# Lower dimensional ${S}^{1}$-invariant Kähler-Einstein metrics via integrable
structures

Filippo Salis

###### Abstract.

We focus on the classical open problem of the classification of Kähler-
Einstein manifolds that can be Kähler immersed into a complex projective space
endowed with the Fubini-Study metric. In particular, we will deal with such
problem in the special case of Kähler-Einstein metrics admitting symmetries of
rotational type. This leads to certain integrable distributions allowing a
classification of such metrics.

###### Contents

  1. 1 Introduction
  2. 2 Proof of Proposition 1.2 and Corollary 1.4
    1. 2.1 Arezzo-Loi-Zuddas’ result
    2. 2.2 Final remark

##  1\. Introduction

The existence and rigidity of holomorphic and isometric immersions (Kähler
immersions) of a given Kähler manifold into a space of constant holomorphic
sectional curvature (called complex space forms) can be considered as a
classical and long-staying problem in complex differential geometry (see e.g.
[4, 6]). Nevertheless, a complete classification of Kähler manifolds admitting
such type of immersions does not exist, even for Kähler manifolds of great
interest such as Kähler-Einstein manifolds.

In [24], M. Umehara proved that Kähler-Einstein manifolds that can be Kähler
immersed into a complex space form with non-positive holomorphic sectional
curvature, are only the totally geodesic submanifolds of the ambient space.
Instead, when the space form has positive holomorphic curvature, only some
partial results are known (see for instance [21, 22, 9, 23, 10, 12, 13]). They
anyway made some authors (see e.g. [16, Chap. 4]) suppose the validity of the
following

######  Conjecture 1.

A Kähler-Einstein manifold that can be Kähler immersed into a complex
projective space, is an open subset of a complex flag manifold, namely a
compact simply-connected Kähler manifold acted upon transitively by its
holomorphic isometry group.

We recall that complex projective spaces $\mathds{C}\mathrm{P}^{N}$ endowed
with metrics111If $c=1$ we have the standard Fubini-Study metric $g_{FS}$.
associated to Kähler forms reading in affine coordinates $z$ as

$\frac{i}{2c}\partial\bar{\partial}\log(1+\|z\|^{2}),$

are the universal covering of an $n$-dimensional complex space form with
holomorphic sectional curvature equal to $4c>0$.

In the present paper, we are going to test the above conjecture in the case of
${S}^{1}$-invariant Kähler manifolds, i.e. those Kähler manifolds admitting
(around a suitable point and in suitable holomorphic coordinates) a local
Kähler potential

(1) $\Phi:0\in U\subseteq\mathbb{C}^{n}\to\mathbb{R}$

of the form $\Phi\left(|z_{1}|^{2},\dots,|z_{n}|^{2}\right)$.

We are going to study only Kähler manifolds $(M,g)$ that can be Kähler
immersed in complex projective spaces endowed with the standard Fubini-Study
metric (in this case we say that $g$ is projectively induced). By introducing
suitable normalization coefficients, our proof can be easily adapted (leading
to similar results) to cases where the ambient space is endowed with a
different multiple of the Fubini-Study metric.

Since complex projective spaces are the only irreducible ${S}^{1}$-invariant
flag manifolds and since only the integer multiples of the Fubini-Study metric
are projectively induced, the above conjecture reads in our setting as

######  Conjecture 2.

The only projectively induced and ${S}^{1}$-invariant Kähler-Einstein
manifolds are open subsets of
$\mathds{C}\mathrm{P}^{n_{1}}\times\mathellipsis\times\mathds{C}\mathrm{P}^{n_{k}}$
endowed with the Kähler metric

$q\left(c_{1}g_{FS}\oplus\mathellipsis\oplus c_{k}g_{FS}\right),$

where $k$ and $q\in\mathds{Z}^{+}$. Let $G$ be the least common multiple
between $(n_{1}+1,\mathellipsis,n_{k}+1)$, then

$c_{i}=\frac{G}{n_{i}+1}\text{ for }i=1,\mathellipsis,k.$

######  Remark 1.1.

The homogeneous spaces
$\big{(}\mathds{C}\mathrm{P}^{n_{1}}\times\dots\times\mathds{C}\mathrm{P}^{n_{k}},q(c_{1}g_{FS}\oplus\dots\oplus
c_{k}g_{FS})\big{)}$ are embedded into
$\mathds{C}\mathrm{P}^{{n_{1}+qc_{1}\choose qc_{1}}\cdots{n_{k}+qc_{k}\choose
qc_{k}}-1}$. A Kähler embedding can be explicitly described through a
composition of suitable normalizations of Veronese embeddings:

