#  Weighted projective embeddings, stability of orbifolds and constant scalar
curvature Kähler metrics

Julius Ross and Richard Thomas

###### Abstract

We embed polarised orbifolds with cyclic stabiliser groups into weighted
projective space via a weighted form of Kodaira embedding. Dividing by the
(non-reductive) automorphisms of weighted projective space then formally gives
a moduli space of orbifolds. We show how to express this as a _reductive_
quotient and so a GIT problem, thus defining a notion of stability for
orbifolds.

We then prove an orbifold version of Donaldson’s theorem: the existence of an
orbifold Kähler metric of constant scalar curvature implies K-semistability.

By extending the notion of slope stability to orbifolds we therefore get an
explicit obstruction to the existence of constant scalar curvature orbifold
Kähler metrics. We describe the manifold applications of this orbifold result,
and show how many previously known results (Troyanov, Ghigi-Kollár, Rollin-
Singer, the AdS/CFT Sasaki-Einstein obstructions of Gauntlett-Martelli-Sparks-
Yau) fit into this framework.

######

  1. 0 Introduction
    1. 1 Extensions
  2. 1 Orbifold embeddings in weighted projective space
    1. 1 Orbibasics
    2. 2 Codimension one stabilisers
    3. 3 Weighted projective spaces
    4. 4 Orbifold line bundles and $\mathbb{Q}$-divisors
    5. 5 Orbifold polarisations
    6. 6 Orbifold Kodaira embedding
    7. 7 OrbiProj
    8. 8 Orbifold Riemann-Roch
      1. 1 Equivariant case
    9. 9 Reducing to the reductive quotient
  3. 2 Metrics and balanced orbifolds
  4. 3 Limits of Fubini-Study metrics
  5. 4 Limits of balanced metrics
  6. 5 K-stability as an obstruction to orbifold cscK metrics
    1. 1 Definition of orbifold K-stability
    2. 2 Orbifold version of Donaldson’s theorem
  7. 6 Slope stability of orbifolds
  8. 7 Applications and further examples
    1. 1 Orbifold Riemann surfaces
    2. 2 Index obstruction to stability
    3. 3 Orbifold ruled surfaces
    4. 4 Slope stability of canonically polarised orbifolds

*

##  Chapter 0 Introduction

The problem of finding canonical Kähler metrics on complex manifolds is
central in Kähler geometry. Much of the recent work in this area centres
around the conjecture of Yau, Tian and Donaldson that the existence of a
constant scalar curvature Kähler (cscK) metric should be equivalent to an
algebro-geometric notion of stability. This notion, called “K-stability”,
should be understood roughly as follows. Suppose we are looking for such a
metric on $X$ whose Kähler form lies in the first Chern class of an ample line
bundle $L$. Then using sections of $L^{k}$ one can embed $X$ in a large
projective space $\mathbb{P}^{N_{k}}$ for $k\gg 0$, and stability is taken in
a Geometric Invariant Theory (GIT) sense with respect to the automorphisms of
these projective spaces as $k\to\infty$. By the Hilbert-Mumford criterion this
in turn can be viewed as a statement about numerical invariants coming from
one-parameter degenerations of $X$. The connection with metrics is through the
Kempf-Ness theorem, that a stable orbit contains a zero of the moment map.
Here this says that a (Chow) stable $X$ can be moved by an automorphism of
$\mathbb{P}^{N_{k}}$ to be _balanced_ , and then the restriction of the
Fubini-Study metric on $\mathbb{P}^{N_{k}}$ approximates a cscK metric for
$k\gg 0$.

In this paper we formulate and study a Yau-Tian-Donaldson correspondence for
orbifolds. On the algebro-geometric side this involves orbifold line bundles,
embeddings in weighted projective space, and a notion of stability for
orbifolds. This is related in differential geometry to orbifold Kähler metrics
(those which pull back to a genuine Kähler metric upstairs in an orbifold
chart; downstairs these are Kähler metrics with cone angles $2\pi/m$ about
divisors with stabiliser group $\mathbb{Z}/m\mathbb{Z}$) and their scalar
curvature. So we restrict to the case of orbifolds with cyclic quotient
singularities, but importantly we do allow the possibility of orbifold
structure in codimension one.

Our motivation is not the study of orbifolds per se, but their applications to
manifolds. Orbifold metrics are often the starting point for constructions of
metrics on manifolds (see for instance GK , and the gluing construction of RS
) or arise naturally as quotients of manifolds (for instance quasi-regular
Sasaki-Einstein metrics on odd dimensional manifolds correspond to orbifold
Kähler-Einstein metrics on the leaf space of their Reeb vector fields). What
first interested us in this subject was the remarkable work of GMSY  finding
new obstructions to the existence of Ricci-flat cone metrics on cones over
singularities, Sasaki-Einstein metrics on the links of the singularities, and
orbifold Kähler-Einstein metrics on the quotient. We wanted to understand
their results in terms of stability. In fact we found that most known results
concerning orbifold cscK metrics could be understood through an extension of
the “slope stability” of RT [1, 2] to orbifolds.

The end product is a theory very similar to that of manifolds, but with a few
notable differences requiring new ideas:

  * •

Embedding an orbifold into projective space loses the information of the
stabilisers, so instead we show how to embed them faithfully into weighted
projective space. This requires the correct notion of ampleness for an orbi-
line bundle $L$, and we are forced to use sections of _more than one_ power
$L^{k}$ – in fact at least as many as the order of the orbifold (defined in
Section 1). Then the relevant stability problem is taken not with respect to
the full automorphism group of weighted projective space (which is not
reductive) but with respect to its reductive part (a product of general linear
groups). This later quotient exactly reflects the ambiguity given by the
choice of sections used in the embedding and, it turns out, gives the same
moduli problem.

  * •

By considering the relevant moment maps we define the Fubini-Study Kähler
metrics on weighted projective space required for stability. A difference
between this and the smooth case is that the curvature of the natural
hermitian metric on the hyperplane line bundle is not the Fubini-Study Kähler
metric, though we prove that the difference becomes negligible asymptotically.

  * •

A key tool connecting metrics of constant scalar curvature to stability is the
asymptotic expansion of the Bergman kernel. To ensure an expansion on
orbifolds similar to that on manifolds we consider not just the sections of
$L^{k}$ but sections of $L^{k+i}$ as $i$ ranges over one or more periods.
Moreover these sections must be taken with appropriate weights to ensure
contributions from the orbifold locus add up to give a global expansion. This
is the topic of the companion paper RT [3], which also contains a discussion
of the exact weights needed.

This choice of weights can also be seen from the moment map framework. The
stability we consider is with respect to a product of unitary groups acting on
a weighted projective space, and since the centraliser of this group is large
the moment map is only defined up to some arbitrary constants. These
correspond exactly to the weights required for the Bergman kernel expansion,
and the main result of RT [3] is that there is a choice of weights (and thus a
choice of stability notion) that connects with scalar curvature.

  * •

The numerical invariants associated to orbifolds and their 1-parameter
degenerations are not polynomial but instead consist of a of polynomial
“Riemann-Roch” term plus periodic terms coming from the orbifold strata. The
definition of the numerical invariants needed for stability (such as the
Futaki invariant) will be made by normalising these periodic terms so they
have average zero, and then only using the Riemann-Roch part. Then
calculations involving stability become identical to the manifold case, only
with the canonical divisor replaced with the orbifold canonical divisor.

After setting up this general framework, our main result is one direction of
the Yau-Tian-Donaldson conjecture for orbifolds.

######  Theorem 1.

Let $(X,L)$ be a polarised orbifold with cyclic quotient singularities. If
$c_{1}(L)$ admits an orbifold Kähler metric of constant scalar curvature then
$(X,L)$ is K-semistable.

Our approach follows the proof given for manifolds by Donaldson in Don [2]. An
improvement by Stoppa Sto  says that, as long as one assumes a discrete
automorphism group, the existence of a cscK metric actually implies
K-stability – it is natural to ask if this too can be extended to orbifolds.

Finally we give an orbifold version of the slope semistability of RT [1, 2],
which we show is implied by orbifold K-semistability. Together with Theorem 1
it gives an obstruction to the existence of orbifold cscK metrics. We use this
to interpret some of the known obstructions in terms of stability, for
instance the work of Troyanov on orbifold Riemann surfaces, Ghigi-Kollár on
orbifold projective spaces, and Rollin-Singer on projectivisations of
parabolic bundles. A particularly important class for this theory is Fano
orbifolds, where cscK metrics are Kähler-Einstein and equivalent to certain
quasi-regular Sasaki-Einstein metrics on odd dimensional manifolds. In this
vein we interpret the Lichnerowicz obstruction of Gauntlett-Martelli-Sparks-
Yau in terms of stability.

###  1 Extensions

Non-cyclic orbifolds. We have restricted our attention purely to orbifold with
cyclic quotient singularities. It should extend easily to orbifolds whose
stabilisers are products of cyclic groups by using several ample (in the sense
of Section 5) line bundles to embed in a product of weighted projective
spaces. To encompass also non-abelian orbifolds one should replace the line
bundle with a bundle of higher rank so that the local stabiliser groups can
act effectively on the fibre over a fixed point, to give a definition of local
ampleness mirroring 7 in the cyclic case. Then one would hope to embed into
weighted Grassmannians. We thank Dror Varolin for this suggestion.

More general cone angles and ramifolds. It would be nice to extend our results
from orbifold Kähler metrics – which have cone angles of the form $2\pi/p,\
p\in\mathbb{N}$, along divisors $D$ – to metrics with cone angles which are
any positive rational multiple of $2\pi$. It should be possible to study these
within the framework of algebro-geometric stability as well.

The one dimensional local model transverse to $D$ is as follows. In this paper
to get cone angle $2\pi/m$ along $x=0$ we introduce extra local functions
$x^{\frac{k}{m}}$ (by passing the local $m$-fold cover and working with
orbifolds). Therefore to produce cone angles $2\pi p$ it makes sense to
discard the local functions $x,x^{2},\ldots x^{p-1}$ and use only
$1,x^{p},x^{p+1},\ldots$ . (We could use $1,x^{p},x^{2p},\ldots$, i.e. pass to
a $p$-fold quotient instead of an $m$-fold cover, but this would be less
general, producing metrics invariant under $\mathbb{Z}/p\mathbb{Z}$ rather
than those with this invariance on only the tangent space at $x=0$.)

The map $(x^{p},x^{p+1})$ from $\mathbb{C}$ to $\mathbb{C}^{2}$ is a set-
theoretic injection with image $\\{v^{p}=u^{p+1}\\}\subset\mathbb{C}^{2}$. For
very small $x$ (so that $x^{p+1}$ is negligible compared to $x^{p}$) it is
very close to the $p$-fold cover $x\mapsto x^{p}$. More precisely,
$\\{v^{p}=u^{p+1}\\}$ has $p$ local branches (interchanged by monodromy) all
tangent to the $u$-axis. Going once round $x=0$ through angle $2\pi$ we go $p$
times round $u=0$ through angle $2\pi p$. Therefore if we restrict a Kähler
metric from $\mathbb{C}^{2}$ to $\\{v^{p}=u^{p+1}\\}$ and pullback to
$\mathbb{C}$ we get a smooth Kähler metric away from $x=0$ which has cone
angle $2\pi p$ at the origin. Similarly the map
$(x^{p},x^{p+1},\ldots,x^{p+k})$ to $\mathbb{C}^{k+1}$ has the same property.

To work globally one has to pick a splitting $H^{0}(X,L^{k})\cong
H^{0}(D,L^{k})\oplus H^{0}(X,L^{k}(-D))$ and discard those functions in the
second summand which do not vanish to at least order $p$ along $D$. That is,
we take the obvious map

$X\to\mathbb{P}\big{(}H^{0}(D,L^{k})^{*}\oplus H^{0}(L^{k}(-pD))^{*}\big{)}.$
(2)

So instead of Kodaira embedding, we take an injection which fails to be an
embedding in the normal directions to $D$ just as in the local model above.
(More generally, to get cone angles $2\pi p/q$ one should apply the above
description to an orbifold with $\mathbb{Z}/q\mathbb{Z}$ stabilisers along $D$
and injections instead into weighted projective spaces.) One might hope for a
relation between balanced injections of $X$ (2) and cscK metrics with
prescribed cone angles along $D$. We thank Dmitri Panov for discussions about
these “ramifolds”. He has also pointed out that it is too ambitious to expect
the full theory for manifolds and orbifolds to carry over verbatim to this
setting since cscK metrics with cone angles greater than $2\pi$ can be non-
unique. We hope to return to this in future work.

Zero cone angles, cuspidal metrics and stability of pairs. It would be
fruitful to consider the limit of large orbifold order. By this we mean fixing
the underlying space $X$ and a divisor $D$, then putting
$\mathbb{Z}/m\mathbb{Z}$-stabilisers along $D$ (as in Section 2) and
considering $m\gg 0$. Then, formally at least, stability in the limit
$m\to\infty$ is the same as stability of the underlying space where the
numerical invariants are calculated with $K_{X}$ replaced with $K_{X}+D$. This
has been studied by Székelyhidi Szé  under the name of “relative stability” of
the pair $(X,D)$, which he conjectures to be linked via a Yau-Tian-Donaldson
conjecture to the existence of complete “cuspidal” cscK metrics on
$X\backslash D$. And indeed one can think of orbifold metrics with cone angle
$2\pi/m$ along a divisor $D$ as tending (as $m\to\infty$) to a complete metric
on $X\backslash D$ (thanks to Simon Donaldson and Dmitri Panov for explaining
this to us).

Pairs. In principle this paper gives many other ways of forming moduli spaces
of pairs $(X,D)$. Initially one should take $X$ smooth projective and $D$ a
simple normal crossings divisor which is a union of smooth divisors $D_{i}$.
Labelling the $D_{i}$ by integers $m_{i}>0$ satisfying the conditions of
Section 2 we get a natural orbifold structure on $X$ from which we recover $D$
as the locus with nontrivial stabiliser group. Taking (for instance) the
orbifold line bundle produced by tensoring a polarisation on $X$ by
$\mathcal{O}(\sum_{i}D_{i}/m_{i})$ gives an orbifold line bundle which is
ample in the sense of Section 5. Embedding in weighted projective space as in
Section 6 and dividing the resulting Hilbert scheme by the reductive group
described in Section 9 gives a natural GIT problem and notion of stability.

One should then analyse which orbischemes appear in the compactification that
this produces (in this paper we mainly study only smooth orbifolds and their
cscK metrics). It is quite possible that the resulting stable pairs will form
a new interesting class. Studying moduli and stability of varieties using GIT
fell out of favour, not least because the singularities it allows are not
those that arise naturally in birational geometry, but interesting recent work
of Odaka suggests a relationship between the newer notion of K-stability
(rather than Chow stability) and semi-log-canonical singularities. It is
therefore natural to wonder if orbifold K-stability of $(X,D)$ is related to
some special types of singularity of pairs (perhaps this is most likely in the
$m\to\infty$ limit of the last section). In fact the recent work of
Abramovich-Hassett AH  precisely studies moduli of varieties and pairs using
orbischemes, birational geometry and the minimal model programme (but not
GIT).

An obvious special case is curves with weighted marked points, as studied by
Hassett Has  and constructed using GIT by Swinarski Swi . It is possible that
Swinarski’s construction can be simplified by using embeddings in weighted
projective space instead of projective space, and even that his (difficult)
stability argument might follow from the existence of an orbifold cscK metric.

##### Acknowledgements

We thank Dan Abramovich, Simon Donaldson, Alessandro Ghigi, Hiroshi Iritani,
Johan de Jong, Dmitri Panov, Miles Reid, Yann Rollin, James Sparks, Balázs
Szendrői and Dror Varolin for useful conversations. Abramovich and Brendan
Hassett have also recently studied moduli of orbifolds and weighted projective
embeddings AH , though from a very different and much more professional point
of view. In particular they do not use GIT and are mainly interested in the
singularities that occur in the compactification; here we are only concerned
with smooth orbifolds for the link to differential geometry. JR received
support from NSF Grant DMS-0700419 and Marie Curie Grant PIRG-GA-2008-230920
and RT held a Royal Society University Research Fellowship while this work was
carried out.

##  Chapter 1 Orbifold embeddings in weighted projective space

The proper way to write this paper would be using Deligne-Mumford stacks, but
this would alienate much of its potential readership (as well as the two
authors). Most of our DM stacks are smooth, so there is an elementary
description in terms of orbifolds, and it therefore makes sense to use it.
However at points (such as when we consider the central fibre of a
degeneration of orbifolds) DM stacks, or orbischemes, are unavoidable. At this
point most of the results we need (such as the appropriate version of Riemann-
Roch) are only available in the DM stacks literature. So we adopt the
following policy. Where possible we phrase things in elementary terms using
only orbifolds. We state the results we need in this language, even when the
only proofs available are in the DM stacks literature. Where we do something
genuinely new we give proofs using the orbifold language, even though they of
course apply more generally to orbischemes or DM stacks.

###  1 Orbibasics

We sketch some of the basics of the theory of orbifolds and refer the reader
to BG [2]; GK  for more details. An orbifold consists of a variety $X$ (either
an algebraic variety or, for us, an analytic space) with only finite quotient
singularities that is covered by orbifold charts of the form $U\to U/G\cong
V\subset X$, where $V$ is an open set in $X$, $U$ is an open set in
$\mathbb{C}^{n}$ and $G$ is a finite group acting effectively on $U$. We also
insist on a minimality condition, that the subgroups of $G$ given by the
stabilisers of points of $U$ generate $G$ (otherwise one should make both $U$
and $G$ smaller – it is important that we are using the analytic topology
here).

The gluing condition on charts is the following. If $V^{\prime}\subset V$ are
open sets in $X$ with charts $U^{\prime}/G^{\prime}\cong V^{\prime}$ and
$U/G\cong V$ then there should exist a monomorphism $G^{\prime}\hookrightarrow
G$ and an injection $U^{\prime}\hookrightarrow U$ commuting with the given
$G^{\prime}$-action on $U^{\prime}$ and its action through
$G^{\prime}\hookrightarrow G$ on $U$.

Notice that these injections are _not_ in general unique, so the charts do not
have to satisfy a cocycle condition upstairs, though of course they do
downstairs where the open sets $V$ glue to give the variety $X$. That is, the
orbifold charts need not glue since an orbifold need not be a global quotient
by a finite group, though we will see in Remark 16 that they _are_ global
$\mathbb{C}^{*}$-quotients under a mild condition.

It follows from the gluing condition that the _order_ of a point $x\in X$ –
the size of the stabiliser of any lift of $x$ is any orbifold chart – is well
defined. The _order of $X$_ is defined to be the least common multiple of the
order of its points (which is finite if $X$ is compact). The _orbifold locus_
is the set of points with nontrivial stabiliser group.

In this paper we will mostly consider only compact orbifolds with cyclic
stabiliser groups, so that each $G$ is always cyclic.

By an _embedding_ $f\colon X\to Y$ of orbifolds we shall mean an embedding of
the underlying spaces of $X$ and $Y$ such that for every $x\in X$ there exist
orbifold charts $U^{\prime}\to U^{\prime}/G\ni x$ and $U\to U/G\ni f(x)$ such
that $f$ lifts to an equivariant embedding $U^{\prime}\hookrightarrow U$. We
say that the orbifold structure on $X$ is pulled back from that on $Y$.
Similarly we get a notion of isomorphism of orbifolds.

