Ann. Glob. Anal. Geom. 21 (2002), 85-110  Annals of Global

Analysis and Geometry

3-Sasakian Geometry, Nilpotent Orbits,

and Exceptional Quotients

Charles P. Boyer Krzysztof Galicki and Paolo Piccinni

Abstract. Using 3-Sasakian reduction techniques we obtain infinite families of new 3-Sasakian manifolds M(p_{1},p_{2},p_{3}) and M(p_{1},p_{2},p_{3},p_{4}) in dimension 11 and 15 respectively. The metric cone on M(p_{1},p_{2},p_{3}) is a generalization of the Kronheimer hyperkähler metric on the regular maximal nilpotent orbit of 𝔰𝔩(3,ℂ) whereas the cone on M(p_{1},p_{2},p_{3},p_{4}) generalizes the hyperkähler metric on the 16-dimensional orbit of 𝔰𝔬(6,ℂ). These are the first examples of 3-Sasakian metrics which are neither homogeneous nor toric. In addition we consider some further U(1)-reductions of M(p_{1},p_{2},p_{3}). These yield examples of non-toric 3-Sasakian orbifold metrics in dimensions

7. As a result we obtain explicit families O(Θ) of compact self-dual positive scalar curvature Einstein metrics with orbifold singularities and with only one Killing vector field.

Introduction

In 1990 Kronheimer showed that the co-adjoint orbits in the complex Lie algebra 𝔤^{ℂ} of both semi-simple and nilpotent elements are hyperkähler [Kr1, Kr2]. In particular these orbits carry Ricci-flat metrics. Later, using Kronheimer’s result, Swann showed that the hyperkähler structure on the nilpotent orbits is very special [Sw]. Such orbits admit an action of ℍ^{∗} with the orbit space being a compact quaternionic Kähler orbifold of positive scalar curvature. Another way of expressing this result is to say that the nilpotent orbits are metric cones C(S) on compact 3-Sasakian orbifolds [BGM1]. The mimimal nilpotent orbit is easily seen to be a metric cone C(G/K), where G/K is a simply-connected 3-Sasakian homogeneous space of [BGM1].

Quaternionic geometry of the regular maximal nilpotent orbit of 𝔰𝔩(3, ℂ) was investigated by Kobak and Swann in great detail [KS1]. This orbit is 12-dimensional and, in the language of 3-Sasakian geometry, it is a cone N = C(S) on the 3-Sasakian orbifold S = ℤ_{3}\G_{2}/Sp(1). Here S is simply an ℤ_{3} quotient of the 3-Sasakian homogeneous space associated to the exceptional Lie group G_{2}, where ℤ_{3} is the center of SU(3) ⊂ G_{2}. This is a typical example of a bi-quotient which has orbifold singularities and the singular locus can be easily identified with a homogeneous 3-Sasakian 7-manifold SU(3)/U(1).

Furthermore, Kobak and Swann [KS1] show that this particular nilpotent orbit can be obtained as a hyperkähler quotient of another nilpotent orbit by a circle action. In the language of 3-Sasakian geometry it is simply a hyperkähler reduction of the metric cone C(\frac{SO(7)}{SO(3)×Sp(1)}) associated to the action of the diagonal U(1) ⊂ U(3) ⊂ SO(7). As C(\frac{SO(7)}{SO(3)×Sp(1)}) can be realized as an Sp(1) reduction of the flat space ℍ^{7} = C(S^{27}) [G] we have the following hyperkähler (3-Sasakian [BGM1], quaternionic Kähler [GL]) quotients:

During the preparation of this work the first two authors were supported by NSF grant DMS-9970904. The third author was supported by MURST and CNR.

(0.1)  \begin{matrix}C(S^{27}) & ⟹^{Sp(1)} & C(\frac{SO(7)}{SO(3)×Sp(1)}) & ⟹^{U(1)} & C(ℤ_{3}\G_{2}/Sp(1)) \\ ↓ & & ↓ & & ↓ \\ S^{27} & ⟹^{Sp(1)} & \frac{SO(7)}{SO(3)×Sp(1)} & ⟹^{U(1)} & ℤ_{3}\G_{2}/Sp(1) \\ ↓ & & ↓ & & ↓ \\ ℍℙ^{6} & ⟹^{Sp(1)} & \frac{SO(7)}{SO(4)×SO(3)} & ⟹^{U(1)} & ℤ_{3}\G_{2}/SO(4). \end{matrix}

In a later paper Kobak and Swann show that any classical nilpotent orbit can be obtained as a hyperkähler quotient of a flat space of an appropriate dimension so that the above diagram is only an example [KS2]. The middle horizontal line of Diagram 0.1 is absent from the discussion in [KS1]as the importance of 3-Sasakian geometry in this context was realized later [BGM1].

The starting point of this paper is to revisit the Kobak-Swann construction in the context of the associated 3-Sasakian geometry. We are going to examine the quotient construction of the orbifold fibration ℤ_{3}\G_{2}/Sp(1) → ℤ_{3}\G_{2}/SO(4) showing that it admits interesting generalizations. More precisely, the Kobak-Swann quotient can be “deformed” by introducing weights in much the same way toric 3-Sasakian manifolds with b_{2} = 1 can all be obtained by “deforming” the classical homogeneous example of the fibration S(1) = SU(n)/S(U(n − 2) × U(1)) → Gr_{2}(ℂ^{n}) [BGM1]. On the other hand our new quotients are quite different from the construction of S(p) considered in [BGM1] as they cannot be interpreted as bi-quotients. In Section 2 we prove Theorem A: Let p = (p_{1}, p_{2}, p_{3}) ∈ ℤ^{3}. For any such non-zero p one can define an isometric action of Sp(1) × U(1)_{p} ⊂ Sp(7) on S^{27} with the following property: If 0 < p_{1} < p_{2} < p_{3} are pairwise relatively prime and gcd(p_{1} ± p_{2}, p_{1} ± p_{3}) = 1 then the reductions M(p) → Z(p) → O(p) of S^{27} → ℂℙ^{13} → ℍℙ^{6} give a compact smooth 11-dimensional 3-Sasakian manifold M(p) together with the (orbifold) leaf spaces of its fundamental foliations Z(p) and O(p). Furthermore, the reduced space M(1, 1, 1) is a 3-Sasakian orbifold ℤ_{3}\G_{2}/Sp(1).

Analysis of the symmetry structure of all the quotients together with the associated foliations gives Theorem B: The manifold M(p) of Theorem A is not toric. The corresponding leaf spaces Z(p) and O(p) are compact Riemannian orbifolds with inhomogeneous Einstein metrics of positive scalar curvature.

In Section 3 we investigate whether M(p_{1}, p_{2}, p_{3}) admits further reduction by an isometric circle action. More generally we consider Sp(1) × S^{1} × S^{1} actions on the 27sphere and ask if one can get any smooth quotients. Surprisingly, no smooth examples can be found but orbifold quotients exist in profusion. These are interesting, since, due to Theorem B, they are necessarily non-toric (more precisely of cohomogeneity 3). More importantly, they yield new explicit self-dual Einstein metrics of positive scalar curvature with only orbifold singularities and with one-dimensional group of isometries. We get Theorem C: Let Θ ∈ M_{2×3}(ℤ) be any integral 2 × 3 matrix such that each of its three 2 × 2 minor determinants does not vanish. In addition suppose that the sum of the all minor determinants is nonvanishing, and none of them is equal to the some of the other two. For any such Θ there exists a compact 4-dimensional orbifold O(Θ) which admits a self-dual Einstein metric of positive scalar curvature with a one-dimensional group of isometries. Moreover, this metric can be constructed explicitly as a quaternionic Kähler reduction of the real Grassmannian Gr_{4}(ℝ^{7}) by an isometric action of the 2-torus T_{Θ}^{2} defined by Θ.

The first examples of positive self-dual Einstein metrics on orbifolds were obtained in [GL]. Later such metrics were considered by Hitchin [Hi1, Hi2]. Hitchin’s examples have large group of isometries. More recently many new orbifold metrics with T^{2}-symmetry group were constructed in [BGMR]. The examples presented here are perhaps the first self-dual Einstein metrics with only one Killing vector field. We are not aware of any self-dual Einstein metrics which have only discrete isometries.

In Section 4 we consider the obvious higher-dimensional extension of the problem. Not surprisingly, once again the new examples involve hyperkähler geometry of a nilpotent variety. This time it is the 16-dimensional nilpotent orbit of 𝔰𝔬(6, ℂ) which is the Swann bundle over the quaternionic Kähler orbifold Gr_{4}(ℝ^{7})/ℤ_{2}. In [KS2] it is shown how all classical nilpotent orbits can be obtained as hyperkähler reductions from flat spaces (typically in more than one way). This orbit can be constructed as a Sp(1) × U(1) reduction of ℍ^{8} [KS3] and thus it appears as part of a diagram similar to the one in (0.1). In this context our construction is a systematic study of the general U(1) actions which, at the 3-Sasakian level, produce smooth metrics. We prove the following analogue of the Theorem A: Theorem D: Let p = (p_{1}, p_{2}, p_{3}, p_{4}) ∈ ℤ^{4}. For any such non-zero p one can define an action of Sp(1) × U(1)_{p} ⊂ Sp(8) on S^{31} with the following property: If 0 ≤ p_{1} < p_{2} < p_{3} < p_{4}, any triple p_{i} < p_{j} < p_{k} satisfies gcd(p_{i}, p_{j}, p_{k}) = 1, and gcd(p_{i} ± p_{j}, p_{i} ± p_{k}) = 1 then the the reductions M(p) → Z(p) → O(p) of S^{31} → ℂℙ^{15} → ℍℙ^{7} give a compact smooth 15-dimensional 3-Sasakian manifold M(p) together with the (orbifold) leaf spaces of its fundamental foliations Z(p) and O(p). Furthermore, the reduced space M(1, 1, 1, 1) is a 3-Sasakian orbifold ℤ_{2}\Spin(7)/Spin(4).

