Quaternionic Kähler and Hyperkähler Manifolds
Quaternionic Kähler and Hyperkähler Manifolds
Abstract and Keywords
This chapter gives an extensive overview of various quaternionic geometries. The main focus is on positive quaternionic Kähler manifolds (orbifolds) and on hyper Kähler manifolds (orbifolds). Various other quaternionic and hypercomplex geometries are introduced along the way. The hyper Kähler and quaternionic Kähler quotient construction is described. Other topics include the theory of toric hyper Kähler manifolds, the classification of positive toric selfdual and Einstein orbifolds, Hitchin's construction of SO(3)invariant orbifold selfdual Einstein metrics on a 4sphere, McKay's correspondence and Kronheimer's construction of ALE gravitational instantons.
Keywords: quaternionic geometry, hypercomplex geometry, quaternionic Kähler orbifolds, hyper Kähler manifolds, twistor spaces, quaternionic Kähler reduction, hyper Kähler reduction, Hitchin orbifolds, Konishi orbibundle, selfdual Einstein orbifolds
Quaternions were first described by the Sir William Rowan Hamilton^{1} in 1843. Hamilton believed that his invention, like complex numbers, should play a fundamental role in mathematics as well as in physics. The jury is perhaps still out on what, if any, importance should quaternions have in describing our physical world. But there is little or no doubt that they have earned an important place in Riemannian and algebraic geometry. Following Hitchin [Hit92] we would like to argue that today's rich theory of quaternionic manifolds, in some sense, vindicates Hamilton's conviction.
In this chapter we will recall some basic results concerning various quaternionic geometries which were introduced briefly from the point of view of Gstructures in Example 1.4.18. Our main focus will be on positive quaternionic Kähler (QK) and hyeprkähler (HK) manifolds, as these two geometries are of special importance in the description and understanding of 3Sasakian structures, the main topic of our next chapter. It would be impossible here, in a single chapter, to give a complete account of what is currently known about QK and HK spaces. Each case would require a separate monograph. Our goal is to describe some of the properties of such manifolds relevant to Sasakian geometry. Quaternionic Kähler geometry is traditionally defined by the reduction of the holonomy group Hol(M,g) to a subgroup of Sp(n)Sp(1)⊂ SO(4n,R). Observe that Sp(1)Sp(1)≃ SO(4) so any oriented Riemannian 4manifold has this property. It is generally accepted and, as we shall see later, quite natural, to extend this definition in dimension 4 via an additional curvature condition: an oriented Riemannian manifold (M ^{4},g) is said to be QK if the metric g is selfdual or antiselfdual and Einstein. Interest in QK manifolds and this holonomy definition dates back to the celebrated Berger Theorem 1.4.8. The Lie group Sp(n)Sp(1) appears on Berger's list of possible restricted holonomy groups of an oriented Riemannian manifold (M,g) which is neither locally a product nor locally symmetric. In particular, the holonomy reduction implies that QK manifolds are always Einstein [Ber66], though their geometric nature very much depends on the sign of the scalar curvature. The model example of a QK manifold with positive scalar curvature (positive QK manifold) is that of the quaternionic projective space H P ^{n}. The model example of a QK manifold with negative scalar curvature (negative QK manifold) is that of the quaternionic hyperbolic ball HH^{n}. The first attempts to study QK manifolds span over a decade and date back to the works of Bonan [Bon64], Kraines [Kra65,Kra66], Wolf [Wol65], Alekseevsky [Ale68,Ale75], Gray [Gra69a], Ishihara, and Konishi [IK72,Ish73,Ish74,Kon75]. (p.422) At this early stage the departing point was the holonomy reduction and efforts to understand what kind of geometric structures on the manifold would naturally lead to such a holonomy reduction. For example, it appears that, independently, Bonan and Kraines were the first to consider the fundamental 4form Ω of the quaternionic structure and deduced some topological information. Wolf and Alekseevsky studied and classified symmetric and some homogeneous examples, respectively. Ishihara and Konishi explored some special geometric properties of such manifolds and their relation to the 3Sasakian spaces. The true revolution, however, came in the early 1980s. Salamon [Sal82] and, independently, Bérard Bergery [BB82] realized that QK manifolds can be studied in the language of algebraic and holomorphic geometry. Their twistor correspondence was a generalization of the beautiful Penrose twistor space construction in dimension 4 to the case of QK manifolds of any quaternionic dimension. The power of the twistor correspondence which allows for applying algebraic geometry when dealing with problems involving positive QK manifolds will be illustrated by many results described in this chapter.
When the scalar curvature vanishes a QK manifold is necessarily locally hyperkähler. In the language of holonomy the hyperkähler manifolds are characterized by the reduction of the holonomy group Hol(M,g) to a subgroup of Sp(n)⊂ SO(4n,R). In this sense HK geometry is a special case of QK geometry and, just as in the QK case, we find the Lie group Sp(n) on Berger's list. The model example of HK geometry is that of a quaternionic vector space H ^{n} with the flat metric. The first study of hyperkähler manifolds appears to be that of Wakakuwa [Wak58] who gave an example in local coordinates, but the name hyperkähler, as well the name hypercomplex, is due to Calabi [Cal79] who constructed complete hyperkähler metrics on the cotangent bundle T ^{*} C P ^{n}. Of course, Yau's famous proof [Yau77] of the Calabi conjecture provides the K3 surface with a hyperkähler structure. Hyperkähler manifolds are special cases of Calabi–Yau manifolds, and so compact examples are important to mirror symmetry. There are several recent books [VK99,GHJ03,NW04] treating hyperkähler manifolds. In this book we are more interested in noncompact hyperkähler manifolds, especially hyperkähler cones.
12.1. Quaternionic Geometry of H ^{n} and H P ^{n}
The purpose of this section is to describe quaternionic geometries of some model examples of quaternionic manifolds. We will do it in considerable detail using terms which, in greater generality, will only be defined later. The quaternions H are the associative, noncommutative real algebra
The imaginary units are often denoted by {i _{1},i _{2},i _{3}} = {i,j,k}. The imaginary quaternions Im(H) = span(i _{1},i _{2},i _{3})≃R ^{3} and the multiplication rules are given by the formula
We define the quaternionic conjugate q¯ and the norm u by
(p.423) The nonzero quaternions H∖{0} = H ^{*} = GL(1,H) from a group isomorphic to R ^{+} × Sp(1), where Sp(1) is the subgroup of unit quaternions and the isomorphism is given explicitly by the map u↦(u,u/u). The group of unit quaternions
as a manifold, is just the unit 3sphere in R ^{4}. Furthermore, we have the group isomorphism f:Sp(1)→ SU(2) explicitly given by
It is known that Spin(4) = Sp(1) × Sp(1) and SO(4)≃ Sp(1)Sp(1), where customarily Sp(1)Sp(1) denotes the quotient of Sp(1) × Sp(1) by the diagonal Z _{2}. This is yet another group isomorphism between classical groups which can be explained using the quaternionic geometry of H≃R ^{4}. Consider the action of G = Sp(1)_{+} × Sp(1)_{−} on H given by
We assume the convention that the Sp(1)_{+} factor acts by the left quaternionic multiplication while the Sp(1)_{−} factor acts from the right. Clearly, the two actions commute and the Z _{2} subgroup generated by (−1,−1) acts trivially. The quotient acts on R ^{4} preserving the Euclidean metric and orientation. This is the special orthogonal group SO(4). It is worthwhile to write this action on R ^{4}. The Sp(1)_{+} part is given by the following group homomorphism A _{+}:Sp(1)→ SO(4):
give a globally defined hypercomplex structure ${I}^{+}=\left\{{I}_{1}^{+},{I}_{2}^{+},{I}_{3}^{+},\right\}$ on R ^{4}. For a purely imaginary τ = τ¯ in Sp(1) one sets I ^{+}(τ) = A _{+}(τ) and gets the whole S ^{2}family of complex structures. We obtain the left hyperkähler structure on H by further setting ${g}_{0}{w}^{+}={g}_{0}{i}_{1}{w}_{1}^{+}{i}_{2}{w}_{2}^{+}{i}_{3}{w}_{3}^{+}=du\otimes d\overline{u},$ where the multiplication in H is used to interpret the left hand side as an Hvalued tensor. This gives the standard Euclidean metric g _{0} and the three symplectic forms
where (a,b,c) is any cyclic permutation of (1,2,3). We can also introduce an Hvalued differential 2form
(p.424) The 2from du 𝛌 du¯ is purely imaginary as $\overline{\alpha \wedge \beta}={\left(1\right)}^{pq}\overline{\beta}\wedge \overline{\alpha},$ where p,q are the respective degrees. The Sp(1)_{−} part is given by A _{−}:Sp(1)→ SO(4) with
The matrices ${I}_{i}^{}={A}_{}\left({e}_{i}\right)$ where
give a globally defined hypercomplex structure ${I}^{}=\left\{{I}_{1}^{},{I}_{2}^{},{I}_{3}^{}\right\}.$ Furthermore, with the Euclidean metric one gets the right hyperkähler structure on H by setting ${g}_{0}+{w}^{}={g}_{0}+{i}_{1}{w}_{1}^{}+{i}_{2}{w}_{2}^{}+{i}_{3}{w}_{3}^{}=d\overline{u}\otimes du.$ This gives
where, as before, we get the three symplectic forms
for each cyclic permutation of (1,2,3). These are clearly fundamental 2forms associated to the complex structures $\left\{{I}_{a}^{}\right\}.$ Note that by construction, for any (σ,λ) one has [A _{+}(σ), A _{−}(λ)] = 0 and the product A _{+}(σ)A _{−}(λ)∈ SO(4). In particular, the two hypercomplex structures I^{+} and I^{−} commute. The hyperkähler structure $\left({g}_{o},{I}_{a}^{},{\omega}_{a}^{}\right)$ is preserved by Sp(1)_{+} (hyperkähler isometries) while Sp(1)_{−} acts by rotating the complex structures on S ^{2}. The role of Sp(1)_{+} and Sp(1)_{−} reverses for $\left({g}_{o},{I}_{a}^{+},{\omega}_{a}^{+}\right)$. With only little extra effort one can “compactify" this example to see that another Lie group U(2) is a compact manifold with two commuting hypercomplex structures, though U(2) admits no hyperkähler metric.
REMARK 12.1.1: Consider the group of integers Z acting on H by translations of the real axis. The action preserves both the hypercomplex structures and the metric, hence, the quotient H/Z≃ S ^{1} × R ^{3} is also a flat hyperkähler manifold with infinite fundamental group π_{1} = Z. To indicate the difference, we will write the flat metric in this case as g _{0} = dθ^{2}+d x· d x replacing x ^{0} with the angle coordinate θ.
EXAMPLE 12.1.1: Quaternionic vector spaces. Much of the above discussion extends to ${H}^{n}=\left\{u=\left({u}_{1},\dots ,{u}_{n}\right){u}_{j}={u}_{j}^{0}+{u}_{j}^{1}{i}_{1}+{u}_{j}^{2}{i}_{2}+{u}_{j}^{3}{i}_{3}\in H,j=1,\dots ,n\right\}.$ Here and from now on we will choose to work with the left hyperkähler structure on H ^{n}, i.e., with the symplectic 2forms given by
so that
(p.425) for any cyclic permutation (a,b,c) of (1,2,3). The corresponding hypercomplex structure is then given by left multiplication by $\left\{{\overline{i}}_{1},{\overline{i}}_{2},{\overline{i}}_{3}\right\}=\left\{i,j,k\right\}$ with the standard basis as in (12.1.10), where 0,1 are now matrices of size n × n.
We associate to g _{0} a quaternionic Hermitian inner product
and define
Now, Sp(n) × Sp(1) acts on H ^{n} by
with Sp(n)Sp(1) acting effectively. Clearly, Sp(n)Sp(1) is now a subgroup of SO(4n). The group Sp(n) assumes the role of Sp(1)_{+} and it acts by hyperkähler isometries, while Sp(1) is the previous Sp(1)_{−} and rotates the complex structures. We will also work with complex coordinates (z,w) on H ^{n}≃C ^{2n} writing
With such conventions we obtain
where
Comparing with Example 3.1.11 we recognize (g _{0},ω_{1}) as the standard Hermitian metric and Hermitian form on C ^{2n}. In addition, the (2,0)from ω_{+} is a complex symplectic form so that
is the standard holomorphic volume form on C ^{2n}.
EXAMPLE 12.1.2: Quaternionic projective space. We now use the left H ^{*}action on H ^{n} to introduce another model space of quaternionic geometry.
Definition 12.1.3: The quaternionic projectivization
defined with respect to the left action of H ^{*} on H ^{n+1} is called the quaternionic projective nspace.
Let S ^{4n+3} = {u∈ H ^{n+1}  F(u,u) = 1} be the unit sphere in H ^{n+1}. The group Sp(n+1) acts on S ^{4n+3} transitively whose isotropy group at every point is Sp(n). Hence,
(p.426) is a homogeneous space and the induced metric is of constant sectional curvature 1. Note, that the Sp(1) subgroup of H ^{*} acts on the sphere and we get the natural identification
so we observe that H P ^{n} is actually a compact rank one symmetric space. If, in addition, we make a choice {±1}⊂R ^{*}⊂C ^{*}⊂H ^{*} we can also define three more projective spaces associated to H P ^{n}.
Definition 12.1.4: Let H ^{n+1} be the quaternionic vector space and H P ^{n} the associated quaternionic projective space. We define

(i) $\mathcal{Z}={\mathbb{P}}_{\u2102}\left({\mathbb{H}}^{n+1}\right)=\left({\mathbb{H}}^{n+1}\backslash \left\{0\right\}/\u2102*\right),$

(ii) $S={\mathbb{P}}_{\mathbb{R}}\left({\mathbb{H}}^{n+1}\right)=\left({\mathbb{H}}^{n+1}\backslash \left\{0\right\}/\mathbb{R}*\right),$

(iii) $\mathcal{U}={\mathbb{P}}_{{\mathbb{Z}}_{2}}\left({\mathbb{H}}^{n+1}\right)=\left({\mathbb{H}}^{n+1}\backslash \left\{0\right\}/{\mathbb{Z}}_{2}\right).$
The spaces 𝒵,𝒮,𝒰 are called the twistor space, the Konishi bundle, and the Swann bundle of H P ^{n}, respectively.
As homogeneous spaces we have
Proposition 12.1.5: Let H P ^{n} be the quaternionic projective space. We have the following natural fiber bundles defined by {±1}⊂R ^{*}⊂C ^{*}⊂H ^{*}

(i) $\mathbb{H}*/\u2102={S}^{2}\to Z\to \mathbb{H}{\mathbb{P}}^{n},$

(ii) $\mathbb{H}*/\mathbb{R}*=SO\left(3\right)\to S\to \mathbb{H}{\mathbb{P}}^{n},$

(iii) $\mathbb{H}*/{\mathbb{Z}}_{2}\to \mathcal{U}\to \mathbb{H}{\mathbb{P}}^{n},$

