Quaternionic Kähler and Hyperkähler Manifolds

CHAPTER 12

Quaternionic K¨ahler and Hyperk¨ahler Manifolds

In this chapter we will recall some basic results concerning various quaternionic geometries. Our main focus will be on the quaternionic Kähler (QK) and the 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 the Sasakian Geometry. Quaternionic Kähler geometry is traditionally defined by the reduction of the holonomy group Hol(M,g)toa 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 and Einstein. Interest in QK manifolds and this holonomy definition dates back to the celebrated Berger Theorem [Ber55]. The Lie group Sp(n)Sp(1) appears on the list of possible restricted holonomy groups of an oriented Riemannian manifold (M,g) which is neither locally product nor locally symmetric [Ber55]. In particular, the holonomy reduction implies that QK manifolds are always Einstein, 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 HPn. The model example of a QK manifold with negative scalar curvature (negative QK manifold) is that of the quaternionic hyperbolic ball HHn. 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). Hence, the HK Geometry is a special case of the QK Geometry and, just as in the previous case, we find the Lie group Sp(n) on the Berger’s list. The model example of the HK geometry is that of the quaternionic vector space Hn with the flat metric. The first non-trivial complete examples of HK metrics, as well as the terminology, is due to Calabi [Cal79] who constructed such metrics on M = T ∗CPn.

12.1. Quaternionic Geometry of Hn, HPn and HHn

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 0 1 2 3 a 4 H = {u | u = u + u i1 + u i2 + u i3,u∈ R} R .

307 308 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS

The imaginary units are often denoted by {i1,i2,i3} = {i, j, k}. The imaginary 3 quaternions Im(H)=span(i1,i2,i3) R and the multiplication rules are given by the formula 3 (12.1.1) iaib = −δab + abcic, c=1 We deﬁne the quaternionic conjugateq ¯ and the norm |u| by 3 3 0 a 2 a 2 u¯ = u − u ia, and |u| = (u ) . a=1 a=0 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) Sp(1) = {σ ∈ H∗ GL(1, H) | σσ¯ =1}, as a manifold, is just the unit 3-sphere in R4. Furthermore, we have the group isomorphism f : Sp(1) → SU(2) explicitly given by 0 1 2 3 σ + σ i1 σ + σ i1 (12.1.3) f(σ)= 2 3 0 1 . −σ + σ i1 σ − σ i1 It is known that Spin(4) = Sp(1) × Sp(1) and SO(4) Sp(1)Sp(1), where custom- arily Sp(1)Sp(1) denotes the quotient by the diagonal Z2. This is yet another group isomorphism between classical groups which can be explained using the quaternionic 4 geometry of H R . Consider the action of G = Sp(1)+ × Sp(1)− on H given by ¯ (12.1.4) ϕ(σ,λ)(u)=σuλ.

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 Z2 subgroup generated by (−1, −1) acts trivially. The quotient acts on R4 preserving the Euclidean metric and orientation. This is the special 4 orthogonal group SO(4). It is worthwhile to write this action on R .TheSp(1)+ part is given by the following group homomorphism A+ : Sp(1) → SO(4): ⎛ ⎞ σ0 −σ1 −σ2 −σ3 ⎜σ1 σ0 −σ3 σ2 ⎟ (12.1.5) A (σ)=⎜ ⎟ = σ01l + σ1I+ + σ2I+ + σ3I+, + ⎝σ2 σ3 σ0 −σ1⎠ 4 1 2 3 σ3 −σ2 σ1 σ0

+ + where the matrices Ii = A (ei) (12.1.6)⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 −10 0 00−10 00 0 −1 ⎜1000⎟ ⎜00 01⎟ ⎜00−10⎟ I+ = ⎜ ⎟ ,I+ = ⎜ ⎟ ,I+ = ⎜ ⎟ , 1 ⎝000−1⎠ 2 ⎝10 00⎠ 3 ⎝01 0 0⎠ 0010 0 −100 10 0 0 { + + +} R4 give a globally defined hypercomplex structure I1 ,I2 ,I3 on . For a purely + 2 imaginary τ = −τ¯ in Sp(1) one sets I (τ)=A+(τ) and gets the whole S -family of complex structures. We obtain the right hyperkähler structure on H by further ω+ + + + ⊗ setting g0 + := g0 + i1ω1 + i2ω2 + i3ω3 = du du¯, where the multiplication 12.1. QUATERNIONIC GEOMETRY OF Hn, HPn AND HHn 309 in H is used to interpret the left hand side as an H-valued tensor. This gives the standard Euclidean metric g0 and the three symplectic forms + b ∧ c 0 ∧ a (12.1.7) ωa = du du + du du where (a, b, c) is any cyclic permutation of (1, 2, 3). We can also introduce an H- valued differential 2-form

1 (12.1.8) ω+ = i ω+ + i ω+ + i ω+ =Im(du ⊗ du¯)= du ∧ du.¯ 1 1 2 2 3 3 2 The 2-from du ∧ du¯ is purely imaginary as α ∧ β =(−1)pqβ¯ ∧ α¯, where p, q are the respective degrees. The Sp(1)− part is given by A− : Sp(1) → SO(4) with ⎛ ⎞ λ0 λ1 λ2 λ3 ⎜− 1 0 − 3 2 ⎟ ⎜ λ λ λ λ ⎟ 0 1 − 2 − 3 − (12.1.9) A−(λ)= = λ 1l + λ I + λ I + λ I . ⎝−λ2 λ3 λ0 −λ1⎠ 4 1 2 3 −λ3 −λ2 λ1 λ0 − − The matrices Ii = A (ei) (12.1.10)⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0100 0010 0001 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − −100 0 − 0001 − 00−10 I = ⎜ ⎟ ,I= ⎜ ⎟ ,I= ⎜ ⎟ , 1 ⎝ 000−1⎠ 2 ⎝−1000⎠ 3 ⎝ 0100⎠ 0010 0 −100 −1000 { − − −} give a globally defined hypercomplex structure I1 ,I2 ,I3 . Furthermore, with the − Euclidean metric one gets the left hyperkähler structure on H by setting g0 −ω := − − − − − − ⊗ g0 iω1 i2ω2 i3ω3 = du¯ du. This gives 1 (12.1.11) ω− = i ω− + i ω− + i ω− = −Im(du ⊗ du¯)=− du ∧ du,¯ 1 1 2 2 3 3 2 where, as before, we get the three symplectic forms − b ∧ c − 0 ∧ a (12.1.12) ωa = du du du du , for each cyclic permutation of (1, 2, 3). These are clearly fundamental 2-forms { −} associated to the complex structures Ia . 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 I1 ,I2 ,I3 and I1 ,I2 ,I3 commute. The − − hyperkähler structure (g0,Ia ,ωa ) is preserved by Sp(1)+ (hypekähler isometry) 2 while Sp(1)− acts by rotating the complex structures on S . The role of Sp(1)+ and + + Sp(1)− reverses for (g0,Ia ,ωa ). 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 hypercomplex structures and the metric, hence, the quotient H/Z S1 × R3 is also a flat hyperkähler manifold with infinite fundamental group π1 = Z. To indicate the difference, we will write the flat metric 2 0 in this case as g0 = dθ + dx · dx replacing x with the angle coordinate θ. Example 12.1.1: Quaternionic vector spaces. Much of the above discussion Hn { | 0 1 2 3 ∈ H } extends to = u =(u1,...,un) uj = uj +uj i1+uj i2+uj i3 ,j=1,...,n . 310 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS

Here and from now on we will choose to work with the left hyperkähler structure on Hn, i.e., with the symplectic 2-forms are given by n (12.1.13) g0 − ω = du¯j ⊗ duj j=1 so that n 3 n a 2 b ∧ c − 0 ∧ a (12.1.14) g0 = (dxj ) ,ωa = dxj dxj dxj dxj j=1 a=0 j=1 for any cyclic permutation (a, b, c)of(1, 2, 3). The corresponding hypercomplex structures is then given by the left multiplication by {¯i1,¯i2,¯i3} = {−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 g0 a quaternionic Hermitian inner product n (12.1.15) F (u, v)= u¯jvj j=1 and define (12.1.16) Sp(n)={A ∈ GL(n, H) |Au,Av = u, v}. Now, Sp(n) × Sp(1) acts on Hn by ¯ (12.1.17) ϕ(A,λ)(u)=A · uλ 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)onHn C2n writing (12.1.18) u := z¯ + jw¯ =(x0 + ix1)+j(x2 − ix3). With such conventions we obtain 1 (12.1.19) ω = iω + ω j = iω +(ω + iω )j = − du¯ ∧ du , 1 + 1 2 3 2 j j j where i n n (12.1.20) ω = dz ∧ dz¯ + dw ∧ dw¯ ,ω= dw ∧ dz . 1 2 j j j j + j j j=1 j=1

Comparing with Example ?? we recognize (g0,ω1) as the standard Hermitian metric 2n and Hemitian form on C . In addition, the (2, 0)-from ω+ is a complex symplectic form so that 1 ωn = dw ∧···∧dw ∧ dz ∧···∧dz n! + 1 n 1 n is the standard holomorphic volume form on C2n. Example 12.1.2: Quaternionic projective space. WenowusetheleftH∗ action on Hn to introduce another model space of quaternionic geometry. Deﬁnition 12.1.3: The quaternionic projectivisation n n+1 n+1 ∗ HP := PH(H )=(H \{0})/H deﬁned with respect to the left action of H∗ on Hn+1 is called the quaternionic projective n-space. 12.1. QUATERNIONIC GEOMETRY OF Hn, HPn AND HHn 311

Let S4n+3 = {u ∈ Hn+1 | F (u, u)=1} be the unit sphere in Hn+1. The group Sp(n+1) acts on S4n+3 transitively with the isotropy at every point Sp(n). Hence, as an Sp(n + 1)-homogeneous space Sp(n +1) (12.1.21) S4n+3 = Sp(n) is a homogeneous space and the induce 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 Sp(n +1 (12.1.22) HPn = S4n+3/Sp(1) , Sp(n) × Sp(1) so we observe that HPn is actually a compact rank one symmetric spaces. If, in addition, we make a choice {±1}⊂R∗ ⊂ C∗ ⊂ H∗ we can also define two more projective spaces associated to PH(n). Definition 12.1.4: Let Hn+1 be be the quaternionic vector space and HPn the associated quaternionic projective space. We define n+1 n+1 ∗ (i) Z = PC(H )=(H \{0})/C , n+1 n+1 ∗ (ii) S = PR(H )=(H \{0})/R , U P Hn+1 Hn+1 \{ } Z (iii) = Z2 ( )=( 0 )/ 2. The spaces Z, S, U are called the twistor space, the Konishi bundle, and the Swann’s bundle of HPn, respectively. As Sp(n + 1)-homogeneous spaces as we have Sp(n +1) Sp(n +1) Z = CP2n+1 , S = RP4n+3 . Sp(n) × U(1) Sp(n) × Z2 Proposition 12.1.5: Let HPn be the quaternionic projective space. We have the following natural fibrations defined by {±1}⊂R∗ ⊂ C∗ ⊂ H∗ (i) H∗/C∗ = S2 →Z→HPn (ii) H∗/R∗ = SO(3) →S→HPn ∗ n (iii) H /Z2 →U→HP , (iv) C∗/R∗ = S1 →S→Z, ∗ (v) C /Z2 →U→Z, ∗ + (vi) R /Z2 = R →U→S. The six fibrations of the previous proposition are the six arrows in the following diagram

n (H \{0})/Z2 ⏐ ⏐ − ⏐ − (12.1.23) CP2n 1 ←−−−−−− − RP4n 1 PHn−1 We shall see later in this chapter that all these fibrations exist and are natural on more general quaternionic manifolds. However, the following is very special property of HPn and has to do with vanishing of H2(HPn, Z). Proposition 12.1.6: With the exception of the first one (S2 is already simply- connected), all fibrations of the previous proposition admit global Z2-lifting. 312 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS

The existence of the bundle Sp(1) → S4n+3 → HPn means that the the structure group of HPn is can be lifted from Sp(n)Sp(1) to Sp(n) × Sp(1). We will now n n construct an atlas on HP . Consider homogeneous coordinates [u0 : ···: un] ∈ HP . These are defined in analogy with homogeneous charts on a complex projective space by the equivalence of non-zero vectors in Hn+1, with u u meaning u = uλ, for some λ ∈ H∗.Let n (12.1.24) Uj = {[u0 : ···: un] ∈ HP | uj =0 } n and consider the maps φj : Uj → H defined by ··· −1 −1 −1 −1 φj([u0 : : un]) = (u0uj ,...,uj−1uj ,uj+1uj ,...,unuj ),j=0,...,n. n Now A = {Uj,φj}j=0,...,n is clearly an atlas on HP giving it a structure of differentiable manifold. Consider the inhomogeneous quaternionic coordinates (j) −1 xi := uiuj ,i= j, j =0,...,n. on Uj and H-valued 1-forms (j) − (j) −1 (12.1.25) dxi =(dui xi duj)uj ,i= j, j =0,...,n. ∈ HPn (j) HPn Hn At each x the forms dxi define an isomorphism Tx and thus a (j) n n n local section η ∈ Γ(Uj,L(HP )) of the principal frame bundle L(HP ) → HP . (k) n Let η ∈ Γ(Uk,L(HP )) be another such local section and consider Uk ∩Uj.An easy computation shows that at any x ∈Uk ∩Uj (k) (j) − (k) (j) (j) −1 (12.1.26) dxi =(dxi xi dxk )[xk ] ,i= k. (j) (j) Note that by convention xj =1,dxj = 0. The equations 12.1.26 imply that (12.1.27) η(k) = η(j)Aq, where q ∈ GL(1, H)andA ∈ GL(n, H) ⊂ GL(4n, R) is the centralizer of GL(1, H). Thus the structure group of HPn reduces to GL(n, H)GL(1, H) and because Aq = (A|q|)(q/|q|) the structure group further reduces to GL(n, H)Sp(1). We are now ready to give HPn a Riemannian metric which is induced by from the flat metric on Hn+1 or, equivalently, from the constant sectional curvature 1 metric on S4n+3 via the appropriate Riemannian submersions. We consider 4 4 (12.1.28) g − ω i − ω i − ω i := duj ⊗ duj − (ujduj) ⊗ (dukuk) 0 1 1 2 2 3 3 c c j j,k Note that the above equation defines the metric on HPn while the three 2-forms {ω1,ω2,ω3} are only local sections in a 3-dimensional vector bundle Q. Using the language of H-valued forms we can introduce ω 4 4 (12.1.29) −ω = duα ∧ duα − (uαduα) ∧ (duβuβ), c c α αβ 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) Ω=Ω=ω ∧ ω. It is real and closed. We have the following 12.1. QUATERNIONIC GEOMETRY OF Hn, HPn AND HHn 313

Theorem 12.1.7: The 4-from Ω is parallel. When n>1 the holonomy group 1 4 Hol(g0) ⊂ Sp(n)Sp(1).Whenn =1HP S and the metric g0 is simply the metric of constant sectional curvature on S4 which is self-dual and Einstein. Example 12.1.8: Quaternionic semi-projective spaces. Our previous example can be generalized to the semi-Riemannian case giving rise to many interesting k,l quaternionic geometries. Let H = {u =(a, b) | a =(u0,...,uk−1), b = (uk,...,uk+l−1)} be the set of all quaternionic (n + 1)-vectors together with the symmetric form k−1 k+l−1 1 2 − 1 2 1 2 − 1 2 1 2 (12.1.31) Fk,l(u , u )= u¯αuα + u¯αuα = a , a + b , b α=0 α=k Here a1, a2 denotes the standard quaternionic-Hermitian inner product on Hk 2 and we shall denote the associated norm by || a|| = a, a.TheformFk,l defines the flat semi-Riemannian metric of signature (4k, 4l)onHk,l. k,l 2 2 Definition 12.1.9: Let Hk,l()={(a, b) ∈ H |−||a|| + || b|| = }. 4k−1 l (i) Hk,l(−1) S × H ,wherek>0, is a semi-Riemannian submanifold of signature (4k − 1, 4l) called the pseudohyperboloid. k 4l−1 (ii) Hk,l(+1) H × S ,wherel>0, is a semi-Riemannian submanifold of signature (4k, 4l − 1) called the pseudosphere. 4k−1 4l−1 4k−1 (iii) Hk,l(0) S × S × R/∼,wherek, l > 0 and ∼ identifies S × S4l−1 ×{0} with a point, is called the null cone.

Let k + l = n +1 and Sp(k, l) ⊂ GL(n +1, H) which preserves the form Fk,l.It is well-known that Hk,l(±1) are spaces of constant curvature and as homogeneous spaces of the semi-symplectic group Sp(k, l) they are

Sp(k, l)/Sp(k, l − 1) when l>0and =+1, Hk,l()= Sp(k, l)/Sp(k − 1,l) when k>0and = −1. Hk,l { ∈ Hk,l ||||| 2 || || 2} Hk,l { ∈ Hk,l ||||| 2 Consider + = (a, b) a < b , and − = (a, b) a > || || 2} Hk,l H b . Also, let us write 0 for k,l(0) as an alternative notation. We can then write Hk,l Hk,l ∪ Hk,l ∪ Hk,l (12.1.32) = − 0 + . After removing 0 ∈ Hk,l we consider the action of H∗ on (12.1.32) by right multiplication. Definition 12.1.10: Let Hk,l be the quaternionic vector space with semi-hyperkähler metric of signature (4k, 4l). We define the following projective spaces. k−1,l k,l k,l ∗ (i) QH := PH(H− )=H− /H = Hk,l(−1)/Sp(1), QHk,l−1 P Hk,l Hk,l H∗ H (ii) := H( + )= + / = k,l(+1)/Sp(1), P Hk,l Hk,l \{ } H∗ 4k−1 × 4l−1 (iii) H( 0 )=( 0 0 )/ = S Sp(1) S . If we make a choice of C∗ ⊂ H∗ we also have complex ‘projective’ spaces. Definition 12.1.11: Let Hk,l be the quaternionic vector space with semi-hyperkähler metric of signature (4k, 4l).LetC∗ ⊂ H∗. We define k,l k,l ∗ (i) PC(H− )=H− /C = Hk,l(−1)/U(1), P Hk,l Hk,l C∗ H (ii) C( + )= + / = k,l(+1)/U(1), P Qk,l Hk,l \{ } C∗ 4k−1 × 4l−1 (iii) C( 0 )=( 0 0 )/ = S U(1) S . 314 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS

Proposition 12.1.12: As homogeneous spaces of the semi-symplectic group (12.1.33) k,l Sp(k, l) k,l Sp(k, l) PH(H )= , PC(H )= ,k>0 − Sp(1) × Sp(k − 1,l) − U(1) × Sp(k − 1,l)

k,l Sp(k, l) k,l Sp(k, l) PH(H )= , PC(H )= ,l>0. + Sp(k, l − 1) × Sp(1) + Sp(k, l − 1) × U(1) Furthermore, we have the natural fibrations Hk,l Hk,l − + ⏐ ⏐ ⏐ ⏐ P Hk,l ⏐ ⏐ P Hk,l (12.1.34) C( − ) C( + ) P Hk,l P Hk,l H( − ) H( + ) which can be glued together along the common boundary C∗ → Hk,l \{ } 0 0 ↓ 2 → P Hk,l 4k−1 × 4l−1 (12.1.35) S C( 0 ) S S1 S ↓ P Hk,l 4k−1 × 4l−1 H( 0 ) S S3 S to give HPk+l−1, its twistor space CP2k+2l−1 and the vector space Hk+l \{0}. Note P Hk,l 4k−1× 4l−1 4k−1 HPl−1 4l−1 that H( 0 ) S S3 S is both S -bundle over and S -bundle over HPk−1. The following proposition is straightforward. k,l Proposition 12.1.13: The manifolds PH(H− ), k>0, are semi-quaternion Kähler with holonomy group Sp(k − 1,l)Sp(1),indexν = k − 1, negative scalar curva- k,l k,l k,l ture, twistor space PC(H− ), and Swann bundle H− ; furthermore, PH(H− ) is the quaternionic Hl-bundle over the standard quaternionic projective space HPk−1 as- P Hk,l sociated to the quaternionic Hopf fibration. The manifolds H( + ), l>0,are semi-quaternion Kähler with holonomy group Sp(k, l − 1)Sp(1),indexν = k,posi- P Hk,l Hk,l tive scalar curvature, twistor space C( + ), and Swann bundle + ; furthermore, P Hk,l Hk H( + ) is the quaternionic -bundle over the standard quaternionic projective space HPl−1 associated to the quaternionic Hopf fibration. Topologically, QHk−1,l = P Hk,l QHk,l−1 P Hk,l HPk+l−1 \ P Hk,l H( − ) and = H( + ) are the components of H( 0 ). k,l The bundle structure of PH(H− ) is the one associated to the right quaternionic multiplication of the quaternionic vector space Hk by the unit quaternions. Explic- k,l 4k−1 l itly, let (a, b) ∈Hk,l(−1) ⊂ H− . Let us identify Hk,l(−1) S (1) × H via a map a f(a, b)=(v, b)= , b . || b|| 2 +1 Then σ ∈ Sp(1) acting on S4k−1(1) × Hl by (v, b) → (vσ, bσ) gives the quo- k,l k,l k tient which can be identified with PH(H− ). Hence, PH(H− )isanH -bundle (quaternionic vector bundle) over HPl−1 associated to the quaternionic Hopf bundle S3 → S4l−1 → HPl−1. 12.1. QUATERNIONIC GEOMETRY OF Hn, HPn AND HHn 315

1,2 Example 12.1.14: Let (k, l)=(1, 2). Then PH(H− ) is simply the unit open 8-ball H2 P H1,2 3 × 7 7 in . The boundary of this cell H( 0 )=S S3 S S is the unit sphere. The P H1,2 H R4 HP1 4 space H( + )isthe bundle over S associated to the quaternionic 3 → 7 → 4 P H1,2 Hopf bundle S S S . Viewed another way H( + ) is a complement of the unit 8-ball in H2 with HP1 S4 added in at inﬁnity. Remark 12.1.2: Note that the map ψ : Hk,l → Hl,k deﬁned by

(12.1.36) ψ(u0,u1,...,uk−1,uk,...,un)=(un,...,uk,uk−1,...,u0) is the anti-isometry (or metric reversal) which induces anti-isometries

k,l l,k ψ : Hk,l() →Hl,k(−), i.e., ψ : QH → QH .

n+1,0 n For example, PH(H− ) is diffeomorphic to HP but has negative-definite metric. P H0,n+1 HPn It can be identified with H( + ) which is obviously the usual definition of by changing the sign of the metric. As a result we can restrict our discussion only k,l to the negative scalar curvature spaces PH(H− ), k>0. This is not natural if one n+1,0 talks about the projective space PH(H− ) but in this paper we will mostly deal with the case k

k,l (12.1.37) Uβ = {u ∈ PH(H− ) | uβ =0 },β=0,...,k− 1.

On Uβ we write β β β −1 −1 −1 −1 ∈ Hn (12.1.38) x =(x1 ,...,xn)=(u0uβ ,...,uβ−1uβ ,uβ+1uβ ...,unuβ ) .

