# A detailed look at the Calabi-Eguchi-Hanson spaces

Jørgen Olsen Lye  Institut für Differentialgeometrie, Leibniz Universität
Hannover<EMAIL_ADDRESS>
###### Abstract.

This article takes a detailed look at the Ricci-flat metrics introduced by
Eguchi-Hanson and Calabi on the canonical line bundle of complex projective
space. We give a description of these spaces as resolutions of certain
orbifold singularities. We then compute the curvature explicitly and show that
all compact, minimal submanifolds are contained in the zero section. This
extends a result by Tsai and Wang.

###### 2020 Mathematics Subject Classification:

Primary 32Q25; Secondary 53C42, 53C25
Key words and phrases: Calabi-Yau, Kähler-Einstein ALE, Minimal submanifolds.

##  1\. Introduction

Finding Einstein (pseudo-) Riemannian metrics has been an interesting project
for more than a century. All the early successes rely on assuming a high
degree of symmetry, reducing the Einstein equations to an ODE. This was
successfully carried out by Eguchi and Hanson in [8], allowing them to find a
Ricci-flat, nowhere flat metric locally defined on
$\mathbb{R}^{4}\setminus\\{0\\}$, which in fact is a complete hyperkähler
metric defined on the total space of the cotangent bundle
$T^{*}\mathbb{C}\mathbb{P}^{1}=\mathcal{O}_{\mathbb{C}\mathbb{P}^{1}}(-2)$ of
$\mathbb{C}\mathbb{P}^{1}$. The fact that the a priori real metric is Kähler
and defined on a certain line bundle was quickly seen by Calabi [7] to not be
a coincidence. Calabi [7] then proceeded to generalize the Eguchi-Hanson
metrics to two families of spaces, giving explicit examples of complete
hyperkähler metrics on $T^{*}\mathbb{C}\mathbb{P}^{n}$ and complete Ricci-flat
Kähler metrics on $\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$. We call the
latter family Calabi-Eguchi-Hanson spaces in this article, even though the
first family would also deserve this name. The Calabi construction was taken
up again by R. Bryant and S. Salamon [6] and M. Stenzel [27], and they
introduced Ricci-flat metrics on certain spin bundles and on
$T^{*}\mathbb{S}^{n}$ respectively. In the Stenzel case $T^{*}\mathbb{S}^{n}$,
the metrics are Kähler metrics.

A common feature of all these special holonomy spaces is that they are the
total spaces of certain vector bundles over compact manifolds
($\mathbb{S}^{n}$ or $\mathbb{C}\mathbb{P}^{n}$, to be precise). These
constructions were taken up again by B. Feix [9] and D. Kaledin [16] for
arbitrary complex spaces $X$. They both independently showed that no matter
which analytic metric on puts on $X$, one can extend it to hyperkähler metric
defined on a neighbourhood of the zero section of $T^{*}X$, but one cannot
hope for completeness. In [4] R. Bielawski proved that for any real analytic
Kähler manifold $X$, one can extend the metric to be a Ricci-flat Kähler
metric in a neighbourhood of the zero section in the canonical line bundle
$\mathcal{K}_{X}$. This extends Calabi’s second family of examples,
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)=\mathcal{K}_{\mathbb{C}\mathbb{P}^{n-1}}$,
but one cannot in general hope for completeness.

Apart from being explicit examples of non-flat, Ricci-flat spaces, the Calabi-
Eguchi-Hanson spaces provide simple but interesting examples of instantons and
asymptotically locally Euclidean (ALE) spaces. For $n=2$ they enter as crucial
building blocks of Kummer K3 surfaces [26, 11, 18], which has a generalization
to $n=3$ [22]. In both cases, one glues in copies of Calabi-Eguchi-Hanson
spaces into a singular quotient of a torus. One equips the Calabi-Eguchi-
Hanson spaces with the Ricci-flat metrics we will discuss in this article,
which one glues together with a flat metric on the torus. If the parameter in
the Calabi-Eguchi-Hanson metric is small enough, this procedure produces a
compact, almost Ricci-flat Calabi-Yau manifold. The Ricci-flat metric of Yau
[29] can be shown [18, 22] to be close to the given almost Ricci-flat metric.

The aims of this article are twofold. The first goal is to present an
interpretation of the spaces $\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)$ as
smooth resolutions of $\mathbb{C}^{n}/\mu_{k}$, where $\mu_{k}$ denotes the
$k$’th roots of unity acting diagonally. The second goal is to give a more
detailed analytic description of the Ricci-flat Calabi-Eguchi-Hanson metric.
We also include an addition to [28, Theorem 5.3]. Much of the material is
known in some form to the experts, but seems to be lacking a clear and
explicit exposition. We hope the present article can serve as a gentle
introduction to these fascinating spaces.

Calabi’s deduction of the Ricci-flat metric in [7] is a bit different. He
instead works directly on the bundle
$p\colon\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)\to\mathbb{C}\mathbb{P}^{n-1}$.
On the trivializing sets $U_{i}$, he makes an ansatz for a Kähler potential on
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)_{|U_{i}}$ of the form

$\Phi_{i}\coloneqq p^{*}\phi_{FS}+v(t),$

where $\phi_{FS}$ is the Fubini-Study Kähler potential on
$U_{i}\subset\mathbb{C}\mathbb{P}^{n-1}$, $t=|y|^{2}_{g_{FS}}$ is the norm
squared of an element in the fibre, and $v$ is an unknown scalar function.
Imposing Ricci-flatness, Calabi finds an ODE for $v$ which he explicitly
solves. This leads to the metric expression (22).

Calabi’s approach makes the completeness of the metric and the role of the
Fubini-Study metric much more transparent. What we do in this paper is to
start looking for rotationally symmetric Kähler potentials on
$\mathbb{C}^{n}\setminus\\{0\\}$. These of course descend to quotients of
$\mathbb{C}^{n}\setminus\\{0\\}$ by $\mu_{k}$ for any $k$, and extend to
Ricci-flat metrics on $\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)$ if and
only if $k=n$. These Ricci-flat metrics are exactly the same as the ones found
by Calabi. We feel that the deduction presented in this paper offers a
fruitful complementary access to these metrics.

###  1.1. Statement of the results

Definitions and proofs of the following results are given in the main text.

Notational: We deal with Kähler manifolds in this article, and we will employ
complex notation for tensors throughout. Good introductions can be found in
[19, Chapter IX.5] and [2, p. 41 & Chapter 4.2-4.4].

Throughout the article, we will use $u\coloneqq|z|^{2}_{\mathbb{C}^{n}}$.
Indices on $z\in\mathbb{C}^{n}$ are raised and lowered using the Euclidean
metric. In particular, $\bar{z}_{\mu}\coloneqq\bar{z}^{\nu}\delta_{\mu\nu}$
etc. Let $\mu_{k}$ denote the group of $k$’th roots of unity. Consider the
diagonal $\mu_{k}$ action on $\mathbb{C}^{n}$,

$\mathbb{C}^{n}\times\mu_{k}\to\mathbb{C}^{n}$

given by

$(z,\zeta)\mapsto\zeta z.$

######  Theorem 1.

For $k,n\geq 2$ the quotient $\mathbb{C}^{n}/\mu_{k}$ is singular, but the
blow-up is a resolution

(1) $\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)\cong
Bl_{0}(\mathbb{C}^{n}/\mu_{k})\cong
Bl_{0}(\mathbb{C}^{n})/\mu_{k}=\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)/\mu_{k}.$

For $n=k=2$, the space $\mathcal{O}_{\mathbb{C}\mathbb{P}^{1}}(-2)$ is the
minimal resolution of $\mathbb{C}^{2}/\mu_{2}$.

######  Remark 2.

This result is well-known to the experts, but we do not know of any reference
for all $k,n\geq 2$.

######  Theorem 3 ([8],[7]).

The spaces ${O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)$ admit Ricci-flat metrics if
and only if $k=n$, in which case one can equip them with an asymptotically
flat, Ricci-flat metric whose expression on
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)\setminus\mathbb{C}\mathbb{P}^{n-1}\cong\left(\mathbb{C}^{n}\setminus\\{0\\}\right)/\mu_{n}$
is given by

(2)
$g_{\mu\bar{\nu}}=\sqrt[n]{1+\frac{a^{n}}{u^{n}}}\left(\delta_{\mu\bar{\nu}}-\frac{a^{n}}{a^{n}+u^{n}}\frac{\bar{z}_{\mu}z_{\nu}}{u}\right),$

for some parameter $a>0$. The associated Kähler potential reads

(3)
$f(u)=\sqrt[n]{a^{n}+u^{n}}+\frac{a}{n}\sum_{j=0}^{n-1}\zeta^{j}\log\left(\sqrt[n]{1+\frac{u^{n}}{a^{n}}}-\zeta^{j}\right)+const.,$

where $\zeta$ is a primitive $n$’th root of unity. Restricting this Ricci-flat
metric to the zero section
$\mathbb{C}\mathbb{P}^{n-1}\subset\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$
gives

(4) $ds^{2}_{|\mathbb{C}\mathbb{P}^{n-1}}=a\cdot ds^{2}_{FS},$

where $ds^{2}_{FS}$ is the Fubini-Study metric on
$\mathbb{C}\mathbb{P}^{n-1}$, given in homogeneous coordinates
$[z^{1}:\dots:z^{n}]$ by

(5) $ds^{2}_{FS}\coloneqq\left(\frac{|dz|^{2}}{|z|^{2}}-\frac{|\bar{z}\cdot
dz|^{2}}{|z|^{4}}\right).$

######  Remark 4.

For $n=2$, the potential (3) coincides with the one derived in [8]. According
to [15, Equation 5], (3) coincides with [7, Equation 4.14].

######  Remark 5.

The spaces $\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$ with metric given by
(2) are examples of ALE (asymptotically locally Euclidean) spaces, and Theorem
1 says that they are resolutions of $\mathbb{C}^{n}/\mu_{n}$.

For $n=2$, [20] and [21] shows that any ALE hyperkähler manifold is the
resolution on $\mathbb{C}^{2}/\Gamma$ for some finite subgroup $\Gamma\subset
SU(2)$. The Eguchi-Hanson space $\mathcal{O}_{\mathbb{C}\mathbb{P}^{1}}(-2)$
is in this sense the simplest, non-trivial hyperkähler ALE space.

