GAUGE-FIXING CONSTANT SCALAR CURVATURE EQUATIONS ON RULED MANIFOLDS AND THE FUTAKI INVARIANTS

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1 j. differential geometry 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 Futai invariants vanish. In this article we will prove that the vanishing of certain natural Futai invariants would imply the existence results for Kaehler metrics on ruled manifolds with constant scalar curvature. This wor 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 bacground 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 hm 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 hm: hm =h o M cm Received September 10, 001; revised September,

2 390 ying-ji hong in which h o M { Z hm :i Z ω M = f for some smooth C-valued function f ΓM : C on M } and cm { } Z hm :i Z ω M H 0:1 M : C. Note that the complex Lie algebra cm 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 M:ωM of h o M with the Lie algebra of the isometry group of M : ω M. Remar. 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. Remar. Note that the Lie algebra of smooth vector fields on M preserving the complex structure of M is isomorphic to hm in the Lie algebra sense. This interpretation of hm 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:ωM denote the compact connected component, containing the identity map of M, of the Lie subgroup of Aut M generated by the intersection M:ωM 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:ωM. Assume that π : E M is a holomorphic vector bundle of ran n over M with Einstein-Hermitian metric H E. Let A denote the Einstein- Hermitian connection on E induced by H E. Let PE denote the projectivization of E over M. Then PE is a compact complex manifold with 1m n dimensions. Let L be the universal line bundle over PE. Then the Einstein-Hermitian metric H E induces a Hermitian metric H L on the dual L of L over PE. Let A L denote the Hermitian connection on L induced by H L. Thus there is a representative i F AL = i log H L = i log H L

3 curvature equations 391 of the Euler class el ofl on PE induced by the Hermitian connection A L. Here H L is the Hermitian metric on L over PE induced by the Einstein-Hermitian metric H E on E over M. Note that the representative i F A L of el onpe induces the Fubini-Study metric on each fiber PC n ofˇπ : PE M. Thus, for each N large enough, i F AL ˇπ ω M is a Kaehler form on PE. ]: Suppose that, for each N large enough, there exists a corresponding Kaehler form on PE, lying in the Kaehler class ] i FAL ˇπ ω M, carrying constant scalar curvature. Then, for each N large enough, the corresponding Futai character must be zero. Let Aut PE denote the group of holomorphic automorphisms of PE. 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 PE preserving the holomorphic projection map ˇπ : PE M. Then we have Aut E G 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 PE preserving the holomorphic projection map ˇπ : PE M. Theorem A. Assume that the elements of h o M can be lifted to holomorphic vector fields on PE preserving the holomorphic projection map ˇπ : PE M. Suppose that, for each N large enough, the corresponding Futai character associated with g E the lifted action of h o M i FAL ] and the Kaehler class ˇπ ω M on PE is zero. Then, for each N large enough, there exists a corresponding Kaehler form on

4 39 ying-ji hong PE, lying in the Kaehler class i FAL carrying constant scalar curvature. ˇπ ω M ], 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 PE preserving the holomorphic projection map ˇπ : PE M. Suppose that, for each N large enough, the corresponding Futai character associated with the lifted action of h o M and the Kaehler i FAL ] class ˇπ ω M on PE is zero. Then, for each N large enough, there exists a corresponding Kaehler form on PE, lying in the Kaehler class ] i FAL ˇπ 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 N large enough, the corresponding Futai character associated with g E and the Kaehler i FAL class on PE is zero. Then, for each N large ] ˇπ ω M enough, there exists a corresponding Kaehler form on PE, lying in the Kaehler class i FAL carrying constant scalar curvature. ˇπ ω M ], In view of Theorem 0 it might seem necessary to add the following invariance assumption of i F A L to Theorem A: i F A L is invariant under the lifted action of M:ωM on PE. Our Theorem B shows that it is unnecessary to mae such extra assumption in Theorem A because the invariance of i F A L can be inferred directly from the vanishing of Futai invariants. In particular, in Corollary A, i F A L is automatically

5 curvature equations 393 invariant under the lifted action of M:ωM on PE 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 PE preserving the holomorphic projection map ˇπ : PE M. Suppose that, for each N large enough, the corresponding Futai invariants associated with g E the lifted action of M:ωM i FAL ] and the Kaehler class ˇπ ω M on PE are zero. Then the lifting of the intersection M:ωM of h o M with the Lie algebra of the isometry group of M : ω M can be properly rearranged such that i F AL is invariant under the rearranged lifted action of M:ωM on PE. Let E denote the maximal compact Lie subalgebra of g E. Actually, in Theorem B, the rearranged lifting of M:ωM is, modulo Hom M:ωM : 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 large enough. The gaugefixing constant scalar curvature equation and its solvability, when the parameter 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 Futai invariants. Actually, by incorporating the vanishing of the corresponding Futai 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 is sufficiently large, in Section VI. I. Futai invariants Here we summarize some basic facts about the Futai Invariants. The reader can find more bacground material in 7, 15].

