## Charles Boyer and Krzysztof Galicki

Print publication date: 2007

Print ISBN-13: 9780198564959

Published to Oxford Scholarship Online: January 2008

DOI: 10.1093/acprof:oso/9780198564959.001.0001

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# Quaternionic Kähler and Hyperkähler Manifolds

Chapter:
(p.421) Chapter 12 Quaternionic Kähler and Hyperkähler Manifolds
Source:
Sasakian Geometry
Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198564959.003.0013

# 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 self-dual and Einstein orbifolds, Hitchin's construction of SO(3)-invariant orbifold self-dual Einstein metrics on a 4-sphere, McKay's correspondence and Kronheimer's construction of ALE gravitational instantons.

Quaternions were first described by the Sir William Rowan Hamilton1 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 G-structures 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 3-Sasakian 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 4-manifold 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 self-dual or anti-self-dual 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 HHn. 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 4-form 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 3-Sasakian 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 non-compact 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, non-commutative real algebra

$Display mathematics$

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

(12.1.1)
$Display mathematics$

We define the quaternionic conjugate q¯ and the norm |u| by

$Display mathematics$

(p.423) The non-zero 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

(12.1.2)
$Display mathematics$

as a manifold, is just the unit 3-sphere in R 4. Furthermore, we have the group isomorphism f:Sp(1)→ SU(2) explicitly given by

(12.1.3)
$Display mathematics$

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

(12.1.4)
$Display mathematics$

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):

(12.1.5)
$Display mathematics$
where the matrices $I i + = A + ( e i )$ where
(12.1.6)
$Display mathematics$

give a globally defined hypercomplex structure $I + = { I 1 + , I 2 + , I 3 + , }$ 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 $g 0 − w + = g 0 − i 1 w 1 + − i 2 w 2 + − i 3 w 3 + = d u ⊗ d u ¯ ,$ where the multiplication in H is used to interpret the left hand side as an H-valued tensor. This gives the standard Euclidean metric g 0 and the three symplectic forms

(12.1.7)
$Display mathematics$

where (a,b,c) is any cyclic permutation of (1,2,3). We can also introduce an H-valued differential 2-form

(12.1.8)
$Display mathematics$

(p.424) The 2-from du 𝛌 du¯ is purely imaginary as $α ∧ β ¯ = ( − 1 ) p q β ¯ ∧ α ¯ ,$ where p,q are the respective degrees. The Sp(1) part is given by A :Sp(1)→ SO(4) with

(12.1.9)
$Display mathematics$

The matrices $I i − = A − ( e i )$ where

(12.1.10)
$Display mathematics$

give a globally defined hypercomplex structure $I − = { I 1 − , I 2 − , I 3 − } .$ Furthermore, with the Euclidean metric one gets the right hyperkähler structure on H by setting $g 0 + w − = g 0 + i 1 w 1 − + i 2 w 2 − + i 3 w 3 − = d u ¯ ⊗ d u .$ This gives

(12.1.11)
$Display mathematics$

where, as before, we get the three symplectic forms

(12.1.12)
$Display mathematics$

for each cyclic permutation of (1,2,3). These are clearly fundamental 2-forms associated to the complex structures ${ I a − } .$ 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 $( g o , I a − , ω a − )$ 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 $( g o , I a + , ω a + )$. 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 π‎1 = Z. To indicate the difference, we will write the flat metric in this case as g 0 = dθ‎2+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 $H n = { u = ( u 1 , … , u n ) | u j = u j 0 + u j 1 i 1 + u j 2 i 2 + u j 3 i 3 ∈ H , j = 1 , … , n } .$ Here and from now on we will choose to work with the left hyperkähler structure on H n, i.e., with the symplectic 2-forms given by

(12.1.13)
$Display mathematics$

so that

(12.1.14)
$Display mathematics$

(p.425) for any cyclic permutation (a,b,c) of (1,2,3). The corresponding hypercomplex structure is then given by left multiplication by ${ i ¯ 1 , i ¯ 2 , i ¯ 3 } = { − i , − j , − k }$ 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

(12.1.15)
$Display mathematics$

and define

(12.1.16)
$Display mathematics$

Now, Sp(n) × Sp(1) acts on H n by

(12.1.17)
$Display mathematics$

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

(12.1.18)
$Display mathematics$

With such conventions we obtain

(12.1.19)
$Display mathematics$

where

(12.1.20)
$Display mathematics$

Comparing with Example 3.1.11 we recognize (g 0,ω‎1) 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

$Display mathematics$

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

$Display mathematics$

defined with respect to the left action of H * on H n+1 is called the quaternionic projective n-space.

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,

(12.1.21)
$Display mathematics$

(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

(12.1.22)
$Display mathematics$

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

1. (i) $𝒵 = ℙ ℂ ( ℍ n + 1 ) = ( ℍ n + 1 \ { 0 } / ℂ * ) ,$

2. (ii) $S = ℙ ℝ ( ℍ n + 1 ) = ( ℍ n + 1 \ { 0 } / ℝ * ) ,$

3. (iii) $𝒰 = ℙ ℤ 2 ( ℍ n + 1 ) = ( ℍ n + 1 \ { 0 } / ℤ 2 ) .$

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

$Display mathematics$

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 *

1. (i) $ℍ * / ℂ = S 2 → Z → ℍ ℙ n ,$

2. (ii) $ℍ * / ℝ * = S O ( 3 ) → S → ℍ ℙ n ,$

3. (iii) $ℍ * / ℤ 2 → 𝒰 → ℍ ℙ n ,$

4. (iv) $ℂ * / ℝ * = S 1 → S → Z ,$

5. (v) $ℂ * / ℤ 2 → 𝒰 → Z ,$

6. (vi) $ℝ * / ℤ 2 = ℝ + → 𝒰 → S .$

The six bundles of this proposition are the six arrows in the following diagram

(12.1.23)

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 non-zero vectors in H n+1, with uu′ meaning u = u′λ‎, for some λ‎∈H *. Let

(12.1.24)
$Display mathematics$

and consider the maps φ‎j:U j→H n defined by

$Display mathematics$

Now A = {(Ujj)}j=0,…,n is clearly an atlas on H P n giving it a structure of differentiable manifold. Consider the inhomogeneous quaternionic coordinates

$Display mathematics$

on U j and H-valued 1-forms

(12.1.25)
$Display mathematics$

At each x∈H P n the forms $d x i ( j )$ 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 kU j. An easy computation shows that at any xU kU j

(12.1.26)
$Display mathematics$

Note that by convention $x j ( j ) = 1 , d x j ( j ) = 0.$ The Equations (12.1.26) imply that pointwise in U kU j

(12.1.27)
$Display mathematics$

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

(12.1.28)
$Display mathematics$

Note that the above equation defines the metric on H P n as well as the three local 2-forms {ω‎1,ω‎2,ω‎3} which are local sections of a 3-dimensional vector subbundle Q ∩ 𝛌2T*H P n. Using the language of H-valued forms we can introduce ω‎ by

(12.1.29)
$Display mathematics$

with ω‎ = -ω‎¯, so that ω‎ is purely imaginary. The constant c is equal to the so-called quaternionic sectional curvature which generalizes the notion of holomorphic sectional curvature in complex geometry. The quaternionic Kähler 4-form Ω‎ is then given by

(12.1.30)
$Display mathematics$

(p.428) It is real and closed. We have the following

Theorem 12.1.7: The 4-form Ω‎ is parallel. When n > 1 the holonomy group Hol(g 0)⊂ Sp(n)Sp(1). When n = 1 H P 1S 4 and the metric g 0 is simply the metric of constant sectional curvature on S 4 which is self-dual and Einstein.

