On Positive Sasakian Geometry

Charles P. Boyer Krzysztof Galicki Michael Nakamaye

Abstract: A Sasakian structure S=(ξ,η,Φ,g) on a manifold M is called positive if its basic first Chern class c_{1}(F_{ξ}) can be represented by a positive (1,1)-form with respect to its transverse holomorphic CR-structure. We prove a theorem that says that every positive Sasakian structure can be deformed to a Sasakian structure whose metric has positive Ricci curvature. This allows us by example to give a completely independent proof of a result of Sha and Yang [SY] that for every nonnegative integer k the 5-manifolds k#(S^{2}×S^{3}) admits metrics of positive Ricci curvature.

Introduction

Let (M, J) is a compact complex manifold and g a Kähler metric on M, with Kähler form M. Suppose that ρ′ is a real, closed (1, 1)-form on M with [ρ′] = 2πc_{1}(M). Then there exists a unique Kähler metric g′ on M with Kähler form ω′, such that [ω] = [ω′] ∈ H^{2}(M, ℝ), and the Ricci form of g′ is ρ′. The above statement is the celebrated Calabi Conjecture which was posed by Eugene Calabi in 1954. The conjecture in its full generality was eventually proved by Yau in 1976. In the Fano case when c_{1}(M) > 0, i.e., when the first Chern class can be represented by a positive-definite real, closed (1, 1)-form ρ′ on M, the conjecture implies that the Kähler form of M can be represented by a metric of positive Ricci curvature.

There are several reasons one might be interested in a more general Calabi Problem when M is not necessarily a smooth manifold but rather a V -manifold or an orbifold [DK]. In the context of Sasakian manifolds one would naturally consider the Kähler geometry of the associated one-dimensional foliation. In this context one can actually prove a “transverse Yau theorem” and this was done by El Kacimi-Alaoui in 1990 [ElK]. In this note we discuss some consequences of the transverse Yau theorem. In particular, we prove the following which can be viewed as a Sasakian version of the “positive Calabi Conjecture” mentioned above:

Theorem A: 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 homologous to S.

Using some of the techniques developed earlier in our studies of circle V-bundles over log del Pezzo surfaces [BG2,BGN1,BGN2] we are able to apply Theorem A to demonstrate

Theorem B: There exist Sasakian metrics with positive Ricci tensor on k#(S^{2} × S^{3}) for every positive integer k. In particular, for each k > 5 there is a 2(k − 5) parameter family of inequivalent Sasakian structures with positive Ricci curvature.

During the preparation of this work the first two authors were partially supported by NSF grant DMS-9970904, and third author by NSF grant DMS-0070190.

From the standpoint of Riemannian metrics with positive Ricci curvature, this result is not new as it was first obtained by Sha and Yang [SY] in 1991; however, the Sasakian nature of such metrics is new. Moreover, our method of proof here is completely different.

In [JK] a list of all anticanonically (i.e. orbifold Fano index = 1) quasi-smooth log del Pezzo surfaces embedded in a weighted projective space ℙ(w) was given, and in [BGN1] it is shown that the largest Picard number occuring is 10. However, as soon as we allow higher Fano index, namely 2, we find:

Corollary C: There exist log del Pezzo surfaces ⊂ ℙ(w) with arbitrary Picard number.

1. Some Transverse Holomorphic Invariants

Recall that associated with every Sasakian structure S = (ξ, η, Φ, g) on a smooth manifold M is its characteristic foliation F_{ξ}, and as described elsewhere (cf. [BGN1]) the transverse geometry of a Sasakian manifold is Kähler. Moreover, there are transverse versions of both the Hodge theory and the Dolbeault theory [ElK] of a Sasakian manifold. In particular if (S, ξ, η, Φ, g) is a compact Sasakian manifold, then dη defines a nontrivial class [dη]_{B} in the basic cohomology group H_{B}^{1,1}(F_{ξ}) ⊂ H_{B}^{2}(F_{ξ}). We also define the set 𝔉(ξ) of all deformed Sasakian structures (ξ, η′, Φ′, g′) on S that are homologous to (ξ, η, Φ, g), that is such that [dη′]_{B} = [dη]_{B}. Now we have the basic Betti numbers

1.1  b_{r}^{B}(F_{ξ}) = dim H_{B}^{r}(F_{ξ}),

and the basic hodge numbers

1.2  h_{B}^{p,q}(F_{ξ}) = dim H_{B}^{p,q}(F_{ξ})

which satisfy the relations

1.3  b_{r}^{B}(F_{ξ}) = ∑_{p+q=r} h^{p,q}(F_{ξ}),  h_{B}^{p,q}(F_{ξ}) = h_{B}^{q,p}(F_{ξ}).

