aa r X i v : . [ m a t h . DG ] O c t EXTREMAL METRICS ON RULED MANIFOLDS
ZHIQIN LU AND REZA SEYYEDALI
Abstract.
In this paper, we consider a compact K¨ahler manifold with extremalK¨ahler metric and a Mumford stable holomorphic bundle over it. We proved that,if the holomorphic vector field defining the extremal K¨ahler metric is liftable to thebundle and if the bundle is relatively stable with respect to the action of automor-phisms of the manifold, then there exist extremal K¨ahler metrics on the projec-tivization of the dual vector bundle.
Contents
1. Introduction 12. Preliminaries 43. Scalar curvature 84. Construction of approximate solutions 125. Proof of Theorem 1.2 196. Hong’s moment map setting and proof of Theorem 1.1 21References 241.
Introduction
Let (
M, ω ) be a K¨ahler manifold of dimension m and L be an ample line bundle over M such that ω ∈ πc ( M ). Let π : E → M be a holomorphic vector bundle of rank r ≥
2. This gives a holomorphic fibre bundle P E ∗ over M with fibre P r − . We denotethe tautological line bundle on P E ∗ by O P E ∗ ( −
1) and its dual bundle by O P E ∗ (1). Bythe Kodaira embedding theorem, for k ≫
0, the line bundles O P E ∗ (1) ⊗ π ∗ L k on P E ∗ are very ample.In [10, 11], Hong proved that if E is Mumford stable; ω has constant scalar curva-ture; and M does not admit any nontrivial holomorphic vector fields, then P E ∗ admits Date : November 14, 2012.2000
Mathematics Subject Classification.
Primary: 53A30; Secondary: 32C16.
Key words and phrases. ruled manifold, extremal metric, stable vector bundle.The first author is partially supported by the DMS-12-06748 of the National Science Foundationof USA. cscK metric in the class of O P E ∗ (1) ⊗ π ∗ L k for k ≫
0. In [12], he generalized the resultto the case that the base manifold has nontrivial automorphism group. He provedthat if all Hamiltonian holomorphic vector fields on M can be lifted to holomorphicvector fields on P E ∗ and the corresponding Futaki invariants vanish, then P E ∗ admitscscK metrics in the class of O P E ∗ (1) ⊗ π ∗ L k for k ≫
0. The result was further general-ized by replacing the liftiblity of holomorphic vector fields by a stability condition(cf.[13]). Hong considered the action of Aut(M) on the space of holomorphic structureson E and showed that if E is stable under this action, then there exist cscK metricson ( P E ∗ , O P E ∗ (1) ⊗ π ∗ L k ) for k ≫
0. The stability assumption is used to perturbapproximation solutions to genuine cscK metrics.In this article, we generalize Hong’s result to the case that the base admits anextremal metric. Our main theorem is the following
Theorem 1.1.
Let ( M, L ) be a compact polarized manifold and ω ∞ ∈ c ( L ) be anextremal K¨ahler metric. Let X s be the gradient vector field of the scalar curvature of ω ∞ , i.e. dS ( ω ∞ ) = ι X s ω ∞ . Let E be a Mumford stable holomorphic vector bundleover M . Suppose that the holomorphic vector field X s can be lifted to a holomorphicvector field on P E ∗ . If E is relatively stable under the action of Aut( M ) in the senseof Definition 6.5, then there exist extremal metrics on ( P E ∗ , O P E ∗ (1) ⊗ π ∗ L k ) for k ≫ . We follow the ideas of [13, 22]. Let G = Ham( M, ω ∞ ) be the group of Hamiltonianisometries of ( M, ω ∞ ) and g be its Lie algebra. Let G E be the subgroup of all Hamil-tonian isometries of ( M, ω ∞ ) that can be lifted to automorphisms of P E ∗ . Let g E bethe Lie algebra of G E , i.e., space of all Hamiltonian holomorphic vector fields X on M that are liftable to holomorphic vector fields ˜ X on P E ∗ . Fix T ⊆ G E a maximaltorus and K ⊆ G the subgroup of all elements in G that commute with T . Let t and k be the Lie algebras of T and K respectively. We denote the space of all Hamiltonianswhose gradient vector fields are in t and k by ¯ t and ¯ k respectively (including constantfunctions). Suppose that E is Mumford stable. Then the Donaldson-Uhlenbeck-YauTheorem implies that E admits a Hermitian-Einstein metric h . The metric h inducesa hermitian metric g = ˆ h on O P E ∗ (1). The restriction of the (1 , ω g = i ¯ ∂∂ log g = i ¯ ∂∂ log b h on fibres are Fubini-Study metrics and therefore ω g | Fiber is non-degenerate. Hencefor k ≫
0, the (1 , ω k = ω g + kω ∞ define K¨ahler metrics. Finding extremalmetrics on ( P E ∗ , O P E ∗ (1) ⊗ L k ) is equivalent to finding φ ∈ C ∞ ( P E ∗ ) T and f ∈ ¯ t such that(1.1) S ( ω k + √− ∂∂φ ) + 12 h∇ f, ∇ φ i = f, where ∇ and h , i are taken with respect to ω k , and C ∞ ( P E ∗ ) T is the space of smoothfunctions on P E ∗ that are invariant under the action of T . To see that ω k + √− ∂∂φ is an extremal metric, we assume that df = ι ( X ) ω k for some holomorphic vector field X . We write X = X + ¯ X for holomorphic (1 , X . Then a straightforward computation shows that¯ ∂S = ι ( X )( ω k + √− ∂∂φ ) . Our strategy is to replace equation (1.1) with the one that is easier to solve and isrelating it to a finite dimensional GIT problem (cf. [13, 22]). The first step is to find φ ∈ C ∞ ( P E ∗ ) T and b ∈ ¯ k such that(1.2) S ( ω k + √− ∂∂φ ) + 12 h∇ l k ( b ) , ∇ φ i = l k ( b ) , where ∇ and h , i are taken with respect to ω k and l k ( b ) is a lift of b to P E ∗ defined inDefinition 4.5. Note that if b ∈ ¯ t , then ω k + √− ∂∂φ is an extremal metric. Allowing b to be in a slightly larger space makes it easier to solve the equation. In order tosolve equation (1.2), we first construct K¨ahler forms ω k,p in the class of ω k for anypositive integer p and k ≫ Theorem 1.2.