6 394 ying-ji hong Theorem I.A. Assume that B : ω B is a b-dimensional compact Kaehler manifold not necessarily with constant scalar curvature. Let hb 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 ωb ] R denote the constant associated with ω B ] satisfying the following equality: ω m c ωb ] B m! = i ρ ω B ω 1m ω ω B ]. 1m! 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 B F : hb ω B ] C be defined as follows: FZ : ω ψ Z:ω c ωb ] ωm m! i ρ ω ω 1m 1m! Z : ω hb ω B ] in which the smooth function ψ Z:ω ΓB : C on B satisfies L Z ω = i ψ Z:ω. Then F only depends on hb: FZ : is constant on ω B ] for each Z hb. Besides we have FZ : W ]:ω =0 Z : W : ω hb hb ω B ]. F is called the Futai character associated with hb :ω B ]. It is obvious that when B carries constant scalar curvature the Futai character associated with hb and B : ω B must be zero. We will apply Theorem I.A to the compact complex manifold PE. 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 PE preserving the holomorphic projection map ˇπ : PE M.

7 curvature equations 395 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 Thus we have the following: 0=F A Z : ] H 1 A M : Hom E : E. 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 1 A 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 PE preserving both the complex structure of PE and the holomorphic projection map ˇπ : PE M.

8 396 ying-ji hong Remar. 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 1 A M : Hom E : E only depends on the holomorphic structure of E over M. Itdoesnot 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 1 M : A Hom E : E there certainly exists a lifting of X Z to a smooth vector field ˇX Z on PE preserving both the complex structure of PE and the holomorphic projection map ˇπ : PE M. Theorem II.B. Suppose that the Futai invariant associated with the lifting ˇX i FAL ] Z of X Z and the Kaehler class ˇπ ω M on PE vanishes for any sufficiently large N. Then there exists a lifting of X Z, preserving the holomorphic projection map ˇπ : PE M, toa smooth vector field ˇX Z on PE preserving both i F A L and the complex structure of PE. Note that the smooth vector field ˇX Z on PE is, modulo the compact Lie subalgebra E of g E, uniquely determined. It is obvious that the converse of Theorem II.B is true. Proving Theorem II.B requires more nowledge 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 bacground material in 1].

9 curvature equations 397 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 AutE 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 n Aut E =GL : C. ran E o 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 ernel identification Since the restriction of i F A L on each fiber P C n ofˇπ : PE M is simply the Fubini-Study Kaehler form there is a well-defined

10 398 ying-ji hong 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 ofpe 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-nown 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 : sln : C sun. On the other hand integration along the fibers of ˇπ : PE M maps a smooth function on PE onto a smooth function on M. Let ΓM : W denote the space of smooth sections of W over M. Then for each smooth R-valued function f ΓPE :R onpe wehavethe following corresponding decomposition: f =ˆσf σf σf in which ˆσf :σf ΓM : R ΓM : W while the restriction of σf oneachfiberp C n ofˇπ : PE 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 PC n f ω 1n F-S 1n! =0. Now we introduce a basic result about a special ind of quadratic combinations of elements of the eigenspace of the lowest nonzero eigenvalue 4πn of the Fubini-Study Laplacian 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 = i 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 F-S on P C n as follows:

11 curvature equations 399 Given X sln : C let f X Γ o PC 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 =0when X sun. Besides we always have F-S f X =4πn f X. We define Q f X : f X through the following equality: ω 1n F-S Q f X : f X 1n! = i f X i f X ω 3n F-S 3n!. Then for each element f X of the eigenspace of the lowest nonzero eigenvalue 4πn of the Fubini-Study Laplacian F-S on P C n the following smooth function: n f X 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 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 Futai character associated with sln : C and the Fubini-Study Kaehler class ω F-S ]on P C n. Proof of Proposition IV.A. Let ρ ωf-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 i ρ ωf-s = i log det H F-S = 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

12 400 ying-ji hong eigenvalue 4πn of the Fubini-Study Laplacian F-S on P C n through the following equality: ω 1n F-S Q f X : f X 1n! = d dt d ω 1n F-S:t n 1n dt 1n! i ρ ω F-S:t It can be checed readily that and thence Q f X : f X ω 1n F-S 1n! = i F-Sf X Q f X : f X = F-S 4π F-Sf X F-Sf X ω n F-S:t. n! t=0 Q f X : f X ω n F-S n! F-Sf X Q f X : f X = F-S 4π n f X n f X Qf X : f X ]. Now for any Y sln : C the Futai invariant associated with Y and the Fubini-Study Kaehler class ω F-S ]onpc n vanishes. Thus we have 0=F Y : ω F-S:t = L Y log H C n t L Y f X PC n ω 1n F-S:t n 1n 1n! i ρ ω F-S:t ω n F-S:t n! for t R with t 0 being small. Let us now consider the equality 0= d dt d dt F Y : ω F-S:t. t=0 Since X sln : C preserves the Einstein-Kaehler condition equivalently the constant scalar curvature condition on P C n : ω L X n 1n F-S 1n 1n! i ρ ω F-S ω n F-S =0 n!

13 curvature equations 401 and L X ω F-S = i f X we have d ω 1n F-S:t n 1n dt 1n! i ρ ω F-S:t Thus 0= d dt d dt F Y : ω F-S:t t=0 = L Y log H C n d dt d dt = = PC n n 1n PC n PC n ω 1n F-S:t 1n! i ρ ω F-S:t L Y log H C n ω 1n F-S Qf X : f X 1n! ω n F-S:t =0. n! t=0 ω n F-S:t n! t=0 f Y F-S 4π n f ω 1n F-S X n f X Qf X : f X ] 1n! for any Y sln : C. Since 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 sln : C F-S f Y ω 1n F-S 0= n f X n f X Qf X : f X ] PC n 4π 1n! ω 1n F-S = n f Y n f X n f X Qf X : f X ] 1n! PC 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 = i log H C n on P Cn. We define a symmetric quadratic operation Q on the eigenspace of the lowest nonzero eigenvalue 4πn of the Fubini-Study Laplacian F-S on P C n as follows:

