Mathematische Annalen Volume 282 (1988), 1-21.

Quaternionic Reduction and Quaternionic Orbifolds

Krzysztof Galicki and Blaine H. Lawson, Jr.

0. Introduction

In recent years there has been a growth of interest in riemannian manifolds with Sp_{1}·Sp_{n} holonomy, or so-called quaternionic Kähler manifolds. One reason for this is that the classical σ-model, which consists of the energy functional for maps f : Σ ⟶ X of a compact riemannian 4-manifold into a riemannian manifold X, admits locally an N = 2 supersymmetric extension only when X is a quaternionic Kähler space with non-zero scalar curvature.

From the classification of Berger [B] we know that there are very few holonomy groups possible on non-locally-symmetric, irreducible riemannian manifolds. The Kähler, hyperkähler and quaternionic Kähler manifolds constitute a large proportion of the interesting cases. Whenever a Kähler or hyperkähler manifold X carries a non-trivial connected group H of automorphisms, there is a process called the Marsden-Weinstein reduction for constructing a new Kähler (or hyperkähler) space [MW, H1].

The main part of this note is to present an analogous reduction process for quaternionic Kähler manifolds. We shall show that there exists in this case a certain “momentum mapping” which reduces to the classical one when the scalar curvature κ of the space is zero (and the metrics becomes hyperKähler). When κ ≠ 0, this momentum mapping is \underline{uniquely} defined for any compact group of automorphisms H. Its zero set Z_{H} is Hinvariant and the quotient Z_{H}/H is again a quaternionic Kähler manifold at all regular points.

It is interesting that the momentum mapping has quite different features in the two cases: κ = 0 and κ ≠ 0. (Since the metric on any quaternionic Kähler manifold is always Einstein, κ is constant). When κ = 0, the map is defined by integration. The constants of integration introduce ambiguities and for non-semi-simple Lie groups the map cannot always be made H-equivariant. When κ ≠ 0, the map is defined by differentiation, and it is always H-equivariant. Moreover, only the zero-momentum level has global meaning in this case.

Beginning with the classical quaternionic Kähler manifolds, namely quaternionic projective space ℙ_{ℍ}^{n}, and considering S^{1}-actions, one can produce by this method many compact spaces with Sp_{1}·Sp_{n} holonomy. These spaces are often non-trivial riemannian orbifolds. (By “non-trivial” we mean that they are not the quotients of the known spaces by finite automorphism groups.)

A \underline{riemannian orbifold} is, by definition, a space whose local structure is that of a riemannian manifold divided by a finite group of isometries. The metric singularities on such spaces are of the simplest kind. (Note that even among 2-dimensional cones only those whose unit circle has length of the form 2π/n for some n ∈ ℤ^{+}, are riemannian orbifolds.)

For each integer k ≥ 1, we show that there exists an infinite family of distinct compact, simply-connected, quaternionic Kähler orbifolds of dimension 4k. This result is of particular interest when k = 1. Here we have an infinite family of compact, simplyconnected riemannian orbifolds of dimension 4 each of which is Einstein, self-dual and of positive scalar curvature. By a well-known theorem of Hitchin [H1], the only compact 4-dimensional manifolds admitting such metrics are the sphere S^{4} and complex projective space ℙ_{ℂ}^{2}. However, if we allow these minor singularities, which regularize by “unfolding”, we see that many examples exist. Each of these spaces has the topological type of the Thom-space of a complex line bundle over ℙ_{ℂ}^{1}. Every line bundle appears infinitely often in the family. These spaces are differentially equivalent, as orbifolds, to weighted projective spaces. It seems that the study of the σ-models for such spaces would be quite interesting.

The construction presented here is an abstraction to the general setting of the work done by the first named author [G1]. The case considered in [G1] is the most basic and already gives all the interesting examples.

The authors gratefully acknowledge many useful and stimulating conversations with Martin Roček during the preparation of this work.

1. Definitions

We begin by recalling the basics of quaternionic Kähler geometry (cf. [I, A, S]). Let X be a smooth 4n-dimensional manifold (n > 1). We say that X is \underline{almost} \underline{quaternionic} if there is a 3-dimensional subbundle G ⊂ Hom(T X, T X) with the following property: At each point x ∈ X there is a basis {J_{1}, J_{2}, J_{3}} of the fibre G_{x} such that:

J_{i} ∘ J_{j} = −δ_{ij}id + ϵ_{ijk}J_{k} .  (1.1)

In other words, ℝ · Id ⊕ G is, at each point, a subalgebra isomorphic to the quaternions.

Suppose now that X carries a riemannian metric \underline{adapted} to the quaternion structure in the sense that each J ∈ G is orthogonal, i.e.,

⟨JV, JW ⟩ = ⟨V, W ⟩  (1.2)

for all J ∈ G and all V, W ∈ T_{x}X at all points x ∈ X. Adapted metric always exists. Given an adapted metric we obtain an isometric bundle embedding

G ⊂ Λ^{2}T^{∗}X  (1.3)

which associates to each J ∈ G_{x} the non-degenerate 2-form ω defined by

ω(V, W) = ⟨JV, W ⟩.  (1.4)

for V, W ∈ T_{x}X.

Suppose {J_{1}, J_{2}, J_{3}} are locally defined smooth sections of G which satisfy (1.1) at each point. Then these form an orthonormal frame field for G in the standard metric ⟨A, B⟩ = \frac{1}{2n}trace(A^{t}B) on Hom(T X, T X). Let \{ω_{i}\}_{i=1,2,3} be the basis of 2-forms corresponding under (1.4). The associated exterior 4-form

Ω = ∑_{i=1}^{3} ω_{i} ∧ ω_{i}  (1.5)

is invariant under change of frame field and thus globally defined on X. It is non-degenerate in the sense that Ω^{n} is nowhere vanishing on X. Thus, if dΩ = 0 there are strong consequences in H^{∗}(X; ℝ).

Definition 1.6: The Riemannian manifold X (n > 1) together with G is \underline{quaternionic} \underline{Kähler} if ∇Ω = 0 where ∇ denotes the Levi-Civita connection.

This is equivalent to the hypothesis that the local holonomy group of the metric at point x is contained in the subgroup:

{g ∈ SO_{4n}(T_{x}X) : g^{∗}Ω_{x} = Ω_{x}} ≅ Sp_{1} × Sp_{n}/ℤ_{2} =^{def} Sp_{1}·Sp_{n}.

The hypothesis ∇Ω = 0 clearly implies that dΩ = 0. It is a basic fact that if X is quaternionic Kähler it is Einstein. We suppose from this point on that X is quaternionic Kähler. Let {ω_{1}, ω_{2}, ω_{3}} be a local orthonormal frame field for G ⊂ Λ^{2}T^{∗}X as above. Then from (1.5) we have that

∑_{i=1}^{3}(∇ω_{i}) ∧ ω_{i} = 0

from which it follows that

∇ω_{i} = ∑_{j=1}^{3} α_{ij} ⊗ ω_{j},  (1.7)

where the α_{ij} are 1-forms which satisfy

α_{ij} ≡ −α_{ji}  ∀i, j = 1, 2, 3 .  (1.8)

This means in particular that the subspace Γ(G) ⊂ Γ(Λ^{2}T^{∗}X) is preserved by the riemannian covariant derivative. The matrix

A ≡ ( \begin{matrix}0 & a_{12} & a_{13} \\ −a_{12} & 0 & a_{23} \\ −a_{13} & −a_{23} & 0 \end{matrix} )  (1.9)

is the connection 1-form with respect to the local frame field {ω_{1}, ω_{2}, ω_{3}}. The curvature of this induced connection represents a component of the Riemann curvature tensor R and is given by

R = dA − A ∧ A.

Using the facts that dω_{i} = ∑_{j=1}^{3} α_{ij} ∧ ω_{j} and d^{2}ω_{i} = 0, one deduces that

R_{ij} = dα_{ij} − ∑_{l} α_{il} ∧ α_{lj} = λ ∑_{k} ϵ_{ijk}ω_{k}  (1.10)

for some constant λ. (One uses here the full structure of the forms {ω_{1}, ω_{2}, ω_{3}} and the fact that n > 1.) Note that this equation means that

⟨R(ω_{i}), ω_{j}⟩ = λ ∑_{k} ϵ_{ijk}ω_{k}  (1.10′)

for all i, j. Since G is an oriented 3-dimensional bundle, there is a canonical identification SkewEnd(G) ≅ G via the cross-product. Using this identification we can consider R as a map

R : Λ^{2}T X ⟶ SkewEnd(G) ≅ G ⊂ Λ^{2}T X

and, as such, equation (1.10′) simply states that

R = λπ,  (1.11)

where π denotes pointwise orthogonal projection of Λ^{2}T X into G.

