The Geometry on Hyper-Kähler Manifolds of Type A _∞

Abstract

Hyper-Kähler manifolds of type A _∞ are noncompact complete Ricci-flat Kähler manifolds of complex dimension 2, constructed by Anderson, Kronheimer, LeBrun (Commun. Math. Phys., 125, 637–642, 1989) and Goto (Geom. Funct. Anal., 4(4), 424–454, 1994). We review the asymptotic behavior, the holomorphic symplectic structures and period maps on these manifolds.

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1 Introduction

Hyper-Kähler manifolds of type A _∞ were first constructed by Anderson, Kronheimer and LeBrun in [1], as the first example of complete Ricci-flat Kähler manifolds with infinite topological type. Here, infinite topological type means that their homology groups are infinitely generated. The construction in [1] is due to Gibbons-Hawking ansatz, and Goto [5] has constructed these manifolds in another way, using hyper-Kähler quotient construction. Some of the topological and geometric properties of hyper-Kähler manifolds of type A _∞ were studied well in the above papers. In this article, we focus on the volume growth of the hyper-Kähler metrics, the holomorphic symplectic structures, and the period maps.

The construction of hyper-Kähler manifolds of type A _∞ is similar to that of ALE spaces of type A _k , where k is a nonnegative integer. Moreover, their topological properties and complex geometric properties are also similar to type A _k . For example, both of the ALE spaces of type A _k and the hyper-Kähler manifolds of type A _∞ have the parameter naturally given by the construction. We review that they correspond to the cohomology classes of three Kähler forms along [8].

On the other hand, one of the essentially different properties between them appears in their asymptotic behaviors. In fact, the volume growth of ALE spaces is Euclidean, but that of hyper-Kähler manifolds of type A _∞ are less than Euclidean volume growth, which is a main result of [7].

Moreover, we will review the independence of the volume growth of hyper-Kähler metrics and the complex structures. More precisely, we review the result in [9] to the effect that the volume growth of the hyper-Kähler metric of type A _∞ can be deformed preserving the complex structure.

2 Hyper-Kähler Manifolds of Type A _∞

2.1 Hyper-Kähler Quotient Construction

In this section, we review shortly the construction of hyper-Kähler manifolds of type A _∞ along [5]. For more details, see [1, 5] or review in Section 2 of [7].

First of all, hyper-Kähler manifolds are defined as follows.

Definition 1.

Let (X, g) be a Riemannian manifold of dimension 4n with three integrable complex structures $I_{1},I_{2},I_{3}$, and g be a hermitian metric with respect to each I _i . Then $(X,g,I_{1},I_{2},I_{3})$ is a hyper-Kähler manifold if $(I_{1},I_{2},I_{3})$ satisfying the relations $I_{1}^{2} = I_{2}^{2} = I_{3}^{2} = I_{1}I_{2}I_{3} = -1$ and each $\omega _{i}:= g(I_{i}\cdot,\cdot )$ being closed.

Denote by $\mathbb{H} = \mathbb{R} \oplus \mathbb{R}i \oplus \mathbb{R}j \oplus \mathbb{R}k = \mathbb{C} \oplus \mathbb{C}j$ the quaternion and denote by $\mathrm{Im}\mathbb{H} = \mathbb{R}i \oplus \mathbb{R}j \oplus \mathbb{R}k$ its Imaginary part. Then an $\mathrm{Im}\mathbb{H}$-valued 2-form $\omega:= i\omega _{1} + j\omega _{2} + k\omega _{3} \in \varOmega ^{2}(X) \otimes \mathrm{ Im}\mathbb{H}$ characterizes the hyper-Kähler structure $(g,I_{1},I_{2},I_{3})$. Accordingly, we call ω the hyper-Kähler structure on X instead of $(g,I_{1},I_{2},I_{3})$.

