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4n Outline of paper. Throughout this article, we denote by (M , σ1, σ2, σ3,gM ) a hyperKähler 4n-manifold with σi denoting the triple of Kähler forms and gM 2 the hyperKähler metric. If [σ1] H (M, R) defines an integral cohomology class i.e. [σ ] H2(M, 2πZ), we denote∈ by M 4n+1 the total space of the S1 bundle 1 ∈ α determined by [σ1] and let α be a connection 1-form such that
(1.1) dα = σ1. If additionally, [ σ ] H2(M, 2πZ), we denote by M 4n+2 the total space of the T2 − 2 ∈ α,ξ bundle determined by [σ ], [ σ ] and let ξ be a connection 1-form such that 1 − 2 (1.2) dξ = σ . − 2 Likewise, if [ σ ] H2(M, 2πZ), we denote by M 4n+3 the T3 bundle determined − 3 ∈ α,ξ,η by [σ ], [ σ ], [ σ ] and endowed with a connection 1-form η such that 1 − 2 − 3 (1.3) dη = σ . − 3 Then we show the following holds Theorem 1.1. 2 2t (1.4) gQ := dt + e gM is an Einstein metric on Q4n+1 = R M 4n. t ×
2 4t 2 2t (1.5) gP := dt +4e (α )+ e gM 4n+2 4n+1 is a Kähler-Einstein metric on P = Rt Mα with associated Kähler 2-form given by × (1.6) ω := 2e2tdt α + e2tσ . P ∧ 1 2 4t 2 2 2t (1.7) gL := dt +4e (α + ξ )+ e gM is an Einstein metric on L4n+3 = R M 4n+2. t × α,ξ 2 4t 2 2 2 2t (1.8) gN := dt +4e (α + ξ + η )+ e gM is a quaternion-Kähler metric on N 4n+4 = R M 4n+3. The associated quaternion- t × α,ξ,η Kähler 4-form is given by 1 (1.9) Ω := (ω ω + ω ω + ω ω ), QK 2 1 ∧ 1 2 ∧ 2 3 ∧ 3 where (1.10) ω := e2tσ + (2e2tξ) (2e2tη)+ dt (2e2tα), 1 1 ∧ ∧ (1.11) ω := e2tσ + (2e2tξ) (dt)+(2e2tα) (2e2tη), 2 2 ∧ ∧ (1.12) ω := e2tσ + (2e2tξ) (2e2tα)+(2e2tη) dt. 3 3 ∧ ∧ Furthermore, the almost complex structures Ii defined by gN (Ii( ), ) = ωi( , ) are all integrable i.e. N 4n+4 is a hypercomplex manifold. In all the cases,· · the Einstein· · metrics have negative scalar curvature. Remark 1.2. The warped product metric (1.4) is well-known and in fact, one can define an Einstein metric of this form by replacing gM by any Ricci-flat metric cf. [6, p. 267]. The reason we include this case here is to highlight the striking similarity with the other examples. EINSTEIN METRICS ON BUNDLES OVER HYPERKÄHLER MANIFOLDS 3
4n+4 It is worth emphasising that N is not a hyperKähler manifold since ωi are not closed. When M = T4, the total spaces P 6, L7 and N 8 in Theorem 1.1 with their Einstein metrics have been identified as certain non-unimodular solvable Lie groups in [27]. The latter QK metric on N 8 was rediscovered in [17, (4.24)], more generally when M = T4n, by studying quaternionic contact structures introduced by Biquard [8]. The fact that the underlying QK structure is hypercomplex is not too surprising since it is well-known that infinitesimal symmetries of QK manifolds correspond to integrable almost complex structures cf. [24, Prop. 3.1] and here 3 we have that ΩQK is T -invariant. In [28] Goldstein and Prokushkin construct complex manifolds which are T2 bundles over suitable Hermitian manifolds. The hypercomplex structure on N 4n+4 can be viewed as a quaternionic extension of their result as follows. By definition, hyperKähler manifolds have holonomy group contained in Sp(n). Thus, the space of 2-forms on M 4n splits into irreducible Sp(n)-modules as (1.13) Λ2(M) = sp(1) sp(n) (sp(1) sp(n))⊥, ∼ ⊕ ⊕ ⊕ where sp(1) ∼= σ1, σ2, σ3 is a trivial vector bundle (in fact, flat with respect to the Levi-Civita connection)h andi sp(n) is the holonomy algebra of M. If γ sp(n) is a closed 2-form such that [γ] H2(M, 2πZ), let ν be the connection 1-form∈ such that ∈ dν = γ on the associated T4 bundle M 4n+4 M 4n. Then we have α,ξ,η,ν → 4n+4 Theorem 1.3. On Mα,ξ,η,ν the data 2 2 2 2 (1.14) gˇ := gM + α + ξ + η + ν , (1.15) ωˇ := σ + ξ η + ν α, 1 1 ∧ ∧ (1.16) ωˇ := σ + ξ ν + α η, 2 2 ∧ ∧ (1.17) ωˇ := σ + ξ α + η ν, 3 3 ∧ ∧ defines a hyper-Hermitian Sp(n + 1)-structure. The hypercomplex structure on N 4n+4 is then locally just a special case when 1 γ =0 with the trivial S bundle determined by [γ] lifted to Rt. We also show that another hypercomplex extension of the result in [28] is given by 4n Theorem 1.4. Let (M , σ1, σ2, σ3) be a hyperKähler manifold such that there exists a quadruple of closed 2-forms γ sp(n) with [γ ] H2(M, 2πZ). Then i ∈ i ∈ 2 2 2 2 (1.18) gˇ := gM + ν1 + ν2 + ν3 + ν4 , (1.19) ωˇ := σ + ν ν + ν ν , 1 1 1 ∧ 2 3 ∧ 4 (1.20) ωˇ := σ + ν ν + ν ν , 2 2 1 ∧ 3 4 ∧ 2 (1.21) ωˇ := σ + ν ν + ν ν , 3 3 1 ∧ 4 2 ∧ 3 where νi are connection 1-forms such that dνi = γi, define a hyper-Hermitian struc- T4 4n+4 ture on the total space of the bundle Mν1,ν2,ν3,ν4 determined by [γi]. Note that by contrast to Theorem 1.1 and 1.3, in Theorem 1.4 we do not require that the hyperKähler triple define integral cohomology classes. Examples of hyper- complex structures of this type have been studied by Fino and Dotti in [18] when M = T4. Theorems 1.1, 1.3, 1.4 are all proved in Section 3. In Sections 4, 5, 6 and 7 we then study the geometry of these examples in detail. In Section 4 we construct a large class of examples that arise from Theorems 1.1, 1.3 and 1.4 by choosing suitable HK manifold M 4n. As already mentioned above 4 U. FOWDAR when M = T4, the total spaces P 6, L7 and N 8 correspond to certain solvable Lie groups and as such these examples admit lots of symmetries. In general, however, the only Killing vector fields present in the examples of Theorem 1.1 correspond to the vertical vector fields of the T3 torus fibres. In Section 5 we show that under suitable hypothesis one can lift Killing vector fields on M 4n to Killing vector fields on the total spaces Q4n+1, P 4n+2, L4n+3 and N 4n+4. Furthermore, homothetic Killing vector fields as well as permuting Killing vector fields can be lifted to Killing vector fields (see Section 5 for definitions). When lifted to N 4n+4, we show that these vector fields in fact preserve the quaternion- Kähler form ΩQK . Analogous to the well-known Marsden-Weinstein symplectic reduction and its hyperKähler extension described in [33], Galicki-Lawson devel- oped the theory of QK reduction in [22]. This gives a way of taking quotients in the realm of QK manifolds. We derive general expressions for the Galicki-Lawson QK moment map for these Killing vector fields on N 4n+4. In particular, when M = R4n endowed with its standard flat hyperKähler structure, we consider the radial homothetic Killing vector field and by performing the QK reduction using Sp(n,1) the lifted action we recover explicitly the QK symmetric space Sp(n)Sp(1) :
4n+4 (N , ΩQK ) R3 R R α,ξ,η× t //// QK
4n Sp(n,1) (R ,ω1,ω2,ω3) Sp(n)Sp(1)
3 3 3 4n+3 Here Rα,ξ,η denotes the R fibres obtained by lifting the T fibres of Mα,ξ,η to the universal cover so that topologically N 4n+4 is just R4n+4. We prove that such a diagram holds more generally when M 4n is a conical HK manifold. We also show that the HK quotient of M 4n by tri-holomorphic Killing vector fields and the QK quotient of N 4n+4 by the lifted action commutes with the construction of Theorem 1.1. Strikingly similar features were demonstrated by Swann in [47], but rather 4n than starting from a HK base Swann starts from a QK base N+ (with positive × scalar curvature) and constructs a HK cone metric on a H /Z2 bundle known as 4n the Swann bundle (N+ ). U 1 4n Swann also shows that one can lift Killing S actions on N+ to triholomorphic 4n ones on (N+ ). Taking the HK reduction this yields a HK manifold of dimension 4n and theU latter inherits a permuting Killing vector field. In [29] Haydys then shows 4n that this reduction can in fact be inverted to recover the original QK manifold N+ , thereby establishing the HK/QK correspondence. This correspondence was further extended by Alekseevsky et al. in [1, 2] and Hitchin gave a twistorial description of Haydys’ construction in [32]. In Section 6 we show that given the HK manifold M 4n with a permuting S1 action we can lift it to a Killing action on N 4n+4. The QK 4n+4 4n quotient of N then yields a QK manifold N− of negative scalar curvature. In the same spirit as Haydys’ result, we show that one can then reconstruct M 4n from 4n 4n+4 N− (see Theorem 6.11 for a precise statement). Thus, N can be thought of 4n as an analogue of (N+ ). As a simple application we show that one can construct explicitly both theU complex hyperbolic space and the quaternion hyperbolic space starting from R4 with suitable permuting vector fields (and hence of course vice- versa). In Section 7 we construct explicitly the twistor spaces Z(N) and Z(Mα,ξ,η,ν) of 4n+4 the QK and hypercomplex manifolds N and Mα,ξ,η,ν , and we describe their relation with the twistor space Z(M) of the HK manifold M 4n: EINSTEIN METRICS ON BUNDLES OVER HYPERKÄHLER MANIFOLDS 5
4 T 3×R T Z(N) Z(M) Z(Mα,ξ,η,ν) Z(M)
2 2 S2 S2 S S T 3×R T 4 NM Mα,ξ,η,ν M
Twistor theory allows us to reinterpret geometric data on the N and Mα,ξ,η,ν in terms of the complex geometry of Z(N) and Z(Mα,ξ,η,ν ). In particular, the afore- mentioned Killing vector fields on N 4n+4 lift to holomorphic vector fields on Z(N) cf. [41]. In Section 8 we recall some known (incomplete) examples of Ricci-flat metrics on 4n+2 4n+4 the bundles of Theorem 1.1 and we also show that the manifolds Mξ,η , N and 4n+4 2n Mν1,ν2,ν3,ν4 admit balanced Hermitian metrics; a Hermitian manifold (M ,g,J,ω) is called balanced if ω is coclosed, or equivalently dωn−1 =0. In particular, we show that the metric gˇ of Theorem 1.4 is balanced for any of the hypercomplex structure. Balanced manifolds generalise the notion of Kähler manifolds which instead have closed ω. They were introduced by Michelsohn in [38] and they also appear in the Strominger system: a system that generalises the complex Monge-Ampère equa- tions and Hermitian-Yang-Mills equations cf. [21, 25, 37, 46]. All of our examples are very explicit and as such we hope will be useful in various applications.
