Negative Sasakian structures on simply-connected 5-manifolds
aa r X i v : . [ m a t h . DG ] J u l NEGATIVE SASAKIAN STRUCTURES ONSIMPLY-CONNECTED -MANIFOLDS VICENTE MU ˜NOZ, MATTHIAS SCH ¨UTT, AND ALEKSY TRALLE
Abstract.
We study several questions on the existence of negative Sasakianstructures on simply connected rational homology spheres and on Smale-Bardenmanifolds of the form k ( S × S ). First, we prove that any simply connectedrational homology sphere admitting positive Sasakian structures also admits anegative one. This result answers the question, posed by Boyer and Galicki intheir book [3], of determining which simply connected rational homology spheresadmit both negative and positive Sasakian structures. Second, we prove that theconnected sum k ( S × S ) admits negative quasi-regular Sasakian structuresfor any k . This yields a complete answer to another question posed in [3]. Introduction
This work deals with Sasakian manifolds. The basic definitions and facts re-garding this structure are given in Section 2. Recall that the Reeb vector field ξ on a co-oriented contact manifold ( M, η ) determines a 1-dimensional foliation F ξ called the characteristic foliation . If we are given a manifold M with a Sasakianstructure ( η, ξ, φ, g ), then one can define basic Chern classes c k ( F ξ ) of F ξ whichare elements of the basic cohomology H kB ( F ξ ) (see [3, Theorem/Definition 7.5.17]).We say that a Sasakian structure is positive (negative) if c ( F ξ ) can be representedby a positive (negative) definite (1 , c ( F ξ ) = 0. If none of these, it is called indefinite.The following problems are formulated in the seminal book on Sasakian geometryof Boyer and Galicki [3]. Question . Which simply connected ra-tional homology 5-spheres admit negative Sasakian structures?To put this question into a broader context, let us recall the foundational resultsof Koll´ar [15], which yield the full structural description of rational homologyspheres M with H ( M, Z ) = 0 and admitting positive Sasakian structures.
Theorem 1 ([15, Theorem 9.1]) . Let M → ( X, ∆) be a -dimensional Seifertbundle, with M smooth, where ∆ = P (cid:0) − m i (cid:1) D i is the branch divisor. Then Mathematics Subject Classification.
Key words and phrases.
Sasakian, Smale-Barden manifold, Seifert bundle, singular surface. (1) If M is a rational homology sphere with H ( M, Z ) = 0 , then (a) X has only cyclic quotient singularities, and H ( X, Z ) = Weil( X ) ∼ = Z , (b) D i are orbismooth curves, intersecting transversally, (c) m i are coprime, and each m i is coprime with the degree of D i . (2) Conversely, given any ( X, ∆) satisfying (a), (b), (c), there is a unique Seifertbundle M → X such that M is a rational homology sphere with H ( M, Z ) =0 . In (1b) above, D i are axes in a cyclic coordinate chart of the form C / Z m , thatis D i is either { z = 0 } or { z = 0 } .There are very few rational homology spheres which admit positive Sasakianstructures. Theorem 2 ([3, Theorem 10.2.19], [15, Theorem 1.4]) . Suppose that a rationalhomology sphere M admits a positive Sasakian structure. Then M is spin and H ( M, Z ) is one of the following: , Z m , Z , Z , Z , Z , Z , Z n , where n > , and m ≥ , m not divisible by . Conversely, all these cases dooccur. Theorem 3 ([3, Theorem 10.3.14]) . Let M be a rational homology sphere. If itadmits a Sasakian structure, then it is either positive, and the torsion in H ( M, Z ) is restricted by Theorem 2, or it is negative. There exist infinitely many such mani-folds which admit negative Sasakian structures but no positive Sasakian structures.There exist infinitely many positive rational homology spheres which also admitnegative Sasakian structures. In particular, any torsion group which is realizable by a simply connected Sasakianrational homology sphere M but which does not appear in the list given by Theo-rem 2 gives an example of an answer to Question 1.In the smaller class of semi-regular Sasakian structures, Question 1 is answeredby the following result. Theorem 4 ([22]) . Let m i ≥ be pairwise coprime, and g i = ( d i − d i − .Assume that gcd( m i , d i ) = 1 . Let M be a Smale-Barden manifold with H ( M, Z ) = r ⊕ i =1 Z g i m i and spin, with the exceptions Z m , Z n , Z . Then M admits a negative semi-regular Sasakian structure. Conversely, if M is a simply-connected rational homol-ogy -sphere admitting a semi-regular Sasakian structure, then it must satisfy theabove assumptions. Theorem 26 in [8] shows that in the quasi-regular case Theorem 4 does not hold.The above discussion motivates the first problem we want to address, whichcertainly contributes to Question 1.
EGATIVE SASAKIAN STRUCTURES ON SIMPLY-CONNECTED 5-MANIFOLDS 3
Question . Which simply connected rational homology spheres admit both nega-tive and positive Sasakian structures?The following is a partial result on this question.
Theorem 5 ([10]) . Any simply connected rational homology sphere from Koll´ar’slist (Theorem 2) except (possibly) Z m , m < , and Z n , n > , admits both negativeand positive Sasakian structures. We give a complete answer to Question 2 in Theorem 25: all positive Sasakiansimply connected rational homology spheres also admit negative Sasakian struc-tures.Simply connected rational homology 5-spheres belong to a wider class of 5-manifolds. A 5-dimensional simply connected manifold M is called a Smale-Bardenmanifold . These manifolds are classified by their second homology group over Z and the so-called Barden invariant [1], [30]. In more detail, let M be a compactsmooth oriented simply connected 5-manifold. Let us write H ( M, Z ) as a directsum of cyclic groups of prime power order H ( M, Z ) = Z k ⊕ (cid:0) ⊕ p,i Z c ( p i ) p i (cid:1) , where k = b ( M ). Choose this decomposition in a way that the second Stiefel-Whitney class map w : H ( M, Z ) → Z is zero on all but one summand Z j .The value of j is unique, it is denoted by i ( M ) and is called the Barden invariant.The fundamental question arises, which Smale-Barden manifolds admit Sasakianstructures?One can ask for a generalization of Question 1 for Smale-Barden manifolds with b > Question . Determine which torsion groups correspondto Smale-Barden manifolds admitting negative Sasakian structures.
