Math. Ann. 325 (2003), 485-524  Mathematische Annalen

On the Geometry of Sasakian-Einstein

5-Manifolds

Charles P. Boyer Krzysztof Galicki Michael Nakamaye

Abstract: On simply connected five manifolds Sasakian-Einstein metrics coincide with Riemannian metrics admitting real Killing spinors which are of great interest as models of near horizon geometry for three-brane solutions in superstring theory [KW]. We expand on the recent work of Demailly and Kollár [DK] and Johnson and Kollár [JK1] who give methods for constructing Kähler-Einstein metrics on log del Pezzo surfaces. By [BG1] circle V-bundles over log del Pezzo surfaces with Kähler-Einstein metrics have Sasakian-Einstein metrics on the total space of the bundle. Here these simply connected 5-manifolds arise as links of isolated hypersurface singularities which by the well known work of Smale [Sm] together with [BG3] must be diffeomorphic to S^{5}#l(S^{2}×S^{3}). More precisely, using methods from Mori theory in algebraic geometry we prove the existence of 14 inequivalent Sasakian-Einstein structures on S^{2}×S^{3} and infinite families of such structures on #l(S^{2}×S^{3}) with 2≤l≤7. We also discuss the moduli problem for these Sasakian-Einstein structures.

0. Introduction

Surprisingly little is known about complete Einstein metrics on compact 5-manifolds. Until now one could list only two constructions of such metrics. The more recent one was developed by Böhm [Bö] who obtained cohomogeneity one complete Einstein metrics on S^{5} and S^{2} ×S^{3}. On the other hand, the well-known irreducible homogeneous Einstein metrics on these two spaces are the simplest examples of what is called a Sasakian-Einstein structure. Roughly, a Sasakian structure on a manifold is a metric contact structure whose Reeb vector field is a unit Killing field and whose transverse geometry is Kähler. In fact, any complex surface whose metric is Kähler-Einstein and of positive scalar curvature admits a unique simply connected circle bundle which is canonically Sasakian-Einstein. Since all such del Pezzo surfaces Z admitting Kähler-Einstein metrics are known [Siu,Ti1-3, TY] one gets the list and the classification statement of the first theorem of the introduction. Purely in the context of Einstein metrics this construction is due to Kobayashi [Be] whereas the realization that S is actually a 5-manifold with real Killing spinors and, hence, Sasakian-Einstein space came later in the work of Friedrich and Kath [FK]. More generally, the importance of Killing spinors was already realized in 1980 in the context of the spectrum of that Dirac operator [F]. A classification of Riemannian manifolds admitting real Killing spinors was finally obtained by Bär [Bä] who observed that the real Killing spinors on M correspond to the parallel spinors on C(M) = (ℝ^{+} × M, dt^{2} + t^{2}g) the metric cone on M. This simple fact allowed for an elegant formulation of the problem in the powerful language of holonomy groups.

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. 2000 Mathematics Classification: 53C25, 53C12, 14E30

The physicists’ interest in five and seven-dimensional manifolds admitting real Killing spinors dates back to the early eighties, where Kaluza-Klein models played a central role in the supergravity theory. Today we witness a renewed interest in these manifolds in the context of p-brane solutions in superstring theory. These so-called p-branes, “near the horizon” are modeled by the pseudo-Riemannian geometry of the product adS_{p+2} × M, where adS_{p+2} is the (p + 2)-dimensional anti-de-Sitter space (a Lorentzian version of a space of constant sectional curvature) and (M, g) is a Riemannian manifold of dimension d = D − p − 2. Here D is the dimension of the original supersymmetric theory. In the most interesting cases of M2-branes, M5-branes, and D3-branes D equals to either 11 (Mp-branes of M-theory) or 10 (Dp-branes in type IIA or type IIB string theory). String theorists are particularly interested in those vacua of the form adS_{p+2} × M that preserve some residual supersymmetry. It turns out that this requirement imposes constraints on the geometry of the Einstein manifold M which is forced to admit real Killing spinors. Depending on the dimension d, the possible geometries of M are as follows: nearly Kähler for d = 6, weak G_{2} holonomy for d = 7, Sasakian-Einstein for d = 2k + 1, and 3-Sasakian for d = 4k + 3 [AFHS, MP]. Furthermore, given a p-brane solution of the above type, the interpolation between adS_{p+2} × M and ℝ^{p,1} × C(M) leads to a conjectured duality between supersymmetric background of the form adS_{p+2} × M and a (p + 1)-dimensional superconformal field theory of n coincident p-branes located at the conical singularity of the ℝ^{p,1} × C(M) vacuum. This is a generalized version of the Maldacena’s conjecture.

In the case of D3-branes of string theory the relevant near horizon geometry is that of adS_{5} ×M, where M is a Sasakian-Einstein 5-manifold. The D3-brane solution interpolates between adS_{5} ×M and ℝ^{3,1} ×C(M), where the cone C(M) is a Calabi-Yau threefold. In its original version the Maladacena conjecture (also known as AdS/CFT duality) states that the ’t Hooft large n limit of N = 4 supersymmetric Yang-Mills theory with gauge group SU(n) is dual to type IIB superstring theory on adS_{5} × S^{5} [Ma, Wi]. This conjecture was further examined by Klebanov and Witten [KW] for the type IIB theory on adS_{5} × T^{1,1}, where T^{1,1} is the other homogeneous Sasakian-Einstein 5-manifold T^{1,1} = S^{2} ×S^{3} and the Calabi-Yau 3-fold C(T^{1,1}) is simply the quadric cone in ℂ^{4}. Using the well-known fact that C(T^{1,1}) is a Kähler quotient of ℂ^{4} (or, equivalently, that S^{2} × S^{3} is a Sasakian-Einstein quotient of S^{7}), a dual super Yang-Mills theory was proposed, representing D3-branes at the conical singularities. In the framework of D3-branes and the AdS/CFT duality the question of what are all the possible near horizon geometries M and C(M) might be of importance. Until this year, the only known examples of such geometries were few and they were exactly the circle bundles over the del Pezzo surfaces (see the comment of Yau in [Y]). This has drastically changed now as we shall show in our paper.

Recently Demailly and Kollár have developed some new techniques to study the existence of Kähler-Einstein metrics on compact Fano orbifolds [DK], and applied their methods to prove the existence of Kähler-Einstein metrics on three log del Pezzo surfaces. Following this, two of the present authors [BG3] showed how one can use the results of [DK] to obtain new Sasakian-Einstein 5-manifolds. Later Johnson and Kollár [JK1] discovered a more efficient way to treat the algebraic equation involved, and with the aid of a computer program gave many more examples of log del Pezzo surfaces in weighted projective 3-spaces, including one infinite series example. All of the [JK1] examples have Fano index equal to one. In the current paper we extend the results of [JK1] to the case of higher index as well as use these results together with the results of [JK1] to construct a plethora of Sasakian-Einstein metrics in dimension five. This is accomplished by realizing the compact simply connected 5-manifolds as links of isolated hypersurface singularities given by weighted homogeneous polynomials in ℂ^{4}. The projectivization then gives the log del Pezzo surfaces Z and proving the existence of a Kähler-Einstein metric Z is then tantamount to proving the existence of an Sasakian-Einstein metric on the 5-manifold [BG1]. It turns out that from the work of Smale [Sm] one can deduce that these 5-manifolds are all of the form S_{l} = #l(S^{2} × S^{3}) for 1 ≤ l ≤ 7. We also show that the moduli of Sasakian-Einstein structures on #l(S^{2} × S^{3}) is exceedingly rich. Recall [BG1] that a Sasakian manifold is quasi-regular if the leaves of the foliation generated by the unit Killing field ξ are all circles, and it is regular if this foliation is a locally trivial fibration. In the latter case the space of leaves Z is a smooth Kähler manifold while generally in the former Z has orbifold singularities. First it is known [FK,BG1] that the regular Sasakian-Einstein structures on #l(S^{2} × S^{3}) satisfy

Theorem [FK,BG1]: Let S_{l} = S^{5}#l(S^{2} × S^{3}).

1) For each l = 0, 1, 3, 4, there is precisely one regular Sasakian-Einstein structure on S_{l}.

2) For each 5 ≤ l ≤ 8 there is a 2(l−4) complex parameter family of inequivalent regular Sasakian-Einstein structures on S_{l}.

3) For l = 2 or l ≥ 9 there are no regular Sasakian-Einstein structures on S_{l}.

The first non-regular Sasakian-Einstein structures on any 5-manifold were given in [BG3]. Here the two of the present authors showed the existence of two inequivalent nonregular Sasakian-Einstein structures on S^{2} × S^{3}, and one non-regular Sasakian-Einstein structure on #2(S^{2} ×S^{3}). The latter was the first example of any Sasakian-Einstein metric on #2(S^{2} × S^{3}). In the present work we use the methods presented in [BG3] as well as those in [JK1] to greatly expand the list of non-regular Sasakian-Einstein structures on simply connected 5-manifolds. Explicitly we prove the following:

Theorem A:

1) S^{2} × S^{3} admits 14 inequivalent non-regular Sasakian-Einstein structures.

2) #2(S^{2} × S^{3}) admits two distinct 1-complex parameter families plus 21 inequivalent non-regular Sasakian-Einstein structures.

3) #3(S^{2} × S^{3}) admits two distinct 2-complex parameter families, and four distinct 1-complex parameter families, and one countably infinite family of inequivalent nonregular Sasakian-Einstein structures.

4) #4(S^{2} × S^{3}) admits two distinct 3-parameter complex families, one 2-complex parameter family, two countably infinite complex families, and two distinct inequivalent non-regular Sasakian-Einstein structures.

5) #5(S^{2} × S^{3}) admits one 4-complex parameter family, one 3-complex parameter family, and two countably infinite families of inequivalent non-regular Sasakian-Einstein structures.

6) #6(S^{2} × S^{3}) admits one 5-complex parameter family, one 3-complex parameter family, and two countably infinite families of inequivalent non-regular Sasakian-Einstein structures.

7) #7(S^{2} × S^{3}) admits a countably infinite series of 5-complex parameter families of inequivalent non-regular Sasakian-Einstein structures.

Furthermore, inequivalent Sasakian structures correspond to inequivalent Riemannian metrics.

In light of these results an outstanding question is:

Question: Are there any Sasakian-Einstein structures on #l(S^{2} × S^{3}) for l ≥ 9?

It is well-known for the regular case that the second Betti number of any smooth del Pezzo surface is less than or equal to 9. However, in stark contrast to this one can construct log del Pezzo surfaces with any desired second Betti number, but as shown in Theorem 4.5 below the methods used in this paper for proving the existence of Kähler-Einstein metrics apply only to log del Pezzo surfaces with second Betti number less than or equal to 10. Indeed, we shall show in a forthcoming work that our methods prove the existence of a 7-complex parameter family of Kähler-Einstein metrics on a log del Pezzo surface whose second Betti number is 10; hence, proving the existence of a 7-complex parameter family of Sasakian-Einstein structures on #9(S^{2} × S^{3}).

Acknowledgments: We would like to thank Alex Buium, for several discussions, and János Kollár for many valuable e-mail communications as well as his interest in our work. We also thank Gang Tian for e-mail communications, and Gueo Grantcharov for telling us about Pontecorvo’s result [Pon] which allowed us to sharpen some results on the moduli spaces which appeared in the first version of this manuscript. We owe special thanks to Jennifer Johnson for letting us use the original C program employed in [JK1]. Finally, we would like to thank Evan Thomas for helping us modify this program and make it more efficient in our calculations for the higher index case.

1. The Transverse Geometry of a Sasakian Manifold

In this section we study the transverse geometry of the Riemannian foliation F_{ξ} of a Sasakian manifold M. Good references for the transverse geometry of foliations are [Ton] and [Mol]. First we recall some basic notation and definitions. Let (M, g) be a Riemannian manifold, ∇ the Levi-Civita connection of g, and let R(X, Y) denote the Riemann curvature tensor of ∇.

Definition-Proposition 1.1: A Riemannian manifold (M, g) is called a Sasakian manifold if any one, hence all, of the following equivalent conditions hold:

(i) There exists a Killing vector field ξ of unit length on M so that the tensor field Φ of type (1, 1), defined by Φ(X) = ∇_{X}ξ, satisfies the condition

(∇_{X}Φ)(Y) = g(ξ, Y)X − g(X, Y)ξ

for any pair of vector fields X and Y on M.

(ii) There exists a Killing vector field ξ of unit length on M so that the Riemann curvature satisfies the condition

R(X, ξ)Y = g(ξ, Y)X − g(X, Y)ξ,

for any pair of vector fields X and Y on M.

(iii) The metric cone on M, (C(M), ḡ) = (ℝ_{+} × M, dr^{2} + r^{2}g) is Kähler with Kähler form d(rη).

We shall refer to the quadruple (ξ, η, Φ, g) as a Sasakian structure on M. Associated to any Sasakian structure is the characteristic foliation F_{ξ} which is the one dimensional foliation generated by the unit vector field ξ, and whose leaves are geodesics with respect to the Sasakian metric g which is bundle-like with respect to F_{ξ}.

Let (ξ, η, Φ, g) be a Sasakian structure on M, and consider the contact subbundle D = ker η. There is an orthogonal splitting of the tangent bundle as

1.2  T M = D ⊕ L_{ξ},

where L_{ξ} is the trivial line bundle generated by the Reeb vector field ξ. The contact subbundle D is just the normal bundle to the characteristic foliation F_{ξ} generated by ξ. It is naturally endowed with both a complex structure J = Φ|D and a symplectic structure dη. Hence, (D, J, dη) gives M a transverse Kähler structure with Kähler form dη and metric g_{D} defined by

1.3  g_{D}(X, Y) = dη(X, JY)

which is related to the Sasakian metric g by

g = g_{D} ⊕ η ⊗ η.