Let p ≥ be an integer. Suppose that X s ∈ t . Then for any k ≫ ,we can find φ ∈ C ∞ ( P E ∗ ) T and b ∈ ¯ k such that S ( ω k,p + √− ∂∂φ ) + 12 h∇ l k,p ( b ) , ∇ φ i = l k,p ( b ) . Here ∇ and h , i are taken with respect to ω k,p . Moreover b has the following expan-sion: b = r ( r −
1) + k − S ( ω ) − k − π N (Σ E ) + O ( k − ) , where π N : C ∞ ( M ) → ker( D ∗ D ) and Σ E = 2 r Λ (Ric( ω ) ∧ tr( iF h )) − r ( r + 1) Λ (tr( iF h ) ∧ tr( iF h ))+ 2 r + 1 Λ tr( iF h ∧ iF h ) − µS ( ω ) + tr( u X s ) r . See Proposition 4.3 and 4.13 for the definition of u X s and l k,p respectively. ZHIQIN LU AND REZA SEYYEDALI
The metric ω k,p + √− ∂∂φ would have been an extremal metric if b were in ¯ t . Ifnot, we perturb the holomorphic structure on E so that the Hamiltonian b lies in ¯ t after the perturbation. This can be done by applying implicit function theorem usingthe stability assumption. In a recent paper, Br¨onnle [1], using the similar method,proved that if the base is cscK without holomorphic vector fields and the bundle isa direct sum of stable bundles with different slopes, then the projectivization admitsextremal metrics.The outline of the paper is as follows: In section 2, we go over some basic factsand definitions. In section 3, we compute an expansion for the scalar curvature ofthe metrics ω k . Section 4 is devoted to the construction of K¨ahler metrics ω k,p . InSection 5, we prove Theorem 1.2. In the last section, we adopt Hong’s moment mapsetting to our situation and prove the main theorem.2. Preliminaries
Let V be a hermitian vector space of dimension r . The projective space P V ∗ canbe identified with the space of hyperplanes in V via f ∈ V ∗ → ker( f ) = V f ⊆ V .There is a natural isomorphism between V and H ( P V ∗ , O P V ∗ (1)) which sends v ∈ V to ˆ v ∈ H ( P V ∗ , O P V ∗ (1)) such that for any f ∈ V ∗ , ˆ v ( f ) = f ( v ). Definition 2.1.
For any hermitian inner product h on V , we use h · , · i h to denotethe hermitian inner product induced by h and we use k · k h to denote the norm withrespect to h on both V and V ∗ . The hermitian inner product h induces a hermitianmetric on O P V ∗ (1), which can be explicitly represented as follows: for v, w ∈ V and f ∈ V ∗ we define(2.1) h ˆ v, ˆ w i ˆ h = f ( v ) f ( w ) k f k h . We denote the induced metric on O P V ∗ (1) by b h .The following is a straightforward computation. Proposition 2.2.
For any v, w ∈ V we have h v, w i h = C − r Z P V ∗ h ˆ v, ˆ w i b h ω r − FS ( r − where C r is a constant defined by (2.2) C r = Z C r − ( √− r − dξ ∧ dξ (1 + P r − j =1 | ξ j | ) r +1 = (2 π ) r − r ! , and ( √− r − dξ ∧ dξ = ( √− dξ ∧ dξ ) ∧ · · · ∧ ( √− dξ r − ∧ dξ r − ) . Definition 2.3.
For any v ∈ V and any hermitian inner product h on V , we definean endomorphism λ ( h ) = λ ( v, h ) of V by λ ( v, h ) = 1 k v k h v ⊗ v ∗ h , where v ∗ h ( · ) = h ( · , v ) is the dual element of v with respect to the inner product h .The above settings can be made into the following family version. Let ( M, ω ) bea K¨ahler manifold of dimension m and E be a holomorphic vector bundle on M ofrank r ≥
2. Let L be an ample line bundle on M endowed with a hermitian metric σ so that i ¯ ∂∂ log σ = ω . The configuration ( M, ω, L, σ ) is called a polarized K¨ahlermanifold. Let P E ∗ be the projectivization of the dual bundle E ∗ of E . A hermitianmetric h on E induces a hermitian metric b h on the line bundle O P E ∗ (1) by (2.1).Let ω g be the (1 , P E ∗ defined by ω g = i ¯ ∂∂ log b h. Let π : P E ∗ → M be the projection map. Define the smooth functions f , . . . f m ∈C ∞ ( P E ∗ ) by(2.3) ω r − jg ( r − j )! ∧ π ∗ ω m − j ( m − j )! = f j ω r − g ( r − ∧ π ∗ ω m m ! . Alternatively, f j ’s can be generated by the following equation(2.4) ω m + r − k = ( m + r − m !( r − m X j =1 k m − j f j ω r − g ∧ π ∗ ω m , where ω k = ω g + kπ ∗ ω. Definition 2.4.
Let X be a compact K¨ahler manifold with the K¨ahler metric ω .Assume that the complex dimension of X is N . For any ( j, j )-form α on X , we definethe contraction Λ jω α of α with respect to the K¨ahler form ω by N ! j !( N − j )! α ∧ ω N − j = (Λ jω α ) ω N . In particular, we define Λ ω α = Λ ω α. Definition 2.5.
We define the vertical subbundle V and the horizontal subbundle H of the holomorphic tangent bundle T ( P E ∗ ) of P E ∗ as follows: let u ∈ P E ∗ and let ZHIQIN LU AND REZA SEYYEDALI π ( u ) = x . V u = T u ( P E ∗ x ); H u = { ξ ∈ T u ( P E ∗ ) | ω g ( ξ, w ) = 0 , ∀ w ∈ V u } . Since the restriction of ω g to the fibre is the Fubini-Study metric of P E ∗ x , ω g | Fibre is non-degenerate. As a result, H is indeed a vector bundle of rank m , and we havethe following (holomorphic bundle) decomposition T ( P E ∗ ) = H ⊕ V. By the dimension consideration, we have H ∗ = π ∗ ( T ∗ M ), where H ∗ is the dual bundleof H . Let V ∗ be the dual bundle of V . Then we have(2.5) T ∗ ( P E ∗ ) = V ∗ ⊕ π ∗ ( T ∗ M ) . Let V ( T ∗ ( P E ∗ )) be the bundle of differential forms of P E ∗ . Write ^ ( T ∗ ( P E ∗ )) = C H ⊕ C V ⊕ C m , where C H , C V and C m are the bundles of horizontal, vertical, and mixed forms, re-spectively. Note that C H = π ∗ ( V ( T ∗ M )) . For any differential form α on P E ∗ , wewrite α = α H + α V + α m , where α H , α V , α m are the horizontal, vertical, and mixedcomponents of α respectively.Using the above notation, we have Lemma 2.6.
There is no mixed component of ω g .Proof. This follows from (2.5). (cid:3)
If we write ω g = ( ω g ) H + ( ω g ) V as its horizontal and vertical parts. Then (2.3) can be written as(2.6) f j π ∗ ( ω m ) = m ! j !( m − j )! ( ω g ) jH ∧ π ∗ ω m − j . Let F h ∈ V , (Hom ( E, E )) be the curvature tensor of hF h = ¯ ∂ ( ∂h · h − ) . From (2.6), we can prove the following
Lemma 2.7.