######  Proposition 6.

Choose standard coordinates on $\mathbb{C}^{n}\setminus\\{0\\}$. Then the
Christoffel symbols associated to the metric (2) take the form

(6)
$\Gamma_{\mu\alpha}^{\lambda}=-\frac{a^{n}}{u(a^{n}+u^{n})}\left(\bar{z}_{\mu}\delta^{\lambda}{}_{\alpha}+\bar{z}_{\alpha}\delta^{\lambda}{}_{\mu}-(n+1)\frac{\bar{z}_{\alpha}\bar{z}_{\mu}}{u}z^{\lambda}\right)$

The curvature tensor of (2) reads

$\displaystyle R_{\mu\bar{\nu}\alpha\bar{\beta}}$
$\displaystyle=\frac{a^{n}}{(a^{n}+u^{n})^{\frac{n+1}{n}}}\Bigg{(}g_{\alpha\bar{\nu}}g_{\mu\bar{\beta}}+g_{\mu\bar{\nu}}g_{\alpha\bar{\beta}}$
$\displaystyle-(n+1)\left(\frac{u}{\sqrt[n]{a^{n}+u^{n}}}\right)^{n-1}\left(\frac{\bar{z}_{\mu}z_{\beta}}{u}g_{\alpha\bar{\nu}}+\frac{\bar{z}_{\alpha}z_{\beta}}{u}g_{\mu\bar{\nu}}+\frac{\bar{z}_{\mu}z_{\nu}}{u}g_{\alpha\bar{\beta}}+\frac{\bar{z}_{\alpha}z_{\nu}}{u}g_{\mu\bar{\beta}}\right)$
(7)
$\displaystyle+(n+1)(n+2)\left(\frac{u}{\sqrt[n]{a^{n}+u^{n}}}\right)^{2n-2}\frac{\bar{z}_{\mu}z_{\nu}\bar{z}_{\alpha}z_{\beta}}{u^{2}}\Bigg{)}.$

and the Kretschmann scalar (norm of the curvature tensor) is

(8) $K(z)\coloneqq
R_{\mu\bar{\nu}\alpha\bar{\beta}}R^{\bar{\mu}\nu\bar{\alpha}\beta}(z)=\frac{a^{2n}}{(a^{n}+u^{n})^{2\left(\frac{n+1}{n}\right)}}n(n+2)(n^{2}-1).$

or, for $n=2$,

(9) $K(z)=\frac{24a^{4}}{(a^{2}+u^{2})^{3}}.$

######  Remark 7.

Again, these are both computed for $n=2$ in [8], but we do not know of a
reference for the higher dimensional results.

The nowhere vanishing, parallel holomorphic volume form is the same as for the
flat metric:

######  Proposition 8.

Let $\eta^{\mathbb{C}}\coloneqq dz^{1}\wedge\dots\wedge dz^{n}$ be the
standard holomorphic $n-$form on $\mathbb{C}^{n}$. This descends to
$\left(\mathbb{C}^{n}\setminus\\{0\\}\right)/\mu_{n}$ and
$\eta\coloneqq\pi^{*}\eta^{\mathbb{C}}$ extends to a parallel (with respect to
the Ricci-flat metric), nowhere vanishing holomorphic volume form on
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$.

######  Theorem 9.

Equip $\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$ with the Ricci-flat
metric described in Theorem 3. Then any compact, minimal submanifold is
contained in
$\mathbb{C}\mathbb{P}^{n-1}\subset\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$.

######  Corollary 10.

The only closed geodesics in $\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$
are the geodesics in $(\mathbb{C}\mathbb{P}^{n-1},g_{FS})$, all of which are
closed, and these are not stable.

######  Remark 11.

Theorem 9 applies the trick of [28, Theorem 5.3], where Chung-Jun Tsai and Mu-
Tao Wang prove a direct analogue of Theorem 9 for $T\mathbb{S}^{n}$ with the
Stenzel metric, $T^{*}\mathbb{C}\mathbb{P}^{n}$ with the Calabi metric111
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{1}}(-2)\cong
T^{*}\mathbb{C}\mathbb{P}^{1}$, and the metrics are the same, but for $n>2$
the spaces considered in this article are different from
$T^{*}\mathbb{C}\mathbb{P}^{n}$., and the total space of certain spinor
bundles with the Bryant-Salamon metric.

The $n=2$ case of the corollary is also covered by [5], which deals with
arbitrary hyperkähler 4-manifolds. [5] applies since (9) in particular says
that $\mathcal{O}_{\mathbb{C}\mathbb{P}^{1}}(-2)$ with its Ricci-flat Kähler
metric is nowhere flat.

Much of this article is based on the author’s PhD thesis [23, Chapter 4],
written under the supervision of Nadine Große and Katrin Wendland. It grew out
of an investigation of Kummer K3 surfaces, which is a companion paper [24].

##  2\. Algebraic aspects

In this section we set out to prove Theorem 1.

######  Lemma 12.

The quotient spaces $\mathbb{C}^{n}/\mu_{k}$ are complex analytic varieties
with isolated singularities at the origin, hence they are complex spaces. The
subspaces $(\mathbb{C}^{n}\setminus\\{0\\})/\mu_{k}$ are complex manifolds.

###### Proof.

To see that $\mathbb{C}^{n}/\mu_{k}$ is a variety, one can argue as follows.
For any finite group $G$ acting on an affine variety $X$, i.e. $X=Spm(R)$, we
have that $X/G\cong Spm(R^{G})$, where $Spm(R)$ has maximal ideals as points
(as opposed to $Spec(R)$, which has all prime ideals as points), and $R^{G}$
means the $G$-invariants of $R$. The ring $R^{G}$ is known to be finitely
generated when $G$ is linearly reductive, which all finite groups are. A
reference for these facts is for instance [25, Theorem 5.9] combined with [25,
Corollary 5.17]. In our case,
$X=\mathbb{C}^{n}=Spm(\mathbb{C}[z_{1},\dots,z_{n}])$, so
$\mathbb{C}^{n}/\mu_{k}\cong Spm(\mathbb{C}[z_{1},\dots,z_{n}]^{\mu_{k}})$,
which is a variety.

To see that $\mathbb{C}^{n}/\mu_{k}$ is not smooth near the origin, notice
that it is not even a topological manifold there. A way to see this is as
follows. Assume for contradiction that there is some neighbourhood
$U\subset\mathbb{C}^{n}/\mu_{k}$ containing $0$ such that
$U\cong\mathbb{C}^{n}$. Any neighbourhood containing $0$ also contains the
boundary of an orbiball $\partial
B_{\delta}(0)/\mu_{k}\coloneqq\left\\{[(z_{1},\dots,z_{n})]\in\mathbb{C}^{n}/\mu_{k}\,\big{|}\,|z|_{\mathbb{C}^{n}}=\delta\right\\}$
for some $\delta>0$, which is a lens space of real dimension $2n-1$, hence
cannot be embedded in codimension 1. This contradicts the existence of the
neighbourhood $U$.

When the origin is removed from $\mathbb{C}^{n}$, the $\mu_{k}$ action is free
and proper. One can then cite a well-known result, for instance [14, Prop.
2.1.13], to conclude that $(\mathbb{C}^{n}\setminus\\{0\\})/\mu_{k}$ is a
complex manifold. ∎

######  Remark 13.

For the Eguchi-Hanson case, $k=n=2$, one can of course easily write out some
more explicit details as follows. First of all,
$\mathbb{C}[u,v]^{\mu_{2}}\xrightarrow{\cong}\mathbb{C}[x,y,z]/(xy-z^{2})$,
where the map sends $(u,v)\mapsto(u^{2},v^{2},uv)$. That the variety
$\mathbb{C}^{2}/\mu_{2}\cong Spm(\mathbb{C}[x,y,z]/(xy-z^{2}))$ is singular at
the origin can be directly checked using the Jacobi criterion.

######  Definition 14 (Resolution of Singularities).

Let $X$ be a complex space with singular set $X_{sing}$. A resolution of $X$
is a complex manifold $\tilde{X}$ and a proper, bimeromorphic morphism
$P\colon\tilde{X}\to X$ such that $P^{-1}(X_{sing})\subset\tilde{X}$ is a
simple normal crossings divisor222Meaning the irreducible components of the
divisor are smooth and intersect transversely, so this is in particular
satisfied if $P^{-1}(X_{sing})$ is a complex submanifold., and

$P\colon\tilde{X}\setminus P^{-1}(X_{sing})\to X_{0}$

is a biholomorphism.

A resolution $\tilde{X}\to X$ is minimal if any other resolution of
singularities $Q\colon Y\to X$, factors through $\tilde{X}$.

######  Remark 15.

Resolutions of singularities always exits (see [13] or, more specific to the
analytic setting, [30, Thm. 2.0.1] and [1, Appendix]), but we will construct
one explicitly for $\mathbb{C}^{n}/\mu_{k}$.

One easily checks that minimal resolutions, if they exist, are unique up to
isomorphisms, so we will talk about the minimal resolution (if it exists). We
need to fix some standard notation.

######  Definition 16.

Define the total space of the tautological line bundle over
$\mathbb{C}\mathbb{P}^{n-1}$ as

$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)\coloneqq\left\\{(z,\ell)\,\big{|}\,z\in\ell\right\\}\subset\mathbb{C}^{n}\times\mathbb{C}\mathbb{P}^{n-1}.$

The blow-up of the origin of $\mathbb{C}^{n}$ is defined as
$Bl_{0}(\mathbb{C}^{n})\coloneqq\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)$
with projection map $P\colon B_{0}(\mathbb{C}^{n})\to\mathbb{C}^{n}$ defined
by $P(z,\ell)=z$.

Next is a well-known result about extending group actions to the blow-up.

######  Lemma 17.

Let $G$ be any group acting on $\mathbb{C}^{n}$ via a representation $G\to
GL(n,\mathbb{C})$. Then one gets an induced action on $Bl_{0}(\mathbb{C}^{n})$
by $G$ given by $g\cdot(z,\ell)\coloneqq(g\cdot z,g\cdot\ell)$, making the
following diagram commute.

${Bl_{0}(\mathbb{C}^{n})}$${Bl_{0}(\mathbb{C}^{n})/G}$${\mathbb{C}^{n}}$${\mathbb{C}^{n}/G}$$\tilde{Q}$$P$$\tilde{P}$$Q$

###### Proof.

Since $G$ acts via linear transformations, $G$ will take lines to lines. So if
$\ell$ is some line through the origin, then $g\cdot\ell$ is as well. This
means we get a well-defined group action

$G\times Bl_{0}(\mathbb{C}^{n})\to Bl_{0}(\mathbb{C}^{n}),$
$(g,(z,\ell))\mapsto(g\cdot z,g\cdot\ell).$

The commutative diagram is then evident. ∎

We are finally ready to attack Theorem 1.