These 15-dimensional quotients are “deformations” of the standard homogeneous 3-Sasakian structure on SO(7)/SO(3) × Sp(1) which projects to the Wolf space Gr_{4}(ℝ^{7}) in the quaternionic Kähler base. In higher dimensions the orbifold Gr_{4}(ℝ^{n+3})/ℤ_{2} can equally be obtained as a quaternionic Kähler reduction of ℍℙ^{n+4} by U(1) × Sp(1). Our construction shows that the n = 4 case, from the standpoint of 3-Sasakian geometry, is somewhat exceptional: In higher dimension we can only get orbifold metrics. The orbifold bundles M^{15}(p) → Z^{15}(p) → O^{15}(p) can be viewed as singular analogues of SO(7)/SO(3) × Sp(1) → SO(7)/SO(3) × U(2) → Gr_{4}(ℝ^{7}). This makes use of the wellknown isometry between Gr_{4}(ℝ^{7}) and the space Spin(7)/(Sp(1) × Sp(1) × Sp(1))/ℤ_{2} of the Cayley 4-planes in ℝ^{8}.

Again, standard analysis of the symmetry structure of all the quotients together with the associated foliations gives Theorem B′: The manifold M(p) of Theorem D is not toric. The corresponding leaf spaces Z(p) and O(p) are compact Riemannian orbifolds with inhomogeneous Einstein metrics of positive scalar curvature.

In Section 5 we give what topological information is available to us. In particular, we show that as long as p satifies the conditions of Theorems A and D (actually this hypothesis can be weakened) the rational cohomology of the corresponding M(p) is independent of p. Finally, in section 6 we briefly mention the construction of hypercomples structures on circle bundles over our new 3-Sasakian manifolds (orbifolds). Acknowledgements: The first named author would like to thank Odense University for its hospitality and support for a short visit there in June of 2000. The second named author would like to thank Università di Roma “La Sapienza”, C.N.R, M.P.I-Bonn, and I.H.E.S-Bures sur Yvette for hospitality and support as parts of this paper were written during his visits there. We would also like to thank Mike Buchner and Andrew Swann for comments and discussion.

1. Quotient construction of 3-Sasakian structure on ℤ_{3}\G_{2}/Sp(1)

There are two homogeneous Sasakian-Einstein geometries that are naturally associated with the exceptional Lie group G_{2} [BG1, BG2]. They both come from the classical Lie group isomorphism between SO(4) ⊂ G_{2} and Sp(1)_{−} · Sp(1)_{+} ⊂ G_{2}. The two Sp(1)_{±} subgroups are very different. One of them has index 1 in G_{2} and the other one has index 3. Consequently, the quotients are not of the same homotopy type as can be seen from the exact sequence in homotopy for the fibration

Sp(1)_{±} ⟶ G_{2} ⟶ G_{2}/Sp(1)_{±}.

In particular, the two spaces can be distinguished by their third homotopy groups being trivial in one case and ℤ_{3} in the other. One of these quotients, which we shall denote by G_{2}/Sp(1)_{−} is diffeomorphic to the real Stiefel manifold V_{7,2}(ℝ) = SO(7)/SO(5) of 2-frames in ℝ^{7} [HL]. As V_{7,2}(ℝ) is 4-connected π_{3}(G_{2}/Sp(1)_{−}) = 0. The other quotient denoted here by G_{2}/Sp(1)_{+} is one of the 11-dimensional 3-Sasakian homogeneous spaces and π_{3}(G_{2}/Sp(1)_{+}) = ℤ_{3}. G_{2}/Sp(1)_{+} fibers as a circle bundle over a generalized flag Z = G_{2}/U(2)_{+}, which in turn is well-known to be the twistor space of the exceptional 8-dimensional Wolf space G_{2}/SO(4). The second homogeneous Sasakian-Einstein manifold is a circle bundle over the complex flag G_{2}/U(2)_{−} which can be identified with the complex quadric in the 6-dimensional complex projective space ℂℙ^{6} or, equivalently, the real Grassmannian Gr_{2}(ℝ^{7}) = SO(7)/SO(2) × SO(5) of oriented 2-planes in ℝ^{7}. Both Sasakian-Einstein metrics are well-known and have been studied in the context of homogeneous Einstein geometries [BG1]. We have the following diagram of Riemannian submersions:

1.1  G_{2} ; ↙  ↘ ; \frac{G_{2}}{Sp(1)_{+}}  \frac{G_{2}}{Sp(1)_{−}} ≃ V_{7,2}(ℝ) ; ↓  ↓ ; Z = \frac{G_{2}}{U(2)_{+}}  \frac{G_{2}}{U(2)_{−}} ≃ Gr_{2}(ℝ^{7}) ; ↘  ↙ ; \frac{G_{2}}{SO(4)}

Poon and Salamon [PS] proved that G_{2}/SO(4) is one of the three possible models of positive quaternionic Kähler manifolds in dimension 8. Later the geometry of G_{2}/SO(4) was examined by Kobak and Swann [KS1] who proved the following remarkable theorem:

Theorem 1.2: The quaternionic Kähler manifold Gr_{4}(ℝ^{7}) admits an action of U(1) such that the quaternionic Kähler quotient is a compact quaternionic Kähler orbifold O = O_{r} ∪ ℂℙ(2) = G_{2}/(SO(4) × ℤ_{3}).

We will first re-examine the Kobak-Swann construction from the point of view of the 3-Sasakian geometry of a certain SO(3) V-bundle over O (or, equivalently, the hyperkähler geometry of the regular nilpotent orbit of 𝔰𝔩(3, ℂ)). In particular, we have the following: Theorem 1.3: The 3-Sasakian homogeneous manifold SO(7)/SO(3) × Sp(1) admits an action of U(1) such that the 3-Sasakian quotient is a compact 3-Sasakian orbifold M = M_{r} ∪ SU(3)/U(1) = ℤ_{3}\G_{2}/Sp(1)_{+}.

Theorem 1.3 is a straightforward translation of Theorem 1.2 into the language of 3-Sasakian geometry and we could leave it at that. However, we will outline a constructive proof of this result as our description of the corresponding quotient differs slightly from the one given in [KS1].

One can think of the homogeneous 3-Sasakian manifold SO(7)/SO(3) × Sp(1) as the 3-Sasakian reduction S^{4n−1}///Sp(1) as follows [G, BGM1]: Let u = (u_{1}, ..., u_{7}) ∈ S^{27}. Consider the Sp(1) action given by multiplication by unit quaternion λ ∈ Sp(1) on the left that is

1.4  φ_{λ}(u) = λu.

In the {i, j, k} basis the 3-Sasakian moment maps for this action read:

1.5  μ_{i}(u) = ∑_{α=1}^{7} \overline{u}_{α}iu_{α},  μ_{j}(u) = ∑_{α=1}^{7} \overline{u}_{α}ju_{α},  μ_{k}(u) = ∑_{α=1}^{7} \overline{u}_{α}ku_{α}.

Then, the common zero-locus of the moment maps

1.6  N = {u ∈ S^{4n−1} : μ_{i}(u) = μ_{j}(u) = μ_{k}(u) = 0}

is the Stiefel manifold N ≃ SO(7)/SO(3) = V_{7,4}(ℝ) of the orthonormal 4-frames in ℝ^{7} and the corresponding 3-Sasakian quotient S = N/Sp(1) is Konishi’s ℝℙ^{3}-bundle over the real Grassmannian of oriented 4-planes in ℝ^{7}. We can combine Theorem 1.3 with this description to get Corollary 1.7: The 3-Sasakian sphere S^{27} admits an action of U(1) × Sp(1) such that the 3-Sasakian quotient is a compact 3-Sasakian orbifold M = M_{r} ∪ SU(3)/U(1) = ℤ_{3}\G_{2}/Sp(1)_{+}.

We now turn to the explicit description of the U(1) quotient. Consider the following subgroups of the group of 3-Sasakian isometries of the 27-sphere:

1.8  Sp(7) ⊃ SO(7) ⊃ 1 × SO(6) ⊃ 1 × U(3),

where U(1) ⊂ U(3) is the central subgroup. Explicitly, we shall write f : [0, 2π) → SO(7)

1.9  f(t) = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & A(t) & 0 & 0 \\ 0 & 0 & A(t) & 0 \\ 0 & 0 & 0 & A(t) \end{matrix} ) ∈ SO(7),

where

1.10  A(t) = ( \begin{matrix}cos t & sin t \\ − sin t & cos t \end{matrix} )

are the real rotations in ℝ^{2}. The homomorphism f(t) yields a circle action on S^{27} or, equivalently, (after performing the “Sp(1)-reduction” first) on the homogeneous 3-Sasakian manifold SO(7)/SO(3)×Sp(1) via left multiplication f(t)u and the associated 3-Sasakian moment map can be written as

1.11  ν(u) = ∑_{α=1,2,3}(\overline{u}_{2α}u_{2α+1} − \overline{u}_{2α+1}u_{2α}).