# 12.2. Quaternionic Kähler Metrics

Let M be a smooth 4n-dimensional manifold (n≥1). Recall from Example 1.4.18 that M is almost quaternionic if there is a 3-dimensional subbundle Q ∩ End(TM) with the property that at each point xM there is a basis of local sections { I 1,I 2,I 3} of Q satisfying the quaternion algebra, i.e.,

(12.2.1)
$Display mathematics$

This definition is equivalent to M admitting a G-structure with G = GL(n,H)Sp(1). Note that any oriented 4-manifold 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.,

(12.2.2)
$Display mathematics$

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* ∩ 𝛌2T*M which associates to each local section I of Q the local non-degenerate 2-form ω‎ defined by

(12.2.3)
$Display mathematics$

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 H1(M,Sp(n)Sp(1)) with coefficients in the sheaf Sp(n)Sp(1) of smooth Sp(n)Sp(1)-valued functions. The short exact sequence

(12.2.4)
$Display mathematics$

gives rise to the homomorphism

(12.2.5)
$Display mathematics$

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

(12.2.6)
$Display mathematics$

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 (M4n,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 8-dimensional 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 8-manifold. If the conditions

$Display mathematics$

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 $< A , B > = 1 2 n T r ( A t B )$ on End(TM). Let { ω‎i }i = 1,2,3 be the basis of 2-forms corresponding under (12.2.3). The associated exterior 4-form

(12.2.7)
$Display mathematics$

is invariant under a change of frame and thus globally defined on M. It is non-degenerate in the sense that Ω‎n 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 4-form 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 (M4n,Q) with n > 1 is 1-integrable if M admits a torsion-free connectionQ preserving the quaternionic structure Q. In such a case (M4n,Q) is called a quaternionic structure on M 4n, and if it has an adapted Riemannian metric g, the triple (M4n,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 1-integrability has been studied by Salamon in [Sal86]. In the 4-dimensional case, there is no obstruction as G = GL(1,H)Sp(1) = R + × SO(4) so that G-structure is equivalent to a choice of orientation and conformal class. In particular, the Levi-Civita connection of any compatible metric preserves the G-structure and has no torsion. But in higher dimensions, there are non-trivial obstructions.

Here we shall be interested in a very special class of quaternionic Hermitian manifolds, namely the case where a torsion-free quaternionic connection ∇Q is also a metric connection. In this case it must be the Levi-Civita connection.

(p.430) Definition 12.2.7: An almost quaternionic Hermitian manifold (M4n,Q,Ω‎,g) of quaternionic dimension n > 1 is called quaternionic Kähler (QK) if ∇Q coincides with the Levi-Civita 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 (M4n,Q,Ω‎,g) n > 1, is quaternionic Kähler if ∇ Ω‎ = 0, where ∇ denotes the Levi-Civita connection of g. In particular, an almost quaternionic Hermitian manifold (M4n,Q,Ω‎,g) n > 1, is quaternionic Kähler if it admits a parallel 4-form which is in the same GL(4n,R)-orbit as Ω‎ at each point xM.

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 (M4n,Q,Ω‎,g) of quaternionic dimension n > 2 whose fundamental 4-form Ω‎ is closed is quaternionic Kähler.

The geometry of almost quaternionic Hermitian 8-manifolds is somewhat richer as there are examples of such spaces for which the fundamental 4-form Ω‎ is closed but not parallel. Swann showed [Swa91] that

Theorem 12.2.10: An almost quaternionic Hermitian 8-manifold is quaternionic Kähler if and only if the fundamental 4-form Ω‎ is closed and the algebraic ideal generated by the subbundle Q* ⊂ 𝛌2T*M is a differential ideal.

We now investigate some curvature properties of quaternionic Kähler manifolds. Let { ω‎1,ω‎2,ω‎3} be a local orthonormal frame field for Q* ⊂ 𝛌2T*M. If Ω‎ is parallel we get

$Display mathematics$

from which it follows that

(12.2.8)
$Display mathematics$

where the α‎ij are 1-forms which satisfy

(12.2.9)
$Display mathematics$

This means in particular that the subspace Γ‎(Q*) ⊂ Γ‎(𝛌2T*M) is preserved by the Levi-Civita connection. The Equations (12.2.8) were considered by Ishihara [Ish74]. The matrix

(12.2.10)
$Display mathematics$

is the connection 1-form with respect to the local frame field { ω‎1,ω‎2,ω‎3}. The curvature of this induced connection represents a component of the Riemann curvature tensor R and is given by

$Display mathematics$

(p.431) Using the facts that $d ω i = ∑ j = 1 3 α i j ∧ ω j and d 2 ω i = 0 ,$ one deduces that

(12.2.11)
$Display mathematics$

for some constant λ‎. Since Q* is an oriented 3-dimensional bundle, there is a canonical identification SkewEnd(Q*) ≅ Q* via the cross-product. Using this identification we can consider F as a map F: 𝛌2TM → SlewEnd (Q*) ≅ Q* ⊂ 𝛌2TM and, as such, Equation (12.2.11) simply states that F=Λπ‎Q*, where π‎Q* denotes pointwise orthogonal projection π‎Q* : 𝛌2TM → Q*. The full Riemann curvature tensor R of a QK manifold viewed as a symmetric endomorphism R: Λ‎2 TM→ Λ‎2 TM (curvature operator), has the property that

(12.2.12)
$Display mathematics$

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 4-dimensional 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 4-dimensional oriented Riemannian geometry. So the problem is caused by a certain low-dimensional 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 Λ‎2(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

(12.2.13)
$Display mathematics$

where $Λ ± 2$ are precisely the ± eigenspaces of the Hodge star operator ⋆. The bundles $Λ ± 2$ and $Λ − 2$ are known as the bundles of self-dual and anti-self-dual 2-forms, respectively. Reversing orientation interchanges the self-dual and anti-self-dual 2-forms. (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 $Λ − 2$ we have

Definition 12.2.11: An oriented Riemannian 4-manifold (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

