Geometry of the space of sections of twistor spaces with circle action
Florian Beck, Indranil Biswas, Sebastian Heller, Markus Röser
aa r X i v : . [ m a t h . DG ] F e b GEOMETRY OF THE SPACE OF SECTIONS OF TWISTOR SPACES WITH CIRCLEACTION
FLORIAN BECK, INDRANIL BISWAS, SEBASTIAN HELLER, AND MARKUS R ¨OSER
Abstract.
We study the holomorphic symplectic geometry of (the smooth locus of) the space of holomor-phic sections of a twistor space with rotating circle action. The twistor space has a meromorphic connectionconstructed by Hitchin. We give an interpretation of Hitchin’s meromorphic connection in the context of theAtiyah–Ward transform of the corresponding hyperholomorphic line bundle. It is shown that the residue ofthe meromorphic connection serves as a moment map for the induced circle action, and its critical pointsare studied. Particular emphasis is given to the example of Deligne–Hitchin moduli spaces.
Contents
Introduction 2Acknowledgements 41. Geometry of the space of holomorphic sections of a twistor space 41.1. Twistor space 41.2. Space of holomorphic sections as a complexified hyperk¨ahler manifold 51.3. Alternative description of the holomorphic symplectic form Ω x on S ′
92. The hyperholomorphic line bundle and the energy functional 102.1. Rotating circle actions and the hyperholomorphic line bundle 102.2. The S -action on S and the energy functional 122.3. Critical Points of E L Z M irr DH C ∗ -fixed sections 284.3. The energy of a C ∗ -fixed section 314.4. The second variation of the Energy at a C ∗ -fixed section 334.5. Sections and the degree of the hyperholomorphic line bundle 34References 35 Mathematics Subject Classification.
Key words and phrases.
Deligne–Hitchin twistor space, self-duality equation, connection, circle action, hyperholomorphicbundle.
Introduction
For a compact Riemann surface Σ of genus at least two, and a complex semisimple Lie group G , themoduli space of G -Higgs bundles on Σ has a hyperk¨ahler structure [26]. The corresponding twistor space M DH (Σ , G ) is known as the Deligne–Hitchin moduli spaces [44]. These spaces M DH (Σ , G ) have been thetopic of many papers in recent years; see for example [10, 32, 33, 15, 16] and [5, 6, 4, 24, 25] and referencestherein.In [6] the last three authors initiated a detailed investigation of the space S of holomorphic sections ofthe natural projection M DH (Σ , G ) −→ C P . Our aim here is to go deeper into the study of S . There aretwo distinct sources of motivation for doing this: One is the integrable systems approach to the theory ofharmonic maps from Riemann surfaces into symmetric spaces, the other being the hyperk¨ahler geometry.We start by explaining the latter source. Let K be a compact real form of a complex semisimple Liegroup G with Lie algebra g . Hitchin’s self-duality equations on Σ are given by F ∇ + [Φ ∧ Φ ∗ ] = 0 = ∂ ∇ Φ (0.1)[26]. Here ∇ is a K -connection and Φ ∈ Ω , (Σ , g ), which is called a Higgs field, while Φ Φ ∗ is givenby the negative of the Cartan involution on g associated with the real form K ⊂ G . Each solution of (0.1)gives rise to a family of flat G -connections parametrized by C ∗ ∇ λ = ∇ + λ − Φ + λ Φ ∗ that satisfies a reality condition which is determined by the symmetric pair ( G, K ). The nonabelian Hodgecorrespondence [46, 26, 12, 11] allows us to invert this process, meaning any (reductive) flat G -connectionoccurs in a suitable C ∗ -family of flat G -connections given by a solution of the self-duality equations in (0.1).Recall from [44] that the Deligne–Hitchin moduli space M DH (Σ , G ) is obtained by gluing the Hodgemoduli spaces M Hod (Σ , G ) and M Hod (Σ , G ) of λ -connections via the Riemann Hilbert correspondence,where Σ is the conjugate of Σ (see § C ∗ -family of flat G -connections associated with a solution of (0.1) as a family of λ -connections, which extendsto all of C P . From this point of view, we may think of the above C ∗ -family as a holomorphic section of thefibration M DH (Σ , G ) −→ C P satisfying an appropriate reality condition. This observation fits naturallyinto the twistor theory of hyperk¨ahler manifolds, as we shall explain next.The moduli space M SD (Σ , G ) of solutions of (0.1) is a (typically singular) hyperk¨ahler manifold whichcomes equipped with a certain rotating circle action given by ( ∇ , Φ) ( ∇ , e iθ Φ), θ ∈ R . Here rotating means that the circle action is isometric, it preserves one of the three K¨ahler forms while rotating theother two (see § § M DH (Σ , G ) is the twistorspace associated with M SD (Σ , G ). The C ∗ -family of flat connections associated with a solution to theself-duality equations can then be interpreted as the twistor line corresponding to the point in M SD (Σ , G )represented by ( ∇ , Φ). In this way, we can view M SD (Σ , G ) in a natural way as a subset of the space S of holomorphic sections of the twistor family M DH (Σ , G ) −→ C P . In fact, S may be interpreted asa natural complexification of the real analytic hyperk¨ahler manifold M SD (Σ , G ). More generally, if Z isthe twistor space of a hyperk¨ahler manifold M , the twistor construction of hyperk¨ahler metrics produces ahyperk¨ahler metric on the space M ′ of all real holomorphic sections (with appropriate normal bundle) ofthe fibration Z −→ C P . From this point of view, it is a natural question whether M = M ′ [44]. In thecontext of the self-duality equations, this question translates into whether every real holomorphic section of M DH (Σ , G ) −→ C P is actually obtained from a solution to the self-duality equations. In [6] an answer tothis question was given in the case G = SL(2 , C ) by showing that in general M SD (Σ , G ) is strictly containedin the space of real sections. Furthermore, in [6] the real holomorphic sections belonging to M SD (Σ , G ) werecharacterized in terms of a certain Z / Z -valued invariant that can be associated with any real section.Hyperk¨ahler manifolds with rotating circle action are an active field of research in hyperk¨ahler geometry,especially their connection with the well-known hyperk¨ahler/quaternion K¨ahler correspondence [23, 1, 29, 30].Haydys [23] has observed that if ( M, ω I , ω J , ω K ) is a hyperk¨ahler manifold with rotating circle action suchthat ω I is integral, then there exists a complex line bundle L −→ M with a unitary connection whosecurvature is ω I + dd cI µ , where µ is the moment map for the S -action with respect to the K¨ahler form ω I .This connection is hyperholomorphic, in the sense that its curvature is of type (1 ,
1) with respect to everycomplex structure in the family of K¨ahler structures on M parametrized by S . The moment map facilitates ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 3 a lift of the S -action to an S -action on the total space of the principal S -bundle P −→ M associated with L Z . Haydys has shown that the quotient Q = P/S for this action carries a quaternionic K¨ahler metric.The natural S -action on the principal bundle S -bundle P descends to an isometric circle action on Q , and M can be recovered from it as a hyperk¨ahler quotient of the Swann bundle of Q by the lift of this isometriccircle action. This is a brief outline of what is known as the hyperk¨ahler/quaternion K¨ahler (HK/QK)correspondence. Hitchin in [29, 30] has described the HK/QK correspondence from a purely twistorial pointof view. A natural starting point for him is the observation that, by the Atiyah–Ward correspondence,the hyperholomorphic line bundle on M produces a holomorphic line bundle L Z on the twistor space Z associated to M . Hitchin has shown how to construct L Z directly on the twistor space Z and explained howthe circle action determines a distinguished meromorphic connection on L Z . This meromorphic connectionplays a key role in the construction, from Z , of the twistor space Z Q of the associated quaternionic K¨ahlermanifold Q in the sense that it determines the contact distribution on Z Q . Therefore, in order to makeprogress towards the construction of the quaternionic K¨ahler manifold associated with M SD (Σ , G ), it isclearly important to obtain information on this meromorphic connection.As mentioned earlier, our other motivation comes from the integrable system approach to harmonic mapsfrom Σ into symmetric spaces. As is well known, Hitchin’s self-duality equations (0.1) can be interpreted asthe gauge-theoretic equations for a (twisted) harmonic map Σ −→ G/K (a so-called harmonic metric ).One observes that the twisted harmonic maps into other (pseudo-Riemannian) symmetric spaces of theform
G/G R and their duals have also analogous gauge-theoretic interpretations and they also give rise to C ∗ -families of flat G -connections satisfying different reality conditions; the reality condition depends on thetarget of the harmonic map. For example, if G = SL(2 , C ) then K = SU(2), and the harmonic maps intothe symmetric spaces SL(2 , C ) / SU(2) , SL(2 , C ) / SU(1 , , SU(2) , SU(1 ,
1) and also into the hyperbolic discSU(1 , / U(1) can all be encoded in C ∗ -families of flat SL(2 , C ) connections satisfying appropriate realityconditions; see for instance [42, 47, 27, 13]. Considering these C ∗ -families as families of λ -connections with λ ∈ C P we can — at least formally — interpret twisted harmonic maps from Σ into various symmetricspaces associated with real forms of G as holomorphic sections of M DH (Σ , G ); see [6]. In this way wemay view the space of holomorphic sections of the fibration M DH (Σ , G ) −→ C P as a master space forthe moduli spaces of twisted harmonic maps from Σ into various (pseudo-Riemannian) symmetric spacesassociated with G .This interaction with the theory of harmonic maps has provided useful guidance and intuition for thepredecessor [6] and [4] of this project. For example, the new real holomorphic sections for the SL(2 , C )-caseconstructed in [6] are closely related to twisted harmonic maps Σ −→ SL(2 , C ) / SU(1 , S of holomorphic sections of the twistor family M DH (Σ , G ) −→ C P there exists an interesting and useful holomorphic functional, called the energy functional, whose evaluationon a real section coming from a harmonic map is (a constant multiple of) the Dirichlet energy of the associatedharmonic map. This functional is also intimately related to the Willmore energy of certain immersions of Σinto the 3-sphere. It was then shown that this functional has a natural interpretation from the hyperk¨ahlerpoint of view. In fact it can be interpreted as a holomorphic extension of the moment map, associated withthe rotating circle action, from the space M of twistor lines to the whole of S . Essentially, it is given byassociating to any s ∈ S the residue of the meromorphic connections along s .In this article, we continue our study of this setup in the context of the twistor space Z of a generalhyperk¨ahler manifold M with a rotating circle action. The natural geometric structure on the space S ofholomorphic sections of Z −→ C P , viewed as a complexification of the hyperk¨ahler manifold M , has beenelucidated by Jardim and Verbitsky [35, 36]. They show that S comes equipped with a certain family ofclosed two-forms called a trisymplectic structure. In particular, the complexifications of the K¨ahler formson M are contained in this family. We describe in this paper how the picture gets enriched by the presenceof a rotating circle action. It turns out that the energy functional E is a moment map for the circle actionon S induced from the circle action on M with respect to a natural holomorphic symplectic form Ω whichis a part of the canonical trisymplectic structure on S (see Theorem 2.3). In particular, the critical pointsof E are exactly the fixed points of the circle action.We also explain how to use the Atiyah–Ward transform to obtain a natural holomorphic extension of theline bundle L −→ M to L −→ S , and how the meromorphic connection on L Z can be described naturallyin terms of L and the data of the circle action (see Theorem 2.14 and § F. BECK, I. BISWAS, S. HELLER, AND M. R ¨OSER work [29], but we believe our point of view in terms of the geometry of the space S of holomorphic sectionsmight shed a new light onto his constructions. Moreover, our results allow us to make conclusions about theglobal structure of S (see Theorem 2.15)We then apply this general framework to the space of holomorphic sections of M DH (Σ , SL( n, C )). Eventhough S is expected to be singular in this case, the general theory tells us that the critical points of theenergy functional are closely related to the fixed points of the C ∗ -action on S . We analyze these C ∗ -invariantsections in detail, building on [10] (see Proposition 4.5). We obtain explicit formulas for the energy of a C ∗ -invariant section (Proposition 4.12) and we describe the second variation of the energy at such a section(Proposition 4.13). Moreover, we find an explicit formula for the degree of the hyperholomorphic line bundlerestricted to a C ∗ -invariant section (Proposition 4.16), from which we are able to show that there existsections along which the hyperholomorphic line bundle has non-zero degree. Using this it can be deducedthat S is in general not connected (Theorem 4.17).The paper is organized as follows. In § ̟ : Z −→ C P associatedwith a hyperk¨ahler manifold M and describe the geometric structure induced on the space S of holomorphicsections of ̟ in a way that is suitable for our purpose. We then go on in § S is enriched by the presence of a rotating circle action on M and hence on Z . In particular, we explain howthe energy functional E ties up naturally with the holomorphic symplectic geometry on S discussed in § § § §
3. In § M DH (Σ , G ) and theresults mentioned above are proved. Acknowledgements.
F.B. is supported by the DFG Emmy Noether grant AL 1407/2-1. I.B. is partiallysupported by a J. C. Bose Fellowship. S.H. is supported by the DFG grant HE 6829/3-1 of the DFG priorityprogram SPP 2026 Geometry at Infinity.1.
Geometry of the space of holomorphic sections of a twistor space
In this section we collect some aspects of the geometry of holomorphic sections of a hyperk¨ahler twistorspace that will be used later. Useful references are [31], [35], [36]. See also [37] for the similar, but different,quaternionic K¨ahler case.1.1.
Twistor space.
Let (
M, g , I, J, K ) be a hyperk¨ahler manifold of complex dimension 2 d , where I, J, K are almost complex structures and g a Riemannian metric on M . The associated K¨ahler formsare ω L = g ( L · , · ), L ∈ { I, J, K } . For convenience, the complex manifolds ( M, I ) and ( M, − I ) willsometimes be denoted by simply M and M respectively; it will be ensured that this abuse of notation doesnot create any confusion.There is a family of K¨ahler structures on M with complex structures { I x := x I + x J + x K | x := ( x , x , x ) ∈ S } (1.1)parametrized by the sphere S := { ( x , x , x ) ∈ R | x + x + x = 1 } . The twistor space Z = Z ( M )of ( M, g , I, J, K ) is a complex manifold whose underlying smooth manifold is S × M ([31, § I Z of Z at any point ( x, m ) ∈ S × M is I Z | ( x,m ) = ( I C P | T x S ) ⊕ ( I x | T m M ) , where I C P is the standard almost complex structure on S = C P and I x is the almost complex structurein (1.1). Here we identify S with C P using the stereographic projection from ( − , ,
0) to the plane in R spanned by the x and x axises. In particular, (1 , , ∈ S corresponds to 0 ∈ C P and ( − , , ∞ ∈ C P . Throughout we shall use an affine coordinate λ on C P , so that C P ∼ = C ∪ {∞} .The hyperk¨ahler structure on M is encoded in the following complex-geometric data on Z . The naturalprojection S × M −→ S corresponds to a holomorphic submersion ̟ : Z −→ C P (1.2) ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 5 with fibers ̟ − ( λ ) =: Z λ = ( M, I λ ) . (1.3)Here I λ is the complex structure on M in (1.1) corresponding to λ ∈ C P ∼ = S . For any complex vectorbundle V on Z and any integer m , we use the notation V ( m ) := V ⊗ ̟ ∗ O C P ( m ). Let T ̟ = T ̟ Z := (ker d̟ ) ⊂ T Z be the relative holomorphic tangent bundle (also called the vertical tangent bundle) for the projection ̟ in(1.2). Then Z carries the twisted relative holomorphic symplectic form ω ∈ H ( Z, (Λ T ∗ ̟ )(2)) given by ω = (cid:0) ω J + i ω K + 2 i λω I + λ ( ω J − i ω K ) (cid:1) ⊗ ∂∂λ ∈ H ( Z, (Λ T ∗ ̟ )(2)) . (1.4)Moreover, Z carries an anti-holomorphic involution (or real structure) τ Z : Z −→ Z (1.5)given by the map S × M −→ S × M , ( x, m ) ( − x, m ), using the diffeomorphism Z ∼ = S × M . Notethat τ Z covers the antipodal map τ C P : C P −→ C P , λ λ − . (1.6)The relative twisted symplectic form ω is real with respect to τ in the sense that τ ∗ ω = ω .Let S denote the space of all holomorphic sections of the projection ̟ in (1.2); this space is discussed indetail in Section 1.2. We have the embedding ι : M ֒ → S (1.7)that sends any m ∈ M to the constant section x ( x, m ) ∈ S × M , which is called the twistor line s m associated to m . By [31, §
3, (F)] it is known that the normal bundle of a twistor line is isomorphic to O C P (1) ⊕ d . The space S has a real structure defined by τ : S −→ S , s τ Z ◦ s ◦ τ C P , (1.8)where τ Z and τ C P are the involutions in (1.5) and (1.6) respectively; we note that while the section τ ( s )is holomorphic for fixed s , the map τ itself is anti-holomorphic. The manifold M , considered as space oftwistor lines, is a component of the fixed point locus S τ ⊂ S , also known as the space of real sections.We have described the complex-geometric data on Z induced by the hyperk¨ahler structure on M . Con-versely, suppose we have a complex manifold Z with a holomorphic submersion ̟ : Z −→ C P , a twistedrelative symplectic form ω ∈ H ( Z, (Λ T ∗ ̟ )(2)) and an antiholomorphic involution τ : Z −→ Z coveringthe antipodal map on C P satisfying τ ∗ ω = ω . Then the parameter space M of real sections of ̟ withnormal bundle isomorphic to O C P (1) ⊕ d is a (pseudo-)hyperk¨ahler manifold of real dimension 4 d , if it isnon-empty; see [31, Theorem 3.3].1.2. Space of holomorphic sections as a complexified hyperk¨ahler manifold.
