EINSTEIN METRICS ON SPHERES

CHARLES P. BOYER, KRZYSZTOF GALICKI AND JÁNOS KOLLÁR

1. Introduction

Any sphere S^{n} admits a metric of constant sectional curvature. These canonical metrics are homogeneous and Einstein, that is the Ricci curvature is a constant multiple of the metric. The spheres S^{4m+3}, m > 1 are known to have another Sp(m + 1)-homogeneous Einstein metric discovered by Jensen [Jen73]. In addition, S^{15} has a third Spin(9)-invariant homogeneous Einstein metric discovered by Bourguignon and Karcher [BK78]. In 1982 Ziller proved that these are the only homogeneous Einstein metrics on spheres [Zil82]. No other Einstein metrics on spheres were known until 1998 when Böhm constructed infinite sequences of nonisometric Einstein metrics, of positive scalar curvature, on S^{5}, S^{6}, S^{7}, S^{8}, and S^{9} [Böh98]. Böhm’s metrics are of cohomogeneity one and they are not only the first inhomogeneous Einstein metrics on spheres but also the first non-canonical Einstein metrics on even-dimensional spheres. Even with Böhm’s result Einstein metrics on spheres appeared to be rare.

The aim of this paper is to demonstrate that on the contrary, at least on odddimensional spheres, such metrics occur with abundance in every dimension. Just as in the case of Böhm’s construction, ours are only existence results. However, we also answer in the affirmative the long standing open question about the existence of Einstein metrics on exotic spheres. These are differentiable manifolds that are homeomorphic but not diffeomorphic to a standard sphere S^{n}.

Our method proceeds as follows. For a sequence a = (a_{1}, …, a_{m}) ∈ ℤ_{+}^{m} consider the Brieskorn–Pham singularity

Y (a) := {∑_{i=1}^{m} z_{i}^{a_{i}} = 0} ⊂ ℂ^{m} and its link L(a) := Y (a) ∩ S^{2m−1}(1).

L(a) is a smooth, compact, (2m − 3)-dimensional manifold. Y (a) has a natural ℂ^{∗}-action and L(a) a natural S^{1}-action (cf. §33). When the sequence a satisfies certain numerical conditions, we use the continuity method to produce an orbifold Kähler-Einstein metric on the quotient (Y (a) \ {0})/ℂ^{∗} which then can be lifted to an Einstein metric on the link L(a). We get in fact more:

• The connected component of the isometry group of the metric is S^{1}.

• We construct continuous families of inequivalent Einstein metrics.

• The Kähler-Einstein structure on the quotient (Y (a) \ {0})/ℂ^{∗} lifts to a Sasakian-Einstein metric on L(a). Hence, these metrics have real Killing spinors [FK90] which play an important role in the context of p-brane solutions in superstring theory and in M-theory.

In each fixed dimension (2m − 3) we obtain a Kähler-Einstein metric on infinitely many different quotients (Y (a) \ {0})/ℂ^{∗}, but the link L(a) is a homotopy sphere only for finitely many of them. Both the number of inequivalent families of Sasakian-Einstein metrics and the dimension of their moduli grow double exponentially with the dimension.

There is nothing special about restricting to spheres or even to Brieskorn-Pham type – our construction is far more general. All the restrictions made in this article are very far from being optimal and we hope that many more cases will be settled in the future. Even with the current weak conditions we get an abundance of new Einstein metrics. Theorem 1. On S^{5} we obtain 68 inequivalent families of Sasakian-Einstein metrics. Some of these admit non-trivial continuous Sasakian-Einstein deformations.

The biggest family, constructed in Example 40 has (real) dimension 10.

The metrics we construct are almost always inequivalent not just as Sasakian structures but also as Riemannian metrics. The only exception is that a hypersurface and its conjugate lead to isometric Riemannian metrics, see §20.

In the next odd dimension the situation becomes much more interesting. An easy computer search finds 8,610 distinct families of Sasakian-Einstein structures on standard and exotic 7-spheres. By Kervaire and Milnor there are 28 oriented diffeomorphism types of topological 7-spheres [KM63]. (15 types if we ignore orientation.) The results of Brieskorn allow one to decide which L(a) corresponds to which exotic sphere [Bri66]. We get: Theorem 2. All 28 oriented diffeomorphism classes on S^{7} admit inequivalent families of Sasakian-Einstein structures, some of them in each diffeomorphism class depending on moduli.

In each case, the number of families is easily computed and they range from 231 to 452. Moreover, there are fairly large moduli. For example, the standard 7-sphere admits an 82-dimensional family of Sasakian-Einstein metrics, see Example 40. Let us mention here that any orientation reversing diffeomorphism takes a Sasakian-Einstein metric into an Einstein metric, but not necessarily a Sasakian-Einstein metric, since the Sasakian structure fixes the orientation.

Since Milnor’s discovery of exotic spheres [Mil56] the study of special Riemannian metrics on them has always attracted a lot of attention. Perhaps the most intriguing question is whether exotic spheres admit metrics of positive sectional curvature. This problem remains open. In 1974 Gromoll and Meyer wrote down a metric of non-negative sectional curvature on one of the Milnor spheres [GM74]. More recently it has been observed by Grove and Ziller that all exotic 7-spheres which are S^{3} bundles over S^{4} admit cohomogeneity one metrics of non-negative sectional curvature [GZ00]. But it is not known if any of these metrics can be deformed to a metric of strictly positive curvature. Another interesting question concerns existence of metrics of positive Ricci curvature on exotic 7-spheres. This question has now been settled by the result of Wraith who proved that all spheres that are boundaries of parallelizable manifolds admit a metric of positive Ricci curvature [Wra97]. A proof of this result using techniques similar to the present paper was recently given in [BGN03b]. In dimension 7 all homotopy spheres have this property. In this context the result of Theorem 2 can be rephrased as to say that all homotopy 7-spheres admit metrics with positive constant Ricci curvature. Lastly, we should add that although heretofore it was unknown whether Einstein metrics existed on exotic spheres, Einstein metrics have been known to exist on manifolds which are homeomorphic but not diffeomorphic. In dimension 7 there are even examples of homogeneous Einstein metrics with this property [KS88]. Kreck and Stolz find that there are 7-dimensional manifolds with the maximal number of 28 smooth structures, each of which admits an Einstein metric with positive scalar curvature. Our Theorem 2 establishes the same result for 7-spheres.

In order to organize the higher dimensional cases, note that every link L(a) bounds a parallelizable manifold (called the Milnor fiber). Homotopy n-spheres that bound a parallelizable manifold form a group, called the Kervaire-Milnor group, denoted by bP_{n+1}. When n ≡ 1 mod 4 the Kervaire-Milnor group has at most 2 elements, the standard sphere and the Kervaire sphere. (It is not completely understood in which dimensions are they different.) Theorem 3. For n ≥ 2, the (4n + 1)-dimensional standard and Kervaire spheres both admit many families of inequivalent Sasakian-Einstein metrics.

A partial computer search yielded more than 3 · 10^{6} cases for S^{9} and more than 10^{9} cases for S^{13}, including a 21300113901610-dimensional family, see Example 45. The only Einstein metric on S^{13} known thus far was the standard one.