By our assumption each of the subbundles in the orthogonal splitting

Λ^{2}T X = G ⊕ G^{⊥}

is preserved by the riemannian connection, and therefore also by the curvature tensor of Λ^{2}T X

R : Λ^{2}T X ⟶ SkewEnd(Λ^{2}T X)

which is given as follows. Let

˜( · ) : Λ^{2}T X ⟶ SkewEnd(T X)

be the canonical identification and let

R : Λ^{2}T X ⟶ SkewEnd(T X)

denote the Riemann curvature tensor of T X. Then for φ, ψ ∈ Λ^{2}T X we have that

R_{φ}(ψ) = [R_{φ} , ψ̃],  (1.12)

where [· , ·] denotes the commutator of endomorphisms. For any φ, R_{φ} maps G into G (and G^{⊥} into G^{⊥}), that is, we have a decomposition

R = R_{G} ⊕ R_{G^{⊥}},

where the first component is just the curvature tensor of G discussed above. Hence, equations (1.11) and (1.12) tell us that

[R_{φ} , ψ̃] = λ[π φ , ψ̃] ∀φ ∈ Λ^{2}T X and ∀ψ ∈ G.

Therefore, for any φ ∈ Λ^{2}T X, we have that

R_{φ} = λπ φ + c_{φ},

where c̃_{φ} commutes with the endomorphisms {J_{1}, J_{2}, J_{3}} of G. This means that

c_{φ} ∈ 𝔰𝔭_{n} ⊂ G^{⊥},

where 𝔰𝔭_{n} denotes the skew-endomorphisms commuting with G. Suppose now that φ ∈ G and ψ = c_{φ}. Then from the symmetry of the Riemann curvature tensor, we have that

∥ψ∥^{2} = ⟨ψ , R_{φ}⟩ = ⟨R_{ψ} , φ⟩ = ⟨λπ ψ , φ⟩ = 0.

Hence c_{φ} = 0, and we conclude that as a symmetric endomorphism R : Λ^{2}T X ⟶ Λ^{2}T X, the Riemann curvature tensor has the property that

R |_{G}= λId_{G} .  (1.13)

It is straightforward to show that λ is a positive multiple of the scalar curvature κ on X.

Consequently, if κ = 0, then G is flat and the metric is \underline{hyperKähler}(Sp_{n}-holonomy). We are interested here in the case where κ ≠ 0, for example, on quaternionic projective space ℙ_{ℍ}^{n}.

Note that in dimension 4, the condition that ∇Ω = 0 is trivially satisfied since Ω is the volume form. Nevertheless, a good extension of the concept of being quaternionic Kähler does exist for dimension 4. In this case the decomposition (1.12) corresponds exactly to the decomposition Λ^{2}T X = Λ_{+} ⊗ Λ_{−} into self-dual and anti-self-dual 2-forms.

Definition 1.14: An oriented riemannian 4-manifold is called \underline{quaternionic} \underline{Kähler} if condition (1.13) holds (where G = Λ_{+}). This condition is equivalent to the assumption that X is Einstein and anti-self-dual.

For further details on quaternionic Kähler manifolds, see [A, S, I, W].

2. The Quaternionic Momentum Mapping

We shall now describe an analogue of the Marsden-Weinstein reduction for the case of quaternionic Kähler manifolds. To do this we first consider the spaces Ω^{p}(G) ≡ Γ(Λ^{p}T^{∗}X⊗ G) of smooth exterior p-forms on X with values in G. The connection given on G induces a “de Rham” sequence

Ω^{0}(G) ⟶^{d^{∇}=∇} Ω^{1}(G) ⟶^{d^{∇}} Ω^{2}(G) ⟶^{d^{∇}} ⋯  (2.1)

such that

d^{∇} ∘ d^{∇}(f) = R(f)  (2.2)

for f ∈ Ω^{0}(G). (See [BL].)

Consider now the Lie group

Aut(X) ≡ {g ∈ Isom(X) : g^{∗}Ω = Ω}

and its Lie algebra

aut(X) ≡ {V ∈ K(X) : L_{V} Ω = 0}

embedded naturally in the space K(X) of Killing vector fields on X. (Here L_{V} denotes the Lie derivative.) To each V ∈ aut(X) we associate the G-valued 1-form

Θ_{V} ∈ Ω^{1}(G)

defined in terms of a local frame ω_{1}, ω_{2}, ω_{3} by

Θ_{V} ≡ ∑_{i}(i_{V} ω_{i}) ⊗ ω_{i}  (2.3)

where i_{V} denotes contraction with V. Clearly Θ_{V} remains invariant under local change of frame field (i.e., under local gauge transformations).

Theorem 2.4: Assume that the scalar curvature of X is not zero. Then to each V ∈ aut(X) there corresponds a unique section f_{V} ∈ Ω^{0}(G) such that

∇f_{V} = Θ_{V} .  (2.5)

In fact, under the canonical bundle isometry σ : SkewEnd(G) ⟶ G, f_{V} is given explicitly by the formula

f_{V} = \frac{1}{λ}σ(L_{V} − ∇_{V}).  (2.6)

Proof: In light of the explicit formula (2.6) the proof could be left as an exercise for the reader. However, there is an approach to this theorem which proceeds in analogy with the classical (Kähler) case. It was by this route that the authors were led to the result. We present it here because it gives insight into the problem and may be of use to people searching for other generalizations of the momentum map. Moreover, in this argument we shall clearly see exactly where the classical and non-classical cases diverge.

Consider the ”de Rham” sequence (2.1). We are given Θ_{V} ∈ Ω^{1}(G) and want to solve the equation d^{∇}f_{V} = Θ_{V}. Differentiating this equation and using (2.2) we find that

R(f_{V}) = d^{∇}d^{∇}f_{V} = d^{∇}Θ_{V} .

The main point now is that if λ ≠ 0, then the map R : G ⟶ Λ^{2}T X ⊗ G is an injective bundle map. Therefore equation (2.5) can be solved uniquely for f_{V} provided that d^{∇}Θ_{V} lies in the subbundle R(G) ⊂ Λ^{2}T X ⊗ G. This “compatibility condition” is guaranteed by the invariance of the form Ω under V.

Let us carry out this argument explicitly in terms of a local orthonormal frame field {ω_{1}, ω_{2}, ω_{3}} for G. We write

f_{V} = ∑_{i} f_{i}ω_{i}

and recall that

∇f_{V} = ∑_{j}(df_{j} + ∑_{i} f_{i}α_{ij}) ⊗ ω_{j} .

Hence, equation (2.5) can be reexpressed as

df_{j} + ∑_{i} f_{i}α_{ij} = i_{V} ω_{j}.  (2.7)

Applying d to this equation and recalling that R_{ij} = dα_{ij} − ∑ α_{il} ∧ α_{lj}, we find that

∑_{i} f_{i}R_{ij} = d(i_{V} ω_{j}) − ∑_{i}(i_{V} ω_{i}) ∧ α_{ij} =^{def} Ψ_{j}

From (1.10) we deduce that for each j

λ ∑_{i,k} f_{i}ϵ_{ijk}ω_{k} = Ψ_{j}.  (2.8)

Since λ ≠ 0 this equation can be solved uniquely for {f_{i}} provided that {Ψ_{j}} is of the form Ψ_{j} = ∑ c_{jk}ω_{k} where c_{jk} = −c_{kj}, i.e., provided that

∑_{j} ω_{j} ∧ Ψ_{j} = 0.

This last equation follows easily from the condition L_{V} (∑ ω_{j}^{2}) = 0.

