Polarized orbifolds associated to quantized Hamiltonian torus actions
aa r X i v : . [ m a t h . S G ] A ug Polarized orbifolds associated to quantizedHamiltonian torus actions
Roberto Paoletti ∗ Abstract
Suppose given an holomorphic and Hamiltonian action of a com-pact torus T on a polarized Hodge manifold M . Assume that theaction lifts to the quantizing line bundle, so that there is an inducedunitary representation of T on the associated Hardy space. If in ad-dition the moment map is nowhere zero, for each weight ν the ν -thisotypical component in the Hardy space of the polarization is finite-dimensional. Assuming that the moment map is transverse to the raythrough ν , we give a gometric interpretation of the isotypical compo-nents associated to the weights k ν , k → + ∞ , in terms of certain po-larized orbifolds associated to the Hamiltonian action and the weight.These orbifolds are generally not reductions of M in the usual sense,but arise rather as quotients of certain loci in the unit circle bundleof the polarization; this construction generalizes the one of weightedprojective spaces as quotients of the unit sphere, viewed as the domainof the Hopf map. Let M be a d -dimensional connected complex projective manifold, with com-plex structure J . Let ( A, h ) be a positive holomorphic line bundle on (
M, J );the curvature of the unique covariant derivative on A compatible with boththe Hermitian metric h and the complex structures has the form Θ = − π ı ω ,where ω is a K¨ahler form on ( M, J ). Let d V M := ω ∧ d /d ! be the associatedvolume form on M .Let A ∨ be the dual line bundle of A , endowed with the dual Hermitianmetric h ∨ . As is well-known, positivity of ( A, h ) is equivalent to the unit ∗ Address:
Dipartimento di Matematica e Applicazioni, Universit`a degli Studi diMilano-Bicocca, Via R. Cozzi 55, 20125 Milano, Italy; e-mail : [email protected] D ⊂ A ∨ being a strictly pseudoconvex domain [Gr]. We shalldenote by X := ∂D ⊂ A ∨ the unit circle bundle of h ∨ , and by α ∈ Ω ( X )the (normalized) connection 1-form on X . Thus, X is a principal S -bundleon M , with the structure S -action ρ X : S × X → X given by clockwisefiber rotation. If π : X → M is the bundle projection, and − ∂ θ ∈ X ( X ) isthe generator of ρ X , thend α = 2 π ∗ ( ω ) , α ( ∂ θ ) = 1 . (1)Let d V X := ( α/ π ) ∧ π ∗ (d V M ) be the associated volume form on X Then α is a contact form on X , and X is a CR manifold, with CRstructure supported by the horizontal tangent bundle Hor ( X ) := ker( α ) ⊂ T X. (2)Let H ( X ) ⊆ L ( X ) denote the Hardy space of X . Since ρ X preserves α and the CR structure, it induces a unitary representation ˆ ρ X of S on H ( X ),given byˆ ρ Xe ı ϑ ( s )( x ) := s (cid:0) ρ Xe − ı ϑ ( x ) (cid:1) = s (cid:0) e ı ϑ x (cid:1) (cid:0) x ∈ X, e ı ϑ ∈ S , s ∈ H ( X ) (cid:1) . The induced isotypical decomposition is the Hilbert space direct sum H ( X ) = + ∞ M k =0 H ( X ) k , (3)where H ( X ) k := (cid:8) s ∈ H ( X ) : s (cid:0) e ı θ x (cid:1) = e ı k θ s ( x ) ∀ x ∈ X, e ı θ ∈ S (cid:9) . It is well-known that there are natural unitary isomorphisms H ( X ) k ∼ = H ( M, A ⊗ k ), the latter being the space of global holomorphic sections of A ⊗ k .Furthermore, let T ∼ = ( S ) r be an r -dimensional compact torus, withLie algebra and coalgebra t and t ∨ , respectively. We shall equivariantlyidentify t ∼ = t ∨ ∼ = ı R r . Suppose given an Hamiltonian and holomorphicaction µ M : T × M → M of T on the K¨ahler manifold ( M, J, ω ). LetΦ : M → t ∨ ∼ = ı R r be the moment map.It is standard that µ M and Φ generate an infinitesimal contact and CRaction of t on X , so defined [Ko]. If ξ ∈ t , let ξ M ∈ X ( M ) be the Hamiltonianvector field induced by ξ on M , and define a vector field ξ X ∈ X ( X ) by setting ξ X := ξ ♯M − h Φ ◦ π, ξ i ∂ θ ∈ X ( X ); (4)2ere V ♯ ∈ X ( X ) denotes the horizontal lift to X of a vector field V ∈ X ( M ),with respect to α . The ξ X ’s are commuting contact vector fields on X , whoseflow preserves the CR structure, and the map ξ ξ X is a morphism of Liealgebras t → X ( X ).Let us make the stronger hypothesis that µ M lifts to an actual contactand CR action of T on X , µ X : T × X → X , and that d µ X ( ξ ) = ξ X for any ξ ∈ t . Then µ X determines a unitary representation ˆ µ X of T on H ( X ), givenby ˆ µ X t ( s )( x ) := s (cid:0) µ X t − ( x ) (cid:1) ( x ∈ X, t ∈ T, s ∈ H ( X )) . (5)By the Peter-Weyl Theorem [St], ˆ µ X induces a unitary and equivariant split-ting of H ( X ) into isotypical components.Let us regard any ν ∈ Z r as an integral weight on T , associated to thecharacter χ ν ( t ) := t ν , where for t = ( t , . . . , t r ) ∈ T we set t ν := Q rj =1 t ν j j . For any ν ∈ Z r , let usconsider the ν -th isotypical component H ( X ) ˆ µ ν := { s ∈ H ( X ) : ˆ µ t ( s ) = χ ν ( t ) · s ∀ t ∈ T } . Then we have an equivariant Hilbert space direct sum H ( X ) = M ν ∈ Z r H ( X ) ˆ µ ν . (6)In the special case where T = S , µ M is trivial, and Φ = ı , ı ∈ t is mappedto − ∂ θ , and so µ X = ρ X ; hence (6) reduces to (3), that is, H ( X ) k = H ( X ) ˆ ρk with k in place of ν .In general, it may happen that H ( X ) ˆ µ ν ∩ H ( X ) k = (0) for several k ’s, sothat H ( X ) ˆ µ ν does not correspond to a space of holomorphic sections of somepower of A . Furthermore, H ( X ) ˆ µ ν may be infinite-dimensional. The lattercircumstance does not occur, however, if Φ( M ) (see § Basic Assumption 1.1.
Φ and ν satisfy the following properties:1. ν = is coprime, that is, l . c . d . ( ν , . . . , ν r ) = 1;2. Φ is nowhere vanishing, that is, Φ( M );3. Φ is transverse to the ray R + · ı ν , and M ν := Φ − ( R + · ı ν ) = ∅ .If the previous properties are satisfied, then µ X is generically locally free [P1];perhaps after replacing T with its quotient by a finite subgroup, we may andwill assume without loss of generality that µ X is generically free.3et us assume that BA holds. Then H ( X ) ˆ µk ν = (0) for all k ≤ § (cid:0) H ( X ) ˆ µk ν (cid:1) + ∞ k =1 associated to the weights on the ray R + · ν . Thecorresponding ‘equivariant Szeg¨o projectors’ Π ˆ µk ν : L ( X ) → H ( X ) ˆ µk ν aresmoothing operators (that is, they have C ∞ integral kernels). Furthemore, M ν ⊆ M is a T -invariant coisotropic connected compact submanifold of realcodimension r − k → + ∞ of theintegral kernels Π ˆ µk ν and their concentration behaviour along M ν were studiedin [P1], [P2], and related variants in the presense of additional symmetrieswhere investigated in [Ca].Our present aim is to clarify the geometric significance of the sequence (cid:0) H ( X ) ˆ µk ν (cid:1) + ∞ k =1 , generalizing the interpretation of the sequence (cid:0) H ( X ) k (cid:1) interms of the spaces H ( M, A ⊗ k ). We shall prove the following: Theorem 1.1.
Assume BA holds. Then there exists a ( d +1 − r ) -dimensionalconnected compact complex orbifold N ν , and a positive holomorphic orbifoldline bundle B ν on N ν , naturally constructed from A , ν and Φ , such that thefollowing holds:1. for k ≥ , there is a natural injection δ k : H ( X ) ˆ µk ν ֒ → H (cid:0) N ν , B ⊗ k ν (cid:1) ;2. δ k is an isomorphism if k ≫ . Corollary 1.1. If k ≫ , dim H ( X ) ˆ µk ν = χ (cid:0) N ν , B ⊗ k ν (cid:1) . Obviously with no pretense of exhaustiveness, discussions of orbifolds andorbifold line bundles (also known as V -manifolds and line V -bundles) can befound in [S1], [S2], [B], [Ka], [ALR], [BG]; specific treatments of Hamiltonianactions on symplectic orbifolds can be found in [LT] and [MS].The geometric significance of the Theorem lies in the relation betweenthe polarized orbifold ( N ν , B ν ) and the ‘prequantum data’ ( A, Φ , ν ). It istherefore in order to outline how the former is constructed from the latter.The following statements will be clarified and proved in § T ∼ = ( C ∗ ) r be the complexification of T . Then µ X extends to an holo-morphic line bundle action ˜ µ A ∨ : ˜ T × A ∨ → A ∨ . Let A ∨ be the complementof the zero section in A ∨ , and let A ∨ ν ⊂ A ∨ be the inverse image of M ν . Let˜ A ∨ ν := ˜ T · A ∨ ν be its saturation under ˜ µ A ∨ .Then ˜ µ A ∨ is proper and locally free on ˜ A ∨ ν , and N ν = ˜ A ∨ ν / ˜ T . Thus theprojection p ν : ˜ A ∨ ν → N ν is a principal V -bundle with structure group ˜ T over N ν [S2]. 4urthermore, χ ν : T → S extends to a character ˜ χ ν : ˜ T → C ∗ ; thedatum of p ν and ˜ χ ν determines the orbifold line bundle B ν . Similarly, B ⊗ k ν (or B k ν ) denotes the orbifold line bundle associated to p ν and ˜ χ k ν = ˜ χ k ν .We can give the following alternative algebro-geometric characterizationof ˜ A ∨ ν . Let ν ⊥ ⊂ R r be the orthocomplement of ν with respect to thestandard scalar product, and consider the (Abelian) subalgebra ı ν ⊥ t . Let T r − ν ⊥ T be the corresponding subtorus, ˜ T r − ν ⊥ ˜ T be its complexification.The restriction of ˜ µ M to ˜ T r − ν ⊥ is an holomorphic action ˜ γ M of ˜ T r − ν ⊥ on ( M, J ),with a built-in complex linearization ˜ γ A ∨ : ˜ T r − ν ⊥ × A ∨ → A ∨ . Let ˜ M ν ⊆ M be the locus of (semi)stable points of ˜ γ M ; then ˜ A ∨ ν is the inverse image of˜ M ν in A ∨ .Up to a natural isomorphism, an alternative description of N ν is asfollows. Let X ν := π − ( M ν ). Then T acts locally freely on X ν , and N ν ∼ = X ν /T . This description is instrumental in describing the positivityof B ν and the K¨ahler structure of N ν .When r = 1, M = P d , and A is the hyperplane line bundle with thestandard metric, we have X ν = X = S d +1 ; thus the previous constructiongeneralizes the one of weighted projective spaces (see also the discussions in[P2] and [P3]). This section is devoted to a closer description of the geometric setting, andto the statement and proof of a series of geometric results that will combineinto the proof of Theorem 1.1.