(iv) $\u2102*/\mathbb{R}*={S}^{1}\to S\to Z,$

(v) $\u2102*/{\mathbb{Z}}_{2}\to \mathcal{U}\to Z,$

(vi) $\mathbb{R}*/{\mathbb{Z}}_{2}={\mathbb{R}}^{+}\to \mathcal{U}\to S.$
The six bundles of this proposition are the six arrows in the following diagram
We shall see later in this chapter that all these bundles exist in a more general setting. However, the following is a very special property of H P ^{n} and has to do with the vanishing of a certain secondary characteristic class, the Marchiafava–Romani class ∈ defined in Definition 12.2.1 below. This class clearly vanishes for H P ^{n} since H ^{2}(H P ^{n},Z) = 0.
Proposition 12.1.6: With the exception of the first one, all the bundles of the previous proposition admit a global Z _{2} lifting.
The existence of the bundle Sp(1)→ S ^{4n+3}→H P ^{n} means that the structure group of H P ^{n} can be lifted from Sp(n)Sp(1) to Sp(n) × Sp(1). We will now construct (p.427) an atlas on H P ^{n}. Consider homogeneous coordinates [u _{0}, … ,u _{n}]∈H P ^{n}. These are defined in analogy with homogeneous charts on a complex projective space by the equivalence of nonzero vectors in H ^{n+1}, with u≃u′ meaning u = u′λ, for some λ∈H ^{*}. Let
and consider the maps φ_{j}:U _{j}→H ^{n} defined by
Now A = {(U_{j};ø_{j})}_{j=0,…,n} is clearly an atlas on H P ^{n} giving it a structure of differentiable manifold. Consider the inhomogeneous quaternionic coordinates
on U _{j} and Hvalued 1forms
At each x∈H P ^{n} the forms $d{x}_{i}^{\left(j\right)}$ define an isomorphism T _{x} H P ^{n}≃H ^{n} of quaternionic vector spaces, and thus a local section η^{(j)}∈Γ(U _{j},L ^{*}(H P ^{n})) of the principal coframe bundle L ^{*}(H P ^{n})→H P ^{n}. Let η^{(k)}∈Γ(U _{k},L ^{*}(H P ^{n})) be another such local section and consider U _{k}∩ U _{j}. An easy computation shows that at any x∈ U _{k}∩ U _{j}
Note that by convention ${x}_{j}^{\left(j\right)}=1,d{x}_{j}^{\left(j\right)}=0.$ The Equations (12.1.26) imply that pointwise in U _{k}∩ U _{j}
where q has its values in GL(1,H) and A in GL(n,H)⊂ GL(4n,R). The group GL(n,H) × GL(1,H) does not act effectively, but there is an effective action of the quotient group (GL(n,H) × GL(1,H))/R ^{*} = GL(n,H)Sp(1). Thus the structure group of H P ^{n} reduces to GL(n,H)Sp(1). We are now ready to give H P ^{n} a Riemannian metric which is induced by the flat metric on H ^{n+1}∖{0}. We can write the quaternionic Hermitian form in homogeneous coordinates as
Note that the above equation defines the metric on H P ^{n} as well as the three local 2forms {ω_{1},ω_{2},ω_{3}} which are local sections of a 3dimensional vector subbundle Q ∩ 𝛌^{2}T*H P ^{n}. Using the language of Hvalued forms we can introduce ω by
with ω = ω¯, so that ω is purely imaginary. The constant c is equal to the socalled quaternionic sectional curvature which generalizes the notion of holomorphic sectional curvature in complex geometry. The quaternionic Kähler 4form Ω is then given by
(p.428) It is real and closed. We have the following
Theorem 12.1.7: The 4form Ω is parallel. When n > 1 the holonomy group Hol(g _{0})⊂ Sp(n)Sp(1). When n = 1 H P ^{1}≃ S ^{4} and the metric g _{0} is simply the metric of constant sectional curvature on S ^{4} which is selfdual and Einstein.
12.2. Quaternionic Kähler Metrics
Let M be a smooth 4ndimensional manifold (n≥1). Recall from Example 1.4.18 that M is almost quaternionic if there is a 3dimensional subbundle Q ∩ End(TM) with the property that at each point x∈ M there is a basis of local sections { I _{1},I _{2},I _{3}} of Q satisfying the quaternion algebra, i.e.,
This definition is equivalent to M admitting a Gstructure with G = GL(n,H)Sp(1). Note that any oriented 4manifold admits such a structure, but in higher dimensions there are obstructions to admitting an almost quaternionic structure as we now describe.
Suppose now that M carries a Riemannian metric g adapted to the quaternionic structure in the sense that each point on M has a neighborhood such that any local section I of Q is a local isometry, i.e.,
for any local vector fields X,Y. Adapted metrics always exists and the resulting triple (M,Q,g) is called an almost quaternionic Hermitian manifold, giving a further reduction of the structure group to Sp(n)Sp(1). Given an adapted metric we obtain a subbundle Q* ∩ 𝛌^{2}T*M which associates to each local section I of Q the local nondegenerate 2form ω defined by
The Sp(n)Sp(1)structure is a principal Sp(n)Sp(1)bundle P over M and as such it can be regarded as an element of the cohomology set H^{1}(M,S^{p}(n)S_{p}(1)) with coefficients in the sheaf S^{p}(n)S_{p}(1) of smooth Sp(n)Sp(1)valued functions. The short exact sequence
gives rise to the homomorphism
We have
Definition 12.2.1: Let ε = δ(P)∈ H ^{2}(M,Z _{2}). Then ε is called the Marchiafava–Romani class of (M,Q,g).
The Marchiafava–Romani class was introduced in [MR75] and it is the obstruction to lifting P to the Sp(n) × Sp(1) bundle. When n = 1 the sequence 12.2.4 becomes
and it follows that ε equals the second Stiefel–Whitney class w _{2}(M). For n > 1 we can identify ε with the second Stiefel–Whitney class w _{2}(Q) of the vector bundle Q. Furthermore, we get [MR75,Sal82]
Proposition 12.2.2: Let (M^{4n},Q,g) be an almost quaternionic Hermitian manifold. Then w _{2}(M) ≡ nε(2).
(p.429) In particular, the Marchiafava–Romani class ε is the second Stiefel–Whitney class of M if its dimension is 4(8). In complementary dimensions we get
Corollary 12.2.3: Any almost quaternionic manifold M of dimension 0(8) is spin.
EXAMPLE 12.2.4: Consider the complex projective spaces C P ^{4n} of real dimension 8n. The first Chern class c _{1}(C P ^{4n}) = (4n+1)Γ where Γ is a positive generator of H ^{2}(C P ^{4n},Z). Since w _{2} is the mod 2 reduction of c _{1}, the manifold C P ^{4n} is not spin, and so by Corollary 12.2.3 C P ^{4n} cannot admit an almost quaternionic structure.
The full obstruction theory for Sp(n)Sp(1)structures (even in the 8dimensional case) is subtle and not completely understood. See, for example, the article by Čadek and Vanžura [ČV98] where they prove
Theorem 12.2.5: Let M be a compact oriented 8manifold. If the conditions
hold, then M admits an almost quaternionic structure.
Not all these conditions are necessary, however. We know that the vanishing of w _{2} is necessary, and Čadek and Vanžura remarked that the middle condition is necessary. However, they also noticed that Borel and Hirzebruch [BH58] had computed the mod 2 cohomology ring of the quaternionic Kähler manifold G _{2}/SO(4) showing that w _{6}≠ 0 which implies that the vanishing of w _{6} is not necessary.
Suppose {I _{1},I _{2},I _{3}} are locally defined smooth sections of Q which satisfy (12.2.1) at each point. Then these form a local orthonormal frame for Q with respect to the standard metric $<A,B>=\frac{1}{2n}Tr\left({A}^{t}B\right)$ on End(TM). Let { ω_{i} }_{i = 1,2,3} be the basis of 2forms corresponding under (12.2.3). The associated exterior 4form
is invariant under a change of frame and thus globally defined on M. It is nondegenerate in the sense that Ω^{n} is nowhere vanishing on M. The group Sp(n)Sp(1) is precisely the stabilizer of the form Ω in GL(4n,R). The form Ω is called the fundamental 4form of the almost quaternion Hermitian structure (M,Q,Ωg). Recall the following from Definition 1.4.6 and Example 1.4.18:
Definition 12.2.6: An almost quaternionic structure (M^{4n},Q) with n > 1 is 1integrable if M admits a torsionfree connection ∇^{Q} preserving the quaternionic structure Q. In such a case (M^{4n},Q) is called a quaternionic structure on M ^{4n}, and if it has an adapted Riemannian metric g, the triple (M^{4n},Q,g) is called a quaternionic Hermitian manifold.
The case of real dimension 4, i.e., n = 1, is given in Definition 12.2.12 below. Recall that the connection ∇^{Q} is not unique. The obstruction to 1integrability has been studied by Salamon in [Sal86]. In the 4dimensional case, there is no obstruction as G = GL(1,H)Sp(1) = R ^{+} × SO(4) so that Gstructure is equivalent to a choice of orientation and conformal class. In particular, the LeviCivita connection of any compatible metric preserves the Gstructure and has no torsion. But in higher dimensions, there are nontrivial obstructions.
Here we shall be interested in a very special class of quaternionic Hermitian manifolds, namely the case where a torsionfree quaternionic connection ∇^{Q} is also a metric connection. In this case it must be the LeviCivita connection.
(p.430) Definition 12.2.7: An almost quaternionic Hermitian manifold (M^{4n},Q,Ω,g) of quaternionic dimension n > 1 is called quaternionic Kähler (QK) if ∇^{Q} coincides with the LeviCivita connection, or alternatively, if the holonomy group Hol(g) lies in Sp(n)Sp(1).
We also refer to such a manifold as one with a quaternionic Kähler structure. We can easily see the holonomy definition to be equivalent to the following
Proposition 12.2.8: An almost quaternionic Hermitian manifold (M^{4n},Q,Ω,g) n > 1, is quaternionic Kähler if ∇ Ω = 0, where ∇ denotes the LeviCivita connection of g. In particular, an almost quaternionic Hermitian manifold (M^{4n},Q,Ω,g) n > 1, is quaternionic Kähler if it admits a parallel 4form which is in the same GL(4n,R)orbit as Ω at each point x∈ M.
The hypothesis ∇ Ω = 0 clearly implies that dΩ = 0. Surprisingly, the following theorem was proved by Swann [Swa89]:
Theorem 12.2.9: An almost quaternionic Hermitian (M^{4n},Q,Ω,g) of quaternionic dimension n > 2 whose fundamental 4form Ω is closed is quaternionic Kähler.
The geometry of almost quaternionic Hermitian 8manifolds is somewhat richer as there are examples of such spaces for which the fundamental 4form Ω is closed but not parallel. Swann showed [Swa91] that
Theorem 12.2.10: An almost quaternionic Hermitian 8manifold is quaternionic Kähler if and only if the fundamental 4form Ω is closed and the algebraic ideal generated by the subbundle Q* ⊂ 𝛌^{2}T*M is a differential ideal.
We now investigate some curvature properties of quaternionic Kähler manifolds. Let { ω_{1},ω_{2},ω_{3}} be a local orthonormal frame field for Q* ⊂ 𝛌^{2}T*M. If Ω is parallel we get
from which it follows that
where the α_{ij} are 1forms which satisfy
This means in particular that the subspace Γ(Q*) ⊂ Γ(𝛌^{2}T*M) is preserved by the LeviCivita connection. The Equations (12.2.8) were considered by Ishihara [Ish74]. The matrix
is the connection 1form with respect to the local frame field { ω_{1},ω_{2},ω_{3}}. The curvature of this induced connection represents a component of the Riemann curvature tensor R and is given by
(p.431) Using the facts that $d{\omega}_{i}={\sum}_{j=1}^{3}{\alpha}_{ij}\wedge {\omega}_{j}\text{}\text{and}\text{}{d}^{2}{\omega}_{i}=0,$ one deduces that
for some constant λ. Since Q* is an oriented 3dimensional bundle, there is a canonical identification SkewEnd(Q*) ≅ Q* via the crossproduct. Using this identification we can consider F as a map F: 𝛌^{2}TM → SlewEnd (Q*) ≅ Q* ⊂ 𝛌^{2}TM and, as such, Equation (12.2.11) simply states that F=ΛπQ*, where πQ* denotes pointwise orthogonal projection πQ* : 𝛌^{2}TM → Q*. The full Riemann curvature tensor R of a QK manifold viewed as a symmetric endomorphism R: Λ^{2} TM→ Λ^{2} TM (curvature operator), has the property that
where λ is a positive multiple of the scalar curvature s on M.
We will now use Equation (12.2.12) to extend our definition of quaternionic Kähler manifolds to 4dimensional spaces. Recall that the problem in dimension 4 is that the structure group Sp(1)Sp(1) is isomorphic to the orthogonal group SO(4) which just describes generic 4dimensional oriented Riemannian geometry. So the problem is caused by a certain lowdimensional isomorphism of Lie groups. Remarkably this same isomorphism of Lie groups provides us with the solution as well. Now the Lie algebra 𝔖𝔬(n) and Λ^{2}(R ^{n}) are isomorphic as SO(n) modules. So in dimension 4 we have a splitting 𝔖𝔬(4) = 𝔖𝔘(2)⊕ 𝔖𝔘(2) giving rise to a splitting
where ${\Lambda}_{\pm}^{2}$ are precisely the ± eigenspaces of the Hodge star operator ⋆. The bundles ${\Lambda}_{\pm}^{2}$ and ${\Lambda}_{}^{2}$ are known as the bundles of selfdual and antiselfdual 2forms, respectively. Reversing orientation interchanges the selfdual and antiselfdual 2forms. (Note also that in dimension 4, the condition that ∇ Ω = 0 is trivially satisfied since Ω is the volume form). Fixing an orientation and identifying Q* with ${\Lambda}_{}^{2}$ we have
Definition 12.2.11: An oriented Riemannian 4manifold (M,g) is called quaternionic Kähler if condition (12.2.12) holds.
With the reverse orientation Q ^{*} is identified with ∇^{2} _{+}. Relative to the decomposition 12.2.13, the curvature operator R can be represented by the matrix
where W _{±} are the selfdual and antiselfdual Weyl curvatures, Ric_{0} is the tracefree part of the Ricci curvature, and s is the scalar curvature. So Equation (12.2.11) implies that Definition 12.2.11 is equivalent to
Definition 12.2.12: A 4dimensional oriented Riemannian manifold (M,g) is quaternionic Kähler if and only if it is selfdual (i.e., W _{−} = 0) or antiselfdual (i.e., W _{+} = 0) and Einstein (i.e., Ric_{0} = 0). More generally, and oriented 4manifold (M,g) is quaternionic if W _{−} = 0 or W _{+} = 0.
REMARK 12.2.1: It follows from Definition 4.2.15 that Definitions 12.2.6, 12.2.7, and 12.2.12 work equally well in the case of orbifolds. Thus, it makes perfect sense to talk about quaternionic or quaternionic Kähler orbifolds. These will play an important role in Sections 12.4 and 12.5 as well as Chapter 13.
(p.432) REMARK 12.2.2: Recall that changing the orientation of a quaternionic 4manifold interchanges W _{+} and W _{−}. We shall stick with the more usual convention by saying that a QK 4manifold or orbifold with nonzero scalar curvature is selfdual and Einstein, thus fixing the orientation. This becomes particularly important when one adds a complex structure, since a complex structure fixes the orientation. The complex manifold C P ^{2} is selfdual and Einstein, not antiselfdual; whereas, a K3 surface is antiselfdual and Einstein. Neither of these manifolds are complex if one reverses the orientation. Indeed, there are very few compact complex surfaces that are complex with respect to the reverse orientation [Kot97].
There are at least two more justifications for adopting the Definition 12.2.12. One is the theory of quaternionic submanifolds of QK manifolds. N⊂ M is called a quaternionic submanifold if for each x∈ N, T _{x} N is an H ^{*}submodule of T _{x} M. Marchiafava observed that a 4dimensional submanifold of a QK manifold is necessarily selfdual and Einstein. The other justification comes in the theory of quaternionic Kähler reduction which will be discussed in Section 12.4.
We now return to the general case. Following Salamon [Sal82] we decompose the Riemannian curvature on a QK manifold in terms of its irreducible pieces under the group Sp(n)Sp(1). We can write the cotangent bundle as T ^{*} M = E⊗ H, where E and H are locally defined vector bundles on M that transform as the standard representations of Sp(n) and Sp(1), respectively. Although E and H are not globally defined bundles their symmetric products S ^{2}(E),S ^{2}(H) and antisymmetric products Λ^{2} E, Λ^{2} H are. Note that S ^{2}(E) and S ^{2}(H) transform as the adjoint representations of Sp(n) and Sp(1), respectively, so the vector bundle A = S ^{2}(E)+S ^{2}(H) transforms as the adjoint representation of Sp(n)Sp(1). So the (4,0) Riemannian curvature tensor R can be thought of as a section of
satisfying the first Bianchi identities.
We already mentioned that the quaternionic projective space is quite special as it is the only example of a compact QK manifold which admits an integrable Sp(n)Sp(1)structure. The curvature tensor of the canonical symmetric metric on H P ^{n} plays a key role in the more general setting. The following result which is due to Alekseevsky [Ale68] is presented in the form of Salamon [Sal82] to which we refer for a proof. This proof can also be found in [Bes87].
Theorem 12.2.13: Let (M ^{4n},Q,Ω,g) be a QK manifold. The Riemann curvature tensor can be written as
where s is the scalar curvature, R _{1} is the curvature tensor of quaternionic projective space H P ^{n}, and R _{0} is a section of S ^{2}(S ^{2}(E)).
The section R _{0} behaves like the curvature tensor of a hyperkähler manifold. In particular, R _{0} has zero Ricci curvature and R _{1} has a traceless Ricci curvature, so we get the following result due to Berger [Ber66]:
Corollary 12.2.14: Any QK manifold is Einstein. A QK manifold with vanishing scalar curvature s is locally hyperkähler, i.e., the restricted holonomy group Hol_{0}(g)⊂ Sp(n).
Theorem 12.2.13 and Corollary 12.2.14 are of fundamental importance to any further study of QK manifolds. They imply that QK geometry splits into three (p.433) cases, positive, negative, and null or the hyperkähler case. We will discuss some basic properties of hyperkähler metrics in the last four sections of this chapter, so until then we assume that scalar curvature is not zero. Furthermore, we almost exclusively discuss the positive QK case, as it is this case that has a strong connection with the main theme of this book. We end this section by defining three important bundles that generalize Definition 12.1.4 and will play an important role in the next chapter.
Definition 12.2.15: Let (M,Q) be an almost quaternionic manifold. Let S(M) be the SO(3)principal bundle associated to Q. This principal bundle is called the Konishi bundle of M. We define the following associated bundles S×s_{0}(3)F

(i) $\mathcal{U}\left(M\right)=S(M)\times s{o}_{\left(3\right)}F,\text{}where\text{}F=H*/{\mathbb{Z}}_{2},$