14 40 ying-ji hong Given X sln : C and Y sln : 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: ω 1n F-S Q f X : f Y 1n! = i f X i f Y ω 3n F-S 3n!. Then for each pair f X : f Y of elements of the eigenspace of the lowest nonzero eigenvalue 4πn of the Fubini-Study Laplacian F-S on P C n the following smooth function: n f X 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 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 PE: T PE = V H. Here V is the subbundle of T PE over PE consisting of tangent vectors which are tangential to the fibers of ˇπ : PE M. Let V ] denote the maximal subbundle of T PE over PE whose action on H is identically zero. Then the decomposition T PE = V H of T PE over PE induces the following corresponding decomposition: T PE = V ] ˇπ T M of T PE over PE. Thus we have the following decomposition: T PE = C V C m C M of T PE over PE. Here C V = V ] and C M = ˇπ T M while C m is the subbundle of T PE over PE consisting of the mixed components of T PE. Thus we have the following diagram: C V Π CV T PE Π Cm Π CM C M C m

15 curvature equations 403 of projection maps over PE such that id = Π CV Π Cm Π CM on T PE. Since the decomposition T PE = V ] ˇπ T M of T PE is defined by the Einstein-Hermitian connection A on E over M we note that the representative i F AL of the Euler class e L ofl on PE has no nontrivial mixed components of T PE: i F AL =Π CV i FAL i FAL Π CM. Now we introduce a Hermitian form metric ˇω on PE by setting ˇω =Π CV i FAL ˇπ ω M. Remar. It should be noted that ˇω 1mn = i FAL lim m ω M 1mn. Actually ˇω can be realized as a modified limit of i F A L ω M,as, with the base directions of PE being properly rescaled. Note that the derivation operator can be expressed as d :ΓPE :R ΓPE :T PE R d = d V d M in which d V :ΓPE :R Γ PE :R V ] and d M :ΓPE :R ΓPE :R ˇπ 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 ˇω on PE. Then we have =d d = V M

16 404 ying-ji hong in which V d V d V and 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 i F AL Π CV and ˇπ ω M on PE with respect to the Hermitian form metric ˇω. We use the symbols V and M to denote respectively Π C V and Π CM : V =Π C V and M =Π C M. Similarly we use the symbol m to denote Π C m : m = Π Cm. Then we have the following results proved in the Appendix of 11]: Proposition IV.B. Given f ΓPE : R we have the equalities i Λ V V f = V f and i Λ M M f = Mf. Proposition IV.C. M V = V M. In particular we have M 4πn id V = 4πn id V M and thence M preserves Γ M : W. In 11] it is shown that the invertibility of the linear partial differential operator 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 checed readily that the smooth R-valued function s on PE satisfies 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 ernel of M acting on Γ M : W is isomorphic to g E E.

17 curvature equations 405 It should be noted that the linear partial differential operator M acting on Γ M : W is both nonnegative and symmetric with respect to the Hermitian form metric ˇω on PE. We can realize this picture more concretely as follows. Let G E denote the natural image of Aut E in Aut PE preserving the holomorphic projection map ˇπ : PE M. Then we have G E = Aut E C. Let g E denote the Lie algebra of G E over C. Let E denote the compact Lie algebra generated by the elements of g E preserving the representative i F AL of the Euler class of L on PE so that g E is the complexification of E. Let K E denote the compact subgroup of G E generated by E. Given a smooth vector field Y g E on PE we denote by f Y ΓM : W the corresponding smooth R-valued function on PE satisfying L Y i FAL = i f Y. Note that when Y E we have f Y = 0. Let N W denote the ernel of M acting on Γ M : W. Then we have N W = { i FAL f Y ΓM : W :L Y = i f Y for some Y g } E. 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 ΓM : W we have f = τ N W f τ NW f in which τ N W f is orthogonal to N W while τ NW f isthen W -component of f. Let V M denote the infinitesimal deformation operator for the con-

18 406 ying-ji hong stant scalar curvature equation on M : ω M : M V M = i 4π Λ M trace ω 1m M 1m! ] i FωM i FωM i trace = M M 8π ωm M m! i FωM i trace i ω 1m M 1m! ω m M m! i FωM Λ M trace ω m M m!. ] M ωm M m! Here F ωm is the curvature form of the holomorphic tangent bundle of M induced by the Kaehler form ω M on M while i FωM Λ M trace is the scalar curvature of M : ω M : ] i FωM Λ M trace ωm M i m! = trace FωM ω 1m M 1m!. Let Γ o M : R denote the space of smooth R-valued functions f on M satisfying f =0. M Here we set ωm M m!. Then the linear partial differential operator V M = V M 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 ernel of the linear partial differential operator V M acting on Γ o M : R is isomorphic to the vector space h o M M:ωM

19 curvature equations 407 over R. Let N VM denote the vector space over R of the ernel 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 VM and the orthogonal complement of N VM in Γ o M : R. Thus for f Γ o M : R we have f = τ N VM f τ NVM f in which τ N VM f is orthogonal to N VM while τ NVM f isthen VM - component of f: V M τ NVM f V M τ NVM f =0 =0. ωm m m! V. Gauge-fixing constant scalar curvature equation In this section we will introduce the gauge-fixing constant scalar curvature equation, depending on the parameter N, and prove its solvability when is large enough. Let o H # denote the Kaehler metric on PE induced by the Kaehler form oω # i F A L ˇπ ω M. Suppose that, for each N large enough, ω # is a Kaehler form on PE lying in the Kaehler class ] oω # so that ω # = o ω # i ψ with ψ ΓPE :R satisfying ψ Ω PE =0 PE M ˆσ ψ =0 i FAL 1n in which = ωm M m! and Ω PE ˇω 1mn 1mn! = 1n! ωm M m!. Let č R, depending on the parameter N, be the topological invariant satisfying the following equality: oω 1mn # č PE 1m n! = i log det o H # PE o ω mn # m n!.

