J. Diff. Geom 57 (2001), 443-463  Journal of Differential

Geometry

New Einstein Metrics in Dimension Five

Charles P. Boyer Krzysztof Galicki

Abstract: The purpose of this note is to prove the existence of new Sasakian-Einstein metrics on S^{2}×S^{3} and on (S^{2}×S^{3})#(S^{2}×S^{3}). These give the first known examples of non-regular Sasakian-Einstein 5-manifolds. Our method involves describing the Sasakian-Einstein structures as links of certain isolated hypersurface singularities, and makes use of the recent work of Demailly and Kollár who obtained new examples of Kähler-Einstein del Pezzo surfaces with quotient singularities.

0. Introduction

Since any three dimensional Einstein manifold has constant curvature, the essential study of Sasakian-Einstein manifolds begins in dimension five. Moreover, since a complete Sasakian-Einstein manifold is necessarily spin with finite fundamental group, Smale’s classification [Sm] of simply connected compact 5-manifolds with spin applies. If there is no torsion in H_{2} then Smale’s theorem says that any such 5-manifold is diffeomorphic to S^{5}#_{k}(S^{2} × S^{3}) for some k. In the classification of Sasakian-Einstein manifolds it is judicious to distinguish between regular Sasakian-Einstein manifolds and non-regular ones. The compact simply connected five dimensional manifolds admitting a regular Sasakian-Einstein structure have been classified by Friedrich and Kath [FK], and it follows from the classification of smooth del Pezzo surfaces admitting Kähler-Einstein metrics due to Tian and Yau [TY]. These 5-manifolds are S^{5} and #_{k}(S^{2} × S^{3}) for k = 1, 3, ⋯, 8. For k = 3, ⋯, 8 they are are circle bundles over ℙ^{2} blown up at k generic points, whereas for k = 1 this is the homogeneous Stiefel manifold V_{4,2}(ℝ) which is a circle bundle over ℙ^{1} × ℙ^{1}. Notice that k = 2 is missing from this list as are the circle bundles over ℙ^{2} blownup at one or two points. The reason for this is Matsushima’s well-known obstruction to the existence of Kähler-Einstein metrics when the complex automorphism group is not reductive. Thus, any Sasakian-Einstein structure on the connected sum of two copies of S^{2} × S^{3} must be non-regular. In this note we prove the existence of such a non-regular Sasakian-Einstein metric thus filling this “k = 2 gap”. It is an interesting question as to whether Sasakian-Einstein structures exist on #_{k}(S^{2} × S^{3}) for k > 8. Of course if such structures exist they must be non-regular.

We also prove the existence of two inhomogeneous non-regular Sasakian-Einstein metrics on S^{2} × S^{3}. Recently it has been shown that the manifold S^{2} × S^{3} admits a quite a few Einstein metrics. First Wang and Ziller [WZ] proved the existence of a countable number of homogeneous Einstein metrics on S^{2} × S^{3}. More recently Böhm [Böh] showed the existence of a countable number of cohomogeneity one Einstein metrics on S^{2} × S^{3}. Our new Sasakian-Einstein metrics are also inhomogeneous and they are not isometric to any of the Böhm’s examples. Explicitly, we prove Theorem A: There exists a non-regular Sasakian-Einstein metric on (S^{2}×S^{3})#(S^{2}×S^{3}).

During the preparation of this work the authors were partially supported by NSF grant DMS-9970904.

Theorem B: There exist two inequivalent inhomogeneous Sasakian-Einstein metrics on S^{2} × S^{3}. These metrics are inequivalent as Riemannian metrics to the inhomogeneous metrics of Böhm. Hence, S^{2} × S^{3} admits at least three distinct Sasakian-Einstein metrics.

A Sasakian structure on a manifold defines several interesting objects. It defines a one-dimensional foliation, a CR-structure, and a contact structure. Indeed, combining all three of these, it defines a Pfaffian structure with a transverse Kähler geometry. Now in each case above the Sasakian-Einstein structure is unique within the CR-structure. Thus, on S^{2} × S^{3} we have three distinct CR-structures. It is interesting to ask the question as to whether the three Sasakian-Einstein structures on S^{2} × S^{3} belong to distinct contact structures. This is a more subtle question as contact geometry has no local invariants. Perhaps there is a connection between certain link invariants and contact invariants as suggested by Arnold [Arn].

Our method of proofs of Theorems A and B is to consider the Sasakian geometry of links of isolated hypersurface singularities defined by weighted homogeneous polynomials. We then make use of a recent result of Demailly and Kollár [DK] proving the existence of Kähler-Einstein metrics on certain del Pezzo orbifolds given as hypersurfaces in certain weighted projective spaces. The links which can then be represented as the total space of V-bundles over these orbifolds admit Sasakian-Einstein metrics. We then use a wellknown algorithm of Milnor and Orlik [MO] to compute the characteristic polynomials of the monodromy maps associated to the links. This allows us to determine the second Betti number of the link. Then using a method of Randell [Ran] we can show that the links have no torsion, and apply Smale’s classification theorem. Acknowledgments: We would like to thank Alex Buium and Michael Nakamaye for helpful discussions. We also want to thank János Kollár for several valuable e-mail communications as well as his interest in our work.