Note here that ν(u) does not depend on the u_{1} quaternionic coordinate. Definition 1.12: Let us define the zero level set of this new moment map intersected with N, that is

1.12  N_{ν} ≡ N ∩ ν^{−1}(0)

First we observe, following Kobak and Swann [KS1] that

Lemma 1.13: The manifold N_{ν} can be identified with U(1) · G_{2} = (S^{1} × G_{2})/ℤ_{3} where U(1) ∩ G_{2} = ℤ_{3}. Proof: The argument is similar here to the one used by Kobak and Swann in [KS1] and it is based on the Proposition 1.10 of [HL]. First, using the basis {i, j, k} of unit imaginary quaternions, we write u_{α} = u_{α}^{0} + iu_{α}^{1} + ju_{α}^{2} + ku_{α}^{3} and introduce the 4 × 7 real matrix

1.14  𝔸 = ( \begin{matrix}u_{1}^{0} & u_{2}^{0} & u_{3}^{0} & u_{4}^{0} & u_{5}^{0} & u_{6}^{0} & u_{7}^{0} \\ u_{1}^{1} & u_{2}^{1} & u_{3}^{1} & u_{4}^{1} & u_{5}^{1} & u_{6}^{1} & u_{7}^{1} \\ u_{1}^{2} & u_{2}^{2} & u_{3}^{2} & u_{4}^{2} & u_{5}^{2} & u_{6}^{2} & u_{7}^{2} \\ u_{1}^{3} & u_{2}^{3} & u_{3}^{3} & u_{4}^{3} & u_{5}^{3} & u_{6}^{3} & u_{7}^{3} \end{matrix} ) ≡ ( \begin{matrix}f^{0} \\ f^{1} \\ f^{2} \\ f^{3} \end{matrix} ) ,

where to make the connection with the notation in [KS1] we also think of the rows of 𝔸 as purely imaginary octonions Im(𝕆). In the standard basis of Im(𝕆) we write f^{a} = u_{1}^{a}i + u_{2}^{a}j + u_{3}^{a}k + u_{4}^{a}e + u_{5}^{a}ie + u_{6}^{a}je + u_{7}^{a}ke. Let ϕ(a, b, c) =< ab, c > denote the 3-form defining the associative calibration [HL] on Im(𝕆) where < ·, · > denotes the standard Euclidean inner product. Then writing ν = ν_{1}i + ν_{2}j + ν_{3}k a straightforward computation shows that for a = 1, 2, 3 and ϵ^{abc} the totally antisymmetric tensor satisfying ϵ^{123} = 1

1.15  ν_{a} = 2 < f^{0}f^{a} + ϵ^{abc}f^{b}f^{c}, i >,

where the summation convention on repeated indices is used. Now, u ∈ N if and only if the rows {f^{0}, f^{1}, f^{2}, f^{3}} of 𝔸 form an orthonormal frame in ℝ^{7} ≃ Im(𝕆), and one can identify G_{2} with a special kind of oriented orthonormal 4-frame, namely those which are co-associative. This means that the 3-plane that is orthogonal to the 4-plane defined by the frame {f^{0}, f^{1}, f^{2}, f^{3}} is spanned by an associative subalgebra of Im(𝕆). Then one shows that these special 4-frames satisfy the U(1)-moment map equation ν(u) = 0 and, hence, U(1) · G_{2} ⊂ N ∩ ν^{−1}(0). As, U(1) ∩ G_{2} = ℤ_{3} it is enough to show that by acting with U(1) one gets the whole N ∩ ν^{−1}(0). The argument is similar to the one presented in [KS1]. (See [KS1] Lemma 5.1 and the discussion that follows.)

Now, Theorems 1.2 and 1.3 and Corollary 1.7 all follow from the above lemma as we get the quotient

1.16  M = \frac{N_{ν}}{U(1) × Sp(1)} ≃ \frac{U(1) · G_{2}}{U(1) × Sp(1)} ≃ ℤ_{3}\G_{2}/Sp(1).

remark 1.17: The U(1)×Sp(1) action on the level set N_{ν} is not locally free. If we divide by Sp(1) first and consider the U(1) action on the orbit space N_{ν}/Sp(1) this circle action is quasi-free. This means that there are only two kinds orbits: regular orbits with the trivial isotropy group and singular orbits (points) where the isotropy group is the whole U(1). In such cases the quotient space is often an orbifold (or even a smooth manifold). The stratification of the Theorem 2 is precisely with respect to the orbit types as will be seen in the next section.

2. Generalizations of the Kobak-Swann Quotient

In this section we will consider the simplest possible family of quotients which generalize the construction of Kobak and Swann via an introduction of weights. Instead of considering the central U(1) ⊂ U(3) in 1.9 we can consider and arbitrary circle subgroup of the maximal torus U(1) ⊂ T^{3} ⊂ U(3). Again, to be more specific, we have the following inclusions:

2.1  Sp(7) ⊃ SO(7) ⊃ 1 × SO(6) ⊃ 1 × SO(2) × SO(2) × SO(2).

We can consider arbitrary circle subgroups of the last 3-torus. Let p = (p_{1}, p_{2}, p_{3}) ∈ ℤ^{3} and define the following homomorphism

2.2  f_{p}(t) = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & A(p_{1}t) & 0 & 0 \\ 0 & 0 & A(p_{2}t) & 0 \\ 0 & 0 & 0 & A(p_{3}t) \end{matrix} ) ∈ SO(7),

where

2.3  A(p_{i}t) = ( \begin{matrix}cos(p_{i}t) & sin(p_{i}t) \\ − sin(p_{i}t) & cos(p_{i}t) \end{matrix} ) ∈ SO(2), i = 1, 2, 3

are 2-dimensional real rotations. Note that for p_{1} = p_{2} = p_{3} = 1 we recover the example of the previous section. The homomorphism f_{p}(t) yields a circle action on the homogeneous 3-Sasakian manifold SO(7)/SO(3) × Sp(1) via left multiplication f_{p}(t)u and the moment map can now be written as

2.4  ν_{p}(u_{1}, ..., u_{7}) = ∑_{α=1,2,3} p_{α}(\overline{u}_{2α}u_{2α+1} − \overline{u}_{2α+1}u_{2α}).

Observe that without loss of generality we can assume all weights to be non-negative as p_{i} can be changed to −p_{i} by renaming the quaternions in the associated pair (u_{2i}, u_{2i+1}). We begin analysis of this quotient by considering the level set of the moment map Definition 2.5: Let N_{ν}(p) ⊂ S^{27} be the level set of the 3-Sasakian moment map of the Sp(1) × U(1)_{p}-action, i.e., N_{ν}(p) ≡ N ∩ {ν_{p}^{−1}(0)}.

We want to consider a stratification of the level set N_{ν}(p) that will allow us to analyze the quotient space

2.6  M(p) = \frac{N_{ν}(p) ≡ V_{7,4}(ℝ) ∩ \{ν_{p}^{−1}(0)\}}{Sp(1) × U(1)_{p}}.

Since N_{ν}(p) is a submanifold of the Stiefel manifold V_{7,4}(ℝ) at most 3 quaternionic coordinates can vanish on N_{ν}(p). So setting various quaternionic coordinates equal to zero determines a stratification of N_{ν}(p) in which the strata of minimal dimension play an important role. We call these strata vertices although, as we shall see, they each have two connected components. Lemma 2.7: Let 0 < p_{1} < p_{2} < p_{3}. At a vertex neither u_{1} nor any of the three pairs of quaternions (u_{2i}, u_{2i+1}), i = 1, 2, 3 can vanish. Thus, there are precisely eight vertices and they are all diffeomorphic to O(4). Proof: Every vertex has precisely 3 quaternionic coordinates vanishing, so the Stiefel manifold becomes V_{4,4}(ℝ) = O(4). Let V be a vertex. Then V = V_{ijkl} can be represented by a matrix of the form

2.8  𝔹 = ( \begin{matrix}u_{i}^{0} & u_{i}^{1} & u_{i}^{2} & u_{i}^{3} \\ u_{j}^{0} & u_{j}^{1} & u_{j}^{2} & u_{j}^{3} \\ u_{k}^{0} & u_{k}^{1} & u_{k}^{2} & u_{k}^{3} \\ u_{l}^{0} & u_{l}^{1} & u_{l}^{2} & u_{l}^{3} \end{matrix} ) ,  𝔹^{t}𝔹 = 𝔹𝔹^{t} = \frac{1}{4}𝕀_{4},

where 1 ≤ i < j < k < l ≤ 7. Here we have written a quaternionic coordinate as u_{i} = u_{i}^{0} + iu_{i}^{1} + ju_{i}^{2} + ku_{i}^{3}. Suppose i > 1, i.e., u_{1} = 0 on on V_{ijkl}. Then there are two possibilities: (1) the quadruple i < j < k < l contains only one quaternionic pair (2α, 2α+1) or (2) it contains two such pairs. Let us examine the first possibility. Without loss of generality we can take i = 2, j = 3, k = 4, l = 6. With this choice the vanishing of the U(1) moment map 2.4 now becomes

Im(ū_{2}u_{3}) = 0.

One can easily check that orthogonality of the basis forces

Re(ū_{2}u_{3}) = u_{2}^{1}u_{3}^{1} + u_{2}^{2}u_{3}^{2} + u_{2}^{3}u_{3}^{3} + u_{2}^{4}u_{3}^{4} = 0.

But this implies ū_{2}u_{3} = 0 which forces either u_{2} or u_{3} to vanish, giving a contradiction. Now assume the second case, that is, that i < j < k < l consists of two quaternionic pairs (2α, 2α + 1). Again, when we take i = 2, j = 3, k = 4, l = 5, the orthogonality of the vectors in the associated 𝔹-matrix forces

2.9  \begin{aligned}Re(ū_{2}u_{3}) & = u_{2}^{1}u_{3}^{1} + u_{2}^{2}u_{3}^{2} + u_{2}^{3}u_{3}^{3} + u_{2}^{4}u_{3}^{4} = 0, \\ Re(ū_{4}u_{5}) & = u_{4}^{1}u_{5}^{1} + u_{4}^{2}u_{5}^{2} + u_{4}^{3}u_{5}^{3} + u_{4}^{4}u_{5}^{4} = 0. \end{aligned}

But orthogonality also implies

|u_{2}|^{2} = |u_{3}|^{2} = |u_{4}|^{2} = |u_{5}|^{2} = \frac{1}{4}.

