CANONICAL METRICS IN KÄHLER GEOMETRY AND STABILITY IN GIT

AKITO FUTAKI DEPARTMENT OF MATHEMATICS, TOKYO INSTITUTE OF TECHNOLOGY

1. Canonical metrics in Kähler geometry

In the early 1950’s E. Calabi proposed to give compact Kähler manifolds canonical metrics, the so-called extremal Kähler metrics. Extremal Kähler metrics include Kähler metrics of constant scalar curvature as special cases, and Kähler metrics of constant scalar curvature include Kähler-Einstein metrics as special cases. It goes without saying that Kähler-Einstein metrics are well investigated because they are the most restrictive ones, and the general theory of extremal Kähler metrics is less studied.

The main issue is to find necessary and sufficient conditions for these three kinds of metrics to exist in a given compact Kähler manifold or given Kähler class.

I. The case of Kähler-Einstein metrics : A Kähler metric g = (g_{ij̄}) is said to be a Kähler-Einstein metric if its Ricci curvature is proportional to g. The Ricci curvature R_{ij̄} is expressed using the Kähler metric g by

R_{ij̄} = −\frac{∂^{2}}{∂z^{i}∂\overline{z}^{j}} log det g,

and the condition for g to be a Kähler-Einstein metric is given by

R_{ij̄} = k g_{ij̄}, k ∈ ℝ (1)

which is a nonlinear partial differential equation of second order in terms of the Kähler metric g. Here z^{1}, ⋯, z^{m} are the local holomorphic coordinates of the Kähler manifold M. (This equation corresponds to the field equation of gravity in relativity.) But in reality the equation (1) is reduced to the so-called complex Monge-Ampère equation in terms of the Kähler potential.

On the other hand, as the Ricci form

ρ_{g} = \frac{i}{2π} ∑_{i=1}^{m} R_{ij̄}dz^{i} ∧ dz̄^{j}

represents as a de Rham class the first Chern class c_{1}(M), according as the sign of k in (1) is positive, zero or negative, c_{1}(M) is represented by a positive definite, zero or negative definite (1.1)-form. Such situations are expressed by saying that c_{1}(M) > 0, c_{1}(M) = 0 or c_{1}(M) < 0. Namely in order for M to admit a Kähler-Einstein metric one of these three conditions has to hold.

A conjecture of Calabi asks the converse in the cases c_{1}(M) < 0 and c_{1}(M) = 0. This was solved affirmatively by Aubin [1] and Yau [32] independently in the negative case and by Yau [32] in the zero case. The manifolds with c_{1}(M) = 0 have physical importance, and such manifolds are called Calabi-Yau manifolds and the Ricci-flat Kähler metrics are called Calabi-Yau metrics. Moreover Yau’s solution to the conjecture gave birth to various applications in complex geometry. For example, Yau obtained an inequality involving the first and the second Chern classes for manifolds with c_{1}(M) < 0, generalizing an inequality previously obtained by Y.Miyaoka for complex surfaces of general type. Moreover Yau proved that a compact complex surface homotopy equivalent to the complex projective plane must be biholomorphic to the complex projective plane as an application of his results.

In the remaining case where c_{1}(M) > 0 Kähler-Einstein metrics do not always exist. Manifolds with c_{1}(M) > 0 are called Fano manifolds in algebraic geometry. First of all, a theorem of Matsushima [24] says that the complex Lie algebra 𝔥(M) of all holomorphic vector fields on M is reductive. Secondly, on Fano manifolds M there is a Lie algebra character f : 𝔥(M) → C such that if M admits a Kähler-Einstein metric then f = 0 ([12]).

On the other hand there is a problem of finding an Hermitian-Einstein metric on holomorphic vector bundles. Such metrics on compact Kähler surfaces give special cases of anti-self-dual connections in gauge theory. For this reason Hermitian-Einstein metrics were studied extensively in 1980’s, and completely solved including higher dimensions ([7], [31]). It turned out that a holomorphic vector bundle E admits an Hermitian-Einstein metric if and only if E is semi-stable in the sense of Mumford-Takemoto, i.e. in a sense of geometric invariant theory. We will not go into the detail here, but it turned out that the issue for Einstein condition is the stability in GIT.

