Projectively induced Kähler cones over regular Sasakian manifolds

Abstract

Motivated by a conjecture in Loi et al. (Math Zeit 290:599–613, 2018) we prove that the Kähler cone over a regular complete Sasakian manifold is Ricci-flat and projectively induced if and only if it is flat. We also obtain that, up to $\mathcal D_a$—homothetic transformations, Kähler cones over homogeneous compact Sasakian manifolds are projectively induced. As main tool we provide a relation between the Kähler potentials of the transverse Kähler metric and of the cone metric.

1 Introduction

In the celebrated work [4], E. Calabi gives an algebraic criterion to check when a Kähler manifold admits a Kähler (i.e. holomorphic and isometric) immersion into a complex space form. Of particular interest is the elliptic case, namely when the complex space form is the complex projective space endowed with the Fubini–Study metric, where many questions are still open. Throughout this paper $\mathbb {C}\textrm{P}^N$ is always implicitly understood, as a Kähler manifold, equipped with the Fubini–Study metric. We say that a Kähler manifold is projectively induced if it admits a local Kähler immersion into the complex projective space $\mathbb {C}\textrm{P}^{N\le \infty }$. Obviously many examples of projectively induced manifolds can be constructed pulling–back the Fubini–Study metric on complex submanifolds. However, it is much more difficult to find examples of projectively induced manifolds with prescribed curvature. In [12] D. Hulin proves that the scalar curvature of a compact Kähler–Einstein submanifold of the complex projective space is forced to be positive. It is important to highlight that when the submanifold is compact the ambient space $\mathbb {C}\textrm{P}^N$ can be taken to be finite dimensional. In fact, when the ambient space is infinite-dimensional, Hulin’s result does not hold, as there are examples of Kähler–Einstein submanifolds of $\mathbb {C}\textrm{P}^\infty$ with negative scalar curvature (see [9]). Furthermore, the complex flat space $\mathbb {C}^n$ is an example of Kähler submanifold of $\mathbb {C}\textrm{P}^\infty$. Observe that by the rigidity result of Calabi [4, Theorem 9], $\mathbb {C}^n$ does not admit a Kähler immersion in $\mathbb {C}\textrm{P}^{N<\infty }$. Actually, in the recent work [1] C. Arezzo, C. Li and A. Loi prove that no $\mathbb {C}\textrm{P}^{N<\infty }$ admits Ricci-flat Kähler submanifolds. On the other hand, when the ambient space is infinite-dimensional, it is still an open problem to understand whether the flat metric is the only example of Ricci-flat projectively induced Kähler metric, as conjectured by Loi et al. in [8] (see also [10] for a class of metrics that confirms such conjecture).

In this paper we address the problem of studying Sasakian manifolds whose Kähler cone is Ricci-flat and projectively induced. We recall that the Kähler cone over a Sasakian manifold is Ricci-flat if and only if the Sasakian manifold is Sasaki–Einstein, and it is flat if and only if the Sasakian manifold is a standard sphere (see Theorem 8 below). More precisely we prove the following.

Theorem 1

The Kähler cone over a regular complete Sasakian manifold is Ricci-flat and projectively induced if and only if it is flat.

Theorem 1 should be compared to the results in [2, 5, 7], where the existence of a Sasakian immersion of a Sasakian manifold into a Sasakian space form is investigated either in terms of the existence of a Kähler immersion of the Kähler cone above or in terms of the transverse Kähler metric below. A key step in their approach consists in observing that a compact Sasakian manifold immersed into a regular Sasakian manifold is itself regular, as follows from [11, Prop. 3.1] (see also [2, Prop. 5] for a generalization to the non–compact case). More precisely, they do not need to assume a priori the regularity of the Sasakian manifold since they obtain it by assuming a Sasakian immersion into a (regular) Sasakian space form. In particular, when the Sasakian space form is elliptic one can lift the Sasakian map to the Kähler cone in a natural way, obtaining that a Sasakian immersion into a sphere is possible if and only if a Kähler immersion of the Kähler cone into the complex Euclidean space occurs. When the Sasakian space form is not elliptic its Kähler cone is not a complex space form, but using regularity it is possible to relate the existence of a Sasakian immersion to a Kähler immersion of the transverse Kähler metric below. In our approach instead, we need to impose the regularity of the Sasakian manifold, since no Sasakian immersion is assumed to exist. More precisely, we cannot deduce the existence of a Sasakian immersion from the Kähler cone being projectively induced, since the complex projective space is not a cone over a Sasakian manifold.

The novelty of our paper consists in relating the existence of a Kähler immersion of the Kähler cone over a regular Sasakian manifold into the (infinite-dimensional) complex projective space, to the existence of a Kähler immersion of the transverse Kähler manifold below into a complex projective space (Proposition 11 below).

Finally, we show that by performing a $\mathcal D_a$–homothetic transformation, namely a rescaling of the structure (see Formula (4) below), we can find many examples of projectively induced Kähler cones over homogeneous Sasakian compact manifolds. More precisely, as a consequence of Proposition 11 we get the following:

Theorem 2

Let $(S,\xi ,\eta ,\Phi ,g)$ be a homogeneous compact Sasakian manifold. Then, up to a $\mathcal D_a$–homothetic transformation, its Kähler cone is projectively induced.