Notation 2.1.
We shall adopt the following notation and conventions.1. If a Lie group G with Lie algebra g acts smoothly on a manifold R , forany ξ ∈ g we shall denote by ξ R ∈ X ( R ) the vector field on R generatedby ξ .2. If r ∈ R and l ⊆ g is a vector subspace, we shall set l R ( r ) := (cid:8) ξ R ( r ) : ξ ∈ l (cid:9) ⊆ T r R.
3. Given an isomorphism T ∼ = ( S ) r , we have t ∼ = ı R r . If we identify theLie algebra ˜ t of ˜ T ∼ = R r + × T with C r ∼ = R r ⊕ ı R r , t corresponds tothe imaginary summand ı R r . For x = (cid:0) x · · · x r (cid:1) ∈ R r , we have e x := (cid:0) e x · · · e x r (cid:1) ∈ R r + ˜ T r , while e ı x := (cid:0) e ı x · · · e ı x r (cid:1) ∈ T r .5. We shall equivariantly identify t ∼ = t ∨ , and view Φ as t -valued.5. If V is any Euclidean vector space and ǫ > V ( ǫ ) ⊂ V will denote theopen ball in V centered at the origin and of radius ǫ .6. g ( · , · ) := ω (cid:0) · , J ( · ) (cid:1) is the Riemannian metric associated to ω .7. J ′ is the complex structure of A ∨ .8. The superscript ♯ will denote horizontal lifts from M to either X or A ∨ ,according to the context, and will be applied to both tangent vectorsand vector subspaces of tangent spaces.9. π : X → M and π ′ : A ∨ → M are the projections.10. If β Z : G × Z → Z is an action of the group G on the set Z , and if S ⊆ Z is G -invariant, we shall often denote by β S : G × S → S therestricted action. Thus, for example, ˜ T acts on A ∨ by ˜ µ A ∨ , on A ∨ ⊂ A ∨ by ˜ µ A ∨ , on ˜ A ∨ ν ⊆ A ∨ by ˜ µ ˜ A ∨ ν . M ν ⊆ M Let γ M : T r − ν ⊥ × M → M be the action induced by restriction of µ M . Then γ M is Hamiltonian with respect to 2 ω , and its moment map Φ ν ⊥ : M → ı ν ⊥ is the composition of Φ with the orthogonal projection t → ı ν ⊥ . AssumingBA, we can draw the following conclusions:1. ∈ ı ν ⊥ is a regular value of Φ ν ⊥ ;2. M ν = Φ − ν ⊥ ( ) is a compact and connected coisotropic submanifold of M , of (real) codimension r − γ M is locally free along M ν ⊥ , that is,dim (cid:0) ı ν ⊥ (cid:1) M ( m ) = r − ∀ m ∈ M ν ⊥ ;4. for every m ∈ M ν , we have T m M ν = (cid:0) ı ν ⊥ (cid:1) M ( m ) ⊥ ωm = J m (cid:0) ( ı ν ⊥ ) M ( m ) (cid:1) ⊥ gm . This implies the following statement. Let us defineΨ : ( x , m ) ∈ ( ı ν ⊥ ) × M ν ˜ µ e x ( m ) ∈ M. (7)6 emma 2.1. Given Basic Assumption 1.1, the following holds:1. ˜ γ M is locally free along M ν ;2. for any sufficiently small ǫ > , Ψ in (7) restricts to a diffeomorphismbetween ( ı ν ⊥ )( ǫ ) × M ν and an open tubular neighborhood U ǫ of M ν in M . X ν ⊆ X and its saturation ˜ X ν in A ∨ Let us set: X ν := π − ( M ν ) ⊆ X. (8)If x ∈ X ν , then in view of (4) (cid:0) ı ν ⊥ (cid:1) X ( x ) = (cid:0) ı ν ⊥ (cid:1) M ( m ) ♯ . (9)We have the following analogue of Lemma 2.1. Lemma 2.2.
Given BA 1.1, the following holds:1. µ X is locally free along X ν ;2. for any x ∈ X ν , T x X ν ∩ J ′ x (cid:0) t A ∨ ( x ) (cid:1) = (0);
3. for all suitably small ǫ > , the map Ψ ′ : ( x , x ) ∈ ( ı t ) × X ν ˜ µ A ∨ e x ( x ) ∈ A ∨ determines a diffemorphism from ( ı t )( ǫ ) × X ν to a tubular neighborhoodof X ν in A ∨ ;4. ˜ µ A ∨ is locally free along X ν .Proof of Lemma 2.2. That µ X is locally free on X ν under the transversalityassumption in BA is proved in § § x ∈ X ν and m = π ( x ) then J m (cid:0) ı ν ⊥ (cid:1) M ( m ) ♯ ⊆ T x X is the normal space of X ν in X at x ; hence given (9) we have T x X ν ∩ J ′ x (cid:0) ı ν ⊥ (cid:1) X ( x ) = T x X ν ∩ J m (cid:0) ı ν ⊥ (cid:1) M ( m ) ♯ = (0) . (10)7urthermore, by definition of X ν there exists a smooth function λ ν : M ν → R + such that( ı ν ) X ( x ) = ( ı ν ) M ( m ) ♯ − λ ν ( m ) k ν k ∂ θ | x Hor ( X ) x , ∀ x ∈ X ν . (11)If r denotes the radial coordinate along the fibers of A ∨ , this implies J ′ x (cid:0) ( ı ν ) X ( x ) (cid:1) = J m (cid:0) ( ı ν ) M ( m ) (cid:1) ♯ + λ ν ( m ) k ν k ∂ r | x ∈ T x A ∨ \ T x X. (12)The second statement follows from (10), (11) and (12).The third statement is an immediate consequence of the second.Since X ν is a T -invariant submanifold of A ∨ , (cid:0) ı ν ⊥ (cid:1) X ( x ) ⊆ T x X ν for any x ∈ X ν . Hence if x ∈ X ν and m = π ( x ) then by (10) (cid:0) ı ν ⊥ (cid:1) X ( x ) ∩ J ′ x (cid:0) ı ν ⊥ (cid:1) X ( x ) = (cid:0) ı ν ⊥ (cid:1) M ( m ) ♯ ∩ J m (cid:0) ı ν ⊥ (cid:1) M ( m ) ♯ = (0) . (13)Together with the first statement, this implies dim C (cid:0) ˜ t A ∨ ( x ) (cid:1) = r for any x ∈ X ν .Let us consider the saturation˜ X ν := ˜ T · X ν ⊆ A ∨ . (14) Corollary 2.1. ˜ X ν is open in A ∨ . Corollary 2.2.
If Basic Assumption 1.1 holds, then µ X is generically freeon X ν .Proof of Corollary 2.2. If the general x ∈ X ν had non-trivial stabilizer in T ,the same would hold of the general ℓ ∈ ˜ X ν ; since the latter is open in A ∨ ,this contradicts the assumption that µ A ∨ is generically free. Corollary 2.3.
If BA 1.1 holds, then N ′ ν := X ν /T is a compact orbifold ofreal dimension d + 1 − r ) , and the projection p ′ ν : X ν → N ′ ν (15) is a principal V -bundle with structure group T . Remark 2.1.