For any v ∈ E ∗ , we have f j ([ v ]) = Λ jω (cid:16) √− (cid:0) λ ( v, h ) F h (cid:1)(cid:17) j , where [ v ] ∈ P E ∗ is the class of v in P E ∗ .Proof. Let(2.7) β = √− (cid:0) λ ( v, h ) F h (cid:1) = √− k v k − h h F h ( v ) , v i h . Let x = π ( u ). We assume that at x , { e , · · · , e r } is a normal frame. That is, underthis frame h i ¯ j ( x ) = δ ij , dh i ¯ j ( x ) = 0 . Since there are no connection terms, by a straightforward computation, we obtain (2.8) ω g = π ∗ ( β ) + ω g | P E ∗ x . Therefore,(2.9) ( ω g ) H = π ∗ β. The lemma follows from Definition 2.4. (cid:3)
Let α be a (1 , P E ∗ . Define ˜Λ ω g α V by˜Λ ω g α V ∧ (( ω g ) V ) r − = ( r − α V ∧ (( ω g ) V ) r − . Therefore, we have( ˜Λ ω g α V ) ω m + r − j − g ∧ π ∗ ω j = ( m + r − j − α V ∧ ω m + r − j − g ∧ π ∗ ω j for j ≥ Definition 2.8.
For any smooth function f ∈ C ∞ ( P E ∗ ), define the operators ∆ V , ∆ H and ˜∆ H (and call them the Laplacians) by the following equations( r − √− ∂∂f ∧ ω r − g ∧ π ∗ ω m = ∆ V f ω r − g ∧ π ∗ ω m ,m √− ∂∂f ∧ ω r − g ∧ π ∗ ω m − = ∆ H f ω r − g ∧ π ∗ ω m , ˜∆ H f = ∆ H f − f ∆ V f. Remark . The Laplacians ∆ H and ∆ V are the same as ones defined in [10]. Strictly speaking, β is a section of the sheaf C ∞ ( P E ∗ ) ⊗ A , ( M ). So the π ∗ operation is onlyacting on the second component. ZHIQIN LU AND REZA SEYYEDALI
Definition 2.9.
For any x ∈ M , we define W x as the space of all eigenfunctions ofthe Laplacian (on functions) on P E x (with respect to the metric ω g | P E x ) associatedto the first nonzero eigenvalue. Define the vector bundle W whose fibers are W x (c.f.[11]).Let End ( E x ) be the space of traceless endomorphisms of E x for any x ∈ M . Thefirst nonzero eigenvalue of the Laplacian is r . As is well-known,Φ ∈ End ( E x ) → Tr (cid:0) λ ( h )Φ (cid:1) ∈ W x is a 1-1 correspondence. Define End ( E ) to be the smooth vector bundle whose fibersare End ( E x ) for any x ∈ M . Thus we have W = End ( E ).3. Scalar curvature
The goal of this section is to find the asymptotic expansion for the scalar curvatureof the K¨ahler form ω k = ω g + kπ ∗ ω . The main result of this section is Theorem 3.1.
Let ω be a K¨ahler metric on M and h be a hermitian metric on E .Let ω k = ω g + kπ ∗ ( ω ) , where k is a large positive integer. Then we have the following expansion of the scalarcurvature Scal( ω k ) of ω k Scal( ω k ) = r ( r −
1) + k − ( π ∗ S ( ω ) + 2 r Λ ω (Tr( λ ( h ) F ◦ h )))+ k − (cid:16) ω (( π ∗ ( Ric ( ω ) − Tr( iF h )) ∧ ω g ) H ) − f ( π ∗ ( S ( ω ) − Λ ω (Tr( iF h ))))+ ∆ V ( f − f ) + ˜∆ H f − rf + 2 rf (cid:17) + O ( k − ) , where S ( ω ) is the scalar curvature of ω and F ◦ h = F h − r tr ( F h ) is the trace-less partof the curvature tensor of h . (For the definition of f , . . . f m , λ ( h ) , ˜∆ H , ∆ V , Λ ω and Λ ω , see (2.3) , Definition 2.3, Definition 2.4 and Definition 2.8). Let α = π ∗ α be a horizontal form of P E ∗ (see footnote 1). Then we defineΛ ω α = π ∗ (Λ ω α ) . First we prove the following purely algebraic lemmas.
Lemma 3.2.
Let α be a (1 , -form on P E ∗ . Then Λ ω k α = ˜Λ ω g α V + k − Λ ω α H + k − (cid:0) ω ( α ∧ ω g ) H − (Λ ω α H ) f (cid:1) + O ( k − ) . In particular if α ∈ V , ( M ) , then Λ ω k α = k − π ∗ (Λ ω α ) + k − (cid:16) ω ( α ∧ ω g ) H − f π ∗ (Λ ω α ) (cid:17) + O ( k − ) . Proof.
By definition, we have(Λ ω k α ) ω m + r − k = ( m + r − α ∧ ω m + r − k . We define g j = g j ( α ) by the equation1( m + r − α ∧ ω m + r − k = 1( r − m ! k m ( m X j =0 k − j g j ) ω r − g ∧ ω m . Let α = α V + α H + α m be the decomposition of α into its vertical, horizontal, and mixed components. Thenwe have ( r − m !( m + r − α ∧ ω m + r − k = ( ˜Λ ω g α V )(( ω g ) H + kπ ∗ ω ) m ∧ (( ω g ) V ) r − + mα H ∧ (( ω g ) H + kπ ∗ ω ) m − ∧ (( ω g ) V ) r − = X j k m − j g j (( ω g ) V ) r − ∧ π ∗ ω m . Simple calculation shows that g = ˜Λ ω g α V ; g = Λ ω α H + ( ˜Λ ω g α V ) f ; g = 2Λ ω ( α ∧ ω g ) H + ( ˜Λ ω g α V ) f . By (2.4), the above equation impliesΛ ω k α = P k − j g j P k − j f j = g + k − ( g − g f ) + k − ( g − g f − g f + g f ) + O ( k − ) . The lemma is proved. (cid:3)
Let ∆ k be the Laplacian with respect to the metric ω k . That is,∆ k f = Λ ω k ( √− ∂∂f )for smooth functions f on P E ∗ . Then we have the following asymptotics: Lemma 3.3.