It has been conjectured (c.f. Yau [34]) that the existence of Kähler-Einstein metric in the case when c_{1}(M) > 0 would also be equivalent to a notion related to stability in geometric invariant theory. The meaning of this conjecture has become clearer by the recent works by G. Tian [30], and S.K. Donaldson [10], [11].

II. The case of Kähler metrics of constant scalar curvature : Given a Kähler metric g = (g_{ij̄}) and its Ricci curvature R_{ij̄},

σ = ∑_{i,j=1}^{m} g^{ij̄} R_{ij̄}

is called the scalar curvature of g, where (g^{ij̄}) is the inverse matrix of (g_{ij̄}). The scalar curvature is the average of the Ricci curvature in all directions as one can see it if one uses local coordinates such that (g_{ij̄}) is the identity matrix. The scalar curvature is a C^{∞}-function M. We are interested in Kähler metrics of constant scalar curvature. Kähler-Einstein metrics are Kähler metrics of constant scalar curvature.

Unlike Kähler-Einstein metrics there are no restrictions on c_{1}(M) for Kähler metrics of constant scalar curvature. So we fix a Kähler class, the de Rham cohomology class of the Kähler form, and look for a Kähler metric of constant scalar curvature. Many of results for Kähler-Einstein metrics can be extended to Kähler metrics of constant scalar curvature. For example Matsushima’s theorem holds true for Kähler manifolds of constant scalar curvature, and the character f can be defined as an invariant depending only on the Kähler class and f vanishes if there exists a constant scalar curvature Kähler form in the Kähler class. The above-mentioned works [10], [11] of Donaldson are carried out when the manifold is not necessarily Fano but the Kähler class is an integral class. In such situations the manifold M can be considered as a projective variety by Kodaira imbedding theorem, and we can work in the framework of algebraic geometry.

On the other hand Donaldson [8], [9], and X.-X. Chen [5] study the space of Kähler metrics with non-rational Kähler class by regarding the geodesic equation in the space of Kähler metrics as a degenerate Monge-Ampère equation. They specifically tries to study the relationship to the stability in GIT sense and to prove the uniqueness of the Kähler metrics of constant scalar curvature. Chen indeed succeeded in proving the uniqueness when c_{1}(M) < 0.

III. The case of extremal Kähler metrics : A Kähler metric g is called an extremal Kähler metric if the gradient vector field of its scalar curvature σ

gradσ = ∑_{i,j=1}^{m} g^{ij̄} \frac{∂σ}{∂zj̄} \frac{∂}{∂z^{i}}

is a holomorphic vector field. Such a metric is a critical point of the functional

g ↦ ∫_{M} |σ|^{2}dV_{g}

on the space of metrics in a fixed Kähler class. If the scalar curvature σ is constant then its gradient vector filed is 0. Therefore a Kḧler metric of constant scalar curvature is an extremal Kähler metric. Hence it is natural to try to extend known results for Kähler-Einstein or constant scalar curvature Kähler metrics to extremal Kähler metrics. In fact Calabi [4] proved a structure theorem of the Lie algebra 𝔥(M) of extremal Kähler manifold M in a form which generalizes Lichnerowicz-Matsushima theorem. On the other hand there is no extension for f which obstructs the existence of extremal Kähler metrics. Mabuchi [23] has results on the stability and the uniqueness for extremal Kähler metrics, extending the results of Donaldson for Kähler metrics of constant scalar curvature.

The purpose of this lecture is to explain the recent result [15] and its background on the relationship between the existence of constant scalar curvature Kähler metrics and the asymptotic Chow stability of polarized algebraic manifolds.

2. Geometric invariant theory

Geometric invariant theory says if one wish to form a moduli space with good property such as Hausdorff property and compactifiability one has to collect only semistable ones ([25]). The definition of stability and semistability is as follows. Let V be a vector space. (Typically the vector space of homogenneous polynomials of degree d with n+1 variables with coefficients in ℂ. These describe the hypersurfaces of degree d in ℙ^{n}(ℂ).) Let G be a subgroup of SL(V). An element x ∈ V is said to be stable if Gx is closed and the stabilizer of x is a finite subgroup of G. An element x ∈ V is said to be semistable if the closure of Gx does not contain the origin o.