Associated to p ′ ν and the character χ ν there is a orbifoldcomplex line bundle B ′ ν on N ′ ν . 8 .3 The K¨ahler structure of A ∨ Let ̺ : A ∨ → R denote the square norm function in the Hermitian metric h ,and set ˜ ω := 2 d (cid:0) ℑ (cid:0) ∂̺ / (cid:1)(cid:1) = 2 ı ∂∂ (cid:0) ̺ / (cid:1) . (16)If π ′ : A ∨ → M is the projection, then˜ ω = 2 ̺ / π ′∗ ( ω ) + ı ̺ / ∂̺ ∧ ∂̺. (17)The contact action µ X : T × X → X extends to an holomorphic unitaryaction µ A ∨ : T × A ∨ → A ∨ . Proposition 2.1. ˜ ω is a µ A ∨ -invariant exact K¨ahler form on A ∨ .Proof. Since µ A ∨ preserves both ̺ and the complex structure, by its definition˜ ω is a µ A ∨ -invariant closed (1 , ω isnon-degenerate.The unique connection compatible with both h and the holomorphicstructure determines an invariant decomposition T A ∨ = Hor ( A ∨ ) ⊕ V er ( A ∨ ) , (18)where Hor ( A ∨ ) := ker( ∂̺ ) , V er ( A ∨ ) := ker(d π ′ ) ⊂ T A ∨ (19)denote the horizontal and vertical tangent bundles. Then Hor ( A ∨ ) and V er ( A ∨ ) are complex vector subbundles of T A ∨ , and by (17) they are ortho-gonal for ˜ ω . Furthermore, the first summand on the right hand side of (17)is symplectic on Hor ( A ∨ ) and vanishes on V er ( A ∨ ), and conversely for thesecond summand. Hence ˜ ω is non-degenerate. Corollary 2.4. µ A ∨ is Hamiltonian on ( A ∨ , ˜ ω ) , with moment map ˜Φ := ̺ / · Φ ◦ π ′ : A ∨ → t , (20) where π ′ : A ∨ → M is the projection.Proof of Corollary 2.4. Given an exact symplectic manifold (
R, η ) with η = − d λ , and a smooth Lie group action ς : G × N → N preserving λ , it is well-known that ς is Hamiltonian, with moment map Υ : R → g ∨ determined bythe relation υ ξ := h Υ , ξ i = ι ( ξ R ) λ ∈ C ∞ ( M ) .
9n our setting, R = A ∨ , ς = µ A ∨ , η = ˜ ω and, in view of (16), λ = − (cid:0) ℑ (cid:0) ∂̺ / (cid:1)(cid:1) = ı (cid:0) ∂̺ / − ∂̺ / (cid:1) ;furthermore, for any ξ ∈ t we have ξ A ∨ = ξ ♯M − h Φ ◦ π ′ , ξ i ∂ ϑ . Since ξ ♯M is a section of Hor ( A ∨ ), it follows from (19) that ι ( ξ ♯M ) ∂̺ = 0.Furthermore, one can verify that ı ( ∂ θ ) ∂ρ = ı ρ . Putting this together, weconclude that µ A ∨ is Hamiltonian, and furthermore the component ˜ φ ξ = h ˜Φ , ξ i of the moment map is˜ φ ξ = ı · ̺ − / ι (cid:16) ξ ♯M − ( ϕ ξ ◦ π ′ ) ∂ θ (cid:17) (cid:0) ∂̺ − ∂̺ (cid:1) = ( ϕ ξ ◦ π ′ ) ̺ / . Let {· , ·} A ∨ denote by Poisson brackets on ( A ∨ , ˜ ω ). Since µ A ∨ is unitary,in view of (20) we conclude the the following. Corollary 2.5. { ˜ φ ξ , ˜ φ η } vanishes, ∀ ξ, η ∈ t . In particular, the orbits of µ A ∨ in A ∨ are isotropic for ˜ ω . Therefore:
Corollary 2.6.
For every ℓ ∈ A ∨ , t A ∨ ( ℓ ) ⊆ T ℓ A ∨ is totally real, that is, t A ∨ ( ℓ ) ∩ J ′ ℓ (cid:0) t A ∨ ( ℓ ) (cid:1) = (0) . By Proposition 1.6 and Theorem 1.12 in [Sj], Corollary 2.4 has the fol-lowing consequences.
Corollary 2.7.
For every ℓ ∈ A ∨ , the following holds:1. the stabilizer ˜ T ℓ ˜ T of ℓ for ˜ µ A ∨ is the complexification of of thestabilizer T ℓ T of ℓ for µ A ∨ ;2. there exists an holomorphic slice for ˜ µ A ∨ at ℓ . Let A ∨ lf ⊆ A ∨ be the open subset where ˜ µ A ∨ is locally free. It followsfrom Proposition 1.6 of [Sj] and Corollary 2.5 above that the stabilizer in ˜ T of any ℓ ∈ A ∨ lf is finite and contained in T .10 efinition 2.1. Following [Sj], we shall call ˜ µ A ∨ proper at ℓ ∈ A ∨ if for allsequences ( ℓ j ) ⊂ A ∨ and (˜ t j ) ⊂ ˜ T such that ℓ j → ℓ and ˜ µ A ∨ ˜ t j ( ℓ j ) converges tosome point in A ∨ , (˜ t j ) is convergent in ˜ T . Remark 2.2.
Let U ⊆ A ∨ be a ˜ T -invariant open set. Then:1. if ˜ µ A ∨ is proper at every ℓ ∈ U , then so is a fortiori ˜ µ U ;2. if ˜ µ U is proper at every ℓ ∈ U , ˜ µ U is (globally) proper. Corollary 2.8.
Given BA, the following holds:1. ˜ µ A ∨ is proper on A ∨ lf (that is, ˜ µ A ∨ lf is proper);2. ˜ X ν ⊆ A ∨ lf ;3. ˜ µ A ∨ is proper on ˜ X ν (that is, ˜ µ ˜ X ν is proper).Proof of Corollary 2.8. We have remarked that the stabilizer in ˜ T of any ℓ ∈ A ∨ lf coincides with the stabilizer of ℓ in T , and therefore it is finite. Inview of Theorem 1.22 of [Sj], ˜ µ A ∨ is proper at any ℓ ∈ A ∨ lf , and therefore itis proper on A ∨ lf . This proves the first statement.We know that µ X ν is locally free; in other words, µ A ∨ is locally free along X ν . Therefore, µ A ∨ is locally free on ˜ X ν = ˜ T · X ν , because ˜ T is Abelian.Therefore, by Corollary 2.7, the stabilizer of any ℓ ∈ ˜ X ν in ˜ T for ˜ µ A ∨ isfinite, since it coincides with the stabilizermof ℓ in T for µ A ∨ . This provesthe second statement.The third statement is a straighforward consequence of the first two.The structure S -action ρ X extends to the holomorphic action˜ ρ A ∨ : ( z, ℓ ) ∈ C ∗ × A ∨ z − ℓ ∈ A ∨ , whose orbits are the fibers of A ∨ over M . Lemma 2.3. ˜ X ν is ˜ ρ A ∨ -invariant.Proof of Lemma 2.3. Since ˜ µ A ∨ and ˜ ρ A ∨ commute, it suffices to show thatfor any x ∈ X ν and z ∈ C ∗ we have z x ∈ ˜ X ν .Let us set C x := n z ∈ C ∗ : z x ∈ ˜ X ν o . Then 1 ∈ C x and C x is open in C ∗ because scalar multiplication is continuousand ˜ X ν is open in A ∨ (Corollary 2.1).11uppose z ∞ ∈ C ∗ is a limit point of C x . Then there exist z , z , . . . ∈ C x such that z i → z ∞ . By definition of C x , for any i = 1 , , . . . we have z i x ∈ ˜ X ν for any i = 1 , , . . . . By definition of ˜ X ν , therefore, there exist ˜ t i ∈ ˜ T and x i ∈ X ν such that z i x = ˜ µ A ∨ ˜ t i ( x i ). Since A ∨ lf in Corollary 2.8 is clearly C ∗ -invariant, we have z ∞ x ∈ A ∨ lf . Thus˜ µ A ∨ ˜ t i ( x i ) = z i x → z ∞ x ∈ A ∨ lf . By Corollary 2.8 and the compactness of X ν , perhaps replacing (˜ t i ) and ( x i )by subsequences, we may assume that ˜ t i → ˜ t ∞ ∈ ˜ T and x i → x ∞ ∈ X ν .Hence z ∞ x = ˜ µ A ∨ ˜ t ∞ ( x ∞ ) ∈ ˜ X ν ⇒ z ∞ ∈ C x . We conclude that C x = C ∗ for any x ∈ X ν .Let us set, as in the Introduction, A ∨ ν := ( π ′ ) − ( M ν ) , ˜ A ∨ ν := ˜ T · A ∨ ν . Corollary 2.9. ˜ X ν = ˜ A ∨ ν .Proof of Corollary 2.9. Since X ν ⊂ A ∨ ν , clearly ˜ X ν ⊆ ˜ A ∨ ν . On the otherhand, ˜ X ν is C ∗ -invariant by Lemma 2.3 and contains X ν , hence ˜ X ν ⊇ A ν .Since ˜ X ν is ˜ T -invariant, we also have ˜ X ν ⊇ ˜ A ∨ ν .It follows from Lemma 2.3 that ˜ X ν is the inverse image of a ˜ T -invariantopen set of M . More precisely, let M ′ ν := ˜ T · M ν ⊆ M. Since π ′ is a submersion and intertwines ˜ µ M and ˜ µ A ∨ , we conclude the fol-lowing: Corollary 2.10. M ′ ν is open in M and ˜ X ν = ( π ′ ) − ( M ′ ν ) . As in the Introduction, let ˜ M ν ⊆ M be the dense open subset of stablepoints for γ M . Obviously ˜ M ν is ˜ γ M -invariant (notation is as in the Introduc-tion and § Lemma 2.4. ˜ M ν = M ′ ν .Proof of Lemma 2.4. Since ∈ ı ν ⊥ is a regular value of Φ ν ⊥ , ˜ M ν = ˜ T r − ν ⊥ · M ν . Hence trivially ˜ M ν = ˜ T r − ν ⊥ · M ν ⊆ ˜ T · M ν = M ′ ν .To prove the converse inclusion it suffices to check that ˜ M ν is ˜ T -invariant.For k = 1 , , . . . , let ˜ˆ µ ( k ) be the representation of ˜ T on H ( M, A ⊗ k ) induced12y ˜ µ A ∨ , and let H ( M, A ⊗ k ) T r − ν ⊥ ⊆ H ( M, A ⊗ k ) be the subspace of thosesections that are invariant under T r − ν ⊥ (equivalently, ˜ T r − ν ⊥ ). Then m ∈ ˜ M ν if and only if for some k = 1 , , . . . there exists σ ∈ H ( M, A ⊗ k ) T r − ν ⊥ suchthat σ ( m ) = 0. Since ˜ T is Abelian, ˜ˆ µ ( k )˜ t ( σ ) ∈ H ( M, A ⊗ k ) T r − ν ⊥ for any ˜ t ∈ ˜ T ;therefore, if m ∈ M is stable for γ M , then so is ˜ µ M ˜ t ( m ), for any ˜ t ∈ ˜ T .In the following, we shall write ˜ A ∨ ν for ˜ X ν . Since ˜ µ A ∨ is holomorphic,proper, effective and locally free on ˜ A ∨ ν , we reach the following conclusion. Corollary 2.11.
