Variation of toric hyperKähler manifolds

Hiroshi Konno

2000 Mathematics Subject Classification: Primary 53C26; Secondary 53D20, 32G05

Dedicated to Professor Takushiro Ochiai for his 60th birthday

Abstract. A toric hyperKähler manifold is defined to be a smooth hyperKähler quotient of the quaternionic vector space ℍ^{N} by a subtorus of T^{N}. It has two parameters corresponding to the de Rham cohomology classes represented by the Kähler form and the complex symplectic form respectively. We study the variation of its complex structure according to these parameters. After the detailed analysis of the stability condition depending on the first parameter, we show that toric hyperKähler manifolds with the same second parameter are related by a sequence of Mukai’s elementary transformations. We also give a complete description of its Kähler cone and discuss when certain rational curves exist.

1 Introduction

A hyperKähler manifold is a 4n-dimensional Riemannian manifold whose holonomy group is contained in Sp(n). That is, the Riemannian metric g is Kähler with respect to three complex structures I_{1}, I_{2}, I_{3}, which satisfies the quaternion relations. It is a fundamental building block in Riemannian geometry, due to Berger’s classification of holonomy groups [2]. Fujiki and Beauville constructed examples of compact irreducible hyperKähler manifolds of higher dimensions [6] [1], which can be considered as higher dimensional analogues of K3 surfaces. On the other hand, many non-compact examples are constructed by the hyperKähler quotient method due to [10].

One of the important problems in hyperKähler geometry is to understand the global structure of the deformation space of hyperKähler manifolds. For a compact irreducible hyperKähler manifold, Beauville introduced the period map and established the local Torelli theorem [1]. Recently Huybrechts proved the surjectivity of the period map, and proved that non-separating points in the moduli space correspond to birational hyperKähler manifolds [11] [12]. In his study the Kähler cone plays an important role to investigate birational transformations between hyperKähler manifolds.

A toric hyperKähler manifold is not a toric manifold in the usual sense, but a corresponding object in hyperKähler geometry. It was introduced by Bielawski and Dancer [4] and further studied in [14] and [9]. It is defined to be a smooth hyperKähler quotient of the quaternionic vector space ℍ^{N} by a subtorus of T^{N}. Therefore it has certain parameters. In this paper, we study the variation of its complex structure according to these parameters and give a global description of this family of hyperKähler manifolds. Bielawski intrinsically characterize a toric hyperKähler manifold as a complete hyperKähler T^{n}-manifold of dimension 4n with Euclidean volume growth [5]. So our family of hyperKähler manifolds can be considered as a component of the moduli space of such manifolds.

Let us explain our results in more detail. Let K be a subtorus of T^{N} acting on ℍ^{N} with the hyperKähler moment map μ = (μ_{1}, μ_{ℂ}): ℍ^{N} → k^{∗} ⊗ ℝ^{3} ≅ k^{∗} × k_{ℂ}^{∗}. If (α, β) ∈ k^{∗} × k_{ℂ}^{∗} is a regular value of μ, then we have a toric hyperKähler manifold X(α, β) = μ^{−1}(α, β)/K. First we show that, under mild conditions, the parameters α ∈ k^{∗}, β ∈ k_{ℂ}^{∗} can be interpreted as its periods. That is, by the natural isomorphism κ^{∗}: k^{∗} → H^{2}(X(α, β); ℝ), the parameters α, β are identified with the de Rham cohomology classes represented by the Kähler form and the complex symplectic form respectively.

Next we show that the parameter β determines the birational class of the complex manifold (X(α, β), I_{1}). More precisely, if we fix the parameter β ∈ k_{ℂ}^{∗}, we have the chamber structure in k^{∗}. If α′ ∈ k^{∗} belongs to the same chamber as α, (X(α′, β), I_{1}) is canonically isomorphic to (X(α, β), I_{1}) as a complex manifold. If α, α′ ∈ k^{∗} belong to the next chamber to each other, two complex manifolds (X(α, β), I_{1}) and (X(α′, β), I_{1}) are related by an elementary transformation introduced by Mukai [16], which is a special birational transformation between complex symplectic manifolds.

In addition, we discuss when (X(α, β), I_{1}) contains a smooth rational curve, whose volume goes to zero as the ‘Kähler class’ α approaches to the wall. This arrows us to give a complete description of the Kähler cone of every toric hyperKähler manifold. We also discuss when (X(α, β), I_{1}) is affine. These results give a good global description of our family of toric hyperKähler manifolds.

The proof of the above results involves the detailed analysis of the stability. The complexification of the subtorus K_{ℂ} acts on μ_{ℂ}^{−1}(β). The stability condition for this action depends on the polarization, which corresponds to α ∈ k^{∗}. The change of the quotient varieties with respect to different polarizations has been studied by many authors [8] [20]. In this case, this change is proved to be expressed by a more specific birational transformation because of their complex symplectic structures. Thaddeus showed similar phenomena in the case of the moduli space of parabolic Higgs bundles [21].

The content of this paper is as follows. In Section 2, we review the definition and basic properties of toric hyperKähler manifolds. The period map is discussed in Section 3. In Section 4, we review an elementary transformation and explain how it appears naturally in our context. Next we study the stability condition and give useful criterions of stable points in Section 5. In Section 6, we study the variation of toric hyperKähler manifolds and prove the main results mentioned above. In the final section, we give an example.

A part of this work has been done during the author’s stay at Max-Planck-Institut für Mathematik in Bonn. The author would like to express his gratitude to the institute for its support and good environment for research.

2 Toric hyperKähler manifolds

The standard hyperKähler structure on the quaternionic vector space ℍ^{N} is the standard flat Riemannian metric g together with three complex structures I_{1}, I_{2}, I_{3}, which are defined by the left multiplication of i, j, k respectively. Throughout this paper we identify ℍ^{N} with ℂ^{N} × ℂ^{N} with respect to I_{2}. Then these complex structures are given by

I_{1}(z, w) = (\sqrt{−1}z, \sqrt{−1}w), I_{2}(z, w) = (−\overline{w}, \overline{z}), I_{3}(z, w) = (−\sqrt{−1}\overline{w}, \sqrt{−1}\overline{z}).

The real torus T^{N} = {ζ = (ζ_{1}, …, ζ_{N}) ∈ ℂ^{N} | |ζ_{i}| = 1 for i = 1, …, N} acts on ℍ^{N} by (z, w)ζ = (zζ, wζ^{−1}), preserving the hyperKähler structure (g; I_{1}, I_{2}, I_{3}) on ℍ^{N}. Let t^{N} be the Lie algebra of T^{N} and {X_{1}, …, X_{N}} its basis such that Exp(∑_{i=1}^{N} s_{i}X_{i}) = (e^{2π\sqrt{−1}s_{1}}, …, e^{2π\sqrt{−1}s_{N}}) ∈ T^{N}. The dual basis is denoted by {u_{1}, …, u_{N}} ⊂ (t^{N})^{∗}.

Let K be a subtorus of T^{N} and T^{n} = T^{N}/K the quotient torus with the Lie algebras k and t^{n} respectively. So we have the following exact sequences;

\begin{matrix}0 & ⟶ & k & ⟶^{ι} & t^{N} & ⟶^{π} & t^{n} & ⟶ & 0, \\ 0 & ⟵ & k^{∗} & ⟵^{ι^{∗}} & (t^{N})^{∗} & ⟵^{π^{∗}} & (t^{n})^{∗} & ⟵ & 0. \end{matrix}

The hyperKähler moment map for the action of K on ℍ^{N}

μ = (μ_{1}, μ_{2}, μ_{3}): ℍ^{N} → k^{∗} ⊗ ℝ^{3} ≅ k^{∗} × k_{ℂ}^{∗}

is given by

μ_{1}(z, w) = π ∑_{i=1}^{N}(|z_{i}|^{2} − |w_{i}|^{2})ι^{∗}u_{i} ∈ k^{∗},

μ_{ℂ}(z, w) = −2π\sqrt{−1} ∑_{i=1}^{N} z_{i}w_{i}ι^{∗}u_{i} ∈ k_{ℂ}^{∗},

where μ_{ℂ} = μ_{2} + \sqrt{−1}μ_{3} and k_{ℂ}^{∗} = k^{∗} ⊗ ℂ.

Definition 2.1. If the subtorus K acts on μ^{−1}(α, β) freely for (α, β) ∈ k^{∗} × k_{ℂ}^{∗}, the hyperKähler quotient X(α, β) = μ^{−1}(α, β)/K is called a toric hyperKähler manifold.

Remark 2.2. A toric hyperKähler manifold X(α, β) is a smooth non-compact complete connected hyperKähler manifold of dimension 4n. The quotient torus T^{n} = T^{N}/K acts on X(α, β), preserving the hyperKähler structure. Note that a toric hyperKähler manifold is not a toric manifold in the usual sense, but a corresponding object in hyperKähler geometry.

Throughout this paper we also denote the hyperKähler structure on X(α, β) by (g; I_{1}, I_{2}, I_{3}). The corresponding Kähler forms are denoted by ω_{1}, ω_{2}, ω_{3} respectively. If we consider X(α, β) as a complex manifold with respect to I_{1}, we denote it by (X(α, β), I_{1}). Since ω_{ℂ} = ω_{2} + \sqrt{−1}ω_{3} is a holomorphic non-degenerate 2-form on (X(α, β), I_{1}), it is called the complex symplectic form of X(α, β).

In [14] we studied when the hyperKähler quotient μ^{−1}(α, β)/K is a toric hyperKähler manifold.

Proposition 2.3. [14] Let {P_{1}, …, P_{l}} be the set of all subspaces of codimension one in k^{∗} which are spanned by subsets of {ι^{∗}u_{i}}_{i=1}^{N}. If we denote the set of regular values of the hyperKähler moment map μ by (k^{∗} × k_{ℂ}^{∗})_{reg}, then we have

(k^{∗} × k_{ℂ}^{∗})_{reg} = (k^{∗} × k_{ℂ}^{∗}) \ ⋃_{s=1}^{l}(P_{s} × P_{sℂ}),

where P_{sℂ} = P_{s} ⊗ ℂ ⊂ k_{ℂ}^{∗}.

Proposition 2.4. [14] The subtorus K acts freely on μ^{−1}(α, β) if and only if the following conditions (R) and (C1) hold: (R) (α, β) ∈ (k^{∗} × k_{ℂ}^{∗})_{reg}. (C1) If {ι^{∗}u_{i}}_{i∈J}, where J ⊂ {1, …, N}, is a basis of k^{∗}, then the following holds;

t_{ℤ}^{N} = k_{ℤ} + ∑_{i∈{1,...,N}\J} ℤX_{i} as a ℤ-module,

where t_{ℤ}^{N} = ∑_{i=1}^{N} ℤX_{i} and k_{ℤ} = k ∩ t_{ℤ}^{N}.

Remark 2.5. The condition (R) ensures that the hyperKähler quotient μ^{−1}(α, β)/K is an orbifold of the expected dimension. The condition (C1) ensures that the orbifold is a manifold. Note that the condition (C1) depends only on the subtorus K, not on (α, β).

Later we will impose the following condition (C2) on the subtorus K:

(C2) X_{i} ∉ k and ι^{∗}u_{i} ≠ 0 for i = 1, …, N.