In the remaining case of n ≡ 3 mod 4 the situation is more complicated. For these values of n the group bP_{n+1} is quite large (see §29) and we do not know how to show that every member of it admits a Sasakian-Einstein structure, since our methods do not apply to the examples given in [Bri66]. We believe, however, that this is true: Conjecture 4. All odd-dimensional homotopy spheres which bound parallelizable manifolds admit Sasakian-Einstein metrics.

Computer searches are under way to check this in dimensions 11 and 15. 5 (Plan of the proof). Our construction can be divided into four main steps, each of quite different character. The first step, dating back to Kobayashi’s circle bundle construction [Kob63], is to observe that a positive Kähler-Einstein metric on the base space of a circle bundle gives an Einstein metric on the total space. This result was generalized to orbifolds giving Sasakian-Einstein metrics in [BG00]. Thus, a positive Kähler-Einstein orbifold metric on (Y (a) \ {0})/ℂ^{∗} yields a Sasakian-Einstein metric on L(a) if ∑ \frac{1}{a_{i}} > 1. In contrast with the cases studied in [BG01, BGN03a], our quotients are not well formed, that is, some group elements have codimension 1 fixed point sets.

The second step is to use the continuity method developed by [Aub82, Siu88, Siu87, Tia87] to construct Kähler-Einstein metrics on orbifolds. With minor modifications, the method of [Nad90, DK01] arrives at a sufficient condition, involving the integrability of inverses of polynomials on Y (a).

The third step is to check these conditions. Reworking the earlier estimates given in [JK01, BGN03a] already gives some examples, but here we also give an improvement. This is still, however, quite far from what one would expect.

The final step is to get examples, partly through computer searches, partly through writing down well chosen sequences.

2. Orbifolds as quotients by ℂ^{∗}-actions

Definition 6 (Orbifolds). An orbifold is a normal, compact, complex space X locally given by charts written as quotients of smooth coordinate charts. That is, X can be covered by open charts X = ∪U_{i} and for each U_{i} there is a smooth complex space V_{i} and a finite group G_{i} acting on V_{i} such that U_{i} is biholomorphic to the quotient space V_{i}/G_{i}. The quotient maps are denoted by ϕ_{i} : V_{i} → U_{i}.

The classical (or well formed) case is when the fixed point set of every nonidentity element of every G_{i} has codimension at least 2. In this case X alone determines the orbifold structure.

One has to be more careful when there are codimension 1 fixed point sets. (This happens to be the case in all our examples leading to Einstein metrics.) Then the quotient map ϕ_{i} : V_{i} → U_{i} has branch divisors D_{ij} ⊂ U_{i} and ramification divisors R_{ij} ⊂ V_{i}. Let m_{ij} denote the ramification index over D_{ij}. Locally near a general point of R_{ij} the map ϕ_{i} looks like

ℂ^{n} → ℂ^{n}, ϕ_{i} : (x_{1}, x_{2}, …, x_{n}) ↦ (z_{1} = x_{1}^{m_{ij}}, z_{2} = x_{2}, …, z_{n} = x_{n}).

Note that

ϕ_{i}^{∗}(dz_{1} ∧ ⋯ ∧ dz_{n}) = m_{ij}x_{1}^{m_{ij}−1} · dx_{1} ∧ ⋯ ∧ dx_{n}.  6.1

The compatibility condition between the charts that one needs to assume is that there are global divisors D_{j} ⊂ X and ramification indices m_{j} such that D_{ij} = U_{i} ∩ D_{j} and m_{ij} = m_{j} (after suitable re-indexing).

It will be convenient to codify these data by a single ℚ-divisor, called the branch divisor of the orbifold,

Δ := ∑(1 − \frac{1}{m_{j}})D_{j}.

It turns out that the orbifold is uniquely determined by the pair (X, Δ). Slightly inaccurately, we sometimes identify the orbifold with the pair (X, Δ).

In the cases that we consider X is algebraic, the U_{i} are affine, V_{i} ≅ ℂ^{n} and the G_{i} are cyclic, but these special circumstances are largely unimportant. Definition 7 (Main examples). Fix (positive) natural numbers w_{1}, …, w_{m} and consider the ℂ^{∗}-action on ℂ^{m} given by λ : (z_{1}, …, z_{m}) ↦ (λ^{w_{1}}z_{1}, …, λ^{w_{m}}z_{m}). Set W = gcd(w_{1}, …, w_{m}). The W th roots of unity act trivially on ℂ^{m}, hence without loss of generality we can replace the action by

λ : (z_{1}, …, z_{m}) ↦ (λ^{w_{1}/W} z_{1}, …, λ^{w_{m}/W} z_{m}).

That is, we can and will assume that the w_{i} are relatively prime, i.e. W = 1. It is convenient to write the m-tuple (w_{1}, ⋯, w_{m}) in vector notation as w = (w_{1}, ⋯, w_{m}), and to denote the ℂ^{∗} action by ℂ^{∗}(w) when we want to specify the action.

We construct an orbifold by considering the quotient of ℂ^{m} \ {0} by this ℂ^{∗} action. We write this quotient as ℙ(w) = (ℂ^{m} \{0})/ℂ^{∗}(w). The orbifold structure is defined as follows. Set V_{i} := {(z_{1}, ⋯, z_{m}) | z_{i} = 1}. Let G_{i} ⊂ ℂ^{∗} be the subgroup of w_{i}-th roots of unity. Note that V_{i} is invariant under the action of G_{i}. Set U_{i} := V_{i}/G_{i}. Note that the ℂ^{∗}-orbits on (ℂ^{m} \ {0}) \ (z_{i} = 0) are in one–to–one correspondence with the points of U_{i}, thus we indeed have defined charts of an orbifold. As an algebraic variety this gives the weighted projective space ℙ(w) defined as the projective scheme of the graded polynomial ring S(w) = ℂ[z_{1}, ⋯, z_{m}], where z_{i} has grading or weight w_{i}. The weight d piece of S(w), also denoted by H^{0}(ℙ(w), d), is the vector space of weighted homogeneous polynomials of weighted degree d. That is, those that satisfy

f(λ^{w_{1}}z_{1}, …, λ^{w_{m}}z_{m}) = λ^{d}f(z_{1}, …, z_{m}).

The weighted degree of f is denoted by w(f).

Let 0 ∈ Y ⊂ ℂ^{m} be a subvariety with an isolated singularity at the origin which is invariant under the given ℂ^{∗}-action. Similarly, we can construct an orbifold on the quotient (Y \ {0})/ℂ^{∗}(w). As a point set, it is the set of orbits of ℂ^{∗}(w) on Y \ {0}. It’s orbifold structure is that induced from the orbifold structure on ℙ(w) obtained by intersecting the orbifold charts described above with Y. In order to simplify notation, we denote it by Y/ℂ^{∗}(w) or by Y/ℂ^{∗} if the weights are clear.

Definition 8. Many definitions concerning orbifolds simplify if we introduce an open set U_{ns} ⊂ X which is the complement of the singular set of X and of the branch divisor. Thus U_{ns} is smooth and we take V_{ns} = U_{ns}.

For the main examples described above U_{ns} is exactly the set of those orbits where the stabilizers are trivial. Every orbit contained in ℂ^{m} \ (∏ z_{i} = 0) is such. More generally, a point (y_{1}, …, y_{m}) corresponds to such an orbit if and only if gcd{w_{i} : y_{i} ≠ 0} = 1.

