Hyperbolic geometry and non-Kahler manifolds with trivial canonical bundle
aa r X i v : . [ m a t h . S G ] D ec Hyperbolic geometry and non-K¨ahler manifoldswith trivial canonical bundle
Joel Fine and Dmitri Panov
Abstract
We use hyperbolic geometry to construct simply-connected symplectic or complexmanifolds with trivial canonical bundle and with no compatible K¨ahler structure.We start with the desingularisations of the quadric cone in C : the smoothing is anatural S -bundle over H , its holomorphic geometry is determined by the hyperbolicmetric; the small-resolution is a natural S -bundle over H with symplectic geometrydetermined by the metric. Using hyperbolic geometry, we find orbifold quotients withtrivial canonical bundle; smooth examples are produced via crepant resolutions. Inparticular, we find the first example of a simply-connected symplectic 6-manifold with c = 0 that does not admit a compatible K¨ahler structure. We also find infinitelymany distinct complex structures on 2( S × S ) S × S ) with trivial canonicalbundle. Finally, we explain how an analogous construction for hyperbolic manifoldsin higher dimensions gives symplectic non-K¨ahler “Fano” manifolds of dimension 12and higher. Contents n . . . . . . . . . . . . . . . . . . 31.4 A hyperbolic picture of the desingularisations . . . . . . . . . . . . . . 31.5 Symplectic non-K¨ahler “Fano” manifolds . . . . . . . . . . . . . . . . . 41.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 π -hyperbolic knots . . . . . . . . . . . . . . . 143.5 Recovering the orbifold from the threefold . . . . . . . . . . . . . . . . 17 Some symplectic “Fano” manifolds 21 c = 0 . . . . . . . . 256.3 Rational curves in hyperbolic twistor spaces . . . . . . . . . . . . . . . 256.4 Miles Reid’s fantasy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 The main goal of this article is to describe how hyperbolic geometry can be used toconstruct simply-connected complex and symplectic manifolds with trivial canonicalbundle , which admit no compatible K¨ahler structure . In the symplectic case, triv-ial means c = 0, whilst in the complex case, trivial means holomorphically trivial.Our examples have real dimension six. Hyperbolic geometry in dimension three givescomplex examples whilst hyperbolic geometry in dimension four leads to symplecticexamples.Using this approach we find a symplectic structure on a certain 6-manifold. This 6-manifold is simply connected, has vanishing first Chern class and admits no compatibleK¨ahler structure, giving the first such 6-dimensional example, at least to the best ofour knowledge. On the complex side we find, amongst other things, infinitely manydistinct complex structures on 2( S × S ) S × S ) all with trivial canonical bundle.Before giving an outline of the constructions, we begin by recalling some back-ground. First consider the case of real dimension four. It follows from the classification ofcompact complex surfaces that the only simply-connected compact complex surfaceswith trivial canonical bundle are K3 surfaces and so, in particular, are K¨ahler. In thesymplectic case the situation is not yet understood, but it is a folklore conjecture thatK¨ahler K3 surfaces again provide the only examples. So, conjecturally, in dimensionfour all simply-connected symplectic manifolds with c = 0 are K¨ahler too. If webelieve this conjecture, to find the first non-K¨ahler examples of simply-connectedmanifolds with trivial canonical bundle, we should look to higher dimensions. In dimension six, much use has been made of ordinary double-points and their desin-gularisations. An ordinary double-point is modelled locally on a neighbourhood ofthe origin in the quadric cone P z j = 0 in C . To desingularise the cone, one cansmooth it to obtain the smooth affine quadric P z j = 1; this replaces the double pointby a three-sphere. Alternatively, one can resolve the double-point by the so-calledsmall resolution, which is the total space of O ( − ⊕ O ( − → CP ; here the doublepoint is replaced by a two-sphere. (We will give more detail about these desingularisa-tions later in § X with trivial canonical bundle and or-dinary double points they considered the manifold X ′ obtained by smoothing all the ouble points. Under certain conditions, the complex structure on X can be smoothedto give a complex structure on X ′ , again with trivial canonical bundle. In this wayone finds a family of complex structures on n ( S × S ) for all n ≥
2, with trivialcanonical bundle. (The article of Lu–Tian [20] explains the details for these particularsmoothings.) These manifolds are clearly not K¨ahler.There is an analogous story on the symplectic side, explained in the article ofSmith–Thomas–Yau [30]. Here, one starts with a symplectic manifold Y with ordi-nary double points and trivial canonical bundle and considers the manifold Y ′ ob-tained by the small resolution of all the double points. Under certain conditions,the symplectic structure on Y can be resolved to give a symplectic structure on Y ′ with c = 0. In this way Smith–Thomas–Yau produced many interesting examples ofsimply-connected symplectic manifolds with c = 0 which are widely believed to benon-K¨ahler. However, in contrast with the complex side, they were unable to find ex-amples which violate the standard K¨ahler topological restrictions. It is still unknownif any of the examples they give actually are non-K¨ahler. n In [17], Guan constructs examples of simply-connected compact complex-symplecticmanifolds of dimension 4 n for n ≥
2, which do not admit a compatible K¨ahler struc-ture. (See also the article [2] of Bogomolov for an alternative exposition of Guan’sconstruction.) Taking the real part of the complex symplectic form, one obtains asymplectic manifold in the real sense which has c = 0. In [17], Guan proves that hisexamples are not diffeomorphic to hyperk¨ahler manifolds. In fact even more is true;whilst it is not proven directly in [17], Guan has informed us that it follows from theresults of [17] that his manifolds admit no K¨ahler structure whatsoever.Guan’s construction uses Kodaira–Thurston surfaces to produce a single exampleof a simply-connected 4 n -manifold for each n ≥ n pointson a hyperk¨ahler surface. In contrast, the construction we present here produceslarge numbers of topologically distinct symplectic manifolds all of the same dimension.Whilst we describe just one in this article, we produce examples with arbitrarily highBetti numbers in a sequel [11]. The idea is outlined here in § As in the articles [5, 13, 32, 30] mentioned above, the construction of symplecticand complex manifolds that we explore in this article relies on the desingularisationsof the threefold quadric cone. However, rather than using the small resolution orsmoothing as local models for desingularising double-points, here we consider their global geometry. H and the small resolution As we explain in § R = O ( − ⊕O ( −
1) is intimately related to hyperbolic geometry in dimension four. R is a natu-ral S -bundle over H , with the symplectic Calabi–Yau structure on R determinedby the metric on H . Indeed, R is symplectomorphic to the twistor space of H ;the symplectic structure on the twistor space was first defined by Reznikov [28] andDavidov–Muˇskarov–Grantcharov [6] and a symplectomorphism with R was given in[12]. In § § R can beseen as a coadjoint orbit of SO(4 , CP → HP ∼ = S . t follows that any hyperbolic 4-manifold carries an S -bundle—its twistor space—whose total space is a symplectic manifold with trivial canonical bundle.To produce a simply connected example, in § z
7→ − z . Resolving the 16 orbifold pointsgives a K3 surface.The role of the abelian surface in our situation is played by a beautiful hyperbolic 4-manifold called the Davis manifold [8] (see also the description in [26]). This manifoldis constructed by gluing opposite 3-faces of a 120-cell in H , a certain regular four-dimensional polyhedron (so-called because it has 120 three-dimensional faces, each aregular dodecahedron).An important point for our purposes is that reflection in the centre of the 120-cellgives an isometry of the Davis manifold M . This isometry gives a quotient M/ Z which is a simply-connected singular manifold with 122 isolated singularities modelledon z
7→ − z , just as in the Kummer construction. In terms of the symplectic S -bundle Z → M , the isometry induces a symplectic action of Z on Z which fixes 122 fibres.The quotient is a symplectic orbifold with trivial canonical bundle, with singularitiesalong 122 S ’s. Making a symplectic crepant resolution of these singularities gives asmooth simply-connected symplectic manifold ˆ Z . As we will see, b ( ˆ Z ) = 0 which cannever happen for a simply-connected K¨ahler manifold with c = 0. H and the smoothing There is an analogous story for the smoothing S = { P z j = 1 } , this time relating thecomplex geometry of S with hyperbolic geometry in dimension three. S is a natural S -bundle over H with the complex geometry of S determined by the metric on H .One way to see this is to note that S ∼ = SL(2 , C ); the action of SU(2) on SL(2 , C )makes it a principal bundle over SL(2 , C ) / SU(2) ∼ = H , the principal spin bundle of H . As a consequence, any hyperbolic 3-manifold carries an S -bundle whose totalspace is a complex manifold with trivial canonical bundle.To produce simply-connected examples, in § S with cone angle 2 π/m along a knot K . It is a standard fact that thereis a smooth hyperbolic manifold M admitting an isometric Z m -action with quotient M → S branched over K . Now, as M is hyperbolic, its spin-bundles are complexmanifolds with trivial canonical bundle. As we will show, provided the spin-structure P → M is well-chosen, the generator of the Z m -action lifts to P where it generates a Z m -action (covering the Z m -action on M via the projection Z m → Z m ).The quotient P/ Z m is just the quotient of the frame bundle Q → M by the lift ofthe action of Z m . It is straightforward to see that Z m acts freely on Q and so Q/ Z m issmooth and has fundamental group equal to the original orbifold fundamental groupof the metric on S . To remedy this, we “twist” the Z m -action on P around theknot K ; this induces fixed points in the fibres over K and results in a singular butsimply-connected quotient of P . Finally, we obtain a smooth manifold by taking acrepant resolution.The examples we find this way have a smooth surjection X → S . Away from theknot, this makes X a trivial fibration S × ( S \ K ). These threefolds can never beK¨ahler; as we explain, they admit a C ∗ -action with no fixed points. This constructionprovides, amongst other things, infinitely many distinct complex structures with trivialcanonical bundle on 2( S × S ) S × S ). In § n is the base of a fibre bundle whose otal space is a symplectic manifold of real dimension n ( n + 1). (The fibre bundle isthe twistor space; these symplectic manifolds were first considered by Reznikov [28].)When n = 1 the fibres are zero-dimensional and the total space is just the originalsurface. In this case, the symplectic manifold is “general type” with symplectic classa negative multiple of c . When n = 2, we have the situation described above, forwhich c = 0. For all higher dimensions, however, it turns out that the symplecticmanifold has symplectic class equal to a positive multiple of c . No compact exampleproduced this way can be K¨ahler (e.g., for fundamental group reasons). To the bestof our knowledge, these examples (found first by Reznikov) are the first symplecticnon-K¨ahler manifolds which have symplectic class equal to a positive multiple of c ,symplectic analogues of Fano varieties.It is interesting to compare this with the situation in dimension four. There, it is aconsequence of the work of Gromov [16], Taubes [31] and McDuff [22] that a symplecticFano 4-manifold must be K¨ahler. Meanwhile, the lowest dimension attained by ourconstruction is twelve. It is natural to ask for the minimal dimension in which non-K¨ahler examples occur. Do they exist in dimension six? The symplectic Fanos comingfrom hyperbolic 2 n -manifolds also have the property that c n can be arbitrarily large(it is essentially the volume of the hyperbolic 2 n -manifold). Again, it seems naturalto ask for the minimal dimension in which Fano manifolds exist for which c n can bemade arbitrarily large. It is a pleasure to thank the following people for discussions held during the courseof this work: Mohammed Abouzaid, Michel Boileau, Alessio Corti, Simon Donald-son, Daniel Guan, Maxim Kontsevich, Federica Pasquotto, Vicente Mu˜noz, BrendanOwens, Simon Salamon, Paul Seidel, Ivan Smith, Burt Totaro, Richard Thomas, HenryWilton, Claire Voisin, Jean-Yves Welschinger and Dominic Wright. We would also liketo thank an anonymous reader who pointed out an error in an earlier version of thiswork.The first author was supported by an FNRS charg´e de recherche fellowship. Thesecond author was supported by EPSRC grant EP/E044859/1.