1. The Sasakian Geometry of Links of Weighted Homogeneous Polynomials

In this section we discuss the Sasakian geometry of links of isolated hypersurface singularities defined by weighted homogeneous polynomials. Consider the affine space ℂ^{n+1} together with a weighted ℂ^{∗}-action 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.

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}. Explicitly, we have

1.3

Φ_{w} = Φ − Φξ_{w} ⊗ η_{w}
g_{w} = \frac{1}{η(ξ_{w})}[g − η_{w} ⊗ ξ_{w}⌋g − ξ_{w}⌋g ⊗ η_{w} + g(ξ_{w}, ξ_{w})η_{w} ⊗ η_{w}] + η_{w} ⊗ η_{w}.

Now, by equation 1.1, the ℂ^{∗}(w) action on ℂ^{n+1} restricts to an action on C_{f}, and the associated S^{1} action restricts to an action on both S^{2n+1} and L_{f}. It follows that ξ_{w} is tangent to the submanifold L_{f} and, by abuse of notation, we shall denote by ξ_{w}, η_{w}, Φ_{w}, g_{w} the corresponding tensor fields on both S^{2n+1} and L_{f}. Now Φ_{w} coincides with Φ on the contact subbundle D on S^{2n+1} which defines an integrable almost complex structure on D. Moreover, since f is a holomorphic function on ℂ^{n+1} the Cauchy-Riemann equations imply that for any smooth section X of D we have Φ_{w}X(f) = 0. Thus, L_{f} is an invariant submanifold of S^{2n+1} with its weighted Sasakian structure. We have arrived at a theorem given by Takahashi [Tak, YK] in the case of Brieskorn-Pham links and we have seen that Takahashi’s proof easily generalizes to the case of arbitrary weighted homogeneous hypersurface singularities. Theorem 1.4: The quadruple (ξ_{w}, η_{w}, Φ_{w}, g_{w}) gives L_{f} a quasi-regular Sasakian structure.

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. To see this let (ξ, η, Φ, g) be a Sasakian structure on a smooth manifold (orbifold) M, and 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

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

This last non-degeneracy condition implies that η_{t} is a contact form on M for all t ∈ [0, 1]. Then by Gray’s Stability Theorem [MS] η_{t} belongs to the same 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

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}

Note that it is not at all clear from this definition that g_{t} is a Riemannian metric, but we shall check this below. We have Theorem 1.7: Let (M, ξ, η, Φ, g) be a Sasakian manifold. Then for all t ∈ [0, 1] and every basic 1-form ζ_{t} such that dζ_{t} is of type (1, 1) and such that 1.5 holds (ξ, η_{t}, Φ_{t}, g_{t}) defines a Sasakian structure on M belonging to the same underlying contact structure as η.

Proof: The conditions of 1.5 guarantee that (ξ, η_{t}, Φ_{t}, g_{t}) defines a Pfaffian structure, i.e. a contact structure with a fixed contact 1-form. We need to check that it is a metric contact structure and that it is normal. It is easy to check that the metric g_{t} of 1.6 can be rewritten as

1.8  g_{t} = dη_{t} ∘ (Φ_{t} ⊗ id) + η_{t} ⊗ η_{t}.

It follows from the fact dη_{t} is type (1, 1) on the contact bundle D_{t} = ker η_{t} that g_{t} is a symmetric bilinear form and then a straightforward computation checks the compatibility

condition g_{t}(Φ_{t}X, Φ_{t}Y) = g_{t}(X, Y) − η_{t}(X)η_{t}(Y).

The positive definiteness of g_{t} follows from the positive definiteness of g and the nondegeneracy condition in 1.5. Moreover, one easily checks the identity Φ_{t}^{2} = −id + ξ ⊗ η_{t}. Next we check normality which amounts to checking two conditions, that the almost CR structure defined by Φ_{t} on the contact bundle D_{t} is integrable, and that ξ is a Killing vector field for the metric g_{t}. The last condition is equivalent to vanishing of the Lie derivative L_{ξ}Φ_{t} for all t which follows immediately from the first of equations 1.6 and the facts that it holds for t = 0 and that ζ_{t} is basic. Integrability follows from the fact that the almost CR structure defined by Φ on D is integrable, and that the first of equations 1.6 is just the projection of the image of Φ onto D_{t}.