Recall [Ton] that a smooth p-form α on M is called basic if

1.4  ξ⌋α = 0,  L_{ξ}α = 0,

and we let Λ_{B}^{p} denote the sheaf of germs of basic p-forms on M, snd by Ω_{B}^{p} the set of global sections of Λ_{B}^{p} on M. The sheaf Λ_{B}^{p} is a module under the ring, Λ_{B}^{0}, of germs of smooth basic functions on M. We let C_{B}^{∞}(M) = Ω_{B}^{0} denote global sections of Λ_{B}^{0}, i.e. the ring of smooth basic functions on M. Since exterior differentiation preserves basic forms we get a de Rham complex

1.5  ⋯ ⟶Ω_{B}^{p} ⟶^{d}Ω_{B}^{p+1}⟶ ⋯

whose cohomology H_{B}^{∗}(F_{ξ}) is called the basic cohomology of (M, F_{ξ}). The basic cohomology ring H_{B}^{∗}(F_{ξ}) is an invariant of the foliation F_{ξ} and hence, of the Sasakian structure on M. It is related to the ordinary de Rham cohomology H^{∗}(M, ℝ) by the long exact sequence [Ton]

1.6 ⋯ ⟶H_{B}^{p}(F_{ξ})⟶H^{p}(M, ℝ)⟶^{j_{p}}H_{B}^{p−1}(F_{ξ})⟶^{δ}H_{B}^{p+1}(F_{ξ})⟶ ⋯

where δ is the connecting homomorphism given by δ[α]_{B} = [dη ∧ α]_{B} = [dη]_{B} ∪ [α]_{B}, and j_{p} is the composition of the map induced by ξ⌋ with the well known isomorphism H^{r}(M, ℝ) ≈ H^{r}(M, ℝ)^{S^{1}} where H^{r}(M, ℝ)^{S^{1}} is the S^{1}-invariant cohomology defined from the S^{1}-invariant r-forms Ω^{r}(M)^{S^{1}}. Here we denote cohomology classes in H_{B}^{p}(F_{ξ}) by [·]_{B} in order to distinquish them from the ordinary cohomology classes. We also note that dη is basic even though η is not.

Next we exploit the fact that the transverse geometry is Kähler [ElK]. Let D_{ℂ} denote the complexification of D, and decompose it into its eigenspaces with respect to J, that is, D_{ℂ} = D^{1,0} ⊕ D^{0,1}. Similarly, we get a splitting of the complexification of the sheaf Λ_{B}^{1} of basic one forms on M, namely

Λ_{B}^{1} ⊗ ℂ = Λ_{B}^{1,0} ⊕ Λ_{B}^{0,1}.

We let Λ_{B}^{p,q} denote the sheaf of germs of basic forms of type (p, q), and as in the usual case there is a splitting

1.7  Λ_{B}^{r} ⊗ ℂ = ⨁_{p+q=r} Λ_{B}^{p,q},

as well as the basic Dolbeault complex

1.8  0⟶Λ_{B}^{p,0} ⟶^{¯∂}Λ_{B}^{p,1} ⟶^{¯∂} ⋯ ⟶Λ_{B}^{p,n}⟶0,

together with its basic Dolbeault cohomology groups H_{B}^{p,q}(F_{ξ}). Most of the usual results about Kähler geometry carry over to transverse Kähler geometry [ElK]. In particular, we have the following Proposition which follows from transverse Kähler geometry:

Proposition 1.9: Let (M, ξ, η, Φ, g) be a compact Sasakian manifold of dimension 2n+1. Then we have

(1) H_{B}^{n,n}(F_{ξ}) ≈ H_{B}^{2n}(F_{ξ}) ≈ ℝ.

(2) The class [dη]_{B} ∈ H_{B}^{1,1}(F_{ξ}) is nontrivial.

(3) H_{B}^{p,p}(F_{ξ}) > 0.

(4) H_{B}^{2p+1}(F_{ξ}) has even dimension.

(5) H^{1}(M, ℝ) ≈ H_{B}^{1}(F_{ξ}).

(6) H_{B}^{r}(F_{ξ}) = ⨁_{p+q=r} H_{B}^{p,q}(F_{ξ}).

(7) Complex conjugation induces an anti-linear isomorphism H_{B}^{p,q}(F_{ξ}) ≈ H_{B}^{q,p}(F_{ξ}).

(8) If ω is a closed real (1, 1) form such that ω ∈ [dη]_{B}, then there exists a smooth basic function ϕ such that ω = dη + i∂ ¯∂ϕ.

Proof: (1): The exact sequence 1.6 gives an isomorphism H^{2n+1}(M)^{{j_{2n+1} \atop ≈}} H_{B}^{2n}(F_{ξ}). (2): dη is a (1,1) form by 1.3, and since η ∧ (dη)^{n} is a volume form on M, the isomorphism j_{2n+1} implies that dη defines a nontrivial class in H_{B}^{2}(F_{ξ}). (3) follows since [dη]_{B} cups to a non-trivial element in H_{B}^{n,n}(F_{ξ}). (4): This follows from transverse Hodge theory in the usual way [ElK].

To prove (5) we notice that the beginning of the exact sequence 1.6 is

0⟶H_{B}^{1}(F_{ξ})⟶H^{1}(M, ℝ)⟶^{j_{1}}ℝ⟶^{δ_{0}}H_{B}^{2}(F_{ξ}).

By (2) δ_{0} is injective, so {0} = ker δ_{0} = im j_{1}; hence, j_{1} is the zero map. (6) and (7) follow from (1) and Theorem 3.4.6 of [ElK], and (8) is Proposition 3.5.1 of [ElK].

Next we define the transverse Ricci tensor Ric_{g}^{T} of g to be the Ricci tensor of g_{D}. It is related to the Ricci tensor Ric_{g} of g by

1.10a  Ric_{g}^{T} = Ric_{g}|_{D×D} + 2g|_{D×D}.

Similarly, we define the Ricci form ρ_{g} and transverse Ricci form ρ_{g}^{T} by

1.10b  ρ_{g}(X, Y) = Ric_{g}(X, ΦY),  ρ_{g}^{T} (X, Y) = Ric_{g}^{T} (X, ΦY)

for X, Y smooth sections of D. It is easy to check that these are anti-symmetric of type (1, 1) and are related by

1.10c  ρ_{g}^{T} = ρ_{g} + 2dη.

Now the contact subbundle D is a complex vector bundle and thus has a first Chern class c_{1}(D) ∈ H^{2}(M, ℤ). Consider the long exact sequence 1.6 together with the natural map H^{2}(M, ℤ)⟶H^{2}(M, ℝ) whose kernel is the torsion part of H^{2}(M, ℤ). From (5) of Proposition 1.9 we have

1.11  H^{2}(M, ℤ) ; ↓ ; 0⟶ℝ⟶^{δ}H_{B}^{2}(F_{ξ})⟶^{ι_{∗}}H^{2}(M, ℝ)⟶H^{1}(M, ℝ)⟶ · · · .

As in 1.6 the map δ is given by δ(c) = c[dη] where c ∈ ℝ. Now on a Sasakian manifold the vector bundle D^{1,0} is holomorphic with respect to the CR-structure, so we can compute the free part of c_{1}(D) = c_{1}(D^{1,0}) from the transverse Kähler geometry in the usual way. That is c_{1}(D) can be represented by a basic real closed (1, 1)-form ρ_{B}. The class c_{1}^{B} = [ρ_{B}] ∈ H_{B}^{2}(F_{ξ}) is independent of the transverse metric and basic connection used to compute it, and depends only on the foliated manifold (M, F_{ξ}) with its CR-structure. It is described in [ElK] and called the basic first Chern class of D there. Alternatively, we can think of c_{1}^{B} as the negative of the first Chern class of the “transverse canonical bundle” K = (Λ_{B}^{1,0})^{n} of M.

We now consider deformations of the Sasakian structure which fix the basic first Chern class c_{1}^{B}. We do this by considering deformations that fix the foliation F_{ξ}, in fact fix the characteristic vector field ξ. Let η_{t} be a continuous one parameter family of real1-forms obtained by adding to η a continuous family of basic 1-forms ζ_{t} so that η_{t} = η +ζ_{t} satisfies the conditions

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

This last non-degeneracy condition implies that η_{t} is a contact form on M for all t ∈ [0, 1]. Then by Gray’s Stability Theorem η_{t} belongs to the same underlying contact structure as η. Moreover, since ζ_{t} is basic ξ is the Reeb (characteristic) vector field associated to η_{t} for all t. Now let us define

1.13  \begin{aligned}Φ_{t} & = Φ − ξ ⊗ ζ_{t} ∘ Φ, \\ g_{t} & = dη_{t} ∘ (id ⊗ Φ_{t}) + η_{t} ⊗ η_{t}. \end{aligned}

In [BG3] it was proved that for each t ∈ [0, 1], (ξ, η_{t}, Φ_{t}, g_{t}) defines a Sasakian structure on M associated to the foliation F_{ξ} and belonging to the same underlying contact structure as η. These new Sasakian structures correspond to a different splitting of the tangent bundle, namely

1.14  T M = L_{ξ} ⊕ D_{t}

which is orthogonal with respect to the metric g_{t}, where D_{t} = ker η_{t}.

Given a Sasakian structure (ξ, η, Φ, g) on a manifold M, we define 𝔉(ξ) to be the family of all Sasakian structures obtained by the deformations above. We are now ready for

Definition 1.15: Two Sasakian structures S = (ξ, η, Φ, g) and S′ = (ξ′, η′, Φ′, g′) on M are said to be homologous if ξ′ = ξ and [dη]_{B} = [dη′]_{B} ∈ H_{B}^{1,1}(F). In this case we also say that the Sasakian metrics g and g′ are homologous.

Notice that if L_{ξ′} = L_{ξ} we can always scale the Sasakian structure by choosing ξ′ = ξ, so there is no loss in generality by choosing the characterstic vector fields to coincide. We have

Proposition 1.16: Any two Sasakian structures in 𝔉(ξ) are homologous and the class c_{1}^{B} ∈ H_{B}^{1,1}(F_{ξ}) ⊂ H_{B}^{2}(F_{ξ}) depends only on the family 𝔉(ξ).

Proof: The first statement follows from (8) of Proposition 1.9, while the second statement follows from the fact that for all t ∈ [0, 1] the complex vector bundles (D_{t}, Φ_{t}) are isomorphic. The isomorphism between D and D_{t} is given by the map

𝕀 − ξ ⊗ ζ_{t} : T M⟶T M,

and the induced map on the exterior bundle ΛD is

𝕀 − ζ_{t} ⊗ ξ

which is the identity on basic forms.

Now we wish to consider the Sasakian analogue of the Calabi problem. Its solution essentially follows from the ‘transverse Yau Theorem’ given by El Kacimi-Alaoui [ElK]:

Theorem 1.17[ElK]: If c_{1}^{B} is represented by a real basic (1, 1) form ρ^{T} , then it is the Ricci curvature form of a unique transverse Kähler form ω^{T} in the same basic cohomology class as dη.

This is given in local CR-coordinates (z_{i}, z̄_{i}, x) on M by solving the “transverse Monge-Ampere equation”

1.18  \frac{det(g_{ij̄}^{T} + ϕ_{ij̄})}{det(g_{ij̄}^{T})} = e^{−kϕ+F} , g_{ij̄}^{T} + ϕ_{ij̄} > 0,

for the k = 0 case. Here g^{T} is the transverse metric, ϕ and F are real basic functions, and ϕ_{ij̄} are the components of i∂ ¯∂ϕ = dζ_{t} with respect to the transverse coordinates (z_{i}, z̄j̄). Then by 1.10c this translates for Sasakian geometry to:

Thereom 1.19: Let (M, ξ, η, Φ, g) be a Sasakian manifold whose basic first Chern class c_{1}^{B} is represented by the real basic (1, 1) form ρ, then there is a unique Sasakian structure (ξ, η_{1}, Φ_{1}, g_{1}) ∈ 𝔉(ξ) homologous to (ξ, η, Φ, g) such that ρ_{g_{1}} = ρ − 2dη_{1} is the Ricci form of g_{1}, and η_{1} = η + ζ_{1}, with ζ_{1} = \frac{1}{2}d^{c}ϕ. The metric g_{1} and endomorphism Φ_{1} are then given by 1.12.

Notice that the Ricci forms are related by

1.20  ρ = ρ_{g} + \frac{1}{2}dd^{c}(F − ϕ).

Next we discuss a positivity requirement on the basic Chern class of a compact Sasakian manifold.

Definition 1.21: A Sasakian manifold M is said to be positive if its basic first Chern class c_{1}^{B} can be represented by a positive definite (1, 1)-form.

Notice that a positive Sasakian manifold does not necessarily have a metric with positive Ricci curvature; however, Theorem 1.19 does imply:

Proposition 1.22: A Sasakian manifold (M, g, ξ, η, Φ) is positive if and only if there is a Sasakian metric g′ homologous to g whose Ricci curvature satisfies the bound Ric_{g′} > −2. In particular, a complete positive Sasakian manifold is compact with finite fundamental group.

Since a Sasakian-Einstein manifold necessarily has positive Ricci tensor, its Sasakian structure is necessarily positive. We are interested in sufficient conditions on a Sasakian manifold that guarentee the existence of a Sasakian-Einstein structure. These conditions are algebraic geometric in nature, so we need to impose another condition on the Sasakian structure, namely that it is quasi-regular. Recall that the toral rank or just rank defined [BG2] is the dimension of the closure of the characteristic foliation and denoted by rk(M). For a Sasakian manifold M^{2n+1} of dimension 2n + 1 we have 1 ≤ rk(M) ≤ n + 1. The case rk(M) = 1 corresponds to the quasi-regular case, and if rk(M) > 1 there are infinitely many rank one structures that are close in an appropriate sense [BG2]. In the remainder of this paper we are essentially interested in the rank one case only. Now the aforementioned algebraic geometric conditions which imply the existence of a Kähler-Einstein metric are described in the works of Nadel [Na] and Demailly and Kollár [DK]. The main point is that the obstructions for finding a solution to the Monge-Ampere equations 1.18 involves the non-triviality of certain multiplier ideal sheaves associated with effective canonical ℚ-divisors on the space of leaves Z. Consequently, if one can show that these multiplier ideal sheaves coincide with the full structure sheaf, one obtains the existence of a Kähler-Einstein metric. Equivalently, and this is the approach of Johnson and Kollár [JK1], this is given in the language of Mori theory by the appellation Kawamata log terminal, whose precise technical definition is given in 2.3 below. It is important to realize that these conditions are only sufficient conditions for the existence of Kähler-Einstein metrics. Indeed, they do not hold in the case of complex projective space. We can easily reformulate the Demailly-Kollár result in terms of Sasakian geometry as

Theorem 1.23: Let (S, g, ξ) be a rank 1 positive Sasakian manifold and let Z denote the space of leaves of the characteristic foliation, and K_{Z} be its canonical bundle. Suppose further that for some ϵ > 0 and every effective ℚ-divisor D on Z numerically equivalent to −K_{Z} the pair (Z, \frac{n+ϵ}{n+1}D) is klt. Then there is a Sasakian-Einstein metric g′ on M homologous to g.