(ii) $\mathcal{Z}\left(M\right)=S(M)\times s{o}_{\left(3\right)}F,\text{}where\text{}F={S}^{2}$ where F = S ^{2} is the unit sphere in Q
The bundles 𝒵(M),𝒵(M) are called the Swann bundle, and the twistor space of M, respectively.
The bundle 𝒮(M) was first described in Konishi [Kon75] for quaternionic Kähler manifolds, and the bundle 𝒰(M) by Swann [Swa91], again for QK manifolds. The twistor space 𝒵(M) takes its name from Penrose's twistor theory, cf. the two volume set [PR87,PR88] and references therein. The twistor space construction used here has its origins in Penrose's “nonlinear graviton” [Pen76]. It plays an important role in understanding the geometry of both quaternionic and quaternionic Kähler manifolds. Here is why. Since the bundle 𝒵 is just the unit sphere in Q, each point 𝒵 ∈ 𝒵 represents an almost complex structure I(τ) = τ_{1} I _{1}+τ_{2} I _{2}+τ_{3} I _{3} as in Section 12.1. Thus, a smooth section s of Z over an open set U⊂ M is an almost complex structure on U. So we can think of the twistor space as a bundle of almost complex structures on M. Global sections do not exist generally, so M is not almost complex. However, if V denotes the vertical subbundle of TZ consisting of tangent vectors to the fibers F = S ^{2}, a choice of quaternionic connection ∇^{Q} determines an equivariant splitting TZ=V⊕H. So we obtain an almost complex structure on Z by adding the standard complex structure I _{0} on S ^{2} to I(τ) at each point z = (π(z),τ) making Z an almost complex manifold. Moreover, the antipodal map on the fibers induces an antiholomorphic involution θ : Z→Z. Then the main result concerning twistor spaces is the following theorem due to Atiyah, Hitchin, and Singer [AHS78] in quaternionic dimension 1, and Salamon [Sal84] for quaternionic dimension greater than 1 which encodes the quaternionic geometry of M in the complex geometry of 𝒵.
Theorem 12.2.16: Let M be a quaternionic manifold (orbifold). Then the twistor space Z(M) is a complex manifold (orbifold). Moreover, the fibers of π: 𝒵→M are rational curves whose normal bundle is 2nO(1) and Z has a free anti holomorphic involution that is the antipodal map on the fibers.
For 4dimensional manifolds (quaternionic dimension 1) the converse is true, i.e., if the induced almost complex structure on 𝒵 is integrable, then the conformal structure is selfdual (W _{−} = 0). However, in higher dimension the integrability of Z(M).only implies the vanishing of a piece of the torsion of ∇^{Q}.
We end this section with a brief discussion of some quaternionic manifolds in dimension 4. The manifolds S ^{4} = H P ^{1},C P ^{2},K3,T ^{4} are all wellknown to be quaternionic. In fact, S ^{4} is selfdual Einstein with one orientation and antiselfdual Einstein with the other, C P ^{2} is selfdual Einstein, and K3 is antiselfdual Einstein with the standard orientation induced by the complex structure. Of course T ^{4}, being (p.434) flat, is selfdual, antiselfdual and Einstein with either orientation. There has been much work on selfdual and antiselfdual structures on 4manifolds over the years, and it is not our purpose here to describe what is known. Indeed many examples of such manifolds as well as orbifolds will make their appearance either explicitly or implicitly in the present monograph. Usually they occur enjoying some other property, such as being Einstein, or Kähler. We mention here only some results of a more general nature. First, there is the existence of selfdual structures on the connected sums k C P ^{2} for k > 1 [Poo86,DF89,Flo91,LeB91b,PP95,Joy95] as well as other simply connected 4manifolds that are neither Einstein nor complex. Second, Taubes [Tau92] has proven a type of stability theorem that says that given any smooth oriented 4manifold M then M#k C P ^{2} admits a selfdual conformal structure for k large enough.
12.3. Positive Quaternionic Kähler Manifolds and Symmetries
All known complete positive QK manifolds are symmetric spaces (see Conjecture 12.3.7 below), and Salamon [Sal82] showed that through dimension 16 any positive QK manifold must have a fairly large isometry group. Moreover, a bit earlier Alekseevsky proved that all homogeneous positive QK manifolds must be symmetric [Ale75]. These spaces had been classified by Wolf [Wol65] and they are often called Wolf spaces. There is precisely one for each simple Lie algebra and we have
Theorem 12.3.1: Let M be a compact homogeneous positive QK manifold. Then M = G/H is precisely one of the following:
Here n≥ 0, Sp(0) denotes the trivial group, m≥ 3, and k≥ 7. In particular, each such M is a symmetric space and, there is a onetoone correspondence between the simple Lie algebras and positive homogeneous QK manifolds.
We remark that the integral cohomology group H ^{2}(M,Z) vanishes for M = H P ^{n} and it is Z for the complex Grassmannian M = Gr_{2}(C ^{n+2}). In all other cases we have H ^{2}(M,Z) = Z _{2}. The main results concerning positive QK manifolds are due to LeBrun and Salamon. Before embarking into a description of their work we give an infinitesimal rigidity theorem due to LeBrun [LeB88].
Theorem 12.3.2: Let (M^{4n},Q,Ω,g) be a compact positive QK manifold. If g _{t} is a family of positive QK metrics of fixed volume depending smoothly on R such that g _{0} = g. Then there exists a family of diffeomorphisms f _{t}:M→ M depending smoothly on t such that g_{t} = f*_{t}g.
This result is far from true for hyperkähler or negative QK manifolds. Moduli in the hyperkähler case is wellknown, and LeBrun has shown that the moduli space of complete negative QK metrics on H ^{n} is infinite dimensional [LeB91c]. LeBrun and Salamon [LeB93,LS94] have strengthen LeBrun's infinitesimal rigidity Theorem 12.3.2 to a strong rigidity result. They give two theorems, the first is a finiteness theorem, and the second severely restricts the topological type of positive (p.435) QK manifolds. We should also mention that the rigidity results given below break down entirely in the case of compact positive QK orbifolds.
Theorem 12.3.3: For each positive integer n there are, up to isometries and rescalings, only finitely many compact positive QK manifolds of dimension 4n.
OUTLINE OF PROOF. The proof of this theorem relies heavily on Mori theory applied to the twistor space 𝒵(M) In the positive QK case Theorem 12.3.16 was strengthened in [Sal82]. We state this as a lemma together with another result from [LS94]. □
Lemma 12.3.4: Let M ^{4n+3} be a positive QK manifold. Then its twistor space Z(M) is a Fano manifold with a positive Kähler–Einstein metric and a complex contact structure. Moreover, two positive QK structures are homothetic if and only if their twistor spaces are biholomorphic.
Since 𝒵(M) has a complex contact structure c _{1}(M) is divisible by n+1. Then by contracting extremal rays Wiśniewski [Wiś91] shows that b_{2}(𝒵(M)) = 1 with the exception of three cases. Only one of these cases admits a complex contact structure, namely the flag variety P(T ^{*} C P ^{n+1}) which is the twistor space of the complex Grassmannian Gr_{2}(C ^{n+2}). So we can conclude from Lemma 12.3.4 that b _{2}(M) = 0 unless M = Gr_{2}(C ^{n+2}) with its symmetric space metric. In the remaining cases we are dealing with Fano manifolds with Picard number one. We can make use of the rational connectedness theorem which implies that there are only a finite number of deformation types of smooth Fano varieties. This was proven for Picard number one in [Cam91,Nad91], and more generally without the Picard number restriction in [Cam92,KMM92]. Then in line with Theorem 12.3.2 LeBrun and Salamon show that there is only a finite number of Fano contact manifolds up to biholomorphism. We refer to [LeB93,LS94] for details. Then the second statement of Lemma 12.3.4 implies that there are only a finite number of QK manifolds up to homotheties. □
This proof says a lot about the topology of compact positive QK manifolds. Here we collect the results of [LS94] together with what was known earlier about the topology of positive QK manifolds.
Theorem 12.3.5: Let M be a compact positive QK manifold. Then

(i) π_{1}(M) = 0;

(ii) ${\pi}_{2}(M)=\{\begin{array}{ll}0& if\text{}M\text{}is\text{}isometric\text{}to\text{}\mathbb{H}{\mathbb{P}}^{n}\\ \mathbb{Z}& if\text{}M\text{}is\text{}isometric\text{}to\text{}{\text{Gr}}_{\text{2}}\text{}({\mathbb{C}}^{n+2}),\\ finite\text{}containing\text{}{\mathbb{Z}}_{2}& otherwise,\end{array}$

(iii) b _{2k+1}(M) = 0 for all k≥0;

(iv) b _{4i}(M) > 0 for 0≤ i ≤ n;

(v) b _{2i}(M)−b _{2i−4}(M)≥0 for 2≤ i≤ n;

(vi) ${\sum}_{r=0}^{n1}\left[6r\left(n1r\right)\left(n1\right)\left(n3\right)\right]{b}_{2r}\left(M\right)=\frac{1}{2}n\left(n1\right){b}_{2n}\left(M\right).$
PROOF. To prove (i) we see that by Lemma 12.3.4 Z(M) is Fano so by Theorem 3.6.9 it is simply connected. Then (i) follows by the long exact homotopy sequence applied to the fibration S^{2}→Z(M)→M. For (ii) we recall in the proof of Theorem 12.3.3 b _{2}(M) = 0 unless M = Gr_{2}(C ^{n+2}) in which case π_{2}(M) = Z. But this together with (i) and universal coefficients imply that if b _{2}(M) = 0 (p.436) then H ^{2}(M,Z _{2}) is the 2torsion of H _{2}(M,Z), and this is nonvanishing when the Marchiafava–Romani class ∈ is nonvanishing. So the remainder of (ii) then follows from the following result of Salamon [Sal82]:
Lemma 12.3.6: Let (M,Q,g) be a compact positive QK manifold with vanishing Marchiafava–Romani class ∈. Then (M,g)≃ (H P ^{n}, g _{0}) with its canonical symmetric metric g _{0}.
Part (iii) which is due to Salamon [Sal82] is a consequence of the fact that the twistor space Z(M) of a QK manifold has only (p,p)type cohomology which in turn is a consequence of the Kodaira–Nakano Vanishing Theorem 3.5.8 and a generalization due to Akizuki and Nagano [AN54]. Part (iv) follows immediately from the nondegeneracy of the closed 4form ω of Equation (12.2.7). Part (v) follows from the quaternionic version of Lefschetz Theory [Kra65,Bon82] as described, for example, in Section 3.3. (vi) follows from index theory computations which we refer to Section 5 of [LS94]. □
REMARKS 12.3.1: Parts (iv) and (v) hold for any compact QK manifold, not only positive ones. However, in the positive case we shall see in Proposition 13.5.5 below that the numbers β_{2i} = b _{2i}−b _{2i−4} are precisely the even Betti numbers of the principal SO(3)bundle associated to the quaternionic bundle Q.
In the absence of any counterexamples, Theorems 12.3.3 and 12.3.5 strongly points towards
Conjecture 12.3.7: All compact positive QK manifolds are symmetric.
The above conjecture was first formulated in [LS94] and we will refer to it as the LeBrun–Salamon Conjecture. Beyond Theorems 12.3.3 and 12.3.5 there are several other results showing the conjecture to be true in some special cases. We shall collect all these results in the following
Theorem 12.3.8: Let (M^{4n},Q,Ω,g) be a compact positive QK manifold. Then M is a symmetric space if

(i) n≤3,

(ii) n = 4 and b _{4} = 1.
PROOF. The statement in (i) dates back to the Hitchin's proof that all compact selfdual and Einstein manifolds of positive scalar curvature must be isometric to either S ^{4} with the standard constant curvature metric or C P ^{2} with the FubiniStudy metric [Hit81] (see also [FK82,Bes87]). For n = 2 the result was proved by Poon and Salamon [PS91]. The proof was greatly simplified in [LS94] using the rigidity results of Theorem 12.3.5. The n = 3 case is a recent result of Herrera and Herrera [HH02a,HH02b]. Their proof uses an old result which estimates the size of the isometry group of M in lower dimension [Sal82]. The dimension of the isometry group of a positive QK 12manifold must be at least 6 and the dimension of the isometry group of a positive QK 16manifold must be at least 8. In particular, when n = 3 the manifold M admits an isometric circle action. Using some deep results concerning the Â(M) genus of nonspin manifolds with finite π_{2}(M) and smooth circle actions Herrera and Herrera prove that
Lemma 12.3.9: Let M be a positive QK 12manifold which is not Gr_{2}(C ^{5}). Then Â(M)=0.
(p.437) The result is then a consequence of the vanishing of Â(M) and Theorem 12.3.5. The result in (ii) follows from the estimate on the dimension of the isometry group in this case and Betti number constraints of Theorem 12.3.5 [GS96]. □
REMARK 12.3.1: The argument of [HH02a] does not work in the 16dimensional case because all QK manifolds of quaternionic dimension 4 are automatically spin. Nevertheless, the estimate on the size of the isometry group together with all the known results can most likely be used to construct a proof of the LeBrun–Salamon Conjecture in this case. However, as pointed out by Salamon in [Sal99], the biggest gap in any potential geometric proof of this conjecture is the conundrum of whether a QK manifold of quaternionic dimension n > 4 has any nontrivial Killing vector fields.
There is another approach to the LeBrun–Salamon Conjecture which proceeds via the algebraic geometry of the twistor space Z(M) and uses Lemma 12.3.4. The following, apparently stronger, conjecture was suggested by Beauville [Bea98,Bea05]
Conjecture 12.3.10: Any compact Fano manifold with a complex contact structure is homogeneous.
This, of course implies the LeBrun–Salamon Conjecture. Several years ago there were some attempts to use algebraic geometry to prove this result. Wiśniewski even briefly claimed the proof of the conjecture but later Campana found a gap in Wiśniewski's argument. Campana briefly claimed to have bridged that gap but later also withdrew the claim. Hence, as of the time of writing this monograph, both the Beauville Conjecture 12.3.10 and LeBrun–Salamon Conjecture 12.3.7 remain open.
Let rk(M) be the symmetry rank of M defined as the rank of its isometry group 𝔦𝔖𝔬𝔪(M,g), i.e., the dimension of the maximal Abelian subgroup in 𝔦𝔖𝔬𝔪(M,g). Bielawski [Bie99] proved that a positive QK manifold of quaternionic dimension n with rk(M)≥ n+1 is isometric to H P ^{n} or to the Grassmannian Gr _{2}(C ^{n+2}). Recently Fang proved several rigidity theorems for positive quaternionic Kähler manifolds in terms its symmetry rank [Fan04]. Fang's result slightly enhances Bielawski's theorem.
Theorem 12.3.11: Let (M^{4n},Q,Ω,g) be a compact positive QK manifold. Then the isometry group 𝔦𝔖𝔬𝔪(M,g) has rank at most (n+1), and M is isometric to H P ^{n} or Gr _{2}(C ^{n+2}) if rk(M) ≥ n−2 and n≥10.
This theorem is quite interesting and apparently rather deep. It follows from several different results. First recall that a quaternionic submanifold is one that preserves the quaternionic structure. It is a wellknown result of Gray that [Gra69a]
Proposition 12.3.12: Any quaternionic submanifold in a QK manifold is totally geodesic and QK.
In [Fan04] Fang proves the following rigidity results for positive QK manifolds:
Theorem 12.3.13: Let (M^{4n},Q,Ω,g) be a positive QK manifold. Assume f = (f _{1},f _{2}): N→ M × M, where N = N _{1} × N _{2} and f _{i}: N _{i}→ M are quaternionic immersions of compact QK manifolds of dimensions 4n _{i}, i = 1,2. Let Δ be the diagonal of M × M and set m = n _{1}+n _{2}. Then

(i) If m≥ n, then f ^{−1}(Δ) is nonempty.