20 408 ying-ji hong Let H # be the Kaehler metric on PE induced by the Kaehler form ω #. Then the Constant Scalar Curvature Equation for ω # is in which S ω # č S ω # =0 ω 1mn # 1m n! i log det H # ω mn # m n!. Based on the wor 10, 11] we might want to solve the constant scalar curvature equation, depending on N large enough, directly. However it is impractical to do so as there exist nontrivial ernels of linear partial differential operators associated with the constant scalar curvature equation. These nontrivial ernels exist simply because the constant scalar curvature equation is invariant under the action of the group Aut PE of holomorphic automorphisms of PE. In order to tacle this difficulty we add the following gauge-fixing term: n τ NW σ ψ τ N VM ˆσ ψ m Ω PE to the constant scalar curvature equation and define the gauge-fixing constant scalar curvature equation as in which S G-F ω# S G-F ω# =0 S n τ NW σ ψ ω # τ N VM ˆσ ψ ω 1mn # = č 1m n! i log det H # n τ NW σ ψ τ N VM ˆσ ψ m Ω PE ω mn # m n! m Ω PE. Let Γ o PE :R denote the space of smooth R-valued functions f on PE satisfying f Ω PE =0 ˆσf =0. PE M

21 curvature equations 409 We will solve the gauge-fixing constant scalar curvature equation for N large enough by considering ψ Γ o PE :R admitting asymptotic expansion of the following form: ψ φ 0 θ N as. Here each φ Γ o PE :R is a smooth R-valued function, independent of the parameter, on PE. Besides the following induction condition: σ φ 0 = σ φ 0 =0= σ φ 1 φ 0 Γ o M : R and φ 1 Γ o M : R ΓM : W is imposed on the leading terms φ 0 and φ 1. Before solving the gauge-fixing constant scalar curvature equation, for large enough, we collect some relevant basic facts which can be checed readily. Note that it is virtually better to rewrite the term φ θ θ i log det H # of the gauge-fixing constant scalar curvature equation S G-F ω# =0 as i log m det Ȟ i ω 1mn # log m ˇω 1mn. Here Ȟ is the Hermitian metric on PE induced by ˇω and thence m det Ȟ is a Hermitian metric on the dual of the canonical line bundle of PE. It can be shown that i log m det Ȟ = n i F A L i ˇπ FA trace i ˇπ FωM trace. Here F A is the curvature form of E induced by the Einstein-Hermitian connection A on E over M while F ωm is the curvature form of the

22 410 ying-ji hong holomorphic tangent bundle of M induced by the Kaehler form ω M on M. Since we have i log det o H # = i log m det Ȟ i log oω 1mn # č PE 1m n! i log det o H = # PE i = = PE PE oω mn # m n!. o ω mn # m n! oω 1mn # m ˇω 1mn o ω mn # log m det Ȟ m n! n i F A L i FA i FωM trace trace ] Thus the topological invariant č is a rational function of the parameter. By using the Einstein-Hermitian condition of the connection A on E over M n Π CM i FAL ω 1m M 1m! trace i FA ω 1m M 1m! =0 it can be shown readily that the power series expansion of č in 1 is č = 1n n i FωM Λ M trace č: higher order terms in which č : is a constant, independent of the parameter, while i FωM Λ M trace is the scalar curvature of M : ω M : ] i FωM Λ M trace ωm M i m! = trace FωM ω 1m M 1m!.

23 curvature equations 411 Now let us consider the asymptotic expansion of ψ as. By substituting ω # = o ω # i ψ o ω # i φ 0 θ N i φ θ θ as into S G-F ω# we have S G-F ω# m B 0 θ N B θ θ as in which each B is independent of the parameter. In order to show that the asymptotic expansion of ψ as exists we simply need to show the solvability of the following system of equations: B θ =0 B θ Ω PE =0 for any integer θ 0. By using the induction condition φ 0 Γ o M : R and φ 1 Γ o M : R ΓM : W it can be checed readily that We can then find through solving the equation B θ Ω PE =ˆσ B 0 =0=B 1. ˆσ φ θ σ φ θ1 σ φ θ Bθ Ω PE σ Bθ Ω PE Bθ σ =0 Ω PE 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 ΓM : W there exists a unique family of smooth R-valued functions φ θ Γ o PE :R on PE, depending on integers θ 0, such that B θ =0 for any integer θ 0.