Accordingly, we have the transverse Euler characteristic of F_{ξ} given by

1.4a  χ(F_{ξ}) = ∑_{p=0}^{2n}(−1)^{p}dim H_{B}^{p}(F),

and the transverse holomorphic Euler characteristic defined by

1.4b  χ_{hol}(F_{ξ}) = ∑_{i}(−1)^{i}dim H_{B}^{0,q}(F_{ξ}).

Now the basic cohomology ring H^{∗}(F_{ξ}) is invariant under foliated homeomorphisms of M, that is homeomorphisms that preserve the foliation [ElKN], so the basic Betti numbers are also invariant under such homeomorphisms. The basic Hodge numbers, however, are only invariant under homemorphisms that preserve the foliation together with its transverse complex structure. Thus, they are not only invariants of the Sasakian structure, but also of its basic deformation class 𝔉(ξ). In the case that S = (ξ, η, Φ, g) is quasi-regular, the basic Betti numbers and basic Hodge numbers coincide with the corresponding Betti numbers and Hodge numbers on the space of leaves Z = S/S^{1} which is a compact Kähler orbifold.

The relationship between the basic cohomology H_{B}^{r}(F_{ξ}) and the ordinary cohomology H^{r}(S, ℝ) is given by the exact sequence [Ton]

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

where δ is the connecting homomorphism given by δ[α]_{B} = [dη ∧ α]_{B} = [dη]_{B} ∪ [α]_{B}, and j_{p} is the composition of the map induced by ξ⌋ with the well known isomorphism H^{r}(M, ℝ) ≈ H^{r}(M, ℝ)^{S^{1}} where H^{r}(M, ℝ)^{S^{1}} is the S^{1}-invariant cohomology defined from the S^{1}-invariant r-forms Ω^{r}(M)^{S^{1}}. Another important transverse invariant of the class 𝔉(ξ) is the basic first Chern class c_{1}(F_{ξ}) ∈ H^{2}(F_{ξ}) of the foliation F_{ξ}. The image ι_{∗}(c_{1}(F_{ξ})) is the real first Chern class of the normal bundle ν(F_{ξ}) to the foliation with its induced transverse complex structure.

If the dimension of S is 4n + 1 we can also define the basic Hirzebruch signature τ_{B}(F_{ξ}) to be the signature of the bilinear form defined by the cup product on the middle basic cohomology group H_{B}^{2n}(F_{ξ}). Using transverse Lefschetz theory [ElK] one can obtain the usual formula:

1.6  τ_{B}(F_{ξ}) = ∑_{p,q}(−1)^{p}h_{B}^{p,q}(F_{ξ}) = ∑_{p≡q(2)}(−1)^{p}h_{B}^{p,q}(F_{ξ}).

As is usual in complex geometry we introduce the geometric genus p_{g}(F_{ξ}) = h_{B}^{0,n}(F_{ξ}), the arithmetic genus p_{a}(F_{ξ}) = (−1)^{n}(χ_{hol}(F_{ξ}) − 1), and the irregularity q = q(S) = h_{B}^{0,1}. We remark that q = \frac{1}{2}b_{1}^{B}(F_{ξ}) = \frac{1}{2}b_{1}(M) is actually a topological invariant by (5) of Proposition 1.9 of [BGN1]. In the case n = 2, things simplify much more. All the basic Hodge numbers are given in terms of the three invariants q, p_{g}(F_{ξ}) and h_{B}^{1,1}(F_{ξ}), and we have the relations

1.7 χ_{B}(F_{ξ}) = 2 + 2p_{g}(F_{ξ}) − 4q + h_{B}^{1,1}(F_{ξ}),  τ_{B}(F_{ξ}) = 2 + 2p_{g}(F_{ξ}) − h_{B}^{1,1}(F_{ξ})
χ_{B}(F_{ξ}) + τ_{B}(F_{ξ}) = 4χ_{hol}(F_{ξ}).

