THE KÄHLER–EINSTEIN METRIC FOR SOME HARTOGS DOMAINS OVER BOUNDED SYMMETRIC DOMAINS

AN WANG, WEIPING YIN, LIYOU ZHANG, AND GUY ROOS

Abstract. We study the complete Kähler-Einstein metric of a Hartogs domain Ω̃ built on an irreducible bounded symmetric domain Ω, using a power N^{μ} of the generic norm of Ω. The generating function of the Kähler-Einstein metric satisfies a complex Monge-Ampère equation with boundary condition. The domain Ω̃ is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by X ∈ [0, 1[. This allows to reduce the Monge-Ampère equation to an ordinary differential equation with limit condition. This equation can be explicitly solved for a special value μ_{0} of μ. We work out the details for the two exceptional symmetric domains. The special value μ_{0} seems also to be significant for the properties of other invariant metrics like the Bergman metric; a conjecture is stated, which is proved for the exceptional domains.

Introduction

Let D be a bounded domain in ℂ^{n}. The complete (normalized) Kähler-Einstein metric on D is the Hermitian metric E

E_{z}(u, v) = ∂_{u}\overline{∂}_{v}g|_{z} ,

whose generating function g is the unique solution of the complex Monge-Ampère equation with boundary condition

det (\frac{∂^{2}g}{∂z^{i}∂\overline{z}^{j}} ) = e^{(n+1)g} (z ∈ D),

g(z) → ∞ (z → ∂D)

(see [1], [2], [3]).

Let Ω be a bounded irreducible symmetric domain in V ≃ ℂ^{d}; we will always consider such a domain in its circled realization. For a real positive number μ, let Ω̃ be the Hartogs type domain defined by

Ω̃ = Ω̃_{k}(μ) = {(z, Z) ∈ Ω × ℂ^{k} | ∥Z∥^{2} < N(z, z)^{μ}} ,

where N(z, z) denotes the generic norm of Ω (see Appendix A.3). The Bergman kernel of Ω̃ has been computed in [4] .

Date: 12th January 2005. 2000 Mathematics Subject Classification. Primary: 32F45, 32M15; secondary: 32A25. Key words and phrases. Hartogs domain, Kähler–Einstein metric, bounded symmetric domain. Research supported in part by NSF of China (Grant 10471097). The present work was completed during the stay of G. Roos at Capital Normal University of

Beijing in August-September 2004.

In this paper, we study the complete Kähler-Einstein metric of Ω̃. The domain Ω̃ is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by X ∈ [0, 1[. This allows to reduce the Monge-Ampère equation to an ordinary differential equation, following a method used in [5] when Ω belongs to one of the four series of classical domains. This equation can be explicitly solved for a special value μ_{0} of μ. We first study the case k = 1, then generalize the results to any integer k. Tables are given in Appendix B, allowing the reader to apply the results to each irreducible bounded symmetric domain. In Section 3, we work out some details for the two exceptional bounded symmetric domains; the construction and the main properties of these domains are recalled in Appendix C. In Section 4, starting from the example of exceptional domains, we state a conjecture which links the critical exponent μ_{0} for the Kähler-Einstein metric and the properties of the Bergman kernel of Ω̃_{k}(μ).

1. Symmetric domains inflated by discs

This is the case k = 1, with

Ω̃ = Ω̃_{1}(μ) = {(z, w) ∈ Ω × ℂ | |w|^{2} < N(z, z)^{μ}} .

1.1. Automorphisms. Let Ω be a bounded irreducible symmetric domain, μ > 0 a real number and

Ω̃ = Ω̃_{1}(μ) = {(z, w) ∈ Ω × ℂ | |w|^{2} < N(z, z)^{μ}} .

Let X be the function X : Ω̃ → [0, 1[ defined by

(1.1) X(z, w) = \frac{|w|^{2}}{N(z, z)^{μ}} .

Denote by Aut′ Ω̃ the subgroup of automorphisms of Ω̃ which leave X invariant.

Let Φ ∈ Aut Ω. Denote by d Φ(z) the differential of Φ at z and by JΦ(z) = det d Φ(z) its Jacobian.

Lemma 1. Let Ω be a bounded irreducible symmetric domain in V ≃ ℂ^{d}, with generic norm N and genus γ (see Appendix A.3). Then the function on Ω × Ω

\frac{N(z, z)N(t, t)}{|N(z, t)|^{2}}

is invariant by each Φ ∈ Aut Ω acting diagonally on Ω × Ω:

(1.2) \frac{N(Φz, Φz)N(Φt, Φt)}{|N(Φz, Φt)|^{2}} = \frac{N(z, z)N(t, t)}{|N(z, t)|^{2}}.

Proof. If Φ ∈ Aut_{0} Ω, the identity component of Aut Ω, we deduce from (A.1):

B(Φz, Φt) = d Φ(z) ∘ B(z, t) ∘ d Φ(t)^{∗}

(see Appendix A.1) and det B(z, t) = N(z, t)^{γ}, that

(1.3) N(Φz, Φt)^{γ} = JΦ(z)N(z, t)^{γ} \overline{JΦ(t)}.

If Φ ∈ Aut Ω, it can be written Φ = Φ_{1}Φ_{2}, where Φ_{1} ∈ Aut_{0} Ω and Φ_{2}(0) = 0; then Φ_{2} is linear, unitary (with respect to the Bergman metric at 0) and leaves N invariant, so that (1.3) holds for Φ_{2}, hence also for every Φ ∈ Aut Ω. The relation (1.2) follows immediately. □

Proposition 1. The group Aut′ Ω̃ consists of all Ψ = (Ψ_{1}, Ψ_{2}):

(1.4) Ψ_{1}(z, w) = Φ(z),
(1.5) Ψ_{2}(z, w) = wψ(z),

such that Φ ∈ Aut Ω and

(1.6) |ψ(z)|^{2} = (\frac{N(Φz, Φz)}{N(z, z)})^{μ} .

For Φ ∈ Aut Ω, let z_{0} = Φ^{−1}(0); then the functions ψ satisfying (1.6) are the functions

(1.7) ψ(z) = e^{i θ} \frac{N (z_{0}, z_{0})^{μ/2}}{N (z, z_{0})^{μ}} .

The orbits of Aut′ Ω̃ are the level sets

Σ_{λ} = {X = λ | λ ∈ [0, 1[} .

Proof. It is easily checked that the maps Ψ of the form (1.4)-(1.5), satisfying (1.6), are automorphisms of Ω̃, leave X invariant and form a subgroup G of automorphisms of Ω̃.

Let z^{∗} = Φ(z). Applying (1.2) to z_{0} = Φ^{−1}(0), we get

(1.8) \frac{N(z^{∗}, z^{∗})}{N(z, z)} = \frac{N (z_{0}, z_{0})}{|N(z, z_{0})|^{2}} ,

as N(x, 0) = 1 for each x. The relation (1.3) also implies

1 = JΦ(z)N (z, z_{0})^{γ} \overline{JΦ (z_{0})};

this means that the holomorphic function z ↦ N (z, z_{0}) never vanishes on the convex domain Ω, and we can define the holomorphic function N (z, z_{0})^{μ} for any real μ assuming it is positive for z = z_{0}. By (1.8), the function

ψ_{0}(z) = \frac{N (z_{0}, z_{0})^{μ/2}}{N (z, z_{0})^{μ}}

satisfies (1.6). If ψ is another function satisfying (1.6), then ψ/ψ_{0} has constant modulus 1 and ψ = e^{i θ} ψ_{0}.

Let Ψ ∈ Aut′ Ω̃; as Ψ preserves X, Ψ(Ω ×{0}) = Ω ×{0} and Ψ(z, 0) = (Φ(z), 0) with Φ ∈ Aut Ω. There exists Ψ^{1} ∈ G such that Ψ^{1}(z, 0) = (Φ(z), 0) and Θ = Ψ ∘ (Ψ^{1})^{−1} is an element of Aut′ Ω̃, such that Θ(z, 0) = (z, 0) for all z ∈ Ω. In particular, Θ(0, 0) = (0, 0); as Ω̃ is bounded and circled, it follows from a lemma of H. Cartan (see [6], [7]) that Θ is linear. Then Θ has the form

Θ_{1}(z, w) = z + wu,
Θ_{2}(z, w) = cw.

The invariance of X under Θ implies

|c|^{2} = \frac{N(z + wu, z + wu)^{μ}}{N(z, z)^{μ}}

for all (z, w) ∈ Ω̃ and in particular

|c|^{2} = N (wu, wu)^{μ}

for all w such that |w| < 1. This implies |c| = 1 and u = 0. Then Θ ∈ G and Ψ = Θ ∘ Ψ^{1} belongs also to G.

If (z_{0}, w_{0}) and (z′_{0}, w′_{0}) belong to the same level set Σ_{λ}, that is X(z_{0}, w_{0}) = X(z′_{0}, w′_{0}) = λ, take Φ ∈ Aut Ω such that Φ(z_{0}) = z′_{0} and ψ : Ω → ℂ such that

|ψ(z)|^{2} = (\frac{N(Φz, Φz)}{N(z, z)})^{μ} .

Then Ψ ∈ Aut′ Ω̃ defined by

Ψ_{1}(z, w) = Φ(z),
Ψ_{2}(z, w) = wψ(z)

maps (z_{0}, w_{0}) to (z′_{0}, w_{0}ψ(z_{0})); we have |w_{0}ψ(z_{0})|^{2} = |w′_{0}|^{2}, so it suffices to change Ψ_{2} in αΨ_{2} for some α ∈ ℂ, |α| = 1, in order to obtain Ψ(z_{0}, w_{0}) = (z′_{0}, w′_{0}). So the group Aut′ Ω̃ acts transitively on the level sets Σ_{λ} of X. □

The orbits Σ_{λ} are real hypersurfaces of Ω̃ when λ > 0; the orbit Σ_{0} is Ω × {0}. Note also that Ψ ∈ Aut′ Ω̃ extends continuously to the boundary ∂Ω̃ of Ω̃, as Φ extends continuously to ∂Ω and

∂Ω̃ = (∂Ω × {0}) ∪ {(z, w) ∈ Ω × ℂ | |w|^{2} = N(z, z)^{μ}} .

The part

∂_{0}Ω̃ = {(z, w) ∈ Ω × ℂ | |w|^{2} = N(z, z)^{μ}}

is an orbit of Aut′ Ω̃ (corresponding to X = 1), and ∂Ω × {0} is a finite union of orbits.

In order to compute the differential of ψ and Ψ, we need the following general result in Jordan triple systems. See Appendix A.1 for the notations. In particular, B is the Bergman operator, N the generic norm, m_{1} the generic trace, y^{x} is the quasi-inverse; if V is simple (if Ω is irreducible), the genus is denoted by γ.

Lemma 2. 1) Let V be a Hermitian Jordan triple system. Then

d_{x} B(x, y) = −D(d x, y^{x})B(x, y), (1.9)

d_{y} B(x, y) = −B(x, y)D(x^{y}, d y), (1.10)

(1.11) \frac{d_{x} det B(x, y)}{det B(x, y)} = − tr D(d x, y^{x}),

(1.12) \frac{d_{y} det B(x, y)}{det B(x, y)} = − tr D(x^{y}, d y).

2) If V is simple,

\frac{d_{x} N(x, y)}{N(x, y)} = −m_{1}(d x, y^{x}), (1.13)

\frac{d_{y} N(x, y)}{N(x, y)} = −m_{1}(x^{y}, d y), (1.14)

∂ (\frac{\overline{∂}N(z, z)}{N(z, z)}) = −m_{1} (B(z, z)^{−1} d z, d \overline{z}) . (1.15)

In particular,

(1.16) −∂ (\frac{\overline{∂}N(z, z)}{N(z, z)})|_{z=0} = m_{1} (d z, d \overline{z}) .

Proof. We start from the addition formula for the Bergman operator

B(x + z, y) = B(z, y^{x})B(x, y)

(see [8], p. 469, (J6.4’)). Using the definition of B, this identity can be written

B(x + z, y) = (id_{V} −D(z, y^{x}) + Q(z)Q(y^{x})) B(x, y).

