Topology 42 (2003), 981-1002  Topology

Sasakian Geometry, Homotopy Spheres and Positive Ricci Curvature

Charles P. Boyer Krzysztof Galicki Michael Nakamaye

Abstract: We discuss the Sasakian geometry of odd dimensional homotopy spheres. In particular, we give a completely new proof of the existence of metrics of positive Ricci curvature on exotic spheres that can be realized as the boundary of a parallelizable manifold. Furthermore, it is shown that on such homotopy spheres Σ^{2n+1} the moduli space of Sasakian structures has infinitely many positive components determined by inequivalent underlying contact structures. We also prove the existence of Sasakian metrics with positive Ricci curvature on each of the known 2^{2m} distinct diffeomorphism types of homotopy real projective spaces ℝℙ^{4m+1}.

0. Introduction

Milnor’s discovery of exotic spheres [Mil1] presented Riemannian geometry with a very natural question. What kind of special metrics or, more generally, geometric structures can exist on exotic spheres? Perhaps the most intriguing example of such a question concerns the existence of metrics with positive sectional curvature. In 1974 Gromoll and Meyer [GM] observed that one of the Milnor 7-spheres admits a metric of non-negative scalar curvature.Only recently Grove and Ziller [GZ] have observed that indeed all Milnor spheres, i.e., 7-spheres that are 3-sphere bundles over the 4-sphere admit metrics of nonnegative sectional curvature. Yet, it is not known whether any of these metrics can be deformed to give a metric of positive sectional curvature. However, it is known that some exotic spheres cannot admit such metrics. This follows from a beautiful result of Hitchin, who observed that some of the exotic spheres, starting in dimension 9, do not even admit metric of positive scalar curvature [Hi].

A somewhat more tractible problem concerns the existence of positive Ricci curvature metrics on exotic spheres. Here, many examples have been known [Ch, Na, Her, Po]. In 1997, using surgery theory, D. Wraith [Wr] proved the existence of Riemannian metrics with positive Ricci curvature on any exotic sphere that can be realized as the boundary of a parallelizable manifold. In particular, in dimension 7 all exotic spheres admit such metric.

The question of the existence of other geometric structures on exotic spheres has been equally important and intriguing. For example, it was realized by many [Abe1-2, AE, LM, SH, Tak, Tho, Vai, YK] that Brieskorn manifolds naturally admit almost contact, contact, and Sasakian structures. The appoach closest to ours is that of Takahashi [Tak] who constructed weighted Sasakian structures on Brieskorn manifolds.

During the preparation of this work the authors were partially supported by NSF grant DMS-9970904, and the third author by NSF grant DMS-0070190. 2000 Mathematics Subject Classification: 53C25, 57D60

The purpose of this paper is to investigate the Sasakian geometry of homotopy spheres with the goal of producing Sasakian metrics with positive Ricci curvature on exotic spheres. The main ingredients of the proof are Brieskorn’s description of homotopy spheres as links of isolated hypersurface singularities in ℂ^{n+1}, a positivity theorem in [BGN2] which is based on the orbifold or equivalently foliation version of the famous Calabi problem, and the algebraic geometry of Fano hypersurfaces in non well-formed weighted projective spaces. The link is the total space of an S^{1} orbifold V-bundle over this Fano variety with a specific orbifold structure. From our point of view the differential invariants of the link are encoded in the orbifold structure, although the precise nature of this encoding still remains a mystery.

The classical Calabi Conjecture, later proved by Yau, states that if (Z, ω) is a smooth Kähler manifold then for any (1,1) form ρ′ such that [ρ′] = c_{1}(Z) there exist a Kähler metric ω′ on Z whose Ricci form equals to ρ′. Moreover, [ω′] = [ω], i.e., ω′ is in the same cohomology class as ω. In particular, in the Fano case when c_{1}(M) > 0 the ω′ has positive Ricci curvature. It has been often noted that the Calabi-Yau Conjecture holds in the situation when M is a Kähler orbifold (cf. [DK] or [Joy]). In the context of foliations El Kacimi-Alaoui [ElK] actually gave a proof of what one can call the “transverse Yau theorem”. In [BGN2] (see Theorem 2.10 below) we showed that this theorem can be adapted to the situation, where the foliation at hand is the characteristic foliation of a Sasakian structure. In the present paper we combine all of these results to prove: Theorem A: For n ≥ 3 let Σ^{2n−1} be a homotopy sphere which can be realized as the boundary of a parallelizable manifold. Then Σ^{2n−1} admits Sasakian metrics with positive Ricci curvature.

We also show that there exists positive Sasakian structures belonging to infinitely many inequivalent contact structures. In particular, this implies that the moduli space of positive Sasakian structures on odd dimensional homotopy spheres has infinitely many components. Specifically, we prove the following: Theorem B: On each odd homotopy sphere Σ^{2n−1} ∈ bP_{2n} there exists countably infinitely many deformation classes of positive Sasakian structures belonging to non-isomorphic underlying contact structures. Hence, the moduli space of Sasakian structures on Σ^{2n−1} has infinitely many positive components.

By a well-known theorem of Smale there are no exotic spheres in dimension 5; however, there are 4 diffeomorphism types and 2 homeomorphism types of homotopy ℝℙ^{5}’s. More generally on the homotopy projective space ℝℙ^{4m+1} there are at least 2^{2m} diffeomorphism types. Homotopy projective spaces have been studied in [AB,Bro,Gi1-2,LdM]. Our methods yield: Theorem C: On each of the known 2^{2m} diffeomorphism types of homotopy projective spaces ℝℙ^{4m+1} there exists deformation types of positive Sasakian structures, and each deformation class contains Sasakian metrics of positive Ricci curvature.

Finally, we employ a method of Savel’ev [Sav] to construct homotopy spheres from rational homology spheres. Combining this with previous work [BGN4] we give positive Sasakian structures on homotopy 9-spheres belonging to deformation classes of Sasakian structures that are inequivalent to the ones constructed previously using Brieskorn spheres. Acknowledgments: The authors would like to thank A. Buium, Y. Eliashberg, H. Hofer, and A. Vistoli for fruitful discussions on various aspects of this work, and Ian Hambleton for a clarification about homotopy projective spaces in the first version of this paper.

1. Exotic Spheres

Let us briefly recall a construction of exotic differential structures on odd dimensional spheres. In 1956 Milnor stunned the mathematical world with the construction of exotic differential structures on S^{7}. Later the work of Milnor and Kervaire [KeMi] (see also [Hz]) showed that associated with each sphere S^{n} with n ≥ 5 there is an Abelian group Θ_{n} consisting of equivalence classes of homotopy spheres S^{n} that are equivalent under oriented h-cobordism. By Smale’s h-cobordism theorem this implies equivalence under oriented diffeomorphism. The group operation on Θ is connected sum. Θ_{n} has a subgroup bP_{n+1} consisting of equivalence classes of those homotopy n-spheres which bound parallelizable manifolds V_{n+1}. Kervaire and Milnor [KeMi] proved that bP_{2k+1} = 0 for k ≥ 1. Moreover, for m ≥ 2, bP_{4m} is cyclic of order

|bP_{4m}| = 2^{2m−2}(2^{2m−1} − 1) numerator (\frac{4B_{m}}{m} ),

where B_{m} is the m-th Bernoulli number. For bP_{4m+2} the situation is still not entirely understood. It entails computing the Kervaire invariant which is hard. It is known (see the recent review paper [La] and references therein) that bP_{4m+2} = 0, or ℤ_{2}, and is ℤ_{2} if 4m + 2 ≠ 2^{i} − 2 for any i ≥ 3. Furthermore, bP_{4m+2} vanishes for m = 1, 3, 7, and 15.

Ten years after Milnor’s landmark paper, Brieskorn [Br] showed how exotic spheres can be obtained as links of isolated hypersurface singularities. In particular if g_{m} denotes the Milnor generator in bP_{4m} (see [Hz]) then Brieskorn showed that for each k ∈ ℤ^{+} the link defined as the locus of points in ℂ^{2m+1} that satisfies

1.1  ∑_{i=0}^{2m} |z_{i}^{2}| = 1,  z_{0}^{6k−1} + z_{1}^{3} + z_{2}^{2} + ⋯ + z_{2m}^{2} = 0,  k ≥ 1

represents an element in bP_{4m}. We shall denote these homotopy (4m − 1)-spheres by Σ_{k}^{4m−1}. The diffeomorphism type of Σ ∈ bP_{4m} is determined [KeMi] by the signature τ of V_{4m} which is necessarily divisible by 8. Two Σ, Σ′ ∈ bP_{4m} are diffeomorphic if and only if τ(V′) ≡ τ(V)(mod 8|bP_{4m}|), and Brieskorn shows that τ(V_{k}) = (−1)^{m}8k, so that Σ_{k}^{4m−1} and Σ_{k′}^{4m−1} are diffeomorphic if and only if

k′ ≡ k(mod |bP_{4m}|).