Acknowledgements. The author would like to thank Simon Salamon for in- troducing to him the beautiful geometry of quaternion-Kähler manifolds and also for many useful comments that helped shape this article. We would also like to thank Vicente Cortés for interesting remarks that led to the results in section 6. This work was partly supported by the São Paulo Research Foundation (FAPESP) [2021/07249-0].
2. Preliminaries The aim of this section to fix some notations and gather some standard facts about Hermitian and quaternion-Kähler geometry that will relevant for us. We refer the reader to [6, 43] for proofs.
2.1. Hermitian structures. Let (M 2n,g,J,ω) be an almost Hermitian manifold. The complexified space of 1-forms splits as a +i and i eigenspace under J which we denote by Λ1,0 and Λ0,1 respectively. We denote by− Λp,q the space of complex (p + q)-forms obtained by wedging p elements of Λ1,0 with q elements of Λ0,1. (M 2n,J) is a complex manifold if and only if d(Γ(Λ1,0)) Γ(Λ1,1 Λ2,0), or equivalently d(Γ(Λn,0)) Γ(Λn,1). (M,g,J,ω) is then called a⊂ Hermitian⊕ manifold. If additionally, ω is closed⊂ then M is Kähler. The Hermitian manifold is said to be balanced if instead the weaker condition d(ωn−1)=0 is satisfied i.e. ω is coclosed. If there exists a global section Υ˜ of Λn,0 then the first Chern class of (M 2n,J) vanishes. If (M, g, J, ω, Υ)˜ is Kähler and Υ˜ is both closed and has constant norm then M is a Calabi-Yau manifold. This is equivalent to saying that the holonomy group of g is contained in SU(n). In particular, g is then Ricci-flat. We should point out that the definition of a Calabi-Yau manifold is not stan- dard. Some people also call a complex manifold (M,J) Calabi-Yau if it admits a holomorphic (n, 0)-form cf. [21, 25]. Henceforth, we shall always assume that M is a 4n-dimensional manifold.
4n 2.2. Quaternionic structures. (M , I1, I2, I3) is called an almost hypercomplex manifold if the almost complex structures I1, I2, I3 satisfy the quaternion relations (2.1) I I = I and I I = I I i ◦ j k i ◦ j − j ◦ i 6 U. FOWDAR where (i, j, k) are cyclic permutation of (1, 2, 3). If Ii are all complex structures then 4n (M , I1, I2, I3) is called hypercomplex. In fact, the quaternion relations imply that if any two of the almost complex structures is integrable then so is the third. A metric g is hyper-Hermitian if it is Hermitian for each Ii. We also have a triple of 2-forms ωi associated to each complex structure Ii by g(Ii( ), )= ωi( , ). 4n · · · · (M ,g,I1, I3, I3) is called hyperKähler if in addition each ωi is closed. This is equivalent to saying that the holonomy group of g is contained in Sp(n). In partic- ular, g is then Ricci-flat. If a Riemannian manifold (M 4n,g) admits locally a triple of compatible almost complex structure satisfying (2.1) then it is called almost quaternion Hermitian. Note that although the almost complex structures are not required to be globally well-defined the 4-form Ω locally defined by the expression 1 Ω := (ω ω + ω ω + ω ω ) 2 1 ∧ 1 2 ∧ 2 3 ∧ 3 is in fact globally well-defined. An equivalent way of phrasing this is saying that although the almost complex structures Ii can only be chosen locally the unit 2- sphere in the rank 3 subbundle = I1, I2, I3 End(TM) is globally well-defined. For n 2, (M 4n, g, Ω) is calledG h a quaternion-Kähleri⊂ manifold if Ω is parallel with respect≥ to the Levi-Civita connection g of g. Note that QK manifolds are not hypercomplex in general, consider for instance∇ HPn. It is well-known that they are Einstein manifolds i.e. Ric(g)= λ g [41]. If λ =0 then M is locally hyperKähler so this is excluded in the definition of· QK manifolds. If λ> 0 then the only known examples are the Wolf symmetric spaces [48]. By contrast, as demonstrated by LeBrun in [36], the λ < 0 case can even occur in infinite families and this will be the case of interest in this paper. In [47] Swann shows that Theorem 2.1. If n > 2 then gΩ=0 if and only if dΩ=0. If n = 2 then in ∇ addition to dΩ=0, we also require that the algebraic ideal generated by ω1,ω2,ω3 is a differential ideal i.e. the algebraic ideal is closed under differentiation.h i In [44] Salamon gave the first example of a compact quaternion Hermitian 8- manifold with dΩ=0 but whose algebraic ideal ω1,ω2,ω3 is not differential. This confirms that indeed closedness of Ω is not sufficienth forin = 2. More examples were constructed in [16]. None of these examples are not Einstein however. When n =1 one defines QK manifolds as 4-manifolds which are Einstein and self-dual i.e. the anti-self-dual part of the Weyl curvature vanishes e.g. S4 and CP2 [4]. The holonomy group of a QK manifold (M 4n, g, Ω) is contained in Sp(n)Sp(1). 2 Since Sp(n)Sp(1) SO(4n) and so(4n) ∼= Λ we have the splitting (1.13) of the space of 2-forms as⊂ Sp(n)Sp(1)-modules. As already mentioned above if M is a HK manifold then the subbundle sp(1) ∼= ω1,ω2,ω3 can be trivialised by parallel sections. For our purpose we will need a moreh concretei description of the subbundle determined by sp(n). Locally with respect to an almost complex structure J we have the decomposition ∈ G Λ2 C =Λ2,0 Λ1,1 Λ0,2 ⊗ J ⊕ J ⊕ J sp u 1,1 1,1 C 1,1 and hence we have (n) (2n) ∼= [ΛJ ], where by definition [ΛJ ] = ΛJ [43]. Since this holds for any⊂ J together with an argument using Schur’s⊗ lemma it follows that ∈ G sp 1,1 (2.2) (n) ∼= [ΛJ ]. J\∈G So the subbundle determined by sp(n) consists of 2-forms which are of type (1, 1) with respect to any almost complex structure J cf. [14, Proposition 1]. With these facts in mind we now proceed to prove Theorems∈ G 1.1, 1.3 and 1.4. EINSTEIN METRICS ON BUNDLES OVER HYPERKÄHLER MANIFOLDS 7
3. Proofs of Theorems 1.1, 1.3 and 1.4 Proof of Theorem 1.1. Let us first show that (1.8) is a QK metric on N 4n+4. In view of Theorem 2.1 we need to show that ΩQK is closed and for the n =1 case we also need to verify that the algebraic ideal generated by ω1,ω2,ω3 is closed under differentiation. A straightforward calculation using (1.1)-(1.3)h showsi that
2t 2t dω1 = 0 +( 4e η) ω2 + (4e ξ) ω3, 2t − ∧ 2t ∧ (3.1) dω2 = (4e η) ω1 + 0 +(4e α) ω3, ∧ ∧ dω = ( 4e2tξ) ω + ( 4e2tα) ω + 0, 3 − ∧ 1 − ∧ 2 which confirms the latter. One also easily sees from (3.1) that
3 dΩ = ω dω =0. QK i ∧ i Xi=1 This concludes the proof that (1.8) is a QK Einstein metric for all n. The fact that the almost complex structures Ii are all integrable will follow from Theorem 1.3 which we prove below. We now show that (1.5) is a Kähler metric on P 4n+2. It is clear from the structure equations that dωP =0. So it suffices to show that the associated almost complex structure is integrable. Observe that the form dt +2ie2tα is of type (1, 0) and σ2 + iσ3 is of type (2, 0) with respect to the associated almost complex structure. We have that 1 d(( dt + iα) (σ + iσ )n)= iσ (σ + iσ )n =0 2e2t ∧ 2 3 1 ∧ 2 3 i.e. d(Γ(Λ2n+1,0)) Γ(Λ2n+1,1) and hence the almost complex structure is inte- ⊂ grable. In fact the above calculation shows that c1(P )=0. Since QK manifolds are Einstein to complete the proof of the theorem it only remains to show that gP and gL are Einstein metrics as well. Consider more generally a metric of the form
2 2 2 2 2 2 2 2 (3.2) g = dt + p(t) gM + q(t) α + r(t) ξ + s(t) η on N 4n+4, then the Einstein equation Ric(g)= λ g is given by the system: · q′′ r′′ s′′ p′′ (3.3) + + +4n = λ, q r s · p − r′ s′ p′qq′ q4 (3.4) qq′′ + qq′( + )+4n n = λ q2, r s · p − · p4 − · q′ s′ p′rr′ r4 (3.5) rr′′ + rr′( + )+4n n = λ r2, q s · p − · p4 − · r′ q′ p′ss′ s4 (3.6) ss′′ + ss′( + )+4n n = λ s2, r q · p − · p4 − · q′ r′ s′ 1 (3.7) pp′′ + pp′( + + )+(4n 1) (p′)2 + (q2 + r2 + s2)= λ p2. q r s − · 2p2 − · For the metric ansatz
2 2 2 2 2 2 (3.8) g = dt + p(t) gM + q(t) α + r(t) ξ on L4n+3, the corresponding Einstein equation is obtained simply by eliminating the terms involving s(t) in (3.3)-(3.7). Likewise, for the metric ansatz
2 2 2 2 (3.9) g = dt + p(t) gM + q(t) α 8 U. FOWDAR on P 4n+2, one eliminates terms involving r(t) and s(t) in (3.3)-(3.7) to get the corresponding Einstein equation and finally for the ansatz 2 2 (3.10) g = dt + p(t) gM on Q4n+1 the whole system reduces to the well-known warped product Einstein equation λ (3.11) (p′)2 + p2 =0. 4n · One verifies directly that p(t) = aebt and q(t) = r(t) = s(t)=2a2be2bt solves the Einstein equations in all the cases. By rescaling the metric we can always set b = 1. The factor a can be interpreted geometrically as follows. If a is an 3 4n+3 4n integer we can instead consider the T bundle Maα,aξ,aη on M determined by [aσ1], [ aσ2], [ aσ3]. Thus, the parameter a essentially corresponds to pullbacking the metric− on covers− of the T3 fibres. As a consequence of this observation we can in 2 fact relax the hypothesis of the theorem to simply requiring that [ωi] H (M, 2πQ). Since the metrics are well-defined for all t R it follows that they∈ are complete, ∈ provided of course that gM is. In view of the Ambrose-Singer theorem cf. [35, Theorem 8.1], one can verify by computing the rank of the curvature operator, say using Maple, that the Einstein metrics gQ, gP , gL and gN have (restricted) holonomy groups equal to SO(4n + 1), U(2n + 1), SO(4n + 3) and Sp(n)Sp(1) respectively. It is also clear that each of the projection maps (N 4n+4,g ) (L4n+3,g ) (P 4n+2,g ) (Q4n+1,g ) (M 4n,g ) N → L → P → Q → M is a Riemannian submersion. Moreover, the T2 bundle N 4n+4 P 4n+2 is a holo- →4n+4 morphic fibration with respect to the complex structure I1 on N . Geometrically, one can interpret the metrics of Theorem 1.1 as twisting the HK metric gM with the hyperbolic metrics. For instance, the metric gL restricted to a fibre R T2 (over a point in M) can be expressed as t × 2 4t 2 2 (3.12) dt +4e (dy1 + dy2). 2 for local coordinates y1,y2 on the T . Thus, gL can be thought of as a twisted sum of gM with (3.12). Likewise the same is applies to gP and gN . Remark 3.1. In the case when we considered ansatz (3.9) on P 4n+2, we define a non-degenerate 2-form (or, equivalently, an almost complex structure) by (3.13) ω = qdt α + p2σ . P ∧ 1 Then the (2n +1, 0)-form (3.14) Ω := (q−1dt + iα) (σ + iσ )n P ∧ 2 3 is automatically closed and hence holomorphic. Thus, P 4n+2 is always a Hermitian manifold. The requirement that ωP is closed becomes q =2pp′. Assuming the latter holds, P is then Kähler and we can compute its Ricci form as Ric(ω )= i∂∂¯(log( Ω 2 )) P k P kgP (3.15) = ddc(log(p2n+1p′)). With p = et we get Ric(ω )= 4(n + 1) ω P − · P EINSTEIN METRICS ON BUNDLES OVER HYPERKÄHLER MANIFOLDS 9 confirming that it is Einstein. We can also easily deduce the Kähler potential from 1 ω = ddc(t). P −2 In this particular situation solving the Einstein equation was rather simple owing to the Kähler structure but in general, one has to solve the more complicated second order system (3.3)-(3.7). We now move to the proofs of Theorems 1.3 and 1.4.