Let L → M be an ample line bundle. Put V_{k} := H^{0}(M, L^{k})^{∗} and let Φ_{|L^{k}|} : M → ℙ(V_{k}) be the Kodaira embedding determined by L^{k}. Let d be the degree of M in ℙ(V_{k}). A point in the product ℙ(V_{k}^{∗}) × ⋯ × ℙ(V_{k}^{∗}) of m + 1 copies of ℙ(V_{k}^{∗}) determines m + 1 hyperplanes H_{1}, ⋯ H_{m+1} in ℙ(V_{k}). The set of all m + 1 hyperplanes H_{1}, ⋯ H_{m+1} such that H_{1} ∩ ⋯ ∩ H_{m+1} ∩ M is not empty defines a divisor in ℙ(V_{k}^{∗}) × ⋯ × ℙ(V_{k}^{∗}). But since the degree of M is d, this divisor is defined by M̂_{k} ∈ (Sym^{d}(V_{k}))^{⊗m+1}. Of course M̂_{k} is defined up to a constant. The point [M̂_{k}] ∈ ℙ((Sym^{d}(V_{k}))^{⊗m+1}) is called the Chow point of (M, L^{k}). M is said to be Chow stable with respect to L^{k} if M̂_{k} is stable under the action of SL(V_{k}) on (Sym^{d}(V_{k}))^{⊗m+1}. M is said to asymptotically Chow stable with respect to L if there exists a k_{0} > 0 such that M̂_{k} is stable for all k ≥ k_{0}. Asymptotic Chow semistability is defined similarly. The stabilizer Ĝ_{k} ⊂ SL(V_{k}) of M̂_{k} is a finite covering of a subgroup G_{k} of the automorphism group Aut(M) of M. If we denote by Aut(M, L) the subgroup of Aut(M) consisting the elements which lift to an action on L, then G_{k} is a subgroup of Aut(M, L). Under the assumption that Aut(M, L) is discrete, Donaldson obtained the following result. The Kähler form of the Fubini-Study metric of ℙ(V_{k}) is denoted by ω_{F S}.

Theorem 2.1 ([10]). (1) Suppose that Aut(M, L) is discrete and that M is asymptotically Chow stable. If the sequence of Kähler forms ω_{k} := \frac{2π}{k} Φ_{k}^{∗}(ω_{F S}) belonging in c_{1}(L) converges in C^{∞} to ω_{∞}, then ω_{∞} has constant scalar curvature.

(2) Suppose that Aut(M, L) is discrete and that ω_{∞} ∈ 2πc_{1}(L) has constant scalar curvature. Then M is asymptotically Chow stable with respect to L, and ω_{k} converges in C^{∞} to ω_{∞}.

(3) Suppose that Aut(M, L) is discrete. Then a Kähler metric of constant scalar curvature in 2πc_{1}(L) is unique.

The case where the assumption that Aut(M, L) is not discrete is treated by T. Mabuchi in [21], [22], [23].

3. K stability

In [30] Tian defined the notion of K-stability for Fano manifolds and proved that if a Fano manifold carries a Kähler-Einstein metric then M is K-stable. Tian K-stability considers the degenerations of M to normal varieties and use the generalized Futaki invariant defined by Ding and Tian. Note that the generalized Futaki invariant is only defined for normal varieties.

Further Donaldson redefined in [11] Futaki invariant for general polarized varieties and also redefined the notion of K-stability for (M, L). The new definition does not require M to be Fano and considers degeneration to non-normal varieties. We now briefly review Donaldson’s definition of K stability.

Let Λ → N be an ample line bundle over an n-dimensional projective scheme. We assume that a ℂ^{∗}-action as bundle isomorphisms of Λ covering the ℂ^{∗}-action on N.

For any positive integer k, there is an induced ℂ^{∗} action on W_{k} = H^{0}(N, Λ^{k}). Put d_{k} = dim W_{k} and let w_{k} be the weight of ℂ^{∗}-action on ∧^{d_{k}}W_{k}. By the Riemann-Roch and the equivariant Riemann-Roch theorems d_{k} and w_{k} are polynomials in k of degree n and n + 1. Therefore w_{k}/kd_{k} is bounded from above as k tends to infinity. For sufficiently large k we expand

\frac{w_{k}}{kd_{k}} = F_{0} + F_{1}k^{−1} + F_{2}k^{−2} + ⋯

and call F_{1} the Futaki invariant of ℂ^{∗}-action on (N, Λ).

For an ample line bundle L over a projective variety M, a test configuration of degree r consists of the following.