(12.2.14)
$Display mathematics$

where W ± are the self-dual and anti-self-dual Weyl curvatures, Ric0 is the trace-free 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 4-dimensional oriented Riemannian manifold (M,g) is quaternionic Kähler if and only if it is self-dual (i.e., W = 0) or anti-self-dual (i.e., W + = 0) and Einstein (i.e., Ric0 = 0). More generally, and oriented 4-manifold (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 4-manifold interchanges W + and W . We shall stick with the more usual convention by saying that a QK 4-manifold or orbifold with non-zero scalar curvature is self-dual 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 self-dual and Einstein, not anti-self-dual; whereas, a K3 surface is anti-self-dual 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. NM is called a quaternionic submanifold if for each xN, T x N is an H *-submodule of T x M. Marchiafava observed that a 4-dimensional submanifold of a QK manifold is necessarily self-dual 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 = EH, 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 anti-symmetric products Λ‎2 E, Λ‎2 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

$Display mathematics$

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

$Display mathematics$

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 Hol0(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×s0(3)F

1. (i) $𝒰 ( M ) = S ( M ) × s o ( 3 ) F , w h e r e F = H * / ℤ 2 ,$

2. (ii) $𝒵 ( M ) = S ( M ) × s o ( 3 ) F , w h e r e F = S 2$ 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 “non-linear 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) = τ‎1 I 1+τ‎2 I 2+τ‎3 I 3 as in Section 12.1. Thus, a smooth section s of Z over an open set UM 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 anti-holomorphic 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 4-dimensional 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 self-dual (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 well-known to be quaternionic. In fact, S 4 is self-dual Einstein with one orientation and anti-self-dual Einstein with the other, C P 2 is self-dual Einstein, and K3 is anti-self-dual Einstein with the standard orientation induced by the complex structure. Of course T 4, being (p.434) flat, is self-dual, anti-self-dual and Einstein with either orientation. There has been much work on self-dual and anti-self-dual structures on 4-manifolds 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 self-dual structures on the connected sums k C P 2 for k > 1 [Poo86,DF89,Flo91,LeB91b,PP95,Joy95] as well as other simply connected 4-manifolds that are neither Einstein nor complex. Second, Taubes [Tau92] has proven a type of stability theorem that says that given any smooth oriented 4-manifold M then M#k C P 2 admits a self-dual 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:

$Display mathematics$

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 one-to-one 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 = Gr2(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 (M4n,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:MM depending smoothly on t such that gt = f*tg.

This result is far from true for hyperkähler or negative QK manifolds. Moduli in the hyperkähler case is well-known, 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 b2(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 Gr2(C n+2). So we can conclude from Lemma 12.3.4 that b 2(M) = 0 unless M = Gr2(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

1. (i) π‎1(M) = 0;

2. (ii) $π 2 ( M ) = { 0 i f M i s i s o m e t r i c t o H P n Z i f M i s i s o m e t r i c t o Gr 2 ( C n + 2 ) , f i n i t e c o n t a i n i n g Z 2 o t h e r w i s e ,$

3. (iii) b 2k+1(M) = 0 for all k≥0;

4. (iv) b 4i(M) > 0 for 0≤ in;

5. (v) b 2i(M)−b 2i−4(M)≥0 for 2≤ in;

6. (vi) $∑ r = 0 n − 1 [ 6 r ( n − 1 − r ) − ( n − 1 ) ( n − 3 ) ] b 2 r ( M ) = 1 2 n ( n − 1 ) b 2 n ( M ) .$

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 S2→Z(M)→M. For (ii) we recall in the proof of Theorem 12.3.3 b 2(M) = 0 unless M = Gr2(C n+2) in which case π‎2(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 2-torsion of H 2(M,Z), and this is non-vanishing when the Marchiafava–Romani class ∈ is non-vanishing. 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 non-degeneracy of the closed 4-form 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 β‎2i = b 2ib 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 (M4n,Q,Ω‎,g) be a compact positive QK manifold. Then M is a symmetric space if

1. (i) n≤3,

2. (ii) n = 4 and b 4 = 1.

PROOF. The statement in (i) dates back to the Hitchin's proof that all compact self-dual 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 Fubini-Study 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 12-manifold must be at least 6 and the dimension of the isometry group of a positive QK 16-manifold 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 non-spin manifolds with finite π‎2(M) and smooth circle actions Herrera and Herrera prove that

Lemma 12.3.9: Let M be a positive QK 12-manifold which is not Gr2(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 16-dimensional 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 non-trivial 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 (M4n,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 well-known 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 (M4n,Q,Ω‎,g) be a positive QK manifold. Assume f = (f 1,f 2): NM × M, where N = N 1 × N 2 and f i: N iM 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

1. (i) If mn, then f −1(Δ‎) is non-empty.

2. (ii) If mn+1, then f −1(Δ‎) is connected.

3. (p.438)
4. (iii) If f is an embedding, then for imn there is a natural isomorphism, π‎i(N 1,N 1N 2)→ π‎i(M,N 2) and a surjection for i = mn+1.

REMARK 12.3.2: The study of homogeneous negative QK manifold is more delicate. There are of course the non-compact duals of the Wolf spaces. Alekseevsky showed that there are also non-symmetric 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 σ‎-models 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 4-dimensional 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 Γ‎p(Q*) ≡ Γ‎(𝛌pT*M⊗Q*) of smooth exterior p-forms on M with values in the bundle Q*. The connection given on Q* induces a “\sl de Rham" sequence

(12.4.1)
$Display mathematics$

such that

(12.4.2)
$Display mathematics$

for f ∈ Γ‎0(Q*).

Consider now the Lie group

$Display mathematics$

and its Lie algebra

$Display mathematics$

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 (M4n,Q,Ω‎,g) be a QK manifold of non-zero scalar curvature. Then aut(M4n,Q,Ω‎,g). It follows that any one parameter subgroup H⊂𝔦𝔖𝔬𝔪(M 4n,g) is also a subgroup of ⌟(M4n,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 1-parameter subgroup and these may not preserve the quaternionic 4-from Ω‎. To each V∈ 𝔦𝔰𝔬𝔪(M,g) we associate the

$Display mathematics$

defined in terms of a local frame ω‎1,ω‎2,ω‎3 by

(12.4.3)
$Display mathematics$

Clearly Θ‎V 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 ∈ Γ‎0(Q*) such that

(12.4.4)
$Display mathematics$

In fact, under the canonical bundle isometry θ‎ : SkewEnd(Q*) → Q*, μ‎ is given explicitly by the formula

(12.4.5)
$Display mathematics$

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 Vμ‎ transforms naturally under the group of automorphisms, specifically for g ∈ ⌟ut (M,Q,Ω‎,g) and V∈ 𝔦𝔰𝔬𝔪(M,g) we have

(12.4.6)
$Display mathematics$

where g*(μ‎)(x)=gη‎(μ‎(g-1(x))) and where gη‎ denotes the map induced by g on the bundle Q* ∩ 𝛌2TM. Note also that g * V = Adg(V). Hence, (12.4.6) means that the diagram

(12.4.7)

commutes.