To examine the localstructure of S , for any s ∈ S let N s = s ∗ T Z/ds ( T C P ) be the normal bundle of s ( C P ) ⊂ Z . Since s isa section of ̟ , we have a canonical isomorphism s ∗ T ̟ Z ∼ = N s , where T ̟ Z ⊂ T Z as before is the kernel of the differential d̟ of the map ̟ . The following proposition iswell-known, however a proof of it is given because parts of the proof will be used later. Proposition 1.1.
Let Z be the twistor space of a hyperk¨ahler manifold M of complex dimension d . Thenthe set S of holomorphic sections of ̟ : Z −→ C P is a complex space in a natural way. The tangent spaceof s ∈ S is H ( C P , N s ) , and S is smooth at a point s ∈ S if H ( C P , N s ) = 0 . If H ( C P , N s ) = 0 ,then dim T s S = 4 d .Proof. For any s ∈ S , the sufficiently small deformations of the complex submanifold s ( C P ) ⊂ Z continueto be the image of a section of ̟ . Consequently, S is an open subset of the corresponding Douady space ofrational curves in Z ; see also [40, Theorem 2]. In particular, T s S = H ( C P , N s ) for all s ∈ S . F. BECK, I. BISWAS, S. HELLER, AND M. R ¨OSER
The complex structure of S around a point s ∈ S is constructed in the following way. There are openneighbourhoods V s ⊂ H ( C P , N s ) and U s ⊂ S of 0 ∈ H ( C P , N s ) and s ∈ S respectively, as well as aholomorphic map k s : H ( C P , N s ) −→ H ( C P , N s ) , which is sometimes called the Kuranishi map. Then there is a natural isomorphism U s ∼ = V s ∩ k − s (0) (1.9)taking s to 0. The complex structure on U s is given by the complex structure of V s ∩ k − s (0) using theisomorphism in (1.9).The statement on the smoothness of S follows from (1.9), because k s is the zero map if H ( C P , N s ) = 0.The dimension of T s S ∼ = H ( C P , N s ) is computed from the Riemann–Roch theorem applied to N s : h ( C P , N s ) = deg( N s ) + rank( N s ) = 2 d + 2 d = 4 d. Here we have used the fact deg N s = 2 d for any holomorphic section s : C P → Z (cf. [18, 39]). To seethis, note that N s = s ∗ T ̟ Z and that the twisted relative symplectic form ω ∈ H ( Z, (Λ T ∗ ̟ )(2)) induce anisomorphism s ∗ ω : N s → N ∗ s ⊗ O C P (2), and thus (det N s ) ⊗ ∼ = O C P (4 d ). (cid:3) As mentioned before, the normal bundle N s m of any twistor line s m , m ∈ M , is isomorphic to O C P (1) ⊕ d so that H ( C P , N s m ) = 0. Consequently, by Proposition 1.1, the image M ⊂ S of the embedding in (1.7)is contained in the smooth locus of S . Define the complex manifold S ′ := { s ∈ S | N s ∼ = O C P (1) ⊕ d } ⊂ S which is of complex dimension dim S ′ = 4 d = 2 dim M by Proposition 1.1. Since the vector bundle O C P (1) ⊕ d is semistable, from the openness of the semistability condition, [38, p. 635, Theorem 2.8(B)], itfollows immediately that S ′ is an open subset of S .We want to transfer geometric objects from Z to S . To this end, it is useful to introduce the correspondencespace F = C P × S (1.10)as well as F ′ = C P × S ′ . Remark 1.2.
The correspondence space is usually described as the space of pairs ( z, ℓ ), where ℓ is a complexline in Z passing through z ∈ Z . In our work, the lines ℓ in Z are treated as sections, meaning ℓ ∈ S . Thenwe obtain an isomorphism from this space of pairs ( z, ℓ ) to F in (1.10) by mapping any ( z, ℓ ) to ( ̟ ( z ) , ℓ ).Consider the evaluation map ev : F −→
Z , ev( λ , s ) = s ( λ ) . For a fixed λ ∈ C P = C ∪ {∞} define the mapev λ : S −→ Z λ := ̟ − ( λ ) , ev λ ( s ) = ev( λ, s ) . (1.11)We will denote the restrictions of ev to F ′ and of ev λ to S ′ by ev and ev λ respectively. Lemma 1.3.
The evaluation map ev : F ′ → Z is a surjective holomorphic submersion.Proof. The surjectivity of ev : F ′ → Z is clear, since there is a twistor line through each point of Z = C P × M . If ( l, V ) ∈ T ( λ,s ) F ′ = T λ C P ⊕ T s S ′ = T λ C P ⊕ H ( s ∗ T ̟ ), then we have( d ev) ( λ,s ) ( l, V ) = ( ds ) λ ( l ) + V ( λ ) . Now, since the evaluation map H ( O (1)) → O (1) λ at λ is surjective, d ev ( λ,s ) is surjective as well. (cid:3) We thus obtain a commutative diagram in which each arrow is a holomorphic submersion: Z F = C P × S C P S . ev ̟ π π (1.12) ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 7 If V −→ F ′ is a holomorphic vector bundle such that H q ′ ( π − ( s ) , V ) = 0 for all s ∈ S ′ and q ′ = q ,then the q -th direct image ( π ) q ∗ V is a holomorphic vector bundle on S ′ (see [21, Ch. 10]) with fibers(( π ) ∗ V ) s = H q ( π − ( s ) , V ). Hence the sheaves V := ( π ) ∗ ev ∗ ( T ̟ Z ( − , H := ( π ) ∗ π ∗ O C P (1) (1.13)are holomorphic vector bundles over S ′ because s ∗ T ̟ Z ( − ∼ = O d C P for any s ∈ S ′ . Note that ( π ) ∗ π ∗ W isin fact trivial with fiber H ( C P , W ). Lemma 1.4.
There is a canonical isomorphism T S ′ = V ⊗ H (defined in (1.13) ), and the bundles V and H carry natural holomorphic symplectic forms ω V and ω H . Thus, S ′ comes naturally equipped with theholomorphic Riemannian metric g = ω V ⊗ ω H .Proof. To prove the first statement observe that T s S ′ = H ( C P , s ∗ T ̟ Z ) = H ( C P , s ∗ T ̟ Z ( − ⊗ H ( C P , O C P (1)) = V s ⊗ H s for every s ∈ S ′ . This produces the identification T S ′ = V ⊗ H .To obtain the symplectic forms note that the twisted relative symplectic structure ω ∈ H ( Z, (Λ T ∗ ̟ Z )(2))induces a natural symplectic form on the vector bundle T ̟ Z ( − ∗ T ̟ Z ( −
1) is trivial on each fiber π − ( s ) = { s } × C P , s ∈ S ′ , the pull-back ev ∗ ω induces a symplectic form ω V | s on V s = H ( π − ( s ) , ev ∗ T ̟ Z ( − H ( C P , s ∗ T ̟ Z ( − . Finally, the symplectic form ω H | s on H s = H ( C P , O C P (1)) is induced from the Wronskian on π ∗ O C P (1).More explicitly, let ψ , ψ ∈ H ( C P , O C P (1)) = H s , and denote by dψ i the derivative defined in termsof local trivialisations. Then we set ω H | s ( ψ , ψ ) := ψ ⊗ ( dψ ) − ( dψ ) ⊗ ψ (1.14)which is a well-defined element of H ( C P , K C P ⊗ O C P (1) ⊗ ) = H ( C P , O C P ) = C . (cid:3) The restriction to M ⊂ S ′ of g in Lemma 1.4 coincides with the Riemannian metric g on M .The above considerations yield natural distributions T ev F ′ := ker( d ev) and T ev x S ′ = ker( d ev x ), x ∈ C P ,on F and S ′ respectively. Their associated leaves are the fibers F z := ev − ( z ) ∼ = S z := ev − x ( z ) , z ∈ Z, where x = ̟ ( z ) ∈ C P . Lemma 1.5.
For any x ∈ C P , the above integrable distribution T ev x S ′ is maximally isotropic with respectto g in Lemma 1.4. For two distinct points x = y ∈ C P , T ev x S ′ ∩ T ev y S ′ = { } . Proof.
We have T ev x S ′ | s = { V ∈ H ( s ∗ T ̟ Z ) | V ( x ) = 0 } , and this subbundle is of rank 2 d = dim S ′ .Now note that ev ∗ T ̟ ( −
1) is trivial on π − ( s ) = { s } × C P . If we choose an affine coordinate λ on C P such that λ ( x ) = 0 and view λ as an element of H ( C P , O C P (1)), we may therefore write any V ∈ T ev x S ′ | s in the form V = v ⊗ λ . Thus, if we evaluate g ( V , V ) at x ∈ C P it follows from the formula for ω H (see(1.14)) that we get zero.Recall that T s S ′ = H ( C P , s ∗ T ̟ Z ) = H ( C P , O C P (1) ⊕ d ). Any holomorphic section of O C P (1)vanishing at two distinct points of C P must be identically zero. This implies the second part of thelemma. (cid:3) Given any x ∈ C P we can thus define an associated non-degenerate holomorphic two-form Ω x on S ′ bytaking the natural skew-form on T S ′ = T ev x S ′ ⊕ T ev τ C P x ) S ′ (1.15)(recall that τ C P ( x ) = x ) induced from the metric g : Write V, W ∈ T S ′ as V = V x + V τ C P ( x ) , W = W x + W τ C P ( x ) with respect to the splitting in (1.15), and putΩ x ( V, W ) := − i ( g ( V x , W τ C P ( x ) ) − g ( V τ C P ( x ) , W x )) . (1.16) We identify a holomorphic vector bundles with its locally free analytic sheaf of sections.
F. BECK, I. BISWAS, S. HELLER, AND M. R ¨OSER
We may now define an endomorphism I x ∈ H ( S ′ , End( T S ′ )) via I x = (cid:18) − i i (cid:19) (1.17)again with respect to the splitting in (1.15). Then we see immediately that I x is orthogonal with respect to g and satisfies the equation Ω x = g ( I x − , − ) . Next consider the map φ x := ev x × ev τ C P ( x ) : S −→ Z x × Z τ C P ( x ) = Z x × Z x . (1.18) Proposition 1.6.
The map φ x in (1.18) restricts to a local biholomorphism φ x : S ′ −→ Z x × Z x .Proof. Clearly, the spaces have the common complex dimension 4 d . The kernel of the differential dφ x isgiven by T ev x S ′ ∩ T ev τ C P x ) S ′ , and T ev x S ′ ∩ T ev τ C P x ) S ′ = { } , by Lemma 1.5. The proposition follows. (cid:3) Now we discuss how the above data interact with the real structure τ on S ′ defined in (1.8). Let M ′ := ( S ′ ) τ (1.19)be the space of real sections so that we have an embedding M ֒ → M ′ induced by (1.7). Hence S ′ is a naturalcomplexification of the real analytic smooth manifolds M ′ . Note that in some examples M is all of M ′ (e.g.for the standard flat hyperk¨ahler manifolds C d ) but not always (see Example 4.11).For any s ∈ M ′ in (1.19), the differential dτ : T s S ′ −→ T s S ′ is C -antilinear, involutive and satisfies theequation dτ ( T ev x S ′ ) = T ev τ C P x ) S ′ . Indeed, we have for V ∈ T s S ′ = H ( s ∗ T ̟ Z ) the formula dτ ( V )( x ) = dτ Z ( V ( τ C P ( x ))) . Thus, if V ( x ) = 0, then dτ ( V )( τ C P ( x )) = 0. This implies in particular that if s ∈ M ′ , and V ∈ ( T s S ′ ) τ is real , then the section V ∈ H ( C P , s ∗ T ̟ Z ) is either identically zero or it is nowhere vanishing (a nonzeroholomorphic section of O C P (1) cannot vanish at two distinct points of C P ). As a consequence, the mapev x in (1.11) gives a local diffeomorphism from M ′ to Z x = ( M, I x ). Moreover, I x is real in the sense that dτ ◦ I x = I x ◦ dτ and therefore preserves T M ′ ∼ = ( T S ′ ) τ , so it defines an almost complex structure on M ′ .Consequently, we obtain a hypercomplex structure { (ev x ) ∗ I x | x ∈ C P } on M ′ . Remark 1.7.
The differential of the inclusion map
M ֒ → M ′ is a R -linear isomorphism T m M ∼ −→ T s m M ′ whose complexification yields an identification T s S ′ ∼ = T s (0) M ⊗ C . Under this identification, the decompo-sition in (1.15) is mapped to the natural decomposition T m M ⊗ C ∼ = T , m M ⊕ T , m M with respect to thecomplex structure I x .The real tangent vectors at s ∈ M ′ can be described as follows: Let V ∈ T ev x S ′ | s . Then the tangentvector V + τ ( V ) ∈ T ev x S ′ | s ⊕ T ev τ C P x ) S ′ | s is obviously real and we have g ( V + τ ( V ) , V + τ ( V )) = 2 g ( V, τ ( V )) . Since the twisted relative symplectic form ω on Z satisfies the condition τ ∗ Z ω = ω , it follows, by workingthrough the definition of g , that g is real in the sense that τ ∗ g = g . Hence g induces a real-valued pseudo-Riemannian metric on M ′ . Note that this immediately forces the restriction of Ω x to M ′ to be real as well.Pulled back to ( M, I x ), the form Ω is just the K¨ahler form associated with the pseudo-Riemannian metricand the hermitian almost complex structure I x .In summary, we have obtained the following result. Proposition 1.8.
For each x ∈ C P the form Ω x constructed in (1.17) defines a holomorphic symplecticform on each component of S ′ that intersects M ′ = ( S ′ ) τ . On M ⊂ M ′ it induces the K¨ahler form ω x . ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 9
Remark 1.9.
