Ann. Inst. Fourier 52 (2002), 1569-1584  Annales l’Institute

Fourier

Einstein Metrics on Rational Homology 7-Spheres

Charles P. Boyer Krzysztof Galicki Michael Nakamaye

Abstract: In this paper we demonstrate the existence of Sasakian-Einstein structures on certain 2-connected rational homology 7-spheres. These appear to be the first non-regular examples of Sasakian-Einstein metrics on simply connected rational homology spheres. We also briefly describe the rational homology 7-spheres that admit regular positive Sasakian structures.

Introduction

Dimension seven appears to be rather special when it comes to examples of compact Einstein manifolds. It is perhaps the prominent rôle such manifolds have played in physics ever since the early days of Kaluza-Klein supergravity that made both theoretical physicists and mathematicians alike particularly interested in them. Discoveries of many different constructions followed as a result of this interest.

Arguably, today a special place among all compact Einstein 7-manifolds is reserved for the so-called Sasakian-Einstein spaces. They are defined to be Riemannian manifolds with the property that the metric cone on them is a Calabi-Yau 4-fold and, in particular, they are always of positive scalar curvature. All regular Sasakian-Einstein manifolds are circle bundles of Fano 3-folds that admit Kähler-Einstein metrics. Non-regular ones fiber over compact Kähler-Einstein Fano 3-folds with orbifold singularities. An interesting subfamily of the family of Sasakian-Einstein 7-manifolds consists of the so-called 3-Sasakian spaces. They are characterized by fact that their metric cone is not only Calabi-Yau, but also hyperkähler and are all orbifold fibrations over compact Kähler-Einstein Fano 3-folds which admit a complex contact structure.

Regular and non-regular examples of both Sasakian-Einstein and 3-Sasakian manifolds are now plentiful and they were extensively studied by the first two authors [BG1, BG2]. There is an example of a regular Sasakian-Einstein (4n+3)-manifold which is worthy of some further discussion. It is the homogeneous Stiefel manifold of 2-frames in (2n+1)-dimensional Euclidean space, V_{2}(ℝ^{2n+1}) = SO(2n + 1)/SO(2n − 1) which is a circle bundle over the oriented Grassmannian, G̃_{2}(ℝ^{2n+1}). From the point of view of an algebraic geometer it is a classical fact that G̃_{2}(ℝ^{2n+1}) is diffeomorphic to the complex quadric Q_{2n−1} in ℂℙ^{2n} which is well-known to be Fano and to admit a Kähler-Einstein metric. It is perhaps less well-known that the quadric Q_{2n−1} has the same cohomology groups as ℂℙ^{2n−1} , but differs in the ring structure. Hence, V_{2}(ℝ^{2n+1}) is a rational homology sphere with H_{n−1}(V_{2}(ℝ^{2n+1}), ℤ) ≈ ℤ_{2}. Now it has been known for quite some time that V_{2}(ℝ^{2n+1}) carries a Sasakian-Einstein structure [BGFK,BG1]. Up to date, apart from S^{2n+1}, V_{2}(ℝ^{2n+1}), and the 3-Sasakian homogeneous 11-manifold G_{2}/Sp(1) we are not aware of any other examples of simply connected rational homology spheres which are also known to admit Sasakian-Einstein structures. In this paper we shall demonstrate that for n = 2, quite to the contrary, there are many examples of such structures, 184 to be precise. These examples are obtained as hypersurfaces in certain weighted projective 4-spaces, but we certainly expect the phenomena to occur in arbitrary dimension.

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.

The key to this construction is a recent paper of Johnson and Kollár [JK2]. There they give a list of 4442 quasi-smooth Fano 3-folds Z anticanonically embedded in weighted projective 4-spaces ℙ(w). Moreover, they show that 1936 of these 3-folds admit Kähler-Einstein metrics. According to our general theory [BG1] such Fano 3-folds give rise to Sasakian-Einstein metrics on smooth 7-manifolds M^{7}. Moreover, these 7-manifolds arise as links of isolated hypersurface singularities associated to certain weighted homogeneous polynomials in ℂ^{5}. As in [JK1] Johnson and Kollár [JK2] only consider the case when the orbifold Fano index is one, and as the authors showed in [BGN1] for log del Pezzo surfaces, there should be many more interesting examples of quasi-smooth Fano 3-folds with higher orbifold Fano index. This is currently under study.

In this note we prove Theorem A: There are1936 distinct Sasakian-Einstein structures on certain 2-connected 7-manifolds M_{w,d}^{7} realized as links of weighted homogeneous polynomials in ℂ^{5} with weight vector w = (w_{0}, w_{1}, w_{2}, w_{3}, w_{4}) and degree d. In particular, there are 184 2-connected rational homology spheres which are listed in the Table below. In addition to the weight vector w and degree d the Table lists the Milnor number μ of the link and the order of H_{3}(M_{w,d}^{7}, ℤ).

We have not answered the question as to whether two distinct or non-conjugate Sasakian-Einstein structures on the same link M^{7} could belong to the same underlying Riemannian metric g. Indeed, this can happen, but if g is not the standard round metric on S^{7} then by a Theorem of Tachibana and Yu [TaYu], the two Sasakian-Einstein structures must belong to a 3-Sasakian structure. But then by a Theorem of Galicki and Salamon [GS], we must have b_{3} = 0, so M^{7} must be a rational homology sphere. However, we do not know whether any of the rational homology 7-spheres discussed here admit 3-Sasakian structures.

2. The Sasakian Geometry of Links of Weighted Homogeneous Polynomials

In this section we briefly review 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 they are ordered w_{0} ≤ w_{1} ≤ ⋯ ≤ w_{n} and that gcd(w_{0}, …, w_{n}) = 1. Let f be a quasi-homogeneous polynomial, that is f ∈ ℂ[z_{0}, …, z_{n}] and satisfies

2.1  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

2.2  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}. The embedding L_{f} ↪ S^{2n+1} induces a Sasakian structure on L_{f} [BG3].

Given a sequence w = (w_{0}, …, w_{n}) of ordered positive integers one can form the graded polynomial ring S(w) = ℂ[z_{0}, …, z_{n}], where z_{i} has grading or weight w_{i}. The weighted projective space [Dol, Fle] ℙ(w) = ℙ(w_{0}, …, w_{n}) is defined to be the scheme Proj(S(w)). It is the quotient space (ℂ^{n+1} − {0})/ℂ^{∗}(w), where ℂ^{∗}(w) is the weighted action defined in 2.1, or equivalently, ℙ(w) is the quotient of the weighted Sasakian sphere S_{w}^{2n+1} = (S^{2n+1}, ξ_{w}, η_{w}, Φ_{w}, g_{w}) by the weighted circle action S^{1}(w) generated by ξ_{w}. As such ℙ(w) is also a compact complex orbifold with an induced Kähler structure. We have from [BG3] Theorem 2.3: The quadruple (ξ_{w}, η_{w}, Φ_{w}, g_{w}) gives L_{f} a quasi-regular Sasakian structure such that there is a commutative diagram

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

where the horizontal arrows are Sasakian and Kählerian embeddings, respectively, and the vertical arrows are principal S^{1} V-bundles and orbifold Riemannian submersions. Moreover, if Z_{f} is Fano, L_{f} is the total space of the principal S^{1} V-bundle over the orbifold Z_{f} whose first Chern class in H_{orb}^{2}(Z_{f}, ℤ) is c_{1}(Z_{f})/I, where I is the index.

