Hamiltonian homogeneity on a complex contact manifold
aa r X i v : . [ m a t h . DG ] N ov Hamiltonian homogeneity on a complex contact manifold
Osami
Yasukura
Abstract.
The notion of c-manifolds is introduced as a generalization ofcomplex contact manifolds as well as the degree of symplectifications. Andthe condition of infinitesimal Hamiltonian homogeneity on a c-manifold isestablished by means of a moment map defined on the symplectification.
0. Introduction.
This paper is concerned with a complex contact manifold ( Z, J Z ; γ ), i.e. acomplex manifold ( Z, J Z ) with a complex contact structure γ in the sense ofS. Kobayashi [11]. W.M. Boothby [4, 5] obtained that any connected simplyconnected compact homogeneous complex contact manifold is biholomor-phic to the adjoint variety Z G of a finitely dimensional complex simple Liealgebra G of rank ≧ a k¨ahlerian C-space of Boothby type . Table of Contents
0. Introduction.1. Principal contact bundles on a c-manifold.2. Hamiltonian vector fields on a principal contact bundle.ReferencesIn §
1, the notion of a c-manifold is defined as a complex contact manifold(of complex dimension ≧
3) or a complex 1-dimensional manifold. And aconcept of a principal contact bundle with a degree δ is introduced on acomplex manifold as well as a notion of a symplectification of degree δ on ac-manifold, which unifies several notions of symplectifications on a (complex)contact manifold [4, 1, 16] such that every c-manifold ( Z, J Z ; γ ) admits aprincipal contact bundle of degree 1, ( P γ , J γ ; ˘ R λ ( λ ∈ C × ) , p γ ; θ γ ), called thestandard symplectification (Lemma 1.2), on which any symplectification ofdegree δ covers as a | δ | -covering space (Corollary 1.7).1n §
2, for a c-manifold (
Z, J Z ; γ ) with a symplectification ( P, J ; θ ) ofdegree δ , it appears that the complex Lie algebra a ( Z, J Z ; γ ) of all con-tactly holomorphic vector fields is C -linear isomorphic to the followingthree C -linear spaces (Theorem 2.3): O δ ( P, J ) of all homogeneous holo-morphic functions of degree δ on ( P, J ), a δ ( P, J ) of the Hamiltonian vec-tor fields of all functions in O δ ( P, J ), and Γ( L γ ) of all holomorphic sec-tions of the contact line bundle L γ on ( Z, J Z ; γ ). In general, for a holo-morphic line bundle L on a complex manifold Z and the dual line bun-dle L ∗ with the canonical projection p L ∗ : L ∗ −→ Z , put P L := L ∗ \ R L ∗ ,λ of λ ∈ C × := C \ P L | ζ ⊂ L ∗ | ζ ( ζ ∈ Z ) with the canonical projection p L ∗ : P L −→ Z , sothat ( P L ; R L ∗ ,λ ( λ ∈ C × ) , p L ∗ ) is a holomorphic principal C × -bundle. Let O ( P L ) be the C -linear space of all holomorphic functions on P L . And Γ( L )the C -linear space of all holomorphic sections of L . For an integer ℓ , put O ℓ ( P L ) := { f ∈ O ( P L ) | f ◦ R L ∗ ,λ = λ ℓ · f ( λ ∈ C × ) } . Let < η, ξ > L denotethe C -value of η ∈ L ∗ | ζ on ξ ∈ L | ζ at ζ ∈ Z . For any ϕ ∈ Γ( L ), put ι L ( ϕ ) : P L −→ C ; η ι L ( ϕ )( η ) := < η, ϕ ( p ( η )) > L . For any f ∈ O ( P L ), thereexists Σ L ( f ) ∈ Γ( L ) such that f ( η ) = < η, Σ L ( f ) | p L ∗ ( η ) > L for all η ∈ P L .Then ι L : Γ( L ) −→ O ( P L ); ϕ ι L ( ϕ ) is a C -linear isomorphism with theinverse Σ L : O ( P L ) −→ Γ( L ); f Σ L ( f ). Put ν m : P L −→ P L ⊗ m ; η η ⊗ m for a positive integer m , which satisfies that ν m ◦ R L ∗ ,λ = R ( L ⊗ m ) ∗ ,λ m ◦ ν m for all λ ∈ C × . Then ι mL : Γ( L ⊗ m ) −→ O m ( P L ); ϕ ι mL ( ϕ ) := ι L ⊗ m ( ϕ ) ◦ ν m is a C -linear mapping such that ι mL ( ϕ )( η ) = < η ⊗ m , ϕ | p ( L ⊗ m ) ∗ ( η ) > L ⊗ m forall η ∈ P L . Lemma
For every positive integer m , ι mL is a C -linear isomorphism.Proof: For any f ∈ O m ( P L ), there exists Σ mL ( f ) ∈ Γ( L ⊗ m ) such that f ( η ) = < η ⊗ m , Σ mL ( f ) | p ( L ⊗ m ) ∗ ( η ) > L ⊗ m for all η ∈ P L . Then ( ι mL ) − = Σ mL : O m ( P L ) −→ Γ( L ⊗ m ); f Σ mL ( f ).For a finite subset { ϕ , · · · , ϕ N } of Γ( L ), the associated map is definedas Φ : P L −→ C N +1 ; η Φ( η ) := ( ι L ( ϕ )( η ) , · · · , ι L ( ϕ N )( η )). Then Φ ◦ R L ∗ ,λ = ˜ R λ ◦ Φ for all λ ∈ C × , where ˜ R λ denotes the scalar multiple by λ on C N +1 . By definition, L is said to be very ample (resp. immersional ) iff thereexists a finite subset { ϕ , · · · , ϕ N } of Γ( L ) such that the projectivization[Φ] : Z −→ P N C of the associated map Φ is well-defined as a holomorphicembedding (resp. immersion), i.e. the associated map Φ is an embedding(resp. an immersion). For a positive integer m , L is said to be ample oforder m (resp. immersionally ample of order m ) iff L ⊗ m is very ample (resp.immersional). By abbreviation, L is said to be ample (or immersionally mple ) iff L is ample (resp. immersionally ample) of order m for somepositive integer m . By constructing a moment map of degree 1, the followingresult is proved in an elementary way (cf. [4], [23, Theorem 2], [2]): Proposition 0.2.
For a connected compact c-manifold, if the contactline bundle is immersional, then it is isomorphic to the adjoint varietyof a finitely dimensional complex simple Lie algebra with the canonical c-structure, so that the contact line bundle is very ample.
1. Principal contact bundles on a c-manifold
Let (
Z, J Z ) be a complex manifold with the real tangent bundle T Z .And X ( Z ) the Lie algebra of all smooth real vector fields on Z . Let Diff( Z )be the group of all smooth diffeomorphisms on Z . And putAut( Z, J Z ) := { α ∈ Diff( Z ) | α ∗ ◦ J Z = J Z ◦ α ∗ } . For X ∈ X ( Z ), let L X be the Lie derivative operator acting on a smoothtensor on Z . Put a ( Z, J Z ) := { X ∈ X ( X ) | L X J Z = 0 } , which is a complexLie algebra with respect to the complex structure J Z : X J Z X . Forthe complex number field C = R ⊕ √− R , put T c Z := C ⊗ R T Z =1 ⊗ T Z + √− ⊗ T Z = T Z + √− T Z as the complexification of
T Z . For X = X + √− X ∈ T c Z , put X := X −√− X , Re( X ) := X = ( X + X ) / X ) := X = ( X − X ) /
2. Let T ′ Z (resp. T ′′ Z ) be the (1, 0) (resp.(0, 1))-tangent bundle of ( Z, J Z ) as the √− −√−
1) eigen subspaceof T c Z with J Z . For X ∈ T c Z , put X ′ := 12 ( X − √− J Z X ) ∈ T ′ Z and X ′′ := 12 ( X + √− J Z X ) ∈ T ′′ Z. Then X = X ′ + X ′′ , 2Re( X ′ ) = X ( X ∈ T Z ), 2Re( Y ) ′ = Y ( Y ∈ T ′ Z ).The holomorphic structure of T Z is induced from T ′ Z by a complex vectorbundle isomorphism: ( T Z, J Z ) −→ ( T ′ Z, √− X X ′ . Put X c ( Z ) := C ⊗ R X ( Z ), which contains a complex Lie subalgebra a ( Z, J Z ) ′ := { X ′ | X ∈ a ( Z, J Z ) } isomorphic to a ( Z, J Z ). Let Λ r ( T ∗ ′ Z ) be the r-alternating tensorproduct of the dual complex vector bundle T ∗ ′ Z of T ′ Z . Put A r, ( Z ) :=Γ(Λ r ( T ∗ ′ Z )) as the set of all holomorphic sections of Λ r ( T ∗ ′ Z ), that is,the set of all holomorphic (r, 0)-forms on ( Z, J Z ). Let O ( Z, J Z ) be the C -algebra of all holomorphic functions on ( Z, J Z ). For any holomorphicmapping p : ( P, J ) −→ ( Z, J Z ), the C -linear extension of the differential3 ∗ : T P −→ T Z is denoted by the same letter p ∗ : T c P −→ T c Z . And therestriction to T ′ P and T ′ Z is denoted as p ∗ ′ : T ′ P −→ T ′ Z .Assume that ( Z, J Z ) is a complex manifold of dim C Z = 2 n + 1 ≧
1, anodd number. Let γ be a set of local holomorphic (1 , γ i : T ′ V i −→ C defined on open subsets V i of Z with Z = ∪ i V i such that(C. 1) ( γ i ∧ ( dγ i ) ∧ n ) | ζ = 0 at all ζ ∈ V i (when 2 n + 1 ≧ γ i | ζ = 0 ∈ T ∗ ′ V i at all ζ ∈ V i (when 2 n + 1 = 1); and(C. 2) γ i = f ij γ j on V i ∩ V j for some f ij ∈ O ( V i ∩ V j , J Z ).In this case, γ is called a c-structure on ( Z, J Z ), and ( Z, J Z ; γ ) is called ac-manifold with the c-distribution E γ := { X ∈ T ′ Z | γ i ( X ) = 0 for some i } .When dim C Z ≧ γ is called a complex contact structure , and ( Z, J Z ; γ )is a complex contact manifold in the sense of S. Kobayashi [11, 12]. Twoc-structures γ , ω on ( Z, J Z ) are said to be equivalent iff E γ = E ω . Whendim C Z = 1, E γ = 0 for any c-structure γ on ( Z, J Z ). Let C ( Z, J Z ) bethe set of the equivalence classes [ γ ] of all c-structures γ on ( Z, J Z ). Amapping α : Z −→ W between two c-manifolds ( Z, J Z ; γ ) and ( W, J W ; ω )is said to be an isomorphism iff α is a biholomorphism such that α ∗ E γ = E ω . And ( Z, J Z ; γ ) and ( W, J W ; ω ) are called isomorphic iff there exists anisomorphism α : Z −→ W . When the dimension of W and Z is non lessthan three, the terminology of “an isomorphism” (resp. “isomorphic”) is alsotermed as “a contact biholomorphism” (resp. “contactly biholomorphic”).Let ( Z, J Z ; γ ) be a c-manifold. PutAut( Z, J Z ; γ ) := { α ∈ Aut(
Z, J Z ) | α ∗ E γ = E γ } ; and a ( Z, J Z ; γ ) := { X ∈ a ( Z, J Z ) | L X γ i = f · γ i for some f ∈ O ( V i , J Z ) } , where L X denotes the Lie derivative by X . And the contact line bundle is defined as the quotient line bundle L γ := T ′ Z/E γ with the canonicalprojection ω γ : T ′ Z −→ L γ . For γ i ∈ γ and ζ ∈ V i , let ˘ γ i ( ζ ) ∈ L ∗ γ | ζ bedefined such that γ i ( X ) = < ˘ γ i ( ζ ) , ̟ γ ( X ) > L γ for all X ∈ T ′ ζ Z . Then˘ γ i : V i −→ L ∗ γ ; ζ ˘ γ i ( ζ ) is a holomorphic local section of L ∗ γ . By thecondition (C. 1), ˘ γ i is non-vanishing. Hence, a holomorphic local section s i of L γ defined on V i is determined by the equation: < ˘ γ i ( ζ ) , s i ( ζ ) > L γ = 1for ζ ∈ V i . Put the canonical line bundle of ( Z, J Z ) as K Z := Λ n +1 ( T ∗ ′ Z ). Proposition 1.1. (i) (
S. Kobayashi [12, p.29]) ̟ γ | T ′ V i = γ i s i ;(ii) ( S. Kobayashi [11, Theorem (2)]) ( L ∗ γ ) ⊗ ( n +1) ∼ = K Z .Proof. (i) For X ∈ T ζ Z , take ν i ( X ) ∈ C such that ̟ γ ( X ) = ν i ( X )( s i | ζ ).Then γ i ( X ) = < ˘ γ i ( ζ ) , ̟ γ ( X ) > L γ = ν i ( X ) < ˘ γ i ( ζ ) , s i ( ζ ) > L γ = ν i ( X ).4ii) Note that (˘ γ i ) ⊗ ( n +1) is a non-vanishing holomorphic local sectionof ( L ∗ γ ) ⊗ ( n +1) . By (C. 1), γ i ∧ ( dγ i ) ∧ n is a non-vanishing holomorphic localsection of K Z . Because of (˘ γ i ) ⊗ ( n +1) = ( f ij ) ( n +1) (˘ γ j ) ⊗ ( n +1) and γ i ∧ ( dγ i ) n =( f ij ) ( n +1) γ j ∧ ( dγ j ) n , the following map is well-defined:( L ∗ γ ) ⊗ ( n +1) −→ K Z ; c (˘ γ i ) ⊗ ( n +1) c ( γ i ∧ ( dγ i ) n ) ( c ∈ R ) , which gives an isomorphism between holomorphic line bundles.Put ( P γ , J γ ; ˘ R λ ( λ ∈ C × ) , p γ ) := ( P L γ ; R L ∗ γ ,λ ( λ ∈ C × ) , p L ∗ γ ) as a holo-morphic principal C × -bundle on ( Z, J Z ) (see, for general notations, §
0. In-troduction). Let π : T ′ P γ −→ P γ be the canonical projection. Put θ γ ( X ) := < π ( X ) , ̟ γ ( p γ ∗ X ) > L γ for X ∈ T ′ P γ , which defines a holomorphic (1 , θ γ on ( P γ , J γ ). Withrespect to local trivializations, ψ i : V i × C × −→ p − γ ( V i ); ( ζ, λ ) ˘ R λ ˘ γ i | ζ , itis represented as follows:(P. γ ) θ γ | ψ i ( ζ,λ ) = λ · p ∗ γ ( γ i | ζ ). Lemma 1.2.