So far we have not shown that Ω x is actually closed. One way to show that is to use theAtiyah–Ward transform which we will discuss in detail in Section 2.4. The bundle V can be seen to arisefrom this transform applied to the bundle T ̟ Z ( − H can be shown to give the Levi-Civita connectionof the holomorphic Riemannian manifold ( T S ′ , g ). The form Ω x can then be shown to be parallel, fromwhich its closedness follows. In the next subsection we will give another, more direct proof that Ω x is closed.1.3. Alternative description of the holomorphic symplectic form Ω x on S ′ . Fix a point x ∈ C P .We now give an alternative description of Ω x which is better suited for computations. This alternativedescription also shows that Ω x can be extended to a holomorphic two-form on S , which is typically strictlylarger than S ′ .Consider the diagram Z F = C P × S C P S . ev ̟ π π We observe that for any k ∈ Z the bundle ( π ) ∗ π ∗ O C P ( k ) on S is trivial with fiber H ( C P , O C P ( k )).Starting with the twisted relative symplectic form ω ∈ H ( Z, (Λ T ∗ ̟ Z )(2)), its pull-back ev ∗ ω defines aholomorphic section of Λ π ∗ ( T ∗ S )(2). Invoking push-forward to S we obtain a vector-valued holomorphictwo-form Ω ∈ H ( S , Λ T ∗ S ) ⊗ H ( C P , O C P (2)) . Now note that H ( C P , O C P (2)) is the space of holomorphic vector fields on C P and therefore has thestructure of a Lie algebra isomorphic to sl ( C ). Fix an affine coordinate λ on C P such that λ ( x ) = 0 and τ ( λ ) = − λ − . Then we obtain the following basis of H ( C P , O C P (2)): e = ∂∂λ , h = − λ ∂∂λ , f = − λ ∂∂λ , which satisfies the standard relations[ h, e ] = 2 e, [ h, f ] = − f, [ e, f ] = h. We can now use the Killing form κ on sl ( C ) = H ( C P , O C P (2)) to define e Ω x := i κ (Ω , h ) ∈ H ( S , Λ T ∗ S ) . (1.20)Note that e Ω x in (1.20) is independent of the affine coordinate λ at x . Using κ ( h, h ) = tr(ad h ◦ ad h ) = 8and κ ( h, e ) = 0 = κ ( h, f ) we may rewrite this as follows. A general element A ∈ H ( C P , O C P (2)) is ofthe form A = A e e + A h h + A f f = ( A e − λA h − λ A f ) ∂∂λ = : A ( λ ) ∂∂λ with A e , A h , A f ∈ C . Then i κ ( A, h ) = − i A h = i ∂∂λ | λ =0 A ( λ ) . With this setup in place, we may therefore write for s ∈ S ′ , and tangent vectors V s , W s ∈ T s S ′ = H ( C P , s ∗ T ̟ Z )the following: e Ω x | s ( V, W ) = i κ (ev ∗ ( ω ( π ∗ V, π ∗ W ) , h ) = i ∂∂λ | λ =0 ω s ( λ ) ( V s ( λ ) , W s ( λ )) . (1.21)Recall the non-degenerate two-form Ω x defined in (1.16). Theorem 1.10.
Let x ∈ C P .a) The two-form e Ω x ∈ H ( S , Λ T ∗ S ) defined in (1.21) restricts to a holomorphic symplectic form on S ′ which is real with respect to τ .b) Over the open subset S ′ ⊂ S , Ω x | S ′ = e Ω x | S ′ . In particular, ( S ′ , Ω ) is a complexification of the (real analytic) K¨ahler manifold ( M ′ , ω I ) , where ω I isthe K¨ahler form associated to ∈ C P .c) The distributions T ev x S ′ and T ev τ C P x ) S ′ are Lagrangian with respect to e Ω x . Proof.
We choose ad fix throughout the proof an affine coordinate λ on C P such that λ ( x ) = 0 and τ ( λ ) = − λ − . To prove the first statement, recall that τ ∗ Z ω = ω . This implies that e Ω x is real with respectto τ . To show that e Ω x is a holomorphic symplectic form on S ′ , fix s ∈ S ′ and let U be a chart around s which is isomorphic to a neighbourhood U ′ ⊂ H ( C P , N s ) of 0 (cf. the proof of Proposition 1.1). Hencewe may work on U ′ where we have the natural isomorphism T U ′ ∼ = U ′ × H ( C P , N s ). In the following weuse this identification throughout.By shrinking U , and hence U ′ , if necessary, we may assume that there is a neighbourhood B x of x ∈ C P such that the image of the map ev : U × B x −→ Z is contained in a relative Darboux chart U ′′ ⊂ Z for ω around s ( x ). Let ( λ, v i , ξ i ), i ∈ { , · · · , d } , be these Darboux coordinates. In the following we restrict tothe case d = 1 because the general case works exactly the same way. Thus we write v = v i , ξ = ξ i . Notethat in these coordinates ω = d ̟ v ∧ d ̟ ξ , where d ̟ is the relative differential with respect to ̟ .Without loss of generality, we may assume that under the isomorphism N s ∼ = s ∗ T ̟ Z ∼ = O C P (1) ⊕ , the sections s ∗ ∂∂v and s ∗ ∂∂ξ form a frame for the first and the second summand respectively. Then any twoholomorphic vector fields V, W on U ′ ⊂ H ( C P , N s ) are expressed as V s ′ ( λ ) = ( a + a λ ) s ∗ ∂∂v + ( b + b λ ) s ∗ ∂∂ξ ,W s ′ ( λ ) = ( c + c λ ) s ∗ ∂∂v + ( d + d λ ) s ∗ ∂∂ξ (1.22)for s ′ ∈ U , where a i , b i , c i , d i are holomorphic functions in s ′ ∈ U . Note that in (1.22) we restrict theglobal sections V s ′ , W s ′ of N s over C P to B x . Now we compute ω λ ( V s ′ , W s ′ ) = dv ∧ dξ (cid:16) ( a + a λ ) ∂∂v + ( b + b λ ) ∂∂ξ , ( a ′ + a ′ λ ) ∂∂v + ( b ′ + b ′ λ ) ∂∂ξ (cid:17) = ( a + a λ )( b ′ + b ′ λ ) − ( a ′ + a ′ λ )( b + b λ ) . Thus we have e Ω x | s ′ ( V s ′ , W s ′ ) = i ∂∂λ | λ =0 ω λ ( V s ′ ( λ ) , W s ′ ( λ )) = i (( a b ′ + b ′ a ) − ( a ′ b + a ′ b )) . Here it is crucial that e Ω is linear in functions on U . If we choose ( a , a , b , b ) as coordinates on U , we seethat e Ω x = i ( da ∧ db + da ∧ db ) . Hence e Ω x is a holomorphic symplectic form on S ′ .By explicitly comparing Ω x | s and e Ω x | s , we see that Ω x = e Ω x . This implies the remaining claims becausethey have been established for Ω x . (cid:3) Remark 1.11.
Both definitions of Ω will be used. For example, (1.21) makes sense on all of S and thuscan be evaluated on any holomorphic section s , even if we do not know the normal bundle of s .2. The hyperholomorphic line bundle and the energy functional
Rotating circle actions and the hyperholomorphic line bundle.
Now assume that M is equippedwith an action of S that preserves the Riemannian metric g and also preserves the family of complexstructures { I x } x ∈ S . We also assume that the resulting action of S on S is nontrivial. This implies thatthe S -action on M preserves the associative algebra structure of R · Id T M ⊕ R · I ⊕ R · J ⊕ R · K . Consequently,the action of S on S is a nontrivial rotation. This means that without loss of generality we may assumethat the S -action on M preserves I (thus also preserves − I ) and rotates the plane spanned by J, K in thestandard way. We therefore call this a rotating circle action . The K¨ahler forms ω I , ω J , ω K and the Killingvector field X on M associated with this circle satisfy the following: L X ω I = 0 , L X ω J = ω K , L X ω K = − ω J . Note that this condition implies that the K¨ahler forms ω J , ω K are exact, and therefore the manifold M mustnecessarily be non-compact. ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 11
The above S -action on M evidently induces a holomorphic S -action on Z . In terms of the identification Z = C P × M of the underlying smooth manifold, this action of S is given by ζ. ( λ, m ) = ( ζλ, ζ.m ) , ζ ∈ S . (2.1)Here λ is an affine coordinate on C P such that I = I . The C ∞ -vector field on Z associated to the S -action in (2.1) will be denoted by Y . Note that Y | Z ∪ Z ∞ (see (1.3)) is actually a holomorphic vector fieldon the divisor Z ∪ Z ∞ = ( M, I ) ∪ ( M, − I ), because the S -action on M preserves both I and − I .We normalize the affine coordinate λ on C P such that the antipodal map S −→ S , x x ,corresponds to the map C P −→ C P , λ λ − . The coordinate λ is then uniquely determined up tomultiplication by a constant phase e i θ , θ ∈ [0 , π ).Clearly, the S -action on Z is compatible with the real structure τ Z in (1.5) in the sense that τ Z ( ζ.z ) = ζ − .τ Z ( z ) = ζ.τ Z ( z ) (2.2)for all z ∈ Z and ζ ∈ S . It is straightforward to check that for any ζ ∈ S the twisted relative symplecticform ω in (1.4) satisfies the equation ζ ∗ ω = ω, i.e., ω is S -invariant.Let µ : M −→ i R = u (1) be a moment map with respect to ω I for the S -action on M . Note that sinceour moment map is complex-valued, the moment map equation takes the form dµ ( − ) = i ω I ( X, − ) . (2.3)Haydys has shown in [23] that the 2-form ω I + i dd cI µ is of type (1 ,
1) with respect to every complex structure I λ , λ ∈ C P . Thus, if [ ω I / π ] ∈ H ( M, Z ), then there exists a C ∞ hermitian line bundle ( L M , h M ) −→ M with a compatible hermitian connection ∇ M whose curvature is ω I + i dd c µ . Consequently, if ∇ (0 , λ M is the (0 , ∇ M with respect to the complex structure I λ , we obtain a holomorphic line bundle( L M , ∇ (0 , λ ) −→ ( M, I λ ). This implies that L Z = ( q ∗ L M , ( q ∗ ∇ M ) , ) −→ Z is a holomorphic linebundle over the twistor space Z of M . Here q : Z −→ M is the C ∞ submersion given by the naturalprojection S × M −→ M . Note that the Chern connection of the hermitian holomorphic line bundle( L, e − µ h M ) −→ ( M, I ) has curvature ω I , i.e., it is a prequantum line bundle on the K¨ahler manifold( M, ω I ).In [29] Hitchin provided a twistorial description of the line bundle L Z exhibiting a natural meromorphicconnection ∇ on L Z . To recall this, observe that the S -action on Z covers the standard S -action on C P ,and therefore the associated holomorphic vector field Y on Z is ̟ -related to σ := i λ ∂∂λ on C P . Viewing σ as a section of ̟ ∗ O C P (2) which vanishes on the divisor D := Z ∪ Z ∞ , we have theshort exact sequence 0 T ∗ Z T ∗ Z (2) T ∗ Z (2) | D · σ (2.4)of sheaves on Z . Hitchin then constructs from the S -action a certain element ϕ ∈ H ( D, T ∗ Z (2) | D ).Explicitly, using the C ∞ -splitting T ∗ Z = T ∗ M ⊕ T ∗ C P we can write ϕ in terms of the data on M as ϕ = (cid:0) ( d c µ + i d c µ ) ⊗ ∂∂λ , ( µ dλ ) ⊗ ∂∂λ (cid:1) = (cid:0) − ι Y ω λ =0 , ( µ dλ ) ⊗ ∂∂λ (cid:1) . Note that ϕ satisfies the equation ϕ | T ̟ Z | D = ι Y ω, (2.5)where ω is the relative symplectic form on Z . Note that since the vector field Y | D is vertical, the formula in(2.5) makes sense. From the long exact sequence of cohomologies associated to (2.4)0 H ( Z, T ∗ Z ) H ( Z, T ∗ Z (2)) H ( D, T ∗ Z (2) | D ) H ( Z, T ∗ Z ) . . . . · σ δ we then obtain that α L := δ ( ϕ ) ∈ H ( Z, T ∗ Z ) . In fact, this element α L lies in the image of H ( Z, Ω Z,cl ), where Ω Z,cl denotes the sheaf of closed 1-formson Z . The above class α L therefore defines an extension0 O E T Z E −→ T Z is a holomorphic Lie algebroid. If α L is integral, this is the Atiyah algebroid of a linebundle L Z . Thus, α L is simply the Atiyah class of the line bundle L Z in this case. Explicitly, relative tosome open cover U = { U i } of Z , we have ( α L ) ij = g − ij dg ij and { g ij ∈ H ( U i ∩ U j , O ∗ Z ) } is a cocycle representing L Z . Since on the other hand α L = δ ( ϕ ), we may rewritethis using the definition of the connecting homomorphism δ . Let ϕ be given by { ϕ i ∈ H ( U i ∩ D, T ∗ Z (2) | D ) } and take extensions { e ϕ i ∈ H ( U i , T ∗ Z (2)) } . Then a representative for α L = δ ( ϕ ) is given by( α L ) ij = e ϕ i − e ϕ j σ . Thus, A i = e ϕ i σ are the local 1-forms of a meromorphic connection ∇ on L Z , which have a simple pole alongthe divisor D . Its residue along D is given by ϕ .Moreover, Hitchin shows that the curvature F of the meromorphic connection ∇ has the following prop-erties: • ι Y F = 0 and Y spans the annihilator of F . • F = ωσ on T ∗ ̟ Z | ̟ − ( C ∗ ) , where ω ∈ H ((Λ T ∗ ̟ Z )(2)) is the relative symplectic form on Z −→ C P .We may thus think of ( L Z , ∇ ) as a “meromorphic relative prequantum data” for the meromorphic relativesymplectic form ωσ .2.2. The S -action on S and the energy functional. The S -action on Z obtained from the S -actionon M produces an S -action on S , which is constructed as follows:( ζ.s )( λ ) = ζ. ( s ( ζ − λ ))for all s ∈ S , ζ ∈ S and λ ∈ C P . The evaluation map ev : C P × S −→ Z is evidently equivariantwith respect to the diagonal action of S on C P × S and the S -action on Z . Moreover, the S -action on S is compatible with τ in the sense that τ ( ζ.s ) = ζ.τ ( s )for all s ∈ S and ζ ∈ S . In particular, the S -action on S preserves the subset S τ fixed by τ . Proposition 2.1.
The holomorphic two-form Ω on S constructed in (1.16) is S -invariant.Proof. We use the description of Ω given in § ω is S -invariant and ev : F −→ S is S -equivariant. Moreover, the Killing form on sl = H ( C P , O C P (2)) and the element h = 2 i σ ∈ H ( C P , O C P (2))are also S -invariant. Hence Ω = i κ (ev ∗ Ω , h ) too must be S -invariant. (cid:3) Let Y be the vector field on Z which is induced by the S -action. Since the S -action commutes with τ Z (see (2.2)), Y is τ Z -invariant, meaning dτ Z ( Y ) = Y ◦ τ Z . Lemma 2.2.
Let s ∈ S ′ , and let X be the vector field associated to the S -action on S ′ . Then X s ∈ T s S ′ ∼ = H ( C P , N s ) is given by X s ( λ ) = Y s ( λ ) − i λ ˙ s ( λ ) . (2.6)Note that X s is indeed vertical because d̟ ( Y s ( λ ) − i λ ˙ s ( λ )) = i λ ∂∂λ − i λ ∂∂λ = 0 . ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 13
Proof.
This is a direct computation: Let s ∈ S ′ and λ ∈ C P . Then X s ( λ ) = ddt (cid:12)(cid:12)(cid:12) t =0 ( e i t .s )( λ ) = ddt (cid:12)(cid:12)(cid:12) t =0 e i t . ( s ( e − i t λ )) = Y s ( λ ) − i λ ˙ s ( λ ) . More invariantly, we can use the vector field σ for the standard S -action on C P , i.e., σ ( λ )( λ ) = i λ ∂∂λ torewrite the relation of Lemma 2.2 as d ev ◦ X = Y ◦ ev − d ev ◦ σ. (2.7)The fundamental vector field Y F for the diagonal S -action on F = C P × S is given by Y F ( λ, s ) = π ∗ σ ( λ ) + π ∗ X ( s ) . Thus, the formulae (2.6) and (2.7) just become d ev ◦ Y F = Y ◦ ev. (cid:3) The recent paper [4] discusses the holomorphic energy function E : S −→ C , E ( s ) = i π res λ =0 ( s ∗ D ) . (2.8)Note that E ( s ) = s ∗ ϕ ∈ T ∗ C P (2) | { , ∞} ∼ = O C P | { , ∞} , so this indeed is a complex number.The function E has the property that ι ∗ E = µ for the natural inclusion ι : M −→ S in (1.7). The residueformula in (2.8) implies that τ ∗ E ( s ) = E ( s ) + deg( s ∗ L Z ) . (2.9)To give an explicit formula for the energy E , we recall the section ϕ = ϕ | Z ∈ Γ( Z , T ∗ Z (2) | Z ) definedabove. Contracting dλ ⊗ ∂∂λ , we therefore obtain that E ( s ) = s ∗ ϕ = − ι Y ω (cid:0) ˙ s (0) − ˙ s s (0) (0) (cid:1) + µ ( s (0))for ˙ s (0) = ds ( ∂∂λ ) etc. The difference ˙ s (0) − ˙ s s (0) (0) accounts for the fact that we are working with the C ∞ -splitting induced by twistor lines; see [4] for the details. Theorem 2.3.
The energy function E : S ′ −→ C is a moment map for the natural S -action and holomor-phic symplectic structure Ω on S ′ , i.e., d E ( − ) = i Ω ( X, − ) . In particular, the S -fixed points in S ′ are precisely the critical points of the function E . Since Ω | M = ω I , this is a holomorphic extension of (2.3). Proof.