To get the explicit form of the solution, note that

L_{V} ω_{j} = d(i_{V} ω_{j}) + i_{V} (dω_{j}) = d(i_{V} ω_{j}) + i_{V} (∑_{k} α_{jk} ∧ ω_{k}) =

= ∑_{k} α_{jk}(V)ω_{k} + d(i_{V} ω_{j}) − ∑_{k} α_{jk} ∧ (i_{V} ω_{k}) = ∇_{V} ω_{j} + Ψ_{j}.

Replacing Ψ_{j} by (L_{V} − ∇_{V})ω_{j} in (2.8) and solving for f_{V} yields the formula (2.6).

We observe now that by the uniqueness in Theorem 2.4, the map V ⟶ f_{V} transforms naturally under the group of automorphisms. This means specifically that for g ∈ Aut(X) and V ∈ aut(X) we have

f_{g_{∗}(V)} = g_{∗}(f_{V})  (2.9)

where g_{∗}(f_{V} )(x) =^{def} g̃(f_{V} (g^{−1}(x))) and where g̃ denotes the map induced by g on the bundle G ⊂ Λ^{2}T X. Note also that g_{∗}V = Ad_{g}(V). Hence, (2.9) means that the diagram

aut(X)  ⟶^{Ad_{g}}  aut(X)
↓^{f}  ↓^{f}
Ω^{0}(G)  ⟶^{g_{∗}}  Ω^{0}(G)

commutes.

Suppose now that H ⊂ Aut(X) is a compact Lie subgroup with corresponding Lie algebra 𝔥.

Definition 2.10: The \underline{momentum map} associated to H is the section f of the bundle 𝔥^{∗} ⊗ G ≅ Hom(𝔥, G) whose value at a point x is the homomorphism V ⟶ f_{V} (x).(Here 𝔥 denotes the trivial 𝔥-bundle over X).

From the equivariance above we see immediately that the \underline{momentum} \underline{map} \underline{is} \underline{H}- \underline{equivariant}. Since the action of H in the bundle 𝔥^{∗} ⊗ G is linear on the fibers, it preserves the zero section. Consequently the set

Z_{H} =^{def} {x ∈ X : f(x) = 0}  (2.11)

is H-invariant.

This momentum mapping (in the case when scalar curvature λ ≠ 0), differs from the standard symplectic one in two essential ways. Here there are no constants of integration; in fact we solve by differentiation rather then by integration. This gives us the group invariance, which does not hold in general in the symplectic case. On the other hand, the only naturally defined “level set” of this mapping is the set Z_{H}, where f = 0.

3. Quaternionic Reduction

The main result of this section is the following:

Theorem 3.1: Let X be a quaternionic Kähler manifold with scalar curvature κ ≠ 0. Let H ⊂ Aut(X) be a compact subgroup with momentum map f. Let Z_{H}^{X} denote the H-invariant subset of Z_{H} = {x ∈ X : f(x) = 0} where f intersects the zero section transversally and where H acts freely. Then Z_{H}^{X}/H equipped with the submersed metric (i.e., the one which makes the projection Z_{H}^{X} ⟶ Z_{H}^{X}/H a riemannian submersion), is again a quaternionic Kähler manifold.

Corollary 3.2: Let X be as above and suppose H ≅ S^{1} ⊂ Aut(X) is a closed 1-parameter subgroup generated by a vector field V ∈ aut(X). If V_{x} ≠ 0 at all points x ∈ Z_{H}, then Z_{H}/H is a compact quaternionic Kähler orbifold.

Proof of 3.1: It will suffice to consider the case when H ≅ S^{1}. The general case is quite similar. Let V be the Killing vector field generating the free S^{1}-action on Z_{H}^{X}. Then V^{⊥} is the field of horizontal planes for the submersion Z_{H}^{X} ⟶^{π} Z_{H}^{X}/H. Furthermore, at any point x ∈ Z_{H}^{X}, we have that

J_{1}V, J_{2}V and J_{3}V are normal to Z_{H}^{X},  (3.3)

where J_{1}, J_{2}, J_{3} are a basis of G_{x} as in §1. To see this, let {ω_{i}} be a local frame field for G as above, and recall that f_{V} = ∑ f_{i}ω_{i} where ∇f_{i} = i_{V} ω_{i}. Consequently

df_{i} = ∑_{j} α_{ij}f_{j} + i_{V} ω_{i} = i_{V} ω_{i} ≅ J_{i}V  (3.4)

at each point of the set where f_{V} = 0. This establishes (3.3) and furthermore shows that the subset of Z_{H} where f_{V} intersects the zero-section transversally is precisely the subset where V ≠ 0.

Let Ω = ∑_{i} ω_{i} ∧ ω_{i} be the parallel 4-form on X where ω_{i}(V, W) ≡ ⟨J_{i}V, W ⟩, and let Ω̃ denotes the restriction of Ω to Z_{H}^{X}. From (3.3) we see that

i_{V} Ω̃ = 0.  (3.5)

Hence, Ω̃ is a horizontal H-invariant 4-form, and so it descends to a form Ω_{H} on Z_{H}^{X}/H with π^{∗}Ω_{H} = Ω̃.

Note from (3.3) that at any point x ∈ Z_{H}^{X} we have an orthogonal decomposition T_{x}X = V_{x}^{⊥}⊕ℝ·V_{x}⊕span{J_{i}V_{x}}. It follows that the plane field V^{⊥} is {J_{1}, J_{2}, J_{3}}-invariant. This gives us a H-invariant almost quaternionic structure on V^{⊥} which descends to an almost quaternion structure on Z_{H}^{X}/H, whose associated 4-form is Ω_{H}. Hence, it is only necessary to prove that

∇Ω_{H} = 0.

To do this, we assume that W_{0}, ..., W_{4} are smooth vector fields on Z_{H}^{X}/H and we let W̃_{0}, ..., W̃_{4} denote their unique horizontal lifts to Z_{H}^{X}. We want to show that

(∇_{W_{0}}Ω_{H})(W_{1}, ..., W_{4}) = 0 .

This can be rewritten as

W_{0}(Ω_{H}(W_{1}, ..., W_{4})) = ∑_{1=i}^{4} Ω_{H}(W_{1}, ..., ∇_{W_{0}}W_{i}, ..., W_{4})

which is equivalent to the equation

W̃_{0}(Ω̃(W̃_{1}, ..., W̃_{4})) = ∑_{i=1}^{4} Ω̃(W̃_{1}, ..., ∇_{W_{0}}W_{i}, ..., W̃_{4}) .  (3.6)

Now it is a consequence of the O’Neill formulas [ON] that

∇_{W_{0}}W_{i} = [ ˜∇_{W_{0}} W̃_{i}]^{H},

where ˜∇ denotes the covariant derivative on Z_{H}^{X} and (·)^{H} denotes orthogonal projection

onto V^{⊥}. Since the form Ω̃ is horizontal we can rewrite (3.6) as

W̃_{0}(Ω̃(W̃_{1}, ..., W̃_{4})) = ∑_{i=1}^{4} Ω̃(W̃_{1}, ..., ˜∇_{W_{0}} W̃_{i}, ..., W̃_{4})

and we may clearly drop the tildas on the Ω’s since Ω̃ is just the restriction of Ω to Z_{H}^{X}. Now it is easy to see from the fact that ω_{i}(W, W′) = ⟨J_{i}W, W′⟩, that the following is true. Fix x ∈ Z_{H}^{X} and choose vectors U_{1}, U_{2}, U_{3} ∈ (V)_{x}^{⊥}.Then for any U ∈ T_{x}X we have that

Ω(U_{1}, U_{2}, U_{3}, U) = Ω(U_{1}, U_{2}, U_{3}, U^{H}).

It is a fundamental fact that

˜∇_{W_{0}} W̃_{i} = (∇_{W_{0}} W̃_{i})^{T},

where ∇ is the riemannian connection on X and (·)^{T} denotes orthogonal projection onto

T_{x}Z_{H}^{X}. Hence, we can rewrite (3.6) as:

W̃_{0}(Ω(W̃_{1}, ..., W̃_{4})) = ∑_{i=1}^{4} Ω(W̃_{1}, ..., ∇_{W_{0}} W̃_{i}, ..., W̃_{4}) .

This equation is a direct consequence of the hypothesis that

∇Ω ≡ 0

on X. This proves the theorem in dimensions greater than 4. In dimension 4 one could compute directly using fundamental equations for submersions and immersions. We shall present an alternative proof which uses the moving frame and works also in dimension four.