###### Proof of Theorem 1.

The action of $\mu_{k}$ on $\mathbb{C}^{n}$ is via linear transformations, so
by Lemma 17 it induces an action on
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)$ as follows.

$\mu_{k}\times\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)\to\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)$
$(\zeta,(z,\ell))\mapsto(\zeta\cdot z,\zeta\cdot\ell)=(\zeta\cdot z,\ell).$

From Lemma 17, we also get the following commutative diagram.

${\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)}$${\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)/\mu_{k}}$${\mathbb{C}^{n}}$${\mathbb{C}^{n}/\mu_{k}}$$\tilde{Q}$$\pi$$\tilde{\pi}$$Q$

The blow-down map is biholomorphisms away from the zero section
$\mathbb{C}\mathbb{P}^{n-1}$,
$\pi\colon\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)\setminus\mathbb{C}\mathbb{P}^{n-1}\xrightarrow{\cong}\mathbb{C}^{n}\setminus\\{0\\}$,
which gives an isomorphism

$\tilde{\pi}\colon\left(\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)\setminus\mathbb{C}\mathbb{P}^{n-1}\right)/\mu_{k}\xrightarrow{\cong}\left(\mathbb{C}^{n}\setminus\\{0\\}\right)/\mu_{k}.$

Furthermore, $\mu_{k}$ acts freely and transitively on
$\mathbb{C}^{n}\setminus\\{0\\}$, so
$\left(\mathbb{C}^{n}\setminus\\{0\\}\right)/\mu_{k}$ is a smooth manifold. It
thus remains to show that
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)/\mu_{k}$ is smooth and that
$\tilde{\pi}^{-1}(\\{0\\})$ is a complex submanifold. We will do this by
exhibiting an isomorphism

$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)/\mu_{k}\xrightarrow{\cong}\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)$

and showing that $\tilde{\pi}^{-1}(\\{0\\})\cong\mathbb{C}\mathbb{P}^{n-1}$.

We use the description
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)\coloneqq\left\\{(z,\mathbf{\xi})\,\Big{|}\,z^{i}(\xi^{j})^{k}=z^{j}(\xi^{i})^{k}\right\\}\subset\mathbb{C}^{n}\times\mathbb{C}\mathbb{P}^{n-1}$.
We can define a map

$f\colon\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)/\mu_{k}\to\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)$

by specifying what happens when restricting both bundles333The total space
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)$ can be thought of as a line
bundle over $\mathbb{C}\mathbb{P}^{n-1}$, and that is the view taken in this
proof. to $U_{i}\coloneqq\\{\xi_{i}\neq
0\\}\subset\mathbb{C}\mathbb{P}^{n-1}$. Let
$\zeta^{j}\coloneqq\frac{\xi^{j}}{\xi^{i}}$ on $U_{i}$. Write
$([z^{1}],\dots,[z^{n}])$ for a point in $\mathbb{C}^{n}/\mu_{k}$. We have

$\left(\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)/\mu_{k}\right)_{|U_{i}}\cong\left\\{\left(\zeta^{1}[z^{i}],\zeta^{2}[z^{i}],\dots,\zeta^{i-1}[z^{i}],[z^{i}],\zeta^{i+1}[z^{i}],\dots\zeta^{n}[z^{i}]\right)\,\Big{|}\,z^{i},\zeta^{j}\in\mathbb{C}\right\\}$

and

$\left(\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)\right)_{|U_{i}}\cong\left\\{\left((\zeta^{1})^{k}z^{i},(\zeta^{2})^{k}z^{i},\dots,(\zeta^{i-1})^{k}z^{i},z^{i},(\zeta^{i+1})^{k}z^{i},\dots(\zeta^{n})^{k}z^{i}\right)\,\Big{|}\,z^{i},\zeta^{j}\in\mathbb{C}\right\\}.$

Define
$f_{i}\colon\left(\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)/\mu_{k}\right)_{|U_{i}}\to\left(\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)\right)_{|U_{i}}$
by

$f_{i}\left(w^{1},\dots,w^{n}\right)=\left((w^{1})^{k},\dots,(w^{n})^{k}\right).$

The important thing happens at slot $i$, where $[z^{i}]\mapsto(z^{i})^{k}$,
which is clearly surjective, and it is injective since if $z^{k}=w^{k}$, then
$\exists\,\zeta\in\mu_{k}$ such that $z=\zeta w\implies[z]=[w]$.

Under $f$, we see that
$\\{([0],\ell)\in\mathbb{C}^{n}/\mu_{k}\times\mathbb{C}\mathbb{P}^{n-1}\\}\subset\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)/\mu_{k}$
corresponds to
$\\{(0,\ell)\in\mathbb{C}^{n}\times\mathbb{C}\mathbb{P}^{n-1}\\}\cong\mathbb{C}\mathbb{P}^{n-1}\subset\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)$.
So the fibre over $\\{0\\}$ is a complex submanifold.

Finally, to prove the fact that
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{1}}(-2)\to\mathbb{C}^{2}/\mu_{2}$ is the
minimal resolution one can use a neat little criterion, namely Lemma 19, along
with the fact that the spaces $\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$
have trivial canonical bundle, which follows from the adjunction formula as we
show below as Corollary 22.

∎

######  Remark 18.

A couple of things should be pointed out at this point. Firstly, it is
important to note that although the blow-down map
$P\colon\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)\to\mathbb{C}^{n}$ is just
a linear projection, the map
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)\to\mathbb{C}^{n}/\mu_{k}$ is not
linear, but behaves instead like a $k$’th root.

Secondly, the $\mu_{k}$-action on
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)$ is not free, so the meaning of
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)/\mu_{k}\cong\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)$
is that the left hand space is given a holomorphic structure by the
identification with the right hand space, which is a complex manifold. This is
analogous to identifying $\mathbb{T}^{2}/\mu_{2}$ with
$\mathbb{C}\mathbb{P}^{1}$ via the Weierstraß $\wp$-function, which makes
$\mathbb{T}^{2}/\mu_{2}$ into a Riemann surface even though the
$\mu_{2}$-action has 4 fixed points.

######  Lemma 19.

Assume $X\to Y$ is a resolution where $X$ is a non-singular, complex surface
and $Y$ is potentially singular. If the canonical line bundle is trivial,
$\mathcal{K}_{X}\cong\underline{\mathbb{C}}$, then $X\to Y$ is the minimal
resolution.

###### Proof.

By [3, (4.3) Lemma, p. 98] and their definition of a minimal resolution, [3,
p. 106], a non-singular, complex surface $X$ is the minimal resolution of
$X\to Y$ for some singular, complex surface $Y$ if and only if $X$ does not
contain any $(-1)$-curves. By [3, (2.2) Prop., p. 91], a curve $C\subset X$ is
a $(-1)$-curve if and only if $C^{2}<0$ and $(\mathcal{K}_{X},C)<0$, both
equations meaning intersection pairing. Since
$\mathcal{K}_{X}\cong\underline{\mathbb{C}}$ by assumption,
$(\mathcal{K}_{X},C)<0$ cannot hold.444This argument is standard. Indeed, [20]
merely remarks that his resolutions of $\mathbb{C}^{2}/\Gamma$, for finite
groups $\Gamma\subset SU(2)$, have $c_{1}(X)=0$ since they are hyperkähler,
and thus have to be the minimal resolution. ∎

We again need some preliminary results and definitions. We will divide the
proof into two parts for readability. Let us first go through a standard
argument to see that $\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)$ cannot
admit a Ricci-flat metric if $k\neq n$. The idea is: if
$X\coloneqq\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)$ admits a Ricci-flat
metric, then Chern-Weil theory says that $c_{1}(X)=0$, which means
$\mathcal{K}_{X}=\underline{\mathbb{C}}$, which will contradict the adjunction
formula for $\mathcal{K}_{X}$ using $Y=\mathbb{C}\mathbb{P}^{n-1}\subset X$ as
submanifold unless $k=n$. The details are as follows. First recall what the
adjunction formula says.

######  Proposition 20.

Let $Y\xrightarrow{\iota}X$ be a submanifold of a complex manifold $X$. Let
$\mathcal{N}_{Y/X}$ denote the normal bundle of $Y$ in $X$. Let
$\mathcal{K}_{X}$ and $\mathcal{K}_{Y}$ be the canonical (aka. determinant)
line bundles of $X$ and $Y$ respectively. Then the adjunction formula is the
statement that

(10)
$\mathcal{K}_{Y}\cong\iota^{*}\left(\mathcal{K}_{X}\right)\otimes\det\left(\mathcal{N}_{Y/X}\right).$

See [14, Prop. 2.2.17] for a proof.

######  Lemma 21.

Let $E\to X$ be a vector bundle and $\iota\colon X\to E$ the inclusion as the
zero section. Then

$\mathcal{N}_{X/E}\cong E.$

######  Corollary 22.

For $Y=\mathbb{C}\mathbb{P}^{n-1}$ and
$X=\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)$, the adjunction formula,
(10), says that

(11)
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)\cong\iota^{*}\left(\mathcal{K}_{X}\right)\otimes\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k).$

###### Proof.

The previous lemma says
$\mathcal{N}_{Y/X}\cong\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)$. This is
a line-bundle, hence equals its determinant line-bundle;
$\det\left(\mathcal{N}_{Y/X}\right)=\mathcal{N}_{Y/X}$. It is a well-known
fact that
$\mathcal{K}_{\mathbb{C}\mathbb{P}^{n-1}}\cong\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$
(see for instance [14, Prop. 2.4.3]). Putting this into the adjunction
formula, (10), gives the conclusion. ∎

######  Proposition 23.

$X\coloneqq\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)$ does not admit a
Ricci-flat metric for $k\neq n$.

###### Proof.

Assume for contradiction that $X$ carries a Ricci-flat metric. Since $X$
deformation retract onto $\mathbb{C}\mathbb{P}^{n-1}$, we know that
$H^{1}(X;\mathbb{C})=0$, so also $H^{1}(X,\mathcal{O}_{X})=0$. Then
$\mathcal{K}_{X}\cong\underline{\mathbb{C}}$ as holomorphic line bundles,
contradicting (11) unless $k=n$, since
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)\cong\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)$
if and only if $k=n$. ∎

We end this section by giving an explicit section of
$\mathcal{K}_{X}\cong\underline{\mathbb{C}}$ when
$X=\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$.