The meaning of the condition (C2) is as follows. Suppose that X_{i_{0}} ∈ k. Then the toric hyperKähler manifold X(α, β) can be also obtained as a hyperKähler quotient of ℍ^{N−1} by K ∩ T^{N−1}, where ℍ^{N−1} = {(z, w) ∈ ℍ^{N} | z_{i_{0}} = w_{i_{0}} = 0} and T^{N−1} = {ζ ∈ T^{N} | ζ_{i_{0}} = 1}. On the other hand, suppose that ι^{∗}u_{i_{0}} = 0. Then the subtorus K is a subgroup of T^{N−1}, and X(α, β) is decomposed into the product of a hyperKähler quotient of ℍ^{N−1} by K with ℍ. These cases will not be essential for our argument. The condition (C2) excludes such cases.

Since the hyperKähler moment map μ: ℍ^{N} → k^{∗} ⊗ ℝ^{3} is surjective, X(α, β) is not empty for all (α, β) ∈ (k^{∗} × k_{ℂ}^{∗})_{reg}. Moreover we have the following.

Theorem 2.6. The diffeomorphism type of the toric hyperKähler manifold X(α, β) is independent of (α, β) ∈ (k^{∗} × k_{ℂ}^{∗})_{reg}.

Remark 2.7. The above result is essentially due to Bielawski and Dancer [4]. They showed that the homeomorphism type is independent by a similar argument due to [15] and [18]. By a slight modification of their argument, the above results can be proved. See Remark 6.2 below for the proof.

Thus we have a family of hyperKähler manifolds parameterized by (α, β) ∈ (k^{∗} × k_{ℂ}^{∗})_{reg}. In [5] Bielawski gave an intrinsic characterization of toric hyperKähler manifolds as follows.

Theorem 2.8. [5] Let M be a complete connected hyperKähler manifold of dimension 4n. Suppose that the torus T^{n} acts on M effectively, preserving the hyperKähler structure, and that M has Euclidean volume growth. Then M is isomorphic to one of toric hyperKähler manifolds in the sense of Definition 2.1 as a hyperKähler T^{n}- manifold.

Thus, roughly speaking, our family of hyperKähler manifolds can be considered as a component of the moduli space of such manifolds. From now on we study the structure of this family of hyperKähler manifolds.

3 The period map

Let X(α, β) = μ^{−1}(α, β)/K be a toric hyperKähler manifold. Define k_{ℤ}^{∗} ⊂ k^{∗} by

k_{ℤ}^{∗} = {v ∈ k^{∗} | ⟨v, Y ⟩ ∈ ℤ for all Y ∈ k_{ℤ}},

where ⟨ , ⟩: k^{∗} × k → ℝ is the natural pairing. Due to the condition (C1), it is easy to see that k_{ℤ}^{∗} = ι^{∗}((t_{ℤ}^{N})^{∗}), where (t_{ℤ}^{N})^{∗} = ∑_{i=1}^{N} ℤu_{i}. Define a K-equivariant line bundle L̃_{ρ} = ℍ^{N} × ℂ for ρ ∈ k_{ℤ}^{∗} by ((z, w), v)ζ = ((zζ, wζ^{−1}), ve^{2π\sqrt{−1}⟨ρ,Y ⟩}), where ζ = ExpY ∈ K.

Let H_{K}^{∗}(ℍ^{N}; ℤ) be the K-equivariant cohomology of ℍ^{N}. Define the map c: k_{ℤ}^{∗} → H_{K}^{2}(ℍ^{N}; ℤ) by c(ρ) = c_{1K}(L̃_{ρ}), where c_{1K}(L̃_{ρ}) ∈ H_{K}^{2}(ℍ^{N}; ℤ) is the K-equivariant first Chern class of L̃_{ρ}. This induces the isomorphism c̃: S^{∗}(k_{ℤ}^{∗}) → H_{K}^{∗}(ℍ^{N}; ℤ) as a graded ring, where S^{∗}(k_{ℤ}^{∗}) is the symmetric product of k_{ℤ}^{∗} and each element of k_{ℤ}^{∗} is of degree 2. Let r: H_{K}^{∗}(ℍ^{N}; ℤ) → H_{K}^{∗}(μ^{−1}(α, β); ℤ) be the restriction map. Since K acts freely on μ^{−1}(α, β), we have the isomorphism j: H_{K}^{∗}(μ^{−1}(α, β); ℤ) → H^{∗}(X(α, β); ℤ). The composition of the above maps

κ_{ℤ} = j ∘ r ∘ c̃: S^{∗}(k_{ℤ}^{∗}) → H^{∗}(X(α, β); ℤ)

is called the Kirwan map. Note that (L̃_{ρ}|_{μ^{−1}(α,β)})/K is a line bundle on X(α, β). Then we have the following. Since the proof is easy, we omit the proof.

Lemma 3.1. The Kirwan map κ_{ℤ} is given by

κ_{ℤ}(ρ) = c_{1}((L̃_{ρ}|_{μ^{−1}(α,β)})/K) ∈ H^{2}(X(α, β); ℤ) for an arbitrary ρ ∈ k_{ℤ}^{∗}.

In [14] we computed the integral cohomology ring of a toric hyperKähler manifold X(α, β) as follows.

Theorem 3.2. [14] Let K be a subtorus of T^{N} satisfying the condition (C1). Let Φ: ℤ[u_{1}, …, u_{N}] → H^{∗}(X(α, β); ℤ) be a ring homomorphism defined by Φ(u_{i}) = κ_{ℤ}(ι^{∗}u_{i}). Then the map Φ is surjective, and the kernel ker Φ ⊂ ℤ[u_{1}, …, u_{N}] is the ideal generated by all

(i) ∑_{i=1}^{N} a_{i}u_{i} ∈ ker ι^{∗} ∩ (t_{ℤ}^{N})^{∗}, (ii) ∏_{b_{i}≠0} u_{i} for ∑_{i=1}^{N} b_{i}X_{i} ∈ k \ {0}. That is, H^{∗}(X(α, β); ℤ) is isomorphic to ℤ[u_{1}, …, u_{N}]/ ker Φ as a ring.

As a corollary, we have the following.

Proposition 3.3. Let K be a subtorus of T^{N} satisfying the conditions (C1) and (C2). Then the restriction of the Kirwan map κ_{ℤ} to k_{ℤ}^{∗} induces the following isomorphism for all (α, β) ∈ (k^{∗} × k_{ℂ}^{∗})_{reg};

κ^{∗} = (κ_{ℤ} ⊗ ℝ)|_{k^{∗}}: k^{∗} → H^{2}(X(α, β); ℝ).

Definition 3.4. Let K be a subtorus of T^{N} satisfying the conditions (C1) and (C2). Then we define the period map P: (k^{∗} × k_{ℂ}^{∗})_{reg} → k^{∗} × k_{ℂ}^{∗} by

P(α, β) = ((κ^{∗})^{−1}([ω_{1}]), (κ^{∗} ⊗ ℂ)^{−1}([ω_{ℂ}])),

where [ω_{1}] ∈ H^{2}(X(α, β); ℝ) and [ω_{ℂ}] ∈ H^{2}(X(α, β); ℂ) denote the de Rham cohomology classes represented by the Kähler form ω_{1} and the complex symplectic form ω_{ℂ} of X(α, β) respectively.

The following theorem implies that, if K be satisfies the conditions (C1) and (C2), the parameters α, β can be interpreted as the periods of X(α, β).

Theorem 3.5. Let K be a subtorus of T^{N} satisfying the conditions (C1) and (C2). Then the period map P: (k^{∗} × k_{ℂ}^{∗})_{reg} → k^{∗} × k_{ℂ}^{∗} is given by P(α, β) = (α, β).

Proof. Let Ω^{p}(ℍ^{N})^{K} be the space of smooth K-invariant differential p-forms on ℍ^{N}. Recall that the complex for the Cartan model of the K-equivariant cohomology is defined by Ω_{K}^{∗}(ℍ^{N}) = S^{∗}(k^{∗}) ⊗ Ω^{∗}(ℍ^{N})^{K}, where the degree of ξ ⊗ ψ ∈ Ω_{K}^{∗}(ℍ^{N}) is given by deg(ξ ⊗ ψ) = deg ξ + deg ψ. Let {Y_{1}, …, Y_{N−n}} be a basis of the Lie algebra k and {v_{1}, …, v_{N−n}} ⊂ k^{∗} the dual basis. The exterior differential operator d_{c}: Ω_{K}^{p}(ℍ^{N}) → Ω_{K}^{p+1}(ℍ^{N}) is defined by

d_{c}(ξ ⊗ ψ) = ξ ⊗ dψ − ∑_{i=1}^{N−n}(ξv_{i}) ⊗ i(Y_{i}^{∗})ψ,

where Y_{i}^{∗} is the vector fields on ℍ^{N} generated by Y_{i} ∈ k and i(Y_{i}^{∗}) denotes the interior product. (See [3] for detail.)

Let ω̃_{j} be the Kähler form on ℍ^{N} with respect to the complex structure I_{j} for j = 1, 2, 3 and ω̃_{ℂ} = ω̃_{2}+\sqrt{−1}ω̃_{3}. Since ω̃_{1}−(μ_{1}−α) ∈ Ω_{K}^{2}(ℍ^{N}) and ω̃_{ℂ}−(μ_{ℂ}−β) ∈ Ω_{K}^{2}(ℍ^{N}) ⊗ ℂ are d_{c}-closed and their restrictions to μ^{−1}(α, β) are ω̃_{1}|μ^{−1}(α, β) and ω̃_{ℂ}|μ^{−1}(α, β) respectively, we have

κ^{∗} ∘ (c̃⊗ ℝ)^{−1}([ω̃_{1} − (μ_{1} − α)]) = [ω_{1}] ∈ H^{2}(X(α, β); ℝ),
(κ^{∗} ⊗ ℂ) ∘ (c̃⊗ ℂ)^{−1}([ω̃_{ℂ} − (μ_{ℂ} − β)]) = [ω_{ℂ}] ∈ H^{2}(X(α, β); ℂ).

On the other hand, if we define 1-forms on ℍ^{N} by

η_{1} = \frac{\sqrt{−1}}{4} ∑_{i=1}^{N}(z_{i}d\overline{z}_{i} − \overline{z}_{i}dz_{i}) + \frac{\sqrt{−1}}{4} ∑_{i=1}^{N}(w_{i}d\overline{w}_{i} − \overline{w}_{i}dw_{i}),

η_{ℂ} = \frac{1}{2} ∑_{i=1}^{N}(z_{i}dw_{i} − w_{i}dz_{i}),

then we have d_{c}η_{1} = ω̃_{1} − μ_{1} and d_{c}η_{ℂ} = ω̃_{ℂ} − μ_{ℂ}. This completed the proof.

4 Elementary transformation

First let us recall a special birational map between complex symplectic manifolds, which was discovered by Mukai [16]. Let X be a complex symplectic manifold of complex dimension 2n, containing a complex submanifold S isomorphic to ℙ^{n}. Since S is a complex Lagrangian submanifold, the normal bundle N_{S/X} is isomorphic to the cotangent bundle T^{∗}S. Let π: X̂ → X be the blowing-up along S. Then the exceptional divisor E = π^{−1}(S) is isomorphic to the projective cotangent bundle ℙ(T^{∗}S). This can be viewed as the variety of pairs (p, h) such that p ∈ S, h ∈ S^{∗} with p ∈ h, where S^{∗} is the space of hyperplanes in S. Therefore we can blow-down ℙ(T^{∗}S) to S^{∗} and get a new complex symplectic manifold X′ and the birational map elm_{S}: X −−→ X′, which is called the elementary transformation along S. This construction can be generalized directly to the case where S is a ℙ^{r}-bundle.