Given a point in the orbifold locus with stabiliser group
$\mathbb{Z}/m\mathbb{Z}$, call its preimage in a chart $p$, with maximal ideal
$\mathfrak{m}_{p}$. Split its cotangent space
$\mathfrak{m}_{p}/\mathfrak{m}_{p}^{2}$ into weight spaces under the group
action (and use the fact that the ring of formal power series about that point
is $\oplus_{i}S^{i}(\mathfrak{m}_{p}/\mathfrak{m}_{p}^{2})$) to see that
locally analytically there is a chart $U\to U/(\mathbb{Z}/m\mathbb{Z})$ of the
form

$(z_{1},z_{2},\ldots,z_{n})\mapsto(z_{1}^{a_{1}},z_{2}^{a_{2}},\cdots
z_{k}^{a_{k}},z_{k+1},\ldots,z_{n}),$ (1)

for some integers $a_{i}$ which divide $m$. We call this an _orbifold point_
of type $\frac{1}{m}(\lambda_{1},\ldots,\lambda_{k})$ if
$\zeta\in\mathbb{Z}/m\mathbb{Z}$ acts111Here $\lambda_{i}$ is a multiple of
$m/a_{i}$, of course. We are disobeying Miles Reid and picking the usual
identification of $\mathbb{Z}/m\mathbb{Z}$ with the $m$th roots of unity. as

$\zeta\cdot(z_{1},\ldots,z_{k})=(\zeta^{\lambda_{1}}z_{1},\ldots,\zeta^{\lambda_{k}}z_{k}).$

The general principle is that any local object (e.g. a tensor) on an orbifold
is defined to be an invariant object on a local chart (rather than an object
downstairs on the underlying space). So an _orbifold Kähler metric_ is an
invariant Kähler metric on $U$ for each orbifold chart $U\to U/G$ which glues:
its pullback under an injection $U^{\prime}\hookrightarrow U$ of charts above
is the corresponding metric on $U^{\prime}$. Such a metric descends to give a
Kähler metric on the underlying space $X$, but with possible singularities
along the orbifold locus.

For instance the standard orbifold Kähler metric on
$\mathbb{C}/(\mathbb{Z}/m\mathbb{Z})$ is given by $\frac{i}{2}dz\,d\bar{z}$,
where $z$ is the coordinate on $\mathbb{C}$ upstairs and $x=z^{m}$ is the
coordinate on the scheme theoretic quotient $\mathbb{C}$. Downstairs this
takes the form $\frac{i}{2}m^{-2}|x|^{\frac{2}{m}-2}dx\,d\bar{x}$, which is a
singular Kähler metric on $\mathbb{C}$. The circumference of the circle of
radius $r$ about the origin is easily calculated to be $2\pi r/m$, so the
metric has cone angle $2\pi/m$ at the origin, whereas usual Kähler metrics
have cone angle $2\pi$. More generally for any divisor $D$ in the orbifold
locus with stabiliser group $\mathbb{Z}/m\mathbb{Z}$, orbifold Kähler metrics
on $X$ have cone angle $2\pi/m$ along $D$. So it is important for us to think
of $\mathbb{C}/(\mathbb{Z}/m\mathbb{Z})$ as an orbifold, and not as its scheme
theoretic quotient $\mathbb{C}$.

Even when the stabilisers have codimension two (so that the orbifold is
determined by the underlying variety with quotient singularities, and one “can
forget” the orbifold structure if only interested in the algebraic or analytic
structure) an orbifold metric is very different from the usual notion of a
Kähler metric over the singularities (i.e. one which is locally the
restriction of a Kähler metric from an embedding in a smooth ambient space).

###  2 Codimension one stabilisers

The cyclic orbifolds which will most interest us will be those for which the
orbifold locus has codimension one. These are the orbifolds whose local model
(1) has coprime weights $a_{i}$.

Therefore globally the orbifold is described by the pair $(X,\Delta)$, where

  * •

$X$ is a _smooth_ variety,

  * •

$\Delta$ is a $\mathbb{Q}$-divisor of the form
$\Delta=\sum_{i}\left(1-\frac{1}{m_{i}}\right)D_{i}$,

  * •

the $D_{i}$ are distinct smooth irreducible effective divisors,

  * •

$D=\sum D_{i}$ has normal crossings, and

  * •

the $m_{i}$ are positive integers such that $m_{i}$ and $m_{j}$ are coprime if
$D_{i}$ and $D_{j}$ intersect.

Then the stabiliser group of points in the intersection of several components
$D_{i}$ will be the product of groups $\mathbb{Z}/m_{i}\mathbb{Z}$, and this
is cyclic by the coprimality assumption.

Here $\Delta$ is the ramification divisor of the orbifold charts; see Example
8 for the expression of this in terms of the orbifold canonical bundle.

Notice that above we are also claiming the converse: that given such a pair
$(X,\Delta)$ it is an easy exercise to construct an orbifold with stabiliser
groups $\mathbb{Z}/m_{i}\mathbb{Z}$ along the $D_{i}$, and this is unique.
This can be generalised to Deligne-Mumford stacks Cad ; we give a global
construction in (15).

Orbifolds with codimension one stabilisers were called “not well formed” in
the days when “we were doing the wrong thing” (Miles Reid, Alghero 2006). Then
orbifolds were studied as a means to produce schemes, so only the quotient was
relevant. The orbifold locus could be removed, since the quotient is smooth.
Hence in much of the literature (e.g. Dol ) the not well formed case is
unfortunately ignored.

More generally, _any_ orbifold can be dealt with in much the same way: it can
be described by a pair $(X,\Delta)$ just as above, but where $X$ has at worst
finite cyclic quotient singularities. This is the point of view taken by GK .

###  3 Weighted projective spaces

The standard source of examples of orbifolds is weighted projective spaces. A
graded vector space $V=\oplus_{i}V^{i}$ is equivalent to a vector space $V$
with a $\mathbb{C}^{*}$-action, acting on $V^{i}$ with weight $i$. Throughout
this paper $V$ will always be finite dimensional, with all weights strictly
positive. We can therefore form the associated weighted projective space
$\mathbb{P}(V):=(V\backslash\\{0\\})/\mathbb{C}^{*}$. This is sometimes
denoted $\mathbb{P}(\lambda_{1},\ldots\lambda_{n})$, where $n=\dim V$ and the
$\lambda_{j}$ are the weights (so the number of $\lambda_{j}$ that equal $i$
is $\dim V^{i}$).

Let $x_{j},\ j=1,\ldots,n$, be coordinates on $V$ such that $x_{j}$ has weight
$-\lambda_{j}$. Then $\mathbb{P}(V)$ is covered by the orbifold charts

$\displaystyle\\{x_{j}=1\\}$ $\displaystyle\\!\\!\cong\mathbb{C}^{n-1}$ (2)
$\displaystyle\downarrow\ \ \ \,$ $\displaystyle\mathbb{P}(V).\ $

The $\lambda_{j}$th roots of unity
$\mathbb{Z}/\lambda_{j}\mathbb{Z}\subset\mathbb{C}^{*}$ act trivially on the
$x_{j}$ coordinate, preserving the above $\mathbb{C}^{n-1}$ slice. The
vertical arrow is the quotient by this $\mathbb{Z}/\lambda_{j}\mathbb{Z}$; the
generator $\exp(2\pi i/\lambda_{j})\in\mathbb{C}^{*}$ acting by

$(x_{i})\mapsto(\exp(2\pi i\lambda_{i}/\lambda_{j})x_{i}).$ (3)

The order of $\mathbb{P}(V)$ is the least common multiple of the weights
$\lambda_{j}$. If the $\lambda_{j}$ have highest common factor $\lambda>1$
then $\mathbb{P}(V)$ has generic stabilisers: every point is stabilised by the
$\lambda$th roots of unity, and we will usually assume that this is not the
case, so $\mathbb{P}(V)$ inherits the structure of an orbifold with cyclic
stabiliser groups.

The orbifold points of $\mathbb{P}(V)$ are as follows. Each vertex
$P_{i}:=[0,\dots,1,\dots,0]$ is of type
$\frac{1}{\lambda_{i}}(\lambda_{1},\ldots,\widehat{\lambda}_{i},\ldots,\lambda_{N})$.
The general points along the line $P_{i}P_{j}$ are orbifold points of type
$\frac{1}{\operatorname{hcf}(\lambda_{i},\lambda_{j})}(\lambda_{1},\ldots,\widehat{\lambda}_{i},\ldots,\widehat{\lambda}_{j},\ldots,\lambda_{N})$,
with similar orbifold types along higher dimensional strata.

Thus if for some $j$ the $\lambda_{i},\,i\neq j$, have highest common factor
$\lambda>1$ then $\mathbb{P}(V)$ is not well formed: it has a divisor of
orbifold points with stabiliser group containing
$\mathbb{Z}/\lambda\mathbb{Z}$ along $x_{j}=0$. Replacing the
$\lambda_{i},\,i\neq j$, by $\lambda_{i}/\lambda$ gives a well formed weighted
projective space Dol ; Fle  which is just the underlying variety without the
divisor of orbifold points. As discussed in the last section, it is important
for us _not_ to mess with the orbifold structure in this way.

Similarly the map
$\mathbb{P}^{n-1}\to\mathbb{P}(\lambda_{1},\ldots,\lambda_{n}),\
[x_{1},\ldots,x_{n}]\mapsto[x_{1}^{\lambda_{1}},\ldots,x_{n}^{\lambda_{n}}]$
exhibits the underlying variety of weighted projective space as a global
finite quotient of ordinary projective space. Again this does _not_ give the
right orbifold structure of (2), so we do not use it.

###  4 Orbifold line bundles and $\mathbb{Q}$-divisors

Locally an orbifold line bundle is simply an equivariant line bundle on an
orbifold chart. This differs from an ordinary line bundle pulled back from
downstairs which satisfies the property that the $G$-action on the line over
any fixed point is trivial. In other words (the pull back to an orbifold chart
of) an ordinary line bundle has a local invariant trivialisation, which an
orbifold line bundle may not. So in general orbifold line bundles are _not_
locally trivial.

To define them globally we need some notation. Suppose that
$V_{i},V_{j},V_{k}$ are open sets in $X$ with charts $U_{i}/G_{i}\cong V_{i}$,
etc. Then by the definition of an orbifold the overlaps $V_{ij}:=V_{i}\cap
V_{j}$, etc, also have charts $U_{ij}/G_{ij}\cong V_{ij}$ and inclusions
$U_{ij}\hookrightarrow U_{i},\ G_{ij}\hookrightarrow G_{i}$, etc.

Given local equivariant line bundles $L_{i}$ over each $U_{i}$, the gluing (or
cocycle) condition to define a global orbifold line bundle is the following.
Pulling back $L_{j}$ and $L_{i}$ to $U_{ij}$ (via its inclusions in
$U_{j},\,U_{i}$ respectively) there should be isomorphisms $\phi_{ij}$ from
the former to the latter, intertwining the actions of $G_{ij}$. Pulling back
further to $U_{ijk}$ we call this isomorphism $\phi_{ij}\in L_{i}\otimes
L_{j}^{*}$ (suppressing the pullback maps for clarity). The cocycle condition
is that over $U_{ijk}$,

$\phi_{ij}\phi_{jk}\phi_{ki}\ \in\ L_{i}\otimes L_{j}^{*}\otimes L_{j}\otimes
L_{k}^{*}\otimes L_{k}\otimes L_{i}^{*}$

should be precisely the identity element $1$.

The standard example is the orbifold canonical bundle $K_{\text{orb}}$, which
is defined to be $K_{U}$ on the chart $U$ (with the obvious $G$-action induced
from that on $U$) and which glues automatically.

######  Example 4.

Take $X$ a smooth space with a smooth divisor $D$ along which we put
$\mathbb{Z}/(m\mathbb{Z})$ stabiliser group to form the orbifold
$(X,(1-1/m)D)$. Then the orbifold line bundle
$\mathcal{O}\big{(}\\!-\\!\frac{1}{m}D\big{)}$ is easily defined as the ideal
sheaf of the reduced pullback of $D$ to any chart. In this way it glues
automatically.

Locally it has generator $z$, a local coordinate upstairs cutting out the
reduced pullback of $D$. But this has weight one under the
$\mathbb{Z}/m\mathbb{Z}$-action; it is not an invariant section, so does _not_
define a section of the orbifold line bundle downstairs ($z^{km-1}$ times this
generator does, for all $k\geq 0$). Therefore this orbifold bundle is _not_
locally trivial: it is locally the trivial line bundle with the weight one
_nontrivial_ $\mathbb{Z}/m\mathbb{Z}$-action.

Away from $D$, the section which is $z^{-1}$ times by this weight one
generator is both regular and invariant, so can be glued to the trivial line
bundle. In this way one can give an equivalent definition of
$\mathcal{O}\big{(}\\!-\\!\frac{1}{m}D\big{)}$ via transition functions, much
as in the manifold case.

Taking tensor powers we can form $\mathcal{O}\big{(}\frac{n}{m}D\big{)}$ for
any integer $n$. This is an ordinary line bundle only for $n/m$ an integer.
The inclusion $\mathcal{O}(-\frac{1}{m}D)\hookrightarrow\mathcal{O}_{X}$
defines a canonical section $s_{D/m}$ of $\mathcal{O}(\frac{1}{m}D)$ which in
the orbifold chart above looks like $z$ vanishing on $D$.

The pushdown to the underlying manifold $X$ of
$\mathcal{O}\big{(}\frac{n}{m}D\big{)}$ is the ordinary line bundle given by
the round down

$\mathcal{O}\left(\left\lfloor\frac{n}{m}\right\rfloor D\right).$ (5)

That is to say that the (invariant) sections of
$\mathcal{O}\big{(}\frac{n}{m}D\big{)}$ are of the form
$s_{D/m}^{\frac{n}{m}-\left\lfloor\\!\frac{n}{m}\\!\right\rfloor}t$, where $t$
is any section of the ordinary line bundle
$\mathcal{O}\big{(}\\!\left\lfloor\frac{n}{m}\right\rfloor\\!\big{)}$ on $X$.

Since tensor product does not commute with round down, we lose information by
pushing down to $X$: the natural consequence of orbifold line bundles not
being locally trivial.

More generally on any orbifold given by a pair $(X,\Delta)$ as in Section 2,
orbifold line bundles and their sections correspond to $\mathbb{Q}$-divisors
such that the denominator of the coefficient of $D_{i}$ must divide $m_{i}$,
and any irreducible divisor $D$ not in the list of $D_{i}$ must have integral
coefficients. The space of global sections of the orbifold line bundle is the
space of sections of the round down. Care must be taken however; for instance
if $D_{1}$ and $D_{2}$ have $\mathbb{Z}/m\mathbb{Z}$-stabilisers along them
and $\mathcal{O}(D_{1})\cong\mathcal{O}(D_{2})$ this certainly does _not_
imply that $\mathcal{O}(D_{1}/m)\cong\mathcal{O}(D_{2}/m)$.

The _tautological line bundle_ $\mathcal{O}_{\mathbb{P}(V)}(-1)$ over the
weighted projective space $\mathbb{P}(V)$ is the orbi-line bundle over
$\mathbb{P}(V)$ with fibre over $[v]$ the union of the orbit
$\mathbb{C}^{*}.v\subset V$ and $0\in V$. (Any two elements in a fibre can be
written $w_{i}=t_{i}.v$ for $t_{i}\in\mathbb{C},\ i=1,2$, so we can define the
linear structure by $aw_{1}+bw_{2}:=(at_{1}+bt_{2}).v$. Ordinarily this is not
the linear structure on $V$ and the fibre $\mathcal{O}_{[v]}(-1)\subset V$ is
not a linear subspace.) Over the orbi-chart (2) this is the trivial line
bundle $\mathbb{C}^{n-1}\times\mathbb{C}$ with the weight one
$\mathbb{Z}/\lambda_{j}\mathbb{Z}$-action on the line $\mathbb{C}$ times by
its action (3) on $\mathbb{C}^{n-1}$. In other words the map

$\displaystyle\mathbb{C}^{n-1}\times\mathbb{C}$ $\displaystyle\to$
$\displaystyle\mathbb{C}^{n}$ (6)
$\displaystyle(x_{1},\ldots,\widehat{x}_{j},\ldots,x_{n},t)$
$\displaystyle\mapsto$
$\displaystyle(t^{\lambda_{1}}x_{1},\ldots,t^{\lambda_{j}},\ldots,t^{\lambda_{n}}x_{n})$

becomes $(\mathbb{Z}/\lambda_{j}\mathbb{Z})$-equivariant when we use the
action (3) on $\mathbb{C}^{n-1}$, the standard weight-one action on
$\mathbb{C}$, and the original weighted $\mathbb{C}^{*}$-action on
$\mathbb{C}^{n}$. The map (6) is defined in order to take the trivialisation
$1$ of $\mathbb{C}$ to the tautological trivialisation of the pullback of the
orbit to the chart (2) (a point of the chart (2) is a point of its own orbit
and so trivialises it).

Note that Dolgachev Dol  uses the same notation
$\mathcal{O}_{\mathbb{P}(V)}(-1)$ to denote the push forward of our
$\mathcal{O}_{\mathbb{P}(V)}(-1)$ to the underlying space, thus rounding down
fractional divisors. Therefore
$\mathcal{O}_{\mathbb{P}(V)}(a+b)=\mathcal{O}_{\mathbb{P}(V)}(a)\otimes\mathcal{O}_{\mathbb{P}(V)}(b)$
does not hold for his sheaves, but is true almost by definition for our
orbifold line bundles.

As a trivial example, consider $\mathcal{O}(k)$ over the weighted projective
line $\mathbb{P}(1,m)$. The first coordinate $x$ on $\mathbb{C}^{2}$ has
weight one, so restricts to a linear functional on orbits (the fibres of
$\mathcal{O}(-1)$). It therefore defines a section of $\mathcal{O}(1)$ which
vanishes at the orbifold point $x=0$. Since $x$ is the coordinate upstairs in
the chart (2) and $x^{m}$ the coordinate downstairs, this is $\frac{1}{m}$
times by a real manifold point. The coordinate $y$ has weight $m$ on the
fibres of $\mathcal{O}(-1)$ so defines a section of $\mathcal{O}(m)$ which
vanishes at the manifold point $y=0$.

The underlying variety is the projective space on the degree $m$ variables
$x^{m},y$, i.e. it is $\mathbb{P}^{1}$ with reduced points $0$ and $\infty$
where these two variables vanish. Thus

$\mathcal{O}_{\mathbb{P}(1,m)}(k)=\mathcal{O}\left(\frac{k}{m}(0)\right)=\mathcal{O}\left(\left\lfloor\frac{k}{m}\right\rfloor(\infty)+\left(\frac{k}{m}-\left\lfloor\frac{k}{m}\right\rfloor\right)(0)\right).$

Similarly on $\mathbb{P}(a,b)$ with $pa+qb=1$, the underlying variety is the
usual Proj of the graded ring on the degree $ab$ generators $x^{b}$ and
$y^{a}$. Denote by $0$ and $\infty$ the zeros of $x^{b}$ and $y^{a}$
respectively. Then it is a nice exercise to check that the orbifold line
bundle $\mathcal{O}_{\mathbb{P}(a,b)}(1)$ is isomorphic to

$\mathcal{O}\left(\frac{p}{b}(0)+\frac{q}{a}(\infty)\right),$

of degree $\frac{1}{ab}$.

###  5 Orbifold polarisations

To define orbifold polarisations we need the right notion of ampleness or
positivity. For manifolds (or schemes) this is engineered to ensure that the
global sections of $L$ generate the local ring of functions at each point. For
orbifolds, this requires also a _local_ condition on an orbifold line bundle
$L$, as we explain using the simplest example. Consider the orbifold
$\mathbb{C}/(\mathbb{Z}/2\mathbb{Z})$ with local coordinate $z$ on
$\mathbb{C}$ acted on by $\mathbb{Z}/2\mathbb{Z}$ via $z\mapsto-z$. Then
$x=z^{2}$ is a local coordinate on the quotient thought of as a manifold. Any
line bundle pulled back from the quotient (i.e. which has trivial
$\mathbb{Z}/2\mathbb{Z}$-action upstairs when considered as a trivial line
bundle there) has invariant sections $\mathbb{C}[x]=\mathbb{C}[z^{2}]$.
Therefore it sees the quotient only as a manifold, missing the extra functions
of $\sqrt{x}=z$ that the orbifold sees. So we do not think of it as locally
ample: if we tried to embed using its sections we would “contract” the
stabilisers, leaving us with the underlying manifold.

Conversely the trivial line bundle upstairs with nontrivial
$\mathbb{Z}/2\mathbb{Z}$-action (acting as $-1$ on the trivialisation) has
invariant sections $\sqrt{x}\,\mathbb{C}[x]=z\mathbb{C}[z^{2}]$. Its square
has trivial $\mathbb{Z}/2\mathbb{Z}$-action and has sections
$\mathbb{C}[x]=\mathbb{C}[z^{2}]$ as above. Therefore its sections and those
of its powers generate the entire ring of functions
$\mathbb{C}[\sqrt{x}]=\mathbb{C}[z]$ upstairs, and see the full orbifold
structure.

