# Weighted Bergman kernels on orbifolds

Julius Ross and Richard Thomas

###### Abstract

We describe a notion of ampleness for line bundles on orbifolds with cyclic
quotient singularities that is related to embeddings in weighted projective
space, and prove a global asymptotic expansion for a weighted Bergman kernel
associated to such a line bundle.

††volume: :††volume: B††volume: missing††volume: missing

##  Chapter 0 Introduction

Let $(X,\omega)$ be a compact $n$-dimensional Kähler manifold and $L$ be a
positive line bundle on $X$ equipped with a hermitian metric $h$ whose
curvature form is $2\pi\omega$. These induce an $L^{2}$-metric on the space of
sections $H^{0}(L^{k})$, and given an orthonormal basis $\\{s_{\alpha}\\}$ the
Bergman kernel is the smooth function

$B_{k}(x)=\sum_{\alpha}|s_{\alpha}(x)|^{2},$

where $|s_{\alpha}(x)|$ is the pointwise norm induced by $h$ (more precisely,
its $k$th power considered as a metric on $L^{k}$). More invariantly, the
$L^{2}$-metric on $H^{0}(L^{k})$ induces a Fubini-Study metric on
$\mathbb{P}(H^{0}(L^{k})^{*})\supset X$ and $\mathcal{O}(1)$ over it;
restricting to $X$ we get a metric $h_{FS}$ on $\mathcal{O}_{X}(1)\cong L^{k}$
which may not equal the first $h$. Then $B_{k}=h/h_{FS}$ is their ratio.

A well known theorem of Tian-Zelditch-Catlin describes the asymptotic
behaviour of this function for large $k$: there exist smooth functions
$b_{1},\ldots,b_{N}$ and a global expansion

$B_{k}=k^{n}+b_{1}k^{n-1}+\cdots+b_{N}k^{n-N}+O(k^{n-N-1})$ (1)

for $k\gg 0$, where $O(k^{n-N-1})$ can even be taken in the $C^{\infty}$ norm
[Rua]. Moreover the $b_{j}$ can be expressed in terms of the derivatives of
the metric. In particular $b_{1}$ is half the scalar curvature of $\omega$,
leading to the importance of $B_{k}$ in the theory of constant scalar
curvature Kähler metrics.

Throughout this paper, $X$ will instead be a orbifold (usually compact, except
when we work locally) with cyclic stabiliser groups and an orbifold line
bundle $L$ over it. We let $\operatorname{ord}_{x}$ denote the order of a
point $x\in X$: the size of the cyclic stabiliser group of any lift of this
point in an orbifold chart. When $X$ is compact, $\operatorname{ord}(X)$
denotes the least common multiple of this finite collection of integers.

Then the same expansion of $B_{k}$ holds away from the orbifold locus [DLM,
Son] but not over it. We illustrate the situation locally using the simplest
example: 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. An ordinary (non-orbifold) line bundle is one pulled back from
the quotient, i.e. one which has trivial $\mathbb{Z}/2\mathbb{Z}$-action
upstairs when considered as a trivial line bundle there. This has invariant
sections $\mathbb{C}[x]=\mathbb{C}[z^{2}]$. We do not consider such line
bundles as their sections only see the quotient manifold
$\mathrm{Spec}\,\mathbb{C}[x]$ (to which the usual Bergman expansion applies),
missing the extra functions of $\sqrt{x}=z$ that the orbifold sees.

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

######  Definition 2.

[RT] 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. (That is, $L^{\operatorname{ord}(X)}$ is ample in the
usual sense when thought of as a line bundle on the underlying space of $X$;
by the Kodaira-Baily embedding theorem [Bai] one can equivalently ask that $L$
admits a hermitian metric with positive curvature.)

In [RT, Sections 2.5–2.6] this notion of ampleness is shown to be equivalent
to the existence of an orbifold embedding

$(X,L)\hookrightarrow(\mathbb{WP},\mathcal{O}(1))$

of $X$ into a weighted projective space $\mathbb{W}\mathbb{P}$. That is, the
orbifold structure of $X$ is pulled back from that of $\mathbb{WP}$, and $L$
is the pullback of the orbifold hyperplane bundle
$\mathcal{O}_{\mathbb{WP}}(1)$.

Just as in our example above, for any orbifold line bundle $L$ which has
nontrivial stabiliser action on the line $L_{x}$ over an orbifold point $x$,
the sections of $L$ (i.e. the invariant sections upstairs) all vanish at $x$.
Therefore for most powers $L^{k}$ of an orbi-ample line bundle, the Bergman
kernel $B_{k}$ will vanish at orbifold points. Conversely for powers divisible
by $\operatorname{ord}_{x}$, $B_{k}(x)$ will be nonzero, so in general
$B_{k}(x)$ will have some periodic behaviour in $k$ at orbifold points. The
purpose of this paper is to get a smooth global expansion by taking weighted
sums of Bergman kernels associated to various powers of $L$, flattening out
the periodicity. In the companion paper [RT] we apply this to orbifold Kähler
metrics of constant scalar curvature and their relationship to stability.

The Bergman kernel measures the local density of holomorphic sections, so to
get an expansion on an orbifold we first discuss the general local situation –
the quotient of an open set in $\mathbb{C}^{n}$ by a linear action of the
cyclic group $\mathbb{Z}/m\mathbb{Z}$. The reader will not lose much by
considering only the $\mathbb{C}/(\mathbb{Z}/2\mathbb{Z})$ example above, with
the orbifold line bundle with trivialisation which has weight $\pm 1$ under
the $\mathbb{Z}/2\mathbb{Z}$-action.

By the local ampleness condition there exists an identification between
$\mathbb{Z}/m\mathbb{Z}$ and the group of $m$th roots of unity with respect to
which there exists a local equivariant (not invariant!) trivialisation of $L$
of weight $-1$. Then, with respect to this trivialisation, sections of $L^{k}$
downstairs, i.e. invariant sections upstairs on $\mathbb{C}^{n}$, are the same
thing as functions upstairs of $\mathbb{Z}/m\mathbb{Z}$-weight $k$ mod $m$.
(In particular they vanish unless $k\equiv 0$ mod $m$.)

We consider the weighted Bergman kernel

$B^{\mathrm{orb}}_{k}:=\sum_{i}c_{i}B_{k+i}\,,$ (3)

where $c_{i}$ are positive constants and $i$ runs over some fixed finite index
set of nonnegative integers. This is the global expression downstairs on the
quotient; upstairs locally we are taking a similar expression, but only
include the sections of $L^{k+i}$ that have $\mathbb{Z}/m\mathbb{Z}$-weight
$0$ mod $m$ (or, using the trivialisation, the functions of weight $k+i$ mod
$m$).

Since functions of nonzero weight vanish at the orbifold point, the ordinary
Bergman kernel at the origin upstairs is equal to the sum of the
$|s_{\alpha}|^{2}$ over an orthonormal basis of _only the invariant sections_
– i.e. it equals the “downstairs” Bergman kernel $B_{k+i}$ at the origin.
(Using the trivialisation, this corresponds to summing over only the functions
of weight $k+i$.) Therefore by (1) applied to the ordinary upstairs Bergman
kernel at the origin, the downstairs kernel $B_{k+i}(0)$ has the expansion

$(k+i)^{n}+\frac{1}{2}\operatorname{Scal}(\omega)(k+i)^{n-1}+\cdots.$

Summing over $i$ we find that in (3) the functions upstairs of
$\mathbb{Z}/m\mathbb{Z}$-weight $u$ contribute, to leading order in $k$,

$\sum_{i\equiv u-k}c_{i}k^{n}$ (4)

to $B_{k}^{\mathrm{orb}}(0)$. (Here and throughout the paper the subscript
means that the sum is taken over all $i$ equal to $u-k$ mod $m$.)

For instance, in our $\mathbb{C}/(\mathbb{Z}/2\mathbb{Z})$ example above with
the nontrivial orbifold line bundle, the functions of
$\mathbb{Z}/2\mathbb{Z}$-weight $0$ (respectively $1$) mod $2$ contribute only
to the $B_{k+i}(0)$s with $k+i$ even (respectively odd). To even things out
and ensure that $B_{k}^{\mathrm{orb}}$ has a smooth global expansion we want
to use _all_ local functions of all $\mathbb{Z}/m\mathbb{Z}$-weights equally
(thus returning us to something close to the original Bergman kernel
_upstairs_). So we choose the $c_{i}$ so that the functions of each
$\mathbb{Z}/m\mathbb{Z}$-weight contribute to the sum (3) with the same
coefficient.

We therefore want for each $k$ for (4) to be equal for all $u$, i.e.