Then we have

|Im(ū_{2}u_{3})|^{2} = |Im(ū_{4}u_{5})|^{2} = \frac{1}{16},

and this contradicts the U(1) moment map equation

2p_{1}Im(ū_{2}u_{3}) + 2p_{2}Im(ū_{4}u_{5}) = 0

if p_{1} ≠ p_{2}. Repeating the argument for the other choices of 1 < i < j < k < l gives the result under the hypothesis p_{1} < p_{2} < p_{3}. Thus, at a vertex we must have u_{1} ≠ 0. This proves the “neither” part of the statement.

Now assume that i = 1 < j < k < l and suppose either j < k or k < l is a quaternionic pair. We can take j = 2, k = 3 and let l be arbitrary. Then the orthogonality of the corresponding 𝔹-matrix again forces

Re(ū_{2}u_{3}) = 0.

But again the U(1) moment map constraint is

Im(ū_{2}u_{3}) = 0

giving a contradiction. This proves the “nor” part of the lemma. It is now clear that the vertices must automatically satisfy the U(1) moment map constraint ν_{p}(u) = 0; hence, they are all diffeomorphic to O(4). Moreover, a simple counting shows that there are precisely eight vertices.

Our analysis suggests the importance of the following strata:

2.10

S_{0} = {u ∈ N_{ν}(p) | u_{1} = 0},
S_{1} = {u ∈ N_{ν}(p) | u_{1} ≠ 0},
S_{2} = {u ∈ N_{ν}(p) | some quaternionic pair (u_{2i}, u_{2i+1}) vanishes},
S_{3} = {u ∈ N_{ν}(p) | no quaternionic pair (u_{2i}, u_{2i+1}) vanishes}.

Then Lemma 2.7 easily implies that Corollary 2.11: Let 0 < p_{1} < p_{2} < p_{3}. Then

(i) S_{0} ∪ S_{1} = S_{2} ∪ S_{3} = N_{ν}(p). (ii) S_{0} ∩ S_{1} = S_{2} ∩ S_{3} = ∅. (iii) S_{0} ∩ S_{2} = ∅. (iv) S_{2} ⊂ S_{1}.

(v) S_{0} ⊂ S_{3}. Notice that (iii) fails if p_{i} = p_{j} for some i ≠ j. In particular it fails for the level set N_{ν}

of 1.12 in the previous section, and this is the reason that the quotient M of 1.16 is not smooth. We now are ready to give necessary conditions to guarantee a smooth quotient. Lemma 2.12: Let p = (p_{1}, p_{2}, p_{3}) ∈ (ℤ_{+})^{3} be pairwise relatively prime. Then the isotropy group of the Sp(1) × U(1)_{p} action at every point of S_{1} is the identity. Proof: The action of Sp(1)×U(1)_{p} on ℍ^{7} is the diagonal action of Sp(1) by quaternionic multiplication by a unit quaternion λ on the left, and the matrix multiplication u ↦ f_{p}(t)u for the U(1)_{p} action. These two actions clearly commute. Since u_{1} ≠ 0 we immediately get that λ = 1. Consider the set where a quaternionic pair (u_{6}, u_{7}) = (0, 0). Then the fixed point equation becomes

2.13  A(p_{1}t) = A(p_{2}t) = 𝕀_{2},

which has only the trivial solution provided that gcd(p_{1}, p_{2}) = 1. Setting the other two quaternionic pairs to be zero gives gcd(p_{1}, p_{3}) = gcd(p_{2}, p_{3}) = 1. As one cannot set more than one quaternionic pair equal to (0, 0) the lemma is proved. Lemma 2.14: Let p = (p_{1}, p_{2}, p_{3}) ∈ (ℤ_{+})^{3} satisfy the four conditions gcd(p_{1}±p_{2}, p_{1}±p_{3}) = 1. Then the isotropy group of the Sp(1) × U(1)_{p} action at every point of S_{0} is the identity. Proof: To determine the conditions for fixed points of the action we consider the following equations

A(p_{i}t) ( \begin{matrix}u_{2i} \\ u_{2i+1} \end{matrix} ) = ( \begin{matrix}a_{i} & b_{i} \\ −b_{i} & a_{i} \end{matrix} ) ( \begin{matrix}u_{2i} \\ u_{2i+1} \end{matrix} ) = λ ( \begin{matrix}u_{2i} \\ u_{2i+1}, \end{matrix} ) i = 1, 2, 3

for λ ∈ Sp(1) and t ∈ [0, 2π). For each i = 1, 2, 3 this reads

a_{i}u_{2i} + b_{i}u_{2i+1} = λu_{2i},
−b_{i}u_{2i} + a_{i}u_{2i+1} = λu_{2i+1}.

For each i = 1, 2, 3 we multiply the first equation from the right by \overline{u}_{2i} and the second by \overline{u}_{2i+1} to get

a_{i}|u_{2i}|^{2} + b_{i}u_{2i+1}\overline{u}_{2i} = λ|u_{2i}|^{2},
2.15  −b_{i}u_{2i}\overline{u}_{2i+1} + a_{i}|u_{2i+1}|^{2} = λ|u_{2i+1}|^{2}.

By adding these two equations we get

a_{i}(|u_{2i}|^{2} + |u_{2i+1}|^{2}) + b_{i}(u_{2i+1}\overline{u}_{2i} − u_{2i}\overline{u}_{2i+1}) = λ(|u_{2i}|^{2} + |u_{2i+1}|^{2}), i = 1, 2, 3.

By (iii) of Lemma 2.10 the term multiplying λ on the right hand side of this equation never vanishes. This gives for each i = 1, 2, 3

2.16  \begin{aligned}Re(λ) & = a_{i}, \\ Im(λ) & = b_{i} \frac{u_{2i+1}\overline{u}_{2i} − u_{2i}\overline{u}_{2i+1}}{|u_{2i}|^{2} + |u_{2i+1}|^{2}} . \end{aligned}

The first of these equations gives

2.17  a_{1} = a_{2} = a_{3},

and combining this with a_{i}^{2} + b_{i}^{2} = 1 implies

2.18  b_{1} = ±b_{2} = ±b_{3}.

Let us write τ = e^{it}. Then 2.17 and 2.18 give

2.19  τ^{p_{1}} = τ^{±p_{2}},  τ^{p_{1}} = τ^{±p_{3}}.

These have only trivial solutions if and only if gcd(p_{1} ± p_{2}, p_{1} ± p_{3}) = 1.

It is convenient to make the following: Definition 2.20: Let p = (p_{1}, p_{2}, p_{3}) ∈ ℤ^{3}. We say that the weight vector p is admissible if 0 < p_{1} < p_{2} < p_{3}, gcd(p_{i}, p_{j}) = 1 for all i < j, and gcd(p_{1} ± p_{2}, p_{1} ± p_{3}) = 1.

It now follows immediately from Lemmas 2.12, 2.14 and Definition 2.20 that Theorem 2.21: The Sp(1) × U(1)_{p} action on N_{ν}(p) is free if and only if p ∈ ℤ_{+}^{3} is admissible.

Note that there are infinitely many admissible weight vectors. For example we can take p = (2k − 1, 2k, 2k + 1), where k ∈ ℤ_{+}. Thus, there are infinite families of smooth quotients M(p) and infinite families of the associated triples M(p) → Z(p) → O(p) with their (orbifold) Einstein metrics.

Theorem A now follows from the Theorem 2.21 and various theorems concerning 3-Sasakian (complex contact, quaternionic Kähler) reductions [BGM1, GL]. The last statement of Theorem A follows from Corollary 1.7.

We briefly return to the p = (1, 1, 1) case of the previous section. As already observed we get singularities here because the corresponding action is not even locally free on the level set of the moment map. But it is easy to see that it is quasi-free. That is there are only two types of isotropy subgroups in the circle: the identity and the whole group. In such a situation Dancer and Swann [DS] observed that 3-Sasakian quotients, stratify as the union of 3-Sasakian manifolds [DS]. This is true in this case in particular, but as we saw in the previous section the two strata nicely fit together and one gets a compact 3-Sasakian orbifold. Let us explicitly describe the singular part M_{1} ⊂ M(1, 1, 1).

Note that if p_{1} = p_{2} = p_{3} = 1 then a_{1} = a_{2} = a_{3} = a and b_{1} = b_{2} = b_{3} = b and we can add equations (2.15) to get

2.22  a + bρ = λ,

where now

2.23  ρ = ∑_{i=1,2,3}(u_{2i+1}\overline{u}_{2i} − u_{2i}\overline{u}_{2i+1}).

When u_{1} ≠ 0 one does not get any fixed points of the action. But when u_{1} = 0 for any imaginary unit ρ there is a U(1) ⊂ Sp(1) × U(1) subgroup

(cos t + ρ sin t, A(t)) ∈ Sp(1) × U(1),

which acts trivially on the following set

2.24  {u ∈ N_{ν} | u_{2i+1} = ρu_{2i}}.

In this case all 4 moment map equations reduce to the same one and it reads

∑_{i=1,2,3} \overline{u}_{2i}ρu_{2i} = 0.

For any fixed ρ we can recognize this set as the complex Stiefel manifold U(3)/U(1) and it follows that the singular stratum M_{1} is precisely the quotient SU(3)/U(1) = S(1, 1, 1).