Fix a non-degenerate complex quadratic form q on C . The conifold is the affinequadric cone Q = { q ( ζ ) = 0 } . We will consider two well-studied ways to removethe singularity at the origin, giving smooth K¨ahler manifolds with trivial canonicalbundle. We describe this briefly here; for more details we refer to [30], from where weoriginally learnt this material.One desingularisation is the smooth affine quadric S = { q ( ζ ) = 1 } , called the smoothing of Q . S is a K¨ahler manifold with trivial canonical bundle, although wewill be concerned only with the complex geometry of S .Another way to desingularise the conifold is to take a resolution. Choose coordi-nates ( x, y, w, z ) so that the quadratic form is given by q ( x, y, z, w, ) = xw − yz . Let R denote the total space of π : O ( − ⊕ O ( − → CP . Each summand of R has anatural map O ( − → C ; these combine to give a map p : R → C ⊕ C ∼ = C . Here,we use the identification (( a, b ) , ( c, d )) ( a, b, c, d ). The image of p consists of points( x, y, z, w ) such that [ x : y ] = [ z : w ], i.e., such that xw − yz = 0, which is the conifold.So p : R → Q is a resolution, called the small resolution , in which the singularity hasbeen replaced by the zero section in R . n this description, we could equally have chosen to identify C ⊕ C ∼ = C by(( a, b ) , ( c, d )) ( a, c, b, d ). With this choice, the image of p : R → C consists ofpoints ( x, y, z, w ) such that [ x : z ] = [ y : w ] which is again the conifold xw − yz = 0.So, in fact, there are two inequivalent ways to view R as the small resolution of Q .Put another way, there are two small resolutions R ± → Q ; each is isomorphic to O ( − ⊕ O ( −
1) abstractly, but there is no isomorphism between them which respectsthe projections to Q .There is an alternative coordinate-free description of the small resolutions. Blowup the origin in C to obtain e C . The proper transform e Q of Q meets the exceptional CP ⊂ e C in a quadric surface. This surface is biholomorphic to CP × CP and eachof the rulings has negative normal bundle. Blowing down one or other of the rulingsgives the two small resolutions R ± → Q of the conifold.Either resolution R is a K¨ahler manifold with trivial canonical bundle, althoughwe will be concerned only with the symplectic geometry of R . The symplectic form isgiven by ω R = π ∗ ω CP + p ∗ ω C where π : R → CP and p : R → Q ⊂ C are the vector-bundle and resolution projec-tions respectively.The first hint of a link with twistor geometry is provided by considering the sym-plectic action of SO(4) on R . The Hermitian metric and complex quadratic form on C define a choice of conjugation map (the real points are those on which the Hermitianand complex forms agree). Then SO(4) is the subgroup of U(4) which commutes withthis conjugation. In this way SO(4) acts by K¨ahler isometries on the conifold Q andthis action lifts to a K¨ahler action on the small resolutions R ± . The action on theexceptional CP in R ± is given by one or other of the projections SO(4) → PSU(2)arising from the exceptional isomorphismSO(4) ∼ = SU(2) × SU(2) ± . (1) To describe the connection between hyperbolic geometry and S the smoothing, usecoordinates ( x, y, z, w ) in which the quadratic form q is given by q ( x, y, z, w ) = xw − yz .Identify C with the set of complex 2 × (cid:18) x yz w (cid:19) ( x, y, z, w ) . When evaluated on a matrix A , the quadratic form is q ( A ) = det A . Hence, in thispicture, S ∼ = SL(2 , C ) is given by matrices with determinant 1. Consider the actionof SU(2) on SL(2 , C ) given by multiplication on the left; this makes SL(2 , C ) into aprincipal SU(2)-bundle over the symmetric space SL(2 , C ) / SU(2) which is preciselyhyperbolic space H . The bundle SL(2 , C ) → H is the principal spin bundle of H .To verify this note that Isom( H ) ∼ = PSL(2 , C ) acts freely and transitively on the framebundle of H ; so PSL(2 , C ) can be identified with the frame bundle of H whilst itsdouble cover SL(2 , C ) is identified with the principal spin bundle.As it is a complex Lie group, SL(2 , C ) has a holomorphic volume form which isinvariant under right-multiplication. It will be essential to us that this same form isalso invariant under left -multiplication; in other words, that SL(2 , C ) is a complex uni-modular group. Given P, Q ∈ SL(2 , C ), the map A P − AQ sets up an isomorphismSO(4 , C ) ∼ = SL(2 , C ) × SL(2 , C ) ± . (2)(This is, of course, just the complexification of (1).) Unimodularity of SL(2 , C ) nowamounts to the following: roposition 1. SO(4 , C ) acts by biholomorphisms on S . Moreover, S admits aninvariant holomorphic volume form. Alternatively, the SO(4 , C )-invariant volume form can be seen directly, withoutreference to SL(2 , C ) and unimodularity. SO(4 , C ) preserves both the Euclidean holo-morphic volume form Ω and the radial vector e = x∂ x + y∂ y + z∂ z + w∂ w , which istransverse to S . Hence the volume form Ω = ι e Ω on S is also invariant.To produce compact quotients of S , let M be an oriented compact hyperbolic 3-manifold with fundamental group Γ ⊂ PSL(2 , C ). Since M is spin (as all oriented3-manifolds are), Γ lifts to SL(2 , C ) and the total space of the principal spin bundleof M is X = SL(2 , C ) / Γ, where Γ acts by right-multiplication. More precisely, eachchoice of lift of Γ to SL(2 , C ) gives a spin structure on M ; this correspondence is one-to-one: lifts and spin structures are both parametrised by H ( M, Z ). Whatever thechoice of lift, the action of Γ preserves the holomorphic volume form on SL(2 , C ) andso X is a compact complex threefold with trivial canonical bundle.Left-multiplication gives us additional freedom. Denote the chosen lift by α : Γ → SL(2 , C ) and let ρ : Γ → SL(2 , C ) be some other homomorphism. From ρ we obtain anew action of Γ on S : γ · A = ρ ( γ ) − A α ( γ ) . We will use this “twisting” to produce orbifold quotients of S , but it was originallyused in a different context by Ghys [14]. Ghys studies infinitesimal deformations ofthe complex manifold X = SL(2 , C ) / Γ, where Γ acts by right-multiplication, i.e.,where ρ is the trivial homomorphism in the above picture. He shows that infinitesimalholomorphic deformations of X are equivalent to infinitesimal deformations of thetrivial homomorphism. It is possible to extend this picture, giving a description ofpart of the space of complex structures on X , something we address in forthcomingwork [10]. This section proves the analogue of Proposition 1 for the small resolution R . Proposition 2.