Definition 9 (Tensors on orbifolds). A tensor η on the orbifold (X, Δ) is a tensor η_{ns} on U_{ns} such that for every chart ϕ_{i} : V_{i} → U_{i} the pull back ϕ_{i}^{∗}η_{ns} extends to a tensor on V_{i}. In the classical case the complement of U_{ns} has codimension at least 2, so by Hartogs’ theorem holomorphic tensors on U_{ns} can be identified with holomorphic tensors on the orbifold. This is not so if there is a branch divisor Δ. We are especially interested in understanding the top dimensional holomorphic forms and their tensor powers.

The canonical line bundle of the orbifold K_{X^{orb}} is a family of line bundles one on each chart V_{i} which is the highest exterior power of the holomorphic cotangent bundle Ω_{V_{i}}^{1} = T_{V_{i}}^{∗}. We would like to study global sections of powers of K_{X^{orb}}. Let U_{i}^{ns} denote the smooth part of U_{i} and V_{i}^{ns} := ϕ_{i}^{−1}U_{i}^{ns}. As shown by (6.1), K_{V_{i}} is not the pull back of K_{U_{i}}, rather

K_{V_{i}^{ns}} ≅ ϕ_{i}^{∗}K_{U_{i}^{ns}}(∑(m_{ij} − 1)R_{ij}).

Since R_{ij} = m_{j}ϕ_{i}^{∗}D_{ij}, we obtain, at least formally, that K_{X^{orb}} is the pull back of K_{X} + Δ, rather than the pull back of K_{X}. The latter of course makes sense only if we define fractional tensor powers of line bundles. Instead of doing it, we state a consequence of the formula:

Claim 10. For s > 0, global sections of K_{X^{orb}}^{⊗s} are those sections of K_{U_{ns}}^{⊗s} which have an at most s(m_{i} −1)/m_{i}-fold pole along the branch divisor D_{i} for every i. For s < 0, global sections of K_{X^{orb}}^{⊗s} are those sections of K_{U_{ns}}^{⊗s} which have an at least s(m_{i} − 1)/m_{i}-fold zero along the branch divisor D_{i} for every i.

Definition 11 (Metrics on orbifolds). A Hermitian metric h on the orbifold (X, Δ) is a Hermitian metric h_{ns} on U_{ns} such that for every chart ϕ_{i} : V_{i} → U_{i} the pull back ϕ^{∗}h_{ns} extends to a Hermitian metric on V_{i}. One can now talk about curvature, Kähler metrics, Kähler-Einstein metrics on orbifolds.

12 (The hypersurface case). We are especially interested in the case when Y ⊂ ℂ^{m} is a hypersurface. It is then the zero set of a polynomial F (z_{1}, …, z_{m}) which is equivariant with respect to the ℂ^{∗}-action. F is irreducible since it has an isolated singularity at the origin, and we always assume that F is not one of the z_{i}. Thus Y \ (∏ z_{i} = 0) is dense in Y.

A differential form on U_{ns} is the same as a ℂ^{∗}-invariant differential form on Y_{ns} and such a form corresponds to a global differential form on X^{orb} iff the corresponding ℂ^{∗}-invariant differential form extends to Y \ {0}.

The (m − 1)-forms

η_{i} := \frac{1}{∂F/∂z_{i}} dz_{1} ∧ ⋯ ∧ d̂z_{i} ∧ ⋯ ∧ dz_{m}|_{Y}

satisfy η_{i} = (−1)^{i−j}η_{j} and they glue together to a global generator η of the canonical line bundle K_{Y \{0}} of Y \ {0}.

Proposition 13. Assume that m ≥ 3 and s(w(F)−∑ w_{i}) > 0. Then the following three spaces are naturally isomorphic:

(1) Global sections of K_{X^{orb}}^{⊗s}.

(2) ℂ^{∗}-invariant global sections of K_{Y}^{⊗s}.

(3) The space of weighted homogeneous polynomials of weight s(w(F) − ∑ w_{i}), modulo multiples of F.

Proof. We have already established that global sections of K_{X^{orb}}^{⊗s} can be identified with ℂ^{∗}-invariant global sections of K_{Y \{0}}^{⊗s}. If m ≥ 3 then Y is a hypersurface of dimension ≥ 2 with an isolated singularity at the origin, thus normal. Hence global sections of K_{Y}^{⊗s} agree with global sections of K_{Y \{0}}^{⊗s}. This shows the equivalence of (1) and (2).

The ℂ^{∗}-action on η has weight ∑ w_{i} − w(F), thus K_{Y}^{⊗s} is the trivial bundle on Y, where the ℂ^{∗}-action has weight s(∑ w_{i} − w(F)). Its invariant global sections are thus given by homogeneous polynomials of weight s(w(F) − ∑ w_{i}) times the generator η. □

In particular, we see that:

Corollary 14. Notation as in §12. K_{X^{orb}}^{−1} is ample iff w(F) < ∑ w_{i}.

15 (Automorphisms and Deformations). If m ≥ 4 and Y ⊂ ℂ^{m} is a hypersurface, then by the Grothendieck–Lefschetz theorem, every orbifold line bundle on Y/ℂ^{∗} is the restriction of an orbifold line bundle on ℂ^{m}/ℂ^{∗} [Gro68]. This implies that every isomorphism between two orbifolds Y/ℂ^{∗}(w) and Y′/ℂ^{∗}(w′) is induced by an automorphisms of ℂ^{m} which commutes with the ℂ^{∗}-actions. This implies that the weight sequences w and w′ are the same (up to permutation) and every such automorphism τ has the form

τ(z_{i}) = g_{i}(z_{1}, …, z_{m}) where w(g_{i}) = w_{i}.  15.1

They form a group Aut(ℂ^{m}, w). For small values of t, maps of the form τ(z_{i}) = z_{i} + tg_{i}(z_{1}, …, z_{m}) where w(g_{i}) = w_{i} are automorphisms, hence the dimension of Aut(ℂ^{m}, w) is ∑_{i} dim H^{0}(ℙ(w), w_{i}). Thus we see that, up to isomorphisms, the orbifolds Y (F)/ℂ^{∗} where w(F) = d form a family of complex dimension at least

dim H^{0}(ℙ(w), d) − ∑_{i} dim H^{0}(ℙ(w), w_{i}),  15.2

and equality holds if the general orbifold in the family has only finitely many automorphisms.

16 (Contact structures). A holomorphic contact structure on a complex manifold M of dimension 2n + 1 is a line subbundle L ⊂ Ω_{M}^{1} such that if θ is a local section of L then θ ∧ (dθ)^{n} is nowhere zero. This forces an isomorphism L^{n+1} ≅ K_{M}. We would like to derive necessary conditions for X^{orb} = Y/ℂ^{∗} to have an orbifold contact structure.

First of all, its dimension has to be odd, so m = 2n+3 and n+1 must divide the canonical class K_{X^{orb}} ≅ O(w(F) − ∑ w_{i}). If these conditions are satisfied, then a contact structure gives a global section of

Ω_{X^{orb}}^{1} ⊗ O (\frac{2}{m−1}(−w(F) + ∑ w_{i})) .

