The geometric quantizations and the measured Gromov-Hausdorff convergences
aa r X i v : . [ m a t h . DG ] N ov THE GEOMETRIC QUANTIZATIONS AND THEMEASURED GROMOV-HAUSDORFF CONVERGENCES
KOTA HATTORI
Abstract.
On a compact symplectic manifold (
X, ω ) with a prequan-tum line bundle ( L, ∇ , h ), we consider the one-parameter family of ω -compatible complex structures which converges to the real polarizationcoming from the Lagrangian torus fibration. There are several researcheswhich show that the holomorphic sections of the line bundle localize atBohr-Sommerfeld fibers. In this article we consider the one-parameterfamily of the Riemannian metrics on the frame bundle of L determinedby the complex structures and ∇ , h , and we can see that their diametersdiverge. If we fix a base point in some fibers of the Lagrangian fibrationwe can show that they measured Gromov-Hausdorff converge to somepointed metric measure spaces with the isometric S -actions, which maydepend on the choice of the base point. We observe that the propertiesof the S -actions on the limit spaces actually depend on whether thebase point is in the Bohr-Sommerfeld fibers or not. Introduction
In this article we introduce a new approach to the geometric quantizationfrom the viewpoint of the convergence of the Riemannian manifolds withrespect to the measured Gromov-Hausdorff topology. On a compact sym-plectic manifold (
X, ω ) of dimension 2 n , a prequantum line bundle ( L, ∇ , h )is a triple of a complex line bundle L , hermitian metric h and connection ∇ preserving h with curvature form F ∇ = −√− ω . By considering thefollowing geometric structures compatible with ω , we can equip L with thefinite dimensional vector subspace consisting of the special sections of L .The first one is an ω -compatible complex structure J , then denote by H ( X J , L ) = { s ∈ C ∞ ( L ); ∇ ∂ J s = 0 } the space of all of the holomorphic sections of L .The second one is a Lagrangian fibration µ : X → Y , where Y is a smoothmanifold, all of the points in Y are regular values of µ , and all fibers arecompact and connected. Then put V µ := M y ∈ Y H (cid:0) µ − ( y ) , L | µ − ( y ) , ∇| µ − ( y ) (cid:1) ,H (cid:0) µ − ( y ) , L | µ − ( y ) , ∇| µ − ( y ) (cid:1) := (cid:8) s ∈ C ∞ ( L | µ − ( y ) ); ∇| µ − ( y ) s ≡ (cid:9) .µ − ( y ) is called a Bohr-Sommerfeld fiber if L | µ − ( y ) has nontrivial parallelsections. Tyurin showed in [19, Proposition 3.2] that if X is compact then Mathematics Subject Classification. there are at most finitely many Bohr-Sommerfeld fibers, accordingly, dim V µ is finite.In many examples of symplectic manifolds with some complex structuresand Lagrangian fibrations, it is observed thatdim V µ = dim H ( X J , L )when the Kodaira vanishing holds, which can be interpreted as the local-ization of the Riemann-Roch index to the Bohr-Sommerfeld fibers, and dis-cussed by Andersen [1], by Fujita, Furuta and Yoshida [8], and by Kubota[14].Moreover, on smooth toric varieties, Baier, Florentino, Mour˜ao and Nuneshave constructed a one parameter family of the pairs of the complex struc-tures and the basis of the spaces of holomorphic sections of L , then showedthat the holomorphic sections localize on the Bohr-Sommerfeld fibers in [3].The similar phenomena were observed in the case of the abelian varieties byBaier, Mour˜ao and Nunes [4] and the flag varieties by Hamilton and Konno[11]. In these examples, the family of complex structures and holomorphicsections are described concretely.In the context of the geometric quantization, the ω -compatible complexstructures and Lagrangian fibrations are treated uniformly by using thenotion of polarizations. The one-parameter families of complex structuresgiven in the above papers are taken such that the K¨ahler polarizations cor-responding to them converge to the real polarization corresponding to theLagrangian fibration.Recently, Yoshida showed the localization of holomorphic sections of pre-quantum line bundle to the Bohr-Sommerfeld fiber if X admits a Lagrangianfibration with a complete base in [22], where the family of complex struc-tures are taken such that it converges to the real polarization correspondingto the Lagrangian fibration.In this article, we also study the behavior of holomorphic sections of L where the family of complex structures converges to the real polariza-tion from the view of the point of the measured Gromov-Hausdorff con-vergence. Fix an ω -compatible complex structure J . Then H ( X J , L ) canbe identified with the eigenspace of a Laplace operator as follows. Put S := { u ∈ L ; h ( u, u ) = 1 } ⊂ L be the orthogonal frame bundle of ( L, h ),then there is the standard identification C ∞ ( X, L ) ∼ = ( C ∞ ( S ) ⊗ C ) ρ , where ρ : S → GL ( C ) is the 1-dimensional unitary representation of S defined by ρ ( e √− t ) := e √− t , and the S -action on C ∞ ( S ) ⊗ C is definedby { e √− t · ( f ⊗ ξ ) } ( u ) := e −√− t f ( ue √− t ) ⊗ ξ for any f ∈ C ∞ ( S ), ξ ∈ C and u ∈ S . The connection ∇ gives the connection1-form on S and the decomposition of T S into the horizontal and verticalsubspaces. Then we have the Riemannian metric ˆ g on S which respects theconnection form and the K¨ahler metric g J := ω ( · , J · ). The precise definitionof ˆ g is given by Section 3. Denote by ∆ ˆ g the Laplace operator of ˆ g . Since S acts on ( S, ˆ g ) isometrically, the C -linear extension of ∆ ˆ g gives the operator HE GEOMETRIC QUANTIZATIONS 3 acting on ( C ∞ ( S ) ⊗ C ) ρ . Then we can see that H ( X J , L ) is identified withthe eigenspace of ∆ ˆ g : ( C ∞ ( S ) ⊗ C ) ρ → ( C ∞ ( S ) ⊗ C ) ρ associate with the eigenvalue n + 1.Now, we suppose that a one-parameter family of the ω -compatible com-plex structures { J s } s> on X is given, then we consider the one-parameterfamily of the operators∆ ˆ g s : ( C ∞ ( S ) ⊗ C ) ρ → ( C ∞ ( S ) ⊗ C ) ρ . There are several research of the spectral convergence of the metric Laplacianon Riemannian manifolds or the connection Laplacians on vector bundlesunder the convergence of the spaces in the sense of the measured Gromov-Hausdorff topology [5][9][13][15][16][17]. Therefore, there should be the sig-nificant relation between the convergence of principal bundle S with theconnection metric ˆ g s and the convergence of holomorphic sections with re-spect to J s . This article focus on the convergence of ( S, ˆ g s , p ) as s → S -equivariant measured Gromov-Hausdorff topologyand we study the metric measure spaces appearing as the limit.Now we explain the main result of this article. Let ( X, ω ) be a sym-plectic manifold of dimension 2 n , which is not necessarily to be compact,( L, ∇ , h ) be a prequantum line bundle and { J s }
0, there is a constant κ ∈ R such that Ric g Js ≥ κg J s for all s . We also suppose additional assumptionswhich are precisely described in ♠ of Section 7.2. Let g m, ∞ and µ ∞ be aRiemannian metric and a measure on R n × S defined by g m, ∞ := 1 m (1 + k y k ) ( dt ) + n X i =1 ( dy i ) ,dµ ∞ := dy · · · dy n dt, where m is a positive integer, y = ( y , . . . , y n ) ∈ R n and e √− t ∈ S . Wedefine the isometric S -action on ( R n × S , g m, ∞ , µ ∞ ) by ( y, e √− t ) · e √− τ :=( y, e √− t + mτ ) ) for e √− τ ∈ S . The followings are the main results of thisarticle. Theorem 1.1.
Let m be a positive integer and u ∈ S | µ − ( y ) . Assume that µ − ( y ) is a Bohr-Sommerfeld fiber of L ⊗ m and not a Bohr-Sommerfeld fiberof L ⊗ m ′ for any < m ′ < m . Then for some positive constant K > , thefamily of pointed metric measure spaces with the isometric S -action (cid:26)(cid:18) S, ˆ g s , µ ˆ g s K √ s n , u (cid:19)(cid:27) s converges to (cid:0) R n × S , g m, ∞ , µ ∞ , (0 , (cid:1) as s → in the sense of the pointed S -equivariant measured Gromov-Hausdorff topology. KOTA HATTORI
Theorem 1.2.
Let u ∈ S | µ − ( y ) and assume that µ − ( y ) is not a Bohr-Sommerfeld fiber of L ⊗ m for any positive integer m . Then { ( S, ˆ g s , µ ˆ gs K √ s n , u ) } s converges to ( R n , t dy · dy, dy · · · dy n , as s → in the sense of the pointed S -equivariant measured Gromov-Hausdorff topology. Here, the S -actionon R n is trivial. Now let S ∞ be the metric measure space appears as the limit in Theorem1.1 or Theorem 1.2 and denote by ∆ ∞ its Laplacian. Denote by W ( n + 1)the eigenspace of∆ ∞ : ( C ∞ ( S ∞ ) ⊗ C ) ρ → ( C ∞ ( S ∞ ) ⊗ C ) ρ associate with the eigenvalue n + 1. Theorem 1.3. If S ∞ be the metric measure space appears as the limit inTheorem 1.1, then dim W ( n + 1) = 1 if m = 1 and dim W ( n + 1) = 0 if m > . If S ∞ be the metric measure space appears as the limit in Theorem1.2, then dim W ( n + 1) = 0 . This article is organized as follows. First of all, we explain how to iden-tify the holomorphic sections of L on ( X, J ) with the eigenfunctions on theframe bundle S equipped with the connection metric in Section 2 and 3. InSection 4, we review the definition of Bohr-Sommerfeld fibers for the pairsof symplectic manifolds and prequantum line bundles. In Section 5, we re-view the notion of Polarizations, which enables us to treat the ω -compatiblecomplex structures and the Lagrangian fibrations. In Section 6 we explainthe notion of the pointed S -equivariant measured Gromov-Hausdorff con-vergence. This notion is the special case of the convergence introduced byFukaya and Yamaguchi [10]. These sections are the preparations for themain argument. In Section 7, we show the pointed S -equivariant measuredGromov-Hausdorff convergence near the Bohr-Sommerfeld fibers. First ofall we obtain the local description of the connection metric ˆ g s on S , thendiscuss the condition equivalent to the existence of the lower bound of theRicci curvatures. Then we show the convergence of ˆ g s to g m, ∞ as s →
0. InSection 9 we consider the limit of ˆ g s near the non Bohr-Sommerfeld fibers,then show that the S -action on the limit space is trivial. In Section 8, westudy the spectral structure of the Laplacian of the metric measure spacesappearing as the limit of ˆ g s .2. Holomorphic line bundles
Let (
X, J, ω ) be a compact K¨ahler manifold. We write X = X J when weregard X as a complex manifold. Let π E : E → X J be a holomorphic linebundle over X J . Suppose h is a hermitian metric on E and ∇ : Γ( E ) → Ω ( E ) is the Chern connection. Under the decomposition Ω = Ω , ⊕ Ω , ,we have the decomposition ∇ = ∇ , + ∇ , . Let ∇ ∗ , ( ∇ , ) ∗ , ( ∇ , ) ∗ are theformal adjoint of ∇ , ∇ , , ∇ , , respectively. HE GEOMETRIC QUANTIZATIONS 5
For a holomorphic coordinate (
U, z , . . . , z n ) on X J , put ω = √− g i ¯ j dz i ∧ d ¯ z j . Then we may write ∇ ∗ = ( ∇ , ) ∗ + ( ∇ , ) ∗ , ( ∇ , ) ∗ = − g i ¯ j ι ∂ i ∇ ¯ ∂ j , ( ∇ , ) ∗ = − g i ¯ j ι ¯ ∂ j ∇ ∂ i , where ∂ i := ∂∂z i . Let F ∈ Ω , ( X J ) be the curvature form. Since we have( ∇ , ) ∗ ∇ , s = ( ∇ , ) ∗ ∇ , s + g i ¯ j F ( ∂ i , ¯ ∂ j ) s, we obtain ∇ ∗ ∇ = 2∆ ¯ ∂ + Λ ω F, ∆ ¯ ∂ := ( ∇ , ) ∗ ∇ , , Λ ω F := g i ¯ j F ( ∂ i , ¯ ∂ j ) ∈ C ∞ ( X ) . Let L → X J be a holomorphic line bundle with hermitian metric h andhermitian connection ∇ such that the curvature form is equal to −√− ω ,and put E = L k . Then for the connection on E determined by ∇ we have F = − k √− ω , then Λ ω F = nk. Now, put H ( X J , L k ) := n s ∈ C ∞ ( L k ); ∇ , s = 0 o . Since X is compact, we can see H ( X J , L k ) = n s ∈ C ∞ ( L k ); ∇ ∗ ∇ s = nks o . Holomorphic sections on line bundles and eigenfunctions onframe bundle
Let (
X, ω ) be a connected symplectic manifold of dimension 2 n and( π : L → X, ∇ , h ) be a prequantum line bundle over ( X, ω ), that is, a com-plex line bundle with a hermitian metric h a connection ∇ preserving h whose curvature form is equal to −√− ω .The complex structure J on X is ω -compatible if ω ( J · , J · ) = ω holds and g J := ω ( · , J · ) is positive definite. If J is ω -compatible, then ω is a K¨ahlerform on X J .Since ω is of type (1 , ∇ determines a holomorphic structure on L ,consequently ∇ is the Chern connection determined by h and J .By the previous section we have ∇ ∗ ∇ = 2∆ ¯ ∂ + n . Put S := S ( L, h ) := { u ∈ L ; | u | h = 1 } , which is a principal S -bundle over X equipped with the S -connection √− ∈ Ω ( S, √− R ) corresponding to ∇ . The S -connection induces the KOTA HATTORI following decomposition T u S := H u ⊕ V u ,H u := Ker (Γ u : T u S → R ) ,V u := { ξ ♯u ∈ T u S ; ξ ∈ √− R } , where ξ ♯u := ddt e tξ | t =0 . Then the connection metric ˆ g = ˆ g ( L, J, h, σ, ∇ ) on S is defined by ˆ g ( L, J, h, σ, ∇ ) := σ · Γ + ( dπ | H ) ∗ g J for σ > Remark 3.1.
