Stability conditions and
canonical metrics

John Benjamin McCarthy

Thesis submitted for the degree of
_Doctor of Philosophy_

Supervised by
Sir Simon Donaldson
&
Dr Ruadhaí Dervan

Department of Mathematics
Imperial College London
&
London School of Geometry and Number Theory

February 2023

_For Diclehan’ım._

$\mathcal{M}$$\mathrm{Met}(E)$$\partial\mathrm{Met}(E)$$\mathcal{M}^{\mathbf{NA}}$$\mathcal{M}^{\mathbf{NA}}>0$$\mathcal{M}^{\mathbf{NA}}<0$$\mathcal{M}^{\mathbf{NA}}=0$

### Abstract

In this thesis we study the principle that _extremal objects in differential
geometry correspond to stable objects in algebraic geometry_. In our
introduction we survey the most famous instances of this principle with a view
towards the results and background needed in the later chapters. In Part I we
discuss the notion of a $Z$-critical metric recently introduced in joint work
with Ruadhaí Dervan and Lars Martin Sektnan [DMS21]. We prove a correspondence
for existence with an analogue of Bridgeland stability in the large volume
limit, and study important properties of the subsolution condition away from
this limit, including identifying the analogues of the Donaldson and
Yang–Mills functionals for the equation. In Part II we study the recent theory
of optimal symplectic connections on Kähler fibrations in the isotrivial case.
We prove a correspondence with the existence of Hermite–Einstein metrics on
holomorphic principal bundles.

### Statement of originality

I declare that the work in this thesis is my own except where explicitly
stated otherwise, and that all content has been appropriately referenced. In
particular the majority of Chapters 4 and 5 is joint work with Ruadhaí Dervan
and Lars Martin Sektnan [DMS21] and has been reproduced with an emphasis on my
own contributions.

### Copyright declaration

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### Acknowledgements

First I wish to thank my supervisors, Simon Donaldson and Ruadhaí Dervan, for
their support and expertise throughout my PhD. In particular I thank Simon for
providing me with some of his deep and expansive views of the subject of
differential geometry and an appreciation of where to find interesting
problems, and Ruadhaí for his remarkable ability to clearly and precisely
explain concepts in complex differential and algebraic geometry. The help and
guidance of my supervisors has been critical for my PhD and for developing my
understanding and appreciation of mathematics as a subject.

I thank Lars Martin Sektnan for many helpful discussions throughout our joint
work, my academic siblings Michael Hallam and Annamaria Ortu for companionship
as we embarked on an exploration of Kähler geometry, and all the attendees of
the complex geometry reading group for interesting and stimulating discussions
and presentations. I wish to thank my fellow students at the LSGNT, including
Jaime, Jordan, Jordan, Qaasim, Wendelin, Federico, Federico, Liam, Corvin, and
many others for the fun and sporadically productive conversations and for
surviving the pandemic as PhD students with me.

I thank Richard Thomas for making me believe algebraic geometry is simple,
even if I haven’t quite figured out how yet, and for directing my attention to
the deformed Hermitian Yang–Mills equation which lead to the major body of
work of this thesis.

I thank my family for providing support and encouragement throughout my PhD,
even if it could only come from across the globe, and for miraculously helping
me find safe passage across the Australian border during the pandemic.

Finally, thank you Diclehan for waiting patiently for me all these years.
Without your kindness, joy, and endless support, my PhD may not have survived
COVID-19 intact. Your commitment to me despite being a literal world apart is
an inspiration and a wonder which I am truly privileged to have.

This PhD was supported by EPSRC grant number EP/L015234/1. I thank the
examiners for many helpful comments and corrections.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Geometric invariant theory
      1. 2.1.1 The Hilbert–Mumford Criterion
      2. 2.1.2 The Kempf–Ness theorem
    2. 2.2 Holomorphic vector bundles
      1. 2.2.1 Stability
        1. 2.2.1.1 Slope stability
        2. 2.2.1.2 Filtrations
        3. 2.2.1.3 Gieseker stability
      2. 2.2.2 Yang–Mills connections
        1. 2.2.2.1 Hermite–Einstein metrics
        2. 2.2.2.2 Hermitian geometry of Chern connections
        3. 2.2.2.3 Almost Hermite–Einstein metrics
    3. 2.3 Varieties
      1. 2.3.1 K-stability
      2. 2.3.2 Canonical Kähler metrics
      3. 2.3.3 Deformations of Fano threefolds with $\operatorname{SL}(2,\mathbb{C})$ actions
        1. 2.3.3.1 Mukai–Umemura threefold
        2. 2.3.3.2 $V_{5}$ manifold
        3. 2.3.3.3 A singular threefold in the class $V_{14}$
        4. 2.3.3.4 General remarks
  3. I $Z$-critical metrics
    1. 3 Background
      1. 3.1 Mirror symmetry and BPS branes
        1. 3.1.1 Large volume limit
        2. 3.1.2 BPS branes
      2. 3.2 $\Pi$-stability and Bridgeland stability
        1. 3.2.1 Polynomial stability conditions
      3. 3.3 Deformed Hermitian Yang–Mills equation
        1. 3.3.1 Higher rank
    2. 4 $Z$-critical connections
      1. 4.1 Asymptotic $Z$-stability
        1. 4.1.1 Examples
      2. 4.2 $Z$-critical connections
      3. 4.3 Subsolutions
        1. 4.3.1 Linearisation and ellipticity
        2. 4.3.2 Stability with respect to subvarieties
      4. 4.4 Existence on surfaces
      5. 4.5 Moment map
        1. 4.5.1 Variational functional
      6. 4.6 Future directions
        1. 4.6.1 Modification due to Freed–Kapustin–Witten anomaly
        2. 4.6.2 Categorical Kähler geometry
        3. 4.6.3 Metrics on complexes and a variational proposal
    3. 5 Correspondence in the large volume limit
      1. 5.1 Existence implies stability
      2. 5.2 Stability implies existence
        1. 5.2.1 Step 1: The slope stable case
        2. 5.2.2 Deformations of complex structure
        3. 5.2.3 Step 2: Approximate solutions below the discrepancy order
        4. 5.2.4 Step 3: Stability at the discrepancy order
        5. 5.2.5 Step 4: Approximate solutions to arbitrary order
        6. 5.2.6 Step 5: Perturbing to a solution
        7. 5.2.7 Remarks on the general case
  4. II Kähler Fibrations
    1. 6 Background
      1. 6.1 Optimal symplectic connections
        1. 6.1.1 Kähler fibrations
        2. 6.1.2 Holomorphy potentials and automorphisms
        3. 6.1.3 The equation
      2. 6.2 Stability of fibrations
      3. 6.3 Principal bundles
        1. 6.3.1 Hermite–Einstein connections
        2. 6.3.2 Stability
      4. 6.4 Product fibrations
    2. 7 Isotrivial fibrations
      1. 7.1 Existence of optimal symplectic connections
        1. 7.1.1 Induced optimal symplectic connections
        2. 7.1.2 Induced Hermite–Einstein structures
          1. 7.1.2.1 Alternative description of $P$
          2. 7.1.2.2 Curvature of the connection on $P$
      2. 7.2 Future directions
        1. 7.2.1 Stability
          1. 7.2.1.1 The case of projective bundles
        2. 7.2.2 Principal bundles for non-isotrivial fibrations

###  Chapter 1 Introduction

The following fundamental principle has guided much of the research in complex
geometry since the late-20th century:

######  Principle 1.1.

_Stable objects in algebraic geometry correspond to extremal objects in
differential geometry._

The origins of this principle go as far back as Riemann’s work on
uniformisation of algebraic curves. What exactly is meant by _stable_ or
_extremal_ depends on the setting, and there are now instances of 1.1
appearing as celebrated theorems in the context of _points_ , _varieties_ ,
_vector bundles_ , and many variations on these geometric themes.

In this thesis we will investigate several new directions which test 1.1. In
Part I we will study the principle in the context of a new and exciting
development in algebraic geometry, that of _Bridgeland stability conditions_.
This is largely a reproduction of joint work of the author with Ruadhaí Dervan
and Lars Martin Sektnan [DMS21] with an emphasis of the contributions of the
author.

The notion of a stability condition arose out of the physics of mirror
symmetry, and key features of the new algebro-geometric theory include the
ability to vary the choice of stability condition (given, to first
approximation, by some algebraic invariant $Z$ called a _central charge_), and
indeed to extend this concept of stability from Abelian to triangulated
categories.

Thus the principle suggests the enticing possibility that we may see several
new features from the theory of stability conditions reflected in differential
geometry: the study of wall-crossing behaviour as we _vary differential
equations_ in interesting ways aligned with the algebraic geometry of the
situation, and _the study of derived phenomena using differential geometry_.

The following is the central folklore conjecture in the study of D-branes on
the B-side of mirror symmetry, and the conjecture aligned with 1.1 in our
setting. It is the mirror analogue of the (mostly) equally vague
Thomas–Yau–Joyce conjecture in symplectic geometry and can be variously
attributed to Douglas, Aspinwall–Douglas, Bridgeland, Leung–Yau–Zaslow,
Mariaño–Minasian–Moore–Strominger, Collins–Jacob–Yau, and the mononymous
“Physics” [DFR05, AD02, Bri07, LYZ00, MnMMS00, JY17, CJY20, HKK+03, ABC+09].

######  Conjecture 1.2 (Folklore).

There exists a Bridgeland stability condition $(Z,\mathcal{A})$ on the derived
category $\mathcal{D}^{b}\operatorname{Coh}(X)$ of any Calabi–Yau threefold
$X$, with complexified Kähler form $B+i\omega$, whose central charge is given
by

$Z(E)=-\int_{X}e^{-i\omega}e^{-B}\operatorname{Ch}(E)\sqrt{\operatorname{Td}(X)},$

and a differential equation $D_{Z}(h)=0$ for a Hermitian metric-type structure
$h$ on an object $E\in\mathcal{A}$ such that $D_{Z}(h)=0$ admits a solution
for $E$ if and only if $E$ is Bridgeland stable with respect to
$(Z,\mathcal{A})$.

Several features of this conjecture are left to be made more precise, and we
summarise the key points here:

  * •

The existence of the Bridgeland stability condition itself is a serious open
problem for general Calabi–Yau threefolds, and much of the difficulty arises
in constructing the Harder–Narasimhan filtrations of elements of the derived
category. A programme for how to resolve this using differential geometry is
being developed by Haiden–Katzarkov–Kontsevich–Pandit [HKKP20] called
“categorical Kähler geometry”, but any such process will in fact rely on a
Donaldson–Uhlenbeck–Yau-type theorem as proposed in the conjecture.

  * •

The exact central charge $Z$ may need to change (either by removing
$\sqrt{\operatorname{Td}(X)}$, changing to $\sqrt{\hat{A}(X)}$, or by
accounting for quantum corrections).

  * •

The exact differential equation is not known, and the current best guess is
the _deformed Hermitian Yang–Mills equation_ identified by Leung–Yau–Zaslow
and Mariaño–Minasian–Moore–Strominger [LYZ00, MnMMS00].111Recent discussion by
Li [Li22] suggests that even a _differential equation_ may be too much to hope
for in general, but physical reasoning assures that such Hermitian metric-type
objects will be critical points of some kind of action functional. This has
only been derived from mirror symmetry or physics in the case of a line
bundle, but natural analogues in higher rank can be justified on general
grounds.

In Part I we make some progress towards 1.2. In particular, guided in part by
1.1 we do not restrict ourselves just to the central charge $Z$ above, but to
a larger class of _polynomial central charges_ introduced by Bayer [Bay09].
Nor do we consider the case of a Calabi–Yau threefold, but instead any compact
Kähler manifold. To any of Bayer’s polynomial central charges $Z$ we associate
a corresponding extremal metric, appearing as the solution of a differential
equation on a holomorphic vector bundle. We call such an extremal metric a
_$Z$ -critical metric_,222“Zed critical” and prove a correspondence between a
stability condition and existence of solutions to the _$Z$ -critical equation_
in an asymptotic limit, the _large volume limit_.333This is the limit where
the Kähler metric $\omega$ is replaced by $k\omega$ and we scale $k\to\infty$.
In the language of physics this is the “quantum limit” of string theory, where
string length goes to zero and strings become well-approximated by point
particles.

The main result of Part I is the following, which justifies _a posteriori_ our
differential equation which has been chosen on general grounds.

######  Theorem 1.3.

A simple, slope semistable holomorphic vector bundle $E\to(X,\omega)$ over a
compact Kähler manifold, with locally-free graded object
$\operatorname{Gr}(E)$, admits a $Z$-critical metric in the large volume limit
if and only if it is asymptotically $Z$-stable.

The existence part of the above result has a particularly simple proof in the
case where $E$ is actually slope stable, covered in Section 5.2.1, and such
bundles give the first examples of solutions to the deformed Hermitian
Yang–Mills equation on higher rank vector bundles. In Section 5.2 we will give
the full details of the proof of Theorem 1.3 in the case where
$\operatorname{Gr}(E)$ has two components, which demonstrates most clearly how
the algebro-geometric condition of asymptotic $Z$-stability enters into the
analysis, and refer the reader to [DMS21, §4] for the details of the more
general case. The case we prove in Section 5.2 is sufficient to produce new
“non-trivial” examples of $Z$-critical metrics on strictly semistable rank 3
vector bundles over $\mathbb{CP}^{2}$ described in Example 4.1.22.

Additionally in Chapter 4 we will study the basic geometric theory of these
new equations away from the large volume limit. This leads to a new notion of
a _subsolution_ which has not previously appeared in the context of non-
Abelian gauge theory in the literature, and we study the geometric
consequences of it. In particular we prove:

######  Theorem 1.4.

When restricted to the locus of subsolutions inside the space of Chern
connections on a holomorphic vector bundle $E\to(X,\omega)$, the $Z$-critical
equation is elliptic and arises from an infinite-dimensional moment map
construction with respect to the action of the gauge group.

Obstructions to existence of solutions to the $Z$-critical equation are
expected to arise from both subsheaves and subvarieties, and the latter is a
new phenomenon in gauge theory, but familiar in the literature of stability
conditions in algebraic geometry. This has been recently studied in the
context of the _deformed Hermitian Yang–Mills equation_ on line bundles, and
the _J-equation_ , both of which can be described as $Z$-critical equations
for specific choices of $Z$.444With degenerate stability vector, in the case
of the J-equation. It has been shown in those cases how algebro-geometric
obstructions due to subvarieties are captured by the notion of subsolution,
and we demonstrate this further in the case of complex surfaces.

######  Theorem 1.5.

For a holomorphic line bundle $L\to(X,\omega)$ over a compact Kähler surface
and a choice of polynomial central charge $Z$ satisfying the volume
hypothesis, the following are equivalent:

  1. (i)

$L$ admits a solution to the $Z$-critical equation,

  2. (ii)

$L$ admits a subsolution to the $Z$-critical equation,

  3. (iii)

$L$ is $Z$-stable with respect to analytic curves $C\subset X$.

We begin the investigation of this phenomenon in higher rank for the first
time, giving the first explanation for how obstructions due to subvarieties
may be captured in a non-Abelian gauge-theoretic context using the subsolution
condition.

######  Theorem 1.6.

If a holomorphic vector bundle $E\to(X,\omega)$ admits a subsolution to the
$Z$-critical equation then $E$ is $Z$-stable with respect to irreducible
analytic divisors $D\subset X$. Moreover if $E$ admits a strong subsolution,
then it is $Z$-stable with respect to all irreducible analytic subvarieties.

In Section 4.5.1 we give further evidence of the importance of the subsolution
condition by constructing Donaldson and Yang–Mills-type functionals for the
$Z$-critical equation, and showing that restricted to the locus of
subsolutions, the Donaldson-type functional is convex and has critical points
precisely given by $Z$-critical metrics, which are also absolute minima of the
Yang–Mills-type functional.

Curiously, the algebro-geometric stability condition implied by the
subsolution condition for the $Z$-critical equation is not the same as the one
predicted by Bridgeland stability, as has been observed by Collins–Yau [CY21].
Indeed due to this discrepancy, a counterexample to 1.2 has been constructed
on (the non-Calabi–Yau) $\operatorname{Bl}_{p}\mathbb{CP}^{2}$ by Collins–Shi
[CS19]. This suggests it may be necessary to modify the definition of
Bridgeland stability in order to accurately reflect this discrepancy, and a
possible (rudimentary) modification implied by the Freed–Kapustin–Witten
anomaly has already been suggested in the physics literature, but appears to
have gone unnoticed in the mathematical literature. In Section 4.6 we briefly
investigate this modification, and discuss several other future directions
including how the notion of quasi-isomorphism may be reflected using the
differential geometry of complexes of vector bundles.

In Part II, our focus is shifted to a completely different geometric context
in which 1.1 appears, _Kähler fibrations_. This new setting interpolates
between the setting of vector bundles (seen in this context as arising from
projective bundles, particular examples of Kähler fibrations) and varieties
(Kähler fibrations over a point). These two settings have been well-studied in
complex geometry, with 1.1 manifesting as the celebrated
Donaldson–Uhlenbeck–Yau theorem and Chen–Donaldson–Sun theorem respectively,
both of which we shall review in Chapter 2.

In the interpolating setting of fibrations, a hybrid of the stability
conditions of bundles and varieties has been introduced by Dervan–Sektnan
[DS21a], who also identified a corresponding notion of canonical metric on the
fibration, an _optimal symplectic connection_ [DS21b].