The fact that q is actually a topological invariant has some nice consequences for n = 2. Proposition 1.8: Let S = (ξ, η, Φ, g) be a Sasakian structure on a 5-manifold M with b_{1}(M) = 0. Then we have

b_{2}(F_{ξ}) = 1 + b_{2}(M),  χ_{B}(F_{ξ}) = 3 + b_{2}(M) ≥ 3,  χ_{hol}(F_{ξ}) = 1 + p_{g}(F_{ξ}) ≥ 1.

Proof: Since b_{1}(M) = 0 Proposition 1.9 of [BGN1] and the exact cohomology sequence 1.5 imply

1.9  H_{B}^{2}(F_{ξ}) ≈ ℝ ⊕ H^{2}(M, ℝ),

and the results easily follow from this and equations 1.7. Corollary 1.10: On a Sasakian 5-manifold M with b_{1}(M) = 0 we must have b_{2}(M) ≥ 2p_{g}(F_{ξ}). In particular, every Sasakian structure S = (ξ, η, Φ, g) on S^{5} or S^{2} × S^{3} satisfies p_{g}(F_{ξ}) = 0 and χ_{hol}(F_{ξ}) = 1. Proof: This follows from Proposition 1.8 since by Proposition 1.9 of [BGN1] we must have h_{B}^{1,1}(F_{ξ}) ≥ 1.

The reader no doubt notices that a classification of Sasakian structures on compact 5-manifolds involves a “transverse Enriques-Kodaira classification” paralleling complex surface theory. Such a classification is currently under study.

2. Positive Sasakian Geometry

Definition 2.1: A Sasakian manifold M is said to be positive if its basic first Chern class c_{1}(F_{ξ}) can be represented by a basic positive definite (1, 1)-form.

A basic ingredient in studying positive Sasakian geometry is the “transverse Yau Theorem” of El Kacimi-Alaoui [ElK]:

Theorem 2.2 [ElK]: If c_{1}(F_{ξ}) is represented by a real basic (1, 1) form ρ^{T} , then it is the Ricci curvature form of a unique transverse Kähler form ω^{T} in the same basic cohomology class as dη.

In [BGN1] this theorem was reformulated in terms of Sasakian geometry in a convenient way, namely Theorem 2.3 [BGN1]: Let (M, ξ, η, Φ, g) be a Sasakian manifold whose basic first Chern class c_{1}(F_{ξ}) is represented by the real basic (1, 1) form ρ, then there is a unique Sasakian structure (ξ, η_{1}, Φ_{1}, g_{1}) ∈ 𝔉(ξ) homologous to (ξ, η, Φ, g) such that ρ_{g_{1}} = ρ − 2dη_{1} is the Ricci form of g_{1}, and η_{1} = η + ζ_{1}, with ζ_{1} = \frac{1}{2}d^{c}ϕ. The metric g_{1} and endomorphism Φ_{1} are then given by

Φ_{1} = Φ − ξ ⊗ ζ_{1} ∘ Φ, g_{1} = dη_{1} ∘ (id ⊗ Φ_{1}) + η_{1} ⊗ η_{1}.

One key ingredient still missing in the study of transverse holomorphic geometry is the transverse analogue of Dolbeault’s Theorem. The reason is that the sheaves in the basic Dolbeault complex are not fine, since they are constant along the leaves of the foliation. We can, however, obtain some partial results. Proposition 2.4: If S = (ξ, η, Φ, g) is a complete (i.e. g is complete) and positive Sasakian structure on a manifold M of dimension 2n + 1, then M is compact and q = 0. If in addition S = (ξ, η, Φ, g) is quasi-regular, then h^{p,0}(F_{ξ}) = 0 for all p > 0, and so χ_{hol}(F_{ξ}) = 1, p_{g}(F_{ξ}) = p_{a}(F_{ξ}) = 0. Moreover, any two quasi-regular positive Sasakian structures on a 5-manifold must have the same basic Hodge number h_{B}^{1,1}, the same basic Euler characteristic χ_{B}, and the same basic Hirzebruch signature τ_{B}. Proof: As mentioned above q = \frac{1}{2}b_{1}(M) follows from Proposition 1.9 of [BGN1]. Since c_{1}(F_{ξ}) > 0 Theorem 2.3 implies there is a Sasakian structure (ξ, η_{1}, Φ_{1}, g_{1}) ∈ 𝔉(ξ) homologous to (ξ, η, Φ, g) whose Ricci curvature is positive and represents c_{1}(F_{ξ}). It then follows from Hasegawa and Seino’s [HS] application of Myers’ Theorem to Sasakian geometry that g_{1} is complete, and hence, M is compact with finite fundamental group. This implies q = 0.