In general these structures are inequivalent and the moduli space of Sasakian structures having the same characteristic vector field is infinite dimensional. Indeed since the link of a hypersurface is determined by the S^{1}(w) action we formulate the following: Definition 1.8: A Sasakian structure (ξ, η, Φ, g) on L_{f} is said to be compatible with the link L_{f} if ξ is a generator of the S^{1} action on L_{f}. We say that ξ is the standard generator if ξ = ξ_{w}, where w is the weight vector of L_{f} satisfying gcd(w_{0}, …, w_{n}) = 1.

For every compatible Sasakian structure there is one with a standard generator, and hereafter we shall always choose the standard generator for a compatible Sasakian structure unless otherwise stated. We are interested in the following question: Problem 1.9: Given 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 for the link L_{f} which is discussed further in [BGN1]. Here we use the fact that the leaf space of the characteristic foliation of a Sasakian structure of a link L_{f} is a compact Kähler orbifold together with recent results of Demailly and Kollár [DK] on the existence of Kähler-Einstein orbifold metrics on certain singular del Pezzo surfaces to construct the Sasakian-Einstein metrics on the corresponding link.

2. Kähler-Einstein Orbifolds and Sasakian-Einstein Manifolds

Given a sequence w = (w_{0}, …, w_{n}) of positive integers one can form the graded polynomial ring S(w) = ℂ[z_{0}, …, z_{n}], where z_{i} has grading or weight w_{i}. The weighted projective space [BR, Del, 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 section 1. Clearly, ℙ(w) is also 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. At times it will be important to distinguish between ℙ(w) as a complex orbifold and ℙ(w) as an algebraic variety.

Now the cone C_{f} in ℂ^{n+1} cuts out a hypersurface Z_{f} of ℙ(w) which is also a compact orbifold with an induced Kähler structure ω_{w}. So there is a commutative diagram

2.1  \begin{matrix}L_{f} & ⟶ & S_{w}^{2n+1} \\ ↓π & & ↓ \\ Z_{f} & ⟶ & ℙ(w), \end{matrix}

where the horizontal arrows are Sasakian and Kählerian embeddings, respectively, and the vertical arrows are orbifold Riemannian submersions. Furthermore, by the inversion theorem of [BG1] L_{f} is the total space of the principal S^{1} V-bundle over the orbifold Z_{f} whose first Chern class is [ω_{w}] ∈ H_{orb}^{2}(Z_{f}, ℤ), and η_{w} is the connection in this V-bundle whose curvature is π^{∗}ω_{w}. For further discussion of the orbifold cohomology groups we refer the reader to [Hae] and [BG1].

Now H_{orb}^{2}(Z_{f}, ℤ) ⊗ ℚ ≈ H^{2}(Z_{f}, ℚ), so [ω_{w}] defines a rational class in the usual cohomology. Thus, by Baily’s [Bai] projective embedding theorem for orbifolds, Z_{f} is a projective algebraic variety with at most quotient singularities. It follows that as an algebraic variety Z_{f} is normal. As with ℙ(w) it is important to distinguish between Z_{f} as an orbifold and Z_{f} as an algebraic variety. For example the singular loci may differ. There are examples where the orbifold singular locus Σ^{orb}(Z_{f}) has codimension one over ℂ, whereas Σ^{alg}(Z_{f}) always has codimension ≥ 2 by normality. Indeed there are examples with Σ^{orb}(Z_{f}) ≠ ∅, but Z_{f} is smooth as an algebraic variety. In these cases an orbifold metric is NOT a Riemannian metric in the usual sense.

We are interested in finding Sasakian-Einstein structures (ξ_{w}, η′_{w}, Φ′_{w}, g′_{w}) on L_{f} with the same Sasakian vector field ξ_{w} as the original Sasakian structure, that is Sasakian-Einstein structures that are compatible with the link L_{f}. Since a Sasakian-Einstein metric necessarily has positive Ricci curvature, we see that a necessary condition for a Sasakian-Einstein metric g to exist on L_{f} is that there be an orbifold metric h on Z_{f} with positive Ricci form ρ_{h} such that g = π^{∗}h + η ⊗ η. The positive definiteness of ρ_{h} follows from the relation

2.2  π^{∗}Ric_{h} = Ric_{g}|_{D×D} + 2g|_{D×D}.

But since ρ_{h} represents the first Chern class c_{1}(K^{−1}) of the anti-canonical line V-bundle K^{−1} we see that Z_{f} must be a Fano orbifold, i.e., some power of K^{−1} is invertible and ample. Here K is the canonical line V-bundle of Baily [Bai]. By the previous section and the well-known theory of Sasakian-Einstein structures [BG1] we have Theorem 2.3: The link L_{f} has a compatible Sasakian-Einstein structure if and only if the Fano orbifold Z_{f} admits a compatible Kähler-Einstein orbifold metric of scalar curvature 4n(n + 1).

