Contact Structures Hypersurface Singularities

and Einstein Metrics

KRZYSZTOF GALICKI Max-Planck-Institut für Mathematik, Bonn

and

University of New Mexico, Albuquerque

Srni, January 2004

CONTENTS

0. Introduction

(i) some problems and questions (ii) some answers and theorems

1. Contact and Sasakian manifolds

(i) contact and almost contact structures (ii) Sasakian manifolds and fundamental foliations (iii) examples of 3-Sasakian manifolds (iv) examples of Sasakian-Einstein manifolds

2. Links of isolated hypersurface singularities

(i) Sasakian structure on links (ii) Milnor fibration theorem (iii) Brieskorn-Pham links (iv) Brieskorn Graph theorem

(v) exotic spheres as Brieskorn-Pham links

3. Kähler-Einstein metrics on Fano orbifolds

(i) Yau’s proof of Calabi Conjecture (ii) Monge-Ampère equations and existence of

Kähler-Einstein metrics (iii) transverse Yau’s theorem (iv) Demailly-Kollár generalization of continuity method to

Fano orbifolds

REFERENCES

1. C. P. Boyer, K. Galicki, Einstein metrics on rational homology spheres, preprint 2003.

2. C. P. Boyer, K. Galicki, J. Kollár, Einstein metric on spheres, preprint 2003.

3. C. P. Boyer, K. Galicki, J. Kollár, E. Thomas, Einstein metrics on exotic spheres in dimension 7,11, and 15, preprint 2003.

4. C. P. Boyer, K. Galicki, M. Nakamaye, On Sasakian-Einstein Geometry of 5-Manifolds, Mathematische Annalen 325 (2003), 485-524.

5. C. P. Boyer, K. Galicki, M. Nakamaye, Sasakian Geometry, Homotopy Spheres, and Positive Ricci Curvature, Topology 42 (2003), 981-1002.

6. C. P. Boyer, K. Galicki, New Einstein Manifolds in Dimension Five, Journal of Differential Geometry 57 (2001), 443-467.

7. J.-P. Demailly, J. Kollár, Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Scient. Ec. Norm. Sup. Paris 34 (2001), 525-556.

8. C. P. Boyer, K. Galicki, Sasakian Manifolds, Oxford University Press (hopefully in 2005).

More, including these slides, at http://www.math.unm.edu/˜ galicki.

0. Introduction

Some Problems and Questions

We want to ask questions regarding the existence of positive Ricci curvature metrics (including the case constant positive Ricci curvature, i.e., Einstein with positive Einstein constant) on contact M^{2n+1}. Let us start with

• dim(M)=3: Well-know result of Martinet says that very orientable 3-manifold admits a contact structure. Another celebrated theorem of Hamilton shows that proving existence of a positive Ricci curvature on every simply connected 3-manifold is equivalent to proving Poincaré Conjecture. On the other hand Every Einstein 3-manifold is of constant curvature.

• dim(M)=5: In 1965 Barden proved the following remarkable theorem

Theorem: The class of simply connected, closed, oriented, smooth, 5-manifolds is classifiable under diffeomorphism. Furthermore, any such M is diffeomorphic to one of the spaces M_{j;k_{1},...,k_{s}} = X_{j}#M_{k_{1}}# ⋯ #M_{k_{s}}, where −1 ≤ j ≤ ∞, s ≥ 0, 1 < k_{1} and k_{i} divides k_{i+1} or k_{i+1} = ∞. A complete set of invariants is provided by H_{2}(M, ℤ) and an additional diffeomorphism invariant i(M) = j which depends only on the second Stiefel-Whitney class w^{2}(M).

In these lectures we will refer to a simply connected, closed, oriented, smooth, 5-manifold as a Barden manifold.

BUILDING BLOCKS X_{−1} = SU(3)/SO(3); H_{2}(X_{−1}, ℤ) = ℤ_{2}, X_{∞} = non − trivial S^{3} bundle over S^{2}; H_{2}(X_{∞}, ℤ) = ℤ, X_{j}, j ∈ ℕ, H_{2}(X_{j}, ℤ) = ℤ_{2^{j}} ⊕ ℤ_{2^{j}} X_{0} = S^{5}, M_{∞} = S^{2} × S^{3} H_{2}(M_{∞}, ℤ) = ℤ, M_{α}, α ∈ ℕ; H_{2}(M_{α}, ℤ) = ℤ_{α} ⊕ ℤ_{α} When M is spin i(M) = j = 0 as is w^{2}(M) = 0 and Barden’s result is the extension of the well-known theorem of Smale for spin 5-manifolds. By an old theorem of John Gray M admits an almost contact structure when j = 0, ∞ and by another result of Geiges M is in such a case necessarily contact. We have the following inclusions:

{Barden Manifolds}

∪ {Contact Manifolds} ⊃ {Sasakian Manifolds}

∪ {Spin Manifolds}

∪ {Positive Sasakian Manifolds}

∪ {Sasakian − Einstein Manifolds} A. Which Barden Manifolds admit metrics of positive Ricci curvature?

The expected answer is ALL but remarkably little is known about the subject. One positive result was proved by Sha and Yang which, in dimension 5, reads

Theorem: #n(S^{2} × S^{3}) admits a positive Ricci curvature metric for all n ∈ ℕ.

In addition, the symmetric metrics on S^{5}, X_{−1} and a bundle metric on X_{∞} are also of positive Ricci curvature.

B. Which Barden Manifolds admit Einstein metrics of positive Ricci curvature?

There are several constructions that lead to such Einstein metrics in dimension 5.

[Kobayashi Bundle Construction] An old construction of Kobayashi asserts that certain unique simply connected circle bundle over any del Pezzo surface with a Kähler-Einstein metric admits an Einstein metric of positive scalar curvature. Existence results of Siu, Tian-Yau, and Tian settle the problem as we have

Theorem: The following del Pezzo surfaces admit Kähler-Einstein metrics: ℂℙ^{2}, ℂℙ^{1} × ℂℙ^{1}, ℂℙ^{2}#n\overline{ℂℙ^{2}}, 3 ≤ n ≤ 8. Furthermore, the moduli space of K-E structures in each case is completely understood.

There are two del Pezzo surfaces which do not admit any K-E metrics due to theorem of Matsushima: the existence is obstructed by holomorphic vector fields. These are blow-ups of ℂℙ^{2} at one or two points.

