EINSTEIN METRICS ON BUNDLES OVER HYPERKÄHLER MANIFOLDS UDHAV FOWDAR Abstract. We give explicit examples of quaternion-Kähler and hypercom- plex structures on bundles over hyperKähler manifolds. We study the in- finitesimal symmetries of these examples and the associated Galicki-Lawson quaternion-Kähler moment map. In particular, this leads us to a new proof a hyperKähler/quaternion-Kähler type correspondence. We also give examples of other Einstein metrics and balanced Hermitian structures on these bundles. Contents 1. Introduction 1 2. Preliminaries 5 3. Proofs of Theorems 1.1, 1.3 and 1.4 7 4. Examples 10 5. Symmetries and moment maps 12 6. A HK/QK type correspondence 17 7. Twistor spaces 25 8. Other examples 27 References 29 1. Introduction In this paper, we study natural geometric structures arising on certain torus bundles over hyperKähler (HK) manifolds. In particular, we find solutions to the Einstein equations on these bundles and in the case when they are of rank 4, we show that they admit quaternion-Kähler (QK) and hypercomplex structures. The infinitesimal symmetries of these examples turn out to be surprisingly rich and we study these in detail. In particular, we give a different proof of a HK/QK type correspondence, described in [2], between a HK manifold with a permuting Killing vector field and a QK manifold of negative scalar curvature with a suitable Killing arXiv:2105.04254v2 [math.DG] 21 Sep 2021 vector field. Many explicit examples are also described. Motivation. When searching for examples of Einstein metrics, a natural start- ing point is to look for examples on bundles over other Einstein manifolds. For instance, this strategy was used by Calabi to find the first examples of Calabi-Yau and hyperKähler metrics on bundles over Kähler-Einstein manifolds [13] and by Bryant-Salamon to find the first examples of G2 and Spin(7) metrics on bundles over S3, S4 and CP2 [12]. In this paper, motivated by examples found in [27], we consider the Einstein equations on certain bundles over hyperKähler manifolds. The rather simple nature of the structure equations also allows us to find other 2010 Mathematics Subject Classification. 53C10, 53C26, 53C55. Key words and phrases. Hermitian geometry, Quaternion-Kähler geometry, Torus action. 1 2 U. FOWDAR geometrically interesting non-Einstein structures. 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.