Next we give a fundamental example, where an elementary transformation appears naturally. Consider the subtorus K of T^{N} whose Lie algebra k is spanned by X_{1} +⋯+X_{N}. Let {v} ⊂ k^{∗} be its dual basis. Then we have ι^{∗}u_{1} = ⋯ = ι^{∗}u_{N} = v. If we identify k^{∗} with ℝ with respect to the basis {v}, the hyperKähler moment map for the action of K on ℍ^{N}

μ = (μ_{1}, μ_{ℂ}): ℍ^{N} → k^{∗} × k_{ℂ}^{∗} ≅ ℝ × ℂ

is given by

μ_{1}(z, w) =π ∑_{i=1}^{N}(|z_{i}|^{2} − |w_{i}|^{2}) ∈ ℝ,

μ_{ℂ}(z, w) = − 2π\sqrt{−1} ∑_{i=1}^{N} z_{i}w_{i} ∈ ℂ.

Moreover (α, β) ∈ ℝ × ℂ is a regular value of μ if and only if (α, β) ≠ (0, 0). Since the condition (C1) is also satisfied, we have a toric hyperKähler manifold

X(α, β) = {(z, w) ∈ ℍ^{N} | μ_{1}(z, w) = α, μ_{ℂ}(z, w) = β}/K

for (α, β) ≠ (0, 0), which is diffeomorphic to the cotangent bundle of the complex projective space ℙ^{N−1}.

Suppose β ≠ 0. If μ_{ℂ}(z, w) = β, there exists i ∈ {1, …, N} such that z_{i} ≠ 0 and w_{i} ≠ 0. Then we have

(X(α, β), I_{1}) = μ_{ℂ}^{−1}(β)/K_{ℂ},

where K_{ℂ} is the complexification of K. Since the right hand side is independent of α ∈ ℝ, (X(α′, β), I_{1}) is canonically isomorphic to (X(α, β), I_{1}) as a complex manifold for an arbitrary α′ ∈ ℝ.

Next suppose α_{+} > 0 and β = 0. If μ_{1}(z, w) = α_{+}, then we have z ≠ 0. Therefore we have

(X(α_{+}, 0), I_{1}) = {(z, w) ∈ μ_{ℂ}^{−1}(0) | z ≠ 0}/K_{ℂ}.

Thus (X(α′_{+}, 0), I_{1}) is canonically isomorphic to (X(α_{+}, 0), I_{1}) as a complex manifold for an arbitrary α′_{+} > 0.

Similarly, in the case that α_{−} < 0 and β = 0, we have

(X(α_{−}, 0), I_{1}) = {(z, w) ∈ μ_{ℂ}^{−1}(0) | w ≠ 0}/K_{ℂ}.

Therefore (X(α′_{−}, 0), I_{1}) is canonically isomorphic to (X(α_{−}, 0), I_{1}) as a complex manifold for an arbitrary α′_{−} < 0.

Moreover, if we set U = {(z, w) ∈ μ_{ℂ}^{−1}(0) | z ≠ 0, w ≠ 0}/K_{ℂ}, then we have

X(α_{+}, 0) \ U = {(z, 0) ∈ ℍ^{N} | z ≠ 0}/K_{ℂ} ≅ ℙ^{N−1},
X(α_{−}, 0) \ U = {(0, w) ∈ ℍ^{N} | w ≠ 0}/K_{ℂ} ≅ ℙ^{N−1}.

Thus we have a birational map f: X(α_{+}, 0) −−→ X(α_{−}, 0) such that f|_{U} is the identity map of U. Since this is the same situation as the one in Section 1 in [21], the birational map f is the elementary transformation along X(α_{+}, 0) \ U.

5 Stability

Let K be a subtorus of T^{N} satisfying the condition (C1). We denote its complexification by K_{ℂ}. Let μ = (μ_{1}, μ_{ℂ}): ℍ^{N} → k^{∗} ⊗ℝ^{3} be the hyperKähler moment map for the action of K on ℍ^{N}. Since a hyperKähler quotient μ^{−1}(α, β)/K is a symplectic quotient of μ_{ℂ}^{−1}(β) by K, it should be identified with a quotient space in geometric invariant theory (GIT for short). The key notion in GIT is stability, which depends on α ∈ k^{∗} in our case. Here we give an ad hoc definition of α-stability.

Definition 5.1. Fix (α, β) ∈ (k^{∗} ×k_{ℂ}^{∗})_{reg}. A point (z, w) ∈ μ_{ℂ}^{−1}(β) is α-stable if and only if

α ∈ ∑_{i∈J_{(z,w)}^{1}} ℝι^{∗}u_{i} + ∑_{i∈J_{(z,w)}^{2}} ℝ_{>0}ι^{∗}u_{i} + ∑_{i∈J_{(z,w)}^{3}} ℝ_{<0}ι^{∗}u_{i},

where ℝ_{>0}, ℝ_{<0} are the sets of positive and negative numbers respectively, and J_{(z,w)}^{1}, J_{(z,w)}^{2}, J_{(z,w)}^{3} are the subsets of {1, …, N} defined as follows:

J_{(z,w)}^{1} ={i | z_{i} ≠ 0 and w_{i} ≠ 0},
J_{(z,w)}^{2} ={i | z_{i} ≠ 0 and w_{i} = 0},
J_{(z,w)}^{3} ={i | z_{i} = 0 and w_{i} ≠ 0}.

(5.1)

Moreover, we denote the set of α-stable points in μ_{ℂ}^{−1}(β) by μ_{ℂ}^{−1}(β)^{α−st}.

It is easy to see that μ_{ℂ}^{−1}(β)^{α−st} is preserved by the action of K_{ℂ} and that μ^{−1}(α, β) ⊂ μ_{ℂ}^{−1}(β)^{α−st}. Therefore we have the following natural map

i: μ^{−1}(α, β)/K → μ_{ℂ}^{−1}(β)^{α−st}/K_{ℂ}.

Theorem 5.2. Let K be a subtorus of T^{N} satisfying the condition (C1). Suppose (α, β) ∈ (k^{∗} × k_{ℂ}^{∗})_{reg}. Then the following holds.

(1) K_{ℂ} acts freely on μ_{ℂ}^{−1}(β)^{α−st}.

(2) The map i is bijective.

Remark 5.3. Here we discuss the relation between the definitions of α-stability in Definition 5.1 and in GIT. To define a GIT quotient, we fix a linearization of the action, which is a lift of the action of K_{ℂ} on μ_{ℂ}^{−1}(β) to a line bundle. This corresponds to α ∈ k_{ℤ}^{∗}, because L̃_{α}|_{μ_{ℂ}^{−1}(β)} is a K_{ℂ}-equivariant line bundle on μ_{ℂ}^{−1}(β). Then we have the notion of α-stable and α-semi-stable points (in the sense of GIT) in μ_{ℂ}^{−1}(β) and also the GIT quotient μ_{ℂ}^{−1}(β)//_{α}K_{ℂ} for α ∈ k_{ℤ}^{∗}. By the theorem due to [13], we have a bijection i′: μ^{−1}(α, β)/K → μ_{ℂ}^{−1}(β)//_{α}K_{ℂ}. See Chapter 8 in [17] for the proof. For non-compact cases, see also Chapter 3 in [19]. In the proof of the theorem due to [13], a similar functional to the one given in (5.3) below plays an important role. The key point of the proof of Theorem 5.2 is to show that the same procedure also works even if we define α-stability as in Definition 5.1. By Theorem 5.2 together with the theorem due to [13], we have the following conclusion. Under the condition (α, β) ∈ (k^{∗} ×k_{ℂ}^{∗})_{reg}, all α-semi-stable points (in the sense of GIT) are α-stable (in the sense of GIT), and α-stability in Definition 5.1 is equivalent to the one in GIT for α ∈ k_{ℤ}^{∗}. Or rather, it is better to say that Definition 5.1 is a criterion of α-stable points (in the sense of GIT) for α ∈ k_{ℤ}^{∗}. Thus our quotient μ_{ℂ}^{−1}(β)^{α−st}/K_{ℂ} is identified with the GIT quotient μ_{ℂ}^{−1}(β)//_{α}K_{ℂ} for α ∈ k_{ℤ}^{∗}. Moreover, Definition 5.1 can be considered as a natural generalization of the notion of α-stability to all α ∈ k^{∗}.

To prove this theorem, we need two lemmas.

Lemma 5.4. Suppose (α, β) ∈ (k^{∗} × k_{ℂ}^{∗})_{reg} and (z, w) ∈ μ_{ℂ}^{−1}(β)^{α−st}. Then there exists a basis {ι^{∗}u_{i}}_{i∈I} of k^{∗} satisfying the following conditions:

(i) I ⊂ J_{(z,w)} = J_{(z,w)}^{1} ∪ J_{(z,w)}^{2} ∪ J_{(z,w)}^{3}, (ii) α ∈ ∑_{i∈I^{1}} ℝι^{∗}u_{i} + ∑_{i∈I^{2}} ℝ_{>0}ι^{∗}u_{i} + ∑_{i∈I^{3}} ℝ_{<0}ι^{∗}u_{i}, (iii) β ∈ ∑_{i∈I^{1}} ℂι^{∗}u_{i}, where I^{j} = I ∩ J_{(z,w)}^{j} for j = 1, 2, 3.

Proof. Let I be the set of subsets of {1, …, N} satisfying the conditions (i), (ii) and (iii). Since (z, w) ∈ μ_{ℂ}^{−1}(β)^{α−st}, we have J_{(z,w)} ∈ I. Moreover, we have the following.

Claim 5.5. If {ι^{∗}u_{i}}_{i∈I} are linearly dependent for I ∈ I, then there exists a subset I′ ⫋ I such that I′ ∈ I.

Proof. Suppose that {ι^{∗}u_{i}}_{i∈I^{1}} are linearly dependent. Then there exists a subset I^{1′} ⫋ I^{1} such that {ι^{∗}u_{i}}_{i∈I^{1′}} is a basis of span_{ℝ}{ι^{∗}u_{i}}_{i∈I^{1}}. Then we have I′ = I^{1′} ∪ I^{2} ∪ I^{3} ∈ I. So Claim 5.5 holds in this case.

Next suppose that {ι^{∗}u_{i}}_{i∈I^{1}} are linearly independent. In this case there exists i_{0} ∈ I^{2} ∪ I^{3} such that ι^{∗}u_{i_{0}} = ∑_{i∈I\{i_{0}}} a_{i}ι^{∗}u_{i}. By the condition (ii) we have α = ∑_{i∈I} b_{i}ι^{∗}u_{i}, where b_{i} > 0 for i ∈ I^{2} and b_{i} < 0 for i ∈ I^{3}. So we have

α = (1 − t)b_{i_{0}}ι^{∗}u_{i_{0}} + ∑_{i∈I\{i_{0}}}(b_{i} + tb_{i_{0}}a_{i})ι^{∗}u_{i}.  (5.2)

Since b_{i_{0}} ≠ 0, the coefficients of ι^{∗}u_{i} vary as the parameter t increases from 0. Define a subset S(t) ⊂ I^{2} ∪ I^{3} for t ≥ 0 by

S(t) = {i ∈ I^{2} ∪ I^{3} | The coefficient of ι^{∗}u_{i} in (5.2) is zero}.