As and immediate consequence, for 5-manifolds, we have the following (Friedrich-Kath, Boyer-Galicki):

Theorem: 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}.

[Homogeneous Einstein Metrics] The symmetric metrics on X_{0} = S^{5}, X_{−1} = SU(3)/SO(3) are Einstein. Furthermore, M_{∞} = S^{2} × S^{3} admits infinitely many homogeneous Einstein metrics (Wang and Ziller).

[Böhm Metrics] Böhm has explicitly constructed cohomogeneity one Einstein metrics on S^{5} and S^{2} × S^{3}.

• dim(M)=2n+1 In higher dimensions there are many constructions of positive Ricci curvature metrics as well as Einstein metrics. We will be more selective in our choice of problems and will restrict our interest to the case when M is homeomorphic (but not necessarily diffeomorphic) to the sphere S^{2n+1}.

Questions A1: Do any exotic 7-spheres admit metrics of positive Ricci curvature?

This question was answered by Wraith who showed that Theorem: All spheres which bound parallelizable manifolds admit metrics of positive Ricci curvature.

By a result of Hitchin it is known that some exotic spheres (first in dimension 9) cannot admit a metric of positive scalar curvature. In particular such spheres will not admit metrics of positive Ricci curvature. It is still not know if any of the exotic spheres admit metrics of positive sectional curvature.

Any standard sphere S^{n} admits a metric of constant sectional curvature. These canonical metrics are homogeneous and Einstein. The spheres S^{4m+3}, m > 1 are known to have another Sp(m+1)-homogeneous Einstein metric discovered by Jensen in 1974. In addition, S^{15} has a third Spin(9)-invariant homogeneous Einstein metric discovered by Bourguignon and Karcher. In 1982 Ziller proved that these are the only homogeneous Einstein metrics on spheres. No other Einstein metrics on spheres were known until 1998 when Böhm constructed infinite sequences of non-isometric Einstein metrics, of positive scalar curvature, on S^{5}, S^{6}, S^{7}, S^{8}, and S^{9}. Böhm’s metrics are of cohomogeneity one and they are not only the first inhomogeneous Einstein metrics on spheres but also the first non-canonical Einstein metrics on even-dimensional spheres. Even with Böhm’s result Einstein metrics on spheres appeared to be rare.

Questions B1: Are there other inhomogeneous Einstein metrics on standard spheres?

Questions B2: Do some exotic spheres admit Einstein metrics of positive scalar curvature?

Some Answers and Theorems

Theorem 1: [Boyer,–] For every integer k > 2 that is either relatively prime to 3 or 2, there exist cohomogeneity 4 Sasakian-Einstein metrics, depending on two real parameters, on a simply connected rational homology 5-sphere M_{k}^{5} with w_{2}(M_{k}^{5}) = 0, and H_{2}(M_{k}^{5}, ℤ) having order k^{2}. Furthermore, in each odd dimensions there are infinitely many rational homology spheres admitting families of Sasakian-Einstein metrics with one dimensional isometry group.

Theorem 2: [Boyer,–,Nakamaye] For every integer l ≥ 1, l#(S^{2} × S^{3}) admits a Sasakian metric with positive Ricci curvature.

Theorem 3: [Boyer,–,Nakamaye] All spheres which bound parallelizable manifolds admit Sasakian metrics of positive Ricci curvature.

Theorem 4: [Boyer,–,Nakamaye] S_{l} = l#(S^{2} × S^{3}) admit infinite families of quasi-regular Sasakian-Einstein metrics for 1 < l ≤ 9. S_{1} admits at least 14 quasi-regular inequivalent Sasakian-Einstein metrics.

Theorem 5: [Boyer,–,Kollár] On S^{5} we obtain at least 68 inequivalent families of Sasakian-Einstein metrics. Some of these admit non-trivial continuous Sasakian-Einstein metrics deformations. The biggest constructed family has has (real) dimension 10.

Theorem 6: [Boyer,–,Kollár] All 28 oriented diffeomorphism classes on S^{7} admit inequivalent families of Sasakian-Einstein metrics structures, some of them (in each diffeomorphism class) depending on moduli.

In each case, the number of families ranges from 231 to 452. Moreover, there are fairly large moduli. For example, the standard 7-sphere admits an 82-dimensional family of Sasakian-Einstein metrics. All these metrics have onedimensional isometry group.

Theorem 7: [Boyer,–,Kollár] The (4n + 1)-dimensional standard and Kervaire spheres both admit many families of inequivalent Sasakian-Einstein metrics for each n ≥ 2.

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

Conjecture: [Boyer,–,Kollár] All odd-dimensional homotopy spheres which bound parallelizable manifolds admit Sasakian-Einstein metrics.

Theorem 8: [Boyer,–,Kollár, Thomas] The conjecture is true in dimension 11 and 15. More precisely, each homotopy sphere (992 possible diffeomorphism types) in dimension 11 admits at least one Sasakian-Einstein metric and each homotopy 15-sphere which bound a parallelizable manifold (8128 possible diffeomorphism types) admits at least one Sasakian-Einstein metric.

Theorem: [Kollár] For every l ≥ 6, there are infinitely many (2l − 2)-dimensional families of Einstein metrics on S_{l} = l#(S^{2} × S^{3}).

1. Contact and Sasakian Manifolds

Contact transformations arose in the theory of Analytical Mechanics developed in the 19th century by Hamilton, Jacobi, Lagrange, and Legendre. But its first systematic treatment was given by Sophus Lie. Consider ℝ^{2n+1} with Cartesian coordinates (x^{1}, ⋯, x^{n}; y^{1}, ⋯, y^{n}; z), and a 1-form η given by

(1.1)  η = dz − ∑_{i} y^{i}dx^{i}.

It is easy to see that η satisfies η ∧ (dη)^{n} ≠ 0. A 1-form on ℝ^{2n+1} that satisfies this equation is called a contact form. Locally we have the following

Theorem 1.2 [Darboux] Let η be a 1-form on ℝ^{2n+1} that satisfies η ∧(dη)^{n} ≠ 0. Then there is an open set U ⊂ ℝ^{2n+1} and local coordinates (x^{1}, ⋯, x^{n}; y^{1}, ⋯, y^{n}; z) such that η has the form (1.1) in U.