Proof of Theorem 1.3. We want to show that the almost complex structures Iˇ1, Iˇ2, Iˇ3 associated to ωˇ1, ωˇ2, ωˇ3 are all integrable. Let us first consider the almost complex structure Iˇ1. The complex form (3.16) Υˇ := (σ + iσ )n (ξ + iη) (ν + iα) 1 2 3 ∧ ∧ is then of type (2n+2, 0) with respect to Iˇ1. The Nijenhuis tensor of Iˇ1 is determined by the (2n 1, 2) component of the exterior differential of Υˇ . Computing we get − 1 dΥˇ = (σ + iσ )n+1 (ν + iα) (σ + iσ )n (ξ + iη) (γ + iσ ) 1 − 2 3 ∧ − 2 3 ∧ ∧ 1 = (σ + iσ )n (ξ + iη) γ − 2 3 ∧ ∧ = 0
n+1 2n+2,0 4n n 2n+1,1 4n where we used that (σ2+iσ3) Λ (M ) and (σ2+iσ3) σ1 Λ (M ) ∈ I1 ∧ ∈ I1 for the second equality and for the last equality we used the fact that γ sp(n), n 2n+1,1 4n ∈ see (2.2), so (σ2 + iσ3) γ Λ (M ). To complete the proof it suffices to ∧ ∈ I1 apply the analogous argument for Iˇ2 and Iˇ3 with (3.17) Υˇ := (σ + iσ )n (ξ + iν) (α + iη) 2 3 1 ∧ ∧ and (3.18) Υˇ := (σ + iσ )n (ξ + iα) (η + iν). 3 1 2 ∧ ∧
Taking 1 ν = e−2tdt 2 and M 4n+4 = R M 4n+4 (by lifting to the universal cover of trivial S1 bundle) α,ξ,η,ν t × α,ξ,η ν we see that Iˇ1, Iˇ2, Iˇ3 agree with I1, I2, I3: on the fibres we have I (ξ)= η, I (ν)= α, 1 − 1 − I (ξ)= ν, I (α)= η, 2 − 2 − I (ξ)= α, I (η)= ν, 3 − 3 − and it is clear that they agree on the horizontal space. Thus, I1, I2, I3 define a hypercomplex structure on N 4n+4. h i
Remark 3.2. The proof of Theorem 1.3 in fact shows that Υˇ i are holomorphic (2n +2, 0)-forms for Iˇi and hence the canonical bundles are trivial. However, if 4n+4 either one of [σi], [ν] is non-trivial in cohomology then Mα,ξ,η,ν cannot even be Kähler. Say [σ1] is non-trivial then a simple calculation using the Gysin sequence 1 shows that the S fibres of Mα are trivial in homology cf. [28]. It follows that a 2 T fibre in Mα,ξ,η,ν determined by [σ1] and [σ2] is a homologically trivial elliptic curve with respect to I3. Now any I3-compatible Kähler form would integrate to zero on this T2 but on the other hand this corresponds to its volume which is a contradiction. This argument can be applied to any of the complex structures by 2 4n+4 4n choosing suitable T . In particular, we deduce that Mα,ξ,η,ν is never Kähler if M is compact. 10 U. FOWDAR
Proof of Theorem 1.4. The proof consists of just repeating the argument in the proof of Theorem 1.3 but replacing α, ξ, η and ν by ν1, ν2, ν3 and ν4 respectively and using (2.2).
4n Remark 3.3. The fact that dνi are curvature forms of type (1, 1) on M is equivalent to saying that the connection forms νi are (abelian) Hermitian Yang- Mills connections, and this is true with respect to any of the hyperKähler triple. In the terminology of [14] these connections are called self-dual instantons. By contrast, the connection forms of Theorem 1.1 are called anti-self-dual instantons. In both cases the connections are critical points of the Yang-Mills functional
2 YM(A) := FA gM volM , ZM k k where A is a connection form and FA its curvature. It is easy to see that one can replace the hypothesis that M 4n is hyperKähler 4n+4 with just hypercomplex and still conclude that Mν1,ν2,ν3,ν4 is hypercomplex. The only thing we lose in doing so is that there is no longer an Sp(n + 1)-structure with trivial canonical bundle but instead just a GL(n+1, H)-structure on the total space 4n+4 Mν1,ν2,ν3,ν4 . 4. Examples In this section we construct various examples that arise from Theorems 1.1, 1.3 and 1.4. We consider primarily the situation when M is 4-dimensional as this is the most well-understood one. Kodaira’s classification states that the only compact complex surfaces with trivial canonical bundle are K3 surfaces and T4 cf. [5]. In the former case one needs to study the K3 lattice to find integral classes. We shall instead consider the latter case as it is more explicit.
4 4.1. Examples with M = T . Denote by (x1, x2, x3, x4) the local coordinates on T4 and consider the standard HK triple:
σ1 = dx12 + dx34,
σ2 = dx13 + dx42,
σ3 = dx14 + dx23, which after suitable rescaling we may assume define integral cohomology classes. 5 6 7 The manifolds Mα, Mα,ξ and Mα,ξ,η then correspond to nilmanifolds with nilpotent Lie algebras (4.1) (0, 0, 0, 0, 12+ 34), (4.2) (0, 0, 0, 0, 12+34, 13+ 42), (4.3) (0, 0, 0, 0, 12+34, 13+42, 14+ 23), in Salamon’s notation cf. [45]; in this convention (0, 0, 0, 0, 12+ 23) represents the Lie algebra with an invariant coframing ei such that de1 = de2 = de3 = de4 = 0 and de5 = e12 + e34. The Lie algebra (4.3) corresponds to the quaternion Heisenberg group, which is the nilpotent part in the Iwasawa decomposition of the isometry group of the quaternionic hyperbolic space [18]. It is thus not surprising that the universal cover 8 Sp(2,1) of N corresponds to the quaternion hyperbolic space Sp(2)Sp(1) as was shown in ˜ 4n+4 Sp(n+1,1) 4n [27]. More generally, we have that N = Sp(n+1)Sp(1) when M = T . Recall from Theorem 1.3 that we need ν sp(n) to construct hypercomplex structures. In dimension 4 this means that ν ∈ Λ2 (M) i.e. ν is an anti-self-dual ∈ − EINSTEIN METRICS ON BUNDLES OVER HYPERKÄHLER MANIFOLDS 11
2-form. So we can construct distinct hypercomplex structures by choosing ν in the Z-module generated by dx dx , dx dx , dx dx . h 12 − 34 13 − 42 14 − 23i For instance, if ν = dx dx then M 8 is the nilmanifold with Lie algebra 12 − 34 α,ξ,η,ν (0, 0, 0, 0, 12, 34, 13+ 42, 14+ 23).