Now we construct hyper-Kähler quotient method introduced in [9]. Put

$$\displaystyle\begin{array}{rcl} (\mathrm{Im}\mathbb{H})_{0}^{\mathbb{N}}:=\{\lambda = (\lambda _{ n})_{n\in \mathbb{N}} \in (\mathrm{Im}\mathbb{H})^{\mathbb{N}};\ \sum _{ n\in \mathbb{N}} \frac{1} {1 + \vert \lambda _{n}\vert } < +\infty \},& & {}\\ \end{array}$$

where $\mathbb{N}$ is the set of positive integers. Here, we denote by $S^{\mathbb{N}}$ the set of all maps from $\mathbb{N}$ to a set S.

Let

$$\displaystyle\begin{array}{rcl} M_{\mathbb{N}}:=\{ v \in \mathbb{H}^{\mathbb{N}};\ \|v\|_{ \mathbb{N}}^{2} < +\infty \},& & {}\\ \end{array}$$

where

$$\displaystyle\begin{array}{rcl} \langle u,v\rangle _{\mathbb{N}}:=\sum _{n\in \mathbb{N}}u_{n}\overline{v}_{n},\quad \|v\|_{\mathbb{N}}^{2}:=\langle v,v\rangle _{ \mathbb{N}}& & {}\\ \end{array}$$

for $u,v \in \mathbb{H}^{\mathbb{N}}$. Here, the quaternionic conjugate of v _n is denoted by $\overline{v}_{n}$.

For each $\lambda \in (\mathrm{Im}\mathbb{H})_{0}^{\mathbb{N}}$, $\varLambda \in \mathbb{H}^{\mathbb{N}}$ can be taken so that $\varLambda _{n}i\overline{\varLambda }_{n} =\lambda _{n}$. Put

$$\displaystyle\begin{array}{rcl} M_{\varLambda }&:=& \varLambda +M_{\mathbb{N}} =\{\varLambda +v;\ v \in M_{\mathbb{N}}\}, {}\\ G_{\lambda }&:=& \{g \in (S^{1})^{\mathbb{N}};\ \sum _{ n\in \mathbb{N}}(1 + \vert \lambda _{n}\vert )\vert 1 - g_{n}\vert ^{2} < +\infty,\ \prod _{ n\in \mathbb{N}}g_{n} = 1\}. {}\\ \end{array}$$

Here, $\prod _{n\in \mathbb{N}}g_{n}$ always converges by the condition

$$\displaystyle\begin{array}{rcl} \sum _{n\in \mathbb{N}} \frac{1} {1 + \vert \lambda _{n}\vert } < +\infty.& & {}\\ \end{array}$$

Then G _λ is an infinite dimensional Lie group, and G _λ acts on M _Λ by $xg:= (x_{n}g_{n})_{n\in \mathbb{N}}$ for x ∈ M _Λ , g ∈ G _λ .

Now G _λ acts on

$$\displaystyle\begin{array}{rcl} N_{\varLambda } =\{ x \in M_{\varLambda };\ x_{n}i\overline{x}_{n} -\lambda _{n} = x_{m}i\overline{x}_{m} -\lambda _{m}\ \mathrm{for\ all\ }n,m \in \mathbb{N}\}& & {}\\ \end{array}$$

and we obtain the quotient space $N_{\varLambda }/G_{\lambda }$ which is called the hyper-Kähler quotient. Here, N _Λ corresponds to the level set of the hyper-Kähler moment map.

Definition 2.

$\lambda \in (\mathrm{Im}\mathbb{H})_{0}^{\mathbb{N}}$ is generic if $\lambda _{n} -\lambda _{m}\neq 0$ for all distinct $n,m \in \mathbb{N}$.

Theorem 1 ([5]).

If $\lambda \in (\mathrm{Im}\mathbb{H})_{0}^{\mathbb{N}}$ is generic, then $N_{\varLambda }/G_{\lambda }$ is a smooth manifold of real dimension 4, and the hyper-Kähler structure on M _Λ induces a hyper-Kähler structure ω _λ on $N_{\varLambda }/G_{\lambda }$.