Let ( Z, J Z ; γ ) be a c-manifold with γ := { ( γ i , V i ) } . And ( P, J ; R λ ( λ ∈ C × ) , p ) a holomorphic principal C × -bundle on ( Z, J Z ) withholomorphic local sections { ( σ i , V i ) } . Let θ be a holomorphic (1, 0)-form θ on ( P, J ) which admits an integer δ such that (P. γ ) δ θ | ψ i ( ζ,λ ) = λ δ · p ∗ γ i | ζ with respect to the local trivialization ψ i : V i × C × −→ P ; ( ζ, λ ) R λ σ i ( ζ ) .Then: (i) dθ = δλ δ − · dλ ∧ p ∗ γ i + λ δ · p ∗ ( dγ i ) on p − ( V i ).(ii) ( P, J ; R λ ( λ ∈ C × ) , p ; θ ) satisfies the following four conditions: (P. 0) ker ′ θ := { X ∈ T ′ P | θ ( X ) = 0 } = p − ∗ ′ E γ ;(P. 1) ker ′ θ k p − ∗ ′ (0);(P. 2) δ R ∗ λ θ = λ δ θ for all λ ∈ C × .(iii) δ = 0 if and only if (P. 3) ( P, J ; dθ ) is a holomorphic symplectic manifold . Proof. (i) It follows by direct calculations from the assumption (P. γ ) δ .(ii) The condition (P. 2) δ follows from (P. γ ) δ , And that X ∈ ker ′ θ iff γ i ( p ∗ X ) = 0 for X ∈ T ′ V i , so that ker ′ θ = p − ∗ ′ (ker ′ γ i ) = p − ∗ ′ E γ on V i ,which is (P. 0). And (P. 1) follows from (P. 0).5iii) If δ = 0, then dθ = p ∗ ( dγ i ) on p − ( V i ), which is degenerate becauseof p ∗ ( ψ i ∗ (0 ζ ⊕ T C × )) = { ζ } . Assume that δ = 0. Put dim C Z = 2 n + 1.Then ( dθ ) ∧ ( n +1) = ( n + 1) δλ ( n +1) δ − · dλ ∧ p ∗ ( γ i ∧ ( dγ i ) ∧ n ) on p − ( V i ), whichis not zero by virtue of the condition (C. 1). Hence, dθ is non-degenerate,that is the condition (P. 3).By definition, a principal contact bundle of degree δ on ( Z, J Z ) is a holo-morphic principal C × -bundle ( P, J ; R λ ( λ ∈ C × ) , p ) on a complex mani-fold ( Z, J Z ) with a holomorphic (1 , θ on P satisfying the conditions(P. 1), (P. 2) δ and (P. 3) defined in Lemma 1.2 (cf. [4, § Z, J Z ; γ ), a principal contact bundle of degree δ on( Z, J Z ) satisfying the condition (P. 0) is called a symplectification of degree δ on ( Z, J Z ; γ ). Lemma 1.3.
Let ( P, J ; R λ ( λ ∈ C × ) , p ); θ ) be a principal contact bundleof degree δ on a complex manifold ( Z, J Z ) . Take a set { σ i : V i −→ P } ofholomorphic local sections of the canonical projection p : P −→ Z such that Z = ∪ i V i . Put γ θ := { γ i ∈ A , ( V i ) } with γ i := σ ∗ i θ : T ′ V i −→ C . Then: (i) { ψ i : V i × C × −→ p − ( V i ); ( ζ, λ ) R λ σ i ( ζ ) } defines a system oflocally trivializing holomorphic coordinates on ( P, J ; R λ ( λ ∈ C × ) , p ) . (ii) At each ψ i ( ζ, λ ) ∈ p − ( V i ) , θ | ψ i ( ζ,λ ) = λ δ · p ∗ ( γ i | ζ ) , so that δ = 0 . (iii) There exists a non-negative integer n such that dim C P = 2 n +2 and dim C Z = 2 n + 1 . Put E := p ∗ (ker ′ θ ) . Then E | V i = ker ′ γ i , ker ′ θ = p − ∗ ′ E . (iv) ( Z, J Z ; γ θ ) is a c-manifold with E γ θ := p ∗ (ker ′ θ ) , so that [ γ θ ] doesnot depend on the choice of a set { σ i } of holomorphic local sections of thecanonical projection p : P −→ Z . And ( P, J ; R λ ( λ ∈ C × ) , p ; θ ) is a sym-plectification of degree δ on ( Z, J Z ; γ θ ) .Proof. (i) The holomorphy of the coordinates follows from the holomor-phy of the principal action R λ ( λ ∈ C × ) on P .(ii) Note that ( σ i ◦ p ) ψ i ( ζ, λ ) = σ i ( ζ ) = R /λ ψ i ( ζ, λ ). By (P. 2), p ∗ γ i = θ ◦ ( σ i ◦ p ) ∗ = θ ◦ R /λ ∗ = θ/λ δ on ψ i ∗ ( ζ,λ ) ( T ζ V i ⊕ λ ). By (P. 1), one hasthat θ = λ δ p ∗ γ i . By (P. 3) and Lemma 1.2 (iii), r = 0.(iii) By (P. 3), dim C P is an even number, say, 2 n + 2 ≧
2. Thendim C Z = dim C P − n + 1 ≧
1. By (ii), θ = λ δ · p ∗ γ i on p − ( V i ). Then E | V i = ker ′ γ i and ker ′ θ | p − ( V i ) = p − ∗ ′ (ker ′ γ i ) = p − ∗ ′ ( E | V i ), as required.(iv) Putting λ i : p − ( V i ) −→ C × ; η = R λ σ i ( p ( η )) λ , one has that η = R λ i ( η ) σ i ( p ( η )) = R λ j ( η ) σ j ( p ( η )) for η ∈ p − ( V i ∩ V j ). And put h ij := λ j /λ i : p − ( V i ∩ V j ) −→ C × . Then there exists a holomorphic function g ij on V i ∩ V j such that p ∗ g ij = h ij . In this case, σ i = R g ij σ j . Hence,6 i = σ ∗ i θ = θ ◦ R g ij ∗ ◦ σ j ∗ = g δij · θ ◦ σ j ∗ = g δij · γ j , that is (C. 2) for f ij := g δij . When dim C P = 2: dγ i = 0 by dim C Z = 1. By (P. 3),0 = dθ | p − ( V i ) = δλ δ − i dλ i ∧ p ∗ γ i . Hence, γ i = 0, that is (C. 1). Whendim C P = 2 n + 2 ≧
4: ( dθ ) n +1 = δλ ( n +1) δ − i dλ i ∧ p ∗ ( γ i ∧ ( dγ i ) n ) = 0 by(P. 3). Then γ i ∧ ( dγ i ) n = 0, that is (C. 1). Hence, ( Z, J Z ; γ θ ) is a c-manifold.And ( P, J ; R λ ( λ ∈ C × ) , p ; θ ) is a symplectification of ( Z, J Z ; γ θ ), because of(ii), E γ θ | V i := ker ′ γ i = E | V i and ker ′ θ = p − ∗ ′ E = p − ∗ ′ E γ θ . Example 1.4. ( Principal contact bundles of degree δ )(i) ( Symplectifications of degree ±
1) Let (
Z, J Z ; γ ) be a c-manifold. Bythe condition (P. γ ) and Lemma 1.2, ( P γ , J γ ; ˘ R λ ( λ ∈ C × ) , p γ ; θ γ ) is a prin-cipal contact bundle of degree 1 on ( Z, J Z ), and also a symplectification ofdegree 1 on ( Z, J Z ; γ ), called the standard symplectification of ( Z, J Z ; γ ).(i-1) Put ˘ R − λ := ˘ R − λ . Then ( P γ , J γ ; ˘ R − λ ( λ ∈ C × ) , p γ ; θ γ ) is a symplecti-fication of degree − Z, J Z ; γ ).(i-2) ( V.I. Arnold [1],
T. Oshima & H. Komatsu [18]) As a complexmanifold, P γ is identified with F γ := { λγ i | ζ ∈ T ∗ ′ Z | λ ∈ C × , ζ ∈ Z, γ i ∈ γ } by µ : F γ −→ P γ ; λγ i | ζ λ ˘ γ i ( ζ ). Hence, ( F γ ; µ − ◦ ˘ R λ ◦ µ ( λ ∈ C × ); µ ∗ θ γ )is a symplectification of degree 1 on ( Z, J Z ; γ ).(i-3) ( T. Nitta & M. Takeuchi [16, Theorem 1.3]) Put L × γ := L γ \ p : L × γ −→ Z and π : T ′ L × γ −→ L × γ ; X π ( X ) as the canonicalprojections. Then ( L × γ ; R L γ ,λ ( λ ∈ C ) , p ) is a holomorphic principal C × -bundle. For any η ∈ L × γ , put ˆ η : C −→ L γ | p ( η ) ; λ R L γ ,λ η , which is a C -linear isomorphism, by which L × γ is identified with the C -linear framebundle of L γ . Put θ − γ ( X ) := ˆ π ( X ) − ( ̟ γ ( p ∗ X )) for X ∈ T ′ L × γ . Then( L × γ ; R L γ ,λ ( λ ∈ C × ) , p ; θ − γ ) is a symplectification of degree − Z, J Z ; γ ).(i-4) ( When dim C Z = 1) Put P := T ∗ ′ Z \ p : P −→ Z , on which the restriction of the canonical complex structure J on T ∗ ′ Z and the scalar multiple R λ ( λ ∈ C × ) on each C -linear fiber of T ∗ ′ Z are well-defined. Then θ ∈ A , ( P ) is defined as θ ( X ) := < π ( X ) , p ∗ X > T ′ Z for X ∈ T ′ P , where π : T ′ P −→ P is the canonical projection, so that θ = λ ( p ∗ dz i ) on p − ( V i ) by the local trivializations given as ψ i : V i × C × −→ P ; ( ζ, λ ) λdz i | ζ with the complex coordinates { ( z i , V i ) } on ( Z, J Z ). And( P, J ; R λ ( λ ∈ C × ) , p ; θ ) is a principal contact bundle of degree 1 on ( Z, J Z ),which is the standard symplectification of ( Z, J Z ; { ( dz i , V i ) } ).(ii) ( Hopf-fibration ) Let ∼ be the equivalence relation on the non-zerorow vectors C × N +1 := C N +1 \{ } defined as follows: For z, w ∈ C × N +1 , z ∼ w
7ff there exists λ ∈ C × such that z = λw (=: ˜ R λ w ). Put P N C := C N +1 / ∼ ,the complex pojective space of complex dimension N with ˜ p : C × N +1 −→ P N C ; z = ( z , · · · , z N ) ˜ p N ( z ) =: [ z : · · · : z N ]. Then ( C × N +1 ; ˜ p, ˜ R λ ( λ ∈ C × )) is a holomorphic principal C × -bundle on P N C . For i ∈ { , · · · , N } ,put ζ i : C N +1 −→ C ; z = ( z , · · · , z N ) z i . Assume that N = 2 n + 1 ≧ ζ , ζ , . . . , ζ n +1 ) on C × n +2 ,put ˜ θ := P nk =0 ( ζ k dζ k + n +1 − ζ k + n +1 dζ k ). Then ˜ θ ∈ A , ( C × n +2 ). And( C × n +2 ; ˜ R λ ( λ ∈ C × ) , ˜ p ; ˜ θ ) is a principal contact bundle of degree 2 on P n +1 C .(ii-2) Put C × n +2 := {{± z }| z ∈ C × n +2 } as the quotient space of C × n +2 by the fixed point free action ˜ R a of a ∈ Z := { , − } on C × n +2 with thecanonical projection µ : C × n +2 −→ C × n +2 ; z
7→ {± z } . Put R λ {± z } := {±√ λz } , p : C × n +2 −→ P n +1 C ; {± z } 7→ ˜ p ( z ). Then R λ ◦ µ = µ ◦ ˜ R λ , p ◦ µ = ˜ p . And ( C × n +2 ; p, R λ ( λ ∈ C × )) is a holomorphic principal C × -bundle on P n +1 C . Because of ˜ R ∗ a ˜ θ = a ˜ θ = ˜ θ ( a ∈ Z ), there exists aholomorphic (1 , θ on C × n +2 such that µ ∗ θ = ˜ θ . In this case, θ satisfies the conditions (P. 1) and (P. 3) because of (P. 1) and (P. 3) on ˜ θ .And µ ∗ ( R ∗ λ θ ) = ˜ R ∗ λ ( µ ∗ θ ) = ˜ R ∗ λ ˜ θ = λ ˜ θ = µ ∗ ( λ θ ) because of (P. 2) on ˜ θ .Since µ ∗ is bijective at each tangent space, R ∗ λ θ = λ θ for λ ∈ C × , that is(P. 2) on θ . Hence, ( C × n +2 ; R λ ( λ ∈ C × ) , p ; θ ) is a principal contact bundleof degree 1 on P n +1 C .(iii) ( Adjoint varieties ) Let G be a complex simple Lie algebra with theproduct [ X, Y ] of
X, Y ∈ G , a Cartan subalgebra H , the set △ of all roots,and the root space G α := { X ∈ G| [ H, X ] = α ( H ) X ( H ∈ H ) } of α ∈ △ .Since the Killing form B G : G × G −→ C is non-degenerate on H [7, (2.2.2)], h α ∈ H for α ∈ △ is defined as B G ( h α , H ) = α ( H ) for all H ∈ H . Then H R := ⊕ α ∈△ R h α is a real form of H , on which B G is positive-definite[7, (2.3.4)]. By H. Weyl, there exists e α ∈ G α such that [ e α , e − α ] = − h α and [ e α , e β ] = N α,β e α + β with R ∋ N α,β = N − α, − β , B G ( H, e α ) = 0 and B G ( e α , e β ) = − r α, − β for α, β ∈ △ , H ∈ H [7, (2.2.5), (2.6.4), (2.6.5),p.171], so that U := { H + P α ∈△ x α e α | H ∈ √− H R , C ∋ x α = x − α } is a compact real form of G . Take a lexicographic ordering on H R with △ + := { α ∈ △| α > } . Let ρ be the highest root in △ + with H ρ := B G ( h ρ ,h ρ ) h ρ . Then G = ⊕ i ∈ Z G i for G i := { X ∈ G| [ H ρ , X ] = iX } with G i = { } ( | i | ≧ G ± = G ± ρ , G ± = ⊕ α ( H ρ )= ± G α and G = H ⊕ ( ⊕ α ( H ρ )=0 G α ) as a Z -graded Lie algebra [22, 4.2.Theorem] (cf. [8]). Put L := { Y ∈ G| [ Y, G ρ ] j G ρ } = ⊕ i =0 , , G i , L := { X ∈ G| [ X, G ρ ] = 0 } and G := G ∩ L . Then G = C H ρ ⊕ G , L = G ⊕ G ⊕ G ✁ L = C H ρ ⊕ L G = { P i [ X − i , X + i ] | X ± i ∈ G ± } (cf. [9, Lemma 1]).Let End( G ) be the associative C -algebra of all C -linear endomorphismson G and GL ( G ) the group of all non-singular elements in End( G ) withthe identity transformation 1 : G −→ G ; X X and the exponentialmapping exp : End( G ) −→ GL ( G ); f e f := P ∞ n =0 f n n ! . Put Aut( G ) := { α ∈ GL ( G ) | α [ X, Y ] = [ αX, αY ] } , G the identity connected componentof Aut( G ), and exp G := exp ◦ ad : G −→ GL ( G ); X e ad( X ) , wheread : G −→
End( G ); X ad( X ) is defined by ad( X ) Y := [ X, Y ] for Y ∈ G .For g ∈ G , put L g , R g : G −→ G such as L g ( f ) := gf , R g ( f ) := f g ( f ∈ G ),and that A g := L g ◦ R g − with exp G ( gY ) = A g (exp G ( Y )). For Y ∈ G , put Y L := ddt (cid:12)(cid:12) t =0 L exp G ( tY ) and Y R := ddt (cid:12)(cid:12) t =0 R exp G ( tY ) as smooth vector fieldson G . Because of exp G ( tY ) g = gA g − (exp G ( tY )) = g exp G ( tg − Y ), Y L | g = ( g − Y ) R | g (1)at each g ∈ G . Moreover, ( dA g ) Y R = ( gY ) R , ( dL g ) Y R = Y R , ( dR g ) Y R =( dA g − )( dL g − ) Y R = ( g − Y ) R , ( dν ) Y R = ( − Y ) L , [ X R , Y R ] = [ X, Y ] R and[ X L , Y L ] = ( − [ X, Y ]) L . Let J G be a smooth section of End R ( T G ) givenby J G X R := ( √− X ) R . Put S J ( ˜ X, ˜ Y ) := [ ˜ X, ˜ Y ] + J [ J ˜ X, ˜ Y ] + J [ ˜ X, J ˜ Y ] − [ J ˜ X, J ˜ Y ] with ˜ X := X, X R ; ˜ Y := Y, Y R ; and J := J G , √−
1; respectively.Then S J G ( X R , Y R ) = S √− ( X, Y ) R = 0 R = 0. Hence, J G is a complexstructure on G . By the equation (1), J G Y L (cid:12)(cid:12) g = J G ( g − Y ) R (cid:12)(cid:12) g = ( √− g − Y ) R (cid:12)(cid:12) g = ( √− Y ) L (cid:12)(cid:12) g , (2)( dL f ) J G Y R = ( √− Y ) R = J G ( dL f ) Y R , ( dR f ) J G Y R = ( √− f − Y ) R = J G ( dR f ) Y R , ( dν ) J G Y R = ( −√− Y ) L = J G ( − Y ) L = J G ( dν ) Y R , so that( G, J G ) is a complex Lie group. By Campbell-Hausdorff-Dynkin formula [19,p.29], Y R | exp G ( X ) = ddt (cid:12)(cid:12) t =0 exp G ( X )exp G ( tY ) = ddt (cid:12)(cid:12) t =0 exp G ( X + tβ X ( Y )) ∈ T exp G ( X ) G for some C -linear mapping β X : G −→ G ; Y β X ( Y ). Then β X is surjective since exp G : G −→ G is locally diffeomorphic. For X, Z ∈ G ,put g = exp G ( X ) and Y ∈ G such as Z = β X ( Y ). Then ddt (cid:12)(cid:12) t =0 exp G ( X + tZ ) = ddt (cid:12)(cid:12) t =0 exp G ( X + tβ X ( Y )) = Y R | g and J G Y R | g = ( √− Y ) R | g = ddt (cid:12)(cid:12) t =0 exp G ( X + tβ X ( √− Y )) = ddt (cid:12)(cid:12) t =0 exp G ( X + t √− Z ). Hence, exp G :( G , √− −→ ( G, J G ) is holomorphic, so that φ g := L g ◦ exp G : G −→ G ; X g · exp G ( X ) = g · e ad X gives holomorphic local coordinates of G around g ∈ G , and that ( G, J G )is a complex submanifold of (End( G ) , √− · ) by the canonical embedding,9 G : G −→ End( G ); g g , since ι G ◦ φ g : G −→
End( G ); X g · e ad X isholomorphic. Then π P : ( G, J G ) −→ ( G , √− · ); g ge ρ and π Z := p G ◦ π P :( G, J G ) −→ C P ( G ); g g G ρ are holomorphic, where C P ( G ) is the complexprojective space of G with the canonical projection p G : G\ −→ C P ( G ).Put P G := π P ( G ) j G and Z G := π Z ( G ) such that Z G = p G ( P G ). In thiscase, P G is a G -homogeneous affine algebraic variety in G with the complexstructure J P := √− · | T P G , and that Z G is a G -homogeneous projectivealgebraic variety with the complex structure J Z such that p ∗G J Z = J P , called the adjoint variety of G (cf. [8, Theorem 3.1]).(iii-0) Put L := { α ∈ G | α G ρ = G ρ } with π G/L : G −→ G/L ; g gL .Then L is a complex algebraic subgroup of G corresponding to L . Put L := { α ∈ G | αe ρ = e ρ } with π G/L : G −→ G/L ; g gL . Because of R f ∗ ◦ J G = J G ◦ R f ∗ ( f ∈ L ), there are complex structures J G/L , J G/L on G/L , G/L such that J G/L ◦ π G/L ∗ = π G/L ∗ ◦ J G , J G/L ◦ π G/L ∗ = π G/L ∗ ◦ J G ,respectively. Put ˆ p : ( G/L , J G/L ) −→ ( G/L, J
G/L ) such that π G/L = ˆ p ◦ π G/L . Then ˆ p ∗ ◦ J G/L = J G/L ◦ ˆ p ∗ . Put ι G/L : ( G/L , J G/L ) −→ ( P G , J P )and ι G/L : (
G/L, J
G/L ) −→ ( Z G , J Z ) such that ι G/L ◦ π G/L = π P and ι G/L ◦ π G/L = π Z . Then ι G/L ∗ ◦ J G/L ◦ π G/L ∗ = π P ∗ ◦ J G = √− · ◦ π P ∗ = J P ◦ ι G/L ∗ ◦ π G/L ∗ , so that ι G/L is biholomorphic. By ι G/L ◦ ˆ p ◦ π G/L = p G ◦ ι G/L ◦ π G/L , one has that ι G/L is also biholomorphic. Put G − := G − ⊕ G − ⊂ N := G − ⊕ C H ρ . (3)Then G = G − ⊕ L = N ⊕ L , so that there is an open neighbourhood N of 0 in N such that the mapping, φ g,L : N −→ G/L ; X g exp G ( X ) L ,gives holomorphic local coordinates around gL ∈ G/L with the frame u g : N −→ T gL ( G/L ); Y π G/L ∗ g Y R . (4)Because of ˆ p ∗ ( u g H ρ ) = ˆ p ∗ ( π G/L ∗ g H Rρ ) = π G/L ∗ g H Rρ = 0, one has thatˆ p − ∗ (0) = π G/L ∗ ( C H Rρ ) . (5)Put χ L : L −→ C × ; f B G ( f e ρ , − e − ρ ), which is a complex Lie groupepimorphism: In fact, because of B ( e ρ , − e − ρ ) = 1, for z ∈ C one has that χ L (exp G ( zH ρ )) = e z . (6)Note that f e ρ = χ L ( f ) e ρ for f ∈ L , so that L = χ − L (1). Hence, χ − L : C × −→ L/L = π G/L ( L ); λ χ − L ( λ ) is a complex Lie group isomorphismsatisfying that χ − L ( χ L ( f )) = f L = π G/L ( f ). Put10 λ : G/L −→ G/L ; gL ( gL ) χ − L ( λ ) (7)for λ ∈ C × , which is well-defined by L f = f L for f ∈ L , and that theaction R L/L : G/L × C × −→ G/L ; ( gL , λ ) R λ ( gL ) of C × on G/L is holomorphic , since the action R L : G × L −→ G ; ( g, f ) gf of L on G isholomorphic and π G/L ◦ R L = R L/L ◦ ( π G/L × χ L ). Note that ˆ p ◦ R λ = ˆ p , andthat the action R L/L is simply transitive on each fiber ˆ p − ( gL ). In fact, if gL = g ′ L for some g ′ ∈ G then g ′ = gf for some f ∈ L , that is, R χ L ( f ) gL = g ′ L , where f ∈ L iff gL = g ′ L . Hence, G/L is a holomorphic principal C × -bundle on G/L with the right action R λ ( λ ∈ C × ) and the canonicalprojection ˆ p : G/L −→ G/L . Let θ G be a smooth C -valued 1-form on G defined by θ G ( X R | g ) := B G ( e ρ , X ) at g ∈ G for X ∈ G . Then θ G ( J G X R ) = θ G (( √− X ) R ) = B G ( e ρ , √− X ) = √− B G ( e ρ , X ) = √− θ G ( X R ), thatis, θ G ( T ′′ G ) = 0, so that θ G is a smooth (1 , G, J G ). For X, Y ∈ G , dθ G ( X R , Y R ) = X R ( θ G ( Y R )) − Y R ( θ G ( X R )) − θ G ([ X R , Y R ]) = − θ G ([ X, Y ] R ) = − B G ( e ρ , [ X, Y ]), since θ G ( Z R ) | g = B G ( e ρ , Z ) is a constantfunction of g for Z = X, Y . Then dθ G ( J G X R , Y R ) = − B G ( e ρ , [ √− X, Y ]) = −√− B G ( e ρ , [ X, Y ]) = √− dθ G ( X R , Y R ), that is, dθ G ( T ′′ G, T G ) = 0.Similarly, dθ G ( T G, T ′′ G ) = 0. Hence, dθ G is a smooth (2 , G, J G ),so that θ G is a holomorphic (1 , G, J G ). If f ∈ L , then( R ∗ f θ G ) X R = θ G ( R f ∗ X R ) = θ G (( f − X ) R ) = B G ( e ρ , f − X )= B G ( f e ρ , X ) = B G ( χ L ( f ) e ρ , X ) = χ L ( f ) θ G ( X R ) , so that R ∗ f θ G = χ L ( f ) θ G . In particular, R ∗ f θ G = θ G for f ∈ L . Thenthere is a holomorphic (1 , θ on G/L such that π ∗ G/L ˆ θ = θ G , whichsatisfies the following properties:(P. 1) G ˆ θ (ˆ p − ∗ (0)) = ˆ θ ( π G/L ∗ ( C H Rρ )) = θ G ( C H Rρ ) = B G ( e ρ , C H ρ ) = 0;(P. 2) G R ∗ λ ˆ θ = λ ˆ θ ( λ ∈ C × ), because of π ∗ G/L ( R ∗ χ L ( f ) ˆ θ ) = R ∗ f ( π ∗ G/L ˆ θ ) = R ∗ f θ G = χ L ( f ) θ G = π ∗ G/L ( χ L ( f )ˆ θ ) for f ∈ L ; and(P. 3) G { X ∈ T ( G/L ) | d ˆ θ ( X, Y ) = 0 for Y ∈ T ( G/L ) } = { } : In fact, { X ∈ G| dθ G ( X R , Y R ) = 0 if Y ∈ G} = { X ∈ G| B G ([ e ρ , X ] , G ) = 0 } = { X ∈ G| [ e ρ , X ] = 0 } = L (cf. [4, (5.6)]). Because of π G/L ∗ X R = 0( X ∈ L ), one has the result.Hence, ( G/L , J G/L ; R λ ( λ ∈ C × ) , ˆ p ; ˆ θ ) is a principal contact bundle ofdegree 1 on ( G/L, J
G/L ). Then (
G/L, J
G/L ; γ ˆ θ ) given as Lemma 1.3 (iv)11s a c-manifold with E γ ˆ θ = ˆ p ∗ (ker ˆ θ ) = ∪ g ∈ G { π G/L ∗ X R | X ∈ G − } ′ j T ′ ( G/L ) as the c-distribution such that g ∗ E γ ˆ θ = E γ ˆ θ for g ∈ G . Notethat ( P G , J P ; p G ) is a holomorphic principal C × -bundle on ( Z G , J Z ) withthe right action ˜ R λ := ι G/L ◦ R λ ◦ ι − G/L . Then ˜ R λ e ρ = ι G/L ( L χ − L ( λ )) = ι G/L (exp G ( log λ H ρ ) L ) = λe ρ , the scalar multiple by λ ∈ C × on G\ θ G is defined by ι ∗ G/L θ G = ˆ θ , so that θ G = π ∗ G/L ˆ θ = ( ι G/L ◦ π G/L ) ∗ θ G = π ∗ P θ G , which gives a principal contactbundle of degree 1 on ( Z, J Z ), ( P G , J P ; p G , ˜ R λ ( λ ∈ C × ); θ G ). Put γ G := γ θ G ,which is called the canonical c-structure on ( Z G , J Z ) with the c-distribution E γ G = p G∗ (ker θ G ) = ∪ g ∈ G { π Z ∗ X R | X ∈ G − } ′ j T ′ Z .(iii-1) If dim C Z G = 1, then G − = 0 and G ∼ = sl C , so that Z G ∼ = Z sl C ∼ = P C , the complex projective line.(iii-2) If dim C Z G ≧
3, then rank( G ) ≧
2. In this case, ( Z G , J Z ) is called a k¨ahlerian C-space of Boothby type after [16, Example 1.1], and γ G is saidto be the canonical complex contact structure on ( Z G , J Z ). Lemma 1.5.
Let ( P, J ; R λ ( λ ∈ C × ) , p ; θ ) be a principal contact bundleof degree δ on a complex manifold ( Z, J Z ) . Put P δ := P/ Z δ as the quotientspace by the action of Z δ := { R ω | ω ∈ C × , ω δ = 1 } with the canonicalprojection µ δ : P −→ P δ . Then: (i) P δ uniquely admits a structure of complex manifold with the complexstructure tensor J δ such that µ δ : ( P, J ) −→ ( P δ , J δ ) is holomorphic; (ii) R λ µ δ ( η ) := µ δ ( R λ /δ η ) defines a holomorphic fixed-point free actionof C × on P δ such that µ δ ◦ R λ = R λ δ ◦ µ δ ; (iii) ( P δ , J δ ; R λ ( λ ∈ C × )) is a holomorphic principal C × -bundle on ( Z, J Z ) with a holomorphic mapping p δ : ( P δ , J δ ) −→ ( Z, J Z ) such that p δ ◦ µ δ = p as the projection. (iv) There exists a (1 , -form θ δ on ( P δ , J δ ) such that µ ∗ δ θ δ = θ . In thiscase, ( P δ , J δ ; R λ ( λ ∈ C × ) , p δ ; θ δ ) is a principal contact bundle of degree on ( Z, J Z ) such that p − δ ∗ E γ θ = ker ′ ( θ δ ) . (v) ( P , J ; R λ ( λ ∈ C × ) , p ; θ ) = ( C × n +2 ; R λ ( λ ∈ C × ) , p ; θ ) when ( P, J ; R λ ( λ ∈ C × ) , p ; θ ) = ( C × n +2 ; ˜ R λ ( λ ∈ C × ) , ˜ p ; ˜ θ ) .Proof. (i) In general, the fixed-point free action of every finite subgroupof the structure group of any (holomorphic) principal fiber bundle is properlydiscontinuous in the sense of [13, p.43], so that the quotient space by thisaction uniquely admits a structure of (complex) manifold such that thecanonical projection is a (holomorphic) covering mapping [13, Propoition12.4.3]. Since Z δ is a finite subgroup of C × , the principal action of Z δ is properly discontinuous, so that the quotient space P δ uniquely admitsa structure of complex manifold such that µ δ is a holomorphic coveringmapping.(ii) For λ ∈ C × , η ∈ P δ and i = 1 ,
2. take z i ∈ C × and η i ∈ P such as λ = z δi and η = µ δ ( η i ). There exists ω , ω ∈ Z δ such that z = z ω and η = R ω η , so that µ δ ( R z η ) = µ δ ( R ω ( R z ( R ω η ))) = µ δ ( R z η ). Hence, R λ is well-defined such that µ δ ◦ R z = µ δ ◦ R ( z δ ) /δ = R z δ ◦ µ δ . Since µ δ is a holomorphic submersion, the holomorphy of the action of C × on P δ : C × × P δ −→ P δ ; ( z, η ) R z η follows from the holomorphy of the action of C × on P : C × × P −→ P ; ( z, η ) R z η .(iii) In the step (ii), put p δ ( η ) := p ( η ) = p ( R ω η ) = p ( η ), which well-defines a mapping p δ : P δ −→ Z ; η p δ ( η ) such as p δ ◦ µ δ = p . Since µ δ is aholomorphic submersion, the holomorphy of p δ follows from the holomorphyof p .(iv) In the step (i), put θ δ ( Y ) := θ ( X ) = ω − δ θ ( R ω ∗ X ) = θ ( X ), whichwell-defines a (1, 0)-form θ δ on ( P δ , J δ ) such that µ ∗ δ θ δ = θ . Since µ δ is aholomorphic submersion, the holomorphy of θ δ follows from the holomorphyof θ . For z ∈ C × , µ ∗ δ ( R ∗ z δ θ δ ) = R ∗ z ( µ ∗ δ θ δ ) = R ∗ z θ = z δ · θ = µ ∗ δ ( z δ · θ δ ),i.e. R ∗ λ θ δ = λ · θ δ for λ = z δ . And ker ′ θ δ = µ δ ∗ (ker ′ θ ) = µ δ ∗ ( p − ∗ E γ θ ) = µ δ ∗ (( p δ ◦ µ δ ) − ∗ E γ θ ) = p − δ ∗ E γ θ .(v) It follows from Example 1.4 (ii).For i = 1 ,
2, let P i = ( P i ; R i,λ ( λ ∈ C × ) , p i ; θ i ) be a principal contactbundle on a complex manifold ( Z, J Z ). By definition, a mapping f : P −→ P is said to be a C × -bundle isomorphism iff f is a biholomorphism suchthat p ◦ f = p and R ,λ ◦ f = f ◦ R ,λ ( λ ∈ C × ). And f is said tobe an isomorphism iff f is a C × -bundle isomorphism such that f ∗ θ = θ . In this case, they have the same non-zero degree. And p ∗ (ker ′ θ ) = p ∗ ( f ∗ (ker ′ θ )) = p ∗ (ker ′ θ ), so that E γ θ = E γ θ (i.e. [ γ θ ] = [ γ θ ]) byLemma 1.3 (iii, iv). By definition, two principal contact bundles are said tobe isomorphic iff there exists an isomorphism between them. For a non-zerointeger δ , let P δ ( Z, J Z ) be the set of the isomorphism classes [ P, J ; θ ] of allprincipal contact bundles ( P, J ; θ ) of degree δ on ( Z, J Z ). Then the followingmapping is well-defined: C δ : P δ ( Z, J Z ) −→ C ( Z, J Z ); [ P, J ; θ ] [ γ θ ] . By virtue of Lemma 1.2, the following mapping is well-defined: P : C ( Z, J Z ) −→ P ( Z, J Z ); [ γ ] [ P γ , J γ ; θ γ ] , P γ , J γ ; p γ , R λ ( λ ∈ C × ); θ γ ) is determined by E γ . Theorem 1.6. (i) C ◦ P = id C ( Z,J Z ) ; (ii) P ◦ C = id P ( Z,J Z ) . Proof. (i) Let (
Z, J Z ; γ ) be a c-manifold with γ = { γ i : V i → C } . Then C ( P ([ γ ])) = C ([ P γ , J γ ; θ γ ]) = [ γ ( θ γ ) ], which is equal to [ γ ]: In fact, forany X ∈ T ζ V i , γ i ( X ) = < ˘ γ i ( ζ ) , ̟ γ ( X ) > L γ = θ γ (˘ γ i ∗ ζ X ) = (˘ γ ∗ i θ γ ) X , so that E γ | V i = ker ′ γ i = ker ′ (˘ γ ∗ i θ γ ) = E γ ( θγ ) | V i by Lemma 1.3 (iv).(ii) Let ( P, J ; p, R λ ( λ ∈ C × ); θ ) be a principal contact bundle of de-gree 1 on a complex manifold ( Z, J Z ). Take a set { σ i : V i → P } ofholomorphic local sections of p such as Z = ∪ i V i . According to Lemma1.3, put γ θ = { γ i } with γ i := σ ∗ i θ . Then P ( C ([ P, J ; θ ])) = P ([ γ θ ]) =[ P γ θ , J γ θ ; θ ( γ θ ) ], which is equal to [ P, J ; θ ]: In fact, for any X ∈ T ζ V i , γ i ( X ) = < ˘ γ i ( ζ ) , ̟ γ θ ( X ) > L γθ = θ ( γ θ ) (˘ γ i ∗ ζ X ) = (˘ γ ∗ i θ ( γ θ ) ) X , i.e. γ i = ˘ γ ∗ i θ ( γ θ ) .By Lemma 1.3 (ii), θ ( γ θ ) | R λ ˘ γ i ( ζ ) = λ · p ∗ γ θ γ i | ζ . Take g ji : V i ∩ V j −→ C × suchas σ j ( ζ ) = R g ji σ i ( ζ ). Then γ j | ζ = θ ◦ σ j ∗ ζ = θ ◦ R g ji ∗ ◦ σ i ∗ ζ = g ji · γ i | ζ and < ˘ γ j ( ζ ) , ̟ γ θ ( X ) > = γ j ( X ) = g ji ( ζ ) · γ i ( X ) = < R g ji ( ζ ) · ˘ γ i ( ζ ) , ̟ γ θ ( X ) > ,so that ˘ γ j ( ζ ) = R g ji ( ζ ) ˘ γ i ( ζ ). Then µ : P −→ P γ θ ; R λ σ i ( ζ ) R λ ˘ γ i ( ζ ) iswell-defined as a mapping such that p γ θ ◦ µ = p , which is holomorphicsince the local trivializations are holomorphic. Because of µ ( R a R λ σ i ( ζ )) = R a R λ ˘ γ i ( ζ ) = R a ( µ ( R λ σ i ( ζ ))), µ ◦ R a = R a ◦ µ ( a ∈ C × ). By Lemma1.3 (ii), µ ∗ ( θ ( γ θ ) | R λ ˘ γ i ( ζ ) ) = λ · ( γ i | ζ ◦ p γ θ ∗ ) ◦ µ ∗ R λ σ i ( ζ ) = λ · γ i | ζ ◦ p ∗ R λ σ i ( ζ ) = θ | R λ σ i ( ζ ) . Hence, µ : ( P, J ; θ ) −→ ( P γ θ , J γ θ ; θ ( γ θ ) ) is an isomorphism. Corollary 1.7. (i)
Let ( P, J ; R λ ( λ ∈ C × ) , p ; θ ) be a principal con-tact bundle of degree δ on a complex manifold ( Z, J Z ) . Then there existsa holomorphic | δ | -covering map µ : P −→ P γ θ such that p = p γ θ ◦ µ, θ = µ ∗ θ ( γ θ ) , R λ δ ◦ µ = µ ◦ R λ ( λ ∈ C × ) . (ii) Let ( P, J ; R λ ( λ ∈ C × ) , p ; θ ) be a symplectification of degree δ ona c-manifold ( Z, J Z ; γ ) . Then there exits a holomorphic | δ | -covering map µ : P −→ P γ such that p = p γ ◦ µ, θ = µ ∗ θ γ , ˘ R λ δ ◦ µ = µ ◦ R λ ( λ ∈ C × ) .Proof. (0) By Lemma 1.5, there exists a holomorphic | δ | -covering µ δ : P −→ P δ such that R λ δ ◦ µ δ = µ δ ◦ R λ ( λ ∈ C × ), p = p δ ◦ µ δ and θ = µ ∗ δ θ δ ,so that [ γ θ ] = [ γ θ δ ] = C ([ P δ , J δ ; θ δ ]).(i) By Theorem 1.6 (ii), [ P δ , J δ ; θ δ ] = P ( C ([ P δ , J δ ; θ δ ])) = P ([ γ θ ])= [ P γ θ , J γ θ ; θ γ θ ]. Hence, there exists an isomorphism µ : ( P δ , J δ ; θ δ ) −→ ( P γ θ , J γ θ ; θ γ θ ) . Then p γ θ ◦ µ ◦ µ δ = p δ ◦ µ δ = p , ( µ ◦ µ δ ) ∗ θ ( γ θ ) = µ ∗ δ ( µ ∗ θ ( γ θ ) ) = µ ∗ δ θ δ = θ and µ ◦ µ δ ◦ R λ = µ ◦ R λ δ ◦ µ δ = ˘ R λ δ ◦ µ ◦ µ δ ( λ ∈ C × ), so thatthe claim (i) follows by putting µ := µ ◦ µ δ .(ii) By (P. 0) and Lemma 1.3 (iv), [ γ ] = [ γ θ ] = C ([ P δ , J δ ; θ δ ]). ByTheorem 1.6 (ii), [ P δ , J δ ; θ δ ] = P ( C ([ P δ , J δ ; θ δ ])) = P ([ γ ]) = [ P γ , J γ ; θ γ ],14hich gives an isomorphism µ : ( P δ , J δ ; θ δ ) −→ ( P γ , J γ ; θ γ ). Then p γ ◦ µ ◦ µ δ = p δ ◦ µ δ = p , ( µ ◦ µ δ ) ∗ θ γ = µ ∗ δ ( µ ∗ θ γ ) = µ ∗ δ θ δ = θ and µ ◦ µ δ ◦ R λ = µ ◦ R λ δ ◦ µ δ = ˘ R λ δ ◦ µ ◦ µ δ ( λ ∈ C × ), so that the claim (ii) follows byputting µ := µ ◦ µ δ .
2. Hamiltonian vector fields on a principal contact bundle.
Let (
P, J ; R λ ( λ ∈ C × ) , p ; θ ) be a principal contact bundle of degree δ = 0 on a complex manifold ( Z, J Z ). For η ∈ P and X ∈ T ′ η P , put X ♮ := − ι X dθ ∈ T ∗ ′ η P such that < X ♮ , Y > T ′ η P := − dθ ( X, Y ) for Y ∈ T ′ η P .By (P. 3), ( dθ ) ♮η : T ′ η P −→ T ∗ ′ η P ; X X ♮ , is a C -linear isomorphism, whichgives a holomorphic vector bundle isomorphism, ( dθ ) ♮ : T ′ P −→ T ∗ ′ P . Thenthe inverse, Ξ := (( dθ ) ♮ ) − : T ∗ ′ P −→ T ′ P ; ω Ξ( ω ), is a holomorphicvector bundle isomorphism such that ι Ξ( ω ) dθ = − ω . For f ∈ O ( W, J ), put X f := 2 Re(Ξ( df )) = Ξ( df ) + Ξ( df ), so that X ′ f = Ξ( df ). Then X f is aholomorphic real vector field on ( W, J ) such that ι X f dθ = − df , which issaid to be the Hamiltonian vector field of f on W .Let ℓ be an integer. And W an open subset of P . Put O ℓ ( W, J ) := { f ∈O ( W, J ) | Ξ( θ ) f = − ℓδ f } and a ℓ ( W, J ) := { X f | f ∈ O ℓ ( W, J ) } . Because of X √− f = J X f for f ∈ O ( W, J ), one has that a ℓ ( W, J ) is a C -linear subspaceof a ( W, J ). For w ∈ C and η ∈ W , put B w | η := ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 R e wt η = 2 Re( w ddz (cid:12)(cid:12)(cid:12)(cid:12) z =1 R z η ) ∈ T η W. (8)Then p − ∗ (0) = { ( a + bJ ) B | a, b ∈ R } ⊆ T P . By the locally trivializingholomorphic coordinates ψ i ( ζ, λ ) := R λ σ i ( ζ ) of W given in Lemma 1.3 (i), ddz (cid:12)(cid:12) z =1 R z ψ i ( ζ, λ ) = ddz (cid:12)(cid:12) z =1 ψ i ( ζ, zλ ) = λ ∂∂λ (cid:12)(cid:12) ψ i ( ζ,λ ) ∈ T ′ ψ i ( ζ,λ ) P , so that B ′ w | ψ i ( ζ,λ ) = (cid:18) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 R e tw ψ i ( ζ, λ ) (cid:19) ′ = w ddz (cid:12)(cid:12)(cid:12)(cid:12) z =1 R z ψ i ( ζ, λ ) = wλ ∂∂λ (cid:12)(cid:12)(cid:12)(cid:12) ψ i ( ζ,λ ) . Lemma 2.1. (i) Ξ( θ ) = − λδ ∂∂λ , so that B w = − wδ Re(Ξ( θ )) and B ′ w = − wδ Ξ( θ ) for w ∈ C , and that p − ∗ (0) = Re( C Ξ( θ )) and O ℓ ( W, J ) = { f ∈ O ( W, J ) | B w f = ℓwf ( w ∈ C ) } . n particular, O ℓ ( P, J ) = { f ∈ O ( P, J ) | R ∗ λ f = λ ℓ f ( λ ∈ C × ) } . (ii) R λ ∗ Ξ( θ ) = Ξ( θ ) for any λ ∈ C × . (iii) θ (Ξ( df )) = θ ( X f ) = mδ f for all f ∈ O ℓ ( W, J ) . When ℓ = 0 , χ ℓ : O ℓ ( W, J ) −→ a ℓ ( W, J ); f χ ℓ ( f ) := X f is a C -linear isomorphism suchthat ( χ ℓ ) − = δℓ θ. (iv) Assume that ω ∈ A , ( W, J ) ; f, g ∈ O ( W, J ) ; and Y ∈ X ( W ) . Then L Ξ( ω ) dθ = − dω , L Ξ( df ) dθ = 0 and [ Y, Ξ( ω )] = Ξ( L Y ω + ι Ξ( ω ) ( L Y dθ )) , sothat [ X f , X g ] = X dθ ( X ′ f , X ′ g ) . (v) O ℓ ( W, J ) j { f ∈ O ( W, J ) | [ B w , X f ] ′ = w ( ℓ − δ ) · X ′ f ( w ∈ C ) } and [ a ℓ ( W, J ) , a ℓ ′ ( W, J )] j a ℓ + ℓ ′ − δ ( W, J ) . In particular, O ℓ ( P, J ) j { f ∈ O ( P, J ) | R λ ∗ X ′ f = λ δ − ℓ X ′ f ( λ ∈ C × ) } .Proof. (i) By Lemmas 1.3 (ii) and 1.2 (i), dθ ( λ ∂∂λ , Y ) = δλ δ ( p ∗ ( γ i | ζ ) Y ) = δθ ( Y ) = − δdθ (Ξ( θ ) , Y ) for Y ∈ T ψ i ( ζ,λ ) ( p − V i ), so that the 1st equationfollows. The 2nd and the 3rd equations follow from the 1st one. The 4thand the 5th equations follow from the 2nd one. For f ∈ O ℓ ( P, J ) and w ∈ C , ddt (cid:12)(cid:12) t =0 f ( R e tw η ) = B w f | η = ℓwf ( η ), so that ddt ( e − ℓwt · f ( R e wt η )) ≡ e − wwt · dds (cid:12)(cid:12) s =0 { e − ℓws · f ( R e wt η )+ f ( R e ws ( R e wt η )) } ≡ e − wwt ·{− ℓw · f ( R e wt η )+ ℓw · f ( R e wt η ) } ≡
0, i.e. e − ℓwt · f ( R e wt η ) ≡ f ( η ), that gives the last equation.(ii) For any X ∈ T η P at each η ∈ P and λ ∈ C × , dθ ( X, R λ ∗ Ξ( θ )) = λ δ dθ ( R − λ ∗ X, Ξ( θ )) = λ δ θ ( R − λ ∗ X ) = θ ( X ) = dθ ( X, Ξ( θ )), as required. It alsofollows from B = − δ Re(Ξ( θ )).(iii) If f ∈ O ℓ ( W, J ), then θ ( X f ) = − dθ (Ξ( θ ) , X f ) = − df (Ξ( θ )) = − Ξ( θ ) f = ℓδ f . By the definition, χ ℓ is a C -linear surjection. If ℓ = 0, then δℓ θ ◦ χ ℓ = id O ℓ ( W,J ) , so that χ ℓ is an injection.(iv) L Ξ( ω ) dθ = d ( ι Ξ( ω ) dθ ) + ι Ξ( ω ) ddθ = − dω , so that L Ξ( df ) dθ = − ddf =0. And ι [ Y, Ξ( ω )] dθ = [ L Y , ι Ξ( ω ) ] dθ = L Y ( ι Ξ( ω ) dθ ) − ι Ξ( ω ) ( L Y dθ ) = − ( L Y ω + ι Ξ( ω ) ( L Y dθ )), so that [ Y, Ξ( ω )] = Ξ( L Y ω + ι Ξ( ω ) ( L Y dθ )). In particular,[ X f , X g ] ′ = [Ξ( df ) , Ξ( dg )] = Ξ( L Ξ( df ) dg + ι Ξ( dg ) ( L Ξ( df ) dθ )) = Ξ( L Ξ( df ) dg ) =Ξ( d ( ι Ξ( df ) dg )) = Ξ( d ( − dθ (Ξ( dg ) , Ξ( df )))) = X ′− dθ ( X ′ g ,X ′ f ) = X ′ dθ ( X ′ f ,X ′ g ) .(v) By (iv), L B ′ w dθ = − wδL Ξ( θ ) dθ = wδdθ , so that ι Ξ( df ) ( L B ′ w dθ ) = ι Ξ( df ) ( wδdθ ) = − wδdf . If f ∈ O ℓ ( W, J ), then [ B w , X f ] ′ = [ B ′ w , Ξ( df )] =Ξ( L B ′ w df + ι Ξ( df ) ( L B ′ w dθ )) = Ξ( d ( B ′ w f ) − wδdf ) = Ξ( w ( ℓ − δ ) df ) = w ( ℓ − δ ) X ′ f , that gives the 1st claim. If g ∈ O ℓ ′ ( W, J ), then B ′ w dθ ( X ′ f , X ′ g ) =( L B ′ w dθ )( X ′ f , X ′ g ) + dθ ([ B ′ w , X ′ f ] , X ′ g ) + dθ ( X ′ f , [ B ′ w , X ′ g ]) = wδdθ ( X ′ f , X ′ g ) + dθ ( X ′ w ( ℓ − δ ) f , X ′ g ) + dθ ( X ′ f , X ′ w ( ℓ ′ − δ ) g ) = w ( ℓ + ℓ ′ − δ ) dθ ( X ′ f , X ′ g ), so that[ X f , X g ] = X dθ ( X ′ f ,X ′ g ) ∈ a ℓ + ℓ ′ − δ ( W, J ) by (iv) and (i), that is the 2ndclaim. If f ∈ O ℓ ( P, J ), then ddt (cid:12)(cid:12) t =0 R e wt ∗ ( X ′ f | R e − wt η ) = − [ B w , X f ] ′ | η =16 w ( ℓ − δ ) X ′ f | η , so that e ( ℓ − δ ) wt · R e wt ∗ ( X ′ f | R e − wt η ) ≡ X ′ f | η , that gives thelast claim. Another proof of the last claim when ℓ = 0 : If λ ∈ C × , then θ ( R λ ∗ X f ) | η = λ δ θ ( X f ) | R − λ η = λ δ ℓδ f | R − λ η = λ δ − ℓ ℓδ f | η = λ δ − ℓ θ ( X f ) | η by(iii), so that δℓ χ ℓ ( θ ( R λ ∗ X f )) = δℓ χ ℓ ( λ δ − ℓ θ ( X f )), i.e. R λ ∗ X ′ f = λ δ − ℓ X ′ f .By Lemma 2.1 (i), each f ∈ O ℓ ( W, J ) is said to be a homogeneousholomorphic function of degree ℓ on W , and that Ξ( θ ) is said to be theEuler operator of ( P, J ; R λ ( λ ∈ C × ) , p ; θ ). Put a ( P, J ; dθ, R C × ) := { X ∈ a ( P, J ; dθ ) | dR λ X = X ( λ ∈ C × ) } ,a ( W, J ; θ ) := { X ∈ a ( W, J ) | L X θ = 0 } ,a ( W, J ; dθ ) := { X ∈ a ( W, J ) | L X ( dθ ) = 0 } ,a ( W, J ; dθ, Ξ( θ )) := { X ∈ a ( W, J ; dθ ) | [Ξ( θ ) , X ] = 0 } . Lemma 2.2. (i)
For X ∈ a ( W, J ; θ ) , θ ( X ) = 0 ∈ O ( W, J ) iff X = 0 . (ii) a δ ( W, J ) = a ( W, J ; θ ) = a ( W, J ; dθ, Ξ( θ )).(iii) a δ ( P, J ) = a ( P, J ; θ ) = a ( P, J ; dθ, R C × ) .Proof. (i) If X = 0, then θ ( X ) = 0 since θ is a linear form. For X ∈ a ( W, J ; θ ), 0 = L X θ = d ( θ ( X )) + ι X dθ . If θ ( X ) = 0 ∈ O ( W, J ), then ι X dθ = 0, so that X = 0 because of the condition (P. 3).(ii-1) If f ∈ O δ ( W, J ), L X f θ = d ( θ ( X f )) + ι X f dθ = d ( δδ f ) + ι X f dθ = 0,by Lemma 2.1 (iii) with ℓ = δ , so that a δ ( W, J ) j a ( W, J ; θ ) . (ii-2) For Y ∈ a ( W, J ; θ ), put f := θ ( Y ) ∈ O ( W, J ). For z ∈ C , θ ( B z ) = 0 by the condition (P. 1). Then L B z θ = d ( θ ( B z )) + ι B z dθ = zδ · θ , [ B z , Y ] = zδ Ξ( L Y θ + ι Ξ( θ ) dL Y θ ) = 0, so that B z f = ( L B z θ )( Y ) + θ ([ B z , Y ]) = ( zδθ )( Y ) = zδf . Hence, f ∈ O δ ( W, J ) and X f ∈ a δ ( W, J ) j a ( W, J ; θ ). By Lemma 2.1 (iii), θ ( Y − X f ) = f − δδ f = 0 so that Y − X f = 0by the claim (i). Hence, a ( W, J ; θ ) j a δ ( W, J ).(ii-3) For X ∈ a ( W, J ; dθ ), Ξ( L X θ ) = [ X, Ξ( θ )]. Hence, [ X, Ξ( θ )] = 0 iff L X θ = 0, that is, a ( W, J ; dθ, Ξ( θ )) = a ( W, J ; θ ).(iii) For z ∈ C and X ∈ a ( W, J ), [ B z , X ] = 2Re( − zδ [Ξ( θ ) , X ]). Hence, a ( W, J ; dθ, Ξ( θ )) = { X ∈ a ( W, J ; dθ ) | [ B z , X ] = 0 ( z ∈ C ) } , so that a ( P, J ; dθ, Ξ( θ )) = a ( P, J ; dθ, R C × ).Let ( Z, J Z ; γ ) be a c-manifold with the contact line bundle L γ and thestandard symplectification ( P γ , J γ ; ˘ R λ ( λ ∈ C × ) , p γ ; θ γ ) defined in Example17.4 (i). For g ∈ Aut(
Z, J Z ; γ ) and η ∈ P γ , < η, ̟ γ ( g − ∗ X ) > L γ ∈ C doesnot depend on the choice of X , because of g ∗ E γ = E γ . Hence, g P ( η ) ∈ P ∗ γ | p γ ( η ) is well-defined by < g P ( η ) , ̟ γ ( X ) > L γ := < η, ̟ γ ( g − ∗ X ) > L γ forall X ∈ T ′ p γ ( η ) Z . For λ ∈ C × and g , g ∈ Aut(
Z, J Z ; γ ), p γ ◦ g P = g ◦ p γ ,˘ R λ ◦ g P = g P ◦ ˘ R λ and ( g ◦ g ) P = g P ◦ g P . By the definition of θ γ , θ γ ( g P ∗ Y ) = < π ( g P ∗ Y ) , ̟ γ ( p γ ∗ g P ∗ Y ) > L γ = < g ( πY ) , ̟ γ ( g ∗ p γ ∗ Y ) > L γ = < π ( Y ) , ̟ γ ( g − ∗ g ∗ ( p γ ∗ Y )) > L γ = θ γ ( Y )for Y ∈ T ′ P γ . so that g P ∈ { f ∈ Aut( P γ , J γ ) | f ∗ θ γ = θ γ , R λ ◦ f = f ◦ R λ ( λ ∈ C × ) } =: Aut( P γ , J γ ; θ γ , R C × ), which is said to be the prolongation of g (cf.[4, (3.1)], [16]). Then Pro : Aut(
Z, J Z ; γ ) −→ Aut( P γ , J γ ; θ γ , R C × ) : g g P , is well-defined as a group homomorphism such that p γ is equivariant.For any X ∈ a ( Z, J Z ; γ ), let exp Z tX be a locally defined one parametergroup determined by X as: ddt (cid:12)(cid:12) t =0 (exp Z tX ) ζ = X | ζ . Let (exp Z tX ) P be theprolongation defined on p − γ ( U ) for an open subset U around each point of Z , which gives X P | η := ddt (cid:12)(cid:12) t =0 (exp Z tX ) P η ∈ T η P γ at each η ∈ P γ . Then X P ∈ a ( P γ , J γ ; θ γ ) = a ( P γ , J γ ) (Lemma 2.2 (ii)) such that ( dp γ ) X P = X ,so that pro : a ( Z, J Z ; γ ) −→ a ( P γ , J γ ); X X P is a complex Lie algebrahomomorphism such that dp γ ◦ pro = id a ( Z,J Z ; γ ) , which is said to be theinfinitesimal prolongation isomorphism because of the following result (cf.[12, Theorem I.7.1], [18, p.74, Theorem 2.4’], [16, Theorem 1.6, Corollary]): Theorem 2.3.
