frac{3}{2}\frac{(\omega_{\eta}-\omega)\wedge
i\partial\overline{\partial}((\varphi_{1}-\varphi_{2})\omega)}{\omega^{3}}.$

From here, taking the norms with respect to $\omega$, we can use the fact that
$\eta\in U$ together with Remark 4.5, to obtain

$|(L_{\eta}-L)(\varphi_{1}-\varphi_{2})|\leq
c|\omega_{\eta}-\omega|_{\omega}|i\partial\overline{\partial}((\varphi_{1}-\varphi_{2})\omega)|_{\omega}\leq
c\varepsilon^{\tau}|i\partial\overline{\partial}((\varphi_{1}-\varphi_{2})\omega)|_{\omega},$

and thus, by multiplying the inequality with $\rho^{b+2}$, get

(12)
$||\hat{Q}(\varphi_{1})-\hat{Q}(\varphi_{2})||_{C_{b+2,\varepsilon}^{0,\alpha}}\leq
c\varepsilon^{\tau}||\varphi_{1}-\varphi_{2}||_{C_{\varepsilon,b}^{2,\alpha}}.$

Focusing instead on the $(E-G)$-term, and again using that
$\varphi_{1},\varphi_{2}\in U$ together with Remark 4.5, we get

$\displaystyle|E(\varphi_{1})-G(\varphi_{1})-(E(\varphi_{2})-G(\varphi_{2}))|=$
$\displaystyle
e^{f}|(\varphi_{2}(x)-\varphi_{1}(x)+e^{\varphi_{1}(x)}-e^{\varphi_{2}(x)}|$
$\displaystyle\leq$ $\displaystyle
c(|\varphi_{1}(x)|+|\varphi_{2}(x)|)|\varphi_{1}(x)-\varphi_{2}(x)|$
$\displaystyle\leq$ $\displaystyle
c\varepsilon^{\tau}|\varphi_{1}(x)-\varphi_{2}(x)|.$

Now, as above, multiplying the inequality with $\rho^{b+2}$ and recalling the
choice of $x\in M$ and the definition of $\rho$, we get

$\rho^{b+2}|E(\varphi_{1})-G(\varphi_{1})-(E(\varphi_{2})-G(\varphi_{2}))|\leq
c\varepsilon^{\tau}\rho^{b+2}(x)|\varphi_{1}(x)-\varphi_{2}(x)|$

and hence

(13)
$||E(\varphi_{1})-G(\varphi_{1})-(E(\varphi_{2})-G(\varphi_{2}))||_{C_{b+2,\varepsilon}^{0,\alpha}}\leq
c\varepsilon^{\tau}||\varphi_{1}-\varphi_{2}||_{C_{\varepsilon,b}^{2,\alpha}}.$

Thus, combining estimates (12) and (13) with Lemma 4.3, we get

(14) $||N(\varphi_{1})-N(\varphi_{2})||_{C_{\varepsilon,b}^{2,\alpha}}\leq
c\varepsilon^{\tau}||\varphi_{1}-\varphi_{2}||_{C_{\varepsilon,b}^{2,\alpha}},$

hence, choosing $\varepsilon$ sufficiently small ensures us that $N$ is a
contraction on $U$.
We are left with proving that $N(U)\subseteq U$. To do this, we shall impose a
restriction on $\tau$, that is $\tau<\frac{2}{5}\lambda_{0}$, and we will use
it in the inequalities to follow. Take then $\varphi\in U$, thanks to estimate
(14) and Lemma 4.3, we have