$\sum_{i\equiv u}c_{i}\quad\text{is independent of }u.$

More generally if $N\leq n$ and all of the coefficients of
$k^{n},\ldots,k^{n-N}$ in

$\sum_{i}c_{i}(k+i)^{n}$ (5)

have equal contributions in each weight (mod $m$), i.e.

$\sum_{i\equiv u}i^{p}c_{i}\quad\text{is independent of }u\text{ for
}p=0,..,N,$ (6)

then the same will be true of each $\sum_{i}c_{i}(k+i)^{n-j}$ for
$j=0,\ldots,N$, and $B_{k}^{\mathrm{orb}}$ will admit a polynomial expansion
of order $N$. In fact we impose a slightly stronger condition than (6) in
order to get the expansion at the $C^{r}$ level.

######  Theorem 7.

Let $(X,\omega)$ be a compact $n$-dimensional Kähler orbifold with cyclic
quotient singularities, and $L$ be an orbi-ample line bundle on $X$ equipped
with a hermitian metric whose curvature form is $2\pi\omega$. Fix $0\leq N\leq
n$ and $r\geq 0$ and suppose $c_{i}$ are a finite number of positive constants
chosen so that if $X$ has an orbifold point of order $m$ then

$\displaystyle\frac{1}{m}\sum_{i}i^{p}c_{i}$ $\displaystyle=$
$\displaystyle\mathop{\sum{i^{p}c_{i}}}_{i\equiv u\text{ mod }m}\quad\text{for
all }u\text{ and all }p=0,\ldots,N+r.$ (8)

Then the function

$B^{\mathrm{orb}}_{k}:=\sum_{i}c_{i}B_{k+i}$

admits a global $C^{r}$-expansion of order $N$. That is, there exist smooth
functions $b_{0},\ldots,b_{N}$ on $X$ such that

$B^{\mathrm{orb}}_{k}=b_{0}k^{n}+b_{1}k^{n-1}+\cdots+b_{N}k^{n-N}+O(k^{n-N-1}),$

where the $O(k^{n-N-1})$ term is to be taken in the $C^{r}$-norm. Furthermore
the $b_{j}$ are universal polynomials in the constants $c_{i}$ and the
derivatives of the Kähler metric $\omega$; in particular

$b_{0}=\sum_{i}c_{i}\quad\text{ and }\quad
b_{1}=\sum_{i}c_{i}\left(ni+\frac{1}{2}\operatorname{Scal}(\omega)\right),$

where $\operatorname{Scal}(\omega)$ is the scalar curvature of $\omega$.

######  Remark 9.

If condition (8) holds for some $m$, then it also holds if $m$ is replaced by
any factor; hence for the Theorem it is sufficient to assume it holds when
$m=\operatorname{ord}(X)$ is the lowest common multiple of the orders of the
stabiliser groups of all points of $X$. So in particular, the $N=r=0$ case
yields a top-order $C^{0}$-expansion for $B_{k}^{\mathrm{orb}}$ when $c_{i}=1$
for $i=1,\ldots,\operatorname{ord}(X)$ and $c_{i}=0$ otherwise. It turns out
(Lemma 5) that (8) is equivalent to asking that the function
$\sum_{i}c_{i}z^{i}$ has a root of order $N+r+1$ at each nontrivial $m$-th
root of unity. Thus there is little loss in assuming the $c_{i}$ are defined
by

$\sum_{i}c_{i}z^{i}:=(z^{m-1}+z^{m-2}+\cdots+1)^{N+r+1}.$

In fact the condition (8) is natural in the sense that it is also necessary
for the existence of an expansion; see Remark 8. Also, if $X$ is a manifold
then there is no condition on the $c_{i}$, and the Theorem is equivalent to
the expansion (1) for manifolds stated above.

######  Remark 10.

By a $C^{r}$-expansion of order $N$ we mean there is a constant $C_{r,N}$ such
that

$\left\|B_{k}^{\mathrm{orb}}-\left(b_{0}k^{n}+b_{1}k^{n-1}+\dots+b_{N}k^{n-N}\right)\right\|_{C^{r}}\leq
C_{r,N}k^{n-N-1},$

where the norm $\|f\|_{C^{r}}$ is the sum of the supremum norms of $f$ and its
first $r$ derivatives over all of $X$ in the orbifold sense (i.e. we measure
the norm of derivatives using the metric upstairs on orbifold charts). The
Theorem generalises to hermitian metrics $h$ on $L$ with curvature form
$2\pi\omega_{h}$ not necessarily equal to $\omega$, the only change being that
the coefficients in the expansion become

$\displaystyle b_{0}$ $\displaystyle=$
$\displaystyle\frac{\omega_{h}^{n}}{\omega^{n}}\sum_{i}c_{i},$ $\displaystyle
b_{1}$ $\displaystyle=$
$\displaystyle\frac{\omega_{h}^{n}}{\omega^{n}}\sum_{i}c_{i}[ni+\operatorname{tr}_{\omega_{h}}(\operatorname{Ric}(\omega))-\frac{1}{2}\operatorname{Scal}(\omega_{h})].$
(11)

Moreover the constants $C_{r,N}$ can be taken to be uniform as $(h,\omega)$
runs over a compact set.

The strategy of our proof is to use known results on manifolds. For global
quotients one can obtain what we want rather easily by averaging the expansion
upstairs. To apply this to general orbifolds we need to be able to work
locally, which is what the approach of Berman-Berndtsson-Sjöstrand [BBS] to
the expansion (1) allows us to to do. This starts by proving a local expansion
for the Bergman kernel, and then uses the Hörmander estimate to pass to a
global expansion. In Section 2 we average these local expansions to get local
orbifold expansion, and use it in Section 3 to get a global expansion in much
the same way as for manifolds.

One consequence is a Riemann-Roch expression for orbi-ample line bundles
obtained by integrating the expansion over $X$. The difference between this
and the general Kawasaki-Riemann-Roch formula for orbifolds [Kaw] is that the
periodic term coming from the orbifold singularities vanishes due to the
weighted sum over $i$. We denote the orbifold canonical bundle by
$K_{\mathrm{orb}}$ (defined, for instance, in [RT, Section 2.4]).

######  Corollary 12.

Let $X$ be an $n$-dimensional orbifold with cyclic quotient singularities and
$L$ be an orbi-ample line bundle. Let $m=\operatorname{ord}(X)$ and suppose

$\frac{1}{m}\sum_{i}c_{i}=\mathop{\sum{c_{i}}}_{i\equiv u\text{ mod
m}}\quad\text{ and
}\,\,\quad\frac{1}{m}\sum_{i}ic_{i}=\mathop{\sum{ic_{i}}}_{i\equiv u\text{ mod
m}}\quad\text{ for all }u.$

Then

$\sum_{i}c_{i}h^{0}(L^{k+i})=a_{0}k^{n}+a_{1}k^{n-1}+O(k^{n-2})$

with

$\displaystyle a_{0}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}c_{i}}{n!}\int_{X}c_{1}(L)^{n},$ $\displaystyle
a_{1}$ $\displaystyle=$ $\displaystyle\frac{\sum
ic_{i}}{(n-1)!}\int_{X}c_{1}(L)^{n}-\frac{\sum_{i}c_{i}}{2(n-1)!}\int_{X}c_{1}(K_{\mathrm{orb}})c_{1}(L)^{n-1}.$
(13)

###### Proof.

This follows from $\int_{X}B_{k}\frac{\omega^{n}}{n!}=h^{0}(L^{k})$ and the
fact that the integral of the scalar curvature of any orbifold Kähler metric
is the topological quantity
$-\frac{1}{(n-1)!}\int_{X}c_{1}(K_{\mathrm{orb}})c_{1}(L)^{n-1}$. ∎

As an indication of the connection with constant scalar curvature metrics we
give an analogue of Donaldson’s Theorem [Don, Theorem 2] for orbifolds (for a
more applicable generalisation see [RT, Theorem 5.1]).

######  Corollary 14.

Suppose that (8) is satisfied for $N=1$ and $r=2$. For each $k\gg 0$ let
$h_{k}$ be a hermitian metric on $L$ with curvature $2\pi\omega_{k}$. Consider
the diagonal sequence of Bergman kernels $B_{k}^{\mathrm{orb}}$ associated to
the metrics $(h_{k},\omega_{k})$ and the power $L^{k}$ of $L$, and suppose
that for each $k$ this function is constant over $X$. If also $\omega_{k}$
converges in $C^{2}$ to a Kähler metric $\omega_{\infty}$ then
$\operatorname{Scal}(\omega_{\infty})$ is constant.