Proof: By Theorem 2.1 of [BG1] Z is a ℚ-factorial Fano variety,and by Demailly and Kollár [DK] as stated in Theorem 7 of [JK1] Z has a Kähler-Einstein metric. Thus, again by (iv) of Theorem 2.1 of [BG1] S has a Sasakian-Einstein metric.

Finally we recall [BG1] that the orbifold Fano index or just index of a positive rank 1 Sasakian manifold S is the largest positive integer I such that \frac{1}{I}K_{Z} is an element of Pic^{orb}(Z) where K_{Z} is the canonical V-bundle on Z. Thus, the index is just the generalization to the orbifold category of the usual notion of Fano index; however, it is important to notice that the index I is an invariant of the Sasakian structure on S; in fact, we have

Proposition 1.25: The orbifold Fano index I is an invariant of the family of Sasakian structures 𝔉(ξ).

2. The Existence of Kähler-Einstein Metrics

In this section we quickly review the basic definitions and theorems from minimal model theory; all of this can be found in [KMM]. Throughout this section X will denote a normal projective variety defined over an algebraically closed field k of characteristic zero. By a variety we mean an integral separated scheme of finite type over k. Since X is normal, codim(X_{sing}, X) ≥ 2, where X_{sing} denotes the singular locus of X. The sheaf of Kähler differentials Ω_{X} is locally free of rank dim X on X_{reg} := X\X_{sing}. This gives a well-defined invertible sheaf ⋀^{dim X} Ω_{X_{reg}} on X_{reg}.

Definition 2.1: A canonical divisor K_{X} is any Weil divisor with

O_{X_{reg}}(K_{X}) ≃ ⋀^{dim X} Ω_{X_{reg}}.

K_{X} is well defined modulo linear equivalence since codim(X_{sing}, X) ≥ 2. Because singularities are essential in the study of minimal models, it is important to note the distinction between Weil divisors and Cartier divisors.

Definition 2.2: A Weil divisor D ⊂ X is called ℚ–Cartier if some integral multiple mD is a Cartier divisor, i.e. if O_{X}(mD) is invertible. We say that X is ℚ–factorial (or that X has at worst ℚ–factorial singularities) if all Weil divisors are ℚ–Cartier.

There is a certain class of singularities called “log–terminal” which play an especially prominent role in the minimal model program.

Definition 2.3: Let X be a normal variety and let Δ = ∑ a_{i}E_{i} be an effective ℚ–Weil divisor such that K_{X} +Δ is ℚ–Cartier. Assume 0 ≤ a_{i} < 1 and that the E_{i} are all distinct. Then the pair (X, Δ) has only log–terminal singularities (also called klt or Kawamata logterminal) if there exists a resolution of singularities π : Y → X such that the union of the π–exceptional locus and π^{−1}(∪E_{i}) is a normal crossing divisor and

K_{Y} ≡ π^{∗}(K_{X} + Δ) + ∑ a_{i}F_{i}

with a_{i} > −1 for all π–exceptional F_{i} (here ≡ denotes numerical equivalence). If a_{i} ≥ −1 then the pair (X, Δ) is said to have log–canonical singularities. In the case where Δ = 0, X is said to have has log–terminal (or log–canonical) singularities. When Δ = 0 and a_{i} > 0 (resp. a_{i} ≥ 0) X is said to have terminal singularities (resp. canonical singularities).

Note that Kollár and Mori, [KM] Definition 2.34, give a different definition of klt singularities, one which does not require the “boundary divisor” Δ to be effective. However, in applications one almost always needs the effectivity hypothesis in order to apply the machinery of the minimal model program. All varieties we encounter will have mild quotient singularities and hence will be ℚ–factorial.

Definition 2.4: Suppose L is a line bundle on a variety X. We write

BS(L) = {x ∈ X : s(x) = 0 for all sections s ∈ H^{0} (X, L)}

for the base locus of L. If I ⊂ O_{X} is an ideal sheaf then we will denote by BS(L ⊗ I) the base locus of those sections σ ∈ H^{0}(X, L) whose zero locus contains the zero locus of I.

We will be interested in the case where X is a projective surface with mild quotient singularities. We will assume that −K_{X} is an ample ℚ–Cartier divisor and we will be interested in showing that for some ϵ > 0 and every effective ℚ–divisor D ∈ | − K_{X}|, the pair (X, \frac{2+ϵ}{3}D) is klt. In particular this is certainly true if the pair (X, D) is log–canonical for all choices of D. Checking whether or not (X, D) is log–canonical will be intricate at singular points of X and we begin by reducing this condition to a much simpler one on a smooth finite cover of X.

We begin by noting that by [KM] Proposition 4.18, the variety X itself is klt at a singular point x if and only if X has a quotient singularity at x. In order to check whether or not the pair (X, D) is klt at x we need to consider a local cover

π : (ℂ^{2}, 0) → (X, x),

where the map π is given by taking the quotient via a finite group action. We let U ⊂ ℂ^{2} denote an open subset on which π is defined and let

p : S → U

be the blow–up of U at 0 with exceptional divisor E. We will show that if (S, p_{∗}^{−1}(π^{∗}D)+E) is log–canonical near E then the pair (X, D) is log–canonical at x. Shokurov’s inversion of adjunction ([KM] Theorem 5.50) states that the pair (S, p_{∗}^{−1}(π^{∗}D) + E) is log–canonical near E if (E, p_{∗}^{−1}π^{∗}(D)|E) is log–canonical. But E ≃ ℙ^{1} and so this is an easy condition to check, satisfied provided the round–up of p_{∗}^{−1}π^{∗}(D)|E is a sum of distinct points, each with multiplicity one.

By [KM] Lemma 2.30 and Proposition 5.20.4 together with the inversion of adjunction, one derives

Lemma 2.5: Suppose that mult_{0}(π^{∗}D) ≤ 2 and that (E, p_{∗}^{−1}π^{∗}D + E) is log–canonical. Then (X, D) is log–canonical at x.

We sketch here the details of Lemma 2.5. Suppose that (S, p_{∗}^{−1}(π^{∗}D) + E) is log– canonical near E and consider the following commutative diagram:

Y
{q \atop ↙}  {q′ \atop ↘}
S  ↓μ  X′.

{↘ \atop p}  {↙ \atop p′}
U

Here we choose μ so that μ^{−1}(π^{∗}D) ∪ Exc(μ) is a divisor with normal crossings. Suppose we write

2.6  K_{Y} ≡ q^{∗}(K_{S} + p_{∗}^{−1}(π^{∗}D) + E) + ∑_{i=1}^{s} a_{i}E_{i}.

Then (S, p_{∗}^{−1}(π^{∗}D)+E) is log–canonical near E provided a_{i} ≥ −1 for all E_{i} contracted by q. We also have, using q^{∗}K_{S} − μ^{∗}K_{U} = q^{∗}E ,

K_{Y} ≡ μ^{∗}(K_{U} + π^{∗}(D)) + ∑_{j=1}^{s} b_{j}E_{j}

≡ q^{∗}(K_{S} − E + p^{∗}π^{∗}D) + ∑_{j=1}^{s} b_{j}E_{j}

≡ q^{∗}(K_{S} + p_{∗}^{−1}(π^{∗}D) + E) − (2 − mult_{0}(π^{∗}D))q^{∗}E + ∑_{j=1}^{s} b_{j}E_{j}. 2.7

Combining 2.6 and 2.7 shows that b_{i} ≥ a_{i} for all i, provided that mult_{0}(π^{∗}D) ≤ 2. Thus, whenever mult_{0}(π^{∗}D) ≤ 2, (U, π^{∗}D) is log–canonical at 0 provided (S, p_{∗}^{−1}(π^{∗}D) + E) is log–canonical near E.

Next we need to relate the pair (X, D) at x to the pair (U, π^{∗}D) at 0. For this, we consider the commutative diagram

W
{g \atop ↙}  {h \atop ↘}
U  X̃.
{↘ \atop π}  {↙ \atop f}
X

Write

\begin{matrix}K_{W} = & g^{∗}K_{U} + ∑_{j=1}^{s} b_{j}G_{j}, \\ KX̃ = & f^{∗}K_{X} + ∑_{i=1}^{r} a_{i}F_{i}. \end{matrix}

Using the logarithmic ramification formula ([Ka] Lemma 1.6) we find

2.8  ∑_{j=1}^{s}(1 + b_{j})G_{j} = h^{∗} (∑_{i=1}^{r}(1 + a_{i})F_{i}) + R,

where all divisorial components of R are contracted by h. We now remove the boundary divisor from both sides of 2.8 giving

2.9  ∑_{j=1}^{s}(1 + b_{j})G_{j} − g^{∗}π^{∗}D = h^{∗} (∑_{i=1}^{r}(1 + a_{i})F_{i} − f^{∗}D) + R.

We rewrite 2.9 as

2.10  ∑_{j=1}^{s′} β_{j}G′_{j} = h^{∗} (∑_{i=1}^{r′} α_{i}F′_{i}) + R.

Suppose now that F is one of the F′_{i} in 2.10 which is f–exceptional and let α_{F} denote the corresponding discrepancy. Let G denote a divisor on W such that h : G → F is finite, with discrepancy β_{G}. Then 2.10 says that α_{F} ≥ −1 provided β_{G} ≥ −1. But (U, π^{∗}D) is log–canonical near E if and only if β_{j} ≥ −1 for all G_{j} contracted by g. Thus α_{F} ≥ −1 for all F contracted by f and this shows that (X, D) is log–canonical at x.

We now give some general criteria for establishing whether or not (X, D) is log– canonical at a point x.

Lemma 2.11: Suppose X is a normal projective surface and D an effective ℚ–divisor on X. Suppose moreover that x ∈ X_{reg}. Then the pair (X, D) is klt at x provided A · D < a for some divisor A such that the linear series |A ⊗ m_{x}^{a}| has only isolated base points.

Proof: Suppose x ∈ supp(D). Choose a representative E ∈ |A ⊗ m_{x}^{a}| such that E ∩ supp(D) is proper. Then we have

i_{x}(D, E) ≤ D · E < a.

On the other hand

i_{x}(D, E) ≥ mult_{x}(D)mult_{x}(E) ≥ a(mult_{x}(D)).

Thus we have

mult_{x}(D) < 1

and Lemma 2.11 follows from [KM] Theorem 4.5 (1).

Note that Lemma 2.11 throws away a lot of information because the estimate

D · E ≥ i_{x}(D, E)

can be very crude; to give a more precise estimate, however, requires information about the specific geometry of X. We will also need a slight refinement of Lemma 2.11 to deal with the singular points of X.

Lemma 2.12: Suppose X is a normal projective surface with a quotient singularity of index m at x. Suppose D is an effective ℚ–divisor and A a divisor on X such that |A ⊗ m_{x}^{a}| has only isolated base points for some positive integer a. Then (X, D) is klt at x provided A · D < \frac{a}{m}.

Proof: Choose a general divisor E ∈ |A ⊗ m_{x}^{a}| so that E ∩ D is proper. Choose a local resolution π : Y → X of X at x with π^{−1}(x) = y. Then

mult_{y}(π^{∗}D) ≤ \frac{mD · A}{a}.

Thus, by hypothesis mult_{y}(π^{∗}D) < 1 so that (Y, π^{∗}D) is klt at y. By Lemma 2.11 this implies that the pair (X, D) is klt at x.

3. The Sasakian Geometry of Links of Weighted Homogeneous Polynomials

In this section we discuss the Sasakian geometry of links of isolated hypersurface singularities defined by weighted homogeneous polynomials. Consider the affine space ℂ^{n+1} together with a weighted ℂ^{∗}-action given by (z_{0}, …, z_{n}) ↦ (λ^{w_{0}}z_{0}, …, λ^{w_{n}}z_{n}), where the weights w_{j} are positive integers. It is convenient to view the weights as the components of a vector w ∈ (ℤ^{+})^{n+1}, and we shall assume that 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

3.1  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

3.2  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 3.1, 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 3.3: 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 will only be interested in the n = 3 case of hypersurfaces in a weighted complex projective 3-space. These conditions, known as quasi-smoothness conditions gaurentee that Z_{f} is smooth in the orbifold sense, that is, at a vertex P_{i} ∈ ℙ(w) the preimage of Z_{f} in the orbifold chart of ℙ(w) is smooth. It is easy to see that one can formulate all these conditions as follows [Fle,JK1]:

Quasi-Smoothness Conditions 3.4:

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

Here j = i is possible.

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

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

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

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

Well-formedness Condition 3.5

IV.  For any triple i, j, k ≠, we have gcd(w_{i}, w_{j}, w_{k}) = 1.

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

3.6  K_{Z} ≃ O(−I) = O(d − |w|),

where |w| = ∑_{i} w_{i}.

Finally we end this section by giving a corollary of Lemma 2.12 which, by Theorem 1.23, gives sufficient conditions to ensure that Z_{f} admits a Kähler-Einstein metric.

Corollary 3.7: Let Z_{f} ⊂ ℙ(w_{0}, w_{1}, w_{2}, w_{3}) be a hypersurface of degree d in weighted projective space with w = (w_{0}, w_{1}, w_{2}, w_{3}) well-formed. Let Δ ∈ | − αK_{Z_{f}} | be an effective representative for some rational α > 0. Writing K_{Z_{f}} = O_{Z_{f}} (−I), suppose x ∈ Z_{f} is a point at which ℙ(w) has a singularity of order ℓ_{x}. If BS(O_{Z_{f}} (w_{0}) ⊗ I_{x}) does not contain any component of Δ then (Z_{f}, Δ) is klt at x provided

αℓ_{x}dI < w_{1}w_{2}w_{3}.

If BS(O_{Z_{f}} (w_{0}w_{1}) ⊗ I_{x}) does not contain Δ then (Z_{f}, Δ) is klt at x provided

αℓ_{x}dI < w_{0}w_{2}w_{3}.