If BA 1.1 holds, then N ν := ˜ A ∨ ν / ˜ T is a compact and con-nected orbifold of complex dimension d + 1 − r , and the projection p ν : ˜ A ∨ ν → N ν (21) is a principal V -bundle with structure group ˜ T .Proof. Since ˜ T acts properly, holomorphically and locally freely on ˜ X ν , N ν is a connected complex orbifold of dimension d + 1 − r . Furthermore, bydefinition of ˜ X ν , p ν ( X ν ) = N ν . Hence N ν is compact. Remark 2.3.
The holomorphic slices in Corollary 2.7 provide local uni-formizing charts for N ν . Associated to p ν and the character ˜ χ ν there is anholomorphic orbifold line bundle B ν on N ν . N ′ ν and N ν We shall see that N ′ ν has a natural complex structure, and that the pairs( N ′ ν , B ′ ν ) and ( N ν , B ν ) in Corollaries 2.3 and 2.11 are naturally isomorphicas complex orbifolds and orbifold line bundles.If F ⊆ A ∨ is an holomorphic slice for ˜ µ A ∨ as in Corollary 2.7, let J F beits complex structure. Then ( F, J F ) is a complex submanifold of ( A ∨ , J ′ ),and provides a local uniformizing chart for the complex orbifold N ν .On the other hand, given x ∈ X ν let F ⊆ X ν be a slice at x for the action µ X ν : T × X ν → X ν induced by µ X . The stabilizer T x T of x in T is afinite subgroup of T , and by Corollary 2.7 T x = ˜ T x (the stabilizer in ˜ T ).If ǫ >
0, let F ǫ ⊆ F be the intersection of F with an open ball centeredat x and radius ǫ , in the K¨ahler metric on A ∨ associated to ˜ ω in (17).The proof of the following will be omitted. Proposition 2.2. If ǫ > is suitably small, F ǫ is a slice for of ˜ µ A ∨ . F is not a complex submanifold of A ∨ , and in fact it does notcontain any complex submanifold of positive dimension. Nonetheless, thereis a natural complex structure J F on it, that may be described as follows.If ℓ ∈ ˜ A ∨ ν , the tangent space to the ˜ T -orbit of ℓ , ˜ t A ∨ ( ℓ ) ⊆ T ℓ A ∨ , is an r -dimensional complex subspace; let S ℓ ⊂ T ℓ A ∨ be the orthocomplement of˜ t A ∨ ( ℓ ) for the Riemannian metric associated to (17). Thus S ℓ is a complexsubspace of T ℓ A ∨ , of dimension d + 1 − r , and we have a smoothly varyingdirect sum decomposition T ℓ A ∨ = ˜ t A ∨ ( ℓ ) ⊕ S ℓ . Globally on ˜ A ∨ ν , this yieldsa vector bundle decomposition T A ∨ = ˜ t A ∨ ⊕ S . Projecting along ˜ t A ∨ , weobtain a morphism of vector bundles Π : T A ∨ → S (on A ∨ ν ).Let F by any slice for ˜ µ A ∨ in ˜ A ∨ ν ; in particular, by Proposition 2.2, F might be a slice for µ X ν . At any ℓ ∈ F , the restriction of Π ℓ is an isomorphismof real vector spaces Π Fℓ : T ℓ F → S ℓ . We may define an almost complexstructure J F on F by declaring Π Fℓ to be an isomorphism of complex vectorspaces for each ℓ ∈ F . If F is an holomorphic slice, J F clearly coincides withthe complex structure of F as a submaifold of A ∨ .It is clear that the same J F would be defined, if instead of S one hadchosen another complementary complex subundle S ′ to ˜ t A ∨ . The followingcharacterization does not involve the choice of a specific sub-bundle. Lemma 2.5. If ℓ ∈ F and v ∈ T ℓ F , then J Fℓ ( v ) is uniquely determined bythe conditions: • J Fℓ ( v ) ∈ T ℓ L ; • J Fℓ ( v ) − J ′ ℓ ( v ) ∈ ˜ t A ∨ ( ℓ ) .Proof of Lemma 2.5. For v ∈ T ℓ A ∨ , let v t ∈ ˜ t A ∨ ( ℓ ) and v s ∈ S ℓ be its com-ponents. As both ˜ t A ∨ ( ℓ ) and S ℓ are complex subspaces for J ′ ℓ , J ′ ℓ ( v s ) = J ′ ℓ ( v ) s , J ′ ℓ ( v t ) = J ′ ℓ ( v ) t . By definition of J F if v ∈ T ℓ A ∨ then J Fℓ ( v ) s = J ′ ℓ ( v s ) = J ′ ℓ ( v ) s . Hence, (cid:0) J Fℓ ( v ) − J ′ ℓ ( v ) (cid:1) s = J ′ ℓ ( v ) s − J ′ ℓ ( v ) s = 0 ⇒ J Fℓ ( v ) − J ′ ℓ ( v ) ∈ ˜ t A ∨ ( ℓ ) . Suppose that I Fℓ : T ℓ F → T ℓ F is another operator such that I Fℓ ( v ) − J ′ ℓ ( v ) ∈ ˜ t A ∨ ( ℓ ) for every v ∈ T ℓ F . Then (by definition of slice) ∀ v ∈ T ℓ F wehave I Fℓ ( v ) − J Fℓ ( v ) = (cid:0) I Fℓ ( v ) − J ′ ℓ ( v ) (cid:1) − (cid:0) J Fℓ ( v ) − J ′ ℓ ( v ) (cid:1) ∈ T ℓ F ∩ ˜ t A ∨ ( ℓ ) = (0) . F , F ⊂ ˜ A ∨ ν for ˜ µ A ∨ such that p ν ( F ) ⊆ p ν ( F ). Let ℓ j ∈ F j be such that p ν ( ℓ ) = p ν ( ℓ ). Hence there exists ˜ t ∈ ˜ T such that ℓ = ˜ µ A ∨ ˜ t ( ℓ ). Perhaps after restricting F , we may find a unique C ∞ function f : F → ˜ t , such that f ( ℓ ) = and ( ℓ ) := ˜ µ A ∨ ˜ t e f ( ℓ ) ( ℓ ) ∈ F , for all ℓ ∈ F .Thus : F → F is an injection in the sense of Satake ([S1], [S2]). Lemma 2.6. : F → F is (cid:0) J F , J F (cid:1) -holomorphic.Proof of Lemma 2.6. By local uniqueness, it suffices to prove thatd ℓ : ( T ℓ F , J F ℓ ) → ( T ℓ F , J F ℓ )is C -linear. If v ∈ T ℓ F , we haved ℓ f ( v ) ∈ t , d ℓ f ( v ) A ∨ ∈ X ( A ∨ ) , d ℓ f ( v ) A ∨ ( ℓ ) ∈ ˜ t A ∨ ( ℓ ) ⊆ T ℓ A ∨ , and d ℓ ( v ) = d ℓ f ( v ) A ∨ ( ℓ ) + d ℓ ˜ µ A ∨ ˜ t ( v ) . (22)If w, w ′ ∈ T ℓ A ∨ , we shall write w ≡ w ′ to mean that w − w ′ ∈ ˜ t A ∨ ( ℓ ).By (22), we have d ℓ ( v ) ≡ d ℓ ˜ µ A ∨ ˜ t ( v ) for every v ∈ T ℓ F . Replacing v with J F ℓ ( v ), in view of Lemma 2.5 we obtaind ℓ (cid:0) J F ℓ ( v ) (cid:1) ≡ d ℓ ˜ µ A ∨ ˜ t (cid:0) J F ℓ ( v ) (cid:1) ≡ d ℓ ˜ µ A ∨ ˜ t (cid:0) J ′ ℓ ( v ) (cid:1) = J ′ ℓ (cid:16) d ℓ ˜ µ A ∨ ˜ t ( v ) (cid:17) ≡ J ′ ℓ (d ℓ ( v )) ≡ J F ℓ (d ℓ ( v )) . (23)The first and the last vector in (23) belong to T ℓ F ; hence by Lemma 2.5d ℓ (cid:0) J F ℓ ( v ) (cid:1) = J F ℓ (d ℓ ( v )), for all v ∈ T ℓ F .In Lemma 2.6, we may assume by Corollary 2.7 that F , say, is holomor-phic; hence Lemma 2.6 implies the following. Corollary 2.12.