Set X′ = Z_{H}^{X}/H and let
i : Z_{H}^{X} ⊂ X and π : Z_{H}^{X} ⟶ X′

denote the inclusion and projection maps respectively. Our first observation here is that the standard restriction on 2-forms gives us a bundle map G ⟶ i^{∗}G which, along the submanifold Z_{H}^{X} is a bundle isometry up to homothety, i.e., an orthonormal basis {ω_{1}, ω_{2}, ω_{3}} of G at x ∈ Z_{H}^{X} restricts to an orthogonal basis {i^{∗}ω_{1}, i^{∗}ω_{2}, i^{∗}ω_{3}} with ∥i^{∗}ω_{j}∥^{2} = (n − 1)/n for all j (where 4n = dimX). The forms {i^{∗}ω_{j}} are horizontal for the submersion π, and by assumption the bundles G and therefore also i^{∗}G are preserved by the S^{1}-action. Hence, there is a bundle G′ of 2-forms which determines a topological quaternionic Kähler structure on X′ and has the property that

π^{∗}G′ = i^{∗}G.  (3.7)

(All of the above is easily verified using the relationship of the ω_{j}’s with the J_{j}’s.) Note that when dim(X′) = 4, we have G′ ≅ Λ_{+}^{2}.

Fix a point x ∈ X′ and choose a smooth orthonormal basis {ω′_{1}, ω′_{2}, ω′_{3}} of G′ in a closed neighborhood U of x. Pulling back by π^{∗} and using the isomorphism (3.7) we see that there is a pointwise orthonormal basis \{ω̃_{1}, ω̃_{2}, ω̃_{3}\} of sections of G defined \underline{along} the set π^{−1}(U) ⊂ Z_{H}^{X} such that i^{∗}ω̃_{j} = π^{∗}ω′_{j} for each j. (We ignore the scale factor \sqrt{n/(n − 1)} here.) A standard and easy argument now shows that there exists an orthonormal basis {ω_{1}, ω_{2}, ω_{3}} of sections of G defined in a neighborhood of π^{−1}(U) in X, such that

π^{∗}ω′_{j} = i^{∗}ω_{j} for j = 1, 2, 3.  (3.8)

At this point we shall specialize to the remaining case where dim(X′) = 4. (The argument can be trivially generalized to all dimensions.)

The Levi-Civita connection on 2-forms preserves the sub-bundle and can be written as

∇ω_{j} = ∑_{k=1}^{3} α_{jk} ⊗ ω_{k}, j = 1, 2, 3,

where the α_{jk} are 1-forms which satisfy: α_{jk} = −α_{kj}. (See (1.7).) The sub-bundle G′ = Λ_{+}^{2} ⊂ Λ^{2} is also preserved by the Levi-Civita connection of X′ and similarly expressed:

∇ω′_{j} = ∑_{k=1}^{3} α′_{jk} ⊗ ω′_{k}, j = 1, 2, 3,

where α′_{jk} = −α′_{kj}.

From the symmetry of the Levi-Civita connection we have the following fundamental identities

dω_{j} = ∑_{k=1}^{3} α_{jk} ∧ ω_{k} and dω′_{j} = ∑_{k=1}^{3} α′_{jk} ∧ ω′_{k}  (3.9)

for each j. Since both d and exterior multiplication commute with i^{∗} and π^{∗}, the equations (3.8) and (3.9) imply that

∑_{k=1}^{3}(π^{∗}α′_{jk} − i^{∗}α_{jk}) ∧ π^{∗}ω′_{k} = 0 for j = 1, 2, 3.  (3.10)

We claim that equation (3.10) implies that

π^{∗}α′_{jk} = i^{∗}α_{jk} for all j, k.  (3.11)

To begin we observe that the 1-forms i^{∗}α_{jk} must be “horizontal”, i.e., their restrictions to the fibres of the map π must be zero. (This follows easily from (3.10) and the linear independence of the forms π^{∗}ω′_{j}.) Hence, at any point x̃ ∈ Z_{H}^{X}, all the forms appearing in (3.10) are supported in the horizontal 4-plane, and we are reduced to the following linear algebraic statement which can be formulated downstairs at x = π(x̃). Let A_{1}, A_{2}, A_{3} be 1-forms on T_{x}X′. We want to prove the following assertion.

∑_{k,l} ϵ_{jkl}A_{l} ∧ ω′_{k} = 0 for 1 ≤ j ≤ 3 ⟹ A_{1} = A_{2} = A_{3} = 0.  (3.12)

This is a system of 12 equations in 12 unknowns. Given a Kähler-type form ω(V, W) = ⟨JV, W ⟩ as above and a 1-form A, we note that: ∗(A ∧ ω) = A ∘ J where (A ∘ J)(V) ≡ A(JV). Hence, the first equation in (3.12) can be rewritten in terms of the endomorphisms {J_{1}, J_{2}, J_{3}} as

J (A) ≡ ∑_{lk} ϵ_{jkl}A_{l} ∘ J_{k} = 0.

Writing out the coefficients of the A_{j}’s in 3-successive blocks, the operator J can be expressed in block matrix form as:

( \begin{matrix}0 & J_{3} & −J_{2} \\ −J_{3} & 0 & J_{1} \\ J_{2} & −J_{1} & 0 \end{matrix} ) .

This is easily seen to have non-zero determinant, and our assertion (3.11) is proved.

The condition that X be quaternionic Kähler implies that

dα_{jk} − ∑_{l} α_{jl} ∧ α_{lk} = λ ∑_{l} ϵ_{jkl}ω_{l}, for all j, k

for some constant λ. (See(1.10).) Restricting to Z_{H}^{X} and using equations (3.8) and (3.11), we see that

d(π^{∗}α′_{jk}) − ∑_{l} π^{∗}α′_{jl} ∧ π^{∗}α′_{lk} = λ ∑_{l} ϵ_{jkl}π^{∗}ω′_{l}, for all j, k.

Since π^{∗} is an injection, this implies that

R′_{jk} ≡ dα′_{jk} − ∑_{l} α′_{jl} ∧ α′_{lk} = λ ∑_{l} ϵ_{jkl}ω′_{l}, for all j, k

As observed in (1.13), this means that the Riemann curvature tensor R : Λ^{2} ⟶ Λ^{2} has the property that R |_{Λ_{+}^{2}}= λId_{Λ_{+}^{2}}. This means precisely that X′ is Einstein and anti-self-dual (or self-dual if one reverses the orientation of X).

Proof of 3.2: It is an immediate consequence of (3.4) that the transversality hypothesis for f_{V} is satisfied at all points x ∈ Z_{H} where V_{x} ≠ 0. Hence, the set

Z′_{H} =^{def} {x ∈ Z_{H} : V_{x} ≠ 0}

is a smooth, H-invariant submanifold. We are assuming here that Z′_{H} = Z_{H}. The action of H ≅ S^{1} on Z_{H} is therefore locally free, and the quotient Z_{H}/H is an orbifold with a smooth quaternionic Kähler metric at all regular points.

In fact, this metric extends across the singularities to make Z_{H}/H a riemannian orbifold. This is a consequence of the following general fact:

Proposition 3.13: Suppose S^{1} acts locally freely and by isometries on a riemannian manifold Z. Then the quotient space Z/S^{1} inherits uniquely the structure of a riemannian orbifold with the property that at non-singular points the projection π : Z ⟶ Z/S^{1} is a riemannian submersion.

Proof: Fix p ∈ Z and let N_{p} denote the normal space to the orbit of S^{1} through p. (Since the action is locally free, every orbit is a smoothly embedded curve.) Fix an orthonormal basis {v_{1}, ..., v_{n−1}} of N_{p} and consider the map Ψ : ℝ^{n−1} ⟶ Z given by

Ψ(x_{1}, ..., x_{n−1}) = exp_{p}(∑_{j=1}^{n−1} x_{j}v_{j}),

where exp_{p} denotes the exponential map of Z at p. For ϵ > 0 sufficiently small, this map gives a smooth embedding of the disc D_{ϵ} ≡ {x ∈ ℝ^{n−1} : ∥x∥ < ϵ}. At regular points p, i.e., points where the isotropy group G_{p} ≡ {g ∈ S^{1} : g(p) = p} is trivial, the composition π ∘ Ψ : D_{ϵ} ⟶ Z/S^{1} gives a smooth local coordinate chart on Z/S^{1}. If p is a singular point, i.e., a point where G_{p} ≠ {e}, then the embedded disk Ψ(D_{ϵ}) is G_{p}-invariant and the projection D_{ϵ} ⟶ Ψ(D_{ϵ})/G_{p} defines the orbifold structure at p.

