j. differential geometry 60 (2002) 389-453

GAUGE-FIXING CONSTANT SCALAR CURVATURE

EQUATIONS ON RULED MANIFOLDS AND THE FUTAKI INVARIANTS

YING-JI HONG

Abstract

In this article we introduce and prove the solvability of the gauge-fixing constant scalar curvature equations on ruled Kaehler manifolds. We prove that when some lifting conditions for holomorphic vector fields on the base manifold are satisfied the solutions for the gauge-fixing constant scalar curvature equations are actually solutions for the constant scalar curvature equations provided the corresponding Futaki invariants vanish.

In this article we will prove that the vanishing of certain natural Futaki invariants would imply the existence results for Kaehler metrics on ruled manifolds with constant scalar curvature. This work extends that of [10, 11] to the case where the base m-dimensional compact Kaehler manifold (M : ω_{M}) with constant scalar curvature may admit nontrivial holomorphic vector fields while the holomorphic vector bundle E over M with Einstein-Hermitian connection may not be simple.

In order to state our results properly we recall some facts about the structure of the groups of holomorphic automorphisms of compact Kaehler manifolds with constant scalar curvature. More background material can be found in [15].

Theorem 0. Assume that (M : ω_{M}) is an m-dimensional compact Kaehler manifold with constant scalar curvature. Here ω_{M} is the Kaehler form of M. Let 𝔥(M) denote the complex Lie algebra of holomorphic vector fields on M. Then we have the following direct sum decomposition (in the Lie algebra sense) of the Lie algebra 𝔥(M):

𝔥(M) = 𝔥_{o}(M) ⊕ 𝔠(M)

Received September 10, 2001; revised September 2, 2002.

in which

𝔥_{o}(M) ≡ {Z ∈ 𝔥(M) : i_{Z}ω_{M} = ∂f for some smooth ℂ-valued function f ∈ Γ(M : ℂ) on M}

and

𝔠(M) ≡ {Z ∈ 𝔥(M) : i_{Z}ω_{M} ∈ H^{(0:1)}(M : ℂ)} .

Note that the complex Lie algebra 𝔠(M) is commutative and is a Lie subalgebra of the Lie algebra of the isometry group of (M : ω_{M}). Also, 𝔥_{o}(M) is the complexification of the intersection 𝔨_{(M:ω_{M})} of 𝔥_{o}(M) with the Lie algebra of the isometry group of (M : ω_{M}).

Remark. Elements of 𝔥_{o}(M) can be characterized intrinsically as the holomorphic vector fields on M with nonempty zero loci. This characterization of 𝔥_{o} is valid for any compact Kaehler manifold (not necessarily with constant scalar curvature) and is due to Andre Lichnerowicz.

Remark. Note that the Lie algebra of smooth vector fields on M preserving the complex structure of M is isomorphic to 𝔥(M) (in the Lie algebra sense). This interpretation of 𝔥(M) is already used implicitly in the statement of Theorem 0 and will be used in the rest of this article.

Let Aut (M) denote the group of holomorphic automorphisms of M and G the connected component, containing the identity map of M, of the Lie subgroup of Aut (M) generated by 𝔥_{o}(M). Let K_{(M:ω_{M})} denote the compact connected component, containing the identity map of M, of the Lie subgroup of Aut (M) generated by the intersection 𝔨_{(M:ω_{M})} of 𝔥_{o}(M) with the Lie algebra of the isometry group of (M : ω_{M}) so that G is the complexification of K_{(M:ω_{M})}.

Assume that π : E ⟶ M is a holomorphic vector bundle of rank n over M with Einstein-Hermitian metric H_{E}. Let A denote the Einstein-Hermitian connection on E induced by H_{E}. Let ℙ(E) denote the projectivization of E over M. Then ℙ(E) is a compact complex manifold with (−1 + m + n) dimensions. Let L be the universal line bundle over ℙ(E). Then the Einstein-Hermitian metric H_{E} induces a Hermitian metric H_{L^{∗}} on the dual L^{∗} of L over ℙ(E). Let A_{L^{∗}} denote the Hermitian connection on L^{∗} induced by H_{L^{∗}}. Thus there is a representative

\frac{i · F_{A_{L^{∗}}}}{2π} = \frac{i}{2π} · ∂∂ log H_{L^{∗}} = −\frac{i}{2π} · ∂∂ log H_{L}

of the Euler class e(L^{∗}) of L^{∗} on ℙ(E) induced by the Hermitian connection A_{L^{∗}}. Here H_{L} is the Hermitian metric on L over ℙ(E) induced by the Einstein-Hermitian metric H_{E} on E over M. Note that the representative \frac{i·F_{A_{L^{∗}}}}{2π} of e(L^{∗}) on ℙ(E) induces the Fubini-Study metric on each fiber ℙ(ℂ^{n}) of π̌ : ℙ(E) ⟶ M. Thus, for each k ∈ ℕ large enough,

\frac{i · F_{A_{L^{∗}}}}{2π} + k · π̌^{∗}ω_{M}

is a Kaehler form on ℙ(E).

[‡]: Suppose that, for each k ∈ ℕ large enough, there exists a corresponding Kaehler form on ℙ(E), lying in the Kaehler class

[\frac{i · F_{A_{L^{∗}}}}{2π} + k · π̌^{∗}ω_{M}] ,

carrying constant scalar curvature. Then, for each k ∈ ℕ large enough, the corresponding Futaki character must be zero.

Let Aut (ℙ(E)) denote the group of holomorphic automorphisms of ℙ(E). Let Aut (E) denote the group of holomorphic automorphisms of E over M. Let G_{E} denote the natural image of Aut (E) in Aut (ℙ(E)) preserving the holomorphic projection map π̌ : ℙ(E) ⟶ M. Then we have

G_{E} = \frac{Aut (E)}{ℂ^{∗}}.

Let 𝔤_{E} denote the Lie algebra of G_{E}. Our Theorem A shows that the converse of [‡] is true when the elements of 𝔥_{o}(M) can be lifted to holomorphic vector fields on ℙ(E) preserving the holomorphic projection map π̌ : ℙ(E) ⟶ M.

Theorem A. Assume that the elements of 𝔥_{o}(M) can be lifted to holomorphic vector fields on ℙ(E) preserving the holomorphic projection map

π̌ : ℙ(E) ⟶ M.

Suppose that, for each k ∈ ℕ large enough, the corresponding Futaki character associated with

𝔤_{E} + (the lifted action of ) 𝔥_{o}(M)

and the Kaehler class [\frac{i·F_{A_{L^{∗}}}}{2π} + k · π̌^{∗}ω_{M}] on ℙ(E) is zero. Then, for each k ∈ ℕ large enough, there exists a corresponding Kaehler form on ℙ(E), lying in the Kaehler class

[\frac{i · F_{A_{L^{∗}}}}{2π} + k · π̌^{∗}ω_{M}] ,

carrying constant scalar curvature.

Corollary A. Assume that the holomorphic vector bundle E with Einstein-Hermitian connection A over M is simple while the elements of 𝔥_{o}(M) can be lifted to holomorphic vector fields on ℙ(E) preserving the holomorphic projection map

π̌ : ℙ(E) ⟶ M.

Suppose that, for each k ∈ ℕ large enough, the corresponding Futaki character associated with (the lifted action of ) 𝔥_{o}(M) and the Kaehler class [\frac{i·F_{A_{L^{∗}}}}{2π} + k · π̌^{∗}ω_{M}] on ℙ(E) is zero. Then, for each k ∈ ℕ large enough, there exists a corresponding Kaehler form on ℙ(E), lying in the Kaehler class

[\frac{i · F_{A_{L^{∗}}}}{2π} + k · π̌^{∗}ω_{M}] ,

carrying constant scalar curvature.

Corollary B. Assume that the compact Kaehler manifold (M : ω_{M}) with constant scalar curvature does not admit nontrivial infinitesimal deformation of Kaehler forms in the Kaehler class [ω_{M}] on M with constant scalar curvature. Suppose that, for each k ∈ ℕ large enough, the corresponding Futaki character associated with 𝔤_{E} and the Kaehler class [\frac{i·F_{A_{L^{∗}}}}{2π} + k · π̌^{∗}ω_{M}] on ℙ(E) is zero. Then, for each k ∈ ℕ large enough, there exists a corresponding Kaehler form on ℙ(E), lying in the Kaehler class

[\frac{i · F_{A_{L^{∗}}}}{2π} + k · π̌^{∗}ω_{M}] ,

carrying constant scalar curvature.

In view of Theorem 0 it might seem necessary to add the following invariance assumption of \frac{i·F_{A_{L^{∗}}}}{2π} to Theorem A: \frac{i·F_{A_{L^{∗}}}}{2π} is invariant under the lifted action of 𝔨_{(M:ω_{M})} on ℙ(E). Our Theorem B shows that it is unnecessary to make such extra assumption in Theorem A because the invariance of \frac{i·F_{A_{L^{∗}}}}{2π} can be inferred directly from the vanishing of Futaki invariants. In particular, in Corollary A, \frac{i·F_{A_{L^{∗}}}}{2π} is automatically invariant under the lifted action of 𝔨_{(M:ω_{M})} on ℙ(E) because there is only one possible lifting of 𝔥_{o}(M).

Theorem B. Assume that the elements of 𝔥_{o}(M) can be lifted to holomorphic vector fields on ℙ(E) preserving the holomorphic projection map

π̌ : ℙ(E) ⟶ M.

Suppose that, for each k ∈ ℕ large enough, the corresponding Futaki invariants associated with

𝔤_{E} + (the lifted action of ) 𝔨_{(M:ω_{M})}

and the Kaehler class [\frac{i·F_{A_{L^{∗}}}}{2π} + k · π̌^{∗}ω_{M}] on ℙ(E) are zero. Then the lifting of the intersection 𝔨_{(M:ω_{M})} of 𝔥_{o}(M) with the Lie algebra of the isometry group of (M : ω_{M}) can be properly rearranged such that

\frac{i · F_{A_{L^{∗}}}}{2π}

is invariant under the rearranged lifted action of 𝔨_{(M:ω_{M})} on ℙ(E).

Let 𝔨_{E} denote the maximal compact Lie subalgebra of 𝔤_{E}. Actually, in Theorem B, the rearranged lifting of 𝔨_{(M:ω_{M})} is, modulo Hom (𝔨_{(M:ω_{M})} : 𝔨_{E}), uniquely determined. Precise version (Theorem II.B) of this result can be found in Section II.

We will prove Theorem A through solving the “gauge-fixing constant scalar curvature equation” depending on k large enough. (The gaugefixing constant scalar curvature equation and its solvability, when the parameter k is sufficiently large, will be introduced in Section V.) With the solvability of the gauge-fixing constant scalar curvature equation we will then show that the solvability of constant scalar curvature equation can be inferred from the vanishing of Futaki invariants. Actually, by incorporating the vanishing of the corresponding Futaki invariants, we will show that the solutions to the gauge-fixing constant scalar curvature equation are actually solutions to the constant scalar curvature equation, when the parameter k is sufficiently large, in Section VI.

I. Futaki invariants

Here we summarize some basic facts about the Futaki Invariants. The reader can find more background material in [7, 15].

Theorem I.A. Assume that (B : ω_{B}) is a b-dimensional compact Kaehler manifold (not necessarily with constant scalar curvature). Let 𝔥(B) denote the complex Lie algebra of holomorphic vector fields on B and [ω_{B}] the Kaehler class associated with the Kaehler form ω_{B} on B. Let c_{[ω_{B}]} ∈ ℝ denote the constant associated with [ω_{B}] satisfying the following equality:

c_{[ω_{B}]} · ∫_{B} \frac{ω^{m}}{m!} = ∫_{B} \frac{i · ρ_{ω}}{2π} ∧ \frac{ω^{(−1+m)}}{(−1 + m)!} ∀ω ∈ [ω_{B}] .

Here ρ_{ω} is the curvature form of the holomorphic line bundle ∧^{b}T^{(1:0)}(B) on B (the highest degree wedge product of the holomorphic tangent bundle T^{(1:0)}(B) of B) defined by ω ∈ [ω_{B}]. Let

𝔉 : 𝔥(B) × [ω_{B}] ⟶ ℂ

be defined as follows:

𝔉(Z : ω) ≡ ∫_{B} ψ_{(Z:ω)} · (−c_{[ω_{B}]} · \frac{ω^{m}}{m!} + \frac{i · ρ_{ω}}{2π} ∧ \frac{ω^{(−1+m)}}{(−1 + m)!})

∀(Z : ω) ∈ 𝔥(B) × [ω_{B}]

in which the smooth function ψ_{(Z:ω)} ∈ Γ (B : ℂ) on B satisfies

L_{Z}ω = i∂∂ ψ_{(Z:ω)}.

Then 𝔉 only depends on 𝔥(B): 𝔉(Z : •) is constant on [ω_{B}] for each Z ∈ 𝔥(B). Besides we have

𝔉([Z : W] : ω) = 0 ∀(Z : W : ω) ∈ 𝔥(B) × 𝔥(B) × [ω_{B}] .

𝔉 is called the Futaki character associated with (𝔥(B) : [ω_{B}]).

It is obvious that when B carries constant scalar curvature the Futaki character associated with 𝔥(B) and (B : ω_{B}) must be zero. We will apply Theorem I.A to the compact complex manifold ℙ(E).

II. Lifting of the elements of 𝔥_{o}(M)

In this section we discuss the lifting of the elements of 𝔥_{o}(M) to holomorphic vector fields on ℙ(E) preserving the holomorphic projection map

π̌ : ℙ(E) ⟶ M.

We will consider the Real aspect of 𝔥_{o}(M): Given a holomorphic vector field Z on M with nonempty zero locus we will consider the lifting of the corresponding smooth vector field X_{Z} on M preserving the complex structure of M. Here the smooth vector field X_{Z} on M is defined as follows:

X_{Z} ≡ Z + Z.

Our immediate purpose is to lift X_{Z} to a smooth vector field on E preserving the vector bundle structure and the holomorphic structure of E over M.

Given the connection A on E there is a convenient lifting of X_{Z} (induced by the distribution of horizontal spaces on E specified by A) to a smooth vector field on E preserving the vector bundle structure of E over M. But this lifting does not necessarily preserve the holomorphic structure of E over M. Thus we add a smooth section s of Hom (E : E) over M to this lifting. Let F_{A} denote the curvature form of E induced by the connection A. Then it is easy to see that this modified lifting of X_{Z} preserves the holomorphic structure of E over M if and only if

−∂_{A}s + F_{A} (Z : ) = 0.

In particular we infer that X_{Z} can be lifted to a smooth vector field on E preserving the vector bundle structure and the holomorphic structure of E over M if and only if

0 = [F_{A} (Z : )] ∈ H_{∂_{A}}^{1}(M : Hom (E : E)).

Thus we have the following:

Theorem II.A. Assume that E is a holomorphic vector bundle with Hermitian connection A over a compact Kaehler manifold (M : ω_{M}). Let F_{A} denote the curvature form of E over M induced by the connection A. Then for any holomorphic vector field Z on M we have

0 = [F_{A} (Z : )] ∈ H_{∂_{A}}^{1}(M : Hom (E : E))

if and only if the corresponding smooth vector field X_{Z} = Z + Z on M, preserving the complex structure of M, can be lifted to a smooth vector field on ℙ(E) preserving both the complex structure of ℙ(E) and the holomorphic projection map

π̌ : ℙ(E) ⟶ M.

Remark. Theorem II.A is valid for any compact Kaehler manifold (not necessarily with constant scalar curvature). Note that for any holomorphic vector field Z on M the equation

0 = [F_{A} (Z : )] ∈ H_{∂_{A}}^{1}(M : Hom (E : E))

only depends on the holomorphic structure of E over M. It does not depend on the Hermitian connection A on E over M.

Now suppose further that the smooth vector field X_{Z} on M actually preserves the Kaehler form ω_{M} on M. Since 0 = [F_{A} (Z : )] ∈ H_{∂_{A}}^{1}(M : Hom (E : E)) there certainly exists a lifting of X_{Z} to a smooth vector field X̌_{Z} on ℙ(E) preserving both the complex structure of ℙ(E) and the holomorphic projection map π̌ : ℙ(E) ⟶ M.

Theorem II.B. Suppose that the Futaki invariant associated with the lifting X̌_{Z} (of X_{Z}) and the Kaehler class [\frac{i·F_{A_{L^{∗}}}}{2π} + k · π̌^{∗}ω_{M}] on ℙ(E) vanishes for any sufficiently large k ∈ ℕ. Then there exists a lifting of X_{Z}, preserving the holomorphic projection map π̌ : ℙ(E) ⟶ M, to a smooth vector field X̌_{Z} on ℙ(E) preserving both \frac{i·F_{A_{L^{∗}}}}{2π} and the complex structure of ℙ(E). Note that the smooth vector field X̌_{Z} on ℙ(E) is, modulo the compact Lie subalgebra 𝔨_{E} of 𝔤_{E}, uniquely determined.

It is obvious that the converse of Theorem II.B is true. Proving Theorem II.B requires more knowledge and will not be given in this section. The reader can find it in Appendix II.

III. Splitting of holomorphic vector bundles with Einstein-Hermitian connections over compact Kaehler manifolds

In this section we will consider the splitting of the holomorphic vector bundle E with Einstein-Hermitian connection A over a compact Kaehler manifold M. Since the assumption that (M : ω_{M}) carries constant scalar curvature will not be used the results of this section are valid for any compact Kaehler manifold.

We begin with some basic facts about the structure of holomorphic vector bundles with Einstein-Hermitian connections over compact Kaehler manifolds. The reader can find more background material in

Note that E can be expressed as the direct sum of certain simple holomorphic vector bundles E_{θ} over M

E = ⊕_{θ}E_{θ}.

Besides the Einstein-Hermitian connection A on E can be expressed as

A = ⊕_{θ}A_{θ}

with each connection A_{θ} on E_{θ} being Einstein-Hermitian. Since any nontrivial holomorphic map between slope-stable holomorphic vector bundles over a compact Kaehler manifold must be an isomorphism it is easy to understand the structure of Aut(E) completely through the following examples:

Example I. When all the bundles E_{θ} are lying in the same isomorphism class of a slope-stable holomorphic vector bundle E_{o} over M we have

Aut (E) = GL (\frac{n}{rank E_{o}} : ℂ) .

Example II. Let d denote the number of the isomorphism classes defined by these slope-stable holomorphic vector bundles E_{θ} over M. When the isomorphism classes defined by these bundles E_{θ} over M are all distinct we have

Aut (E) = ℂ^{∗} × ⋯ × ℂ^{∗}

in which there are d copies of the multiplicative group

ℂ^{∗} = {z ∈ ℂ : z ≠ 0} .

Actually let d denote the number of the isomorphism classes defined by these slope-stable holomorphic vector bundles E_{θ} over M. Then Aut (E) is the product of d complex general linear groups. Each complex linear group is acting on the direct sum of those slope-stable holomorphic vector bundles E_{θ} over M lying in the same isomorphism class.

IV. Some basic facts and kernel identification

Since the restriction of \frac{i·F_{A_{L^{∗}}}}{2π} on each fiber ℙ (ℂ^{n}) of π̌ : ℙ(E) ⟶ M is simply the Fubini-Study Kaehler form there is a well-defined smooth vector bundle W over M whose fiber (vector space over ℝ) W_{z} over z ∈ M is the eigenspace of the lowest nonzero eigenvalue of the (Fubini-Study) Laplacian on the fiber ℙ (ℂ^{n}) of ℙ(E) over M. Note that the eigenspace of the lowest nonzero eigenvalue 4πn of the (Fubini-Study) Laplacian on ℙ (ℂ^{n}) simply consists of the quotients of traceless Hermitian quadratic functions on ℂ^{n} by the usual Hermitian metric ∑ δ_{αβ} · w_{α} · w_{β} on ℂ^{n}. It is well-known in Kaehler geometry that this eigenspace represents the tangent space at the Hermitian metric ∑ δ_{αβ} · w_{α} · w_{β} of the moduli space of Einstein-Kaehler metrics on ℙ (ℂ^{n}):

\frac{𝔰𝔩(n : ℂ)}{𝔰𝔲(n)}.

On the other hand integration along the fibers of π̌ : ℙ(E) ⟶ M maps a smooth function on ℙ(E) onto a smooth function on M. Let Γ (M : W) denote the space of smooth sections of W over M. Then for each smooth ℝ-valued function f ∈ Γ (ℙ(E) : ℝ) on ℙ(E) we have the following corresponding decomposition:

f = σ̂(f) ⊕ σ(f) ⊕ σ̃(f)

in which (σ̂(f) : σ(f)) ∈ Γ (M : ℝ) ⊕ Γ (M : W) while the restriction of σ̃(f) on each fiber ℙ (ℂ^{n}) of π̌ : ℙ(E) ⟶ M over z ∈ M is orthogonal to both the space W_{z} and the space of constant functions on that fiber (over z ∈ M).