Taking the linear part in z proves (1.9).

In GL(V) we have the well-known relation

\frac{d(det A)}{det A} = tr (A^{−1} d A) = tr (d A.A^{−1}) .

This implies, using (1.9),

\frac{d_{x} det B(x, y)}{det B(x, y)} = tr (d_{x} B(x, y).B(x, y)^{−1}) = − tr D(d x, y^{x}),

which proves (1.11). The “dual” formulas (1.10) and (1.12) are proved in the same way.

If V is simple, det B(x, y) = N(x, y)^{γ} and tr D(x, y) = γm_{1}(x, y); so (1.13)-(1.14) immediately follow from (1.11)-(1.12).

As N(x, y) is holomorphic in x and anti-holomorphic in y, we have

\frac{\overline{∂}N(z, z)}{N(z, z)} = −m_{1}(z^{z}, d z)

and

∂ (\frac{\overline{∂}N(z, z)}{N(z, z)}) = −m_{1}(∂ (z^{z}) , d z).

The differential of the quasi-inverse x^{y} with respect to x is (see [8], p. 471, relation (D2))

d_{x} (x^{y}) = B(x, y)^{−1} d x;

as x^{y} is holomorphic in x and anti-holomorphic in y, this implies

∂ (z^{z}) = B(z, z)^{−1} d z

and (1.15). □

Lemma 3. 1) Let Φ ∈ Aut_{0} Ω, Φ(z_{0}) = 0 and let ψ : Ω → ℂ defined by (1.7):

ψ(z) = e^{i θ} \frac{N (z_{0}, z_{0})^{μ/2}}{N (z, z_{0})^{μ}} .

Then

(1.17) d ψ(z) = μψ(z)m_{1} (d z, z_{0}^{z}) .

2) Let Ψ = (Ψ_{1}, Ψ_{2}) ∈ Aut′ Ω̃ be defined by

Ψ_{1}(z, w) = Φ(z),
Ψ_{2}(z, w) = wψ(z).

Then the differential of Ψ is given by

(1.18) d Ψ_{1}(z, w) = d Φ(z),
(1.19) d Ψ_{2}(z, w) = μwψ(z)m_{1} (d z, z_{0}^{z}) + ψ(z) d w.

The Jacobian of Ψ is

(1.20) JΨ(z, w) = ψ(z)JΦ(z)

and satisfies

(1.21) |JΨ(z, w)|^{2} = (\frac{N(Φz, Φz)}{N(z, z)})^{γ+μ} .

The relation (1.17) follows immediately from (1.13), and implies (1.19). The triangular form of (1.18)-(1.19) yields (1.20). The relation (1.21) follows then from (1.3) (for z = t) and (1.6).

Remark 1. The relation (1.15) expresses the well-known fact that

−∂ (\frac{\overline{∂}N(z, z)}{N(z, z)})

is equal, up to the factor \frac{1}{γ}, to the Bergman metric of Ω at z. This could also be established directly, as the Bergman kernel of Ω is

K(z) = \frac{1}{vol Ω} N(z, z)^{−γ}.

In this paper, we will only make use of the special case (1.16) at z = 0.

1.2. Reduction of the Monge-Ampère equation.

1.2.1. Let Ω be a bounded irreducible symmetric domain, μ > 0 a real number and

Ω̃ = Ω̃_{1}(μ) = {(z, w) ∈ Ω × ℂ | |w|^{2} < N(z, z)^{μ}} .

We denote by d the complex dimension of Ω and by n = d + 1 the dimension of Ω̃. Let g be a C^{2} function in Ω̃, which is a solution of the Monge-Ampère equation

(1.22) det (\frac{∂^{2}g}{∂z^{i}∂\overline{z}^{j}} ) = e^{(n+1)g}

and which generates an invariant form ∂\overline{∂}g. For Ψ ∈ Aut′ Ω̃, the invariance of the metric and the Monge-Ampère equation (1.22) imply

e^{(n+1)g(z,w)} = |JΨ(z, w)|^{2} e^{(n+1)g(Ψ(z,w))}

and, using (1.21),

e^{(n+1)g(z,w)} N(z, z)^{γ+μ} = e^{(n+1)g(Ψ(z,w))} N(Φz, Φz)^{γ+μ}.

In other words, the function

(1.23) g(z, w) + \frac{γ + μ}{n + 1} log N(z, z)

is constant on the orbits of Aut′ Ω̃. Recall that these orbits are also parameterized by X ∈ [0, 1[ and define h(X) as the value of the function (1.23) on the orbit X = |w|^{2} N(z, z)^{−μ}. The function g can then be written

(1.24) g(z, w) = −\frac{γ + μ}{n + 1} log N(z, z) + h (\frac{|w|^{2}}{N(z, z)^{μ}} );

as N(0, 0) = 1, the function h can be obtained from g by

h (|w|^{2}) = g(0, w).

We will show that the Monge-Ampère equation (1.22) is equivalent to an ordinary differential equation for the function h.

1.2.2. Let (z^{1}, …, z^{d}, z^{d+1} = w) be linear coordinates on Ω × ℂ and let

ω = d z^{1} ∧ d \overline{z}^{1} ∧ ⋯ ∧ d z^{d} ∧ d \overline{z}^{d} ∧ d w ∧ d \overline{w}.

We choose for (z^{1}, …, z^{d}) orthonormal coordinates w.r. to the Hermitian metric m_{1} relative to Ω. By Lemma 2, we have

−∂ (\frac{\overline{∂}N(z, z)}{N(z, z)})|_{z=0} = m_{1} (d z, d \overline{z})

and

(1.25) \frac{(−∂\overline{∂} log N(z, z))^{d}}{d!} |_{z=0} = d z^{1} ∧ d \overline{z}^{1} ∧ ⋯ ∧ d z^{d} ∧ d \overline{z}^{d}.

For any C^{2} function f on Ω̃, we have

\frac{1}{(d + 1)!} (∂\overline{∂}f)^{d+1} = det (\frac{∂^{2}f}{∂z^{i}∂\overline{z}^{j}} ) ω.

Lemma 4. Let g : Ω̃ → ℝ be a C^{2} function and let h : [0, 1[→ ℝ be related to g by (1.24). Then, for X = |w|^{2} > 0,

(1.26) \frac{(∂\overline{∂}g)^{d+1}}{(d + 1)!} (0, w) = (μXh′(X) + \frac{γ + μ}{d + 2})^{d} (Xh′(X))′ ω.

Proof. Let X = |w|^{2} N(z, z)^{−μ}. Then

\frac{∂X}{X} = \frac{d w}{w} − μ\frac{∂N}{N} ,
\frac{\overline{∂}X}{X} = \frac{d \overline{w}}{\overline{w}} − μ\frac{\overline{∂}N}{N} .

We have

\overline{∂} (h(X)) = h′(X)\overline{∂}X = Xh′(X) (\frac{d \overline{w}}{\overline{w}} − μ\frac{\overline{∂}N}{N} )

and

∂\overline{∂} (h(X)) = (Xh′(X))′ X (\frac{d w}{w} − μ\frac{∂N}{N} ) ∧ (\frac{d \overline{w}}{\overline{w}} − μ\frac{\overline{∂}N}{N} )

− μXh′(X)∂ (\frac{\overline{∂}N}{N} ) .

We conclude that

∂\overline{∂}g = − (μXh′(X) + \frac{γ + μ}{d + 2}) ∂\overline{∂} log N(z, z)

+ (Xh′(X))′ X (\frac{d w}{w} − μ\frac{∂N}{N} ) ∧ (\frac{d \overline{w}}{\overline{w}} − μ\frac{\overline{∂}N}{N} )

and

(∂\overline{∂}g)^{d+1} = (d + 1) (μXh′(X) + \frac{γ + μ}{d + 2})^{d} (Xh′(X))′ (−∂\overline{∂} log N(z, z))^{d}

∧ \frac{X}{|w|^{2}} d w ∧ d \overline{w}.

At z = 0, we have X = |w|^{2} and (1.25):

\frac{(−∂\overline{∂} log N(z, z))^{d}}{d!} |_{z=0} = d z^{1} ∧ d \overline{z}^{1} ∧ ⋯ ∧ d z^{d} ∧ d \overline{z}^{d},

which implies the result (1.26). □

Lemma 5. Let g be a C^{2} function on Ω̃, which is a solution of the Monge-Ampère equation

det (\frac{∂^{2}g}{∂z^{i}∂\overline{z}^{j}} ) = e^{(n+1)g}

and which generates an invariant form ∂\overline{∂}g. Let

g(z, w) = −\frac{γ + μ}{d + 2} log N(z, z) + h (X) ,
X = \frac{|w|^{2}}{N(z, z)^{μ}} .

Then h satisfies on ]0, 1[ the differential equation

(μXh′(X) + \frac{γ + μ}{d + 2})^{d} (Xh′(X))′ = e^{(d+2)h}, (1.27)

Xh′(X) → 0 (X → 0). (1.28)

The boundary condition

g(z) → ∞ (z → ∂Ω̃)

implies

(1.29) h(X) → ∞ (X → 1).

Proof. The differential equation (1.27) results directly from (1.26). The limit condition (1.29) results from the boundary condition on g, as

h (|w|^{2}) = g(0, w)

and (0, 1) ∈ ∂Ω̃. From

\overline{w}h′ (|w|^{2}) = \frac{∂g}{∂w} (0, w),

we deduce that X^{1/2}h′(X) → 0 as X → 0. □

1.2.3. Let

(1.30) β = \frac{γ + μ}{μ (d + 2)},
(1.31) Y = Xh′(X) + β.

It results from the previous lemmas that if g is a solution of the Monge-Ampère equation

det (\frac{∂^{2}g}{∂z^{i}∂\overline{z}^{j}} ) = e^{(d+2)g}

with the boundary condition

g(z) → ∞ (z → ∂Ω̃),

the function Y defined by (1.31) satisfies the differential equation

(1.32) (μY)^{d} Y′ = e^{(d+2)h}

with the initial condition Y (0) = β.

This shows that the derivative of Y^{d+1} is positive and tends to ∞ when X → 1, as h(X) → ∞ when X → 1. So the function Y is strictly increasing and maps [0, 1[ onto [β, +∞[.

Taking logarithmic derivatives of both sides of (1.32), we get

(1.33) \frac{(Y^{d}Y′)′}{Y^{d}Y′} = (d + 2)h′

and, using the definition of Y,

(1.34) \frac{(Y^{d}Y′)′}{Y^{d}Y′} = (d + 2)\frac{Y − β}{X} .

This can be written

X (Y^{d}Y′)′ = (d + 2)Y^{d} (Y − β) Y′

or

(XY^{d}Y′)′ = Y^{d}Y′ + (d + 2)Y^{d} (Y − β) Y′

= (d + 2)Y^{d+1}Y′ − \frac{γ}{μ}Y^{d}Y′,

using the definition (1.30) of β. From (1.28), we deduce that Y (0) = β; integrating with this initial condition yields

(1.35) XY^{d}Y′ = Y^{d+2} − β^{d+2} − \frac{γ}{μ (d + 1)} (Y^{d+1} − β^{d+1}) .

Let us denote by P the polynomial

(1.36) P (Y) = Y^{d+2} − β^{d+2} − \frac{γ}{μ (d + 1)} (Y^{d+1} − β^{d+1}) .

By construction,

P′(Y) = Y^{d} + (d + 2)Y^{d} (Y − β) ,

which shows that P′(t) > 0 for t > β and consequently P (t) > 0 for t > β. Let R be the polynomial defined by

P (Y) = (Y − β) R(Y);

then R(β) = β^{d} and R is strictly positive on [β, ∞[.

1.2.4. Now we prove that the resolution of the ordinary differential equation (1.35) allows to construct the generating function g for the Kähler-Einstein metric of Ω̃ = Ω̃_{1}(μ).

Lemma 6. The differential equation

XY^{d}Y′ = P (Y), (1.37)

Y → ∞ (X → 1), (1.38)

where the polynomial P is defined by (1.36), has a unique solution

Y : [0, 1[→ [β, +∞[.

This solution is C^{∞} at 0.