(1) A family of schemes π : M → ℂ:

(2) ℂ^{∗}-action on M covering the usual ℂ^{∗}-action on ℂ:

(3) ℂ^{∗}-equivariant line bundle L → M such that • for t ≠ 0 one has M_{t} = π^{−1}(t) ≅ M and (M_{t}, L|_{M_{t}}) ≅ (M.L^{r}),

• the Hilbert polynomial ∑_{p=0}^{n}(−1)^{p} dim H^{p}(M_{t}, L_{t}^{r}) does not depend on t, in particular for r sufficiently large dim H^{0}(M_{t}, L_{t}) = dim H^{0}(M, L) for all t ∈ ℂ.

ℂ^{∗}-action induces a ℂ^{∗}-action on the central fiber L_{0} → M_{0} = π^{−1}(0). Moreover if (M, L) admits a ℂ^{∗}-action, then one obtains a test configuration by taking the direct product L × ℂ. This is called a trivial configuration.

Definition 3.1. (M, L) is said to be K stable if the Futaki invariant of the central fiber (M_{0}, L_{0}) is greater than or equal to 0 for all test configurations, and the equality occurs only if the test configuration is trivial.

Conjecture : A Kähler metric of constant scalar curvature will exist in the Kähler class c_{1}(L) if and only if (M, L) is K stable.

Lemma 3.2 ([11]). If N is a nonsingular projective variety then the above definition of the Futaki invariant coincides with the one given in [13], [4].

4. The integral invariants

Suppose we are given a Kähler class [ω_{0}] on a compact Kähler manifold M. For any Kähler form ω ∈ [ω_{0}] let us denote its scalar curvature by

σ_{ω} := g^{ij̄}R_{ij̄},

and put

μ = m ∫_{M} ρ_{ω} ∧ ω^{m−1}/ ∫_{M} ω^{m} = ∫_{M} σ_{ω}ω^{m}/ ∫_{M} ω^{m}.

Then there is a C^{∞} function F uniquely up to constant such that

ΔF = σ_{ω} − μ.

For a holomorphic vector field X ∈ 𝔥(M) we define

f(X) = ∫_{M} XF ω^{m}.

It can be proved that f : 𝔥(M) → ℂ is independent of the choice of ω in the fixed Kähler class [ω_{0}], and becomes a Lie algebra character. This is the invariant defined in [13], [4] as mentioned in Lemma 3.2. If there exists a constant scalar curvature Kähler metric in the Kähler class [ω_{0}] then we have an ω ∈ [ω_{0}] with F = 0 and thus it turns out that f = 0. Namely f is an obstruction to the existence of Kähler metrics in [ω_{0}] of constant scalar curvature.

If M is a Fano manifold and take c_{1}(M) to be the Kähler class, then being a Kähler metric of constant scalar curvature is equivalent to being a Kähler-Einstein metric (this is easily proved if one uses the Bianchi identity and the proof is left to the reader). Thus, in this Fano case f becomes an obstruction to the existence of Kähler-Einstein metrics.

By the way this obstruction to the existence of Kähler-Einstein metrics has two extensions.

Extension 1) This extension includes the obstruction to the existence of constant scalar curvature Kähler metrics, and was obtained by S. Bando [2]. Let M be a compact Kähler manifold and [ω_{0}] an arbitrary Kähler class. For any Kähler form ω ∈ [ω_{0}] let c_{k}(ω) be its k-th Chern form, and Hc_{k}(ω) the harmonic part of c_{k}(ω). Then there is a (k − 1, k − 1)-form F_{k} such that

c_{k}(ω) − Hc_{k}(ω) = \frac{i}{2π} ∂\overline{∂}F_{k}.

We define f_{k} : 𝔥(M) → ℂ by

f_{k}(X) = ∫_{M} L_{X}F_{k} ∧ ω^{m−k+1}.

Just like f, f_{k} is independent of the choice of ω ∈ [ω_{0}], and becomes a Lie algebra character. If there exits an ω ∈ [ω_{0}] such that c_{k}(ω) is a harmonic (k, k)-form, then we have a Kähler form with F_{k} = 0. Thence we have f_{k} = 0. Namely f_{k} obstructs the existence of Kähler form in [ω_{0}] with harmonic k-th Chern form. In the case when k = 1, being a Kähler metric with harmonic first Chern form is equivalent to being Kähler metric of constant scalar curvature (this also follows from the Biachi identity). Therefore f_{1} is exactly the one we mentioned before.