Let Γ_{o} (ℙ (ℂ^{n}) : ℝ) denote the space of smooth ℝ-valued functions f on ℙ (ℂ^{n}) satisfying

∫_{ℙ(ℂ^{n})} f · \frac{ω_{F-S}^{(−1+n)}}{(−1 + n)!} = 0.

Now we introduce a basic result about a special kind of quadratic combinations of elements of the eigenspace of the lowest nonzero eigenvalue (4πn) of the Fubini-Study Laplacian Δ_{F-S} on ℙ (ℂ^{n}).

Proposition IV.A. Assume that ℂ^{n} and ℙ (ℂ^{n}) are respectively endowed with the standard Hermitian metric H_{ℂ^{n}} = ∑ δ_{αβ} · w_{α} · w_{β} on ℂ^{n} and the Fubini-Study Kaehler form ω_{F-S} = −\frac{i}{2π} ∂∂ log H_{ℂ^{n}} on ℙ (ℂ^{n}). We define a symmetric quadratic operation Q on the eigenspace of the lowest nonzero eigenvalue (4πn) of the Fubini-Study Laplacian Δ_{F-S} on ℙ (ℂ^{n}) as follows:

Given X ∈ 𝔰𝔩(n : ℂ) let f_{X} ∈ Γ_{o}(ℙ(ℂ^{n}) : ℝ) denote the smooth ℝ-valued function on ℙ (ℂ^{n}) satisfying

L_{X}ω_{F-S} = i∂∂f_{X}.

Then we have f_{X} = 0 when X ∈ 𝔰𝔲(n). Besides we always have

Δ_{F-S}f_{X} = 4πn · f_{X}.

We define Q (f_{X} : f_{X}) through the following equality:

Q (f_{X} : f_{X}) · \frac{ω_{F-S}^{(−1+n)}}{(−1 + n)!} = i∂∂f_{X} ∧ i∂∂f_{X} ∧ \frac{ω_{F-S}^{(−3+n)}}{(−3 + n)!}.

Then for each element f_{X} of the eigenspace of the lowest nonzero eigenvalue (4πn) of the Fubini-Study Laplacian Δ_{F-S} on ℙ (ℂ^{n}) the following smooth function:

− (2π · n · f_{X}) · (2π · n · f_{X}) + Q (f_{X} : f_{X})

on ℙ (ℂ^{n}) is orthogonal to the eigenspace of the lowest nonzero eigenvalue (4πn) of the Fubini-Study Laplacian Δ_{F-S} on ℙ (ℂ^{n}).

This result has been proved in [11] through direct computation. Actually it can be inferred from the vanishing of the Futaki character associated with 𝔰𝔩(n : ℂ) and the Fubini-Study Kaehler class [ω_{F-S}] on ℙ (ℂ^{n}).

Proof of Proposition IV.A. Let ρ_{ω_{F-S}} denote the curvature form of the dual of the canonical line bundle of ℙ (ℂ^{n}) defined by ω_{F-S}. Then we have

\frac{i · ρ_{ω_{F-S}}}{2π} = \frac{i · ∂∂ log det H_{F-S}}{2π} = n · ω_{F-S}.

Here H_{F-S} is the Einstein-Kaehler metric on ℙ (ℂ^{n}) induced by ω_{F-S}. Let

ω_{F-S:t} ≡ ω_{F-S} + t · i∂∂f_{X} ∀t ∈ ℝ.

Then for each t ∈ ℝ with |t| ≥ 0 being small ω_{F-S:t} is a Kaehler form on ℙ (ℂ^{n}) lying in the Fubini-Study Kaehler class [ω_{F-S}]. We define a symmetric quadratic operation Q on the eigenspace of the lowest nonzero eigenvalue (4πn) of the Fubini-Study Laplacian Δ_{F-S} on ℙ (ℂ^{n}) through the following equality:

Q (f_{X} : f_{X}) · \frac{ω_{F-S}^{(−1+n)}}{(−1 + n)!}

= \frac{d}{dt} ∘ \frac{d}{dt} (−n · (−1 + n) · \frac{ω_{F-S:t}^{(−1+n)}}{(−1 + n)!} + \frac{i · ρ_{ω_{F-S:t}}}{2π} ∧ \frac{ω_{F-S:t}^{(−2+n)}}{(−2 + n)!})|_{t=0} .

It can be checked readily that

Q (f_{X} : f_{X}) · \frac{ω_{F-S}^{(−1+n)}}{(−1 + n)!}

= \frac{i}{2π} ∂∂ (−\frac{Δ_{F-S}f_{X}}{2} · \frac{Δ_{F-S}f_{X}}{2} + Q (f_{X} : f_{X})) ∧ \frac{ω_{F-S}^{(−2+n)}}{(−2 + n)!}

and thence

Q (f_{X} : f_{X}) =\frac{Δ_{F-S}}{4π} (−\frac{Δ_{F-S}f_{X}}{2} · \frac{Δ_{F-S}f_{X}}{2} + Q (f_{X} : f_{X}))
=\frac{Δ_{F-S}}{4π} [−(2πn · f_{X}) · (2πn · f_{X}) + Q (f_{X} : f_{X})] .

Now for any Y ∈ 𝔰𝔩(n : ℂ) the Futaki invariant associated with Y and the Fubini-Study Kaehler class [ω_{F-S}] on ℙ (ℂ^{n}) vanishes. Thus we have

0 = 𝔉 (Y : ω_{F-S:t})

= ∫_{ℙ(ℂ^{n})} (−\frac{L_{Y} log H_{ℂ^{n}}}{2π} + t · L_{Y} f_{X})

· (−n · (−1 + n) · \frac{ω_{F-S:t}^{(−1+n)}}{(−1 + n)!} + \frac{i · ρ_{ω_{F-S:t}}}{2π} ∧ \frac{ω_{F-S:t}^{(−2+n)}}{(−2 + n)!})

for t ∈ ℝ with |t| ≥ 0 being small. Let us now consider the equality

0 = \frac{d}{dt} ∘ \frac{d}{dt} 𝔉 (Y : ω_{F-S:t})|_{t=0} .

Since X ∈ 𝔰𝔩(n : ℂ) preserves the Einstein-Kaehler condition (equivalently the constant scalar curvature condition) on ℙ (ℂ^{n}):

L_{X} (−n · (−1 + n) · \frac{ω_{F-S}^{(−1+n)}}{(−1 + n)!} + \frac{i · ρ_{ω_{F-S}}}{2π} ∧ \frac{ω_{F-S}^{(−2+n)}}{(−2 + n)!}) = 0

and

L_{X}ω_{F-S} = i∂∂f_{X}

we have

\frac{d}{dt} (−n · (−1 + n) · \frac{ω_{F-S:t}^{(−1+n)}}{(−1 + n)!} + \frac{i · ρ_{ω_{F-S:t}}}{2π} ∧ \frac{ω_{F-S:t}^{(−2+n)}}{(−2 + n)!})|_{t=0} = 0.

Thus

0 = \frac{d}{dt} ∘ \frac{d}{dt} 𝔉 (Y : ω_{F-S:t})|_{t=0}

= ∫_{ℙ(ℂ^{n})} −\frac{L_{Y} log H_{ℂ^{n}}}{2π} · \frac{d}{dt} ∘ \frac{d}{dt}

· (−n · (−1 + n) · \frac{ω_{F-S:t}^{(−1+n)}}{(−1 + n)!} + \frac{i · ρ_{ω_{F-S:t}}}{2π} ∧ \frac{ω_{F-S:t}^{(−2+n)}}{(−2 + n)!})|_{t=0}

= ∫_{ℙ(ℂ^{n})}−\frac{L_{Y} log H_{ℂ^{n}}}{2π} · Q (f_{X} : f_{X}) · \frac{ω_{F-S}^{(−1+n)}}{(−1 + n)!}

= ∫_{ℙ(ℂ^{n})}f_{Y} · \frac{Δ_{F-S}}{4π} [−(2πn · f_{X}) · (2πn · f_{X}) + Q (f_{X} : f_{X})] · \frac{ω_{F-S}^{(−1+n)}}{(−1 + n)!}

for any Y ∈ 𝔰𝔩(n : ℂ). Since Δ_{F-S} is symmetric with respect to the Fubini-Study Kaehler form ω_{F-S} on ℙ (ℂ^{n}) we conclude from the last equality that for any Y ∈ 𝔰𝔩(n : ℂ)

0 = ∫_{ℙ(ℂ^{n})}\frac{Δ_{F-S}f_{Y}}{4π} · [−(2πn · f_{X}) · (2πn · f_{X}) + Q (f_{X} : f_{X})] · \frac{ω_{F-S}^{(−1+n)}}{(−1 + n)!}

= ∫_{ℙ(ℂ^{n})} n · f_{Y} · [−(2πn · f_{X}) · (2πn · f_{X}) + Q (f_{X} : f_{X})] · \frac{ω_{F-S}^{(−1+n)}}{(−1 + n)!}

and thence the assertion of Proposition IV.A is true. q.e.d.

Corollary IV.A. Assume that ℂ^{n} and ℙ (ℂ^{n}) are respectively endowed with the standard Hermitian metric H_{ℂ^{n}} = ∑ δ_{αβ} ·w_{α} ·w_{β} on ℂ^{n} and the Fubini-Study Kaehler form ω_{F-S} = −\frac{i}{2π} ∂∂ log H_{ℂ^{n}} on ℙ (ℂ^{n}). We define a symmetric quadratic operation Q on the eigenspace of the lowest nonzero eigenvalue (4πn) of the Fubini-Study Laplacian Δ_{F-S} on ℙ (ℂ^{n}) as follows:

Given X ∈ 𝔰𝔩(n : ℂ) and Y ∈ 𝔰𝔩(n : ℂ) let f_{X} ∈ Γ_{o} (ℙ (ℂ^{n}) : ℝ) and f_{Y} ∈ Γ_{o} (ℙ (ℂ^{n}) : ℝ) denote the smooth ℝ-valued functions on ℙ (ℂ^{n}) satisfying

L_{X}ω_{F-S} = i∂∂f_{X} and L_{Y} ω_{F-S} = i∂∂f_{Y} .

We define Q (f_{X} : f_{Y}) through the following equality:

Q (f_{X} : f_{Y}) · \frac{ω_{F-S}^{(−1+n)}}{(−1 + n)!} = i∂∂f_{X} ∧ i∂∂f_{Y} ∧ \frac{ω_{F-S}^{(−3+n)}}{(−3 + n)!}.

Then for each pair (f_{X} : f_{Y}) of elements of the eigenspace of the lowest nonzero eigenvalue (4πn) of the Fubini-Study Laplacian Δ_{F-S} on ℙ (ℂ^{n}) the following smooth function:

− (2π · n · f_{X}) · (2π · n · f_{Y}) + Q (f_{X} : f_{Y})

on ℙ (ℂ^{n}) is orthogonal to the eigenspace of the lowest nonzero eigenvalue (4πn) of the Fubini-Study Laplacian Δ_{F-S} on ℙ (ℂ^{n}).

Note that the Einstein-Hermitian connection A on E over M defines a smooth distribution H of horizontal spaces on ℙ(E):

T (ℙ(E)) = V ⊕ H.

Here V is the subbundle of T (ℙ(E)) over ℙ(E) consisting of tangent vectors which are tangential to the fibers of π̌ : ℙ(E) ⟶ M. Let V^{[∗]} denote the maximal subbundle of T^{∗} (ℙ(E)) over ℙ(E) whose action on H is identically zero. Then the decomposition T (ℙ(E)) = V ⊕ H of T (ℙ(E)) over ℙ(E) induces the following corresponding decomposition:

T^{∗} (ℙ(E)) = V^{[∗]} ⊕ π̌^{∗} (T^{∗}(M))

of T^{∗} (ℙ(E)) over ℙ(E). Thus we have the following decomposition:

∧^{∗}T^{∗} (ℙ(E)) = C_{V} ⊕ C_{m} ⊕ C_{M}

of ∧^{∗}T^{∗} (ℙ(E)) over ℙ(E). Here C_{V} = ∧^{∗}V^{[∗]} and C_{M} = ∧^{∗}π̌^{∗}T^{∗}(M) while C_{m} is the subbundle of ∧^{∗}T^{∗} (ℙ(E)) over ℙ(E) consisting of the mixed components of ∧^{∗}T^{∗} (ℙ(E)). Thus we have the following diagram:

C_{V} ⟵^{Π_{C_{V}}} ∧^{∗}T^{∗} (ℙ(E)) ⟶^{Π_{C_{M}}} C_{M}
↓^{Π_{Cm}}
C_{m}

of projection maps over ℙ(E) such that id = Π_{C_{V}} ⊕ Π_{C_{m}} ⊕ Π_{C_{M}} on ∧^{∗}T^{∗} (ℙ(E)). Since the decomposition

T^{∗} (ℙ(E)) = V^{[∗]} ⊕ π̌^{∗} (T^{∗}(M))

of T^{∗} (ℙ(E)) is defined by the Einstein-Hermitian connection A on E over M we note that the representative

\frac{i · F_{A_{L^{∗}}}}{2π}

of the Euler class e (L^{∗}) of L^{∗} on ℙ(E) has no nontrivial mixed components of ∧^{∗}T^{∗} (ℙ(E)):

\frac{i · F_{A_{L^{∗}}}}{2π} = Π_{C_{V}} (\frac{i · F_{A_{L^{∗}}}}{2π} ) ⊕ Π_{C_{M}} (\frac{i · F_{A_{L^{∗}}}}{2π} ) .

Now we introduce a Hermitian form (metric) ω̌ on ℙ(E) by setting

ω̌ = Π_{C_{V}} (\frac{i · F_{A_{L^{∗}}}}{2π} ) + π̌^{∗}ω_{M}.

Remark. It should be noted that

ω̌^{(−1+m+n)} = lim_{k→+∞} k^{−m} · (\frac{i · F_{A_{L^{∗}}}}{2π} + k · ω_{M})^{(−1+m+n)} .

Actually ω̌ can be realized as a modified limit of \frac{i·F_{A_{L^{∗}}}}{2π} + k · ω_{M}, as k → +∞, with the base directions of ℙ(E) being properly rescaled.

Note that the derivation operator

d : Γ (ℙ(E) : ℝ) ⟶ Γ (ℙ(E) : T^{∗} (ℙ(E)) ⊗ ℝ)

can be expressed as

d = d_{V} + d_{M}

in which d_{V} : Γ (ℙ(E) : ℝ)⟶Γ (ℙ(E) : ℝ ⊗ V^{[∗]}) and d_{M} : Γ (ℙ(E) : ℝ) ⟶ Γ (ℙ(E) : ℝ ⊗ π̌^{∗} (T^{∗}(M))). Let d_{V}^{∗} and d_{M}^{∗} be respectively the adjoint operators of d_{V} and d_{M} with respect to the Hermitian form (metric) ω̌ on ℙ(E). Then we have

Δ = d^{∗} ∘ d = Δ_{V} + Δ_{M}

in which Δ_{V} ≡ d_{V}^{∗} ∘ d_{V} and Δ_{M} ≡ d_{M}^{∗} ∘ d_{M}. Similarly we have ∂ = ∂_{V} + ∂_{M} and ∂ = ∂_{V} + ∂_{M}. Let Λ_{V} and Λ_{M} be respectively the adjoint operators of

• ⟼ Π_{C_{V}} \frac{i · F_{A_{L^{∗}}}}{2π} ∧ •

and

• ⟼ π̌^{∗}ω_{M} ∧ •

on ℙ(E) with respect to the Hermitian form (metric) ω̌. We use the symbols (∂∂)_{V} and (∂∂)_{M} to denote respectively Π_{C_{V}} ∘ (∂∂) and Π_{C_{M}} ∘ (∂∂):

(∂∂)_{V} = Π_{C_{V}} ∘ (∂∂) and (∂∂)_{M} = Π_{C_{M}} ∘ (∂∂).

Similarly we use the symbol (∂∂)_{m} to denote Π_{C_{m}} ∘ (∂∂): (∂∂)_{m} = Π_{C_{m}} ∘(∂∂). Then we have the following results (proved in the Appendix of [11]):

Proposition IV.B. Given f ∈ Γ (ℙ(E) : ℝ) we have the equalities

i · Λ_{V} ∘ (∂∂)_{V} f = \frac{Δ_{V} f}{2}

and

i · Λ_{M} ∘ (∂∂)_{M} f = \frac{Δ_{M}f}{2} .

Proposition IV.C. Δ_{M} ∘ Δ_{V} = Δ_{V} ∘ Δ_{M}.

In particular we have

Δ_{M} ∘ (−4πn · id + Δ_{V}) = (−4πn · id + Δ_{V}) ∘ Δ_{M}

and thence Δ_{M} preserves Γ (M : W). In [11] it is shown that the invertibility of the linear partial differential operator Δ_{M} acting on Γ (M : W) is equivalent to the simplicity of the holomorphic vector bundle E over M. Actually each smooth section s of W over M can be realized as a smooth Hermitian section of Hom (E : E) over M. Using the Einstein-Hermitian condition of A on E over M it can be checked readily that the smooth ℝ-valued function s on ℙ(E) satisfies Δ_{M}s = 0 if and only if its corresponding smooth Hermitian section of Hom (E : E) over M is harmonic (and thence holomorphic by the Einstein-Hermitian condition of A on E over M). Thus the kernel of Δ_{M} acting on Γ (M : W) is isomorphic to

\frac{𝔤_{E}}{𝔨_{E}} .

It should be noted that the linear partial differential operator Δ_{M} acting on Γ (M : W) is both nonnegative and symmetric (with respect to the Hermitian form (metric) ω̌ on ℙ(E)).

We can realize this picture more concretely as follows. Let G_{E} denote the natural image of Aut (E) in Aut (ℙ(E)) preserving the holomorphic projection map π̌ : ℙ(E) ⟶ M. Then we have

G_{E} = \frac{Aut (E)}{ℂ^{∗}}.

Let 𝔤_{E} denote the Lie algebra of G_{E} over ℂ. Let 𝔨_{E} denote the compact Lie algebra generated by the elements of 𝔤_{E} preserving the representative

\frac{i · F_{A_{L^{∗}}}}{2π}

of the Euler class of L^{∗} on ℙ(E) so that 𝔤_{E} is the complexification of 𝔨_{E}. Let K_{E} denote the compact subgroup of G_{E} generated by 𝔨_{E}. Given a smooth vector field Y ∈ 𝔤_{E} on ℙ(E) we denote by f_{Y} ∈ Γ (M : W) the corresponding smooth ℝ-valued function on ℙ(E) satisfying

L_{Y} (\frac{i · F_{A_{L^{∗}}}}{2π} ) = i∂∂f_{Y} .

(Note that when Y ∈ 𝔨_{E} we have f_{Y} = 0.) Let N_{W} denote the kernel of Δ_{M} acting on Γ (M : W). Then we have

N_{W} = \{f_{Y} ∈ Γ (M : W) : L_{Y} (\frac{i · F_{A_{L^{∗}}}}{2π} ) = i∂∂f_{Y} for some Y ∈ \frac{𝔤_{E}}{𝔨_{E}} \} .

We can now decompose the function space Γ (M : W) into the direct sum of N_{W} and the orthogonal complement of N_{W} in Γ (M : W). Thus for f ∈ Γ (M : W) we have

f = τ_{N_{W}}^{+} (f) ⊕ τ_{N_{W}} (f)

in which τ_{N_{W}}^{+} (f) is orthogonal to N_{W} while τ_{N_{W}} (f) is the N_{W} -compo-nent of f.

Let V_{M} denote the infinitesimal deformation operator for the constant scalar curvature equation on (M : ω_{M}):

V_{M}• = i∂∂ (\frac{Δ_{M}•}{4π}) ∧ \frac{ω_{M}^{(−1+m)}}{(−1 + m)!}

+ [−Λ_{M}trace (\frac{i · F_{ω_{M}}}{2π} )] · i∂∂ • ∧\frac{ω_{M}^{(−1+m)}}{(−1 + m)!}

+ i · ∂∂ • ∧trace (\frac{i · F_{ω_{M}}}{2π} ) ∧ \frac{ω_{M}^{(−2+m)}}{(−2 + m)!}

= \frac{Δ_{M} ∘ Δ_{M}•}{8π} · \frac{ω_{M}^{m}}{m!} + [−Λ_{M}trace (\frac{i · F_{ω_{M}}}{2π} )] · \frac{Δ_{M}•}{2} · \frac{ω_{M}^{m}}{m!}

+ i · ∂∂ • ∧trace (\frac{i · F_{ω_{M}}}{2π} ) ∧ \frac{ω_{M}^{(−2+m)}}{(−2 + m)!}.

