# Quot-scheme limit of Fubini–Study metrics and Donaldson’s functional for
bundles

Yoshinori Hashimoto<EMAIL_ADDRESS>Dipartimento di Matematica e
Informatica “U. Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A
50134 Firenze, Italy. Department of Mathematics, Tokyo Institute of
Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan. and  Julien
Keller<EMAIL_ADDRESS>Aix Marseille Université, CNRS, Centrale
Marseille, Institut de Mathématiques de Marseille, UMR 7373, 13453 Marseille,
France. Département de Mathématiques, Université du Québec à Montréal (UQÀM),
C.P. 8888, Succ. Centre-Ville, Montréal (Québec) H3C 3P8 Canada

###### Abstract.

For a holomorphic vector bundle $\mathcal{E}$ over a polarised Kähler
manifold, we establish a direct link between the slope stability of
$\mathcal{E}$ and the asymptotic behaviour of Donaldson’s functional, by
defining the Quot-scheme limit of Fubini–Study metrics. In particular, we
provide an explicit estimate which proves that Donaldson’s functional is
coercive on the set of Fubini–Study metrics if $\mathcal{E}$ is slope stable,
and give a new proof of Hermitian–Einstein metrics implying slope stability.

###### Key words and phrases:

Quot-scheme, vector bundles, slope stability, Hermitian-Einstein metric.

Both authors would like to thank Ruadhaí Dervan for helpful conversations. The
first named author also thanks Yanbo Fang, Nicholas Lindsay, and Benjamin
Sibley for helpful discussions.
The work of both authors has been carried out in the framework of the Labex
Archimède (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02),
funded by the “Investissements d’Avenir” French Government programme managed
by the French National Research Agency (ANR). The first named author was
supported by the Post-doc position in Complex and Symplectic Geometry “Bando
de Bartolomeis 2017”, at Dipartimento di Matematica e Informatica U. Dini,
Università di Firenze, dedicated to the memory of Professor Paolo de
Bartolomeis and funded by his family, and also by Università di Firenze,
Progetto di ricerca scientifica d’ateneo (ex quota 60%) Anno 2017: “Geometria
differenziale, algebrica, complessa e aritmetica”. The second named author was
also supported by the ANR project EMARKS, decision No ANR-14-CE25-0010.

## Introduction

Overview. Let $\mathcal{E}$ be a holomorphic vector bundle over a smooth
complex projective variety. A fundamental result connecting differential and
algebraic geometry states that $\mathcal{E}$ admits Hermitian–Einstein metrics
if and only if it is slope polystable (this is also called the
Donaldson–Uhlenbeck–Yau theorem or the Kobayashi–Hitchin correspondence in the
literature).

One direction of this correspondence was proved by Kobayashi [Kobookjp] and
Lübke [Luebke], and the other, arguably harder, direction was proved by
Donaldson [Do-1983, Don1985, Don1987]; it was also extended to compact Kähler
manifolds by Uhlenbeck and Yau [U-Y, U-Y2]. Fundamental as they are, their
argument connecting the differential geometry of Hermitian–Einstein metrics
and the algebraic geometry of slope stability is rather technical, as we shall
briefly review later.

The aim of this paper is to provide the first step towards establishing a more
direct link between these two concepts, from the variational point of view of
a functional defined by Donaldson [Don1985], whose critical point, if exists,
is the Hermitian–Einstein metric. More precisely, we study the asymptotic
behaviour of Donaldson’s functional on the set of Fubini–Study metrics, which
is dense in the space of all hermitian metrics on $\mathcal{E}$, and prove
that it is controlled by the algebro-geometric slope of certain filtrations of
$\mathcal{E}$ by subsheaves; see Theorems 1 and 2 for the precise statements.
In particular, we prove that the slope stability of $\mathcal{E}$ implies that
Donaldson’s functional is coercive on the set of Fubini-Study metrics
(Corollary 6.3). Moreover, we do so by making practically no use of the PDE
theory.

An application of the above results provided in this paper is a new proof of
the theorem of Kobayashi [Kobookjp] and Lübke [Luebke], which states that the
existence of Hermitian–Einstein metrics implies the slope stability of
$\mathcal{E}$ (Section 7). The converse, probably more interesting, direction
of the correspondence will be treated in the sequel to this paper [HK2]. More
specifically, in [HK2] we provide an alternative proof of the
Donaldson–Uhlenbeck–Yau theorem, which asserts the existence of
Hermitian–Einstein metrics on slope stable vector bundles, by assuming that a
“uniform” version of the inequality in Theorem 1 holds; the novelty of this
alternative proof is that it relies much less on analysis and PDE theory by
using instead the methods established in this paper (see below and [HK2] for
more details).

The key new definitions in achieving the results of the current paper are the
renormalised Quot-scheme limit of Fubini–Study metrics (Section 2) and the
non-Archimedean Donaldson functional (Section 4), which relate certain limit
of Fubini–Study metrics to the flat limit in the Quot-scheme, thereby
providing an algebro-geometric description of it. We believe that they are
interesting in their own right and have a range of potential applications,
e.g. the metric study of Giesekier stability notion for Higgs bundles
initiated in [GF-K-R].

Comparison to other works. In order to clarify the novelty of our method, we
compare our work to the well-established results concerning the correspondence
between the Hermitian–Einstein metrics and the slope stability.

The crux of the theorem of Kobayashi [Kobookjp] and Lübke [Luebke] is an
application of the Gauss–Codazzi formula, together with a sheaf-theoretic
argument which is rather technical (see also [Kobook, Section 5.8]). While our
alternative proof (in Section 7) is perhaps no simpler in terms of
technicality, we believe that it is more geometric, as the proof reduces to
studying the asymptotic behaviour of a convex energy functional.

The other direction of the correspondence, established by Donaldson [Do-1983,
Don1985, Don1987] and Uhlenbeck–Yau [U-Y, U-Y2], rests on a sophisticated
application of the nonlinear PDE theory. Both of their work can be described
as a “continuous deformation” from an initial metric to the Hermitian–Einstein
metric. In both cases, it aims to obtain the Hermitian–Einstein metric at the
limit of such continuous deformations, where the deformation process and the
limiting behaviour are analysed by means of the nonlinear PDE theory. When the
bundle $\mathcal{E}$ is slope unstable, both methods prove the existence of a
destabilising subsheaf. Donaldson uses the Yang–Mills flow and argues by
applying the Mehta–Ramanathan theorem and the induction on the dimension of
the variety (see also the recent complementary results on the Yang–Mills flow
[DasWent04, Jacob15, Jacob16, Sibley, SibWent] and references therein). The
continuity method of Uhlenbeck–Yau relies on a certain compactness result for
treating the limit, but it is well-known that the argument is highly technical
(see also [Popo1]).

Our work, on the other hand, relies much less on analysis. Indeed, in this
paper and the sequel [HK2], the analysis that we use is mostly elementary,
such as the Arzelà–Ascoli theorem. The only exception is the asymptotic
expansion of the Bergman kernel (Theorem 1.16), which we rely on as a
foundational result, since its well-known consequence of the set of
Fubini–Study metrics being dense in the space of hermitian metrics plays a
crucially important role in our work. It seems interesting to point out that
we establish a more direct relationship between slope stability and
Donaldson’s functional by not relying much on analysis, especially the PDE
techniques.

While our work has an obvious relationship to many classical results on
holomorphic vector bundles, it is important to note that it is also inspired
by the recent advances on the Yau–Tian–Donaldson conjecture, relating the
existence of constant scalar curvature Kähler (cscK) metrics to $K$-stability.
More precisely, our work can be regarded as a vector-bundle analogue of the
well-known results due to Paul [Paul], Paul–Tian [PT1, PT2] and
Phong–Ross–Sturm [PRS] on the asymptotic behaviour of the Mabuchi energy on
the set of Fubini–Study metrics, leading to the work of
Boucksom–Hisamoto–Jonsson [BHJ1, BHJ2]. Recall that Boucksom–Hisamoto–Jonsson
[BHJ1, BHJ2] identified the asymptotic slope of various energy functionals as
a “non-Archimedean” functional that is defined in terms of algebro-geometric
intersection formulae (see also e.g. [Boucksom-icm]). Their method was applied
to the problem of Kähler–Einstein metrics by Berman–Boucksom–Jonsson [BBJ].
While their influence on our work should be clear, it is perhaps worth
mentioning that we can draw parallels between many results and concepts in
cscK metrics (varieties) and Hermitian–Einstein metrics (vector bundles),
which will be explained in more details in a forthcoming paper.

Likewise, our proof of Hermitian–Einstein metrics implying slope stability is
similar to many established results in cscK metrics, such as Berman–Darvas–Lu
[BDLu], Dervan–Ross [DerRoss], and Sjöström Dyrefelt [Sj]; the existence of
canonical metrics implying stability is proved by evaluating the asymptotic
behaviour of appropriate energy functionals (Mabuchi energy for cscK metrics,
Donaldson’s functional for Hermitian–Einstein metrics).

Finally, we note that a specific filtration of the vector bundle
$\mathcal{E}$, called the Harder–Narasimhan filtration, played an important
role in the study of Hermitian–Einstein metrics since the seminal work of
Atiyah–Bott [A-B]. A novel point of our work could be that we consider
arbitrary filtrations associated to the Quot-schemes, which can be compared to
[DonCalabi, Section 2].

Organisation of the paper In Section 1, we introduce some background materials
on Quot-schemes, Fubini–Study metrics, and the Donaldson functional. In
Section 2, we define the renormalised Quot-scheme limit of Fubini–Study
metrics, which provides us with a link between differential and algebraic
geometry. This machinery is applied in Section 3, in which we evaluate the
asymptotic behaviour of the Donaldson functional; in particular, we prove that
it can be controlled by the algebro-geometric data. In Section 4, we define
the non-Archimedean Donaldson functional in terms of filtrations of
$\mathcal{E}$ by subsheaves. In Section 5, we look at a particular class of
filtrations, which we call the two-step filtrations, defined by a saturated
subsheaf of $\mathcal{E}$. In Section 6, we summarise the main results of the
paper (Theorems 1, 2, and Corollary 6.3). In Section 7, we apply them to
provide an alternative proof of the theorem of Kobayashi and Lübke.

###### Contents

  1. 1 Background
    1. 1.1 Slope stability
    2. 1.2 Filtrations and Quot-scheme limit
    3. 1.3 Fubini–Study metrics on vector bundles
    4. 1.4 $Q^{*}$ as a $C^{\infty}$-quotient map
    5. 1.5 Donaldson’s functional and the Hermitian–Einstein equation
  2. 2 Quot-scheme limit of Fubini–Study metrics
    1. 2.1 Filtrations and metric duals
    2. 2.2 Renormalised Quot-scheme limit of Fubini–Study metrics
  3. 3 Estimates for Donaldson’s functional
    1. 3.1 Lower bound for Donaldson’s functional
    2. 3.2 Upper bound for Donaldson’s functional
  4. 4 The non-Archimedean Donaldson functional
  5. 5 Two-step filtrations associated to saturated subsheaves
  6. 6 Summary of the main results
  7. 7 From Hermitian–Einstein metrics to slope stability

## Notation

We shall consistently write $(X,L)$ for a polarised smooth projective variety
over $\mathbb{C}$ of complex dimension $n$, where we further assume that $L$
is very ample. We work with a fixed Kähler metric $\omega\in c_{1}(L)$ on $X$
defined by a hermitian metric $h_{L}$ on $L$. We often write
$\mathcal{O}_{X}(k)$ for $L^{\otimes k}$.

We write $\mathcal{O}_{X}$ for the sheaf of rings of holomorphic functions on
$X$, $C^{\infty}_{X}$ for the one of $\mathbb{C}$-valued
$C^{\infty}$-functions on $X$.

Throughout in this paper, coherent sheaves of $\mathcal{O}_{X}$-modules will
be denoted by calligraphic letters (e.g. $\mathcal{E}$). Block letters (e.g.
$E$) will be reserved for complex $C^{\infty}$-vector bundles. This leads to
an unavoidable clash of notation for a holomorphic vector bundle, but we use
the following convention: both $\mathcal{E}$ and $E$ will be used to denote a
holomorphic vector bundle, but $E$ will be treated as a $C^{\infty}$-object
whereas $\mathcal{E}$ is meant to be a sheaf. $C^{\infty}$-vector bundles are
clearly locally free sheaf of $C^{\infty}_{X}$-modules, but when we say
“locally free”, it is always understood that it is locally free as a sheaf of
$\mathcal{O}_{X}$-modules.

$\vee$ is used for the dual; for example, for a vector space $V$, we will
write $V^{\vee}=\mathrm{Hom}_{\mathbb{C}}(V,\mathbb{C})$ and for a coherent
sheaf $\mathcal{F}$ of $\mathcal{O}_{X}$-modules, we write
$\mathcal{F}^{\vee}=\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},\mathcal{O}_{X})$.
Also, for a $C^{\infty}$-vector bundle $F$, we write
$F^{\vee}:=\mathcal{H}om_{C^{\infty}_{X}}(F,C^{\infty}_{X})$. $*$ is reserved
for the metric dual, with respect to some metric which will be specified in
the text.

Given a coherent sheaf $\mathcal{F}$ of $\mathcal{O}_{X}$-modules,
$H^{0}(X,\mathcal{F})=H^{0}(\mathcal{F})$ denotes the set of global sections
of $\mathcal{F}$; for example, we shall often write $H^{0}(\mathcal{F}(k))$
for $H^{0}(X,\mathcal{F}\otimes L^{\otimes k})$ for any coherent sheaf
$\mathcal{F}$ on $X$. We write
$\Gamma_{C^{\infty}_{X}}(X,F)=\Gamma_{C^{\infty}_{X}}(F)$ for the one for the
$C^{\infty}$-vector bundle $F$. Following the standard notation, we also write
$\mathcal{E}nd_{\mathcal{O}_{X}}(\mathcal{F}):=\mathcal{F}^{\vee}\otimes_{\mathcal{O}_{X}}\mathcal{F}$
and
$\mathrm{End}_{\mathcal{O}_{X}}(\mathcal{F}):=H^{0}(X,\mathcal{E}nd_{\mathcal{O}_{X}}(\mathcal{F}))$.
When $\mathcal{F}$ is locally free, by regarding it as a $C^{\infty}$-complex
vector bundle, we write $\mathrm{End}_{C^{\infty}_{X}}(F)$ for the set of
$C^{\infty}$-sections of the vector bundle $F^{\vee}\otimes F$.

##  1\. Background

A large part of the materials presented in this section can be found in the
references [H-L, Kobook, Siu87, L-T1, M-M, P-S03, Thomas, Wang1].

###  1.1. Slope stability

Let $(X,L)$ be a polarised smooth projective variety of complex dimension $n$,
and $\mathcal{E}$ be a holomorphic vector bundle over $X$ of rank $r$.

######  Definition 1.1.

Let $\mathcal{F}$ be a coherent sheaf on $X$. Its slope
$\mu(\mathcal{F})=\mu_{L}(\mathcal{F})\in\mathbb{Q}$ is defined as

$\mu(\mathcal{F}):=\frac{\deg(\mathcal{F})}{\mathrm{rk}(\mathcal{F})},$

where $\mathrm{rk}(\mathcal{F})\in\mathbb{N}$ is the rank of $\mathcal{F}$
where it is locally free (cf. [H-L, p.11]), and
$\deg(\mathcal{F})\in\mathbb{Z}$ is defined as
$\int_{X}c_{1}(\det\mathcal{F})c_{1}(L)^{n-1}/(n-1)!$ (where $\det\mathcal{F}$
is a line bundle defined in terms of the locally free resolution of
$\mathcal{F}$, see [detdiv], [Kobook, V.6] for details).

In general we should define the rank and degree in terms of the coefficients
of the Hilbert polynomial [H-L, Definitions 1.2.2 and 1.2.11] and hence they
are a priori rational numbers. For us, however, they are defined as integers
and the above definition suffices since $X$ is smooth.

The following stability notion was first introduced by Mumford for Riemann
surfaces, which was generalised to higher dimensional varieties by Takemoto by
choosing a polarisation $L$.

######  Definition 1.2 (Slope stability).

A holomorphic vector bundle $\mathcal{E}$ is said to be slope stable (or
Mumford–Takemoto stable) if for any coherent subsheaf
$\mathcal{F}\subset\mathcal{E}$ with
$0<\mathrm{rk}(\mathcal{F})<\mathrm{rk}(\mathcal{E})$ we have
$\mu(\mathcal{E})>\mu(\mathcal{F})$. $\mathcal{E}$ is said to be slope
semistable if the same condition holds with non-strict inequality, and slope
polystable if it is a direct sum of slope stable bundles with the same slope.

######  Definition 1.3.

A holomorphic vector bundle $\mathcal{E}$ is said to be

  1. (1)

reducible if it can be written as a direct sum of two or more nontrivial
holomorphic subbundles as $\mathcal{E}=\bigoplus_{j}\mathcal{E}_{j}$;

  2. (2)

irreducible if it is not reducible;

  3. (3)

simple if $\mathrm{End}_{\mathcal{O}_{X}}(\mathcal{E})=\mathbb{C}$.

It follows immediately that simple vector bundles are irreducible. The
following fact is well-known.

######  Lemma 1.4.

(cf. [Kobook, Corollaries 5.7.14]) If $\mathcal{E}$ is slope stable, then it
is simple.

######  Definition 1.5.

A coherent sheaf $\mathcal{F}$ is called reflexive if it is isomorphic to its
reflexive hull $\mathcal{F}^{\vee\vee}$.

######  Definition 1.6.

A subsheaf $\mathcal{F}$ of $\mathcal{E}$ is called saturated if
$\mathcal{E}/\mathcal{F}$ is torsion-free or zero. Any subsheaf $\mathcal{F}$
of $\mathcal{E}$ has a minimal saturated subsheaf containing $\mathcal{F}$,
called the saturation of $\mathcal{F}$ in $\mathcal{E}$, defined as the kernel
of the surjection
$\mathcal{E}\to\mathcal{E}/\mathcal{F}\to(\mathcal{E}/\mathcal{F})/T(\mathcal{E}/\mathcal{F})$,
where $T(\mathcal{E}/\mathcal{F})$ is the maximal torsion subsheaf of
$\mathcal{E}/\mathcal{F}$ [H-L, Section 1.1].

It is well-known that saturated subsheaves of a locally free sheaf
$\mathcal{E}$ are reflexive (cf. [Kobook, Proposition 5.5.22]). An important
fact is the following.

######  Lemma 1.7.

_(cf.[Kobook, Corollaries 5.5.11, 5.5.15, and 5.5.20])_ Every coherent sheaf
is locally free outside of a Zariski closed subset of codimension at least 1.
Moreover, torsion-free (resp. reflexive) sheaves are locally free outside of a
Zariski closed subset of codimension at least 2 (resp. 3).

We have the following lemma, which follows from the well-known fact that, in
order to test slope stability of a vector bundle $\mathcal{E}$, it is
sufficient to consider its saturated subsheaves [Kobook, Proposition 5.7.6].

######  Lemma 1.8.

$\mathcal{E}$ is slope stable if and only if
$\mu(\mathcal{E})>\mu(\mathcal{F})$ for any saturated subsheaf
$\mathcal{F}\subset\mathcal{E}$ with
$0<\mathrm{rk}(\mathcal{F})<\mathrm{rk}(\mathcal{E})$.

###  1.2. Filtrations and Quot-scheme limit

We recall some basic definitions and results that concern the Quot-schemes. We
refer to the reference book of Huybrechts and Lehn [H-L] for more details.

######  Definition 1.9.

A coherent sheaf $\mathcal{F}$ is said to be $k$-regular if
$H^{i}(\mathcal{F}(k-i))=0$ for all $i>0$. The Castelnuovo–Mumford regularity
of $\mathcal{F}$ is the integer defined by

$\mathrm{reg}(\mathcal{F}):=\inf_{k\in\mathbb{Z}}\\{\mathcal{F}\text{ is
$k$-regular.}\\}.$

Recall [H-L, Lemma 1.7.2] that if $\mathcal{F}$ is $k$-regular, then
$\mathcal{F}$ is $k^{\prime}$-regular for all $k^{\prime}\geq k$ and
$\mathcal{F}(k)$ is globally generated. Moreover, the natural homomorphism

$H^{0}(\mathcal{F}(k))\otimes H^{0}(\mathcal{O}_{X}(l))\to
H^{0}(\mathcal{F}(k+l))$

is surjective for $l\geq 0$.

In particular, if $\mathcal{F}$ is $k$-regular, $\mathcal{F}$ can be written
as a quotient

(1.1) $\rho:V_{k}\otimes\mathcal{O}_{X}(-k)\twoheadrightarrow\mathcal{F}$

where the vector space $V_{k}:=H^{0}(\mathcal{F}(k))$ has dimension
$P_{\mathcal{F}}(k)$, the Hilbert polynomial of $\mathcal{F}$.

