# Continuity of HYM connections with respect to metric variations

Rémi Delloque

###### Abstract

We investigate the set of (real Dolbeault classes of) balanced metrics
$\Theta$ on a balanced manifold $X$ with respect to which a torsion-free
coherent sheaf $\mathcal{E}$ on $X$ is slope stable. We prove that the set of
all such $[\Theta]\in H^{n-1,n-1}(X,\mathbb{R})$ is a convex cone defined by a
finite number of linear inequalities. For this, we prove that the set of all
$c_{1}(\mathcal{S})$ for $\mathcal{S}\subset\mathcal{E}$ saturated and
coherent is finite.

When $\mathcal{E}$ is a Hermitian vector bundle, the Kobayashi–Hitchin
correspondence provides associated Hermitian Yang–Mills connections, which we
show depend continuously on the metric, even around classes with respect to
which $\mathcal{E}$ is only semi-stable. In this case, the holomorphic
structure induced by the connection is the holomorphic structure of the
associated graded object. The method relies on semi-stable perturbation
techniques for geometric PDEs with a moment map interpretation and is quite
versatile, and we hope that it can be used in other similar problems.

##  1 Introduction

The famous Kobayashi–Hitchin correspondence [13, 17, 5, 27] gives the
equivalence between the existence of Hermitian Yang–Mills (HYM) connections on
a vector bundle $\mathcal{E}\rightarrow X$ and slope stability [19, 26]. HYM
connections correspond to solutions of a differential equation while slope
stability is a purely algebraic notion. These tools are used to construct and
study the moduli space of vector bundles on a given complex manifold.

Both of these notions depend on the balanced metric on the balanced manifold
$X$ (balanced metrics are defined below). It is a natural question to ask how
$\mathcal{E}$ may become stable or unstable when this metric varies, and to
study the behaviour of the associated HYM connections. It is easy to see that
stability is an open condition in the space of balanced metrics and that we
can locally build a smooth family of associated HYM connections around a
metric $\Theta_{0}$ with respect to which $\mathcal{E}$ is stable. However,
when $\mathcal{E}$ is only semi-stable with respect to $\Theta_{0}$, problems
occur and wall-crossing phenomena appear. The method used here is inspired
from [25, 4, 3], and relies on semi-stable perturbation techniques for
geometric PDEs with a moment map interpretation.

Let $g$ be a metric on $X$ and $\omega$ be its Kähler form. Then, by
definition, $X$ is Kähler if and only if $d\omega=0$. However, slope stability
of $\mathcal{E}$ is determined by the inequalities
$\frac{c_{1}(\mathcal{S})\cup[\omega^{n-1}]}{\mathrm{rk}(\mathcal{S})}<\frac{c_{1}(\mathcal{E})\cup[\omega^{n-1}]}{\mathrm{rk}(\mathcal{E})}$
($n=\dim(X)$) for all sub-sheaves $\mathcal{S}$ of $\mathcal{E}$ with
$0<\mathrm{rk}(\mathcal{S})<\mathrm{rk}(\mathcal{E})$. We see that these
inequalities are linear in $\omega^{n-1}$ and $\omega$ is not required to be
closed, only $\omega^{n-1}$ is. When $\omega^{n-1}$ is closed (but not
necessarily $\omega$), we say that $X$ is balanced and this is enough to get
the Kobayashi–Hitchin correspondence. This is inspired from the idea of Greb,
Ross and Toma in [9] where wall-crossing phenomena are studied with respect to
multipolarisations to make the equations linear instead of polynomial. We
recall general definitions and results about balanced manifolds in Section 2.

From here, the inequalities that define slope stability are linear in
$\Theta=\frac{\omega^{n-1}}{(n-1)!}$. As in the Kähler case, we define the
balanced cone as the set of all real Dolbeault classes $[\Theta]\in
H^{n-1,n-1}(X,\mathbb{R})=H^{n-1,n-1}(X,\mathbb{C})\cap
H^{2n-1}(X,\mathbb{R})$ of positive $(n-1,n-1)$ forms $\Theta$. In Section 3,
we prove that there is only a finite number of these inequalities to verify in
order to obtain all of them. This is a consequence of the following result
(Corollary 3.6 below).

######  Proposition 1.

Let $M$ be a compact complex manifold and $\mathcal{E}\rightarrow M$ a
torsion-free coherent sheaf. Then, the set of all $c_{1}(\mathcal{S})$ with
$\mathcal{S}\subset\mathcal{E}$ saturated and coherent is finite with a
cardinality bounded by $2^{\mathrm{rk}(\mathcal{E})}$.

In particular, it gives the structure of the set of metrics with respect to
which $\mathcal{E}$ is stable (resp. semi-stable).

######  Theorem 2.

The set $\mathcal{C}_{s}^{\mathrm{D}}(\mathcal{E})$ (resp.
$\mathcal{C}_{ss}^{\mathrm{D}}(\mathcal{E})$) of balanced classes of metrics
$[\Theta]$ with respect to which $\mathcal{E}$ is stable (resp. semi-stable)
is the intersection of the balanced cone and an open (resp. closed) convex
polyhedral cone defined by at most $2^{\mathrm{rk}(\mathcal{E})}-2$
inequalities.

A more precise version of this theorem with topological properties is given by
Theorem 3.7 below.

Let $\mathcal{E}=(E,\overline{\partial}_{E},h)$ be a Hermitian holomorphic
vector bundle. When $\mathcal{E}$ is $[\Theta]$-stable (i.e. stable with
respect to $\Theta$), the Kobayashi–Hitchin correspondence tells us that we
can find a $\Theta$-HYM connection $\nabla=\partial+\overline{\partial}$ (i.e.
HYM with respect to $\Theta$) such that $(E,\overline{\partial})$ is
biholomorphic to $\mathcal{E}$. In other words, the holomorphic structure is
preserved when we replace $\overline{\partial}_{E}$ by $\overline{\partial}$.
Concretely, it is characterised by the fact that $\overline{\partial}$ is in
the $\mathcal{G}^{\mathbb{C}}$-orbit of $\overline{\partial}_{E}$, where
$\mathcal{G}^{\mathbb{C}}$ is the complex gauge group, whose action on complex
structures is defined below.

In Section 4, we investigate the behaviour of the HYM connections when
$\Theta$ approaches a metric $\Theta_{0}$ with respect to which $\mathcal{E}$
is only semi-stable. When the graded object of $\mathcal{E}$ with respect to
$\Theta_{0}$, namely $\mathrm{Gr}_{[\Theta_{0}]}(\mathcal{E})$, is locally
free, we show a convergence result when $\Theta\rightarrow\Theta_{0}$ (Theorem
4.7 below).

######  Theorem 3.

For all $\Theta_{\varepsilon}=\Theta_{0}+\varepsilon$ $L_{1}^{2}$-close to
$\Theta_{0}$ such that $\mathcal{E}$ is $[\Theta_{\varepsilon}]$-semi-stable,
there is an associated connection
$\nabla_{\varepsilon}=\partial_{\varepsilon}+\overline{\partial}_{\varepsilon}$
that has the three following properties,

  1. 1.

$\nabla_{\varepsilon}$ is HYM with respect to $\Theta_{\varepsilon}$.

  2. 2.

$(E,\overline{\partial}_{\varepsilon})$ is biholomorphic to
$\mathrm{Gr}_{[\Theta_{\varepsilon}]}(\mathcal{E})$, which is locally free. In
particular, if $\mathcal{E}$ is $\Theta_{\varepsilon}$-stable, then
$(E,\overline{\partial}_{\varepsilon})\cong\mathcal{E}$.

  3. 3.

For all integer $d\geqslant 2$, the bound
$\left\|\overline{\partial}_{\varepsilon}-\overline{\partial}_{0}\right\|_{L_{d}^{2}}=\mathrm{O}\left(\left\|\varepsilon\right\|_{L_{d+1}^{2}}+\sqrt{\left\|\varepsilon\right\|_{L_{1}^{2}}}\right)$
holds, where $\left\|\cdot\right\|_{L_{d}^{2}}$ is an $L_{d}^{2}$ Sobolev
norm. Therefore, the same bound holds for
$\left\|\partial_{\varepsilon}-\partial_{0}\right\|_{L_{d}^{2}}$ and
$\left\|\nabla_{\varepsilon}-\nabla_{0}\right\|_{L_{d}^{2}}$. In particular,
$\nabla_{\varepsilon}\rightarrow\nabla_{0}$ for the $\mathcal{C}^{\infty}$
topologies.

The HYM connection, when it exists in some gauge orbit, is not unique but it
is exactly one $\mathcal{G}$-orbit where $\mathcal{G}$ is the group of smooth
sections $u$ of endomorphisms of $E$ which verify $uu^{*}=\mathrm{Id}_{E}$. A
consequence of the previous theorem is the following one (Theorem 4.10 below).

######  Theorem 4.

The function that maps each balanced metric $\Theta$ with respect to which
$\mathcal{E}$ is semi-stable and sufficiently smooth to the class modulo
$\mathcal{G}$ of the $\Theta$-HYM connections
$\nabla=\partial+\overline{\partial}$ such that
$(E,\overline{\partial})\cong\mathrm{Gr}_{[\Theta]}(\mathcal{E})$, is a
continuous function with respect to the $\mathcal{C}^{\infty}$ topologies.

Theorem 2 is a generalisation of [3, Proposition 1.1] and Theorem 3 is a
generalisation of [3, Theorem 1.2], where in both cases, we remove the
hypothesis that the group of automorphisms of the graded object is abelian.
Theorem 4 is a natural consequence of Theorem 3 and may be useful to study the
variations of the moduli space of vector bundles when the metric of the
manifold varies. Proposition 1 may also be more generally useful to study
wall-crossing phenomena involving the slope stability. For example, [20,
Theorem 4.1.9] could be generalised to a more general situation.

The method used in this article are classical semi-stable perturbation
techniques. When $\mathcal{E}$ is a $[\Theta_{0}]$-semi-stable vector bundle,
it can be viewed as a small deformation of its graded object
$\mathrm{Gr}(\mathcal{E})$. Then, by Kuranishi theory [14], all the small
deformations of $\mathrm{Gr}(\mathcal{E})$ belong, up to a gauge
transformation, in the Kuranishi slice which is a germ of finite dimensional
complex manifold, on which the group $G$ of automorphisms of
$\mathrm{Gr}(\mathcal{E})$ acts naturally. Moreover, for all $\Theta$
$L_{1}^{2}$-close to $[\Theta_{0}]$, there is a Kähler form on this germ
depending of $\Theta$ such that the action of $G$ is hamiltonian with respect
to this form. Additionally, the zeroes of the associated moment map correspond
to connections $\nabla$ such that the contraction of the curvature with
respect to $\Theta$ is orthogonal to the Lie algebra $\mathfrak{k}$ of the
maximal compact subgroup of $G$. We want this contraction to be a constant
homothety in order to obtain a HYM connection.

From here, there are two methods. The first one is to find a zero of this
moment map using GIT theory, and then perturb the induced connection to obtain
a new connection whose contraction of the curvature is a constant homothety.
This method is used in [4] in the case of $Z$-critical connections for
example. The second method is to first perturb the Kuranishi slice so the
holomorphic structures in the perturbed slice induce connections whose
contraction of the curvature belongs to $\mathfrak{k}$. Then, we find a zero
of the associated perturbed moment map, whose induced connection is directly
HYM. The main issue with this second method is that the perturbed slice is no
more a germ of complex manifold, it is only real symplectic. Nevertheless,
this second method is used in [22] in the context of $K$-stability and in [3]
with HYM equations. It is also used in [21] together with the moment map flow
in the context of constant scalar curvature metrics.

#### Acknowledgement

I thank my PhD supervisor Carl Tipler for pointing out this problem, giving me
the idea of the perturbed Kuranishi slice and for his feedbacks on the paper.
I also thank Lars Martin Sektnan for his feedbacks on this paper and the talk
we had about it and about his work with Annamaria Ortu on constant scalar
curvature metrics.

##  2 Generalities about balanced metrics and their Laplace-Beltrami operators

###  2.1 Operations on compact complex manifolds

Let $(X,g)$ be a Hermitian compact complex manifold of dimension $n\geqslant
2$ and let $\omega$ be its Kähler form (not necessarily closed). Then
$\Theta=\frac{\omega^{n-1}}{(n-1)!}$ is a positive $(n-1,n-1)$ form in the
following sens. In a local frame $(z_{1},\ldots,z_{n})$, we can define
$\widehat{dz_{i}\wedge d\overline{z_{j}}}$ for each $1\leqslant i,j\leqslant
n$ as $dz_{1}\wedge d\overline{z_{1}}\wedge\cdots\wedge dz_{n}\wedge
d\overline{z_{n}}$ where we removed the terms $dz_{i}$ and
$d\overline{z_{j}}$. Then, the local expression of $\Theta$ has the form,

$\Theta=\sum_{1\leqslant i,j\leqslant n}\Theta_{ij}\widehat{dz_{i}\wedge
d\overline{z_{j}}},$

Positivity of $\Theta$ means that the matrix formed of the $\Theta_{ij}$ is
Hermitian positive definite at each point, and this does not depend on the
choice of the local frame. By [18, Equation (4.8)], $\omega$ (thus $g$) is
entirely determined by $\Theta$ and depends smoothly on it. In other words,
any positive $(n-1,n-1)$ form $\Theta$ gives rise to a metric $g$ on $X$ which
depends smoothly on $\Theta$. Let
$\mathrm{Vol}_{\Theta}=\frac{\omega^{n}}{n!}$ be the natural associated volume
form. For all $\Theta>0$, we can define the trace operator $\Lambda_{\Theta}$
on complex forms as $\star^{-1}\circ(\omega\wedge\cdot)\circ\star$ where
$\star$ is the Hodge star. In particular, on $(1,1)$ forms $\alpha$, it is
characterised by the equality,

$(\Lambda_{\Theta}\alpha)\mathrm{Vol}_{\Theta}=\alpha\wedge\Theta.$

This operator naturally extends to forms with values in a complex vector
bundle $E$ on $X$. Let $(E,h)$ be a complex Hermitian vector bundle on $X$.
Following for example [10, Definition 4.1.11], let us recall the natural
Hermitian product on $\Omega^{p,q}(X,E)$ with respect to the metric $\Theta$,

$\left<\alpha,\beta\right>_{\Theta}=\int_{X}(\alpha|\beta)_{\Theta,h}\mathrm{Vol}_{\Theta},$
(1)

where $(\cdot|\cdot)_{\Theta,h}$ is the natural Hermitian product on
$\Lambda^{p,q}T^{*}X\otimes E$ depending on $\Theta$ and $h$, and we see
$\alpha$ and $\beta$ as sections of this bundle. It makes the decomposition
$\Omega^{*}(X,E)=\bigoplus_{0\leqslant p,q\leqslant n}\Omega^{p,q}(X,E)$
orthogonal. In particular, on sections $\xi,\eta$ and on $1$-forms
$\alpha,\beta$, we have,

$\left<\xi,\eta\right>_{\Theta}=\int_{X}h(\xi,\eta)\mathrm{Vol}_{\Theta},\qquad\left<\alpha,\beta\right>_{\Theta}=\int_{X}h(\alpha,J\beta)\wedge\Theta.$

Here, $J$ is the natural complex structure on $\Omega^{*}(X,E)$ given by
$J=\boldsymbol{\mathrm{i}}^{p-q}$ on $(p,q)$-forms. It enables us to consider
adjoints of operators with respect to these Hermitian products. Let
$\nabla=\partial+\overline{\partial}$ be an integrable unitary connection on
$E$ (i.e. $\overline{\partial}^{2}=0$ and $\nabla$ is the Chern connection
associated to $(\overline{\partial},h)$). Then, we can define the Laplacian
operators with respect to $\Theta$ and $\nabla$,

$\Delta_{\Theta,\partial}=\partial\partial^{*}+\partial^{*}\partial,\qquad\Delta_{\Theta,\overline{\partial}}=\overline{\partial}\overline{\partial}^{*}+\overline{\partial}^{*}\overline{\partial},\qquad\Delta_{\Theta,\nabla}=\nabla\nabla^{*}+\nabla^{*}\nabla,$

The second one is called the Dolbeault Laplacian and the third one the
Laplace-Dolbeault Laplacian. These three operators are real self-adjoint and
elliptic of order $2$. See for example [11, Chapter VI, Section 5.4].

###  2.2 When $X$ is balanced

The Hermitian metric $g$ on $X$ is said to be a balanced metric if the
associated Kähler form $\omega$ is such that $\omega^{n-1}$ is closed.
Balanced metrics where introduced by Michelsohn in [18], see [6] for a
reference. Gauduchon showed that $\Theta=\frac{\omega^{n-1}}{(n-1)!}$ is
closed (i.e. $\omega$ is balanced) if and only if $\omega$ is co-closed [7,
Proposition 1]. We also call abusively $\Theta$ a balanced metric in this
case. When such a metric exists, we say that $X$ is balanced. A
characterisation of balanced manifolds is given by [18, Theorem A]. Similarly
to the Kähler cone, we can introduce the balanced cone,

$\mathcal{B}_{X}=\\{[\Theta]\in H^{n-1,n-1}(X,\mathbb{R})|\Theta\textrm{
positive $(n-1,n-1)$-form}\\},$

where the $H^{p,p}(X,\mathbb{R})=H^{p,p}(X,\mathbb{C})\cap
H^{2p}(X,\mathbb{R})$ are the real Dolbeault cohomology groups.

######  Remark.

In the literature about balanced manifolds, the balanced cone usually contains
elements of the Bott-Chern cohomology groups instead of the real Dolbeault
ones, where the boundary morphisms are
$\boldsymbol{\mathrm{i}}\partial\overline{\partial}$ instead of $d$. Here, we
only need to work modulo $d$. Since
$\partial\overline{\partial}=\frac{1}{2}(\partial-\overline{\partial})d$, the
real Dolbeault cohomology groups are naturally quotients of the Bott-Chern
ones. Therefore, this article’s balanced cone can be seen as a projection of
the classical balanced cone on a quotient space.

From now on, we assume that $\Theta$ is balanced.

######  Lemma 2.1.

The following Kähler identities hold on the space $\Omega^{1}(X,E)$,

$\displaystyle\Lambda_{\Theta}\partial$
$\displaystyle=\boldsymbol{\mathrm{i}}\overline{\partial}^{*},$
$\displaystyle\Lambda_{\Theta}\overline{\partial}$
$\displaystyle=-\boldsymbol{\mathrm{i}}\partial^{*},$

and the following hold on the space of smooth sections $\Omega^{0}(X,E)$,

$\displaystyle\Delta_{\Theta,\partial}$
$\displaystyle=\boldsymbol{\mathrm{i}}\Lambda_{\Theta}\overline{\partial}\partial,$
$\displaystyle\Delta_{\Theta,\overline{\partial}}$
$\displaystyle=-\boldsymbol{\mathrm{i}}\Lambda_{\Theta}\partial\overline{\partial},$
$\displaystyle\Delta_{\Theta,\partial}+\Delta_{\Theta,\overline{\partial}}$
$\displaystyle=\Delta_{\Theta,\nabla}=\boldsymbol{\mathrm{i}}\Lambda_{\Theta}(\overline{\partial}\partial-\partial\overline{\partial}),$
$\displaystyle\Delta_{\Theta,\partial}-\Delta_{\Theta,\overline{\partial}}$
$\displaystyle=\Lambda_{\Theta}\boldsymbol{\mathrm{i}}F_{\nabla},$

where
$F_{\nabla}=\nabla^{2}=\overline{\partial}\partial+\partial\overline{\partial}$
is the curvature form of $\nabla$.