Let t_{0} ≥ 0 be the smallest non-negative number such that S(t_{0}) ≠ ∅. Since S(1) ≠ ∅, we have t_{0} ≤ 1. Thus we have I′ = I \ S(t_{0}) ∈ I. Therefore Claim 5.5 also holds in this case.

By Claim 5.5 there exists I ∈ I such that {ι^{∗}u_{i}}_{i∈I} is linearly independent. Since (α, β) ∈ (k^{∗} ×k_{ℂ}^{∗})_{reg}, {ι^{∗}u_{i}}_{i∈I} is a basis of k^{∗}. Thus we finished the proof of Lemma 5.4.

Define a functional l_{α,(z,w)} : k → ℝ for α ∈ k^{∗} and (z, w) ∈ ℍ^{N} by

l_{α,(z,w)}(X) = ⟨α, X⟩ + \frac{1}{4} ∑_{i=1}^{N} |z_{i}|^{2}e^{−4π⟨ι^{∗}u_{i},X⟩} + \frac{1}{4} ∑_{i=1}^{N} |w_{i}|^{2}e^{4π⟨ι^{∗}u_{i},X⟩},  (5.3)

where ⟨ , ⟩: k^{∗} × k → ℝ is the natural pairing.

Remark 5.6. This functional is a slight generalization of the one in Appendix 1 in [7]. It was also used in [4]. See Remark 6.13 below.

Lemma 5.7. Suppose (α, β) ∈ (k^{∗} × k_{ℂ}^{∗})_{reg} and (z, w) ∈ μ_{ℂ}^{−1}(β)^{α−st}. Then the functional l_{α,(z,w)} has the unique critical point, where it attains its minimum.

Proof. First we show the following.

Claim 5.8. The functional l_{α,(z,w)} is strictly convex.

Proof. It is enough to show that \frac{d^{2}}{dt^{2}} l_{α,(z,w)}(X + tY)|_{t=0} > 0 for all X ∈ k and all Y ∈ k \ {0}. Take a basis {ι^{∗}u_{i}}_{i∈I} of k^{∗} in Lemma 5.4. Note that z_{i} ≠ 0 or w_{i} ≠ 0 for all i ∈ I. Since Y ≠ 0, there exists i ∈ I such that ⟨ι^{∗}u_{i}, Y ⟩ ≠ 0. Therefore we have

\frac{d^{2}}{dt^{2}} l_{α,(z,w)}(X + tY)|_{t=0} = 4π^{2} ∑_{i=1}^{N}⟨ι^{∗}u_{i}, Y ⟩^{2}(|z_{i}|^{2}e^{−4π⟨ι^{∗}u_{i},X⟩} + |w_{i}|^{2}e^{4π⟨ι^{∗}u_{i},X⟩}) > 0.

Thus we proved Claim 5.8.

The next claim is the key to prove Lemma 5.7.

Claim 5.9. lim_{t→∞} l_{α,(z,w)}(X + tY) = +∞ for all X ∈ k and all Y ∈ k \ {0}.

Proof. Take a basis {ι^{∗}u_{i}}_{i∈I} of k^{∗} in Lemma 5.4 again. Since {ι^{∗}u_{i}}_{i∈I} is a basis of k^{∗}, we can express α = ∑_{i∈I} a_{i}ι^{∗}u_{i} uniquely. Due to the condition (ii), we have z_{i} ≠ 0 if a_{i} > 0; w_{i} ≠ 0 if a_{i} < 0; z_{i} ≠ 0, w_{i} ≠ 0 if a_{i} = 0 respectively.

Since

l_{α,(z,w)}(X + tY) = ⟨α, X + tY ⟩ + \frac{1}{4} ∑_{i=1}^{N} |z_{i}|^{2}e^{−4π⟨ι^{∗}u_{i},X+tY ⟩} + \frac{1}{4} ∑_{i=1}^{N} |w_{i}|^{2}e^{4π⟨ι^{∗}u_{i},X+tY ⟩},

the claim holds if ⟨α, Y ⟩ > 0.

So we have to prove Claim 5.9 in the case that 0 ≥ ⟨α, Y ⟩ = ∑_{i∈I} a_{i}⟨ι^{∗}u_{i}, Y ⟩. Suppose, in addition, that there exists i ∈ I such that a_{i}⟨ι^{∗}u_{i}, Y ⟩ < 0. If a_{i} > 0,

then z_{i} ≠ 0 and ⟨ι^{∗}u_{i}, Y ⟩ < 0. If a_{i} < 0, then w_{i} ≠ 0 and ⟨ι^{∗}u_{i}, Y ⟩ > 0. So Claim 5.9 holds in this case.

Next, suppose that a_{i}⟨ι^{∗}u_{i}, Y ⟩ = 0 for all i ∈ I. Since Y ≠ 0, there exists i_{0} ∈ I such that ⟨ι^{∗}u_{i_{0}}, Y ⟩ ≠ 0. So we have a_{i_{0}} = 0, which implies z_{i_{0}} ≠ 0 and w_{i_{0}} ≠ 0. Therefore Claim 5.9 holds in this case.

Thus Claim 5.9 holds in all cases.

By Claim 5.9, there exists X ∈ k where the functional l_{α,(z,w)} attains its minimum. Then, by Claim 5.8, X ∈ k is the unique critical point of the functional l_{α,(z,w)}. Thus we finished the proof of Lemma 5.7.

Proof of Theorem 5.2. First we prove (2). Fix (z, w) ∈ μ_{ℂ}^{−1}(β)^{α−st}. Then, by Lemma 5.7, there exists a critical point X ∈ k of the functional l_{α,(z,w)}. Since

\frac{d}{dt}l_{α,(z,w)}(X + tY)|_{t=0} = ⟨α − μ_{1}((z, w)Exp(\sqrt{−1}X)), Y ⟩,

we have (z, w)Exp(\sqrt{−1}X) ∈ μ_{1}^{−1}(α). This implies the surjectivity of the map i. The uniqueness of the critical point implies the injectivity of the map i.

Next we prove (1). Suppose that (z, w) ∈ μ_{ℂ}^{−1}(β)^{α−st} is fixed by ζ = Exp(X + \sqrt{−1}Y) ∈ K_{ℂ}, where X, Y ∈ k. By (2), we may assume (z, w) ∈ μ^{−1}(α, β). Since (z, w), (z, w)Exp\sqrt{−1}Y ∈ μ_{1}^{−1}(α), 0 and Y are the critical points of the functional l_{α,(z,w)}. Since the critical point is unique, we have Y = 0 ∈ k. That is, ζ = ExpX ∈ K fixes (z, w) ∈ μ^{−1}(α, β). Since K acts freely on μ^{−1}(α, β), we have ζ = 1 ∈ K.

Next we give another useful criterion of α-stable points. Let {P_{1}, …, P_{l}} be the set of subspaces of k^{∗} introduced in Proposition 2.3. Fix Y_{s} ∈ k \ {0} such that P_{s} = {v ∈ k^{∗} | ⟨v, Y_{s}⟩ = 0} for s = 1, …, l. Moreover, define a subset of {1, …, N} for s = 1, …, l by

J_{s}^{0} = {i ∈ {1, …, N} | ⟨ι^{∗}u_{i}, Y_{s}⟩ = 0}.  (5.4)

Then we have the following theorem.

Theorem 5.10. Let K be a subtorus of T^{N} satisfying the condition (C1). Suppose (α, β) ∈ (k^{∗} × k_{ℂ}^{∗})_{reg} and (z, w) ∈ μ_{ℂ}^{−1}(β). Define a subset of {1, … l} by

Λ_{β} = {s ∈ {1, … l} | β ∈ P_{s} ⊗ ℂ}.  (5.5)

Moreover, define subsets of {1, …, N} for s ∈ Λ_{β} by

\begin{matrix}J_{s}^{+} = & \{i ∈ \{1, …, N\} | ⟨ι^{∗}u_{i}, Y_{s}⟩⟨α, Y_{s}⟩ > 0\}, \\ J_{s}^{−} = & \{i ∈ \{1, …, N\} | ⟨ι^{∗}u_{i}, Y_{s}⟩⟨α, Y_{s}⟩ < 0\}. \end{matrix}  (5.6)

Then the following (i) and (ii) are equivalent.

(i) (z, w) ∈ μ_{ℂ}^{−1}(β)^{α−st}. (ii) For each s ∈ Λ_{β}, there exists i ∈ J_{s}^{+} ∪ J_{s}^{−} such that z_{i} ≠ 0 if i ∈ J_{s}^{+} or w_{i} ≠ 0 if i ∈ J_{s}^{−}.

Remark 5.11. If (α, β) ∈ (k^{∗} × k_{ℂ}^{∗})_{reg}, then α ∉ P_{s}, therefore ⟨α, Y_{s}⟩ ≠ 0 for all s ∈ Λ_{β}. Therefore we have {1, …, N} = J_{s}^{0} ∪ J_{s}^{+} ∪ J_{s}^{−}, where the right hand side is disjoint union. Moreover, the subsets J_{s}^{+}, J_{s}^{−} depends on α, but not on the sign of Y_{s}.

Remark 5.12. If β ∈ k_{ℂ}^{∗} \ ⋃_{s=1}^{l}(P_{s} ⊗ ℂ), then we have Λ_{β} = ∅. By Theorem 5.10 we have μ_{ℂ}^{−1}(β)^{α−st} = μ_{ℂ}^{−1}(β) in this case. This is also proved in [4]. See Remark 6.13.

Proof. First we show that (i) ⟹ (ii). Suppose the contrary, that is, there exists (z, w) ∈ μ_{ℂ}^{−1}(β)^{α−st} and s ∈ Λ_{β} such that z_{i} = 0 for all i ∈ J_{s}^{+} and w_{i} = 0 for all i ∈ J_{s}^{−}. Since (z, w) is α-stable, we have

α ∈ ∑_{i∈J_{s}^{0}} ℝι^{∗}u_{i} + ∑_{i∈J_{s}^{+}} ℝ_{≤0}ι^{∗}u_{i} + ∑_{i∈J_{s}^{−}} ℝ_{≥0}ι^{∗}u_{i},

where ℝ_{≤0}, ℝ_{≥0} are the sets of non-positive and non-negative numbers respectively. Since ⟨α, Y_{s}⟩ ≠ 0, we have

0 < ⟨α, Y_{s}⟩^{2} ∈ ∑_{i∈J_{s}^{+}} ℝ_{≤0}⟨ι^{∗}u_{i}, Y_{s}⟩⟨α, Y_{s}⟩ + ∑_{i∈J_{s}^{−}} ℝ_{≥0}⟨ι^{∗}u_{i}, Y_{s}⟩⟨α, Y_{s}⟩ ⊂ ℝ_{≤0}.

This is a contradiction.