Take any s ∈ S ′ , and set m := s (0). We first compute d s E ( V ) for V ∈ T ev S ′ | s ⊂ T s S ′ = H ( C P , N s ) . By definition V (0) = 0, so V ( λ ) = v ⊗ λ for some v ∈ H ( C P , s ∗ T ̟ Z ( − V is representable by a family s t of sections with s t (0) = m and ∂ t =0 s t ( λ ) = V ( λ ). Note that ∂ λ =0 V ( λ ) iswell-defined because V (0) = 0. Then a local computation shows that ∂ t =0 ˙ s t (0) = ∂ λ =0 V ( λ ) = v ∈ T m M .
Hence we conclude that d s E ( V ) = ∂∂t (cid:12)(cid:12)(cid:12) t =0 (cid:0) − ι Y ω ( ˙ s t (0) − ˙ s m (0)) + µ ( m ) (cid:1) = − ι Y ω ( v ) . Next we consider X s ∈ H ( C P , N s ), the fundamental vector field X evaluated at s . Since Y ( m ) = X s (0) (see Lemma 2.2), we conclude thatΩ | s ( X s , V ) = i ∂∂λ | λ =0 ω s ( λ ) ( X s ( λ ) , V ( λ )) = ω ( Y (0) , v ) = i ι Y ω ( v ) . It remains to prove the claim for V ∈ T ev ∞ S ′ | s . To this end, observe that d ( τ ∗ E ) = d E by (2.9) because s deg( s ∗ L Z ) is locally constant. Further we know dτ ( T ev ∞ S ′ ) = τ ∗ T ev S ′ , so that any V ∈ T ev ∞ S ′ | s is of the form dτ τ ( s ) ( W ) for W ∈ T ev S ′ | τ ( s ) . Then we compute using the previous result and τ ∗ Ω = Ω, d E s ( V ) = τ ∗ d E s ( V )= d E τ ( s ) ( W )= Ω τ ( s ) ( X τ ( s ) , dτ τ ( s ) ( V ))= ( τ ∗ Ω) s ( X s , V )= Ω s ( X s , V ) . This completes the proof. (cid:3)
Critical Points of E . In this subsection we assume that the S -action on Z extends to a holomorphicaction of C ∗ . That is, the vector field I Z Y on Z is complete. Since E is holomorphic and S -invariant, itfollows that it is in fact C ∗ -invariant in this case. Thus the critical points of E are the C ∗ -fixed points in S ′ .We first examine the C ∗ -fixed points in S . Any C ∗ -fixed point s ∈ S is characterized by s ( ζλ ) = ζ.s ( λ )for all ζ ∈ C ∗ and λ ∈ C P . In particular, s ∈ S C ∗ is determined by its value at λ = 1. Indeed, s ( λ ) = λ.s (1) for λ ∈ C ∗ , and by continuity we have s (0) = lim λ → λ.s (1) , s ( ∞ ) = lim λ →∞ λ.s (1) . (2.10)Hence the closures of the C ∗ -orbits in Z lying over C ∗ ⊂ C P correspond precisely to the C ∗ -fixed pointsin S . Conversely, any point z ∈ Z potentially determines a C ∗ -invariant section s z : C ∗ −→ Z of ̟ asfollows. For λ ∈ C ∗ , set s z ( λ ) = λ.z which is clearly a section over C ∗ . If the limitslim λ → λ.z, lim λ →∞ λ.z, exist in Z , cf. (2.10), then the section extends to a C ∗ -invariant section s z : C P −→ Z of ̟ . The existenceof these limits has been investigated in detail in [45] for M DH ; see also Section 4.2 below.Clearly, for any fixed point s ∈ S C ∗ , we have s (0) ∈ Z = M and s ( ∞ ) ∈ Z ∞ ∼ = M are fixed points ofthe C ∗ -actions on M and M respectively. The following gives the converse on S ′ ⊂ S (also see [18, 19, 20]). Proposition 2.4.
Let s ∈ S ′ be such that s (0) ∈ Z C ∗ and s ( ∞ ) ∈ Z C ∗ ∞ . Then s ∈ ( S ′ ) C ∗ .Proof. We see from X s ( λ ) = Y s ( λ ) − i λ ˙ s ( λ ) that X s (0) = 0 = X s ( ∞ ). Since N s ∼ = O C P (1) ⊕ d , thisimplies that X s = 0, and thus s is a fixed point. (cid:3) Proposition 2.4 has the following consequence.
Corollary 2.5.
Let s ∈ S be a section such that s (0) ∈ Z C ∗ and s ( ∞ ) ∈ Z C ∗ ∞ . If s is not a fixed point ofthe C ∗ -action, then s cannot be contained in S ′ , i.e., the normal bundle of s is not isomorphic to O ⊕ d C P . We end this subsection with the observation that C ∗ -fixed points s ∈ S are also the fixed points underthe twisting procedure that was introduced in [4]; see also [6]. Recall that a section s ∈ S is twistable if thesection e s ( λ ) := λ − .s ( λ ) over C ∗ extends to a section on all of C P . Proposition 2.6.
Let s be a fixed point of the C ∗ -action, then s is twistable, and the twist e s satisfies theequation e s = s .Proof. If s is fixed, then s ( λ ) = λ.s (1) for all λ ∈ C ∗ . If follows that e s ( λ ) = λ − .s ( λ ) = λ − .λ .s (1) = λ.s (1) = s ( λ ) , completing the proof. (cid:3) ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 15
The Atiyah–Ward transform of L Z . Let S ⊂ S ′ for the space of all sections s ∈ S such that • the normal bundle is isomorphic to O C P (1) ⊕ d , and • s ∗ L Z trivial.Since s ∗ L Z trivial if and only if deg s ∗ L Z = 0, we conclude that S is an open subset of S , and it is a unionof some connected components of S ′ . Consider the space F = C P × S and restrict the diagram (1.12)to F : Z F = C P × S C P S . ev ̟ π π Clearly, we have the identification T F = π ∗ T C P ⊕ π ∗ T S . Since d ev : π ∗ T S ′ −→ ev ∗ T ̟ Z is surjective by Proposition 1.6, we obtain the following commutativediagram with exact rows 0 T ev F T F ev ∗ T Z T ev F π ∗ T S ev ∗ T ̟ Z . d evid d ev (2.11)We next describe the Atiyah–Ward transform of L Z [2, 3, 34] and study how it interacts with the meromorphicconnection on L Z . The Atiyah–Ward construction is of course valid in a more general context, but for theconvenience of the reader we spell out the details relevant for our discussion in the case of a line bundle.The Atiyah–Ward transform of L Z is a holomorphic line bundle L over S with a holomorphic connection ∇ AW . To construct this line bundle, recall that L Z is trivial along s ( C P ) for any s ∈ S . Hence ev ∗ L Z istrivial along C P × { s } = π − ( s ), and the (0-th) direct image construction yields the line bundle L := ( π ) ∗ ev ∗ L Z . We observe that there is a natural isomorphism π ∗ L ∼ = ev ∗ L Z . Over ( λ, s ) ∈ F it is given by evaluatingan element of π ∗ L ( λ,s ) = H ( C P , s ∗ L Z ) at λ .We next equip L with a holomorphic connection. First consider the relative exterior differential d ev : O F −→ T ∗ ev ( F ) which is defined as the composition d ev : O F T ∗ F T ∗ ev ( F ) . d By construction, we have ev ∗ O Z ⊂ ker d ev , and for any f ∈ O F : d ev f = df | T ev F . Next we tensor the diagram defining d ev with ev ∗ L Z and observe that this gives a well-defined ”relativeconnection” ∇ ev : ev ∗ L Z −→ T ∗ ev F ⊗ ev ∗ L Z , ∇ ev ( f · ev ∗ ν ) := ( d ev f ) ⊗ ev ∗ ν, for a locally defined trivializing section ν of L Z and f ∈ O F . This operator is well-defined because thetransition functions for ev ∗ L Z lie in ev ∗ O Z and hence in the kernel of d ev . The next lemma will enable usto construct ∇ AW from ∇ ev . Lemma 2.7.
There is a natural isomorphism of vector bundles over S : T ∗ S ∼ = ( π ) ∗ T ∗ ev F , A π ∗ A | T ev F . In particular, there is an isomorphism T ∗ S ⊗ L ∼ = ( π ) ∗ ( T ∗ ev F ⊗ ev ∗ L Z ) . Here we slightly abuse notation since the restriction of the maps will be evident in the following.
Proof.
Dualizing the second row of (2.11) yields0 ev ∗ T ∗ ̟ Z π ∗ T ∗ S T ∗ ev F . r Here r denotes the restriction of 1-forms to T ∗ ev F . For this short exact sequence, consider the correspondinglong exact sequence of direct image sheaves with respect to π :0 ( π ) ∗ ev ∗ ( T ∗ ̟ Z ) ( π ) ∗ ( π ∗ T ∗ S ) ( π ) ∗ ( T ∗ ev F ) ( π ) ∗ ev ∗ ( T ∗ ̟ Z ) · · · . π ∗ r The fibers of the vector bundles ( π ) q ∗ ev ∗ ( T ∗ ̟ Z ), q = 0 ,
1, over any s ∈ S satisfy H q ( C P , s ∗ T ∗ ̟ Z ) ∼ = H q ( C P , O ( − ⊕ d ) = 0 . Thus the above long exact sequence of cohomologies gives the isomorphism π ∗ r : ( π ) ∗ ( π ∗ T ∗ S ) ∼ = ( π ) ∗ ( T ∗ ev F )of vector bundles on S .By the projection formula (see for example [22, Chapter 3]) or direct computation, we have( π ) ∗ ( π ∗ S ⊗ O F ) ∼ = T ∗ S ⊗ ( π ) ∗ O F ∼ = T ∗ S . The last isomorphism follows from ( π ) ∗ O F ∼ = O S , which in turn follows from the fact that the fibers of π are connected. Since ev ∗ L Z ∼ = π ∗ L over S , the second statement in the lemma is again derived fromthe projection formula. (cid:3) Proposition 2.8.
The operator ∇ AW = ( π ) ∗ ( ∇ ev ) : L −→ T ∗ S ⊗ L ∼ = ( π ) ∗ (cid:0) T ∗ ev F ⊗ ev ∗ L Z (cid:1) induces a natural holomorphic connection on L . This connection is trivial on the submanifolds S z = π (ev − ( z )) for all z ∈ Z . This property determines ∇ AW completely as long as S z is connected. We call the above connection ∇ AW on L the Atiyah–Ward connection . Proof.
The O S -module structure on the sheaf of sections of the vector bundle L = ( π ) ∗ ev ∗ L Z is as follows.Let S ⊂ S be an open subset, and take f ∈ O S ( S ). Then f acts on L ( S ) = H ( C P × S, ev ∗ L Z ) bymultiplication with π ∗ f . Hence for any ψ ∈ L ( S ) we have ∇ AW ( f ψ ) = ∇ ev ( π ∗ f ψ ) = d ev π ∗ f ⊗ ψ + f ∇ ev ψ. We see that d ev π ∗ f ⊗ ψ + f ∇ ev ψ is obtained from π ∗ ( df ⊗ ψ + f ∇ AW ψ ) by restricting to the subbundle T ev F . Note that this is uniquely determined by Lemma 2.7, so we conclude that ∇ AW ( f ψ ) = df ⊗ ψ + f ∇ ψ. Therefore, ∇ AW is indeed a connection.To show that this connection is trivial on S z for each z ∈ Z , observe that any section of the form ev ∗ ψ is actually parallel (covariant constant) for the relative connection ∇ ev by the formula ∇ ev ( f · ev ∗ ψ ) = ( d ev f ) ⊗ ev ∗ ψ . Now the fibers of ev are of the form ev − ( z ) = { ̟ ( z ) } × S z . Then we get a frame ψ of ev ∗ L Z | ev − ( z ) by putting ψ ( ̟ ( z ) , s ) = l for any fixed 0 = l ∈ ( L Z ) z . Note that this construction makes sense because ev( ̟ ( z ) , s ) = s ( ̟ ( z )) = z for all s ∈ S z . The frame ψ induces a natural trivialization ψ L of L along S z . Since ∇ ev ψ = 0, we have ∇ AW ψ L = 0 along S z . Thus ∇ AW is trivial on S z .Now let e ∇ = ∇ AW + A be another holomorphic connection on S which is trivial along each S z . Thenthe holomorphic 1-form A restricted to S z is exact for all z ∈ Z . A result of Buchdahl in [9] implies that ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 17 we can find f ∈ O ( F ) such that d ev f = ( π ∗ A ) | T ev F . Note that f is constant along the compact fibers of π . Therefore there exists e f ∈ O ( S ) such that f = e f ◦ π if S z is connected. It now follows that π ∗ ( A − d e f ) | T ev F = 0 , and this implies that A = d e f . Thus, ∇ AW and e ∇ = ∇ AW + d e f are gauge-equivalent. (cid:3) Corollary 2.9.
On the real submanifold M ⊂ S the connection ∇ AW coincides with the hyperholomorphicconnection on L M . Thus its curvature F AW of ∇ AW satisfies F AW = Ω + i dI d E on every component of S that meets M .Proof. The evaluation map ev : F −→ Z restricts to a diffeomorphism ǫ : M × C P −→ Z . Thus wehave a natural map q = π ◦ ǫ − : Z −→ M . Note that q ∗ L = q ∗ ( π ) ∗ ev ∗ L Z = ( ǫ − ) ∗ π ∗ ( π ) ∗ ev ∗ L Z = ( ǫ − ) ∗ ev ∗ L Z = L Z . On the other hand, we have, by definition, that q ∗ L M = L Z .Now the hyperholomorphic connection ∇ M is real analytic. Thus we can complexify and extend it toa holomorphic connection e ∇ on L −→ S at least locally in a neighbourhood of M ⊂ S . Note that itscurvature is Ω + i dI d E , i.e., the complexification of the curvature of ∇ M .Since there is a unique twistor line through each point in z , we see that S z intersects M in a uniquepoint, namely q ( z ). Moreover, as the twistor line q ( z ) through z also passes through τ Z ( z ), we see that q ( z ) = S z ∩ S τ ( z ) ∩ M . We obtain a splitting T q ( z ) S = T q ( z ) S z ⊕ T q ( z ) S τ ( z ) , which can be identified with the splitting T q ( z ) M ⊗ C = T , ,λ M ⊕ T , ,λ M . Since ∇ M has curvature oftype (1 ,
1) on (
M, I λ ), (here λ = ̟ ( z )), its complexification e ∇ is flat along S z (and S τ ( z ) ) for all z ∈ Z .By Proposition 2.8 the connections ∇ and e ∇ must be gauge equivalent in a neighbourhood of M in S .The curvature F AW is a holomorphic two-form on S which is compatible with the real structure. Onthe other hand, Ω + dI d E is also holomorphic and real and coincides with F AW on the real submanifold M . Thus, they must coincide on every component of S that meets M . (cid:3) Corollary 2.10.
The curvature F ∇ of the holomorphic connection ∇ = ∇ AW − i I d E satisfies theequation F ∇ = Ω on every component of S that intersects M . The Atiyah–Ward transform and the meromorphic connection.
We want to describe the rela-tionship between the meromorphic connection ∇ on L Z and the Atiyah–Ward transform ( L , ∇ AW ) of L Z .As the first step, we observe that the Atiyah class ev ∗ α L of ev ∗ L Z −→ F vanishes because ev ∗ L Z = π ∗ L admits the holomorphic connection π ∗ ∇ AW . This has the following implications: on F considerˆ σ = ev ∗ σ ∈ H ( π ∗ O (2))which vanishes on the divisor b D = ev − ( D ) = ( { } × S ) ∪ ( {∞} × S ) . We have the short exact sequence of sheaves0 T ∗ F T ∗ F (2) T ∗ F (2) | b D · b σ Note that there exists a neighbourhood M ⊂ U ⊂ S such that S z ∩ U is connected for each z ∈ Z , because M ∩ S z = { q ( z ) } . with the corresponding long exact sequence0 H ( F , T ∗ F ) H ( F , T ∗ F (2)) H ( b D, T ∗ F (2) | b D ) H ( F , T ∗ F ) . . . . · ˆ σ b δ (2.12)The construction of L Z implies that ev ∗ α L = b δ (ev ∗ ϕ ), which vanishes by the previous observation. Hencethere exists φ ∈ H ( F , T ∗ F (2)) such that φ | b D = ev ∗ ϕ by the long exact sequence in (2.12). Proposition 2.11.