The principle predicts a correspondence between these notions, and we
investigate this in the case of _isotrivial Kähler fibrations_ , where the
complex structure of the fibres does not vary over the base. All such
fibrations arise as associated bundles to holomorphic principal bundles, and
in this geometric setting we prove the following.

######  Theorem 1.7.

Suppose an isotrivial Kähler fibration $(X,\omega_{X})\to(B,\omega_{B})$
arises through the associated bundle construction from a holomorphic principal
bundle $P\to(B,\omega_{B})$ with connection $A$. Then the natural induced
symplectic connection $\omega_{X}$ is an optimal symplectic connection if and
only if $A$ is a Hermite–Einstein connection on $P$.

In particular this implies a version of 1.1 where we utilise the already-
existing theory of stability of holomorphic principal bundles:

######  Corollary 1.8.

If a holomorphic principal bundle $P\to(B,\omega_{B})$ is polystable, then the
associated isotrivial Kähler fibration $(X,\omega_{X})\to(B,\omega_{B})$
admits an optimal symplectic connection.

This statement becomes an _if and only if_ when the principal bundle under
consideration is the bundle of relative automorphisms of the isotrivial Kähler
fibration.

Through the work of Dervan–Sektnan about existence of constant scalar
curvature Kähler metrics in adiabatic Kähler classes on the total space of
Kähler fibrations $(X,\omega_{X})\to(B,\omega_{B})$, the existence result for
optimal symplectic connections above allows us to construct many new examples
of cscK metrics on the total space of isotrivial Kähler fibrations.

######  Corollary 1.9.

Suppose $P\to(B,\omega_{B})$ is a simple, stable holomorphic principal
$G$-bundle. Suppose that $(B,\omega_{B})$ is cscK with discrete automorphisms.
If $G$ acts holomorphically on a cscK manifold $(Y,\omega_{Y})$ and the
maximal compact subgroup $K\subset G$ acts by holomorphic isometries, then the
total space of the associated bundle $X=P\times_{G}Y$ admits cscK metrics in
adiabatic Kähler classes.

We describe an example of such a such a fibre bundle admitting a cscK metric
on the total space, using a principal $\operatorname{SL}(2,\mathbb{C})$-bundle
and model fibre given by the Mukai–Umemura threefold, a Kähler–Einstein Fano
threefold admitting an action of $\operatorname{SL}(2,\mathbb{C})$. Many such
principal bundles exist, so this gives a large class of new examples of
fibrations admitting cscK metrics in adiabatic Kähler classes.

Further to this isotrivial setting, in Section 7.2 we will discuss the
stability of isotrivial fibrations, particularly in the case of projective
bundles which has been understood by Ross–Thomas using slope K-stability
[RT06], and we will discuss a description of non-isotrivial fibrations in
terms of principal bundles.

###  Chapter 2 Preliminaries

In this chapter we recall some of the preliminaries necessary to understand
both $Z$-critical connections in Part I and Kähler fibrations in Part II, and
indulge in a survey of the most famous instances of 1.1.

We will give a brief overview of geometric invariant theory and its
relationship to Kähler geometry through the Kempf–Ness theorem, which will not
be used but serves as the philosophical background for the principle.

We will then recall the theory of slope stability and canonical metrics on
holomorphic vector bundles over compact Kähler manifolds, where 1.1 is made
precise in the celebrated Donaldson–Uhlenbeck–Yau theorem (Theorem 2.2.21),
emphasising the necessary Hermitian geometry required in later chapters. This
is important in understanding the behaviour of $Z$-critical metrics _at_ the
large volume limit $k=\infty$, and for understanding the existence of
canonical metrics on isotrivial Kähler fibrations.

Finally we detour slightly into the corresponding stability theory for
varieties, K-stability, and work towards stating the correspondence there, the
Yau–Tian–Donaldson conjecture (2.3.7). This theory places the notion of a
_degeneration_ as a focal point, and this language is used to understand the
stability of fibrations in terms of so-called _fibration degenerations_ as we
will describe in Part II. The same language has also been used in the study of
the deformed Hermitian Yang–Mills equation by Collins–Yau [CY21].

As part of our exposition of K-stability, we will investigate an interesting
new example of a singular variety whose deformation theory within the Fano
deformation class $V_{14}$ can be explicitly understood in terms of the
representations of $\operatorname{SL}(2,\mathbb{C})$.

####  2.1 Geometric invariant theory

We will begin with the simplest, finite-dimensional manifestation of 1.1,
which has been either directly applied or used as a guide for most subsequent
instances of the principle (except, perhaps, for Bridgeland stability). The
theory of stability is known as _geometric invariant theory_ (GIT), and the
principle becomes the Kempf–Ness theorem.

Geometric invariant theory is the language developed by Mumford to define
quotients of algebraic varieties by group actions [MFK94]. In accordance with
the revolution of algebraic geometry in the mid 20th century, which placed the
ring of functions of a variety at the centre of focus, GIT concerns itself
with invariant elements of commutative rings under group actions. This problem
was considered to varying extents by Hilbert in his _invariant theory_ ,
whence the name comes.

The basic observation of Mumford111To an extent, this was already implicit in
the work of Hilbert, where the concept of a semi-stable point (a ”Nullform”)
was characterised using a weight inequality [Hil93, p. 359]. was the
following: in order for the quotient of an algebraic variety $X$ by a group
action of $G$ to be a well-behaved algebraic space, it is necessary to
restrict to a locus $X^{s}\subset X$ of well-behaved, _stable_ points.

Let us restrict to the setting of a reductive group $G$ acting on a polarised
variety $(X,L)$. We will only consider varieties over $\mathbb{C}$, although
the theory has been developed over more general fields. We require that the
$G$ action on $X$ lifts to an action on the ample line bundle $L\to
X$,222Equivalently if we view $X\hookrightarrow\mathbb{P}^{N}$ for some $N$
then $G\subset\operatorname{PGL}(N,\mathbb{C})$ must lift to
$G\subset\operatorname{GL}(N+1,\mathbb{C})$ and act on $\mathbb{C}^{N+1}$. and
hence acts on all its powers $L^{k}$ and on the spaces of global sections
$H^{0}(X,L^{k})$.

We have a homogeneous coordinate ring

$R(X,L)=\bigoplus_{k\geq 0}H^{0}(X,L^{k})$

upon which $G$ acts, and the ring of invariants is

$R(X,L)^{G}=\bigoplus_{k\geq 0}H^{0}(X,L^{k})^{G}.$

This is a graded $\mathbb{C}$-algebra, and when $G$ is reductive it is
finitely generated [Hil90, Nag64]. Thus we can define a projective variety

$X\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}G=\operatorname{Proj}R(X,L)^{G},$

the geometric invariant theory quotient of $(X,L)$ by $G$. The first key
observation is that this quotient variety misses some points of $X$. That is,
there is a natural rational map $X\dashrightarrow
X\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}G$
induced by the inclusion $R^{G}\hookrightarrow R$, and if $x\in X$ is a point
such that $f(x)=0$ for every invariant section $f\in H^{0}(X,L^{k})^{G}$ for
every $k$, then the morphism $X\dashrightarrow
X\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}G$
is not defined at $x$. Such points are called _unstable_ by Mumford (or
“nullforms” by Hilbert). We have the following further characterisations of
points in $X$:

######  Definition 2.1.1.

A point $x\in(X,L)$ is called:

  * •

_semistable_ if there exists some $f\in H^{0}(X,L^{k})^{G}$ such that
$f(x)\neq 0$. Call the locus of semistable points $X^{ss}$.

  * •

_polystable_ if the orbit $G\cdot x$ is closed in $X$. Call the locus of
polystable points $X^{ps}$.

  * •

_stable_ if the orbit is closed and $x$ has finite stabiliser $G_{x}$. Call
the locus of stable points $X^{s}$.

  * •

_unstable_ if it is not semistable.

By definition, the rational morphism $X\dashrightarrow
X\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}G$
becomes a genuine morphism when restricting to the semistable locus, $X^{ss}$.
Furthermore if two semistable points $x,y\in X^{ss}$ have orbits whose
closures intersect, $\overline{G\cdot x}\cap\overline{G\cdot y}\neq\emptyset$,
then every $G$-invariant section takes the same value on $x$ and $y$ so they
become identified in
$X\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}G$.
This relation is called _S-equivalence_ (after Seshadri, who investigated its
consequences for the construction of moduli spaces of vector bundles).
Conversely the invariant functions will separate points whose closures do not
intersect, and thus the GIT quotient $X^{ps}\to
X^{ps}\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}G$
is actually an orbit space
$X^{ps}\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}G=X^{ps}/G$.
In fact it will be a consequence of the Kempf–Ness theorem that the orbit
space $X^{ps}/G$ is equal to the GIT quotient
$X\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}G$.333We
note however that $X^{ps}$ does not admit any nice description as an algebraic
subvariety of $X$, but is instead only a constructible subset of $X$.
Therefore to obtain an algebraic quotient which is also an orbit space, one
must further restrict to the stable locus $X^{s}$. Thus we obtain the famous
diagram of GIT quotients:

${X}$${X\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}G}$${X^{ss}}$${X^{ss}\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}G}$${X^{ps}}$${X^{ps}/G}$${X^{s}}$${X^{s}/G}$

It can be shown that the GIT quotient
$X\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}G$
satisfies the axioms of a _categorical quotient_ for the action of $G$ on $X$,
and therefore is the optimal solution to taking quotients by reductive groups
in the category of algebraic varieties (or schemes).

######  Remark 2.1.2.

There are alternative notions of GIT quotient when $X$ is affine or non-
projective and still twisted by a line bundle $L\to X$. This is important in,
for example, the construction of $\mathbb{CP}^{n}$ as a twisted affine GIT
quotient of $\mathbb{C}^{n+1}$, or the construction of moduli of
representations as in the work of King [Kin94].

#####  2.1.1 The Hilbert–Mumford Criterion

The criteria to determine the stability of a point $x\in X$ above are often
difficult to verify in practice, as they require information about the entire
$G$-orbit of $x$. Mumford identified a simpler criterion which only requires
one to know about the one parameter subgroups (1-PS)
$\lambda:\mathbb{C}^{*}\hookrightarrow G$ of $G$, called the _Hilbert–Mumford
criterion_.

Consider again the setting of a reductive group $G$ acting on a polarised
variety $(X,L)$. Given a point $x\in X$ and a 1-PS $\lambda$, the limit point
$y:=\lim_{t\to 0}\lambda(t)\cdot x$ is a fixed point of the $\mathbb{C}^{*}$
action of $\lambda$ on $X$. Thus since the $G$ action lifts to $L$, the fibre
$L_{y}$ over $y$ is also fixed by $\lambda$, and so comes with a weight
$\mu(x,\lambda)$ such that $\lambda(t)\cdot v=t^{\mu(x,\lambda)}v$ for every
$v\in L_{y}$. This is the _Hilbert–Mumford weight_ of $(x,\lambda)$.

The Hilbert–Mumford criterion is the following [MFK94, Tho06]:

######  Theorem 2.1.3 (Hilbert–Mumford criterion).

A point $x\in X$ is:

  * •

semistable if for every 1-PS $\lambda$ in $G$, $\mu(x,\lambda)\geq 0$.

  * •

polystable if $x$ is semistable and whenever $\mu(x,\lambda)=0$, $\lambda$ is
in the stabiliser $G_{x}$ of $x$.

  * •

stable if $x$ is polystable and has finite stabiliser, so that
$\mu(x,\lambda)=0$ only if $\lambda$ is the trivial 1-PS.

  * •

unstable if for at least one 1-PS $\lambda$ in $G$, $\mu(x,\lambda)<0.$

$x$$v$$\lambda\rightarrow 0$$\lambda\cdot
x$$y$$0$$\infty$$L$$X$$\mu(x,\lambda)>0$$\mu(x,\lambda)=0$$\mu(x,\lambda)<0$
Figure 2.1: The Hilbert–Mumford criterion. If $v$ lies over $x$, then as
$\lambda\to 0$, $\lambda(t)\cdot v\in L_{\lambda(t)\cdot x}$ will go to
infinity for $\mu(x,\lambda)>0$ and zero for $\mu(x,\lambda)<0$. Notice that
the orbit of $\tilde{x}$ is half-closed if $\mu(x,\lambda)>0$, but half-open
when $\mu(x,\lambda)=0$. One can check the other ($\lambda\to\infty$) end of
the orbit by replacing the 1-PS with its inverse. Combining the statement for
these two 1-PS gives the Hilbert-Mumford criterion relating the asymptotic
weight of $\lambda$ to the closedness of the orbit $\lambda\cdot x$.

A famous pictorial proof of the Hilbert–Mumford criterion as it relates to the
closedness of $\mathbb{C}^{*}$-orbits of points is given in Figure 2.1.

We will not utilize the Hilbert–Mumford criterion directly following this,
however the notion of a 1-parameter degeneration of an object is central to
the understanding of stability for both bundles and varieties as we will
subsequently explain.

#####  2.1.2 The Kempf–Ness theorem

According to 1.1 corresponding to the GIT stability of points there should be
an extremal notion, which we now explain.

Consider now the setting of $(X,\omega)$ a compact symplectic manifold, and
suppose a real compact Lie group $K$ acts on $X$ by Hamiltonian
symplectomorphisms. That is, for each $\xi\in\mathfrak{k}$, the induced vector
field $v_{\xi}$ is Hamiltonian.

######  Definition 2.1.4.

A _moment map_ $\mu:X\to\mathfrak{k}^{*}$ for the Hamiltonian action of $K$ on
$(X,\omega)$ is a $K$-equivariant map with respect to the coadjoint action on
$\mathfrak{k}^{*}$ such that

$d\langle\mu,\xi\rangle=-i_{v_{\xi}}\omega$

for all $\xi\in\mathfrak{k}$, where $\langle-,-\rangle$ is the natural pairing
between $\mathfrak{k}^{*}$ and $\mathfrak{k}$.. The map $\mathfrak{k}\to
C^{\infty}(X)$ sending $\xi\mapsto\langle\mu,\xi\rangle$ is called the
_comoment map_ , which is sometimes denoted $\mu^{*}$.

The _symplectic_ or _Marsden–Weinstein quotient_ of $(X,\omega)$ by $K$ is
given by

$X\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}K:=\mu^{-1}(0)/K.$

If $0$ is a regular value of $\mu$ and $K$ acts freely on $\mu^{-1}(0)$, then
the symplectic quotient is a smooth manifold and naturally inherits a
symplectic structure $\omega_{\mathrm{red}}$ from $X$ such that

$\pi^{*}\omega_{\mathrm{red}}=\iota^{*}\omega$

where $\iota:\mu^{-1}(0)\hookrightarrow X$ is the inclusion and
$\pi:\mu^{-1}(0)\to
X\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}K$
the projection.

Let us now return to the case of a projective variety $(X,L)$. Fix a Kähler
metric $\omega\in c_{1}(L)$ and suppose a group $G$ acts linearly on $(X,L)$.
Furthermore, suppose a maximal compact subgroup $K\subset G$ acts by
Hamiltonian symplectomorphisms on $(X,\omega)$. Then we now have two possible
quotients of $X$, one arising from GIT and the other from symlectic geometry.
The Kempf–Ness theorem relates these.

######  Theorem 2.1.5 (Kempf–Ness [KN79], Guillemin–Sternberg [GS84], Kirwan
[Kir84]).

Let $G$ be a reductive group with maximal compact subgroup $K$ acting linearly
on a projective variety $(X,L)$, such that $K$ acts by Hamiltonians on
$(X,\omega)$ where $\omega\in c_{1}(L)$ is a Kähler form. Suppose
$\mu:X\to\mathfrak{k}^{*}$ is a moment map for the $K$ action. A point $x\in
X$ is GIT polystable for the $G$ action if and only if $G\cdot x$ intersects
$\mu^{-1}(0)$.444Exactly which level set of the moment map one takes is
critical in the statement of the Kempf–Ness theorem, as shifting the level set
to some $\xi\in\mathfrak{z}(\mathfrak{k}^{*})$ corresponds to twisting the
$\mathbb{C}^{*}$ linearisation on the ample line bundle $L$ by a character.
See [Szé14, Rmk. 5.22]. Furthermore, if $x$ is polystable then $G\cdot x$
intersects $\mu^{-1}(0)$ is a unique $K$-orbit.

It was first proved by Kirwan [Kir84] that this gives a homeomorphism of
quotient spaces. Indeed further to Theorem 2.1.5 one has:

  * •

$G\cdot\mu^{-1}(0)=X^{ps}$,

  * •

A point $x\in X$ is semistable if and only if $\overline{G\cdot x}$ meets
$\mu^{-1}(0)$ in a single $K$-orbit.

  * •

$0$ is a regular value of $\mu$ if and only if $X^{ss}=X^{s}$.

  * •

The inclusion $\mu^{-1}(0)\hookrightarrow X^{ss}$ induces a homeomorphism of
quotients

$X\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}_{0}K\to(X,L)\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}G.$

The homeomorphism between quotients follows from the fact that a continuous
bijection from a compact space to a Hausdorff space is a homeomorphism. The
continuous inverse

$(X,L)\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}G\to
X\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}_{0}K$

can be found by following the gradient flow of the norm squared of the moment
map,

$V=\|\mu\|^{2}.$

In particular the gradient flow of $V$ takes a semistable point $x\in X^{ss}$
and flows down to a point $y\in X^{ps}$ in the unique polystable orbit in the
closure of $G\cdot x$ [Kir84].