######  Proposition 24.

Let
$\pi\colon\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)\to\mathbb{C}^{n}/\mu_{n}$
be the resolution described in Theorem 1, $n\geq 2$. Let
$\eta^{\mathbb{C}}\coloneqq dz^{1}\wedge dz^{2}\wedge\dots\wedge dz^{n}$
denote a nowhere vanishing, holomorphic $n$-form on $\mathbb{C}^{n}$. Then
$\eta^{\mathbb{C}}$ descends to a holomorphic $n$-form (also denoted
$\eta^{\mathbb{C}}$) on $\left(\mathbb{C}^{n}\setminus\\{0\\}\right)/\mu_{n}$,
and this $n$-form extends to a nowhere vanishing, holomorphic $n$-form $\eta$
on all of $\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$.

###### Proof.

That $\eta^{\mathbb{C}}$ descends to
$\left(\mathbb{C}^{n}\setminus\\{0\\}\right)/\mu_{n}$ follows from $\mu_{n}$
acting as a $SU(n)$-representation.

Let
$U_{i}\coloneqq\left\\{(\xi^{1}\colon\dots\colon\xi^{n})\,\Big{|}\,\xi^{i}\neq
0\right\\}\subset\mathbb{C}\mathbb{P}^{n-1}$ for $1\leq i\leq n$. Then, as in
the proof of Theorem 1,

$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)_{|U_{i}}\cong\left\\{\left((\zeta^{1})^{n}z^{i},\dots,(\zeta^{i-1})^{n}z^{i},z^{i},(\zeta^{i+1})^{n}z^{i},\dots(\zeta^{n})^{n}z^{i})\right)\,\Big{|}\,z^{i},\zeta^{j}\in\mathbb{C}\right\\},$

with $\zeta^{k}\coloneqq\frac{\xi^{k}}{\xi^{i}}$. The blow-down map
$\pi\colon\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)_{|U_{i}}\to\mathbb{C}^{n}/\mu_{n}$
has the description

$\displaystyle\pi\left(\left((\zeta^{1})^{n}z^{i},\dots,(\zeta^{i-1})^{n}z^{i},z^{i},(\zeta^{i+1})^{n}z^{i},\dots(\zeta^{n})^{n}z^{i})\right)\right)$
$\displaystyle=\left[\left(\sqrt[n]{z^{i}}\zeta^{1},\dots\sqrt[n]{z^{i}}\zeta^{i-1},\sqrt[n]{z^{i}},\sqrt[n]{z^{i}}\zeta^{i+1},\dots,\sqrt[n]{z^{i}}\zeta^{n}\right)\right].$

The pullback of $\eta^{\mathbb{C}}$ by $\pi$ can be described in two steps.

$\pi^{*}(dz^{k})=\begin{cases}\frac{1}{n}\zeta^{k}(z^{i})^{\frac{1}{n}-1}dz^{i}+\sqrt[n]{z^{i}}d\zeta^{k}&i\neq
k\\\ \frac{1}{n}(z^{i})^{\frac{1}{n}-1}dz^{i}&i=k,\end{cases}$

so

(12)
$\pi^{*}(\eta^{\mathbb{C}})=\bigwedge_{k=1}^{n}\pi^{*}(dz^{k})=\frac{1}{n}d\zeta^{1}\wedge
d\zeta^{2}\wedge\dots d\zeta^{i-1}\wedge dz^{i}\wedge
d\zeta^{i+1}\wedge\dots\wedge d\zeta^{n}.$

From (12), one sees that $\eta=\pi^{*}(\eta^{\mathbb{C}})$ is a nowhere
vanishing holomorphic $n$-form on
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)_{|U_{i}}$. Since $i$ was
arbitrary, this shows that $\eta^{\mathbb{C}}$ pulls back to a nowhere
vanishing holomorphic $n$-form on all of
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$. ∎

##  3\. Metric aspects

Here we prove Theorem 3, Proposition 6 and Theorem 9.

It remains to deduce that $\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$
indeed possesses the Ricci-flat metric claimed in Theorem 3. We will show how
this comes about by starting with the ansatz that one has a rotationally
symmetric Kähler potential.

######  Lemma 25.

Assume a Kähler metric on $\mathbb{C}^{n}\setminus\\{0\\}$ is given by

$g_{\mu\bar{\nu}}=\partial_{\mu}\partial_{\bar{\nu}}f,$

and assume that the Kähler potential $f$ is $SO(2n)$-symmetric. Then, with
$u\coloneqq r^{2}\coloneqq\sum_{\mu=1}^{n}z^{\mu}\bar{z}^{\mu}$, we have that

(13)
$g_{\mu\bar{\nu}}=\delta_{\mu\nu}f^{\prime}(u)+\overline{z}_{\mu}z_{\nu}f^{\prime\prime}(u).$

This may also be written

$g=\mathds{1}f^{\prime}(u)+\overline{z}\otimes zf^{\prime\prime}(u).$

Furthermore,

(14) $\det(g)=f^{\prime}(u)^{n-1}(uf^{\prime}(u))^{\prime}.$

###### Proof.

The first bit is a trivial computation. For the determinant, notice that we
may write the determinant as555Note that $f^{\prime}(u)$ is an eigenvalue of
$g$ which we assume is a metric, hence $f^{\prime}(u)$ cannot be 0.

$\det(g)=f^{\prime}(u)^{n}\det\left(\delta_{\mu\bar{\nu}}+\frac{f^{\prime\prime}(u)}{f^{\prime}(u)}\overline{z}_{\mu}z_{\nu}\right).$

Define the hermitian $n\times n$ matrix

$B_{\mu\bar{\nu}}\coloneqq\frac{f^{\prime\prime}(u)}{f^{\prime}(u)}\overline{z}_{\mu}z_{\nu},$

or, equivalently,

$B=\frac{f^{\prime\prime}(u)}{f^{\prime}(u)}\overline{z}\otimes z.$

The matrix $B$ has eigenvalue $u\frac{f^{\prime\prime}(u)}{f^{\prime}(u)}$
with multiplicity 1 and eigenvalue 0 with multiplicity $n-1$, as is seen by
choosing eigenvectors parallel and orthogonal (with respect to Euclidean norm
on $\mathbb{C}^{n}$) to any given $z$. From this little discussion, it follows
that

$\det(\mathds{1}+B)=\left(1+u\frac{f^{\prime\prime}(u)}{f^{\prime}(u)}\right),$

and thus

$\det(g)=f^{\prime}(u)^{n}\det(\mathds{1}+B)=f^{\prime}(u)^{n}\left(1+u\frac{f^{\prime\prime}(u)}{f^{\prime}(u)}\right),$

which one can of course write

$\displaystyle\det(g)=$ $\displaystyle
f^{\prime}(u)^{n}\left(1+u\frac{f^{\prime\prime}(u)}{f^{\prime}(u)}\right)$
$\displaystyle=$ $\displaystyle
f^{\prime}(u)^{n-1}\left(f^{\prime}(u)+uf^{\prime\prime}(u)\right)$
$\displaystyle=$ $\displaystyle
f^{\prime}(u)^{n-1}\left(uf^{\prime}(u)\right)^{\prime}.$

∎

The requirement that the Ricci-curvature vanishes will determine the function
$f$ up to a non-trivial constant. We will explain this with two lemmas.

######  Lemma 26.

Assume we have a Kähler metric $g$ on $\mathbb{C}^{n}\setminus\\{0\\}$ with
Kähler potential $f$, and assume that $f$ is spherically symmetric. Then $g$
is Ricci-flat if and only if the $det(g)$ is constant, which one might assume
is 1 after scaling.

###### Proof.

It’s well known that on a Kähler manifold, one may write the Ricci curvature
locally as

$R_{\mu\bar{\nu}}=-\partial_{\mu}\partial_{\bar{\nu}}\ln\left(\det(g)\right).$

See for instance [2, Equation 4.63] for this fact.

Since the potential is assumed to be spherically symmetric, (14) tells us that
the determinant of the metric is spherically symmetric. Hence the Ricci-tensor
takes the form

(15)
$R_{\mu\bar{\nu}}=-\left(\delta_{\mu\bar{\nu}}\frac{d}{du}\ln\left(\det(g)\right)+\overline{z}_{\mu}z_{\nu}\frac{d^{2}}{du^{2}}\left(\ln\left(\det(g)\right)\right)\right).$

One direction is clear; if $\det(g)$ is constant, the Ricci curvature
vanishes.

For the converse, observe that (15) has eigenvalue
$\frac{d}{du}\ln\left(\det(g)\right)$ (with multiplicity $(n-1)$), hence
$R_{\mu\bar{\nu}}=0$ implies $\frac{d}{du}\ln\left(\det(g)\right)=0$.

∎

######  Lemma 27 ([7]).

Assume, as before, that we have a Kähler metric $g$ on
$\mathbb{C}^{n}\setminus\\{0\\}$ with Kähler potential $f$, and assume that
$f$ is spherically symmetric. Assume in addition that the metric is Ricci-flat
and non-flat. Then the metric takes the form

(16)
$g_{\mu\bar{\nu}}=\left(1+\left(\frac{a}{u}\right)^{n}\right)^{\frac{1}{n}}\left(\delta_{\mu\bar{\nu}}-\frac{a^{n}}{a^{n}+u^{n}}\frac{\bar{z}_{\mu}z_{\nu}}{u}\right)$

for some constant $a>0$. In terms of the potential $f$, we have

(17)
$f^{\prime}(u)=\left(1+\left(\frac{a}{u}\right)^{n}\right)^{\frac{1}{n}},$

which integrates to

(18)
$f(u)=\sqrt[n]{a^{n}+u^{n}}+\frac{a}{n}\sum_{j=0}^{n-1}\zeta^{j}\log\left(\sqrt[n]{1+\frac{u^{n}}{a^{n}}}-\zeta^{j}\right)+const.$

for some primitive $n$’th root of unity $\zeta$.