For other results concerning immersions of homogeneous Sasakian manifolds the reader is referred to [5, Theorem 4], where it is proved that up to a $\mathcal D_a$-homothetic deformation a compact homogeneous Sasakian manifold admits a Sasakian immersion into the Sasakian sphere if and only if its fundamental group is cyclic (see also [7, Theorem 1.5] for a noncompact version). It is worth pointing out that from Theorem 2 we obtain examples of $\eta$–Einstein Sasakian manifolds whose Kähler cone is projectively induced, in contrast to the Sasaki–Einstein case studied in Theorem 1. This occurs since the Sasaki–Einstein condition is very rigid and it transforms to $\eta$–Einstein under $\mathcal D_a$-homothetic transformations.

The paper is organized as follows. Sections 2 and 3 are devoted to recalling the main definitions and results we need, respectively in the Kähler and the Sasakian contexts. In Sect. 4 we describe the relation between the Kähler potential on the cone over a Sasakian manifold and that of the transverse metric. Finally in Sect. 5 we prove the main theorems.

2 Projectively induced Kähler manifolds

We start by recalling some well-known facts about projectively induced Kähler manifolds and by fixing some notation. Let (M, g) be an n-dimensional Kähler manifold and denote by $\omega$ its associated (1, 1)-form. Recall that, given any point $p\in M$, the condition of being Kähler is equivalent to the existence, in a neighborhood $U_p$ of p, of a real valued function $\varphi \!:U_p\rightarrow \mathbb {R}$ called Kähler potential for $\omega$ such that $\omega _{|_{U_p}}=\frac{i}{2}\partial \bar{\partial }\varphi$. A Kähler potential is not unique, as any other function obtained from $\varphi$ by adding the real part of a holomorphic function is also a Kähler potential. We say that (M, g) is projectively induced if and only if for all $p\in M$ there exists a holomorphic and isometric (i.e. Kähler) immersion $f\!:U_p\rightarrow \mathbb {C}\textrm{P}^N$, $N\le \infty$, such that $f^*\omega _{FS}=\omega$. Here $\mathbb {C}\textrm{P}^\infty$ denotes the infinite-dimensional complex projective space, namely the set of equivalence classes with the usual equivalence relation of points $(Z_0,Z_1,\dots )\in l^2(\mathbb {C})\setminus \{0\}$, endowed with the Fubini–Study metric $\omega _{FS}$ (see [4]), where by $l^2(\mathbb {C})$ we denote the Hilbert space of sequences of complex numbers $z=(z_0,z_1,z_2,\dots )$ limited in norm, i.e. $\sum _{j=0}^\infty |z_j|^2<\infty$. In the sequel we will also denote by $\omega _0$ the flat metric on $\ell ^2(\mathbb {C})$, that is $\omega _0=\frac{i}{2}\partial \bar{\partial }||z||^2=\frac{i}{2}\sum _{j=0}^\infty dz_j\wedge d\bar{z}_j$. Recall that if $[Z_0:\dots :Z_j:\dots ]$ are homogeneous coordinates and $(z_1,\dots , z_j,\dots )$, $z_j:=\frac{Z_j}{Z_0}$, are affine coordinates on $U_0:=\{Z_0\ne 0\}$, the Fubini–Study metric reads on $U_0$ as

$$\begin{aligned} \omega _{FS{}_{|_{U_0}}}:=\frac{i}{2}\partial \bar{\partial }\log (1+||z||^2). \end{aligned}$$

We can restrict ourselves to considering real analytic Kähler metrics, since the pull-back by a holomorphic map of the Fubini–Study metric is necessarily real analytic and Kähler. Thus, one can define the diastasis function $D\!:U_p\times {U_p}\rightarrow \mathbb {R}$ by:

$$\begin{aligned} D(z,w):=\tilde{\varphi }(z,\bar{z})+\tilde{\varphi }(w,\bar{w})-\tilde{\varphi }(z,\bar{w})-\tilde{\varphi }(w,\bar{z}), \end{aligned}$$

where $\tilde{\varphi }$ is the function obtained by extending the Kähler potential $\varphi$ on a neighborhood of the diagonal in $U_p\times \overline{U_p}$ (here $\overline{U}_p$ denotes the neighborhood defined by conjugated local coordinates). It is easy to see that fixing one of the two variables, the function $D_w(z):=D(z,w)$ is a Kähler potential for g, called diastasis function centered at w. The diastasis function has been introduced by Calabi in [4] for its property of being—once one of the variables is fixed—a Kähler potential invariant by pull-backs by holomorphic maps, i.e. if $(\tilde{M},\tilde{\omega })$ is a second Kähler manifold and $f\!:U_p\rightarrow \tilde{M}$ is holomorphic and such that $f^*\tilde{\omega }=\omega$ we have:

$$\begin{aligned} \frac{\partial ^2 D_p(z)}{\partial z_j\partial \bar{z}_k}=\frac{\partial ^2 (\tilde{D}_{f(p)}\circ f)(z)}{\partial z_j\partial \bar{z}_k} \end{aligned}$$

where we denote by $D_p(z)$ the diastasis function of M on $U_p$ centered at p and by $\tilde{D}_{f(p)}$ the diastasis function for $\tilde{M}$ in a neighborhood of f(p).