If one applies Theorem 1.4 with all νi =0 then we just get the product hypercom- plex structure on M T4. If instead one sets × ν = dx dx , ν = dx dx , ν = dx dx , ν =0 1 12 − 34 2 13 − 42 3 14 − 23 4 then we recover the abelian hypercomplex structure studied in [18] on (0, 0, 0, 0, 0, 12 34, 13 42, 14 23). − − − Note that although we get an isomorphic Lie algebra from Theorem 1.3 with ν =0, the hypercomplex structures are different. These examples all generalise naturally to the case when M = T4n, of course the space sp(n) is then much larger so one can construct more examples. Next we consider the case when M 4 is non-compact. 4.2. Examples from the Gibbons-Hawking ansatz. Consider an open set B 3 ⊂ R with coordinates u = (u1,u2,u3) and endowed with the flat Euclidean structure. Suppose that V : B R+ is a positive harmonic function such that [ dV ] H2(B, 2πZ), where →is the Hodge star operator on B, and denote by−∗M 4 the∈ associated S1-bundle.∗ The Gibbons-Hawking ansatz then asserts that σ = θ du + V du , 1 ∧ 1 23 σ = θ du + V du , 2 ∧ 2 31 σ = θ du + V du 3 ∧ 3 12 define a HK triple on the M 4, where θ is a connection 1-form such that dθ = dV [26]. If we take −∗ k 1 V = c + , 2 u p Xi=1 | − i| 3 where pi are k distinct points in R and c 0 then we get complete HK metrics called multi-Taub-NUT if c = 0 and multi-Eguchi-Hanson≥ if c = 0. Euclidean R4 is a special case of the latter6 when k = 1. Although the harmonic function V is 3 singular at pi the HK metric gM extends smoothly to a fibration over all of R with 1 the S fibres collapsing to points over pi. Suppose now that pi are chosen to lie on the line u2 = u3 =0 with pi In the next section we study the infinitesimal symmetries of the Einstein mani- folds constructed in Theorem 1.1. 5. Symmetries and moment maps Just like the symplectic reduction can be used to construct symplectic manifolds starting from suitable symplectomorphic group actions, one can also reduce QK manifolds with certain Killing vector fields to produce new QK manifolds. This is so-called the QK reduction [22]. In this section we describe how one can apply this technique to the QK manifolds N 4n+4 of Theorem 1.1 to construct more examples. 5.1. The QK reduction. We recall briefly the QK reduction. Let (N,gN , ΩQK ) be a general QK manifold and let X˜ be a Killing vector field such that ˜ Ω =0. LX QK Then Galicki-Lawson show that there exists a unique map f ˜ : N such that X → G ˜ y dfX˜ = X ΩQK called the QK moment map for the action generated by X˜. This applies more generally when we have a G-action preserving gN and ΩQK but we shall restrict to the case when G is 1-dimensional i.e. generated by a single vector field X˜ . Theorem 5.1 ([22]). If G acts freely and properly on f −1(0) then the quotient X˜ space N := f −1(0)/G red X˜ is a QK manifold. If the G-action is only locally free then Nred is instead an orbifold. Note that in contrast to the symplectic reduction, however, one can only take the quotient at the zero section in . This is essentially due to the fact that there are no “constants” in to add. G G 5.2. Reduction using tri-holomorphic Killing vector fields. Let X be a tri- holomorphic Killing vector field on M 4n i.e. σ = 0 for i = 1, 2, 3. Working LX i locally there exists 1-forms κi such that (5.1) σi = dκi, 3 4n+3 and by choosing local coordinates y1,y2,y3 on the T fibres of Mα,ξ,η we can write (5.2) α = dy1 + κ1, (5.3) ξ = dy κ , 2 − 2 (5.4) η = dy κ . 3 − 3 By Poincaré Lemma we can choose κ so that κ =0 and hence the natural lift i LX i X˜ of X (satisfying dyi(X˜)=0) to N preserves α, ξ and η. The functions (5.5) µ := X˜ y α, α − y (5.6) µξ := X˜ ξ, y (5.7) µη := X˜ η, then define the HK moment maps for the action generated by X on M since from Cartan’s formula we have y y y dµ = d(X˜ α)= ˜ α + X dα = X σ , α − −LX 1 and likewise for µη and µη. It is clear by construction that X˜ is Killing and preserves ΩQK . We compute the QK moment map for the action generated by X˜ on N as 2t (5.8) fX˜ = e (µαω1 + µξω2 + µηω3). EINSTEIN METRICS ON BUNDLES OVER HYPERKÄHLER MANIFOLDS 13 Indeed one verifies using (3.1) that X˜y Ω = (e2tdµ +4e4tµ η 4e4tµ ξ +2e2tµ dt) ω + QK α ξ − η α ∧ 1 (e2tdµ 4e4tµ η 4e4tµ α +2e2tµ dt) ω + ξ − α − η ξ ∧ 2 (e2tdµ +4e4tµ α +4e4tµ ξ +2e2tµ dt) ω η ξ α η ∧ 3 = dfX˜ The zero set f −1(0) N corresponds to the bundle N M restricted to the X˜ ⊂ → set (µ ,µ ,µ )−1(0) M. Taking the quotient we see that N is just the QK α ξ η ⊂ red manifold we get from Theorem 1.1 by taking the HK base to be Mred, the HK quotient of M by the action generated by X [33]. In other words, we have: Proposition 5.2. The construction of Theorem 1.1 commutes with the QK reduc- tion of N and the HK reduction of M: /QK N Nred 3 T ×R T 3×R /HK MMred As a trivial example consider the case when M is a HK 4-manifold. The HK moment map then corresponds to the functions (x1, x2, x3) in the notation of sub- section 4.2. Mred can be viewed as a discrete set of points in M (say by fixing 1 a point in each S orbit) and Nred is just the fibres over these points with the hyperbolic QK metric 2 4t 2 2 2 gred = dt +4e (dy1 + dy2 + dy3). 5.3. Reduction using permuting vector fields. Let us now assume that X is a permuting Killing vector field on M 4n i.e. it preserves one of Kähler forms and rotates the other two. Without loss of generally we can assume X σ1 = 0 while σ = 2σ and σ = +2σ . We choose κ as before and setL LX 2 − 3 LX 3 2 1 1 (5.9) κ := + (Xy σ ), 2 2 3 1 (5.10) κ := (Xy σ ). 3 −2 2 This implies that (5.1) holds and that (5.11) κ =0, κ = 2κ , κ = +2κ . LX 1 LX 2 − 3 LX 3 2 We now define a lift of X to N by (5.12) X˜ = X 2 (y ∂ y ∂ ) − · 3 y2 − 2 y3 and define α,ξ,η,µα,µξ,µη by expressions (5.2)-(5.7) as before. An analogous computation shows that X˜ is Killing and ˜ Ω =0 (since by construction X˜ is LX QK also permuting on ω1,ω2,ω3). We compute the QK moment map for X˜ as 2t 1 −2t (5.13) f ˜ = e ((µ e )ω + µ ω + µ ω ). X α − 2 1 ξ 2 η 3 Note that unlike before the functions µξ,µη cannot be interpreted as moment maps 4n but µα is still a symplectic moment map on M for X. We refer the reader to [30] for several examples of permuting Killing vector fields. We shall re-examine this construction in greater detail in section 6 below. 14 U. FOWDAR 5.4. Reduction using homothetic Killing vector fields. Let us now assume that X is a homothetic Killing vector field so that σ = 2σ and define LX i − i 1 (5.14) κ := (Xy σ ). i −2 i This implies that (5.1) holds and that (5.15) κ = 2κ . LX i − i We define a lift of X to N by (5.16) X˜ = X 2 (y ∂ + y ∂ + y ∂ )+ ∂ − · 1 y1 2 y2 3 y3 t and define α,ξ,η,µα,µξ,µη by expressions (5.2)-(5.7). A simple computation shows that ˜ Ω =0 and we find the QK moment map for X˜ is LX QK 2t (5.17) fX˜ = e (µαω1 + µξω2 + µηω3). Remark 5.3. It’s worth pointing out that we only used that κi satisfy relations (5.1) and (5.15) to compute the moment map. In particular, one can choose κi different from (5.14). This amounts to modifying κi by a suitable closed 1-form. Lastly note that the vertical Killing vector fields ∂y1 , ∂y2 , ∂y3 always preserve ΩQK . The corresponding QK moment map for (5.18) Y = a∂ + b∂ + c∂ , − y1 y2 y3 where a,b,c R is given by ∈ 2t (5.19) fY = e (aω1 + bω2 + cω3). This moment map is nowhere vanishing so Nred would be an empty set in this case. However, by taking linear combinations of Y with X˜ in either of the previous 3 cases we may add constants to the corresponding moment maps. Remark 5.4. It is also easy to see that the vector fields Y and X˜ (modified in the obvious way) are also Killing for the metrics gQ,gP ,gL of Theorem 1.1. More- over, we get a Hamiltonian action on P 4n+2 so one can investigate the symplectic reduction in this case. 5.5. An explicit example. We illustrate the above reduction explicitly in the case when M = R4 endowed with a homothetic Killing vector field. 4 Denoting by (x1, x2, x3, x4) the coordinates on R we consider the standard HK triple as in section 4.1. We take the homothetic Killing vector field U = x ∂ x ∂ x ∂ x ∂ , − 1 x1 − 2 x2 − 3 x3 − 4 x4 which corresponds to scaling along the radial direction. From the definition of κi we find that the connection forms are given by 1 α = dy (x dx x dx + x dx x dx ), 1 − 2 2 1 − 1 2 4 3 − 3 4 1 ξ = dy + (x dx x dx x dx + x dx ), 2 2 3 1 − 4 2 − 1 3 2 4 1 η = dy + (x dx + x dx x dx x dx ), 3 2 4 1 3 2 − 2 3 − 1 4 and the moment map becomes 2t f ˜ =2e (y ω y ω y ω ). U 1 1 − 2 2 − 3 3 4 The reduction now yields a QK metric on Nred R . In general, identifying the quotient metric explicitly is quite hard, however,≃ in this rather simple situation we can write 1 2 − (5.20) gN f 1(0) = y dt d(ln(y)) + gred U˜ − 2 EINSTEIN METRICS ON BUNDLES OVER HYPERKÄHLER MANIFOLDS 15 −1 2 2t 2 2 2 2 where y := cosh (√1+ r e ) and r := x1 + x2 + x3 + x4. The 1-form dt 1 − 2 d(ln(y)) is just the canonical Riemannianp connection for the action generated by U˜ i.e. ˜ 1 gΩ −1 (U, ) dt d(ln(y)) = f (0) · . − 2 ˜ ˜ gΩ f −1(0)(U, U) and the reduced QK metric gred can be explicitly expressed as 2 2 2 (5.21) gred = dy + sinh (y)cosh (y)gS3 . The above calculation is easily carried out using spherical coordinates on R4. This Sp(1,1) quotient corresponds to the QK symmetric space Sp(1)Sp(1) . Remark 5.5. In view of remark 5.3 one can take say κ1 to be x2dx1 x4dx3 instead. Repeating the above calculation one can show that the− quotient− metric gred remains unchanged although the QK moment fU˜ changes. An example of a permuting Killing action in this situation is given by the diagonal C2 R4 action of U(1) on ∼= generated by the vector field V = x ∂ + x ∂ x ∂ + x ∂ − 2 x1 1 x2 − 4 x3 3 x4 and an example of a tri-holomorphic Killing vector field is given by W = x ∂ + x ∂ + x ∂ x ∂ − 2 x1 1 x2 4 x3 − 3 x4 corresponding to the action of the diagonal U(1) in SU(2) on C2. Taking linear combination of these vector fields one can investigate more general reductions of N 8. The QK moment map for the Killing vector field X˜ = uU˜ + vV˜ + wW˜ + Y , where u,v,w R is given by ∈ 2t v 2 −2t w 2 2 2 2 (5.22) f ˜ = e (a +2uy (r + e ) (x + x x x ))ω + X 1 − 2 − 2 1 2 − 3 − 4 1 (b 2uy 2vy w(x x + x x ))ω + − 2 − 3 − 1 4 2 3 2 (c 2uy +2vy + w(x x x x ))ω . − 3 2 1 3 − 2 4 3 Identifying the reduced metric explicitly appears to be rather hard in this more general situation. 