For any slice F ⊂ ˜ A ∨ ν for ˜ µ A ∨ , J F is integrable. We may also take F = F = F be a slice at ℓ ∈ A ∨ , and consider theself-injections of F induced by the stabilizer T ℓ T of ℓ . Corollary 2.13. If ℓ ∈ A ∨ ν and F ⊂ A ∨ ν is a slice for ˜ µ A ∨ at ℓ , then ˜ T ℓ actsholomorphically on ( F, J F ) . If we apply these considerations to the slices F ⊆ X ν for µ X ν , we concludethe following. 15 orollary 2.14. The V-manifold N ′ ν in Corollary 2.3 is complex. Since every T -orbit in X ν is obviously contained in a unique ˜ T -orbit in˜ A ν , there is a well-defined map ψ : T · x ∈ N ′ ν ˜ T · x ∈ N ν . Let J N ′ ν and J N ν be the orbifold complex structures of N ′ ν and N ν , respec-tively. Proposition 2.3. ψ is an isomorphism of complex orbifolds (cid:0) N ′ ν , J N ′ ν (cid:1) → (cid:0) N ν , J N ν (cid:1) .Proof of Proposition 2.3. By Corollary 2.9, any ˜ T -orbit in ˜ A ∨ ν intersects X ν ;thus ψ is surjective.To prove that ψ is injective, suppose by contradiction that there exist x , x ∈ X ν such that x ∈ ˜ T · x (i.e., ψ ( T · x ) = ψ ( T · x )), but x T · x (i.e., T · x = T · x ). Perhaps after replacing x with another point in T · x ,we may assume that x = ˜ µ A ∨ e − ξ ( x ) for some ξ ∈ R r \ { } . We may writeuniquely ξ = ξ ′ + a ν , where ξ ′ ∈ ν ⊥ and a ∈ R . Perhaps interchanging x and x , we may assume without loss that a ≥ η := ı ξ ∈ t . Considering the associated vector fields ξ A ∨ , η A ∨ ∈ X ( A ∨ ) we have − ξ A ∨ = J ′ ( η A ∨ ); hence − ξ A ∨ is the gradient vector field ofthe Hamiltonian function ˜Φ η = h ˜Φ , η i , where ˜Φ is as in (20).Since x ∈ X ν , we have ˜Φ( x ) = ı λ ν for some λ >
0, hence ˜Φ η ( x ) = λ a k ν k ≥
0. Since ˜Φ η is strictly increasing along its gradient flow where thegradient is non-vanishing,˜Φ η (cid:16) ˜ µ A ∨ e − t ξ ( x ) (cid:17) > ˜Φ η ( x ) ≥ ∀ t > . (24)On the other hand, we have η A ∨ = η ♯M − ˜Φ η ∂ θ ⇒ − ξ A ∨ = (cid:0) J η M ) ♯ + ˜Φ η r ∂ r . Here r ∂ r is the generator of the 1-parameter group of diffeomorphisms ℓ e t ℓ . With ̺ as in (16), for every t > − ξ A ∨ ( ̺ ) (cid:16) ˜ µ A ∨ e − t ξ ( x ) (cid:17) = ˜Φ η (cid:16) ˜ µ A ∨ e − t ξ ( x ) (cid:17) r ∂ r ̺ (cid:16) ˜ µ A ∨ e − t ξ ( x ) (cid:17) > . It follows that ̺ (cid:16) ˜ µ A ∨ e − t ξ ( x ) (cid:17) > ̺ ( x ) = 1 for t >
0; taking t = 1, we concludethat x X , a contradiction. Hence ψ is a bijection.16et us verify that ψ is a homeomorphism. The open sets of N ′ ν havethe form U/T , where U ⊆ X ν is open and T -invariant, and the open setsof N ν have the form ˜ U / ˜ T , where ˜ U ⊆ A ∨ ν is open and ˜ T -invariant. Theprevious argument shows that each ˜ T -orbit in ˜ A ∨ ν intersects X ν in a single T -orbit. One can see from this (and the definition of ˜ A ∨ ν ) that there is abijection between the family of ˜ T -invariant open sets ˜ U in ˜ A ∨ ν and the familyof T -invariant open sets U in X ν given by ˜ U U := ˜ U ∩ X ν , with inverse U ˜ U := ˜ T · U .Given any such ˜ U , we have ψ − ( ˜ U / ˜ T ) = U/T ⊆ N ′ ν , implying that ψ iscontinuous. Similarly, given any such U we have ψ ( U/T ) = ˜
U / ˜ T , implyingthat ψ is open. Hence ψ is a homeomorphism.To conclude that ψ is an isomorphism of complex orbifolds, it suffices toverify that its local expressions in uniformizing charts are biholomorphisms;actually, it suffices to do so for corresponding defining families in the senseof [S1] and [S2] that cover N ′ ν and N ν . Let F be a slice at x for µ X ν at some x ∈ X ν ; by Proposition 2.2, perhaps after shrinking F if necessary, we mayassume that F is also a slice at x for ˜ µ ˜ A ∨ ν . Hence ( F, J F ) is a uniformizingchart of both N ′ ν and N ν . By definition of ψ and the previous considerations,the identity id F : F → F is a local representative map of ψ , and it is obviouslybiholomorphic ( F, J F ) → ( F, J F ).The sheaf of holomorphic functions on N ν is defined equivalently by the˜ T -invariant holomorphic functions on ˜ A ∨ ν or the T ℓ -invariant holomorphicfunctions on the slices ( F, J F ). Let us briefly clarify this point.Since ( F, J F ) is generally not a complex submanifold of ( A ∨ , J ′ ), arbitraryholomorphic functions on the saturation ˜ T · F needn’t restrict to holomorphicfunctions on ( F, J F ). However, this does happens if we restrict to invariantholomorphic functions. Definition 2.2.
Suppose that ℓ ∈ A ∨ ν and that F ⊆ A ∨ ν is a slice at ℓ for˜ µ ˜ A ∨ ν . Let us adopt the following notation.1. O ( F ) is the ring of J F -holomorphic functions on F ;2. O ( F ) T ℓ ⊆ O ( F ) is the subring of T ℓ -invariant functions in O ( F );3. O ( ˜ T · F ) is the ring of J ′ -holomorphic functions on the saturation of F under ˜ µ A ∨ ;4. O ( ˜ T · F ) ˜ T ⊆ O ( ˜ T · F ) is the subring of ˜ µ ˜ A ∨ ν -invariant functions.Then we have the following, whose prooof will be omitted (see the argu-ment for Proposition 2.4). 17 emma 2.7. In the situation of Definition 2.2, restriction yields an isomor-phism O ( ˜ T · F ) ˜ T → O ( F ) T ℓ . ˜ A ∨ ν and X ν M ν is a CR submanifold of M , and the maximal complex sub-bundle H ( M ν ) ⊆ T M ν has complex dimension d + 1 − r , and is as follows. If m ∈ M ν ,(˜ t r − ν ⊥ ) M ( m ) ⊆ T m M ν is a complex subspace of dimension r −
1, since ˜ γ M islocally free at m . Then H ( M ν ) m = (˜ t r − ν ⊥ ) M ( m ) ⊥ hm , where h m = g m − ı ω m is the Hermitian product on T m M associated to theK¨ahler metric.Similarly, X ν is a CR submanifold of A ∨ . The maximal complex sub-bundle H ( X ν ) ⊂ T X ν is as follows. If x ∈ X ν and m = π ( x ), then H ( X ν ) x = H ( M ν ) ♯m . (25) Definition 2.3.
Let be given λ ∈ Z r .For any ˜ T -invariant open subset ˜ U ⊆ ˜ A ∨ ν , let O ( ˜ U ) λ be the ring of holo-morphic functions ˜ S : ˜ U → C such that˜ S (cid:16) ˜ µ A ∨ ˜ t − ( ℓ ) (cid:17) = ˜ χ λ (˜ t ) ˜ S ( ℓ ) (˜ t ∈ ˜ T , ℓ ∈ ˜ U ) . (26)For any T -invariant open subset U ⊆ X ν , let let CR ( U ) λ be the ring ofCR functions on U satisfying S ( µ c t − ( x )) = χ λ ( t ) S ( x ) ( t ∈ T, x ∈ X ν ) . (27) Proposition 2.4.
With notation in Definition 2.3, suppose that U = ˜ U ∩ X ν .Then restriction yields a ring isomorphism O ( ˜ U ) λ → CR ( U ) λ . Corollary 2.15.