Theorem 6 ([10]) . For any pair ( n, s ) , n > , s > there exists a Smale–Bardenmanifold M which admits a negative Sasakian structure such that H ( M ) tors =( Z n ) s . Note that the manifold M provided in Theorem 6 is not a rational homologysphere.It is expected that simply connected 5-manifolds with negative Sasakian struc-tures should be little constrained in terms of what kind of H ( M, Z ) is. However,there are only few known examples: • circle bundles over the Fermat hypersurfaces of degree d in C P , d ≥ regular negative Sasakian structures on k ( S × S ) , k = ( d − d − d + 2) + 1 , thus k begins with k = 52 (see [3, Example 5.4.1]); V. MU ˜NOZ, M. SCH ¨UTT, AND A. TRALLE • some links yield negative Sasakian structures on ( S × S ) , ( S × S ) , ( S × S )([3, Corollary 10.3.18] – note the typo in ibid. listing instead of ); • S admits infinitely many inequivalent negative Sasakian structures [3,Proposition 10.3.13].Motivated by this, Boyer and Galicki pose the following question that pertains toSmale-Barden manifolds with torsion-free homology. Question . Show that all k ( S × S ) admit negativeSasakian structures or determine precisely for which k this holds true.In Theorem 29 we give a complete answer to this question: any k ( S × S )admits negative Sasakian structures. In this work we consider only spin 5-manifolds M . The reason for this can be explained as follows. A Sasakian structure ( η, ξ, φ, g )is said to be η -Einstein if the Ricci curvature tensor of the metric g satisfies theequation Ric g = λg + νη ⊗ η for some constants λ and ν . It is known [5] thatby the orbifold version of the theorem of Aubin and Yau every negative Sasakianstructure can be deformed to a Sasakian η -Einstein structure. On the other hand,the following theorem holds. Theorem 7 ([5, Theorem 14]) . Let M be a non-spin manifold with H ( M, Z ) torsion free. Then M does not admit a Sasakian η -Einstein structure. This shows that negative Sasakian Smale-Barden manifolds should be spin (thatis, their Barden invariant is i ( M ) = 0). This can also be checked by the proof of [6,Proposition 2.6] that also works for simply connected negative Sasakian manifolds.Finally, let us mention that negative Sasakian structures are an important toolin constructing Lorentzian Sasaki-Einstein metrics and, therefore, receive extraattention in physics [5]. Acknowledgment . The first author was partially supported by Project MINECO(Spain) PGC2018-095448-B-I00. The third author was supported by the NationalScience Center (Poland), grant. no. 2018/31/B/ST1/000532.
Sasakian manifolds and Seifert bundles
Let M be a smooth manifold of dimension 2 n + 1. A contact metric structure on M consists of a quadruplet ( η, ξ, φ, g ), where η is contact form, ξ is the Reebvector field of this form, φ is a C ∞ -section of End( T M ) and g is a Riemannianmetric on M , satisfying the following conditions: φ = − Id + ξ ⊗ η, g ( φV, φW ) = g ( V, W ) − η ( V ) η ( W ) , for any vector fields V, W on M . Given a contact metric structure ( η, ξ, φ, g ) on M , one defines the fundamental 2-form F on M by the formula F ( V, W ) = g ( φV, W ) . EGATIVE SASAKIAN STRUCTURES ON SIMPLY-CONNECTED 5-MANIFOLDS 5
One can check that F ( φV, φW ) = F ( V, W ) and that η ∧ F n = 0 everywhere. Acontact metric structure is K-contact if L ξ g = 0 for the Lie derivative L ξ . Definition 8.
A contact metric structure ( η, ξ, φ, g ) on M is called normal, if theNijenhuis tensor N φ given by the formula N φ ( V, W ) = φ [ V, W ] + [ φV, φY ] − φ [ φV, W ] − φ [ V, φW ]satisfies the equation N φ = − dη ⊗ ξ. A Sasakian structure is a K-contact structure which is normal.
Definition 9.
A Sasakian structure on a compact manifold M is called quasi-regular if there is a positive integer δ satisfying the condition that each point of M has a neighbourhood U such that each leaf of the foliation F ξ passes through U atmost δ times. If δ = 1 the structure is called regular .We freely use the notion of a cyclic orbifold referring to [3], [16], [20], [21]. Notethat in this work we need only 4-dimensional cyclic orbifolds, so our exposition willbe restricted only to this case and simplified accordingly. Also we refer for sometechnical results to [20], where the symplectic versions are stated. Note that aK¨ahler orbifold is in particular an almost-K¨ahler and hence a symplectic orbifold,hence the results of [20] hold also.For a singular K¨ahler manifold X with cyclic singularities, a local model arounda singular point x ∈ X is of the form C / Z d , where θ = exp(2 πi/d ) ∈ S acts onthe neighbourhood of x by the formula(1) exp(2 πi/d )( z , z ) = ( e πie /d z , e πie /d z ) , where gcd( e , d ) = gcd( e , d ) = 1. We will write d = d ( x ). An orbismooth curve(as in (1b) of Theorem 1) is a complex curve D ⊂ X such that around x it is ofthe form D = { z = 0 } or D = { z = 0 } . Two orbismooth curves D , D intersectnicely if at every intersection point x ∈ D ∩ D , there is a chart C / Z d at x suchthat D = { ( z , } and D = { (0 , z ) } .Here is a method of constructing cyclic K¨ahler orbifolds. Proposition 10 ([20]) . Let X be a singular K¨ahler -manifold with cyclic singu-larities, and set of singular points P . Let D i be embedded orbismooth curves inter-secting nicely. Take coefficients m i > such that gcd( m i , m j ) = 1 , if D i and D j intersect. Then there exists a K¨ahler orbifold structure X with isotropy surfaces D i of multiplicities m i , and singular points x ∈ P of multiplicity m = d ( x ) Q i ∈ I x m i ,where I x = { i | x ∈ D i } . If P = ∅ , then the family D i with multiplicities m i as in Proposition 10 definesa structure of a smooth cyclic K¨ahler orbifold.An important basic tool of constructing cyclic K¨ahler orbifolds in this work is byblowing-down complex surfaces along chains of smooth rational curves of negative V. MU ˜NOZ, M. SCH ¨UTT, AND A. TRALLE self-intersection < − − Proposition 11 ([2]) . Consider the action of the cyclic group Z m on C given by ( z , z ) ( ηz , η r z ) , where η = e πi/m , < r < m and gcd( r, m ) = 1 . Then writea continuous fraction mr = [ b , . . . , b l ] = b − b − b − ... The resolution of C / Z m has an exceptional divisor formed by a chain of smoothrational curves of self-intersection numbers − b , − b , . . . , − b l . Proposition 12 ([8, Lemma 15]) . Conversely, let X be a smooth complex sur-face containing a chain of smooth rational curves E , . . . , E l of self-intersections − b , − b , . . . , − b l , with all b i ≥ , intersecting transversally (so that E i ∩ E i +1 arenodes, i = 1 , . . . , l − ). Let π : X → ¯ X be the contraction of E = E ∪ . . . ∪ E l .Then ¯ X has a cyclic singularity at p = π ( E ) , with an action given by Proposition11. Moreover, if D is a curve intersecting transversally a tail of the chain (that is,either E or E l at a non-nodal point), then the push down curve ¯ D = π ( D ) is anorbismooth curve in ¯ X . We also need the self-intersection of the orbismooth curve of Proposition 12.Take D intersecting E and ¯ D = π ( D ) (the case where D intersects E l can betreated by reversing the chain of rational curves). Let [ b , . . . , b l ] = mr . Then(2) ¯ D = D + rm . We check this as follows. Let β i = [ b i , . . . , b l ], so that β i = b i − β i +1 . We provethe assertion by induction on l . Blow-down the chain of l − E , . . . , E l ,getting a blow-down map ̟ : X → ˆ X with a singular point q . The curve ˆ E = ̟ ( E ) has self-intersection ˆ E = − b + β = − β by the induction hypothesis while ˆ D = ϕ ( D ) ∼ = D , so ˆ D = D .Contracting ˆ E yields another map ψ : ˆ X → ¯ X such that π = ψ ◦ ̟ , and ¯ D = ψ ( ˆ D ). Take the pull-back ψ ∗ ( ¯ D ) = ˆ D + x ˆ E , andcompute x ∈ Q by using 0 = ˆ E · ψ ∗ ( ¯ D ) = 1 − xβ , so x = β and ψ ∗ ( ¯ D ) = D + β ˆ E .Now ¯ D = ψ ∗ ( ¯ D ) = ˆ D · ψ ∗ ( ¯ D ) = ˆ D + x = D + β , as stated.The basic method of constructing Sasakian structures is the method of Seifertbundles [3], [15], [16]. Definition 13.