(ii) If m≥ n+1, then f ^{−1}(Δ) is connected.
(p.438)

(iii) If f is an embedding, then for i≤ m−n there is a natural isomorphism, π_{i}(N _{1},N _{1}∩ N _{2})→ π_{i}(M,N _{2}) and a surjection for i = m−n+1.
REMARK 12.3.2: The study of homogeneous negative QK manifold is more delicate. There are of course the noncompact duals of the Wolf spaces. Alekseevsky showed that there are also nonsymmetric homogeneous examples. He obtained a classification of such spaces under the assumption that the symmetry group is completely solvable[Ale75]. We will not describe these spaces here referring the interested reader to an extensive review on this subject by Cortes [Cor00]. All these are typically called Alekseevskian spaces. Much later de Wit and Van Proyen discovered a gap in Alekseevsky's classification [dWVP92] while considering some supersymmetric σmodels coupled to supergravity. They filled in the gap and also claimed that there should be no other homogeneous examples. Inspired by this work Cortes [Cor96] provided a Lie algebraic proof filling the gap in Alekseevsky's original paper. A proof that all negative QK manifolds are the known Alekseevskian spaces is, however, still lacking.
12.4. Quaternionic Kähler Reduction
In this section we introduce a quaternionic analogue of the symmetry reduction method described in Section 8.4 for symplectic manifolds. Just as in the case of hyperkähler quotients which are introduced later in Section 12.8, the ideas originated in the physics of supersymmetric field theories. In 1983 Witten and Bagger observed that matter coupled to 4dimensional supergravity theory with N = 2 supersymmetries requires the scalar fields of the coupling to be local coordinates on a negative scalar curvature QK manifold [BW83]. Later more general Lagrangians of such theories were constructed and studied. In particular, some elements of the symmetry reduction can be found in [dWLP^{+}84,dWLVP85]. However, mathematical formulation of the theory of quaternionic Kähler quotients and its application to the case of positive QK manifolds was developed later in [Gal87a,GL88].
To begin we consider the spaces Γ^{p}(Q*) ≡ Γ(𝛌^{p}T*M⊗Q*) of smooth exterior pforms on M with values in the bundle Q*. The connection given on Q* induces a “\sl de Rham" sequence
such that
for f ∈ Γ^{0}(Q*).
Consider now the Lie group
and its Lie algebra
which is a Lie subalgebra of the Lie algebra 𝔦𝔖𝔬𝔪(M,g) of Killing vector fields on M. We have the following immediate consequence
Proposition 12.4.1: Let (M^{4n},Q,Ω,g) be a QK manifold of nonzero scalar curvature. Then aut(M^{4n},Q,Ω,g). It follows that any one parameter subgroup H⊂𝔦𝔖𝔬𝔪(M ^{4n},g) is also a subgroup of ⌟(M^{4n},Q,Ω,g).
(p.439) PROOF. When M is symmetric all statements follow by inspection. When M is not locally symmetric the holonomy Lie algebra 𝔥𝔬≷ = 𝔰𝔭(n)⊕𝔰𝔭(1). Since M is irreducible, by a theorem of Kostant [Kos55] any Killing vector field normalizes the holonomy algebra and in particular the 𝔰𝔭(1)factor which defines the quaternionic structure Q. Hence 𝔦𝔰𝔬𝔪(M,g) normalizes Q and therefore any Killing vector field V preserves Ω. The rest follows from the fact that both groups are compact Lie groups. □
The full isometry group may contain discrete isometries which do not lie on any 1parameter subgroup and these may not preserve the quaternionic 4from Ω. To each V∈ 𝔦𝔰𝔬𝔪(M,g) we associate the
defined in terms of a local frame ω_{1},ω_{2},ω_{3} by
Clearly Θ_{V} remains invariant under a local change of frame field (i.e., under local gauge transformations). We have [GL88]
Theorem 12.4.2: Assume that the scalar curvature of V ∈ isom(M,g) is not zero. Then to each V∈𝔦𝔰𝔬𝔪(M,g) there corresponds a unique section μ ∈ Γ^{0}(Q*) such that
In fact, under the canonical bundle isometry θ : SkewEnd(Q*) → Q*, μ is given explicitly by the formula
where λ is the constant positive multiple of the scalar curvature defined by (12.2.12).
We observe now that by the uniqueness in Theorem 12.4.2, the map V → μ transforms naturally under the group of automorphisms, specifically for g ∈ ⌟ut (M,Q,Ω,g) and V∈ 𝔦𝔰𝔬𝔪(M,g) we have
where g_{*}(μ)(x)=gη(μ(g^{1}(x))) and where gη denotes the map induced by g on the bundle Q* ∩ 𝛌^{2}TM. Note also that g _{*} V = Ad_{g}(V). Hence, (12.4.6) means that the diagram
Suppose now that G⊂ 𝔦𝔰𝔬𝔪(M,g) is a compact connected Lie subgroup with corresponding Lie algebra 𝔰.
(p.440) Definition 12.4.3: The moment map associated to G is the section μ of the bundle g*⊗Q* ≅Hom(g,Q*) whose value at a point x is the homomorphism V→1.2μ(x).
From the equivariance above we see immediately that the moment map is Gequivariant. Since the action of G in the bundle g* ⊗ Q* is linear on the fibers, it preserves the zero section. Consequently the set
is Ginvariant. We have the following reduction theorem due to Galicki and Lawson [GL88].
Theorem 12.4.4: Let (M,Q,Ω,g) be a QK manifold with nonzero scalar curvature. Let G⊂ 𝔦𝔰𝔬𝔪(M,g) be a compact connected subgroup with moment map μ. Let N_{0} denote the Ginvariant subset of N = { x∈ M: μ(x) = 0}, where μ intersects the zero section transversally and where G acts locally freely. Then Mˆ = N_{0}/G is a QK orbifold.
The statement here is a slight generalization of the main theorem in [GL88]. The proof of the theorem proceeds along the lines discussed in [GL88] with the exception that the locally free action allows us to use Molino’s Theorem 2.5.11 to conclude that the quotient has an orbifold structure. We rephrase an important special case of Theorem 12.4.4 as:
Corollary 12.4.5: Let M be as above and suppose G ≅ T^{k}⊂ 𝔦𝔰𝔬𝔪(M,g) is a ktorus subgroup generated by vector fields V_{k}∈ 𝔦𝔰𝔬𝔪(M,g). If V_{1}∧…∧ V_{k} is a kplane field at all points x∈ N, then N/G is a compact QK orbifold.
EXAMPLE 12.4.6: Consider the S ^{1}action defined on H P ^{n} in homogeneous coordinates as follows:
where λ is a complex unit. Recall that the projectivization is by right multiplication u↦ u q as described in Section 12.1. In a local trivialization we can identify the local frame ω_{1},ω_{2},ω_{3} with the imaginary quaternions i,j,k, respectively, in which case the moment map μ for the S ^{1}action can be written as
So the zero set μ ^{−1}(0) is invariant under right multiplication by H ^{*} and so cuts out a real codimension 3 subvariety of H P ^{n}. Moreover, one easily checks that it is a smooth embedded submanifold that is invariant under the left action of U(n+1) by u↦ A u. The isotropy subgroup of the point [u] = [1,j,0, … ,0]∈ μ ^{−1}(0) is identified with SU(2) × U(n−1). So μ ^{−1}(0) is the homogeneous space U(n+1)/(SU(2) × U(n−1)), which identifies the quotient μ ^{−1}(0)/S ^{1} with the complex Grassmannian Mˆ = Gr_{2}(C ^{n+1}).
This can easily be generalized to the case of a weighted circle action
(p.441) in which case the moment map becomes
In this case the quotients μ ^{−1}(0)/S ^{1} are QK orbifolds. The orbifold stratification is analyzed for the special case p _{0} = q and p _{1} = ··· p _{n} = p in [GL88].
Theorem 12.4.4 can be used to obtain many examples of compact QK orbifolds. When the reduced space is 4dimensional it is automatically selfdual and Einstein. The only complete positive QK manifolds in dimension 4 are H P ^{1}≃ S ^{4} or C P ^{2} with their standard symmetric space metrics. In this context the more interesting quotients are those with orbifold singularities. They will be discussed in the next section.
Just as in the symplectic case one can study more singular QK quotients without assuming that the action of the quotient group G on N = μ ^{−1}(0) is locally free. A detailed study of this more general situation was done by Dancer and Swann [DS97a]. Let (M,Q,Ω,g) be a QK manifold and let G be a connected Lie group acting smoothly and properly on M preserving the QK structure with moment map μ. For any subgroup H⊂ G we denote by M ^{H} the set of points in M fixed by H and M _{H} the set of points whose isotropy subgroups are exactly H. Further we write M _{(H)} for the set of points whose isotropy subgroups are conjugate to H in G. Now, if M _{H} is not empty then H must be compact. It follows that both M _{H}⊂ M ^{H}⊂ M are smooth manifolds. If N(H) is the normalizer of H in G then L = N(H)/H acts freely and properly on M _{H} with the quotient M _{H}/L = M _{(H)}/G a smooth manifold. Thus M decomposes into the union of M _{(H)}/G, where (H) runs over all conjugacy classes of stabilizers. We define
One can first show that the stratification of M by orbit types induces the stratification of Mˆ into a union of smooth manifolds, i.e.,
Theorem 12.4.7: Let H⊂ G be a subgroup so that M _{H} is notempty. Then Mˆ is a smooth manifold.
However, not all the pieces Mˆ_{H} have QK structures. Their geometry depends on the way H acts on Q. To be more precise, let x∈ M _{H} and consider the differential action of H on T _{x} M. Since H acts preserving the quaternionic structure we have the representation H→1.1Sp(n)Sp(1) which induces the representation φ:H→1.1SO(3). The group H acts on Q ∼R ^{3} via the composition of φ with the standard 3dimensional representation. If x,y∈ M _{H} are on the same pathcomponent then parallel transport along any path joining x to y defines an Hequivariant isomorphism T _{x} M≃ T _{y} M. It follows that the representation φ:H→1.1SO(3) is equivalent at all points on a path component of M _{H}. Hence, the image φ(H)⊂ SO(3), up to isomorphism, is the same on Lorbits. There are four possibilities [DS97a]:

• φ(H) is trivial ⇒ Mˆ_{H} is a QK manifold,

• φ(H) = Z _{k}, k > 1 ⇒ each path component of Mˆ_{H} is covered by a Kähler manifold,

• φ(H) is finite but not cyclic ⇒ M _{H} is totally real in M (Mˆ_{H} is “real"),

• φ(H) = SO(2) or SO(3) ⇒ Mˆ_{H} is empty.
(p.442) Even simple examples show that the stratification of the quotient by orbit type can include all of these pieces. However, there is a coarser stratification of Mˆ in which all pieces are in fact QK. Let M _{[H]} be the set of points in M, where the identity component of the stabilizer equals H and M _{([H])} the set of points whose stabilizer has identity component conjugate to H in G.
Theorem 12.4.8: The union
taken over all compact connected subgroups H⊂ G induces a canonical decomposition of Mˆ into a union of QK orbifolds each of which is a finite quotient of a QK manifold.
PROOF. The key observation is that M _{[H]} is an open submanifold in M ^{H}, where M ^{H} is a smooth QK submanifold of M. Hence, M _{[H]} itself is a smooth QK manifold. The restriction of μ to M _{[H]} is the moment map for the locally free action of L, hence (μ ^{−1}(0)∩ M _{[H]})/L = (μ ^{−1}(0)∩ M _{([H])})/G is a QK orbifold by Theorem 12.4.4. □
We finish this chapter with a brief discussion of Morse theory on QK manifolds. The idea to consider f = \!μ\!^{2} as the Morse function is quite natural as suggested in analogy with Kirwan’s work on symplectic quotients [Kir84,Kir98]. The function f was first introduced by Battaglia in [Bat96b,Bat99] and more recently also in [ACDVP03]. The motivation behind [ACDVP03] was the fact that the socalled BPS sates in 5dimensional supergravity theory correspond to gradient flows on a product M × N, where M is a negative QK manifold and N is a special Kähler space. Such flows are generated by a certain “energy function" f which is nothing but the square of the moment map f = μ^{2}. Battaglia was interested mostly in the positive QK case and, it appears the authors of [ACDVP03] were not aware of her work. We now describe some of the Battaglia’s results. Recall that a Morse function f is called equivariantly perfect over Q if the equivariant Morse equalities hold, i.e., if
where the sum ranges over the set of connected components of the fixed point set, λ_{F} is the index of F, and Pˆ_{t} is the equivariant Poincaré polynomial for the equivariant cohomology with coefficients in Q. Battaglia proves
Theorem 12.4.9: Let (M^{4n},Q,Ω,g) be a positive QK manifold acted on isometrically by S^{1}. Then the nondegenerate Morse function f = ‖μ ‖^{2} is equivariantly perfect over Q. The critical set of f is the union of the zero set f^{−1}(0) = μ ^{−1}(0) and the fixed point set of the circle action.
Moreover, the zero set μ ^{−1}(0) is connected, and a fixed point component is either contained in μ ^{−1}(0) or does not intersect with μ ^{−1}(0).
Proposition 12.4.10: Let M^{4n} be a positive QK manifold acted on isometrically by S^{1}. Then every connected component of the fixed point set, not contained in μ ^{−1}(0), is a Kähler submanifold of M∖ μ ^{−1}(0) of real dimension less than or equal to 2n whose Morse index is at least 2n, with respect to the function f.
(p.443) In [Bat99] Battaglia uses Morse theory to improve the results obtained earlier in [Bat96b]. She shows that the quotient in Example 12.4.6 is unique in the following sense.
Theorem 12.4.11: Let (M^{4n},Q,Ω,g) be a positive QK manifold acted on isometrically by S^{1}. Suppose S^{1} acts freely on N = μ ^{−1}(0). Then M^{4n} is homotopic to H P ^{n} with the quotient Mˆ=Gr_{2}(C ^{n+1}).
12.5. Compact Quaternionic Kähler Orbifolds
As already indicated the method of QK reduction enjoys much success if one allows the quotient M of be a QK orbifold. The price that is paid is the loss of the rigidity described in Section 12.3. Perhaps a more interesting observation about such a generalization is that when M has orbifold singularities the total space of the orbifold Konishi bundle S(M), which is a principal orbibundle with structure group G = SO(3) or G = Sp(1), may actually be smooth. That should not come as a surprise to the reader familiar with earlier chapters of our book. In fact, this happens exactly (just as it does in the case of orbifold circle Vbundles) when the orbifold uniformizing groups are subgroups of the structure group G. In this section we introduce some examples of compact positive QK orbifolds and discuss some obvious classification problems.
The first examples of positive QK orbifolds were introduced in 1987 by Galicki and Lawson [Gal87a,GaLa88]. We will briefly describe the construction slightly generalizing the original example. The key to the construction is Corollary 12.4.5 of the previous section. Consider (M,g) = (H P ^{n},g _{can}) and an arbitrary reduction of M by a kdimensional Abelian subgroup of the isometry group. Such a reduction is associated to a choice
where Sp(n+1) = 𝔦𝔰𝔬𝔪(H P ^{n},g _{can}) and 𝓣_{max} = T ^{n+1} is the maximal torus subgroup. One can always choose T ^{n+1} to be the set of diagonal matrices in the unitary group U(n+1). Any rational subtorus H is then determined by a collection of a nonzero integer vectors {θ _{1}, … ,θ _{n+1}} generating R ^{n+1−k}. These can be put together as a matrix Θ∈M_{n+1k,n}(Z). Dually, we can consider a matrix Ω ∈ M_{k,n+1}(Z) whose column vectors {ω _{1}, … ,ω _{n+1}} generate R ^{k}. This gives the exact sequence of Lie algebras
and its dual
where θ(e _{i}) = Θe _{i} = θ _{i}∈R ^{n+1−k} and Γ^{*}(e _{i}) = Ωe _{i} = ω _{i}∈R ^{k} with {e _{1}, … ,e _{n+1}} being the standard basis in R ^{n+1}. There is a corresponding exact sequence at the group level 1→1 H→1 T ^{n+1}→1 T ^{n+1−k}→11 and the subtorus H is identified with the image of the homomorphism f _{Ω}:T ^{k}→ T ^{n+1}
(p.444) where $\left({a}_{i}^{j}\right)$ Now, with each Ω we associate H acting on H P ^{n}, the QK moment map μ_{Ω}, and the zero level set $N(\Omega )={\mu}_{\Omega}^{1}\left(0\right)\subset H{P}^{n}.$ It is elementary to check when the condition of Corollary 12.4.5 is satisfied. We have arrived at
Theorem 12.5.1: Suppose all k × k minor determinants of Ω do not vanish, i.e., any collection of k column vectors of Ω are linearly independent. Then the reduced space is a compact positive QK orbifold. Furthermore, the Lie algebra isom,(O(Ω),ĝ(Ω)) contains (n+1−k) commuting Killing vector fields. In particular, when k = n−1, O(Ω) is a compact selfdual Einstein orbifold of positive scalar curvature and 2 commuting Killing vector fields.
The rational cohomology of these orbifolds were computed independently in [BGMR98] and [Bie97]. In particular, we have
Corollary 12.5.2: There exist compact toric positive selfdual Einstein orbifolds with arbitrary second Betti number.
REMARK 12.5.1: The case originally considered in [Gal87a,GL88] corresponds to k = 1 and Ω = (q,p, … ,p) with Ω = (1, … ,1) being the canonical quotient of Example 12.4.6. More general cases were analyzed only much later in [BGM94a] (Ω = (p _{1}, … ,p _{n+1})) and [BGMR98] (an arbitrary Ω), where it was realized that the Konishi orbibundle of such orbifolds can often be a smooth manifold carrying a natural Einstein metric. We shall return to a detailed analysis of these examples in the next chapter after we define 3Sasakian manifolds.
We now specialize to the case of compact 4dimensional QK orbifolds (M,g), i.e., 4orbifolds with selfdual conformal structure with an Einstein metric g of positive scalar curvature. Recall that when M is smooth it must be isomorphic to S ^{4} or C P ^{2}. On the other hand, Theorem 12.5.1 alone provides plenty of examples of such spaces [GL88]. As orbifolds, some of them are the familiar examples of weighted projective spaces introduced in Chapter 4.
Proposition 12.5.3: Let O_{(P)} be O(Ω) of Theorem 12.5.1 with Ω = (p _{1},p _{2},p _{3}) = p, i.e., O(P) is a QK reduction of H P ^{2} by the isometric circle action with weights p. In addition assume that all p _{i}'s are positive integers such that gcd(p _{1},p _{2},p _{3}) = 1. Then

(i) there is smooth orbifold equivalence
$${\mathcal{O}}_{(\text{P})}\simeq \{\begin{array}{ll}\mathbb{C}{\mathbb{P}}_{\frac{{p}_{1}+{p}_{2}}{2},\frac{{p}_{2}+{p}_{3}}{2},\frac{{p}_{3}+{p}_{1}}{2},\text{}}^{2}& when\text{}{p}_{i}\text{}is\text{}odd\text{}for\text{}all\text{}i,\\ \mathbb{C}{\mathbb{P}}_{{p}_{1}+{p}_{2},{p}_{2}+{p}_{3},{p}_{3}+{p}_{1}\text{}}^{2}& otherwise;\end{array}$$ 
(ii) the metrics g(p) defined by the QK reduction are inhomogeneous unless p = (1,1,1) in which case we get the Fubini–Study metric on C P ^{2};