$\displaystyle||N(\varphi)|||_{C_{\varepsilon,b}^{2,\alpha}}\leq$
$\displaystyle||N(\varphi)-N(0)|||_{C_{\varepsilon,b}^{2,\alpha}}+||N(0)|||_{C_{\varepsilon,b}^{2,\alpha}}$
$\displaystyle\leq$ $\displaystyle
c\varepsilon^{\tau}||\varphi|||_{C_{\varepsilon,b}^{2,\alpha}}+||\tilde{L}^{-1}(1-e^{f})|||_{C_{\varepsilon,b}^{2,\alpha}}$
$\displaystyle\leq$ $\displaystyle
c\varepsilon^{\tau}||\varphi|||_{C_{\varepsilon,b}^{2,\alpha}}+||f||_{C_{\varepsilon,b}^{0,\alpha}}$
$\displaystyle\leq$ $\displaystyle
c(\varepsilon^{b+2+2\tau}+\varepsilon^{b+2+(2/5)\lambda_{0}})$
$\displaystyle\leq$ $\displaystyle
c\varepsilon^{\min\\{\tau,(2/5)\lambda_{0}-\tau\\}}\varepsilon^{b+2+\tau}$
$\displaystyle\leq$ $\displaystyle\tilde{c}\varepsilon^{b+2+\tau},$

implying that $N(U)\subseteq U$.
This shows that everything is into place to apply Banach’s Lemma on the open
set $U$ and obtain $\hat{\omega}$ a Chern-Ricci flat balanced metric
$\hat{\omega}$ on $M$, thus proving most of Theorem 1.1.

######  Remark 4.6.

It is known that for a manifold which is Calabi-Yau with holomorphic volume
$\Omega$, the existence of a Chern-Ricci flat balanced metric implies that
$\Omega$ is parallel with respect to the Bismut connection associated to said
metric. Among the other things, this implies that the restricted holonomy of
the Bismut connection of Chern-Ricci flat balanced metrics is contained in
$SU(n)$.

######  Remark 4.7.

Recalling Remark 4.5, we have that the couple $(M,\hat{\omega})$ Gromov-
Hausdorff converges to the singular Calabi-Yau metric on $M_{reg}$ and, up to
rescaling, to the Candelas-de la Ossa metrics nearby the exceptional curve.

######  Remark 4.8.

Notice that en passant we have proved that, on our small resolutions, for
certain classes nearby the boundary of the balanced cone, the Gauduchon
conjecture for balanced metrics (see [STW]) holds, i.e. in said balanced
classes we can always find a Chern-Ricci flat balanced metric.

###  4.3. Cohomology classes

We end the discussion on the construction by analyzing the cohomology class
naturally associated to the metric $\hat{\omega}$ just obtained. Indeed,
balanced metrics naturally induce an $(n-1,n-1)$-class, thus in our case we
get

$[\hat{\omega}^{2}]\in H_{dR}^{2,2}(M).$

We are then interested in understanding if this cohomology class can be
described in terms of the cohomology classes of the objects we used for the
gluing construction. The first thing to observe is that the deformation
introduced in the previous section preserves the balanced class, thus we have
that $[\omega^{2}]=[\hat{\omega}^{2}]$, thus we shall work with $\omega$.
Let’s then introduce two cut off functions
$\theta_{1},\theta_{2}:[0,+\infty)\rightarrow[0,1]$ defined as follows:

$\theta_{1}(x):=\begin{cases}1&\text{if}\>\>x\leq\frac{1}{8}\varepsilon^{-3/5}\\\
\text{non increasing}&\text{if}\>\>\frac{1}{8}\varepsilon^{-3/5}\leq
x\leq\frac{1}{4}\varepsilon^{-3/5}\\\
0&\text{if}\>\>x\geq\frac{1}{4}\varepsilon^{-3/5}\end{cases}$

and

$\theta_{2}(x):=\begin{cases}0&\text{if}\>\>x\leq 8\varepsilon^{-3/5}\\\
\text{non decreasing}&\text{if}\>\>8\varepsilon^{-3/5}\leq x\leq
16\varepsilon^{-3/5}\\\ 1&\text{if}\>\>x\geq 16\varepsilon^{-3/5};\end{cases}$

and since for sufficiently small $\varepsilon$ we have that $\omega$ is exact
on $K:=\\{\frac{1}{8}\varepsilon^{-3/5}\leq r(\zeta)\leq
8\varepsilon^{-3/5}\\}$, it exists a $3$-form $\beta$ such that

$\omega^{2}=d\beta\quad\text{on}\>K.$

Introduce then the form

$\Omega_{c}:=d(\theta_{1}(r(\zeta))+\theta_{2}(r(\zeta)))\beta),$

which is a smooth compactly supported form. Moreover, the form

$\beta-(\theta_{1}(r)+\theta_{2}(r))\beta$

can be extended as zero to the whole $M$, thanks to the definition of the cut-
offs, and thus get that