Here F_{ω_{M}} is the curvature form of the holomorphic tangent bundle of M induced by the Kaehler form ω_{M} on M while

Λ_{M}trace (\frac{i · F_{ω_{M}}}{2π} )

is the scalar curvature of (M : ω_{M}):

[Λ_{M}trace (\frac{i · F_{ω_{M}}}{2π} )] · \frac{ω_{M}^{m}}{m!} = trace (\frac{i · F_{ω_{M}}}{2π} ) ∧ \frac{ω_{M}^{(−1+m)}}{(−1 + m)!}.

Let Γ_{o} (M : ℝ) denote the space of smooth ℝ-valued functions f on M satisfying

∫_{M} f · Ω_{M} = 0.

Here we set Ω_{M} ≡ \frac{ω_{M}^{m}}{m!} . Then the linear partial differential operator

\frac{V_{M}}{Ω_{M}} = \frac{V_{M}}{\frac{ω_{M}^{m}}{m!}}

acting on Γ_{o} (M : ℝ) is both nonnegative and symmetric (with respect to the Kaehler form (metric) ω_{M} on M). Note that the kernel of the linear partial differential operator V_{M} acting on Γ_{o} (M : ℝ) is isomorphic to the vector space

\frac{𝔥_{o}(M)}{𝔨_{(M:ω_{M})}}

over ℝ. Let N_{V_{M}} denote the vector space over ℝ of the kernel of V_{M} acting on Γ_{o} (M : ℝ). We can now decompose the function space Γ_{o} (M : ℝ) into the direct sum of N_{V_{M}} and the orthogonal complement of N_{V_{M}} in Γ_{o} (M : ℝ). Thus for f ∈ Γ_{o} (M : ℝ) we have

f = τ_{N_{V_{M}}}^{+} (f) ⊕ τ_{N_{V_{M}}} (f)

in which τ_{N_{V_{M}}}^{+} (f) is orthogonal to N_{V_{M}} while τ_{N_{V_{M}}} (f) is the N_{V_{M}} - component of f:

V_{M} (τ_{N_{V_{M}}} (f)) = 0 ⟺ \frac{V_{M} (τ_{N_{V_{M}}} (f))}{\frac{ω_{M}^{m}}{m!}} = 0.

V. Gauge-fixing constant scalar curvature equation

In this section we will introduce the gauge-fixing constant scalar curvature equation, depending on the parameter k ∈ ℕ, and prove its solvability when k is large enough.

Let _{o}H_{#k} denote the Kaehler metric on ℙ(E) induced by the Kaehler form

oω_{#k} ≡ \frac{i · F_{A_{L^{∗}}}}{2π} + k · π̌^{∗}ω_{M}.

Suppose that, for each k ∈ ℕ large enough, ω_{#k} is a Kaehler form on ℙ(E) lying in the Kaehler class [_{o}ω_{#k}] so that

ω_{#k} = _{o}ω_{#k} + i · ∂∂ψ_{k}

with ψ_{k} ∈ Γ (ℙ(E) : ℝ) satisfying

∫_{ℙ(E)} ψ_{k} · Ω_{ℙ(E)} = 0 ⟺ ∫_{M} σ̂ (ψ_{k}) · Ω_{M} = 0

in which Ω_{M} = \frac{ω_{M}^{m}}{m!} and Ω_{ℙ(E)} ≡ \frac{ω̌^{(−1+m+n)}}{(−1+m+n)!} = ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{(−1+n)}}{(−1+n)!}} ∧ \frac{ω_{M}^{m}}{m!} . Let č_{k} ∈ ℝ, depending on the parameter k ∈ ℕ, be the topological invariant satisfying the following equality:

č_{k} · ∫_{ℙ(E)} \frac{_{o}ω(−1+m+n)_{#k}}{(−1 + m + n)!} = ∫_{ℙ(E)} \frac{i · ∂∂ log det _{o}H_{#k}}{2π} ∧ \frac{_{o}ω(−2+m+n)_{#k}}{(−2 + m + n)!}.

Let H_{#k} be the Kaehler metric on ℙ(E) induced by the Kaehler form ω_{#k}. Then the Constant Scalar Curvature Equation for ω_{#k} is

S (ω_{#k}) = 0

in which

S (ω_{#k}) ≡ −č_{k} · \frac{ω(−1+m+n)_{#k}}{(−1 + m + n)!} + \frac{i · ∂∂ log det H_{#k}}{2π} ∧ \frac{ω(−2+m+n)_{#k}}{(−2 + m + n)!}.

Based on the work [10, 11] we might want to solve the constant scalar curvature equation, depending on k ∈ ℕ large enough, directly. However it is impractical to do so as there exist nontrivial kernels of linear partial differential operators associated with the constant scalar curvature equation. These nontrivial kernels exist simply because the constant scalar curvature equation is invariant under the action of the group Aut (ℙ(E)) of holomorphic automorphisms of ℙ(E). In order to tackle this difficulty we add the following “gauge-fixing” term:

(\frac{n · τ_{N_{W}} ∘ σ (ψ_{k})}{k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ (ψ_{k})}{k · k}) · k^{m} · Ω_{ℙ(E)}

to the constant scalar curvature equation and define the “gauge-fixing constant scalar curvature equation” as

S_{G-F} (ω_{#k}) = 0

in which

S_{G-F} (ω_{#k})

≡ S (ω_{#k}) + (\frac{n · τ_{N_{W}} ∘ σ (ψ_{k})}{k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ (ψ_{k})}{k · k}) · k^{m} · Ω_{ℙ(E)}

= −č_{k} · \frac{ω(−1+m+n)_{#k}}{(−1 + m + n)!} + \frac{i · ∂∂ log det H_{#k}}{2π} ∧ \frac{ω(−2+m+n)_{#k}}{(−2 + m + n)!}

+ (\frac{n · τ_{N_{W}} ∘ σ (ψ_{k})}{k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ (ψ_{k})}{k · k}) · k^{m} · Ω_{ℙ(E)}.

Let Γ_{o} (ℙ(E) : ℝ) denote the space of smooth ℝ-valued functions f on ℙ(E) satisfying

∫_{ℙ(E)} f · Ω_{ℙ(E)} = 0 ⟺ ∫_{M} σ̂(f) · Ω_{M} = 0.

We will solve the gauge-fixing constant scalar curvature equation for k ∈ ℕ large enough by considering ψ_{k} ∈ Γ_{o} (ℙ(E) : ℝ) admitting asymptotic expansion of the following form:

ψ_{k} ∼ ϕ_{0} + ∑_{θ∈ℕ} \frac{ϕ_{θ}}{k^{θ}}

as k → +∞. Here each ϕ_{•} ∈ Γ_{o} (ℙ(E) : ℝ) is a smooth ℝ-valued function, independent of the parameter k, on ℙ(E). Besides the following induction condition:

σ (ϕ_{0}) = σ̃ (ϕ_{0}) = 0 = σ̃ (ϕ_{1}) ⟺ ϕ_{0} ∈ Γ_{o} (M : ℝ) and ϕ_{1} ∈ Γ_{o} (M : ℝ) ⊕ Γ (M : W)

is imposed on the leading terms ϕ_{0} and ϕ_{1}.

Before solving the gauge-fixing constant scalar curvature equation, for k large enough, we collect some relevant basic facts which can be checked readily. Note that it is virtually better to rewrite the term

\frac{i · ∂∂ log det H_{#k}}{2π}

of the gauge-fixing constant scalar curvature equation S_{G-F} (ω_{#k}) = 0 as

\frac{i}{2π} · ∂∂ log (k^{m} · det Ȟ) + \frac{i}{2π} · ∂∂ log (\frac{ω(−1+m+n)_{#k}}{k^{m} · ω̌^{(−1+m+n)}} ) .

Here Ȟ is the Hermitian metric on ℙ(E) induced by ω̌ and thence

k^{m} · det Ȟ

is a Hermitian metric on the dual of the canonical line bundle of ℙ(E). It can be shown that

\frac{i}{2π} · ∂∂ log (k^{m} · det Ȟ)

= n · \frac{i · F_{A_{L^{∗}}}}{2π} + π̌^{∗}trace (\frac{i · F_{A}}{2π} ) + π̌^{∗}trace (\frac{i · F_{ω_{M}}}{2π} ) .

Here F_{A} is the curvature form of E induced by the Einstein-Hermitian connection A on E over M while F_{ω_{M}} is the curvature form of the holomorphic tangent bundle of M induced by the Kaehler form ω_{M} on M. Since

\frac{i · ∂∂ log det _{o}H_{#k}}{2π}

= \frac{i}{2π} · ∂∂ log (k^{m} · det Ȟ) + \frac{i}{2π} · ∂∂ log (\frac{_{o}ω(−1+m+n)_{#k}}{k^{m} · ω̌^{(−1+m+n)}} )

we have

č_{k} · ∫_{ℙ(E)} \frac{_{o}ω(−1+m+n)_{#k}}{(−1 + m + n)!}

= ∫_{ℙ(E)} \frac{i · ∂∂ log det _{o}H_{#k}}{2π} ∧ \frac{_{o}ω(−2+m+n)_{#k}}{(−2 + m + n)!}

= ∫_{ℙ(E)} \frac{i}{2π} · ∂∂ log (k^{m} · det Ȟ) ∧ \frac{_{o}ω(−2+m+n)_{#k}}{(−2 + m + n)!}

= ∫_{ℙ(E)} [n · \frac{i · F_{A_{L^{∗}}}}{2π} + trace (\frac{i · F_{A}}{2π} ) + trace (\frac{i · F_{ω_{M}}}{2π} )] ∧

\frac{_{o}ω(−2+m+n)_{#k}}{(−2 + m + n)!}.

Thus the topological invariant č_{k} is a rational function of the parameter k. By using the Einstein-Hermitian condition of the connection A on E over M

n · Π_{C_{M}} (\frac{i · F_{A_{L^{∗}}}}{2π} ) ∧ \frac{ω_{M}^{(−1+m)}}{(−1 + m)!} + trace (\frac{i · F_{A}}{2π} ) ∧ \frac{ω_{M}^{(−1+m)}}{(−1 + m)!} = 0

it can be shown readily that the power series expansion of č_{k} in \frac{1}{k} is

č_{k} = (−1 + n) · n + \frac{Λ_{M}trace (\frac{i·F_{ω_{M}}}{2π} )}{k} + \frac{č_{k:2}}{k · k} + higher order terms

in which č_{k:2} is a constant, independent of the parameter k, while Λ_{M}trace (\frac{i·F_{ω_{M}}}{2π} ) is the scalar curvature of (M : ω_{M}):

[Λ_{M}trace (\frac{i · F_{ω_{M}}}{2π} )] · \frac{ω_{M}^{m}}{m!} = trace (\frac{i · F_{ω_{M}}}{2π} ) ∧ \frac{ω_{M}^{(−1+m)}}{(−1 + m)!}.

Now let us consider the asymptotic expansion of ψ_{k} as k → +∞. By substituting

ω_{#k} = _{o}ω_{#k} + i∂∂ψ_{k} ∼ _{o}ω_{#k} + i∂∂ϕ_{0} + ∑_{θ∈ℕ} \frac{i∂∂ϕ_{θ}}{k^{θ}} as k → +∞

into S_{G-F} (ω_{#k}) we have

S_{G-F} (ω_{#k}) ∼ k^{m} · (B_{0} + ∑_{θ∈ℕ} \frac{B_{θ}}{k^{θ}} ) as k → +∞

in which each B_{•} is independent of the parameter k. In order to show that the asymptotic expansion of ψ_{k} (as k → +∞) exists we simply need to show the solvability of the following system of equations:

B_{θ} = 0 ⟺ \frac{B_{θ}}{Ω_{ℙ(E)}} = 0

for any integer θ ≥ 0. By using the induction condition

ϕ_{0} ∈ Γ_{o} (M : ℝ) and ϕ_{1} ∈ Γ_{o} (M : ℝ) ⊕ Γ (M : W)

it can be checked readily that

B_{0} = 0 = B_{1}.

We can then find

σ̂ (ϕ_{θ}) ⊕ σ (ϕ_{θ+1}) ⊕ σ̃ (ϕ_{θ+2})

through solving the equation

\frac{B_{θ+2}}{Ω_{ℙ(E)}} = σ̂ (\frac{B_{θ+2}}{Ω_{ℙ(E)}}) ⊕ σ (\frac{B_{θ+2}}{Ω_{ℙ(E)}}) ⊕ σ̃ (\frac{B_{θ+2}}{Ω_{ℙ(E)}}) = 0

by induction on integers θ ≥ 0. Actually we have the following result (proved in Appendix I):

Proposition V.A. By choosing the induction condition

ϕ_{0} ∈ Γ_{o} (M : ℝ) and ϕ_{1} ∈ Γ_{o} (M : ℝ) ⊕ Γ (M : W)

there exists a unique family of smooth ℝ-valued functions ϕ_{θ} ∈ Γ_{o} (ℙ(E) : ℝ) on ℙ(E), depending on integers θ ≥ 0, such that B_{θ} = 0 for any integer θ ≥ 0.

Now for each large N ∈ ℕ we define a Kaehler form _{N} ω_{#k} on ℙ(E), depending on k ∈ ℕ large enough, as follows:

N ω_{#k} ≡ _{o}ω_{#k} + i∂∂ϕ_{0} + ∑_{θ∈ℕ with θ≤N} \frac{i∂∂ϕ_{θ}}{k^{θ}}

= \frac{i · F_{A_{L^{∗}}}}{2π} + k · ω_{M} + i∂∂ϕ_{0} + ∑_{θ∈ℕ with θ≤N} \frac{i∂∂ϕ_{θ}}{k^{θ}} .

Here each ϕ_{•} is taken from the unique family of smooth ℝ-valued functions on ℙ(E) of Proposition V.A. Then we have the following result:

Corollary V.A. Given γ ≥ 0 we denote by ∥ • ∥_{C^{γ}(ℙ(E):ω̌)} the C^{γ}- norm of • with respect to the Hermitian form (metric) ω̌ on ℙ(E). Given p ∈ ℕ there exists a corresponding constant C_{(γ:p)} > 0 such that for each N ≥ p we have

∥\frac{S_{G-F} (_{N} ω_{#k})}{k^{m} · Ω_{ℙ(E)}} ∥_{C^{γ}(ℙ(E):ω̌)} ≤ \frac{C_{(γ:p)}}{k^{p}}

whenever k ≥ k_{(γ:p:N)}. Here the choice of k_{(γ:p:N)} ∈ ℕ depends on N.

Actually when N ≥ p we have, by Proposition V.A,

B_{0} = ⋯ = B_{p} = 0

and therefore the smooth ℝ-valued function ^{\frac{S_{G-F}(_{N} ω_{#k} )}{k^{m}·Ω_{ℙ(E)}}} on ℙ(E) must carry the factor \frac{1}{k·k^{p}} intrinsically when k > 0 is large enough (equivalently when \frac{1}{k} > 0 is small enough). Corollary V.A then follows immediately from standard results of calculus.

We define for each large k ∈ ℕ a functional R on the Kaehler class [_{o}ω_{#k}] = [\frac{i·F_{A_{L^{∗}}}}{2π} + k · ω_{M}] as follows:

R (_{o}ω_{#k} + i∂∂•) ≡ \frac{(_{o}ω_{#k} + i∂∂•)^{(−1+m+n)}}{k^{m} · ω̌^{(−1+m+n)}} = \frac{\frac{(_{o}ω_{#k} +i∂∂•)^{(−1+m+n)}}{(−1+m+n)!}}{k^{m} · Ω_{ℙ(E)}}

for any Kaehler form (_{o}ω_{#k} + i∂∂•) on ℙ(E) lying in the Kaehler class [_{o}ω_{#k}]. Then the gauge-fixing constant scalar curvature equation for ω_{#k} = _{o}ω_{#k} + i∂∂ψ_{k} can be expressed equivalently as follows:

\frac{S_{G-F} (ω_{#k})}{k^{m} · Ω_{ℙ(E)}} = 0

in which

\frac{S_{G-F} (ω_{#k})}{k^{m} · Ω_{ℙ(E)}}

≡ −č_{k} · R (ω_{#k}) + \frac{^{\frac{i·∂∂ log R(ω_{#k} )}{2π}} ∧ ^{\frac{ω_{#k}^{(−2+m+n)}}{(−2+m+n)!}}}{k^{m} · Ω_{ℙ(E)}}

+ \frac{[n · \frac{i·F_{A_{L^{∗}}}}{2π} + π̌^{∗}trace (\frac{i·F_{A}}{2π} ) + π̌^{∗}trace (\frac{i·F_{ω_{M}}}{2π} )] ∧ ^{\frac{ω_{#k}^{(−2+m+n)}}{(−2+m+n)!}}}{k^{m} · Ω_{ℙ(E)}}

+ (\frac{n · τ_{N_{W}} ∘ σ (ψ_{k})}{k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ (ψ_{k})}{k · k}) .

Now for each large N ∈ ℕ we define a corresponding 4^{th} order (elliptic) linear partial differential operator L_{N}, depending on the parameter k, acting on ψ ∈ Γ_{o} (ℙ(E) : ℝ) as follows:

L_{N} (ψ) = −č_{k} · _{A}L_{N} (ψ) + \frac{\frac{i∂∂}{2π} (_{\begin{matrix}A & L & N & (ψ) \\ R & (_{N} ω_{#k} ) \end{matrix}}) ∧ ^{\frac{_{N} ω_{#k}^{(−2+m+n)}}{(−2+m+n)!}}}{k^{m} · Ω_{ℙ(E)}}

+ \frac{^{\frac{i·∂∂ log R(_{N} ω_{#k} )}{2π}} ∧ i∂∂ψ ∧ ^{\frac{_{N} ω_{#k}^{(−3+m+n)}}{(−3+m+n)!}}}{k^{m} · Ω_{ℙ(E)}} + _{B}L_{N} (ψ)

+ (\frac{n · τ_{N_{W}} ∘ σ (ψ)}{k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ (ψ)}{k · k}) .

Here _{A}L_{N} is the corresponding 2^{nd} order linear partial differential operator (without the 0^{th} order part) acting on ψ ∈ Γ_{o} (ℙ(E) : ℝ) defined as follows:

AL_{N} (ψ) ≡ \frac{i∂∂ψ ∧ ^{\frac{_{N} ω_{#k}^{(−2+m+n)}}{(−2+m+n)!}}}{k^{m} · Ω_{ℙ(E)}}

while _{B}L_{N} is the corresponding 2^{nd} order linear partial differential operator (without the 0^{th} order part) acting on ψ ∈ Γ_{o} (ℙ(E) : ℝ) defined as follows:

BL_{N} (ψ)

≡ \frac{i∂∂ψ ∧[n · \frac{i·F_{A_{L^{∗}}}}{2π} +π̌^{∗}trace (\frac{i·F_{A}}{2π} )+π̌^{∗}trace (\frac{i·F_{ω_{M}}}{2π} )]∧ ^{\frac{_{N} ω_{#k}^{(−3+m+n)}}{(−3+m+n)!}}}{k^{m} · Ω_{ℙ(E)}} .

Proposition V.B. Let ∥ • ∥_{L^{2}(ℙ(E):ω̌)} denote the L^{2}-norm of • with respect to the Hermitian form (metric) ω̌ on ℙ(E). Then for each large N ∈ ℕ there exists a corresponding k_{N} ∈ ℕ such that for any ψ ∈ Γ_{o} (ℙ(E) : ℝ) we have

C · ∥L_{N} (ψ)∥_{L^{2}(ℙ(E):ω̌)} ≥ ∥(Δ_{V} + \frac{Δ_{M}}{k} ) ∘ (Δ_{V} + \frac{Δ_{M}}{k} ) σ̃ψ∥_{L^{2}(ℙ(E):ω̌)}

+ ∥\frac{(Δ_{M} + id)}{k} ∘ (Δ_{V} + \frac{Δ_{M}}{k} ) σψ∥_{L^{2}(ℙ(E):ω̌)}

+ ∥\frac{(Δ_{M} + id) ∘ (Δ_{M} + id) σ̂ψ}{k · k} ∥_{L^{2}(ℙ(E):ω̌)}

whenever k ≥ k_{N}. Here the constant C > 0 depends on (M : ω_{M}) and the Einstein-Hermitian structure of the holomorphic vector bundle E over M but not on N ∈ ℕ.