By regarding − Γ as a contact structure and −√− ♯ as theReeb vector field, ( S, ˆ g ( L, J, h, , ∇ )) becomes a Sasakian manifold.Now we can recover L by S as the associate bundle as follows. Let ρ k : S → GL ( C ) be defined by ρ k ( λ ) = λ k for k ∈ Z , then we have theidentification L k ∼ = S × ρ k C . Then there are natural isomorphisms C ∞ ( X, L k ) ∼ = ( C ∞ ( S ) ⊗ C ) ρ k , where the action of S on C ∞ ( S ) ⊗ C is defined by ( λ · f )( u ) := λ k f ( uλ ).By applying the argument in the previous section for E = L k we have ∇ ∗ ∇ = 2∆ ¯ ∂ + kn . Note that we may regard ∇ ∗ ∇ and ∆ ¯ ∂ as operatorsacting on ( C ∞ ( S ) ⊗ C ) ρ k , then by [13, Section 3] we have ∇ ∗ ∇ = ∆ ˆ g − k σ ,therefore we obtain2∆ ¯ ∂ = ∆ ˆ g − (cid:18) k σ + kn (cid:19) : ( C ∞ ( S ) ⊗ C ) ρ k → ( C ∞ ( S ) ⊗ C ) ρ k . On some open set U ⊂ X , suppose that L | U is trivial as C ∞ complexbundles, then there exists a global smooth section E ∈ C ∞ ( U, L ) such that h ( E, E ) ≡
1. Let γ ∈ Ω ( U, R ) be defined by ∇ E = √− γ ⊗ E . Underthe diffeomorphism U × S → S ( L | U , h ) defined by ( z, e √− t ) e √− t E z ,one can obtain the following identification as Riemannian manifolds withisometric S -action;( S | U , ˆ g ) ∼ = ( U × S , g J | U + σ ( dt + γ ) ) . (1) 4. Bohr-Sommerfeld fibers
Let ( π : L → X, ∇ ) be a prequantum line bundle over a symplectic mani-fold ( X, ω ). A Lagrangian fibration over (
X, ω ) is a smooth map µ : X → B ,where B is a smooth manifold of dimension dim X , such that X b := µ − ( b ) isa Lagrangian submanifold for every b ∈ B \ B sing and B \ B sing is open densein B . We suppose that B and all of the fibers X b are path-connected. Thenevery X b is diffeomorphic to a compact torus by Liouville-Arnold theorem.For a subset Y ⊂ X , the holonomy Hol( L | Y , ∇ ) is defined byHol( L | Y , ∇ ) := { e √− t ∈ S ; ˜ c (1) = ˜ c (0) e √− t , c ∈ P ( a ) } , where P ( a ) consists of piecewise smooth curve c : [0 , → X with c (0) = c (1) = a ∈ Y , Im( c ) ⊂ Y and ˜ c is the horizontal lift of c . Note thatHol( L | Y , ∇ ) does not depend on a ∈ Y if Y is path-connected. HE GEOMETRIC QUANTIZATIONS 7
Definition 4.1. (i) X b is a Bohr-Sommerfeld fiber of µ : X → B if Hol( L | X b , ∇ ) is trivial.(ii) X b is an m -BS fiber of µ : X → B if Hol( L | X b , ∇ ) is a subgroup of Z /m Z . X b is a strict m -BS fiber of µ : X → B if Hol( L | X b , ∇ ) ∼ = Z /m Z . Remark 4.2. X b is a m -BS fiber of µ : X → B iff Hol( L m | X b , ∇ ) is trivial. Remark 4.3.
In this article we suppose that B m := { b ∈ B ; X b is an m -BS fiber } are discrete in B for all m >
0. This condition always holds if µ : X → B comes from completely integrable system or B sing = ∅ by the results in[19][20]. If we put B ′ m := { b ∈ B ; X b is a strict m -BS fiber } , then B m = F l | m B ′ l holds. 5. Polarizations
In this section we review the notion of polarizations in the sense of [21]to treat complex structures and Lagrangian fibrations uniformly.Let V R be a real vector space of dimension 2 n with symplectic form α ∈ V V ∗ and put V = V R ⊗ C . Then α extends C -linearly to a complexsymplectic form on V . A Lagrangian subspace W of V is a complex vectorsubspace of V such that dim C W = n and α ( u, v ) = 0 for all u, v ∈ W . PutLag( V, α ) := { W ⊂ V ; W is a Lagrangian subspace } , which is a submanifold of Grassmannian Gr( n, V ).For a symplectic manifold ( X, ω ), putLag ω := G x ∈ X Lag( T x X ⊗ C , ω x ) . This is a fiber bundle over X , and a section P of Lag ω is a subbundle of T X ⊗ C . P is said to be integrable if[Γ( P| U ) , Γ( P| U )] ⊂ Γ( P| U )holds for any open set U ⊂ X , and we call such P a polarization of X . Inthis article we consider the following two types of polarizations. K¨ahler polarizations.
Let J be an ω -compatible complex structure. Thesubbundle P J := T , J X ⊂ T X ⊗ C is called a K¨ahler polarization. KOTA HATTORI
Real polarizations.
Let Y be a smooth manifold of dimension n , µ : X → Y be a smooth map such that all b ∈ µ ( X ) are regular values and µ − ( b )are Lagrangian submanifolds. Then P µ := Ker( dµ ) ⊗ C ⊂ T X ⊗ C is called a real polarization.Define l : Lag( V, α ) → { , , . . . , n } by l ( W ) := dim C ( W ∩ W ). Thenfor any K¨ahler polarization P J we have l (( P J ) x ) = 0, and for any realpolarization P µ we have l (( P µ ) x ) = n .Conversely, for a polarization P such that l ( P x ) = 0 for all x ∈ X , there isa unique complex structure J such that ω ( J · , J · ) = ω and P = T , J X . For apolarization P such that l ( P x ) = n for all x ∈ X , we obtain the Lagrangianfoliation.Next we observe the local structure of Lag( V, α ). For W ∈ Lag(
V, α ), wecan take a basis { w , . . . , w n } ⊂ W and vectors u , . . . , u n ∈ V such that { w , . . . , w n , u , . . . , u n } is a basis of V and α ( w i , w j ) = α ( u i , u j ) = 0 , α ( u i , w j ) = δ ij hold. Put W ′ := span C { u , . . . , u n } and take A ∈ Hom(
W, W ′ ). Then thesubspace W A := { w + Aw ∈ V ; w ∈ W } is Lagrangian iff the matrix ( A ij ) defined by Aw i = A ij u j is symmetric.Consequently, we have the identification T W Lag(
V, α ) = (cid:8) A ∈ Hom(
W, W ′ ); A ij = A ji (cid:9) . (2)Now, we fix W such that l ( W ) = n . Then w , . . . , w n , u , . . . , u n can betaken to be real vectors, hence l ( W A ) = dim Ker( A − A ) = n − rank( A − A )holds. Moreover W A comes form an almost complex structure which makes α the positive hermitian iff Im A ∈ M n ( R ) is the positive definite symmetricmatrix. We define T W Lag(
V, α ) + := (cid:8) A ∈ Hom(
W, W ′ ); A ij = A ji , Im A > (cid:9) under the identification (2). If W t is a smooth curve in Lag( V, α ) such that l ( W ) = n and ddt W t | t =0 ∈ T W Lag(
V, α ) + , then there is δ > l ( W t ) = 0 and α ( w, ¯ w ) > w ∈ W t \ { } and 0 < t ≤ δ . Conversely,even if W t satisfies l ( W ) = n and l ( W t ) = 0 , α ( w, ¯ w ) > w ∈ W t \ { } for all t > ddt W t | t =0 is not necessary to be in T W Lag(
V, α ) + since theclosure of positive definite symmetric matrices contains semi-positive definitesymmetric matrices. HE GEOMETRIC QUANTIZATIONS 9 Topology
In this section we explain the notion of the S -equivariant measuredGromov-Hausdorff topology. The following notion is the special case of[10, Definition 4.1]. Definition 6.1.