We should also mention that c_{1}(Z_{f}) pulls back to the basic first Chern class c_{1}^{B} ∈ H_{B}^{2}(F_{ξ_{w}}) and η_{w} is the connection in this V-bundle whose curvature is dη = \frac{2ni}{I}π^{∗}ω_{w}, where ω_{w} is the Kähler form on Z_{f}.

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 become more complicated as the dimension increases [Fle,JK2]; however, in this paper we are only interested in the n = 4 case of hypersurfaces in a weighted complex projective 4-space. 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,JK2]:

Quasi-Smoothness Conditions 2.4:

I.  For each i = 0, ⋯, 4 there is a j and a monomial z_{i}^{m_{i}}z_{j} ∈ O(d).

Here j = i is possible.

II.  For all distinct i, j either there is a monomial z_{i}^{b_{i}}z_{j}^{b_{j}} ∈ O(d).

or there exist monomials z_{i}^{n_{1}}z_{j}^{m_{1}}z_{k}, z_{i}^{n_{2}}z_{j}^{m_{2}}z_{l} ∈ O(d) with {k, l} ≠ {i, j} and k ≠ l.

III.  For every i, j there exists a monomial of degree d that does not involve either z_{i}

or z_{j}. 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 2.5

IV.  For each i we have gcd(w_{0}, ⋯, ŵ_{i}, ⋯, w_{4}) = 1.

Here theˆ means skip that element.

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

2.6  K_{Z} ≃ O(−I) = O(d − |w|),

where |w| = ∑_{i} w_{i}.

In this note we shall only consider the anticanonically embedded Fano 3-folds of [JK2], that is, we shall assume hereafter that I = |w| − d = 1. The examples we consider are from the list sporadic.txt of Johnson and Kollár [JK2] which is found at: http://www.math.princeton.edu/˜ jmjohnso.

3. The Topology of the Link M_{w,d}^{7}

The topology of a link L_{f} of an isolated hypersurface singularity is encoded in the characteristic polynomial Δ(t) of the monodromy map. Δ(t) is an important link invariant that generalizes the Alexander polynomial of a knot, and is often called the “Alexander polynomial” of the link [HZ]. 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).

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^{μ}. Using the Wang sequence of the Milnor fibration together with Alexander-Poincare duality gives the exact sequence [Mil]

3.2  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_{∗}) of the monodromy map. The Betti number b_{n}(L_{f}) = b_{n−1}(L_{f}) equals the number of factors of (t−1) in Δ(t). Generally, finding the torsion is much more difficult. However, in the case of rational homology spheres, b_{n}(L_{f}) = b_{n−1}(L_{f}) = 0, the group H_{n−1}(M, ℤ) is a torsion group of order Δ(1).

It is not our purpose in this note to give a systematic study of the Johnson-Kollár list. This requires a computer program for computing the Betti numbers which is currently under study. Here we are content with giving algorithm for finding special cases when rational homology spheres occur. We have written a MAPLE program which allows us to search the JK list, sporatic.txt and pick out certain rational homology spheres. We emphasize that this procedure does not necessarily find all rational homology spheres, but only all of those that satisfy certain additional conditions. Actually there are two distinct types of conditions on the weights that allow us to find rational homology spheres and they are described in the lemmas below. The first and simplest is that the weights are all relatively prime to the degree. Lemma 3.4: Let w = (w_{0}, w_{1}, w_{2}, w_{3}, w_{4}) be the weights of a quasi-smooth Fano 3-fold, Z_{f} of degree d and index 1. Suppose further that gcd(w_{i}, d) = 1 for all i = 0, ⋯, 4. Then there exists an integer N(w) such that the Alexander polynomial Δ(t) of the link M_{w,d}^{7} has the form

Δ(t) = \frac{(t^{d} − 1)^{N(w)}}{t − 1}

Hence, the Betti number b_{3} of the link M_{w,d}^{7} is given by b_{3}(M_{w,d}^{7}) = N(w) − 1.

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. The divisor div Δ is given by

3.6  div Δ = (\frac{Λ_{u_{0}}}{v_{0}} − 1) ⋯ (\frac{Λ_{u_{4}}}{v_{4}} − 1)

which can be reduced to the form

3.7  div Δ(t) = ∑_{j} a_{j}Λ_{j} − 1

for some integers a_{j} upon using the relations Λ_{a}Λ_{b} = gcd(a, b)Λ_{lcm(a,b)}. The characteristic polynomial Δ(t) is then determined from its divisor by

3.8  Δ(t) = \frac{∏(t^{j} − 1)^{a_{j}}}{t − 1} ,

and the third Betti number is given by

b_{3}(M_{w,d}^{7}) = ∑_{j} a_{j} − 1.

In our case we have gcd(w_{i}, d) = 1 so equation 3.7 must take the form

3.9  div Δ(t) = N(w)Λ_{d} − 1

where N(w) is an integer.

The integer N(w) can be computed explicitly from the above procedure in terms of the weights and index. We find

3.10  N(w) = \frac{d(dr_{01}r_{23} + r_{01} + r_{23})}{w_{4}} + \frac{1}{w_{4}} − (dr_{01}r_{23} + r_{01} + r_{23})

where

d = |w| − 1, r_{ij} = \frac{d}{w_{i}w_{j}} − \frac{1}{w_{i}} − \frac{1}{w_{j}} .

We should remark here that although it is far from manifest in equation 3.10, under the hypothesis of Lemma 3.4 the function N(w) is invariant under a permutation of the weights, i.e. if Σ_{5} denotes the permutation group on 5 letters, then N(σ(w)) = N(w) for any σ ∈ Σ_{5}. We have an immediate Corollary 3.11: Let M_{w,d}^{7} be the link of an isolated hypersurface singularity defined by a weighted homogeneous polynomial f with well-formed weights w = (w_{0}, w_{1}, w_{2}, w_{3}, w_{4}) and degree d which satisfy gcd(w_{i}, d) = 1 for all i = 0, ⋯, 4. Then M_{w,d}^{7} is a rational homology sphere if and only if N(w) = 1. Furthermore, in this case the Milnor number μ = d − 1, and the order of H_{3}(M_{w,d}^{7}, ℤ) equals the degree d.

Proof: The only part that we need to compute is the order of H_{3}. Since for a 2-connected rational homology sphere H_{4}(M_{w,d}^{7}, ℤ) = 0, the exact sequence 3.2 shows [Mil] that the order of H_{3}(M_{w,d}^{7}, ℤ) equals Δ(1). But from 3.8 and 3.9 we see that in our special case the characteristic polynomial takes the form

Δ(t) = \frac{t^{d} − 1}{t − 1} = t^{d−1} + ⋯ + t + 1

from which the result follows.