Of course, if we find a Kähler-Einstein metric of positive scalar curvature on the orbifold Z_{f}, we can always rescale the metric so that the scalar curvature in 4n(n + 1).

We want conditions that guarantee that the hypersurfaces be Fano, but first we restrict somewhat the hypersurfaces that we treat. Definition 2.3 [Fle]:

(1) A weighted projective space ℙ(w_{0}, …, w_{n}) is said to be well-formed if

gcd(w_{0}, …, ŵ_{i}, …, w_{n}) = 1 for all i = 1, …, n.

(2) A hypersurface in a weighted projective space ℙ(w_{0}, …, w_{n}) is said to be well-formed if in addition it contains no codimension 2 singular stratum of ℙ(w_{0}, …, w_{n}),

where the hat means delete that element.

In (1) of definition 2.3, we prefer the terminology introduced in [BR] for an equivalent condition which seems to have first appeared in [Dol]. So if a weighted projective space satisfies (1) of definition 2.3 we say that w is normalized. In [Fle] it is shown that (2) of definition 2.3 can be formulated solely in terms of the weighted homogeneous polynomial f. That is, a hypersurface in ℙ(w) defined by a weighted homogeneous polynomial f is well-formed if and only if w is normalized and

2.4  gcd(w_{0}, …, ŵ_{i}, …, ŵ_{j}, …, w_{n}) | d  for all distinct i, j = 0, …, n.

In this case we shall also say that the weighted homogeneous polynomial f defining the hypersurface is well-formed. The following proposition is essentially an exercise in [Kol]: Proposition 2.5: Let f_{w} be a well-formed weighted homogeneous polynomial with weights w and degree d. Then the hypersurface Z_{f} in ℙ(w_{0}, …, w_{n}) is Fano if and only if the following condition holds (∗) d < |w| = ∑_{i=0}^{n} w_{i}. Thus if w is normalized and satisfies 2.4, a necessary condition for the link L_{f} to admit a compatible Sasakian-Einstein metric is that (∗) be satisfied. Proof: Following [Mo,BR] in any weighted projective space ℙ(w) we define the Morisingular locus as follows: Let m = lcm(w_{0}, …, w_{m}) and consider for each prime divisor p of m the subscheme S_{p} cut out by the ideal I_{p} generated by those indeterminates z_{i} such that p |̸w_{i}, and define the Mori-singular locus by

S(w) = ∪_{p|m}S_{p},

and the Mori-regular locus by

ℙ^{0}(w) = ℙ(w) − S(w).

Now ℙ^{0}(w) is contained in the smooth locus of ℙ(w) but can be a proper subset of it. Thus, S(w) contains the singular locus Σ(w) of ℙ(w) but in general is larger. However, when w is normalized the local uniformizing groups of the orbifold ℙ(w) contain no quasireflections as in [Fle]. It follows in this case that the singular locus Σ(w) coincides with the Mori-singular locus S(w), and ℙ^{0}(w) is precisely the smooth locus of ℙ(w).

Now the sheaves O_{ℙ(w)}(n) for n ∈ ℤ are not in general invertible, but they are reflexive [BR]. Moreover, the Mori-regular locus ℙ^{0}(w) is the largest open subset U of ℙ(w) on which

(1) O_{ℙ(w)}(1)|U is invertible, and

(2) the natural map O_{ℙ(w)}(1)^{⊗n}|U⟶O_{ℙ(w)}(n)|U is an isomorphism for each n ∈ ℤ^{+} [Mo].

Furthermore, if w is normalized ℙ^{0}(w) is characterized by condition (1) alone. As an orbifold ℙ(w) has a canonical V-bundle K_{ℙ(w)} whose associated sheaf, the dualizing sheaf κ_{ℙ(w)}, is isomorphic to O_{ℙ(w)}(−|w|). Generally the adjunction formula does not hold, but if w is normalized we have

2.6  κ_{Z_{f}} ≅ O_{Z_{f}} (d − |w|).

These sheaves are invertible on ℙ^{0}(w) ∩ Z_{f} and we have

κ_{Z_{f} ∩ℙ^{0}(w)}^{−1} ≅ O_{Z_{f} ∩ℙ^{0}(w)}(|w| − d).

Moreover, since Z_{f} is well-formed, Z_{f} ∩ S(w) = Z_{f} ∩ Σ(w) has codimension ≥ 2 in Z_{f}. It follows by a standard result (cf. [KMM] Lemma 0-1-10) that

κ_{Z_{f}}^{−1} = ι_{∗}(κ_{Z_{f} ∩ℙ^{0}(w)}^{−1}), O_{Z_{f}} (|w| − d) = ι_{∗}(O_{Z_{f} ∩ℙ^{0}(w)}(|w| − d)),

where ι : Z_{f} ∩ ℙ^{0}(w)⟶Z_{f} is the natural inclusion. We then have

2.7  κ_{Z_{f}}^{−1} ≅ O_{Z_{f}} (|w| − d).