Definition 1.3: A (2n + 1)-dimensional manifold M is a contact manifold if there exists a 1-form η, called a contact 1-form, on M such that

η ∧ (dη)^{n} ≠ 0

everywhere on M. A contact structure on M is an equivalence class of such 1-forms, where η′ ∼ η if there is a nowhere vanishing function f on M such that η′ = fη.

Lemma 1.4: On a contact manifold (M, η) there is a unique vector field ξ, called the Reeb vector field, satisfying the two conditions

ξ⌋η = 1, ξ⌋dη = 0.

Definition 1.5: An almost contact structure on a differentiable manifolds M is a triple (ξ, η, Φ), where Φ is a tensor field of type (1, 1) (i.e. an endomorphism of T M), ξ is a vector field, and η is a 1-form which satisfy

η(ξ) = 1 and Φ ∘ Φ = −𝕀 + ξ ⊗ η,

where 𝕀 is the identity endomorphism on T M. A smooth manifold with such a structure is called an almost contact manifold.

Let (M, η) be a contact manifold with a contact 1-form η and consider D = ker η ⊂ T M. The subbundle D is maximally non-integrable and it is called the contact distribution. The pair (D, ω), where ω is the restriction of dη to D gives D the structure of a symplectic vector bundle. We denote by J (D) the space of all almost complex structures J on D that are compatible with ω, that is the subspace of smooth sections J of the endomorphism bundle End(D) that satisfy

(1.6) J^{2} = −𝕀, dη(JX, JY) = dη(X, Y), dη(X, JX) > 0

for any smooth sections X, Y of D. Notice that each J ∈ J (D) defines a Riemannian metric g_{D} on D by setting

(1.7)  g_{D}(X, Y) = dη(X, JY).

One easily checks that g_{D} satisfies the compatibility condition g_{D}(JX, JY) = g_{D}(X, Y). Furthermore, the map J ↦ g_{D} is one-to-one, and the space J (D) is contractible. A choice of J gives M an almost CR structure.

Moreover, by extending J to all of T M one obtains an almost contact structure. There are some choices of conventions to make here. We define the section Φ of End(T M) by Φ = J on D and Φξ = 0, where ξ is the Reeb vector field associated to η. We can also extend the transverse metric g_{D} to a metric g on all of M by

1.8 g(X, Y) = g_{D}+η(X)⊗η(Y) = dη(X, ΦY)+η(X)⊗η(Y)

for all vector fields X, Y on M. One easily sees that g satisfies the compatibility condition g(ΦX, ΦY) = g(X, Y) − η(X)η(Y).

Definition 1.9: A contact manifold M with a contact form η, a vector field ξ, a section Φ of End(T M), and a Riemannian metric g which satisfy the conditions

η(ξ) = 1, Φ^{2} = −𝕀 + ξ ⊗ η,
g(ΦX, ΦY) = g(X, Y) − η(X)η(Y)

is known as a metric contact structure on M.

Definition-Proposition 1.10: 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.

We refer to the quadruple S = (g, η, ξ, Φ) as a Sasakian structure on M, where η is the 1-form dual vector field ξ. It is easy to see that η is a contact form whose Reeb vector field is ξ. In particular S = (g, η, ξ, Φ) is a special type of metric contact structure.

The vector field ξ is nowhere vanishing, so there is a 1-dimensional foliation F_{ξ} associated with every Sasakian structure, called the characteristic foliation. We will denote the space of leaves of this foliation by Z. Each leaf of F_{ξ} has a holonomy group associated to it. The dimension of the closure of the leaves is called the rank of S. We shall be interested in the case rk(S) = 1.

Definition 1.11: When rk(S) = 1 we say that Sasakian structure S is quasi-regular. If F_{ξ} defines a principal S^{1} bundle, we say that S is regular.

When S is compact and quasi-regular then then Z has a structure of a Riemannian orbifold (or a V-manifold). Z is a smooth manifold in the regular case.

Definition-Proposition 1.12: A Sasakian space (M, g) is Sasakian-Einstein if the metric g is also Einstein. For any 2n+1-dimensional Sasakian manifold Ric(X, ξ) = 2nη(X) implying that any Sasakian-Einstein metric must have positive scalar curvature. Thus any complete Sasakian-Einstein manifold must have a finite fundamental group. Furthermore the metric cone on M (C(M), ḡ) = (ℝ_{+}×M, dr^{2}+r^{2}g) is Kähler Ricc-flat (Calabi-Yau).

The following are the two fundamental structure theorems for Sasakian and Sasakian-Einstein manifolds:

Theorem 1.13: [Boyer,–] Let (M, g) be a compact quasiregular Sasakian manifold of dimension 2n+1, and let Z denote the space of leaves of the characteristic foliation. Then

(i) The leaf space Z is a Hodge orbifold with Kähler metric h and Kähler form ω which defines an integral class [ω] in H_{orb}^{2}(Z, ℤ) so that π : (M, g)⟶(Z, h) is an orbifold Riemannian submersion. The fibers of π are totally geodesic submanifolds of M diffeomorphic to S^{1}.

(ii) (M, g) is Sasakian-Einstein iff (Z, h) is Kähler-Einstein with scalar curvature 4n(n + 1).

Theorem 1.14: [Boyer,–] Let (Z, h) be a Hodge orbifold. Let π : M⟶Z be the S^{1} V-bundle whose first Chern class is [ω], and let η be a connection 1-form in M whose curvature is 2π^{∗}ω, then M with the metric π^{∗}h + η ⊗ η is a Sasakian orbifold. Furthermore, if all the local uniformizing groups inject into the group of the bundle S^{1}, the total space M is a smooth Sasakian manifold.

Definition 1.15: A Sasakian manifold (M, g) is called a positive if the transverse geometry Z is Fano.

Positivity, or the Fano condition just as in the smooth case can be expressed in terms of orbifold Chern classes of the Z (first Chern class c_{1}(Z)), or in terms of basic Chern classes of the characteristic foliation F_{ξ} (first Chern class c_{1}(F_{ξ})).

Theorem 1.16: [Boyer,–, Nakamaye] Any positive Sasakian manifold (M, g) admits a Sasakian metric g′ of positive Ricci curvature.

The following is an orbifold version of the famous Kobayashi bundle construction of Einstein metrics on bundles over positive Kähler-Einstein manifolds.