###### Proof.

The proof is the same as the standard proof of Kähler identities, but only on
sections and $1$-forms. Indeed, if $\alpha$ is a $(0,1)$ form and $s$ a
$0$-form with values in $E$,

$\displaystyle\left<\Lambda_{\Theta}\overline{\partial}\alpha,s\right>_{\Theta}$
$\displaystyle=\int_{X}h(\Lambda_{\Theta}\overline{\partial}\alpha,s)\mathrm{Vol}_{\Theta}$
$\displaystyle=\int_{X}h(\overline{\partial}\alpha,s)\wedge\Theta$
$\displaystyle=\int_{X}dh(\alpha,s)\wedge\Theta+\int_{X}h(\alpha,\partial
s)\wedge\Theta$
$\displaystyle=\int_{X}h(\alpha,-J\boldsymbol{\mathrm{i}}\partial
s)\wedge\Theta\textrm{ because }d\Theta=0,$
$\displaystyle=\left<\alpha,-\boldsymbol{\mathrm{i}}\partial
s\right>_{\Theta}.$

We obtain the second identity. The first one is similar. We deduce that, on
smooth sections,

$\Delta_{\Theta,\partial}=\partial\partial^{*}+\partial^{*}\partial=\partial^{*}\partial=\boldsymbol{\mathrm{i}}\Lambda_{\Theta}\overline{\partial}\partial,$

and similarly,

$\Delta_{\Theta,\overline{\partial}}=\overline{\partial}\overline{\partial}^{*}+\overline{\partial}^{*}\overline{\partial}=\overline{\partial}^{*}\overline{\partial}=-\boldsymbol{\mathrm{i}}\Lambda_{\Theta}\partial\overline{\partial}.$

The equality
$\Delta_{\Theta,\nabla}=\Delta_{\Theta,\partial}+\Delta_{\Theta,\overline{\partial}}$
always hold by definition even if we don’t assume $d\Theta=0$. The third
equality involving Laplacians is the sum of the two first and the fourth one
is their difference. ∎

######  Proposition 2.2.

On smooth sections, $\Delta_{\Theta,\nabla}$ verifies
$\ker(\Delta_{\Theta,\nabla})=\ker(\nabla)$ is the space of $\nabla$-parallel
sections and if $\Lambda_{\Theta}\boldsymbol{\mathrm{i}}F_{\nabla}=0$, then
$\ker(\Delta_{\Theta,\nabla})=\ker(\overline{\partial})$ is the space of
$\overline{\partial}$-holomorphic sections.

###### Proof.

The first equality is a direct consequence of the fact that
$\Delta_{\Theta,\nabla}=\nabla^{*}\nabla$ on smooth forms. The second one
comes from Lemma 2.1, which implies that
$\Delta_{\Theta,\nabla}=\Delta_{\Theta,\partial}+\Delta_{\Theta,\overline{\partial}}$
and if $\Lambda_{\Theta}\boldsymbol{\mathrm{i}}F_{\nabla}=0$,
$\Delta_{\Theta,\partial}=\Delta_{\Theta,\overline{\partial}}$ hence
$\Delta_{\Theta,\nabla}=2\Delta_{\Theta,\overline{\partial}}=2\overline{\partial}^{*}\overline{\partial}$.
∎

By abuse of notation, we still call $\nabla=\partial+\overline{\partial}$ the
natural connection on $\mathrm{End}(E)$ that arises from $\nabla$ on $E$.

###  2.3 Gauge action and its derivative

Let us to introduce the complex gauge group $\mathcal{G}^{\mathbb{C}}$ of
smooth sections of $\mathrm{End}(E)$ which are isomorphisms on each fibre. It
is an infinite dimensional complex Lie group and
$\mathrm{Lie}(\mathcal{G}^{\mathbb{C}})=\Omega^{0}(X,\mathrm{End}(E))$. This
group acts on connections on $E$ with,

$f\cdot\partial=f^{-1*}\circ\partial\circ f^{*},\quad
f\cdot\overline{\partial}=f\circ\overline{\partial}\circ f^{-1},\quad
f\cdot\nabla=f\cdot\partial+f\cdot\overline{\partial}.$

The adjoints are computed with respect to the metric $h$. Notice that this
action preserves the integrability condition of the Dolbeault operator
$\overline{\partial}^{2}=0$ and it preserves the compatibility with the metric
$h$ of the connection. When
$\overline{\partial}_{1}=f\cdot\overline{\partial}_{2}$, the holomorphic
vector bundles $\mathcal{E}_{1}=(E,\overline{\partial}_{1})$ and
$\mathcal{E}_{2}=(E,\overline{\partial}_{2})$ are isomorphic because
$f:\mathcal{E}_{2}\rightarrow\mathcal{E}_{1}$ is a biholomorphism. Conversely,
such an isomorphism gives rise to a gauge equivalence between the Dolbeault
operators of $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$.

Similarly, we call $\mathcal{G}\subset\mathcal{G}^{\mathbb{C}}$ the unitary
gauge group defined as the set of all $u\in\mathcal{G}^{\mathbb{C}}$ such that
$uu^{*}=\mathrm{Id}_{E}$. It is an infinite dimensional real Lie group with
$\mathrm{Lie}(\mathcal{G})=\boldsymbol{\mathrm{i}}\Omega^{0}(X,\mathrm{End}_{H}(E,h))$
where $\mathrm{End}_{H}(E,h)\subset\mathrm{End}(E)$ is the real smooth bundle
of Hermitian endomorphisms with respect to $h$. Notice that the action of
$\mathcal{G}$ on connections is given by $u\cdot\nabla=u\circ\nabla\circ
u^{*}$, thus the associated curvature verifies
$F_{u\cdot\nabla}=uF_{\nabla}u^{*}$.

We will be interested in finding a Dolbeault operator $\overline{\partial}$ on
$E$ which satisfy the property of being Hermitian Yang–Mills (see Definition
3.8 in Section 3) such that $(E,\overline{\partial})$ is isomorphic to a given
holomorphic vector bundle $\mathcal{E}=(E,\overline{\partial}_{E})$.
Therefore, we shall search $\overline{\partial}$ in the
$\mathcal{G}^{\mathbb{C}}$-orbit of $\overline{\partial}_{E}$. For this, it
will be useful to compute infinitesimal variations of $f\mapsto
f\cdot\overline{\partial}_{E}$.

######  Proposition 2.3.

For any $\nabla$, the map

$\Psi:\left\\{\begin{array}[]{rcl}\displaystyle\Omega^{0}(X,\mathrm{End}_{H}(E,h))&\rightarrow&\displaystyle\Omega^{0}(X,\mathrm{End}_{H}(E,h))\\\
\displaystyle
s&\mapsto&\displaystyle\Lambda_{\Theta}\boldsymbol{\mathrm{i}}F_{\boldsymbol{\mathrm{e}}^{s}\cdot\nabla}\end{array}\right.,$

verifies $d\Psi(0)=\Delta_{\Theta,\nabla}$ (on $\mathrm{End}(E)$). Notice that
since $\Delta_{\Theta,\nabla}$ is a real operator, it preserves
$\mathrm{End}_{H}(E,h)$.

###### Proof.

We have for all smooth Hermitian sections $s$ of $\mathrm{End}(E)$,

$\boldsymbol{\mathrm{e}}^{s}\cdot\nabla=\nabla+\boldsymbol{\mathrm{e}}^{s}\overline{\partial}(\boldsymbol{\mathrm{e}}^{-s})+\boldsymbol{\mathrm{e}}^{-s}\partial(\boldsymbol{\mathrm{e}}^{s}),$

hence

$F_{\boldsymbol{\mathrm{e}}^{s}\cdot\nabla}=F_{\nabla}+\nabla\alpha_{s}+\alpha_{s}\wedge\alpha_{s},$

where
$\alpha_{s}=\boldsymbol{\mathrm{e}}^{s}\overline{\partial}(\boldsymbol{\mathrm{e}}^{-s})+\boldsymbol{\mathrm{e}}^{-s}\partial(\boldsymbol{\mathrm{e}}^{s})\in\boldsymbol{\mathrm{i}}\Omega^{1}(X,\mathrm{End}_{H}(E,h))$.
We have $\alpha_{0}=0$ and if $v\in
T_{s}\Omega^{0}(X,\mathrm{End}_{H}(E,h))=\Omega^{0}(X,\mathrm{End}_{H}(E,h))$,

$\frac{\partial}{\partial
s}|_{s=0}(\boldsymbol{\mathrm{e}}^{s}\overline{\partial}(\boldsymbol{\mathrm{e}}^{-s}))\cdot
v=\frac{\partial}{\partial s}|_{s=0}(\boldsymbol{\mathrm{e}}^{s})\cdot
v\overline{\partial}(\boldsymbol{\mathrm{e}}^{-0})+\boldsymbol{\mathrm{e}}^{0}\overline{\partial}\left(\frac{\partial}{\partial
s}|_{s=0}(\boldsymbol{\mathrm{e}}^{-s})\cdot v\right)=-\overline{\partial}v.$

Hence $\frac{\partial}{\partial s}|_{s=0}\alpha_{s}\cdot v=\partial
v-\overline{\partial}v$ symmetrically. Since $\alpha_{s}\wedge\alpha_{s}$ is
quadratic and $\alpha_{0}=0$, its derivative at $0$ vanishes thus

$\frac{\partial}{\partial
s}|_{s=0}\boldsymbol{\mathrm{i}}F_{\boldsymbol{\mathrm{e}}^{s}\cdot\nabla}=\boldsymbol{\mathrm{i}}\nabla(\partial
v-\overline{\partial}v)=\boldsymbol{\mathrm{i}}(\overline{\partial}\partial-\partial\overline{\partial})v.$

After applying $\Lambda_{\Theta}$, by Lemma 2.1, we deduce that
$d\Psi(0)=\Delta_{\Theta,\nabla}$. ∎

##  3 Hermitian Yang–Mills equation and finiteness results about slope
stability

###  3.1 Slope stability

Any holomorphic vector bundle can be seen as the locally free coherent sheaf
of its holomorphic sections. In this subsection, we study more generally
coherent sheaves on $X$ and their slope stablity with respect to a positive
real Dolbeault class $[\Theta]$.

######  Definition 3.1.

We define the $[\Theta]$-slope $\mu_{[\Theta]}$ of a non-zero torsion-free
coherent sheaf $\mathcal{E}$ as

$\mu_{[\Theta]}(\mathcal{E})=\frac{c_{1}(\mathcal{E})\cup[\Theta]}{\mathrm{rk}(\mathcal{E})}.$

Following Mumford [19], we say that,

  * •

$\mathcal{E}$ is $[\Theta]$-stable if for all coherent sub-sheaf
$\mathcal{S}\subset\mathcal{E}$ with
$0<\mathrm{rk}(\mathcal{S})<\mathrm{rk}(\mathcal{E})$,
$\mu_{[\Theta]}(\mathcal{S})<\mu_{[\Theta]}(\mathcal{E})$,

  * •

$\mathcal{E}$ is $[\Theta]$-semi-stable if for all coherent sub-sheaf
$\mathcal{S}\subset\mathcal{E}$ with
$0<\mathrm{rk}(\mathcal{S})<\mathrm{rk}(\mathcal{E})$,
$\mu_{[\Theta]}(\mathcal{S})\leqslant\mu_{[\Theta]}(\mathcal{E})$,

  * •

$\mathcal{E}$ is $[\Theta]$-polystable if $\mathcal{E}$ is a direct sum of
stable coherent sheaves of the same $[\Theta]$-slope.

Clearly, stability implies polystability. When $\mathcal{E}$ can be written as
a direct sum $\bigoplus_{i=1}^{m}\mathcal{E}_{i}$ (or more generally as an
extension of the $\mathcal{E}_{i}$), then
$c_{1}(\mathcal{E})=\sum_{i=1}^{m}c_{1}(\mathcal{E}_{i})$ thus
$\mu_{[\Theta]}(\mathcal{E})$ is a weighted average of the
$\mu_{[\Theta]}(\mathcal{E}_{i})$ where the weights are the
$\mathrm{rk}(\mathcal{E}_{i})$. Therefore, polystability implies semi-
stability.

We say that a coherent sub-sheaf $\mathcal{S}\subset\mathcal{E}$ is saturated
in $\mathcal{E}$ when $\mathcal{E}/\mathcal{S}$ is torsion-free. It is enough
in order to get stability (resp. semi-stability) to verify the above slope
inequalities with all saturated coherent sub-sheaves
$0\subsetneq\mathcal{S}\subsetneq\mathcal{E}$ by [12, Proposition 5.7.6]. It
is proven for Kähler manifolds but the proof is the same in the balanced case.
It relies on the fact that torsion coherent sheaves have a non-negative first
Chern class.

In the case where $\mathcal{E}$ is only $[\Theta]$-semi-stable, we may use a
Jordan-Hölder filtration to reduce to the polystable case.

######  Proposition 3.2.

If $\mathcal{E}$ is a $[\Theta]$-semi-stable vector bundle, then it admits a
Jordan-Hölder filtration, i.e. a filtration by coherent sub-sheaves
$0=\mathcal{F}_{0}\subsetneq\mathcal{F}_{1}\subsetneq\cdots\subsetneq\mathcal{F}_{m}=\mathcal{E}$
such that for all $k$,
$\mu_{[\Theta]}(\mathcal{F}_{k})=\mu_{[\Theta]}(\mathcal{E})$ and each
$\mathcal{G}_{k}=\mathcal{F}_{k}/\mathcal{F}_{k-1}$ is torsion-free and
$[\Theta]$-stable. Moreover, the graded object,

$\mathrm{Gr}_{[\Theta]}(\mathcal{E})=\bigoplus_{k=1}^{m}\mathcal{G}_{k},$

is unique up to isomorphism.

###### Proof.

This is a standard categorical result. See for example [24, Lemma 12.9.7]. ∎

When $\mathcal{E}$ is locally free i.e. a vector bundle, it is said to be
sufficiently smooth if its graded object $\mathrm{Gr}_{[\Theta]}(\mathcal{E})$
is locally free too, or equivalently, each $\mathcal{G}_{i}$ is locally free.

######  Remark.

When $F\subset E$ are smooth vector bundles, a Hermitian metric $h$ on $E$
gives rise to a diffeomorphism between $E/F$ and $F^{\bot}$. In other words,
any exact sequence of smooth vector bundles splits. In particular,
$\mathcal{E}$ and $\mathrm{Gr}_{[\Theta]}(\mathcal{E})$ have the same smooth
structure when $\mathcal{E}$ is a sufficiently smooth vector bundle.

######  Remark.

$\mathcal{E}$ is $[\Theta]$-polystable if and only if it is $[\Theta]$-semi-
stable and $\mathrm{Gr}_{[\Theta]}(\mathcal{E})\cong\mathcal{E}$.

###  3.2 Stable and semi-stable polyhedral cones

Let $\mathcal{E}$ be a torsion-free coherent sheaf. We are interested here in
the structure of the set of $\Theta$ such that $\mathcal{E}$ is
$[\Theta]$-stable (resp. semi-stable). Let us introduce,

$\mathcal{C}_{s}(\mathcal{E})=\\{\Theta>0|d\Theta=0\textrm{ and $\mathcal{E}$
is
$[\Theta]$-stable}\\},\qquad\mathcal{C}_{ss}(\mathcal{E})=\\{\Theta>0|d\Theta=0\textrm{
and $\mathcal{E}$ is $[\Theta]$-semi-stable}\\},$

and their real Dolbeault projection counterparts,

$\mathcal{C}_{s}^{\mathrm{D}}(\mathcal{E})=\\{[\Theta]|\Theta\in\mathcal{C}_{s}(\mathcal{E})\\}=\\{[\Theta]\in\mathcal{B}_{X}|\mathcal{E}\textrm{
is $[\Theta]$-stable}\\},$
$\mathcal{C}_{ss}^{\mathrm{D}}(\mathcal{E})=\\{[\Theta]|\Theta\in\mathcal{C}_{ss}(\mathcal{E})\\}=\\{[\Theta]\in\mathcal{B}_{X}|\mathcal{E}\textrm{
is $[\Theta]$-semi-stable}\\}.$

When $0\subsetneq\mathcal{S}\subsetneq\mathcal{E}$ is a saturated coherent
sub-sheaf, let,

$l_{\mathcal{S}}:\left\\{\begin{array}[]{rcl}\displaystyle
H^{n-1,n-1}(X,\mathbb{R})&\rightarrow&\displaystyle\mathbb{R}\\\
\displaystyle[\Theta]&\mapsto&\displaystyle\mu_{[\Theta]}(\mathcal{E}/\mathcal{S})-\mu_{[\Theta]}(\mathcal{S})=\left(\frac{c_{1}(\mathcal{E}/\mathcal{S})}{\mathrm{rk}(\mathcal{E}/\mathcal{S})}-\frac{c_{1}(\mathcal{S})}{\mathrm{rk}(\mathcal{S})}\right)\wedge[\Theta]\end{array}\right..$

Each $l_{\mathcal{S}}$ is a linear form on $H^{n-1,n-1}(X,\mathbb{R})$ and the
see-saw property of slopes implies that for each $[\Theta]\in\mathcal{B}_{X}$,
$\mathcal{E}$ is $[\Theta]$-stable (resp $[\Theta]$-semi-stable) if and only
if for all $\mathcal{S}$, $l_{\mathcal{S}}([\Theta])>0$ (resp. $\geqslant 0$).
In other words, we have,

$\mathcal{C}_{s}^{\mathrm{D}}(\mathcal{E})=\mathcal{B}_{X}\cap\hat{\mathcal{C}}_{s}^{\mathrm{D}}(\mathcal{E}),\qquad\mathcal{C}_{ss}^{\mathrm{D}}(\mathcal{E})=\mathcal{B}_{X}\cap\hat{\mathcal{C}}_{ss}^{\mathrm{D}}(\mathcal{E}),$
(2)

where,

$\hat{\mathcal{C}}_{s}^{\mathrm{D}}(\mathcal{E})=\bigcap_{0\subsetneq\mathcal{S}\subsetneq\mathcal{E}}\\{l_{\mathcal{S}}>0\\},\qquad\hat{\mathcal{C}}_{s}^{\mathrm{D}}(\mathcal{E})=\bigcap_{0\subsetneq\mathcal{S}\subsetneq\mathcal{E}}\\{l_{\mathcal{S}}\geqslant
0\\}.$

In particular, these six sets all are convex cones. The purpose of the next
subsection is to show that the number of linear equations $l_{\mathcal{S}}$ is
actually finite, so $\hat{\mathcal{C}}_{ss}^{\mathrm{D}}(\mathcal{E})$ and
$\hat{\mathcal{C}}_{s}^{\mathrm{D}}(\mathcal{E})$ are polyhedral convex cones.

###  3.3 Abstract nonsens

######  Lemma 3.3.

In an abelian category, if we have a commutative diagram of the form,

${0}$${A^{\prime}}$${B^{\prime}}$${C^{\prime}}$${0}$${0}$${A}$${B}$${C}$${0}$$\scriptstyle{i}$$\scriptstyle{j}$$\scriptstyle{k}$

and if we assume that the rows are exact, that $i$ and $j$ are monomorphisms
and that $A^{\prime}=A\times_{B}B^{\prime}$, then $k$ is a monomorphism too.