5.5.1. 3-Sasakian manifolds. The above simple example can more generally be ap- plied to any 3-Sasakian manifold 4n−1. There are several equivalent definitions of 3-Sasakian manifolds cf. [10, 23],S the simplest to state is that they are Riemann- ian manifolds whose cone metric is hyperKähler. More concretely, this means that they admit 3 unit Killing vector fields, or equivalently using the metric a triple of 1-forms γ , spanning a rank 3 subbundle of T such that i F S dγ =2γ γ +2ˇσ , i j ∧ k i where (i, j, k) are cyclic permutations of (1, 2, 3) and with σˇi defining a Sp(n 1)- ⊥ 4n + − structure on . The hyperKähler triple on the cone M := Rr are then given by F × S 1 σ := d(r2γ ). i 2 i The homothetic Killing vector field X = r∂r (i.e. the Euler vector field) satisfies the hypothesis of subsection 5.4. Repeating− the exact calculation as in case with M = R4 shows that 2 2 2 2 2 2 2 (5.23) gred = dy + sinh (y)cosh (y)(γ1 + γ2 + γ3 ) + sinh (y)gF ⊥ , ⊥ where g ⊥ denotes the 3-Sasakian metric restricted to . F F Note that this metric is complete only if is the round sphere in which case gred corresponds to the quaternionic hyperbolicS metric whereas the HK cone metric on 16 U. FOWDAR M corresponds to the flat Euclidean metric. Otherwise, for general we have an isolated singularity at y =0. S More generally, if X is a homothetic Killing vector field on HK M 4n then fol- 4 lowing the same strategy as in the M = R case we can express gred as 3 (5.24) g = dt2 +4e4t( κ2)+ e2tg (1 + e2t X 2 )−1(dt + e2tg (X, ))2. red i M − k kgM M · Xi=1 5.5.2. Local examples from the Gibbons-Hawking ansatz. If one chooses V = u1 in the Gibbons-Hawking ansatz, see subsection 4.2, so that locally the connection form 1 is θ = dy + u3du2, where y denotes the coordinate of the S fibre then we still get a HK metric but it is incomplete. We leave it to the reader to verify that 2 X = (2y∂ + u ∂ + u ∂ + u ∂ ) −3 y 1 u1 2 u2 3 u3 is a homothetic Killing vector field satisfying the hypothesis of section 5.4. Unlike in the previous case however this is not a gradient vector field. One can find many such examples that by choosing suitable V and thus one can apply the above quotient construction. However, as X is not a gradient vector field it appears to be much harder to identify the reduced metric (5.24) explicitly. 5.6. QK products. In general the product of two QK manifolds is not QK. In [47] Swann gave a way of joining two QK manifolds N1,N2 of positive scalar curvature to construct a QK manifold of dimension dim(N1)+ dim(N2)+4. We now describe a similar join construction which applies to the quotient manifolds of section 5.4. Let M1,M2 be HK manifolds with homothetic Killing vector fields X1 and X2 respectively as in section 5.4. The above quotient construction then produces two (negative scalar curvature) QK manifolds M1red and M2red. Of course the product metric on M1red M2red is not QK but one can still define a “QK product” of these spaces as follows.× Observe that M M is a HK manifold with the standard 1 × 2 product structure and the vector field X1 +X2 defines a diagonal homothetic action. Applying the construction of section 5.4 to the data (M1 M2,X1 +X2) yields a QK manifold of dimension dim(M ) + dim(M ) which× we denote by (M ,M ): 1red 2red J 1 2 the QK product of M1red and M2red. R+ Here is an example. Let Mi := xi i be the HK cone of the 3-Sasakian manifold for i =1, 2. Then product HK× metric S is given by Si 2 2 2 2 gM1 + gM2 = dx1 + dx2 + x1gS1 + x2gS2 2 2 2 2 2 (5.25) = dr + r (dθ + cos (θ)gS1 + sin (θ)gS2 ), where we use polar coordinates x1 = r cos(θ) and x2 = r sin(θ). The metric 2 2 2 (5.26) dθ + cos (θ)gS1 + sin (θ)gS2 is of course again a 3-Sasakian metric. For instance, if one takes both i to be the standard 3-sphere S3 then (5.26) corresponds to the round metric on SS7. The QK metric on (M1,M2) can be explicitly worked out from (5.23). J 4n 4k When M1 = R and M2 = R with the homothetic radial Killing vector fields Sp(n,1) Sp(k,1) as above we already saw that M1red = Sp(n)Sp(1) and M2red = Sp(k)Sp(1) . Thus we deduce that the QK product is Sp(n + k, 1) (5.27) (R4n, R4k)= . J Sp(n + k)Sp(1) This simple example shows that the name QK product is rather apt. EINSTEIN METRICS ON BUNDLES OVER HYPERKÄHLER MANIFOLDS 17 Remark 5.6. The metric (5.26) is closely related to the so-called sine-cone con- struction: if is a positive scalar curvature Einstein metric then its sine-cone is the product spaceM [0, π] endowed with the metric ×M 2 2 (5.28) dθ + sin (θ)gM . Although these metrics are singular at θ = 0, π (except for round spheres) the singularities are rather mild as they are only conical singularities. In [19] Foscolo- Haskins found exotic Einstein metrics on S6 and S3 S3 by desingularising suitable sine-cones. The sine-cone metric (5.28) can be viewed× as a special case of (5.26) when is taken to be a point. S1 6. A HK/QK type correspondence The goal of this section is to prove that the QK reduction using a permuting Killing vector field X as in subsection 5.3 can in fact be inverted. More precisely, 4n we can reconstruct the hyperKähler manifold (M , σ1, σ2, σ3) together with the vector field X from suitable data on N 4n := N 4n+4////S1 . This is closely related to red X˜ the construction of Haydys in [29] whereby he shows that the HK quotient of the Swann bundle by a (lifted) S1 action can be inverted to recover the original (positive scalar curvature) QK manifold. By contrast our construction gives a negative scalar curvature version of this correspondence. 6.1. The permuting quotient. Before describing the inverse construction we first 4n need to understand better the induced geometric structure on the quotient Nred. Given a permuting Killing vector field X on M 4n we consider the more general lift a X˜ = X 2 (y ∂ y ∂ ) ∂ − · 3 y2 − 2 y3 − 2 y1 to N 4n+4. Then from the results of the previous section we know that the QK moment map is given by 2t a 1 −2t y (6.1) f ˜ = e (( e X κ )ω 2y ω +2y ω ) X 2 − 2 − 1 1 − 3 2 2 3 and hence f −1(0) = y =0,y =0,t = 1 log(a 2(Xyκ )) . It is worth pointing X˜ { 2 3 − 2 − 1 } out that the presence of the constant a is important as we shall see in subsection 6.2. For the rest of this section we shall work on f −1(0) and pullback the data on X˜ N 4n+4 to the submanifold f −1(0). It is worth highlighting that f −1(0) is diffeomor- X˜ X˜ 4n+1 phic to Mα . By abuse of notation we denote the pullbacked differential forms and metric by the same expression. The S1 action generated by the Killing vector field X˜ on f −1(0) can be identified with X˜ a ˜ − X f 1(0) = X ∂y1 . X˜ − 2 We shall also denote the latter by X˜ to ease notation. 4n Proposition 6.1. The differential forms ω1 and Ω descend to the quotient Nred. Proof. Using the fact that t = 1 log(a 2(Xyκ )) on f −1(0), a simple computation − 2 − 1 X˜ shows that X˜ y ω = e2t(Xy σ )+ e2t(a 2Xy κ )dt 1 1 − 1 = e2td(Xy κ )+ e2td(Xy κ ) − 1 1 =0. 18 U. FOWDAR A similar calculation shows that y y (6.2) X˜ ω2 = X˜ ω3 =0. Together with the first equation of (3.1) one deduces that X˜ ω1 = 0 and hence 4n L ω1 passes to Nred. Since dΩ=0, it follows that Ω is also a basic form and hence passes to the quotient as well. 4n The induced QK metric gred on Nred is determined by (6.3) g = g (X,˜ X˜) ξ2 + g , Ω Ω · X red where ξ is the Riemannian connection 1-form on π : f −1(0) N 4n defined by X X˜ → red gΩ(X,˜ ) (6.4) ξX ( ) := · . · gΩ(X,˜ X˜) ∗ Strictly speaking one should write π gred in (6.3) but since π is a Riemannian ∗ submersion we can identify gred with π gred. The key observation now is that there is another S1 action on f −1(0) generated X˜ by the Killing vector field ∂y1 which also preserves ξX . Moreover it commutes with ˜ 1 4n X and as such it defines an S action on Nred. This will be the crucial ingredient 4n to inverting this quotient. We denote by Z the vector field on Nred generated by this action. Proposition 6.2. The Killing vector field ∂y1 can be expressed as (6.5) ∂ = Z 2(a 2p + g (X,X))−1 X,˜ y1 h − − M · y where Zh denotes the horizontal lift of Z and p := X κ1 (the negative of the 4n moment map for X associated to σ1 on M ). Proof. Since by definition π∗(∂y1 ) = π∗(Zh) = Z we only need to check that ξ (∂ )= 2(a 2(Xy κ )+ g (X,X))−1. Observe that on f −1(0) we have X y1 − − 1 M X˜ dp2 (6.6) g = + 4(a 2p)−2((dy + κ )2 + κ2 + κ2) + (a 2p)−1g Ω (a 2p)2 − 1 1 2 3 − M − and from this we compute the connection form explicitly as X♭ 2dy 2κ (6.7) ξ = − 1 − 1 , X a 2p + g (X,X) − M where X♭ := g (X, ). It is now easy to see that M · ξ (∂ )= 2(a 2p + g (X,X))−1 X y1 − − M and this concludes the proof. 