This proves the result.

There are further obstructions to the existence of a Sasakian-Einstein structure on L_{f}, namely those coming from the well-known failure of the zeroth order a priori estimate for solving the Monge-Ampere equation for the Calabi problem on Z_{f}. Now the link L_{f} admits a compatible Sasakian-Einstein structure if and only if Z_{f} has a Kähler-Einstein structure ω′ in the same cohomology class as ω_{w}. So the well-known obstructions in the Kähler-Einstein case, such as the non-reductiveness of the connected component of the complex automorphism group, or the vanishing of the Futaki invariant become obstructions to the existence of Sasakian-Einstein metrics. The problem of solving the Calabi problem in the positive case has drawn much attention in the last decade [Sui, Ti1, Ti2, Ti3, TY, DT, Nad] but still remains open. Here it suffices to consider three examples given recently by Demailly and Kollar [DK].

3. The Demailly-Kollár Examples

In a recent preprint Demailly and Kollár [DK] give a new derivation of Nadel’s existence criteria for positive Kähler-Einstein metrics [Nad] which is valid for orbifolds. Furthermore their method of implementing Nadel’s theorem does not depend on the existence of a large finite group of symmetries, but rather uses intersection inequalities. In this sense it is complementary to the work of Nadel [Nad] and others [Siu, Ti1-Ti4]. As an application of their method Demailly and Kollár [DK] construct three new del Pezzo orbifolds which admit Kähler-Einstein metrics. Explicitly, they prove Theorem 3.1 [DK]: Let Z_{f} be a del Pezzo orbifold, and let f be given by one of the following three quasi-homogeneous polynomials of degree d whose zero set Z_{f} is a surface in the weighted projective space ℙ(w_{0}, w_{1}, w_{2}, w_{3}) :

(1) f = z_{0}^{5}z_{1} + z_{0}z_{2}^{3} + z_{1}^{4} + z_{3}^{3} with w = (9, 15, 17, 20) and d = 60.

(2) f = z_{0}^{17}z_{2} + z_{0}z_{1}^{5} + z_{1}z_{2}^{3} + z_{3}^{2} with w = (11, 49, 69, 128) and d = 256.

(3) f = z_{0}^{17}z_{1} + z_{0}z_{2}^{3} + z_{1}^{5}z_{2} + z_{3}^{2} with w = (13, 35, 81, 128) and d = 256.

Then Z_{f} admits a Kähler-Einstein orbifold metric.

We denote these del Pezzo surfaces by Z_{60}, Z_{256}^{(1)}, Z_{256}^{(2)}, respectively. Actually more is true. By a result of Bando and Mabuchi [BM] (which also holds in the case of orbifolds) the Kähler-Einstein metric is unique up to complex automorphisms. Let us denote the corresponding links L_{f} of these del Pezzo surfaces by L_{60}, L_{256}^{(1)}, L_{256}^{(1)}, respectively. Due to the form of the polynomials as f = g+z_{3}^{m}, there is a well-known description of both Z_{f} and L_{f} in terms of branched covers (cf. [DuKa]). We briefly describe this geometry here. Z_{60} is a 3-fold cover of ℙ(9, 15, 17) branched over the curve C_{60} = {z_{0}^{5}z_{1}+z_{0}z_{2}^{3}+z_{1}^{4} = 0}. Similarly, the surfaces Z_{256}^{(i)} are 2-fold covers of ℙ(11, 49, 69) and ℙ(13, 35, 81), respectively, branched over the curves C_{256}^{(1)} = {z_{0}^{17}z_{2} + z_{0}z_{1}^{5} + z_{1}z_{2}^{3} = 0}, and C_{256}^{(2)} = {z_{0}^{17}z_{1} + z_{0}z_{2}^{3} + z_{1}^{5}z_{2} = 0}, respectively. In section 5 we show that in each case the genus of these curves is zero, so they are all ℙ^{1}’s (there are no quotient singularities in dimension one). Similarly, L_{60} is a 3-fold cyclic cover of S^{5} branched over the Seifert manifold K(4, 3, 5; IV), and L_{256}^{(i)} are double covers of S^{5} branched over the Seifert manifolds K(17, 3, 5; V) and K(17, 5, 3; V) for i = 1, 2, respectively. The notation K(a_{0}, a_{1}, a_{2}; ·) is that of Orlik [Or1]. These Seifert manifolds are quite complicated. For example, they all have infinite fundamental group, but have finite abelianization H_{1}, and they all are Seifert fibrations over the Riemann sphere. As we shall see shortly the links L_{256}^{(1)} and L_{256}^{(2)} have the same characteristic polynomial although their Sasakian structures are distinct since their leaf holonomy groups are different. In particular they can be distinguished by their orders [BG1]. One easily finds by analyzing the orbifold singularity structure that ord(L_{256}^{(1)}) = 37191 = 3·7^{2} ·11·23, while ord(L_{256}^{(2)}) = 36855 = 3^{4} ·5·7·13. Now combining the Demailly-Kollár Theorem with Theorem 2.3 gives

Theorem 3.2: The simply connected 5-manifolds L_{60}, L_{256}^{(i)} admit compatible Sasakian-Einstein metrics. Furthermore, these Sasakian-Einstein metrics are unique within the CR structure up to a CR automorphism.