$[\omega^{2}]=[\Omega_{c}],$

i.e. the class $[\omega^{2}]$ admits a compactly supported representative. In
addition, the two cut-offs introduced also allow us to decompose
$\Omega_{c}=\Omega_{c}^{\prime}+\Omega_{c}^{\prime\prime}$, such that on $K$
hold

$\Omega_{c}^{\prime}=d(\theta_{1}(r)\beta)\quad\text{and}\quad\Omega_{c}^{\prime\prime}=d(\theta_{2}(r)\beta),$

and both $\Omega_{c}^{\prime}$ and $\Omega_{c}^{\prime\prime}$ are compactly
supported and closed; in particular said compact supports are respectively
contained in $\hat{X}$ and $M_{reg}$ (via the obvious identifications), and
from their definition it is straightforward to see that

$[\Omega_{c}^{\prime}]=\varepsilon^{4}[\omega_{co,1}^{2}]\in
H_{c}^{4}(\hat{X})$

and

$[\Omega_{c}^{\prime\prime}]=[\tilde{\omega}^{2}]\in H_{c}^{4}(M_{reg}),$

where $H_{c}$ denotes the compactly supported cohomology group. Also,
recalling that $\hat{X}\simeq\mathcal{O}_{\mathbb{P}^{1}}(-1)^{\oplus 2}$, it
is clear that $\hat{X}$ is homotopically equivalent to $\mathbb{P}^{1}$; hence
applying Poincaré duality we get

$H_{c}^{4}(\hat{X})\simeq H_{2}^{c}(\hat{X})\simeq
H_{2}^{c}(\mathbb{P}^{1})=H_{2}(\mathbb{P}^{1})=\langle[\mathbb{P}^{1}]\rangle,$

which means that the non-zero class $[\omega_{co,1}^{2}]$, up to
multiplicative constants, is the Poincaré dual of the generator of
$H_{2}(\mathbb{P}^{1})$ (thus we can "confuse" them with each other), and thus
we can write

$[\omega^{2}]=[\tilde{\omega}^{2}]+\varepsilon^{4}[\mathbb{P}^{1}]\quad\text{in}\>\>H_{dR}^{2,2}(M).$

Finally, we also notice that

$\int_{\mathbb{P}^{1}}\omega=\varepsilon^{2}\int_{\mathbb{P}^{1}}\omega_{co,1}\underset{\varepsilon\rightarrow
0}{\longrightarrow}0,$

hence the balanced class $[\omega^{2}]$, as $\varepsilon\rightarrow 0$,
converges to a nef class, i.e. to the boundary of the balanced cone.

This concludes the proof of Theorem 1.1.

######  Remark 4.9.

Even though this construction is done to address the situation of a non-Kähler
small resolution, it can also be applied when said small resolution $M$ is
instead Kähler. In this case we know from Yau’s theorem that $M$ admits a
Kähler Calabi-Yau metric $\omega_{1}$, hence together with the balanced class
induced by our Chern-Ricci flat balanced metric $\hat{\omega}$ we also have
the one induced by $\omega_{1}$. This two balanced classes need not be the
same, however, even if they are to coincide, there is no uniqueness result
that would guarantee that the two metrics have to be the same; moreover, the
deformation we used in our construction does not cover the whole balanced
class, hence in this case we are not even guaranteed that the two metrics are
linked by our chosen deformation.

###  4.4. Relation to the Hull-Strominger system

As a conclusion of the paper, we would like to briefly relate our construction
to the Hull-Strominger system and how we intend to develop our research in
this direction, hence we shall first quickly recall the definition of said
system (for more details we refer to the notes [GF]).