SO(4 , acts symplectomorphically on R , extending the action of SO(4) . Moreover, R admits an invariant compatible almost complex structure andinvariant complex volume form. Before proving this, we give three different descriptions of the hyperbolic geometryof R . The twistor space of H carries a natural symplectic structure (a fact noticed indepen-dently by Reznikov [28] and Davidov–Muˇskarov–Grantcharov [6]) and this was shownto be symplectomorphic to the small resolution in [12]. We very briefly recall the ideahere.Choose coordinates z j on C in which the conifold is { P z j = 0 } . The map Q → R given by z Re z exhibits Q \ S -bundle over R \
0. The fibre over a point x is all points of the form x + iy where y ∈ h x i ⊥ with | y | = | x | .The twistor space of R is the bundle of unit-length self-dual two forms. Given x ∈ R \
0, interior contraction with x gives an isomorphism Λ + ∼ = h x i ⊥ . In this waywe can identify Q \ R \
0. This identification extends overthe small resolution to give an SO(4)-equivariant identification of R with the twistorspace of R . (More precisely it extends over one of the two small resolutions, to obtainthe other small resolution we should consider Λ − .) t was proved in [12] that, up to homotheties and rescaling, there is a unique SO(4)-invariant symplectic form on the twistor space R of R with infinite volume and whosesign changes under the antipodal map. From this twistorial view-point, it comesnaturally from hyperbolic geometry. The Levi–Civita connection of the hyperbolicmetric on R ∼ = H induces a metric connection in the vertical tangent bundle V → R .Its curvature − πiω determines the symplectic form. It follows from this that there isa symplectic action of the hyperbolic isometry group SO(4 ,
1) extending that of SO(4).In [7], Davidov–Muˇskarov–Grantcharov compute the total Chern form of the tan-gent bundle of the twistor space. Combining their calculation with the symplectomor-phism of the twistor space and the small resolution in [12] gives a proof of Proposition2. We give a separate proof later, using an alternative description of R . The next description, involving quaternions, is analogous to the standard picture ofthe twistor fibration CP → S which we recall first. Begin by identifying C ∼ = H .Each complex line in C determines a quaternionic line in H , giving a map t : CP → HP ∼ = S . The fibre t − ( p ) is the CP of all complex lines in the quaternionic line p .A choice of positive-definite Hermitian form on C gives a Fubini–Study metric on CP and a round metric on S . The isometries SO(5) of S are identified with thoseisometries of CP which preserve the fibres of t , giving an injection SO(5) → PSU(4).To describe the twistor fibration of H in an analogous way we use an indefinite Hermitian form: let C , denote C together with the Hermitian form h ( w ) = | w | + | w | − | w | − | w | and consider the space N = { w ∈ C , : h ( w ) < } / C ∗ of negative lines in C , ; N is an open set in CP . Transverse to a negative line in C , , h is indefinite with signature (2 , N inherits a pseudo -K¨ahler metric ofsignature (2 , CP . The pseudo-K¨ahler metric makes N into a symplecticmanifold. Alternatively, one can see the N as the symplectic reduction of C , by thediagonal circle action; the Hamiltonian for this action is just h and the reduction at h = − N .It is standard that N is symplectomorphic to R . (One way to prove this is to usethe fact that both admit Hamiltonian T -actions with equivalent moment polytopes.)To use this picture to relate R to H , let H , denote H together with the indefiniteform h ′ ( p ) = | p | − | p | . The map( w , w , w , w ) ( w + jw , w + jw )identifies the Hermitian spaces C , ∼ = H , . There is a natural projection from N tothe space of negative quaternionic lines in H , , i.e., to the quaternionic-hyperbolicspace: H H = { p ∈ H , : h ′ ( p ) < } / H ∗ . In general, the space of negative lines in H n, is the quaternionic analogue of hyperbolicor complex-hyperbolic n -space. When n = 1, however, this is isometric to H , four-dimensional real-hyperbolic space. (In the lowest dimension, the symmetric spacesassociated to complex or quaternionic geometry coincide with their equidimensionalreal analogues.)So, analogous to t : CP → S , we have a projection t : R → H ; the fibre t − ( p ) isthe CP of all complex lines in the quaternionic line p . The pseudo-K¨ahler isometriesof R are PSU(2 ,
2) whilst the isometries SO(4 ,
1) of H can be identified with thoseisometries of R which preserve the fibres of t , giving an injection SO(4 , → PSU(2 , ,
1) on R by symplectomorphisms. .3.3 A coadjoint description Let G be a Lie group and ξ ∈ g ∗ ; denote the orbit of ξ under the coadjoint actionby O ( ξ ). It is a standard fact that there is a G -invariant symplectic structure on O ( ξ ). We will show how the small resolution fits into this general theory as a certaincoadjoint orbit of SO(4 , so (4 ,
1) is 5 × (cid:18) u t u A (cid:19) , (3)where u is a column vector in R and A ∈ so (4). Those elements with u = 0 generateSO(4) ⊂ SO(4 , so (4 ,
1) and so gives an equivariant isomor-phism so (4 , ∼ = so (4 , ∗ . We consider the orbit of ξ = (cid:18) J (cid:19) where J ∈ so (4) is a choice of almost complex structure on R (i.e., J = − h of matrices commuting with ξ is those with u = 0 and [ A, J ] = 0,i.e., h = u (2) ⊂ so (4) ⊂ so (4 , ξ is U(2) and so O ( ξ ) ∼ = SO(4 , / U(2).
Lemma 3.
There is an isomorphism of
U(2) -representation spaces: so (4 , ∼ = u (2) ⊕ Λ ( C ) ∗ ⊕ C . Proof.
There is a U(2)-equivariant isomorphism so (4) ∼ = u (2) ⊕ Λ ( C ) ∗ . To see this,write so (4) ∼ = Λ ( R ) ∗ . Given a choice of almost complex structure on R , any real2-form a can be written uniquely as a = α + β + ¯ β where α ∈ Λ , R is a real (1 , β ∈ Λ , . Identifying a with ( α, β ) gives a U(2)-equivariant decompositionΛ R ∼ = Λ , R ⊕ Λ , . But, via the Hermitian form, Λ , R is identified with skew-Hermitianmatrices u (2) and this gives the claimed isomorphism.There is also an SO(4)-equivariant isomorphism so (4 , ∼ = so (4) ⊕ R . In theform (3), the so (4) summand is given by u = 0 whilst the R summand by A = 0.Combining these two isomorphisms completes the proof. Lemma 4.
Up to scale, there is a unique
SO(4 , -invariant symplectic form on SO(4 , / U(2) .Proof.
The existence follows from coadjoint orbit description. For uniqueness, we be-gin by describing all invariant non-degenerate 2-forms. This amounts to describing allnon-degenerate 2-forms at a point which are invariant under the stabiliser U(2). FromLemma 3 the tangent space at a point is isomorphic as a U(2)-representation space toΛ ( C ) ∗ ⊕ C . Up to scale, there is a unique invariant 2-form on each summand, givingtwo SO(4 , a, b on SO(4 , / U(2). It follows that, up to scale, allnon-degenerate invariant 2-forms have the form a + tb for t = 0. At most one of thesecan be closed, t being fixed by the requirement that d a = t d b . Corollary 5. O ( ξ ) is symplectomorphic to R .Proof. This follows from the previous lemma and the transitive action of SO(4 ,
1) on R which can be seen in either the twistorial or quaternionic pictures.We make a small digression to point out a simple consequence of the coadjointorbit description of R which is, perhaps, less apparent from its K¨ahler description asthe total space of O ( − ⊕ O ( − R isnot relevant to what follows. roposition 6. R is symplectomorphic to O ( − × R (where O ( − → CP hassymplectic form given by adding the pull-backs of the standard forms via the projectionto CP and the resolution O ( − → C / Z ).Sketch proof. As R is a coadjoint orbit of SO(4 , ,
1) is Hamil-tonian. It follows that R admits a free Hamiltonian R -action given as follows: fixa point p at infinity on H and consider the subgroup R ⊂ SO(4 ,
1) which acts bylinear translations on the horospheres centred at p . The action on R admits a sectiongiven by considering the restriction of R → H to a geodesic through p ; this showsthat the orbit space is S × R . If the section were Lagrangian, we would be dealingwith the cotangent bundle of S × R , but the S -factor has symplectic area 1 and so R is symplectomorphic to O ( − × R . Corollary 7. R contains no Lagrangian 3-spheres.Proof. Since O ( −
2) is convex at infinity, this follows from recent work of Welschinger[34] (see, for example, Corollary 4.13).
SO(4 , -invariant complex volume form We now give the proof of Proposition 2, which says that the small resolution admits anSO(4 , Proof of Proposition 2.