Let P_{#k} denote the 4^{th} order (elliptic) linear partial differential operator, depending on the parameter k, acting on ψ ∈ Γ_{o} (ℙ(E) : ℝ) defined as follows:

P_{#k} (ψ) = \frac{Δ_{V}}{8π} ∘ (−4πn · id + Δ_{V}) ψ

+ \frac{\frac{Δ_{M} ∘Δ_{V} ψ}{8π} + \frac{Δ_{V} ∘Δ_{M} ψ}{8π}}{k} + \frac{n · τ_{N_{W}} ∘ σψ}{k}

+ \frac{\frac{Δ_{M} ∘Δ_{M} ψ}{8π} + [−Λ_{M}trace (\frac{i·F_{ω_{M}}}{2π} )] · \frac{Δ_{M} ψ}{2}}{k · k}

+ \frac{\frac{i(∂∂)_{M} ψ∧[trace( \frac{i·Fω_{M}}{2π} )]∧^{\frac{ω{(−2+m) \atop M}}{(−2+m)!}}}{Ω_{M}}}{k · k}

+ \frac{τ_{N_{V_{M}}} ∘ σ̂ψ}{k · k} .

Actually, for each large N ∈ ℕ, it can be shown that L_{N} is dominated by P_{#k} when the parameter k is sufficiently large.

Corollary V.B. Given integer γ ≥ 0 we define the Sobolev norm ∥ • ∥_{H^{[2γ]}(ℙ(E):ω̌)} of • as follows:

∥ • ∥_{H^{[2γ]}(ℙ(E):ω̌)} ≡ ∥ • ∥_{L^{2}(ℙ(E):ω̌)} + ∥(Δ_{V} + Δ_{M}) •∥_{L^{2}(ℙ(E):ω̌)}
+ ⋯ + ∥(Δ_{V} + Δ_{M})^{γ} •∥_{L^{2}(ℙ(E):ω̌)} .

Then for each large N ∈ ℕ there exists a corresponding k_{(γ:N)} ∈ ℕ such that for any ψ ∈ Γ_{o} (ℙ(E) : ℝ) we have

C_{γ} · ∥L_{N} (ψ)∥_{H^{[2γ]}(ℙ(E):ω̌)}

≥ ∥(Δ_{V} + \frac{Δ_{M}}{k} ) ∘ (Δ_{V} + \frac{Δ_{M}}{k} ) σ̃ψ∥_{H^{[2γ]}(ℙ(E):ω̌)}

+ ∥\frac{(Δ_{M} + id)}{k} ∘ (Δ_{V} + \frac{Δ_{M}}{k} ) σψ∥_{H^{[2γ]}(ℙ(E):ω̌)}

+ ∥\frac{(Δ_{M} + id) ∘ (Δ_{M} + id) σ̂ψ}{k · k} ∥_{H^{[2γ]}(ℙ(E):ω̌)}

whenever k ≥ k_{(γ:N)}. Here the constant C_{γ} > 0 depends on γ but not on N. In particular we have for any ψ ∈ Γ_{o} (ℙ(E) : ℝ) the following estimate:

C_{γ} · ∥L_{N} (ψ)∥_{H^{[2γ]}(ℙ(E):ω̌)} ≥ \frac{∥(Δ_{V} + Δ_{M}) ∘ (Δ_{V} + Δ_{M}) ψ∥_{H^{[2γ]}(ℙ(E):ω̌)}}{k · k}

whenever k ≥ k_{(γ:N)}.

Remark. Note that the linear 2^{nd} order elliptic linear partial differential operator Δ_{M} is coercive when acting on Γ_{o} (M : ℝ). Besides the linear 2^{nd} order elliptic linear partial differential operator Δ_{M} + Δ_{V} is coercive when acting on Γ_{o} (ℙ(E) : ℝ).

We will prove these results in Appendix III.

Given γ ≥ 0 we denote by H_{o}^{[2γ]} (ℙ(E) : ω̌) the Sobolev space consisting of ℝ-valued functions f ∈ H^{[2γ]} (ℙ(E) : ω̌) on ℙ(E) satisfying

∫_{ℙ(E)} f · Ω_{ℙ(E)} = 0.

Note that Corollary V.B implies the invertibility of the 4^{th} order (elliptic) linear partial differential operator

L_{N} : H_{o}^{[2γ+4]} (ℙ(E) : ω̌) ⟶ H_{o}^{[2γ]} (ℙ(E) : ω̌)

whenever k is sufficiently large. Let I_{N} denote the inverse of L_{N}. Then we have, for any f ∈ H_{o}^{[2γ]} (ℙ(E) : ω̌), the following estimate:

∥I_{N}f∥_{H_{o}^{[2γ+4]}(ℙ(E):ω̌)} ≤ C_{γ} · k · k · ∥f∥_{H_{o}^{[2γ]}(ℙ(E):ω̌)}

whenever k ≥ k_{(γ:N)}. Note that in this estimate for I_{N} the constant C_{γ} > 0 does not depend on N ∈ ℕ though k_{(γ:N)} must be chosen larger for large N.

Given ψ ∈ Γ_{o}(ℙ(E) : ℝ) we note that, for t ∈ ℝ,

\frac{d}{dt} [\frac{S_{G-F} (_{N} ω_{#k} + t · i∂∂ψ)}{k^{m} · Ω_{ℙ(E)}}]

= −č_{k} · _{A}L_{N}^{t} (ψ) + _{B}L_{N}^{t} (ψ)

+ \frac{\frac{i∂∂}{2π} (\begin{matrix}A & L & ^{t}_{N} & (ψ) \\ R & (_{N} ω_{#k} +t·i∂∂ψ) \end{matrix}) ∧ ^{\frac{(_{N} ω_{#k} +t·i∂∂ψ)^{(−2+m+n)}}{(−2+m+n)!}}}{k^{m} · Ω_{ℙ(E)}}

+ \frac{^{\frac{i·∂∂ log R(_{N} ω_{#k} +t·i∂∂ψ)}{2π}} ∧ i∂∂ψ ∧ ^{\frac{(_{N} ω_{#k} +t·i∂∂ψ)^{(−3+m+n)}}{(−3+m+n)!}}}{k^{m} · Ω_{ℙ(E)}}

+ (\frac{n · τ_{N_{W}} ∘ σ (ψ)}{k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ (ψ)}{k · k})

in which _{A}L_{N}^{t} and _{B}L_{N}^{t} are the corresponding 2^{nd} order nonlinear partial differential operators acting on ψ ∈ Γ_{o} (ℙ(E) : ℝ) defined respectively as follows:

AL_{N}^{t} (ψ) ≡ \frac{i∂∂ψ ∧ ^{\frac{(_{N} ω_{#k} +t·i∂∂ψ)^{(−2+m+n)}}{(−2+m+n)!}}}{k^{m} · Ω_{ℙ(E)}}

and

BL_{N}^{t} (ψ)

≡ \frac{i∂∂ψ∧[n · \frac{i·F_{A_{L^{∗}}}}{2π} +π̌^{∗}trace ( \frac{i·F_{A}}{2π} )+π̌^{∗}trace ( \frac{i·F_{ω_{M}}}{2π} )]∧^{\frac{(_{N} ω_{#k} +t·i∂∂ψ)^{(−3+m+n)}}{(−3+m+n)!}}}{k^{m} · Ω_{ℙ(E)}} .

From the shape of \frac{d}{dt} [^{\frac{S_{G-F}(_{N} ω_{#k} +t·i∂∂ψ)}{k^{m}·Ω_{ℙ(E)}}}] it is easy to see that, for

bounded t ∈ ℝ, the nonlinear partial differential operator

\frac{d}{dt} ∘ \frac{d}{dt} [\frac{S_{G-F} (_{N} ω_{#k} + t · i∂∂ψ)}{k^{m} · Ω_{ℙ(E)}}] = \frac{d}{dt} ∘ \frac{d}{dt} [\frac{S (_{N} ω_{#k} + t · i∂∂ψ)}{k^{m} · Ω_{ℙ(E)}}]

is genuinely nonlinear. Actually, by the Sobolev Embedding Theorem, for each sufficiently large γ ∈ ℕ there exists a corresponding k_{(γ:N)} ∈ ℕ such that for any pair (f : g) of elements of H_{o}^{[2γ]} (ℙ(E) : ω̌) satisfying ∥f∥_{H^{[2γ+4]}(ℙ(E):ω̌)} ≤ \frac{1}{k} and ∥g∥_{H^{[2γ+4]}(ℙ(E):ω̌)} ≤ \frac{1}{k} we have the following estimate:

∥−\frac{d}{dt} ∘ \frac{d}{dt} [\frac{S_{G-F} (_{N} ω_{#k} + t · i∂∂f)}{k^{m} · Ω_{ℙ(E)}}]

+\frac{d}{dt} ∘ \frac{d}{dt} [\frac{S_{G-F} (_{N} ω_{#k} + t · i∂∂g)}{k^{m} · Ω_{ℙ(E)}}]∥_{H^{[2γ]}(ℙ(E):ω̌)}

≤ C_{γ} · (∥f∥_{H^{[2γ+4]}(ℙ(E):ω̌)} + ∥g∥_{H^{[2γ+4]}(ℙ(E):ω̌)}) · ∥−f + g∥_{H^{[2γ+4]}(ℙ(E):ω̌)}

whenever k ≥ k_{(γ:N)}. Here the constant C_{γ} > 0 can be chosen to be independent of N ∈ ℕ.

Now we note that for ψ ∈ Γ_{o} (ℙ(E) : ℝ) the gauge-fixing constant scalar curvature equation for (_{N} ω_{#k} + i∂∂ψ) lying in the Kaehler class [_{o}ω_{#k}] can be expressed as

0 = \frac{S_{G-F} (_{N} ω_{#k} + i∂∂ψ)}{k^{m} · Ω_{ℙ(E)}} = \frac{S_{G-F} (_{N} ω_{#k})}{k^{m} · Ω_{ℙ(E)}} + L_{N} (ψ) + GN_{N} (ψ)

in which GN_{N} is the genuinely nonlinear partial differential operator acting on ψ ∈ Γ_{o} (ℙ(E) : ℝ) defined as follows:

GN_{N} (ψ) ≡ ∫_{0}^{1} (−t + 1) · \frac{d}{dt} ∘ \frac{d}{dt} [\frac{S_{G-F} (_{N} ω_{#k} + t · i∂∂ψ)}{Ω_{ℙ(E)}}] · dt

= ∫_{0}^{1} (−t + 1) · \frac{d}{dt} ∘ \frac{d}{dt} [\frac{S (_{N} ω_{#k} + t · i∂∂ψ)}{Ω_{ℙ(E)}}] · dt.

Since for any Kaehler form (_{N} ω_{#k} + i∂∂•) lying in the Kaehler class [_{o}ω_{#k}] the integral

∫_{ℙ(E)} \frac{S_{G-F} (_{N} ω_{#k} + i∂∂•)}{Ω_{ℙ(E)}} · Ω_{ℙ(E)} = ∫_{ℙ(E)} S_{G-F} (_{N} ω_{#k} + i∂∂•)

always vanishes we can apply the inverse I_{N} of L_{N} to the last expression of the gauge-fixing constant scalar curvature equation and obtain the following equivalent

0 = I_{N} (\frac{S_{G-F} (_{N} ω_{#k})}{k^{m} · Ω_{ℙ(E)}}) + ψ + I_{N} ∘ GN_{N} (ψ) .

We will now solve this equation through the Contraction Mapping Theorem. Given sufficiently large γ ∈ ℕ we may choose large q ∈ ℕ such that the genuinely nonlinear operator I_{N} ∘ GN_{N}, when acting on the complete metric space

\{ψ ∈ H_{o}^{[2γ+4]} (ℙ(E) : ω̌) : ∥ψ∥_{H^{[2γ+4]}(ℙ(E):ω̌)} ≤ \frac{1}{k^{q}} \} ,

is contractive with contraction constant ≤ \frac{1}{2} whenever the parameter k is large enough. On the other hand, by Corollary V.A and Corollary V.B, we may choose sufficiently large N ∈ ℕ such that

∥I_{N} (\frac{S_{G-F} (_{N} ω_{#k})}{k^{m} · Ω_{ℙ(E)}})∥_{H^{[2γ+4]}(ℙ(E):ω̌)} ≤ \frac{1}{k · k^{q}}

whenever the parameter k is large enough. Thus by the Contraction Mapping Theorem we conclude that for each suitable choice of (γ : q : N) ∈ ℕ × ℕ × ℕ the gauge-fixing constant scalar curvature equation can be solved uniquely, whenever the parameter k is large enough, by a Kaehler form (_{N} ω_{#k} +i∂∂ψ) on ℙ(E) with ψ ∈ H_{o}^{[2γ+4]} (ℙ(E) : ω̌) satisfying

∥ψ∥_{H^{[2γ+4]}(ℙ(E):ω̌)} ≤ \frac{1}{k^{q}} .

With standard results of partial differential equations it can be shown readily that the solutions ψ ∈ H_{o}^{[2γ+4]} (ℙ(E) : ω̌) to the gaugefixing constant scalar curvature equation, depending on sufficiently large k, found in this way are actually smooth because we already have high regularity and good approximation results. Hence we have:

Theorem V.A. When the parameter k ∈ ℕ is sufficiently large the corresponding gauge-fixing constant scalar curvature equation

\frac{S_{G-F} (_{o}ω_{#k} + i∂∂ψ_{k})}{k^{m} · Ω_{ℙ(E)}} = 0

can be solved by some smooth ℝ-valued function ψ_{k} ∈ Γ_{o} (ℙ(E) : ℝ) on ℙ(E). Besides this family of smooth ℝ-valued functions ψ_{k} ∈ Γ_{o}(ℙ(E) : ℝ) on ℙ(E) admits asymptotic expansion of the following form

ψ_{k} ∼ ϕ_{0} + ∑_{θ∈ℕ} \frac{ϕ_{θ}}{k^{θ}}

as k → +∞. Here each ϕ_{•} is taken from the unique family of smooth ℝ-valued functions on ℙ(E) of Proposition V.A. Actually, for each pair (γ : q) ∈ ℕ × ℕ of large enough integers, we may even require, when N ∈ ℕ is chosen sufficiently large, that

oω_{#k} + i∂∂ψ_{k} = _{N} ω_{#k} + i∂∂ψ_{(k:N)}

with ψ_{(k:N)} ∈ Γ_{o} (ℙ(E) : ℝ) satisfying

∥ψ_{(k:N)}∥_{H^{[2γ+4]}(ℙ(E):ω̌)} ≤ \frac{1}{k^{q}}

whenever k is large enough. In this case the choice of the solution

N ω_{#k} + i∂∂ψ_{(k:N)}

to the gauge-fixing constant scalar curvature equation

\frac{S_{G-F} (_{N} ω_{#k} + i∂∂ψ_{(k:N)})}{k^{m} · Ω_{ℙ(E)}} = 0

with ψ_{(k:N)} ∈ Γ_{o} (ℙ(E) : ℝ) satisfying ∥ψ_{(k:N)}∥_{H^{[2γ+4]}(ℙ(E):ω̌)} ≤ \frac{1}{k^{q}} is, for each sufficiently large k, unique.

Now we conclude this section with a remark. Suppose that, for each k ∈ ℕ large enough, the corresponding Futaki invariants associated with

𝔤_{E} + (the lifted action of) 𝔨_{(M:ω_{M})}

and the Kaehler class [\frac{i·F_{A_{L^{∗}}}}{2π} + k · π̌^{∗}ω_{M}] on ℙ(E) are zero. Then, by Theorem II.B, we may assume that

\frac{i · F_{A_{L^{∗}}}}{2π}

is invariant under the lifted action of 𝔨_{(M:ω_{M})} on ℙ(E). Thus both \frac{i·F_{A_{L^{∗}}}}{2π} and ω_{M} are invariant under the action of (𝔨_{E} + the lifting of 𝔨_{(M:ω_{M})}) on ℙ(E). In particular the gauge-fixing constant scalar curvature equation is invariant under the action of

(𝔨_{E} + the lifting of 𝔨_{(M:ω_{M})})

on ℙ(E). (Note that [𝔤_{E} : the lifting of 𝔨_{(M:ω_{M})}] ⊂ 𝔤_{E}.) Hence, by Proposition V.A and the uniqueness result of Theorem V.A, the solutions

oω_{#k} + i∂∂ψ_{k} = _{N} ω_{#k} + i∂∂ψ

to the gauge-fixing constant scalar curvature equation, depending on sufficiently large k, are invariant under the action of (𝔨_{E}+ the lifting of 𝔨_{(M:ω_{M})}) on ℙ(E).

VI. Solving the constant scalar curvature equation

In this section our main purpose is to prove Theorem A based on the solvability results for the gauge-fixing constant scalar curvature equation — Theorem V.A.

By Theorem II.B we may assume that the lifting of 𝔨_{(M:ω_{M})} on ℙ(E)

preserves \frac{i·F_{A_{L^{∗}}}}{2π}. Moreover such lifting of 𝔨_{(M:ω_{M})} on ℙ(E) is, modulo the compact Lie subalgebra 𝔨_{E} of 𝔤_{E}, uniquely determined. Thus, by complexification, there is a preferred lifting of 𝔥_{o}(M) on ℙ(E) which is essentially uniquely determined. In this section will fix one such preferred lifting of 𝔥_{o}(M). Besides, for each smooth vector field X on M preserving the complex structure of M, we will use the same symbol X to denote the lifting of X on ℙ(E) when there is no confusion.

Now we fix respectively a K_{(M:ω_{M})}-invariant metric on 𝔥_{o}(M) and a K_{E}-invariant metric on 𝔤_{E}. Here K_{E} is the maximal compact subgroup of G_{E}. By doing so the ℝ-linear subspace of 𝔥_{o}(M), orthogonal to 𝔨_{(M:ω_{M})}, is isomorphic to the ℝ-linear space N_{V_{M}} while the ℝ-linear subspace of 𝔤_{E}, orthogonal to 𝔨_{E}, is isomorphic to the ℝ-linear space N_{W}. In particular, for each f ∈ N_{V_{M}}, ∥f∥_{C^{0}(ℙ(E):ω̌)} is comparable with the C^{0}-norm of its correspondent in the orthogonal complement of 𝔨_{(M:ω_{M})} in 𝔥_{o}(M). Moreover this comparability is uniform on N_{V_{M}}. Actually fixing a K_{(M:ω_{M})}-invariant metric on 𝔥_{o}(M) simply means that we have fixed the uniform comparability between N_{V_{M}} and

\frac{𝔥_{o}(M)}{𝔨_{(M:ω_{M})}} .

Similar results are valid for N_{V_{M}}.

Let ω_{#k} = _{o}ω_{#k} + i∂∂ψ_{k} denote the solution of Theorem V.A to the gauge-fixing constant scalar curvature equation. We will denote by ⟨ : ⟩ the inner product on L^{2} (ℙ(E) : ω̌) defined by the Hermitian (metric) form ω̌ on ℙ(E):

⟨f : g⟩ ≡ ∫_{ℙ(E)} f · g · Ω_{ℙ(E)}

∀(f : g) ∈ L^{2} (ℙ(E) : ω̌) × L^{2} (ℙ(E) : ω̌). Besides we will use the symbol c to denote a sufficiently large constant > 0 independent of the parameter k.

Suppose that X_{k} ∈ Γ (M : T (M)), orthogonal to 𝔨_{(M:ω_{M})}, is the smooth vector field on M, preserving the complex structure of M, such that

L_{X_{k}}ω_{M} = i∂∂ (τ_{N_{V_{M}}} ∘ σ̂ψ_{k}) .

Let f_{X_{k}} ∈ Γ_{o} (ℙ(E) : ℝ) be the corresponding smooth ℝ-valued function on ℙ(E) satisfying

L_{X_{k}} (\frac{i · F_{A_{L^{∗}}}}{2π} ) = i∂∂f_{X_{k}}.

Then we have

L_{X_{k}}ω_{#k} = i∂∂ (k · τ_{N_{V_{M}}} ∘ σ̂ψ_{k} + f_{X_{k}} + L_{X_{k}}ψ_{k}) .

Since the Futaki invariant, corresponding to X_{k} and the Kaehler class [\frac{i·F_{A_{L^{∗}}}}{2π} + k · π̌^{∗}ω_{M}] on ℙ(E), vanishes we have, by incorporating the gauge-fixing constant scalar curvature equation, the following equality:

∫_{ℙ(E)} (k · τ_{N_{V_{M}}} ∘ σ̂ψ_{k} + f_{X_{k}} + L_{X_{k}}ψ_{k})

· [\frac{n · τ_{N_{W}} ∘ σψ_{k}}{k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ψ_{k}}{k · k} ] · Ω_{ℙ(E)} = 0.