Let X be a cyclic oriented 4-orbifold. A Seifert bundle over X isan oriented 5-manifold M endowed with a smooth S -action and a continuous map EGATIVE SASAKIAN STRUCTURES ON SIMPLY-CONNECTED 5-MANIFOLDS 7 π : M → X such that for an orbifold chart ( U, ˜ U , Z m , ϕ ) there is a commutativediagram ( S × ˜ U ) / Z m ∼ = −−−→ π − ( U ) y π y ˜ U / Z m ∼ = −−−→ U where the action of Z m on S is by multiplication by exp(2 πi/m ), and the topdiffeomorphism is S -equivariant.The relation between quasi-regular Sasakian manifolds and cyclic orbifolds isgiven by the following result. Theorem 14 ([3, Theorems 7.5.1 and 7.5.2]) . Let M be a manifold endowed with aquasi-regular Sasakian structure ( η, ξ, φ, g ) . Then the space of leaves of the foliation F ξ determined by the Reeb vector field has a natural structure of a cyclic K¨ahlerorbifold, and the projection M → X is a Seifert bundle. Conversely, if ( X, ω ) is acyclic K¨ahler orbifold and M is the total space of the Seifert bundle determined bythe class [ ω ] of the K¨ahler form, then M admits a quasi-regular Sasakian structure. In particular, regular Sasakian structures correspond to circle bundles M → ( X, ω ) over K¨ahler manifolds with integral K¨ahler form ω . These bundles are de-termined by the first Chern class equal to [ ω ]. They are often called the Boothby-Wang fibrations. Note that by a theorem of Rukimbira [27], any manifold whichadmits a Sasakian structure, admits a quasi-regular one. In this work we areinterested in the existence questions, and all Sasakian structures constructed inthis article will be quasi-regular. Therefore, our approach amounts to construct-ing Seifert bundles over cyclic K¨ahler orbifolds with prescribed properties. It isalso useful to have an intermediate class of Sasakian structures [16], [21], [22]. ASasakian structure is called semi-regular , if it is determined by a Seifert bundle M → X whose base orbifold is smooth.Let us mention the following. For the convenience of references, we will inter-changebly follow the terminology and notation of [3], [15], [16]. If X is a cyclicorbifold with singular set P and a family of surfaces D i we will say that we aregiven a divisor ∪ D i with multiplicities m i >
1. The formal sum(3) ∆ = X i (cid:18) − m i (cid:19) D i will be called the branch divisor .Given a cyclic K¨ahler orbifold X , with singular points P , and branch divisor(3), we need an extra piece of information in order to determine a Seifert bundle M → X . For each point x ∈ P with multiplicity m = dm m as in Proposition10, we have an adapted chart U ⊂ C / Z m with action(4) exp(2 πi/m )( z , z ) = ( e πij /m z , e πij /m z ) , V. MU ˜NOZ, M. SCH ¨UTT, AND A. TRALLE so that the local model of the Seifert fibration is as in Definition 13. We call j x = ( m, j , j ) the local invariants at x ∈ P . Note that this relates to the originalK¨ahler structure of X given by (1) by the formulas j = m e and j = m e .Assume that D = { ( z , } is one of the isotropy surfaces with multiplicity m .The local invariant of D is by definition, j D = ( m , j ), where j is consideredmodulo m . This also yields a compatibility condition for the local invariants ofsingular points and isotropy surfaces.In order to construct Seifert bundles, we need to assign local invariants to each ofthe singular points x ∈ P and each of the isotropy surfaces D i ⊂ X , in a compatibleway. For this we need to choose for each D i some j i with gcd( j i , m i ) = 1. Thenwe can assign local invariants at the singular points using the following result. In[20] it is stated in the case that the isotropy surfaces D i are disjoint, but it is alsovalid in the singular points which do not lie in the intersection of two surfaces. Proposition 15 ([20, Proposition 25]) . Suppose that X is a cyclic -orbifold suchthat each singular point x ∈ P lies in a single isotropy surface D i , if any. Takeintegers j i with gcd( m i , j i ) = 1 for each D i . Then there exist local invariants for X . Once compatible local invariants are fixed, a Seifert bundle is determined by itsorbifold Chern class. Given a Seifert bundle π : M → X , the order of a stabilizer(in S ) of any point p in the fiber over x ∈ X is denoted by m = m ( x ), as in (4). Definition 16.
For a Seifert bundle M → X define the first Chern class as follows.Let l = lcm( m ( x ) | x ∈ X ). Denote by M/l the quotient of M by Z l ⊂ S . Then M/l → X is a circle bundle with the first Chern class c ( M/l ) ∈ H ( X, Z ). Define c ( M ) = 1 l c ( M/l ) ∈ H ( X, Q ) . In the proofs of our results we basically need to construct a Seifert bundle M → X as in Proposition 17 such that H ( M, Z ) = 0. In order to explain the con-struction, we recall some results from [20]. As before, X is a 4-dimensional cyclicorbifold. Let p j be a cyclic isotropy point of some order d j > P be the set ofall such points. Consider small balls around all the points p j ∈ P , say, B = ⊔ B j .Every L j = ∂B j is a lens space of order d j . Let L = ⊔ L j . From the Mayer-Vietorisexact sequence of X − B and ¯ B and the equalities H ( L j , Z ) = 0 , H ( L j , Z ) = Z d j one derives an exact sequence0 → H ( X, Z ) → H ( X − P, Z ) → ⊕ j Z d j . Let X o = X − B . It is a manifold with boundary L . There is the Poincar´e duality H ( X o , Z ) × H ( X o , L, Z ) → H ( X o , L, Z ) . There are isomorphisms H ( X − P, Z ) = H ( X o , Z ) EGATIVE SASAKIAN STRUCTURES ON SIMPLY-CONNECTED 5-MANIFOLDS 9 and H k ( X o , L, Z ) = H k ( X, B, Z ) = H k ( X, P, Z ) = H k ( X, Z ) , k ≥ , since the dimension of P is zero. Hence the Poincar´e duality is a perfect pairing H ( X − P, Z ) × H ( X, Z ) → Z . In particular, the class [ D i ] ∈ H ( X, Z ) yields a map H ( X − P, Z ) → Z , and,therefore, a cohomology class in H ( X, Q ) via the inclusion H ( X, Z ) ⊂ H ( X − P, Z ) . The first orbifold Chern class of the Seifert bundle can be calculated using theformula given by the following result, that we state only for complex orbifolds.