###### Proof.

Let $=\int_{X}\frac{c_{1}(L)^{n}}{n!}$ be the volume of $L$. Notice that since
the Bergman kernel is constant, by the previous corollary and integrating over
$X$ we have ${}_{k}^{\mathrm{orb}}=a_{0}k^{n}+a_{1}k^{n-1}+O(k^{n-2})$ with
$a_{i}$ as in (12) . Hence applying Theorem 7 to the pair
$(h_{k},\omega_{k})$,

$\frac{1}{}(a_{0}k^{n}+a_{1}k^{n-1}+O(k^{n-2}))=B_{k}^{\mathrm{orb}}=b_{0}k^{n}+b_{1}k^{n-1}+O(k^{n-2}),$

where the $O(k^{n-2})$ term is independent of $(h_{k},\omega_{k})$ since they
lie in a compact set. Furthermore $b_{0}=\sum_{i}c_{i}=\frac{a_{0}}{}$ and
$b_{1}=\sum_{i}c_{i}(ni+\frac{1}{2}\operatorname{Scal}(\omega_{k}))$. Thus the
$k^{n}$ terms cancel, so $b_{1}$ tends to $a_{1}{-1}$ uniformly over $X$ as
$k$ tends to infinity. Since $\omega_{k}$ tends to $\omega_{\infty}$ in
$C^{2}$, $\operatorname{Scal}(\omega_{k})$ tends to
$\operatorname{Scal}(\omega_{\infty})$ and thus
$\operatorname{Scal}(\omega_{\infty})$ is constant. ∎

### Interpretation using Peaked Sections

Intuitively one can understand what is going on as follows. On a manifold
$B_{k}(x)$ is the pointwise norm square of a _peaked section_ $s_{x}$ at $x$
of unit $L^{2}$-norm. This is glued from zero outside a small ball of radius
$O(k^{-1/2})$ centred on $x$ and the standard local model (with Gaussian norm)
inside the ball. (If $s_{x}$ is constructed to have zero $L^{2}$-inner product
with any section vanishing at $x$ then the Bergman kernel at $x$ is
_precisely_ $|s_{x}(x)|^{2}$; if not then this is still a good enough
approximation for the discussion here.)

To describe the orbifold case consider the simple model consisting of a single
orbifold chart $\mathbb{C}\to\mathbb{C}/(\mathbb{Z}/m\mathbb{Z})$. Suppose
that $x$ is a smooth point, which in this model case is any point other than
the origin. Then for large $k$ we can pick a ball of radius $O(k^{-1/2})$
centred at $x$ and disjoint from the origin. It therefore admits a copy
upstairs in $\mathbb{C}$, i.e. a ball around one of its preimages. Then just
as in the smooth case, there is a peaked section supported in this ball, and
moving it around $\mathbb{C}$ by the group action gives an invariant section
with $m$ peaks. This can then be equivariantly glued to zero, giving an
invariant section that has $L^{2}$-norm approximately $m$, but considered as a
section downstairs it has norm $1$ (since the volume downstairs is defined to
be the volume upstairs divided by the order of the chart). Therefore its
pointwise norm squared at $x$ is approximately the value of the Bergman
kernel.

Now instead consider an orbifold point, which here is the origin. A section
upstairs on $\mathbb{C}$, peaked about the origin, can be equivariant _only_
when $k\equiv 0$ mod $m$. Therefore for most $k$, the Bergman kernel will have
value zero at the origin. But when $k$ is a multiple of $m$, we get an
invariant section with $L^{2}$-norm 1 upstairs, and therefore $L^{2}$-norm
only $\frac{1}{m}$ when considered as an orbifold section downstairs.
Therefore we multiply the section by $\sqrt{m}$ to get one of unit
$L^{2}$-norm, and find that the Bergman kernel of the orbifold at the origin
is approximately $m$ times as big as at manifold points.

Thus when we sum the Bergman kernels over a period $L^{k},L^{k+1},\ldots$ of
length at least $m$ we stand a chance of them averaging out to something which
is again approximately constant to top order in $k$. The condition that they
do is that the average of the coefficients $c_{i}$ with which we make each
$L^{k+i}$ contribute is equal to the sum of the coefficients $c_{i}$ which
contribute at the origin (i.e. those for which $k+i\equiv 0$ mod $m$). But
this is precisely the $p=0$ condition in (8). A similar analysis for $p=1$
means that we can ensure that we smooth out all of the $i$-dependence of the
orbifold Bergman kernel except the part coming from scalar curvature.

Conventions: We refer to [RT] for detailed definitions and conventions. For
this paper we need only that an orbifold with cyclic stabiliser groups is an
analytic space covered by charts of the form $U\to U/G\subset X$, where $U$ is
an open set in $\mathbb{C}^{n}$ and $G$ is a finite cyclic group acting
effectively and _linearly_ on $U$; see for example [RT, Section 2.1]. It is
important for our applications that we allow orbifold structure in codimension
one. Thus if $m\geq 2$ then $\mathbb{C}/(\mathbb{Z}/m\mathbb{Z})$ is
considered a distinct orbifold from $\mathbb{C}$ even though they have the
same underlying space. Quantities on $X$ (hermitian metrics, Kähler metrics,
sections, supremum norms etc.) are always taken in the orbifold sense, as an
invariant quantity upstairs in orbifold charts. An orbifold line bundle is an
equivariant line bundle on orbifold charts; the gluing condition is described
in [RT].

Acknowledgements. We thank Florin Ambro, Bo Berndtsson and Yanir Rubenstein
for helpful conversations. We also wish to acknowledge our debt to the paper
[BBS], which allowed for a significant improvement of ours.

##  Chapter 1 Bergman Kernels

From now on let $(X,\omega)$ be a compact $n$-dimensional Kähler orbifold and
$L$ be an orbifold line bundle with a hermitian metric $h$ whose curvature
form is $2\pi\omega$. By abuse of notation we denote the induced metric on
$L^{k}$ also by $h$, which along with the volume form defined by $\omega$
yields an $L^{2}$-inner product on $H^{0}(L^{k})$ given by

$(s,t)_{L^{2}}=\int_{X}(s,t)_{h}\frac{\omega^{n}}{n!}\quad\text{for }s,t\in
H^{0}(L^{k}).$

The Bergman kernel of $L^{k}$ is defined to be
$B_{k}(x)=\sum_{\alpha}|s_{\alpha}(x)|_{h}^{2}$, where $\\{s_{\alpha}\\}$ is
any $L^{2}$-orthonormal basis for $H^{0}(L^{k})$. Recall that the _reproducing
kernel_ of $L^{k}$ is the section $K=K_{k}$ of $L^{k}\boxtimes\bar{L}^{k}$ on
$X\times X$ given by

$K(y,x)=\sum_{\alpha}s_{\alpha}(y)\boxtimes\overline{s}_{\alpha}(x),$

and therefore

$B_{k}(x)=|K(x,x)|_{h}.$

More invariantly, for $x\in X$ let $K_{x}$ be the section of
$L^{k}\otimes\bar{L}_{x}^{k}$ on $X$ given by $K_{x}(y):=K(y,x)$, so that
$(s,K_{x})_{L^{2}}$ is an element of the line $L_{x}^{k}$ for any section $s$
of $L^{k}$. Then the defining property of the reproducing kernel is that this
should equal $s(x)\in L_{x}^{k}$:

$s(x)=(s,K_{x})_{L^{2}}\quad\text{ for all }s\in H^{0}(L^{k}).$

As discussed in the introduction, when $X$ is a manifold the functions $B_{k}$
admit a global $C^{\infty}$-expansion as $k$ tends to infinity. This has been
worked out and extended in various ways, e.g. [Fef, Tia, Zel, Cat, Rua, DLM];
moreover the associated functions $b_{j}$ are geometric and depend on the
curvature of $\omega$ [Lu]. Our proof is based on the approach [BBS] that
starts by constructing a local expansion, as we now describe.

Let $U$ be an open ball in $\mathbb{C}^{n}$ centred at the origin, and
$\phi\colon U\to\mathbb{R}$ be smooth and strictly plurisubharmonic. Setting
$\omega=\frac{i}{2\pi}\partial\bar{\partial}\phi$, define an inner product on
the space of smooth functions by

$(u,v)^{2}_{\phi}=\int_{U}u\bar{v}\,e^{-\phi}\frac{\omega^{n}}{n!},$

and let $H(U)_{\phi}$ be the space of holomorphic functions of finite norm.
(The reader should of course think of this situation as arising when $U$ is a
chart of a manifold over which the hermitian line bundle $L|_{U}$ has
curvature $2\pi\omega$ and a holomorphic trivialisation of norm
$e^{-\phi/2}$.) Let $\chi$ be a smooth cutoff function supported on $U$ that
takes the value $1$ on $\frac{1}{2}U$.