###### Proof.

We prove it by diagram chasing since we only need this result in the category
of coherent sheaves on a complex manifold. We let the formal proof to the
reader. Let $a^{\prime}=A^{\prime}\rightarrow B^{\prime}$,
$b^{\prime}=B^{\prime}\rightarrow C^{\prime}$, $a=A\rightarrow B$ and
$b=B\rightarrow C$. Let $x\in\ker(k)$. $b^{\prime}$ is surjective thus
$x=b^{\prime}(y)$ for some $y\in B^{\prime}$.
$b(j(y))=k(b^{\prime}(y))=k(x)=0$. We deduce that
$j(y)\in\ker(b)=\mathrm{im}(a)$ so we may write $j(y)$ as $a(z)$ for some
$z\in A$. Recall that $A^{\prime}=A\times_{B}B^{\prime}$ and $a(z)=j(y)$ hence
there is a unique $t\in A^{\prime}$ such that $y=a^{\prime}(t)$ and $z=i(t)$.
In particular, $x=b^{\prime}(y)=b^{\prime}(a^{\prime}(t))=0$. $k$ is indeed a
monomorphism. ∎

######  Lemma 3.4.

In an abelian category, if $i:A\rightarrow B$ and $j:B^{\prime}\rightarrow B$
are monomorphisms, there is a commutative diagram of the following form, where
all the rows and the columns are exact.

${0}$${0}$${0}$${0}$${A^{\prime}}$${B^{\prime}}$${C^{\prime}}$${0}$${0}$${A}$${B}$${C}$${0}$${0}$${A^{\prime\prime}}$${B^{\prime\prime}}$${C^{\prime\prime}}$${0}$${0}$${0}$${0}$$\scriptstyle{j}$$\scriptstyle{i}$

Moreover, $A^{\prime}=A\times_{B}B^{\prime}$.

###### Proof.

Let $A^{\prime}=A\times_{B}B^{\prime}$. In particular,
$\ker(A^{\prime}\rightarrow B^{\prime})=\ker(A\rightarrow B)=0$ and
$\ker(A^{\prime}\rightarrow A)=\ker(B^{\prime}\rightarrow B)=0$. Let
$C=\mathrm{coker}(i)$ and $C^{\prime}=\mathrm{coker}(A^{\prime}\rightarrow
B^{\prime})$. We have $(B\rightarrow C)\circ(B^{\prime}\rightarrow
B)\circ(A^{\prime}\rightarrow B^{\prime})=(B\rightarrow C)\circ(A\rightarrow
B)\circ(A^{\prime}\rightarrow A)=0$ hence by universal property of the
cokernel, there exists a unique morphism $C^{\prime}\rightarrow C$ that makes
the diagram commutative. Now, set
$A^{\prime\prime}=\mathrm{coker}(A^{\prime}\rightarrow A)$,
$B^{\prime\prime}=\mathrm{coker}(j)$ and
$C^{\prime\prime}=\mathrm{coker}(C^{\prime}\rightarrow C)$. By the snake
lemma, we have the following commutative diagram, where
$K=\ker(C^{\prime}\rightarrow C)$.

${0}$${0}$${K}$${0}$${A^{\prime}}$${B^{\prime}}$${C^{\prime}}$${0}$${0}$${A}$${B}$${C}$${0}$${K}$${A^{\prime\prime}}$${B^{\prime\prime}}$${C^{\prime\prime}}$${0}$${0}$${0}$${0}$$\scriptstyle{j}$$\scriptstyle{i}$

Moreover, all the horizontal and vertical sequences are exact.
$A^{\prime}=A\times_{B}B^{\prime}$ hence by Lemma 3.3, $K=0$. ∎

Let $M$ be a complex manifold and $\mathrm{Coh}(M)$ its abelian category of
coherent sheaves. We set $K(\mathrm{Coh}(M))$ to be the associated
Grothendieck group and $K_{0}(\mathrm{Coh}(M))$ the quotient of
$K(\mathrm{Coh}(M))$ by the subgroup generated by torsion sheaves whose
support has codimension at least $2$ in $M$. When
$\mathcal{E}\in\mathrm{Coh}(M)$, we denote by $[\mathcal{E}]$ its class in
$K_{0}(\mathrm{Coh}(M))$.

######  Proposition 3.5.

Let $\mathcal{E}\in\mathrm{Coh}(M)$ be torsion-free and
$K_{0}^{\subset\mathcal{E}}(\mathrm{Coh}(M))$ be the set of all
$[\mathcal{S}]\in K_{0}(\mathrm{Coh}(M))$ for $\mathcal{S}\subset\mathcal{E}$
saturated. This set is finite of cardinality bounded by
$2^{\mathrm{rk}(\mathcal{E})}$.

###### Proof.

We prove it by induction on the rank of $\mathcal{E}$. If
$\mathrm{rk}(\mathcal{E})=0$, then $\mathcal{E}=0$ because it is torsion-free.
Thus, $K_{0}^{\subset\mathcal{E}}(\mathrm{Coh}(M))=\\{[0]\\}$ has cardinality
$1=2^{\mathrm{rk}(\mathcal{E})}$.

Assume now that $\mathrm{rk}(\mathcal{E})\geqslant 1$. If there is no
saturated coherent sub-sheaf $0\subsetneq\mathcal{F}\subsetneq\mathcal{E}$,
then $K_{0}^{\subset\mathcal{E}}(\mathrm{Coh}(M))=\\{[0],[\mathcal{E}]\\}$ has
cardinality $2\leqslant 2^{\mathrm{rk}(\mathcal{E})}$. Else, let
$0\subsetneq\mathcal{F}\subsetneq\mathcal{E}$ be saturated and coherent. In
particular, $0<\mathrm{rk}(\mathcal{F})<\mathrm{rk}(\mathcal{E})$. Let
$\mathcal{S}\subset\mathcal{E}$ be saturated. By Lemma 3.4, we have the
following commutative diagram with exact rows and columns,

${0}$${0}$${0}$${0}$${\mathcal{F}^{\prime}}$${\mathcal{S}}$${\mathcal{Q}^{\prime}}$${0}$${0}$${\mathcal{F}}$${\mathcal{E}}$${\mathcal{Q}}$${0}$${0}$${\mathcal{F}^{\prime\prime}}$${\mathcal{E}^{\prime\prime}}$${\mathcal{Q}^{\prime\prime}}$${0}$${0}$${0}$${0}$

Recall that $\mathcal{F}\subset\mathcal{E}$ and
$\mathcal{S}\subset\mathcal{E}$ are saturated. In particular, all the sheaves
on this diagram are torsion-free except maybe $\mathcal{Q^{\prime\prime}}$.
Let $\mathcal{T}$ be the torsion sheaf of $\mathcal{Q^{\prime\prime}}$ and
$\mathcal{R}=\ker(\mathcal{Q}\rightarrow\mathcal{Q}^{\prime\prime}/\mathcal{T})$.
Clearly,
$\mathcal{Q^{\prime}}=\ker(\mathcal{Q}\rightarrow\mathcal{Q}^{\prime\prime})$
is included in $\mathcal{R}$ hence we obtain the following commutative diagram
where the rows and columns are exact,

${0}$${0}$${0}$${\mathcal{Q}^{\prime}}$${\mathcal{Q}}$${\mathcal{Q}^{\prime\prime}}$${0}$${0}$${\mathcal{R}}$${\mathcal{Q}}$${\mathcal{Q}^{\prime\prime}/\mathcal{T}}$${0}$${0}$${0}$

By the snake lemma,
$\mathrm{coker}(\mathcal{Q}^{\prime}\rightarrow\mathcal{R})=\ker(\mathcal{Q}^{\prime\prime}\rightarrow\mathcal{Q}^{\prime\prime}/\mathcal{T})=\mathcal{T}$.
We deduce that in $K_{0}(\mathrm{Coh}(M))$,

$[\mathcal{S}]=[\mathcal{F}^{\prime}]+[\mathcal{Q}^{\prime}]=[\mathcal{F}^{\prime}]+[\mathcal{R}]-[\mathcal{T}].$

Now, let,

$S=\mathrm{Sing}(\mathcal{F}^{\prime})\cup\mathrm{Sing}(\mathcal{S})\cup\mathrm{Sing}(\mathcal{Q}^{\prime})\cup\mathrm{Sing}(\mathcal{F})\cup\mathrm{Sing}(\mathcal{E})\cup\mathrm{Sing}(\mathcal{Q})\cup\mathrm{Sing}(\mathcal{F}^{\prime\prime})\cup\mathrm{Sing}(\mathcal{E}^{\prime\prime}),$

where $\mathrm{Sing}(\mathcal{A})\subset\mathcal{E}$ denotes the singular
locus of a coherent sheaf $\mathcal{A}$ on $M$ i.e. the analytic sub-variety
of $M$ where $\mathcal{A}$ is not locally free. By [12, Corollary 5.5.15], the
singular locus of a torsion-free sheaf has codimension at least $2$. In
particular, $S$ has codimension at least $2$. When $x\in M\backslash S$, we
have the following commutative diagram in the category of
$\mathcal{O}_{M,x}$-modules of finite rank.

${0}$${0}$${0}$${0}$${\mathcal{F}_{x}^{\prime}}$${\mathcal{S}_{x}}$${\mathcal{Q}_{x}^{\prime}}$${0}$${0}$${\mathcal{F}_{x}}$${\mathcal{E}_{x}}$${\mathcal{Q}_{x}}$${0}$${0}$${\mathcal{F}_{x}^{\prime\prime}}$${\mathcal{E}_{x}^{\prime\prime}}$${\mathcal{Q}_{x}^{\prime\prime}}$${0}$${0}$${0}$${0}$

Since $x\notin S$, all the objects in the diagram are free except maybe
$\mathcal{Q}_{x}^{\prime\prime}$. In particular, $\mathcal{Q}_{x}$ is
projective hence the middle row sequence splits so
$\mathcal{E}_{x}=\mathcal{F}_{x}\oplus\mathcal{Q}_{x}$. The middle column
sequence restricts to two exact sequences $0\rightarrow
A\rightarrow\mathcal{F}_{x}\rightarrow C\rightarrow 0$ and $0\rightarrow
B\rightarrow\mathcal{Q}_{x}\rightarrow D\rightarrow 0$ for some
$\mathcal{O}_{M,x}$-modules $A$, $B$, $C$ and $D$ that verify $A\oplus
B=\mathcal{S}_{x}$ and $C\oplus D=\mathcal{E}_{x}^{\prime\prime}$. Since
$\mathcal{S}_{x}$ and $\mathcal{E}_{x}^{\prime\prime}$ are projective, so are
$A$, $B$, $C$ and $D$. In particular, the previous two sequences split and we
get that $\mathcal{F}_{x}=A\oplus C$ and $\mathcal{Q}_{x}=B\oplus D$. We
deduce that
$\mathcal{F}_{x}^{\prime}=\mathcal{F}_{x}\times_{\mathcal{E}_{x}}\mathcal{S}_{x}=A$,
$\mathcal{Q}_{x}=\mathcal{S}_{x}/\mathcal{F}_{x}^{\prime}=B$,
$\mathcal{F}_{x}^{\prime\prime}=\mathcal{F}_{x}/\mathcal{F}_{x}^{\prime}=C$
and
$\mathcal{Q}_{x}^{\prime\prime}=\mathcal{E}_{x}^{\prime\prime}/\mathcal{F}_{x}^{\prime\prime}=D$
is projective. Since $\mathcal{O}_{M,x}$ is local,
$\mathcal{Q}_{x}^{\prime\prime}$ free. It proves that
$\mathrm{Sing}(\mathcal{Q}^{\prime\prime})\subset S$ has codimension at least
$2$. In particular, $[\mathcal{T}]=0$ in $K_{0}(\mathrm{Coh}(M))$.

In particular, we have the equality
$[\mathcal{S}]=[\mathcal{F}^{\prime}]+[\mathcal{R}]$, which implies that
$[\mathcal{S}]\in
K_{0}^{\subset\mathcal{F}}(\mathrm{Coh}(M))+K_{0}^{\subset\mathcal{Q}}(\mathrm{Coh}(M))$.
It is true for all $\mathcal{S}$ hence
$K_{0}^{\subset\mathcal{E}}(\mathrm{Coh}(M))\subset
K_{0}^{\subset\mathcal{F}}(\mathrm{Coh}(M))+K_{0}^{\subset\mathcal{Q}}(\mathrm{Coh}(M))$.
By induction, $K_{0}^{\subset\mathcal{E}}(\mathrm{Coh}(M))$ is finite with
cardinality bounded by,

$\\#(K_{0}^{\subset\mathcal{E}}(\mathrm{Coh}(M)))\leqslant\\#(K_{0}^{\subset\mathcal{F}}(\mathrm{Coh}(M)))\\#(K_{0}^{\subset\mathcal{Q}}(\mathrm{Coh}(M)))\leqslant
2^{\mathrm{rk}(\mathcal{F})}2^{\mathrm{rk}(\mathcal{Q})}=2^{\mathrm{rk}(\mathcal{E})}.$

∎

######  Corollary 3.6.

Let $M$ be a compact complex manifold and $\mathcal{E}$ be a torsion-free
coherent sheaf on $M$. The set of all $c_{1}(\mathcal{S})$ for
$\mathcal{S}\subset\mathcal{E}$ saturated and coherent is finite with at most
$2^{\mathrm{rk}(\mathcal{E})}$ elements. We go down to at most
$2^{\mathrm{rk}(\mathcal{E})}-2$ elements if $\mathcal{E}\neq 0$ and if we
only consider the non-zero strict sub-sheaves.

###### Proof.

By [12, Proposition 5.6.14] and additivity of $c_{1}$ classes, the morphism
$[\mathcal{E}]\mapsto(\mathrm{rk}(\mathcal{E}),c_{1}(\mathcal{E}))$ from
$K_{0}(\mathrm{Coh}(M))$ to $\mathbb{Z}\times H^{1,1}(M,\mathbb{R})$ is well
defined. The result follows immediately from Proposition 3.5. ∎

######  Remark.

The bound of $2^{\mathrm{rk}(\mathcal{E})}$ is sharp whenever there exists a
non flat line bundle $\mathcal{L}$ on $M$. Indeed, if $\mathcal{L}$ is non
flat, $c_{1}(\mathcal{L})\neq 0$. Consider
$\mathcal{E}=\bigoplus_{i=0}^{r-1}\mathcal{L}^{\otimes 2^{i}}$ for some
$r\geqslant 0$. Then $\mathrm{rk}(\mathcal{E})=r$ and any integer $0\leqslant
k\leqslant 2^{r}-1$ can be uniquely decomposed as $k=\sum_{i\in I}2^{i}$ for
some $I\subset[\\![0,r-1]\\!]$. The saturated sub-sheaf
$\mathcal{S}_{k}=\bigoplus_{i\in I}\mathcal{L}^{\otimes 2^{i}}$ of
$\mathcal{E}$ verifies $c_{1}(\mathcal{S}_{k})=kc_{1}(\mathcal{L})$. In
particular, the $c_{1}(\mathcal{S}_{k})$ are pairwise disjoint.

This last corollary enables us to understand better the geometric properties
of the stable and semi-stable cones. For the topological structures, we endow
the space of $(n-1,n-1)$ closed forms with the $L_{1}^{2}$ topology. We call
$\left\|\cdot\right\|_{L_{1}^{2}}$ an $L_{1}^{2}$ norm (computed thanks to
some background metric on $X$). In particular, the $L_{1}^{2}$ topology makes
the subset of exact forms closed hence its projection on
$H^{n-1,n-1}(X,\mathbb{R})$ continuous.

Clearly, $\mathcal{C}_{s}(\mathcal{E})$ is an $L_{1}^{2}$-open convex cone in
the space of closed $(n-1,n-1)$ forms.
$\hat{\mathcal{C}}_{s}^{\mathrm{D}}(\mathcal{E})$ and
$\mathcal{C}_{s}^{\mathrm{D}}(\mathcal{E})$ are open convex cones in
$H^{n-1,n-1}(X,\mathbb{R})$. $\mathcal{C}_{ss}(\mathcal{E})$ is an
$L_{1}^{2}$-closed convex cone in the space of balanced metrics.
$\hat{\mathcal{C}}_{ss}^{\mathrm{D}}(\mathcal{E})$ and
$\mathcal{C}_{ss}^{\mathrm{D}}(\mathcal{E})$ are convex cones in
$H^{n-1,n-1}(X,\mathbb{R})$,
$\hat{\mathcal{C}}_{ss}^{\mathrm{D}}(\mathcal{E})$ is closed in
$H^{n-1,n-1}(X,\mathbb{R})$ and $\mathcal{C}_{ss}(\mathcal{E})$ is closed in
$\mathcal{B}_{X}$. All of these are an trivial consequences of the finiteness
result of Corollary 3.6 and of the continuity of the natural projection on
$H^{n-1,n-1}(X,\mathbb{R})$. Moreover, we have the more precise version of
Introduction’s Theorem 2.

######  Theorem 3.7.

On a balanced manifold $X$, for all torsion-free coherent sheaf $\mathcal{E}$,
the stable and semi-stable cones satisfy the following properties,

  * •

$\hat{\mathcal{C}}_{s}^{\mathrm{D}}(\mathcal{E})$ and
$\hat{\mathcal{C}}_{ss}^{\mathrm{D}}(\mathcal{E})$ are polyhedral convex
cones, defined by at most $2^{\mathrm{rk}(\mathcal{E})}-2$ linear
inequalities.

  * •

If there is a saturated coherent $0\subsetneq\mathcal{S}\subsetneq\mathcal{E}$
such that
$\frac{c_{1}(\mathcal{S})}{\mathrm{rk}(\mathcal{S})}=\frac{c_{1}(\mathcal{E})}{\mathrm{rk}(\mathcal{E})}$,
$\mathcal{C}_{s}(\mathcal{E})=\hat{\mathcal{C}}_{s}^{\mathrm{D}}(\mathcal{E})=\mathcal{C}_{s}^{\mathrm{D}}(\mathcal{E})=\emptyset$.

  * •

If there is no saturated coherent
$0\subsetneq\mathcal{S}\subsetneq\mathcal{E}$ such that
$\frac{c_{1}(\mathcal{S})}{\mathrm{rk}(\mathcal{S})}=\frac{c_{1}(\mathcal{E})}{\mathrm{rk}(\mathcal{E})}$,
$\overset{\mathrm{o}}{\mathcal{C}_{ss}(\mathcal{E})}=\mathcal{C}_{s}(\mathcal{E})$,
$\overset{\mathrm{o}}{\hat{\mathcal{C}}_{ss}^{\mathrm{D}}(\mathcal{E})}=\hat{\mathcal{C}}_{s}^{\mathrm{D}}(\mathcal{E})$
and
$\overset{\mathrm{o}}{\mathcal{C}_{ss}^{\mathrm{D}}(\mathcal{E})}=\mathcal{C}_{s}^{\mathrm{D}}(\mathcal{E})$.

  * •

If $\hat{\mathcal{C}}_{s}^{\mathrm{D}}(\mathcal{E})\neq\emptyset$,
$\overline{\mathcal{C}_{s}(\mathcal{E})}\cap\mathcal{B}_{X}=\mathcal{C}_{ss}(\mathcal{E})$,
$\overline{\hat{\mathcal{C}}_{s}^{\mathrm{D}}(\mathcal{E})}=\hat{\mathcal{C}}_{ss}^{\mathrm{D}}(\mathcal{E})$
and
$\overline{\mathcal{C}_{s}^{\mathrm{D}}(\mathcal{E})}\cap\mathcal{B}_{X}=\mathcal{C}_{ss}^{\mathrm{D}}(\mathcal{E})$.

  * •

For all $[\Theta_{0}]\in\mathcal{C}_{ss}(\mathcal{E})$, if
$[\Theta]\in\mathcal{B}_{X}$ is close enough to $[\Theta_{0}]$,
$[\Theta]\in\mathcal{C}_{s}(\mathcal{E})$ (resp.
$[\Theta]\in\mathcal{C}_{ss}(\mathcal{E})$) if and only if for all
$0\subsetneq\mathcal{S}\subsetneq\mathcal{E}$ such that
$l_{\mathcal{S}}([\Theta_{0}])=0$, $l_{\mathcal{S}}([\Theta])>0$ (resp.
$l_{\mathcal{S}}([\Theta])\geqslant 0$).