Let G be a compact topological group.(1) Let ( P ′ , d ′ ) and ( P, d ) be metric spaces with isometric G -action. Amap φ : P ′ → P is an G -equivariant ε -approximation if φ is G -equivariant and ε -approximation. Here, ε -approximation means that | d ′ ( x ′ , y ′ ) − d ( φ ( x ′ ) , φ ( y ′ )) | < ε holds for all x ′ , y ′ ∈ P ′ and P ⊂ B ( φ ( P ′ ) , ε ). Moreover if φ is a Borel map then it is called a Borel G -equivariant ε -approximation .(2) Let { ( P i , d i , ν i , p i ) } i be a sequence of pointed metric measure spaceswith isometric G -action. ( P ∞ , d ∞ , ν ∞ , p ∞ ) is said to be the pointed G -equivariant measured Gromov-Hausdorff limit of { ( P i , d i , ν i , p i ) } i if G acts on P ∞ isometrically and for any R > { ε i } i , { R i } i withlim i →∞ ε i = 0 , lim i →∞ R i = R, and Borel G -equivariant ε i -approximation φ i : ( π − i ( B ( x i , R i )) , p i ) → ( π − ∞ ( B ( x ∞ , R )) , p ∞ )for every i such that φ i ∗ ( ν i | π − i ( B ( x i ,R i )) ) → ν ∞ | π − ∞ ( B ( x ∞ ,R )) vaguely.Here, π : P i → P i /G is the quotient map and x i = π i ( p i ).7. Convergence
Throughout of this section let ( X n , ω ) be a symplectic manifold, Y n asmooth manifold and µ : X → Y be a smooth surjective map such that µ − ( y ) are smooth compact connectedLagrangian submanifolds for all regular value y ∈ Y . Assume that y ∈ Y isa regular value of µ . Then by [2][7][18], there are open neighborhoods U ⊂ X of X := µ − ( y ), B ′ ⊂ Y of y , B ⊂ R n of the origin 0, diffeomorphisms˜ f : B × T n ∼ = → U and f : B ′ ∼ = → B such that ˜ f ∗ ω = P ni =1 dx i ∧ dθ i , and f ( y ) =0, where x = ( x , . . . , x n ) = f ◦ µ ◦ ˜ f and θ = ( θ . . . , θ n ) ∈ T n = R n / Z n .Therefore, we may suppose U = B × T n , µ = ( x , . . . , x n ) , ω = dx i ∧ dθ i ,B = (cid:26) x = ( x , . . . , x n ) ∈ R n ; k x k = q x + · · · + x n < R (cid:27) ,X = { } × T n for some 0 < R ≤ L, ∇ ) be the prequantum line bundle on ( X, ω ) and h be a hermitianmetric such that ∇ h = 0. Since [ ω | U ] = 0 ∈ H ( U ), then the 1st Chernclass of ( L, ∇ ) | U vanishes, hence L | U is trivial as C ∞ complex line bundleby [6, Section 5]. From now on we consider some covering spaces of U given by the follow-ings. Let Φ : Z n → Z /m Z be a homomorphism of Z -modules. Then Ker Φis of rank n , hence R n / Ker Φ is diffeomorphic to the n -dimensional torus.Now we have the natural projection R n / Ker Φ → T n ∈ ∈ θ mod Ker Φ θ mod Z n which give a covering space and a covering map U Φ := B × ( R n / Ker Φ) , p Φ : U Φ → U. From now on we denote by θ the element of R n / Ker Φ or T n for the sim-plicity, if there is no fear of confusion. If we take w ∈ Z n then β (Φ( w )) : U Φ → U Φ ∈ ∈ ( x, θ ) ( x, θ + w )gives the action of Im Φ on U Φ , which is the deck transformations of p Φ . Proposition 7.1.
Let X be a strict m -BS fiber. Then there are surjectivehomomorphism Φ : Z n → Z /m Z and E ∈ C ∞ ( p ∗ Φ L ) such that h ( E, E ) ≡ and ∇ E = −√− x i dθ i ⊗ E . Moreover, the deck transformations of p Φ satisfies β ( k ) ∗ E = e k √− πm E for k ∈ Z /m Z .Proof. Since X is the m -BS fiber, one can obtain the flat section ˆ E of( L m | U ) | x =0 ) such that h ⊗ m ( ˆ E, ˆ E ) ≡
1. Then ˆ E can be extended to thenowhere vanishing section of C ∞ ( L m | U ) with h ⊗ m ( ˆ E, ˆ E ) ≡
1. Define γ ∈ Ω ( U ) by ∇ ˆ E = √− γ ⊗ ˆ E . By computing the curvature form of ∇ oneobtain dγ = − mω | U = − mdx i ∧ dθ i which implies that γ + mx i dθ i is a closed1-form on U . Denote by α the cohomology class represented by γ + mx i dθ i and let ι : { } × T n → U be the natural embedding. Since ˆ E | x =0 is flat, thenone can see that ι ∗ γ = 0 and ι ∗ α = 0. Since ι ∗ : H ( B × T n ) → H ( { }× T n )is isomorphic, one can see that α = 0, therefore there exists τ ∈ C ∞ ( U, R )such that γ + mx i dθ i = dτ .Then one have ∇ ( e −√− τ ˆ E ) = √− − dτ + γ ) ⊗ e −√− τ ˆ E = − m √− x i dθ i ⊗ e −√− τ ˆ E, accordingly, by replacing e −√− τ ˆ E by ˆ E , we may suppose ∇ ˆ E = − m √− x i dθ i ⊗ ˆ E. Let ˜ p : ˜ U = B × R n → B × T n be the universal cover of U . Then thereis a nowhere vanishing section E ∈ C ∞ (˜ p ∗ L ) such that E ⊗ m = ˜ p ∗ ˆ E , con-sequently we obtain the homomorphism Φ : π ( U ) = Z n → Z /m Z defiendby E ( x,θ + k ) = e π √− k ) E ( x,θ ) for k ∈ Z n . Therefore, E descends to the section of p ∗ Φ L , then ∇ E = −√− x i dθ i ⊗ E holds. Since X is the strict m -BS fiber, Φ is surjective and p Φ is an m -foldcovering. (cid:3) HE GEOMETRIC QUANTIZATIONS 11
Local description of the complex structures and the metrics.
We assume that an ω -compatible complex structure J on X is given suchthat P J | U is close to P µ | U , as sections of Lag ω | U → U . Define P ′ µ by( P ′ µ ) p := span C ((cid:18) ∂∂x (cid:19) p , . . . , (cid:18) ∂∂x n (cid:19) p ) ⊂ T p U ⊗ C , then we have the direct decomposition T U ⊗ C = P µ ⊕ P ′ µ . Since P J | U isclose to P µ | U , the identification (2) gives A = ( A ij ( x, θ )) i,j ∈ C ∞ ( U ) ⊗ M n ( C )such that A ij = A ji , Im A > ∂∂θ i + A ij ( x, θ ) ∂∂x j , i = 1 , . . . , n is a frame of P J | U . Moreover the integrability of J gives ∂A jk ∂θ i − ∂A ik ∂θ j + A il ∂A jk ∂x l − A jl ∂A ik ∂x l = 0 . (3)Conversely, if a complex matrix valued function A satisfies above prop-erties then we can recover J | U . Therefore, the ω -compatible J complexstructure close to P µ is identified with the matrix valued function A on U .If we put A ij = P ij + √− Q ij , where P ij , Q ij ∈ R , and denote by ( Q ij )the inverse of ( Q ij ), then one can see J (cid:18) ∂∂θ i (cid:19) = − P ij Q jk ∂∂θ k − ( Q ik + P ij Q jl P lk ) ∂∂x k , (4) J (cid:18) ∂∂x i (cid:19) = Q ik ∂∂θ k + Q ij P jk ∂∂x k , (5) J dθ k = − P ij Q jk dθ i + Q ik dx i , (6) J dx k = − ( Q ik + P ij Q jl P lk ) dθ i + Q ij P jk dx i , (7)therefore we obtain g J | U = g A := ( Q ij + P ik Q kl P lj ) dθ i dθ j − P ik Q jk dθ i dx j + Q ij dx i dx j . Denote by d g the Riemannian distance of a Riemannian metric g . Then g J | U = g A , d g J | U ≤ d g A always holds, however, the opposite inequality doesnot hold in general since the shortest path connecting two points in U neednot be included in U . Here we consider the lower estimate of d g J and theupper estimate of d g A .For a real symmetric positive definite matrix valued function S ( x, θ ) =( S ij ( x, θ )) i,j depending on ( x, θ ) ∈ U continuously, let λ ( x, θ ) , . . . , λ n ( x, θ )be the eigenvalues of S ( x, θ ). Define U r := { ( x, θ ) ∈ R n × T n ; k x k < r } ⊂ U ( r ≤ R ) , sup S := sup i, ( x,θ ) ∈ U R λ i ( x, θ ) , inf S := inf i, ( x,θ ) ∈ U R λ i ( x, θ ) . Since U R is compact, 0 < inf S ≤ sup S < ∞ holds. Proposition 7.2.
Put
Θ := Q + P Q − P for A = P + √− Q . The following inequalities p inf(Θ − ) k x − x ′ k ≤ d g J ( u, u ′ ) ,d g A ( u, u ′ ) ≤ p sup(Θ − ) k x − x ′ k + √ n sup Θ2 hold for any u = ( x, θ ) , u ′ = ( x ′ , θ ′ ) ∈ U R .Proof. First of all we show the first equality. If we write dθ = dθ ... dθ n , dx = dx ... dx n , x = x ... x n , then we may write g A = t dθ · Θ · dθ − t dx · Q − P · dθ − t dθ · P Q − · dx + t dx · Q − · dx = t (cid:16) √ Θ dθ − √ Θ − P Q − dx (cid:17) · (cid:16) √ Θ dθ − √ Θ − P Q − dx (cid:17) + t dx · (cid:0) Q − − Q − P Θ − P Q − (cid:1) · dx. Since we haveΘ (cid:0) Q − − Q − P Θ − P Q − (cid:1) = 1 + P Q − P Q − − P Θ − P Q − − P Q − P Q − P Θ − P Q − = 1 + P Q − P Q − − P Q − (cid:0) Q + P Q − P (cid:1) Θ − P Q − = 1 + P Q − P Q − − P Q − P Q − = 1 , we can see Θ − = Q − − Q − P Θ − P Q − . Therefore, g A = t (cid:16) √ Θ dθ − √ Θ − P Q − dx (cid:17) · (cid:16) √ Θ dθ − √ Θ − P Q − dx (cid:17) + t dx · Θ − · dx (8)holds. Now let c : [0 , → X be a path connecting u, u ′ ∈ U R , and put u = ( x, θ ) and u ′ = ( x ′ , θ ′ ) with k x k , k x ′ k < R . Note that the image of c isnot always contained in U R . If Im( c ) ⊂ U R does not hold, then let τ := inf { τ ∈ [0 , c ( τ ) / ∈ U R } . If Im( c ) ⊂ U R holds, then put τ := 1. Put c ( τ ) = ( x ( τ ) , θ ( τ )). Then by(8) we can see L ( c ) ≥ Z τ p t x ′ ( τ ) · Θ − · x ′ ( τ ) dτ ≥ p inf(Θ − ) Z τ | x ′ ( τ ) | dτ ≥ p inf(Θ − ) k x − x ′ k . HE GEOMETRIC QUANTIZATIONS 13
Next we show the second inequality. To show it, we compute the lengthof two types of paths in U R .For θ ∈ R n put c ( τ ) := ( x, τ θ ), then (8) gives L ( c ) = Z | c ′ ( τ ) | g A dτ = Z q Θ ij θ i θ j dτ ≤ p sup Θ k θ k . If c ( τ ) := ( τ x + (1 − τ ) x ′ , θ ), where k x k ≤ R , then L ( c ) = Z | c ′ ( τ ) | ˆ g A dτ = Z q Θ ij ( x i − x ′ i )( x j − x ′ j ) dτ ≤ p sup(Θ − ) k x − x ′ k . Connecting these two types of paths one can see d A ( u, u ′ ) ≤ p sup(Θ − ) k x k + p sup Θ · diam( T n )= p sup(Θ − ) k x k + √ n sup Θ2 . (cid:3) Now, we describe Riemannian metric ˆ g ( L | U , J, h, σ, ∇ ) using the identifi-cation (1) in the case of X is a strict m -BS fiber. First of all we considerthe connection metric with respect to the pullback of g J and L | U by thecovering map p Φ : U Φ → U , which is obtained in Proposition 7.1. We alsodenote by p Φ : p Φ ∗ L → L | U the lift of the covering map, then the followingcommutative diagram is obtained; p ∗ Φ L → L | U ↓ (cid:9) ↓ U Φ → U Let p ∗ Φ J be the complex structure on U Φ inherited from U by the coveringmap. Then one can see S ( p Φ ∗ L, p Φ ∗ h ) = p − ( S ( L, h ))and ˆ g ( p Φ ∗ L, p Φ ∗ J, p Φ ∗ h, σ, p Φ ∗ ∇ ) = p Φ ∗ ˆ g ( L | U , J, h, σ, ∇ ) . Since p Φ ∗ L is trivial as C ∞ complex line bundle, there is the identification U Φ × S → S ( p Φ ∗ L, p Φ ∗ h ) ∈ ∈ ( x, θ, e √− t ) e √− t · E ( x,θ ) by (1), where E ∈ C ∞ ( p Φ ∗ L ) is taken as in Proposition 7.1. Under theidentification we haveˆ g ( p Φ ∗ L, p Φ ∗ J, p Φ ∗ h, σ, p Φ ∗ ∇ ) = σ ( dt − x i dθ i ) + ( Q ij + P ik Q kl P lj ) dθ i dθ j − P ik Q jk dθ i dx j + Q ij dx i dx j . By Proposition 7.1, the deck transformation of p Φ : ( S ( p Φ ∗ L, p Φ ∗ h )) → S ( L | U , h )is identified with k · ( x, θ, e √− t ) := ( x, θ + k w , e √− t − kπm ) ) ( k ∈ Z /m Z ) , (9)where w ∈ Z n is taken such that Φ( w ) = 1 ∈ Z /m Z . Thus we obtain thenext proposition. Proposition 7.3.