$\displaystyle(\mathds{C}\mathrm{P}^{n},cg_{FS})$
$\displaystyle\to(\mathds{C}\mathrm{P}^{\binom{n+c}{c}-1},g_{FS})$
$\displaystyle[Z_{i}]_{0\leq i\leq n}$
$\displaystyle\mapsto\sqrt{\frac{(c-1)!}{c^{c-2}}}\left[\frac{Z_{0}^{c_{0}}\mathellipsis
Z_{n}^{c_{n}}}{\sqrt{c_{0}!\mathellipsis
c_{n}!}}\right]_{c_{0}+\mathellipsis+c_{n}=c},$

together with a Segre embedding (cfr. [6, 16]).

Our main result is

######  Proposition 1.2.

A $n$-complex dimensional Kähler-Einstein manifold endowed with a
${S}^{1}$-invariant projectively induced metric is an open subset of a compact
toric Kähler manifold (i.e. a Kähler manifold equipped with an effective
Hamiltonian action of the $n$-dimensional real torus).

######  Remark 1.3.

A simple example of non-toric compact ${S}^{1}$-invariant Kähler-Einstein
manifold can be the compact complex torus $\mathbb{C}^{n}/\mathds{Z}^{n}$
together with the metric induced by the Euclidean metric on $\mathbb{C}^{n}$.

From Arezzo-Loi-Zuddas’ classification of lower dimensional compact Fano toric
manifolds admitting a Kähler-Einstein projectively induced metric (see
Subsection 2.1), we get the following corollary, namely we proved that
Conjecture 2 holds true for manifolds with dimension less than five.

######  Corollary 1.4.

Let $M$ be a projectively induced and ${S}^{1}$-invariant Kähler-Einstein
manifold with complex dimension less than 5. Then $M$ is an open subsets of
one of the following manifolds endowed with an integer multiple of the metric
below indicated:

  * •

if $\dim M=2$, $(\mathds{C}\mathrm{P}^{2},g_{FS})\text{ or
}(\mathds{C}\mathrm{P}^{1}\times\mathds{C}\mathrm{P}^{1},g_{FS}\oplus
g_{FS});$

  * •

if $\dim M=3$, $(\mathds{C}\mathrm{P}^{3},g_{FS}),\
(\mathds{C}\mathrm{P}^{2}\times\mathds{C}\mathrm{P}^{1},2g_{FS}\oplus
3g_{FS})\text{ or }$
$(\mathds{C}\mathrm{P}^{1}\times\mathds{C}\mathrm{P}^{1}\times\mathds{C}\mathrm{P}^{1},g_{FS}\oplus
g_{FS}\oplus g_{FS});$

  * •

if $\dim M=4$, $(\mathds{C}\mathrm{P}^{4},g_{FS}),\
(\mathds{C}\mathrm{P}^{3}\times\mathds{C}\mathrm{P}^{1},g_{FS}\oplus
2g_{FS}),\
(\mathds{C}\mathrm{P}^{2}\times\mathds{C}\mathrm{P}^{2},g_{FS}\oplus g_{FS}),\
(\mathds{C}\mathrm{P}^{2}\times\mathds{C}\mathrm{P}^{1}\times\mathds{C}\mathrm{P}^{1},2g_{FS}\oplus
3g_{FS}\oplus 3g_{FS}),\text{ or
}(\mathds{C}\mathrm{P}^{1}\times\mathds{C}\mathrm{P}^{1}\times\mathds{C}\mathrm{P}^{1}\times\mathds{C}\mathrm{P}^{1},g_{FS}\oplus
g_{FS}\oplus g_{FS}\oplus g_{FS}).$

The paper is organized as follow. We are going to recall the definition and
some basic facts about Calabi’s diastasis function and Bochner’s coordinates
that will be used in the proof of Lemmas 2.1 and 2.2. This two auxiliary
lemmas are particularly important in the proof of compactness and simple
connectedness of complete Kähler-Einstein ${S}^{1}$-invariant projectively
induced manifolds (Lemma 2.3), topological properties that turn out to be
fundamental in the construction of an integrable distribution, used in the
proof of Proposition 1.2. Finally, the proof of Corollary 1.4 is given.