We now describe the second type of condition.

Lemma 3.12: Let w = (w_{0}, w_{1}, w_{2}, w_{3}, w_{4}) be the weights of a quasi-smooth Fano 3-fold, Z_{f} of degree d and index 1. Suppose further that the degree can be written as d = m_{3}m_{2}, where m_{2} and m_{3} are relatively prime, and that the “rational weights” \frac{d}{w_{i}} take the form \frac{m_{3}}{v_{i}} for 3 values of i and \frac{m_{2}}{v_{i}} for the other 2 values of i. Then there exist positive integers l = l(w), and n = n(w), depending on the weights w, such that the Alexander polynomial Δ(t) of the link M_{w,d}^{7} takes the form

Δ(t) = \frac{(t^{d} − 1)^{ln}(t^{m_{3}} − 1)^{l}}{(t − 1)(t^{m_{2}} − 1)^{n}} .

Hence,

b_{3}(M_{w,d}^{7}) = (n(w) + 1)(l(w) − 1).

Proof: Computing as in the proof of Lemma 3.4, we see that from the Milnor-Orlik procedure [MO] that the divisor of the Alexander polynomial must take the form

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

for some positive integers l(w) and n(w) depending on the weights. The above form of the Alexander polynomial then follows from equations 3.7 and 3.8. The explicit form of the functions l(w) and n(w) are also easily calculated. Let i_{1}, i_{2}, i_{3} denote the 3 indices whose rational weights take the form \frac{m_{3}}{v_{i}} and similarly let j_{1}, j_{2} denote the indices corresponding to the rational weights \frac{m_{2}}{v_{j}} . Then one finds

3.14  l(w) = \frac{m_{3}^{2}}{v_{i_{1}}v_{i_{2}}v_{i_{3}}} − m_{3}(\frac{1}{v_{i_{1}}v_{i_{2}}} + \frac{1}{v_{i_{1}}v_{i_{3}}} + \frac{1}{v_{i_{2}}v_{i_{3}}} ) + \frac{1}{v_{i_{1}}} + \frac{1}{v_{i_{2}}} + \frac{1}{v_{i_{3}}}

3.15  n(w) = \frac{m_{2}}{v_{j_{1}}v_{j_{2}}} − \frac{1}{v_{j_{1}}} − \frac{1}{v_{j_{2}}}

The expression for b_{3} follows directly from the expression for Δ(t). Corollary 3.16: Let w = (w_{0}, w_{1}, w_{2}, w_{3}, w_{4}) be the weights of a quasi-smooth Fano 3-fold, Z_{f} of degree d and index 1. Suppose further that the hypothesis of Lemma 3.12 is satisfied, then M_{w,d}^{7} is a rational homology sphere if and only if l(w) = 1. Furthermore, in this case the Milnor number μ = (m_{3} − 1)(n(w)m_{2} + 1), and the order of H_{3}(M_{w,d}^{7}, ℤ)

is m_{3}^{n(w)+1}. Proof: As in the proof of Corollary 3.11 this follows from Lemma 3.12 by cancelling the t − 1 factors in Δ(t) and evaluating at t = 1. Remark 3.17: One can also write d = m_{4}m_{1} or d = m_{2,1}m_{2,2}m_{1} where in each case the m_{s} are pairwise relatively prime positive integers. Also in the first case d/w_{i} = m_{4}/v_{i} for 4 values of i and d/w_{j} = m_{1}/v_{j} for the remaining index. In the second case d/w_{i} = m_{2,1}/v_{i} and d/w_{j} = m_{2,2}/v_{j} for two pairs of index and d/w_{k} = m_{1}/v_{k} for the remaining index. In both cases one finds rational homology spheres without any further conditions; however, one can also easily show that the weights are not well-formed in either case.

Table: ℚ-Homology 7-Spheres M_{w,d}^{7} admitting S-E Structures

w = (w_{0}, w_{1}, w_{2}, w_{3}, w_{4})  d  μ  Order of H_{3}(M_{w,d}^{7}, ℤ)
(17, 34, 75, 125, 175)  425  4416  582622237229761 = 17^{12}
(17, 238, 381, 635, 889)  2159  16272  118587876497 = 17^{9}
(19, 57, 100, 125, 175)  475  3168  16983563041 = 19^{8}
(49, 334, 525, 668, 763)  2338  4995  37259704 = 2^{3} · 167^{3}
(49, 573, 1862, 2483, 4393)  9359  36720  282475249 = 7^{10}
(50, 65, 73, 73, 105)  365  1152  28398241 = 73^{4}
(52, 127, 381, 533, 559)  1651  5040  260144641 = 127^{4}
(55, 160, 373, 373, 905)  1865  5932  19356878641 = 373^{4}
(87, 558, 687, 1331, 1331)  3993  4320  1771561 = 11^{6}
(93, 459, 780, 1331, 1331)  3993  4320  1771561 = 11^{6}
(93, 1011, 2298, 3401, 3401)  10203  13600  11566801 = 19^{2} · 179^{2}
(97, 1531, 2201, 2775, 3253)  9856  9855  9856 = 2^{7} · 7 · 11
(99, 318, 465, 881, 881)  2343  3520  776161 = 881^{2}
(101, 439, 559, 579, 619)  2296  2295  2296 = 2^{3} · 7 · 41
(101, 1597, 1996, 2695, 3693)  10081  10080  10081 = 17 · 593
(101, 1597, 2096, 2495, 3793)  10081  10080  10081 = 17 · 593
(101, 1697, 2296, 2695, 4093)  10881  10880  10881 = 3^{3} · 13 · 31
(103, 1321, 2337, 2845, 3251)  9856  9855  9856 = 2^{7} · 7 · 11
(108, 267, 507, 881, 881)  2643  3520  776161 = 881^{2}
(109, 1616, 2047, 2693, 4417)  10881  10880  10881 = 3^{3} · 13 · 31
(111, 329, 407, 423, 470)  1739  1728  1369 = 37^{2}
(111, 658, 2303, 3071, 6031)  12173  24600  35611289 = 7^{3} · 47^{3}
(111, 768, 2523, 3401, 3401)  10203  13600  11566801 = 19^{2} · 179^{2}
(113, 1115, 6021, 8362, 9589)  25199  50064  1142897 = 113^{3}
(113, 1561, 3345, 8362, 11819)  25199  50064  1142897 = 113^{3}
(115, 341, 523, 591, 727)  2296  2295  2296 = 2^{3} · 7 · 41
(115, 797, 949, 987, 2050)  4897  4896  4897 = 59 · 83
(125, 1732, 4577, 5567, 6433)  18433  18432  18433
(125, 2599, 4208, 5569, 9901)  22401  22400  22401 = 3^{2} · 19 · 131
(127, 1888, 2643, 4657, 6671)  15985  15984  15985 = 5 · 23 · 139
(127, 2266, 3651, 6043, 8435)  20521  20520  20521
(127, 2392, 3399, 6043, 8561)  20521  20520  20521
(127, 2770, 4407, 7429, 10325)  25057  25056  25057
(129, 511, 1192, 1235, 1831)  4897  4896  4897 = 59 · 83
(133, 346, 379, 527, 857)  2241  2240  2241 = 3^{3} · 83
(136, 337, 421, 455, 893)  2241  2240  2241 = 3^{3} · 83
(136, 2023, 3237, 5395, 7553)  18343  17280  289 = 17^{2}
(137, 1223, 1427, 2786, 4349)  9921  9920  9921 = 3 · 3307
(137, 1495, 1699, 3466, 5301)  12097  12096  12097
(138, 171, 393, 701, 701)  2103  2800  149401 = 701^{2}
(139, 2343, 3721, 6202, 10061)  22465  22464  22465 = 5 · 4493