Define the Riemannian metric ˆ g A on U Φ × S by ˆ g A = σ ( dt − x i dθ i ) + ( Q ij + P ik Q kl P lj ) dθ i dθ j − P ik Q jk dθ i dx j + Q ij dx i dx j , which is invariant under the Z /m Z action defined by (9) . If X is a strict m -BS fiber, then p Φ ∗ ˆ g ( L | U , J | U , h, σ, ∇ ) = ˆ g A holds. Boundedness of the Ricci curvatures.
First of all we compute theRicci curvature of g J | U . Since ω is the K¨ahler form on ( U, J ), it suffices tocompute the Ricci form of ω . First of all we can see that ∂θ i (cid:18) ∂∂θ j + A jk ∂∂x k (cid:19) = dθ i (cid:18) ∂∂θ j + A jk ∂∂x k (cid:19) = δ ij , hence ∂θ , . . . , ∂θ n forms the dual frame of Ω , . Proposition 7.4.
The K¨ahler form ω | U and the Ricci form ρ ω | U are givenby ω | U = 2 √− Q ij ∂θ i ∧ ∂θ j ,ρ ω | U = √− ∂∂ log det( Q ij ) − √− ∂α + √− ∂α, where α := ∂ ¯ A ij ∂x i ∂θ j ∈ Ω , ( U ) . Proof.
Since dx i − A ij dθ j is of type (0 , ∂x i = A ij ∂θ j . Thenwe have ω | U = dx i ∧ dθ i = ∂x i ∧ ∂θ i + ∂x i ∧ ∂θ i = 2 √− Q ij ∂θ i ∧ ∂θ j . Take f ∈ C ∞ ( U ′ , C × ) such that Ω := f ∂θ ∧ · · · ∧ ∂θ n is a nowherevanishing holomorphic section of the canonical bundle K X | U ′ on some openset U ′ ⊂ U . If we put β = f − ∂f , then the Ricci form ρ ω | U ′ is given by −√− ∂∂ log ω | nU ′ Ω ∧ Ω = −√− ∂∂ log det( Q ij ) + √− ∂∂ log | f | = −√− ∂∂ log det( Q ij ) + √− ∂β − √− ∂β. Since we have0 = f − ∂ Ω = β ∧ ∂θ ∧ · · · ∧ ∂θ n + ∂ ( ∂θ ∧ · · · ∧ ∂θ n ) , (10) HE GEOMETRIC QUANTIZATIONS 15 it suffices to compute ∂∂θ i to describe β . Now, we have ∂∂θ i (cid:18) ∂∂θ k + A kj ∂∂x j , ∂∂θ l + ¯ A lh ∂∂x h (cid:19) = d∂θ i (cid:18) ∂∂θ k + A kj ∂∂x j , ∂∂θ l + ¯ A lh ∂∂x h (cid:19) = − ∂θ i (cid:18)(cid:20) ∂∂θ k + A kj ∂∂x j , ∂∂θ l + ¯ A lh ∂∂x h (cid:21)(cid:19) = − (cid:18) ∂ ¯ A lh ∂θ k + A kj ∂ ¯ A lh ∂x j (cid:19) ∂θ i (cid:18) ∂∂x h (cid:19) + (cid:18) ∂A kj ∂θ l + ¯ A lh ∂A kj ∂x h (cid:19) ∂θ i (cid:18) ∂∂x j (cid:19) . Since ∂∂x h = Q hl √− (cid:18) ∂∂θ l + A lk ∂∂x k − ∂∂θ l − ¯ A lk ∂∂x k (cid:19) holds, we have ∂θ i (cid:16) ∂∂x h (cid:17) = Q hi √− , which gives ∂∂θ i = − Q hi √− (cid:18) ∂ ¯ A lh ∂θ k + A kj ∂ ¯ A lh ∂x j − ∂A kh ∂θ l − ¯ A lj ∂A kh ∂x j (cid:19) ∂θ k ∧ ∂θ l . Moreover, the integrability of J implies ∂ ¯ A lh ∂θ k + A kj ∂ ¯ A lh ∂x j = ∂ ¯ A lh ∂θ k + ¯ A kj ∂ ¯ A lh ∂x j + 2 √− Q kj ∂ ¯ A lh ∂x j = ∂ ¯ A kh ∂θ l + ¯ A lj ∂ ¯ A kh ∂x j + 2 √− Q kj ∂ ¯ A lh ∂x j , accordingly one can see that ∂∂θ i = Q hi (cid:18) ∂Q kh ∂θ l + ¯ A lj ∂Q kh ∂x j − Q kj ∂ ¯ A lh ∂x j (cid:19) ∂θ k ∧ ∂θ l . (11)By combining (10), we have β = (cid:18) Q ih ∂Q ih ∂θ l + ¯ A lj Q ih ∂Q ih ∂x j − ∂ ¯ A lj ∂x j (cid:19) ∂θ l . Now the Jacobi’s formula yields ∂ (log det( Q ij )) = Q ih ∂Q ih = Q ih (cid:18) ∂Q ih ∂θ l ∂θ l + ∂Q ih ∂x j ∂x j (cid:19) = Q ih (cid:18) ∂Q ih ∂θ l + ¯ A jl ∂Q ih ∂x j (cid:19) ∂θ l , therefore, we obtain β = ∂ (log det( Q ij )) − ∂ ¯ A lj ∂x j ∂θ l , which gives the assertion. (cid:3) Proposition 7.5.
Let α be as in Proposition . . Then we have ∂α = (cid:18) ∂ ¯ A il ∂θ k ∂x i + A kh ∂ ¯ A il ∂x h ∂x i (cid:19) ∂θ k ∧ ∂θ l − Q mh ∂ ¯ A im ∂x i (cid:18) ∂Q kh ∂θ l + ¯ A lj ∂Q kh ∂x j − Q kj ∂ ¯ A lh ∂x j (cid:19) ∂θ k ∧ ∂θ l . Proof.
Since ∂α = ∂ (cid:18) ∂ ¯ A il ∂x i ∂θ l (cid:19) = (cid:18) ∂ ¯ A il ∂θ k ∂x i + A kh ∂ ¯ A il ∂x h ∂x i (cid:19) ∂θ k ∧ ∂θ l + ∂ ¯ A il ∂x i ∂∂θ l , the assertion follows from (11). (cid:3) From now on we consider the one parameter family of ω -compatible com-plex structures { J s } P ( · ,
0) = P µ | U and dds P ( x, s ) (cid:12)(cid:12)(cid:12) s =0 ∈ T P µ ( x ) Lag( T x X ⊗ C , ω x ) + . By assuming ♠ , there are a constant K > A ∈ C ∞ ( U ) ⊗ M n ( C )such that sup i,j k A ij ( s, · ) − sA ij k C ( U ) ≤ Ks , Im( A ) is a positive definitesymmetric matrix and sup i,j k A ij k C ( U ) < ∞ .For a function f ( s, x, θ ) and f ( s, x, θ ) we write f ( s, x, θ ) = f ( s, x, θ ) + O C l ( s k )if there exists a constant K > k f ( s, x, θ ) − f ( s, x, θ ) k C l ( U ) ≤ Ks k . For instance, if { J s } s satisfies ♠ , then we may write A ij = sA ij + O C ( s ) . Proposition 7.6.
Assume that { J s } s satisfies ♠ . Put A ij = P ij + √− Q ij for P ij , Q ij ∈ C ∞ ( U, R ) . (i) ∂A ij ∂θ k = ∂A ik ∂θ j hold for any i, j, k . (ii) Let
Ric g Js be the Ricci curvature of g J s . There exists a constant κ ∈ R such that Ric g Js ≥ κg J s hold for all < s < δ , if and only if Q ij ( x, θ ) are independent of θ ∈ T n . HE GEOMETRIC QUANTIZATIONS 17
Proof.
We have ∂A ij ∂θ k = s ∂A ij ∂θ k + O C ( s ) and ∂A ij ∂x k = s ∂A ij ∂x k + O C ( s ), thenby (3) and taking s → κ such that ρ ω ≥ κω holds. To show it, we write ρ ω = √− ρ kl ∂θ k ∧ ∂θ l for ρ kl ∈ R ,then we expand ρ kl about s = 0.We have det Q ij = s n (cid:0) det Q ij + O C ( s ) (cid:1) , log det Q ij = log( s n ) + log det Q ij + O C ( s ) , where A ij = P ij + √− Q ij , and Q ij = s − Q ,ij + O C (1) , where ( Q ,ij ) i,j is the inverse of ( Q ij ) i,j . Since ∂∂θ i + A ij ∂∂x j forms the dualbasis of ∂θ i , we have ∂∂ log det Q ij = ∂ (log det Q ij ) ∂θ k ∂θ l + O C ( s ) ! ∂θ k ∧ ∂θ l ,∂α − ∂α = ( O C ( s )) ∂θ k ∧ ∂θ l . Set H = log det Q ij . We obtain ρ ω = √− (cid:18) ∂ H∂θ k ∂θ l + O C ( s ) (cid:19) ∂θ k ∧ ∂θ l . Put Q = ( Q ij ) ij , Q = ( Q ij ) ij and Hess θ H = ( ∂ H∂θ i ∂θ j ) ij , and let √ Q bethe symmetric matrix such that √ Q = Q . Since ω = 2 √− Q kl ∂θ k ∧ ∂θ l , then ρ ω ≥ κω holds for some κ ∈ R if and only if the eigenvalues of p Q − (Hess θ H + O C ( s )) p Q − are bounded from the below by a constant.Since p Q − = √ s − p ( Q ) − + O C ( s ) = √ s − (cid:16)p ( Q ) − + O C ( s ) (cid:17) , we obtain p Q − (Hess θ H + O C ( s )) p Q − = s − (cid:16)p ( Q ) − + O C ( s ) (cid:17) (Hess θ H + O C ( s )) (cid:16)p ( Q ) − + O C ( s ) (cid:17) = s − p ( Q ) − Hess θ H p ( Q ) − + O C (1) . Therefore, the existence of the lower bound of the Ricci curvatures of { g J s } is equivalent to p ( Q ) − Hess θ H p ( Q ) − ≥ , moreover, it is equivalent to Hess θ H ≥
0. Consequently, H should be con-stant by the maximum principle.By the imaginary part of (i), we can see that Q ij dθ j is a closed 1-formon { x } × T n , hence there exists a constant ¯ Q ij depends only on x such that[ ¯ Q ij dθ j ] = [ Q ij dθ j ] ∈ H ( { x } × T n ). Consequently, there are F i ( x, · ) ∈ C ∞ ( { x } × T n ) such that Q ij = ¯ Q ij + ∂F i ∂θ j holds. Integrating this equalityover { x } × T n , we have Z { x }× T n Q ij ( x, θ ) dθ · · · dθ n = ¯ Q ij ( x ) , which implies that ( ¯ Q ij ) i,j is a positive definite symmetric matrix. Since ∂F i ∂θ j = ∂F j ∂θ i holds, one can see that F i dθ i is a closed 1-form on { x } × T n , thenby repeating the above argument, there are F ( x, · ) ∈ C ∞ ( { x } × T n ) and¯ Q i ( x ) ∈ R such that F i = ¯ Q i + ∂F∂θ i , hence we may write Q ij = ¯ Q ij + ∂ F∂θ i ∂θ j . Since ¯ Q ij can be obtained by integrating Q ij along some cycles of H ( { x } × T n , Z ), ( ¯ Q ij ) i,j is also a positive definite symmetric matrix. Now we takeanother torus T n copy = R n / Z n and the coordinate τ , . . . , τ n coming from R n . Next we regard M x := { x } × T n × T n copy as a complex manifold whoseholomorphic coordinate is given by z := θ + √− τ , . . . , z n := θ n + √− τ n . Define the K¨ahler form ˆ ω x on M x by ˆ ω x := √− Q ij ( x ) dz i ∧ d ¯ z j . Since ¯ Q ij is constant on M x , it is a Ricci-flat K¨ahler metric. Moreoverˆ ω x + 4 √− ∂∂F = √− Q ij ( x, θ ) dz i ∧ d ¯ z j is also a Ricci-flat K¨ahler metric since det Q is constant. By the uniquenessof the Ricci-flat K¨ahler metric in the fixed K¨ahler class, we obtain Q ij =¯ Q ij . (cid:3) Convergence.