######  Definition 7.

An orbifold line bundle $L$ over a cyclic orbifold $X$ is _locally ample_ if
in an orbifold chart around $x\in X$, the stabiliser group acts faithfully on
the line $L_{x}$. We say $L$ is _orbi-ample_ if it is both locally ample and
globally positive. (By globally positive here we mean
$L^{\operatorname{ord}(X)}$ is ample in the usual sense when thought of as a
line bundle on the underlying space of $X$; from the Kodaira-Baily embedding
theorem Bai  one can equivalently ask that $L$ admits a hermitian metric with
positive curvature.)

By a _polarised_ orbifold we mean a pair $(X,L)$ where $L$ is an orbi-ample
line bundle on $X$.

Note that ordinary line bundles on the underlying space are never ample on
genuine orbifolds. Some care needs to be taken when applying the usual theory
to orbi-ample line bundles. For instance it is not necessarily the case that
the tensor product of locally ample line bundles remain locally ample, but if
$L$ is locally ample then so is $L^{-1}$. One can easily check that $L$ is
orbi-ample if and only if $L^{k}$ is ample for one (or all) $k>0$ coprime to
$\operatorname{ord}(X)$.

######  Example 8.

The orbifold canonical bundle $K_{\text{orb}}$ is locally ample along divisors
of orbifold points, but not necessarily at codimension two orbifold points.
For instance the quotient of $\mathbb{C}^{2}$ by the scalar action of $\pm 1$
has trivial canonical bundle, so local ampleness is not determined in
codimension one.

Suppose that $X$ is smooth but with a divisor $D$ with stabiliser group
$\mathbb{Z}/m\mathbb{Z}$. Locally write $D$ as $x=0$ and pick a chart with
coordinate $z$ such that $z^{m}=x$. Then the identity
$dx=mz^{m-1}dz=mx^{1-\frac{1}{m}}dz$ shows that $X$ has orbifold canonical
bundle

$K_{\text{orb}}\ =\ K_{X}+\left(1-\frac{1}{m}\right)D\ =\ K_{X}+\Delta,$

where $K_{X}$ is the canonical divisor of the variety underlying $X$. More
generally if the orbifold locus is a union of divisors $D_{i}$ with stabiliser
groups $\mathbb{Z}/m_{i}\mathbb{Z}$ then $K_{\text{orb}}=K_{X}+\Delta$, where
$\Delta=\sum_{i}\big{(}1-\frac{1}{m_{i}}\big{)}D_{i}$ as in Section 2.

######  Example 9.

The hyperplane bundle $\mathcal{O}_{\mathbb{P}(V)}(1)$ on any weighted
projective space $\mathbb{P}(V)$ is locally ample, and it is actually orbi-
ample since some power is ample [Dol, , Proposition 1.3.3] (we shall also show
below that it admits a hermitian metric with positive curvature). The pullback
of an orbi-ample bundle along an orbifold embedding is also orbi-ample, and
thus any orbifold embedded in weighted projective space admits an orbi-ample
line bundle. If $(X,\Delta)$ is an orbifold, $X$ is smooth and $H$ is an ample
divisor on $X$ then the orbifold bundle $H+\Delta$ of Section 2 is orbi-ample
if and only if $H+\Delta$ is an ample $\mathbb{Q}$-divisor on $X$.

###  6 Orbifold Kodaira embedding

Now fix a polarised orbifold $(X,L)$ and $k\gg 0$. Let $i$ run throughout a
fixed indexing set ${0,1,\ldots,M}$, where $M\geq\operatorname{ord}(X)$, and
let $V$ be the graded vector space

$V=\bigoplus_{i}V^{k+i}:=\bigoplus_{i}H^{0}(L^{k+i})^{*}.$

We give the $i$th summand weight $k+i$. Map $X$ to the weighted projective
space $\mathbb{P}(V)$ by

$\phi_{k}(x):=\big{[}\oplus_{i}\operatorname{ev}^{k+i}_{x}\big{]}.$ (10)

Here we fix a trivialisation of $L_{x}$ on an orbifold chart, inducing
trivialisations of all powers $L_{x}^{k+i}$, and then
$\operatorname{ev}^{k+i}_{x}$ is the element of $H^{0}(L^{k+i})^{*}$ which
takes a section $s\in H^{0}(L^{k+i})$ to $s(x)\in L^{k+i}_{x}\cong\mathbb{C}$.
The weights are chosen so that a change in trivialisation induces a change in
$\oplus_{i}\operatorname{ev}^{k+i}_{x}$ that differs only by the action of
$\mathbb{C}^{*}$ on $V$.

Picking a basis $s_{j}^{k+i}$ for $H^{0}(L^{k+i})$, then, the map can be
described by

$\phi_{k}(x)=\big{[}(s_{j}^{k+i}(x))_{i,j}\big{]}.$

This map is well defined at all points $x$ for which there exists a global
section of some $L^{k+i}$ not vanishing at $x$.

######  Proposition 11.

If $(X,L)$ is a polarised orbifold then for $k\gg 0$ the map (10) is an
embedding of orbifolds (i.e. the orbifold structure on $X$ is pulled back from
that on the weighted projective space $\mathbb{P}(V)$) and

$\phi_{k}^{*}\mathcal{O}_{\mathbb{P}(V)}(1)\cong L.$

###### Proof.

Fix $x\in X$. It has stabiliser group $\mathbb{Z}/m\mathbb{Z}$ for some $m\geq
1$, and a local orbifold chart $U/(\mathbb{Z}/m\mathbb{Z})$. Let $y\in U$
(with maximal ideal $\mathfrak{m}_{y}$) map to $x$, and decompose
$\mathfrak{m}_{y}/\mathfrak{m}_{y}^{2}=\oplus_{l}V^{l}$ into weight spaces.
Since we have chosen the indexing set for $i$ to range over at least a full
period of length $m$, at least one of the $L^{k+i}_{y}$ has weight 0 and, for
each $l$, there is at least one $i_{l}$ in the indexing set such that
$L^{k+i}_{y}\otimes V^{i_{l}}$ has weight 0.

Therefore each of these $\mathbb{Z}/m\mathbb{Z}$-modules has invariant local
generators, defining local sections of the appropriate power of $L$ on $X$.
For $k\gg 0$ these extend to global sections, by ampleness. (The pushdowns of
the powers of $L$ from the orbifold to the underlying scheme give sheaves
which all come from a finite collection of sheaves tensored by a line bundle.
For $k\gg 0$ this line bundle becomes very positive, and so eventually has no
cohomology. This value of $k$ can be chosen uniformly for all $y$ by
cohomology vanishing for a _bounded_ family of sheaves on a scheme.)

Therefore, trivialising $L$ locally, the sections generate $\mathcal{O}_{y}$
and $\mathfrak{m}_{y}/\mathfrak{m}_{y}^{2}$, so the pullback of the local
functions on $\mathbb{P}(V)$ (the polynomials in $(x_{i})_{i\neq j}$ on the
orbifold chart (2)) generate the local functions on $U$. It follows that the
map is an embedding for large $k$.

Invariantly, the map (10) can be described as follows. Any lift $\tilde{x}\in
L^{-1}_{x}$ of $x$ is a linear functional on $L_{x}$. Similarly
$\tilde{x}^{\otimes(k+1)}$ is a linear functional on $L_{x}^{k+i}$. Composed
with the evaluation map $\operatorname{ev}_{x}^{k+i}\colon H^{0}(L^{k+i})\to
L^{k+i}_{x}$ gives

$\tilde{x}^{\otimes(k+1)}\circ\operatorname{ev}_{x}^{k+i}\,\colon\
H^{0}(L^{k+i})\to\mathbb{C}.$

Therefore

$\oplus_{i}\big{(}\tilde{x}^{\otimes(k+1)}\circ\operatorname{ev}_{x}^{k+i}\big{)}\,\in\
\bigoplus_{i}H^{0}(L^{k+i})^{*}=V$

is a well defined point, with no $\mathbb{C}^{*}$-scaling ambiguities or
choices. In other words (10) lifts to a natural $\mathbb{C}^{*}$-equivariant
embedding of the orbi-line

$L^{-1}_{x}\ \hookrightarrow\ \bigoplus_{i}H^{0}(L^{k+i})^{*}$ (12)

onto the $\mathbb{C}^{*}$-orbit over the point (10). This makes it clear that
under this weighted Kodaira embedding, the pullback of the
$\mathcal{O}_{\mathbb{P}(V)}(-1)$ orbifold line bundle over $\mathbb{P}(V)$ is
$L^{-1}$. ∎

######  Remark 13.

That $\phi_{k}^{*}\mathcal{O}_{\mathbb{P}(V)}(-1)=L^{-1}$, even though the
embedding uses the sections of $L^{k},\ldots,L^{k+M}$ and not those of $L$,
follows from the fact that we give $H^{0}(L^{k+i})^{*}$ weight $k+i$. This
might come as a surprise and appear to contradict what we know about Kodaira
embedding for manifolds. For instance, suppose we embed the _manifold_
$\mathbb{P}^{1}$ using $\mathcal{O}(2)$. Under the normal Kodaira embedding we
get a conic in
$\mathbb{P}^{2}=\mathbb{P}(H^{0}(\mathcal{O}_{\mathbb{P}^{1}}(2)^{*})$ such
that the pullback of $\mathcal{O}_{\mathbb{P}^{2}}(-1)$ is
$\mathcal{O}_{\mathbb{P}^{1}}(-2)$.

However, from the above orbifold perspective, this is _not_ an embedding of
$\mathbb{P}^{1}$, but of the orbifold
$\mathbb{P}^{1}/(\mathbb{Z}/2\mathbb{Z})$, where the
$\mathbb{Z}/2\mathbb{Z}$-action is trivial. We see this as follows. At the
level of line bundles (12), it is an embedding of
$\mathcal{O}_{\mathbb{P}^{1}}(-1)\big{/}(\mathbb{Z}/2\mathbb{Z})$ into
$\mathcal{O}_{\mathbb{P}^{2}}(-2)$, where the $\mathbb{Z}/2\mathbb{Z}$-action
is by $-1$ on each fibre. As a manifold this quotient is indeed
$\mathcal{O}_{\mathbb{P}^{1}}(-2)$, but as an orbifold it is instead an
orbifold line bundle over the orbifold
$\mathbb{P}^{1}/(\mathbb{Z}/2\mathbb{Z})$, where the
$\mathbb{Z}/2\mathbb{Z}$-action is trivial.

######  Remark 14.

When we began this project in early 2006 we were using a different, perhaps
more natural, weighted projective embedding. We embedded in the same way in

$\mathbb{P}\Big{(}\bigoplus_{i}H^{0}(L^{ik})^{*}\Big{)},$

where we give $H^{0}(L^{ik})^{*}$ weight $i$ (_not_ $ik$). (Notice how this
cures the problem with Veronese embeddings described in Remark 13 above.) This
can also be shown to pull back the orbifold structure of weighted projective
space to that of $X$ when $L$ is ample, and to pull $\mathcal{O}(1)$ back to
$L^{k}$. However the corresponding Bergman kernel turns out not to be relevant
to constant scalar curvature orbifold Kähler metrics. We learnt about the
related alternative embedding (10) from Dan Abramovich; see AH . The idea of
using weighted projective embeddings certainly goes back further to Miles
Reid; see for instance Rei [1].

###  7 OrbiProj

It is similarly simple to write down an orbifold version of the Proj
construction, using the whole graded ring $\oplus_{k}H^{0}(L^{k})$ at once.
Given a finitely generated graded ring $R=\oplus_{k\geq 0}R_{k}$ (not
necessarily generated in degree 1!) we can form the scheme
$\operatorname{Proj}\,R$ in the usual way [Har, , Proposition II.2.5]. However
this loses information (for instance we could throw away all the graded pieces
except the $R_{nk},\,k\gg 0$, and get the same result).

We endow $\operatorname{Proj}\,R$ with an orbischeme structure by describing
the orbischeme charts. Fix a homogeneous element $r\in R_{+}$ and consider the
Zariski-open subset
$\operatorname{Spec}\,R_{(r)}=(\operatorname{Proj}\,R)\backslash\\{r=0\\}$.
(As usual $R_{(r)}$ is the degree zero part of the localised ring $r^{-1}R$.)
Then

$\operatorname{Spec}\,\,\frac{R}{(r-1)}\ \longrightarrow\
\operatorname{Spec}\,R_{(r)}$

is our orbichart. Here $R/(r-1)$ is the quotient of $R$ (thought of as a ring
and forgetting the grading) by the ideal $(r-1)$. The map from $R_{(r)}$ sets
$r$ to $1$.

More simply but less invariantly, pick homogeneous generators and relations
for the graded ring $R$. Then $\operatorname{Proj}\,R$ is embedded in the
weighted projective space on the generators, cut out by the equations defined
by the relations.

Given a projective scheme $(X,L)$ and a Cartier divisor $D\subset X$, this
gives a very direct way to produce Cadman’s $r$th root orbischeme
$\big{(}X,\big{(}1-\frac{1}{r}\big{)}D\big{)}$ Cad . This has underlying
scheme $X$ but with stabilisers $\mathbb{Z}/r\mathbb{Z}$ along $D$, and in the
above notation it is simply

$\left(X,\Big{(}1-\frac{1}{r}\Big{)}D\right)\ =\ \operatorname{Proj}\,\
\bigoplus_{k\geq
0}H^{0}\Big{(}X,\mathcal{O}\left(\left\lfloor\frac{k}{r}\right\rfloor
D\right)\otimes L^{k}\Big{)}.$ (15)

The hyperplane line bundle $\mathcal{O}(1)\otimes L^{-1}$ on this Proj is
$\mathcal{O}\big{(}\frac{1}{r}D\big{)}$. Picking generators and relations for
the above graded ring we see the $r$th root orbischeme very concretely, cut
out by equations in weighted projective space.

######  Remark 16.

Although orbifolds need not be global quotients by _finite_ groups, we see
that polarised orbifolds _are_ global quotients of varieties by
$\mathbb{C}^{*}$-actions. In terms of the weighted Kodaira embedding of
Proposition 11, we take the total space of $L^{-1}$ over $X$, minus the zero
section, and divide by the natural $\mathbb{C}^{*}$-action on the fibres.
Equivalently, we express the orbifold Proj of the graded ring $R$ as the
quotient of $\operatorname{Spec}\,(R)\backslash\\{0\\}$ by the action of
$\mathbb{C}^{*}$ induced by the grading.

###  8 Orbifold Riemann-Roch

Suppose that $L$ is an orbifold polarisation on $X$. We will need the
asymptotics of $h^{0}(L^{k})$ for $k\gg 0$. These follow from Kawazaki’s
orbifold Riemann-Roch theorem Kaw , or Toën’s for Deligne-Mumford stacks Toë ,
and some elementary algebra (see for example Rei [2] in the well formed case).
Alternatively they follow from the weighted Bergman kernel expansion (see [RT,
3, Corollary 1.12]), or by embedding in weighted projective space and taking
hyperplane sections in the usual way. The result is that

$h^{0}(L^{k})\ =\ \frac{\int_{X}c_{1}(L)^{n}}{n!}\,k^{n}\ -\
\frac{\int_{X}c_{1}(L)^{n-1}.c_{1}(K_{\text{orb}})}{2(n-1)!}\,k^{n-1}\ +\
\tilde{o}(k^{n-1}).$ (17)

Here and in what follows we define $\tilde{o}(k^{n-1})$ to mean functions of
$k$ that can be written as $r(k)\delta(k)+O(k^{n-2})$, where $r(k)$ is a
polynomial of degree $n-1$ and $\delta(k)$ is periodic in $k$ with period
$m=\operatorname{ord}(X)$ and _average zero_ :

$\delta(k)=\delta(k+m),\qquad\sum_{u=1}^{m}\delta(u)=0.$

Therefore the average of $\tilde{o}(k^{n-1})$ over a period is in fact
$O(k^{n-2})$, and we think of it as being a lower order term than the two
leading ones of (17).

Here we are also using integration of Chern-Weil forms on orbifolds (or
intersection theory on DM stacks). Of course integration works for orbifolds
just as it does for manifolds; it is defined in local charts, but then the
local integral is divided by the size of the group. It also extends easily to
orbischemes, just as usual integration works on schemes once we weight by
local multiplicities.

We give a simple example which nonetheless illustrates a number of the issues
we have been considering.

######  Example 18.

Let $\mathbb{Z}/m\mathbb{Z}$ act on ordinary $\mathbb{P}^{1}$ and the
tautological line bundle over it by $\lambda\cdot[x,y]=[\lambda x,y]$. Then
the quotient $X$ is naturally an orbifold with $\mathbb{Z}/m\mathbb{Z}$
stabilisers at the two points $x=0$ and $y=0$. And the quotient of
$\mathcal{O}(-1)$ is naturally an orbifold line bundle $L_{X}^{-1}$ over $X$.

However $L_{X}$ is not locally ample at $x=0$, since the above action is
trivial on the fibre over $x=0$. So we “contract” the orbifold structure of
$X$ at this point to produce another orbifold $Y$ by ignoring the stabiliser
group at $x=0$ and thinking of it locally as a manifold. Only the orbifold
point $y=0$ survives, and $L_{X}$ automatically descends to an ample orbifold
line bundle $L_{Y}$ on $Y$, to which orbifold Riemann-Roch (17) should
therefore apply.

The sections of $L_{Y}^{k}$ (or those of $L_{X}^{k}$; they are the same) are
the invariant sections of $\mathcal{O}_{\mathbb{P}^{1}}(k)$, a basis for which
is
$y^{k},y^{k-m}x^{m},\ldots,y^{k-m\left\lfloor\\!\frac{k}{m}\\!\right\rfloor}x^{m\left\lfloor\\!\frac{k}{m}\\!\right\rfloor}$.
In particular $Y=\mathbb{P}\langle x^{m},y\rangle=\mathbb{P}(m,1)$ and
$h^{0}(L^{k})=\left\lfloor\frac{k}{m}\right\rfloor+1$.

Writing this as $\frac{k}{m}+1-\frac{m-1}{2m}+\delta(k)$, where $\delta$ is
periodic with average zero, we find

$h^{0}(L^{k})=\frac{k}{m}-\frac{1}{2}\left(-2+\left(1-\frac{1}{m}\right)\right)+\delta(k)=k\deg
L-\frac{1}{2}\deg K_{\text{orb}}+\delta(k).$

Hence, as expected, the single $\mathbb{Z}/m\mathbb{Z}$-orbifold point of $Y$
adds $1-1/m$ to the degree of $K_{\text{orb}}$, and the other orbifold point
of $X$ does not show up.

####  1 Equivariant case

Fix a polarised orbifold $(X,L)$ as above, but now with a
$\mathbb{C}^{*}$-action on $L$ linearising one on $X$. We need a similar
expansion for the weight of a $\mathbb{C}^{*}$-action on $H^{0}(L^{k})$.
Instead of using the full equivariant Riemann-Roch theorem we follow Donaldson
in deducing what we need by using $\mathbb{P}^{1}$ to approximate
$B\mathbb{C}^{*}=\mathbb{P}^{\infty}$ and applying the above orbifold Riemann-
Roch asymptotics to the total space of the associated bundle over
$\mathbb{P}^{1}$.