By pull back, this corresponds to a global section of Ω_{Y \{0}}^{1} on which ℂ^{∗} acts with weight \frac{2}{m−1}(−w(F) + ∑ w_{i}).

Next we claim that every global section of Ω_{Y \{0}}^{1} lifts to a global section of Ω_{ℂ^{m}}^{1}. As a preparatory step, it is easy to compute that H^{i}(ℂ^{m} \ {0}, O_{ℂ^{m}\{0}}) = 0 for 0 < i < m − 1. (This is precisely the computation done in [Har77, III.5.1].) Using the exact sequence

0 → O_{ℂ^{m}\{0}} ⟶^{F} O_{ℂ^{m}\{0}} → O_{Y \{0}} → 0,

these imply that H^{i}(Y \ {0}, O_{Y \{0}}) = 0 for 0 < i < m − 2. Next apply the i = 1 case to the co-normal sequence (cf. [Har77, II.8.12])

0 → O_{Y \{0}} ⟶^{dF} Ω_{ℂ^{m}\{0}}^{1}|_{Y \{0}} → Ω_{Y \{0}}^{1} → 0

to conclude that for m ≥ 4, every global section of Ω_{Y \{0}}^{1} lifts to a global section of Ω_{ℂ^{m}\{0}}^{1}|_{Y \{0}}. The latter is the restriction of the free sheaf Ω_{ℂ^{m}}^{1}|_{Y} to Y \ {0}; hence, we can extend the global sections to Ω_{ℂ^{m}}^{1}|_{Y} since Y is normal. Finally these lift to global sections of Ω_{ℂ^{m}}^{1} since ℂ^{m} is affine. Ω_{ℂ^{m}}^{1} = ∑_{i} dz_{i}O_{ℂ^{m}}, hence there every ℂ^{∗}-eigenvector has weight at least min_{i}{w_{i}}. So we obtain: Lemma 17. The hypersurface Y/ℂ^{∗} has no holomorphic orbifold contact structure if m ≥ 4 and \frac{2}{m−1}(−w(F) + ∑ w_{i}) < min_{i}\{w_{i}\}.

This condition is satisfied for all the orbifolds considered in Theorem 34.

3. Sasakian-Einstein structures on links

18 (Brief review of Sasakian geometry). For more details see [BG00] and references therein. Roughly speaking a Sasakian structure on a manifold M is a contact metric structure (ξ, η, Φ, g) such that the Reeb vector field ξ is a Killing vector field of unit length, and whose structure transverse to the flow of ξ is Kähler. Here η is a contact 1-form, Φ is a (1, 1) tensor field which defines a complex structure on the contact subbundle ker η which annihilates ξ, and the metric is g = dη ∘ (Φ ⊗ id) + η ⊗ η.

We are interested in the case when both M and the leaves of the foliation generated by ξ are compact. In this case the Sasakian structure is called quasi-regular, and the space of leaves X^{orb} is a compact Kähler orbifold [BG00]. M is the total space of a circle orbi-bundle (also called V-bundle) over X^{orb}. Moreover, the 2-form dη pushes down to a Kähler form ω on X^{orb}. Now ω defines an integral class [ω] of the orbifold cohomology group H^{2}(X^{orb}, ℤ) which generally is only a rational class in the ordinary cohomology H^{2}(X, ℚ).

This construction can be inverted in the sense that given a Kähler form ω on a compact complex orbifold X^{orb} which defines an element [ω] ∈ H^{2}(X^{orb}, ℤ) one can construct a circle orbi-bundle on X^{orb} whose orbifold first Chern class is [ω]. Then the total space M of this orbi-bundle has a natural Sasakian structure (ξ, η, Φ, g), where η is a connection 1-form whose curvature is ω. The tensor field Φ is obtained by lifting the almost complex structure I on X^{orb} to the horizontal distribution ker η and requiring that Φ annihilates ξ. Furthermore, the map (M, g)⟶(X^{orb}, h) is an orbifold Riemannian submersion.

The Sasakian structure constructed by the inversion process is not unique. One can perform a gauge transformation on the connection 1-form η and obtain a distinct Sasakian structure. However, a straightforward curvature computation shows that there is a unique Sasakian-Einstein metric g with scalar curvature necessarily 2n(2n−1) if and only if the Kähler metric h is Kähler-Einstein with scalar curvature 4(n − 1)n, see [Bes87, BG00]. Hence, the correspondence between orbifold Kähler-Einstein metrics on X^{orb} with scalar curvature 4(n − 1)n and Sasakian-Einstein metrics on M is one-to-one.

19 (Sasakian structures on links of isolated hypersurface singularities). Let F be a weighted homogeneous polynomial as in Definition 7, and consider the subvariety Y := (F = 0) ⊂ ℂ^{n+1}. Suppose further that Y has only an isolated singularity at the origin. Then the link L_{F} = F^{−1}(0) ∩ S^{2m−1} of F is a smooth compact (m − 3)- connected manifold of dimension 2m − 3 [Mil68]. So if m ≥ 4 the manifold L_{F} is simply connected. L_{F} inherits a circle action from the circle subgroup of the ℂ^{∗} group described in Definition 7. We denote this circle group by S_{w}^{1} to emphasize its dependence on the weights.

As noted in §18 the Kähler structure on Y/ℂ^{∗} induces a Sasakian structure on the link L_{F} such that the infinitesimal generator of the weighted circle action defined on ℂ^{m} restricts to the Reeb vector field of the Sasakian structure, which we denote by ξ_{w}. This Sasakian structure (ξ_{w}, η_{w}, Φ_{w}, g_{w}), which is induced from the weighted Sasakian structure on S^{2m−1}, was first noticed by Takahashi [Tak78] for Brieskorn manifolds, and is discussed in detail in [BG01].

The quotient space of the link L_{F} by this circle action is just the orbifold X^{orb} = Y/ℂ^{∗} introduced in Definition 7. It has a natural Kähler structure. In fact, all of this fits nicely into a commutative diagram [BG01]:

(1)  \begin{matrix}L_{F} & ⟶ & S_{w}^{2m−1} \\ ↓π & & ↓ \\ X^{orb} & ⟶ & ℙ(w), \end{matrix}

where S_{w}^{2m−1} emphasizes the weighted Sasakian structure described for example in [BG01], the horizontal arrows are Sasakian and Kählerian embeddings, respectively, and the vertical arrows are orbifold Riemannian submersions. In particular, the Sasakian metric g satisfies g = π^{∗}h + η ⊗ η, where h is the Kähler metric on X^{orb}.

20 (Isometries of Sasakian structures). Let (X_{1}^{orb}, h_{1}) and (X_{2}^{orb}, h_{2}) be two Kähler-Einstein orbifolds and M_{1} and M_{2} the corresponding Sasakian-Einstein manifolds. As explained in §18, M_{1} and M_{2} are isomorphic as Sasakian structures iff (X_{1}^{orb}, h_{1}) and (X_{2}^{orb}, h_{2}) are biholomorphically isometric. Here we are interested in understanding isometries between M_{1} and M_{2}. As we see, with two classes of exceptions, isometries automatically preserve the Sasakian structure as well.