Proposition 17 ([20], Proposition 35) . Let X be a cyclic -orbifold with a complexstructure and D i ⊂ X complex curves of X which intersect transversally. Let m i > such that gcd( m i , m j ) = 1 if D i and D j intersect. Suppose that there aregiven local invariants ( m i , j i ) for each D i and j p for every singular point p , whichare compatible. Choose any < b i < m i such that j i b i ≡ m i ) . Let B be acomplex line bundle over X . Then there exists a Seifert bundle M → X with thegiven local invariants and the first orbifold Chern class c ( M ) = c ( B ) + X i b i m i [ D i ] . The set of all such Seifert bundles forms a principal homogeneous space under H ( X, Z ) , where the action corresponds to changing B .Moreover, if X is a K¨ahler cyclic orbifold and c ( M ) = [ ω ] for the orbifoldK¨ahler form, then M is Sasakian. Note that [ D i ] are understood as cohomology classes in H ( X, Q ) according tothe explanation before Proposition 17. Proposition 18 ([20, Lemma 34]) . Assume that we are given a Seifert bundle M → X over a cyclic orbifold with the set of isotropy points P and branch divisor P (1 − m i ) D i . Let µ = lcm( m i ) . Then c ( M/µ ) = µ c ( M ) is integral in H ( X − P, Z ) . Recall that an element a in a free abelian group A is called primitive if it cannotbe represented as a = kb with non-trivial b ∈ A , k ∈ N , k > Theorem 19 ([20]) . Suppose that π : M → X is a quasi-regular Seifert bundleover a cyclic orbifold X with isotropy surfaces D i and set of singular points P . Let µ = lcm( m i ) . Then H ( M, Z ) = 0 if and only if (1) H ( X, Z ) = 0 , (2) H ( X, Z ) → ⊕ H ( D i , Z m i ) is onto, (3) c ( M/µ ) ∈ H ( X − P, Z ) is primitive. Moreover, H ( M, Z ) = Z k ⊕ ( ⊕ i Z g i m i ) , g i = genus of D i , k + 1 = b ( X ) . Thus, if one wants to check the assumptions of Theorem 19, one in particularcalculates H ( X − P, Z ) and checks the primitivity of c ( M/µ ) in H ( X − P, Z ).The first orbifold Chern class of the Seifert bundle can be calculated using theformula in Proposition 17.Let us explain the notion of a definite Sasakian structure in more detail. Recallthat a Sasakian structure is positive (negative), if c ( F ξ ) can be represented by apositive (negative) definite (1 , c ( F ξ ) is an element ofthe basic cohomology of the foliated manifold ( M, F ξ ). A differential form α on M is called basic if i ξ α = 0 and i ξ dα = 0. Consider the spaces of basic k -forms Ω kB ( M ).Clearly, the de Rham differential takes basic forms to basic forms, the restrictionof the differential d onto Ω ∗ B ( M ) yields a differential complex (Ω ∗ B , d B ) and a basiccohomology H ∗ B (Ω ∗ B , d B ) = H ∗ B ( M ). In the Sasakian case, the transverse geometryis K¨ahler, and the basic cohomology inherits the bigrading H p,qB ( M ). If π : M → X is a Seifert bundle determined by a quasi-regular Sasakian structure, then by [3,Proposition 7.5.23], c ( F ξ ) = π ∗ c orb1 ( X ). Since c ( F ξ ) is represented by a basicform (see [3, Section 7.5.2]), it follows that c ( F ξ ) is represented by a negativedefinite (1 , c orb1 ( X ) < , Proposition 20. A -manifold M admits a quasi-regular negative Sasakian struc-ture if and only if the base X of the corresponding Seifert bundle M → X has theproperty that the canonical class K orb X is ample. In the case of rational homology spheres Sasakian structures fall into two classes.
Definition 21.
A Sasakian structure is called canonical (anti-canonical) , if c orb1 ( X )is a positive (negative) multiple of [ dη ] B .Clearly, an anti-canonical Sasakian structure is negative. Note that in generalthere are positive or negative Sasakian structures on M which are not canonicalor anti-canonical. If b ( M ) >
0, then the condition of being canonical or anti-canonical chooses a ray in the K¨ahler cone K ( X ). Thus, there are many positive(negative) Sasakian structures determined by Seifert bundles M → X whose firstChern class c ( M ) is not proportional to c ( F ξ ). However, if M is a rationalhomology sphere, then any Sasakian structure is either canonical or anti-canonical. Proposition 22 ([3, Proposition 7.5.29]) . Let M be a (2 n +1) -dimensional rationalhomology sphere. Any Sasakian structure on M satisfies c ( F ξ ) = a [ dη ] B for somenon-zero constant a . Hence, any Sasakian structure on M is either positive ornegative. Finally, recall that we are interested on Smale-Barden manifolds obtained asparticular Seifert bundles M → X . Therefore, we need to check that π ( M ) = 1.We will systematically use the following. EGATIVE SASAKIAN STRUCTURES ON SIMPLY-CONNECTED 5-MANIFOLDS 11
Definition 23.
The orbifold fundamental group π orb1 ( X ) is defined as π orb1 ( X ) = π ( X − (∆ ∪ P )) / h γ m i i = 1 i , where h γ m i i = 1 i denotes the following relation on π ( X − (∆ ∪ P )): for any smallloop γ i around a surface D i in the branch divisor, one has γ m i i = 1.We will use without further notice the following exact sequence · · · → π ( S ) = Z → π ( M ) → π orb1 ( X ) → . It can be found in [3, Theorem 4.3.18]. It is easy to see that if H ( M, Z ) = 0 and π orb1 ( X ) is abelian, then π ( M ) must be trivial. This holds since if H ( M, Z ) = 0,then π ( M ) has no abelian quotients. As π orb1 ( X ) is assumed abelian, we find that π ( M ) is a quotient of Z , hence again abelian. This implies that π ( M ) = 0.The following result will be used in the calculations of the orbifold fundamentalgroups. Proposition 24 ([23]) . If Z is a smooth simply-connected projective surface withsmooth complex curves C i intersecting transversally and satisfying C i > , then π ( Z − ( C ∪ . . . ∪ C r )) is abelian. Rational homology spheres with positive and negative Sasakianstructures
The aim of this section is to prove the following result.
Theorem 25.
Any simply connected rational homology sphere admitting a positiveSasakian structure admits also a negative Sasakian structure.