(iii) the QK metrics g(p) are Hermitian with respect to the standard complex structure on the corresponding weighted projective space.
PROOF. To proof (i) one needs to identify the level set of the moment map with S ^{5} which is easily done. We refer to Section 13.7.4, where it is shown that the level set of the 3Sasakian moment map is diffeomorphic to the Stiefel manifold V _{2}(C ^{3}) = U(3)/U(1). Now, in terms of the QK quotient, the level set must be U(3)/U(2)≃ S ^{5}. The result follows by observing that the circle action on V=Z¯×W (which can be thought of as a coordinate on the 5sphere) has weights (p _{2}+p _{3},p _{3}+p _{1},p _{1}+p _{2}). (ii) can be proved by using the relation between the local form of any positive toric metric given in Theorem 12.5.5 and QK toric quotients. (p.445) This relation was established in [CP02]. In particular, it easily follows that the metric is of cohomogeneity 2 for all distinct weights, and of cohomogeneity 1 if exactly two weights are equal. The case of equal weight gives the Fubini–Study metric which is symmetric. Finally (iii) follows from a result of Apostolov and Gauduchon [AG02]. □
The above orbifolds are also quite interesting for another reason. In addition to being selfdual and Einstein the metric g(p) is often of positive sectional curvature. This curvature property of O(P) was discovered by Dearricott [Dea04,Dea05] and by Blažić and Vukmirović [BV04]. First we have the following result of Dearricott:
Theorem 12.5.4: Let O(P) be the Galicki–Lawson orbifold with p _{1}≤ p _{2}≤ p _{3}. Then the selfdual Einstein metric g(p) is of positive sectional curvature if and only if
where σ_{3} denotes the third symmetric polynomial in 4 variables.
The paper of Blažić and Vukmirović uses quite different methods. In fact their main theorem is a generalization of the Galicki–Lawson examples to the case of pseudoRiemannian metrics of split signature (+,+,−,−), where the quotient construction involves paraquaternions. However, the curvature calculations apply to the Riemannian case as well. For the U(2)symmetric orbifolds O(P) =O(p,q,q) case they calculate the pinching constants and get the following result.
Theorem 12.5.5: The selfdual Einstein metric on O(P) = O(p,q,q) has positive sectional curvature if ${p}^{2}<{q}^{2}\sqrt{2}<2{p}^{2}\sqrt{2}$ and then at every point x ∈ O(p,q,q) the sectional curvature is kpinched with 0 < k < 1 and
Clearly, $k=\frac{1}{4}$ when p→ q. Furthermore, $k=\frac{1}{4}$ if and only if p = q = 1 in which case O ∼ C P ^{2} is symmetric.
When Ω ∈ M_{n1,n+1}(Z) the orbifold structure of O(Ω)is more involved. However, as each O(Ω) has two commuting Killing vectors, locally these metrics are described by the following results of Calderbank and Pedersen [CP02]
Theorem 12.5.6: Let F(ρ,η) be a solution of the linear differential equation
on some open subset of the halfspace ρ > 0, and consider the metric g(ρ,η,φ,ψ) given by
(p.446) where $\alpha =\sqrt{p}d\phi \text{}\text{and}\text{}\beta =\left(d\psi +\eta d\phi \right)/\sqrt{p}.$ Then

(i) On the open set where ${F}^{2}>4{p}^{2}\left({F}_{p}^{2}+{F}_{\eta}^{2}\right)$, g is a selfdual Einstein metric of positive scalar curvature, whereas on the open set where ${F}^{2}>4{p}^{2}\left({F}_{p}^{2}+{F}_{\eta}^{2}\right)$, −g is a selfdual Einstein metric of negative scalar curvature.

(ii) Any selfdual Einstein metric of nonzero scalar curvature with two linearly independent commuting Killing fields arises locally in this way (i.e., in a neighborhood of any point, it is of the form 12.5.6 up to a constant multiple).
This theorem, together with the explicit construction of Theorem 12.5.1 leads quite naturally to the question: Are all compact positive QK orbifolds admitting two commuting Killing vectors obtained via some QK reduction O(Ω) of H P ^{n}? A partial answer to this question in the case when the Konishi bundle of O(Ω) is smooth (which is an extra condition on Ω) was provided by Bielawski in [Bie99]. We shall discuss his result later in the context of smooth “toric" 3Sasakian manifold. More recently, via a more careful analysis of orbifold singularities Calderbank and Singer proved the following [CS06a]
Theorem 12.5.7: Let (O,g) be a compact selfdual Einstein 4orbifold of positive scalar curvature whose isometry group contains a 2torus. Then, up to orbifold coverings, (O,g) is isometric to a quaternionic Kähler quotient of quaternionic projective space HP ^{n}, for some n≥ 1, by a (n−1)dimensional subtorus of Sp(n+1).
There is yet another family of orbifold metrics due to Hitchin [Hit95a,Hit95b,Hit96]. These metrics come from solutions of the Painlevé VI equation and as such were also introduced by Tod [Tod94]. We describe these metrics in some detail here and come back to them once more in the next chapter. Consider the space V defined by
that is, of traceless symmetric 3 × 3 matrices with inner product 〈B _{1},B _{2}〉 = tr(B _{1} B _{2}). Clearly, V≃R ^{5} and SO(3) acts on V by conjugation B↦ g ^{−1} B g, g∈ SO(3) and the unit sphere M = S ^{4} in V can be described as matrices in V whose eigenvalues {λ_{1},λ_{2},λ_{3}} satisfy
This action has cohomogeneity 1 and is the Z _{2}quotient of the first case described in Example 1.6.34. The associated group diagram has the structure
Explicitly, we can parameterize the conic (12.5.8) by observing that (λ_{1}−λ_{2})^{2}+3(λ_{1}+λ_{2})^{2} = 2 so that
where α^{2}+β^{2} = 1. Choose the standard rational parameterization
Note that t = 0 gives ${\lambda}_{1}={\lambda}_{2}=\frac{1}{\sqrt{6}}$ and t = +∞ gives the other degenerate orbit ${\lambda}_{1}={\lambda}_{2}=\frac{1}{\sqrt{6}}$ so that we have an explicit diffeomorphism S ^{4}∖{B _{−},B _{+}}≃ (0,+∞) × SO(3)/D given by (t,g)↦ g ^{−1}Δ(t)g, where g∈ SO(3) and Δ(t) = diag(λ_{1}(t),λ_{2}(t),λ_{3}(t)).
Any SO(3)invariant metric on S ^{4}∖{B _{+},B _{−}} defines an invariant metric on each orbit SO(3)/D. It follows that any such metric must be of the form
where {σ_{1},σ_{2},σ_{3}} is the basis of Maurer–Cartan invariant oneforms dual to the standard basis of the Lie algebra 𝔰𝔬(3). The equations for the most general selfdual Einstein metric with nonzero scalar curvature Λ and in the diagonal form (12.5.12) has been derived by Tod [Tod94]. It follows that
where {F _{1}(x),F _{2}(x),F _{3}(x)} satisfy the following firstorder system of ODEs
and the conformal factor
The expression for the conformal factor is algebraic in x,F _{1},F _{2},F _{3} so that the problem reduces to solving the system (12.5.14). It turns out that this system can be reduced to a single secondorder ODE: the Painlevé VI equation
(p.448) which then allows one to express the original functions {F _{1},F _{2},F _{3}} in terms of any solution y = y(x) of the Equation (12.5.16):
In a series of papers Hitchin analyzed the Painlevé VI equation giving an algebrogeometric description of the solutions in terms of isomonodromic deformations [Hit95a,Hit95b,Hit96]. In particular, any such solution can be described in terms of a meromorphic function on an elliptic curve C˜ with a zero of order k at a chosen point P and a pole of order k at a point −P. Hitchin’s description gives explicit formulas for the coefficients of the metric {F _{1},F _{2},F _{3}} in terms of the elliptic functions. In particular, Hitchin shows
Theorem 12.5.8: Choose an integer k≥3 and consider the SO(3)invariant metric g _{k} defined on (1,∞) × SO(3)/D by the formula (12.5.13) via the corresponding solution of the Painlevé VI equation with the metric coefficients F _{i} = F _{i}(x,k), i = 1,2,3.

(i) The metric g _{k} is a positive definite selfdual Einstein of positive scalar curvature for all 1 < x < ∞.

(ii) The metric g _{k} extends smoothly over at x = 1 over B _{−} = R P ^{2} and as x→∞ g _{k} acquires an orbifold singularity with angle 2π/k2 around B _{+} = R P ^{2}.
Hence, for any integer k≥3 the metric g _{k} can be interpreted as an orbifold metric on O_{k} = B_{−}∧((1,∞)×SO(3)/D)∧B_{+}∈S^{4} where (O_{k},g_{k}) is a compact selfdual Einstein orbifold of positive scalar curvature.
We shall return to these cohomogeneity 1 orbifold metrics in the next chapter when we consider the Konishi bundle over O_{k} and the twistor space Z(O_{k}) Here we discuss the metric for some lower values of k. To each k one associates a solution of the Painlevé VI equationy = y(s) and we write the metric g _{k} as in (12.5.12)
Explicit computation shows that for k = 3 the metric is given by the following solution of the Painlevé VI equation
The components of g _{3} can easily be calculated with f(s;3) = 3(1+s+s ^{2})^{−2} and
In the arc length coordinates this metric can be easily transformed to
which shows that g _{3} is the standard metric (p.449) on S ^{4} written in triaxial form. Hence, the orbifold (O_{3},g_{3}) is actually nonsingular and the metric is the standard one on S ^{4}. For k = 4 the metric comes from the following solution of the Painlevé VI equation
so that
In an arc length parameterization g _{4} becomes
which is indeed locally the FubiniStudy metric on C P ^{2}. The orbifold (O_{4},g_{k}) has ${\pi}_{1}^{orb}={\mathbb{Z}}_{2}$ and its universal cover is (C P ^{2}, g _{FS}). This corresponds to the second case of Example 1.6.34.
Just to illustrate how complicated the metric coefficients get for larger values of k, following Hitchin [Hit96] we also give explicit formulas for k = 6,8. For k = 6 (the orbifold singularity at angle π/2) one gets
This yields
This gives the metric g _{6} of Equation (12.5.19) for the range 1 < s < ∞.
For k = 8 (the orbifold singularity at angle π/3) we get
This yields
This metric g _{8} of Equation (12.5.19) becomes positive definite for the range $\sqrt{21}<s<.$
The Theorems of Hitchin [Hit95b], Calderbank and Pedersen [CP02], and Calderbank and Singer [CS06a] are milestones in the broader problem of the (p.450) classification of all compact selfdual Einstein 4orbifolds. It seems plausible that these are the only cohomogeneity 1 compact positive selfdual Einstein 4orbifolds, but a proof is lacking so far.
OPEN PROBLEM 12.5.1: Classify all compact positive selfdual Einstein 4orbifolds with a cohomogeneity one action of a Lie group.
If one adds to this classifying the cohomogeneity 2 actions of SU(2) one arrives at
OPEN PROBLEM 12.5.2: Classify all compact positive selfdual Einstein 4orbifolds with at least a 2dimensional isometry group.
The problem of finding examples of compact positive selfdual Einstein 4orbifolds without any assumption about symmetries seems quite intractable. No compact orbifolds without any Killing vector fields are known at this time. However, there are two examples of positive QK 4orbifolds with one Killing vector, both obtained via symmetry reduction.
EXAMPLE 12.5.9: QK Extension of Kronheimer Quotients. The first orbifold examples of positive QK orbifold metrics which are not toric were constructed in [GN92]. The construction is a quaternionic Kähler modification of the Kronheimer construction of hyperkähler ALE spaces discussed later in Section 12.10. We use the notation there to explain the result. Let Γ⊂ Sp(1) be a discrete subgroup. Consider the quaternionic projective space P _{h}(H ^{Γ} × H). The Kronheimer group K(Γ) acts on H ^{Γ} as in (12.10.3) below. Let us consider a homomorphism b:K(Γ)→1.2 Sp(1). Such a homomorphism extends the action of K(Γ) to P _{h}(H ^{Γ} × H) via
where u↦ g·u is the Kronheimer's action discussed in (12.10.3). Consider the map db:𝔨→1.2𝔰𝔭(1) of Lie algebras. We can think of db∈ 𝔨^{*}⊗ sp(1) by setting 〈 db,X〉 = db(X), ≠X∈𝔨. Let us denote the new action by K(Γ;b) and the QK reduction of P _{h}(H ^{Γ} × H) by K(Γ;b) We have the following [GN92]
Theorem 12.5.10: The QK reduced space O(Γ,b) is a compact positive selfdual Einstein orbifold if db∈ 𝔨^{*}⊗ sp(1) is in Kronheimer's good set of Definition 12.10.4.
Note that in the case of Γ = Z _{n} the construction and Theorem 12.5.10 merely give subfamilies of the toric examples discussed earlier. However, in all cases when Γ is nonAbelian we get families of positive selfdual Einstein orbifold metrics which are not toric.
EXAMPLE 12.5.11: Abelian Quotients of Real Grassmannians. Consider the positive QK structure on the Grasmannian of oriented 4planes in R ^{n}. The isometry group SO(n) of the symmetric space $G{r}_{4}^{+}\left({\mathbb{R}}^{n}\right)$ contains a torus and one can examine possible QK reductions of $G{r}_{4}^{+}\left({\mathbb{R}}^{n}\right)$. These are described in detail at the level of the Konishi bundle in Section 13.9. In particular, there are only two possible Abelian quotients of $G{r}_{4}^{+}\left({\mathbb{R}}^{n}\right)$ for which the reduced space is 4dimensional: (i) T ^{3}reduction of $G{r}_{4}^{+}\left({\mathbb{R}}^{8}\right)$ and (ii) T ^{2}reduction of $G{r}_{4}^{+}\left({\mathbb{R}}^{7}\right)$ (see Proposition 13.9.2). Both lead to nontrivial examples of positive selfdual Einstein orbifold metrics which are not toric and they were first introduced in [BGP02] and later studied in [Bis07]. In particular, with the notation of Definition 13.9.1 and Proposition 13.9.2 we have (p.451) the following
Theorem 12.5.12: Let the weight matrices ${\Theta}_{2,3}^{1}\in {M}_{2,3}\left(\mathbb{Z}\right)\text{}and\text{}{\Theta}_{3,4}^{0}\in {M}_{3,4}\left(\mathbb{Z}\right)$ describe the choices of T ^{2}⊂ T ^{3}⊂ SO(7) and T ^{3}⊂ T ^{4}⊂ SO(8), respectively. Let $O\left({\Theta}_{2,3}^{1}\right)\text{}\text{and}\text{}\text{O}\text{}\left({\Theta}_{3,4}^{0}\right)$ denote the corresponding QK reductions of $G{r}_{4}^{+}\left({\mathbb{R}}^{7}\right)\text{}and\text{}G{r}_{4}^{+}\left({\mathbb{R}}^{8}\right)\text{}$ We have:

(i) If all three 2 × 2 minor determinants of ${\Theta}_{2,3}^{1}$ are nonzero then $O\left({\Theta}_{2,3}^{1}\right)$ is a compact 4orbifold.

(ii) If all four 3 × 3 minor determinants of ${\Theta}_{3,4}^{0}$ are nonzero then $O\left({\Theta}_{3,4}^{0}\right)$
With all the available examples one can naturally begin asking questions about geometric properties of such metrics. It turns out that a pivotal role in understanding such metrics is played by the nonlinear PDE
This equation was first described in [BF82] as providing solutions to the selfdual Einstein equations with zero scalar curvature and one Killing vector field of “rotational type”. In [BF82] it was shown that the zero scalar curvature (or vacuum) selfdual Einstein equations admitting one Killing vector field amounts to solving either the wellknown 3dimensional Laplace equation or Equation (12.5.22). Those Killing fields that led to the 3dimensional Laplace equation were called translational Killing fields, whereas, those leading to Equation (12.5.22) were called rotational. The translational Killing fields have selfdual covariant derivative and are well understood [TW79]. For example, they give rise to the wellknown GibbonsHawking Ansatz [GH78a]. On the other hand Equation (12.5.22) has proven to be very resistent in offering up explicit solutions [Fin01]. Nevertheless, it has appeared in a variety of settings, for example, LeBrun [LeB91b] used it in his construction of selfdual metrics on the connected sums of C P ^{2}. (See also [PP98b,AG02] for further development in terms of Hermitian–Einstein geometry). Moreover, it can be viewed as an infinitedimensional version of the better known Toda lattice equation associated with the Lie algebra of type A _{n}, and so it has become known as the SU(∞)Toda field equation [Sav89,War90] or alternatively the Boyer–Finley equation [FKS02]. Its importance for us at this stage lies in the remarkable observation made by Tod [Tod97] that finding solutions to the selfdual Einstein equations with nonzero scalar curvature can be reduced to solving Equation (12.5.22). This equation as with the full selfdual or antiselfdual Einstein equations is related to integrability questions, infinite sequences of conservations laws, and twistor theory, which we briefly discuss in Section 12.7. Actually there are several cases where there are known implicit solutions to Equation (12.5.22). It would be interesting to see if one could turn implicit solutions of Theorem 12.5.10 and Theorem 12.5.12 into explicit solutions of Equation (12.5.22).
Most 4dimensional Einstein metrics appear as Riemannian metrics adapted to some other geometric structure, selfdual (or antiselfdual) metrics, and/or Kähler metrics are perhaps the best known examples. Below is an interesting Venn diagram taken from Tod [Tod97], where the special intersecting regions deserve some comment.
Let us now discuss the overlapping areas of Figure 12.1. All of the labelled areas can be related to Sasakian geometry by taking an appropriate S ^{1} or SO(3) bundle or (p.452)
(p.453) 12.6. Hypercomplex and Hyperhermitian Structures
Recall from Example 1.4.19 the following
Definition 12.6.1: A smooth manifold M is said to be almost hypercomplex if it admits a GL(n,H)structure.
Alternatively, an almost hypercomplex structure is an almost quaternionic structure such that the subbundle Q ⊂ End(TM) is trivial. Thus, Q has a global orthonormal frame {I _{1},I _{2},I _{3}} whose elements satisfy Equation (12.2.1). Such an orthonormal frame can be viewed as a map I:R ^{3} → Q satisfying I(e _{a}) = I _{a}, where ${\left\{{e}_{a}\right\}}_{a=1}^{3}$ is the standard basis for R ^{3}. So given any two points τ,τ′∈ S ^{2} we can write Equation (12.2.1) in terms of arbitrary frames of Q as
where 〈τ,τ′〉 is the standard inner product in R ^{3} and τ × τ′ is the crossproduct. So a hypercomplex structure provides M with an S ^{2}'s worth of complex structures. We denote the family of complex structures satisfying Equation (12.6.1) by I and refer to it as a hypercomplex structure.
Let g be a metric on M such that
for any τ∈ S ^{2} and X,Y∈ χ(M). Such a metric is said to be adapted to the hypercomplex structure I and the pair (I,g) is called a hyperhermitian structure. It is easy to see that such a metric always exists.
Definition 12.6.2: An almost hypercomplex manifold (M,I,g) with an adapted metric g is called an almost hyperhermitian manifold.
As discussed in Chapter 1 an almost hyperhermitian structure is equivalent to a reduction of the GL(n,H)bundle to the subgroup Sp(n). Obata showed that every almost hypercomplex manifold M ^{4n} admits a canonical GL(n,H)invariant connection [Ob65], called the Obata connection.
Definition 12.6.3: An almost hypercomplex manifold (M,I) is called hypercomplex if all complex structures I(τ), τ∈ S ^{2} are integrable. A hypercomplex manifold with an adapted metric is called hyperhermitian.
Actually, if any two orthogonal almost complex structures in the almost hypercomplex structure are integrable then all the complex structures I(τ) are integrable [Oba66,Sal89]. In the hypercomplex case integrability can be expressed in several different ways. For example, the Obata connection in general has nontrivial torsion. But on a hypercomplex manifold this unique connection is torsionfree. So an alternative definition of a hypercomplex structure is that it is an almost hyp ercomplex structure such that the Obata connection is torsionfree. In the lowest dimension compact hyperhermitian 4manifolds were classified by Boyer [Boy88a] who proved
Theorem 12.6.4: Let (M,I,g) be a compact hyperhermitian 4manifold. Then (M,I,g) is conformally equivalent to one of the following