Theorem 1.17: Let (Z, h) be a compact Fano orbifold with π_{1}^{orb}(Z) = 0 and Kähler-Einstein metric h. Let π : M⟶Z be the S^{1} V-bundle whose first Chern class is \frac{c_{1}(Z)}{Ind(Z)}. Suppose further that the local uniformizing groups of Z inject into S^{1}. Then with the metric g = π^{∗}h + η ⊗ η, M is a compact simply connected Sasakian-Einstein manifold.

A very special class of Sasakian-Einstein spaces is naturally related to several quaternionic geometries.

Definition 1.18: Let (M, g) be a Riemannian manifold of dimension m. We say that (M, g) is 3-Sasakian if the metric cone (C(M), ḡ) = (ℝ_{+} × S, dr^{2} + r^{2}g) on M is hyperkähler.

1.19  \begin{aligned}C(M) & ↩ & M \\ ↓ & π \\ & & Z \end{aligned}

C(M): CONE  M/F_{ξ}: TRANSVERSE
GEOMETRY  M  GEOMETRY
SYMPLECTIC CONTACT  SYMPLECTIC

KÄHLER  SASAKIAN  KÄHLER

POSITIVE

KÄHLER  SASAKIAN  FANO

CALABI-YAU SASAKIAN  FANO
EINSTEIN  KÄHLER-EINSTEIN

HYPER-  FANO, ℂ-CONTACT
KÄHELER  3-SASAKIAN KÄHLER-EINSTEIN

HYPERKAHLER GEOMETRY

TWISTOR GEOMETRY

ℝ_{+} × M_{↙  ↘}_{Z  ⟵ ↓  M}_{↘  ↙} O

3−SASAKIAN GEOMETRY

QUATERNIONIC KAHLER GEOMETRY

EXAMPLES OF S-E MANIFOLDS

Example 1.20: 3-Sasakian Manifolds:

(a) “flat example”

ℍ^{n} \ {0}
↙  ↘
ℂℙ^{2n−1}  ⟵ ↓  S^{4n−1}
↘  ↙
ℙℍ^{n−1}

(b) homogeneous examples

C(G/L)

↙  ↘
G/L · U(1)  ⟵ ↓  G/L
↘  ↙

G/L · Sp(1)

(c) toric examples [Boyer,–, Mann, Rees’98]

ℍ^{n} \ {0}
↙  ↘
ℂℙ^{2n−1}  ⟵ ↓  S^{4n−1}
↘  ↙
ℙℍ^{n−1}

T^{k}  ↓QUOT IENT

ℝ_{+} × S(Ω)
↙  ↘
Z(Ω)  ⟵ ↓  S(Ω)
↘  ↙
O(Ω)

Here, Ω is an integral matrix which defines a homomorphism f_{Ω} : T^{k}⟶U(n). In the case dim(S(Ω)) = 7, there are choices of Ω for any k ≥ 1 which make S(Ω) smooth. Since b_{2}(S(Ω)) = k we conclude that there exist Einstein manifolds with arbitrarily large second Betti number. These were first such examples.

(d) first non-toric examples [Boyer,–, Piccinni’02] (in dimension 11,15) were obtained by non-Abelian reduction of the flat example, where instead of the torus G = T^{k} one takes G = Sp(1) × T^{k}.

(e) more recently non-toric examples in dimension 7 were obtained by Grove, Wilking, and Ziller. They use orbifold bundle construction with the examples of orbifold twistor space and self-dual Einstein metrics Z_{k}⟶O_{k} discovered by Nigel Hitchin in 1996. the self-dual Einstein metric on O_{k} is defined on S^{4} \ ℝℙ^{2} and it has ℤ_{k} orbifold singularity along ℝℙ^{2}. However, it turns out that the bundle M_{k}⟶Z_{k} is actually smooth. In particular, one can compute integral cohomology ring of M_{k}. For odd k the 3-Sasakian manifold M_{k} is a rational homology 7-sphere with non-zero torsion depending on k. Hence, there exist infinitely many rational homology 7-spheres which have 3-Sasakian metrics.

Example 1.21: Other Sasakian-Einstein Manifolds:

(a) complex Hopf fibration:

ℂ^{n} \ {0}
↘
↓  S^{2n−1}
↙
ℙℂ^{n−1}

(b) There are many homogeneous examples; all compact homogeneous Kähler-Einstein spaces are classified and they are of the form Z = G/P. Hence, one can replace the complex projective space with Z = G/P and apply the usual Kobayashi construction.

(c) There are also many inhomogeneous examples. One can take any smooth compact Fano variety Z which admits a Kähler-Einstein metric. For example, in the case of complex surfaces it is know exactly which Fano (del Pezzo) surfaces admit a K-E metric. These were already mentioned in the introduction.

Until recently, examples of smooth irregular Sasakian-Einstein manifolds were rare (with the exception of inhomogeneous 3-Sasakian spaces mentioned above). On the other there are plentiful examples of compact irregular Sasakian, or even positive Sasakian manifolds. In these lectures we will concentrate on one particular example.

2. Links of isolated hypersurface singularities

[The basic reference on links is Milnor’s beautiful book: Singular Points of Complex Hypersurfaces.]

Consider the affine space ℂ^{n+1} together with a weighted ℂ^{∗}-action given by (z_{0}, …, z_{n}) ↦ (λ^{w_{0}}z_{0}, …, λ^{w_{n}}z_{n}), where the weights w_{j} are positive integers. It is convenient to view the weights as the components of a vector w ∈ (ℤ^{+})^{n+1}, and we shall assume that gcd(w_{0}, …, w_{n}) = 1. Let f be a weighted homogeneous polynomial, that is f ∈ ℂ[z_{0}, …, z_{n}] and satisfies

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 shall assume that the origin in ℂ^{n+1} is an isolated singularity or to further simplify matters that it is the only singularity.

We are interested in the link L_{f} defined by

2.1  L_{f} = {f = 0} ∩ S^{2n+1},

where

2.2  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}. L_{f} is endowed with a natural quasi-regular Sasakian structure inherited as a Sasakian submanifold of the sphere S^{2n+1} with its “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 2.3

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 quotient of S^{2n+1} by the “weighted S^{1}-action” generated by the vector field ξ_{w} is the weighted projective space ℙ(w) = ℙ(w_{0}, …, w_{n}), and we have a commutative diagram:

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

where the horizontal arrows are Sasakian and Kählerian embeddings, respectively, and the vertical arrows are orbifold Riemannian submersions. L_{f} is the total space of the principal S^{1} V-bundle over the orbifold Z_{f} whose first Chern class is \frac{c_{1}(Z)}{I} ∈ H^{2}(Z_{f}, ℚ), and η_{w} is a connection in this V-bundle whose curvature is dη.