Although the hyper-Kähler quotient $N_{\varLambda }/G_{\lambda }$ seems to depend on the choice of $\varLambda \in \mathbb{H}^{\mathbb{N}}$, the induced hyper-Kähler structure on $N_{\varLambda }/G_{\lambda }$ depends only on λ by the argument of Section 2 of [7]. Accordingly we may put

$$\displaystyle\begin{array}{rcl} X(\lambda )&:=& N_{\varLambda }/G_{\lambda } {}\\ & =& \{x \in M_{\varLambda };x_{n}i\bar{x}_{n} -\lambda _{n}\ \mathrm{is\ independent\ of}\ n \in \mathbb{N}\}/G_{\lambda }, {}\\ \end{array}$$

and call it a hyper-Kähler manifold of type A _∞

If $\mathbb{N}$ is replaced by a finite set in the above construction, (X(λ), ω _λ ) becomes an ALE hyper-Kähler manifold of type A _k [4].

2.2 S ¹-actions and Moment Maps

An S ¹-action on X(λ) preserving the hyper-Kähler structure is defined as follows. (See also [5].) Let $[x] \in N_{\varLambda }/G_{\lambda }$ be the equivalence class represented by x ∈ N _Λ . Take $m \in \mathbb{N}$ arbitrarily and let

$$\displaystyle\begin{array}{rcl} [x]g:= [x_{m}g,(x_{n})_{n\in \mathbb{N}\setminus \{m\}}]& & {}\\ \end{array}$$

for $x = (x_{m},(x_{n})_{n\in \mathbb{N}\setminus \{m\}}) \in N_{\varLambda }$ and g ∈ S ¹. This definition does not depend on the choice of $m \in \mathbb{N}$. Then we obtain the hyper-Kähler moment map

$$\displaystyle\begin{array}{rcl} \mu _{\lambda }([x]):= x_{n}i\bar{x}_{n} -\lambda _{n} \in \mathrm{ Im}\mathbb{H}.& & {}\\ \end{array}$$

The right hand side is independent of the choice of $n \in \mathbb{N}$ since x is an element of N _Λ .

We have a principal S ¹-bundle $\mu _{\lambda }\big\vert _{X(\lambda )^{{\ast}}}: X(\lambda )^{{\ast}}\rightarrow Y (\lambda )$, where

$$\displaystyle\begin{array}{rcl} X(\lambda )^{{\ast}}&:=& \{[x] \in X(\lambda );\ x_{ n}\neq 0\ \mathrm{for\ all\ }n \in \mathbb{N}\}, {}\\ Y (\lambda )&:=& \mathrm{Im}\mathbb{H}\setminus \{-\lambda _{n};\ n \in \mathbb{N}\}. {}\\ \end{array}$$

By the Gibbons-Hawking construction [1], we can check easily that X(λ) and X(λ′) are isomorphic as hyper-Kähler manifolds if λ and λ′ satisfy one of the following conditions; (i) $\lambda '_{n} -\lambda _{n} \in \mathrm{ Im}\mathbb{H}$ is independent of n, (ii) $\lambda '_{n} =\lambda _{a(n)}$ for some bijective maps $a: \mathbb{N} \rightarrow \mathbb{N}$, (iii) $\lambda = -\lambda '$.

3 The Volume Growth

Here we focus on the Riemannian geometric aspects of X(λ), especially their volume growth.

For a Riemannian manifold (X, g), denote by V _g (p, r) the volume of the geodesic ball of radius r > 0 centered at p ∈ X. By the volume comparison theorem [2, 6], we can deduce that

$$\displaystyle\begin{array}{rcl} \lim _{r\rightarrow \infty }\frac{V _{g}(p_{0},r)} {V _{g}(p_{1},r)} = 1& & {}\\ \end{array}$$

for any Ricci flat manifold (X, g) and any p ₀, p ₁ ∈ X. Thus the volume growth of g is the invariant for Ricci flat manifolds.

Theorem 2 ([7]).