4. Link Invariants and the Milnor Fibration

Recall the well-known construction of Milnor [Mil] for 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]

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

One immediately has Proposition 4.2: The Milnor numbers of the simply connected Sasakian-Einstein 5manifolds L_{60}, L_{256}^{(1)}, L_{256}^{(2)} are

μ(L_{60}) = 86, μ(L_{256}^{(i)}) = 255.

The closure F̄ of F has the same homotopy type as F and is a compact manifold with boundary 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^{μ}. Using the Wang sequence of the Milnor fibration together with Alexander-Poincare duality gives the exact sequence [Mil]

4.3  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_{∗}). So the topology of L_{f} is encoded in the monodromy map 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_{∗}), namely 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 examples. Proposition 4.4: The characteristic polynomials Δ(t) of the simply connected Sasakian-Einstein 5-manifolds L_{60}, L_{256}^{(1)}, L_{256}^{(2)} are given by

\begin{aligned}Δ(t) = & (t − 1)^{2}(t^{14} + t^{13} + ⋯ + 1)(t^{15} + 1)(t^{30} + 1)(t^{4} + t^{3} + t^{2} + t + 1) \\ & ×(t^{5} + 1)(t^{10} + 1)(t^{4} − t^{2} + 1)(t^{2} − t + 1) \end{aligned}

for L_{60} and

Δ(t) = (t − 1)(t^{2} + 1)(t^{4} + 1)(t^{8} + 1)(t^{16} + 1)(t^{32} + 1)(t^{64} + 1)(t^{128} + 1)

for L_{256}^{(1)} and L_{256}^{(2)}. 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

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 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 for the surface of degree 60 we have

w′_{0} = \frac{20}{3}, w′_{1} = 4, w′_{2} = \frac{60}{17}, w′_{3} = 3.

Then by Theorem 4 of [MO] the divisor of the characteristic polynomial is

div Δ(t) = (\frac{Λ_{20}}{3} − 1)(Λ_{4} − 1)(\frac{Λ_{60}}{17} − 1)(Λ_{3} − 1).

Using the relations Λ_{a}Λ_{b} = gcd(a, b)Λ_{lcm(a,b)} we find in this case

4.5  div Δ(t) = Λ_{60} + Λ_{20} + Λ_{12} − Λ_{4} − Λ_{3} + 1.

Then Δ(t) is obtained from the formula

div Δ(t) = 1 − ∑ \frac{s_{j}}{j} Λ_{j}

yielding

Δ(t) = \frac{(t − 1)}{∏(t^{j} − 1)^{s_{j}/j}} .

We see from 4.5 that the only nonzero s_{j}’s are

s_{60} = −60, s_{20} = −20, s_{12} = −12, s_{4} = 4, s_{3} = 3,

and Δ(t) becomes

Δ(t) = \frac{(t − 1)(t^{60} − 1)(t^{20} − 1)(t^{12} − 1)}{(t^{4} − 1)(t^{3} − 1)}

which we see easily reduces the the expression given.

Similar computations for the two surfaces of degree 256 show that

4.6  div Δ(t) = Λ_{256} − Λ_{2} + 1,

giving the characteristic polynomial noted above.

5. The Topology of L_{f}

Since the multiplicity of the root 1 in Δ(t) is precisely the second Betti number of L_{f} we have an immediate corollary of Proposition 4.4: Corollary 5.1: The second Betti numbers of the 5-manifolds L_{60}, L_{256}^{(i)} are b_{2}(L_{60}) = 2, and b_{2}(L_{256}^{(i)}) = 1 for i = 1, 2.

One can give another proof of this corollary by making use of a formula due to Steenbrink [Dim]. One considers the Milnor algebra

5.2  M(f) = \frac{ℂ[z_{0}, z_{1}, z_{2}, z_{3}]}{θ(f)},

where θ(f) is the Jacobi ideal. M(f) has a natural ℤ grading

M(f) = ⊕_{i≥0}M(f)_{i},

and Steenbrink’s formula determines the primitive Hodge numbers

5.3  h_{0}^{i,n−i−1} = dim_{ℂ}M(f)_{(i+1)d−w}

of the projective surface f = 0 in the weighted projective space ℙ(w). Then b_{2}(L_{f}) = h_{0}^{1,1} can be computed by finding a basis of residue classes in M(f)_{2d−|w|}.