We now consider the holomorphic vector bundle $\mathcal{E}$. We take
$k\geq\mathrm{reg}(\mathcal{E})$ so that it is given as a quotient
$\rho:V_{k}\otimes\mathcal{O}_{X}(-k)\to\mathcal{E}$, where

$V_{k}=H^{0}(\mathcal{E}(k)),\quad\dim
V_{k}=P_{\mathcal{E}}(k)=h^{0}(\mathcal{E}(k)).$

In other words, $\rho:V_{k}\otimes\mathcal{O}_{X}(-k)\to\mathcal{E}$ define a
point in the Quot-scheme $\mathrm{Quot}(\mathcal{Q},P_{\mathcal{E}})$ where
$\mathcal{Q}:=V_{k}\otimes\mathcal{O}_{X}(-k)$. The reader is referred to
[H-L, Chapter 2] for more details on the Quot-schemes.

Suppose that we are given a 1-parameter subgroup (1-PS for short)
$\sigma_{T}:\mathbb{C}^{*}\ni T\mapsto\sigma_{T}\in SL(V_{k})$. This defines a
$\mathbb{C}^{*}$-orbit in $\mathrm{Quot}(\mathcal{Q},P_{\mathcal{E}})$ defined
by $\\{\rho\circ\sigma_{T}\\}_{T\in\mathbb{C}^{*}}$. We can give an explicit
description of the limit of this orbit as $T\to 0$, following [H-L, Section
4.4].

Recall that giving a 1-PS $\sigma_{T}:\mathbb{C}^{*}\to SL(V_{k})$ is
equivalent to giving a weight space decomposition
$V_{k}=\bigoplus_{i\in\mathbb{Z}}V_{i,k}$ of $V_{k}$, where $\sigma_{T}$ acts
by $\sigma_{T}\cdot v=T^{i}v$ for $v\in V_{i,k}$. Note that $V_{i,k}\neq 0$
for only finitely many $i$’s. Equivalently, this defines a filtration

$V_{\leq i,k}:=\bigoplus_{j\leq i}V_{j,k}$

which in turn induces a subsheaf $\mathcal{E}_{\leq i}$ of $\mathcal{E}$ by

$\mathcal{E}_{\leq i}:=\rho(V_{\leq i,k}\otimes\mathcal{O}_{X}(-k)),$

defining a filtration $\\{\mathcal{E}_{\leq i}\\}_{i\in\mathbb{Z}}$ of
$\mathcal{E}$ by subsheaves. Then $\rho$ induces surjections
$\rho_{i}:V_{i,k}\otimes\mathcal{O}_{X}(-k)\to\mathcal{E}_{i}:=\mathcal{E}_{\leq
i}/\mathcal{E}_{\leq i-1}$, and summing up these graded pieces we get a point
in $\mathrm{Quot}(\mathcal{Q},P_{\mathcal{E}})$ represented by the sheaf

(1.2)
$\hat{\rho}:=\bigoplus_{i\in\mathbb{Z}}\rho_{i}:V_{k}\otimes\mathcal{O}_{X}(-k)\to\hat{\mathcal{E}}:=\bigoplus_{i\in\mathbb{Z}}\mathcal{E}_{i}.$

The importance of this sheaf is that it is, up to equivalence, the limit of
the 1-PS $\sigma_{T}$ in $\mathrm{Quot}(\mathcal{Q},P_{\mathcal{E}})$; recall
that two quotient sheaves $\rho_{i}:\mathcal{Q}\to\mathcal{E}_{i},\ i=1,2$ are
equivalent if and only if there exists an isomorphism
$\psi:\mathcal{E}_{1}\to\mathcal{E}_{2}$ of holomorphic vector bundles such
that $\rho_{2}=\psi\circ\rho_{1}$. Writing $[\rho]$ for the equivalence class
of $\rho$, the above result can be stated as follows.

######  Lemma 1.10.

(cf. [H-L, Lemma 4.4.3])
$[\hat{\rho}]\in\mathrm{Quot}(\mathcal{Q},P_{\mathcal{E}})$ is the limit of
$[\rho:V_{k}\otimes\mathcal{O}_{X}(-k)\to\mathcal{E}]$ as $T\to 0$ under the
1-parameter subgroup defined by $\sigma_{T}:\mathbb{C}^{*}\to SL(V_{k})$.

We shall review an outline of the proof of this result, following [H-L], for
the reader’s convenience.

###### Sketch of the proof.

We construct a family of sheaves

$\theta:V_{k}\otimes\mathcal{O}_{X}(-k)\otimes\mathbb{C}[T]\to\tilde{\mathcal{E}}$

over $\mathbb{C}=\mathrm{Spec}(\mathbb{C}[T])$, such that we have
$[\theta_{\alpha}]=[\rho]\circ\sigma_{T}$ for the fibre $\theta_{\alpha}$ over
$\alpha\in\mathbb{C}^{*}$ and look at its behaviour under the limit $\alpha\to
0$; $\alpha\in\mathbb{C}^{*}$ corresponds to the maximal ideal
$(T-\alpha)\in\mathrm{Spec}(\mathbb{C}[T])$, and the limit $\alpha\to 0$
corresponds to $T\to 0$. Throughout in the proof, the tensor product is meant
to be over $\mathbb{C}$.

Given the decomposition $V_{k}=\bigoplus_{i\in\mathbb{Z}}V_{i,k}$, we define a
map

$\gamma:V_{k}\otimes\mathbb{C}[T]=\bigoplus_{i\in\mathbb{Z}}V_{i,k}\otimes\mathbb{C}[T]\to\bigoplus_{i\in\mathbb{Z}}V_{\leq
i,k}\otimes T^{i}$

by $\gamma(v_{j}\otimes p):=v_{j}\otimes(T^{j}\cdot p)$ for $v_{j}\otimes p\in
V_{j}\otimes\mathbb{C}[T]$. Note that $\gamma$ is an isomorphism of vector
spaces which coincides with the action of $\sigma_{T}$. In other words, the
action of $\sigma_{T}$ on $V_{k}$ defines an isomorphism

$\gamma:V_{k}\otimes\mathcal{O}_{X}(-k)\otimes\mathbb{C}[T]\stackrel{{\scriptstyle\sim}}{{\to}}\bigoplus_{i\in\mathbb{Z}}V_{\leq
i,k}\otimes\mathcal{O}_{X}(-k)\otimes T^{i}=:\tilde{\mathcal{V}}_{k},$

where we used the same symbol $\gamma$ by abuse of notation.

Note also that $\rho$ induces the following surjection

$\tilde{\rho}:\tilde{\mathcal{V}}_{k}\to\tilde{\mathcal{E}}:=\bigoplus_{i\in\mathbb{Z}}\mathcal{E}_{\leq
i}\otimes T^{i},$

defining a family $\tilde{\mathcal{E}}$ of sheaves over $\mathbb{C}$. Thus the
desired family $\theta$ can be given by $\tilde{\rho}\circ\gamma$; restricting
to the central fibre at $T=0$, we have

$\tilde{\mathcal{E}}_{0}=\tilde{\mathcal{E}}/T\cdot\tilde{\mathcal{E}}=\bigoplus_{i\in\mathbb{Z}}\mathcal{E}_{i},$

while the noncentral fibre is equivalent to $\rho\circ\sigma_{T}$ since
$\gamma$ coincides with the action of $\sigma_{T}$. ∎

We shall later develop a “differential-geometric” version of this
construction, in terms of the Fubini–Study metrics. For this purpose, we shall
re-phrase the above lemma as follows. Suppose that we write

$\hat{\rho}=\bigoplus_{i\in\mathbb{Z}}\rho_{i},$

with

$\rho_{i}:V_{i,k}\otimes\mathcal{O}_{X}(-k)\to\mathcal{E}_{i}=\mathcal{E}_{\leq
i}/\mathcal{E}_{\leq i-1}$

for the limit in the Quot-scheme by $\sigma_{T}$ as in (1.2). The proof of
Lemma 1.10 shows that we find

$\rho(\sigma_{T}\cdot V_{i,k})=T^{i}\rho_{i}(V_{i,k})+T^{i}(\text{elements of
$V_{k,i}$ mapped to $\mathcal{E}_{\leq i-1}$}).$

Suppose that we write $\rho_{j\to i}(V_{j,k})$ for the elements of $V_{j,k}$
that are mapped to $\mathcal{E}_{i}$. With this notation, we collect the above
for all $i$’s by using $V_{k}=\bigoplus_{i\in\mathbb{Z}}V_{i,k}$, to get

(1.3) $\rho(\sigma_{T}\cdot V_{k})=\sum_{i\in\mathbb{Z}}T^{i}\left(\rho_{i\to
i}(V_{i,k})+\sum_{j>i}T^{j-i}\rho_{j\to i}(V_{j,k})\right).$

Thus, we may write the limit of $\rho\circ\sigma_{T}$ as $T\to 0$ as

(1.4) $\rho(\sigma_{T}\cdot
V_{k})=\bigoplus_{i\in\mathbb{Z}}T^{i}\left(\rho_{i}(V_{i,k})+o(T)\right),$

where $o(T)$ stands for the terms that go to zero as $T\to 0$, and $T^{i}$ is
the weight of $\sigma_{T}$ on $\mathcal{E}_{i}$.

######  Remark 1.11.

It is important to note that $\mathcal{E}_{i}$ above is a priori just a
coherent sheaf and in general not locally free; it may even have a torsion.
However, it is still possible to define a Zariski open subset in $X$ such that
all $\mathcal{E}_{\leq i}$ and $\mathcal{E}_{i}$ are locally free (torsion
sheaves are treated as zero), recalling that $X$ is smooth (by Lemma 1.7 or
[H-L, p.11]).

###  1.3. Fubini–Study metrics on vector bundles

Suppose that a holomorphic vector bundle $\mathcal{E}$ is $k$-regular. Then
$\mathcal{E}(k)$ is globally generated, and hence there exists a holomorphic
map $\Phi:X\to\mathrm{Gr}(r,H^{0}(\mathcal{E}(k))^{\vee})$ to the Grassmannian
such that the pullback under $\Phi$ of the universal bundle is isomorphic to
$\mathcal{E}(k)$. More precisely, writing $\mathcal{U}$ for the tautological
bundle over the Grassmannian $\mathrm{Gr}(r,H^{0}(\mathcal{E}(k))^{\vee})$ of
$r$-planes (rather than quotients), we have
$\Phi^{*}\mathcal{U}^{\vee}\cong\mathcal{E}(k)$.

Recall that a positive definite hermitian forms on $H^{0}(\mathcal{E}(k))$
define a hermitian metric on (the complex vector bundle) $U^{\vee}$ on
$\mathrm{Gr}(r,H^{0}(\mathcal{E}(k))^{\vee})$, called the Fubini–Study metrics
on the Grassmannian (see e.g. [M-M, Section 5.1.1]). By pulling them back by
$\Phi$, we have hermitian metrics on $E$ that are also called Fubini–Study
metrics. We recall the details of this construction. Noting that the sections
of $\mathcal{U}^{\vee}$ can be naturally identified with
$H^{0}(\mathcal{E}(k))$, hermitian forms on $U^{\vee}$ are the elements of
$H^{0}(\mathcal{E}(k))^{\vee}\otimes\overline{H^{0}(\mathcal{E}(k))^{\vee}}$,
where the bar stands for the complex conjugate.

Recall that giving a basis $\mathbb{S}_{k}=\\{s_{1},\dots,s_{N_{k}}\\}$,
$N_{k}:=\dim H^{0}(\mathcal{E}(k))$, for $H^{0}(\mathcal{E}(k))$ is equivalent
to defining a hermitian form $I_{\mathbb{S}_{k}}$ on $H^{0}(\mathcal{E}(k))$
that has $\mathbb{S}_{k}$ as its orthonormal basis, up to unitary equivalence.
Similarly, giving a local frame $\\{e_{1}(x),\dots,e_{r}(x)\\}$ for $E(k)$ at
$x\in X$ is equivalent to defining a hermitian metric on $E(k)$, up to unitary
gauge transforms. Recalling that we have fixed the hermitian metric $h_{L}$ on
$L$, we have a hermitian metric $h_{\mathrm{ref}}$ on $E$, such that
$\\{e_{1}(x),\dots,e_{r}(x)\\}$ is an $h_{\mathrm{ref}}\otimes h_{L}^{\otimes
k}$-local orthonormal frame.

We can also define the metric duals of the above. Fixing $\mathbb{S}_{k}$ as
above, we have a basis $\\{s^{*}_{1},\dots,s^{*}_{N_{k}}\\}$ for
$\overline{H^{0}(\mathcal{E}(k))^{\vee}}$ that is $I_{\mathbb{S}_{k}}$-metric
dual of $\mathbb{S}_{k}$, and a local frame
$\\{e^{*}_{1}(x),\dots,e^{*}_{r}(x)\\}$ for $\overline{E(k)^{\vee}}$ that is
$h_{\mathrm{ref}}\otimes h_{L}^{\otimes k}$-metric dual to the one
$\\{e_{1}(x),\dots,e_{r}(x)\\}$ for $E(k)$. We also write $s_{i}(x)$ (resp.
$s^{*}_{i}(x)$) for the evaluation of $s_{i}$ (resp. $s^{*}_{i}$) at $x\in X$
with respect to the trivialisation given by $\\{e_{1}(x),\dots,e_{r}(x)\\}$.

We first define $C^{\infty}$-maps

$Q_{\mathbb{S}_{k},\mathrm{ref}}:E(k)\to H^{0}(\mathcal{E}(k))\otimes
C^{\infty}_{X},$

and

$Q^{*}_{\mathbb{S}_{k},\mathrm{ref}}:\overline{H^{0}(\mathcal{E}(k))^{\vee}}\otimes\overline{C^{\infty}_{X}}\to\overline{E(k)^{\vee}},$

with respect to $\mathbb{S}_{k}$ and $\\{e_{1}(x),\dots,e_{r}(x)\\}$ as fixed
above. The bar, again, denotes the complex conjugate. A well-known result is
the following, whose proof can be found in [Wang1, Remark 3.5] or [M-M,
Theorem 5.1.16].

######  Lemma 1.12.

(cf. [Wang1, Remark 3.5] or [M-M, Theorem 5.1.16]) Suppose that we are given a
hermitian metric $h_{\mathrm{ref}}\otimes h_{L}^{\otimes k}$ on $E(k)$ with
$\\{e_{1}(x),\dots,e_{r}(x)\\}$ as its local orthonormal frame. Let
$\mathbb{S}_{k}=\\{s_{1},\dots,s_{N_{k}}\\}$ be an orthonormal basis for
$H^{0}(\mathcal{E}(k))$ with respect to the $L^{2}$-hermitian form
$I_{\mathbb{S}_{k}}$ defined by $h_{\mathrm{ref}}\otimes h_{L}^{\otimes k}$
and $\omega$. Then there exists a local $N_{k}\times r$ matrix
$Q_{\mathbb{S}_{k},\mathrm{ref}}(x)$ which smoothly depends on $x\in X$ such
that

$s_{i}(x)=\sum_{\alpha=1}^{r}Q_{\mathbb{S}_{k},\mathrm{ref}}(x)_{i\alpha}e_{\alpha}(x),\
\ \ \sum_{i=1}^{N_{k}}Q^{*}_{\mathbb{S}_{k},\mathrm{ref}}(x)_{\alpha
i}s^{*}_{i}(x)=e^{*}_{\alpha},$

where the metric dual $*$ is defined with respect to $h_{\mathrm{ref}}\otimes
h_{L}^{\otimes k}$ and $I_{\mathbb{S}_{k}}$.

The hermitian metric on $E(k)$ defined by

$h_{\mathbb{S}_{k},\mathrm{ref}}\otimes h_{L}^{\otimes
k}(x):=Q_{\mathbb{S}_{k},\mathrm{ref}}^{*}(x)\circ I_{\mathbb{S}_{k}}\circ
Q_{\mathbb{S}_{k},\mathrm{ref}}(x)$

agrees with the pullback of the Fubini–Study metric on $U^{\vee}$ with respect
to $I_{\mathbb{S}_{k}}$ over the Grassmannian
$\mathrm{Gr}(r,H^{0}(\mathcal{E}(k))^{\vee})$ by the Kodaira embedding
$\Phi:X\hookrightarrow\mathrm{Gr}(r,H^{0}(\mathcal{E}(k))^{\vee})$.

######  Remark 1.13.

In the above, the $L^{2}$-hermitian form $I_{\mathbb{S}_{k}}$ (defined by
$h_{\mathrm{ref}}\otimes h_{L}^{\otimes k}$ and $\omega$) in fact involves a
scaling by $\dim H^{0}(\mathcal{E}(k))$; see e.g. [M-M, Section 5.2.3].

Suppose that we now change the basis $\mathbb{S}_{k}$ to
$\sigma\cdot\mathbb{S}_{k}$ with $\sigma\in SL(H^{0}(\mathcal{E}(k)))$. This
defines another hermitian metric on the universal bundle $U^{\vee}$ over the
Grassmannian, which we can pull back by $\Phi$ to get a hermitian metric on
$E(k)$, exactly as before. Then, noting that $(\sigma^{-1})^{*}\circ
I_{\mathbb{S}_{k}}\circ\sigma^{-1}$ has $\sigma\cdot\mathbb{S}_{k}$ as its
orthonormal basis, we may write the hermitian metric
$h_{\sigma\cdot\mathbb{S}_{k},\mathrm{ref}}\otimes h_{L}^{\otimes k}$ as

$h_{\sigma\cdot\mathbb{S}_{k},\mathrm{ref}}\otimes h_{L}^{\otimes
k}=Q_{\mathbb{S}_{k},\mathrm{ref}}^{*}\circ((\sigma^{-1})^{*}\circ
I_{\mathbb{S}_{k}}\circ\sigma^{-1})\circ Q_{\mathbb{S}_{k},\mathrm{ref}},$

which is also well-known in the literature (see e.g. [P-S03, Section 2]). The
appearance of the inverse in $\sigma^{-1}$ can also be seen from the fact that
$I_{\mathbb{S}_{k}}$ is an element of
$H^{0}(\mathcal{E}(k))^{\vee}\otimes\overline{H^{0}(\mathcal{E}(k))^{\vee}}$
and hence $SL(H^{0}(\mathcal{E}(k)))$ acts by the dual action.

In what follows, we shall adopt the following convention to simplify the
notation. We fix $h_{\mathrm{ref}}$ and $h_{L}$ once and for all, which also
fixes $\mathbb{S}_{k}$ as its $L^{2}$-orthonormal basis and
$I_{\mathbb{S}_{k}}$ associated to it. Moreover, choosing a local orthonormal
frame $e_{L}(x)$ of $L$ with respect to $h_{L}$, we identify
$\\{e_{1}(x),\dots,e_{r}(x)\\}$ with a local orthonormal frame of $E$ (rather
than $E(k)$), which defines a hermitian metric
$h_{\mathbb{S}_{k},\mathrm{ref}}$ on $E$. Recalling
$Q_{\mathbb{S}_{k},\mathrm{ref}}:E(k)\to H^{0}(\mathcal{E}(k))\otimes
C^{\infty}_{X}$, we thus get a $C^{\infty}$-map

$Q:E\to H^{0}(\mathcal{E}(k))\otimes C^{\infty}_{X}(-k).$

Similarly,
$Q_{\mathbb{S}_{k},\mathrm{ref}}^{*}:\overline{H^{0}(\mathcal{E}(k))^{\vee}}\otimes\overline{C^{\infty}_{X}}\to\overline{E(k)^{\vee}}$
yields a $C^{\infty}$-map

$Q^{*}:\overline{H^{0}(E(k))^{\vee}}\otimes\overline{C^{\infty}_{X}(k)}\to\overline{E^{\vee}},$

by noting
$\overline{E(k)^{\vee}}\cong\overline{E^{\vee}}\otimes\overline{C^{\infty}_{X}(-k)}$.
With these definitions, we have $h_{\mathbb{S}_{k},\mathrm{ref}}=Q^{*}\circ
I_{\mathbb{S}_{k}}\circ Q$. In what follows, $\mathbb{S}_{k}$ will be fixed
once and for all, and we shall omit $I_{\mathbb{S}_{k}}$ and simply write
$Q^{*}Q$ for $h_{\mathbb{S}_{k},\mathrm{ref}}$. Also, in order not to be
bothered by the inverse in $\sigma^{-1}$, we shall consistently write

$h_{\sigma}:=Q^{*}\sigma^{*}\sigma Q,$

with $\sigma\in SL(H^{0}(\mathcal{E}(k))^{\vee})$, to denote
$h_{\sigma^{-1}\cdot\mathbb{S}_{k},\mathrm{ref}}$.

We summarise the above argument to define the Fubini–Study metric on the
bundle $E$.

######  Definition 1.14.

Given a reference metric $h_{\mathrm{ref}}$ on $E$ and a basis
$\mathbb{S}_{k}$ for $H^{0}(\mathcal{E}(k))$ that is $L^{2}$-orthonormal with
respect to $h_{\mathrm{ref}}\otimes h_{L}^{\otimes k}$, there exist
$C^{\infty}$-maps $Q$ and $Q^{*}$ as above, such that the hermitian metric

$h:=Q^{*}Q\in\Gamma_{C^{\infty}_{X}}(E^{\vee}\otimes\overline{E^{\vee}})$

agrees with the pullback of the Fubini–Study metric on the universal bundle
over the Grassmannian defined by the hermitian form $I_{\mathbb{S}_{k}}$ on
$H^{0}(\mathcal{E}(k))$. $h$ is called the (reference) Fubini–Study metric on
$E$ defined by $I_{\mathbb{S}_{k}}$.

Moreover, given $\sigma\in SL(H^{0}(\mathcal{E}(k))^{\vee})$, the hermitian
metric

$h_{\sigma}:=Q^{*}\sigma^{*}\sigma
Q\in\Gamma_{C^{\infty}_{X}}(E^{\vee}\otimes\overline{E^{\vee}})$

agrees with the one defined by the hermitian form
$I_{\sigma^{-1}\cdot\mathbb{S}_{k}}$ on $H^{0}(\mathcal{E}(k))$. $h_{\sigma}$
is called the Fubini–Study metric on $E$ defined by the positive hermitian
form $\sigma^{*}\sigma=\sigma^{*}\circ I_{\mathbb{S}_{k}}\circ\sigma$ on
$H^{0}(\mathcal{E}(k))$. .

When it is necessary to make the exponent $k$ more explicit, we also write
$h_{k}$ for $h$, for example. The construction above gives us a way of
associating a hermitian metric $FS(H)$ on $E$ to a positive definite hermitian
form $H$ on $H^{0}(\mathcal{E}(k))$; we simply choose $\sigma\in
GL(H^{0}(\mathcal{E}(k))^{\vee})$ so that $H=\sigma^{*}\circ
I_{\mathbb{S}_{k}}\circ\sigma=\sigma^{*}\sigma$, and define $FS(H)=h_{\sigma}$
as above.

Recall that there is an alternative definition ([Wang1], see also [M-M,
Theorem 5.1.16]) of $FS(H)$ by means of the equation

(1.5) $\sum_{i=1}^{N_{k}}s_{i}\otimes s_{i}^{*_{FS(H)}}=\mathrm{Id}_{E}$

where $\\{s_{i}\\}$ is an $H$-orthonormal basis for $H^{0}(\mathcal{E}(k))$
and $s_{i}^{*_{FS(H)}}$ is the $FS(H)$-metric dual of $s_{i}$. Defining

$\mathcal{H}_{\infty}:=\\{\text{smooth hermitian metrics on }E\\},$

we thus have a map $FS$ that maps a positive definite hermitian form $H$ on
$H^{0}(\mathcal{E}(k))^{\vee}$ to an element in $\mathcal{H}_{\infty}$.
Writing $N_{k}=\dim H^{0}(\mathcal{E}(k))$ and fixing the isomorphism
$H^{0}(\mathcal{E}(k))\cong\mathbb{C}^{N_{k}}$, recall that the homogeneous
space $GL(N_{k},\mathbb{C})/U(N_{k})$ can be identified with the set
$\mathcal{B}_{k}$ of all positive definitive hermitian forms on the vector
space $H^{0}(\mathcal{E}(k))$, by means of the isomorphism
$GL(N_{k},\mathbb{C})/U(N_{k})\ni\sigma\mapsto\sigma^{*}\sigma\in\mathcal{B}_{k}$.

These classical definitions will be important for us later, and are summarised
as follows.

######  Definition 1.15.

The Bergman space $\mathcal{B}_{k}$ at level $k$ is defined to be the set of
all positive definite hermitian forms on the vector space
$H^{0}(\mathcal{E}(k))$, which can be identified with the homogeneous space
$\mathcal{B}_{k}=GL(N_{k},\mathbb{C})/U(N_{k})$. The Fubini–Study map
$FS:\mathcal{B}_{k}\to\mathcal{H}_{\infty}$ is defined by the equation (1.5).
The image of $\mathcal{B}_{k}$ by the Fubini–Study map is a subset of
$\mathcal{H}_{\infty}$ denoted by

$\mathcal{H}_{k}:=FS(\mathcal{B}_{k}).$

We thus have a natural inclusion $\mathcal{H}_{k}\subset\mathcal{H}_{\infty}$
for each $k$, but it turns out that the union of $\mathcal{H}_{k}$’s is dense
in $\mathcal{H}_{\infty}$, i.e.

$\mathcal{H}_{\infty}=\overline{\cup_{k>>0}\mathcal{H}_{k}}.$

This fact follows from the following foundational result in Kähler geometry.

######  Theorem 1.16 (Asymptotic expansion of the Bergman kernel).

Suppose that we fix $h\in\mathcal{H}_{\infty}$ and a positively curved
hermitian metric $h_{L}$ on $\mathcal{O}_{X}(1)$, which defines a positive
definite $L^{2}$-hermitian form $H$ on $H^{0}(\mathcal{E}(k))$. The Bergman
kernel $B_{k}(h)\in\mathrm{End}_{C^{\infty}_{X}}(E)$, defined by the equation
$B_{k}(h)\circ FS(H)=h$, satisfies the asymptotic expansion
$B_{k}(h)=\mathrm{Id}_{E}+O(k^{-1})$; more precisely, there exists a constant
$C(h,h_{L},p)>0$ such that

$\|B_{k}(h)-\mathrm{Id}_{E}\|_{C^{p}}\leq C(h,h_{L},p)/k,$

where $\|\cdot\|_{C^{p}}$ is the $C^{p}$-norm on
$\mathrm{End}_{C^{\infty}_{X}}(E)$.

In particular, for any $h\in\mathcal{H}_{\infty}$ and $p\in\mathbb{N}$ there
exists a sequence $\\{h_{k,p}\\}_{k\in\mathbb{N}}$ of Fubini–Study metrics
with $h_{k,p}\in\mathcal{H}_{k}$ such that $h_{k,p}\to h$ as $k\to+\infty$ in
the $C^{p}$-norm.

The above expansion for the Bergman kernel is called the Tian–Yau–Zelditch
expansion, and is essentially a theorem in analysis. Several proofs have been
written especially when $E$ has rank one. For higher ranks, we refer to
[Catlin], [Wang2]; see also the book [M-M] and references therein. An
elementary proof can be found in [BBS2008].

Finally, recall that the homogeneous space
$\mathcal{B}_{k}=GL(N_{k},\mathbb{C})/U(N_{k})$ has a distinguished Riemannian
metric called the bi-invariant metric, with respect to which geodesics can be
written as $t\mapsto e^{\zeta t}$ for some
$\zeta\in\mathfrak{gl}(N_{k},\mathbb{C})$. Recalling the isomorphism
$GL(N_{k},\mathbb{C})/U(N_{k})\ni\sigma\mapsto\sigma^{*}\sigma\in\mathcal{B}_{k}$,
we thus have a 1-PS in $\mathcal{B}_{k}$ defined for each
$\zeta\in\mathfrak{gl}(H^{0}(\mathcal{E}(k))^{\vee})$ by
$\exp(\zeta^{*}t+\zeta t)$ (note that we may choose the isomorphism
$H^{0}(\mathcal{E}(k))\cong\mathbb{C}^{N_{k}}$ so that the standard basis for
$\mathbb{C}^{N_{k}}$ is $I_{\mathbb{S}_{k}}$-orthonormal, and hence
$\zeta^{*}$ is the $I_{\mathbb{S}_{k}}$-hermitian conjugate of $\zeta$). This
defines an important 1-PS of Fubini–Study metrics as follows.

######  Definition 1.17.

The Bergman 1-parameter subgroup (or Bergman 1-PS)
$\\{h_{\sigma_{t}}\\}_{t\geq 0}\subset\mathcal{H}_{k}$ generated by
$\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$ and emanating from
$h_{\sigma_{0}}=Q^{*}Q$ is a path of Fubini–Study metrics defined by

$h_{\sigma_{t}}:=Q^{*}\sigma_{t}^{*}\sigma_{t}Q,$

where $\sigma_{t}:=e^{\zeta t}\in SL(H^{0}(\mathcal{E}(k))^{\vee})$, in the
notation of Definition 1.14. When $\zeta$ has rational eigenvalues, we call
$\\{h_{\sigma_{t}}\\}_{t\geq 0}$ a rational Bergman 1-PS.

The Bergman 1-PS’s can be regarded as an analogue of geodesics in
$\mathcal{B}_{k}=GL(N_{k},\mathbb{C})/U(N_{k})$, and play an important role
later.

######  Remark 1.18.

Throughout in what follows, we shall only consider $\zeta$’s that have
rational eigenvalues, and hence any Bergman 1-PS will be meant to be rational
(see e.g. Sections 2 and 4).

We also recall that, in fact, $\mathcal{H}_{\infty}$ can be regarded as an
“infinite dimensional homogeneous space”, with a well-defined notion of
geodesics, as explained in [Kobook, Section 6.2].

######  Definition 1.19.

A path $\\{h_{s}\\}_{s\in\mathbb{R}}\subset\mathcal{H}_{\infty}$ is called a
geodesic in $\mathcal{H}_{\infty}$ if it satisfies

(1.6) $\partial_{s}(h_{s}^{-1}\partial_{s}h_{s})=0,$

as an equation in $\mathrm{End}_{C^{\infty}_{X}}(E)$; an overall constant
scaling $h_{s}:=e^{bs}h_{0}$ for some $b\in\mathbb{R}$ will be called a
trivial geodesic.

An elementary yet crucially important fact is that $\mathcal{H}_{\infty}$ is
geodesically complete; for any $h_{0},h_{1}\in\mathcal{H}_{\infty}$ there
exists a geodesic path $\\{h_{s}\\}_{0\leq s\leq 1}$ connecting them; we can
write it explicitly as $h_{s}=\exp(s\log h_{1}h^{-1}_{0})h_{0}$.

######  Remark 1.20.

Although the Bergman 1-PS $\\{h_{\sigma_{t}}\\}_{t\geq 0}$ does not satisfy
the geodesic equation (1.6), we can show that it is in fact a subgeodesic in
$\mathcal{H}_{\infty}$, i.e.
$\partial_{t}(h_{\sigma_{t}}^{-1}\partial_{t}h_{\sigma_{t}})$ is positive
semidefinite as an element in $\mathrm{End}_{C^{\infty}_{X}}(E)$. While we do
not use this result in this paper, it seems interesting to point out an
analogy to the case of varieties, in which test configurations define
subgeodesics (see e.g. [BHJ2, Section 3]). The proof of this fact will appear
in a forthcoming paper.

###  1.4. $Q^{*}$ as a $C^{\infty}$-quotient map

In this section, we give an interpretation of $Q^{*}$ as a
“$C^{\infty}$-analogue” of the quotient map
$\rho:H^{0}(\mathcal{E}(k))\otimes\mathcal{O}_{X}(-k)\to\mathcal{E}$ as
defined in (1.1). While the argument in this section is elementary, it will
provide a geometric intuition for the later arguments.

We start with some observations. Suppose that $\mathcal{E}$ is a $k$-regular
holomorphic vector bundle with the quotient map

(1.7) $\rho:H^{0}(\mathcal{E}(k))\otimes\mathcal{O}_{X}(-k)\to\mathcal{E}.$

Forgetting the holomorphic structure, we get a map

(1.8) $\rho_{C^{\infty}}:H^{0}(\mathcal{E}(k))\otimes C^{\infty}_{X}(-k)\to
E.$

Given the above definitions, the following is immediate.

######  Lemma 1.21.

Writing $V_{k}:=H^{0}(\mathcal{E}(k))$, we have the following commutative
diagram:

$\textstyle{SL(V_{k})\times V_{k}\otimes
C^{\infty}_{X}(-k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{V_{k}\otimes
C^{\infty}_{X}(-k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{C^{\infty}}}$$\textstyle{E}$$\textstyle{SL(V_{k})\times
V_{k}\otimes\mathcal{O}_{X}(-k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$forget$\textstyle{V_{k}\otimes\mathcal{O}_{X}(-k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$forget$\textstyle{\mathcal{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$forget

where $a:SL(V_{k})\times V_{k}\to V_{k}$ is the linear action on $V_{k}$, and
the vertical arrows are the forgetful maps forgetting the holomorphic
structures.

Choosing hermitian metrics for $H^{0}(\mathcal{E}(k))$ and $E(k)$, we can take
the “metric dual” of (1.8). More precisely, recalling $I_{\mathbb{S}_{k}}$ and
$h_{\mathrm{ref}}$ that appeared in the previous section, we have the
following $C^{\infty}$-maps

$\displaystyle I_{\mathbb{S}_{k}}$
$\displaystyle:H^{0}(\mathcal{E}(k))\stackrel{{\scriptstyle\sim}}{{\to}}\overline{H^{0}(\mathcal{E}(k))^{\vee}},$
$\displaystyle h_{\mathrm{ref}}$
$\displaystyle:E\stackrel{{\scriptstyle\sim}}{{\to}}\overline{E^{\vee}},\ \ \
h_{L}:C^{\infty}_{X}(-k)\stackrel{{\scriptstyle\sim}}{{\to}}\overline{C^{\infty}_{X}(k)},$

by means of the metric dual. Composing with these maps, we get

(1.9)
$\rho^{*}_{C^{\infty}}:\overline{H^{0}(\mathcal{E}(k))^{\vee}}\otimes\overline{C^{\infty}_{X}(k)}\to\overline{E^{\vee}},$

which looks similar to the definition of $Q^{*}$ that we had before.

We now state the relationship between $\rho^{*}_{C^{\infty}}$ and $Q^{*}$.

######  Lemma 1.22.

Suppose that $\mathcal{E}$ is a $k$-regular holomorphic vector bundle with the
quotient map $\rho$. Fix a hermitian metric $I_{\mathbb{S}_{k}}$ for
$H^{0}(\mathcal{E})$, $h_{\mathrm{ref}}$ for $E$, and $h_{L}$ for $L$, so that
we have $Q=Q_{\mathbb{S}_{k},\mathrm{ref}}$ as in Lemma 1.12. Then there
exists a $C^{\infty}$-gauge transform of $\overline{E^{\vee}}$, written
$f_{\mathbb{S}_{k},\mathrm{ref}}\in\mathrm{End}_{C^{\infty}_{X}}(\overline{E^{\vee}})$,
which is fibrewise invertible, such that

(1.10) $Q^{*}=f_{\mathbb{S}_{k},\mathrm{ref}}\circ\rho^{*}_{C^{\infty}}.$

###### Proof.

Since both $Q^{*}$ and $\rho^{*}_{C^{\infty}}$ can be written locally as an
$N_{k}\times r$ matrix of fibrewise full-rank which depends smoothly on $x$
(cf. Lemma 1.12), it is immediate that there exists a fibrewise invertible
smooth linear map
$f_{\mathbb{S}_{k},\mathrm{ref}}:\overline{E^{\vee}}\to\overline{E^{\vee}}$ so
that (1.10) holds. ∎

######  Remark 1.23.

$f_{\mathbb{S}_{k},\mathrm{ref}}$ above should be regarded as performing a
“fibrewise Gram–Schmidt process” with respect to $I_{\mathbb{S}_{k}}$ and
$h_{\mathrm{ref}}$, so that $\\{s^{*}_{1},\dots,s^{*}_{N_{k}}\\}$ are mapped
to the frame $\\{e^{*}_{1}(x),\dots,e^{*}_{r}(x)\\}$ as in Lemma 1.12.

While the above lemma looks rather trivial, we observe the following
consequence that can be obtained from it combined with Lemma 1.21; when we
consider the Quot-scheme limit $\hat{\mathcal{E}}$ of $\mathcal{E}$ generated
by a $\mathbb{C}^{*}$-action given by
$\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$, the limit of the
Bergman 1-PS $Q^{*}e^{\zeta^{*}t}e^{\zeta t}Q$ as $t\to\infty$ must be in a
certain sense “compatible” with this Quot-scheme limit. Indeed, this will be
the main topic to be discussed in the next section.

###  1.5. Donaldson’s functional and the Hermitian–Einstein equation

As before, $(X,L)$ stands for a polarised smooth complex projective variety,
$E$ a holomorphic vector bundle of rank $r$. We write
$\mathrm{Vol}_{L}:=\int_{X}c_{1}(L)^{n}/n!$. The following functional plays a
central role in our paper.

######  Definition 1.24.

Given two hermitian metrics $h_{0}$ and $h_{1}$ on $E$, the Donaldson
functional
${\mathcal{M}}^{Don}:\mathcal{H}_{\infty}\times\mathcal{H}_{\infty}\to\mathbb{R}$
is defined as

${\mathcal{M}}^{Don}(h_{1},h_{0}):={\mathcal{M}}_{1}^{Don}(h_{1},h_{0})-\mu(E){\mathcal{M}}_{2}^{Don}(h_{1},h_{0}),$

where

${\mathcal{M}}_{1}^{Don}(h_{1},h_{0}):=\int_{0}^{1}\int_{X}\mathrm{tr}\left(h_{t}^{-1}\partial_{t}h_{t}\cdot
F_{t}\right)\frac{\omega^{n-1}}{(n-1)!}dt$

and

${\mathcal{M}}_{2}^{Don}(h_{1},h_{0}):=\frac{1}{\mathrm{Vol}_{L}}\int_{X}\log\det(h_{0}^{-1}h_{1})\frac{\omega^{n}}{n!}.$

In the above, $\\{h_{t}\\}_{0\leq t\leq 1}\subset\mathcal{H}_{\infty}$ is a
smooth path of hermitian metrics between $h_{0}$ to $h_{1}$ and $F_{t}$
denotes ($\sqrt{-1}/2\pi$) times the Chern curvature of $h_{t}$, with respect
to the fixed holomorphic structure of $E$ (such that the sheaf of germs of
holomorphic sections of $E$ are identified with $\mathcal{E}$).

######  Remark 1.25.

Throughout in what follows, we shall fix the second argument of
${\mathcal{M}}^{Don}$ as a reference metric. Thus
${\mathcal{M}}^{Don}(h,h_{0})$ is regarded as a function of $h$ with a fixed
reference $h_{0}$.

We recall some well-known properties of this functional as established in
[Don1985]; the reader is also referred to [Kobook, Section 6.3] for more
details. First of all it is well-defined, i.e. does not depend on the path
$\\{h_{t}\\}_{0\leq t\leq 1}$ chosen to connect $h_{0}$ and $h_{1}$ (cf.
[Kobook, Lemma 6.3.6]). This easily implies the following cocycle property

(1.11)
$\mathcal{M}^{Don}(h_{2},h_{0})=\mathcal{M}^{Don}(h_{2},h_{1})+\mathcal{M}^{Don}(h_{1},h_{0}),$

for any $h_{0},h_{1},h_{2}\in\mathcal{H}_{\infty}$.

######  Remark 1.26.

Recalling $\mathcal{M}^{Don}(e^{c}h,h)=0$ for any constant $c\in\mathbb{R}$
[Kobook, Lemma 6.3.23], (1.11) implies

${\mathcal{M}}^{Don}(e^{c}h,h_{1})={\mathcal{M}}^{Don}(e^{c}h,h)+{\mathcal{M}}^{Don}(h,h_{1})={\mathcal{M}}^{Don}(h,h_{1})$

for any $c\in\mathbb{R}$, i.e. $\mathcal{M}^{Don}(-,h_{1})$ is invariant under
an overall constant scaling.

Second, the Euler–Lagrange equation for
$\mathcal{M}^{Don}(-,h_{\mathrm{ref}})$ is the Hermitian–Einstein equation.
More precisely, a critical point of $\mathcal{M}^{Don}(-,h_{\mathrm{ref}})$ is
the following object.

######  Definition 1.27.

A hermitian metric $h\in\mathcal{H}_{\infty}$ is called a Hermitian–Einstein
metric if it satisfies

$\Lambda_{\omega}F_{h}=\frac{\mu(E)}{\mathrm{Vol}_{L}}\mathrm{Id}_{E},$

where $\Lambda_{\omega}$ is the contraction with respect to the Kähler metric
$\omega$ on the manifold (i.e. the dual Lefschetz operator).

Critical points of $\mathcal{M}^{Don}$ being the Hermitian–Einstein metric is
a consequence of the following lemma.

######  Lemma 1.28.

(cf. [Don1985, Section 1.2]) Fixing a reference metric
$h_{\mathrm{ref}}\in\mathcal{H}_{\infty}$, we have

$\displaystyle\frac{d}{dt}\mathcal{M}^{Don}_{1}(h_{t},h_{\mathrm{ref}})$
$\displaystyle=\int_{X}\mathrm{tr}\left(h_{t}^{-1}\partial_{t}h_{t}\cdot
F_{t}\right)\frac{\omega^{n-1}}{(n-1)!},$
$\displaystyle\frac{d}{dt}\mathcal{M}^{Don}_{2}(h_{t},h_{\mathrm{ref}})$
$\displaystyle=\frac{1}{\mathrm{Vol}_{L}}\int_{X}\mathrm{tr}(h^{-1}_{t}\partial_{t}h_{t})\frac{\omega^{n}}{n!},$

for a path $\\{h_{t}\\}_{0\leq t\leq 1}\subset\mathcal{H}_{\infty}$ of smooth
metrics with $h_{0}=h_{\mathrm{ref}}$, and hence

$\frac{d}{dt}\mathcal{M}^{Don}(h_{t},h_{\mathrm{ref}})=\int_{X}\mathrm{tr}\left(h_{t}^{-1}\partial_{t}h_{t}\left(\Lambda_{\omega}F_{t}-\frac{\mu(E)}{\mathrm{Vol}_{L}}\mathrm{Id}_{E}\right)\right)\frac{\omega^{n}}{n!}.$

The final important point is that $\mathcal{M}^{Don}$ is convex along
geodesics in $\mathcal{H}_{\infty}$, as defined in (1.6). Recalling the
important fact that $\mathcal{H}_{\infty}$ is geodesically complete, these
properties together imply that a critical point of $\mathcal{M}^{Don}$ is
necessarily the global minimum. We summarise the above in the following
proposition.

######  Proposition 1.29.

(cf. [Kobook, Section 6.3]) The functional $\mathcal{M}^{Don}$ is convex along
geodesics in $\mathcal{H}_{\infty}$, and its critical point (if exists)
attains the global minimum. Moreover, $\mathcal{M}^{Don}$ is strictly convex
along nontrivial geodesics if $E$ is irreducible (in particular if $E$ is
simple) and in this case the critical point (if exists) is unique up to an
overall constant scaling.

##  2\. Quot-scheme limit of Fubini–Study metrics

###  2.1. Filtrations and metric duals

Suppose that we fix a basis $\mathbb{S}_{k}$ for $H^{0}(\mathcal{E}(k))$,
which fixes $Q^{*}$ as in Section 1.3. We choose a hermitian
$\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$ with rational
eigenvalues, and hence a rational 1-PS $\\{\sigma_{t}\\}_{t}\subset
SL(H^{0}(\mathcal{E}(k))^{\vee})$ associated to it by $\sigma_{t}:=e^{\zeta
t}$. We may change $\mathbb{S}_{k}$ (without affecting $Q^{*}Q$) to a
unitarily equivalent basis $\tau\cdot\mathbb{S}_{k}$ ($\tau\in U(N_{k})$),
say, so that it is compatible with the weight space decomposition

(2.1) $H^{0}(\mathcal{E}(k))^{\vee}=\bigoplus_{i=1}^{\nu}V^{\vee}_{w_{i},k}$

where $\zeta$ acts on $V^{\vee}_{w_{i},k}$ with weight $w_{i}\in\mathbb{Q}$
for all $i=1,\dots,\nu$. Note that $\nu$ depends on $\zeta$. Henceforth we
shall assume the ordering

(2.2) $w_{1}>\cdots>w_{\nu}.$

######  Remark 2.1.

We shall assume without loss of generality that the operator norm (i.e. the
modulus of the maximum eigenvalue) of $\zeta$ is at most 1; this can be
achieved by an overall constant multiple of $\zeta$, which just results in a
different scaling of the parameter $t$.

Although $\sigma_{t}$ can be written as $e^{\zeta t}$, we shall introduce an
auxiliary variable

$T:=e^{-t},$

so that the limit $t\to+\infty$ corresponds to $T\to 0$, which makes it easier
for us to compare the limit $t\to+\infty$ to the description of the Quot-
scheme limit as a central fibre, as presented in Lemma 1.10. In this
description, we have

(2.3) $\sigma_{t}=T^{-\zeta},$

and the weight space decomposition (2.1) defines a filtration

(2.4) $V^{\vee}_{\leq-w_{i},k}:=\bigoplus_{j=1}^{i}V^{\vee}_{w_{j},k},$

where we note that the minus sign appears because (2.3) gives the ordering of
the weight

(2.5) $-w_{1}<\dots<-w_{\nu}$

with respect to $T$.

We now consider a filtration that
$\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$ induces on
$H^{0}(\mathcal{E}(k))$. It turns out that the natural dual action is not the
one relevant for our purpose. Recall that Lemma 1.22 implies that we have the
following diagram

(2.6)
$\textstyle{\overline{H^{0}(\mathcal{E}(k))^{\vee}}\otimes\overline{C^{\infty}_{X}(k)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q^{*}}$$\scriptstyle{*}$$\textstyle{\overline{E^{\vee}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{*,\
\textup{gauge}}$$\textstyle{H^{0}(\mathcal{E}(k))\otimes\mathcal{O}_{X}(-k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\mathcal{E}}$

where the left vertical arrow stands for the metric duality isomorphism given
by $I_{\mathbb{S}_{k}}$ and $h_{L}$, and the right vertical arrow for the one
given by $h_{\mathrm{ref}}$ and the $C^{\infty}$-gauge transformation (written
$f_{\mathbb{S}_{k},\mathrm{ref}}$ in Lemma 1.22). The fact that the right
vertical arrow involves taking the dual of the bundle $\mathcal{E}$ will have
nontrivial consequences concerning the sign (cf. Remark 3.4).

We decree that the $I_{\mathbb{S}_{k}}$-metric dual gives a weight-preserving
isomorphism between $H^{0}(\mathcal{E}(k))$ and
$\overline{H^{0}(\mathcal{E}(k))^{\vee}}$, and hence define a weight
decomposition

$H^{0}(\mathcal{E}(k))=\bigoplus_{i=1}^{\nu}V_{w_{i},k}$

in which $\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$ acts with
weight $w_{i}$ on $V_{w_{i},k}$. We thus get a filtration

$V_{\leq-w_{i},k}:=\bigoplus_{j=1}^{i}V_{w_{j},k},$

as in (2.4), which defines a filtration

(2.7) $0\neq\mathcal{E}_{\leq-w_{1}}\subset\cdots\subset\mathcal{E}_{\leq-
w_{\nu}}=\mathcal{E}$

of $\mathcal{E}$ by subsheaves of $\mathcal{E}$, where $\mathcal{E}_{\leq-
w_{i}}$ is defined by the quotient map

$\rho:V_{\leq-w_{i},k}\otimes\mathcal{O}_{X}(-k)\to\mathcal{E}_{\leq-w_{i}}.$

Recall, as in Lemma 1.10, that the Quot-scheme limit associated to the above
filtration is given by

$\hat{\mathcal{E}}:=\bigoplus_{i=1}^{\nu}\mathcal{E}_{-w_{i}},$

where $\mathcal{E}_{-w_{i}}=\mathcal{E}_{-w_{i}}/\mathcal{E}_{-w_{i-1}}$ for
$i=1,\dots,\nu$ (in which we assume $\mathcal{E}_{-w_{0}}=0$).

The components of $\hat{\mathcal{E}}$ with nontrivial rank are the objects
that we need later.

######  Definition 2.2.

We define a subset $\\{\hat{1},\dots,\hat{\nu}\\}$ of $\\{1,\dots,\nu\\}$,
with ordering $\hat{1}<\dots<\hat{\nu}$, so that
$\mathrm{rk}(\mathcal{E}_{-w_{\alpha}})>0$ if and only if
$\alpha\in\\{\hat{1},\dots,\hat{\nu}\\}$.

The above definition, together with the invariance of the rank under the flat
limit in Quot-schemes, implies

$\sum_{\alpha=\hat{1}}^{\hat{\nu}}\mathrm{rk}(\mathcal{E}_{-w_{\alpha}})=\mathrm{rk}(\mathcal{E}).$

######  Remark 2.3.

Since $\mathcal{E}_{\leq-w_{1}}$ is a non-zero torsion-free subsheaf of
$\mathcal{E}$, we find $\hat{1}=1$ in the above notation. In particular, note
that $\mathcal{E}_{\leq-w_{1}}\subset\mathcal{E}_{\leq-w_{\alpha}}$ for all
$\alpha\in\\{\hat{1},\dots,\hat{\nu}\\}$.

We shall mainly work over an open dense subset of $X$ where each
$\mathcal{E}_{\leq-w_{i}}$ is a holomorphic subbundle of $\mathcal{E}$. We
make the following definition, by recalling the standard definition of the
singular locus.

######  Definition 2.4.

The singular locus $\mathrm{Sing}(\mathcal{E}_{\leq-w_{i}})$ of
$\mathcal{E}_{\leq-w_{i}}$ is the Zariski closed set in $X$ where
$\mathcal{E}_{\leq-w_{i}}$ is not locally free. We define the regular locus to
be the complement of

$\bigcup_{i=1}^{\nu}\mathrm{Sing}(\mathcal{E}_{\leq-
w_{i}})\cup\mathrm{Sing}(\mathcal{E}/\mathcal{E}_{\leq-w_{i}})$

in $X$, and denote it by $X^{reg}(\zeta)$ or $X^{reg}$.

Thus each $\mathcal{E}_{\leq-w_{i}}$ is a holomorphic subbundle of
$\mathcal{E}$ over $X^{reg}$, by recalling that a subsheaf $\mathcal{F}$ of a
locally free sheaf $\mathcal{E}$ is a holomorphic subbundle of $\mathcal{E}$
if and only if both $\mathcal{F}$ and $\mathcal{E}/\mathcal{F}$ are locally
free. Observe that $X\setminus X^{reg}$ has codimension at least 1 in $X$
Lemma 1.7.

We define a $C^{\infty}$ complex vector bundle $\widehat{E}$ over $X^{reg}$ as

$\widehat{E}:=\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}E_{\leq-
w_{\alpha}}/E_{\leq-w_{\alpha-1}},$

where $E_{\leq-w_{\alpha}}$ is the $C^{\infty}$ complex vector bundle over
$X^{reg}$ defined by $\mathcal{E}_{\leq-w_{\alpha}}$, and the quotient is the
one of the $C^{\infty}$ complex vector bundle over $X^{reg}$. Note that we
have an isomorphism $\widehat{E}\stackrel{{\scriptstyle\sim}}{{\to}}E$ as
$C^{\infty}$-vector bundles over $X^{reg}$. Combining this with another
$C^{\infty}$-isomorphism
$E\stackrel{{\scriptstyle\sim}}{{\to}}\overline{E^{\vee}}$ (afforded by the
$h_{\mathrm{ref}}$-metric dual, say), we get

(2.8)
$\overline{E^{\vee}}\stackrel{{\scriptstyle\sim}}{{\to}}\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}E_{-w_{\alpha}}$

where we wrote $E_{-w_{\alpha}}=E_{\leq-w_{\alpha}}/E_{\leq-w_{\alpha-1}}$.

It turns out that what we need later is the modification of the filtration
$0\neq\mathcal{E}_{\leq-w_{1}}\subset\cdots\subset\mathcal{E}_{\leq-
w_{\nu}}=\mathcal{E}$ on a Zariski closed subset of $X$, given by the
following lemma.

######  Lemma 2.5.

There exists a filtration

(2.9) $0\neq\mathcal{E}^{\prime}_{\leq-
w_{1}}\subset\cdots\subset\mathcal{E}^{\prime}_{\leq-w_{\nu}}=\mathcal{E}$

of $\mathcal{E}$ by subsheaves of $\mathcal{E}$, such that

  1. (1)

$\mathcal{E}^{\prime}_{\leq-w_{i}}$ is saturated in $\mathcal{E}$ and agrees
with $\mathcal{E}_{\leq{-w_{i}}}$ on a Zariski open subset of $X$ for all
$i=1,\dots,\nu$,

  2. (2)

setting $\mathcal{E}^{\prime}_{-w_{i}}:=\mathcal{E}^{\prime}_{\leq-
w_{i}}/\mathcal{E}^{\prime}_{\leq-w_{i-1}}$,
$\mathrm{rk}(\mathcal{E}^{\prime}_{-w_{i}})>0$ if and only if
$i\in\\{\hat{1},\dots,\hat{\nu}\\}$.

###### Proof.

We define (2.9) inductively as follows. $\mathcal{E}^{\prime}_{\leq-
w_{\hat{\nu}}}$ is defined as the saturation of $\mathcal{E}_{\leq-
w_{\hat{\nu}}}$ in $\mathcal{E}$; note that this implies
$\mathcal{E}^{\prime}_{\leq-w_{\hat{\nu}}}=\mathcal{E}$, since
$\mathcal{E}^{\prime}_{\leq-w_{\hat{\nu}}}$ has full rank. We then replace
$\mathcal{E}_{\leq-w_{\hat{\nu}+1}},\dots,\mathcal{E}_{\leq-w_{\nu}}$ by
$\mathcal{E}^{\prime}_{\leq-w_{\hat{\nu}}}=\mathcal{E}$. Continuing
inductively, for $\alpha\in\\{\hat{1},\dots,\widehat{\nu-1}\\}$ we define
$\mathcal{E}^{\prime}_{\leq-w_{\alpha}}$ to be the saturation of
$\mathcal{E}_{\leq-w_{\alpha}}$ in $\mathcal{E}$. Defining $\beta$ to be the
smallest element of $\\{\hat{1},\dots,\hat{\nu}\\}$ that is larger than
$\alpha\in\\{\hat{1},\dots,\widehat{\nu-1}\\}$, we replace $\mathcal{E}_{\leq-
w_{\alpha+1}},\dots,\mathcal{E}_{\leq-w_{\beta-1}}$ by
$\mathcal{E}^{\prime}_{\leq-w_{\alpha}}$. The end product that we get is a
filtration $0\neq\mathcal{E}^{\prime}_{\leq-
w_{1}}\subset\cdots\subset\mathcal{E}^{\prime}_{\leq-w_{\nu}}=\mathcal{E}$ of
$\mathcal{E}$ by saturated subsheaves, in which strict inclusion holds if and
only if $i\in\\{\hat{1},\dots,\hat{\nu}\\}$ (recall also $\hat{1}=1$ as in
Remark 2.3).

Since the above construction does not change the rank of each
$\mathcal{E}_{\leq-w_{i}}$ for all $i=1,\dots,\nu$, and hence does not change
$\mathcal{E}_{\leq-w_{i}}$ where it is locally free, we see that
$\mathcal{E}_{\leq-w_{i}}$ and $\mathcal{E}^{\prime}_{\leq-w_{i}}$ agree on
the (Zariski open) locus where they are locally free. ∎

###  2.2. Renormalised Quot-scheme limit of Fubini–Study metrics

Our aim in this section is to find an expansion for
$h_{\sigma_{t}}:=Q^{*}\sigma^{*}_{t}\sigma_{t}Q$ as $t\to+\infty$. Although
this limit is divergent, we can find an appropriate re-scaling (in terms of
the constant gauge transformation) to get a 1-PS of hermitian metrics
$\\{\hat{h}_{\sigma_{t}}\\}_{t\geq 0}$ which is convergent. The limit
$\hat{h}$, which we call the renormalised Quot-scheme limit of
$h_{\sigma_{t}}$, is well-defined only on the Zariski open subset $X^{reg}$ of
$X$, and it may tend to a degenerate metric as it approaches the singular
locus $X\setminus X^{reg}$. In particular, its curvature may blow up. In spite
of this problem, it will play an important role later. A combinatorial
argument making use of the ordering $w_{1}>\cdots>w_{\nu}$ (2.2) is of crucial
importance for the definition of $\hat{h}$.

######  Definition 2.6.

Suppose that we fix a $C^{\infty}$-isomorphism
$\overline{E^{\vee}}\stackrel{{\scriptstyle\sim}}{{\to}}\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}E_{-w_{\alpha}}$
of $C^{\infty}$-vector bundles over $X^{reg}$ as in (2.8), and let
$\mathrm{pr}_{\alpha}:\overline{E^{\vee}}\twoheadrightarrow E_{-w_{\alpha}}$
be the surjection to each factor.

  1. (1)

A $C^{\infty}$-map
$\tilde{Q}^{*}_{\alpha}:\overline{V^{\vee}_{k}}\otimes\overline{C^{\infty}_{X^{reg}}(k)}\to
E_{-w_{\alpha}}$ of locally free sheaves of $C^{\infty}_{X^{reg}}$-modules is
defined by the composition of $Q^{*}$ and the projection
$\mathrm{pr}_{\alpha}:\overline{E^{\vee}}\twoheadrightarrow E_{-w_{\alpha}}$.

  2. (2)

Let $j\geq\alpha$. A map
$Q_{j\to\alpha}^{*}:\overline{V^{\vee}_{-w_{j},k}}\otimes\overline{C^{\infty}_{X^{reg}}(k)}\to
E_{-w_{\alpha}}$ of locally free sheaves of $C^{\infty}_{X^{reg}}$-modules is
defined as the composition of
$Q^{*}|_{V^{\vee}_{-w_{j},k}}:\overline{V^{\vee}_{-w_{j},k}}\otimes\overline{C^{\infty}_{X^{reg}}(k)}\to\overline{E^{\vee}}$
and $\mathrm{pr}_{\alpha}$ (where the $C^{\infty}$-isomorphism (2.8) is
understood), i.e.

$Q^{*}_{j\to\alpha}:=\mathrm{pr}_{\alpha}\circ Q^{*}|_{V^{\vee}_{-w_{j},k}}.$

In diagram,

$\textstyle{\overline{V^{\vee}_{-w_{j},k}}\otimes\overline{C^{\infty}_{X^{reg}}(k)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q^{*}|_{V^{\vee}_{-w_{j},k}}}$$\scriptstyle{Q_{j\to\alpha}^{*}}$$\textstyle{\overline{E^{\vee}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pr}_{\alpha}}$$\textstyle{E_{-w_{\alpha}}.}$

Recalling $E_{-w_{\alpha}}:=E_{\leq-w_{\alpha}}/E_{\leq-w_{\alpha-1}}$, we
find $Q^{*}_{j\to\alpha}=0$ for $j<\alpha$ (see e.g. (1.3) and (2.5)).

######  Remark 2.7.

In what follows, we assume a convention in which $Q^{*}_{j\to\alpha}$ is
defined on the whole $V^{\vee}_{k}$, with
$Q^{*}_{j\to\alpha}|_{V^{\vee}_{-w_{l},k}}=0$ if $l\neq j$.

We define $\tilde{Q}_{\alpha}$ and $Q_{\alpha\to j}$ analogously, taking the
fibrewise hermitian conjugate of $\tilde{Q}^{*}_{\alpha}$ and
$Q^{*}_{j\to\alpha}$, which is equivalent to taking the hermitian metric dual
with respect to the reference metrics defined in Section 1.3.

The main technical result that we establish in this section is the following.

######  Proposition 2.8.

Suppose that we write

(2.10) $e^{wt}:=\mathrm{diag}(e^{w_{\hat{1}}t},\cdots,e^{w_{\hat{\nu}}t}),$

with respect to the block decomposition (2.8). Then the Fubini–Study metric
$h_{\sigma_{t}}=Q^{*}e^{\zeta^{*}t+\zeta t}Q$ can be written as
$h_{\sigma_{t}}=e^{wt}\hat{h}_{\sigma_{t}}e^{wt}$, where
$\hat{h}_{\sigma_{t}}$ is defined as

$\hat{h}_{\sigma_{t}}:=\begin{pmatrix}Q^{*}_{\hat{1}\to\hat{1}}Q_{\hat{1}\to\hat{1}}&\cdots&0\\\
\vdots&\ddots&\vdots\\\
0&\cdots&Q^{*}_{{\hat{\nu}}\to\hat{\nu}}Q_{\hat{\nu}\to{\hat{\nu}}}\end{pmatrix}+\bar{I}_{\sigma_{t}},$

over $X^{reg}$, in terms of the decomposition
$\bigoplus_{\alpha=1}^{\hat{\nu}}E_{\alpha}$; $\bar{I}_{\sigma_{t}}$ is a
fibrewise hermitian form of rank $r$, whose $(\alpha,\beta)$-th block is given
by

$\sum_{j>\alpha,\beta}e^{-(w_{\alpha}-w_{j})t}e^{-(w_{\beta}-w_{j})t}Q^{*}_{j\to\alpha}Q_{\beta\to
j},$

which decays exponentially as $t\to+\infty$ (since $w_{\alpha}-w_{j}>0$ for
$j>\alpha$ by (2.2)).

Thus, we can “separate” the part $\hat{h}_{\sigma_{t}}$ of $h_{\sigma_{t}}$
which converges as $t\to+\infty$; moreover, we have an explicit description of
the rate of convergence. Since they play an important role in what follows, we
give it a definition.

######  Definition 2.9.

The path of hermitian metrics $\\{\hat{h}_{\sigma_{t}}\\}_{t\geq 0}$, where
$\hat{h}_{\sigma_{t}}$ is a hermitian metric defined on the regular locus
$X^{reg}$ by

$\hat{h}_{\sigma_{t}}:=e^{-wt}h_{\sigma_{t}}e^{-wt}\in\Gamma_{C^{\infty}_{X^{reg}}}(E^{\vee}\otimes\overline{E^{\vee}}),$

is called the renormalised Bergman 1-parameter subgroup associated to
$\sigma_{t}$, where the $C^{\infty}$-isomorphism
$\overline{E^{\vee}}\stackrel{{\scriptstyle\sim}}{{\to}}\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}E_{-w_{\alpha}}$
and the notation (2.10) are understood. We also call its limit in
$\Gamma_{C^{\infty}_{X^{reg}}}(E^{\vee}\otimes\overline{E^{\vee}})$

$\hat{h}:=\lim_{t\to+\infty}e^{-wt}h_{\sigma_{t}}e^{-wt}=\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}Q^{*}_{{\alpha}\to\alpha}Q_{\alpha\to{\alpha}}$

the renormalised Quot-scheme limit of $h_{\sigma_{t}}$, and each component of
$\hat{h}$ is written as
$\hat{h}_{\alpha}:=Q^{*}_{{\alpha}\to\alpha}Q_{\alpha\to{\alpha}}$, so that
$\hat{h}=\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}\hat{h}_{\alpha}$, where
$\hat{h}_{\alpha}\in\Gamma_{C^{\infty}_{X^{reg}}}(E_{-w_{\alpha}}\otimes
E_{-w_{\alpha}})$, with
$\overline{E^{\vee}}\stackrel{{\scriptstyle\sim}}{{\to}}\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}E_{-w_{\alpha}}$
understood.

Thus, “renormalised” in the above means that we apply the constant gauge
transform over $X^{reg}$ to $h_{\sigma_{t}}$, by ($-1$ times) the weight of
$\zeta$ that acts on
$\widehat{E}=\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}E_{-w_{\alpha}}$; we then
get a component $\hat{h}_{\sigma_{t}}$ of $h_{\sigma_{t}}$ which converges as
$t\to+\infty$.

It turns out that the renormalised Quot-scheme limit $\hat{h}$ defines a
hermitian metric on the vector bundle $E$ over a Zariski open subset
$X^{reg}=X^{reg}(\zeta)$ of $X$. In general, however, $\hat{h}$ fails to be
positive definite on the Zariski closed subset $X\setminus X^{reg}$ of $X$,
which fundamentally comes from the singularity of the sheaves
$\mathcal{E}_{\leq-w_{i}}$ and $\mathcal{E}/\mathcal{E}_{\leq-w_{i}}$.

We now start the proof of Proposition 2.8 by proving the following result,
which can be regarded as a $C^{\infty}$-analogue of the expansion (1.3).

######  Lemma 2.10.

We have the following expansion over $X^{reg}$

$Q^{*}\circ e^{\zeta
t}=\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}e^{w_{\alpha}t}\left(Q^{*}_{{\alpha}\to\alpha}+\sum_{j>\alpha}e^{-(w_{\alpha}-w_{j})t}Q^{*}_{j\to\alpha}\right),$

where the direct sum is in terms of the $C^{\infty}$-isomorphism
$\overline{E^{\vee}}\stackrel{{\scriptstyle\sim}}{{\to}}\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}E_{-w_{\alpha}}$.

###### Proof.

Definition 2.6 immediately yields

$\tilde{Q}^{*}_{\alpha}=Q^{*}_{{\alpha}\to\alpha}+\sum_{j>\alpha}Q^{*}_{j\to\alpha}.$

With the $C^{\infty}$-isomorphism
$\overline{E^{\vee}}\cong\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}E_{\alpha}$, we
can write

$Q^{*}=\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}\tilde{Q}^{*}_{\alpha}=\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}\left(Q^{*}_{{\alpha}\to\alpha}+\sum_{j>\alpha}Q^{*}_{j\to\alpha}\right).$

Recalling that $e^{\zeta t}=T^{-\zeta}$ acts on $V_{-w_{j},k}$ as
$e^{w_{j}t}=T^{-w_{j}}$, and hence

$Q^{*}_{j\to\alpha}\circ e^{\zeta t}=e^{w_{j}t}Q^{*}_{j\to\alpha},$

we get the claim. ∎

###### Proof of Proposition 2.8.

We start by re-writing Lemma 2.10 in a block-matrix notation, according to the
decomposition $\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}E_{-w_{\alpha}}$ and
$\bigoplus_{j=1}^{\nu}V_{-w_{j},k}$.

We can write $Q^{*}$ in the following (block) row echelon form

$Q^{*}=\begin{pmatrix}Q^{*}_{1\to\hat{1}}&Q^{*}_{2\to\hat{1}}&\cdots&\cdots&\cdots&\cdots&Q^{*}_{{\nu}\to\hat{1}}\\\
0&\cdots&0&Q^{*}_{\hat{2}\to\hat{2}}&\cdots&\cdots&Q^{*}_{{\nu}\to\hat{2}}\\\
\vdots&\vdots&\vdots&\cdots&\ddots&\ddots&\vdots\\\
0&0&0&\cdots&Q^{*}_{\hat{\nu}\to\hat{\nu}}&\cdots&Q^{*}_{{\nu}\to\hat{\nu}}\end{pmatrix},$

where we recall $Q^{*}_{j\to\alpha}=0$ for $j<\alpha$, and also $\hat{1}=1$
(as in Remark 2.3).

Thus, recalling that $\zeta$ is hermitian and acts on $V_{-w_{j},k}$ as
$T^{-w_{j}}$ with $w_{j}\in\mathbb{Q}$, we compute

$\displaystyle Q^{*}e^{\zeta^{*}t}=Q^{*}\circ T^{-\zeta}=$
$\displaystyle=\begin{pmatrix}T^{-w_{1}}Q^{*}_{1\to\hat{1}}&T^{-w_{2}}Q^{*}_{2\to\hat{1}}&\cdots&\cdots&\cdots&\cdots&T^{-w_{\nu}}Q^{*}_{{\nu}\to\hat{1}}\\\
0&\cdots&0&T^{-w_{\hat{2}}}Q^{*}_{\hat{2}\to\hat{2}}&\cdots&\cdots&T^{-w_{\nu}}Q^{*}_{{\nu}\to\hat{2}}\\\
\vdots&\vdots&\vdots&\cdots&\ddots&\ddots&\vdots\\\
0&0&0&\cdots&T^{-w_{\hat{\nu}}}Q^{*}_{\hat{\nu}\to\hat{\nu}}&\cdots&T^{-w_{\nu}}Q^{*}_{{\nu}\to\hat{\nu}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}T^{-w_{1}}&0&\cdots&0\\\
0&T^{-w_{\hat{2}}}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&T^{-w_{\hat{\nu}}}\end{pmatrix}$ $\displaystyle\,\hskip
5.69046pt\times\begin{pmatrix}Q^{*}_{1\to\hat{1}}&T^{-(w_{2}-w_{1})}Q^{*}_{2\to\hat{1}}&\cdots&\cdots&\cdots&\cdots&T^{-(w_{\nu}-w_{1})}Q^{*}_{{\nu}\to\hat{1}}\\\
0&\cdots&0&Q^{*}_{\hat{2}\to\hat{2}}&\cdots&\cdots&T^{-(w_{\nu}-w_{\hat{2}})}Q^{*}_{{\nu}\to\hat{2}}\\\
\vdots&\vdots&\vdots&\cdots&\ddots&\ddots&\vdots\\\
0&0&0&\cdots&Q^{*}_{\hat{\nu}\to\hat{\nu}}&\cdots&T^{-(w_{\nu}-w_{\hat{\nu}})}Q^{*}_{{\nu}\to\hat{\nu}}\end{pmatrix}.$

By taking the conjugate transpose, we obtain a similar formula for $e^{\zeta
t}Q$. Recalling $\hat{1}=1$ (cf. Remark 2.3) and $T^{-w_{j}}=e^{w_{j}t}$, we
thus get

$\displaystyle Q^{*}e^{\zeta^{*}t+\zeta t}Q$
$\displaystyle=e^{wt}\left(\begin{pmatrix}Q^{*}_{\hat{1}\to\hat{1}}Q_{\hat{1}\to\hat{1}}&\cdots&0\\\
\vdots&\ddots&\vdots\\\
0&\cdots&Q^{*}_{\hat{\nu}\to\hat{\nu}}Q_{\hat{\nu}\to\hat{\nu}}\end{pmatrix}+\bar{I}_{\sigma_{t}}\right)e^{wt},$

where the $(\alpha,\beta)$-th block of $\bar{I}_{\sigma_{t}}$ in the above,
with $\alpha,\beta\in\\{\hat{1},\dots,\hat{\nu}\\}$, can be written as

$\sum_{j>\alpha,\beta}T^{-(w_{j}-w_{\alpha})}T^{-(w_{j}-w_{\beta})}Q^{*}_{j\to\alpha}Q_{\beta\to
j}.$

Recalling again $T=e^{-t}$, we get the result as claimed. ∎

######  Lemma 2.11.

The metric $\hat{h}$ is strictly positive definite over $X^{reg}$.

###### Proof.

It suffices to show that the map
$Q_{\alpha\to\alpha}^{*}:\overline{V^{\vee}_{-w_{\alpha},k}}\otimes\overline{C^{\infty}_{X^{reg}}(k)}\to
E_{-w_{\alpha}}$ defined in Definition 2.6 is surjective, since it implies
that $Q^{*}_{\alpha\to\alpha}Q_{\alpha\to\alpha}$ is strictly nondegenerate.

First consider the map
$\rho_{\alpha\to\alpha}:V_{-w_{\alpha}}\otimes\mathcal{O}_{X}(-k)\to\mathcal{E}_{-w_{\alpha}}$,
which is defined by the composition of
$\rho|_{V_{-w_{\alpha}}}:V_{-w_{\alpha}}\otimes\mathcal{O}_{X}(-k)\to\mathcal{E}_{\leq-
w_{\alpha}}$ and the projection $\mathcal{E}_{\leq-
w_{\alpha}}\to\kern-7.74998pt\to\mathcal{E}_{\leq-
w_{\alpha}}/\mathcal{E}_{\leq-w_{\alpha-1}}=\mathcal{E}_{-w_{\alpha}}$.
Recalling the definition $\mathcal{E}_{-w_{\alpha}}=\rho(V_{\leq-
w_{\alpha}})/\rho(V_{\leq-w_{\alpha-1}})$, we see that
$\rho_{\alpha\to\alpha}$ is surjective.

Lemma 1.22 and the diagram (2.6) imply that the surjectivity of
$Q_{\alpha\to\alpha}^{*}$ follows from that of $\rho_{\alpha\to\alpha}$. ∎

While $\hat{h}$ may degenerate as it approaches $X\setminus X^{reg}$ and hence
has no uniform lower bound, it has a uniform upper bound given by $Q^{*}Q$,
stated as follows.

######  Lemma 2.12.

$|Q^{*}_{j\to\alpha}|^{2}=\mathrm{tr}(Q^{*}_{j\to\alpha}Q_{\alpha\to j})$ can
be bounded by $|Q^{*}|^{2}$ for any $j=1,\dots,\nu$ and
$\alpha=\hat{1},\dots,\hat{\nu}$.

###### Proof.

This is straightforward, by noting $Q^{*}_{j\to\alpha}Q_{\beta\to m}=0$ if
$j\neq m$ and so on. ∎

Note also that $\sum_{j>\alpha,\beta}Q^{*}_{j\to\alpha}Q_{\beta\to j}$ can be
bounded by $\sum_{j}|Q_{\alpha\to j}|^{2}$, by Cauchy–Schwarz.

##  3\. Estimates for Donaldson’s functional

We establish several technical estimates, involving the renormalised
Fubini–Study metrics and the renormalised Quot-scheme limit. Much of their
proofs is elementary, but we shall also need to use the resolution of
singularities for saturated subsheaves, as explained in [Jacob, Sibley].

###  3.1. Lower bound for Donaldson’s functional

The main result of this section is the following.

######  Proposition 3.1.

Let $\\{h_{\sigma_{t}}\\}_{t\geq 0}$ be a path of Fubini–Study metrics
emanating from $h_{k}$ generated by
$\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$. Then there exists a
constant $c(h_{k},k)>0$ which depends only on $h_{k}$ and $k$ such that

$\mathcal{M}^{Don}(h_{\sigma_{t}},h_{k})\geq-\sum_{i=1}^{\nu}2w_{i}\left(\deg(\mathcal{E}^{\prime}_{-w_{i}})-\mu(\mathcal{E})\mathrm{rk}(\mathcal{E}^{\prime}_{-w_{i}})\right)t-c(h_{k},k)$

holds for all $t\geq 0$, where
$\mathcal{E}^{\prime}_{-w_{i}}:=\mathcal{E}^{\prime}_{\leq-
w_{i}}/\mathcal{E}^{\prime}_{\leq-w_{i-1}}$ is defined as in Lemma 2.5 by
using the filtration $0\neq\mathcal{E}^{\prime}_{\leq-
w_{1}}\subset\cdots\subset\mathcal{E}^{\prime}_{\leq-w_{\nu}}=\mathcal{E}$.

The rest of this section concerns the proof of the above proposition, which
follows from Lemmas 3.2, 3.3, 3.6, 3.7, and Proposition 3.8 that are proved
below.

######  Lemma 3.2.

Writing $h_{\sigma_{t}}=e^{wt}\hat{h}_{\sigma_{t}}e^{wt}$ by using the
renormalised Fubini–Study metric $\hat{h}_{\sigma_{t}}$ (as in Definition
2.9), we have

$h_{\sigma_{t}}^{-1}\partial_{t}h_{\sigma_{t}}=e^{-wt}\left(2w+\hat{h}_{\sigma_{t}}^{-1}[w,\hat{h}_{\sigma_{t}}]+\hat{h}_{\sigma_{t}}^{-1}\partial_{t}\hat{h}_{\sigma_{t}}\right)e^{wt},$

where $w=\mathrm{diag}(w_{\hat{1}},\dots,w_{\hat{\nu}})$ and $[,]$ is the
commutator.

###### Proof.

We can write $h_{\sigma_{t}}=e^{wt}\hat{h}_{\sigma_{t}}e^{wt}$ by using the
renormalised Fubini–Study metric $\hat{h}_{\sigma_{t}}$, and note

$\frac{d}{dt}h_{\sigma_{t}}=e^{wt}\left(w\hat{h}_{\sigma_{t}}+\frac{d}{dt}\hat{h}_{\sigma_{t}}+\hat{h}_{\sigma_{t}}w\right)e^{wt},$

which immediately gives us the formula that we claimed. ∎

Note that the connection 1-form of $h_{\sigma_{t}}$, which is
$h_{\sigma_{t}}^{-1}\partial h_{\sigma_{t}}$, can be evaluated as
$e^{-wt}(\hat{h}_{\sigma_{t}}^{-1}\partial\hat{h}_{\sigma_{t}})e^{wt}$. This
immediately implies that, writing $F(\hat{h}_{\sigma_{t}})$ for
($\sqrt{-1}/2\pi$ times) the curvature of $\hat{h}_{\sigma_{t}}$, we have the
following pointwise equality over $X^{reg}$:

$F(h_{\sigma_{t}})=e^{-wt}F(\hat{h}_{\sigma_{t}})e^{wt}.$

We thus get the following result.

######  Lemma 3.3.

Writing $\hat{h}_{\sigma_{t}}$ for the renormalised Fubini–Study metric, we
have

$\displaystyle\mathcal{M}^{Don}(h_{\sigma_{t}},h_{k})=$
$\displaystyle-\int_{0}^{t}dt\int_{X^{reg}}\mathrm{tr}\left((2w+\hat{h}_{\sigma_{t}}^{-1}[w,\hat{h}_{\sigma_{t}}])(F(\hat{h}_{\sigma_{t}})-\mu(E)\mathrm{Id}_{E})\right)\frac{\omega^{n}}{n!}$
$\displaystyle+\mathcal{M}^{Don}(\hat{h}_{\sigma_{t}},h_{k}).$

######  Remark 3.4.

An important remark about the sign is in order. We considered the renormalised
Bergman 1-PS as a hermitian metric on $E$ by means of the
$C^{\infty}$-isomorphism
$\overline{E^{\vee}}\stackrel{{\scriptstyle\sim}}{{\to}}\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}E_{-w_{\alpha}}$
which involves the metric dual. In other words, the (renormalised) Bergman
1-PS is a priori a hermitian metric on $E^{\vee}$, which is identified with
the one on $E$ by means of the metric dual. Although $h_{\sigma_{t}}$ is a
hermitian metric on $E$, when we write it as
$h_{\sigma_{t}}=e^{wt}\hat{h}_{\sigma_{t}}e^{wt}$ as we do here, it is tacitly
assumed that the $E$ is identified with $\overline{E^{\vee}}$ with the metric
duality isomorphism as in the diagram (2.6).

With this metric duality isomorphism understood, we should consider
$h_{\sigma_{t}}=e^{wt}\hat{h}_{\sigma_{t}}e^{wt}$ as a hermitian metric on
$E^{\vee}$, and hence we should consider the Donaldson functional for
$E^{\vee}$, which is nothing but $-1$ times the usual definition. Thus, given
$h_{\sigma_{t}}\in\mathcal{H}_{\infty}(\mathcal{E})$, we apply the metric
duality isomorphism
$\mathcal{H}_{\infty}(\mathcal{E})\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{H}_{\infty}(\mathcal{E}^{\vee})$,
$h_{\sigma_{t}}\mapsto
e^{wt}\hat{h}_{\sigma_{t}}e^{wt}=:h^{\vee}_{\sigma_{t}}$ to get

$\displaystyle\mathcal{M}^{Don}(e^{wt}\hat{h}_{\sigma_{t}}e^{wt},h_{k})=\mathcal{M}^{Don}(h^{\vee}_{\sigma_{t}},h_{k})$
$\displaystyle=-\int_{0}^{t}dt\int_{X}\mathrm{tr}((h^{\vee}_{\sigma_{t}})^{-1}\partial_{t}h^{\vee}_{\sigma_{t}}(\Lambda_{\omega}F(h^{\vee}_{\sigma_{t}})-\mu(\mathcal{E})\mathrm{Id}_{E}))\frac{\omega^{n}}{n!},$

in which $F(h^{\vee}_{\sigma_{t}})$ is meant to be the curvature of
$\mathcal{E}$, defined by composing
$h^{\vee}_{\sigma_{t}}=e^{wt}\hat{h}_{\sigma_{t}}e^{wt}$ with the metric
duality isomorphism. We shall not make distinctions between $h_{\sigma_{t}}$
and $h^{\vee}_{\sigma_{t}}=e^{wt}\hat{h}_{\sigma_{t}}e^{wt}$ in what follows
to avoid further complication in the notational convention, but it is
important to note that the sign in Lemma 3.3 is consistent with the metric
duality isomorphism that is implicit in this convention. Finally, it is
perhaps worth pointing out that $\mathcal{E}$ admits the Hermitian–Einstein
metric if and only if $\mathcal{E}^{\vee}$ does, and that subbundles of
$\mathcal{E}$ correspond to the quotient bundles of $\mathcal{E}^{\vee}$.

Lemma 3.3 indicates an interesting role played by the renormalised
Fubini–Study metric $\hat{h}_{\sigma_{t}}$, and indeed evaluating the
asymptotic behaviour of $\hat{h}_{\sigma_{t}}$ will be the key ingredient of
the proof of Proposition 3.1.

Before we start evaluating the terms in Lemma 3.3, we recall the following
important theorem that will be used in the proof.

######  Theorem 3.5 (Chern–Weil formula [Sibley, SimpYM]).

Let $\mathcal{S}$ be a saturated subsheaf of a holomorphic vector bundle
$\mathcal{E}$ endowed with a hermitian metric $h$. Then we have

$\int_{X}\mathrm{tr}(F(h)|_{\mathcal{S}})\frac{\omega^{n-1}}{(n-1)!}=\deg(\mathcal{S})+\int_{X}|\mathrm{II}(h)|^{2}\frac{\omega^{n-1}}{(n-1)!},$

where $\mathrm{II}(h)$ is the second fundamental form defined by $h$
associated to $\mathcal{S}\subset\mathcal{E}$.

In the above, $F(h)|_{\mathcal{S}}$ is meant to be the composition of $F(h)$
with the orthogonal projection $\pi:\mathcal{E}\to\mathcal{S}$ defined by $h$.
We shall use the following observation concerning this theorem. Suppose that
$\mathcal{S}$ is a subsheaf of $\mathcal{E}$ that is not necessarily
saturated, and let $\mathcal{S}^{\prime}$ be its saturation in $\mathcal{E}$.
Since $\mathcal{S}$ and $\mathcal{S}^{\prime}$ agree on a Zariski open subset
in $X$, we find

$\int_{X}\mathrm{tr}(F(h)|_{\mathcal{S}})\frac{\omega^{n-1}}{(n-1)!}=\int_{X}\mathrm{tr}(F(h)|_{\mathcal{S}^{\prime}})\frac{\omega^{n-1}}{(n-1)!}.$

Thus, we get

(3.1)
$\int_{X}\mathrm{tr}(F(h)|_{\mathcal{S}})\frac{\omega^{n-1}}{(n-1)!}=\deg(\mathcal{S}^{\prime})+\int_{X}|\mathrm{II}^{\prime}(h)|^{2}\frac{\omega^{n-1}}{(n-1)!},$

where the second fundamental form that appears on the right hand side is
defined with respect $h$ and the saturation $\mathcal{S}^{\prime}$ of
$\mathcal{S}$.

Using this result, we estimate the first term of Lemma 3.3.

######  Lemma 3.6.

There exists a constant $c_{1}(h_{k},k)>0$ which depends on the reference
metric $h_{k}$ and $k$ such that

$\displaystyle-\sum_{i=1}^{\nu}2w_{i}\left(\deg(\mathcal{E}^{\prime}_{-w_{i}})-\mu(\mathcal{E})\mathrm{rk}(\mathcal{E}^{\prime}_{-w_{i}})\right)t-c_{1}(h_{k},k)$
$\displaystyle\leq-\int_{0}^{t}dt\int_{X^{reg}}\mathrm{tr}\left((2w+\hat{h}_{\sigma_{t}}^{-1}[w,\hat{h}_{\sigma_{t}}])(\Lambda_{\omega}F(\hat{h}_{\sigma_{t}})-\mu(E)\mathrm{Id}_{E})\right)\frac{\omega^{n}}{n!}$
$\displaystyle\leq-\sum_{i=1}^{\nu}2w_{i}\left(\deg(\mathcal{E}^{\prime}_{-w_{i}})-\mu(\mathcal{E})\mathrm{rk}(\mathcal{E}^{\prime}_{-w_{i}})\right)t,$

uniformly for all $t\geq 0$ and
$\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$, where
$\mathcal{E}^{\prime}_{-w_{i}}:=\mathcal{E}^{\prime}_{\leq-
w_{i}}/\mathcal{E}^{\prime}_{\leq-w_{i-1}}$ (cf. Lemma 2.5).

###### Proof.

Recall that we have the $C^{\infty}$-isomorphism
$E\cong\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}E_{\alpha}$ over $X^{reg}$. We
compose this with another $C^{\infty}$-isomorphism so that each $E_{\alpha}$
is pairwise orthogonal with respect to $\hat{h}_{\sigma_{t}}$. Thus,

$\displaystyle\mathrm{tr}\left(wF(\hat{h}_{\sigma_{t}})\right)$
$\displaystyle=\sum_{\alpha=\hat{1}}^{\hat{\nu}}w_{\alpha}\mathrm{tr}(F(\hat{h}_{\sigma_{t}})|_{E_{-w_{\alpha}}}),$
$\displaystyle=\sum_{\alpha=\hat{1}}^{\hat{\nu}}w_{\alpha}\left(\mathrm{tr}(F(\hat{h}_{\sigma_{t}})|_{E_{\leq-
w_{\alpha}}})-\mathrm{tr}(F(\hat{h}_{\sigma_{t}})|_{E_{\leq-
w_{\alpha-1}}})\right).$

We now recall the Chern–Weil formula (3.1), and get

$\int_{X^{reg}}\mathrm{tr}(F(\hat{h}_{\sigma_{t}})|_{E_{\leq-
w_{\alpha}}})\frac{\omega^{n-1}}{(n-1)!}=\deg(\mathcal{E}^{\prime}_{\leq-
w_{\alpha}})+\int_{X}|\mathrm{II}_{\alpha}^{\prime}(\hat{h}_{\sigma_{t}})|^{2}\frac{\omega^{n-1}}{(n-1)!},$

where $\mathrm{II}_{\alpha}^{\prime}(\hat{h}_{\sigma_{t}})$ is the second
fundamental form associated to the saturated subsheaf
$\mathcal{E}^{\prime}_{\leq-w_{\alpha}}$ defined with respect to
$\hat{h}_{\sigma_{t}}$. Thus

$\displaystyle-\int_{X^{reg}}\mathrm{tr}\left(wF(\hat{h}_{\sigma_{t}})\right)\frac{\omega^{n-1}}{(n-1)!}$
$\displaystyle=-\sum_{\alpha=\hat{1}}^{\hat{\nu}}w_{\alpha}\deg(\mathcal{E}^{\prime}_{-w_{\alpha}})-\sum_{\alpha=\hat{1}}^{\hat{\nu}}(w_{\alpha-1}-w_{\alpha})\int_{X}|\mathrm{II}_{\alpha-1}^{\prime}(\hat{h}_{\sigma_{t}})|^{2}\frac{\omega^{n-1}}{(n-1)!}.$

In Lemma 3.7 that follows, we prove the existence of a positive constant
$c_{1}(h_{k},k)$ which depends only on the reference metric $h_{k}$ and $k$
such that

(3.2)
$0\leq\sum_{\alpha=\hat{1}}^{\hat{\nu}}(w_{\alpha-1}-w_{\alpha})\int_{0}^{t}dt\int_{X}|\mathrm{II}_{\alpha-1}^{\prime}(\hat{h}_{\sigma_{t}})|^{2}\frac{\omega^{n-1}}{(n-1)!}\leq
c_{1}(h_{k},k),$

uniformly for all $t>0$ and
$\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$.

Similarly, we find

$\int_{X^{reg}}\mathrm{tr}(w\cdot\mathrm{Id}_{E})=\sum_{\alpha=\hat{1}}^{\hat{\nu}}w_{\alpha}\mathrm{rk}(E_{\alpha})=\sum_{\alpha=\hat{1}}^{\hat{\nu}}w_{\alpha}\mathrm{rk}(\mathcal{E}^{\prime}_{\alpha}),$

by noting that saturation does not change the rank.

Finally, since $\bigoplus_{\alpha=1}^{\hat{\nu}}E_{\alpha}$ is assumed to be
orthogonal with respect to $\hat{h}_{\sigma_{t}}$, by our choice of
$C^{\infty}$-isomorphism, $w$ must commute with $\hat{h}_{\sigma_{t}}$, i.e.
$[w,\hat{h}_{\sigma_{t}}]=0$. Recalling (3.2), we thus get

$\displaystyle-\sum_{\alpha=\hat{1}}^{\hat{\nu}}2w_{\alpha}\left(\deg(\mathcal{E}^{\prime}_{-w_{\alpha}})-\mu(\mathcal{E})\mathrm{rk}(\mathcal{E}^{\prime}_{-w_{\alpha}})\right)t-c_{1}(h_{k},k,\zeta)$
$\displaystyle\leq-\int_{0}^{t}dt\int_{X^{reg}}\mathrm{tr}\left((2w+\hat{h}_{\sigma_{t}}^{-1}[w,\hat{h}_{\sigma_{t}}])(\Lambda_{\omega}F(\hat{h}_{\sigma_{t}})-\mu(E)\mathrm{Id}_{E})\right)\frac{\omega^{n}}{n!}$
$\displaystyle\leq-\sum_{\alpha=\hat{1}}^{\hat{\nu}}2w_{\alpha}\left(\deg(\mathcal{E}^{\prime}_{-w_{\alpha}})-\mu(\mathcal{E})\mathrm{rk}(\mathcal{E}^{\prime}_{-w_{\alpha}})\right)t.$

Recall from Lemma 2.5 that each $\mathcal{E}^{\prime}_{-w_{i}}$ is torsion-
free or zero for each $i=1,\dots,\nu$, and that
$\mathrm{rk}(\mathcal{E}^{\prime}_{-w_{i}})>0$ if and only if
$i\in\\{\hat{1},\dots,\hat{\nu}\\}$. This means that
$\mathcal{E}^{\prime}_{-w_{i}}$ is simply zero for all
$i\notin\\{\hat{1},\dots,\hat{\nu}\\}$, and hence
$\deg(\mathcal{E}^{\prime}_{-w_{i}})=0$ if
$i\notin\\{\hat{1},\dots,\hat{\nu}\\}$. We thus get

$\sum_{\alpha=\hat{1}}^{\hat{\nu}}2w_{\alpha}\left(\deg(\mathcal{E}^{\prime}_{-w_{\alpha}})-\mu(\mathcal{E})\mathrm{rk}(\mathcal{E}^{\prime}_{-w_{\alpha}})\right)=\sum_{i=1}^{\nu}2w_{i}\left(\deg(\mathcal{E}^{\prime}_{-w_{i}})-\mu(\mathcal{E})\mathrm{rk}(\mathcal{E}^{\prime}_{-w_{i}})\right),$

as required. Thus, granted (3.2) that we prove below, we complete the proof of
Lemma 3.6. ∎

We now prove (3.2).

######  Lemma 3.7.

Using the notation as in the proof of Lemma 3.6, there exists a constant
$c_{1}(h_{k},k)>0$ which depends only on $k$ and the reference metric $h_{k}$
such that

$0\leq\sum_{\alpha=\hat{1}}^{\hat{\nu}}(w_{\alpha-1}-w_{\alpha})\int_{0}^{t}dt\int_{X}|\mathrm{II}_{\alpha-1}^{\prime}(\hat{h}_{\sigma_{t}})|^{2}\frac{\omega^{n-1}}{(n-1)!}\leq
c_{1}(h_{k},k),$

uniformly for all $t\geq 0$ and
$\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$ (with
$\|\zeta\|_{op}\leq 1$, as in Remark 2.1).

###### Proof.

First note that $w_{\alpha}-w_{\alpha-1}<0$ by (2.2), which establishes the
first inequality. Note also that the integral
$\int_{0}^{t}dt\int_{X}|\mathrm{II}_{\alpha-1}^{\prime}(\hat{h}_{\sigma_{t}})|^{2}\frac{\omega^{n-1}}{(n-1)!}$
is monotonically increasing in $t$, and hence it suffices to establish the
bound for $t=+\infty$.

We start by proving a weaker estimate

(3.3)
$\sum_{\alpha=1}^{\hat{\nu}}(w_{\alpha}-w_{\alpha-1})\int_{0}^{+\infty}dt\int_{X}|\mathrm{II}_{\alpha-1}^{\prime}(\hat{h}_{\sigma_{t}})|^{2}\frac{\omega^{n-1}}{(n-1)!}\leq
c_{1}(h_{k},k,\zeta)$

where the constant also depends on
$\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$. It suffices to show

(3.4)
$\int_{0}^{+\infty}dt\int_{X}|\mathrm{II}_{\alpha-1}^{\prime}(\hat{h}_{\sigma_{t}})|^{2}\frac{\omega^{n-1}}{(n-1)!}<+\infty$

for each $\alpha=\hat{1},\dots,\hat{\nu}$. Note that, by the monotone
convergence theorem, we may exchange the order of the integrals.

Step 1: the case when $\mathcal{E}_{\leq-w_{i}}$’s are all holomorphic
subbundles of $\mathcal{E}$. In this case, $X^{reg}=X$, and it is easy to see
that $|\mathrm{II}_{\alpha-1}^{\prime}(\hat{h}_{\sigma_{t}})|^{2}$ decays
exponentially as $t\to+\infty$ with the decay rate at least
$e^{-|w_{\alpha}-w_{\alpha-1}|t}$, by Proposition 2.8. Recalling Lemmas 2.11
and 2.12, this immediately establishes (3.3).

Step 2: the case when $\mathcal{E}_{<-w_{i}}$’s are all saturated, i.e.
$\mathcal{E}_{\leq-w_{i}}=\mathcal{E}^{\prime}_{\leq-w_{i}}$. We apply the
regularisation (or the resolution of singularities) for the saturated sheaves,
as in Jacob [Jacob] and Sibley [Sibley], to reduce to the case of holomorphic
subbundles. We closely follow the argument by Jacob [Jacob, Sections 3 and 4].
After a finite sequence of blowups $\pi:\tilde{X}\to{X}$, we have a locally
free subsheaf $\tilde{\mathcal{E}}_{\leq-w_{\alpha}}$ of $\pi^{*}\mathcal{E}$
with a locally free quotient $\pi^{*}\mathcal{E}/\tilde{\mathcal{E}}_{\leq-
w_{\alpha}}$ over $\tilde{X}$ (see [Jacob, Proposition 3] and [Sibley,
Proposition 4.3]). Now, the pullback $\pi^{*}\hat{h}_{\sigma_{t}}$ of
$\hat{h}_{\sigma_{t}}$ can be regarded as defining the induced metric on
$\tilde{E}_{\leq-w_{\alpha}}\oplus\pi^{*}E/\tilde{E}_{\leq-
w_{\alpha}}\cong\pi^{*}E$ (as $C^{\infty}$-vector bundles), as in [Jacob,
Section 2]. As in [Jacob, Proposition 4], we construct hermitian metrics
$\tilde{h}_{\sigma_{t}}|_{\tilde{E}_{\leq-w_{\alpha}}}$ on $\tilde{E}_{\leq-
w_{\alpha}}$ and $\tilde{h}_{\sigma_{t}}|_{\pi^{*}E/\tilde{E}_{\leq-
w_{\alpha}}}$ on $\pi^{*}E/\tilde{E}_{\leq-w_{\alpha}}$, by removing the
factors that vanish (or blow up) on the exceptional divisor of $\pi$.
Moreover, we can do so without altering the $L^{2}$-norm of the second
fundamental form [Jacob, Proof of Proposition 1] (we use a degenerate Kähler
metric on $\tilde{X}$ but this does not matter). At the limit $t=+\infty$,
since $\hat{h}$ as in Definition 2.9 may be degenerate on $X\setminus
X^{reg}$, $\tilde{h}_{\sigma_{t}}$ may develop further degeneracy on the
exceptional divisor of $\pi$, as $t\to+\infty$. Recall also that Lemma 1.22
implies that the degeneracy of $\hat{h}$ is exactly of the form as described
in [Jacob, equation (3.8)], and after pulling back by $\pi$, we may further
remove these factors that vanish on the exceptional divisors, to get a well-
defined metric $\tilde{h}$, say, on $\pi^{*}E\cong\tilde{E}_{\leq-
w_{\alpha}}\oplus\pi^{*}E/\tilde{E}_{\leq-w_{\alpha}}$. Since $\hat{h}$ has no
off-diagonal block (cf. Proposition 2.8 and Definition 2.9), the $L^{2}$-norm
of the second fundamental form of the metric $\tilde{h}$ thus constructed
(which agrees with the one defined by $\hat{h}$, again by [Jacob, Proof of
Proposition 1]) is zero. Thus, by also recalling that the decay (as
$t\to+\infty$) of $|\mathrm{II}_{\alpha-1}^{\prime}(\hat{h}_{\sigma_{t}})|$ at
each $x\in X^{reg}$ is exponential, as we saw in Step 1, we prove (3.4).
Hence, summarising the argument above, after a finite sequence of blowups
$\pi:\tilde{X}\to{X}$, we may reduce to the case in Step 1. This establishes
(3.3) when all $\mathcal{E}_{\leq-w_{i}}$’s are all saturated.

Step 3: we now consider the general case where $\mathcal{E}_{\leq-w_{i}}$’s
are (not necessarily saturated) subsheaves of $\mathcal{E}$. We first apply a
sequence of blowups $\pi$ so that
$\bigcup_{\alpha=\hat{1}}^{\hat{\nu}}\mathrm{Sing}(\mathcal{E}_{\leq-
w_{\alpha}})\cup\mathrm{Sing}(\mathcal{E}/\mathcal{E}_{\leq-w_{\alpha}})$,
where $\hat{h}_{\sigma_{t}}$ may be degenerate as $t\to+\infty$, is contained
in the union of normal crossing divisors (see [Sibley, Theorem 4.4]). As in
Step 2, we may remove the factors of $\hat{h}$ that vanish on the exceptional
divisors of $\pi$, without changing the $L^{2}$-norm of the second fundamental
form at $t=+\infty$ (as in [Jacob, Proof of Proposition 1], which holds for an
arbitrary blowup). Note that we do not claim that, by such sequence of
blowups, the subsheaves $\mathcal{E}_{\leq-w_{\alpha}}$ can be pulled back to
holomorphic subbundles; we only remove the factors that vanish on
$\mathrm{Sing}(\mathcal{E}_{\leq-
w_{\alpha}})\cup\mathrm{Sing}(\mathcal{E}/\mathcal{E}_{\leq-w_{\alpha}})$ that
makes $\hat{h}_{\sigma_{t}}$ degenerate as $t\to+\infty$. We then apply the
argument in Step 2 to the saturation $\mathcal{E}^{\prime}_{\leq-w_{\alpha}}$
of $\mathcal{E}_{\leq-w_{\alpha}}$, by composing $\pi$ with a further sequence
of blowups, where we note that $\mathcal{E}_{\leq-w_{\alpha}}$ and its
saturation $\mathcal{E}^{\prime}_{\leq-w_{\alpha}}$ differ only on a Zariski
closed subset and hence does not affect the value of the integral (3.4), and
that the second fundamental form in Step 2 is defined with respect to the
saturation $\mathcal{E}^{\prime}_{\leq-w_{\alpha}}$. Thus, we can reduce to
the case in Step 2, to establish (3.3) in general.

We finally note that the map defined by

$\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})\ni\zeta\mapsto\sum_{\alpha=\hat{1}}^{\hat{\nu}}(w_{\alpha}-w_{\alpha-1})\int_{0}^{+\infty}dt\int_{X}|\mathrm{II}_{\alpha-1}^{\prime}(\hat{h}_{\sigma_{t}})|^{2}\frac{\omega^{n-1}}{(n-1)!}\in\mathbb{R}$

is continuous, where we recall that
$|\mathrm{II}_{\alpha-1}^{\prime}(\hat{h}_{\sigma_{t}})|^{2}$ decays
exponentially as $t\to+\infty$ at each $p\in X$, with decay rate at least (and
in fact faster than) $e^{-|w_{\alpha}-w_{\alpha-1}|t}$ by Proposition 2.8.
(Note in particular that, when we consider the limit of $w_{\alpha-1}$ tending
to $w_{\alpha}$, the integral over $t$ of
$(w_{\alpha}-w_{\alpha-1})e^{-|w_{\alpha}-w_{\alpha-1}|t}$ remains bounded.)
By recalling $\|\zeta\|_{op}\leq 1$ (cf. Remark 2.1), we obtain that the bound
that does not depend on $\zeta$, as claimed. ∎

We now prove that the Donaldson functional for renormalised Bergman 1-PS is
bounded from below.

######  Proposition 3.8.

Let $\\{\hat{h}_{\sigma_{t}}\\}_{t\geq 0}$ be a renormalised Bergman 1-PS
associated to $\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$. Then
there exists a constant $c_{2}(h_{k},k)>0$ which depends only on $k$ and the
reference metric $h_{k}$ such that

$\mathcal{M}^{Don}(\hat{h}_{\sigma_{t}},h_{k})>-c_{2}(h_{k},k)$

irrespectively of $t\geq 0$ and
$\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$.

######  Remark 3.9.

The sign convention concerning the metric duality isomorphism, as mentioned in
Remark 3.4, still applies, so $\hat{h}_{\sigma_{t}}$ is meant to be a
hermitian metric on $\mathcal{E}^{\vee}$.

###### Proof.

We first consider the following general situation. Suppose that we have a
geodesic segment $\gamma(s):=e^{sv}$ ($0\leq s\leq 1$), with $v:=\log
hh^{-1}_{0}$, which connects $h_{0}$ and $h$, where $h_{0}$ is a fixed
reference metric. The convexity of the Donaldson functional (Proposition 1.29)
implies

(3.5)
$\mathcal{M}^{Don}(h,h_{0})\geq\left.\frac{d}{ds}\right|_{s=0}\mathcal{M}^{Don}(\gamma(s),h_{0}),$

where we recall

$\left.\frac{d}{ds}\right|_{s=0}\mathcal{M}^{Don}(\gamma(s),h_{0})=\int_{X}\mathrm{tr}\left(v\left(\Lambda_{\omega}F(h_{0})-\frac{\mu(E)}{\mathrm{Vol}_{L}}\mathrm{Id}_{E}\right)\right)\frac{\omega^{n}}{n!}.$

Now define

(3.6)
$\bar{v}:=\frac{1}{r\mathrm{Vol}_{L}}\int_{X}\mathrm{tr}(v)\frac{\omega^{n}}{n!}\mathrm{Id}_{E},$

so that $v-\bar{v}$ has average 0. Since $\bar{v}$ is a constant multiple of
the identity, we have

$\left.\frac{d}{ds}\right|_{s=0}\mathcal{M}^{Don}(\gamma(s),h_{0})=\int_{X}\mathrm{tr}\left((v-\bar{v})\left(\Lambda_{\omega}F(h_{0})-\frac{\mu(E)}{\mathrm{Vol}_{L}}\mathrm{Id}_{E}\right)\right)\frac{\omega^{n}}{n!}.$

Thus, by using Cauchy–Schwarz, we have

$\left.\frac{d}{ds}\right|_{s=0}\mathcal{M}^{Don}(\gamma(s),h_{0})\geq-\|v-\bar{v}\|_{L^{2}}\left\|\Lambda_{\omega}F(h_{0})-\frac{\mu(E)}{\mathrm{Vol}_{L}}\mathrm{Id}_{E}\right\|_{L^{2}}$

where the norm $\|\cdot\|_{L^{2}}$ is defined as

$\|v-\bar{v}\|^{2}_{L^{2}}:=\int_{X}\mathrm{tr}\left((v-\bar{v})\cdot(v-\bar{v}))\right)\frac{\omega^{n}}{n!}=\int_{X}\mathrm{tr}(v^{2})-\mathrm{tr}(\bar{v}^{2})\frac{\omega^{n}}{n!}\geq
0.$

Defining a constant

$\underline{C}(h_{0})=\left\|\Lambda_{\omega}F(h_{0})-\frac{\mu(E)}{\mathrm{Vol}_{L}}\mathrm{Id}_{E}\right\|_{L^{2}}\geq
0,$

we get

(3.7)
$\mathcal{M}^{Don}(h,h_{0})\geq-\underline{C}(h_{0})\|v-\bar{v}\|_{L^{2}},$

by recalling (3.5).

Let us consider $\\{\hat{h}_{\sigma_{t}}\\}_{t\geq 0}$, a renormalised Bergman
1-PS associated to $\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$
emanating from $h_{k}$. We apply the above argument to
$h=\hat{h}_{\sigma_{t}}$ and $h_{0}=h_{k}$ (in which we replace $X$ by
$X^{reg}$); more precisely the curvature $F(\hat{h}_{\sigma_{t}})$ should be
defined with respect to $\hat{h}_{\sigma_{t}}$ composed with the metric
duality isomorphism (as in Remark 3.4), but in any case this does not affect
the above argument. Writing $v_{t}:=\log\hat{h}_{\sigma_{t}}h^{-1}_{k}$ for
the geodesic segment connecting them. (3.7) implies that we have

$\mathcal{M}^{Don}(\hat{h}_{\sigma_{t}},h_{k})\geq-\underline{C}(h_{k})\|v_{t}-\bar{v}_{t}\|_{L^{2}}.$

We prove that $\|v_{t}-\bar{v}_{t}\|_{L^{2}}$ can be bounded by a constant
depending only on $\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$ and
$k$, irrespectively of $t$.

We choose an orthonormal frame with respect to which
$\hat{h}_{\sigma_{t}}h^{-1}_{k}$ is diagonalised as

$\hat{h}_{\sigma_{t}}h^{-1}_{k}=\mathrm{diag}(\lambda_{1}(x,t),\dots,\lambda_{r}(x,t))$

at each point $x\in X^{reg}$. By the definition of $\hat{h}_{\sigma_{t}}$,
each $\lambda_{\alpha}(x,t)$ can be written as

$\lambda_{\alpha}(x,t):=\lambda_{\alpha,1}(x)+e^{-c_{\alpha}t}\lambda_{\alpha,2}(x),$

for some smooth functions $\lambda_{\alpha,1}(x)$ and $\lambda_{\alpha,2}(x)$,
and some $c_{\alpha}>0$. Note that $\lambda_{1,1}(x),\dots,\lambda_{r,1}(x)$
are the eigenvalues of the renormalised Quot-scheme limit $\hat{h}$ at $x\in
X^{reg}$. Thus $\lambda_{\alpha,1}(x)>0$ for $x\in X^{reg}$, but it may tend
to zero as it approaches the singular locus $X\setminus X^{reg}$ that is
Zariski closed in $X$. Since $\hat{h}_{\sigma_{t}}=h_{k}$ at $t=0$, observe
that

(3.8) $\lambda_{\alpha,1}(x)+\lambda_{\alpha,2}(x)=1$

at all $x\in X^{reg}$. Moreover, since
$\mathrm{tr}(\hat{h}_{k})\leq\mathrm{tr}(h_{k})$ for all $t\geq 0$ (by
Proposition 2.8 and Definition 2.9), we have

(3.9)
$\lambda_{\alpha,1}(x)+e^{-c_{\alpha}t}\lambda_{\alpha,2}(x)\leq\mathrm{tr}(h_{k})$

for all $t\geq 0$ and all $x\in X^{reg}$.

We evaluate

$\int_{X^{reg}}\mathrm{tr}(v_{t}^{2})\frac{\omega^{n}}{n!}=\sum_{\alpha=1}^{r}\int_{X^{reg}}\left(\log(\lambda_{\alpha,1}(x)+e^{-c_{\alpha}t}\lambda_{\alpha,2}(x))\right)^{2}\frac{\omega^{n}}{n!}.$

We fix $\alpha$ and consider the integral

$\int_{X^{reg}}\left(\log(\lambda_{\alpha,1}(x)+e^{-c_{\alpha}t}\lambda_{\alpha,2}(x))\right)^{2}\frac{\omega^{n}}{n!}.$

The estimates (3.8) and (3.9) imply that, if $\lambda_{\alpha,2}(x)\leq 0$,
then $1\leq\lambda_{\alpha,1}(x)\leq\mathrm{tr}(h_{k})$, and hence

$\left(\log(\lambda_{\alpha,1}(x)+e^{-c_{\alpha}t}\lambda_{\alpha,2}(x))\right)^{2}\leq\left(\log\mathrm{tr}(h_{k})\right)^{2}.$

Thus, without loss of generality, we may restrict to a subset of $X^{reg}$ on
which we have $0<\lambda_{\alpha,2}(x)\leq 1$. We may further assume that
$\lambda_{\alpha,1}(x)\leq 1/2$, since otherwise

$\left(\log(\lambda_{\alpha,1}(x)+e^{-c_{\alpha}t}\lambda_{\alpha,2}(x))\right)^{2}\leq\left(\log
2\right)^{2}.$

We thus evaluate the integral of
$\left(\log(\lambda_{\alpha,1}(x)+e^{-c_{\alpha}t}\lambda_{\alpha,2}(x))\right)^{2}$
over the subset of $X^{reg}$ where $\lambda_{\alpha,1}(x)\leq 1/2$ and
$0<\lambda_{\alpha,2}(x)\leq 1$. With this assumption we have

$\left(\log(\lambda_{\alpha,1}(x)+e^{-c_{\alpha}t}\lambda_{\alpha,2}(x))\right)^{2}\leq
2\left(\log(\lambda_{\alpha,1}(x))\right)^{2}$

for all $t\geq 0$.

We may assume that the singular locus is contained in a finite union of normal
crossing divisors $D:=\cup_{j}D_{j}$, by e.g. pulling back to the resolution
of $X\setminus X^{reg}$ (see [Sibley, Theorem 4.4]). We first consider the
case where we have a neighbourhood $U_{x}$ of $x$ such that $U_{x}$ intersects
with $D$ only at the smooth locus of $D$. Then, writing $z$ for the local
holomorphic coordinate which defines $D$ as $z=0$ in such neighbourhood, we
find that $\lambda_{\alpha,1}(x)$ tends to zero along the singular locus at a
polynomial order in $|z|$, by recalling Lemma 1.22 and that the quotient map
$\rho$ (defined in (1.1)) is algebraic. Thus we get, for all $1\leq\alpha\leq
r$,

$\int_{U_{x}\setminus
D}\left(\log(\lambda_{\alpha,1}(x))\right)^{2}\frac{\omega^{n}}{n!}\leq\text{const.}\int_{0}^{c}\mathrm{r}\left(\log\mathrm{r}^{m_{\alpha}}\right)^{2}d\mathrm{r}<+\infty,$

where $\mathrm{r}:=|z|$, $m_{\alpha}\in\mathbb{N}$, and $c$ is some constant.
The case when $U_{x}$ contains a (normal crossing) singularity of $D$ can be
treated by the same argument.

Thus, the conclusion is that there exists a constant $c_{2}(\zeta,h_{k},k)>0$
such that

(3.10) $\int_{X^{reg}}\mathrm{tr}(v_{t}^{2})\frac{\omega^{n}}{n!}\leq
c_{2}(\zeta,h_{k},k)$

for all $t\geq 0$. Since

$\|v_{t}-\bar{v}_{t}\|^{2}_{L^{2}}=\int_{X^{reg}}\mathrm{tr}(v_{t}^{2})\frac{\omega^{n}}{n!}-\int_{X^{reg}}\mathrm{tr}(\bar{v}^{2}_{t})\frac{\omega^{n}}{n!}\geq
0$

implies
$\int_{X^{reg}}\mathrm{tr}(v_{t}^{2})\frac{\omega^{n}}{n!}\geq\int_{X^{reg}}\mathrm{tr}(\bar{v}^{2}_{t})\frac{\omega^{n}}{n!}$,
we get

(3.11) $\|v_{t}-\bar{v}_{t}\|^{2}_{L^{2}}\leq
2\int_{X^{reg}}\mathrm{tr}(v_{t}^{2})\frac{\omega^{n}}{n!}\leq
2c_{2}(\zeta,h_{k},k).$

Moreover, the above argument shows that the integrand
$\mathrm{tr}(v_{t}^{2})\frac{\omega^{n}}{n!}$ extends continuously to the
whole of $X$. Thus, henceforth in the proof we may replace $X^{reg}$ by $X$.

We now prove that the constant $c_{2}(\zeta,h_{k},k)$ can be bounded uniformly
in $\zeta$. To simplify the notation, we write $v(\zeta t)$ for
$v_{t}=\log\hat{h}_{\sigma_{t}}h^{-1}_{k}$, by noting that $\zeta$ always
appears in the form $\zeta t$. Suppose by contradiction that there exists a
sequence
$\\{\zeta_{j}\\}_{j\in\mathbb{N}}\subset\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$,
$||\zeta_{j}||_{op}\leq 1$ (cf. Remark 2.1) such that

$\sup_{t}\int_{X}\mathrm{tr}(v(\zeta_{j}t)^{2})\frac{\omega^{n}}{n!}\to+\infty$

as $j\to+\infty$. We may assume that there exists a sequence
$\\{t_{j}\\}_{j\in\mathbb{N}}\subset\mathbb{R}_{\geq 0}$ such that

(3.12)
$\int_{X}\mathrm{tr}(v(\zeta_{j}t_{j})^{2})\frac{\omega^{n}}{n!}\to+\infty$

as $j\to+\infty$, and also assume that $\\{\zeta_{j}\\}_{j}$ contains a
subsequence which converges to
$\zeta_{\infty}\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$, by
$||\zeta_{j}||_{op}\leq 1$ (cf. Remark 2.1). By arguing as in the proof of
(3.10), we find

(3.13)
$\sup_{t}\int_{X}\mathrm{tr}(v(\zeta_{\infty}t)^{2})\frac{\omega^{n}}{n!}\leq
c_{2}(\zeta_{\infty},h_{k},k)<+\infty.$

If $\\{t_{j}\\}_{j}$ is bounded in $\mathbb{R}_{\geq 0}$, we see that (3.12)
contradicts (3.13) since

$\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})\ni\zeta\mapsto\int_{X}\mathrm{tr}(v(\zeta
t)^{2})\frac{\omega^{n}}{n!}\in\mathbb{R}$

is continuous for each fixed $t$. Thus, by taking a further subsequence in
$j$, we may assume that $\\{t_{j}\\}_{j}$ monotonically increases to
$+\infty$. Recalling Definition 2.9, the Lebesgue integration theorem implies

(3.14)
$\lim_{j\to\infty}\int_{X}\mathrm{tr}(v(\zeta_{j}t_{j})^{2})\frac{\omega^{n}}{n!}=\int_{X}\mathrm{tr}\left((\log\hat{h}h_{k}^{-1})^{2}\right)\frac{\omega^{n}}{n!}$

where we wrote

$\hat{h}:=\lim_{j\to+\infty}e^{-w_{j}t_{j}}h_{\sigma_{j,t_{j}}}e^{-w_{j}t_{j}},$

$w_{j}$ being the eigenvalues of $\zeta_{j}$ (as in (2.10)) and
$\sigma_{j,t_{j}}:=\exp(\zeta_{j}t_{j})$. However, the right hand side of
(3.14) is finite by arguing as in the proof of (3.10) (the limit depends on
the choice of the subsequence in $\\{\zeta_{j}\\}_{j}$ and $\\{t_{j}\\}_{j}$,
but the point is that it is finite for any choice of subsequence). This is a
contradiction, and hence there exists a constant $c_{2}(h_{k},k)>0$ that
depends only on $k$ and the reference metric $h_{k}$, and not on $\zeta$, such
that

$\sup_{t}\int_{X}\mathrm{tr}(v(\zeta t)^{2})\frac{\omega^{n}}{n!}\leq
c_{2}(\zeta,h_{k},k)\leq c_{2}(h_{k},k),$

as claimed.

Combining (3.7) and (3.11), we establish all the claims stated in the
proposition. ∎

######  Remark 3.10.

Suppose that the reference metrics $\\{h_{k}\\}_{k\in\mathbb{N}}$ converge
(say in $C^{p}$, $p\geq 2$) to $h_{\mathrm{ref}}\in\mathcal{H}_{\infty}$ as
$k\to+\infty$. It seems tempting to speculate that in fact we have a uniform
constant $c_{\mathrm{ref}}>0$ which depends only on $h_{\mathrm{ref}}$ such
that

$\mathcal{M}^{Don}(\hat{h}_{\sigma_{t}},h_{\mathrm{ref}})>-c_{\mathrm{ref}}$

and

$\sum_{\alpha=\hat{1}}^{\hat{\nu}}(w_{\alpha}-w_{\alpha-1})\int_{0}^{+\infty}dt\int_{X}|\mathrm{II}_{\alpha-1}^{\prime}(\hat{h}_{\sigma_{t}})|^{2}\frac{\omega^{n-1}}{(n-1)!}>-c_{\mathrm{ref}},$

for all $t\geq 0$ and $\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$.
In our forthcoming paper, we shall prove that the above uniform bounds lead to
an alternative proof of the theorem of Donaldson [Do-1983, Don1985, Don1987].

###  3.2. Upper bound for Donaldson’s functional

We prove the following estimate which complements Proposition 3.1.

######  Proposition 3.11.

With the notation as in Proposition 3.1, there exists a constant
$c(\zeta,k,h_{k})>0$ depending on $h_{k}$, $k$, $\zeta$ such that

$\mathcal{M}^{Don}(h_{\sigma_{t}},h_{k})\leq-\sum_{i=1}^{\nu}2w_{i}\left(\deg(\mathcal{E}^{\prime}_{-w_{i}})-\mu(\mathcal{E})\mathrm{rk}(\mathcal{E}^{\prime}_{-w_{i}})\right)t+c(\zeta,h_{k},k)$

holds for all sufficiently large $t>0$.

Recalling the argument in the previous section (the proof of Lemmas 3.3 and
3.6 in particular), it suffices to prove the following.

######  Lemma 3.12.

Let $\\{\hat{h}_{\sigma_{t}}\\}_{t\geq 0}$ be a renormalised Bergman 1-PS
associated to $\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$. Then

$\lim_{t\to+\infty}\left|\mathcal{M}^{Don}(\hat{h}_{\sigma_{t}},h_{k})\right|<+\infty.$

###### Proof.

We argue in three steps, as in the proof of Lemma 3.7: when each
$\mathcal{E}_{<w_{i}}$ is locally free, saturated, and the general case.

Step 1: when each $\mathcal{E}_{\leq-w_{i}}$ is locally free, and hence
$X^{reg}=X$, $\hat{h}_{\sigma_{t}}$ converges exponentially to the product
metric on the $C^{\infty}$-vector bundle
$\bigoplus_{\alpha=\hat{1}}^{\hat{\nu}}E_{-w_{\alpha}}$ over $X$. The claimed
estimate immediately follows from Lemmas 2.11 and 2.12.

Step 2: when each $\mathcal{E}_{\leq-w_{i}}$ is saturated, we again argue as
in [Jacob, Sections 3 and 4], as we did in Lemma 3.7. We pull back by a finite
sequence of blowups, and remove the factors that vanish (or blow up) on the
exceptional divisors of $\pi$ (as in the proof of Lemma 3.7) without altering
the value of the Donaldson functional [Jacob, Proof of Propositions 1 and 5].
We thus reduce to the case in Step 1.

Step 3: the general case, where $\mathcal{E}_{\leq-w_{i}}$’s are not
necessarily saturated, can be treated by a further sequence of blowups, as in
the proof of Lemma 3.7. By removing the factors that vanish on the exceptional
divisors of $\pi$, we reduce to the case in Step 2. ∎

##  4\. The non-Archimedean Donaldson functional

Recall that $\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$ defines the
filtration $0\neq\mathcal{E}_{\leq-
w_{1}}\subset\cdots\subset\mathcal{E}_{\leq-w_{\nu}}=\mathcal{E}$ as in (2.7).
We define the following integer.

######  Definition 4.1.

Let $w_{1},\dots,w_{\nu}\in\mathbb{Q}$ be the weights of
$\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$. We define
$j(\zeta,k)\in\mathbb{N}$ to be the minimum positive integer such that

$j(\zeta,k)w_{i}\in\mathbb{Z}$

for all $i=1,\dots,\nu$.

Thus, by clearing up the denominators, we may assume that the filtration
$0\neq\mathcal{E}_{\leq-w_{1}}\subset\cdots\subset\mathcal{E}_{\leq-
w_{\nu}}=\mathcal{E}$ is graded by integers; more precisely, defining
$j(\zeta,k)w_{i}=:\bar{w}_{i}\in\mathbb{Z}$, we define the filtration

(4.1)
$0\neq\mathcal{E}_{\leq-\bar{w}_{1}}\subset\cdots\subset\mathcal{E}_{\leq-\bar{w}_{\nu}}=\mathcal{E}$

in an obvious way. We also have the filtration
$0\neq\mathcal{E}^{\prime}_{\leq-\bar{w}_{1}}\subset\cdots\subset\mathcal{E}^{\prime}_{\leq-\bar{w}_{\nu}}=\mathcal{E}$
of $\mathcal{E}$ by saturated subsheaves as in Lemma 2.5. With this convention
understood, we prove the following result.

######  Lemma 4.2.

Recalling $\mathcal{E}^{\prime}_{-w_{i}}=\mathcal{E}^{\prime}_{\leq-
w_{i}}/\mathcal{E}^{\prime}_{\leq-w_{i-1}}$ as in Lemma 2.5, we have

$-\sum_{i=1}^{\nu}w_{i}\left(\deg(\mathcal{E}^{\prime}_{-w_{i}})-\mu(\mathcal{E})\mathrm{rk}(\mathcal{E}^{\prime}_{-w_{i}})\right)=\frac{1}{j(\zeta,k)}\sum_{q\in\mathbb{Z}}\mathrm{rk}(\mathcal{E}^{\prime}_{\leq
q})(\mu(\mathcal{E})-\mu(\mathcal{E}^{\prime}_{\leq q})).$

###### Proof.

Note

$\mathrm{rk}(\mathcal{E}^{\prime}_{-w_{i}})=\mathrm{rk}\left(\mathcal{E}^{\prime}_{\leq-
w_{i}}/\mathcal{E}^{\prime}_{\leq-
w_{i-1}}\right)=\mathrm{rk}(\mathcal{E}^{\prime}_{\leq-
w_{i}})-\mathrm{rk}(\mathcal{E}^{\prime}_{\leq-w_{i-1}})$

and

$\deg(\mathcal{E}^{\prime}_{-w_{i}})=\deg\left(\mathcal{E}^{\prime}_{\leq-
w_{i}}/\mathcal{E}^{\prime}_{\leq-
w_{i-1}}\right)=\deg(\mathcal{E}^{\prime}_{\leq-
w_{i}})-\deg(\mathcal{E}^{\prime}_{\leq-w_{i-1}}).$

Thus, defining $\bar{w}_{i}:=j(\zeta,k)w_{i}\in\mathbb{Z}$, we have

$\displaystyle-\sum_{i=1}^{\nu}w_{i}\left(\deg(\mathcal{E}^{\prime}_{-w_{i}})-\mu(\mathcal{E})\mathrm{rk}(\mathcal{E}^{\prime}_{-w_{i}})\right)$
$\displaystyle=\frac{-1}{j(\zeta,k)}\sum_{i=1}^{\nu}\bar{w}_{i}\left(\deg(\mathcal{E}^{\prime}_{\leq-\bar{w}_{i}})-\deg(\mathcal{E}^{\prime}_{\leq-\bar{w}_{i-1}})-\mu(\mathcal{E})(\mathrm{rk}(\mathcal{E}^{\prime}_{\leq-\bar{w}_{i}})-\mathrm{rk}(\mathcal{E}^{\prime}_{\leq-\bar{w}_{i-1}}))\right),$
$\displaystyle=\frac{1}{j(\zeta,k)}\sum_{q\in\mathbb{Z}}q\left(\deg(\mathcal{E}^{\prime}_{\leq
q})-\deg(\mathcal{E}^{\prime}_{\leq
q-1})-\mu(\mathcal{E})(\mathrm{rk}(\mathcal{E}^{\prime}_{\leq
q})-\mathrm{rk}(\mathcal{E}^{\prime}_{\leq q-1}))\right),$

by recalling the ordering $w_{1}>\cdots>w_{\nu}$ as in (2.2).