Since Γ (M : ℝ) is orthogonal to Γ (M : W) with respect to the inner product ⟨ : ⟩ on L^{2} (ℙ(E) : ω̌) we note that the term

∫_{ℙ(E)} k · τ_{N_{V_{M}}} ∘ σ̂ (ψ_{k}) · \frac{n · τ_{N_{W}} ∘ σψ_{k}}{k} · Ω_{ℙ(E)}

of the above equality vanishes. Thus, when the parameter k is sufficiently large, we infer from Theorem V.A that there exists a constant c > 0, independent of k, such that

∥f_{X_{k}}∥_{C^{0}(ℙ(E):ω̌)} + ∥L_{X_{k}}ψ_{k}∥_{C^{0}(ℙ(E):ω̌)} ≤ c · ∥X_{k}∥_{C^{0}(ℙ(E):ω̌)}.

On the other hand

∥X_{k}∥_{C^{0}(ℙ(E):ω̌)}

is comparable with ∥τ_{N_{V_{M}}} ∘σ̂ (ψ_{k}) ∥_{C^{0}(ℙ(E):ω̌)}. Thus, by using the Schwarz inequality, we infer from the above equality

∫_{ℙ(E)} (k · τ_{N_{V_{M}}} ∘ σ̂ψ_{k} + f_{X_{k}} + L_{X_{k}}ψ_{k})

· [\frac{n · τ_{N_{W}} ∘ σψ_{k}}{k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ψ_{k}}{k · k} ] · Ω_{ℙ(E)} = 0

that, when k is sufficiently large, there exists a constant ĉ > 0, independent of k, such that

∥τ_{N_{V_{M}}} ∘ σ̂ (ψ_{k}) ∥_{C^{0}(ℙ(E):ω̌)}^{2} ≤ ĉ · ∥τ_{N_{W}} ∘ σ (ψ_{k}) ∥_{C^{0}(ℙ(E):ω̌)}^{2}.

Suppose that Y_{k} ∈ 𝔤_{E}, orthogonal to 𝔨_{E}, is the smooth vector field on ℙ(E) such that

L_{Y_{k}} (\frac{i · F_{A_{L^{∗}}}}{2π} ) = i∂∂ (τ_{N_{W}} ∘ σψ_{k}) .

Then we have

L_{Y_{k}}ω_{#k} = i∂∂ (τ_{N_{W}} ∘ σψ_{k} + L_{Y_{k}}ψ_{k}) .

Since the Futaki invariant, corresponding to Y_{k} and the Kaehler class [\frac{i·F_{A_{L^{∗}}}}{2π} + k · π̌^{∗}ω_{M}] on ℙ(E), vanishes we have, by incorporating the gauge-fixing constant scalar curvature equation, the following equality:

∫_{ℙ(E)} (τ_{N_{W}} ∘ σψ_{k} + L_{Y_{k}}ψ_{k})

· [\frac{n · τ_{N_{W}} ∘ σψ_{k}}{k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ψ_{k}}{k · k} ] · Ω_{ℙ(E)} = 0.

Since Γ (M : ℝ) is orthogonal to Γ (M : W) with respect to the inner product ⟨ : ⟩ on L^{2} (ℙ(E) : ω̌) we note that the term

∫_{ℙ(E)} τ_{N_{W}} ∘ σ (ψ_{k}) · \frac{τ_{N_{V_{M}}} ∘ σ̂ψ_{k}}{k · k} · Ω_{ℙ(E)}

of the above equality vanishes. Since the zeroth order term ϕ_{0} in the asymptotic expansion of ψ_{k}, as k → +∞,

ψ_{k} ∼ ϕ_{0} + ∑_{θ=1}^{+∞} \frac{ϕ_{θ}}{k^{θ}}

satisfies

σ (ϕ_{0}) = σ̃ (ϕ_{0}) = 0 ⟺ ϕ_{0} ∈ Γ_{o} (M : ℝ)

we infer from Theorem V.A that, when the parameter k is sufficiently large, there exists a constant c > 0, independent of k, such that

∥L_{Y_{k}}ψ_{k}∥_{C^{0}(ℙ(E):ω̌)} ≤ c · \frac{∥Y_{k}∥_{C^{0}(ℙ(E):ω̌)}}{k}

because L_{Y_{k}}ϕ_{0} = 0. On the other hand

∥Y_{k}∥_{C^{0}(ℙ(E):ω̌)}

is comparable with ∥τ_{N_{W}} ∘σ (ψ_{k}) ∥_{C^{0}(ℙ(E):ω̌)}. Thus, by using the Schwarz inequality, we infer from the above equality

∫_{ℙ(E)}(τ_{N_{W}} ∘ σψ_{k} + L_{Y_{k}}ψ_{k}) · [\frac{n · τ_{N_{W}} ∘ σψ_{k}}{k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ψ_{k}}{k · k} ] · Ω_{ℙ(E)} = 0

that, when k is sufficiently large, there exists a constant c̃ > 0, independent of k, such that

∥τ_{N_{W}} ∘ σ (ψ_{k}) ∥_{C^{0}(ℙ(E):ω̌)}^{2} ≤ c̃ · \frac{∥τ_{N_{V_{M}}} ∘ σ̂ (ψ_{k}) ∥_{C^{0}(ℙ(E):ω̌)}^{2}}{k · k} .

By comparing this inequality

∥τ_{N_{W}} ∘ σ (ψ_{k}) ∥_{C^{0}(ℙ(E):ω̌)}^{2} ≤ c̃ · \frac{∥τ_{N_{V_{M}}} ∘ σ̂ (ψ_{k}) ∥_{C^{0}(ℙ(E):ω̌)}^{2}}{k · k}

with the previous inequality

∥τ_{N_{V_{M}}} ∘ σ̂ (ψ_{k}) ∥_{C^{0}(ℙ(E):ω̌)}^{2} ≤ ĉ · ∥τ_{N_{W}} ∘ σ (ψ_{k}) ∥_{C^{0}(ℙ(E):ω̌)}^{2}

we conclude immediately that

τ_{N_{V_{M}}} ∘ σ̂ (ψ_{k}) = 0 = τ_{N_{W}} ∘ σ (ψ_{k})

when the parameter k is sufficiently large.

Hence the (uniquely determined, for each k) smooth ℝ-valued solutions ψ_{k} of Theorem V.A to the gauge-fixing constant scalar curvature equation are actually solutions to the constant scalar curvature equation, without gauge-fixing, when the parameter k is sufficiently large.

q.e.d.

Appendix I. Induction scheme

We will prove Proposition V.A by induction on integers θ ≥ 0. Before proceeding we note that the integral of S_{G-F}(•) on ℙ(E) is always zero:

∫_{ℙ(E)} S_{G-F}(•) = ∫_{ℙ(E)} \frac{S_{G-F}(•)}{Ω_{ℙ(E)}} · Ω_{ℙ(E)} = ∫_{ℙ(E)} σ̂ (\frac{S_{G-F}(•)}{Ω_{ℙ(E)}}) · Ω_{ℙ(E)} = 0.

Let us start with solving the equation

B_{2} = 0 ⟺ \frac{B_{2}}{Ω_{ℙ(E)}} = 0.

Assume that

\frac{ω(−1+m+n)_{#k}}{k^{m} · ω̌^{(−1+m+n)}} ∼ \frac{_{o}ω(−1+m+n)_{#k}}{k^{m} · ω̌^{(−1+m+n)}} + \frac{R_{1}}{k} + \frac{R_{2}}{k · k} + higher order terms

in which R_{1} and R_{2} are independent of the parameter k. Since the connection A on E over M is Einstein-Hermitian we have

n · Π_{C_{M}} (\frac{i · F_{A_{L^{∗}}}}{2π} ) ∧ \frac{ω_{M}^{(−1+m)}}{(−1 + m)!} + trace (\frac{i · F_{A}}{2π} ) ∧ \frac{ω_{M}^{(−1+m)}}{(−1 + m)!} = 0

and thence

\frac{_{o}ω(−1+m+n)_{#k}}{k^{m} · ω̌^{(−1+m+n)}} = \frac{Ω_{ℙ(E)}}{Ω_{ℙ(E)}} + \frac{−^{\frac{Λ_{M} trace( \frac{i·F_{A}}{2π} )}{n}}}{k}

+ \frac{\frac{^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{(n+1)}}{(n+1)!}} ∧^{\frac{ω{(−2+m) \atop M}}{(−2+m)!}}}{Ω_{ℙ(E)}}}{k · k} + higher order terms

in which Λ_{M}trace (\frac{i·F_{A}}{2π} ) is the constant associated with the Einstein-Hermitian connection A on E over M:

[Λ_{M}trace (\frac{i · F_{A}}{2π} )] · \frac{ω_{M}^{m}}{m!} = trace (\frac{i · F_{A}}{2π} ) ∧ \frac{ω_{M}^{(−1+m)}}{(−1 + m)!}.

Besides it can be checked readily that

R_{1} = \frac{Δ_{M}ϕ_{0}}{2} + \frac{Δ_{V} ϕ_{1}}{2} = \frac{Δ_{M}ϕ_{0}}{2} + \frac{Δ_{V} σ (ϕ_{1})}{2} = \frac{Δ_{M}ϕ_{0}}{2} + 2πn · σ (ϕ_{1})

and

R_{2} = \frac{Δ_{V} ϕ_{2}}{2} + \frac{Q (σ (ϕ_{1}) : σ (ϕ_{1}))}{2!} + \frac{Δ_{M}ϕ_{0}}{2} · \frac{Δ_{V} ϕ_{1}}{2}

+ \frac{\frac{i∂∂ϕ_{0}∧i∂∂ϕ_{0}}{2!} ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}} + \frac{i∂∂ϕ_{0} ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{n}}{n!}} ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{ℙ(E)}}

+ \frac{i∂∂ϕ_{1} ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{(−1+n)}}{(−1+n)!}} ∧ \frac{ω_{M}^{(−1+m)}}{(−1+m)!}}{Ω_{ℙ(E)}} .

Here the symmetric quadratic operator Q (• : •) is defined along each fiber ℙ (ℂ^{n}) of the holomorphic projection map π̌ : ℙ(E) ⟶ M as in Proposition IV.A and Corollary IV.A. Thus

\frac{B_{2}}{Ω_{ℙ(E)}} = − ((−1 + n) · n · R_{2} + [Λ_{M}trace (\frac{i · F_{ω_{M}}}{2π} )] · R_{1})

+ n · (−2 + n) · \frac{Δ_{V} ϕ_{2}}{2}

+ n · (−1 + n) · \frac{i∂∂ϕ_{1} ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{(−1+n)}}{(−1+n)!}} ∧ \frac{ω_{M}^{(−1+m)}}{(−1+m)!}}{Ω_{ℙ(E)}}

+ n · n · \frac{i∂∂ϕ_{0} ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{n}}{n!}} ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{ℙ(E)}}

+ n · (−3 + n) · \frac{Q (σ (ϕ_{1}) : σ (ϕ_{1}))}{2!}

+ n · (−2 + n) · \frac{Δ_{M}ϕ_{0}}{2} · \frac{Δ_{V} ϕ_{1}}{2}

+ n · (−1 + n) · \frac{\frac{i∂∂ϕ_{0}∧i∂∂ϕ_{0}}{2!} ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ [Λ_{M}trace (\frac{i · F_{A}}{2π} ) + Λ_{M}trace (\frac{i · F_{ω_{M}}}{2π} )] · \frac{Δ_{V} ϕ_{1}}{2}

+ \frac{[trace (\frac{i·F_{A}}{2π} ) + trace (\frac{i·F_{ω_{M}}}{2π} )] ∧ i∂∂ϕ_{0} ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ \frac{\frac{i∂∂R_{1}}{2π} ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{(−1+n)}}{(−1+n)!}} ∧ \frac{ω_{M}^{(−1+m)}}{(−1+m)!}}{Ω_{ℙ(E)}}

+ Q (\frac{σ (R_{1})}{2π} : σ (ϕ_{1})) + \frac{Δ_{V} R_{1}}{4π} · \frac{Δ_{M}ϕ_{0}}{2}

+ \frac{Δ_{V} R_{2}}{4π} + \frac{−Δ_{V}}{8π} (−\frac{Λ_{M}trace (\frac{i·F_{A}}{2π} )}{n} + R_{1})^{2}

+ n · τ_{N_{W}} ∘ σ (ϕ_{1}) + τ_{N_{V_{M}}} ∘ σ̂ (ϕ_{0}) + known terms.

By substituting the formulae for R_{1} and R_{2} into the above expression of \frac{B_{2}}{Ω_{ℙ(E)}} we see that

\frac{B_{2}}{Ω_{ℙ(E)}} = − [Λ_{M}trace (\frac{i · F_{ω_{M}}}{2π} )] · \frac{Δ_{M}ϕ_{0}}{2} + (−n) · \frac{Δ_{V} ϕ_{2}}{2}

+ n · \frac{i∂∂ϕ_{0} ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{n}}{n!}} ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{ℙ(E)}}

+ [Λ_{M}trace (\frac{i · F_{A}}{2π} )] · \frac{Δ_{V} ϕ_{1}}{2}

+ \frac{[trace (\frac{i·F_{A}}{2π} ) + trace (\frac{i·F_{ω_{M}}}{2π} )] ∧ i∂∂ϕ_{0} ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ \frac{Δ_{M} ∘ Δ_{M}ϕ_{0}}{8π} + \frac{n · i∂∂σ (ϕ_{1}) ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{(−1+n)}}{(−1+n)!}} ∧ \frac{ω_{M}^{(−1+m)}}{(−1+m)!}}{Ω_{ℙ(E)}}

+ \frac{Δ_{V} R_{2}}{4π} + [Λ_{M}trace (\frac{i · F_{A}}{2π} )] · \frac{Δ_{V} σ (ϕ_{1})}{2}

+ (−n) · \frac{Δ_{M}ϕ_{0}}{2} · \frac{Δ_{V} σ (ϕ_{1})}{2}

+ (−n · n · π) · \frac{Δ_{V}}{2} [σ (ϕ_{1}) · σ (ϕ_{1})]

+ n · τ_{N_{W}} ∘ σ (ϕ_{1}) + τ_{N_{V_{M}}} ∘ σ̂ (ϕ_{0}) + known terms.

Einstein-Hermitian condition of A on E over M)

\frac{i∂∂σ (ϕ_{1}) ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{(−1+n)}}{(−1+n)!}} ∧ \frac{ω_{M}^{(−1+m)}}{(−1+m)!}}{Ω_{ℙ(E)}}

= \frac{Δ_{M}σ (ϕ_{1})}{2} + \frac{Δ_{V} σ (ϕ_{1})}{2} · \frac{Π_{C_{M}} (\frac{i·F_{A_{L^{∗}}}}{2π} ) ∧ \frac{ω_{M}^{(−1+m)}}{(−1+m)!}}{Ω_{M}}

= \frac{Δ_{M}σ (ϕ_{1})}{2} + \frac{[−Λ_{M}trace (\frac{i·F_{A}}{2π} )]}{n} · \frac{Δ_{V} σ (ϕ_{1})}{2}

and thence

\frac{B_{2}}{Ω_{ℙ(E)}} = − [Λ_{M}trace (\frac{i · F_{ω_{M}}}{2π} )] · \frac{Δ_{M}ϕ_{0}}{2} + (−n) · \frac{Δ_{V} ϕ_{2}}{2}

+ n · \frac{i∂∂ϕ_{0} ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{n}}{n!}} ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{ℙ(E)}}

+ \frac{[trace (\frac{i·F_{A}}{2π} ) + trace (\frac{i·F_{ω_{M}}}{2π} )] ∧ i∂∂ϕ_{0} ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ \frac{Δ_{M} ∘ Δ_{M}ϕ_{0}}{8π} + n · \frac{Δ_{M}σ (ϕ_{1})}{2}

+ \frac{Δ_{V} R_{2}}{4π} + [Λ_{M}trace (\frac{i · F_{A}}{2π} )] · \frac{Δ_{V} σ (ϕ_{1})}{2}

+ (−n) · \frac{Δ_{M}ϕ_{0}}{2} · \frac{Δ_{V} σ (ϕ_{1})}{2}

+ (−n · n · π) · \frac{Δ_{V}}{2} [σ (ϕ_{1}) · σ (ϕ_{1})]

+ n · τ_{N_{W}} ∘ σ (ϕ_{1}) + τ_{N_{V_{M}}} ∘ σ̂ (ϕ_{0}) + known terms.

Now we note that

∫_{ℙ(ℂ^{n})} n · \frac{i∂∂ϕ_{0} ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{n}}{n!}} ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{ℙ(E)}}

= ∫_{ℙ(ℂ^{n})} n · \frac{i∂∂ϕ_{0} ∧ Π_{C_{M}} (\frac{i·F_{A_{L^{∗}}}}{2π} ) ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

= ∫_{ℙ(ℂ^{n})} − \frac{i∂∂ϕ_{0} ∧ trace (\frac{i·F_{A}}{2π} ) ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

along each fiber ℙ (ℂ^{n}) of π̌ : ℙ(E) ⟶ M. Thus

σ̂ (\frac{B_{2}}{Ω_{ℙ(E)}}) = −\frac{[Λ_{M}trace (\frac{i·F_{ω_{M}}}{2π} )] · Δ_{M}ϕ_{0}}{2}

+ \frac{trace (\frac{i·F_{ω_{M}}}{2π} ) ∧ i∂∂ϕ_{0} ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}} + \frac{Δ_{M} ∘ Δ_{M}ϕ_{0}}{8π}

+ τ_{N_{V_{M}}} ϕ_{0} + σ̂ (known terms)

= (\frac{V_{M}}{Ω_{M}} + τ_{N_{V_{M}}} ) ϕ_{0} + σ̂ (known terms) .

Remark. Let ξ_{A} denote the smooth (−1 + m + n)-form on ℙ(E) defined by A as follows:

ξ_{A} ≡ (n + n) · \frac{(\frac{i·F_{A_{L^{∗}}}}{2π} )(n+1)}{(n + 1)!} ∧ \frac{ω_{M}^{(−2+m)}}{(−2 + m)!}

+ π̌^{∗}trace (\frac{i · F_{A}}{2π} ) ∧ \frac{(\frac{i·F_{A_{L^{∗}}}}{2π} )n}{n!} ∧ \frac{ω_{M}^{(−2+m)}}{(−2 + m)!}

+ π̌^{∗}trace (\frac{i · F_{ω_{M}}}{2π} ) ∧ \frac{(\frac{i·F_{A_{L^{∗}}}}{2π} )n}{n!} ∧ \frac{ω_{M}^{(−2+m)}}{(−2 + m)!}.

Let \frac{ξ_{A}}{Ω_{ℙ(E)}} denote the smooth ℝ-valued function on ℙ(E) satisfying

ξ_{A} = \frac{ξ_{A}}{Ω_{ℙ(E)}} · Ω_{ℙ(E)}.

Then, using the constancy of the scalar curvature of (M : ω_{M}) and the Einstein-Hermitian condition satisfied by A, it can be shown that

σ̂ (\frac{B_{2}}{Ω_{ℙ(E)}}) = (\frac{V_{M}}{Ω_{M}} + τ_{N_{V_{M}}} ) ϕ_{0} + σ̂ (−c_{ξ_{A}} + \frac{ξ_{A}}{Ω_{ℙ(E)}})

in which c_{ξ_{A}} ∈ ℝ is the constant satisfying

c_{ξ_{A}} · ∫_{ℙ(E)} Ω_{ℙ(E)} = ∫_{ℙ(E)} ξ_{A}.

Thus in the equality

σ̂ (\frac{B_{2}}{Ω_{ℙ(E)}}) = −\frac{[Λ_{M}trace (\frac{i·F_{ω_{M}}}{2π} )] · Δ_{M}ϕ_{0}}{2}

+ \frac{trace (\frac{i·F_{ω_{M}}}{2π} ) ∧ i∂∂ϕ_{0} ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}} + \frac{Δ_{M} ∘ Δ_{M}ϕ_{0}}{8π}

+ τ_{N_{V_{M}}} ϕ_{0} + σ̂ (known terms)

= (\frac{V_{M}}{Ω_{M}} + τ_{N_{V_{M}}} ) ϕ_{0} + σ̂ (known terms) .

the phrase “known terms” simply means

−c_{ξ_{A}} + \frac{ξ_{A}}{Ω_{ℙ(E)}}

which is, given A, obviously a known smooth ℝ-valued function on M.

Since the elliptic linear partial differential operator (\frac{V_{M}}{Ω_{M}} + τ_{N_{V_{M}}} ) acting on Γ_{o} (M : ℝ) is both symmetric and positive we infer that

ϕ_{0} ∈ Γ_{o} (M : ℝ)

can be uniquely solved from the equation σ̂ (\frac{B_{2}}{Ω_{ℙ(E)}}) = 0.