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 G-equivariant. Since the action of G in the bundle g* ⊗ Q* is linear on the fibers, it preserves the zero section. Consequently the set

(12.4.8)
$Display mathematics$

is G-invariant. 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 non-zero scalar curvature. Let G⊂ 𝔦𝔰𝔬𝔪(M,g) be a compact connected subgroup with moment map μ‎. Let N0 denote the G-invariant subset of N = { x∈ M: μ‎(x) = 0}, where μ‎ intersects the zero section transversally and where G acts locally freely. Then Mˆ = N0/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 ≅ Tk⊂ 𝔦𝔰𝔬𝔪(M,g) is a k-torus subgroup generated by vector fields Vk∈ 𝔦𝔰𝔬𝔪(M,g). If V1∧…∧ Vk is a k-plane 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:

$Display mathematics$

where λ‎ is a complex unit. Recall that the projectivization is by right multiplication uu q as described in Section 12.1. In a local trivialization we can identify the local frame ω‎1,ω‎2,ω‎3 with the imaginary quaternions i,j,k, respectively, in which case the moment map μ‎ for the S 1-action can be written as

(12.4.9)
$Display mathematics$

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 uA 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 = Gr2(C n+1).

This can easily be generalized to the case of a weighted circle action

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(p.441) in which case the moment map becomes

(12.4.10)
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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 4-dimensional it is automatically self-dual and Einstein. The only complete positive QK manifolds in dimension 4 are H P 1S 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 HG 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 HM HM 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

(12.4.11)
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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 HG be a subgroup so that M H is not-empty. Then Mˆ is a smooth manifold.

However, not all the pieces MˆH have QK structures. Their geometry depends on the way H acts on Q. To be more precise, let xM 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 3-dimensional representation. If x,yM H are on the same path-component then parallel transport along any path joining x to y defines an H-equivariant isomorphism T x MT 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 L-orbits. There are four possibilities [DS97a]:

• φ‎(H) is trivial ⇒ MˆH is a QK manifold,

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

• φ‎(H) is finite but not cyclic ⇒ M H is totally real in M (MˆH is “real"),

• φ‎(H) = SO(2) or SO(3) ⇒ MˆH 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

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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 so-called BPS sates in 5-dimensional 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

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where the sum ranges over the set of connected components of the fixed point set, λ‎F is the index of F, and Pˆt is the equivariant Poincaré polynomial for the equivariant cohomology with coefficients in Q. Battaglia proves

Theorem 12.4.9: Let (M4n,Q,Ω‎,g) be a positive QK manifold acted on isometrically by S1. Then the non-degenerate Morse function f = ‖μ‎2 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 M4n be a positive QK manifold acted on isometrically by S1. 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 (M4n,Q,Ω‎,g) be a positive QK manifold acted on isometrically by S1. Suppose S1 acts freely on N = μ‎ −1(0). Then M4n is homotopic to H P n with the quotient Mˆ=Gr2(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 V-bundles) 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 k-dimensional Abelian subgroup of the isometry group. Such a reduction is associated to a choice

(12.5.1)
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where Sp(n+1) = 𝔦𝔰𝔬𝔪(H P n,g can) and 𝓣max = 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 non-zero integer vectors {θ‎ 1, … ,θ‎ n+1} generating R n+1−k. These can be put together as a matrix Θ‎∈Mn+1-k,n(Z). Dually, we can consider a matrix Ω‎ ∈ Mk,n+1(Z) whose column vectors {ω‎ 1, … ,ω‎ n+1} generate R k. This gives the exact sequence of Lie algebras

(12.5.2)
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and its dual

(12.5.3)
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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 kT n+1

(12.5.4)
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(p.444) where $( a i j )$ Now, with each Ω‎ we associate H acting on H P n, the QK moment map μ‎Ω‎, and the zero level set $N ( Ω ) = μ Ω − 1 ( 0 ) ⊂ H P n .$ 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 self-dual 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 self-dual 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 3-Sasakian manifolds.

We now specialize to the case of compact 4-dimensional QK orbifolds (M,g), i.e., 4-orbifolds with self-dual 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

1. (i) there is smooth orbifold equivalence

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2. (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;

3. (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 3-Sasakian 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 5-sphere) 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 self-dual 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 1p 2p 3. Then the self-dual Einstein metric g(p) is of positive sectional curvature if and only if

(12.5.5)
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where σ‎3 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 pseudo-Riemannian 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 self-dual Einstein metric on O(P) = O(p,q,q) has positive sectional curvature if $p 2 < q 2 2 < 2 p 2 2$ and then at every point x ∈ O(p,q,q) the sectional curvature is k-pinched with 0 < k < 1 and

$Display mathematics$

Clearly, $k = 1 4$ when pq. Furthermore, $k = 1 4$ if and only if p = q = 1 in which case O ∼ C P 2 is symmetric.

When Ω‎ ∈ Mn-1,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

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on some open subset of the half-space ρ‎ > 0, and consider the metric g(ρ‎,η‎,φ‎,ψ‎) given by

(12.5.6)
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(p.446) where $α = p d φ and β = ( d ψ + η d φ ) / p .$ Then

1. (i) On the open set where $F 2 > 4 p 2 ( F p 2 + F η 2 )$, g is a self-dual Einstein metric of positive scalar curvature, whereas on the open set where $F 2 > 4 p 2 ( F p 2 + F η 2 )$, −g is a self-dual Einstein metric of negative scalar curvature.

2. (ii) Any self-dual Einstein metric of non-zero 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" 3-Sasakian 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 self-dual Einstein 4-orbifold of positive scalar curvature whose isometry group contains a 2-torus. 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

(12.5.7)
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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, gSO(3) and the unit sphere M = S 4 in V can be described as matrices in V whose eigenvalues {λ‎1,λ‎2,λ‎3} satisfy

(12.5.8)
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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

(12.5.9)

with the generic orbit SO(3)/D, where D = Z 2 × Z 2SO(3) is the subgroup of diagonal matrices which is the stabilizer of any generic point. The two degenerate orbits B ± = SO(3)/K ±≃ R P 2 are both Veronese surfaces in S 4 that correspond to (p.447) the subset of matrices with two equal eigenvalues. If two eigenvalues are equal, then they must be equal to $± 1 6 ,$ and the two signs correspond to the two orbits B ± and the subgroups K ± = O(2). The diagram (12.5.9) gives rise to the decomposition
(12.5.10)
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Explicitly, we can parameterize the conic (12.5.8) by observing that (λ‎1−λ‎2)2+3(λ‎1+λ‎2)2 = 2 so that

(12.5.11)
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where α‎2+β‎2 = 1. Choose the standard rational parameterization