The connection ev ∗ ∇ − φ ˆ σ on ev ∗ L Z = π ∗ L is holomorphic.Proof. From the construction of ev ∗ ∇ , we know that it has connection 1-forms b A i = ev ∗ e ϕ i b σ with respect to the open covering ev ∗ U = { ev − ( U i ) } . Here U = { U i } and e ϕ i are as in the construction of ∇ in Section 2.1. The connection ev ∗ ∇ − φ ˆ σ has the connection 1-forms b A i − φ i b σ . But φ i | b D and ev ∗ ϕ i coincide on ev − ( U i ) ∩ b D and the forms b A i have no other poles. Hence the propositionfollows. (cid:3) We next explicitly construct such a section φ ∈ H ( F , T ∗ F (2)) and describe the difference form B between the holomorphic connections ev ∗ ∇ − φ b σ and π ∗ ∇ AW . To do so, we first observe that T ∗ F (2) ∼ = ( π ∗ T ∗ C P ⊕ π ∗ T ∗ S )(2) ∼ = O F ⊕ π ∗ T ∗ S (2) . (2.13)We describe φ component-wise, i.e., φ = ( β, γ ) with respect to the splitting (2.13). Given any V ∈ T s S ◦ ,construct the following γ ∈ H ( F , π ∗ T ∗ S )(2)) γ ( s,λ ) ( V ) := ω ( X s ( λ ) , V ( λ )) = ( ι X ev ∗ ω ) ( λ,s ) ( V ) . The first component β of φ is β := E id π ∗ O (2) ∈ H ( F , π ∗ T ∗ C P (2)) = H ( F , End( π ∗ O (2)) ∼ = H ( F , O F ) . Lemma 2.12.
The element φ = ( β, γ ) ∈ H ( F , T ∗ F (2)) constructed above has the desired properties.Moreover, φ | T ev F = 0 .Proof. We may write β = E dλ ⊗ ∂∂λ on { λ = ∞} , and hence at λ = 0:ev ∗ ϕ ( s, ( ∂∂λ ,
0) = ϕ s (0) ( ˙ s (0)) = E ( s ) ∂∂λ = β ( s, (cid:0) ∂∂λ (cid:1) . Similarly, at λ = ∞ , ev ∗ ϕ ( s, ∞ ) ( ∂∂ e λ ,
0) = ϕ s ( ∞ ) ( ˙ s ( ∞ )) = E ( s ) ∂∂ e λ = β ( s, ∞ ) (cid:0) ∂∂λ (cid:1) . Now we check the component γ . Let V ∈ T s S . Then, by (2.5),ev ∗ ϕ ( s, ( V ) = ϕ s (0) ( V (0)) = ( ι Y ω ) s (0) ( V (0)) . On the other hand, by (2.6),( ι X ev ∗ ω ) ( s, ( V ) = ω ( X (0) , V (0)) = ω ( Y (0) , V (0)) = ι Y ω ( V (0)) = ev ∗ ϕ ( s, ( V ) . Similar considerations apply at λ = ∞ . Since γ = ι X (ev ∗ ω ) | T S and T ev F ⊂ π ∗ T S , it follows that φ | T ev F = 0. (cid:3) Hence the connection b ∇ := ev ∗ ∇ − φ b σ is holomorphic. To understand its relationship with the Atiyah–Ward connection, we need the following: Lemma 2.13.
There is a unique holomorphic connection ∇ L on L such that π ∗ ∇ L = b ∇ on π ∗ L = ev ∗ L Z . ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 19
Proof.
We will prove the more general statement that any holomorphic connection on π ∗ L is pulled backfrom a unique holomorphic connection on L . Let F be the curvature of such a connection which is aholomorphic two-form on F . Further let V be a vector field on S which we view as a vector field on F = C P × S . Then ι V F D pulls back to a holomorphic 1-form on C P × { s } , which must vanish, sincethere are no non-trivial holomorphic 1-forms on C P . This shows that F is purely horizontal with respectto the map π : C P × S −→ S .We can thus find local frames for π ∗ L which are parallel along the fibers of π (which are C P ). Moreprecisely, there exists an open cover of C P × S by open subsets of the form C P × U , where U ⊂ S isopen, together with holomorphic frames for ev ∗ L Z over C P × U that are parallel (covariant constant) inthe C P -direction. The transition functions with respect to these local frames are holomorphic functions C P × ( U ∩ V ) −→ C ∗ , which are thus constant along the C P -fibers. Hence the transition functions are ofthe form g UV ◦ π , where g UV define the line bundle L on S . Moreover, the holomorphic connection uniquelydescends to L , concluding the proof. (cid:3) Theorem 2.14.
The holomorphic connections ∇ L and ∇ AW on L coincide.Proof. Clearly, ∇ L = ∇ AW + B for a unique holomorphic 1-form B on S . Consider the pulled backconnection b ∇ = π ∗ ∇ L = π ∗ ∇ AW + π ∗ B. (2.14)Let U ⊂ Z be an open subset such that L Z has a local frame ν on U . Hence ψ := ev ∗ ν is a local frame ofev ∗ L Z on b U := ev − ( U ). Observe that along S z , for z ∈ U , the frame ψ ( ̟ ( z ) , − ) coincides with the parallelframe (with respect to ∇ AW ) constructed in the proof of Proposition 2.8. Since ev − ( z ) = { ̟ ( z ) } × S z , itfollows that π ∗ ∇ AW ψ restricted to T ev F = ker d ev vanishes on all of b U .Next let ( λ, s ) ∈ b U − b D , and X ∈ ( T ev F ) ( λ,s ) . Then we compute b ∇ X ψ = (ev ∗ ∇ ) X ψ − φ b σ ( X ) ψ = 0 . Here we made use of the definition of T ev F and Lemma 2.12. By continuity, the restriction of b ∇ ψ to T ev F vanishes on all of b U as well. Since F can be covered by open subsets b U as above, from (2.14) it follows that π ∗ B | T ev F = 0. But the isomorphism T ∗ S ∼ = π ∗ T ev F is given by A π ∗ A | T ev F (cf. Lemma 2.7), sowe have B = 0. (cid:3) The existence of a holomorphic connection on S with curvature Ω together with Theorem 1.10 allow usto obtain the following results on the global structure of S . Denote S z = ev − ̟ ( p ) ( z ) ∩ S for z ∈ Z . Theorem 2.15.
Suppose = [ ω ] ∈ H ( M ) , and let z ∈ Z ∞ . Then S z cannot be isomorphic to Z . Inparticular, the local diffeomorphism ev × ev ∞ : S −→ Z × Z ∞ cannot extend to a global diffeomorphism.Proof. Since T S z is Lagrangian for the symplectic form Ω , the Atiyah-Ward connection ∇ AW pulls back toa flat connection on L ( z ) := L| ev − ∞ ( z ) . Consequently, c ( L ( z ) , C ) = 0 ∈ H (ev − ∞ ( z )). But the first Chernclass of L is [ ω ] = 0 ∈ H ( Z , C ). (cid:3) Remark 2.16. If Z = M is not simply-connected, then ev : S p −→ Z might be a covering map. Infact, this happens for the rank 1 Deligne–Hitchin moduli space; see [5]. It would be interesting to understandwhat happens if Z is simply-connected, for example in the case of SL(2 , C )-Deligne-Hitchin moduli spaces.2.6. An alternative construction of Hitchin’s meromorphic connection.
We can reverse the aboveconstructions to obtain the meromorphic Hitchin connection on a hyperholomorphic bundle over the twistorspace Z of a hyperk¨ahler manifold with a rotating circle action. We only sketch the steps, as the technicaldetails are already explained above. The necessary twistor data are: • the twistor space Z of a hyperk¨ahler manifold with a rotating circle action; • the twisted relative holomorphic symplectic form ω ∈ H ( Z, (Λ T ∗ ̟ )(2)) . Adding the additional structure of a “prequantum” hyperholomorphic line bundle L , we obtain from theAtiyah-Ward construction that ev ∗ L −→ S is equipped with the holomorphic connection π ∗ ∇ AW . FromTheorem 2.14 we have ev ∗ ∇ = π ∗ ∇ AW + φ ˆ σ . Note that not only ev ∗ ∇ is the pull-back of a meromorphic connection on the twistor space but also π ∗ ∇ AW and the meromorphic 1-form φ ˆ σ are pull-backs from Z as well. Clearly, the corresponding objects on Z areneither a holomorphic connection nor a meromorphic 1-form but only real analytic objects.We shall give a local construction of the holomorphic connection ∇ AW on S which only involves the abovelisted twistor data: As explained in Section 1, we obtain from the twistor data the holomorphic symplecticform Ω . Locally, S is identified with Z × Z ∞ by evaluation, and there is the moment map E of the circleaction on S ; see Theorem 2.3. Define the bundle automorphism η : T ∗ ( Z × Z ∞ ) −→ T ∗ ( Z × Z ∞ )that acts on T ∗ Z (respectively, T ∗ Z ∞ ) as multiplication by +1 (respectively, − α U with η ( α U ) = α U and dα U = Ω + i d I d E . (2.15)We may think of the forms α U as holomorphic connection 1-forms. Note that the corresponding cocycle isthe pull-back of a cocycle on Z . By Corollary 2.10, the connection defined by (2.15) is flat on S z for every z ∈ Z, and therefore it is locallytrivial. Hence, we recover the Atiyah–Ward connection at least locally. By adding the 1-form φ ˆ σ , which isconstructed from the circle action and from the twisted symplectic form via Lemma 2.12, we obtain (thepullback of) the meromorphic connection on Z .2.6.1. τ -sesquilinear forms and a generalized Chern connection. It seems appropriate to put the above localconstruction into a global context. We need an additional structure on
L −→ S , which can be regarded asthe complexification of the hermitian metric h M on L M . Definition 2.17.
Let
L −→ S be a holomorphic line bundle, and let τ : S −→ S be an anti-holomorphicinvolution. A non-degenerate pairing h· , ·i : L × τ ∗ L −→ C is called a holomorphic τ -sesquilinear form on L if for all local holomorphic sections v, w ∈ Γ( U , L ) definedon some τ -invariant open subset U ⊂ S , the function hh v, w ii on U defined by hh v, w ii ( s ) := h v ( s ) , τ ∗ w ( s ) i is holomorphic and it satisfies the identity τ ∗ hh v, v ii = hh v, v ii . Let M ′ ⊂ S the set of real points. It follows that h v s , v s i ∈ R for all s ∈ M ′ and v s ∈ L s . Note thatfor functions f, g : U −→ C , we have hh f v, gv ii = f ( τ ∗ g ) hh v, v ii . Thus, up to sign, a holomorphic τ -sesquilinear form is a complexification of a hermitian metric over the locusof real points.Assume that there is a holomorphic involution η ∈ H ( S , End( T ∗ S ))such that η ( τ ∗ α ) = − τ ∗ η ( α ) for any α ∈ Ω , ( S ) . (2.16)Further, assume that the holomorphic line bundle L −→ S is given by a cocycle of the form { f i,j : U i,j −→ C ∗ } (2.17)where the derivatives df i,j are in the +1 eigenspaces of η , i.e., df i,j = ( df i,j + η ( df i,j ) ) . Under these assumptions we have the following:
ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 21
Lemma 2.18.
There is a unique holomorphic
Chern -connection ∇ on L −→ S with d ( hh σ, σ ii ) = hh∇ σ, σ ii + hh σ, ∇ σ ii (2.18) for any holomorphic section σ ∈ Γ( U , L ) and any τ -invariant open subset U ⊂ S .Proof. Take a collection of local trivializations by nowhere-vanishing holomorphic sections σ i ∈ Γ( U i , L )satisfying the condition that locally σ i = ev ∗ v i , where each open subset U i is τ -invariant, such that thetransition functions satisfy df i,j = η ( df i,j ), where f i,j = σ i σ j . Then, writing ∇ σ i = A i σ i , we have η ( A i ) = A i . The condition in (2.18) then implies that d log hh σ i , σ i ii = ( A i + τ ∗ A i ) hh σ i , σ i ii . By (2.16), we have η ( τ ∗ A i ) = − τ ∗ A i and thus A i = ( d log hh σ i , σ i ii + ηd log hh σ i , σ i ii ) . This shows the uniqueness of the connection.For the existence, set ∇ σ i = ( d log hh σ i , σ i ii + ηd log hh σ i , σ i ii ) ⊗ σ i . Arguing analogously as for the usual Chern connection, we see that this defines a connection with the desiredproperties. (cid:3)
The main example we have in mind is given by the space S of sections of a twistor space Z of a hyperk¨ahlermanifold M equipped with a hyperholomorphic hermitian line bundle L −→ M . Assume that there existsa global τ -sesquilinear form on L −→ S . Consider the decomposition in (1.15) for x = 0 ∈ C P , andthe involution which is − Id on T ev S and +Id on T ev ∞ S . Denote its dual endomorphism by η . It satisfies(2.16). Clearly, L admits a cocycle as in (2.17). Applying (2.18) we obtain a natural connection, which canbe shown to be the Atiyah–Ward connection. It would be interesting to find natural expressions for theholomorphic τ -sesquilinear form in concrete examples such as the Deligne–Hitchin moduli spaces; see alsoSection 3.3 below and the following remark. Remark 2.19.
The existence of a τ -sesquilinear form on L −→ S follows almost automatically from theexistence of an Atiyah–Ward type connection ∇ on L −→ S : assume that there exists a hermitian metric h on L | M −→ M over the real points M ⊂ S . Consider a local holomorphic frame σ ∈ Γ( U , L ) on asimply-connected τ -invariant open subset U ⊂ S such that there exists s ∈ M ∩ U . Write ∇ σ = α ⊗ σ. Then α + τ ∗ α is closed; its exterior derivative is a (2 , M , hence on its complexification S .Integrating this closed form on the simply connected set U produces the τ -sesquilinear form via hh ( µ σ ( p ) , µ σ ( τ ( p )) ii = µ µ h ( σ ( p ) , σ ( p )) exp (cid:18)Z ps α + τ ∗ α (cid:19) for all µ , µ ∈ C and p ∈ U .3. Space of holomorphic sections of the Deligne–Hitchin moduli space
In this section we illustrate the general theory described in § § Hitchin’s self-duality equations.