The geometry of the smooth families M(p) is rather interesting. First we observe that these spaces cannot be toric. This can be seen in several different ways, for example by careful analysis of the associated foliations. One can also generalize the analysis of [BGM2] to show that the only isometries of the level set of the moment map N_{ν}(p) ⊂ S^{27} ⊂ ℝ^{28} can come from the restriction of the isometries of the Euclidean space ℝ^{28}. From this we conclude Theorem 2.25: Let p be admissible so that M(p) is a smooth compact 3-Sasakian 11manifold. Then the Lie algebra Isom^{0}(M(p), g(p)) of the group of 3-Sasakian isometries of M(p) is isomorphic to ℝ^{2} ⊕ 𝔰𝔭(1). In particular, all such quotients are non-toric.

This proves part of Theorem C of the introduction which relates to the 11-dimensional quotients.

Finally, observe that M(p) contains a special 7-manifold which is embedded as a 3-Sasakian submanifold. Define

2.26  S_{0}(p) = \frac{N_{ν}(p) ∩ \{u_{1} = 0\}}{Sp(1) × U(1)_{p}}.

One can see that S_{0}(p) is a submanifold and, as it is itself a 3-Sasakian reduction, it must be a 3-Sasakian submanifold. One can even identify this space. Observe that the classical group isomorphism Spin(6) ≃ SU(4) implies that the Sp(1) quotient yields the homogeneous 3-Sasakian 11-dimensional manifold S(1, 1, 1, 1). Hence, S_{0}(p) is either a U(1)_{p}-reduction of S(1, 1, 1, 1) or, equivalently, a T^{2}-reduction of S^{15}. Hence, there exists an admissible integer weight matrix Ω ∈ A_{2×4}(ℤ) (see [BGMR]) such that S_{0}(p) ≃ S(Ω). Hence S_{0}(p) is toric with second Betti number equal to 2.

3. Further Reductions of M(p_{1}, p_{2}, p_{3}) by a Circle

In this section we will examine reductions of M(p_{1}, p_{2}, p_{3}) by isometric circle actions. More generally we shall consider an arbitrary 2-torus subgroup of the maximal torus T^{2} ⊂ T^{3} ⊂ SO(7). Let

3.1  Θ = ( \begin{matrix}p_{1} & p_{2} & p_{3} \\ q_{1} & q_{2} & q_{3} \end{matrix} ) ∈ M_{2×3}(ℤ)

be an arbitrary integral 2 × 3 matrix and define the homomorphism f_{Θ} : T^{2} → SO(7)

3.2  f_{Θ}(t, s) = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & A(p_{1}t + q_{1}s) & 0 & 0 \\ 0 & 0 & A(p_{2}t + q_{2}s) & 0 \\ 0 & 0 & 0 & A(p_{3}t + q_{3}s) \end{matrix} ) ∈ SO(7),

where A is the SO(2) rotations as in 2.3. Note that if p = (p_{1}, p_{2}, p_{3}) were admissible then we would be considering arbitrary isometric circle actions on the quotient M(p); however, so far we assume nothing about Θ. The homomorphism f_{Θ}(t, s) yields a 2-torus action on the homogeneous 3-Sasakian manifold SO(7)/SO(3) × Sp(1) via left multiplication f_{Θ}(t, s)u and the moment map can now be written as

3.3  ν_{Θ}(u_{1}, ..., u_{7}) = ( \begin{matrix}∑_{α=1,2,3} p_{α}(\overline{u}_{2α}u_{2α+1} − \overline{u}_{2α+1}u_{2α}) \\ ∑_{α=1,2,3} q_{α}(\overline{u}_{2α}u_{2α+1} − \overline{u}_{2α+1}u_{2α}) \end{matrix} ) ∈ ℝ^{2} ⊗ 𝔰𝔭(1).

We begin our analysis of this quotient by considering the level set of the moment map. Definition 3.4: Let N_{ν}(Θ) ⊂ S^{27} denote the level set of the 3-Sasakian moment map of the Sp(1) × T_{Θ}^{2}-action, i.e., N_{ν}(Θ) ≡ N ∩ {ν_{Θ}^{−1}(0)} and let

3.5  M(Θ) = N_{ν}(Θ)/Sp(1) × T_{Θ}^{2}.

We want to determine for which Θ ∈ M_{2×3}(ℤ) the 7-dimensional quotient M(Θ) is an orbifold and, if possible which weight matrices yield smooth quotients. Let us define

3.6  Δ_{ij} = Δ_{ij}(Θ) = det ( \begin{matrix}p_{i} & p_{j} \\ q_{i} & q_{j} \end{matrix} ) ,  1 ≤ i < j ≤ 3,

the three minor determinants of Θ. Lemma 3.7: The action of Sp(1) × T_{Θ}^{2} on N_{ν}(Θ) ∩ {u_{1} ≠ 0} is

(i) locally free if and only if Δ_{ij}(Θ) ≠ 0, ∀ 1 ≤ i < j ≤ 3, (ii) free if and only if |Δ_{ij}(Θ)| = 1, ∀ 1 ≤ i < j ≤ 3.

Proof: Since u_{1} ≠ 0 we must have λ = 1 and, hence, it is enough to consider the T_{Θ}^{2}- action. Now, suppose that a quaternionic pair, say (u_{6}, u_{7}) vanishes. Then the fixed point equation reads:

3.8  A(p_{i}t + q_{i}s) ( \begin{matrix}u_{2i} \\ u_{2i+1} \end{matrix} ) = ( \begin{matrix}u_{2i} \\ u_{2i+1} \end{matrix} ) , i = 1, 2

for t, s ∈ [0, 2π), or, equivalently,

3.9  A(p_{i}t + q_{i}s) = 𝕀_{2}, i = 1, 2.

Let τ = e^{it} and ρ = e^{is}. Then we can rewrite 3.9 as

3.10  τ^{p_{i}}ρ^{q_{i}} = 1, i = 1, 2.

This has only discrete solutions provided Δ_{12}(Θ) ≠ 0. Furthermore, the isotropy group at all such points will be trivial provided Δ_{12}(Θ) = ±1. This proves the lemma.

Note that the second condition is already very restrictive as it says that any 2 × 2 submatrix of Θ must be an element of P SL(2, ℤ). However, there are many matrices which satisfy both conditions, for example

Θ_{1} = ( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 1 \end{matrix} ) , Θ_{2} = ( \begin{matrix}9 & 2 & 7 \\ 40 & 9 & 31 \end{matrix} ) .

It remains to analyze the fixed point equations on N_{ν}(Θ) ∩ {u_{1} = 0}. We now prove

Lemma 3.11: The action of Sp(1) × T_{Θ}^{2} on N_{ν}(Θ) is locally free if and only if Δ_{ij}(Θ) ≠ 0, ∀ 1 ≤ i < j ≤ 3 and

3.12  □_{∓}^{∓} = det ( \begin{matrix}p_{1} ∓ p_{2} & q_{1} ∓ q_{2} \\ p_{1} ∓ p_{3} & q_{1} ∓ q_{3} \end{matrix} ) ≠ 0

Furthermore, there is no weight matrix Θ ∈ M_{2×3}(ℤ) for which the actions is free.

Proof: First let us clarify that what we mean in 3.12 is that four determinants □_{+}^{+}, □_{+}^{−}, □_{−}^{+}, and □_{−}^{−} must vanish (in any row we can choose either upper or lower signs). Since the Δ_{ij}(Θ) ≠ 0, by the previous Lemma, we know that the action is locally free on the u_{1} ≠ 0 part. Hence, it is enough to consider u_{1} = 0. Here the analysis is similar to the one presented in Lemma 2.14 and it is entirely based on the fact that no quaternionic pair can vanish. As a result we get the following analogue of the fixed point equations 2.19:

3.13  τ^{p_{1}}ρ^{q_{1}} = (τ^{p_{2}}ρ^{q_{2}})^{±1},  τ^{p_{1}}ρ^{q_{1}} = (τ^{p_{3}}ρ^{q_{3}})^{±1}.

We can rewrite these as

3.14  τ^{p_{1}∓p_{2}} = ρ^{−q_{1}±q_{2}},  τ^{p_{1}∓p_{3}} = ρ^{−q_{1}±q_{3}}.

These are four systems of two equations in (τ, ρ) variables. We want all four of them to have at most discrete solutions. This requires that the four determinants

3.15  det ( \begin{matrix}p_{1} ∓ p_{2} & −q_{1} ± q_{2} \\ p_{1} ∓ p_{3} & −q_{1} ± q_{3} \end{matrix} ) = −det ( \begin{matrix}p_{1} ∓ p_{2} & q_{1} ∓ q_{2} \\ p_{1} ∓ p_{3} & q_{1} ∓ q_{3} \end{matrix} )

do not vanish and gives 3.12.

The fact that orbifold singularities are always present requires more subtle analysis. To have smooth quotients we must assume

3.16  Δ_{ij}(Θ) = ±1,  ∀ 1 ≤ i < j ≤ 3,

one one hand, and

3.17  (□_{∓}^{∓}) = | \begin{matrix}p_{1} ∓ p_{2} & q_{1} ∓ q_{2} \\ p_{1} ∓ p_{3} & q_{1} ∓ q_{3} \end{matrix} | = ±1

on the other. A simple computation relates all of these four determinants to the 3 minor determinants Δ_{ij}(Θ) and we get

□_{−}^{−} = det ( \begin{matrix}p_{1} − p_{2} & q_{1} − q_{2} \\ p_{1} − p_{3} & q_{1} − q_{3} \end{matrix} ) = Δ_{12}(Θ) + Δ_{23}(Θ) − Δ_{13}(Θ),

□_{+}^{−} = det ( \begin{matrix}p_{1} − p_{2} & q_{1} − q_{2} \\ p_{1} + p_{3} & q_{1} + q_{3} \end{matrix} ) = Δ_{12}(Θ) − Δ_{23}(Θ) + Δ_{13}(Θ),

□_{−}^{+} = det ( \begin{matrix}p_{1} + p_{2} & q_{1} + q_{2} \\ p_{1} − p_{3} & q_{1} − q_{3} \end{matrix} ) = −Δ_{12}(Θ) − Δ_{23}(Θ) − Δ_{13}(Θ),

3.18  □_{+}^{+} = det ( \begin{matrix}p_{1} + p_{2} & q_{1} + q_{2} \\ p_{1} + p_{3} & q_{1} + q_{3} \end{matrix} ) = −Δ_{12}(Θ) + Δ_{23}(Θ) + Δ_{13}(Θ).