4n Using (6.6) and (6.7) we can express gred explicitly in terms of the data on M as dp2 4g (X,X) (6.8) g = + M (dy + κ )2 red (a 2p)2 (a 2p)2(a 2p + g (X,X)) 1 1 − − − M 4 1 + (κ2 + κ2)+ g (a 2p)2 2 3 (a 2p) M − − 1 + (2X♭ (dy + κ ) (X♭)2). (a 2p)(a 2p + g (X,X)) ⊙ 1 1 − − − M 4n In contrast to ω , ˜ ω = 0 for i = 2, 3 and hence these do not descend to N . 1 LX i 6 red However by choosing new local coordinates on f −1(0) we can write X˜ = ∂ , where X˜ x x denoting the fibre coordinate and we define (6.9) ω¯ := h ω f ω and ω¯ := f ω + h ω 2 · 2 − · 3 3 · 2 · 3 EINSTEIN METRICS ON BUNDLES OVER HYPERKÄHLER MANIFOLDS 19 where f := cos(2x) and h := sin(2x). It follows that 2 2 2 2 ω2 + ω3 =ω ¯2 +¯ω3 and moreover we have: 4n Proposition 6.3. The 2-forms ω¯2 and ω¯3 descend to Nred. Proof. From (6.2) we already know that ω¯2 and ω¯3 are horizontal forms, and by −1 construction we have that ˜ ω = 2ω and ˜ ω = +2ω on f (0). Hence we X 2 3 X 3 2 X˜ compute L − L ˜ ω¯ = ( ˜ h)ω ( ˜ f)ω + h( ˜ ω ) f( ˜ ω ) LX 2 LX 2 − LX 3 LX 2 − LX 3 =2fω +2hω 2hω 2fω 2 3 − 3 − 2 =0 and likewise ˜ ω¯ =0. LX 3 Remark 6.4. The intuition behind definition (6.9) is indeed to rotate ω2 and ω3 in the opposite direction to that determine by the permuting vector field X˜ and as such to obtain invariant forms. 4n ¯ In summary, we have shown that Nred inherits the data (gred, Ω := Ω, ω¯1 := ω1, ω¯2, ω¯3), the curvature 2-form dξX and the Killing vector field Z. It is clear 4n by construction that the triple ω¯1, ω¯2, ω¯3 determines the QK structure on Nred. Moreover we have: 4n Proposition 6.5. The sp(1)-component of the Levi-Civita connection of (Nred,gred) is determined by dω¯ = 0 + α ω¯ + α ω¯ , 1 − 2 ∧ 2 3 ∧ 3 (6.10) dω¯2 = α2 ω¯1 + 0 + β ω¯3, ∧ ∧ dω¯ = α ω¯ + β ω¯ + 0, 3 − 3 ∧ 1 − ∧ 2 where α := 4e2t(fκ + hκ ), α := 4e2t( hκ + fκ ) and β := 2dx +4e2tα. 2 − 2 3 3 − 2 3 Proof. First note that from (6.9) we can write (6.11) ω = h ω¯ + f ω¯ and ω = f ω¯ + h ω¯ . 2 · 2 · 3 3 − · 2 · 3 So from the first equation of (3.1) (again pullbacked to f −1(0)) we have X˜ dω = (4e2tκ ) (h ω¯ + f ω¯ ) (4e2tκ ) ( f ω¯ + h ω¯ ) 1 3 ∧ · 2 · 3 − 2 ∧ − · 2 · 3 =4e2t(fκ + hκ ) ω¯ +4e2t( hκ + fκ ) ω¯ 2 3 ∧ 2 − 2 3 ∧ 3 and this proves the first line of (6.10). Likewise from (3.1) we have dω = 4e2tκ ω +4e2tα ( f ω¯ + h ω¯ ) 2 − 3 ∧ 1 ∧ − · 2 · 3 and dω =4e2tκ ω 4e2tα (h ω¯ + f ω¯ ). 3 2 ∧ 1 − ∧ · 2 · 3 Hence we get dω¯ =2fdx ω +2hdx ω + hdω fdω 2 ∧ 2 ∧ 3 2 − 3 =2fdx (h ω¯ + f ω¯ )+2hdx ( f ω¯ + h ω¯ ) ∧ · 2 · 3 ∧ − · 2 · 3 + h( 4e2tκ ω +4e2tα ( f ω¯ + h ω¯ )) − 3 ∧ 1 ∧ − · 2 · 3 f(4e2tκ ω 4e2tα (h ω¯ + f ω¯ )) − 2 ∧ 1 − ∧ · 2 · 3 = 4e2t(fκ + gκ ) ω + (2dx +4e2tα) ω¯ − 2 3 ∧ 1 ∧ 3 and a similar computation for ω3 yields the result. 20 U. FOWDAR −1 4n It is worth noting that β(Z)=4(a 2p) . Since Nred inherits a Killing vector field Z, the natural next step is to understand− the properties of this action. Proposition 6.6. The vector field Z preserves the data (gred, ω¯1, ω¯2, ω¯3, dξX ) on 4n Nred. Proof. From (6.6) it is easy to see that gΩ = 0. As X˜ and ∂ commute, we L∂y1 y1 have that ∂y1 ξX = 0 and ∂y1 gred = 0. It is also clear that ∂y1 ω¯1 = ∂y1 ω¯2 = L L 4n L L ω¯3 =0. Since for any 1-form γ on N L∂y1 red π∗( γ)= (π∗γ)= (π∗γ), LZ LZh L∂y1 the result follows. Next we derive an explicit expression for the connection form ξX in terms of the 4n data on Nred. Proposition 6.7. On f −1(0) we have X˜ 1 1 (6.12) ξ = dx + (a 2p)Z♭ β, X 2 − − 2 ♭ where Z := gred(Z, ). In particular, the curvature form dξX is completely deter- · 4n mined by Z and the QK structure on Nred. Proof. The result follows from the definition of β =2dx + 4(a 2p)−1α, − g (X,˜ X˜) = (a 2p)−1(a 2p + g (X,X)) Ω − − M and Z♭ = g (Z , ) Ω h · = g (∂ , )+2(a 2p + g (X,X))−1g (X,˜ X˜)ξ Ω y1 · − M Ω X = 4(a 2p)−2α + 2(a 2p + g (X,X))−1g (X,˜ X˜)ξ . − − M Ω X ♭ y 4n Now observe that X := gM (X, ) = gM (IiX, Ii ) = Ii(X σi) on M . Thus, we have · · (6.13) X♭ = I (dp)= 2I (κ )=+2I (κ ) − 1 − 2 3 3 2 and in particular, dX♭ Ω1,1(M 4n). The analogous calculation on N 4n shows that ∈ I1 red 2 (6.14) Z♭ = I¯ (dp) = (a 2p)−1I¯ (α )= (a 2p)−1I¯ (α ), −(a 2p)2 1 − 2 2 − − 3 3 − or equivalently, (6.15) 4Z♭ = I¯ (d(β(Z))) = β(Z) I¯ (α )= β(Z) I¯ (α ), − 1 · 2 2 − · 3 3 where I¯1, I¯2, I¯3 denote the almost complex structures associated to ω¯1, ω¯2, ω¯3. We then have that 4n ¯ Proposition 6.8. (Nred, I1) is a complex manifold. Proof. From (6.10) we have d(¯ω + iω¯ ) = (α iα ) ω iβ (¯ω + iω¯ ). 2 3 2 − 3 ∧ 1 − ∧ 2 3 Since α3 = I¯1(α2), it follows that d(¯ω2 + iω¯3) is of type (2, 1)+(3, 0) with respect to I¯1 and hence I¯1 is an integrable almost complex structure. EINSTEIN METRICS ON BUNDLES OVER HYPERKÄHLER MANIFOLDS 21 ♭ 1,1 4n Thus, we deduce that dZ Ω ¯ (N ) and using (6.12) we have that ∈ I1 red 1 1 dξ = dβ ddc¯1 (log(β(Z))) X −2 − 2 1 1 c¯1 1,1 4n = 2¯ω1 (α2 α3) dd (log(β(Z))) Ω ¯ (Nred) − − 2 ∧ − 2 ∈ I1 c¯1 ¯ 4n where d f := I1(df) on functions on Nred. In fact we can be a bit more precise about the former. The results of [24] assert that up to a constant factor involving the ♭ sp(1) ♭ scalar curvature d( (dZ ) ) is equal to I¯1(Z ), and by verifying in an example (for instance see subsectionk 6.2k below) we find that 4 (6.16) (dZ♭)sp(1) = ω¯ . (a 2p) 1 − 4n Moreover the QK moment map associated to the action generated by Z on Nred is given by 1 (6.17) Zy Ω=¯ d ω¯ . − (a 2p) 1 − 4n Remark 6.9. Since a 2p> 0, from equation (6.17) the QK quotient of Nred with respect to Z is the empty− set. So far we have shown how to define gred starting from gM and suitable data on M 4 (see (6.8)). We now give a somewhat converse result. Proposition 6.10. The HK metric on M 4n can be expressed as (6.18) g = 4β(Z)−1g 16β(Z)−3(Z♭)2 +8β(Z)−2(ξ Z♭) M red − X ⊙ 16 Z 2 (Zy dβ)2 β(Z)−1(α2 + α2) k k (ξ )2 , − 2 3 − 4β(Z) Z 2 β(Z)3 X − β(Z)3 k k − where Z 2 := g (Z,Z). k k red Proof. From Proposition 6.2 we have Z = (1 a(a 2p + g (X,X))−1)∂ + 2(a 2p + g (X,X))−1X h − − M y1 − M and from Proposition 6.7 we have 1 ξ = 2(a 2p)−1α + (a 2p)Z♭. X − − 2 − Since ξX (Zh)=0, one computes (6.19) g (X,X) = (a 2p)3 Z 2(4 (a 2p)2 Z 2)−1. M − k k − − k k Using the latter, (6.6) and (6.20) α2 + α2 = 16(a 2p)−2(κ2 + κ2) 2 3 − 2 3 one can rewrite (6.8) as (6.18). Propositions 6.5, 6.7 and 6.10 are all suggestive that one should be able recover 4n 4n the HK structure of M from the QK structure of Nred and Z. Before proving that this is indeed the case, we first describe a couple of concrete examples illustrating the above reduction. 6.2. Examples. 22 U. FOWDAR 6.2.1. Example 1. As in subsection 5.5 we again take M = R4 with its standard HK structure but we now consider the permuting vector field X = 2x ∂ +2x ∂ . − 4 x3 3 x4 2 This corresponds to rotation on the R factor spanned by x3, x4. We can choose the connection 1-form 1 α = dy (x dx x dx + x dx x dx ) 1 − 2 2 1 − 1 2 4 3 − 3 4 as before, since it is also preserved by X, and by definition (see subsection 5.3) we have ξ = dy + x dx x dx , 2 3 1 − 4 2 η = dy3 + x4dx1 + x3dx2. Applying the above construction, from (6.4) one finds that ξ = (2x2 +2x2 + a)−1(2dy x dx + x dx + x dx x dx ). X − 3 4 1 − 2 1 1 2 4 3 − 3 4 y 2 2 x3 Writing p := X κ1 = x + x and q := 4y1 a arctan( ), we find after a long 3 4 − x4 computation that (2p + a) (2p + a) g = (dx2 + dx2)+ (dp2) red (2p a)2 1 2 4p(2p a)2 − p − + (dq 2(x dx x dx ))2. (2p a)2(2p + a) − 2 1 − 1 2 − 4 1 3 From this expression we see that Nred can be viewed an Sq bundle over R with coordinates (x1, x2,p). The vector field ∂q essentially corresponds to the Killing vector field Z. One can now verify directly that gred is indeed a self-dual Einstein metric (with scalar curvature 48). Note that although the QK reduction is only − 1 valid for a 2p > 0 (since we require t = 2 log(a 2p)) the metric gred is well- defined even− if a =0. When a =0, a change− of coordinates− shows that g =4es(dx2 + dx2)+ ds2 + e2s(dq 2(x dx x dx ))2 red 1 2 − 2 1 − 1 2 which one recognises is the complex hyperbolic metric. In particular, the latter is a Kähler metric but when a =0 the metrics gred are not Kähler (this can be checked directly by computing the6 rank of the curvature operator which is 6 = dim(so(4)) rather than 4 = dim(u(2))). 