The exceptional cases are easy to describe:

(1) M_{1} and M_{2} are both the sphere S^{2n+1} with its round metric. By a theorem of Boothby and Wang, the corresponding circle action is fixed point free [BW58] with weights (1, …, 1). This happens only in the uninteresting case when Y ⊂ ℂ^{m} is a hyperplane.

(2) M_{1} and M_{2} have a 3–Sasakian structure. This means that there is a 2sphere’s worth of Sasakian structures with a transitive action of SU(2) (cf. [BG99] for precise definitions). This happens only if the X_{i}^{orb} admit holomorphic contact orbifold structures, see [BG97].

Theorem 21. Let (X_{1}^{orb}, h_{1}) and (X_{2}^{orb}, h_{2}) be two Kähler-Einstein orbifolds and M_{1} and M_{2} the corresponding Sasakian-Einstein manifolds. Assume that we are not in either of the exceptional cases enumerated above.

Let ϕ : M_{1} → M_{2} be an isometry. Then there is an isometry ϕ̄ : X_{1}^{orb} → X_{2}^{orb} which is either holomorphic or anti-holomorphic, such that the following digram commutes:

M_{1}  ⟶^{ϕ} M_{2}
↓π_{1}  ↓π_{2}
X_{1}^{orb} ⟶^{ϕ̄} X_{2}^{orb}

Moreover, ϕ̄ determines ϕ up to the S^{1}-action given by the Reeb vector field.

Proof. Let S_{i} denote the Sasakian structure on M_{i}. Then S_{1} and ϕ^{∗}S_{2} are Sasakian structures on M_{1} sharing the same Riemannian metric. Since neither g_{1} nor g_{2} are of constant curvature nor part of a 3-Sasakian structure, the proof of Proposition 8.4 of [BGN03a] implies that either ϕ^{∗}S_{2} = S_{1} or ϕ^{∗}S_{2} = S_{1}^{c} the conjugate Sasakian structure, S_{1}^{c} := (−ξ_{1}, −η_{1}, −Φ_{1}, g_{1}). Thus, ϕ intertwines the foliations and gives rise to an orbifold map ϕ̄: X_{1}^{orb}⟶X_{2}^{orb} as required.

Conversely, any such biholomorphism or anti-biholomorphism ϕ̄ lifts to an orbibundle map ϕ : M_{1}⟶M_{2} uniquely up to the S^{1}-action given by the Reeb vector field. □

Putting this together with §15 we obtain: Corollary 22. Let Y_{1} ⊂ ℂ^{m} (resp. Y_{2} ⊂ ℂ^{m}) be weighted homogeneous hypersurfaces with isolated singularities at the origin with weights w_{1} (resp. w_{2}). Assume that

(1) m ≥ 4.

(2) Y_{1}, Y_{2} have isolated singularities at the origin.

(3) Y_{1}/ℂ^{∗}(w_{1}) and Y_{2}/ℂ^{∗}(w_{2}) both have Kähler-Einstein metrics.

(4) Neither Y_{1}/ℂ^{∗}(w_{1}) nor Y_{2}/ℂ^{∗}(w_{2}) has a holomorphic contact structure. Let (L_{1}, g_{1}) and (L_{2}, g_{2}) be the corresponding Einstein metrics on the links. Then

(5) The connected component of the isometry group of (L_{i}, g_{i}) is the circle S^{1}.

(6) (L_{1}, g_{1}) and (L_{2}, g_{2}) are isometric iff w_{1} = w_{2} (up to permutation) and there is an automorphism τ ∈ Aut(ℂ^{m}, w_{1}) as in (15.1) such that τ(Y_{1}) is either Y_{2} or its conjugate Ȳ_{2}.

4. Kähler-Einstein metrics on orbifolds

23 (Continuity method for finding Kähler-Einstein metrics). Let (X, Δ) be a compact orbifold of dimension n such that K_{X^{orb}}^{−1} is ample. The continuity method for finding a Kähler-Einstein metric on (X, Δ) was developed by [Aub82, Siu88, Siu87, Tia87, Nad90, DK01].

We start with an arbitrary smooth Hermitian metric h_{0} on K_{X^{orb}}^{−1} with positive definite curvature form θ_{0}. Choose a Kähler metric ω_{0} such that Ricci(ω_{0}) = θ_{0}.

Since θ_{0} and ω_{0} represent the same cohomology class, there is a C^{∞} function f such that

ω_{0} = θ_{0} + \frac{i}{2π} ∂ ¯∂f.

Our aim is to find a family of functions ϕ_{t} and numbers C_{t} for t ∈ [0, 1], normalized by the condition ∫_{X} ϕ_{t}ω_{0}^{n} = 0, such that they satisfy the Monge–Ampère equation

log \frac{(ω_{0} + \frac{i}{2π} ∂ ¯∂ϕ_{t})^{n}}{ω_{0}^{n}} + t(ϕ_{t} + f) + C_{t} = 0.

We start with ϕ_{0} = 0, C_{0} = 0 and if we can reach t = 1, we get a Kähler-Einstein metric

ω_{1} = ω_{0} + \frac{i}{2π} ∂ ¯∂ϕ_{1}.

It is easy to see that solvability is an open condition on t ∈ [0, 1], the hard part is closedness. It turns out that the critical step is a 0th order estimate. That is, as the values of t for which the Monge–Ampère equation is solvable approach a critical value t_{0} ∈ [0, 1], a subsequence of the ϕ_{t} converges to a function ϕ_{t_{0}} which is the sum of a C^{∞} and of a plurisubharmonic function. We only need to prove that

∫_{X} e^{−γϕ_{t_{0}}} ω_{0}^{n} < +∞ for some γ > \frac{n}{n+1}.  23.1

We view h_{0}e^{−ϕ_{t_{0}}} as a singular metric on K_{X^{orb}}^{−1}. Its curvature current θ_{0} + \frac{i}{2π} ∂ ¯∂ϕ_{t_{0}} is easily seen to be semi positive.

The method is thus guaranteed to work if there is no singular metric with semi positive curvature on K_{X^{orb}}^{−1} for which the integral in (23.1) is divergent.

A theorem of Demailly and Kollár establishes how to approximate a plurisubharmonic function by sums of logarithms of absolute values of holomorphic functions [DK01]. This allows us to replace an arbitrary plurisubharmonic function ϕ_{t_{0}} by \frac{1}{s} log |τ_{s}|, where τ_{s} is holomorphic. This gives the following criterion: Theorem 24. [DK01] Let X^{orb} be a compact, n-dimensional orbifold such that K_{X^{orb}}^{−1} is ample. The continuity method produces a Kähler-Einstein metric on X^{orb} if the following holds:

There is a γ > \frac{n}{n+1} such that for every s ≥ 1 and for every holomorphic section τ_{s} ∈ H^{0}(X^{orb}, K_{X^{orb}}^{−s}) the following integral is finite:

∫ |τ_{s}|^{− \frac{2γ}{s}} ω_{0}^{n} < +∞.

For the hypersurface case considered in §12 we can combine this with the description of sections of H^{0}(X^{orb}, K_{X^{orb}}^{−s}) given in Proposition 13 to make the condition even more explicit: Corollary 25. Let Y = (F (z_{1}, …, z_{m}) = 0) be as in §12. Assume that w(F) < ∑ w_{i}. The continuity method produces a Kähler-Einstein metric on Y/ℂ^{∗} if the following holds:

There is a γ > \frac{n}{n+1} such that for every weighted homogeneous polynomial g of weighted degree s(∑ w_{i} − w(F)), not identically zero on Y, the function

|g|^{−γ/s} is locally L^{2} on Y \ {0}.