In general, unless Z_{f} is the hypersurface {z_{0} = 0}, one always has that (Z_{f}, Δ) is klt at

x if αℓ_{x}dI < w_{0}w_{1}w_{3}.

Proof: This corollary follows immediately from Lemma 2.12 by looking at the linear systems |O_{Z_{f}} (w_{0})|, |O_{X}(w_{0}w_{1})|, and |O_{Z_{f}} (w_{0}w_{1}w_{2})| respectively. For example, taking A = |O_{Z_{f}} (w_{0}w_{1})| in Lemma 2.12, we find that we can take a = w_{0} since one of the sections z_{0}^{w_{1}}, z_{1}^{w_{0}} will not vanish along Δ by hypothesis and w_{0} ≤ w_{1}. Then Lemma 2.12 states that (Z_{f}, αΔ) is klt at x provided A · αΔ < \frac{w_{0}}{ℓ_{X}}. But

A · αΔ = \frac{w_{0}w_{1}dαI}{w_{0}w_{1}w_{2}w_{3}} = \frac{dαI}{w_{2}w_{3}} .

The middle formula follows immediately and the others are obtained similarly.

4. The Algebraic Equations and Their Solutions

We want to repeat the analysis of [JK1] for the case of an arbitrary index I. Following their approach we consider the linear system

4.1  m_{i}w_{i} + w_{j(i)} = d = w_{0} + w_{1} + w_{2} + w_{3} − I

where m_{i} is a positive integer and both i and j(i) are integers ranging over 0, 1, 2, 3. Note we can have j(i) = i. This is an obvious translation of the quasi-smoothness condition 3.4.I Lemma 4.2: Assuming conditions 3.4.I and 3.4.IV, the following bounds hold

(1) 1 ≤ m_{3} ≤ 2.

(2) Either 2 ≤ m_{2} ≤ 4, or 2w_{0} ≤ I.

(3) Either 2 ≤ m_{1} ≤ 10, or 2I = w_{0} + w_{1}, or 2I ≥ 3w_{0}, or condition 3.4.II is violated. Proof: Since the weights are ordered w_{0} ≤ w_{1} ≤ w_{2} ≤ w_{3}, (1) follows immediately from the linear system with i = 3. To prove (2) we consider i = 1 in the system 4.1. There are two cases: w_{j(2)} = w_{3} and w_{j(2)} ≠ w_{3}. If w_{j(2)} = w_{3} we immediately have

m_{2}w_{2} = w_{0} + w_{1} + w_{2} − I ≤ 3w_{2} − I

which implies m_{2} ≤ 2.

Now consider w_{j(2)} ≠ w_{3}, then there are two subcases depending on whether m_{3} = 2 or 1. If m_{3} = 2 our system gives

w_{0} + w_{1} + w_{2} = w_{3} + w_{j(3)} + I,

and here there are two subcases giving:

1. If w_{j(3)} = w_{3}, then 2w_{3} < 3w_{2}.

2. If w_{j(3)} ≠ w_{3}, then w_{3} < 2w_{2}, and in both cases we have w_{3} < 2w_{2}. Thus since w_{j(2)} ≠ w_{3}, we get

m_{2}w_{2} ≤ 2w_{2} + w_{3} − I < 4w_{2} − I

implying m_{2} ≤ 3.

Finally we consider the case m_{3} = 1. Here we obtain

4.3  w_{j(3)} + I = w_{0} + w_{1} + w_{2}.

If w_{3} = w_{j(3)} = w_{0} + w_{1} + w_{2} − I, then d = 2w_{3}, so m_{2}w_{2} + w_{j(2)} = 2w_{0} + 2w_{1} + 2w_{2} − 2I. This gives

m_{2}w_{2} ≤ 5w_{2} − 2I,

implying that m_{2} ≤ 4, whereas if w_{j(3)} ≠ w_{3}, then we have I = w_{i} + w_{j} for some distinct i, j = 0, 1, 2. We have thus shown that either 1 ≤ m_{2} ≤ 4 or I = w_{i} + w_{j} for some distinct i, j = 0, 1, 2. It is now easy to see that if m_{2} = 1 then we must have 2w_{0} ≤ I. Thus, in all cases (2) of the lemma holds.

Finally we consider (3). We proceed along the lines of (2). If possible we solve the system 4.1 for i = 2, 3 for w_{3} and w_{2}; however, there are exceptional cases, that is, values of m_{3} and m_{2} such w_{3} and/or w_{2} are free. We shall show that in these cases when m_{1} is unbounded that either 2I ≥ 3w_{0}, or 2I = w_{0} + w_{1}, or condition 3.4.II is violated. Moreover, when we can solve for w_{2} and w_{3} and substitute into 4.1 with i = 1 we obtain an equation for m_{1} in terms of m_{2}, m_{3}, w_{1}, w_{2}, w_{3}. This gives a bound for m_{1} which is certainly maximal for I = 1 (in fact for I = 0) and one can see that m_{1} ≤ 10 as in [JK]. To see the lower bound on m_{1} we show that if m_{1} = 1 then I ≥ 2w_{0}. There are two cases to consider, w_{j(1)} = w_{1} and w_{j(1)} ≠ w_{1}, and one easily sees that both cases give the desired estimate.

We now consider the exceptional cases. For the i = 3 equation of the system 4.1 this occurs for m_{3} = 1, and one easily sees that there are 3 exceptional cases all which imply the estimate I ≥ 2w_{0}. Now assuming that w_{3} is determined in terms of the remaining weights and index, we consider the i = 2 equation in 4.1. There are 4 subcases. Since m_{2} > 1 the case w_{j(2)} = w_{3} gives no exceptional solution. Now if w_{j(2)} = w_{2} we have m_{2}w_{2} = w_{0} + w_{1} + w_{3} − I and we must plug in the possibilities for w_{3} determined by the i = 3 equation. But since all of these possibilities are of the form w_{3} = aw_{2} +b with a = 0, a = \frac{1}{2} or a = 1, we see that there are no exceptional solutions in this case either.

Next we put w_{j(2)} = w_{1}, and consider the 5 possibilities for w_{3}. The first is w_{3} = w_{0} + w_{1} + w_{2} − I, and this leads to the exceptional case with m_{2} = 2 and 2I = 2w_{0} + w_{1} ≥ 3w_{0}. The next case with 2w_{3} = w_{0} + w_{1} + w_{2} − I is not exceptional, and the third with w_{3} = w_{0} + w_{1} − I again gives 2I ≥ 3w_{0}. The fourth case w_{3} = w_{0} + w_{2} − I, however, gives the exceptional case m_{2} = 2 and I = w_{0}, and this requires a bit more analysis. Here we have that w_{3} = w_{2} and so the weights are (I, w_{1}, w_{2}, w_{2}) with degree d = 2w_{2} + w_{1}. But this fails the quasi-smoothness condition (2) of section 3 the w_{2} must divide d, and this can occur only if w_{2} = w_{1} which violates the well-formedness condition. Finally the fifth subcase with w_{3} = w_{1} + w_{2} − I leads to m_{2} = 2 and 2I = w_{0} + w_{1}.

Finally we have the case w_{j(2)} = w_{0}. The first possibility for w_{3} gives an exceptional case with 2I ≥ 3w_{0}, while the next two do not lead to any exceptional cases. Subcase 4, however, with w_{3} = w_{0} + w_{2} − I give m_{2} = 2 and 2I = w_{0} + w_{1}. This implies the inequalities w_{0} ≤ I ≤ w_{1} and the equation w_{3} = w_{2} − w_{1} + I which violate the order w_{2} ≤ w_{3} unless w_{3} = w_{2} and I = w_{1} = w_{0}. But in this case the degree is d = 2w_{2} + I, and condition 3.4.II then implies that w_{2} = w_{3} = I which contradicts the well-formedness condition 3.4.IV. The last possibility for w_{3} is w_{3} = w_{1} + w_{2} − I and this leads to the exceptional case m_{2} = 2 and I = w_{1}. But then we have w_{3} = w_{2} and d = 2w_{2} + w_{0} which violates condition 3.4.II above again since w_{2} cannot divide w_{0}.

Remarks 4.4: There are solutions with arbitrary m_{1} that are quasi-smooth, but our lemma implies that these must satisfy 2I ≥ 3w_{0} or 2I = w_{0} + w_{1}. As we shall see in the next section, in both cases the sufficient condition for the existence of a Kähler-Einstein metric fails. Examples of such a series are: (1, 1, k, k) of degree 2k and index 2, (1, 2, 2k + 1, 2k + 1) of degree 2(2k + 1) and index 3, and (2, 2, 2k + 1, 2k + 1) of degree 2(2k+1) and index 4, as well as the general series (I −n, I +n, w, w+n) of degree 2w+n+I and index I. There are also double series such as (1, 1, m, m + k) of degree 2m + k and index 2.

We have modified the computer program of [JK1] to solve the system 4.1 for any index I as well as to discard some solutions that are not quasi-smooth. In light of Remarks 4.4, we shall now consider all solutions for which 1 ≤ I ≤ 10, 2I < 3w_{0}, and 2I ≠ w_{0} + w_{1}. The last column of the tables indicate whether the klt condition (which implies that Z_{w} admits a Kähler-Einstein metric) holds (Y) or is unknown (?). The computer search gives the following complete list:

Theorem 4.5: Let Z_{w} be a well-formed quasi-smooth log del Pezzo surface of index I ≤ 10 and degree d embedded in the weighted projective space ℙ(w) = ℙ(w_{0}, w_{1}, w_{2}, w_{3}). If 2I ≥ 3w_{0} or 2I = w_{0} + w_{1} then for every ϵ > 0 there exists a divisor D ∈ | − K_{Z_{w}}| such that (Z_{w}, \frac{2+ϵ}{3}D) is not klt. If 2I < 3w_{0} and 2I ≠ w_{0} + w_{1} then Z_{w} must belong to one of the following cases:

(1) If I = 1 [JK1] then Z_{w} is either one of the series solution below

\begin{matrix}w & d & b_{2} & K-E \\ (2, 2k + 1, 2k + 1, 4k + 1) & 8k + 4 & 8 & Y \end{matrix}

or it is one of the 19 sporadic solutions listed in Table 1.

(2) If I = 2 then Z_{w} is one of the 6 infinite series solutions with w and d given by:

\begin{matrix}w & d & b_{2} & K-E \\ (4, 2k + 1, 2k + 1, 4k) & 8k + 4 & 7 & Y \\ (3, 3k + 1, 6k + 1, 9k + 3) & 18k + 6 & 6 & Y \\ (3, 3k + 1, 6k + 1, 9k) & 18k + 3 & 5 & ? \\ (3, 3k, 3k + 1, 3k + 1) & 9k + 3 & 7 & Y \\ (3, 3k + 1, 3k + 2, 3k + 2) & 9k + 6 & 5 & Y \\ (4, 2k + 1, 4k + 2, 6k + 1) & 12k + 6 & 6 & ? \end{matrix}

or it is one of the 25 sporadic solutions listed in Table 1.

(3) If I = 3 then Z_{w} is one of the 7 sporadic solutions listed in Table 1.

(4) If I = 4 then Z_{w} is one of the 3 infinite series with w and d given by:

\begin{matrix}w & d & b_{2} & K-E \\ (6, 6k + 3, 6k + 5, 6k + 5) & 18k + 15 & 5 & Y \\ (6, 6k + 5, 12k + 8, 18k + 9) & 36k + 24 & 3 & ? \\ (6, 6k + 5, 12k + 8, 18k + 15) & 36k + 30 & 4 & Y \end{matrix}

or it is one of the 10 sporadic solutions of Table 1.

(5) If I = 5 then Z_{w} is one of the 3 sporadic solutions listed in Table 2.

(4) If I = 6 then Z_{w} is one of the 2 infinite series given by:

\begin{matrix}w & d & b_{2} & K-E \\ (8, 4k + 1, 4k + 3, 4k + 5) & 12k + 11 & 3 & ? \\ (9, 3k + 2, 3k + 5, 6k + 1) & 12k + 11 & 3 & ? \end{matrix}

or it is one of the 3 sporadic solutions of Table 1.

(5) If I = 7, 8, 9, 10 then Z_{w} is one of the 6 sporadic solutions listed in Table 1.

