Trans. of the AMS 354 (2002), 2983-2996  Transations of the American

Mathematical Society

Sasakian-Einstein Structures on 9#(S^{2} × S^{3})

Charles P. Boyer Krzysztof Galicki Michael Nakamaye

Abstract: We show that #9(S^{2}×S^{3}) admits an 8-dimensional complex family of inequivalent non-regular Sasakian-Einstein structures. These are the first known Einstein metrics on this 5-manifold. In particular, the bound b_{2}(M)≤8 which holds for any regular Sasakian-Einstein M does not apply to the non-regular case. We also discuss the failure of the Hitchin-Thorpe inequality in the case of 4-orbifolds and describe the orbifold version.

Introduction

Recently Demailly and Kóllar have developed some new techniques to study the existence of Kähler-Einstein metrics on compact Fano orbifolds [DK]. Johnson and Kóllar applied these techniques to study Kähler-Einstein metrics on certain log del Pezzo surfaces in weighted projective 3-spaces [JK1] as well as anti-canonically embedded orbifold Fano 3-folds in weighted projective 4-spaces [JK2]. In [BG3, BGN1] we have extended some of the results of [JK1] to the case of higher index and have studied their implications in the realm of Sasakian-Einstein metrics on simply connected smooth 5-manifolds. These arise as links of isolated hypersurface singularities given by quasi-homogeneous polynomials in ℂ^{4} [BG3]. In [BGN1] we showed that there are many families of non-regular Sasakian-Einstein structures on k-fold connected sums of S^{2} × S^{3} for k = 1, 2, 3, 4, 5, 6, 7. When k = 8 we have not found any non-regular Sasakian-Einstein structures. However, 8#(S^{2} ×S^{3}) viewed as a circle bundle over a blow-up of ℂℙ^{2} at eight points comes with an eight complex parameter family of regular Sasakian-Einstein structures. It follows from the well-known classification of smooth del Pezzo surfaces that for regular Sasakian-Einstein 5-manifolds S we must have b_{2}(S) ≤ 8. Moreover, it follows from the Hitchin-Thorpe inequality that ℂℙ^{2} blown-up at 9 or more points in general position is excluded from admitting any Einstein metric whatsoever. Neither of these bounds hold in the orbifold case. For the Hitchin-Thorpe type result the topological Euler number and Hirzebruch signature must be replaced by their corresponding orbifold analogue. Hence, no such topological bound holds in the orbifold case. Nevertheless, all the non-regular examples found in [BGN1] still satisfy this bound. The main purpose of this note is to show that the bound does not hold for non-regular Sasakian-Einstein 5-manifolds. We achieve this by a careful analysis of the question of the existence of Kähler-Einstein metrics for the log del Pezzo surface Z_{16} ⊂ ℂℙ^{2}(1, 3, 5, 8) of index 1 and degree 16 found by Johnson and Kóllar [JK1]. Johnson and Kóllar leave the question of existence of Kähler-Einstein metrics on Z_{16} open. Using well-known techniques of Milnor and Orlik [MO], one can easily compute the characteristic polynomial of the associated link L_{16} ⊂ ℂ^{4} and from this that the second Betti number b_{2}(L_{16}) = 9. As a consequence of this, a general argument concerning the absence of torsion in H_{2} [BG3], and a classification theorem of Smale [Sm], the link L_{16}, which is naturally a V -bundle over L_{16} → Z_{16}, must be diffeomorphic to 9#(S^{2} × S^{3}). In section 3, we show that Z does satisfy the sufficient conditions of [DK] and does admit a Kähler-Einstein metric. This is the key to establishing the following

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. 1991 Mathematics Subject Classification: 53C25,53C12,14E30

Theorem A: Let S = #9(S^{2}×S^{3}). Then S admits an 8-dimensional family of inequivalent non-regular Sasakian-Einstein structures. Furthermore, the Einstein metrics in the family are inequivalent as Riemannian metrics.

It follows that the metric cone on S = #9(S^{2} × S^{3}) is a singular Calabi-Yau 3-fold [BG1-2]. Such cones arise naturally in the context of the supersymmetric string theory (see, for example, [Y]). In view of Theorem A and Smale’s classification result [Sm] one can ask the following important questions:

Problem B: Suppose M is a compact simply-connected Sasakian-Einstein 5-manifold. Can b_{2}(M) be arbitrarily large? And, if so, is there a Sasakian-Einstein structure on any connected sum of k copies of S^{2} × S^{3}?

We are unable to answer this question here; however, in a forthcoming note [BGN2] the authors show that there is a positive Sasakian structure on k#(S^{2} × S^{3}) for k ≥ 1.

Acknowledgments: We would like to thank János Kollár for discussions and his interest in our work, and Carolina Araujo for pointing out some minor computational errors in the first draft of this work.

1. Sasakian Structures on Links of Isolated Hypersurface Singularities

Although the purpose of this note is to describe a solitary example of the Sasakian-Einstein geometry of 9#(S^{2} × S^{3}) we shall begin with a very brief summary of the Sasakian and Sasakian-Einstein geometry of links of isolated hypersurface singularities defined by weighted homogeneous polynomials. For more details we refer the reader to [BG3, BGN1]. Consider the affine space ℂ^{n+1} together with a weighted ℂ^{∗}-action ℂ_{w}^{∗} given by (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 gcd(w_{0}, …, w_{n}) = 1. Let f be a quasi-homogeneous polynomial, that is f ∈ ℂ[z_{0}, …, z_{n}] and satisfies

1.1  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

1.2  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 [Mil] that the link L_{f} is (n − 2)-connected.