The framework is given by $(X,\Omega)$ a (not necessarily Kähler) Calabi-Yau
manifold, and the first equation of the system, for $\omega$ a hermitian
metric, is known as the dilatino equation and is given by

$d^{*}\omega=d^{c}\log||\Omega||_{\omega},$

which easily seen to be equivalent to the conformally balanced equation

$d\left(||\Omega||_{\omega}\omega^{n-1}\right)=0.$

Hence, this last equation tells us that we need to work with balanced
manifolds, and thus we see a first relation to our scenario.
To complete the system we need to pair the dilatino equation with two Hermite-
Einstein equations for holomorphic vector bundles, thus, in the same fashion
as what we had with the dilatino equation, the presence of the Hermite-
Einstein equation in the Hull-Strominger system will limit us to consider only
polystable bundles. Finally, adding one last equation, known as the Bianchi
identity, we can introduce the system.

######  Definition 4.10.

Given a Calabi-Yau manifold $(X,\Omega)$ and a holomorphic vector bundle $E$
on $X$, we say that the triple $(\omega,\>h,\>\overline{\partial}_{T})$ is a
solution of the Hull-Strominger system if it satisfies

$\displaystyle\Lambda_{\omega}F_{h}$ $\displaystyle=0,$
$\displaystyle\Lambda_{\omega}R$ $\displaystyle=0,$ $\displaystyle
d^{*}\omega-d^{c}\log||\Omega||_{\omega}$ $\displaystyle=0,$ $\displaystyle
dd^{c}\omega-\alpha(\text{tr}R\wedge R-\text{tr}F_{h}\wedge F_{h})$
$\displaystyle=0;$

where, $\alpha$ is a non-vanishing constant, $\omega$ is a hermitian metric on
$X$, $h$ is a hermitian metric along the fibers of $E$,
$\overline{\partial}_{T}$ is a holomorphic structure on the tangent bundle of
$X$, and $R$ is the Chern curvature tensor of $\omega$, read as a hermitian
metric on the holomorphic vector bundle $(TX,J,\overline{\partial}_{T})$.

The Bianchi identity (also known as anomaly cancellation equation), is the
hardest and least understood equation of the system and also the one we wish
to address in the development of our research.
It is significant to notice that if we choose $(E,h)$ to be the holomorphic
tangent bundle with the metric $\omega$, and take $\omega$ a Kähler Ricci-flat
metric, we see that this satisfies the system, thus being a solution of the
Hull-Strominger system is a condition that generalizes being Kähler Calabi-
Yau, hence a very promising candidate class of special metrics.
Also, thanks to the equivalence between Hermite-Einstein metrics and Hermite-
Yang-Mills connections it is possible to rewrite the system from a gauge-
theoretical point of view.

######  Definition 4.11.

Given a Calabi-Yau manifold $(X,\Omega)$ and a hermitian vector bundle $(E,h)$
(with a fixed holomorphic structure) on $M$, the triple
$(\omega,\>A,\>\nabla)$ is a solution of the Hull-Strominger system if it
satisfies

$\displaystyle\Lambda_{\omega}F_{A}$ $\displaystyle=0,\quad F_{A}^{0,2}=0,$
$\displaystyle\Lambda_{\omega}R_{\nabla}$ $\displaystyle=0,\quad
R_{\nabla}^{0,2}=0,$ $\displaystyle
d\left(||\Omega||_{\omega}\omega^{n-1}\right)$ $\displaystyle=0,$
$\displaystyle dd^{c}\omega-\alpha(\text{tr}R\wedge R-\text{tr}F_{h}\wedge
F_{h})$ $\displaystyle=0;$

where $\alpha$ is a non-vanishing constant, $\omega$ a hermitian metric on
$X$, $A$ is unitary connection on $(E,h)$ and $\nabla$ is a unitary connection
on $(TM,J,g)$.

This second description is useful to notice a series of necessary conditions
when $X$ is compact. Indeed, as already observed we have that $M$ has to be
necessarily balanced, and given the natural balanced class $\tau$ given by the
dilatino equation, we have that $E$ and $TM$ have to be $\tau$-polystable.
Morever, we have that $c_{1}(E)\cdot\tau=0$ and $c_{1}(M)=0$, and also

(15) $\displaystyle c_{1}(E)\cdot\tau$ $\displaystyle=0$ $\displaystyle
ch_{2}(E)$ $\displaystyle=ch_{2}(X)\in H_{BC}^{2,2}(X,\mathbb{R}),$

where $ch_{2}$ denotes the second Chern character.
Although several examples of solutions have been studied over time (several
can be found, for example, in [GF]), there is still a very poor understanding
on the existence of solutions, even on threefolds, where however it was
conjectured by Yau in [Y] that conditions (15) are not only a necessary
condition but even a sufficient one to the existence of solutions in the case
of threefolds.