Proof. Let Y :]c, 1[→ [β, ∞[ satisfy (1.37)-(1.38). As P is positive on ]β, ∞[, the function Y is monotone and its inverse function satisfies the differential equation

\frac{1}{X} \frac{d X}{d Y} = \frac{Y^{d}}{P (Y)},

X → 1 (Y → ∞).

The solution of this equation is given by

(1.39) − log X = ∫_{Y}^{∞}\frac{y^{d} d y}{P (y)} .

This gives X as a function of Y, and Y as an implicit function of X. It is defined

on ]β, ∞[ and maps ]β, ∞[ on [0, 1[, as ∫_{β}^{∞}\frac{y^{d} d y}{P (y)} = +∞. So the maximal solution

of (1.37)-(1.38) is defined on ]0, 1[; it is C^{∞} on ]0, 1[ and extends continuously to [0, 1[, with Y (0) = β.

The relation (1.39) can be written

(1.40) − log X = − log(Y − β) + log β + ∫_{Y}^{2β}\frac{(y^{d} − R(y)) d y}{(y − β)R(y)} + ∫_{2β}^{∞}\frac{y^{d} d y}{P (y)} .

The polynomial R is positive on [β, ∞[ and R(β) = P′(β) = β^{d}. Let S be defined by

y^{d} − R(y) = (y − β)S(y).

Then (1.40) may be written, for Y > β,

− log X = − log(Y − β) + C_{0} − ∫_{β}^{Y} \frac{S(y) d y}{R(y)} ,

with

C_{0} = log β + ∫_{β}^{2β}\frac{(y^{d} − R(y)) d y}{(y − β)R(y)} + ∫_{2β}^{∞}\frac{y^{d} d y}{P (y)} .

For Y > β, we have

X = e^{−C_{0}}(Y − β) exp ∫_{β}^{Y} \frac{S(y) d y}{R(y)} ,

which shows that X is a C^{∞} invertible function of Y ∈ [β, ∞[, and Y a C^{∞} function of X ∈ [0, 1[. □

Theorem 1. The generating function g for the Kähler-Einstein metric of

Ω̃ = Ω̃(μ) = {(z, w) ∈ Ω × ℂ | |w|^{2} < N(z, z)^{μ}}

is given by

(1.41) g(z, w) = −\frac{γ + μ}{d + 2} log N(z, z) + h (\frac{|w|^{2}}{N(z, z)^{μ}} ) ,

where

(1.42) e^{(d+2)h} = (μY)^{d} Y′

and the function Y : [0, 1[→ [β, +∞[ is the solution of (1.37)-(1.38).

Proof. We have already seen that, if g is the generating function for the Kähler-Einstein metric of Ω̃(μ), the functions h and Y satisfy the conditions of the theorem.

Let now Y : [0, 1[→ [β, ∞[ be the solution of (1.37)-(1.38).and let h and g be defined from Y by (1.42) and (1.41). As Y verifies (1.37), which is equivalent to (1.35), this implies (1.34). Comparing with the logarithmic derivative of (1.42), we get for X > 0

\frac{Y − β}{X} = h′(X).

This can be written

h′(X) = ∫_{0}^{1} Y′(tX) d t (X > 0),

which implies

h′′(X) = ∫_{0}^{1} XY′′(tX) d t (X > 0)

and the existence of h′(0) = Y′(0) and h′′(0) = \frac{1}{2}Y′′(0) follows. So h is C^{2} on [0, 1[ and g is C^{2} on Ω̃.

For X ∈ [0, 1[, we have then Y = Xh′(X) + β and h satisfies the differential equation (1.27)

(μXh′(X) + \frac{γ + μ}{d + 2})^{d} (Xh′(X))′ = e^{(d+2)h} .

Using Lemma 4, we obtain

(1.43) det (\frac{∂^{2}g}{∂z^{i}∂\overline{z}^{j}} ) (0, w) = e^{(d+2)h(|w|^{2})} = e^{(d+2)g(0,w)},

which means that g satisfies the Monge-Ampère equation at the points (0, w). Let (z, w) ∈ Ω̃ and Ψ ∈ Aut′ Ω̃ such that Ψ (z, w) = (0, w′). Then

det (\frac{∂^{2}g}{∂z^{i}∂\overline{z}^{j}} ) (z, w) = det (\frac{∂^{2}g}{∂z^{i}∂\overline{z}^{j}} ) (0, w′) |JΨ(z, w)|^{2}

= det (\frac{∂^{2}g}{∂z^{i}∂\overline{z}^{j}} ) (0, w′)\frac{1}{N(z, z)^{γ+μ}} ,

|w′|^{2} = \frac{|w|^{2}}{N(z, z)^{μ}} ,
g(z, w) = −\frac{γ + μ}{d + 2} log N(z, z) + g(0, w′),

so that (1.43) implies that g satisfies the Monge-Ampère equation on Ω̃.

It remains to prove that g(z, w) → ∞ as (z, w) → (z_{0}, w_{0}) ∈ ∂Ω̃. From Y = Xh′(X) + β and Y → ∞ as X → 1, we see that h′(X) > 0, h′(X) → ∞ and h(X) → ∞ as X → 1. The boundary points (z_{0}, w_{0}) of Ω̃ are of two different types:

• z_{0} ∈ Ω, |w_{0}|^{2} = N (z_{0}, z_{0})^{μ}. Then X (z_{0}, w_{0}) = 1 and

g(z, w) = −\frac{γ + μ}{d + 2} log N(z, z) + h (X(z, w)) → ∞

as (z, w) → (z_{0}, w_{0}).

• z_{0} ∈ ∂Ω, w_{0} = 0. In this case, N(z, z) → 0 and h(X) ≥ h(0), which shows

again that g(z, w) → ∞ as (z, w) → (z_{0}, w_{0}).

□

Remark 2. It is easy to check directly that ∂\overline{∂}g defines a Kähler metric. We have

∂\overline{∂}g = − (μXh′(X) + \frac{γ + μ}{d + 2}) ∂\overline{∂} log N(z, z)

+ (Xh′(X))′ X (\frac{d w}{w} − μ\frac{∂N}{N} ) ∧ (\frac{d \overline{w}}{\overline{w}} − μ\frac{\overline{∂}N}{N} )

= − μY ∂\overline{∂} log N(z, z) + XY′ (\frac{d w}{w} − μ\frac{∂N}{N} ) ∧ (\frac{d \overline{w}}{\overline{w}} − μ\frac{\overline{∂}N}{N} ) .

The associated Hermitian form is

H(ζ, ω) = − μY ∑ \frac{∂^{2}}{∂z^{j}∂\overline{z}^{k}} log N(z, z)ζ^{j}\overline{ζ}^{k}

+ XY′ |\frac{ω}{w} − \frac{μ}{N(z, z)} ∑ \frac{∂}{∂z^{j}} N(z, z)ζ^{j}|^{2} .

The term

B(ζ) = − ∑ \frac{∂^{2}}{∂z^{j}∂\overline{z}^{k}} log N(z, z)ζ^{j}\overline{ζ}^{k}

is, up to a constant factor, the Bergman metric of Ω at z. Hence H(ζ, ω) ≥ 0, as Y > 0 and Y′ > 0. If H(ζ, ω) = 0, then B(ζ) = 0, which implies ζ = 0 and then ω = 0.

1.3. The critical exponent. If μ = μ_{0}, we have C = 0 and (1.39) has a very simple form. We call

μ_{0} = \frac{γ}{d + 1}

the critical exponent for the bounded symmetric domain Ω.

If μ = μ_{0}, we have C = 0 and (1.39) is

− log X = ∫_{Y}^{∞}\frac{d y}{y^{2} − y} = log \frac{y − 1}{y}|_{Y}^{∞} = − log \frac{Y − 1}{Y},

which gives X = \frac{Y −1}{Y} or

Y = \frac{1}{1 − X} .

From e^{(d+2)h} = (μY)^{d} Y′, we obtain

e^{(d+2)h} = (μ_{0})^{d} \frac{1}{(1 − X)^{d+2}} ,

that is

h = \frac{d}{d + 2} log μ_{0} + log (\frac{1}{1 − X} ) ,

= \frac{d}{d + 2} log μ_{0} + log (\frac{N(z, z)^{μ_{0}}}{N(z, z)^{μ_{0}} − |w|^{2}} ) .

Here

\frac{γ + μ_{0}}{d + 2} = μ_{0};

applying (1.41), we have

g(z, w) = −μ_{0} log N(z, z) + \frac{d}{d + 2} log μ_{0} + log (\frac{N(z, z)^{μ_{0}}}{N(z, z)^{μ_{0}} − |w|^{2}} )

= .\frac{d}{d + 2} log μ_{0} + log (\frac{1}{N(z, z)^{μ_{0}} − |w|^{2}} ) .

The Kähler-Einstein metric of Ω̃_{1} (μ_{0}) is associated to the Kähler form

∂\overline{∂}g = −∂\overline{∂} log (N(z, z)^{μ_{0}} − |w|^{2}) .

2. Inflation by Hermitian balls

The above results can be extended to the domain

Ω̃ = Ω̃_{k}(μ) = {(z, Z) ∈ Ω × ℂ^{k} | ∥Z∥^{2} < N(z, z)^{μ}} ,

where ℂ^{k} is endowed with the standard Hermitian norm

∥Z∥^{2} = ∑_{j=1}^{k} |Z^{j}|^{2} .

We outline the results, omitting the proofs when they are entirely analogous to the case k = 1.

14 AN WANG, WEIPING YIN, LIYOU ZHANG, AND GUY ROOS

2.1. Automorphisms. Let X be the function X : Ω̃ → [0, 1[ defined by

X(z, Z) = \frac{∥Z∥^{2}}{N(z, z)^{μ}} .

Denote by Aut′ Ω̃ the subgroup of automorphisms of Ω̃ which leave X invariant. Proposition 2. The group Aut′ Ω̃ consists of all Ψ = (Ψ_{1}, Ψ_{2}):

Ψ_{1}(z, Z) = Φ(z),
Ψ_{2}(z, Z) = ψ(z)U(Z),

where Φ ∈ Aut Ω, U : ℂ^{k} → ℂ^{k} is special unitary and ψ satisfies

|ψ(z)|^{2} = (\frac{N(Φz, Φz)}{N(z, z)})^{μ} .

The orbits of Aut′ Ω̃ are the level sets

Σ_{λ} = {X = λ | λ ∈ [0, 1[} . The construction of the functions ψ is given in Proposition 1.

Lemma 7. Let Ψ = (Ψ_{1}, Ψ_{2}) ∈ Aut′ Ω̃ be defined as above by

Ψ_{1}(z, Z) = Φ(z),
Ψ_{2}(z, Z) = ψ(z)U(Z),

where U : ℂ^{k} → ℂ^{k} is special unitary. The Jacobian of Ψ is

JΨ(z, Z) = ψ^{k}(z)JΦ(z) and satisfies

(2.1) |JΨ(z, Z)|^{2} = (\frac{N(Φz, Φz)}{N(z, z)})^{γ+kμ} .

2.2. Reduction of the Monge-Ampère equation. Let Ω be a bounded irreducible symmetric domain, μ > 0 a real number and

Ω̃ = Ω̃_{k}(μ) = {(z, Z) ∈ Ω × ℂ^{k} | ∥Z∥^{2} < N(z, z)^{μ}} . We denote by d the complex dimension of Ω and by n = d + k the dimension of Ω̃.

Let g be a C^{2} function in Ω̃, which is a solution of the Monge-Ampère equation

det (\frac{∂^{2}g}{∂z^{i}∂\overline{z}^{j}} )_{1≤i,j≤d+k} = e^{(n+1)g}

and which generates an invariant form ∂\overline{∂}g. For Ψ ∈ Aut′ Ω̃, the invariance of the metric and the Monge-Ampère equation imply

e^{(n+1)g(z,Z)} = |JΨ(z, Z)|^{2} e^{(n+1)g(Ψ(z,Z))} and, using (2.1),

e^{(n+1)g(z,Z)} N(z, z)^{γ+kμ} = e^{(n+1)g(Ψ(z,Z))} N(Φz, Φz)^{γ+kμ}. The function

g(z, Z) + \frac{γ + kμ}{d + k + 1} log N(z, z)

is then constant on the orbits of Aut′ Ω̃. For X ∈ [0, 1[, we define h(X) as the value of this function on the orbit X = ∥Z∥^{2} N(z, z)^{−μ}. The function g can then be written

(2.2) g(z, Z) = −\frac{γ + kμ}{d + k + 1} log N(z, z) + h (\frac{∥Z∥^{2}}{N(z, z)^{μ}} );

the function h can be obtained from g by

h (∥Z∥^{2}) = g(0, Z)

or

h(X) = g (0, (X^{1/2}, 0, …, 0)) .

