Levi-Civita Ricci-Flat Metrics on Non-Kähler Calabi-Yau Manifolds

Abstract

In this paper, we provide new examples of Levi-Civita Ricci-flat Hermitian metrics on certain compact non-Kähler Calabi-Yau manifolds, including compact Hermitian Weyl-Einstein manifolds, compact locally conformal hyperKähler manifolds, certain suspensions of Brieskorn manifolds, and every generalized Hopf manifold provided by suspensions of exotic spheres. These examples generalize previous constructions on Hopf manifolds. Additionally, we also construct new examples of compact Hermitian manifolds with nonnegative first Chern class that admit constant strictly negative Riemannian scalar curvature. Further, we remark some applications of our main results in the study of the Chern-Ricci flow on compact Hermitian Weyl-Einstein manifolds. In particular, we describe the Gromov-Hausdorff limit for certain explicit finite-time collapsing solutions which generalize previous constructions on Hopf manifolds.

Appendix A: Remarks on Chern-Ricci flow

On a Hermitian manifold $(M,\Omega _{0},J)$, a solution of the Chern-Ricci flow starting at $\Omega _{0}$ is given by a smooth family of Hermitian metrics $\Omega = \Omega (t)$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \frac{\partial }{\partial t} \Omega = -\mathrm{{Ric}}^{(1)}(\Omega ), \ \ 0 \le t < T, \\ \Omega (0) = \Omega _{0}, \end{array}\right. } \end{aligned}$$

(A.1)

for $T \in (0,\infty ]$, see, for instance, [22] and [37]. Inspired by the results on Hopf manifolds provided in [23, 37], and [38], we observe that the ideas involved in the proof of Theorem 3.3 can be used to obtain explicit solutions of the Chern-Ricci flow on compact Hermitian Weyl-Einstein manifolds of complex dimension at least 3. More precisely, we have the following result:

Theorem A.1

Let (M, g, J) be a compact Hermitian Weyl-Einstein manifold of complex dimension at least 3, then there exists an explicit solution g(t) of the Chern-Ricci flow on M for $t \in [0,\frac{2}{n})$, starting at g, satisfying the following properties:

• (1) $\mathrm{{Vol}}(M,g(t)) \rightarrow 0$ as $t \rightarrow \frac{2}{n}$ (i.e., g(t) is finite-time collapsing);

• (2) $\lim _{t \rightarrow \frac{2}{n}}g(t) = h_{T}$, where $h_{T}$ is a nonnegative symmetric tensor on M;

• (3) The Chern scalar curvature of g(t) blows up like $(n-1)/(\frac{2}{n}- t)$;

• (4) $\mathrm{{scal}}(g(t)) \rightarrow -\infty$ as $t \rightarrow \frac{2}{n}$;

• (5) $\lim _{t \rightarrow \frac{2}{n}}d_{GH}\big ((M,d_{t}),(S^{1},d_{S^{1}}) \big ) = 0$,

where $d_{t}$ is the distance induced by g(t) on M and $d_{S^{1}}$ is the distance on the unit circle $S^{1}$ induced by a suitable scalar multiple of the standard Riemannian metric.

Remark A.2

In order to prove Theorem A.1, we proceed highlighting the background on Sasaki-Einstein geometry underlying all our main results. In fact, in what follows we shall prove the results of Theorem A.1 for compact Hermitian manifolds as in the proof of Theorem 3.3, i.e., for compact Hermitian manifolds of the form $(\Sigma _{\phi ,c}(Q),g,J)$, such that g is a Hermitian Weyl-Einstein metric induced by the Calabi-Yau structure of $\mathcal {C}(Q)$. We also include in the proof some comments relating our results with previous known results on Hopf manifolds.