Note that (12.1.31) implies that on Uβ we have

k−1 n − β β | β |2 − | β |2 | |2 (12.1.39) 1 Fk−1,l(x , x )=1+ xα xα =1/ uβ > 0. α=1 α=k

Let us denote Fk−1,l simply by ∗, ∗k−1,l with the associated semi-norm || ∗ || k−1,l. U || β|| − Then, on β, x k−1,l < 1 and the semi-quaternion K¨ahler metric gk,l reads − 1 β 2 1 β β 2 (12.1.40) g = || dx || + |dx , x − | . k,l −|| β|| 2 k−1,l −|| β|| 2 k 1,l 1 x k−1,l 1 x k−1,l

k,l We will often refer to u =(u0,u1,...un) as homogeneous coordinates on PH(H− ). n 1,n Example 12.1.15: Quaternionic hyperbolic space HH . The space PH(H− )= QHn is simply the unit ball in Hn with the quaternionic hyperbolic metric. In this case U0 is the only chart so we have global inhomogeneous coordinates −1 −1 ∈ Hn (12.1.41) x =(x1,...,xn)=(u1u0 ,...,unu0 ) . with the positive deﬁnite hyperbolic metric 1 1 (12.1.42) g = |dx|2 + |dx, x|2 . 1 −|x|2 1 −|x|2 316 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS

12.2. Quaternionic K¨ahler Metrics

Let M be a smooth 4n-dimensional manifold (n ≥ 1). We say that M is almost quaternionic if there is a 3-dimensional subbundle Q⊂End(TM) with the following property: At each point x ∈ M there is a basis {I1,I2,I3} of the ﬁbre Qx such that:

(12.2.1) Ii ◦ Ij = −δij 1l + ijkIk. In other words, R · 1l ⊕Qis, at each point, a subalgebra isomorphic to the algebra of quaternions H. As remarked in Example 1.3.15 this deﬁnition is equivalent to M admitting a G-structure with G = GL(n, H)H∗ (or equivalently with G = GL(n, H)Sp(1)). Any oriented 4-manifold admits such a structure. But in higher dimensions the almost quaternionic structures are obstructed not just by w1(M). Suppose now that M carries a Riemannian metric adapted to the quaternion structure in the sense that each I ∈Qis orthogonal, i.e., (12.2.2) g(IX,IY )=g(X, Y ), for all I ∈Qand all X, Y ∈ TxM at all points x ∈ M. Adapted metric always exists and the resulting triple (M,Q,g) is called an almost quaternionic Hermitian manifold, giving further reduction of the structure group to Sp(n)Sp(1). Given an adapted metric we obtain an isometric bundle embedding Q⊂Λ2T ∗M which associates to each I ∈Qx the non-degenerate 2-form ω deﬁned by

(12.2.3) ωI (V,W)=g(IX,Y ),X,Y∈ TxM. 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 group H1(M,Sp(n)Sp(1)) with coeﬃcients in the sheaf of smooth Sp(n)Sp(1)-valued functions. The short exact sequence

(12.2.4) 0 → Z2 → Sp(n) × Sp(1) → Sp(n)Sp(1) → 0 gives rise to the homomorphism 1 2 (12.2.5) δ : H (M,Sp(n)Sp(1)) → H (M,Z2) We have 2 Deﬁnition 12.2.1: Let = δ(P ) ∈ H (M,Z2).Then 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) 0 → Z2 → Spin(4) → SO(4) → 0 and it follows that equals to the second Stiefel-Whitney class w2(M). For n>1 we can identify = w2(Q) with the second Stiefel-Whitney class of the vector bundle Q. Furthermore, we get [Sal82, MR75] Proposition 12.2.2: Let (M 4n, Q,g) be an almost quaternionic Hermitian manifold. Then w2(M)=n. In particular, the Marchiafava-Romani class is the second Stiefel-Whitney class of M if its dimension is 4 mod 8. In complementary dimensions we get Corollary 12.2.3: Any almost quaternionic manifold M of dimension 0mod8is spin. 12.2. QUATERNIONIC KAHLER¨ METRICS 317

The full obstruction theory for Sp(n)Sp(1)-structures (even in the simples 8- dimensional case) is subtle and not completely understood. See, for example, the article by Cadekˇ and Vanˇzura [CV98ˇ ]. Suppose {I1,I2,I3} are locally deﬁned smooth sections of Q which satisfy 12.2.1 at each point. Then these form an orthonormal frame ﬁeld for Q in the standard 1 t { } metric A, B = 2n Tr(A 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

3 (12.2.7) Ω = ωi ∧ ωi i=1 is invariant under change of frame field and thus globally defined on M.Itisnon- 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 from Ω is called the fundamental 4-form of the almost quaternion Hermitian structure (M,Q, Ω,g). Definition 12.2.4: An almost quaternionic structure (M,Q) is 1-integrable if M admits a torsion-free connection ∇Q preserving quaternionic structure Q.Insuch acase(M,Q) is called a quaternionic structure on M and with adapted metric (M,Q,g) is called a quaternionic Hremitian manifold. The connection ∇Q, called the Oproiu connection, if it exists, 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)H∗ = R+ × SO(4) so that G-structure is equivalent to orientation and conformal class. In particular, Levi-Civita connection of any compatible metric preserves the G-structure and has no torsion. But in higher dimensions, there are non-trivial obstructions. Remark 12.2.1: To complete this definition it is logical to define a quaternionic manifold of real dimension 4 to be one with self-dual conformal structure. See for example [Sal99]. Here we shall be interested in a very special class of quaternionic Hermitian manifolds Definition 12.2.5: An almost quaternionic Hermitian (M 4n, Q, Ω,g) of quaternionic dimension n>1 is called quaternionic Kähler (QK) if the holonomy group Hol(g) ⊂ Sp(n)Sp(1). Naturally, a QK manifold is quaternionic as the Levi-Civita connection is torsion free and it preserves the quaternionic structure. We can easily see the holonomy definition to be equivalent to the following Proposition 12.2.6: An almost quaternionic Hermitian manifold (M 4n, 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 (M 4n, Q, Ω,g), n>1, is quatenionic Kähler if it admits a parallel 4-form which is in the same GL(4n, R)-orbit as Ω at each point x ∈ M. The hypothesis ∇Ω = 0 clearly implies that dΩ = 0. Surprisingly, the following theorem was proved by Swann [Swa89]: Theorem 12.2.7: An almost quaternionic Hermitian (M 4n, Q, Ω,g) of quaternionic dimension n>2 whose fundamental 4-form Ω is closed is quaternionic Kähler. 318 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS

The geometry of almost quaternionic Hermitian 8-manifolds is 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 12.2.8: An almost quaternionic Hermitian 8-manifold is quaternionic K¨ahler 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. Consider a QK manifold (M 4n, Q, Ω,g). As observed by Kraines wedging with the fundamental 4-form Ω determines an injection [Kra65, Fuj87]. Hk(M,R) → Hk+4(M,R),k≤ n − 1. Since Ωn = 0 there are very strict topological consequences: the Betti numbers b4i(M) > 0 cannot vanish for 0 ≤ i ≤ n. On a compact positive QK manifold we get more information [Sal82, Fuj87] Theorem 12.2.9: Let (M 4n, Q, Ω, ) be a compact positive QK manifold. Then

(i) b2i+1 =0, (ii) β2i = b2i − b2i−4 ≥ 0, ∀i ≤ n.

In fact, as we shall see in the next chapter β2i are simply even Betti numbers of the principal SO(3)-bundle associated to the quaternionic bundle Q. 2 ∗ Let {ω1,ω2,ω3} be a local orthonormal frame ﬁeld for Q⊂Λ T M.IfΩis parallel we get 3 (∇ωi) ∧ ωi =0 i=1 from which it follows that 3 (12.2.8) ∇ωi = αij ⊗ ωj, j=1 where the αij are 1-forms which satisfy

(12.2.9) αij ≡−αji ∀i, j =1, 2, 3 . 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 ⎛ ⎞ 0 a12 a13 ⎝ ⎠ (12.2.10) A = −a12 0 a23 −a13 −a23 0 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 F = dA − A ∧ A. 3 ∧ 2 Using the facts that dωi = j=1 αij ωj and d ωi = 0, one deduces that (12.2.11) Fij = dαij − αil ∧ αlj = λ ijkωk l k for some constant λ. Since Q is an oriented 3-dimensional bundle, there is a canon- ∼ ical identification SkewEnd(Q) = Q via the cross-product. Using this identification ∼ we can consider F as a map F :Λ2TM −→ SkewEnd(Q) = Q⊂Λ2TM and, as 12.2. QUATERNIONIC KAHLER¨ METRICS 319 such, equation 12.2.11 simply states that F = λπQ, where πQ denotes pointwise 2 orthogonal projection πQ :Λ TM →Q. The full Riemann curvature tensor R of QK manifold viewed as a symmetric endomorphism R :Λ2TM −→ Λ2TM (curvature operator), has the property that

(12.2.12) R |Q= λIdQ, where λ is a positive multiple of the scalar curvature s on M. We will now use the equation 12.2.12 to extend our definition of quaternionic Kähler manifolds to 4-dimensional spaces. Note that in dimension 4, the condition that ∇Ω=0is trivially satisfied since Ω is the volume form. Nevertheless, a natural extension of the concept of being quaternionic Kähler does exist for dimension 4. In this case we have the decomposition 2 2 ⊕ 2 (12.2.13) Λ TM =Λ+ Λ−

2 into self-dual and anti-self-dual 2-forms. If one identifies Q with Λ− we have Definition 12.2.10: An oriented Riemannian 4-manifold (M,g) is called quaternionic Kähler if condition 12.2.12 holds. Recall that, relative to the decomposition 12.2.13 the curvature operator has the following form ⎛ ⎞ s W+ + 12 1l R i c 0 (12.2.14) R = ⎝ ⎠ , s Ric0 W− + 12 1l where W± are trace-free pieces of the appropriate blocks and are called 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. We have Corollary 12.2.11: Aa 4-dimensional manifold is quaternionic Kähler if and only if it is self-dual (i.e., W− =0) and and Einstein (i.e., Ric0 =0). There are at least two more justifications for adopting the Definition 12.2.10. One is the theory of quaternionic submanifolds of QK manifolds. N ⊂ M is called ∗ a quaternionic submanifold if for each x ∈ NTxN is an H -submodule of TxM. Marchiafava observed that a 4-dimensional submanifold of a QK manifod is necessarily self-dual and Einstein. The other justification comes in the theory of quaternionic Kähler reduction which will be discussed in a later section. We already mentioned that the quaternionic projective space is quite special as it is the only example of a compact quaternionic kähler manifold which admits an integrable Sp(n)Sp(1)-structure. The curvature tensor of the canonical symmetric metric on HPn plays a key role in the more general setting. The following result is due to Alekseevsky [Ale68] Theorem 12.2.12: Let (M 4n, Q, Ω,g) be a quaternionic Kähler manifold. The the Riemann curvature tensor can be written as

R = sR1 + R0, where R1 has is the curvature of the quaternionic projective n-space and R0 a curvature of a hyperk¨ahler manifold. In particular we get the following 320 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS

Corollary 12.2.13: Any QK manifold is automatically Einstein. A QK manifold with vanishing scalar curvature s is locally HK, i.e., the restricted holonomy group Hol0(g) ⊂ Sp(n). We will discuss some basic properties of hyperk´ahler metrics in the second half of this chapter so for now we assume that scalar curvature is not zero. Then we have positive and negative QK manifolds to consider. One of the fundamental properties of positive QK manifold is that they are necessarily simply connected. Moreover, Theorem 12.2.14: [Sal82] Let (M,Q, Ω,g) be a compact positive QK manifold. Then π1(M)=0and π2(M)=0if and only if (M,g) (HP(n),g0) with its canonical symmetric metric g0. In addition, positive QK manifold are rigid. We have the following theorem due to LeBrun [LeB88].

Theorem 12.2.15: Let (M,Q, Ω,g) be a compact positive QK manifold. If gt is a family of positive QK metrics of fixed volume depending smoothly on R such that g0 = g. Then there exist family of diffeomorphisms ft : M → M depending ∗ smoothly on t such that gt = ft g. None of these results holds for negative QK manifolds. LeBrun showed, for example, that the moduli space of complete negative QK metrics on Hn is infinite- dimensional [LeB91]. The following result was also proved by Salamon [Sal82] and gives yet another characterization of canonical example. Theorem 12.2.16: Let (M,Q,g) be a compact positive QK manifold with vanish- n ing Marchiafava-Romani class. Then (M,g) (HP ,g0) with its canonical symmetric metric g0. Definition 12.2.17: Let (M,Q, Ω,g) be a quaternionic Kähler manifold with non- zero scalar curvature. 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×SO(3) F ∗ (i) U(M)=S(M) ×SO(3) F ,whereF = H /Z2, 2 (ii) Z(M)=S(M) ×SO(3) F ,whereF = S is the unit sphere in Q. The spaces U(M), Z(M) are called the Swann’s bundle, and the twistor space of M respectively. When M is a positive QK manifold all these spaces have special geometric structures which will be discussed in great detail in Chapter 13.

12.3. Symmetries of Quternionic K¨ahler Manifolds

Alekseevsky proved that all homogeneous positive QK manifolds must be symmetric [Ale75]. These were classiﬁed by Wolf [Wol65] and they are often called Wolf space. There is 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:

Sp(n +1) SU(m) SO(k) , , , Sp(n) × Sp(1) S U(m − 2)×U(2) SO(k − 4)×SO(4) G F E E E 2 , 4 , 6 , 7 , 8 . SO(4) Sp(3)Sp(1) SU(6)Sp(1) Spin(12)Sp(1) E7Sp(1) 12.3. SYMMETRIES OF QUTERNIONIC KAHLER¨ MANIFOLDS 321

Here n ≥ 0, Sp(0) denotes the trivial group, m ≥ 3,andk ≥ 7. In particular, each such M is a symmetric space and, there is one-to-one correspondence between the simple Lie algebras and positive homogeneous QK manifolds. The integral cohomology group H2(M,Z) vanishes for M = HPn and it is n+2 Z for the complex Grassmannian M =Gr2(C ). In all other cases we have 2 H (M,Z)=Z2. LeBrun and Salamon proved the following strong rigidity result [LS94]. Theorem 12.3.2: Let (M 4n, Q, Ω,g) be a compact positive QK manifold. Then π1(M)=0and n (i) π2(M)=0if and only if M is isometric to HP , n+2 (ii) π2(M)=Z if and only if M is isometric to Gr2(C ), (iii) π2(M) if ﬁnite with 2-torsion otherwise Furthermore, up to isometries and rescalings there are only ﬁnitely many positive QK manifold in each quaternionic dimension. In addition, in the same paper LeBrun and Salamon show that there are linear relations between Betti numbers of any compact positive QK manifold [LS94] Theorem 12.3.3: Let (M 4n, Q, Ω,g) be a compact positive QK manifold. Then

(i) b2k+1(M)=0for all k ≥ 0, ≥ Cn+2 (ii) b2(M) 1 if and only if M is isometric to Gr2( ) n−1 − − − − − 1 − (iii) r=0 [6r(n 1 r) (n 1)(n 3)]b2r(M)= 2 n(n 1)b2n. In the absence of any examples, Theorem 12.3.2 strongly points towards Conjecture 12.3.4: 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 Theorem 12.3.2 there are several other results showing the conjecture to be true in some special cases. We will collect all these results in the following Theorem 12.3.5: Let (M 4n, Q, Ω,g) be a compact positive QK manifold. Then M is a symmetric space if (i) n ≤ 3, (ii) n =4and b4 =1, ≥ n (iii) rank of I(M,g) denoted by rank(M) 2 +3.

Proof. The statement in (i) dates back to the Hitchin’s proof that all compact self-dual and Einstein spaces of positive scalar curvature must be isometric to either S4 with the standard constant curvature metric or CP2 with the Fubini-Study metric [Hit81](seealso[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 12.3.2. The n = 3 case is a recent result of Herrera and Herrera [HH02a, HH02b]. Their prof uses and 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 Aˆ(M) genus of non-spin manifolds with ﬁnite π2(M) and smooth circle actions Herrera and Herrera prove that 322 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS

5 Lemma 12.3.6: Let M be a positive QK 12-manifold which is not Gr2(C ).Then Aˆ(M)=0. The result is then a consequence of the vanishing of Aˆ(M) = 0 and the Theorem 12.3.2. The result in (ii) follows from the estimate on the dimension of the isometry group in this case and Betti number constraints of Theorem 12.3.3 [GS96]. Finally, (iii) is a recent valuable estimate on the rank obtained by Fang in [Fan04, Fan05] which greatly improves an earlier estimate by Bielawski [Bie99].

We remark that the argument of [HH02a] does not work in 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 this conjecture which via algebraic geometry of the twistor space Z(M). The twistor space of any positive QK manifold M is actually a Fano manifold with Kähler-Einstein metric and a complex contact structure. The following, apparently stronger, conjecture was suggested by Beauville [Bea05, Bea98] Conjecture 12.3.7: 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. Wisniewski even briefly claimed the proof of the conjecture but later Campana found a gap in Wisniewski’s argument. Camapna briefly claimed to have bridged that gap but later also withdrew the claim. Hence, as of the time of writing this monograph, both the Baeuville Conjecture 12.3.7 and LaBrun-Salamon Conjecture 12.3.4 remain open. Let the symmetry rank of M be defined as the rank of its isometry group I(M,g), i.e, the dimension of the maximal Abelian subgroup in I(M,g). Bielawski [Bie99] proved that a positive QK manifold of quaternionic dimension n of sym- n n+2 metry rank at least n + 1 is isometric to HP or to the Grassmannian Gr2(C ). Recently Fang proved several a rigidity theorems for positive quaternionic Kähler manifolds in terms its symmetry rank [Fan04, Fan05]. Fang’s result much en- hances Bielawski’s theorem. Theorem 12.3.8: Let (M 4n, Q, Ω,g) be a compact positive quaternionic Kähler manifold. Then the isometry group I(M,g) has rank at most (n +1),andM is HPn Cn+2 ≥ n isometric to or Gr2( ) if rank(M) 2 +3. This theorem is quite interesting and apparently rather deep. It follows from several different results. First recall Definition 12.3.9: A submanifold N→ M in a quaternionic Kähler (M,Q, Ω,g) manifold is called a quaternionic submanifold if the quaternionic structure Q preserves the tangent bundle of N. It is an well-known result of Gray that [Gra69] Proposition 12.3.10: Any quaternionic submanifold in a quaternionic Kähler manifold is totally geodesic and quaternionic Kähler. 12.4. QUTERNIONIC KAHLER¨ REDUCTION 323

In [Fan04] Fang proves the following rigidity results for positive QK manifold Theorem 12.3.11: Let (M 4n, Q, Ω,g) be a positive QK manifold. Assume f = (f1,f2):N → M × M,whereN = N1 × N2 and fi : Ni → M are quaternionic immersions of compact quaternionic Kähler manifolds of dimensions 4ni, i =1, 2. Let Δ be the diagonal of M × M and set m = n1 + n2.Then (i) If m ≥ n, then f −1(Δ) is nonempty. (ii) If m ≥ n +1, then f −1(Δ) is connected. (iii) If f is an embedding, then for i ≤ m − n there is a natural isomorphism, πi(N1,N1 ∩ N2) → πi(M,N2) and a surjection for i = m − n +1. The proof of the rank theorem 12.3.8 uses the above theorem together with some Morse theoretic results concerning the quaternionic momentum mapping obtained earlier by Battaglia [Bat99]. These will be discussed in the next section. Remark 12.3.1: The study of homogeneous negative QK manifold is more deli- cate. 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 consider- ing some supersymmetric σ-model 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 the Alekseevsky’s original paper. A proof that all negative QK manifolds are the known Alekseevskian spaces is, however, still lacking.

12.4. Quternionic K¨ahler Reduction

In this section wee will describe an analogue of the Marsden-Weinstein reduction for quaternionic K¨ahler manifolds [Gal87, GL88]. To do this we ﬁrst 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 “de Rham” sequence

∇ ∇ ∇ ∇ (12.4.1) Ω0(Q) d−→= Ω1(Q) −→d Ω2(Q) −→···d such that (12.4.2) d∇ ◦ d∇(f)=R(f) for f ∈ Ω0(Q). Consider now the Lie group Aut(M,Q, Ω,g) ≡{g ∈ I(M,g): g∗Ω=Ω} and its Lie algebra

aut(M,Q, Ω,g) ≡{V ∈ i(M,g): £V Ω=0} embedded naturally in the space i(M,g) of Killing vector ﬁelds on M.Wehavethe following immediate consequence 324 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS

Proposition 12.4.1: Let (M,Q, Ω,g) be a QK manifold of non-zero scalar curvature. Then aut(M 4n, Q, Ω,g)=i(M 4n,g). It follows that any one parameter subgroup H ⊂ I(M 4n,g) is also a subgroup of Aut(M 4n, Q, Ω,g). Proof. When M is symmetric all statements follow by inspection. When M is not locally symmetric the holonomy Lie algebra hol = sp(n) ⊕ sp(1). Since M is irreducible, by Kostant [Kos55] any Killing vector field normalizes the holonomy algebra and in particular the sp(1)-factor which defines the quaternionic structure Q. Hence i(M,g) normalizes Q and therefore any Killing vector 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 one-parameter subgroup and these may not preserve the quaternionic 4-from Ω. To each V ∈ i(M,g) we associate the Q-valued 1-form 1 ΘV ∈ Ω (Q) defined in terms of a local frame ω ,ω ,ω by 1 2 3 (12.4.3) ΘV ≡ (V ωi) ⊗ ωi. i

Clearly ΘV remains invariant under local change of frame ﬁeld (i.e, under local gauge transformations). We have [GL88] Theorem 12.4.2: Assume that the scalar curvature of (M,Q, Ω,g) is not zero. Then to each V ∈ i(M,g) there corresponds a unique section μ ∈ Ω0(Q) such that

(12.4.4) ∇μ =ΘV . In fact, under the canonical bundle isometry σ : SkewEnd(Q) −→ Q , μ is given explicitly by the formula 1 (12.4.5) μ = σ(£ −∇ ), λ V V where λ is the constant positive multiple of scalar curvature deﬁned by 12.2.12. We observe now that by the uniqueness in Theorem 12.4.2, the map V −→ μ transforms naturally under the group of automorphisms. This means speciﬁcally that for g ∈ Aut(M,Q, Ω,g)andV ∈ i(M,g)wehave μ μ (12.4.6) g∗(V ) = g∗( ),

−1 where g∗ μ (x):=˜g μ(g (x)) and whereg ˜ denotes the map induced by g on the 2 bundle Q⊂Λ TM. Note also that g∗V =Adg(V ). Hence, 12.4.6 means that the diagram

Adg i(M,g⏐ ) −−−−→ i(M,g⏐ ) ⏐ ⏐ ⏐ ⏐ (12.4.7) ⏐μ ⏐μ g∗ Ω0(Q) −−−−→ Ω0(Q), commutes. Suppose now that H ⊂ I(M,g)) is a compact connected Lie subgroup with corresponding Lie algebra h. 12.4. QUTERNIONIC KAHLER¨ REDUCTION 325

Deﬁnition 12.4.3: The momentum 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 −→μ(x).(Here g denotes the trivial g-bundle over M). From the equivariance above we see immediately that the momentum map is G-equivariant. Since the action of G in the bundle g∗ ⊗Qis linear on the ﬁbers, it preserves the zero section. Consequently the set