For any f ∈ C ∞ ( P E ∗ ) , we have ∆ k f = ∆ V f + k − ˜∆ H f + k − (cid:16) − f ˜∆ H f + 2Λ ω ( √− ∂∂f ∧ ω g ) H (cid:17) + O ( k − ) as k → ∞ .Proof. Let α = √− ∂∂f . Then we have∆ k f = Λ ω k α. By Lemma 3.2, we have∆ k f = ˜Λ ω g α V + k − Λ ω α H + k − (cid:0) ω ( α ∧ ω g ) H − (Λ ω α H ) f (cid:1) + O ( k − ) . By Definition 2.8, we have( ˜Λ ω g α V ) ω r − g ∧ π ∗ ω m = ( r − α ∧ ω r − g ∧ π ∗ ω m = (∆ V f ) ω r − g ∧ π ∗ ω m . Thus ˜Λ ω g α V = ∆ V f. Similarly, we have ˜Λ ω g α V f + Λ ω α H = ∆ H f. Thus we have Λ ω α H = ˜∆ H f. The lemma is proved. (cid:3)
Proof of Theorem 3.1. we have the following exact sequence of holomorphic vectorbundles on P E ∗ . 0 → V → T P E ∗ → π ∗ T M → . The hermitian metric h on E induces a Fubini-Study metric h F S on V . The positive(1 , ω k and ( ω g ) H + kπ ∗ ω induce hermitian metrics on vector bundles T P E ∗ and π ∗ T M respectively. As holomorphic hermitian vector bundles, the above exactsequence splits in the smooth category:( T P E ∗ , ω k ) = ( V, h
F S ) M ( π ∗ T M, ( ω g ) H + kπ ∗ ω )and in addition, we haveRic( ω k ) = Tr( iF h F S ) + Ric(( ω g ) H + kπ ∗ ω ) . On the other hand, we have the following Euler sequence of holomorphic vector bun-dles on P E ∗ . 0 → C → π ∗ E ∗ ⊗ O P E ∗ (1) → V → . This gives the following isometric isomorphism of holomorphic line bundles on P E ∗ .(det( V ) , det( h F S )) ∼ = (det( π ∗ E ⊗ O P E ∗ (1)) , det( π ∗ h ⊗ ˆ h )) . Therefore (cf. (2.8)), Tr( iF h F S ) = rω g − π ∗ Tr( iF h ), and we haveRic( ω k ) = rω g + Ric(( ω g ) H + kπ ∗ ω ) − π ∗ Tr( iF h ) . On the other hand, by (2.6), we have k − m (cid:0) ( ω g ) H + kπ ∗ ω (cid:1) m = (1 + k − f + · · · + k − m f m ) π ∗ ω m . As a result, Ric(( ω g ) H + kπ ∗ ω ) = √− ∂∂ log(( ω g ) H + kπ ∗ ω (cid:1) m = √− ∂∂ log( m X j =0 k − j f j ) + π ∗ (Ric( ω )) . Consequently,(3.1) Ric ( ω k ) = rω g − π ∗ Tr( iF h ) + π ∗ (Ric( ω )) + √− ∂∂ log( m X j =0 k − j f j ) . Taking trace of (3.1) with respect to ω k , we getScal( ω k ) = Λ ω k α + ∆ k log( m X j =0 k − j f j ) , where α = π ∗ (Ric( ω ) − Tr( iF h )) + rω g . Let b = π ∗ ( S ( ω ) − Λ ω (Tr( iF h ))) . Using Lemma 3.2, we getΛ ω k α = r ( r −
1) + k − ( b + rf )+ k − (2Λ ω ( π ∗ (Ric( ω ) − Tr( iF h )) ∧ ω g ) H − f b − rf + 2 rf ) + O ( k − ) . By Lemma 3.3, we have∆ k log( m X j =0 k − j f j ) = k − (∆ V f ) + k − (∆ V ( f − f ) + ˜∆ H f ) + O ( k − ) . Therefore, we haveScal( ω k ) = r ( r −
1) + k − ( b + rf + ∆ V f )+ k − (2Λ ω ( π ∗ (Ric( ω ) − Tr( iF h )) ∧ ω g ) H − f b + ∆ V ( f − f ) + ˜∆ H f − rf + 2 rf ) + O ( k − ) . On the other hand, by the discussion at the end of the last section, we have ∆ V f = rf − Λ ω Tr( iF h ) . This concludes the proof. (cid:3)
An easy computation shows the following
Corollary 3.4 (c.f. [13]) . Suppose that h is a Hermitian-Einstein metric on E withrespect to ω , i.e. Λ ω ( iF h ) = µI E , where µ is the ω − slope of the bundle E . Then forany x ∈ M , we have π ) r − Z P E ∗ x Scal( ω k ) ω r − g = C ( k ) + k − S ( ω ) + k − (cid:16) r Λ ω (Ric( ω ) ∧ Tr( iF h )) − r ( r + 1) Λ ω (Tr( iF h ) ∧ Tr( iF h )) + 2 r + 1 Λ ω Tr( iF h ∧ iF h ) − µS ( ω ) (cid:17) + O ( k − ) , where C ( k ) is a constant depends on k . Construction of approximate solutions
In this section, we first compute the linearization of the scalar curvature operatorat the K¨ahler metrics ω k . Proposition 4.1. [9]
Let ( Y, ω ) be a K¨ahler manifold of dimension n . Then thelinearization of the scalar curvature operator at the K¨ahler metric ω is given by thefollowing formula. L ( φ ) = (∆ − S ( ω )∆) φ + n ( n − √− ∂∂φ ∧ Ric( ω ) ∧ ω n − ω n , where φ is a smooth function on Y . Applying the above proposition to ( P E ∗ , ω k ), we obtain the following. Proposition 4.2.
Let L k be the linearization of the scalar curvature operator atK¨ahler metrics ω k . Then we have the following L k = ∆ V (∆ V − r ) + O ( k − ) . Proof.
By (3.1), we have √− ∂∂φ ∧ Ric( ω k ) ∧ ω n − k = C nr − n √− ∂∂φ ∧ ω g ∧ π ∗ ω n + O ( k n − ) . Since Scal( ω k ) = r ( r −
1) + O ( k − ) by Theorem 3.1, we have( n + r − n + r − √− ∂∂φ ∧ Ric( ω k ) ∧ ω n + r − k ω n + r − k = r ( r − V + O ( k − ) . The result follows from Proposition 4.1. (cid:3)
We make the following definition of a holomorphic vector field. Let X be a (1 , ∂X = 0. Then X + ¯ X is a real vector field and it is called aholomorphic vector field. A holomorphic vector field generates a one-parameter groupof holomorphic automorphisms.Let ω ∞ be an extremal metric on M and X s be the holomorphic vector field suchthat dS ( ω ∞ ) = ι X s ω ∞ .Let G = Ham( M, ω ∞ ) be the group of Hamiltonian isometries of ( M, ω ∞ ) and g beits Lie algebra. Let G E be the subgroup of all Hamiltonian isometries of ( M, ω ) thatcan be lifted to automorphisms of P E ∗ and let g E be its Lie algebra. g E is the spaceof holomorphic vector fields X on M such that(1) there exist holomorphic vector fields ˜ X of P E ∗ such that π ∗ ˜ X = X ;(2) there exist real valued functions f such that df = ι X ω ∞ .Let h be a Lie sub algebra of g . We denote the space of all Hamiltonians (includingconstant functions) whose gradient vector fields are in h by ¯ h . Fix T ⊆ G E a maximaltorus and K ⊆ G the subgroup of all elements in G that commute with T . Let t and k be the Lie algebras of T and K respectively. Suppose that b ∈ ¯ k . By definition,there exists a holomorphic vector field X on M such that db = ι X ω ∞ . If we furtherassume that b ∈ ¯ t , then there exists a unique holomorphic vector field ˜ X on P E ∗ such that π ∗ ˜ X = X . As a result, we are able to define the Hamiltonian functions l k ( b ) on P E ∗ such that kd ( l k ( b )) = ι ˜ X ω k . However, if b does not belong to ¯ t , then thecorresponding holomorphic vector field does not lift to a holomorphic vector field on P E ∗ . Nevertheless, we are still able to define l k ( b ). In order to do that, we use thefollowing proposition proved in [13]. Proposition 4.3.