Let us denote by $(b_{jk})$ the $\infty \times \infty$ matrix of coefficients of the following power expansion:

$$\begin{aligned} e^{D_p(z)}-1:=\sum _{j,k=0}^\infty b_{jk}z^{m_j}\bar{z}^{m_k}, \end{aligned}$$

(1)

where we use a multi–index notation $m_j=(m_{j,1},\dots , m_{j,n})$, $z^{m_j}=z_1^{m_{j,1}}\cdots z_n^{m_{j,n}}$, and, setting $|m_j|:=m_{j,1}+\cdots +m_{j,n}$, the strings $m_j$’s are ordered in such a way that $m_0=(0,\dots ,0)$, $|m_j|\le |m_k|$ for $j<k$, and when $|m_j|=|m_k|$ the order can be taken e.g. to be lexicographic. Furthermore,

$$\begin{aligned} b_{jk}:=\frac{1}{m_j!m_k!}\frac{\partial ^{|m_j|+|m_k|}}{\partial z^{m_j}\partial \bar{z}^{m_k}}\left( e^{D_p(z)}-1\right) , \end{aligned}$$

(2)

where $m_j!=m_{j,1}!\cdots m_{j,n}!$.

The importance of the diastasis function relies on the following theorem, which summarizes the results for projectively induced metrics proved by Calabi in his celebrated work [4].

Theorem 3

(Calabi [4, Theorems 8 and 9]) Let (M, g) be a real analytic Kähler manifold. Then, (M, g) admits a local Kähler immersion into $\mathbb {C}\textrm{P}^N$ if and only if the matrix $(b_{jk})$ defined by (1) is positive semidefinite of rank at most N. Furthermore, the Kähler immersion is unique up to unitary transformations of the ambient space.

Remark 4

It is worth pointing out that the condition of being positive semidefinite as well as the rank of $(b_{jk})$ do not depend on the point p but only on the metric g. Furthermore, the rank of $(b_{jk})$ is exactly N when the immersion is full, i.e. when the image is not contained in any lower dimensional complex projective space. Observe also that when M is simply connected, the local Kähler immersions glue together to a global Kähler immersion of the whole M (see [4]). Finally a Kähler immersion of a compact Kähler manifold into $\mathbb {C}\textrm{P}^\infty$ is never full, as the immersion map is constructed using a basis of the space of global holomorphic sections of some holomorphic line bundle, and when M is compact such space is finite dimensional.

In this paper we are interested in the case when $(M,\omega )$ is compact and Kähler–Einstein (from now on KE). If we denote by $\rho$ the Ricci form associated to $\omega$, which can be written as $\rho =-i\partial \bar{\partial }\log \det g$, then the KE condition reads $\rho =\lambda \omega$, for $\lambda \in \mathbb {R}$.

In [12] D. Hulin studied compact projectively induced Kähler–Einstein manifolds proving the following:

Theorem 5

([12, Theorem 7.1]) Let (M, g) be a compact projectively induced Kähler–Einstein manifold of complex dimension n. Then M is simply connected, the immersion is global and the Einstein constant $\lambda$ is rational. Furthermore, if we write $\lambda = 2p/q > 0$, where p/q is irreducible, then $p \le n + 1$, and:

• (i) if $p = n + 1$, then $(M,g) = (\mathbb {C}\textrm{P}^n,qg_{FS})$;

• (ii) if $p = n$, then $(M,g) = (Q_n,qg_{FS})$, where $Q_n$ denotes the quadric in $\mathbb {C}\textrm{P}^{n+1}$ of homogeneous equation $Z_0^2+\cdots +Z_{n+1}^2=0$.

3 Preliminaries on Sasakian geometry

We devote this section to recall the main definitions and results about Sasakian manifolds that we will need in the following. We mainly refer to [3, 13] and references therein.

Definition 6

A Riemannian manifold (S, g) is Sasakian if and only if its metric cone $(C(S), \bar{g})$ is Kähler, where $C(S):=S\times \mathbb {R}^+$ and $\bar{g}:= dt^2+t^2g$, for $t\in \mathbb {R}^+$.

It follows that $\dim _{\mathbb {R}}S=2n+1$, where $\dim _{\mathbb {C}}(C(S))=n+1$, and the Kähler structure on the cone induces on S:

• (1) a contact 1-form $\eta$, i.e. $\eta \wedge (d\eta )^n\ne 0$, that is the restriction to S of the form $d^c\log t$ on the cone.

• (2) a vector field $\xi$, uniquely defined by the conditions $\eta (\xi )=1$ and $d\eta (\xi ,\,\cdot \,)=0$, named Reeb vector field of $\eta$;

• (3) a $(1,1)-$tensor field $\Phi$ which satisfies $\Phi ^2=-\mathbb {I}+\xi \otimes \eta$, that is the restriction to the contact distribution $\ker \eta$ of the complex structure of the cone.

It follows that for any vector fields X, Y on S we have

$$\begin{aligned} g(\Phi (X),\Phi (Y))=g(X,Y)-\eta (X)\eta (Y), \end{aligned}$$

and

$$\begin{aligned} d\eta (\Phi X,\Phi Y)=d\eta (X,Y),\quad d\eta (\Phi X,X)>0,\;\forall X\ne 0. \end{aligned}$$

Let us denote by $\mathcal F$ the Reeb foliation defined by the integral curves of the Reeb vector field $\xi$. If we denote by $T_\mathcal {F}$ the tangent bundle to $\mathcal F$, the tangent bundle to S splits canonically as $TS=\mathcal D\oplus T_{\mathcal F}$, where $\mathcal D:=\text {ker}\,\eta$ is the codimension one subbundle, with natural almost complex structure defined by $J:=\Phi |_\mathcal {D}$. Accordingly, the metric g decomposes as $g=g^T+ \eta \otimes \eta$, where $g^T$ is a metric on the contact distribution called transverse Kähler metric, which can be globally identified with $g^T(\cdot ,\cdot )=d\eta (\cdot ,\Phi \cdot )$.