Recall (cf. [BG1]) that a Riemannian manifold (M, g) of dimension m is Sasakian if the holonomy group of the metric cone (C(M), ḡ) = (M × ℝ^{+}, dr^{2} + r^{2}g) is contained in U(\frac{m+1}{2}), i.e. if (C(M), ḡ) is Kähler. The symplectic form ω on a Kähler cone is exact and takes the form ω = d(rη), where η is a contact 1-form on M. So if I is the almost complex structure on C(M), the Kähler structure (ω, I) on C(M) together with the ‘Liouville vector field’ Ψ = r \frac{∂}{∂r} induces a Sasakian structure S = (ξ, η, Φ, g) on M ≃ M × \{1\}, where the Reeb vector field is ξ = IΨ, and the (1, 1) tensor field Φ is the restriction of I to T M.

On S^{2n+1} there is a well-known “weighted” Sasakian structure S_{w} = (ξ_{w}, η_{w}, Φ_{w}, g_{w}), where the vector field ξ_{w} is the infinitesimal generator of the circle subgroup S_{w}^{1} ⊂ ℂ_{w}^{∗}. This Sasakian structure on S^{2n+1} induces a Sasakian structure, also denoted by S_{w}, on the link L_{f}. (See [YK, BG3, BGN1] for details. The quotient space Z_{f} of S^{2n+1} by S_{w}^{1}, or equivalently the space of leaves of the characteristic foliation F_{ξ} of S_{w}, is a compact Kähler orbifold which is a projective algebraic variety embedded in the weighted projective ℙ(w) = ℙ(w_{0}, w_{1}, ⋯, w_{n}), in such a way 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.

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. Such deformations of a given Sasakian structure S = (ξ, η, Φ, g) are obtained by adding to η a continuous one parameter family of basic1-forms ζ_{t}. We require that the 1-form η_{t} = η + ζ_{t} satisfy the conditions

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

Since ζ_{t} is basic ξ is the Reeb (characteristic) vector field associated to η_{t} for all t. Now let us define

1.6  \begin{aligned}Φ_{t} & = Φ − ξ ⊗ ζ_{t} ∘ Φ, \\ g_{t} & = g + dζ_{t} ∘ (Φ ⊗ id) + ζ_{t} ⊗ η + η ⊗ ζ_{t} + ζ_{t} ⊗ ζ_{t}. \end{aligned}

Then S_{t} = (ξ, η_{t}, Φ_{t}, g_{t}) is a Sasakian structure for all t ∈ [0, 1] that has the same underlying contact structure and the same characteristic foliation. In general these structures are inequivalent and the moduli space of Sasakian structures having the same characteristic vector field is infinite dimensional.

Suppose now we have a link L_{f} with a given Sasakian structure (ξ, η, Φ, g). When can we find a 1-form ζ such that the deformed structure (ξ, η + ζ, Φ′, g′) is Sasakian-Einstein ? This is a Sasakian version of the Calabi problem and its solution is equivalent to solving the corresponding Calabi problem on the space of leaves Z_{f}. Since a Sasakian-Einstein manifold necessarily has positive Ricci tensor, its Sasakian structure is necessarily positive. This also implies that the Kähler structure on Z_{f} be positive, i.e. c_{1}(Z_{f}) can be represented by a positive definite (1, 1) form. In this case there are well-known obstructions to solving to solving the Calabi problem. These obstructions for finding a solution to the Monge-Ampere equations involves the non-triviality of certain multiplier ideal sheaves [Na, DK] associated with effective canonical ℚ-divisors on the space of leaves Z_{f}. Consequently, if one can show that these multiplier ideal sheaves coincide with the full structure sheaf, one obtains the existence of a positive Kähler-Einstein metric on Z_{f} and hence, a Sasakian-Einstein metric on L_{f}. Sufficient conditions that guarantee this come from Mori theory and can be phrased as follows [JK1]:

Sufficient Conditions for a K-E metric on Z_{f} and S-E metric on L_{f} 1.7:

I. For some ϵ > 0 and every effective ℚ-divisor D on Z_{f} numerically equivalent to −K_{Z_{f}} , the pair (Z, \frac{n+ϵ}{n+1}D) is Kawamata log terminal (klt).

Now conditions on the weights that guarantee that the hypersurface C_{f} ⊂ ℂ^{n+1} have only an isolated singularity at the origin are well-known [Fle, JK1]. These conditions, known as quasi-smoothness conditions guarantee that Z_{f} is smooth in the orbifold sense, that is, at a vertex P_{i} ∈ ℙ(w) the preimage of Z_{f} in the orbifold chart of ℙ(w) is smooth. It is easy to see that one can formulate all these conditions as follows [Fle, JK1]:

Quasi-Smoothness Conditions 1.8:

II.  For each i = 0, ⋯, 3 there is a j and a monomial z_{i}^{m_{i}}z_{j} ∈ O(d).

Here j = i is possible.

III.  If gcd(w_{i}, w_{j}) > 1 then there is a monomial z_{i}^{b_{i}}z_{j}^{b_{j}} ∈ O(d).
IV.  For every i, j either there is a monomial z_{i}^{b_{i}}z_{j}^{b_{j}} ∈ O(d),

or there are monomials z_{i}^{c_{i}}z_{j}^{c_{j}}z_{k} and z_{i}^{d_{i}}z_{j}^{d_{j}}z_{l} ∈ O(d) with {k, l} ≠ {i, j}.

In condition I the i = j case corresponds to the case when Z_{f} does not pass through the point P_{i}. The second condition is equivalent to Z_{f} not containing any of the singular lines in ℙ(w). If Z_{f} contains a coordinate axis (say z_{j} = z_{j} = 0) then the condition III forces Z_{f} to be smooth along it, except possibly at the vertices.