Restriction yields an isomorphism O ( ˜ A ∨ ν ) λ → CR ( X ν ) λ .Proof of Proposition 2.4. Clearly if ˜ S ∈ O ( ˜ U ) λ then S := ˜ S (cid:12)(cid:12)(cid:12) U ∈ CR ( U ) λ .Thus the ring homomorphim in the statement is well-defined and obviouslyinjective.To prove surjectivity, suppose conversely that S ∈ CR ( U ) λ . Let us define˜ S : ˜ U = ˜ T · U → C by setting˜ S (cid:16) ˜ µ A ∨ ˜ t ( x ) (cid:17) = ˜ χ − λ (˜ t ) S ( x ) = ˜ χ λ (˜ t ) − S ( x ) (˜ t ∈ ˜ T , x ∈ X ν ) . (28)18o verify that (28) is well-defined, suppose that ˜ µ A ∨ ˜ t ( x ) = ˜ µ A ∨ ˜ t ( x ) with˜ t j ∈ ˜ T and x j ∈ U . By the argument in the proof of Proposition 2.3,˜ t − ˜ t ∈ T . Therefore S ( x ) = χ − λ (cid:0) ˜ t − ˜ t (cid:1) S ( x ) = ˜ χ − λ (cid:0) ˜ t (cid:1) − ˜ χ − λ (cid:0) ˜ t (cid:1) S ( x ) . By construction, ˜ S satisfies (26) and restricts to S on U . To prove that S ˜ S inverts restriction it remains to verify that ˜ S is holomorphic, i.e. thatd ℓ ˜ S is C -linear for any ℓ ∈ ˜ U .By Corollary 2.9, the map F : (˜ t , x ) ∈ ˜ T × X ν ˜ µ A ∨ ˜ t ( x ) ∈ ˜ A ∨ ν is surjective. In fact, F exhibits ˜ A ∨ ν as the quotient of ˜ T × X ν by the freeaction of T given by t · (cid:0) ˜ t , x (cid:1) := (cid:0) ˜ t t − , µ X t ( x ) (cid:1) . (29)Furthermore, F ( ˜ T × U ) = ˜ U .For every ˜ t ∈ ˜ T let us set X ˜ t ν := F (cid:0)(cid:8) ˜ t (cid:9) × X ν (cid:1) = ˜ µ A ∨ ˜ t ( X ν ) . Again by the proof of Proposition 2.3 we have X ˜ t ν = X ˜ t ν if ˜ t − ˜ t ∈ T , and X ˜ t ν ∩ X ˜ t ν = ∅ otherwise. Clearly X ˜ t ν is a CR submanifold of A ∨ , and its CRbundle H ( X ˜ t ν ) is as follows. If ℓ = F (˜ t , x ) ∈ X ˜ t ν , then H ( X ˜ t ν ) ℓ = d x ˜ µ A ∨ ˜ t (cid:0) H ( X ν ) x (cid:1) . If ˜ t ∈ ˜ T , let us identify T ˜ t ˜ T ∼ = ˜ t in the standard manner. For (˜ t , x ) ∈ ˜ T × X ν , let us consider the vector subspace K (˜ t , x ) := ˜ t × H ( X ν ) x ⊆ T ˜ t ˜ T × T x X ν ∼ = T (˜ t ,x ) ( ˜ T × X ν ) . The distribution
K ⊆ T ( ˜ T × X ν ) is invariant under (29) and is naturallya complex vector bundle; furthermore, d F yields an isomorphism of complexvector bundles K → F ∗ ( T ˜ A ∨ ν ). More explicitly, if ℓ = F (˜ t , x ) thend (˜ t ,x ) F (cid:12)(cid:12) K (˜ t,x ) : ˜ t × H ( X ν ) x → ˜ t A ∨ ( ℓ ) ⊕ H ( X ˜ t ν ) ℓ = T ℓ A ∨ (30)is an isomorphism of complex vector spaces, respecting the direct sum de-compositions on both sides. 19iven S ∈ CR ( U ) λ , let us consider the complex function ˆ S on ˜ T × X ν given by ˆ S (cid:0) ˜ t , x (cid:1) := ˜ χ ν (˜ t ) − S ( x ) . (31)Then ˆ S = ˜ S ◦ F .Let us assume that ℓ = F (˜ t , x ). We haved (˜ t ,x ) ˆ S (cid:12)(cid:12)(cid:12) H (˜ t ,x ) = d ℓ ˜ S ◦ d (˜ t ,x ) F (cid:12)(cid:12) K (˜ t ,x ) : K (˜ t , x ) → C . Hence to prove that d ℓ ˜ S : T ℓ A ∨ ν → C is C -linear it suffices to show that d (˜ t ,x ) ˆ S is C -linear on K (˜ t ,x ) = ˜ t × H ( X ν ) x ; to do so, in turn it is sufficient to verify C -linearity on each summand ˜ t and H ( X ν ) x separately. This follows from(31), since ˜ χ ν is holomorphic (implying C -linearity on the first summand)and S is CR (implying C -linearity on the second summand). B ν and B ′ ν We have seen that the restrictions of µ X to X ν and of ˜ µ A ∨ to ˜ A ∨ ν are lo-cally free, effective and proper actions of T and ˜ T , respectively, and thatthe corresponding quotients N ′ ν := X ν /T and N ν := ˜ A ∨ ν / ˜ T are naturallyisomorphic complex orbifolds. Furthermore, the projections p ′ ν : X ν → N ′ ν and p ν : ˜ A ν ν → N ν are principal V -bundles with structure group T and ˜ T ,respectively.Associated to the characters χ ν : T → S and ˜ χ ν : ˜ T → C ∗ , we have 1-dimensional representations of T and ˜ T , respectively; we shall denote eitherone by C ν . The product actions µ X × C ν and ˜ µ A ∨ × C ν are therefore also locallyfree, effective and proper on X ν × C ν and ˜ A ∨ ν × C ν respectively. Hence thequotients B ′ ν := X ν × T C ν and B ν := ˜ A ∨ ν × ˜ T C ν are orbifold line bundles on N ′ ν and N ν . Let us denote by P ′ ν : B ′ ν → N ′ ν and P ν : B ν → N ν (32)the respective projections. Lemma 2.8.
Suppose x ∈ X ν and let F ⊆ X ν be a slice for the restrictionof µ X to X ν . Then F × C ν is a slice at ( x, for the restriction of µ X × C ν to X ν × C ν . The collection of all these slices yields a defining family for B ′ ν .Similarly, suppose ℓ ∈ ˜ A ∨ ν and let F ⊆ ˜ A ∨ ν be a slice at ℓ for the restrictionof ˜ µ A ∨ to ˜ A ∨ ν . Then F × C ν is a slice at ( ℓ, for the restriction of ˜ µ A ∨ × C ν to ˜ A ∨ ν × C ν . The collection of all these slices yields a defining family for B ν . roof of Lemma 2.8. Let us consider the former statement, the proof of thelatter being similar.Since F × C ν is transverse to the T -orbits in X ν × C ν , the map h : T × ( F × C ν ) → X ν × C ν induced by the diagonal action is a local diffeomorphismonto the open saturation T · ( F × C ν ) ⊆ X ν × C .Clearly we have the equality of stabilizers T ( x, = T x . Furthermore,suppose ( y, w ) ∈ F × C ν , t ∈ T . Then µ X × C ν t ( y, w ) = (cid:0) µ X t ( y ) , χ ν ( t ) w (cid:1) ∈ F × C ν if and only if µ X t ( y ) ∈ F , that is, if and only if t ∈ T x . Hence h descends toa diffeomorphism h : (cid:0) T × ( F × C ν ) (cid:1) /T x → T · ( F × C ν ) , where T x acts antidiagonally on T × ( F × C ν ). Corollary 2.16. F × C ν with the diagonal action of T x uniformizes the openset ( F × C ν ) /T x ⊆ B ν ′ . The collection of all these uniformizing charts is adefnining family for the orbifold line bundle B ′ ν . A similar statement holdsfo B ν . Given the complex structure J F on each F (Lemma 2.5), we obtain aproduct complex structure on F × C ν . Hence both B ′ ν and B ν are complexorbifolds of complex dimension d + 2 − r .Since any T -orbit in X ν × C is contained in a unique ˜ T -orbit in ˜ A ∨ ν × C ,there is a natural continuous map ˜ ψ : B ′ ν → B ν . The proof of Proposition2.3 can be adapted to yield the following: Proposition 2.5. ˜ ψ is an isomorphism of complex orbifolds, and ψ ◦ P ′ ν = P ν ◦ ˜ ψ . Y ν We need an alternative description of B ′ ν . Consider the intermediate quotient Y ν := X ν /T r − ν ⊥ ;then Y ν is compact orbifold, of (real) dimension 2 ( d + 1 − r ) + 1, and theintegrable and invariant CR structure on X ν descends to an integrable CRstructure on Y ν . We shall denote by H ( Y ν ) the CR bundle of Y ν .21et T ν T be the connected compact subgroup of T associated to theLie subalgebra span( ı ν ) ⊆ t . Given that ν is coprime, we have a Lie groupisomorphism κ ν : e ı ϑ ∈ S e ı ϑ ν := (cid:0) e ı ϑ ν , . . . , e ı ϑ ν r (cid:1) ∈ T ν . (33)Let us set T ν := T /T r − ν ⊥ ∼ = T ν / ( T ν ∩ T r − ν ⊥ ).Suppose x ∈ X ν , and let F ⊆ X ν be a slice at x for the restriction of γ X to X ν . We can view T × F as a uniformizing chart for the smooth orbifold X ν , with uniformized open set T · F = ( T × F ) /T x . Then T ν × F is auniformizing chart for Y ν , covering the open set ( T · F ) /T r − ν ⊥ .Explicitly, T x act effectively on T ν × F by t · (cid:0) t, f (cid:1) := (cid:16) t t − , µ Xt ( f ) (cid:17) , where for any t ∈ T we have set t = t T r − ν ⊥ ∈ T ν . Then the map γ : ( t, f ) ∈ T ν × F T r − ν ⊥ · µ Xt ( f ) ∈ ( T · F ) /T r − ν ⊥ ⊆ Y ν induces a homeomorphism ( T · F ) /T r − ν ⊥ = ( T × F ) /T ℓ . Letting F vary, weobtain a defining family for Y ν .Furthermore, T ν acts effectively on Y ν , and N ′ ν = Y ν /T ν ; let σ ν : Y ν → N ′ ν be the projection. For each slice F ⊆ X ν , as above, the local represen-tation of σ ν is the projection T ν × F → F . Thus Y ν is a principal V -bundleover N ′ ν , with structure group T ν .Being trivial on T r − ν ⊥ , χ ν descends to a character χ ′ ν : T ν → S . Lemma 2.9.
Given that ν is coprime, χ ′ ν is a Lie group isomorphism.Proof of Lemma 2.9. Since T ν ∼ = T ν / (cid:0) T ν ∩ T r − ν ⊥ (cid:1) , the statement is equiva-lent to the equality ker( χ ν | T ν ) = T ν ∩ T r − ν ⊥ ; (34)since clearly T r − ν ⊥ ⊆ ker( χ ν ), we need only prove that ker( χ ν | T ν ) ⊆ T ν ∩ T r − ν ⊥ .Since ν is coprime, there exists k = (cid:0) k · · · k r (cid:1) ∈ Z r such that h ν , k i = P rj =1 k j ν j = 1.Let κ ν be as in (33). Then χ ν ◦ κ ν (cid:0) e ı ϑ (cid:1) = χ ν (cid:0) e ı ϑ ν (cid:1) = e ı ϑ k ν k (cid:0) e ı ϑ ∈ S (cid:1) . (35)22ence if e ı ϑ ν ∈ ker( χ ν ), then we may assume ϑ = ϑ j := 2 π j/ k ν k forsome j = 0 , . . . , k ν k −
1. We have h ϑ j ν , ν i = 2 π j k ν k h ν , ν i = 2 π j = 2 π j h k , ν i , so that ϑ j ν − π j k ∈ ν ⊥ . Thus e ı ϑ j ν = e ı [ ϑ j ν − π j k ] ∈ T ν ∩ T r − ν ⊥ . Since k ν is an isomorphism, (35) implies the following. Corollary 2.17.