(12.4.8) N = μ−1(0) := {x ∈ M : μ(x)=0} 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 quaternionic Kähler manifold with non- zero scalar curvature. Let G ⊂ I(M,g) be a compact connected subgroup with momentum map μ.LetN denote the G-invariant subset of N = {x ∈ M : μ(x)= 0} where μ intersects the zero section transversally and where G acts freely. Then Mˆ := N/G equipped with the submersed metric is again a quaternionic Kähler manifold. ∼ Corollary 12.4.5: Let M be as above and suppose G = T k ⊂ I(M,g) is a k-torus subgroup generated by a vector fields Vk ∈ i(M,g).IfV1 ∧ ...∧ Vk is a k-plane field at all points x ∈ N, then N/G is a compact quaternionic Kähler orbifold. Example 12.4.6: Consider the S1-action defined on HPn in homogeneous coordinates as follows

(12.4.9) ϕλ([u0,...,un]) = [λu0,...,λun], where λ is a complex unit. One easily shows that the reduced space by this action n+1 is smooth and Mˆ =Gr2(C ). The quaternionic reduction method has been used to obtain many examples of compact quaternionic K¨ahler orbifolds. When the reduced space is 4-dimensional it is automatically self-dual and Einstein. The only complete positive QK metrics in dimension 4 are the standard symmetric metrics on HP1 S4 or CP2. In this context the only interesting quotients are in fact the ones which end up giving rise to orbifold singularities. They will be discussed in the next section. Just as in the symplectic case one can study singular quaternionic K¨ahler quotients without assuming that the action of the quotient group G on N = μ−1(0) is free. A detailed study of this more general situation was done by Dancer and Swann [DS97]. Let (M,Q, Ω,g) be a QK manifold and let G be a connected Lie group act smoothly and properly on M preserving QK structure with moment map μ. For any subgroup H

One can first show that stratification of M by the orbit type induces the stratification of Mˆ into a union of smooth manifolds, i.e.,

Theorem 12.4.7: Let H

However, not all the pieces Mˆ H are QK manifolds. Their geometry depends on the way H acts on Q. To be more precise, let x ∈ MH and consider the linear action of H on TxM. Since H acts preserving the quaternionic structure we have the representation H−→Sp(n)Sp(1) which induces the representation φ : H−→SO(3). The group H acts on Q R3 via the composition of φ with the standard three- dimensional representation. If x, y ∈ MH are on the same path-component then parallel transport along any path joining x to y deﬁnes H-equivariant isomorphism of TxM TyM. It follows that the representation φ : H−→SO(3) is equivalent at all points on a path component of MH . Hence, the image φ(H)

• φ(H) is trivial ⇒ Mˆ H is a QK manifold, • φ(H)=Zk, k>1 ⇒ each path component of Mˆ H is covered by a Kähler manifold, • φ(H) is finite but not cyclic ⇒ MH is totally real in M (Mˆ H is “real”), • φ(H)=SO(2) or SO(3) ⇒ Mˆ H is empty. Even simple examples show that the stratification of the quotient by orbit type can include all of the three possible pieces. However, there is a coarser stratification of ˆ M 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 to H and M([H]) the set of points whose stabilizer has identity component conjugate to H in G. Theorem 12.4.8: The union (μ−1(0) ∩ M ) Mˆ = ([H]) G H

H Proof. The key observation is that M[H] is an open submanifold in M , H where M is a smooth QK submanifold of M. Hence, itself M[H] is a smooth QK manifold. The restriction of μ to M[H] is the moment map for the locally free −1 −1 action of L, hence (μ (0) ∩ M[H])/L =(μ (0) ∩ M([H]))/G is a QK orbifold. We finish this chapter with a brief discussion of Morse theory on quaternionic Kähler manifolds. The idea to consider f = || μ||2 as the Morse function is quite natural as suggested by analogy with Kirwan’s work on symplectic quotients [Kir84, Kir98]. The function f was first introduced by Battaglia in [Bat96, Bat99] and more recently also in [ACDVP03]. The motivation behind [ACDVP03] was the fact that the so-called BPS sates in the 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 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 [ACDVP03] were not aware of her work. We describe some of the 12.5. COMPACT QUTERNIONIC KAHLER¨ ORBIFOLDS 327 results. Recall that a Morse function f is called equivariantly perfect over Q if the equivariant Morse equalities hold, that is if −1 λF Pˆt(M)=Pˆt(μ (0)) + t Pˆt(F ) where the sum ranges over the set of connected components of the fixed point set, λF is the index of F ,andPˆt is the equivariant Poincaré polynomial for the equivariant cohomology with coefficients in Q. Battaglia proves Theorem 12.4.9: Let (M 4n, Q, Ω,g) be a positive QK manifold acted on isometrically by S1. Then the non-degenerate Morse function f = *μ*2 is equivariantly perfect over Q. The critical set of f is the union of the zero set f −1(0) = μ−1(0) and the fixed point set of the circle action. Moreover, the zero set μ−1(0) is connected, and a fixed point component is either contained in μ−1(0) or does not intersect with μ−1(0). Proposition 12.4.10: Let M 4n be a positive quaternionic Kähler manifold acted on isometrically by S1. Then every connected component of the fixed point set, not contained in μ−1(0),isaKähler 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. Battagila uses Morse theory to improve the results obtained earlier in [Bat96]. She shows that the quotient example 12.4.6 is unique in the following sense. Theorem 12.4.11: Let (M 4n, Q, Ω,g) be a positive QK manifold acted on isometrically by S1.SupposeS1 acts freely on N = μ−1(0).ThenM 4n is homotopic to n n+1 HP with the quotient Mˆ =Gr2(C ).

12.5. Compact Quternionic K¨ahler Orbifolds

Much of the discussion of this chapter can be generalized in the case when M is a QK orbifold. Perhaps the most interesting observation about such a generalization is that when M has orbifold singularities the orbifold Konishi bundle S(M) which, more generally now, is a principal V -bundle 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 by with earlier chapters of our book. In fact, this must happen 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. This puts quite a lot of rigidity if one looks for compact positive QK orbifolds whose associated Konishi bundle is smooth. In particular, all the orbifold uniformizing groups of M must be subgroups of SO(3) or Sp(1). We will not develop a theory of QK orbifold yet, postponing this task to the following chapter where they will become essential as part of the detailed study of the 3-Sasakian geometry of the Konishi V-bundle. In this section we will 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 [Gal87, GL88]. We will briefly describe the construction slightly generalizing the original example. The key to the construction is the Corollary n 12.4.5 of the previous section. Consider (M,g)=(HP ,gcan) and an arbitrary reduction of M be a k-dimensional Abelian subgroup of the isometry group. Such 328 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS a reduction is associated to a choice

k n+1 (12.5.1) H = T ⊂ Tmax = T ⊂ U(n +1)⊂ Sp(n +1), n n+1 where Sp(n +1)=I(HP ,gcan)andTmax = T 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 matrix Ω ∈Mk,n+1(Z) whose column vectors generate Rk. This gives the exact sequences of Lie algebras

ι Ω 0 −−→ h−−→ Rn+1 −−→ Rk −−→ 0,

Ω∗ ι∗ 0 −−→ Rk −−→ Rn+1 −−→ h∗ −−→ 0, where the map Ω : Rn → Rk is given by the matrix multiplication with Ω. There is corresponding exact sequence at the group level 1−→H−→T n+1−→T k−→1 and the k n+1 subtorus H is identiﬁed with the image of the homomorphism fΩ : T → T k k j aj a1 n+1 (12.5.2) fΩ(τ1,...,τk) := diag τj ,..., τj , j=1 j=1

n where (aij ) = Ω. Now, with each Ω we associate H acting on HP , the QK −1 ⊂ HPn momentum map μΩ, and the zero level set N(Ω) = μΩ (0) . It is elementary to check when the condition of the Corollary 12.4.5 is satisfied and we have 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 O(Ω) := Mˆ (Ω) is a compact positive QK orbifold. Furthermore, the Lie algebra i(O(Ω), gˆ(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. Remark 12.5.1: The case originally considered in [Gal87, GL88] corresponds to k =1andΩ=(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 [BGM94] (Ω = (p1,...,pn+1)) and [BGMR98] (an arbitrary Ω), where it was realized that the Konishi V-bundle of such orbifolds can often be a smooth manifold carrying a natural Einstein metric. We will return to a detailed analysis of these examples in the next chapter after we define 3-Sasakian manifolds. We will now specialize to the case of compact 4-dimensional QK orbifolds (M,g), i.e., 4-orbifolds with self-dual conformal structure, and an Einstein metric g of positive scalar curvature. Recall that when M is smooth M must be isomorphic to S4 or CP2. On the other hand, Theorem 12.5.1 alone provides plenty of examples of such spaces. As orbifolds, some of them are the familiar examples of Chapter 3 [GL88].

Proposition 12.5.2: Let O(p) be O(Ω) of Theorem 12.5.1 with Ω=(p1,p2,p3)= p, i.e., O(p) is a QK reduction of HP2 by the isometric circle action with weights p. In addition assume that all pi’s are positive integers such that gcd(p1,p2,p3)=1. Then there is smooth orbifold equivalence

CP2 p +p p +p p +p , when p1 + p2 + p3 is odd, 1 2 , 2 3 , 3 1 (12.5.3) O(p) 2 2 2 CP2 p1+p2,p2+p3,p3+p1 , when p1 + p2 + p3 is even. 12.5. COMPACT QUTERNIONIC KAHLER¨ ORBIFOLDS 329

Furthermore, the metrics g(p) defined by QK reduction are not locally symmetric, unless p =(1, 1, 1) in which case we get the Fubini-Study metric on CP2. 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) has been discovered by Dearricott [Dea04, Dea05] and by Blaˇzić and Vukmirović[BV04]. First we have the following result Dearricott

Theorem 12.5.3: Let O(p) be the Galicki-Lawson orbifold with p1 ≤ p2 ≤ p3. Then the self-dual Einstein metric g(p) is of positive sectional curvature if and only if − − − − − − 3 (12.5.4) σ3(p1 + p2 + p3, p1 p2 + p3, p1 + p2 p3,p1 p2 p3) > 4p3, where σ3 denotes the third symmetric polynomial in 4 variables. The paper of Blaˇzić 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 12.5.4: The self-dual√ Einstein√ metric on O(p)=O(p, q, q) has positive sectional curvature if p2

When Ω ∈Mn−1,n+1(Z) the orbifold structure of O(Ω) is more involved. How- ever, as each O(Ω) has two commuting Killing vectors, locally these metrics are described by the following remarkable results of Calderbank and Pedersen [CP02] Theorem 12.5.5: Let F (ρ, η) be a solution of the linear diﬀerential equation 3F F + F = ρρ ηη 4ρ2 on some open subset of the half-space ρ>0, and consider the metric g(ρ, η, φ, ψ) given by F 2 − 4ρ2(F 2 + F 2) dρ2 + dη2 g = ρ η 4F 2 ρ2

(12.5.5) 2 2 (F − 2ρFρ)α − 2ρFηβ + −2ρFηα +(F +2ρFρ)β + , 2 2 − 2 2 2 F F 4ρ (Fρ + Fη ) √ √ where α = ρdφ and β =(dψ + ηdφ)/ ρ.Then: 2 2 2 2 (i) On the open set where F > 4ρ (Fρ + Fη ), g is a self-dual Einstein metric of positive scalar curvature, whereas on the open set where 0 < 2 2 2 2 − F < 4ρ (Fρ + Fη ), g is a self-dual Einstein metric of negative scalar curvature. 330 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS

(ii) Any self-dual Einstein metric of nonzero scalar curvature with two linearly independent commuting Killing fields is arises locally in this way (i.e., in a neighborhood of any point, it is of the form (12.5.5) up to a constant multiple). The above 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 HPn? 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 [?]. 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 [CS05] Theorem 12.5.6: 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 cov- erings, (O,g) is isometric to a quaternion Kähler quotient of quaternionic projective space HPn,forsomen ≥ 1,bya(n − 1)-dimensional subtorus of Sp(n +1). There is yet another family of orbifold metrics due to Hitchin [Hit95a, Hit96, Hit95b]. These metrics come from solutions of the Painlevé VI equation and as such were also introduced by Tod [Tod94]. We will describe these metrics in some details here and come back to them once more in the next chapter. Consider the space V T (12.5.6) V = {B ∈M3,3(R) |B = B, Tr(B)=0} of traceless symmetric 3 × 3 matrices with inner product A1, A2 =tr(B1B2). Clearly, V R5 and SO(3) acts on V by conjugation B → g−1Bg, g ∈ SO(3) and the unit sphere M = S4 in V can be described as matrices in V whose eigenvalues {λ1,λ2,λ3} satisfy 3 3 − 2 (12.5.7) λi =0=1 λi i=1 i=1 As SO(3)-action on this model of S4 is of cohomogeneity one and the orbit space M/SO(4) is an interval [−1, +1]. The associated group diagram has the structure

(12.5.8) SO(3) . n7 PgP nn PP + j− nn PPj nnn PPP nnn PP K− = O(2) K = O(2) PgP n7+ PPP nnn PPP nnn PPP nnn h− P nn h+ D = Z2 × Z2 with the generic orbit SO(3)/D, where D = Z2 × Z2 ⊂ SO(3) is the subgroup of diagonal matrices which is the stablizer of any generic point. Two degenerate 2 orbits are B± := SO(3)/K± RP . The degenerate obits correspond to the subset of matrices with two equal eigenvalues. If two eigenvalues are equal, then they 1 must be equal to ± √ , and the two signs correspond to the two orbits B± and 6 the subgroups K± = O(2). The diagram 12.5.8 allows for the decomposition (see 12.5. COMPACT QUTERNIONIC KAHLER¨ ORBIFOLDS 331 discussion at the beginning of section 13.8 for the structure of cohomogeneity one manifolds)

(12.5.9) S4 = RP2 ∪ (0, +∞) × (SO(3)/D) ∪ RP2. 2 Explicitly, we can parameterize the conic 12.5.7 by observing that (λ1 − λ2) + 2 3(λ1 + λ2) = 2 so that (12.5.10) √ √ 2 1 2 1 λ1(t)= α(t)+√ β(t),λ2(t)=− α(t)+√ β(t),λ3(t)=−2β(t), 2 6 2 6 where α2 + β2 = 1. Choose the standard rational parameterization 2t 1 − t2 α(t)= ,β(t)= . t2 +1 t2 +1 Note that t = 0 gives λ = λ = √1 and t =+∞ the other degenerate orbit 1 2 6 1 4 λ = λ = − √ so that we have an explicit diffeomorphism S \{B−,B } 2 2 6 + (0, +∞) × SO(3)/D given via (t, g) → g−1Δ(t)g, where g ∈ SO(3) and Δ(t)= diag(λ1(t),λ2(t),λ3(t)). 4 Any SO(3)-invariant metric on S \{B+,B−} defines and invariant metric on each orbit SO(3)/D. It follows that any such metric must be of the form 2 2 2 2 2 2 2 (12.5.11) g = f(t)dt +[T1(t)] σ1 +[T2(t)] σ2 +[T3(t)] σ3, where {σ1,σ2σ3} is the basis of invariant one-forms. The equations for the most general self-dual and Einstein metric with non-zero scalar curvature Λ and in the diagonal form 12.5.11 has been derived by Tod [Tod94]. It follows that dx2 σ2 (1 − x)σ2 xσ2 (12.5.12) Fg = + 1 + 2 + 3 , − 2 2 2 x(x 1) F1 F2 F3 where {F1(x),F2(x),F3(x)} satisfy the following first order system of ODEs dF F F dF F F dF F F (12.5.13) 1 = − 2 3 , 2 = − 3 3 , 3 = − 1 2 dx x(1 − x) dx x dx x(1 − x) and the conformal factor (12.5.14) 8xF 2F 2F 2 +2F F F (x(F 2 + F 2) − (1 − 4F 2)(F 2 − (1 − x)F 2)) −4ΛF = 1 2 3 1 2 3 1 2 3 2 1 2 − − 2 2 (xF1F2 +2F3(F2 (1 x)F1 ))

The expression for the conformal factor is algebraic in x, F1,F2,F3 so that the problem reduces to solving the system 12.5.13. It turns out that this system can be reduced to a single second order ODE: the Painlevé VI equation d2y 1 1 1 1 dy 2 1 1 1 dy = + + − + + + dx2 2 y y − 1 y − x dx x x − 1 y − x dx (12.5.15) y(y − 1)(y − x) x x − 1 x(x − 1) + α + β + γ + δ , x2(x − 1)2 y2 (y − 1)2 (y − x)2 where (α, β, γ, δ)=(1/8, −1/8, 1/8, 3/8). One can define an auxiliary variable z by dy y(y − 1)(y − x) 1 1 1 (12.5.16) = 2z − + + dx x(x − 1) 2y 2(y − 1) 2(y − x) 332 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS which then allows to express the original functions {F1,F2,F3} in terms of any solution y = y(x) of the equation 12.5.15: y(y − 1)(y − x)2 1 1 F 2 = z − z − 1 x(1 − x) 2(y − 1) 2y y2(y − 1)(y − x) 1 1 (12.5.17) F 2 = z − z − 1 x 2(y − x) 2(y − 1) y(y − 1)2(y − x) 1 1 F 2 = z − z − . 1 1 − x 2y 2(y − x) In a series of papers Hitchin analyzed the Painlevé VI equation giving an algebraic geometry description of the solutions in terms of isomonodromic deformations [Hit95a, Hit96, Hit95b]. 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 {F1,F2,F3} in terms of the elliptic functions. In particular, Hitchin shows Theorem 12.5.7: Choose an integer k ≥ 3 and consider the SO(3)-invariant metric gk defined on (1, ∞)×SO(3)/D by the formula 12.5.12 via the corresponding solution of the Painlevé VI equation with the metric coefficients Fi = Fi(x, k), i =1, 2, 3.

(i) The metric gk is a positive deﬁnite self-dual Einstein of positive scalar curvature for all 1

In an arc length parametrization g4 becomes 2 2 2 2 2 2 2 (12.5.19) g4 = dt +sin tσ1 +cos 2tσ2 +cos tσ3 , 2 which is indeed locally the Fubini-Study metric on CP . The orbifold (O4,gk)has orb Z CP2 π1 = 2 and its universal cover is ( ,gFS). Just to illustrate how complicated the metric coeﬃcients get for larger values of k we also give explicit formulas for k =6, 8. For k = 6 (the orbifold singularity at angle π/2)) one gets s(s2 + s +1) s3(2s +2) y = , with x = . (2s +1) s +1 This yields 3(1 + s + s2) f(s;6) = , s (s +2)2 (2s +1)2 3(1 + s + s2) [T (s; 6)]2 = , 1 (s +2)(2s +1)2 3(s2 − 1)2 [T (s; 6)]2 = , 2 (1 + s + s2)(s +2)(2s +1) 3s (1 + s + s2) [T (s; 6)]2 = . 3 (s +2)2 (2s +1) For k = 8 (the orbifold singularity at angle π/3)) we get 4s(3s2 − 2s +1) 2s 4 y = , with x = . (s + 1)(1 − s)3(s2 +2s +3) 1 − s2 This yields 4(1 + s)(3 − 2s + s2)(1 − 2s +3s2)(1 + 2s +3s2) f(s;8) = , (1 − s) s (1 + s2)(1 + 2s − s2)2 (3 + 2s + s2)2 4(1 − s)(1 + s)3 (3 − 2s + s2)(1 − 2s +3s2) [T (s; 8)]2 = , 1 (1 + 2s − s2)(3 + 2s + s2)2 (1 + 2s +3s2) 4(1 + s2)(3 − 2s + s2)(1 − 2s − s2)2(1 + 2s +3s2) [T (s; 8)]2 = , 2 (1 + 2s − s2)2(3 + 2s + s2)2(1 − 2s +3s2) 16s (1 − 2s +3s2)(1 + 2s +3s2) [T (s; 8)]2 = . 3 (1 + 2s − s2)(3 − 2s + s2)(3 + 2s + s2)2 We summarize the above discussion with the following statement Theorem 12.5.8: Let (O,g) be a compact positive QK 4-orbifold and suppose that dimension of i(O,g) is at least 2. Then, up to orbifold cover, (O,g) is isometric to one of the following (i) the Galicki-Lawson toric orbifold O(Ω) obtained vas a QK reduction of some HPn via an action of some (n − 1)-torus subgroup of Sp(n +1), 334 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS

(ii) the Hitchin orbifold O(k) with its SO(3)-invariant metrics given by a solution of the Painlevé VI equation. Theorems of Hitchin [Hit95b] Calderbank and Pedersen [CP02] and Calder- bank and Singer [CS05] are milestones in the broader problem of the classification of all compact self-dual Einstein 4-orbifolds without any assumption about the symmetries. This more general question appears to be very hard. No compact orbifolds without any Killing vector fields are known at the moment. Just as in the case of smooth positive QK manifolds of dimension greater than 16 here as well the biggest puzzle is Question 12.5.1: Are there any compact positive QK 4-orbifolds (O,g) with no continuous symmetries, i.e., i(O,g)=0? We remark that several quotient constructions of compact positive QK 4- orbifolds with 1-dimensional isometry group have been studied [BGP02]. We will discuss some of them in the next chapter. It is logical to attempt a classification of such orbifold metrics first but even such a classification does not appear to be in sight.