######  Definition 1.

We say that a sequence $\mathcal{K}_{k},\ k\gg 0,$ of smooth functions on
$U\times U$ are _local reproducing kernels_ mod $O(k^{-N-1})$ for
$H(U)_{k\phi}$ if there is a neighbourhood $U_{0}\subset U$ of the origin such
that for any $x\in U_{0}$ and any $u\in H(U)_{k\phi}$,

$u(x)=(\chi
u,\mathcal{K}_{k,x})_{k\phi}+O(k^{-N-1}e^{k\phi(x)/2})\|u\|_{k\phi},$

where $\mathcal{K}_{k,x}(y):=\mathcal{K}_{k}(y,x)$, and the
$O(k^{-N-1}e^{k\phi(x)/2})$ term is uniform on $U_{0}$.

So whereas the reproducing kernels $K_{k}$ are globally and uniquely defined,
the local kernels $\mathcal{K}_{k}$ are far from unique. The next theorem
exhibits preferred ones, which will be shown later to be local approximations
to the global kernels.

We need the notion of a symmetric almost sesqui-holomorphic extension
$\psi\colon U\times U\to\mathbb{C}$ of $\phi$. That is,
$\psi(x,y)=\overline{\psi(y,x)},\ \psi(x,x)=\phi(x),$ and
$\bar{\partial}(\psi(x,\bar{y}))$ vanishes to all orders on $\\{x=y\\}$, i.e.
$D^{\alpha}(\bar{\partial}(\psi(x,\bar{y})))|_{\\{x=y\\}}=0$ for all $\alpha$.
(This notion is discussed in [BBS, Section 2.6] and [Rub, Section 3.3.3.1]; if
$\phi$ is analytic then one can take $\psi$ to be holomorphic in the first
variable and anti-holomorphic in the second.) By shrinking $U$ if necessary we
can ensure there is a $\delta>0$ such that

$\operatorname{Re}(\psi(x,y))\leq\phi(x)/2+\phi(y)/2-\delta\|x-y\|^{2}$ (2)

for $x,y\in U$ [BBS, Equation 2.7], [Rub, Lemma 3.8]. We write
$f=O(k^{-\infty})$ to mean $f=O(k^{-M})$ for any $M$.

######  Theorem 3 (Berman-Berndtsson-Sjöstrand).

For fixed $N,r\geq 0$ there exist smooth functions $\tilde{b}_{j}$ defined on
$U\times U$ such that

$\mathcal{K}_{k}(y,x)=\left(k^{n}+\tilde{b}_{1}(y,x)k^{n-1}+\dots+\tilde{b}_{N+r}(y,x)k^{n-N-r}\right)e^{k\psi(y,x)}$

is a local reproducing kernel mod $O(k^{-N-r-1})$ for $H(U)_{k\phi}$. Each
$\tilde{b}_{j}$ can be written as a polynomial in the derivatives
$\partial_{x}^{\alpha}\bar{\partial}^{\beta}_{y}\psi(x,y)$, and in particular
$\tilde{b}_{1}(x,x)=\frac{1}{2}\operatorname{Scal}(\omega)$.

Moreover, if $D_{x,y}$ is a differential operator of any order in $x$ and $y$
then $e^{-k(\phi(x)/2+\phi(y)/2)}D_{x,y}\bar{\partial}_{x}\mathcal{K}_{k}$ and
$e^{-k(\phi(x)/2+\phi(y)/2)}D_{x,y}\partial_{y}\mathcal{K}_{k}$ are both
$O(k^{-\infty})$.

###### Proof.

This is [BBS, Proposition 2.5 and Section 2.4] when $\phi$ is analytic, and
generalised to the smooth case in _loc. cit._ Proposition 2.7. (The cited work
is stated in the case that $U$ is a ball of radius $1$, but case for general
radius follows immediately on rescaling; there is also an extra factor of
$\pi^{-n}$ due to differences in conventions for the curvature form.) ∎

Observe that if a finite group $G$ acts on $U$ preserving $\phi$ then we can
take $\psi$ and $\chi$ to be $G$-invariant, in which case
$\mathcal{K}_{k}(\zeta x,\zeta x)=\mathcal{K}_{k}(x,x)$ for all $\zeta\in G$.

##  Chapter 2 Local expansion

We next extend the local expansion of the previous section to orbifold charts,
which begins by averaging the local reproducing kernels $\mathcal{K}_{k}$.
Building on the notation of the previous section, suppose that a cyclic group
$G$ of order $m$ acts linearly and faithfully on $U$ and that our
plurisubharmonic function $\phi$ and cutoff function $\chi$ are invariant. The
inner product on the space of functions is replaced with

$(u,v)_{\phi,m}=\frac{1}{m}\int_{U}u\bar{v}\,e^{-\phi}\frac{\omega^{n}}{n!},$

and $H(U)_{\phi,m}$ is the space of holomorphic functions of finite norm. Fix
a generator $\zeta$ of $G$ and suppose we are given a character of $G$ which
maps $\zeta$ to a primitive $m$-th root of unity $\lambda$. We say a function
$u$ on $U$ _has weight $j$_ if $u(\zeta x)=\lambda^{j}u(x)$ for all $x\in U$.
(The reader should now think of $U$ as arising from an orbifold chart $U\to
U/G$, and the character as describing the action of $G$ on $L|_{U}$ under some
(noninvariant) trivialisation; the condition that $\lambda$ is primitive comes
from the local condition for orbi-ample line bundles, and the sections of
$L^{k}|_{U}$ now arise from functions on $U$ of weight $k$ mod $m$.)

Let
$\mathcal{K}_{k}=(k^{n}+\tilde{b}_{1}k^{n-1}+\cdots+\tilde{b}_{N+r}k^{n-N-r})e^{k\psi}$
be the preferred local reproducing kernel for $H(U)_{k\phi}$ from Theorem 3
and define

$\mathcal{K}_{k}^{\mathrm{av}}(y,x):=\frac{1}{m}\sum_{u,v=0}^{m-1}\lambda^{k(v-u)}\mathcal{K}_{k}(\zeta^{u}y,\zeta^{v}x).$
(1)

One sees that $\mathcal{K}_{k}^{\mathrm{av}}$ has weight $k$ in the first
variable and weight $-k$ in the second variable (as indeed it must if it is to
be a section of $L^{k}|_{U}\boxtimes\bar{L}^{k}|_{U}$).

######  Lemma 2.

$\mathcal{K}_{k}^{\mathrm{av}}$ is a local reproducing kernel mod
$O(k^{-N-r-1})$ for the subspace of $H(U)_{k\phi,m}$ consisting of functions
that have weight $k$.

###### Proof.

Let $u$ have weight $k$. Then by the change of variables
$y^{\prime}=\zeta^{u}y$ and the invariance of $\phi$ and $\chi$ we have, up to
terms of order $O(e^{k\phi(x)/2}k^{-N-r-1})\|u\|_{k\phi}$,

$\int_{U}\chi(y)u(y)\overline{\lambda^{k(v-u)}\mathcal{K}_{k,\zeta^{v}x}(\zeta^{u}y)}e^{-k\phi(y)}\frac{\omega^{n}}{n!}=\lambda^{-kv}u(\zeta^{v}x)=u(x),$

so

$(\chi
u,\mathcal{K}_{k,x}^{\mathrm{av}})_{k\phi,m}=\frac{1}{m}\int_{U}\chi(y)u(y)\overline{\mathcal{K}_{k,x}^{\mathrm{av}}}(y)e^{-k\phi(y)}\frac{\omega^{n}}{n!}=\frac{1}{m^{2}}\sum_{u,v=0}^{m-1}u(x)=u(x).$

∎

We show in the next section that downstairs on the orbifold, the Bergman
kernel $B_{k}$ is locally approximated by
$\mathcal{K}_{k}^{\mathrm{av}}(x,x)e^{-k\phi(x)}$, and thus the weighted
Bergman kernel $B^{\mathrm{orb}}_{k}=\sum_{i}c_{i}B_{k+i}$ is approximated by

$\mathcal{B}_{k}^{\mathrm{orb}}(x):=\sum_{i}c_{i}\mathcal{K}_{k+i}^{\mathrm{av}}(x,x)e^{-(k+i)\phi(x)},$

where, we recall, the sum is over a fixed finite index of nonnegative integers
$i$.

######  Theorem 3.

Suppose the $c_{i}$ satisfy

$\frac{1}{m}\sum_{i}i^{p}c_{i}=\sum_{i\equiv u}i^{p}c_{i}\quad\text{for }0\leq
u\leq m-1,0\leq p\leq N+r.$ (4)

Then there is a local $C^{r}$-expansion of $\mathcal{B}_{k}^{\mathrm{orb}}$ of
order $N$

$\mathcal{B}_{k}^{\mathrm{orb}}(x)=b_{0}(x)k^{n}+b_{1}(x)k^{n-1}+\cdots+b_{N}(x)k^{n-N}+O(k^{n-N-1})$

on $U$. Moreover the $b_{j}$ depend only on $c_{i}$ and the derivatives of the
metric; in particular $b_{0}=\sum_{i}c_{i}$ and
$b_{1}=\sum_{i}c_{i}(ni+\frac{1}{2}\operatorname{Scal}(\omega))$.