The Kempf–Ness theorem may be proven working in a fixed orbit of $G$ and
considering the so-called _Kempf–Ness functional_ ,

$\mathcal{M}:G/K\to\mathbb{R}.$

The Kempf–Ness functional $\mathcal{M}_{x}$ on an orbit $G\cdot x$ satisfies
(or may be defined as the unique functional satisfying) the characteristic
variational property that

$\left.\frac{d}{dt}\right|_{t=0}\mathcal{M}([\exp(it\xi)g])=-2\langle\mu(g\cdot
x),\xi\rangle$

for $\xi\in\mathfrak{k}$. In particular a point $g\cdot x\in G\cdot x$ is a
zero of the moment map $\mu(g\cdot x)=0$ if and only if $[g]$ is a critical
point of the Kempf–Ness functional. To prove the theorem, one goes further to
establish that $\mathcal{M}$ is convex along geodesics in $G/K$, which is a
complete negatively curved metric space with respect to the induced Riemannian
metric after choosing a bi-invariant metric on $G$. One deduces that the orbit
of $g\cdot v$ is closed for $v$ some lift of $x$ if and only if
$\mathcal{M}_{x}$ is proper on $G/K$, which by the convexity of $\mathcal{M}$
occurs precisely when it has a critical point [Szé14, GRS21].

The relationship to the GIT picture can be made even more explicit. Indeed one
has the following.

######  Lemma 2.1.6 (See for example [GRS21, Lem. 5.2]).

Let $\xi\in\mathfrak{k}$ be rational and let
$\lambda:\mathbb{C}^{*}\hookrightarrow G$ be the 1-PS generated by $\xi$. Then
the Hilbert–Mumford weight $\mu(x,\lambda)$ is the slope at infinity of the
Kempf–Ness functional:

$\mu(x,\lambda)=\lim_{t\to\infty}\frac{\mathcal{M}_{x}([\lambda(t)])}{t}.$

The density of rational directions in
$\mathfrak{g}/\mathfrak{k}\cong\mathfrak{k}$ and the above lemma produce the
following philosophical statement: Provided $\mathcal{M}$ has good enough
behaviour, it suffices to verify properness over all of $G/K$ just by checking
the limiting slopes are positive along rationally defined geodesics. This is
the fundamental fact which underpins 1.1, and explains how just _algebraic_
(i.e. _rational_) information is strong enough to imply _analytical_ existence
theorems (see for example [Bou18, §1] for a discussion of this principle in
the finite-dimensional setting and how it relates to the variational approach
to the study of K-stability).

####  2.2 Holomorphic vector bundles

The study of stability and extremal metrics on holomorphic vector bundles
began with the work of Narasimhan and Seshadri relating the slope stability of
an algebraic vector bundle over a compact Riemann surface, in the sense of
Mumford–Takemoto, with existence of a projective unitary representation of the
fundamental group of the surface [NS65]. This work was a development in
relation to ideas of André Weil, who had already understood this
correspondence for line bundles and suggested a similar picture in higher
rank.

The theorem of Narasimhan and Seshadri was placed within a much broader
context by Atiyah and Bott [AB83], who emphasized the role played by
_Yang–Mills connections_ , which in the case of compact Riemann surfaces are
just projectively flat connections, whose holonomy therefore produces the
representations of the fundamental group. Variously, Atiyah–Bott introduced an
interpretation of the Yang–Mills equation as an infinite-dimensional moment
map, bringing the conceptual picture of geometric invariant theory into focus,
as well as providing new techniques in equivariant Morse theory to compute the
topology of these symplectic quotients — the moduli of stable vector bundles
on a curve (a theory subsequently developed by Kirwan for many types of
symplectic quotients). At the same time an independent proof of the
Narasimhan–Seshadri theorem using the language of Yang–Mills connections was
provided by Donaldson [Don83], and interpreted in the language of Atiyah and
Bott this proof could be interpreted as an infinite-dimensional Kempf–Ness
theorem for the curvature as a moment map.

Over the 1980s this theory was generalised to higher dimensional bases by
Donaldson, Uhlenbeck and Yau, and others, culminating in a correspondence
bearing their names. We now describe this theory.

#####  2.2.1 Stability

Let us first turn to the stability theory of holomorphic vector bundles, and
in particular slope stability and (briefly) Gieseker stability.

######  2.2.1.1 Slope stability

The notion of slope stability of a vector bundle was determined by Mumford
shortly after his development of geometric invariant theory [Mum63]. Mumford
considered the problem over a curve, and the condition for higher dimensional
bases was determined by Takemoto [Tak72], so this notion is often called
Mumford–Takemoto stability. We will state the definition in the setting of
compact Kähler manifolds, which includes the projective settings of Mumford
and Takemoto.

######  Definition 2.2.1.

The slope of a coherent analytic sheaf $\mathcal{E}$ over a compact Kähler
manifold $(X,\omega)$ is defined as

$\mu(\mathcal{E})=\frac{\operatorname{deg}\mathcal{E}}{\operatorname{rk}\mathcal{E}}=\frac{(c_{1}(\mathcal{E}).[\omega]^{n-1})[X]}{\operatorname{rk}\mathcal{E}}$

whenever $\operatorname{rk}\mathcal{E}>0$, and $\mu(\mathcal{E})=+\infty$ if
$\operatorname{rk}\mathcal{E}=0$.

######  Definition 2.2.2.

A coherent analytic sheaf $\mathcal{E}$ on a compact Kähler manifold
$(X,\omega)$ is _slope semistable_ if for all non-trivial coherent analytic
subsheaves $\mathcal{S}\hookrightarrow\mathcal{E}$, we have the inequality

$\mu(\mathcal{S})\leq\mu(\mathcal{E}).$

Furthermore we say $\mathcal{E}$ is _slope stable_ if, whenever
$\operatorname{rk}\mathcal{S}<\operatorname{rk}\mathcal{E}$, we have

$\mu(\mathcal{S})<\mu(\mathcal{E}).$

We say the sheaf is _slope polystable_ if it is a direct sum of slope stable
sheaves of the same slope. The sheaf is _slope unstable_ if it is not slope
semistable.

######  Remark 2.2.3.

Typically stability is defined for a _torsion-free_ coherent analytic sheaf.
It is an instant consequence of our definition that a slope (semi)stable
coherent sheaf of positive rank is torsion-free. In general a well-behaved
theory exists for pure coherent sheaves of a fixed dimension, as detailed in
[HL10, §1.6], but the slope of torsion sheaves must be defined as the quotient
of leading order coefficients of the Hilbert polynomial in that case.

Slope stability can be intuitively understood when $\mathcal{E}$ is a
holomorphic vector bundle $E$. In particular, it is a topological condition on
the holomorphic geometry of $E$, which asks that $E$ does not have sitting
inside it holomorphic subsheaves which are “more twisted” than $E$ itself
(after appropriate normalization by dividing by ranks). See Figure 2.2.

$0$$E$$F$$X$$s$$t$$t=0$$t=0$$s=0$$0$ Figure 2.2: An unstable bundle $E$ has
holomorphic subbundles $F$ which are more twisted than $E$.

This criterion limiting the complexity of the holomorphic geometry of $E$ has
various consequences, in particular for the existence of automorphisms of $E$.
We recall:

######  Proposition 2.2.4 (See [Kob87, Prop. 5.7.11, Cor. 5.7.12]).

Suppose $\mathcal{E},\mathcal{E}^{\prime}$ are slope semistable coherent
sheaves over a compact Kähler manifold $(X,\omega)$. Then

  1. (i)

If $\mu(\mathcal{E})>\mu(\mathcal{E}^{\prime})$ then
$\operatorname{Hom}(\mathcal{E},\mathcal{E}^{\prime})=0$.

  2. (ii)

Suppose $\mathcal{E},\mathcal{E}^{\prime}$ are slope stable,
$\mu(\mathcal{E})=\mu(\mathcal{E}^{\prime})$, and
$u:\mathcal{E}\to\mathcal{E}^{\prime}$ is a non-zero morphism. Then $u$ is
injective and
$\operatorname{rk}\operatorname{im}u=\operatorname{rk}\mathcal{E}^{\prime}$
(that is, $u$ is generically surjective). In particular $u$ is an isomorphism
away from a locus of codimension $\geq 2$. If $\mathcal{E}$ and
$\mathcal{E}^{\prime}$ have the same Chern numbers, that is

$c_{i}(\mathcal{E}).[\omega]^{n-i}=c_{i}(\mathcal{E}^{\prime}).[\omega]^{n-i}$

for all $i$, then $u$ is in fact an isomorphism, since
$\mathcal{E}^{\prime}/\operatorname{im}u$ must have vanishing Chern numbers
but if it is not zero then
$\operatorname{Ch}_{k}(\mathcal{E}^{\prime}/\operatorname{im}u).[\omega]^{n-k}>0$
where $k$ is the codimension of the support of
$\mathcal{E}^{\prime}/\operatorname{im}u$.555When
$\mathcal{E},\mathcal{E}^{\prime}$ are locally free it is sufficient that
$\operatorname{rk}\mathcal{E}=\operatorname{rk}\mathcal{E}^{\prime}$ and
$\operatorname{deg}\mathcal{E}=\operatorname{deg}\mathcal{E}^{\prime}$. See
[Kob87, Cor. 5.7.12] for the proof of this case. Furthermore, in this case
after making an identification $\mathcal{E}=\mathcal{E}^{\prime}$ then $u$ is
always a constant multiple of the identity morphism
$u=\lambda{\boldsymbol{1}}_{\mathcal{E}}$.

The second point above gives a strong restriction on the automorphisms of
slope stable coherent sheaves. We recall the following definition:

######  Definition 2.2.5.

A coherent sheaf $\mathcal{E}$ on $X$ is called _simple_ if

$H^{0}(X,\operatorname{End}(\mathcal{E}))\cong\mathbb{C}\cdot{\boldsymbol{1}}_{\mathcal{E}}.$

Thus the second point of Proposition 2.2.4 asserts that a slope stable
coherent sheaf is simple. Every such sheaf has automorphisms given by the
constant multiples of the identity morphism, so a simple sheaf is one with
automorphism group _as small as possible_.666One perspective is to consider
the projectivised gauge group
$\mathcal{G}(E)/\mathbb{C}^{*}\cdot{\boldsymbol{1}}_{E}$, under which stable
bundles have trivial automorphism group. As observed in our discussion of
geometric invariant theory, it is often the automorphisms of an object (the
stabilisers of the corresponding point in the Quot scheme) which are
problematic for the construction of well-behaved quotients. In particular, one
expects that after restricting to (semi)stable sheaves one should obtain a
better-behaved moduli space. The moduli space of slope (semi)stable sheaves is
particularly well understood in one and two dimensions [HL10, §8.2], and has
more recently been understood in general [GT17].

A considerable theory of the algebraic geometry of stable bundles and stable
sheaves now exists, and a good survey is the text of Huybrechts and Lehn
[HL10]. We will emphasise just a few further points about stability of bundles
which we will need later. Firstly, we will recall the following see-saw
property for short exact sequences of sheaves. We state the see-saw lemma
explicity in the case of torsion sheaves, in order to emphasize the comparison
with the notion of $Z$-stability with respect to subvarieties appearing in
Chapter 4.

######  Lemma 2.2.6 (See-saw lemma; See [Kob87, Prop. 5.7.6]).

Let

${0}$${\mathcal{S}}$${\mathcal{E}}$${\mathcal{Q}}$${0}$

be a short exact sequence of coherent sheaves. Suppose
$\operatorname{rk}\mathcal{E}>0$. Then

$\mu(\mathcal{S})\leq\mu(\mathcal{E})\quad\iff\quad\mu(\mathcal{E})\leq\mu(\mathcal{Q})$

and

$\mu(\mathcal{S})\geq\mu(\mathcal{E})\quad\iff\quad\mu(\mathcal{E})\geq\mu(\mathcal{Q}).$

Furthermore:

  * •

If $0<\operatorname{rk}\mathcal{S}<\operatorname{rk}\mathcal{E}$ then

$\mu(\mathcal{S})<\mu(\mathcal{E})\quad\iff\quad\mu(\mathcal{E})<\mu(\mathcal{Q})$

and

$\mu(\mathcal{S})>\mu(\mathcal{E})\quad\iff\quad\mu(\mathcal{E})>\mu(\mathcal{Q}).$

  * •

If $\operatorname{rk}\mathcal{S}=0$ then $\mu(\mathcal{S})>\mu(\mathcal{E})$
and $\mu(\mathcal{E})\geq\mu(\mathcal{Q})$ with equality whenever
$\mathcal{S}$ is supported in codimension $\geq 2$.

  * •

If $\operatorname{rk}\mathcal{S}=\operatorname{rk}\mathcal{E}$ then
$\mu(\mathcal{S})\leq\mu(\mathcal{E})$ with equality whenever $\mathcal{Q}$ is
supported in codimension $\geq 2$, and $\mu(\mathcal{E})<\mu(\mathcal{Q})$.

###### Proof.

The proof follows quickly from noting that rank $\operatorname{rk}$ and degree
$\operatorname{deg}$ are additive in short exact sequences of coherent sheaves
and we omit it here. We give a similar proof for asymptotic $Z$-stability in
Lemma 4.1.18. The final considerations follow from the fact that if
$\mathcal{T}$ is a torsion sheaf, then $\operatorname{deg}\mathcal{T}\geq 0$,
and furthermore if $\mathcal{T}$ is supported in codimension $\geq 2$, then
$\operatorname{deg}\mathcal{T}=0$ (see for example [Kob87, Lem. 5.7.5]). ∎

The see-saw property for the slope of sheaves allows us to rephrase the
characteristic inequality for stability in several ways: Suppose
$\mathcal{S}\subset\mathcal{E}$ is a coherent subsheaf with quotient
$\mathcal{E}\to\mathcal{Q}$. Then the following notions of stability are
equivalent (see [Kob87, Prop. 5.7.6]):

  1. (i)

Stable if $\mu(\mathcal{S})\leq\mu(\mathcal{E})$ for all coherent subsheaves
$\mathcal{S}\hookrightarrow\mathcal{E}$, with equality only if
$\operatorname{rk}\mathcal{S}=\operatorname{rk}\mathcal{E}$.

  2. (ii)

Stable if $\mu(\mathcal{E})\leq\mu(\mathcal{Q})$ for all coherent quotients
$\mathcal{E}\twoheadrightarrow\mathcal{Q}$, with equality only if
$\operatorname{rk}\mathcal{Q}=\operatorname{rk}\mathcal{E}$.

  3. (iii)

Stable if $\mu(\mathcal{S})<\mu(\mathcal{Q})$ for all short exact sequences
$S\hookrightarrow\mathcal{E}\twoheadrightarrow\mathcal{Q}$.

We will see later in Part I that $Z$ stability is often more naturally stated
in terms of condition (ii) above, as obstructions due to subvarieties
$V\subset X$ appear via quotient sheaves $E\to E\otimes\mathcal{O}_{V}$.
Furthermore we will see in Chapter 5 that it is actually condition (iii) which
naturally manifests in the analysis of the $Z$-critical equation, as opposed
to the equivalent but less symmetric condition (i).

We will see a geometric manifestation of Lemma 2.2.6 in the next section, in
terms of the principle of Hermitian geometry that _curvature (generically)
decreases in subbundles and increases in quotients_.

######  2.2.1.2 Filtrations

It will be useful to us to recall several kinds of filtrations which naturally
appear for holomorphic vector bundles.

First, given a semistable holomorphic vector bundle $E\to X$ there are a
natural class of so-called _Jordan–Hölder filtrations_ of $E$.

######  Definition 2.2.7.

A _Jordan–Hölder filtration_ $\mathscr{E}$ of a semistable vector bundle $E\to
X$ is a filtration

$\mathscr{E}:\quad
0=\mathcal{E}_{0}\subset\mathcal{E}_{1}\subset\cdots\subset\mathcal{E}_{k}=E$

of $E$ by torsion-free coherent subsheaves such that for every $i$,
$\mathcal{E}^{\prime}_{i}:=\mathcal{E}_{i}/\mathcal{E}_{i-1}$ is slope stable
and $\mu(\mathcal{E}^{\prime}_{i})=\mu(E)$. The polystable sheaf

$\operatorname{Gr}(E):=\bigoplus_{i}\mathcal{E}^{\prime}_{i}$

is called the _associated graded object of $E$ with respect to $\mathscr{E}$_.

There is a basic existence and uniqueness result for Jordan–Hölder filtrations
of $E$. This comes with a subtlety that the graded object is not strictly
unique. Since slope stability is ignorant to torsion sheaves of codimension
$\geq 2$, the graded object $\operatorname{Gr}(E)$ is only unique up to
codimension 2. This may be resolved by passing to the _reflexive hull_
$\operatorname{Gr}(E)^{**}$ of $\operatorname{Gr}(E)$.777A _reflexive sheaf_
is a sheaf $\mathcal{E}$ isomorphic to its double dual
$\mathcal{E}\cong\mathcal{E}^{**}$. The _reflexive hull_ of a sheaf
$\mathcal{E}$ is its double dual $\mathcal{E}^{**}$, which is always
reflexive. Reflexive sheaves are torsion-free and are close enough to being
locally free to do analysis with.

######  Proposition 2.2.8 (See [HL10, Thm. 1.6.7, Prop. 1.6.10] or [Kob87,
Thm. 5.7.18]).

Any semistable vector bundle admits a Jordan–Hölder filtration. The graded
objects of any two Jordan–Hölder filtrations are isomorphic outside a set of
codimension $\geq 2$. The reflexive hulls $\operatorname{Gr}(E)^{**}$ of any
two Jordan–Hölder filtrations are isomorphic, and the hull is therefore
uniquely defined.