12.6. Hypercomplex and Hyperhermitian Structures

Deﬁnition 12.6.1: A smooth manifold M is called almost hypercomplex if it admits a smooth action of the algebra H of quaternions on the tangent bundle TM. More precisely, for any quaternion q ∈ H we have a smooth section I(q)of End(TM), which satisﬁes (12.6.1) I(q) ◦I(q)=I(qq)

Clearly I(1) = 1l and we introduce the triple of almost complex structures I(ea)= Ia, a =1, 2, 3 to recover the conventional deﬁnition. Furthermore, for any τ ∈ S2 = {q ∈ H | q¯ = −q and |q|2 =1} the endomorphism I(τ ) is an almost complex structure. Thus an almost hypercomplex manifold comes with an S2- worth of almost complex structures. If τ is an imaginary quaternion of norm 1 then we can think of it as a vector on a unit sphere in R3. We can rewrite 12.6.1 as (12.6.2) I(τ ) ◦I(τ )=− <τ , τ > 1l + I(τ × τ ), where <τ , τ > is the standard inner product in R3 and τ × τ is the cross-product. It is often convenient to choose a particular hypercomplex structure, that is a triple {I1,I2,I3} such that I1 and I2 anticommute and I3 = I1 ◦ I2. However, this is only a convention and more generally we allow such a choice to be speciﬁed 3 by any orthonormal basis in R . Choosing the standard right-handed basis e1 = (1, 0, 0),e2 =(0, 1, 0),e3 =(0, 0, 1) and setting Ia = I(ea) we can rewrite 12.6.2 relative to basis {e1,e2,e3} as

(12.6.3) Ia ◦ Ib = −δab1l + abcIc, Conversely, let M be an almost complex manifold with two anti-commuting almost complex structures I1,I2. Then we can deﬁne I3 = I1 ◦ I2 and then the triple {I1,I2,I3} together with the identity endomorphism induces a smooth action of the algebra of quaternions H on TM. The action of H makes each tangent space into a left H-module isomorphic to Hn. In particular the dimension of any such M equals 4n and that the structure 12.6. HYPERCOMPLEX AND HYPERHERMITIAN STRUCTURES 335 group of the frame bundle reduces to GL(n, H). Hence, we can equivalently say that Deﬁnition 12.6.2: A smooth manifold M is called almost hypercomplex if it admits a GL(n, H)-structure. Let g be a metric on M such that

(12.6.4) g(I(τ )X, I(τ )Y )=g(X, Y ), for any τ ∈ S2 nd X, Y ∈ Γ(TM). Such a metric is called adapted to the hypercomplex structure I(τ )orhyperhermitian. It is easy to see that such a metric always exists. Definition 12.6.3: An almost hypercomplex manifold (M,I(τ ),g) with an adapted metric g is called an almost hyperhermitian manifold. Equivalently, a smooth manifold M 4n is called almost hyperhermitian if it admits an Sp(n)-structure. Obata showed that every almost hyperhermitian manifold M 4n admits a can- nonical GL(n, H)-invariant connection [Oba66]. This connection is often called Obata connection. Definition 12.6.4: An almost hypercomplex manifold (M,I(τ ) is called hypercomplex if all complex structures I(τ ), τ ∈ S2 are integrable. A hypercomplex manifold with and adapted metric is called hyperhermitian. 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. Hence, one often defines a hypercomplex manifold as Definition 12.6.5: A smooth manifold M 4n is called hypercomplex if it admits an GL(n, H)-structure with torsion-free connection. Compact hyperhermitian 4-manifolds were classified by Boyer [Boy88]who proved Theorem 12.6.6: Let (M,I(τ ),g) be a compact hyperhermitian 4-manifold. Then (M,I(τ ),g) is conformally uquivalent to one of the following (i) a 4-torus with its flat metric, (ii) aK3surfacewithaKähler Ricci-flat metric, (iii) a coordinate quaternionic Hopf surface with its standard locally conformally flat metric. In higher dimensions there are only partial classification results. For example, it is known which Lie groups admit such structures [SSTVP88, Joy92]. The simplest example here is G = U(2) which is also a Hopf surface and it actually admits two commuting hypercomplex structures. More generally Theorem 12.6.7: Let G be a compact Lie group. Then there exist and integer 0 ≤ k ≤ max{3, rk(G)} such that U(1)k × G has a homogeneous hypercomplex structure. We shall discuss some examples of compact hypercomplex manifolds in the next chapter. Here we recall the quotient construction of Joyce [Joy91] which we shall use later. Let (M,I(τ )) be a hypercomplex manifold. We consider any compact Lie group H ⊂ Aut(M,I1,I2,I3) of hypercomplex automorphisms of M, i.e, H acts smoothly on M preserving each complex structure in {I1,I2,I3}. In particular, H 336 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS acts on M as complex automorphisms with respect to any of the complex structure in {I(τ )}τ ∈S2 .

Deﬁnition 12.6.8: Let (M,I1,I2,I3) be a hypercomplex manifold. Given a compact Lie subgroup H ⊂ Aut(M,I1,I2,I3) a hypercomplex moment map is any ∗ H-equivariant function μ = i1μ1 + i2μ2 + i3μ3 : M → h ⊗ sp(1) satisfying both of the following condition ∗ ∗ (i) I1dμ1 = I2dμ2 = I3dμ3, where Ia acts on sections Γ(T M ⊗ h ). (ii) For any non-zero element ζ ∈ h and the induced vector ﬁeld Xζ ∈ Γ(TM) I1dμ1(Xζ ) =0 on M. Note that the condition (i) of 12.6.8 is equivalent to requiring that the complex valued function μa + iμb on a complex manifold (M,Ic) be holomorphic function with respect to the complex structure Ic, this for any cyclic permutation (a, b, c)of (1, 2, 3). Joyce proves the following [Joy91]

Theorem 12.6.9: Let (M,I1,I2,I3) be a hypercomplex manifold H aanycompact Lie group in Aut(M,I1,I2,I3). Choose any hypercomplex moment map μ and let ∗ ∗ ζ = i1ζ1 +i2ζ2 +i3ζ3 ∈ h ⊗sp(1),where all three ζi are in the center of h .Suppose −1 H-action on Nζ := μ (ζ) has only finite isotropy groups and Mˆ (ζ)=Nζ/H is an orbifold. Then Mˆ (ζ) 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 following sections the hypercomplex reduction is much more flexible in the way one chooses the associated moment map.

12.7. Hyperk¨ahler Manifolds

Given a an almost hyperhermitian manifold (M,I(τ ),g) we can use the metric to deﬁne the 2-forms (12.7.1) ω(τ )(X, Y )=g(I(τ )X, Y ),X,Y∈ Γ(TM).

In particular, using the basis {I1,I2,I3} we get the three fundamental 2-forms {ω1,ω2,ω3}. By analogy with the almost Kähler case consider Definition 12.7.1: An almost hyperhermitian manifold (M,I(τ ),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 in the Kähler case almost HK manifold must automatically be HK [Hit87]. In fact we have the following equivalent characterization of hyperkähler manifolds Theorem 12.7.2: Let (M 4n, I(τ ),g) be an almost hyperhermitian manifold with the fundamental 2-forms ωa(X, Y )=g(IaX, Y ), a =1, 2, 3 defined for any basis {I1,I2,I3}. Then the following conditions are equivalent (i) (M 4n, I(τ ),g) is HK, (ii) (M 4n, I(τ ),g) is hyperhermitian, (iii) ∇I1 = ∇I2 = ∇I3 =0, (iv) Hol(g) ⊂ Sp(n). 12.7. HYPERKAHLER¨ MANIFOLDS 337

In particular, n HK manifold is K¨ahler with respect to any choice of complex structure in I(τ ). The holonomy reduction implies that HK manifolds must be ∧ Ricci-ﬂat. Of course the 4-form Ω = a ωa 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

Quaternionic K¨ahler Quaternionic Sp(n)Sp(1) −−−−→ GL(n, H)H∗ ∇Ω=0 Oproiu connection ∇ ⏐ ⏐ ⏐ ⏐ ⏐ n>1 ⏐

Hyperk¨ahler Hypercomplex Sp(n) −−−−→ GL(n, H) ∇ω1 =∇ω2 =∇ω3 =0 Obata connection ∇

Figure 1. Qauternionic geometries in dimension ≥ 8.

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

Self-Dual Einstein −−−−→ Self-Dual Conformal Sp(1)Sp(1) R∗ × SO(4) ⏐ ⏐ ⏐ ⏐ ⏐ n =1 ⏐

Hyperk¨ahler −−−−→ Hypercomplex Sp(1) H∗

Figure 2. Qauternionic 4-manifolds.

When M 4 is a compact HK manifold then, up to cover, it must be either the K3 surface or a flat torus. On the other hand, if we do not insist on compactness the question of classification of such metrics remains wide open. Only partial classification results are known. All these metrics are important in the General Relativity Theory as they are vacuum solutions (Ricci-flat) of the Euclidean Einstein equations. Such solutions (not just in Riemannian signature) are called gravitational instantons. In the Riemannian case, the only known complete gravitational instanton metrics are in fact HK. They will all be described as certain quotients in the next section. 338 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS

Proposition 12.7.3: Let (M 4n, I(τ ),g) be an HK manifold and consider the K¨ahler structure (I1,g,ω1). The from ω+ = ω2 + iω3 is of type (2, 0), i.e., it is a holomorphic symplectic form on M.

Proof. Let {z1,...,z2n} be a holomorphic local chart on (I1,g,ω1). Consider the 2-form ω+(X, Z)=g(I2X, Y )+ig(I3X, Y ) and extend it by linearity to the ∂ complexiﬁed tangent bundle TCM. Setting X = we have compute ω+(∂z¯ ,Y) ∂z¯j j ∂ ∂ ∂ ∂ g I2 ,Y + ig I3 ,Y = ig I2I1 ,Y + ig I3 ,Y =0, ∂z¯j ∂z¯j ∂z¯j ∂z¯j since I2I1 = −I3.

An important tool in the study of HK geometry, just as in the QK case is the twistor space introduced by Hitchin in [HKLR87]. Let (M 4n, I(τ ),g)beanHK manifold and consider the product Z = CP1 × M, where CP1 S2 is a 2-sphere with its standard complex structure J0. A point z in Z as a pair z =(τ ,x), where x ∈ M and τ ∈ S2, defining a complex structure I(τ )onM. The tangent space 1 TzZ = Tτ CP ⊕ TxM so we can define an almost complex structure on TzZ by setting J = J0 ⊕I(τ ). It is a simple computation to show Theorem/Definition 12.7.4: Let (M 4n, I(τ ),g) be an HK manifold with (Z,J) defined as above. Then (Z,J) is integrable so that Z is a complex manifold of complex dimension 2n +1. It is called the twistor space of M. Note that Z is a product as a differentiable manifold but its complex structure is not a product. Let p : Z→CP1 and π : Z→M be the projection maps. Then p is a holomorphic map whose fiber over any point τ ∈ S2 is a complex manifold (M,I(τ )). We define σ : Z→Zby σ(τ ,x)=(−τ ,x) which is identity on M and the antipodal map on S2. Such σ is a free antiholomorphic involution on Z. The fibers of π are holomorphic curves in Z isomorphic to CP1 with normal bundle 2nO(1) invariant under σ. Finally, the fibers p−1(τ ) which are complex manifolds (M,I(τ )) for each τ ∈ S2 have a complex symplectic 2-form holomorphic with respect to I(τ ). For example, ω2 + iω3 is a complex symplectic 2-form on (M,I1) holomorphic with respect to I1. These 2-forms can be assembled together to give a non-degenerate holomorphic section of certain bundle. More precisely we have [HKLR87] Theorem 12.7.5: Let (M 4m, I(τ ),g) be an HK manifold with (Z,J) its twistor space. Let p : Z→CP1 and π : Z→M the projections and σ : Z→Zthe involution defined above. Then (i) π ◦ σ = π, (ii) p is holomorphic and p ◦ σ = σ ◦ p,whereσ is the antipodal map on CP1. (iii) there exists a holomorphic section ω ∈ Γ(p∗(O(2)) ⊗ Λ2D∗) with D = ker(dp : T Z→T CP1) and σ∗(ω)=ω. Conversely, Hitchin shows that one can reconstruct HK metric out of such holomorphic data. More precisely [HKLR87, Joy00] Theorem 12.7.6: Let (Z,J) be a complex manifold of complex dimension (2n+1) equipped with the following data (i) a holomorphic projection p : Z→CP2, 12.8. HYPERKAHLER¨ QUOTIENTS 339

(ii) a holomoprhic section ω of p∗O(2) ⊗ Λ2D∗, which is symplectic on the ﬁbers of D = ker(dp : T Z→T CP1), (iii) a free antiholomorphic involution σ : Z→Zsuch that σ∗(ω)=ω,and p ◦ ω = ω ◦ p,withω : CP1 → CP1 the antipodal map. Let M be the set of all holomorphic curves C in Z isomorphic to CP1, with normal bundle 2nO(1) and σ(C)=C (so-called twistor lines). Then M is a hypercomplex manifold with a natural pseudo-HK metric g.Ifg is positive deﬁnite the M is HK.

12.8. Hyperk¨ahler Quotients

In this section we review the generalization of the Marsden-Weinstein construction to HK manifolds with hyperholomorphic isometries. Such reduction was first considered by Lindström and Roˇcek as early as in 1983 [LR83], while in- vestigating 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 Roˇcek observed that one can “gauge away” hyperholomorphic symmetries. In the process one introduces auxiliary gauge fields without kinetic terms in the La- grangian, 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 quotient or hyperkähler reduction [HKLR87]. We will describe this construction and some of the basic examples as it provides blueprint for much of the material of the next chapter. Let (M,I(τ ),g) be an HK manifold and G ⊂ Aut(M,I(τ ),g) ⊂ I(M,g) be a Lie group acting smoothly and properly on M by preserving the metric and the hypercompelx structure. Then G acts by sym- plectomorphism preserving symplectic forms ωa, a =1, 2, 3. Suppose the G-action is Hamiltonian with respect to each symplectic form ωa. We will call such an action hyperhamiltonian. Definition 12.8.1: A hyperkähler manifold (M,I(τ ),g) together with an effective hyperhamiltonian G-action is called a hyperhamiltonian G-manifold. As discussed in section 7.4.1 such action gives raise to three G-equivariant ∗ symplectic moment maps μa : M → h . We can assemble these maps together to get Definition 12.8.2: Let (M,I(τ ),g) be a hyperkähler H-manifold. The map μ = (μ1,μ2,μ3)=i1μ1 + i2μ2 + i3μ3 (12.8.1) μ = M → g∗ ⊗ sp(1) 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(τ ),g) be HK and G be a hyperhamiltonian action on ∗ ∗ M with the HK moment map μ : M → g ⊗ sp(1).Letλ =(λ1,λ2λ3) ∈ h ⊗ sp(1) be any element fixed by the co-adjoint action of G on its Lie co-algebra g∗.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 340 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS

HK manifold (orbifold) with the HK structure induced from M via inclusion and projection maps.

Proof. We only sketch proof here. The manifold (M,I1,ω1,g)) is a K¨ahler manifold. The HK reduction can be seen as a two step process: First, we consider the function

(12.8.2) μ+ := μ2 + i1μ3, −1 which is easily seen to be holomorphic on (M,I1,g). Thus the set N+ := μ+ (0) is a complex subspace of (M,I1,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 N N+ N+. The action of G restricts to ∗ N+ with the Kähler moment map μ1 : N+ → g . 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 Iˆ1 induced from M by the quotient construction. Now, the result follows by observing that the same argument applies to I2 and I3, and Mˆ is therefore Kähler with respect to all three complex structures {Iˆ1, Iˆ2, Iˆ3}. 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. That was done by Dancer and Swann [DS97, Swa97]who showed Theorem 12.8.4: Let (M,I(τ ),g) be a hyperhamiltonian G-manifold with the moment map μ : M−−→h∗ ⊗ sp(1). Further suppose G is acts smoothly and properly on M.LetM(H) denote the stratum consisting of orbits of type H

In complex charts we get n 2 2 (12.8.4) μ1(w, z)=i (|wj | −|zj| ),μ+(w, z)=−2i wjzj, j=1 j and the circle action reads (w, z) → (eitw,e−itz). One could consider an arbitrary level set of the moment map. As Sp(1)+ is the symmetry of the ﬂat HK metric and one can use it to reset ζ to be a constant multiple of i. Further scaling the metric μ−1 −1 ⊂ shows that it is suﬃcient to consider N = (i)andN+ = μ+ (i) N which are described as n n n 2 2 (12.8.5) N = (w, z) ∈ C × C | (|wj | −|zj| )=1, wjzj =0 , j=1 j n n (12.8.6) N+ = (w, z) ∈ C × C | wjzj =0 . j

−1 1 Let Mˆ = μ (i)/S . 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 N+ its Kähler ⊂ reduction N+ will be an algebraic quotient of N+ N+ by the complexification C∗ 1 of G = S . Note, however, that N+ is not compact so we cannot relay on the Kirwan’s theorems in this setting setting[Kir84]1. Nevertheless, we get the following identification (12.8.7) Mˆ = μ−1(i)/S1 N /C∗, + ss { ∈ Cn × Cn | } C∗ where N+ = (w, z) j wjzj =0, w = 0 ,and acts by (w, z) → (λw, λ¯z). This is clearly the holomorphic cotangent bundle T ∗CPn−1.It turns out that the HK metric obtained on T ∗CPn−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 Roˇcek [LR83] and, in the above language it appears in [HKLR87]. Locally, in dimension 4, this metric was discovered by Eguchi and Hanson [EH79] and it is called the Eguchi-Hanson gravitational instanton. Example 12.8.6: This example involves a non-compact hyperhamiltonian group n n+1 n action of G = R on H × H H defined for any p(p1,...,pn) ∈ R by

p1it pnit (12.8.8) φp(u0,u1,...,un)=(u0 − t, e u1,...,e un), with the moment map n (12.8.9) μp(u)=iμ1 + μ+j =2Im(u0) − ipku¯kiuk. j=1 We can always shift to zero-level set of the moment map and then the moment map equations can be “solved” by writing n (12.8.10) Im(u0)= pku¯kiuk. j=1

1This is not a special feature of this example. On the contrary, this is what typically happens with HK reductions of Hn by compact hyperhamiltonian G-actions. 342 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS

This action is free and proper on H × Hn. In particular it is free and proper on −1 −1 μp (0) := Np. Denote the quotient manifold by M(p):=μp (0)/R. First, note that M(p) is diffeomorphic to Hn. This follows from the observation that the set −1 S = {u ∈ μp (0) | Re(u0)=0} is the global slice for this action. The induced HK metricg ˆ(p) can easily be calculated and g(0)=g0 is the flat metric. In dimension 4, this metricg ˆ(Λ) depends on one parameter Λ and when Λ = 0 we get M(0) isomorphic to H C2 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) is called the Taub-NUT 2 gravitational instanton and it has interesting history. Just as the famous Schwarzschield metric, or Kerr solution, the metric appears first in the Lorenzian signature. One can always perform the so-called “Wick rotation” changing t → it which locally gives a Riemannian metric with similar properties. However, there is no reason for the Riemannian metric to extend globally to a complete metric on some manifold. This is rare and happens only in very special situation. It was Hawking who observed that this indeed is the case of the Lorenzian Taub-NUT solution, giving rise to a complete Ricci flat metric on R4 [Haw77]. For some time after that the metric was not really understood and literature is full of wrong assertions about this metric being not Kähler. We should point out that this is the only complete Ricci-flat Kähler metric on C2 apart from the standard one. If one imposes Euclidean volume growth condition the only known example of such complete Ricci-flat Kähler (or just Ricci-flat) metric on C2 is the Euclidean one.

12.9. Toric Hyperk¨ahler Metrics

It is easy to see that previous two examples fall into a special category of complete hyperkähler manifolds: they admit n commuting hyperholomorphic Killing vector fields, where n is the quaternionic dimension. Following Bielawski and Dancer we consider [BD00, Bie99] Definition 12.9.1: An HK manifold (M 4n, I(τ ),g) is locally toric if it admits n commuting hyperholomorphic Killing vector fields, linearly independent at each point x ∈ M 4n, i.e., locally M admits a free action of Rn by hyperholomorphic isometries. Furthermore, (M 4n, I(τ ),g) is said to be a toric HK manifold if it is a hyperhamiltonian T n-space. A local description of such metrics in dimension 4 is due to Gibbons and Hawk- ing [GH78a] and in arbitrary dimension 4n it has been generalized by Lindström and Roˇcek [LR83]. The so-called Legendre transform method developed by Lind- strOm´ and Roˇcek associates a 4n-dimensional hyperkähler metric with n commuting hyperholomorphic Killing vectors to every real-valued function F on an open subset U⊂R3 ⊗ Rn which is harmonic on any affine 3-dimensional subspace L of the form R3 ⊗ Rv, v ∈ Rn (such functions are sometimes called polyharmonic). The construction proceeds as follows: Let us identify R3 ⊗Rn with Rn ×C2n and let (x, z) ∈ Rn × C2n be coordinates on U. Given any polyharmonic function F (z, z¯) on U we consider a real-valued function n (12.9.1) K(u, u¯, z, z¯)=F (x, z, z¯) − 2 (ui +¯ui)xi, i=1

2The acronym NUT stands for Newmann-Unti-Tamburino though it is also a standard term describing singularities of solutions of the Einstein’s gravitational equations. 12.9. TORIC HYPERKAHLER¨ METRICS 343 where the xi are determined by ∂F (12.9.2) =2(ui +¯ui). ∂xi 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(z¯ − z) then Xi = ∂/∂yi,i=1,...,n yield n commuting hyperholomorphic Killing vector fields with the corresponding hyperkähler moment maps μi i i (12.9.3) =(μ1,μ+)=(xi,zi),i=1,...,n. One can show that, relative to local coordinates (y, x, z), the HK metric takes the form [PP88] −1 (12.9.4) g = Φij (dxidxj + dzidz¯j)+(Φ )ij (dyi + Ai)(dyj + Aj) , i,j √ 1 −1 − where Φij = 4 Fxixj and Aj = 2 l(Fxj z¯l dz¯l Fxj zl dzl). The functions Φij are also polyharmonic. The n × n matrix [Φij ] locally determines the hyperkähler and hyperamiltonian structure. When n = 1 this is well-known Gibbons-Hawking Ansatz. Renaming (y1,x1,z1):=(t, x1,x2 + ix3)=(t, x), A1 = α · dx,Φ11 = V we can write the metric in a more familiar form (12.9.5) g = V (dx · dx)+V −1(dt + α · dx)2, where gradV = curlα. In particular, V is a solution of the Laplace equation so that we can write k m (12.9.6) V (x ,x ,x )=δ + i . 1 2 3 |x − a | i=1 i When δ = 0 these metrics are called k-center gravitational multi-instantons. First two values of 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 is the manifold Mk on which it is defined. When δ = 1 we get the so-called k-center Taub-NUT gravitational multi- instantons with k = 1 corresponing to the Eunclidean Taub-NUT metric discussed in Example 12.8.6. The two basic examples of this construction are flat S1-invariant metrics on S1 × R3 and on H. In the first case we have (12.9.7) F (x, z, z¯)=2x2 − zz¯ and consequently Φ ≡ 1, while in the second case (12.9.8) F (x, z, z¯)=x ln(x + r) − r where r2 = x2 + zz¯. This time Φ = 1/4r. More general forms are given [BD00]. In the latter, the functions F and the metrics for hyperkähler 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) (see also the proof of Theorem 1 in section ??). Our aim is to show 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 (see definition 2). For a metric of the form (12.5.5) taking hyperkähler quotients by subtori is very simple. Indeed, the moment map equations are now linear (in xi,zi), and the 344 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS hyperkähler quotient corresponds to restricting the function F to an appropriate affine subspace of R3 ⊗ Rn. In fact, the requirement that F be polyharmonic is a consequence of the fact that we must be able to take hyperkähler quotients by any subtorus. An explanation of this construction in terms of twistors was given by Hitchin, Karlhede, Lindström and Roˇcek [HKLR87]. In particular, they have shown that any hyperkähler 4n-manifold with a free tri-Hamiltonian Rn-action which extends to a Cn-action with respect to each complex structure and such that the moment map is surjective is given by the Legendre transform. In the next section we shall show that any hyperkähler 4n-manifold with a free local tri-Hamiltonian Rn-action is locally given by the Legendre transform. One can show that the Legendre transform provied a complete local description of such metric, i.e., [Bie99] Proposition 12.9.2: Let M 4n be a locally toric HK manifold. Then M is locally given by the Legendre transform of Lindström and Roˇcek. The key to the understanding of the global properties of such metrics is the HK quotient construction. This problem was firstmetrics as hyperkähler quotients were studied later by Bielawski and Dancer [BD00] culminating in a complete classification by Bielawski [Bie99]. We will discuss this classification here. Definition 12.9.3: Let M,M be two hyperhamiltonian G-manifolds and let μ, μ be the chosen moment maps. We say that M and M are isomorphic hyperkähler G-manifolds, if there is a hyperholomorphic G-equivariant isometry f : M → M such that μ = μ ◦ f. The previous corollary suggests the following definition. Definition 12.9.4: Let M 4n be as in Theorem 1. A Taub-NUT deformation (of order m)ofM is the hyperkähler quotient of M × Hm by Rm where Rm acts on M via an injective linear map ρ : Rm → Lie (T n)=Rn. Such a deformation M is canonically T n-equivariantly diffeomorphic to M by a diffeomorphism f which respects the hyperkähler moment maps μ, μ, i.e. μ = μ◦f. Theorem 12.9.5: Let M 4n be a connected complete hyperkähler manifold of finite topological type with an effective tri-Hamiltonian action of G = Rp × T n−p.Then