Our proof requires a reformulation of the condition made on the $c_{i}$.

######  Lemma 5.

The constants $c_{i}$ satisfy (4) if and only if the function
$\sum_{i}c_{i}z^{i}$ has a root of order $N+r+1$ at every $m$-th root of unity
other than $1$.

###### Proof.

If (4) holds then for each $0\leq p\leq N+r$ the quantity $c=\sum_{i\equiv
u}i^{p}c_{i}$ is independent of $u$. Hence if $\sigma^{m}=1$ with $\sigma\neq
1$ then

$\sum_{i}i^{p}c_{i}\sigma^{i}=\sum_{u=0}^{m-1}\sum_{i\equiv
u}i^{p}c_{i}\sigma^{i}=\sum_{u=0}^{m-1}\sigma^{u}\sum_{i\equiv
u}i^{p}c_{i}=c\sum_{u=0}^{m-1}\sigma^{u}=0,$

proving that $\sum_{i}c_{i}z^{i}$ has a root of order $N+r+1$ at $\sigma$. For
the converse let $\sigma$ be a primitive root of unity, so the hypothesis is
that $\sum_{i}i^{p}c_{i}\sigma^{si}=0$ for $1\leq s\leq m-1$ and $0\leq p\leq
N+r$, so for any given $u$,

$\frac{1}{m}\sum_{i}i^{p}c_{i}=\frac{1}{m}\sum_{s=0}^{m-1}\sigma^{-su}\sum_{i}i^{p}c_{i}\sigma^{si}=\frac{1}{m}\sum_{i}i^{p}c_{i}\sum_{s=0}^{m-1}\sigma^{(i-u)s}=\sum_{i\equiv
u}i^{p}c_{i}.$

∎

###### Proof of Theorem 3.

Write

$\displaystyle\mathcal{B}_{k}^{\mathrm{orb}}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{m}\sum_{i}c_{i}\sum_{u,v=0}^{m-1}\lambda^{(k+i)(v-u)}\mathcal{K}_{k+i}(\zeta^{u}x,\zeta^{v}x)e^{-(k+i)\phi(x)}$
$\displaystyle=$ $\displaystyle S_{1}+S_{2},$

where $S_{1}$ consists of the terms with $u=v$ and $S_{2}$ consists of the
terms with $u\neq v$. We show below that $S_{2}$ is $O(k^{n-N-1})$. Observe
that

$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{m}\sum_{i}c_{i}\sum_{u=0}^{m-1}\mathcal{K}_{k+i}(\zeta^{u}x,\zeta^{u}x)e^{-(k+i)\phi(x)}$
$\displaystyle=$
$\displaystyle\sum_{i}c_{i}\mathcal{K}_{k+i}(x,x)e^{-(k+i)\phi(x)},$

by the invariance of $\mathcal{K}_{k+i}$, and since each
$\mathcal{K}_{k+i}e^{-(k+i)\phi}$ is a polynomial in $k$ the same is true of
$S_{1}$. In fact,

$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\sum_{i}c_{i}\sum_{j=0}^{N}(k+i)^{n-j}\tilde{b}_{j}(x,x)$
$\displaystyle=$ $\displaystyle\sum_{j=0}^{N}b_{j}(x)k^{n-j},$

where

$b_{j}(x)=\sum_{q=0}^{j}\left(\tilde{b}_{q}(x,x)\binom{n-q}{j-q}\sum_{i}c_{i}i^{j-q}\right).$

In particular $b_{0}$ and $b_{1}$ are as claimed in the statement of the
Theorem.

Now to bound $S_{2}$, write $S_{2}=\frac{1}{m}\sum_{u\neq v}S_{u,v}$ where

$S_{u,v}=\sum_{i}c_{i}\mathcal{K}_{k+i}(\zeta^{u}x,\zeta^{v}x)\lambda^{(k+i)(v-u)}e^{-(k+i)\phi(x)}.$

Fixing $u\neq v$, let $\sigma=\lambda^{v-u}$ and note that since $\lambda$ is
primitive, $\sigma\neq 1$. Furthermore set

$\eta=\eta(x)=e^{\psi(\zeta^{u}x,\zeta^{v}x)-\phi(x)},$

so

$\displaystyle S_{u,v}$ $\displaystyle=$
$\displaystyle\sum_{i}c_{i}\sum_{j=0}^{N}(k+i)^{n-j}\tilde{b}_{j}(\zeta^{u}x,\zeta^{v}x)\sigma^{k+i}\eta^{k+i}$
$\displaystyle=$
$\displaystyle\sigma^{k}\sum_{j=0}^{N}k^{n-j}\sum_{q=0}^{j}\tilde{b}_{q}(\zeta^{u}x,\zeta^{v}x)\binom{n-q}{j-q}\sum_{i}c_{i}i^{j-q}\sigma^{i}\eta^{k+i}.$

Therefore to prove the theorem it is sufficient to show that for $0\leq l\leq
N$,

$u_{l,k}:=\sum_{i}c_{i}i^{l}\sigma^{i}\eta^{k+i}=O(k^{l-N-1})\text{ in
}C^{r}.$ (7)

In fact since $du_{l,k}=\frac{d\eta}{\eta}(ku_{l,k}+u_{l+1,k})$, one sees by
induction that it is sufficient to prove $u_{l,k}=O(k^{l-N-r-1})$ in $C^{0}$.
To this end write

$\displaystyle u_{l,k}$ $\displaystyle=$
$\displaystyle\left[\frac{\sum_{i}c_{i}i^{l}\sigma^{i}\eta^{i}}{(\eta-1)^{N+r-l+1}}\right](\eta-1)^{N+r-l+1}\eta^{k}.$

From Lemma 5 the function $\eta\mapsto\sum_{i}c_{i}i^{l}\sigma^{i}\eta^{i}$
has a root of order $N+r-l+1$ at $\eta=1$, and so the term is square brackets
is bounded. So it is sufficient to prove the following:

_Claim:_ Let $s\geq 1$ and $u\neq v$. Then

$(\eta-1)^{s}\eta^{k}=O(k^{-s}).$

_Proof:_ Since $G$ acts linearly on $U$, so we can write $U=U_{1}\times U_{2}$
where the action is faithful on $U_{1}$ and trivial on $U_{2}$. For $x\in U$
write $x=(x_{1},x_{2})$ under this decomposition. Then there exist positive
constants $c,c^{\prime}$ such that
$c\|x_{1}\|^{2}\leq\|\zeta^{u}x-\zeta^{v}x\|^{2}\leq c^{\prime}\|x_{1}\|^{2}$
for all $x\in U$. Hence from (2), there is a $\delta^{\prime}>0$ such that

$\operatorname{Re}(\psi(\zeta^{u}x,\zeta^{v}x)-\phi(x))\leq-\delta^{\prime}\|x_{1}\|^{2}$

for all $x\in U$. If $x_{1}=0$ then $\eta=1$ and we are done. Assuming
$x_{1}\neq 0$, recall that if $z$ is a complex number with
$\operatorname{Re}z<0$ then
$|e^{kz}|\leq\frac{s!}{(-k\operatorname{Re}z)^{s}}$ for all $k$. Thus there is
a constant $C$ such that

$\left|(\eta(x)-1)^{s}\eta(x)^{k}\right|\leq\frac{C}{k^{s}}\left|\frac{\eta(x)-1}{\|x_{1}\|^{2}}\right|^{s}.$

But $\psi(\zeta^{u}x,\zeta^{v}x)-\phi(x)=O(\|x_{1}\|^{2})$, so
$\frac{\eta(x)-1}{\|x_{1}\|^{2}}$ is bounded and the claim follows. ∎

######  Remark 8.

Conversely, suppose that $\mathcal{B}_{k}^{\mathrm{orb}}$ admits an asymptotic
expansion in $C^{0}$ of order $N$ at the point $x=0$ in $U$ which is fixed by
the group action. We will show that the $c_{i}$ must satisfy the conditions
(4) for $r=0$.

Since $\lambda$ is primitive and sections of orbi-ample line bundles vanish at
orbifold points,

$\mathcal{K}_{k+i}^{\mathrm{av}}(0,0)=\left\\{\begin{array}[]{ll}m\mathcal{K}_{k+i}(0,0)&\text{
if }k+i\equiv 0\text{ mod }m,\\\ 0&\text{ otherwise.}\end{array}\right.$

Therefore

$\displaystyle\mathcal{B}_{k}^{\mathrm{orb}}(0)$ $\displaystyle=$
$\displaystyle
m\sum_{i\equiv-k}c_{i}\mathcal{K}^{\mathrm{av}}_{k+i}(0,0)=m\sum_{i\equiv-k}c_{i}\sum_{p=0}^{N}\tilde{b}_{p}(0,0)(k+i)^{n-p}$
$\displaystyle=$ $\displaystyle\sum_{p=0}^{N}b_{p}k^{n-p},$

where

$b_{p}=\sum_{q=0}^{p}\left(\tilde{b}_{q}(0,0)\binom{n-q}{p-q}\sum_{i\equiv-k}c_{i}i^{p-q}\right).$
(9)

Each $b_{p}$ is periodic in $k$; the existence of an asymptotic expansion of
$\mathcal{B}_{k}^{\mathrm{orb}}(0)$ implies that in fact each $b_{p}$ is
independent of $k$.