Before we start the proof, we recall that the Gromov-Hausdorff distance of metric spaces can be defined as follows (see, for instance, [12]). Given two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$, a correspondence between the underlying sets X and Y is a subset $R \subseteq X \times Y$ satisfying the following property: for every $x \in X$, there exists at least one $y \in Y$, such that $(x,y) \in R$, and similarly for every $y \in Y$ there exists an $x \in X$, such that $(x,y) \in R$. Let us denote by $\mathcal {R}(X,Y)$ the set of all correspondences between X and Y. Now we consider the following definition

Definition A.3

Let $R \in \mathcal {R}(X,Y)$ be a correspondence between two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$. The distortion of R is defined by

$$\begin{aligned} \mathrm{{dis}}(R) = \sup \Big \{ |d_{X}(x,x') - d_{Y}(y,y')| \ \ \big | \ \ (x,y), (x',y') \in R \ \Big \}. \end{aligned}$$

(A.2)

From above, we can define the Gromov-Hausdorff distance between two metric spaces as follows.

Definition A.4

We define the Gromov-Hausdorff distance of any two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$ as being

$$\begin{aligned} d_{GH}\big ((X,d_{X}),(Y,d_{Y}) \big ) = \frac{1}{2}\inf \Big \{ \mathrm{{dis}}(R) \ \ \Big | \ \ R \in \mathcal {R}(X,Y) \Big \}. \end{aligned}$$

(A.3)

Now we can prove Theorem A.1.

Proof

(Theorem A.1) As we have seen, given a compact Hermitian Weyl-Einstein manifold of the form $(\Sigma _{\phi ,c}(Q),g,J)$, it follows that

$$\begin{aligned} \Omega = -\mathrm{{d}}(J\theta ) + \theta \wedge J\theta \ \ \ {\text {and}} \ \ \ \mathrm{{Ric}}^{(1)}(\Omega ) = - \frac{n}{2}\mathrm{{d}}(J\theta ). \end{aligned}$$

(A.4)

Therefore, we set $\Omega _{0} := \Omega$ and

$$\begin{aligned} \Omega (t) := \Omega _{0} - t\mathrm{{Ric}}^{(1)}(\Omega _{0}) = -\Big (1-\frac{n}{2}t \Big )\mathrm{{d}}(J\theta ) + \theta \wedge J\theta . \end{aligned}$$

(A.5)

Since $-\mathrm{{d}}(J\theta ) \ge 0$, we have that $\Omega (t)$ is a Hermitian metric for all $t \in [0,\frac{2}{n})$. Moreover, a straightforward computation shows that

$$\begin{aligned} \Omega (t)^{n} = \Big (1-\frac{n}{2}t \Big )^{n-1}\Omega _{0}^{n}. \end{aligned}$$

(A.6)

Thus, we have $\mathrm{{Ric}}^{(1)}(\Omega (t)) = \mathrm{{Ric}}^{(1)}(\Omega _{0})$, $\forall t \in [0,\frac{2}{n})$. From this, we conclude that

$$\begin{aligned} \frac{\partial }{\partial t} \Omega (t) = - \mathrm{{Ric}}^{(1)}(\Omega _{0}) = - \mathrm{{Ric}}^{(1)}(\Omega (t)), \end{aligned}$$

(A.7)

for all $t \in [0,\frac{2}{n})$. Therefore, the family of Hermitian metrics $\Omega (t) = \Omega _{0} - t\mathrm{{Ric}}^{(1)}(\Omega _{0})$ on $\Sigma _{\phi ,c}(Q)$ gives a solution of the Chern-Ricci flow on the maximal existence interval [0, T), such that $T = \frac{2}{n}$. Let us observe that

$$\begin{aligned} \mathrm{{d}}\Omega (t) = \theta (t) \wedge \Omega (t), \ \ {\text {such that }} \ \ \theta (t) = \frac{T}{T-t}\theta , \end{aligned}$$