Let ( P, J ; R λ ( λ ∈ C × ) , p ; θ ) be a symplectification ofdegree δ on a c-manifold ( Z, J Z ; γ ) with the contact line bundle L γ . Thenthe following is a commutative diagram of C -linear isomorphisms: dp γ dµa ( Z, J Z ; γ ) ⇆ a ( P γ , J γ ) ←− a δ ( P, J ) pro ̟ γ ↓ θ γ ↓↑ χ θ ↓↑ χ δ Σ L γ Γ( L γ ) ⇆ O ( P γ , J γ ) −→ O δ ( P, J ) ι L γ µ ∗ where µ : P −→ P γ is the one given in Corollary 1.7 (ii).Proof. By Lemma 2.1 (iii), χ δ = θ − and χ = θ − γ . By Corollary 1.7(ii), µ ◦ ˘ R λ = R λ δ ◦ µ ( λ ∈ C × ), so that µ ∗ : O ( P γ , J γ ) −→ O δ ( P, J ); f f ◦ µ , is well-defined as a C -linear isomorphism. And dθ γ ( µ ∗ η X f , µ ∗ η Y ) =18 θ ( X f , Y ) | η = df ( Y ) = d ( µ ∗ f )( µ ∗ η Y ), so that µ ∗ η X f = X µ ∗ f | µ ( η ) . Hence, dµ : a δ ( P, J ) −→ a ( P γ , J γ ); X f dµ ( X f ) = X µ ∗ f , is well-defined as a C -linear mapping. For X ∈ a δ ( P, J ), θ ( X | η ) = θ δ ( µ ∗ ( X | η )) = θ δ ( dµ ( X ) | µ ( η ) ),so that θ = µ ∗ ◦ θ δ ◦ dµ . Hence, dµ = θ − δ ◦ ( µ ∗ ) − ◦ θ , is a C -linear isomor-phism, so that the right hand of the diagram gives a commutative diagramof C -linear isomorphisms. By Lemma 2.2 (iii), Y := dp γ ( X ) ∈ a ( Z, J Z )is well-defined for X ∈ a δ ( P, J ). And exp( tY ) ∗ E γ = p ∗ (exp( tX ) ∗ ker ′ θ ) = p ∗ (ker ′ exp( tX ) ∗ θ ) = p ∗ (ker ′ θ ) = E γ , so that Y ∈ a ( Z, J Z ; γ ). Hence, dp γ : a δ ( P, J ) −→ a ( Z, J Z ; γ ); X dp γ ( X ), is well-defined as a C -linearsurjection since dp γ ◦ pro = id a ( Z,J Z ; γ ) is surjective. By the definition of θ γ , θ γ = ι L γ ◦ ̟ γ ◦ dp γ , which completes the commutativity of the dia-gram. Since θ γ is bijective, dp γ is injective, so that dp γ is bijective and that pro = ( dp γ ) − . By Lemma 0.1, Σ L γ = ( ι L γ ) − . Then ̟ γ = Σ L γ ◦ θ ◦ pro isa C -linear isomorphism.In general, put V | ξ := { X | ξ | X ∈ V } for any subset V of real vectorfields on a manifold M and a point ξ ∈ M as a subset of the real tangentspace T ξ M at ξ ∈ M . By definition, a c-manifold ( Z, J Z ; γ ) is said to be homogeneous iff there exists ζ ∈ Z such that Aut( Z, J Z ; γ ) · ζ = Z . And( Z, J Z ; γ ) is said to be infinitesimally homogeneous iff T ζ Z = a ( Z, J Z ; γ ) | ζ atall ζ ∈ Z . By definition, ( Z, J Z ; γ ) is said to be infinitesimally Hamiltonianhomogeneous iff T η P = a δ ( P, J ) | η at all η ∈ P for some symplectification( P, J ; θ ) of degree δ on ( Z, J Z ; γ ). In this case, ( Z, J Z ; γ ) is infinitesimallyhomogeneous by Theorem 2.3. Proposition 2.4. (i) Aut(
Z, J Z ; γ ) of a connected compact c-manifold ( Z, J Z ; γ ) is a finitely dimensional complex Lie group acting holomorphicallyon ( Z, J Z ) with the Lie algebra a ( Z, J Z ; γ ) . (ii) Let ( Z, J Z ; γ ) be an infinitesimally homogeneous connected compactc-manifold. Then it is homogeneous. If Z is simply connected, then thereexists a conected complex semisimple Lie subgroup G Z of Aut(
Z, J Z ; γ ) suchthat Z is one orbit of G Z .Proof. (i) By S. Bochner & D. Montgomery [12, Theorem III.1.1],Aut( Z, J Z ) is a complex Lie transformation group with the Lie algebra a ( Z, J Z ). And Aut( Z, J Z ; γ ) is a closed subgroup of Aut( Z, J Z ) with theLie algebra a ( Z, J Z ; γ ). Let Γ local ( E γ ) be the set of all holomorphic lo-cal sections of E γ on Z . Then X ∈ a ( Z, J Z ; γ ) iff X ∈ a ( Z, J Z ) and[ X, Y ] ∈ Γ local ( E γ ) for any Y ∈ Γ local ( E γ ) stabilizing E γ . In this case,[ J Z X, Y ] = J Z [ X, Y ] ∈ E γ . Hence, a ( Z, J Z ; γ ) is a complex Lie subalge-bra of ( a ( Z, J Z ) , J Z ), so that Aut( Z, J Z ; γ ) is a complex Lie subgroup of19ut( Z, J Z ) with the Lie algebra a ( Z, J Z ; γ ).(ii) For ζ ∈ Z , put ˜ ζ : Aut( Z, J Z ; γ ) −→ Z ; f f ( ζ ), which is a smoothmapping such that ˜ ζ ∗ ( T f Aut(
Z, J Z ; γ )) = a ( Z, J Z ; γ ) | f ( ζ ) by (i). Because of a ( Z, J Z ; γ ) | f ( ζ ) = T f ( ζ ) Z for all ζ ∈ Z , ˜ ζ (Aut( Z, J Z ; γ )) is open in Z . Notethat Z = ∪ ζ ∈ Z ˜ ζ (Aut( Z, J Z ; γ )) with ˜ ζ (Aut( Z, J Z ; γ )) ∩ ˜ ζ ′ (Aut( Z, J Z ; γ )) = ∅ or ˜ ζ (Aut( Z, J Z ; γ )). Since Z is connected, ˜ ζ (Aut( Z, J Z ; γ )) = Z . If Z issimply connected, by H.C. Wang [21, (2.3)], the connected simply connectedcompact complex coset space Z is one orbit of a maximal connected complexsemisimple Lie subgroup G Z of Aut( Z, J Z ; γ ).For a non-zero integer ℓ and a finite subset { f , · · · , f N } of O ℓ ( P, J ), theassociated map , the associated projective map and the associated Hamilto-nian vector bundle are defined as F := ( f , · · · , f N ) : P −→ C N +1 . [ F ] :=[ f : · · · : f N ] : Z −→ P N C and V ( f , · · · , f N ) := { P Nj =0 c j X f j | c j ∈ C } ,where [ F ] is well-defined as a mapping iff F ( P ) j C N +1 \ Lemma 2.5.
Let ( P, J ; R λ ( λ ∈ C × ) , p ; θ ) be a symplectification of degree δ on a c-manifold ( Z, J Z ; γ ) . And ℓ, m be integers with ℓ = 0 , m > . Then: (i) For a fixed η ∈ P and a finite subset { f , · · · , f N } of O ℓ ( P, J ) , V ( f , · · · , f N ) | η = T η P iff F ∗ η : T η P −→ T F ( η ) C N +1 is injective iff [ F ] is well-defined around p ( η ) and that [ F ] ∗ p ( η ) is injective. (ii) L γ is immersionally ample of order m (or ample of degree m ) iff thereexists a finite subset { f , · · · , f N } of O m ( P γ , J γ ) such that the associatedprojective map [ F ] : Z −→ P N C is well-defined as an immersion (resp. anembedding ). In these cases, V ( f , · · · , f N ) | η = T η P γ at all η ∈ P γ . (iii) L γ is immersional (or very ample ) iff there exists a finite subset { f , · · · , f N } of O δ ( P, J ) such that the associated projective map [ F ] : Z −→ P N C is well-defined as an immersion (resp. embedding ). In these cases, ( p γ ∗ V ( f , · · · , f N )) | ζ = T ζ Z at all ζ ∈ Z .Proof. (i) Put ζ j : C N +1 −→ C ; ( z , · · · , z N ) z j for j ∈ { , , · · · , N } ,so that W F ( η ) := { d F ( η ) ζ j | j ∈ { , · · · , N }} spans T ∗ ′ F ( η ) C N +1 . Then F ∗ η isinjective iff the dual map F ∗ η : T ∗ ′ η C N +1 −→ T ∗ ′ η P is surjective iff F ∗ η W F ( η ) = { d η f j = F ∗ η dζ | j ∈ { , · · · , N }} spans T ∗ ′ η P iff { Ξ( df j ) | η | j ∈ { , · · · , N }} spans T ′ η P iff V ( f , · · · , f N ) | η = T η P . In this case, F ( η ) = 0 because of F ∗ η B = ℓ ˜ B | F ( η ) = 0 iff F ( η ) = 0 for 0 = B and ˜ B | F ( η ) = ddt (cid:12)(cid:12) t =0 ˜ R e t F ( η ).If [ F ] ∗ ( p ∗ η X ) = 0 then ˜ p ∗ ( F ∗ η X ) = 0 and that F ∗ η X = B z | F ( η ) for some z ∈ C so that X = ℓ B z | η and p ∗ X = 0, i.e. [ F ] ∗ p ( η ) is injective. Conversely,assume that F = 0 on an open neighbourhood of η and that [ F ] ∗ p ( η ) isinjective. Take X ∈ T η P \
0. If p ∗ X = 0 then ˜ p ∗ ( F ∗ X ) = [ F ] ∗ ( p ∗ X ) = 020nd F ∗ X = 0. If p ∗ X = 0 then X = B z | η for some z ∈ C × and F ∗ X = ℓ ˜ B z | F ( η ) = 0. Hence, F ∗ η is injective.(ii) The last claim follows from (i). L γ is immersionally ample (or ample)of order m iff there exists a finite subset { ϕ , · · · , ϕ N } of Γ( L ⊗ mγ ) such that[Φ] : Z −→ P N C of Φ := ( ι L ⊗ γ ( ϕ ) , · · · , ι L ⊗ γ ( ϕ N )) : P L ⊗ mγ −→ C N +1 is well-defined as an immersion (resp. embedding). In this case, put F := Φ ◦ ν m : P γ −→ C × N +1 . Then [ F ] = [Φ], which is well-defined as an immersion (resp.embedding). Put ( f , · · · , f N ) := F , so that f j = ι mL γ ( ϕ j ) ∈ O m ( P γ , J γ )( j ∈ { , , · · · , N } ) by Lemma 0.1, as required. Coversely, assume that [ f : · · · : f N ] : Z −→ P N C is well-defined as an immersion (resp. embedding)for some f j ∈ O m ( P γ , J γ ). Then f j = ι mL γ ( ϕ j ) = ι L ⊗ mγ ( ϕ j ) ◦ ν m for some ϕ j := Σ mL γ ( f j ) ∈ Γ( L ⊗ mγ ) by Lemma 0.1, so that [ ι L ⊗ mγ ( ϕ ) , · · · , ι L ⊗ mγ ( ϕ N )] =[ f : · · · : f N ] : Z −→ P N C , which is well-defined as an immersion (resp.embedding), as required.(iii) As the case of m = 1 for (ii), L γ is immersional (or very ample) iff[ g : · · · : g N ] : Z −→ P N C is an immersion (resp. embedding) for somefinite subset { g , · · · , g N } of O ( P γ , J γ ). In this case, taking µ : P −→ P γ given in Corollary 1.7 (ii), put f j := µ ∗ g j ∈ O δ ( P, J ), so that [ f , · · · , f N ] =[ g , · · · , g N ] : Z −→ P N C , which is an immersion (resp. embedding). Co-versely, assume that [ f : · · · : f N ] : Z −→ P N C is an immersion (resp.embedding) for some f j ∈ O δ ( P γ , J γ ). Then ( f , · · · , f N ) = ( g , · · · , g N ) ◦ µ for some g j ∈ O ( P γ , J γ ), so that [ g , · · · , g N ] = [ f , · · · , f N ] : Z −→ P N C isan immersion (resp. embedding). Hence, the first claim is proved. And thelast claim follows from the last claim of (ii) and Theorem 2.3. Proposition 2.6.