There is another condition apart from quasi-smoothness that assures us that the adjunction theory behaves correctly, and that ℙ(w) does not have any orbifold singularities of codimension 1. It is [Dol, Fle]

Well-formedness Condition 1.9

V.  For any triple of distinct integers i, j, k we have gcd(w_{i}, w_{j}, w_{k}) = 1.

Condition IV guarantees that the canonical V-bundle K_{Z} is determined in terms of the degree and index by

1.10  K_{Z} ≃ O(−I) = O(d − |w|),

where |w| = ∑_{i} w_{i}.

2. Sufficient Conditions for Kähler-Einstein Metrics

In this section we consider hypersurfaces Z_{16} of degree 16 and weights (1, 3, 5, 8). As noted in [BGN1] there are 20 monomials in H^{0}(ℙ(1, 3, 5, 8), O(16)). To begin we consider the hypersurface Z_{16} given by the zero set of

f(z_{0}, z_{1}, z_{2}, z_{3}) = z_{0}^{16} + z_{1}^{5}z_{0} + z_{2}^{3}z_{0} + z_{1}z_{2}z_{3} + z_{3}^{2}

in the weighted projective space ℙ(1, 3, 5, 8). Thus Z_{16} is a hypersurface of degree 16 and index 1. This hypersurface passes through only two singular points of ℙ(1, 3, 5, 8), namely P_{1} = (0, 0, 1, 0) and P_{2} = (0, 1, 0, 0). We will show

Theorem 2.1 For any effective divisor

D ≡ O_{Z_{16}} (−\frac{11}{16}K_{Z_{16}})

the pair (Z_{16}, D) is klt.

Proof: Suppose D ≡ −K_{Z_{16}}. If Q ≠ P_{1}, P_{2} then we see, since the linear series O_{Z_{16}}(5) has only isolated base points, that

mult_{Q}(D) ≤ D · O_{Z_{16}}(5) = \frac{5 · 16}{3 · 5 · 8} = \frac{2}{3} < 1

and hence D is klt at Q (see [BGN1] Lemma 2.11).

We next consider the point P_{1} which is the most difficult. Let π : (ℂ^{2}, 0) → (Z_{16}, P_{1}) be a local cover of the index 5 quotient singularity at P_{1}. Intersecting with a general member of π^{∗}O_{Z_{16}}(3), all of which pass through the point P_{1}, we see that

mult_{0}(π^{∗}D) ≤ \frac{5 · 3 · 16}{3 · 5 · 8} = 2.

Thus we may apply Shokurov’s inversion of adjunction ([KM] Theorem 5.50) which says that the pair (Z_{16}, D) is log–canonical at P_{1} provided the pair p_{∗}^{−1}π^{∗}D is a sum of points with multiplicity at most one: here p : S → ℂ^{2} is the blow-up at the origin.

In order to analyze p_{∗}^{−1}(π^{∗}D) we first look at the natural candidate for “worst case scenario,” namely D = Z(z_{0}). This is a curve C ⊂ ℙ(3, 5, 8) with two components, namely C_{1} = Z(z_{3}) and C_{2} = Z(z_{3} + z_{1}z_{2}). Let D_{1} = π^{∗}C_{1} and D_{2} = π^{∗}C_{2}. We claim that D_{1} and D_{2} are both smooth at 0. To see this, consider F = Z(z_{1}). Then F ∩ C is proper and

hence π^{∗}F · π^{∗}C ≥ 2

with equality holding if and only if π^{∗}F meets D_{1} and D_{2} transversally at 0 (and at no other point). It is easy to check that F and C meet only at P_{1} and hence π^{∗}F meets D_{1} and D_{2} only at 0. We compute the intersection multiplicity

i_{0}(π^{∗}F · π^{∗}C) = \frac{5 · 3 · 16}{3 · 5 · 8} = 2.

Thus π^{∗}F meets D_{1} and D_{2} transversally at 0 and so D_{1} and D_{2} are both smooth at 0.

We now show that D_{1} and D_{2} meet transversally at 0. We already know that D_{1} meets π^{∗}F transversally. Let f_{1} be a local defining equation of D_{1} near 0 and f_{2} a local defining equation of D_{2}. Also, let m_{0} denote the maximal ideal of the local ring of ℂ^{2} at 0. Then f_{1} and f_{2} generate m_{0} if and only if f_{1} and f_{1}−f_{2} generate m_{0}. But Z(f_{1}) = π^{∗}Z(z_{3}) and Z(f_{2}) = π^{∗}Z(z_{1}z_{2} + z_{3}). Thus Z(f_{1} − f_{2}) = π^{∗}Z(z_{1}z_{2}). Since 0 ∉ π^{∗}Z(z_{2}), we see that D_{1} and D_{2} meet transversally at 0 if and only if D_{1} and π^{∗}(Z(z_{1})) = π^{∗}F meet transversally at 0. Thus, using the previous paragraph, we have established that D_{1} and D_{2} are both smooth at 0 and meet transversally, i.e. the divisor D = Z(z_{0}) is klt at P_{1}. Note that the same argument also holds with the point P_{1} replaced by the other singular point P_{2}.

Now, following [JK1] and [BGN1], we deal with an arbitrary (effective) choice of D ≡ −K_{Z_{16}}. Write

2.2  D = aC_{1} + bC_{2} + D′,

where D′ meets C_{1} and C_{2} properly. We compute intersection numbers:

2.3  O_{Z_{16}}(1) · (C_{1} + C_{2}) = \frac{16}{3 · 5 · 8} = \frac{2}{15},
2.4  O_{Z_{16}}(1) · C_{1} = \frac{8}{3 · 5 · 8} = \frac{1}{15},
2.5  O_{Z_{16}}(1) · C_{2} = \frac{1}{15},
2.6  C_{1} · C_{2} = \frac{8}{15},
2.7  C_{1}^{2} = C_{2}^{2} = \frac{−7}{15},
2.8  C_{1} · D′ ≤ \frac{2 − a}{15}.