A Sasakian manifold is said to be regular, quasi-regular or irregular if the foliation defined by the Reeb vector field has the corresponding property, i.e. respectively each leaves of the Reeb foliation intersect every neighborhood of a point exactly one time, a finite number of times or neither of the previous two cases. In this paper we are interested in compact regular Sasakian manifolds, for which we recall the following fundamental result (see e.g. [3, Theorem 7.5.1]):

Theorem 7

(Structure Theorem) Let $(S,\xi ,\eta ,\Phi ,g)$ be a compact regular Sasakian manifold. Then the space of leaves of the Reeb foliation $(X,\omega )$ is a compact Kähler manifold with integral Kähler form $\frac{1}{2\pi }\omega$ so that the projection $p:(S,g)\rightarrow (X,g_\omega )$ is a Riemannian submersion, where $g_\omega$ is the metric associated to $\omega$.

Viceversa, any principal $S^1$-bundle S with Euler class $-\frac{1}{2\pi }[\omega ]\in H^2(X,\mathbb {Z})$ over a compact Kähler manifold $(X,\omega )$ admits a Sasakian structure.

We will call $(X,g_\omega )$ in Theorem 7 the Kähler manifold associated to S.

This theorem allows us to describe the Kähler structure of the cone over S through the dual of a positive line bundle over X in the following way (see e.g. [7, p. 6]). Assume that S is a principal circle bundle $p\!:S\rightarrow X$ over a compact Hodge Kähler manifold $(X,\omega )$, namely a compact manifold X with integral Kähler form $\omega$, such that $p^*\omega =\frac{1}{2}d\eta$. The Kähler class of $\omega$ defines an ample line bundle L over X such that $C(S)=L^{-1}\setminus \{0\}$, where $L^{-1}$ is the line bundle dual to L. Furthermore, there exists a Hermitian metric h on L such that $\omega =-i\partial \bar{\partial }\log h$ and the dual metric $h^{-1}$ on $L^{-1}$ defines the radial coordinate of the cone in the following way:

$$\begin{aligned} t\!:L^{-1}\setminus \{0\}\rightarrow \mathbb {R}^+,\quad (x,v)\mapsto |v|_{h_x^{-1}} \ \ (v\in L_x^{-1}). \end{aligned}$$

(3)

The Kähler metric $\bar{g}$ on the cone has Kähler form $\Omega =\frac{i}{2}\partial \bar{\partial }t^2$.

We recall that for $a>0$, by a $\mathcal {D}_a$-homothetic trasformation of $(S,\xi ,\eta ,\Phi ,g)$ we mean a change of the structure tensors in the following way (see [15]):

$$\begin{aligned} \eta _a=a\eta ,\quad \xi _a=\frac{1}{a}\xi ,\quad \Phi _a=\Phi , \quad g_a=ag+a(a-1)\eta \otimes \eta . \end{aligned}$$

(4)

It is important to notice that if we perform a $\mathcal D_a$-homothetic transformation of the Sasakian structure $(\xi ,\eta ,\Phi ,g)$ over S, then the new Sasakian structure is still regular and the complex manifold below is still X, but its Kähler form $\omega _a$ is $\omega$ rescaled by a, so that $p^*\omega _a=ap^*\omega =\frac{a}{2}d\eta$.

Observe that setting a new coordinate ${t'}=t^a$ on the Kähler cone induces the same structure $(\xi _a,\eta _a,\Phi _a,g_a)$ on S.

We conclude this section with the following theorem (see for instance [3, Th. 11.1.3, Lemma 11.1.5, Cor. 11.1.8]) that summarizes what we need about Sasakian–Einstein manifolds, namely Sasakian manifolds whose metric g is Einstein in the Riemannian sense, i.e. $\textrm{Ric}_g=\lambda g$, for a constant $\lambda$.

Theorem 8

Let $(S,\xi ,\eta ,\Phi ,g)$ be a Sasakian manifold, then the following are equivalent:

• (1) the metric g is Sasakian–Einstein with Einstein constant 2n;

• (2) the metric $\bar{g}$ on the Kähler cone is Ricci-flat.

In addition, if S is compact with associated Kähler manifold $(X, \omega )$, then the previous conditions are equivalent to $\omega$ being Kähler–Einstein with Einstein constant $2n+2$.

4 Kähler potentials related to regular Sasakian manifolds

Consider a compact regular Sasakian manifold $(S,\xi ,\eta ,\Phi ,g)$. By Theorem 7, S is a principal circle bundle $p\!:S\rightarrow X$ over a compact Hodge Kähler manifold $(X,\omega )$ such that $p^*\omega =\frac{a}{2}d\eta$, for some $a>0$.