Set U Φ ,r := B r × ( R n / Ker Φ) = p − ( U r ) ,S r := S ( L | U r , h ) ,S Φ ,r := U Φ ,r × S = p − ( S r )for 0 < r ≤ R .For the brevity, put˜ d A := the Riemannian distance of ˆ g A on S Φ ,R , ˆ g J := ˆ g ( L, J, h, σ, ∇ ) ,d J := the Riemannian distance of ˆ g J on S ( L, h ) ,d A := the Riemannian distance of ˆ g J | S ( L | U ,h ) on S ( L | U , h ) , then d A ( p Φ ( u ) , p Φ ( v )) = inf k =0 , ,...,m − ˜ d A ( k · u, v ) ,d J ( p Φ ( u ) , p Φ ( v )) ≤ d A ( p Φ ( u ) , p Φ ( v ))hold for all u, v ∈ S Φ ,R . HE GEOMETRIC QUANTIZATIONS 19
Denote by B g J ( p, r ) the geodesic ball in ( X, g J ) of radius r centered at p ,and denote by B g A ( p, r ) the geodesic ball in ( U, g A ). Put := (0 , ∈ U, and B d A ( r ) := { p ∈ S ( L | U , h ); d A ( p Φ ( u ) , p ) < r } ,B d J ( r ) := { u ∈ X ; d J ( p Φ ( u ) , u ) < r } . The the connection metric ˆ g A given in Proposition 7.3 is written asˆ g A = σ ( dt − x i dθ i ) + Θ ij dθ i dθ j − P ik Q jk dθ i dx j + Q ij dx i dx j . Proposition 7.7. (i) B g J (cid:16) , p inf(Θ − ) R ′ (cid:17) ⊂ U R ′ holds for any < R ′ ≤ R . (ii) Take R > such that p sup(Θ − ) p inf(Θ − ) ! R + √ n sup Θ + 2 √ σπ p inf(Θ − ) ≤ R. Then d J ( p, p ′ ) = d A ( p, p ′ ) holds for any p, p ′ ∈ S R . (iii) Assume that { J s } s satisfies ♠ . Then there are constants s > , < R < R and C > such that B g Js (cid:18) , CR ′ √ s (cid:19) ⊂ U R ′ , d J s | S R = d A ( s, · ) | S R hold for any < s ≤ s and < R ′ ≤ R .Proof. (i) Let p ∈ B g J (cid:16) , p inf(Θ − ) R ′ (cid:17) and suppose p / ∈ U R ′ . Then thereis a piecewise smooth path c : [0 , → X such that c (0) = , c (1) = p and the length L ( c ) is less than p inf(Θ − ) R ′ . Let τ := inf { τ ∈ [0 , c ( τ ) / ∈ U R ′ } ≤ . Then by the first inequality of Proposition 7.2, L ( c ) ≥ L ( c | [0 ,τ ] ) ≥ d g J ( , c ( τ )) ≥ p inf(Θ − ) R ′ holds, hence we have the contradiction.(ii) Take R > p, p ′ ∈ S R and sup-pose d J ( p, p ′ ) < d A ( p, p ′ ). Then there is a piecewise smooth path c : [0 , → S ( L, h ) connecting p and p ′ such that Im( c ) is not contained in S R and L ( c ) is less than d A ( p, p ′ ). Put τ := inf { τ ∈ [0 , c ( τ ) / ∈ S R } , then L ( c ) ≥ L ( c | [0 ,τ ] ) ≥ d J ( c (0) , c ( τ ))holds. Since π : ( S ( L, h ) , ˆ g J ) → ( X, g J ) is a Riemannian submersion, d J ( c (0) , c ( τ )) ≥ d g J ( π ( c (0)) , π ( c ( τ ))) holds, then we can see L ( c ) ≥ d g J ( π ( c (0)) , π ( c ( τ ))) ≥ p inf(Θ − ) (cid:18) R − R (cid:19) , by the first inequality of Proposition 7.2. The second inequality of Proposi-tion 7.2 gives p inf(Θ − ) (cid:18) R − R (cid:19) < d A ( p, p ′ ) ≤ p sup(Θ − ) R + √ n sup Θ2 + √ σπ, therefore we obtain R < p sup(Θ − ) p inf(Θ − ) ! R + 1 p inf(Θ − ) (cid:18) √ n sup Θ2 + √ σπ (cid:19) , which contradicts the assumption.(iii) Since we have p inf(Θ − ) = 1 √ s (cid:16)p inf((Θ ) − ) + O ( s ) (cid:17) , p sup(Θ − ) = 1 √ s (cid:16)p sup((Θ ) − ) + O ( s ) (cid:17) , p sup(Θ) = √ s (cid:16)p sup(Θ ) + O ( s ) (cid:17) by the Hoffman-Wielandt’s inequality [12], there exists s > p sup(Θ − ) p inf(Θ − ) ≤ p sup((Θ ) − ) p inf((Θ ) − ) , √ n sup Θ + 2 √ σπ p inf(Θ − ) ≤ R s ≤ s . If we take 0 < R < R such that2 p sup((Θ ) − ) p inf((Θ ) − ) ! R ≤ R , then the assumption of (ii) is satisfied for s ≤ s , hence we have d J s | S R = d A ( s, · ) | S R . Moreover, if we put C := inf , then we can see p inf(Θ − ) R ′ ≥ CR ′ √ s for R ′ ≤ R , hence we have B g Js (cid:16) , CR ′ √ s (cid:17) ⊂ U R ′ by (i). (cid:3) Next we consider ω -compatible complex structures J, J ′ , and compare theRiemannian distances of g J and g J ′ . We will show that if g J and g J ′ areclose to each other in some sense then their Riemannian distances are alsoclose to each other. HE GEOMETRIC QUANTIZATIONS 21
Now, we define the distance d Sym + ( R N ) onSym + ( R N ) := { g ∈ M N ( R ); g ij = g ji , g > } as follows. For g ∈ Sym + ( R N ), take v , . . . , v N ∈ R N such that g ( v i , v j ) = δ ij . For g ′ ∈ Sym + ( R N ) let λ , . . . , λ N ∈ R be eigenvalues of ( g ′ ( v i , v j )) i,j .Then define d Sym + ( R N ) ( g, g ′ ) := max i | log λ i | . Moreover, if g, g ′ are Riemannian metrics on M , then define d Sym + ( M ) ( g, g ′ ) := sup x ∈ M d Sym + ( T x M ) ( g x , g ′ x ) . Lemma 7.8.
Let M be a smooth manifold of dimension N , g, g ′ be Rie-mannian metrics on M and d, d ′ be the Riemannian distances of g, g ′ , re-spectively. If we assume d Sym + ( M ) ( g, g ′ ) ≤ , then | d ( p , p ) − d ′ ( p , p ) | ≤ d Sym + ( M ) ( g, g ′ ) d ′ ( p , p ) holds. Moreover, for any f ∈ C ( M ) (cid:12)(cid:12)(cid:12)(cid:12)Z M f dµ g − Z M f dµ g ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ N sup | f | · µ g ′ (supp( f )) · d Sym + ( M ) ( g, g ′ ) holds if d Sym + ( M ) ( g, g ′ ) ≤ log 2 N .Proof. Let c : [ a, b ] → M be a piecewise smooth path, and denote by L g ( c )be the length of c with respect to g . Since we have g ( c ′ ( t ) , c ′ ( t )) ≤ exp (cid:16) d Sym + ( T c ( t ) M ) ( g c ( t ) , g ′ c ( t ) ) (cid:17) g ′ ( c ′ ( t ) , c ′ ( t )) ≤ exp (cid:16) d Sym + ( M ) ( g, g ′ ) (cid:17) g ′ ( c ′ ( t ) , c ′ ( t ))then we can see L g ( c ) ≤ exp d Sym + ( M ) ( g, g ′ )2 ! L g ′ ( c )and d ( p , p ) ≤ exp d Sym + ( M ) ( g, g ′ )2 ! d ′ ( p , p ) . By the symmetry we also haveexp − d Sym + ( M ) ( g, g ′ )2 ! d ′ ( p , p ) ≤ d ( p , p ) . Therefore, we obtain d ′ ( p , p ) − d ( p , p ) ≤ − exp − d Sym + ( M ) ( g, g ′ )2 !! d ′ ( p , p )and d ( p , p ) − d ′ ( p , p ) ≤ exp d Sym + ( M ) ( g, g ′ )2 ! − ! d ′ ( p , p ) . Since 1 − e − t ≤ t and e t − ≤ t holds for any 0 ≤ t ≤ f ∈ C ( M ) and denote by dµ g the Riemannian measure of g . Then we have (cid:12)(cid:12)(cid:12)(cid:12)Z M f dµ g − Z M f dµ g ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z M | f | (cid:12)(cid:12)(cid:12)(cid:12) det g det g ′ − (cid:12)(cid:12)(cid:12)(cid:12) dµ g ′ . Since | log det g det g ′ | ≤ N d
Sym + ( M ) ( g, g ′ ) holds and | e t − | ≤ | t | holds for | t | ≤ log 2, we can see (cid:12)(cid:12)(cid:12)(cid:12)Z M f dµ g − Z M f dµ g ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ N sup | f | · µ g ′ (supp( f )) · d Sym + ( M ) ( g, g ′ )if d Sym + ( M ) ( g, g ′ ) ≤ log 2 N . (cid:3) Lemma 7.9.