In general it is not easy to decide if a given function |g|^{−c} is locally L^{2} or not, but we at least have the following easy criterion. (See, for instance, [Kol97, 3.14, 3.20].)

Lemma 26. Let M be a complex manifold and h a holomorphic function on M. If c · mult_{p} h < 1 for every p ∈ M then |h|^{−c} is locally L^{2}.

For g as in Corollary 25 it is relatively esay to estimate the multiplicities of its zeros via intersection theory, and we obtain the following generalization of [JK01, Prop.11].

Proposition 27. Let Y = (F (z_{1}, …, z_{m}) = 0) be as in §12. Assume that the intersections of Y with any number of hyperplanes (z_{i} = 0) are all smooth outside the origin. Let g be a weighted homogeneous polynomial and pick δ_{i} > 0. Then

|g|^{−c} ∏_{i} |z_{i}|^{δ_{i}−1} is locally L^{2} on Y \ {0}

if c · w(F) · w(g) < min_{i,j}{w_{i}w_{j}}.

Proof. The case when every δ_{i} = 1 is [JK01, Prop.11] combined with Lemma 26. These also show that in our case the L^{2}-condition holds away from the hyperplanes (z_{i} = 0).

We still need to check the L^{2} condition along the divisors H_{i} := (z_{i} = 0)∩Y \{0}. This is accomplished by reducing the problem to an analogous problem on H_{i} and using induction.

In algebraic geometry, this method is called inversion of adjunction. Conjectured by Shokurov, the following version is due to Kollár [Kol92, 17.6]. It was observed by [Man93] that it can also be derived from the L^{2}-extension theorem of Ohsawa and Takegoshi [OT87]. See [Kol97] or [KM98] for more detailed expositions.

Theorem 28 (Inversion of adjunction). Let M be a smooth manifold, H ⊂ M a smooth divisor with equation (h = 0) and g a holomorphic function on M. Let g_{H} denote the restriction of g to H and assume that it is not identically zero. The following are equivalent:

(1) |g|^{−c}|h|^{δ−1} is locally L^{2} near H for every δ > 0.
(2) |g_{H}|^{−c} is locally L^{2} on H.
□

5. Differential Topology of Links

In this section we briefly describe the differential topology of odd dimensional spheres that can be realized as links of Brieskorn–Pham singularities and discuss methods for determining their diffeomorphism type.

29 (The group bP_{2n}). The essential work here is that of Kervaire and Milnor [KM63] who showed that associated with each sphere S^{n} with n ≥ 5 there is an Abelian group Θ_{n} consisting of equivalence classes of homotopy spheres S^{n} that are equivalent under oriented h-cobordism. By Smale’s h-cobordism theorem this implies equivalence under oriented diffeomorphism. The group operation on Θ_{n} is connected sum. Θ_{n} has a subgroup bP_{n+1} consisting of equivalence classes of those homotopy n-spheres which bound parallelizable manifolds V_{n+1}. Kervaire and Milnor [KM63] proved that bP_{2k+1} = 0 for k ≥ 1. Moreover, for m ≥ 2, bP_{4m} is cyclic of order

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

where B_{m} is the m-th Bernoulli number. Thus, for example |bP_{8}| = 28, |bP_{12}| = 992, |bP_{16}| = 8128. In the first two cases these include all exotic spheres; whereas, in the last case |bP_{16}| is precisely half of the homotopy spheres.

For bP_{4m+2} the situation is still not entirely understood. It entails computing the Kervaire invariant, which is hard. It is known (see the recent review paper [Lan00] and references therein) that bP_{4m+2} = 0 or ℤ_{2} and is ℤ_{2} if m ≠ 2^{i} − 1 for any i ≥ 3. Furthermore, bP_{4m+2} vanishes for m = 1, 3, 7, and 15.

To a sequence a = (a_{1}, …, a_{m}) ∈ ℤ_{+}^{m} Brieskorn associates a graph G(a) whose m vertices are labeled by a_{1}, …, a_{m}. Two vertices a_{i} and a_{j} are connected if and only if gcd(a_{i}, a_{j}) > 1. Let G(a)_{ev} denote the connected component of G(a) determined by the even integers. Note that all even vertices belong to G(a)_{ev}, but G(a)_{ev} may contain odd vertices as well. Theorem 30. [Bri66] The link L(a) (with m ≥ 3) is homeomorphic to the (2m−3)- sphere if and only if either of the following hold.

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

(2) G(a) contains a unique odd isolated point and G(a)_{ev} has an odd number

of vertices with gcd(a_{i}, a_{j}) = 2 for any distinct a_{i}, a_{j} ∈ G(a)_{ev}. 31 (Diffeomorphism types of the links L(a)). In order to distinguish the diffeomorphism types of the links L(a) we need to treat the cases m = 2k + 1 and m = 2k separately.

By [KM63], the diffeomorphism type of a homotopy sphere Σ in bP_{4k} is determined by the signature (modulo 8|bP_{4k}|) of a parallelizable manifold M whose boundary is Σ. By the Milnor Fibration Theorem [Mil68], if Σ = L(a), we can take M to be the Milnor fiber M^{4k}(a) which for links of isolated singularities coming from weighted homogeneous polynomials is diffeomorphic to the hypersurface {z ∈ ℂ^{m} | f(z_{1}, ⋯, z_{m}) = 1}.

Brieskorn shows that the signature of M^{4k}(a) can be written combinatorially as

τ(M^{4k}(a)) = #\{x ∈ ℤ^{2k+1} | 0 < x_{i} < a_{i} and 0 < ∑_{j=0}^{2k} \frac{x_{i}}{a_{i}} < 1 mod 2\}

− #\{x ∈ ℤ^{2k+1} | 0 < x_{i} < a_{i} and 1 < ∑_{j=0}^{2k} \frac{x_{i}}{a_{i}} < 2 mod 2\}.

Using a formula of Eisenstein, Zagier (cf. [Hir71]) has rewritten this as:

τ(M^{4k}(a)) = \frac{(−1)^{k}}{N} ∑_{j=0}^{N−1} cot \frac{π(2j + 1)}{2N} cot \frac{π(2j + 1)}{2a_{1}} ⋯ cot \frac{π(2j + 1)}{2a_{2k+1}},  31.1

where N is any common multiple of the a_{i}’s. It is formula 31.1 that our computer program uses to determine the diffeomorphism type.

For the case of bP_{4k−2} the diffeomorphism type is determined by the so-called Arf invariant C(M^{4k−2}(a)) ∈ {0, 1}. Brieskorn then proves the following: Proposition 32. C(M^{4k−2}(a)) = 1 holds if and only if condition 2 of Theorem 30 holds and the one isolated point, say a_{0}, satisfies a_{0} ≡ ±3 mod 8.

Following conventional terminology we say that L(a) is a Kervaire sphere if C(M^{4k−2}(a)) = 1. A Kervaire sphere is not always exotic, but it is exotic when bP_{4k−2} = ℤ_{2}.