We use the coadjoint orbit description. Lemma 3 says thatat each point z there is a U(2)-equivariant isomorphism of the tangent space T z ∼ =Λ ( C ) ∗ ⊕ C . Accordingly, there is a natural U(2)-invariant almost complex struc-ture on T z which is ω -compatible and, hence, an SO(4 , J on O ( ξ ).For the complex volume form, note that U(2) acts trivially on Λ T ∗ z ∼ = Λ ( C ) ⊗ Λ ( C ) ∗ hence any non-zero element of Λ T ∗ z can be extended in a unique way to anSO(4 , R → H is a CP -bundle and that ω is non-degenerate on the fibres. Hence T R = V ⊕ H where V is the vertical tangent bundle and H its complement with respect to ω . Thiscorresponds to the splitting in Lemma 3. The pseudo-K¨ahler metric is negative definiteon V and positive definite on H . Let J = − J int | V + J int | H where J int is the integrable complex structure on R ⊂ CP . J is an SO(4 , § Complex examples
Compact complex manifolds will be built starting from hyperbolic orbifold metrics on S with cone angle 2 π/m along a knot K ⊂ S . When such a metric exists, K is saidto be 2 π/m -hyperbolic. Such knots are well-known to be plentiful. As is explainedin [3], it is a consequence of Thurston’s orbifold Dehn surgery theorem that when K is a hyperbolic knot—i.e., when S \ K admits a complete finite-volume hyperbolicmetric— K is also 2 π/m -hyperbolic for all m ≥ K is the figure eight knot and m = 3. There are also infinitely many π -hyperbolicknots. (We are grateful to Michel Boileau for advice on this matter.)Let H / Γ be a hyperbolic orbifold metric on S with cone angle 2 π/m along aknot K where Γ ⊂ PSL(2 , C ) is the orbifold fundamental group. It is standard thatthere a is smooth hyperbolic manifold M which is an m -fold cyclic cover M → S branched along the knot K . (By Mayer–Vietoris, H ( S \ K ) = Z ; hence there is ahomomorphism π ( S \ K ) → Z which in turn induces a homomorphism ψ : Γ → Z m ;the kernel Γ ′ of ψ has no fixed points on H and is the fundamental group of M .)Since M is a hyperbolic manifold, its spin-bundles are complex manifolds withtrivial canonical bundle (as is described in § P → M iswell-chosen, the generator of the Z m -action lifts to P where it generates a Z m -action.To produce a simply-connected quotient of P , we then “twist” this action around theknot; this induces fixed points in the fibres over K and results in a singular quotientfor which all the fibres are simply-connected, giving a simply-connected total space.Finally, we obtain a smooth manifold by taking a crepant resolution. We begin by considering the model situation. Take a geodesic γ in H and let Z m acton H by fixing γ pointwise and rotating perpendicular to γ by 2 π/m . Let ϕ = e πi/m .In appropriate coordinates, Z m ⊂ PSL(2 , C ) is generated by the class [ U ] of thediagonal matrix U with entries ϕ, ϕ − . Z m ∈ PSL(2 , C ) is covered by the copy of Z m ⊂ SL(2 , C ) generated by U . Theaction of Z m on H is covered by the action of Z m by right multiplication on SL(2 , C ).However, this action on SL(2 , C ) is free and so the quotient has non-trivial fundamentalgroup. To produce a singular but simply-connected quotient, we “twist” to introducefixed points. This additional twist is given by simultaneously multiplying on the left .Consider the action of Z m on SL(2 , C ) generated by conjugation A U − AU .Explicitly, in coordinates, the generator is (cid:18) x yz w (cid:19) (cid:18) x ϕ − yϕ z w (cid:19) . (4)The points of SL(2 , C ) fixed by Z m are the C ∗ subgroup of diagonal matrices. On atangent plane normal to the fixed points, the Z m -action is that of the A m -singularity(i.e., C / Z m with action generated by U ∈ SU(2)). Left multiplication on SL(2 , C )by the C ∗ subgroup of diagonal matrices is free and commutes with the action of Z m ;hence it descends to a free C ∗ -action on the quotient. The C ∗ -action is transitive onthe fixed locus and so gives an identification of the a neighbourhood of the orbifoldpoints with the product of C ∗ × V , where V is a neighbourhood of the singular pointin the A m -singularity.The Z m -action preserves the fibres of SL(2 , C ) → H and covers the Z m action on H —the fibres are right cosets of SU(2) and U ∈ SU(2)—hence there is a projectionSL(2 , C ) / Z m → H / Z m . The Z m -fixed points form a circle bundle over the geodesic γ . We have implicitly oriented γ in our choice of coordinates (4); the image of the fixedpoints in the frame bundle PSL(2 , C ) are those frames whose first vector is positivelytangent to γ . ince the A m -singularity admits a crepant resolution, it follows that SL(2 , C ) / Z m does too. The exceptional divisor of the resolution W → SL(2 , C ) / Z m maps to the C ∗ of singular points with fibre a chain of m − A m singularity.In our model example, the Z m fixed locus in H is a geodesic γ . In our compactexamples, the branch locus will be a closed geodesic loop. The geodesic γ is “closed up”by the action of Z on SL(2 , C ) generated by right-multiplication by the diagonal matrixwith entries a, a − , where a ∈ C has | a | >
1. This action is free and commutes with the Z m action described above, hence it induces an action of Z on SL(2 , C ) / Z m and on thecrepant resolution W . The quotient W/ Z is a crepant resolution of SL(2 , C ) / ( Z m ⊕ Z ),a complex orbifold whose singular locus is an elliptic curve. This is the singularitywhich will actually appear in our compact examples.Finally note that the free C ∗ -action on SL(2 , C ) / Z m described above commuteswith the action of Z and so descends to SL(2 , C ) / ( Z m ⊕ Z ). It induces a free C ∗ -actionon the resolution, a feature which will be shared by our compact examples. We now consider a hyperbolic orbifold metric on S with cone angle 2 π/m along aknot K . It is a standard fact that there a is smooth hyperbolic manifold M which isan m -fold cyclic cover M → S branched along the knot K . Z m acts by isometrieson M , fixing the geodesic branch locus pointwise and rotating the normal bundle by2 π/m . The action of Z m on the universal cover H of M is precisely the situationconsidered in § Z m ⊂ PSL(2 , C ) is generated (in appropriate coordinates) by theclass [ U ] of a matrix U which is diagonal with entries ϕ and ϕ − , where ϕ = e πi/m .In order to produce a complex orbifold we first try to lift the Z m -action to a spinbundle of M . Let Q denote the frame bundle of M and P a choice of spin bundle.The generator of the Z m -action on M induces a diffeomorphism f of Q and we aim tochoose the spin structure so that f lifts to P . For this we state the following standardresult: Lemma 8.
Let Y be a connected manifold, f : Y → Y a diffeomorphism and Y h → Y the double cover corresponding to the element h ∈ H ( Y, Z ) . Then f lifts to Y h if andonly if f ∗ h = h . Double covers of Q are parametrised by H ( Q, Z ) and spin structures correspondto elements which are non-zero on restriction to the fibres of Q → M . So to lift thegenerator f of the Z m -action to P we must find a suitable invariant h ∈ H ( Q, Z ).For certain values of m , this can always be done. Lemma 9. If m is odd or m = 2 r then there is a Z m -invariant element of H ( Q, Z ) corresponding to a spin structure P → Q .Proof. If m is odd, take any h ∈ H ( Q, Z ) whose restriction to a fibre is non-zero andaverage over the group Z m to give an invariant element. Since m is odd, the fibrewiserestriction remains non-zero.When m = 2 r we use a lemma from knot theory: the 2 r -fold cover M ′ → S branched along a knot has H ( M, Z ) = 0 (see, e.g., page 16 of [15]). It follows that M has a unique spin structure, H ( Q, Z ) ∼ = Z and the generator of H ( Q, Z ) is Z m -invariant.So, when m is odd or m = 2 r , there is a choice of spin structure P → Q for whichgenerator of the Z m -action lifts. Upstairs in P , the lifted action has order 2 m . Thiscan be seen by considering the Z m -action on a fibre of Q over a point in K : herewe have lifted the standard action of rotation by 2 π/m on SO(3) to its double coverSU(2); it is straightforward to check the order upstairs is 2 m . The action of Z m on he universal cover SL(2 , C ) of P is generated (again, in appropriate coordinates) byright multiplication by the matrix U which is diagonal with entries ϕ and ϕ − , where ϕ = e πi/m .Now we can “twist” the action around the knot, to introduce a singularity justas in the model case. We define an action of Z m on P by sending the generator toconjugation by U . More correctly, this defines an action of Z m on the universal coverof P ; but we have only altered the original Z m -action by left-multiplication and alldeck transformations come from right-multiplication, so the Z m -action on SL(2 , C )commutes with the action of π ( M ) by deck transformations and hence descends toa Z m -action on P . We consider the quotient P/ Z m . Just as the model singularityprojects SL(2 , C ) / Z m → H / Z m there is a projection P/ Z m → S . Away from theknot K this is a locally trivial S -bundle, over a small neighbourhood of K we haveprecisely the model singularity considered above in § X → P/ Z m , giving a complex manifold with trivial canonicalbundle. Lemma 10.
The resolution X admits a fixed-point free C ∗ -action.Proof. The Z m -action on P is given (on the universal cover SL(2 , C )) by conjugationby a diagonal matrix with determinant 1. Hence, on the universal cover, it commuteswith left-multiplication by the whole C ∗ of such diagonal matrices. Since the remainingdeck transformations are all given by right-multiplication, they also commute withleft-multiplication by C ∗ , hence this C ∗ -action descends to P/ Z m and then lifts tothe resolution X . The action is free on X since the same is true of the C ∗ -action onSL(2 , C ). To show that our examples are simply connected, we begin with the following standardlemma.
Lemma 11.
Let X and Y be two finite dimensional CW complexes and let f : X → Y be a surjective map with connected fibres. Suppose that Y has an open cover by sets U i such that for any y ∈ U i the inclusion homomorphism π ( f − ( y )) → π ( f − ( U i )) is an isomorphism. Then the following sequence is right-exact: π ( f − ( y )) → π ( X ) → π ( Y ) → . Lemma 12.
Let X be the complex threefold associated to a π/m -hyperbolic knot, asdescribed above. Then X is simply-connected.Proof. First, we consider the projection f : P/ Z m → S . Away from the knot K this is a locally trivial S -fibration. The fibre of f over a point in K is the quotientSU(2) / Z m via the action (4) which is easily seen to be homeomorphic to S . Hence, toapply Lemma 11 to f , we just need to check that each point of the knot K is containedin an open set U ⊂ S such that π ( f − ( U )) = 0. It suffices to do this in the localmodel of Z m acting on H fixing a geodesic (as in § B centred at a point p on the fixed geodesic. Parallel transport in the radial directionsof B gives a retraction of the frame bundle over B to the fibre over p and, hence, aretraction of the portion S | B of SL(2 , C ) lying over B to the copy of SU(2) over p .This retraction is Z m -equivariant, hence the portion ( S | B ) / Z m of SL(2 , C ) / Z m lyingover B/ Z m retracts onto SU(2) / Z m ∼ = S . In particular, it is simply connected asrequired.It follows that π ( P/ Z m ) = 0. To deduce that the resolution X is also simplyconnected we apply Lemma 11 again, this time to the map r : X → P/ Z m . Away rom the singular locus, r is one-to-one; meanwhile each point on the singular locushas preimage a chain of m − S . So to apply the lemma we must show thateach point of the singular locus has a neighbourhood U for which π ( r − ( U )) = 0. Thisfollows from the fact that, near any point in the singular locus of P/ Z m , the singularitylooks locally like the product D × ( C / Z m ) of a disc with the A m -singularity and theresolution looks locally like the product D × A m of a disc with the A m -resolution. Lemma 13.