With ϕ_{0} ∈ Γ_{o} (M : ℝ) being known we have (by the Einstein-Hermi-tian condition of A on E over M)

R_{2} = \frac{Δ_{V} ϕ_{2}}{2} + \frac{Q (σ (ϕ_{1}) : σ (ϕ_{1}))}{2!} + \frac{Δ_{M}ϕ_{0}}{2} · \frac{Δ_{V} σ (ϕ_{1})}{2}

+ \frac{Δ_{M}σ (ϕ_{1})}{2} + \frac{[−Λ_{M}trace (\frac{i·F_{A}}{2π} )]}{n} · \frac{Δ_{V} σ (ϕ_{1})}{2} + \frac{Δ_{M} σ̂ (ϕ_{1})}{2}

+ known terms

and thence

\frac{B_{2}}{Ω_{ℙ(E)}} = \frac{Δ_{V}}{8π} ∘ (−4πn · id + Δ_{V}) ϕ_{2}

+ \frac{Δ_{V}}{8π} [−(2πn · σϕ_{1}) · (2πn · σϕ_{1}) + Q (σϕ_{1} : σϕ_{1})]

+ n · (Δ_{M} + 8π · τ_{N_{W}}) σ (ϕ_{1}) + known terms.

Now by Proposition IV.A we have

σ (\frac{B_{2}}{Ω_{ℙ(E)}}) = n · (Δ_{M} + 8π · τ_{N_{W}}) σ (ϕ_{1}) + σ (known terms) .

Since the elliptic linear partial differential operator (Δ_{M} + 8π · τ_{N_{W}}) acting on Γ (M : W) is both symmetric and positive we infer that σ (ϕ_{1}) ∈ Γ (M : W) can be uniquely solved from the equation σ (\frac{B_{2}}{Ω_{ℙ(E)}}) = 0. With both ϕ_{0} ∈ Γ_{o} (M : ℝ) and σ (ϕ_{1}) ∈ Γ (M : W) being known we have

σ̃ (\frac{B_{2}}{Ω_{ℙ(E)}}) = \frac{Δ_{V}}{8π} ∘ (−4πn · id + Δ_{V}) σ̃ (ϕ_{2}) + σ̃ (known terms)

and thence σ̃ (ϕ_{2}) can be uniquely solved from the equation σ̃ (\frac{B_{2}}{Ω_{ℙ(E)}}) = 0 fiberwisely.

Now given θ ∈ ℕ we will solve the equation

B_{θ+2} = 0 ⟺ \frac{B_{θ+2}}{Ω_{ℙ(E)}} = 0

under the hypothesis that

σ̂ (ϕ_{μ}) ⊕ σ (ϕ_{μ+1}) ⊕ σ̃ (ϕ_{μ+2})

is already known for any integer 0 ≤ μ < θ. In particular σ̂ (ϕ_{0}) ⊕ σ (ϕ_{1}) ⊕ σ̃ (ϕ_{2}) and R_{1} = \frac{Δ_{M} ϕ_{0}}{2} + \frac{Δ_{V} σ(ϕ_{1})}{2} = \frac{Δ_{M} ϕ_{0}}{2} + 2πn · σ (ϕ_{1}) are already known.

Suppose that

\frac{ω(−1+m+n)_{#k}}{k^{m} · ω̌^{(−1+m+n)}} ∼ \frac{_{o}ω(−1+m+n)_{#k}}{k^{m} · ω̌^{(−1+m+n)}} + \frac{R_{1}}{k}

+ ⋯ + \frac{R_{θ+1}}{k^{θ} · k} + \frac{R_{θ+2}}{k^{θ} · k · k} + higher order terms

in which each R_{•} is independent of the parameter k. Then by our induction hypothesis we have R_{μ} being known for any μ ∈ ℕ satisfying μ ≤ θ. It can be checked readily that

R_{θ+1} = \frac{Δ_{M} σ̂ (ϕ_{θ})}{2} + \frac{Δ_{V} σ (ϕ_{θ+1})}{2} + known terms

curvature equations 431

and

R_{θ+2} = \frac{Δ_{V} ϕ_{θ+2}}{2} + \frac{i∂∂ϕ_{θ+1} ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{(−1+n)}}{(−1+n)!}} ∧ \frac{ω_{M}^{(−1+m)}}{(−1+m)!}}{Ω_{ℙ(E)}}

+ \frac{i∂∂σ̂ (ϕ_{θ}) ∧ Π_{C_{M}} (\frac{i·F_{A_{L^{∗}}}}{2π} ) ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}} + Q (σϕ_{1} : σϕ_{θ+1})

+ \frac{Δ_{V} σ (ϕ_{1})}{2} · \frac{Δ_{M} σ̂ (ϕ_{θ})}{2} + \frac{Δ_{M}ϕ_{0}}{2} · \frac{Δ_{V} σ (ϕ_{θ+1})}{2}

+ \frac{i∂∂ϕ_{0} ∧ i∂∂σ̂ (ϕ_{θ}) ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}} + known terms.

Note that σ̃ (R_{θ+1}) is already known. Thus

\frac{B_{θ+2}}{Ω_{ℙ(E)}} = − ((−1 + n) · n · R_{θ+2} + [Λ_{M}trace (\frac{i · F_{ω_{M}}}{2π} )] · R_{θ+1})

+ n · (−2 + n) · \frac{Δ_{V} ϕ_{θ+2}}{2}

+ n · (−1 + n) · \frac{i∂∂ϕ_{θ+1} ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{(−1+n)}}{(−1+n)!}} ∧ \frac{ω_{M}^{(−1+m)}}{(−1+m)!}}{Ω_{ℙ(E)}}

+ n · n · \frac{i∂∂σ̂ (ϕ_{θ}) ∧ Π_{C_{M}} (\frac{i·F_{A_{L^{∗}}}}{2π} ) ∧ \frac{ω^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ n · (−3 + n) · Q (σϕ_{1} : σϕ_{θ+1})

+ n · (−2 + n) · \frac{Δ_{M}ϕ_{0}}{2} · \frac{Δ_{V} σ (ϕ_{θ+1})}{2}

+ n · (−2 + n) · \frac{Δ_{V} σ (ϕ_{1})}{2} · \frac{Δ_{M} σ̂ (ϕ_{θ})}{2}

+ n · (−1 + n) · \frac{i∂∂ϕ_{0} ∧ i∂∂σ̂ (ϕ_{θ}) ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ [Λ_{M}trace (\frac{i · F_{A}}{2π} ) + Λ_{M}trace (\frac{i · F_{ω_{M}}}{2π} )] · \frac{Δ_{V} σ (ϕ_{θ+1})}{2}

+ \frac{[trace (\frac{i·F_{A}}{2π} ) + trace (\frac{i·F_{ω_{M}}}{2π} )] ∧ i∂∂σ̂ (ϕ_{θ}) ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ Q (\frac{σ (R_{1})}{2π} : σ (ϕ_{θ+1})) + \frac{Δ_{V} R_{1}}{4π} · \frac{Δ_{M} σ̂ (ϕ_{θ})}{2}

+ \frac{\frac{i∂∂R_{θ+1}}{2π} ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{(−1+n)}}{(−1+n)!}} ∧ \frac{ω_{M}^{(−1+m)}}{(−1+m)!}}{Ω_{ℙ(E)}} + Q (σ (ϕ_{1}) : \frac{σ (R_{θ+1})}{2π})

+ \frac{Δ_{M}ϕ_{0}}{2} · \frac{Δ_{V} R_{θ+1}}{4π} + \frac{Δ_{V} R_{θ+2}}{4π}

+ \frac{−Δ_{V}}{4π} ([−\frac{Λ_{M}trace (\frac{i·F_{A}}{2π} )}{n} + R_{1}] · R_{θ+1})

+ n · τ_{N_{W}} ∘ σ (ϕ_{θ+1}) + τ_{N_{V_{M}}} ∘ σ̂ (ϕ_{θ}) + known terms.

Substituting the formulae for R_{θ+1} and R_{θ+2} into this expression of \frac{B_{θ+2}}{Ω_{ℙ(E)}} we have

\frac{B_{θ+2}}{Ω_{ℙ(E)}} = − [Λ_{M}trace (\frac{i · F_{ω_{M}}}{2π} )] · \frac{Δ_{M} σ̂ (ϕ_{θ})}{2} + (−n) · \frac{Δ_{V} ϕ_{θ+2}}{2}

+ n · \frac{i∂∂σ̂ (ϕ_{θ}) ∧ Π_{C_{M}} (\frac{i·F_{A_{L^{∗}}}}{2π} ) ∧ \frac{ω^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ [Λ_{M}trace (\frac{i · F_{A}}{2π} )] · \frac{Δ_{V} σ (ϕ_{θ+1})}{2}

+ \frac{[trace (\frac{i·F_{A}}{2π} ) + trace (\frac{i·F_{ω_{M}}}{2π} )] ∧ i∂∂σ̂ (ϕ_{θ}) ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ \frac{Δ_{M} ∘ Δ_{M} σ̂ (ϕ_{θ})}{8π}

+ \frac{n · i∂∂σ (ϕ_{θ+1}) ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{(−1+n)}}{(−1+n)!}} ∧ \frac{ω_{M}^{(−1+m)}}{(−1+m)!}}{Ω_{ℙ(E)}}

+ \frac{Δ_{V} R_{θ+2}}{4π} + \frac{−Δ_{V}}{4π} ([−\frac{Λ_{M}trace (\frac{i·F_{A}}{2π} )}{n} + R_{1}] · R_{θ+1})

+ n · τ_{N_{W}} ∘ σ (ϕ_{θ+1}) + τ_{N_{V_{M}}} ∘ σ̂ (ϕ_{θ}) + known terms.

Einstein-Hermitian condition of A on E over M)

\frac{i∂∂σ (ϕ_{θ+1}) ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{(−1+n)}}{(−1+n)!}} ∧ \frac{ω_{M}^{(−1+m)}}{(−1+m)!}}{Ω_{ℙ(E)}}

= \frac{Δ_{M}σ (ϕ_{θ+1})}{2} + \frac{[−Λ_{M}trace (\frac{i·F_{A}}{2π} )]}{n} · \frac{Δ_{V} σ (ϕ_{θ+1})}{2}

and thence

\frac{B_{θ+2}}{Ω_{ℙ(E)}} = − [Λ_{M}trace (\frac{i · F_{ω_{M}}}{2π} )] · \frac{Δ_{M} σ̂ (ϕ_{θ})}{2} + (−n) · \frac{Δ_{V} ϕ_{θ+2}}{2}

+ n · \frac{i∂∂σ̂ (ϕ_{θ}) ∧ Π_{C_{M}} (\frac{i·F_{A_{L^{∗}}}}{2π} ) ∧ \frac{ω^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ \frac{[trace (\frac{i·F_{A}}{2π} ) + trace (\frac{i·F_{ω_{M}}}{2π} )] ∧ i∂∂σ̂ (ϕ_{θ}) ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ \frac{Δ_{M} ∘ Δ_{M} σ̂ (ϕ_{θ})}{8π} + n · \frac{Δ_{M}σ (ϕ_{θ+1})}{2}

+ \frac{Δ_{V} R_{θ+2}}{4π} + \frac{−Δ_{V}}{4π} ([−\frac{Λ_{M}trace (\frac{i·F_{A}}{2π} )}{n} + R_{1}] · R_{θ+1})

+ n · τ_{N_{W}} ∘ σ (ϕ_{θ+1}) + τ_{N_{V_{M}}} ∘ σ̂ (ϕ_{θ}) + known terms.

Since

∫_{ℙ(ℂ^{n})} ( \frac{n · i∂∂σ̂ (ϕ_{θ}) ∧ Π_{C_{M}} (\frac{i·F_{A_{L^{∗}}}}{2π} ) ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ \frac{i∂∂σ̂ (ϕ_{θ}) ∧ trace (\frac{i·F_{A}}{2π} ) ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}} ) = 0

along each fiber ℙ (ℂ^{n}) of π̌ : ℙ(E) ⟶ M we have:

σ̂ (\frac{B_{θ+2}}{Ω_{ℙ(E)}}) = −\frac{[Λ_{M}trace (\frac{i·F_{ω_{M}}}{2π} )] · Δ_{M} σ̂ (ϕ_{θ})}{2}

+ \frac{trace (\frac{i·F_{ω_{M}}}{2π} ) ∧ i∂∂σ̂ (ϕ_{θ}) ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ \frac{Δ_{M} ∘ Δ_{M} σ̂ (ϕ_{θ})}{8π} + τ_{N_{V_{M}}} ∘ σ̂ (ϕ_{θ}) + σ̂ (known terms)

= (\frac{V_{M}}{Ω_{M}} + τ_{N_{V_{M}}} ) σ̂ (ϕ_{θ}) + σ̂ (known terms) .

Thus σ̂ (ϕ_{θ}) ∈ Γ_{o} (M : ℝ) can be uniquely solved from the equation σ̂ (\frac{B_{θ+2}}{Ω_{ℙ(E)}}) = 0.

With σ̂ (ϕ_{θ}) ∈ Γ_{o} (M : ℝ) being known we have (by the Einstein-Hermitian condition of A on E over M)

R_{θ+2} = \frac{Δ_{V} ϕ_{θ+2}}{2} + \frac{Δ_{M}σ (ϕ_{θ+1})}{2} + \frac{Δ_{M} σ̂ (ϕ_{θ+1})}{2}

+ \frac{[−Λ_{M}trace (\frac{i·F_{A}}{2π} )]}{n} · \frac{Δ_{V} σ (ϕ_{θ+1})}{2}

+ Q (σϕ_{1} : σϕ_{θ+1}) + \frac{Δ_{M}ϕ_{0}}{2} · \frac{Δ_{V} σ (ϕ_{θ+1})}{2} + known terms

and thence

\frac{B_{θ+2}}{Ω_{ℙ(E)}} = \frac{Δ_{V}}{8π} ∘ (−4πn · id + Δ_{V}) ϕ_{θ+2} + n · Δ_{M}σ (ϕ_{θ+1})

+ \frac{Δ_{V}}{4π} [−(2πn · σϕ_{1}) · (2πn · σϕ_{θ+1}) + Q (σϕ_{1} : σϕ_{θ+1})]

+ n · τ_{N_{W}} ∘ σ (ϕ_{θ+1}) + known terms.

Now by Corollary IV.A we have

σ (\frac{B_{θ+2}}{Ω_{ℙ(E)}}) = n · (Δ_{M} + 8π · τ_{N_{W}}) σ (ϕ_{θ+1}) + σ (known terms) .

Thus σ (ϕ_{θ+1}) ∈ Γ (M : W) can be uniquely solved from the equation σ (\frac{B_{θ+2}}{Ω_{ℙ(E)}}) = 0. With both σ̂ (ϕ_{θ}) ∈ Γ_{o} (M : ℝ) and σ (ϕ_{θ+1}) ∈ Γ (M : W) being known we have

σ̃ (\frac{B_{θ+2}}{Ω_{ℙ(E)}}) = \frac{Δ_{V}}{8π} ∘ (−4πn · id + Δ_{V}) ϕ_{θ+2} + σ̃ (known terms)

and thence σ̃ (ϕ_{θ+2}) can be uniquely solved from the equation σ̃ (\frac{B_{θ+2}}{Ω_{ℙ(E)}}) = 0 fiberwise. q.e.d.

Appendix II. Proof of Theorem II.B

Since

0 = [F_{A} (Z : )] ∈ H∂_{A}^{1} (M : Hom (E : E))

there exists, by Theorem II.A, a smooth vector field X̌_{Z} on ℙ(E) preserving both the complex structure of ℙ(E) and the holomorphic projection map π̌ : ℙ(E) ⟶ M such that

π̌_{∗}X̌_{Z} = X_{Z}.

Since X_{Z} preserves the Kaehler form ω_{M} on M we have

LX̌_{Z}ω_{M} = 0.

Let fX̌_{Z} ∈ Γ_{o} (ℙ(E) : ℝ) denote the smooth ℝ-valued function on ℙ(E) satisfying

LX̌_{Z} (\frac{i · F_{A_{L^{∗}}}}{2π} ) = i∂∂fX̌_{Z}.

Since the Futaki invariant associated with the lifting X̌_{Z} (of X_{Z}) and the Kaehler class [\frac{i·F_{A_{L^{∗}}}}{2π} + k · π̌^{∗}ω_{M}] on ℙ(E) vanishes for any sufficiently large k ∈ ℕ we have, for any smooth ℝ-valued function g on ℙ(E),

∫_{ℙ(E)} (fX̌_{Z} + σ̂g + \frac{σg}{k} + \frac{σ̃g}{k · k} )

· S (_{o}ω_{#k} + i∂∂ ∘ σ̂g + \frac{i∂∂ ∘ σg}{k} + \frac{i∂∂ ∘ σ̃g}{k · k} ) = 0

whenever the parameter k ∈ ℕ is large enough. Thus, by the expansion results of Appendix I, we have, by choosing σ̂g = 0 = σg, the following

power series expansion result for ^{\frac{S(_{o}ω_{#k} + \frac{i∂∂∘σ̃g}{k·k} )}{Ω_{ℙ(E)}}} in \frac{1}{k}:

\frac{S (_{o}ω_{#k} + \frac{i∂∂∘σ̃g}{k·k} )}{Ω_{ℙ(E)}} = \frac{\frac{Δ_{V}}{8π} ∘ (−4πn · id + Δ_{V}) σ̃g}{k · k} + \frac{known terms}{k · k}

+ higher order terms for any smooth ℝ-valued function g on ℙ(E) with σ̂g = 0 = σg. Hence we infer from the vanishing of Futaki invariants:

∫_{ℙ(E)} (fX̌_{Z} + \frac{σ̃g}{k · k} ) · S (_{o}ω_{#k} + \frac{i∂∂ ∘ σ̃g}{k · k} ) = 0

for any sufficiently large k ∈ ℕ

that

\frac{Δ_{V}}{8π} ∘ (−4πn · id + Δ_{V}) fX̌_{Z} = 0 ⟺ σ̃ (fX̌_{Z}) = 0.

Similarly, by using the expansion results of Appendix I, we have, by choosing σ̂g = 0 = σ̃g, the following power series expansion result for

\frac{S(_{o}ω_{#k} + \frac{i∂∂∘σg}{k} )}{Ω_{ℙ(E)}} in \frac{1}{k}:

\frac{S (_{o}ω_{#k} + \frac{i∂∂∘σg}{k} )}{Ω_{ℙ(E)}} = \frac{n · Δ_{M}σg}{k · k}

+ \frac{\frac{Δ_{V}}{8π} [−(2πn · σg) · (2πn · σg) + Q (σg : σg)]}{k · k}

+ \frac{known terms}{k · k} + higher order terms

for any smooth section g ∈ Γ (M : W). Thus we infer from Proposition IV.A and the vanishing of Futaki invariants:

∫_{ℙ(E)} (fX̌_{Z} + \frac{σg}{k} ) · S (_{o}ω_{#k} + \frac{i∂∂ ∘ σg}{k} ) = 0

for any sufficiently large k ∈ ℕ

that

Δ_{M} σ (fX̌_{Z}) = 0 ⟺ σ (fX̌_{Z}) ∈ Γ (M : W) is a harmonic section.

Now we consider the isometry condition of X_{Z}:

L_{X_{Z}}ω_{M} = 0 ⟺ LX̌_{Z}ω_{M} = 0.

Since Δ_{M} σ (fX̌_{Z}) = 0 there exists a corresponding element YX̌_{Z} ∈ 𝔤_{E} such that

L_{YX̌_{Z}} (\frac{i · F_{A_{L^{∗}}}}{2π} ) = i∂∂ ∘ σ (fX̌_{Z}) .

curvature equations 437

(Note that YX̌_{Z} ∈ 𝔤_{E} is uniquely determined modulo the compact Lie subalgebra 𝔨_{E} of 𝔤_{E}.) Let X̌_{Z} ≡ (−YX̌_{Z} + X̌_{Z}). Then we have

L_{X̌_{Z}} (\frac{i · F_{A_{L^{∗}}}}{2π} ) = −L_{YX̌_{Z}} (\frac{i · F_{A_{L^{∗}}}}{2π} ) + LX̌_{Z} (\frac{i · F_{A_{L^{∗}}}}{2π} )
= i∂∂ ∘ σ̂ (fX̌_{Z}) .