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Note that t = 0 gives $λ 1 = λ 2 = 1 6$ and t = +∞ gives the other degenerate orbit $λ 1 = λ 2 = 1 6$ so that we have an explicit diffeomorphism S 4∖{B ,B +}≃ (0,+∞) × SO(3)/D given by (t,g)↦ g −1Δ‎(t)g, where gSO(3) and Δ‎(t) = diag(λ‎1(t),λ‎2(t),λ‎3(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

(12.5.12)
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where {σ‎1,σ‎2,σ‎3} is the basis of Maurer–Cartan invariant one-forms dual to the standard basis of the Lie algebra 𝔰𝔬(3). The equations for the most general self-dual Einstein metric with non-zero scalar curvature Λ‎ and in the diagonal form (12.5.12) has been derived by Tod [Tod94]. It follows that

(12.5.13)
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where {F 1(x),F 2(x),F 3(x)} satisfy the following first-order system of ODEs

(12.5.14)
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and the conformal factor

(12.5.15)
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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 second-order ODE: the Painlevé VI equation

(12.5.16)
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where (α‎,β‎,γ‎,δ‎) = (1/8,−1/8,1/8,3/8). One can define an auxiliary variable z by
(12.5.17)
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(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):

(12.5.18)
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In a series of papers Hitchin analyzed the Painlevé VI equation giving an algebro-geometric 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.

1. (i) The metric g k is a positive definite self-dual Einstein of positive scalar curvature for all 1 < x < ∞.

2. (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/k-2 around B + = R P 2.

Hence, for any integer k≥3 the metric g k can be interpreted as an orbifold metric on Ok = B∧((1,∞)×SO(3)/D)∧B+∈S4 where (Ok,gk) is a compact self-dual 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 Ok and the twistor space Z(Ok) 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)

(12.5.19)
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Explicit computation shows that for k = 3 the metric is given by the following solution of the Painlevé VI equation

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The components of g 3 can easily be calculated with f(s;3) = 3(1+s+s 2)−2 and

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In the arc length coordinates this metric can be easily transformed to

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which shows that g 3 is the standard metric (p.449) on S 4 written in triaxial form. Hence, the orbifold (O3,g3) is actually non-singular 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

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so that

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In an arc length parameterization g 4 becomes

(12.5.20)
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which is indeed locally the Fubini-Study metric on C P 2. The orbifold (O4,gk) has $π 1 o r b = ℤ 2$ 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

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This yields

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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

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This yields

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This metric g 8 of Equation (12.5.19) becomes positive definite for the range $2 − 1 < s < .$

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 self-dual Einstein 4-orbifolds. It seems plausible that these are the only cohomogeneity 1 compact positive self-dual Einstein 4-orbifolds, but a proof is lacking so far.

OPEN PROBLEM 12.5.1: Classify all compact positive self-dual Einstein 4-orbifolds 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 self-dual Einstein 4-orbifolds with at least a 2-dimensional isometry group.

The problem of finding examples of compact positive self-dual Einstein 4-orbifolds 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 4-orbifolds 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

(12.5.21)
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where ug·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 self-dual 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 non-Abelian we get families of positive self-dual 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 4-planes in R n. The isometry group SO(n) of the symmetric space $G r 4 + ( ℝ n )$ contains a torus and one can examine possible QK reductions of $G r 4 + ( ℝ n )$. 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 $G r 4 + ( ℝ n )$ for which the reduced space is 4-dimensional: (i) T 3-reduction of $G r 4 + ( ℝ 8 )$ and (ii) T 2-reduction of $G r 4 + ( ℝ 7 )$ (see Proposition 13.9.2). Both lead to non-trivial examples of positive self-dual 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 $Θ 2 , 3 1 ∈ M 2 , 3 ( ℤ ) a n d Θ 3 , 4 0 ∈ M 3 , 4 ( ℤ )$ describe the choices of T 2T 3SO(7) and T 3T 4SO(8), respectively. Let $O ( Θ 2 , 3 1 ) and O ( Θ 3 , 4 0 )$ denote the corresponding QK reductions of $G r 4 + ( ℝ 7 ) a n d G r 4 + ( ℝ 8 )$ We have:

1. (i) If all three 2 × 2 minor determinants of $Θ 2 , 3 1$ are non-zero then $O ( Θ 2 , 3 1 )$ is a compact 4-orbifold.

2. (ii) If all four 3 × 3 minor determinants of $Θ 3 , 4 0$ are non-zero then $O ( Θ 3 , 4 0 )$

In both cases we get compact orbifold families of positive self-dual Einstein metrics with a 1-dimensional isometry group.

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 non-linear PDE

(12.5.22)
$Display mathematics$

This equation was first described in [BF82] as providing solutions to the self-dual 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) self-dual Einstein equations admitting one Killing vector field amounts to solving either the well-known 3-dimensional Laplace equation or Equation (12.5.22). Those Killing fields that led to the 3-dimensional Laplace equation were called translational Killing fields, whereas, those leading to Equation (12.5.22) were called rotational. The translational Killing fields have self-dual covariant derivative and are well understood [TW79]. For example, they give rise to the well-known Gibbons-Hawking 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 self-dual 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 infinite-dimensional 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 self-dual Einstein equations with non-zero scalar curvature can be reduced to solving Equation (12.5.22). This equation as with the full self-dual or anti-self-dual 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 4-dimensional Einstein metrics appear as Riemannian metrics adapted to some other geometric structure, self-dual (or anti-self-dual) 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)

Figure 12.1 Tod's Venn diagram of special metrics in dimension 4.

orbibundle. In the area labelled C are the generic Kähler–Einstein 4-manifolds that were treated in Chapter 5. In the regions labelled A and D the complex structure chooses an orientation which breaks the equivalence between being self-dual or anti-self-dual. Thus, in both region A and region D one must consider the two cases separately. The two cases for region A consists of self-dual Kähler–Einstein, and anti-self-dual Kähler Einstein. In the category of compact manifolds the former consists of C P 2 with the Fubini-Study metric and compact quotients of Hermitian hyperbolic space with the Bergman metric [Kod87,Boy88b,KS93b,ADM96], whereas, the later consists only of flat tori, K3 and Enriques surfaces, which are all locally hyperkähler and have zero scalar curvature. Many more examples of both self-dual Kähler–Einstein and anti-self-dual Kähler–Einstein appear when one allows orbifold singularities [Bry01,ACG06], for example, in the hyperkähler case Reid's list of 95 singular K3 given in Appendix B.1 and discussed in Section 5.4.2 occur. Likewise, complete metrics on non-compact 4-manifolds are plentiful in all regions of the above diagram. We discuss hyperkähler geometry more fully in Sections 12.7 through 12.10 with many 4-dimensional examples. In region D many researchers [Che78,Bou81,Der83,Ito84,Bry01] have obtained results concerning self-dual Kähler manifolds. Anti-self-dual Kähler metrics automatically have zero scalar curvature [Der83]. Moreover, which possible compact complex surfaces can admit such metrics have been delineated [Boy86]. There has been much work [LeB91d,LS93,KP95,Tod95,Dan96,KLP97,RS05,DF06] within the last 15 years or so in proving the existence of scalar flat Kähler metrics on 4-manifolds. Finally region B are the self-dual metrics which are not Kähler. Here we are interested mainly in the case of positive scalar curvature, since only these admit SO(3) or SU(2)-orbibundles whose total space has a 3-Sasakian structure. These have been discussed in detail above. The world of negative scalar curvature self-dual Einstein metrics is fascinating with a spectacular abundance of complete metrics on non-compact manifolds [Ped86,Gal87b,Gal91,LeB91c,Hit95b,Biq00,Biq02,CP02,CS04,BCGP05,Duc06]. The subject is worthy of a separate book. We concentrate on the positive QK manifolds here as they form the base space of Konishi orbibundles of the 3-Sasakian spaces considered in the next chapter. However, negative QK manifolds are also related to 3-Sasakian (and not just semi-Riemannian 3-Sasakian geometry) as first observed by Biquard [Biq99].

