Transverse Fano Structures

and

Einstein Metrics on Exotic Spheres

KRZYSZTOF GALICKI Max-Planck-Institut für Mathematik, Bonn

and

University of New Mexico, Albuquerque

Bonn, March 2004

“WHY?” and “WHAT?”

0. Introduction

(i) what’s known, some problems and questions (ii) some answers and theorems

“HOW?”

1. Riemannian Geometry

(i) Sasakian manifolds: basic definitions and properties (ii) fundamental foliations and transverse geometry

2. Differential Topology

(i) Sasakian geometry of links of isolated hypersurface singularities

(ii) Differential topology of the Brieskorn-Pham links

3. Global Analysis ⟶ Algebraic Geometry

(i) Calabi Conjecture I (ii) Calabi Conjecture II (iii) Kähler-Einstein metrics on Brieskorn-Pham orbifolds

REFERENCES and CO-AUTHORS

1. J. Kollár, Einstein metrics on connected sums of S^{2} × S^{3}, preprint 2004.

2. C. P. Boyer, K. Galicki, Einstein metric on rational homology spheres, preprint 2003.

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

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

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

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 at http://www.math.unm.edu/˜ galicki.

• Any standard sphere S^{n}, n > 1, admits a metric of constant positive sectional curvature. These canonical metrics are SO(n+1)-homogeneous and Einstein, i.e., the Ricci curvature tensor is a constant positive multiple of the metric.

•[1974] The spheres S^{4m+3}, m > 1 are known to have another Sp(m+1)-homogeneous Einstein metric discovered by Jensen. The metric is obtained from the “quaternionic Hopf fibration” S^{3} → S^{4m+3} → ℍℙ^{m}. Since both base and fiber are Einstein spaces with positive Einstein constant we obtain two Einstein metrics in the canonical variation. The second metric is also called “squashed sphere” metric in some physics literature.

•[1978] In addition, S^{15} has a third Spin(9)-invariant homogeneous Einstein metric discovered by Bourguignon and Karcher in 1978. The existence of such metric has to do with the fact that S^{15}, in addition to fibering over ℍℙ^{2}, also fibers over S^{8} with fiber S^{7}. Thus 15-sphere admits 3 different homogeneous Einstein metrics.

•[1982] Ziller proved that these are the only homogeneous Einstein metrics on spheres.

•[1998] Böhm obtained 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.

Remark: Even with Böhm’s result Einstein metrics on spheres appear to be rather rare. But are they really?

Questions 1: Are there other inhomogeneous Einstein metrics on standard spheres?

Questions 2: Do some exotic spheres admit Einstein metrics of positive scalar curvature?

In odd dimensions we will provide some surprising answers to these questions.

Theorem 1 [Boyer-Galicki-Kollár’2003]: On S^{5} there are at least 68 inequivalent families of Sasakian-Einstein metrics. Some of these families admit non-trivial continuous Sasakian-Einstein deformations. The biggest constructed family has real dimension 10.

Remarks:

• Our metrics differ from Böhm’s. His have rather large isometry group while all ours (and this comment applies to all dimensions) are, by construction, of “maximal” cohomogeneity, i.e., they have one-dimensional isometry group.

• Böhm gets infinitely many non-isometric Einstein metrics on S^{5} but he does not get any continuous families as is typically the case with our metrics.

• We note that the numbers “68” or “10” in Theorem 1 no special significance: more families and higher-dimensional moduli can most likely be obtained using the same basic techniques. Nonetheless, in each odd dimension, they are to illustrate the abundance of metrics we get: both the number of families as well as the dimension of the moduli of the largest family increases doubly exponentially in each next odd dimension.

Theorem 2 [Boyer-Galicki-Kollár’2003]: Let Σ_{i}^{7}, be a homotopy 7-sphere corresponding to the element i ∈ bP_{8} ≃ ℤ_{28} ≃ Θ_{7} in the Kervaire-Milnor group. Σ_{i}^{7} admits at least n_{i} inequivalent families of Sasakian-Einstein metrics, where (n_{1}, …, n_{28}) =(376, 336, 260, 294, 231, 284, 322, 402, 317, 309, 252, 304, 258, 390,409, 352, 226, 260, 243, 309, 292, 452, 307, 298, 230, 307, 264, 353), giving a total of 8610 cases. In each oriented diffeomorphism class some of the families depend on a moduli. In particular, the standard 7sphere Σ_{28}^{7} admits an 82-dimensional family of inequivalent Sasakian-Einstein metrics.

Theorem 3 [Boyer-Galicki-Kollár’2003]: In dimensions (4n+1) both the standard and the so-called Kervaire spheres admit many families of inequivalent Sasakian-Einstein metrics for each n ≥ 2.

Remarks:

• In dimension 9, for example, the Kervaire sphere is exotic, i.e., bP_{10} = ℤ_{2}. A computer search yields more than 3 · 10^{6} inequivalent families of Sasakian-Einstein metrics on both spheres.

• In dimension 13 the Kervaire sphere is diffeomorphic to the standard sphere as bP_{14} = 0. In sharp contrast with the fact that the only Einstein metric known on S^{13} is the standard one we find

Corollary: The standard sphere S^{13} admits over 10^{9} distinct families of Sasakian-Einstein metrics one of them depending on 21,300,113,901,610 parameters.

Conjecture [Boyer-Galicki-Kollár’2003]: In any odd dimension, all homotopy spheres which bound parallelizable manifolds admit families of Sasakian-Einstein metrics.

Theorem 4 [Boyer-Galicki-Kollár-Thomas’2003]: The conjecture is true in dimensions 11 and 15. More precisely, each homotopy sphere (992 possible oriented diffeomorphism types) in dimension 11 admits at least one Sasakian-Einstein metric and each homotopy 15-sphere which bounds a parallelizable manifold (8128 possible oriented diffeomorphism types) admits at least one Sasakian-Einstein metric.

Remark:

• Theorem 4 required some considerable computing time. For example, using the same machines and the same codes would require about 3000 years to extend the result to the case of 19-spheres (bP_{20} has 130,816 elements).

Important Remark: The methods of proving such existence results apply to a wide range of smooth contact manifolds. In this talks I have chosen homotopy spheres to illustrate the power of the technique. To give another example here are two more recent theorems one can get in 5 dimensions:

Theorem 5 [Boyer-Galicki’03]: Let M_{k}^{5} be a simply connected, smooth, oriented, compact, spin manifold such that H_{2}(M_{k}^{5}, ℤ) = ℤ_{k} ⊕ℤ_{k}. Then, for any k such that gcd(k, 3) = 1, the rational homology 5-sphere M_{k}^{5} admits continuous families of Sasakian-Einstein metrics.

Theorem 6 [Kollár’04]: Let M be a simply connected, smooth, oriented, compact, spin 5-manifold with b_{2}(M) = n and such that H_{2}(M, ℤ) = ℤ^{n}. Then for any n ≥ 6, M admits infinitely many (2n − 2)-dimensional families of Sasakian-Einstein metrics.

Remarks:

• By Smale’s theorem simply connected, smooth, oriented, compact, spin 5-manifold is determined by its second homology.

• Theorem 6, without the statement about the moduli, is also true for n = 0, 1, …, 5.

1. RIEMANNIAN GEOMETRY

Sasakian Manifolds

Definition-Proposition: A Riemannian manifold (M, g) is called Sasakian 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 onedimensional foliation F_{ξ} associated to any 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: 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: A Sasakian manifold (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 finite fundamental group. Furthermore the metric cone (C(M), ḡ) = (ℝ_{+} × M, dr^{2} + r^{2}g) is Kähler Ricci-flat (Calabi-Yau).

Definition-Proposition:(M, g) is 3-Sasakian if the metric cone (C(M), ḡ) = (ℝ_{+} × S, dr^{2} + r^{2}g) on M is hyperkähler. In particular, any 3-Sasakian manifold is Einstein.

The following slide presents some fundamental structure theorems for Sasakian and Sasakian-Einstein manifolds:

Characteristic Foliation

and

Transverse Kähler Geometry

Theorem: Let (M, g) be a compact quasi-regular 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: 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.

Standard Example:

ℂ^{n} \ {0}

↘

↓  S^{2n−1}

↙

ℙℂ^{n−1}

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

Remarks:

• A complex orbifold Z is Fano if suitably defined orbifold canonical bundle K_{Z^{orb}} is anti ample. In the case Z is wellformed, that is when orbifold singularities are of codimension at least 2, this is the same as saying that K_{Z} (viewed simply as a complex variety) is anti ample. However, in the presence of codimension 1 singularities one needs to be more careful as it is not true in general.

• Just as in the smooth case positivity can also be be expressed in terms of orbifold Chern classes of Z (first Chern class c_{1}(Z) > 0), or in terms of basic Chern classes of the characteristic foliation F_{ξ} (first Chern class c_{1}(F_{ξ}) > 0).

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

Theorem: 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 smooth compact simply connected Sasakian-Einstein manifold.

C(M) ↩ M
↓π

Z

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
KÄHLER  3-SASAKIAN KÄHLER-EINSTEIN

2. DIFFERENTIAL TOPOLOGY Sasakian Structures on Links

Let w = (w_{0}, …, w_{n}) and consider a weighted ℂ^{∗}-action on ℂ^{n+1} given by (z_{0}, …, z_{n}) ↦ (λ^{w_{0}}z_{0}, …, λ^{w_{n}}z_{n}), where the weights w_{j} are all positive integers. We assume that gcd(w_{0}, …, w_{n}) = 1.

Definition f ∈ ℂ[z_{0}, …, z_{n}] is a weighted homogeneous polynomial of weight w and degree d if

f(λ^{w_{0}}z_{0}, …, λ^{w_{n}}z_{n}) = λ^{d}f(z_{0}, …, z_{n}).

Assume that the origin in ℂ^{n+1} is an isolated singularity.

Definition L_{f} = {f = 0} ∩ S^{2n+1}, where S^{2n+1} is the unit sphere in ℂ^{n+1} is called the link of f.

• Fact 1: The well-known construction of Milnor for isolated hypersurface singularities shows that L_{f} is (n-2)-connected. Hence, he only non-vanishing integral homology groups are H_{n−1}(L_{f}, ℤ) and H_{n}(L_{f}, ℤ) and they, at least in principle, can be calculated using Milnor’s fibration theorem.

• Fact 2: L_{f} is endowed with a quasi-regular Sasakian structure inherited as a Sasakian submanifold of the sphere S^{2n+1} with a “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 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:

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 orbifold Riemannian submersions. L_{f} is the total space of the principal S^{1} V-bundle over the orbifold Z_{f}.

Proposition: The orbifold Z_{f} is Fano iff d − ∑ w_{i} < 0.

Milnor Fibration Theorem

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}. 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 by the formula 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

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).

Brieskorn-Pham Links

In his famous work, in 1966 Brieskorn considered special links often called Brieskorn-Pham links (or BP for short) L(a) defined by

∑_{i=0}^{n} |z_{i}^{2}| = 1, f_{a}(z) = z_{0}^{a_{0}} + ⋯ + 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: The link L(a) (with n ≥ 3) is homeomorphic to the (2n-1)-sphere if and only if either of the following conditions hold

(i) G(a) contains at least two isolated points,

(ii) 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.

Homotopy Groups of Spheres

Recall that by seminal work of Milnor, Kervaire and Milnor and Smale, for each k ≥ 5, differentiable homotopy spheres of dimension k form an Abelian group Θ_{k}, where the group operation is connected sum. Θ_{k} has a subgroup bP_{k+1} consisting of those homotopy k-spheres which bound parallelizable manifolds V_{k+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

|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

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.

In the case of BP links, by the Milnor Fibration Theorem we can take V to be the Milnor fiber which is diffeomorphic to V_{a}^{2n} = {z ∈ ℂ^{n+1} | f_{a}(z_{0}, ⋯, z_{n}) = 1}. Brieskorn shows that the signature of V_{a}^{2n} can be written combinatorially as

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

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

where n = 2m.

Example: Let us consider the Brieskorn-Pham link L(6k − 1, 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(6k − 1, 3, 2, 2, 2)) = 8k. 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).

Question: Suppose we take a link L_{f} so that the transverse geometry 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}.

J.-P. Demailly and J. Kollár, Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, preprint AG/9910118, Ann. Scient. Ec. Norm. Sup. Paris 34 (2001), 525-556.

3. GLOBAL ANALYSIS

Calabi Yau Conjecture I

Let (M, g, J, ω_{g}) be a Kähler manifold. In a local complex chart the metric g is simply a Hermitian matrix g_{ij̄}. It was Kähler who discovered that locally the Ricci tensor of a Kähler metric can be written as

R_{ij̄} = −\frac{∂}{∂z_{i}} \frac{∂}{∂z̄_{j}} ln(det(g_{ij̄})).

Only much later Calabi realized the importance of this remarkable formula. If one defines the associated 2-form

ρ_{ω} = −\frac{i}{2}R_{ij̄}dz^{i} ∧ dz̄^{j}

which is called the Ricci form one realizes that the cohomology class of ρ_{ω} does not depend on the choice of the Kähler metric within Kähler class. In fact, it is easy to see that Ricci form represents the cohomology class 2πc_{1}(M). Naturally Calabi asked if the converse was also true. More specifically 50 years ago he conjectured that

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

Calabi was right and his conjecture in its full generality was eventually proved by Yau in 1976.

Monge-Ampère Problem

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

Ω − ρ_{g} = −i∂ ¯∂f.

Appropriately, f may be called a discrepancy potential function for the Calabi problem.

Now, supposed the desired solution of the problem is a metric h with [ω_{h}] = [ω_{g}] and ρ_{h} = 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). In a local, complex chart (and with appropriate choice of constants)

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

Remarks

• This equation is called the Monge-Ampère equation.

• It is “folklore” that Calabi-Yau Conjecture is also true for compact orbifolds. 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.

Calabi Yau Conjecture II

Kähler-Einstein Condition

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 such that [ω_{h}] = [ω_{g}]. Recall that on a Kähler-Einstein manifold

ρ_{g} = λω_{g},

where the metric can be normalized so that λ = ±1 or 0. Since 2πc_{1}(M) = λ[ω_{g}] we have

• When c_{1}(M) > 0 we must have λ = +1.

• When c_{1}(M) < 0 we must have λ = −1.

• When c_{1}(M) = 0 we must have λ = 0.

Suppose there exists an Einstein metric h with Kähler from in the same cohomology class. We have globally defined functions ϕ, f ∈ C^{∞}(M, ℝ) such that

ρ_{g} − λω_{g} = i∂ ¯∂f, ω_{h} − ω_{g} = i∂ ¯∂ϕ.

After fixing appropriate normalization we get, in local coordinates, the following Monge-Ampère equation

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

The case of λ = −1 is actually the simplest and was solved by Aubin and independently by Yau.

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

When λ = +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)].

Calabi Conjecture II Let (M, g, J, ω_{g}) be a Kähler manifold with c_{1}(M) > 0. Suppose 𝔥(M) = 0. Then there exists an Einstein metric h with [ω_{h}] = [ω_{g}].

Remark This time Calabi was wrong. This conjecture is true for smooth del Pezzo surfaces but it already breaks down for del Pezzo surfaces with orbifold singularities. This was established by Ding and Tian in 1992. Later Tian showed that there exist smooth Fano 3-folds which have 𝔥(M) = 0 but do not admit any Kähler-Einstein metric.

Continuity Method

Here one tries to solve the Monge-Ampère equation

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

We start with ϕ_{0} and C_{0} = 0 and we try to reach 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. It turns out that the critical step is a 0th order estimate. That is, as the values of t for which the Monge-Ampère equation is solvable approach a critical value t_{0} ∈ [0, 1], a subsequence of the ϕ_{t} converges to a function ϕ_{t_{0}} which is the sum of a C^{∞} and of a plurisubharmonic function. We only need to prove that

∫_{Z} e^{−γtϕ_{t_{0}}} ω_{0}^{n} < +∞

for some γ ∈ (\frac{n}{n+1}, 1), where ω_{0} is the Kähler form of g_{0} = g.

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.

Kähler Einstein Metrics on

Brieskorn-Pham Orbifolds

Consider a Brieskorn–Pham singularity

Y (a) := (∑_{i=0}^{n} z_{i}^{a_{i}} = 0) ⊂ ℂ^{n+1}.

Set C = lcm(a_{i} : i = 0, …, n). Y (a) is invariant under the ℂ^{∗}-action

(z_{0}, …, z_{n}) ↦ (λ^{C/a_{0}}z_{0}, …, λ^{C/a_{n}}z_{n}).

In the notation of a previous slide we have w_{i} = C/a_{i} and d = C. Furthermore, the quotient Y (a)/ℂ^{∗} can be identified with the transverse space Z(a) of the associated BP link L(a). One can easily see that Y (a)/ℂ^{∗} is a Fano orbifold iff

1 < ∑_{i=0}^{n} \frac{1}{a_{i}} .

More generally, we consider weighted homogeneous perturbations

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

where weighted degree(p) = C. The genericity condition we need, which is always satisfied by p ≡ 0 is: The intersections of Y (a, p) with any number of hyperplanes (z_{i} = 0) are all smooth outside the origin.

The continuity methods produces the following sufficient conditions on Y (a, p)/ℂ^{∗} to admit a Kähler-Einstein metric:

Theorem Let Z(a, p) = Y (a, p)/ℂ^{∗} be the transverse space of the BP link L(a). Then Z(a, p) is Fano and it has a Kähler-Einstein metric if

(i) 1 < ∑_{i=0}^{n} \frac{1}{a_{i}},

(ii) ∑_{i=0}^{n} \frac{1}{a_{i}} < 1 + \frac{n}{n−1} min_{i}\{\frac{1}{a_{i}} \}, and

(iii) ∑_{i=0}^{n} \frac{1}{a_{i}} < 1 + \frac{n}{n−1} min_{i,j}\{\frac{1}{b_{i}b_{j}} \}.

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

Remark All but one theorems mentioned the introduction are special cases of the above result.

Example Consider sequences of the form a = (2, 3, 7, m). By explicit calculation, the corresponding link L(a) gives a Sasakian-Einstein metric on S^{5} if 5 ≤ m ≤ 41 and m is relatively prime to at least two of 2, 3, 7. This is satisfied in 27 cases.

Example The sequence a = (2, 3, 7, 35) is especially noteworthy. If C(u, v) is any sufficiently general homogeneous septic polynomial, then the link of

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

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

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

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

Example The sequence a = (2, 3, 7, 43, 43 · 31) gives a standard 7-sphere with a 2(43 − 2) = 82-dimensional family of Sasakian-Einstein metrics on S^{7}.

Consider the sequence defined by the recursion relation

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

It starts as

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

We have

∑_{i=1}^{m} \frac{1}{c_{i}} = 1 − \frac{1}{c_{m+1} − 1} = 1 − \frac{1}{c_{1} ⋯ c_{m}} .

Example

(i) a = (2, 3, 7, 43, 1807, 3263443, 10650056950807, m) (ii) L(a) is just the standard 13-sphere for any suitably cho-

sen m. (iii) If we choose m = (10650056950807−2)·10650056950807

we get 2(10650056950807−2)-dimensional family of deformations.