###### Proof.

By Corollary 3.6, the set of all $c_{1}(\mathcal{S})$ when
$0\subsetneq\mathcal{S}\subsetneq\mathcal{E}$ is saturated and coherent sub-
sheaf is finite with cardinality bounded by $2^{\mathrm{rk}(\mathcal{E})}$. We
go down to $2^{\mathrm{rk}(\mathcal{E})}-2$ when we remove $0$ and
$\mathcal{E}$. Therefore, there exists an integer $p\leqslant
2^{\mathrm{rk}(\mathcal{E})}-2$ and a finite $p$-tuple
$0\subsetneq\mathcal{S}_{1},\ldots,\mathcal{S}_{p}\subsetneq\mathcal{E}$ of
such sheaves such that for all $\mathcal{S}$,
$c_{1}(\mathcal{S})=c_{1}(\mathcal{S}_{k})$ for some $k$. In particular, in
this case, $l_{\mathcal{S}}=l_{\mathcal{S}_{k}}$ so,

$\hat{\mathcal{C}}_{s}^{\mathrm{D}}(\mathcal{E})=\bigcap_{k=1}^{p}\\{l_{\mathcal{S}_{k}}>0\\},\qquad\hat{\mathcal{C}}_{ss}^{\mathrm{D}}(\mathcal{E})=\bigcap_{k=1}^{p}\\{l_{\mathcal{S}_{k}}\geqslant
0\\}.$

Moreover, since the pairing $H^{n-1,n-1}(X,\mathbb{R})\times
H^{1,1}(X,\mathbb{R})\rightarrow\mathbb{R}$ is non-degenerate by Serre
duality, for each $\mathcal{S}$,
$l_{\mathcal{S}}=0\Leftrightarrow\frac{c_{1}(\mathcal{S})}{\mathrm{rk}(\mathcal{S})}=\frac{c_{1}(\mathcal{E})}{\mathrm{rk}(\mathcal{E})}$.
From here, the results about the finite dimensional cones follow from basic
topology and convex geometry, and the results about the infinite dimensional
cones follow from the continuity of the projection onto
$H^{n-1,n-1}(X,\mathbb{R})$. ∎

###  3.4 Hermitian Yang–Mills equation

######  Definition 3.8.

Following [12, Section 4.1], we say that $\nabla$ is a $\Theta$-Hermitian
Yang–Mills connection (or satisfies the Einstein condition) if

$\Lambda_{\Theta}\boldsymbol{\mathrm{i}}F_{\nabla}=c\mathrm{Id}_{E},$

for some constant $c$.

######  Remark.

When it exists, $c$ is entirely determined by the topology of $E$ and the real
Dolbeault class $[\Theta]\in H^{n-1,n-1}(X,\mathbb{R})$. Indeed, by Chern-Weil
theory, we have $c_{[\Theta]}=\frac{2\pi
c_{1}(E)\cup[\Theta]}{\mathrm{rk}(E)\mathrm{Vol}(X)}$ by integrating over $X$
the above equality.

Since the Hermitian metric $h$ on $E$ is fixed, $\nabla$ depends only on
$\overline{\partial}$ as it is the Chern connection associated with
$(\overline{\partial},h)$. We write $F_{\overline{\partial}}=F_{\nabla}$ and
we also say abusively that $\overline{\partial}$ is $\Theta$-HYM.

A powerful analytic tool to study stability is the Kobayashi–Hitchin
correspondence, which was originally proven for Kähler metrics [27] and
extended to more general cases, including balanced metrics [15], and on Higgs
bundles [16].

######  Proposition 3.9.

If $\mathcal{E}=(E,\overline{\partial}_{E})$ is a holomorphic vector bundle of
positive rank, then it is $[\Theta]$-polystable if and only if it admits a
$\Theta$-HYM operator $\overline{\partial}$ in the gauge orbit of
$\overline{\partial}_{E}$. In this case, it is unique up to a unitary gauge
transformation.

Our purpose from now on is to extend locally the algebraic results of Theorem
3.7 to analytic results. Concretely, we want to show that when
$\Theta\in\mathcal{C}_{s}(\mathcal{E})$ approaches a point
$\Theta_{0}\in\mathcal{C}_{ss}(\mathcal{E})$, we can build some $\Theta$-HYM
$\overline{\partial}$ in the gauge orbit of $\overline{\partial}_{E}$ that
approaches some $\Theta_{0}$-HYM $\overline{\partial}_{0}$, which is in the
gauge orbit of the Dolbeault operator of
$\mathrm{Gr}_{[\Theta_{0}]}(\mathcal{E})$ (under the additional condition that
$\mathcal{E}$ is sufficiently smooth with respect to $[\Theta_{0}]$).

##  4 Local analytic results

Let $\Theta_{0}$ be a balanced metric on $X$ and
$\mathcal{E}=(E,\overline{\partial}_{E})$ a $[\Theta_{0}]$-semi-stable
sufficiently smooth vector bundle. Assume that
$\mathcal{C}_{s}(\mathcal{E})\neq\emptyset$ so
$\Theta_{0}\in\overline{\mathcal{C}_{s}(\mathcal{E})}$ by Theorem 3.7 and
$\mathcal{E}$ is simple. We associate to $\mathcal{E}$ a Jordan-Hölder
filtration
$0=\mathcal{F}_{0}\subsetneq\mathcal{F}_{1}\subsetneq\cdots\subsetneq\mathcal{F}_{m}=\mathcal{E}$
given by Proposition 3.2 and we set
$\mathrm{Gr}(\mathcal{E})=\mathrm{Gr}_{[\Theta_{0}]}(\mathcal{E})=\bigoplus_{k=1}^{m}\mathcal{G}_{k}$
to be its graded object where the
$\mathcal{G}_{k}=\mathcal{F}_{k}/\mathcal{F}_{k-1}$ are its stable components.
We call $\nabla_{0}=\partial_{0}+\overline{\partial}_{0}$ the connection on
$\mathrm{Gr}(\mathcal{E})$. We may assume it is $\Theta_{0}$-HYM by
Proposition 3.9.

Being a balanced metric is an open condition in the space of closed
$(n-1,n-1)$ forms endowed with the $L_{1}^{2}$ topology. During all this
article, $U$ will be an $L_{1}^{2}$ open neighbourhood of $0$ such that for
all $\varepsilon\in U$, $\Theta_{\varepsilon}=\Theta_{0}+\varepsilon$ is
positive. With the $L_{1}^{2}$ topology, all the natural operations involving
$\Theta$ like $\Theta\mapsto\mathrm{Vol}_{\Theta}$ or
$\Theta\mapsto\Lambda_{\Theta}\alpha$ for some $(1,1)$-form $\alpha$, are
smooth. In particular, up to shrinking $U$, the $L^{2}$ norms
$\left\|\cdot\right\|_{\varepsilon}$ are uniformly equivalent. From now on,
when we consider objects that depend on the balanced metric, we replace the
subscript $\Theta=\Theta_{\varepsilon}$ by $\varepsilon$ for simplicity.

###  4.1 Perturbed Kuranishi slice and moment map

We call
$G=\mathrm{Aut}_{0}(\mathrm{Gr}(\mathcal{E}))\subset\mathcal{G}^{\mathbb{C}}$
the group of automorphisms of $\mathrm{Gr}(\mathcal{E})$ of determinant $1$
and $K=\mathrm{Aut}_{0}(\mathrm{Gr}(\mathcal{E}),h)=\mathcal{G}\cap G$ the
group of unitary automorphisms of $\mathrm{Gr}(\mathcal{E})$ of determinant
$1$. They are both finite dimensional, $K$ is compact and $G=K^{\mathbb{C}}$,
hence $G$ is a reductive Lie group with maximal compact subgroup $K$. Let
$\mathfrak{g}=\mathrm{Lie}(G)\subset\mathrm{Lie}(\mathcal{G}^{\mathbb{C}})$
and $\mathfrak{k}=\mathrm{Lie}(K)=\mathrm{Lie}(\mathcal{G})\cap\mathfrak{g}$.
Notice that the condition of having a determinant that equals $1$ on $G$ and
$K$ implies that all elements of $\mathfrak{g}$ and $\mathfrak{k}$ have a
trace that equals $0$. Each of these Lie algebras is identified with its dual
space thanks to the scalar products $\left<\cdot,\cdot\right>_{\varepsilon}$
on $\mathrm{Lie}(\mathcal{G}^{\mathbb{C}})=\Omega^{0}(X,\mathrm{End}(E))$. We
must be careful that this identification depends on $\varepsilon$. Notice also
that $\mathrm{Aut}(\mathrm{Gr}(\mathcal{E}))=\mathbb{C}^{*}G$ and its Lie
algebra is $\mathfrak{g}\oplus\mathbb{C}\mathrm{Id}_{E}$. Similarly,
$\mathrm{Aut}(\mathrm{Gr}(\mathcal{E}),h)=\mathbb{S}^{1}K$ and
$\mathrm{Lie}(\mathrm{Aut}(\mathrm{Gr}(\mathcal{E}),h))=\mathfrak{k}\oplus\boldsymbol{\mathrm{i}}\mathbb{R}\mathrm{Id}_{E}$.
The reason why we restrict ourselves to the automorphisms that have a
determinant equal to $1$ is that
$\mathbb{C}^{*}\mathrm{Id}_{E}\subset\mathrm{Aut}(\mathrm{Gr}(\mathcal{E}))$
acts trivially on connections.

Define
$\Omega_{\varepsilon}^{D}:(\alpha,\beta)\mapsto\left<J\alpha,\beta\right>_{\varepsilon}$
to be the symplectic form associated to the Hermitian product introduced at
(1) with respect to $\Theta_{\varepsilon}$ on $\mathrm{End}(E)$. We will
mostly be interested in the case where $\alpha$ and $\beta$ are $(0,1)$-forms.
In this case,

$\Omega_{\varepsilon}^{D}(\alpha,\beta)=\int_{X}\mathrm{tr}(\alpha\wedge\beta^{*})\wedge\Theta_{\varepsilon}.$

The set of (not necessarily integrable) Dobleault operators is an affine space
of direction $\Omega^{0,1}(X,\mathrm{End}(E))$. Thus it is in particular an
infinite dimensional manifold and $\Omega_{\varepsilon}^{D}$ is formally a
Kähler form on it. Moreover, by [1, 5], the gauge group acts by Hamiltonian
actions on this manifold and the associated equivariant moment map is
$\nu_{\infty,\varepsilon}:\overline{\partial}\mapsto\Lambda_{\varepsilon}F_{\overline{\partial}}$.
Concretely, it means that for all $\overline{\partial}$, for all $v$ tangent
to $\overline{\partial}$ and all
$a\in\mathrm{Lie}(\mathcal{G}^{\mathbb{C}})=\Omega^{0}(X,\mathrm{End}(E))$,

$\left<d\nu_{\infty,\varepsilon}(\overline{\partial})v,a\right>_{\varepsilon}=\Omega_{\varepsilon}^{D}(L_{\overline{\partial}}a,v),$
(3)

where $\overline{\partial}\mapsto
L_{\overline{\partial}}a=\frac{\partial}{\partial
t}|_{t=0}\boldsymbol{\mathrm{e}}^{ta}\cdot\overline{\partial}$ is the vector
field induced by the infinitesimal action of $a$. We use here the same
notation as in [8]. We want now to reduce ourselves to the same kind of moment
map, but for finite dimensional Lie groups. For this, we use first the same
method as in [4, 3, 25] involving the Kuranishi slice.

Let $V$ be the finite dimensional complex space
$H^{0,1}(X,\mathrm{End}(\mathrm{Gr}(\mathcal{E})),\Theta_{0})$ of harmonic
$(0,1)$-forms with values in $\mathrm{End}(\mathrm{Gr}(\mathcal{E}))$ with
respect to $\Theta_{0}$. Let
$\Phi:B\rightarrow\Omega^{0,1}(X,\mathrm{End}(\mathrm{Gr}(\mathcal{E})))$ be
the Kuranishi slice [14] where $B\subset V$ is a ball around $0$. It is
explicitly given by $\Phi:b\mapsto b+\overline{\partial}_{0}^{*}A(b\wedge b)$
where $A$ is the Green operator associated to
$\Delta_{0}=\Delta_{\Theta_{0},\nabla_{0}}$. In particular, it is holomorphic.
Recall that $\Phi$ verifies the following properties,

  1. 1.

$\Phi(0)=0$ and $d\Phi(0):v\mapsto v$ for $v\in
T_{0}V=V\subset\Omega^{0,1}(X,\mathrm{End}(E))$ is injective hence, up to
shrinking $B$, $\Phi$ is an embedding.

  2. 2.

$B$ is $K$-invariant and $\Phi$ is $G$-equivariant where it is defined in the
sens that for all $b\in B$ and all $g\in G$ such that $g\cdot b\in B$,
$\overline{\partial}_{g\cdot b}=g\cdot\overline{\partial}_{b}$ where
$\overline{\partial}_{b}$ is defined as $\overline{\partial}_{0}+\Phi(b)$.

  3. 3.

For any small enough deformation $\overline{\partial}$ of
$\overline{\partial}_{0}$, there is a gauge transformation
$f\in\mathcal{G}^{\mathbb{C}}$ such that
$f\cdot\overline{\partial}\in\Phi(B)$.

  4. 4.

$Z=\left\\{b\in B\middle|\overline{\partial}_{b}^{2}=0\right\\}$ is a complex
subspace of $B$.

Since $\mathcal{E}$ is a small holomorphic deformation of
$\mathrm{Gr}(\mathcal{E})$ (see for example [3, Section 3]), we can find some
$b_{0}\in B$ such that
$\overline{\partial}_{b_{0}}=\overline{\partial}_{0}+\Phi(b_{0})$ is gauge
equivalent to $\overline{\partial}_{E}$. From now on, we call
$\mathcal{O}=G\cdot b_{0}\cap B$ the $G$-orbit of $b_{0}$ in $B$. Any
$b\in\mathcal{O}$ is such that $\overline{\partial}_{b}$ is gauge equivalent
to $\overline{\partial}_{E}$.

Then, we deform $\Phi$ because we are interested in Dolbeault operators
$\overline{\partial}$ such that
$\nu_{\infty,\varepsilon}(\overline{\partial})$ belongs to the Lie algebra
$\mathfrak{k}$ of $K$ as in [3, Proposition 3.2]. In the following, we build
functions that depend on $\varepsilon\in U$. To understand accurately the
smoothness of these objects with respect to $\varepsilon$, let us introduce
for all integer $d\geqslant 1$, an $L_{d}^{2}$ Sobolev norm
$\left\|\cdot\right\|_{L_{d}^{2}}$ on $\Omega^{n-1,n-1}(X,\mathbb{C})$. For
example, we can compute them with respect to a background metric on $X$ and
the resulting norm doesn’t depend on the choice of this metric up to
equivalence.

######  Proposition 4.1.

Up to shrinking $U$ and $B$, there exists a map $\sigma:U\times
B\rightarrow\Omega^{0}_{0}(X,\mathrm{End}_{H}(E,h))$ such that for all integer
$d\geqslant 1$, if $U$ is endowed with an $L_{d}^{2}$ norm and
$\Omega^{0}_{0}(X,\mathrm{End}_{H}(E,h))$ with an $L_{d+2}^{2}$ norm, $\sigma$
is smooth and we have the following properties : $\sigma(0,0)=0$,
$\frac{\partial}{\partial b}|_{b=0}\sigma(0,b)=0$ and if we set
$\tilde{\Phi}:(\varepsilon,b)\mapsto\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,b)}\cdot\overline{\partial}_{b}-\overline{\partial}_{0}\in\Omega^{0,1}(X,\mathrm{End}(E))$,
$\overline{\partial}_{\varepsilon,b}=\overline{\partial}_{0}+\tilde{\Phi}(\varepsilon,b)$
and $F_{\varepsilon,b}=F_{\overline{\partial}_{\varepsilon,b}}$, we have,

  1. 1.

For all $b\in\mathcal{O}$, $\overline{\partial}_{\varepsilon,b}$ is gauge
equivalent to $\overline{\partial}_{E}$.

  2. 2.

For all $\varepsilon\in U$, $\tilde{\Phi}(\varepsilon,\cdot)$ is
$K$-equivariant.

  3. 3.

For all $(\varepsilon,b)\in U\times B$,
$\Lambda_{\varepsilon}F_{\varepsilon,b}+\boldsymbol{\mathrm{i}}c_{\varepsilon}\mathrm{Id}_{E}\in\mathfrak{k}$.

  4. 4.

$\frac{\partial}{\partial b}|_{b=0}\tilde{\Phi}(0,b):v\mapsto v$. In
particular, up to shrinking $U\times B$, each
$\tilde{\Phi}(\varepsilon,\cdot)$ is a smooth embedding.

###### Proof.

Let
$\Pi_{\bot}:\Omega_{0}^{0}(X,\mathrm{End}_{H}(E,h))\rightarrow\boldsymbol{\mathrm{i}}\mathfrak{k}^{\bot}$
be the orthogonal projection on $\boldsymbol{\mathrm{i}}\mathfrak{k}^{\bot}$
(thus parallel to $\boldsymbol{\mathrm{i}}\mathfrak{k}$) with respect to
$\left<\cdot,\cdot\right>_{0}$ and,

$\Psi:\left\\{\begin{array}[]{rcl}\displaystyle U\times
B\times\boldsymbol{\mathrm{i}}\mathfrak{k}^{\bot}&\rightarrow&\displaystyle\boldsymbol{\mathrm{i}}\mathfrak{k}^{\bot}\\\
\displaystyle(\varepsilon,b,s)&\mapsto&\displaystyle\Pi_{\bot}(\Lambda_{\varepsilon}\boldsymbol{\mathrm{i}}F_{\boldsymbol{\mathrm{e}}^{s}\cdot\overline{\partial}_{b}}-c_{\varepsilon}\mathrm{Id}_{E})\end{array}\right..$

By Proposition 2.3, $\frac{\partial}{\partial
s}_{|s=0}\Psi(0,0,s)=\Pi_{\bot}\Delta_{0}$ and $\Delta_{0}$ is a symmetric
semi-definite positive operator and it is continuous from the $L_{3}^{2}$
completion of $\Omega_{0}^{0}(X,\mathrm{End}_{H}(E,h))$ to its $L_{1}^{2}$
completion. In particular, $\Psi$ is smooth when we endow the starting
$\boldsymbol{\mathrm{i}}\mathfrak{k}^{\bot}$ with an $L_{3}^{2}$ norm, the
arrival one with an $L_{1}^{2}$ norm and $U$ with an $L_{1}^{2}$ norm.
According to Proposition 2.2, its kernel (hence its cokernel) is
$\boldsymbol{\mathrm{i}}\mathfrak{k}$. In particular,
$\Pi_{\bot}\Delta_{0}=\Delta_{0}$ and $\frac{\partial}{\partial
s}_{|s=0}\Psi(0,0,s)=\Delta_{0}:L_{3}^{2}(\boldsymbol{\mathrm{i}}\mathfrak{k}^{\bot})\rightarrow
L_{1}^{2}(\boldsymbol{\mathrm{i}}\mathfrak{k}^{\bot})$ is an isomorphism. With
the same argument, for any integer $d\geqslant 1$, $\Psi:L_{d}^{2}(U)\times
B\times L_{d+2}^{2}(\boldsymbol{\mathrm{i}}\mathfrak{k}^{\bot})\rightarrow
L_{d}^{2}(\boldsymbol{\mathrm{i}}\mathfrak{k}^{\bot})$ is smooth. Here,
$L_{d}^{2}(W)$ is the completion of a space $W$ endowed with an $L_{d}^{2}$
norm.