For the adjoint variety Z G of a complex simple Liealgebra G with the canonical c-structure γ G , the contact line bundle L γ G isvery ample, so that Z G is simply connected.Proof. By Example 1.4 (iii), P G in G is a symplectifications of degree δ =1 on ( Z, J Z ; γ G ) as well as the standard symplectification P γ G . By Corollary1.7 (ii), there exists an isomorphism µ : P G −→ P γ G . Let X , · · · , X N be a C -linear basis of G , and ω , · · · , ω N the dual basis of G ∗ , so that ( ω , · · · , ω N ) : G −→ C N +1 is a C -linear isomorphism. For i ∈ { , · · · , N } , put f i := ω i ◦ µ − : P γ G −→ C . By Example 1.4 (iii), ˘ R ∗ λ f i = µ ∗ ( ˜ R ∗ λ ω i ) = µ ∗ ( λ · ω i ) = λ · f i for λ ∈ C × . By Theorem 2.3 (or Lemma 0.1), there exists ϕ i ∈ Γ( L γ G ) suchthat f i = ι L γ ( ϕ i ), so that ( ι L γ ( ϕ ) , · · · , ι L γ ( ϕ N )) = ( f , · · · , f N ) : P γ G −→ C N +1 is an embedding. Hence, L γ G is very ample, especially it is positive.By Proposition 1.1 (ii), the first Chern class of ( Z, J Z ) is positive, so that Z is simply connected by a theorem of S. Kobayashi [3, 11.26]21 roof of Proposition 0.2. (0) Assume that ( Z, J Z ; γ ) is a connectedcompact c-manifold with immersional contact line bundle L γ . By Lemma 2.5(iii), ( Z, J Z ; γ ) is infinitesimally homogeneous. Since L γ is immersional, it ispositive, so that the first Chern class of ( Z, J Z ) is positive by S. Kobayashi(Proposition 1.1 (ii)). Then Z is simply connected by S. Kobayashi [3,11.26]. By H.C. Wang (Proposition 2.4 (ii)), ( Z, J Z ; γ ) is homogeneous suchthat Z is one orbit of some connected complex semisimple Lie subgroup G Z of Aut( Z, J Z ; γ ). By an argument of W.M. Boothby [4, (2.2), Theorem A],if P γ is not one orbit of G P , then P γ admits a global holomorphic section,so that c ( L γ ) = 0, which contradicts with L γ is positive. Hence, P γ is oneorbit of G P := { g P | g ∈ G Z } , so that T η P γ = { X | η | X ∈ G P } at each η ∈ P γ .(1) Let G Z be the complex Lie subalgebra of ( a ( Z, J Z ; γ ) , J Z ) correspond-ing to G Z . Put G P := { X P | X ∈ G Z } j a ( P γ , J γ ), which is isomorphic to G Z = dp γ ( G P ) by dp γ (Theorem 2.3). Let G ∗ P be the dual C -linear spaceof G P with < ω, X > ∈ C as the value of ω ∈ G ∗ P on X ∈ G P . Accord-ing to A.A. Kirillov [10, p.234, (11); p.301], B. Kostant [14, (5.4.2)] andJ.-M. Souriau [20, (11.10)] with Lemma 2.1 (iii), a moment map of degreeone is defined as follows: µ G P : P γ −→ G ∗ P ; η µ G ( η ); < µ G P ( η ) , X > := θ γ ( X | η ) , where µ G P ◦ ˘ R λ = λµ G P ( λ ∈ C × ) by θ γ ( X | ˘ R λ η ) = θ γ ( ˘ R λ ∗ ( X | η )) = λθ γ ( X | η )( X ∈ G P j a ( P γ , J γ )). For g ∈ G P , put δg : G ∗ P −→ G ∗ P ; ω ( δg ) ω such that < ( δg ) ω, X > := < ω, ( dg − ) X > , which defines the coadjointaction. Note that µ G P ( gη ) = ( δg ) µ G P ( η ) ( η ∈ P γ , g ∈ G P ) by θ ( X | gη ) = θ ( g − ∗ ( X | gη )) = θ ((( dg − ) X ) | η ), so that the image µ G P ( P γ ) is one coadjointorbit of G P . Take a C -linear basis { X i | i = 1 , · · · , r } of G P with the dualbasis { ω i | i = 1 , · · · , r } of G ∗ P . For i ∈ { , · · · , r } , put f i := θ γ ( X i ) ∈O ( P γ , J γ ), so that µ G P ( η ) = P ri =1 f i ( η ) ω i . By Theorem 2.3, X i = X f i and df i = − ι X i dθ γ . If µ G P ∗ ( X ) = 0, then 0 = df i ( X ) = dθ γ ( X i , X ) for all i ∈ { , · · · , r } . Note that dθ γ is non-degenerate and that X i ’s span T P γ bythe step (0), so that X = 0. Hence, µ G P is an immersion. For X ∈ G P ,put X ♭ ∈ G ∗ P such that < X ♭ , Y > = B ( X, Y ) for Y ∈ G P w.r.t. the Killingform B of G P , which is non-degenerate since G P is semisimple in the step(0). Then ♭ : G P −→ G ∗ P ; X X ♭ is a C -linear isomorphism such that( δg ) ◦ ♭ = ♭ ◦ ( dg ) ( g ∈ G P ), because of < ( δg ) X ♭ , Y > = < X ♭ , ( dg − ) Y > = B ( X, ( dg − ) Y ) = B (( dg ) X, Y ) = < (( dg ) X ) ♭ , Y > for all Y ∈ G P . Put κ := ♭ − ◦ µ G P : P γ −→ G P . Then κ ◦ g = ♭ − ◦ µ G P ◦ ( δg ) = ♭ − ◦ ( δg ) ◦ µ G P =( dg ) ◦ ♭ − ◦ µ G P = ( dg ) ◦ κ for all g ∈ G P . Hence, κ is a holomorphicimmersion onto one adjoint orbit of G P by the inner automorphism group˘ G := { dg | g ∈ G P } on the complex semisimple Lie algebra G P .22ut κ Z : Z −→ C P ( G P ); p γ ( η ) p G P ( κ ( η )), which is well-defined as aholomorphic immersion onto one adjoint ray orbit κ Z ( Z ) of ˘ G on C P ( G P ).Then the restriction to the image, κ Z | ( Z, J Z ) −→ κ Z ( Z ) is a covering map.Since Z is compact, the image κ Z ( Z ) is a closed complex submanifold of C P ( G P ), which is a projective algebraic variety by a theorem of W.L. Chow(cf. [6, p.217, p.36]).(2) G P is simple (cf. [4, (6.3)]): Let G P = ⊕ ki =1 G P,i be the direct-sumdecomposition into simple ideals G P,i with the correspoding Lie subgroups˘ G i of ˘ G , where g i ∈ G i acts as g i ( P j X j ) = g i X i + P j = i X j for X j ∈ G P,j .Then ˘ G = Π ki =1 ˘ G i . Take some η ∈ P γ . Then κ ( η ) = 0 since κ is a C × -equivariant immersion. And there exist κ i ( η ) ∈ G P,i ( i = 1 , · · · , k ) suchthat κ ( η ) = P ki =1 κ i ( η ). Take j such that κ j ( η ) = 0. Suppose that thereis another j ′ such that κ j ′ ( η ) = 0. Then ˘ η := P i = j ′ κ i ( η ) = 0. And˘ η ˘ Gκ ( η ) = κ ( P γ ), since the G P,j ′ -component of ˘ η is zero. On the otherhand, for any λ ∈ C × , λκ ( η ) = κ ( ˘ R λ η ) ∈ P G P = Gκ ( η ), so that there exists˘ G ∋ g ( λ ) =: ( g ( λ ) , · · · , g k ( λ )) such that λκ ( η ) = g ( λ ) κ ( η ), i.e. λκ i ( η ) = g i ( λ ) κ i ( η ) ( i = 1 , · · · , k ). With respect to the Euclidian metric topology of G P , G P \ ∋ ˘ η = P i = j ′ κ i ( η ) + lim λ → λκ j ′ ( η ) = lim λ → g j ′ ( λ ) κ ( η ), where g j ′ ( λ ) κ ( η ) ∈ ˘ G j ′ κ ( η ) j ˘ Gκ ( η ) = κ ( P γ ). With the canonical projection p G P : G P \ −→ C P ( G P ), ˘ ζ := p G P (˘ η ) = lim λ → p G P ( g j ′ ( λ ) κ ( η )) is containedin the closure of κ Z ( Z ) in C P ( G P ), so that ˘ ζ ∈ κ Z ( Z ) since κ Z ( Z ) is closed in C P ( G P ). Hence, p G P (˘ η ) ∈ p G P ( κ ( P γ )), so that ˘ η ∈ κ ( P γ ), which contradictswith the first observation. Then κ ( η ) = κ j ( η ) and κ ( P γ ) = ˘ Gκ ( η ) = ˘ Gκ j ( η ),so that ˘ G i ( i = j ) acts trivially on κ ( P γ ). Hence, G P = G P,j , which is acomplex simple Lie algebra.(3) (
Z, J Z ) is biholomorphic to Z G P (cf. [4, Theorem B]): Take a Cartansubalgebra H of G P and the set △ + of all positive roots with respect tosome lexicographic ordering on H ∗ R . Then B := H ⊕ ( ⊕ α ∈△ + C e α ) is a Borelsubalgebra of G P such that ˘ B := { α ∈ Aut( G P ) | α B = B} is an algebraicsubgroup of a complex algebraic group Aut( G P ) with the Lie algebra B ,so that the identity connected component ˘ B ◦ of ˘ B in the real topology issolvable. Note that ˘ B ◦ equals the identity irreducible component of ˘ B inthe Zariski topology [17, Chapter 3, §
3, 1 ◦ ]. By a theorem of A. Borel[17, p.117, Theorem 9], ˘ B ◦ admits a fixed point on the projective algebraicvariety κ Z ( Z ), so that there exists ˘ ζ ∈ κ Z ( Z ) such that ˘ B ◦ ˘ ζ = ˘ ζ . Take η ∈ P γ such that ˘ ζ = p G P ( κ ( η )), where κ ( η ) = h + P α ∈△ c α e α for some h ∈ H , c α ∈ C . Then [ B , κ ( η )] j C κ ( η ), so that h = 0 and c α = 0 for any α ∈ △\{ ρ } with the highest root ρ . Hence, κ ( η ) = c ρ e ρ with c ρ = 0 because23f P γ = ∅ . Put η ′ := ˘ R c − ρ η ∈ P γ . Then e ρ = c − ρ κ ( η ) = κ ( η ′ ) ∈ κ ( P γ ),so that κ ( P γ ) = ˘ Ge ρ = P G P and κ Z ( Z ) = Z G P , where the notations P G P and Z G P are given in Example 1.4 (iii). Since Z G P is simply connected(Propostion 2.6), the covering map κ Z | ( Z, J Z ) −→ Z G P is biholomorphic.(4) For X ∈ G P , put X R | dg := ddt (cid:12)(cid:12) t =0 ( dg ) ◦ e t ad( X ) ∈ T dg ˘ G with g ∈ G P .Then θ γ ( X | η ′ ) = < µ G P ( η ′ ) , X > = B ( ♭ − ( µ G P ( η ′ )) , X ) = B ( κ ( η ′ ) , X ) = B ( e ρ , X ) = θ ˘ G ( X R ) at the point η ′ in the step (3). By Example 1.4 (iii-0), θ ˘ G ( X R ) = θ G P ( π P ∗ X R ) = θ G P ( ddt (cid:12)(cid:12) t =0 e t ad( X ) e ρ ) = θ G P ( κ ∗ ( X | η ′ )). Hence, θ γ ( X | η ′ ) = θ G P ( κ ∗ ( X | η ′ )). Because of { X | η ′ | X ∈ G P } = T η ′ P γ , θ γ | η ′ =( κ ∗ θ G P ) | η ′ . For any g ∈ G P , θ γ | gη ′ = θ γ | η ′ ◦ g − ∗ gη ′ = ( κ ∗ θ G P ) | η ′ ◦ g − ∗ gη ′ = θ G P | κ ( η ′ ) ◦ κ ∗ η ′ ◦ g − ∗ gη ′ = θ G P | κ ( η ′ ) ◦ ( dg ) − ∗ κ ( gη ′ ) ◦ κ ∗ gη ′ = θ G P | κ ( gη ′ ) ◦ κ ∗ gη ′ = κ ∗ θ G P | gη ′ , so that κ ∗ (ker θ γ ) = ker θ G P . Then κ Z ∗ E γ = κ Z ∗ ( p γ ∗ (ker θ γ )) = p G P ∗ ( κ ∗ (ker θ γ )) = p G P ∗ (ker θ G P ) = E γ G P , i.e. κ Z | ( Z, J ; γ ) −→ ( Z G P ; γ G P ) isisomorphic, so that L γ and L γ G P are isomorphic. Since L γ G P is very ample(Proposition 2.6), so is L γ . Acknowledgement . The author is indebted to Masataka Tomari, ClaudeLeBrun, the late Masaru Takeuchi, Yasuyuki Nagatomo, Ryoichi Kobayashi,an anonymous referee of JMSJ, Jaroslaw Buczynski and Bogdan Alexandrov,who very kindly pointed out some crucial mistakes in the previous versions.The author is grateful to Shigeru Mukai, Tomonori Noda and Hajime Kajifor their advice in differential or algebraic geometry on the previous versions.
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