24 41 ying-ji hong Now for each large N N we define a Kaehler form N ω # on PE, depending on N large enough, as follows: N ω # oω # i φ 0 = i F A L θ N with θ N ω M i φ 0 i φ θ θ θ N with θ N i φ θ θ. Here each φ is taen from the unique family of smooth R-valued functions on PE of Proposition V.A. Then we have the following result: Corollary V.A. Given γ 0 we denote by C γ PE:ˇω the C γ - norm of with respect to the Hermitian form metric ˇω on PE. Given p N there exists a corresponding constant C γ:p > 0 such that for each N p we have S G-F ω N # m C γ:p Ω PE p C γ PE:ˇω whenever γ:p:n. Here the choice of 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 S G-F N ω # m Ω on PE must PE 1 carry the factor intrinsically when >0is large enough equivalently when 1 p > 0 is small enough. Corollary V.A then follows immediately from standard results of calculus. We define for each large N a functional R on the Kaehler class ] i FAL ] oω # = ω M as follows: R oω # i oω # i 1mn m ˇω 1mn = 1mn oω # i 1mn! m Ω PE for any Kaehler form oω # i on PE lying in the Kaehler class oω # ]. Then the gauge-fixing constant scalar curvature equation for ω # = o ω # i ψ can be expressed equivalently as follows: S G-F ω# m Ω PE =0

25 curvature equations 413 in which S G-F ω# m Ω PE ω # č R i log R ω # m Ω PE n i F A L ˇπ trace i FA ω mn # mn! ˇπ trace i FωM m Ω PE n τ NW σ ψ τ N VM ˆσ ψ. ] ω mn # mn! Now for each large N N we define a corresponding 4 th order elliptic linear partial differential operator L N, depending on the parameter, acting on ψ Γ o PE :R as follows: L N ψ = č AL N ψ i log R N ω # i AL N ψ R N ω # m Ω PE i ψ N ω 3mn # 3mn! m Ω PE n τ NW σ ψ τ N VM ˆσ ψ. N ω mn # mn! B L N ψ Here A L N is the corresponding nd order linear partial differential operator without the 0 th order part acting on ψ Γ o PE :R defined as follows: AL N ψ i ψ N ω mn # mn! m Ω PE while B L N is the corresponding nd order linear partial differential operator without the 0 th order part acting on ψ Γ o PE :R defined as follows: BL N ψ i ψ n i F A L ˇπ trace i FA ˇπ trace i FωM ] N ω 3mn # 3mn! m Ω PE.

26 414 ying-ji hong Proposition V.B. Let L PE:ˇω denote the L -norm of with respect to the Hermitian form metric ˇω on PE. Then for each large N N there exists a corresponding N N such that for any ψ Γ o PE :R we have C L N ψ L PE:ˇω V M V M σψ L PE:ˇω M id V M σψ L PE:ˇω M id M id ˆσψ L PE:ˇω whenever 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 # denote the 4 th order elliptic linear partial differential operator, depending on the parameter, acting on ψ Γ o PE :R defined as follows: P # ψ = V 8π 4πn id V ψ M V ψ 8π V M ψ 8π M M ψ 8π n τ N W σψ Λ M trace i FωM i FωM ] i M ψ trace ω m M m! ] M ψ Ω M τ N VM ˆσψ. Actually, for each large N N, it can be shown that L N is dominated by P # when the parameter is sufficiently large. Corollary V.B. Given integer γ 0 we define the Sobolev norm H γ] PE:ˇω of as follows: H γ] PE:ˇω L PE:ˇω V M L PE:ˇω V M γ L PE:ˇω.

27 curvature equations 415 Then for each large N N there exists a corresponding γ:n N such that for any ψ Γ o PE :R we have C γ L N ψ H γ] PE:ˇω V M M id V M V M M id M id ˆσψ σψ σψ H γ] PE:ˇω H γ] PE:ˇω H γ] PE:ˇω whenever γ:n. Here the constant C γ > 0 depends on γ but not on N. In particular we have for any ψ Γ o PE :R the following estimate: C γ L N ψ H γ] PE:ˇω V M V M ψ H γ] PE:ˇω whenever γ:n. Remar. Note that the linear nd order elliptic linear partial differential operator M is coercive when acting on Γ o M : R. Besides the linear nd order elliptic linear partial differential operator M V is coercive when acting on Γ o PE :R. We will prove these results in Appendix III. Given γ 0 we denote by H o γ] PE : ˇω the Sobolev space consisting of R-valued functions f H γ] PE :ˇω onpe satisfying f Ω PE =0. PE Note that Corollary V.B implies the invertibility of the 4 th order elliptic linear partial differential operator L N : H γ4] o PE :ˇω H o γ] PE :ˇω whenever is sufficiently large. Let I N denote the inverse of L N. Then we have, for any f H o γ] PE : ˇω, the following estimate: I N f H γ4] o PE:ˇω C γ f H γ] o PE:ˇω

28 416 ying-ji hong whenever γ:n. Note that in this estimate for I N the constant C γ > 0 does not depend on N N though γ:n must be chosen larger for large N. Given ψ Γ o PE :R we note that, for t R, d dt SG-F ω t i ψ ] N # m Ω PE = č AL t N ψ B L t N ψ AL t N ψ N ω # t i ψ i R i log R N ω # t i ψ m Ω PE i ψ m Ω PE n τ NW σ ψ τ N VM ˆσ ψ N ω # t i ψ mn mn! N ω # t i ψ 3mn 3mn! in which A L t N and BL t N are the corresponding nd order nonlinear partial differential operators acting on ψ Γ o PE :R defined respectively as follows: and BL t N ψ i ψ AL t N ψ i ψ n i F A L ˇπ trace N ω # t i ψ mn mn! m Ω PE i FA ˇπ i FωM ] 3mn N trace ω # t i ψ 3mn!. m Ω PE S G-F N ω # t i ψ d From the shape of dt m Ω it is easy to see that, for PE bounded t R, the nonlinear partial differential operator d dt d dt SG-F ω t i ψ ] N # m = d Ω PE dt d S ω t i ψ ] N # dt m Ω PE ]