Steenbrink’s formula can also be used to compute the Hirzebruch signature τ of the orbifolds Z_{f} as well as the genus of the curves C_{60} and C_{256}^{(i)} discussed at the end of the last section. We have Proposition 5.4: The Fano orbifold Z_{60} has τ = −1 while the orbifolds Z_{256}^{(i)} have τ = 0. The curves C_{60} and C_{256}^{(i)} are all isomorphic to the projective line ℙ^{1}. Proof: The formula for the signature is [Dim]

τ = 1 + 2dim_{ℂ}M(f)_{d−|w|} − dim_{ℂ}M(f)_{2d−|w|},

whereas the genus of the curves is given by

g = h_{0}^{0,1} = dim_{ℂ}M(f_{0})_{d−|w|},

where f_{0} is a weighted homogeneous polynomial in z_{0}, z_{1}, z_{2} which is related to f by f = f_{0} + z_{3}^{a_{3}}. Now d − |w| = −1 for all the surfaces and dim_{ℂ}M(f)_{2d−|w|} = h_{0}^{1,1} = b_{2}(L_{f}) which is 2 for Z_{60} and 1 for Z_{256}^{(i)}. Now for the curve C_{60} we have d − |w| = 19, and one easily sees that g = dim_{ℂ}M(f_{0})_{19} = 0. For the curves C_{256}^{(i)} we have d − |w| = 127, so g = dim_{ℂ}M(f_{0})_{127}. For example, for C_{256}^{(1)} one easily checks that there are no monomials of the form z_{0}^{a}z_{1}^{b}z_{2}^{c} such that 11a + 49b + 69c = 127. In both cases we find g = 0.

Next we turn to the calculation of the torsion in H_{2}(L_{f}, ℤ). Actually we show that it is zero. We follow a method due to Randell [Ran] for computing the torsion of generalized Brieskorn manifolds (complete intersections of Brieskorn’s) where he verified a conjecture due to Orlik [Or2] for this class of manifolds. Randell’s methods apply to our more general weighted homogeneous polynomials case. For any x ∈ L_{f} let Γ_{x} denote the isotropy subgroup of the S^{1}(w)-action on L_{f} and by |Γ_{x}| its order. Following [Ran] for each prime p we define the p-singular set by

5.5  S_{p} = {x ∈ L_{f}| p divides |Γ_{x}|}.

Then we have Lemma 5.6: Suppose that for each prime p the p-singular set S_{p} is contained in a submanifold S of codimension 4 in L_{f}. Then the isomorphism holds:

5.7  Tor(H_{n−1}(L_{f}, ℤ)) ≈ Tor(H_{n−3}(S, ℤ)).

In particular, if the corresponding projective hypersurface Z_{f} is well-formed, the isomorphism 5.7 holds. Proof: The first statement is due to Randell [Ran]. To prove the second statement we notice that the set ∪_{p}S_{p}, where the union is over all primes, is precisely the subset of L_{f}, where the leaf holonomy groups are non-trivial and its projection to Z_{f} is just the orbifold singular locus Σ^{orb}(Z_{f}). Now Z_{f} ⊂ ℙ(w) and w is normalized, so the fact that Z_{f} is well-formed says that the orbifold singular locus Σ^{orb}(Z_{f}) has complex codimension at least two in Z_{f}; hence, for each prime p, S_{p} has real codimension at least four in L_{f}.

For the case at hand, n = 3 so Randell’s lemma says that H_{2}(L_{f}, ℤ) is torsion free. Thus, we have arrived at:

Lemma 5.8: Let L_{f} ⊂ ℂ^{4} be the link of an isolated singularity defined by a weighted homogeneous polynomial f in four complex variables. Suppose further that the del Pezzo surface Z_{f} ⊂ ℙ(w) is well-formed, then Tor (H_{2}(L_{f}, ℤ)) = 0.

Now a well-known theorem of Smale [Sm] says that any simply connected compact 5manifold which is spin, and whose second homology group is torsion free, is diffeomorphic to S^{5}#_{k}(S^{2} × S^{3}) for some non-negative integer k. Furthermore, it is known [BG2,Mor] that any simply connected Sasakian-Einstein manifold is spin. Combining this with the development above gives Theorem 5.9: Let L_{f} be the link associated to a well-formed weighted homogeneous polynomial f in four complex variables. Suppose also that L_{f} admits a Sasakian-Einstein metric. Then L_{f} is diffeomorphic to S^{5}#_{k}(S^{2} × S^{3}), where k is the multiplicity of the root 1 of the characteristic polynomial Δ(t) of L_{f}.