We now use induction on $p$ to show the conditions (4), i.e. that
$\sum_{i\equiv-k}c_{i}i^{p}$ is also independent of $k$. For $p=0$ the sum (9)
reduces to $b_{0}=\sum_{i\equiv-k}c_{i}$ since $\tilde{b}_{0}(0,0)=1$.
Therefore $b_{0}$’s independence of $k$ implies the $p=0$ case of (4). For
general $p$ the sum (9) is $\binom{n}{p}\sum_{i\equiv-k}c_{i}i^{p}$ plus terms
shown inductively to be independent of $k$. Therefore we recover the
conditions (4) for $r=0$.

##  Chapter 3 Global Expansion

We will require the following standard estimate, which we prove for
completeness. At first we do not require $X$ to be compact.

######  Lemma 1.

Let $X$ be a Kähler manifold of dimension $n$. There exists a constant $C$
such that for any $p\in X$ we have $|f(p)|\leq
C(a\|\bar{\partial}f\|_{C^{0}}+a^{-n}\|f\|_{L^{2}})$ for all smooth functions
$f$ and all sufficiently small $a>0$. Moreover if $X$ is compact then we can
choose $a>0$ uniformly over all $p\in X$.

###### Proof.

We first prove the one dimensional case. Let $\epsilon=\sup|\bar{\partial}f|$,
pick a local coordinate $z=re^{i\theta}$ about the point $p=0$, and let
$D_{a}$ denote the ball $|z|=r\leq a$ in this coordinate. By the fundamental
theorem of calculus,

$\frac{1}{2\pi a}\int_{\partial D_{a}}\frac{\partial
f}{\partial\theta}d\theta=0.$ (2)

We know that

$\bar{\partial}f=\frac{1}{2}\left(\frac{\partial f}{\partial
r}+\frac{i}{r}\frac{\partial f}{\partial\theta}\right)(dr-ird\theta)$

has a pointwise bound on its norm of $O(\epsilon)$. Here we measure norms of
1-forms in the standard metric, which by the compactness of $X$ is boundedly
close to the Kähler metric in $D_{a}$.

Then $\frac{1}{r}\frac{\partial f}{\partial\theta}$ is within $\epsilon$ of
$i\frac{\partial f}{\partial r}$, and (2) gives

$\frac{1}{2\pi}\left|\int_{\partial D_{a}}\frac{\partial f}{\partial
r}d\theta\right|=\left|\frac{1}{2\pi}\frac{d}{da}\int_{\partial
D_{a}}fd\theta\right|\leq\epsilon.$

Integrating with respect to $a$ gives the estimate

$|f(0)|\leq\frac{1}{2\pi}\left|\int_{\partial
D_{a}}fd\theta\right|+a\epsilon.$

Multiplying by $a$ and integrating again yields

$\frac{a^{2}}{2}|f(0)|\leq\frac{1}{2\pi}\int_{D_{a}}|f|rdrd\theta+\frac{a^{3}}{3}\epsilon.$

By the Cauchy-Schwarz inequality this gives the desired bound on $|f(0)|$ in
terms of the $L^{2}$-norm of $f$ and the pointwise supremum of
$|\bar{\partial}f|$.

Now we pass to the general case. Pick local coordinates $z_{i}$ in which $p$
is the origin, and apply the above argument over one dimensional discs in the
$z_{n}$ direction to bound

$|f(z_{1},\ldots,z_{n-1},0)|\leq\frac{1}{\pi a^{2}}\int_{|z_{n}|\leq
a}|f(z_{1},\ldots,z_{n_{1}},z_{n})|+\frac{2a}{3}\epsilon.$ (3)

Similarly

$|f(z_{1},\ldots,z_{n-2},0,0)|\leq\frac{1}{\pi a^{2}}\int_{|z_{n-1}|\leq
a}|f(z_{1},\ldots,z_{n_{2}},z_{n-1},0)|+\frac{2a}{3}\epsilon,$

which by (3) can be bounded by

$|f(z_{1},\ldots,z_{n-2},0,0)|\leq\frac{1}{\pi^{2}a^{4}}\int_{|z_{n}|,|z_{n-1}|\leq
a}|f(z_{1},\ldots,z_{n})|+\frac{2a}{3}\epsilon+\frac{2a}{3}\epsilon.$

Inductively we find that

$|f(0)|\leq\frac{1}{\pi^{n}a^{2n}}\int_{|z_{i}|\leq
a}|f(z_{1},\ldots,z_{n})|+\frac{2na}{3}\epsilon.$

Again an application of Cauchy-Schwartz gives the result. ∎

######  Corollary 4.

Let $X$ be a compact Kähler orbifold and $L$ be an orbifold line bundle with
hermitian metric. There exists a constant $C$ such that for any section
$s\in\Gamma(L^{k})$ and $x\in X$ we have $|s(x)|\leq
C(k^{-\frac{1}{2}}\|\bar{\partial}s\|_{C^{0}}+k^{\frac{n}{2}}\|s\|_{L^{2}})$.

###### Proof.

This follows from the previous result, since we can work locally in an
orbifold chart where upstairs we have a smooth Kähler metric. We trivialise
$L$ (and so each $L^{k}$) upstairs with a holomorphic section of norm $1$ at
$x$ which is possibly not invariant under $G$. In a ball of radius
$a=k^{-1/2}$ about $x$, the hermitian metric is then boundedly close to taking
absolute values. Picking the coordinates in the above proof to be invariant
under the finite group gives, for $k\gg 0$, a bound in terms of an integral
upstairs over a $G$-invariant ball. The actual integral on the orbifold
differs from this by dividing by the order of the group; since this is finite
we get the same order bound. ∎

We now apply the results from the previous section to orbifold charts. As
usual, $X$ is an orbifold with cyclic quotient singularities, and $L$ is an
orbi-ample line bundle with hermitian metric with curvature $2\pi\omega$.
Suppose that $U\to U/G$ is a small orbifold chart in $X$, where $U$ is a small
ball centred at the origin in $\mathbb{C}^{n}$ and $G$ is cyclic of order $m$.
By the orbi-ampleness condition, $L$ has a trivialisation with weight $-1$
under the action of $G$ via an identification between $G$ and the $m$th roots
of unity. Then sections of $L^{k}|_{U}$ are given locally by holomorphic
functions $f$ of weight $k$. The pointwise norm of such a section is
$|f(z)|e^{-k\phi(z)/2}$, where the plurisubharmonic function $\phi$ is the
norm squared of the trivialising section.

As previously mentioned, the local reproducing kernel
$\mathcal{K}_{k}^{\mathrm{av}}(y,x)$ on $U\times U$ defined in (1) has weight
$k$ in the first variable, and weight $-k$ in the second and $\chi$ is an
invariant cutoff function supported on $U$ and identically 1 on
$\frac{1}{2}U$. Thus multiplying by the local trivialisation, we think of
$\chi\mathcal{K}_{k,x}^{\mathrm{av}}$ as a smooth section of
$L^{k}\otimes\bar{L}^{k}_{x}$. The local reproducing property for
$\mathcal{K}_{k}^{\mathrm{av}}$ proved in Lemma 2 says there is a
neighbourhood $U_{0}\subset U$ of the origin such that if $x\in U_{0}$ and
$t\in H^{0}(L^{k})\otimes\bar{L}^{k}_{x}$ then

$t(y)=(t,\chi\mathcal{K}^{\mathrm{av}}_{k,y})_{L^{2}}+O(k^{-N-r-1})\|t\|_{L^{2}}\quad\text{for
}y\in U_{0},$

where the error term is measured using the hermitian metric on
$L_{y}^{k}\otimes\bar{L}_{x}^{k}$. In what follows $U$ will be fixed, but
$U_{0}$ will be allowed to shrink as required. If $s\in
H^{0}(L\boxtimes\bar{L})$ then applying this to $t=s_{x}=s(\ \cdot\ ,x)$ gives

$s(y,x)=(s_{x},\chi\mathcal{K}^{\mathrm{av}}_{k,y})_{L^{2}}+O(k^{-N-r-1})\|s_{x}\|_{L^{2}}.$
(5)

Now recall that $K_{k}=\sum_{\alpha}s_{\alpha}\boxtimes\overline{s}_{\alpha}$
denotes the global reproducing kernel discussed at the start of Section 1. The
next proposition shows how this is approximated by the local reproducing
kernels $\mathcal{K}^{\mathrm{av}}_{k}$.