Proposition 2.5: The orbifold Z_{f} is Fano if and only if d − ∑ w_{i} < 0.

Now, recall the well-known construction of Milnor 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

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

The closure F̄ of F has the same homotopy type as F and is a compact manifold with boundary precisely the link L_{f}. So the reduced homology of F and F̄ is only non-zero in dimension n and H_{n}(F, ℤ) ≈ ℤ^{μ}. Using the Wang sequence of the Milnor fibration together with Alexander-Poincare duality gives the exact sequence 2.7

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

where h_{∗} is the monodromy map (or characteristic map) induced by the S_{w}^{1} action. From this we see that H_{n}(L_{f}, ℤ) = ker(𝕀 − h_{∗}) is a free Abelian group, and H_{n−1}(L_{f}, ℤ) = Coker(𝕀 − h_{∗}) which in general has torsion, but whose free part equals ker(𝕀 − h_{∗}). So the topology of L_{f} is encoded in the monodromy map h_{∗}. There is a well-known algorithm due to Milnor and Orlik for computing the free part of H_{n−1}(L_{f}, ℤ) in terms of the characteristic polynomial Δ(t) = det(t𝕀 − h_{∗}), namely the Betti number b_{n}(L_{f}) = b_{n−1}(L_{f}) equals the number of factors of (t − 1) in Δ(t). First we mention an important immediate consequence of the exact sequence (2.7) which is due to Milnor:

Proposition 2.8: The following hold:

(i) L_{f} is a rational homology sphere if and only if Δ(1) ≠ 0. (ii) L_{f} is a homology sphere if and only if |Δ(1)| = 1. (iii) If L_{f} is a rational homology sphere, then the order of

H_{n−1}(L_{f}, ℤ) equals |Δ(1)|.

Example 2.9:

(a) w = (1, 1, 1, 1), f = z_{0} + z_{1} + z_{2} + z_{3}, d = 1. Z_{f} = ℂℙ(2), L_{f} = S^{5}.

(b) w = (1, 1, 1, 1), f = z_{0}^{2} + z_{1}^{2} + z_{2}^{2} + z_{3}^{2}, d = 2. Z_{f} = ℂℙ(1) × ℂℙ(1), L_{f} = S^{2} × S^{3}.

(c) w = (1, 1, 1, 1), f = z_{0}^{3} + z_{1}^{3} + z_{2}^{3} + z_{3}^{3}, d = 3.

Z_{f} = ℂℙ(2)#6\overline{ℂℙ(2)} , L_{f} = 6#(S^{2} × S^{3}).

(d) w = (1, 1, 1, 2), f = z_{0}^{4} + z_{1}^{4} + z_{2}^{4} + z_{3}^{2}, d = 4.

Z_{f} = ℂℙ(2)#7\overline{ℂℙ(2)} , L_{f} = 7#(S^{2} × S^{3}).

(e) w = (1, 1, 2, 3), f = z_{0}^{6} + z_{1}^{6} + z_{2}^{3} + z_{3}^{2}, d = 6.

Z_{f} = ℂℙ(2)#8\overline{ℂℙ(2)} , L_{f} = 8#(S^{2} × S^{3}).

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

L_{f} = k#(S^{2} × S^{3}).

(g) w = (1, 2, 3, 5), f = z_{0}^{10} + z_{1}^{5} + z_{2}^{3}z_{1} + z_{3}^{2}, d = 10

L_{f} = 9#(S^{2} × S^{3}).

(h) w = (11, 29, 39, 49), f = z_{0}^{8}z_{2}+z_{1}^{4}z_{0}+z_{2}^{3}+z_{3}^{2}z_{1}, d = 127

L_{f} = 2#(S^{2} × S^{3}).

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

L_{f} = S^{2} × S^{3}.

(j) w = (17, 34, 75, 125, 175), f = homework, d = 425

|H_{3}(L_{f}, ℤ)| = 17^{12}.

(k) w = (127, 2266, 3651, 6043, 8435), f = homework, d = 20521

|H_{3}(L_{f}, ℤ)| = 20521.

(l) w = (127, 2392, 3399, 6043, 8561), f = homework, d = 20521

|H_{3}(L_{f}, ℤ)| = 20521.

(m) w = (6, 2(6k − 1), 3(6k − 1), 3(6k − 1), 3(6k − 1)), f = z_{0}^{6k−1} + z_{1}^{3} + z_{2}^{2} + z_{3}^{2} + z_{4}^{2}, d = 6(6k − 1)

Z_{f} ≃ ℙ(2, 1, 1, 1), L_{f} ≃ S^{7}.

But the smooth structure depends on k.

(n) w = (2, 2p + 1, 2p + 1, 2p + 1, 2p + 1, 2p + 1), f = z_{0}^{2p+1} + z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + z_{4}^{2} + z_{5}^{2}, d = 2(2p + 1)

Z_{f} ≃ ℙ(1, 1, 1, 1, 1), L_{f} ≃ S^{9}.

But the smooth structure depends on p.

In some of the above examples the homogeneous polynomial f contains no “mixed” monomial terms of the form z_{i}^{c_{i}}z_{j}^{c_{j}}. Such an f is called of Brieskorn-Pham type (abbreviated BP type). In his famous work, in 1966 Brieskorn considered links L(a) defined by

2.10  ∑_{i=0}^{n} |z_{i}^{2}| = 1,  f_{a}(z) = z_{0}^{a_{1}} + ⋯ + z_{n}^{a_{n}} = 0.

To the vector a = (a_{0}, ⋯, a_{n}) ∈ ℤ_{+}^{n+1} one associates a graph G(a) whose n+1 vertices are labeled by a_{0}, ⋯, a_{n}. Two vertices a_{i} and a_{j} are connected if and only if gcd(a_{i}, a_{j}) > 1. Let C_{ev} denote the connected component of G(a) determined by the even integers. Note that all even vertices belong to C_{ev}, but C_{ev} may contain odd vertices as well. Then we have the so-called Brieskorn Graph Theorem

Theorem 2.11: The following hold:

(i) The link L(a) is a rational homology sphere if and only if either G(a) contains at least one isolated point, or C_{ev} has an odd number of vertices and for any distinct a_{i}, a_{j} ∈ C_{ev}, gcd(a_{i}, a_{j}) = 2.