If we then take our construction, and we view it in the system’s scenario, we
can make the following final remark in which we explain our ideas on how to
expand our construction in this direction.

######  Remark 4.12.

Given $\hat{\beta}$ a Chern-Ricci flat balanced metric on a Calabi-Yau
threefold $(Y,\Psi)$, it holds

$||\Psi||_{\hat{\beta}}\equiv const.,$

showing that our metric $\hat{\omega}$ gives a solution of the conformally
balanced/dilatino equation on our small resolutions $(M,\Omega)$. Thus our
construction gives us two solutions of the dilatino equation on $(M,\Omega)$,
that are $\hat{\omega}$ and
$\omega^{\prime}:=||\hat{\Omega}||_{\omega}^{-2}\omega$, where this last one
is the dilatino equation solution associated to the balanced metric $\omega$
obtained in the first part of the gluing construction. From here, thanks to
the fact that this metrics are nearby a Kähler Ricci-flat metric, our idea
(currently in devolopment in the upcoming [Gi]) is to adapt the stretegy in
[CPY1] to construct a Hermite-Einstein metric on the tangent bundle with
respect to the above metrics, and eventually from there try and extend it to a
whole solution of the Hull-Strominger system, using - for example - some
version of the approach of [AGF].

## References

  * [AB] L. Alessandrini, G. Bassanelli, Modifications of compact balanced manifolds, C. R. Acad. Sci. Paris Sér. I Math. 320, 1517-1522 (1995).
  * [AGF] B. Andreas, M Garcia-Fernandez, Solutions of the Strominger System via Stable Bundles on Calabi-Yau Threefolds, Communications in Mathematical Physics 315, 153–168 (2012)
  * [AI] B. Alexandrov, S. Ivanov, Vanishing theorems on Hermitian manifolds, Differential Geom. Appl.14, 251–265 (2001).
  * [AP] C. Arezzo, F. Pacard, _Blowing up and desingularizing constant scalar curvature Kähler manifolds_ , Acta mathematica 196.2 (2006): 179-228.
  * [BM] O. Biquard, V. Minerbe. _A Kummer construction for gravitational instantons_ , Communications in mathematical physics 308.3 (2011) 773-794.
  * [C] E. Calabi, _Extremal Kähler metrics_ , Seminar on Differential Geom., Ann. of Math. Stud. 16 (1982) 259-290.
  * [Ch] I. Cheltsov, _Factorial threefold hypersurfaces_ , J. Algebraic Geometry 19 (2010) 781-791.
  * [CO] P. Candelas, X. de la Ossa, _Comments on conifolds_ , Nuclear Phys. B 342 (1990), no. 1, 246-268.
  * [CPY1] T. C. Collins, S. Picard, S.-T. Yau, _Stability of the tangent bundle through conifold transitions_ , arXiv:2102.11170v2, to appear in Comm. Pure and Appl. Math.
  * [CPY2] T. C. Collins, S. Picard, S.-T. Yau, _The Strominger system in the square of a Kähler class_ , arXiv:2211.03784
  * [F] R. Friedman, _Simultaneous resolution of threefold double points_ , Math. Ann. 274 (1986) 671-689.
  * [Fe] T. Fei, _Some torsional local models of heterotic strings_ , Communications in Analysis and Geometry Volume 25, Number 5, 941–968, 2017.
  * [FeY] T. Fei, S.-T. Yau, Invariant solutions to the Strominger system on complex Lie groups and their quotients, Comm. Math. Phys. 338, 1183–1195 (2015)
  * [FLY] J.-X. Fu, J. Li and S.-T. Yau, _Constructing balanced metrics on some families of non- Kähler Calabi-Yau threefolds_ , J. Diff. Geom. 90 (2012) 81–129.