######  Remark 2.2.9.

Due to our notion of slope stability implying torsion-freeness, it is an
automatic consequence of the stability of the factors of the graded object
$\operatorname{Gr}(E)$ is torsion-free for any semistable sheaf $E$. The
existence of a filtration with this property is automatic when $E$ is torsion-
free as the maximal destabilising subsheaf of a semistable torsion-free sheaf
is always saturated. Moreover, a filtration with the the property that the
successive quotients are only destabilised by torsion sheaves of codimension
$\geq 2$ can be upgraded to a Jordan–Hölder filtration simply by replacing
$\mathcal{E}_{i}$ with the saturation of $\mathcal{E}_{i}$ inside
$\mathcal{E}_{i+1}$ at every step. The resulting filtration has the same
numerical properties, and each successive quotient is slope stable in our
sense.

The same remark applies to the existence of the Harder–Narasimhan filtration
in Theorem 2.2.13 (except that the summands are semistable and torsion-free).
In that case the full details that the maximal destabilising sheaf is torsion-
free can be found in [Kob87, Thm. 5.7.15].

The key properties of slope stability which are used in the construction and
uniqueness of Jordan–Hölder filtrations are those of Proposition 2.2.4 and
Lemma 2.2.6. In particular any slope-type stability function which satisfies
these properties will produce an analogous Jordan–Hölder filtration theory for
semistable sheaves (for example one could use Gieseker or $Z$-stability
instead of slope stability).

######  Remark 2.2.10.

Even if $E$ is a vector bundle, there is no guarantee that the polystable
degeneration $\operatorname{Gr}(E)$ is locally free, or even reflexive.
However in Chapter 5 we will work under the assumption that
$\operatorname{Gr}(E)$ is locally free.

We will make use of the existence of a Hermite–Einstein metric on
$\operatorname{Gr}(E)$ in the proof of the correspondence for $Z$-critical
metrics. The Donaldson–Uhlenbeck–Yau theorem admits an extension to polystable
reflexive sheaves (see [BS94]) and so the proof of the correspondence in
Chapter 5 should generalise to the setting where $\operatorname{Gr}(E)$ is not
necessarily locally free, and a singular Hermite–Einstein metric on the
polystable reflexive sheaf $\operatorname{Gr}(E)^{**}$ is utilized in the
perturbation argument instead. The key difficulty then is to understand the
behaviour of the HE metric on the singular set of $\operatorname{Gr}(E)^{**}$
(i.e. the analytic subvariety forming the support of the torsion sheaf
$\operatorname{Gr}(E)^{**}/\operatorname{Gr}(E)$, which has been well-studied
in the theory of bubbling phenomena in gauge theory).

One may obtain the vector bundle $E$ from its graded object
$\operatorname{Gr}(E)$ by a sequence of extensions. After fixing a
Jordan–Hölder filtration $\mathscr{E}$ of $E$, we note that $E$ fits into a
short exact sequence

${0}$${\mathcal{E}_{k-1}}$${E}$${\mathcal{E}^{\prime}_{k}}$${0}$

and so $E$ corresponds to an extension of the graded factor
$\mathcal{E}^{\prime}_{k}$ by the subsheaf $\mathcal{E}_{k-1}$, classified by
a class
$e_{k}(\mathscr{E})\in\operatorname{Ext}^{1}(\mathcal{E}^{\prime}_{k},\mathcal{E}_{k-1})$.
Indeed we can repeat this process to build a sequence of extension classes
$e_{i}(\mathscr{E})\in\operatorname{Ext}^{1}(\mathcal{E}^{\prime}_{i},\mathcal{E}_{i-1})$
which reproduce $E$ from $\operatorname{Gr}(E)$ given $\mathscr{E}$.

######  Remark 2.2.11.

This process should be seen as a deformation of complex structure from
$\operatorname{Gr}(E)$ to $E$, or conversely a degeneration from $E$ to
$\operatorname{Gr}(E)$, through the process of “turning off an extension” (see
[RT06, Rmk. 5.14]). In particular given any one-step filtration
$0\subset\mathcal{F}\subset E$ of a vector bundle $E$ (not necessarily the
Jordan–Hölder filtration of a semistable bundle), we obtain a short exact
sequence

${0}$${\mathcal{F}}$${E}$${E/\mathcal{F}}$${0}$

which defines an extension class
$e\in\operatorname{Ext}^{1}(E/\mathcal{F},\mathcal{F})$. By scaling $\lambda
e$ for $\lambda\in\mathbb{C}$ we obtain a family $\mathcal{E}\to\mathbb{C}$
for which the generic fibre $\mathcal{E}_{t}\cong E$ for $t\neq 0$, and the
central fibre splits into the direct sum $\mathcal{F}\oplus E/\mathcal{F}$.

In Part I we will take the viewpoint of deformation of complex structure for
this construction, in particular in the proof of the correspondence in Chapter
5.

In Part II we will take the viewpoint of bundle degenerations, and this gives
a different interpretation of the stability of $E$ as an object, which is more
aligned with the language used in the study of K-stability of varieties and of
geometric invariant theory. See for example [Don05] for a discussion of this
perspective.

For the sake of completeness of discussion, we will also mention the existence
of a filtration for _any_ holomorphic vector bundle $E\to X$, where $E$ need
not be slope semistable. This is the famous _Harder–Narasimhan filtration_ of
$E$ [HN75].

######  Definition 2.2.12.

A Harder–Narasimhan filtration of a holomorphic vector bundle $E\to X$ is a
filtration

$0=\mathcal{E}_{0}\subset\mathcal{E}_{1}\subset\cdots\subset\mathcal{E}_{k}=E$

by coherent subsheaves such that each quotient
$\mathcal{E}^{\prime}_{i}=\mathcal{E}_{i}/\mathcal{E}_{i-1}$ is slope
semistable and the slopes strictly decrease:
$\mu(\mathcal{E}^{\prime}_{i})>\mu(\mathcal{E}^{\prime}_{i+1})$ for all $i$.

In contrast to Jordan–Hölder filtrations which only have uniqueness of the
associated graded object, the Harder–Narasimhan filtration of $E$ _is_ unique.
Indeed we have:

######  Theorem 2.2.13 (See for example [Kob87, Thm. 5.7.15].).

Any holomorphic vector bundle $E\to X$ over a compact Kähler manifold admits a
Harder–Narasimhan filtration, and there is a unique filtration with each
successive quotient $\mathcal{E}_{i}^{\prime}$ torsion-free.

Associated to the HN filtration of $E$ is a graded object
$\operatorname{Gr}^{\mathrm{HN}}(E)$ whose summands are torsion-free and slope
semistable of decreasing slope. For each summand $\mathcal{E}^{\prime}_{i}$ we
may then take the Jordan–Hölder filtration
$\operatorname{Gr}^{\mathrm{JH}}(\mathcal{E}^{\prime}_{i})$ to obtain a double
filtration of $\mathcal{E}$. Since the JH filtrations of each summand of
$\operatorname{Gr}^{\mathrm{HN}}(E)$ are not unique, the double filtration of
$E$ is not unique, but its double graded object associated to the
_Harder–Narasimhan–Seshadri filtration_ , denoted
$\operatorname{Gr}^{\mathrm{HNS}}(E)$, _is_ unique.888More precisely, the
reflexive hull of the double graded object is unique.

######  2.2.1.3 Gieseker stability

Let us briefly review the notion of Gieseker stability, which will be useful
by comparison to $Z$-stability as it will appear in Part I. Gieseker stability
was introduced by Gieseker and Maruyama [Gie77, Mar77, Mar78] and is an
alternative to slope stability which is more closely related to the geometric
invariant theory of sheaves. First let us recall the definition of the Hilbert
polynomial of a coherent sheaf.

######  Definition 2.2.14.

Let $(X,L)$ be a smooth polarised variety. The _Hilbert polynomial_ of a
coherent sheaf $\mathcal{E}$ on $X$ is given by

$\mathcal{P}_{\mathcal{E}}(k):=\raisebox{0.0pt}{$\chi$}(\mathcal{E}\otimes
L^{k})=\sum_{i>0}(-1)^{i}\dim H^{i}(X,\mathcal{E})$

where $\chi$ is the holomorphic Euler characteristic of $\mathcal{E}\otimes
L^{k}$.

By the Riemann–Roch formula we can compute the Euler characteristic as

$\raisebox{0.0pt}{$\chi$}(\mathcal{E}\otimes
L^{k})=\int_{X}\operatorname{Ch}(\mathcal{E}\otimes
L^{k})\operatorname{Td}(X)$

which is manifestly a polynomial in $k$. If $L$ is ample (by definition for
$(X,L)$ polarised), then by Serre vanishing we see $H^{i}(X,\mathcal{E}\otimes
L^{k})=0$ for $i>0$ and $k\gg 0$. Thus for $k$ sufficiently large, we have

$\raisebox{0.0pt}{$\chi$}(\mathcal{E}\otimes L^{k})=\dim
H^{0}(X,\mathcal{E}\otimes L^{k}).$

Note that the latter is not a polynomial for all $k$, but for $k\gg 0$ the
Riemann–Roch formula implies that it is polynomial.

######  Definition 2.2.15.

A torsion-free coherent sheaf $\mathcal{E}$ on $(X,L)$ is called _Gieseker
stable_ (resp. _Gieseker semistable_) if for all proper, non-zero coherent
subsheaves $\mathcal{F}\subset\mathcal{E}$ we have

$\frac{\mathcal{P}_{\mathcal{F}}(k)}{\operatorname{rk}\mathcal{F}}<\frac{\mathcal{P}_{\mathcal{E}}(k)}{\operatorname{rk}\mathcal{E}}\quad(\text{resp.
}\leq)$

for all $k\gg 0$.

The quantity
$\frac{\mathcal{P}_{\mathcal{E}}(k)}{\operatorname{rk}\mathcal{E}}$ is the
_Gieseker slope_ of $\mathcal{E}$. Gieseker stability is an asymptotic form of
stability for a sheaf near a “large volume” limit for $(X,L)$ (where $L$ is
replaced by $L^{k}$), and the first key property of Gieseker stability is the
following, which follows readily from the Riemann–Roch formula applied to the
Gieseker slope.

######  Lemma 2.2.16.

To leading order in $k$, the Gieseker slope is given (up to unimportant
geometric factors depending on $(X,L)$) by the slope $\mu(E)$. Thus a slope
stable vector bundle is Gieseker stable. In particular there are implications

Slope stable $\implies$ Gieseker stable $\implies$ Gieseker semistable
$\implies$ Slope semistable.

One can go on to observe that the analogues of Proposition 2.2.4 and Lemma
2.2.6 hold for the Gieseker slope also, and so the discussion of Section
2.2.1.2 could have been equally carried out for Gieseker stability. For a
detailed presentation of this theory, see [HL10, Ch. 1].

#####  2.2.2 Yang–Mills connections

In view of 1.1, corresponding to the preceding notion of a slope stable vector
bundle there should be a notion of an extremal object in differential
geometry. The first indications of what this extremal object is were provided
by Donaldson and Atiyah–Bott in their rephrasing of the theorem of
Narasimhan–Seshadri in terms of gauge theory [Don83, AB83].

In particular Atiyah and Bott made the following remarkable insight: If $A$ is
a Chern connection on a Hermitian vector bundle $(E,h)$ over a compact Riemann
surface then the map

$\mu:A\mapsto F(A)$

is a moment map for the action of the unitary gauge group $\mathcal{G}$ acting
on $E$. Indeed we recall that $A$ lives in the space $\mathcal{A}(h)$ of
integrable $h$-unitary connections, which is an affine space modelled on
$\Omega^{1}(X,\operatorname{End}_{SH}(E,h))$ where
$\operatorname{End}_{SH}(E,h)$ denotes the skew-Hermitian endomorphisms of $E$
with respect to $h$. The curvature
$F_{A}\in\Omega^{1,1}(X,\operatorname{End}_{SH}(E,h))$ may be viewed as an
element of the formal dual of the Lie algebra
$\mathfrak{g}=\Omega^{0}(X,\operatorname{End}_{SH}(E,h))$ of $\mathcal{G}$
under the natural pairing

$(\varphi,\psi)\mapsto-\int_{X}\operatorname{tr}\varphi\psi$

for $\varphi\in\mathfrak{g}$,
$\psi\in\Omega^{2}(X,\operatorname{End}_{SH}(E,h))$. Here the underlying
symplectic structure on $\mathcal{A}(h)$ is given by the _Atiyah–Bott
symplectic form_ ,

$\Omega_{AB}(a,b)=-\int_{X}\operatorname{tr}(a\wedge b).$

Primarily through observations of Donaldson [Don85, §4], this understanding of
the curvature as a moment map on compact Riemann surfaces admits an upgrade to
compact Kähler manifolds.999Here the Atiyah–Bott symplectic form simply picks
up a $\wedge\omega^{n-1}$ in the integrand. Associated to this moment map
construction are two natural functionals which have finite-dimensional
analogues we have seen in Section 2.1. The first is the _Yang–Mills
functional_ $\|\mu\|^{2}$, given by

$\operatorname{YM}(A)=\int_{X}|F_{A}|^{2}d\text{vol}.$ (Eq. 2.1)

To identify the second functional, let us now recall that any Chern connection
$A$ for $h$ is determined uniquely by a Dolbeault operator
$\nabla_{A}^{0,1}=\overline{\partial}_{A}$ satisfying the integrablity
condition $\overline{\partial}_{A}^{2}=F_{A}^{0,2}=0$. Such Dolbeault
operators are in one-to-one correspondence with holomorphic structures on the
underlying smooth vector bundle $E$, so using this identification of the space
of Chern connections $\mathcal{A}(h)$ with holomorphic structures we transfer
the action of the complex gauge group $\mathcal{G}^{\mathbb{C}}$ onto
$\mathcal{A}(h)$, and we obtain the formal Kempf–Ness picture from finite
dimensions analogous to Section 2.1.

On a fixed $\mathcal{G}^{\mathbb{C}}$ orbit inside $\mathcal{A}(h)$ we may
consider the “Kempf–Ness functional” of our problem, just as in Section 2.1.2.
Indeed in this setting we switch our perspective from varying the holomorphic
structure of $(E,h)$ to fixing a holomorphic structure
$\overline{\partial}_{E}$ and varying the metric, as follows: In a fixed
complex gauge orbit $\mathcal{G}^{\mathbb{C}}\cdot\overline{\partial}_{E}$
inside $\mathcal{A}(h)$, pulling back the fixed Hermitian metric $h$ from
$(E,g\cdot\overline{\partial}_{E})$ to $(E,\overline{\partial}_{E})$ gives a
new Hermitian metric $g^{*}h$ on $(E,\overline{\partial}_{E})$ (modulo the
unitary gauge transformations $g\in\mathcal{G}$), which produces a Chern
connection $A(g^{*}h)$. This construction gives a one-to-one correspondence
between Dolbeault operators
$\overline{\partial}_{A}\in\mathcal{G}^{\mathbb{C}}\cdot\overline{\partial}_{E}$
and Hermitian metrics $h$ on $(E,\overline{\partial}_{E})$, and the orbit in
$\mathcal{A}(h)$ may now be described as the quotient
$\mathcal{G}^{\mathbb{C}}/\mathcal{G}\cong\operatorname{Herm}(E)$.

The Kempf–Ness functional in this setting is known as the _Donaldson
functional_ , and was introduced by Donaldson in [Don85]. Given a fixed
reference metric $h$ on $E$, the Donaldson functional is the unique functional

$\mathcal{M}:\mathcal{G}^{\mathbb{C}}/\mathcal{G}\to\mathbb{R}$

satisfying $\mathcal{M}(h)=0$ and with first variation given by

$\frac{d}{dt}\mathcal{M}(h_{t})=\int_{X}\operatorname{tr}(h_{t}^{-1}\partial_{t}h_{t}\circ(\Lambda_{\omega}iF(h_{t})-\lambda{\boldsymbol{1}}_{E}))\omega^{n}.$

######  2.2.2.1 Hermite–Einstein metrics

As is typical in a Kempf–Ness formalism for 1.1, there are two types of
extremal objects we can study:

  1. (i)

Critical points of the functional $\|\mu\|^{2}$ on $\mathcal{A}(h)$.

  2. (ii)

Critical points of $\mathcal{M}$ on
$\mathcal{G}^{\mathbb{C}}/\mathcal{G}\cong\operatorname{Herm}(E)$.

The latter critical points correspond to the _absolute minima_ of
$\|\mu\|^{2}$ on $\mathcal{A}(h)$ inside $\mathscr{G}^{\mathbb{C}}$ orbits,
but one also has higher critical points of $\|\mu\|^{2}$ which we will remark
on later.

An analysis of the variation of the Yang–Mills functional $\operatorname{YM}$
or Donaldson functional $\mathcal{M}$ reveals that the absolute minima of the
Yang–Mills functional $\operatorname{YM}=\|\mu\|^{2}$ are given by so-called
_Hermite–Einstein metrics_ (see [Kob87, Thm. 4.3.9]).

######  Definition 2.2.17.

A Hermitian metric $h$ on a holomorphic vector bundle $E\to(X,\omega)$ over a
compact Kähler manifold is called _Hermite–Einstein_ (or _Hermitian
Yang–Mills_) if

$iF(h)\wedge\omega^{n-1}=\lambda{\boldsymbol{1}}_{E}\otimes\omega^{n}$ (Eq.
2.2)

for some constant $\lambda\in\mathbb{R}$.

If a metric $h$ satisfies the Hermite–Einstein equation where $\lambda=f\in
C^{\infty}(X)$ is a non-constant function, we call $h$ a _weak
Hermite–Einstein metric with function $f$_.