By Theorem 5, it only remains to check that the homology groups H ( M, Z ) = Z n , n >
0, and Z m , m <
5, from Theorem 2 can be covered. We shall work outeach of them separately.We begin with some known facts on Hirzebruch surfaces [2], [11], [13]. Fix n ≥ F n is the projectivization of the vector bundle O C P ( n ) ⊕ O C P . For a holomorphic section σ : C P → O C P ( n ), we denote by E σ the image of( σ,
1) : C P → O C P ( n ) ⊕ O C P in F n . The curve E ⊂ F n is called the zero sectionof F n . As E ≡ E σ for all sections σ , we have that E = n . Let C denote the fiberof the fibration F n → C P . Then C = 0 , E · C = 1 . It is known [32] that for n > F n contains a unique irreducible curve E ∞ of negative self-intersection: E ∞ · E ∞ = − n. This curve is called the section at infinity since it is given by the image of ( σ,
0) : C P → O C P ( n ) ⊕ O C P . One can calculate E ∞ ≡ E − nC, K F n ≡ ( n − C − E , where K F n is the canonical divisor. Clearly H ( F n , Z ) = Z h C, E ∞ i . Now we are ready to prove our first main result.
Proposition 26.
The simply connected rational homology sphere with H ( M, Z ) = Z n , n ≥ , and spin admits a negative Sasakian structure.Proof. Consider F n with the zero section E , the section at infinity E ∞ , and thefiber C of F n → C P . Let β ≥ D ≡ C + βE . This exists because C + βE is very ample (it can be represented by a K¨ahler form)and Bertini’s theorem is applicable. By the genus formula D + K F n · D = 2 g ( D ) − g ( D ) = β − β · n. One also calculate D · E ∞ = ( C + βE ) · E ∞ = 1 , as E · E ∞ = 0. Consider also rational curves D i = E σ i , i = 1 , . . . , s , which aresections, D i ≡ E . These can be taken to intersect transversally (each other andalso to D ).Let X be the orbifold obtained by the blow-down of F n along E ∞ . By Proposition12, X is a cyclic orbifold with a cyclic singularity p of degree d ( p ) = n . Forsimplicity we denote curves on F n and on X by the same letters. Endow X withan orbifold structure with the branch divisor(6) ∆ = (cid:18) − m (cid:19) D + s X i =1 (cid:18) − m i (cid:19) D i , where m , . . . , m s are pairwise coprime, and the m i are coprime to m and n (butpossibly gcd( n, m ) > H ( X, Z ) = H ( X − { p } , Z ) = Z h C i , (7) H ( X − { p } , Z ) = H ( X, Z ) = Z h E i . (8)Note that C is a divisor passing through the singular point p . Since the (co)homologygroups in (7), (8) are dual, we can calculate the intersection numbers over Q andobtain C = 1 n EGATIVE SASAKIAN STRUCTURES ON SIMPLY-CONNECTED 5-MANIFOLDS 13 in H ( X, Q ). Indeed, we can write E ≡ a C , for some a ∈ Z , and substitutingthis to n = E = a E · C = a , we get E ≡ n C . Then C = n E = n . The samefollows from (2). We also have D ≡ C + βE ≡ C + βnC = (1 + βn ) C. Considering X as an algebraic variety, we calculate the canonical divisor K X from the adjunction formula (noting that C is orbismooth): K X · C + C = − χ orb ( C ) = 2 g ( C ) − (cid:18) − n (cid:19) . Since g ( C ) = 0 we get K X · C + n = − − n ). Writing K X = b C , b ∈ Q , andsubstituting in the above, we get b = − ( n + 2). So finally K X = − ( n + 2) C. Since all D i are homologous to E , we obtain the following formula for the orbifoldcanonical class:(9) K orb X = K X + (cid:18) − m (cid:19) D + s X i =1 (cid:18) − m i (cid:19) D i = − ( n + 2) C + (cid:18) − m (cid:19) (1 + βn ) C + s X i =1 (cid:18) − m i (cid:19) nC = − ( n + 2) + (cid:18) − m (cid:19) (1 + βn ) + n s X i =1 (cid:18) − m i (cid:19)! C. Now consider a Seifert bundle M → X determined by the orbifold data (6). We need to choose local invariants 0 < j i 32 + n s X i =1 (cid:18) − m i (cid:19) . Choosing m i and s large, we can get positivity for any n . As we said, we take m, m i all of them pairwise coprime, and also gcd( m i , n ) = 1 for all i . Note that we have done this calculation assuming that the assumptions of The-orem 19 are satisfied. The assumption (2) holds for D i since H ( X, Z ) = Z h E i , D i ≡ E , and gcd( m i , n ) = 1. It holds for D since D ≡ (1 + nβ ) C , so the map H ( X, Z ) → H ( D, Z m ) is [ E ] nβ , and gcd(1 + nβ, m ) = 1 for β = m = 2.Let us now check assumption (3) of Theorem 19. By Proposition 17, c ( M ) = c ( B ) + bm [ C + βE ] + X i b i m i [ E ] , for a line bundle B , so c ( B ) = q [ E ], for some integer q ∈ Z . As we take all m, m , . . . , m s pairwise coprime, we have µ = m · m · · · m s . We need to check thatwe can arrange for µ c ( M ) to be integral and primitive in H ( X − P, Z ), whichmeans that we need c ( M/µ ) = µ c ( M ) = [ C ] . Note also that as c ( M ) > 0, we have by Proposition 17 that M is Sasakian. Wecompute µ c ( M ) = b · m · · · m s [ C + βE ] + X i b i · m · m · · · ˆ m i · · · m s [ E ] + µ c ( B ) . In H ( X, Z ) we have [ E ] = n [ C ], so c ( M/µ ) = k [ C ] with(10) k = (1 + nβ ) bm · · · m s + n X i b i mm · · · ˆ m i · · · m s ! + µq. Since we already fixed m = β = 2, we need to arrange b, b i , m i , q to get k = 1.First note that(11) gcd((1 + nβ ) m · · · m s , nmm · · · ˆ m i · · · m s ) = 1 , by the choice of m, m i . Then there are ¯ b, ¯ b i ∈ Z such that1 = (1 + nβ )¯ bm · · · m s + n X i ¯ b i mm · · · ˆ m i · · · m s ! . Take ¯ b = b + mq , ¯ b i = b i + m i q i , where 0 ≤ b < m and 0 ≤ b i < m i . Then1 = (1 + nβ ) bm · · · m s + n X i b i mm · · · ˆ m i · · · m s ! + (cid:16) (1 + nβ ) q + X nq i (cid:17) µ, as required. Finally note that gcd( b, m ) = 1 and gcd( b i , m i ) = 1, so that we cantake local invariants j, j i with bj ≡ m ) and b i j i ≡ m i ) and applyProposition 15.It remains to show that π ( M ) = 1. By Proposition 24 applied to π ( F n − S ), S = D ∪ D ∪ . . . ∪ D s , we get that this group is abelian. Now take the curve E ∞ , that intersects transversally S at one point. Take a loop δ around E ∞ , then π ( F n − ( S ∪ E ∞ )) is generated by δ and π ( F n − S ). Let γ, γ , . . . , γ s be loopsaround each of the divisors D, D , . . . D s . Take a general fiber C of F n → C P , EGATIVE SASAKIAN STRUCTURES ON SIMPLY-CONNECTED 5-MANIFOLDS 15 and intersecting with S ∪ E ∞ , we get a homotopy δ = γ β Q si =1 γ i , since β = C · D and 1 = C · D i , i = 1 , . . . , s . Note also that δ, γ commute since the curves