6. Brieskorn–Pham singularities

Notation 33. Consider a Brieskorn–Pham singularity Y (a) := (∑_{i=1}^{m} z_{i}^{a_{i}} = 0) ⊂ ℂ^{m}. Set C = lcm(a_{i} : i = 1, …, m). Y (a) is invariant under the ℂ^{∗}-action

(z_{1}, …, z_{m}) ↦ (λ^{C/a_{1}}z_{1}, …, λ^{C/a_{m}}z_{m}).  33.1

In the notation of Definition 7 we have w_{i} = C/a_{i} and w = w(F) = C. Thus Y (a)/ℂ^{∗} is a Fano orbifold iff 1 < ∑_{i=1}^{m} \frac{1}{a_{i}} .

More generally, we consider weighted homogeneous perturbations

Y (a, p) := (∑_{i=1}^{m} z_{i}^{a_{i}} + p(z_{1}, …, z_{m}) = 0) ⊂ ℂ^{m}, where w(p) = C.

The genericity condition we need, which is always satisfied by p ≡ 0 is: (GC) The intersections of Y (a, p) with any number of hyperplanes (z_{i} = 0)

are all smooth outside the origin. In order to formulate the statement, we further set

C^{j} = lcm(a_{i} : i ≠ j), b_{j} = gcd(a_{j}, C^{j}) and d_{j} = a_{j}/b_{j}.

Theorem 34. The orbifold Y (a, p)/ℂ^{∗} is Fano and has a Kähler-Einstein metric if it satisfies condition (GC) and

1 < ∑_{i=1}^{m} \frac{1}{a_{i}} < 1 + \frac{m − 1}{m − 2} min_{i,j}\{\frac{1}{a_{i}} , \frac{1}{b_{i}b_{j}} \}.

Note that if the a_{i} are pairwise relatively prime then all the b_{i}’s are 1 and we get the simpler bounds 1 < ∑_{i=1}^{m} \frac{1}{a_{i}} < 1 + \frac{m−1}{m−2} min_{i}\{\frac{1}{a_{i}} \}.

Proof. By Corollary 25 we need to show that for every s > 0 and for every weighted homogeneous polynomial g of weighted degree s(∑ w_{i} − w(F)) = sC(∑ a_{i}^{−1} − 1), the function

|g|^{−γ/s} is locally L^{2} on Y \ {0}.

Our aim is to reduce this to a problem on a perturbation of the simpler Brieskorn– Pham singularity Y (b). Lemma 35. Let g be a weighted homogeneous polynomial with respect to the ℂ^{∗}- action (33.1). Then there is a polynomial G such that

g(z_{1}, …, z_{m}) = ∏ z_{i}^{e_{i}} · G(z_{1}^{d_{1}}, …, z_{m}^{d_{m}}).

Proof. Note that C = d_{i}C^{i}. Thus d_{i} divides C/a_{j} = d_{i}C^{i}/a_{j} for j ≠ i but C/a_{i} is relatively prime to d_{i}. Write g = ∏ z_{i}^{e_{i}} · g^{∗} where g^{∗} is not divisible by any z_{i}. Thus g^{∗} has a monomial which does not contain z_{i}, and so its weight is divisible d_{i}. Thus every time z_{i} appears, its exponent must be divisible by d_{i}. □

Applying this to the defining equation of Y (a, p) we obtain that p(z_{1}, …, z_{m}) = p^{∗}(z_{1}^{d_{1}}, …, z_{m}^{d_{m}}) for some polynomial p^{∗}. Set

Y (b, p^{∗}) := (∑_{i=1}^{m} x_{i}^{b_{i}} + p^{∗}(x_{1}, …, x_{m}) = 0) ⊂ ℂ^{m}.

We have a map π : Y (a, p) → Y (b, p^{∗}) given by π^{∗}x_{i} = z_{i}^{d_{i}} and

|g| = π^{∗} ∏ |x_{i}|^{e_{i}/d_{i}} · |G(x_{1}, …, x_{m})|.

The Jacobian of π has (d_{i} − 1)-fold zero along (z_{i} = 0). Thus

|g|^{−γ/s} is locally L^{2} on Y \ {0}

iff

|G|^{−γ/s} · ∏ |x_{i}|^{− \frac{γe_{i}}{sd_{i}} + \frac{1}{d_{i}} −1} is locally L^{2} on Y^{∗} \ \{0\}.

The latter condition is guaranteed by Proposition 27. Indeed, first we need that each x_{i} has exponent bigger than −1. This is equivalent to e_{i} < γ^{−1}s. We know that e_{i}C/a_{i} ≤ wdeg g = sC(∑ a_{i}^{−1}−1) and so it is enough to know that ∑ a_{i}^{−1}−1 < \frac{m−1}{m−2} \frac{1}{a_{i}}. The latter is one of our assumptions.

Note that w(∑ x_{i}^{b_{i}}) = B, where B := lcm(b_{1}, …, b_{m}) and going from G(x_{1}, …, x_{n}) to G(z_{1}^{d_{1}}, …, z_{m}^{d_{m}}) multiplies the weighted degree by C/B. Thus w(G) ≤ \frac{B}{C} w(g) = sB(∑ \frac{1}{a_{i}} − 1). Therefore the last condition

c · w(F) · w(g) < min{w_{i}, w_{j}}

of Proposition 27 becomes

\frac{γ}{s} · B · sB (∑ \frac{1}{a_{i}} − 1) < min\{\frac{B}{b_{i}} , \frac{B}{b_{i}} \}.

After dividing by B^{2}, this becomes our other assumption. □ Note 36. As algebraic varieties, Y (a)/ℂ^{∗} is the same as Y (b)/ℂ^{∗}. In particular, when the a_{i} are pairwise relatively prime then all the b_{i} = 1 hence, as a variety, Y (a)/ℂ^{∗} ≅ ℂℙ^{m−2}. The orbifold structure is given by the divisor

∑_{i=1}^{m−1}(1 − \frac{1}{a_{i}} )(y_{i} = 0) + (1 − \frac{1}{a_{m}} )(∑ y_{i} = 0).

It would be very interesting to write down the corresponding Kähler-Einstein metric explictly. This form would then hopefully give a Kähler-Einstein metric without the required upper bound in Theorem 34.

For most cases we get orbifolds with finite automorphism group: Proposition 37. Assume that m ≥ 4 and all but one of the a_{i} is at least 3. Then the automorphism group of {∑_{i} x_{i}^{a_{i}} = 0}/ℂ^{∗} is finite.

Proof. It is enough to prove that there are no continuous families of isomorphisms of the form

τ_{t}(x_{i}) = x_{i} + ∑_{j≥1} t^{j}g_{ij}(x_{1}, …, x_{m}).

By assumption

∑_{i} τ_{t}(x_{i})^{a_{i}} = ∑_{i} x_{i}^{a_{i}}.

Let j_{0} be the smallest j such that g_{ij_{0}} ≠ 0 for some i and look at the t^{j_{0}} term in the Taylor expansion of the left hand side:

∑_{i} x_{i}^{a_{i}−1}g_{ij_{0}}(x_{1}, …, x_{m}) = 0.

Note that w(g_{ij}) = w(x_{i}) so as long as a_{i} ≥ 3 for all but one i, the terms coming from different values of i do not cancel. Thus every g_{ij_{0}} = 0, a contradiction. □

Remark 38. More generally, the automorphism group of any (F = 0)/ℂ^{∗} is finite as long as w_{i} < \frac{1}{2}w(F) for all but one of the w_{i}s and (F = 0) is smooth outside the origin. Indeed, in this case we would get a relation ∑(∂F/∂x_{i}) · g_{ij_{0}} = 0. By assumption, the ∂F/∂x_{i} form a regular sequence, and so linear relationships with polynomial coefficients between them are generated by the obvious “Koszul” relations (∂F/∂x_{i}) · (∂F/∂x_{j}) = (∂F/∂x_{j}) · (∂F/∂x_{i}). We get a contradiction by degree considerations.

7. Numerical examples

Our ultimate aim is to obtain a complete enumeration of all sequences that yield a Sasakian-Einstein metric on some homotopy sphere. We can accomplish this in low dimensions via a computer program, to be published elsewhere. Here we content ourselves with obtaining some examples which show the double exponential growth of the number of cases. Example 39. Consider sequences of the form a = (2, 3, 7, m). By explicit calculation, the corresponding link L(a) gives a Sasakian-Einstein metric on S^{5} if 5 ≤ m ≤ 41 and m is relatively prime to at least two of 2, 3, 7. This is satisfied in 27 cases. Example 40. Among the above cases, the sequence a = (2, 3, 7, 35) is especially noteworthy. If C(u, v) is any sufficiently general homogenous septic polynomial, then the link of

x_{1}^{2} + x_{2}^{3} + C(x_{3}, x_{4}^{5})

also gives a Sasakian-Einstein metric on S^{5}. The relevant automorphism group of ℂ^{4} is

(x_{1}, x_{2}, x_{3}, x_{4}) ↦ (x_{1}, x_{2}, α_{3}x_{3} + βx_{4}^{5}, α_{4}x_{4}).

Hence we get a 2(8 − 3) = 10 real dimensional family of Sasakian-Einstein metrics on S^{5}.

Similarly, the sequence a = (2, 3, 7, 43, 43 · 31) gives a standard 7-sphere with a 2(43 − 2) = 82-dimensional family of Sasakian-Einstein metrics on S^{7}. Example 41 (Euclid’s or Sylvester’s sequence). (See [GKP89, Sec.4.3] or [Slo03, A000058].)

Consider the sequence defined by the recursion relation

c_{k+1} = c_{1} ⋯ c_{k} + 1 = c_{k}^{2} − c_{k} + 1

beginning with c_{1} = 2. It starts as

2, 3, 7, 43, 1807, 3263443, 10650056950807, ...

It is easy to see (cf. [GKP89, 4.17]) that

c_{k} ≥ (1.264)^{2^{k}} − \frac{1}{2} and ∑_{i=1}^{m} \frac{1}{c_{i}} = 1 − \frac{1}{c_{m+1} − 1} = 1 − \frac{1}{c_{1} ⋯ c_{m}} .

Example 42. Consider sequences of the form a = (a_{1} = c_{1}, …, a_{m−1} = c_{m−1}, a_{m}). The troublesome part of the condition of Theorem 34 is the computation of the b_{i}. However b_{i} ≤ a_{i} thus it is sufficient to satisfy the following stronger restriction:

1 < ∑_{i=1}^{m} \frac{1}{a_{i}} < 1 + \frac{m − 1}{m − 2} min_{i,j}\{\frac{1}{a_{i}a_{j}} \} = 1 + \frac{m − 1}{m − 2} · \frac{1}{a_{m−1}a_{m}} .

By direct computation this is satisfied if c_{m} − c_{m−1} < a_{m} < c_{m}. At least a third of these numbers are relatively prime to a_{1} = 2 and to a_{2} = 3, thus we conclude: Proposition 43. Our methods yield at least \frac{1}{3}(c_{m−1} − 1) ≥ \frac{1}{3}(1.264)^{2^{m−1}} − 0.5 inequivalent families of Sasakian-Einstein metrics on (standard and exotic) (2m − 3)-spheres.

If 2m−3 ≡ 1 mod 4 then by Proposition 32, all these metrics are on the standard sphere. If 2m − 3 ≡ 3 mod 4 then all these metrics are on on both standard and exotic spheres but we cannot say anything in general about their distribution. Example 44. We consider the sequences a = (a_{1} = c_{1}, …, a_{m−1} = c_{m−1}, a_{m} = (c_{m−1}−2)c_{m−1}). Any two of them are relatively prime, except for gcd(a_{m−1}, a_{m}) = c_{m−1}, and the conditions of Theorem 34 are satisfied.

The Brieskorn–Pham polynomial has weighted homogeneous perturbations

x_{1}^{a_{1}} + ⋯ + x_{m−2}^{a_{m−2}} + G(x_{m−1}, x_{m}^{c_{m−1}−2})

where G is any homogeneous polynomial of degree c_{m−1}. Up to coordinate changes, these form a family of complex dimension c_{m−1} − 2. Thus we conclude:

Proposition 45. Our methods yield an at least 2(c_{m−1} − 2) ≥ 2((1.264)^{2^{m−1}} − 2.5)-dimensional (real) family of pairwise inequivalent Sasakian-Einstein metrics on some (standard or exotic) (2m − 3)-sphere.

As before, if 2m − 3 ≡ 1 mod 4 then these metrics are on the standard sphere. Example 46. Consider sequences of the form a = (a_{1} = 2c_{1}, …, a_{m−2} = 2c_{m−2}, a_{m−1} = 2, a_{m}) where a_{m} is relatively prime to all the other a_{i}s. By easy computation, the condition of Theorem 34 is satisfied if 2c_{m−2} < a_{m} < 2c_{m−1} − 2.

The relatively prime condition is harder to pin down, but it certainly holds if in addition a_{m} is a prime number. By the prime number theorem, the number of primes in the interval [c_{m−1}, 2c_{m−1}] is about

\frac{c_{m−1}}{log c_{m−1}} ≥ \frac{(1.264)^{2^{m−2}}}{2^{m−1} log 1.264} ≥ (1.264)^{2^{m−1}−4(m−1)},

so it is still doubly exponential in m.

By Proposition 32, for even m, L(a) the standard sphere if a ≡ ±1 mod 8 and the Kervaire sphere if a ≡ ±3 mod 8. It is easy to check for all values of m that we get at least one solution of both types. Thus we conclude: Proposition 47. Our methods yield a doubly exponential number of inequivalent families of Sasakian-Einstein metrics on both the standard and the Kervaire (4m − 3)-spheres. Acknowledgments . We thank Y.-T. Siu for answering several questions, and E. Thomas for helping us with the computer programs. CPB and KG were partially supported by the NSF under grant number DMS-0203219 and J.K. was partially supported by the NSF under grant number DMS-0200883. The authors would also like to thank Università di Roma “La Sapienza” for partial support where discussions on this work initiated.

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CPB and KG: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131.

E-mail address: cboyer@math.unm.edu
E-mail address: galicki@math.unm.edu

JK: Department of Mathematics, Princeton University, Princeton, NJ 08544-1000.

E-mail address: kollar@math.princeton.edu