So let $\mathcal{O}_{\mathbb{P}^{1}}(1)^{*}$ denote the principal
$\mathbb{C}^{*}$-bundle over $\mathbb{P}^{1}$ given by the complement of the
zero-section in $\mathcal{O}(1)$. Form the associated $(X,L)$-bundle

$(\mathcal{X},\mathcal{L}):=\mathcal{O}_{\mathbb{P}^{1}}(1)^{*}\times^{\
}_{\mathbb{C}^{*}}(X,L).$

Let $\pi\colon\mathcal{X}\to\mathbb{P}^{1}$ denote the projection. Then it is
clear that $\pi_{*}\mathcal{L}^{k}$ is the associated bundle of the
$\mathbb{C}^{*}$-representation $H^{0}(X,L^{k})$. Splitting the latter into
one dimensional weight spaces splits the former into line bundles. A line with
weight $i$ becomes the line bundle $\mathcal{O}(i)$. It follows that the
_total weight_ (i.e. the weight of the induced action on the determinant) of
the $\mathbb{C}^{*}$-action on $H^{0}(X,L^{k})$ is the first Chern class of
$\pi_{*}\mathcal{L}^{k}$. Therefore

$w(H^{0}(X,L^{k}))=\chi(\mathbb{P}^{1},\pi_{*}\mathcal{L}^{k})-\operatorname{rank}(\pi_{*}\mathcal{L}^{k})=\chi(\mathcal{X},\mathcal{L}^{k})-\chi(X,L^{k}).$

In particular orbifold Riemann-Roch on $\mathcal{X}$ and $X$ show that this
has an expansion $b_{0}k^{n+1}+b_{1}k^{n}+\tilde{o}(k^{n})$, where
$b_{0}=\int_{\mathcal{X}}\frac{c_{1}(\mathcal{L})^{n}}{(n+1)!}$ .

We can express $b_{0}$ as an integral over $X$ as follows. Take a hermitian
metric $h$ on $L$ which is invariant under the action of
$S^{1}\subset\mathbb{C}^{*}$ and which has positive curvature $2\pi\omega$.
Let $\sigma$ denote the resulting connection 1-form on the principal
$S^{1}$-bundle given by the unit sphere bundle $S(L)$ of $L$.

Differentiating the $S^{1}$-action gives a vector field $v$ on $S(L)$. Then
$\sigma(v)$ is the pullback of a function $H$ on $X$. With respect to the
symplectic form $\omega$, this $H$ is a Hamiltonian for the $S^{1}$-action on
$X$.

Write $(\mathcal{X},\mathcal{L})$ as the associated bundle to the
$S^{1}$-principal bundle $S(\mathcal{O}_{\mathbb{P}^{1}}(1))$ as follows,

$(\mathcal{X},\mathcal{L})=S(\mathcal{O}_{\mathbb{P}^{1}}(1))\times^{\
}_{S^{1}}(X,L).$

The Fubini-Study connection on $\mathcal{O}_{\mathbb{P}^{1}}(1)$ and the
connection $\sigma$ on $L$ induce natural connections on
$\mathcal{X}\to\mathbb{P}^{1}$ and on $\mathcal{L}\to\mathcal{X}$. In [Don, 2,
Section 5.1] Donaldson shows that the latter has curvature
$H\omega_{FS}+\omega$. (Here $\omega_{FS}$ is pulled back from
$\mathbb{P}^{1}$, and we think of $\omega$ as a form on $\mathcal{X}$ by using
its natural connection over $\mathbb{P}^{1}$ to split its tangent bundle as
$T\mathcal{X}=T\mathbb{P}^{1}\oplus TX$.) Therefore $b_{0}$ equals

$\frac{1}{(n+1)!}\int_{\mathcal{X}}(H\omega_{FS}+\omega)^{n+1}=\frac{n+1}{(n+1)!}\int_{\mathbb{P}^{1}}\omega_{FS}\int_{X}H\omega^{n}=\int_{X}H\frac{\omega^{n}}{n!}.$

This proves

######  Proposition 19.

The total weight of the $\mathbb{C}^{*}$-action on $H^{0}(L^{k})$ is

$w(H^{0}(X,L^{k}))=b_{0}k^{n+1}+b_{1}k^{n}+\tilde{o}(k^{n}),\quad\text{where
}\ b_{0}=\int_{X}H\,\frac{\omega^{n}}{n!}\,.$

We will apply this to weighted projective space $X=\mathbb{P}(V)$ and also to
its sub-orbischemes, where the integral on the right must then take into
account scheme-theoretic multiplicities and the possibility of generic
stabiliser (so if an irreducible component of $X$ has generic stabiliser
$\mathbb{Z}/m\mathbb{Z}$ then the integral over it is $\frac{1}{m}$ times the
integral over the underlying scheme).

Finally we remark that working with $\mathcal{O}_{\mathbb{P}^{2}}(1)$ in place
of $\mathcal{O}_{\mathbb{P}^{1}}(1)$ replaces the trace of the infinitesimal
action on $H^{0}(X,L^{k})$ (i.e. the total weight) by the trace of the
_square_ of the infinitesimal action on $H^{0}(X,L^{k})$, proving it equals

$c_{0}k^{n+2}+O(k^{n+1})\quad\text{where}\
c_{0}=\int_{X}H^{2}\frac{\omega^{n}}{n!}\,.$ (20)

###  9 Reducing to the reductive quotient

To form a moduli space of polarised varieties $(X,L)$ one first embeds $X$ in
projective space $\mathbb{P}$ with a high power of $L$, thus identifying $X$
with a point of the relevant Hilbert scheme of subvarieties of $\mathbb{P}$.
It is easy to see that two points of the Hilbert scheme correspond to
abstractly isomorphic polarised varieties if and only if they differ by an
automorphism of $\mathbb{P}$. Therefore a moduli space of varieties can be
formed by taking the GIT quotient of the Hilbert scheme by the special linear
group. (Different choices of linearisations of the action give different
notions of stability of varieties.)

By Proposition 11 we can now mimic this for polarised orbifolds, first
embedding in a _weighted_ projective space $\mathbb{P}$. The Hilbert scheme of
suborbischemes of $\mathbb{P}$ has been constructed in OS . Therefore we are
left with the problem of quotienting this by the action of
$\operatorname{Aut}(\mathbb{P})$.

At first sight this seems difficult because $\operatorname{Aut}(\mathbb{P})$
is not reductive. Classical GIT works only for reductive groups (though a
remarkable amount of the theory has now been pushed through in the
nonreductive case DK ).

As a trivial example consider $\mathbb{P}(1,2)$ embedded by the identity map
in itself. The automorphisms contain a nonreductive piece $\mathbb{C}$ in
which $t\in\mathbb{C}$ acts by

$[x,y]\ \mapsto\ [x,y+tx^{2}].$ (21)

However this arises because $\mathbb{P}(1,2)$ has not been Kodaira embedded as
described in Section 6. Using _all_ sections of $H^{0}(\mathcal{O}(1))=\langle
x\rangle$ and $H^{0}(\mathcal{O}(2))=\langle x^{2},y\rangle$ (not just $x$ and
$y$) we embed instead via

$\mathbb{P}(1,2)\hookrightarrow\mathbb{P}(1,2,2),\qquad[x,y]\mapsto[x,x^{2},y].$

Then the nonreductive $\mathbb{C}$ lies in a bigger, reductive subgroup of
$\operatorname{Aut}(\mathbb{P}(1,2,2))$. Namely (21) can be realised as the
restriction to $\mathbb{P}(1,2)$ of the automorphism

$[A,B,C]\mapsto[A,B,C+tB]$

lying in the reductive subgroup
$SL(H^{0}(\mathcal{O}(2)))\subset\operatorname{Aut}\mathbb{P}(1,2,2)$. Of
course it can also be seen as the restriction of
$[A,B,C]\mapsto[A,B,C+tA^{2}]$, another nonreductive $\mathbb{C}$ subgroup,
but the point is that our embedding has a stabiliser in
$\operatorname{Aut}(\mathbb{P}(1,2,2))$, and this causes the two copies of
$\mathbb{C}$ restrict to the same action.

Having seen an example, the general case is actually simpler. Given a
polarised variety $(X,L)$, pick an isomorphism from $H^{0}(X,L^{k+i})$ to a
fixed vector space $V^{k+i}$. Then from Section 6 we get an embedding of $X$
into $\mathbb{P}(\oplus_{i}(V^{i+k})^{*})$. This embedding is _normal_ – the
restriction map $H^{0}(\mathcal{O}_{\mathbb{P}}(k+i))\to
H^{0}(\mathcal{O}_{X}(k+i))$ is an isomorphism by construction. The next
result says that the resulting point of the Hilbert scheme of $\mathbb{P}$ is
unique up to the action of the _reductive_ group $\prod_{i}GL(V^{k+i})$.

######  Proposition 22.

Two normally embedded orbifolds
$X_{j}\subset\mathbb{P}(\oplus_{i}(V^{i+k})^{*})$ are abstractly isomorphic
polarised varieties if and only if there is $g\in\prod_{i}GL(V^{k+i})$ such
that $g.X_{1}=X_{2}$.

###### Proof.

If the $(X_{j},\mathcal{O}_{X_{j}}(1)))$ are abstractly isomorphic then their
spaces of sections $H^{0}(\mathcal{O}_{X_{j}}(k+i))$ are isomorphic vector
spaces. Under this isomorphism, the two identifications
$H^{0}(\mathcal{O}_{X_{j}}(k+i))\cong V^{i+k},\ j=1,2$, therefore differ by an
element $g_{k+i}\in GL(V^{k+i})$. Then $g:=\oplus_{i\,}g_{k+i}$ takes
$X_{1}\subset\mathbb{P}$ to $X_{2}$.

The converse is of course trivial, needing only the fact that the action of
$\prod_{i}GL(V^{k+i})$ preserves the polarisation
$\mathcal{O}_{\mathbb{P}}(1)$. ∎

Therefore one can set up a GIT problem to form moduli of orbifolds, just as
Mumford did for varieties.

Firstly one needs Matsusaka’s big theorem for orbifolds, to ensure that for a
fixed $k\gg 0$, uniform over all smooth polarised orbifolds of the same
topological type, the orbifold line bundles $L^{k+i}$ have the number of
sections predicted by orbifold Riemann-Roch. This follows by pushing down to
the underlying variety, which has only quotient, and so rational,
singularities, to which [Mat, , Theorem 2.4] applies.

We can thus embed them all in the same weighted projective space. Then one can
remove those suborbifolds of weighted projective space whose embedding is non-
normal, since they are easily seen to be unstable for the action of
$\prod_{i}GL(V^{k+i})$. Thus by the above result, orbits on the Hilbert scheme
really corresponds to isomorphism classes of polarised orbifolds. Finally one
should compactify with orbischemes (or Deligne-Mumford stacks) to get proper
moduli spaces of stable objects. We do not pursue this here as only smooth
orbifolds and their stability are relevant to cscK metrics, but many of the
foundations are worked out in AH . (Their point of view is slightly different
from ours – their notion of stability is related to the minimal model
programme rather than GIT, and they form moduli using the machinery of
stacks.)

##  Chapter 2 Metrics and balanced orbifolds

Our next point of business is to generalise the Fubini-Study metric to
weighted projective space. Anticipating the application we have in mind, fix
some $k\geq 0$ and let $V=\oplus_{i=1}^{M}V^{k+i}$ be a finite dimensional
graded vector space. By a metric $|\cdot|_{V}$ on a $V$ we will mean a
hermitian metric which makes the vector spaces $V^{p}$ and $V^{q}$ orthogonal
for $p\neq q$. Thus a metric on $V$ is simply given by a hermitian metric
$|\cdot|_{V^{p}}$ on each $V^{p}$. By a graded orthonormal basis
$\\{t_{\alpha}^{p}\\}$ for $V$ we mean an orthonormal basis
$\\{t^{p}_{1},\ldots,t^{p}_{\dim V^{p}}\\}$ for $V^{p}$ for each
$p=k+1,\ldots,k+M$.

As usual let $\mathbb{P}(V)$ be the weighted projective space obtained by
declaring that $V^{k+i}$ has weight $k+i$. The unitary group
$U:=\prod_{i}U(V^{k+i})$ acts on $V$ with moment map

$\mu^{\ }_{U}(v)=\frac{1}{2}\left(v\otimes
v^{*}-\bigoplus_{i}c_{i}\operatorname{Id}_{V^{k+i}}\right)\ \in\
\oplus_{i}\,\mathfrak{u}(V^{k+i})^{*}.$ (1)

Here the $c_{i}$ are arbitrary real constants, which we will take to be
positive, and $v^{*}\in V^{*}$ is the linear functional corresponding to $v$
under the hermitian inner product. Therefore the $U(1)$ action on $V$ which
acts on $V^{k+i}$ with weight $k+i$ has moment map
$\mu_{U(1)}=\operatorname{Tr}_{w}\circ\,\mu_{U}$, where
$\operatorname{Tr}_{w}\colon\mathfrak{u}^{*}\to\mathfrak{u}(1)^{*}$ is the
projection
$\operatorname{Tr}_{w}(\oplus_{i}A^{i})=\sum_{i}(k+i)\operatorname{tr}(A^{i})$.
Thus if $v=\oplus_{i}v_{k+i}$, then

$\mu^{\
}_{U(1)}(v)=\frac{1}{2}\\!\left(\sum_{i}(k+i)|v_{k+i}|^{2}-c\right)\\!\\!,\quad\text{where
}\ c:=\sum_{i}(k+i)c_{i}\dim V^{k+i}.$ (2)

######  Definition 3.

The _Fubini-Study_ orbifold Kähler metric $\omega_{FS}$ associated to
$|\cdot|_{V}$ is $\frac{1}{c}$ times the metric on $\mathbb{P}(V)$ which
results from viewing it as the symplectic quotient $\mu_{U(1)}^{-1}(0)/U(1)$
and taking the Kähler reduction of the metric $|\cdot|_{V}$ under the
isometric action of $U$.

This is an orbifold Kähler metric: on the orbifold chart (2) it pulls back to
a genuine Kähler metric on $\mathbb{C}^{n-1}$. In fact it follows from Lemma 6
below that it is the curvature of a hermitian metric $h_{1}$ on the orbifold
line bundle $\mathcal{O}_{\mathbb{P}(V)}(1)$. The dual of this hermitian
metric is the one of three natural candidates for the name of Fubini-Study
metric on $\mathcal{O}_{\mathbb{P}(V)}(-1)$. A second natural choice $h_{2}$
is given by $|v|^{2}_{h_{2}}=\sum_{i}|v_{k+i}|^{\frac{2}{k+i}}$ (note that
$|v|^{2}=\sum_{i}|v_{k+i}|^{2}$ does not scale correctly under the action of
$\mathbb{C}^{*}$ to define a hermitian metric). However it is the third
candidate $h_{3}=h_{FS}$ below that we choose. It should be noted that only on
an unweighted projective space do all three agree and metrics. It seems that
$h_{FS}$ is a special case of the more general metrics on line bundles over
toric varieties constructed by Batyrev-Tschinkel [BT, , Section 2.1].

######  Definition 4.

The _Fubini-Study metric_ $h_{FS}$ on $\mathcal{O}_{\mathbb{P}(V)}(-1)$ is the
hermitian metric defined by setting the points of $\mu_{U(1)}^{-1}(0)$ to have
norm 1\. Therefore

$|v|_{h_{FS}}:=\frac{1}{\lambda(v)}\,,$

where $\lambda(v).v$ is the unique point of $\mu_{U(1)}^{-1}(0)$ in the orbit
$(0,\infty).v$. That is, by (2), $\lambda(v)$ is the unique positive real
solution to

$\sum_{i}(k+i)\lambda(v)^{2(k+i)}|v_{k+i}|^{2}=c.$ (5)

We also use $h_{FS}$ to denote the induced metrics on
$\mathcal{O}_{\mathbb{P}(V)}(i)$.

The discrepancy between $\omega_{FS}$ and the curvature
$2\pi\omega_{h_{FS}}:=i\partial\overline{\partial}\log h_{FS}$ of the metric
$h_{FS}$ on $\mathcal{O}_{\mathbb{P}(V)}(1)$ can be deduced from a result in
BG [1].

######  Lemma 6.

We have

$\omega_{FS}=\omega_{h_{FS}}+\frac{i}{2c}\partial\overline{\partial}f,$ (7)

where $f\colon\mathbb{P}(V)\to\mathbb{R}$ is the function

$f:=\sum_{i}\sum_{\alpha}|t^{i}_{\alpha}|^{2}_{h_{FS}}.$ (8)

Here $\\{t_{\alpha}^{i}\\}$ is a $|\cdot|_{V}$-orthonormal basis of $V^{*}$,
so each $t_{\alpha}^{i}$ defines a section of
$\mathcal{O}_{\mathbb{P}(V)}(i)$, whose pointwise $h_{FS}$-norm is what
appears in (8).

###### Proof.

Let $p\colon V\backslash\\{0\\}\to\mathbb{P}(V)$ be projection to the
quotient. We use [BG, 1, 3.1]; in their notation we set $\chi$ to be the $c$
th power homomorphism from $S^{1}$ to itself and shift our moment map by
$\frac{c}{2}$ to agree with theirs. The result is that the pullback of the
Kähler form produced by symplectic reduction is

$p^{*}(c\,\omega_{FS})=\frac{i}{2}\partial\overline{\partial}\,|\lambda(v).v|_{V}^{2}+\frac{i}{2\pi}\partial\overline{\partial}\log\lambda(v)^{c},$
(9)

where $\lambda(v)\in(0,\infty)$ is defined as in (5) so that
$\lambda(v).v\in\mu_{U(1)}^{-1}(0)$.

Over an open set of $\mathbb{P}(V)$ pick a holomorphic section, or
multisection, of $p$, lifting $x$ to $v=v(x)$. Then the curvature of $h_{FS}$
on $\mathcal{O}_{\mathbb{P}(V)}(-1)$ is
$i\partial\overline{\partial}\log|v|_{h_{FS}}$, which by Definition 4 is
$i\partial\overline{\partial}\log\lambda(v)^{-1}$. Therefore the curvature of
$\mathcal{O}_{\mathbb{P}(V)}(1)$ is
$i\partial\overline{\partial}\log\lambda(v)$ and we can rewrite (9) (divided
through by $c$) as

$p^{*}(\omega_{FS})=\frac{i}{2c}\partial\overline{\partial}\,|\lambda(v).v|_{V}^{2}+p^{*}\omega_{h_{FS}}.$

Then at $v\in V\backslash\\{0\\}$ lying over a point $x\in\mathbb{P}(V)$ we
calculate $|\lambda(v).v|_{V}^{2}$ as

$\sum_{i}|\lambda(v).v|_{V_{i}}^{2}=\sum_{i}\sum_{\alpha}|t^{i}_{\alpha}(\lambda(v).v)|^{2}=\sum_{i}\sum_{\alpha}|t^{i}_{\alpha}|^{2}_{h_{FS},x}\,.$

The last equality follows from the definition of $h_{FS}$ (4), since
$\lambda(v).v$ lies in $\mu_{U(1)}^{-1}(0)$. ∎

The restriction of $\mu_{U}$ (1) to $\mu_{U(1)}^{-1}(0)$ descends to
$\mathbb{P}(V)$ as the moment map $m$ for the induced action of $U/U(1)$ on
$\mathbb{P}(V)$:

$m([v])=\frac{1}{2}\bigoplus_{i}\left(\lambda^{2(k+i)}(v)\,v_{k+i}\otimes
v_{k+i}^{*}-c_{i}\operatorname{Id}_{V^{k+i}}\right),$ (10)

with $\lambda(v)$ is defined in (5). Integrating this allows us to define a
notion of balanced orbifolds.

######  Definition 11.

Given an orbifold embedding $X\subset\mathbb{P}(V)$ define

$M(X)=\int_{X}m\,\frac{\omega_{FS}^{n}}{n!}\,,$

where $m$ is the moment map from (10). We say that an orbifold
$X\subset\mathbb{P}(V)$ is _balanced_ if $M(X)=0$.

######  Remark 12.

The balanced condition depends on $|\cdot|_{V}$ and on the choice of constants
$c_{i}$. Later we will choose specific constants to ensure a connection with
scalar curvature.