For each $\lambda \in (\mathrm{Im}\mathbb{H})_{0}^{\mathbb{N}}$ and p ₀ ∈ X(λ), the function $V _{g_{\lambda }}(p_{0},r)$ satisfies

$$\displaystyle\begin{array}{rcl} 0 <\liminf _{r\rightarrow +\infty }\frac{V _{g_{\lambda }}(p_{0},r)} {r^{2}\tau _{\lambda }^{-1}(r^{2})} \leq \limsup _{r\rightarrow +\infty }\frac{V _{g_{\lambda }}(p_{0},r)} {r^{2}\tau _{\lambda }^{-1}(r^{2})} < +\infty,& & {}\\ \end{array}$$

where the function $\tau _{\lambda }: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ is defined by

$$\displaystyle\begin{array}{rcl} \tau _{\lambda }(R):=\sum _{n\in \mathbb{N}} \frac{R^{2}} {R + \vert \lambda _{n}\vert }& & {}\\ \end{array}$$

for R ≥ 0. Moreover, we have

$$\displaystyle\begin{array}{rcl} \lim _{r\rightarrow +\infty }\frac{V _{g_{\lambda }}(p_{0},r)} {r^{4}} = 0,\quad \lim _{r\rightarrow +\infty }\frac{V _{g_{\lambda }}(p_{0},r)} {r^{3}} = +\infty.& & {}\\ \end{array}$$

Next we see some examples computed in [7].

Example 1.

Fix γ > 1 and put $\lambda _{n}^{\gamma }:= i \cdot n^{\gamma } \in \mathrm{ Im}\mathbb{H}$. Then there exist positive constants A, B > 0 such that

$$\displaystyle\begin{array}{rcl} Ar^{4- \frac{2} {^{\gamma }+1} } \leq V _{g_{\lambda ^{\gamma }}}((p_{0},r) \leq Br^{4- \frac{2} {^{\gamma }+1} }.& & {}\\ \end{array}$$

Example 2.

Put $\lambda _{n}:= i \cdot e^{n} \in \mathrm{ Im}\mathbb{H}$. Then there exist positive constants A, B > 0 such that

$$\displaystyle\begin{array}{rcl} A\frac{r^{4}} {\log r} \leq V _{g_{\lambda }}(p_{0},r) \leq B\frac{r^{4}} {\log r} & & {}\\ \end{array}$$

for any α < 4.

4 Period Maps

4.1 Holomorphic Curves

In this subsection, we see that there are several compact minimal submanifolds in X(λ) following [8].

Definition 3.

(i) Let X be a complex manifold of dimension 2n and $\omega _{\mathbb{C}}$ be a holomorphic 2-form on X. Then $(X,\omega _{\mathbb{C}})$ is called a holomorphic symplectic manifold if $d\omega _{\mathbb{C}} = 0$ and $\omega _{\mathbb{C}}^{n}$ is nowhere vanishing. (ii) An n dimensional complex submanifold L of a holomorphic symplectic manifold $(X,\omega _{\mathbb{C}})$ is holomorphic Lagrangian submanifold if $\omega _{\mathbb{C}}\vert _{L} = 0$.

Let (X, ω) be a hyper-Kähler manifold of real dimension 4n. For each $y \in \mathrm{ Im}\mathbb{H}$ with | y |  = 1, $\mathrm{Im}\mathbb{H}$ is decomposed into y-component and its orthogonal complement. Then we denote by $\omega _{y} \in \varOmega ^{2}(X)$ the y-component of $\omega \in \varOmega ^{2}(X) \otimes \mathrm{ Im}\mathbb{H}$. Let I _y be the complex structure corresponding to the Kähler form ω _y .

Let $\eta = (\eta _{1}\ \eta _{2}\ \eta _{3}) \in SO(3)$, where $\langle \eta _{1},\eta _{2},\eta _{3}\rangle$ is an orthonormal basis of $\mathbb{R}^{3}$. Then η gives the orthogonal decomposition $\mathrm{Im}\mathbb{H} = \mathbb{R}^{3} = \mathbb{R}\eta _{1} \oplus \mathbb{R}\eta _{2} \oplus \mathbb{R}\eta _{3}$, and the hyper-Kähler structure $\omega \in \varOmega ^{2}(X) \otimes \mathrm{ Im}\mathbb{H}$ can be written as $\omega =\eta _{1}\omega _{\eta _{1}} +\eta _{2}\omega _{\eta _{2}} +\eta _{3}\omega _{\eta _{3}}$ for every η ∈ SO(3). Now we regard $(X,I_{\eta _{1}})$ as a complex manifold. Then a holomorphic symplectic structure on X is given by $\omega _{\eta _{\mathbb{C}}}:=\omega _{\eta _{2}} + i\omega _{\eta _{3}}$.