###### Proof.

From Lemma 26, our metric is Ricci-flat if and only if, $\det(g)=1$. Inserting
this into (14) gives us the following ODE to solve

(19) $f^{\prime}(u)^{n-1}(uf^{\prime}(u))^{\prime}=1.$

This is readily solved by multiplying each side by $u^{n-1}$ and integrating
in $u$, yielding

$(uf^{\prime}(u))^{n}=u^{n}+a^{n}$

as a first step, where $a^{n}$ is some constant of integration. This
solves666We remark again that $f^{\prime}$ is an eigenvalue of $g$, which we
want to be a metric. As such, one has to take the real, positive root when
solving for $f^{\prime}(u)$. for $f^{\prime}(u)$ as

$f^{\prime}(u)=\left(1+\left(\frac{a}{u}\right)^{n}\right)^{\frac{1}{n}}.$

To see that the function $f(u)$ given in (3) is the correct integral of
$f^{\prime}(u)$ requires a little computation. Introduce $x=\frac{u}{a}$, and
observe that $f^{\prime}(u)=\frac{1}{a}\frac{d}{dx}f(x)$, and that

$\frac{f(x)}{a}=\sqrt[n]{1+x^{n}}+\frac{1}{n}\sum_{j=0}^{n-1}\zeta^{j}\log\left(\sqrt[n]{1+x^{n}}-\zeta^{j}\right).$

The derivative of this is easily computed to be

(20)
$\frac{1}{a}f^{\prime}(x)=\frac{x^{n-1}\sqrt[n]{x^{n}+1}}{x^{n}+1}\left(1+\frac{1}{n}\sum_{j=0}^{n-1}\frac{\zeta^{j}}{\sqrt[n]{x^{n}+1}-\zeta^{j}}\right).$

To progress further, we need the fact that for any complex number $\alpha$
with $|\alpha|\neq 1$, and integer $n\geq 1$, the following identity holds.

(21)
$\frac{1}{n}\sum_{j=0}^{n-1}\frac{\zeta^{j}}{\alpha-\zeta^{j}}=\frac{1}{\alpha^{n}-1}.$

Inserting this with $\alpha\coloneqq\sqrt[n]{1+x^{n}}$ into (20) yields

$f^{\prime}(u)=\sqrt[n]{1+x^{-n}}=\sqrt[n]{1+\frac{a^{n}}{u^{n}}},$

which proves the claim. ∎

For many purposes, the metric is best expressed as a metric on
$\mathbb{C}^{n}\setminus\\{0\\}$ or its quotient by $\mu_{n}$. To study the
behaviour near the blown-up point, the standard coordinates on
$\mathbb{C}^{n}$ are no longer suitable, and one should instead pull back the
metric to $\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$ using the blow-down
map $\pi$. Using the coordinates from the standard trivialization
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)_{|U_{i}}\cong
U_{i}\times\mathbb{C}\cong\mathbb{C}^{n}$, we get tractable expressions for
the metric which both shows that the metric extends across the zero section
$\mathbb{C}\mathbb{P}^{n-1}$ and that $g_{|\mathbb{C}\mathbb{P}^{n-1}}=a\cdot
g_{FS}$. The details are as follows.

######  Lemma 28.

Identify $\left(\mathbb{C}^{n}\setminus\\{0\\}\right)/\mu_{n}$ as
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)\setminus\mathbb{C}\mathbb{P}^{n-1}$
(see Theorem 1), and give it the Ricci-flat Kähler metric described as (2),

$g_{\mu\bar{\nu}}=\left(1+\left(\frac{a}{u}\right)^{n}\right)^{\frac{1}{n}}\left(\delta_{\mu\bar{\nu}}-\frac{a^{n}}{a^{n}+u^{n}}\frac{\bar{z}_{\mu}z_{\nu}}{u}\right).$

Then this metric extends to all of
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$, where $u=0$ corresponds to the
zero-section
$\mathbb{C}\mathbb{P}^{n-1}\subset\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$,
and the Ricci-flat metric induces a metric on $\mathbb{C}\mathbb{P}^{n-1}$
which is precisely the parameter $a$ times the Fubini-Study metric.

###### Proof.

Indeed, let $U_{i}=\\{\xi_{i}\neq 0\\}$ as before and let
$\zeta_{k}\coloneqq\frac{\xi_{k}}{\xi_{i}}$. We write a point in
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)_{|U_{i}}\cong\mathbb{C}^{n}$ as
$(z\zeta_{1}^{n},\dots,z\zeta_{i-1}^{n},z,z\zeta_{i+1}^{n},\dots
z\zeta_{n}^{n})$. The blow-down map then takes the form

$\pi_{i}((z\zeta_{1}^{n},\dots,z\zeta_{i-1}^{n},z,z\zeta_{i+1}^{n},\dots
z\zeta_{n}^{n}))=(\sqrt[n]{z}\zeta_{1},\dots,\sqrt[n]{z},\dots,\sqrt[n]{z}\zeta_{n}).$

We claim that the Calabi-Eguchi-Hanson metric takes the form

$\displaystyle\pi_{i}^{*}g_{EH}$
$\displaystyle=\left(\frac{1+|\zeta|^{2}}{\sqrt[n]{a^{n}+u^{n}}}\right)^{n-1}\left(\frac{(1+|\zeta|^{2})}{n^{2}}|dz|^{2}+|z|^{2}|d\zeta|^{2}+\frac{1}{n}\sum_{k}(\overline{z}\zeta_{k}dzd\overline{\zeta}_{k}+z\overline{\zeta}_{k}d\overline{z}d\zeta_{k}\right)$
(22)
$\displaystyle+\left(\frac{a}{\sqrt[n]{a^{n}+u^{n}}}\right)^{n-1}ag_{FS}.$

Here

$|\zeta|^{2}=\sum_{k\neq i}|\zeta_{k}|^{2},$
$|d\zeta|^{2}=\sum_{k}d\zeta_{k}d\overline{\zeta}_{k},$

and

$u=|z|^{2/n}(1+|\zeta|^{2}).$

This metric expression is clearly regular down to $z=0$, in which case it
reduces to

$\pi_{i}^{*}{g_{EH}}_{|z=0}=\frac{1}{a^{n-1}n^{2}}(1+|\zeta|^{2})^{n}|dz|^{2}+ag_{FS}.$

To deduce (22), start by writing

$w_{k}\coloneqq\begin{cases}\sqrt[n]{z}\zeta_{k}&k\neq i\\\
\sqrt[n]{z}&k=i.\end{cases}$

One then finds

$|dw|^{2}\coloneqq\sum_{k=1}^{n}|dw_{k}|^{2}=|z|^{2/n}\left(\frac{(1+|\zeta|^{2})}{n^{2}|z|^{2}}|dz|^{2}+|d\zeta|^{2}+\frac{1}{n|z|^{2}}\sum_{k}\left(\overline{z}\zeta_{k}dzd\overline{\zeta}_{k}+z\overline{\zeta}_{k}d\overline{z}d\zeta_{k}\right)\right)$

and

$\overline{w}\cdot
dw\coloneqq\sum_{k}\overline{w}_{k}dw_{k}=|z|^{2/n}\left(\frac{(1+|\zeta|^{2})}{nz}dz+\sum_{k}\overline{\zeta}_{k}d\zeta_{k}\right).$

Hence

$\displaystyle|dw|^{2}-\frac{a^{n}}{a^{n}+u^{n}}\frac{|\overline{w}\cdot
dw|^{2}}{u}$
$\displaystyle=|z|^{2/n}\Bigg{(}\frac{(1+|\zeta|^{2})}{n^{2}|z|^{2}}\frac{u^{n}}{a^{n}+u^{n}}|dz|^{2}+\left(|d\zeta|^{2}-\frac{a^{n}}{a^{n}+u^{n}}\frac{|\overline{\zeta}\cdot
d\zeta|^{2}}{1+|\zeta|^{2}}\right)$
$\displaystyle+\frac{u^{n}}{n|z|^{2}(a^{n}+u^{n})}\sum_{k}\left(\left(\overline{z}\zeta_{k}dzd\overline{\zeta}_{k}+z\overline{\zeta}_{k}d\overline{z}d\zeta_{k}\right)\right)\Bigg{)}.$

We now use $u=\sum_{k}|w_{k}|^{2}=|z|^{2/n}(1+|\zeta|^{2})$ and

$g_{FS}=\frac{(1+|\zeta|^{2})|d\zeta|^{2}-|\overline{\zeta}\cdot
d\zeta|^{2}}{(1+|\zeta|^{2})^{2}}$

to deduce the statement.

∎

###### Proof of Proposition 6.

The starting point will be equation [2, Equation 4.39]), which says that

$\Gamma_{\alpha\mu}^{\lambda}=g_{\mu\bar{\nu},\alpha}g^{\bar{\nu}\lambda}.$

We return to the setting of Lemma 25 and introduce some shorthand notation,

$e^{\psi(u)}\coloneqq f^{\prime}(u)$

and

$\phi(u)\coloneqq-uf^{\prime\prime}(u)e^{-\psi(u)}.$

Lemma 25 tells us that we may write

(23)
$g_{\mu\bar{\nu}}=e^{\psi(u)}\left(\delta_{\mu\bar{\nu}}-\phi(u)\frac{\bar{z}_{\mu}z_{\nu}}{u}\right),$

One readily checks that the inverse of $g$ is given as

(24)
$g^{\bar{\nu}\lambda}=e^{-\psi(u)}\left(\delta^{\bar{\nu}\lambda}+\frac{\phi(u)}{1-\phi(u)}\frac{\bar{z}^{\nu}z^{\lambda}}{u}\right).$

Furthermore, the definition of $e^{\psi}$ tells us that
$\psi^{\prime}(u)=-\frac{\phi(u)}{u}$. One can thus compute that

$\displaystyle g_{\mu\bar{\nu},\alpha}$
$\displaystyle=-\frac{\phi(u)}{u}g_{\mu\bar{\nu}}\bar{z}_{\alpha}+e^{\psi(u)}\left(\frac{\phi(u)-u\phi^{\prime}(u)}{u^{2}}\bar{z}_{\mu}z_{\nu}\bar{z}_{\alpha}-\frac{\phi(u)}{u}\delta_{\alpha\bar{\nu}}\bar{z}_{\mu}\right)$
$\displaystyle=-\frac{\phi(u)}{u}(g_{\mu\bar{\nu}}\bar{z}_{\alpha}+g_{\alpha\bar{\nu}}\bar{z}_{\mu})+e^{\psi(u)}\frac{\phi(u)(1-\phi(u))-u\phi^{\prime}(u)}{u}\frac{\bar{z}_{\mu}z_{\nu}\bar{z}_{\alpha}}{u}.$