The first equality 2.3 follows since z_{0} = 0 has C_{1} + C_{2} as divisor of zeroes; 2.4 follows since C_{1} is the intersection of the two hypersurfaces z_{0} = 0 and z_{3} = 0. The third formula 2.5 follows from 2.3 and 2.4. Since C_{1} and C_{2} meet only at the two singular points P_{1} and P_{2} and since the corresponding pull–backs meet transversally at the origin, this implies 2.6. The next equation, 2.7, follows from 2.4, 2.5, and 2.6, using the fact that O_{Z_{16}}(1)·Z_{16} is represented by C_{1} + C_{2}. Finally, 2.8 follows since

C_{1} · D′ ≤ D · D′ ≤ D · D − aD · C_{1}.

Intersecting 2.2 with C_{1} and using 2.4, 2.6, and 2.7 gives

\frac{1}{15} = C_{1} · O_{Z_{16}}(1)
= C_{1}(aC_{1} + bC_{2} + D′)

= \frac{−7a}{15} + \frac{8b}{15} + C_{1} · D′. 2.9

Applying 2.8 and using the fact that a + b ≤ 2 (since mult_{0}(π^{∗}D) ≤ 2), 2.9 implies

2.10  \frac{7a}{15} ≤ \frac{8(2 − a)}{15} + \frac{2 − a}{15}.

Rearranging terms in 2.10 gives

2.11  a ≤ \frac{9}{8}.

An identical argument shows that b ≤ \frac{9}{8}.

It remains to deal with the final term D′. Since D · D′ ≤ \frac{2−a}{15}, it follows that

2.12  mult_{0}(π^{∗}D′) ≤ \frac{2 − a}{3}.

Thus, in the tangent direction of D_{1} at 0 one obtains a divisor with coefficient at most

a + \frac{2 − a}{3} = \frac{2a + 2}{3} < \frac{16}{11}.

An identical bond holds with b in place of a and so we conclude that (Z_{16}, \frac{11}{16}D) is klt at P_{1}.

We now turn to the point P_{2} which will be simple as we have already performed the relevant computations. Let π : (ℂ^{2}, 0) → Z_{16} be a local cover of the quotient singularity at P_{2}. Shokurov’s inversion of adjunction still applies, as one sees by intersecting with a general member of |O_{Z_{16}}(5)|. We again let Z(z_{0}) = C_{1} ∪ C_{2}. Taking F = Z(z_{2}) and arguing exactly as above establishes that π^{∗}(C_{1}) and π^{∗}(C_{2}) meet transversally at 0.

If D ≡ −K_{Z_{16}} is an effective divisor then as above we write

D = aC_{1} + bC_{2} + D′,

where a, b ≤ \frac{9}{8}. The bound 2.12 still holds for mult_{0}(π^{∗}(D′)) (in fact, one can do better since the ramification index of π is only 3) and so exactly as before we find that (Z_{16}, \frac{11}{16}D) is klt at P_{2}.

Note that in the proof of Theorem 2.1 the same argument works for a much more general hypersurface Y. In particular, any of the numerous monomials with a z_{0} term can be added without changing the proof. The only monomial which presents a problem is z_{1}^{2}z_{2}^{2}. Consider the hypersurface Y defined by

f(z_{0}, z_{1}, z_{2}, z_{3}) = z_{0}^{16} + z_{1}^{5}z_{0} + z_{2}^{3}z_{0} + z_{3}^{2} + z_{1}^{2}z_{2}^{2}.

The divisor z_{0} = 0 again splits into two irreducible components, C_{1} = Z(z_{1}z_{2} + iz_{3}) and C_{2} = Z(z_{1}z_{2} − iz_{3}). Arguing precisely as above shows that the divisor z_{1} = 0 meets C_{1} and C_{2} transversally on the local cover π : (ℂ^{2}, 0) → (Y, P_{1}). Hence π^{∗}(C′_{1}) meets π^{∗}(C_{2}) transversally, where C′_{1} = Z(z_{1}z_{2}). Adding the equations for C_{1} and C_{2} then establishes that π^{∗}(C_{1}) and π^{∗}(C_{2}) meet transversally. An identical argument applies for the point P_{2} as above. Thus the inversion of adjunction formula applies and the proof of Theorem 2.1 can be copied to establish that (Y, \frac{11}{16}D) is klt for any choice of effective D ≡ −K_{Y}. This argument always applies provided the quadratic form as^{2} + bst + ct^{2} has distinct roots, where a is the coefficient of z_{3}^{2}, b is the coefficient of z_{1}z_{2}z_{3} and c is the coefficient of (z_{1}z_{2})^{2}. This then guarantees the existence of Kähler-Einstein metrics on the hypersurfaces Z_{16} which depend on 20 complex parameters. We thus obtain induced Sasakian-Einstein metrics on the link L_{16} depending on 20 complex parameters. However, these parameters are not all effective, and in section 5 we discuss the moduli problem for these hypersurfaces and links.

3. The Topology of the Link L_{16}

The main purpose of this section is to prove

Theorem 3.1: The Sasakian-Einstein manifold L_{16} is diffeomorphic to 9#(S^{2} × S^{3}).

In the process of proving this theorem we study the geometry of the link, and in particular compute the characteristic polynomial which is an important link invariant that generalizes the Alexander polynomial of a knot. Let us recall the well-known construction of Milnor [Mil] concerning isolated hypersurface singularities: There is a fibration of (S^{2n+1} − L_{f})⟶S^{1} whose fiber F is an open manifold that is homotopy equivalent to a bouquet of n-spheres S^{n} ∨ S^{n} ⋯ ∨ S^{n}. The Milnor number μ of L_{f} is the number of S^{n}’s in the bouquet. It is an invariant of the link which can be calculated explicitly in terms of the degree d and weights (w_{0}, …, w_{n}) by the formula [MO]

3.1  μ = μ(L_{f}) = ∏_{i=0}^{n}(\frac{d}{w_{i}} − 1).