(i) a 4torus with its flat metric,

(ii) a K3 surface with a Kähler Ricci flat metric,

(iii) a coordinate quaternionic Hopf surface with its standard locally conformally flat metric.
(p.454) In higher dimensions there are many examples of hypercomplex structures, but no classification results so far. For example, it is known which Lie groups admit such structures [SSTVP88,Joy92,BDM96]. The simplest example here is G = U(2) which as a compact complex surface is a Hopf surface, and it actually admits two commuting hypercomplex structures. More generally [Joy92]
Theorem 12.6.5: Let G be a compact Lie group. Then there exist an integer 0≤ k≤max{3,rk(G)} such that U(1)^{k} × G has a homogeneous hypercomplex structure.
There is a natural construction of hypercomplex structures on the total space of circle bundles over any 3Sasakian manifold M [BGM98a] (See the next chapter for a description of 3Sasakian structures). For example, any trivial bundle S ^{1} × M then admits locally conformally hyperkähler structures that are automatically hypercomplex. However, nontrivial circle bundles give more interesting results. For example, large families of hypercomplex structures were shown [BGM94b,BGM96a] to exist on the complex Stiefel manifolds ${\text{V}}_{\text{n,2}}^{\u2102}$ of 2frames in C ^{n}. (See also [Bat96a]). There is also a good deformation theory for hypercomplex structures [PP98a]. Further discussion of hypercomplex structures and their relation to quaternionic geometry can be found in [Joy92,AM96a,AM96b,PPS98].
Here we recall the quotient construction of Joyce [Joy91] which we shall use later. Let (M,I) be a hypercomplex manifold. We define the automorphism group ⌟ut(M,I) of (M,I) by
Since a hypercomplex structure is a Gstructure of finite type, it follows that the group ⌟ut(M,I) is a Lie group. Let H be a Lie subgroup of H ⊂ ⌟ut(M,I). Then H acts on M as complex automorphisms with respect to any of the complex structures in {I(τ)}_{τ∈ S 2}.
Definition 12.6.6: Let (M,I) be a hypercomplex manifold. Given a compact Lie subgroup H ∩ ⌟ut(M,I) a hypercomplex moment map is any Hequivariant map μ = i _{1}μ_{1}+i _{2}μ_{2}+i _{3}μ_{3}:M→𝔥^{*}⊗𝔰𝔭(1) satisfying both of the following conditions:

(i) I _{1} dμ_{1} = I _{2} dμ_{2} = I _{3} dμ_{3}, where I _{a} acts on sections Γ(T ^{*} M⊗𝔥^{*}).

(ii) For any nonzero element ζ∈𝔥 and its induced vector field X _{ζ}∈Γ(TM) I _{1} dμ_{1}(X _{ζ})≠ 0 on M.
Note that the condition (i) of Definition 12.6.6 is equivalent to requiring that the complex valued function μ_{a}+iμ_{b} on the complex manifold (M,I _{c}) be a holomorphic function with respect to the complex structure I _{c} for any cyclic permutation (a,b,c) of (1,2,3). Joyce proves the following [Joy91]:
Theorem 12.6.7: Let (M,I) be a hypercomplex manifold, and H any compact Lie subgroup of ⌟ut(M,I) Choose any hypercomplex moment map μ and let ζ = ζ_{1} i _{1}+ζ_{2} i _{2}+ζ_{3} i _{3}∈𝔥^{*}⊗𝔰𝔭(1), where all three ζ_{i} are in the center of 𝔥^{*}. Suppose the Haction on N _{ζ} = μ ^{−1}(ζ) has only finite isotropy groups and Mˆ(Ξ) = N_{Ξ}/H is an orbifold. Then Mˆ(ξ) has a naturally induced hypercomplex structure.
The hypercomplex quotient construction can be used to build many examples of hypercomplex manifolds as we shall see in Chapter 13. The main point here is that unlike in the case of hyperkähler reduction which will be defined in the (p.455) following sections the hypercomplex reduction is much more flexible in the way one chooses the associated moment map.
Since a hypercomplex manifold M is quaternionic, it has a twistor space Z(M) which satisfies all the properties of Theorem 12.3.16. But also in the hypercomplex case the trivialization of Q gives a trivialization of the twistor space Z(M) = S^{2}×M as smooth manifolds, but not as complex manifolds. Nevertheless, the projection p onto the first factor is holomorphic, and we have a double fibration
12.7. Hyperkähler Manifolds
Given a an almost hyperhermitian manifold (M,I,g) we can use the metric to define the 2forms
In particular, given a hypercomplex structure I and choosing the basis {I _{1},I _{2},I _{3}} we get the three fundamental 2forms {ω_{1},ω_{2},ω_{3}} which trivializes the subbundle Q*. By analogy with the almost Kähler case consider
Definition 12.7.1: An almost hyperhermitian manifold (M,I,g) is called almost hyperkähler if the associated fundamental 2forms are closed and it is called hyperkähler (HK) if the associated 2forms are parallel with respect to the LeviCivita connection of g.
Unlike the Kähler case an almost HK manifold must automatically be HK [Hit87a]. In fact, we have the following equivalent characterization of HK manifolds
Theorem 12.7.2: Let (M^{4n},I,g) be an almost hyperhermitian manifold with the fundamental 2forms ω_{a}(X,Y) = g(I _{a} X,Y), a = 1,2,3. Then the following conditions are equivalent:

(i) (M^{4n},I,g) is hyperkähler,

(ii) (M^{4n},I,g) is almost hyperkähler,

(iii) (M^{4n},I,g) is 1integrable.

(iv) ∇ I _{1} = ∇ I _{2} = ∇ I _{3} = 0,

(v) Hol(g)⊂ Sp(n).
In particular, an HK manifold is Kähler with respect to any choice of complex structure in I(τ), and the holonomy reduction implies that HK manifolds must be Ricci flat. Of course the 4form Ω = ∑_{a}ω_{a}∧ω_{a} is parallel so that any HK manifold is also QK, only the quaternionic bundle Q on M is trivial and the scalar curvature vanishes. The following diagrams describes how HK geometry relates to other quaternionic geometries discussed in previous sections
In dimension 4 the situation is special. Since Sp(1)≃ SU(2) the HK condition is equivalent to asking that M ^{4} be Kähler and Ricci flat. From the decomposition (p.456)
When M ^{4} is a compact HK manifold then, up to cover, it must be either a K3 surface or a flat torus. On the other hand, if we do not insist on compactness the question of the classification of such metrics remains wide open. Only partial classification results are known. All these metrics are important in General Relativity Theory as they are vacuum solutions (Ricci flat) of the Euclidean Einstein equations. Such solutions are called gravitational instantons. They will all be described as certain quotients in the next section.
Proposition 12.7.3: Let (M^{4n},I,g) be an HK manifold and consider the Kähler structure (I_{1},g,ω_{1}). The complex 2form ω_{+} = ω_{2}+iω_{3} is of type (2,0) and holomorphic, i.e., it is a holomorphic symplectic form on M.
PROOF. Let (U; z _{1}, … ,z _{2n}) be a holomorphic local chart on M with respect to the complex structure I _{1} . Consider the 2form ω_{+}(X,Z) = g(I _{2} X,Y)+ig(I _{3} X,Y) and extend it by linearity to the complexified tangent bundle TM⊗ C. Setting $X=\frac{\partial}{\partial {\overline{z}}_{j}}$ we compute for any vector field Y
(p.457) since I _{2} I _{1} = −I _{3} which implies that ω_{+} is a (2,0)form. It is holomorphic since it is closed. □
Note that the (0,2)form conjugate under I _{1} to ω_{+} is ω_{−} = ω_{2}−iω_{3}. Proposition 12.7.3 can easily be generalized to an arbitrary oriented orthonormal 3frame {τ _{1}, τ _{2},τ _{3}}, namely
Proposition 12.7.4: Let (M^{4n},I,g) be an HK manifold and consider the Kähler structure (I(τ _{1}),g,ω(τ _{1})). The complex 2form ω_{+}(τ _{1}) = ω(τ _{2})+iω(τ _{3}) is a holomorphic symplectic form on M with respect to the complex structure I(τ _{1}).
We now consider briefly the twistor space Z(M) of an HK manifold M. For some references here see [Sal86,HKLR87,Joy00]. Since an HK structure is hypercomplex, the twistor space Z(M) of an HK manifold satisfies all the properties of Theorem 12.3.16 as well as the correspondence given by diagram (12.6.4). But when M is HK the complex manifold (M,I(τ _{1})), which is the fiber over τ _{1}∈ S ^{2}≃ C P ^{1} of the holomorphic fibration p, can be viewed as a divisor in Z(M) with a holomorphic symplectic form ω_{+}(τ _{1}). This gives a twisted holomorphic 2form ϖ on Z(M) that is a section of P*O(2) ⊗ Qη* where Qη* is the bundle Q* on M pulled back to Z(M) under the right hand projection in diagram (12.6.4). To express this holomorphic data explicitly we choose the standard basis {e _{1},e _{2},e _{3}} of R ^{3}, and write a point τ∈ S ^{2} as τ = ∑_{a}τ_{a} e _{a}. These standard coordinates are related to the complex affine coordinate t∈ C ⊂ C P ^{1}≃ S ^{2} by [HKLR87]
Then the complex structure I_{Z} on Z(M) as an endomorphism of the tangent space T_{(x,t)}Z(M)=T_{x}M ⊕T_{t}S^{2} becomes
where I _{0} denotes the standard complex structure on the tangent space T _{t} S ^{2} given by multiplication by i. Of course, there is a similar expression for the other chart with affine coordinate $s=\frac{1}{t}$ centered about (−1,0,0). Now the twisted holomorphic 2form ϖ on 𝒵(M) is written as
There is a converse to this twistor space construction [HKLR87], but first some notation. Given a complex manifold 𝒵 with a holomorphic fibration p :𝒵 → CP ^{1} we let T _{F} denote the vertical subbundle of T𝒵, i.e., the kernel of the differential p_{*} :T𝒵 → TCP ^{1}
Theorem 12.7.5: Let (𝒵,J) be a complex manifold of complex dimension (2n+1) equipped with the following data

(i) a holomorphic projection p :𝒵 → CP ^{1}

(ii) a holomorphic section ϖ of p*O(2)⊗𝛌^{2}T_{F*} which restricts to a holomorphic symplectic form on the fibers of p.