Let (z^{1}, …, z^{d}) be coordinates on V ⊃ Ω, which are orthonormal w.r. to the Hermitian metric m_{1} relative to Ω and let (Z^{1}, …, Z^{k}) be orthonormal coordinates for the Hermitian space ℂ^{k}. Let

(z^{1}, …, z^{d+k}) = (z^{1}, …, z^{d}, Z^{1}, …, Z^{k}) and

ω(z, Z) = ω_{d}(z) ∧ ω_{k}(Z)
= d z^{1} ∧ d \overline{z}^{1} ∧ ⋯ ∧ d z^{d} ∧ d \overline{z}^{d} ∧ d Z^{1} ∧ d \overline{Z}^{1} ∧ ⋯ ∧ d Z^{k} ∧ d \overline{Z}^{k}.

For any C^{2} function f on Ω̃, we have

\frac{1}{(d + k)!} (∂\overline{∂}f)^{d+k} = det (\frac{∂^{2}f}{∂z^{i}∂\overline{z}^{j}} ) ω.

Lemma 8. Let g : Ω̃ → ℝ and h : [0, 1[→ ℝ be C^{2} functions related by (2.2). Then, for ∥Z∥^{2} = X > 0,

(2.3) \frac{(∂\overline{∂}g)^{d+k}}{(d + k)!} (0, Z) = (h′(X))^{k−1} (μXh′(X) + \frac{γ + kμ}{d + k + 1})^{d} (Xh′(X))′ ω.

Proof. Let X = ∥Z∥^{2} N(z, z)^{−μ}. Then

\frac{∂X}{X} = \frac{∂ ∥Z∥^{2}}{∥Z∥^{2}} − μ\frac{∂N}{N} ,
\frac{\overline{∂}X}{X} = \begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} − μ\frac{\overline{∂}N}{N} .

We have

\overline{∂} (h(X)) = h′(X)\overline{∂}X = Xh′(X) (\begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} − μ\frac{\overline{∂}N}{N} )

and

(2.4) ∂\overline{∂} (h(X)) = (Xh′(X))′ X (\frac{∂ ∥Z∥^{2}}{∥Z∥^{2}} − μ\frac{∂N}{N} ) ∧ (\begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} − μ\frac{\overline{∂}N}{N} )

+ Xh′(X)∂ (\begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} ) − μXh′(X)∂ (\frac{\overline{∂}N}{N} ) .

16 AN WANG, WEIPING YIN, LIYOU ZHANG, AND GUY ROOS

As

g(z, Z) = −\frac{γ + kμ}{d + k + 1} log N(z, z) + h (X) ;

we have

∂\overline{∂}g =Xh′(X)∂ (\begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} ) − (μXh′(X) + \frac{γ + kμ}{d + k + 1}) ∂\overline{∂} log N(z, z)

+ XY′ (\frac{∂ ∥Z∥^{2}}{∥Z∥^{2}} − μ\frac{∂N}{N} ) ∧ (\begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} − μ\frac{\overline{∂}N}{N} ) .

Let

β = \frac{γ + kμ}{μ (d + k + 1)},
Y_{0} = Xh′(X),
Y = Xh′(X) + \frac{γ + kμ}{μ (d + k + 1)} = Y_{0} + β.

Then

∂\overline{∂}g = Y_{0}∂ (\begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} ) − μY ∂\overline{∂} log N(z, z)

+ X Y′ (\frac{∂ ∥Z∥^{2}}{∥Z∥^{2}} − μ\frac{∂N}{N} ) ∧ (\begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} − μ\frac{\overline{∂}N}{N} )

and

(∂\overline{∂}g)^{d+k} = (Y_{0}∂ (\begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} ) − μY ∂\overline{∂} log N(z, z))^{d+k}

+ (d + k) (Y_{0}∂ (\begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} ) − μY ∂\overline{∂} log N(z, z))^{d+k−1}

∧ XY′ (\frac{∂ ∥Z∥^{2}}{∥Z∥^{2}} − μ\frac{∂N}{N} ) ∧ (\begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} − μ\frac{\overline{∂}N}{N} )

= \binom{d + k}{k}μ^{d}Y_{0}^{k}Y^{d} (−∂\overline{∂} log N(z, z))^{d} ∧ (∂ (\begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} ))^{k}

+ (d + k)\binom{d + k − 1}{k − 1}μ^{d}XY′Y_{0}^{k−1}Y^{d}

(−∂\overline{∂} log N(z, z))^{d} ∧ (∂ (\begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} ))^{k−1} ∧ \frac{∂ ∥Z∥^{2}}{∥Z∥^{2}} ∧ \begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix}

+ (d + k)\binom{d + k − 1}{k}μ^{d+2}XY′Y_{0}^{k}Y^{d−1}

(−∂\overline{∂} log N(z, z))^{d−1} ∧ \frac{∂N}{N} ∧ \frac{\overline{∂}N}{N} ∧ (∂ (\begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} ))^{k} .

KÄHLER–EINSTEIN METRIC 17

One checks easily that

(∂ (\begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} ))^{k} = 0

and

(∂ (\begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} ))^{k−1} ∧ \frac{∂ ∥Z∥^{2}}{∥Z∥^{2}} ∧ \begin{matrix}\overline{∂} & ∥Z∥ & 2 \\ ∥Z∥ & 2 \end{matrix} = \frac{(k − 1)!ω_{k}(Z)}{∥Z∥^{2k} .}.

So the expression of (∂\overline{∂}g)^{d+k} is reduced to

(∂\overline{∂}g)^{d+k} = \frac{(d + k)!}{d!}μ^{d}XY′Y_{0}^{k−1}Y^{d} (−∂\overline{∂} log N(z, z))^{d} ∧ \frac{ω_{k}(Z)}{∥Z∥^{2k} .}.

At z = 0, we have X = ∥Z∥^{2} and (1.25):

\frac{(−∂\overline{∂} log N(z, z))^{d}}{d!} |_{z=0} = ω_{d}(z).

Finally, we obtain

\frac{1}{(d + k)!} (∂\overline{∂}g)^{d+k} (0, Z) = μ^{d}X^{1−k}Y′Y_{0}^{k−1}Y^{d}ω,

which is equivalent to (2.3). □ Lemma 9. Let g be a C^{2} function in Ω̃, which is a solution of the Monge-Ampère equation

det (\frac{∂^{2}g}{∂z^{i}∂\overline{z}^{j}} ) = e^{(n+1)g}

and which generates an invariant form ∂\overline{∂}g. Let

g(z, Z) = −\frac{γ + kμ}{d + k + 1} log N(z, z) + h (X) ,

X = \frac{∥Z∥^{2}}{N(z, z)^{μ}} .

Then h satisfies on ]0, 1[ the differential equation

(h′(X))^{k−1} (μXh′(X) + \frac{γ + kμ}{d + k + 1})^{d} (Xh′(X))′ = e^{(d+k+1)h},

Xh′(X) → 0 (X → 0). The boundary condition

g(z) → ∞ (z → ∂Ω̃) implies

h(X) → ∞ (X → 1).

Let

(2.5) β = \frac{γ + kμ}{μ (d + k + 1)},
(2.6) Y = Y_{0} + β = Xh′(X) + \frac{γ + kμ}{μ (d + k + 1)}.

It results from the previous lemmas that if g is a solution of the Monge-Ampère equation

det (\frac{∂^{2}g}{∂z^{i}∂\overline{z}^{j}} ) = e^{(d+k+1)g}

on Ω̃, with the boundary condition

g(z) → ∞ (z → ∂Ω̃),

then the function Y defined by (2.6) satisfies the differential equation

(2.7) (\frac{Y − β}{X} )^{k−1} (μY)^{d} Y′ = e^{(d+k+1)h}

with the initial condition Y (0) = β.

Writing (2.7) as

(2.8) μ^{d}Y_{0}^{k−1} (Y_{0} + β)^{d} Y′_{0} = X^{k−1} e^{(d+k+1)h}

shows that there exists a polynomial R of degree d+k, with non negative coefficients, such that the derivative of R(Y_{0}) is positive on ]0, 1[ and tends to ∞ when X → 1. So the function Y_{0} is strictly increasing and maps [0, 1[ onto [0, +∞[ and the function Y is strictly increasing and maps [0, 1[ onto [β, +∞[.

Taking logarithmic derivatives of both sides of (2.8), we get

\frac{(Y_{0}^{k−1} (Y_{0} + β)^{d} Y′_{0})′}{Y_{0}^{k−1} (Y_{0} + β)^{d} Y′_{0}} = (d + k + 1)\frac{Y_{0}}{X} + \frac{k − 1}{X},

which is equivalent to

X (Y_{0}^{k−1} (Y_{0} + β)^{d} Y′_{0})′ = (Y_{0}^{k−1} (Y_{0} + β)^{d} Y′_{0}) ((d + k + 1)Y_{0} + k − 1)

or to

(2.9) (XY_{0}^{k−1} (Y_{0} + β)^{d} Y′_{0})′ = ((d + k + 1)Y_{0} + k) Y_{0}^{k−1} (Y_{0} + β)^{d} Y′_{0}.

Let S be the polynomial of degree d + 1 defined by

(2.10) T^{k}S(T) = ∫_{0}^{T} ((d + k + 1)t + k) t^{k−1} (t + β)^{d} d t.

Integrating (2.9) with the initial condition Y_{0}(0) = 0 yields

XY_{0}^{k−1} (Y_{0} + β)^{d} Y′_{0} = Y_{0}^{k}S(Y_{0})

or

X (Y_{0} + β)^{d} Y′_{0} = Y_{0}S(Y_{0}).

Integrating by parts in (2.10), we get

T^{k}S(T) = T^{k}(T + β)^{d+1} + k(1 − β) ∫_{0}^{T} t^{k} (t + β)^{d} d t;

KÄHLER–EINSTEIN METRIC 19

let (2.11) ∫_{0}^{T} t^{k} (t + β)^{d} d t = T^{k+1}S_{1}(T). Then S_{1} is a polynomial of degree d, and

(−1)^{k+d} ∫_{0}^{−β} t^{k} (t + β)^{d} d t > 0

shows that S_{1}(−β) ≠ 0. Finally, (2.12) S(T) = (T + β)^{d+1} + k(1 − β)T S_{1}(T), S_{1}(−β) ≠ 0. Lemma 10. The differential equation

X (Y_{0} + β)^{d} Y′_{0} = Y_{0}S(Y_{0}),

Y_{0} → ∞ (X → 1),

where the polynomial S is defined by (2.10), has a unique solution

Y_{0} : [0, 1[→ [0, +∞[. This solution is C^{∞} at 0. Theorem 2. The generating function g for the Kähler-Einstein metric of

Ω̃ = Ω̃_{k}(μ) = {(z, Z) ∈ Ω × ℂ^{k} | ∥Z∥^{2} < N(z, z)^{μ}} is given by

g(z, Z) = −\frac{γ + kμ}{d + k + 1} log N(z, z) + h (\frac{∥Z∥^{2}}{N(z, z)^{μ}} ) ,

where

e^{(d+k+1)h} = μ^{d}X^{1−k}Y′_{0}Y_{0}^{k−1} (Y_{0} + β)^{d}

and the function Y_{0} : [0, 1[→ [0, +∞[ satisfies X (Y_{0} + β)^{d} Y′_{0} = Y_{0}S(Y_{0}), (2.13) Y_{0} → ∞ (X → 1) (2.14) with

T^{k}S(T) = ∫_{0}^{T} ((d + k + 1)t + k) t^{k−1} (t + β)^{d} d t.