(ii) The link L(a) is an integral homology sphere if and only if either G(a) contains at least two isolated points, or G(a) contains one isolated point and C_{ev} has an odd number of vertices and for any distinct a_{i}, a_{j} ∈ C_{ev}, gcd(a_{i}, a_{j}) = 2.

Recall that by seminal work of Milnor, Kervaire and Milnor and Smale, for each n ≥ 5, differentiable homotopy spheres of dimension n form an Abelian group Θ_{n}, where the group operation is connected sum. Θ_{n} has a subgroup bP_{n+1} consisting of those homotopy n-spheres which bound parallelizable manifolds V_{n+1}. Kervaire and Milnor proved that bP_{2m+1} = 0 for m ≥ 1, bP_{4m+2} = 0, or ℤ_{2} and is ℤ_{2} if 4m + 2 ≠ 2^{i} − 2 for any i ≥ 3. The most interesting groups are bP_{4m} for m ≥ 2. These are cyclic of order

2.12 |bP_{4m}| = 2^{2m−2}(2^{2m−1} − 1) numerator ( \begin{matrix}4B_{m} \\ m \end{matrix} ),

where B_{m} is the m-th Bernoulli number. Thus, for example |bP_{8}| = 28, |bP_{12}| = 992, |bP_{16}| = 8128 and |bP_{20}| = 130, 816. In the first two cases these include all exotic spheres. The correspondence is given by

2.13  KM : Σ ↦ \frac{1}{8}τ(V_{4m}(Σ))mod|bP_{4m}|,

where V_{4m}(Σ) is any parallelizable manifold bounding Σ and τ is its signature. Let Σ_{i} denote the exotic sphere with KM(Σ_{i}) = i. Now, Brieskorn Graph theorem tells us for which a the BP link L(a) is a homotopy sphere. By (2.13) we need to be able to compute the signature to determine the diffeomorphism types of various links. We restrict our interest just to the case when m = 2k + 1.

In this case, the diffeomorphism type of a homotopy sphere L(a) ∈ bP_{2m−2} is determined by the signature τ(M) of a parallelizable manifold M whose boundary is Σ_{a}^{2m−3}. By the Milnor Fibration Theorem we can take M to be the Milnor fiber M_{a}^{2m−2} which, for links of isolated singularities coming from weighted homogeneous polynomials is diffeomorphic to the hypersurface {z ∈ ℂ^{m} | f_{a}(z_{0}, ⋯, z_{m−1}) = 1}.

Brieskorn shows that the signature of M_{a}^{2m−2} can be written combinatorially as

2.14  τ(M_{a}^{4k}) =

#\{x ∈ ℤ^{2k+1} | 0 < x_{i} < a_{i} and 0 < ∑_{j=0}^{2k} \frac{x_{i}}{a_{i}} < 1mod2\}

−#\{x ∈ ℤ^{2k+1} | 0 < x_{i} < a_{i} and 1 < ∑_{j=0}^{2k} \frac{x_{i}}{a_{i}} < 2mod2\},

where m = 2k + 1.

Using a formula of Eisenstein, Zagier has rewritten this formula as:

2.15  τ(M_{a}^{4k}) =

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

where N is any common multiple of the a_{i}’s. Both formulas are quite well suited to computer use. We wrote a C code which we call sig.c. For any m-tuple with m = 2k + 1 = 5, 7, 9 sig.c computes the signature τ(a) := τ(M_{a}^{4k}) and the diffeomorphism type of the link using either of the above formulas.

Example 2.16: Let us consider the Breiskorn-Pham link L(5, 3, 2, 2, 2). By Brieskorn Graph Theorem this is a homotopy 7-sphere. One can easily compute the signature using (2.14) to find out that τ(L(5, 3, 2, 2, 2)) = 8. Hence L(5, 3, 2, 2, 2) = Σ_{1}^{7} is an exotic 7-sphere and it is called Milnor generator (all others can be obtained from it by taking connected sums). Similarly one does not need a computer to find that the signature of any L(6k − 1, 3, 2, 2, 2) to see that these so-called Brieskorn spheres realize all 28 diffeomorphism types of 7-spheres.

Question 2.17: Suppose that Z_{f} is Fano. Could one perhaps prove existence of an Kähler-Einstein metric on Z_{f}? Every time this can successfully be done we automatically get a Sasakian-Einstein metric on the link L_{f}.

3. Kähler-Einstein Metrics on Fano Orbifolds

Recall that on a Kähler manifold Ricci curvature 2-form ρ_{ω} of any Kähler metric represents the cohomology class 2πc_{1}(M). The well-known Calabi Conjecture is the question whether or not the converse is also true. To be more specific we begin with a couple of definitions

Definition 3.1 Let (M, J, g, ω_{g}) be a compact Kähler manifold. The Kähler cone of M

K(M) = \{ \begin{matrix}[ω] ∈ H^{1,1}(M, ℂ) ∩ H^{2}(M, ℝ) | [ω] = [ω_{h}] \\ for some Kaehler metric h \end{matrix} \}

is the set of all possible Kähler classes on M.

It is easy to show that K(M) is a convex open set in H^{1,1}(M, ℂ) ∩ H^{2}(M, ℝ).

Definition 3.2 Let (M, J, g, ω_{g}) be a compact Kähler manifold and K(M) it Kähler cone. For any fixed Kähler class [ω] ∈ K(M) we define

K_{[ω]} = {h ∈ Γ(⊙^{2}(T M)) | h his Kaehler and [ω] = [ω_{h}]}

to be the space of all Kähler metrics in a given cohomology class.

Global i∂ ¯∂-lemma provides for a very simple description of the space of Kähler metrics K_{[ω]}. Suppose we have a Kähler metric g in a with Kähler class [ω_{g}] = [ω] ∈ K(M). If h ∈ K_{[ω]} is another Kähler metric then, up to a constant, there exists a global function ϕ ∈ C^{∞}(M, ℝ) such that ω_{h} − ω_{g} = i∂ ¯∂ϕ. We could fix constant by requiring, for example, that ∫_{M} ϕω_{g}^{n} = 0. Hence, we have

Corollary 3.3 Let (M, J, g, ω_{g}) be a compact Kähler manifold with [ω_{g}] = [ω] ∈ K(M). Then, relative to the metric g the space of all Kähler metric K_{[ω]} can be described as

K_{[ω]} = {ϕ ∈ C^{∞}(M, ℝ) | ω_{h} = ω_{g}+i∂ ¯∂ϕ > 0, ∫_{M} ϕω_{g}^{n} = 0},

where the 2-form ω_{h} > 0 means that ω_{h}(X, JY) is a Hermitian metric on M.

We have the following theorem

Theorem 3.4 Let (M, J, g, ω_{g}) be a compact Kähler manifold, [ω_{g}] = ω ∈ K(M) the corresponding Kähler class and ρ_{g} the Ricci form. Consider any (1, 1)-form Ω on M such that [Ω] = 2πc_{1}(M). Then there exists a unique Kähler metric h ∈ K_{[ω]} such that Ω = ρ_{h}.

The above statement is the celebrated Calabi Conjecture which was posed by Eugene Calabi in 1954. The conjecture in its full generality was eventually proved by Yau in 1976. In the Fano case when c_{1}(M) > 0, i.e., when the first Chern class can be represented by a positive-definite real, closed (1, 1)-form ρ′ on M, the conjecture implies that the Kähler form of M can be represented by a metric of positive Ricci curvature.

Let us reformulate the problem using the global i∂ ¯∂- lemma. We start with a given Kähler metric g on M in Kähler class [ω_{g}] = [ω]. Since ρ_{g} also represents 2πc_{1}(M) there exists a globally defined function f ∈ C^{∞}(M, ℝ) such that

ρ_{g} − Ω = i∂ ¯∂f.

Appropriately, f may be called a discrepancy potential function for the Calabi problem and we could fix the constant by asking that ∫_{M}(e^{f} − 1)ω_{g} = 0.

Now, supposed the desired solution of the problem is a metric h ∈ K_{[ω]}. We know that the Kähler form of h can be written as

ω_{h} = ω_{g} + i∂ ¯∂ϕ,

for some smooth function ϕ ∈ C^{∞}(M, ℝ). We normalize ϕ as in previous corollary. Combining these two equations we see that

ρ_{h} − ρ_{g} = i∂ ¯∂f.

If we define a smooth function F ∈ C^{∞}(M, ℝ) relating the volume forms of the two metrics ω_{h}^{n} = e^{F} ω_{g}^{n} then the lefthand side of the above equation takes the following form

i∂ ¯∂F = ρ_{h} − ρ_{g} = i∂ ¯∂f,

and, hence, simply i∂ ¯∂(F − f) = 0. Hence, F = f + c. But since we normalized ∫_{M}(e^{f} − 1)ω_{g} = 0 we must have c = 0. Hence, F = f, or ω_{h}^{n} = e^{f}ω_{g}^{n}. We can now give two more equivalent formulations of the Calabi Problem.

Theorem 3.5 Let (M, J, g, ω_{g}) be a compact Kähler manifold, [ω_{g}] = ω ∈ K(M) the corresponding Kähler class and ρ_{g} the Ricci form. Consider any (1, 1)-form on M such that [Ω] = 2πc_{1}(M). Let ρ_{g} − Ω = i∂ ¯∂f, with ∫_{M}(e^{f} − 1)ω_{g} = 0.

(i) There exists a unique Kähler metric h ∈ K_{[ω]} whose volume form ω_{h}^{n} equals to e^{f}ω_{g}^{n}.

(ii) Let (U, z_{1}, …, z_{n}) be a local complex chart on M with respect to which the metric g = (g_{ij̄}). Then, up to a constant, there exists a unique smooth function ϕ in K_{[ω]}, which satisfies the following equation

\frac{det(g_{ij̄} + \frac{∂^{2}ϕ}{∂z_{i}∂z̄_{j}} )}{det(g_{ij̄})} = e^{f}.

The equation in (ii) is called the Monge-Ampére equation. Part (i) gives a very simple geometric characterization of the Calabi-Yau theorem. On a compact Kähler manifold one can always find a metric with arbitrarily prescribed volume form. The uniqueness part of this theorem was already proved by Calabi. This part involves only Maximum Principle. The existence proof uses continuity methods and it involves several difficult a priori estimates. These were found by Yau in 1978. We have

Corollary 3.6 Let (M, J, g, ω_{g}) be a compact Kähler manifold with c_{1}(M) = 0. Then M admits a unique Kähler Ricci-flat metric.

It is “folklore” that Calabi-Yau Conjecture is also true for compact orbifold. In the context of Sasakian geometry and fundamental foliations the transverse space Z is typically a compact Kähler orbifold. In the context of foliations a transverse Yau theorem was proved by El Kacimi-Alaoui in 1990.

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

3.7  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

3.8  g_{D}(X, Y) = dη(X, JY)

which is related to the Sasakian metric g by

g = g_{D} ⊕ η ⊗ η.

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

3.9  ξ⌋α = 0,  L_{ξ}α = 0,

and we let Λ_{B}^{p} denote the sheaf of germs of basic p-forms on M, and 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

3.10  ⋯ ⟶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 3.11

→ 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 distinguish 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. 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

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

as well as the basic Dolbeault complex

3.13  0⟶Λ_{B}^{p,0} ⟶^{¯∂}Λ_{B}^{p,1} ⟶^{¯∂} ⋯ ⟶Λ_{B}^{p,n}⟶0,

together with basic Dolbeault cohomology groups H_{B}^{p,q}(F_{ξ}). Most of the usual results about Kähler geometry carry over to transverse Kähler geometry.

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 (3.11) together with the natural map H^{2}(M, ℤ)⟶H^{2}(M, ℝ) whose kernel is the torsion part of H^{2}(M, ℤ). We have

3.14  H^{2}(M, ℤ) ; ↓ ; 0⟶ℝ⟶^{δ}H_{B}^{2}(F_{ξ})⟶^{ι_{∗}}H^{2}(M, ℝ)⟶ ⋯

As in (3.11) 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 by El Kacimi-Alaoui 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.

Theorem 3.15: [ El Kacimi-Alaoui] 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.

In the language of positive Sasakian manifolds this theorem can be used to show

Theorem 3.16: [Boyer,–,Nakamaye] Any positive Sasakian manifold admits a metric of positive Ricci curvature.