It now remains to observe that the riemannian structure on Z/S^{1} can be defined as follows. Let g′ be the degenerate metric on Z obtained from the given metric g by “killing” the direction of the vector field V_{0} which generates the S^{1}-action. Specifically, we set

g′(V, W) = g(V, W) − \frac{g(V, V_{0})g(W, V_{0})}{g(V_{0}, V_{0})}.

This new pseudo-metric is still S^{1}-invariant and has the following property. For any two smooth embeddings transversal to the orbits Ψ_{1}, Ψ_{2} : D_{ϵ} ⟶ Z, such that π∘Ψ_{1} = π∘Ψ_{2}, we have that

Ψ_{1}^{∗}g′ = Ψ_{2}^{∗}g′.

These induced metrics are non-degenerate (since the maps are transversal to the orbits) and give us the desired metric structure on Z/S^{1}. This completes the proof of Proposition 3.13 and Corollary 3.2.

4. Some Examples

In this section we shall examine a large family of examples of compact simplyconnected riemannian orbifolds which arise from the process of quaternionic reduction. Many of the examples constructed will be mutually homeomorphic as topological spaces, but mutually distinct as orbifolds. We shall examine this phenomenon in detail. Using the global orbifold structure, we shall then prove that these quaternionic Kähler spaces are never locally symmetric.

Let (u_{o}, ..., u_{n}) denote linear coordinates on the quaternionic vector space bbh^{n+1}, where scalar multiplication is defined from the right. We consider these to be “homogeneous coordinates” for the quaternionic projective space ℙ_{ℍ}^{n} = (ℍ^{n+1} − {0})/ℍ^{∗}. When equipped with Sp_{n+1}-invariant metric

ds^{2} = \frac{1}{∥u∥^{2}} ∑_{α} d\overline{u}^{α} ⊗ du^{α} − \frac{1}{∥u∥^{4}} ∑_{αβ}(\overline{u}^{α}du^{α}) ⊗ (d\overline{u}^{β}u^{β})

ℙ_{ℍ}^{n} is a model example of a compact symmetric manifold with holonomy Sp_{n} ·Sp_{1}. For notational convenience we shall write our homogeneous coordinates as u = (u_{o}, u) where u = (u_{1}, ..., u_{n}). For each pair of integers p, q ∈ ℤ^{+} with (p, q) = 1 and 0 < q/p ≤ 1 , we shall consider the action on ℙ_{ℍ}^{n} defined in homogeneous coordinates by

φ_{t}(u_{o}, u) = (e^{2πiqt}u_{o}, e^{2πipt}u),  (4.1)

where t ∈ [0, 1) if (p + q) is odd and where t ∈ [0, 1/2) if (p + q) is even.

Remark 4.2: The circle action described in (4.1) is an isometry of ℙ_{ℍ}^{n} and it preserves its quaternionic structure, i.e.,

g^{∗}Ω = Ω, g ∈ S^{1}.

Note that all isometries of ℙ_{ℍ}^{n} preserve its quaternionic structure, i.e.,

Aut(ℙ_{ℍ}^{n}) ≡ {g ∈ Isom(ℙ_{ℍ}^{n}) : g^{∗}Ω = Ω} = Isom(ℙ_{ℍ}^{n}).

Consequently, we can consider the quaternionic momentum mapping for this action. The zero level set is given by an algebraic submanifold in ℙ_{ℍ}^{n}:

Z_{H} = \{(u_{o}, u) ∈ ℙ_{ℍ}^{n} : q\overline{u}_{o}iu_{o} + p\overline{u}iu = 0\}.  (4.3)

Theorem 4.4: The circle action given by (4.1) is locally free on Z_{H} for all p, q ∈ ℤ^{+}, q/p < 1 and is \underline{free} for q = p = 1.

Proof: This action on ℍ^{n+1} is generated by the ℍ^{∗}-invariant vector field

V (u_{o}, u) = (iqu_{o}, ipu).

The projection of this vector field on ℙ_{ℍ}^{n} is zero at a point with homogeneous coordinates (u_{o}, u) if and only if

V (u_{o}, u) = (u_{o}ν, uν)

for some quaternion ν. Taking the inner product with (u_{o}, u) and assuming that |u_{o}|^{2} + |u|^{2} = 1 we find that

q\overline{u}_{o}iu_{o} + p\overline{u}iu = ν

which, on Z_{H}, implies that ν = 0. However, this is impossible since V ≠ 0 outside the origin. Hence, the projection of V is non-zero on Z_{H}, and thus the action on Z_{H} is locally free as claimed. It is trivial to see that the action is free if p = q = 1.

It follows that all conditions of Theorem 3.1-2 apply and that the quaternionic reduction gives a compact riemannian quaternionic orbifold O_{q,p}(n − 1)

O_{q,p}(n − 1) =^{def} Z_{H}/S^{1}.

In the case where p = q = 1 we see that the set Z_{H} is invariant under transformations of the homogeneous coordinates (with matrix multiplication from the left). The resulting space is the Grassmann manifold

O_{1,1}(n − 1) = \frac{U(n + 1)}{U(n − 1) × U(2)}

with its symmetric quaternionic Kähler metric.

For q/p < 1, O_{q,p}(n − 1) is an orbifold with a quaternionic Kähler metric. Explicit calculation shows that as q/p → 1 the metric on O_{q,p}(n−1) converges locally to the metric on O_{1,1}. Similarly, as q/p → 0 these metrics converge locally to a hyperKähler metric on T ℙ_{ℂ}^{n−1} (constructedby Eguchi-Hanson for n = 2 and by Calabi in general dimensions). This limiting metric on T ℙ_{ℂ}^{n−1} is not locally symmetric, and the convergence is uniform in three derivatives. Since the condition ∇R ≡ 0 does not hold in the limit, it does not hold on O_{q,p}(n − 1) for all q, p with q/p sufficiently small. This argument, which is presented in detail in [G1], shows that O_{q,p}(n − 1) is not locally symmetric whenever q/p is sufficiently close to zero. We shall present here an alternative argument which shows that O_{q,p}(n − 1) is not locally symmetric for all q, p with 0 < q/p < 1. This argument will be based on an analysis of the large scale architecture of these spaces.

We shall now examine the global structure of O_{q,p}(n − 1). We shall see that it has two connected singular sets: a “nut” and a “bolt”, and we shall analyze the structure of O_{q,p}(n − 1) in a neighborhood of each. To begin we introduce affine coordinates w = (w_{1}, ..., w_{n}) on the open set U_{0} ≡ {[(u_{o}, u)] ∈ ℙ_{ℍ}^{n} : u_{o} ≠ 0} by setting

w_{α} = u_{α}u_{o}^{−1}, for α = 1, ..., n.

In these coordinates the action (4.1) becomes

φ_{t}(w) = e^{2πipt}we^{−2πiqt},

and the set Z_{H} is given as

Z_{H} ∩ U_{0} ≅ \{w ∈ ℍ^{n} : iq + p\overline{w}iw = 0\}.