Let us now consider the links L_{60}, L_{256}^{(i)} of Theorem 3.2 which admit Sasakian-Einstein metrics. Theorem 5.10: The link L_{60} is diffeomorphic to (S^{2} × S^{3})#(S^{2} × S^{3}) while the links L_{256}^{(i)} for i = 1, 2 are diffeomorphic to the Stiefel manifold S^{2} × S^{3}. In particular, S^{2} × S^{3} admits 3 distinct Sasakian-Einstein structures. Proof: This follows from Smale’s Theorem [Sm], Theorems 3.2, 5.9, and Corollary 5.1 as soon as we check that the weighted homogeneous polynomials are well-formed. But this follows easily from the definition and equation 2.4.

6. Proofs of the Main Theorems and Further Discussion

Theorem A of the Introduction now follows immediately from Theorems 3.2 and 5.10. Similarly for Theorem B these two theorems give the existence of Sasakian-Einstein metrics on S^{2} ×S^{3}. Furthermore, since the Sasakian structures are non-regular the metrics are inhomogeneous. The statement that the Sasakian-Einstein structures are inequivalent follows from the fact that their characteristic foliations are inequivalent (indeed, they have different orders). It remains to show that our Sasakian-Einstein metrics are inequivalent as Riemannian metrics to any of the metrics of Böhm. To see this we notice that for all of Böhm’s metrics on S^{2} × S^{3}, the connected component of the isometry group ℑ_{0}(g) is SO(3) × SO(3) (see [Wa] Theorem 2.19). But a Theorem of Tanno [Tan] says that the connected component of the group of Sasakian automorphisms, 𝔄_{0}(g) coincides with the connected component of the group of isometries ℑ_{0}(g), and 𝔄_{0}(g) has the form 𝔊 × S^{1}, where the S^{1} is generated by the Sasakian vector field ξ. Thus, our metrics are not isometric to the any of Böhm’s metrics.

Smale’s Theorem actually says more than we have mentioned. It says that the torsion in H_{2} must be of the form

5.11  Tor(H_{2}(M, ℤ)) ≈ ⨁(ℤ_{q_{i}} ⊕ ℤ_{q_{i}}).

Thus we have Theorem 6.1: Let M be a complete simply connected Sasakian-Einstein 5-manifold. Then the torsion group Tor(H_{2}(M, ℤ)) must be of the form 5.11. In particular, the number of torsion generators must be even.

It is a very interesting question whether the form of the torsion actually provides an obstruction to the existence of a Sasakian-Einstein structure or whether it is always satisfied. The general form of Randell’s proof suggests that the torsion may always vanish, but we do not yet have a proof of this even in the case of hypersurfaces defined by weighted homogeneous polynomials when we drop the well-formed assumption. If there were an honest obstruction, it would provide a new type of obstruction to the existence of positive Kähler-Einstein metrics on del Pezzo orbifolds.

Finally we mention that many new Sasakian-Einstein structures can be constructed in higher dimension by applying the join construction introduced in [BG1] to our new examples. For example,

Proposition 6.2: The joins S^{3} ⋆ L_{60} and S^{3} ⋆ L_{256}^{(i)} are all smooth Sasakian-Einstein 7-manifolds. S^{3} ⋆ L_{60} has the rational cohomology type of S^{2} × ((S^{2} × S^{3})#(S^{2} × S^{3})), while S^{3} ⋆ L_{256}^{(i)} have the rational cohomology type of S^{3} ⋆ S^{3} ⋆ S^{3}. Proof: By Proposition 4.6 of [BG1] these joins are smooth when the Fano indices of the links are one. But it follows from the proof of Proposition 2.5 that κ^{−1}(L_{f}) ≈ O(|w| − d) and in all three cases we have |w| − d = 1. The rational cohomology types can easily be seen from Theorem 5.22 of [BG1].

Many other examples can be worked out along the lines of [BG1]. However, what is perhaps a more interesting question is, for example, whether S^{3} ⋆ L_{256}^{(1)}, S^{3} ⋆ L_{256}^{(2)} and S^{3}⋆S^{3}⋆S^{3} have the same integral cohomology type, and if they do, are they homeomorphic (diffeomorphic)? We plan to study these types of questions in the future. Note Added: The ideas of this paper have been developed much further in [BGN1,BGN2] where the existence of infinite families of Sasakian-Einstein metrics on the k-fold connected sum k#(S^{2}×S^{3}) is proven for k = 2, ⋯, 9. The moduli of such structures is also discussed.

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Department of Mathematics and Statistics March 2000 University of New Mexico Albuquerque, NM 87131 email: cboyer@math.unm.edu, galicki@math.unm.edu web pages: http://www.math.unm.edu/˜ cboyer, http://www.math.unm.edu/˜ galicki