Proposition 1.

Let (X,ω) be a hyper-Kähler manifold and take η ∈ SO(3). Then each holomorphic Lagrangian submanifold L ⊂ X with respect to $\omega _{\eta _{\mathbb{C}}}$ gives the minimum volume in their homology class.

Proof.

The pair of a Kähler form $\omega _{\eta _{3}}$ and a holomorphic volume form $(\omega _{\eta _{1}} + i\omega _{\eta _{2}})^{n}$ gives the Calabi-Yau structure on $(X,I_{\eta _{3}})$. Here, n is the half of the complex dimension of X. Now, assume that L ⊂ X is a holomorphic Lagrangian submanifold with respect to $\omega _{\eta _{\mathbb{C}}}$. Then $\omega _{\eta _{2}}\vert _{L} =\omega _{\eta _{3}}\vert _{L} = 0$, hence L is lagrangian with respect to $\omega _{\eta _{3}}$. Since $\mathrm{Im}(\omega _{\eta _{1}} + i\omega _{\eta _{2}})^{n}$ is the multiplication of $\omega _{\eta _{2}}$ and some differential forms, we also have $\mathrm{Im}(\omega _{\eta _{1}} + i\omega _{\eta _{2}})^{n}\vert _{L} = 0$, which means L is a special Lagrangian submanifold. The volume minimizing property of special Lagrangian submanifolds [11] gives the assertion. □ 

Take a generic $\lambda \in (\mathrm{Im}\mathbb{H})_{0}^{\mathbb{N}}$ and consider the hyper-Kähler manifold (X(λ), ω _λ ) as constructed in Sect. 2. Put

$$\displaystyle\begin{array}{rcl} [a,b]&:=& \{ta + (1 - t)b \in \mathrm{ Im}\mathbb{H};\ 0 \leq t \leq 1\}, {}\\ (a,b]&:=& \{ta + (1 - t)b \in \mathrm{ Im}\mathbb{H};\ 0 \leq t < 1\}, {}\\ \ [a,b)&:=& \{ta + (1 - t)b \in \mathrm{ Im}\mathbb{H};\ 0 < t \leq 1\}, {}\\ (a,b)&:=& \{ta + (1 - t)b \in \mathrm{ Im}\mathbb{H};\ 0 < t < 1\} {}\\ \end{array}$$

for $a,b \in \mathrm{ Im}\mathbb{H}$.

Proposition 2.

Let $n,m \in \mathbb{N}$ satisfy n ≠ m and $(-\lambda _{n},-\lambda _{m}) \subset Y (\lambda )$ . The inverse image $\mu _{\lambda }^{-1}([-\lambda _{n},-\lambda _{m}])\mathop{\cong}\mathbb{C}P^{1}$ is a complex submanifold of X(λ) with respect to I _y and gives the minimum volume in its homology class, where $y:= \frac{\lambda _{n}-\lambda _{m}} {\vert \lambda _{n}-\lambda _{m}\vert }$.

Proof.

Let η ∈ SO(3) satisfies η i = y. If we write $\mu _{\lambda } = (\mu _{\lambda,1},\mu _{\lambda,2},\mu _{\lambda,3})$ with respect to the decomposition $\mathrm{Im}\mathbb{H} = \mathbb{R}\eta _{1} \oplus \mathbb{R}\eta _{2} \oplus \mathbb{R}\eta _{3}$, then μ _λ, 2 and μ _λ, 3 are constant on $\mu _{\lambda }^{-1}([-\lambda _{n},-\lambda _{m}])$. Hence we have $d\mu _{\lambda,\alpha }\vert _{\mu _{\lambda }^{-1}([-\lambda _{n},-\lambda _{m}])} = 0$ for α = 2, 3, which gives $\omega _{\lambda,\eta _{\mathbb{C}}}\vert _{\mu _{\lambda }^{-1}([-\lambda _{n},-\lambda _{m}])} = 0$. □ 

4.2 Topology

In this subsection we review the construction of the deformation retracts of X(λ) following [3, 5]. See also [8]. In the case of toric hyper-Kähler varieties, the deformation retracts are constructed in [3].