By the implicit functions theorem, up to shrinking $U\times B$, there is a
unique smooth $\sigma:U\times B\rightarrow
L_{3}^{2}(\boldsymbol{\mathrm{i}}\mathfrak{k}^{\bot})$ such that
$\sigma(0,0)=0$ and for all $(\varepsilon,b)\in U\times B$,
$\Psi(\varepsilon,b,\sigma(\varepsilon,b))=0$. When we set
$\tilde{\Phi}(\varepsilon,b)=\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,b)}\cdot\overline{\partial}_{b}-\overline{\partial}_{0}$,
the first point is immediate and the third point is verified. For the second
one, notice that for all $u\in K$ and all $(\varepsilon,b)\in U\times B$

$\displaystyle\Psi(\varepsilon,b,u^{*}\sigma(\varepsilon,u\cdot b)u)$
$\displaystyle=\Pi_{\bot}(\Lambda_{\varepsilon}\boldsymbol{\mathrm{i}}F_{\boldsymbol{\mathrm{e}}^{u^{*}\sigma(\varepsilon,u\cdot
b)u}\cdot\overline{\partial}_{b}}-c_{\varepsilon}\mathrm{Id}_{E})$
$\displaystyle=\Pi_{\bot}(\Lambda_{\varepsilon}\boldsymbol{\mathrm{i}}F_{u^{*}\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,u\cdot
b)}\cdot\overline{\partial}_{u\cdot b}u}-c_{\varepsilon}\mathrm{Id}_{E})$
$\displaystyle=\Pi_{\bot}(u^{*}(\Lambda_{\varepsilon}\boldsymbol{\mathrm{i}}F_{\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,u\cdot
b)}\cdot\overline{\partial}_{u\cdot b}}-c_{\varepsilon}\mathrm{Id}_{E})u)$
$\displaystyle=0.$

By uniqueness of $\sigma$, it implies that $u^{*}\sigma(\varepsilon,u\cdot
b)u=\sigma(\varepsilon,b)$. Therefore,

$\overline{\partial}_{\varepsilon,u\cdot
b}=\overline{\partial}_{0}+\tilde{\Phi}(\varepsilon,u\cdot
b)=\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,u\cdot
b)}\cdot\overline{\partial}_{u\cdot
b}=u\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,b)}u^{*}u\overline{\partial}_{b}u^{*}=u(\overline{\partial}_{0}+\tilde{\Phi}(\varepsilon,b))u^{*}=u\overline{\partial}_{\varepsilon,b}u^{*}.$

It proves the second point. For the fourth point, we have for all $b$,

$\displaystyle\Psi(0,b,0)$
$\displaystyle=\Pi_{\bot}(\Lambda_{0}F_{\overline{\partial}_{b}}+\boldsymbol{\mathrm{i}}c_{0}\mathrm{Id}_{E})$
$\displaystyle=-\boldsymbol{\mathrm{i}}\Pi_{\bot}(\Lambda_{0}\boldsymbol{\mathrm{i}}F_{\overline{\partial}_{0}}+\nabla_{0}(\Phi(b)-\Phi(b)^{*})+(\Phi(b)-\Phi(b)^{*})\wedge(\Phi(b)-\Phi(b)^{*})-c_{0}\mathrm{Id}_{E})$
$\displaystyle=-\boldsymbol{\mathrm{i}}\Pi_{\bot}\nabla_{0}(\Phi(b)-\Phi(b)^{*})+\mathrm{o}(b).$

Therefore, $\frac{\partial}{\partial
b}_{|b=0}\Psi(0,b,0)=-\boldsymbol{\mathrm{i}}\Pi_{\bot}\nabla_{0}(d\Phi(0)-d\Phi(0)^{*})$.
Now, recall that the image of $d\Phi(0)$ is the space $V$ of $(0,1)$-harmonic
forms. In particular, it is included in the kernel of $\nabla_{0}$. Same thing
for $d\Phi(0)^{*}$ since $\nabla_{0}$ is unitary. Thus
$\frac{\partial}{\partial b}_{|b=0}\Psi(0,b,0)=0$. Therefore, when we
differentiate the equality $\Psi(0,b,\sigma(0,b))$ at $b=0$,

$0=\frac{\partial}{\partial b}|_{b=0}\Psi(0,b,0)+\frac{\partial}{\partial
s}|_{s=0}\Psi(0,0,s)\frac{\partial}{\partial
b}|_{b=0}\sigma(0,b)=\Delta_{0}\frac{\partial}{\partial b}|_{b=0}\sigma(0,b).$

Since $\frac{\partial}{\partial b}|_{b=0}\sigma(0,b)$ belongs to
$\boldsymbol{\mathrm{i}}\mathfrak{k}^{\bot}$ where $\Delta_{0}$ is injective,
we have $\frac{\partial}{\partial b}|_{b=0}\sigma(0,b)=0$ hence,

$\tilde{\Phi}(0,b)-\overline{\partial}_{0}=\boldsymbol{\mathrm{e}}^{\sigma(0,b)}\cdot\overline{\partial}_{b}-\overline{\partial}_{0}=\boldsymbol{\mathrm{e}}^{\sigma(0,b)}\overline{\partial}_{0}(\boldsymbol{\mathrm{e}}^{-\sigma(0,b)})+\boldsymbol{\mathrm{e}}^{\sigma(0,b)}\Phi(b)\boldsymbol{\mathrm{e}}^{-\sigma(0,b)}=\Phi(b)+\mathrm{o}(b).$

We deduce the fourth point and we showed along the way that
$\frac{\partial}{\partial b}|_{b=0}\sigma(0,b)=0$. All we have left to show is
that the $\sigma(\varepsilon,b)$ are smooth as sections and that $\sigma$ has
the wanted regularity. The first part follows from elliptic regularity. For
the second one let $d\geqslant 1$ be an integer. Then,
$\Psi:L_{d}^{2}(U)\times B\times
L_{d+2}^{2}(\boldsymbol{\mathrm{i}}\mathfrak{k}^{\bot})\rightarrow
L_{d}^{2}(\boldsymbol{\mathrm{i}}\mathfrak{k}^{\bot})$ is smooth. When we
differentiate $m$ times the equality
$\Psi(\varepsilon,b,\sigma(\varepsilon,b))=0$, we can, by the Faà di Bruno
formula, express $\frac{\partial}{\partial
s}\Psi(\varepsilon,b,s)d^{m}\sigma(\varepsilon,b)$ as a polynomial expression
of the successive derivatives of $\Psi$ and the lower order derivatives of
$\sigma$, which are all smooth by induction on $m$. $\frac{\partial}{\partial
s}|_{s=0}\Psi(0,0,s)=\Delta_{0}:L_{d+2}^{2}(\boldsymbol{\mathrm{i}}\mathfrak{k}^{\bot})\rightarrow
L_{d}^{2}(\boldsymbol{\mathrm{i}}\mathfrak{k}^{\bot})$ is an isomorphism and
this condition is open, hence $\sigma:L_{d}^{2}(U)\times B\rightarrow
L_{d+2}(\boldsymbol{\mathrm{i}}\mathfrak{k}^{\bot})$ is smooth. ∎

Following [3, Section 3.2], we define for all $\varepsilon\in U$, the closed
$2$-form
$\Omega_{\varepsilon}=\tilde{\Phi}(\varepsilon,\cdot)^{*}\Omega_{\varepsilon}^{D}$
on $B$ and,

$\nu_{\varepsilon}:\left\\{\begin{array}[]{rcl}\displaystyle
B&\rightarrow&\displaystyle\mathfrak{k}\\\ \displaystyle
b&\mapsto&\displaystyle\nu_{\infty,\varepsilon}(\overline{\partial}_{\varepsilon,b})=\Lambda_{\varepsilon}F_{\varepsilon,b}+\boldsymbol{\mathrm{i}}c_{\varepsilon}\mathrm{Id}_{E}\end{array}\right..$

By Proposition 4.1, this function indeed takes values in $\mathfrak{k}$. It is
easy to compute that since $\nu_{\infty,\varepsilon}$ is a $K$-equivariant
moment map for $\Omega_{\varepsilon}^{D}$ and
$\tilde{\Phi}(\varepsilon,\cdot)$ is a $K$-equivariant embedding,
$\nu_{\varepsilon}$ is a $K$-equivariant moment map for
$\Omega_{\varepsilon}$, which is a simplectic form. We must be careful about
the fact that $\Omega_{\varepsilon}$ is not compatible with the complex
structures because $\tilde{\Phi}(\varepsilon,\cdot)$ is a priori not
holomorphic, but we still have the following positivity condition.

######  Lemma 4.2.

Up to shrinking $U$ and $B$, for all $b\in B$ and $v\in
T_{b}B\backslash\\{0\\}$, $\Omega_{\varepsilon}(v,\boldsymbol{\mathrm{i}}v)$
has a positive real part.

###### Proof.

This is due to the fact that $d\tilde{\Phi}(0,0):v\mapsto v$ is
$\mathbb{C}$-linear, hence, for all $v\in T_{0}B=V$,

$\Omega_{0}(v,\boldsymbol{\mathrm{i}}v)=\Omega_{0}^{D}(d\tilde{\Phi}(0,0)v,d\tilde{\Phi}(0,0)(\boldsymbol{\mathrm{i}}v))=\Omega_{0}^{D}(v,\boldsymbol{\mathrm{i}}v)>0.$

It means that $\Omega_{0}(\cdot,\boldsymbol{\mathrm{i}}\cdot)$ at $b=0$ is a
positive quadratic form. By openness of positivity, up to shrinking $U$ and
$B$, $\Re\Omega_{\varepsilon}$ is positive at any point $b$ of $B$, hence the
result. ∎

###  4.2 Bound on the norm of $b$

Now, when $\Theta_{\varepsilon}$ is in the stable cone of $\mathcal{E}$, we
want to find a $\Theta_{\varepsilon}$-HYM connection in the image of
$\overline{\partial}_{0}+\tilde{\Phi}(\varepsilon,\cdot)$ restricted to the
orbit $\mathcal{O}$, because if
$\overline{\partial}=\overline{\partial}_{\varepsilon,b}=\overline{\partial}_{0}+\tilde{\Phi}(\varepsilon,b)$,
then all the Sobolev norms of $\overline{\partial}-\overline{\partial}_{0}$
can be controlled by the norm of $b$ for any euclidean norm (see the later
Proposition 4.3). The first thing to do is the following estimate on the norm
of $b$ in term of the norm of $\nu_{\varepsilon}(b)$. Let
$\left\|\cdot\right\|$ be an euclidean norm on the finite dimensional vector
space $V$ that contains $B$.

######  Proposition 4.3.

Up to shrinking $U$ and $B$, for all $\varepsilon\in U$ and
$b\in\overline{\mathcal{O}}$, $\left\|b\right\|^{2}\leqslant
C(\left\|\nu_{\varepsilon}(b)\right\|_{\varepsilon}+\left\|\varepsilon\right\|_{L_{1}^{2}})$
for some positive constant $C$ independent from $\varepsilon$ and $b$.

###### Proof.

By density, it is enough to verify it for $b\in\mathcal{O}$.

Step 1 : Finding an orthogonal decomposition of $E$ where $\Phi(b)$ is
strictly upper triangular.

If $b\in\mathcal{O}$, $\overline{\partial}_{b}$ is gauge equivalent to
$\overline{\partial}_{E}$ thus the holomorphic vector bundle
$\mathcal{E}_{b}=(E,\overline{\partial}_{b})$ admits a Jordan-Hölder
filtration,

$0\subsetneq\mathcal{F}_{1,b}\subsetneq\cdots\subsetneq\mathcal{F}_{m,b}=\mathcal{E}_{b},$

For all $k$, the smooth structure of $\mathcal{F}_{k,b}$ is $F_{k}$ and the
smooth structure of $\mathcal{G}_{k,b}=\mathcal{F}_{k,b}/\mathcal{F}_{k-1,b}$
is $G_{k}$. The $(0,1)$-form $\Phi(b)$ can be decomposed as
$\Phi(b)=\sum_{i,j=1}^{m}\gamma_{ij}$ with
$\gamma_{ij}\in\Omega^{0,1}(X,\mathrm{Hom}(G_{j},G_{i}))$. Since each
$\mathcal{F}_{k,b}$ is a sub-bundle of $\mathcal{E}_{b}$ (isomorphic to
$\mathcal{F}_{k}$), $\overline{\partial}_{b}$ preserves it, meaning that
$\gamma_{ij}=0$ whenever $i>j$. Therefore, $\Phi(b)=\sum_{i\leqslant
j}\gamma_{ij}$ is upper triangular.

Now, let for all $t\geqslant 1$,
$g_{t}=\sum_{k=1}^{m}a_{t}t^{k}\mathrm{Id}_{G_{k}}$ where
$a_{t}\in\mathbb{R}_{+}^{*}$ is chosen such that $g_{t}$ has determinant $1$.
In particular, $g_{t}\in G$ and for all $t$, when $g_{t}\cdot b\in B$,
$g_{t}\cdot\overline{\partial}_{0}=\overline{\partial}_{0}$ so,

$\overline{\partial}_{g_{t}\cdot
b}=g_{t}\cdot\overline{\partial}_{b}=\overline{\partial}_{0}+g_{t}\Phi(b)g_{t}^{-1}=\overline{\partial}_{0}+\sum_{i\leqslant
j}t^{i-j}\gamma_{ij}.$

Moreover, the decomposition $E=\bigoplus_{k=1}^{m}G_{k}$ is orthogonal, which
implies that the decomposition
$\Omega^{0,1}(X,\mathrm{End}(E))=\bigoplus_{i,j=1}^{m}\Omega^{0,1}(X,\mathrm{Hom}(G_{j},G_{i}))$
is orthogonal too for any $\left<\cdot,\cdot\right>_{\varepsilon}$. In
particular,

$\left\|\Phi(g_{t}\cdot b)\right\|_{\varepsilon}^{2}=\sum_{i\leqslant
j}t^{2i-2j}\left\|\gamma_{ij}\right\|_{\varepsilon}^{2}\leqslant\left\|\Phi(b)\right\|_{\varepsilon}^{2},$

because $t\geqslant 1$. Since $\Phi$ is a smooth embedding, there are, up to
shrinking $B$, positive constants $C_{1},C_{2}$ such that for all
$b^{\prime}\in B$,
$\frac{1}{C_{1}}\left\|b\right\|\leqslant\left\|\Phi(b)\right\|_{0}\leqslant
C_{2}\left\|b\right\|$. Let $r>0$ such that the closed ball of centre $0$ and
radius $r$ is included in $B$. Let $B^{\prime}$ be the open ball of centre $0$
and radius $\frac{r}{C_{1}C_{2}}$ and assume that $b\in B^{\prime}$. Let
$T=\sup\\{t\geqslant 1|g_{t}\cdot b\in B\\}$. $g_{1}=\mathrm{Id}_{E}$ so
$T\geqslant 1$ is well defined. If $T$ is finite, we have, for all $t<T$,
$\left\|g_{t}\cdot b\right\|\leqslant C_{1}\left\|\Phi(g_{t}\cdot
b)\right\|_{0}^{2}\leqslant C_{1}\left\|\Phi(b)\right\|_{0}^{2}\leqslant
C_{1}C_{2}\left\|b\right\|\leqslant r$. We stay in the closed ball of radius
$r$, included in $B$. When $t\rightarrow T$, we reach a point of $B$ and we
contradict the maximality of $T$ by openness of $B$. It proves that
$T=+\infty$ and,

$\lim_{t\rightarrow+\infty}\Phi(g_{t}\cdot b)=\sum_{k=1}^{m}\gamma_{kk}.$

This convergence implies that $g_{t}\cdot b\rightarrow b_{\infty}$ for some
$b_{\infty}$ in the closed ball of radius $r$ because $\Phi$ is a closed
embedding. In particular, $b_{\infty}\in B$. Notice that for all $k$,
$(G_{k},\overline{\partial}_{b_{\infty}|_{G_{k}}})=(G_{k},\overline{\partial}_{0}+\gamma_{kk})=\mathcal{G}_{k,b}\cong\mathcal{G}_{k}$
is $[\Theta_{0}]$-stable. It means that
$\mathcal{E}^{\prime}=(E,\overline{\partial}_{b_{\infty}})$ is
$[\Theta_{0}]$-polystable and by [2, Theorem 1, 3, 4]111In this paper, the
results are proven in the Kähler case. However, in the third point of their
conclusion and in the remark following [2, Proposition 3.2], the authors
notice that the only obstruction for it to work on any compact complex
manifold equipped with a Gauduchon metric is that in this case, the slope
might not be of topological nature. In the balanced case, the slope is of
topological nature since it only depends on the first Chern class of the
bundle and not on the choice of its representative.,
$\overline{\partial}_{b_{\infty}}$ is in the $G$-orbit of
$\overline{\partial}_{0}$ i.e. it equals $\overline{\partial}_{0}$. It means
that for all $k$, $\gamma_{kk}=0$. Up to replacing $B$ by $B^{\prime}$, it
means that $\Phi(b)=\sum_{i<j}\gamma_{ij}$ is strictly upper diagonal in the
orthogonal decomposition $E=\bigoplus_{k=1}^{m}G_{k}$ whenever $b\in B$.

Step 2 : Computing the scalar product between $\nu_{\infty,\varepsilon}(b)$
and each
$a_{S}=\frac{\boldsymbol{\mathrm{i}}}{\mathrm{rk}(S)}\mathrm{Id}_{S}-\frac{\boldsymbol{\mathrm{i}}}{\mathrm{rk}(S^{\bot})}\mathrm{Id}_{S^{\bot}}\in\mathfrak{k}$.

Let $\mathcal{S}\subset\mathcal{E}_{b}$ be a sub-bundle with
$0<\mathrm{rk}(\mathcal{S})<\mathrm{rk}(\mathcal{E})$ and $S$ its smooth
structure. Then, in the decomposition $E=S\oplus S^{\bot}$, we can write
$\overline{\partial}_{b}=\partial_{b|S}+\beta_{S,b}$ and by [12, Equation
(1.6.12)],

$\boldsymbol{\mathrm{i}}F_{b}=\begin{pmatrix}\boldsymbol{\mathrm{i}}F_{S}-\boldsymbol{\mathrm{i}}\beta_{S,b}\wedge\beta_{S,b}^{*}&\boldsymbol{\mathrm{i}}\partial_{\varepsilon,b}\beta_{S,b}\\\
-\boldsymbol{\mathrm{i}}\overline{\partial}_{b}\beta_{S,b}^{*}&\boldsymbol{\mathrm{i}}F_{S^{\bot}}-\boldsymbol{\mathrm{i}}\beta_{S,b}^{*}\wedge\beta_{S,b}\end{pmatrix},$

where $F_{b}$ is the curvature of the Chern connection associated to
$\overline{\partial}_{b}$ on $E$, $F_{S}$ is some curvature form on
$\mathcal{S}$ and $F_{S^{\bot}}$ is some curvature form on
$\mathcal{E}_{b}/\mathcal{S}$ (they depend on $b$). By Chern-Weil theory,

$\int_{X}\mathrm{tr}(\boldsymbol{\mathrm{i}}F_{S})\wedge\Theta_{\varepsilon}=2\pi\mathrm{rk}(S)\mu_{\varepsilon}(S),\qquad\int_{X}\mathrm{tr}(\boldsymbol{\mathrm{i}}F_{S^{\bot}})\wedge\Theta_{\varepsilon}=2\pi\mathrm{rk}(E/S)\mu_{\varepsilon}(E/S),$

and for any $(0,1)$-form $\beta$,
$\displaystyle\int_{X}\mathrm{tr}(\boldsymbol{\mathrm{i}}\beta^{*}\wedge\beta)\wedge\Theta_{\varepsilon}=\left\|\beta\right\|_{\varepsilon}^{2}$
and
$\displaystyle\int_{X}\mathrm{tr}(\boldsymbol{\mathrm{i}}\beta\wedge\beta^{*})\wedge\Theta_{\varepsilon}=-\left\|\beta\right\|_{\varepsilon}^{2}$.
We can now compute that,

$\displaystyle\left<\nu_{\infty,\varepsilon}(\overline{\partial}_{b}),a_{S}\right>_{\varepsilon}$
$\displaystyle=\left<\Lambda_{\varepsilon}F_{b}+\boldsymbol{\mathrm{i}}c_{\varepsilon}\mathrm{Id}_{E},a_{S}\right>_{\varepsilon}$
$\displaystyle=\int_{X}\mathrm{tr}\left(\Lambda_{\varepsilon}\boldsymbol{\mathrm{i}}F_{b}\left(\frac{1}{\mathrm{rk}(S)}\mathrm{Id}_{S}-\frac{1}{\mathrm{rk}(S^{\bot})}\mathrm{Id}_{S^{\bot}}\right)\right)\mathrm{Vol}_{\varepsilon}$
$\displaystyle=\frac{1}{\mathrm{rk}(S)}\int_{X}\mathrm{tr}(\boldsymbol{\mathrm{i}}F_{b}\mathrm{Id}_{S})\wedge\Theta_{\varepsilon}-\frac{1}{\mathrm{rk}(S^{\bot})}\int_{X}\mathrm{tr}(\boldsymbol{\mathrm{i}}F_{b}\mathrm{Id}_{S^{\bot}})\wedge\Theta_{\varepsilon}$
$\displaystyle=\frac{1}{\mathrm{rk}(S)}\int_{X}\mathrm{tr}(\boldsymbol{\mathrm{i}}F_{S}-\boldsymbol{\mathrm{i}}\beta_{S,b}\wedge\beta_{S,b}^{*})\wedge\Theta_{\varepsilon}-\frac{1}{\mathrm{rk}(S^{\bot})}\int_{X}\mathrm{tr}(\boldsymbol{\mathrm{i}}F_{S^{\bot}}-\boldsymbol{\mathrm{i}}\beta_{S,b}^{*}\wedge\beta_{S,b}\mathrm{Id}_{S^{\bot}})\wedge\Theta_{\varepsilon}$
$\displaystyle=2\pi\mu_{\varepsilon}(S)+\frac{1}{\mathrm{rk}(S)}\left\|\beta_{S,b}\right\|_{\varepsilon}^{2}-2\pi\mu_{\varepsilon}(E/S)+\frac{1}{\mathrm{rk}(S^{\bot})}\left\|\beta_{S,b}\right\|_{\varepsilon}^{2}$
$\displaystyle=-2\pi
l_{\mathcal{S}}(\varepsilon)+\left(\frac{1}{\mathrm{rk}(S)}+\frac{1}{\mathrm{rk}(S^{\bot})}\right)\left\|\beta_{S,b}\right\|_{\varepsilon}^{2}.$

Step 3 : Inequality with $\nu_{\infty,\varepsilon}$.

Applying step 2 with the particular case
$\mathcal{S}=\mathcal{F}_{k,b}\subset\mathcal{E}$ with $k<m$, we obtain
$\beta_{S,b}=\mathrm{Id}_{F_{k,b}}\overline{\partial}_{b|F_{k,b}^{\bot}}=\sum_{i\leqslant
k\leqslant j}\gamma_{ij}$. Moreover, all the
$l_{\mathcal{S}}:L_{1}^{2}(U)\rightarrow\mathbb{R}$ are continuous thus,

$\displaystyle\sum_{k=1}^{m}\left<\nu_{\infty,\varepsilon}(\overline{\partial}_{b}),a_{F_{k,b}}\right>_{\varepsilon}$
$\displaystyle=\sum_{k=1}^{m}-2\pi
l_{F_{k,b}}(\varepsilon)+\left(\frac{1}{\mathrm{rk}(F_{k,b})}+\frac{1}{\mathrm{rk}(F_{k,b}^{\bot})}\right)\left\|\mathrm{Id}_{F_{k,b}}\Phi(b)_{|F_{k,b}^{\bot}}\right\|_{\varepsilon}^{2}$
$\displaystyle\geqslant\sum_{k=1}^{m}C_{3}\sum_{1\leqslant i\leqslant
k\leqslant j\leqslant
m}\left\|\gamma_{ij}\right\|_{0}^{2}-C_{4}\left\|\varepsilon\right\|_{L_{1}^{2}}$
$\displaystyle\geqslant
C_{3}\left\|\Phi(b)\right\|_{0}^{2}-C_{4}\left\|\varepsilon\right\|_{L_{1}^{2}}$
$\displaystyle\geqslant\frac{C_{3}}{C_{1}}\left\|b\right\|^{2}-C_{4}\left\|\varepsilon\right\|_{L_{1}^{2}},$

where $C_{3}$ and $C_{4}$ are positive constants that only depend on
$\mathrm{rk}(\mathcal{E})$, $\max_{1\leqslant k\leqslant
m}\left\\{\left\|l_{\mathcal{F}_{k,b}}\right\|\right\\}$,
$\max_{\varepsilon\in U}\\{\mathrm{Vol}_{\varepsilon}(X)\\}$ (which is finite
up to shrinking $U$) and the fact that the
$\left\|\cdot\right\|_{\varepsilon}$ are uniformly equivalent. Clearly, it
exists a constant $C_{5}$ independent from $\varepsilon$ and $b$ such that for
all $k$ and $\varepsilon$, $\left\|a_{F_{k,b}}\right\|_{\varepsilon}\leqslant
C_{5}$. Let for all $\varepsilon$, $\Pi_{\varepsilon}$ be the orthogonal
projection on $\mathfrak{k}$ with respect to
$\left<\cdot,\cdot\right>_{\varepsilon}$. We have,

$\sum_{k=1}^{m}\left<\nu_{\infty,\varepsilon}(\overline{\partial}_{b}),a_{F_{k,b}}\right>_{\varepsilon}=\sum_{k=1}^{m}\left<\Pi_{\varepsilon}\nu_{\infty,\varepsilon}(\overline{\partial}_{b}),a_{F_{k,b}}\right>_{\varepsilon}\leqslant
C_{5}\sum_{k=1}^{m}\left\|\Pi_{\varepsilon}\nu_{\infty,\varepsilon}(\overline{\partial}_{b})\right\|_{\varepsilon}=mC_{5}\left\|\Pi_{\varepsilon}\nu_{\infty,\varepsilon}(\overline{\partial}_{b})\right\|.$

We deduce that
$\left\|b\right\|^{2}=\mathrm{O}\left(\left\|\Pi_{\varepsilon}\nu_{\infty,\varepsilon}(\overline{\partial}_{b})\right\|_{\varepsilon}+\left\|\varepsilon\right\|_{L_{1}^{2}}\right)$
where the $\mathrm{O}$ is independent from $\varepsilon$ and $b$. From now on,
all the asymptotic developments are with respect to
$(\varepsilon,b)\rightarrow(0,0)$.

Step 4 : Conclusion.

By Proposition 4.1, we have
$\nu_{\varepsilon}(b)=\nu_{\infty,\varepsilon}(\overline{\partial}_{\varepsilon,b})$
with
$\overline{\partial}_{\varepsilon,b}=\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,b)}\cdot\overline{\partial}_{b}$
for some smooth $\sigma:L_{1}^{2}(U)\times B\rightarrow
L_{3}^{2}(\Omega_{0}^{0}(X,\mathrm{End}_{H}(E,h)))$ which verifies
$\sigma(0,0)=0$ and $\frac{\partial}{\partial b}|_{b=0}\sigma(0,b)=0$.
Therefore
$\left\|\sigma(\varepsilon,b)\right\|_{L_{3}^{2}}=\mathrm{O}(\left\|\varepsilon\right\|_{L_{1}^{2}}+\left\|b\right\|^{2})$
and using the fact that $\nu_{\varepsilon}$ takes values in $\mathfrak{k}$,

$\displaystyle\left\|\nu_{\varepsilon}(b)-\Pi_{\varepsilon}\nu_{\infty,\varepsilon}(\overline{\partial}_{b})\right\|_{\varepsilon}$
$\displaystyle=\left\|\Pi_{\varepsilon}(\nu_{\varepsilon}(b)-\nu_{\infty,\varepsilon}(\overline{\partial}_{b}))\right\|_{\varepsilon}$
$\displaystyle=\left\|\Pi_{\varepsilon}(\nu_{\infty,\varepsilon}(\overline{\partial}_{\varepsilon,b})-\nu_{\infty,\varepsilon}(\overline{\partial}_{b}))\right\|_{\varepsilon}$
$\displaystyle=\left\|\Pi_{\varepsilon}(\nu_{\infty,\varepsilon}(\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,b)}\cdot\overline{\partial}_{b})-\nu_{\infty,\varepsilon}(\overline{\partial}_{b}))\right\|_{\varepsilon}$
$\displaystyle=\left\|\Pi_{\varepsilon}\Delta_{\varepsilon,\overline{\partial}_{b}}\sigma(\varepsilon,b)\right\|_{\varepsilon}+\mathrm{O}(\left\|\sigma(\varepsilon,b)\right\|_{L_{3}^{2}}^{2})\textrm{
by Proposition \ref{PRO:Calcul dPsi(0)},}$
$\displaystyle=\left\|\Pi_{0}\Delta_{0}\sigma(\varepsilon,b)\right\|_{\varepsilon}+\mathrm{O}((\left\|\varepsilon\right\|_{L_{1}^{2}}+\left\|b\right\|)\left\|\sigma(\varepsilon,b)\right\|_{L_{3}^{2}}^{2})+\mathrm{O}(\left\|\sigma(\varepsilon,b)\right\|_{L_{3}^{2}}^{2})$
$\displaystyle=\mathrm{O}(\left\|\varepsilon\right\|_{L_{1}^{2}}^{2}+\left\|b\right\|^{3}).$

The last equality holds because $\Delta_{0}$ takes values in
$\mathfrak{k}^{\bot}$ (for $\left<\cdot,\cdot\right>_{0}$) hence
$\Pi_{0}\Delta_{0}=0$. Finally,

$\left\|b\right\|^{2}=\mathrm{O}\left(\left\|\Pi_{\varepsilon}\nu_{\infty,\varepsilon}(\overline{\partial}_{b})\right\|_{\varepsilon}+\left\|\varepsilon\right\|_{L_{1}^{2}}\right)=\mathrm{O}(\left\|\nu_{\varepsilon}(b)\right\|_{\varepsilon}+\left\|\varepsilon\right\|_{L_{1}^{2}}+\left\|b\right\|^{3}),$

so
$\left\|b\right\|^{2}=\mathrm{O}(\left\|\nu_{\varepsilon}(b)\right\|_{\varepsilon}+\left\|\varepsilon\right\|_{L_{1}^{2}})$,
which is the wanted bound. ∎

###  4.3 Finding a zero of the moment map

The method now consists in defining a vector field whose flow converges toward
a zero of the moment map. When $b\in B$ and $a\in\mathfrak{g}$, we denote by
$L_{b}a=\frac{\partial}{\partial t}|_{t=0}\boldsymbol{\mathrm{e}}^{ta}\cdot
b\in T_{b}B$ the infinitesimal action of $a$ on $b$. Following [8, Chapter 3],
we define the vector field,

$V_{\varepsilon}:b\mapsto-\boldsymbol{\mathrm{i}}L_{b}\nu_{\varepsilon}(b).$

Let $t\mapsto b_{\varepsilon}(t)$ be the flow on $V_{\varepsilon}$ whose
starting point will be defined by the following proposition.

######  Proposition 4.4.

Up to shrinking $U$, for all $\varepsilon\in U$, if we choose
$b_{\varepsilon}(0)$ close enough to $0$, $b_{\varepsilon}$ is defined for all
non-negative times,
$t\mapsto\left\|\nu_{\varepsilon}(b_{\varepsilon}(t))\right\|_{\varepsilon}$
decreases and
$\left\|b_{\varepsilon}(t)\right\|=\mathrm{O}\left(\sqrt{\left\|\varepsilon\right\|_{L_{1}^{2}}}\right)$
uniformly with respect to $t$.

###### Proof.

For all $t$ where the flow is defined, whatever is the choice of the starting
point,

$\displaystyle\frac{\partial}{\partial
t}\left(\frac{1}{2}\left\|\nu_{\varepsilon}(b_{\varepsilon}(t))\right\|_{\varepsilon}^{2}\right)$
$\displaystyle=\left<d\nu_{\varepsilon}(b_{\varepsilon}(t))(-\boldsymbol{\mathrm{i}}L_{b}\nu_{\varepsilon}(b_{\varepsilon}(t))),\nu_{\varepsilon}(b_{\varepsilon}(t))\right>_{\varepsilon}$
$\displaystyle=-\Omega_{\varepsilon}(L_{b}\nu_{\varepsilon}(b_{\varepsilon}(t)),\boldsymbol{\mathrm{i}}L_{b}\nu_{\varepsilon}(b_{\varepsilon}(t)))\textrm{
because $\nu_{\varepsilon}$ is a moment map,}$ $\displaystyle\leqslant
0\textrm{ by Lemma \ref{LEM:Omega positive}}.$

It proves that
$t\mapsto\left\|\nu_{\varepsilon}(b_{\varepsilon}(t))\right\|_{\varepsilon}$
decreases while the flow is defined. In particular, by Proposition 4.3,

$\left\|b_{\varepsilon}(t)\right\|^{2}\leqslant
C(\left\|\nu_{\varepsilon}(b_{\varepsilon}(t))\right\|_{\varepsilon}+\left\|\varepsilon\right\|_{L_{1}^{2}})\leqslant
C(\left\|\nu_{\varepsilon}(b_{\varepsilon}(0))\right\|_{\varepsilon}+\left\|\varepsilon\right\|_{L_{1}^{2}})$
(4)

Let $r>0$ such that the closed ball of centre $0$ and radius $r$ is included
in $B$. Assume, up to shrinking $U$, that for all $\varepsilon$,
$\left\|\varepsilon\right\|_{L_{1}^{2}}\leqslant\frac{r^{2}}{2C}$ and
$\left\|\nu_{\varepsilon}(0)\right\|_{\varepsilon}\leqslant\frac{r^{2}}{4C}$.
Then choose $b_{\varepsilon}(0)$ close enough to $0$ so
$\left\|\nu_{\varepsilon}(b_{\varepsilon}(0))\right\|_{\varepsilon}\leqslant
2\left\|\nu_{\varepsilon}(0)\right\|_{\varepsilon}\leqslant\frac{r^{2}}{2C}$.
It implies by (4) that $\left\|b_{\varepsilon}(t)\right\|\leqslant r$ hence
$b_{\varepsilon}$ stays in a compact set included in $B$. By the finite time
explosion theorem, the flow is defined at all time. Moreover,
$\left\|b_{\varepsilon}(t)\right\|^{2}\leqslant
C(\left\|\nu_{\varepsilon}(b_{\varepsilon}(0))\right\|_{\varepsilon}+\left\|\varepsilon\right\|_{L_{1}^{2}})\leqslant
C(2\left\|\nu_{\varepsilon}(0)\right\|_{\varepsilon}+\left\|\varepsilon\right\|_{L_{1}^{2}})=\mathrm{O}(\left\|\varepsilon\right\|_{L_{1}^{2}})$.
∎

Let $(g_{\varepsilon}(t))$ in $G$ be the smooth function defined by,

$g_{\varepsilon}(0)=\mathrm{Id}_{E},\qquad
g_{\varepsilon}(t)^{-1}g_{\varepsilon}^{\prime}(t)=\boldsymbol{\mathrm{i}}\nu_{\varepsilon}(b_{\varepsilon}(t)).$

Similarly to [8, Lemma 3.2], we have for all $t$,

$\frac{\partial}{\partial t}(g_{\varepsilon}(t)^{-1}\cdot
b_{\varepsilon}(0))=-g_{\varepsilon}(t)^{-1}g_{\varepsilon}^{\prime}(t)g_{\varepsilon}^{-1}(t)\cdot
b_{\varepsilon}(0)=-\boldsymbol{\mathrm{i}}L_{g_{\varepsilon}(t)^{-1}\cdot
b_{\varepsilon}(0)}\nu_{\varepsilon}(b_{\varepsilon}(t)).$

By uniqueness of the flow, it proves that for all $t$,
$g_{\varepsilon}(t)^{-1}\cdot b_{\varepsilon}(0)=b_{\varepsilon}(t)$. In
particular, $\mathcal{O}$ is preserved by the flow. Let for all $t$,
$\tilde{g}_{\varepsilon}(t)=\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,b_{\varepsilon}(0))}g_{\varepsilon}(t)\boldsymbol{\mathrm{e}}^{-\sigma(\varepsilon,b_{\varepsilon}(t))}\in\mathcal{G}^{\mathbb{C}}$.
We have,

$\displaystyle\overline{\partial}_{\varepsilon,b_{\varepsilon}(t)}$
$\displaystyle=\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,b_{\varepsilon}(t))}\cdot\overline{\partial}_{b_{\varepsilon}(t)}$
$\displaystyle=\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,b_{\varepsilon}(t))}g_{\varepsilon}(t)^{-1}\cdot\overline{\partial}_{b_{\varepsilon}(0)}$
$\displaystyle=\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,b_{\varepsilon}(t))}g_{\varepsilon}(t)^{-1}\boldsymbol{\mathrm{e}}^{-\sigma(\varepsilon,b_{\varepsilon}(0))}\cdot\overline{\partial}_{\varepsilon,b_{\varepsilon}(0)}$
$\displaystyle=\tilde{g}_{\varepsilon}(t)^{-1}\cdot\overline{\partial}_{\varepsilon,b_{\varepsilon}(0)}.$

Now, when $\overline{\partial}$ is a Dolbeault operator on $E$, we call
$M_{\varepsilon,\overline{\partial}}$ the Donaldson functional [5, Proposition
6] associated with $\Theta_{\varepsilon}$ and $\overline{\partial}$. It takes
two Hermitian metrics on $E$ as argument and is characterised by,

$\forall
k_{1},k_{2},k_{3},M_{\varepsilon,\overline{\partial}}(k_{1},k_{2})+M_{\varepsilon,\overline{\partial}}(k_{2},k_{3})=M_{\varepsilon,\overline{\partial}}(k_{1},k_{3}),$
(5) $\frac{\partial}{\partial
s}|_{s=0}M_{\varepsilon,\overline{\partial}}(k,\boldsymbol{\mathrm{e}}^{-s/2}\cdot
k)=\int_{X}\mathrm{tr}\left(v(\Lambda_{\varepsilon}\boldsymbol{\mathrm{i}}F_{\overline{\partial},k}-c_{\varepsilon}\mathrm{Id}_{E})\right)\mathrm{Vol}_{\varepsilon}.$
(6)

Here, $s\in\Omega^{0}(X,\mathrm{End}_{H}(E,k))$ and $v\in
T_{s}\Omega^{0}(X,\mathrm{End}_{H}(E,k))$ are $k$-Hermitian,
$F_{\overline{\partial},k}$ is the curvature associated to the Chern
connection of $(\overline{\partial},k)$ and $\mathcal{G}^{\mathbb{C}}$ acts on
metrics by $(f\cdot k)(\xi,\eta)=k(f^{-1}\xi,f^{-1}\eta)$222In literature, it
is more common to only consider the action of a positive definite Hermitian
$\boldsymbol{\mathrm{e}}^{s}$ with respect to the metric $k$. In this case,
the notation used is $k\boldsymbol{\mathrm{e}}^{s}:(\xi,\eta)\mapsto
k(\boldsymbol{\mathrm{e}}^{s}\xi,\eta)$. By symmetry, we obtain that
$\boldsymbol{\mathrm{e}}^{-s/2}\cdot k=k\boldsymbol{\mathrm{e}}^{s}$. To avoid
confusion, we only the use in this case the notation
$\boldsymbol{\mathrm{e}}^{-s/2}\cdot k$.. Ulhenbeck and Yau showed that on a
stable bundle, $M(h,\boldsymbol{\mathrm{e}}^{-s/2}\cdot h)$ uniformly bounds
$s$ [27]. We refer the reader to Simpson’s proof of this result [23,
Proposition 5.3]333Simpson proves it in the Kähler case but the closedness of
$\omega$ is not used in the proof of [23, Proposition 5.3]. Only the
closedness of $\omega^{n-1}$ is necessary for [23, Proposition 5.1]. He also
only proves it when $\mathrm{tr}(s)=0$ identically, but this hypothesis is
only useful to get that the $u_{\infty}$ built in [23, Lemma 5.4] is not a
homothety (see [23, Lemma 5.5]). For this, it is enough to know that
$\int_{X}\mathrm{tr}(s)\mathrm{Vol}_{0}=0$ so the inequality holds in this
more general case.. Concretely, when $(E,\overline{\partial})$ is
$[\Theta_{\varepsilon}]$-stable, there exists positive constants
$C_{1,\varepsilon},C_{2,\varepsilon}$ that depend on $\varepsilon$ (and
$\overline{\partial}$) such that for all
$s\in\Omega_{0}^{0}(X,\mathrm{End}_{H}(E,h))$,

$\left\|s\right\|_{\varepsilon}\leqslant
C_{1,\varepsilon}+C_{2,\varepsilon}M_{\varepsilon,\overline{\partial}}(h,\boldsymbol{\mathrm{e}}^{-s/2}\cdot
h).$ (7)

We could even get this inequality with a $\mathcal{C}^{0}$ norm instead of an
$L^{2}$ norm on $s$. Now, let,

$\varphi_{\varepsilon}:b\mapsto
M_{\varepsilon,\overline{\partial}_{E}}(h,f_{\varepsilon,b}\cdot h),$

where $f_{\varepsilon,b}$ is defined as the gauge transformation up to
constant homothety such that
$\overline{\partial}_{\varepsilon,b}=f_{\varepsilon,b}^{-1}\cdot\overline{\partial}_{E}$.
It is unique by simplicity of $\mathcal{E}$. We want now to show that
$\varphi$ decreases with the flow and use the inequality (7) to show that
$g_{\varepsilon}$ is bounded, thus converges up to extraction.

In the classical case where we consider the heat equation flow, the derivative
of $t\mapsto\varphi(b_{\varepsilon}(t))$ is given by
$-2\left\|\nu_{\varepsilon}(b_{\varepsilon}(t))\right\|_{\varepsilon}^{2}$
(the constant $2$ may disapear in function of the conventions). See for
reference [12, Proposition 6.9.1], [23, Lemma 7.1], [5, Section 1.2]. It is a
natural consequence of (6). Here, the fact that $\tilde{g}_{\varepsilon}(t)$
is not exactly $g_{\varepsilon}(t)$ makes this equality false in general.
However, since these two gauge transformations are close when
$\varepsilon\rightarrow 0$ uniformly with respect to $t$, we are still able to
show the wanted decrease. The issue is that we fix the metric and we make the
complex structure vary to find a HYM connection. However, Donaldson’s
functional works better with variations of the metric when the holomorphic
structure is fixed. This issue however, is only technical and we can move from
one to the other. The proof of the next proposition is the only part of this
article where we need to consider an other metric on $\mathcal{E}$.

######  Proposition 4.5.

We have for all $t$,

$\frac{\partial}{\partial
t}\varphi_{\varepsilon}(b_{\varepsilon}(t))=-2\left\|\nu_{\varepsilon}(b_{\varepsilon}(t))\right\|_{\varepsilon}^{2}+\mathrm{o}(\left\|\nu_{\varepsilon}(b_{\varepsilon}(t))\right\|_{\varepsilon}^{2}),$

where the $\mathrm{o}$ is when $\varepsilon\rightarrow 0$ and is uniform with
respect to $t$. In particular, up to shrinking $U$, for all $\varepsilon$,
$t\mapsto\varphi_{\varepsilon}(b_{\varepsilon}(t))$ decreases.

###### Proof.

Step 1 : Derivative of the Hermitian part of a gauge transformation.

$\varphi_{\varepsilon}$ is defined intrinsically in the sens that it doesn’t
depend on the choice of the starting point $b_{\varepsilon}(0)$. Since we have
moreover a uniform bound on the norm of $b_{\varepsilon}(t)$ given by
Proposition 4.4, it is enough to verify the wanted equality at $t=0$ and the
uniform bound follows.

For all $t$ near $0$, we have
$\overline{\partial}_{\varepsilon,b_{\varepsilon}(t)}=\tilde{g}_{\varepsilon}(t)^{-1}\cdot\overline{\partial}_{\varepsilon,b_{\varepsilon}(0)}=\tilde{g}_{\varepsilon}(t)^{-1}f_{\varepsilon,b_{\varepsilon}(0)}^{-1}\cdot\overline{\partial}_{E}$
hence
$f_{\varepsilon,b_{\varepsilon}(t)}=f_{\varepsilon,b_{\varepsilon}(0)}\tilde{g}_{\varepsilon}(t)$.
In particular, if we set $k=f_{\varepsilon,b_{\varepsilon}(0)}\cdot h$, we
have,

$f_{\varepsilon,b_{\varepsilon}(t)}\cdot
h=f_{\varepsilon,b_{\varepsilon}(0)}\tilde{g}_{\varepsilon}(t)f_{\varepsilon,b_{\varepsilon}(0)}^{-1}\cdot
k.$

Let us write the polar decomposition of
$f_{\varepsilon,b_{\varepsilon}(0)}\tilde{g}_{\varepsilon}(t)f_{\varepsilon,b_{\varepsilon}(0)}^{-1}$
with respect to $k$ as $\boldsymbol{\mathrm{e}}^{-s(t)/2}u(t)$. In particular,

$\boldsymbol{\mathrm{e}}^{-s(t)}=f_{\varepsilon,b_{\varepsilon}(0)}\tilde{g}_{\varepsilon}(t)f_{\varepsilon,b_{\varepsilon}(0)}^{-1}(f_{\varepsilon,b_{\varepsilon}(0)}\tilde{g}_{\varepsilon}(t)f_{\varepsilon,b_{\varepsilon}(0)}^{-1})^{\dagger},$

where $\dagger$ is the adjoint with respect to $k$. $s$ is smooth with respect
to any Sobolev norm and we have $s(0)=0$ because
$\tilde{g}_{\varepsilon}(0)=0$ so,

$\displaystyle-s(t)$
$\displaystyle=\boldsymbol{\mathrm{e}}^{-s(t)}-\mathrm{Id}_{E}+\mathrm{o}(t)$
$\displaystyle=f_{\varepsilon,b_{\varepsilon}(0)}\tilde{g}_{\varepsilon}(t)f_{\varepsilon,b_{\varepsilon}(0)}^{-1}f_{\varepsilon,b_{\varepsilon}(0)}^{-1\dagger}\tilde{g}_{\varepsilon}(t)^{\dagger}f_{\varepsilon,b_{\varepsilon}(0)}^{\dagger}-\mathrm{Id}_{E}+\mathrm{o}(t)$
$\displaystyle=(f_{\varepsilon,b_{\varepsilon}(0)}\tilde{g}_{\varepsilon}^{\prime}(0)f_{\varepsilon,b_{\varepsilon}(0)}^{-1}+f_{\varepsilon,b_{\varepsilon}(0)}^{-1\dagger}\tilde{g}_{\varepsilon}^{\prime}(0)^{\dagger}f_{\varepsilon,b_{\varepsilon}(0)}^{\dagger})t+\mathrm{o}(t)$
$\displaystyle=2\Re_{k}(f_{\varepsilon,b_{\varepsilon}(0)}\tilde{g}_{\varepsilon}^{\prime}(0)f_{\varepsilon,b_{\varepsilon}(0)}^{-1})t+\mathrm{o}(t),$

where $\Re_{k}$ is the Hermitian part with respect to $k$. We deduce that
$s^{\prime}(0)=-2\Re_{k}(f_{\varepsilon,b_{\varepsilon}(0)}\tilde{g}_{\varepsilon}^{\prime}(0)f_{\varepsilon,b_{\varepsilon}(0)}^{-1})$.
Now, recall that for all $t$,
$\tilde{g}_{\varepsilon}(t)=\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,b_{\varepsilon}(0))}g_{\varepsilon}(t)\boldsymbol{\mathrm{e}}^{-\sigma(\varepsilon,b_{\varepsilon}(t))}$.
Therefore, since $\left\|b_{\varepsilon}(0)\right\|=\mathrm{o}(1)$ when
$\varepsilon\rightarrow 0$,

$\displaystyle\tilde{g}_{\varepsilon}^{\prime}(0)$
$\displaystyle=d\exp(\sigma(\varepsilon,b_{\varepsilon}(0)))\frac{\partial}{\partial
b}|_{b=b_{\varepsilon}(0)}\sigma(\varepsilon,b)b_{\varepsilon}^{\prime}(0)+\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,b_{\varepsilon}(0))}g_{\varepsilon}^{\prime}(0)\boldsymbol{\mathrm{e}}^{-\sigma(\varepsilon,b_{\varepsilon}(0))}$
$\displaystyle=g_{\varepsilon}^{\prime}(0)+\mathrm{o}(b_{\varepsilon}^{\prime}(0))+\mathrm{o}(g_{\varepsilon}^{\prime}(0))\textrm{
because $\sigma(0,0)=0$ and $\frac{\partial}{\partial
b}|_{b=0}\sigma(0,b)=0$,}$
$\displaystyle=\boldsymbol{\mathrm{i}}\nu_{\varepsilon}(b_{\varepsilon}(0))+\mathrm{o}(L_{b_{\varepsilon}(0)}\nu_{\varepsilon}(b_{\varepsilon}(0)))+\mathrm{o}(\nu_{\varepsilon}(b_{\varepsilon}(0)))$
$\displaystyle=\boldsymbol{\mathrm{i}}\nu_{\varepsilon}(b_{\varepsilon}(0))+\mathrm{o}(\nu_{\varepsilon}(b_{\varepsilon}(0))).$

$\boldsymbol{\mathrm{i}}\nu_{\varepsilon}(b_{\varepsilon}(0))$ is Hermitian
with respect to $h$ and $k=f_{\varepsilon,b_{\varepsilon}(0)}\cdot h$ so
$\boldsymbol{\mathrm{i}}f_{\varepsilon,b_{\varepsilon}(0)}\nu_{\varepsilon}(b_{\varepsilon}(0))f_{\varepsilon,b_{\varepsilon}(0)}^{-1}$
is Hermitian with respect to $k$. We deduce that
$s^{\prime}(0)=-2\boldsymbol{\mathrm{i}}f_{\varepsilon,b_{\varepsilon}(0)}\nu_{\varepsilon}(b_{\varepsilon}(0))f_{\varepsilon,b_{\varepsilon}(0)}^{-1}+\mathrm{o}(f_{\varepsilon,b_{\varepsilon}(0)}\nu_{\varepsilon}(b_{\varepsilon}(0))f_{\varepsilon,b_{\varepsilon}(0)}^{-1})$.

Step 2 : Relation between curvatures.

Since $f_{\varepsilon,b_{\varepsilon}(t)}\cdot
h=\boldsymbol{\mathrm{e}}^{-s(t)/2}u(t)\cdot
k=\boldsymbol{\mathrm{e}}^{-s(t)/2}\cdot k$, we have, by (5),

$\varphi_{\varepsilon}(b_{\varepsilon}(t))-\varphi_{\varepsilon}(b_{\varepsilon}(0))=M_{\varepsilon,\overline{\partial}_{E}}(h,\boldsymbol{\mathrm{e}}^{-s(t)/2}\cdot
k)-M_{\varepsilon,\overline{\partial}_{E}}(h,k)=M_{\varepsilon,\overline{\partial}_{E}}(k,\boldsymbol{\mathrm{e}}^{-s(t)/2}\cdot
k).$

Moreover, if we call $\partial_{E,k}$ the $(1,0)$ part of the Chern connection
associated with $(\overline{\partial}_{E},k)$ and we set
$\partial_{\varepsilon,b_{\varepsilon}(0)}=f_{\varepsilon,b_{\varepsilon}(0)}^{-1}\circ\partial_{E,k}\circ
f_{\varepsilon,b_{\varepsilon}(0)}$, we can compute that the connection
$\nabla_{\varepsilon,b_{\varepsilon}(0)}=\partial_{\varepsilon,b_{\varepsilon}(0)}+\overline{\partial}_{\varepsilon,b_{\varepsilon}(0)}$
is unitary with respect to $h$. Thus, by uniqueness,
$\nabla_{\varepsilon,b_{\varepsilon}(0)}$ is the Chern connection associated
with $(\overline{\partial}_{\varepsilon,b_{\varepsilon}(0)},h)$. Therefore,

$\displaystyle F_{\overline{\partial}_{E},k}$
$\displaystyle=\partial_{E,k}\overline{\partial}_{E,k}+\overline{\partial}_{E,k}\partial_{E,k}$
$\displaystyle=f_{\varepsilon,b_{\varepsilon}(0)}\circ\partial_{\varepsilon,b_{\varepsilon}(0)}\circ
f_{\varepsilon,b_{\varepsilon}(0)}^{-1}\circ
f_{\varepsilon,b_{\varepsilon}(0)}\circ\overline{\partial}_{\varepsilon,b_{\varepsilon}(0)}\circ
f_{\varepsilon,b_{\varepsilon}(0)}^{-1}+f_{\varepsilon,b_{\varepsilon}(0)}\circ\overline{\partial}_{\varepsilon,b_{\varepsilon}(0)}\circ
f_{\varepsilon,b_{\varepsilon}(0)}^{-1}\circ
f_{\varepsilon,b_{\varepsilon}(0)}\circ\partial_{\varepsilon,b_{\varepsilon}(0)}\circ
f_{\varepsilon,b_{\varepsilon}(0)}^{-1}$
$\displaystyle=f_{\varepsilon,b_{\varepsilon}(0)}F_{\overline{\partial}_{\varepsilon,b_{\varepsilon}(0)},h}f_{\varepsilon,b_{\varepsilon}(0)}^{-1}.$

Thus,
$\Lambda_{\varepsilon}\boldsymbol{\mathrm{i}}F_{\overline{\partial}_{E},k}-c_{\varepsilon}\mathrm{Id}_{E}=\boldsymbol{\mathrm{i}}f_{\varepsilon,b_{\varepsilon}(0)}\nu_{\varepsilon}(b_{\varepsilon}(0))f_{\varepsilon,b_{\varepsilon}(0)}^{-1}$.

Step 3 : Conclusion.

Finally, by (6),

$\displaystyle\frac{\partial}{\partial
t}\varphi_{\varepsilon}(b_{\varepsilon}(t))$
$\displaystyle=\int_{X}\mathrm{tr}\left(s^{\prime}(0)(\Lambda_{\varepsilon}\boldsymbol{\mathrm{i}}F_{\overline{\partial}_{E},k}-c_{\varepsilon}\mathrm{Id}_{E})\right)\mathrm{Vol}_{\varepsilon}$
$\displaystyle=-2\int_{X}\mathrm{tr}\left((f_{\varepsilon,b_{\varepsilon}(0)}\boldsymbol{\mathrm{i}}\nu_{\varepsilon}(b_{\varepsilon}(0))f_{\varepsilon,b_{\varepsilon}(0)}^{-1}+\mathrm{o}(f_{\varepsilon,b_{\varepsilon}(0)}\boldsymbol{\mathrm{i}}\nu_{\varepsilon}(b_{\varepsilon}(0))f_{\varepsilon,b_{\varepsilon}(0)}^{-1}))f_{\varepsilon,b_{\varepsilon}(0)}\boldsymbol{\mathrm{i}}\nu_{\varepsilon}(b_{\varepsilon}(0))f_{\varepsilon,b_{\varepsilon}(0)}^{-1}\right)\mathrm{Vol}_{\varepsilon}$
$\displaystyle=-2\left\|\nu_{\varepsilon}(b_{\varepsilon}(0)\right\|_{\varepsilon}^{2}+\mathrm{o}(\left\|\nu_{\varepsilon}(b_{\varepsilon}(0)\right\|_{\varepsilon}^{2}).$

∎

We are now ready to conclude this subsection.

######  Proposition 4.6.

If $\Theta_{\varepsilon}$ is in the semi-stable cone of $\mathcal{E}$, there
exists a $b_{\infty,\varepsilon}\in B\cap\overline{\mathcal{O}}$ such that
$\nu_{\varepsilon}(b_{\infty,\varepsilon})=0$. If moreover, we assume that
$\Theta_{\varepsilon}$ is in the stable cone of $\mathcal{E}$, then
$b_{\infty,\varepsilon}\in\mathcal{O}$. Moreover,
$\left\|b_{\infty,\varepsilon}\right\|=\mathrm{O}\left(\sqrt{\left\|\varepsilon\right\|_{L_{1}^{2}}}\right)$.

###### Proof.

Assume first of all that $\mathcal{E}$ is $[\Theta_{\varepsilon}]$-stable. Let
for all $t$, $f_{\varepsilon,b_{\varepsilon}(t)}\cdot
h=\boldsymbol{\mathrm{e}}^{-s(t)/2}\cdot h$ for some $s(t)$ Hermitian with
respect to $h$. We may assume that for all $t$,
$\displaystyle\int_{X}\mathrm{tr}(s(t))\mathrm{Vol}_{0}=0$. Then, by the
inequality (7) and Proposition 4.5, for all $t$,

$\left\|s(t)\right\|_{\varepsilon}\leqslant
C_{1,\varepsilon}+C_{2,\varepsilon}M(h,\boldsymbol{\mathrm{e}}^{-s(t)/2}\cdot
h)=C_{1,\varepsilon}+C_{2,\varepsilon}\varphi_{\varepsilon}(b_{\varepsilon}(t))\leqslant
C_{1,\varepsilon}+C_{2,\varepsilon}\varphi_{\varepsilon}(b_{\varepsilon}(0)).$

It means that $(s(t))$ is $L^{2}$ bounded. We have by definition
$\boldsymbol{\mathrm{e}}^{-s(t)}=f_{\varepsilon,b_{\varepsilon}(0)}\tilde{g}_{\varepsilon}(t)f_{\varepsilon,b_{\varepsilon}(0)}^{-1}$
hence $\left(\tilde{g}_{\varepsilon}(t)\right)$ and
$\left(\tilde{g}_{\varepsilon}(t)^{-1}\right)$ are $L^{2}$ bounded. We also
have
$g_{\varepsilon}(t)=\boldsymbol{\mathrm{e}}^{-\sigma(\varepsilon,b_{\varepsilon}(0))}\tilde{g}_{\varepsilon}(t)\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,b_{\varepsilon}(t))}$
thus $(g_{\varepsilon}(t))$ and $(g_{\varepsilon}(t)^{-1})$ are bounded too
(for any norm since they live in a finite dimensional vector space).
Therefore, we can find a sequence $t_{m}\rightarrow+\infty$ such that
$g_{\varepsilon}(t_{m})\rightarrow g_{\varepsilon}\in G$ is invertible and in
particular, $b_{\varepsilon}(t_{m})\rightarrow
b_{\infty,\varepsilon}=g_{\varepsilon}\cdot b_{\varepsilon}(0)\in\mathcal{O}$.
Let $l=\nu_{\varepsilon}(b_{\infty,\varepsilon})\in\mathfrak{k}$. Since the
norm of the moment map decreases by Proposition 4.4,
$\left\|\nu_{\varepsilon}(b_{\varepsilon}(t))\right\|_{\varepsilon}\rightarrow\left\|l\right\|_{\varepsilon}$.
By Proposition 4.5, up to shrinking $U$, $\frac{\partial}{\partial
t}\varphi_{\varepsilon}(b_{\varepsilon}(t))\leqslant-\left\|l\right\|_{\varepsilon}^{2}+\mathrm{o}(t)$
and inequality (7) implies that $\varphi$ is bounded from below hence $l=0$.

In the semi-stable case, we have by Theorem 3.7 that
$\varepsilon=\lim_{m\rightarrow+\infty}\varepsilon_{m}$ for some
$\varepsilon_{m}$ such that $\mathcal{E}$ is
$[\Theta_{\varepsilon_{m}}]$-stable. The $b_{\infty,\varepsilon_{m}}$ remain
in a compact set included in $B$ so, up to extraction, we may build
$b_{\infty,\varepsilon}\in B\cap\overline{\mathcal{O}}$ as
$\lim_{m\rightarrow+\infty}b_{\infty,\varepsilon_{m}}$. The bound on the norm
of $b_{\infty,\varepsilon}$ is given by Proposition 4.3. ∎

###  4.4 Convergence when $\varepsilon\rightarrow 0$ and continuity results

A consequence of Proposition 4.6 is Introduction’s Theorem 3.

######  Theorem 4.7.

Let $(X,\Theta_{0})$ be a compact balanced manifold and $\mathcal{E}$ a
$[\Theta_{0}]$-semi-stable sufficiently smooth holomorphic vector bundle.
There is an $L_{1}^{2}$ open neighbourhood $U$ of $0$ in the space of closed
$(n-1,n-1)$ forms and a family
$(\overline{\partial}_{\varepsilon})_{\varepsilon\in
U|\Theta_{\varepsilon}\in\mathcal{C}_{ss}(\mathcal{E})}$ of integrable
Dolbeault operators on $E$ such that for all $\varepsilon$,
$\overline{\partial}_{\varepsilon}$ is $\Theta_{\varepsilon}$-HYM and
$(E,\overline{\partial}_{\varepsilon})\cong\mathrm{Gr}_{[\Theta_{\varepsilon}]}(\mathcal{E})$.
Moreover, for all integer $d\geqslant 2$,

$\left\|\overline{\partial}_{\varepsilon}-\overline{\partial}_{0}\right\|_{L_{d}^{2}}=\mathrm{O}\left(\left\|\varepsilon\right\|_{L_{d+1}^{2}}+\sqrt{\left\|\varepsilon\right\|_{L_{1}^{2}}}\right)$

In particular,
$\overline{\partial}_{\varepsilon}\underset{\varepsilon\rightarrow
0}{\longrightarrow}\overline{\partial}_{0}$ for the $\mathcal{C}^{\infty}$
topology on $U$ and on the space of Dolbeault operators.

###### Proof.

Let for all $\varepsilon\in U$ such that $\mathcal{E}$ is
$[\Theta_{\varepsilon}]$-semi-stable,
$\overline{\partial}_{\varepsilon}=\overline{\partial}_{0}+\tilde{\Phi}(\varepsilon,b_{\infty,\varepsilon})$
where $b_{\infty,\varepsilon}$ is given by Proposition 4.6. The HYM condition
is fulfilled and when $\Theta_{\varepsilon}$ is in the stable cone,
$b_{\infty,\varepsilon}\in\mathcal{O}$ so
$(E,\overline{\partial}_{\varepsilon})\cong\mathcal{E}\cong\mathrm{Gr}_{[\Theta_{\varepsilon}]}(\mathcal{E})$.
When $\Theta_{\varepsilon}$ is only in the semi-stable cone, by [2, Theorem
4],
$(E,\overline{\partial}_{\varepsilon})\cong(E,\overline{\partial}_{0}+\Phi(b_{\infty,\varepsilon}))\cong\mathrm{Gr}_{[\Theta_{\varepsilon}]}(\mathcal{E})$.

By smoothness of $\sigma$ given by Proposition 4.1 and the bound on the norm
of $b$ given by Proposition 4.6, we have for all $d\geqslant 2$,

$\displaystyle\left\|\overline{\partial}_{\varepsilon}-\overline{\partial}_{0}\right\|_{L_{d}^{2}}$
$\displaystyle=\left\|\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,b_{\infty,\varepsilon})}\cdot(\overline{\partial}_{0}+\Phi(b_{\infty,\varepsilon}))-\overline{\partial}_{0}\right\|_{L_{d}^{2}}$
$\displaystyle=\left\|\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,b_{\infty,\varepsilon})}\overline{\partial}_{0}(\boldsymbol{\mathrm{e}}^{-\sigma(\varepsilon,b_{\infty,\varepsilon})})+\boldsymbol{\mathrm{e}}^{\sigma(\varepsilon,b_{\infty,\varepsilon})}\Phi(b_{\infty,\varepsilon})\boldsymbol{\mathrm{e}}^{-\sigma(\varepsilon,b_{\infty,\varepsilon})}\right\|_{L_{d}^{2}}$
$\displaystyle=\mathrm{O}(\left\|\sigma(\varepsilon,b_{\infty,\varepsilon})\right\|_{L_{d-1}^{2}})+\mathrm{O}(\left\|\Phi(b_{\infty,\varepsilon})\right\|_{L_{d}^{2}})$
$\displaystyle=\mathrm{O}(\left\|\varepsilon\right\|_{L_{d+1}^{2}}+\left\|b_{\infty,\varepsilon}\right\|^{2})+\mathrm{O}(\left\|b_{\infty,\varepsilon}\right\|)$
$\displaystyle=\mathrm{O}\left(\left\|\varepsilon\right\|_{L_{d+1}^{2}}+\sqrt{\left\|\varepsilon\right\|_{L_{1}^{2}}}\right).$

∎

######  Remark.

Clearly, the same bound holds for
$\left\|\partial_{\varepsilon}-\partial_{0}\right\|_{L_{d}^{2}}$ and
$\left\|\nabla_{\varepsilon}-\nabla_{0}\right\|_{L_{d}^{2}}$, where
$\nabla_{\varepsilon}=\partial_{\varepsilon}+\overline{\partial}_{\varepsilon}$
is the Chern connection associated to $(\overline{\partial}_{\varepsilon},h)$.

######  Lemma 4.8.

Let $\mathcal{E}$ be a torsion-free coherent sheaf. For all
$[\Theta]\in\mathcal{C}_{ss}(\mathcal{E})$, there is a unique face
$F_{[\Theta]}$ in the cone $\hat{\mathcal{C}}_{ss}^{\mathrm{D}}(\mathcal{E})$
such that $[\Theta]$ belongs to the relative interior of $F_{[\Theta]}$. If
$[\Theta_{1}]$ and $[\Theta_{2}]$ are in
$\mathcal{C}_{ss}^{\mathrm{D}}(\mathcal{E})$ and $F_{[\Theta_{1}]}\subset
F_{[\Theta_{2}]}$, then $\mathrm{Gr}_{[\Theta_{2}]}(\mathcal{E})$ is
$[\Theta_{1}]$-semi-stable and
$\mathrm{Gr}_{[\Theta_{1}]}(\mathrm{Gr}_{[\Theta_{2}]}(\mathcal{E}))\cong\mathrm{Gr}_{[\Theta_{1}]}(\mathcal{E})$.
In particular, if $F_{[\Theta_{1}]}=F_{[\Theta_{2}]}$,
$\mathrm{Gr}_{[\Theta_{1}]}(\mathcal{E})\cong\mathrm{Gr}_{[\Theta_{2}]}(\mathcal{E})$.

###### Proof.

Let $0\subsetneq\mathcal{S}_{1},\ldots,\mathcal{S}_{p}\subsetneq\mathcal{E}$
be a finite family of saturated coherent sheaves with pairwise distinct first
Chern classes such that for all $\mathcal{S}$,
$c_{1}(\mathcal{S})=c_{1}(\mathcal{S}_{k})$ for some $k$, given by Corollary
3.6. Up to rearranging the $\mathcal{S}_{k}$, we may assume that the faces
$F_{[\Theta_{2}]}$ and $F_{[\Theta_{1}]}\subset F_{[\Theta_{2}]}$ are defined
by the equations,

$F_{[\Theta_{1}]}=\bigcap_{k=1}^{j}\\{l_{\mathcal{S}_{k}}=0\\}\cap\bigcap_{k=j+1}^{p}\\{l_{\mathcal{S}_{k}}\geqslant
0\\},\qquad
F_{[\Theta_{2}]}=\bigcap_{k=1}^{i}\\{l_{\mathcal{S}_{k}}=0\\}\cap\bigcap_{k=i+1}^{p}\\{l_{\mathcal{S}_{k}}\geqslant
0\\},$

for some $0\leqslant i\leqslant j\leqslant p$. Moreover, since $[\Theta_{1}]$
(resp. $[\Theta_{2}]$) belongs to the relative interior of $F_{[\Theta_{1}]}$
(resp. $F_{[\Theta_{2}]}$), we have $l_{\mathcal{S}_{k}}([\Theta_{1}])>0$ when
$k>j$ (resp. $l_{\mathcal{S}_{k}}([\Theta_{2}])>0$ when $k>i$).

Let
$0\subsetneq\mathcal{F}_{1}\subsetneq\cdots\subsetneq\mathcal{F}_{m}=\mathcal{E}$
be a Jordan-Hölder filtration for $\mathcal{E}$ with respect to
$[\Theta_{2}]$. We have
$\mathrm{Gr}_{[\Theta_{2}]}(\mathcal{E})=\bigoplus_{k=1}^{m}\mathcal{G}_{k}$
where $\mathcal{G}_{k}=\mathcal{F}_{k}/\mathcal{F}_{k-1}$. By definition, for
all $k$, $l_{\mathcal{F}_{k}}([\Theta_{2}])=0$. Thus, the $q$ such that
$c_{1}(\mathcal{F}_{k})=c_{1}(\mathcal{S}_{q})$ is less or equal than $i$. In
particular, $q\leqslant j$ so for all $k$,
$l_{[\Theta_{1}]}(\mathcal{F}_{k})=0$. It implies that each $\mathcal{F}_{k}$
is $[\Theta_{1}]$-semi-stable in this case, by $[\Theta_{1}]$-stability of
$\mathcal{E}$. Therefore, for all $k$, $l_{\mathcal{G}_{k}}([\Theta_{1}])=0$
and $\mathcal{G}_{k}$ is $[\Theta_{1}]$-semi-stable as a quotient of
$[\Theta_{1}]$-semi-stable sheaves of the same $[\Theta_{1}]$-slope.

It means that the $\mathcal{F}_{k}$ form a filtration of $\mathcal{E}$ by
semi-stable sub-sheaves with torsion-free semi-stable quotients with respect
to $[\Theta_{1}]$. Let us consider a maximal refinement of this filtration
with for all $1\leqslant k\leqslant m$,

$\mathcal{F}_{k-1}=\mathcal{F}_{k-1,0}\subsetneq\mathcal{F}_{k-1,1}\subsetneq\cdots\subsetneq\mathcal{F}_{k-1,p_{k}}=\mathcal{F}_{k}=\mathcal{F}_{k,0}.$

Then, the $\mathcal{F}_{k,q}$ form a Jordan-Hölder filtration of $\mathcal{E}$
with respect to $[\Theta_{1}]$ hence on the one hand,

$\mathrm{Gr}_{[\Theta_{1}]}(\mathcal{E})=\bigoplus_{k=1}^{m}\bigoplus_{q=1}^{p_{k}}\mathcal{F}_{k-1,q}/\mathcal{F}_{k-1,q-1},$

and on the other hand, for each $k$, the
$\mathcal{F}_{k-1,q}/\mathcal{F}_{k-1}$ form a Jordan-Hölder filtration of
$\mathcal{G}_{k}=\mathcal{F}_{k}/\mathcal{F}_{k-1}$ with respect to
$[\Theta_{1}]$ so,

$\mathrm{Gr}_{[\Theta_{1}]}(\mathrm{Gr}_{[\Theta_{2}]}(\mathcal{E}))=\bigoplus_{k=1}^{m}\mathrm{Gr}_{[\Theta_{1}]}(\mathcal{G}_{k})=\bigoplus_{k=1}^{m}\bigoplus_{q=1}^{p_{k}}\mathcal{F}_{k-1,q}/\mathcal{F}_{k-1,q-1}.$

It proves the lemma. ∎

Let
$\underline{\mathcal{C}}_{ss}(\mathcal{E})\subset\mathcal{C}_{ss}(\mathcal{E})$
be the set of all $\Theta$ with respect to which $\mathcal{E}$ is semi-stable
and sufficiently smooth. Let
$\underline{\mathcal{C}}_{ss}^{\mathrm{D}}(\mathcal{E})\subset\mathcal{C}_{ss}^{\mathrm{D}}(\mathcal{E})$
be its projection on the real Dolbeault cohomology group. Lemma 4.8 has the
following consequence,

######  Corollary 4.9.

$\underline{\mathcal{C}}_{ss}^{\mathrm{D}}(\mathcal{E})$ is an open subset of
$\mathcal{C}_{ss}^{\mathrm{D}}(\mathcal{E})$ whose complement is formed of a
union of closed faces of $\hat{\mathcal{C}}_{ss}^{\mathrm{D}}(\mathcal{E})$
intersected with $\mathcal{B}_{X}$. Similarly,
$\underline{\mathcal{C}}_{ss}(\mathcal{E})$ is an open subset of
$\mathcal{C}_{ss}(\mathcal{E})$.

###### Proof.

This is a direct consequence of Lemma 4.8 and the fact that the graded object
of a sheaf which is not locally free is itself not locally free (in
particular, if $\mathcal{E}$ is not locally free, these cones are empty). ∎

We come back to the case where $\mathcal{E}$ is locally free. Let
$\mathcal{H}(E)$ be the set of holomorphic structures on $E$ i.e. the set of
the integrable Dolbeault operators. We endow it with the
$\mathcal{C}^{\infty}$ topology. By [12, Corollary 7.1.15], the action of
$\mathcal{G}$ on $\mathcal{H}(E)$ is proper hence the space
$\mathcal{H}(E)/\mathcal{G}$ is Hausdorff. We endow
$\underline{\mathcal{C}}_{ss}(\mathcal{E})\subset\Omega^{n-1,n-1}(X,\mathbb{R})$
with the $\mathcal{C}^{\infty}$ topology too. We are now ready to prove
Introduction’s Theorem 4.

######  Theorem 4.10.

The function,

$\Gamma_{\mathcal{E}}:\left\\{\begin{array}[]{rcl}\displaystyle\underline{\mathcal{C}}_{ss}(\mathcal{E})&\rightarrow&\displaystyle\mathcal{H}(E)/\mathcal{G}\\\
\displaystyle\Theta&\mapsto&\displaystyle\\{\overline{\partial}\in\mathcal{H}(E)|(\mathcal{E},\overline{\partial})\cong\mathrm{Gr}_{[\Theta]}(\mathcal{E})\textrm{
and $\overline{\partial}$ is $\Theta$-HYM}\\}\end{array}\right.,$

is well-defined and continuous.

###### Proof.

For all $\Theta\in\underline{\mathcal{C}}_{ss}(\mathcal{E})$,
$\mathrm{Gr}_{[\Theta]}(\mathcal{E})$ is a $[\Theta]$-polystable vector bundle
so Proposition 3.9 tells us that the set
$\\{\overline{\partial}\in\mathcal{H}(E)|(\mathcal{E},\overline{\partial})\cong\mathrm{Gr}_{[\Theta]}(\mathcal{E})\textrm{
and $\overline{\partial}$ is $\Theta$-HYM}\\}$ is exactly one
$\mathcal{G}$-orbit hence $\Gamma_{\mathcal{E}}$ is well defined and indeed
takes values in $\mathcal{H}(E)/\mathcal{G}$. For the continuity, we may
assume that $\underline{\mathcal{C}}_{ss}(\mathcal{E})\neq\emptyset$ and we
have to consider three cases.

First case : $\mathcal{C}_{s}(\mathcal{E})\neq\emptyset$. In this case,
continuity of $\Gamma$ at each point $\Theta_{0}$ is a consequence of Theorem
4.7.

Second case : It exists $\Theta_{0}$ such that $F_{[\Theta_{0}]}$ is the whole
cone $\hat{\mathcal{C}}_{ss}^{\mathrm{D}}(\mathcal{E})$ and $\mathcal{E}$ is
$[\Theta_{0}]$-polystable. In this case, let us write
$\mathcal{E}=\bigoplus_{k=1}^{m}\mathcal{E}_{k}$ with the $\mathcal{E}_{k}$
$[\Theta_{0}]$-stable with the same $[\Theta_{0}]$-slope. The unitary gauge
group of $E$ is the direct product of the unitary gauge groups of the smooth
structures $E_{k}$ of $\mathcal{E}_{k}$. For all
$\Theta\in\underline{\mathcal{C}}_{ss}(\mathcal{E})$, each $\mathcal{E}_{k}$
is $[\Theta]$-semi-stable and have the same $[\Theta]$-slope as $\mathcal{E}$
because $[\Theta]\in F_{[\Theta_{0}]}$. It implies that
$\mathrm{Gr}_{[\Theta]}(\mathcal{E})=\bigoplus_{k=1}^{m}\mathrm{Gr}_{[\Theta]}(\mathcal{E}_{k})$.
Moreover, any $[\Theta]$-HYM connection on
$\mathrm{Gr}_{[\Theta]}(\mathcal{E})$ makes this decomposition holomorphic and
induces a $[\Theta]$-HYM connection on each
$\mathrm{Gr}_{[\Theta]}(\mathcal{E}_{k})$. Therefore, it makes sens to write
that $\Gamma_{\mathcal{E}}=\sum_{k=1}^{m}\Gamma_{\mathcal{E}_{k}}$. The fact
that each $\Gamma_{\mathcal{E}_{k}}$ is continuous is the first case because
for all $k$, $\Theta_{0}\in\mathcal{C}_{s}(\mathcal{E}_{k})$.

Third case : General case. $\underline{\mathcal{C}}_{ss}(\mathcal{E})$ is
assumed to be non-empty so there is a $\Theta_{0}$ such that
$F_{[\Theta_{0}]}=\hat{\mathcal{C}}_{ss}^{\mathrm{D}}(\mathcal{E})$. Thus, for
all $\Theta\in\mathcal{C}_{ss}(\mathcal{E})$, $F_{[\Theta]}\subset
F_{[\Theta_{0}]}$. Therefore, a consequence of Lemma 4.8 is that
$\Gamma_{\mathcal{E}}=\Gamma_{\mathrm{Gr}_{[\Theta_{0}]}(\mathcal{E})}$ and
$\mathrm{Gr}_{[\Theta_{0}]}(\mathcal{E})$ is $[\Theta_{0}]$-polystable. We
conclude thanks to the second case. ∎

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