E ∞ , D intersect transversally at one point. Also γ i , γ j commute since D i , D j intersecttransversally (just take the link of the complement of the two curves around anintersection point, which is a 2-torus containing both loops; hence they commute inthis T ). Analogously γ i , γ commute. Finally, δ = γ β Q sj =1 γ j , so it also commuteswith γ i in π ( F n − ( S ∪ E ∞ )).Now recall that π orb1 ( X ) is the quotient of π ( X − S ) = π ( F n − ( S ∪ E ∞ )) withthe relations γ m = 1, γ m i i = 1. Then we have that π orb1 ( X ) is also abelian. Itfollows that π ( M ) is abelian. Since H ( M, Z ) = 0 we get necessarily π ( M ) = 1.By Theorem 19, M is a rational homology sphere with H ( M, Z ) = Z n . (cid:3) The case Z m , m < m = 2 , , Proposition 27. The simply connected rational homology sphere with H ( M, Z ) = Z m , gcd( m, 7) = 1 , and spin, admits a negative Sasakian structure.Proof. Consider a smooth cubic C ⊂ C P and a line L ⊂ C P , tangent to thecubic at some point p ∈ L ∩ C and intersecting transversally at another point p ∈ L ∩ C . We blow-up at p , and let ˜ L , ˜ C denote the proper transforms.Since the initial curves were tangent (intersecting with multiplicity two), ˜ L and ˜ C intersect transversally at a point ˜ p . Blowing up again at ˜ p , we get two curves ˆ L and ˆ C in C P C P with ˆ L ∩ ˆ C = { p } . The second blow-up yields a ( − B (obtained by a blow-up of the exceptional divisor) and a new exceptional divisor E . Since E and ˜ L intersect, one can blow up a third time and get new curvesˇ L, ˇ E = A, B, E , in X = C P C P , where E is the last exceptional divisor. Note that now ˇ L = − 2, and A ∪ B is achain of two ( − A ∪ B and ˇ L and applying Proposition 12we get a cyclic orbifold ¯ X . Let ̟ : X → ¯ X denote the blow-down map. Let q = ̟ ( A ∪ B ) and q = ̟ ( ˇ L ) be the two singularpoints P = { q , q } of orders d ( q ) = 3 and d ( q ) = 2.We have that ¯ C = ̟ ( ˆ C ) is a genus 1 orbismooth curve with two singular points q , q . Let also ¯ E = ̟ ( E ), which is a genus 0 orbismooth curve through q , q ,although the intersection ¯ C ∩ ¯ E is not nice at q .The cohomology of the blow-up is H ( X, Z ) = Z h L, A, B, E i . As we contract3 curves, we have H ( ¯ X, Q ) = Q h ¯ E i . Moreover π ( X ) = 1, hence H ( ¯ X, Z ) is torsion-free. We compute the intersection numbers using (2) to get¯ C = ˆ C + 12 + 23 = 7 + 76 = 496 , ¯ E = E + 12 + 23 = − , ¯ E · ¯ C = E · C + 12 + 23 = 0 + 76 = 76 , hence ¯ C = 7 ¯ E . It is clear that the cycle x = ¯ C − ¯ E can be pushed-off P (since x · A = x · B = x · ˇ L = 0), so x ∈ H ( ¯ X − P, Z ). Since x = 6 ¯ E and ¯ E · x = 1, both x, ¯ E are generators of their respective groups: H ( ¯ X, Z ) = H ( ¯ X − P, Z ) = Z h ¯ E i ,H ( ¯ X, Z ) = H ( ¯ X − P, Z ) = Z h x i . For a generic line L ′ of C P , we have L ′ = ˇ L + B + 2 A + 3 E , so ¯ L ′ = ̟ ( L ′ ) ≡ E .Take a collection of generic lines L , . . . , L s in C P that we transfer to ¯ X . Endow¯ X with the structure of a cyclic K¨ahler orbifold determined by the set of singularpoints P = { p , p } and a branch divisor∆ = (cid:18) − m (cid:19) ¯ C + s X i =1 (cid:18) − m i (cid:19) L i , where m i are integers pairwise coprime and coprime to m . We construct a Seifertbundle M → ¯ X determined by local data associated to this branch divisor. Westart by arranging the Chern class of the Seifert bundle c ( M ) = c ( B ) + bm [ ¯ C ] + P b i m i L i , where c ( B ) = q [ ¯ E ] for some integer q . Then µ = mm · · · m s and(12) c ( M/µ ) = mm · · · m s c ( B ) + bm · · · m s [ ¯ C ] + X b i mm · · · ˆ m i · · · m s [ L ′ ]= (cid:16) mm · · · m s q + 7 bm · · · m s + X b i mm · · · ˆ m i · · · m s (cid:17) [ ¯ E ] . As we are assuming gcd( m, 7) = 1, there isgcd( mm · · · m s , m · · · m s , mm · · · ˆ m i · · · m s ) = 1 , where we also choose m i coprime to 7 and to 3. So we can solve (12) to have c ( M/µ ) = [ ¯ E ]. We can arrange that 0 < b < m , 0 < b i < m i and gcd( b, m ) = 1,gcd( b i , m i ) = 1. This implies first that c ( M ) > M is Sasakian, and secondthat c ( M/m ) ∈ H ( ¯ X − P, Z ) is primitive. To check the assumptions of Theorem19, we need to show the surjectivity of the natural map H ( ¯ X, Z ) → H ( ¯ C, Z m ).But this sends x x · ¯ C = 7, so the map is surjective because gcd( m, 7) = 1. Alsothe maps H ( ¯ X, Z ) → H ( ¯ L i , Z m i ) are surjective (since gcd( m i , 3) = 1), and so isthe map to the direct sum of these. EGATIVE SASAKIAN STRUCTURES ON SIMPLY-CONNECTED 5-MANIFOLDS 17 Now let us compute K orb¯ X . We start by writing K ¯ X = b [ ¯ E ] and computing K ¯ X · ¯ C + ¯ C = − χ orb ( ¯ C ). This gives b 76 + 496 = 1 − 12 + 1 − 13 = ⇒ b = − . Therefore K orb¯ X = (cid:18) − (cid:16) − m (cid:17) + X (cid:16) − m i (cid:17)(cid:19) [ ¯ E ] > , for s and m i large enough (e.g. s ≥ M has a negative Sasakian structure.Finally, we have to calculate π orb1 ( ¯ X ). First we compute π ( ¯ X − P ), which isgenerated by loops α , α around the points q , q . Note that α = 1 , α = 1. Thesphere E going through q , q gives a homotopy α = α . This implies that both α = α = 1, and hence π ( ¯ X − P ) = π ( X − ( A ∪ B ∪ ˇ L )) = 1. We removeˇ C ∪ L ∪ . . . ∪ L s , hence(13) π ( X − ( A ∪ B ∪ ˇ L ∪ ˇ C ∪ L ∪ . . . ∪ L s )) = π ( ¯ X − ( ¯ C ∪ L ∪ . . . ∪ L s ))is generated by loops γ, γ , . . . , γ s , where γ surrounds ˇ C and γ i surrounds L i .As all these curves intersect transversally, we have that all these loops commute.Therefore (13) is abelian. This implies that π orb1 ( ¯ X ) is abelian, and hence M issimply-connected. Again, by Theorem 19, M is a rational homology sphere with H ( M, Z ) = Z m . (cid:3) Remark . In recent work [26] a complete classification of Sasaki-Einstein rationalhomology spheres was obtained. This result is a positive type counterpart to thepresent work which deals with the negative case. One can find more results on thepositive case in [7].4. Quasi-regular negative Sasakian structures on k ( S × S )In this section we give a complete answer to Question 4: Theorem 29. Any k ( S × S ) admits a quasi-regular negative Sasakian structure. The proof of this result will follow from Propositions 32, 33 and 34.In the proof of Proposition 32 (for most part of Theorem 29) we will use ellipticsurfaces with section. For background, the reader is referred to [28]. Let f : Y → C P be an elliptic surface with section O . Denote the general fiber by F . The classifica-tion of singular fibers goes back to Kodaira, but we will only need the semi-stablefibers, denoted by I n , n ∈ N . These are either a nodal cubic ( n = 1) or cycles of n ≥ , Θ , . . . , Θ n − such that Θ meets O , and among fibercomponents, Θ i intersects exactly Θ i − and Θ i +1 , indices taken modulo n . The invariants of Y are determined by the Euler characteristic χ ( Y ) = 12 N .Namely we have q = 0 , p g = N − , b = 12 N − , and h , = 10 N. Note that the components of a fiber of type I n together with the zero sectiongenerate a hyperbolic sublattice L of the N´eron-Severi group NS( Y ), of rank n + 1.Hence Lefschetz’ theorem on (1 , n ≤ h , − N − ρ denote the Picard number of Y . Proposition 30. Let N > . For any n ∈ N such that n ≤ N − , there isan elliptic surface Y → C P with section and χ ( Y ) = 12 N such that there is asingular fiber of type I n and ρ = n + 1 .Remark . By the theory of Mordell-Weil lattices [29], the assumption on thePicard number implies that there are no other reducible fibers and only sectionsof finite order. Proof of Proposition 30. By [18], the elliptic surfaces with χ ( Y ) = 12 N lie insidea (10 N − M N . For N = 1, any such Y is rationalwith ρ = 10. For N > 1, however, a very general Y has ρ = 2. This follows from[14] where all irreducible components of the higher Noether-Lefschetz lociNL r = { Y ∈ M N | ρ ( Y ) ≥ r } , ≤ r ≤ N, are shown to have dimension ≥ N − r , with equality attained outside the isotriviallocus. But an isotrivial fibration admits only additive singular fibers while a generalsurface in M N has only fibers of type I , so the claim follows.Along the same lines, any elliptic surface Y ∈ M N with a singular fiber of type I n lies in the non-isotrivial locus of an irreducible component Z n +1 ⊂ NL n +1 . As soon as this component is shown to be non-empty, it follows from the abovediscussion that a very general member will have ρ = n + 1. The non-emptinesscan be proved in a way similar to the argument in [29, Thm. 8.39]: the ellipticsurface Y with I n fiber can be degenerated to one with I n +1 and so on all the wayto a terminal object with I N − (the maximum possible by Lefschetz as discussedbefore). The existence of this terminal object (for any N ∈ N ) follows from workof Davenport [9] and Stothers [31]. In turn, this implies that all the intermediatestrata of elliptic surfaces with an I n fiber, n < N − 1, are non-empty as well.(Compare the deformation style argument in [19, Lemma 2.4]). (cid:3) We are now in the position to return to Theorem 29. We start by the case oflarge k . Proposition 32. Any k ( S × S ) , k ≥ , admits a quasi-regular negative Sasakianstructure. EGATIVE SASAKIAN STRUCTURES ON SIMPLY-CONNECTED 5-MANIFOLDS 19 Proof. Consider first the case k > 1. Fix N > n ≤ N − Y bean elliptic surface as in Proposition 30. Then Y is simply connected and we have O = − χ ( O Y ) = − N . Thus O together with the fibre components Θ , . . . , Θ n − form a chain D of n smooth rational curves where all the Θ i are ( − D to a point p of order d =( N − n + 1 on a singular surface X . Note that[ N, , . . . , | {z } ( n − 1) times ] = dn = ( N − n + 1 n . By construction, we have b ( X ) = 12 N − − n, and ρ ( X ) = 1 , where NS( X ) = Z h F i . Since the fiber F contains p , we have by (2) that F = nd . We continue to provethat K orb X > 0. To this end, it suffices to prove that K orb X · F > 0, since ρ ( X ) = 1.But the latter can be computed using adjunction through K orb X · F + F = − χ orb ( F ) = 1 − d , since F is orbismooth. Therefore K orb X · F = d − n − d = ( N − n ( N − n + 1 , so the claim follows as soon as N > 2. For such N , and 1 ≤ n ≤ N − 1, wehave b ( X ) = k + 1 = 12 N − − n ∈ [2 N − , N − N ≥ k + 1 ≥ 5, that is k ≥ M → X according to the general procedure asin Proposition 17. Note that there is no branch divisor and a singular point oforder d = ( N − n + 1. We need to ensure that the assumptions of Theorem19 are satisfied. In this case, we only need to check that c ( M ) is primitive in H ( X − P, Z ). We just take a line bundle M → X − P , whose first Chern class isample and primitive, and extend it over X as a Seifert bundle.Finally we claim that π orb1 ( X ) = 1. To see this, note that π orb1 ( X ) is a quotientof π ( X − { p } ) = π ( Y − D ). Here the last group is generated by a small loop β around D , so π orb1 ( X ) is cyclic generated by β , and hence abelian, so the claimfollows as before. Hence we have a quasi-regular negative Sasakian structures on M , which is diffeomorphic to k ( S × S ), by the classification of Smale-Bardenmanifolds and Theorem 7. By construction, this can be done for all k ≥ (cid:3) In order to complete Theorem 29, we are left with the cases k = 1 , , 3. We startwith k = 1. Proposition 33. The -manifold S × S admits a negative Sasakian structure. Proof. Let X = C P C P be the blow-up of C P at a point p . The cohomology H ( X, Z ) = Z h L, E i is generated by a line L ⊂ C P and an exceptional divisor E .A standard calculation yields K X = − L + E . The ample cone is generated by L − E, L , and the effective cone is generated by L − E, E . Take a collection of generalconics in C P through p , and denote C , . . . , C s ⊂ X the proper transforms. Inhomology, [ C i ] = 2 L − E and C i , C j intersect transversally at 3 points (outside theexceptional divisor). Introduce a structure of a smooth orbifold on X by choosinga branch divisor ∆ = s X i =1 (cid:18) − m i (cid:19) C i , where m i are pairwise coprime integer numbers. By the general formula for theorbifold canonical class we get(14) K orb X = − L + E + s X i =1 (cid:18) − m i (cid:19) (2 L − E ) . As in all previous considerations we use Proposition 17 and Theorem 19, and con-struct a Seifert bundle determined by the orbifold structure on X and an orbifoldChern class c ( M ) = [ ω ]. This will prove the Proposition provided that K orb X > K X withelements of the effective cone: K orb X · ( L − E ) > , K orb X · E > . Substituting (14) to these inequalities, we see that they are satisfied for sufficientlylarge s and m i . For example, the first inequality yields − s P i =1 (1 − m i ) > 0, so s ≥ m i will do. The second equality says − s P i =1 (1 − m i ) > M → X determined by the local invariants ( m i , b i )and the orbifold first Chern class c ( M ) = c ( B ) + s X i =1 b i m i [ C i ] . We need to check assumptions (2) and (3) of Theorem 19. Since C i · E = 1,the canonical map H ( X, Z ) → H ( C i , Z ) sends [ E ] → E · C i = 1, therefore theinduced map H ( X, Z ) → H ( C i , Z m i ) is onto. As m i are coprime, we see that(2) is satisfied. For checking assumption (3), note that the K¨ahler forms are [ ω ] = a ( L − E )+ a L , a , a > 0. Take a line bundle B and write c ( B ) = β ( L − E )+ β L , β , β ∈ Z , and calculate(15) c ( M ) = (cid:18) β + X b i m i (cid:19) [ L − E ] + (cid:18) β + X b i m i (cid:19) [ L ] . This is a K¨ahler form if we arrange both coefficients positive. For the first one wetake β = 0. Now µ = m · · · m s and so the second coefficient of c ( M/µ ) = µc ( M ) EGATIVE SASAKIAN STRUCTURES ON SIMPLY-CONNECTED 5-MANIFOLDS 21 is adjusted to be(16) β m · · · m s + s X i =1 b i m · · · ˆ m i · · · m s = 1 . This can be solved since the m i are pairwise coprime. Note that we can take0 < b i < m i , gcd( b i , m i ) = 1. Hence c ( M ) can be represented by a K¨ahler formand c ( M/µ ) is primitive.Finally, we need to show that π orb1 ( X ) is trivial, which is proved applying Propo-sition 24, because C i > 0. We complete the proof noting that M = S × S byTheorem 19, the classification of Smale-Barden manifolds and Theorem 7. (cid:3) Proposition 34. The Smale-Barden manifolds ( S × S ) and ( S × S ) admitnegative Sasakian structures.Proof. The steps of the proof are similar to those in Proposition 33, so we onlyexplain the necessary modifications. For the case ( S × S ), we take X = C P C P , by blowing-up C P at three different non-collinear points. We write H ( X, Z ) = Z h L, E, E ′ , E ′′ i , where E, E ′ , E ′′ denote the exceptional divisors. Nowthe effective cone is generated by E, E ′ , E ′′ and L − E − E ′ , L − E − E ′′ , L − E ′ − E ′′ (these are classes of the proper transforms of the lines through each two of thechosen blow-up points). Thus, the conditions ensuring ampleness of a cohomologyclass aL − bE − cE ′ − dE ′′ can be calculated by taking the inner product with gen-erators of the effective cone. We get the ampleness conditions as the inequalities:(17) b > , c > , d > , a > b + c, a > c + d, a > d + b. Take the strict transforms C , C , C of 3 conics forming a configuration suchthat each conic passes through two of the three chosen points, but no two conicspass through exactly the same points (and the other 3 intersections are transverseand generic). This yields(18) [ C ] = 2 L − E − E ′ , [ C ] = 2 L − E ′ − E ′′ , [ C ] = 2 L − E − E ′′ . Endow X with a structure of a smooth orbifold with branch divisor∆ = (cid:18) − m (cid:19) C + (cid:18) − m (cid:19) C + (cid:18) − m (cid:19) C . As K orb X = − L + E + E ′ + E ′′ + ∆, a direct calculation using (18) yields K orb X = (cid:16) − X i =1 m i (cid:17) L − (cid:16) − m − m (cid:17) E − (cid:16) − m − m (cid:17) E ′ − (cid:16) − m − m (cid:17) E ′′ . Comparing this with (17) we see that the ampleness condition is satisfied for suf-ficiently large m i so that1 m + 1 m < , m + 1 m < , m + 1 m < . The coprimality of m i ensures that H ( X, Z ) → H ( C i , Z m i ) is surjective, in thesame way as in the proof of Proposition 33.Next we need to arrange that µc ( M ) is primitive and ample (represented by aK¨ahler form). For this take c ( B ) = β [ L ], and solve so that the coefficient of L in µ c ( M ) is 1 in a similar vein to (16). Finally, the orbifold fundamental group iscalculated as before, using C i > 0. As before, we know that M = ( S × S ).The case ( S × S ) is similar but easier, and it is left to the reader. (cid:3) Finally, we just recall that Propositions 32, 33 and 34 complete the proof ofTheorem 29.The constructions of Propositions 33 and 34 yield semi-regular Sasakian struc-tures. It is likely that an extension of this construction may cover also the cases k ≥ 4, although it can be more involved. On the other hand, the construction inProposition 32 yields quasi-regular Sasakian structures with no branch divisor.It is natural to ask the following: Question . Do the manifolds k ( S × S ) admit regular negative Sasakian struc-tures?As mentioned in the introduction, the values k = ( d − d − d + 2) + 1, d ≥ k = 7 , , 20 are known. We have some more cases. Theorem 35. The -manifold k ( S × S ) admits a regular negative Sasakianstructures for k = 5 , , , and k = 13 , ≤ k ≤ .Proof. By [12, Lemma 9.10] and [12, Theorem 9.12], if X is a simply connected4-manifold and M → X be a circle bundle over X with primitive Euler class e and w ( X ) ≡ , e (mod 2), then M = k ( S × S ), where k = b ( X ) − 1. Thereforeit is sufficient to construct a simply connected smooth complex surface of generaltype and with Betti numbers b ( X ) = k + 1 (compare the discussion in [3, Example10.4.6]). This is because K X > 0, and Theorem 7 applies. In particular, there isno need to check the condition w ( X ) ≡ , e (mod 2). The case k = 8 is alreadyknown: X is the Barlow surface (see [3, Example 10.4.6]). The surfaces with b ( X ) = 6 , , Q -Gorenstein smoothing theory. In more detail: in [17] and [24]the authors construct simply connected smooth complex surfaces of general typewith p g = 0 and K X = 1 , , , 4. 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Teicher, Hirzebruch surfaces: degenerations, related braid monodromy, Galois cov-ers , Contemp. Math. 241 (1999), 305-326. Departamento de ´Algebra, Geometr´ıa y Topolog´ıa, Facultad de Ciencias, Uni-versidad de M´alaga, Campus de Teatinos s/n, 29071 Mlaga, Spain E-mail address : [email protected] Institut f¨ur Algebraische Geometrie, Gottfried Wilhelm Leibniz Universit¨atHannover, Welfengarten 1, 30167 Hannover, Germany E-mail address : [email protected] Faculty of Mathematics and Computer Science, University of Warmia and Mazury,S loneczna 54, 10-710 Olsztyn, Poland E-mail address ::