##  2\. Proof of Proposition 1.2 and Corollary 1.4

It is worth to point out that the condition of being a Kähler-Einstein metric
reads locally, with respect to any local holomorphic coordinates, as a
nonlinear overdetermined system of fourth-order elliptic PDEs. From regularity
results for elliptic PDEs, we have the real-analyticity of any Kähler
potential (see e.g. [3]). We easily see that we can assume without loss of
generality that the ${S}^{1}$-invariant Kähler potential $\Phi$ is the so-
called Calabi’s diastasis function (cfr. [4, 6, 16]), namely the only Kähler
potential whose Taylor expansion does not contain zero and first order terms.
Moreover, in our case, coordinates $z=(z_{1},\mathellipsis,z_{n})$ can be
chosen in order to have the following

$\Phi(z)=\sum_{\alpha=1}^{n}|z_{\alpha}|^{2}+\psi_{2,2},$

where $\psi_{2,2}$ is a power series with degree $\geq 2$ in both $z$ and
$\bar{z}$. Such coordinates are uniquely defined up to unitary transformations
(see [16]) and they are called Bochner’s coordinates with respect to the
origin of the open set $U\subseteq\mathbb{C}^{n}$, where is defined $\Phi$
(cfr. (1)).

######  Lemma 2.1.

Let $\Phi$ be a ${S}^{1}$-invariant Kähler potential of a projectively induced
${S}^{1}$-invariant Kähler metric. Then

(2)
$\Phi=\log\left(1+\sum_{j=1}^{n}|z_{j}|^{2}+\sum_{j=n+1}^{N}a_{j}|z^{m_{j}}|^{2}\right)$

where $a_{j}\in\mathbb{R}^{+}$, $m_{i}\in\mathbb{N}^{n}$ and $m_{j}\neq m_{i}$
for $j\neq i$.

###### Proof.

Let $f:(M,g)\to(\mathds{C}\mathrm{P}^{N},g_{FS})$ be a Kähler immersion. We
are going to assume without loss of generality that $f$ is full, namely $f(M)$
is not contained into any $\mathds{C}\mathrm{P}^{h}$ with $h<N$. Let
$w=(z_{1},\mathellipsis,z_{\dim M})$ be Bochner’s coordinates on $V\subseteq
M$. By [6], there exists a Bochner’s coordinate system
$(w_{1},\mathellipsis,w_{N})$ on $\mathds{C}\mathrm{P}^{N}$ such that

$w_{1}|_{f(V)}=z_{1},\mathellipsis,w_{\dim M}|_{f(V)}=z_{\dim M},$

and

$w_{\dim M+1}|_{f(V)}=f_{\dim M+1}(z),\mathellipsis,w_{N}|_{f(V)}=f_{N}(z),$

where $f_{j}$ are holomorphic functions that vanish at the second order at the
origin.

The hereditary property of Calabi’s diastasis (see [6]) states that Calabi’s
diastasis function $D^{g}_{p}$ for $g$ around a point $p\in M$ needs to
satisfy

$D^{g}_{p}=D^{g_{FS}}_{f(p)}\circ f.$

Therefore, we have that $\Phi$ needs to assume the following form

$\Phi=\log\left(1+\sum_{j=1}^{n}|z_{j}|^{2}+\sum_{j=n+1}^{N}|f_{j}(z)|^{2}\right).$

Since $\Phi$ is ${S}^{1}$-invariant and $f$ is full, we get our statement. ∎

######  Lemma 2.2.

Let $(M,g)$ be a Kähler-Einstein ${S}^{1}$-invariant manifold. Its
${S}^{1}$-invariant Kähler potential satisfy the following Monge-Ampère
equation

(3) $\det g=e^{-\frac{\lambda}{2}\Phi},$

for a suitable real constant $\lambda$.