Lemma 9

Let $(S,\xi ,\eta ,\Phi ,g)$ be a compact regular Sasakian manifold and let $(X,\omega _a)$ be its associated Kähler manifold. Assume $p^*\omega =\frac{a}{2}d\eta$, $a>0$. If $\omega =\frac{i}{2}\partial \bar{\partial }\psi$ on a open set U around $x\in X$, then the Kähler metric $\Omega$ on the cone over S is given by $\Omega =\frac{i}{2}\partial \bar{\partial }\!\left( |z_0|^{\frac{2}{a}}e^{\frac{1}{a} \psi }\right)$ on $\mathbb {C}{\setminus }\{0\}\times U\subset C(S)$, where $z_0$ is the coordinate on $\mathbb {C}\setminus \{0\}$.

Proof

Following the line bundle construction in the previous section, S is a principal circle bundle over a compact Hodge Kähler manifold $(X,\omega )$ and we can construct an ample line bundle $\pi \!: L\rightarrow X$ over X endowed with a Hermitian metric h such that $\omega =-i\partial \bar{\partial }\log h$. Observe that $\Omega _a=\frac{i}{2}\partial \bar{\partial }t^{2}$ is the Kähler metric on the cone over the Sasakian manifold $(S,\xi _a, \eta _a,\Phi ,g_a)$, where t is given by (3). Since to move from $\eta _a$ to $\eta$ we need a $\mathcal D_{1/a}$ homothetic transformation, which corresponds on the cone to the change of variable $t'=t^{1/a}$, the Kähler metric on the cone over $(S,\xi ,\eta ,\Phi ,g)$ is $\Omega =\frac{i}{2}\partial \bar{\partial }t^{2/a}$.

Consider a trivialization $\{U,\sigma \}$ of the line bundle L, where U is an open neighborhood of $x\in X$ and $\sigma \!:U\rightarrow \pi ^{-1}(U)$, $x\mapsto (x,\sigma (x))$, is a trivializing section. Recall that $\sigma$ defines an isomorphism $\pi ^{-1}(U)\rightarrow \mathbb {C}\times U$ by $(w,x)\mapsto (\alpha _w,x)$, with $w\in L_x$, and $\alpha _w\in \mathbb {C}$ defined by $w=\alpha _w\sigma (x)$. Thus, any other section $s\!:U\rightarrow \pi ^{-1}(U)$, $x\mapsto (s(x),x)$ can be written as $s(x)=f(x)\sigma (x)$ with $f\!:U\rightarrow \mathbb {C}$ holomorphic.

On the dual bundle $\hat{\pi }\!:L^{-1}\rightarrow X$ denote by $\sigma ^{-1}$ the section dual to $\sigma$, i.e. $\sigma ^{-1}\!:U\rightarrow \hat{\pi }^{-1}(U)$ with $\sigma ^{-1}(x)\!:L_x\rightarrow \mathbb {C}$ satisfying $\sigma ^{-1}(x)(\sigma (x))=1$. By linearity, for any $w=\alpha _w\sigma (x)\in L_x$, $\sigma ^{-1}(x)(w)=\sigma ^{-1}(x)(\alpha _w\sigma (x))=\alpha _w$. As before any other section can be written as $\hat{s}(x)=\hat{f}(x)\sigma ^{-1}(x):L_x\rightarrow \mathbb {C}$. Observe that evaluating at $\sigma (x)$ we get $\hat{f}(x)=\hat{s}(x)(\sigma (x))$. In other words, in the coordinate system defined by $\sigma ^{-1}$, the coordinate of $\hat{s}(x)$ is $\hat{s}(x)(\sigma (x))$.

The dual Hermitian metric $h^{-1}$ is defined on $x\in U$ by:

$$\begin{aligned} h_x^{-1}\!:L_x^{-1}\times L_x^{-1}\rightarrow \mathbb {C},\quad h_x^{-1}(\hat{s}(x),\hat{s}(x)):=h_x(h_x^{\sharp }(\hat{s}(x)),h_x^{\sharp }(\hat{s}(x))), \end{aligned}$$

where $h^\sharp \!:L_x^{-1}\rightarrow L_x$ is the inverse of the canonical isomorphism induced by h on the fibers, $h_x^\flat \!:L_x\rightarrow L_x^{-1}$, $w\mapsto h_x(w,\cdot )$. Since $h_x^{\sharp }(\hat{s}(x))\in L_x$ we have for some $\beta \in \mathbb {C}$, $h_x^{\sharp }(\hat{s}(x))=\beta \sigma (x)$. Furthermore, the function $h_x(w,\cdot )\!: L_x\rightarrow \mathbb {C}$ applied to $\sigma (x)$ gives:

$$\begin{aligned} h_x(w,\sigma (x))=h_x(\alpha _w\sigma (x),\sigma (x))=\alpha _w h_x(\sigma (x),\sigma (x)), \end{aligned}$$

so applying $h_x^\flat (h_x^{\sharp }(\hat{s}(x)))=\hat{s}(x)$ to $\sigma (x)$ we get:

$$\begin{aligned} h_x^\flat (h_x^{\sharp }(\hat{s}(x)))(\sigma (x))=\beta h_x(\sigma (x),\sigma (x))=\hat{s}(x)(\sigma (x)). \end{aligned}$$

Thus,

$$\begin{aligned} h_x^{\sharp }(\hat{s}(x))=\frac{\hat{s}(x)(\sigma (x))}{h_x(\sigma (x),\sigma (x))}\sigma (x), \end{aligned}$$

which implies:

$$\begin{aligned} h_x^{-1}(\hat{s}(x),\hat{s}(x))=\frac{|\hat{s}(x)(\sigma (x))|^2}{h_x(\sigma (x),\sigma (x))}. \end{aligned}$$