There is no compatible K¨ahler structure on the complex threefold asso-ciated to a π/m -hyperbolic knot.Proof. The fact that our examples are simply-connected (Lemma 12) and admit a free C ∗ -action (Lemma 10) implies that they have no compatible K¨ahler metric. For ifthey admitted a compatible K¨ahler metric, by averaging it would be possible to findan S ⊂ C ∗ invariant K¨ahler form. Now b = 0 implies that the symplectic S -actionwould, in fact, be Hamiltonian and hence have fixed points. π -hyperbolic knots We now turn to the question of the diffeomorphism type of our examples in the topo-logically most simple case, that of a π -hyperbolic knot. We proceed via Wall’s clas-sification theorem [33]. Wall’s result states that oriented, smooth, simply-connected,spin, 6-manifolds with torsion free cohomology are determined up to oriented-diffeo-morphism by: • the integer b ; • the symmetric trilinear map H × H × H → Z given by cup-product • the homomorphism H → Z given by cup-product with the first Pontrjagin class.The goal of this section is to prove Theorem 14.
Given a π -hyperbolic knot, the resulting complex threefold constructedabove is diffeomorphic to S × S ) S × S ) . To compute the cohomology of the complex threefold, we begin with the topologyof the orbifold P/ Z . In fact, the results here hold for P/ Z m with any choice of m .Let K ⊂ S be a 2 π/m -hyperbolic knot (with m odd or m = 2 r ) and let U be asmall tubular neighbourhood of K . We write the boundary of U as S × S where S is a meridian circle, which is contractible in U , and S is a longitudinal circle, whichis contractible in S \ K . Let f : P/ Z m → S denote the projection from the complexorbifold to the hyperbolic orbifold. Write X = f − ( S \ K ) and X = f − ( U ).In what follows, (co)homology groups are taken with coefficients in Z unless ex-plicitly stated. We begin with a couple of standard topological lemmas. Lemma 15.
Given any knot K ⊂ S , the knot complement has H ( S \ K ) ∼ = Z ,generated by the class of S , whilst H ( S \ K ) ∼ = 0 ∼ = H ( S \ K ) . Lemma 16.
Every
SO(4) -bundle over a 3-manifold with H ∼ = 0 ∼ = H is trivial. We apply these results to show that for j = 1 ,
2, the subset f − ( S j ) carries all thehomology of X j . Lemma 17. f : X → S \ K is a trivial S -fibration. The inclusion f − ( S ) → X induces an isomorphism on homology.Proof. SL(2 , C ) → H is an SU(2)-bundle so, in particular is an S -bundle withstructure group SO(4). Moreover, this SO(4) structure is preserved by the imageof SU(2) × SL(2 , C ) in SO(4 , C ). Accordingly, away from K , the map f : X → S \ K has structure group SO(4). The result now follows from the previous two lemmas. emma 18. X retracts to f − ( K ) . The inclusion f − ( S ) → X induces an iso-morphism on homology.Proof. To prove that X retracts as claimed it suffices to consider the model of § , C ) / ( Z ⊕ Z m ). The orthogonal retraction of H onto the ( Z ⊕ Z m )-fixed geodesicinduces a retraction of a small tubular neighbourhood of the geodesic loop in thequotient H / ( Z ⊕ Z m ). This lifts to give the claimed retraction of the pre-image of thetubular neighbourhood in SL(2 , C ) / ( Z ⊕ Z m ). It follows from this that the embedding f − ( S ) → X induces an isomorphism on homology.Now we are in a position to compute the homology of the complex orbifold. Proposition 19. P/ Z m has the integral homology of S × S .Proof. We compute the cohomology of P/ Z m by applying the Mayer–Vietoris sequenceto the pair ( X , X ). For this we need the maps ( i ) ∗ and ( i ) ∗ induced on homologyby the inclusions i : X ∩ X → X and i : X ∩ X → X . It follows from Lemma 17that X ∩ X is the product f − ( U \ K ) × S . In particular, it retracts to f − ( S × S ) = S × S × S .Let i ′ and i ′ denote the compositions of the retraction of X ∩ X to S × S × S with a consecutive projection to S × S and S × S respectively. The precedingLemmas 17 and 18 show that the images of the maps ( i ) ∗ and ( i ) ∗ coincide withthe images of the maps ( i ′ ) ∗ and ( i ′ ) ∗ respectively. It follows from this that theMayer–Vietoris sequence of the pair ( X , X ) can be identified with the Mayer–Vietorissequence of the pair (( S \ K ) × S , U × S ). Since this second sequence calculatesthe homology of S × S , the proposition is proved.The singular locus of P/ Z m is an elliptic curve C . The next step is to computethe homology of the pair ( P/ Z m , C ). Lemma 20.
The non-zero relative homology groups H j ( P/ Z m , C ) are H ( P/ Z m , C ) ∼ = Z , H ( P/ Z m , C ) ∼ = Z , H ( P/ Z m , C ) ∼ = Z . Proof.
This follows immediately from the exact sequence of the pair ( P/ Z m , C ) alongwith the fact that P/ Z m has the integral homology of S × S .To convert this into information about the homology of the resolution X we restrictto the case when K is π -hyperbolic. Proposition 21.
Let X be the complex threefold constructed from a π -hyperbolic knot,as described above. Then X has the integral homology of S × S ) S × S ) .Proof. Let E ⊂ X denote the exceptional divisor of the resolution X → P/ Z . Thesingularity in P/ Z is locally the product of the elliptic curve C with the A -singularityin C . So E ∼ = CP × C , with normal bundle O ( −
2) pulled back from CP Since
X/E (where E is crushed to a point) is homeomorphic to ( P/ Z ) /C , itfollows that the relative groups H j ( X, E ) ∼ = H j ( P/ Z , C ) are given by Lemma 20. Wenow consider the long exact sequence of the pair ( Z, E ). Since H ( E ) = 0 = H ( Z, E ),we have that H ( Z ) = 0. Meanwhile, since H ( Z, E ) = 0 = H ( Z, E ) we have that H ( Z ) ∼ = H ( E ) ∼ = Z .From here we can compute all the Betti numbers. Indeed, by Poincar´e duality, b ( Z ) = b ( Z ) = 1; now considering the alternating sum of the ranks of the groupsin the part of the sequence from H ( Z, E ) = 0 to H ( Z ) = 0 gives b = 4. So therational homology of Z coincides with that of the connected sum. It remains to showthat H ( Z ) and H ( Z ) are torsion-free. ince H ( Z, E ) = 0, the long exact sequence gives0 → H ( E ) → H ( Z ) j → H ( Z, E ) → · · · As H ( Z, E ) ∼ = Z is torsion-free, all torsion in H ( Z ) must be contained in ker j .However, ker j ∼ = H ( E ) ∼ = Z is torsion-free, hence so is H ( Z ).The argument for H ( Z ) is more involved. First, note that since H ( Z ) = 0and H ( Z, E ) ∼ = Z ∼ = H ( E ), the map H ( Z, E ) → H ( E ) in the sequence of thepair ( Z, E ) is an isomorphism. Now the preceding part of the sequence gives that H ( E ) → H ( Z ) is surjective. E ∼ = C × CP , so H ( E ) ∼ = Z is generated by theclass of a CP -fibre and the class of a C -fibre. Since H ( Z ) has rank 1, to show that H ( Z ) ∼ = Z it suffices to show that the C -fibre in E is null-homologous in Z .For this it suffices to consider the model case of the resolution X ′ → SL(2 , C ) / ( Z ⊕ Z ) as in § C in X ′ , then theretraction in H onto the fixed geodesic pushes this 3-chain into an arbitrarily smallneighbourhood of the exceptional divisor; hence there is such a 3-chain in the resolu-tion X .SL(2 , C ) / ( Z ⊕ Z ) is the quotient of SL(2 , C ) / Z by an involution with fixed locusan elliptic curve C . Away from C , there is a two-to-one map SL(2 , C ) / Z X ′ .This extends to the blow-up ˆ X of SL(2 , C ) / Z along C to give a ramified double coverˆ X → X ′ . (This is analogous to the fact that, for bundles over CP , squaring O ( − →O ( −
2) gives a ramified double cover of the A -resolution.) The ramification locus inˆ X is the exceptional divisor ˆ E of the blow up ˆ X → SL(2 , C ) / Z . ˆ E is identifiedwith branch locus in X ′ , which is the exceptional divisor E of the resolution X ′ → SL(2 , C ) / ( Z ⊕ Z ). So ˆ E ∼ = CP × C and to show that a C -fibre of E is null-homologousin X ′ it suffices to show that a C -fibre of ˆ E is null-homologous in ˆ X .To prove this statement, first note that SL(2 , C ) / Z is homeomorphic to S × S × R .The elliptic curve C corresponds to taking the product of a Hopf circle S ⊂ S with S × { pt } . Choosing a different Hopf circle gives another copy C ′ of C whichlies entirely inside (SL(2 , C ) / Z ) \ C . On the one hand, C ′ is null-homologous in(SL(2 , C ) / Z ) \ C (simply move it in the R -direction so that it lies in a complete copyof S × S ). On the other hand, when thought of as a 2-cycle in the blow-up ˆ X , C ′ is homologous to a C -fibre of ˆ E . Hence the C -fibre of ˆ E is zero in homology and theproposition is proved. Lemma 22.
Let X be the complex threefold constructed from a π -hyperbolic knot, asdescribed above. Then all the Chern classes of X vanish.Proof. We already know that c = 0 and c = 0 (as it is the Euler characteristic).Since H ( X ) is generated by the exceptional divisor E , to prove c = 0 it suffices toshow that h c , E i = 0. Now, the normal bundle to E ∼ = CP × C is O ( −
2) pulled-backfrom CP . Combining this with the fact that c ( X ) = 0 gives that over E we have atopological isomorphism T X | E ∼ = O ( − ⊕ O (2) ⊕ O . Since all of these bundles arepulled back to E from CP we see that h c , E i = 0, as claimed.We are now ready to prove that X is diffeomorphic to 2( S × S ) S × S ). Proof of Theorem 14.