ℚ-Homology 7-Spheres M_{w,d}^{7} admitting S-E Structures (cont.)

w = (w_{0}, w_{1}, w_{2}, w_{3}, w_{4})  d  μ  Order of H_{3}(M_{w,d}^{7}, ℤ)
(139, 3171, 5101, 8548, 13787)  30745  30744  30745 = 5 · 11 · 13 · 43
(141, 652, 1909, 2375, 2701)  7777  7776  7777 = 7 · 11 · 101
(141, 1259, 1492, 3031, 4663)  10585  10584  10585 = 5 · 29 · 73
(141, 2224, 3475, 4031, 9729)  19599  19600  19881 = 3^{2} · 47^{2}
(141, 2363, 3058, 4309, 9729)  19599  19600  19881 = 3^{2} · 47^{2}
(143, 194, 209, 231, 291)  1067  1152  9409 = 97^{2}
(143, 1135, 2057, 3476, 5675)  12485  13608  1288225 = 5^{2} · 227^{2}
(143, 2981, 6530, 9795, 16467)  35915  39168  10660225 = 5^{2} · 363^{2}
(145, 2157, 3451, 5752, 9347)  20851  20160  841 = 29^{2}
(146, 869, 1955, 2969, 3983)  9921  9920  9921 = 3 · 3307
(147, 207, 230, 245, 299)  1127  1152  2401 = 7^{4}
(147, 255, 1056, 1457, 1457)  4371  5824  2122849 = 31^{2} · 47^{2}
(154, 535, 739, 1427, 2115)  4969  4968  4969
(154, 763, 1297, 2975, 3891)  9079  10368  1682209 = 1297^{2}
(155, 921, 1612, 2149, 4681)  9517  9792  94249 = 307^{2}
(155, 1075, 3532, 5835, 7064)  17660  21186  12475024 = 3532^{2}
(155, 2309, 7543, 10006, 12469)  32481  32480  32481 = 3^{4} · 401
(155, 2617, 8467, 11392, 14163)  36793  36792  36793
(157, 269, 637, 665, 1090)  2817  2816  2817 = 3^{2} · 313
(157, 436, 1401, 1775, 1993)  5761  5760  5761 = 7 · 823
(157, 545, 1051, 1401, 2608)  5761  5760  5761 = 7 · 823
(157, 883, 1558, 2597, 4311)  9505  9504  9505 = 5 · 1901
(157, 1195, 2182, 3689, 6027)  13249  13248  13249
(157, 2339, 5146, 7641, 12943)  28225  28224  28225 = 5^{2} · 1129
(157, 2651, 5770, 8733, 14659)  31969  31968  31969 = 7 · 4567
(159, 2365, 3784, 6307, 12455)  25069  25488  223729 = 11^{2} · 43^{2}
(163, 1939, 2747, 6786, 8887)  20521  20520  20521
(163, 1939, 3070, 5171, 10179)  20521  20520  20521
(163, 2101, 3394, 5657, 11151)  22465  22464  22465 = 5 · 4493
(166, 1237, 4371, 5773, 7175)  18721  18720  18721 = 97 · 193
(166, 1399, 2551, 5513, 7077)  16705  16704  16705 = 5 · 13 · 257
(167, 717, 1324, 1765, 3805)  7777  7776  7777 = 7 · 11 · 101
(169, 2015, 6549, 8732, 10915)  28379  26208  169 = 13^{2}
(170, 1267, 3041, 4477, 7687)  16641  16640  16641 = 3^{2} · 43^{2}
(171, 247, 556, 695, 973)  2641  2520  361 = 19^{2}
(175, 271, 299, 306, 751)  1801  1800  1801
(175, 289, 925, 2312, 3525)  7225  14688  180124137569 = 17^{6}
(175, 2434, 10605, 13387, 15995)  42595  43776  1481089 = 1217^{2}
(176, 295, 317, 481, 973)  2241  2240  2241 = 3^{3} · 83
(176, 1135, 1397, 4103, 5675)  12485  13608  1288225 = 5^{2} · 227^{2}
(177, 997, 3695, 5044, 6217)  16129  16128  16129 = 3^{6}

ℚ-Homology 7-Spheres M_{w,d}^{7} admitting S-E Structures (cont.)