Then in (3), if we denote by $z_0$ the coordinate of a vector $v\in L_x^{-1}$ with respect to $\sigma ^{-1}$ we have:

$$\begin{aligned} |v|^{2}_{h_x^{-1}}=h_x^{-1}(v,v)=\frac{|v(\sigma (x))|^2}{h_x(\sigma (x),\sigma (x))}=\frac{|z_0|^2}{h_x(\sigma (x),\sigma (x))}, \end{aligned}$$

and on U and $\mathbb {C}\setminus \{0\}\times U$, we have:

$$\begin{aligned} \Omega =\frac{i}{2}\partial \bar{\partial }\frac{|z_0|^{2/a}}{h_x(\sigma (x),\sigma (x))^{1/a}},\quad \omega =-\frac{i}{2}\partial \bar{\partial }\log h_x(\sigma (x),\sigma (x)). \end{aligned}$$

Notice that this construction is independent of the choice of the trivializing section, as the linear bundle is Hermitian and thus the transition functions are unitary. The conclusion follows by noticing that $\psi = \log h_x(\sigma (x),\sigma (x))^{-1}$ implies $|z_0|^{\frac{2}{a}}e^{\frac{1}{a}\psi }=\frac{|z_0|^{\frac{2}{a}}}{h_x(\sigma (x),\sigma (x))^{\frac{1}{a}}}$.$\square$

Example 1

If $\omega =\omega _{FS}$ is the Fubini–Study metric on $X=\mathbb {C}P^n$, then $\psi =\log (1+||z||^2)$ lifts to $|z_0|^2(1+||z||^2)$. The Kähler cone corresponding to $(\mathbb {C}\textrm{P}^n,\omega _{FS})$ is $(\mathbb {C}^{n+1}{\setminus }\{0\},\omega _0)$. In our notation one obtains the canonical potential for the flat metric, i.e. $|z_0|^2+||z||^2$, after performing the holomorphic change of variables $(z_0,z_1,\dots , z_n)\mapsto \left( z_0,\frac{z_1}{z_0},\dots , \frac{z_n}{z_0}\right)$.

Remark 10

It is worth pointing out that a similar construction holds in the noncompact setting. In fact, due to [7, Sec. 5] a regular noncompact Sasakian manifold is a principal bundle over a noncompact Kähler manifold $(X,\omega )$. More precisely, the fiber of a noncompact Sasakian manifold over a compact X would be $\mathbb {R}$, the principal bundle is forced to be trivial and the Kähler form on X must be exact, which is impossibile if X is compact. Thus a Sasaki noncompact manifold is either a circle bundle over a noncompact Kähler manifold $(X,\omega )$ or it is the product $X\times \mathbb {R}$. In both cases, if $\omega$ is integral, we construct a positive Hermitian line bundle (L, h) over X such that $\omega =-i\partial \bar{\partial }h$. Then Lemma 9 holds for noncompact Sasakian manifolds.

5 Projectively induced Kähler cones

We begin by proving the following proposition, interesting in its own sake and a key step in the proof of our main theorems.

Proposition 11

Let U be a contractible domain in $\mathbb {C}^n$, endowed with a Kähler metric $\omega =\frac{i}{2}\partial \bar{\partial }\psi$. Consider on $\mathbb {C}\setminus \{0\}\times U$ the Kähler metric $\Omega _c=\frac{i}{2}\partial \bar{\partial }(|z_0|^{2c}e^{c\psi })$ for a constant $c>0$, where $z_0$ is the coordinate on $\mathbb {C}{\setminus } \{0\}$. Then for any $c>0$, $\Omega _c$ is projectively induced if and only if $c\,\omega$ is.

Proof

Observe that $\Omega _c$ is real analytic if and only if $c\omega$ is. Set coordinates z around a point $p\in U$ such that $z(p)=0$.

Assume first that $c\omega$ is projectively induced, i.e. there exist holomorphic functions $f_j\!:U\rightarrow \mathbb {C}$ such that $e^{c\psi }=\sum _{j=0}^\infty |f_j|^2$, with $f_0\equiv 1$ and for all $j\ge 1$, $f_j(0)=0$. Then we have,

$$\begin{aligned} |z_0|^{2c}e^{c\psi }=|z_0|^{2c}\sum _{j=0}^\infty |f_j|^2=\sum _{j=0}^\infty |z_0^cf_j|^2, \end{aligned}$$

and thus:

$$\begin{aligned} e^{|z_0|^{2c}e^{c\psi }}=\sum _{j=0}^\infty |k_j(z_0,z)|^2 \end{aligned}$$

for some holomorphic functions $k_j$ defined on $\mathbb {C}{\setminus }\{0\}\times U$ and such that $k_0=e^{|z_0|^{2c}}$. It follows that $k\!:\mathbb {C}{\setminus }\{0\}\times U\rightarrow \mathbb {C}\textrm{P}^\infty$, $k(z_0,z)=\left[ k_0:k_1:\dots :k_j:\dots \right]$ is a Kähler immersion and thus $(\mathbb {C}{\setminus }\{0\}\times U,\Omega _c)$ is projectively induced.