Now, because of 3.16, all three minor determinants must be ±1. This gives 8 possible combinations of the values of Δ_{ij}(Θ). It is trivial to check that for any one out of these eight at least two determinants □_{∓}^{∓} will be equal to ±3 (the other six all being equal to ±1). Hence, even if we choose Θ so that 3.16 holds, the quotient will necessarily have orbifold singularities of type ℤ_{3}. This concludes the proof of Lemma 3.11

Using the calculation in the proof of the above lemma we restate condition 3.12 to get Theorem 3.19: The action of Sp(1) × T_{Θ}^{2} on N_{ν}(Θ) is locally free if and only if

(1) all their determinants Δ_{12}(Θ), Δ_{23}(Θ), Δ_{13}(Θ) do not vanish, and

(2) their sum does not vanish, and

(3) none of the determinants is equal to the sum of the other two. In such a case the quotient M(Θ) is a compact 7-dimensional 3-Sasakian orbifold. Furthermore, there is no weight matrix Θ for which M(Θ) is a smooth manifold.

Now, Theorem C of the introduction follows from Theorem 3.19, and the fact that the fundamental 3-dimensional foliation M(Θ) → O(Θ), in the case M(Θ) is a compact orbifold, yields a compact self-dual Einstein orbifold with a positive scalar curvature (orbifold) metric as the space of leaves. The fact that this metric has only one Killing vector field follows from an appropriate generalization of Theorem B. Remark 3.20: Note that both M(Θ) and O(Θ) depend only on the three minor determinants Δ_{12}(Θ), Δ_{23}(Θ), Δ_{13}(Θ) rather that on Θ itself. Different weight matrices can certainly lead to equivalent quotients. One could compute the self-dual Einstein metrics g(Θ) on O(Θ) explicitly. Locally we can change variables so that

( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & A(p_{1}t+q_{1}s) & 0 & 0 \\ 0 & 0 & A(p_{2}t+ q_{2}s) & 0 \\ 0 & 0 & 0 & A(p_{3}t+q_{3}s) \end{matrix} )=( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & A(λ_{1}) & 0 & 0 \\ 0 & 0 & A(λ_{2}) & 0 \\ 0 & 0 & 0 & A(aλ_{1}+bλ_{2}) \end{matrix} ) ,

where a = −\frac{Δ_{23}(Θ)}{Δ_{12}(Θ)} and b = \frac{Δ_{13}(Θ)}{Δ_{12}(Θ)} are now non-zero rational numbers. Such a choice simplifies both the 2-torus action as well as the moment map equations. The self-dual Einstein quotient metric in question (up to scale) will depend on these two parameters g(Θ) = g(a, b).

4. 15-Dimensional Examples

At first, it may appear that there are obvious higher dimensional analogues of our construction. However, a simple parity argument shows that such actions do not yield 3-Sasakian metrics other than in dimensions 7, 11, and 15. One can have orbifold metrics in any 3-Sasakian dimension. On the other hand the existence of these smooth quotient in dimension 11 and 15 is closely related to the geometry of G_{2} and Spin(7).

We consider two separate cases of U(1)_{p} ⊂ SO(2k + 1) and U(1)_{p} ⊂ SO(2k). In the SO(2k + 1) case it suffices to take k = 4. Then p = (p_{1}, p_{2}, p_{3}, p_{4}) and the new U(1)_{p} action is defined by adding an SO(2) matrix A(p_{4}t) and let it act by rotating the additional quaternionic coordinates (u_{8}, u_{9}). To get a smooth quotient any triple p_{i} < p_{j} < p_{k} would have to be admissible according to the definition 2.31, as is easily seen from the analysis given in Section 2. However, this is clearly impossible as admissibility implies that each triple contains two odd and one even number.

In the SO(2k) case we have already seen in the previous section that k = 3 leads to the toric 7-manifolds S(Ω) with b_{2} = 2. Now we will show that, in fact, k = 4 leads to smooth quotients. Let p = (p_{1}, p_{2}, p_{3}, p_{4}) ∈ ℤ^{4} and define the following homomorphism

4.1  f_{p}(t) = ( \begin{matrix}A(p_{1}t) & 0 & 0 & 0 \\ 0 & A(p_{2}t) & 0 & 0 \\ 0 & 0 & A(p_{3}t) & 0 \\ & 0 & 0 & A(p_{4}t) \end{matrix} ) ∈ SO(8),

where

4.2  A(p_{i}t) = ( \begin{matrix}cos(p_{i}t) & sin(p_{i}t) \\ − sin(p_{i}t) & cos(p_{i}t) \end{matrix} ) ∈ SO(2), i = 1, 2, 3, 4

are 2-dimensional real rotations. As before we can choose all weights to be non-negative. Further note that at most one of the weights can vanish. We prove the following Lemma 4.3: Let (p_{1}, p_{2}, p_{3}, p_{4}) ∈ ℤ^{4}. Then the action of U(1)_{p} × Sp(1) on N_{ν}(p) is free if and only if 0 ≤ p_{1} < p_{2} < p_{3} < p_{4}, gcd(p_{i}, p_{j}, p_{k}) = 1 and gcd(p_{i} ± p_{j}, p_{i} ± p_{k}) = 1 for any triple in p. Proof: First, let us assume that all of the weights are non-negative. Consider a triple, say (p_{1}, p_{2}, p_{3}). Set u_{7} = u_{8} = 0. Then the analysis of the previous section shows that the three must be distinct and that we must have gcd(p_{1} ± p_{2}, p_{1} ± p_{2}) = 1. However, we no longer need the three weights to be pairwise relatively prime as one cannot set two of the quaternionic pairs (u_{2i−1}, u_{2i}), i = 1, 2, 3, 4 equal to (0, 0) at the same time. One such quaternionic pair can vanish; hence, we need gcd(p_{1}, p_{2}, p_{3}) = 1 to get a free action. The analysis in the case when p_{1} = 0 is similar. Then one sees that the triple (p_{2}, p_{3}, p_{4}) has to be admissible in the sense of the Definition 2.31. But that is what Lemma 2.9 says in this case.

It is immediately clear that one cannot extend this construction for k > 4 without admitting orbifold singularities in the quotient.

Theorem B follows from the above lemma except for its last statement. When p = (1, 1, 1, 1) the quotient is singular. It easy to see that the action is quasi-regular just as in the 11-dimensional case. The fact that the two strata fit together giving the 3-Sasakian orbifold ℤ_{2}\Spin(7)/Spin(4) can be seen by first identifying the zero level set N_{ν}(1, 1, 1, 1) with the U(1) · (Spin(7)/Sp(1)) and observing that U(1) ∩ Spin(7) ≃ U(1) ∩ SU(4) ≃ ℤ_{2} [BGOP].

Actually, p = (1, 1, 1, 1) is not quite the reduction considered by Kobak and Swann [KS3]. Instead they use p = (0, 0, 0, 1) for the U(1)-action. The latter is easily seen to give Gr_{4}(ℝ^{7})/ℤ_{2} as the quaternionic Kähler quotient. This is a simple consequence of the fact that a ”zero momentum” hyperkähler reduction of ℍ^{2} by any circle action is isometric to ℍ/Γ (Γ = ℤ_{2} for the standard action). The identification of the cases p = (1, 1, 1, 1) and p = (0, 0, 0, 1) owes to the isomorphism between 𝔰𝔬(6) and 𝔰𝔲(4).

The fact that these quotients cannot be toric follows from the same type of argument as the one used for 11-dimensional quotients. The Lie algebra of the of the group of the 3-Sasakian isometries is ℝ^{3} ⊕ 𝔰𝔭(1).

The smooth 15-dimensional manifolds M(p) for p_{1} = 0 and p_{1} > 0 are geometrically different. In the first case M(p) contains two copies of the 11-dimensional 3-Sasakian manifold M(p_{2}, p_{3}, p_{4}) which intersect in the 7-dimensional toric 3-Sasakian submanifold S(Ω(p_{2}, p_{3}, p_{4})). When p_{1} > 0 then M(p) does not have any obvious 11-dimensional 3-Sasakian submanifolds. However, we do get 4 disjoint toric 7-dimensional 3-Sasakian submanifolds S(Ω(p_{1}, …, p̂_{i}, …, p_{4})) by setting one of the quaternionic pairs (u_{2i−1}, u_{2i}) = (0, 0).

We can also consider non-zero momentum ξ ∈ 𝔰𝔭(1)-deformations of the hyperkähler metric on the cones C(M(p)). (Up to scale one can set ξ to be any imaginary quaternion, so we really have just one parameter family). In some sense they are all deformations of Kronheimer’s hyperkähler metrics on the two nilpotent orbits in 𝔰𝔩(3, ℂ) and 𝔰𝔬(6, ℂ). Unfortunately such hyperkähler quotients are rarely free from orbifold singularities. In fact, in the 12-dimensional case we never get any complete metrics. In 16 dimensions we do get a complete metric only when p = (1, 1, 1, 1) (or equivalently p = (0, 0, 0, 1)). The metric is SU(4)-invariant and it gives the Kronheimer metric on the 16-dimensional nilpotent orbit of 𝔰𝔬(6, ℂ) in the ξ → 0 scaling limit. More generally, cohomogeneity 2 metrics were studied by Kobak and Swann [KS4].

As each classical nilpotent orbit is a hyperkähler reduction of some quaternionic vector space [KS2] it is tempting to undertake a more systematic study of the following problem: Which of the nilpotent orbits can give rise to compact 3-Sasakian manifolds? Certainly, any time a quotient involves some U(1)-factor one can introduce weights. However, as demonstrated here, requiring smoothness often puts very severe restrictions on the weights. We plan to address some of these questions in a future work [BGOP]. Remark 4.4: Note that all of the quotients considered in this paper are examples of toric reductions of the 3-Sasakian homogeneous space associated to the real Grassmanian Gr_{4}(ℝ^{n}). This space is SO(n)/SO(n − 4) × Sp(1) and it has SO(n) as the group of isometries preserving the 3-Sasakian structure. Consider the maximal torus T^{l} ⊂ SO(n). Then the relevant question is: Which subgroups T^{m} ⊂ T^{l} yield smooth 3-Sasakian quotients? Sections 2 and 3 give the complete analysis of the n = 2k + 1 = 7 case. In Section 2 m = 1 and in Section 3 m = 2 which exhaust all interesting possibilities. This section considers n = 2k = 8 case with m = 1. One can see that m > 1 does not yield any smooth quotient but we leave the analysis of this to a future work where we plan to give a complete answer in the most general case of arbitrary (m, n) [BGP].

5. Comments on the Topology of M(p) and Related Spaces

In this section we denote by M(p) either one of the 11 or 15 dimensional 3-Sasakian manifolds discussed in sections 2 and 4 with p admissible. Actually as discussed below M(p) will denote a component of the manifolds discussed previously. It would be interesting to know the topology of our quotients, most importantly π_{1}(M(p)) and H_{2}(M(p), ℤ). For this one needs to understand the topology of the level set of the moment map N_{ν}(p). Of course, we do know that π_{1}(M(p)) is finite and that the odd Betti numbers of M(p) vanish up to the middle dimension [GS]. However, beyond this not much explicit topological information can be obtained. For example, so far we have been unable to determine whether M(p) and N_{ν}(p) are even connected. This presents no real problem as we shall always mean by these spaces connected components such that N_{ν}(p) is a S^{1} × Sp(1) bundle over M(p). Generally, the determination of the topology of an intersection of real quadrics such as N_{ν}(p) is quite complicated. The analysis in previous work [BGM1] and [BGMR] relied heavily on very specialized information. In the former case the level sets in question were diffeomorphic to certain Stiefel manifolds whose topology is completely understood, whereas in the later case there was a large symmetry group whose quotient was a two dimensional space whose topology could be analysized. In the present case we meet with no such good fortune. Nevertheless, we are able to obtain a small amount of general information about the topology of N_{ν}(p). Our first result is that for p and p′ admissible the level sets N_{ν}(p) and N_{ν}(p′) are diffeomorphic. This does not hold generally as the case of N_{ν}(1, 1, 1) shows. In this case the Jacobian matrix drops rank making the quotient singular. Lemma 5.1: For p and p′ admissible, the level sets N_{ν}(p) and N_{ν}(p′) are diffeomorphic. Proof: For p admissible there are no fixed points of the U(2) action so by the 3-Sasakian version of a well known result in symplectic geometry, zero is a regular value of the U(2) moment map

ν̃_{p} : S^{27}⟶ℝ^{12} = ℝ^{3} × ℝ^{3} × ℝ^{3} × ℝ^{3}

defined by ν̃_{p} = (ν_{p}, μ_{i}, μ_{j}, μ_{k}). Now the level sets N_{ν}(p) are defined for all p ∈ ℝ^{3}. Moreover, for any non-zero ρ ∈ ℝ we see that level sets of ν̃_{ρp} and ν̃_{p} coincide, that is N_{ν}(ρp) = N_{ν}(p). Thus, by scaling we can choose ρ such that ν̃_{ρp′} is in an ϵ-neighborhood of ν̃_{ρp} in C^{∞}(S^{27}, ℝ^{12}) with the C^{∞} compact-open topology. Since zero is a regular value of both ν̃_{ρp} and ν̃_{ρp′}, it follows from a well known theorem (cf. [BCR], 14.1.1) that N_{ν}(p) and N_{ν}(p′) are diffeotopic, that is there is a one parameter family of diffeomorphisms ϕ_{t} : S^{27}⟶S^{27} parameterized by the unit interval such that ϕ_{0} is the identity and ϕ_{1} takes N_{ν}(p′) diffeomorphically to N_{ν}(p).

This immediately implies that the homotopy groups as well as the cohomology rings of N_{ν}(p) and N_{ν}(p′) are isomorphic.

Now once and for all we shall choose a component of N_{ν}(p) and the corresponding component of M(p) so that N_{ν}(p) is an S^{1} × Sp(1) bundle over M(p) with both base space and total space connected. We also denote by H(p) the circle bundle over M(p) that coincides with the quotient of the corresponding component of N_{ν}(p) by the Sp(1) action. We now study the latter as a fibration, namely S^{3}⟶N_{ν}(p)⟶^{π}H(p). Note that since ϕ_{1} in Lemma 5.1 is not necessarily a bundle map, we cannot claim that the manifolds H(p) and H(p′) are diffeomorphic. Nevertheless, we can obtain some useful information about their cohomology and homotopy groups. The manifolds H(p) are of interest in their own right since as discussed briefly in the next section they admit hypercomplex structures.

Consider the following commutative diagram of Gysin sequences with ℤ coefficients: 5.2

H^{r+3}(N_{ν}(p))⟶ H^{r}(H(p))⟶^{∪χ} H^{r+4}(H(p))⟶^{π^{∗}} H^{r+4}(N_{ν}(p))⟶ H^{r+1}(H(p))
↓  ↓
H^{r+3}(N_{ν}(p′))⟶ H^{r}(H(p′))⟶^{∪χ} H^{r+4}(H(p′))⟶^{π^{∗}} H^{r+4}(N_{ν}(p′))⟶ H^{r+1}(H(p′)),

where the two vertical arrows are isomorphisms by Lemma 5.1, and ∪χ denotes cupping by the Euler class of the bundle. The idea is to construct, as best as possible, the missing vertical arrows and relate the cohomology of H(p) and H(p′) by the Five Lemma. First we notice that setting r = −3 and −2 and using Lemma 5.1 gives isomorphisms

H^{1}(H(p), ℤ) ≈ H^{1}(N_{ν}(p), ℤ) ≈ H^{1}(N_{ν}(p′), ℤ) ≈ H^{1}(H(p′), ℤ)
H^{2}(H(p), ℤ) ≈ H^{2}(N_{ν}(p), ℤ) ≈ H^{2}(N_{ν}(p′), ℤ) ≈ H^{2}(H(p′), ℤ)

Next by setting r = −1 we have

5.3  \begin{matrix}0 & ⟶ & H^{3}(H(p))⟶^{π^{∗}} & H^{3}(N_{ν}(p)) & ⟶ & H^{0}(H(p)) \\ ↓= & & & ↓ϕ^{∗} & & ↓≈ \\ 0 & ⟶ & H^{3}(H(p′))⟶^{(π′)^{∗}} & H^{3}(N_{ν}(p′)) & ⟶ & H^{0}(H(p′)) \end{matrix}

Since both π^{∗} and (π′)^{∗} are injective, ϕ^{∗} is an isomorphism, and the diagram is exact and commutative, we can define the missing vertical arrow by ψ = ((π′)^{∗})^{−1} ∘ ϕ^{∗} ∘ π^{∗} and it is an isomorphism. We thus have

H^{3}(H(p), ℤ) ≈ H^{3}(H(p′), ℤ).

Now generally we cannot construct the missing vertical maps; however, we can construct them if the groups are free. We thus change to rational coefficients ℚ.

Consider now diagram 5.2 for r = 0. We can now fill in the second and last columns with vertical arrows that are isomorphisms. Now considering the diagram with rational coefficients, we can split the middle groups, for all admissable p, as

H^{4}(H(p), ℚ) ≈ im(∪χ) ⊕ coker(∪χ).

Choosing bases for these groups we can define the middle vertical map simply by sending a basis element in im(∪χ) ⊂ H^{4}(H(p), ℚ) to a basis element in im(∪χ) ⊂ H^{4}(H(p′), ℚ), and a basis element in coker(∪χ) ⊂ H^{4}(H(p), ℚ) to a basis element in coker(∪χ) ⊂ H^{4}(H(p′), ℚ). That these subspaces have the same dimension making this possible follows from exactness and commutativity of the diagram. It then follows from the Five Lemma that this middle arrow is an isomorphism. This argument is general and using a simple induction we arrive at Theorem 5.4: For admissible p and p′ we have

(i) H^{r}(H(p), ℤ) ≈ H^{r}(H(p′), ℤ) for r = 0, 1, 2, 3.

(ii) b_{r}(H(p)) = b_{r}(H(p′)) for all r. In particular, H(p) and H(p′) have isomorphic rational cohomology groups.

Here b_{r}(M) denotes the rth Betti number of M. Similar arguments can be used for the homotopy groups, and combining these results with well known facts about circle bundles over 3-Sasakian manifolds [BGM3] we obtain Proposition 5.5: For p and p′ admissible, we have isomorphisms:

(i) π_{i}(H(p)) ≈ π_{i}(N_{ν}(p)) ≈ π_{i}(H(p′)) for i = 0, 1, 2.

(ii) H^{1}(H(p), ℤ) ≈ H^{1}(N_{ν}(p), ℤ) ≈ H^{1}(H(p′), ℤ) ≈ 0 or ℤ.