Table 1. Sporadic Examples of Z_{w} of Index 1 ≤ I ≤ 10

Index  w  Monomials of f_{w}  d  b_{2} K-E
1  (1,2,3,5)  z_{0}^{10}, z_{1}^{5}, z_{2}^{3}z_{1}, z_{3}^{2}, … (17)^{∗}  10  9  ?
1  (1,3,5,7)  z_{0}^{15}, z_{1}^{5}, z_{2}^{3}, z_{3}^{2}z_{0}, … (19)  15  9  ?
1  (1,3,5,8)  z_{0}^{16}, z_{1}^{5}z_{0}, z_{2}^{3}z_{0}, z_{3}^{2}, … (20)  16  10  ?
1  (2,3,5,9)  z_{0}^{9}, z_{1}^{6}, z_{2}^{3}z_{1}, z_{3}^{2}, … (13)  18  7  Y
1  (3,3,5,5)  g_{5}(z_{0}, z_{1}), f_{3}(z_{2}, z_{3})  15  5  Y
1  (3,5,7,11)  z_{0}^{6}z_{2}, z_{1}^{5}, z_{2}^{2}z_{3}, z_{3}^{2}z_{0}, … (8)  25  5  Y
1  (3,5,7,14)  z_{0}^{7}z_{2}, z_{1}^{5}z_{0}, g_{2}(z_{2}^{2}, z_{3}), … (9)  28  6  Y
1  (3,5,11,18)  g_{2}(z_{0}^{6}, z_{3}), z_{1}^{5}z_{2}, z_{2}^{3}z_{0}, … (10)  36  6  Y
1  (5,14,17,21)  z_{0}^{7}z_{3}, z_{1}^{4}, z_{2}^{3}z_{0}, z_{3}^{2}z_{1}, z_{0}^{5}z_{1}z_{2}  56  4  Y
1  (5,19,27,31)  z_{0}^{10}z_{3}, z_{1}^{4}z_{0}, z_{2}^{3}, z_{3}^{2}z_{1}, z_{0}^{7}z_{1}z_{2}  81  3  Y
1  (5,19,27,50)  z_{0}^{20}, z_{0}^{10}z_{3}, z_{3}^{2}, z_{1}^{5}z_{0}, z_{2}^{3}z_{1}, z_{0}^{7}z_{1}^{2}z_{3}  100  4  Y
1  (7,11,27,37)  z_{0}^{10}z_{1}, z_{1}^{4}z_{3}, z_{2}^{3}, z_{3}^{3}z_{0}  81  3  Y
1  (7,11,27,44)  z_{0}^{11}z_{1}, z_{2}^{3}z_{0}, z_{1}^{8}, z_{1}^{4}z_{3}, z_{3}^{2}, z_{0}^{4}z_{1}^{3}z_{2}  88  4  Y
1  (9,15,17,20)  z_{0}^{5}z_{1}, z_{1}^{4}, z_{2}^{3}z_{0}, z_{3}^{3}  60  3  Y
1  (9,15,23,23)  z_{0}^{6}z_{1}, z_{1}^{4}z_{0}, z_{2}^{3}, z_{2}^{2}z_{3}, z_{2}z_{3}^{2}, z_{3}^{3}  69  5  Y
1  (11,29,39,49)  z_{0}^{8}z_{2}, z_{1}^{4}z_{0}, z_{2}^{3}, z_{3}^{2}z_{1}  127  3  Y
1  (11,49,69,128)  z_{0}^{17}z_{2}, z_{1}^{5}z_{0}, z_{2}^{4}, z_{2}^{2}z_{3}, z_{3}^{2}  256  2  Y
1  (13,23,35,57)  z_{0}^{8}z_{1}, z_{1}^{4}z_{2}, z_{2}^{2}z_{3}, z_{3}^{2}z_{0}  127  3  Y
1  (13,35,81,128)  z_{0}^{17}z_{1}, z_{1}^{5}z_{2}, z_{2}^{3}z_{0}, z_{3}^{2}  256  2  Y
2  (2,3,4,5)  z_{0}^{6}, z_{1}^{4}, z_{2}^{3}, z_{3}^{2}z_{0}, … (10)  12  5  ?
2  (2,3,4,7)  z_{0}^{7}, z_{1}^{4}z_{0}, z_{2}^{3}z_{0}, z_{3}^{2}, … (11)  14  6  ?
2  (3,4,5,10)  z_{0}^{5}z_{2}, z_{1}^{5}, z_{2}^{4}, z_{3}^{2}, … (9)  20  5  Y
2  (3,4,6,7)  g_{3}(z_{0}^{2}, z_{2}), z_{1}^{3}z_{2}, z_{3}^{2}z_{1}, z_{0}z_{3}z_{1}^{2}, z_{0}^{2}z_{1}^{3}  18  6  ?
2  (3,4,10,15)  z_{0}^{10}, z_{1}^{5}z_{3}, z_{2}^{3}, z_{3}^{2}, … (10)  30  7  Y
2  (3,7,8,13) z_{0}^{7}z_{2}, z_{1}^{3}z_{2}, z_{2}^{2}z_{3}, z_{3}^{2}z_{0}, z_{0}^{5}z_{1}, z_{0}^{3}z_{1}z_{3}, z_{0}^{2}z_{1}z_{2}^{2} 29  5  ?
2  (3,10,11,19) z_{0}^{10}z_{3}, z_{1}^{3}z_{2}, z_{2}^{2}z_{3}, z_{3}^{2}z_{0}, z_{0}^{7}z_{1}^{2}, z_{0}^{4}z_{1}z_{3}, z_{0}^{3}z_{1}z_{2}^{3} 41  5  ?
2  (5,13,19,22)  z_{0}^{7}z_{3}, z_{1}^{4}z_{0}, z_{2}^{3}, z_{3}^{2}z_{1}, z_{0}^{5}z_{1}z_{2}  57  3  Y
2  (5,13,19,35)  z_{0}^{14}, z_{0}^{7}z_{3}, z_{3}^{2}, z_{1}^{5}z_{0}, z_{2}^{3}z_{1}, z_{0}^{5}z_{1}^{2}z_{2}  70  3  Y
2  (6,9,10,13)  z_{0}^{6}, z_{1}^{4}, z_{2}^{3}z_{0}, z_{3}^{2}z_{2}, z_{0}^{3}z_{1}^{2}  36  4  Y
2  (7,8,19,25)  z_{0}^{7}z_{1}, z_{1}^{4}z_{3}, z_{2}^{3}, z_{3}^{2}z_{0}, z_{0}^{2}z_{1}^{3}z_{2}  57  3  Y
2  (7,8,19,32)  z_{0}^{8}z_{1}, z_{1}^{8}, z_{1}^{4}z_{3}, z_{3}^{2}, z_{2}^{3}z_{0}, z_{0}z_{2}^{3}, z_{0}^{3}z_{1}^{3}z_{2}  64  4  Y
2  (9,12,13,16)  z_{0}^{4}z_{1}, z_{1}^{4}, z_{2}^{3}z_{0}, z_{3}^{3}  48  3  Y
2  (9,12,19,19)  z_{0}^{5}z_{1}, z_{1}^{4}z_{0}, z_{2}^{3}, z_{2}^{2}z_{3}, z_{2}z_{3}^{3}, z_{3}^{3}  57  5  Y
2  (9,19,24,31)  z_{0}^{9}, z_{1}^{3}z_{2}, z_{2}^{3}z_{0}, z_{3}^{2}z_{1}  81  3  Y
2  (10,19,35,43)  z_{0}^{7}z_{2}, z_{1}^{5}z_{0}, z_{2}^{3}, z_{3}^{2}z_{1}  105  3  Y
2  (11,21,28,47)  z_{0}^{7}z_{2}, z_{1}^{5}, z_{2}^{3}z_{1}, z_{3}^{2}z_{0}  105  3  Y
2  (11,25,32,41)  z_{0}^{6}z_{3}, z_{1}^{3}z_{2}, z_{2}^{3}z_{0}, z_{3}^{2}z_{1}  107  3  Y
2  (11,25,34,43)  z_{0}^{10}, z_{1}^{4}z_{0}, z_{2}^{2}z_{3}, z_{3}^{2}z_{1}  111  3  Y
2  (11,43,61,113)  z_{0}^{15}z_{2}, z_{1}^{5}z_{0}, z_{2}^{3}z_{1}, z_{3}^{2}  226  2  Y
2  (13,18,45,61)  z_{0}^{9}z_{1}, z_{1}^{5}z_{2}, z_{2}^{3}, z_{3}^{2}z_{0}  135  3  Y

Table 1. (cont.) Sporadic Examples of Z_{w} of Index 1 ≤ I ≤ 10

Index  w  Monomials of f_{w}  d  b_{2} K-E
2  (13,20,29,47)  z_{0}^{6}z_{3}, z_{1}^{3}z_{3}, z_{2}^{3}z_{1}, z_{3}^{2}z_{0}  107  3  Y
2  (13,20,31,49)  z_{0}^{7}z_{1}, z_{1}^{4}z_{2}, z_{2}^{3}z_{3}, z_{3}^{2}z_{0}  111  3  Y
2  (13,31,71,113)  z_{0}^{15}z_{1}, z_{1}^{5}z_{2}, z_{2}^{3}z_{0}, z_{3}^{2}  226  2  Y
2  (14,17,29,41)  z_{0}^{5}z_{3}, z_{1}^{5}z_{0}, z_{2}^{2}z_{3}, z_{3}^{2}z_{1}  99  3  Y
3  (5,7,11,13)  z_{0}^{4}z_{3}, z_{1}^{4}z_{0}, z_{2}^{3}, z_{3}^{2}z_{1}, z_{0}^{3}z_{1}z_{2}  33  3  ?
3  (5,7,11,20)  z_{0}^{8}, z_{1}^{5}z_{0}, z_{2}^{3}z_{1}, z_{3}^{2}, z_{0}^{4}z_{3}, z_{0}^{3}z_{1}^{2}, z_{2}^{2}  40  4  Y
3  (11,21,29,37)  z_{0}^{6}z_{2}, z_{1}^{4}z_{0}, z_{2}^{2}z_{3}, z_{3}^{2}z_{1}  95  3  Y
3  (11,37,53,98)  z_{0}^{13}z_{2}, z_{1}^{5}z_{0}, z_{2}^{3}z_{1}, z_{3}^{2}  196  2  Y
3  (13,17,27,41)  z_{0}^{6}z_{1}, z_{1}^{4}z_{2}, z_{2}^{2}z_{3}, z_{3}^{2}z_{0}  95  3  Y
3  (13,27,61,98)  z_{0}^{13}z_{1}, z_{1}^{5}z_{2}, z_{2}^{3}z_{0}, z_{3}^{2}  196  2  Y
3  (15,19,43,74)  z_{0}^{7}z_{2}, z_{1}^{7}z_{0}, z_{2}^{3}z_{1}, z_{3}^{2}  148  2  Y
4  (5,6,8,9)  z_{0}^{3}z_{3}, z_{1}^{4}, z_{2}^{3}, z_{3}^{2}z_{1}, z_{0}^{2}z_{1}z_{2}  24  3  ?
4  (5,6,8,15)  z_{0}^{6}, z_{1}^{5}, z_{2}^{3}z_{1}, z_{3}^{2}, z_{0}^{3}z_{3}, z_{0}^{3}z_{3}, z_{0}^{2}z_{1}^{2}z_{2}  30  4  ?
4  (9,11,12,17)  z_{0}^{5}, z_{1}^{3}z_{2}, z_{2}^{3}z_{0}, z_{3}^{2}z_{1}  45  3  ?
4  (10,13,25,31)  z_{0}^{5}z_{2}, z_{1}^{5}z_{0}, z_{2}^{3}, z_{3}^{2}z_{1}  75  3  Y
4  (11,17,20,27)  z_{0}^{4}z_{3}, z_{1}^{3}z_{2}, z_{2}^{3}z_{0}, z_{3}^{2}z_{1}  71  3  ?
4  (11,17,24,31)  z_{0}^{5}z_{2}, z_{1}^{4}z_{0}, z_{2}^{2}z_{3}, z_{3}^{2}z_{1}  79  3  Y
4  (11,31,45,83)  z_{0}^{11}z_{2}, z_{1}^{5}z_{0}, z_{2}^{3}z_{1}, z_{3}^{2}  166  2  Y
4  (13,14,19,29)  z_{0}^{4}z_{2}, z_{1}^{3}z_{3}, z_{2}^{3}z_{1}, z_{3}^{2}z_{0}  71  2  ?
4  (13,14,23,33)  z_{0}^{5}z_{1}, z_{1}^{4}z_{2}, z_{2}^{2}z_{3}, z_{3}^{2}z_{0}  79  3  Y
4  (13,23,51,83)  z_{0}^{11}z_{1}, z_{1}^{5}z_{2}, z_{2}^{3}z_{0}, z_{3}^{2}  166  2  Y
5  (11,13,19,25)  z_{0}^{4}z_{2}, z_{1}^{4}z_{0}, z_{2}^{2}z_{3}, z_{3}^{2}z_{1}  63  3  ?
5  (11,25,37,68)  z_{0}^{9}z_{2}, z_{1}^{5}z_{0}, z_{2}^{3}z_{1}, z_{3}^{2}  136  2  Y
5  (13,19,41,68)  z_{0}^{9}z_{1}, z_{1}^{5}z_{2}, z_{2}^{3}z_{0}, z_{3}^{2}  136  2  Y
6  (7,10,15,19)  z_{0}^{5}z_{1}, z_{1}^{3}z_{2}, z_{2}^{3}, z_{3}^{2}z_{0}  45  3  ?
6  (11,19,29,53)  z_{0}^{7}z_{2}, z_{1}^{5}z_{0}, z_{2}^{4}, z_{3}^{2}  106  2  Y
6  (13,15,31,53)  z_{0}^{7}z_{1}, z_{1}^{5}z_{2}, z_{2}^{3}z_{0}, z_{3}^{2}  106  2  Y
7  (11,13,21,38)  z_{0}^{5}z_{2}, z_{1}^{5}z_{0}, z_{2}^{3}z_{1}, z_{3}^{2}  76  2  Y
8  (7,11,13,23)  z_{0}^{5}z_{1}, z_{1}^{3}z_{2}, z_{2}^{3}z_{0}, z_{3}^{2}  46  2  ?
8  (7,18,27,37)  z_{0}^{9}z_{1}, z_{1}^{3}z_{2}, z_{2}^{3}, z_{3}^{2}z_{0}  81  3  ?
9  (7,15,19,32)  z_{0}^{7}z_{1}, z_{1}^{3}z_{2}, z_{2}^{3}z_{0}, z_{3}^{2}  64  2  ?
10  (7,19,25,41)  z_{0}^{9}z_{1}, z_{1}^{3}z_{2}, z_{2}^{3}z_{0}, z_{3}^{2}  82  2  ?
10  (7,26,39,55)  z_{0}^{13}z_{1}, z_{1}^{3}z_{2}, z_{2}^{3}, z_{3}^{2}z_{0}  117  3  ?

* (for lack of space only the total number of monomial terms in f_{w} is indicated)

The computer program indicates that there are neither series solutions nor sporadic solutions satisfying the hypothesis of Theorem 4.5 for I > 10. In fact, an easy argument shows that there are no such solutions for sufficiently large I. * Remarks 4.6: As already mentioned when 2I = w_{0} + w_{1} or 2I < 3w_{0} the sufficient condition for the existence of the Kähler-Einstein metric fails. In all such cases techniques of [DK] say nothing about the corresponding Z_{w}. Table 2 below gives the classical examples of Z_{w} that can be written as smooth hypersurfaces in weighted projective spaces and which by other methods admit K-E metrics [TY]. We only indicate a particular polynomial as the most general such hypersurface involves many such monomials which for reasons of space we do not include. We are not aware of any non-smooth orbifold examples for which the conditions of Theorem 4.5 fail, but are known to admit Kähler-Einstein metrics by another method.

* The code for the C program used to generate the tables of the Theorem 4.5 are available at the following URL http://www.math.unm.edu/˜ galicki/papers/publications.html.