For $(-\lambda _{n},-\lambda _{m}) \subset Y (\lambda )$, the orientation of $\mu _{\lambda }^{-1}([-\lambda _{n},-\lambda _{m}])$ is determined as follows. By taking a smooth section $(-\lambda _{n},-\lambda _{m}) \rightarrow \mu _{\lambda }^{-1}((-\lambda _{n},-\lambda _{m}))$ of μ _λ , a coordinate (s, t) on $\mu _{\lambda }^{-1}((-\lambda _{n},-\lambda _{m}))$ is naturally given where $t \in \mathbb{R}/2\pi \mathbb{Z}$ is the parameter of S ¹-action and a function $s:\mu _{ \lambda }^{-1}((-\lambda _{n},-\lambda _{m})) \rightarrow \mathbb{R}$ is given by

$$\displaystyle\begin{array}{rcl} s(p):= \frac{\lambda _{n} +\mu _{\lambda }(p)} {\lambda _{n} -\lambda _{m}} & & {}\\ \end{array}$$

for $p \in \mu _{\lambda }^{-1}((-\lambda _{n},-\lambda _{m}))$. Then the orientation of $\mu _{\lambda }^{-1}([-\lambda _{n},-\lambda _{m}])$ is given by ds ∧ dt. Therefore, $\mu _{\lambda }^{-1}([-\lambda _{n},-\lambda _{m}])$ and $\mu _{\lambda }^{-1}([-\lambda _{m},-\lambda _{n}])$ are same as manifolds but have opposite orientations.

For $n,m,l \in \mathbb{N}$, $\mu _{\lambda }^{-1}([-\lambda _{n},-\lambda _{m}]) \cup \mu _{\lambda }^{-1}([-\lambda _{m},-\lambda _{l}])$ and $\mu _{\lambda }^{-1}([-\lambda _{n},-\lambda _{l}])$ determines the same homology class since the boundary of $\mu _{\lambda }^{-1}(\bigtriangleup _{n,m,l})$ is given by $\mu _{\lambda }^{-1}([-\lambda _{n},-\lambda _{m}]) \cup \mu _{\lambda }^{-1}([-\lambda _{m},-\lambda _{l}]) \cup \mu _{\lambda }^{-1}([-\lambda _{l},-\lambda _{n}])$, where

$$\displaystyle\begin{array}{rcl} \bigtriangleup _{n,m,l}:=\{ -\alpha \lambda _{n} -\beta \lambda _{m} -\gamma \lambda _{l} \in \mathrm{ Im}\mathbb{H};\ \alpha +\beta +\gamma = 1,\ \alpha,\beta,\gamma \geq 0\}.& & {}\\ \end{array}$$

We denote by C _n, m the homology class determined by $\mu _{\lambda }^{-1}([-\lambda _{n},-\lambda _{m}])$. Then the above observation implies

$$\displaystyle\begin{array}{rcl} C_{n,m} + C_{m,l} + C_{l,n} = C_{n,m} + C_{m,n} = 0& & {}\\ \end{array}$$

for $n,m,l \in \mathbb{N}$.

If $n,m,l,h \in \mathbb{N}$ satisfies n ≠ h, n ≠ m and l ≠ h then the intersection number $C_{n,m} \cdot C_{l,h}$ is given by

$$\displaystyle{C_{n,m}\cdot C_{l,h} = \left \{\begin{array}{ccc} 1&(m = l)\\ 0 & (m\neq l) \end{array} \right.}$$

and $C_{n,m} \cdot C_{n,m} = -2$.