# (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 ${ e a } a = 1 3$ 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

(12.6.1)
$Display mathematics$

where 〈τ‎,τ‎′〉 is the standard inner product in R 3 and τ‎ × τ‎′ is the cross-product. 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

(12.6.2)
$Display mathematics$

for any τ‎S 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), τ‎S 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 non-trivial torsion. But on a hypercomplex manifold this unique connection is torsion-free. So an alternative definition of a hypercomplex structure is that it is an almost hyp- ercomplex structure such that the Obata connection is torsion-free. In the lowest dimension compact hyperhermitian 4-manifolds were classified by Boyer [Boy88a] who proved

Theorem 12.6.4: Let (M,I,g) be a compact hyperhermitian 4-manifold. Then (M,I,g) is conformally equivalent to one of the following

1. (i) a 4-torus with its flat metric,

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

3. (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 3-Sasakian manifold M [BGM98a] (See the next chapter for a description of 3-Sasakian structures). For example, any trivial bundle S 1 × M then admits locally conformally hyperkähler structures that are automatically hypercomplex. However, non-trivial circle bundles give more interesting results. For example, large families of hypercomplex structures were shown [BGM94b,BGM96a] to exist on the complex Stiefel manifolds $V n,2 ℂ$ of 2-frames 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

(12.6.3)
$Display mathematics$

Since a hypercomplex structure is a G-structure 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)}τ‎S 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 H-equivariant map μ‎ = i 1μ‎1+i 2μ‎2+i 3μ‎3:M→𝔥*⊗𝔰𝔭(1) satisfying both of the following conditions:

1. (i) I 1 dμ‎1 = I 2 dμ‎2 = I 3 dμ‎3, where I a acts on sections Γ‎(T * M⊗𝔥*).

2. (ii) For any non-zero element ζ‎∈𝔥 and its induced vector field X ζ‎∈Γ‎(TM) I 1 dμ‎1(X ζ‎)≠ 0 on M.

Note that the condition (i) of Definition 12.6.6 is equivalent to requiring that the complex valued function μ‎a+iμ‎b 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 = ζ‎1 i 1+ζ‎2 i 2+ζ‎3 i 3∈𝔥*⊗𝔰𝔭(1), where all three ζ‎i are in the center of 𝔥*. Suppose the H-action on N ζ‎ = μ‎ −1(ζ‎) has only finite isotropy groups and Mˆ(Ξ‎) = NΞ‎/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) = S2×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

(12.6.4)

which gives a correspondence: points τ‎S 2≃ C P 1 correspond to complex structures I(τ‎) on M in the given hypercomplex structure I(τ‎) points xM correspond to rational curves in Z(M) with normal bundle 2nO(1) called twistor lines.

# 12.7. Hyperkähler Manifolds

Given a an almost hyperhermitian manifold (M,I,g) we can use the metric to define the 2-forms

(12.7.1)
$Display mathematics$

In particular, given a hypercomplex structure I and choosing the basis {I 1,I 2,I 3} we get the three fundamental 2-forms {ω‎1,ω‎2,ω‎3} 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 2-forms are closed and it is called hyperkähler (HK) if the associated 2-forms are parallel with respect to the Levi-Civita 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 (M4n,I,g) be an almost hyperhermitian manifold with the fundamental 2-forms ω‎a(X,Y) = g(I a X,Y), a = 1,2,3. Then the following conditions are equivalent:

1. (i) (M4n,I,g) is hyperkähler,

2. (ii) (M4n,I,g) is almost hyperkähler,

3. (iii) (M4n,I,g) is 1-integrable.

4. (iv)I 1 = ∇ I 2 = ∇ I 3 = 0,

5. (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 4-form Ω‎ = ∑aω‎a∧ω‎a 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)

Figure 12.2 Quaternionic geometries in dimension ≥ 8.

of the Riemann curvature in (12.2.14) we see that the only non-zero component is W and such manifolds are also sometimes called half-flat.

Figure 12.3 Quaternionic 4-manifolds.

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 (M4n,I,g) be an HK manifold and consider the Kähler structure (I1,g,ω‎1). The complex 2-form ω‎+ = ω‎2+iω‎3 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 2-form 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 $X = ∂ ∂ z ¯ j$ we compute for any vector field Y

$Display mathematics$

(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 = ω‎2iω‎3. Proposition 12.7.3 can easily be generalized to an arbitrary oriented orthonormal 3-frame {τ‎ 1, τ‎ 2,τ‎ 3}, namely

Proposition 12.7.4: Let (M4n,I,g) be an HK manifold and consider the Kähler structure (I(τ‎ 1),g,ω‎(τ‎ 1)). The complex 2-form 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 1S 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 2-form 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 = ∑aτ‎a e a. These standard coordinates are related to the complex affine coordinate t∈ C ⊂ C P 1S 2 by [HKLR87]

$Display mathematics$

Then the complex structure IZ on Z(M) as an endomorphism of the tangent space T(x,t)Z(M)=TxM ⊕TtS2 becomes

(12.7.2)
$Display mathematics$

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 $s = 1 t$ centered about (−1,0,0). Now the twisted holomorphic 2-form ϖ‎ on 𝒵(M) is written as

(12.7.3)
$Display mathematics$

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

1. (i) a holomorphic projection p :𝒵 → CP 1

2. (ii) a holomorphic section ϖ‎ of p*O(2)⊗𝛌2TF* which restricts to a holomorphic symplectic form on the fibers of p.

3. (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 pseudo-HK 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 so-called 4-dimensional N = 2 globally supersymmetric σ‎-model theories. It is known that target manifolds of such σ‎-models 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 ω‎a, a = 1,2,3. Suppose the G-action is Hamiltonian with respect to each symplectic form ω‎a. We call such an action hyperhamiltonian.