Next we show that (ii) ⟹ (i). Suppose that (z, w) ∈ μ_{ℂ}^{−1}(β) satisfies the condition (ii). For each s ∈ Λ_{β}, fix i_{s} ∈ J_{s}^{+} ∪ J_{s}^{−} such that z_{i_{s}} ≠ 0 if i_{s} ∈ J_{s}^{+} or w_{i_{s}} ≠ 0 if i_{s} ∈ J_{s}^{−}. Set Λ_{β}^{+} = {s ∈ Λ_{β} | i_{s} ∈ J_{s}^{+}} and Λ_{β}^{−} = {s ∈ Λ_{β} | i_{s} ∈ J_{s}^{−}}. Moreover, note that, if s ∈ Λ_{β}^{c} = {1, …, l}\Λ_{β}, then there exists i_{s} ∈ {1, …, N}\J_{s}^{0} such that z_{i_{s}} ≠ 0 and w_{i_{s}} ≠ 0. Fix such i_{s} for each s ∈ Λ_{β}^{c}. Thus we have fixed i_{s} ∈ {1, …, N} \ J_{s}^{0} for each s = 1, …, l. Note that k^{∗} is spanned by ι^{∗}u_{i_{1}}, …, ι^{∗}u_{i_{l}}. In fact, if not, there exists t ∈ {1, …, l} such that ι^{∗}u_{i_{1}}, …, ι^{∗}u_{i_{l}} ∈ P_{t}. However, ι^{∗}u_{i_{t}} ∉ P_{t}. This is a contradiction.

Define an open convex cone C ⊂ k^{∗} by

C = ∑_{s∈Λ_{β}^{c}} ℝι^{∗}u_{i_{s}} + ∑_{s∈Λ_{β}^{+}} ℝ_{>0}ι^{∗}u_{i_{s}} + ∑_{s∈Λ_{β}^{−}} ℝ_{<0}ι^{∗}u_{i_{s}}.  (5.7)

Claim 5.13. α ∈ C.

Proof. If C = k^{∗}, the claim holds. Suppose C ⫋ k^{∗}. Note that C is an open convex cone in k^{∗} and that each codimension one boundary of C is an open subset of P_{t} for some t ∈ {1, …, l}. Define a subset Λ′_{β} ⊂ {1, …, l} such that t ∈ Λ′_{β} if and only if there exists a codimension one boundary of C which is an open subset of P_{t}. Since the range of the coefficient ι^{∗}u_{i_{s}} in (5.7) is ℝ for s ∈ Λ_{β}^{c}, any open subset of the hyperplane P_{s} cannot be a boundary of C for s ∈ Λ_{β}^{c}. Therefore we have Λ′_{β} ⊂ Λ_{β}.

Fix an arbitrary t ∈ Λ′_{β}. By the same reason as above, we have ι^{∗}u_{i_{s}} ∈ P_{t} for all s ∈ Λ_{β}^{c}. Since ⟨ι^{∗}u_{i_{t}}, Y_{t}⟩⟨α, Y_{t}⟩ > 0 for t ∈ Λ_{β}^{+} and ⟨ι^{∗}u_{i_{t}}, Y_{t}⟩⟨α, Y_{t}⟩ < 0 for t ∈ Λ_{β}^{−}, there exists a point v ∈ C such that ⟨v, Y_{t}⟩⟨α, Y_{t}⟩ > 0. Since an open subset of the hyperplane P_{t} is the boundary of C, we also have ⟨ι^{∗}u_{i_{s}}, Y_{t}⟩⟨α, Y_{t}⟩ ≥ 0 for all s ∈ Λ_{β}^{+}, and ⟨ι^{∗}u_{i_{s}}, Y_{t}⟩⟨α, Y_{t}⟩ ≤ 0 for all s ∈ Λ_{β}^{−}. Thus, if we set

C′ = {v ∈ k^{∗} | ⟨v, Y_{t}⟩⟨α, Y_{t}⟩ > 0 for all t ∈ Λ′_{β}},

we have C ⊂ C′. Since C is an open convex cone in k^{∗}, by the definition of Λ′_{β}, we have C = C′. Therefore we have α ∈ C. Thus we proved the claim.

Since C is open in k^{∗}, it is easy to see that (z, w) is α-stable. Thus we finished the proof of Theorem 5.10.

6 Variation of complex structures

Let K be a subtorus of T^{N} satisfying the condition (C1). Let {P_{1}, …, P_{l}} be the set of subspaces of k^{∗} introduced in Proposition 2.3. Fix an arbitrary β ∈ k_{ℂ}^{∗}. Define a subset Λ_{β} ⊂ {1, … l} by (5.5). Each connected component of k^{∗} \ ⋃_{s∈Λ_{β}} P_{s} is called a chamber. In this case the hyperplane P_{s} for s ∈ Λ_{β} is called a wall. Fix Y_{s} ∈ k such that P_{s} = {v ∈ k^{∗} | ⟨v, Y_{s}⟩ = 0} for s = 1, …, l. Recall that a vector Y ∈ k_{ℤ} is primitive if and only if Y = aZ for a ∈ ℤ, Z ∈ k_{ℤ} implies a = ±1. We may assume that Y_{s} is contained in k_{ℤ} and primitive. Then Y_{s} is determined up to sign.

Due to Theorems 5.2 and 5.10, we have

(X(α, β), I_{1}) =μ_{ℂ}^{−1}(β)^{α−st}/K_{ℂ}
={(z, w) ∈ μ_{ℂ}^{−1}(β) | the condition (ii) in Theorem 5.10}/K_{ℂ}.

Since the condition (ii) in Theorem 5.10 depends only on the chamber, to which α belongs, we have the following.

Theorem 6.1. Let K be a subtorus of T^{N} satisfying the condition (C1). Suppose that α′ belong to the same chamber as α given by {P_{s}}_{s∈Λ_{β}} for a fixed β ∈ k_{ℂ}^{∗}. Then (X(α′, β), I_{1}) is canonically isomorphic to (X(α, β), I_{1}) as a complex manifold.

Remark 6.2. Here we give a proof of Theorem 2.6. Fix an arbitrary (α, β) ∈ (k^{∗} × k_{ℂ}^{∗})_{reg}. Since (k^{∗} × k_{ℂ}^{∗})_{reg} is connected, we have only to show that there exists an open neighborhood V of (α, β) such that X(α′, β′) is diffeomorphic to X(α, β) for all (α′, β′) ∈ V. Here we use the notation (α_{1}, α_{2}, α_{3}) instead of (α, β) if α = α_{1}, β = α_{2} + \sqrt{−1}α_{3} for α_{1}, α_{2}, α_{3} ∈ k^{∗}. Since (k^{∗} × k_{ℂ}^{∗})_{reg} is open, there exists an open neighborhood U_{i} of α_{i} in k^{∗} for i = 1, 2, 3 such that V = U_{1} × U_{2} × U_{3} ⊂ (k^{∗} × k_{ℂ}^{∗})_{reg}. Fix an arbitrary (α′_{1}, α′_{2}, α′_{3}) ∈ V. By Theorem 6.1, X(α′_{1}, α′_{2}, α′_{3}) is diffeomorphic to X(α_{1}, α′_{2}, α′_{3}). By Theorem 6.1 with respect to the complex structure I_{2}, X(α_{1}, α′_{2}, α′_{3}) is diffeomorphic to X(α_{1}, α_{2}, α′_{3}). By Theorem 6.1 with respect to the complex structure I_{3}, X(α_{1}, α_{2}, α′_{3}) is diffeomorphic to X(α_{1}, α_{2}, α_{3}). Thus we proved that X(α′_{1}, α′_{2}, α′_{3}) is diffeomorphic to X(α_{1}, α_{2}, α_{3}). This completed the proof of Theorem 2.6.

Thus we have seen that, if we fix β ∈ k_{ℂ}^{∗}, the complex structure I_{1} depends only on the chamber. Next we will discuss the relation between two complex structures corresponding to different chambers. Namely, we study the so called wall crossing phenomena.

Theorem 6.3. Let K be a subtorus of T^{N} satisfying the conditions (C1) and (C2). Suppose that α_{+}, α_{−} ∈ k^{∗} belong to the next chamber to each other. Then two complex manifolds (X(α_{+}, β), I_{1}) and (X(α_{−}, β), I_{1}) are related by Mukai’s elementary transformation.

Proof. Suppose that α_{+}, α_{−} ∈ k^{∗} belong to the chambers C_{+}, C_{−} respectively and that the two chambers C_{+}, C_{−} have the common wall, say P_{s_{0}}, as their codimension one boundary. So we assume s_{0} ∈ Λ_{β}. Define J_{s}^{0} ⊂ {1, …, N} by (5.4) and J_{s}^{+}, J_{s}^{−} ⊂ {1, …, N} by (5.6) with respect to α_{+} for s ∈ Λ_{β} respectively. Note that J_{s}^{+}, J_{s}^{−} is independent of the sign of Y_{s}.

We set M̃_{+} = μ_{ℂ}^{−1}(β)^{α_{+}−st}, M̃_{−} = μ_{ℂ}^{−1}(β)^{α_{−}−st}, and Ũ = M̃_{+} ∩ M̃_{−}. To describe these sets explicitly, we introduce the four conditions as follows;

(s) For each s ∈ Λ_{β} \ {s_{0}}, there exists i ∈ J_{s}^{+} ∪ J_{s}^{−} such that z_{i} ≠ 0 if i ∈ J_{s}^{+} or w_{i} ≠ 0 if i ∈ J_{s}^{−}; (s_{0}+) There exists i ∈ J_{s_{0}}^{+} ∪ J_{s_{0}}^{−} such that z_{i} ≠ 0 if i ∈ J_{s_{0}}^{+} or w_{i} ≠ 0 if i ∈ J_{s_{0}}^{−}; (s_{0}−) There exists i ∈ J_{s_{0}}^{+} ∪ J_{s_{0}}^{−} such that w_{i} ≠ 0 if i ∈ J_{s_{0}}^{+} or z_{i} ≠ 0 if i ∈ J_{s_{0}}^{−}; (not s_{0}−) w_{i} = 0 for all i ∈ J_{s_{0}}^{+} and z_{i} = 0 for all i ∈ J_{s_{0}}^{−}.

Due to Theorem 5.10, we have

M̃_{+} ={(z, w) ∈ μ_{ℂ}^{−1}(β) | the conditions (s) and (s_{0}+)},
M̃_{−} ={(z, w) ∈ μ_{ℂ}^{−1}(β) | the conditions (s) and (s_{0}−)},
Ũ ={(z, w) ∈ μ_{ℂ}^{−1}(β) | the conditions (s), (s_{0}+) and (s_{0}−)}.

Due to Theorem 5.2, we have (X(α_{+}, β), I_{1}) = M̃_{+}/K_{ℂ}, (X(α_{−}, β), I_{1}) = M̃_{−}/K_{ℂ}. We will show that the subset X(α_{+}, β) \ U, where U = Ũ/K_{ℂ}, is a ℙ^{r}-bundle over a certain toric hyperKähler manifold. To do this, we need several lemmas.

Set t\overline{N} = span\{X_{i}\}_{i∈J_{s_{0}}^{0}}, where \overline{N} is the number of elements of J_{s_{0}}^{0}. Define the projection p: t^{N} → t\overline{N} by p(∑_{i=1}^{N} a_{i}X_{i}) = ∑_{i∈J_{s_{0}}^{0}} a_{i}X_{i}. Let \overline{k} be the Lie subalgebra

of t\overline{N} defined to be the image of k by p. Then we have the dual map (p|_{k})^{∗}: \overline{k}^{∗} → k^{∗}. Let \overline{ι}: \overline{k} → t\overline{N} be the inclusion. Since (p|_{k})^{∗}(\overline{ι}^{∗}u_{i}) = ι^{∗}u_{i} for i ∈ J_{s_{0}}^{0}, we can identify \overline{k}^{∗} with P_{s_{0}} canonically. So we can consider ι^{∗}u_{i} ∈ \overline{k}^{∗} for i ∈ J_{s_{0}}^{0} and also β ∈ \overline{k}_{ℂ}^{∗}. Moreover, we have dim \overline{k} = dim k − 1 = N − n − 1 and ker p|_{k} = ℝY_{s_{0}}.

Let \overline{K} be the subtorus of T \overline{N} corresponding to the Lie algebra \overline{k}. Then we have the following.

Lemma 6.4. The subtorus \overline{K} of T \overline{N} satisfies the condition (C1).

Proof. Suppose that \{\overline{ι}^{∗}u_{i}\}_{i∈\overline{J}}, where \overline{J} ⊂ J_{s_{0}}^{0}, be a basis of \overline{k}^{∗}. Fix an arbitrary i_{0} ∈ J_{s_{0}}^{+} ∪J_{s_{0}}^{−}. If we set J = \overline{J} ∪\{i_{0}\}, then \{ι^{∗}u_{i}\}_{i∈J} is a basis of k^{∗}. By the condition (C1) for K, we have

t_{ℤ}^{N} = k_{ℤ} + ∑_{i∈{1,...,N}\J} ℤX_{i} as a ℤ-module.

Applying the projection p: t^{N} → t\overline{N} to the both hand sides, we have

t\overline{N} ; ℤ = \overline{k}_{ℤ} + ∑_{i∈J_{s_{0}}^{0}\\overline{J}} ℤX_{i} as a ℤ-module,

where t\overline{N} ; ℤ = ∑_{i∈J_{s_{0}}^{0}} ℤX_{i} and \overline{k}_{ℤ} = \overline{k} ∩ t\overline{N} ; ℤ. This is the condition (C1) for \overline{K}.

The action of \overline{K} on ℍ\overline{N} induces the hyperKähler moment map

\overline{μ} = (\overline{μ}_{1}, \overline{μ}_{ℂ}): ℍ\overline{N} → \overline{k}^{∗} ⊗ ℝ^{3} ≅ \overline{k}^{∗} × \overline{k}_{ℂ}^{∗}.

Denote the set of regular values of \overline{μ} by (\overline{k}^{∗} × \overline{k}_{ℂ}^{∗})_{reg}.

Fix an arbitrary \overline{α} in the interior (with respect to the topology of P_{s_{0}}) of ∂C_{+}∩P_{s_{0}}, where ∂C_{+} is the boundary of C_{+} in k^{∗}. Then we have the following.

Lemma 6.5. (\overline{α}, β) ∈ (\overline{k}^{∗} × \overline{k}_{ℂ}^{∗})_{reg}.

Proof. Let \{\overline{P}_{1}, …, \overline{P} \overline{l}\} be the set of all subspaces of codimension one in \overline{k}^{∗} which are spanned by subsets of \{ι^{∗}u_{i}\}_{i∈J_{s_{0}}^{0}}. Define a subset of \{1, …, \overline{l}\} by \overline{Λ}_{β} = \{s | β ∈ \overline{P}_{s} ⊗ ℂ\}. To prove the lemma, we have to show that \overline{α} ∉ \overline{P}_{s} for all s ∈ \overline{Λ}_{β}.

Suppose the contrary, that is, there exists s ∈ \overline{Λ}_{β} such that \overline{α} ∈ \overline{P}_{s}. By the definition of \overline{P}_{s}, there exists a subset \{i_{1}, …, i_{N−n−2}\} ⊂ J_{s_{0}}^{0} such that \overline{P}_{s} is spanned by ι^{∗}u_{i_{1}}, …, ι^{∗}u_{i_{N−n−2}}. Fix an arbitrary i_{N−n−1} ∈ J_{s_{0}}^{+} (resp. J_{s_{0}}^{−}). For small ϵ > 0 we have \overline{α} + ϵι^{∗}u_{i_{N−n−1}} ∈ C_{+} (resp. C_{−}). Therefore we have (\overline{α} + ϵι^{∗}u_{i_{N−n−1}}, β) ∈ (k^{∗} × k_{ℂ}^{∗})_{reg}.

On the other hand, there exists t ∈ {1, …, l} such that P_{t} is spanned by ι^{∗}u_{i_{1}}, …, ι^{∗}u_{i_{N−n−1}}. So we have (\overline{α} + ϵι^{∗}u_{i_{N−n−1}}, β) ∈ P_{t} × (P_{t} ⊗ ℂ). This is a contradiction. Thus we proved Lemma 6.5.

Due to Lemmas 6.4 and 6.5, we have a toric hyperKähler manifold \overline{X}(\overline{α}, β) = \overline{μ}^{−1}(\overline{α}, β)/\overline{K}. By Lemma 6.5, the complex structure I_{1} on \overline{X}(\overline{α}, β) is independent of \overline{α}.

When we write J_{s_{0}}^{0} = \{i_{1}, …, i\overline{N}\}, define a map θ̃: ℍ^{N} → ℍ\overline{N} by θ̃(z, w) = (\overline{z}, \overline{w}), where \overline{z} = (z_{i_{1}}, …, z_{i\overline{N}}) and \overline{z} = (w_{i_{1}}, …, w_{i\overline{N}}). Then we have the following.

Lemma 6.6. θ̃(z, w) ∈ \overline{μ}_{ℂ}^{−1}(β)^{\overline{α}−st} for all (z, w) ∈ M̃_{+} \ Ũ.

Proof. Fix an arbitrary (z, w) ∈ M̃_{+} \ Ũ. Since (z, w) satisfies the condition (not s_{0}−), we have θ̃(z, w) ∈ \overline{μ}_{ℂ}^{−1}(β). So we have to show that θ̃(z, w) is \overline{α}-stable.

Let \{α_{n}\}_{n=1}^{∞} ⊂ C_{+} be a sequence which converges to \overline{α}. Since (z, w) is α_{+}-stable, we have (z, w) is α_{n}-stable for all n.

If we define the subsets J_{(z,w)}^{1}, J_{(z,w)}^{2}, J_{(z,w)}^{3} ⊂ {1, …, N} by (5.1), then we can express α_{n} = ∑_{i∈J_{(z,w)}} a_{i}^{(n)}ι^{∗}u_{i} (not uniquely), where J_{(z,w)} = J_{(z,w)}^{1} ∪ J_{(z,w)}^{2} ∪ J_{(z,w)}^{3}

and a_{i}^{(n)} > 0 for i ∈ J_{(z,w)}^{2}, a_{i}^{(n)} < 0 for i ∈ J_{(z,w)}^{3} respectively.

Claim 6.7. lim_{n→∞} a_{i}^{(n)} = 0 for i ∈ J_{(z,w)} ∩ (J_{s_{0}}^{+} ∪ J_{s_{0}}^{−}).

Proof. Since (z, w) satisfies the condition (not s_{0}−), we have J_{(z,w)} ∩ J_{s_{0}}^{+} ⊂ J_{(z,w)}^{2} and J_{(z,w)} ∩ J_{s_{0}}^{−} ⊂ J_{(z,w)}^{3}. Therefore we have ⟨a_{i}^{(n)}ι^{∗}u_{i}, Y_{s_{0}}⟩⟨α_{+}, Y_{s_{0}}⟩ > 0 for all i ∈ J_{(z,w)} ∩ (J_{s_{0}}^{+} ∪ J_{s_{0}}^{−}). On the other hand, we have

lim_{n→∞} ∑_{i∈J_{(z,w)}∩(J_{s_{0}}^{+}∪J_{s_{0}}^{−})}⟨a_{i}^{(n)}ι^{∗}u_{i}, Y_{s_{0}}⟩⟨α_{+}, Y_{s_{0}}⟩ = lim_{n→∞} ∑_{i∈J_{(z,w)}}⟨a_{i}^{(n)}ι^{∗}u_{i}, Y_{s_{0}}⟩⟨α_{+}, Y_{s_{0}}⟩

= lim_{n→∞}⟨α_{n}, Y_{s_{0}}⟩⟨α_{+}, Y_{s_{0}}⟩ = ⟨\overline{α}, Y_{s_{0}}⟩⟨α_{+}, Y_{s_{0}}⟩ = 0.

Therefore the claim follows.

By Claim 6.7, if we set \overline{α}_{n} = ∑_{i∈J_{(z,w)}∩J_{s_{0}}^{0}} a_{i}^{(n)}ι^{∗}u_{i} ∈ P_{s_{0}}, we have lim_{n→∞} \overline{α}_{n} = \overline{α}.

By Lemma 6.5, \overline{α} is in a chamber given by the hyperplanes \{\overline{P}_{s}\}_{s∈\overline{Λ}_{β}}. Therefore \overline{α}_{n} belongs to the same chamber as \overline{α} for sufficiently large n > 0. Since θ̃(z, w) is \overline{α}_{n}-stable for all n, it is also \overline{α}-stable. Thus we proved Lemma 6.6.

Due to Lemma 6.6, the map θ̃|_{M̃_{+}\Ũ}: M̃_{+} \ Ũ → ℍ\overline{N} induces a map θ_{+} from

X(α_{+}, β) \ U = (M̃_{+} \ Ũ)/K_{ℂ} to (\overline{X}(\overline{α}, β), I_{1}) = \overline{μ}_{ℂ}^{−1}(β)^{\overline{α}−st}/\overline{K}_{ℂ}. Then we have the following.

Lemma 6.8. (1) θ_{+} : X(α_{+}, β) \ U → \overline{X}(\overline{α}, β) is a ℙ^{N−\overline{N}−1}-bundle.

(2) The primitive vector Y_{s_{0}} ∈ k_{ℤ}, which satisfies the condition ⟨α_{+}, Y_{s_{0}}⟩ > 0, is given by Y_{s_{0}} = ∑_{i∈J_{s_{0}}^{+}} X_{i} − ∑_{i∈J_{s_{0}}^{−}} X_{i}.

(3) N − \overline{N} − 1 ≥ 1.

Proof. Fix an arbitrary (\overline{z}, \overline{w}) ∈ \overline{μ}_{ℂ}^{−1}(β)^{\overline{α}−st}. Define a subset S̃_{(\overline{z},\overline{w})}^{+} ⊂ ℍ^{N} by

S̃_{(\overline{z},\overline{w})}^{+} = \{(z, w) ∈ θ̃^{−1}(\overline{z}, \overline{w}) | the conditions (s_{0}+) and (not s_{0}−) \}.

Then we have the following.

Claim 6.9. θ̃^{−1}(\overline{z}, \overline{w}) ∩ (M̃_{+} \ Ũ) = S̃_{(\overline{z},\overline{w})}^{+}.

Proof. Since ⊂ is trivial, it is enough to show ⊃. Fix an arbitrary (z, w) ∈ S̃_{(\overline{z},\overline{w})}^{+}. Since (z, w) ∉ Ũ by the condition (not s_{0}−), it is enough to show that (z, w) ∈ M̃_{+} = μ_{ℂ}^{−1}(β)^{α_{+}−st}. Since (\overline{z}, \overline{w}) ∈ \overline{μ}_{ℂ}^{−1}(β) and (z, w) satisfies the condition (not s_{0}−), we have (z, w) ∈ μ_{ℂ}^{−1}(β).