To prove the second part we note that any nontrivial element of H_{B}^{p,0}(F_{ξ}) is represented by a basic (p, 0)-form in the kernel of ¯∂, i.e. by a transversely holomorphic p-form α. Since S = (ξ, η, Φ, g) is quasi-regular, M is the total space of an orbifold S^{1} V-bundle over a compact Kähler orbifold Z. Moreover, since α is basic, it descends to a nontrivial element α̌ ∈ H^{p,0}(Z). Now since the Sasakian structure S is positive so is the Kähler structure on Z, i.e. c_{1}(Z) > 0, and this implies h^{p,0}(Z) = 0 by the Kodaira-Baily vanishing Theorem. Proof of Theorem A: c_{1}(F_{ξ}) can be represented by a positive definite basic (1, 1) form ρ, so by Theorem 2.3 there is a Sasakian structure S_{1} = (ξ, η_{1}, Φ_{1}, g_{1}) ∈ 𝔉(ξ) homologous to (ξ, η, Φ, g) such that ρ_{g_{1}} = ρ−2dη_{1} is the Ricci form of g_{1}. Let g_{1}^{T} denote the transverse Kähler metric of this Sasakian structure. Then the Ricci curvatures of g_{1}^{T} and g_{1} are related by

2.3  Ric_{g_{1}}|_{D_{1}×D_{1}} = Ric_{g_{1}^{T}} − 2g_{1}^{T} .

Next for any real number a > 0 we can perform a homothetic deformation [YK] of the Sasakian structure by defining

2.4  g_{2}^{T} = \frac{1}{a}g_{1}^{T} , η_{2} = \frac{1}{a}η_{1}, ξ_{2} = aξ, Φ_{2} = Φ_{1},

in which case S_{2} = (ξ_{2}, η_{2}, Φ_{2}, g_{2}) is a Sasakian structure where g_{2} = g_{2}^{T} + η_{2} ⊗ η_{2}. Notice also that S_{1} and S_{2} both have the same contact subbundles with the same underlying transverse complex structures, and the same characteristic foliations. Now since the Ricci tensor is invariant under homothety we find

2.5  Ric_{g_{2}}|_{D_{2}×D_{2}} = Ric_{g_{2}^{T}} − \frac{2}{a}g_{2}^{T} .

But since Ric_{g_{2}^{T}} > 0 and M is compact there exists a_{0} ∈ ℝ^{+} such that for all a > a_{0} we have Ric_{g_{2}}|_{D_{2}×D_{2}} > 0. But also for any Sasakian metric we have Ric_{g_{2}}(X, ξ_{2}) = 2nη_{2}(X) which proves the result.

We shall make use of Proposition 2.6: Any simply connected compact positive Sasakian manifold M is spin. Proof: M is spin if and only if its second Stiefel-Whitney class w_{2}(M) vanishes. But if D is the contact subbundle of a Sasakian structure S = (ξ, η, Φ, g) on M, we have a natural splitting

T M = D ⊕ L_{ξ}

where L_{ξ} is the trivial real line bundle generated by ξ. Thus,

w_{2}(M) = w_{2}(T M) = w_{2}(D)

which is the mod 2 reduction of c_{1}(D) ∈ H^{2}(M, ℤ).

Now suppose that S is a positive Sasakian structure so that c_{1}(F_{ξ}) > 0. Choose a transverse Kähler metric g′_{T} whose Kähler form ω′_{T} ∈ c_{1}(F_{ξ}). Then both ω′_{T} and its transverse Ricci form ρ′_{T} represent c_{1}(F_{ξ}). This transverse Kähler structure defines a positive Sasakian structure S′ = (ξ, η′, Φ′, g′) on M such that g′ = g′_{T} + η′ ⊗ η′, and ω′_{T} represents c_{1}(F_{ξ}). The exact sequence 1.11 of [BGN1] becomes

2.7  H^{2}(M, ℤ) ; ↓ ; 0⟶ℝ⟶^{δ}H_{B}^{2}(F_{ξ})⟶^{ι_{∗}} H^{2}(M, ℝ) ⟶

where δ(c) = c[ω′_{T}]_{B}. But since c_{1}(F_{ξ}) is represented by ω′_{T} we have c_{1}(D′) = ι_{∗}c_{1}(F_{ξ}) = ι_{∗}δ(1) = 0 in H^{2}(M, ℝ). But since M is simply connected c_{1}(D′) = 0, so this implies that w_{2}(M) = w_{2}(D′) = 0.