In order to apply Wall’s Theorem we must first check that X is spin, which follows from the fact that it is complex with trivial canonical bundle.Next, we must show the cup-product is trivial on H . In the proof of Proposition21, we saw that H ( X ) is generated by the Poincar´e dual e to the exceptional divisor E . Since H ( X ) is generated by E , it suffices to check that h e , E i = 0. The normalbundle of E ∼ = CP × C is pulled back from CP . Hence, when restricted to E , e isalso pulled back from CP and so squares to zero. inally, we must show that p ( X ) = 0, but this follows from the formula p = c − c combined with c = 0 = c . In this section we prove that the hyperbolic orbifold can be recovered from the complexthreefold, so distinct orbifolds lead to distinct threefolds.
Theorem 23.
Let X and X ′ be the complex threefolds constructed from hyperbolicorbifolds N and N ′ as above. If X and X ′ are biholomorphic then N and N ′ areisometric.Proof. Suppose that X is constructed from an orbifold metric on S with cone angle2 π/m , whilst X ′ involves the cone angle 2 π/m ′ . Away from the elliptic curve C ⊂ P/ Z m of singular points, r : X → P/ Z m is an isomorphism. The exceptional locus E = r − ( C ) is biholomorphic to the product of C with a chain Θ m of m − CP s asin the A m -resolution.First, we claim that any analytic surface in X is contained in E . Any surface notcontained in E would project to a surface in P/ Z m ; its preimage would then be ananalytic surface in P = SL(2 , C ). However, it is known that the quotient of SL(2 , C )by a cocompact subgroup never contains an analytic surface (Theorem 9.2 of [18]). Itfollows that E is the union of all surfaces in X . Since X and X ′ are biholomorphic,the union of all surfaces in X ′ is also biholomorphic to C × Θ m . It follows that theexceptional loci E ′ , E are biholomorphic and so m = m ′ .Next, note that a neighbourhood of E is canonically biholomorphic to the productof C with a neighbourhood of Θ m in the A m -resolution. The canonical biholomorphismis provided by the C ∗ -action on P/ Z m which acts by translations on C and which liftsto X . It follows that there is a canonical choice of contraction of the exceptionalloci E and E ′ which produces P/ Z m and P ′ / Z m . Since X and X ′ are biholomorphic, P/ Z m ∼ = P ′ / Z m . By construction, the orbifold universal cover of P/ Z m is SL(2 , C ) andthat of N is H and both are obtained as quotients by the action of the same group.Hence the orbifold fundamental groups of P/ Z m and N coincide. So P/ Z m ∼ = P ′ / Z m induces an isomorphism between the orbifold fundamental groups of N and N ′ andhence, by Mostow rigidity, an isometry between N and N ′ . In this section we will explain how a similar approach—passing from a hyperbolicorbifold to a symplectic manifold via a crepant resolution—can be used to build asimply-connected symplectic manifold with c = 0 which does not admit a compatibleK¨ahler structure.We content ourselves here with a single example for which it is not hard to finda crepant resolution because of the simple nature of the singularities. In § Our construction is based on a beautiful hyperbolic 4-manifold called the Davis man-ifold M (see [8] as well as the description in [26] which also computes the homologygroups of M ). The key fact for us is that M admits an isometric involution whichkills the fundamental group. This is analogous to the role played in the Kummerconstruction by the involution z
7→ − z on an abelian surface. is built using a regular polytope called the 120-cell (or hecatonicosachoron).The 120-cell is a four-dimensional regular solid with 120 three-dimensional faces, the“cells”, each of which is a solid dodecahedron. Each edge is shared by 3 dodecahedraand each vertex by 4 dodecahedra. In total, the 120-cell has 600 vertices, 1200 edgesand 720 pentagonal faces. Take a hyperbolic copy P ⊂ H of the 120-cell in which thedihedral angles are 2 π/
5. For each pair of opposite dodecahedral faces of P there isa unique hyperbolic reflection which identifies them. Gluing opposite faces via thesereflections gives the hyperbolic four-manifold M .The central involution of H which fixes the centre of P preserves both P and theidentifications of opposite faces, hence it gives an isometric involution σ of M . Oursymplectic construction will begin with the resulting orbifold M/σ , which we call the
Davis orbifold .To analyse the fixed points of σ it is helpful to use the so-called “inside-out”isometry of M (defined in [26]). To describe this, note that P can be divided up into14400 hyperbolic Coxeter simplexes. The vertices of a simplex are given by takingfirst the centre of P , then the centre of one of its 120 3-faces F , then the centre of oneof the 12 2-faces f of F , then the centre of one of the 5 edges e of f and, finally, one ofthe two vertices of e . Denote by v , v , v , v , v one such choice. The correspondingsimplex has a isometry that exchanges v (the centre of P ) with v (a vertex of e ), v (the centre of F ) with v (the centre of an edge of F ′ ) and fixes v (the centre of f ).This isometry of the simplex extends to define the inside-out isometry of M , whichcommutes with σ . Lemma 24.
The fixed set of σ consists of points. The quotient M/σ is simplyconnected as a topological space.Proof.
In the interior of the 120-cell there is only one fixed point, the centre, all otherfixed points of σ lie on the image in M of the boundary of P . Let F denote the imagein M of a three-dimensional face of P ; σ preserves F and induces on it the symmetryof the dodecahedron given by inversion x
7→ − x with respect to its centre. So, onceagain, in the interior of F there is only one fixed point, its centre. Considering allopposite pairs of three-dimensional faces of M this gives 60 more fixed points of σ .All remaining fixed points are contained in the image in M of the union of the 2-facesof P .The symmetry σ takes 2-faces to 2-faces. We claim next that σ does not fix aninterior point of any pentagonal 2-face. Assume for a moment that it does fix sucha point. Then it would give an involution of the pentagon which would hence fix avertex and so also the line joining the vertex to the centre of the polygon. The Davismanifold has two distinguished points, the centre and the image of all the vertices ofthe 120-cell. The assumption that σ fixes an interior point of a pentagonal 2-face givesa σ -fixed tangent direction at the vertex point in M . However, the inside-out isometryexchanges the centre and vertex of M . Since σ acts as x
7→ − x at the centre it doesso also at the vertex and hence acts freely on the tangent space there. It follows that σ does not fix an interior point of any 2-face.The remaining fixed points are contained in the image in M of the union of the edgesof P . Under the inside-out involution of M , the middles of all edges are exchangedwith centres of all 3-faces whilst the centre of P is exchanged with the image in M of the vertices of P . Since the inside-out isometry commutes with σ , this give anadditional 61 fixed points of σ making 122 in total.We now turn to the (topological, not orbifold) fundamental group π ( M/σ ). Themap π ( M ) → π ( M/σ ) is surjective so we need to show its image is trivial. Considerthe 60 closed geodesics γ i in M going through the centre of P and joining the centresof opposite faces. The deck transformations corresponding to these geodesics generatethe whole of π ( M ). Indeed, these deck-transformations take the fundamental domain to all its 120 neighbours. Now the result follows from the fact that every loop σ ( γ i )is contractible. Locally, the singularities of
M/σ are modelled on the quotient of H by x
7→ − x . (Here x is the coordinate provided by the Poincar´e ball model of H .) From Proposition 2we know that the corresponding symplectomorphism of R is given by z
7→ − z in thevector-bundle fibres of O ( − ⊕ O ( − O ( − ⊕ O ( − / Z is a K¨ahlerCalabi–Yau orbifold with singular locus CP corresponding to the zero section. Wenext describe a crepant resolution of this singularity. Lemma 25.
There is a crepant resolution O ( − , − → O ( − ⊕ O ( − / Z where O ( − , − → CP × CP is the tensor product of the two line bundles given bypulling back O ( − → CP from either factor.Proof. Blow up the zero section of O ( − ⊕ O ( −
1) to obtain the total space of O ( − , − → CP × CP . The Z -action lifts to this line bundle where it again hasfixed locus the zero section and acts by z
7→ − z in the fibres. For such an involutionon any line bundle L , the square gives a resolution L → L/ Z . Hence O ( − , − → O ( − , − / Z → O ( − ⊕ O ( − / Z gives the claimed resolutionIt must be emphasised that this is resolution is holomorphic . The total space of O ( − , −
2) is a K¨ahler manifold with trivial canonical bundle and, away from theexceptional divisor, the map in Lemma 25 is a biholomorphism when we consider O ( − ⊕ O ( − / Z with its holomorphic complex structure. However, when con-structing symplectic six-manifolds from hyperbolic four-manifolds, the relevant almostcomplex structure and volume form on O ( − ⊕ O ( −
1) (and its Z -quotient) are notthe holomorphic ones; rather we use the SO(4 , holomorphic geometryof O ( − ⊕ O ( − / Z and not the SO(4 , O ( − ⊕ O ( −
1) to the SO(4 , Lemma 26.