Let Σ be a compact Riemann surface of genus at least two; denoteits holomorphic cotangent bundle by K Σ . Consider a smooth complex rank n vector bundle E of degree zerowith structure group in SU( n ).Denote by A ( E ) the space of SU( n )-connections on E . Sending any ∇ ∈ A ( E ) to its (0 , ∂ ∇ wemay identify A with the space of holomorphic structures on E inducing the trivial holomorphic structure ∂ on the determinant bundle det E = V n E of E , i.e., (det E, ∂ ∇ ) = ( O Σ , ∂ ). Thus, A is an affine spacemodelled on Ω , (Σ , sl ( E )), where sl ( E ) is the bundle of trace-free endomorphisms of E . The product T ∗ A = A × Ω , (Σ , sl ( E ))can be thought of as the cotangent bundle of A . We will denote its elements by ( ∂ ∇ , Φ) and call Φ the
Higgsfield of the pair ( ∂, Φ). Formally, T ∗ A carries a flat hyperk¨ahler structure that can be described as follows.The Riemannian metric is given by the L -inner product on Ω (Σ , sl ( E )). The almost complex structures I, J, K = IJ act on a tangent vector ( α, φ ) ∈ T ( ∂ ∇ , Φ) T ∗ A = Ω , (Σ , sl ( E )) ⊕ Ω , (Σ , sl ( E )) as I ( α, φ ) = ( i α, i φ ) J ( α, φ ) = ( − φ ∗ , α ∗ ) . The K¨ahler forms are given by ω I (( α, φ ) , ( β, ψ )) = Z Σ tr( α ∗ ∧ β − β ∗ ∧ α + φ ∧ ψ ∗ − ψ ∧ φ ∗ ) (3.1)( ω J + i ω K )(( α, φ ) , ( β, ψ )) = 2 i Z Σ tr( ψ ∧ α − φ ∧ β ) . The group G := Γ(Σ , SU( E )) of unitary gauge transformations of E acts (on the right) on T ∗ A as( ∂ ∇ , Φ) .g = ( g − ◦ ∂ ∇ ◦ g, g − Φ g ) . (3.2)This action preserves the flat hyperk¨ahler structure, and, formally, the vanishing condition for the associatedhyperk¨ahler moment map yields Hitchin’s self-duality equations : F ∇ + [Φ ∧ Φ ∗ ] = 0 (3.3) ∂ ∇ Φ = 0 . The second equation implies that Φ ∈ H (Σ , sl ( E ) ⊗ K Σ ) with respect to the holomorphic structure ∂ ∇ .Note that these equations imply that the connection ∇ + Φ + Φ ∗ is flat.Let H ⊂ T ∗ A be the set of solutions of (3.3). The moduli space of solutions to the self-duality equationsis the quotient M SD := M SD (Σ , SL( n, C )) := H / G . Formally, it is the hyperk¨ahler quotient of T ∗ A by the action of G .A solution ( ∂ ∇ , Φ) to the self-duality equations (3.3) is called irreducible if its stabilizer in G is the centerof SU( n ). We write H irr ⊂ H for the set of irreducible solutions. It is known that M irr SD = H irr / G is the smooth locus of M SD . It is equipped with a hyperk¨ahler metric induced by the flat hyperk¨ahlerstructure on T ∗ A described above. Moreover, there is a rotating circle action on M SD given by ζ. ( ∂ ∇ , Φ) = ( ∂ ∇ , ζ Φ) , (3.4) ζ ∈ S . It is Hamiltonian with respect to ω I , and the map µ I : M SD −→ i R , µ I ( ∂ ∇ , Φ) = − Z Σ tr(Φ ∧ Φ ∗ ) (3.5)restricts to a natural moment map on M irr SD .The complex manifold ( M irr SD , I ) can be described as follows. An SL( n, C ) -Higgs bundle is a pair consistingof a holomorphic vector bundle ( E, ∂ E ), such that det( E, ∂ E ) = O Σ , together with a Higgs field Φ ∈ H (Σ , sl ( E ) ⊗ K Σ ). The group G C = Γ(Σ , SL( E )) acts on the set of Higgs bundles by the same ruleas in (3.2). A Higgs bundle ( ∂ E , Φ) is called stable if every Φ-invariant holomorphic subbundle E ′ has ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 23 negative degree, and it is called polystable if it is a direct sum of stable Higgs bundles of degree zero. Wedenote by M st Higgs := M st Higgs (Σ , SL( n, C )) and M ps Higgs := M ps Higgs (Σ , SL( n, C )) the moduli spaces of stableand polystable Higgs bundles respectively. It is known that the map ( ∂ ∇ , Φ) (( E, ∂ ∇ ) , Φ) induces abiholomorphism ( M irr SD , I ) = M st Higgs that extends to a homeomorphism M SD = M ps Higgs . The circle action described in (3.4) extends to a holomorphic C ∗ -action on M Higgs : ζ · ( ∂, Φ) = ( ∂, ζ Φ) . (3.6)While the complex structure I is induced from the complex structure on Σ, the complex structure J has amore topological origin. Consider the space A C of SL( n, C ) connections on E . The group G C acts on A C as ∇ .g = g − ◦ ∇ ◦ g. A connection ∇ ∈ A C is called irreducible if its stabilizer in G C is the center of SL( n, C ), and ∇ is called reductive if it is isomorphic to a direct sum of irreducible connections, i.e., if any ∇ -invariant subbundle E ′ ⊂ E admits a ∇ -invariant complement. Let M dR := M dR (Σ , SL( n, C ))be the moduli space of reductive flat SL( n, C )-connections on E . Its smooth locus is given by M irr dR , themoduli space of irreducible flat SL( n, C )-connections. Then the map ( ∂ ∇ , Φ) + Φ + Φ ∗ is a biholo-morphism ( M irr SD , J ) ∼ = M irr dR that extends to a homeomorphism M SD ∼ = M dR . The Riemann–Hilbert correspondence gives a biholomorphism M dR ∼ = Hom( π (Σ) , SL( n, C )) red / SL( n, C ) =: M B (Σ , SL( n, C )) =: M B ;the space M B is known as the Betti moduli space.3.2. The Deligne–Hitchin moduli space.
We now recall Deligne’s construction of the twistor space Z ( M SD ) via λ -connections; see [44].Let λ ∈ C . A holomorphic SL( n, C ) λ –connection on the SU( n )-vector bundle E −→ Σ is a pair ( ∂, D ),where ∂ is a holomorphic structure on E and D : Γ(Σ , E ) −→ Ω , (Σ , E ) is a differential operator such that • for any f ∈ C ∞ (Σ) we have D ( f s ) = λs ⊗ ∂f + f Ds , • D is holomorphic, i.e., ∂ ◦ D + D ◦ ∂ = 0, and • as a holomorphic vector bundle we have (Λ n E, ∂ ) = ( O Σ , ∂ Σ ), and the holomorphic differentialoperator on Λ n E induced by D coincides with λ∂ Σ .Therefore, an SL( n, C ) 0–connection is an SL( n, C )–Higgs bundle, and an SL( n, C ) 1–connection is an usualholomorphic SL( n, C )–connection. More generally, the operator ∂ E + λ − D ( λ ) is a holomorphic connectionfor every λ = 0.The group G C of complex gauge transformations of E acts on the set of holomorphic λ –connections in theusual way: ( ∂, D ) · g = ( g − ◦ ∂ ◦ g, g − ◦ D ◦ g ) . We again call ( ∂, D ) stable if any D -invariant holomorphic subbundle E ′ ⊂ E has negative degree, and( ∂, D ) is called polystable if it is isomorphic to a direct sum of stable λ -connections of degree zero (if λ = 0, then the degree of any λ -connection is automatically zero). Equivalently, ( ∂, D ) is stable if it isirreducible, i.e., its stabilizer in G C given by the center of SL( n, C ). The Hodge–moduli space M Hod := M Hod (Σ , SL( n, C )) is the moduli space of polystable holomorphic λ -connections, where λ varies over C : M Hod := M Hod (Σ , SL( n, C )) = { ( ∂, D, λ ) | λ ∈ C , ( ∂, D ) polystable holomorphic λ -connection } / G C . It has a natural holomorphic projection to C given by ̟ : M Hod −→ C , [ ∂, D, λ ] λ. (3.7) The smooth locus M irr Hod of the Hodge moduli space M Hod coincides with the locus of irreducible λ -connections.The C ∗ -action in (3.6) extends to a natural C ∗ -action on M Hod covering the standard C ∗ -action on C by ζ · ( ∂, D, λ ) = ( ∂, ζD, ζλ ) . The map ( ∂, D, λ ) ( ∂ + λ − D, λ ) induces a biholomorphism ̟ − ( C ∗ ) ∼ = M dR × C ∗ ∼ = M B × C ∗ , where ̟ is the map in (3.7).The Deligne–Hitchin moduli space M DH := M DH (Σ , SL( n, C )) is obtained by gluing M Hod and M Hod ∼ = M Hod (Σ , SL( n, C )) over C ∗ via the Riemann–Hilbert correspondence: M DH := M DH (Σ , SL( n, C )) = ( M Hod (Σ , SL( n, C )) ˙ ∪ M Hod (Σ , SL( n, C ))) / ∼ , where [( ∂, D, λ )] ∼ [( λ − D, λ − ∂, λ − )]for any [( ∂, D, λ )] ∈ M Hod with λ = 0.The projections from the respective Hodge moduli spaces to C glue to give a holomorphic projection ̟ : M DH −→ C P . The smooth locus of M DH coincides with the locus M irr DH of irreducible λ -connections which in turn coincideswith the twistor space Z ( M irr SD ) of the hyperk¨ahler manifold M irr SD .The space M B × C ∗ admits an anti-holomorphic involution τ M B covering the antipodal involution λ λ − of C ∗ which is constructed as follows. For any ρ ∈ Hom( π (Σ) , SL( n, C )), define τ M B ( ρ, λ ) = ( ρ − T , − λ − ) . This produces the following antiholomorphic involution τ M DH of M DH : τ M DH : M DH −→ M DH , [( ∂, D, λ )] [( λ − D ∗ , − λ − ∂ ∗ , − λ − )] = [( ∂ ∗ , − D ∗ , − λ )] . This involution τ M DH is compatible with the C ∗ -action in the following sense. For ζ ∈ C ∗ , τ M DH ( ζ. ( ∂, D, λ )) = ζ − .τ M DH ( ∂, D, λ ) . The C ∗ -action on M Hod extends to a C ∗ -action on M DH . For any ζ ∈ C ∗ , we have ζ · [ ∂, D ( λ ) , λ ] = [ ∂, ζD ( λ ) , ζλ ] . Clearly, this action covers the natural C ∗ -action on C P . Therefore the C ∗ -fixed points of M DH are givenby M C ∗ DH ∼ = M Higgs (Σ , SL( n, C )) C ∗ ∐ M Higgs (Σ , SL( n, C )) C ∗ , i.e., by the locus of complex variations of Hodge structures on Σ and Σ (see [46]).The twisted relative symplectic form on M irr DH can be described as follows. Let V i = ( ˙ ∂ i , ˙ D i ), i = 1 , ̟ − ( λ ). Then ω λ ( V , V ) = 2 i Z Σ tr( ˙ D ∧ ˙ ∂ − ˙ D ∧ ˙ ∂ ) . (3.8)Note that at λ = 0 this exactly resembles ω J + i ω K as defined in (3.1). ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 25
The line bundle on M irr DH . Let us describe the holomorphic line bundle L Z for Z = M irr DH alongeach fiber of ̟ . First consider the moduli space of holomorphic SL( n, C )-connections M dR . Let M ′ dR bethe Zariski open subset of M irr dR such that the underlying holomorphic bundle is stable. So we have a map f : M ′ dR −→ N , where N is the moduli space of stable SL( n, C )-bundles, that sends any (( E, ∂ E ) , ∇ E ) to ( E, ∂ E ).The pullback map f ∗ : Pic( N ) −→ Pic( M ′ dR ) is an isomorphism [8]. On the other hand, Pic( M ′ dR ) = Z ,and holomorphic line bundles on M ′ dR are uniquely determined by their first class [14]. Also the restrictionmap Pic( M irr dR ) −→ Pic( M ′ dR ) is an isomorphism, because the codimension of the complement of M ′ dR inside M irr dR is at least two. Therefore, we have Pic( M irr dR ) = Pic( N ) ∼ = Z , i.e., the holomorphic linebundles on M irr dR are uniquely determined by their first Chern class.Let L denote the restriction of L Z to Z = M irr dR . On L , the meromorphic connection ∇ on L Z inducesa holomorphic connection and its curvature on Z is the holomorphic symplectic form 2 ω J + 2 i ω I , which iscohomologous to 2 i ω I , since ω J is exact. So by Chern–Weil theory, we have c ( L ) = [ ω I ], the cohomologyclass of ω I .It follows that L is the holomorphic line bundle on Z determined by [ ω I ]. To calculate [ ω I ], note thatthe above projection f has a C ∞ -section that sends any stable bundle ( E, ∂ E ) to the unique unitary flatconnection on E [41]. The restriction of ω I to the image of that section coincides with the standard K¨ahlerform on N . On the other hand, the first Chern class of the determinant line bundle ξ on N is the K¨ahlerform [43]. In fact, the curvature for the Quillen metric on ξ is the K¨ahler form.We conclude that L is holomorphically isomorphic to the determinant bundle on Z , which coincideswith the pullback of the determinant line bundle on N because the determinant line bundle is functorial.Similarly, the line bundles on the moduli space of stable Higgs bundles, namely Z = M st Higgs , are uniquelydetermined by their first Chern class. Let L denote the restriction of L Z to Z . Since L is isomorphic to thedeterminant line bundle, and since the first Chern class of the family of line bundles L t −→ Z t , t ∈ C P ,is independent of t , it follows that L is isomorphic to the determinant line bundle. For a description of L Z on all of Z = M irr DH , see [29, § Irreducible and admissible sections.
In this subsection we recall some concepts and definitionsfrom [6] and [4] on sections of the twistor projection ̟ : M DH −→ C P . We write S M DH for the space ofholomorphic sections. Since M DH is a complex space, a similar argument as in Proposition 1.1 shows that S M DH is a complex space as well. It is equipped with an antiholomorphic involution τ defined in (1.8), with τ M DH playing the role of τ Z . Definition 3.1.
A holomorphic section s ∈ S M DH is irreducible if the image of s is contained in M irr DH .These are precisely the sections of the twistor space of M irr SD .If s : C P −→ M DH is an irreducible section, then by [6, Lemma 2.2] (see also [4, Remark 1.11]) itadmits a holomorphic lift b s ( λ ) = ( ∂ ( λ ) , D ( λ ) , λ ) = ( ∂ + ∞ X j =1 λ j Ψ j , λ∂ + Φ + ∞ X j =2 λ j Φ j , λ ) , λ ∈ C (3.9)to the space of holomorphic λ -connections of class C k . Here Ψ , Ψ k ∈ Ω , ( sl ( E )) , Φ , Φ k ∈ Ω , ( sl ( E )).There is also a lift − b s on C P \ { } .The lifts b s, − b s over C and C P \{ } respectively, allow us to interpret s | C ∗ as a C ∗ -family of flat connections + ∇ ∇ = ∂ ( λ ) + λ − D ( λ ) = λ − Φ + ∇ + . . . , and − ∇ λ defined similarly over C P \ { , ∞} . Here we write ∇ = ∂ + ∂ using the notation of equation(3.9). There exists a holomorphic C ∗ -family g ( λ ) of GL( n, C )-valued gauge transformations, unique up tomultiplication by a holomorphic scalar function, such that + ∇ λ .g ( λ ) = − ∇ λ (see [6]). Definition 3.2.
We call a holomorphic section s ∈ S M DH admissible if it admits a lift b s on C of the form b s ( λ ) = ( ∂ + λ Ψ , λ∂ + Φ , λ )for a Dolbeault operator ∂ of type (0 , ∂ of type (1 , , , ∂, Φ) and ( ∂, Ψ) are semi-stable Higgs pairs on Σ and Σ respectively.
Now suppose that s ∈ S τ M DH is a real section, i.e., s = τ M DH ◦ s ◦ τ C P . If we have a lift ∇ λ on C ⊂ C P of s , then for every λ ∈ C ∗ there is a gauge transformation g ( λ ) ∈ G C such that ∇ λ .g ( λ ) = ∇ − λ − ∗ . Remark 3.3.
If ( ∂ ∇ , Φ) is a solution to the self-duality equations, then the associated twistor line is givenby the C ∗ -family of flat SL( n, C )-connections ∇ λ = λ − Φ + ∇ + λ Φ ∗ h . By the non-abelian Hodge correspondence, this family gives rise to an equivariant harmonic map f : e Σ −→ SL( n, C ) / SU( n ) from the universal cover. If the solution ( ∂ ∇ , Φ) is irreducible, then so is the associatedsection in S τ M DH .In the next example we describe a large class of irreducible sections for n = 2. Example 3.4.
Let ∇ be an irreducible flat SL(2 , C )-connection on the rank two bundle V −→ Σ. Weassume that ∂ ∇ is not strictly semi-stable, but either stable or strictly unstable. We also assume theanalogous condition for the holomorphic structure ∂ ∇ on Σ. Due to [7] this assumption does not hold forall irreducible flat SL( n, C )-connections. Under the assumption, we obtain a section s = s ∇ as follows. If ∂ ∇ is stable λ ( λ, ∂ ∇ , λ∂ ∇ )is an irreducible section over C ⊂ C P . If ∂ ∇ is unstable, we consider its destabilizing subbundle L ⊂ V of positive degree. The connection induces a nilpotent Higgs field Φ on the holomorphic vector bundle L ⊕ ( V /L ) = L ⊕ L ∗ via Φ = π V/L ◦ ∇ | L . This is a special case of [45] and can be interpreted from a gauge theoretic point of view (see also [4, §
4] fordetails): Consider a complementary bundle e L ⊂ V of L , and the family of gauge-transformations g ( λ ) = (cid:18) λ (cid:19) . The family λ ( λ, ∂ ∇ .g ( λ ) , ∂ ∇ .g ( λ ) )extends to an irreducible (stable) Higgs pair at λ = 0 which identifies with ( L ⊕ L ∗ , Φ)4.
Energy functional on sections of the Deligne–Hitchin moduli space
The energy as a moment map.