This is a second order elliptic partial differential equation in the metric
$h$, which is linear in the curvature $F(h)$. Let us make several remarks:

######  Remark 2.2.18.

We note that the Einstein constant $\lambda$ is purely topological, and by
integrating and using Chern–Weil theory we deduce

$\lambda=\frac{2\pi}{n!\operatorname{vol}(X)}\mu(E).$

When instead we have a non-constant Einstein function $\lambda=f$ the above
calculation simply determines the topological average of $f$ over $X$.

######  Remark 2.2.19.

After a conformal change of metric any weak Hermite–Einstein metric may be
transformed into a genuine Hermite–Einstein metric, so without loss of
generality we may always take $\lambda$ to be constant (see [Kob87, §4.2]). In
particular the relationship of stability with existence applies to weak
Hermite–Einstein metrics also.

Weak Hermite–Einstein metrics will naturally appear in the large volume limit
of the $Z$-critical equation in Part I.

######  Remark 2.2.20.

The higher critical points of the Yang–Mills functional $\operatorname{YM}$
correspond to Hermite–Einstein-type metrics with a block diagonal matrix whose
coefficients depend on the Harder–Narasimhan–Seshadri type of $E$ (see [Kob87,
Thm. 4.3.27]). In particular suppose $E\to(X,\omega)$ is a holomorphic vector
bundle with HNS type $\nu(E)$. If a Hermitian metric $h$ on $E$ is Yang–Mills,
that is $h$ is a critical point of the Yang–Mills functional 2.1, then $E$
admits a holomorphic orthogonal decomposition

$E=E^{\prime}_{1}\oplus\cdots\oplus E^{\prime}_{k}$

with summands $E^{\prime}_{i}$ each vector subbundles precisely so
$E\cong\operatorname{Gr}^{\mathrm{HNS}}(E)$ biholomorphically. One can
characterise when the Hermitian metric $h$ on $E$ is Yang–Mills by a
differential equation similar to the Hermite–Einstein equation. The metric is
Yang–Mills if and only if

$iF(h)\wedge\omega^{n-1}=\frac{2\pi}{n!\operatorname{vol}(X)}\operatorname{diag}(\nu(E))\otimes\omega^{n}$

with respect to this orthogonal decomposition of $E$, where
$\operatorname{diag}(\nu(E))$ has value $\mu(E^{\prime}_{i})$ on the summand
$E^{\prime}_{i}$.

We will not delay any longer in stating the celebrated correspondence between
Hermite–Einstein metrics and slope stability.

######  Theorem 2.2.21 (Donaldson–Uhlenbeck–Yau).

A holomorphic vector bundle $E\to(X,\omega)$ over a compact Kähler manifold
admits a Hermite–Einstein metric if and only if it is slope polystable.

This correspondence was conjectured independently by Kobayashi and Hitchin
[Kob82, Hit79] at the beginning of the 1980s. The “easy” direction that
existence implies stability was proven by Kobayashi and Lübke [Kob80, Lüb83].
In the reverse direction, the case of compact Riemann surfaces is essentially
the theorem of Narasimhan–Seshadri [NS65] reinterpreted using Donaldson’s
proof [Don83]. The case of algebraic surfaces was proven by Donaldson [Don85],
and the full correspondence for compact Kähler manifolds using a continuity
method by Uhlenbeck and Yau the following year [UY86]. Donaldson afterwards
gave a new proof of the theorem for all projective manifolds using a different
technique to Uhlenbeck and Yau, with an inductive argument along the lines of
the case of algebraic surfaces [Don87].

There are generalisations of Theorem 2.2.21 known for non-Kähler manifolds,
proved for surfaces by Buchdahl [Buc88] and in general by Li–Yau [LY87].
Additionally we recall that a version of the correspondence is known when $E$
is instead a reflexive sheaf over $X$ due to Bando–Siu, which in particular
applies in some settings to the graded object $\operatorname{Gr}(E)$ of a
semistable bundle $E$, which is always polystable but not necessarily locally
free [BS94].

As is true for the slope stability of vector bundles, we note that there is a
significant body of work using the Hermite–Einstein equation to construct
moduli of metrics on bundles, and indeed a rich interplay between the
algebraic and differential-geometric techniques to describe the same moduli
spaces. We will not comment further on these results, and refer for example to
[GSTW21] for more details.

######  2.2.2.2 Hermitian geometry of Chern connections

In this section we will analyse the Hermite–Einstein equation and in doing so
collect a number of facts in the Hermitian geometry of complex vector bundles
which will be important in Parts I and II. As part of this we will observe
many of the basic properties of stability which appeared in Section 2.2.1
manifesting in differential geometry, and in particular we will prove the easy
direction of the Donaldson–Uhlenbeck–Yau theorem (at least when the subobject
is also locally free).

Let us begin by linearising the Hermite–Einstein equation 2.2. To do so, let
us recall the tangency structure of the space of Hermitian metrics
$\operatorname{Herm}(E)$ on a holomorphic vector bundle. The isomorphism
$\operatorname{Herm}(E)\cong\mathcal{G}^{\mathbb{C}}/\mathcal{G}(E,h)$ for a
fixed reference metric $h$ shows that any nearby metric (equivalently, any
nearby Chern connection in the same gauge orbit of $A(h)$) is given by

$h_{t}=\exp(tV)\cdot h=h(-,\exp(tV)-)$

where $V\in\Omega^{0}(X,\operatorname{End}_{H}(E,h))$ is a Hermitian
endomorphism with respect to $h$. This has the effect of transforming

$\overline{\partial}_{h_{t}}=\overline{\partial}_{h}=\overline{\partial}_{E}$

and

$\partial_{h_{t}}=\exp(-tV)\circ\partial_{h}\circ\exp(tV).$

Indeed we have:

######  Lemma 2.2.22.

The curvature $F(h)$ of a Hermitian metric transforms as

$F(h_{t})=F(h)+\overline{\partial}\partial_{h}Vt+O(t^{2}).$

###### Proof.

Recall in local coordinates the $(1,0)$-part of the Chern connection takes the
form

$A=H^{-1}\partial H$

where $H$ is the local Hermitian matrix representing $h$. In local coordinates
$H_{t}=H\exp(tV)$ so $H_{t}^{-1}=\exp(-tV)H^{-1}$. Then we compute

$\displaystyle A_{t}$ $\displaystyle=H_{t}^{-1}\partial H_{t}$
$\displaystyle=\exp(-tV)H^{-1}(\partial H\exp(tV)+Ht\partial V\exp(tV))$
$\displaystyle=\exp(-tV)A\exp(tV)+t\exp(-tV)\partial V\exp(tV)$
$\displaystyle=A+(-VA+AV+\partial V)t+O(t^{2})$
$\displaystyle=A+\partial_{h}Vt+O(t^{2}).$

The Chern curvature is further computed as $\overline{\partial}A_{t}$ which
produces

$F(h_{t})=F(h)+\overline{\partial}\partial_{h}Vt+O(t^{2}).\qed$

Instead of working with a fixed holomorphic structure on $E$ and varying $h$
inside $\operatorname{Herm}(E)$, one could instead fix $h$ and vary the Chern
connection $A$ within its complex gauge orbit inside $\mathcal{A}(h)$. Acting
by a unitary gauge transformation $g\in\mathscr{G}^{\mathbb{C}}$ (i.e. such
that $g^{*}g={\boldsymbol{1}}_{E}$) conjugates the Chern connection, and in
particular $F_{A}$ will satisfy a gauge-invariant equation if and only if
$F_{g\cdot A}$ will. Therefore if we wish to capture a genuine change of the
curvature inside a complex gauge orbit, we should work orthogonal to the
unitary gauge transformations. The orthogonal complement to the unitary
transformations are the Hermitian transformations
$g\in\mathscr{G}^{\mathbb{C}}$ such that $g^{*}=g$ (in the sense that the
Hermitian endomorphisms are the orthogonal complement to the unitary Lie
algebra of $\mathscr{G}$ with respect to the natural trace pairing). The well-
known formula for the complex gauge group acting on Chern connections is given
by

$\displaystyle g\cdot d_{A}$
$\displaystyle=g^{*}\circ\overline{\partial}_{A}\circ
g^{*-1}+g^{-1}\circ\partial_{A}\circ g$ (Eq. 2.3)
$\displaystyle=g\circ\overline{\partial}_{A}\circ
g^{-1}+g^{-1}\circ\partial_{A}\circ g$

where in the second line we have used that $g=g^{*}$ in our case.101010Some
sources use the convention that
$g\cdot\overline{\partial}_{E}=g^{-1}\circ\overline{\partial}_{E}\circ g$
which produces a sign difference in the computation of the linearisation, but
no other differences. Our convention matches the transformation of the
Hermitian metric and follows for example [Don85], at the cost of treating the
action on $\operatorname{Herm}(E)$ as a right action and on $\mathcal{A}(h)$
as a left action. To produce a small perturbation of a Chern connection $A$
within its complex gauge orbit, we act by $g_{t}=\exp(tV)$ where $V$ is a
Hermitian endomorphism to produce $d_{A_{t}}=g_{t}\cdot d_{A}$. Then we have
the following.

######  Lemma 2.2.23.

The curvature $F_{A}$ of a Chern connection for a fixed Hermitian metric $h$
transforms as

$F_{A_{t}}=F_{A}+\left(\overline{\partial}_{A^{\operatorname{End}E}}\partial_{A^{\operatorname{End}E}}-\partial_{A^{\operatorname{End}E}}\overline{\partial}_{A^{\operatorname{End}E}}\right)Vt+O(t^{2}).$

###### Proof.

Using the formula $F_{A_{t}}=d_{A_{t}}^{2}$ and the fact that
$\overline{\partial}_{A}^{2}=\partial_{A}^{2}=0$ since the Chern curvature has
type $(1,1)$, we have

$\displaystyle F_{A_{t}}$
$\displaystyle=(g_{t}\circ\overline{\partial}_{A}\circ
g_{t}^{-1}+g_{t}^{-1}\circ\partial_{A}\circ
g_{t})\circ(g_{t}\circ\overline{\partial}_{A}\circ
g_{t}^{-1}+g_{t}^{-1}\circ\partial_{A}\circ g_{t})$
$\displaystyle=g_{t}\overline{\partial}_{A}^{2}g_{t}^{-1}+g_{t}\overline{\partial}_{A}g_{t}^{-2}\partial_{A}g_{t}$
$\displaystyle\quad+g_{t}^{-1}\partial_{A}g_{t}^{2}\overline{\partial}_{A}g_{t}^{-1}+g_{t}^{-1}\partial_{A}^{2}g_{t}$
$\displaystyle=g_{t}\overline{\partial}_{A}g_{t}^{-2}\partial_{A}g_{t}+g_{t}^{-1}\partial_{A}g_{t}^{2}\overline{\partial}_{A}g_{t}^{-1}$
$\displaystyle=\overline{\partial}_{A}\partial_{A}+\partial_{A}\overline{\partial}_{A}+t(V\overline{\partial}_{A}\partial_{A}-2\overline{\partial}_{A}V\partial_{A}+\overline{\partial}_{A}\partial_{A}V$
$\displaystyle\quad-V\partial_{A}\overline{\partial}_{A}+2\partial_{A}V\overline{\partial}_{A}-\partial_{A}\overline{\partial}_{A}V)+O(t^{2}).$
(Eq. 2.4)

Recalling the endomorphism connections

$\partial_{A^{\operatorname{End}E}}V=\partial
V+[A^{1,0},V],\quad\overline{\partial}_{A^{\operatorname{End}E}}V=\overline{\partial}V+[A^{0,1},V]$

and expanding $\partial_{A}=\partial+A^{1,0}$ and
$\overline{\partial}_{A}=\overline{\partial}+A^{0,1}$ we have

$\partial_{A}V=\partial_{A^{\operatorname{End}E}}V-A^{1,0}V,\quad\overline{\partial}_{A}V=\overline{\partial}_{A^{\operatorname{End}E}}V-A^{0,1}V,$

and substituting these into 2.4 one may compute

$F_{A_{t}}=F_{A}+\left(\overline{\partial}_{A^{\operatorname{End}E}}\partial_{A^{\operatorname{End}E}}-\partial_{A^{\operatorname{End}E}}\overline{\partial}_{A^{\operatorname{End}E}}\right)Vt+O(t^{2}).\qed$

######  Remark 2.2.24.

One can see that the two ways of linearising the curvature are equivalent.
Indeed recall that the isomorphism between $\mathscr{G}^{\mathbb{C}}\cdot
d_{A}$ and $\operatorname{Herm}(E,\overline{\partial}_{E})$ is given by
sending $(h,g\cdot\overline{\partial}_{E})$ to $(g\cdot
h,\overline{\partial}_{E})$ where $g$ is a Hermitian endomorphism. One can
show

$F(h,g^{1/2}\cdot\overline{\partial}_{E})=g^{1/2}\circ F(g\cdot
h,\overline{\partial}_{E})\circ g^{-1/2}$

and setting $g=\exp(tV)$ and computing the first order expressions in $t$, one
observes that the expression in Lemma 2.2.23 appearing on the left-hand side
is transformed into the expression from Lemma 2.2.22 appearing on the right.

As a consequence of these computations, we can now compute the linearisation
of the Hermite–Einstein equation.

######  Proposition 2.2.25.

The linearisation $P$ of the Hermite–Einstein operator

$D:A\mapsto
iF_{A}\wedge\omega^{n-1}-\lambda{\boldsymbol{1}}_{E}\otimes\omega^{n}$

under the action of the Hermitian gauge transformations111111Instead we could
work with Hermitian metrics on a fixed bundle in an orbit. is (up to a
constant factor) given by the $\operatorname{End}E$-Laplacian
$\Delta_{A^{\operatorname{End}E}}=d_{A}^{*}d_{A}$.

###### Proof.

Let $A_{t}=\exp(tV)\cdot A$ be a small perturbation of a Chern connection,
where $V$ is Hermitian. Then by Lemma 2.2.23 we have

$\displaystyle\left.\frac{d}{dt}\right|_{t=0}D(A_{t})$
$\displaystyle=i\omega^{n-1}\wedge\left(\overline{\partial}_{A^{\operatorname{End}E}}\partial_{A^{\operatorname{End}E}}-\partial_{A^{\operatorname{End}E}}\overline{\partial}_{A^{\operatorname{End}E}}\right)V$
$\displaystyle=\frac{i}{n}\Lambda_{\omega}\left(\left(\overline{\partial}_{A^{\operatorname{End}E}}\partial_{A^{\operatorname{End}E}}-\partial_{A^{\operatorname{End}E}}\overline{\partial}_{A^{\operatorname{End}E}}\right)V\right)\omega^{n}$
$\displaystyle=\frac{1}{n}(\Delta_{A^{\operatorname{End}E}}V)\omega^{n}.$

The last equality follows from the Nakano identities

$\Lambda_{\omega}\partial_{A^{\operatorname{End}E}}=i\overline{\partial}_{A^{\operatorname{End}E}}^{*},\quad\Lambda_{\omega}\overline{\partial}_{A^{\operatorname{End}E}}=-i\partial_{A^{\operatorname{End}E}}^{*}.$

See for example [Bal06, Prop. 5.22]. ∎

Let us explicitly record some useful corollaries of our computation of the
linearisation, which we will use in Part I. These follow immediately from
well-known properties of the bundle Laplacian.

######  Corollary 2.2.26.

The Hermite–Einstein equation is elliptic, and the kernel of the linearisation
is given by the global holomorphic endomorphisms

$\ker P_{A}=H^{0}(X,\operatorname{End}(E,d_{A}^{0,1})).$

######  Remark 2.2.27.

The linearisation of the Hermite–Einstein operator as a map

$D:h\mapsto
iF(h)\wedge\omega^{n-1}-\lambda{\boldsymbol{1}}_{E}\otimes\omega^{n}$

produces the $\partial_{h}$-Laplacian $\Delta_{\partial_{h}}$ by the Nakano
identity

$-i\partial_{h}^{*}=\Lambda\overline{\partial}.$

This differs from the bundle Laplacian $\Delta_{A^{\operatorname{End}E}}$ in
Proposition 2.2.25 by a commutator involving the contraction of the curvature:

$2\Delta_{\partial_{h}}V=\Delta_{A^{\operatorname{End}E}}V+[i\Lambda_{\omega}F_{h},V].$

Thus _at a solution_ to the Hermite–Einstein equation, the linearisations
agree and are both the standard bundle Laplacian. We will only need the
linearisation at a solution, so there is no subtlety introduced here.

We will now briefly recall the theory of second fundamental forms and the
Hermitian geometry of subbundles, which produces the fundamental principle
that “curvature (generically) decreases in holomorphic subbundles” which
underpins the easy direction of the Donaldson–Uhlenbeck–Yau theorem, and which
we will make extensive use of in Theorem 5.1.4.