###### Proof.

Since $g$ is a Kähler-Einstein metric, there exists a suitable
$\lambda\in\mathbb{R}$ such that

$-i\partial\bar{\partial}\log\det
g=\frac{i\lambda}{2}\partial\bar{\partial}\Phi.$

Hence, there exists a holomorphic function $f:U\to\mathbb{C}$, defined on a
suitably small open subset $U$ of $M$, such that

$\det g=e^{-\frac{\lambda}{2}\Phi+f+\bar{f}}.$

Once Bochner’s coordinates are set, by comparing the series expansions of both
sides of the previous equation, it follows $f+\bar{f}\equiv 0$. ∎

######  Lemma 2.3.

A Kähler-Einstein manifold endowed with a ${S}^{1}$-invariant projectively
induced metric is an open subset of a compact and simply connected manifold.

###### Proof.

Since the previous equation (3) can be seen as an equality between polynomials
in $z$ and $\bar{z}$, by comparing their degrees, we get $\lambda$ needs to be
positive (for details see [17]), namely $M$ is a Fano manifold.

D. Hulin proved in [12] that every Kähler-Einstein manifold Kähler immersed
into a complex projective space can be extended to a complete Kähler-Einstein
manifold, Kähler immersed into the same complex projective space. Therefore,
by Bonnet-Myers’ theorem, $M$ needs to be compact. Moreover, every compact
Kähler manifold with positive definite Ricci tensor is simply connected by
[14]. ∎

From now on, we are going to consider without loss generality only metrics
defined on complete, compact and simply connected manifolds.

###### Proof of Proposition 1.2.

Let $\phi:U\cap(\mathbb{C}\setminus\\{0\\})^{n}\to\mathbb{R}$ such that

$2\phi(\log{|z_{1}|^{2}},\mathellipsis,\log{|z_{n}|^{2}})=\Phi\left(|z_{1}|^{2},\dots,|z_{n}|^{2}\right).$

Chosen any branch of the complex logarithm, we set holomorphic coordinates
$w_{i}=\log z_{i}$. Hence, the Kähler form reads locally as222We are going to
use the Einstein convention on repeated indices.

$\omega=i\partial\bar{\partial}\phi=i\frac{\partial^{2}\phi}{\partial
w_{k}\partial\bar{w}_{j}}dw_{k}\wedge
d\bar{w}_{j}=\frac{1}{2}\frac{\partial^{2}\phi}{\partial r_{k}\partial
r_{j}}dr_{k}\wedge d\theta_{j},$

where $w_{k}=r_{k}+i\theta_{k}$. Therefore,

$g=\phi^{kj}(r)dr_{k}dr_{j}+\phi^{kj}(r)d\theta_{k}d\theta_{j},$

where $\phi^{kj}(r)=\frac{\partial^{2}\phi}{\partial r_{k}\partial r_{j}}$. We
easily see that $\partial_{\theta_{k}}$ are Killing vector fields on $U$.
Being $M$ simply connected and real analytic, each Killing vector field can be
extended to a unique Killing vector field defined on the whole manifold $M$
(see [19] theorems 1 and 2). We denote with $X_{k}$ the global extension of
$\partial_{\theta_{k}}$. We recall that every Killing vector field on a
compact Kähler manifold is real holomorphic (see e.g. [18] prop. 9.5), i.e. if
we identify $TM$ with $T^{(1,0)}M\subset TM\otimes\mathbb{C}$ in the usual way
(namely by identifying $Z\in TM$ with $\tilde{Z}=(Z-iJZ)/2\in T^{(1,0)}M$), we
have that $\tilde{X}_{k}=(X_{k}-iJX_{k})/2$ is holomorphic.

Since $M$ is complete, we get a holomorphic and isometric action of
$\mathbb{R}$ on $M$ by means of the flows of the each Killing vector field
$X_{k}$. Furthermore, for any $1\leq k,j\leq n$, $[X_{k},X_{j}]$ is a Killing
vector field vanishing on $U$. Since Killing vector fields (different from the
identically zero vector field) vanish on totally geodesic submanifolds of real
codimension at least 2 (cfr. [15]), the commutator $[X_{k},X_{j}]$ needs to
vanish everywhere on $M$. Therefore, we have a transitive holomorphic and
isometric action $\mathcal{G}$ of $\mathbb{R}^{n}$ on $M$.