Let g, g ′ ∈ Sym + ( R N ) and { v , · · · , v N } be a basis of R N .Put g = ( g ( v i , v j )) i,j and g ′ = ( g ′ ( v i , v j )) i,j . Denote by α , . . . , α N be theeigenvalues of g ′ g − . Then α i ∈ R and d Sym + ( R N ) ( g, g ′ ) = max i | log α i | .Proof. Let √ g be the square root of g . If we put e i = P j √ g − ij v j , then e , · · · , e N is an orthonormal basis of ( R N , g ), therefore we have d Sym + ( R N ) ( g, g ′ ) = max i | log λ i | , where λ i are the eigenvalues of( g ′ ( e i , e j )) ij = √ g − g ′ √ g − . Since we have √ g − · (cid:0) g ′ g − (cid:1) · √ g = √ g − g ′ √ g − , { α , . . . , α N } = { λ , . . . , λ N } holds. (cid:3) Suppose that X is a strict m -BS fiber and fix a small s > dt − x i dθ i , √ sdθ , . . . , √ sdθ n , √ s dx , . . . , √ s dx n of T ∗ ( U Φ × S ). Then the matrix representation of ˆ g A is given by g A := σ s − Θ − P Q − − Q − P sQ − , and its inverse is g − A = σ − sQ − Q − P P Q − s − Θ . Suppose that { A ( s, · ) } s corresponds to { J s } which satisfies ♠ . Fix r ≥ K > { A ( s, · ) } s such that | A ( s, x, θ ) − sA ( x, θ ) | ≤ Ks | A ( x, θ ) − A (0 , θ ) | ≤ K k x k HE GEOMETRIC QUANTIZATIONS 23 for any ( x, θ ) ∈ U Φ . If ( x, θ ) ∈ U Φ , √ sr for r ≥ s > √ sr ≤ R ,then we have s ≤ √ s Rr ≤ √ sr since R ≤
1, hence we obtain (cid:12)(cid:12) s − A ( s, x, θ ) − A (0 , θ ) (cid:12)(cid:12) ≤ K √ sr. Here we write f ( s, x, θ ) ∼ = √ sr f ( s, x, θ )if there is a constant K > | f ( s, x, θ ) − f ( s, x, θ ) | ≤ K √ sr holdsfor any ( x, θ ) ∈ U Φ , √ sr .Now A ′ ( s, x, θ ) := sA (0 , θ ) gives another family of complex structures { J ′ s } s which satisfies ♠ , by (i) of Proposition 7.6. Since we have s − Θ ∼ = √ sr Θ (0 , θ ) ,P Q − ∼ = √ sr P (0 , θ ) Q (0 , θ ) − ,Q − P ∼ = √ sr Q (0 , θ ) − P (0 , θ ) ,sQ − ∼ = √ sr Q (0 , θ ) − , where Θ (0 , θ ) = Q (0 , θ ) + P (0 , θ ) Q (0 , θ ) − P (0 , θ ), then we obtain g − A ′ g A ∼ = √ sr I n +1 . By Lemma 7.9, the eigenvalues of g − A ′ g A are real. If 1 + λ ∈ R is one of theeigenvalues, then f ( λ ) := det (cid:0) (1 + λ ) I n +1 − g − A ′ g A (cid:1) = 0holds. Since we have f ( λ ) = det (cid:8) λI n +1 + ( I n +1 − g − A ′ g A ) (cid:9) , there exists a constant K > { A ( s, · ) } s , and there exist c , c , . . . , c n ∈ R such that max i | c i | ≤ K and f ( λ ) = λ n +1 + n X i =0 c i ( √ sr ) n +1 − i λ i . Lemma 7.10.
For any n ∈ Z ≥ , K > and r ≥ there is a sufficientlylarge N > depending only on n and K such that for any c , c , . . . , c n ∈ [ − K, K ] and ε > , the solution λ of the equation f ( λ ) = λ n +1 + n X i =0 c i ε n +1 − i λ i = 0 always satisfies | λ | ≤ N ε .Proof.
Put λ = εt . Then f ( λ ) = ε n +1 (cid:16) t n +1 + P ni =0 c i t i (cid:17) . If f ( λ ) = 0then we have t n = − P ni =0 c i t i . Suppose | t | ≥
1. Then | t | n +1 ≤ n X i =0 | c i || t | i ≤ n X i =0 K | t | n = (2 n + 1) K | t | n holds, hence | t | ≤ (2 n + 1) K is obtained. Consequently we can see | λ | ≤ max { , (2 n + 1) K } ε . (cid:3) By Lemma 7.10 we can see | log(1 + λ ) | ≤ N √ sr for the eigenvalue 1 + λ of g − A ′ g A , where N is the constant depending only on K . Therefore, weobtain the following proposition by Lemma 7.9. Proposition 7.11.
Let
A, A ′ be as above and let r ≥ , s > with √ sr ≤ R .Then there exists a constant C > depending only on A such that d Sym + ( U Φ , √ sr × S ) (ˆ g A , ˆ g A ′ ) ≤ C √ sr. From now on we assume
R > CR ≤ , where C is the constant in Proposition 7.11. Then Lemma 7.8 holds for M = S Φ ,R , g = ˆ g A , g ′ = ˆ g A ′ . and for M = S Φ ,R , g = ˆ g A ′ , g ′ = ˆ g A . Proposition 7.12.
Let { J s } s satisfy ♠ and A ′ ( s, x, θ ) := sA (0 , θ ) . (i) There are positive constants C ′ , C ′ depending only on A (0 , · ) and σ such that B g A ′ ( , C ′ r ) ⊂ U √ sr ⊂ B g A ′ ( , C ′ r ) for any r ≥ and s > with √ sr ≤ R . (ii) Suppose X is a strict m -BS fiber. Then there are constants C > and < R < R depending only on A and σ such that | d J s ( p, q ) − d A ′ ( p, q ) | < C √ sr holds for any r ≥ , s > with √ sr ≤ R and p, q ∈ S √ sr . (iii) There are positive constants C , C and < R < R depending onlyon A and σ such that B g Js ( , C r ) ⊂ U √ sr ⊂ B g Js ( , C r ) for any r ≥ , s > with √ sr ≤ R .Proof. (i) Apply Proposition 7.2 for A ′ . Then there are positive constants C , C , C depending only on A (0 , · ) and σ such that C √ s − k x k ≤ d g A ′ ( , u ) ≤ C √ s − k x k + C for any u = ( x, θ ) and s >
0. If k x k < √ sr then d g A ′ ( , u ) < C r + C ≤ ( C + C ) r holds since r ≥
1, which implies U √ sr ⊂ B g A ′ ( , ( C + C ) r ). On the otherhand if d g A ′ ( , u ) < C r holds then C √ s − k x k ≤ d g A ′ ( , u ) < C r gives k x k < √ sr , hence B g A ′ ( , C r ) ⊂ U √ sr holds.(ii) By applying Proposition 7.11, there is a constant C > d Sym + ( S Φ , √ sr ) (ˆ g A , ˆ g A ′ ) ≤ C √ sr HE GEOMETRIC QUANTIZATIONS 25 holds if √ sr ≤ R . Now take R ≤ min { C , R } and assume √ sr ≤ R ,then we may apply Lemma 7.8 and we have | ˜ d A ( u, v ) − ˜ d A ′ ( u, v ) | ≤ d Sym + ( M ) ( g, g ′ ) ˜ d A ′ ( u, v ) ≤ C √ sr ˜ d A ′ ( u, v )for all u, v ∈ S Φ , √ sr . By the same argument in the proof of Proposition 7.2,we have the upper estimate˜ d A ′ ( u, v ) ≤ p sup( Q ) − k x − x ′ k√ s + √ s p σr + · sup Θ · diam( R n / KerΦ)+ √ σπ, where u = ( x, θ, e √− t ) , v = ( x ′ , θ ′ , e √− t ′ ). Since k x − x ′ k ≤ √ sr , √ sr ≤ R ≤ r ≥
1, we have k x − x ′ k√ s ≤ r and s ≤ R r ≤
1, then there isa constant C > A , σ, Φ such that ˜ d A ′ ( u, v ) ≤ C r ,which gives | ˜ d A ( u, v ) − ˜ d A ′ ( u, v ) | < C C √ sr . Therefore, we can see d A ( p Φ ( u ) , p Φ ( v )) = inf k =0 , ,...,m − ˜ d A ( k · u, v ) < inf k =0 , ,...,m − n ˜ d A ′ ( k · u, v ) + 2 C C √ sr o = d A ′ ( k · u, v ) + 2 C C √ sr and similarly d A ′ ( p Φ ( u ) , p Φ ( v )) < d A ( k · u, v ) + 2 C C √ sr is obtained. By(iii) of Proposition 7.7, we can take 0 < R ′ < R and s > d J s | S R ′ = d A | S R ′ holds for any 0 < s ≤ s . If we put C = 2 C C and R = min { C , R ′ , √ s } , then √ sr ≤ R implies s ≤ s , hence we have(ii).(iii) Take C, s , R as in (iii) of Proposition 7.7 and replace R by thesmaller one such that R ≤ √ s . Then we have B g Js ( , Cr ) ⊂ U √ sr if √ sr ≤ R . Next we assume u ∈ U √ sr . By (i), we have u ∈ B g A ′ ( , C ′ r ).Since π : ( S R , ˆ g J s ) → ( U R , g J s ) and π : ( S R , ˆ g A ′ ) → ( U R , g A ′ ) are Riemanniansubmersions, therefore (ii) gives d g Js ( π ( u ) , π ( u ′ )) = inf e √− t ∈ S d J s ( ue √− t , u ′ ) ≤ inf e √− t ∈ S d A ′ ( ue √− t , u ′ ) + C √ sr = d g A ′ ( π ( u ) , π ( u ′ )) + C √ sr . Consequently we obtain d g Js ( , u ) ≤ d g A ′ ( , u ) + C √ sr < C ′ r + CR r, which implies U √ sr ⊂ B g Js ( , C r ) by putting C = C ′ + C . (cid:3) Proposition 7.13.