(i) If M is simply connected and p =0, then M is isomorphic, as a tri- Hamiltonian hyperkähler T n-manifold, to a hyperkähler quotient of some flat Hd × Hm, m ≤ n,byT d−n × Rm. (ii) If M is simply connected and p>0, then M is isomorphic, as a tri- Hamiltonian hyperkähler G-manifold, to the product of a flat Hp and a − 4(n p)-dimensional manifold described in part (i). l (iii) If M is not simply connected, then M is the product of a flat (S1 × R3 , 1 ≤ l ≤ n,anda4(n − l)-dimensional manifold described in part (ii). Corollary 12.9.6: Let M be a simply connected 4-dimensional complete hyperkähler manifold with a nontrivial tri-Hamiltonian vector field. If b2(M)=k>0, then M is isometric either to an ALE-space of type Ak (i.e. a multi-Eguchi-Hanson space) or to its Taub-NUT-like deformation (i.e. to the hyperkähler quotient by R of the product of such a space with H). If b2(M)=0, then M is the flat H 12.9. TORIC HYPERKAHLER¨ METRICS 345

Theorem 12.9.7: Complete connected HK hyperhamiltonian T n-manifolds of finite topological type and dimension 4n are classified, up to Taub-NUT deformations, by arrangements of codimension 3 affine subspaces in R3 ⊗ Rn of the form { 1 2 3 ∈ R3 ⊗ Rn i i } Hk = (x ,x ,x ) ; x ,uk = λk,i=1, 2, 3 Rn i for some finite collection of vectors uk in and scalars λk, i =1, 2, 3, such that, 3 n n for any p ∈ R ⊗ R , the set {uk; p ∈ Hk} is part of a Z-basis of Z . Definition 12.9.8: Bibliography

[AB68] M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 (1968), 451–491. MR MR0232406 (38 #731) [Abb84] E. Abbena, An example of an almost Kähler manifold which is not Kählerian, Boll. Un. Mat. Ital. A (6) 3 (1984), no. 3, 383–392. MR 86a:53036 [ABC+96] J. Amorós, M. Burger, K. Corlette, D. Kotschick, and D. Toledo, Fundamental groups of compact Kähler manifolds, Mathematical Surveys and Monographs, vol. 44, Amer- ican Mathematical Society, Providence, RI, 1996. MR 97d:32037 [Abr01] M. Abreu, Kähler metrics on toric orbifolds,J.DifferentialGeom.58 (2001), no. 1, 151–187. MR MR1895351 (2003b:53046) [ACDVP03] D. V. Alekseevsky, V. Cortés, C. Devchand, and A. Van Proeyen, Flows on quaternionic-Kähler and very special real manifolds, Comm. Math. Phys. 238 (2003), no. 3, 525–543. MR MR1993384 (2004m:53080) [ADM03] V. Apostolov, T. Draghici, and A. Moroianu, The odd-dimensional Goldberg Con- jecture, arXiv:math.DG/0308241; preprint, 2003. [AGnZV85] V. I. Arnold, S. M. Guse˘ı n Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. I, Monographs in Mathematics, vol. 82, Birkhäuser Boston Inc., Boston, MA, 1985, The classification of critical points, caustics and wave fronts, Translated from the Russian by Ian Porteous and Mark Reynolds. MR 86f:58018 [AGV72] M. Artin, A. Grothendieck, and J. L. Verdier, Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Springer-Verlag, Berlin, 1972, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics, Vol. 269. MR 50 #7130 [AK94] N. A’Campo and D. Kotschick, Contact structures, foliations, and the fundamental group, Bull. London Math. Soc. 26 (1994), no. 1, 102–106. MR 94j:57027 [Akh90] D. N. Akhiezer, Homogeneous complex manifolds, Several Complex Variables IV (G.M. Khenkin, ed.), Encyclopedia of mathematical Sciences, Vol. 10, Springer- Verlag, Berlin, 1990, Algebraic aspects of complex analysis, A translation of Sovre- mennye problemy matematiki. Fundamentalnye napravleniya, Tom 10, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1986 [MR 87m:32003], Translation by J. Leiterer and J. Nunemacher, Translation edited by S. G. Gindikin and G. M. Khenkin, pp. 223–275, 283. MR 88i:32047 [Akh95] , Lie group actions in complex analysis, Aspects of Mathematics, E27, Friedr. Vieweg & Sohn, Braunschweig, 1995. MR MR1334091 (96g:32051) [Alb89] C. Albert, Le théorème de réduction de Marsden-Weinstein en géométrie cosym- plectique et de contact,J.Geom.Phys.6 (1989), no. 4, 627–649. MR 91k:58033 [Ale68] D. V. Alekseevski˘ı, Riemannian spaces with unusual holonomy groups, Funkcional. Anal. i Priloˇzen 2 (1968), no. 2, 1–10. MR 37 #6868 [Ale75] , Classification of quaternionic spaces with transitive solvable group of motions, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 2, 315–362, 472. MR 53 #6465 [AM78] R. Abraham and J. E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, Mass., 1978, Second edition, revised and enlarged, With the assistance of Tudor Rat¸iu and Richard Cushman. MR 81e:58025 [AMP97] L. Astey, E. Micha, and G. Pastor, Homeomorphism and diffeomorphism types of Eschenburg spaces, Differential Geom. Appl. 7 (1997), no. 1, 41–50. MR 98h:53072

429 430 BIBLIOGRAPHY

[And90] M. T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990), no. 2, 429–445. MR 92c:53024 [Ano69] D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature., Proceedings of the Steklov Institute of Mathematics, No. 90 (1967). Translated from the Russian by S. Feder, American Mathematical Society, Providence, R.I., 1969. MR 39 #3527 [Arn78] V. I. Arnold, Mathematical methods of classical mechanics, Springer-Verlag, New York, 1978, Translated from the Russian by K. Vogtmann and A. Weinstein, Grad- uate Texts in Mathematics, 60. MR 57 #14033b [Ati82] M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1–15. MR MR642416 (83e:53037) [Aub82] T. Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sci- ences], vol. 252, Springer-Verlag, New York, 1982. MR 85j:58002 [Aub98] , Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR MR1636569 (99i:58001) [AW75] S Aloff and N. R. Wallach, An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures, Bull. Amer. Math. Soc. 81 (1975), 93–97. MR 51 #6851 [Bai56] W. L. Baily, The decomposition theorem for V -manifolds, Amer. J. Math. 78 (1956), 862–888. MR 20 #6537 [Bai57] , On the imbedding of V -manifolds in projective space, Amer. J. Math. 79 (1957), 403–430. MR 20 #6538 [Ban90] Augustin Banyaga, A note on Weinstein’s conjecture, Proc. Amer. Math. Soc. 109 (1990), no. 3, 855–858. MR 90m:58170 [Ban97] A. Banyaga, The structure of classical diffeomorphism groups, Mathematics and its Applications, vol. 400, Kluwer Academic Publishers Group, Dordrecht, 1997. MR MR1445290 (98h:22024) [Bar65] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR 32 #1714 [Bär93] Ch. Bär, Real Killing spinors and holonomy, Comm. Math. Phys. 154 (1993), no. 3, 509–521. MR 94i:53042 [Bat96] F. Battaglia, S1-quotients of quaternion-Kähler manifolds, Proc. Amer. Math. Soc. 124 (1996), no. 7, 2185–2192. MR MR1307492 (96i:53045) [Bat99] , Circle actions and Morse theory on quaternion-Kähler manifolds, J. London Math.Soc.(2)59 (1999), no. 1, 345–358. MR MR1688506 (2000h:53060) [BD00] R. Bielawski and A. S. Dancer, The geometry and topology of toric hyperkähler manifolds, Comm. Anal. Geom. 8 (2000), no. 4, 727–760. MR 2002c:53078 [Bea98] A. Beauville, Fano contact manifolds and nilpotent orbits, Comment. Math. Helv. 73 (1998), no. 4, 566–583. MR MR1639888 (99e:14046) [Bea05] , Riemannian holonomy and algebraic geometry, preprint, math.AG/9902110 (2005). [Bel00] F. A. Belgun, On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), no. 1, 1–40. MR 2002c:32027 [Bel01] , Normal CR structures on compact 3-manifolds,Math.Z.238 (2001), no. 3, 441–460. MR 2002k:32065 [Ber55] M. Berger, Sur les groupes d’holonomie homogène des variétésa ` connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955), 279–330. MR 18,149a [Bes87] A. L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 88f:53087 [BFGK91] H. Baum, Th. Friedrich, R. Grunewald, and I. Kath, Twistors and Killing spinors on Riemannian manifolds, Teubner-Texte zur Mathematik [Teubner Texts in Math- ematics], vol. 124, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1991, With German, French and Russian summaries. MR 94a:53077 [BG67] D. E. Blair and S. I. Goldberg, Topology of almost contact manifolds, J. Differential Geometry 1 (1967), 347–354. MR 37 #2128 BIBLIOGRAPHY 431

[BG97] C. P. Boyer and K. Galicki, The twistor space of a 3-Sasakian manifold,Internat.J. Math. 8 (1997), no. 1, 31–60. MR 98e:53072 [BG00a] , A note on toric contact geometry,J.Geom.Phys.35 (2000), no. 4, 288–298. MR 2001h:53124 [BG00b] , On Sasakian-Einstein geometry, Internat. J. Math. 11 (2000), no. 7, 873– 909. MR 2001k:53081 [BG01a] , Einstein manifolds and contact geometry, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2419–2430 (electronic). MR 2001m:53071 [BG01b] , New Einstein metrics in dimension five, J. Differential Geom. 57 (2001), no. 3, 443–463. MR 2003b:53047 [BG03] Charles P. Boyer and Krzysztof Galicki, New Einstein metrics on 8#(S2 × S3), Differential Geom. Appl. 19 (2003), no. 2, 245–251. MR 2 002 662 [BG04] D. Burns and V. Guillemin, Potential functions and actions of tori on Kähler manifolds, Comm. Anal. Geom. 12 (2004), no. 1-2, 281–303. MR MR2074879 (2005e:53133) [BG05] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, preprint, arXiv:math.DG/0509189 (2005). [BGK05] C. P. Boyer, K. Galicki, and J. Kollár, Einstein metrics on spheres, Ann. of Math. (2) 162 (2005), no. 1, 557–580. [BGKT05] C. P. Boyer, K. Galicki, J. Kollár, and E. Thomas, Einstein metrics on exotic spheres in dimensions 7, 11, and 15, Experiment. Math. 14 (2005), no. 1, 59–64. MR MR2146519 [BGM93] C. P. Boyer, K. Galicki, and B. M. Mann, Quaternionic reduction and Einstein manifolds, Comm. Anal. Geom. 1 (1993), no. 2, 229–279. MR 95c:53056 [BGM94] , The geometry and topology of 3-Sasakian manifolds, J. Reine Angew. Math. 455 (1994), 183–220. MR 96e:53057 [BGM96a] , Hypercomplex structures on Stiefel manifolds, Ann. Global Anal. Geom. 14 (1996), no. 1, 81–105. MR 97e:53084 [BGM96b] , On strongly inhomogeneous Einstein manifolds, Bull. London Math. Soc. 28 (1996), no. 4, 401–408. MR 97a:53069 [BGM98a] , Hypercomplex structures from 3-Sasakian structures, J. Reine Angew. Math. 501 (1998), 115–141. MR 99g:53049 [BGM98b] , A note on smooth toral reductions of spheres, Manuscripta Math. 95 (1998), no. 2, 149–158. MR 99c:53042 [BGMR98] C. P. Boyer, K. Galicki, B. M. Mann, and E. G. Rees, Compact 3-Sasakian 7- manifolds with arbitrary second Betti number, Invent. Math. 131 (1998), no. 2, 321–344. MR 99b:53066 [BGN02] C. P. Boyer, K. Galicki, and M. Nakamaye, Sasakian-Einstein structures on 9#(S2 × S3), Trans. Amer. Math. Soc. 354 (2002), no. 8, 2983–2996 (electronic). MR 2003g:53061 [BGN03a] , On the geometry of Sasakian-Einstein 5-manifolds, Math. Ann. 325 (2003), no. 3, 485–524. MR 2004b:53061 [BGN03b] , Sasakian geometry, homotopy spheres and positive Ricci curvature, Topol- ogy 42 (2003), no. 5, 981–1002. MR 1 978 045 [BGN03c] Charles P. Boyer, Krzysztof Galicki, and Michael Nakamaye, On positive Sasakian geometry, Geom. Dedicata 101 (2003), 93–102. MR MR2017897 (2005a:53072) [BGO05] C. P. Boyer, K. Galicki, and L. Ornea, Constructions in Sasakian Geometry,in preparation (2005). [BGP02] C. P. Boyer, K. Galicki, and P. Piccinni, 3-Sasakian geometry, nilpotent orbits, and exceptional quotients, Ann. Global Anal. Geom. 21 (2002), no. 1, 85–110. MR 2003d:53076 [BH58] A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I,Amer. J. Math. 80 (1958), 458–538. MR MR0102800 (21 #1586) [Bie96] R. Bielawski, On the hyper-Kähler metrics associated to singularities of nilpotent varieties, Ann. Global Anal. Geom. 14 (1996), no. 2, 177–191. MR 97e:53083 [Bie97] , Betti numbers of 3-Sasakian quotients of spheres by tori, Bull. London Math. Soc. 29 (1997), no. 6, 731–736. MR 98j:53051 432 BIBLIOGRAPHY

[Bie99] , Complete hyper-Kähler 4n-manifolds with a local tri-Hamiltonian Rn- action, Math. Ann. 314 (1999), no. 3, 505–528. MR 2001c:53058 [Bla70] D. E. Blair, Geometry of manifolds with structural group U(n) × O(s), J. Differential Geometry 4 (1970), 155–167. MR MR0267501 (42 #2403) [Bla76a] , Contact manifolds in Riemannian geometry, Springer-Verlag, Berlin, 1976, Lecture Notes in Mathematics, Vol. 509. MR 57 #7444 [Bla76b] , On the non-existence of flat contact metric structures,Tôhoku Math. J. (2) 28 (1976), no. 3, 373–379. MR 55 #4029 [Bla77] , Two remarks on contact metric structures,Tôhoku Math. J. (2) 29 (1977), no. 3, 319–324. MR 57 #4043 [Bla02] , Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, vol. 203, Birkhäuser Boston Inc., Boston, MA, 2002. MR 2002m:53120 [BM93] A. Banyaga and P. Molino, Géométrie des formes de contact complètement intégrables de type toriques,Séminaire Gaston Darboux de Géométrie et Topolo- gie Différentielle, 1991–1992 (Montpellier), Univ. Montpellier II, Montpellier, 1993, pp. 1–25. MR 94e:53029 [BM96] , Complete integrability in contact geometry, 1996, pp. 1–25. [Bon67] E. Bonan, Sur les G-structures de type quaternionien, Cahiers Topologie Géom. Différentielle 9 (1967), 389–461. MR 38 #1624 [Boo80] W. M. Boothby, On compact homogeneous contact manifolds, Bull. Inst. Math. Acad. Sinica 8 (1980), no. 2-3, part 2, 341–351. MR MR595542 (82a:53052) [Bor54] A. Borel, Kählerian coset spaces of semisimple Lie groups, Proc. Nat. Acad. Sci. U. S. A. 40 (1954), 1147–1151. MR 17,1108e [Bor60] , Density properties for certain subgroups of semi-simple groups without compact components, Ann. of Math. (2) 72 (1960), 179–188. MR MR0123639 (23 #A964) [Bou89] N. Bourbaki, Commutative algebra. Chapters 1–7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989, Translated from the French, Reprint of the 1972 edition. MR MR979760 (90a:13001) [Bou02] F. Bourgeois, Odd dimensional tori are contact manifolds, Int. Math. Res. Not. (2002), no. 30, 1571–1574. MR 2003f:53157 [Boy88] C. P. Boyer, A note on hyper-Hermitian four-manifolds, Proc. Amer. Math. Soc. 102 (1988), no. 1, 157–164. MR MR915736 (89c:53049) [BPVdV84] W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 86c:32026 [BR62] A. Borel and R. Remmert, Uber¨ kompakte homogene Kählersche Mannigfaltigkeiten, Math. Ann. 145 (1961/1962), 429–439. MR MR0145557 (26 #3088) [BR86] M. Beltrametti and L. Robbiano, Introduction to the theory of weighted projective spaces, Exposition. Math. 4 (1986), no. 2, 111–162. MR 88h:14052 [Bre72] G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972, Pure and Applied Mathematics, Vol. 46. MR 54 #1265 [Bre93] , Topology and geometry, Graduate Texts in Mathematics, vol. 139, Springer- Verlag, New York, 1993. MR 94d:55001 [Bre97] Glen E. Bredon, Sheaf theory, second ed., Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, 1997. MR MR1481706 (98g:55005) [Bri66] E. Brieskorn, Beispiele zur Differentialtopologie von Singularitäten, Invent. Math. 2 (1966), 1–14. MR 34 #6788 [Bri68] , Rationale Singularitäten komplexer Flächen, Invent. Math. 4 (1967/1968), 336–358. MR MR0222084 (36 #5136) [Bri00] , Singularities in the work of Friedrich Hirzebruch, Surveys in differential geometry, Surv. Differ. Geom., VII, Int. Press, Somerville, MA, 2000, pp. 17–60. MR MR1919421 (2003g:32002) [Bro73] W. Browder, Cobordism invariants, the Kervaire invariant and fixed point free involutions, Trans. Amer. Math. Soc. 178 (1973), 193–225. MR MR0324717 (48 #3067) [Bro82] K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1982. MR MR672956 (83k:20002) [Bru70] G. Brumfiel, The homotopy groups of BPL and PL/O. III, Michigan Math. J. 17 (1970), 217–224. MR MR0271938 (42 #6819) BIBLIOGRAPHY 433

[Bry87] R. L. Bryant, Metrics with exceptional holonomy, Ann. of Math. (2) 126 (1987), no. 3, 525–576. MR 89b:53084 [BV04] N. Blaˇzić and S. Vukmirović, Examples of self-dual, Einstein metrics of (2, 2)- signature, Math. Scand. 94 (2004), no. 1, 63–74. MR MR2032336 (2004j:53056) [BW58] W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math. (2) 68 (1958), 721–734. MR 22 #3015 [Cal56] E. Calabi, The space of kähler metrics, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Amsterdam 2, 1954) (Amsterdam), North-Holland, 1956, pp. 206–207. [Cal57] , On Kähler manifolds with vanishing canonical class, Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N. J., 1957, pp. 78–89. MR 19,62b [Cal79] , Métriques kählériennes et fibrés holomorphes, Ann. Sci. Ecole´ Norm. Sup. (4) 12 (1979), no. 2, 269–294. MR 83m:32033 [Cam04] F. Campana, Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 3, 499–630. MR MR2097416 [Car84a] Y. Carrière, Flots riemanniens,Astérisque (1984), no. 116, 31–52, Transversal structure of foliations (Toulouse, 1982). MR 86m:58125a [Car84b] , Les propriétés topologiques des flots riemanniens retrouvéesa ` l’aide du théorème des variétés presque plates,Math.Z.186 (1984), no. 3, 393–400. MR 86f:53048 [CC00] A. Candel and L. Conlon, Foliations. I, Graduate Studies in Mathematics, vol. 23, American Mathematical Society, Providence, RI, 2000. MR 2002f:57058 [CC03] , Foliations. II, Graduate Studies in Mathematics, vol. 60, American Mathe- matical Society, Providence, RI, 2003. MR 1 994 394 [CDG02] D. M. J. Calderbank, L. David, and P. Gauduchon, The Guillemin formula and Kähler metrics on toric symplectic manifolds, J. Symplectic Geom. 1 (2002), no. 4, 767–784. MR MR2039163 (2005a:53145) [CdS01] A. Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, vol. 1764, Springer-Verlag, Berlin, 2001. MR MR1853077 (2002i:53105) [CdS03] , Symplectic toric manifolds, Symplectic geometry of integrable Hamiltonian systems (Barcelona, 2001), Adv. Courses Math. CRM Barcelona, Birkhäuser, Basel, 2003, pp. 85–173. MR MR2000746 [CdSW99] A. Cannas da Silva and A. Weinstein, Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, vol. 10, American Mathematical Society, Providence, RI, 1999. MR 2001m:58013 [CFdL85] L. A. Cordero, M. Fernández, and M. de León, Examples of compact non-Kähler almost Kähler manifolds, Proc. Amer. Math. Soc. 95 (1985), no. 2, 280–286. MR 86m:53049 [CFG86] L. A. Cordero, M. Fernández, and A. Gray, Symplectic manifolds with no Kähler structure, Topology 25 (1986), no. 3, 375–380. MR 87j:53051 [CFG93] , The failure of complex and symplectic manifolds to be Kählerian, Differential geometry: geometry in mathematical physics and related topics (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 107–123. MR 94e:53071 [Che55] C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782. MR MR0072877 (17,345d) [CL85] C. Camacho and N. A. Lins, Geometric theory of foliations,Birkhäuser Boston Inc., Boston, MA, 1985, Translated from the Portuguese by Sue E. Goodman. MR 87a:57029 [CLOT03] O. Cornea, G. Lupton, J. Oprea, and D. Tanré, Lusternik-Schnirelmann category, Mathematical Surveys and Monographs, vol. 103, American Mathematical Society, Providence, RI, 2003. MR 1 990 857 [CLS90] P. Candelas, M. Lynker, and R. Schimmrigk, Calabi-Yau manifolds in weighted P4, Nuclear Phys. B 341 (1990), no. 2, 383–402. MR 91m:14062 [CM00] M. Crainic and I. Moerdijk, A homology theory for étale groupoids, J. Reine Angew. Math. 521 (2000), 25–46. MR MR1752294 (2001f:58039) 434 BIBLIOGRAPHY