######  Proposition 6.

There is a neighbourhood $U_{0}\subset\frac{1}{4}U$ of the origin such that

$K_{k}|_{U_{0}\times U_{0}}=\mathcal{K}_{k}^{\mathrm{av}}+O(k^{n-N-r-1}).$

###### Proof.

This is identical to the manifold case [BBS, Theorem 3.1], and for convenience
we sketch the details. The Bergman kernel has the extremal characterisation
$B_{k}(x)=\sup_{\|s\|=1}|s(x)|^{2}$ where the supremum is over all sections of
$L^{k}$ of unit $L^{2}$-norm. Using this, it is shown in [Ber, Theorem 1.1]
that there is a constant $C$ such that $B_{k}(x)\leq Ck^{n}$ uniformly on $X$
(the cited work is for noncompact manifolds; what is important is that it is
local, so the inequality only improves if we apply it upstairs on an orbifold
chart and restrict to invariant sections).

By the Cauchy-Schwarz inequality applied to
$K_{k}=\sum_{\alpha}s_{\alpha}\boxtimes\bar{s}_{\alpha}$, we have
$|K_{k}(y,x)|_{h}\leq\sqrt{B_{k}(x)B_{k}(y)}=O(k^{n})$, so in particular
$\|K_{k,x}\|_{L^{2}}=O(k^{n})$ uniformly in $x$. Putting $s:=K_{k}$ (which is
holomorphic) into the local reproducing property, (5) gives

$\displaystyle K_{k}(y,x)$ $\displaystyle=$
$\displaystyle(K_{k,x},\chi\mathcal{K}_{k,y}^{\mathrm{av}})_{L^{2}}+O(k^{n-N-r-1})$

on $U_{0}\times U_{0}$. Then for $x,y\in U_{0}$ we can write

$\displaystyle K_{k}(y,x)$ $\displaystyle=$
$\displaystyle\mathcal{K}^{\mathrm{av}}_{k}(y,x)-\overline{w(y,x)}+O(k^{n-N-r-1}),$
(7)

where

$\displaystyle w(y,x)$ $\displaystyle:=$
$\displaystyle\overline{\mathcal{K}^{\mathrm{av}}_{k}(y,x)}-(\chi\mathcal{K}^{\mathrm{av}}_{k,y},K_{k,x})_{L^{2}}$
$\displaystyle=$
$\displaystyle\chi(x)\mathcal{K}^{\mathrm{av}}_{k,y}(x)-(\chi\mathcal{K}^{\mathrm{av}}_{k,y},K_{k,x})_{L^{2}},$

as $\mathcal{K}_{k}^{\mathrm{av}}$ is hermitian. With the aim of bounding $w$,
define $u_{y}(\,\cdot\,):=w(y,\,\cdot\,)$ so

$u_{y}(\,\cdot\,)=\chi\mathcal{K}^{\mathrm{av}}_{k,y}-(\chi\mathcal{K}^{\mathrm{av}}_{k,y},K_{k,\,\cdot\,})_{L^{2}}.$

Notice that the inner product in this expression is precisely the projection
of $\chi\mathcal{K}^{\mathrm{av}}_{k,y}$ onto the space of holomorphic
sections of $L^{k}$. So said another way, $u_{y}$ is the $L^{2}$-minimal
solution of the equation

$\bar{\partial}u_{y}=\bar{\partial}(\chi\mathcal{K}^{\mathrm{av}}_{k,y}).$ (8)

Now

$\bar{\partial}(\chi\mathcal{K}^{\mathrm{av}}_{k,y})(t)=\mathcal{K}^{\mathrm{av}}_{k,y}(t)\bar{\partial}\chi(t)+\chi(t)\bar{\partial}\mathcal{K}^{\mathrm{av}}_{k,y}(t),$

which we claim is $O(k^{-\infty})$ in $C^{0}$. Since we are assuming that
$U_{0}\subset\frac{1}{4}U$, $\chi$ is identically $1$ on $2U_{0}$ and
supported on $U$ so $\bar{\partial}\chi(t)$ vanishes for $t\in 2U_{0}$ and for
$t\notin U$. On the other hand by (2) there is a $\delta_{1}>0$ such that

$\operatorname{Re}[2\psi(t,y)-\phi(t)-\phi(y)]\leq-\delta_{1}\|t-y\|^{2}$

on $U\times U$. If $t\in U\backslash 2U_{0}$, so that $t$ is a bounded
distance away from $y$, then by Theorem 3

$\displaystyle|\mathcal{K}_{k,y}(t)|_{h}^{2}$ $\displaystyle=$ $\displaystyle
O(k^{2n})e^{2k\psi(t,y)}e^{-\phi(t)}e^{-\phi(y)}$ $\displaystyle=$
$\displaystyle O(k^{2n})e^{k[2\psi(t,y)-\phi(t)-\phi(y)]}$ $\displaystyle\leq$
$\displaystyle O(k^{2n})e^{-\delta_{2}k}\ =\ O(k^{-\infty}).$

Thus $\mathcal{K}_{k,y}\bar{\partial}\chi=O(k^{-\infty})$ in $C^{0}$ on all of
$X$, and furthermore the $O(k^{-\infty})$ term is independent of $y\in U_{0}$.
Applying this to $\xi^{v}y$ as $v$ ranges over a period we get
$\mathcal{K}^{\mathrm{av}}_{k,y}\bar{\partial}\chi=O(k^{-\infty})$ as well.
Now from the second statement in Theorem 3,
$\chi\bar{\partial}\mathcal{K}^{\mathrm{av}}_{k,y}=O(k^{-\infty})$, hence
$\bar{\partial}(\chi\mathcal{K}^{\mathrm{av}}_{k,y})=O(k^{-\infty})$ in
$C^{0}$, and therefore in $L^{2}$ as well.

Thus applying the Hörmander estimate to sections of $L\otimes\bar{L}_{y}$ we
conclude that $\|u_{y}\|_{L^{2}}=O(k^{-\infty})$. Using this and that the
bound on $\bar{\partial}u_{y}$ is uniform in $x$ and $y$, Corollary 4 gives
the pointwise estimate

$|w(y,x)|=|u_{y}(x)|=O(k^{-\infty})\quad\text{ on }U_{0}\times U_{0}.$

Therefore (7) becomes $K_{k}=\mathcal{K}^{\mathrm{av}}_{k}+O(k^{n-N-r-1})$ on
$U_{0}\times U_{0}$, as required. ∎

So $\mathcal{K}_{k}^{\mathrm{av}}$ approximates $K_{k}$ to order
$O(k^{n-N-r-1})$ in the $C^{0}$ norm. We next use this to get a $C^{r}$
expansion at the expense of a factor of $O(k^{r})$.

######  Lemma 9.

Suppose $f_{k}(x,y)$ is a sequence of functions on $U_{0}\times U_{0}$ such
that $(\bar{\partial}_{x}f_{k})e^{-k(\phi(x)/2+\phi(y)/2)}$ and
$(\partial_{y}f_{k})e^{-k(\phi(x)/2+\phi(y)/2)}$ are both of order
$O(k^{-\infty})$. Suppose also that
$f_{k}(x,y)e^{-k(\phi(x)/2+\phi(y)/2)}=O(k^{q})$ uniformly on $U_{0}\times
U_{0}$. Then for any differential operator in $x$ and $y$ of total order $p$,

$(Df_{k})e^{-k(\phi(x)/2+\phi(y)/2)}=O(k^{q+p})\text{ on
}\frac{1}{2}U_{0}\times\frac{1}{2}U_{0}.$

###### Proof.

Fix $(x,y)\in\frac{1}{2}U_{0}\times\frac{1}{2}U_{0}$. Consider first the case
that $f(x,y)=f_{k}(x,y)$ is holomorphic in the first variable and anti-
holomorphic in the second. Suppose $x=(x_{1},\ldots x_{n})$,
$y=(y_{1},\ldots,y_{n})$ and let
$F(t)=f(x_{1},\ldots,t,\ldots,x_{n},y_{1},\ldots y_{n})$ and similarly
$\Phi(t)=\phi(x_{1},\ldots,t,\ldots,x_{n})$, where the $t$ lies in the $i$-th
coordinate. If $B$ denotes the ball of radius $k^{-1}$ around $x_{i}$, then
for $k$ sufficiently large $(x_{1},\ldots,t,\ldots,x_{n})\in U$ for all $t\in
B$. So applying the Cauchy formula to $B$ gives

$\displaystyle\left.\frac{\partial f}{\partial
x_{i}}\right|_{(x,y)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!e^{-k(\phi(x)/2+\phi(y)/2)}$
$\displaystyle=\\!\\!$ $\displaystyle\frac{1}{2\pi i}\int_{\partial
B}\\!\\!\\!\frac{F(t)e^{-k(\Phi(t)/2+\phi(y)/2)}}{(t-x_{i})^{2}}e^{-k(\phi(x)/2-\Phi(t)/2)}dt$
$\displaystyle=$ $\displaystyle\int_{\partial
B}\frac{O(k^{q})}{(t-x_{i})^{2}}dt$ $\displaystyle=$ $\displaystyle
O(k^{q+1}),$

where the second inequality uses $\phi(x)-\Phi(t)=O(|x_{i}-t|)=O(k^{-1})$ on
$\partial B$, so the exponential term is $O(1)$. The other derivatives are
treated similarly.