Let { J s } s satisfy ♠ and A ′ ( s, x, θ ) := sA (0 , θ ) . Thereexist constants R , C > such that id : (cid:0) π − ( B g Js ( , r − C √ sr )) , d J s (cid:1) → (cid:16) π − ( B g A ′ ( s, · ) ( , r )) , d A ′ ( s, · ) (cid:17) is a Borel C √ sr - S -equivariant Hausdorff approximation for any r ≥ and s ≤ R Cr . Moreover, if f : S R → R is a Borel function such that supp( f ) ⊂ S R ′ for some R ′ ≤ R and sup | f | < ∞ , then (cid:12)(cid:12)(cid:12)(cid:12)Z S R f dµ ˆ g A ′ − Z S R f dµ ˆ g Js (cid:12)(cid:12)(cid:12)(cid:12) ≤ C sup | f | ( R ′ ) n +1 holds.Proof. Fix r ≥
1. Take R , C , C , C ′ , C ′ , C such that Proposition 7.12holds. We may suppose C > C = C ′ = C − , C = C ′ = C . Then B g Js ( , r ) ⊂ U C √ sr and | d J s ( p, q ) − d A ′ ( p, q ) | < C √ sr hold for any p, q ∈ S C √ sr and 0 < s ≤ R C r . If u ∈ B g Js ( , r − C √ sr ),then d g A ′ ( , u ) < d g Js ( , u ) + C √ sr < r, which implies B g Js ( , r − C √ sr ) ⊂ B g A ′ ( , r ).Now, B g A ′ ( , r ) ⊂ U C √ sr holds. We also have B g A ′ ( , r ) ⊂ B g Js ( , r + C √ sr ) . Since d g Js is an intrinsic metric, we have B g Js ( , r + C √ sr ) = B g Js (cid:0) B g Js ( , r − C √ sr ) , C √ sr (cid:1) . hence we can see thatid : (cid:0) π − ( B g Js ( , r − C √ sr )) , d J s (cid:1) → (cid:0) π − ( B g A ′ ( , r )) , d A ′ ( s, · ) (cid:1) is a Borel ε i - S -equivariant Hausdorff approximation.Let f : S R → R be a Borel function such that supp( f ) ⊂ S R ′ for some R ′ ≤ R and sup | f | < ∞ . Then one can see (cid:12)(cid:12)(cid:12)(cid:12)Z S R f dµ ˆ g A ′ − Z S R f dµ ˆ g Js (cid:12)(cid:12)(cid:12)(cid:12) ≤ n sup | f | · µ ˆ g A ′ ( S R ′ ) · CR ′ by Lemma 7.8 and Proposition 7.11. Since dµ g A ′ = det( g A ′ ) dtdθ · · · dθ n dx · · · dx n and det( g A ′ ) = σ det (cid:18) Θ − P ( Q ) − − ( Q ) − P ( Q ) − (cid:19) , one can see that µ g A ′ ( S R ′ ) = σC ′ ( R ′ ) n , which gives the assertion. (cid:3) Let { J s } s satisfy ♠ and A ′ ( s, x, θ ) := sA (0 , θ ). By Proposition 7.6, P ij (0 , θ ) dθ j is a closed 1-form on T n . Then there are constants ¯ P ij ∈ R suchthat [ P ij (0 , · ) dθ j ] = [ ¯ P ij dθ j ] ∈ H ( T n , R ) , hence there are H i ∈ C ∞ ( T n ) such that P ij (0 , · ) dθ j = ¯ P ij dθ j + d H i . HE GEOMETRIC QUANTIZATIONS 27
Since P ij = P ji and ¯ P ij = Z T n P ij (0 , θ ) dθ · · · dθ n , we have ¯ P ij = ¯ P ji and ∂ H i ∂θ j = ∂ H j ∂θ i . Consequently, H i dθ i is closed, thereforethere are ¯ P i ∈ R and H ∈ C ∞ ( T n ) such that H i = ¯ P i + ∂ H ∂θ i , which gives P ij (0 , · ) = ¯ P ij + ∂ H ∂θ i ∂θ j . If Ric g Js has the lower bound, then by Proposition 7.6 we have Q ij (0 , · ) =¯ Q ij ∈ R and g A ′ = s ( ¯ Q ij + P ik ¯ Q kl P lj ) dθ i dθ j − P ik ¯ Q jk dθ i dx j + s − ¯ Q ij dx i dx j = s ¯ Q ij dθ i dθ j + ¯ Q ij s (cid:16) dx i − sP ik dθ k (cid:17) (cid:16) dx j − sP jl dθ l (cid:17) = s ¯ Q ij dθ i dθ j + ¯ Q ij s (cid:26) d (cid:18) x i − s ∂ H ∂θ i (cid:19) − s ¯ P ik dθ k (cid:27) (cid:26) d (cid:18) x j − s ∂ H ∂θ j (cid:19) − s ¯ P jl dθ l (cid:27) . Now, define F s : R n × T n → R n × T n by F s ( x, θ ) := (cid:18) x + s ∂ H ∂θ , . . . , x n + s ∂ H ∂θ n , θ (cid:19) . Then F − s is the inverse of F s and F ∗ s g A ′ = s ¯ Q ij dθ i dθ j + ¯ Q ij s (cid:16) dx i − s ¯ P ik dθ k (cid:17) (cid:16) dx j − s ¯ P jl dθ l (cid:17) holds. Moreover, we can lift F s toˆ F s : R n × ( R n / KerΦ) × S → R n × ( R n / KerΦ) × S by ˆ F s ( x, θ, e √− t ) := (cid:18) x + s ∂ H ∂θ , . . . , x n + s ∂ H ∂θ n , θ, e √− t + s H ( θ )) (cid:19) . One can easy to check that ˆ F s is Z /m Z -equivariant and S -equivariant map,and ˆ F ∗ s ˆ g A ′ = σ ( dt − x i dθ i ) + s ¯ Q ij dθ i dθ j + ¯ Q ij s (cid:16) dx i − s ¯ P ik dθ k (cid:17) (cid:16) dx j − s ¯ P jl dθ l (cid:17) . Put ¯ P = ( ¯ P ij ) i,j , ¯ Q = ( ¯ Q ij ) i,j , ¯Θ = ¯ Q + ¯ P ¯ Q − ¯ P and y = √ s ¯Θ − x , τ = √ s ¯Θ θ . Then we may writeˆ F ∗ s ˆ g A ′ = σ ( dt − t x · dθ ) + t dθ · s ¯ Q · dθ + t (cid:0) dx − s ¯ P dθ (cid:1) · ¯ Q − s · (cid:0) dx − s ¯ P dθ (cid:1) = σ ( dt ) − σ ( t y · dτ ) dt + t dτ · (cid:0) σy · t y (cid:1) · dτ + t dy · p ¯Θ ¯ Q − p ¯Θ · dy − · t dy · p ¯Θ ¯ Q − ¯ P p ¯Θ − · dτ. Since K y := 1 + σy · t y is positive definite, it has the inverse and the squareroot. Accordingly, we haveˆ F ∗ s ˆ g A ′ = σ ( dt ) − σ ( t y · dτ ) dt + t dτ · K y · dτ + t dy · p ¯Θ ¯ Q − p ¯Θ · dy − · t dy · p ¯Θ ¯ Q − ¯ P p ¯Θ − · dτ = t (cid:18)p K y dτ − σ p K y − ydt − p K y − p ¯Θ − ¯ P ¯ Q − p ¯Θ dy (cid:19) · (cid:18)p K y dτ − σ p K y − ydt − p K y − p ¯Θ − ¯ P ¯ Q − p ¯Θ dy (cid:19) + (cid:0) σ − ( σ ) t yK − y y (cid:1) ( dt ) − σ t y · K − y p ¯Θ − ¯ P ¯ Q − p ¯Θ dydt + t dy · p ¯Θ (cid:18) ¯ Q − − ¯ Q − ¯ P p ¯Θ − K − y p ¯Θ − ¯ P ¯ Q − (cid:19) p ¯Θ · dy. Here, we have t y · K − y = t y σ k y k , t y · K − y · y = k y k σ k y k , and the similar computation as in the proof of Proposition 7.2 gives¯Θ − = ¯ Q − − ¯ Q − ¯ P ¯Θ − ¯ P ¯ Q − . Put T := dτ − σK − y · ydt − K − y p ¯Θ − ¯ P ¯ Q − p ¯Θ dy, ¯ S := p ¯Θ − ¯ P ¯ Q − p ¯Θ . Then we may writeˆ F ∗ s ˆ g A ′ = t T · K y · T + σ σ k y k ( dt ) − σ σ k y k t y · ¯ Sdydt + t dy · (cid:0) t ¯ S (cid:0) − K − y (cid:1) ¯ S (cid:1) · dy. Since we have1 − K − y = ( K y − K − y = σy · t y · K − y = σy · t y σ k y k , we can see thatˆ F ∗ s ˆ g A ′ = t T · K y · T + σ σ k y k ( dt ) − σ σ k y k t y · ¯ Sdydt + t dy · (cid:18) t ¯ S (cid:18) σy · t y σ k y k (cid:19) ¯ S (cid:19) · dy = t T · K y · T + σ σ k y k (cid:0) dt − t y · ¯ Sdy (cid:1) + t dy · dy Define φ m,s : S Φ ,R → R n × S by φ m,s ( x, θ, e √− t ) := ( p s ¯Θ − x, e √− t ) . HE GEOMETRIC QUANTIZATIONS 29 and define Z /m Z -action on R n × S by k · ( y, e √− t ) := ( y, e √− t − kπm ) ).Then φ m,s is Z /m Z -equivariant map and φ m,s : ( R n × ( R n / Ker Φ) × S , ˆ F ∗ s ˆ g A ′ ) → ( R n × S , g ∞ )is a Riemannian submersion, where g ∞ = σ σ k y k (cid:0) dt − t y · ¯ Sdy (cid:1) + t dy · dy. Denote by µ ∞ the measure on R n × S defined by dµ ∞ = dy · · · dy n dt . Proposition 7.14.
Let f ∈ C ( R n × S ) . Then there is a constant K > depending only on Φ , σ, ¯Θ such that Z R n × ( R n / Ker Φ) × S f ◦ φ m,s dµ ˆ F ∗ s ˆ g A ′ = K √ s n Z R n × S f dµ ∞ . Proof.
Since dµ ˆ F ∗ s ˆ g A ′ = (cid:18) σ σ k y k det( K y ) (cid:19) dtdτ · · · dτ n dy · · · dy n = (cid:18) σ σ k y k (1 + σ k y k ) (cid:19) dtdτ · · · dτ n dy · · · dy n = √ σdtdτ · · · dτ n dy · · · dy n , we have Z R n × ( R n / Ker Φ) × S f ◦ φ m,s dµ ˆ F ∗ s ˆ g A ′ = Z R n × ( R n / Ker Φ) × S f ◦ φ m,s √ σdtdτ · · · dτ n dy · · · dy n = √ σ √ s n p det ¯Θ Z R n × ( R n / Ker Φ) × S f ◦ φ m,s dtdθ · · · dθ n dy · · · dy n = √ σ Vol ( R n / Ker Φ) p det ¯Θ √ s n Z R n × S f dtdy · · · dy n . (cid:3) Now, we put S Φ := R n × ( R n / Ker Φ) × S Z /m Z , then ˆ g A ′ and ˆ F ∗ s ˆ g A ′ induces the Riemannian metrics on S Φ such that p Φ islocal isometry. We also denote by ˆ g A ′ and ˆ F ∗ s ˆ g A ′ , respectively if there is nofear of confusion.Since φ m,s is Z /m Z -equivariant, we have the following commutative dia-gram; ( R n × ( R n / Ker Φ) × S , ˆ F ∗ s ˆ g A ′ ) φ m,s → ( R n × S , g ∞ ) p Φ ↓ p m ↓ ( S Φ , ˆ F ∗ s ˆ g A ′ ) φ s → ( R n × S , g m, ∞ ) where p m is the quotient map defined by p m ( y, e √− t ) := ( y, e √− mt ) and g m, ∞ is defined by g m, ∞ = σ σ k y k (cid:18) dtm − t y · ¯ Sdy (cid:19) + t dy · dy (12)such that p m ∗ g m, ∞ = g ∞ and φ s is the Riemannian submersion. Proposition 7.15.
Let { J s } s satisfy ♠ and A ′ ( s, x, θ ) := sA (0 , θ ) and put p = p Φ (0 , , ∈ S Φ . Assume that there are constants s > and κ ∈ R such that Ric g Js ≥ κg J s for any < s ≤ s . Then the family of pointedmetric measure spaces with the isometric S -action (cid:26)(cid:18) S Φ , d A ′ , µ ˆ g A ′ K √ s n , p (cid:19)(cid:27) s converges to (cid:0) R n × S , d g m, ∞ , µ ∞ , (0 , (cid:1) as s → in the sense of the pointed S -equivariant measured Gromov-Hausdorff topology.Proof. Since ˆ F s is an S -equivariant isometry, it suffices to show that (cid:26)(cid:18) S Φ , d ˆ F ∗ s ˆ g A ′ , µ ˆ F ∗ s ˆ g A ′ K √ s n , p (cid:19)(cid:27) s converges to (cid:0) R n × S , d g m, ∞ , µ ∞ , (0 , (cid:1) as s → S -equivariant measured Gromov-Hausdorff topology. Sinceˆ F ∗ s ˆ g A ′ = t T K y T + g ∞ , one can see that φ s is a Riemannian submersion and the diameters of thefibers φ − s ( y, t ) are at most C p s (1 + σ k y k ), where C >
P , ¯ Q and Φ, hence the pointed Gromov-Hausdorff conver-gence follows. Moreover, Proposition 7.14 implies that ( φ m,s ) ∗ µ ˆ F ∗ s hatg A ′ = K √ s n µ ∞ , especially we also have the vague convergence of the measures. (cid:3) Theorem 7.16.