[CMS96] F. M. Cabrera, M. D. Monar, and A. F. Swann, Classification of G2-structures,J. London Math. Soc. (2) 53 (1996), no. 2, 407–416. MR 97e:53047 [Con74] L. Conlon, Transversally parallelizable foliations of codimension two, Trans. Amer. Math. Soc. 194 (1974), 79–102, erratum, ibid. 207 (1975), 406. MR 51 #6844 [Cor96] V. Cortés, Alekseevskian spaces, Differential Geom. Appl. 6 (1996), no. 2, 129–168. MR MR1395026 (97m:53079) [Cor00] , A new construction of homogeneous quaternionic manifolds and related geometric structures,Mem.Amer.Math.Soc.147 (2000), no. 700, viii+63. MR MR1708628 (2001e:53047) [CP02] D. M. J. Calderbank and H. Pedersen, Selfdual Einstein metrics with torus symmetry, J. Differential Geom. 60 (2002), no. 3, 485–521. MR MR1950174 (2003m:53065) [CR00] W. Chen and Y. Ruan, A New Cohomology Theory for Orbifold, 2000, electronic preprint; arXiv:math.AG/0004129. [Cro01] D. Crowley, The classification of highly connected manifolds in dimension 7 and 15, Indiana University Thesis (2001). [CS05] D. M. J. Calderbank and M. A. Singer, Toric selfdual Einstein metrics on compact orbifolds, arXiv:math.DG/0405020 (2005). [CV98]ˇ M. Cadekˇ and J. Vanˇzura, Almost quaternionic structures on eight-manifolds,Osaka J. Math. 35 (1998), no. 1, 165–190. MR MR1621248 (99f:57031) [Dea04] O. Dearricott, Positive sectional curvature on 3-Sasakian manifolds, Ann. Global Anal. Geom. 25 (2004), no. 1, 59–72. MR MR2033140 (2005f:53060) [Dea05] , Positively curved self-dual Einstein metrics on weighted projective planes, Ann. Global Anal. Geom. 27 (2005), no. 1, 79–86. MR MR2130534 [Del75] Ch. Delorme, Erratum: “Espaces projectifs anisotropes” (Bull. Soc. Math. France 103 (1975), no. 2, 203–223), Bull. Soc. Math. France 103 (1975), no. 4, 510. MR 53 #8080b [Del88] Th. Delzant, Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France 116 (1988), no. 3, 315–339. MR MR984900 (90b:58069) [DGMS75] P. Deligne, P. Griffiths, J. Morgan, and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274. MR 52 #3584 [Dim92] A. Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992. MR 94b:32058 [DK75] A. Durfee and L. Kauffman, Periodicity of branched cyclic covers, Math. Ann. 218 (1975), no. 2, 157–174. MR 52 #6731 [DK79] T. Duchamp and M. Kalka, Deformation theory for holomorphic foliations,J.Dif- ferential Geom. 14 (1979), no. 3, 317–337 (1980). MR 82b:57019 [DK80] , Stability theorems for holomorphic foliations, Trans. Amer. Math. Soc. 260 (1980), no. 1, 255–266. MR 81f:57022 [DK84] , Holomorphic foliations and deformations of the Hopf foliation,PacificJ. Math. 112 (1984), no. 1, 69–81. MR 85m:57019 [DK90] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds,Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1990, Oxford Science Publications. MR MR1079726 (92a:57036) [DK00] J. J. Duistermaat and J. A. C. Kolk, Lie groups, Universitext, Springer-Verlag, Berlin, 2000. MR 2001j:22008 [DMR82] A. D´ıaz Miranda and A. Reventós, Homogeneous contact compact manifolds and homogeneous symplectic manifolds, Bull. Sci. Math. (2) 106 (1982), no. 4, 337–350. MR MR688194 (84d:53034) [dMT97] S. F. B. de Moraes and C. Tomei, Moment maps on symplectic cones,PacificJ. Math. 181 (1997), no. 2, 357–375. MR 98m:58038 [DO04] O. Dragulete and L. Ornea, Non-zero contact and Sasakian reduction, arXiv:math.DG/0407434 (2004). [Dol82] I. Dolgachev, Weighted projective varieties, Group actions and vector fields (Vancou- ver, B.C., 1981), Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 34–71. MR 85g:14060 [DOR02] O. Dr˘agulete, L. Ornea, and T. S. Ratiu, Cosphere bundle reduction in contact geometry, J. Symplectic Geom. 1 (2002), no. 4, 695–714. MR MR2039161 (2004m:53141) BIBLIOGRAPHY 435

[DP04] J.-P. Demailly and M. Paun, Numerical characterization of the Kähler cone of acompactKähler manifold, Ann. of Math. (2) 159 (2004), no. 3, 1247–1274. MR MR2113021 [DS97] A. Dancer and A. Swann, The geometry of singular quaternionic Kähler quotients, Internat. J. Math. 8 (1997), no. 5, 595–610. MR MR1468352 (98m:53062) [Dur71] A. Durfee, Diffeomorphism Classification of Isolated Hypersurfaces Singularities, Cornell University Thesis (1971). [Dur77] A. H. Durfee, Bilinear and quadratic forms on torsion modules, Advances in Math. 25 (1977), no. 2, 133–164. MR MR0480333 (58 #506) [dWVP92] B. de Wit and A. Van Proeyen, Special geometry, cubic polynomials and homogeneous quaternionic spaces, Comm. Math. Phys. 149 (1992), no. 2, 307–333. MR MR1186031 (94a:53079) [EH79] T. Eguchi and A. J. Hanson, Self-dual solutions to Euclidean gravity, Ann. Physics 120 (1979), no. 1, 82–106. MR MR540896 (80h:83011) [Ehr53] Ch. Ehresmann, Introductionalath´ ` eorie des structures infinitésimales et des pseudogroupes de Lie, Colloque de topologie et géométrie différentielle, Strasbourg, 1952, no. 11, La Bibliothèque Nationale et Universitaire de Strasbourg, 1953, p. 16. MR 15,828e [EKA90] A. El Kacimi-Alaoui, Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, Compositio Math. 73 (1990), no. 1, 57–106. MR 91f:58089 [EKAH86] A. El Kacimi-Alaoui and G. Hector, Décomposition de Hodge basique pour un feuilletage riemannien, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 3, 207–227. MR 87m:57029 [EKAN93] A. El Kacimi-Alaoui and M. Nicolau, On the topological invariance of the basic cohomology, Math. Ann. 295 (1993), no. 4, 627–634. MR MR1214951 (94c:57046) [Eli98] Y. Eliashberg, Invariants in contact topology, Proceedings of the International Con- gress of Mathematicians, Vol. II (Berlin, 1998), no. Extra Vol. II, 1998, pp. 327–338 (electronic). MR 2000a:57068 [Eps72] D. B. A. Epstein, Periodic flows on three-manifolds, Ann. of Math. (2) 95 (1972), 66–82. MR MR0288785 (44 #5981) [Esc82] J.-H. Eschenburg, New examples of manifolds with strictly positive curvature,Invent. Math. 66 (1982), no. 3, 469–480. MR 83i:53061 [Esc84] , Freie isometrische Aktionen auf kompakten Lie-Gruppen mit positiv gekrümmten Orbiträumen, Schriftenreihe des Mathematischen Instituts der Univer- sität Münster, 2. Serie [Series of the Mathematical Institute of the University of Münster, Series 2], vol. 32, Universität Münster Mathematisches Institut, Münster, 1984. MR 86a:53045 [Esc92] , Cohomology of biquotients, Manuscripta Math. 75 (1992), no. 2, 151–166. MR 93e:57070 [Fan04] F. Fang, Positive quaternionic Kähler manifolds and symmetry rank,J.Reine Angew. Math. 576 (2004), 149–165. MR MR2099202 (2005i:53060) [Fan05] , Positive quaternionic Kähler manifolds and symmetry rank, ii., Math. Res. Lett. (2005). [FG82] M. Fernández and A. Gray, Riemannian manifolds with structure group G2, Ann. Mat. Pura Appl. (4) 132 (1982), 19–45 (1983). MR 84e:53056 [FGG88] M. Fernández, M. J. Gotay, and A. Gray, Compact parallelizable four-dimensional symplectic and complex manifolds, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1209– 1212. MR 90a:53039 [Fin77] R. Fintushel, Circle actions on simply connected 4-manifolds, Trans. Amer. Math. Soc. 230 (1977), 147–171. MR MR0458456 (56 #16659) [Fin78] , Classification of circle actions on 4-manifolds, Trans. Amer. Math. Soc. 242 (1978), 377–390. MR MR496815 (81e:57036) [FIP04] M. Falcitelli, S. Ianus, and A. M. Pastore, Riemannian submersions and related topics, World Scientific Publishing Co. Inc., River Edge, NJ, 2004. MR MR2110043 [FK82] Th. Friedrich and H. Kurke, Compact four-dimensional self-dual Einstein manifolds with positive scalar curvature, Math. Nachr. 106 (1982), 271–299. MR 84b:53043 [FK90a] Th. Friedrich and I. Kath, 7-dimensional compact Riemannian manifolds with Killing spinors, Comm. Math. Phys. 133 (1990), no. 3, 543–561. MR 91j:58170 436 BIBLIOGRAPHY

[FK90b] , Compact 5-dimensional Riemannian manifolds with parallel spinors,Math. Nachr. 147 (1990), 161–165. MR 92m:53080 [FKMS97] Th. Friedrich, I. Kath, A. Moroianu, and U. Semmelmann, On nearly parallel G2- structures,J.Geom.Phys.23 (1997), no. 3-4, 259–286. MR 98j:53053 [Fri80] Th. Friedrich, Der erste Eigenwert des Dirac-Operators einer kompakten, Rie- mannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr. 97 (1980), 117–146. MR 82g:58088 [Fuj66] T. Fujitani, Complex-valued differential forms on normal contact Riemannian manifolds,Tôhoku Math. J. (2) 18 (1966), 349–361. MR 35 #3596 [Fuj72] A. Fujimoto, Theory of G-structures, Study Group of Geometry, Department of Applied Mathematics, College of Liberal Arts and Science, Okayama University, Okayama, 1972, English edition, translated from the original Japanese, Publications of the Study Group of Geometry, Vol. 1. MR 50 #1161 [Fuj87] A. Fujiki, On the de Rham cohomology group of a compact Kähler symplectic manifold, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North- Holland, Amsterdam, 1987, pp. 105–165. MR 90d:53083 [Ful84] W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR MR732620 (85k:14004) [Ful93] , Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993, The William H. Roever Lectures in Geometry. MR 94g:14028 [Fur86] M. Furushima, Singular del Pezzo surfaces and analytic compactifications of 3-dimensional complex affine space C3, Nagoya Math. J. 104 (1986), 1–28. MR MR868434 (88m:32051) [Gal87] K. Galicki, A generalization of the momentum mapping construction for quaternionic Kähler manifolds, Comm. Math. Phys. 108 (1987), no. 1, 117–138. MR 88f:53088 [Gei91] H. Geiges, Contact structures on 1-connected 5-manifolds,Mathematika38 (1991), no. 2, 303–311 (1992). MR 93e:57042 [Gei97a] , Constructions of contact manifolds, Math. Proc. Cambridge Philos. Soc. 121 (1997), no. 3, 455–464. MR MR1434654 (98f:53027) [Gei97b] , Normal contact structures on 3-manifolds, Tohoku Math. J. (2) 49 (1997), no. 3, 415–422. MR 98h:53046 [GH78a] G. W. Gibbons and S. W. Hawking, Gravitational multi-instantons, Phys. Lett. B 78 (1978), 430–432. MR MR0465052 (57 #4965) [GH78b] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978, Pure and Applied Mathematics. MR 80b:14001 [GH03] K. Guruprasad and A. Haefliger, Closed geodesics on orbifolds, University of Geneve preprint (2003). [GHS83] J. Girbau, A. Haefliger, and D. Sundararaman, On deformations of transversely holomorphic foliations, J. Reine Angew. Math. 345 (1983), 122–147. MR 84j:32026 [GHV73] W. Greub, S. Halperin, and R. Vanstone, Connections, curvature, and cohomology. Vol. II: Lie groups, principal bundles, and characteristic classes, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973, Pure and Applied Mathematics, Vol. 47-II. MR 49 #1424 [Gif69a] C. H. Giffen, Smooth homotopy projective spaces, Bull. Amer. Math. Soc. 75 (1969), 509–513. MR MR0239607 (39 #964) [Gif69b] , Weakly complex involutions and cobordism of projective spaces, Ann. of Math. (2) 90 (1969), 418–432. MR MR0253346 (40 #6561) [GL88] K. Galicki and H. B. Lawson, Jr., Quaternionic reduction and quaternionic orbifolds, Math. Ann. 282 (1988), no. 1, 1–21. MR 89m:53075 [GL02] V. Guillemin and E. Lerman, Melrose-Uhlmann projectors, the metaplectic representation and symplectic cuts, J. Differential Geom. 61 (2002), no. 3, 365–396. MR 2004e:53126 [GLS96] V. Guillemin, E. Lerman, and S. Sternberg, Symplectic fibrations and multiplicity diagrams, Cambridge University Press, Cambridge, 1996. MR MR1414677 (98d:58074) [GM80] X. Gómez-Mont, Transversal holomorphic structures, J. Differential Geom. 15 (1980), no. 2, 161–185 (1981). MR 82j:53065 BIBLIOGRAPHY 437

[GM88] M. Goresky and R. MacPherson, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988. MR MR932724 (90d:57039) [GN92] K. Galicki and T. Nitta, Nonzero scalar curvature generalizations of the ALE hyper- Kähler metrics,J.Math.Phys.33 (1992), no. 5, 1765–1771. MR 93d:53056 [GO98] P. Gauduchon and L. Ornea, Locally conformally Kähler metrics on Hopf surfaces, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 4, 1107–1127. MR 2000g:53088 [GO01] G/ Grantcharov and L. Ornea, Reduction of Sasakian manifolds,J.Math.Phys.42 (2001), no. 8, 3809–3816. MR MR1845220 (2002e:53060) [God91] C. Godbillon, Feuilletages, Progress in Mathematics, vol. 98, Birkhäuser Verlag, Basel, 1991, Etudes´ géométriques. [Geometric studies], With a preface by G. Reeb. MR 93i:57038 [Gol69] S. I. Goldberg, Integrability of almost Kaehler manifolds, Proc. Amer. Math. Soc. 21 (1969), 96–100. MR 38 #6514 [Gol86] , Nonnegatively curved contact manifolds, Proc. Amer. Math. Soc. 96 (1986), no. 4, 651–656. MR 87d:53068 [Gol02] R. F. Goldin, An effective algorithm for the cohomology ring of symplectic reductions, Geom. Funct. Anal. 12 (2002), no. 3, 567–583. MR MR1924372 (2003m:53148) [Gom95] R. E. Gompf, A new construction of symplectic manifolds, Ann. of Math. (2) 142 (1995), no. 3, 527–595. MR 96j:57025 [GR84] H. Grauert and R. Remmert, Coherent analytic sheaves, Grundlehren der Mathema- tischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 265, Springer-Verlag, Berlin, 1984. MR MR755331 (86a:32001) [Gra59] J. W. Gray, Some global properties of contact structures, Ann. of Math. (2) 69 (1959), 421–450. MR 22 #3016 [Gra67] A. Gray, Pseudo-Riemannian almost product manifolds and submersions,J.Math. Mech. 16 (1967), 715–737. MR 34 #5018 [Gra69] , A note on manifolds whose holonomy group is a subgroup of Sp(n) · Sp(1), Michigan Math. J. 16 (1969), 125–128. MR 39 #6226 [Gra71] , Weak holonomy groups,Math.Z.123 (1971), 290–300. MR 45 #2614 [Gro61] A. Grothendieck, El´´ ements de géométrie algébrique. II. Etude´ globale élémentaire de quelques classes de morphismes, Inst. Hautes Etudes´ Sci. Publ. Math. (1961), no. 8, 222. MR 36 #177b [Gro81] M. Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1981), no. 2, 179–195. MR 82k:53062 [GS66] V. Guillemin and S. Sternberg, Deformation theory of pseudogroup structures,Mem. Amer. Math. Soc. No. 64 (1966), 80. MR 35 #2302 [GS82a] , Convexity properties of the moment mapping, Invent. Math. 67 (1982), no. 3, 491–513. MR MR664117 (83m:58037) [GS82b] , Homogeneous quantization and multiplicities of group representations,J. Funct. Anal. 47 (1982), no. 3, 344–380. MR 84d:58034 [GS84] , Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984. MR MR770935 (86f:58054) [GS96] K. Galicki and S. Salamon, Betti numbers of 3-Sasakian manifolds, Geom. Dedicata 63 (1996), no. 1, 45–68. MR 98e:53075 [GS99] R. E. Gompf and A. I. Stipsicz, 4-manifolds and Kirby calculus, Graduate Stud- ies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999. MR MR1707327 (2000h:57038) [Gui94a] V. Guillemin, Kaehler structures on toric varieties, J. Differential Geom. 40 (1994), no. 2, 285–309. MR MR1293656 (95h:32029) [Gui94b] , Moment maps and combinatorial invariants of Hamiltonian T n-spaces, Progress in Mathematics, vol. 122, Birkhäuser Boston Inc., Boston, MA, 1994. MR MR1301331 (96e:58064) [GY70] S. I. Goldberg and K. Yano, On normal globally framed f-manifolds,Tôhoku Math. J. (2) 22 (1970), 362–370. MR MR0305295 (46 #4425) [GY71] , Globally framed f-manifolds, Illinois J. Math. 15 (1971), 456–474. MR MR0278247 (43 #3978) 438 BIBLIOGRAPHY

[GZ02] K. Grove and W. Ziller, Cohomogeneity one manifolds with positive Ricci curvature, Invent. Math. 149 (2002), no. 3, 619–646. MR MR1923478 (2004b:53052) [Hae58] A. Haefliger, Structures feuilletées et cohomologiea ` valeur dans un faisceau de groupo¨ıdes,Comment.Math.Helv.32 (1958), 248–329. MR 20 #6702 [Hae62] , Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa (3) 16 (1962), 367–397. MR 32 #6487 [Hae70] , Feuilletages sur les variétés ouvertes, Topology 9 (1970), 183–194. MR 41 #7709 [Hae71] , Homotopy and integrability, Manifolds–Amsterdam 1970 (Proc. Nuffic Sum- mer School), Lecture Notes in Mathematics, Vol. 197, Springer, Berlin, 1971, pp. 133– 163. MR 44 #2251 [Hae84] , Groupo¨ıdes d’holonomie et classifiants,Astérisque (1984), no. 116, 70–97, Transversal structure of foliations (Toulouse, 1982). MR 86c:57026a [Har77] R. Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, No. 52. MR 57 #3116 [Haw77] S. W. Hawking, Gravitational instantons, Phys. Lett. A 60 (1977), no. 2, 81–83. MR MR0465052 (57 #4965) [HD84] A. Haefliger and Quach Ngoc Du, Appendice: une présentation du groupe fonda- mental d’une orbifold,Astérisque (1984), no. 116, 98–107, Transversal structure of foliations (Toulouse, 1982). MR MR755164 (86c:57026b) [HE73] S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge University Press, London, 1973, Cambridge Monographs on Mathematical Physics, No. 1. MR 54 #12154 [Her96] G. Hernandez, On hyper f-structures, Math. Ann. 306 (1996), no. 2, 205–230. MR MR1411345 (97m:53057) [HH86] G. Hector and U. Hirsch, Introduction to the geometry of foliations. Part A, second ed., Aspects of Mathematics, vol. 1, Friedr. Vieweg & Sohn, Braunschweig, 1986, Fo- liations on compact surfaces, fundamentals for arbitrary codimension, and holonomy. MR 88a:57048 [HH02a] H. Herrera and R. Herrera, Classification of positive quaternion-Kähler 12-manifolds, C. R. Math. Acad. Sci. Paris 334 (2002), no. 1, 43–46. MR 2003e:53056 [HH02b] , Aˆ-genus on non-spin manifolds with S1 actions and the classification of positive quaternion-Kähler 12-manifolds, J. Differential Geom. 61 (2002), no. 3, 341– 364. MR 1 979 364 [Hir67] F. Hirzebruch, Singularities and exotic spheres,Séminaire Bourbaki, Vol. 10, Soc. Math. France, Paris, 1967, pp. Exp. No. 314, 13–32. MR 1 610 436 [Hir71] , Pontrjagin classes of rational homology manifolds and the signature of some affine hypersurfaces, Proceedings of Liverpool Singularities Symposium, II (1969/1970) (Berlin), Springer, 1971, pp. 207–212. Lecture Notes in Math., Vol. 209. MR 49 #1525 [Hit74] N. J. Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1–55. MR 50 #11332 [Hit81] , Kählerian twistor spaces, Proc. London Math. Soc. (3) 43 (1981), no. 1, 133–150. MR 84b:32014 [Hit87] , The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126. MR MR887284 (89a:32021) [Hit95a] , Poncelet polygons and the Painlevéequations, Geometry and analysis (Bom- bay, 1992), Tata Inst. Fund. Res., Bombay, 1995, pp. 151–185. MR 97d:32042 [Hit95b] , Twistor spaces, Einstein metrics and isomonodromic deformations,J.Dif- ferential Geom. 42 (1995), no. 1, 30–112. MR 96g:53057 [Hit96] , A new family of Einstein metrics, Manifolds and geometry (Pisa, 1993), Sympos. Math., XXXVI, Cambridge Univ. Press, Cambridge, 1996, pp. 190–222. MR 97d:53050 [HKLR87] N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roˇcek, Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), no. 4, 535–589. MR 88g:53048 [HL82] R. Harvey and H. B. Lawson, Jr., Calibrated geometries,ActaMath.148 (1982), 47–157. MR MR666108 (85i:53058) [HM04] Andre Henriques and David S. Metzler, Presentations of noneffective orbifolds, Trans. Amer. Math. Soc. 356 (2004), no. 6, 2481–2499 (electronic). MR MR2048526 BIBLIOGRAPHY 439

[HOT63] Y. Hatakeyama, Y. Ogawa, and S. Tanno, Some properties of manifolds with contact metric structure,Tôhoku Math. J. (2) 15 (1963), 42–48. MR 26 #4295 [HS91] A. Haefliger and E.´ Salem, Actions of tori on orbifolds, Ann. Global Anal. Geom. 9 (1991), no. 1, 37–59. MR 92f:57047 [HS97] A. D. Hwang and S. R. Simanca, Extremal Kähler metrics on Hirzebruch surfaces which are locally conformally equivalent to Einstein metrics, Math. Ann. 309 (1997), no. 1, 97–106. MR 98f:58056 [HS82] I. Hasegawa and M. Seino, Some remarks on sasakian geometry—applications of Myers’ theorem and the canonical affine connection,J.HokkaidoUniv.Ed.Sect.II A 32 (1981/82), no. 1, 1–7. MR 84j:53055 [Hus66] D. Husemoller, Fibre bundles, McGraw-Hill Book Co., New York, 1966. MR 37 #4821 [HZ74] F. Hirzebruch and D. Zagier, The Atiyah-Singer theorem and elementary number theory, Publish or Perish Inc., Boston, Mass., 1974, Mathematics Lecture Series, No. 3. MR 58 #31291 [IF00] A. R. Iano-Fletcher, Working with weighted complete intersections, Explicit bira- tional geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 101–173. MR 2001k:14089 [IK72] S. Ishihara and M. Konishi, Fibred Riemannian spaces with Sasakian 3-structure, Differential geometry (in honor of Kentaro Yano), Kinokuniya, Tokyo, 1972, pp. 179– 194. MR 48 #4964 [Ish73] S. Ishihara, Quaternion Kählerian manifolds and fibred Riemannian spaces with Sasakian 3-structure,K¯odai Math. Sem. Rep. 25 (1973), 321–329. MR 48 #2943 [Ish74] , Quaternion Kählerian manifolds, J. Differential Geometry 9 (1974), 483– 500. MR 50 #1184 [Ito97] M. Itoh, Odd-dimensional tori and contact structure, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 4, 58–59. MR 98h:53048 [IY82] H. Inoue and K. Yano, The Gromov invariant of negatively curved manifolds, Topol- ogy 21 (1982), no. 1, 83–89. MR 82k:53091 [Jac90] H. Jacobowitz, An introduction to CR structures, Mathematical Surveys and Monographs, vol. 32, American Mathematical Society, Providence, RI, 1990. MR 93h:32023 [JK01] J. M. Johnson and J. Kollár, Fano hypersurfaces in weighted projective 4-spaces, Experiment. Math. 10 (2001), no. 1, 151–158. MR 2002a:14048 [Joy91] D. Joyce, The hypercomplex quotient and the quaternionic quotient, Math. Ann. 290 (1991), no. 2, 323–340. MR 92f:53052 [Joy92] , Compact hypercomplex and quaternionic manifolds, J. Differential Geom. 35 (1992), no. 3, 743–761. MR MR1163458 (93g:53064) [Joy00] , Compact manifolds with special holonomy, Oxford Mathematical Mono- graphs, Oxford University Press, Oxford, 2000. MR 2001k:53093 [Kas71] T. Kashiwada, A note on a Riemannian space with Sasakian 3-structure,Natur.Sci. Rep. Ochanomizu Univ. 22 (1971), 1–2. MR 46 #2586 [Kaw78] T. Kawasaki, The signature theorem for V -manifolds, Topology 17 (1978), no. 1, 75–83. MR 57 #14072 [Kaw79] , The Riemann-Roch theorem for complex V -manifolds, Osaka J. Math. 16 (1979), no. 1, 151–159. MR 80f:58042 [Kaw81] , The index of elliptic operators over V -manifolds, Nagoya Math. J. 84 (1981), 135–157. MR 83i:58095 [Kir84] F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31, Princeton University Press, Princeton, NJ, 1984. MR MR766741 (86i:58050) [Kir98] , Momentum maps and reduction in algebraic geometry, Differential Geom. Appl. 9 (1998), no. 1-2, 135–171, Symplectic geometry. MR MR1636303 (99e:58072) [KKS78] D. Kazhdan, B. Kostant, and S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31 (1978), no. 4, 481–507. MR MR0478225 (57 #17711) [KM63] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR 26 #5584 440 BIBLIOGRAPHY