Table 2. Classical del Pezzo surfaces as hypersurfaces in ℂℙ^{3}(w)

Index  w  f_{w}  d  Z_{w}
1  (1,1,1,1)  z_{0}^{3} + z_{1}^{3} + z_{2}^{3} + z_{3}^{3}  3  ℂℙ(2)#6\overline{ℂℙ(2)}
1  (1,1,1,2)  z_{0}^{4} + z_{1}^{4} + z_{2}^{4} + z_{3}^{2}  4  ℂℙ(2)#7\overline{ℂℙ(2)}
1  (1,1,2,3)  z_{0}^{6} + z_{1}^{6} + z_{2}^{3} + z_{3}^{2}  6  ℂℙ(2)#8\overline{ℂℙ(2)}
2  (1,1,1,1)  z_{0}^{2} + z_{1}^{2} + z_{2}^{2} + z_{3}^{2}  2  ℂℙ(1) × ℂℙ(1)
3  (1,1,1,1)  z_{0} + z_{1} + z_{2} + z_{3}  1  ℂℙ(2)

5. Proof of Theorem 4.5

The proof goes as follows: we begin by proving two lemmas which by Lemma 4.2 will reduce the problem to one in which the integers m_{2} and m_{1} of 4.1 are bounded. The computer programs then give a printout of solutions of 4.1 which give all log del Pezzo orbifolds Z_{w} such that (Z_{w}, D) might be klt for all D ≡ −\frac{2+ϵ}{3}K_{Z_{w}}. In the case that m_{0} is unbounded, these are the series solutions, and the case when m_{0} is bounded, they are the sporadic solutions. Corollary 3.7 is sufficient, for most of the sporatic solutions, to check that (Z_{w}, D) is klt for all D ≡ −\frac{2+ϵ}{3}K_{Z_{w}}. However, for the series solutions the klt condition is much more tedious. Indeed, in several cases we are unable to give an answer. The results are collected in tables. In the last column of the tables we indicate whether the klt condition holds (Y) or is unknown (?). Since there exists a Kähler-Einstein orbifold metric whenever (Z_{w}, D) is klt for all D ≡ −\frac{2+ϵ}{3}K_{Z_{w}}, this column is labeled K-E. The second to the last column gives the second Betti number of Z_{w} which is computed by the method described in section 6. Also, where space permits, we give the monomials that make up the defining weighted homogeneous polynomial. This is indicated in the tables for most of the sporadic solutions.

Lemma 5.1: Suppose 2I ≥ 3w_{0}. Then for any ϵ > 0, there exists D ∈ | − K_{Z_{w}}| such that (Z_{w}, \frac{2+ϵ}{3}D) is not klt.

Proof: Consider the section z_{0} ∈ H^{0}(Z_{w}, O_{Z_{w}}(w_{0})) and let D be its zero divisor. Then by hypothesis rD ∈ |O_{Z_{w}}(−K_{Z_{w}})| for r ≥ \frac{3}{2}. But then \frac{2+ϵ}{3}D can never be klt at a generic point y ∈ D, for ϵ > 0, since it has multiplicity greater than 1.

Lemma 5.2: Suppose Z_{w} ⊂ ℙ(w_{0}, w_{1}, w_{2}, w_{3}) is a hypersurface of index I with 2I = w_{0} + w_{1}. Then for any ϵ > 0, there exists D ∈ | − K_{Z_{w}}| such that (Z_{w}, \frac{2+ϵ}{3}D) is not klt.

Proof: First we notice in the proof of Lemma 4.2 that 2I = w_{0} + w_{1} occurs in precisely one case and in this case we have w_{3} = w_{1} + w_{2} − I. But then w must have the form

w = (I − n, I + n, w, w + n)

for some w ∈ ℤ^{+} with w ≥ I + n, and some non-negative integer n < I and in this case the degree d = 2w + n + I. Now suppose that z_{0}^{a_{0}}z_{1}^{a_{1}}z_{2}^{a_{2}}z_{3}^{a_{3}} is a monomial occurring with nonzero coefficient in the polynomial defining Z_{w}. We claim that if a_{0} = 0 then a_{1} ≠ 0. Indeed, if a_{0} = a_{1} = 0 then

a_{2}w + a_{3}w = 2w + 2I + n.

But w ≥ I +n so the only possible solutions would require a_{2} +a_{3} = 3 and w = 2I +n and one readily checks that these hypersurfaces are never quasi-smooth. Thus the divisor D = {z_{0} = 0} ∩ Z_{w} has at least two components, E and F, where E is the line z_{0} = z_{1} = 0 and F is defined, inside the weighted projective plane {z_{0} = 0}, by a polynomial f(z_{1}, z_{2}, z_{3}). Moreover, f(z_{1}, z_{2}, z_{3}) = z_{2}^{2} + z_{1}^{2}(g(z_{1}, z_{2}, z_{3})). Note that the point P = (0, 0, 0, 1) ∈ Z_{w} since w + n does not divide the degree of Z_{w}. Thus if π : ℂ^{2}⟶Z_{w} is a local cover of the quotient singularity at P = (0, 0, 0, 1) then π^{∗}D = π^{∗}E + π^{∗}F has multiplicity at least 1 + mult_{0}(π^{∗}F) at the origin. To compute the multiplicity of π^{∗}F at the origin, let Y = Z(f(z_{1}, z_{2}, z_{3})). Then Y ∩Z_{w} = F ∪G, where G does not contain the point (0, 0, 0, 1). Thus

mult_{0}(π^{∗}F) = mult_{0}(π^{∗}Z(f)) ≥ 2.

Consequently, (Z_{w}, \frac{2+ϵ}{3}D) is never klt as it always has multiplicity > 2 at 0. But \frac{I}{I−n}D ∈ | − K_{Z_{w}}| so this completes the proof of the lemma.

The analysis of most of the sporadic examples of Table 1 is easily done with help of Corollary 3.7 which can restated for this purpose as:

Corollary 5.3: Let w = (w_{0}, w_{1}, w_{2}, w_{3}) and Z_{w} ⊂ ℙ(w) be a quasi-smooth surface of degree d = w_{0} + w_{1} + w_{2} + w_{3} − I. Then Z_{w} admits a Kähler-Einstein metric if 2Id < 3w_{0}w_{1}. If the line (z_{0} = z_{1} = 0) ⊄ Z_{w} then 2Id < 3w_{0}w_{2} is also sufficient. If the point (0, 0, 0, 1) ∉ Z_{w} then 2Id < 3w_{0}w_{3} is also sufficient.

We begin with the more complicated analysis of the infinite series examples. In what follows ℙ will frequently denote the appropriate weighted projective space, with the weights w being understood; similarly Z_{w} ⊂ ℙ will denote a hypersurface in the weighted projective space, depending on an integer parameter k for all of the series examples.

• We consider now the hypersurface Z_{w} ⊂ ℙ(4, 2k + 1, 2k + 1, 4k) given by the zero set of the homogeneous polynomial of degree d = 8k + 4

f_{w}(z_{0}, z_{1}, z_{2}, z_{3}) = z_{0}^{2k+1} + z_{0}z_{3}^{2} + g(z_{1}, z_{2}),

where g is of degree 4 and we assume that the polynomial g has 4 distinct roots on the projective line. Thus −K_{Z_{w}} = O_{Z_{w}}(2). The only singularities of Z_{w} lie at the singular points of ℙ(w). We can apply Corollary 3.7 to see that (Z_{w}, D) is klt at all smooth points of ℙ(w), for all k > 0, for all D ≡ −K_{Z_{w}} since

d · κ = (8k + 4)2 < 4(2k + 1)(2k + 1).

We now pass to the singular points of Z_{w} which are slightly more involved as we need to consider a desingularization. We first treat the point P = (0, 0, 0, 1). We would like to apply the preceding discussion and use Shokurov’s inversion of adjunction (Lemma 2.5) for which we need to verify that mult_{0}(π^{∗}D) ≤ 2 for any effective ℚ–divisor D ≡ −K_{Z_{w}}: here π : (ℂ^{2}, 0) → (Z_{w}, P) is a local cover of the quotient singularity at P. We will consider the linear series |z_{1}, z_{2}| on Z_{w}. This has only isolated base points on Z_{w}, including the point P in question, and consequently if E is a general member of this system then D ∩ E will be proper. Pulling back to ℂ^{2} to compute the intersection multiplicity at 0 we find, since the ramification of π over P is of degree 4k,

mult_{0}(π^{∗}D) ≤ 4kD · E = \frac{2(2k + 1)(8k + 4)4k}{4(2k + 1)^{2}4k} = 2,

and thus we may apply Lemma 2.5. Hence it is sufficient, in order to establish that (Z_{w}, D) is log–canonical at P, to show that

p_{∗}^{−1}(π^{∗}D)|E

is a sum of points each with coefficient at most one.

Consider the hypersurface z_{0} = 0 on Z_{w}. This is a union of four lines, L_{1}, L_{2}, L_{3}, L_{4}, one for each (projective) zero of the polynomial g(z_{1}, z_{2}). Thus we have

5.4  L_{1} + L_{2} + L_{3} + L_{4} ∈ |O_{Z_{w}}(4)|.

Note that if an effective divisor D ≡ −K_{Z_{w}} does not contain any of the lines L_{i}, then computing (L_{1} +L_{2} +L_{3} +L_{4})D will immediately establish that (Z_{w}, D) is klt at P. Thus the representatives of −K_{Z_{w}} which are of concern are those containing some or all of the lines L_{i}. So suppose

5.5  D ≡ a_{1}L_{1} + a_{2}L_{2} + a_{3}L_{3} + a_{4}L_{4} + D′,

where D′ meets each of the four lines properly. We are interested in bounding the a_{i} which can be accomplished intersection theoretically. In particular we compute

5.6  L_{i} · L_{j} = \frac{1}{4k}  for  i ≠ j.

To see this, note that the four lines L_{i} are algebraically equivalent. Thus, using 5.4 above gives

5.7  Z_{w} · O_{ℙ}(4) · O_{ℙ}(2k + 1) = 4(L_{i} · O_{ℙ}(2k + 1))  for all i.

The left hand side of 5.7 is \frac{4(8k+4)(2k+1)}{4(2k+1)^{2}4k} = \frac{1}{k}. On the other hand one can check, choosing the appropriate representative for O_{ℙ}(2k + 1) that O_{ℙ}(2k + 1) · Z_{w} = L_{j} + C, where C does not meet L_{i}; 5.6 follows immediately.

Intersecting 5.4 with O_{ℙ}(1) gives

5.8  O_{ℙ}(1) · L_{i} = \frac{1}{4(2k + 1)k} ,  ∀ i.

Using 5.6 and 5.8 we can compute L_{i}^{2}:

5.9  L_{i}^{2} = L_{i} · O_{ℙ}(4) − ∑_{j≠i} L_{i} · L_{j} = \frac{1 − 6k}{(2k + 1)4k} .

We have, by 5.8

5.10  D · L_{i} = \frac{1}{2k(2k + 1)}.

Using 5.5 we obtain from 5.10

5.11  a_{i}L_{i}^{2} + ∑_{j≠i} a_{j}L_{j} · L_{i} + D′ · L_{i} = \frac{1}{2k(2k + 1)}.

To compute the terms in 5.11 observe first that ∑_{i=1}^{4} a_{i} ≤ 2; this follows from the fact that mult_{0}(π^{∗}D) ≤ 2. Next, note that

D′ · L_{i} ≤ D · D = O_{Z_{w}}(2) · O_{Z_{w}}(2) = \frac{4(8k + 4)}{4(2k + 1)^{2}4k} = \frac{1}{k(2k + 1)}.

Finally, using 5.6 and 5.9, 5.11 becomes

\frac{a_{i}(1 − 6k)}{(2k + 1)4k} + \frac{1}{4k} ∑_{j≠i} a_{j} + \frac{1}{k(2k + 1)} ≥ \frac{1}{2k(2k + 1)}.

Clearing denominators then yields

a_{i} ≤ \frac{k + 1}{2k}.

Thus if k ≥ 2 we have a_{i} ≤ \frac{3}{4}.

Next we bound mult_{0}(π^{∗}D′). Since L_{i} ∩ D′ is proper for any i we compute, using 5.8

mult_{0}(π^{∗}D′) ≤ 4kL_{i} · D ≤ \frac{2}{2k + 1}.

Returning to the inversion of adjunction set–up, we see that p_{∗}^{−1}(π^{∗}(D)) is a sum of four points (corresponding to the pull–backs of the four of the lines L_{i}) having total multiplicity at most \frac{3}{4} and another divisor with total degree at most \frac{2}{2k+1}. A simple computation then shows that \frac{11}{13}D is klt at P = (0, 0, 0, 1) for any D ≡ −K_{Z_{w}}.

Next we turn to the points P_{i} = (0, a_{i}, b_{i}, 0) on Z_{w}; there are exactly four of these by our assumption on the homogeneous polynomial f(z_{1}, z_{2}) and the singularities have index 2k + 1. Since BS(O_{Z_{w}}(w_{0}w_{1}w_{2}) ⊗ I_{P_{i}}) is a finite set of points, we will apply Lemma 2.12 with A = O_{ℙ}(8k + 4). Then we have

A · D = \frac{(8k + 4)(2)(8k + 4)}{4(2k + 1)(2k + 1)4k} = \frac{2}{k} .

We can take a = 4 in Lemma 2.12 since the linear series O_{ℙ}(8k + 4) allows for an isolated zero at P_{i} of multiplicity 4. Thus for k ≥ 2 we see that (Z_{w}, \frac{3}{4}D) is klt at P_{i}.

Finally, we deal with the points Q_{j} = (a_{j}, 0, 0, b_{j}), where the quotient singularities are of index 4. We again apply Lemma 2.12, this time taking A = O_{ℙ}(2k + 1). This linear series has sections with an isolated singularity of multiplicity 1 at any of the Q_{j}. We compute

A · D = \frac{2(2k + 1)(8k + 4)}{4(2k + 1)^{2}4k} = \frac{1}{2k}

Thus for k ≥ 2 we see that (Z_{w}, \frac{3}{4}D) is klt at Q_{j}. Putting together all of our computations establishes that (Z_{w}, \frac{3}{4}D) is klt for k ≥ 2 and for any D ≡ −K_{Z_{w}}.