In view of (2.12), the differential equation (2.13) may also be written as

(2.15) XY′_{0} = Y_{0}(Y_{0} + β) + k(1 − β)\frac{Y_{0}^{2}S_{1}(Y_{0})}{(Y_{0} + β)^{d}},

where S_{1} is the polynomial defined by (2.11).

2.3. The critical exponent. The expression (2.12) shows that the polynomial S defined by (2.10) is divisible by (T + β)^{d} if and only if β = 1. In this case, S = (T + β)^{d+1}. The equation (2.13) is then

XY′_{0} = Y_{0} (Y_{0} + 1) .

With the limit condition (2.14), it integrates as

− log X = ∫_{Y_{0}}^{∞} \frac{1}{y(y + 1)} d y,

which gives

Y_{0} = \frac{X}{1 − X}

and

Y = Y_{0} + 1 = \frac{1}{1 − X} .

As

β = \frac{γ + kμ}{μ (d + k + 1)},

the value of μ corresponding to β = 1 is again

μ_{0} = \frac{γ}{d + 1},

that is, the same value as for k = 1.

For μ = μ_{0}, it is again possible to compute explicitly the Kähler-Einstein metric of Ω̃_{k}(μ). Actually,

e^{(d+k+1)h} = μ^{d}X^{1−k}Y′_{0}Y_{0}^{k−1} (Y_{0} + β)^{d}

yields

e^{(d+k+1)h} = μ_{0}^{d} \frac{1}{(1 − X)^{d+k+1}} ,

h(X) = \frac{d}{d + k + 1} log μ_{0} + log \frac{1}{1 − X} .

As β = 1, we have \frac{γ+kμ_{0}}{d+k+1} = μ_{0},

g(z, Z) = −μ_{0} log N(z, z) + h (\frac{∥Z∥^{2}}{N(z, z)^{μ_{0}}} )

= \frac{d}{d + k + 1} log μ_{0} + log \frac{1}{N(z, z)^{μ_{0}} − ∥Z∥^{2}}

and finally

∂\overline{∂}g = −∂\overline{∂} log (N(z, z)^{μ_{0}} − ∥Z∥^{2}) .

3. The exceptional cases

Here we study the Kähler-Einstein metric and the Bergman metric of Ω̃_{k} (μ_{0}), when Ω is one of the two exceptional domains Ω_{V} and Ω_{V I} (see Appendix C).

3.1. The Bergman kernel of Ω̃_{k} (μ) has been computed (for all μ > 0) in [4]. If the polynomial k ↦ χ(kμ) (which has degree d = dim Ω) is decomposed as

(3.1) \frac{χ(kμ)}{χ(0)} = ∑_{j=0}^{d} c_{μ,j} \frac{(k + 1)_{j}}{j!} ,

where (k + 1)_{j} = \frac{Γ (k+j)}{Γ (k)} denotes the raising factorial, let the function F_{χ,μ} be defined by

(3.2) F_{χ,μ}(t) = ∑_{j=0}^{d} c_{μ,j} (\frac{1}{1 − t})^{j} .

Then the Bergman kernel K̃_{k}(z, Z) of Ω̃_{k} (μ) is given by the relations

(3.3) K̃_{k}(z, Z) = L_{k} (z, ∥Z∥^{2}) ,
(3.4) L_{k}(z, r) = \frac{1}{k!} \frac{∂^{k}}{∂r^{k}} L_{0}(z, r),
(3.5) L_{0} (z, r) = K(z)F_{χ,μ} (\frac{r}{N(z, z)^{μ}} ) ,

where

K(z) = \frac{1}{vol Ω} \frac{1}{N (z, z)^{γ}}

is the Bergman kernel of Ω.

Applying these results, we get

L_{0} (z, r) = \frac{1}{vol Ω} \frac{1}{N (z, z)^{γ}} F_{χ,μ} (\frac{r}{N(z, z)^{μ}} ) ,
L_{k}(z, r) = \frac{1}{k!} \frac{1}{vol Ω} \frac{1}{N (z, z)^{γ+kμ}} F_{χ,μ}^{(k)} (\frac{r}{N(z, z)^{μ}} )

and

K̃_{k}(z, Z) = \frac{1}{k!} \frac{1}{vol Ω} \frac{1}{N (z, z)^{γ+kμ}} F_{χ,μ}^{(k)}(X),

where

X = \frac{∥Z∥^{2}}{N(z, z)^{μ}} .

Hence the Bergman metric of Ω̃_{k} (μ) is associated to the (1, 1) form

(3.6) ϕ = − (γ + kμ) ∂\overline{∂} log N(z, z) + ∂\overline{∂} log F_{χ,μ}^{(k)}(X)..

If μ = μ_{0}, the critical exponent, then γ + kμ_{0} = μ_{0}(d + k + 1) and the Bergman metric of Ω̃_{k} (μ_{0}) is associated to the (1, 1) form

(3.7) ϕ_{0} = −μ_{0}(d + k + 1)∂\overline{∂} log N(z, z) + ∂\overline{∂} log F_{χ,μ_{0}}^{(k)}(X).

For the critical exponent μ = μ_{0}, the Kähler-Einstein metric of Ω̃_{k} (μ_{0}) is associated to the (1, 1) form

(3.8) Ψ_{0} = ∂\overline{∂}g = −∂\overline{∂} log (N(z, z)^{μ_{0}} − ∥Z∥^{2}) .

3.2. The exceptional case of dimension 16. See Appendix C.2 for definitions, notations, and basic results.

Let

V = M_{2,1}(𝕆_{ℂ}) = {(a_{2}, a_{3}) | a_{2}, a_{3} ∈ 𝕆_{ℂ}}

and consider the exceptional symmetric domain of dimension 16

Ω = Ω_{V} = {x ∈ M_{2,1}(𝕆_{ℂ}) | 1 − (x|x) + (x^{♯} | x^{♯}) > 0, 2 − (x|x) > 0} .

Here d = 16 and γ = 12. The generic norm is

N(x, y) = 1 − (x|y) + (x^{♯}|y^{♯}).

The critical exponent is

μ_{0} = \frac{12}{17}.

The inflated Hartogs domain is Ω̃ = Ω̃_{k}(μ_{0}) ⊂ ℂ^{16+k}, consisting in (z, Z) ∈ 𝕆_{ℂ}×ℂ^{k} which verify

2 − (z|z) > 0,
1 − (z|z) + (z^{♯} | z^{♯}) > 0,
∥Z∥^{2} < (1 − (z|z) + (z^{♯}|z^{♯}))^{12/17} .

The Kähler-Einstein metric of Ω̃_{k}(μ_{0}) is associated to the (1, 1)-form

(3.9) Ψ_{0} = −∂\overline{∂} log ((1 − (z|z) + (z^{♯}|z^{♯}))^{12/17} − ∥Z∥^{2}) .

The Hua polynomial χ is

χ(s) = (s + 1)_{8}(s + 4)_{8}.

For μ = μ_{0} = \frac{12}{17}, the Bergman metric of Ω̃_{k} (μ_{0}) is associated to the (1, 1) form

(3.10) ϕ_{0} = −μ_{0}(d + k + 1)∂\overline{∂} log N(z, z) + ∂\overline{∂} log F_{χ,μ_{0}}^{(k)}(X).

Recall that

F_{χ,μ_{0}}(t) = ∑_{j=0}^{d} c_{μ_{0},j} (\frac{1}{1 − t})^{j} ,

where the c_{μ,j} are determined by

\frac{χ(kμ_{0})}{χ(0)} = ∑_{j=0}^{16} c_{μ_{0},j} \frac{(k + 1)_{j}}{j!} .

A computation with Maple gives

χ (\frac{12}{17}s) = (\frac{12}{17})^{16} ∑_{j=0}^{16} c_{j}(s + 1)_{j},

with

c_{16} = 1,  c_{15} = 0,  c_{14} = \frac{595}{12},  c_{13} = \frac{4165}{6},  c_{12} = \frac{30042145}{3456},
c_{11} = \frac{14448385}{144},  c_{10} = \frac{790269316375}{746496},  c_{9} = \frac{1259425781075}{124416},

c_{8} = \frac{12447571001586875}{143327232}, c_{7} = \frac{2957566710311675}{4478976},

c_{6} = \frac{11300125622942496725}{2579890176}, c_{5} = \frac{10677213117341703625}{429981696},

c_{4} = \frac{65190770448545396318125}{557256278016}, c_{3} = \frac{10209484788366056549125}{23219011584},

c_{2} = \frac{114818904611324955416375}{92876046336}, c_{1} = \frac{35779252854815307462625}{15479341056},

c_{0} = \frac{33368892412222545303125}{15479341056}.

This shows that all coefficients c_{μ_{0},j} (0 ≤ j ≤ 16) in

F_{χ,μ_{0}}(t) = ∑_{j=0}^{16} c_{μ_{0},j} (\frac{1}{1 − t})^{j}

are strictly positive, except c_{μ_{0},15} = 0.

3.3. The exceptional case of dimension 27. See Appendix C.1 for definitions, notations, and basic results.

Let

V = H_{3}(𝕆_{ℂ}),

the space of Cayley-Hermitian 3 × 3 matrices with entries in 𝕆_{ℂ} and consider the exceptional symmetric domain Ω = Ω_{V} of dimension 27, defined by the inequalities

1 − (z|z) + (z^{♯}|z^{♯}) − |det z|^{2} > 0,
3 − 2(z|z) + (z^{♯}|z^{♯}) > 0,
3 − (z|z) > 0.

The generic minimal polynomial is

m(T, x, y) = T^{3} − (x|y)T^{2} + (x^{♯}|y^{♯})T − det x det \overline{y}.

Here d = 27 and γ = 18. The generic norm is

N(x, y) = 1 − (x | y) + (x^{#} | y^{#}) − det x det \overline{y}.

The critical exponent is

μ_{0} = \frac{9}{14}..

The inflated Hartogs domain is Ω̃ = Ω̃_{k}(μ_{0}) ⊂ ℂ^{16+k}, consisting in (z, Z) ∈ 𝕆_{ℂ}×ℂ^{k} which verify

3 − (z|z) > 0,
3 − 2(z|z) + (z^{♯} | z^{♯}) > 0,
1 − (z|z) + (z^{♯}|z^{♯}) − |det z|^{2} > 0,
∥Z∥^{2} < (1 − (z|z) + (z^{♯}|z^{♯}) − |det z|^{2})^{9/14} .

The Kähler-Einstein metric of Ω̃_{k}(μ_{0}) is associated to the (1, 1)-form

(3.11) Ψ_{0} = −∂\overline{∂} log ((1 − (z|z) + (z^{♯}|z^{♯}) − |det z|^{2})^{9/14} − ∥Z∥^{2}) .

24 AN WANG, WEIPING YIN, LIYOU ZHANG, AND GUY ROOS

The Hua polynomial χ is

χ(s) = (s + 1)_{9}(s + 5)_{9}(s + 9)_{9}. For μ = μ_{0} = \frac{9}{14}, the Bergman metric of Ω̃_{k} (μ_{0}) is associated to the (1, 1) form (3.12) ϕ_{0} = −μ_{0}(d + k + 1)∂\overline{∂} log N(z, z) + ∂\overline{∂} log F_{χ,μ_{0}}^{(k)}(X), with

F_{χ,μ_{0}}(t) = ∑_{j=0}^{27} c_{μ_{0},j} (\frac{1}{1 − t})^{j} ,

where the c_{μ,j} are determined by

\frac{χ(kμ_{0})}{χ(0)} = ∑_{j=0}^{27} c_{μ_{0},j} \frac{(k + 1)_{j}}{j!} .