(A.8)

for all $t \in [0,T)$. Moreover, one can easily verify that $\theta (t)$ is parallel with respect to the Levi-Civita connection induced by $\Omega (t)$. Therefore, we have that $\Omega (t)$ is Vaisman for all $t \in [0,T)$. Also, from Eq. (A.6) and Eq. (A.5), it follows that

$$\begin{aligned} \lim _{t \rightarrow T}\mathrm{{Vol}}\big (\Sigma _{\phi ,c}(Q), \Omega (t)\big ) = 0 \ \ \ {\text {and}} \ \ \ \lim _{t \rightarrow T}\Omega (t) = \theta \wedge J\theta , \end{aligned}$$

(A.9)

i.e., $\Omega (t)$ is finite-time collapsing [38]. Observing that $\Omega _{T}:= \theta \wedge J\theta$ is a nonnegative (1, 1)-form, we conclude that the behavior of the Chern-Ricci flow $\Omega (t)$ is quite similar to the behavior of the Chern-Ricci flow on Hopf manifolds provided in [37, Proposition 1.8]. Further, as it was shown in [23], finite-time singularities are characterized by the blow-up of the Chern scalar curvature. This last fact can be easily verified for the Chern-Ricci flow describe in Eq. (A.5). In fact, by a straightforward computation, one can show that

$$\begin{aligned} \mathrm{{d}}(J \theta ) \wedge \Omega (t)^{n-1} = -(n-1)!\bigg ( \frac{n-1}{1-\frac{n}{2}t}\bigg ) \frac{\Omega (t)^{n}}{n!}. \end{aligned}$$

(A.10)

Thus, since $\mathrm{{Ric}}^{(1)}(\Omega (t)) = \mathrm{{Ric}}^{(1)}(\Omega _{0})$, $\forall t \in [0,T)$, we obtain the following

$$\begin{aligned} \frac{s_{C}(\Omega (t))}{n} \Omega (t)^{n} = -\frac{n}{2}\mathrm{{d}}(J \theta ) \wedge \Omega (t)^{n-1} = \frac{1}{2}\bigg ( \frac{n-1}{1-\frac{n}{2}t}\bigg ) \Omega (t)^{n}. \end{aligned}$$

(A.11)

From above, we have

$$\begin{aligned} s_{C}(\Omega (t)) = \frac{n}{2}\bigg ( \frac{n-1}{1-\frac{n}{2}t} \bigg ) = \frac{n-1}{T - t}, \end{aligned}$$

(A.12)

for all $0 \le t < T$. It follows that $s_{C}(\Omega (t)) \rightarrow +\infty$ as $t \rightarrow T$ (cf. [23]). Also, denoting by g(t) the underlying Riemannian metric associated with $\Omega (t)$, it follows from Eq. (3.31) that

$$\begin{aligned} \mathrm{{scal}}(g(t)) = \frac{n(n-1)}{(1-\frac{n}{2}t)^{2}} \bigg [\frac{2n-1}{2n} - \frac{n}{2}t\bigg ] = \frac{2(n-1)}{(T-t)^{2}} \bigg [T - \frac{1}{n^{2}} - t\bigg ], \end{aligned}$$

(A.13)

for all $0 \le t < T$. Hence, we obtain $\mathrm{{scal}}(g(t)) \rightarrow -\infty$ as $t \rightarrow T$. In particular, we have:

• (a) $0 \le t < T-\frac{1}{n^{2}} \Longrightarrow \mathrm{{scal}}(g(t)) > 0$;

• (b) $t = T-\frac{1}{n^{2}} \Longrightarrow \mathrm{{scal}}(g(t)) = 0$;

• (c) $T-\frac{1}{n^{2}}< t< T \Longrightarrow \mathrm{{scal}}(g(t)) < 0$.