  * [FuY] J. Fu, S.-T. Yau, _The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation_ , J. Differential Geom. 78 (2008), no. 3, 369-428.
  * [FV] A. Fino, L. Vezzoni, _Special Hermitian metrics on compact solvmanifolds_ , J. Geom. Phys. 91 (2015), 40-53.
  * [FWW] J. Fu, Z. Wang, D. Wu, _Form-type Calabi-Yau equations_ , Math. Res. Lett. 17 (2010), no. 05, 887-903.
  * [G] P. Gauduchon, Fibré hermitiens à endomorphisme de Ricci non négativ, Bulletin de la S.M.F. 105, 113–140 (1977).
  * [Gi] F. Giusti, Hermite-Einstein metrics on non-Kähler flops, Upcoming
  * [GF] M. Garcia-Fernandez, Lectures on the Strominger system, Trav. Math. 24, 7–61 (2016).
  * [Hi] H. Hironaka, _Flattening theorems in complex analytic geometry_ , Amer. J. Math. 97 (1975) 503-547.
  * [HS] H. J. Hein, S. Sun, _Calabi-Yau manifolds with isolated conical singularities_ , Publications mathématiques de l’IHES 126.1 (2017), 73-130.
  * [Hu] C. Hull, _Superstring compactifications with torsion and space-time supersimmetry_ , In Turin 1985 Proceedings "Superunification and Extra Dimensions" (1986) 347-375.
  * [J] D. D. Joyce, _Compact manifolds with special holonomy_ , Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.
  * [LY1] J. Li, S.-T. Yau, _The existence of supersymmetric string theory with torsion_ , J. Diff. Geom. 70 (2005) 143-181.
  * [LY2] J. Li, S.-T. Yau, _Hermitian-Yang-Mills connections on non-Kähler manifiolds_ , Mathematical aspects of string theory (San Diego, Calif., 1986) 560-573, Adv. Ser. Math. Phys., 1, Worlds Sci. Publishing.
  * [LY3] J. Li, S.-T. Yau, _The existence of supersymmetric string theory with torsion_ , J. Differential Geom. 70 (2005), no. 1, 143–181.
  * [M] M.L. Michelsohn, On the existence of special metrics in complex geometry, Acta Math. 149, 261–295 (1982).
  * [NS] Y. Namikawa, J. H. M. Steenbrink. _Global smoothing of Calabi-Yau threefolds_ , Inventiones Mathematicae 122 (1995) 403-419.
  * [P] D.H. Phong, _Moduli in geometry and physics, in Geometry and Physics in Cracow_ , Acta Phys. Polon. B Proc. Suppl. 4 (2011), no.3, 351–378.
  * [PPZ] D. H. Phong, S. Picard, and X. Zhang. _Geometric flows and Strominger systems_ , Mathematische Zeitschrift 288.1 (2018): 101-113.
  * [R] M. Reid, The moduli space of 3-folds with $K=0$ may nevertheless be irreducible, Math. Ann. 278 (1987), no. 1-4, 329-334.
  * [S] A. Strominger, _Superstrings with torsion_ , Nucl. Phys. B 274 (2) (1986) 253-284.
  * [STW] G. Székelyhidi, V. Tosatti, B. Weinkove, _Gauduchon metrics with prescribed volume form_ , Acta Math., 219 (2017), 181-211.
  * [Sz] G. Székelyhidi, _An introduction to extremal Kähler metrics_ , Graduate Studies in Mathematics. Vol. 152. American Mathematical Soc. (2014).
  * [TW] V. Tosatti, B. Weinkove, _The complex Monge-Ampère equation on compact Hermitian manifolds_ , J. Amer. Math. Soc. 23 (2010), no.4, 1187–1195.
  * [TY] L.-S. Tseng, S.-T. Yau, _Non-Kähler Calabi-Yau manifolds_ , in String-Math 2011, 241– 254, Proc. Sympos. Pure Math., 85, Amer. Math. Soc., Providence, RI, 2012.
  * [W] J. Werner _Kleine Auflösungen spezieller dreidimensionaler Varietäten_ , Bonn, Univ., Diss., 1987 (Nicht f.d. Austausch).
  * [Y] S.-T. Yau, _Metrics on complex manifolds_ , Science in China Series A Mathematics 53 (2010) 565-572.