For our link L_{16} with weights w = (1, 3, 5, 8), one immediately obtains

Proposition 3.2: The Milnor number of the simply connected Sasakian-Einstein 5manifold L_{16} is μ(L_{16}) = 143.

The closure F̄ of F has the same homotopy type as F and is a compact manifold whose boundary is precisely the link L_{f}. So the reduced homology of F and F̄ is only non-zero in dimension n and H_{n}(F, ℤ) ≈ ℤ^{143}. Using the Wang sequence of the Milnor fibration together with Alexander-Poincare duality gives the exact sequence [Mil]

3.3  0⟶H_{n}(L_{f}, ℤ)⟶H_{n}(F, ℤ)⟶^{𝕀−h_{∗}}H_{n}(F, ℤ)⟶H_{n−1}(L_{f}, ℤ)⟶0,

where h_{∗} is the monodromy map (or characteristic map) induced by the S_{w}^{1} action. From this we see that H_{n}(L_{f}, ℤ) = ker(𝕀 − h_{∗}) is a free Abelian group, and H_{n−1}(L_{f}, ℤ) = coker(𝕀 − h_{∗}) which in general has torsion, but whose free part equals ker(𝕀 − h_{∗}). There is a well-known algorithm due to Milnor and Orlik [MO] for computing the free part of H_{n−1}(L_{f}, ℤ) in terms of the characteristic polynomial Δ(t) = det(t𝕀 − h_{∗}), but finding the torsion is, in general, much more difficult. However, in the case at hand, links of dimension five, under a mild assumption a method of Randell [Ran] shows that the torsion vanishes [BG3]. The Betti number b_{n}(L_{f}) = b_{n−1}(L_{f}) equals the number of factors of (t − 1) in Δ(t).

We now compute the characteristic polynomials Δ(t) for our example.

Proposition 3.4: The characteristic polynomial Δ(t) of the link L_{16} is given by

Δ(t) = (t − 1)^{9}(t + 1)^{8}(t^{2} + 1)^{9}(t^{4} + 1)^{9}(t^{8} + 1)^{9}.

Hence, the second Betti number is b_{2}(L_{16}) = 9.

Proof: The Milnor and Orlik [MO] algorithm for computing the characteristic polynomial of the monodromy operator for weighted homogeneous polynomials is as follows: First associate to any monic polynomial F with roots α_{1}, …, α_{k} ∈ ℂ^{∗} its divisor

÷F =< α_{1} > + ⋯ + < α_{k} >

as an element of the integral ring ℤ[ℂ^{∗}] and let Λ_{n} = ÷(t^{n} − 1). The rational weights w′_{i} used in [MO] are related to our integer weights w_{i} by w′_{i} = \frac{d}{w_{i}} , and we write the w′_{i} = \frac{u_{i}}{v_{i}} in irreducible form. So we have w′_{0} = 16, w′_{1} = \frac{16}{3}, w′_{2} = \frac{16}{5}, w′_{3} = 2. Then the divisor of the characteristic polynomial is

3.5  ÷Δ(t) = (\frac{Λ_{16}}{5} − 1)(\frac{Λ_{16}}{3} − 1)(Λ_{16} − 1)(Λ_{2} − 1)

which upon using the relations Λ_{a}Λ_{b} = gcd(a, b)Λ_{lcm(a,b)} reduces to

÷Δ(t) = 9Λ_{16} − Λ_{2} + 1.

The characteristic polynomial Δ(t) is then determined from its divisor by

3.6  Δ(t) = (t − 1) ∏(t^{j} − 1)^{a_{j}},

where ÷Δ(t) = 1 + ∑_{j} a_{j}Λ_{j}. This gives the result.

Now the proof of Theorem 3.1 follows by the Smale Classification Theorem [Sm], since L_{16} has no torsion by Lemma 5.8 of [BG3], and any Sasakian-Einstein manifold is spin.

4. Topological versus Orbifold Invariants

Dimensions three and four are the only dimensions where there are known topological obstructions to the existence of Einstein metrics on smooth manifolds. The first example of this in dimension four is due to Marcel Berger [Be] who noticed that a compact four dimensional Einstein manifold must have non-negative Euler characteristic. Later Thorpe [Be] and then Hitchin [Hit] noticed a much sharper inequality obstructing Einstein metrics.

In this section we discuss both topological and orbifold invariants of compact complex four dimensional orbifolds with isolated orbifold singularities, and the failure of the Hitchin-Thorpe (cf. [Be, Hit, LeB]) inequality in the singular case. Of course, we certainly don’t expect this to hold as is in the orbifold case. There are orbifold correction terms as indicated by Satake [Sat] for the Euler characteristic. We note that for any log del Pezzo surface Z, both the topological Euler characteristic and Hirzebruch signature are determined in terms of the second Betti number b_{2}(Z) by

4.1  χ_{top}(Z) = 2 + b_{2}(Z),  τ_{top}(Z) = 2 − b_{2}(Z).

This gives

4.2  2χ_{top}(Z) + 3τ_{top}(Z) = 10 − b_{2}(Z).

In the smooth case this is positive for del Pezzo surfaces and any smooth rational surface with b_{2} ≥ 10 cannot admit any Einstein metric. However, for our singular Z_{16} we have

4.3  b_{2}(Z_{16}) = 10,  χ_{top}(Z_{16}) = 12,  τ_{top}(Z_{16}) = −8.