We claim that Δ_{M} σ̂ (fX̌_{Z}) = 0. To see this we note that

L_{X̌_{Z}}ω_{M} = −L_{YX̌_{Z}} ω_{M} + LX̌_{Z}ω_{M} = 0 + 0 = 0

and thence

\frac{Δ_{M} σ̂ (fX̌_{Z})}{2} · Ω_{ℙ(E)}

= \frac{(\frac{i·F_{A_{L^{∗}}}}{2π} )(−1+n)}{(−1 + n)!} ∧ i∂∂ ∘ σ̂ (fX̌_{Z}) ∧ \frac{ω_{M}^{(−1+m)}}{(−1 + m)!}

= \frac{(\frac{i·F_{A_{L^{∗}}}}{2π} )(−1+n)}{(−1 + n)!} ∧ L_{X̌_{Z}} (\frac{i · F_{A_{L^{∗}}}}{2π} ) ∧ \frac{ω_{M}^{(−1+m)}}{(−1 + m)!}

= L_{X̌_{Z}} [ \frac{(\frac{i·F_{A_{L^{∗}}}}{2π} )n}{n!} ∧ \frac{ω_{M}^{(−1+m)}}{(−1 + m)!}] .

However, by the Einstein-Hermitian condition of A on E over M, we have

\frac{(\frac{i·F_{A_{L^{∗}}}}{2π} )n}{n!} ∧ \frac{ω_{M}^{(−1+m)}}{(−1 + m)!}

= \frac{(\frac{i·F_{A_{L^{∗}}}}{2π} )(−1+n)}{(−1 + n)!} ∧ Π_{C_{M}} (\frac{i · F_{A_{L^{∗}}}}{2π} ) ∧ \frac{ω_{M}^{(−1+m)}}{(−1 + m)!}

= − \frac{[Λ_{M}trace (\frac{i·F_{A}}{2π} )]}{n} · \frac{(\frac{i·F_{A_{L^{∗}}}}{2π} )(−1+n)}{(−1 + n)!} ∧ \frac{ω_{M}^{m}}{m!} .

Thus

\frac{Δ_{M} σ̂ (fX̌_{Z})}{2} · Ω_{ℙ(E)}

= − \frac{[Λ_{M}trace (\frac{i·F_{A}}{2π} )]}{n} · L_{X̌_{Z}} [\frac{(\frac{i·F_{A_{L^{∗}}}}{2π} )(−1+n)}{(−1 + n)!} ∧ \frac{ω_{M}^{m}}{m!} ] = 0

because

L_{X̌_{Z}} [\frac{(\frac{i·F_{A_{L^{∗}}}}{2π} )(−1+n)}{(−1 + n)!} ∧ \frac{ω_{M}^{m}}{m!} ]

= \frac{(\frac{i·F_{A_{L^{∗}}}}{2π} )(−2+n)}{(−2 + n)!} ∧ L_{X̌_{Z}} (\frac{i · F_{A_{L^{∗}}}}{2π} ) ∧ \frac{ω_{M}^{m}}{m!}

= \frac{(\frac{i·F_{A_{L^{∗}}}}{2π} )(−2+n)}{(−2 + n)!} ∧ i∂∂ ∘ σ̂ (fX̌_{Z}) ∧ \frac{ω_{M}^{m}}{m!}

naturally vanishes. Hence Δ_{M} σ̂ (fX̌_{Z}) = 0, as claimed, and so σ̂ (fX̌_{Z})

= 0. That is : \frac{i·F_{A_{L^{∗}}}}{2π} is invariant under the action of X̌_{Z} = −YX̌_{Z} +X̌_{Z}. q.e.d.

Appendix III. Large k behavior of L_{N}

In this section we will investigate, for each large N ∈ ℕ, the behavior of the 4^{th} order elliptic linear partial differential operator L_{N} as the parameter k goes to infinity. Here the 4^{th} order (elliptic) linear partial differential operator L_{N}, depending on the parameter k, acting on ψ ∈ Γ_{o} (ℙ(E) : ℝ) is defined as follows:

L_{N} (ψ) = −č_{k} · _{A}L_{N} (ψ) + \frac{\frac{i∂∂}{2π} (_{\begin{matrix}A & L & N & (ψ) \\ R & (_{N} ω_{#k} ) \end{matrix}}) ∧ ^{\frac{_{N} ω_{#k}^{(−2+m+n)}}{(−2+m+n)!}}}{k^{m} · Ω_{ℙ(E)}}

+ \frac{^{\frac{i·∂∂ log R(_{N} ω_{#k} )}{2π}} ∧ i∂∂ψ ∧ ^{\frac{_{N} ω_{#k}^{(−3+m+n)}}{(−3+m+n)!}}}{k^{m} · Ω_{ℙ(E)}} + _{B}L_{N} (ψ)

+ (\frac{n · τ_{N_{W}} ∘ σ (ψ)}{k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ (ψ)}{k · k}) .

in which the 2^{nd} order linear partial differential operators _{A}L_{N} and BL_{N} (both without the 0^{th} order parts), acting on ψ ∈ Γ_{o} (ℙ(E) : ℝ), are respectively defined as follows:

AL_{N} (ψ) ≡ \frac{i∂∂ψ ∧ ^{\frac{_{N} ω_{#k}^{(−2+m+n)}}{(−2+m+n)!}}}{k^{m} · Ω_{ℙ(E)}}

and

BL_{N} (ψ)

≡ \frac{i∂∂ψ ∧ [n · \frac{i·F_{A_{L^{∗}}}}{2π} +π̌^{∗}trace (\frac{i·F_{A}}{2π} )+π̌^{∗}trace (\frac{i·F_{ω_{M}}}{2π} )]∧ ^{\frac{_{N} ω_{#k}^{(−3+m+n)}}{(−3+m+n)!}}}{k^{m} · Ω_{ℙ(E)}} .

Note that

R (_{N} ω_{#k}) = \frac{_{o}ω(−1+m+n)_{#k}}{k^{m} · ω̌^{(−1+m+n)}} + \frac{R_{1}}{k} + \frac{R_{2}}{k · k} + higher order terms
= \frac{Ω_{ℙ(E)}}{Ω_{ℙ(E)}} + \frac{R_{1}}{k} + \frac{R_{2}}{k · k} + higher order terms

in which

R_{1} = \frac{[−Λ_{M}trace (\frac{i·F_{A}}{2π} )]}{n} + \frac{Δ_{M}ϕ_{0}}{2} + \frac{Δ_{V} σ (ϕ_{1})}{2}

and

R_{2} = \frac{^{\frac{Π_{C_{M}} ( \frac{i·F_{A_{L^{∗}}}}{2π} )∧Π_{C_{M}} ( \frac{i·F_{A_{L^{∗}}}}{2π} )}{2!}} ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}} + R_{2}

are smooth ℝ-valued functions, independent of the parameter k, on ℙ(E). (More expansion results for R (_{N} ω_{#k}) can be found in Appendix I.) We define a symmetric operator Q_{V} (• : •) as follows:

Q_{V} (f : g) ≡ \frac{i∂∂f ∧ i∂∂g ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{(−3+n)}}{(−3+n)!}} ∧ \frac{ω_{M}^{m}}{m!}}{Ω_{ℙ(E)}}

for any pair (f : g) of smooth ℝ-valued functions on ℙ(E). In particular we have

Q_{V} (f : g) = Q (f : g)

when both f and g are smooth sections of W over M. It should be noted that Q_{V} (f : g) only depends on the fiber-directional differentiation of f and g:

Q_{V} (f : g) = Q_{V} (σf + σ̃f : σg + σ̃g) .

Let _{E}L_{N} denote the 2^{nd} order linear partial differential operator (without the 0^{th} order part) acting on ψ ∈ Γ_{o} (ℙ(E) : ℝ) defined as follows:

EL_{N} (ψ) ≡ \frac{^{\frac{i·∂∂ log R(_{N} ω_{#k} )}{2π}} ∧ i∂∂ψ ∧ ^{\frac{_{N} ω_{#k}^{(−3+m+n)}}{(−3+m+n)!}}}{k^{m} · Ω_{ℙ(E)}} .

Then _{E}L_{N} depends on \frac{1}{k} in the polynomial manner and we have

EL_{N} (ψ) = \frac{Q_{V} (\frac{Δ_{V} σϕ_{1}}{4π} : ψ)}{k} + \frac{_{E}L_{N:2} (ψ)}{k · k}

+ those terms of _{E}L_{N} (ψ) carrying higher order powers of \frac{1}{k}

= \frac{n · Q_{V} (σϕ_{1} : ψ)}{k} + \frac{_{E}L_{N:2} (ψ)}{k · k}

+ those terms of _{E}L_{N} (ψ) carrying higher order powers of \frac{1}{k} in which

EL_{N:2} (ψ)

= \frac{\frac{i∂∂R_{1}}{2π} ∧ i∂∂ψ ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{(−2+n)}}{(−2+n)!}} ∧ \frac{ω_{M}^{(−1+m)}}{(−1+m)!}}{Ω_{ℙ(E)}}

+ \frac{\frac{i∂∂R_{1}}{2π} ∧ i∂∂ψ ∧ i∂∂ϕ_{1} ∧ ^{\frac{( \frac{i·F_{A_{L^{∗}}}}{2π} )^{(−4+n)}}{(−4+n)!}} ∧ \frac{ω_{M}^{m}}{m!}}{Ω_{ℙ(E)}}

+ \frac{Δ_{M}ϕ_{0}}{2} · Q_{V} (\frac{Δ_{V} σϕ_{1}}{4π} : ψ) + \frac{Q_{V}}{2π} (−\frac{R_{1} · R_{1}}{2} + R_{2} : ψ) .

Thus

L_{N} (ψ) = −č_{k} · _{A}L_{N} (ψ) + \frac{\frac{i∂∂_{A}L_{N} (ψ)}{2π} ∧ ^{\frac{_{N} ω_{#k}^{(−2+m+n)}}{(−2+m+n)!}}}{k^{m} · Ω_{ℙ(E)}} + _{B}L_{N} (ψ)

+ \frac{n · Q_{V} (σϕ_{1} : ψ)}{k} + \frac{\frac{i∂∂}{2π} [−R_{1} · _{A}L_{N} (ψ)] ∧ ^{\frac{_{N} ω_{#k}^{(−2+m+n)}}{(−2+m+n)!}}}{k · k^{m} · Ω_{ℙ(E)}}

+ \frac{n · τ_{N_{W}} ∘ σ (ψ)}{k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ (ψ)}{k · k}

+ \frac{\frac{i∂∂}{2π} [(−R_{2} + R_{1} · R_{1}) · _{A}L_{N} (ψ)] ∧ ^{\frac{_{N} ω_{#k}^{(−2+m+n)}}{(−2+m+n)!}}}{k · k · k^{m} · Ω_{ℙ(E)}}

+ \frac{_{E}L_{N:2} (ψ)}{k · k}

+ those terms of L_{N} (ψ) carrying

higher order powers of \frac{1}{k} intrinsically

= −č_{k} · _{A}L_{N} (ψ) + \frac{_{A}L_{N}}{2π} ∘ _{A}L_{N} (ψ) + _{B}L_{N} (ψ)

+ \frac{n · Q_{V} (σϕ_{1} : ψ)}{k} + \frac{\frac{_{A}L_{N}}{2π} [−R_{1} · _{A}L_{N} (ψ)]}{k}

+ \frac{n · τ_{N_{W}} ∘ σ (ψ)}{k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ (ψ)}{k · k}

+ \frac{\frac{_{A}L_{N}}{2π} [(−R_{2} + R_{1} · R_{1}) · _{A}L_{N} (ψ)]}{k · k} + \frac{_{E}L_{N:2} (ψ)}{k · k}

+ those terms of L_{N} (ψ) carrying higher order powers of \frac{1}{k} intrinsically.

Now we note that the 2^{nd} order linear partial differential operators AL_{N} and _{B}L_{N} both depend on \frac{1}{k} in the polynomial manner. Actually we have

AL_{N}• = \frac{Δ_{V} •}{2} + \frac{\frac{Δ_{M} •}{2} + ^{\frac{[−Λ_{M} trace( \frac{i·F_{A}}{2π} )]}{n}} · \frac{Δ_{V} •}{2}}{k}

+ \frac{Q_{V} (σϕ_{1} : •)}{k} + \frac{\frac{Δ_{M} ϕ_{0}}{2} · \frac{Δ_{V} •}{2}}{k} + \frac{_{A}L_{N:2}•}{k · k}

+ those terms of _{A}L_{N} • carrying higher order powers of \frac{1}{k}

and

BL_{N} (ψ) = n · (−2 + n) · \frac{Δ_{V} ψ}{2}

+ n · (−1 + n) · \frac{\frac{Δ_{M} ψ}{2} + ^{\frac{[−Λ_{M} trace( \frac{i·F_{A}}{2π} )]}{n}} · \frac{Δ_{V} ψ}{2}}{k}

+ \frac{n · (−3 + n) · Q_{V} (σϕ_{1} : ψ)}{k} + \frac{n · (−2 + n) · \frac{Δ_{M} ϕ_{0}}{2} · \frac{Δ_{V} ψ}{2}}{k}

+ \frac{[Λ_{M}trace (\frac{i·F_{A}}{2π} ) + Λ_{M}trace (\frac{i·F_{ω_{M}}}{2π} )] · \frac{Δ_{V} ψ}{2}}{k}

+ \frac{_{B}L_{N:2} (ψ)}{k · k} + those terms of _{B}L_{N} (ψ) carrying

higher order powers of \frac{1}{k}

in which _{A}L_{N:2} and _{B}L_{N:2} are 2^{nd} order linear partial differential operators (both without the 0^{th} order parts), independent of the parameter k, acting on Γ_{o} (ℙ(E) : ℝ). Let _{C-A}L denote the 2^{nd} order linear partial differential operator acting on Γ_{o} (ℙ(E) : ℝ) defined as follows:

C-AL• ≡ \frac{Δ_{M}•}{2}+\frac{[−Λ_{M}trace (\frac{i·F_{A}}{2π} )]}{n}·\frac{Δ_{V} •}{2}+Q_{V} (σϕ_{1} : •)+\frac{Δ_{M}ϕ_{0}}{2} ·\frac{Δ_{V} •}{2}

so that

AL_{N}• = \frac{Δ_{V} •}{2} + \frac{_{C-A}L•}{k} + \frac{_{A}L_{N:2}•}{k · k}

+ those terms of _{A}L_{N} • carrying higher order powers of \frac{1}{k} .

curvature equations 443

Then (by substituting

• = [−R_{1} · _{A}L_{N} (ψ)] and • = [(−R_{2} + R_{1} · R_{1}) · _{A}L_{N} (ψ)] respectively into the last formula for _{A}L_{N}•) we have

L_{N} (ψ) = −č_{k} · _{A}L_{N} (ψ) + \frac{_{A}L_{N}}{2π} ∘ _{A}L_{N} (ψ) + _{B}L_{N} (ψ)

+ \frac{n · τ_{N_{W}} ∘ σ (ψ)}{k} + \frac{n · Q_{V} (σϕ_{1} : ψ)}{k}

+ \frac{−\frac{Δ_{V}}{4π} [R_{1} · _{A}L_{N} (ψ)]}{k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ (ψ)}{k · k}

+ \frac{− \frac{_{C-A}L}{2π} [R_{1} · _{A}L_{N} (ψ)]}{k · k} + \frac{\frac{Δ_{V} [(−R_{2}+R_{1}·R_{1})·_{A}L_{N} (ψ)]}{4π}}{k · k}

+ \frac{_{E}L_{N:2} (ψ)}{k · k} + those terms of L_{N} (ψ) carrying

higher order powers of \frac{1}{k} intrinsically

and thence (by substituting • = _{A}L_{N} (ψ) into the last formula for AL_{N}•)

L_{N} (ψ) = −č_{k} · _{A}L_{N} (ψ) + \frac{Δ_{V} [_{A}L_{N} (ψ)]}{4π} + _{B}L_{N} (ψ) + \frac{n · τ_{N_{W}} ∘ σ (ψ)}{k}

+ \frac{n · Q_{V} (σϕ_{1} : ψ)}{k} + \frac{\frac{_{C-A}L}{2π} ∘ _{A}L_{N} (ψ)}{k}

+ \frac{−\frac{Δ_{V}}{4π} [R_{1} · \frac{Δ_{V} ψ}{2} ]}{k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ (ψ)}{k · k}

+ \frac{\frac{_{A}L_{N:2}}{2π} ∘ _{A}L_{N} (ψ)}{k · k} + \frac{−\frac{Δ_{V}}{4π} [R_{1} · _{C-A}L_{N} (ψ)]}{k · k}

+ \frac{− \frac{_{C-A}L}{2π} [R_{1} · \frac{Δ_{V} ψ}{2} ]}{k · k} + \frac{\frac{Δ_{V} [(−R_{2}+R_{1}·R_{1})·Δ_{V} ψ]}{8π}}{k · k} + \frac{_{E}L_{N:2} (ψ)}{k · k}

+ those terms of L_{N} (ψ) carrying

higher order powers of \frac{1}{k} intrinsically

= −č_{k} · _{A}L_{N} (ψ) + \frac{Δ_{V} ∘ Δ_{V} ψ}{8π} + _{B}L_{N} (ψ) + \frac{n · τ_{N_{W}} ∘ σψ}{k}

+ \frac{n · Q_{V} (σϕ_{1} : ψ)}{k} + \frac{\frac{Δ_{V}}{4π} ∘ _{C-A}L (ψ)}{k} + \frac{_{C-A}L ∘ \frac{Δ_{V}}{4π} (ψ)}{k}

+ \frac{−\frac{Δ_{V}}{4π} [R_{1} · \frac{Δ_{V} ψ}{2} ]}{k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ψ}{k · k} + \frac{\frac{Δ_{V}}{4π} ∘ _{A}L_{N:2} (ψ)}{k · k}

+ \frac{\frac{_{C-A}L}{2π} ∘ _{C-A}L (ψ)}{k · k} + \frac{\frac{_{A}L_{N:2}}{4π} ∘ Δ_{V} (ψ)}{k · k} + \frac{\frac{−Δ_{V}}{4π} [R_{1} · _{C-A}L (ψ)]}{k · k}

+ \frac{− \frac{_{C-A}L}{2π} [R_{1} · \frac{Δ_{V} ψ}{2} ]}{k · k} + \frac{\frac{Δ_{V} [(−R_{2}+R_{1}·R_{1})·Δ_{V} ψ]}{8π}}{k · k} + \frac{_{E}L_{N:2} (ψ)}{k · k}

+ those terms of L_{N} (ψ) carrying higher order powers of \frac{1}{k} intrinsically.

Hence by using the detailed formulae for _{A}L_{N} (ψ) and _{B}L_{N} (ψ) we have

L_{N} (ψ) = \frac{Δ_{V}}{8π} ∘ (−4πn · id + Δ_{V}) ψ + \frac{\frac{Δ_{M} ∘Δ_{V} ψ}{8π} + \frac{Δ_{V} ∘Δ_{M} ψ}{8π}}{k}

+ \frac{n · τ_{N_{W}} ∘ σψ}{k} + \frac{_{U}L_{N:1} (ψ)}{k} + \frac{L_{N:2} (ψ)}{k · k}

+ those terms of L_{N} (ψ) carrying

higher order powers of \frac{1}{k} intrinsically

in which _{U}L_{N:1} and L_{N:2} are the 4^{th} order linear partial differential operators (both without the 0^{th} order parts), independent of the parameter k, acting on Γ_{o} (ℙ(E) : ℝ) defined respectively as follows:

\frac{_{U}L_{N:1} (ψ)}{k} = \frac{\frac{Δ_{V}}{4π} (−\frac{Δ_{V} σϕ_{1}}{2} · \frac{Δ_{V} ψ}{2} + Q_{V} (σϕ_{1} : ψ))}{k}

+ \frac{Q_{V} (σϕ_{1} : \frac{(−4πn·id+Δ_{V} )ψ}{4π} )}{k}

+ \frac{^{\frac{[−Λ_{M} trace( \frac{i·F_{A}}{2π} )]}{n}} · \frac{Δ_{V}}{8π} ∘ (−4πn · id + Δ_{V}) ψ}{k}

+ \frac{\frac{Δ_{M} ϕ_{0}}{2} · \frac{(−4πn·id+Δ_{V} )}{4π} ∘ \frac{Δ_{V} ψ}{2}}{k}

and

\frac{L_{N:2} (ψ)}{k · k} = \frac{τ_{N_{V_{M}}} ∘ σ̂ψ}{k · k} + \frac{\frac{Δ_{V}}{4π} ∘ _{A}L_{N:2} (ψ)}{k · k} + \frac{\frac{_{C-A}L}{2π} ∘ _{C-A}L (ψ)}{k · k}

+ \frac{\frac{_{A}L_{N:2}}{4π} ∘ Δ_{V} ψ}{k · k} + \frac{\frac{−Δ_{V}}{4π} [R_{1} · _{C-A}L (ψ)]}{k · k}

+ \frac{− \frac{_{C-A}L}{2π} [R_{1} · \frac{Δ_{V} ψ}{2} ]}{k · k} + \frac{\frac{Δ_{V} [(−R_{2}+R_{1}·R_{1})·Δ_{V} ψ]}{8π}}{k · k}

+ \frac{_{E}L_{N:2} (ψ)}{k · k} + \frac{−(−1 + n) · n · _{A}L_{N:2} (ψ)}{k · k} + \frac{_{B}L_{N:2} (ψ)}{k · k}

+ \frac{− [Λ_{M}trace (\frac{i·F_{ω_{M}}}{2π} )] · _{C-A}L (ψ)}{k · k} + \frac{−č_{k:2} · \frac{Δ_{V} ψ}{2}}{k · k} .