Actually we have proven more. Proposition 2.8: For every positive Sasakian structure S = (ξ, η, Φ, g) on M the first Chern class c_{1}(D) ∈ H^{2}(M, ℤ) is torsion.

So on a Sasakian manifold the free part of the first Chern class of the contact bundle D is an obstruction to positivity, and in particular to the existence of a compatible Sasakian-Einstein metric. However, the authors know of no example of a simply connected non-spin manifold admitting a Sasakian structure, or more generally a Sasakian manifold with the free part of c_{1}(D) nontrivial. It should be mentioned that Sasakian manifolds are known to admit Spin^{ℂ}-structures [Mor]. Now combining our results with Smale’s [Sm] well-known classification of simply connected spin 5-manifolds, we arrive at Theorem 2.9: Let M^{5} be a complete positive Sasakian 5-manifold. Then the universal cover M̃^{5} is diffeomorphic to one of the following:

M^{5}(k, α_{i}) = S^{5}#k#(S^{2} × S^{3})#M_{α_{1}}^{5}# ⋯ #M_{α_{r}}^{5}

for some nonnegative integers k, r where M_{α}^{5} is a compact simply connected 5-manifold with H_{2}(M_{α}^{5}, ℤ) ≈ ℤ_{α} ⊕ ℤ_{α}, and α_{i} are positive integers satisfying α_{1}|α_{2}| ⋯ |α_{r}.

The problem of existence of positive Sasakian structures on the above 5-manifolds is still open in general. However, in the next section we prove existence in the case r = 0 for all k > 0. We should mention that for r = 0 and 0 ≤ k ≤ 9 the manifolds are known to have Sasakian-Einstein structures [BG2, BGN1, BGN2] which are automatically positive. We should also mention that generally there is an a priori obstruction for M^{2n+1} to admit a Sasakian structure, namely the top Stiefel-Whitney class w_{2n+1}. However, it follows from Smale’s Theorem 3.2 [Sm] that w_{5}(M^{5}(k, α_{i})) = 0; hence, all M^{5}(k, α_{i}) are candidates for admitting Sasakian structures. These 5-manifolds with α_{i} nontrivial can never be realized as hypersurfaces in a well-formed weighted projective space since a result of [BG2] says that such hypersurfaces necessarily have no torsion in H_{2}(M, ℤ). Nevertheless, other methods are available to show that M^{5}(k, α_{i}) can admit Sasakian structures. Such examples with k = 0 and r > 0 are given in [BL].

3. Positive Sasakian Structures on k#(S^{2} × S^{3})

We consider weighted homogeneous polynomials f of degree d in four complex variables z_{0}, z_{1}, z_{2}, z_{3} which describe hypersurfaces in ℂ^{4} with only an isolated singularity at the origin. Here we are interested in the case that f has degree d = k + 1 with weights w = (1, 1, 1, k), k ≥ 2 and Fano index I = 2. Explicitly f is given by a general polynomial of the form

3.1  f(z_{0}, z_{1}, z_{2}, z_{3}) = g_{(k+1)}(z_{0}, z_{1}, z_{2}) + g_{(1)}(z_{0}, z_{1}, z_{2})z_{3}

where g_{(k+1)}(z_{0}, z_{1}, z_{2}) is a homogeneous polynomial of degree (k+1) which is not divisible by g_{(1)}(z_{0}, z_{1}, z_{2}) ≠ 0. This guarantees that f has only an isolated singularity at the origin. The zero locus of f cuts out a cone C_{f} ⊂ ℂ^{4} and the link L_{f} is defined by intersecting with the unit sphere S^{7}, viz.

3.2  L_{f} = C_{f} ∩ S^{7}.

The link L_{f} is a smooth simply connected 5-manifold which for k ≥ 5 depends on 2(k − 5) effective parameters. To see this, note that the group 𝔊(1, 1, 1, k) of complex automorphisms of the projective space ℂℙ(1, 1, 1, k) is obtained by the projectivisation of the following group of transformations [BGN1]

φ_{𝔸,λ,ϕ} ( \begin{matrix}z_{0} \\ z_{1} \\ z_{2} \\ z_{3} \end{matrix} ) = ( \begin{matrix}𝔸 ( \begin{matrix}z_{0} \\ z_{1} \\ z_{2} \end{matrix} ) \\ λz_{3} + ϕ_{(k)}(z_{0}, z_{1}, z_{2}) \end{matrix} ) ,

where 𝔸 ∈ GL(3, ℂ), λ ∈ ℂ^{∗}, and ϕ_{(k)}(z_{0}, z_{1}, z_{2}) is an arbitrary homogeneous polynomial of degree k.