From above, we obtain a complete picture of the behavior of $\mathrm{{scal}}(g(t))$, $t \in [0,T)$. Further, by means of a suitable change in the argument presented in [38, $\S$ 4] one can show that

$$\begin{aligned} \big (\Sigma _{\phi ,c}(Q),d_{t}\big ) \xrightarrow {\text {G.H.}} \big (S^{1},d_{S^{1}}\big ), \ \ {\text {as}} \ \ t \rightarrow T, \end{aligned}$$

(A.14)

where $d_{t}$ is the distance induced by g(t) and $d_{S^{1}}$ is the distance on the unit circle $S^{1}$ induced by a suitable scalar multiple of the standard Riemannian metric. From Definition A.3, we say that $\big (\Sigma _{\phi ,c}(Q),d_{t}\big ) \xrightarrow {\text {G.H.}} \big (S^{1},d_{S^{1}}\big )$, as $t \rightarrow T$, if

$$\begin{aligned} \lim _{t\rightarrow T}d_{GH}\big ((\Sigma _{\phi ,c}(Q),d_{t}),(S^{1},d_{S^{1}}) \big ) =0. \end{aligned}$$

(A.15)

In order to conclude the proof, consider $F :Q \times \mathbb {R} \rightarrow \mathbb {R}$, such that $F(x,\varphi ) = -2\varphi$, $\forall (x,\varphi ) \in Q \times \mathbb {R}$. From this, we set $F_{c} :\Sigma _{\phi ,c}(Q) \rightarrow S^{1}$, such that

$$\begin{aligned} F_{c}([x,\varphi ]) := \exp \bigg ( \frac{2\pi \sqrt{-1}F(x,\varphi )}{\log (c^{2})}\bigg ), \end{aligned}$$

(A.16)

for all $[x,\varphi ] \in \Sigma _{\phi ,c}(Q)$. Since

$$\begin{aligned} F(\phi (x),\varphi + n \log (c)) = F(x,\varphi ) - n\log (c^{2}), \end{aligned}$$

(A.17)

it follows that $F_{c}$ is a well-defined smooth map. Considering the canonical angular 1-form $\mathrm{{d}}\sigma$ on $S^{1}$, a straightforward computation shows us that

$$\begin{aligned} F_{c}^{*}(\mathrm{{d}}\sigma ) = \frac{\theta }{\log (c^{2})}. \end{aligned}$$

(A.18)

From above, since $\theta$ is a non-vanishing 1-form, it follows that $F_{c}$ is a submersion. Also, we notice that $\ker ((F_{c})_{*}) = \ker (\theta (t))$, $\forall 0 \le t < T$. Since

$$\begin{aligned} g(t) = \underbrace{-\Big (1-\frac{n}{2}t \Big )\mathrm{{d}}(J\theta )(\mathbb {1} \otimes J)}_{h(t)} + \underbrace{\theta \otimes \theta + J\theta \otimes J\theta }_{h_{T}}, \end{aligned}$$

(A.19)

see Eq. (A.5), by considering the g(t)-orthogonal complement $\ker ((F_{c})_{*})^{\perp }$, it follows that

$$\begin{aligned} g(t)|_{\ker ((F_{c})_{*})^{\perp }} = \theta \otimes \theta = \lambda ^{2} F_{c}^{*}\big (\mathrm{{d}}\sigma \otimes \mathrm{{d}}\sigma \big ), \end{aligned}$$

(A.20)

where $\lambda = \log (c^{2})$, i.e., $F_{c} :(\Sigma _{\phi ,c}(Q),g(t)) \rightarrow (S^{1},\lambda ^{2}\mathrm{{d}}\sigma \otimes \mathrm{{d}}\sigma )$ is a Riemannian submersion. Since $\Sigma _{\phi ,c}(Q)$ is compact, we have that $F_{c} :\Sigma _{\phi ,c}(Q) \rightarrow S^{1}$ is in fact a locally trivial fiber bundle with typical fiber diffeomorphic to Q (e.g., [4]). From $h_{T}$ given in Eq. (A.19), we set

$$\begin{aligned} \mathcal {D}:= \big \{X \in T\Sigma _{\phi ,c}(Q) \ \big | \ h_{T}(X,Y) = 0, \ \forall Y \big \}. \end{aligned}$$