The group operation in bP_{4m} is connected sum and since the signature is additive under connected sum, the element (−1)^{m}Σ_{1}^{4m−1} is a generator g_{m} ∈ bP_{4m}, and (−1)^{m}|bP_{4m}|g_{m} is diffeomorphic to the standard sphere S^{4m−1}.

More generally Brieskorn [Br,Di] considered the links L(a) defined by

1.2  ∑_{i=0}^{n} |z_{i}^{2}| = 1,  z_{0}^{a_{0}} + ⋯ + z_{n}^{a_{n}} = 0.

To the vector a = (a_{0}, ⋯, a_{n}) ∈ ℤ_{+}^{n+1} one associates a graph G(a) whose n+1 vertices are labeled by a_{0}, ⋯, a_{n}. Two vertices a_{i} and a_{j} are connected if and only if gcd(a_{i}, a_{j}) > 1.

Let C_{ev} denote the connected component of G(a) determined by the even integers. Note that all even vertices belong to C_{ev}, but C_{ev} may contain odd vertices as well. Then Brieskorn proved that L(a) is a ℤ-homology sphere if and only if either

1. G(a) contains at least two isolated points, or

2. G(a) contains one isolated point and C_{ev} has an odd number of vertices and for any distinct a_{i}, a_{j} ∈ C_{ev}, gcd(a_{i}, a_{j}) = 2.

Hence, if n ≥ 3 such an L(a) is homeomorphic to the sphere S^{2n−1}, and by Milnor’s fibration theorem [Mil2] describes an element of bP_{2n}.

Clearly, the links in bP_{4m} described by 1.1 are of type 1 above, whereas, examples of type 2 links in bP_{4m+2} are given by

1.3  ∑_{i=0}^{2m} |z_{i}^{2}| = 1,  z_{0}^{p} + z_{1}^{2} + z_{2}^{2} + ⋯ + z_{2m+1}^{2} = 0,  p ≥ 1.

We shall denote these spheres by Σ_{p}^{4m+1}. If p ≡ ±1mod 8 then Σ_{p}^{4m+1} is diffeomorphic to the standard sphere S^{4m+1}, whereas assuming that 4m + 2 ≠ 2^{i} − 2 if p ≡ ±3mod 8 then Σ_{p}^{4m+1} is the exotic Kervaire sphere in bP_{4m+2}. This follows from Levine’s [Lev] computation of the Arf invariant [Br].

The polynomials given in 1.1 and 1.3 are by no means unique. Any other Brieskorn-Pham polynomial 1.2 satisfying either of the two conditions on G(a) above will do as well. For example, in leau of 1.1 we can consider

1.4  ∑_{i=0}^{2m} |z_{i}^{2}| = 1,  z_{0}^{10k−1} + z_{1}^{5} + z_{2}^{2} + ⋯ + z_{2m}^{2} = 0,  k ≥ 1

In this case we find that Σ_{k}^{4m−1} and Σ_{k′}^{4m−1} are diffeomorphic if and only if 3k′ ≡ 3k(mod |bP_{4m}|).

2. Sasakian Geometry on Homotopy Spheres

Recall [Bl,YK] that a Sasakian structure on a manifold M of dimension 2n + 1 is a metric contact structure (ξ, η, Φ, g) such that the Reeb vector field ξ is a Killing field and whose underlying almost CR structure is integrable. Briefly, let (M, D) be a contact manifold, and choose a 1-form η so that η ∧ (dη)^{n} ≠ 0 and D = ker η. The pair (D, ω), where ω is the restriction of dη to D gives D the structure of a symplectic vector bundle. Choose an almost complex structure J on D that is compatible with ω, that is J is a smooth section of the endomorphism bundle End D that satisfies

2.1  J^{2} = −𝕀,  dη(JX, JY) = dη(X, Y),  dη(X, JX) > 0

for any smooth sections X, Y of D. Notice that J defines a Riemannian metric g_{D} on D by setting g_{D}(X, Y) = dη(X, JY). One easily checks that g_{D} satisfies the compatibility condition g_{D}(JX, JY) = g_{D}(X, Y). Now we can extend J to an endomorphism Φ on all of T M by putting Φ = J on D and Φξ = 0. Likewise we can extend the metric g_{D} on D to a Riemannian metric g on M by setting

2.2  g = g_{D} + η ⊗ η.

The quadruple (ξ, η, Φ, g) is called a metric contact structure on M. If in addition ξ is a Killing vector field and the almost complex structure J on D is integrable the underlying almost contact structure is said to be normal and (ξ, η, Φ, g) is called a Sasakian structure. The fiduciary examples of compact Sasakian manifolds are the odd dimensional spheres S^{2n+1} with the standard contact structure and standard round metric g.

Actually as with Kähler structures there are many Sasakian structures on a given Sasakian manifold. In fact there are many Sasakian structures which have ξ as its characteristic vector field. If (ξ, η, Φ, g) is a Sasakian structure on a smooth manifold (orbifold) M, we consider a deformation of this structure by adding to η a continuous one parameter family of 1-forms ζ_{t} that are basic with respect to the characteristic foliation. We require that the 1-form η_{t} = η + ζ_{t} satisfy the conditions

2.3  η_{0} = η,  ζ_{0} = 0,  η_{t} ∧ (dη_{t})^{n} ≠ 0 ∀ t ∈ [0, 1].

This last nondegeneracy condition implies that η_{t} is a contact form on M for all t ∈ [0, 1] which by Gray’s Stability Theorem belongs to the same underlying contact structure as η. Moreover, since ζ_{t} is basic ξ is the Reeb (characteristic) vector field associated to η_{t} for all t. Now let us define

2.4  \begin{aligned}Φ_{t} & = Φ − ξ ⊗ ζ_{t} ∘ Φ \\ g_{t} & = dη_{t} ∘ (Φ_{t} ⊗ id) + η_{t} ⊗ η_{t}. \end{aligned}

In [BG3] it was proved that for all t ∈ [0, 1] and every basic 1-form ζ_{t} such that dζ_{t} is of type (1, 1) and such that 2.3 holds (ξ, η_{t}, Φ_{t}, g_{t}) defines a continuous 1-parameter family of Sasakian structures on M belonging to the same underlying contact structure as η. Given a Sasakian structure S = (ξ, η, Φ, g) on a manifold M, we define 𝔉(ξ) to be the family of all Sasakian structures obtained by the deformations above.

Any Sasakian structure S = (ξ, η, Φ, g) has a 1-dimensional foliation F_{ξ} associated to it, defined by the flow of the Reeb vector field ξ and called the characteristic foliation. Every Sasakian structure S ∈ 𝔉(ξ) defines the same basic cohomology class [dη]_{B} ∈ H_{B}^{2}(F) in the basic cohomology of the foliation F(ξ), and conversely, any two homologous Sasakian structure S = (ξ, η, Φ, g) and S′ = (ξ, η′, Φ′, g′) lie in 𝔉(ξ). Moreover, if (M, S) is a compact Sasakian manifold then the cohomology class [dη]_{B} is non-trivial, and cups to a non-trivial class in H_{B}^{2n}(F_{ξ}). We are interested in the set of Sasakian structures which correspond to the same foliation F_{ξ}. This set contains 𝔉(ξ), but is slightly larger. We define 𝔉(F_{ξ}) to be the set of all Sasakian structures whose characteristic foliation is F_{ξ}. Clearly, we have 𝔉(ξ) ⊂ 𝔉(F_{ξ}). For any Sasakian structure S = (ξ, η, Φ, g) there is the “conjugate Sasakian structure” defined by S^{c} = (ξ^{c}, η^{c}, Φ^{c}, g) = (−ξ, −η, −Φ, g) ∈ 𝔉(F_{ξ}). Moreover, there is a type of homothety [YK] defined by

2.5  ξ′ = a^{−1}ξ, η′ = aη, Φ′ = Φ, g′ = ag + (a^{2} − a)η ⊗ η

for a ∈ ℝ^{+}. So fixing S we define

2.6  𝔉^{+}(F_{ξ}) = ⋃_{a∈ℝ^{+}} 𝔉(a^{−1}ξ),

and 𝔉^{−}(F_{ξ}) to be the image of 𝔉^{+}(F_{ξ}) under conjugation. It is then easy to see that we have a decomposition 𝔉(F_{ξ}) = 𝔉^{+}(F_{ξ}) ⊔ 𝔉^{−}(F_{ξ}). Notice that for n even conjugation reverses orientation; whereas, for n odd it preserves the orientation. This discussion shows that for each F_{ξ} the subset of homology classes represented by Sasakian structures forms a line minus the origin in H_{B}^{2}(F_{ξ}). Since conjugation interchanges the positive and negative rays, we often restrict our considerations to the positive ray, and hence to 𝔉^{+}(F_{ξ}). We shall refer to elements of 𝔉^{±}(F_{ξ}) as a-deformation classes of Sasakian structures. We need Definition 2.7: Two Sasakian structures S = (ξ, η, Φ, g) and S′ = (ξ′, η′, Φ′, g′) in 𝔉(F_{ξ}) on a smooth manifold M are said to be a-homologous if there is an a ∈ ℝ^{+} such that ξ′ = a^{−1}ξ and [dη′]_{B} = a[dη]_{B}.