Thus, we have

4.4  2χ_{top}(Z_{16}) + 3τ_{top}(Z_{16}) = 0

even though Z_{16} has a positive Kähler-Einstein metric. Thus, the Hitchin-Thorpe inequality clearly fails in the singular category.

The point is, of course, that for orbifolds both the Gauss-Bonnet [Sat] and signature theorems [Kaw] relate integrals of curvature invariants to rational numbers that are orbifold invariants. In particular, the orbifold Euler characteristic χ_{orb}(Z) for a compact 4-orbifold Z with at most isolated orbifold singularities is given by [Sat]

4.5  χ_{orb}(Z) = \frac{1}{8π^{2}} ∫_{Z}(|W_{+}|^{2} + |W_{−}|^{2} + \frac{s^{2}}{24} − \frac{|Ric^{0}|^{2}}{2} ),

where W_{±}, s, Ric^{0} are the self-dual (anti-self-dual) pieces of the Weyl tensor, the scalar curvature, and traceless Ricci curvature, respectively. In the case that Z is a compact complex surface χ_{orb}(Z) equals [Bla] the top Chern class c_{2}(Z). Moreover, χ_{orb} is related to the topological Euler characteristic χ_{top}(Z) by [Bla, Sat]

4.6  χ_{orb}(Z) = χ_{top}(Z) − ∑_{x}(1 − \frac{1}{|Γ_{x}|}),

where |Γ_{x}| denotes the order of the local uniformizing group Γ_{x} at the singular point x and the sum is taken over all singular points. Notice that for orbifolds the analogue of Berger’s result is a bit stronger:

Proposition 4.7: Let Z be a compact 4-dimensional orbifold with precisely N isolated orbifold singularities. Suppose further that Z admits an orbifold Einstein metric, then

χ_{top}(Z) ≥ ∑_{x}(1 − \frac{1}{|Γ_{x}|}) ≥ \frac{N}{2} .

A similar analysis can be done for the signature. The Hirzebruch signature theorem says that for smooth manifolds whose dimension is a multiple of four the signature τ can be obtained by evaluating the L-function of the tangent bundle on the fundamental class of the manifold. In the orbifold case this is replaced by Kawasaki’s signature theorem [Kaw], which gives

4.8  τ_{top}(Z) =< L(T Z), [Z] > + ∑_{x} \frac{L_{x}}{|Γ_{x}|},

where again the sum is over all singular points, and L_{x} denotes the evaluation of the residual L-class at x. Here we have

4.9  < L(T Z), [Z] >= \frac{1}{12π^{2}} ∫_{Z}(|W_{+}|^{2} − |W_{−}|^{2}).

We can view this expression as an “orbifold signature” which is a rational number, and write τ_{orb}(Z) =< L(T Z), [Z] > . Then we rewrite equation 4.8 as

τ_{top}(Z) = τ_{orb}(Z) + τ_{res}(Z)

and the analogue for orbifolds of the Hitchin-Thorpe curvature integral becomes

4.10  2χ_{orb}(Z) ± 3τ_{orb}(Z) = \frac{1}{4π^{2}} ∫_{Z}(2|W_{±}|^{2} + \frac{s^{2}}{24} − \frac{|Ric^{0}|^{2}}{2} ),

so we obtain

Proposition 4.11: Let Z be a compact 4-dimensional orbifold with only isolated orbifold singularities. Suppose further that Z admits an orbifold Einstein metric, then

χ_{orb}(Z) ≥ \frac{3}{2}|τ_{orb}(Z)|.

Thus, we have

χ_{top}(Z) ≥ \frac{3}{2}|τ_{orb}(Z)| + ∑_{x}(1 − \frac{1}{|Γ_{x}|}) ≥ \frac{3}{2}|τ_{orb}(Z)| + \frac{N}{2} .

In general Proposition 4.11 contains much less information than the Hitchin-Thorpe result, since it involves orbifold invariants and not topological invariants. Nevertheless, it does give an improvement of Proposition 4.7. We also have

Proposition 4.12: Let Z be a compact complex 4-orbifold. Then

c_{1}^{2}(Z) = 2χ_{orb}(Z) + 3τ_{orb}(Z).

Thus, if Z admits an Einstein metric compatible with the orbifold structure, then c_{1}^{2} ≥ 0.

Proof: The Chern number c_{1}^{2} is obtained by evaluating the square of the Ricci form ρ on the fundamental class of the orbifold, and the usual proof holds once we replace the topological invariants χ_{top} and τ_{top} by their orbifold counterparts. The last statement then follows immediately from Proposition 4.11.

Remark 4.13: If Z is a compact complex 4-orbifold embedded as a hypersurface of degree d in the weighted projective space ℙ(w), then it is easy to see that

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

Thus, we see that if a compact complex 4-orbifold Z has 2χ_{orb} + 3τ_{orb} = c_{1}^{2} < 0 it cannot admit a compatible Einstein metric nor be embedded as a hypersurface in ℙ(w). We also mention that the case that the canonical V-bundle is trivial, so c_{1}^{2} = 0 and Z is simply connected gives Reid’s list [Fle] of 95 mostly singular K3 surfaces.

Let us now evaluate the orbifold invariants for the case currently under study, namely the singular complex surface Z_{16}. In this case it is easy to evaluate both c_{1}^{2} and χ_{orb} directly giving

c_{1}^{2}(Z_{16}) = \frac{16}{3 · 5 · 8} = \frac{2}{15}, χ_{orb}(Z_{16}) = 12 − \frac{2}{3} − \frac{4}{5} = 10 + \frac{8}{15}.

We have not evaluated τ_{orb}(Z_{16}) directly, but 4.12 gives τ_{orb}(Z_{16}) = −7 + \frac{1}{45}.