More generally if we assume only $\bar{\partial}_{x}f=O(k^{-\infty})$ then the
calculation above holds up to a term
$e^{-k(\phi(x)/2+\phi(y)/2)}\int_{B}\partial F/\partial\bar{t}\,dtd\bar{t}$
which is of order $O(k^{-\infty})$, so the conclusion still holds. ∎

Recall that the Bergman kernel is $B_{k}(x)=|K_{k}(x,x)|_{h}$.

######  Corollary 10.

For $k$ sufficiently large,
$B_{k}(x)=\mathcal{K}^{\mathrm{av}}_{k}(x,x)e^{-k\phi(x)}+O(k^{n-N-1})$ in
$C^{r}$ on $\frac{1}{2}U_{0}$. Moreover if $D$ is a differential operator of
order $p$ then

$DB_{k}=O(k^{n+p})\text{ in }C^{0}$ (11)

uniformly on $X$.

###### Proof.

Let $g_{k}$ be the local expression of $K_{k}$ in our trivialisation of
$L|_{U}$, so $B_{k}(x)=|K_{k}(x,x)|_{h}=g_{k}(x,x)e^{-k\phi(x)}$. Now
Proposition 6 restricted to the diagonal implies
$(g_{k}-\mathcal{K}_{k}^{\mathrm{av}})e^{-k\phi(x)}=O(k^{n-N-r-1})$ in $C^{0}$
on $U_{0}$. So we can apply Lemma 9 to
$f_{k}:=g_{k}-\mathcal{K}_{k}^{\mathrm{av}}$ to deduce that if $D$ is a
differential operator of order $q$ in $x$ and $y$ then

$D(g_{k}-\mathcal{K}_{k}^{\mathrm{av}})e^{-k(\phi(x)/2+\phi(x)/2)}=O(k^{n-N-r-1+q})$

on $\frac{1}{2}U_{0}$. Therefore taking the first $r$ derivatives of

$B_{k}(x)-\mathcal{K}_{k}^{\mathrm{av}}(x,x)e^{-k\phi(x)}=\big{(}g_{k}(x,x)-\mathcal{K}_{k}^{\mathrm{av}}(x,x)\big{)}e^{-k\phi(x)}$

with respect to $x$ using the Leibnitz rule proves the first statement.

For the second statement, Proposition 6 implies
$g_{k}e^{-k(\phi(x)/2+\phi(y)/2)}=O(k^{n})$ on $U_{0}\times U_{0}$. Thus from
Lemma 9 we get that if $D$ has order at most $q$ then
$(Dg_{k})e^{-k(\phi(x)/2+\phi(x)/2)}=O(k^{n+q})$ on $\frac{1}{2}U_{0}$. Hence
taking the derivatives of $B_{k}(x)=g_{k}(x,x)e^{-k\phi(x)}$ with the Leibnitz
rule gives $DB_{k}=O(k^{n+p})$ on $\frac{1}{2}U_{0}$ for any differential
operator of order $p$. So by compactness the same holds uniformly on all of
$X$. ∎

The proof of our main theorem is now immediate. Each $x$ is contained in some
open set $U_{0}$ for which Theorem 3 and Corollary 10 hold, and by compactness
there is a finite cover by such sets. Thus

$\displaystyle B_{k}^{\mathrm{orb}}(x)$ $\displaystyle=$
$\displaystyle\sum_{i}c_{i}B_{k+i}(x)=\sum_{i}c_{i}\mathcal{K}_{k+i}^{\mathrm{av}}(x,x)e^{-(k+i)\phi(x)}+O(k^{n-N-1})$
$\displaystyle=$ $\displaystyle\sum_{j=0}^{N}b_{j}(x)k^{n-j}+O(k^{n-N-1})$

in $C^{r}$, as required.

In [RT] we use a variant of our main theorem, which for convenience we record
here.

######  Corollary 12.

Let $\gamma$ be a homogeneous polynomial in two variables of degree $d\leq N$,
and suppose that as usual the $c_{i}$ are chosen to satisfy (8). Then there is
a $C^{r}$-expansion

$\sum_{i}c_{i}\gamma(k,i)B_{k+i}=b_{0}k^{n+d}+b_{1}k^{n+d-1}+\cdots+O(k^{n-N+d-1}).$

Moreover if $\gamma(i,k)=Ak^{d}+Bk^{d-1}i+\cdots$ then

$b_{0}=A\sum_{i}c_{i}\quad\text{ and }\quad
b_{1}=\sum_{i}c_{i}(A[ni+\operatorname{Scal}(\omega)/2]+iB).$

###### Proof.

By linearity we may suppose $\gamma(i,k)=i^{a}k^{d-a}$ for some $a\leq d$.
Since $c_{i}$ satisfy (8) for $p=0,\ldots,N+r$, we get that
$c_{i}^{\prime}:=i^{a}c_{i}$ satisfy this condition for $p=0,\ldots,N+r-a$.
Applying our main Theorem with the constants $c_{i}^{\prime}$ proves the
corollary. ∎

######  Remark 13.

For example

$\sum_{i}(k+i)c_{i}B_{k+i}=b_{0}k^{n+1}+b_{1}k^{n}+\cdots$

where $b_{0}=\sum_{i}c_{i}$ and
$b_{1}=\sum_{i}c_{i}[(n+1)i+\operatorname{Scal}(\omega)/2]$.

## References

  * [Bai] W. L. Baily.  On the imbedding of $V$-manifolds in projective space.  Amer. J. Math., 79:403–430, 1957.
  * [BBS] R. Berman, B. Berndtsson, and J. Sjöstrand.  A direct approach to Bergman kernel asymptotics for positive line bundles.  Arkiv för Matematik, 46(2):197–217, 2008.  arXiv:math/0506367.
  * [Ber] R. Berman.  Bergman kernels and local holomorphic Morse inequalities.  Math. Z., 248(2):325–344, 2004.  arXiv:math/0211235.
  * [Cat] D. Catlin.  The Bergman kernel and a theorem of Tian.  In Analysis and geometry in several complex variables (Katata, 1997), Trends Math., pages 1–23. Birkhäuser Boston, Boston, MA, 1999.
  * [DLM] X. Dai, K. Liu, and X. Ma.  On the asymptotic expansion of Bergman kernel.  J. Differential Geom., 72(1):1–41, 2006.  arXiv:math/0404494.
  * [Don] S. Donaldson.  Scalar curvature and projective embeddings. I.  J. Differential Geom., 59(3):479–522, 2001.
  * [Fef] C. Fefferman.  The Bergman kernel and biholomorphic mappings of pseudoconvex domains.  Invent. Math., 26:1–65, 1974.
  * [Kaw] T. Kawasaki.  The Riemann-Roch theorem for complex $V$-manifolds.  Osaka J. Math., 16(1):151–159, 1979.
  * [Lu] Z. Lu.  On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch.  Amer. J. Math., 122(2):235–273, 2000.  arXiv:math/9811126.
  * [RT] J. Ross and R. P. Thomas.  Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kähler metrics.  2009, arXiv:0907.5214.
  * [Rua] W.-D. Ruan.  Canonical coordinates and Bergmann metrics.  Comm. Anal. Geom., 6(3):589–631, 1998.  arXiv:dg-ga/9610011.
  * [Rub] Y. Rubenstein.  Geometric quantization and dynamical constructions on the space of Kähler metrics.  2008, Ph.D. Thesis. Massachusetts Institute of Technology, Dept. of Mathematics.
  * [Son] J. Song.  The Szegö kernel on an orbifold circle bundle.  arXiv:math/0405071.
  * [Tia] G. Tian.  On a set of polarized Kähler metrics on algebraic manifolds.  J. Differential Geom., 32(1):99–130, 1990.
  * [Zel] S. Zelditch.  Szegő kernels and a theorem of Tian.  Internat. Math. Res. Notices, (6):317–331, 1998.  arXiv:math-ph/0002009.