The a-homology classes form a set of two elements that can be identified with positive and negative rays in H_{B}^{2}(F_{ξ}). So every Sasakian structure in 𝔉(F_{ξ}) is a-homologous to S or its conjugate S^{c}.

Other important invariants of 𝔉(ξ) are the basic Chern classes c_{i}(F_{ξ}) of the symplectic vector bundle (D, dη) as elements of the basic cohomology ring H_{B}^{∗}(F_{ξ}). In particular we are interested in the basic first Chern class c_{1}(F_{ξ}) ∈ H_{B}^{2}(F_{ξ}). Recall [BGN2] that a Sasakian structure S = (ξ, η, Φ, g) is positive (negative) if its basic first Chern class c_{1}(F_{ξ}) ∈ H^{2}(F_{ξ}) can be represented by a positive (negative) definite (1, 1)-form. On rational homology spheres there are are only two possibilities, viz. Proposition 2.8: Let S = (ξ, η, Φ, g) be a Sasakian structure on a rational homology sphere M^{2n−1}. Then either c_{1}(F_{ξ}) > 0 or c_{1}(F_{ξ}) < 0. Proof: For rational homology spheres M the basic long exact sequence

⋯ ⟶H_{B}^{p}(F_{ξ})⟶H^{p}(M, ℝ)⟶^{j_{p}}H_{B}^{p−1}(F_{ξ})⟶^{δ}H_{B}^{p+1}(F_{ξ})⟶ ⋯

implies the isomorphisms

H_{B}^{p−1}(F_{ξ}) ≈ H_{B}^{p+1}(F_{ξ})

for p = 1, ⋯, 2n−1, and since H_{B}^{1}(F_{ξ}) = 0 and H_{B}^{2}(F_{ξ}) is generated by [dη]_{B}, this implies the ring isomorphism

2.9  H_{B}^{∗}(F_{ξ}) ≈ ℝ[[dη]_{B}]/([dη]_{B})^{n}.

Thus, C(F_{ξ}) = ℝ^{+} ∪ ℝ^{−}, and c_{1}(F_{ξ}) = a[dη]_{B} for some a ∈ ℝ. Since dη is the transverse Kähler form it is positive definite, and a cannot vanish since this would imply that the basic geometric genus [BGN2] p_{g}(F_{ξ}) ≠ 0 implying b_{n−1}(M) ≠ 0. Thus, either a > 0 or a < 0 implying either c_{1}(F_{ξ}) > 0 or c_{1}(F_{ξ}) < 0.

In the next two sections we shall discuss how to construct positive Sasakian structures on homotopy spheres, and prove Theorem A of the Introduction. Recall that a positive Sasakian structure [BGN2] is a Sasakian structure whose basic first Chern class can be represented by a positive definite basic (1, 1)-form. Once positive Sasakian structures are obtained Sasakian structures with positive Ricci curvature follow from the following theorem of the authors. Theorem 2.10 [BGN2]: Let S = (ξ, η, Φ, g) be a positive Sasakian structure on a compact manifold M of dimension 2n + 1. Then M admits a Sasakian structure S′ = (ξ′, η′, Φ′, g′) with positive Ricci curvature a-homologous to S for some a > 0.

3. The Characteristic Foliation and Kähler Orbifolds

This foliation encodes important invariants of the Sasakian structure. For example, (M, ξ, η, Φ, g) is said to be quasi-regular if all the leaves of F_{ξ} are compact. In this case the leaves are all circles, but in general they can have nontrivial holonomy. If the leaf holonomy group is trivial for every leaf, the Sasakian manifold (M, ξ, η, Φ, g) is called regular. It is well-known that if (M, ξ, η, Φ, g) is compact and quasi-regular, then the space of leaves Z is a compact Kähler orbifold, and a smooth manifold in the regular case, whose Kähler class [ω] is represented by an integral class in H_{orb}^{2}(Z, ℤ) and a rational class in H^{2}(Z, ℚ). The leaf holonomy groups of F_{ξ} are the local uniformizing groups of the orbifold Z and these are invariants of the Sasakian structure on M. A somewhat courser invariant is the order υ(M) of a compact Sasakian manifold defined to be the least common multiple of the orders of the leaf holonomy groups. So (M, ξ, η, Φ, g) is regular if and only if υ_{M} = 1.

For a quasi-regular Sasakian structure S on a compact manifold M the space of leaves Z is also a projective algebraic variety, and it is important to distinguish between Z as an orbifold and Z as an algebraic variety. In general the singular sets are different. In the case of the Brieskorn-Kervaire exotic spheres considered in Proposition 4.3 below, we shall see that Z as an algebraic variety is smooth, in fact a projective space, whereas, its orbifold singular set is actually a divisor. For hypersurfaces (or more generally complete intersections) in weighted projective spaces, the two singular sets coincide precisely when the hypersurfaces are well-formed (cf. [Fl]); however, the orbifolds Z associated to the Brieskorn spheres are never well-formed. In the non well-formed case certain pathologies can occur. For, example if Z ⊂ ℙ(w_{0}, ⋯, w_{n}) is not well-formed, there can be isomorphic coherent sheaves O_{Z}(m) and O_{Z}(n) with n ≠ m.

In our previous work [BG1,BG2,BGN1,BGN2,BGN3] we have always worked with the well-formed case, and there was little need to distinguish between the orbifold context and the algebraic geometric context. In the present paper this is no longer the case. In algebraic geometry one uses a sort of poetic license not to distinguish between vector bundles and locally free sheaves. There is a similar situation in the category of orbifolds though much less well-known. First, the notion of a vector V-bundle was introduced by Satake [Sat] and essentially consists of vector bundles defined on the local uniformizing covers (Ũ_{i}, Γ_{i}, ϕ_{i}) that patch together in a nice way, together with group homomorphisms Γ_{i}⟶GL(n, ℂ) that satisfy certain compatibility conditions (cf. [BG1] for details). On the other hand the notion of a locally V-free sheaf on a normal projective variety with only quotient singularities has been defined ([Bla], see also [Kaw] where it is referred to as a ℚ-vector bundle), and proves to be convenient in our context. A coherent sheaf 𝔉 of O_{Z}-modules is locally V-free if on each uniformizing neighborhood Ũ_{i} there is a free sheaf 𝔉̃ together with an action of Γ_{i} on 𝔉̃ such that 𝔉 = ((ϕ_{i})_{∗}𝔉̃)^{Γ_{i}} where (·)^{Γ} denotes the Γ-invariant subsheaf. Notice that a locally V-free sheaf is not necessarily locally free; however, a locally free sheaf is locally V-free. Now it is straightforward to see that the usual correspondence between vector V-bundles and locally V-free sheaves still holds. We shall also refer to a locally V-free sheaf of rank one as a V-invertible sheaf. V-invertible sheaves correspond to holomorphic line V-bundles. In our previous work [BG1,BG2] we introduced the group Pic^{orb}(Z) of holomorphic line V-bundles on Z. We also refer to Pic^{orb}(Z) as the group of V-invertible sheaves on Z. The dualizing sheaf ω_{Z} on Z, defined by

3.1  ω_{Z} = ι_{∗}(Ω^{n}Z_{reg}),

where Ω^{p}Z_{reg} denotes the sheaf of differential p-forms on the algebro-geometric regular set Z_{reg}, is of particular importance. For a given orbifold structure on Z, ω_{Z} is V-invertible with respect to that structure, and corresponds to the canonical V-bundle K_{Z} as defined by Baily [Ba]. However, Z may admit several orbifold structures, and ω_{Z} can take different forms as V-invertible sheaves.