29 curvature equations 417 is genuinely nonlinear. Actually, by the Sobolev Embedding Theorem, for each sufficiently large γ N there exists a corresponding γ:n N such that for any pair f : g of elements of H o γ] PE : ˇω satisfying f H γ4] PE:ˇω 1 and g H γ4] PE:ˇω 1 we have the following estimate: d dt d SG-F ω t i f ] N # dt m Ω PE SG-F ω t i g ] N # d dt d dt m Ω PE H γ] PE:ˇω C γ f H γ4] PE:ˇω g H γ4] PE:ˇω f g H γ4] PE:ˇω whenever 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 PE :R the gauge-fixing constant scalar curvature equation for ] ω i ψ N lying in the Kaehler class # oω # can be expressed as 0= S G-F ω i ψ N # m = S G-F ω N # Ω PE m L N ψgn N ψ Ω PE in which GN N is the genuinely nonlinear partial differential operator acting on ψ Γ o PE :R defined as follows: GN N ψ = t 1 d dt d dt t 1 d dt d dt SG-F ω t i ψ ] N # dt Ω PE S ω t i ψ ] N # dt. Ω PE Since for any Kaehler form ] ω i N lying in the Kaehler class # oω # the integral S G-F ω i N # Ω Ω PE = S G-F ω i N # PE PE PE 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 SG-F 0=I ω N # N m ψ I N GN N ψ. Ω PE

30 418 ying-ji hong 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 γ4] PE :ˇω : ψ H γ4] PE:ˇω 1 } q, is contractive with contraction constant 1 whenever the parameter is large enough. On the other hand, by Corollary V.A and Corollary V.B, we may choose sufficiently large N N such that SG-F I ω N # N m 1 Ω PE q H γ4] PE:ˇω whenever the parameter 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 is large enough, by a Kaehler form ω i ψ N on PE with ψ H γ4] # o PE :ˇω satisfying ψ H γ4] PE:ˇω 1 q. With standard results of partial differential equations it can be shown readily that the solutions ψ H o γ4] PE : ˇω to the gaugefixing constant scalar curvature equation, depending on sufficiently large, 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 N is sufficiently large the corresponding gauge-fixing constant scalar curvature equation S G-F oω # i ψ m Ω PE =0 can be solved by some smooth R-valued function ψ Γ o PE :R on PE. Besides this family of smooth R-valued functions ψ Γ o PE : R on PE admits asymptotic expansion of the following form ψ φ 0 θ N φ θ θ

31 curvature equations 419 as. Here each φ is taen from the unique family of smooth R-valued functions on PE 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 # i ψ = N ω # i ψ :N with ψ :N Γ o PE :R satisfying ψ H :N γ4] PE:ˇω 1 q whenever is large enough. In this case the choice of the solution N ω # i ψ :N to the gauge-fixing constant scalar curvature equation S G-F ω i ψ N # :N m =0 Ω PE with ψ :N Γ o PE :R satisfying ψ:n H γ4] PE:ˇω 1 is, for q each sufficiently large, unique. Now we conclude this section with a remar. Suppose that, for each N large enough, the corresponding Futai invariants associated with g E the lifted action of M:ωM i FAL ] and the Kaehler class ˇπ ω M on PE are zero. Then, by Theorem II.B, we may assume that i F AL is invariant under the lifted action of M:ωM on PE. Thus both i F A L and ω M are invariant under the action of E the lifting of M:ωM on PE. In particular the gauge-fixing constant scalar curvature equation is invariant under the action of E the lifting of M:ωM on PE. Note that g E : the lifting of M:ωM ] ge. Hence, by Proposition V.A and the uniqueness result of Theorem V.A, the solutions oω # i ψ = N ω # i ψ

32 40 ying-ji hong to the gauge-fixing constant scalar curvature equation, depending on sufficiently large, are invariant under the action of E the lifting of M:ωM on PE. 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 M:ωM on PE preserves i F A L. Moreover such lifting of M:ωM on PE is, modulo the compact Lie subalgebra E of g E, uniquely determined. Thus, by complexification, there is a preferred lifting of h o M onpe 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 PE when there is no confusion. Now we fix respectively a K M:ωM -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 M:ωM, is isomorphic to the R-linear space N VM while the R-linear subspace of g E, orthogonal to E, is isomorphic to the R-linear space N W. In particular, for each f N VM, f C 0 PE:ˇω is comparable with the C 0 -norm of its correspondent in the orthogonal complement of M:ωM in h o M. Moreover this comparability is uniform on N VM. Actually fixing a K M:ωM -invariant metric on h o M simply means that we have fixed the uniform comparability between N VM and h o M. M:ωM Similar results are valid for N VM. Let ω # = o ω # i ψ denote the solution of Theorem V.A to the gauge-fixing constant scalar curvature equation. We will denote by : the inner product on L PE :ˇω defined by the Hermitian metric form ˇω on PE: f : g f g Ω PE PE f : g L PE :ˇω L PE :ˇω. Besides we will use the symbol c to denote a sufficiently large constant > 0 independent of the