w = (w_{0}, w_{1}, w_{2}, w_{3}, w_{4})  d  μ  Order of H_{3}(M_{w,d}^{7}, ℤ)
(177, 2275, 5950, 7175, 15399)  30975  30976  31329 = 3^{2} · 59^{2}
(178, 883, 2031, 3973, 5033)  12097  12096  12097
(181, 1613, 4301, 5197, 11110)  22401  22400  22401 = 3^{2} · 19 · 131
(181, 1618, 7011, 8809, 10607)  28225  28224  28225 = 5^{2} · 1129
(181, 2338, 10251, 12949, 15467)  41185  41184  41185 = 5 · 8237 · 131
(183, 787, 1453, 2422, 4661)  9505  9504  9505 = 5 · 1901
(183, 1031, 2608, 4003, 6793)  14617  14616  14617 = 47 · 311
(185, 477, 991, 1321, 2788)  5761  5760  5761 = 7 · 823
(187, 409, 539, 1320, 2045)  4499  4896  167281 = 409^{2}
(187, 631, 1299, 2746, 3563)  8425  8424  8425 = 5^{2} · 337
(187, 781, 2306, 3459, 5951)  12683  13824  1329409 = 1153^{2}
(187, 1853, 2594, 6485, 10931)  22049  23328  1682209 = 1297^{2}
(187, 2416, 8177, 10965, 19328)  41072  43470  5837056 = 2^{8} · 151^{2}
(191, 235, 433, 509, 904)  2241  2240  2241 = 3^{3} · 83
(191, 2467, 7990, 10627, 18787)  40041  40040  40041 = 3^{3} · 1483
(191, 3607, 11770, 15757, 27717)  59041  59040  59041 = 17 · 23 · 151
(193, 3247, 4202, 10877, 18335)  36863  36864  37249 = 193^{2}
(194, 539, 1155, 2425, 3157)  7469  7488  9409 = 97^{2}
(194, 693, 847, 2425, 3311)  7469  7488  9409 = 97^{2}
(195, 484, 613, 1291, 2387)  4969  4968  4969
(196, 681, 827, 2383, 3259)  7345  7344  7345 = 5 · 13 · 113
(196, 1119, 1411, 4135, 5741)  12601  12600  12601
(196, 2337, 7595, 10127, 17917)  38171  37440  2401 = 7^{4}
(197, 881, 4111, 5188, 6265)  16641  16640  16641 = 3^{2} · 43^{2}
(197, 1273, 6071, 7736, 9205)  24481  24480  24481
(199, 305, 377, 628, 1309)  2817  2816  2817 = 3^{2} · 313
(199, 376, 831, 1603, 2177)  5185  5184  5185 = 5 · 17 · 61
(199, 673, 2811, 3880, 4751)  12313  12312  12313 = 7 · 1759
(199, 1973, 3157, 7300, 12429)  25057  25056  25057
(203, 1409, 1912, 4931, 8251)  16705  16704  16705 = 5 · 13 · 257
(205, 389, 1093, 1389, 2686)  5761  5760  5761 = 7 · 823
(205, 1629, 7127, 10588, 12421)  31969  31968  31969 = 7 · 4567
(206, 247, 259, 319, 771)  1801  1800  1801
(206, 1331, 4607, 6143, 10955)  23241  23240  23241 = 3 · 61 · 127
(207, 2468, 8021, 10695, 21183)  42573  43120  380689 = 617^{2}
(208, 439, 1007, 2091, 2737)  6481  6480  6481
(209, 707, 2038, 3161, 5407)  11521  11520  11521 = 41 · 281
(209, 1351, 5092, 6859, 12159)  25669  27000  1825201 = 7^{2} · 193^{2}
(211, 339, 436, 985, 1631)  3601  3600  3601 = 13 · 277
(211, 1886, 6077, 8173, 16135)  32481  32480  32481 = 3^{4} · 401
(211, 2306, 7547, 10063, 19915)  40041  40040  40041 = 3^{3} · 1483

ℚ-Homology 7-Spheres M_{w,d}^{7} admitting S-E Structures (cont.)

w = (w_{0}, w_{1}, w_{2}, w_{3}, w_{4})  d  μ  Order of H_{3}(M_{w,d}^{7}, ℤ)
(214, 489, 659, 1849, 2551)  5761  5760  5761 = 7 · 823
(217, 731, 1075, 2752, 4557)  9331  9504  47089 = 7^{2} · 31^{2}
(217, 817, 946, 2795, 4557)  9331  9504  47089 = 7^{2} · 31^{2}
(217, 2795, 7310, 13115, 23219)  46655  46656  47089 = 7^{2} · 31^{2}
(217, 3440, 5375, 14405, 23219)  46655  46656  47089 = 7^{2} · 31^{2}
(220, 237, 1021, 1367, 1477)  4321  4320  4321 = 29 · 149
(221, 317, 805, 1073, 2194)  4609  4608  4609 = 11 · 419
(221, 2416, 5491, 13617, 19328)  41072  43470  5837056 = 2^{8} · 151^{2}
(223, 256, 563, 1041, 1519)  3601  3600  3601 = 13 · 277
(223, 2212, 3539, 9511, 15261)  30745  30744  30745 = 5 · 11 · 13 · 43
(223, 2437, 13071, 18166, 20825)  54721  54720  54721
(223, 4211, 9087, 22606, 31915)  68041  68040  68041
(226, 2245, 12123, 16837, 19307)  50737  50400  12769 = 113^{2}
(226, 3143, 6735, 16837, 23797)  50737  50400  12769 = 113^{2}
(227, 721, 856, 2523, 4099)  8425  8424  8425 = 5^{2} · 337
(227, 901, 1051, 3228, 5179)  10585  10584  10585 = 5 · 29 · 73
(227, 1015, 3496, 4737, 9247)  18721  18720  18721 = 97 · 193
(227, 1241, 4400, 5867, 11507)  23241  23240  23241 = 3 · 61 · 127
(229, 2503, 5461, 13652, 19341)  41185  41184  41185 = 5 · 8237
(235, 323, 334, 891, 1459)  3241  3240  3241 = 7 · 463
(237, 275, 766, 1021, 2023)  4321  4320  4321 = 29 · 149
(238, 1301, 7451, 10289, 11827)  31105  31104  31105 = 5 · 6221
(241, 639, 3275, 4792, 5671)  14617  14616  14617 = 47 · 311
(242, 385, 409, 1419, 2045)  4499  4896  167281 = 409^{2}
(243, 457, 2339, 3280, 3979)  10297  10296  10297 = 7 · 1471
(245, 2434, 5355, 13387, 21175)  42595  46776  1481089 = 1217^{2}
(247, 292, 799, 1583, 2121)  5041  5040  5041 = 71^{2}
(247, 1351, 3439, 8474, 12159)  25669  27000  1825201 = 1361^{2}
(247, 3190, 6871, 17177, 27237)  54721  54720  54721
(250, 275, 393, 917, 1441)  3275  3168  625 = 5^{4}
(250, 1367, 3231, 8077, 11557)  24481  24480  24481
(251, 561, 592, 1963, 3115)  6481  6480  6481
(253, 311, 1096, 1377, 2725)  5761  5760  5761 = 7 · 823
(253, 1507, 6530, 9795, 17831)  35915  39168  10660225 = 5^{2} · 653^{2}
(255, 844, 1519, 4135, 6497)  13249  13248  13249
(259, 285, 407, 950, 1615)  3515  3456  1369 = 37^{2}
(259, 643, 1993, 3536, 6171)  12601  12600  12601
(261, 287, 410, 957, 1653)  3567  3520  1681 = 41^{2}
(261, 491, 1762, 2773, 4795)  10081  10080  10081 = 17 · 593
(262, 291, 331, 883, 1475)  3241  3240  3241 = 7 · 463
(262, 443, 469, 1641, 2371)  5185  5184  5185 = 5 · 17 · 61

ℚ-Homology 7-Spheres M_{w,d}^{7} admitting S-E Structures (cont.)