Consider the affine space ℂ^{n+1} together with a weighted ℂ^{∗}-action given by

3.2  (z_{0}, …, z_{n}) ↦ (λ^{w_{0}}z_{0}, …, λ^{w_{n}}z_{n}),

where the weights w_{j} are positive integers. It is convenient to view the weights as the components of a vector w ∈ (ℤ^{+})^{n+1}, and we shall assume that they are ordered w_{0} ≤ w_{1} ≤ ⋯ ≤ w_{n} and that gcd(w_{0}, …, w_{n}) = 1. Let f be a weighted homogeneous polynomial, that is f ∈ ℂ[z_{0}, …, z_{n}] and satisfies

3.3  f(λ^{w_{0}}z_{0}, …, λ^{w_{n}}z_{n}) = λ^{d}f(z_{0}, …, z_{n}),

where d ∈ ℤ^{+} is the degree of f. We are interested in the weighted affine cone C_{f} defined by the equation f(z_{0}, …, z_{n}) = 0. We shall assume that the origin in ℂ^{n+1} is an isolated singularity, in fact the only singularity, of f. Then the link L_{f} defined by

L_{f} = C_{f} ∩ S^{2n+1},

where

S^{2n+1} = {(z_{0}, …, z_{n}) ∈ ℂ^{n+1}| ∑_{j=0}^{n} |z_{j}|^{2} = 1}

is the unit sphere in ℂ^{n+1}, is a smooth manifold of dimension 2n − 1. Furthermore, it is well-known [Mil2] that the link L_{f} is (n − 2)-connected.

On S^{2n+1} there is a well-known [YK] “weighted” Sasakian structure (ξ_{w}, η_{w}, Φ_{w}, g_{w}) which in the standard coordinates {z_{j} = x_{j} + iy_{j}}_{j=0}^{n} on ℂ^{n+1} = ℝ^{2n+2} is determined by

η_{w} = \frac{∑_{i=0}^{n}(x_{i}dy_{i} − y_{i}dx_{i})}{∑_{i=0}^{n} w_{i}(x_{i}^{2} + y_{i}^{2})}, ξ_{w} = ∑_{i=0}^{n} w_{i}(x_{i}∂_{y_{i}} − y_{i}∂_{x_{i}}),

and the standard Sasakian structure (ξ, η, Φ, g) on S^{2n+1}. The embedding L_{f} ↪ S^{2n+1} induces a Sasakian structure on L_{f} [BG3].

Given a sequence w = (w_{0}, …, w_{n}) of ordered positive integers one can form the graded polynomial ring S(w) = ℂ[z_{0}, …, z_{n}], where z_{i} has or weight w_{i}. The weighted projective space [Dol, Fle] ℙ(w) = ℙ(w_{0}, …, w_{n}) is defined to be the scheme Proj(S(w)). It is the quotient space (ℂ^{n+1} − {0})/ℂ^{∗}(w), where ℂ^{∗}(w) is the weighted action defined in 3.2, or equivalently, ℙ(w) is the quotient of the weighted Sasakian sphere S_{w}^{2n+1} = (S^{2n+1}, ξ_{w}, η_{w}, Φ_{w}, g_{w}) by the weighted circle action S^{1}(w) generated by ξ_{w}. As such ℙ(w) is also a compact complex orbifold with an induced Kähler structure. Furthermore, the hypersurface Z_{f} ⊂ ℙ(w) defined as the zero set of the section f ∈ O_{ℙ(w)}(d) is the S^{1}(w) quotient of the link L_{f}. The assumption that the origin of ℂ^{n+1} is the only singularity of f guarantees that the quotient space Z_{f} is an orbifold. In this case we also say the Z_{f} is quasi-smooth. We have from [BG3] Theorem 3.4: The quadruple (ξ_{w}, η_{w}, Φ_{w}, g_{w}) gives L_{f} a quasi-regular Sasakian structure S_{w} such that there is a commutative diagram

L_{f} ⟶ S_{w}^{2n+1}
↓π  ↓
Z_{f} ⟶ ℙ(w),

where the horizontal arrows are Sasakian and Kählerian embeddings, respectively, and the vertical arrows are principal S^{1} V-bundles and orbifold Riemannian submersions. Moreover, the orbifold first Chern class in c_{1}(Z_{f}) ∈ H_{orb}^{2}(Z_{f}, ℤ) is related to the basic first Chern class c_{1}(F_{ξ_{w}}) of S_{w} by c_{1}(F_{ξ_{w}}) = π^{∗}c_{1}(Z_{f}).

Hence, to prove Theorem A we need to compute the orbifold first Chern class c_{1}(Z_{f}). This requires determining the canonical V-bundle K_{Z_{f}} by an adjunction formula. The following is well-known [Dol,BR]: Lemma 3.5: [Dol] On ℙ(w) the dualizing sheaf ω_{ℙ(w)} is isomorphic to O_{ℙ(w)}(−|w|).

Warning: This does not imply that the anti-dualizing sheaf ω_{ℙ(w)}^{−1} is isomorphic to O_{ℙ(w)}(|w|). We note that this isomorphism does hold if the weighted projective space ℙ(w) is well-formed, i.e. the gcd of any n components of w = (w_{0}, ⋯, w_{n}) is one (cf. [BG3]), but here we are interested in the non well-formed case.

Next we are interested in an adjunction formula. This is known to hold when the hypersurface is well-formed as defined by Fletcher [Fl]; that is a hypersurface Z_{f} ⊂ ℙ(w) is well-formed if ℙ(w) is well-formed and Z_{f} contains no codimension 2 singular stratum of ℙ(w). However, the condition that guarantees the adjunction formula is that the hypersurface Z_{f} contain no codimension 2 singular stratum of ℙ(w), and no further condition on ℙ(w) is needed. Indeed, in the sequel we shall make use of the adjunction formula for hypersurfaces in weighted projective spaces that are not well-formed. Lemma 3.6 [Adjunction formula]: Let Z_{f} be a quasi-smooth hypersurface in ℙ(w) defined by a weighted homogeneous polynomial f of degree d. Suppose that codim(Z_{f} ∩ ℙ_{w}^{sing}, Z_{f}) ≥ 2. Then the adjunction formula

ω_{Z_{f}} ≈ O_{Z_{f}} (d − |w|)

holds. Proof: By Proposition 5.75 of [KM], ω_{X} ≃ O_{X}(K_{X}) for any normal variety X. In particular, for the hypersurface Z_{f} in ℙ(w) we have

ω_{Z_{f}} ≈ O_{Z_{f}} (K_{Z_{f}}).

In general, Z_{f} will not be a Cartier divisor and so we cannot apply the adjunction formula directly. On the other hand, if U ⊂ ℙ(w) denotes the smooth locus and U_{f} = Z_{f} ∩ U then the adjunction formula says that

K_{U_{f}} = K_{U} + U_{f}|U_{f} = O_{U_{f}} (d − |w|).

Consequently

ω_{U_{f}} ≈ O_{U_{f}} (d − |w|)

The lemma then follows from Proposition 0-1-10 of [KMM] since ω_{Z_{f}} and O_{Z_{f}} (d − |w|) are reflexive rank one sheaves which agree off a set of codimension at least 2. Remark 3.7: In several of our applications the hypersurface Z_{f} is a Cartier divisor on ℙ(w), and in this case one can directly apply the adjunction formula in [KM].

The dualizing sheaf ω_{Z_{f}} is a V-invertible sheaf with respect to the orbifold structure on Z_{f}, and as such corresponds to the canonical V-bundle K_{Z_{f}} , or equivalently to the canonical V-divisor of Baily [Ba], i.e.

3.8  ω_{Z_{f}} = O(K_{Z_{f}}).