Next we show (z, w) is α_{+}-stable. By the condition (s_{0}+), there exists i_{0} ∈ J_{s_{0}}^{+} such that z_{i_{0}} ≠ 0 or there exists i_{0} ∈ J_{s_{0}}^{−} such that w_{i_{0}} ≠ 0. Suppose the former. Then we have \overline{α}+ϵι^{∗}u_{i_{0}} ∈ C_{+} for sufficiently small ϵ > 0. Since (\overline{z}, \overline{w}) is \overline{α}-stable and C_{+} is open in k^{∗}, it is easy to see that (z, w) is \overline{α} + ϵι^{∗}u_{i_{0}}-stable. So it is α_{+}-stable. If the latter holds, we can also show that (z, w) is α_{+}-stable by a similar argument. Thus we proved Claim 6.9.

Fix an arbitrary point [\overline{z}, \overline{w}] ∈ \overline{X}(\overline{α}, β), which is a point represented by (\overline{z}, \overline{w}) ∈ \overline{μ}_{ℂ}^{−1}(β)^{\overline{α}−st}. By Theorem 5.2, \overline{K} acts freely on \overline{μ}_{ℂ}^{−1}(β)^{\overline{α}−st}. Therefore we have θ_{+}^{−1}([\overline{z}, \overline{w}]) = S̃_{(\overline{z},\overline{w})}^{+}/ExpℂY_{s_{0}}, where ExpℂY_{s_{0}} = \{ExptY_{s_{0}} | t ∈ ℂ\} ≅ ℂ^{∗}. If we write Y_{s_{0}} = ∑_{i=1}^{N} a_{i}X_{i}, then we have

J_{s_{0}}^{0} = {i | a_{i} = 0}, J_{s_{0}}^{+} = {i | a_{i} > 0}, J_{s_{0}}^{−} = {i | a_{i} < 0}.

We have assumed that a_{i} ∈ ℤ for all i = 1, …, N and that Y_{s_{0}} is primitive. Since, by Theorem 5.2 again, ExpℂY_{s_{0}} acts freely on S̃_{(\overline{z},\overline{w})}^{+} ≅ ℂ^{N−\overline{N}} \ \{0\}, we have Y_{s_{0}} = ∑_{i∈J_{s_{0}}^{+}} X_{i} − ∑_{i∈J_{s_{0}}^{−}} X_{i}. Moreover the action of ExpℂY_{s_{0}} on S̃_{(\overline{z},\overline{w})}^{+} is isomorphic to

the standard action of ℂ^{∗} on ℂ^{N−\overline{N}−1} \ \{0\}. Therefore we have S̃_{(\overline{z},\overline{w})}^{+}/ExpℂY_{s_{0}} ≅ ℙ^{N−\overline{N}−1}.

Finally we show (3). Denote the number of elements of J_{s_{0}}^{+} ∪ J_{s_{0}}^{−} by #(J_{s_{0}}^{+} ∪ J_{s_{0}}^{−}). First we have #(J_{s_{0}}^{+} ∪ J_{s_{0}}^{−}) = N − \overline{N}. On the other hand, due to the condition (C2), Y_{s_{0}} ≠ ±X_{i} for any i = 1, …, N. Therefore, by (2), we have #(J_{s_{0}}^{+} ∪ J_{s_{0}}^{−}) ≥ 2. Thus we finish the proof of Lemma 6.8.

Now we are in the final stage of the proof of Theorem 6.3. Since the codimension of the complex submanifold X(α_{+}, β) \ U is N − \overline{N} − 1 ≥ 1, we have a birational map F: X(α_{+}, β) −−→ X(α_{−}, β) such that F |_{U} is the identity map of U.

Next we describe a neighborhood of X(α_{+}, β) \ U. Fix an arbitrary (\overline{z}, \overline{w}) ∈ \overline{μ}_{ℂ}^{−1}(β)^{\overline{α}−st}. Define a subset W̃_{(\overline{z},\overline{w})}^{+} ⊂ ℍ^{N} by

W̃_{(\overline{z},\overline{w})}^{+} = \{(z, w) ∈ θ̃^{−1}(\overline{z}, \overline{w}) | the conditions (s_{0}+) and ∑_{i∈J_{s_{0}}^{+}∪J_{s_{0}}^{−}} z_{i}w_{i} = 0 \}.

Then, by a similar argument in the proof of Claim 6.9, we have W̃_{(\overline{z},\overline{w})}^{+} ⊂ μ_{ℂ}^{−1}(β)^{α_{+}−st}. If we set W̃^{+} = ⋃_{(\overline{z},\overline{w})∈\overline{μ}_{ℂ}^{−1}(β)^{\overline{α}−st}} W̃_{(\overline{z},\overline{w})}^{+}, then we have a neighborhood W^{+} = W̃^{+}/K_{ℂ} of X(α_{+}, β) \ U. It is easy to see that the neighborhood W^{+} has the structure of a fiber bundle Θ_{+} : W^{+} → \overline{X}(\overline{α}, β), whose fiber Θ_{+}^{−1}([\overline{z}, \overline{w}]) is isomorphic to W̃_{(\overline{z},\overline{w})}^{+}/ExpℂY_{s_{0}} ≅ T^{∗}ℙ^{N−\overline{N}−1}.

Similarly, we have the neighborhood W^{−} of X(α_{−}, β)\U which has the structure of a fiber bundle over the same toric hyperKähler manifold \overline{X}(\overline{α}, β). Then the restriction of the birational map F: X(α_{+}, β) −−→ X(α_{−}, β) to W^{+} is a family of the birational map f in Section 4 parameterized \overline{X}(\overline{α}, β). Therefore the birational map F: X(α_{+}, β) −−→ X(α_{−}, β) is the elementary transformation along X(α_{+}, β)\U. Thus we finished the proof of Theorem 6.3.

In Proposition 3.3 we have an isomorphism κ^{∗}: k^{∗} → H^{2}(X(α, β); ℝ) for a subtorus K satisfying the conditions (C1) and (C2). Now we define the dual map κ_{∗}: H_{2}(X(α, β); ℝ) → k by

⟨γ, κ_{∗}([C])⟩ = ⟨κ^{∗}(γ), [C]⟩′ for γ ∈ k^{∗} and [C] ∈ H_{2}(X(α, β); ℝ),

where ⟨ , ⟩: k^{∗} × k → ℝ and ⟨ , ⟩′: H^{2}(X(α, β); ℝ) × H_{2}(X(α, β); ℝ) → ℝ are the natural pairings.

Recall that the Kähler cone of a complex manifold M is the set of de Rham cohomology classes represented by its Kähler classes. It is a convex cone in H^{2}(M; ℝ). The Kähler cone of a toric hyperKähler manifold is described as follows.

Theorem 6.10. Let K be a subtorus of T^{N} satisfying the conditions (C1) and (C2). Fix an arbitrary β ∈ k_{ℂ}^{∗} and an arbitrary chamber C in k^{∗} given by the hyperplanes {P_{s}}_{s∈Λ_{β}}. Fix also an arbitrary α ∈ C. Then the following holds.

(1) Suppose, in addition, that P_{s_{0}} ∩ ∂C is a codimension one boundary of C. Then the complex manifold (X(α, β), I_{1}) contains a smooth rational curve C_{s_{0}} such that κ_{∗}([C_{s_{0}}]) = Y_{s_{0}}, where Y_{s_{0}} ∈ k_{ℤ} is the primitive vector satisfying P_{s_{0}} = {v ∈ k^{∗} | ⟨v, Y_{s_{0}}⟩ = 0} and ⟨α, Y_{s_{0}}⟩ > 0.

(2) The Kähler cone KC of the complex manifold (X(α, β), I_{1}) is given by KC = κ^{∗}(C) ⊂ H^{2}(X(α, β); ℝ).

Remark 6.11. The volume of the rational curve C_{s_{0}} in X(α, β) is given by

vol(C_{s_{0}}) = ∫_{C_{s_{0}}} ω_{1} = ⟨κ^{∗}(α), [C_{s_{0}}]⟩′ = ⟨α, κ_{∗}([C_{s_{0}}])⟩ = ⟨α, Y_{s_{0}}⟩.

Therefore vol(C_{s_{0}}) goes to zero as the ‘Kähler class’ α approaches to the wall P_{s_{0}}.

Proof. (1) We continue to use the notation in the proof of Lemma 6.8 except that α_{+}, C_{+}, S̃_{(\overline{z},\overline{w})}^{+} in the proof of Lemma 6.8 should be replaced by α, C, S̃_{(\overline{z},\overline{w})} respectively. In the proof of Lemma 6.8, we showed that there exists a subset S̃_{(\overline{z},\overline{w})} ⊂ μ_{ℂ}^{−1}(β)^{α−st} such that S_{(\overline{z},\overline{w})} = S̃_{(\overline{z},\overline{w})}/ExpℂY_{s_{0}} ≅ ℙ^{N−\overline{N}−1} is a complex submanifold in (X(α, β), I_{1}). So there exists a smooth rational curve C_{s_{0}} in X(α, β), which is the intersection of N − \overline{N} − 2 generic hyperplanes in S_{(\overline{z},\overline{w})} ≅ ℙ^{N−\overline{N}−1}.

We have to show κ_{∗}([C_{s_{0}}]) = Y_{s_{0}}. Recall that we have defined the K-equivariant line bundle L̃_{ρ} on ℍ^{N} in Section 3. Let i: S_{(\overline{z},\overline{w})} → X(α, β) be the natural inclusion. Then we have

i^{∗}((L̃_{ρ}|_{μ^{−1}(α,β)})/K) = (L̃_{ρ}|S̃_{(\overline{z},\overline{w})})/ExpℂY_{s_{0}} = H^{⊗⟨ρ,Y_{s_{0}}⟩},

where H is the hyperplane bundle on S_{(\overline{z},\overline{w})} ≅ ℙ^{N−\overline{N}−1}. By Lemma 3.1 we have i^{∗} ∘ κ_{ℤ}(ρ) = c_{1}(H^{⊗⟨ρ,Y_{s_{0}}⟩}). Therefore we have

⟨ρ, κ_{∗}([C_{s_{0}}])⟩ = ⟨κ^{∗}(ρ), [C_{s_{0}}]⟩′ = ⟨ρ, Y_{s_{0}}⟩ for an arbitrary ρ ∈ k_{ℤ}^{∗}.

Thus we finished the proof of (1).

(2) First we prove KC ⊃ κ^{∗}(C). By Theorem 6.1 (X(α′, β), I_{1}) is isomorphic to (X(α, β), I_{1}) as a complex manifold for an arbitrary α′ ∈ C. Since κ^{∗}(α′) is the Kähler class of (X(α′, β), I_{1}) ≅ (X(α, β), I_{1}), we have KC ⊃ κ^{∗}(C).

Next we show KC ⊂ κ^{∗}(C). Fix an arbitrary α′ ∈ k^{∗} \ C. Then there exists s ∈ Λ_{β} such that P_{s} ∩ ∂C is a codimension one boundary of C and ⟨α′, Y_{s}⟩ ≤ 0, where Y_{s} ∈ k_{ℤ} is the primitive vector satisfying P_{s} = {v ∈ k^{∗} | ⟨v, Y_{s}⟩ = 0} and ⟨α, Y_{s}⟩ > 0. By (1) there exists a smooth rational curve C_{s} such that κ_{∗}([C_{s}]) = Y_{s}. Then we have

⟨κ^{∗}(α′), [C_{s}]⟩′ = ⟨α′, κ_{∗}([C_{s}])⟩ = ⟨α′, Y_{s}⟩ ≤ 0.

Therefore κ^{∗}(α′) ∈ H^{2}(X(α, β); ℝ) cannot be a Kähler class of (X(α, β), I_{1}). So we have KC ⊂ κ^{∗}(C). Thus we proved (2).