Let $V\subset M$ be the subset where at least a Killing vector field $X_{k}$
vanishes. By considering that $M\setminus V$ consists of and only of maximal
dimensional orbits of the action $\mathcal{G}$, such action can be restricted
to $M\setminus V$. We easily see that $\mathds{Z}^{n}$ is the stabilizer of
$\mathcal{G}$ in $(M\setminus V)\cap U$. Hence, we have a holomorphic and
isometric action of the real torus $\mathbb{R}^{n}/\mathds{Z}^{n}$ on
$M\setminus V$. Since stabilizers of $\mathcal{G}$ in points belonging to $V$
contain $\mathds{Z}^{n}$, this toric action can be extended to the whole
manifold $M$.

Every Killing vector field on a compact and simply connected Kähler manifold
$M$, is Hamiltonian. Indeed, since these vector fields are also real
holomorphic because $M$ is compact, they need to be symplectic too, i.e.
$i_{X_{k}}\omega$ is closed. Being $M$ simply connected,
$H^{1}_{\text{dR}}(M)=0$. Therefore, $i_{X_{i}}\omega$ needs to be also exact.
Therefore, the action of $\mathbb{R}^{n}/\mathds{Z}^{n}$ is Hamiltonian333 We
easily see that the action is effective. Indeed, if $p\in(M\setminus V)\cap
U$, then $g\cdot p\neq p$ for every $g\in\mathbb{R}^{n}/\mathds{Z}^{n}$..

The existence of such action allow us to conclude that $M$ is a toric Kähler
manifold. ∎

###### Proof of Corollary 1.4.

C. Arezzo, A. Loi, F. Zuddas proved in [1] that the only compact toric Fano
manifolds with dimension less than five, that can be endowed with a Kähler-
Einstein projectively induced metric, are (or can be decomposed as a product
of) complex projective spaces with multiples of the Fubini-Study metric. By
[6], only integer multiples of the Fubini-Study metric are projectively
induced. Therefore, we get the statement of Corollary 1.4 by straightforward
computations of normalization coefficients necessary to obtain an Einstein
metric on the product of complex projective spaces.∎

In the following section we provide a sketch of the technique used by C.
Arezzo, A. Loi, F. Zuddas to get their result.

###  2.1. Arezzo-Loi-Zuddas’ result

Let $H_{k}$ be a real function on $M$ such that

$i_{X_{k}}\omega=-dH_{k},$

where $X_{k}$ are symplectic vector fields related to the toric action on $M$
as described in the proof of Proposition 1.2. A moment map
$\mu:M\to\mathbb{R}^{n}$ for such action can be written as

$\mu(x)=\left(H_{1}(x),\mathellipsis,H_{n}(x)\right).$

By Atiyah and Guillemin-Sternberg convexity theorems [2, 11], the imagine of a
compact symplectic manifold through a moment map for a Hamiltonian action of a
torus, is the convex hull444The smallest convex set of $\mathbb{R}^{n}$
containing such points. of the images of the fixed points of the action.
Moreover, T. Delzant proved (see [7] or [8]) that momentum images of toric
symplectic manifolds need to satisfy some additional properties, namely they
turn out to be the so-called Delzant polytopes. We recall that a polytope is a
convex subset of $\mathbb{R}^{n}$ that can be represented as intersection of a
finite number of half-spaces.

######  Definition 1.

Let $\Gamma$ be the group of maps $\mathbb{R}^{n}\to\mathbb{R}^{n}$ of the
form $x\mapsto Ax+b$, where $A\in\mathrm{SL}_{n}(\mathds{Z})$ and
$b\in\mathbb{R}^{n}$. A Delzant polytope $\Delta$ is a convex polytope such
that a neighbourhood of any vertex of $\Delta$ is equivalent via $\Gamma$ to a
neighbourhood of the origin of the infinite polytope
$(\mathbb{R}^{+}_{0})^{n}$. Moreover, all vertices of the polytope are
integers.

Figure 1. Examples of Delzant polytopes related to compact complex toric
surfaces.