Just as in the manifold situation Don [1]; Wan , $M$ is the moment map for the
action of $U/U(1)$ on Olsson and Starr’s Hilbert scheme OS  of sub-orbischemes
of $\mathbb{P}(V)$ endowed with its natural $L^{2}$-symplectic form. To make
sense of this statement one can either work purely formally, make a precise
statement at smooth points, or restrict attention to a single orbit of
Aut$(\mathbb{P}(V))$; the latter is smooth and all we will need in the
application to constant scalar curvature. For $X\subset\mathbb{P}(V)$ and
$v,w$ sections of $T\mathbb{P}(V)|_{X}$ their pairing with the symplectic form
is defined to be

$\Omega(v,w):=\int_{X}v\lrcorner\left(w\lrcorner\frac{\omega^{n+1}}{(n+1)!}\right).$

The moment map calculation is the following. We let $A=\oplus_{i}A^{k+i}$ be a
graded Hermitian matrix generating the 1-parameter subgroup $\exp(tA)$ of
automorphisms of $\mathbb{P}(V)$, inducing the vector field $v_{A}$ on
$\mathbb{P}(V)$. Since $m_{A}:=\operatorname{tr}(mA)$ is a hamiltonian for
$v_{A}$, we have $v_{A}\lrcorner\,\omega=dm_{A}$. Moving in the Hilbert scheme
down a vector field $v$ on $\mathbb{P}(V)$ we have

$\displaystyle\left.\frac{d}{dt}\right|_{t=0}\\!\\!\operatorname{tr}(M(X)A)$
$\displaystyle=$
$\displaystyle\int_{X}\mathcal{L}_{v}\left(m_{A}\frac{\omega^{n}}{n!}\right)$
(13) $\displaystyle=$
$\displaystyle\int_{X}v(m_{A})\frac{\omega^{n}}{n!}+\int_{X}m_{A}d\left(v\lrcorner\frac{\omega^{n}}{n!}\right)$
$\displaystyle=$
$\displaystyle\int_{X}\omega(v,v_{A})\frac{\omega^{n}}{n!}-\int_{X}d(m_{A})\wedge\left(v\lrcorner\frac{\omega^{n}}{n!}\right)$
$\displaystyle=$
$\displaystyle\int_{X}\omega(v,v_{A})\frac{\omega^{n}}{n!}-\int_{X}(v_{A}\lrcorner\,\omega)\wedge\left(v\lrcorner\frac{\omega^{n}}{n!}\right)$
$\displaystyle=$
$\displaystyle\int_{X}v\lrcorner\left(v_{A}\lrcorner\left(\frac{\omega^{n+1}}{(n+1)!}\right)\right)=\Omega(v,v_{A}).$

To express the balanced condition in terms of sections of line bundles, fix a
polarised orbifold $(X,L)$ with cyclic stabiliser groups. Embed $X$ in
weighted projective space with $k\gg 0$ as in Section 6:

$\phi_{k}\colon X\hookrightarrow\mathbb{P}(V)\ \text{ where }\
V=\bigoplus_{i=1}^{M}H^{0}(L^{k+i})^{*}\text{ and
}L=\phi_{k}^{*}\mathcal{O}_{\mathbb{P}(V)}(1).$ (14)

A metric $|\cdot|_{V}$ on $V$ induces by Definition 4 a Fubini-Study metric on
$\mathcal{O}(1)$, and so one on $L$ which we also denote by $h_{FS}$. The next
Lemma expresses the balanced condition in terms of coordinates on $V$ given by
a graded $|\cdot|_{V}$-orthonormal basis $\\{t_{\alpha}^{i}\\}$, where
$t^{i}_{\alpha}\in H^{0}(L^{k+i})$. To ease notation we write

$\operatorname{vol}:=\int_{X}\frac{c_{1}(L)^{n}}{n!}\,.$

######  Lemma 15.

With respect to these coordinates the matrix $M(X)=\oplus_{i}M^{i}(X)$ has
entries

$(M^{i}(X))_{\alpha\beta}=\frac{1}{2}\left(\int_{X}(t^{i}_{\alpha},t^{i}_{\beta})_{h_{FS}}\,\frac{\omega_{FS}^{n}}{n!}-c_{i}\operatorname{vol}\delta_{\alpha\beta}\right).$

###### Proof.

Given a point $x$ in $X$ let $\tilde{x}\in L_{x}^{-1}$ be any non-zero lift,
and write $t_{\alpha}^{i}(\tilde{x})$ for the complex number
$(\tilde{x}^{\otimes(k+i)},t_{\alpha}^{i}(x))$. Then

$(t_{\alpha},t_{\beta})_{h_{FS}}=\lambda(\tilde{x})^{2(k+i)}t_{\alpha}^{i}(\tilde{x})\overline{t_{\beta}^{i}(\tilde{x})}$

where $\lambda(\tilde{x})$ is the positive solution to
$\sum_{i}(k+i)\lambda(\tilde{x})^{2(k+i)}\sum_{\alpha}|t_{\alpha}^{i}(\tilde{x})|^{2}=c.$
Now the embedding of $X$ maps $x$ to the point with coordinates
$[t_{\alpha}^{i}(x)]$, so in these coordinates $m(x)=\oplus_{i}m^{i}(x)$ where

$(m^{i}(x))_{\alpha\beta}=\frac{1}{2}\left(\lambda(\tilde{x})^{2(k+i)}t_{\alpha}^{i}(\tilde{x})\overline{t_{\beta}^{i}(\tilde{x})}-c_{i}\delta_{\alpha\beta}\right)=\frac{1}{2}\left((t^{i}_{\alpha},t^{i}_{\beta})_{h_{FS}}-c_{i}\delta_{\alpha\beta}\right)$

and the result follows by integrating over $X$. ∎

The balanced condition can also be expressed in terms of hermitian metrics on
$L$. Let $\mathcal{K}(c_{1}(L))$ denote the orbifold Kähler metrics on $X$
which are $(2\pi)^{-1}$ times the curvature of an orbifold hermitian metric on
$L$. Define maps as follows:

  * •

If $|\cdot|_{V}$ is a metric on $V:=\oplus_{i}H^{0}(L^{k+i})^{*}$ then

$FS(|\cdot|_{V})=(\phi_{k}^{*}h_{FS},\phi_{k}^{*}\omega_{FS}),$

where $h_{FS}$ and $\omega_{FS}$ are Fubini-Study metrics associated to
$|\cdot|_{V}$.

  * •

If $h$ is a hermitian metric on $L$ and $\omega$ a Kähler metric in
$\mathcal{K}(c_{1}(L))$ the metric $\operatorname{Hilb}(h,\omega)$ on $V$ is
defined by requiring that for $s\in H^{0}(L^{k+i})$

$|s|^{2}_{\operatorname{Hilb}(h,\omega)}=\frac{1}{c_{i}\operatorname{vol}}\int_{X}|s|^{2}_{h}\frac{\omega^{n}}{n!}\,.$
(16)

Notice this differs from the usual $L^{2}$-metric by the
$c_{i}\operatorname{vol}$ factors. Obviously $V$ and the maps
$\operatorname{Hilb}$ and $\operatorname{FS}$ depend on $k$, and this will
always be clear from the context.

######  Definition 17.

We say that the pair $(h,\omega)$ is _balanced_ at level $k$ if it is a fixed
point of $\operatorname{FS}\circ\operatorname{Hilb}$. A metric $|\cdot|_{V}$
is said to be _balanced_ if it is a fixed point of
$\operatorname{Hilb}\circ\operatorname{FS}$.

######  Proposition 18.

A metric $|\cdot|_{V}$ on $V$ is balanced if and only if $\phi_{k}\colon
X\subset\mathbb{P}(V)$ is a balanced orbifold.

###### Proof.

Given a metric $|\cdot|_{V}$ let $(h_{FS},\omega_{FS})=FS(|\cdot|_{V})$ and
$\\{t_{\alpha}^{i}\\}$ be a graded $|\cdot|_{V}$-orthonormal basis for $V$.
Then by Lemma 15, $M(X)=0$ if and only if

$\frac{1}{c_{i}\operatorname{vol}}\int_{X}(t_{\alpha}^{i},t^{i}_{\beta})_{h_{FS}}\,\frac{\omega_{FS}^{n}}{n!}=\delta_{\alpha\beta}\quad\text{
for all }i,\alpha,\beta,$

if and only if $\\{t_{\alpha}^{i}\\}$ is orthonormal with respect to the
$\operatorname{Hilb}(h_{FS},\omega_{FS})$ metric, if and only if it is the
same metric as $|\cdot|_{V}$. ∎

Another way to express the balanced condition is through Bergman kernels.

######  Definition 19.

Let $h$ be a hermitian metric on $L$ and $\omega$ be a Kähler metric on $X$.
The _weighted Bergman kernel_ is the function

$B_{k}=B_{k}(h,\omega):=\operatorname{vol}\sum_{i}c_{i}(k+i)\sum_{\alpha}|s_{\alpha}^{i}|_{h}^{2}$

where $\\{s_{\alpha}^{i}\\}$ is graded basis of $\oplus_{i}H^{0}(L^{k+i})$
that is orthonormal with respect to the $L^{2}$-metric defined by
$(h,\omega)$. Equivalently

$B_{k}=\sum_{i}(k+i)\sum_{\alpha}|t_{\alpha}^{i}|_{h}^{2}$

where $\\{t_{\alpha}^{i}\\}$ is orthonormal with respect to the
$\operatorname{Hilb}(h,\omega)$ metric. Of course $B_{k}$ is independent of
these choices of basis.

If $B_{k}$ is constant over $X$, then we see by integrating over $X$ that this
constant is necessarily $c=\sum_{i}c_{i}(k+i)h^{0}(L^{k+i})$. In the
unweighted case, $B_{k}$ can be written invariantly in terms of the ratio of
the hermitian metrics $h$ and $h_{FS}$ on $L$. We have the following analogue
here.

######  Proposition 20.

Fix a hermitian metric $h$ on $L$ and a Kähler metric
$\omega\in\mathcal{K}(c_{1}(L))$ and let
$(h_{FS},\omega_{FS})=FS\circ\operatorname{Hilb}(h,\omega)$. Then $h=h_{FS}$
if and only if $B_{k}(h,\omega)\equiv c$ is constant on $X$.

###### Proof.

Let $\\{t_{\alpha}^{i}\\}$ be a graded basis for $\oplus_{i}H^{0}(L^{k+i})$
that is orthonormal with respect to the
$\operatorname{Hilb}(h,\omega)$-metric. For $x\in X$ let $\tilde{x}$ be any
non-zero lift in $L^{-1}|_{x}$. Then

$\displaystyle\sum_{i}(k+i)\sum_{\alpha}|t_{\alpha}^{i}(x)|^{2}_{h_{FS}}$
$\displaystyle=$
$\displaystyle\sum_{i}(k+i)\sum_{\alpha}|(\tilde{x},t_{\alpha}^{i}(x))|^{2}|\tilde{x}|^{-2(k+i)}_{h_{FS}}$
(21) $\displaystyle=$
$\displaystyle\sum_{i}(k+i)\sum_{\alpha}\lambda(\tilde{x})^{2(k+i)}|(\operatorname{ev}_{\tilde{x}}^{i},t_{\alpha}^{i})|^{2}$
$\displaystyle=$
$\displaystyle\sum_{i}(k+i)\lambda(\tilde{x})^{2{i}}|\operatorname{ev}_{\tilde{x}}^{i}|^{2}=c,$

from the definition of dual norms, the fact that
$\lambda(\tilde{x})=|\tilde{x}|_{h_{FS}}^{-1}$ and the defining equation for
$\lambda(\tilde{x})$ (5). Thus if $h=h_{FS}$ then $B_{k}$ is constant.
Conversely, if $\beta:=h_{FS}/h$ we have

$c=\sum_{i}(k+i)\sum_{\alpha}|t_{\alpha}^{i}(x)|^{2}_{h_{FS}}=\sum_{i}^{m}(k+i)\sum_{\alpha}\beta(x)^{(k+i)}|t_{\alpha}^{i}(x)|^{2}_{h}.$
(22)

Now note that for fixed $x$, the quantity
$u_{i}=(k+i)\sum_{\alpha}|t_{\alpha}^{i}(x)|^{2}_{h}$ is nonnegative for each
$i$, so there is a unique positive real solution to the equation
$\sum_{i=1}\beta^{2(k+i)}(x)u_{i}=c$. If $B_{k}\equiv c$ is constant then
$\beta(x)=1$ is one solution, and thus the unique solution, so $h=h_{FS}$. ∎

##  Chapter 3 Limits of Fubini-Study metrics

The connection between constant scalar curvature metrics and stability comes
through the asymptotics of Fubini-Study metrics. The crucial ingredient is the
asymptotics, as $k\to\infty$, of the weighted Bergman kernel of Definition 19:

$B_{k}=\operatorname{vol}\,\sum_{i}c_{i}(k+i)\sum_{\alpha}|s^{i}_{\alpha}|^{2}_{h}.$

Here $\\{s_{\alpha}^{i}\\}$ is a basis of $H^{0}(L^{k+i})$ that is orthonormal
with respect to the $L^{2}$-metric induced by $h$ and $\omega$. Ensuring that
this is related to scalar curvature requires a particular choice of $c_{i}$,
so for concreteness assume from now on they are chosen by requiring

$\sum_{i}c_{i}t^{i}:=(t^{\operatorname{ord}(X)-1}+t^{\operatorname{ord}(X)-2}+\cdots+1)^{p+1}$
(1)

for some sufficiently large integer $p$. We prove in [RT, 3, 1.7 and 4.13]
that with this choice of $c_{i}$ there is an asymptotic expansion

$B_{k}=b_{0}k^{n+1}+b_{1}k^{n}+\cdots\quad\mathrm{as}\ k\to\infty$ (2)

for some smooth functions $b_{i}$. Taking larger values of $p$ yields a
stronger expansion: in fact if $p\geq r+q$ for integers $r,q\geq 0$ then (2)
holds up to terms of order $O(k^{n+1-r})$ in the $C^{q}$-norm. By this we mean
that there is a constant $C$ such that for all $k$,

$\left|\\!\left|B_{k}-b_{0}k^{n+1}-b_{1}k^{n}-\cdots-
b_{r-1}k^{n+1-(r-1)}\right|\\!\right|\leq Ck^{n+1-r},$

where the norm is the $C^{q}$-norm taken over $X$ in the orbifold sense, with
the pointwise norm of the derivatives measured with respect to the metric
defined by $\omega$. Moreover the constant $C$ can be taken to be uniform for
$(h,\omega)$ in a compact set.

To achieve what we need in this paper it is sufficient to select $p=5$, so in
particular there is a $C^{2}$-expansion involving the top two terms $b_{0}$
and $b_{1}$; however nothing is lost if the reader prefers to take a larger
$p$ for simplicity. Moreover if $2\pi\omega_{h}$ denotes the curvature
$i\partial\overline{\partial}\log h$ of $h$, the top two coefficients are
given by [RT, 3, 1.11]

$\displaystyle b_{0}$ $\displaystyle=$
$\displaystyle\operatorname{vol}\,\frac{\omega_{h}^{n}}{\omega^{n}}\sum_{i}c_{i},$
$\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle\operatorname{vol}\,\frac{\omega_{h}^{n}}{\omega^{n}}\sum_{i}c_{i}\left((n+1)i+\operatorname{tr}_{\omega_{h}}(\operatorname{Ric}(\omega))-\frac{1}{2}\operatorname{Scal}(\omega_{h})\right).$

In particular if $2\pi\omega$ is in fact the curvature of $h$ this simplifies
to

$b_{0}=\operatorname{vol}\,\sum_{i}c_{i},\qquad
b_{1}=\operatorname{vol}\,\sum_{i}c_{i}\left((n+1)i+\frac{1}{2}\operatorname{Scal}(\omega)\right).$
(3)

Observe that in this case the top order term, $b_{0}$, is constant over $X$.

Now integrating the expansion over $X$ shows the quantity
$c=\sum_{i}c_{i}(k+i)h^{0}(L^{k+i})$ is polynomial modulo small terms (this is
shown directly in Lemma 5). In fact

$c=\operatorname{vol}\sum_{i}c_{i}\left[k^{n+1}+\left((n+1)i+\frac{\,\overline{\\!S}}{2}\right)k^{n}\right]+O(k^{n-1}),$
(4)

where $\overline{\\!S}$ denotes the average of the scalar curvature of any
Kähler metric in $\mathcal{K}(c_{1}(L))$.

Similarly [RT, 3, Remark 4.13] there is also an asymptotic expansion

$\operatorname{vol}\,\sum_{i}c_{i}\sum_{\alpha}|s^{i}_{\alpha}|^{2}_{h}\ =\
b_{0}k^{n}+b_{1}^{\prime}k^{n-1}+\cdots$ (5)

for some function $b_{1}^{\prime}$, and where $b_{0}$ is as above. Here the
choice of $c_{i}$ is as above (1), and if $p\geq r+q$ the expansion is in the
$C^{q}$-norm up to terms of order $O(k^{n-r})$.

In what follows fix a hermitian metric $h$ on $L$ and Kähler metric
$\omega\in\mathcal{K}(c_{1}(L))$, and let $(h_{FS,k},\omega_{FS,k})$ be the
pair $FS\circ\operatorname{Hilb}(h,\omega)$ coming from the embedding
$X\subset\mathbb{P}(\oplus_{i}H^{0}(L^{k+i})^{*})$. For embeddings of
manifolds in ordinary projective space, the asymptotics of $h/h_{FS,k}$ are
those of the Bergman kernel. For orbifolds, the fact that the Fubini-Study
fibre metric is defined implicitly in Definition 4 means that we have to work
harder.

######  Theorem 6.

Suppose that $2\pi\omega$ is the curvature of $h$. Then
$(h_{FS,k},\omega_{FS,k})$ converges to $(h,\omega)$ as $k$ tends to infinity.
In fact if $\,\overline{\\!S}$ denotes the average of the scalar curvature
then

$\frac{h_{FS,k}}{h}=1+\frac{\,\overline{\\!S}-\operatorname{Scal}(\omega)}{2}k^{-2}+O({k^{-3}})$
(7)

in the $C^{2}$-norm, and

$\omega=\omega_{FS,k}+O(k^{-2})$ (8)

in $C^{0}$. In particular the set of Fubini-Study Kähler metrics is dense in
$\mathcal{K}(c_{1}(L))$.

######  Remark 9.

The Theorem can be generalised to the case that $\omega$ is not the curvature
of $h$, in which case there will be an additional $O(k^{-1})$ term appearing
in the expansion of $h_{FS,k}/h$.

###### Proof of (7).

The aim is to find an asymptotic expansion of

$\displaystyle\alpha_{k}:=\frac{h_{FS,k}}{h}\,.$

Set $\mathcal{B}_{r}:=\sum_{\alpha}|t^{r}_{\alpha}|_{h}^{2}$ where
$\\{t^{r}_{\alpha}\\}$ is a basis of $H^{0}(L^{r})$ that is orthonormal with
respect to the $\operatorname{Hilb}(h,\omega)$-norm from (16), so that
$B_{k}=\sum_{i}(k+i)\mathcal{B}_{k+i}$. Then if $0\neq\tilde{x}\in
L_{x}^{-1}$,

$\displaystyle\sum_{i}(k+i)\alpha_{k}^{k+i}\mathcal{B}_{{k+i}}$
$\displaystyle=$
$\displaystyle\sum_{i}(k+i)\|\tilde{x}^{k+i}\|^{-2}_{h_{FS,k}}\sum_{\alpha}|t_{\alpha}^{k+i}(\tilde{x})|_{h}^{2}$
(10) $\displaystyle=$
$\displaystyle\sum_{i}(k+i)\|\tilde{x}^{k+i}\|_{h_{FS,k}}^{-2}\sum_{\alpha}\|\tilde{x}\|^{2}_{\operatorname{Hilb}(h,\omega)}$
$\displaystyle=$ $\displaystyle c,$

where the second equality uses the fact that the $t_{\alpha}^{k+i}$ are
orthonormal, the third inequality comes from the definition of the $FS$-norm
(5), and as in (2), $c=\sum_{i}c_{i}(k+i)h^{0}(L^{k+i})$ is constant over $X$.

We aim first for an asymptotic expansion of $\alpha_{k}$ that holds in
$C^{0}$. Say a sequence $a_{k}$ of real numbers is of order $\Omega(k^{p})$ if
there is a $\delta>0$ such that $a_{k}\geq\delta k^{p}$ for $p\gg 0$. A
sequence of real-valued functions $f_{k}$ on $X$ is of order $\Omega(k^{p})$
if there is a $\delta>0$ with $f_{k}\geq\delta k^{p}$ uniformly on $X$ for all
$p\gg 0$.