Now K_{Z_{f}} is an element of Pic^{orb}(Z_{f}) and as such has an inverse, namely, the anti-canonical V-bundle K_{Z_{f}}^{−1}. We recall [BG1] that a compact complex orbifold Z is Fano if its anticanonical V-bundle K_{Z_{f}}^{−1} is ample, and in this case the orbifold Fano index of Z is defined to be the largest positive integer I such that \frac{1}{I}K_{Z}^{−1} is an element of Pic^{orb}(Z). Now the group Pic^{orb}(Z) is isomorphic [BG4] to the divisor class group Cl(Z). Since Cl(Z) is an algebraic invariant, the orbifold Fano index is also a purely algebraic invariant. Furthermore, on a weighted projective space ℙ(w) we have that Cl(ℙ(w)) ≈ ℤ and Cl(ℙ(w)) is generated by O_{ℙ(w)}(a_{w}) [BR] where a_{w} = lcm(d_{0}, ⋯, d_{n}) and d_{i} = gcd(w_{0}, ⋯, ŵ_{i}, ⋯, w_{n}). We have Lemma 3.9: The following hold:

a) There is an isomorphism Pic^{orb}(ℙ(w)) ≈ ℤ and as a V-bundle (or equivalently Vinvertible sheaf) O_{ℙ(w)}(a_{w}) is the generator of Pic^{orb}(ℙ(w)) corresponding to 1 ∈ ℤ.

b) The (orbifold) Fano index I is independent of the orbifold structure and on ℙ(w) equals |w̄| where w̄ is the normalization of w.

c) For any integers m, n there is a V-isomorphism of V-invertible sheaves

O_{ℙ(w)}(na_{w}) ⊗ O_{ℙ(w)}(ma_{w})⟶O_{ℙ(w)}((n + m)a_{w})

describing the group structure of Pic^{orb}(ℙ(w)).

d) Pic(ℙ(w)) is the subgroup of Pic^{orb}(ℙ(w)) generated by O_{ℙ(w)}(υ_{w}) where υ_{w} = lcm(w_{0}, ⋯, w_{n}) is the order of ℙ(w) as an orbifold.

Remark. It is clear from the definitions that a_{w} divides υ_{w}.

Lemma 3.9 follows from the discussion above and the next lemma which is an orbifold version of a result of Delorme [Del]. Lemma 3.10: For any j ∈ ℤ there is a unique J ∈ ℤ and a V-isomorphism of V-invertible sheaves

O_{ℙ(w)}(j) ≈ O_{ℙ(w)}(Ja_{w}).

Proof: As an isomorphism of reflexive sheaves this is a result of Delorme [Del]. This isomorphism can be described as follows (cf. [BR]): first notice that O_{ℙ(w)}(j) = (S(w)(j)) where the graded S(w)-module S(w)(j) is defined by S(w)(j)_{l} = S(w)_{j+l}. Now there is an equality of schemes Proj(S(w)) = Proj(S′(w)) where S′(w) is the subring of S(w) defined by

S′(w) = ⨁_{k∈ℤ} S(w)_{ka_{w}}.

The isomorphism in 3.10 is then induced by the equality of graded S′(w)-modules

⨁_{k∈ℤ} S(w)(j)_{ka_{w}} = ⨁_{k∈ℤ} z_{0}^{b_{0}(j)} ⋯ z_{n}^{b_{n}(j)}(S(w)(j − ∑_{i} b_{i}(j)w_{i}))_{ka_{w}}

where b_{i}(j) are the unique integers 0 ≤ b_{i}(j) < d_{i} satisfying j = b_{i}(j)w_{i} + c_{i}d_{i} for some c_{i} ∈ ℤ. So the integer J is j − ∑_{i} b_{i}(j)w_{i}.

Let {U, ϕ, Γ} be a local uniformizing system. But since the isomorphic sheaves O_{ℙ(w)}(j) and O_{ℙ(w)}(Ja_{w}) are locally V-free, they pull-back to isomorphic sheaves ϕ^{∗}O_{ℙ(w)}(j)^{∨∨} and ϕ^{∗}O_{ℙ(w)}(Ja_{w})^{∨∨} on the local cover U.

Our next task is to compute the orbifold Fano index. It appears to be difficult to get a general formula, so we content ourselves with less. Actually all that is needed for this paper is the positivity of the index, but with a little extra effort we can do better. All of the weighted hypersurfaces considered in this paper are branched covers of the form

3.11  g = z_{0}^{p} + f(z_{1}, ⋯, z_{n}) = 0,

where f is a weighted homogeneous polynomial of degree d and weights w = (w_{1}, ⋯, w_{n}) that are reduced, but not necessarily well-formed. We also assume that f is quasi-smooth and that gcd(d, p) = 1. Under these conditions we have Lemma 3.12: Let Z_{f} be a quasi-smooth hypersurface in a weighted projective space ℙ(w). Suppose also that Z_{f} is cut out by the zero locus of the weighted homogeneous polynomial f(z_{1}, ⋯, z_{n}) of degree d. Let p be a positive integer such that gcd(p, d) = 1. Then the hypersurface Z_{g} of degree d_{g} = dp cut out by equation 3.11 is quasi-smooth and Fano with orbifold Fano index |w̄| where w̄ is the normalization of w, and as an algebraic variety Z_{g} is isomorphic to the weighted projective space ℙ(w̄). Proof: It is easy to see that Z_{g} is quasi-smooth whenever Z_{f} is. To show that Z_{g} is Fano we first show that the adjunction formula holds and then compute its index. Now the hypersurface Z_{g} has degree dp in the weighted projective space ℙ(d, pw) which is not well-formed. The map z_{0}^{p} ↦ z_{0} gives an isomorphism of embeddings of algebraic varieties

3.13  \begin{matrix}Z_{g} & ↪ & ℙ(d, pw) \\ ↓≈ & & ↓≈ \\ ℙ(w) & ↪ & ℙ(d, w). \end{matrix}

Here the bottom inclusion is the hyperplane z_{0} = 0, and the isomorphism Z_{g} ≈ ℙ(w) is given by the map z_{0}^{p} ↦ z_{0} ↦ z_{0} − f. But ℙ(w) is isomorphic as an algebraic variety to ℙ(w̄). This gives an isomorphism of algebraic varieties Z_{g} ≈ ℙ(w̄). Since the Fano index is an algebric invariant, we have I(Z_{g}) = |w̄|. Remark 3.14: In Section 7 we shall see that the link associated to the hypersurface Z_{g} of Lemma 3.12 is always a rational homology sphere.

4. The Kähler Orbifolds Associated to Homotopy Spheres

In this section we discuss the Kähler orbifolds associated to the characteristic foliation of the two types of homotopy spheres Σ_{k}^{4m−1} and Σ_{p}^{4m+1} discussed in section 1. We shall denote by Σ^{2n+1} these when there is no need to distinguish between them. We treat the case of Σ_{k}^{4m−1} first.

Proposition 4.1: For all k ≥ 1 and m ≥ 2, the Sasakian structure on Σ_{k}^{4m−1} given by Theorem 3.5 is positive, quasi-regular of order 6(6k − 1) and the space of leaves Z_{k} is a Fano orbifold given as a hypersurface in the weighted projective space ℙ(6, 2(6k−1), 3(6k− 1), ⋯, 3(6k − 1)) cut out by the equation

z_{0}^{6k−1} + z_{1}^{3} + z_{2}^{2} + ⋯ + z_{2m}^{2} = 0.

Moreover, the natural projection π_{k} : Σ_{k}^{4m−1}⟶Z_{k} is a principal S^{1} V-bundle over the Fano orbifold Z_{k} with orbifold Fano index I = 2m+1. The orbifold first Chern class of this V-bundle is (c_{1}(Z_{k})/I) ∈ H_{orb}^{2}(Z_{k}, ℤ) which is associated to the generator O_{Z_{k}}(3(6k − 1)) in Pic^{orb}(Z_{k}). Proof: With the exception of the statements about the order and the Fano condition this follows immediately from the commutative diagram

4.2  \begin{matrix}Σ_{k}^{4m−1} & ⟶ & S_{w}^{4m+1} \\ ↓π & & ↓ \\ Z_{k} & ⟶ & ℙ(w) \end{matrix}

where the weight vector w is given by 2.5, the horizontal arrows are Sasakian and Kählerian embeddings, respectively, and the vertical arrows are the natural projections. Furthermore, assuming that Z_{k} is Fano, it follows by the inversion theorem of [BG1] Σ_{k}^{4m−1} is the total space of the principal S^{1} V-bundle over the orbifold Z_{k} whose first Chern class is c_{1}(Z_{k})/I. So we need to compute the index I. First we note that adjunction holds since by the computation of the order in Lemma 4.3 below, Z_{k} is a Cartier divisor. Now by Lemma 3.12 Z_{k} is isomorphic as an algebraic variety to a well-formed weighted projective space. This isomorphism is determined by the sequence of transformations

(z_{0}^{6k−1}, z_{1}^{3}, z_{2}^{2}, ⋯, z_{2m}^{2}) ↦ (z_{0}, z_{1}^{3}, z_{2}^{2}, ⋯, z_{2m}^{2}) ↦ (z_{0}, z_{1}, z_{2}^{2}, ⋯, z_{2m}^{2}).

giving the sequence of isomorphisms

ℙ(6, 2(6k − 1), 3(6k − 1), ⋯, 3(6k − 1)) ≈ ℙ(6, 2, 3, ⋯, 3) ≈ ℙ(2, 2, 1, ⋯, 1).