We then write

w = w_{+} + j\overline{w}_{−}

where w_{+}, w_{−} ∈ ℂ^{n} and observe that

φ_{t}(w_{+}, w_{−}) = (e^{2πi(p−q)t}w_{+}, e^{2πi(p+q)t}w_{−}),
Z_{H} ∩ U_{0} = \{(w_{+}, w_{−}) ∈ U_{0} : ∥w_{−}∥^{2} − ∥w_{+}∥^{2} = q/p, \overline{w}_{−} · w_{+} = 0\}.  (4.6)

It is not difficult now to see that the action has non-trivial isotropy in Z_{H} ∩ U_{0} exactly at the points where w_{+} = 0. This gives us a singular set Σ_{0} ⊂ O_{q,p}(n − 1) described explicitly as

Σ_{0} = {(w_{+}, w_{−}) ∈ Z_{H} : w_{+} = 0}/S^{1}
= {(0, w_{−}) ∈ ℂ^{2n} : ∥w_{−}∥^{2} = 1}/S^{1} ≅ ℙ_{ℂ}^{n−1}.  (4.7)

One easily checks that the isotropy group Γ_{0} = {t ∈ S^{1} : φ_{t}(x) = x}, for points x corresponding to Σ_{0}, is

Γ_{0} ≅ \{ \begin{matrix}ℤ_{p+q}, & if (p + q) is odd; \\ ℤ_{\frac{p+q}{2}} , & if (p + q) is even. \end{matrix}  (4.8)

To describe the second singular set of O_{q,p}(n − 1) we introduce affine coordinates v = (v_{o}, ...v_{n−1}) on the open set

U_{1} ≡ {[(u_{o}, u)] ∈ ℙ_{ℍ}^{n} : u_{n} ≠ 0}

by setting

v_{o} = \sqrt{\frac{p}{q}}u_{o}u_{n}^{−1}, and v_{α} = u_{α}u_{n}^{−1} for α = 1, ..., n − 1.

If we write v ≡ (v_{1}, ..., v_{n−1}), the action becomes

φ_{t}(v_{o}, v) = (e^{2πiqt}v_{o}e^{−2πipt}, e^{2πipt}ve^{−2πipt}).

If, as before, we write v = v_{+} + j\overline{v}_{−}, then Z_{H} is described here as:

Z_{H} ∩ U_{1} ≅ \{(v_{+}, v_{−}) ∈ ℂ^{n} × ℂ^{n} : ∥v_{+}∥^{2} + 1 = ∥v_{−}∥^{2} and \overline{v}_{−} · v_{+} = 0\}.

It is now easy to see that the action has non-trivial isotropy in Z_{H} ∩ U_{1} exactly at the points where v_{o} = 0. Hence, the remaining singular points form a set

Σ_{1} ≡ {[u] ∈ Z_{H} : u_{o} = 0}/S^{1}.

The action of S^{1} on the hyperplane {u_{o} = 0} ⊂ ℙ_{ℍ}^{n} is of type (1,1) and Σ_{1} is just quaternionic reduction with respect to this action. Hence, using our observation above,

Σ_{1} ≅ O_{1,1}(n − 2) ≅ \frac{U(n)}{U(n − 2) × U(2)}.  (4.9)

An easy calculation shows that the isotropy group

Γ_{1} = {t ∈ S^{1} : φ_{t}(x) = x} for points x above Σ_{1}, is

Γ_{1} ≅ \{ \begin{matrix}ℤ_{2p}, & if (p + q) is odd; \\ ℤ_{p}, & if (p + q) is even. \end{matrix}  (4.10)

In summation we have the following. Let 𝔾_{k,n} denote the Grassmann manifold of complex k-planes in ℂ^{n}.

Proposition 4.11: The singular points of the orbifold O_{q,p}(n − 1) are the two disjoint manifolds Σ_{0} ≅ 𝔾_{1,n−1} and Σ_{1} ≅ 𝔾_{2,n} described above, and the local group of the orbifold structure of Σ_{k} is Γ_{k} for each k.

If we set O_{q,p}^{0}(n − 1) ≡ O_{q,p}(n − 1) − Σ_{1} and O_{q,p}^{1}(n − 1) ≡ O_{q,p}(n − 1) − Σ_{0}, then our orbifold is the union

O_{p,q}(n − 1) = O_{p,q}^{0}(n − 1) ∪ O_{p,q}^{1}(n − 1)  (4.12)

of these two open sets. Note that

Σ_{k} ⊂ O_{q,p}^{k}(n − 1) for k = 0, 1.

We shall see that topologically O_{q,p}^{k}(n − 1) is the finite quotient of a vector bundle over Σ_{k} by a fibre-wise linear action. The orbifold structure is, however, more subtle.

For simplicity of exposition we shall confine ourselves from this point on to the case where n = 2 (dimO = 4) and where (p + q) is odd. We write O_{q,p}^{k}(1) ≡ O_{q,p}^{k}. The simpler component of the singular set here is Σ_{1} = a point, and the orbifold structure in a neighborhood is straightforward.

Lemma 4.13: There is a smooth orbifold equivalence O_{q,p}^{1} ≅ ℂ^{2}/ℤ_{2p} where the action of ℤ_{2p} on ℂ^{2} is generated by scalar multiplication by the complex number e^{πiq/p}.

Proof: We have from above that

Z_{H} ∩ U_{1} ≅ \{(v_{+}, v_{−}) ∈ ℂ^{2} × ℂ^{2} : ∥v_{+}∥^{2} + 1 = ∥v_{−}∥^{2} and \overline{v}_{−} · v_{+} = 0\},
φ_{t}(v_{o+}, v_{1+}) = (e^{2πi(p−q)t}v_{o+}, v_{1+}),

and

φ_{t}(v_{o−}, v_{1−}) = (e^{2πi(p+q)t}v_{o−}, v_{1−}).

There is a map of the complex 2-disc transversal to the orbit at v_{o} = 0 in Z_{H} given by

v_{+} = (t, −\frac{t\overline{r}}{g}), v_{−} = (r, g), (t, r) ∈ ℂ^{2},

where g is a real analytic function of t and r of the form g = 1 + ϵ(∥r∥, ∥t∥).(The function g is determined by the equation ∥v_{+}∥^{2} + 1 = ∥v_{−}∥^{2}.) Set ω = e^{πi/p}. The image disc is invariant under the transformation φ_{ω} which generates the isotropy group at the point v_{+} = (0, 0), v_{−} = (0, 1). The induced transformations of the disc in our (t, r)-coordinates is

(t, r) ⟶ (ω^{q}t , ω^{q}r).

Hence, O_{q,p}^{1} is of the form ℂ^{2}/ℤ_{2p} as claimed.

To understand the structure of the component O_{q,p}^{0} ⊃ Σ_{0} ≅ S^{2}, we introduce the following spaces. Set S^{2n−1} = {z ∈ ℂ^{n} : ∥z∥ = 1} and let r, s be a pair of relatively prime positive integers. Then we set

H_{r,s} ≡ S^{3} × ℂ/Φ^{r,s},  (4.14)

where Φ^{r,s} is the action of S^{1} on S^{3} × ℂ given by

Φ_{τ}^{r,s}(z, α) = (τ^{r}z, τ^{s}α).

The natural projection H_{r,s} ⟶ S^{2} ≅ S^{3}/S^{1} makes H_{r,s} a complex line bundle over S^{2}. When r = 1, this is one of standard definitions of the complex line bundle of Chern class 1 over S^{2}, so we take as given that c_{1}(H_{1,s}) = s.

Lemma 4.15: c_{1}(H_{r,s}) = s.

Proof: Consider the map F : S^{3} × ℂ ⟶ S^{3} × ℂ given by F (z, α) = (z, α^{r}), and note that

F ∘ Φ_{τ}^{r,s} = Φ_{τ^{r}}^{1,s} ∘ F

for all τ ∈ S^{1}. It follows that F induces a smooth map H_{r,s} ⟶ H_{1,s} which is easily shown to be a complex line bundle isomorphism.

This lemma shows that the structure of H_{r,s} as a topological manifold is completely determined by the integer s. However, the structure of H_{r,s} as an orbifold ( determined by the S^{1}-action) depends on both r and s.

Theorem 4.16: For (p + q) odd, there is a smooth orbifold equivalence

O_{q,p}^{0} ≅ H_{p+q,2p}.

In particular as a topological space O_{q,p}^{0} is simply a complex line bundle over S^{2} of Chern class 2p.

Proof: Let (w_{+}, w_{−}) ∈ ℂ^{2} × ℂ^{2} be the affine coordinates on ℙ_{ℍ}^{2} defined above. We make a slight change of these coordinates by setting

z_{+} = w_{+} and z_{−} = \sqrt{(q/p + ∥w_{+}∥^{2})}w_{−} .