A computation with Maple gives

χ (\frac{9}{14}s) = (\frac{9}{14})^{27} ∑_{j=0}^{27} c_{j}(s + 1)_{j},

with

c_{27} = 1, c_{26} = 0, c_{25} = \frac{2275}{9},
c_{24} = \frac{56875}{9},  c_{23} = \frac{38591735}{243},
c_{21} = 15425515970150/3^{11},
c_{20} = 37061881356500/3^{9},
c_{19} = 184328710104188650/3^{14},
c_{18} = 3564334218619774600/3^{14},
c_{17} = 584735324681177419750/3^{16},
c_{16} = 10020732894163060819750/3^{16},
c_{15} = 352001611351295587864253500/3^{23},
c_{14} = 586664566244061492395923000/3^{21},
c_{13} = 1988637252859632373297511212000/3^{26},
c_{12} = 25672251717038124392289396301000/3^{26},
c_{11} = 8233663487061605972803486331644375/3^{29},
c_{10} = 89384793443821000370862374382625000/3^{29},
c_{9} = 1923754293102540042201539198326959366875/3^{36},
c_{8} = 209778908005712588859591649123533801875/3^{32},
c_{7} = 399192552377373476318550395682751432975625/3^{37},
c_{6} = 2728484170046421839052459199725228012518750/3^{37},
c_{5} = 47840351197962492631409316902739852226831250/3^{38},
c_{4} = 232468257762517753158460641861539626710125000/3^{38},

c_{3} = 73037107041363504672642146434776735686797778125/3^{42},
c_{2} = 23557400955895564936769134062033297681918662500/3^{40},
c_{1} = 409456797752799914624225389536137199476376953125/3^{42},
c_{0} = 394594700340674453245747775040231797415576953125/3^{42}.

This shows again that all coefficients c_{μ_{0},j} (0 ≤ j ≤ 27) in

F_{χ,μ_{0}}(t) = ∑_{j=0}^{27} c_{μ_{0},j} (\frac{1}{1 − t})^{j}

are strictly positive, except c_{μ_{0},26} = 0.

4. A conjecture

It has been shown above that for the critical exponent μ = μ_{0}, the Kähler-Einstein metric of Ω̃_{k} (μ_{0}) is associated to the (1, 1) form

Ψ_{0} = ∂\overline{∂}g = −∂\overline{∂} log (N(z, z)^{μ_{0}} − ∥Z∥^{2}) .

On the other hand, the Bergman metric of Ω̃_{k} (μ) is associated to the (1, 1) form

ϕ_{0} = −μ(d + k + 1)∂\overline{∂} log N(z, z) + ∂\overline{∂} log F_{χ,μ}^{(k)}(X),

with

F_{χ,μ}(t) = ∑_{j=0}^{d} c_{μ,j} (\frac{1}{1 − t})^{j} ,

where the c_{μ,j} are determined by

(4.1) \frac{χ(kμ)}{χ(0)} = ∑_{j=0}^{d} c_{μ,j} \frac{(k + 1)_{j}}{j!} .

Conjecture. Let Ω be an irreducible circled bounded symmetric domain of genus γ and dimension d, μ_{0} = \frac{γ}{d+1} its critical exponent, χ its Hua polynomial. The coefficients c_{μ,j} in (4.1) are all strictly positive if and only if

μ < μ_{0}.

For μ = μ_{0}, all coefficients c_{μ,j} in (4.1) are strictly positive, except c_{μ,d−1} = 0 and except for the rank 1 type I_{1,n} (where c_{μ,d} > 0 and c_{μ,j} = 0 for all j < d = n).

The values of the critical exponent are

\begin{matrix}Type & I_{m,n} & II_{n} & III_{n} & IV_{n} & V & V I \\ μ_{0} & \frac{m+n}{mn+1} & \frac{2}{n+ \frac{2}{n−1}} & \frac{2}{n+ \frac{2}{n+1}} & \frac{n}{n+1} & \frac{12}{17} & \frac{9}{14} \end{matrix}

We have always μ_{0} < 1, except in the rank 1 case I_{1,n}. For μ = 1, the signs of c_{μ,j} are alternating, starting with c_{μ,d} > 0 and ending with c_{μ,j} = 0 for j < j_{0}, where j_{0} is a positive integer depending on Ω.

Remark 3. The conjecture has been checked with help of computer algebra software in many significant cases:

• for μ = μ_{0} and the types I_{3,3}, IV_{3}, IV_{4}, IV_{6}, V, V I;

• for type V and all values of μ.

Remark 4. As the function F_{χ,μ} is related to the Bergman kernel of Ω̃_{k} (μ), all derivatives F_{χ,μ}^{(k)}, k > 0, of this function are strictly positive on [0, 1[ for all μ > 0.

Remark 5. If the conjecture is true, it would help to compare the Bergman metric and the Kähler-Einstein metric of Ω̃_{k} (μ) for μ = μ_{0}. The exponent μ_{0} seems also to be a limit case for other comparisons: in [12], a comparison theorem is given between the Kähler-Einstein metric and the Kobayashi metric of Ω̃_{1} (μ) when Ω is a symmetric domain of type I_{m,n} and μ < μ_{0}.

Appendix A. Bounded symmetric domains and Jordan triple systems

Hereunder we give a review of properties of the Jordan triple structure associated to a complex bounded symmetric domain (see [7], [8]).

A.1. Jordan triple system associated to a bounded symmetric domain. Let Ω be an irreducible bounded circled homogeneous domain in a complex vector space V. Let K be the identity component of the (compact) Lie group of (linear) automorphisms of Ω leaving 0 fixed. Let ω be a volume form on V, invariant by K and by translations. Let K be the Bergman kernel of Ω with respect to ω, that is, the reproducing kernel of the Hilbert space H^{2}(Ω, ω) = Hol(Ω) ∩ L^{2}(Ω, ω). The Bergman metric at z ∈ Ω is defined by

h_{z}(u, v) = ∂_{u}\overline{∂}_{v} log K(z).

The Jordan triple product on V is characterized by

h_{0}(\{uvw\}, t) = ∂_{u}\overline{∂}_{v}∂_{w}\overline{∂}_{t} log K(z) |_{z=0} .

The triple product (x, y, z) ↦ {xyz} is complex bilinear and symmetric with respect to (x, z), complex antilinear with respect to y. It satisfies the Jordan identity

{xy{uvw}} − {uv{xyw}} = {{xyu}vw} − {u{vxy}w}.

The space V endowed with the triple product {xyz} is called a (Hermitian) Jordan triple system. For x, y, z ∈ V, denote by D(x, y) and Q(x, z) the operators defined by

{xyz} = D(x, y)z = Q(x, z)y.

The Bergman metric at 0 is related to D by

h_{0}(u, v) = tr D(u, v).

A Jordan triple system is called Hermitian positive if (u|v) = tr D(u, v) is positive definite. As the Bergman metric of a bounded domain is always definite positive, the Jordan triple system associated to a bounded symmetric domain is Hermitian positive.

The quadratic representation

Q : V ⟶ End_{ℝ}(V)

is defined by Q(x)y = \frac{1}{2}\{xyx\}. The following fundamental identity for the quadratic representation is a consequence of the Jordan identity:

Q(Q(x)y) = Q(x)Q(y)Q(x).

The Bergman operator B is defined by

B(x, y) = I − D(x, y) + Q(x)Q(y),

where I denotes the identity operator in V. It is also a consequence of the Jordan identity that the following fundamental identity holds for the Bergman operator:

Q(B(x, y)z) = B(x, y)Q(z)B(y, x).

The Bergman operator gets its name from the following property:

h_{z} (B(z, z)u, v) = h_{0}(u, v) (z ∈ Ω; u, v ∈ V).

If Φ ∈ (Aut Ω)_{0}, the identity component of the automorphism group of Ω, the relation

(A.1) B(Φx, Φy) = d Φ(x) ∘ B(x, y) ∘ d Φ(y)^{∗}

holds for x, y ∈ Ω, where ^{∗} denotes the adjoint with respect to the Hermitian metric h_{0}. As a consequence, the Bergman kernel of Ω is given by

(A.2) K(z) = \frac{1}{vol Ω} \frac{1}{det B(z, z)}.

The quasi-inverse x^{y} is defined, for each pair (x, y) such that B(x, y) is invertible, by

x^{y} = B (x, y)^{−1} (x − Q(x)y) .

A.2. Spectral theory. An Hermitian positive Jordan triple system is always semisimple, that is, the direct sum of a finite family of simple subsystems, with component-wise triple product.

As the domain Ω is assumed to be irreducible, the associated Jordan triple system V is simple, that is V is not the direct sum of two non trivial subsystems.

An automorphism f : V → V of the Jordan triple system V is a complex linear isomorphism preserving the triple product : f{u, v, w} = {fu, fv, fw}. The automorphisms of V form a group, denoted Aut V, which is a compact Lie group; we will denote by K its identity component.

An element c ∈ V is called tripotent if {ccc} = 2c. If c is a tripotent, the operator D(c, c) annihilates the polynomial T (T − 1)(T − 2).

Let c be a tripotent. The decomposition V = V_{0}(c)⊕V_{1}(c)⊕V_{2}(c), where V_{j}(c) is the eigenspace V_{j}(c) = {x ∈ V ; D(c, c)x = jx}, is called the Peirce decomposition of V (with respect to the tripotent c).

Two tripotents c_{1} and c_{2} are called orthogonal if D(c_{1}, c_{2}) = 0. If c_{1} and c_{2} are orthogonal tripotents, then D(c_{1}, c_{1)} and D(c_{2}, c_{2}) commute and c_{1} + c_{2} is also a tripotent.

A non zero tripotent c is called primitive if it is not the sum of non zero orthogonal tripotents. A tripotent c is maximal if there is no non zero tripotent orthogonal to c. The set of maximal tripotents is equal to the Shilov boundary of the domain Ω.

A frame of V is a maximal sequence (c_{1}, …, c_{r}) of pairwise orthogonal primitive tripotents. The frames of V form a manifold F, which is called the Satake-Furstenberg boundary of Ω.

Let c = (c_{1}, …, c_{r}) be a frame. For 0 ≤ i ≤ j ≤ r, let

V_{ij}(c) = {x ∈ V | D(c_{k}, c_{k})x = (δ_{i}^{k} + δ_{j}^{k})x, 1 ≤ k ≤ r} :

the decomposition V = ⨁_{0≤i≤j≤r} V_{ij}(c) is called the simultaneous Peirce decomposition with respect to the frame c.

Let V be a simple Hermitian positive Jordan triple system. Then there exist frames for V. All frames have the same number of elements, which is the rank r of V. The subspaces V_{ij} = V_{ij}(c) of the simultaneous Peirce decomposition have the following properties: V_{00} = 0 ; V_{ii} = ℂe_{i} (0 < i); all V_{ij}’s (0 < i < j) have the same dimension a; all V_{0i}’s (0 < i) have the same dimension b.

The numerical invariants of V (or of Ω) are the rank r and the two integers

a = dim V_{ij} (0 < i < j),
b = dim V_{0i} (0 < i).

The genus of V is the number γ defined by

γ = 2 + a(r − 1) + b;

the genus is generally denoted by g or p, but we denote it here by γ as g stands for the generating function of the Kähler-Einstein metric. The HPJTS V and the domain Ω are said to be of tube type if b = 0.

Let V be a simple Hermitian positive Jordan triple system. Then any x ∈ V can be written in a unique way

(A.3) x = λ_{1}c_{1} + λ_{2}c_{2} + ⋯ + λ_{p}c_{p},

where λ_{1} > λ_{2} > ⋯ > λ_{p} > 0 and c_{1}, c_{2} …, c_{p} are pairwise orthogonal tripotents. The element x is regular iff p = r; then (c_{1}, c_{2}, …, c_{r}) is a frame of V. The decomposition (A.3) is called the spectral decomposition of x.

A.3. The generic minimal polynomial. Let V be a Jordan triple system of rank r. There exist polynomials m_{1}, …, m_{r} on V × \overline{V}, homogeneous of respective bidegrees (1, 1), …, (r, r), such that for each regular x ∈ V, the polynomial

m(T, x, y) = T^{r} − m_{1}(x, y)T^{r−1} + ⋯ + (−1)^{r}m_{r}(x, y)

satisfies

m(T, x, x) = ∏_{i=1}^{r}(T − λ_{i}^{2}),

where x = λ_{1}c_{1} + λ_{2}c_{2} + ⋯ + λ_{r}c_{r} is the spectral decomposition of x. Here \overline{V} denotes the space V with the conjugate complex structure. The polynomial

m(T, x, y) = T^{r} − m_{1}(x, y)T^{r−1} + ⋯ + (−1)^{r}m_{r}(x, y)

is called the generic minimal polynomial of V (at (x, y)). The (inhomogeneous) polynomial N : V × \overline{V} → ℂ defined by

N(x, y) = m(1, x, y)

is called the generic norm. The following identities hold:

(A.4) det B(x, y) = N(x, y)^{γ},
tr D(x, y) = γ m_{1}(x, y).