It was proven in [4, Corollary 3.11] that the energy of an irreduciblesection s with lift b s as in (3.9) is given by E ( s ) = π i Z Σ tr(Φ ∧ Ψ) . (4.1)In particular, this integral is independent of the lift b s . The reader should be aware of the different prefactorsin (4.1) and in (3.5). In particular, if we think of E as the energy of a harmonic map, it should be real-valued,while we want a moment map for the S -action to be i R -valued. Working with the prefactor π i also hasthe advantage that we get fewer factors of 2 π i in the statements of the results below. Remark 4.1.
As pointed out in [6, Remark 2.3], the energy in the present example is defined for all localsections around λ = 0 which admit a lift as in (3.9).Let us write again S ′ = S ′M DH for the space of irreducible sections whose normal bundle is isomorphicto O C P (1) ⊕ d . Take any s ∈ S ′ . In terms of lifts of sections, a tangent vector V ∈ T s S ′ is expressed asfollows. Let b s be a lift of s as in (3.9), and denote the curvature of the connection ∂ + ∂ by F ∂ + ∂ = ∂∂ + ∂∂ .Expanding the integrability condition ∂ ( λ ) D ( λ ) + D ( λ ) ∂ ( λ ) = 0 (4.2) ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 27 in powers of λ , the zeroth and first order coefficients yield ∂ Φ = 0 (4.3) F ∂ + ∂ + [Φ ∧ Ψ] = 0 . Consider a family of sections ( s t ∈ S M DH ) t with s = s which represents V ∈ T s S ′ . The corresponding(lifted) infinitesimal variation ˙ b s = ( ˙ ∂ ( λ ) , ˙ D ( λ ) , λ ) satisfies the linearisation of (4.2), i.e., ∂ ( λ )( ˙ D ( λ )) + D ( λ )( ˙ ∂ ( λ )) = 0 . (4.4)Expanding ˙ b s into a power series ˙ b s ( λ ) = ∞ X k =0 ψ k λ k , ∞ X k =0 ϕ k λ k , λ ! , for ϕ k ∈ Ω , ( sl ( E )) , ψ k ∈ Ω , ( sl ( E )), the linearisation of (4.3) becomes ∂ϕ + [ ψ ∧ Φ] = 0 ∂ϕ + ∂ψ + [ ϕ ∧ Ψ] + [Φ ∧ ψ ] = 0 . Variations along the gauge orbit of b s are determined by infinitesimal gauge transformations C ∋ λ ξ ( λ ) ∈ Γ(Σ , sl ( E )) and are of the form ( ∂ ( λ ) ξ ( λ ) , D ( λ ) ξ ( λ ) , λ ) . (4.5)By expanding ξ ( λ ) = P ∞ k =0 ξ k λ k , we get with (4.5) and (3.9) ∂ ( λ ) ξ ( λ ) = ∂ξ + ( ∂ξ + [Ψ , ξ ]) λ + O ( λ ) D ( λ ) ξ ( λ ) = [Φ , ξ ] + ( ∂ξ + [Φ , ξ ]) λ + O ( λ ) . Now let s ∈ S ′ with lift b s over C , and consider V j ∈ T s S ′ , j = 1 ,
2, represented by˙ b s j = ( ˙ ∂ j ( λ ) , ˙ D j ( λ ) , λ ) = ( ψ ( j )0 + ψ ( j )1 λ, ϕ ( j )0 + ϕ ( j )1 λ, λ ) + O ( λ ) . Then we define, recalling the definition of ω λ given in (3.8), b Ω b s ( V , V ) = − i ∂∂λ | λ =0 ω λ ( V ( λ ) , V ( λ ))= − i ∂∂λ | λ =0 i Z Σ tr (cid:16) − ˙ D ( λ ) ∧ ˙ ∂ ( λ ) + ˙ D ( λ ) ∧ ˙ ∂ ( λ ) (cid:17) = Z Σ tr (cid:16) − ϕ (1)0 ∧ ψ (2)1 + ϕ (2)0 ∧ ψ (1)1 − ϕ (1)1 ∧ ψ (2)0 + ϕ (2)1 ∧ ψ (1)0 (cid:17) . We view b Ω as a two-form on the infinite-dimensional space of germs of sections of ̟ at λ = 0. Note thatthe formula for b Ω is exactly (1.21) in the present context.
Proposition 4.2.
The two-form b Ω descends to a holomorphic two-form on the space of irreducible sections,which on S ′M DH coincides with the holomorphic symplectic form Ω defined in (1.21) .Proof. We will show that b Ω b s is degenerate along the gauge orbits. To this end, let b s be a germ of a sectionnear λ = 0, and let ξ ( λ ) = P ∞ k =0 ξ k λ k be an infinitesimal gauge transformation. The corresponding tangentvector V is represented by ˙ b s = ( ∂ ( λ ) ξ ( λ ) , D ( λ ) ξ ( λ ) , λ ) . Then for an arbitrary tangent vector V represented by ˙ b s = ( ˙ ∂ ( λ ) , ˙ D ( λ ) , λ ), we find b Ω b s ( V , V ) = ∂∂λ | λ =0 Z Σ tr (cid:16) − ˙ D ( λ ) ∧ ∂ ( λ ) ξ ( λ ) + D ( λ ) ξ ( λ ) ∧ ˙ ∂ ( λ ) (cid:17) (Stokes) = ∂∂λ | λ =0 Z Σ tr (cid:16) ∂ ( λ )( ˙ D ( λ )) + D ( λ )( ˙ ∂ ( λ ))) ξ ( λ ) (cid:17) = 0;we used (4.4). This shows that b Ω descends to S ′ . (cid:3) Theorem 2.3 thus allows us to make the following conclusion.
Corollary 4.3.
The restriction of π i E : S ′M DH −→ C is a holomorphic moment map for the natural C ∗ -action on S ′M DH with respect to the holomorphic symplectic form Ω . In particular, the C ∗ -orbits in S ′M DH are exactly the critical points of E| S ′M DH . Explicit description of some C ∗ -fixed sections. Corollary 4.3 shows a close relationship between C ∗ -orbits in S M DH and the energy functional. We therefore examine the C ∗ -orbits more closely in thissection. Before explicitly determining the C ∗ -fixed irreducible sections, we first observe: Lemma 4.4.
The set S C ∗ M DH of all C ∗ -fixed sections is in a natural bijection with M dR , the moduli space offlat completely reducible SL( n, C ) -connections.In particular, the critical points of E : S ′M DH −→ C correspond to an open subset of M irr dR , the modulispace of flat irreducible SL( n, C ) -connections.Proof. Let ∇ ∈ M dR . As in Section 2.3, we obtain the following C ∗ -invariant section s ∇ : C ∗ −→ M DH : s ∇ ( λ ) = [( ∂ ∇ , λ∂ ∇ , λ )] , ∂ ∇ = ∇ , , ∂ ∇ = ∇ , . (4.6)By a crucial result of Simpson ([44] for existence and [45] for a more explicit approach), the limits of s ∇ ( λ )for λ → λ ∞ always exist in M Higgs (Σ , SL( n, C )) and M Higgs (Σ , SL( n, C )) respectively. The resultingsection, also denoted by s ∇ ∈ M DH , is C ∗ -invariant by continuity. Evaluation of sections s : C P −→ M DH at λ = 1 gives the inverse of the map ∇ 7−→ s ∇ .The last statement in the lemma is a direct consequence of Theorem 2.3 and Corollary 4.3. (cid:3) We next determine explicitly the C ∗ -fixed sections s ∈ S M DH such that s is irreducible over C , by usingsome results of [10]. In terms of Lemma 4.4, these are precisely the sections s ∇ such that s ∇ (0) is stable.Indeed, since irreducibility is an open condition, s ∇ ( λ ) is an irreducible λ -connection for λ close to 0. Usingthe C ∗ -invariance, we see that s ( λ ) is irreducible for every λ ∈ C .For any C ∗ -fixed sections s ∇ , its values at 0 and ∞ are C ∗ -fixed Higgs bundles on Σ and Σ respectively.These are called complex variations of Hodge structures (VHS). Let ( ∂, Φ) be any VHS on Σ. The fact that( ∂, Φ) is a C ∗ -fixed point yields a splitting E = l M j =1 E j (4.7)into a direct sum of holomorphic bundles. With respect to this splitting, ∂ and Φ are given in the followingblock form ∂ = ∂ E . . . . . . ∂ E . . . ...... . . . . . . . . . ...... . . . . . . 00 . . . . . . ∂ E l , Φ = . . . . . . . . . (1) . . . ...0 Φ (2) . . . ...... . . . . . . . . . ...0 . . . ( l − . (4.8)where Φ ( j ) ∈ H (Σ , Hom( E j , E j +1 ) ⊗ K Σ ). The sheaf sl ( E ) of trace-free holomorphic endomorphisms of E further decomposes into sl ( E ) = M k ∈ Z sl ( E ) k , sl ( E ) k = { ψ ∈ sl ( E ) | ψ ( E i ) ⊂ E i − k } . By construction, Φ ∈ H (Σ , K Σ ⊗ sl ( E ) − ). To define the next notion, let N + = M k> sl ( E ) k , N − = M k< sl ( E ) k , L = sl ( E ) . (4.9)Note that N + (respectively, N − ) is the subspace of sl ( E ) consisting of endomorphisms of E that are strictlyupper (respectively, lower) block-triangular with respect to the splitting (4.7), while L is the space of block-diagonal elements of sl ( E ). ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 29
Now let ( ∂, Φ) ∈ M Higgs (SL( n, C )) be a stable complex variation of Hodge structures. Then the BB-slice([10, Definition 3.7]) through ( ∂, Φ) is defined by B +( ∂, Φ) = { ( β, φ ) ∈ Ω , ( N + ) ⊕ Ω , ( L ⊕ N + ) | D ′′ ( β, φ ) + [ β ∧ φ ] = 0 , D ′ ( β, φ ) = 0 } . (4.10)Here we denote by D := ∂ + ∂ h + Φ + Φ ∗ h the non-abelian Hodge connection attached to ( ∂, Φ) with harmonic metric h , and D ′′ := ∂ + Φ , D ′ := ∂ h + Φ ∗ h . Hence the equations in (4.10) are explicitly given by D ′′ ( β, φ ) + [ β ∧ φ ] = ∂φ + [(Φ + φ ) ∧ β ] = 0 , D ′ ( β, φ ) = ∂ h β + [Φ ∗ h ∧ φ ] = 0 . (4.11)Note that B +( ∂, Φ) is a finite-dimensional affine space. Then, [10, Theorem 1.4 (3)] states that the map p : B +( ∂, Φ) × C −→ M Hod , (( β, φ ) , λ ) [ λ, ∂ + λ Φ ∗ + β, λ∂ h + Φ + φ ]is a holomorphic embedding onto the “attracting set” W ( ∂, Φ) = { m ∈ M irr Hod | lim ζ → ζ · m = ( ∂, Φ) } and is compatible with the obvious projections to C . In particular, if W λ ( ∂, Φ) denotes the intersectionof W ( ∂, Φ) with the fiber ̟ − ( λ ), then W λ ( ∂, Φ) is biholomorphic to the affine space B +( ∂, Φ) via the map p λ := p ( • , λ ). Thus, M irr Hod is stratified by affine spaces.Given ( β, φ ) ∈ B +( ∂, Φ) , we can use Lemma 4.4 and (4.6) to define the C ∗ -fixed section s ( β,φ ) := s p ( β,φ ) ∈ S M DH . As observed earlier, s ( β,φ ) is an irreducible section over C ⊂ C P but not necessarily over all of C P . Proposition 4.5.
Over C , the C ∗ -fixed section s ( β,φ ) may be expressed as s ( β,φ ) ( λ ) = λ, ∂ + λ (Φ ∗ h + β ) + l X j =2 λ j β j , Φ + λ∂ h + l X j =0 λ j +1 φ j , (4.12) where β = P lj =1 β j , with β j ∈ Ω , ( sl ( E ) j ) and φ = P j =0 φ j with φ j ∈ Ω , ( sl ( E ) j ) .Proof. Let ∇ = p ( β, φ ) = [ D + β + φ ] so that ∂ ∇ = ∂ + Φ ∗ h + β, ∂ ∇ = ∂ h + Φ + φ. Hence s ( β,φ ) = s ∇ is given by s ( β,φ ) ( λ ) = [ λ, ∂ + (Φ ∗ h + β ) , λ∂ h + λ Φ + λφ ]for λ ∈ C ∗ (see (4.6)). This does not give a lift of s ( β,φ ) over all of C , unless the holomorphic bundle ( E, ∂ )is stable, in which case we must have β = 0 and Φ = 0.To construct a lift over all of C we use the C ∗ -family of gauge transformations g ( λ ) = λ m λ − l id E . . . . . . λ − l id E . . . ...... 0 . . . . . . ...... . . . . . . 00 . . . . . . λ id E l , (4.13)where m = n P lj =1 ( l − j )rk( E j ) in order to ensure det g ( λ ) = 1. Then any ξ ∈ sl ( E ) j satisfies g ( λ ) − ξg ( λ ) = λ j ξ. Let β = P lj =1 β j , with β j ∈ Ω , ( sl ( E ) j ), and similarly φ = P j =0 φ j with φ j ∈ Ω , ( sl ( E ) j ). Then usingΦ ∈ H ( K ⊗ sl ( E ) − ) and Φ ∗ h ∈ Ω , ( K ⊗ sl ( E ) ), we get that( ∂ + (Φ ∗ h + β ) , λ∂ h + λ Φ + λφ ) .g ( λ ) = ∂ + λ (Φ ∗ h + β ) + l X j =2 λ j β j , Φ + λ∂ h + l X j =0 λ j +1 φ j . The result follows. (cid:3)
We next discuss the implications for the C ∗ -fixed leaves of the foliation F + on S ′ = S ′M DH . Recall thatthese leaves consist, in particular, of irreducible sections (on all of C P ) by definition. We denote by S ′ ( ∂, Φ) all sections in S ′ which pass through the stable complex variation of Hodge structure ( ∂, Φ) ∈ M C ∗ Higgs at λ = 0. Proposition 4.6.
The C ∗ -fixed point locus ( S ′ ( ∂, Φ) ) C ∗ is isomorphic to an open and non-empty subset of theaffine space B +( ∂, Φ) .Proof. Consider the section s ( β,φ ) : C P −→ M DH for ( β, φ ) ∈ B +( ∂, Φ) which is irreducible over C . Sincethe complement of M irr Higgs (Σ , SL( n, C )) in M Higgs (Σ , SL( n, C )) is closed and of codimension at least two(cf. [17]), it follows that s ( β,φ ) is an irreducible section for ( β, φ ) ∈ B +( ∂, Φ) in an open and dense subset of B +( ∂, Φ) .Note that ( β, φ ) = (0 ,
0) corresponds to the twistor line s ( ∂, Φ) through ( ∂, Φ), which lies in S ′ . Since S ′ is open and non-empty in the space of all irreducible sections, we therefore see that the irreducible and C ∗ -fixed section s ( β,φ ) has the desired normal bundle for ( β, φ ) in an open and non-empty subset U ⊂ B +( ∂, Φ) .Altogether we obtain the isomorphism p − ◦ ev : ( S ′ ( ∂, Φ) ) C ∗ ∼ = −→ U . (cid:3) From Theorem 2.3, we immediately obtain:
Corollary 4.7.
The locus of critical points s ∈ S ′ of E : S ′ −→ C is isomorphic to an open and non-emptysubset in M irr dR . It is foliated by leaves which are isomorphic to open and non-empty subsets of affine spaces.Proof. The first statement follows, by a genericity argument, from Lemma 4.4. The second one is a conse-quence of Proposition 4.6. (cid:3)
Remark 4.8.