For any holomorphic vector field X on M , there exists a uniquesmooth u X ∈ Γ(End( E )) such that Λ ω ∞ ∂ ( ¯ ∂u X − ι X F h ) = 0 , Z M tr( u X ) ω m ∞ = 0 . Moreover, there exists a holomorphic vector field ˜ X on P E ∗ such that π ∗ ˜ X = X ifand only if ¯ ∂u X − ι X F h = 0 . If f ∈ ¯ g E and X be the gradient vector field corresponding to b , we can explicitlycompute l k ( b ) in terms of u X . Indeed, we have the following. Lemma 4.4.
Suppose that holomorphic vector field X has a holomorphic lift ˜ X to P E ∗ , then ι ˜ X ω g = dθ X , where θ X = Tr( u X λ ( h )) . Moreover, if f ∈ ¯ g E such that df = ι X ω ∞ , then d ( θ X + kf ) = ι ˜ X ω k . Inspired by the proceeding lemma, we define the lift of elements of ¯ g to P E ∗ . Definition 4.5.
We define l k : ¯ g → C ∞ ( P E ∗ ) f ∈ ¯ g l k ( f ) = f + k − θ X , where X is the holomorphic vector field on M such that ι X ω = df and θ X =Tr( u X λ ( h )).Suppose that X s ∈ t . Then there exists a holomorphic vector field ˜ X s on P E ∗ sothat π ∗ ˜ X s = X s . Moreover from the definition of the function f → l k ( f ), we concludethat dl k ( S ( ω ∞ )) = k − ι ˜ X s ω k .Let A be a vector space on which the group T acts. Let A T be the subspace of T invariant elements of A . The main goal of this section is to prove the followingproposition. Proposition 4.6.
Let h HE be the Hermitian-Einstein metric on E with respect to ω ∞ , i.e. Λ ω ∞ F ( E,h HE ) = µI E , where µ is the slope of the bundle E . Then thereexist η , η , · · · ∈ C ∞ ( M ) T , Φ , Φ , · · · ∈ Γ( M, W ) T , ϕ , ϕ , · · · ∈ C ∞ ( P E ∗ ) T and b , b , · · · ∈ ¯ k such that for any positive integer p , if ϕ k,p = p X j =2 η j k − j +2 + p X j =2 Φ j k − j +1 + p X j =2 ϕ j k − j , and b k,p = p X j =0 k − j b j , then S ( ω k + √− ∂∂ϕ k,p ) + 12 h∇ l k ( b k,p ) , ∇ ϕ k,p i − l k ( b k,p ) = O ( k − p − ) . Here the gradient and inner product are computed with respect to the K¨ahler metrics ω k . Moreover b = r ( r − and b = S ( ω ∞ ) . Define A ( ω, h ) = S ( ω ) I E + i π Λ ω F h and S ( ω, h ) = Tr( A ( h, ω ) λ ( h )), where F h isthe traceless part of F h . Proposition 4.7.
Suppose that ω ∞ ∈ πc ( L ) is an extremal K¨ahler metric on M and h HE is a Hermitian-Einstein metric on E with the ω ∞ -slope µ . Then we have A , := ddt (cid:12)(cid:12)(cid:12) t =0 A ( ω ∞ + it∂∂η, h HE ( I + tφ ))= (cid:0) D ∗ D η − h∇ ω ∞ S ( ω ∞ ) , ∇ ω ∞ η i ω ∞ (cid:1) I E + i π n (Λ ω ∞ ∂∂ Φ + 2Λ ω ∞ ( F h HE ∧ ( i∂∂η ))) o , where D ∗ D is Lichnerowicz operator (cf. [7, Page 515] ) and { Σ } is the traceless partof Σ , i.e. { Σ } = Σ − r tr (Σ) . Note that we use the operator ∂ to denote the covariantderivative of sections of the bundle End( E ) .Proof. Define f ( t ) = Λ ω ∞ + it∂∂η F ( h HE ( I + tφ )) . Then we have mF ( h HE ( I + tφ )) ∧ ( ω ∞ + it∂∂η ) m − = f ( t )( ω ∞ + it∂∂η ) m . Differentiating with respect to t at t = 0, we obtain m∂∂φ ∧ ω m − ∞ + m ( m − F h HE ∧ ( i∂∂η ) ∧ ω m − ∞ = f ′ (0) ω m ∞ + mf (0)( i∂∂η ) ∧ ω m − ∞ . Since f (0) = µI E , we get f ′ (0) = Λ ω ∞ ∂∂φ + 2Λ ω ∞ ( F h HE ∧ ( i∂∂η )) − µ Λ ω ∞ ( i∂∂η ) I E .On the other hand (cf. [7, pp. 515, 516].) ddt (cid:12)(cid:12)(cid:12) t =0 S ( ω ∞ + it∂∂η ) = D ∗ D η − h∇ ω ∞ S ( ω ∞ ) , ∇ ω ∞ η i ω ∞ . The proposition follows from the above two equations. (cid:3)
Lemma 4.8.
Suppose that ω ∞ ∈ πc ( L ) is an extremal metric on M and h HE be aHermitian-Einstein metric on E , i.e. Λ ω ∞ F ( E,h HE ) = µI E , where µ is the ω ∞ -slope of the bundle E . We have S , := ddt (cid:12)(cid:12)(cid:12) t =0 S ( ω ∞ + it∂∂η, h HE ( I + tφ ))= D ∗ D η − h∇ ω ∞ S ( ω ∞ ) , ∇ ω ∞ η i ω ∞ + i π Tr (cid:16)(cid:8) Λ ω ∞ ∂Dφ + 2Λ ω ∞ ( F h HE ∧ ( i∂∂η )) (cid:9) λ ( h HE ) (cid:17) . Proof.