6.2.2. Example 2. If instead we take the permuting Killing vector field to be Xˇ = x ∂ + x ∂ x ∂ + x ∂ , − 2 y1 1 x2 − 4 x3 3 x4 R2 R2 which corresponds to rotation on both x1,x2 and x3,x4 , then one finds that 2 ξ ˇ = dy . X −a 1 After another long calculation we have be able to show that gred corresponds to the QK hyperbolic metric on R4. More concretely, the metric takes the form a g = gR4 , red (a r2)2 · − where R4 is viewed as an S1 bundle over R3 as in the Gibbons-Hawking ansatz with the S1 action generated by 2a−1Z i.e. θ(2a−1Z)=1, 1 2 2 2 2 u = (x1 + x2 x3 x4), x1x3 + x2x4, x1x4 x2x3 2 − − − 1 1 and the harmonic function V = 2|u| = r2 . The metric becomes singular for a =0 but nonetheless it converges (in the pointed Cheeger-Gromov sense) to the flat metric on R4 as a 0. → EINSTEIN METRICS ON BUNDLES OVER HYPERKÄHLER MANIFOLDS 23 One can naturally extend these examples to the case M = R4n by extending the permuting Killing vector fields X and Xˇ to R4n in the obvious way. On R4n defining a permuting Killing vector field amounts to choosing a U(1) subgroup of Sp(n)U(1) ∼= U(2n) Sp(n)Sp(1), that is not properly contained in the Sp(n) factor. Observe that this∩ was the case in the above examples. So in general, there is an Sp(n) family of such U(1) actions. In particular, by choosing 1-parameter families of such U(1) actions one should be able to construct 2-parameter families (1-parameter coming from constant a) of QK metrics. Concretely in the above case this means that one can consider the permuting Killing vector field s X + (1 s) Xˇ · − · for any s R. This again satisfies the hypothesis of subsection 5.3 and hence we can apply∈ the QK reduction as described above. We expect this will give a family of QK metrics (possibly including incomplete ones) connecting the complex hyperbolic metric to the quaternion hyperbolic one, though finding an explicit expression for gred(s) seems rather hard. It is worth emphasising that the definitions of κi will also depend on s. 6.3. The inverse construction. Suppose now that (N¯ 4n, Ω¯, g¯) is an arbitrary QK manifold. Then by choosing locally ω¯i we have that (6.10) holds for some 1-forms β, α2, α3. Moreover, we have that dβ = 4sω¯1 + α2 α3, − ∧ (6.21) dα2 = +4sω¯3 + α3 β, ∧ dα = +4sω¯ + β α , 3 2 ∧ 2 where by rescaling (Ω¯, g¯) we can set s = +1 if the scalar curvature is positive and s = 1 if it is negative cf. [24]. The above is essentially a consequence of the fact − that the induced Levi-Civita connection on ω¯1, ω¯2, ω¯3 has constant curvature. Note that differentiating (6.21) one indeed recoversh (6.10)i . We will henceforth assume that s = 1 i.e. scal(¯g) < 0. − 4n Let Z be a Killing vector field on N¯ such that Z Ω=0¯ and such that the projection (dZ♭)sp(1) does not vanish everywhere. WeL can then define (dZ♭)sp(1) (6.22) ω¯ := (2n)1/2 1 · (dZ♭)sp(1) k k on a suitable open set. It follows that ω¯ =0. We shall further assume that we LZ 1 can choose ω¯2, ω¯3 such that Z ω¯2 = Z ω¯3 =0 and that β(Z) > 2 Z . Applying to (6.10) andL using theL quaternions relations we getk k LZ (6.23) (β)= (α )= (α )=0. LZ LZ 2 LZ 3 From this one deduces that (6.15) holds and that I¯1 is integrable (this was also −1 ♭ 1 2 ¯ shown in [24, Sec. 3]). If [d(2β(Z) Z 2 β)] H (N, Z), we define the connection 1-form − ∈ 1 ξ := dx + 2(β(Z)−1)Z♭ β X − 2 on the total space of this S1 bundle, where x denotes the coordinate on the fibre. This is consistent with (6.12) . We denote by Z˜ the lift of Z to the total space satisfying dx(Z˜)=0 and we define M 4n as the quotient obtained by the action generated by Z˜. Observe that ˜ ξ = 0 by construction. The reader should think of Z˜ as corresponding to the LZ X vector field ∂y1 of subsection 6.1. 24 U. FOWDAR Theorem 6.11. The closed 2-forms (6.24) σ := d(β(Z)−1(2 dx β)), 1 − · − (6.25) σ := d(β(Z)−1(h α + f α )), 2 − · 3 · 2 (6.26) σ := d(β(Z)−1(h α f α )), 3 − · 2 − · 3 where f := cos(2x) and h := sin(2x), descend to M 4n to define a HK triple. More- 1 4n over, the vector field ∂x induces an S action on M preserving σ1 and permuting σ2 and σ3. Proof. It is easy to see that σ = 0, σ = 2σ and σ = +2σ . The L∂x 1 L∂x 2 − 3 L∂x 3 2 fact that σi satisfy the quaternionic relation is an algebraic condition and can be easily deduced by reversing the steps in subsection 6.1 with β(Z) = 4(a 2p)−1. y − So the only remaining thing to check is that Z˜ σi =0. First we have that y −1 −1 Z˜ σ = ˜ (β(Z) (2 dx β)) d(β(Z) β(Z))=0. 1 −LZ · − − y y Secondly note that from (6.15) we know that Z α2 = Z α3 =0 and hence y −1 Z˜ σ = ˜ (β(Z) (h α + f α ))=0 2 −LZ · 3 · 2 and y −1 Z˜ σ = ˜(β(Z) (h α f α ))=0. 3 −LZ · 2 − · 3 Thus, σi are basic 2-forms (with respect to the action generated by Z˜) and hence descend to M 4n. Remark 6.12. The above definition of σi was chosen to match the previous con- dition that σi = dκi. So this indeed corresponds to the inverting the construction described in subsection 6.1. It also follows now that the induced HK metric on M 4n is given by (6.18). One can verify directly that indeed g (Z,˜ ) vanishes. The hypothesis that β(Z) > 2 Z M · k k ensures that gM is positive definite i.e. gM (I¯i(Z), I¯i(Z)),gM (X,X) > 0 (see (6.18) and (6.19)). Remark 6.13. It was brought to our attention by Vicente Cortés that the corre- spondence established by Theorem 6.11 was also demonstrated in [2] albeit from a rather different approach. The latter work was motivated by the c-map construc- tion, originating from theoretical physics, and involves making the correspondence via a certain pseudo-HK manifold whereas our approach here stays in the realm of Riemannian geometry and goes via the QK manifold N 4n+4. The authors of [2] considered more generally the case when gM and gred also have mixed signature (though our construction can naturally be extended to these cases as well). Analogy with the HK/QK correspondence. 4n In [47] Swann shows that given a QK manifold N+ with positive scalar curva- ture, one can construct an associated bundle (N+) using the action of Sp(n)Sp(1) × U on H /Z2 and moreover it admits a natural HK structure (as well as QK one with positive scalar curvature). The HK metric is in fact the cone of a 3-Sasakian mani- fold (N+). Now if N+ admits a Killing vector field Y preserving the QK structure thenS he shows that one can lift this action to a triholomorphic Killing action on 3 × (N+). On the other hand there is also an isometric S H action on the fibre of U(N ) commuting with the lifted S1 action and this descends⊂ to the HK quotient U + Y˜ M 4n := (N )////S1 to give a permuting Killing vector field X. In [29] Haydys then U + Y˜ shows that one can in fact invert this construction to recover N+ from M. The construction described in this section shows that (M,X) is also in a one-to-one correspondence with (N,Z¯ ). This is summed up in the following diagram: EINSTEIN METRICS ON BUNDLES OVER HYPERKÄHLER MANIFOLDS 25 S1 y (N ) x S3 S1 y N x T3 Y˜ U + X˜ yi 4n 4n ¯ 4n (N+ , Y ) (M ,X) (N ,Z) Remark 6.14. It is likely that (N+) and N are also related by our correspondence say by fixing a permuting S1 US3 on (N ) and a corresponding S1 Sp(1, 1) on ⊂ U + ⊂ N (recall that Sp(1,1) is the fibre of N M). Furthermore we expect that this Sp(1)Sp(1) → correspondence can more generally be extended to the hypercomplex manifolds of Theorems 1.3 and 1.4, and certain complex manifolds using Joyce’s hypercomplex reduction [34]. 4n Remark 6.15. It was shown in [11] that associated to N+ are three other positive scalar curvature Einstein manifolds, namely (N+), Z(N+) and (N+). By analogy associated to M 4n, there are three negativeU scalar curvature EinsteinS manifolds, namely N, P and L of Theorem 1.1. Moreover in each case Z(N+) and P are in fact Kähler-Einstein. It would be interesting to find directly a correspondence between these Einstein manifolds as well. 7. Twistor spaces In this section we construct the twistor spaces of the QK manifolds N 4n+4 of 4n+4 Theorem 1.1 and that of the hypercomplex manifolds Mα,ξ,η,ν of Theorem 1.3. The twistor space is topologically defined as the unit sphere bundle in the rank 3 vector bundle . In general this bundle is not trivial; for instance the twistor space of HPn is CP2Gn+1. However in our case we do have globally well-defined hyper- complex structures so that topologically Z(N) and Z(Mα,ξ,η,ν) are diffeomorphic 2 2 to N S and Mα,ξ,η,ν S respectively. Both the twistor space of a QK manifold and hypercomplex× manifold× admit a distinguished complex structure (not a prod- uct one). The idea of twistor theory is essentially to encode the data on N and Mα,ξ,η,ν in terms of the complex geometry of their twistor spaces [4, 42]. We shall first consider the hypercomplex twistor space then the QK one following closely [33] and [41]. 7.1. Twistor space of hypercomplex examples. We now define the complex 2 structure on the twistor space Z(Mα,ξ,η,ν ). Consider the unit 2-sphere S = 2 2 2 2 (a,b,c) a + b + c =1 , then at the point p = (m, (a,b,c)) Mα,ξ,η,ν S we { | } ˜ ∈ × 2 define an almost complex structure I on TpZ(Mα,ξ,η,ν ) ∼= TmMα,ξ,η,ν T(a,b,c)S by ⊕ I˜ = (aI1 + bI2 + cI3, I0) 1 zz¯ z +¯z i(z z¯) = − I1 + I2 + − I3, I0 1+ zz¯ 1+ zz¯ 1+ zz¯ 2 CP1 where z denotes a local complex coordinate on S ∼= and I0 is the associated complex structure. To prove that I˜ is integrable we first need to identify the (1, 0) forms on Z(Mα,ξ,η,ν ) aside from dz. As in the hyperKähler case [33], it is easy to check that if θ is a (1, 0) form with respect to I1 then θ + zI3(θ) is a (1, 0) form with respect to I˜ on Z(Mα,ξ,η,ν). Thus, a basis of (1, 0) forms for I1 together with dz give a basis of (1, 0) form for I˜. Integrability of I˜ will follow if we can show that d(Γ(Λ1,0)) Γ(Λ2,0 Λ1,1). For any torsion-free connection on M we have that ⊂ ⊕ ∇ α,ξ,η,ν d(θ + zI3(θ)) = dxi ∂ (θ + zI3(θ)) + dz I3(θ), ∧ ∇ xi ∧ 26 U. FOWDAR where xi denote local coordinates on Mα,ξ,η,ν , since d( ) = Alt( ( )) on differ- ential forms{ } cf. [35, Corollary 8.6]. Now let be the Obata· connection∇ · i.e. the ∇ unique torsion-free connection which preserves I1, I2, I3 [39] so that I˜ = I˜ . Since ∇◦ ◦ ∇ I˜( ∂ (θ + zI3(θ))) = i( ∂ (θ + zI3(θ))) ∇ xi ∇ xi we deduce that dz and (θ + zI (θ)) are both of type (1, 0) with respect to I˜ ∂xi 3 ∇ 2,0 1,1 and it follows that d(θ + zI (θ)) Λ Λ i.e. N ˜ 0. 