Let $A$ be a Chern connection for a Hermitian metric $h$ on a holomorphic
vector bundle $E\to X$. Suppose $S\subset E$ is a holomorphic subbundle,
producing a short exact sequence

${0}$${S}$${E}$${Q}$${0}$

where $Q=E/S$ is the quotient. With respect to the smooth (but not necessarily
holomorphic!) splitting

$E\cong S\oplus Q$

induced by the metric $h$, the Chern connection may be written in block matrix
form as

$A_{E}=\begin{pmatrix}A_{S}&\beta\\\ -\beta^{*}&A_{Q}\end{pmatrix}$ (Eq. 2.5)

where $\beta\in\Omega^{0,1}(X,\operatorname{Hom}(Q,S))$ is the _second
fundamental form_ of the subbundle $S\subset E$. Further, with respect to this
same splitting the curvature $F$ of $A$ may be computed as

$F_{E}=\begin{pmatrix}F_{S}-\beta\wedge\beta^{*}&\partial_{A^{\operatorname{Hom}(Q,S)}}\beta\\\
-\overline{\partial}_{A^{\operatorname{Hom}(S,Q)}}\beta^{*}&F_{Q}-\beta^{*}\wedge\beta\end{pmatrix}.$
(Eq. 2.6)

######  Remark 2.2.28.

The second fundamental form can be given an algebraic interpretation using the
Dolbeault isomorphism theorem, for example following a computation similar to
[Ati57, Prop. 4].121212Atiyah’s calculation is for the particular problem of
defining a holomorphic connection on a principal bundle, but the particular
sequence (the “Atiyah sequence”) being split is unimportant to his proof.
Namely, if $e\in\operatorname{Ext}^{1}(Q,S)$ is the extension class of the
short exact sequence, then using the identification
$\operatorname{Ext}^{1}(Q,S)\cong H^{1}(X,\operatorname{Hom}(Q,S))$ when $Q$
and $S$ are locally free, and the Dolbeault isomorphism

$H^{1}(X,\operatorname{Hom}(Q,S))\cong
H_{\overline{\partial}_{E}}^{0,1}(X,\operatorname{Hom}(Q,S))$

one has $e=[\beta]$. In particular if $\beta$ is zero in Dolbeault cohomology,
then the short exact sequence splits. Indeed in that case the Chern connection
$A$ can be transformed by a change of gauge to one in which $\beta=0$, and
then $S$ is invariant under the covariant derivative $d_{A}$, so the
orthogonal complement is too (since $d_{A}$ is unitary), making $Q$ a
holomorphic subbundle of $E$.

To finish off our discussion of Hermitian geometry of subbundles, let us
demonstrate the principle that curvature (generically) decreases in subbundles
and increases in quotients. This is the analogue in Hermitian geometry of the
see-saw property Lemma 2.2.6 for slope stability, and it quickly produces a
proof of one direction of Theorem 2.2.21 (at least when the subobject is a
subbundle).

######  Proposition 2.2.29.

Suppose $E\to(X,\omega)$ is indecomposable and admits a Hermite–Einstein
metric $h$. Then $E$ is slope stable with respect to holomorphic subbundles
$S\subset E$.

###### Proof.

Here we use the decomposition of the Chern curvature and the properties of the
Hermite–Einstein metric. We know

$iF(h)\wedge\omega^{n-1}=\lambda{\boldsymbol{1}}_{E}\otimes\omega^{n}.$

Suppose $S\subset E$ is any proper non-zero holomorphic subbundle and $Q=E/S$
is the quotient bundle. Then with respect to the $C^{\infty}$ direct sum
decomposition $E=S\oplus Q$ the $\operatorname{End}(S)$ component of the
Hermite–Einstein equation is

$i(F_{S}-\beta\wedge\beta^{*})\wedge\omega^{n-1}=\frac{2\pi}{n!\operatorname{vol}(X)}\mu(E){\boldsymbol{1}}_{S}\otimes\omega^{n}.$

Taking trace and integrating we note that

$-\int_{X}\operatorname{tr}(\beta\wedge\beta^{*})\wedge\omega^{n-1}=\frac{i}{2\pi}\|\beta\|^{2}$

(suitably normalised) and thus we obtain

$2\pi(\operatorname{deg}S+\|\beta\|^{2})=2\pi\mu(E)\operatorname{rk}S$

and using that $\|\beta\|^{2}>0$ since $E$ is indecomposable (otherwise
$E=S\oplus Q$ holomorphically as $\beta=0$ implies the sequence splits) we
obtain

$\mu(S)<\mu(E).$

Thus $E$ is slope stable with respect to $S$. ∎

######  2.2.2.3 Almost Hermite–Einstein metrics

In order to complete our background on holomorphic vector bundles, we will
discuss the differential-geometric analogue of Gieseker stability discussed in
Section 2.2.1.3. This theory was developed by Leung in [Leu97, Leu98], and
shares a number of ideas with those in Part I.

Let us take the moment map approach to the problem. As was observed by
Atiyah–Bott and Donaldson, the Hermite–Einstein equation can be obtained as
the moment map equation for the symplectic form

$\Omega_{AB}(a,b)=-\int_{X}\operatorname{tr}(a\wedge b)\wedge\omega^{n-1}$

on the space of integrable unitary connections $\mathcal{A}(h)$ on a Hermitian
vector bundle $(E,h)\to(X,\omega)$. Donaldson constructed a determinant line
bundle $\mathcal{L}\to\mathcal{A}(h)$ for which the Atiyah–Bott symplectic
form is the curvature of a Hermitian metric [Don87]. Donaldsons construction
uses a local version of the index theorem and ideas due to Bismut–Freed, and
indeed going back to Quillen, to identify the correct determinant bundle
[Qui85, BF86a, BF86b]. As part of this construction, one considers the
universal bundle

$\mathbb{E}\to\mathcal{A}(h)\times X$

which has a universal connection $\mathbb{A}$ with the property that
$\left.\mathbb{A}\right|_{\\{A\\}\times X}=A$. This connection has a universal
curvature $F_{\mathbb{A}}\in\Omega^{2}(\mathcal{A}(h)\times X)$. It follows
from Bismut–Freed’s local version of the Atiyah–Singer index theorem for
families that

$\Omega:=c_{1}(\operatorname{ind}\mathbb{E})=\left\\{\int_{X}\exp\left(\frac{i}{2\pi}F_{\mathbb{A}}\right)\operatorname{Td}(X)\right\\}_{(2)}$

where on the right-hand side one takes the degree two component of the form on
$\mathcal{A}(h)$.

The symplectic structure which arises from this form on $\mathcal{A}(h)$ was
studied by Leung [Leu98]. In particular Leung considers the setting where
$E\to X$ is replaced by $E\otimes L^{k}$ where $L\to X$ is an ample line
bundle and $\omega\in c_{1}(L)$ is a representative Kähler form. Leung
computes

$\Omega_{k}(A)(a,b)=\int_{X}\operatorname{tr}\left[\exp\left(\frac{i}{2\pi}F_{A}+k\omega\otimes{\boldsymbol{1}}_{E}\right),a,b\right]_{\mathrm{sym}}\operatorname{Td}(X)$
(Eq. 2.7)

where the symmetric bracket is defined as in Definition 4.3.1. With respect to
the action of the gauge group $\mathcal{G}^{\mathbb{C}}$ on $\mathcal{A}(h)$,
one obtains a moment map equation

$\left\\{\exp\left(\frac{i}{2\pi}F_{A}+k\omega\otimes{\boldsymbol{1}}_{E}\right)\widetilde{\operatorname{Td}}(X)\right\\}^{(2n)}=\frac{1}{\operatorname{rk}(E)}\raisebox{0.0pt}{$\chi$}(E\otimes
L^{k})\frac{\omega^{n}}{n!}\otimes{\boldsymbol{1}}_{E}$ (Eq. 2.8)

where $\widetilde{\operatorname{Td}}(X)$ is given by the Chern–Weil
representatives of the Chern classes of $X$ with respect to the Levi-Civita
connection induced by the Kähler metric $\omega$. The form of this moment map
is highly suggestive of a link to Gieseker stabiliy. A connection $A$
satisfying 2.8 is called an _almost Hermite–Einstein connection_ (or for a
Hermitian metric $h$, an _almost Hermite–Einstein metric_).131313Not to be
confused with an “approximate Hermite–Einstein metric” as defined by Kobayashi
[Kob87], which is any sequence of Hermitian metrics which are
$\varepsilon$-close to being Hermite–Einstein in an appropriate norm. Indeed
almost HE metrics are examples of approximate HE metrics, as are the
$Z_{k}$-critical metrics for $k\gg 0$ which will appear in Part I. The almost
Hermite–Einstein equation converges to the Hermite–Einstein equation in the
large volume limit, and Leung proves the following.

######  Theorem 2.2.30 ([Leu97]).

Let $E\to(X,\omega)$ be a simple, slope semistable vector bundle with locally
free graded object $\operatorname{Gr}(E)$. Then $E$ admits an almost
Hermite–Einstein metric for all $k\gg 0$ if and only if $E$ is Gieseker
stable.

######  Remark 2.2.31.

Theorem 2.2.30 is the direct analogue of our main theorem Theorem 1.3 for
Gieseker stability as opposed to asymptotic $Z$-stability. The idea of the
proof is similar, in that Leung applies a perturbation result around the limit
$k\to\infty$ utilising the Donaldson–Uhlenbeck–Yau theorem and the existence
of a Hermite–Einstein metric on the graded object. However the details of the
proof are different. In particular Leung’s analysis of the linearised operator
does not take into account possible automorphisms arising from symmetries of
the graded object, which introduces difficulties in bounding the inverse of
the linearisation in the proof of Theorem 5.1.1 in the general case.141414We
note that these terms do not appear in the proof in Chapter 5 where we
restrict to the simpler case where $\operatorname{Gr}(E)$ has two components.

####  2.3 Varieties

Having described in some detail the background of stability on holomorphic
vector bundles, we will now discuss the corresponding theory for varieties.
Here the notion of _degenerations_ takes the central role over the _subobject_
perspective which was most geometrically meaningful in the case of bundles.
Such degenerations are called _test configurations_ , and we will make use of
this language in Part II to describe the notion of a _fibration degeneration_.

#####  2.3.1 K-stability

The theory of stability of varieties grew out of the success of the
Donaldson–Uhlenbeck–Yau theorem Theorem 2.2.21 in predicting the existence of
solutions to geometric PDEs on bundles and a search for a criterion which
would imply the existence of Kähler–Einstein metrics on Fano manifolds (the
cases of Calabi–Yau and general type having been resolved by Yau’s proof of
the Calabi conjecture). Indeed it was known by work of Matsushima and
Lichnerowicz that the Lie algebra of the automorphism group of any
Kähler–Einstein Fano manifold must be reductive [Mat57, Lic58]. However the
example of $X=\operatorname{Bl}_{p}\mathbb{CP}^{2}$ is Fano with non-reductive
automorphism group.151515The automorphism group is isomorphic to
$\operatorname{Aff}(2,\mathbb{C})$ which is not reductive as it contains a
factor of the additive group $\mathbb{C}_{+}^{2}$.

Yau conjectured that the existence of a Kähler–Einstein metric on a smooth
Fano variety should be equivalent to a stability condition analogous to slope
stability of bundles, and suggested perhaps that existence should be
equivalent to the slope polystability of the tangent bundle [Yau93]. The
correct criterion was identified by Tian, who called this condition _K-
stability_ [Tia97]. The “K” was labelled after the K-energy functional
introduced by Mabuchi [Mab86], and stands for _Kinetic_ or _Kanonisch_ (and
not _Kähler_!).161616This was confirmed in a private correspondence between
Ruadhaí Dervan and Mabuchi. Tian’s stability condition was rephrased purely
algebro-geometrically and expanded to include the case of all polarised
varieties by Donaldson [Don02]. We will describe the notion of K-stability in
Donaldson’s language.

K-stability is defined in direct analogy with the Hilbert–Mumford criterion
for stability of points on a variety with group action. In order to make this
analogy precise, one must therefore define:

  * •

a notion of one-parameter degeneration of a polarised variety $(X,L)$, and

  * •

a notion of weight computed on the limiting fibre of that one-parameter
degeneration.

First we will describe the one-parameter degenerations.

######  Definition 2.3.1 (Test configuration).

A test configuration (see Figure 2.3) $(\mathcal{X},\mathcal{L})$ with
exponent $r$ of a polarised variety $(X,L)$ is a scheme $\mathcal{X}$ with a
flat morphism $\pi:\mathcal{X}\to\mathbb{C}$ and a relatively ample line
bundle $\mathcal{L}$ over it, such that

  * •

There is a $\mathbb{C}^{*}$ action on the polarised family
$(\mathcal{X},\mathcal{L})$ covering the standard $\mathbb{C}^{*}$ action on
$\mathbb{C}$ making $\pi$ equivariant.

  * •

For every $t\in\mathbb{C}^{*}$,
$(\mathcal{X}_{t},\mathcal{L}_{t})\cong(X,L^{r})$.

We say a test configuration is a _product_ if $\mathcal{X}\cong
X\times\mathbb{C}$ and _trivial_ if it is a product and the $\mathbb{C}^{*}$
action on the $X$ factor is trivial.

$\mathcal{X}$$\mathbb{C}$$0$$\mathcal{X}_{0}$$\mathcal{X}_{t}\cong X$$t$
Figure 2.3: A test configuration. The generic fibre $\mathcal{X}_{t}$ is
isomorphic to $X$, but the central fibre $\mathcal{X}_{0}$ may have singular
points or multiple components.

One consequence of the flatness of the morphism $\pi$ is that the Hilbert
polynomial
$\raisebox{0.0pt}{$\chi$}(\mathcal{X}_{t},\mathcal{L}_{t})=\raisebox{0.0pt}{$\chi$}(X,L)$
agrees with $\raisebox{0.0pt}{$\chi$}(\mathcal{X}_{0},\mathcal{L}_{0})$.
Indeed, through more than just an analogy, test configurations _literally_
arise as (closures of) orbits of one-parameter subgroups applied to
$(X,L^{r})$ as a point inside a Hilbert scheme
$\operatorname{Hilb}_{\mathcal{P}(r)}$ for
$\mathcal{P}(r)=\raisebox{0.0pt}{$\chi$}(X,L^{r})$ (see [RT07, Prop. 3.7]).

The weight associated to a test configuration is defined as follows. The
central fibre $(\mathcal{X}_{0},\mathcal{L}_{0})$ is fixed by the
$\mathbb{C}^{*}$ action on $(\mathcal{X},\mathcal{L})$ and therefore this
action lifts to $H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{k})$ for $k\gg 0$. A
$\mathbb{C}^{*}$ action on a vector space splits it into a sum of weight
spaces, and the _total weight_ is the sum of the weights on each factor.
Equivalently we define the total weight function $w(k)$ as the weight of the
induced $\mathbb{C}^{*}$ action on $\det
H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{k})$. Let us explicitly write

$\mathcal{P}(k)=\dim
H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{k})=a_{0}k^{n}+a_{1}k^{n-1}+O(k^{n-2}),\quad
w(k)=b_{0}k^{n+1}+b_{1}k^{n}+O(k^{n-2}).$

######  Definition 2.3.2 (Donaldson–Futaki invariant).

The weight or _Donaldson–Futaki invariant_ of a test configuration
$(\mathcal{X},\mathcal{L})$ for $(X,L^{r})$ is given by

$\operatorname{DF}(\mathcal{X},\mathcal{L}):=\frac{a_{0}b_{1}-b_{0}a_{1}}{a_{0}}.$

An alternative computation of the Donaldson–Futaki weight in terms of
intersection theory is now known. Indeed an intersection-theoretic formula was
already understood by Tian in terms of the CM line bundle when he introduced
his original condition for K-stability [Tia97], and the equivalence of these
notions was later proven [PT09, PRS08] (see also [Wan12, Oda13]). Compactify
the test configuration trivially at $\infty$ to a family
$(\bar{\mathcal{X}},\bar{\mathcal{L}})\to\mathbb{CP}^{1}$. Then the DF
invariant or $\mathrm{CM}$-weight is given by

$\operatorname{DF}(\mathcal{X},\mathcal{L}):=\frac{1}{2(n+1)L^{n}}\left(n\mu(X,L)(\bar{\mathcal{L}})^{n+1}+(n+1)K_{\bar{\mathcal{X}}/\mathbb{CP}^{1}}.\bar{\mathcal{L}}^{n}\right)$
(Eq. 2.9)

where $\mu(X,L)=-(L^{n-1}.K_{X})/L^{n}$ is the _slope_ of $(X,L)$. The
intersection-theoretic formula will be useful in the definition of fibration
stability in Part II.

######  Definition 2.3.3.

Let $(X,L)$ be a polarised variety. We say the pair is

  * •

_K-semistable_ if $\operatorname{DF}(\mathcal{X},\mathcal{L})\geq 0$ for any
test configuration $(\mathcal{X},\mathcal{L})$.

  * •

_K-polystable_ if it is K-semistable and
$\operatorname{DF}(\mathcal{X},\mathcal{L})=0$ only if
$(\mathcal{X},\mathcal{L})$ normalises to a product test configuration.

  * •

_K-stable_ if $\operatorname{DF}(\mathcal{X},\mathcal{L})>0$ for any test
configuration which does not normalise to the trivial test configuration.

  * •

_K-unstable_ if it is not K-semistable.

######  Remark 2.3.4.

An attempt to define a notion of K-stability more in line with Yau’s original
prediction of a stability condition analogous to slope stability was carried
out by Ross–Thomas [RT06, RT07]. The notion of _slope K-stability_ is useful
for finding obstructions to K-stability, but except in the case of curves it
is not equivalent to K-stability.171717However in that case it essentially
provides the first purely algebraic proof of the K-stability of smooth curves,
which can only be slope K-destabilised by singular points, therefore allowing
the uniformization theorem to take on an interpretation as an instance of 1.1.
Indeed Panov–Ross showed that $\operatorname{Bl}_{p,q}\mathbb{CP}^{2}$ is
slope K-stable but not K-stable [PR09, Ex. 7.8]. The notion of slope
K-stability considers test configurations arising from the deformation to the
normal cone of a subscheme $Z\subset X$ (see [RT06, §4]), but to get a theory
equivalent to K-stability one must instead consider _flags_ of subschemes.
This was already noted by Mumford [Mum77] and has been discussed in detail by
Ross–Thomas [RT07, §3]. We will discuss certain test configurations arising
out of flags of subschemes in the study of stability of fibrations in Part II.