The proof follows from the previous proposition and the fact that { Λ ω ∞ F ( E,h HE ) } = 0 . Note that ddt (cid:12)(cid:12)(cid:12) t =0 Tr (cid:16) { Λ ω ∞ + it∂∂η F ( h HE ( I + tφ )) } λ ( h HE ( I + tφ )) (cid:17) = Tr (cid:16) ddt (cid:12)(cid:12)(cid:12) t =0 { Λ ω ∞ + it∂∂η F ( h HE ( I + tφ )) } λ ( h HE ) (cid:17) + Tr (cid:16) { Λ ω ∞ F h HE } ddt (cid:12)(cid:12)(cid:12) t =0 λ ( h HE ( I + tφ )) (cid:17) = Tr (cid:16) A , ( η, φ ) λ ( h HE ) (cid:17) . (cid:3) Since T is a compact group, by the uniqueness of the Hermitian-Einstein metric, h is invariant under T . Lemma 4.9.
Suppose that E is Mumford stable and h is a Hermitian-Einstein metricwith respect to ω ∞ , i.e. Λ ω ∞ F h = µI E . Then h is invariant under the action of T . Corollary 4.10.
The scalar curvature of ω k is invariant under the action of T . The above two results follows from the uniqueness of the Hermitian-Einstein metric.
Corollary 4.11.
The map C ∞ ( M ) ⊕ Γ( M, W ) ⊕ ¯ g → C ∞ ( M ) ⊕ Γ( M, W )( η, Φ , b ) S , ( η, Φ) + 12 h∇ ω ∞ S ∞ , ∇ ω ∞ η i ω ∞ − b is surjective. Here S ∞ = S is the scalar curvature of ω ∞ and C ∞ ( M ) is the space ofsmooth functions η on M such that Z M ηω m ∞ = 0 . Moreover, the equivariant version is also valid, that is, C ∞ ( M ) T ⊕ Γ( M, W ) T ⊕ ¯ k → C ∞ ( M ) T ⊕ Γ( M, W ) T ( η, Φ , b ) S , ( η, Φ) + 12 h∇ ω ∞ S ∞ , ∇ ω ∞ η i ω ∞ − b is surjective. Lemma 4.12.
Let η ∈ C ∞ ( M ) and ϕ ∈ C ∞ ( P E ∗ ) . Then h∇ ω k ϕ, ∇ ω k η i ω k = O ( k − ) . Moreover if ϕ ∈ C ∞ ( M ) , then h∇ ω k ϕ, ∇ ω k η i ω k = k − h∇ ω ∞ ϕ, ∇ ω ∞ η i ω ∞ + O ( k − ) . Before we give the proof of proposition, we explain how to find ϕ k, and b k, . Wecan write S ( ω k ) = r ( r −
1) + S k − + S k − + . . . . Note that Corollary 4.10 implies that S ( ω k ) is invariant under the action of T . Thus, S i ∈ C ∞ ( P E ∗ ) T , where S = S ( ω ∞ , h HE ). For any smooth function ϕ on P E ∗ , wehave S ( ω k + k − √− ∂∂ϕ ) = r ( r −
1) + ( S + △ V ( △ V − r ) ϕ ) k − + O ( k − ) ,S ( ω k + k − √− ∂∂ϕ ) = r ( r −
1) + S k − + ( S + △ V ( △ V − r ) ϕ ) k − + O ( k − ) . Hence for η ∈ C ∞ ( M ), Φ ∈ Γ( M, E ) and ϕ ∈ C ∞ ( P E ∗ ), we have S ( ω k + √− ∂∂η + k − √− ∂∂ Φ + k − √− ∂∂ϕ )= r ( r −
1) + S k − + ( S + S , ( η, Φ) + △ V ( △ V − r ) ϕ ) k − + O ( k − ) . Therefore S ( ω k + √− ∂∂η + k − √− ∂∂ Φ + k − √− ∂∂ϕ ) − l ( r ( r −
1) + k − S ( ω ) + k − b )= k − (cid:16) S + △ V ( △ V − r ) ϕ + S , ( η, Φ) − b − Θ s (cid:17) + O ( k − )for some smooth function Θ s . On the other hand h∇ l ( r ( r −
1) + k − S ( ω ∞ ) + k − b ) , ∇ ( η + k − Φ + k − ϕ ) i = k − h∇ ω ∞ S ( ω ∞ ) , ∇ ω ∞ η i ω ∞ + O ( k − ) . Now we can find ϕ ∈ C ∞ ( P E ∗ ) T such that △ V ( △ V − r ) ϕ − b − Θ s ∈ C ∞ ( M ) ⊕ Γ( M, W ) . Lemma 4.12 implies that Θ s is invariant under the action of T . Applying Lemma 4.8implies that there exist η ∈ C ∞ ( M ) T and φ ∈ Γ( M, W ) T and b ∈ such that S , ( η , Φ ) + 12 h∇ ω ∞ S ( ω ∞ ) , ∇ ω ∞ η i ω ∞ = △ V ( △ V − r ) ϕ − b − Θ s . Hence S ( ω k ) − l ( r ( r −
1) + k − S ( ω )) = O ( k − ) ,S ( ω k + √− ∂∂ϕ k, ) + 12 h∇ l ( b k, ) , ∇ ( ϕ k, ) i − l ( b k, ) = O ( k − ) , where ϕ k, = η + k − Φ + k − ϕ and b k, = r ( r −
1) + k − S ( ω ) + k − b . Note that ϕ k, ∈ C ∞ ( P E ∗ ) T , since η , φ and ϕ are invariant under the action of T . Proof of Proposition 4.6.
We prove it by induction on p . Suppose that we have cho-sen η , . . . η p − ∈ C ∞ ( M ) T , Φ , . . . Φ p − ∈ Γ( M, W ) T , ϕ , . . . ϕ p − ∈ C ∞ ( P E ∗ ) T and b , . . . b p − ∈ such that S ( ω k + √− ∂∂ϕ k,p − ) + h∇ l ( b k,p − ) , ∇ ϕ k,p − i − l ( b k,p − ) = k − p ǫ p + O ( k − p − ) . We have S ( ω k,p − + k − p +2 √− ∂∂η p + k − p +1 √− ∂∂ Φ p + k − p √− ∂∂ϕ p )= S ( ω k,p − ) + k − p ( △ V ( △ V − r ) ϕ p + S , ( η p , Φ p )) + O ( k − p − ) . On the other hand, h∇ l ( b k,p − + k − p b p ) , ∇ ( ϕ k,p − + k − p +2 η p + k − p +1 Φ p + k − p ϕ p ) i − l ( b k,p − + k − p b p )= h∇ l ( b k,p − ) , ∇ ϕ k,p − i − l ( b k,p − ) + k − p ( h∇ ω ∞ S ( ω ∞ ) , ∇ ω ∞ η i ω ∞ − b p ) + O ( k − p − ) . Corollary implies that we there exist η p ∈ C ∞ ( M ) T , Φ p ∈ Γ( M, W ) T , ϕ p ∈ C ∞ ( P E ∗ ) T and b p ∈ such that △ V ( △ V − r ) ϕ p + S , ( η p , Φ p ) + 12 h∇ ω ∞ S ( ω ∞ ) , ∇ ω ∞ η i ω ∞ − b p − ǫ p = Constant . This concludes the proof. (cid:3)
Definition 4.13.