w = (w_{0}, w_{1}, w_{2}, w_{3}, w_{4})  d  μ  Order of H_{3}(M_{w,d}^{7}, ℤ)
(263, 1699, 3921, 9802, 15421)  31105  31104  31105 = 5 · 6221
(268, 511, 911, 2599, 3777)  8065  8064  8065 = 5 · 1613
(271, 377, 673, 1696, 2745)  5761  5760  5761 = 7 · 823
(275, 729, 2734, 4465, 7927)  16129  16128  16129 = 127^{2}
(277, 441, 2591, 3748, 4465)  11521  11520  11521 = 41 · 281
(299, 325, 1869, 2492, 3115)  8099  7488  169 = 13^{2}
(301, 363, 1294, 2257, 3851)  8065  8064  8065 = 5 · 1613
(301, 3289, 17342, 24219, 44849)  89999  90000  90601 = 7^{2} · 43^{2}
(301, 3887, 12259, 28704, 44849)  89999  90000  90601 = 7^{2} · 43^{2}
(307, 2441, 13120, 18307, 33867)  68041  68040  68041
(309, 349, 2279, 3244, 3901)  10081  10080  10081 = 17 · 593
(311, 495, 2104, 3403, 6001)  12313  12312  12313 = 7 · 1759
(311, 2473, 8037, 18856, 29365)  59041  59040  59041 = 17 · 23 · 151
(316, 1727, 9577, 13345, 24648)  49612  49770  49856 = 2^{4} · 79^{2}
(316, 2041, 6751, 15857, 24648)  49612  49770  49856 = 2^{4} · 79^{2}
(328, 347, 449, 1571, 2347)  5041  5040  5041 = 71^{2}
(339, 383, 1780, 2839, 4957)  10297  10296  10297 = 7 · 1471
(341, 407, 2306, 3459, 6171)  12683  13824  1329409 = 1153^{2}
(356, 387, 2225, 2967, 5547)  11481  11440  7921 = 89^{2}
(357, 388, 2231, 2975, 5593)  11543  11520  9409 = 97^{2}

4. Discussion of the Table

In this section we give a discussion about the representatives listed in the Table. It is easy to notice the existence of twins in the Table. These are rational homology 7spheres with the same degree d, Milnor number μ and order of H_{3}. Twins often occur as adjacent listings with the same w_{0}, but this is not always the case as with twins d = |H_{3}| = 10881, μ = 10880 and w_{0} = 101 and 109, and the twins d = |H_{3}| = 7777 with w_{0} = 141 and w_{0} = 167. Twins may also be members of a larger set, such as the septuplets with d = |H_{3}| = 5761 and μ = 5760. These have w_{0} = 157, 157, 185, 205, 214, 253, 271, respectively. Since twins have the same Milnor number, it is tempting to conjecture that twins correspond to homeomorphic or even diffeomorphic links, but we have no proof as of yet. In fact, except for cases where the order of H_{3} contains no primes of order larger than one in its prime decomposition, we don’t even know that twins have isomorphic H_{3}’s. Notice that the order of H_{3} tends to be quite large varying from 169 = 13^{2} to 17^{12} a number over 500 trillion.

Another interesting fact is that of the 184 rational homology 7-spheres listed in the Table, only 10 have even degree, while the remaining 174 have odd degree, and the degree is even if and only if the order of H_{3} is even. But even more intriguing is the fact that for all 174 rational homology 7-spheres with odd degree, the order |H_{3}| ≡ 1 mod(8). In [BGN4] we construct positive Sasakian structures on homotopy 9-spheres using the rational homology 7-spheres listed in the Table. There we show that the exotic Kervaire sphere can only occur when the degree of the rational homology sphere is even.

Also of interest are invariants of the underlying contact, and almost contact structures. The underlying almost contact structures are classified [Sa] by homotopy classes of maps [M^{7}, SO(8)/U(4)], and Morita [Mo] shows that for Brieskorn spheres this is a function of the Milnor number μ. It seems reasonable that a similar result holds true in our case. There are candidates for this in the table. For example the rational homology 7-spheres with weights w = (196, 2337, 7595, 10127, 17917) and degree d = 38171, and with weights w = (147, 207, 230, 245, 299) and degree d = 1127 both have |H_{3}| = 7^{4}, so they could be diffeomorphic. But they have very different Milnor numbers, namely, 37440 and 1152, respectively, so they could belong to distinct almost contact structures. Similarly there are 4 rational homology 7-spheres with |H_{3}| = 97^{2}, two are twins having the same Milnor number, but the other two have different Milnor numbers both different than the Milnor number of the twins. Moreover, twins probably belong to the same underlying almost contact structures, but could possibly belong to distinct contact structures. It appears that nothing is known beyond homotopy spheres [Us1, Us2] about distinct contact structures within the same underlying almost contact structures.

5. Some Comments on Regular Rational Homology Spheres

In this section we discuss some rational homology spheres that are regular, in particular the homogeneous ones. The following result follows easily from previous work [BG1] together with the well-known classification of del Pezzo surfaces: Proposition 5.1: Let S = (g, ξ, η, Φ) be a regular positive Sasakian structure on a smooth compact 5-manifold M^{5}. Then M^{5} is a rational homology sphere if and only if M^{5} is covered by S^{5} and S is homologous to the standard Sasakian structure with the round metric g_{0}.

It is well-known that the standard Sasakian structure is a homogeneous Sasakian-Einstein structure. Dimension seven is a bit more interesting: Theorem 5.2: Let S = (g, ξ, η, Φ) be a regular positive Sasakian structure on a smooth compact 7-manifold M^{7}. Then M^{7} is a rational homology sphere if and only if it is one of the following:

1. M^{7} = S^{7} and S is homologous to the standard Sasakian structure with the round metric.

2. M^{7} = V_{2}(ℝ^{5}) the Stiefel manifold of 2-frames in ℝ^{5} and S is homologous to the standard homogeneous Sasakian-Einstein structure on V_{2}(ℝ^{5}) [BG1,BG2].

3. M^{7} is a circle bundle over a smooth variety V_{5} of degree 5 in ℂℙ^{6} with a compatible Sasakian structure S.

4. M^{7} is a circle bundle over a smooth variety V_{22} of degree 22 in ℂℙ^{13} with a compatible Sasakian structure S.

Furthermore, M^{7} admits a homogeneous Sasakian-Einstein structure if and only if M^{7} = S^{7} or V_{2}(ℝ^{5}).

Proof: By [BG1] M^{7} is a regular rational homology sphere with a Sasakian structure S if and only if it is the total space of an S^{1} bundle over a smooth projective 3-fold Z with the same rational homology groups as projective space ℂℙ^{3}. Furthermore, S is positive [BGN3] if and only if Z is Fano. Thus, Z must occur on Iskovskikh’s list [Isk] (see Remark 5.3 below) of smooth Fano 3-folds of the first kind, and there are precisely four which have the same rational cohomology groups as ℂℙ^{3}. This gives the four cases above. The last statement follows from Corollary 4.1.3 of [BG2].

Remarks 5.3: (1) Case 4 in Theorem 5.2 has an interesting history. The 3-fold V_{22} was missed by Fano in his original classification of smooth 3-folds with an ample anti-canonical line bundle. It was then found by Iskovskikh [Ish] in his study of Fano’s work, but a mistake was made and not all were found. Mukai and Umemura [MU] (See also [IsPr]) produced a V_{22} that is an equivariant compactification of SL(2, ℂ)/𝕀 that was missed by Iskovskikh. Here 𝕀 is the icosahedral group. Later Prokhorov (see Proposition 4.3.11 of [IsPr]) showed that the Mukai-Umemura V_{22} completes the Fano-Iskovskikh classification of Fano 3-folds. Recently Tian [Ti1,Ti2] showed that there are deformations P_{a} of the Mukai-Umemura V_{22} which do not admit a Kähler-Einstein structure, giving a counterexample to the folklore conjecture that every every compact Kähler manifold with no holomorphic vector fields admits a compatible Kähler-Einstein metric. Thus, the Sasakian circle bundle over P_{a} does not admit a compatible Sasakian-Einstein metric. (2) In the four cases of Theorem 5.2, the corresponding Fano 3-folds are precisely those Fano 3-folds that are almost homogeneous with respect to the group SL(2, ℂ). (See [IsPr], pg 116).

There is a straightforward procedure for finding all rational homology spheres M^{2n+1} that admit a homogeneous Sasakian-Einstein structure. By Theorem 3.2 of [BG1] M^{2n+1} must fibre over a generalized flag manifold G/P, where G is a complex semi-simple Lie group, and P is a parabolic subgroup. In order that M^{2n+1} be a rational homology sphere, it is necessary that G/P have the rational homology of a projective space. Hence, we may restrict ourselves to the case where G is simple and P is maximal parabolic. The procedure for computing the cohomology ring of G/P is outlined in Baston and Eastwood [BE]. All G/P ’s with G simple are realized by crossing out nodes in each Dynkin diagram of G. When P is maximal parabolic only one node is crossed out. The rank of the cohomology groups is determined by the Hasse diagram W 𝔭 which is the coset space W𝔤/W𝔭 where W𝔤 is the Weyl group of the Lie algebra 𝔤 of G, and W𝔭 is the Weyl group of the Levi factor of the Lie algebra 𝔭 of P. Then the cohomology groups of G/P will have the same rank as ℂℙ^{n} if and only if W 𝔭 has precisely one element of length l for each l = 1, ⋯, n. One then needs to check all maximal parabolics for all Dynkin diagrams, and compute the Hasse diagram for each case. There are many cases and repetitions can and do occur. Here we mention the Stiefel manifolds V_{2}(ℝ^{2n+1}) which are circle bundles over the odd quadrics Q_{2n−1} and the homogeneous 3-Sasakian rational homology sphere G_{2}/Sp(1)_{+} (cf. [BGP] and Remark 5.6(2) below). A Gysin sequence or spectral sequence argument shows that

H^{p}(V_{2}(ℝ^{2n+1}), ℤ) ≈ \{ \begin{matrix}ℤ & if p = 0, 4n − 1; \\ ℤ_{2} & if p = 2n; \\ 0 & otherwise. \end{matrix}

5.4  H^{p}(G_{2}/Sp(1)_{+}, ℤ) ≈ \{ \begin{matrix}ℤ & if p = 0, 11; \\ ℤ_{3} & if p = 4, 8; \\ 0 & otherwise. \end{matrix}

We have Proposition 5.5: The Stiefel manifold V_{2}(ℝ^{2n+1}) and G_{2}/Sp(1)_{+} are simply connected rational homology spheres which admit homogeneous Sasakian-Einstein structures. Remarks 5.6: (1) Since V_{2}(ℝ^{2n+1}) can be represented as the link of the quadric hypersurface singularity, 5.3 can be derived from the Milnor-Orlik algorithm described in section 3. (2) There are two non-conjugate Sp(1) subgroups of the exceptional Lie group G_{2}, denoted in [BGP] as Sp(1)_{±}. The quotient G_{2}/Sp(1)_{+} has a homogeneous 3-Sasakian structure, whereas the quotient G_{2}/Sp(1)_{−} does not. It does, however, have a homogeneous Sasakian-Einstein structure, and as homogeneous Sasakian-Einstein manifolds G_{2}/Sp(1)_{−} ≈ V_{2}(ℝ^{7}).

There is an obvious corollary of Theorem 4.2.6 and Proposition 5.4.4 of [BG2], viz.

Corollary 5.7: Let M^{4n+3} be a rational homology sphere that admits a 3-Sasakian homogeneous structure. Then M^{4n+3} is either S^{4n+3}, ℝℙ^{4n+3}, or G_{2}/Sp(1)_{+}.

Appendix

In this appendix, we generalize a result of Johnson and Kollár (Proposition 11 and Corollary 13 of [JK1]) to arbitrary orbifold Fano index I. While this generalization is straightforward, we give more detail than [JK1]. We refer to our previous work [BGN1] and the book [KM] for background material.

Suppose X ⊂ ℙ(w) = ℙ(w_{0}, …, w_{n}) is an irreducible hypersurface in weighted projective space. We assume that X has at worst quotient singularities (at singular points of ℙ(w)) and that X is Fano. We would like to study under what conditions (X, D) is klt whenever D ≡ −αK_{X} where α > \frac{n−1}{n} and D is effective. Under these circumstances X admits a Kähler-Einstein metric of positive scalar curvature [DK,JK1]. We shall prove Theorem A.1: Suppose X ⊂ ℙ(w) = ℙ(w_{0}, …, w_{n}) is a normal Fano variety of index I satisfying

deg(X) < \frac{nw_{0}w_{1}}{(n − 1)I} .

Then there exists ϵ > 0 such that (X, D) is klt for every effective D ≡ −\frac{n−1+ϵ}{n}K_{X}. Proof: By induction on dimension, which reduces to the surface case already handled in [JK1] and [BGN1]. Suppose for a moment that X is a surface with at worst quotient singularities. Johnson and Kollar [JK1] give sufficient conditions for (X, D) to be klt:

1. If D = ∑_{i=1}^{r} α_{i}C_{i} then α_{i} < 1 for all i,

2. for all smooth points x ∈ X, mult_{x}(D) ≤ 1,

3. if P ⊂ X is a singular point and π : Y → X a local finite cover resolving the singularity at P then π^{∗}D has multiplicity at most one at Q = π^{−1}(P). Condition 1 is also necessary, while neither 2 nor 3 is necessary. We will now formulate

analogues of 1,2, and 3 above, for an arbitrary (X, D), designed to guarantee that 1,2, and 3 will hold when we cut X with the appropriate number of hypersurfaces, allowing us to apply the inversion of adjunction formula to conclude that (X, D) is klt.

(i) If D = ∑_{i=1}^{r} α_{i}D_{i} then α_{i} < 1 for all i, and (ii) for all smooth points x ∈ X, mult_{x}(D) ≤ 1, and (iii) suppose x ∈ X is a singular point of X with local group G_{x} and that H_{1}, H_{2}, …, H_{n−1}

are general hypersurfaces through x; then we ask that the intersection number at x, i(x, X · H_{1} · . . . · H_{n−1}; ℙ(w)) satisfies

i(x, X · H_{1} · . . . · H_{n−1}; ℙ(w)) ≤ \frac{1}{|G_{x}|}.

Suppose that conditions i, ii, and iii are satisfied by a divisor D on X and suppose x ∈ X. Choose general hypersurfaces H_{1}, H_{2}, …, H_{n−1} through x and let V = X ∩ H_{1} ∩ … ∩ H_{n−3} with D_{V} = D ∩ V. Consider the pair (V, D_{V}). We see that V is a surface and we will show that (V, D_{V}) satisfies 1,2, and 3 above and hence (V, D_{V}) is klt at x. Using the inversion of adjunction, (X, D) is klt at x as well.

Write D_{V} = ∑ b_{j}C_{j}. Since the hypersurfaces H_{i} are general, we can use Remark 8.2 of [Ful] to see that b_{j} < 1 for all j as each b_{j} is equal to one of the α_{i}. Similarly, the multiplicity of D at x will be preserved under intersection by general hyperplanes and so mult_{x}(D_{V}) ≤ 1 (see [Ful] Corollary 12.4). Finally, under hypothesis iii above, if π : Y → V is a local cover of the quotient singularity at x, we see, using [Ful] 8.3.12, that 3 is satisfied. More precisely, letting y = π^{−1}(x),

mult_{y}(π^{∗}D_{V}) ≤ π^{∗}D_{V} · π^{∗}H_{n−2} · π^{∗}H_{n−1}
≤ |G_{x}|i(x, D_{V} · H_{n−2} · H_{n−1}; ℙ(w))
≤ 1,

the last inequality coming from iii. Thus it is sufficient, in order to prove Theorem 0.1, to verify conditions i, ii, and iii above where D ≡ −\frac{n−1+ϵ}{n}K_{X} is an effective divisor.

Suppose then that we write

D = ∑_{i=1}^{r} α_{i}D_{i}.

We will show that α_{1} < 1, the other cases being identical. Suppose x ∈ D_{1} is a smooth point of D_{1} and X; here we use the fact that X is normal and hence smooth along D_{1}. To simplify notation in what follows, we let E = α_{1}D_{1}. We will assume for simplicity, rearranging the coordinates if necessary, that

x = (x_{0}, …, x_{k}, 0, …, 0).

Consider the hyperplane H_{1} given by z_{n} = 0. If E ⊂ H_{1} then replace H_{1} with H_{2}, given by z_{n−1} = 0 and start over. Assuming that H_{1} meets E properly, write

H_{1} · E = ∑ β_{j}V_{j}.

We then repeat the above procedure, replacing E with H_{1} · E and intersecting with H_{2}. We continue up to and including H_{n−k} given by z_{k+1} = 0. Next consider the hypersurface D_{i} for 1 ≤ i ≤ k defined by

c_{i−1}z_{i−1}^{w_{i}} − c_{i}z_{i}^{w_{i−1}}

where the c_{i} are chosen so that x ∈ D_{i}. The divisors H_{1}, …, H_{n−k}, D_{1}, …, D_{k} cut out the point x set theoretically. Thus we can continue the intersection process above, using the ℚ-divisors D_{i}/min(w_{i−1}, w_{i}), until we obtain a cycle Z of dimension zero. Bounding the degree of Z by the degree of the total intersection class we have

deg(Z) ≤ w_{n} … w_{3} deg(E)
≤ w_{n} … w_{3} deg(D),

since the worst case scenario for the degree of Z occurs when all possible intersections are proper and where no reordering of variables has been necessary; note that the degrees in this formula are relative to O_{ℙ(w)}(1). On the other hand since mult_{x}(H_{i}) ≥ 1 for all i and

\frac{mult_{x}(D_{i})}{min(w_{i−1}, w_{i})} ≥ 1,

it follows by [Ful] Corollary 12.4 that

deg(Z) ≥ mult_{x}(E).

Hence

mult_{x}(E) ≤ w_{n} … w_{3} deg(D).

But

deg(D) = \frac{n − 1 + ϵ}{n ∏_{i=0}^{n} w_{i}} I deg(X),

where deg(X) denotes the homogeneous degree of the polynomial defining X. So if we choose ϵ so that

\frac{n − 1 + ϵ}{n}I deg(X) < w_{0}w_{1}

we see that

α_{1}mult_{x}(D_{1}) = mult_{x}(E) < 1/w_{2} ≤ 1.

Since we have assumed that X and D_{1} are smooth at x it follows that mult_{x}(D_{1}) = 1 and hence α_{1} < 1 as desired.

Next we address property ii. The argument is essentially identical to the verification of property i. In particular, let

x = (x_{0}, x_{1}, …, x_{n});

again we will assume that x_{0}, …, x_{k} are non–zero and that the rest of the coordinates are zero, possibly after reordering the coordinates. Intersecting with the divisors H_{i} and \frac{D_{j}}{w_{j}} as above will show that

mult_{x}(D) < \frac{1}{w_{2}} ≤ 1.

Again, this result holds with the original w_{2} since the initial ordering of the w_{i}’s is increasing.

Finally we establish property iii. We begin with a specific example so that the coordinates are simple and the argument transparent. Suppose x = (0, …, 0, 1) ∈ X. Thus |G_{x}| ≤ w_{n}. To see that

i(x, X · H_{1} · . . . · H_{n−1}; ℙ(w)) ≤ \frac{1}{w_{n}}

we consider the intersection theoretic argument above. In this case, x is cut out by H_{i} = {z_{i} = 0} for 0 ≤ i ≤ n − 1. Thus we never need to intersect with {z_{n} = 0} and consequently instead of

deg(Z) ≤ w_{n} … w_{3} deg(D),

we will find

deg(Z) ≤ w_{n−1} … w_{2} deg(D).

Writing Z = m[x] + Z′ where Z′ is a zero cycle not supported at x, we find, plugging in the value of deg(D), that

m ≤ \frac{1}{w_{n}} .

But i(x, X ·H_{1} ·. . .·H_{n−1}; ℙ(w)) is the minimal intersection number supported on x which one can obtain but cutting out x and thus

i(x, X · H_{1} · . . . · H_{n−1}; ℙ(w)) ≤ \frac{1}{w_{n}}

as desired.

More generally, if P is some other singular point of X, with coordinates (x_{0}, ⋯, x_{n}) then some of the homogeneous coordinates {z_{i}} must be zero. If j is the smallest index such that z_{j} ≠ 0 then |G_{x}| ≤ w_{j}. The argument then procedes as in case ii, the w_{j} in the denominator coming from taking the worst case scenario in computing the intersection product.

Note that the most difficult of the three conditions to satisfy is definitely the third. This is the same as in the surface case but the higher dimensional case is more difficult to deal with as, in the surface case, condition (iii) was dealt with using the inversion of the adjunction formula. Again as in the surface case, we can weaken the condition somewhat if X does not contain certain planes.

Acknowledgments: We would like to thank Jennifer Johnson and János Kollár for making their computer list available, as well as for valuable comments and their interest in our work. We would also like to thank Alexi Kobalev and Gang Tian for helpful discussions. The second author would like to thank Max-Planck-Institute für Mathematik in Bonn for support and hospitality during the summer of 2001 when this paper was being completed.

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Department of Mathematics and Statistics August 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