(iii) H^{2}(H(p), ℤ) ≈ H^{2}(N_{ν}(p), ℤ).

Next we consider our manifolds of primary interest, namely M(p). First, it is easy to see the following relations: Proposition 5.6: For admissible p we have:

(i) π_{i}(H(p)) ≈ π_{i}(M(p)) for all i > 2.

(ii) b_{2}(N_{ν}(p)) = \{ \begin{matrix}b_{2}(M(p)) − 1 & if b_{1}(N_{ν}(p)) = 0; \\ b_{2}(M(p)) & if b_{1}(N_{ν}(p)) = 1. \end{matrix}

Next we have the analogue of Theorem 5.4 for our 3-Sasakian manifolds M(p). Theorem 5.7: For admissible p and p′ we have

(i) H^{r}(M(p), ℤ) ≈ H^{r}(M(p′), ℤ) for r = 0, 1, 2, 3. (ii) b_{r}(M(p)) = b_{r}(M(p′)) for all r. In particular, M(p) and M(p′) have isomorphic rational cohomology groups. Proof: The proof of this Theorem is analagous to the proof of Theorem 5.4 with diagram 5.2 replaced by the following commutative diagram of Gysin sequences:

H^{r+1}(H(p))⟶ H^{r}(M(p))⟶^{∪χ} H^{r+2}(M(p))⟶ H^{r+2}(H(p))⟶ H^{r+1}(M(p))
↓  ↓
H^{r+1}(H(p′))⟶ H^{r}(M(p′))⟶^{∪χ} H^{r+2}(M(p′))⟶ H^{r+2}(H(p′))⟶ H^{r+1}(M(p′)),

where we have used Theorem 5.4 and its proof to construct the isomorphisms indicated by the vertical arrows. Remark 5.8: Actually we can weaken the hypothesis that p be admissible by noting any of the results of this section concerning rational cohomology hold in the cases when M(p) is an orbifold obtained as the quotient by a locally free action. It follows from the analysis in section 2, that the action is locally free precisely when the components of p are all distinct, and in this case the level sets H(p) are smooth manifolds; hence, Theorem 5.4 and Proposition 5.5 hold in this case as well. It is interesting to note that in the case that M(p) is an orbifold, but not a smooth manifold, the smooth manifold H(p) cannot be the trivial V-bundle. The above remarks apply equally as well to the 7-dimensional orbifolds M(Θ) constructed in section 3 with the condition for a locally free action being that all the minor determinants Δ_{ij}(Θ) are nonvanishing.

Finally we briefly discuss the two singular cases

M(1, 1, 1) ≃ ℤ_{3}\G_{2}/Sp(1) and M(1, 1, 1, 1) ≃ ℤ_{2}\Spin(7)/Spin(4).

Since these are biquotients of Lie groups the topology is more accessible. In particular, their rational cohomology is that of the corresponding 3-Sasakian homogeneous space, G_{2}/Sp(1) and Spin(7)/Spin(4), respectively, which is well known [GS,BG2]. Thus, M(1, 1, 1) has the rational cohomology of S^{11}, whereas M(1, 1, 1, 1) has the rational cohomology of S^{4}×S^{11}. In both cases b_{2} vanishes, and we do not expect this in the non-singular cases.

6. Hypercomplex Structures on Circle Bundles over M(p)

According to the general theory described in [BGM3] 3-Sasakian manifolds (orbifolds) give rise to hypercomplex structures on circle bundles over them. In this short section we give new hypercomplex structures in dimensions 12 and 16 constructed as circle V-bundles over the 3-Sasakian orbifolds M(p) constructed in sections 2 and 4. Of course, there is the trivial bundle M(p)×S^{1} over M(p) which always admits a locally conformally hyperkähler structure, but here we concentrate on the level sets H(p). As discussed in Remark 5.8 these level sets will be smooth manifolds as long as 0 < p_{1} < p_{2} < p_{3} in the 12 dimensional case and 0 ≤ p_{1} < p_{2} < p_{3} < p_{4} in the 16 dimensional case. We now have from our previous results [BGM3]: Theorem 6.1: Let p have components satisfying the inequalities above, then H(p) is a compact hypercomplex manifold of dimension 12 or 16. Furthermore, the connected component of the Lie group of hypercomplex automorphisms is T^{3} in the 12 dimensional case and T^{4} in the 16 dimensional case.

The last statement of Theorem 6.1 implies that these hypercomplex structures are distinct from any of those known previously. We do not know whether for different p the manifolds H(p) are diffeomorphic or not; however, arguments similar to those in [BGM2] show that the hypercomplex structures are distinct. In fact, each smooth manifold H(p) has a real one parameter family of distinct hypercomplex structures on them given by sending p ↦ λp for any real λ > 0.

We can also construct hypercomplex structures on the total space H(Θ) of circle Vbundles over the 7-dimensional 3-Sasakian orbifolds M(Θ) constructed in Section 3. In this case as in [BGM3] there should be gcd conditions on the entries of the matrix Θ that gaurentee that H(Θ) be a smooth manifold. These then give new hypercomplex manifolds in dimension 8 with a two-dimensional group of hypercomplex automorphisms.

Bibliography

[BCR] J. Bochnak, M. Coste, and M.-F. Roy, Géométrie algébrique réele, Springer-Verlag, Berlin, 1987.

[BG1] C.P. Boyer and K. Galicki, On Sasakian-Einstein geometry, mathDG/9811098 to appear in Int. J. Math.

[BG2] C.P. Boyer and K. Galicki, 3-Sasakian Manifolds, Surveys in Differential Geometry, Volume VI, Essays on Einstein Manifolds, A supplement to the Journal of Differential Geometry, pp. 123-184, C. LeBrun and M. Wang, Eds., International Press, Cambridge 1999.

[BGM1] C.P. Boyer, K. Galicki, and B.M. Mann, The geometry and topology of 3-Sasakian manifolds, J. reine angew. Math., 455 (1994), 183-220.

[BGM2] C.P. Boyer, K. Galicki, and B.M. Mann, Hypercomplex structures on Stiefel manifolds, Ann. Global Anal. Geom. 14 (1996), 81-105.

[BGM3] C.P. Boyer, K. Galicki, and B.M. Mann, Hypercomplex structures from 3-Sasakian structures, J. reine angew. Math., 501 (1998), 115-141.

[BGMR] C.P. Boyer, K. Galicki, B.M. Mann, and E. Rees, Compact 3-Sasakian 7-Manifolds with

Arbitrary Second Betti Number, Invent. Math. 131 (1998), 321-344.

[BGP] C.P. Boyer, K. Galicki, and P. Piccinni, Torus actions on Gr_{4}(ℝ^{n}) and 3-Sasakian manifolds, in preparation.

[BGOP] C.P. Boyer, K. Galicki, L. Ornea, and P. Piccinni, Geometry of Exceptional Quotients,

in preparation.

[DS] A. Dancer and A. Swann, The Geometry of Singular Quaternionic Kähler Quotients, Int. J. Math., 8 (1997), 595-610.

[G] K. Galicki, A generalization of the momentum mapping construction for quaternionic Kähler manifolds, Commun. Math. Phys 108 (1987), 117-138.

[GL] K. Galicki and B. H. Lawson, Jr., Quaternionic Reduction and Quaternionic Orbifolds, Math. Ann., 282 (1988), 1-21.

[GS] K. Galicki and S. Salamon, On Betti numbers of 3-Sasakian manifolds, Geom. Ded. 63 (1996), 45-68.

[Hi1] N. J. Hitchin, A new family of Einstein metrics. Manifolds and geometry (Pisa, 1993),

190–222, Sympos. Math., XXXVI, Cambridge Univ. Press, Cambridge, 1996.

[Hi2] N. J. Hitchin, Twistor spaces, Einstein metrics and isomonodromic deformations, J. Differential Geom. 42 (1995), 30–112.

[HL] R. Harvey and B. H. Lawson, Jr., Calibrated geometries, Acta Math. 148 (1982), 47–157.

[Kr1] P. Kronheimer, A hyperkähler structure on coadjoint orbits of a semisimple complex group, J. Lond. Math. Soc., II. Ser. 42 (1990), 193-208.

[Kr2] P. Kronheimer, Instantons and geometry of the nilpotent variety, J. Differential Geom. 32 (1990), 473-490.

[KS1] P. Kobak and A. Swann, Quaternionic geometry of a nilpotent variety, Math. Ann. 297 (1993), 747-764.

[KS2] P. Kobak and A. Swann, Classical nilpotent orbits as hyperkähler quotients, Internat. J. Math. 7 (1996), 193-210.

[KS3] P. Kobak and A. Swann, Exceptional hyperkähler reductions, Twistor Newsletter 44 (1998), 23-26.

[KS4] P. Kobak and A. Swann, Hyperkähler Potentials in Cohomogeneity Two, math.DG/0001024.

[OP] L. Ornea and P. Piccinni, On some moment maps and induced Hopf bundles on the quaternionic projective plane, mathDG/0001066, to appear in Int. J. Math.

[PS] Y. S. Poon and S. Salamon, Eight-dimensional quaternionic Kähler manifolds with positive scalar curvature, J. Differential Geom. 33 (1990), 363-378.

[Sw] A. F. Swann, Hyperkähler and quaternionic Kähler geometry, Math. Ann., 289 (1991), 421-450.

Department of Mathematics and Statistics July 2000 University of New Mexico Albuquerque, NM 87131 email: cboyer@math.unm.edu, galicki@math.unm.edu

Università degli Studi di Roma, “La Sapienza” Piazzale Aldo Moro 2 I-00185 Roma, Italia email: piccinni@mat.uniroma1.it