A.4. The spectral norm. Let V be an HPJTS. The map x ↦ λ_{1}, where x = λ_{1}c_{1} +λ_{2}c_{2} +⋯+λ_{p}c_{p} is the spectral decomposition of x (λ_{1} > λ_{2} > ⋯ > λ_{p} > 0) is a norm on V, called the spectral norm. The bounded symmetric domain Ω is the unit ball of V for the spectral norm. It is also characterized by the set of polynomial inequalities

\frac{∂^{j}}{∂T^{j}} m(T, x, x)|_{T =1} > 0, 0 ≤ j ≤ r − 1.

A.5. The Hua integral. Let Ω be an irreducible bounded circled homogeneous domain, N its generic norm. The Hua integral relative to Ω is

∫_{Ω} N(x, x)^{s}α^{n},

where α = \frac{i}{2π} ∂\overline{∂}m_{1}. It converges for s > −1 and has be computed for the four classical series of symmetric domains by Hua Lookeng in [9].

For a general irreducible bounded circled homogeneous domain with numerical invariants a, b and rank r, the Hua integral is given by

∫_{Ω} N(x, x)^{s}α^{n} = \frac{χ(0)}{χ(s)} ∫_{Ω} α^{n},

where χ is the Hua polynomial

χ(s) = ∏_{j=1}^{r} (s + 1 + (j − 1)\frac{a}{2})_{1+b+(r−j)a}

(see [4], Theorem 2.3). See the tables below for the expression of this polynomial for the six cases of bounded irreducible symmetric domains.

Appendix B. Tables for bounded symmetric domains

The following examples exhaust the list of simple Hermitian positive Jordan triple systems (see [10]). The HPJTS occurring in the four infinite series I_{p,q}, II_{n}, III_{n}, IV_{n} are called classical; the two HPJTS of type V and V I are called exceptional. There is some overlapping between the classical series, due to a finite number of isomorphisms in low dimension. We give hereunder for each type:

• the definition of the space V, its Jordan triple product, and the corresponding bounded circled homogeneous domain;

• the generic norm;

• the numerical invariants r, a, b, γ = 2 + a(r − 1) + b;

• the Hua polynomial

χ(s) = ∏_{j=1}^{r} (s + 1 + (j − 1)\frac{a}{2})_{1+b+(r−j)a} .

B.1. Type I_{m,n} (1 ≤ m ≤ n). V = M_{m,n}(ℂ) (space of m × n matrices with complex entries), endowed with the triple product

\{xyz\} = x^{t}\overline{y}z + z^{t}\overline{y}x.

The domain Ω is the set of m × n matrices x such that I_{m} − x^{t}\overline{x} is definite positive. The generic minimal polynomial is

m(T, x, y) = Det(T I_{m} − x^{t}\overline{y}),

where Det is the usual determinant of square matrices. The numerical invariants are r = m, a = 2, b = n − m, γ = m + n. These HPJTS are of tube type only for m = n.

The polynomial χ is

χ(s) = ∏_{j=1}^{m} (s + j)_{m+n+1−2j} = ∏_{j=1}^{m}(s + j)_{n}.

B.2. Type II_{n} (n ≥ 2). V = A_{n}(ℂ) (space of n × n alternating matrices) with the same triple product as for Type I. The domain Ω is the set of n × n alternating matrices x such that I_{n} + x\overline{x} is definite positive.

B.2.1. Type II_{2p} (n = 2p even). The generic minimal polynomial is here given by

m(T, x, y)^{2} = Det(T I_{n} + x\overline{y}).

The numerical invariants are r = \frac{n}{2} = p, a = 4, b = 0, γ = 2 (n − 1); these HPJTS are of tube type.

The polynomial χ is

χ(s) = ∏_{j=1}^{p} (s + 2j − 1)_{1+4(p−j)} = ∏_{j=1}^{p}(s + 2j − 1)_{2p−1}.

B.2.2. Type II_{2p+1} (n = 2p + 1 odd). The generic minimal polynomial is given by

T m(T, x, y)^{2} = Det(T I_{n} + x\overline{y}).

The numerical invariants are r = [\frac{n}{2}] = p, a = 4, b = 2, γ = 2(n − 1); these HPJTS are not of tube type.

The polynomial χ is

χ(s) = ∏_{j=1}^{p} (s + 2j − 1)_{3+4(p−j)} = ∏_{j=1}^{p}(s + 2j − 1)_{2p+1}.

B.3. Type III_{n} (n ≥ 1). V = S_{n}(ℂ) (space of n × n symmetric matrices) with the same triple product as for Type I. The domain Ω is the set of n × n symmetric matrices x such that I_{n} − x\overline{x} is definite positive. The generic minimal polynomial is

m(T, x, y) = Det(T I_{n} − x\overline{y}).

The numerical invariants are r = n, a = 1, b = 0, γ = n + 1. These HPJTS are of tube type.

The polynomial χ is

χ(s) = ∏_{j=1}^{n} (s + \frac{j + 1}{2})_{1+n−j} .

B.4. Type IV_{n} (n ≠ 2). V = ℂ^{n} with the quadratic operator defined by

Q(x)y = q(x, \overline{y})x − q(x)\overline{y},

where q(x) = ∑ x_{i}^{2}, q(x, y) = 2 ∑ x_{i}y_{i}. The domain Ω is the set of points x ∈ ℂ^{n} such that

1 − q(x, \overline{x}) + |q(x)|^{2} > 0, 2 − q(x, \overline{x}) > 0.

The generic minimal polynomial is

m(T, x, y) = T^{2} − q(x, \overline{y}) + q(x)q(\overline{y}).

The numerical invariants are r = 2, a = n − 2, b = 0, γ = n. These HPJTS are of tube type.

The polynomial χ is

χ(s) = (s + 1)_{n−1} (s + \frac{n}{2}) .

B.5. Type V. V = M_{2,1}(𝕆_{ℂ}), the subspace of H_{3}(𝕆_{ℂ}) consisting in matrices of the form

( \begin{matrix}0 & a_{3} & ã_{2} \\ ã_{3} & 0 & 0 \\ a_{2} & 0 & 0 \end{matrix} )

with the same quadratic operator as for type VI (see below). Here ã denotes the Cayley conjugate of a ∈ 𝕆_{ℂ}. The generic minimal polynomial is

m(T, x, y) = T^{2} − (x|y)T + (x^{♯}|y^{♯}).

The domain Ω is the “exceptional domain of dimension 16” defined by

1 − (x|x) + (x^{♯} | x^{♯}) > 0, 2 − (x|x) > 0.

The numerical invariants are r = 2, a = 6, b = 4, γ = 12. This HPJTS is not of tube type.

The polynomial χ is

χ(s) = (s + 1)_{11} (s + 4)_{5} ;

it can also be written

χ(s) = (s + 1)_{8}(s + 4)_{8}.

B.6. Type VI. V = H_{3}(𝕆_{ℂ}), the space of 3 × 3 matrices with entries in the space 𝕆_{ℂ} of octonions over ℂ, which are Hermitian with respect to the Cayley conjugation; the quadratic operator is defined by

Q(x)y = (x|y)x − x^{♯} × \overline{y},

where × denotes the Freudenthal product, x^{♯} the adjoint matrix in H_{3}(𝕆_{ℂ}) and (x|y) the standard Hermitian product in H_{3}(𝕆_{ℂ}) (see Appendix C.1). The domain Ω is the “exceptional domain of dimension 27” defined by

1 − (x|x) + (x^{♯}|x^{♯}) − |det x|^{2} > 0,
3 − 2(x|x) + (x^{♯}|x^{♯}) > 0,
3 − (x|x) > 0.

The generic minimal polynomial is

m(T, x, y) = T^{3} − (x|y)T^{2} + (x^{♯}|y^{♯})T − det x det \overline{y},

where det denotes the determinant in H_{3}(𝕆_{ℂ}). The numerical invariants are r = 3, a = 8, b = 0, γ = 18. This HPJTS is of tube type.

The polynomial χ is

χ(s) = (s + 1)_{17}(s + 5)_{9}(s + 9)
= (s + 1)_{9}(s + 5)_{9}(s + 9)_{9}.

Appendix C. The exceptional bounded symmetric domains

In this appendix, we recall without proofs the construction and the main results about the two exceptional bounded symmetric domains. For details and proofs, see [11].

C.1. The exceptional Jordan triple system H_{3}(𝕆_{ℂ}) and the exceptional bounded symmetric domain of dimension 27. Let 𝕆_{ℂ} denote the 8-dimen-sional algebra of complex octonions, with Cayley conjugation a ↦ ã, trace t(a) = a + ã, Cayley norm n(a) = aã. If 𝕆 denotes the real Cayley division algebra, we consider 𝕆_{ℂ} as its complexification 𝕆_{ℂ} = ℂ⊗_{ℝ}𝕆; the complex conjugate of a = λα (λ ∈ ℂ, α ∈ 𝕆) is \overline{a} = \overline{λ}α. (Recall that the algebras 𝕆 and 𝕆_{ℂ} are neither commutative nor associative). The (complex bilinear) scalar product of a, b ∈ 𝕆_{ℂ} is defined by

(C.1) (a : b) = ab̃ + ãb = b̃a + bã;

the Hermitian scalar product of a, b ∈ 𝕆_{ℂ} is defined by

(C.2) (a | b) = (a : \overline{b}) = a˜\overline{b} + ã\overline{b} = ˜ \overline{b}a + \overline{b}ã.

We denote by H_{3}(𝕆_{ℂ}) the ℂ-vector space (with the natural operations) of 3 × 3 matrices with entries in 𝕆_{ℂ}, which are Hermitian with respect to the Cayley conjugation in O_{ℂ}. An element a ∈ H_{3}(𝕆_{ℂ}) will be written

(C.3) a = ( \begin{matrix}α_{1} & a_{3} & ã_{2} \\ ã_{3} & α_{2} & a_{1} \\ a_{2} & ã_{1} & α_{3} \end{matrix} ) ,

with α_{1}, α_{2}, α_{3} ∈ ℂ and a_{1}, a_{2}, a_{3} ∈ 𝕆_{ℂ}. Instead of (C.3), we will also write

(C.4) a = ∑_{j=1}^{3} α_{j}e_{j} + ∑_{j=1}^{3} F_{j}(a_{j}),

with the obvious definitions for e_{j} and F_{j}(a_{j}). The vector space H_{3}(𝕆_{ℂ}) decomposes into the direct sum

(C.5) H_{3}(𝕆_{C}) = ℂe_{1} ⊕ ℂe_{2} ⊕ ℂe_{3} ⊕ F_{1} ⊕ F_{2} ⊕ F_{3},

where F_{j} = {F_{j}(a); a ∈ 𝕆_{ℂ}}. The subspaces F_{j} are 8-dimensional and

dim_{ℂ} H_{3}(𝕆_{C}) = 27.

On H_{3}(𝕆_{ℂ}), define a bilinear form by

(C.6) (a : b) = ∑_{j=1}^{3} α_{j}β_{j} + ∑_{j=1}^{3}(a_{j} : b_{j})

for a = ∑_{j=1}^{3} α_{j}e_{j} + ∑_{j=1}^{3} F_{j}(a_{j}), b = ∑_{j=1}^{3} β_{j}e_{j} + ∑_{j=1}^{3} F_{j}(b_{j}). Here (a_{j} : b_{j}) denotes the scalar product (C.16) in 𝕆_{ℂ}. The form defined by (C.6) is clearly nonsingular and the decomposition (C.5) is orthogonal with respect to it. We will refer to (a : b) as the (complex bilinear) scalar product of a and b in H_{3}(𝕆_{ℂ}).