5. The Moduli of Sasakian-Einstein Structures on L_{16}

To analyze the moduli problem in the non-regular case, we begin by describing the group of complex automorphism 𝔊_{w} of the weighted projective 3-space ℙ(w). We shall assume that ℙ(w) is well-formed. Recall that ℙ(w) can be defined as a scheme Proj(S(w)), where

S(w) = ⨁_{d} S^{d}(w) = ℂ[z_{0}, z_{1}, z_{2}, z_{3}].

The ring of polynomials ℂ[z_{0}, z_{1}, z_{2}, z_{3}] is graded with grading defined by the weights w = (w_{1}, w_{1}, w_{2}, w_{3}). As a projective variety we can embed ℙ(w) ⊂ ℂℙ^{N} and then the group 𝔊_{w} is a subgroup of P GL(N, ℂ). ℙ(w) is a toric variety and we can describe 𝔊_{w} explicitly as follows: Let w = (w_{0}, w_{1}, w_{2}, w_{3}) be ordered as before. We consider the group G(w) of automorphisms of the graded ring S(w) defined on generators by

5.1  φ_{w} ( \begin{matrix}z_{0} \\ z_{1} \\ z_{2} \\ z_{3} \end{matrix} ) = ( \begin{matrix}f_{0}^{(w_{0})}(z_{0}, z_{1}, z_{2}, z_{3}) \\ f_{1}^{(w_{1})}(z_{0}, z_{1}, z_{2}, z_{3}) \\ f_{2}^{(w_{2})}(z_{0}, z_{1}, z_{2}, z_{3}) \\ f_{3}^{(w_{3})}(z_{0}, z_{1}, z_{2}, z_{3}) \end{matrix} ) ,

where f_{i}^{(w_{i})}(z_{0}, z_{1}, z_{2}, z_{3}) is an arbitrary weighted homogeneous polynomial of degree w_{i} in (z_{0}, z_{1}, z_{2}, z_{3}). This is a finite dimensional Lie group and it is a subgroup of GL(N, ℂ). Projectivising, we get 𝔊_{w} = ℙ_{ℂ}(G(w)).

Note that when w = (1, 1, 1, 1) then G(w) = GL(4, ℂ). Other than this case three weights are never the same if ℙ(w) is well-formed. If two weights coincide then G(w) contains GL(2, ℂ) as a subgroup. Finally, when all weights are distinct we can write

5.2  φ_{w} ( \begin{matrix}z_{0} \\ z_{1} \\ z_{2} \\ z_{3} \end{matrix} ) = ( \begin{matrix}a_{0}z_{0} \\ a_{1}z_{1} + f_{1}^{(w_{0})}(z_{1}) \\ a_{2}z_{2} + f_{2}^{(w_{2})}(z_{0}, z_{1}) \\ a_{3}z_{3} + f_{3}^{(w_{3})}(z_{0}, z_{1}, z_{2}) \end{matrix} ) ,

where (a_{0}, a_{1}, a_{2}, a_{3}) ∈ (ℂ^{∗})^{4} and f_{i}^{(w_{i})}, i = 1, 2, 3 are weighted homogeneous polynomials of degree w_{i}. The simplest situation occurs when f_{1} = f_{2} = f_{3} are forced to vanish. Then 𝔊_{w} = (ℂ^{∗})^{3} is the smallest it can possibly be as ℙ(w) is toric.

Let S_{w}^{d} ⊂ S(w) be the vector subspace spanned by all monomials in (z_{0}, z_{1}, z_{2}, z_{3}) of degree d = |w|−I, and let Ŝ^{d}(w) ⊂ S^{d}(w) denote subset all quasi-smooth elements. Then we define m_{w}^{d} to be the dimension of the subspace generated by Ŝ_{w}^{d}. Now the automorphism group G(w) acts on S_{w}^{d} leaving the subset Ŝ^{d}(w) of quasi-smooth polynomials invariant. Thus, for each log del Pezzo surface we define the moduli space

5.3  M_{w}^{d} = Ŝ_{w}^{d}/G(w) = ℙ(Ŝ_{w}^{d})/𝔊_{w},

with n_{w}^{d} = dim(M_{w}^{d}). Now there is an injective map

5.4  M_{w}^{d}⟶M^{ℂ}(Z_{w}),

and each element in M_{w}^{d} corresponds to a unique homothety class of Kähler-Einstein metrics modulo 𝔊_{w} and hence, to a unique Sasakian-Einstein structure on the corresponding 5-manifold S_{l} modulo the group 𝔊_{w} acting as CR automorphisms.

Let us now consider our Z_{16}. It is easy to see that S^{16}(1, 3, 5, 8) is isomorphic to ℂ^{20} and it is spanned by the monomials z_{3}^{2}, z_{1}z_{2}z_{3}, z_{1}^{2}z_{2}^{2}, z_{0}z_{2}^{3}, z_{0}z_{1}^{5}, z_{0}^{2}z_{1}^{2}z_{3}, z_{0}^{2}z_{1}^{3}z_{2}, z_{0}^{3}z_{2}z_{3}, z_{0}^{3}z_{1}z_{2}^{2}, z_{0}^{4}z_{1}^{4}, z_{0}^{5}z_{1}z_{3}, z_{0}^{5}z_{1}^{2}z_{2}, z_{0}^{6}z_{2}^{2}, z_{0}^{7}z_{1}^{3}, z_{0}^{8}z_{3}, z_{0}^{8}z_{1}z_{2}, z_{0}^{10}z_{1}^{2}, z_{0}^{11}z_{2}, z_{0}^{13}z_{1}, z_{0}^{16}. We take the open submanifold Ŝ^{16}(1, 3, 5, 8) ⊂ S^{16}(1, 3, 5, 8). This is acted on by the complex automorphism group, namely the group G(1, 3, 5, 8) generated by