(iii) a free antiholomorphic involution σ :𝒵 → 𝒵 that satisfies σ^{*}(ϖ) = ϖ, and p○σ = a˜○ p, where a˜ is the antipodal map on S ^{2}.
(p.458) Let M be the set of all rational curves C in Z with normal bundle 2nO(1) and σ(C) = C. Then M is a hypercomplex manifold with a natural pseudoHK metric g. If g is positive definite then M is HK.
12.8 Hyperkähler Quotients
In this section we review the generalization of the Marsden–Weinstein construction described in Section 8.4 to HK manifolds with hyperholomorphic isometries. Such reductions were first considered by Lindström and Roček as early as in 1983 [LR83], while investigating the socalled 4dimensional N = 2 globally supersymmetric σmodel theories. It is known that target manifolds of such σmodels are HK. Lindström and Roček observed that one can “gauge away" hyperholomorphic symmetries. In the process one introduces auxiliary gauge fields without kinetic terms in the Lagrangian, i.e., hyperholomorphic Killing vectors. The Euler–Lagrange equation for such fields are algebraic (moment map equations) and eliminating these fields leads to a new N = 2 supersymmetric σ model theory, hence, a new HK metric. A few years later Hitchin gave the rigorous mathematical description of what is now known as hyperkähler reduction [HKLR87]. We will describe this construction and some of the basic examples as it provides a blueprint for much of the material of the next chapter. Let (M,I,g) be an HK manifold and G ∩ ⌟(M,I,g) ∩ Isom(M,g) be a Lie group acting smoothly and properly on M by preserving the HK structure. Then G acts by symplectomorphism preserving the symplectic forms ω_{a}, a = 1,2,3. Suppose the Gaction is Hamiltonian with respect to each symplectic form ω_{a}. We call such an action hyperhamiltonian.
Definition 12.8.1: An HK manifold (M,I,g) together with an effective hyperhamiltonian Gaction is called a hyperhamiltonian Gmanifold.
As discussed in Section 8.4.1 such an action gives rise to three Gequivariant symplectic moment maps μ_{a}:M→ ≫ ^{*}. We can assemble these maps together to get
Definition 12.8.2: Let (M,I,g) be a hyperhamiltonian Gmanifold. The map μ = (μ_{1},μ_{2},μ_{3}) = i _{1}μ_{1}+i _{2}μ_{2}+i _{3}μ_{3}
is called the hyperkähler moment map for the action of G.
We have the following natural generalization of the Marsden–Weinstein symplectic reduction theorem [HKLR87]
Theorem 12.8.3: Let (M,I,g) be HK and G be a hyperhamiltonian action on M with the HK moment map μ:M→ ≫ ^{*}⊗𝔰𝔭(1). Let λ = (λ_{1},λ_{2}λ_{3})∈ ≫ ^{*}⊗𝔰𝔭(1) be any element fixed by the coadjoint action of G on its Lie coalgebra ≫ ^{*}. Suppose λ is a regular value of μ so that N = μ ^{−1}(λ)⊂ M is a manifold. Suppose further that the orbit space Mˆ(𝛌) = μ^{1}(𝛌)/G is a manifold (orbifold). Then Mˆ(𝛌) is an HK manifold (orbifold) with the HK structure induced from M via inclusion and projection maps.
PROOF. We only sketch the proof here. The manifold (M,I _{1},ω_{1},g) is a Kähler manifold. The HK reduction can be seen as a two step process: First, we consider the function
(p.459) which is easily seen to be holomorphic on (M,I _{1},g). Thus the set ${N}_{+}={\mu}_{+}^{1}\left(0\right)$ is a complex subspace of (M,I _{1},g), in particular it must be Kähler. Note that N _{+} need not be a smooth manifold, it is sufficient that it be smooth in some Hinvariant open neighborhood N'_{+} such that μ_{1} : N_{+} → g*. The action of G restricts to N _{+} with the Kähler moment map μ_{1}:N _{+}→ ≫ ^{*}. Hence, the reduced space Mˆ is nothing but a Kähler reduction of N _{+} by the action of G. In particular, Mˆ is Kähler with the complex structure Î_{1} induced from M by the quotient construction. Now, the result follows by observing that the same argument applies to I _{2} and I _{3}, and Mˆ is therefore Kähler with respect to all three complex structures {Î_{1},Î_{2},Î_{3}}. One can easily check that the induced complex structures satisfy the quaternionic relations. □
We remark that, just as in the symplectic case, one can consider more general “singular" quotients. This was done by Dancer and Swann [DS97a,Swa97] who showed
Theorem 12.8.4: Let (M,I(τ),g) be a hyperhamiltonian Gmanifold with the moment map μ:M → ≫ ^{*}⊗ 𝔰𝔭(1). Furthermore, suppose G acts smoothly and properly on M. Let M _{(H)} denote the stratum consisting of orbits of type H ⊂ G. Then N _{H} = μ ^{−1}(0)∩ M _{(H)} is a manifold and the orbit space
has a natural HK structure. Consequently, the reduced space Mˆ = μ^{1}(0)/G is a disjoint union of HK manifolds
The proof is a corollary of the Sjamaar–Lerman Theorem 8.4.3. In particular, each stratum is a smooth HK manifold but unlike the symplectic case it is not clear whether the quotient is decomposable in the sense of Goretsky–MacPherson.
The rest of this section is devoted to introducing some basic examples of HK reduction of the flat model H ^{n}. We will often use the pair μ = (μ_{1},μ_{+}) to describe the moment map in complex coordinates on (M,I _{1},ω_{1},g) relative to I _{1}. Since we single out I _{1} we will also use {i,j,k} for the quaternions {i _{1},i _{2},i _{3}}.
EXAMPLE 12.8.5: Calabi metrics on T ^{*} C P ^{n}. Let u = Z¯+wj ∈ ∈ H ^{n} ∼ C _{n} × C _{n} and consider the diagonal action of G = S ^{1}⊂ Sp(n) given by left multiplication with g(t) = e ^{it}. The moment map for this action is
In complex charts we get
and the circle action reads (w,z)↦(e ^{it} w,e ^{−it} z). One could consider an arbitrary level set of the moment map. However, as Sp(1)_{+} is a symmetry of the flat HK metric, one can use it to choose the value of μ to be a constant multiple of i. (p.460) Further scaling the metric shows that it is sufficient to consider N = μ ^{−1}(−i) and ${N}_{+}={\mu}_{+}^{1}\left(1\right)\subset N$ which are described as
Let Mˆ=μ^{1}(i)/S^{1}. We first want to identify Mˆ with the Kähler reduction of N _{+} (or a Ginvariant open set N'_{+} ∩N_{+}). For an appropriate choice of G=S^{1}. its Kähler reduction N _{+} will be an algebraic quotient of G = S^{1} by the complexification C ^{*} of G = S ^{1}. Note, however, that N'_{+} is not compact so we cannot rely on Kirwan’s theorems in this setting[Kir84].^{2} Nevertheless, we get the following identification
where N'_{+} = {(W,z) ∈ C _{n} × C _{n}  Σ_{j}w_{j}z_{j}= 0, w ≠ 0}, and C ^{*} acts by (w,z) → (𝛌w, 𝛌¯z). Thus, Mˆ is the holomorphic cotangent bundle T ^{*} C P ^{n−1}. It turns out that the HK metric obtained on T ^{*} C P ^{n−1} via this reduction is isometric to the Calabi metric [Cal79], the first nonflat example of a complete HK manifold. An N = 2 supersymmetric σmodel description of the metric is due to Lindström and Roček [LR83] and, in the above language it appears in [HKLR87]. An explicit expression for this metric in dimension 4 was discovered by Eguchi and Hanson [EH79] and it is called the Eguchi–Hanson gravitational instanton.
EXAMPLE 12.8.6: TaubNUT Metrics. This example involves a noncompact hyperhamiltonian group action of G = R on H × H ^{n}≃H ^{n+1} defined for any p = (p _{1}, … ,p _{n})∈R ^{n} by
with the moment map
We can always shift to the zerolevel set and then the moment map equations can be “solved" by writing
This action is free and proper on H × H ^{n}. In particular it is free and proper on μ^{1} _{p}(0) = N_{P}. Denote the quotient manifold by M(P) = μ^{1} _{P}(0)/R. First, note that M(p) is diffeomorphic to H ^{n}. This follows from the observation that the set S = {u ∈μ^{1} _{p}(0)Re(u_{0})=0} is a global slice for this action. The induced HK metric ĝ(𝛌) can easily be calculated and g(0) = g _{0} is the flat metric. In dimension 4, this metric ĝ(P) depends on 1parameter Λ and when Λ = 0 we get M(0) isomorphic to H≃C ^{2} with the standard flat metric. Hence, (M(Λ),g(Λ)) is a smooth 1parameter family of HK deformations of the Euclidean metric. The metric g(Λ) (p.461) is called the TaubNUT ^{3} gravitational instanton and it has an interesting history. Just as the famous Schwarzschild metric, or Kerr solution, the metric appears first in the Lorenzian signature. One can always perform the socalled “Wick rotation" changing t↦ it which locally gives a Riemannian metric with similar properties. However, there is no reason for the Riemannian metric to extend globally to a complete metric on some manifold. This is fairly rare and happens only in special situations. It was Hawking who observed that this indeed is the case for the Lorenzian TaubNUT solution, giving rise to a complete Ricci flat metric on R ^{4} [Haw77]. For some time thereafter the metric was not really fully understood, there being claims in the literature is that this metric was not Kähler. We should point out that this is the only known complete Ricci flat Kähler metric on C ^{2} apart from the standard one [LeB91a]. If one imposes a Euclidean volume growth condition the only known example of such complete Ricci flat Kähler (or just Ricci flat) metric on C ^{2} is the Euclidean metric.
12.9 Toric Hyperkähler Metrics
It is easy to see that previous two examples fall into a special category of complete HK manifolds: they admit n commuting hyperholomorphic Killing vector fields, where n is the quaternionic dimension. Following Bielawski and Dancer we consider [Bie99,BD00]
Definition 12.9.1: An HK manifold (M^{4n},I,g) is locally toric if it admits n commuting hyperholomorphic Killing vector fields, linearly independent at each point x∈ M ^{4n}, i.e., locally M admits a free action of R ^{n} by hyperholomorphic isometries. Furthermore, (M^{4n},I,g) is said to be a toric HK manifold if it is a hyperhamiltonian T ^{n} space.
We emphasize that “toric” is understood here to be in the quaternionic sense. A local description of such metrics in dimension 4 is due to Gibbons and Hawking [GH78a] and in arbitrary dimension 4n it has been generalized by Lindström and Roček [LR83]. The socalled Legendre transform method developed by Lindström and Roček associates a 4ndimensional HK metric with n commuting hyperholomorphic Killing vectors to every realvalued function F on an open subset u ∩ R ^{3} ⊗ R _{n} which is harmonic on any affine 3dimensional subspace L of the form R ^{3}⊗ R v, v∈R ^{n} (such functions are sometimes called polyharmonic). The construction proceeds as follows: Let us identify R ^{3}⊗ R ^{n} with R ^{n} × C ^{n} and let (x,z)∈ R ^{n} × C ^{n} be coordinates on u. Given any polyharmonic function F(Z,Z¯) on u we consider a realvalued function
where the x _{i} are determined by
It is an elementary exercise to check that polyharmonicity of F turns K into a Kähler potential of an HK metric. Furthermore, if we set y = i(u¯  u) then X _{i} = (p.462) ∂/∂ y _{i}, i = 1, … ,n yield n commuting hyperholomorphic Killing vector fields with the corresponding HK moment maps
One can show that, relative to local coordinates (y,x,z), the HK metric takes the form [PP88]
where ${\Phi}_{ij}=\frac{1}{4}{F}_{{x}_{i}{x}_{j}}\text{}\text{and}\text{}{A}_{j}=\frac{\sqrt{1}}{2}{\sum}_{l}\left({F}_{{x}_{i}{\overline{z}}_{l}}d{\overline{z}}_{l}{F}_{{x}_{i}{z}_{l}}d{z}_{l}\right).$ The functions Φ_{ij} are also polyharmonic. The n × n matrix [Φ_{ij}] locally determines the hyperkähler and hyperhamiltonian structure. When n = 1 this is the wellknown Gibbons–Hawking Ansatz. Renaming (y _{1},x _{1},z _{1}) = (t,x _{1},x _{2}+ix _{3}) = (t,x), A _{1} = α· d x, Φ_{11} = V we can write the metric in a more familiar form
where grad V = curlα. In particular, V is a solution of the Laplace equation so that we can write
When δ = 0 these metrics are called kcenter gravitational multiinstantons. The first two values k = 1,2 correspond to the Euclidean and Eguchi–Hanson metrics, respectively. For larger values of k it is not easy to determine when the metric is actually complete and even harder to see what the manifold M _{k} on which it is defined is. When δ = 1 we get the socalled kcenter TaubNUT gravitational multiinstantons with k = 1 corresponding to the Euclidean TaubNUT metric discussed in Example 12.8.6.
The two basic examples of this construction are the flat S ^{1}invariant metrics on S ^{1} × R ^{3} and on H. In the first case we have
and, consequently, Φ ≡ 1, while in the second case
where r^{2} = x^{2} + zz¯ with Φ = 1/4r. More general forms are given in [BD00]. In the latter, the functions F and the metrics for HK quotients of flat vector spaces are computed. They are essentially obtained by taking linear combinations and compositions with linear maps of the solution (12.9.8). Bielawski shows [Bie99] that, in the case of a complete metric, the only other possibility is adding a linear combination of (12.9.7), which corresponds to a TaubNUT deformation of Definition 12.9.4.
For an HK metric of the form (12.5.6) taking HK quotients by subtori is simple. The moment map equations are now linear (in x _{i},z _{i}), and the HK quotient corresponds to restricting the function F to an appropriate affine subspace of R ^{3}⊗ R ^{n}. In fact, the requirement that F be polyharmonic is a consequence of the fact that we must be able to take HK quotients by any subtorus. An explanation of this construction in terms of twistors was given by Hitchin, Karlhede, Lindström, and Roček [HKLR87]. In particular, they have shown that any HK 4nmanifold with (p.463) a free hyperhamiltonian R ^{n}action which extends to a C ^{n}action with respect to each complex structure and such that the moment map is surjective is given by the Legendre transform. In fact, one can show [Bie99] that the Legendre transform provides a complete local description of such metrics, i.e.,
Proposition 12.9.2: Let (M ^{4n},g) be a locally toric HK manifold. Then g is locally given by Equation (12.9.4).
The key to the understanding of the global properties of such metrics is the HK quotient construction. The relevant spaces were first introduced in [HKLR87], but the global properties of such metrics were studied only much later by Bielawski and Dancer [BD00] culminating in a complete classification result by Bielawski [Bie99].
We will discuss this classification here as it will be important in the next chapter. First, we would like to identify hyperhamiltonian Gmanifolds which are the same in the sense of the following definition.
Definition 12.9.3: Let M,M′ be two hyperhamiltonian Gmanifolds and let μ,μ′ be the chosen moment maps. We say that M and M′ are isomorphic as hyperkähler Gmanifolds, if there is a hyperholomorphic Gequivariant isometry f:M→ M′ such that μ = μ′○ f.
Second, there is a natural relation between the flat metric and the TaubNUT metric on H. This is, however, only an example of a more general correspondence. The construction of the Example 12.8.6 suggest the following definition.
Definition 12.9.4: Let M ^{4n} be a connected complete HK manifold of finite topological type with an effective hyperhamiltonian action of G = R ^{p} × T ^{n−p}. A TaubNUT deformation (of order m) of M is the HK quotient of M × H ^{m} by R ^{m}, where R ^{m} acts on M via an injective linear map ρ:R ^{m}→ Lie (T ^{n}) = R ^{n}.
Note that such a deformation M′ is canonically T ^{n}equivariantly diffeomorphic to M by a diffeomorphism f which respects the HK moment maps μ,μ′, i.e., μ = μ′○ f. Bielawski shows that up to Gequivariant isometries and TaubNUT deformations one can restrict attention to the following HK quotients studied in detail in [BD00]. We consider a T ^{k}⊂ T ^{m+1}⊂ Sp(m+1) action on H ^{m+1} defined via the exact sequence $0\to h\stackrel{\omega}{\to}{\mathbb{R}}^{m+1}\stackrel{\theta}{\to}{\mathbb{R}}^{m+11}\to 0,$ and its dual, as in (12.5.2, 12.5.3), with m+1−k = n. This is a hyperhamiltonian T ^{k}action for any choice of the weight matrix Ω ∈M_{k,m+1}(Z) or, alternatively, Θ ∈ M_{n,m+1}(Z). We can consider the HK moment map for this action μ_{Ω}:H ^{m+1}→1.5R ^{3} × R ^{k}
where
and ${\lambda}_{\alpha}={\lambda}_{\alpha}^{1}i+{\lambda}_{\alpha}^{2}j+{\lambda}_{\alpha}^{3}k$ are purely imaginary quaternions (or vectors in R ^{3}) for α = 1, … ,m+1. In particular, c∈R ^{3}⊗R ^{k} is simply an arbitrary choice of the constant in the definition of the HK moment map. Let us denote by λ the “moment (p.464) level set" data, i.e.,
Consider now the HK quotient space, i.e.,
Clearly, the HK quotient M(Θ,λ) locally inherits a hyperhamiltonian T ^{n}action. However, it can be singular and the stratification depends on the choice of the quotient data. Note that either choosing (Θ,λ) or (Ω,c) completely determines both the T ^{k}action and the associated HK quotient. In fact, these two descriptions are dual to one another and they are both useful. Consider the following codimension three affine subspaces in R ^{3}⊗R ^{n}:
α = 1, … ,m+1 and θ _{α}∈R ^{n} is the α^{th} column of Θ.
Bielawski and Dancer prove the following theorem [BD00]:
Theorem 12.9.5: Suppose the column vectors θ _{α} of Θ ∈ M_{n,m+1}(Z) are primitive integer vectors spanning R ^{n}. Suppose λ∈R ^{3}⊗R ^{m+1} is such that H _{α} are all distinct. Then the HK quotient M(Θ,λ) is smooth if and only if

(i) every collection of n+1 of the H _{α} have empty intersection, and

(ii) whenever some n of the H_{k1},…,H_{kn} have nonempty intersection, then the set {θ_{k1},…,θ_{kn}} is a Z basis for Z ^{n}.
The first condition is sufficient for M(Θ,λ) to be an HK orbifold with at worst Abelian quotient singularities.
If condition (i) of Theorem 12.9.5 holds the orbifolds M(Θ,λ) and M(Θ,λ′) are homeomorphic, and if λ′ = (λ ^{1},0,0), they are diffeomorphic. In particular, one can always set λ ^{2} = λ ^{3} = 0 while investigating the topology of such quotients. Bielawski and Dancer have given a formula for the Betti numbers of M(Θ,λ) in terms of arrangements of certain hyperplanes. Consider the collection of hyperplanes
Theorem 12.9.6: Let M = M(Θ,λ) be a toric HK orbifold of dimension 4n.

(i) Then M is simply connected and H ^{j}(M,Q) = 0 for j odd;

(ii) ${b}_{{2}_{p}}\left(M\right)={\sum}_{i=p}^{n}{\left(1\right)}^{ip}\left(\begin{array}{c}i\\ p\end{array}\right){d}_{i},$ where the integer d _{i} denotes the number of idimensional elements of the complex C

(iii) if M = M(Θ,λ) is a smooth manifold, there is a ring isomorphism H ^{*}(M,Z)≈ Z[u _{1}, … ,u _{N}], where u _{i} are the first Chern classes of certain complex line bundles on M.
Parts (i) and (ii) of this theorem are due to Bielawski and Dancer [BD00] while (iii) is due to Konno [Kon00]. In [Bie99] it is shown that the quotients M = M(Θ,λ) are the essential part of the classification of locally toric HK manifolds. (p.465) One can show that
Theorem 12.9.7: Let M ^{4n} be a connected complete HK manifold of finite topological type with an effective hyperhamiltonian action of G = R ^{p} × T ^{n−p}. Then

(i) if M is simply connected and p = 0, then M is isomorphic, as a hyperhamiltonian T ^{n} manifold, to a hyperkähler quotient of some flat H ^{d} × H ^{m}, m≤ n, by T ^{d−n} × R ^{m};

(ii) if M is simply connected and p > 0, then M is isomorphic, as a hyperhamiltonian Gmanifold, to the product of a flat H ^{p} and a 4(n−p)dimensional manifold described in part (i);