Let R δ denote the part of R lying over a geodesic ball in H of radius δ . For any δ > , there is an SO(4) -invariant compatible almost complex structure J on R and an SO(4) -invariant nowhere-vanishing section Ω of the J -canonical-bundlesuch that: • Over R δ , J and Ω agree with the standard holomorphic structures. • Over R \ R δ , J and Ω agree with the SO(4 , -invariant structures from Propo-sition 2Proof. As is standard, an SO(4)-invariant interpolation between the “inside” and “out-side” Hermitian metrics gives the existence of J .To produce Ω we start with a description of the SO(4)-action away from the zero-section R . The stabiliser of a point p ∈ R \ R is a circle S p ⊂ SO(4) and the orbit of is 5-dimensional (in fact, isomorphic as an SO(4)-space to the unit tangent bundleof S ). The lift of a geodesic ray out of the origin in H meets each SO(4)-orbit in aunique point, giving a section for the action. We interpolate between the holomorphicand hyperbolic complex volume forms along the relevant portion of this lifted ray andthen use the SO(4)-action to extend the resulting 3-form to the whole of R . In orderfor this to work it is sufficient that at every point p ∈ R \ R the action of S p on thefibre of the J -canonical-bundle at p is trivial. But since the weight is integer valuedand continuous it is constant on R \ R so we can compute it for some p outside of R r where everything agrees with the hyperbolic picture. Here we already have anSO(4)-invariant (hence S p -invariant) complex volume-form so the weight is zero asrequired. With Lemmas 25 and 26 in hand, we can now take a crepant resolution of the twistorspace of the Davis orbifold
M/σ . Let Z → M denote the twistor space of the Davismanifold The involution σ lifts to an involution of Z which we still denote σ . Z/σ is a symplectic orbifold with singularities along 122 CP s, each modelled on O ( − ⊕O ( − / Z .Let δ be a positive number small enough that the geodesic balls in M of radius2 δ centred on the σ -fixed points are embedded and disjoint. Then, by Lemma 26, on Z we can find a new almost complex structure J and complex volume form Ω suchthat outside the geodesic 2 δ -balls they agree with the hyperbolic structures comingfrom Proposition 2, whilst inside the balls of radius δ they agree with the holomorphicstructures coming from the holomorphic geometry of O ( − ⊕ O ( − J and Ω are σ -invariant.In this way the quotient Z/σ is a symplectic orbifold with an almost complexstructure and complex volume form which are modelled near the singular curves onthe holomorphic geometry of O ( − ⊕O ( − / Z . It follows from Lemma 25 that thereis a resolution ˆ Z → Z/σ in which the singular curves have been replaced by copiesof CP × CP with normal bundle O ( − , − Z carries an almost complexstructure ˆ J and complex volume form ˆΩ so that c ( ˆ Z, ˆ J ) = 0.Finally we need to define the symplectic structure on ˆ Z . Pulling back the sym-plectic form via ˆ Z → Z gives a symplectic form on the complement of the exceptionaldivisors. To extend it we use a standard fact about resolutions in K¨ahler geometry.Given any neighbourhood U of the zero locus in O ( − , − O ( − , −
2) for which the projection to O ( − ⊕ O ( − / Z is an isometry on thecomplement of U . (This amounts to the fact that the zero locus has negative normalbundle.)So, in the model, the pull-back of the symplectic form extends over the exceptionaldivisor in a way compatible with holomorphic complex structure. Taking U sufficientlysmall and doing this near all 122 exceptional divisors defines a symplectic form ω onˆ Z which is compatible with ˆ J . This section proves that ˆ Z is simply connected and admits no compatible integrablecomplex structure. The second fact will follow from the first and the fact that b ( ˆ Z ) =0. Lemma 27. ˆ Z is simply connected.Proof. We first apply Lemma 11 to the map
Z/σ → M/σ . The fibres are S s and wesee that π ( Z/σ ) = 1. Next we apply Lemma 11 to ˆ Z → Z/σ . This time the fibresare points or S s and we deduce that π ( ˆ Z ) = 1. o prove that b ( ˆ Z ) = 0 we invoke a lemma of McDuff on the cohomology ofmanifolds obtained by symplectic blow-ups. Lemma 28 (McDuff [21]) . Let X be a symplectic manifold and C ⊂ X a smoothsymplectic submanifold of codimension k . Let ˜ X denote the blow-up of X along C .Then the real cohomology of ˜ X fits into a short exact sequence of graded vector spaces → H ∗ ( X ) → H ∗ ( ˜ X ) → A ∗ → where the first arrow is pull-back via ˜ X → X and where A ∗ is free module over H ∗ ( C ) with one generator in each dimension j , ≤ j ≤ k − . Lemma 29. b ( ˆ Z ) = 0 .Proof. Recall that Z → M is the twistor space of the Davis manifold. We first blowup the 122 fibres which lie over the fixed points of σ to obtain the new manifold˜ Z . It follows from Lemma 28 that pulling back cohomology via ˜ Z → Z induces anisomorphism H ( ˜ Z ) ∼ = H ( Z ).Next, notice that σ lifts to ˜ Z and that ˆ Z = ˜ Z/σ . We now show that σ acts as − H ( ˜ Z ). To see this, consider the action of σ on the Davis manifold M . It acts on H ( M ) as − − H ( M ). Now Z → M is a sphere-bundle so, byLeray–Hirsch, H ∗ ( Z ) is a free module over H ∗ ( M ) with a single generator in degree 2corresponding to the first Chern class of the vertical tangent bundle. This generatoris preserved by σ , so σ acts as − H ( Z ) and hence also as − H ( ˜ Z ). Fromthis we deduce that H ( ˆ Z ) = 0. For if it contained a non-zero element, the pull-backto ˜ Z would be a σ -invariant element of H ( ˜ Z ). Corollary 30.
There is no K¨ahler structure on ˆ Z with c = 0 . In particular, thesymplectic structure on ˆ Z described above admits no compatible complex structure.Proof. For a K¨ahler manifold, the vanishing of b implies the Picard torus is trivial.Now c = 0 implies the existence of a holomorphic volume form, hence b ≥ Z , it is not clear, to us at least, whether or not ˆ Z admits a K¨ahler structure when onedoes not place a restriction on c . In this section we explain how hyperbolic and complex-hyperbolic geometry in higherdimensions leads to symplectic manifolds for which the first Chern class is a positivemultiple of the symplectic class, non-K¨ahler analogues of Fano manifolds.
The passage from hyperbolic 4-manifolds to symplectic 6-manifolds can be generalisedto every even dimension, with hyperbolic 2 n -manifolds giving symplectic n ( n + 1)-manifolds. This was first explained via twistors, by Reznikov [28] (although Reznikovdid not consider the first Chern class of his examples). Here we give an alternativedescription in terms of coadjoint orbits, as in § so (2 n,
1) consists of (2 n + 1) × (2 n + 1) matrices of the form (cid:18) u t u A (cid:19) , here u is a column vector in R n and A ∈ so (2 n ). Those elements with u = 0generate so (2 n ) ⊂ so (2 n, so (2 n,
1) and so gives anequivariant isomorphism so (2 n, ∼ = so (2 n, ∗ . As in § ξ = (cid:18) J (cid:19) where J ∈ so (2 n ) is a choice of almost complex structure on R n (i.e., J = − ξ is U( n ). We write Z n for the coadjoint orbit of ξ .Note that Z n ∼ = SO(2 n, / U(2 n ) is symplectic of dimension n ( n + 1). It fibresover hyperbolic space H n ∼ = SO(2 n, / SO(2 n ) with fibre isomorphic to the spaceSO(2 n ) / U( n ) of orthogonal complex structures on R n inducing a fixed orientation.In other words, it is the twistor space of H n .The same proof as in Lemma 3 gives the following result. Lemma 31.
There is an isomorphism of U ( n ) -representation spaces: so (2 n, ∼ = u ( n ) ⊕ Λ ( C n ) ∗ ⊕ C n . Given a point z ∈ Z n with stabiliser U( n ) ⊂ SO(2 n, there is a U ( n ) -equivariantisomorphism T z ∼ = Λ ( C n ) ∗ ⊕ C n , (5) in which the Λ ( C n ) ∗ summand is tangent to the fibre of the projection Z n → H n . As in Lemma 4, by U( n )-equivariance, the symplectic form on T z is proportionalunder (5) to the form induced by the Euclidean structure on C n . To show this constantof proportionality is positive, first check that the forms are genuinely equal in the case n = 1, where so (2 , ∼ = u (1) ⊕ C as a U(1)-representation. This amounts to the factthat Z = H with symplectic form the hyperbolic area form. Next, use inductionand the fact that the decompositions of so (2 n,
1) and so (2 n + 2 ,
1) from Lemma 31are compatible with the obvious inclusions of the summands induced by a choice of C n ⊂ C n +1 .Having seen that the isomorphism of (5) is symplectic, we define a compatibleSO(2 n, Z n by declaring (5) to be a complexlinear isomorphism. With this almost complex structure, T Z = V ⊕ H splits as a sumof complex bundles with V ∼ = Λ H ∗ , corresponding to (5).In the twistorial picture, this is simply the decomposition of T Z induced by theLevi–Civita connection of H n . The almost complex structure here is the (non-integrable) “Eells–Salamon” structure, given by reversing the (integrable) “Atiyah–Hitchin–Singer” structure in the vertical directions. Lemma 32. c ( Z n ) = (2 − n ) c ( H ) .Proof. This follows from
T Z = Λ H ∗ ⊕ H along with the fact that for any complexrank n vector bundle E , c (Λ E ) = ( n − c ( E ).We now determine the symplectic class of Z n . First, consider the restrictionof the symplectic structure to the fibres of Z n → H n . It follows from (5) thatthe fibres are symplectic and almost-complex submanifolds. Moreover, the stabiliserSO(2 n ) ⊂ SO(2 n,
1) of a point x ∈ H n acts on the fibre F x over x preserving boththese structures. As mentioned above, F x ∼ = SO(2 n ) / U( n ) is the space of orthogonalcomplex structures on T x H n inducing a fixed orientation. The standard theory ofsymmetric spaces gives F x a symmetric K¨ahler structure. It follows from SO(2 n )-equivariance that this must agree with the restriction of the symplectic and almostcomplex structures from Z n . t is also standard that the symmetric K¨ahler structure on F = SO(2 n ) / U( n )comes from a projective embedding. Let E → F denote the “tautological” bundle:each point of F is a complex structure on R n ; the fibre of E at a point J ∈ F is thecomplex vector space ( R n , J ). The bundle det E ∗ is ample and c (det E ∗ ) = − c ( E )is represented by the symmetric symplectic form on F .In our situation, the splitting (5) tells us that the tautological bundle of the fibre F x is simply H | F x . It follows that on restriction to a fibre, the symplectic class agreeswith − c ( H ). However, topologically, Z n ∼ = F × H n is homotopic to F . Since theirfibrewise restrictions agree, it follows that − c ( H ) is equal to the symplectic class of Z n . Hence, writing ω for the symplectic form on Z n , we have: Proposition 33. c ( Z n ) = ( n − ω ] . The symplectic form and almost complex structure on Z n as well as the splitting T Z n = V ⊕ H and identification V ∼ = Λ H ∗ are SO(2 n, Z n by subgroups of SO(2 n, c is a positive multiply of [ ω ]. Let Γ ⊂ SO(2 n,
1) be thefundamental group of a compact hyperbolic 2 n -manifold M . Γ acts by symplectomor-phisms on Z n to give as quotient a symplectic manifold of dimension n ( n + 1). Itfibres Z n / Γ → H n / Γ over M as the twistor space of M . When n ≥
3, these arecompact symplectic manifolds for which c is a positive multiple of [ ω ] (for n ≥ M has fibres CP ∼ = SO(6) / U(3).