• We consider now the hypersurface Z_{w} ⊂ ℙ(3, 3k + 1, 6k + 1, 9k + 3) given by the zero set of the homogeneous polynomial of degree d = 18k + 6

f_{w}(z_{0}, z_{1}, z_{2}, z_{3}) = z_{0}^{6k+2} + z_{1}^{3}z_{3} + z_{2}^{3}z_{0} + z_{3}^{2}.

Here −K_{Z_{w}} = O_{Z_{w}}(2). The worst singularity in this case is at x = (0, 0, 1, 0), where the index is 6k + 1. Since BS(O_{Z_{w}}(w_{0}w_{1}) ⊗ I_{x}) consists of finitely many points, Corollary 3.7 says that (Z_{w}, \frac{5}{7}D) is klt at x for k ≥ 1 for any D ≡ −K_{Z_{w}}.

Next we consider the singular line z_{0} = z_{2} = 0, where the index is 3k + 1. This intersects Z_{w} in the finitely many points. Applying the third formula of Corollary 3.7 shows that (Z_{w}, \frac{5}{7}D) is klt at each of these points for k ≥ 1.

Finally, there are index 3 singularities along the line z_{1} = z_{2} = 0 which again intersects Z_{w} in finitely many points. Applying Corollary 3.7 to these shows that (Z_{w}, \frac{7}{8}D) is klt at these points for k ≥ 1. Putting everything together we see that (Z_{w}, \frac{5}{7}D) is klt for any D ≡ −K_{Z_{w}} provided k ≥ 1.

• Next we consider the hypersurface Z_{w} ⊂ ℙ(6, 6k + 5, 12k + 8, 18k + 15) given by the zero set of the homogeneous polynomial of degree d = 36k + 30

f_{w}(z_{0}, z_{1}, z_{2}, z_{3}) = z_{0}^{6k+5} + z_{1}^{6} + z_{1}^{3}z_{3} + z_{0}z_{2}^{3} + z_{3}^{2}.

Here we have O_{Z_{w}}(−K_{Z_{w}}) = O_{Z_{w}}(4). This variety contains fewer of the singular points of ℙ than the previous example and asymptotically in k, the proportions of these weights are identical and thus this also must be α-klt for all k sufficiently large and for appropriate α. More specifically, we must deal with one point at infinity, x = (0, 0, 1, 0) which has index 12k + 8. Using O_{Z_{w}}(6(6k + 5)) in Corollary 3.7 we see that (Z_{w}, \frac{5}{7}D) is klt at x.

Next there is the line z_{0} = z_{2} = 0, where the index of ℙ is 6k + 5. Applying Corollary 3.7 as above gives (Z_{w}, \frac{5}{7}D) klt for k ≥ 1. Finally, there are is the line z_{1} = z_{2} = 0 of index 3. It follows once more from Corollary 3.7 that (Z_{w}, \frac{7}{8}D) is klt at each of these points for all k ≥ 2. There are singularities of index 2 along the line z_{1} = z_{3} = 0; (Z_{w}, D) is klt at these points for all k ≥ 1. Thus we find that for all k ≥ 2, (Z_{w}, \frac{5}{7}D) is klt for any D ≡ −K_{Z_{w}}.

• Next we consider the hypersurface Z_{w} = ℙ(3, 3k + 1, 3k + 2, 3k + 2) given by the zero set of the homogeneous polynomial of degree d = 9k + 6

f_{w}(z_{0}, z_{1}, z_{2}, z_{3}) = z_{0}^{3k+2} + z_{0}z_{1}^{3} + g(z_{2}, z_{3}),

where g is a homogeneous polynomial of degree three with distinct zeroes. The only singular points to be analyzed in this case are (0, 1, 0, 0) and (0, 0, a_{i}, b_{i}), where (a_{i}, b_{i}) are the zeroes of g. We first examine P = (0, 1, 0, 0), where ℙ has a singularity of index 3k + 1. Let π : (ℂ^{2}, 0) → (Z_{w}, P) be a local cover and let D ≡ −K_{Z_{w}}. Intersecting D with a general member of O_{Z_{w}}(3k + 2) establishes that

mult_{0}(π^{∗}D) ≤ 2

so that we can apply Shokurov’s inversion of adjunction as before. As before we write L_{1} + L_{2} + L_{3} for the zero scheme of z_{0} on Z_{w}. We compute

\begin{aligned}L_{i} · L_{j} & = \frac{1}{3k + 1}, i ≠ j, \\ O_{P}(1) · L_{i} & = \frac{1}{(3k + 1)(3k + 2)}, \\ L_{i}^{2} & = \frac{−1 − 6k}{(3k + 1)(3k + 2).} \end{aligned}

For D ≡ −K_{Z_{w}} effective we write D = a_{1}L_{1} + a_{2}L_{2} + a_{3}L_{3} + D′ as before, with D′ meeting the three lines properly, and we find, expanding D · L_{i} and using the estimates D · D′ ≤ D · D, a_{1} + a_{2} + a_{3} ≤ 2

\frac{(6k + 1)a_{i}}{(3k + 1)(3k + 2)} + \frac{a_{i} − 2}{3k + 1} ≤ \frac{2}{(3k + 1)(3k + 2)}.

Expanding yields a_{i} ≤ \frac{6k+6}{9k+3}. Moreover, intersecting with z_{0} = 0 shows that

mult_{0}(π^{∗}D′) ≤ \frac{2}{3(3k + 2)}.

Thus, with notation as before, p_{∗}^{−1}π^{∗}D is a sum of points with multiplicity at most \frac{6k+6}{9k+3} + \frac{2}{3(3k+2)} and inversion of adjunction applies to show that (Z_{w}, \frac{3}{4}D) is klt at P for all k ≥ 1.

The other singular points of Z_{w} like along the line z_{0} = z_{1} = 0 and these have index 3k + 2. Let Q_{i} = (0, 0, a_{i}, b_{i}) ∈ Z_{w} be one of these three points. The analysis of D above for P = (0, 1, 0, 0) applies at these points as well, unchanged though exactly one of the lines L_{i} passes through Q_{i}. In order to justify the use of Lemma 2.5 we need to show that mult_{0}(π^{∗}D) ≤ 2 for a local cover of ℙ at Q_{i}. This can be done with Lemma 2.12 using the linear series O_{Z_{w}}(3(3k + 1)) allowing for a singularity of multiplicity 3 at Q_{i}. Thus we conclude that (Z_{w}, \frac{3}{4}D) is klt for all k ≥ 1 and for all D ≡ −K_{Z_{w}}.

• Next we consider the hypersurface Z_{w} ⊂ ℙ(6, 6k + 3, 6k + 5, 6k + 5) given by the zero set of the homogeneous polynomial of degree d = 18k + 15

f_{w}(z_{0}, z_{1}, z_{2}, z_{3}) = z_{0}z_{1}^{3} + z_{0}^{2k+2}z_{1} + g(z_{2}, z_{3}),

where again g is homogeneous of degree 3 and Z_{w}. This passes through two points at infinity (1, 0, 0, 0) and (0, 1, 0, 0) and also intersects the singular lines z_{0} = z_{1} = 0 and z_{2} = z_{3} = 0 in finitely many points. The analysis of (0, 1, 0, 0) and (0, 0, a_{i}, b_{i}) is identical to the prior case, yielding (Z_{w}, \frac{5}{7}D) is klt at these points for all k ≥ 1. For the points along the line z_{2} = z_{3} = 0, other than (0, 1, 0, 0),the index does not depend on k, and hence these are certainly klt for k sufficiently large; more specifically, (Z_{w}, \frac{5}{7}D) is also klt at these points for k ≥ 2. Hence (Z_{w}, \frac{5}{7}D) is klt for all k ≥ 2 and for all D ≡ −K_{Z_{w}}.

• Lastly we consider the hypersurface Z_{w} ⊂ ℙ(3, 3k, 3k + 1, 3k + 1) given by the zero set of the homogeneous polynomial of degree d = 9k + 3

f_{w}(z_{0}, z_{1}, z_{2}, z_{3}) = z_{0}^{3k+1} + z_{0}z_{1}^{3} + g(z_{2}, z_{3}),

where g is homogeneous of degree 3. This contains only one singular point at infinity (0, 1, 0, 0) as well as the points (0, 0, a_{i}, b_{i}) and the points (c_{i}, d_{i}, 0, 0). Routine computations exactly as above show that (Z_{w}, \frac{3}{4}D) is klt for k ≥ 2 and for all D ≡ −K_{Z_{w}}.

Remark 5.12: Let us make a few comments for the series which we have not analyzed. For the two series of index 6, the simple intersection theoretic argument which establishes that mult_{0}(π^{∗}D) ≤ 2 on a local cover of one of the singular points, where D ≡ −K_{Z_{w}}, fails. The desired bound may well still hold but more detailed analysis of the other points of intersection of D and the appropriate divisor of Corollary 3.7 would be necessary to establish this. In the other cases which we have not analyzed, the singularities of “bad” divisor D = {z_{0} = 0} ∩ Z_{w} require a more subtle analysis because in these cases the remaining three weights are distinct and hence it is more difficult to compute the contribution of D to the tangent cone at the origin of the appropriate local cover; when two of the remaining three weights are the same, one quickly reduces to ℙ(1, 1, k) which is singular only at (0, 0, 1) and it is easy to check which tangent directions D gives on the resolution. In principle, however, each of these cases could be decided with more involved computations.

6. The Topology of the Link L_{f}

Recall the well-known construction of Milnor [Mil] for isolated hypersurface singularities: There is a fibration of (S^{2n+1} − L_{f})⟶S^{1} whose fiber F is an open manifold that is homotopy equivalent to a bouquet of n-spheres S^{n} ∨ S^{n} ⋯ ∨ S^{n}. The Milnor number μ of L_{f} is the number of S^{n}’s in the bouquet. It is an invariant of the link which can be calculated explicitly in terms of the degree d and weights (w_{0}, …, w_{n}) by the formula [MO]

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

The closure F̄ of F is a manifold with boundary that is homotopy equivalent to F, and whose boundary is precisely the link L_{f}. Then the topology of L_{f} is determined by the monodromy map induced by the S_{w}^{1} action. Milnor and Orlik [MO] use these facts to give an algorithm for computing the Betti number b_{n−1}(L_{f}) from the characteristic polynomial Δ(t) of the monodromy map. The procedure is this. Associate to any monic polynomial f with roots α_{1}, …, α_{k} ∈ ℂ^{∗} its divisor

6.2  div f =< α_{1} > + ⋯ + < α_{k} >

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

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

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

6.4  b_{2}(L_{f}) = 1 + ∑_{j} a_{j}.

Furthermore, the following proposition was proved in [BG3]:

Proposition 6.5: Let L_{f} be the link of an isolated singularity defined by a weighted homogeneous polynomial f in four complex variables with weights w. Suppose further that the weights w are well-formed, then Tor (H_{2}(L_{f}, ℤ)) = 0.

Now a well-known theorem of Smale [Sm] says that any simply connected compact 5manifold which is spin, and whose second homology group is torsion free, is diffeomorphic to S^{5}#l(S^{2} × S^{3}) for some non-negative integer l. Furthermore, it is known [BG2,Mor] that any simply connected Sasakian-Einstein manifold is spin. Combining this with the development above gives

Theorem 6.6: Let L_{f} be the link associated to a well-formed weighted homogeneous polynomial f in four complex variables. Suppose also that L_{f} is spin, in particular, if L_{f} admits an Sasakian-Einstein metric. Then L_{f} is diffeomorphic to S^{5}#l(S^{2} × S^{3}), where l = b_{2}(L_{f}) = 1 + ∑_{j} a_{j}.

7. The Moduli Problem for Sasakian-Einstein 5-Manifolds

In this section we discuss the moduli problem for Sasakian-Einstein manifolds. It is not our intention here to discuss the general moduli problem for Sasakian-Einstein structures, but rather to present some results regarding the 5 dimensional non-regular examples described in the previous sections together with the known regular Sasakian-Einstein 5-manifolds. In what follows by moduli space we shall mean certain sets of objects (sections of vector bundles) modulo the action of the group of diffeomorphisms that are diffeotopic to the identity. In the case of orbifolds diffeomorphism means diffeomorphism in the sense of orbifolds. Thus, in the case of complex moduli any two representatives are deformation equivalent.

We begin by briefly discussing the case complex structures on the del Pezzo surfaces obtained by blowing up ℂℙ^{2} at l distinct points for 0 ≤ l ≤ 8. We shall always assume that no two of these points lie on a line nor any three lie on a conic which we refer to as in general position. The complex structures on Z_{l} = ℂℙ^{2}#l\overline{ℂℙ}^{2} are then determined by the complex coordinates of the l points modulo the action of the complex automorphism group 𝔊(Z_{l}). Since we can fix precisely four points in general position with the action of GL(3, ℂ), the moduli space of complex structures M_{l}^{ℂ} on Z_{l} for l = 0, …, 4 is a single point, whereas for l = 5, 6, 7, 8 it is an open connected manifold of complex dimension 2(l − 4).

It is known that when l > 2, there exists [Siu,TY,Ti1] a unique [BM], up to complex automorphism, Kähler-Einstein structure associated to each complex structure. It follows that in this case (l > 2) there is a 1-1 correspondence between the complex structures on Z_{l} and the homothety classes of positive Kähler-Einstein metrics modulo complex automorphisms. This gives the identification of moduli spaces

7.1  M_{l}^{ℂ} ≃ M_{l}^{KE}, for l > 2

where M_{l}^{KE} denotes the moduli space of homothety classes of Kähler-Einstein metrics on Z_{l}. Moreover, when l = 1, 2 the space M_{l}^{KE} is well-known to be empty, whereas, M_{l}^{ℂ} is a single point space. Furthermore, for ℂℙ^{2} and ℂℙ^{1} × ℂℙ^{1} both M_{l}^{ℂ} and M_{l}^{KE} are single point spaces. When 4 < l < 9, it is possible that for two inequivalent complex structures J_{1}, J_{2} ∈ M_{l}^{ℂ} the Einstein metrics g_{1}, g_{2} solving the corresponding Monge-Ampere equations coincide. However, it follows from a theorem of Pontecorvo [Pon] that up to complex conjugation this can happen only when the metric is anti-self-dual, that is, the anti-self-dual part W_{+} of the Weyl conformal tensor must vanish. But it is well-known (cf. [Boy]) that a compact anti-self-dual Kähler surface has zero scalar curvature. So this cannot happen for del Pezzo surfaces. The precise result is given below in Proposition 7.13, and the argument works equally well for compact Kähler orbifolds.