33 curvature equations 41 parameter. Suppose that X ΓM : T M, orthogonal to M:ωM, is the smooth vector field on M, preserving the complex structure of M, such that L X ω M = i τ NVM ˆσψ. Let f X Γ o PE :R be the corresponding smooth R-valued function on PE satisfying Then we have L X ω # = i L X i FAL = i f X. τ NVM ˆσψ f X L X ψ. Since the Futai invariant, corresponding to X and the Kaehler i FAL ] class ˇπ ω M on PE, vanishes we have, by incorporating the gauge-fixing constant scalar curvature equation, the following equality: τ NVM ˆσψ f X L X ψ PE n τnw σψ τ ] N VM ˆσψ Ω PE =0. Since Γ M : R is orthogonal to Γ M : W with respect to the inner product : on L PE :ˇω we note that the term PE τ NVM ˆσ ψ n τ N W σψ Ω PE of the above equality vanishes. Thus, when the parameter is sufficiently large, we infer from Theorem V.A that there exists a constant c > 0, independent of, such that f X C 0 PE:ˇω L X ψ C 0 PE:ˇω c X C 0 PE:ˇω. On the other hand X C 0 PE:ˇω

34 4 ying-ji hong is comparable with τ NVM ˆσ ψ C 0 PE:ˇω. Thus, by using the Schwarz inequality, we infer from the above equality τ NVM ˆσψ f X L X ψ PE n τnw σψ τ ] N VM ˆσψ Ω PE =0 that, when is sufficiently large, there exists a constant ĉ > 0, independent of, such that τ NVM ˆσ ψ C 0 PE:ˇω ĉ τ N W σ ψ C 0 PE:ˇω. Suppose that Y g E, orthogonal to E, is the smooth vector field on PE such that i FAL L Y = i τ NW σψ. Then we have L Y ω # = i τ NW σψ L Y ψ. Since the Futai invariant, corresponding to Y and the Kaehler class i FAL ] ˇπ ω M on PE, vanishes we have, by incorporating the gauge-fixing constant scalar curvature equation, the following equality: τ NW σψ L Y ψ PE n τnw σψ τ ] N VM ˆσψ Ω PE =0. Since Γ M : R is orthogonal to Γ M : W with respect to the inner product : on L PE :ˇω we note that the term τ NW σ ψ τn VM ˆσψ Ω PE PE of the above equality vanishes. Since the zeroth order term φ 0 in the asymptotic expansion of ψ,as, ψ φ 0 θ=1 φ θ θ

35 curvature equations 43 satisfies σ φ 0 = σ φ 0 =0 φ 0 Γ o M : R we infer from Theorem V.A that, when the parameter is sufficiently large, there exists a constant c > 0, independent of, such that L Y ψ C 0 PE:ˇω c Y C 0 PE:ˇω because L Y φ 0 = 0. On the other hand Y C 0 PE:ˇω is comparable with τ NW σ ψ C 0 PE:ˇω. Thus, by using the Schwarz inequality, we infer from the above equality n τnw σψ τ NW σψ L Y ψ τ ] N VM ˆσψ Ω PE =0 PE that, when is sufficiently large, there exists a constant c > 0, independent of, such that τ NW σ ψ C 0 PE:ˇω c τ N VM By comparing this inequality τ NW σ ψ C 0 PE:ˇω c τ N VM with the previous inequality ˆσ ψ C 0 PE:ˇω. ˆσ ψ C 0 PE:ˇω τ NVM ˆσ ψ C 0 PE:ˇω ĉ τ N W σ ψ C 0 PE:ˇω we conclude immediately that τ NVM ˆσ ψ =0=τ NW σ ψ when the parameter is sufficiently large. Hence the uniquely determined, for each smooth R-valued solutions ψ 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 is sufficiently large. q.e.d.

36 44 ying-ji hong 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 onpe isalways zero: PE S G-F = PE S G-F Ω Ω PE = ˆσ PE PE Let us start with solving the equation Assume that ω 1mn # m ˇω 1mn B =0 B Ω PE =0. SG-F Ω PE Ω PE =0. ω 1mn o # m ˇω 1mn R 1 R higher order terms in which R 1 and R are independent of the parameter. Since the connection A on E over M is Einstein-Hermitian we have i FAL n Π CM ω 1m M i 1m! trace FA ω 1m M 1m! =0 and thence oω 1mn # m ˇω 1mn = Ω PE Ω PE i FAL n1! i FA Λ M trace n n1 ω m M m! Ω PE higher order terms in which Λ M trace i FA is the constant associated with the Einstein- Hermitian connection A on E over M: Λ M trace i FA ] ωm M m! = trace Besides it can be checed readily that i FA ω 1m M 1m!. R 1 = Mφ 0 V φ 1 = Mφ 0 V σ φ 1 = Mφ 0 n σ φ 1

37 curvature equations 45 and R = V φ Q σ φ 1:σ φ 1! i φ 0 i φ 0! ω m M m! i FAL 1n i φ 1 1n! Mφ 0 i φ 0 ω 1m M 1m! Ω PE. V φ 1 i FAL n n! ω m M m! Ω PE Here the symmetric quadratic operator Q : is defined along each fiber P C n of the holomorphic projection map ˇπ : PE M as in Proposition IV.A and Corollary IV.A. Thus ] B i FωM = 1n n R Λ M trace R 1 Ω PE n n V φ n 1n n n i φ 0 i φ 1 i FAL 1n 1n! Ω PE i FAL n n! ω m M m! Ω PE n 3n Q σ φ 1:σ φ 1! n n Mφ 0 n 1n i FA Λ M trace trace i FA V φ 1 i φ 0 i φ 0! ω m M m! Λ M trace trace i FωM i FωM ω 1m M 1m! ] V φ 1 ] i φ 0 ω m M m!