The adjoint a^{#} of an element

a = ∑_{j=1}^{3} α_{j}e_{j} + ∑_{j=1}^{3} F_{j}(a_{j}) ∈ H_{3}(𝕆_{ℂ})

is defined by

(C.7) a^{#} = ∑_{i} (α_{j}α_{k} − n(a_{i})) e_{i} + ∑_{i} F̃_{i} (a_{j}a_{k} − α_{i}ã_{i}) .

In (C.7) and below, ∑_{i} means ∑_{i=1}^{3} and j, k are defined by (i, j, k) being an even permutation of (1, 2, 3); F̃_{i}(c) stands for F_{i}(c̃). The symmetric bilinear map, associated to the quadratic map a ↦ a^{#} by

a × b = (a + b)^{#} − a^{#} − b^{#}, a × a = 2a^{#},

is called the Freudenthal product in H_{3}(𝕆_{ℂ}). It follows directly from the definitions that

(C.8) a×b = ∑_{i} (α_{j}β_{k} + α_{k}β_{j} − (a_{i} : b_{i})) e_{i} +∑_{i} F̃_{i} (a_{j}b_{k} + b_{j}a_{k} − α_{i}b̃_{i} − β_{i}ã_{i}) .

In particular, we have

e_{i} × e_{i} = 0, e_{i} × e_{j} = e_{k},

e_{i} × F_{i}(b) = −F_{i}(b), e_{i} × F_{j}(b) = 0, (C.9)

F_{i}(a) × F_{i}(b) = −(a : b)e_{i}, F_{i}(a) × F_{j}(b) = F̃_{k}(ab).

In these relations, (i, j, k) is always an even permutation of (1, 2, 3). The Freudenthal product verifies

(C.10) (a × b : c) = (a : b × c) (a, b, c ∈ H_{3}(𝕆_{ℂ})).

Let T denote the trilinear symmetric form on H_{3}(𝕆_{ℂ}) defined by

T (a, b, c) = (a × b : c) .

The determinant in H_{3}(𝕆_{ℂ}) is the associated polynomial of degree 3, defined by

(C.11) det a = \frac{1}{3!}T (a, a, a) = \frac{1}{3}(a^{#} : a).

If a ∈ H_{3}(𝕆_{ℂ}) is written as in (C.4), then

(C.12) det a = α_{1}α_{2}α_{3} − ∑_{i} α_{i}n(a_{i}) + a_{1}(a_{2}a_{3}) + (ã_{3}ã_{2})ã_{1}.

The following identities hold in H_{3}(𝕆_{ℂ}):

(C.13) (a^{#})^{#} = (det a)a,
(C.14) det(a^{#}) = (det a)^{2}.

The complex conjugate of a = ∑_{j=1}^{3} α_{j}e_{j} + ∑_{j=1}^{3} F_{j}(a_{j}) ∈ H_{3}(𝕆_{ℂ}) is defined as \overline{a} = ∑_{j=1}^{3} \overline{α_{j}}e_{j} + ∑_{j=1}^{3} F_{j}(\overline{a_{j}}). On H_{3}(𝕆_{ℂ}), we consider the Hermitian scalar product defined by

(C.15) (a | b) = (a : \overline{b}) .

The Jordan triple product on H_{3}(𝕆_{ℂ}) is defined by

(C.16) Q(x)y = (x | y)x − x^{#} × \overline{y},
(C.17) D(x, y)z = \{xyz\} = (x | y)z + (z | y)x − (x × z) × \overline{y}.

With this triple product, H_{3}(𝕆_{ℂ}) is an Hermitian positive Jordan triple system. The space H_{3}(𝕆_{ℂ}) endowed with the triple product defined by (C.17) will be referred to as the Hermitian Jordan triple system H_{3}(𝕆_{ℂ}), or the Hermitian Jordan triple system of type VI, or the exceptional Hermitian triple system of dimension 27.

The generic minimal polynomial of the Jordan triple system H_{3}(𝕆_{ℂ}) is

(C.18) m(T, x, y) = T^{3} − (x | y)T^{2} + (x^{#} | y^{#})T − det x det \overline{y};

the rank of H_{3}(𝕆_{ℂ}) is 3. The generic norm is

N(x, y) = 1 − (x | y) + (x^{#} | y^{#}) − det x det \overline{y}

The set E of tripotents of H_{3}(𝕆) is the disjoint union E = E_{0} ∪ E_{1} ∪ E_{2} ∪ E_{3}, where

E_{0} = {0}, E_{1} = {x; (x | x) = 1, x^{#} = 0} ,
E_{2} = {x; (x | x) = 2, (x^{#} | x^{#}) = 1, det x = 0} ,
E_{3} = {x; (x | x) = 3, (x^{#} | x^{#}) = 3, |det x|^{2} = 1} .

The elements e_{1}, e_{2}, e_{3} belong to E_{1} and are therefore minimal tripotents. It is easily checked that the Peirce spaces for e_{1} are

V_{0}(e_{1}) = ℂe_{2} ⊕ ℂe_{3} ⊕ F_{1},
V_{1}(e_{1}) = F_{2} ⊕ F_{3}, V_{2}(e_{1}) = ℂe_{1}.

Similar results hold for the Peirce decomposition w.r. to e_{2} and e_{3}. As e_{2} and e_{3} belong to V_{0}(e_{1}), they are orthogonal to e_{1}; also, e_{2} is orthogonal to e_{3}. So (e_{1}, e_{2}, e_{3}) is a frame for the Jordan triple system H_{3}(𝕆_{ℂ}). It is then easily checked that the simultaneous Peirce decomposition w.r. to the frame (e_{1}, e_{2}, e_{3}) is

H_{3}(𝕆_{ℂ}) = ⨁_{1≤i≤j≤3} V_{ij},

with V_{ii} = ℂe_{i}, V_{ij} = F_{k}. As all V_{ij} (1 ≤ i < j ≤ 3) for this frame are non-zero, the Hermitian Jordan triple system H_{3}(𝕆_{ℂ}) is simple. Hence the numerical invariants are

a = dim F_{i} = 8, b = dim V_{0i} = 0, r = 3, g = 2 + a(r − 1) = 18.

In particular, H_{3}(𝕆_{ℂ}) is of tube type. In H_{3}(𝕆_{ℂ}),

(C.19) tr D(x, y) = 18(x | y),
(C.20) det B(x, y)) = (1 − (x | y) + (x^{#} | y^{#}) − det x det \overline{y})^{18} .

The bounded circled symmetric domain Ω_{V I} corresponding to the Jordan triple system H_{3}(𝕆_{ℂ}) is defined by the set of inequalities

1 − (x|x) + (x^{♯}|x^{♯}) − |det x|^{2} > 0,
3 − 2(x|x) + (x^{♯}|x^{♯}) > 0,
3 − (x|x) > 0.

C.2. The exceptional Jordan triple and the exceptional domain of dimension 16. The 16-dimensional vector space

W = M_{2,1}(𝕆_{ℂ}) = {(a_{2}, a_{3}) | a_{2}, a_{3} ∈ 𝕆_{ℂ}}

is identified with the subspace of H_{3}(𝕆_{ℂ}) consisting in matrices of the form

( \begin{matrix}0 & a_{3} & ã_{2} \\ ã_{3} & 0 & 0 \\ a_{2} & 0 & 0 \end{matrix} ) .

This is the Peirce subspace V_{1}(e_{1}) of H_{3}(𝕆_{ℂ}) with respect to the tripotent e_{1} and hence a Jordan subsystem of the Jordan triple system H_{3}(𝕆_{ℂ}). The Peirce subspace V_{0}(e_{1}) = ℂe_{2} ⊕ ℂe_{3} ⊕ F_{1} will be identified with the space

H_{2}(𝕆_{ℂ}) = \{( \begin{matrix}α_{2} & ã_{1} \\ a_{1} & α_{3} \end{matrix} ); α_{2}, α_{3} ∈ ℂ, a_{1} ∈ 𝕆_{ℂ}\}

of 2 × 2 Cayley-Hermitian matrices with coefficients in 𝕆_{ℂ}; the space H_{2}(𝕆_{ℂ}) is also a Jordan subsystem of the Jordan triple system H_{3}(𝕆_{ℂ}), and it can be shown it is isomorphic to the classical Hermitian Jordan triple of type IV_{10}.

For x = (x_{2}, x_{3}) ∈ M_{2,1}(𝕆_{ℂ}), the adjoint x^{♯} ∈ H_{2}(𝕆_{ℂ}) is then defined by

x^{♯} = ( \begin{matrix}−n (x_{2}) & x̃_{3}x̃_{2} \\ x_{2}x_{3} & −n (x_{3}) \end{matrix} ) .

The Freudenthal product of x = (x_{2}, x_{3}) , y = (y_{2}, y_{3}) ∈ M_{2,1}(𝕆_{ℂ}) is then

x × y = (x + y)^{♯} − x^{♯} − y^{♯}.

The spaces M_{2,1}(𝕆_{ℂ}) and H_{2}(𝕆_{ℂ}) are endowed with the following Hermitian scalar products, inherited from the Hermitian scalar product in H_{3}(𝕆_{ℂ}):

• if x = (x_{2}, x_{3}) , y = (y_{2}, y_{3}) ∈ M_{2,1}(𝕆_{ℂ}),

(C.21) (x|y) = (x_{2}|y_{2}) + (x_{3}|y_{3});

• if u = ( \begin{matrix}λ_{2} & u_{1} \\ ũ_{1} & λ_{3} \end{matrix} ) , v = ( \begin{matrix}μ_{2} & v_{1} \\ ṽ_{1} & μ_{3} \end{matrix} ) ∈ H_{2}(𝕆_{ℂ}), (λ_{2}, λ_{3}, μ_{2}, μ_{3} ∈ ℂ,

u_{1}, v_{1} ∈ 𝕆_{ℂ}),

(C.22) (u|v) = λ_{2}\overline{μ_{2}} + λ_{3}\overline{μ_{3}} + (u_{1}|v_{1}).

The quadratic operator of the Jordan triple system of type V is defined, for x = (x_{2}, x_{3}) , y = (y_{2}, y_{3}) ∈ M_{2,1}(𝕆_{ℂ}), by

(C.23) Q(x)y = (x_{2} ˜\overline{y_{2}}x_{2} + (x_{2}\overline{y_{3}})x̃_{3}, x̃_{2}(\overline{y_{2}}x_{3}) + x_{3} ˜\overline{y_{3}}x_{3}) .

The generic minimal polynomial of W is

m(T, x, y) = T^{2} − (x | y)T + (x^{#} | y^{#});

the generic norm is

N(x, y) = 1 − (x | y) + (x^{#} | y^{#}).

The set of tripotents of W is E′ = E′_{0} ∪ E′_{1} ∪ E′_{2}, with

E′_{0} = {0}, E′_{1} = {x ∈ W | (x | x) = 1, x^{#} = 0} ,
E′_{2} = {x ∈ W | (x | x) = 2, (x^{#} | x^{#}) = 1} .

The triple system W is simple. Its numerical invariants are

a = 6, b = 4, r = 2, g = 12.

(The numerical invariants and the simplicity of W are obtained by computing the simultaneous Peirce decomposition w.r. to a set of two orthogonal tripotents). In W = M_{2,1}(𝕆_{ℂ}), we have

tr D(x, y) = 12(x | y),
det B(x, y) = (1 − (x | y) + (x^{#} | y^{#}))^{12} .

The exceptional domain of dimension 16 is

(C.24) Ω_{V} = {x ∈ M_{2,1}(𝕆_{ℂ}) | 1 − (x|x) + (x^{♯} | x^{♯}) > 0, 2 − (x|x) > 0} .

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A.W.: Dept. of Math., Capital Normal Univ., Beijing 100037, China

E-mail address: wangancn@sina.com

W.Y.: Dept. of Math., Capital Normal Univ., Beijing 100037, China

E-mail address: wyin@mail.cnu.edu.cn

L.Zh.: Dept. of Math., Capital Normal Univ., Beijing 100037, China

E-mail address: zhangly@mail.cnu.edu.cn

G.R.: Nevski prospekt 113/4-53, 191024 St Petersburg, Russian Federation

E-mail address: guy.roos@normalesup.org