From (4.6) we see that, in these coordinates, Z_{H} is given by

Z_{H} ∩ U_{0} ≅ \{(z_{+}, z_{−}) ∈ ℂ^{2} × ℂ^{2} : ∥z_{−}∥^{2} = 1 and \overline{z}_{−} · z_{+} = 0\}

and the action is given by

φ_{τ} (z_{−}, z_{+}) = (τ^{p+q}z_{−}, τ^{p−q}z_{+})  (4.17)

for τ ∈ S^{1}. Consequently, we have that as an orbifold O_{q,p}^{0} is equivalent to the quotient of the set

Z_{H}^{0} ≡ {(z_{−}, z_{+}) ∈ S^{3} × ℂ^{2} : z_{+}⊥z_{−}}

by the action (4.17). We define a map f : S^{3} × ℂ ⟶ Z_{H}^{0} by setting f(z, α) = (z, αz^{†}) where if z = (z_{o}, z_{1}) then z^{†} = (−\overline{z}_{1}, \overline{z}_{o}). Note that for any τ ∈ S^{1} we have the diagram commutativity:

(z, α)  ⟶  (z, αz^{†})
↓  ↓
(τ^{p+q}z, τ^{2p}α)  ⟶  (τ^{p+q}z, τ^{p−q}αz^{†})

i.e., we have that f ∘ Φ_{τ}^{p+q,2p} = φ_{τ} ∘ f. Thus f is an S^{1}-equivariant diffeomorphism and therefore f induces a smooth equivalence of the quotient spaces as orbifolds.

Remark 4.19: If (p + q) is even, analogous considerations show that

O_{q,p}^{0} ≅ H_{\frac{p+q}{2} ,p} and O_{q,p}^{1} ≅ ℂ^{2}/ℤ_{p}.

Remark 4.20: In higher dimensions the analysis is not dissimilar. Note that the set

Z_{H}^{0} ≡ {(z_{−}, z_{+}) ∈ ℂ^{n} × ℂ^{n} : ∥z_{−}∥^{2} = 1 and z_{+}⊥z_{−}}

divided by the action Ψ_{τ} (z_{−}, z_{+}) = (τz_{−}, τz_{+}) is just the tangent bundle of ℙ_{ℂ}^{n−1}. Using (4.6) we find that if (p + q) is odd, then as a topological space

O_{q,p}^{0}(n − 1) ≅ T ℙ_{ℂ}^{n−1}/ℤ_{2p},

where ℤ_{2p} acts on T ℙ_{ℂ}^{n−1} by scalar multiplication in the fibers by the 2p^{th}-roots of unity. (The orbifold structure is more complicated.) Similarly, we have that

O_{q,p}^{1}(n − 1) ≅ E/ℤ_{2p},

where E is the canonical complex vector bundle of rank 2 over the Grassmannian 𝔾_{2,n}.

In particular, we see that Σ_{k} is a deformation retract of O_{q,p}^{k}. Thus, from the Van-Kampen Theorem we know the following.

Theorem 4.21: O_{q,p}(n) is simply-connected for all q, p and n.

Using the detailed structure presented above we can now prove the following

Theorem 4.22: O_{q,p}(n) is not locally symmetric for any q, p, n with q/p < 1.

Proof: Consider first the case of dimension 4 (n = 1), with p + q odd and > 4. Suppose that the metric on O_{q,p}(1) were locally symmetric for some relatively prime pair p, q with q/p < 1. Then from the classification (cf. [W] for example) we know that O_{q,p}(1) must be locally isometric to either S^{4} or ℙ_{ℂ}^{2} ( with the standard metrics). In fact every point of O_{q,p}(1) must have a neighborhood which is isometric to a neighborhood of a point x_{0} in X/Γ, where X is one of the two spaces above and where Γ is a finite group of isometries of X which fixes the point x_{0}. The action of Γ is entirely determined up to linear equivalence by the orbifold structure of O_{q,p}(1).

Consider the singular point Σ_{1} and its neighborhood O_{q,p}^{1}. By (4.13) we know that as a smooth orbifold, we have O_{q,p}^{1} = ℂ^{2}/ℤ_{2p} where ℤ_{2p} is generated by scalar multiplication by ω = e^{iπ/p} and where Σ_{1} corresponds to the origin in ℂ^{2}. This linear ℤ_{2p}-action is linearly equivalent to a unique conjugacy class of ℤ_{2p} subgroups in SO_{4}. Therefore, under our assumption that O_{q,p}(1) is locally symmetric, we conclude the following. There is a neighborhood U of Σ_{1} in O_{q,p}(1) which is isometric to a neighborhood of x_{0} in X/ℤ_{2p} where X = S^{4} or ℙ_{ℂ}^{2} and ℤ_{2p} acts on X by exponentiation of its orthogonal action on T_{x_{0}}X described above.

We now know quite explicitly what our metric is in a neighborhood of Σ_{1}. Therefore, by the analyticity of the metric, we know it essentially everywhere. To make this precise, let Õ_{q,p}^{1} be the non-singular riemannian manifold with ℤ_{2p}-action so that O_{q,p}^{1} = Õ_{q,p}^{1}/ℤ_{2p}. Now Õ_{q,p}^{1} is locally isometric to the symmetric space X, and we have given an explicit ℤ_{2p}-equivariant isometry defined in a neighborhood of the fixed point Σ̃_{1}. From the standard arguments of Cartan-Ambrose-Hicks, there is a uniquely defined ”developing map” extending this isometry along arbitrary curves in Õ_{q,p}^{1}. Since Õ_{q,p}^{1} is simply-connected we thereby obtain a uniquely defined isometric immersion

ψ̃ : Õ_{q,p}^{1} ⟶ X

which, because of its uniqueness, is ℤ_{2p}-equivariant.

For each r > 0, consider the ball

B_{r} = {v ∈ TΣ̃_{1} Õ_{q,p}^{1} : ∥v∥ < r}

and identify this ball with the corresponding set {v ∈ T_{x_{0}}X : ∥v∥ < r} via ψ̃_{∗}. Then for all r sufficiently small we have a commutative diagram

\begin{aligned}B_{r} & ⟶^{exp} & Õ_{q,p}^{1} \\ & ↘ & ↓ ψ̃ \\ & & X \end{aligned}  (4.23)

of ℤ_{2p}-equivariant maps. For simplicity, let us renormalize our metric so that X has injectivity radius π. We then consider the number

ρ ≡ sup{r | exp : B_{r} ⟶ Õ_{q,p}^{1} is injective}.

The map exp : B_{π} ⟶ X degenerates on ∂B_{π}, and the map ψ̃ is an immersion. Hence, by (4.23) we conclude that ρ ≤ π. Since exp : B_{r} ⟶ X is an embedding for r ≤ π, we conclude that ρ is also the radius of the largest ball on which the immersion ψ̃ is invertible. From this it follows easily that there must exist some unit vector v ∈ \overline{B_{1}} so that exp(ρv) ∈ Õ_{q,p}^{1} is not defined, i.e., so that the geodesic γ = exp(tv) in Õ_{q,p}^{1} cannot be extended beyond the interval (0, ρ) in the positive direction. Cosequently, γ(t), 0 ≤ t < ρ, \underline{must} \underline{be} \underline{the} \underline{lift} \underline{of} \underline{a} \underline{minimal} \underline{geodesic} γ_{0} \underline{in} O_{q,p}^{1} \underline{joining} Σ_{1} \underline{to} Σ_{0}.

Let C̃ = Õ_{q,p}^{1} −{Σ̃_{1}} and note that projection η : C̃ ⟶ C̃/ℤ_{2p} = O_{q,p}(1)−{Σ_{0} ∪Σ_{1}} is the universal covering of the regular set O_{q,p}(1) − {Σ_{0} ∪ Σ_{1}}. The preimage of the geodesic γ_{0} in C̃ consists exactly of the 2p translations gγ for g ∈ ℤ_{2p}. Let us suppose that ρ < π. Then the limits e_{g} ≡ lim_{t→ρ} gγ(t) for g ∈ ℤ_{2p}, correspond to distinct points on the boundary of the ball \overline{exp(B_{ρ})} ⊂ X. In particular, no two of these geodesics will reconverge in exp(B_{ρ}) as t → ρ.