Step 1: We show $\alpha_{k}=1+O(k^{-1})$ in $C^{0}$. Observe that from (3) and
(4),

$B_{k}=\sum_{i}(k+i)\mathcal{B}_{k+i}=\operatorname{vol}\sum_{i}c_{i}k^{n+1}+O(k^{n})=c+O(k^{n}).$

Taking the difference with (10) gives

$\sum_{i}(k+i)\Big{(}\alpha_{k}^{k+i}-1\Big{)}\mathcal{B}_{k+i}=O(k^{n}),$

and so

$(\alpha_{k}-1)\sum_{i}(k+i)\Big{[}1+\alpha_{k}+\alpha_{k}^{2}+\dots+\alpha_{k}^{k+i-1}\Big{]}\mathcal{B}_{k+i}=O(k^{n}).$
(11)

Now $\alpha_{k}$ is pointwise positive, so the term in square brackets is at
least $1$, and $\sum_{i}(k+i)\mathcal{B}_{k+i}=\Omega(k^{n+1})$, so the sum on
the left hand side is $\Omega(k^{n+1})$. Thus $\alpha_{k}-1=O(k^{-1})$ as
claimed.

Step 2: There are positive constants $C_{1},C_{2}$ such that

$\displaystyle C_{1}$ $\displaystyle\leq$
$\displaystyle\alpha_{k}^{j}\quad\text{ for all }\frac{k}{2}\leq j\leq k.$
$\displaystyle\alpha_{k}^{j}$ $\displaystyle\leq$ $\displaystyle
C_{2}\quad\text{ for all }0\leq j\leq k.$ (12)

###### Proof.

As $\alpha_{k}=1+O(k^{-1})$ we have $C_{1}^{2}\leq\alpha_{k}^{k}\leq C_{2}$
for some $C_{1}\in(0,1),\ C_{2}>1$ and all $k\gg 0$. Thus for
$j\geq\frac{k}{2}$ we have $\alpha_{k}^{j}\geq C_{1}$ and for $j\leq k$ we
have $\alpha_{k}^{j}\leq C_{2}$. ∎

Using this we can improve on Step 1 by observing that the term in square
brackets in (11) is of order $\Omega(k)$ since each power of $\alpha_{k}$ is
nonnegative, and there are at least $k/2$ terms bounded from below by $C_{1}$.
Hence

$\alpha_{k}-1=O(k^{-2})\quad\text{in }C^{0}.$

Step 3: Next define

$\beta_{k}=1+\frac{\overline{\\!S}-\operatorname{Scal}(\omega)}{2}k^{-2}.$
(13)

We claim that

$\displaystyle\sum_{i}(k+i)\beta_{k}^{k+i}\mathcal{B}_{k+i}=c+O(k^{n-1})\quad\text{
in }C^{0}.$ (14)

That is, the $\beta_{k}$ satisfy an implicit equation very close to the one
(10) satisfied by the $\alpha_{k}$, which we shall use to deduce they are
approximately equal.

###### Proof.

Note

$\beta_{k}^{k+i}=1+\frac{\overline{\\!S}-\operatorname{Scal}(\omega)}{2}k^{-1}+O(k^{-2}).$

So using the asymptotic expansion (2, 3) of the weighted Bergman kernel
$B_{k}=\sum(k+i)\mathcal{B}_{k+i}$,

$\displaystyle\sum_{i}$
$\displaystyle(k+i)\beta_{k}^{k+i}\mathcal{B}_{{k+i}}=\sum_{i}(k+i)\left(1+\frac{\overline{\\!S}-\operatorname{Scal}(\omega)}{2k}+O(k^{-2})\right)\mathcal{B}_{k+i}$
$\displaystyle=\operatorname{vol}\sum_{i}c_{i}\left[k^{n+1}+\left(\frac{\overline{\\!S}-\operatorname{Scal}(\omega)}{2}+(n+1)i+\frac{\operatorname{Scal}(\omega)}{2}\right)k^{n}\right]+O(k^{n-1})$
$\displaystyle=\operatorname{vol}\sum_{i}c_{i}\left[k^{n+1}+\left((n+1)i+\frac{\overline{\\!S}}{2}\right)k^{n}\right]+O(k^{n-1})\quad\text{in
}C^{0},$ (15)

since $\mathcal{B}_{k+i}=O(k^{n})$ in $C^{0}$. Comparing with (4) proves the
claim. ∎

Step 4: To simplify notation set

$\gamma_{k}:=\alpha_{k}^{k+i-1}+\alpha_{k}^{k+i-2}\beta_{k}+\dots+\beta_{k}^{k+i-1}.$

Taking the difference between the implicit equations (10) and (14) for
$\alpha_{k}$ and $\beta_{k}$ yields

$(\alpha_{k}-\beta_{k})\sum_{i}(k+i)\gamma_{k}\mathcal{B}_{k+i}=O(k^{n-1})\quad\text{in
}C^{0}.$ (16)

From (12) and the definition (13) of $\beta_{k}$ we see that
$\gamma_{k}=\Omega(k)$. Therefore by (16),

$\alpha_{k}=\beta_{k}+O(k^{-3})=1+\frac{\overline{\\!S}-\operatorname{Scal}(\omega)}{2}k^{-2}+O(k^{-3}),$
(17)

which is the expansion we wanted at the level of $C^{0}$-norms.

Step 5: To extend this to the $C^{2}$-norm we actually require an expansion in
the $C^{0}$-norm to higher order (this is because although the pieces of the
Bergman kernel $\mathcal{B}_{k+i}$ are of order $O(k^{n})$, their derivatives
$D^{p}\mathcal{B}_{k+i}$ are of order $O(k^{n+p})$ [RT, 3, Corollary 4.10],
resulting in a loss of a factor of $k$ for each derivative we take). To
achieve this replace $\beta_{k}$ with

$\beta_{k}=1+\frac{\overline{\\!S}-\operatorname{Scal}(\omega)}{2}k^{-2}+\tau_{1}k^{-3}+\tau_{2}k^{-4},$

where the $\tau_{i}$ are smooth functions independent of $k$. Then the
coefficient of $k^{n-1}$ in (15) is $b_{0}\tau_{1}+f$, where $f$ is
independent of $k$ and the $\tau_{i}$. Similarly the coefficient of $k^{n-2}$
is $b_{0}\tau_{2}+g$, where $g$ is independent of $k$ and $\tau_{2}$.

So setting $\tau_{1}=-f/b_{0}$ and $\tau_{2}=-g/b_{0}$ we may assume that the
$k^{n-1}$ and $k^{n-2}$ terms in (15) vanish. Therefore

$\sum_{i}(k+i)\beta_{k}^{k+i}\mathcal{B}_{k+i}=c+O(k^{n-3+p})\text{ in
}C^{p}\text{ for }p=0,1,2,$ (18)

where we have used $\mathcal{B}_{k+i}=O(k^{n+p})$ in $C^{p}$ in place of the
original argument using $\mathcal{B}_{k+i}=O(k^{n})$ in $C^{0}$. Thus

$(\alpha_{k}-\beta_{k})\sum_{i}(k+i)\gamma_{k}\mathcal{B}_{k+i}=O(k^{n-3+p})\text{
in }C^{p},\ p=0,1,2.$ (19)

In particular, $\alpha_{k}=\beta_{k}+O(k^{-5})$ in $C^{0}$.

Now to bound $D\alpha_{k}$, differentiate (10) to get

$D{\alpha_{k}}\sum_{i}(k+i)^{2}\alpha_{k}^{k+i-1}\mathcal{B}_{k+i}=-\sum_{i}(k+i)\alpha_{k}^{k+i}D\mathcal{B}_{k+i}.$
(20)

Since powers of $\alpha_{k}$ are bounded above uniformly (12) and
$D\mathcal{B}_{k+i}=O(k^{n+1})$, the sum on the right hand side is
$O(k^{n+2})$. On the other hand, using the lower bound in (12), the sum on the
left hand side is of order $\Omega(k^{n+2})$, and hence $D\alpha_{k}=O(1)$. We
claim that $D\gamma_{k}=O(k^{2})$. In fact both $\alpha_{k}^{j}$ and
$\beta_{k}^{j}$ are uniformly bounded from above for all $k$ and all $j\leq
k+i$. Thus if $u+v\leq k+i$,

$D(\alpha_{k}^{u}\beta_{k}^{v})=u\alpha_{k}^{u-1}\beta_{k}^{v}D{\alpha_{k}}+v\alpha_{k}^{u}\beta_{k}^{v-1}D\beta_{k}=O(k),$
(21)

since $D\alpha_{k}=O(1)$ and $D\beta_{k}=O(k^{-2})$. Thus $D\gamma_{k}$ is a
sum of $O(k)$ terms each of order $O(k)$ and so $D\gamma_{k}=O(k^{2})$ as
claimed.

So we know $\gamma_{k}\mathcal{B}_{k+i}=O(k^{n+1})$ and
$D(\gamma_{k}\mathcal{B}_{k+i})=O(k^{n+2})$. Differentiating the $p=1$
statement of (19) and using $\gamma_{k}=\Omega(k)$ yields

$(D\alpha_{k}-D\beta_{k})\Omega(k^{n+2})=-(\alpha_{k}-\beta_{k})\sum_{i}(k+i)D(\gamma_{k}\mathcal{B}_{k+i})+O(k^{n-2})=O(k^{n-2})$

as $\alpha_{k}-\beta_{k}=O(k^{-5})$. Hence $D\alpha_{k}=D\beta_{k}+O(k^{-4})$,
and thus we have $\alpha_{k}-\beta_{k}=O(k^{-4})$ in $C^{1}$. In particular
$D\alpha_{k}=O(k^{-2})$.

A similar argument applies to the second derivative. Differentiating (20)
yields

$\displaystyle(D^{2}\alpha_{k})\Omega(k^{n+2})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!-2\sum_{i}(k+i)^{2}\alpha_{k}^{k+i-1}D\alpha_{k}D\mathcal{B}_{k+i}-\sum_{i}(k+i)\alpha_{k}^{k+i}D^{2}\mathcal{B}_{k+i}$
$\displaystyle-\sum_{i}(k+i)^{2}(k+i-1)\alpha_{k}^{k+i-2}(D{\alpha_{k}})^{2}\mathcal{B}_{k+i}$

which is $O(k^{n+3})$. Thus $D^{2}\alpha_{k}=O(k)$. If $u+v\leq k+i$ then

$D^{2}(\alpha_{k}^{u}\beta_{k}^{v})=u\alpha_{k}^{u-1}\beta_{k}^{v}D^{2}\alpha_{k}+v\alpha_{k}^{u}\beta_{k}^{v-1}D^{2}\beta_{k}+O(k^{-2})$

since $D\alpha_{k}$ and $D\beta_{k}$ are both $O(k^{-2})$. Therefore
$D^{2}(\alpha_{k}^{u}\beta_{k}^{v})=O(k^{2})$ which implies that
$D^{2}\gamma_{k}=O(k^{3})$ and hence
$D^{2}(\gamma_{k}\mathcal{B}_{k+i})=O(k^{n+3})$.

Now taking the second derivative of the $p=2$ statement in (19),

$\displaystyle(D^{2}\alpha_{k}-D^{2}\beta_{k})\Omega(k^{n+2})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!-(\alpha_{k}-\beta_{k})\sum_{i}(k+i)D^{2}(\gamma_{k}\mathcal{B}_{k+i})$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!2(D\alpha_{k}-D\beta_{k})\sum_{i}(k+i)D(\gamma_{k}\mathcal{B}_{k+i})+O(k^{n-1}).$

Since $\alpha_{k}-\beta_{k}=O(k^{-5})$, $D\alpha_{k}-D\beta_{k}=O(k^{-4})$ and
$D(\gamma_{k}\mathcal{B}_{k+i})=O(k^{n+2})$, this is $O(k^{n-1})$. Hence
$D^{2}\alpha_{k}=D^{2}\beta_{k}+O(k^{-3})$ as required. ∎

###### Proof of (8).

From Lemma 6 we have
$\omega_{FS,k}=\omega_{h_{FS,k}}+\frac{i}{2c}\partial\overline{\partial}f_{k}$,
where

$f_{k}=\operatorname{vol}\,\sum_{i}c_{i}\sum_{\alpha}|s^{i}_{\alpha}|_{h_{FS,k}}^{2}$

and the $\\{s^{i}_{\alpha}\\}$ is a graded basis of $\oplus_{i}H^{0}(L^{k+i})$
that is orthonormal with respect to the $L^{2}$-norm defined by $(h,\omega)$.
(So $t^{i}_{\alpha}:=\sqrt{c_{i}\operatorname{vol}\,}s^{i}_{\alpha}$ is an
orthonormal basis with respect to the $\operatorname{Hilb}(h,\omega)$ metric.)

Applying $\partial\overline{\partial}\log$ to (7) shows that
$\omega_{h_{FS,k}}=\omega_{h}+O(k^{-2})=\omega+O(k^{-2})$ in $C^{0}$, so
$\omega_{FS,k}=\omega+\frac{i}{2c}\partial\overline{\partial}f_{k}+O(k^{-2})$.
So since $c$ is of order $\Omega(k^{n+1})$, to prove (8) it will be sufficient
to show that $f_{k}$ is constant on $X$ to $O(k^{n-1})$ in $C^{2}$-norm.
Applying the expansion (7),

$\displaystyle f_{k}(x)$ $\displaystyle=$
$\displaystyle\operatorname{vol}\,\sum_{i}c_{i}\frac{h_{FS,k}^{k+i}}{h^{k+i}}\sum_{\alpha}|s_{\alpha}(x)|_{h}^{2}$
$\displaystyle=$
$\displaystyle\operatorname{vol}\,\sum_{i}c_{i}(1+\frac{\operatorname{Scal}(\omega)-\overline{\\!S}}{2k}+O(k^{-2}))\sum_{\alpha}|s_{\alpha}(x)|_{h}^{2}$
$\displaystyle=$ $\displaystyle b_{0}k^{n}+O(k^{n-1}),$

by (5), where $b_{0}$ is constant. ∎

##  Chapter 4 Limits of balanced metrics

We digress in this section from our proof of Donaldson’s Theorem to give
another application of the weighted Bergman kernel that illustrates the
connection between balanced metrics and metrics of constant scalar curvature.

######  Theorem 1.

Let $(h_{k},\omega_{k})$ be a pair that is balanced for the embedding
$X\subset\mathbb{P}(\oplus_{i}H^{0}(L^{k+i})^{*})$, and suppose this sequence
converges in $C^{2}$ to a limit $(h,\omega)$. Then $2\pi\omega$ is the
curvature of $h$ and $\operatorname{Scal}(\omega)$ is constant.

###### Proof.

Letting $2\pi\omega_{h_{k}}$ denote the curvature of $h_{k}$, by Lemma 6 we
have

$\omega_{k}=\omega_{h_{k}}+\frac{i}{2c}\partial\overline{\partial}f_{k},$ (2)

where

$f_{k}(x)=\operatorname{vol}\,\sum_{i}c_{i}\sum_{\alpha}|s^{i}_{\alpha}(x)|^{2}_{h_{k}}$

and $\\{s^{i}_{\alpha}\\}$ is a graded orthonormal basis of
$\oplus_{i}H^{0}(L^{k+i})$ with respect to the $L^{2}$-metric defined by
$(h_{k},\omega_{k})$. (Here we are using the balanced condition: that
$(h_{k},\omega_{k})$ is the Fubini-Study metric induced from this
$L^{2}$-metric.)

By (5) we have the $C^{4}$-estimate

$f_{k}=\operatorname{vol}\frac{\omega_{h_{k}}^{n}}{\omega_{k}^{n}}\sum_{i}c_{i}k^{n}+O(k^{n-1})$
(3)

(The estimate is in $C^{4}$ rather than $C^{2}$ since we only require it to
top order. Moreover we have used here that the sequence $(h_{k},\omega_{k})$
converges so lies in a compact set, and thus the $O(k^{n-1})$ can be taken
uniformly.) Since $c=\sum_{i}c_{i}(k+i)h^{0}(L^{k+i})$ is of order
$\Omega(k^{n+1})$, we deduce from (2) that
$\omega_{k}=\omega_{h_{k}}+O(k^{-1})$ in $C^{2}$.

In turn this implies that $\omega_{h_{k}}^{n}/\omega_{k}^{n}=1+O(k^{-1})$,
which we can feed back into (3) to give
$\partial\overline{\partial}f_{k}=O(k^{n-1})$. Hence in fact

$\omega_{k}=\omega_{h_{k}}+O(k^{-2}).$

In particular, taking the limit as $k\to\infty$ implies that
$\omega=\omega_{h}$, i.e. that $2\pi\omega$ is the curvature of $h$.

Therefore $\omega_{h_{k}}^{n}/\omega_{k}^{n}=1+O(k^{-2})$ and

$\operatorname{tr}_{\omega_{h_{k}}}(\operatorname{Ric}(\omega_{k}))=\operatorname{tr}_{\omega_{k}}(\operatorname{Ric}(\omega_{k}))+O(k^{-2})=\operatorname{Scal}(\omega_{k})+O(k^{-2}).$

Thus the asymptotic expansion (2) for the weighted Bergman kernel becomes

$B_{k}=\operatorname{vol}\sum_{i}(k+i)c_{i}\sum_{\alpha}|s^{i}_{\alpha}|^{2}=\operatorname{vol}\sum_{i}c_{i}k^{n+1}+b_{1}k^{n}+O(k^{n-1}),$
(4)

where
$b_{1}=\operatorname{vol}\sum_{i}c_{i}\left((n+1)i+\frac{1}{2}\operatorname{Scal}(\omega_{k})\right)$.
But by Proposition 20 the balanced condition implies that this weighted
Bergman kernel is the constant

$c=\operatorname{vol}\sum_{i}c_{i}k^{n+1}+O(k^{n}).$

So the coefficient of $k^{n+1}$ agrees with that of (4). Taking coefficients
of $k^{n}$ gives, after some rearranging, a constant $\,\overline{\\!S}$
independent of $k$ such that

$\operatorname{Scal}(\omega_{k})-\,\overline{\\!S}=O(k^{-1}).$

Taking $k$ to infinity yields $\operatorname{Scal}(\omega)=\,\overline{\\!S}$
as required. ∎

######  Remark 5.

The previous theorem was first observed by Donaldson Don [1] in the case of
manifolds embedded in projective space. In the same paper Donaldson also
proves a much harder converse: a cscK metric implies the existence of balanced
metrics for large $k$. We expect that this converse can also be generalised to
orbifolds embeddings in weighted projective space, but have not attempted to
prove it.

##  Chapter 5 K-stability as an obstruction to orbifold cscK metrics

We now have the tools required to prove the orbifold version of Donaldson’s
Theorem, and start with the precise definition of stability.

###  1 Definition of orbifold K-stability

Fix a compact $n$-dimensional polarised orbifold with cyclic quotient
singularities $(X,L)$.

######  Definition 1.

A test configuration for $(X,L)$ consists of a pair
$(\pi\colon\mathcal{X}\to\mathbb{C},\mathcal{L})$ where $\mathcal{X}$ is an
orbischeme, $\pi$ is flat and $\mathcal{L}$ is an ample orbi-line bundle along
with a $\mathbb{C}^{*}$-action such that (1) the action is linear and covers
the usual action on $\mathbb{C}$ and (2) the general fibre $\pi^{-1}(t)$ of
the test configuration is $(X,L)$.

Test configurations arise from the action of a one parameter
$\mathbb{C}^{*}$-subgroup of the automorphisms of weighted projective space
$\mathbb{P}$ on an orbifold embedded in $\mathbb{P}$. In general the limit
$\mathcal{X}_{0}=\pi^{-1}(0)$ will not itself be an orbifold, as it may have
scheme structure or entire components consisting of points with nontrivial
stabilisers. In general one should allow $\mathcal{X}$ to be a Deligne-Mumford
stack, but for most of the applications in this paper $\mathcal{X}$ will
itself be an orbifold.

Conversely, we can realise an abstract test configuration via a
$\mathbb{C}^{*}$-action on weighted projective space, just as in the manifold
case [RT, 2, Proposition 3.7]. Using the orbi-ampleness of $\mathcal{L}$, we
can embed $\mathcal{X}$ into the weighted projective bundle
$\mathbb{P}(\oplus_{i}(\pi_{*}\mathcal{L}^{k+i})^{*})$ over the base curve
$\mathbb{C}$ for $k\gg 0$, such that the pullback of
$\mathcal{O}_{\mathbb{P}}(1)$ is $\mathcal{L}$. Pick a trivialisation of the
bundle, making it isomorphic to $\mathbb{P}(V)\times\mathbb{C}$, where
$V=\oplus_{i}H^{0}(\mathcal{X}_{0},\mathcal{L}^{k+i}|_{\mathcal{X}_{0}})^{*}$.
Thus the $\mathbb{C}^{*}$-action on $V$ arising from the one on the central
fibre $(\mathcal{X}_{0},\mathcal{L}_{0})$ induces a diagonal
$\mathbb{C}^{*}$-action on $\mathbb{P}(V)\times\mathbb{C}\supset\mathcal{X}$
giving the original test configuration.

By Proposition 19 we can write the total weight of the $\mathbb{C}^{*}$-action
on $H^{0}(L^{k})$ as

$w(H^{0}(L^{k}))=w(k)+\tilde{o}(k^{n}),$ (2)

where $w(k)$ is a polynomial $b_{0}k^{n+1}+b_{1}k^{n}$ of degree $n+1$.
Similarly

$h^{0}(L^{k})=h(k)+\tilde{o}(k^{n-1}),$ (3)

where $h(k)=a_{0}k^{n}+a_{1}k^{n-1}$.

######  Definition 4.

The _Futaki invariant_ of the test configuration $(\mathcal{X},\mathcal{L})$
is the $F_{1}=\frac{a_{0}b_{1}-a_{1}b_{0}}{a_{0}^{2}}$ term in the expansion

$\frac{w(k)}{kh(k)}=F_{0}+\frac{F_{1}}{k}+O\left(\frac{1}{k^{2}}\right).$

We say $(X,L)$ is _K-semistable_ if $F_{1}\geq 0$ for any test configuration
with general fibre $(X,L)$. We say it is _K-polystable_ if in addition
$F_{1}=0$ only if the test configuration is a product
$\mathcal{X}=X\times\mathbb{C}$, i.e. it arises from a $\mathbb{C}^{*}$-action
on $X$.

In other words we are simply ignoring the non-polynomial terms in the Hilbert
and weight functions, and then defining stability exactly as for manifolds.

One reason this is a sensible stability notion related to scalar curvature is
given by our next result. This shows that taking a weighted sum with our
choice of $c_{i}$ kills the periodic terms, a result we will apply later to
both $w$ (2) and $h$ (3).

######  Lemma 5.

Let $H$ be a function of the form

$H(k)=h(k)+\epsilon_{h}(k),$

where $h$ is a polynomial of degree $n$ and $\epsilon_{h}$ is a sum of terms
of the form $r(k)\delta(k)$ where $r$ is a polynomial of degree $n-1$ and
$\delta(k)$ is periodic with period $m$ and average zero. Then

$\sum_{i}c_{i}H(k+i)=\sum_{i}c_{i}h(k+i)+O(k^{n-4}).$

###### Proof.

First we claim that if $0\leq p\leq 3$ then $\sum_{i\equiv u}c_{i}i^{p}$ is
independent of $u$. To see this, let $m=\operatorname{ord}(X)$ and observe
that by (1), $\sum_{i}c_{i}t^{i}$ has a root of order at least 4 at every non-
trivial $m$th root of unity. Thus if $\sigma^{m}=1$ with $\sigma\neq 1$ then
$\sum_{i}c_{i}i^{p}\sigma^{ri}=0$ for $1\leq r\leq m-1$. So given any $u$,

$\sum_{i}i^{p}c_{i}=\sum_{r=0}^{m-1}\sigma^{-ru}\sum_{i}i^{p}c_{i}\sigma^{ri}=\sum_{i}i^{p}c_{i}\left(\sum_{r=0}^{m-1}\sigma^{(i-u)r}\right)=m\sum_{i\equiv
u}i^{p}c_{i},$

which proves the claim.

We have to show that $\sum_{i}c_{i}r(k+i)\delta(k+i)=O(k^{n-4})$. By the
claim,

$\displaystyle\sum_{i}c_{i}i^{p}\delta(k+i)$ $\displaystyle=$
$\displaystyle\sum_{u=1}^{m}\sum_{i\equiv u-k\text{ mod
}m}c_{i}i^{p}\delta(u)$ $\displaystyle=$
$\displaystyle\frac{1}{m}\sum_{u=1}^{m}\delta(u)\sum_{i}c_{i}i^{p}\ =0$

for $0\leq p\leq 3$. Hence the $k^{d},\ldots,k^{d-3}$ terms in
$\sum_{i}c_{i}(k+i)^{d}\delta(k+i)$ vanish, and the sum is $O(k^{n-4})$ if
$d\leq n$. The result for general polynomials $r$ follows by linearity. ∎

###  2 Orbifold version of Donaldson’s theorem

To recall the general setup, let $h$ be a hermitian metric on $L$ with
positive curvature $2\pi\omega$ and for $k\gg 0$ consider the
$\operatorname{Hilb}(h,\omega)$ metric on $\oplus_{i}H^{0}(L^{k+i})$ from
(16). From the embedding $X\subset\mathbb{P}(\oplus_{i}H^{0}(L^{k+i})^{*})$ we
produced in Definition 11 a hermitian matrix $M(X)=M_{k}(X)$ (and defined the
embedding to be balanced at level $k$ when $M_{k}(X)$ vanishes). Using the
norm $\|A\|^{2}=\operatorname{tr}(AA^{*})$ on hermitian matrices, the
following is the key estimate.

######  Theorem 6.

There is a constant $C$ such that

$\lVert M_{k}(X)\rVert\leq
Ck^{\frac{n-2}{2}}\lVert\operatorname{Scal}(\omega)-\,\overline{\\!S}\rVert_{L^{2}}+O(k^{\frac{n-4}{2}}),$

where the $L^{2}$-norm is taken with respect to the volume form determined by
$\omega$.

###### Proof.

To ease notation we write $M=M_{k}(X)$. Since $\|M\|$ is unchanged by a
unitary transformation we may pick $\operatorname{Hilb}(h,\omega)$-orthonormal
coordinates $\\{t_{\alpha}^{i}\\}$ such that $M=\oplus M^{i}$ with each
$M^{i}$ diagonal. Thus $M^{i}$ has entries (15)

$M^{i}_{\alpha\alpha}=\frac{1}{2}\left(\int_{X}|t^{i}_{\alpha}|_{h_{FS,k}}^{2}\frac{\omega^{n}_{FS,k}}{n!}-c_{i}\operatorname{vol}\right)=\frac{1}{2}\left(\int_{X}|t^{i}_{\alpha}|_{h}^{2}\frac{h_{FS,k}^{k+i}}{h^{k+i}}\frac{\omega_{FS,k}^{n}}{n!}-c_{i}\operatorname{vol}\right)\\!,$

where $h_{FS,k}$ and $\omega_{FS,k}$ are the induced Fubini-Study metrics.
Using the expansion of $h_{FS,k}/h=1+(\,\overline{\\!S}-S)/2k^{2}+O(k^{-3})$
of Theorem 6 we can write $M=A+B$ where $B^{i}_{\alpha\alpha}=O(k^{-2})$ and

$\displaystyle A^{i}_{\alpha\alpha}$
$\displaystyle=\frac{1}{2}\left(\int_{X}|t^{i}_{\alpha}|_{h}^{2}\left(1+\frac{\,\overline{\\!S}-S}{2k}\right)\frac{\omega^{n}}{n!}-c_{i}\operatorname{vol}\right)=\frac{1}{4k}\int_{X}|t^{i}_{\alpha}|_{h}^{2}(\,\overline{\\!S}-S)\frac{\omega^{n}}{n!}\,.$

Here we have used $\|t^{i}_{\alpha}\|^{2}_{L^{2}}=c_{i}\operatorname{vol}$
from the definition of the $\operatorname{Hilb}(h,\omega)$-norm. Using the
Cauchy-Schwarz inequality,

$\displaystyle|A^{i}_{\alpha\alpha}|^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{16k^{2}}\int_{X}|t^{i}_{\alpha}|_{h}^{2}\frac{\omega^{n}}{n!}\int_{X}|t^{i}_{\alpha}|_{h}^{2}|\,\overline{\\!S}-S|^{2}\frac{\omega^{n}}{n!}$
$\displaystyle\leq$
$\displaystyle\frac{C^{\prime}}{k^{2}}\int_{X}|t^{i}_{\alpha}|_{h}^{2}|\,\overline{\\!S}-S|^{2}\frac{\omega^{n}}{n!},$

for some constant $C^{\prime}$. Thus from the weak form of the expansion
$\sum_{i}\sum_{\alpha}|t^{i}_{\alpha}|_{h}^{2}=O(k^{n})$,

$\displaystyle\|A\|^{2}$ $\displaystyle\leq$
$\displaystyle\frac{C^{\prime}}{k^{2}}\int_{X}\sum_{i,\alpha}|t^{i}_{\alpha}|_{h}^{2}|\,\overline{\\!S}-S|^{2}\frac{\omega^{n}}{n!}$
$\displaystyle\leq$ $\displaystyle
C^{\prime\prime}k^{n-2}\|S-\,\overline{\\!S}\|_{L^{2}}^{2}.$

Therefore $\|M\|\leq\|A\|+\|B\|\leq
Ck^{\frac{n-2}{2}}\|S-\,\overline{\\!S}\|_{L^{2}}+\|B\|$, where
$C=\sqrt{C^{\prime\prime}}$. But $B$ is diagonal with $O(k^{n})$ entries of
size $O(k^{-2})$, so $\|B\|^{2}=O(k^{n-4})$. ∎

Now let $(\mathcal{X},\mathcal{L})$ be a nontrivial test configuration for
$(X,L)$, embedded in $\mathbb{P}(V)\times\mathbb{C}$ (with
$V=\oplus_{i}H^{0}(X_{0},L_{0}^{k+i})^{*}$) as before, induced by a
$\mathbb{C}^{*}$-action on $\mathbb{P}(V)$ that takes $X$ to the limit
$X_{0}$. Suppose that $L$ has a metric $h$ with positive curvature
$2\pi\omega$, inducing the Hilb$(h,w)$-metric on $\oplus_{i}H^{0}(X,L^{k+i})$.
Applying [Don, 2, Lemma 2] to each of the spaces $H^{0}(X,L^{k+i})$, we get a
metric on $V$ such that the induced $S^{1}$-action is unitary. Therefore the
infinitesimal generator $A^{k+i}$ of the induced action on
$H^{0}(L_{0}^{k+i})$ is hermitian. We set

$A:=\bigoplus_{i}A^{k+i}.$ (7)

As in Section 1, we get a hamiltonian $H_{A}$ for the $S^{1}$-action on
$\mathbb{P}(V)$ by contracting its vector field on the circle bundle
$S(\mathcal{O}_{\mathbb{P}(V)}(1))$ with the connection 1-form whose curvature
is $\omega_{FS}$. It is

$H_{A}\colon\mathbb{P}(V)\to\mathbb{R},\qquad
H_{A}([v])=\frac{1}{c}\sum_{i}\lambda^{2(k+i)}(v)\langle
A^{k+i}v_{k+i},v_{k+i}\rangle,$

using the inner product $\langle\cdot,\cdot\rangle$ on $V$ and $\lambda$ as
defined in (5). This differs from our usual hamiltonian
$m_{A}=\operatorname{tr}(mA)$ of (10) by the additive constant
$\sum_{i}c_{i}\dim V^{k+i}$, and by the multiplicative factor $\frac{1}{c}$.
(The latter scaling compensates for the fact that $m_{A}$ is the hamiltonian
for $c\omega_{FS}$; see Definition 3.)

By Proposition 19, then, the polynomial part of the total weight of the
$\mathbb{C}^{*}$-action on $V^{*}$ is $w(k)=b_{0}k^{n+1}+b_{1}k^{n}$, where
$b_{0}=\int_{X_{0}}H_{A}\frac{\omega_{FS}^{n}}{n!}$. From this we can define
the Futaki invariant $F_{1}(\mathcal{X},\mathcal{L})$ of the test
configuration $(\mathcal{X},\mathcal{L})$ as in Definition 4.

######  Theorem 8.

In the set-up as above, suppose that $\omega$ has constant scalar curvature.
Then $F_{1}(\mathcal{X},\mathcal{L})\geq 0$.

###### Proof.

In the notation above, set $s=\log t$ and let $X_{t}=\exp(sA).X$ denote the
fibre of the given test configuration over $t\in\mathbb{C}$, with central
fibre $X_{0}$ the limit of $\exp(sA).X$ as $s\to-\infty$.

For fixed $k$, $\operatorname{tr}(M_{k}(X_{s})A)$ is an increasing function of
$s\in\mathbb{R}$, because $\operatorname{tr}(M(X)A)$ is a hamiltonian for the
action of $\exp(sA)$ on the space of sub-orbifolds of $\mathbb{P}(V)$.
Explicitly, substituting $v=Jv_{A}$ into (13) shows that the derivative of
$\operatorname{tr}(M_{k}(X_{s})A)$ is $\Omega(Jv_{A},v_{A})>0$. Therefore

$\operatorname{tr}(AM_{k}(X))\ =\ \operatorname{tr}(AM_{k}(X_{1}))\ \geq\
\lim_{s\to-\infty}\operatorname{tr}(AM_{k}(X_{t}))\ =\
\operatorname{tr}(AM_{k}(X_{0})).$

Recalling the definition of $M_{k}(X)$ (11), this gives

$\displaystyle\|A\|\|M_{k}(X)\|\ \geq\
\int_{X_{0}}m_{A}\frac{\omega^{n}_{FS}}{n!}$ $\displaystyle=$
$\displaystyle\int_{X_{0}}cH_{A}\frac{\omega^{n}_{FS}}{n!}-\operatorname{vol}\sum_{i}c_{i}\operatorname{tr}(A^{k+i})$
(9) $\displaystyle=$ $\displaystyle
cb_{0}-a_{0}\sum_{i}c_{i}w(H^{0}(L^{k+i})),$

by Proposition 19. Here we are writing
$h^{0}(L^{k})=a_{0}k^{n}+a_{1}k^{n-1}+\widetilde{o}(k^{n-1})$ and
$w(H^{0}(L^{k}))=b_{0}k^{n+1}+b_{1}k^{n}+\widetilde{o}(k^{n})$. Lemma 5 then
gives

$\displaystyle c=\sum_{i}c_{i}(k+i)h^{0}(L^{k+i})$ $\displaystyle=$
$\displaystyle\tilde{a}_{0}k^{n+1}+\tilde{a}_{1}k^{n}+O(k^{n-1}),$
$\displaystyle\text{and}\quad\sum_{i}c_{i}w(H^{0}(L^{k+i}))$ $\displaystyle=$
$\displaystyle\tilde{b}_{0}k^{n+1}+\tilde{b}_{1}k^{n}+O(k^{n-1}),$

where

$\displaystyle\tilde{a}_{0}=a_{0}\sum_{i}c_{i}\ $ and $\displaystyle\
\tilde{a}_{1}=\sum_{i}c_{i}(a_{0}i(n+1)+a_{1}),$
$\displaystyle\tilde{b}_{0}=b_{0}\sum_{i}c_{i}\ $ and $\displaystyle\
\tilde{b}_{1}=\sum_{i}c_{i}(b_{0}i(n+1)+b_{1}).$

Therefore (9) becomes

$\displaystyle\lVert A\rVert\lVert M_{k}(X)\rVert$ $\displaystyle\geq$
$\displaystyle
c\left(b_{0}-a_{0}\frac{\tilde{b}_{0}k^{n+1}+\tilde{b}_{1}k^{n}+O(k^{n-1})}{\tilde{a}_{0}k^{n+1}+\tilde{a}_{1}k^{n}+O(k^{n-1})}\right)$
$\displaystyle=$ $\displaystyle
ca_{0}\left(k^{-1}\frac{\tilde{b}_{0}\tilde{a}_{1}-\tilde{b}_{1}\tilde{a}_{0}}{\tilde{a}_{0}^{2}}+O(k^{-2})\right)$
$\displaystyle=$ $\displaystyle
ca_{0}\left(k^{-1}\frac{b_{0}a_{1}-b_{1}a_{0}}{a_{0}^{2}}+O(k^{-2})\right)$
$\displaystyle=$ $\displaystyle ca_{0}\left(-k^{-1}F_{1}+O(k^{-2})\right).$

Now $\|A\|^{2}=|\operatorname{tr}A^{2}|=O(k^{n+2})$ by (20), and $c$ is
strictly of order $O(k^{n+1})$. So Theorem 6 now gives

$k^{\frac{n-2}{2}}\lVert\operatorname{Scal}(\omega)-\,\overline{\\!S}\rVert_{L^{2}}+O(k^{\frac{n-4}{2}})\geq
Ck^{\frac{n}{2}}\left(-k^{-1}F_{1}+O(k^{-2})\right)$

for some constant $C>0$. Hence when $\operatorname{Scal}(\omega)$ is constant
(and therefore equal to $\,\overline{\\!S}$) we see that $F_{1}\geq 0$. ∎

######  Corollary 10.

Let $(X,L)$ be a polarised orbifold with cyclic stabiliser groups. If $X$
admits an orbifold Kähler metric $\omega\in\mathcal{K}(c_{1}(L))$ with
constant scalar curvature then $(X,L)$ is K-semistable.

##  Chapter 6 Slope stability of orbifolds

To get examples where K-stability obstructs the existence of constant scalar
curvature metrics we need a supply of test configurations for which we can
calculate the Futaki invariant. To this end we briefly describe the notion of
slope stability. The detailed descriptions in RT [1, 2] extend easily from
manifolds to orbifolds with a few minor changes.

Fix an $n$-dimensional polarised orbifold $(X,L)$ and a sub-orbischeme (or
substack) $Z\subset X$: an invariant subscheme $Z_{U}$ in each orbifold chart
$U\to U/G\subset X$ such that for each injection of charts
$U^{\prime}\hookrightarrow U$, the subscheme $Z_{U^{\prime}}$ is the scheme-
theoretic intersection $Z_{U}\cap U^{\prime}$. In most of our examples $Z$
will be smooth but with generic stabilisers. Working equivariantly in charts
one can produce a new orbischeme, the blowup $\pi\colon\hat{X}\to X$ of $X$
along $Z$. Locally this is the blow up of $U$ in $Z_{U}$ divided by the
induced action of the Galois group on this blow up. The exceptional divisors
glue to give an orbifold exceptional divisor $E\subset\hat{X}$.

For large $N$, $\pi^{*}L^{N}(-E)$ is positive. (From now on we will suppress
$\pi^{*}$.) Thus we can define the _Seshadri constant_ by

$\epsilon_{\text{orb}}(Z)=\text{sup}\left\\{x\in\mathbb{Q}_{+}\colon(L(-xE))^{M}\text{
is ample for some }M\in\mathbb{N}\right\\}.$

For example, if we put $\mathbb{Z}/m\mathbb{Z}$ stabilisers along a smooth
divisor $D\subset X$ then as in Section 4 there is a well defined orbi-divisor
$D/m$ whose Seshadri constant $\epsilon_{\text{orb}}(D/m)=m\epsilon(D)$ is
$m$-times the usual Seshadri constant of $D$ in the underlying space of $X$.

To get a test configuration from $Z$, consider the suborbifold
$Z\times\\{0\\}\subset X\times\mathbb{C}$. Blowing this up gives the
degeneration $\mathcal{X}\to X\times\mathbb{C}\to\mathbb{C}$ to the normal
cone of $Z$ with exceptional divisor $P$. As shown in [RT, 2, Proposition 4.1]
for schemes (and the same results go through easily for orbifolds),
$\epsilon_{\text{orb}}(Z\times\\{0\\})=\epsilon_{\text{orb}}(Z)$. Let
$p\colon\mathcal{X}\to X$ be the projection. Then for generic
$c\in(0,\epsilon_{\text{orb}}(Z))\cap\mathbb{Q}$ general integer powers of
$\mathcal{L}_{c}:=p^{*}L(-cP)$ define a polarisation of $\mathcal{X}$. The
natural action of $\mathbb{C}^{*}$ on $X\times\mathbb{C}$ (trivial on $(X,L)$,
weight one on $\mathbb{C}$) lifts naturally to a linearised action on
$(\mathcal{X},\mathcal{L})$, and thus for such $c$ we have a test
configuration $(\mathcal{X},\mathcal{L}_{c})$ with general fibre $(X,L)$. The
central fibre is $\mathcal{X}_{0}=\hat{X}\cup_{E}P$ consisting of the blowup
$\hat{X}\to X$ along $Z$ glued to $P$ along $E$, and the induced
$\mathbb{C}^{*}$ action is trivial on $\hat{X}$ and acts by scaling $P$ along
the normal to $E$.

As usual we write

$h^{0}(L^{k})=a_{0}k^{n}+a_{1}k^{n-1}+\tilde{o}(k^{n-1}),$ (1)

and then define the _slope_ of $(X,L)$ to be

$\mu(X,L)=\frac{a_{1}}{a_{0}}=-\frac{n\int_{X}c_{1}(K_{\text{orb}}).c_{1}(L)^{n-1}}{2\int_{X}c_{1}(L)^{n}}\,,$

by (17). To define the slope of $Z\subset X$, we work on the orbifold blowup
$\pi\colon\hat{X}\to X$ along $Z$ with exceptional divisor $E$. Then orbifold
Riemann-Roch to $L^{k}(-\frac{j}{k}E)$ for fixed $j$ (and $k=jK$ for some
integer $K$) takes the form

$h^{0}(L^{k}(-jE))=p(k,j)+\epsilon_{p}(k,j).$ (2)

Here $p$ is a polynomial of two variables of total degree $n$ and
$\epsilon_{p}$ is a sum of terms of the form $r_{p}\delta^{\prime}$ where
$r_{p}$ is a polynomial of two variables of total degree $n-1$ and
$\delta^{\prime}=\delta^{\prime}(k,j)$ is periodic in each variable with
average $\sum_{k,j=1}^{M}\delta^{\prime}(k,j)=0$. Define polynomials
$a_{i}(x)$ by

$p(k,xk)=a_{0}(x)k^{n}+a_{1}(x)k^{n-1}+O(k^{n-2})\quad\text{for
}kx\in\mathbb{N}.$

Then the _slope_ of $Z$ (with respect to $c$) is

$\mu_{c}(\curly
I_{Z}):=\frac{\int_{0}^{c}a_{1}(x)+\frac{a^{\prime}_{0}(x)}{2}dx}{\int_{0}^{c}a_{0}(x)dx}\,.$
(3)

The only difference from the manifold case is that we ignored the periodic
terms in the relevant Hilbert functions. This amounts to replacing $K_{X}$ by
$K_{\text{orb}}$.

######  Definition 4.

We say that $(X,L)$ is _slope semistable with respect to $Z$_ if

$\mu_{c}(\curly I_{Z})\leq\mu(X)\quad\text{ for all
}0<c<\epsilon_{\text{orb}}(Z).$

We say that $X$ is _slope semistable_ if it is slope semistable with respect
to all sub-orbischemes $Z\subset X$.

Alternatively, just as in the manifold case [RT, 1, Definition 3.13] we can
put $\tilde{a}_{i}(x):=a_{i}-a_{i}(x)$ and define the _quotient slope_ of $Z$
as

$\mu_{c}(\mathcal{O}_{Z}):=\frac{\int_{0}^{c}\tilde{a}_{1}(x)+\frac{\tilde{a}^{\prime}_{0}(x)}{2}dx}{\int_{0}^{c}\tilde{a}_{0}(x)dx}\,,$
(5)

and $(X,L)$ is slope semistable with respect to $Z$ if and only if
$\mu(X)\leq\mu_{c}(\mathcal{O}_{Z})$ for all $0<c<\epsilon_{\text{orb}}(Z)$.

One can check easily that slope semistability is invariant upon replacing $L$
by a positive power. The point of these definitions is that the sign of the
Futaki invariant of the test configuration given by deformation to the normal
cone of $Z$ is the same as the sign of $\mu(X)-\mu_{c}(\curly I_{Z})$,
resulting in the following slope obstruction to stability.

######  Theorem 6.

If $(X,L)$ is K-semistable then it is slope semistable.

###### Proof.

The argument is essentially the same as that in the smooth case [RT, 2,
Section 4]; only the Riemann-Roch formula changes. Since being not slope
semistable is an open condition, we may without loss of generality assume that
$c<\epsilon_{\text{orb}}(Z)$ is general, and so by rescaling $L$ be may assume
that $c$ is integral and coprime to $m$, making
$(\mathcal{X},\mathcal{L}_{c})$ a test configuration. The space of sections on
the central fibre of the test configuration splits as

$H^{0}_{\mathcal{X}_{0}}(\mathcal{L}_{c}^{k})=H^{0}_{X}(L^{k}\otimes\mathcal{I}_{Z}^{ck})\
\oplus\,\bigoplus_{j=1}^{ck}t^{j}\frac{H^{0}_{X}(L^{k}\otimes\mathcal{I}_{Z}^{ck-j})}{H^{0}_{X}(L^{k}\otimes\mathcal{I}_{Z}^{ck-j+1})}\,.$
(7)

Here $H^{0}_{X}(L^{k}\otimes\mathcal{I}_{Z}^{j})$ is the space of sections of
$L^{k}$ which vanish to order $j$ on $Z$ (in an orbi chart). The coordinate
$t$ is pulled back from the base $\mathbb{C}$, and is acted on by
$\mathbb{C}^{*}$ with weight $-1$. Therefore (7) is the weight space
decomposition of $H^{0}_{\mathcal{X}_{0}}(\mathcal{L}_{c}^{k})$, with total
weight

$\displaystyle w(H^{0}_{\mathcal{X}_{0}}(\mathcal{L}_{c}^{k}))$
$\displaystyle=$
$\displaystyle-\sum_{j=1}^{ck}j\Big{(}h^{0}_{X}(L^{k}\otimes\mathcal{I}_{Z}^{ck-j})-h^{0}_{X}(L^{k}\otimes\mathcal{I}_{Z}^{ck-j+1})\Big{)}.$

Some manipulation, and the vanishing of higher cohomology of the pushdowns of
these sheaves to the underlying scheme, give

$\displaystyle\sum_{j=1}^{ck}h_{X}^{0}(L^{k}\otimes\mathcal{I}_{Z}^{j})-ckh_{X}^{0}(L^{k})$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{ck}h^{0}_{\hat{X}}(L^{k}(-jE))-ckh^{0}(L^{k})$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{ck}p(k,j)+\epsilon_{p}(k,j)-ck[h(k)+\tilde{o}(k^{n-1})].$

By Lemma 8 below, the periodic terms to not contribute to the top two order
parts of this sum, so the leading order polynomial parts of the weight are

$w(k)=\sum_{j=1}^{ck}p(j,k)-ckh(k)+\tilde{o}(k^{n}).$

The calculation of the Futaki invariant is now exactly as in the smooth case
[RT, 2, Proposition 4.14 and Equation 4.19], yielding

$F_{1}(\mathcal{X},\mathcal{L}_{c})=(\mu(X)-\mu_{c}(\mathcal{I}_{Z}))\frac{\int_{0}^{c}a_{0}(x)dx}{a_{0}}\,.$

This is nonnegative if and only if $X$ is slope semistable with respect to
$Z$. ∎

######  Lemma 8.

Suppose $\delta(k,j)$ is periodic in each variable with period $m$ and average
$\sum_{k,j=1}^{m}\delta(k,j)=0$. Suppose also that $r(k,j)$ is a polynomial of
two variables of total degree $n-1$ and $c$ is a fixed integer. Then

$\sum_{j=1}^{ck}r(k,j)\delta(k,j)=\epsilon(k)+O(k^{n-1}),$

where $\epsilon(k)$ is a sum of terms of the form $r(k)\delta^{\prime}(k)$,
with $r$ a polynomial of degree $n$ and $\delta^{\prime}$ periodic of average
zero.

###### Proof.

By linearity it is sufficient to consider the case where $r(k,j)=j^{n-1}$. Set
$j_{0}=\left\lfloor\frac{ck}{m}\right\rfloor m$ and split the sum into two
pieces depending on whether $j\leq j_{0}$ or $j\geq j_{0}+1$. In the first
case, writing $j=mu+v$,

$\sum_{j=0}^{j_{0}}j^{n-1}\delta(k,j)=\sum_{v=1}^{m}\delta(k,v)\sum_{u}(um+v)^{n-1}.$

This splits into pieces of the form $P(k)\epsilon(k)$ where $\epsilon(k)$ is
periodic and $P$ is a polynomial of degree at most $n$, with the degree being
equal to $n$ only when $\epsilon(k)=\sum_{v=1}^{m}\delta(v,k)$ in which case
$\epsilon$ has average zero. Then note that if $P$ has degree $n-1$ there is a
constant $a$ so that $\epsilon-a$ has average zero, and
$P\epsilon=P(\epsilon-a)+aP=P(\epsilon-a)+O(k^{n-1})$. Thus, after some
rearrangement, this part of the sum is of the form claimed. The sum for $j\geq
j_{0}+1$ immediately splits into terms of the form $P(k)\epsilon(k)$ where $P$
has degree at most $n-1$ and $\epsilon$ is periodic, so by the same argument
these terms are also of the required form. ∎

We can calculate the slope of (sufficiently nice) suborbifolds much as in the
manifold case. For instance let $\hat{X}\stackrel{{\scriptstyle\pi}}{{\to}}X$
be the orbifold blowup along a smooth $Z$ of codimension $r\geq 2$, with
orbifold exceptional divisor $E$. Then
$K_{\text{orb},\hat{X}}=\pi^{*}K_{\text{orb},X}+(r-1)E$, and

$\displaystyle a_{0}(x)$ $\displaystyle=$
$\displaystyle-\frac{\int_{\hat{X}}c_{1}(L(-xE))^{n}}{n!}\,,$ $\displaystyle
a_{1}(x)$ $\displaystyle=$
$\displaystyle-\frac{\int_{\hat{X}}c_{1}(K_{\text{orb},\hat{X}})c_{1}(L(-xE))^{n-1}}{2(n-1)!}\,.$
(9)

So the formulae only differ from those in RT [1] in replacing $K_{X}$ by
$K_{\text{orb}}$. For example if $Z$ is as small as possible – the invariant
subvariety defined by a reduced fixed point upstairs in an orbifold chart –
then these quickly imply

$\mu_{c}(\mathcal{O}_{Z})=\frac{n(n+1)}{2c}\,.$ (10)

Notice the order of the stabiliser group at this point does not feature;
however it enters into the Seshadri constant of $Z$ and so does affect slope
stability.

Similarly if $Z$ is an orbifold divisor in an orbifold surface then

$\mu_{c}(\mathcal{O}_{Z})=\frac{3(2L.Z-c(K_{\text{orb}}.Z+Z^{2})}{2c(3L.Z-cZ^{2})}\,.$

##  Chapter 7 Applications and further examples

###  1 Orbifold Riemann surfaces

By an orbifold Riemann surface we mean an orbifold of complex dimension one.
This is equivalent to the data of a Riemann surface of genus $g\geq 0$ and $r$
points $p_{1},\dots,p_{r}\in X$ marked by orders of stabiliser groups
$m_{1},\ldots,m_{r}\geq 2$. We assume $r\geq 1$.

######  Theorem 1.

A polarised Riemann surface $(X,L)$ is slope semistable if and only if

$2g+\sum_{i=1}^{r}\left(\frac{m_{i}-1}{m_{i}}\right)\geq
2\max_{i=1,\ldots,r}\left\\{\frac{m_{i}-1}{m_{i}}\right\\}$ (2)

###### Proof.

The orbifold canonical bundle of $X$ is
$K_{\text{orb}}=K_{X}+\sum_{i=1}^{r}\left(1-\frac{1}{m_{i}}\right)p_{i}$ so
the slope of $X$ is

$\mu(X,L)=-\frac{\deg K_{\text{orb}}}{2\deg
L}=\frac{1-g-\frac{1}{2}\sum_{i=1}^{r}\left(1-\frac{1}{m_{i}}\right)}{\deg
L}\,.$ (3)

Without loss of generality assume $m_{1}\geq m_{2}\geq\dots\geq m_{r}$. Let
$Z=\\{p_{1}/m\\}$ be the orbifold point of order $m_{1}$ with a reduced lift
upstairs. Then

$\mu_{c}(\mathcal{O}_{Z})=c^{-1}\quad\text{and}\quad\epsilon_{\text{orb}}(Z,X,L)=m_{1}\deg
L.$

Now if $(X,L)$ is semistable then $\mu_{\epsilon}(\mathcal{O}_{Z})\geq\mu(X),$
so

$\frac{1}{m_{1}}\geq 1-g-\frac{1}{2}\sum_{i=1}^{r}\frac{m_{i}-1}{m_{i}}$

which rearranges to give the inequality (2).

For the converse suppose that (2) holds and consider an orbifold subspace
$Z\subset X$. This $Z$ is an orbifold divisor whose degree is a rational
number $q\geq\frac{1}{m_{1}}$. Thus
$\epsilon:=\epsilon_{\text{orb}}(Z,L)=\frac{1}{q}\deg L\leq m_{1}\deg L$. As
$\mu_{c}(\mathcal{O}_{Z})=c^{-1}$ is decreasing with respect to $c$, we get
$\mu_{c}(\mathcal{O}_{Z})\geq\epsilon^{-1}\geq(m_{1}\deg L)^{-1}$. But (2)
implies that this is greater than or equal to $\mu(X,L)$ (3), so $X$ is slope
semistable. ∎

######  Remark 4.

  1. 1.

It is hard to either violate or achieve equality in the inequality (2). Either
would imply that $2g+0\leq 2\max(1-\frac{1}{m_{i}})<2$ and so $g=0$. Then
since each integer $m_{i}\geq 2$, we find there are only three cases in which
an orbifold Riemann surface is not strictly slope stable:

    1. (a)

$g=0,\ r=1$ (this gives $\mathbb{P}(1,m)$),

    2. (b)

$g=0,\ r=2,\ m_{1}\neq m_{2}$ (giving $\mathbb{P}(m_{1},m_{2})$ if
$\operatorname{hcf}(m_{1},m_{2})=1$), and

    3. (c)

$g=0,\ r=2,\ m_{1}=m_{2}$.

In the first two cases $(X,L)$ is not slope semistable and so not cscK. In the
third case $(X,L)$ is actually slope polystable, as we now describe. The only
way in which $\mu_{c}(\mathcal{O}_{Z})=\mu(X)$ can occur in the proof of (1)
is if $Z=\\{p_{1}/m_{1}\\}$ or $Z=\\{p_{2}/m_{2}\\}$ and $c=\epsilon(Z)=2$. In
this case deformation to the normal cone $\mathcal{X},\mathcal{L}_{c})$ has
$\mathcal{L}_{c}$ only semi-ample, pulled back from the contraction of the
proper transform of the central fibre $X\times\\{0\\}$. This contraction is in
fact a product configuration $X\times\mathbb{C}$ (with a nontrivial
$\mathbb{C}^{*}$-action).

  2. 2.

The stability condition (2) is actually a special case of (10) and thus an
manifestation of the index obstruction which we discuss below.

Slope stability of orbifold Riemann surfaces fits perfectly into the known
theory of orbifold cscK metrics which has been studied by several authors
including Picard Pic , McOwen McO  and Troyanov Tro [1, 2]. In the terminology
of this paper Troyanov’s results can be paraphrased as follows:

######  Corollary 5 (Troyanov).

Let $(X,L)$ be a polarised orbifold Riemann surface. Then $c_{1}(L)$ admits an
orbifold cscK metric if and only if it is slope polystable.

###### Proof.

The main theorems in Tro [2] imply that $X$ admits a cscK metric when strict
inequality holds in (2) i.e. as long as $(X,L)$ is slope stable. (To compare
our notation with Troyanov’s set $\theta_{i}=\frac{2\pi}{m_{i}}$ and
$\chi_{\text{orb}}=-\deg K_{\text{orb}}$. Then this is Theorem A in Tro [2]
when $\chi_{\text{orb}}<0$, Proposition 2 when $\chi_{\text{orb}}=0$ and
Theorem C when $\chi_{\text{orb}}>0$.) The only way that $(X,L)$ can be slope
polystable and not slope stable is case c): if $g=0,r=2$ and $m_{1}=m_{2}$. In
this case $X$ is the global quotient $\mathbb{P}^{1}/(\mathbb{Z}/m\mathbb{Z})$
with orbifold cscK metric descended from the Fubini-Study metric on
$\mathbb{P}^{1}$.

For the converse, if $(X,L)$ is not slope polystable then $g=0$ and either a)
$r=1$ or b) $r=2$ and $m_{1}\neq m_{2}$. In these two cases $(X,L)$ is not
slope semistable, which by Corollary 10 and Theorem 6 implies that $X$ does
not admit an orbifold cscK metric. ∎

######  Remark 6.

The statement that if $g=0$ and a) $r=1$ or b) $r=2$ and $m_{1}\neq m_{2}$
then $X$ does not admit a cscK metric has also been proved by Troyanov [Tro,
1, Theorem I]. Troyanov’s work applies much more generally to cone angles not
necessarily of the form $2\pi/m$, and this is also studied further in Che ; CL
; LT . We hope to return to cone angles in $2\pi\mathbb{Q}_{+}$ using the
method described in the Introduction.

###  2 Index obstruction to stability

Recall that the index $\operatorname{ind}(X)$ of a Fano manifold $X$ is
defined to be the largest integer $r$ such that $K^{-1}_{X}$ is linearly
equivalent to $rD$ for some Cartier divisor $D\subset X$, and it is well known
that if $X$ is smooth then $\operatorname{ind}(X)\leq n+1$ with equality if
and only if $X=\mathbb{P}^{n}$. By contrast for an Fano orbifold $(X,\Delta)$
it is possible that $K^{-1}_{\text{orb}}\cong\mathcal{O}(rD)$ where $D$ is an
orbi divisor and $r\in\mathbb{N}$ is larger than $n+1$. We will show that this
prevents $(X,\Delta)$ from being K-stable. In fact the same is true under the
weaker condition that $K_{\text{orb}}^{k}\cong\mathcal{O}(krD)$ for some
$k\in\mathbb{N}$ and $n+1\leq r\in\mathbb{Q}$.

######  Theorem 7.

(Index Obstruction) Let $(X,K^{-1}_{\text{orb}})\supset D$ be a Fano orbifold
and an orbi divisor. Suppose that $K^{-k}_{\text{orb}}\cong\mathcal{O}(krD)$
for some $k>0$ and $r\in\mathbb{Q}_{+}$. If $r>n+1$ then
$(X,K^{-1}_{\text{orb}})$ is slope unstable, and thus does not admit an
orbifold Kähler-Einstein metric.

###### Proof.

Set $L=K^{-1}_{\text{orb}}$. Using (6) to calculate the slope,

$\displaystyle a_{0}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{n!}\int_{X}(c_{1}(L)-xc_{1}(D))^{n}=\frac{1}{n!}(r-x)^{n}\int_{X}c_{1}(D)^{n},$
$\displaystyle a_{1}(x)+\frac{a_{0}^{\prime}(x)}{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2(n-1)!}(r-1)(r-x)^{n-1}\int_{X}c_{1}(D)^{n}.$

Now $a_{0}=\frac{r^{n}}{n!}\int_{X}c_{1}(D)^{n}$ and
$a_{1}=\frac{r^{n}}{2(n-1)!}\int_{X}c_{1}(D)^{n}$, so $\mu(X,L)=\frac{n}{2}$.
The Seshadri constant of $D$ is $r$. Using the definition of the slope (5),

$\mu_{r}(\mathcal{O}_{D})=\frac{(n+1)((n-1)r+1)}{2nr}\,,$

which is less that $\mu(X)=n/2$ if and only if $r>n+1$. ∎

######  Remark 8.

At the level of Kähler-Einstein metrics the analogous result has already been
proved by Gauntlett-Martelli-Sparks-Yau using the “Lichnerowicz obstruction”
to the existence of Sasaki-Einstein metrics with non-regular Reeb vector
fields [GMSY, , Section 2.2]. In fact it was their work that originally
motivated this paper. In the same paper the authors discuss the “Bishop