To prove Theorem B of the introduction we give several lemmas.

Lemma 3.2: The 5-manifold L_{f} is spin. Proof: We show that L_{f} has natural positive Sasakian structures. The result will then follow by Proposition 2.6. It is shown in [BG2] that links of isolated hypersurface singularities have natural Sasakian structures S whose characteristic foliation F_{ξ} is determined by the weights. Let Z_{f} denote the space of leaves of F_{ξ} which is embedded in the weighted projective space ℂℙ(1, 1, 1, k). The Sasakian structures will be positive if the basic first Chern class c_{1}(F_{ξ}) is positive. But since c_{1}(F_{ξ}) is the pullback of the first Chern class c_{1}(Z_{f}) of Z_{f} by the natural projection, it suffices to show that c_{1}(Z_{f}) is positive. But

c_{1}(Z_{f}) ≃ c_{1}(K_{Z_{f}}^{−1}) ≃ c_{1}(O(2)_{Z_{f}})

by 3.6 of [BGN1] which is clearly positive.

Lemma 3.3: b_{2}(L_{f}) = k. Proof: The procedure for computing the Betti numbers of links is given by Milnor and Orlik [MO]. Associate to any monic polynomial f with roots α_{1}, …, α_{k} ∈ ℂ^{∗} its divisor

3.4  div f =< α_{1} > + ⋯ + < α_{k} >

as an element of the integral ring ℤ[ℂ^{∗}] and let Λ_{n} = div(t^{n} − 1). The ‘rational weights’ used in [MO] are just \frac{d}{w_{i}} , and are written in irreducible form, \frac{d}{w_{i}} = \frac{u_{i}}{v_{i}}. The divisor of the characteristic polynomial Δ(t) is then determined by

3.5.  div Δ(t) = ∏_{i}(\frac{Λ_{u_{i}}}{v_{i}} − 1) = 1 + ∑ a_{j}Λ_{j},

where a_{j} ∈ ℤ and the second equality is obtained by using the relations Λ_{a}Λ_{b} = gcd(a, b)Λ_{lcm(a,b)}. The second Betti number of the link is then given by

3.6  b_{2}(L_{f}) = 1 + ∑_{j} a_{j}.

In our case we have

div Δ(t) = (\frac{Λ_{k+1}}{k} −1)(Λ_{k+1}−1)^{3} = (\frac{Λ_{k+1}}{k} −1)(Λ_{k+1}−1)((k−1)Λ_{k+1}+1) = (k−1)Λ_{k+1}+1

implying b_{2}(L_{f}) = k.

To prove Theorem B of the introduction it suffices to prove Lemma 3.7: L_{f} is diffeomorphic to k#(S^{2} × S^{3}). Proof: This will follow from a Theorem of Smale [Sm] and Lemmas 3.2 and 3.3 if we can show that H_{2}(L_{f}, ℤ) has no torsion. But since the gcd of any three weights is one this follows from Lemma 5.8 of [BG2].

Finally, we mention the failure of the sufficient conditions described in [JK,BGN1] for Z_{f} to admit a Kähler-Einstein metric, and hence, for L_{f} to admit a Sasakian-Einstein metric. Indeed, it follows from Lemma 5.1 of [BGN1] that for any k ≥ 2, (Z_{f}, \frac{2+ϵ}{3}D) cannot be Kawamata log terminal for every effective ℚ-divisor that is numerically equivalent to K_{Z_{f}}^{−1}. Furthermore, the Hitchin-Thorpe inequality (c_{1}^{2} ≥ 0) which prohibits smooth compact complex surfaces with q = p_{g} = 0 and b_{2} ≥ 9 from admitting any Einstein metrics whatsoever, is ineffective for hypersurfaces in the weighted projective space ℙ(w) since any such hypersurface Z_{f} of degree d satisfies

c_{1}^{2}(Z_{f}) = \frac{d(|w| − d)^{2}}{w_{0}w_{1}w_{2}w_{3}} ≥ 0.

In our case we have c_{1}^{2} = 4(1 + \frac{1}{k}).

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Department of Mathematics and Statistics March 2001 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