We are particularly interested in those parts of L_{N:2} acting nontrivially on Γ_{o} (M : ℝ). Note that

C-AL• = \frac{Δ_{M}•}{2} + _{C-A}L^{⋆}•

in which the 2^{nd} order linear partial differential operator

C-AL^{⋆}• = \frac{[−Λ_{M}trace (\frac{i·F_{A}}{2π} )]}{n} · \frac{Δ_{V} •}{2} + Q_{V} (σϕ_{1} : •) + \frac{Δ_{M}ϕ_{0}}{2} · \frac{Δ_{V} •}{2}

has trivial action on Γ (M : ℝ):

C-AL^{⋆} (f) = 0

for any f ∈ Γ (M : ℝ). Similarly for the 2^{nd} order linear partial differential operators _{A}L_{N:2} and _{B}L_{N:2} we have

AL_{N:2}• = \frac{i · (∂∂)_{M} • ∧Π_{C_{M}} (\frac{i·F_{A_{L^{∗}}}}{2π} ) ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}} + \frac{Δ_{V} σϕ_{1}}{2} · \frac{Δ_{M}•}{2}

+ \frac{i · (∂∂)_{M} • ∧i∂∂ϕ_{0} ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}} + _{A}L_{N:2}^{⋆} •

and

BL_{N:2}• = n · n · \frac{i · (∂∂)_{M} • ∧Π_{C_{M}} (\frac{i·F_{A_{L^{∗}}}}{2π} ) ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ n · (−2 + n) · \frac{Δ_{V} σϕ_{1}}{2} · \frac{Δ_{M}•}{2}

+ n · (−1 + n) · \frac{i · (∂∂)_{M} • ∧i∂∂ϕ_{0} ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ \frac{i (∂∂)_{M} • ∧ [trace (\frac{i·F_{A}}{2π} ) + trace (\frac{i·F_{ω_{M}}}{2π} )] ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ _{B}L_{N:2}^{⋆} •

in which both _{A}L_{N:2}^{⋆} and _{B}L_{N:2}^{⋆} act trivially on Γ (M : ℝ):

AL_{N:2}^{⋆} (f) = 0 = _{B}L_{N:2}^{⋆} (f)

for any f ∈ Γ (M : ℝ). Besides we note that

EL_{N:2}• = \frac{Δ_{V} R_{1}}{4π} · \frac{Δ_{M}•}{2} + _{E}L_{N:2}^{⋆} •

= n · \frac{Δ_{V} σϕ_{1}}{2} · \frac{Δ_{M}•}{2} + _{E}L_{N:2}^{⋆} •

in which _{E}L_{N:2}^{⋆} acts trivially on Γ (M : ℝ). Thus

L_{N:2} (ψ) = \frac{Δ_{M} ∘ Δ_{M}ψ}{8π} + \frac{i (∂∂)_{M} ψ ∧ trace (\frac{i·F_{ω_{M}}}{2π} ) ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ [−Λ_{M}trace (\frac{i · F_{ω_{M}}}{2π} )] · \frac{Δ_{M}ψ}{2} + τ_{N_{V_{M}}} ∘ σ̂ψ + _{U}L_{N:2} (ψ)

in which

UL_{N:2} (ψ)

= \frac{Δ_{V}}{4π} ∘ _{A}L_{N:2} (ψ) + \frac{Δ_{M} ∘ _{C-A}L^{⋆} (ψ)}{4π} + \frac{_{C-A}L^{⋆} ∘ Δ_{M}ψ}{4π}

+ \frac{_{C-A}L^{⋆} ∘ _{C-A}L^{⋆} (ψ)}{2π} + \frac{_{A}L_{N:2}}{4π} ∘ Δ_{V} ψ + \frac{−Δ_{V}}{4π} [R_{1} · _{C-A}L (ψ)]

+ \frac{−_{C-A}L}{2π} [R_{1} · \frac{Δ_{V} ψ}{2} ] + \frac{Δ_{V} [(−R_{2} + R_{1} · R_{1}) · Δ_{V} ψ]}{8π}

+ \frac{i (∂∂)_{M} ψ ∧ [n · Π_{C_{M}} (\frac{i·F_{A_{L^{∗}}}}{2π} ) + trace (\frac{i·F_{A}}{2π} )] ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

+ [−(−1 + n) · n · _{A}L_{N:2}^{⋆} (ψ) + _{E}L_{N:2}^{⋆} (ψ) + _{B}L_{N:2}^{⋆} (ψ)]

+ [−Λ_{M}trace (\frac{i · F_{ω_{M}}}{2π} )] · _{C-A}L^{⋆} (ψ) + (−č_{k:2} · \frac{Δ_{V} ψ}{2} ) .

Let P_{#k} denote the 4^{th} order elliptic linear partial differential operator, depending on the parameter k, acting on ψ ∈ Γ_{o} (ℙ(E) : ℝ) defined as follows:

P_{#k} (ψ)

= \frac{Δ_{V}}{8π} ∘ (−4πn · id + Δ_{V}) ψ + \frac{\frac{Δ_{M} ∘Δ_{V} ψ}{8π} + \frac{Δ_{V} ∘Δ_{M} ψ}{8π}}{k} + \frac{n · τ_{N_{W}} ∘ σψ}{k}

+ \frac{\frac{Δ_{M} ∘Δ_{M} ψ}{8π} + [−Λ_{M}trace (\frac{i·F_{ω_{M}}}{2π} )] · \frac{Δ_{M} ψ}{2}}{k · k}

+ \frac{\frac{i(∂∂)_{M} ψ∧[trace( \frac{i·Fω_{M}}{2π} )]∧^{\frac{ω{(−2+m) \atop M}}{(−2+m)!}}}{Ω_{M}}}{k · k} + \frac{τ_{N_{V_{M}}} ∘ σ̂ψ}{k · k} .

Then we have

L_{N} (ψ) = P_{#k} (ψ) + \frac{_{U}L_{N:1} (ψ)}{k} + \frac{_{U}L_{N:2} (ψ)}{k · k}

+ those terms of L_{N} (ψ) carrying

higher order powers of \frac{1}{k} intrinsically.

We will show that L_{N} is dominated by P_{#k} as the parameter k is sufficiently large. Before doing so let us look at P_{#k} more closely. Note that, for any ψ ∈ Γ_{o} (ℙ(E) : ℝ), we have

P_{#k} (ψ) = P_{#k} (σ̂ψ) + P_{#k} (σψ) + P_{#k} (σ̃ψ)

448 ying-ji hong

and thence

P_{#k} (ψ)

= \frac{Δ_{V}}{8π} ∘ (−4πn · id + Δ_{V}) σ̃ψ + \frac{\frac{Δ_{M} ∘Δ_{V} σ̃ψ}{8π} + \frac{Δ_{V} ∘Δ_{M} σ̃ψ}{8π}}{k} + \frac{\frac{Δ_{M} ∘Δ_{M} σ̃ψ}{8π}}{k · k}

+ \frac{n · Δ_{M}σψ}{k} + \frac{n · τ_{N_{W}} ∘ σψ}{k} + \frac{\frac{Δ_{M} ∘Δ_{M} σψ}{8π}}{k · k} + \frac{(\frac{V_{M}}{Ω_{M}} + τ_{N_{V_{M}}} ) σ̂ψ}{k · k}

+ \frac{[−Λ_{M}trace (\frac{i·F_{ω_{M}}}{2π} )] · \frac{Δ_{M} σ̃ψ}{2} + ^{\frac{i(∂∂)_{M} σ̃ψ∧[trace( \frac{i·Fω_{M}}{2π} )]∧^{\frac{ω{(−2+m) \atop M}}{(−2+m)!}}}{Ω_{M}}}}{k · k}

+ \frac{[−Λ_{M}trace (\frac{i·F_{ω_{M}}}{2π} )] · \frac{Δ_{M} σψ}{2} + ^{\frac{i(∂∂)_{M} σψ∧[trace( \frac{i·Fω_{M}}{2π} )]∧^{\frac{ω{(−2+m) \atop M}}{(−2+m)!}}}{Ω_{M}}}}{k · k}

in which V_{M} is the infinitesimal deformation operator for the constant scalar curvature equation on (M : ω_{M}). Now we set

oP_{#k} (ψ) ≡ \frac{Δ_{V}}{8π} ∘ (−4πn · id + Δ_{V}) σ̃ψ + \frac{\frac{Δ_{M} ∘Δ_{V} σ̃ψ}{8π} + \frac{Δ_{V} ∘Δ_{M} σ̃ψ}{8π}}{k}

+ \frac{\frac{Δ_{M} ∘Δ_{M} σ̃ψ}{8π}}{k · k} + \frac{n · Δ_{M}σψ}{k} + \frac{n · τ_{N_{W}} ∘ σψ}{k}

+ \frac{\frac{Δ_{M} ∘Δ_{M} σψ}{8π}}{k · k} + \frac{(\frac{V_{M}}{Ω_{M}} + τ_{N_{V_{M}}} ) σ̂ψ}{k · k}

so that

−_{o}P_{#k} (ψ) + P_{#k} (ψ) = \frac{\frac{(i(∂∂)_{M} σ̃ψ+i(∂∂)_{M} σψ)∧[trace( \frac{i·Fω_{M}}{2π} )]∧^{\frac{ω{(−2+m) \atop M}}{(−2+m)!}}}{Ω_{M}}}{k · k}

+ \frac{[−Λ_{M}trace (\frac{i·F_{ω_{M}}}{2π} )] · \frac{Δ_{M} σ̃ψ+Δ_{M} σψ}{2}}{k · k} .

Let Č > 0 denote a sufficiently large constant independent of k (and N ∈ ℕ). Then it is obvious that

∥−_{o}P_{#k} (ψ) + P_{#k} (ψ)∥_{L^{2}(ℙ(E):ω̌)} ≤ Č · \frac{∥(Δ_{M} + Δ_{V}) σ̃ψ∥_{L^{2}(ℙ(E):ω̌)}}{k · k}

+ Č · \frac{∥(Δ_{V} + Δ_{M}) σψ∥_{L^{2}(ℙ(E):ω̌)}}{k · k} .

Let ⟨ : ⟩ denote the inner product on L^{2} (ℙ(E) : ω̌) defined by the Hermitian form (metric) ω̌ on ℙ(E):

⟨f : g⟩ = ∫_{ℙ(E)} f · g · Ω_{ℙ(E)}

∀(f : g) ∈ L^{2} (ℙ(E) : ω̌) × L^{2} (ℙ(E) : ω̌). Note that the decomposition

Γ_{o} (ℙ(E) : ℝ) = σ̂Γ_{o} (ℙ(E) : ℝ) ⊕ σΓ_{o} (ℙ(E) : ℝ) ⊕ σ̃Γ_{o} (ℙ(E) : ℝ)

of Γ_{o} (ℙ(E) : ℝ) is orthogonal with respect to this inner product ⟨: ⟩ on L^{2} (ℙ(E) : ω̌). Moreover, by Proposition IV.C, this orthogonal decomposition of Γ_{o} (ℙ(E) : ℝ) is preserved by Δ_{M} (and Δ_{V}). Now, by using the Stokes Theorem, it can be shown readily that

Č · ∥σ̃ ∘ _{o}P_{#k} (ψ)∥_{L^{2}(ℙ(E):ω̌)}

≥ ∥(Δ_{V} + \frac{Δ_{M}}{k} ) ∘ (Δ_{V} + \frac{Δ_{M}}{k} ) σ̃ψ∥_{L^{2}(ℙ(E):ω̌)}

and

Č · ∥σ ∘ _{o}P_{#k} (ψ)∥_{L^{2}(ℙ(E):ω̌)}

≥ ∥\frac{(Δ_{M} + id)}{k} ∘ (Δ_{V} + \frac{Δ_{M}}{k} ) σψ∥_{L^{2}(ℙ(E):ω̌)} .

Remark. Note that, on each fiber ℙ (ℂ^{n}) of π̌ : ℙ(E) ⟶ M, the L^{2} norm of Δ_{V} σ̃ψ is always bounded by some universal multiple of the L^{2} norm of (−4πn · id + Δ_{V}) σ̃ψ. Besides we always have Δ_{V} σψ = 4πn·σψ.

Similarly, by standard results of Partial Differential Equations or the Stokes Theorem, we have

Č · ∥σ̂ ∘ _{o}P_{#k} (ψ)∥_{L^{2}(ℙ(E):ω̌)} ≥ ∥\frac{(Δ_{M} + id) ∘ (Δ_{M} + id) σ̂ψ}{k · k} ∥_{L^{2}(ℙ(E):ω̌)} .

Thus, by the Schwarz inequality, there exists a constant C > 0, independent of the parameter k, such that, when k is sufficiently large,

C · ∥P_{#k} (ψ)∥_{L^{2}(ℙ(E):ω̌)}≥ ∥(Δ_{V} + \frac{Δ_{M}}{k} )∘(Δ_{V} + \frac{Δ_{M}}{k} ) σ̃ψ∥_{L^{2}(ℙ(E):ω̌)}

+ ∥\frac{(Δ_{M} + id)}{k} ∘ (Δ_{V} + \frac{Δ_{M}}{k} ) σψ∥_{L^{2}(ℙ(E):ω̌)}

+ ∥\frac{(Δ_{M} + id) ∘ (Δ_{M} + id) σ̂ψ}{k · k} ∥_{L^{2}(ℙ(E):ω̌)}

is true for any ψ ∈ Γ_{o} (ℙ(E) : ℝ). We can now use similar ideas to derive estimates for ∥L_{N} (ψ)∥_{L^{2}(ℙ(E):ω̌)}. Actually, by Corollary IV.A, we have

\frac{_{U}L_{N:1} (ψ)}{k} = \frac{_{U}L_{N:1} (σψ)}{k} + \frac{_{U}L_{N:1} (σ̃ψ)}{k}

= \frac{\frac{Δ_{V}}{4π} (−\frac{Δ_{V} σϕ_{1}}{2} · \frac{Δ_{V} σψ}{2} + Q_{V} (σϕ_{1} : σψ))}{k} + \frac{_{U}L_{N:1} (σ̃ψ)}{k}

= \frac{σ̃ ∘ _{U}L_{N:1} (σψ)}{k} + \frac{_{U}L_{N:1} (σ̃ψ)}{k}

in which

∥\frac{σ̃ ∘ _{U}L_{N:1} (σψ)}{k}∥_{L^{2}(ℙ(E):ω̌)} ≤ Č · \frac{∥Δ_{V} ∘ (Δ_{V} + id) σψ∥_{L^{2}(ℙ(E):ω̌)}}{k}

= C_{∗} · \frac{∥σψ∥_{L^{2}(ℙ(E):ω̌)}}{k}

while

∥ \frac{_{U}L_{N:1} (σ̃ψ)}{k}∥_{L^{2}(ℙ(E):ω̌)} ≤ Č · \frac{∥Δ_{V} ∘ (Δ_{V} + id) σ̃ψ∥_{L^{2}(ℙ(E):ω̌)}}{k} .

Here the constant C_{∗} is defined as C_{∗} ≡ Č · (4πn · 4πn + 4πn). On the other hand we have

\frac{_{U}L_{N:2} (ψ)}{k · k} = \frac{_{U}L_{N:2} (σ̂ψ)}{k · k} + \frac{_{U}L_{N:2} (σψ + σ̃ψ)}{k · k}

= \frac{σ ∘ _{U}L_{N:2} (σ̂ψ)}{k · k} + \frac{σ̃ ∘ _{U}L_{N:2} (σ̂ψ)}{k · k} + \frac{_{U}L_{N:2} (σψ + σ̃ψ)}{k · k}

because σ̂ ∘ _{U}L_{N:2} (σ̂ψ) = 0. (Actually we have

UL_{N:2} (σ̂ψ)

= \frac{Δ_{V}}{4π} ∘ _{A}L_{N:2} (σ̂ψ) + \frac{−Δ_{V}}{4π} [R_{1} · \frac{Δ_{M} σ̂ψ}{2} ]

+ \frac{i (∂∂)_{M} σ̂ψ ∧ [n · Π_{C_{M}} (\frac{i·F_{A_{L^{∗}}}}{2π} ) + trace (\frac{i·F_{A}}{2π} )] ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}} .

However the integral of the term

\frac{i (∂∂)_{M} ψ ∧ [n · Π_{C_{M}} (\frac{i·F_{A_{L^{∗}}}}{2π} ) + trace (\frac{i·F_{A}}{2π} )] ∧ \frac{ω_{M}^{(−2+m)}}{(−2+m)!}}{Ω_{M}}

along each fiber ℙ (ℂ^{n}) of π̌ : ℙ(E) ⟶ M simply vanishes.) Note that

∥\frac{σ ∘ _{U}L_{N:2} (σ̂ψ)}{k · k}∥_{L^{2}(ℙ(E):ω̌)} + ∥\frac{σ̃ ∘ _{U}L_{N:2} (σ̂ψ)}{k · k}∥_{L^{2}(ℙ(E):ω̌)}

≤ Č · \frac{∥Δ_{M} σ̂ψ∥_{L^{2}(ℙ(E):ω̌)}}{k · k}

while

∥ \frac{_{U}L_{N:2} (σψ + σ̃ψ)}{k · k}∥_{L^{2}(ℙ(E):ω̌)}

≤ Č · \frac{∥(Δ_{V} + id) ∘ (Δ_{M} + Δ_{V}) (σψ + σ̃ψ)∥_{L^{2}(ℙ(E):ω̌)}}{k · k} .

Let Ĉ ≡ 2Č · (2Č · C_{∗}) and C̃ = 2Č · [Č · (2Č + Ĉ)]. We claim that, when the parameter k is sufficiently large, the following estimate:

2Č · ∥σ̃ ∘ L_{N} (ψ)∥_{L^{2}(ℙ(E):ω̌)} + Ĉ · ∥σ ∘ L_{N} (ψ)∥_{L^{2}(ℙ(E):ω̌)}

+ C̃ · ∥σ̂ ∘ L_{N} (ψ)∥_{L^{2}(ℙ(E):ω̌)}

≥ ∥(Δ_{V} + \frac{Δ_{M}}{k} ) ∘ (Δ_{V} + \frac{Δ_{M}}{k} ) σ̃ψ∥_{L^{2}(ℙ(E):ω̌)}

+ ∥\frac{(Δ_{M} + id)}{k} ∘ (Δ_{V} + \frac{Δ_{M}}{k} ) σψ∥_{L^{2}(ℙ(E):ω̌)}

+ ∥\frac{(Δ_{M} + id) ∘ (Δ_{M} + id) σ̂ψ}{k · k} ∥_{L^{2}(ℙ(E):ω̌)}

is valid for any ψ ∈ Γ_{o} (ℙ(E) : ℝ). To see this we simply note that

2Č · ∥\frac{σ̃ ∘ _{U}L_{N:1} (σψ)}{k}∥_{L^{2}(ℙ(E):ω̌)} ≤ \frac{Ĉ}{2} · ∥σ ∘ _{o}P_{#k} (ψ)∥_{L^{2}(ℙ(E):ω̌)}

while

2Č · ∥\frac{σ̃ ∘ _{U}L_{N:2} (σ̂ψ)}{k · k}∥_{L^{2}(ℙ(E):ω̌)} + Ĉ · ∥\frac{σ ∘ _{U}L_{N:2} (σ̂ψ)}{k · k}∥_{L^{2}(ℙ(E):ω̌)}

≤ \frac{C̃}{2} · ∥σ̂ ∘ _{o}P_{#k} (ψ)∥_{L^{2}(ℙ(E):ω̌)} .

With these estimates it can be shown readily, by using the Schwarz inequality, that our claim is true. Proposition V.B then follows immediately.

To prove Corollary V.B we simply apply (Δ_{V} + Δ_{M}) iteratively to L_{N} (ψ). Since the orthogonal decomposition

Γ_{o} (ℙ(E) : ℝ) = σ̂Γ_{o} (ℙ(E) : ℝ) ⊕ σΓ_{o} (ℙ(E) : ℝ) ⊕ σ̃Γ_{o} (ℙ(E) : ℝ)

of Γ_{o} (ℙ(E) : ℝ) is preserved by Δ_{M} (and Δ_{V}) we can establish Corollary V.B easily through the same method as demonstrated above.

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