Multiply this by $g^{\bar{\nu}\lambda}$ and sum over $\nu$, observing that
$z_{\nu}$ is an eigenvector of $g^{\bar{\nu}\lambda}$ with eigenvalue
$\frac{e^{-\psi}}{1-\phi}$. This leads to

(25)
$\Gamma^{\lambda}_{\mu\alpha}=g_{\mu\bar{\nu},\alpha}g^{\bar{\nu}\lambda}=-\frac{\phi(u)}{u}(\delta_{\mu}{}^{\lambda}\bar{z}_{\alpha}+\delta_{\alpha}{}^{\lambda}\bar{z}_{\mu})+\frac{\phi(u)(1-\phi(u))-u\phi^{\prime}(u)}{u(1-\phi(u))}\cdot\frac{\bar{z}_{\mu}\bar{z}_{\alpha}z^{\lambda}}{u}.$

Equation (25) holds without assuming that the metric is Ricci-flat. One ”only”
needs to assume that the Kähler potential is spherically symmetric777Two
special examples would be $e^{\psi}=1$, corresponding to the Euclidean metric,
or $e^{\psi}=\frac{a}{a+u}$ and $\phi=\frac{u}{a+u}$, corresponding to the
Fubini-Study metric.. When we in addition assume that the metric is Ricci-
flat, we have seen in Lemma (27) that

$\phi=\frac{a^{n}}{a^{n}+u^{n}},$

hence

(26)
$\phi^{\prime}(u)=-n\frac{u^{n-1}a^{n}}{(a^{n}+u^{n})^{2}}=-\frac{n}{u}\phi(u)(1-\phi(u)).$

Inserting this into equation (25) yields (6).

To compute the Riemann curvature, we will employ equation [2, Equation 4.49],
which says that for Kähler manifolds, the curvature can be written as

$R^{\lambda}{}_{\mu\bar{\beta}\alpha}=-\frac{\partial\Gamma^{\lambda}_{\mu\alpha}}{\partial\bar{z}^{\beta}}.$

Using this fact, one can differentiate (6), to find that as a first step,

$\displaystyle R^{\lambda}{}_{\mu\bar{\beta}\alpha}=$
$\displaystyle\left(\frac{\phi(u)}{u}\right)^{\prime}z_{\beta}\left(\bar{z}_{\mu}\delta^{\lambda}{}_{\alpha}+\bar{z}_{\alpha}\delta^{\lambda}{}_{\mu}-(n+1)\frac{\bar{z}_{\alpha}\bar{z}_{\mu}}{u}z^{\lambda}\right)$
$\displaystyle+\frac{\phi(u)}{u}\left(\delta_{\mu\bar{\beta}}\delta^{\lambda}{}_{\alpha}+\delta_{\alpha\bar{\beta}}\delta^{\lambda}{}_{\mu}-(n+1)\frac{z^{\lambda}}{u}\left(\delta_{\alpha\bar{\beta}}\bar{z}_{\mu}+\delta_{\mu\bar{\beta}}\bar{z}_{\alpha}-\frac{\bar{z}_{\mu}\bar{z}_{\alpha}z_{\beta}}{u}\right)\right).$

Here we are still using the shorthand $\phi=\frac{a^{n}}{a^{n}+u^{n}}$. Using
(26), we find

$\left(\frac{\phi(u)}{u}\right)^{\prime}=\frac{u\phi^{\prime}(u)-\phi}{u^{2}}=-n\frac{\phi(1-\phi)}{u^{2}}-\frac{\phi}{u^{2}},$

which allows us to write

$\displaystyle R^{\lambda}{}_{\mu\bar{\beta}\alpha}=$
$\displaystyle\frac{\phi(u)}{u}\Bigg{(}\left(\delta_{\mu\bar{\beta}}-\frac{\bar{z}_{\mu}z_{\beta}}{u}\right)\delta^{\lambda}{}_{\alpha}+\left(\delta_{\alpha\bar{\beta}}-\frac{\bar{z}_{\alpha}z_{\beta}}{u}\right)\delta^{\lambda}{}_{\mu}$
$\displaystyle-(n+1)\left(\frac{\bar{z}_{\mu}z^{\lambda}}{u}\delta_{\alpha\bar{\beta}}+\frac{\bar{z}_{\alpha}z^{\lambda}}{u}\delta_{\mu\bar{\beta}}-2\frac{\bar{z}_{\mu}z^{\lambda}\bar{z}_{\alpha}z_{\beta}}{u^{2}}\right)$
$\displaystyle-n(1-\phi)\left(\frac{\bar{z}_{\mu}z_{\beta}}{u}\delta^{\lambda}{}_{\alpha}+\frac{\bar{z}_{\alpha}z_{\beta}}{u}\delta^{\lambda}{}_{\mu}-(n+1)\frac{\bar{z}_{\mu}z^{\lambda}\bar{z}_{\alpha}z_{\beta}}{u^{2}}\right)\Bigg{)}.$

This can be rewritten by observing that
$g_{\mu\bar{\nu}}e^{-\psi}=\delta_{\mu\bar{\nu}}-\phi\frac{\bar{z}_{\mu}z_{\nu}}{u}$,
resulting in

$\displaystyle R^{\lambda}{}_{\mu\bar{\beta}\alpha}=$
$\displaystyle\frac{\phi}{u}\Bigg{(}e^{-\psi}\left(g_{\mu\bar{\beta}}\delta^{\lambda}{}_{\alpha}+g_{\alpha\bar{\beta}}\delta^{\lambda}{}_{\mu}\right)$
$\displaystyle-(n+1)e^{-\psi}\left(\frac{\bar{z}_{\mu}z^{\lambda}}{u}g_{\alpha\bar{\beta}}+\frac{\bar{z}_{\alpha}z^{\lambda}}{u}g_{\mu\bar{\beta}}\right)$
$\displaystyle-(n+1)(1-\phi)\left(\frac{\bar{z}_{\mu}z_{\beta}}{u}\delta^{\lambda}{}_{\alpha}+\frac{\bar{z}_{\alpha}z_{\beta}}{u}\delta^{\lambda}{}_{\mu}\right)$
(27)
$\displaystyle+(n+1)(n+2)(1-\phi)\frac{\bar{z}_{\mu}z^{\lambda}\bar{z}_{\alpha}z_{\beta}}{u^{2}}\Bigg{)}.$

Finally, we need a small observation. The inverse of the metric, expressed
using $\phi$ and $\psi$, is given in equation (24). Using that equation, and
recalling that we raise and lower indices on $z$ using the Euclidean metric in
this article, one may write

$z^{\lambda}=e^{\psi}(1-\phi)g^{\bar{\sigma}\lambda}z_{\sigma}.$

Inserting this into (27), we arrive at

$\displaystyle R^{\lambda}{}_{\mu\bar{\beta}\alpha}$
$\displaystyle=\frac{\phi}{u}e^{-\psi}g^{\bar{\sigma}\lambda}\Bigg{(}(g_{\alpha\bar{\sigma}}g_{\mu\bar{\beta}}+g_{\mu\bar{\sigma}}g_{\alpha\bar{\beta}})$
$\displaystyle-e^{\psi}(1-\phi)(n+1)\left(\frac{\bar{z}_{\mu}z_{\beta}}{u}g_{\alpha\bar{\sigma}}+\frac{\bar{z}_{\alpha}z_{\beta}}{u}g_{\mu\bar{\sigma}}+\frac{\bar{z}_{\mu}z_{\sigma}}{u}g_{\alpha\bar{\beta}}+\frac{\bar{z}_{\alpha}z_{\sigma}}{u}g_{\mu\bar{\beta}}\right)$
$\displaystyle+e^{2\psi}(1-\phi)^{2}(n+1)(n+2)\frac{\bar{z}_{\mu}z_{\sigma}\bar{z}_{\alpha}z_{\beta}}{u^{2}}\Bigg{)}.$

This is almost (7). What remains is to multiply by $g_{\sigma\bar{\nu}}$, sum
over $\sigma$, insert $\phi(u)=\frac{a^{n}}{a^{n}+u^{n}}$ and
$\exp(\psi)=\sqrt[n]{1+\frac{a^{n}}{u^{n}}}$, and observe that
$R_{\mu\bar{\nu}\alpha\bar{\beta}}=R_{\bar{\nu}\mu\bar{\beta}\alpha}$.

To compute the Kretschmann scalar, start with (7) and raise the indices using
$g$. Then multiply the result with (7) and contract. This gives 6 different
kinds of terms. Adding them up gives (8).

∎

######  Remark 29.

As a test, one can check directly by summing over $\lambda=\alpha$ or
$\lambda=\mu$ in (27) that the Ricci-tensor vanishes identically, as it
should.

###### Proof of Proposition 8.

We have shown in Proposition 24 that $\pi^{*}\eta^{\mathbb{C}}$ extends to a
nowhere vanishing holomorphic form on
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$, so we only need to argue that
it is parallel. We may work on $\mathbb{C}^{n}\setminus\\{0\\}$ with the
metric given by (2) One way to argue is to use the Christoffel symbols (6) and
simply compute. Indeed,
$\eta^{\mathbb{C}}=\frac{1}{n!}\epsilon_{\mu_{1},\dots,\mu_{n}}dz^{\mu_{1}}\wedge\dots\wedge
dz^{\mu_{n}}$ where $\epsilon$ denotes the completely anti-symmetric Levi-
Civita symbol and $\epsilon_{1,2,\dots,n}=1$. Then

$\nabla_{\alpha}\epsilon_{\mu_{1},\dots,\mu_{n}}=-\Gamma^{\lambda}_{\alpha\mu_{1}}\epsilon_{\lambda,\mu_{2},\dots,\mu_{n}}-\dots-\Gamma^{\lambda}_{\alpha\mu_{n}}\epsilon_{\mu_{1},\dots,\lambda}.$

Due to the anti-symmetry, it suffices to compute with
$(\mu_{1},\dots,\mu_{n})=(1,2,\dots,n)$. We then have

$\displaystyle-\Gamma^{\lambda}_{\alpha\mu_{1}}\epsilon_{\lambda,\mu_{2},\dots,\mu_{n}}-\dots-\Gamma^{\lambda}_{\alpha\mu_{n}}\epsilon_{\mu_{1},\dots,\lambda}$
$\displaystyle=\frac{a^{n}}{u(a^{n}+u^{n})}\left(n\overline{z}_{\alpha}+(\overline{z}_{1}\delta_{\alpha
1}+\dots+\overline{z}_{n}\delta_{\alpha
n})-(n+1)\frac{\overline{z}_{\alpha}}{u}(\overline{z}_{1}z^{\lambda}\delta_{\lambda
1}+\dots+\overline{z}^{n}z^{\lambda}\delta_{\lambda n}\right)$
$\displaystyle=\frac{a^{n}}{u(a^{n}+u^{n})}\left((n+1)\overline{z}_{\alpha}-(n+1)\overline{z}_{\alpha}\right)=0.$

An alternative argument is first to observe
$|\eta^{\mathbb{C}}|^{2}_{g}=\frac{\det(g)}{n!}=\frac{1}{n!}$ is constant. If
$\tilde{\eta}$ is the parallel, holomorphic $n$-form on
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$, then $\tilde{\eta}=f\cdot\eta$
for some holomorphic function $f$. But a parallel form has a constant norm,
hence $|f|$ is constant. So $f$ is a constant and $\eta$ is parallel. ∎

###  3.1. Minimal submanifolds

Before we prove Theorem 9, we need to state a well-known result about minimal
submanifolds.