Define ω k,p = ω k + √− ∂∂ϕ k,p . For any positive integer p and any b ∈ ¯ g , we define l k,p ( b ) = l k ( b ) − h∇ l k ( b ) , ∇ ϕ k,p i ω k,p .The following lemma is straightforward. Lemma 4.14.
Let b ∈ ¯ g E and X be the holomorphic vector fields on M such that db = ι X ω . Suppose that ˜ X is the holomorphic lift of X to P E ∗ . Then k − dl k,p = ι ˜ X ω k,p . Corollary 4.15.
We have S ( ω k,p ) − l k,p ( b k,p ) = O ( k − p − ) . Proof of Theorem 1.2
The goal of this section is to prove Theorem 1.2. We closely follow [1, 12, 22]. Beforewe give the proof, we go over some estimates from Hong and Br¨onnle. Let’s fix alarge positive integer p . In this section, the operators l = l k,p , D ∗ D and ∇ and innerproducts are with respect to the metrics ω k,p . Proposition 5.1 (c.f. [1]) . Let L = H , be the Sobolev space of functions whose upto 4-th derivatives are in L and ( L ) T is the subspace of T -invariant functions. (1) Let p be a fixed positive integer. There exists a constant C independent of k such that the operators G k,p : ( L ) T × ¯ k → ( L ) T ,G k ( φ, b ) = D ∗ D φ − h∇ S ( ω k,p ) , ∇ φ i − h∇ l k,p ( b k,p )) , ∇ φ i − l k,p ( b ) has right sided inverses P k satisfying || P k || op ≤ Ck . Note that G k,p is thelinearization of the extremal operator at ( ω k,p , b k,p ) . (2) There exists a constant C independent of k such that || Q k,p ( φ, b ) − Q k,p ( ψ, b ′ ) || L ≤ C max( || ( φ, b ) || L , || ( ψ, b ′ ) || L ) || ( φ, b ) − ( ψ, b ′ ) || L , where Q k,p ( φ, b ) = S ( ω k,p + √− ∂∂φ ) + 12 h∇ l k,p ( b k,p + b ) , ∇ φ i − l k,p ( b k,p + b ) − G k,p is the nonlinear part of the extremal operator at ( ω k,p , b k,p ) .Remark . Our setting is slightly different from the setting in [1]. In [1], Br¨onnlestudied non-simple bundles over a base that does not admit nontrivial holomorphicvector field. However, the same proof as in [1] works in our setting.
Proof of Theorem 1.2 .
We want to solve the following equation for φ ∈ C ∞ ( P E ∗ ) T and b ∈ ¯ k . S ( ω k,p + √− ∂∂φ ) + 12 h∇ l ( b k,p + b ) , ∇ φ i = l ( b k,p + b ) . We can write it as the sum of linear and non-linear parts. G k,p ( φ, b ) + Q k,p ( φ, b ) = 0 . Then in order to solve the equation, it suffices to solve the fixed point problem Q k ( φ, b ) = ( φ, b ) , where Q ( φ, b ) = − P k ( Q k,p ( φ, b )). We prove that the map Q is a contraction on theset B := { ( φ, b ) ∈ L × ¯ k | || ( φ, b ) || L ≤ C k − p +2 } for p ≥ k ≫
0. First note that ||Q (0 , || L = || P k ( Q k,p (0 , || L ≤ Ck || Q k,p (0 , || L = Ck || S ( ω k,p ) − l k,p ( b k,p ) || L ≤ C k − p +2 . Let ( φ, b ) , ( φ ′ , b ′ ) ∈ B . We have ||Q ( φ, b ) − Q ( φ ′ , b ′ ) || L ≤ || P k || op || Q k,p ( φ, b ) − Q k,p ( φ ′ , b ′ ) || L ≤ Ck || Q k,p ( φ, b ) − Q k,p ( φ ′ , b ′ ) || L ≤ Ck − p || ( φ, b ) , ( φ ′ − b ′ ) || L . Therefore, ||Q ( φ, b ) − Q (0 , || L ≤ Ck − p || ( φ, b ) || L . This implies that ||Q ( φ, b ) || L ≤ ||Q (0 , || L + Ck − p || ( φ, b ) || L ≤ C k − p +2 , for k ≫ p ≥
6. Hence Q ( φ, b ) : B → B is a contraction for k ≫ p ≥ φ ∈ L and b ∈ ¯ k . Now elliptic regularityimplies that φ is smooth. (cid:3) An immediate consequence of Theorem 1.2 is the following.
Corollary . Let (
M, L ) be a compact polarized manifold and ω ∞ ∈ c ( L ) be anextremal K¨ahler metric. Let X s be the gradient vector field of the scalar curvatureof ω ∞ , i.e. dS ( ω ∞ ) = ι X s ω ∞ . Let E be a Mumford stable holomorphic vector bundleover M . Suppose that all holomorphic vector fields on M can be lifted to holomorphicvector fields on P E ∗ . Then there exist extremal metrics on ( P E ∗ , O P E ∗ (1) ⊗ L k ) for k ≫ Hong’s moment map setting and proof of Theorem 1.1
In this section, we follow [13] to prove Theorem 1.1. As before, let (
M, ω ∞ ) be aK¨ahler manifold of dimension m and G be the group of Hamiltonian isometries of( M, ω ∞ ). Note that the Lie algebra of G is the space of Hamiltonian vector fields on( M, ω ∞ ). Define N = { f ∈ C ∞ ( M ) | ι X ω ∞ = df for some X ∈ g } = Ker( D ∗ D ) . Any ξ ∈ g defines a holomorphic vector field ξ on M . For any ξ ∈ g , there existsa unique smooth function f ξ ∈ C ∞ ( M ) such that(6.1) ι ξ ω ∞ = df ξ and Z M f ξ ω m ∞ = 0 . The following is a straightforward computation.
Proposition 6.1.
The map ξ ∈ g → f ξ is an isomorphism of Lie algebras. Moreover,for any g ∈ G and ξ ∈ g , we have f Ad ( g ) ξ = f ξ ◦ σ g − , where σ g : M → M is defined by σ g ( x ) = g.x . Corollary 6.2.