Assuming that ν is coprime, (cid:12)(cid:12) T r − ν ⊥ ∩ T ν (cid:12)(cid:12) = k ν k . Given the isomorphism χ ′ ν : T ν ∼ = S , we shall view Y ν as a principal V -bundle over N ′ ν with structure group S . Let us denote by σ Y ν : S × Y ν → Y ν (36)the corresponding action.Let Q ν : X ν → Y ν (37)be the projection. Then U is a T -invariant open subset of X ν if and onlyif its image Q ν ( U ) is a T ν ∼ = S -invariant open subset of Y ν . It follows(recall the proof of Proposition 2.3) that there is a bijective correspondencebetween ˜ T -invariant open subsets ˜ U ⊆ ˜ A ∨ ν , T -invariant open subsets U ⊆ X ν , S -invariant open subsets U ⊆ Y ν , given by U ; = ˜ U ∩ X ν , U := Q ν ( U ).The character χ ′ k ν = ( χ ′ ν ) k : T ν → S corresponds to the endomorphism χ k : g ∈ S g k ∈ S . Let us denote by CR ( Y ν ) the collection of all CRfunctions on Y ν , and for any k ∈ Z let us set CR ( Y ν ) k = (cid:8) f ∈ CR ( Y ν ) : f ◦ σ Y ν e − ı θ = e ı k θ f, ∀ e ı θ ∈ S (cid:9) . (38)Using that the CR structure of Y ν is obtained by descending the invariantCR structure of X ν , we can complement Proposition 2.4 and Corollary 2.15by the following isomorphisms induced by pull-back: O (cid:0) ˜ U (cid:1) k ν ∼ = CR ( U ) k ν ∼ = CR (cid:0) U (cid:1) k . (39)Letting H ( N ν , B k ν ) denote the space of holomorphic sections of theorbifold line bundle B k ν , we conclude that H ( N ν , B k ν ) ∼ = O (cid:0) ˜ A ∨ ν (cid:1) k ν ∼ = CR ( X ν ) k ν ∼ = CR ( Y ν ) k . (40)23 .8 The induced K¨ahler structure of N ν We shall see that P ν : B ν → N ν is a positive holomorphic V -line bundle. Inview of Propositions 2.3 and 2.5 we may equivalently consider P ′ ν : B ′ ν → N ′ ν .With α as in (1), let α X ν := ∗ ν ( α ), where ν : X ν ֒ → X (41)is the inclusion. Then α X ν is T -invariant, and by definition of X ν for any ξ ∈ ν ⊥ we have ι (cid:0) ( ı ξ ) X ν (cid:1) α X ν = ∗ ν (cid:0) ι (cid:0) ( ı ξ ) X (cid:1) α (cid:1) = − h Φ , ı ξ i ◦ ν = 0 . Hence α X ν is the pull-back of an orbifold 1-form α Y ν on Y ν . Similarly, being T -invariant, Φ descends to a smooth function Φ : Y ν → t ∨ ; hence Φ ν = h Φ , ı ν i descends to a smooth function Φ ν : Y ν → R .Clearly, ι (cid:0) ( ı ν ) Y ν (cid:1) α Y ν = − Φ ν . (42)Let us define β ν := k ν k Φ ν α Y ν , and let − δ Y ν ∈ X ( Y ν ) be the infintesimal generator of σ Y ν in (36). Thus byCorollary 2.17 − δ Y ν = 1 k ν k ( ı ν ) Y ν . (43)Given (43) and (42), we conclude the following. Corollary 2.18. β ν is σ Y ν -invariant, and β ν ( δ Y ν ) = 1 . Hence β ν is a connection 1-form for the principal V -bundle P ′ ν : B ′ ν → N ′ ν .Explicitly, d β ν = k ν k (cid:20) ν d α Y ν − ν ) dΦ ν ∧ α Y ν (cid:21) , (44)and one can also verify that ι ( δ Y ν ) d β ν = 0 by direct inspection using (43)and (44). Furthermore, the kernel of β ν is the CR bundle of Y ν :ker( β ν ) = ker( α Y ν ) = H ( Y ν ) . Hence we reach the following conclusion. Let π ν : Y ν → N ′ ν be the projection. Lemma 2.10.
There exists a (1 , -form η ′ ν on N ′ ν such that d( β ν ) = 2 π ∗ ν ( η ′ ν ) .
24e shall denote by η ν the corresponding form on N ν . With the notationof Proposition 2.3, we have the following. Proposition 2.6. ( N ′ ν , J N ′ ν , η ′ ν ) and ( N ν , J N ν , η ν ) are isomorphic K¨ahlerorbifolds. In particular, ( Y ν , β ν ) is a contact orbifold.Proof of Proposition 2.6. It suffices to prove that ( N ′ ν , J N ′ ν , η ′ ν ) is a K¨ahlerorbifold, since the other statements follow directly.The uniformized tangent space of Y ν splits as the direct sum V ( Y ν ) ⊕ H ( Y ν ), where V ( Y ν ) is the tangent space to the orbits of σ Y ν . To check that η ν is K¨ahler, it suffices therefore to verify that the restriction of d β ν to H ( Y ν )is compatible with the complex structure. In view of (44) and T -invariance,we need only check that the form Q ∗ ν (d β ν ) = k ν k ∗ ν (cid:18) ν d α − ν ) dΦ ν ∧ α (cid:19) , (45)where Q ν is as in (37) and ν as in (41), is compatible with the complexstructure of the CR bundle H ( X ν ).Suppose x ∈ X ν and let m := π ( x ) ∈ M ν . The general vector in H ( X ν ) x has the form v ♯ for some v ∈ H ( M ν ) m (see (25)), and then J ′ x ( v ♯ ) = J m ( v ) ♯ .By (45), for any v, w ∈ H ( M ν ) m Q ∗ ν (d β ν ) x ( v ♯ , w ♯ ) = k ν k Φ ν ( m ) d x α ( v ♯ , w ♯ ) = k ν k Φ ν ( m ) 2 ω m ( v, w ) . (46)The statement follows, since Φ ν ( m ) > M ν , H m ( M ν ) ⊆ T m M is a complex subspace, and ω is K¨ahler. Corollary 2.19. ( N ν , B ν ) is polarized K¨ahler orbifold. Here notation is as in Chaper 4 of [BG]. By the Kodaira-Baily VanishingTheorem ([B], [BG]), we obtain the following conclusion.
Corollary 2.20. H i ( N ν , B k ν ) = 0 , ∀ i > , k ≫ . N ν The action ρ X : S × X → X with infinitesimal generator − ∂ θ in (1) is thecontact lift of the trivial circle action on M corresponding to the momentmap Φ = 1 (recall (4)). We shall see that ρ X determines a contact actionon ( Y ν , β ν ) and an holomorphic Hamiltonian action on ( N ′ ν , J N ′ ν , η ′ ν ), such25hat the former is the contact lift of the latter by (the orbifold version of)the procedure in (4), when we regard Y ν as an orbifold circle bundle on X ν .Clearly, ρ X commutes with µ X . Hence ρ X leaves X ν invariant and de-termines a restricted action ρ X ν : S × X ν → X ν . For the same reason ρ X ν passes to the quotients Y ν and N ν . In other words, ρ X ν descends toactions ρ Y ν : S × Y ν → Y ν and ρ N ′ ν : S × N ′ ν → N ′ ν , so that the projections Q ν : X ν → Y ν and π ν : Y ν → N ′ ν are equivariant.In particular, if − ∂ X ν θ is the restriction of − ∂ θ to X ν , − ∂ Y ν θ is the in-finitesimal generator of ρ Y ν , and − ∂ N ν θ is the infinitesimal generator of ρ N ν ,then ∂ X ν θ and ∂ Y ν θ are Q ν -related, and similarly ∂ Y ν θ and ∂ N ν θ are π ν -related. Lemma 2.11. ρ N ′ ν is Hamiltonian on ( N ′ ν , η ′ ν ) , with moment map k ν k / Φ ν + c , for any c ∈ R .Proof of Lemma 2.11. By T -invariance of all terms involved, and the previ-ous remark about the correlations of the generating vector fields, we needonly prove that − ι (cid:0) ∂ X ν θ (cid:1) Q ∗ ν (d β ν ) = d (cid:0) k ν k / Φ ν ◦ ν (cid:1) , where ν is as in (41) and Q ∗ ν (d β ν ) as in (45). We have on a neighborhoodof X ν : − ι ( ∂ θ ) k ν k (cid:18) ν d α − ν ) dΦ ν ∧ α (cid:19) = − k ν k (Φ ν ) dΦ ν = d (cid:18) k ν k Φ ν (cid:19) , establishing the claim.Thus − ∂ N ′ ν θ is a Hamiltonian vector field on ( N ′ ν , η ′ ν ), and every choiceof c ∈ R in Lemma 2.11 determines a contact lift − g ∂ N ′ ν θ (implicitly dependingon c ) to ( Y ν , β ν ) of − ∂ N ′ ν θ , as in (4).Here Y ν plays the role of X , β ν the role of α , and − ∂ N ′ ν θ the one of ξ M .The role of − ∂ θ (the infinitesimal generator of ρ X ) is played by − δ Y ν (theinfinitesimal generator of σ Y ν ).We need to determine the ‘correct choice’ of c that determines ρ Y ν as thecontact lift of ρ N ′ ν . Lemma 2.12.