5.5  φ_{w} ( \begin{matrix}z_{0} \\ z_{1} \\ z_{2} \\ z_{3} \end{matrix} ) = ( \begin{matrix}a_{0}^{1}z_{0} \\ a_{1}^{1}z_{1} + a_{1}^{2}z_{0}^{3} \\ a_{2}^{1}z_{2} + a_{2}^{2}z_{0}^{5} + a_{2}^{3}z_{0}^{2}z_{1} \\ a_{3}^{1}z_{3} + a_{3}^{2}z_{0}^{8} + a_{3}^{3}z_{1}z_{0}^{5} + a_{3}^{4}z_{1}^{2}z_{0}^{2} + a_{3}^{5}z_{2}z_{0}^{3} + a_{3}^{6}z_{2}z_{1} \end{matrix} ) ,

where a_{i}^{1} ∈ ℂ^{∗} and all other coefficients are in ℂ. G(1, 3, 5, 8) is a 12-dimensional complex Lie group acting on the open submanifold Ŝ^{16}(1, 3, 5, 8) ⊂ S^{16}(1, 3, 5, 8) ≈ ℂ^{20}. It follows that the quotient is an 8-dimensional complex manifold which by the Bando-Mabuchi Theorem [BM] is the moduli space of positive Kähler-Einstein metrics on the underlying compact orbifold Z_{16}.

Theorem A now follows from these result and Proposition 7.14 of [BGN1].

Bibliography

[Be] A. Besse, Einstein manifolds, Springer-Verlag, Berlin-New York, 1987.

[BG1] C. P. Boyer and K. Galicki, On Sasakian-Einstein Geometry, Int. J. of Math. 11 (2000), 873-909.

[BG2] C. P. Boyer and K. Galicki, 3-Sasakian Manifolds, Surveys in Differential Geometry VI: Essays on Einstein Manifolds A supplement to the Journal of Differential Geometry, pp. 123-184, (eds. C. LeBrun, M. Wang); International Press, Cambridge 1999.

[BG3] C. P. Boyer and K. Galicki, New Einstein Metrics in Dimension Five, J. Diff. Geom., to appear; math.DG/0003174.

[BGN1] C. P. Boyer, K. Galicki, and M. Nakamaye, On the Geometry of Sasakian-Einstein 5-Manifolds, submitted for publication; math.DG/0012047.

[BGN2] C. P. Boyer, K. Galicki, and M. Nakamaye, On Positive Sasakian Geometry, submitted

for publication; math.DG/0104126.

[Bla] R. Blache, Chern classes and Hirzebruch-Riemann-Roch theorem for coherent sheaves on complex projective orbifolds with isolated singularities, Math. Z. 222 (1996), 7-57.

[BM] S. Bando and T. Mabuchi, Uniqueness of Einstein Kähler Metrics Modulo Connected Group Actions, Adv. Stud. Pure Math. 10 (1987), 11-40.

[DK] J.-P. Demailly and J. Kollár, Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, preprint AG/9910118, to appear in Ann. Scient. Ec. Norm. Sup. Paris

[Dol] I. Dolgachev, Weighted projective varieties, in Proceedings, Group Actions and Vector Fields, Vancouver (1981) LNM 956, 34-71.

[Fle] A.R. Fletcher, Working with weighted complete intersections, Preprint MPI/89-95, revised version in Explicit birational geometry of 3-folds, A. Corti and M. Reid, eds., Cambridge Univ. Press, 2000, pp 101-173.

[Hit] N. Hitchin, On compact four-dimensional Einstein manifolds, J. Diff. Geom. 9 (1974), 435-442.

[JK1] J.M. Johnson and J. Kollár, Kähler-Einstein metrics on log del Pezzo surfaces in weighted projective 3-space, Ann. Inst. Fourier 51(1) (2001) 69-79.

[JK2] J.M. Johnson and J. Kollár, Fano hypersurfaces in weighted projective 4-spaces, Experimental Math. 10(1) (2001) 151-158.

[Kaw] Y. Kawamata, The cone of curves of algebraic varieties, Annals of Math 119, 1984, pp. 603-33.

[KM] J. Kollár, and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge University Press, 1998.

[LeB] C. LeBrun, Four-Dimensional Einstein Manifolds, and Beyond, Surveys in Differential Geometry VI: Essays on Einstein Manifolds; A supplement to the Journal of Differential Geometry, pp. 247-285, (eds. C. LeBrun, M. Wang); International Press, Cambridge (1999).

[Mil] J. Milnor, Singular Points of Complex Hypersurfaces, Ann. of Math. Stud. 61, Princeton Univ. Press, 1968.

[MO] J. Milnor and P. Orlik, Isolated singularities defined by weighted homogeneous polynomials, Topology 9 (1970), 385-393.

[Na] A.M. Nadel, Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature, Ann. Math. 138 (1990), 549-596.

[Ran] R.C. Randell The homology of generalized Brieskorn manifolds, Topology 14 (1975), 347-355.

[Sm] S. Smale, On the structure of 5-manifolds, Ann. Math. 75 (1962), 38-46.

[Sat] I. Satake, The Gauss-Bonnet theorem for V -manifolds, J. Math. Soc. Japan V.9 No 4. (1957), 464-476.

[Y] S. -T. Yau, Einstein manifolds with zero Ricci curvature, Surveys in Differential Geometry VI: Essays on Einstein Manifolds; A supplement to the Journal of Differential Geometry, pp.1-14, (eds. C. LeBrun, M. Wang); International Press, Cambridge (1999).

[YK] K. Yano and M. Kon, Structures on manifolds, Series in Pure Mathematics 3, World Scientific Pub. Co., Singapore, 1984.

Department of Mathematics and Statistics September 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