Definition 12.8.1: An HK manifold (M,I,g) together with an effective hyperhamiltonian G-action is called a hyperhamiltonian G-manifold.

As discussed in Section 8.4.1 such an action gives rise to three G-equivariant symplectic moment maps μ‎a:M→ ≫ *. We can assemble these maps together to get

Definition 12.8.2: Let (M,I,g) be a hyperhamiltonian G-manifold. The map μ‎ = (μ‎1,μ‎2,μ‎3) = i 1μ‎1+i 2μ‎2+i 3μ‎3

(12.8.1)
$Display mathematics$

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 = (λ‎1,λ‎2λ‎3)∈ ≫ *⊗𝔰𝔭(1) be any element fixed by the co-adjoint action of G on its Lie co-algebra*. 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,ω‎1,g) is a Kähler manifold. The HK reduction can be seen as a two step process: First, we consider the function

(12.8.2)
$Display mathematics$

(p.459) which is easily seen to be holomorphic on (M,I 1,g). Thus the set $N + = μ + − 1 ( 0 )$ 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 H-invariant open neighborhood N'+ such that μ‎1 : N+ → g*. The action of G restricts to N + with the Kähler moment map μ‎1: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 Î1 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 {Î123}. 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 G-manifold 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 HG. Then N H = μ‎ −1(0)∩ M (H) is a manifold and the orbit space

$Display mathematics$

has a natural HK structure. Consequently, the reduced space Mˆ = μ‎-1(0)/G is a disjoint union of HK manifolds

$Display mathematics$

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 = (μ‎1,μ‎+) to describe the moment map in complex coordinates on (M,I 1,ω‎1,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 1Sp(n) given by left multiplication with g(t) = e it. The moment map for this action is

(12.8.3)
$Display mathematics$

In complex charts we get

(12.8.4)
$Display mathematics$

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 $N + = μ + − 1 ( − 1 ) ⊂ N$ which are described as

(12.8.5)
$Display mathematics$
(12.8.6)
$Display mathematics$

Let Mˆ=μ‎-1(-i)/S1. We first want to identify Mˆ with the Kähler reduction of N + (or a G-invariant open set N'+ ∩N+). For an appropriate choice of G=S1. its Kähler reduction N + will be an algebraic quotient of G = S1 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

(12.8.7)
$Display mathematics$

where N'+ = {(W,z) ∈ C n × C n | Σ‎jwjzj= 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 non-flat example of a complete HK manifold. An N = 2 supersymmetric σ‎-model 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: Taub-NUT Metrics. This example involves a non-compact hyperhamiltonian group action of G = R on H × H n≃H n+1 defined for any p = (p 1, … ,p n)∈R n by

(12.8.8)
$Display mathematics$

with the moment map

(12.8.9)
$Display mathematics$

We can always shift to the zero-level set and then the moment map equations can be “solved" by writing

(12.8.10)
$Display mathematics$

This action is free and proper on H × H n. In particular it is free and proper on μ‎-1 p(0) = NP. 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(u0)=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 1-parameter 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 1-parameter family of HK deformations of the Euclidean metric. The metric g(Λ‎) (p.461) is called the Taub-NUT 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 so-called “Wick rotation" changing tit 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 Taub-NUT 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 (M4n,I,g) is locally toric if it admits n commuting hyperholomorphic Killing vector fields, linearly independent at each point xM 4n, i.e., locally M admits a free action of R n by hyperholomorphic isometries. Furthermore, (M4n,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 so-called Legendre transform method developed by Lindström and Roček associates a 4n-dimensional HK metric with n commuting hyperholomorphic Killing vectors to every real-valued function F on an open subset u ∩ R 3 ⊗ R n which is harmonic on any affine 3-dimensional 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 real-valued function

(12.9.1)
$Display mathematics$

where the x i are determined by

(12.9.2)
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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

(12.9.3)
$Display mathematics$

One can show that, relative to local coordinates (y,x,z), the HK metric takes the form [PP88]

(12.9.4)
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where $Φ i j = 1 4 F x i x j and A j = − 1 2 ∑ l ( F x i z ¯ l d z ¯ l − F x i z l d z l ) .$ The functions Φ‎ij are also polyharmonic. The n × n matrix [Φ‎ij] locally determines the hyperkähler and hyperhamiltonian structure. When n = 1 this is the well-known Gibbons–Hawking Ansatz. Renaming (y 1,x 1,z 1) = (t,x 1,x 2+ix 3) = (t,x), A 1 = α‎· d x, Φ‎11 = V we can write the metric in a more familiar form

(12.9.5)
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where grad V = curlα‎. In particular, V is a solution of the Laplace equation so that we can write

(12.9.6)
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When δ‎ = 0 these metrics are called k-center gravitational multi-instantons. 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 so-called k-center Taub-NUT gravitational multi-instantons with k = 1 corresponding to the Euclidean Taub-NUT 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

(12.9.7)
$Display mathematics$

and, consequently, Φ‎ ≡ 1, while in the second case

(12.9.8)
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where r2 = x2 + 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 Taub-NUT 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 4n-manifold 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 G-manifolds which are the same in the sense of the following definition.

Definition 12.9.3: Let M,Mbe two hyperhamiltonian G-manifolds and let μ‎,μ‎′ be the chosen moment maps. We say that M and Mare isomorphic as hyperkähler G-manifolds, if there is a hyperholomorphic G-equivariant isometry f:MMsuch that μ‎ = μ‎′○ f.

Second, there is a natural relation between the flat metric and the Taub-NUT 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 np. A Taub-NUT 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 mLie (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 G-equivariant isometries and Taub-NUT deformations one can restrict attention to the following HK quotients studied in detail in [BD00]. We consider a T kT m+1Sp(m+1) action on H m+1 defined via the exact sequence $0 → h → ω ℝ m + 1 → θ ℝ m + 1 − 1 → 0 ,$ 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 ∈Mk,m+1(Z) or, alternatively, Θ‎ ∈ Mn,m+1(Z). We can consider the HK moment map for this action μ‎Ω‎:H m+1→1.5R 3 × R k

(12.9.9)
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where

(12.9.10)
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and $λ α = λ α 1 i + λ α 2 j + λ α 3 k$ 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.,

$Display mathematics$

Consider now the HK quotient space, i.e.,

(12.9.11)
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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:

(12.9.12)
$Display mathematics$

α‎ = 1, … ,m+1 and θ‎ α‎∈R n is the α‎th column of Θ‎.