3 ∈ ⊕ I ≡ Remark 7.1. For hyperKähler manifolds the Obata connection coincides with the Levi-Civita connection whereas for QK manifolds the difference between these two connections is essentially determined by (3.1). The projection map (7.1) p : Z(M ) CP1 α,ξ,η,ν → is clearly holomorphic and each fibre p−1(z) M can be viewed as a complex ≃ α,ξ,η,ν manifold endowed with the complex structure aI1 + bI2 + cI3. The antipodal map 2 on S induces an anti-holomorphic involution on Z(Mα,ξ,η,ν ) compatible with p. There is also a globally well-defined (2, 0)-form on Z(Mα,ξ,η,ν) given by (7.2) Ωˇ := (ˇω + iωˇ )+2zωˇ z2(ˇω iωˇ ). 2 3 1 − 2 − 3 Unlike in the hyperKähler case however Ωˇ is not holomorphic on the fibres p−1(z) but the (2n +2, 0) form Ωˇ n+1 is. In particular, this implies that each fibre of (7.1) has trivial first Chern class. Consider the (1, 0) forms ξ + iη and ν + iα with respect to I1 on Mα,ξ,η,ν, then from the above argument we have that χ := ξ + iη + zI3(ξ + iη)+ iz(ν + iα + zI3(ν + iα)) = (ξ + iη) 2zα z2(ξ iη) − − − is a (1, 0) form with respect to I˜. Globally χ can be interpreted as an (2)-valued O 1-form on Z(Mα,ξ,η,ν) since it depends quadratically on z. We claim that χ is a holomorphic connection 1-form on p−1(z). Computing fibrewise we have dχ = (σ + iσ ) 2zσ + z2(σ iσ ), − 2 3 − 1 2 − 3 which is indeed a holomorphic (2, 0) form on the fibres of Z(M) CP1, in fact it → covariantly constant with respect to the Levi-Civita connection of gM . Moreover, the form dχ on Z(M) completely determines the HK metric gM [33, Theorem 3.3]. To sum up we have shown that Theorem 1.3 can be described in terms of the twistor theory: Theorem 7.2. The T4-bundle Z(M ) Z(M) α,ξ,η,ν → is holomorphic with vertical (1, 0) forms spanned by z1(ξ + iη) z2(α + iν) and 1 − z1(ν + iα) + z2(η + iξ) for [z1 : z2] CP . Furthermore, χ is a holomorphic 1 ∈ 2 section of Λ Z(Mα,ξ,η,ν ) (2) such that dχ Ω (Z(M))(2) defines a holomorphic symplectic form on the fibres⊗O of Z(M) CP∈1. → EINSTEIN METRICS ON BUNDLES OVER HYPERKÄHLER MANIFOLDS 27 7.2. Twistor space of QK examples. We now construct the complex structure on the QK twistor space Z(N) of N. It will be more convenient to work on the C2 H C2 bundle := N whose projectivisation is just Z(N) [41]. Since su(2) ∼= so(3) we can lift the SO× (3) connection form (3.1) on ω ,ω ,ω to the SU(2) connection h 1 2 3i iα ξ + iη (7.3) A := 2e2t − − ξ + iη iα on H. This is simply the connection induced by the Levi-Civita connection of gN on the associated vector bundle H. Indeed one verifies directly that the curvature 2-form F := dA + A A is given by A ∧ iω ω + iω (7.4) 2 1 − 2 3 . − ω2 + iω3 iω1 − Salamon shows that H admits an integrable complex structure with respect to which the vertical 1-forms (7.5) θ := dz +2e2t( iz α + z ( ξ + iη)), 1 1 − 1 2 − 2t (7.6) θ2 := dz2 +2e (+iz2α + z1(+ξ + iη)) 2 are of type (1, 0) [41], where z1,z2 denote the coordinates of the fibre C . This complex structure is C×-invariant and hence passes down to Z(N). Furthermore, the (1, 0) form Θ := z θ z θ 2 1 − 1 2 = z dz z dz 2e2t(z2(ξ + iη)+2iz z α + z2(ξ iη)) 2 1 − 1 2 − 1 1 2 2 − is holomorphic and defines a complex contact structure on Z(N). Denoting by −1 1 z = iz1 z2 a local coordinate on CP we can express the induced contact structure Θ on−Z(N) as idz 2e2tχ. − − Standard twistor theory now asserts that: Theorem 7.3. Θ together with the data of the anti-holomorphic involution gener- ated by the antipodal map on the S2 fibres of Z(N) completely determine the QK manifold (N, ΩQK ,gN ). 4n Sp(n+1,1) As already seen before when M = R we have that N = Sp(n+1)Sp(1) . In this case a concrete description of (Z(N), Θ) was given by LeBrun in [36, Sec. 2] as an open set of CP2n+3. Using this LeBrun was able to deform the quaternion hyperbolic metric to show the existence of infinitely many QK metrics of negative scalar curvature. In general however it is hard to identify the complex manifold Z(N) explicitly. 8. Other examples The purpose of this section is to describe other closely related geometric struc- tures that arise on the topological spaces that figure in Theorems 1.1, 1.3 and 1.4, and to highlight some links between these various geometries. 8.1. Ricci-flat examples. We recall some known solutions to the system (3.3)- (3.7) when λ =0. All of these examples have special holonomy. We first consider metrics on P 4n+2. From (3.15) we see that if we set (8.1) p(t)= t1/(2n+2) + 4n+1 then we get a Ricci-flat metric on Rt Mα . This metric degenerates as t 0 but is complete as t + . This is a× special case of the so-called Calabi ansatz→ metrics on complex line→ bundles∞ [13]. 28 U. FOWDAR Next we consider metrics on L7 and N 8. When n =1 so that M is a hyperKähler 4-manifold then Apostolov-Salamon found the solution 2 2 2 −2 2 −2 2 (8.2) g = t (t + b) dt + t α + (t + b) ξ + t(t + b)gM 7 on L with holonomy group G2 SO(7) [3] and in [20] we extend their construction and found the solution ⊂ 2 2 2 2 −2 2 −2 2 −2 2 (8.3) g = t (t + b) (t + c) dt + t α + (t + b) ξ + (t + c) η + t(t + b)(t + c)gM on N 8 with holonomy group Spin(7) SO(8), where b,c are positive constants. Like the above Calabi-Yau metrics, these⊂ examples are only complete when t + . We are unaware of any other Ricci-flat solution to (3.3)-(3.7). → ∞ 8.2. Balanced Hermitian examples. We now construct balanced Hermitian 4n+2 4n+4 2n+4 structures on Mξ,η , N and Mν1,ν2,ν3,ν4 . In the latter two cases we show that the Hermitian metric is balanced with respect to the 2-sphere of complex structures. 4n+2 8.2.1. Balanced structures on Mξ,η . Consider the natural U(2n + 1)-structure on 4n+2 Mξ,η defined by 2 2 (8.4) g = ξ + η + gM , (8.5) ω = ξ η + σ . ∧ 1 Then the (2n +1, 0)-form defined by (8.6) Υ = (ξ + iη) (σ + iσ )n. ∧ 2 3 is closed, and hence holomorphic, since dΥ= (σ + iσ )n+1 =0. − 2 3 4n+2 So (Mξ,η , Υ) is a complex manifold with vanishing first Chern class. We also have that 2n 2n−1 d(ω )=(2n)( σ2 η + σ3 ξ) σ1 =0, − ∧ ∧ ∧ 2n−1 2n−1 where the last equality follows from the fact that σ2 σ1 and σ3 σ1 are differential forms of type (2n +1, 2n 1)+(2n 1, 2n ∧+ 1) on M 4n. Thus,∧ we have shown − − 4n+2 Proposition 8.1. (Mξ,η , g, ω, Υ) is a balanced Hermitian manifold with trivial canonical bundle. 2 4n+2 It is clear from (8.4) and (8.6) that the T fibres of Mξ,η are elliptic curves and the restricted metric on the fibres is flat. Remark 8.2. The metric g is of course not Calabi-Yau i.e. the holonomy group of the Levi-Civita connection is not a subgroup of SU(2n + 1). However, there exists a unique Hermitian connection with totally skew-symmetric torsion form called the Bismut connection [9] whose (restricted) holonomy group contained in SU(2n + 1) cf. [25, Proposition 3.6]. 6 Remark 8.3. When M is a hyperKähler 4-manifold, the SU(3)-structure on Mξ,η determined by (ω, Υ) is both complex and half-flat i.e. d(ω2)=0 and dRe(Υ) = 0. If we reverse the complex structure on the T2-fibres so that the (2n +1, 0)-form is now given by (8.7) Υ˜ = (η + iξ) (σ + iσ ) ∧ 2 3 and (8.8) ω˜ = η ξ + σ ∧ 1 EINSTEIN METRICS ON BUNDLES OVER HYPERKÄHLER MANIFOLDS 29 then the SU(3)-structure is still half-flat but it is no longer complex; this bears a certain resemblance to the theory of twistor spaces of self-dual 4-manifolds cf. [43, Proposition 7.5]. In [3, 15] the SU(3)-structure defined by (8.7) and (8.8), 4 when M = T , was evolved via the Hitchin flow [31] to construct the G2 holonomy metrics (8.2). 8.2.2. Balanced structures on N 4n+4. Consider the Sp(n + 1)-structure on N 4n+4 defined by − 4 t − 4 t − 4 t (8.9) ω˜1 := e 2n+1 ω1, ω˜2 := e 2n+1 ω2, ω˜3 := e 2n+1 ω3, where ωi are as in Theorem 1.1. This simply corresponds to conformally rescaling the quaternion-Kähler structure in Theorem 1.1. Since complex structures are − 4 t conformally invariant it follows that (N, g˜ := e 2n+1 gN , ω˜i) is a Hermitian manifold for i =1, 2, 3. We leave it to the reader to verify, using (3.1), that d(ektω2n+1) = 4(2n + 1)(k + 4)e4(n+1)t+atσ2n ξ η dt 1 1 ∧ ∧ ∧ and that one gets analogous expressions for ω2 and ω3. The desired result then follows by taking k = 4: − 4n+4 Proposition 8.4. (N , g,a˜ ω˜1 + bω˜2 + cω˜3) is a balanced Hermitian manifold for (a,b,c) S2. ∈ 4n+4 8.2.3. Balanced structures on Mν1,ν2,ν3,ν4 . Consider the hyper-Hermitian structure 4n+4 on Mν1,ν2,ν3,ν4 as defined in Theorem 1.4. We claim that it is in fact balanced: 4n+4 Proposition 8.5. 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