Extension 2) Another extension was obtained by a joint work [18] by S. Morita and myself. This was motivated by the similarity of f to the Godbillon-Vey invariant in the foliation theory. It turned out that f can be regarded as an equivariant cohomology described by Berline-Vergne [3].

To explain this we return to the problems of Calabi. Let us consider the following question. Given a closed real (1, 1)-form ρ that represents the first Chern class, does there exists a Kähler metric (in a fixed Kähler class) whose first Chern form is ρ? This is the famous Calabi conjecture. Solving this conjecture was reduced to the same type of complex Monge-Ampère equation as the existence of Kähler-Einstein metric with c_{1}(M) = 0. Thus Yau’s work which proved the existence of the solution to this Monge-Ampère equation proves the Calabi conjecture and the existence of Ricci-flat Kähler metric on manifolds with c_{1}(M) = 0 at the same time.

Returning to the situation where M is a Fano manifold, let us regard c_{1}(M) as a Kähler class. Suppose the integral invariant f is written in terms of a Kähler form ω ∈ c_{1}(M). Then, by the Calabi conjecture, there exits a Kähler form η ∈ c_{1}(M) such that its first Chern form is equal to ω. If we rewrite f in terms of η then we get

f(X) = ∫_{M} div_{η}(X) ρ_{η}^{m}. (2)

where div_{η} and ρ_{η} respectively denote the divergence and the Ricci form with respect to the Kähler form η. If η is Kähler-Einsrein then we have ρ_{η}^{m} = η^{m}, which is the volume form. Hence, from the divergence theorem, we obtain f(X) = 0. From this too we see that f obstructs the existence of Kähler-Einstein metrics.

By the way, we may as well rewrite the right hand side of (2) as

f(X) = ∫_{M} div_{Ω}(X) ρ_{Ω}^{m} (3)

where Ω is an arbitrary volume form and

ρ_{Ω} := −\frac{i}{2π} ∂\overline{∂} log Ω

is the Ricci form of Ω, and div_{Ω}(X) is the divergence of X defined by

div_{Ω}(X)Ω = d(i(X)Ω).

Of course (2) is recovered by putting Ω = ω^{m}. It can be proved that the right hand of (3) does not depend on the choice of Ω, and therefore we need not even assume that M is Kähler. Since the extension 1) really depends on the Kähler class, the extension given by (3) is indeed different from the extension 1).

We can further extend (3) as follows. Let P_{G} → M be a holomorphic principal bundle where a complex Lie G acts from the right as the structure group and a complex Lie group H acts from the left as bundle maps commuting with the action of G. Therefore H also acts on M as biholomorphic automorphisms.

Let θ be a type (1, 0)-connection of the principal bundle P_{G}, and Θ its curvature form. An element X in the Lie algebra 𝔥 of H defines a G-invariant vector field on P_{G}. By the abuse of notation we will denote it by the same letter X. Let I^{p}(G) be set of all G-invariant polynomials on 𝔤 of degree p. For any ϕ ∈ I^{p}(G) with p ≥ m we define f_{ϕ} by

f_{ϕ}(X) = ∫_{M} ϕ(θ(X) + Θ).

Then it is not difficult to see f_{ϕ} is independent of the choice of type (1, 0)-connection θ. From this one sees that f_{ϕ} defines an element of I^{p−m}(H). This is one way of expressing the equivariant cohomology.

Let us consider the special case where G = GL(m, ℂ) and P_{G} is a frame bundle of the holomorphic tangent bundle of the complex manifold M, and where H is a subgroup of the automorphism group of M. then I^{∗}(G) is generated by the i-th elementary symmteric functions c_{i}, and f_{c_{1}^{m+1}} coincides with (3).

Now we introduce a new family of integral invariants including the two extensions 1) and 2), c.f. [15]. This is closely related to the asymptotic Chow semistability. Put 𝔥_{0}(M) = {X ∈ 𝔥(M) | X has a zero.}. Then it is well-known that for X ∈ 𝔥_{0}(M) there is a smooth complex valued function u_{X} uniquely up to a constant such that

i(X)ω = \overline{∂}u_{X}.