######  Remark 2.3.5.

The definition of K-stability is expected to need modification in the case
where $(X,L)$ is not Fano in order to be strong enough to imply the existence
of a constant scalar curvature Kähler metric. A modification was proposed by
Székelyhidi based on ideas of Witt-Nyström [Szé15, WN12], known as _filtration
K-stability_. This notion has been expanded upon greatly in the wake of work
by Berman–Boucksom–Jonsson (see [BBJ21]) on a variational approach to
K-stability, and the exciting work of Li on the YTD conjecture for polarised
varieties [Li20].

#####  2.3.2 Canonical Kähler metrics

Repetitiously, guided by 1.1 we now introduce the corresponding notion of
extremal object to K-stability. Indeed K-stability arose directly from its
relationship to the existence of canonical Kähler metrics, and we emphasise
Donaldson’s point of view on _constant scalar curvature Kähler metrics_. The
Ricci form associated to a Kähler manifold $(X,\omega)$ is given by

$\operatorname{Ric}(\omega)=-\frac{i}{2\pi}\partial\overline{\partial}\log\det\omega$

and the scalar curvature by

$S(\omega)=\Lambda_{\omega}\operatorname{Ric}(\omega).$

The average value of $S$ is determined topologically and given by

$\hat{S}=\frac{nc_{1}(X).[\omega]^{n-1}}{[\omega]^{n}}=n\mu(X,L)$

where $[\omega]=c_{1}(L)$ if $\omega$ is integral and a polarising ample line
bundle has been chosen.

######  Definition 2.3.6.

A Kähler metric is _Kähler–Einstein_ if

$\operatorname{Ric}(\omega)=\lambda\omega$

for some $\lambda\in\mathbb{R}$. It is _constant scalar curvature Kähler_
(cscK) if

$S(\omega)=\hat{S}.$

It was proposed by Calabi to search for examples of canonical Kähler metrics
satisfying the above property [Cal82], which ultimately lead to Calabi’s
notion of an extremal metric (which we will not comment on further). The
instance of 1.1 in this setting is a conjecture, proposed for Fano manifolds
by Yau and Tian, and for polarised manifolds by Donaldson (although the idea
that such a correspondence should hold in this generality was already implicit
in the work of Fujiki [Fuj92]).

######  Conjecture 2.3.7 (Yau–Tian–Donaldson).

A smooth polarised variety $(X,L)$ is K-polystable if and only if it admits a
constant scalar curvature Kähler metric $\omega\in c_{1}(L)$.

The case of Fano manifolds, where a cscK metric is Kähler–Einstein, is known
in the smooth case due to Chen–Donaldson–Sun [CDS15a, CDS15b, CDS15c], with
later proofs provided by Tian, and using a variety of different approaches by
Datar–Székelyhidi, Chen–Sun–Wang, and Berman–Boucksom–Jonsson [Tia15, DS16,
CSW18, BBJ21]. There are now versions of the YTD conjecture known for singular
Fanos in great generality due to the work of Li–Tian–Wang, Li, and
Liu–Xu–Zhuang, whose work combines to show any log Fano pair $(X,\Delta)$
admits a weak Kähler–Einstein metric if and only if it is K-polystable [LTW21,
Li22, LXZ22].

Let us complete our short survey of 1.1 for varieties by emphasising the
importance of the Kempf–Ness-type functional in this setting also. Here the
functional was introduced by Mabuchi, and is otherwise known as the K-energy
[Mab86]. As usual, we specify the functional by its first variation. If
$\omega$ is a Kähler metric and
$\omega_{\varphi}=\omega+i\partial\overline{\partial}\varphi$ is any metric in
the same Kähler class, then the Mabuchi functional is the unique functional
$\mathcal{M}$ on the space of Kähler metrics in the class $[\omega]$ such that
$\mathcal{M}(\omega)=0$ and

$\frac{d}{dt}\mathcal{M}(\varphi_{t})=\int_{X}\dot{\varphi}_{t}(\hat{S}-S(\omega_{t}))\omega_{t}^{n}.$

By construction, a critical point of the Mabuchi functional is necessarily a
cscK metric. A version of Lemma 2.1.6 for the Mabuchi functional on a Fano
manifold, computing the limit slope as the Donaldson–Futaki invariant of a
test configuration corresponding to a geodesic ray, is now understood well in
terms of the variational framework (see [BHJ19] and the references therein). A
proof of the YTD conjecture utilising this functional, which follows the
approach of the original Kempf–Ness theorem, was subsequently carried out by
Berman–Boucksom–Jonsson [BBJ21]. There is now a great body of work proving the
analogous properties to the Kempf–Ness functional for arbitrary polarised
manifolds, particularly by Chen and Cheng [CC21a, CC21b].

#####  2.3.3 Deformations of Fano threefolds with
$\operatorname{SL}(2,\mathbb{C})$ actions

In this section we will briefly describe the explicit K-stability problem for
the Mukai–Umemura threefold as exposited by Donaldson [Don08, §5], and
describe a new, singular threefold with large automorphism group for which the
same techniques may apply. This singular threefold has not appeared in the
literature, and we suggest how its deformation theory may interact with
K-stability.

We will make use of the Mukai–Umemura threefold to construct examples of
isotrivial fibrations associated to principal
$\operatorname{SL}(2,\mathbb{C})$-bundles which admit optimal symplectic
connections in Chapter 7.

These threefolds with automorphism group $\operatorname{SL}(2,\mathbb{C})$ are
constructed as common zero-loci of sections of bundles over Grassmannians, and
we will describe three such examples. In order to find threefolds with
automorphism group $G=\operatorname{SL}(2,\mathbb{C})$, we make use of the
basic representation theory of $\operatorname{SL}(2,\mathbb{C})$ which we now
recall.

######  Theorem 2.3.8 (See for example [FH91, Ch. 11]).

For each non-negative integer $k\in\mathbb{Z}_{\geq 0}$, there exists a
unique, irreducible, finite-dimensional representation of
$\operatorname{SL}(2,\mathbb{C})$ of dimension $k+1$, denoted $s^{k}$. This
representation can be explicitly described as

$s^{k}=\mathbb{C}[X,Y]_{k},$

the homogeneous polynomials of degree $k$ in two variables acted upon in the
standard way.

######  Lemma 2.3.9.

Given two irreducible representations $s^{k},s^{l}$ of
$\operatorname{SL}(2,\mathbb{C})$ with $k\geq l$, we have decompositions

$\displaystyle s^{k}\otimes s^{l}$ $\displaystyle\cong s^{k+l}\oplus
s^{k+l-2}\oplus\cdots\oplus s^{k-l+2}\oplus s^{k-l},$
$\displaystyle\operatorname{Sym}^{2}s^{k}$ $\displaystyle\cong s^{2k}\oplus
s^{2k-4}\oplus\cdots\oplus s^{4}\oplus s^{0},$
$\displaystyle\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}s^{k}$
$\displaystyle\cong s^{2k-2}\oplus s^{2k-6}\oplus\cdots\oplus s^{6}\oplus
s^{2}.$

These decompositions can be obtained by analysing the highest weights of the
tensor products of irreducible representations, and using the fact that each
irreducible representation $s^{k}$ of $\operatorname{SL}(2,\mathbb{C})$ is
acted on with highest weight $k$. One may also use the Cayley–Sylvester
formula by viewing $s^{k}$ as the symmetric product
$\operatorname{Sym}^{k}s^{1}$ of the standard representation
$s^{1}=\mathbb{C}^{2}\cong\mathbb{C}[X,Y]_{1}.$

######  2.3.3.1 Mukai–Umemura threefold

In order to construct the Mukai–Umemura threefold as a subset of a
Grassmannian, one considers the 7-dimensional vector space $V=s^{6}$, and the
Grassmannian $\operatorname{Gr}_{3}(V)$ of 3-planes in $V$. This is
$3\cdot(7-3)=12$ dimensional. Begin by choosing a 3-dimensional subspace
$\Pi\subset\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}V^{*}$,
and define

$X_{\Pi}:=\\{P\in\operatorname{Gr}_{3}(V)\mid\left.\omega\right|_{P}=0\text{
for all }\omega\in\Pi\\}.$

We can alternatively view $X_{\Pi}$ as a zero locus of sections of bundles on
the Grassmannian as follows. Denote by

$\mathcal{V}\to\operatorname{Gr}_{3}(V)$

the tautological bundle of 3-planes over $\operatorname{Gr}_{3}(V)$, and let
$\omega_{1},\omega_{2},\omega_{3}$ be a basis of
$\Pi\subset\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}V^{*}$.
Then define sections $\sigma_{1},\sigma_{2},\sigma_{3}$ of
$\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}\mathcal{V}^{*}$
by $\sigma_{i}(P):=\left.\omega_{i}\right|_{P}$ and observe

$X_{\Pi}=Z(\sigma_{1})\cap Z(\sigma_{2})\cap
Z(\sigma_{3})\subset\operatorname{Gr}_{3}(V).$

The resulting variety only depends on the three-plane $\Pi$, and a dimension
count shows $X_{\Pi}$ is a threefold for a generic choice of $\Pi$. Indeed by
perturbing the 2-forms $\omega_{i}$ we have a Zariski-open subspace
$\mathcal{U}\subset\operatorname{Gr}_{3}(\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}V^{*})$
of 3-planes $\Pi$ such that $X_{\Pi}$ is a smooth threefold, and a result of
Mukai shows that the inequivalent 3-planes lie in separate orbits of the
action of $\operatorname{SL}(V)$ on $\mathcal{U}$. Therefore the possible such
threefolds $X_{\Pi}$ are classified by the quotient space

$\mathcal{M}=\mathcal{U}/\operatorname{SL}(V).$

######  Lemma 2.3.10.

The threefold $X_{\Pi}$ is Fano for any $\Pi\in\mathcal{U}$.

###### Proof.

Let
$H=\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}\mathcal{V}^{*}$.
Then $H=\mathcal{V}\otimes L$ where $L=\det\mathcal{V}^{*}$. Therefore we have

$\det H=L^{\otimes 2}.$

The tangent bundle of $\operatorname{Gr}_{3}(V)$ is given

$T\operatorname{Gr}_{3}(V)=\mathcal{V}^{*}\otimes((V\otimes\mathcal{O})/\mathcal{V}),$

so we see $\det T\operatorname{Gr}_{3}(V)=L^{\otimes 7}$. Given some
$\Pi\in\mathcal{U}$, the tangent bundle to $X_{\Pi}$ is the kernel of a
surjective map $T\operatorname{Gr}_{3}(V)\to H\oplus H\oplus H$. Therefore we
obtain

$\det TX_{\Pi}\cong L^{\otimes 7}\otimes 3L^{\otimes-2}\cong L$

so $K_{X_{\Pi}}^{-1}\cong L>0$ is ample. ∎

The Mukai–Umemura threefold arises as a particular choice of $\Pi$. Namely if
$V=s^{6}$ then $V^{*}\cong V$ since the 7-dimensional irreducible
representation of $\operatorname{SL}(2,\mathbb{C})$ is unique, and
consequently

$\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}(V^{*})=s^{10}\oplus
s^{6}\oplus s^{2}.$

The $s^{2}$ summand defines a 3-plane $\Pi_{0}$ in
$\operatorname{Gr}_{3}(\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}V^{*})$,
and the corresponding threefold $X_{0}=X_{\Pi_{0}}$ is the _Mukai–Umemura
threefold_. It was shown by Mukai–Umemura that $X_{0}$ is non-singular, or in
other words that
$s^{2}\in\mathcal{U}\subset\operatorname{Gr}_{3}(\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}V^{*})$
[MU83, §2]. Since $\Pi_{0}$ is $\operatorname{SL}(2,\mathbb{C})$-invariant,
one obtains a natural $\operatorname{SL}(2,\mathbb{C})$ action on $X_{0}$.

Donaldson computed explicitly Tian’s alpha invariant for the Mukai–Umemura
threefold, which is a criterion implying the existence of a Kähler–Einstein
metric on a smooth Fano manifold [Tia87]. Indeed one obtains:

######  Proposition 2.3.11 ([Don08]).

The Mukai–Umemura threefold $X_{0}$ is smooth and admits a Kähler–Einstein
metric. In particular it is K-polystable.

Further to this, one has an $\operatorname{SL}(2,\mathbb{C})$-equivariant
embedding

$X_{0}\hookrightarrow\mathbb{P}(H^{0}(X_{0},K_{X_{0}}^{-1}))$

where $H^{0}(X_{0},K_{X_{0}}^{-1})\cong s^{0}\oplus s^{12}.$ The divisor at
$\infty$ of $X_{0}$ is given by the zero set of the section $\sigma\in
H^{0}(X_{0},K_{X_{0}}^{-1})$ generating the $s^{0}\cong\mathbb{C}$ factor. The
geometry of $X_{0}$ can be explicitly understood in terms of this divisor, and
for example one may observe that $X_{0}\backslash Z(\sigma)$ is isomorphic to
$\operatorname{SL}(2,\mathbb{C})/\Gamma$ where $\Gamma$ is the group of
symmetries of an icosahedron, and $X_{0}$ is therefore a natural
compactification of this space.

Furthermore the deformation theory of $X_{0}$ can be computed in terms of the
representation theory of $\operatorname{SL}(2,\mathbb{C})$. Recall that the
variety $X_{0}$ is identified with a point
$\Pi_{0}\in\mathcal{U}\subset\operatorname{Gr}_{3}(\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}V^{*})$.
The versal deformation space at
$[\Pi_{0}]\in\mathcal{M}=\mathcal{U}/\operatorname{SL}(V)$ can be computed in
the following way.181818By a result of Mukai all smooth Fano threefolds of
rank 1 and degree 22 appear as sections of the Grassmannian and are
parametrised by $\mathcal{U}$, so $\mathcal{M}$ is a representative for the
moduli of Fanos in this deformation family. We have
$\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}V^{*}=s^{10}\oplus
s^{6}\oplus s^{2}$ so

$T_{\Pi_{0}}\operatorname{Gr}_{3}(\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}V^{*})=(s^{10}\oplus
s^{6})\otimes s^{2}=s^{12}\oplus s^{10}\oplus 2s^{8}\oplus s^{6}\oplus s^{4}.$

There is an action of $\operatorname{SL}(V)=\operatorname{SL}(s^{6})$ on
$\operatorname{Gr}_{3}(\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}V^{*})$
whose kernel is given by the stabiliser at $\Pi_{0}=s^{2}$, which is
$\operatorname{SL}(2,\mathbb{C})$. The Lie algebra of
$\operatorname{GL}(s^{6})\cong\operatorname{End}(s^{6})\cong s^{6}\otimes
s^{6}$ is given by

$\mathfrak{gl}(s^{6})=s^{6}\otimes s^{6}=s^{12}\oplus s^{10}\oplus s^{8}\oplus
s^{6}\oplus s^{4}\oplus s^{2}\oplus s^{0}$

and therefore $\mathfrak{sl}(s^{6})\cong s^{12}\oplus\cdots\oplus s^{2}$ after
quotienting out the factor $s^{0}$ generating
$\mathbb{C}^{*}\cdot{\boldsymbol{1}}\in\operatorname{GL}(s^{6})$. The induced
action on $T_{\Pi_{0}}\operatorname{Gr}_{3}(s^{6})$ does not see the
stabiliser $s^{2}$ factor. One therefore obtains that the Zariski tangent
space $T_{[\Pi_{0}]}\mathcal{M}$ is given by the quotient of the morphism

$\frac{\mathfrak{sl}(s^{6})}{s^{2}}\to
T_{\Pi_{0}}\operatorname{Gr}_{3}(s^{6})$

which by the above computation is

$T_{[\Pi_{0}]}\mathcal{M}=s^{8}.$

Note that the generic expected dimension of the smooth locus of $\mathcal{M}$
is 6, and the point $[\Pi_{0}]\in\mathcal{M}$ is expected to be a singular
point, corresponding to the fact that $X_{0}$ has 3-dimensional automorphisms
in comparison to the generic member of $\mathcal{M}$ which has trivial
automorphisms (and so the Zariski tangent space is 6+3=9 dimensional, as
computed).

The GIT stability of points in a representation $s^{k}$ of
$\operatorname{SL}(2,\mathbb{C})$ is well-understood and can be phrased in
terms of the zeroes of the associated homogenous polynomials on
$\mathbb{CP}^{1}$ (see for example [Tho06]). This lead to the enticing
conjecture of Donaldson relating this GIT stability to the K-stability of
deformations of $X_{0}$ in $\mathcal{U}$ [Don08]. This conjecture was resolved
due to the work of Székelyhidi and Brönnle [Szé10, Brö11], whose work applies
in far greater generality to deformations of cscK manifolds with
automorphisms.

######  Theorem 2.3.12.

A small deformation $X_{\Pi}$ of $X_{0}$ in $\mathcal{U}$ is
K-(semi/poly)stable if and only if the associated point $v_{\Pi}\in
T_{[\Pi_{0}]}\mathcal{M}$ is GIT (semi/poly)stable for the action of
$\operatorname{SL}(2,\mathbb{C})=\operatorname{Aut}(X_{\Pi_{0}},L)$ on
$T_{[\Pi_{0}]}\mathcal{M}$.