Vice versa, we have to recall that there exists a canonical way to associate
every Delzant polytope with a compact smooth projective variety, containing a
non-compact complex torus $(\mathbb{C}^{*})^{n}$ as an open dense subset and
whose natural action of $(\mathbb{C}^{*})^{n}$ on itself can be
holomorphically extended to the whole manifold (see e.g. [8]). Moreover,
Delzant showed in [7] how such manifolds can be canonically endowed with a
symplectic structure so that the trivial action of the real torus on that
projective variety is Hamiltonian and moment imagine is the given Delzant
polytope. Furthermore, he proved the existence of a bijective correspondence
between compact symplectic toric manifolds (up to symplectomorphisms) and
Delzant polytopes (up to transformations in $\Gamma$).

Hence, from now on, we can consider without loss of generality _only Delzant
polytopes having a vertex in the origin of $\mathbb{R}^{n}$ and its edges
meeting at the origin agreeing with the positive semi axes of
$\mathbb{R}^{n}$_ (see. e.g. Figure 1).

The following lemma has a key role in Arezzo-Loi-Zuddas’ proof.

######  Lemma 2.4 (C. Arezzo, A. Loi, F. Zuddas [1]).

Let $M$ be a compact toric Kähler-Einstein manifold with Delzant polytope
$\Delta$ and endowed with a projectively induced metric. Then, on the open
dense subset of $M$ consisting of all maximal dimensional orbits of the toric
action, is defined a local Kähler potential reading as (2), i.e.

$\log\left(1+\sum_{j=1}^{n}|z_{j}|^{2}+\sum_{j=n+1}^{N}a_{j}|z^{m_{j}}|^{2}\right).$

A coefficient $a_{i}$ is different from zero if and only if the corresponding
multi-index $m_{i}$ belongs to $\Delta\cap\mathds{Z}^{n}$.

Compact toric Fano manifolds with dimension less than five have been
completely classified (see e.g. [5]). Up to homotheties, Delzant polytopes of
such manifolds admitting a Kähler-Einstein metric (different from complex
projective spaces with the Fubini-Study metric and that cannot be decomposed
as a product of lower dimensional manifolds) are the following

  * •

$\\{(x,y)\in\mathbb{R}^{2}|\ 0\leq x,y\leq 2,\ -1\leq x-y\leq 1\\}$;

  * •

$\\{(x,y,z)\in\mathbb{R}^{3}|\ x,y\geq 0,\ 0\leq z\leq 2,\ x+z\leq 3,\ y-z\leq
1\\}$;

  * •

$\\{(x,y,z,w)\in\mathbb{R}^{4}|\ 0\leq x,y,z,w\leq 2,\ -1\leq x+y-z-w\leq
1\\}$;

  * •

$\\{(x,y,z,w)\in\mathbb{R}^{4}|\ x,y,\geq 0,\ 0\leq z,w\leq 2,\ -1\leq z-w\leq
1,\ x-z\leq 1,\ y+z\leq 3\\}$;

  * •

$\\{(x,y,z,w)\in\mathbb{R}^{4}|\ x,y,z,w\geq 0,\ x-z\leq 1,\ y-w\leq 1,\
z+w\leq 3,\ x+y\leq 1,\ z+w-x-y\leq 1\\}$.

On one hand, the Calabi’s diastasis function of a projectively induced Kähler-
Einstein metric on the previous toric manifold needs to assume the particular
form stated by Lemma 2.4. On the other, it should also be a solution of the
Monge-Ampère equation (3). In [1], the authors showed via direct computations
that in all the previous cases, these two fact lead to a contradiction.

Moreover, the authors proved that manifolds related to Delzant polytopes
$k\Delta$, differing from the polytope $\Delta$ for a homothety of ratio $k$,
can be endowed with a Kähler-Einstein projectively induced metric if and only
if the manifold related to Delzant polytope $\Delta$ can. Hence, compact toric
Fano manifold with a Kähler-Einstein projectively induced metric cannot be
different from a product of complex projective spaces with Fubini-Study
metrics.

###  2.2. Final remark

The absence of a classification of compact Fano toric manifolds with complex
dimension greater than four, precludes the possibility of extending the
Arezzo-Loi-Zuddas’ approach to higher dimensional cases.

In [17], G. Manno and F. Salis proposed a different approach, based on the
classification of special polynomial solutions of the Monge-Ampère equation
describing the Einstein condition. In this way, the authors classified
${S}^{1}$-invariant projectively induces Kähler-Einstein manifolds with
complex dimension two, independently from Arezzo, Loi and Zuddas. It is an
open problem whether this approach can be generalized to any dimension.

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