Corollary 6.12. Let K be a subtorus of T^{N} satisfying the conditions (C1) and (C2). Then the following (i) and (ii) are equivalent.

(i) (X(α, β), I_{1}) is affine. (ii) β ∈ k_{ℂ}^{∗} \ ⋃_{s=1}^{l}(P_{s} ⊗ ℂ).

Remark 6.13. (ii) ⟹ (i) is proved in [4]. The outline of their argument is as follows. Using the functional given in (5.3), they showed that μ_{ℂ}^{−1}(β) = μ_{ℂ}^{−1}(β)^{α−st} for any α ∈ k^{∗} under the condition (ii). So they have (X(α, β), I_{1}) = μ_{ℂ}^{−1}(β)/K_{ℂ}, which implies (i).

Proof. We have only to show (i) ⟹ (ii). Suppose the contrary. That is, there exists s ∈ {1, …, l} such that β ∈ P_{s} ⊗ ℂ. Then, due to Theorem 6.10, there exists a smooth rational curve in (X(α, β), I_{1}). This contradicts to (i).

7 Example

Consider the subtorus K of T^{5} whose Lie algebra k is spanned by X_{1} + X_{2} + X_{4} and X_{1} + X_{3} + X_{5}. Let {v_{1}, v_{2}} be the dual basis. Then we have ι^{∗}u_{1} = v_{1} + v_{2}, ι^{∗}u_{2} = ι^{∗}u_{4} = v_{1} and ι^{∗}u_{3} = ι^{∗}u_{5} = v_{2}. So we have subspaces P_{1} = ℝv_{1}, P_{2} = ℝ(v_{1} + v_{2}) and P_{3} = ℝv_{2} in k^{∗}. If we set Y_{1} = X_{1} + X_{3} + X_{5}, Y_{2} = X_{2} − X_{3} + X_{4} − X_{5} and Y_{3} = X_{1} + X_{2} + X_{4}, then P_{s} = {v ∈ k^{∗} | ⟨v, Y_{s}⟩ = 0} and Y_{s} ∈ k_{ℤ} is primitive for s = 1, 2, 3.

The hyperKähler moment map for the action of K on ℍ^{5}

μ = (μ_{1}, μ_{ℂ}): ℍ^{5} → k^{∗} ⊗ ℝ^{3} ≅ k^{∗} × k_{ℂ}^{∗}

is given by

μ_{1}(z, w) =π ∑_{i=1,2,4}(|z_{i}|^{2} − |w_{i}|^{2})v_{1} + π ∑_{i=1,3,5}(|z_{i}|^{2} − |w_{i}|^{2})v_{2} ∈ k^{∗},
μ_{ℂ}(z, w) = − 2π\sqrt{−1}( ∑_{i=1,2,4} z_{i}w_{i})v_{1} − 2π\sqrt{−1}( ∑_{i=1,3,5} z_{i}w_{i})v_{2} ∈ k_{ℂ}^{∗}.

If β ∈ k_{ℂ}^{∗} \ ⋃_{s=1,2,3} P_{s}, then we have Λ_{β} = ∅. Therefore we have no wall in k^{∗} and only one chamber in this case.

If β ∈ P_{s} \ {O} for some s = 1, 2, 3, then we have Λ_{β} = {s}. Therefore we have only one wall P_{s} and two chambers in k^{∗} in this case.

If β = 0, then Λ_{β} = Λ_{0} = {1, 2, 3}. So we have chambers C_{1}, …, C_{6} in k^{∗} as in Figure 1.

[figure: ι^{∗}u_{2} = ι^{∗}u_{4}; ι^{∗}u_{3} = ι^{∗}u_{5}; ι^{∗}u_{1}; C_{1}; C_{2}; C_{3}; C_{4}; C_{5}; C_{6}; Figure 1; P_{1}; P_{3}; P_{2}]

Let us describe wall crossing phenomena in detail. Take an arbitrary α_{i} ∈ C_{i} for i = 1, …, 6. Set M_{i} = (X(α_{i}, 0), I_{1}) and M̃_{i} = μ_{ℂ}^{−1}(0)^{α_{i}−st}. We also set Ũ_{ij} = M̃_{i} ∩ M̃_{j} and U_{ij} = Ũ_{ij}/K_{ℂ} for i ≠ j. By Theorem 5.2 we have M_{i} = M̃_{i}/K_{ℂ}. By Theorem 5.10 we have the following;

(z, w) ∈ M̃_{1} ⟺ \{ \begin{matrix}z_{1}w_{1} + z_{2}w_{2} + z_{4}w_{4} = 0, z_{1}w_{1} + z_{3}w_{3} + z_{5}w_{5} = 0, \\ (z_{1}, z_{2}, z_{4}) ≠ (0, 0, 0), (z_{1}, z_{3}, z_{5}) ≠ (0, 0, 0), \\ (z_{2}, z_{4}, w_{3}, w_{5}) ≠ (0, 0, 0, 0), \end{matrix} (7.1)

(z, w) ∈ M̃_{2} ⟺ \{ \begin{matrix}z_{1}w_{1} + z_{2}w_{2} + z_{4}w_{4} = 0, z_{1}w_{1} + z_{3}w_{3} + z_{5}w_{5} = 0, \\ (z_{1}, z_{2}, z_{4}) ≠ (0, 0, 0), (z_{1}, z_{3}, z_{5}) ≠ (0, 0, 0), \\ (w_{2}, w_{4}, z_{3}, z_{5}) ≠ (0, 0, 0, 0). \end{matrix} (7.2)

(z, w) ∈ M̃_{6} ⟺ \{ \begin{matrix}z_{1}w_{1} + z_{2}w_{2} + z_{4}w_{4} = 0, z_{1}w_{1} + z_{3}w_{3} + z_{5}w_{5} = 0, \\ (z_{1}, z_{2}, z_{4}) ≠ (0, 0, 0), (w_{1}, w_{3}, w_{5}) ≠ (0, 0, 0), \\ (z_{2}, z_{4}, w_{3}, w_{5}) ≠ (0, 0, 0, 0). \end{matrix} (7.3)

First we describe the relation between M_{1} and M_{2}. By (7.1) and (7.2) we have

(z,w) ∈ M̃_{1} \ Ũ_{12}

⟺ \{ \begin{matrix}z_{1}w_{1} + z_{2}w_{2} + z_{4}w_{4} = 0, z_{1}w_{1} + z_{3}w_{3} + z_{5}w_{5} = 0, \\ (z_{1}, z_{2}, z_{4}) ≠ (0, 0, 0), (z_{1}, z_{3}, z_{5}) ≠ (0, 0, 0), \\ (z_{2}, z_{4}, w_{3}, w_{5}) ≠ (0, 0, 0, 0), (w_{2}, w_{4}, z_{3}, z_{5}) = (0, 0, 0, 0), \end{matrix}

⟺ (z_{2}, z_{4}, w_{3}, w_{5}) ≠ (0, 0, 0, 0), (w_{1}, w_{2}, w_{4}, z_{3}, z_{5}) = (0, 0, 0, 0, 0), z_{1} ≠ 0.

Since we can normalize z_{1} to be 1 by the action of ExpℂY_{1}, we have

M_{1} \ U_{12} = {(z_{2}, z_{4}, w_{3}, w_{5}) ≠ (0, 0, 0, 0), (z_{1}, w_{1}, w_{2}, w_{4}, z_{3}, z_{5}) = (1, 0, 0, 0, 0, 0)}/ExpℂY_{2}, which is isomorphic to ℙ^{3}. Therefore the birational transformation f_{12} : M_{1} −−→ M_{2} is the elementary transformation along M_{1} \ U_{12} ≅ ℙ^{3}.

Next we describe the relation between M_{1} and M_{6}. By (7.1) and (7.3) we have

(z,w) ∈ M̃_{1} \ Ũ_{16}

⟺ \{ \begin{matrix}z_{1}w_{1} + z_{2}w_{2} + z_{4}w_{4} = 0, z_{1}w_{1} + z_{3}w_{3} + z_{5}w_{5} = 0, \\ (z_{1}, z_{2}, z_{4}) ≠ (0, 0, 0), (z_{1}, z_{3}, z_{5}) ≠ (0, 0, 0), \\ (z_{2}, z_{4}, w_{3}, w_{5}) ≠ (0, 0, 0, 0), (w_{1}, w_{3}, w_{5}) = (0, 0, 0), \end{matrix}

⟺ \{ \begin{matrix}z_{2}w_{2} + z_{4}w_{4} = 0, (z_{2}, z_{4}) ≠ (0, 0), \\ (z_{1}, z_{3}, z_{5}) ≠ (0, 0, 0), (w_{1}, w_{3}, w_{5}) = (0, 0, 0). \end{matrix}

Thus we can define a map θ̃: ℍ^{5} → ℍ^{2} by θ̃(z, w) = (\overline{z}, \overline{w}), where \overline{z} = (z_{2}, z_{4}), \overline{w} = (w_{2}, w_{4}). Let \overline{K} be the subtorus of T^{2} whose Lie algebra \overline{k} is spanned by X_{2} + X_{4}. Then the hyperKähler moment map for the action of \overline{K} on ℍ^{2}

\overline{μ} = (\overline{μ}_{1}, \overline{μ}_{ℂ}): ℍ^{2} → \overline{k}^{∗} ⊗ ℝ^{3} ≅ ℝ × ℂ

is given by

\overline{μ}_{1}(\overline{z}, \overline{w}) =π(|z_{2}|^{2} + |z_{4}|^{2} − |w_{2}|^{2} − |w_{4}|^{2}) ∈ ℝ,
\overline{μ}_{ℂ}(\overline{z}, \overline{w}) = − 2π\sqrt{−1}(z_{2}w_{2} + z_{4}w_{4}) ∈ ℂ.

Then we have a hyperKähler quotient \overline{X}(\overline{α}, 0) = \overline{μ}^{−1}(\overline{α}, 0)/\overline{K} for \overline{α} > 0. We also have (\overline{X}(\overline{α}, 0), I_{1}) = \{(\overline{z}, \overline{w}) ∈ \overline{μ}_{ℂ}^{−1}(0) | \overline{z} ≠ 0\}/\overline{K}_{ℂ}, which we denote by \overline{M}. Since this is the same as the example discussed in Section 4, \overline{M} is isomorphic to the cotangent bundle of ℙ^{1}. Since the image of M̃_{1} \ Ũ_{16} by the map θ̃ is \{(\overline{z}, \overline{w}) ∈ \overline{μ}_{ℂ}^{−1}(0) | \overline{z} ≠ 0\}, we have a surjective map θ: M_{1} \ U_{16} → \overline{M}. Then we have

θ^{−1}([\overline{z}, \overline{w}])

= \{(z, w) ∈ θ̃^{−1}(\overline{z}, \overline{w}) | (z_{1}, z_{3}, z_{5}) ≠ (0, 0, 0), (w_{1}, w_{3}, w_{5}) = (0, 0, 0)\}/ExpℂY_{1},

which is isomorphic to ℙ^{2}. Therefore M_{1} \ U_{16} is a ℙ^{2}-bundle over \overline{M} ≅ T^{∗}ℙ^{1}. Thus the birational map f_{16} : M_{1} −−→ M_{6} is the elementary transformation along M_{1} \U_{16}.

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Hiroshi Konno

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo, 153-8914, Japan e-mail: konno@ms.u-tokyo.ac.jp