(iii) if M is not simply connected, then M is the product of a flat (S ^{1} × R ^{3})^{l}, 1≤ l≤ n, and a 4(n−l)dimensional manifold described in part (ii).
In the case of dimension 4 we can be more specific.
Corollary 12.9.8: Let M be a simply connected 4dimensional complete HK manifold with a nontrivial hyperhamiltonian vector field. If b _{2}(M) = k > 0, then M is isometric either to an ALEspace of type A _{k} (i.e., a multiEguchi–Hanson space) or to its TaubNUTlike deformation (i.e., to the HK quotient by R of the product of such a space with H). If b _{2}(M) = 0, then M is either the flat H or it is the TaubNUT metric on R ^{4}.
In particular, Bielawski concludes the following HK analog of the Delzant’s theorem for complex toric manifolds:
Theorem 12.9.9: Complete connected hyperhamiltonian T ^{n}−manifolds of finite topological type and dimension 4n are classified, up to TaubNUT deformations, by arrangements of codimension 3 affine subspaces H _{α} in R ^{3}⊗ R ^{n} defined as in (12.9.12) by
We end this section by briefly mentioning some fascinating recent work by Hausel, Nakajima, Sturmfels, and others that relates hyperkähler geometry, and toric hyperkähler structures in particular, to combinatorics and representation theory [Nak98,Nak99,HS02] as well as to the famous ADHM construction of instanton moduli spaces [BM93b,Nak99,Hau06] and to number theory [Hau05,Hau06].
12.10. ALE Spaces and Other Hyperkähler Quotients
In this last section we shall describe some other examples of hyperkähler metrics with particular focus on examples relevant to 3Sasakian geometry. Our first goal is to describe the quotient construction of ALE spaces discovered by Kronheimer [Kro89a,Kro89b].
12.10.1. Classical McKay Correspondence
We begin by recalling some elementary facts about discrete subgroups of SU(2), their representation, and the classical McKay correspondence. These were already discussed in the proof of Theorem 10.1.5, where they were denoted by: Z _{n},D*_{n},Z*,Z*,Z*. Let H = C ^{2} be the standard complex 2dimensional representation of SU(2). In particular, H gives a (p.466) representation of each finite subgroup Γ⊂ SU(2). Let {ρ_{0},ρ_{1}, … ,ρ_{r}} be the set of irreducible representations of Γ with ρ_{0} the trivial representation. Then
McKay observed that there is a relation between the matrix A = (a _{ij}), whose entries are all either 0 or 1, and the adjacency matrix of extended Dynkin diagrams of type ADE. If C˜ is the Cartan matrix of the extended Dynkin diagram then A + C˜ = 21_{r+1}. Let {θ_{1}, … ,θ_{r}} be the simple roots of the root system of the associated Lie algebra. Let θ_{0} be the negative of the highest root. McKay further noticed that
where n _{i} is the dimension of the representation ρ_{i} [McK80,McK81]. The regular representation ρ of Γ decomposes as
Separately for each Γ, all of this information can now be encoded in a “labelled" extended simplylaced Dynkin graph.
The vertices of the diagram correspond to the irreducible representations ρ_{i} with the numbers in each vertex giving the dimension of that representation n _{i}. The usual Dynkin diagram is obtained from the extended one by removing the one vertex which corresponds to the trivial representation ρ_{0}. In particular, McKay’s observations show that
(p.467) EXAMPLE 12.10.1: Consider the example of Γ = Z _{n+1}. Let τ^{n+1} = 1. Any irreducible representation of Z _{n+1} is 1dimensional and ρ_{i}(x) = τ^{i} x, x∈C. Now, the 2dimensional representation H gives H(τ)·(x,y) = (τx,τY¯). Clearly, H = ρ_{1}⊕ρ_{n} so that ρ_{i}⊗ H = ρ_{i+1}⊕ρ_{i−1}. The vertex ρ_{i} of the McKay graph is joined by an edge to the vertices ρ_{i+1} and ρ_{i−1}. This defines the extended Dynkin diagram ˜A _{n}.
EXERCISE 12.1: Consider the example of Γ = D*_{n}. As a subgroup of SU(2) the binary dihedral group^{4} is generated by two matrices
Note that for n = 3 we simply get the cyclic group Z _{4}. This gives the representation H. In particular, D*_{4} is the group of quaternions Q = {±1,± i,± j,± k}. D*_{4} has three nontrivial 1dimensional irreducible representation {ρ_{1},ρ_{2},ρ_{3}} and one 2dimensional representation ρ_{4}. “Derive" McKay’s graph for D*_{4} as in Example 12.10.1. Repeat this for any binary dihedral group D*_{n}.
12.10.2. Geometric McKay Correspondence and Kleinian Singularities
McKay's observation is closely related to the algebraic geometry of Kleinian singularities which we shall explain next. This is often referred to in the literature as the geometric McKay correspondence.
Definition 12.10.2: For Γ⊂ Sp(1) a finite subgroup, the quotient variety X = C ^{2}/Γ = SpecC[x,y]^{Γ} is called a Kleinian singularity (also known as a simple surface singularity, or a rational double point, or an ADE type singularity).
The quotient can be embedded as a hypersurface X⊂ C ^{3} with an isolated singularity at the origin with the defining equation f(z _{0},z _{1},z _{2}) = 0 determined by the conjugacy class of Γ. These polynomials have already appeared in the table of Remark 10.1.1. Suppose now π:M→1.2X = C ^{2}/Γ is a crepant resolution. Then the divisor Δ = π^{−1}(0) is the dual of the associated Dynkin diagram in the following sense: the vertices of the Dynkin diagram correspond to rational curves D _{i} with selfintersection −2. Two curves intersect transversally at one point if and only if the corresponding vertices are joined by an edge in the Dynkin diagram. Otherwise they do not intersect. The collection of these curves {D _{1}, … , D _{r}} forms a basis for H _{2}(M,Z). The intersection form with respect to this basis is the negative of the Cartan matrix C.
12.10.3 Kronheimer–McKay Correspondence and Hyperkähler ALE Spaces
Hitchin observed that in the case of Γ = Z _{n} the crepant resolution π:M→1.2C ^{2}/Z _{n} admits a family of complete HK metrics [Hit79]. In fact, locally these metrics are produced via the Gibbons–Hawking Ansatz. Using twistor methods Hitchin showed that the Gibbons–Hawking gravitational instantons, as hyperkähler ALE spaces, are the minimal resolution of singularity C ^{2}/Z _{n}. In particular, the minimal resolution of the singularity C ^{2}/Z _{2} is the cotangent bundle T ^{*} C P ^{1} and the HK metric is the Eguchi–Hansom metric. Hitchin then conjectured that such metrics should exist for all other spaces π:M→1.2C ^{2}/Γ. It was only after the discovery of the HK reduction and description of its mathematical foundations in [HKLR87] that the conjecture was finally proved by Kronheimer [Kro89a,Kro89b] who generalizes the quotient construction described in [HKLR87] for Γ = Z _{n} to an arbitrary (p.468) finite group Γ⊂ Sp(1). Again, the key is the McKay correspondence. Not surprisingly, the quotient can be completely described by the extended Dynkin diagram associated to Γ. In a way, Kronheimer's quotient construction is “the third McKay correspondence.” To make our statement more precise let us begin with a precise definition of an ALE space.
Definition 12.10.3: Let Γ⊂ Sp(1) be a finite subgroup and let (H, I^{+},go) be the standard flat (left) HK structure on H defined in Section 12.1. Let r:H/Γ→1.5R _{≥0} be the radius function on H/Γ. We say that an HK manifold (M,I,g) is asymptotically locally Euclidean (ALE), and asymptotic to H/Γ, if there exists a compact subset X⊂ M and a map π:M∖ X→1.2H/Γ that is a diffeomorphism between M∖ X and {x∈ H/Γ r(x) > R} for some R > 0 such that
as r→∞ and k≥0, where ∇ is the LeviCivita connection of the flat metric g _{0}.
Consider any extended Dynkin diagram Δ˜(Γ). With each vertex of Δ˜(Γ) we associate the unitary group U(n _{i}) and with each edge the vector space H ^{ninj} ∼ Hom(C ^{ni},C ^{nj}) ⊕ Hom(C ^{ni},C ^{nj}). One can think of each edge as the vector space M_{ni,nj}(H) of quaternionic matrices. For each Γ, we define
The group G(Γ) acts naturally on H ^{Γ} and the action is hyperhamiltonian with the appropriately defined flat HK structure on each “edge.” However, the action is not effective, hence, we take the quotient K(Γ) = G(Γ)/T, where T is the central U(1)⊂ G(Γ). The action of K(Γ) is then effective and it defines the HK moment map
Definition 12.10.4: We say that a K(Γ)invariant element ξ∈ 𝔨^{*}⊗𝔰𝔭(1) is in the good set if the K(Γ)action on the ξ level set ${\mu}_{\Gamma}^{1}\left(\xi \right)$ of the moment map is free.
The notion of the good set is generic. The set Z of K(Γ)invariant elements in 𝔨^{*} can be identified with the dual of the center. Kronheimer shows that ξ is not in the good set if ξ∈ D _{θ} ⊗𝔰𝔭(1)⊂ Z ⊗𝔰𝔭(1), where D _{θ} are the walls of the Weyl chamber. The reduction gives the following theorem [Kro89a,Kro89b]
Theorem 12.10.5: Let ξ∈𝔨^{*}⊗𝔰𝔭(1) be a G(Γ)invariant element. Let M(Γ,ξ) be the HK reduction of H ^{Γ} by the action of K(Γ) with the momentum level set ξ. Then M(Γ,0)≃ C ^{2}/Γ and M(Γ,ξ) is a HK orbifold for any ξ. In addition, when ξ is in the good set ${\mu}_{\Gamma}^{1}\left(\xi \right)$ the smooth manifold M(Γ,ξ) gives a family of complete ALE HK metrics on the crepant resolution of singularity C ^{2}/Γ.
When ξ is not in the good set, the HK orbifold M(Γ,ξ)→1.2C ^{2}/Γ is a partial resolution of the quotient singularity. In [Kro89b] Kronheimer shows that his construction is also complete. That is any hyperkähler ALE space of Definition 12.10.3 is obtained as such a quotient. The quotient metrics are known in local charts for Γ = Z _{n} (the Gibbons–Hawking Ansatz, see 12.9.5) as they always have nontrivial isometries. For the nonAbelian Γ the ALE metrics have no Killing (p.469) vectors. Recently, Cherkis and Hitchin gave explicit formulas for the ALE gravitational instantons in the binary dihedral case [CH05].
REMARK 12.10.1: The Kleinian singularities and discrete groups of SU(2) also give a onetoone correspondence with all compact 3Sasakian manifolds in dimension 3. We shall discuss this in the next chapter.
12.10.4. Other Hyperkähler Metrics
In the last 25 years HK geometry has become an important field of Riemannian geometry. Already over a decade ago, in a Séminaire Bourbaki review article Hitchin points out that the richness of the theory of HK manifolds, in some sense, vindicates Hamilton's conviction that quaternions should play a fundamental role in mathematics and physics [Hit92]. As it happens, many new ideas in this field have come from mathematical physics. In this chapter we have only covered a small number of selected topics most relevant to the material of Chapter 13. We would like to end it with a brief discussion of several subjects we were unable to introduce. We refer the interested reader to several books and review articles about the subject [Hit87b,AH88,Hit95a,Dan99,VK99].

• Instantons, Monopoles, and Stable Pairs. Many HK metrics emerge in the description of the geometry of various moduli spaces. In 1983 Atiyah and Bott made a fundamental observation that the moduli space of selfdual Yang–Mills connections over a Riemann surface can be described as an infinitedimensional Kähler quotient [AB83]. In such a picture one first equips the space of all connections with a structure of an infinitedimensional Kähler manifold. On it acts an infinitedimensional group of gauge transformations and the image of the moment map for this action is given by the selfduality equations. Hence, the Kähler quotient, which turns out to be finite dimensional, is the space of selfdual Yang–Mills connections modulo the gauge equivalence. When a Riemann surface is replaced by a 4dimensional manifold (such as R ^{4} or S ^{4}, for example) the space of connections can be given the structure of an infinitedimensional HK manifold. Since the HK moment map for the gauge group action produces the selfduality equations, the HK quotient can be naturally identified with the moduli space of instantons. This picture explains why the kinstanton moduli spaces over S ^{4} carry a natural HK structure [AHDM78]. Yang–Mills connections on other 4manifolds were then studied. For example, Kronheimer and Nakajima considered instantons over the ALE spaces discussed earlier [KN90,Nak90]. The infinite dimensional HK quotient picture is inherently present in a variety of different moduli problems. One important case is the moduli space of solutions of the Bogomolny equations on a 3manifold. These are known as monopoles. They can be viewed as Yang–Mills connections which is a translational symmetry. Many complete HK metrics have been constructed as solutions of the Bogomolny equations or the related the Nahm equations [Nah82,AH88,Dan93,Dan94]. For instance, in the SU(2) case, the universal cover ${\tilde{M}}_{k}^{0}$ of the moduli space of charge k monopoles with a fixed center is a complete HK manifold of dimension 4(k−1) [AH88]. The 4dimensional case of ${\tilde{M}}_{k}^{0}$ gives the famous SO(3)invariant Atiyah–Hitchin monopole metric [AH85]. Like in the TaubNUT case the SO(3)action acts by rotations on the 2sphere of complex structures I but the metric is quite different as it is not toric: it has no hyperhamiltonian Killing vectors. Finally, we mention the moduli space of (p.470) R ^{2}invariant Yang–Mills equation. Such a reduction naturally leads to the moduli space of the socalled stable pairs or Yang–Mills–Higgs fields over an arbitrary Riemann surface and was considered by Hitchin in [Hit87b].

• Hyperkähler Manifolds of Type A _{∞} and D _{∞}. The first example of a complete HK manifold of infinite topological type was obtained by Anderson, Kronheimer, and LeBrun [AKL89]. They showed that one can take the k→∞ limit in the Gibbons–Hawking Ansatz (12.9.5) under an appropriate assumption about the distribution of the mass centers. Later these metrics were considered from an algebraic viewpoint by Goto who also showed that a similar limit can be taken in the D*_{n} ALE case [Got98].

• Hyperkähler Deformations of ALE Spaces. TaubNUT deformations of the ALE gravitational instantons corresponding to Γ = Z _{n} were discussed in Section 12.9. These metrics are no longer ALE but the are asymptotically locally flat. Other ALE spaces do not admit any TaubNUT deformations in the sense of Definition 12.9.4 as they have no hyperholomorphic isometries. But the ALE spaces of the binary dihedral group admit deformations similar to TaubNUT deformations. Recall, that we can view the TaubNUT deformation of the Z _{n} ALE space as follows: consider the space H ^{Γ} and replace one Hfactor with the flat HK manifold S ^{1} × R ^{3} modifying the action of K(Γ) to be the translation on S ^{1}. Consider a nonAbelian Γ and suppose we could replace the Euclidean metric on some edge H ^{ninj} with a nonEuclidean HK metric on H ^{ninj} which, however, admits hyperhamiltonian action of U(n _{i}) × U(n _{j}). Dancer observed that there is one such case: a complete U(2) × U(2)invariant HK metric on H ^{4} considered as T ^{*} GL(2,C) [Dan93]. The metric is obtained as a monopole moduli space by solving Nahm's equations. Dancer shows that replacing the flat H ^{4} with one copy of M = T ^{*} GL(2,C) together with its monopole metric and then performing the HK quotient of Kronheimer gives nontrivial deformations of the ALE metrics for each D*_{n},n≥4. n≥4. It is not clear if similar deformations exist for any of the ALE spaces of the Eseries.

• Hyperkähler Metric on Coadjoint Orbits. Kronheimer showed that there is a natural HK metric on regular semisimple coadjoint orbits of a complex group Lie group G _{c} [Kro90a]. Kronheimer also proved that nilpotent orbits admit HK structures [Kro90b]. Later Biquard [Biq96] and independently Kovalev [Kov96] showed that there is an HK structure on any coadjoint orbit of G _{c}.

• Hyperkähler Metrics on Cotangent Bundles. The first explicit nontrivial example of an HK metric is the Calabi metric on the cotangent bundle T ^{*} C P ^{n} as described in Example 12.8.5. Already in 1983 Lindström and Roček constructed a HK metric on the cotangent bundle T ^{*}Gr_{n}(C ^{n+m}) which is realized as a reduction of the flat space H ^{m(n+m)} by a hyperhamiltonian action of U(m) [LR83]. In fact, Lindström and Roček derived an explicit formula for the Kähler potential of this metric generalizing the formula given by Calabi [Cal79]. It is not surprising that these metrics are only a special case. The HK metrics on coadjoint orbits are complete if and only if the orbit is semisimple. Such orbits are then diffeomorphic to the cotangent bundle of flag manifolds for G. In such cases one can write down the metric and Kähler potentials explicitly [BG97a,DS97b].

• Compact Hyperkähler Manifolds. Much work has been done on the geometry of compact HK manifolds with many new examples in higher dimensions. We (p.471) refer the reader to several extensive reviews on the subject and references therein [VK99,GHJ03,NW04].

• Quaternionic Geometries with Torsion. These structures arose from the attempts of physicists to incorporate the socalled bosonic Wess–Zumino–Witten term [WZ71,Wit93] in σmodels with (extended) supersymmetry. The idea of considering torsion connections in such σmodels dates back to the early 1980s (see [HS84] (Kähler with torsion) and [GHR84] (hyperkähler with torsion)). The manifolds involved are not Kähler or hyperkähler, but Hermitian and hyperhermitian. In 1996 Howe and Papadopoulos introduced a formal definition of the socalled HKT geometry (hyperkähler with torsion) and studied the twistor spaces of such manifolds [HP96]. This sparked a considerable interest in such models also among mathematicians (cf. [GP00,GGP03]). Today physicists and mathematicians alike continue studying KT (Kähler with torsion), CYT (Calabi–Yau with torsion), HKT, and even QKT (quaternionic Kähler with torsion) geometries. This subject, although very interesting, goes far beyond the scope and the main focus of our book. We refer the reader to a couple of extensive review articles on the mathematical foundations, the history, and the bibliography [Gra04,Agr06].
Notes:
(1) As a tribute, in this chapter, as in most literature on quaternionic structures, H stands for Sir William Hamilton.
(2) This is not a special feature of this example. On the contrary, this is what typically happens with HK reductions of H ^{n} by compact hyperhamiltonian Gactions.
(3) The acronym NUT stands for Newmann–Unti–Tamburino and has become standard terminology for describing a certain type coordinate singularity of the Einstein equations.
(4) Recall that our notation is not completely standard. Our binary dihedral group ${D}_{n}^{*}={\mathbb{Z}}_{2\left(n2\right)}\times {\mathbb{Z}}_{2}$ has order 4(n−2) and not 4n.