Let s : C P −→ M DH be a C ∗ -fixed section such that s (0) = ( ∂, Φ) and s ( ∞ ) = ( ∂, Ψ)are stable
VHS on Σ and Σ respectively, with respective splittings of the underlying smooth bundle E of theform E = l M j =1 E j , E = l ′ M j =1 E ′ j . With respect to these splittings the respective holomorphic structures are diagonal and the Higgs fields Φand Ψ are lower triangular as in (4.8). Then we have the BB-slices B +( ∂, Φ) (Σ) and B +( ∂, Ψ) (Σ). By Proposition4.6 and its analog on Σ we see that, on the one hand, s corresponds to ( β, φ ) ∈ B +( ∂, Φ) (Σ), and on the otherhand to ( e β, e φ ) ∈ B +( ∂, Ψ) (Σ). Therefore, we obtain two distinguished lifts of s over C and C ∗ ∪ {∞} of theform s ( λ ) = [ λ, b s ( β,φ ) ( λ )] Σ = λ, ∂ + λ (Φ ∗ h + β ) + l X j =2 λ j β j , Φ + λ∂ h + l X j =0 λ j +1 φ j Σ ,s ( λ ) = [ λ − , b s ( e β, e φ ) ( λ − )] Σ = λ − , ∂ + λ − (Ψ ∗ e h + e β ) + l ′ X j =2 λ − j e β j , Ψ + λ − ∂ e h + l ′ X j =0 λ − ( j +1) e φ j Σ . Let g be a gauge transformation such that( ∂ + ∂ e h + Ψ + Ψ ∗ e h + e β + e φ ) .g = ∂ + ∂ h + β + φ. ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 31
Going through the proof of Proposition 4.5 and writing g ( λ ) and e g ( λ − ) for the respective C ∗ -families ofgauge transformations we get that b s ( β,φ ) ( λ ) = b s ( e β, e φ ) ( λ − ) . e g ( λ − ) − g g ( λ )for any λ ∈ C ∗ .In general, starting only with the lift b s ( β,φ ) over C obtained above, it seems hard to determine explicitlythe lift b s ( e β, e φ ) ( λ − ) over C P \ { } or even the limiting VHS s ( β,φ ) ( ∞ ). The next two examples discuss somesituations in which the limit can be computed. Example 4.9.
Suppose the holomorphic structure ∂ h + Φ + φ is stable on Σ. Then we can argue as follows.For λ ∈ C ∗ we can write, using the Deligne gluing: s ( β,φ ) ( λ ) = [ λ, ∂ + Φ ∗ h + β, λ ( ∂ h + Φ + φ )] Σ = [ λ − , ∂ h + Φ + φ, λ − ( ∂ + Φ ∗ h + β )] Σ . Under our assumption that ∂ h + Φ + φ is stable, this allows us to conclude s ( β,φ ) ( ∞ ) = ( ∂ h + Φ + φ, Example 4.10.
Consider the rank two case, n = 2. If s is the twistor line through a VHS ( ∂, Φ) on Σ,then we have E = V ⊕ V ∗ , where V is a line bundle with 0 < deg V ≤ g − V ∗ = ker Φ. Then s ( ∞ ) = ( ∂ h , Φ ∗ h ) and the corresponding splitting is E = V ∗ ⊕ V . Note that, since Σ and Σ come withopposite orientations, we have deg V ∗ >
0, as a bundle on Σ. Then e g ( λ − ) = g ( λ ) in this case, as the orderis reversed. The associated lifts are thus just the lifts of s over C and C ∗ given by the harmonic metric, i.e.the associated solution of the self-duality equations. Example 4.11 (Grafting sections) . In [24] a special class of C ∗ -invariant sections of M DH (Σ , SL(2 , C )),called grafting sections , have been constructed by using grafting of projective structures on Σ. We recoverthem from the previous proposition as follows.Consider the C ∗ -fixed stable Higgs bundle ( ∂, Φ) with E = K Σ ⊕ K − Σ , Φ = (cid:18) (cid:19) where K Σ is a square root of the canonical bundle K Σ . To determine (4.9) in this example, we define E := K Σ , E := K − Σ . Then we see that N + ∼ = K Σ , N − = K − , L ∼ = O Σ By (4.11), (0 , φ ) ∈ B +( ∂, Φ) if and only if ∂φ = 0 and [ φ ∧ Φ ∗ h ] = 0. Hence φ is of the form φ = (cid:18) q (cid:19) , q ∈ H (Σ , K ⊗ ) , with respect to the splitting E = E ⊕ E . For those q such that the monodromy of the correspondingflat connection at λ = 1 is real, the sections s (0 ,φ ) are precisely the grafting sections of [24, § β = 0 in this case, we see that the energy of a grafting section is the same as the energy of the twistor lineassociated with the stable Higgs pair ( ∂, Φ). If the monodromy of the corresponding flat connection is real,then [24] shows that the section s (0 ,φ ) is real and defines an element of ( S ′M DH ) τ , in particular it has thecorrect normal bundle O C P (1) ⊕ d . But the section s (0 ,φ ) is not admissible and thus cannot correspond toa solution of the self-duality equations. This shows that M SD (Σ , SL(2 , C )) ( ( S ′M DH ) τ .4.3. The energy of a C ∗ -fixed section. Proposition 4.5 gives concrete formulas for all C ∗ -fixed points s ∈ S C ∗ M DH such that s (0) is a stable VHS. We next compute the energy of such sections. Proposition 4.12.
Let ( ∂, Φ) be a stable C ∗ -fixed SL( n, C ) -Higgs bundle, and let s ( β,φ ) be the C ∗ -fixedsection corresponding to ( β, φ ) ∈ B +( ∂, Φ) . Its energy is given by E ( s ( β,φ ) ) = E ( s ) = l X k =2 ( k −
1) deg( E k ) , where s is the twistor line through ( ∂, Φ) . Proof.
Write s ( β,φ ) in a form as in (4.12). Then the definition of E immediately implies that E ( s ( β,φ ) ) = E ( s ) + π i Z Σ tr(Φ ∧ β ) . Next we will show that R Σ tr(Φ ∧ β ) = 0. To this end, let us writeΦ = l − X k =1 Φ ( k ) , β = l − X k =1 β ( k ) , where Φ ( k ) ∈ Ω , (Hom( E k , E k +1 ) , β ( k ) ∈ Ω , (Hom( E k +1 , E k ); see the block form in (4.8). It followsthat tr(Φ ∧ β ) = l X k =1 tr E k (Φ ( k − ∧ β ( k − ) . Note that each summand Φ ( k − ∧ β ( k − ) belongs to Ω , (End( E k )) and we have adopted the conventionthat Φ ( k ) = 0 = β ( k ) if k = 0 , l .Now, equation (4.11) implies that ∂φ + [Φ ∧ β ] = 0and we can write [Φ ∧ β ] = l − X k =1 Φ ( k − ∧ β ( k − + β ( k ) ∧ Φ ( k ) . Thus, for each k = 1 , · · · , l , ∂φ ( k )0 + Φ ( k − ∧ β ( k − + β ( k ) ∧ Φ ( k ) = 0 . Consider the case of k = l : ∂φ ( l )0 + Φ ( l − ∧ β ( l − = 0 . Taking the trace of this equation and integrating over Σ, we find, using Stokes’ theorem, that Z Σ tr E l (Φ ( l − ∧ β ( l − ) = 0 . Now assume that R Σ tr E k +1 (Φ ( k ) ∧ β ( k ) ) = 0 for all k ≥ k . Then we have ∂φ ( k )0 + Φ ( k − ∧ β ( k − + β ( k ) ∧ Φ ( k ) = 0 . Taking the trace and integrating yields0 = Z Σ tr E k (Φ ( k − ∧ β ( k − + β ( k ) ∧ Φ ( k ) )= Z Σ tr E k (Φ ( k − ∧ β ( k − − Z Σ tr E k (Φ ( k ) ∧ β ( k ) )= Z Σ tr E k (Φ ( k − ∧ β ( k − ) . It follows inductively that R Σ tr(Φ ∧ β ) = 0.It remains to compute the energy of the twistor line s . To this end, we observe that E ( s ) = π i Z Σ tr(Φ ∧ Φ ∗ h ) = π i Z Σ l X k =2 tr E k (Φ ( k − ∧ (Φ k − ) ∗ h ) = l X k =2 E k ( s ) , where we put E k ( s ) = π i R Σ tr E k (Φ ( k − ∧ (Φ k − ) ∗ h ) for k ≥
2. The equation F ∇ h + [Φ ∧ Φ ∗ h ] = 0 isblock-diagonal with respect to the splitting E = L lk =1 E k , with components F ∇ hEk + Φ ( k − ∧ (Φ ( k − ) ∗ h + (Φ ( k ) ) ∗ h ∧ Φ ( k ) = 0 . ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 33
This gives the following recursive relations: E k ( s ) = π i Z Σ tr E k (Φ ( k − ∧ (Φ ( k − ) ∗ h )= i π Z Σ tr E k ( F ∇ hEk ) + π i Z Σ tr E k +1 (Φ ( k ) ∧ (Φ ( k ) ) ∗ h )= deg( E k ) + E k +1 ( s ) . Thus, if k = l , we find that E l ( s ) = deg( E l ) , and for general k we get that E k ( s ) = l − X j = k deg( E j ) + E l ( s ) = l X j = k deg( E j ) . Therefore, E ( s ) = l X k =2 E k ( s ) = l X k =2 l X j = k deg( E j ) = l X k =2 ( k −
1) deg( E k ) , and this completes the proof. (cid:3) The second variation of the Energy at a C ∗ -fixed section. Next we study the second variationof the energy functional E at a C ∗ –fixed point.Examining the proof of Proposition 4.5, we can check explicitly that the sections s ( β,φ ) satisfy for any ζ ∈ C ∗ the relation ζ. b s ( β,φ ) = b s ( β,φ ) .g ( ζ ) − . Moreover, if we use the notation of equation (4.13) and put ξ = ( m + 1 − l )id E . . . . . .
00 ( m + 2 − l )id E . . . ...... 0 . . . . . . ...... . . . . . . 00 . . . . . . m id E l , then [ ξ, · ] acts as multiplication by k on sl ( E ) k and we see that − i λ ddλ ∂ ( λ ) = ∂ ( λ ) ξ ( λ ) , i D ( λ ) − i λ ddλ D ( λ ) = D ( λ ) ξ ( λ ) (4.14)with ( ∂ ( λ ) , D ( λ )) = ∂ + λ (Φ ∗ h + β ) + l X j =2 λ j β j , Φ + λ∂ h + l X j =0 λ j +1 φ j . For ξ ( λ ) = P ∞ k =0 ξ k λ k we deduce from (4.14) the following equations0 = ∂ξ Φ = [Φ , ξ ] (4.15) − Ψ = ∂ξ + [Ψ , ξ ] 0 = [Φ , ξ ] + ∂ξ . We can now compute the second variation of E at such fixed points. Proposition 4.13.
The second variation of E at a C ∗ –fixed point s with lift b s as in (3.9) is given by d E ( ˙ s ) = 12 π i Z Σ tr( ψ ∧ [ ϕ , ξ ] + ϕ ∧ [ ψ , ξ ] + ψ ∧ [ ϕ , ξ ]) + ϕ ∧ [ ψ , ξ ] + 2 ϕ ∧ ψ ) . Proof.
Let ( s t ) be a family of sections with s = s . We compute, using the notation for b s and ˙ s as in Section4 2 π i d dt | t =0 E ( s t ) = Z Σ tr(Φ ∧ ˙ ψ + ˙ ϕ ∧ Ψ + 2 ϕ ∧ ψ ) . Since s = s is fixed by the action of C ∗ , we can use (4.15) (with ξ = ξ , ξ = 0) to write2 π i d dt | t =0 E ( s t ) = Z Σ tr([Φ , ξ ] ∧ ˙ ψ − ˙ φ ∧ [Ψ , ξ ] + 2 φ ∧ ψ )= Z Σ tr( − ξ ([Φ ∧ ˙ ψ ] + [ ˙ φ ∧ Ψ]) + 2 φ ∧ ψ )= Z Σ tr( ξ ( ∂ ˙ ϕ + ∂ ˙ ψ + 2[ ψ ∧ ϕ ] + 2[ ϕ ∧ ψ ]) + 2 ϕ ∧ ψ )(using ∂ξ = 0 = ∂ξ ) = Z Σ tr( ξ (2[ ψ ∧ ϕ ] + 2[ ϕ ∧ ψ ]) + 2 ϕ ∧ ψ )= Z Σ tr( ψ ∧ [ ϕ , ξ ] + ϕ ∧ [ ψ ∧ ξ ] + ψ ∧ [ ϕ , ξ ]) + ϕ ∧ [ ψ ∧ ξ ] + 2 ϕ ∧ ψ )In the third equation from above we made use of the second linearisation of (4.3). (cid:3) Proposition 4.13 shows that the second variation is closely related to the infinitesimal C ∗ -action on thetangent space. The following proposition is obtained. Proposition 4.14.
Let ˙ s ( λ ) = ( ˙ ∂ ( λ ) , ˙ D ( λ ) , λ ) = ( P ∞ k =0 ψ k λ k , P ∞ k =0 ϕ k λ k , λ ) be an infinitesimal defor-mation of the critical point s ∈ S . Suppose that ˙ s satisfies [ ψ , ξ ] = n ψ , [ ψ , ξ ] = n ψ , [ ϕ , ξ ] = m ϕ , [ ϕ , ξ ] = m ϕ for some m i , n i ∈ Z . Then d E ( ˙ s ) = 12 π i Z Σ tr(( m + n ) ψ ∧ ϕ + ( m + n + 2) ψ ∧ ϕ ) . Remark 4.15.
Note that this resembles the discussion surrounding Eq. (8.10) in [28]. In fact, it doesreproduce Hitchin’s result in the case that s is the twistor line corresponding to a C ∗ -fixed point in M Higgs and the deformation ˙ s is real, so that ψ = ϕ ∗ , ψ = − ϕ ∗ .4.5. Sections and the degree of the hyperholomorphic line bundle.
Our previous results togetherwith the energy can be used to show that the space of irreducible sections is not connected. We begin withthe following
Proposition 4.16.
Let ( ∂, Φ) be a stable C ∗ -fixed Higgs bundle and let s ( β,φ ) be a C ∗ -fixed section cor-responding to ( β, φ ) ∈ B +( ∂, Φ) . If s β,φ ( ∞ ) is given by a VHS on Σ with underlying holomorphic bundle E = L l ′ k =1 E ′ k , then we have deg( s ∗ ( β,φ ) L Z ) = l X k =1 ( k −
1) deg( E k ) + l ′ X k =1 ( k −
1) deg( E ′ k ) . Proof.
Proposition 4.12 allows us to compute E ( s ) and E ∞ ( s ). The assertion now follows from the formuladeg( s ∗ ( β,φ ) L Z ) = E ( s ( β,φ ) ) + E ∞ ( s ( β,φ ) ) . (cid:3) Theorem 4.17.
There exist irreducible sections s of ̟ : M DH (Σ , SL(2 , C )) −→ C P such that the pullback s ∗ L Z of the holomorphic line bundle L Z −→ M DH (Σ , SL(2 , C )) has non-zero degree. In particular, the spaceof irreducible sections is not connected.Proof. Let K Σ be a square-root of the canonical line bundle K Σ . Consider the uniformization (Fuchsian)flat connection ∇ F uchs = ∇ K ∗ ∇ K − ECTIONS OF TWISTOR SPACES WITH ROTATING ACTION 35 on the rank two bundle K Σ ⊕ K − Σ . For generic holomorphic quadratic differential q ∈ H (Σ , K ), theanti-holomorphic structure ∂ K q ∂ K − is stable (i.e., it defines a stable holomorphic bundle on Σ). Then, ∇ := ∇ F uchs + (cid:18) q (cid:19) gives a C ∗ -invariant section s ∇ ∈ S M DH by the construction of Lemma 4.4. In view of Proposition 4.12, theenergy at λ = 0 is given by deg K − Σ = 1 − g = 0 . By assumption, ∂ ∇ is stable, so the anti-Higgs field of s at λ = ∞ vanishes, and the energy at λ = ∞ isgiven by E ∞ = 0 . Finally, we have deg( s ∗ L Z ) = E ( s ) + E ∞ ( s ) = 0by the residue formula for the pull-back under s of the meromorphic connection to C P (see Section 3 of[4]). (cid:3) Given a(n irreducible) section s ∈ S M DH , it is in general very difficult to compute its normal bundle N s .However, by using the methods of [24], it can be shown that the C ∗ -fixed points considered in the proofof Theorem 4.17 do not have normal bundles of generic type, i.e., their normal bundles admit holomorphicsections with double zeros. References [1] D. V. Alekseevsky, V. Cort´es and T. Mohaupt, Conification of K¨ahler and hyper-K¨ahler manifolds,
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Email address , (Beck): [email protected]
Email address , (Biswas): [email protected]
Email address , (Heller): [email protected]
Email address , (R¨oser): [email protected] (Beck, R¨oser)
Fachbereich Mathematik, Universit¨at Hamburg, 20146 Hamburg, Germany (Biswas)
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005,India (Heller)(Heller)