######  Lemma 30 ([28, Lemma 5.1]).

Let $(M,g)$ be a Riemannian manifold. Suppose that $\varphi$ is a smooth
function on $M$ whose Hessian is positive semidefinite. Then any compact,
minimal submanifold $K$ of $M$ must be contained in the set where
$Hess_{M}\varphi$ degenerates. In addition, $\varphi$ is constant on connected
components of $K$.

We refer to [28] for the proof. As Tsai and Wang point out, the lemma is older
and not originally due to them, but neither they nor the author know of
another reference with a proof.

###### Proof of Theorem 9.

As in [28], we choose $\varphi$ to be the distance squared to the zero section
$\mathbb{C}\mathbb{P}^{n-1}\subset\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$,
and by Lemma 30 we just need to show that the Hessian of $\varphi$ is positive
definite at any $z\in\left(\mathbb{C}^{n}\setminus\\{0\\}\right)/\mu_{n}$. We
may compute on the $n$-fold cover $\mathbb{C}^{n}\setminus\\{0\\}$. Let
$\gamma(t)=\alpha(t)z$ be a radial geodesic, where $\alpha(0)=0$,
$\alpha(t),\dot{\alpha}(t)>0$ for $t\in(0,1]$, and888One can in principle find
$\alpha$ by solving the equation $|\dot{\gamma}|^{2}_{g}=C^{2}=const.$, which
combines with (2) to yield
$\dot{\alpha}(t)=C\left(\frac{a^{n}}{\alpha(t)^{2n}}+|z|_{\mathbb{C}^{n}}^{2n}\right)^{\frac{n-1}{2n}},$
but we will not need the explicit solution. $\alpha(1)=1$. By (2) we find

$|\dot{\gamma}|^{2}_{g}=u\dot{\alpha}(t)^{2}\left(u\frac{\alpha(t)^{2}}{\sqrt[n]{\alpha(t)^{2n}u^{n}+a^{n}}}\right)^{n-1}$

and

$\displaystyle\sqrt{\varphi}(z)$
$\displaystyle=\int_{0}^{1}|\dot{\gamma}|_{g}\,dt=\sqrt{u}\int_{0}^{1}\dot{\alpha}(t)\left(\frac{u\alpha(t)^{2}}{\sqrt[n]{\alpha(t)^{2n}u^{n}+a^{n}}}\right)^{\frac{n-1}{2}}\,dt$
$\displaystyle=\sqrt{u}\int_{0}^{1}\left(\frac{u\alpha^{2}}{\sqrt[n]{\alpha^{2n}u^{n}+a^{n}}}\right)^{\frac{n-1}{2}}\,d\alpha$
$\displaystyle=\frac{\sqrt{a}}{n}\int_{0}^{\left(\frac{u}{a}\right)^{\frac{n}{2}}}\frac{1}{(\tau^{2}+1)^{\frac{n-1}{2n}}}\,d\tau$
(28) $\displaystyle\eqqcolon\sqrt{\psi}(u(z)).$

where we have substituted $\tau=\left(\sqrt{\frac{u}{a}}\alpha\right)^{n}$.

We use first compute the components of the Hessian when $\mathbb{C}$-linearly
extended to the complexified tangent bundle. Writing $H\coloneqq\nabla
d\varphi$, we have 4 block matrices to consider. They read

(29) $H_{\mu\overline{\alpha}}\coloneqq
H\left(\partial_{z^{\mu}},\partial_{\overline{z}^{\alpha}}\right)=\partial_{\mu}\partial_{\overline{\alpha}}\varphi=\delta_{\mu\alpha}\psi^{\prime}(u)+\overline{z}_{\mu}z_{\alpha}\psi^{\prime\prime}(u),$
$H_{\mu\alpha}\coloneqq
H\left(\partial_{z^{\mu}},\partial_{z^{\alpha}}\right)=\partial_{\mu}\partial_{\alpha}\varphi-\Gamma^{\lambda}_{\mu\alpha}\partial_{\lambda}\varphi=\overline{z}_{\mu}\overline{z}_{\alpha}\psi^{\prime\prime}(u)-\Gamma_{\mu\alpha}^{\lambda}\overline{z}_{\lambda}\psi^{\prime}(u),$
$H_{\overline{\alpha}\mu}\coloneqq
H\left(\partial_{\overline{z}^{\alpha}},\partial_{z^{\mu}}\right)=\overline{H_{\alpha\overline{\mu}}},$
$H_{\overline{\mu}\overline{\alpha}}\coloneqq
H\left(\partial_{\overline{z}^{\mu}},\partial_{\overline{z}^{\alpha}}\right)=\overline{H_{\mu\alpha}}.$

The Christoffel symbols for the Calabi-Eguchi-Hanson metric (2) are given by
(6). From that equation, we easily find

$\Gamma_{\mu\alpha}^{\lambda}\overline{z}_{\lambda}=(n-1)\frac{a^{n}}{u(a^{n}+u^{n})}\overline{z}_{\mu}\overline{z}_{\alpha}\eqqcolon\Upsilon(u)\overline{z}_{\mu}\overline{z}_{\alpha}.$

So

(30)
$H_{\mu\alpha}=\left(\psi^{\prime\prime}(u)-\Upsilon(u)\psi^{\prime}(u)\right)\overline{z}_{\mu}\overline{z}_{\alpha}.$

To translate into the components of the real Hessian, we write
$z^{\mu}=x^{\mu}+iy^{\mu}$ and use

$\frac{\partial}{\partial z^{\mu}}=\frac{1}{2}\left(\frac{\partial}{\partial
x^{\mu}}-i\frac{\partial}{\partial y^{\mu}}\right).$

to write

(31) $H^{xx}_{\mu\alpha}\coloneqq
H(\partial_{x^{\mu}},\partial_{x^{\alpha}})=H(\partial_{z^{\mu}}+\partial_{\overline{z}^{\mu}},\partial_{z^{\alpha}}+\partial_{\overline{z}^{\alpha}})=H_{\mu\alpha}+H_{\overline{\mu}\overline{\alpha}}+H_{\overline{\mu}\alpha}+H_{\mu\overline{\alpha}}.$

Similarly,

(32) $H^{xy}_{\mu\alpha}\coloneqq
H(\partial_{x^{\mu}},\partial_{y^{\alpha}})=H\left(\partial_{z^{\mu}}+\partial_{\overline{z}^{\mu}},i\left(\partial_{z^{\alpha}}-\partial_{\overline{z}^{\alpha}}\right)\right)=i(H_{\mu\alpha}-H_{\overline{\mu}\overline{\alpha}})+i(H_{\overline{\mu}\alpha}-H_{\mu\overline{\alpha}}),$
(33) $H^{yx}_{\mu\alpha}\coloneqq
H(\partial_{y^{\mu}},\partial_{x^{\alpha}})=H\left(i\left(\partial_{z^{\mu}}-\partial_{\overline{z}^{\mu}}\right),\partial_{z^{\alpha}}+\partial_{\overline{z}^{\alpha}}\right)=i(H_{\mu\alpha}-H_{\overline{\mu}\overline{\alpha}})+i(H_{\mu\overline{\alpha}}-H_{\overline{\mu}\alpha}),$

and

(34) $H^{yy}_{\mu\alpha}\coloneqq
H(\partial_{y^{\mu}},\partial_{y^{\alpha}})=H\left(i\left(\partial_{z^{\mu}}-\partial_{\overline{z}^{\mu}}\right),i\left(\partial_{z^{\alpha}}-\partial_{\overline{z}^{\alpha}}\right)\right)=-(H_{\mu\alpha}+H_{\overline{\mu}\overline{\alpha}})+(H_{\overline{\mu}\alpha}+H_{\mu\overline{\alpha}}).$

The Hessian thus takes the block-matrix form

$\begin{pmatrix}H^{xx}&H^{xy}\\\ H^{yx}&H^{yy}\end{pmatrix}$

in these coordinates.

By (31), (29) and (30), we find999Recall that $x\otimes x$ means the matrix
with entries $(x\otimes x)_{k\ell}=x_{k}x_{\ell}$.

$H^{xx}=2\psi^{\prime}\mathds{1}_{n}+2(2\psi^{\prime\prime}-\Upsilon\psi^{\prime})x\otimes
x+2\Upsilon\psi^{\prime}y\otimes y.$

Similarly, we have

$H^{yy}=2\psi^{\prime}\mathds{1}_{n}+2\Upsilon\psi^{\prime}x\otimes
x+2(2\psi^{\prime\prime}-\Upsilon\psi^{\prime})y\otimes y,$
$H^{xy}=2(2\psi^{\prime\prime}-\Upsilon\psi^{\prime})x\otimes
y-2\Upsilon\psi^{\prime}y\otimes x,$

and

$H^{yx}=2(2\psi^{\prime\prime}-\Upsilon\psi^{\prime})y\otimes
x-2\Upsilon\psi^{\prime}x\otimes y.$

Writing

$A\coloneqq 2\frac{\psi^{\prime\prime}}{\psi^{\prime}}-\Upsilon$
$B\coloneqq\Upsilon,$

the real Hessian takes the form

$H=2\psi^{\prime}\left(\mathds{1}_{2n}+\begin{pmatrix}Ax\otimes x+By\otimes
y&Ax\otimes y-By\otimes x\\\ Ay\otimes x-Bx\otimes y&Bx\otimes x+Ay\otimes
y\end{pmatrix}\right).$

The eigenvalues of the matrix

$\begin{pmatrix}Ax\otimes x+By\otimes y&Ax\otimes y-By\otimes x\\\ Ay\otimes
x-Bx\otimes y&Bx\otimes x+Ay\otimes y\end{pmatrix}$

are given by

  * •

0 with eigenvector $V=\begin{pmatrix}V_{1}\\\ V_{2}\end{pmatrix}$ with
$V_{1},V_{2}$ orthogonal to both $x$ and $y$;

  * •

$A(|x|^{2}+|y|^{2})=Au$ with eigenvector $\begin{pmatrix}x\\\ y\end{pmatrix}$;

  * •

$Bu$ with eigenvector $\begin{pmatrix}y\\\ -x\end{pmatrix}$.