38 46 ying-ji hong i R 1 i FAL 1n 1n! Ω PE σ R1 Q : σ φ 1 V R 4π V 8π ω 1m M 1m! V R 1 4π Λ M trace i FA n Mφ 0 R 1 n τ NW σ φ 1 τ NVM ˆσ φ 0 nown terms. By substituting the formulae for R 1 and R into the above expression of we see that B Ω PE B i FωM = Λ M trace Ω PE n i φ 0 Λ M trace Since i F A L trace i FA i FAL i FA ] Mφ 0 n n! ω m M m! Ω PE ] V φ 1 i FωM trace M M φ 0 n i σ φ 1 8π V R i FA Λ M trace 4π n Mφ 0 V σ φ 1 n V φ ] i φ 0 ω m M m! i FAL 1n! Ω PE ] V σ φ 1 1n n n π V σ φ 1 σ φ 1 ] n τ NW σ φ 1 τ NVM ˆσ φ 0 nown terms. = Π CV i FAL Π CM i FAL ω 1m M 1m! we have by using the

39 curvature equations 47 Einstein-Hermitian condition of A on E over M i σ φ 1 i FAL 1n! 1n ω 1m M 1m! and thence = Mσ φ 1 = Mσ φ 1 B Ω PE = Ω PE V σ φ 1 Λ M trace i FA i FωM Λ M trace n Now we note that i φ 0 trace i FA i FAL n Π CM i FAL ] ] Mφ 0 n n! ω m M m! Ω PE trace i FωM ω 1m M 1m! V σ φ 1 n V φ ] i φ 0 ω m M m! M M φ 0 n Mσ φ 1 8π ] V R i FA Λ M trace V σ φ 1 4π n Mφ 0 V σ φ 1 n n π V σ φ 1 σ φ 1 ] n τ NW σ φ 1 τ NVM ˆσ φ 0 nown terms. PC n = n PC n i φ 0 n i FAL n n! ω m M m! Ω PE i φ 0 Π CM i FAL ω m M m!

40 48 ying-ji hong = PC n i φ 0 trace i FA ω m M m! along each fiber P C n ofˇπ : PE M. Thus ] i FωM Λ M trace M φ 0 B ˆσ = Ω PE = trace i FωM i φ 0 ω m M m! M M φ 0 8π τ NVM φ 0 ˆσ nown terms VM τ NVM φ 0 ˆσ nown terms. Remar. Let ξ A denote the smooth 1m n-form on PE defined by A as follows: Let ξ A Ω PE ξ A n n i FAL n1 n 1! i ˇπ FA trace i ˇπ FωM trace ω m M m! i FAL n! i FAL n! n n ω m M m! ω m M m!. denote the smooth R-valued function on PE satisfying ξ A = ξ A Ω PE Ω PE. 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 B VM ˆσ = τ NVM φ 0 ˆσ c ξa ξ A Ω PE in which c ξa R is the constant satisfying c ξa Ω PE = PE PE ξ A. Ω PE

41 curvature equations 49 Thus in the equality B ˆσ = Ω PE = Λ M trace i FωM trace i FωM ] M φ 0 i φ 0 ω m M m! M M φ 0 8π τ NVM φ 0 ˆσ nown terms VM τ NVM φ 0 ˆσ nown terms. the phrase nown terms simply means c ξa ξ A Ω PE which is, given A, obviously a nown smooth R-valued function on M. Since the elliptic linear partial differential operator VM ΩM τ NVM 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 ˆσ B Ω PE =0. With φ 0 Γ o M : R being nown we have by the Einstein-Hermitian condition of A on E over M R = V φ Q σ φ 1:σ φ 1 Mφ 0! ] Mσ φ 1 Λ M trace i FA n nown terms and thence B = V Ω PE 8π 4πn id V φ V σ φ 1 V σ φ 1 M ˆσ φ 1 V 8π n σφ 1 n σφ 1 Q σφ 1 : σφ 1 ] n M 8π τ NW σ φ 1 nown terms.

42 430 ying-ji hong Now by Proposition IV.A we have B σ = n M 8π τ NW σ φ 1 σ nown terms. Ω PE Since the elliptic linear partial differential operator M 8π τ NW acting on Γ M : W is both symmetric and positive we infer that σ φ 1 ΓM : W can be uniquely solved from the equation σ B Ω =0. PE With both φ 0 Γ o M : R and σ φ 1 ΓM : W being nown we have σ B Ω PE = V 8π 4πn id V σ φ σ nown terms and thence σ φ can be uniquely solved from the equation σ B Ω PE = 0 fiberwisely. Now given θ N we will solve the equation under the hypothesis that B θ =0 B θ Ω PE =0 ˆσ φ µ σ φ µ1 σ φ µ is already nown for any integer 0 µ<θ. σ φ 1 σ φ and R 1 = M φ 0 V σφ 1 already nown. Suppose that ω 1mn # m ˇω 1mn ω 1mn o # m ˇω 1mn R 1 = M φ 0 In particular ˆσ φ 0 n σ φ 1 are R θ1 θ R θ θ higher order terms in which each R is independent of the parameter. Then by our induction hypothesis we have R µ being nown for any µ N satisfying µ θ. It can be checed readily that R θ1 = M ˆσ φ θ V σ φ θ1 nown terms