We have g ∂ N ′ ν θ = ∂ Y ν θ if and only if c = 0 . Given an (orbifold) vector field V on N ν , we shall denote by V ♮ ∈ X ( Y ν )its horizontal lift to Y ν with respect to β ν .26 roof of Lemma 2.12. On X ν we have by (4)1 k ν k ν X ν = 1 k ν k ν ♯M ν − Φ ν k ν k ∂ X ν θ . (47)Here ν M ν is the restriction of ν M to M ν (a vector field on M ν ), and ν ♯M ν isits horizontal lift to X ν .Given that ρ X and µ X commute, [ ν X , ∂ θ ] = 0 on X ; this implies [ ν ♯M , ν X ] =[ ν ♯M , ∂ θ ] = 0. Furthermore, one has [ ν X , γ X ] = 0 for every γ ∈ t , and thisimplies also [ ν ♯M , γ X ] = 0.Being horizontal and T r − ν ⊥ -invariant, ν ♯M ν / Φ ν is π ν -related to a horizontalvector field on Y ν ; the latter is σ Y ν -invariant by the above, and therefore itis the horizontal lift − υ ♮ to Y ν of a vector field − υ on N ν .Multiplying both sides of (47) by k ν k / Φ ν and then pushing down to Y ν we obtain υ ♮ − k ν k Φ ν δ Y ν = − ν ν ♯M ν − k ν k Φ ν δ Y ν = − ∂ Y ν θ , (48)and pushing down to N ν this yields υ = − ∂ N ν θ . (49)In view of Lemma 2.11, (49) implies that υ is the Hamiltonian vector fieldon ( N ′ ν , η ′ ν ) of k ν k / Φ ν + c ; then (48) implies that − ∂ Y ν θ is its contact liftcorresponding to c = 0. It is clear that any other choice of c yields a differentlift.In the following, we shall identify the pairs ( N ′ ν , B ′ k ν ) ∼ = ( N ν , B k ν ). O ( ˜ A ∨ ν ) k ν Consider the holomorphic action ρ A ∨ : ( e ı θ , ℓ ) ∈ S × A ∨ → e − ı θ ℓ ∈ A ∨ . Thus ρ A ∨ extends ρ X . Similarly, let µ A ∨ : T × A ∨ → A ∨ be the holomorphicaction extending µ X . Clearly, ρ A ∨ and µ A ∨ commute.The dense open subset ˜ A ∨ ν ⊆ A ∨ is invariant under both ρ A ∨ and µ A ∨ ,which therefore restrict to commuting holomorphic actions ρ ˜ A ∨ ν and µ ˜ A ∨ ν on˜ A ∨ ν . 27herefore, ρ ˜ A ∨ ν and µ ˜ A ∨ ν determine commuting representations ˆ ρ ˜ A ∨ ν of S and ˆ µ ˜ A ∨ ν of T on the space O ( ˜ A ∨ ν ) of holomorphic functions on ˜ A ∨ ν , given byˆ ρ ˜ A ∨ ν e ıθ ( s ) := s ◦ ρ ˜ A ∨ ν e − ıθ , ˆ µ ˜ A ∨ ν t ( s ) := s ◦ µ ˜ A ∨ ν t − (cid:0) s ∈ O ( ˜ A ∨ ν ) , e ıθ ∈ S , t ∈ T. (cid:1) . For every l ∈ Z and λ ∈ Z r , let O ( ˜ A ∨ ν ) l and O ( ˜ A ∨ ν ) λ be the l -th and λ -thisotypical components of O ( ˜ A ∨ ν ), respectively, for ˆ ρ ˜ A ∨ ν and ˆ µ ˜ A ∨ ν , respectively.Hence ˆ ρ ˜ A ∨ ν restricts to a subrepresentation on O ( ˜ A ∨ ν ) λ . In particular, forevery k = 1 , , . . . the vector space O ( ˜ A ∨ ν ) k ν is finite dimensional by (40),and we have an S × T -equivariant decomposition O ( ˜ A ∨ ν ) k ν = M l ∈ Z O ( ˜ A ∨ ν ) k ν , l , (50)where O ( ˜ A ∨ ν ) k ν , l = O ( ˜ A ∨ ν ) k ν ∩ O ( ˜ A ∨ ν ) l . Since the isomorphisms in (40)are by construction S -equivariant, (50) may be interpreted in terms of H ( N ν , B k ν ): H ( N ν , B k ν ) = M l ∈ Z H ( N ν , B k ν ) l . (51) Lemma 2.13. If k ≫ , H ( N ν , B k ν ) l = 0 for all l ≤ .Proof of Lemma 2.13. In the terminology of [MS], the datum of the Hamilto-nian action ρ N ν , with moment map k ν k / Φ ν , makes B k ν into a prequantum S -equivariant orbibundle, hence into a moment line bundle. By Corollary2.11 of [MS], and given that k ν k / Φ ν >
0, we conclude that the Fourierdecomposition of RR( N ν , B k ν ) (viewed as a virtual character of S ) has theform RR( N ν , B k ν ) = X l> RR( N ν , B k ν ) l · χ l , (52)where where χ l ( e ı θ ) = e ı l θ . In view of Corollary 2.20 this means that, as arepresentation of S , H ( N ν , B k ν ) = M l> H ( N ν , B k ν ) l ∀ k ≫ . (53)By the S -equivariance in (40), we can now sharpen (50) as follows. Corollary 2.21. If k ≫ , then O ( ˜ A ∨ ν ) k ν = M l> O ( ˜ A ∨ ν ) k ν , l . (54)28 Proof of Theorem 1.1
We can now give the proof of Theorem 1.1. First, however, let us considerthe following statement.
Lemma 3.1.
For every λ ∈ Z , restriction yields an isomorphism O ( A ∨ ) λ ∼ = H ( X ) ˆ µ λ .Proof of Lemma 3.1. Clearly, restriction yields a morphism ζ λ : O ( A ∨ ) λ → H ( X ) ˆ µ λ . If f ∈ O ( A ∨ ) is non-zero, then the locus where its differentialvanishes has real codimension ≥
2; if it vanishes on X , therefore, f = 0.Hence ζ λ is injective.Since by assumption Φ( M ), we have dim H ( X ) ˆ µ λ < + ∞ for every λ .Hence we have a finite direct sum H ( X ) ˆ µ λ = b ( λ ) M l = a ( λ ) H ( X ) ˆ µ λ ,l , where 0 ≤ a ( λ ) ≤ b ( λ ) < + ∞ , H ( X ) ˆ µ λ ,l := H ( X ) ˆ µ λ ∩ H ( X ) l . Hence, to verify that ζ λ is surjective, it suffices to show that any s ∈ H ( X ) ˆ µ λ ,l is the restriction of some ˜ s ∈ O ( A ∨ ) λ . Any s ∈ H ( X ) ˆ µl is the restriction ofan holomorphic homogeneous function of degree l , ˜ s ∈ O ( A ∨ ) l . Since ρ A ∨ and γ A ∨ commute, one sees that ˜ s is in the λ -th isotype for T , and thereforefor ˜ T as well. Hence ζ λ is surjective. Proof of Theorem 1.1.
By Lemma 3.1, for every k = 1 , , . . . we have a na-tural equivariant injective linear map F k ν := res k ν ◦ ζ − k ν : H ( X ) ˆ µk ν → O ( ˜ A ∨ ν ) k ν ∼ = H ( N ν , B k ν ) , (55)where res k ν : O ( A ∨ ) k ν → O ( ˜ A ∨ ν ) k ν denotes restriction, and is obviouslyinjective since ˜ A ∨ ν is open and dense in A ∨ ; this proves the first statement ofTheorem 1.1.To prove the second statement, it suffices to verify that res k ν is surjectivefor k ≫
0. We have for some c ( k, ν ) , d ( k, ν ) ∈ Z with c ( k, ν ) ≤ d ( k, ν ): O ( ˜ A ∨ ν ) k ν = d ( k, ν ) M l = c ( k, ν ) O ( ˜ A ∨ ν ) k ν ,l ;29ence res k ν = d ( k, ν ) M l = c ( k, ν ) res k ν ,l , where res k ν ,l : O ( A ∨ ) k ν ,l → O ( ˜ A ∨ ν ) k ν ,l and we need to check that res k ν ,l is surjective for every l = c ( k, ν ) , . . . , d ( k, ν )and k ≫ c ( k, ν ) >
0. Furthermore, byLemma 2.4 ˜ A ∨ ν = ( π ′ ) − ( ˜ M ν ) and therefore res k ν ,l may canonically reinter-preted in terms of the restriction of holomorphic sections: f res k ν ,l : H (cid:0) M, A ⊗ l (cid:1) k ν → H (cid:0) ˜ M ν , A ⊗ l (cid:1) k ν . (56)Hence we are reduced to proving that f res k ν ,l in (56) is surjective for all l > s ∈ H (cid:0) M, A ⊗ l (cid:1) . Then s ∈ H (cid:0) M, A ⊗ l (cid:1) k ν if and only if thefollowing two conditions hold:1. s is γ X -invariant, i.e., s ∈ H ( M, A ⊗ l ) T r − ν ⊥ ;2. for any e ı ϑ ∈ S , ˆ µ e ı ϑ ν ( s ) = e ı k k ν k ϑ s. In other words, we can identify H (cid:0) M, A ⊗ l (cid:1) k ν with the k k ν k -isotypicalcomponent for the representation of T ν ∼ = S on H ( M, A ⊗ l ) T r − ν ⊥ . The sameconsiderations apply to H (cid:0) ˜ M ν , A ⊗ l (cid:1) k ν . We shall express this by writing H ( M, A ⊗ l ) k ν = H ( M, A ⊗ l ) T r − ν ⊥ k k ν k , H ( ˜ M ν , A ⊗ l ) k ν = H ( ˜ M ν , A ⊗ l ) T r − ν ⊥ k k ν k . It is well-known that the restriction map f l, ν : H (cid:0) M, A ⊗ l (cid:1) T r − ν ⊥ → H (cid:0) ˜ M ν , A ⊗ l (cid:1) T r − ν ⊥ (57)is an isomorphism ( § T ν -equivariant. The claim follows, since f res k ν ,l is the restriction of f l, ν to the k k ν k -isotypical component, hence by equivariance it induces an isomor-phism H ( M, A ⊗ l ) T r − ν ⊥ k k ν k ∼ = H ( ˜ M ν , A ⊗ l ) T r − ν ⊥ k k ν k .30 eferences [ALR] A. Adem, J. Leida, Y. Ruan, Orbifolds and stringy topology , Cam-bridge Tracts in Mathematics, , Cambridge University Press,Cambridge, 2007[B] W. L. Baily,
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