Since the subset of $(\mathrm{Im}\mathbb{H})_{0}^{\mathbb{N}}$ consisting of generic elements is connected in $(\mathrm{Im}\mathbb{H})_{0}^{\mathbb{N}}$, the topological structure of X(λ) does not depend on λ. Consequently, it suffices to study $X(\hat{\lambda })$ for investigating the topology of X(λ), where $\hat{\lambda }$ is the special one defined by $\hat{\lambda }_{n}:= (n^{2},0,0) \in \mathrm{ Im}\mathbb{H}$.

Proposition 3.

There exists a deformation retract of $\mu _{\hat{\lambda }}^{-1}(\bigcup _{n\in \mathbb{N}}[-\hat{\lambda }_{n},-\hat{\lambda }_{n+1}]) \subset X(\hat{\lambda })$.

Proof.

There is a deformation retract

$$\displaystyle\begin{array}{rcl} F:\mathrm{ Im}\mathbb{H} \times [0,1] \rightarrow \mathrm{ Im}\mathbb{H}& & {}\\ \end{array}$$

which satisfy $F(\cdot,0) = id_{\mathrm{Im}\mathbb{H}}$, $F(\mathrm{Im}\mathbb{H},1) =\bigcup _{n\in \mathbb{N}}[-\hat{\lambda }_{n},-\hat{\lambda }_{n+1}]$ and F(ζ, 1) = ζ for $\zeta \in \bigcup _{n\in \mathbb{N}}[-\hat{\lambda }_{n},-\hat{\lambda }_{n+1}]$. Then we have the horizontal lift $\tilde{F}: X(\hat{\lambda }) \times [0,1] \rightarrow X(\hat{\lambda })$ of F by using the S ¹-connection on $X(\hat{\lambda })^{{\ast}}$ naturally induced from the hyper-Kähler metric on $X(\hat{\lambda })^{{\ast}}$. The map $\tilde{F}$ is a deformation retract as we expect. □ 

Corollary 1.

The second homology group $H_{2}(X(\lambda ), \mathbb{Z})$ is generated by $\{C_{n,m};\ n,m \in \mathbb{N}\}$.

Thus we obtain the followings.

Theorem 3.

Let $\lambda \in (\mathrm{Im}\mathbb{H})_{0}^{\mathbb{N}}$ be generic. Then $H_{2}(X(\lambda ), \mathbb{Z})$ is a free $\mathbb{Z}$ -module generated by $\{C_{n,m};\ n,m \in \mathbb{N}\}$ with relations

$$\displaystyle\begin{array}{rcl} C_{n,m} + C_{m,l} + C_{l,n} = 0,C_{n,m} + C_{m,n} = 0& & {}\\ \end{array}$$

for all $n,m,l \in \mathbb{N}$ . Moreover the intersection form on $H_{2}(X(\lambda ), \mathbb{Z})$ is given by

$$\displaystyle{C_{n,m}\cdot C_{l,h} = \left \{\begin{array}{ccc} 1&(m = l)\\ 0 & (m\neq l) \end{array} \right.}$$

and $C_{n,m} \cdot C_{n,m} = -2$ for $n,m,l,h \in \mathbb{N}$ taken to be n ≠ h, n ≠ m and l ≠ h.

4.3 Period Maps

Let $[\omega _{\lambda }] \in H^{2}(X(\lambda ), \mathbb{R}) \otimes \mathrm{ Im}\mathbb{H}$ be the cohomology class of ω _λ . In this subsection we compute [ω _λ ], that is, compute the value of $\langle [\omega _{\lambda }],C_{n,m}\rangle:=\int _{C_{n,m}}\omega _{\lambda } \in \mathrm{ Im}\mathbb{H}$ for all $n,m \in \mathbb{N}$ along [8]. In the case of finite topological type of toric hyper-Kähler varieties, the period maps are computed in [12].

Theorem 4.

Let $\lambda \in (\mathrm{Im}\mathbb{H})_{0}^{\mathbb{N}}$ be generic. Then

$$\displaystyle\begin{array}{rcl} \langle [\omega _{\lambda }],C_{n,m}\rangle =\lambda _{n} -\lambda _{m}& & {}\\ \end{array}$$

for all $n,m \in \mathbb{N}$.

Proof.