(A.21)

Let us denote by $S \subset \Sigma _{\phi ,c}(Q)$ a generic fiber of $F_{c}$. Denote also by $\mathcal {D}_{S}$ the distribution $\mathcal {D}$ restricted to S. By construction, since $\ker ((F_{c})_{*}) = \ker (\theta )$, we have that $(S,\eta _{S}:=i^{*}(J\theta ))$ is a contact manifold, where $i :S \hookrightarrow \Sigma _{\phi ,c}(Q)$ is the natural inclusion, see, for instance, [15]. Hence, we obtain the following description

$$\begin{aligned} \mathcal {D}_{S}= \big \{X \in TS \ \big | \ \eta _{S}(X) = 0 \big \}, \end{aligned}$$

(A.22)

i.e., $\mathcal {D}_{S}$ is the contact distribution on S induced by $\eta _{S}$. Since every contact distribution is a bracket-generating distribution, it follows from Chow’s theorem [34] that any two points of S can be connected by a smooth path tangent to $\mathcal {D}_{S}$. It is worth observing that $g(t)|_{\mathcal {D}_{S}} = h(t)|_{\mathcal {D}_{S}}$, where h(t) is given as in Eq. (A.19). Given $p,q \in \Sigma _{\phi ,c}(Q)$, let $\alpha$ be a path connecting $F_{c}(p)$ and $F_{c}(q)$ in $S^{1}$. Since $(\Sigma _{\phi ,c}(Q),g(t))$ is complete, there exists a lift $\widetilde{\alpha }$ of $\alpha$ starting at p and tangent to $\ker ((F_{c})_{*})^{\perp }$. Let us denote by $z \in \Sigma _{\phi ,c}(Q)$ the endpoint of the path $\widetilde{\alpha }$ (Figure 2). From this, we have $d_{t}(p,z) \le d_{S^{1}}(F_{c}(p),F_{c}(z)) = d_{S^{1}}(F_{c}(p),F_{c}(q))$, where $d_{S^{1}}$ denotes the distance induced by $\lambda ^{2}\mathrm{{d}}\sigma \otimes \mathrm{{d}}\sigma$ on $S^{1}$. Hence, we obtain

$$\begin{aligned} d_{t}(p,q) \le d_{t}(p,z) + d_{t}(z,q) \le d_{S^{1}}(F_{c}(p),F_{c}(q)) + d_{t}(z,q). \end{aligned}$$

(A.23)

Since $z,q \in F_{c}^{-1}(F_{c}(q)) = S$, we have a smooth path $\gamma :[0,1] \rightarrow \Sigma _{\phi ,c}(Q)$ tangent to $\mathcal {D}_{S}$ connecting z and q (Figure 2).

Fig. 2

Representation of horizontal and vertical paths used to estimate the distance between p and q.

Therefore, since $g(t)|_{\mathcal {D}_{S}} = h(t)|_{\mathcal {D}_{S}}$ and $\Sigma _{\phi ,c}(Q)$ is compact, we obtain

$$\begin{aligned} d_{t}(z,q) \le \int _{0}^{1}||\gamma '(s)||_{g(t)}ds = \Big (1-\frac{n}{2}t \Big )\int _{0}^{1}||\gamma '(s)||_{g(0)}ds\le C \Big (1-\frac{n}{2}t \Big ), \end{aligned}$$

(A.24)

where $C := \mathrm{{diam}}\big (\Sigma _{\phi ,c}(Q),g(0)\big )$. Hence, it follows that

$$\begin{aligned} d_{t}(p,q) - d_{S^{1}}(F_{c}(p),F_{c}(q)) \le C \Big (1-\frac{n}{2}t \Big ), \end{aligned}$$