Thus, as algebraic varieties the Z_{k} are isomorphic for each k = 1, ⋯ to the weighted quadric

z_{0} + z_{1} + z_{2}^{2} + ⋯ + z_{2m}^{2} = 0

in ℙ(2, 2, 1, ⋯, 1) which can be transformed to z_{0} = 0 giving the weighted projective space ℙ(2, 1, ⋯, 1). There are 2m − 1 ones, so the Fano index is 2m + 1. Furthermore, a_{w} = lcm(d_{0}, ⋯, d_{2m}) = 3(6k − 1), so O_{Z_{k}}(3(6k − 1)) is the positive generator in Pic^{orb}(Z_{k}).

The local uniformizing groups can be computed by analyzing the singular stratum of Z_{k} and this is done in the lemma below where it is shown that the order of Z_{k} is υ(Z_{k}) = 6(6k−1). The positivity of the Sasakian structure is implied by the Fano condition, namely I > 0.

The computation of the order is given in: Lemma 4.3: The (orbifold) singular stratum of Z_{k} is the union Y_{0} ∪ Y_{1} ∪ Y_{2} where

Y_{0} = {[z] ∈ Z_{k}| z_{0} = 0}, Y_{1} = {[z] ∈ Z_{k}| z_{1} = 0}, Y_{2} = {[z] ∈ Z_{k}| z_{2} = ⋯ = z_{2m} = 0}.

The local uniformizing groups Γ_{i} on Y_{i} are

Γ_{0} ≈ ℤ_{6k−1},  Γ_{1} ≈ ℤ_{3},  Γ_{2} ≈ ℤ_{2},

so the order υ(Z_{k}) is 6(6k − 1).

Proof: The condition for determining the fixed point set of Σ_{k}^{4m−1} under the action of finite subgroups of the weighted circle S_{w}^{1} generated by the Reeb vector field ξ_{w} is determined by the equations

4.4  ζ^{6}z_{0} = z_{0},  ζ^{2(6k−1)}z_{1} = z_{1},  ζ^{3(6k−1)}z_{i} = z_{i} for i = 2, ⋯, 2m.

If z_{0}z_{1}z_{i} ≠ 0 for some i = 2, ⋯, 2m there are no fixed points, so we look at the three cases:

1. z_{0} = 0 : This gives the hypersurface z_{1}^{3} + z_{2}^{2} + ⋯ + z_{2m}^{2} = 0 with isotropy group Γ_{0} ≈ ℤ_{6k−1}.

2. z_{1} = 0 : This is another hypersurface z_{0}^{6k−1} + z_{2}^{2} + ⋯ + z_{2m}^{2} = 0 with isotropy group Γ_{1} ≈ ℤ_{3}.

3. z_{i} = 0 for all i = 2, ⋯, 2m : This gives the set of points determined by z_{0}^{6k−1} +z_{1}^{3} = 0 with isotropy group Γ_{2} ≈ ℤ_{2}. For the case of Σ_{p}^{4m+1} we have

Proposition 4.5: For all p ≥ 0 and m ≥ 1, the Sasakian structure on Σ_{p}^{4m+1} given by Theorem 3.5 is positive, quasi-regular of order 2p + 1 and the space of leaves Z_{p} is a Fano orbifold given as a hypersurface in the weighted projective space ℙ(2, 2p + 1 ⋯, 2p + 1) cut out by the equation

z_{0}^{2p+1} + z_{1}^{2} + z_{2}^{2} + ⋯ + z_{2m+1}^{2} = 0.

Moreover, the natural projection π_{k} : Σ_{k}^{4m+1}⟶Z_{p} is a principal S^{1} V-bundle over the Fano orbifold Z_{p} of orbifold Fano index I = 2m + 1. The orbifold first Chern class of this V-bundle is (c_{1}(Z_{p})/I) ∈ H_{orb}^{2}(Z_{p}, ℤ) which is associated to the locally V-free sheaf O_{Z_{p}}(2p + 1). Proof: The proof of this proposition together with the following lemma are quite similar to that of Proposition 4.1 and Lemma 4.3. So we only give the computation of the index I. In this case as given in Lemma 4.4 below, the order υ is 2p+1 and the degree d = 2(2p+1), so the order divides the degree, and adjunction holds. Here we have d_{0} = 2p + 1, and d_{j} = 1 for j > 0, so a_{w} = 2p+1. Thus, O_{Z_{p}}(2p+1) is the positive generator of Pic^{orb}(Z_{p}).

The isomorphism Z_{p} ≈ ℙ(w̄) is given by Lemma 4.12 where w̄ = (2, 1, ⋯, 1) with 2m + 1 ones, implying that the Fano index of Z_{p} is I = 2m + 1.

The order υ(Z_{p}) is determined by: Lemma 4.6: The (orbifold) singular stratum of Z_{p} is the quadric Q_{2m−1} given by

z_{1}^{2} + ⋯ + z_{2m+1}^{2} = 0

with the isotropy group ℤ_{2p+1}. So the order υ(Z_{p}) is 2p + 1. Remarks 4.7: The orbifolds Z_{k} of Proposition 4.1 are, as mentioned in the proof, isomorphic as algebraic varieties to the weighted projective space ℙ(2, 1, ⋯, 1). Likewise, the orbifolds Z_{p} of Proposition 4.5 are isomorphic as algebraic varieties to the complex projective space ℙ^{2m}. This last fact was noticed previously by Thomas [Tho].

5. Some Exotic Projective Spaces

López de Medrano [LdM] has shown that the quotient space of any fixed point free involution on a homotopy sphere is homotopy equivalent to a real projective space ℝℙ^{n}. Here we are interested in fixed point free involutions on Σ^{2n+1} that also preserve the Sasakian structure. In particular, we consider the well known involution T on Σ_{p}^{4m+1} defined by (z_{0}, z_{1}, ⋯, z_{2m+1}) ↦ (z_{0}, −z_{1}, ⋯, −z_{2m+1}). One easily sees that this defines a free action of ℤ_{2} on Σ^{4m+1}, so by López de Medrano’s theorem the quotient manifold Σ_{p}^{4m+1}/T is homotopy equivalent to ℝℙ^{4m+1}. Furthermore, it is clear that this T ≈ ℤ_{2} is a subgroup of the weighted circle group S_{w}^{1} with weights w = (2, 2p + 1, ⋯, 2p + 1). Thus, the deformation class of Sasakian structures passes to the quotient Σ_{p}^{4m+1}/T to give a deformation class of Sasakian structures together with their characteristic foliation F_{ξ}. Furthermore, by Proposition 4.5 the Sasakian structures are positive, and so by [BGN2] the homotopy real projective spaces Σ_{p}^{4m+1}/T admit Sasakian metrics of positive Ricci curvature. We have arrived at: Proposition 5.1: For each (p, m) ∈ ℤ^{+}×ℤ^{+} the homotopy projective space Σ_{p}^{4m+1}/ℤ_{2} admits a deformation class of positive Sasakian structures; hence, Σ_{p}^{4m+1}/T admits Sasakian structures with positive Ricci curvature.

It is known [AB,Bro,Gi1-2] that there are at least 2^{2m} diffeomorphism types on a homotopy real projective space of dimension 4m + 1. Atiyah and Bott [AB] and Browder [Bro] obtained the bound 2^{2m−1} which was then extended to 2^{2m} by Giffen [Gi1-2]. Furthermore, for 0 ≤ p < p′ < 2^{2m}. the homotopy real projective spaces Σ_{p′}^{4m+1}/T and Σ_{p}^{4m+1}/T are not diffeomorphic. Thus, each of the diffeomorphism types is represented by deformation classes of positive Sasakian structures. This proves Theorem C.

6. Exotic Contact Structures

With the exception of the last two sentences of Theorem C of the Introduction, the inequivalences of the contact structures refer to inequivalences of the underlying almost contact structures. The material discussed here is well-known and taken from Sato [Sa] and Morita [Mo]. An almost contact structure on an orientable manifold M^{2n−1} can be defined as a reduction of the oriented orthonormal frame bundle with group SO(2n − 1) to the group U(n − 1) × 1. Furthermore, the set A(M) of homotopy classes of almost contact structures on M is in one-to-one correspondence with the set of homotopy classes of almost complex structures on M × ℝ [Sa], and when M = Σ^{2n−1} is a homotopy sphere, the latter is known [Mo]

6.1  A(Σ^{2n−1}) = π_{2n−1}(SO(2n)/U(n)) ≅ \{ \begin{matrix}ℤ ⊕ ℤ_{2} & if n ≡ 0(mod 4); \\ ℤ_{(n−1)!} & if n ≡ 1(mod 4); \\ ℤ & if n ≡ 2(mod 4); \\ ℤ_{\frac{(n−1)!}{2}} & if n ≡ 3(mod 4); \end{matrix}

Proof of Theorem B: The proof for the homotopy spheres Σ^{4m−1} and Σ^{4m+1} are somewhat different. The case Σ^{4m−1} uses the proof of Theorem 4.1(i) of Morita [Mo]. For each homotopy sphere Σ^{4m−1} ∈ bP_{4m} the underlying almost contact structures are distinguished by Morita’s invariant ∂, and he finds that

6.2  ∂(Σ_{k}^{4m−1}) = (\frac{2}{S_{m}} − 6)k,

where S_{m} is a certain rational number. Moreover, there are an infinite number of k that correspond to the same homotopy sphere Σ^{4m−1} ∈ bP_{4m}. Thus, we will have a countably infinite number of distinct underlying almost contact structures on the same homotopy sphere as long as S_{m} ≠ \frac{1}{3}. If S_{m} = \frac{1}{3} we simply use another description of elements in bP_{4m} by Brieskorn manifolds, for example that of 1.4. This completes the proof for this case.

Representing the homotopy spheres Σ^{4m+1} by polynomials of the form 1.3, we know that Σ^{4m+1} is the standard sphere if p ≡ ±1(mod 8), and the Kervaire sphere if p ≡ ±3(mod 8) which is exotic when 4m + 2 ≠ 2^{i} − 2. In this case the Morita invariant is simply given in terms of the Milnor number μ of the link, viz.

6.3  ∂(Σ_{p}^{4m+1}) = \frac{1}{2}μ(Σ_{p}^{4m+1}) = \frac{1}{2}(p − 1).

Thus, as p varies through ±1(mod 8) or ±3(mod 8), the invariant ∂(Σ_{p}^{4m+1}) varies through 0(mod 4) and 3(mod 4), or 1(mod 4) and 2(mod 4), respectively. Since for m > 1 a(m) ≡ 0(mod 4) we can choose p such that ∂(Σ_{p}^{4m+1}) is any number mod a(m). Thus, there are a(m) distinct underlying almost contact structures which is finite in this case. To obtain an infinite number of inequivalent contact structures we use the recent results of Ustilovsky [Us] who showed by explicitly computing the contact homology of Eliashberg, Giventhal, and Hofer (cf. [Eli]) that for the standard sphere S^{4m+1} distinct p’s satisfying p ≡ ±1(mod 8) give inequivalent contact structures. However, Ustilovsky’s proof works equally well for any homotopy sphere Σ^{4m+1}. This completes the proof. Remarks 6.4: (1) It is interesting to note that for the 5-sphere S^{5} (m = 1 above), Ustilovsky’s results distinguish infinitely many inequivalent contact structures, whereas, there is a unique underlying almost contact structure. (2) It is tempting to adapt the computations in [Us] to the case of the homotopy projective spaces ℝℙ^{4m+1}. However, in this case complications arise, not the least of which is the fact that the simplification of the contact homology arising from the fact that all the Conley-Zehnder indices are even no longer holds.

7. Homotopy Spheres as Branched Covers

It is clear that the Brieskorn spheres described above can be thought of as cyclic branched covers of standard spheres. For example, the Kervaire spheres Σ_{p}^{4m+1} given by 1.3 is a p-fold cyclic cover of S^{4m+1} branched over the Stiefel manifold V_{2}(ℝ^{2m+1}) realized as an S^{1} bundle over an odd complex quadric. In particular, the exotic Kervaire sphere Σ_{3}^{4m+1} can be realized as a 3-fold cover of S^{4m+1} branched over the Stiefel manifold V_{2}(ℝ^{2m+1}) ⊂ S^{4m+1}. Similarly, the homotopy spheres Σ_{k}^{4m−1} ∈ bP_{4m} are 6k−1-fold covers of S^{4m−1} branched over the Kervaire sphere Σ_{3}^{4m−3}. In both cases the branching locus is a rational homology sphere. More generally Savel’ev [Sav] noticed that under certain conditions the cyclic branched cover of a rational homology sphere is a homotopy sphere. Here we formulate Savel’ev’s result in a way that is more convenient for our purpose. We make use of the work of Milnor and Orlik [MO]. Theorem 7.1: Let f(z_{1}, ⋯, z_{n}) be a weighted homogeneous polynomial of degree d and weights w = (w_{1}, ⋯, w_{n}) in ℂ^{n} with an isolated singularity at the origin. Let p ∈ ℤ^{+} and consider the link L_{g} of the equation

g = z_{0}^{p} + f(z_{1}, ⋯, z_{n}) = 0.

Write the numbers \frac{d}{w_{i}} in irreducible form \frac{u_{i}}{v_{i}} , and suppose that gcd(p, u_{i}) = 1 for each i = 1, ⋯, n. Then the link L_{g} is a rational homology sphere with weights \frac{(d,pw)}{gcd(p,d)} and degree lcm(p, d). Moreover, L_{g} is a homotopy sphere if and only if the link L_{f} is a rational homology sphere. Proof: Let us briefly recall the construction of the Alexander (characteristic) polynomial Δ_{f}(t) in [MO] associated to a link L_{f} of dimension 2n−3. It is the characteristic polynomial of the monodromy map 𝕀 − h_{∗} : H_{n−1}(F, ℤ)⟶H_{n−1}(F, ℤ) induced by the S_{w}^{1} action on the Milnor fibre F. Thus, Δ(t) = det(t𝕀−h_{∗}). Now both F and its closure F̄ are homotopy equivalent to a bouquet of n-spheres, and the boundary of F̄ is the link L_{f} which is n − 3connected. The Betti numbers b_{n−2}(L_{f}) = b_{n−1}(L_{f}) equal the number of factors of (t−1) in Δ_{f}(t). Thus, L_{f} is a rational homology sphere if and only if Δ(1) ≠ 0, and a homotopy sphere if and only if Δ_{f}(1) = ±1. Following Milnor and Orlik we let Λ_{j} denote the divisor of t^{j} − 1. Then the divisor of Δ_{f}(t) is given by

7.2  div Δ_{f} = ∏_{i=1}^{n}(\frac{Λ_{u_{i}}}{v_{i}} − 1)

Using the relations Λ_{a}Λ_{b} = gcd(a, b)Λ_{lcm(a,b)}, equation 7.2 takes the form

(−1)^{n} + ∑ a_{j}Λ_{j},

where a_{j} ∈ ℤ and the sum is taken over the set of all least common multiples of all combinations of the u_{1}, ⋯, u_{n}. The Alexander polynomial is then given by

7.3  Δ_{f}(t) = (t − 1)^{(−1)^{n}} ∏_{j}(t^{j} − 1)^{a_{j}},

and

7.4  b_{n−2}(L_{f}) = (−1)^{n} + ∑_{j} a_{j}.

Now we compute the divisor div Δ_{g}. We have

div Δ_{g} = (Λ_{p} − 1)div Δ_{f} = (Λ_{p} − 1)((−1)^{n} + ∑ a_{j}Λ_{j})

= ∑_{j} gcd(p, j)a_{j}Λ_{lcm(p,j)} − ∑_{j} a_{j}Λ_{j} + (−1)^{n}Λ_{p} + (−1)^{n+1}.

Since the j’s run through all the least common multiples of the set {u_{1}, ⋯, u_{n}} and gcd(p, u_{i}) = 1 for all i, we see that for all j, gcd(p, j) = 1. This implies

b_{n−1}(L_{g}) = ∑_{j} a_{j} − ∑_{j} a_{j} + (−1)^{n} + (−1)^{n+1} = 0.

Thus, L_{g} is a rational homology sphere. Next we compute the Alexander polynomial for L_{g}.

Δ_{g}(t) = (t − 1)^{(−1)^{n+1}}(t^{p} − 1)^{(−1)^{n}} ∏_{j}(t^{pj} − 1)^{a_{j}}(t^{j} − 1)^{−a_{j}}

7.5  = (t^{p−1} + ⋯ + t + 1)^{(−1)^{n}} ∏_{j}(\frac{t^{pj−1} + ⋯ + t + 1}{t^{j−1} + ⋯ + t + 1})^{a_{j}}.

This gives

Δ_{g}(1) = p^{Σ_{j}a_{j}+(−1)^{n}} = 1

where by 7.4 the last equality holds if and only if L_{f} is a rational homology sphere.

An easy way to insure the condition gcd(p, u_{i}) = 1 for all i = 1, ⋯, n is to assume that gcd(p, d) = 1 which we shall do henceforth.

Theorem 7.6: Let L_{g} be the link of a branched cover of S^{2n+1} branched over a smooth rational homology sphere L_{f} that is the link of an isolated hypersurface singularity given by a weighted homogeneous polynomial f of weights w_{f} and degree d_{f} as in Theorem 7.1. Suppose also that gcd(p, d_{f}) = 1, and that the corresponding hypersurface Z_{f} ⊂ ℙ(w_{f}) is well-formed. Then L_{g} is a homotopy sphere admitting a Sasakian structure S with positive Ricci curvature. Proof: Since gcd(p, d_{f}) = 1 the conditions of Theorem 7.1 are satisfied, so L_{g} is a homotopy sphere. Furthermore, since Z_{f} is well-formed it follows from Lemma 3.12 that the index I_{g} = w_{f} > 0, so the induced Sasakian structure is positive. The result then follows by Theorem 2.14.

This theorem can be immediately applied to the rational homology spheres constructed in [BGN4]. We have Theorem 7.7: Let L_{g} and L_{f} be as above, in particular gcd(p, d_{f}) = 1, and take L_{f} to be one of the 184 rational homology 7-spheres listed in [BGN4]. Then L_{g} admits a deformation class 𝔉(p, w) of Sasakian structures S ∈ 𝔉(p, w) containing Sasakian metrics with positive Ricci curvature. Furthermore, L_{g} is one of the following:

1. If p and d_{f} are both odd, then L_{g} is diffeomorphic to the standard 9-sphere S^{9}.

2. If p is odd and d_{f} is even, then for the 10 cases listed in the table in [BGN4] with d_{f} even we have: a. For

\begin{matrix}w = (w_{1}, w_{2}, w_{3}, w_{4}, w_{5}) & d_{f} \\ (97, 1531, 2201, 2775, 3253) & 9856 \\ (101, 439, 559, 579, 619) & 2296 \\ (103, 1321, 2337, 2845, 3251) & 9856 \\ (115, 341, 523, 591, 727) & 2296 \end{matrix}

L_{g} is diffeomorphic to the standard 9-sphere S^{9} if p ≡ ±1(mod 8), and to the exotic Kervaire 9-sphere Σ^{9} if p ≡ ±3(mod 8).

b. For

\begin{matrix}w = (w_{1}, w_{2}, w_{3}, w_{4}, w_{5}) & d_{f} \\ (155, 1075, 3532, 5835, 7064) & 17660 \\ (187, 2416, 8177, 10965, 19328) & 41072 \\ (221, 2416, 5491, 13617, 19328) & 41072 \\ (316, 1727, 9577, 13345, 24648) & 49612 \\ (316, 2041, 6751, 15857, 24648) & 49612 \end{matrix}

L_{g} is diffeomorphic to the standard 9-sphere S^{9} if p^{2} ≡ ±1(mod 8), and to the exotic Kervaire 9-sphere Σ^{9} if p^{2} ≡ ±3(mod 8).

c. For

\begin{matrix}w = (w_{1}, w_{2}, w_{3}, w_{4}, w_{5}) & d_{f} \\ (49, 334, 525, 668, 763) & 2338 \end{matrix}

L_{g} is diffeomorphic to the standard 9-sphere S^{9} if p^{3} ≡ ±1(mod 8), and to the exotic Kervaire 9-sphere Σ^{9} if p^{3} ≡ ±3(mod 8).

3. If p is even and d_{f} is odd, then L_{g} is diffeomorphic to the

\{ \begin{matrix}S^{9} & if |H_{3}(L_{f}, ℤ)| ≡ ±1(mod 8); \\ Σ^{9} & if |H_{3}(L_{f}, ℤ)| ≡ ±3(mod 8). \end{matrix}

However, the 174 examples given in [BGN4] with odd degree all have |H_{3}(L_{f}, ℤ)| ≡ 1(mod 8); hence, in this case L_{g} is always a standard sphere.

In items 1 and 2 above there is a countable infinity of deformation classes of Sasakian structures with positive Ricci curvature on each of the homotopy 9-spheres. Proof: The first part follows immediately from Theorem 7.6 by choosing p such that gcd(p, d_{f}) = 1. To compute the differential invariant we use a Theorem of Levine [Lev, Mil2] which asserts that L_{g} is the standard 9-sphere if Δ_{g}(−1) ≡ ±1(mod 8) and the exotic Kervaire sphere if Δ_{g}(−1) ≡ ±3(mod 8). So we need to compute Δ_{g}(−1). First, by factoring we can reduce 7.5 further:

7.7  Δ_{g}(t) = (t^{p−1} + ⋯ + t + 1)^{−1} ∏_{j}(t^{(p−1)j} + ⋯ + t^{j} + 1)^{a_{j}},

where j ranges through the set of all least common multiples of the u_{i}’s.

Case 1: Here we have both p and d_{f} odd. Since d_{f} is odd all the u_{i}’s are odd. Thus, j in the product in 7.7 is always odd. This implies that Δ_{g}(−1) = 1 so L_{g} is the standard 9-sphere.

Case 2: Again p is odd, but now d is even, so both even and odd j’s occur in 7.7. The odd j’s contribute a 1 to the product. Thus, we have

Δ_{g}(−1) = ∏_{j even} p^{a_{j}} = p^{Σ_{even j}a_{j}}.

To proceed further we must look at the list of rational homology 7-spheres in [BGN4]. There are precisely 10 on the list with even degree. Moreover, there are 2 types that occur as indicated by Lemmas 3.4 and 3.12 of [BGN4], and they are distinguished by the divisor of their Alexander polynomials. For the first type we have div Δ = Λ_{d} − 1, and 4 of the 10 are of this type, precisely the ones occurring in 2a of the Theorem. In this case there is one even j, namely j = d and a_{d} = 1. Thus, we have Δ_{g}(−1) = p, so 2a follows by Levine’s result. The remaining 6 rational homology 7-spheres with even degree are of type two and for these we have

7.8  div Δ = n(w)Λ_{d} + Λ_{m_{3}} − n(w)Λ_{m_{2}} − 1

where n(w) ∈ ℤ^{+}, and d = m_{2}m_{3} with m_{2} and m_{3} relatively prime. One can see from the definition of m_{2} and m_{3} in [BG4] that m_{3} must be even and m_{2} odd. So if follow from 7.8 that ∑_{j even} a_{j} = n(w) + 1. Now n(w) can be read off from the table in [BGN4], since

the order of the third homology group H_{3}(L_{f}, ℤ) is m_{3}^{n(w)+1}; hence, 2b and 2c follow.

Case 3: Here p is even and d_{f} is odd. In this case the form of 7.7 is indeterminant at t = −1, so we need some further manipulation. We also need to treat the two types of rational homology 7-spheres separately. For type 1 we have precisely one j which is odd and equal to d, and a_{d} = 1. Thus, from 7.5 we see that the denominator in the product evaluates to 1 at t = −1 since j is odd. Then we have

Δ_{g}(−1) = lim_{t→−1}(t^{p−1}+⋯+t+1)^{−1}(t^{pd_{f} −1}+⋯+t+1) = lim_{t→−1} t^{p(d_{f} −1)}+⋯+t^{p}+1 = d_{f}.

For rational homology 7-spheres L_{f} of type 1 the order of H_{3}(L_{f}, ℤ) is precisely the degree d_{f}. For rational homology 7-spheres L_{f} of type 2, equation 7.8 above holds. Moreover, since d_{f} is odd, so are m_{2} and m_{3}. So we have 3 odd j’s, namely, d, m_{2}, m_{3} and a_{d} = n(w) = −a_{m_{2}}, a_{m_{3}} = 1. Again the terms in the denominator inside the product in 7.5 evaluate to 1, and we find

Δ_{g}(−1) = lim_{t→−1}(\frac{t^{pm_{3}−1} + ⋯ + t + 1}{t^{p−1} + ⋯ + t + 1)})(\frac{t^{pd_{f} −1} + ⋯ + t + 1}{t^{pm_{2}−1} + ⋯ + t + 1})^{n(w)}

= lim_{t→−1}(t^{p(m_{3}−1)} + ⋯ + t^{p} + 1)(t^{p(d_{f} −1)} + ⋯ + t^{pm_{2}} + 1)^{n(w)} = m_{3}^{n(w)+1}.

But m_{3}^{n(w)+1} is the order of H_{3}(L_{f}, ℤ) for L_{f} of type 2. Combining this with Levine’s result mentioned above proves 3 and the theorem.

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Department of Mathematics and Statistics January 2002 University of New Mexico Albuquerque, NM 87131 email: cboyer@math.unm.edu, galicki@math.unm.edu, nakamaye@math.unm.edu web pages: http://www.math.unm.edu/˜ cboyer, http://www.math.unm.edu/˜ galicki