The constructions presented here seem, to us at least, to lead several natural questions.We describe some of these below.
Whilst we only give one example of a simply-connected symplectic manifold with c =0, we plan to exploit a similar construction to produce simply connected 6-dimensionalexamples with arbitrarily large Betti numbers (see the forthcoming article [11]). Bycomparison, note that it is still unknown if there are infinitely many topologicallydistinct K¨ahler
Calabi–Yau manifolds of fixed dimension.First we describe an infinite sequence of simply-connected compact symplecticorbifolds with c = 0. They come from hyperbolic orbifold metrics on S , built usingCoxeter polytopes. Recall that a hyperbolic Coxeter polytope is a convex polytopewith totally geodesic boundary in H n and whose dihedral angles are π/k for k ∈ N ;the polytope is said to be right-angled if all dihedral angles are π/ Lemma 34.
Let P be an n -dimensional compact hyperbolic Coxeter polytope. Dou-bling P gives a hyperbolic orbifold metric on S n .Proof. Let G be the group of isometries of H n generated by reflections in the faces of P and let G ′ be the subgroup of G of index two consisting of all orientation preservingelements. Then H n /G ′ is the double of P . Indeed the fundamental domain of G is P itself, whilst the fundamental domain of G ′ is P ∪ P ′ ; where P ′ is the reflection of P n a face. Identifying further the faces of P ∪ P ′ via G ′ we obtain the double of P .The double is homeomorphic to S n , since P is homeomorphic to a closed n -ball.Up to dimension 6, compact hyperbolic Coxeter polytopes are known to be abun-dant. Theorem 35 (Potyagilo–Vinberg [25], Allcock [1]) . In all dimensions up to 6 thereexist infinitely many compact hyperbolic Coxeter polytopes. Moreover, in dimensionsup to 4 there are infinitely many compact hyperbolic right-angled polytopes.
We are interested in the case of 4 and 6 dimensions. The right angled 4-dimensionalpolytopes are constructed by Potyagilo–Vinberg [25]. Here one uses that there is ahyperbolic 120-cell with dihedral angles π/
2. Gluing it to itself along a 3-face (viathe reflection in that face) produces a new right-angled polytope. This procedurecan be repeated giving infinitely many examples. An infinite family of 6-dimensionalpolytopes is constructed by Allcock [1].Doubling the 4-dimensional hyperbolic Coxeter polytopes gives infinitely manysimply-connected symplectic 6-orbifolds with c = 0.When the polytope is doubled, the hyperbolic metric extends smoothly across the3-faces; the singularities correspond to the 2-skeleton of the polytope. To understandthe singularities in the twistor space, consider the positive “octant” x , x , x , x ≥ R , i.e., the infinitesimal model for the vertex of a right-angled Coxeter polytope.The ( x , x )-plane lifts to two planes L, L ′ in the twistor space, corresponding to thetwo compatible complex structures on R for which the ( x , x )-plane is a complex line.Reflection in the ( x , x )-plane lifts to the twistor space where it fixes L, L ′ pointwise.The other coordinate 2-planes behave similarly. The lifts of coordinate 2-planes aresome of the points with non-trivial stabiliser under the action on the twistor spacegenerated by the reflections. The only other points with non-trivial stabiliser are thoseon the twistor line over the origin. This line is fixed pointwise by the composition ofreflection in one 2-plane with reflection in the orthogonal 2-plane.It follows that the orbifold singularities of the twistor space of a doubled right-angled Coxeter polytope are of two sorts. The generic orbifold point has structuregroup Z and is modelled on the quotient of C by ( z , z , z ) ( z , − z , − z ). Thesecorrespond either to the lifted coordinate 2-planes or the central twistor line in theabove picture. Then there are isolated points in the singular locus, where three surfacesof generic orbifold points meet. Here the structure group is ( Z ) . This is mostsymmetrically described via the action of ( Z ) on C where each generator changesthe sign on one of the three coordinate 2-planes in C ; the diagonal Z acts trivially sothe action factors through ( Z ) . Such points correspond in the picture above to theintersection of the twistor line at the origin with the lifts of two orthogonal coordinate2-planes.The concrete description of these singularities means that it is possible to findsymplectic crepant resolutions “by hand”, as it was for the Davis orbifold. Thisgives an infinite collection of simply-connected symplectic manifolds with c = 0 [11].Meanwhile, the twistor spaces of the 6-orbifolds of Allcock give an infinite collectionof simply-connected symplectic Fano 12-orbifolds. It is natural to ask if these admitsymplectic Fano resolutions, although this looks much harder to answer than thecorresponding question for the 6-orbifolds.On the subject of symplectic resolutions, we mention the recent work of Nieder-kr¨uger–Pasquotto [23, 24], which gives a systematic approach to the resolution ofsymplectic orbifolds arising via symplectic reduction, although they do not considerdiscrepancy. .2 Possible diversity of symplectic 6-manifolds with c = 0 On the subject of hyperbolic orbifolds, there is a much more general existence question,which we learnt from Gromov:
Question 36.
Is there any restriction on the manifolds which can be obtained asquotients of H n by a cocompact discrete subgroup of O( n, ? (The subgroup is allowed to have torsion, of course.) For n = 2 and 3 all compactmanifolds can be obtained as quotients ( n = 2 is straightforward whilst n = 3 usesgeometrization). The relationship between hyperbolic and symplectic geometry out-lined here gives additional motivation to try to answer the question in dimension four.For example, can any finitely presented group be the fundamental group of such aquotient? The 4-dimensional quotients give rise to 6-dimensional symplectic orbifoldswith c = 0 and in this way one might ambitiously hope to approach the problemof which groups can appear as the fundamental group of a symplectic manifold with c = 0. Of course, even if one had the orbifolds, they would still need to be resolved. Inalgebraic geometry, crepant resolutions of threefolds always exist, thanks to the workof Bridgeland–King–Reid [4]. Even independently of the hyperbolic orbifold approachdescribed here, it would be interesting to know what holds in the symplectic setting. We conclude our discussion of the symplectic examples with a brief look at the genuszero Gromov–Witten invariants of hyperbolic twistor spaces.
Lemma 37.
Let CP → Z n be pseudoholomorphic with respect to the SO(2 n, -invariant almost complex structure defined above. Then the image lies entirely in afibre of Z n → H n .Proof. A theorem of Salamon [29] (Eells–Salamon [9] in the case n = 2) shows thatthe image in H n of a pseudoholomorphic curve in Z n is a minimal surface. Since H n contains no minimal spheres the result follows.It follows that there are pairs of points in Z n (or any of its compact quotients)which do not lie on a pseudoholomorphic rational curve. This is in stark contrastto the situation for K¨ahler Fano manifolds. Of course, we consider here a particular almost complex structure. The symplectic definition of rational connectivity (see Li–Ruan [19]) involves the non-vanishing of certain genus zero Gromov–Witten invariants.A weaker property than rational connectivity is that of being uniruled. In algebraicgeometry a variety is called uniruled if each point is contained in a rational curve.Again, the definition in symplectic geometry involves a statement about genus zeroGromov–Witten invariants. Li–Ruan [19] have asked if all symplectic Fano manifoldsare uniruled in this sense.We content ourselves here by mentioning that the above restriction on the imageof rational curves suggests an approach to computing the genus zero Gromov–Witteninvariants of the twistor spaces of hyperbolic 2 n -manifolds. Whilst the invariant almostcomplex structure is not generic, the obstruction bundle should be describable insimple terms. Indeed, using this lemma it should be possible to localise calculationsto the case of a curve in the fibre of Z n → H n and exploit the action of the SO(2 n )of isometries fixing the point in H n . In [27] Miles Reid asked a question, which is now referred to as
Reid’s fantasy : uestion 38 (Reid [27]) . Consider the moduli space of complex structures on ( S × S ) N with K ∼ = O which are deformations of Moishezon spaces. Is it true that forlarge N this space is irreducible?Do all simply-connected K¨ahler Calabi-Yau threefolds appear as small resolutionsof 3-folds with double points lying on the boundary of these moduli spaces? The complex threefolds that we obtain do not appear directly in Reid’s fantasy andindeed seem to be of a very different nature. There is no visible mechanism that wouldenable one to connect these examples in any way to Moishezon manifolds. It seemsmore reasonable that the structures we construct on 2( S × S ) S × S ) belongto an infinite family of disconnected components of complex structures with K ∼ = O .If this were true, it would show that Reid was wise to limit himself in his fantasieswhen he wrote “I aim to consider only analytic threefolds which are deformations ofMoishezon spaces” ([27], page 331). Of course we don’t know how one could try toprove (or disprove) the existence of this infinite number of connected components. References [1] D. Allcock. Infinitely many hyperbolic Coxeter groups through dimension 19.
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D´epartment de Math´ematique, Universit´e Libre de Bruxelles CP218, Boulevard duTriomphe, Bruxelles 1050, Belgique. [email protected]