So the eigenvalues of the real Hessian $H$ are

  * •

$\lambda_{1}=2\psi^{\prime}$ with multiplicity $2n-2$;

  * •

$\lambda_{2}=2\psi^{\prime}+2\psi^{\prime}Au=2\psi^{\prime}+2u(2\psi^{\prime\prime}-\Upsilon\psi^{\prime})$
with multiplicity 1;

  * •

$\lambda_{3}=2\psi^{\prime}+2\psi^{\prime}Bu=2\psi^{\prime}(1+u\Upsilon)$ with
multiplicity 1.

We therefore need to check that these three eigenvalues are positive for
$u>0$.

From the explicit expression for the length squared (28) we find

$\psi^{\prime}(u)=\frac{\sqrt{\psi}}{\sqrt{u}}\left(\frac{u^{n}}{a^{n}+u^{n}}\right)^{\frac{n-1}{2n}}.$

From this and the definition of $\Upsilon=(n-1)\frac{a^{n}}{u(a^{n}+u^{n})}$,
the eigenvalues $\lambda_{1}$ and $\lambda_{3}$ are positive. Differentiating
again, a straight forward computation gives

$\psi^{\prime\prime}(u)=\frac{\psi^{\prime}}{2\psi}\left(\psi^{\prime}+\frac{\psi}{u(a^{n}+u^{n})}\left((n-2)a^{n}-u^{n}\right)\right).$

So

$2\psi^{\prime\prime}-\Upsilon\psi^{\prime}=\frac{\psi^{\prime}}{u\psi}(u\psi^{\prime}-\psi)=\frac{(\psi^{\prime})^{2}}{\psi}-\frac{\psi^{\prime}}{u}.$

Hence
$\lambda_{2}=2\psi^{\prime}+2(2\psi^{\prime\prime}-\Upsilon\psi^{\prime})u=2u\frac{(\psi^{\prime})^{2}}{\psi}>0$.

∎

Geodesics are of course minimal submanifolds, so the corollary follows. For
completeness’ sake, we offer an alternative, direct proof. The spirit is the
same as the above proof.

###### Proof of Corollary 10.

Let us first prove that $\mathbb{C}\mathbb{P}^{n-1}$ sits inside
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$ as a totally geodesic
submanifold. Consider a diagonal $U(1)$-action on $\mathbb{C}^{n}$,
$(\zeta,z)\mapsto\zeta z$. According to Lemma 17, we get an induced
$U(1)$-action on $\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$. This
$U(1)$-action is isometric (since it obviously preserves (2) and (5)). For
$\zeta\in U(1)$, $\zeta\notin\mu_{n}$, no point of
$\left(\mathbb{C}^{n}\setminus\\{0\\}\right)/\mu_{n}$ is left fixed, but all
of $\mathbb{C}\mathbb{P}^{n-1}$ is fixed. So
$Fix(\zeta)=\mathbb{C}\mathbb{P}^{n-1}$. and
$\mathbb{C}\mathbb{P}^{n-1}\subset\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$
is totally geodesic.

To prove the corollary, assume
$\gamma\colon[0,1]\to\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$ is a closed
geodesic. There are two cases. In the first case
$\gamma(t)\in\mathbb{C}\mathbb{P}^{n-1}$ for all $t$ in some interval. Then
$\gamma$ is a geodesic on $\mathbb{C}\mathbb{P}^{n-1}$ with the Fubini-Study
metric by Theorem 3, and hence remains on $\mathbb{C}\mathbb{P}^{n-1}$ for all
time. Then one quotes for instance101010This is according to Klingenberg [17,
p.178] due to Cartan. [12, Prop. 5.3, p. 356] has a proof. [17, Theorem 5.2.1,
p. 178] to say that all geodesics (when their domain is extended to all of
$\mathbb{R}$) in $\mathbb{C}\mathbb{P}^{n-1}$ with the Fubini-Study metric are
closed. The stability statement comes from the fact that any geodesic in
$(\mathbb{C}\mathbb{P}^{n-1},g_{FS})$ lies on some
$\mathbb{S}^{2}\cong\mathbb{C}\mathbb{P}^{1}\subset\mathbb{C}\mathbb{P}^{n-1}$
(this statement is also part of [17, Theorem 5.2.1, p. 178]), and is therefore
not stable.

The second case is that $\gamma([0,1])$ is not contained in
$\mathbb{C}\mathbb{P}^{n-1}$. Then there has to be some maximal interval
$(t_{0},t_{1})\subset[0,1]$ with
$\gamma(t)\in\left(\mathbb{C}^{n}\setminus\\{0\\}\right)/\mu_{n}$. Let
$u(t)\coloneqq|\gamma(t)|^{2}_{\mathbb{C}^{n}}$ for $t\in(t_{0},t_{1})$. Since
$\gamma$ is closed, $u(t)$ has to have a maximum, meaning there is some
$T\in(t_{0},t_{1})$ such that $\dot{u}(T)=0$ and $\ddot{u}(T)\leq 0$. This
will lead to a contradiction unless $u(t)$ is constant, as we explain next. On
$\left(\mathbb{C}^{n}\setminus\\{0\\}\right)/\mu_{n}$, the Christoffel symbols
of the metric are given by (6). The geodesic equation on a Kähler manifold
reads111111To see this, take the usual geodesic equation and observe that
$\Gamma_{\mu\bar{\alpha}}^{\lambda}=\Gamma_{\bar{\mu}\alpha}^{\lambda}=0$ for
a Kähler manifold.

$\ddot{\gamma}(t)^{\lambda}+\Gamma^{\lambda}_{\mu\alpha}\dot{\gamma}(t)^{\alpha}\dot{\gamma}(t)^{\mu}=0.$

In local coordinates with $z(t)\coloneqq\gamma(t)$, we get

(35) $\ddot{z}^{\lambda}=\frac{a^{n}}{a^{n}+u^{n}}\left(2\frac{\left\langle
z,\dot{z}\right\rangle}{u}\dot{z}^{\lambda}-(n+1)\frac{\left\langle
z,\dot{z}\right\rangle^{2}}{u^{2}}z^{\lambda}\right)$

upon inserting (6) (the inner products in (35) are all $\mathbb{C}^{n}$ inner
products).

Directly from the definition of $u(t)$, we have

$\dot{u}(t)=\left\langle
z,\dot{z}\right\rangle(t)+\left\langle\dot{z},z\right\rangle(t)=2\text{Re}\left(\left\langle
z,\dot{z}\right\rangle\right)(t),$

hence $\dot{u}(T)=0$ if and only if $\left\langle
z,\dot{z}\right\rangle(T)=i\alpha\cdot u(T)$ for some $\alpha\in\mathbb{R}$
(the extra factor of $u(T)$ is just for later convenience). Inserting
$\left\langle z,\dot{z}\right\rangle(T)=i\alpha\cdot u(T)$ into (35) gives us

(36)
$\ddot{z}^{\lambda}(T)=\frac{a^{n}}{a^{n}+u(T)^{n}}\left(2i\alpha\dot{z}(T)^{\lambda}+(n+1)\alpha^{2}z(T)^{\lambda}\right).$

Consider now the second derivative of $u$,

$\ddot{u}(t)=2\text{Re}\left(\left\langle
z,\ddot{z}\right\rangle\right)(t)+2|\dot{z}|^{2}_{\mathbb{C}^{n}}(t).$

At the time $T$, we can use the expression for $\ddot{z}(T)$ in (36) to deduce

$\displaystyle\left\langle z,\ddot{z}\right\rangle(T)$
$\displaystyle=\frac{a^{n}}{a^{n}+u(T)^{n}}\left(2i\alpha\left\langle
z,\dot{z}\right\rangle(T)+(n+1)\alpha^{2}u(T)\right)$
$\displaystyle=(n-1)\frac{a^{n}}{a^{n}+u(T)^{n}}u(T)\alpha^{2},$

hence

(37)
$\ddot{u}(T)=2(n-1)\frac{a^{n}}{a^{n}+u(T)^{n}}u(T)\alpha^{2}+2|\dot{z}|^{2}_{\mathbb{C}^{n}}(T),$

where both terms are non-negative, ensuring $\ddot{u}(T)\geq 0$. Since we’re
demanding that $\ddot{u}(T)\leq 0$, we deduce $\ddot{u}(T)=0$. Referring back
to (37), we see that $\dot{z}(T)=0$. One can now conclude that we have a
geodesic with vanishing velocity at a point, hence, by uniqueness of
geodesics, we have a constant geodesic. So there are no non-constant, closed
geodesic which are not contained in $\mathbb{C}\mathbb{P}^{n-1}$.

∎

###  3.2. Homothety

The Calabi-Eguchi-Hanson metrics have an interesting scaling property. Let
$\alpha>0$ be any fixed number, and consider the homothety
$h_{\alpha}^{\mathbb{C}}\colon\mathbb{C}^{n}\to\mathbb{C}^{n}$ given by
$h_{\alpha}^{\mathbb{C}}(z)=\alpha z$. This clearly fixes the origin and
commutes with both the $\mu_{k}$-action and the blow-up, giving us a map
$h_{\alpha}\colon\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)\to\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)$.

######  Lemma 31.

Let $g_{a}$ denote the Calabi-Eguchi-Hanson metric on
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-n)$ with parameter $a>0$. Let
$\alpha>0$. Then

$h_{\alpha}^{*}g_{\alpha^{2}a}=\alpha^{2}g_{a}.$

###### Proof.

Either observe that the potential (18) satisfies $f_{\alpha^{2}a}(\alpha
z)=\alpha^{2}f_{a}(z)$, or consider how the metric (16) changes. ∎

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