[KM99] S. Keel and J. McKernan, Rational curves on quasi-projective surfaces,Mem.Amer. Math. Soc. 140 (1999), no. 669, viii+153. MR MR1610249 (99m:14068) [KN63] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I,Inter- science Publishers, a division of John Wiley & Sons, New York-Lond on, 1963. MR 27 #2945 [KN69] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, In- terscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR MR0238225 (38 #6501) [KO73] S. Kobayashi and T. Ochiai, Characterizations of complex projective spaces and hy- perquadrics, J. Math. Kyoto Univ. 13 (1973), 31–47. MR 47 #5293 [Kob56] S. Kobayashi, Principal fibre bundles with the 1-dimensional toroidal group,Tôhoku Math.J.(2)8 (1956), 29–45. MR 18,328a [Kob61] , On compact Kähler manifolds with positive definite Ricci tensor, Ann. of Math. (2) 74 (1961), 570–574. MR 24 #A2922 [Kob63] , Topology of positively pinched Kaehler manifolds,Tôhoku Math. J. (2) 15 (1963), 121–139. MR 27 #4185 [Kob72] , Transformation groups in differential geometry, Springer-Verlag, New York, 1972, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70. MR 50 #8360 [Kob87] , Differential geometry of complex vector bundles, Publications of the Math- ematical Society of Japan, vol. 15, Princeton University Press, Princeton, NJ, 1987, Kanô Memorial Lectures, 5. MR 89e:53100 [Kod60a] K. Kodaira, On compact complex analytic surfaces. I, Ann. of Math. (2) 71 (1960), 111–152. MR 24 #A2396 [Kod60b] , On deformations of some complex psuedo-group structures, Ann. of Math. (2) 71 (1960), 224–302. MR 22 #5992 [Kod86] , Complex manifolds and deformation of complex structures, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 283, Springer-Verlag, New York, 1986, Translated from the Japanese by Kazuo Akao, With an appendix by Daisuke Fujiwara. MR 87d:32040 [Kol92] J. Kollár, Flips and abundance for algebraic threefolds,SociétéMathématique de France, Paris, 1992, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211 (1992). MR 94f:14013 [Kol96] , Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Math- ematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. MR 98c:14001 [Kol04a] J. Kollár, Einstein metrics on 5-dimensional Seifert bundles, arXiv:math.DG/0408184 (2004). [Kol04b] , Einstein metrics on connected sums of S2 × S3, arXiv:math.DG/0402141 (2004). [Kol04c] , Seifert Gm-bundles, arXiv:math.AG/0404386 (2004). [Kol05] , Circle actions on simply connected 5-manifolds, arXiv:math.GT/0505343 (2005). [Kon75] M. Konishi, On manifolds with Sasakian 3-structure over quaternion Kaehler manifolds,K¯odai Math. Sem. Rep. 26 (1974/75), 194–200. MR 51 #13951 [Kos55] B. Kostant, Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold,Trans.Amer.Math.Soc.80 (1955), 528–542. MR MR0084825 (18,930a) [Kra65] V. Y. Kraines, Topology of quaternionic manifolds, Bull. Amer. Math. Soc. 71 (1965), 526–527. MR 33 #6653 [Kru96] B. Kruggel, Kreck-stolz invariants, normal invariants and the homotopy classification of generalized wallach space, 1996, preprint (unpublished). [Kru97] , A homotopy classification of certain 7-manifolds, Trans. Amer. Math. Soc. 349 (1997), no. 7, 2827–2843. MR 97m:55012 [Kru98] , Kreck-Stolz invariants, normal invariants and the homotopy classification of generalised Wallach spaces, Quart. J. Math. Oxford Ser. (2) 49 (1998), no. 196, 469–485. MR 2000a:55018 BIBLIOGRAPHY 441

[KS58] K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. I, II, Ann. of Math. (2) 67 (1958), 328–466. MR 22 #3009 [KS60] , On deformations of complex analytic structures. III. Stability theorems for complex structures, Ann. of Math. (2) 71 (1960), 43–76. MR 22 #5991 [KS72] A. Kumpera and D. Spencer, Lie equations. Vol. I: General theory, Princeton Uni- versity Press, Princeton, N.J., 1972, Annals of Mathematics Studies, No. 73. MR 52 #1805 [KS88] M. Kreck and S. Stolz, A diffeomorphism classification of 7-dimensional homogeneous Einstein manifolds with SU(3) × SU(2) × U(1)-symmetry, Ann. of Math. (2) 127 (1988), no. 2, 373–388. MR 89c:57042 [KS96] P. Z. Kobak and A. Swann, Classical nilpotent orbits as hyper-Kähler quotients, Internat. J. Math. 7 (1996), no. 2, 193–210. MR 97f:53078 [KT70] Y. Y. Kuo and S. Tachibana, On the distribution appeared in contact 3-structure, Taita J. Math. 2 (1970), 17–24. MR 46 #8115 [KT87] Franz W. Kamber and Philippe Tondeur, de Rham-Hodge theory for Riemannian foliations, Math. Ann. 277 (1987), no. 3, 415–431. MR 89d:53070 [Kuo70] Ying-yan Kuo, On almost contact 3-structure,Tôhoku Math. J. (2) 22 (1970), 325– 332. MR 43 #3956 [Kur71] M. Kuranishi, Deformations of compact complex manifolds, Les Presses de l’UniversitédeMontréal, Montreal, Que., 1971, Séminaire de Mathématiques Supérieures, No. 39 (Et´´ e 1969). MR 50 #7588 [Lan00] T. Lance, Differentiable structures on manifolds, Surveys on surgery theory, Vol. 1, Ann. of Math. Stud., vol. 145, Princeton Univ. Press, Princeton, NJ, 2000, pp. 73– 104. MR 2001d:57035 [LB92] H. Lange and Ch. Birkenhake, Complex abelian varieties, Grundlehren der Mathe- matischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 1992. MR 94j:14001 [LdM71] S. López de Medrano, Involutions on manifolds, Springer-Verlag, New York, 1971, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 59. MR MR0298698 (45 #7747) [LeB88] C. LeBrun, A rigidity theorem for quaternionic-Kähler manifolds, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1205–1208. MR 89h:53105 [LeB91] , On complete quaternionic-Kähler manifolds, Duke Math. J. 63 (1991), no. 3, 723–743. MR 92i:53042 [LeB93] , A finiteness theorem for quaternionic-Kähler manifolds with positive scalar curvature, The Penrose transform and analytic cohomology in representation theory (South Hadley, MA, 1992), Contemp. Math., vol. 154, Amer. Math. Soc., Providence, RI, 1993, pp. 89–101. MR 1 246 379 [LeB95] , Fano manifolds, contact structures, and quaternionic geometry,Internat.J. Math. 6 (1995), no. 3, 419–437. MR 96c:53108 [Ler02a] E. Lerman, Contact toric manifolds, J. Symplectic Geom. 1 (2002), no. 4, 785–828. MR 2 039 164 [Ler02b] Eugene Lerman, A convexity theorem for torus actions on contact manifolds, Illinois J. Math. 46 (2002), no. 1, 171–184. MR 2003g:53146 [Ler03] E. Lerman, Geodesic flows and contact toric manifolds, Symplectic geometry of integrable Hamiltonian systems (Barcelona, 2001), Adv. Courses Math. CRM Barcelona, Birkhäuser, Basel, 2003, pp. 175–225. MR 2 000 747 [Ler04a] , Contact fiber bundles,J.Geom.Phys.49 (2004), no. 1, 52–66. MR MR2077244 [Ler04b] , Homotopy groups of K-contact toric manifolds, Trans. Amer. Math. Soc. 356 (2004), no. 10, 4075–4083 (electronic). MR 2 058 839 [Lic53] A. Lichnerowicz, Sur les espaces homogènes kählériens,C.R.Acad.Sci.Paris237 (1953), 695–697. MR MR0066016 (16,519b) [Lic63] , Théorèmes de réductivitésurdesalgèbres d’automorphismes, Rend. Mat. e Appl. (5) 22 (1963), 197–244. MR MR0157329 (28 #564) [Lie77] S. Lie, Geometrie der Berührungs transformationen, corrected ed., Chelsea Pub- lishing Co., New York, 1977, With editorial assistance by Georg Scheffers. MR MR0460049 (57 #45) 442 BIBLIOGRAPHY

[LM87] P. Libermann and Ch.-M. Marle, Symplectic geometry and analytical mechanics, Mathematics and its Applications, vol. 35, D. Reidel Publishing Co., Dordrecht, 1987, Translated from the French by Bertram Eugene Schwarzbach. MR 88c:58016 [LM89] H. B. Lawson, Jr. and M.-L. Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 91g:53001 [Loo84] E. J. N. Looijenga, Isolated singular points on complete intersections, London Mathe- matical Society Lecture Note Series, vol. 77, Cambridge University Press, Cambridge, 1984. MR MR747303 (86a:32021) [Loo01] F. Loose, Reduction in contact geometry,J.LieTheory11 (2001), no. 1, 9–22. MR MR1828281 (2002g:53147) [LR83] U. Lindström and M. Roˇcek, Scalar tensor duality and N =1, 2 nonlinear σ-models, Nuclear Phys. B 222 (1983), no. 2, 285–308. MR MR710273 (85d:81113) [LS94] C. LeBrun and S. Salamon, Strong rigidity of positive quaternion-Kähler manifolds, Invent. Math. 118 (1994), no. 1, 109–132. MR 95k:53059 [LT97] E. Lerman and S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997), no. 10, 4201–4230. MR 98a:57043 [Lut79] R. Lutz, Sur la géométrie des structures de contact invariantes, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 1, xvii, 283–306. MR 82j:53067 [LW99] Claude LeBrun and McKenzie Wang (eds.), Surveys in differential geometry: essays on Einstein manifolds, Surveys in Differential Geometry, VI, International Press, Boston, MA, 1999, Lectures on geometry and topology, sponsored by Lehigh Univer- sity’s Journal of Differential Geometry. MR MR1798603 (2001f:53003) [LW01] E. Lerman and Ch. Willett, The topological structure of contact and symplectic quotients, Internat. Math. Res. Notices (2001), no. 1, 33–52. MR 2001j:53112 [Mac87] K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, vol. 124, Cambridge University Press, Cambridge, 1987. MR 89g:58225 [Mal99] Juan Maldacena, The large-N limit of superconformal field theories and supergravity, Internat. J. Theoret. Phys. 38 (1999), no. 4, 1113–1133, Quantum gravity in the southern cone (Bariloche, 1998). MR 2001h:81246 [Mar58] A. Markov, Unsolvability of certain problems in topology, Dokl. Akad. Nauk SSSR 123 (1958), 978–980. MR MR0103455 (21 #2224) [Mar60] A. A. Markov, Insolubility of the problem of homeomorphy, Proc. Internat. Congress Math. 1958, Cambridge Univ. Press, New York, 1960, pp. 300–306. MR MR0115160 (22 #5962) [Mar70] S. Marchiafava, Sulle varietà a struttura quaternionale generalizzata, Rend. Mat. (6) 3 (1970), 529–545. MR MR0276892 (43 #2632) [Mat02] K. Matsuki, Introduction to the Mori program, Universitext, Springer-Verlag, New York, 2002. MR MR1875410 (2002m:14011) [Mey73] K. R. Meyer, Symmetries and integrals in mechanics, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973, pp. 259– 272. MR MR0331427 (48 #9760) [MFK94] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory,thirded., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR MR1304906 (95m:14012) [Mil56a] J. Milnor, Construction of universal bundles. II, Ann. of Math. (2) 63 (1956), 430– 436. MR 17,1120a [Mil56b] , On manifolds homeomorphic to the 7-sphere, Ann. of Math. (2) 64 (1956), 399–405. MR 18,498d [Mil63] , Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR 29 #634 [Mil65] , Lectures on the h-cobordism theorem, Notes by L. Siebenmann and J. Son- dow, Princeton University Press, Princeton, N.J., 1965. MR MR0190942 (32 #8352) [Mil68] , Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J., 1968. MR 39 #969 BIBLIOGRAPHY 443

[Mil75] , On the 3-dimensional Brieskorn manifolds M(p, q, r), Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), Princeton Univ. Press, Princeton, N. J., 1975, pp. 175–225. Ann. of Math. Studies, No. 84. MR 54 #6169 [Mil00] R. J. Milgram, The classification of Aloff-Wallach manifolds and their generalizations, Surveys on surgery theory, Vol. 1, Ann. of Math. Stud., vol. 145, Princeton Univ. Press, Princeton, NJ, 2000, pp. 379–407. MR 2000m:57056 [MM03] I. Moerdijk and J. Mrˇcun, Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, vol. 91, Cambridge University Press, Cambridge, 2003. MR 2 012 261 [MM82] S. Mori and S. Mukai, Classification of Fano 3-folds with B2 ≥ 2, Manuscripta Math. 36 (1981/82), no. 2, 147–162. MR 83f:14032 [MO70] J. Milnor and P. Orlik, Isolated singularities defined by weighted homogeneous polynomials, Topology 9 (1970), 385–393. MR 45 #2757 [Moe02] I. Moerdijk, Orbifolds as groupoids: an introduction, Orbifolds in mathematics and physics (Madison, WI, 2001), Contemp. Math., vol. 310, Amer. Math. Soc., Provi- dence, RI, 2002, pp. 205–222. MR 2004c:22003 [Mol75] P. Molino, Sur la géométrie transverse des feuilletages, Ann. Inst. Fourier (Grenoble) 25 (1975), no. 2, xiv, 279–284. MR 52 #11945 [Mol77] , Théorie des G-structures: le problème d’équivalence, Springer-Verlag, Berlin, 1977, Notes rédigées avec la collaboration de F. Toupine, Lecture Notes in Mathematics, Vol. 588. MR 58 #24419 [Mol79] , Feuilletages riemanniens sur les variétés compactes; champs de Killing transverses,C.R.Acad.Sci.ParisSér. A-B 289 (1979), no. 7, A421–A423. MR 80j:57026 [Mol88] , Riemannian foliations, Progress in Mathematics, vol. 73, Birkhäuser Boston Inc., Boston, MA, 1988, Translated from the French by Grant Cairns, With appendices by Cairns, Y. Carrière, E.´ Ghys, E. Salem and V. Sergiescu. MR 89b:53054 [Mor75] S. Morita, A topological classification of complex structures on S1 ×S2n−1, Topology 14 (1975), 13–22. MR MR0405444 (53 #9237) [Mor97] A. Moroianu, Parallel and Killing spinors on Spinc manifolds, Comm. Math. Phys. 187 (1997), no. 2, 417–427. MR 98i:58245 [Mor05] J. W. Morgan, Recent progress on the Poincaréconjectureandtheclassification of 3-manifolds, Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 1, 57–78 (electronic). MR MR2115067 [Mos65] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294. MR 32 #409 [MP97] I. Moerdijk and D. A. Pronk, Orbifolds, sheaves and groupoids, K-Theory 12 (1997), no. 1, 3–21. MR 98i:22004 [MR75] S. Marchiafava and G. Romani, Sui fibrati con struttura quaternionale generalizzata, Ann. Mat. Pura Appl. (4) 107 (1975), 131–157 (1976). MR 53 #6558 [MS74] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974, Annals of Mathematics Studies, No. 76. MR MR0440554 (55 #13428) [MS85] P. Molino and V. Sergiescu, Deux remarques sur les flots riemanniens,Manuscripta Math. 51 (1985), no. 1-3, 145–161. MR 86h:53035 [MS88] C. C. Moore and C. Schochet, Global analysis on foliated spaces, Mathematical Sci- ences Research Institute Publications, vol. 9, Springer-Verlag, New York, 1988, With appendices by S. Hurder, Moore, Schochet and Robert J. Zimmer. MR 89h:58184 [MS98] D. McDuff and D. Salamon, Introduction to symplectic topology, second ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1998. MR 2000g:53098 [Mum61] D. Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Etudes´ Sci. Publ. Math. (1961), no. 9, 5–22. MR MR0153682 (27 #3643) 2 [Mum79] , An algebraic surface with K ample, (K )=9, pg = q = 0, Amer. J. Math. 101 (1979), no. 1, 233–244. MR MR527834 (80j:14032) [MW74] J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5 (1974), no. 1, 121–130. MR MR0402819 (53 #6633) 444 BIBLIOGRAPHY

[MW88] K. Mikami and A. Weinstein, Moments and reduction for symplectic groupoids, Publ. Res. Inst. Math. Sci. 24 (1988), no. 1, 121–140. MR MR944869 (90c:58060) [MZ88] M. Miyanishi and D.-Q. Zhang, Gorenstein log del Pezzo surfaces of rank one,J. Algebra 118 (1988), no. 1, 63–84. MR MR961326 (89i:14033) [Nag92] Y. Nagatomo, Rigidity of c1-self-dual connections on quaternionic Kähler manifolds, J. Math. Phys. 33 (1992), no. 12, 4020–4025. MR 93m:53028 [NN57] A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65 (1957), 391–404. MR 19,577a [Oba56] M. Obata, Affine connections on manifolds with almost complex, quaternion or Her- mitian structure,Jap.J.Math.26 (1956), 43–77. MR MR0095290 (20 #1796a) [Oba66] , Riemannian manifolds admitting a solution of a certain system of differential equations, Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965), Nippon Hyoronsha, Tokyo, 1966, pp. 101–114. MR 35 #7263 [Oda88] T. Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer- Verlag, Berlin, 1988, An introduction to the theory of toric varieties, Translated from the Japanese. MR MR922894 (88m:14038) [Oku62] M. Okumura, Some remarks on space with a certain contact structure,Tôhoku Math. J. (2) 14 (1962), 135–145. MR 26 #708 [O’N83] B. O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983, With applications to relativity. MR 85f:53002 [OP97] L. Ornea and P. Piccinni, Locally conformal Kähler structures in quaternionic geometry,Trans.Amer.Math.Soc.349 (1997), no. 2, 641–655. MR 97e:53091 [Opr84] V. Oproiu, Integrability of almost quaternal structures, An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si Sect¸. I a Mat. 30 (1984), no. 5, 75–84. MR MR800155 (86k:53055) [OR77] P. Orlik and R. C. Randell, The monodromy of weighted homogeneous singularities, Invent. Math. 39 (1977), no. 3, 199–211. MR 57 #314 [OR04] J.-P. Ortega and T. S. Ratiu, Momentum maps and Hamiltonian reduction, Progress in Mathematics, vol. 222, Birkhäuser Boston Inc., Boston, MA, 2004. MR MR2021152 (2005a:53144) [Orl70] P. Orlik, Weighted homogeneous polynomials and fundamental groups, Topology 9 (1970), 267–273. MR 41 #6251 [Orl72a] , On the homology of weighted homogeneous manifolds, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971), Part I (Berlin), Springer, 1972, pp. 260–269. Lecture Notes in Math., Vol. 298. MR 55 #3312 [Orl72b] , Seifert manifolds, Springer-Verlag, Berlin, 1972, Lecture Notes in Mathe- matics, Vol. 291. MR 54 #13950 [OW71a] P. Orlik and P. Wagreich, Singularities of algebraic surfaces with C∗ action,Math. Ann. 193 (1971), 121–135. MR 45 #1914 [OW71b] Peter Orlik and Philip Wagreich, Isolated singularities of algebraic surfaces with C∗ action, Ann. of Math. (2) 93 (1971), 205–228. MR 44 #1662 [OW75] P. Orlik and P. Wagreich, Seifert n-manifolds, Invent. Math. 28 (1975), 137–159. MR 50 #13596 [OW77] , Algebraic surfaces with k∗-action,ActaMath.138 (1977), no. 1-2, 43–81. MR 57 #336 [Pal57] R. S. Palais, A global formulation of the Lie theory of transformation groups,Mem. Amer. Math. Soc. No. 22 (1957), iii+123. MR MR0121424 (22 #12162) [Pal61] , On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295–323. MR MR0126506 (23 #A3802) [Per02] G. Pereleman, The entropy formula for the ricci flow and its geometric applications, arXiv:math.DG/0211159 (2002). [Per03a] , Finite extinction time for the solutions to the Ricci flow on certain three- manifolds, arXiv:math.DG/0307245 (2003). [Per03b] , Ricci flow with surgery on three-manifolds, arXiv:math.DG/0303109 (2003). [Pet98] P. Petersen, Riemannian geometry, Graduate Texts in Mathematics, vol. 171, Springer-Verlag, New York, 1998. MR 98m:53001 BIBLIOGRAPHY 445

[Pic98] P. Piccinni, The geometry of positive locally quaternion Kähler manifolds, Ann. Global Anal. Geom. 16 (1998), no. 3, 255–272. MR 99f:53047 [PP88] H. Pedersen and Y. S. Poon, Hyper-Kähler metrics and a generalization of the Bo- gomolny equations, Comm. Math. Phys. 117 (1988), no. 4, 569–580. MR MR953820 (89f:53069) [PP98] H. Pedersen and Y.-S. Poon, Deformations of hypercomplex structures,J.Reine Angew. Math. 499 (1998), 81–99. MR 99h:32009 [PP99] , A note on rigidity of 3-Sasakian manifolds, Proc. Amer. Math. Soc. 127 (1999), no. 10, 3027–3034. MR 2000a:53075 [Pri67] D. Prill, Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. J. 34 (1967), 375–386. MR MR0210944 (35 #1829) [PS91] Y. S. Poon and S. M. Salamon, Quaternionic Kähler 8-manifolds with positive scalar curvature, J. Differential Geom. 33 (1991), no. 2, 363–378. MR 92b:53071 [Ran75] R. C. Randell, The homology of generalized Brieskorn manifolds, Topology 14 (1975), no. 4, 347–355. MR 54 #1270 [Rav86] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres,Pure and Applied Mathematics, vol. 121, Academic Press Inc., Orlando, FL, 1986. MR MR860042 (87j:55003) [Rei59a] B. L. Reinhart, Foliated manifolds with bundle-like metrics, Ann. of Math. (2) 69 (1959), 119–132. MR 21 #6004 [Rei59b] , Harmonic integrals on foliated manifolds, Amer. J. Math. 81 (1959), 529– 536. MR MR0107280 (21 #6005) [Rei80] M. Reid, Canonical 3-folds,Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, pp. 273–310. MR 82i:14025 [Rei83] B. L. Reinhart, Differential geometry of foliations, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 99, Springer- Verlag, Berlin, 1983, The fundamental integrability problem. MR 85i:53038 [Ruk93] P. Rukimbira, Some remarks on R-contact flows, Ann. Global Anal. Geom. 11 (1993), no. 2, 165–171. MR 94h:53043 [Ruk94] , The dimension of leaf closures of K-contact flows, Ann. Global Anal. Geom. 12 (1994), no. 2, 103–108. MR 95f:53072 [Ruk95a] , Chern-Hamilton’s conjecture and K-contactness, Houston J. Math. 21 (1995), no. 4, 709–718. MR 96m:53032 [Ruk95b] , Topology and closed characteristics of K-contact manifolds, Bull. Belg. Math.Soc.SimonStevin2 (1995), no. 3, 349–356. MR 96g:53044 [Ruk99] , On K-contact manifolds with minimal number of closed characteristics, Proc.Amer.Math.Soc.127 (1999), no. 11, 3345–3351. MR 2000d:53049 [Sae87] O. Saeki, Knotted homology spheres defined by weighted homogeneous polynomials, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 1, 43–50. MR 88c:57021 [Sal82] S. M. Salamon, Quaternionic Kähler manifolds, Invent. Math. 67 (1982), no. 1, 143–171. MR 83k:53054 [Sal86] , Differential geometry of quaternionic manifolds, Ann. Sci. Ecole´ Norm. Sup. (4) 19 (1986), no. 1, 31–55. MR 87m:53079 [Sal89] , Riemannian geometry and holonomy groups, Pitman Research Notes in Mathematics Series, vol. 201, Longman Scientific & Technical, Harlow, 1989. MR 90g:53058 [Sal93] , Index theory and quaternionic Kähler manifolds, Differential geometry and its applications (Opava, 1992), Math. Publ., vol. 1, Silesian Univ. Opava, Opava, 1993, pp. 387–404. MR 95a:58120 [Sal99] , Quaternion-Kähler geometry, Surveys in differential geometry: essays on Einstein manifolds, Surv. Differ. Geom., VI, Int. Press, Boston, MA, 1999, pp. 83– 121. MR MR1798608 (2001m:53078) [Sas65] S. Sasaki, Almost Contact Manifolds, Part 1, Lecture Notes, Mathematical Institute, Tôhoku University (1965). [Sas67] , Almost Contact Manifolds, Part 2, Lecture Notes, Mathematical Institute, Tôhoku University (1967). 446 BIBLIOGRAPHY

[Sas72] S. Sasaki, Spherical space forms with normal contact metric 3-structure, J. Differen- tial Geometry 6 (1971/72), 307–315. MR 46 #831 [Sat56] I. Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363. MR 18,144a [Sat57] , The Gauss-Bonnet theorem for V -manifolds, J. Math. Soc. Japan 9 (1957), 464–492. MR 20 #2022 [Sat77] H. Sato, Remarks concerning contact manifolds,Tôhoku Math. J. 29 (1977), no. 4, 577–584. MR MR0458334 (56 #16537) [Sav79] I. V. Savelev, Structure of the singularities of a class of complex hypersurfaces,Mat. Zametki 25 (1979), no. 4, 497–503, 635. MR MR534292 (80g:32018) [Sav02] N. Saveliev, Invariants for homology 3-spheres, Encyclopaedia of Mathematical Sciences, vol. 140, Springer-Verlag, Berlin, 2002, Low-Dimensional Topology, 1. MR 2004c:57026 [Sco83] P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 84m:57009 [Sco05] A. Scorpan, The wild world of 4-manifolds, American Mathematical Society, Provi- dence, RI, 2005. MR MR2136212 [Sei32] H. Seifert, Topologie dreidimensionaler gefaserte Räume,ActaMath.60 (1932), 148–238. [SH62] S. Sasaki and Y. Hatakeyama, On differentiable manifolds with contact metric structures, J. Math. Soc. Japan 14 (1962), 249–271. MR 25 #4458 [Siu87] Y. T. Siu, Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics, DMV Seminar, vol. 8, Birkhäuser Verlag, Basel, 1987. MR 89d:32020 [Sja95] R. Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. of Math. (2) 141 (1995), no. 1, 87–129. MR MR1314032 (96a:58098) [SL91] R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction, Ann. of Math. (2) 134 (1991), no. 2, 375–422. MR MR1127479 (92g:58036) [Slo03] N. J. A. Sloane, The on-line encyclopedia of integer sequences, AT&T Research, http://www.research.att.com/ njas/sequences/, 2003. [Sma61] S. Smale, Generalized Poincaré’s conjecture in dimensions greater than four, Ann. of Math. (2) 74 (1961), 391–406. MR 25 #580 [Sma62a] , On the structure of 5-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR 25 #4544 [Sma62b] , On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR MR0153022 (27 #2991) [Som75] A. J. Sommese, Quaternionic manifolds, Math. Ann. 212 (1974/75), 191–214. MR MR0425827 (54 #13778) [Sou70] J.-M. Souriau, Structure des systèmes dynamiques, Maitrises de mathématiques, Dunod, Paris, 1970. MR MR0260238 (41 #4866) [Sou97] , Structure of dynamical systems, Progress in Mathematics, vol. 149, Birkhäuser Boston Inc., Boston, MA, 1997, A symplectic view of physics, Trans- lated from the French by C. H. Cushman-de Vries, Translation edited and with a preface by R. H. Cushman and G. M. Tuynman. MR MR1461545 (98i:58103) [Spe62] D. C. Spencer, Deformation of structures on manifolds defined by transitive, continuous pseudogroups. I. Infinitesimal deformations of structure, Ann. of Math. (2) 76 (1962), 306–398. MR 27 #6287a [SS65] I. M. Singer and S. Sternberg, The infinite groups of Lie and Cartan. I. The transitive groups,J.AnalyseMath.15 (1965), 1–114. MR 36 #911 [SSTVP88] Ph. Spindel, A. Sevrin, W. Troost, and A. Van Proeyen, Extended supersymmetric σ- models on group manifolds. I. The complex structures, Nuclear Phys. B 308 (1988), no. 2-3, 662–698. MR MR967938 (89m:53138) [ST80] H. Seifert and W. Threlfall, Seifert and Threlfall: a textbook of topology, Pure and Applied Mathematics, vol. 89, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980, Translated from the German edition of 1934 by Michael A. Goldman, With a preface by Joan S. Birman, With “Topology of 3- dimensional fibered spaces” by Seifert, Translated from the German by Wolfgang Heil. MR MR575168 (82b:55001) BIBLIOGRAPHY 447

[Ste51] N. Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 12,522b [Ste83] S. Sternberg, Lectures on differential geometry, second ed., Chelsea Publishing Co., New York, 1983, With an appendix by Sternberg and Victor W. Guillemin. MR 88f:58001 [Sto74] R. E. Stong, Contact manifolds, J. Differential Geometry 9 (1974), 219–238. MR 49 #6258 [SW99] B. Steer and A. Wren, Grothendieck topology and the Picard group of a complex orbifold, Homotopy invariant algebraic structures (Baltimore, MD, 1998), Con- temp. Math., vol. 239, Amer. Math. Soc., Providence, RI, 1999, pp. 251–262. MR MR1718086 (2000j:14029) [Swa89] A. F. Swann, Quaternionic Kähler geometry and the fundamental 4-form, Proceed- ings of the Workshop on Curvature Geometry (Lancaster, 1989) (Lancaster), ULDM Publ., 1989, pp. 165–173. MR MR1089891 (91m:53053) [Swa91] A. Swann, Hyper-Kähler and quaternionic Kähler geometry, Math. Ann. 289 (1991), no. 3, 421–450. MR 92c:53030 [Swa97] , Singular moment maps and quaternionic geometry, Symplectic singularities and geometry of gauge fields (Warsaw, 1995), Banach Center Publ., vol. 39, Polish Acad. Sci., Warsaw, 1997, pp. 143–153. MR MR1458656 (98m:53066) [Tac65] S. Tachibana, On harmonic tensors in compact Sasakian spaces,Tôhoku Math. J. (2) 17 (1965), 271–284. MR 32 #8288 [Tak78] T. Takahashi, Deformations of Sasakian structures and its application to the Brieskorn manifolds,Tôhoku Math. J. (2) 30 (1978), no. 1, 37–43. MR 81e:53024 [Tak00] S. Takayama, Simple connectedness of weak Fano varieties, J. Algebraic Geom. 9 (2000), no. 2, 403–407. MR MR1735807 (2001d:14042) [Tam92] I. Tamura, Topology of foliations: an introduction, Translations of Mathematical Monographs, vol. 97, American Mathematical Society, Providence, RI, 1992, Trans- lated from the 1976 Japanese edition and with an afterword by Kiki Hudson, With a foreword by Takashi Tsuboi. MR 93c:57021 [Tan67] Shûkichi Tanno, Harmonic forms and Betti numbers of certain contact Riemannian manifolds, J. Math. Soc. Japan 19 (1967), 308–316. MR 35 #3604 [Tan69a] S. Tanno, The automorphism groups of almost contact Riemannian manifolds, Tôhoku Math. J. (2) 21 (1969), 21–38. MR 39 #3428 [Tan69b] , Sasakian manifolds with constant φ-holomorphic sectional curvature, Tôhoku Math. J. (2) 21 (1969), 501–507. MR 40 #4894 [Tan70] , On the isometry groups of Sasakian manifolds, J. Math. Soc. Japan 22 (1970), 579–590. MR 42 #6755 [Tan71] , Killing vectors on contact Riemannian manifolds and fiberings related to the Hopf fibrations,Tôhoku Math. J. (2) 23 (1971), 313–333. MR 44 #4681 3 2 [Tan79] , Geodesic flows on CL-manifolds and Einstein metrics on S × S , Minimal submanifolds and geodesics (Proc. Japan-United States Sem., Tokyo, 1977), North- Holland, Amsterdam, 1979, pp. 283–292. MR 81g:58027 [Tho69] E. Thomas, Vector fields on manifolds, Bull. Amer. Math. Soc. 75 (1969), 643–683. MR MR0242189 (39 #3522) [Tho76] C. B. Thomas, Almost regular contact manifolds, J. Differential Geometry 11 (1976), no. 4, 521–533. MR 57 #4195 [Thu76] W. P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), no. 2, 467–468. MR 53 #6578 [Thu79] W. Thurston, The geometry and topology of 3-manifolds, Mimeographed Notes, Princeton University, 1979. [Thu97] W. P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Math- ematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997, Edited by Silvio Levy. MR 97m:57016 [Tia00] G. Tian, Canonical metrics in Kähler geometry, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2000, Notes taken by Meike Akveld. MR 2001j:32024 [Tod94] K. P. Tod, Self-dual Einstein metrics from the PainlevéVIequation, Phys. Lett. A 190 (1994), no. 3-4, 221–224. MR MR1285788 (95c:83020) 448 BIBLIOGRAPHY

[Ton97] P. Tondeur, Geometry of foliations, Monographs in Mathematics, vol. 90, Birkhäuser Verlag, Basel, 1997. MR 98d:53037 [Tsu88] H. Tsuji, Logarithmic Fano manifolds are simply connected, Tokyo J. Math. 11 (1988), no. 2, 359–362. MR MR976571 (90b:14050) [TW03] S. Tolman and J. Weitsman, The cohomology rings of symplectic quotients, Comm. Anal. Geom. 11 (2003), no. 4, 751–773. MR MR2015175 (2004k:53140) [TY70] S. Tachibana and W. N. Yu, On a Riemannian space admitting more than one Sasakian structures,Tôhoku Math. J. (2) 22 (1970), 536–540. MR 43 #1086 [Udr69] C. Udri¸ste, Structures presque coquaternioniennes, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 13(61) (1969), no. 4, 487–507 (1970). MR 45 #5908 [Ust99] I. Ustilovsky, Infinitely many contact structures on S4m+1, Internat. Math. Res. Notices (1999), no. 14, 781–791. MR MR1704176 (2000f:57028) [Voi04] C. Voisin, On the homotopy types of compact Kähler and complex projective manifolds, Invent. Math. (2004), no. 157, 329–343. [Wad75] A. W. Wadsley, Geodesic foliations by circles, J. Differential Geometry 10 (1975), no. 4, 541–549. MR 53 #4092 [Wal62] C. T. C. Wall, Classification of (n − 1)-connected 2n-manifolds, Ann. of Math. (2) 75 (1962), 163–189. MR MR0145540 (26 #3071) [Wal67a] F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II,Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117. MR MR0235576 (38 #3880) [Wal67b] C. T. C. Wall, Classification problems in differential topology. VI. Classification of (s − 1)-connected (2s +1)-manifolds, Topology 6 (1967), 273–296. MR 35 #7343 [Wan54a] H.-C. Wang, Closed manifolds with homogeneous complex structure, Amer. J. Math. 76 (1954), 1–32. MR MR0066011 (16,518a) [Wan54b] , Closed manifolds with homogeneous complex structure, Amer. J. Math. 76 (1954), 1–32. MR 16,518a [Wan82] M. Y. Wang, Some examples of homogeneous Einstein manifolds in dimension seven, Duke Math. J. 49 (1982), no. 1, 23–28. MR 83k:53069 [Wan89] , Parallel spinors and parallel forms, Ann. Global Anal. Geom. 7 (1989), no. 1, 59–68. MR 91g:53053 [Wan95] , On non-simply connected manifolds with non-trivial parallel spinors, Ann. Global Anal. Geom. 13 (1995), no. 1, 31–42. MR 96a:53066 [Web78] S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), no. 1, 25–41. MR 80e:32015 [Wei77a] A. Weinstein, Lectures on symplectic manifolds, American Mathematical Society, Providence, R.I., 1977, Expository lectures from the CBMS Regional Conference held at the University of North Carolina, March 8–12, 1976, Regional Conference Series in Mathematics, No. 29. MR 57 #4244 [Wei77b] , Symplectic V -manifolds, periodic orbits of Hamiltonian systems, and the volume of certain Riemannian manifolds, Comm. Pure Appl. Math. 30 (1977), no. 2, 265–271. MR 56 #13260 [Wei79] , On the hypotheses of Rabinowitz’ periodic orbit theorems, J. Differential Equations 33 (1979), no. 3, 353–358. MR 81a:58030b [Wei80] , Fat bundles and symplectic manifolds, Adv. in Math. 37 (1980), no. 3, 239– 250. MR 82a:53038 [Wel80] R. O. Wells, Jr., Differential analysis on complex manifolds, second ed., Graduate Texts in Mathematics, vol. 65, Springer-Verlag, New York, 1980. MR MR608414 (83f:58001) [Wil72] D. L. Wilkens, Closed (s − 1)-connected (2s +1)-manifolds, s =3, 7, Bull. London Math. Soc. 4 (1972), 27–31. MR 46 #6378 [Wil02] Ch. Willett, Contact reduction, Trans. Amer. Math. Soc. 354 (2002), no. 10, 4245– 4260 (electronic). MR MR1926873 (2003m:53152) [Win83] H. E. Winkelnkemper, The graph of a foliation, Ann. Global Anal. Geom. 1 (1983), no. 3, 51–75. MR 85j:57043 [Wol65] J. A. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces,J.Math.Mech.14 (1965), 1033–1047. MR 32 #3020 [WZ90] M. Y. Wang and W. Ziller, Einstein metrics on principal torus bundles, J. Differential Geom. 31 (1990), no. 1, 215–248. MR 91f:53041 BIBLIOGRAPHY 449

[Yam01] T. Yamazaki, On a surgery of K-contact manifolds,KodaiMath.J.24 (2001), no. 2, 214–225. MR 2002c:57048 [Yan63] K. Yano, On a structure defined by a tensor field f of type (1, 1) satisfying f 3+f =0, Tensor (N.S.) 14 (1963), 99–109. MR MR0159296 (28 #2513) [Yau77] S.-T. Yau, Calabi’s conjecture and some new results in algebraic geometry,Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, 1798–1799. MR MR0451180 (56 #9467) [Yau78] , On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411. MR 81d:53045 [YK84] K. Yano and M. Kon, Structures on manifolds, Series in Pure Mathematics, vol. 3, World Scientific Publishing Co., Singapore, 1984. MR 86g:53001 [Zhe00] F. Zheng, Complex differential geometry, AMS/IP Studies in Advanced Mathematics, vol. 18, American Mathematical Society, Providence, RI, 2000. MR MR1777835 (2001i:32035)

Index

G-manifold, 23 distribution, 92 Γ-atlas, 22 form, 89, 92 3-Sasakian, 347 Hamiltonian, 97 line bundle, 91 action manifold, 91 local type, 27 co-oriented, 92 locally free, 24 metric manifold, 106 locally proper, 26 metric structure, 106 proper, 24 pseudogroup, 90 transitive, 25 strict transformation group, 97 Alekseevskian spaces, 323 structure, 92 subbundle, 92 Barden invariant, 264 transformation, 90 Barden theorem, 265 infinitesimal, 90 Beauville Conjecture, 322 strict, 90 Boothby-Wang Theorem, 137 transformation group, 96 contact type, 95 Calabi conjecture, 131 convex polytope, 204 Calabi metrics on T ∗CPn, 340 cuplength, 157 characteristic curvature, 149 orbifold Euler, 73 Φ-sectional, 153 characteristic hyperplane, 210 2-form, 8 Chern classes, 169 Ricci, 150 Chern classes, basic, 169 Riemannian, 12, 149 class scalar, 286 orbifold characteristic, 72 orbifold Chern class, 73 sectional, 149, 152, 160 orbifold Euler, 73 orbifold Pontrjagin, 73 divisor, 75 Q real characteristic, 73 ,77 classifying space, 69, 420 absolute, 77 cocyle Baily, 77 Haefliger, 37 branch, 77 cohomology Cartier, 77 group, 70 class group, 77 commuting sheaf, 140 ramification, 78 cone, 107 Weil, 77 metric, 108 symplectic, 109, 110 Eguchi-Hanson metric, 341 connection, 7 Eilenberg-MacLane space, 69, 70 Obata, 20 Einstein metric, 288 Oproiu, 20 elliptic fibration, 261 contact exotic almost, 98–100, 102, 109, 348, 349 contact, 251

451 452 INDEX fan, 207 local uniformizing system, 59 foliated atlas, 17 Lusternik-Schnirelman theory, 159 foliated coordinate chart, 17 foliation, 17 manifold characteristic, 96, 102, 104, 106, 107 homogeneous, 25 Riemannian, 46 almost hyperkähler, 336 simple, 35, 39 hyperkähler, 336 singular Riemannian, 52 quaternionic Kähler, 317 fundamental basic class, 144 Marchiafava-Romani class, 316 metric G-structure, 10 Lorentzian, 15 gravitational instantons, 337 moment map, 202 gravitational multi-instantons, 343 contact, 209 groupoid, 23, 37 hypercomplex, 336 étale, 23, 67, 419 hyperkähler, 339 action, 418 quaternionic Kähler, 324 holonomy, 38 symplectic cone, 208 Lie, 418 Monge-Ampére equation, 132 translation, 419 Nijenhuis tensor, 13, 110, 112, 116 Heisenberg group, 91, 156, 186, 200 normal, 110 holonomy orbifold, 59 covering, 40 classifying space, 69 leaf, 38 developable, 61 pseudogroup, 39 orbifold charts, 59 holonomy group (of a leaf), 39 orbisheaf, 62 holonomy groupoid, 38 structure, 62 homologous orbit, 24 a, 169, 173 exceptional, 29 hypercomplex, 334 principal, 29 hyperhamiltonian action, 339 regular, 29 singular, 29 index type, 27 Fano, 81 order ramification, 77 K-contact, 135 injections, 59, 62 of an orbifold, 61 integrable, 10 irregular, 135 Painlevé VI equation, 330 isotropy subgroup, 60 Picard group, 77 of an orbifold, 80 K-contact, 107, 110, 113, 135, 139, 149 Poincaré homology sphere, 259 structure theorem, 140 Poisson bracket, 202 K-contact manifold, 107 polyhedral cone, 208 Kähler, 16 polytope, 204, 206 almost, 110 Delzant, 205 Kähler cone, 131 lt, 205 Kähler form, 16 pseudo-Riemannian metric, 15 Kirwan map, 203 pseudogroup, 21 transitive, 21 leaf closure, 139 quasi-regular, 135 generic, 39 quotient leaf holonomy, 61 hypercomplex, 336 leaves, 17 hyperkähler, 340 space of, 17 LeBrun-Salamon Conjecture, 321 reduction Legendre transform, 342 contact, 215 lens space, 190 hypercomplex, 336 local uniformizing groups, 60 hyperkähler, 340 INDEX 453

quaternionic Kähler, 325 almost quaternionic, 20, 316 singular symplectic, 202, 340 almost quaternionic Hermitian, 20, 316 symplectic, 201, 202 complex, 6 Reeb flow, 107 conformal, 15 Reeb type, 210 conformal symplectic, 16 Reeb vector field, 96, 105, 136 contact, 89 reflection, 76 contact metric, 102 reflection group, 76 CR, 18, 112, 113 regular, 135 Haefliger, 17, 36 Ricci form, 124, 170 Hermitian, 14 Ricci form, transverse, 170 homogeneous contact, 197 Ricci tensor, 123, 150, 170, 288 homogeneous K-contact, 198 Ricci tensor, transverse, 152, 170 homogeneous Sasakian, 198 Riemann curvature tensor, 122 hyper f-, 21 Riemannian hyperhermitian, 20 curvature, 12 hyperkähler, 336 Riemannian metric, 11 K-contact, 149 ringed space, 75 Kähler, 16 Lorentzian, 15 Sasaki group, 163 pseudo-Riemannian, 15 Sasakian, 113 pseudogroup, 21 Sasakian manifold, 113 quaternionic, 317 Sasakian structure, 113 quaternionic Hermitian, 20, 317 indefinite, 171 Riemannian, 6 negative, 171 Sasakian, 113, 153, 164 null, 171 Seifert fibred, 30 positive, 171, 173, 245 symplectic, 6, 15 Seifert bundles, 86 submersion, 17, 35 Seifert fibered 3-manifold, 86 Taub-NUT metrics, 341 sheaf, 62 taut, 102 derived functor, 70 toral rank, 53, 55, 140 sheaf of groupoids, 23 transversal, 38 singular locus transverse geometry, 17 orbifold, 60, 76, 77 transverse holonomy groupoid, 42 slice, 25 transverse homothety, 155 Smale theorem, 265 type Smale-Barden classification, 264 of a Sasakian structure, 171 stabilizer, 24 startification UFD, 76 smooth, 26 stratification V-manifold, 59 Whitney, 26 Wang-Ziller manifolds, 179 structure weighted sphere, 141, 165 f−,19 Weinstein conjecture, 159 almost contact 3-structure, 349 Wolf spaces, 320 almost contact metric, 102 almost hypercontact, 348 almomst hypercontact, 21 almost complex, 12 almost contact, 18, 109 normal, 110 almost CR, 18, 112 almost Hermitian, 14 almost hypercomplex, 20 almost hyperhermitian, 20 almost hyperkähler, 336 almost product, 17 almost product Riemannian metric, 18