(A.25)

for all $t \in [0,T)$ and for all $p,q \in \Sigma _{\phi ,c}(Q)$. From $F_{c} :\Sigma _{\phi ,c}(Q) \rightarrow S^{1}$, we set

$$\begin{aligned} R(F_{c}) := \big \{ (p,F_{c}(p)) \in \Sigma _{\phi ,c}(Q) \times S^{1} \ \big | \ p \in \Sigma _{\phi ,c}(Q) \big \}. \end{aligned}$$

(A.26)

Since $F_{c}$ is a surjective map, it follows that $R(F_{c}) \in \mathcal {R}(\Sigma _{\phi ,c}(Q),S^{1})$. From Eq. (A.25), we conclude that

$$\begin{aligned} d_{GH}\big ((\Sigma _{\phi ,c}(Q),d_{t}),(S^{1},d_{S^{1}}) \big ) \le \frac{1}{2}\mathrm{{dis}}(R(F_{c})) < \frac{nC}{2} \Big (T-t \Big ). \end{aligned}$$

(A.27)

Therefore, it follows that

$$\begin{aligned} \lim _{t \rightarrow T}d_{GH}\big ((\Sigma _{\phi ,c}(Q),d_{t}),(S^{1},d_{S^{1}}) \big ) = 0. \end{aligned}$$

(A.28)

As it can be seen, the arguments provided above generalize certain ideas introduced in [38, §4] for Hopf manifolds. Following Theorem 2.26 and the above constructions, we conclude the proof of Theorem A.1. $\square$

Example A.5

Consider ${\varvec{\Sigma }}^{7} \times S^{1}$, where ${\varvec{\Sigma }}^{7}$ is any one of the 28 homotopy 7-spheres. Fixed a Hermitian Weyl-Einstein metric $\Omega _{WE}$ on ${\varvec{\Sigma }}^{7} \times S^{1}$, it follows from Theorem A.1 that there exists a family of Hermitian metrics $\Omega (t)$, $t \in [0,\frac{1}{2})$, satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \frac{\partial }{\partial t} \Omega = -\mathrm{{Ric}}^{(1)}(\Omega ), \ \ 0 \le t < \frac{1}{2}, \\ \Omega (0) = \Omega _{WE}. \end{array}\right. } \end{aligned}$$

(A.29)

Moreover, it follows form Eq. (A.12) and from Eq. (A.13) that

$$\begin{aligned} s_{C}(\Omega (t)) = \frac{6}{1 - 2t}, \ \ \ \ {\text {and}} \ \ \ \ \mathrm{{scal}}(g(t)) = \frac{12}{(1 - 2t)^{2}} \bigg [ \frac{7}{16} - t \bigg ]. \end{aligned}$$

(A.30)

From above, we obtain the following:

• (a) $0 \le t < \frac{7}{16} \Longrightarrow \mathrm{{scal}}(g(t)) > 0$;

• (b) $t = \frac{7}{16} \Longrightarrow \mathrm{{scal}}(g(t)) = 0$;

• (c) $\frac{7}{16}< t< \frac{1}{2} \Longrightarrow \mathrm{{scal}}(g(t)) < 0$.

Regarding ${\varvec{\Sigma }}^{7} \times S^{1}$ as a suspension by $(\mathrm{{id}},\sqrt{\mathrm{{e}}})$ of ${\varvec{\Sigma }}^{7}$, from Theorem A.1, we conclude that

$$\begin{aligned} \lim _{t \rightarrow \frac{1}{2}}d_{GH}\big (({\varvec{\Sigma }}^{7} \times S^{1},d_{t}),(S^{1},d_{S^{1}}) \big ) = 0, \end{aligned}$$

(A.31)

where $d_{t}$ is the distance induced by g(t) on ${\varvec{\Sigma }}^{7} \times S^{1}$ and $d_{S^{1}}$ is the distance on the unit circle $S^{1}$ induced by the standard Riemannian metric.