A function f ∈ N is in the center of N if and only if f is G -invariant. Moreover, if f ∈ C ∞ ( M ) is a G -invariant function, then π N ( f ) is in thecenter of N , where π N : C ∞ ( M ) → N is the orthogonal projection. Let Y be a K¨ahler manifold (open or compact without boundary). Suppose thatthe Lie algebra g acts on Y . Then [ ξ ♯ , ξ ♯ ] = [ ξ , ξ ] ♯ for all ξ , ξ ∈ g . Integratingthe action of g , we obtain an action of G (an open neighborhood of identity in G )on Y . Therefore, there exists an equivariant moment map µ Y : Y → g . Compose µ Y with the map ξ ∈ g → f ξ defined by (6.1), we have an equivariant moment map µ Y : Y → N . We apply this setting to the case when Y is the smooth locus of modulispace of Hermitian-Einstein connections M on a smooth complex vector bundle E and the action of G on Y . More precisely, let E be a smooth complex vector bundle ofrank r on M and h be a fixed hermitian metric on E . We fix a holomorphic structureon det( E ). Let G be the group of unitary gauge transformations of ( E , h ) . Let A be the space of Hermitian-Einstien connections A on E such that ( E , ¯ ∂ A ) is a simpleholomorphic vector bundle and A induces the fixed holomorphic structure on det( E )modulo the action of the unitary gauge group det( E ). Now we define the moduli space of simple Hermitian-Einstein metrics on E asfollows: M = AG . One can compute the tangent space to A and the moduli space M (c.f. [15]): for any A ∈ A , we have T A A = { α ∈ Ω ( M, End( E , h )) | ∇ A α ∈ Ω , and Λ ∇ A α = 0 } . Moreover if [ A ] ∈ M is a smooth point of M , then T [ A ] M = { α ∈ Ω ( M, End( E , h )) | ∇ A α ∈ Ω , and Λ ∇ A α = 0 }{∇ A s | s ∈ Γ( M, End( E , h )) } . Note that the moduli space M is not smooth in general. However, one can define anaction of g on A as follows: Proposition 4.3 tells that for any A ∈ A and any X ∈ g ,there exists a unique u X ∈ Γ( M, End( E )), depending on A , such thatΛ ∂ A ( ¯ ∂ A u X − ι X F A ) = 0 and Z M tr( u X ) ω m = 0 . For any A ∈ A and X ∈ g , define θ X ( A ) = − ( − ∂ A g ∗ X + ¯ ∂ A g X − ι X F A ) ∈ T A A . Note that the vector field θ X is the infinitesimal vector field on A induced by theaction of X . Hong proved that the vector field θ X can be descended to the modulispace M . Moreover, he proved that[ θ X , θ Y ] − θ [ X,Y ] ∈ d A Γ( M, End( E )) . This implies that on the moduli space M , we have [ θ X , θ Y ] = θ [ X,Y ] . Therefore wehave an action of the Lie algebra g on M . Proposition 6.3. ( [13] ) The map µ e : M → N given by µ e ([ A ]) = π N (cid:0) Λ tr( iF A ∧ iF A ) (cid:1) is an equivariant moment map for the action of G on M . By the definition of A , any connection A ∈ A induces the fixed holomorphic struc-ture on det( E ) up to the action of the gauge group. Therefore the function2 r Λ (Ric( ω ) ∧ tr( iF A )) − r ( r + 1) Λ (tr( iF A ) ∧ tr( iF A )) − µS ( ω ) + tr( u X s ) r is independent of the choice of A ∈ A . We define the moment map µ : M → N asfollows: µ ([ A ]) = µ e ([ A ])+ π N (cid:16) r Λ (Ric( ω ) ∧ tr( iF A )) − r ( r + 1) Λ (tr( iF A ) ∧ tr( iF A )) − µS ( ω ) + tr( u X s ) r (cid:17) . Lemma 6.4.
The moment map µ is equivarent if c ( L ) = λc ( E ) for some constant λ ∈ Z .Proof. Since c ( L ) = λc ( E ) , there exists a smooth function ϕ on M such that λ tr( iF A )) − ω ∞ = √− ∂∂ϕ. Taking the trace with respect to ω ∞ , we have rµλ − m =∆ ϕ, where µ is the slope of E . Using the Hermitian-Einstein condition, we obtainthat ϕ is constant and therefore λ tr( iF A )) = ω ∞ . Therefore1 r Λ (Ric( ω ∞ ) ∧ tr( iF A )) − r ( r + 1) Λ (tr( iF A ) ∧ tr( iF A )) − µS ( ω ∞ ) + tr( u X s ) r is invariant under the action of G since ω ∞ is invariant under the action of G . (cid:3) Following [13], we define the following.
Definition 6.5.
A holomorphic structure A is called stable relative to the maximaltorus T if there exists a connection A ∞ in the orbit of [ A ] ∈ M such that [ A ∞ ] is asmooth point of M , µ ([ A ∞ ]) ∈ t and ∂µ∂A ([ A ∞ ]) : T [ A ∞ ] → ¯ k ¯ t is surjective. Remark . The moment map µ is not equivariant in general. If c ( L ) = λc ( E ), thenthe notion of relative stability defined above is a GIT notion of stability introduced bySz´ekelyhidi ([21]). However, it is not clear how the notion of stability defined aboveis related to a GIT notion of stability in general . Proof of Theorem 1.1.
Theorem 1.2 implies that for any A ∈ A , we can find φ A ∈C ∞ ( P E ∗ A ) and b A ∈ ¯ k such that S ( ω Ak,p + √− ∂∂φ A ) + 12 h∇ l ( b A ) , ∇ φ A i = l ( b A ) . Moreover b A has the following expansion. b A = r ( r −
1) + k − S ( ω ) − k − µ ( A ) + k − R A . One can easily see through the computation that ω Ak,p , φ A and b A depend smoothlyon A . Suppose that A ∞ is in the orbit of E and µ ( A ∞ ) ∈ t . Define Φ( A, t ) = µ ( A ) + tR A . Then Φ( A ∞ , ∈ ¯ t and p ( ∂ Φ ∂A ( A ∞ , p : ¯ k → ¯ k ¯ t isthe quotient map. Therefore applying the implicit function theorem, we find A t forsmall t such that µ ( A t ) + tR A ∈ ¯ t for t small enough; hence for k = t − ≫
0, we have b A t = r ( r −
1) + tS ( ω ) − t µ ( A t ) + t R A t = r ( r −
1) + tS ( ω ) ∈ ¯ t . This implies that ω A k − k,p + √− ∂∂φ A k − are extremal metrics for k ≫
0. Note that A k − are compatiblewith the holomorphic structure of E since they are all in the orbit of A ∞ . (cid:3) An holomorphic vector bundle E is called projectively flat, if the curvature of E isof the form c · Id ⊗ ω , where c is a constant and ω is the K¨ahler metric of the basemanifold ( M, L ). Assume that (
M, L ) is an extremal K¨ahler manifold. Then by ourresult, for k ≫
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University of California, Irvine, Department of MathematicsDepartment of Mathematics, University of California, Irvine, Irvine, CA 92697,USA
E-mail address , Zhiqin Lu: [email protected]
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario,N2L 3G1, Canada
E-mail address , Reza Seyyedali:, Reza Seyyedali: