RECONSTRUCTION OF HAMILTONIAN T^{n}-SPACES IN QUATERNIONIC RIEMANNIAN GEOMETRY

Roger Bielawski

Abstract. Let M be a connected hyperkähler 4n-manifold with an effective and tri-Hamiltonian T^{n}-action. We show that, under some additional assumptions, the equivariant homeomorphism type of M is determined by the image μ(M) of the hyperkähler moment map and by the image μ(M_{sing}) of the union of singular T^{n}- orbits. It follows that, if μ is proper and the action has a fixed point, then M is T^{n}- equivariantly homeomorphic to a hyperkähler quotient of a flat quaternionic vector space by a torus. Similarily we show that a compact connected (4n − 1)-dimensional 3-Sasakian manifold with an effective action of T^{n} preserving the structure is T^{n}- equivariantly homeomorphic to a 3-Sasakian quotient of a sphere by an abelian group. Finally, a compact connected 4m-dimensional quaternion-Kähler manifold of positive scalar curvature admitting an effective and isometric action of T^{m+1} is isometric to ℍP^{m} or Gr_{2}(ℂ^{m+2}).

A 4n-dimensional manifold is hyperkähler if it possesses a Riemannian metric g which is Kähler with respect to three complex structures J_{1}, J_{2}, J_{3} satisfying the quaternionic relations J_{1}J_{2} = −J_{2}J_{1} = J_{3} etc. A construction of hyperkähler manifolds admitting an isometric action of the torus T^{n}, Hamiltonian with respect to all three Kähler forms, was given in [LR] and in [HKLR]. This construction, based on Legendre transform, is purely local and the global properties of resulting manifolds have not been investigated in general. Particular cases of complete hyperkähler 4n-manifolds with an isometric and tri-Hamiltonian T^{n}-action include ALE-spaces of type A_{n} [GH,Hi], toric hyperkähler manifolds [Go1,2,BD], the Taub-NUT metric and its generalizations [GR].

Let M be a 4n-dimensional hyperkähler manifold with an effective tri-Hamiltonian action of T^{n}. In this work we shall address the following question:

To what extent does the image of the hyperkähler moment map determine the manifold M?

The corresponding question in symplectic geometry was considered by Delzant [De] who showed that a compact symplectic manifold of dimension 2n with an effective Hamiltonian action of T^{n} is determined, as a Hamiltonian T^{n}-space, by

Typeset by AMS-TEX

the image of the moment map (which is a convex polytope). In the hyperkähler case, a tri-Hamiltonian action is automatically isometric and so we cannot expect the tri-symplectic rigidity. In addition, as we show, the hyperkähler moment map μ : M → ℝ^{3n} is open and so its image does not carry much information. In the symplectic case, however, μ(M) carries additional information - its boundary is the image of the subset:

M_{sing} = {m ∈ M; dim Stab_{T^{n}}(m) ≥ 1}.

In the hyperkähler case μ(M_{sing}) is union of codimension 3 affine subspaces and we shall show that (μ(M), μ(M_{sing})) determines (M, μ):

Theorem 1. Let M, M′ be two connected 4n-dimensional hyperkähler manifolds with an effective action of the torus T^{n}, Hamiltonian with respect to all three Kähler forms, and let μ, μ′ be the corresponding hyperkähler moment maps. Suppose that μ(M) = μ(M′), μ(M_{sing}) = μ(M′_{sing}), μ(M) is 2-connected and the maps μ, μ′ : M → μ(M) are proper. Then there exists a T^{n}-equivariant homeomorphism h : M → M′ such that μ = μ′ ∘ h.

Remark. The examples of ℍ^{∗}/ℤ_{k} shows that H^{2}(μ(M), ℤ) = 0 is necessary. Example 4.2 below shows that both π_{1}(μ(M)) = 0 and the properness of μ are necessary.

The homeomorphism type of an M satisfying assumptions of Theorem 1 is that of an open subset of a (possibly singular) hyperkähler quotient of a (possibly infinitedimensional) flat quaternionic vector space by a compact abelian group. While we shall not prove this statement in full generality, we shall show:

Theorem 2. Let M be a connected 4n-dimensional hyperkähler manifold with an effective tri-Hamiltonian action of T^{n} such that the moment map μ : M → ℝ^{3n} is proper. Then M is T^{n}-equivariantly homeomorphic to the product of T^{i} × ℝ^{3i}, for some i ≤ n, and a hyperkähler quotient of a flat quaternionic vector space by a torus.

Remarks. 1. This vector space will be infinite-dimensional if M has infinite topological type. The torus, by which we quotient, will then be a Hilbert group (see [Go2] and section 4 below).

2. The topology of nonsingular toral hyperkähler quotients of flat quaternionic vector spaces has been investigated in detail in [BD]. Such a quotient is T^{n}-equivariantly homeomorphic to the product of an ℍ^{i} and union of cotangent bundles of projective toric varieties glued together.

RECONSTRUCTION OF HAMILTONIAN T^{n}-SPACES 3

Theorems 1 and 2 have a number of consequences. For one they determine the topology of several hyperkähler manifolds constructed in [GR] as hyperkähler quotients of flat quaternionic spaces by (non-compact) abelian groups. These manifolds admit a T^{n}-action as in Theorem 1 and the moment map is proper to ℝ^{3n}. Theorem 2 implies that these manifolds are homeomorphic to the product of one of the manifolds considered in [BD] and (S^{1} × ℝ^{3})^{i} for some i.

Theorem 1 also leads quickly to a topological classification of compact 3-Sasakian (4n−1)-dimensional manifolds with an action of T^{n} preserving the 3-Sasakian structure. We recall that a (4n − 1)-dimensional manifold S is 3-Sasakian if it possesses a Riemannian metric g with three orthonormal Killing fields defining a local SU(2)- action and satisfying a curvature condition. An equivalent condition is that the Riemannian cone C^{∗}(S) = (ℝ_{>0} × S, dr^{2} + r^{2}g) is hyperkähler. Such a manifold is necessarily Einstein with scalar curvature 2(2n − 1)(2n + 1). Moreover the local action extends to a global action of SO(3) or Sp(1) and the quotient of S is a quaternion-Kähler orbifold. Many examples of compact 3-Sasakian manifolds were constructed in [BGM] using the 3-Sasakian quotient construction, analogous to the hyperkähler quotient construction. The toral quotients of spheres were investigated in [BGMR,BD,Bi]. We now have:

Theorem 3. Let S be a compact connected 3-Sasakian manifold of dimension 4n−1 with an effective action of T^{n} preserving the structure. Then there is an integer d such that S is T^{n}-equivariantly homeomorphic to a 3-Sasakian quotient of the (4d − 1)-dimensional sphere by a subgroup of T^{d}. □

As we explain in section 5, the homeomorphism type of S is determined by the image of singular orbits under the 3-Sasakian moment map. Finally we have a much stronger statement for quaternion-Kähler manifolds:

Theorem 4. Let M be a compact connected 4m-dimensional quaternion-Kähler manifold of positive scalar curvature. If M admits an effective and isometric action of T^{m+1}, then M is isometric to ℍP^{m} or Gr_{2}(ℂ^{m+2}) with their symmetric quaternion-Kähler metrics.

1. A local-global principle for classifying T^{n}-spaces

In this section we shall extend the usual classification of locally trivial T^{n}-bundles to a class of T^{n}-spaces. Let M be a T^{n}-space and let π : M → B be a continuous map which factors through M/T^{n} so that the induced map π̄ : M/T^{n} → B is a homeomorphism. We shall refer to π : M → B as a piecified T^{n}-bundle (cf. [GM]). Two piecified T^{n}- bundles π : M → B and π′ : M′ → B over the same B are equivalent (resp. locally equivalent) if there exists an equivariant homeomorphism ϕ : M → M′ such that π′ ∘ ϕ = π (resp. every point in B has a neighbourhood U such that π^{−1}(U) and (π′)^{−1}(U) are equivalent). Recall that an action is locally smooth if it has linear slices (i.e. it is locally smoothable). We have:

Theorem 1.1. Let M be a topological manifold with an effective locally smooth T^{n}-action which has connected stabilizers and suppose that M_{sing})/T^{n} does not disconnect M/T^{n} locally. Then the equivalence classes of piecified T^{n}-bundles which are locally equivalent to a given π : M → B are parameterized by H^{2}(B, ℤ^{n}) (Čech cohomology).

Example 1.2. If M is a 2n-dimensional compact Hamiltonian T^{n}-space, then the moment map μ : M → μ(M) is a piecified T^{n}-bundle. Here μ(M) is a convex polytope Δ ⊂ ℝ^{n}. Furthermore, the isotropy groups and isotropy representations are determined by Δ, and so, by the existence of normal form for the moment map [GSt, Ch.41], any two such Hamiltonian T^{n}-spaces are locally equivalent. Therefore, by Theorem 1.1, the T^{n}-equivariant homeomorphism type of M (and the moment map μ) is determined by Δ. This is the topological content of Delzant’s theorem [De].

The remainder of the section is devoted to a proof of Theorem 1.1. First we establish some terminology.

Let π : X → B be a piecified torus bundle. Then B together with the canonical map τ(b) = Stab_{T^{n}}(x), where x is any element of X with π(x) = b, will be called the T^{n}-structure of the bundle X. More abstractly, let T (n) denote the set of all closed subgroups of T^{n}. We topologize T (n) by taking the smallest topology for which the sets {N; H ≤ N}, H ≤ T^{n}, are closed. Then (cf. [Br]) a T^{n}-structure (B, τ) is a Hausdorff space B together with a continuous map τ : B → T (n). If π : X → B is a piecified T^{n}-bundle and (B, τ) is its T^{n}-structure, we shall refer to X as a T^{n}-bundle over (B, τ). We shall denote the group of auto-equivalences of π : X → B by Aut(X).

Definition 1.3. Let (B, τ) be a T^{n}-structure. The trivial T^{n}-bundle over (B, τ) is the T^{n}-space X = B × T^{n}/ ∼, where ∼ is an equivalence relation defined by (b, t) ∼ (b′, t′) if and only if b = b′ and t′t^{−1} ∈ τ(b). We shall write θ(B, τ) for this X.

We observe that X is a trivial T^{n}-space if and only if there exists a global section σ : B → X to the T^{n}-action. We have the simple

Lemma 1.4. The group of automorphisms of a trivial T^{n}-bundle is given by: Aut(θ(B, τ)) = {{f_{H} : τ^{−1}(H) → T^{n}/H; H ∈ T (n)}} where each f_{H} is continuous and if b_{k} → b, with b_{k} ∈ τ^{−1}(H), b ∈ τ^{−1}(K), then π(f_{H}(b_{k})) → f_{K}(b). Here π : T^{n}/H → T^{n}/K denotes the natural projection. □

The space of all f = {f_{H}} defined above will be denoted by Map((B, τ); T^{n}). Observe that if f = {f_{H}} ∈ Map((B, τ); T^{n}), then f defines an automorphism of any T^{n}-bundle X over (B, τ) by

(1.1)  x ↦ f(π(x)) · x

where f(π(x)) is any element of T^{n} projecting to f_{H}(π(x)), H = τ(π(x)). The natural notion of locally trivial T^{n}-bundles is too strong - for example the diagonal S^{1}-action on ℂ^{2} does not admit local sections at the origin. Instead, we adopt the following definitions:

Definition 1.5. A T^{n}-bundle X over (B, τ) is weakly trivial if the map F : Map((B, τ); T^{n}) → Aut(X), given by (1.1), is a bijection; X is weakly locally trivial if the family of locally finite coverings 𝔘 = {U_{i}}_{i∈I} of B, such that π^{−1}(U) is weakly trivial for each U ∈ N(𝔘) (the nerve of 𝔘), is cofinal in the family of all coverings.

Implicit in the definition of a weakly locally trivial T^{n}-bundle is paracompactness of B. The reason for the above definition is the following

Proposition 1.6. Let X be a weakly locally trivial T^{n}-bundle over (B, τ) such that τ^{−1}(1) is dense and B−τ^{−1}(1) does not disconnect B locally. Then the isomorphism classes of T^{n}-spaces over (B, τ) which are locally equivalent to X are parameterized by H^{2}(B, ℤ^{n}).

Proof. For any locally finite covering 𝔘 = {U_{i}}_{i∈I} of B, the equivalence classes of T^{n}-bundles over (B, τ) which are equivalent over each U_{i} ∈ 𝔘 to π^{−1}(U_{i}) are parameterized by H^{1}(B, A(𝔘)), where A(𝔘) is the presheaf on 𝔘 which attaches to each face U_{i_{1}}∩…∩U_{i_{s}} of N(𝔘) the (abelian) group A(U_{i_{1}}∩…∩U_{i_{s}}) of automorphisms of π^{−1} (U_{i_{1}} ∩ … ∩ U_{i_{s}}). We shall construct a fine resolution B(𝔘) → A(𝔘) → 0. Choose 𝔘 to satisfy the condition of weak local triviality. Then, for any U ∈ N(𝔘), A(U) = Map((U, τ); T^{n}) and is described by Lemma 1.4. We define the presheaf B(𝔘) on 𝔘 by replacing tori in this description with their Lie algebras, viewed as groups under addition. Choosing a partition of unity subordinate to 𝔘 shows that H^{m}(B, B(𝔘)) vanishes for m ≥ 1. The kernel of the resolution is the sheaf Z defined as A or B but with corresponding sublattices of ℤ^{n} replacing the subgroups of T^{n} or ℝ^{n}. The functions are now locally constant and as U ∩ τ^{−1}(1) is open dense and connected, the sheaf Z is just the constant sheaf ℤ^{n}. Taking a direct limit over coverings of B proves the proposition. □

To prove Theorem 1.1 it is now enough to show that any T^{n}-manifold with a locally smooth action, satisfying the assumptions, is weakly locally trivial. First of all, by the definition of local smoothness, the T^{n}-action can be linearized in a neighbourhood of any point p ∈ M. More precisely, since the isotropy group H_{p} of p is a connected subtorus of T^{n}, there exists another subtorus Ĥ_{p} such that T^{n} = H_{p} × Ĥ_{p}. The orbit T^{n}p is isomorphic to Ĥ_{p} and an invariant neighbourhood S of p in M is equivariantly homeomorphic to a T^{n}-vector bundle ζ over Ĥ_{p}. Since the group is abelian, the action of Ĥ_{p} provides a trivialization and so ζ ≃ Ĥ_{p} × V, where V is a vector space with a linear action of H_{p} on it. Theorem 1.1 follows now from

Proposition 1.7. Let V be a vector space equipped with a linear T^{n}-action which has connected stabilizers. Then every invariant open subset of V is a weakly trivial T^{n}-bundle.

Let us show first

Lemma 1.8. Let X be a T^{n}-space over (B, τ) and suppose that there is a finite closed covering {C_{i}} of B such that π^{−1}(C_{i}) is a trivial T^{n}-bundle for each i. Then every open or closed subset of X is weakly trivial.

Proof. The assumption is hereditary and so it is enough to show that X itself is a weakly trivial T^{n}-bundle. Let g_{ij} ∈ Aut(θ(C_{i} ∩ C_{j}, τ)) be the transition functions. Let h be an automorphism of X. Then, like for ordinary locally trivial principal bundles, h is uniquely defined by a collection of u_{i,a} ∈ Map((C_{a} ∩ C_{i}, τ); T^{n}) which satisfy

(1.2)  u_{j,b}(z) = g_{ji}(z)u_{i,a}(z)g_{ab}(z)

for any z ∈ C_{a} ∩ C_{b} ∩ C_{i} ∩ C_{j} and any i, j, a, b. Observe that all objects in (1.2) are elements of T^{n}/τ(z). Now (1.2) implies that u_{j,j}(z) = u_{i,i}(z) for any z ∈ C_{j} ∩ C_{i}, and so h is given by a globally defined u ∈ Map((B, τ); T^{n}). □

To finish the proof of Theorem 1.1 we have to show that, if T^{n} acts linearly and with connected stabilizers on a vector space V, then the assumption of Lemma 1.8 is satisfied. We can assume that V is complex, the action is unitary and the only fixed point is 0. We proceed by induction on n. The statement is trivial for n = 0. Choose a point p ≠ 0 in V and let H_{0} be the isotropy group of p. Let V_{0} be the subspace of V fixed by H_{0} and let V_{1} be the orthogonal complement to V_{0}. By the inductive assumption we have a finite closed H_{0}-invariant covering P_{i} of V_{1} which trivializes the H_{0}-action on V_{1}. Let H_{1} be such that T^{n} = H_{0} × H_{1}. We can find an (open or closed) neighborhood P of p in the sphere in V which is T^{n}-equivariantly homeomorphic to H_{1} ×U ×V_{1}, where U is an (open or closed) ball in some Euclidean space and H_{0} acts linearly on V_{1}. The T^{n}-action is trivial on each H_{1} ×U ×P_{i}. Let ϕ be the homeomorphism between P and H_{1} × U × V_{1} and let Q_{i} = ϕ^{−1}(H_{1} × U × P_{i}). The Q_{i} form a closed invariant covering of P trivializing the T^{n}-action. The action is then also trivial on each of the sets ⋃_{r≥0} rQ_{i}. Thus we can find a finite closed invariant covering of V trivializing the T^{n}-action. This proves Theorem 1.1.

2. Local structure of moment map

We recall that a complex-symplectic Kähler manifold is a Kähler manifold (M, J, ω_{1}) together with a non-degenerate closed complex-valued 2-form ω which is of type (2, 0) for the complex structure J. Such a form ω can be written as ω_{2} + \sqrt{−1}ω_{3} for some real symplectic forms ω_{2}, ω_{3} and we say that a group action is Hamiltonian for ω if it is Hamiltonian for ω_{2} and ω_{3}. Such an action must be holomorphic and the moment map μ_{ω} = μ_{2} +\sqrt{−1}μ_{3} for ω is then also holomorphic.

The purpose of this section is the following result:

Proposition 2.1. Let (M, J, ω_{1}, ω) be a complex-symplectic Kähler manifold of dimension 4n equipped with an effective action of T^{n}, Hamiltonian for ω_{1} and ω, with μ_{1}, μ_{ω} = μ_{2} + \sqrt{−1}μ_{3} the corresponding moment maps. Then the moment map μ = (μ_{1}, μ_{2}, μ_{3}) : M → ℝ^{3n} is open. Furthermore, every point p ∈ M has a T^{n}-invariant neighbourhood U such that:

(i) μ_{|_{U}} has connected fibers and induces a homeomorphism between U/T^{n} and μ(U).

(ii) μ(U_{sing}) is the intersection of μ(U) with the union of codimension 3 affine subspaces H_{k} of the form H_{k} = {x = (x^{1}, x^{2}, x^{3}) ∈ ℝ^{n} ⊗ ℝ^{3}; ⟨x^{i}, u_{k}⟩ = λ_{k}^{i}, i = 1, 2, 3}, passing through μ(m), for some finite collection of vectors {u_{k}} ⊂ ℝ^{n} and some scalars λ_{k}^{i}, i = 1, 2, 3;

(iii) the isotropy group T_{p} of p is the connected subtorus of T^{n} whose Lie algebra is spanned by {u_{k}}. Furthermore, the set {u_{k}} (after rescaling) is part of a ℤ-basis of ℤ^{n} and the isotropy representation of T_{p} is determined by this set.

We shall need the following isotropic embedding theorem for complex-symplectic forms:

Proposition 2.2. Let (M, J, ω_{1}, ω) be a complex-symplectic Kähler manifold equipped with an action of T^{n}, Hamiltonian for ω_{1}. Suppose further that the action extends to a holomorphic action of (ℂ^{∗})^{n}, Hamiltonian for ω. Let H be the isotropy group of a point p ∈ M and let X be the (ℂ^{∗})^{n}-orbit through p, which we identify with (ℂ^{∗})^{n}/H^{ℂ}. Then there exists a (ℂ^{∗})^{n}-invariant neighbourhood of p which is isomorphic, as a complex-symplectic (ℂ^{∗})^{n}-manifold, to an invariant neighbourhood of X in T^{∗}X × W where W ≃ (T_{p}X)^{⊥} /T_{p}X.

Here the orthogonal is taken with respect to ω. The above result has an appropriate extension for reductive group actions, providing that the G-orbit through p is isotropic with respect to ω_{1} and that the G^{ℂ}-orbit is isotropic with respect to ω.

Proof. Since X is an ω-isotropic (as the action is Hamiltonian and the group is abelian) submanifold of M, we have that, as a complex-symplectic vector space, T_{p}M = ((T_{p}X)^{⊥} /T_{p}X) ⊕ (T_{p}X ⊕ T_{p}X^{∗}). In addition, since ω is H^{ℂ}-invariant and H^{ℂ} acts trivially on T_{p}X, it acts trivially on T_{p}X^{∗}. Therefore we can equivariantly identify T_{X}M, as a complex-symplectic bundle with T_{X} (T^{∗}X × W) = (T X ⊕ T^{∗}X) × W. Hence, over X, ω agrees with the canonical complex symplectic form of T^{∗}X × W. Since the T^{n}-orbit through p is isotropic for ω_{1}, a result of Sjamaar [Sj] implies that an open (ℂ^{∗})^{n}-invariant neighbourhood of p is equivariantly biholomorphic to X ×S, where S = H^{ℂ}B with B an open ball in T_{p}M/T_{p}X. Therefore an open (ℂ^{∗})^{n}-invariant neighbourhood of X is equivariantly biholomorphic to an invariant neighbourhood of X in T^{∗}X × W. To finish the proof we have to show that the complex-symplectic form pulled back from M agrees with the standard form in a neighbourhood of X in T^{∗}X ×W, i.e. we need the equivariant Darboux theorem for complex-symplectic forms. For this we observe that, since this neighbourhood U can be taken to be a convex open neighbourhood of the zero section in a vector bundle, there is a holomorphic retraction of U onto X which commutes with the action of (ℂ^{∗})^{n}. Now the proof of the ordinary Darboux theorem, as given in [GSt], goes through for complex symplectic forms. □

As we do not assume that the T^{n}-action extends to a global (ℂ^{∗})^{n}-action we need a slight modification of this result.

Proposition 2.3. Let (M, J, ω_{1}, ω) be as in the previous proposition with the exception that we do not assume that the action of T^{n} extends to a global holomorphic action of (ℂ^{∗})^{n}. Let Y be the T^{n}-orbit through p, Y ≃ T^{n}/H and let X = (ℂ^{∗})^{n}/H^{ℂ} ≃ T^{∗}Y. Then a T^{n}-invariant neighbourhood of p is isomorphic, as a complex-symplectic T^{n}-manifold, to a neighbourhood of Y in T^{∗}X × W where W ≃ (T_{p}X)^{⊥} /T_{p}X.

Proof. Examination of Sjamaar’s proof shows that the construction of a slice can be done for a local (ℂ^{∗})^{n}-action in a neighborhood of a T^{n}-orbit isotropic for ω_{1}. The remainder of the proof of 2.2 also goes through. □

We shall also need the following two lemmata.

Lemma 2.4. Let U be a neighbourhood of T^{n} in (ℂ^{∗})^{n} which is of the form T^{n} × exp (B(0, ϵ)), where B(0, ϵ) is an open ball in ℝ^{n} of radius ϵ, 0 < ϵ ≤ +∞. Let ω be a T^{n}-invariant Kähler form on U such that the T^{n}-action is Hamiltonian. Then any moment map for ω has connected fibers and induces a homeomorphism between B(0, ϵ) and μ(U).

Proof. Cf. [Gu, section A2.3]. Since the action is Hamiltonian for ω, there exists a globally defined invariant Kähler potential K : B(0, ϵ) → ℝ for ω which is a strictly convex function. The moment map μ is then (up to addition of a constant) the Legendre transform of K, i.e. μ(x) = \frac{∂K}{∂x} |_{x}. Since K is strictly convex, the claim follows. □

Lemma 2.5. Let T^{n} act on ℍ^{n} via an isomorphism with the maximal torus of Sp(n). Then the orbit space ℍ^{n}/T^{n} is homeomorphic to ℝ^{3n}.

Proof. It is enough to consider the standard action of the maximal torus of Sp(n). In coordinates z_{1}, …, z_{n}, w_{1}, …, w_{n} on ℂ^{n} × ℂ^{n} ≃ ℍ^{n}, this action of T^{n} = {t = (t_{1}, …, t_{n}) ∈ ℂ^{n}; |t_{k}| = 1} is given by:

(2.1)  t · (z, w) = (t_{1}z_{1}, …, t_{n}z_{n}, t_{1}^{−1}w_{1}, …, t_{n}^{−1}w_{n}).

The ring of invariant polynomials is generated by p_{k} = |z_{k}|^{2}, q_{k} = |w_{k}|^{2}, r_{k} = z_{k}w_{k} with relations p_{k}q_{k} = |r^{k}|^{2}. Hence ℍ^{n}/T^{n} is diffeomorphic (by a theorem of Schwarz [Sch] which is the C^{∞} version of a classical theorem of Hilbert on invariant theory) to the subspace of (ℝ^{4})^{n} of the form X^{n}, where X = {(x_{1}, x_{2}, x_{3}, x_{4}) ∈ ℝ^{4}; x_{1} ≥ 0, x_{2} ≥ 0, x_{1}x_{2} = x_{3}^{2} + x_{4}^{2}}. Now the map X → ℝ^{3} given by (x_{1}, x_{2}, x_{3}, x_{4}) ↦ (x_{1} − x_{2}, x_{3}, x_{4}) defines a homeomorphism. □

We now can prove Proposition 2.1. First, we shall show that T^{n} acts with connected stabilizers. Let p be a point in M and H_{p} its stabilizer, dim H_{p} = i. Let H_{0} be the identity component of H_{p}. There is an (n − i)-dimensional subtorus H_{1} of T^{n} such that T^{n} = H_{0} × H_{1}. Let Γ = H_{p}/H_{0} = H_{1} ∩ H_{p}, so that the T^{n}-orbit of p is isomorphic to H_{1}/Γ. By Proposition 2.3, an invariant neighbourhood of p can be identified with a neighbourhood of H_{1}/Γ in T^{∗} (H_{1}/Γ)^{ℂ} × W. It follows that dim_{ℂ} W ≤ 2i. Moreover (H_{p})^{ℂ} acts on W via a homomorphism into Sp(i, ℂ), and, as the action is effective, the representation is faithful, i.e. (H_{0})^{ℂ} acts via an isomorphism with the maximal complex torus of Sp(i, ℂ). Thus Γ = 1.

We observe that this argument shows that the isotropy representation of H_{p} is given by an isomorphism with the standard maximal torus of Sp(i) acting on ℍ^{i}. For the purpose of showing part (i) we can assume, without loss of generality, that this isomorphism is the identity. From Proposition 2.3 we can identify (U, J, ω) with a neighbourhood of H_{1} in T^{∗}(H_{1}^{ℂ}) × ℂ^{2i}. The complex moment map is of the form μ_{ω}(u, v) = (ϕ_{1}(u), ϕ_{2}(v)), where u ∈ T^{∗}(H_{1}^{ℂ}) and v ∈ ℂ^{2i}, ϕ_{1} is the moment map for H_{1}^{ℂ} and ϕ_{2} for the complexification (ℂ^{∗})^{i} of T^{i}. If we trivialize T^{∗}(H_{1}^{ℂ}), using the H_{1}^{ℂ}-action, as H_{1}^{ℂ} × ℂ^{n−i}, then ϕ_{1} is just the projection on the second factor. On the other hand ϕ_{2}(z, w) = ∑_{k=1}^{i}(z_{k}w_{k})e_{k}, where z = ∑ z_{k}e_{k} and w = ∑ w_{k}e_{k} (the formulas for ϕ_{1} and ϕ_{2} are valid up to addition of constants).

Consider a point (u, v) ∈ U_{1} × U_{2}, where v ∈ ℂ^{2i} is of the form v = (z, w). We observe that as long as ϕ_{2}(z, w) has all coordinates nonzero, the set of points mapping to μ_{ω}(u, v) is the (ℂ^{∗})^{n}-orbit through (u, v). This orbit X (or at least a neighbourhood of T^{n} in it) is a Kähler submanifold of U and, by Lemma 3.4, μ_{1} separates T^{n}-orbits on X. Hence μ induces a 1 − 1 mapping between V/T^{n} and μ(V), where V = {(u, (z, w)) ∈ U; z_{k}w_{k} ≠ 0, k = 1, …, i}. Let μ̄ denote the induced mapping on U/T^{n}. We shall show that it is a homeomorphism. By changing U if necessary we can identify U/T^{n} with an open ball B in ℝ^{3n} (Lemma 2.5) and write μ̄ as μ̄(x^{1}, x^{2}, x^{3}) = (μ̄_{1}(x^{1}, x^{2}, x^{3}), x^{2}, x^{3}). Then, since μ̄ restricted to V/T^{n} is a continuous 1 − 1 map from an open subset of ℝ^{3n} into ℝ^{3n}, it must be a homeomorphism (on V/T^{n}). The complement S of V/T^{n} are the hyperplanes x_{k}^{2} = x_{k}^{3} = 0, k > n − i. The map μ̄ sends each of them into itself. Moreover μ_{ω}^{−1}(x), x ∈ μ_{ω}(U) − μ_{ω}(V), is a finite union of (ℂ^{∗})^{n}-orbits and Lemma 2.4 implies that μ̄ is finite-to-one on B. Suppose, in order to get a contradiction, that μ̄ is not 1 − 1. Let x be a point in B such that {x} ≠ μ̄^{−1}(μ̄(x)) and such that the dimension of the T^{n}-orbit projecting to x is maximal among all such x. We consider intersections of B with (ℝ_{>0})^{n}-orbits (i.e. projections of (ℂ^{∗})^{n}-orbits). The (ℝ_{>0})^{n}-orbits intersecting some neighbourhood W of x are of one of three types: 1) the orbit of x, 2) those contained in V, 3) those that have dimension higher than the orbit through x. It follows, because of our choice of x and Lemma 2.4, that μ̄ is 1 − 1 on W and so a homeomorphism into. Thus μ̄(B) contains arbitrarily small balls around μ̄(x). Since S is union of codimension 2 hyperplanes, there is an arbitrarily small neighbourhood Y of μ̄(x) (in μ̄(B)) such that Y − S is connected. Since μ̄ maps S to itself and is a homeomorphism on V = B − S, μ̄^{−1}(Y − S) must be connected. This is possible only if x is the only point mapping to μ̄(x) (as μ̄ is finite-to-one) and so we obtain a contradiction. Therefore μ̄ is a continuous 1 − 1 map from an open subset of ℝ^{3n} into ℝ^{3n}, and so a homeomorphism.

To finish the proof of 2.1 we have to show (ii) and (iii). Let ρ ∈ ℝ^{n} and let M_{ρ} be any component of the fixed point set of the 1-PS of T^{n} generated by ρ, i.e. X_{ρ} = 0 on M_{ρ}. Then M_{ρ} is a complex-symplectic Kähler submanifold of M. It follows that, for m ∈ M_{ρ} and v ∈ T_{m}M, we have ⟨dμ_{i}(v), ρ⟩ = 0, i = 1, 2, 3. Therefore ⟨μ_{i}, ρ⟩ is constant on M_{ρ}, i = 1, 2, 3, and this proves (ii) and the first part of (iii). Let us return to an identification of a T^{n}-invariant neighbourhood of m with a neighbourhood of T^{n−i} in ((ℂ^{∗})^{n−i} × ℂ^{n−i}) × ℂ^{2i}. As before, we can demand that the equivariant biholomorphism preserves the complex symplectic form and so the moment map μ_{ω}. If the weights of the representation of T^{i} are α_{1}, …, α_{i} (these must form a ℤ-basis of ℤ^{i}, as T^{i} acts via an isomorphism with the maximal torus of Sp(i)), then the canonical form of μ_{ω} is (π, μ_{ω}^{0}) where π is the projection onto ℂ^{n−i} and μ_{ω}^{0} : ℂ^{i} × ℂ^{i} → ℂ^{i} is given by:

μ_{ω}^{0} (∑ z_{k}e_{k}, ∑ w_{k}e_{k}) = ∑(z_{k}w_{k})α_{k}.

Therefore the vector u_{k} is orthogonal to α_{1}, …, α̂_{k}, …, α_{i} and the second claim of 2.1(iii) follows.

3. Local equivalence of Hamiltonian hyperkähler T^{n}-manifolds

We are now ready to show that the image of M_{sing} determines M locally.

Proposition 3.1. Let M, M′ be 4n-dimensional hyperkähler manifolds with effective tri-Hamiltonian actions of T^{n}, μ, μ′ denoting the respective hyperkähler moment maps. Suppose that there are neighbourhoods U, U′ of points m ∈ M and m′ ∈ M such that μ(m) = μ′(m′) and μ(U_{sing}) = μ′(U′_{sing}). Then there exist smaller invariant neighbourhoods V and V′ of m and m′ and an equivariant homeomorphism h : V → V′ such that μ′ ∘ h = μ.

Proof. We can consider both U and U′ to be neighbourhoods of (T^{n−i}, 0, 0) in T^{n−i}× ℝ^{3n−3i} × ℍ^{i} with the same T^{n}-action (thanks to Proposition 2.1). The point m is identified with (1, 0, 0). We have the standard flat hyperkähler structure on T^{n−i} × ℝ^{3n−3i} × ℍ^{i} which is T^{n−i}×Sp(i)-invariant. Let ω_{1}^{0}, ω_{2}^{0}, ω_{3}^{0} be the three Kähler forms for this structure. It is enough to show that μ is equivalent to the moment map μ^{0} for the flat hyperkähler structure. By Proposition 2.3 we can assume that ω_{2} + \sqrt{−1}ω_{3} = ω_{2}^{0} + \sqrt{−1}ω_{3}^{0} and μ_{2} = μ_{2}^{0}, μ_{3} = μ_{3}^{0}. We can even assume that ω_{1} coincides with ω_{1}^{0} at the point m (and so at all points of (T^{n−i}, 0, 0)). Indeed, we can assume that the hyperkähler structures coincide at the point m to start with. The local diffeomorphism f which gives us the standard complex-symplectic form in a neighbourhood of m does not have to have, a priori, df_{|_{m}} = 1; it does, however, satisfy df_{|_{m}}(v) = v for any v ∈ T_{X}U, where X is a neighbourhoood of the T^{n−i}-orbit of m in the (ℂ^{∗})^{n−i}-orbit. In other words df_{|_{m}} = 1 ⊕ L, where L is an element of Sp(i, ℂ) commuting with the maximal torus. If we put L̂ = 1 ⊕ L, then L̂^{−1} is a global automorphism of the standard complex-symplectic structure on T^{∗}(ℂ^{∗})^{n−i} × ℂ^{2i} and L̂^{−1} ∘ f has all the properties.

Hence our hyperkähler structure is of the form (ω_{1}, ω_{2}^{0}, ω_{3}^{0}) on U with ω_{1} = ω_{1}^{0} at m. There is a smaller neighbourhood of m, such that the form ω_{1}^{t} = tω_{1} + (1 − t)ω_{1}^{0} is Kähler (for J_{1}) for all t ∈ [0, 1]. We use the hyperkähler moment map μ^{0} to identify U/T^{n} with an open ball around 0 in ℝ^{3n} (Proposition 2.1). In other words the induced map μ̄^{0} is the identity on ℝ^{3n} and we consider the induced map μ̄^{t} = tμ̄ + (1 − t)Id for (ω_{1}^{t}, ω_{2}^{0}, ω_{3}^{0}). Proposition 2.1 implies that each μ̄^{t} is a homeomorphism (perhaps after further shrinking of U). In addition, since {μ̄^{t}(x); t ∈ [0, 1]} is a segment, x ∈ H_{k} implies that μ̄^{t}(x) ∈ H_{k} for all t. Therefore, as μ̄^{t} is a homeomorphism and μ̄^{t}(0) = 0, μ̄^{t}(x) ∈ H_{k} implies that x ∈ H_{k} (again in some neighbourhood of 0). In other words, μ̄^{t} provides a homotopy which preserves the orbit structure (i.e. the T^{n}-structure of section 1). By a theorem of Palais [Pa] (see also [Br, Thm.II.7.3]), μ̄^{t} can be lifted to an equivariant homotopy P_{t} of T^{n−i} × ℝ^{3n−3i} × ℍ^{i} with P_{0} being the identity map. Now taking h = P_{1} proves Proposition 3.1. □

4. Proof of Theorems 1 and 2

In view of Theorem 1.1 and Proposition 3.1, Theorem 1 will be proved if we can show that under its assumptions μ has connected fibers, i.e. it induces a homeomorphism between M/T^{n} and μ(M). This follows from the following

Lemma 4.1. Let f : X → Y be a proper local homeomorphism from a connected manifold X to a smooth manifold Y. Then f is a covering map.

Proof. Using f we can give X a smooth structure in which f is smooth. Let us fix a complete Riemannian metric on Y (this can be done because of a theorem of Whitney which says that any differentiable manifold can be embedded as a closed submanifold into some ℝ^{N}). Since f is proper, the pulled back metric on X is complete. Now f is a local isometry from a complete Riemannian manifold and so a covering. □

Applying this lemma with X = M/T^{n}, Y = μ(M) and f = μ̄ proves Theorem 1. We turn to Theorem 2. If μ is proper as a map to ℝ^{3n}, then μ(M) is all of ℝ^{3n} (since μ is open). It follows that the image of the union of singular orbits is the union of codimension 3 affine subspaces H_{k}, k ∈ I, (the fixed point set of any subgroup is a hyperkähler submanifold again satisfying the assumptions). The subspaces H_{k} must necessarily be discrete, i.e. every point has a neighbourhood intersecting only finitely many of them. Furthermore, it follows from Proposition 2.1 that H_{k} is of the form

(4.1)  H_{k} = {x = (x^{1}, x^{2}, x^{3}) ∈ ℝ^{n} ⊗ ℝ^{3}; ⟨x^{i}, u_{k}⟩ = λ_{k}^{i}, i = 1, 2, 3}

for some vectors {u_{k}} ⊂ ℝ^{n} and some scalars λ_{k}^{i}, i = 1, 2, 3, k ∈ I. Moreover, the following condition is satisfied (Proposition 2.1(iii)):

(4.2)  for each x ∈ ℝ^{3n} the set {u_{k}; x ∈ H_{k}} is part of a ℤ-basis of ℤ^{n}.

The results of [BD] show that as long as I is finite and the vectors {u_{k}} generate ℝ^{n}, there exists a 4n-dimensional hyperkähler manifold M′ equipped with an effective tri-Hamiltonian action of T^{n} and such that μ(M′_{sing}) is the union of the H_{k}. This hyperkähler manifold can be realized as a hyperkähler quotient of a flat quaternionic space by a torus. It follows from Theorem 1 that M is equivariantly homeomorphic to M′. In fact, the assumption that I is finite is not necessary. It is not difficult to see that the constructions of [BD] and of [Go2] can be put together to give hyperkähler quotients of certain flat quaternionic Hilbert spaces by infinite dimensional tori (Hilbert groups). These quotients will produce 4n-dimensional hyperkähler manifolds M with arbitrary {H_{k}}_{k∈I} as long as the H_{k} are discrete, the vectors u_{k} generate ℝ^{n} and (4.2) is satisfied.

Finally we have to consider the case when the vectors u_{k} do not span ℝ^{n}. Then ℝ^{3n} = ℝ^{3i} × ℝ^{3n−3i} where the H_{k} are of the form ℝ^{3i} × H′_{k} and the vectors u_{k} span ℝ^{3n−3i}. Let M′ be a (4n − 4i)-dimensional hyperkähler manifold with homeomorphism type determined by the H′_{k}. Then the product of M′ and T^{i} × ℝ^{3i} with its flat hyperkähler metric has the H_{k} as the image of singular orbits. Therefore, by Theorem 1, M and M′ × T^{i} × ℝ^{3i} are equivariantly homeomorphic. This finishes the proof of Theorem 2.

We shall now give examples showing the necessity of assumptions in Theorem 1.

Example 4.2. We consider ℝ^{3} ×S^{1} with its flat S^{1}-invariant hyperkähler structure for which the hyperkähler moment map μ is simply the projection onto ℝ^{3}. Let U_{1}, U_{2} be open subsets of ℝ^{3} such that U_{1} is contractible and π_{1}(U_{2}) ≠ 0, π_{n}(U_{2}) = 0 for n > 1. Let f_{1} : Ũ_{1} → U_{1} be a local diffeomorphism but not a diffeomorphism (such a Ũ_{1} can be found in ℝ^{4}). Let f_{2} : Ũ_{2} → U_{2} be a double covering. Consider M_{i} = Ũ_{i} × S^{1}, i = 1, 2, and extend naturally the maps f_{i}. Now pull back the hyperkähler structure from U_{i} × S^{1} to M_{i} using f_{i}. Then the hyperkähler moment map on M_{i} is μ_{i} = μ ∘ f_{i}. We observe that (M_{1}, μ_{1}) satisfies all assumptions of Theorem 1 except the properness, while (M_{2}, μ_{2}) satisfies all of them except the simple-connectedness of the image of the moment map. The conclusion of Theorem 1 (with M′_{i} = U_{i} × S^{1}) fails in both cases.

5. Compact 3-Sasakian and quaternion-Kähler manifolds

Theorem 4.1 quickly leads to the topological classification of (4n−1)-dimensional 3-Sasakian manifolds with an action of T^{n} preserving the 3-Sasakian structure. Such an action is automatically Hamiltonian [BGM], i.e. there is a 3-Sasakian moment map μ : S → ℝ^{3n}. This moment map extends to a hyperkähler moment map on the cone C^{∗}(S): μ(r, s) = r^{2}μ(s) for s ∈ S, r ∈ ℝ_{>0}. For n = 1 the 3-Sasakian manifolds are spherical space forms and so we can assume that n > 1. Let Z = μ(C^{∗}(S)). We observe that, by Proposition 2.1, 0 ∉ Z and μ(Z) is open. Furthermore, since S is compact, Z is closed (in ℝ^{3n} − {0}). Hence Z is both open and closed in ℝ^{3n} − {0} and so Z = ℝ^{3n} − {0}. In addition, the equivariance of μ under the ℝ_{>0}-action implies that μ is proper (as a map to ℝ^{3n} − {0}). Therefore (as n > 1) the assumptions of Theorem 1 are satisfied for C^{∗}(S) and the homeomorphism type of C^{∗}(S) is determined by the image of singular orbits. However, the fixed point set of a 1-PS of T^{n} is of the form C^{∗}(S′) for a 3-Sasakian submanifold S′ and so its image H_{k} is a codimension 3 subspace of ℝ^{3n}. By Proposition 2.1, the subspaces H_{k} satisfy the condition (4.2) and they determine the homeomorphism type of S. It was shown in [BD] that, as long as #{H_{k}} = d ≥ n, then for every choice of subspaces H_{k} satisfying (4.2) (with λ_{k}^{i} = 0) there exists a compact 3-Sasakian manifold S with a T^{n}-action such that μ(C(S)_{sing}) is the union of the H_{k}. This manifold S was given as 3-Sasakian quotient of the standard sphere S^{4d−1} by a (d − n)-dimensional subgroup of T^{d}. Theorem 3 of the introduction will be therefore proved if we can show that #{H_{k}} ≥ n. Suppose that #{H_{k}} = m < n. Then the H_{k} satisfy (4.2) also at the origin of ℝ^{3n}. By the argument at the end of the previous section there is a hyperkähler manifold of the form M = M′ × T^{n−m} × ℝ^{3n−3m} such that μ(M_{sing}) is union of the H_{k}. Since all H_{k} pass through the origin, M′ is homeomorphic to ℍ^{m}. It follows from Theorem 1 that C^{∗}(S) is T^{n}-equivariantly homeomorphic to M − T^{n−m}. However, by the results of [GS], the odd Betti numbers of C^{∗}(S) vanish which is not the case for M − T^{n−m} if m < n. We have proved Theorem 3.

Remark 5.1. The above argument shows that μ(S) is the boundary of a star-shaped region in ℝ^{3n}. Furthermore the homeomorphism type of S is determined by μ(S_{sing}) or, equivalently, by the linear subspaces H_{k}.

We now turn to quaternion-Kähler manifolds of dimension 4n−4 with an effective isometric action of T^{n}. Such a manifold M is an SO(3) or SU(2) quotient of a (4n−1)-dimensional 3-Sasakian manifold S with an effective action of T^{n} preserving the 3-Sasakian structure. By the above argument, S is determined by a collection of linear subspaces H_{k} of ℝ^{3n} with #{H_{k}} ≥ n. It follows from the results of [Bi], that there can be only n or n + 1 subspaces H_{k} (the Betti numbers of S are determined by the number d of the H_{k}; the relation of [GS], which is necessary for Q to be smooth, is satisfied only for d = n or d = n + 1). Furthermore, if d = n + 1, then b_{2}(M) = b_{2}(S) = 1 and it follows from [LS] that M is isometric to Gr_{2}(ℂ^{n+1}) with its symmetric quaternion-Kähler metric. In the d = n case, we want to show that M is isometric to ℍP^{n−1} with its symmetric metric. By results of [Sa] it is enough to show that M is homeomorphic to ℍP^{n−1}. We know that S is homeomorphic to S^{4n−1}/Γ, for some finite group Γ, and so the result will follow if we can show that the Sp(1) action on S is the standard one. This, however, is the case for any S. Indeed, the Sp(1) or SO(3) action rotates the complex structures of C^{∗}(S) and so induces the standard SO(3)-action on ℝ^{n} ⊗ ℝ^{3}. Hence we know the induced action on S/T^{n}. If there were two actions of Sp(1) (we can always lift an SO(3) action to one by Sp(1)), π, π′ : Sp(1) × S → S, inducing the same SO(3) action on S/T^{n}, then the composite action ρ(g, s) = π′ (g^{−1}, π(g, s)) would map fibers to fibers. On each fiber, which is a torus, we obtain an action of Sp(1) which commutes with the action of the torus on itself. Therefore ρ is the trivial action and the claim follows. We have proved Theorem 4.

6. Possible generalizations and refinements

There are several refinements of Theorems 1-3 that one would expect to hold true.

1. The most obvious is to replace “homeomorphism” with “diffeomorphism” in all the statements. In our approach the difficulty with this stems from the fact that the hyperkähler moment map μ on M does not induce a diffeomorphism between M/T^{n} and μ(M), but only a homeomorphism (Lemma 2.5).

2. In Theorem 3 we even expect “isometric” in place of “homeomorphic”.

3. Delzant’s theorem [De] gives classification up to a symplectomorphism. Perhaps it is possible that, in the situation of Theorem 2, the manifolds are actually isomorphic as complex-symplectic manifolds (with respect to one complex structure), as it is the case for the flat metric and the Taub-NUT metric on ℍ. If one could show this, one could probably show 2., since the natural SO(3)-action rotates the complex structures on C^{∗}(S) and so the isomorphism would preserve all three symplectic forms.

4. By analogy with four dimensions (ALE-spaces), one should ask under what conditions “isometric” can be put in Theorem 2. The metric on manifolds in [BD] is asymptotic (in some sense) to the cone metric on C^{∗}(S), for a singular 3-Sasakian space S. Perhaps one could show that a hyperkähler manifold satisfying assumptions of Theorem 2 and a suitable curvature decay (in L^{2}-sense?) is isometric to one of the manifolds in [BD].

5. Finally, it should be possible to extend the results to hyperkähler and 3-Sasakian orbifolds.

References

[Bi] R. Bielawski Betti numbers of 3-Sasakian quotients of spheres by tori. preprint (1996).

[BD] R. Bielawski and A.S. Dancer The geometry and topology of toric hyperkähler manifolds. preprint (1996).

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

[BGMR] C.P. Boyer, K. Galicki, B.M. Mann and E.G. Rees Compact 3-Sasakian 7-manifolds with arbitrary second Betti number. preprint (1996).

[Br] G.E. Bredon Introduction to compact transformation groups. Academic Press, New York (1972).

[De] T. Delzant Hamiltoniens périodiques et images convexe de l’application moment. Bull. Soc. Math. France 116, 315-339 (1988).

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

[GH] G.W. Gibbons and S.W. Hawking Gravitational multi-instantons. Phys. Lett. 78B, 430-432 (1978).

[GR] G.W. Gibbons and P. Rychenkova HyperKähler quotient construction of BPS monopole moduli spaces. Cambridge DAMTP preprint (1996).

[GM] M. Goresky and M. MacPherson On the topology of algebraic torus actions. in: “Algebraic groups, Utrecht 1986”, Lecture Notes in Mathematics 1271, Springer, Berlin 1987.

[Go1] R. Goto On toric hyper-Kähler manifolds given by the hyper-Kähler quotient method. in: “Infinite Analysis” (Kyoto 1991) part A, Adv. Ser. Math. Phys. 16, World Scientific, (1992).

[Go2] R. Goto On the hyper-Kähler manifolds of type A_{∞}. Geom. Fun. Anal. 4, 424-454 (1994).

[Gu] V. Guillemin Moment maps and combinatorial invariants of Hamiltonian T^{n}- spaces. Birkhäuser, Boston (1994).

[GSt] V. Guillemin and S. Sternberg Symplectic techniques in physics. Cambridge University Press, Cambridge (1984).

[Hi] N.J. Hitchin Polygons and gravitons. Math. Proc. Camb. Phil. Soc. 85, 465-476 (1979).

[HKLR] N.J. Hitchin, A. Karlhede, U. Lindström and M. Roček Hyperkähler metrics and supersymmetry. Commun. Math. Phys. 108, 535-589 (1987).

[LS] C. LeBrun and S. Salamon Strong rigidity of positive quaternion-Kähler manifolds. Invent. Math. 118, 109-132 (1994).

[LR] U. Lindström and M. Roček Scalar tensor duality and N = 1, 2 nonlinear σ-models. Nucl. Phys. B 222, 285-308 (1983).

[Pa] R.S. Palais The classification of G-spaces. Mem. Amer. Math. Soc. 36, (1960).

[Sa] S. Salamon Quaternionic Kähler manifolds. Invent. Math. 67, 143-171 (1982).

[Sch] G.W. Schwarz Smooth functions invariant under the action of a compact Lie group. Topology 14, 63-68 (1975).

Max-Planck-Institut fur Mathematik, Gottfried-Claren-Strasse 26, 53225 Bonn, Germany E-mail address: rogerb@mpim-bonn.mpg.de