Assume now $\Omega _c$ to be projectively induced. We can assume without loss of generality that $\psi$ is the diastasis function for $\omega$ centered at p, i.e. $\psi (0)=0$. By Theorem 3, the matrix of coefficients $(B_{jk})$ in the power expansion around the origin of the diastasis function of $\Omega _c$ is positive semidefinite. We need to show that $(B_{jk})\ge 0$ implies $(b_{jk})\ge 0$, where $(b_{jk})$ is the matrix of coefficients in the power expansion of $e^{c\psi }-1$ (cf. (1)).

Let $q=(\epsilon ,p)\in \mathbb {C}\setminus \{0\}\times U$, with $\epsilon \in \mathbb {R}^+$. Perform a change of coordinates $z_0\rightarrow z_0-\epsilon$ (we avoid to change the name of the variable to simplify the notation), so that q is the point of coordinates (0, 0). Observe that in this coordinates a potential for $\Omega _c$ reads:

$$\begin{aligned} \Phi (z_0,z)= |z_0+\epsilon |^{2c}e^{c\psi (z)}, \end{aligned}$$

and the diastasis for $\Omega _c$ centered at q is given by:

$$\begin{aligned} D_q(z_0,z)= |z_0+\epsilon |^{2c}e^{c\psi \left( z\right) }- \epsilon ^c(z_0+\epsilon )^c-\epsilon ^c(\bar{z}_0+\epsilon )^c+\epsilon ^2. \end{aligned}$$

Observe that if $(B_{jk})$ is positive semidefinite, then any submatrix obtained from $(B_{jk})$ collecting the $a_1,\dots , a_l$-th rows and columns also is. In particular the matrix $(u_{jk})$ has to be positive semidefinite, where

$$\begin{aligned} u_{jk}=\left[ \frac{1}{\hat{m}_j!\hat{m}_k!}\frac{\partial ^{|\hat{m}_j|+|\hat{m}_k|}}{\partial z^{\hat{m}_j}\bar{z}^{\hat{m}_k}}\left( \frac{\partial ^{2}}{\partial z_0\partial \bar{z}_0}\left( e^{D_q(z_0,z)}-1\right) \right) _{z_0=0}\right] _{z=0}, \end{aligned}$$

(5)

where comparing with the multi-index notation in (2) we are setting $m_j=(1,\hat{m}_j)$, since we are isolating the derivatives with respect to $z_0$ and $\bar{z}_0$. By direct computation:

$$\begin{aligned} {\begin{matrix} &{}\left[ \frac{\partial ^{2}}{\partial z_0\partial \bar{z}_0}\left( e^{D_q(z_0,z)}-1\right) \right] _{{z_0=0}}\\ &{}\quad =\left[ \frac{\partial }{\partial z_0}\left\{ e^{D_q(z_0,z)}\left( c(z_0+\epsilon )^c(\bar{z}_0+\epsilon )^{c-1}e^{c\psi \left( z\right) }- c\epsilon ^c(\bar{z}_0+\epsilon )^{c-1}\right) \right\} \right] _{{z_0=0}}\\ &{}=\left[ e^{D_q(z_0,z)}\left\{ c^2(z_0+\epsilon )^{c-1}(\bar{z}_0+\epsilon )^{c-1}e^{c\psi \left( z\right) }+\right. \right. \\ &{}\quad \quad \left. \left. +\left( (z_0+\epsilon )^cc(\bar{z}_0+\epsilon )^{c-1}e^{c\psi \left( z\right) }-\epsilon ^cc(\bar{z}_0+\epsilon )^{c-1}\right) \right. \right. \\ &{}\quad \quad \times \left. \left. \left( (\bar{z}_0+\epsilon )^cc( z_0+\epsilon )^{c-1}e^{c\psi \left( z\right) }-\epsilon ^cc( z_0+\epsilon )^{c-1}\right) \right\} \right] _{z_0=0}\\ &{}\quad =c^2\epsilon ^{2c-2}e^{D_q(0,z)}\left\{ \epsilon ^{2c}\left( e^{c\psi \left( z\right) }- 1\right) ^2 +e^{c\psi (z)} \right\} . \end{matrix}} \end{aligned}$$

The matrix $(u_{jk})$ is positive semidefinite if and only if $(v_{jk}):=c^{-2}\epsilon ^{2-2c}(u_{jk})$ is, for any $\epsilon >0$. Since

$$\begin{aligned} v_{jk}=\left[ \frac{1}{\hat{m}_j!\hat{m}_k!}\frac{\partial ^{|\hat{m}_j|+|\hat{m}_k|}}{\partial z^{\hat{m}_j}\bar{z}^{\hat{m}_k}}e^{D_q(0,z)}\left\{ \epsilon ^{2c}\left( e^{c\psi \left( z\right) }- 1\right) ^2 +e^{c\psi (z)} \right\} \right] _{z=0}, \end{aligned}$$

is positive semidefinite for any $\epsilon >0$, it must remain positive semidefinite as $\epsilon$ goes to 0. The conclusion follows observing that for $\epsilon =0$ one has

$$\begin{aligned} (v_{jk})= \left[ \frac{1}{\hat{m}_j!\hat{m}_k!}\frac{\partial ^{|\hat{m}_j|+|\hat{m}_k|}}{\partial z^{\hat{m}_j}\bar{z}^{\hat{m}_k}}e^{c\psi (z)}\right] _{z=0}=\left[ \frac{1}{\hat{m}_j!\hat{m}_k!}\frac{\partial ^{|\hat{m}_j|+|\hat{m}_k|}}{\partial z^{\hat{m}_j}\bar{z}^{\hat{m}_k}}\left( e^{c\psi (z)}-1\right) \right] _{z=0}=(b_{jk}). \end{aligned}$$