In the more general case of log del Pezzo surfaces Z that is complex compact surfaces with positive first Chern class and at most quotient singularities much less is known. Here we are dealing with Kähler orbifolds and as discussed earlier the existence problem of finding Kähler-Einstein orbifold metrics is still open. However, the Bando-Mabuchi uniqueness theorem carries over to the orbifold case, so when a positive Kähler-Einstein orbifold metric exists it is unique up to homothety and complex automorphisms. Thus, generally, if M^{KE} denotes the moduli space of homothety classes of Kähler-Einstein metrics on Z, and M^{ℂ} denotes the moduli space of complex structures on Z, then there is an injective map

7.2  M^{KE}⟶M^{ℂ}.

Next we relate the moduli of Kähler-Einstein structures on Z to the moduli of Sasakian-Einstein structures on S. First we give a result in general dimension from our previous work [BG1], and then specialize to dimension five.

Proposition 7.3: Two 2n + 1 dimensional rank one Sasakian structures S = (ξ, η, Φ, g) and S′ = (ξ′, η′, Φ′, g′) are equivalent if and only if their corresponding spaces of leaves (Z, ω, J, h) and (Z′, ω′, J′, h′) are equivalent as Kähler orbifolds. Furthermore, S = (ξ, η, Φ, g) is Sasakian-Einstein if and only if (Z, ω, J, h) is Kähler-Einstein with scalar curvature 4n(n + 1).

Remark 7.4: Any Sasakian structure S = (ξ, η, Φ, g) has a canonically equivalent structure, namely the conjugate Sasakian structure S̄ = (−ξ, −η, −Φ, g). This corresponds on the space of leaves to the complex conjugate Kähler structure, namely (Z, −ω, −J, h).

Notice that the correspondence in Proposition 7.4 is between rank one Sasakian-Einstein structures and Kähler-Einstein structures with a fixed scalar curvature, or equivalently homothety classes of Kähler-Einstein structures. Now there are two possible types of deformations of Sasakian-Einstein structures, those that deform the foliation and those that do not. The latter all lie in one of the families 𝔉(ξ) discussed in section 1, and the former correspond to different base orbifolds Z (assuming both the original and deformed Sasakian-Einstein structures are rank one). We believe that Sasakian-Einstein structures lying in different families 𝔉(ξ) correspond to distinct components of the moduli space of Sasakian-Einstein structures, but we do not prove this here.

Now let us fix some notation. Let S_{l} = S^{5}#l(S^{2} × S^{3}) denote by M_{l}^{SE} the moduli space of Sasakian-Einstein structures on S_{l}, and let M_{l}^{reg,SE} and M_{l}^{1,SE} denote the moduli space of regular Sasakian-Einstein structures, and rank one Sasakian-Einstein structures on S_{l}, respectively. Then we have natural inclusions

7.5  M_{l}^{reg,SE} ⊂ M_{l}^{1,SE} ⊂ M_{l}^{SE}.

Combining our discussion above with [FK,BG1,Ti1-Ti3] we obtain for the regular case:

Proposition 7.6: The following hold:

(i) M_{l}^{reg,SE} is not empty if and only if l = 0, 1, 3, 4, 5, 6, 7, 8.

(ii) M_{l}^{reg,SE} is a single point when l = 0, 1, 3, 4.

(iii) M_{l}^{reg,SE} is a connected complex manifold of dimension 2(l − 4) when l = 5, 6, 7, 8. Furthermore, up to conjugation there is precisely one Sasakian structure sharing the same Einstein metric g.

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

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

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

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

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

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

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

where (a_{0}, a_{1}, a_{2}, a_{3}) ∈ (ℂ^{∗})^{4} and f_{i}^{(w_{i})}, i = 1, 2, 3 are weighted homogeneous polynomials of degree w_{i}. The simplest situation occurs when f_{1} = f_{2} = f_{3} are forced to vanish. Then 𝔊_{w} = (ℂ^{∗})^{3} is the smallest it can possibly be as ℙ(w) is toric. This is, in fact, common to many examples of the log del Pezzo suraces of Table 1. More precisely, we have Lemma 7.9: Let w_{0} < w_{1} < w_{2} < w_{3}. If for each i = 1, 2, 3 the weight w_{i} is not a ℤ^{+}-linear combination of smaller weights with non-negative integer coefficients than 𝔊_{w} = (ℂ^{∗})^{3}.

Let S_{w}^{d} ⊂ S(w) be the vector subspace spanned by all monomials in (z_{0}, z_{1}, z_{2}, z_{3}) of degree d = |w| − I, and let Ŝ^{d}(w) ⊂ S^{d}(w) denote subset all quasi-smooth elements, i.e. those polynomials f_{w} ∈ S_{w}^{d} such that conditions 3.4 hold. Then we define m_{w}^{d} to be the dimension of the subspace generated by Ŝ_{w}^{d}. Now the automorphism group G(w) acts on S_{w}^{d} leaving the subset Ŝ^{d}(w) of quasi-smooth polynomials invariant. Thus, for each log del Pezzo surface that is a solution of Theorem 4.5 we define the moduli space

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

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

7.11  M_{w}^{d}⟶M^{ℂ}(Z_{w}),

and by our results of sections 4 and 5, each element in M_{w}^{d} correponds to a unique homothety class of Kähler-Einstein metrics modulo 𝔊_{w} and hence, to a unique Sasakian-Einstein structure on the corresponding 5-manifold S_{l} modulo the group 𝔊_{w} acting as CR automorphisms. The results with non-trivial moduli are collected in Table 3 below.

Example 7.12: As an illustration, we shall calculate the moduli spaces of complex structures of the three classical del Pezzo surfaces of Table 2. We begin with the cubic in ℙ(1, 1, 1, 1). The subset Ŝ_{(1,1,1,1)}^{3} ⊂ ℂ^{20} is a dense open complex submanifold. In this case G(w) = GL(4, ℂ) and the quotient M_{(1,1,1,1)}^{3} is a complex manifold of dimension 4. It is well-known that in this case we have the identification Z_{f} ≃ Z_{6} so that M_{(1,1,1,1)}^{3} can be identified with M_{6}^{ℂ}.

The second example is a degree 4 surface in ℙ(1, 1, 1, 2). Here Ŝ^{4}(1, 1, 1, 2) ⊂ ℂ^{22} since the general weighted polynomial f(z) of degree 4 can be written as the sum

f(z) = g^{(4)}(z_{0}, z_{1}, z_{2}) + g^{(2)}(z_{0}, z_{1}, z_{2})z_{3} + λz_{3}^{2}.

The group G((1, 1, 1, 2)) is defined by

φ_{w} ( \begin{matrix}z_{0} \\ z_{1} \\ z_{2} \\ z_{3} \end{matrix} ) = ( \begin{matrix}𝔸 ( \begin{matrix}z_{0} \\ z_{1} \\ z_{2} \end{matrix} ) \\ αz_{3} + ϕ^{(2)}(z_{0}, z_{1}, z_{2}) \end{matrix} ) , 𝔸 ∈ GL(3, ℂ), α ∈ ℂ^{∗},

where ϕ^{(2)}(z_{0}, z_{1}, z_{2}) is an arbitrary homogeneous polynomial of degree 2 in (z_{0}, z_{1}, z_{2}). One can see that the action is free and, hence, M_{(1,1,1,2)}^{4} is a complex manifold of dimension n_{(1,1,1,2)}^{4} = 6. It is known that the smooth member Z_{f} ⊂ ℙ(1, 1, 1, 2) is diffeomorphic to Z_{7} so that M_{(1,1,1,2)}^{4} can be identified with M_{7}^{ℂ}.

The third example is a degree 6 surface in ℙ(1, 1, 2, 3). Here Ŝ^{6}(1, 1, 2, 3) ⊂ ℂ^{23} as the general weighted polynomial f(z) of degree 6 can be written as the sum

f(z) = g^{(6)} + g^{(4)}z_{2} + g^{(2)}z_{2}^{2} + λ_{1}z_{2}^{3} + g^{(3)}z_{3} + g^{(1)}z_{2}z_{3} + λ_{2}z_{3}^{2},

where g^{(i)} = g^{(p)}(z_{0}, z_{1}) is a homogeneous polynomial of degree p = 1, 2, 3, 4, 6. The group G((1, 1, 2, 3)) in defined by

φ_{w} ( \begin{matrix}z_{0} \\ z_{1} \\ z_{2} \\ z_{3} \end{matrix} ) = ( \begin{matrix}𝔸 ( \begin{matrix}z_{0} \\ z_{1} \end{matrix} ) \\ α_{2}z_{2} + ϕ_{2}^{(2)}(z_{0}, z_{1}) \\ α_{3}z_{3} + ϕ_{3}^{(3)}(z_{0}, z_{1}) + ϕ_{3}^{(1)}(z_{0}, z_{1})z_{2} \end{matrix} ) , 𝔸 ∈ GL(2, ℂ), α_{2}, α_{3} ∈ ℂ^{∗},

and ϕ_{i}^{(p)}, i = 1, 2, 3 are homogeneous polynomials in (z_{0}, z_{1}) of degree p. One can see that dim(G(1, 1, 2, 3)) = 15 and M_{(1,1,2,3)}^{6} is a connected complex manifold of dimension n_{(1,1,2,3)}^{6} = 8. Again, the general smooth member Z_{f} ⊂ ℙ(1, 1, 2, 3) is diffeomorphic to Z_{8} so that M_{(1,1,2,3)}^{4} can be identified with M_{8}^{ℂ}.

One can carry out similar calculations for any of the log del Pezzo surfaces. In Table 1 below we tabulate all the examples of log del Pezzo surfaces of Theorem 4.5 which admit Kähler-Einstein metrics and for which n_{w}^{d} ≥ 1 and indicate the corresponding link L_{f} in S^{7} :

Table 3. Log del Pezzo surfaces with n_{w}^{d} ≥ 1

Index  w  d  m_{w}^{d}  n_{w}^{d}  S_{l} = L_{f} ⊂ S^{7}
1  (2,2k+1,2k+1,4k+1)  8k+4  12  5  #7(S^{2} × S^{3})
1  (2,3,5,9)  18  13  5  #6(S^{2} × S^{3})
2  (3,4,10,15)  30  10  3  #6(S^{2} × S^{3})
1  (3,5,7,14)  28  9  4  #5(S^{2} × S^{3})
1  (3,5,11,18)  36  10  3  #5(S^{2} × S^{3})
2  (3,4,5,10)  20  9  3  #4(S^{2} × S^{3})
1  (3,5,7,11)  25  8  3  #4(S^{2} × S^{3})
1  (3,3,5,5)  15  10  2  #4(S^{2} × S^{3})
2  (7,8,19,32)  64  7  2  #3(S^{2} × S^{3})
3  (5,7,11,20)  40  7  2  #3(S^{2} × S^{3})
1  (5,14,17,21)  56  5  1  #3(S^{2} × S^{3})
1  (5,19,27,50)  100  6  1  #3(S^{2} × S^{3})
1  (7,11,27,44)  88  6  1  #3(S^{2} × S^{3})
2  (6,9,10,13)  36  5  1  #3(S^{2} × S^{3})
1  (5,19,27,31)  81  5  1  #2(S^{2} × S^{3})
2  (7,8,19,25)  57  5  1  #2(S^{2} × S^{3})

Next we turn to the equivalence problem for the Einstein metrics. As mentioned previously for the log del Pezzo surfaces with a Y in the last column of Table 1 and the tables of Theorem 4.5, there is a unique homothety class of Kähler-Einstein metrics corresponding to each point of M_{w}^{d}. But the question remains whether two inequivalent Kähler-Einstein structures can share the same Riemannian metric. This is answered by a Theorem of Pontecorvo [Pon] which can be restated for our purposes as:

Proposition 7.13: Let (J, J′) be two distinct log del Pezzo structures on the same underlying orbifold Z that are both compatible with the same Einstein metric g. Then J and J′ are complex conjugates, i.e. J′ = −J.

Proof: By Proposition 3.1 of [Pon] if J′ ≠ −J then the metric g is anti-self-dual, i.e. W_{+} = 0. From [Boy] anti-self-dual Kähler metrics must have vanishing scalar curvature, and this cannot happen for log del Pezzo surfaces which are positive.

Conjugate Kähler-Einstein structures correspond to the same point of the moduli space M_{w}^{d}, so each point of M_{w}^{d} corresponds to a distinct diffeomorphism class of Kähler-Einstein metrics with scalar curvature 4n(n+1). There are similar results on the Sasakian level which are essentially due to Tanno [Tan] and Tachibana and Yu [TaYu]. We reformulate this result as follows:

Proposition 7.14: Let M be a (4n+1)-dimensional compact manifold. Let S = (ξ, η, Φ, g) and S′ = (ξ′, η′, Φ′, g) be two distinct Sasakian structures on M sharing the same Riemannian metric g. Suppose further that (M, g) is not a sphere with the standard round Sasakian metric. Then S′ and S are conjugate Sasakian structures, i.e. S′ = (−ξ, −η, −Φ, g).

Proof: Since (M, g) is not a round sphere, a theorem of Tachibana and Yu [TaYu] says that g(ξ, ξ′) is a constant, say a. By Schwarz inequality |a| ≤ 1 and since the Sasakian structures are distinct we must have −1 ≤ a < 1. If a = −1 then S and S′ are conjugate Sasakian structures. So assume |a| < 1, then following Tanno [Tan] we define

ξ′′ = \frac{ξ′ − aξ}{\sqrt{1 − a^{2}}} .

Now we have

g(ξ, ξ′′) = 0, g(ξ′′, ξ′′) = 1,

and it follows from the characterization of Sasakian structures in terms of the Riemannian curvature (cf. [BG2]) that ξ′′ defines a Sasakian structure S′′ which is orthogonal to S. But then this would define a 3-Sasakian structure [Tan] which cannot exist in dimension 4n + 1. Thus, S and S′ are conjugate Sasakian structures.

Now combining Theorems 1.23, 4.5, 6.6, and Proposition 7.14 with the information in Tables 1 and 3 proves Theorem A of the Introduction.

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Department of Mathematics and Statistics December 2000 University of New Mexico Revised, February 2001 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