Take a smooth path $\gamma: [0,1] \rightarrow \mathrm{ Im}\mathbb{H}$ such that $\gamma (0) = -\lambda _{n}$, $\gamma (1) = -\lambda _{m}$ and γ(s) ∈ Y (λ) for s ∈ (0, 1). Since the homology class represented by $\mu _{\lambda }^{-1}(\gamma ([0,1]))$ is C _n, m , we have

$$\displaystyle\begin{array}{rcl} \langle [\omega _{\lambda }],C_{n,m}\rangle =\int _{\mu _{\lambda }^{-1}(\gamma ([0,1]))}\omega _{\lambda }.& & {}\\ \end{array}$$

Take the local coordinate $(t,\mu _{\lambda,1},\mu _{\lambda,2},\mu _{\lambda,3})$ of an open subset of X(λ)^∗, where $\mu _{\lambda } = (\mu _{\lambda,1},\mu _{\lambda,2},\mu _{\lambda,3})$ and t is the coordinate of S ¹-action. Then the local coordinate (s, t) on $\mu _{\lambda }^{-1}(\gamma ([0,1]))$ is given by $(t,\mu _{\lambda,1} \circ \gamma (s),\mu _{\lambda,2} \circ \gamma (s),\mu _{\lambda,3} \circ \gamma (s))$. By using this, we can see that

$$\displaystyle\begin{array}{rcl} \omega _{\lambda,\alpha } =\gamma _{\alpha }'(s)\frac{1} {2\pi }ds \wedge dt& & {}\\ \end{array}$$

for α = 1, 2, 3, where $\gamma (s) = (\gamma _{1}(s),\gamma _{2}(s),\gamma _{3}(s)) \in \mathrm{ Im}\mathbb{H} = \mathbb{R}^{3}$. Hence we have

$$\displaystyle\begin{array}{rcl} \int _{\mu _{\lambda }^{-1}(\gamma ([0,1]))}\omega _{\lambda,\alpha }& =& \int _{\mu _{\lambda }^{-1}(\gamma ([0,1]))}\gamma _{\alpha }'(s)\frac{1} {2\pi }ds \wedge dt {}\\ & =& \int _{0}^{2\pi }\frac{1} {2\pi }dt\int _{0}^{1}\gamma _{ \alpha }'(s)ds {}\\ & =& \gamma _{\alpha }(1) -\gamma _{\alpha }(0) =\lambda _{n,\alpha } -\lambda _{m,\alpha }. {}\\ \end{array}$$

 □ 

5 Holomorphic Symplectic Structures

In this section we regard a hyper-Kähler manifold $(X,g,I_{1},I_{2},I_{3})$ as a complex manifold by I ₁. Then the holomorphic 2-form $\omega _{\mathbb{C}} =\omega _{2} + \sqrt{-1}\omega _{3}$ is called the holomorphic symplectic structure, and the cohomology class $[\omega _{2} + \sqrt{-1}\omega _{3}]$ is called the holomorphic symplectic class.

Let λ ^γ as in Example 1 of Sect. 3. Then, we can see that the holomorphic symplectic class $[\omega _{\lambda ^{\gamma },\mathbb{C}}]$ is independent of γ by Theorem 4.

Theorem 5 ([9]).

The holomorphic symplectic structures $\omega _{\lambda ^{\gamma },\mathbb{C}}$ are independent of γ. In particular, X(λ ^γ ) and $X(\lambda ^{\hat{\gamma }})$ are biholomorphic for all $\gamma,\hat{\gamma }> 1$.

Since the function $4 - \frac{2} {^{\gamma }+1}$ gives one-to-one correspondence between open intervals (1, ∞) and (3, 4), we have the following conclusion by combining Theorems 2 with 5.

Theorem 6.

Let α ∈ (3,4). Then there is a complex manifold X and the family of Ricci-flat Kähler metrics $\{g_{\alpha }\}_{3<\alpha <4}$ whose volume growth satisfies

$$\displaystyle\begin{array}{rcl} Ar^{\alpha } \leq V _{g_{\alpha }}(p_{0},r) \leq Br^{\alpha }& & {}\\ \end{array}$$

for some positive constants A,B.