Proof of Theorem 1

If the metric is flat it is trivially Ricci-flat, furthermore by a result of Calabi [4] it is also projectively induced. Assume now that C(S) is Ricci-flat and projectively induced. By Theorem 8 the metric cone $(C(S),\bar{g}, \Omega )$ is Ricci-flat if and only if $(S,\xi ,\eta ,\Phi ,g)$ is Sasaki Einstein and by Myers’ Theorem, a complete Sasaki–Einstein manifold is forced to be compact. By the Structure Theorem 7 a compact regular Sasakian manifold is a principal circle bundle over a compact Hodge manifold $(X,\omega )$. Denote by $p\!:S\rightarrow X$ the bundle projection. Recall that the transverse Kähler metric $g^T$ is Kähler–Einstein with Einstein constant $\lambda ^T=2(n+1)$ (cf. Theorem 8). Since for some $a>0$, $p^*\omega =\frac{a}{2}d\eta$, and $\frac{1}{2}d\eta$ is the Kähler form associated to $g^T$, then $(X,\frac{1}{a}\omega )$ is a compact Kähler–Einstein manifold with Einstein constant $2(n+1)$.

By Proposition 11, for any $c>0$, $c\omega$ is projectively induced if and only if $\Omega _c$ is. Since by Lemma 9, $\Omega =\Omega _{1/a}$, we obtain that $\frac{1}{a}\omega$ is projectively induced if and only if $\Omega$ is. Observe that the Kähler immersion of $(X,\frac{1}{a}\omega )$ in $\mathbb {C}\textrm{P}^N$ is also global since the manifold is Fano and thus simply connected (see Remark 4).

By Hulin’s Theorem 5, a projectively induced Kähler-Einstein compact manifold with Einstein constant $2(n+1)$ is forced to be the complex projective space with the Fubini–Study metric, i.e. $(X,\frac{1}{a}\omega )\simeq (\mathbb {C}\textrm{P}^n,\omega _{FS})$, and the Sasakian manifold is Sasaki equivalent to the sphere $\mathbb {S}^{2n+1}$ with the standard Sasakian structure, whose cone is the complex Euclidean space with its flat metric $(C(S),\Omega )\simeq (\mathbb {C}^{n+1}{\setminus } \{0\},\omega _0)$ (see [5, Ex. 5] and Example 1 above).

Remark 12

Observe that instead of applying Hulin’s Theorem, in the last part of the proof of Theorem 1 we could have argued in the following way. The global Kähler immersion of $(X,\frac{1}{a} \omega )$ into $\mathbb {C}\textrm{P}^{N<\infty }$ induces by [5, Proposition 3] a Sasakian immersion of S in the Sasakian sphere $\mathbb {S}^{2N+1}$. Thus, by [5, Theorem 1], S is forced to be Sasakian equivalent to a standard Sasakian sphere and its cone to be biholomorphically isometric to $\mathbb {C}^{n+1}\setminus \{0\}$ with the flat metric.

We conclude the paper with the proof of Theorem 2 showing that, up to $\mathcal D_a$–homothetic transformation, Kähler cones over homogeneous compact Sasakian manifolds are projectively induced.

Proof of Theorem 2

By [3, Theorem 8.3.6], a compact homogeneous Sasakian manifold is regular and fibers over a simply connected homogeneous Kähler manifold $(X,\omega )$. By Theorem 7, $\omega$ is Hodge and $p^*\omega =\frac{a}{2}d\eta$, where $p\!:S\rightarrow X$ is the fiber projection. By Lemma 9, if locally $\omega =\frac{i}{2}\partial \bar{\partial }\psi$, then locally the Kähler form on the cone reads $\Omega =\frac{i}{2}\partial \bar{\partial }|z_0|^{\frac{2}{a}} e^{\frac{1}{a}\psi }$.

Recall that in [14, Section 2] Takeuchi proved that if the Kähler form $\omega$ of a compact and simply connected homogeneous Kähler manifold is integral, then there exists a positive integer k such that $k\omega$ is projectively induced. The flag manifold $(X,\omega )$ satisfies the hypothesis of Takeuchi’s theorem, thus $k\omega$ admits a global Kähler immersion in $\mathbb {C}\textrm{P}^N$.

Performing a $\mathcal D_{ak}$–homothetic transformation on $(S,\eta )$ the potential $t^2=|z_0|^{\frac{2}{a}} e^{\frac{1}{a}\psi }$ on the Kähler cone changes accordingly in $t^{2ak}=|z_0|^{2k} e^{k\psi }$. By Proposition 11, $\frac{i}{2}\partial \bar{\partial }|z_0|^{2k} e^{k\psi }$ is projectively induced if and only if $k\omega$ is, and we are done.

Remark 13

Observe that in both the proofs of Theorems 1 and 2, the local arguments given by Lemma 9 and Proposition 11 could be extended to quasi regular Sasakian manifolds. However, in the quasi-regular case X is a Hodge Kähler orbifold and in this setting we do not have the techniques developed for Kähler immersions that are crucial in the concluding steps of our proofs.