One of the consequences of the Yau’s theorem is that a compact Kähler manifold with c_{1}(M) = 0 must admit a Ricci-flat, hence, Einstein metric. More generally, we can consider existence of Kähler-Einstein metrics with arbitrary Einstein constant λ. By scaling we can assume that λ = 0, ±1. Specifically, let (M, J, g, ω_{g}) be a compact Käher manifold. We would like to know if one can always find a Kähler-Einstein metric h ∈ K_{[ω_{g}]}. Recall that on a Kähler-Einstein manifold ρ_{g} = λω_{g}. This implies that 2πc_{1}(M) = λ[ω_{g}]. Now, if c_{1}(M) > 0 we must have λ = +1 because [ω_{g}] is the Kähler class. Similarly, when c_{1}(M) < 0 the only allowable sign of a Kähler-Einstein metric on M is λ = −1. Clearly, when c_{1}(M) = 0 we must have λ = 0 as [ω_{g}] ≠ 0. As we have already pointed out the λ = 0 case follows from Yau’s solution to the Calabi conjecture. For the reminder of this lecture we shall assume that λ = ±1.

Let (M, J, g, ω_{g}) be a Kähler manifold and [ω_{g}] = [ω] ∈ K(M) the Kähler class. Let us reformulate the existence problem using the global i∂ ¯∂-lemma. Suppose there exists an Einstein metric h ∈ K_{[ω]}. Starting with the original Kähler metric g on M we have a globally defined function f ∈ C^{∞}(M, ℝ) such that

ρ_{g} − λω_{g} = i∂ ¯∂f.

As before we will call f a discrepancy potential function. We also fix the constant by asking that ∫_{M}(e^{f} − 1)ω_{g} = 0. Let h ∈ K_{[ω]} be an Einstein metric for which ρ_{h} = λω_{h}. Using global i∂ ¯∂-lemma once again we have a globally defined function ϕ ∈ C^{∞}(M, ℝ) such that ω_{h} −ω_{g} = i∂ ¯∂ϕ. We shall fix the constant in ϕ later. Using these two equations we easily get

ρ_{g} − ρ_{h} = i∂ ¯∂(f − λϕ).

Defining F so that ω_{h}^{n} = e^{F} ω_{g}^{n} we can write this equation as

i∂ ¯∂F = i∂ ¯∂(f − λϕ).

This implies that F = f − λϕ + c. We have already fixed the constant in f so c depends only on the choice of ϕ. We can make c = 0 by choosing ϕ such that ∫_{M}(e^{f−λϕ} − 1)ω_{g}^{n} = 0. Hence, we have the following

Proposition 3.17 Let (M, J) be a compact Kähler manifold with λc_{1}(M) > 0, where λ = ±1. Let [ω] ∈ K(M) be a Kähler class and g, h two Kähler metrics in K_{[ω]} with Ricci forms ρ_{g}, ρ_{h}. Let f, ϕ ∈ C^{∞}(M, ℝ) be defined by ρ_{g} − λω_{g} = i∂ ¯∂f, ω_{h} − ω_{g} = i∂ ¯∂ϕ. Fix the relative constant of f − λϕ by setting ∫_{M}(e^{f−λϕ} − 1)ω_{g} = 0. Then the metric h is Einstein with Einstein constant λ if and only if ϕ satisfies the following Monge-Ampére equation

ω_{h}^{n} = e^{f−λϕ}ω_{g}^{n},

in a local complex chart (U, z_{1}, …, z_{n}) written as

\frac{det(g_{ij̄} + \frac{∂^{2}ϕ}{∂z_{i}∂z̄_{j}} )}{det(g_{ij̄})} = e^{f−λϕ}.

Note that by setting λ = 0 we get the Monge-Ampére equation for the original Calabi problem. The character of the Monge-Ampére equation above very much depends on the choice of λ. The case of λ = −1 is actually the simplest as the necessary a priori C^{0}-estimates can be derived using the Maximum Principle. This was done by Aubin and independently by Yau. We have

Theorem 3.18 Let (M, J, g, ω_{g}) be a compact Kähler manifold with c_{1}(M) < 0. Then there exists a unique Kähler metric h ∈ K_{[ω_{g}]} such that ρ_{h} = −ω_{h}.

When k = +1 the problem is much harder. It has been known for quite some time that there are actually non-trivial obstructions to the existence. Let 𝔥(M) be the complex Lie algebra of all holomorphic vector fields on M. Matsushima proved that one a compact Kähler-Einstein manifold with c_{1}(M) > 0 𝔥(M) must be reductive, i.e., 𝔥(M) = Z(𝔥(M) ⊕ [𝔥(M), 𝔥(M)].

Here one tries to solve the Monge-Ampére equation

\frac{det(g_{ij̄} + ∂_{i} ¯∂j̄ϕ_{t})}{det(g_{ij̄})} = e^{−tϕ_{t}+f}, g_{ij̄} + ∂_{i} ¯∂j̄ϕ_{t} > 0

for t ∈ [0, 1]. Yau’s Theorem tells us that this has a solution for t = 0, and we try to solve this for t = 1, where the metric will be Kähler-Einstein The so called “continuity method” sets out to show that the interval where solutions exist is both open and closed. Openness follows from the Implicit Function Theorem, but there are well known obstructions to closedness. This problem has been studied by many people; Yau, Tian, Siu, Nadel, and most recently by Demailly and Kollár who work in the orbifold category. Closedness is equivalent to the uniform boundedness of the integrals

∫_{Z} e^{−γtϕ_{t}}ω_{0}^{n}

for any γ ∈ (\frac{n}{n+1}, 1), where ω_{0} is the Kähler form of h_{0}. This means that the multiplier ideal sheaf J (γϕ) = O_{Z} for all γ ∈ (\frac{n}{n+1}, 1).

Here is an example of the estimates used to obtained many (but not all) theorems mentioned in the introduction

Theorem 3.19 [Boyer,–,Kollár] Let Z(a) be the transverse space of the BP link L(a). Let C_{i} = lcm(a_{1}, …, â_{i}, …, a_{n}), b_{i} = gcd(C_{i}, a_{i}). Then Z(a) is Fano and it has a Kähler-Einstein metric if

(i) 1 < ∑_{i=1}^{m} \frac{1}{a_{i}},

(ii) ∑_{i=1}^{m} \frac{1}{a_{i}} < 1 + \frac{m−1}{m−2} min_{i}\{\frac{1}{a_{i}} \}, and

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

In this case the link L(a) admits a Sasakian-Einstein metric with one-dimensional isometry group.