However, suppose we consider the orbifold structure of O_{q,p}(1) in a neighborhood U of the point σ_{0} = γ_{0}(ρ) ∈ Σ_{0}. This neighborhood can be written as Ũ/ℤ_{p+q} where Ũ is a neighborhood of a point σ̃_{0} ∈ X and where ℤ_{p+q} ⊂ Isom(X) is a cyclic group fixing a totally geodesic surface Σ̃_{0} ⊂ X which contains σ̃_{0}. For any choice of σ̃_{0} ∈ Σ̃_{0} ⊂ X, the normal plane Nσ̃_{0} to Σ̃_{0} at σ̃_{0} has the property that exp(Nσ̃_{0}) is totally geodesic in X. Under this ramified covering Ũ ⟶ Ũ/ℤ_{p+q} = U, the geodesic γ_{0} lifts to (p+q) geodesics which lie in the surface exp(Nσ̃_{0}) and converge to the point σ̃_{0} in a regular pattern with successive angles 2π/(p + q). Now this desingularized geometry can be matched to the geometry of exp(B_{ρ}) ⊂ X at the endpoint of the geodesic γ. Since 2π/(p + q) < π/2, we see that there are at least two more geodesics which correspond to local lifts of γ_{0} and which enter into the ball exp(B_{r}). We recall that exp(B_{r} − {0}) is actually a subset of C̃, and that C̃ ⟶ C̃/ℤ_{2p} is the universal cover of O_{q,p}^{1} − {Σ_{0} ∪ Σ_{1}}. It is clear then that these three geodesic rays near γ(ρ) must descend to the minimal geodesic γ_{0}. hence, the geodesics gγ, g ∈ ℤ_{2p} in C \underline{do} reconverge in contradiction with our conclusion above. Hence, we have proved that ρ = π.

Recall now that

X − exp(B_{π}) = \{ \begin{matrix}point, & if X = S^{4}, \\ ℙ_{ℂ}^{1}, & if X = ℙ_{ℂ}^{2}. \end{matrix}

In either case it is clear that Õ_{q,p}^{1} ≡ exp(B_{π}) since the immersion ψ̃ cannot be extended further. Hence, there is an isometry

O_{q,p}^{1} = (S^{4} − point)/ℤ_{2p} or (ℙ_{ℂ}^{2} − ℙ_{ℂ}^{1})/ℤ_{2p}

which immediately yields an isometry

O_{q,p} = S^{4}/ℤ_{2p} or (ℙ_{ℂ}^{2} − ℙ_{ℂ}^{1})/ℤ_{2p}  (4.24)

of the metric space completions. As before we easily see that the riemannian structure of the spaces (4.24) are not compatible with the given ℤ_{p+q} structure acting locally isometrically on S^{4} or ℙ_{ℂ}^{2}. We conclude that O_{q,p} is not locally symmetric. The case where (p + q) is even and > 8 is entirely similar. The remaining 10 cases can be handled in the same set up with a little extra work. The arguments in higher dimensions proceed analogously.

As we have seen above, the space O_{q,p}(1) (for (p + q) odd) can be obtained by gluing the cone on the standard lens space L_{2p,1} = S^{3}/ℤ_{2p} together with the line bundle of Chern class 2p over S^{2}. This space is not equivalent as an orbifold to ℙ_{ℂ}^{2}/ℤ_{2p}. It is, however, equivalent to a so-called “weighted projective space”. These are defined as follows. Fix positive integers p, q and r with (p, q, r) = 1 and define the \underline{weighted} \underline{projective} \underline{plane} ℙ_{p,q,r}^{2} to be the quotient of S^{5} ⊂ ℂ^{3} by the S^{1}-action:

(z_{o}, z_{1}, z_{2}) ⟶ (τ^{p}z_{o}, τ^{q}z_{1}, τ^{r}z_{2}).

Proposition 4.25: For each pair of positive integers q ≤ p with (p, q) = 1 there is a smooth orbifold equivalence

O_{q,p}(1) ≅ \{ \begin{matrix}ℙ_{2p,p+q,p+q}^{2}, & if (p + q) is odd, \\ ℙ_{p, \frac{p+q}{2} , \frac{p+q}{2}}^{2} , & if (p + q) is even. \end{matrix}

Proof: We use the affine coordinates (w_{+}, w_{−}) ∈ ℂ^{2} × ℂ^{2} and write Z_{H} ∩ U_{0} as in (4.6). Let V_{0} ≡ {z ∈ S^{5} : ∥z_{1}∥^{2} + ∥z_{2}∥^{2} ≠ 0} and map V_{0} ⟶ Z_{H} ∩ U_{0} by setting

w_{+} = \frac{1}{∥z_{1}∥^{2} + ∥z_{2}∥^{2}} (z_{o}\overline{z}_{2}, −z_{o}\overline{z}_{1}) ; w_{−} = \frac{1}{∥z_{1}∥^{2} + ∥z_{2}∥^{2}} (z_{1}, z_{2}).

Note that

(τ^{2p}z_{o}, τ^{p+q}z_{1}, τ^{p+q}z_{2}) ⟶ ((τ^{p−q}w_{+}(z_{1}, z_{2}), τ^{p+q}w_{−}(z_{1}, z_{2}))

for any τ ∈ S^{1}, i.e., this map is S^{1}-equivariant. It is straightforward to check that the induced map on quotients extends to give the orbifold equivalence desired.

As a consequence, we have the following result.

Theorem 4.26: Each of the weighted projective planes ℙ_{2p,p+q,p+q}^{2} for p and q as above and (p + q) odd carries a self-dual Einstein orbifold metric with positive scalar curvature. (This is also true of the weighted projective planes ℙ_{p, \frac{p+q}{2} , \frac{p+q}{2}}^{2} with p, q as above and (p+q) even.) At most finitely many of these metrics are locally symmetric. As q/p → 1, these metrics converge locally to the Fubini-Study metric on ℙ_{1,1,1}^{2}. Hence, there are infinitely many which are mutually non-isometric even locally.

Remark 4.27: Choosing other weights for the S^{1}-action on ℙ_{ℂ}^{2} gives similar metrics on ℙ_{q,p,r}^{2} for other values of p, q and r.

Concluding our paper we would like to mention that the quaternionic reduction method described here can be applied to the case of non-compact quaternionic Kähler manifolds, and in particular to the non-compact dual of ℙ_{ℍ}^{n}. Unlike the compact case, one obtains here new smooth manifolds with Sp_{n} ·Sp_{1} holonomy. Hitchin [H2] used the method to quotient quaternionic hyperbolic space by ℝ and constructed 4n-dimensional generalizations of the Pedersen metric [P]. However, it is possible to construct many other new non-compact quaternionic Kähler metrics which are not locally symmetric. All of them are given by quaternionic reduction in quaternionic hyperbolic space. In particular, in dimension 4 one obtains many new self-dual Einstein metrics with negative cosmological constant [G2].

References

[A] D. V. Alekseevskii, Classification of quaternionic spaces with transitive solvable group of motions, Math. USSR-Izv. 9 (1975), 297.

[B] M. Berger, Sur les groupes d’holonomie homogène des variétés à connexion affines et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955), 279.

[BL] J. P. Bourguignon and H. B. Lawson, Jr., Stability and isolation phenomena for Yang-Mills fields, Commun. Math. Phys. 79 (1980), 169.

[G1] K. Galicki, A generalization of the momentum mapping construction for quaternionic Kähler manifolds, Commun. Math. Phys. 108 (1987), 117.

[G2] K. Galicki, New Matter Couplings in N = 2 Supergravity, Nucl. Phys. B289 (1987), 573.

[H1] N. J. Hitchin, Kählerian twistor spaces, Proc. London Math. Soc.(3) 43 (1981), 133.

[H2] N. J. Hitchin, not published.

[I] S. Ishihara, Quaternion Kählerian manifolds, J. Differential Geometry 9 (1974), 483.

[MW] J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121.

[ON] B. O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459

[P] H. Pedersen, Einstein Metrics, Spinning Top Motions and Monopoles, Math. Ann. 274 (1986), 35.

[S] S. Salamon, Quaternionic Kähler manifolds Invent. Math. 67 (1982), 143.

[W] J. A. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. Math. Mech. 14 (1965), 1033.

Institute for Theoretical Physics February 1987 State University of New York at Stony Brook Stony Brook, NY 11794-3840

Department of Mathematics State University of New York at Stony Brook Stony Brook, NY 11794-3840 email: blawson@math.sunysb.edu