In this sense 𝔥_{0}(M) coincides with the set of all Hamiltonian vector fields. We may normalize u_{X} so that

∫_{M} u_{X} ω^{m} = 0. (4)

In fact we may repalce 0 by any constant. For ϕ ∈ I^{k}(GL(m, ℂ)) we put

F_{ϕ}(X) = (m − k + 1) ∫_{M} ϕ(Θ) ∧ u_{X} ω^{m−k}

+ ∫_{M} ϕ(θ(X) + Θ) ∧ ω^{m−k+1}, (5)

where θ is the Levi-Civita connection of the Kähler form ω and Θ its curvature form.

Theorem 4.1 ([15]). F_{ϕ}(X) is independent of the choice of ω ∈ [ω_{0}]. The case when k = m + 1 and H is the Lie group corresponding to 𝔥_{0}(M) coincides with the extension 2), and the case when ϕ = c_{k} coincides with the extension 1)．

A formulation of F_{ϕ} for general principal bundles P_{G} is given in [15].

5. Asymptotic Chow semistability and integral invariants

Let L → M be an ample line bundle and Aut(M, L) the group of all automorphisms of M which lift to bundle automorphisms of L. In Theorem 4.1 of the previous section we choose the Kähler class [ω_{0}] to to be c_{1}(L).

Suppose that a holomorphic vector field X is so chosen that it generates a oneparameter subgroup ℂ^{∗} in Aut(M, L), and a lifting of the ℂ^{∗}-action on L is chosen. This means that a normalization of the Hamiltonian function u_{X} is given, with the average not being necessarily zero. Note that the lifting of the ℂ^{∗}-action on L is not unique, and a different choice of the lifting gives rise to a normalization of the Hamiltonian function with a an integer’s difference [16]. Put d_{k} = H^{0}(M, L^{k}), let w_{k} be the weight of the ℂ^{∗}-action on ∧^{d_{k}}H^{0}(M, L^{k}). Then by the equivariant Riemann-Roch theorem w_{k} is equal to the degree 1 term in t of the integral of the following [11]．

e^{k(ω+tu_{X})}T d(tL(X) + Θ) = ∑_{p=0}^{∞} \frac{k^{p}}{p!} (ω + tu_{X})^{p} ∑_{q=0}^{∞} T d^{(q)}(tL(X) + Θ).

Here T d^{(q)} denotes the q-th Todd polynomial and L(X) = ∇_{X} −L_{X}. By expressing this degree 1 term explicitly we get

w_{k} = ∑_{p=0}^{m+1} \frac{k^{p}}{p!} ∫_{M}(ω^{p} ∧ T d^{(m−p+1)}(L(X) + Θ) + p ω^{p−1} ∧ u_{X} T d^{(m−p+1)}(Θ)). (6)

First of all from [17] one sees

∫_{M} T d^{(m+1)}(L(X) + Θ) = 0. (7)

Hence the term p = 0 in (6) vanishes.

Next, if (M, L) is asymptotically semistable then a suitable lift X̃ of X to L induces a subgroup ℂ^{∗} in SL(H^{0}(M, L^{k})), and moreover

f_{c_{1}^{m+1}}(X̃) = 0, (8)

see [15] for the detail. This means that for a Kähler form ω ∈ c_{1}(L) if we choose a connection θ of the associated bundle P_{ℂ^{∗}} of L so that its Chern form is equal to ω and put u_{X} = θ(X̃) then we have

∫_{M} u_{X}ω^{m} = 0. (9)

Namely (8) is equivalent to choosing the normalization of u_{X} so that its average is 0. Thus the term with p = m + 1 in (6) also vanishes.

As we mentioned above the ℂ^{∗}-action induces a subgroup of SL(H^{0}(M, L^{k})), which means w_{k} = 0. Combining (7) and (9) with rhis we get the following．

Theorem 5.1 ([15]). If (M, L) is asymptotically Chow semistable, then for 1 ≤ ℓ ≤ m we have

F_{T d^{(ℓ)}}(X) = 0. (10)

The case ℓ = 1 implies the vanishing of the Futaki invariant.

So far we discussed about necessary conditions for the existence of canonical Kähler metrics. On the other hand sufficient conditions for the existence of Kähler-Einstein metrics were obtained by Y.T.Siu [27], A.Nadel [26], G,Tian [28]. At the moment of this writing no necessary and sufficient condition seem to have been obtained. The right answer would be expressed in terms of stability in geometric invariant theory.

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Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro, Tokyo 152-8551, Japan

   E-mail address: futaki@math.titech.ac.jp