Since $q\deg(\mathcal{E}^{\prime}_{\leq
q-1})=(q-1)\deg(\mathcal{E}^{\prime}_{\leq
q-1})+\deg(\mathcal{E}^{\prime}_{\leq q-1})$, we get

$\sum_{q\in\mathbb{Z}}q\left(\deg(\mathcal{E}^{\prime}_{\leq
q})-\deg(\mathcal{E}^{\prime}_{\leq
q-1})\right)=-\sum_{q\in\mathbb{Z}}\deg(\mathcal{E}^{\prime}_{\leq
q-1})=-\sum_{q\in\mathbb{Z}}\deg(\mathcal{E}^{\prime}_{\leq q}),$

and similarly for the rank

$\sum_{q\in\mathbb{Z}}q\left(\mathrm{rk}(\mathcal{E}^{\prime}_{\leq
q})-\mathrm{rk}(\mathcal{E}^{\prime}_{\leq
q-1})\right))=-\sum_{q\in\mathbb{Z}}\mathrm{rk}(\mathcal{E}^{\prime}_{\leq
q}).$

Although the above infinite sums are divergent, we substitute them in

$\displaystyle\sum_{q\in\mathbb{Z}}q\left(\deg(\mathcal{E}^{\prime}_{\leq
q})-\deg(\mathcal{E}^{\prime}_{\leq
q-1})-\mu(\mathcal{E})(\mathrm{rk}(\mathcal{E}^{\prime}_{\leq
q})-\mathrm{rk}(\mathcal{E}^{\prime}_{\leq q-1}))\right),$
$\displaystyle=-\sum_{q\in\mathbb{Z}}\left(\deg(\mathcal{E}^{\prime}_{\leq
q})-\mu(\mathcal{E})\mathrm{rk}(\mathcal{E}^{\prime}_{\leq q})\right),$
$\displaystyle=\sum_{q\in\mathbb{Z}}\mathrm{rk}(\mathcal{E}^{\prime}_{\leq
q})(\mu(\mathcal{E})-\mu(\mathcal{E}^{\prime}_{\leq q})),$

to get a finite sum; since the filtration $0\neq\mathcal{E}^{\prime}_{\leq-
w_{1}}\subset\cdots\subset\mathcal{E}^{\prime}_{\leq-w_{\nu}}=\mathcal{E}$ is
finite, there are only finitely many nonzero terms in the above sum. ∎

We now define the non-Archimedean Donaldson functional, following the
terminology of [BHJ2, Definition 3.4].

######  Definition 4.3.

Suppose that $\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$ gives rise
to the filtration $\\{\mathcal{E}_{\leq q}\\}_{q\in\mathbb{Z}}$ of
$\mathcal{E}$ as in (2.9), which we may assume is graded by integers (as in
(4.1)). Taking the saturation $\mathcal{E}^{\prime}_{\leq q}$ of
$\mathcal{E}_{\leq q}$ in $\mathcal{E}$, the non-Archimedean Donaldson
functional $\mathcal{M}^{\mathrm{NA}}(\zeta,k)$ is defined as

$\mathcal{M}^{\mathrm{NA}}(\zeta,k):=\frac{2}{j(\zeta,k)}\sum_{q\in\mathbb{Z}}\mathrm{rk}(\mathcal{E}^{\prime}_{\leq
q})\left(\mu(\mathcal{E})-\mu(\mathcal{E}^{\prime}_{\leq q})\right),$

where $j(\zeta,k)\in\mathbb{N}$ is as in Definition 4.1.

The name “non-Archimedean” originally comes from the analogy with a theory on
Berkovich spaces, as explained in [BHJ1, Section 6.8]. Recall also that a
filtration on a vector space $V_{k}$ corresponds one-to-one with a non-
Archimedean norm on $V_{k}$ [BHJ1, Section 1.1]; see also [Boucksom-icm,
Section 1].

##  5\. Two-step filtrations associated to saturated subsheaves

We now consider the special case when the filtration is given in two steps. A
particularly important case is the one associated to saturated subsheaves of
$\mathcal{E}$, which defines a natural two-step filtration for
$H^{0}(\mathcal{E}(k))=:V_{k}$.

######  Definition 5.1.

Suppose that we are given a $k$-regular saturated subsheaf $\mathcal{F}$ of
$\mathcal{E}$. Then it defines a two-step filtration $0\neq
H^{0}(\mathcal{F}(k))\subset H^{0}(\mathcal{E}(k))=V_{k}$ (whose weights will
be specified later in (5.1) and (5.2)). We write
$\zeta_{\mathcal{F}}\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$ for the
element defining this filtration.

Note that the filtration of $\mathcal{E}$ generated by $\zeta_{\mathcal{F}}$
as in (2.9) is exactly $0\neq\mathcal{F}\subset\mathcal{E}$.

We now specify the weights of $\zeta_{\mathcal{F}}$ defined in Definition 5.1.
Recall that the weights $w_{1},w_{2}$ of $\zeta_{\mathcal{F}}$ are determined
by $\mathrm{tr}(\zeta_{\mathcal{F}})=0$; more precisely (cf. [GF-K-R, Section
4]), we get

(5.1) $\displaystyle w_{1}$
$\displaystyle=\frac{h^{0}(\mathcal{F}(k))}{h^{0}(\mathcal{E}(k))},$ (5.2)
$\displaystyle w_{2}$
$\displaystyle=-\frac{h^{0}(\mathcal{E}(k))-h^{0}(\mathcal{F}(k))}{h^{0}(\mathcal{E}(k))},$

where the factor of $1/h^{0}(\mathcal{E}(k))$ is inserted so as to make the
operator norm of $\zeta$ to be less than 1 (cf. Remark 2.1).

We can re-state some of the results in Section 3 for the two-step filtrations
as follows.

######  Proposition 5.2.

Suppose that
$\zeta_{\mathcal{F}}\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$ defines a
two-step filtration associated to a $k$-regular saturated subsheaf
$\mathcal{F}\subset\mathcal{E}$, as in Definition 5.1. For the Bergman 1-PS
$\\{h_{\sigma_{t}}\\}_{t\geq 0}$ defined by $\zeta_{\mathcal{F}}$ emanating
from $h_{k}$, there exist a constant $c(h_{k},k)>0$ such that

$\mathcal{M}^{Don}(h_{\sigma_{t}},h_{k})\geq\mathrm{rk}(\mathcal{F})(\mu(\mathcal{E})-\mu(\mathcal{F}))\cdot
2t-c(h_{k},k)$

for all $t\geq 0$, and a constant $c(h_{k},\zeta_{\mathcal{F}},k)>0$ such that

$\mathcal{M}^{Don}(h_{\sigma_{t}},h_{k})\leq\mathrm{rk}(\mathcal{F})(\mu(\mathcal{E})-\mu(\mathcal{F}))\cdot
2t+c(h_{k},\zeta_{\mathcal{F}},k)$

holds for all sufficiently large $t>0$.

###### Proof.

We simply write down Propositions 3.1 and 3.11 for the specific filtration
defined by $\zeta_{\mathcal{F}}$. Direct computation immediately yields the
claimed results. ∎

Finally, we end this section by proving the following rather technical result
which we need later. Recall that the distance $\mathrm{dist}(h_{1},h_{2})$
between two hermitian metrics $h_{1},h_{2}\in\mathcal{H}$ can be defined by
using the geodesic between them, as in Section 1.3.

######  Lemma 5.3.

Given a Bergman 1-PS $\\{h_{\sigma_{t}}\\}_{t\geq 0}$ emanating from $h_{k}$,
generated by $\zeta_{\mathcal{F}}\in\mathfrak{sl}(H^{0}(E(k)))$ associated to
a saturated subsheaf $\mathcal{F}\subset\mathcal{E}$, we have
$\mathrm{dist}(h_{\sigma_{t}},h_{k})\to+\infty$ as $t\to+\infty$.

###### Proof.

We can explicitly write down the geodesic $\\{\gamma_{s}(t)\\}_{0\leq s\leq
1}$ connecting $h_{k}$ and $h_{\sigma_{t}}$ as $e^{sv(\sigma_{t})}h_{k}$ where
$v(\sigma_{t}):=\log(h_{\sigma_{t}}h^{-1}_{k})$. We compute its Riemannian
length as

$|\partial_{s}\gamma_{s}(t)|=\int_{X}\mathrm{tr}(v(\sigma_{t})\cdot
v(\sigma_{t}))^{1/2}\frac{\omega^{n}}{n!},$

where we recall that $\gamma_{s}(t)$ being a geodesic is equivalent to
$\partial_{s}\gamma_{s}(t)$ being independent of $s$. We show that this length
segment tends to $+\infty$ as $t\to+\infty$.

We take an $h_{k}$-orthonormal frame. We now recall Proposition 2.8, which
states

$v(\sigma_{t})=\log e^{wt}\left(\hat{h}+o(t)\right)e^{wt},$

where $w:=\mathrm{diag}(w_{1},w_{2})$ (as a block-diagonal matrix), where
$w_{1},w_{2}$ are as defined in (5.1) and (5.2). We also wrote
$\hat{h}:=\bigoplus_{\alpha=1}^{2}Q^{*}_{\alpha\to\alpha}Q_{\alpha\to\alpha}>0$
for the renormalised Quot-scheme limit. Observe that we have

$\log e^{wt}\left(\hat{h}/2\right)e^{wt}<v(\sigma_{t})<\log
e^{wt}\left(2\hat{h}\right)e^{wt},$

for all large enough $t$, and hence

$2wt+\log\left(\hat{h}/2\right)<v(\sigma_{t})<2wt+\log\left(2\hat{h}\right),$

noting that $e^{wt}$ commutes with the renormalised Quot-scheme limit
$\hat{h}$. Thus each eigenvalue of $v(\sigma_{t})$ can be estimated as
$2w_{\alpha}t(1+o(t))$, $\alpha=1,2$. Hence we get

$\mathrm{tr}(v(\sigma_{t})v(\sigma_{t}))>4t^{2}\sum_{\alpha=1}^{2}w_{\alpha}^{2}(1+o(t))>2t^{2}\sum_{\alpha=1}^{2}w_{\alpha}^{2},$

which tends to $+\infty$ as $t\to+\infty$ at each point $x\in X^{reg}$.

Thus we can compute the geodesic length between $h_{k}$ and $h_{\sigma_{t}}$,
as we did in the proof of Proposition 3.8, as

$\mathrm{dist}(h_{\sigma_{t}},h_{k})=\int_{0}^{1}ds\int_{X}\mathrm{tr}(v(\sigma_{t})\cdot
v(\sigma_{t}))^{1/2}\frac{\omega^{n}}{n!}>\text{const}.t^{2}\sum_{\alpha=1}^{2}w_{\alpha}^{2},$

which tends to $+\infty$ as $t\to+\infty$. ∎

##  6\. Summary of the main results

We recall the following terminology (cf. [BHJ2, Definition 5.1]).

######  Definition 6.1.

Let $\mathcal{N}$ be a function

$\mathcal{N}:SL(N,\mathbb{C})\to\mathbb{R}$

that is continuous with respect to the topology on $SL(N,\mathbb{C})$ defined
by a submultiplicative matrix norm $||\cdot||$ (e.g. operator norm).

  1. (1)

$\mathcal{N}$ is said to be coercive if there exist two constants
$c_{0},c_{1}\in\mathbb{R}$ with $c_{0}>0$ such that

$\mathcal{N}(\sigma)\geq c_{0}\log\|\sigma\|-c_{1}$

holds for any $\sigma\in SL(N,\mathbb{C})$;

  2. (2)

$\mathcal{N}$ is said to have log norm singularities if it can be written as

$\mathcal{N}(\sigma)=c_{0}\log\|\sigma\|+O(1)$

for some constant $c_{0}\in\mathbb{R}$, where $O(1)$ stands for the terms that
remain bounded over $SL(N,\mathbb{C})$.

Observe that by means of Bergman 1-PS’s, we can naturally define
$\mathcal{M}^{Don}$ as a function on $SL(H^{0}(\mathcal{E}(k))^{\vee})$ as

$\mathcal{M}^{Don}(-,h_{k}):SL(H^{0}(\mathcal{E}(k))^{\vee})\ni\sigma_{t}\mapsto\mathcal{M}^{Don}(h_{\sigma_{t}},h_{k})\in\mathbb{R},$

with the fixed reference metric $h_{k}$.

Important functionals concerning the constant scalar curvature Kähler metrics
have log norm singularities, as proved by Paul [Paul]; see also [BHJ2, Section
5]. Note also that the property of having log norm singularities is
independent of the choice of a matrix norm, up to changing the constants
$c_{0}$ and $c_{1}$.

Particularly important role is played by coercive functions; they are in
particular bounded from below. If $\mathcal{N}$ is coercive then it is proper,
i.e. preimage of compact set is compact, which is equivalent in this setting
to saying that $\mathcal{N}(\sigma)$ tends to $+\infty$ as $\|\sigma\|$ tends
to $+\infty$.

We now summarise what we have established so far. Suppose that we have a
sequence $\\{h_{k}\\}_{k\in\mathbb{N}}\subset\mathcal{H}_{\infty}$,
$h_{k}\in\mathcal{H}_{k}$, converging to the reference metric
$h_{\mathrm{ref}}\in\mathcal{H}_{\infty}$ in $C^{p}$ for $p\geq 2$, afforded
by Theorem 1.16, and that we write $\\{h_{\sigma_{t}}\\}_{t\geq 0}$ for the
Bergman 1-PS generated by
$\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$ emanating from $h_{k}$.

######  Theorem 1.

There exists a constant $c_{k}>0$ that depends only on $h_{\mathrm{ref}}$ and
$k\in\mathbb{N}$ such that

$\mathcal{M}^{Don}(h_{\sigma_{t}},h_{\mathrm{ref}})\geq\mathcal{M}^{\mathrm{NA}}(\zeta,k)t-c_{k}$

holds for all $t\geq 0$ and all
$\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$, where

$\mathcal{M}^{\mathrm{NA}}(\zeta,k)=\frac{2}{j(\zeta,k)}\sum_{q\in\mathbb{Z}}\mathrm{rk}(\mathcal{E}^{\prime}_{\leq
q})\left(\mu(\mathcal{E})-\mu(\mathcal{E}^{\prime}_{\leq q})\right)$

is as defined in Definition 4.3.

###### Proof.

This follows form what we have established so far. By the cocycle property
(1.11) of the Donaldson functional, we have

$\mathcal{M}^{Don}(h_{\sigma_{t}},h_{\mathrm{ref}})=\mathcal{M}^{Don}(h_{\sigma_{t}},h_{k})+\mathcal{M}^{Don}(h_{k},h_{\mathrm{ref}}).$

By the convergence $h_{k}\to h_{\mathrm{ref}}$ in $C^{p}$, we may assume
$|\mathcal{M}^{Don}(h_{k},h_{\mathrm{ref}})|<1$ for all $k$. The result now
follows from Proposition 3.1 and Lemma 4.2, by replacing $c(h_{k},k)$ by
$c_{k}$. ∎

Similar argument also yields the following result.

######  Theorem 2.

There exists a constant $c(\zeta,k,h_{\mathrm{ref}})>0$ depending on
$h_{\mathrm{ref}}$, $k\in\mathbb{N}$, and
$\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$, such that

$\mathcal{M}^{Don}(h_{\sigma_{t}},h_{\mathrm{ref}})\leq\mathcal{M}^{\mathrm{NA}}(\zeta,k)t+c(\zeta,k,h_{\mathrm{ref}})$

holds for all large enough $t>0$, with $\mathcal{M}^{\mathrm{NA}}(\zeta,k)$ as
defined in Definition 4.3.

Note the following important observation for $\mathcal{M}^{\mathrm{NA}}$
(where a filtration is called trivial if it is $0\subsetneq\mathcal{E}$),
which follows immediately by considering the two-step filtrations as in
Definition 5.1.

######  Proposition 6.2.

The non-Archimedean Donaldson functional $\mathcal{M}^{\mathrm{NA}}(\zeta,k)$
is positive (resp. nonnegative) for all
$\zeta\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$,
$k\geq\mathrm{reg}(\mathcal{E})$ whose associated filtration by saturated
subsheaves (2.9) is nontrivial, if and only if $\mathcal{E}$ is slope stable
(resp. semistable).

We finally summarise particularly important consequences of our main results
as follows.

######  Corollary 6.3.

With notation as in Theorems 1 and 2, we have the following results.

  1. (1)

If $\mathcal{E}$ is slope stable (resp. semistable), then the Donaldson
functional is coercive (resp. bounded from below) along any rational Bergman
1-PS in $\mathcal{H}_{k}\subset\mathcal{H}_{\infty}$,
$k\geq\mathrm{reg}(\mathcal{E})$.

  2. (2)

The Donaldson functional has log norm singularities along any rational Bergman
1-PS in $\mathcal{H}_{k}\subset\mathcal{H}_{\infty}$,
$k\geq\mathrm{reg}(\mathcal{E})$, i.e. we have

$\mathcal{M}^{Don}(h_{\sigma_{t}},h_{\mathrm{ref}})=\mathcal{M}^{\mathrm{NA}}(\zeta,k)t+O(1),$

where $O(1)$ stands for the terms that remain bounded as $t\to+\infty$. In
particular, it is not bounded from below if $\mathcal{E}$ is slope unstable.

  3. (3)

We have

$\lim_{t\to+\infty}\frac{\mathcal{M}^{Don}(h_{\sigma_{t}},h_{\mathrm{ref}})}{t}=\mathcal{M}^{\mathrm{NA}}(\zeta,k).$

##  7\. From Hermitian–Einstein metrics to slope stability

In this section we provide a more geometric proof of the theorem of Kobayashi
[Kobookjp] and Lübke [Luebke], as an application of Theorem 2 and the geodesic
convexity of the Donaldson functional. We start by considering an irreducible
holomorphic vector bundle $\mathcal{E}$ endowed with a Hermitian–Einstein
metric.

Suppose that there exists a critical point for the Donaldson functional, which
is necessarily the global minimum, say $h_{\mathrm{min}}$, over
$\mathcal{H}_{\infty}$ by strict geodesic convexity (cf. Proposition 1.29).
Now we take $h_{\mathrm{min}}$ as the reference metric $h_{\mathrm{ref}}$, and
consider the Donaldson functional

$\mathcal{M}^{Don}(h):=\mathcal{M}^{Don}(h,h_{\mathrm{min}})$

for $h\in\mathcal{H}_{k}\subset\mathcal{H}_{\infty}$.

Using the results of Section 5, consider
$\zeta_{\mathcal{F}}\in\mathfrak{sl}(H^{0}(\mathcal{E}(k))^{\vee})$ associated
to a saturated subsheaf $\mathcal{F}$ of $\mathcal{E}$, where we take $k$ to
be large enough so that $k\geq\mathrm{reg}(\mathcal{F})$. If
$\mathcal{M}^{\mathrm{NA}}(\zeta_{\mathcal{F}},k)<0$, then from Theorem 2, we
have

$\mathcal{M}^{Don}(h_{\sigma_{t}})\to-\infty$

as $t\to+\infty$, for the path $\\{h_{\sigma_{t}}\\}_{t\geq 0}$ in
$\mathcal{H}_{k}$. In particular, for large enough $t$, we have
$\mathcal{M}^{Don}(h_{\sigma_{t}})<\mathcal{M}^{Don}(h_{\mathrm{min}})$,
contradicting the definition of the minimum $h_{\mathrm{min}}$. Thus we get
$\mathcal{M}^{\mathrm{NA}}(\zeta_{\mathcal{F}},k)\geq 0$ for any saturated
subsheaf $\mathcal{F}$ with $k\geq\mathrm{reg}(\mathcal{F})$.

We now prove $\mathcal{M}^{\mathrm{NA}}(\zeta_{\mathcal{F}},k)>0$. Assume for
contradiction $\mathcal{M}^{\mathrm{NA}}(\zeta_{\mathcal{F}},k)=0$. Using the
geodesic completeness of $\mathcal{H}_{\infty}$, we consider the (nontrivial)
geodesic between $h_{\sigma_{t}}$ and $h_{\mathrm{min}}$ (it may be not
contained in $\mathcal{H}_{k}$, but this does not matter), and the geodesic
distance $\mathrm{dist}(h_{\sigma_{t}},h_{\mathrm{min}})$ between them. Then,
from Lemma 5.3, we have
$\mathrm{dist}(h_{\sigma_{t}},h_{\mathrm{min}})\to+\infty$ as $t\to+\infty$;
recall that the sequence
$\\{h_{k}\\}_{k\in\mathbb{N}}\subset\mathcal{H}_{\infty}$,
$h_{k}\in\mathcal{H}_{k}$, converges to the reference metric
$h_{\mathrm{min}}$ in $C^{p}$ for $p\geq 2$, as in Section 6. Thus, using the
strict geodesic convexity of $\mathcal{M}^{Don}$ we obtain

$\mathcal{M}^{Don}(h_{\sigma_{t}})\to+\infty,$

as $t\to+\infty$. But Theorem 2 implies that this contradicts the assumption
$\mathcal{M}^{\mathrm{NA}}(\zeta_{\mathcal{F}},k)=0$. Consequently, we get
$\mathcal{M}^{\mathrm{NA}}(\zeta_{\mathcal{F}},k)>0$, which implies (by
recalling Proposition 5.2 or 6.2)

$\mu(\mathcal{E})>\mu(\mathcal{F}),$

and hence the slope stability of $\mathcal{E}$ by Lemma 1.8.

Finally, the case when $\mathcal{E}$ is reducible can be treated similarly. By
applying the previous argument to each irreducible component of $\mathcal{E}$,
we find that $\mathcal{E}$ is a direct sum of slope stable bundles. The
Hermitian–Einstein equation implies that the slopes of these components are
equal, and hence $\mathcal{E}$ is slope polystable.

######  Remark 7.1.

The above proof uses Theorem 2, which depends on the convergence $h_{k}\to
h_{\mathrm{ref}}$ ($k\to\infty$) afforded by Theorem 1.16. Recalling the
cocycle property (1.11), a straightforward inspection reveals that above proof
does not require the convergence $h_{k}\to h_{\mathrm{ref}}$ and we can in
fact choose $h_{k}$ to be any reference metric in $\mathcal{H}_{k}$. Hence the
above does not depend on Theorem 1.16.

## References