Computations of Tian show that manifolds arising from GIT unstable or strictly
GIT semistable orbits above are not Kähler–Einstein [Tia87]. Deformations of
$X_{0}$ will break the $\operatorname{SL}(2,\mathbb{C})$ action, but some may
preserve the $\mathbb{C}^{*}$ action, and there is a one-parameter family
$X(\tau)$ of such threefolds near $X_{0}$, with $X_{0}$ corresponding to
$\tau=1$. Donaldson made a refined conjecture in this case [Don18, §4.1] where
K-polystability should be characterised by properties of the singularities of
the divisor, and Cheltsov–Shramov have resolved this conjecture showing that
$X(\tau)$ is K-polystable except for a small number of values of $\tau$
[CS18].

######  2.3.3.2 $V_{5}$ manifold

One can construct another example of a smooth Fano threefold with an action of
$\operatorname{SL}(2,\mathbb{C})$ as a zero-locus of sections inside a
Grassmannian, this time in the deformation family $V_{5}$. In this case one
uses the irreducible representation $V=s^{4}$ and considers a triple
intersection in $\operatorname{Gr}_{2}(V)$, which is
$2\cdot(5-2)=6$-dimensional. Indeed if
$\mathcal{V}\to\operatorname{Gr}_{2}(V)$ is the tautological bundle, then
$\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}\mathcal{V}^{*}$
is a line bundle and a triple of sections defined by a plane
$\Pi\subset\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}V^{*}$
produces a Fano threefold, and a similar formal theory exists as in the case
of $V_{22}$ above and the Mukai–Umemura threefold.

Indeed one may compute

$\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}V^{*}=s^{6}\oplus
s^{2}$

and setting $\Pi_{0}=s^{2}$ one obtains a smooth Fano threefold $Y_{0}$ with
$\operatorname{SL}(2,\mathbb{C})$ action. The algebraic geometry of the
manifold $Y_{0}$ has been described in [PCS19, §6] [San14]. In particular the
alpha invariant of $Y_{0}$ can be computed and is greater than $\frac{5}{6}$,
so $Y_{0}$ admits a Kähler–Einstein metric and is K-polystable [CS09].

However a computation of the versal deformation space for $Y_{0}$ reveals
$\mathcal{M}=\\{Y_{0}\\}$ and $Y_{0}$ is rigid. Therefore the same deformation
picture which exists for the Mukai–Umemura threefold $X_{0}$ cannot be studied
for $Y_{0}$.

######  2.3.3.3 A singular threefold in the class $V_{14}$

There is a third deformation family of Fano threefolds which may be
constructed as zeroes of sections of a Grassmannian in a straight-forward way,
class $V_{14}$. Here one takes 5-fold intersections of forms in
$\operatorname{Gr}_{2}(\mathbb{C}^{6})$. There are 15 dimensions of local
moduli of such smooth Fano threefolds. Their K-stability is completely
understood.

######  Theorem 2.3.13 ([ACC+21, Ex. 4.1.7]).

Every smooth Fano threefold in the deformation class $V_{14}$ has discrete
automorphism group and is K-stable.

We will now describe a _singular_ threefold in this deformation class with
large automorphism group. Consider now $V=s^{5}$ the 6-dimensional irreducible
representation of $\operatorname{SL}(2,\mathbb{C})$. As usual let
$\mathcal{V}\to\operatorname{Gr}_{2}(V)$ denote the tautological bundle, and
consider a quintuple of sections of
$\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}\mathcal{V}^{*}$
defined by a 5-plane
$\Pi\subset\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}V^{*}$.
We have

$\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}s^{5}=s^{8}\oplus
s^{4}\oplus s^{0}$

and take
$\Pi_{0}:=s^{4}\subset\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}s^{5}$.
Then set

$Z_{0}:=Z_{\Pi_{0}}\subset\operatorname{Gr}_{2}(V).$

Since $\Pi_{0}$ is $\operatorname{SL}(2,\mathbb{C})$-invariant, $Z_{0}$ comes
with an $\operatorname{SL}(2,\mathbb{C})$ action. Unlike in the cases of the
Mukai–Umemura threefold and $V_{5}$ manifold (which can be shown to be smooth
due to explicit descriptions as compactifications of
$\operatorname{SL}(2,\mathbb{C})/\Gamma$ for $\Gamma$ a finite subgroup), the
above classification of smooth $V_{14}$ manifolds shows that $Z_{0}$ must be
singular.191919It would be interesting to understand if $Z_{0}$ may be
identified as a singular compactification of
$\operatorname{SL}(2,\mathbb{C})/\Gamma$ for some finite subgroup $\Gamma$.
The subgroups admitting smooth compactifications were identified by Mukai and
Umemura. Indeed we will explicitly describe $Z_{0}$ and identify two singular
points.

######  Remark 2.3.14.

Whilst we cannot prove $Z_{0}$ itself is Fano, we expect this to be the case.
A similar argument to the smooth case of $V_{5}$ or $V_{22}$ shows that the
ample bundle $\det\mathcal{V}^{*}$ is generically isomorphic to
$K_{Z_{0}}^{-1}$, so what remains is a more detailed understanding of how the
singular locus of $Z_{0}$ changes its anticanonical bundle. Nevertheless
$Z_{0}$ is a degeneration of smooth Fano threefolds, and therefore has nef
anticanonical bundle.

Consider the Plücker embedding

$\operatorname{Gr}_{2}(V)\hookrightarrow\mathbb{P}(\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}V).$

Under the canonical identification
$\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}(s^{5})^{*}\cong\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}s^{5}$
the $s^{4}$-summand is sent to itself. Indeed $Z_{0}$ can be identified with
the 2-planes $P\in\operatorname{Gr}_{2}(V)$ which are orthogonal to
$s^{4}\subset\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}s^{5}$.
Thus the Plücker embedding of $\operatorname{Gr}_{2}(s^{5})$ restricts to an
$\operatorname{SL}(2,\mathbb{C})$-equivariant embedding

$Z\hookrightarrow\mathbb{P}(s^{0}\oplus s^{8}).$

The $s^{0}$ summand defines an $\operatorname{SL}(2,\mathbb{C})$-invariant
divisor $D$ at $\infty$ of $Z_{0}\subset\mathbb{P}(s^{8}\oplus s^{8})$
corresponding to the intersection
$D=Z_{0}\cap\mathbb{P}(s^{8})\subset\mathbb{P}(s^{0}\oplus s^{8})$. This
summand is generated by a section of the ample line bundle
$\det\mathcal{V}^{*}$ restricted to $Z_{0}$, which we will explicitly
identify.

Let us now explicitly describe the embedding. First we will describe the
representations of $\operatorname{SL}(2,\mathbb{C})$ explicitly. Fix a basis
$x^{i}y^{5-i}$ of $s^{5}=\mathbb{C}[x,y]_{5}$. Then the generators of
$\mathfrak{sl}(2)$ are given by

$X=x\frac{\partial}{\partial y},\quad Y=y\frac{\partial}{\partial x},\quad
H=[X,Y]=x\frac{\partial}{\partial x}-y\frac{\partial}{\partial y}.$

The subspace $s^{4}\subset s^{8}\oplus s^{4}\oplus
s^{0}=\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}s^{5}$
is a 5-dimensional space spanned by the forms

$x^{2}y^{3}\wedge x^{5}-3x^{3}y^{2}\wedge x^{4}y,\quad 2xy^{4}\wedge
x^{5}-4x^{2}y^{3}\wedge x^{4}y,\quad y^{5}\wedge x^{5}+xy^{4}\wedge
x^{4}y-8x^{2}y^{3}\wedge x^{3}y^{2},\\\ 3y^{5}\wedge x^{4}y-6xy^{4}\wedge
x^{3}y^{2},\quad y^{5}\wedge x^{3}y^{2}-3xy^{4}\wedge x^{2}y^{3}.$

The 10-dimensional complement $s^{8}\oplus s^{0}$ is spanned by the forms

$x^{4}y\wedge x^{5},\quad x^{3}y^{2}\wedge x^{5},\quad 3x^{2}y^{3}\wedge
x^{5}+5x^{3}y^{2}\wedge x^{4}y,\\\ xy^{4}\wedge x^{5}+5x^{2}y^{3}\wedge
x^{4}y,\quad y^{5}\wedge x^{5}+15xy^{4}\wedge x^{4}y+20x^{2}y^{3}\wedge
x^{3}y^{2},\\\ y^{5}\wedge x^{5}-5xy^{4}\wedge x^{4}y+10x^{2}y^{3}\wedge
x^{3}y^{2},\quad y^{5}\wedge x^{4}y+5xy^{4}\wedge x^{3}y^{2},\quad
3y^{5}\wedge x^{3}y^{2}+5xy^{4}\wedge x^{2}y^{3},\\\ y^{5}\wedge
x^{2}y^{3},\quad y^{5}\wedge xy^{4},$ (Eq. 2.10)

where the irreducible component $s^{0}$ is spanned by the form

$\tilde{\sigma}=y^{5}\wedge x^{5}-5xy^{5}\wedge x^{4}y+10x^{2}y^{3}\wedge
x^{3}y^{2}.$

Thus the invariant section $\sigma$ can be identified with the induced section
of the ample line bundle
$\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}\mathcal{V}^{*}$
from the form $\tilde{\sigma}$ defined by sending

$P\mapsto\left.\tilde{\sigma}\right|_{P}$

for all $P\in\operatorname{Gr}_{2}(s^{5})$ such that
$\left.\omega\right|_{P}=0$ for all $\omega$ in the span of $s^{4}$. The
points of $Z_{0}\subset\mathbb{P}(s^{0}\oplus s^{8})$ can be identified with
the elements $P=a\wedge b$ for $a,b,\in s^{5}$ with are orthogonal to every
2-form in $\Pi_{0}$. In particular we may single out four points which all lie
along the $\operatorname{SL}(2,\mathbb{C})$-invariant divisor
$D=Z(\sigma)=\mathbb{P}(s^{8})\cap Z_{0}\subset Z_{0}$ at $\infty$ of $Z_{0}$:

$x^{4}y\wedge x^{5},\quad x^{3}y^{2}\wedge x^{5},\quad y^{5}\wedge
x^{2}y^{3},\quad y^{5}\wedge xy^{4}.$ (Eq. 2.11)

We may now translate these representations into an explicit embedding
$Z_{0}\hookrightarrow\operatorname{Gr}_{2}(s^{5})\hookrightarrow\mathbb{P}(\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}s^{5}))$
of the variety $Z_{0}$. Let us change notation slightly to denote $x^{5}\in
s^{5}$ by $e_{1}$, $x^{4}y=e_{2},\dots,y^{5}=e_{6}$. Then if $W=a\wedge
b\in\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}s^{5}$
is a 2-plane in $s^{5}$ with $a=\sum a_{i}e_{i},b=\sum b_{i}e_{i}$, the
condition that $W$ lies in $Z_{0}$ is given by the following relations:

$\displaystyle(a_{4}b_{1}-a_{1}b_{4})-3(a_{3}b_{2}-a_{2}b_{3})$
$\displaystyle=0$ $\displaystyle
2(a_{5}b_{1}-a_{1}b_{5})-4(a_{4}b_{2}-a_{2}b_{4})$ $\displaystyle=0$
$\displaystyle(a_{6}b_{1}-a_{1}b_{6})+(a_{5}b_{2}-a_{2}b_{5})-8(a_{4}b_{3}-a_{3}b_{4})$
$\displaystyle=0$ $\displaystyle
3(a_{6}b_{2}-a_{2}b_{6})-6(a_{5}b_{3}-a_{3}b_{5})$ $\displaystyle=0$
$\displaystyle(a_{6}b_{3}-a_{3}b_{6})-3(a_{5}b_{4}-a_{4}b_{5})$
$\displaystyle=0.$

In Plücker coordinates for $W$ these relations become

$\displaystyle-W_{14}+3W_{23}$ $\displaystyle=0$
$\displaystyle-2W_{15}+4W_{24}$ $\displaystyle=0$ $\displaystyle-
W_{16}-W_{25}+8W_{34}$ $\displaystyle=0$ $\displaystyle-3W_{26}+6W_{35}$
$\displaystyle=0$ $\displaystyle-W_{36}+3W_{45}$ $\displaystyle=0.$

Combined with the Plücker relations, this produces an explicit representation
of $Z_{0}$ embedded in
$\mathbb{P}(\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}s^{5})$
as the intersection of the Grassmannian $\operatorname{Gr}_{2}(s^{5})$ with 5
hyperplanes in
$\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}s^{5}$.

######  Remark 2.3.15.

At this point it becomes clear that despite the non-genericity of
$s^{4}\subset\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}V$
(e.g. $Z_{0}$ is singular), the variety $Z_{0}$ has the expected dimension 3,
being the intersection of the 8-dimensional
$\operatorname{Gr}_{2}(\mathbb{C}^{6})$ with 5 hyperplanes.

Let us now describe the $\operatorname{SL}(2,\mathbb{C})$-equivariant
embedding $Z_{0}\hookrightarrow\mathbb{P}(s^{8}\oplus s^{0})$. The variety
$Z_{0}$ can be identified with the locus of totally decomposable two-forms
$a\wedge b\in s^{8}\oplus s^{0}$. Using the basis of 2.10 we can distil the
embedding of $Z_{0}$ from the above embedding as follows. Let $\alpha\in
s^{8}\oplus s^{0}$ be a two-form. Then $\alpha$ is specified as a vector

$\alpha=\alpha_{i}v_{i}$

where $v_{1},\dots,v_{10}$ are the basis vectors listed in 2.10. Such a vector
is totally decomposable if it satisfies the Plücker relations for
$\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{2}s^{5}$.
We express the basis of $s^{8}\oplus s^{0}$ in terms of the standard basis of
$s^{5}$:

$e_{2}\wedge e_{1},\quad e_{3}\wedge e_{1},\quad 3e_{4}\wedge
e_{1}+5e_{3}\wedge e_{2},\\\ e_{5}\wedge e_{1}+5e_{4}\wedge e_{2},\quad
e_{6}\wedge e_{1}+15e_{5}\wedge e_{2}+20e_{4}\wedge e_{3},\\\ e_{6}\wedge
e_{1}-5e_{5}\wedge e_{2}+10e_{4}\wedge e_{3},\quad e_{6}\wedge
e_{2}+5e_{5}\wedge e_{3},\quad 3e_{6}\wedge e_{3}+5e_{5}\wedge e_{4},\\\
e_{6}\wedge e_{4},\quad e_{6}\wedge e_{5},$ (Eq. 2.12)

Then the element $\alpha$ has Plücker coordinates given by

$\displaystyle W_{12}$ $\displaystyle=-\alpha_{1}$ $\displaystyle W_{13}$
$\displaystyle=-\alpha_{2}$ $\displaystyle W_{14}$
$\displaystyle=-3\alpha_{3}$ $\displaystyle W_{15}$
$\displaystyle=-\alpha_{3}$ $\displaystyle W_{16}$
$\displaystyle=-\alpha_{5}-\alpha_{6}$ $\displaystyle W_{23}$
$\displaystyle=-5\alpha_{3}$ $\displaystyle W_{24}$
$\displaystyle=-5\alpha_{4}$ $\displaystyle W_{25}$
$\displaystyle=-15\alpha_{5}+5\alpha_{6}$ $\displaystyle W_{26}$
$\displaystyle=-\alpha_{7}$ $\displaystyle W_{34}$
$\displaystyle=-20\alpha_{5}-10\alpha_{6}$ $\displaystyle W_{35}$
$\displaystyle=-5\alpha_{7}$ $\displaystyle W_{36}$
$\displaystyle=-3\alpha_{8}$ $\displaystyle W_{45}$
$\displaystyle=-5\alpha_{8}$ $\displaystyle W_{46}$
$\displaystyle=-\alpha_{9}$ $\displaystyle W_{56}$
$\displaystyle=-\alpha_{10}.$

Using the Plücker relations gives the homogeneous ideal defining $Z_{0}$

$\alpha_{8}^{2}-5\,\alpha_{7}\alpha_{9}+20\,\alpha_{5}\alpha_{10}+10\,\alpha_{6}\alpha_{10},\\\
\,5\,\alpha_{7}\alpha_{8}-15\,\alpha_{5}\alpha_{9}+5\,\alpha_{6}\alpha_{9}+5\,\alpha_{4}\alpha_{10},\\\
\,5\,\alpha_{7}^{2}-45\,\alpha_{5}\alpha_{8}+15\,\alpha_{6}\alpha_{8}+5\,\alpha_{3}\alpha_{10},\\\
\,20\,\alpha_{5}\alpha_{7}+10\,\alpha_{6}\alpha_{7}-15\,\alpha_{4}\alpha_{8}+5\,\alpha_{3}\alpha_{9},\\\
\,5\,\alpha_{5}\alpha_{8}+5\,\alpha_{6}\alpha_{8}-\alpha_{3}\alpha_{9}+3\,\alpha_{3}\alpha_{10},\\\
\,5\,\alpha_{5}\alpha_{7}+5\,\alpha_{6}\alpha_{7}-3\,\alpha_{3}\alpha_{8}+\alpha_{2}\alpha_{10},\\\
\,15\,\alpha_{5}^{2}+10\,\alpha_{5}\alpha_{6}-5\,\alpha_{6}^{2}-\alpha_{3}\alpha_{7}+\alpha_{1}\alpha_{10},\\\
\,20\,\alpha_{5}^{2}+30\,\alpha_{5}\alpha_{6}+10\,\alpha_{6}^{2}-9\,\alpha_{3}\alpha_{8}+\alpha_{2}\alpha_{9},\\\