Bielawski and Dancer prove the following theorem [BD00]:

Theorem 12.9.5: Suppose the column vectors θ‎ α‎ of Θ‎ ∈ Mn,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

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

2. (ii) whenever some n of the Hk1,…,Hkn have non-empty intersection, then the set {θ‎k1,…,θ‎kn} 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

$Display mathematics$
i.e., $H α 1$ in R n defined by restricting H α‎ to the first coordinate in the R 3 factor. These hyperplanes divide R n into a finite family of closed convex polyhedra. Let A be the polyhedral complex consisting of all faces of all dimensions of these polyhedra and let C be the polyhedral complex consisting of all bounded polyhedra in A. We have the following

Theorem 12.9.6: Let M = M(Θ‎,λ‎) be a toric HK orbifold of dimension 4n.

1. (i) Then M is simply connected and H j(M,Q) = 0 for j odd;

2. (ii) $b 2 p ( M ) = ∑ i = p n ( − 1 ) i − p ( i p ) d i ,$ where the integer d i denotes the number of i-dimensional elements of the complex C

3. (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 np. Then

1. (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, mn, by T dn × R m;

2. (ii) if M is simply connected and p > 0, then M is isomorphic, as a hyperhamiltonian G-manifold, to the product of a flat H p and a 4(np)-dimensional manifold described in part (i);

3. (iii) if M is not simply connected, then M is the product of a flat (S 1 × R 3)l, 1≤ ln, and a 4(nl)-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 4-dimensional complete HK manifold with a non-trivial hyperhamiltonian vector field. If b 2(M) = k > 0, then M is isometric either to an ALE-space of type A k (i.e., a multi-Eguchi–Hanson space) or to its Taub-NUT-like 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 Taub-NUT 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 nmanifolds of finite topological type and dimension 4n are classified, up to Taub-NUT deformations, by arrangements of codimension 3 affine subspaces H α‎ in R 3⊗ R n defined as in (12.9.12) by

$Display mathematics$
for some finite collection of vectors θ‎ α‎ in R n and scalars $λ α i , i = 1 , 2 , 3 ,$ such that, for any p∈R 3⊗R n, the set {θ‎ α‎; pH α‎} is part of a Z-basis of Z n.

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 3-Sasakian 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 2-dimensional representation of SU(2). In particular, H gives a (p.466) representation of each finite subgroup Γ‎⊂ SU(2). Let {ρ‎0,ρ‎1, … ,ρ‎r} be the set of irreducible representations of Γ‎ with ρ‎0 the trivial representation. Then

(12.10.1)
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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 A-D-E. If C˜ is the Cartan matrix of the extended Dynkin diagram then A + C˜ = 21r+1. Let {θ‎1, … ,θ‎r} be the simple roots of the root system of the associated Lie algebra. Let θ‎0 be the negative of the highest root. McKay further noticed that

$Display mathematics$

where n i is the dimension of the representation ρ‎i [McK80,McK81]. The regular representation ρ‎ of Γ‎ decomposes as

$Display mathematics$

Separately for each Γ‎, all of this information can now be encoded in a “labelled" extended simply-laced Dynkin graph.

The vertices of the diagram correspond to the irreducible representations ρ‎i 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 ρ‎0. In particular, McKay’s observations show that

$Display mathematics$

(p.467) EXAMPLE 12.10.1: Consider the example of Γ‎ = Z n+1. Let τ‎n+1 = 1. Any irreducible representation of Z n+1 is 1-dimensional and ρ‎i(x) = τ‎i x, x∈C. Now, the 2-dimensional representation H gives H(τ‎)·(x,y) = (τ‎x,τ‎Y¯). Clearly, H = ρ‎1⊕ρ‎n so that ρ‎iH = ρ‎i+1⊕ρ‎i−1. The vertex ρ‎i of the McKay graph is joined by an edge to the vertices ρ‎i+1 and ρ‎i−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 group4 is generated by two matrices

(12.10.2)
$Display mathematics$

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,± ijk}. D*4 has three non-trivial 1-dimensional irreducible representation {ρ‎1,ρ‎2,ρ‎3} and one 2-dimensional representation ρ‎4. “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 A-D-E 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 self-intersection −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 XM and a map π‎:MX→1.2H/Γ‎ that is a diffeomorphism between MX and {x∈ H/Γ‎ |r(x) > R} for some R > 0 such that

$Display mathematics$

as r→∞ and k≥0, where ∇ is the Levi-Civita 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 Mni,nj(H) of quaternionic matrices. For each Γ‎, we define

(12.10.3)
$Display mathematics$

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

(12.10.4)
$Display mathematics$

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 $μ Γ − 1 ( ξ )$ 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 ξ‎D 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 $μ Γ − 1 ( ξ )$ 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 non-Abelian 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 one-to-one correspondence with all compact 3-Sasakian 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 self-dual Yang–Mills connections over a Riemann surface can be described as an infinite-dimensional Kähler quotient [AB83]. In such a picture one first equips the space of all connections with a structure of an infinite-dimensional Kähler manifold. On it acts an infinite-dimensional group of gauge transformations and the image of the moment map for this action is given by the self-duality equations. Hence, the Kähler quotient, which turns out to be finite dimensional, is the space of self-dual Yang–Mills connections modulo the gauge equivalence. When a Riemann surface is replaced by a 4-dimensional manifold (such as R 4 or S 4, for example) the space of connections can be given the structure of an infinite-dimensional HK manifold. Since the HK moment map for the gauge group action produces the self-duality equations, the HK quotient can be naturally identified with the moduli space of instantons. This picture explains why the k-instanton moduli spaces over S 4 carry a natural HK structure [AHDM78]. Yang–Mills connections on other 4-manifolds 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 3-manifold. 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 $M ˜ k 0$ of the moduli space of charge k monopoles with a fixed center is a complete HK manifold of dimension 4(k−1) [AH88]. The 4-dimensional case of $M ˜ k 0$ gives the famous SO(3)-invariant Atiyah–Hitchin monopole metric [AH85]. Like in the Taub-NUT case the SO(3)-action acts by rotations on the 2-sphere 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 so-called 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. Taub-NUT 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 Taub-NUT 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 Taub-NUT deformations. Recall, that we can view the Taub-NUT deformation of the Z n ALE space as follows: consider the space H Γ‎ and replace one H-factor with the flat HK manifold S 1 × R 3 modifying the action of K(Γ‎) to be the translation on S 1. Consider a non-Abelian Γ‎ and suppose we could replace the Euclidean metric on some edge H ninj with a non-Euclidean 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 non-trivial 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 E-series.

• 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 non-trivial 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 *Grn(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,N-W04].

• Quaternionic Geometries with Torsion. These structures arose from the attempts of physicists to incorporate the so-called bosonic Wess–Zumino–Witten term [WZ71,Wit93] in σ‎-models with (extended) supersymmetry. The idea of considering torsion connections in such σ‎-models 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 so-called 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].

(p.472)

## 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 G-actions.

(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 $D n * = ℤ 2 ( n − 2 ) × ℤ 2$ has order 4(n−2) and not 4n.