Let { J s } s satisfies ♠ and suppose that there there are con-stants s > and κ ∈ R such that Ric g Js ≥ κg J s for any < s ≤ s . Put p = p Φ (0 , , ∈ S ( L | U , h ) . Then the family of pointed metric measurespaces with the isometric S -action (cid:26)(cid:18) S ( L, h ) , d J s , µ ˆ g Js K √ s n , p (cid:19)(cid:27) s converges to (cid:0) R n × S , d g m, ∞ , µ ∞ , (0 , (cid:1) as s → in the sense of the pointed S -equivariant measured Gromov-Hausdorff topology.Proof. Put A ′ ( s, x, θ ) := sA (0 , θ ). By Proposition 7.13, there exist con-stants R , C > (cid:0) π − ( B g Js ( , r − C √ sr )) , d J s (cid:1) → (cid:16) π − ( B g A ′ ( s, · ) ( , r )) , d A ′ ( s, · ) (cid:17) is a Borel C √ sr - S -equivariant Hausdorff approximation for any r ≥ s ≤ R Cr . Since C √ sr → s → r , therefore, { ( S ( L, h ) , d J s , p ) } s S -GH −→ (cid:0) R n × S , d g m, ∞ , (0 , (cid:1) HE GEOMETRIC QUANTIZATIONS 31 as s → S ( L, h ) , d J s , p ) to (cid:0) R n × S , d g m, ∞ , (0 , (cid:1) is induced by the Z /m Z -equivariant maps ψ s := φ m,s ◦ ˆ F − s . Take f ∈ C ( R n × S ). ThenProposition 7.14 gives Z R n × S f dµ ∞ = 1 K √ s n Z R n × ( R n / Ker Φ) × S f ◦ φ m,s dµ ˆ F ∗ s ˆ g A ′ = 1 K √ s n Z R n × ( R n / Ker Φ) × S f ◦ ψ s dµ ˆ g A ′ . Note that sup | f ◦ ψ s | ≤ sup | f | < ∞ . By the definition of φ m,s , there is r > s such that supp( f ◦ ψ s ) ⊂ S √ sr holds for any 0 < s ≤ s .Then Proposition 7.13 gives some constants C > K √ s n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R n × ( R n / Ker Φ) × S f ◦ ψ s dµ ˆ g Js − Z R n × ( R n / Ker Φ) × S f ◦ ψ s dµ ˆ g A ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C sup | f | ( √ sr ) n +1 K √ s n → s → (cid:3) The spectral structures on the limit spaces
In this section we consider the metric measure space ( R n × S , g m, ∞ , µ ∞ )defined by (12). Now, note that¯ S = p ¯Θ − ¯ P ¯ Q − p ¯Θ = p ¯Θ − (cid:0) ¯ P ¯ Q − ¯Θ (cid:1) p ¯Θ − = p ¯Θ − (cid:0) ¯ P + ¯ P ¯ Q − ¯ P ¯ Q − ¯ P (cid:1) p ¯Θ − , which implies ¯ S is symmetric. Consequently, we can see dtm − t y · ¯ S · dy = d (cid:18) tm − t y · ¯ S · y (cid:19) . Here, by taking the pullback of g m, ∞ by the diffeomorphism R n × S → R n × S ∈ ∈ (cid:16) y, e √− t (cid:17) (cid:18) y, e √− (cid:16) t + m · ty · ¯ S · y (cid:17) (cid:19) , we may suppose g m, ∞ = σm (1 + σ k y k ) ( dt ) + t dy · dydµ ∞ = dy · · · dy n dt and the isometric S -action on ( R n × S , g m, ∞ ) is given by e √− τ · (cid:16) y, e √− t (cid:17) = · (cid:16) y, e √− t + mτ ) (cid:17) . Then the Laplace operator ∆ m, ∞ on ( R n × S , g m, ∞ , µ ∞ ) is defined suchthat Z R n × S (∆ m, ∞ f ) f dµ ∞ = Z R n × S h df , df i g m, ∞ dµ ∞ holds for any f , f ∈ C ∞ ( R n × S ), therefore we have∆ m, ∞ f = ∆ R n f − m (1 + σ k y k ) σ ∂ f∂t , where ∆ R n = − P ni =1 ∂ ∂y i .Let ρ k be the representation of S defined in Section 3, then we have (cid:0) L ( R n × S ) ⊗ C (cid:1) ρ ml = n ϕ ( y ) e −√− lt ; ϕ ∈ L ( R n ) o and (cid:0) L ( R n × S ) ⊗ C (cid:1) ρ k = { } if k / ∈ m Z . Now we consider the operator∆ m, ∞ − (cid:18) k σ + kn (cid:19) : (cid:0) C ∞ ( R n × S ) ⊗ C (cid:1) ρ k → (cid:0) C ∞ ( R n × S ) ⊗ C (cid:1) ρ k for k = ml , which corresponds to the limit of2∆ ∂ Js : C ∞ ( X, L k ) → C ∞ ( X, L k )as s →
0. Let ( R n , t dy · dy, e − k k y k d L R n ) be the Gaussian space, where L R n is the Lebesgue measure on R n and denote by ∆ R n ,k the Laplacian of thismetric measure space. Note that we have∆ R n ,k ϕ = ∆ R n ϕ + 2 k n X i =1 y i ∂ϕ∂y i . Then we can see that the following linear isomorphism C ∞ ( R n ) ⊗ C → (cid:0) C ∞ ( R n × S ) ⊗ C (cid:1) ρ k ∈ ∈ ϕ ϕ · e − k k y k √− lt induces the isomorphism L ( R n , e − k k y k d L R n ) ⊗ C ∼ = (cid:0) L ( R n × S , dµ ∞ ) ⊗ C (cid:1) ρ k and the identification of the operators∆ R n ,k ∼ = ∆ m, ∞ − (cid:18) k σ + kn (cid:19) . Next we construct the eigenfunctions of ∆ R n ,k by the hermitian polynomials.For ξ ∈ R the hermitian polynomials are defined by H k,N ( ξ ) := e kξ d N dξ N e − kξ , which is a polynomial in ξ of degree N , then it is known that H k,N solves − d dξ H k,N + 2 kξ ddξ H k,N = 2 kN H k,N HE GEOMETRIC QUANTIZATIONS 33 and { H k,N } ∞ N =0 is a complete orthonormal system of L ( R , e − kξ d L R ). Let N = ( N , . . . , N n ) ∈ Z n ≥ and put (cid:18) ∂∂y (cid:19) N := ∂ N ∂y N · · · ∂ N n ∂y N n n , | N | := n X i =1 N i . Then ϕ ( y ) = n Y i =1 H k,N i ( y i ) = e k k y k (cid:18) ∂∂y (cid:19) N ( e − k k y k )solves ∆ R n ,k ϕ = 2 k | N | ϕ and { Q ni =1 H k,N i ( y i ); ( N , . . . , N n ) ∈ Z ≥ } is a complete orthonormal sys-tem of L ( R n , e − k k y k d L R n ). Thus we have the following theorem. Theorem 8.1.
Let l ∈ Z > , k = ml and W ( k, λ ) := (cid:26) f ∈ (cid:0) C ∞ ( R n × S ) ⊗ C (cid:1) ρ k ; (cid:18) ∆ m, ∞ − k σ − kn (cid:19) f = 2 λf (cid:27) . Then there is an orthogonal decomposition ( L ( R n × S ) ⊗ C ) ρ k = M d ∈ Z ≥ W ( k, kd ) , where W ( k, kd ) = span C ( e k k y k −√− lt (cid:18) ∂∂y (cid:19) N ( e − k k y k ); N ∈ Z ≥ , | N | = d ) . As a consequence of Theorem 8.1, we obtain the former part of Theorem1.3.9.
The fibers which are not m -BS fibers for any positive m In this section we suppose ( X n , ω ) is a symplectic manifold with a pre-quantum line bundle ( L, ∇ , h ), and assume that there is a continuous map µ : X → Y to a topological space Y . Moreover we fix b ∈ Y such that µ − ( b ) is not an m -BS fiber for any m ∈ Z .Let { J s } B ⊂ Y of b there is s r,B > µ ( B g Js ( p , r )) ⊂ B holds for any s ≤ s r,B . ⋆ c b : [0 , → X there exist anopen neighborhood B of b and a continuous map c : B × [0 , → X such that µ ◦ c ( b, t ) = b , c ( b,
0) = c ( b, c ( b , · ) = c b and c ( b, · ) arepiecewise smooth. ⋆ B of b and a continuous map c : B × [0 , → X such that µ ◦ c ( b, t ) = b and c ( b, · ) are piecewise smooth,lim s → sup b ∈ B L g Js ( c ( b, · )) = 0holds.Let π : S ( L, h ) → X be the natural projection. By the connection ∇ wehave the unique horizontal lift ˜ c : [0 , → S ( L, h ) with ˜ c (0) = u for anypair of a piecewise smooth path c : [0 , → X and u ∈ π − ( c (0)). Proposition 9.1.
Assume that µ − ( b ) is not an m -BS fiber for any m ∈ Z .For any p ∈ µ − ( b ) , e √− t ∈ S and δ > , there is a piecewise smoothpath c : [0 , → µ − ( b ) with c (0) = c (1) = p such that its horizontal lift ˜ c satisfying ˜ c (1) = ˜ c (0) e √− t ′ and | t ′ − t | < δ . In particular, if we assume ⋆ ,then lim s → diam ˆ g Js ( π − ( p )) = 0 holds.Proof. Since µ − ( b ) is not an m -BS fiber for any m ∈ Z , the holonomygroup of ∇| µ − ( b ) may not contained in any proper closed subgroup of S ,hence we obtain the path c which satisfies the assertion. By ⋆ s → L ˆ g Js (˜ c ) = lim s → L g Js ( c ) = 0holds, hence d J s (˜ c (0) , ˜ c (1)) → s →
0. Therefore, for any u ∈ π − ( p ), e √− t ∈ S and δ we havelim s → d J s ( u , u e √− t ) ≤ lim s → d J s ( u , u e √− t ′ ) + σ | t − t ′ | < σδ, which implies lim s → d J s ( u , u e √− t ) = 0, hence we havelim s → diam ˆ g Js ( π − ( p )) = 0 . (cid:3) Let B ⊂ Y be open and ˜ c : B × [0 , → S ( L, h ) be a map such that˜ c y := ˜ c ( y, · ) is one of the horizontal lift of c y := c ( y, · ) with respect to ∇ . Let t y ∈ R be defined by ˜ c ( y,
1) = ˜ c ( y, e √− t y , which is determinedindependent of the choice of the initial point of ˜ c ( y, · ). Then the map y e √− t y is continuous.For a sufficiently large integer N >
0, put t = πN and δ = t = πN and take c and t ′ as in Proposition 9.1, then we extend c to c : B × [0 , → X by ⋆ B is an open neighborhood of b . Then by the continuity of e √− t y ,there is an open neighborhood B N ⊂ Y of b such that πN < t y < πN holdsfor any y ∈ B N . If we consider the path obtained by connecting k copies of c y , we can see that d J s (˜ c y (0) , ˜ c y (0) e √− kt y ) ≤ k L g Js ( c y ) . If we consider the path along the fiber of S ( L, h ) → X , we have d J s (˜ c y (0) e √− a , ˜ c y (0) e √− b ) ≤ σ | a − b | . HE GEOMETRIC QUANTIZATIONS 35
Combining these estimates, we can see d J s (˜ c y (0) , ˜ c y (0) e √− θ ) ≤ N L g Js ( c y ) + 3 πσN for any θ ∈ R , which givesdiam ˆ g Js ( π − (˜ c y (0))) ≤ N L g Js ( c y ) + 3 πσN . Now we can take s N > ⋆ L g Js ( c y ) ≤ N for any 0
Proposition 9.2.
Assume ⋆ , µ − ( b ) is not an m -BS fiber for any m and let u ∈ π − ( p ) . Then for any r > and ε > there is < s r,ε ≤ s such that diam ˆ g Js ( π − ( x )) ≤ ε for all x ∈ B g Js ( p , r ) and < s ≤ s r,ε . Before we prove Theorem 1.2, we describe the relation between the con-vergence of principal G -bundles and the convergence of the base spaces. Let G be a compact Lie group, ( P, d, ν ) be a metric measure space with anisometric G -action. Put X := P/G and define the distance ¯ d on X by¯ d (¯ x, ¯ y ) := inf γ ∈ G d ( x, yγ ) , where ¯ x ∈ X is the equivalence class represented by x ∈ P . Proposition 9.3.
Let { ( P i , d i , ν i , p i ) } i ∈ N be a sequence of pointed metricmeasure spaces with isometric G actions and denote by π i : P i → X i = P i /G be the quotient maps. Suppose that for any r, ε > there is i r,ε ∈ N suchthat sup x ∈ B ( p i ,r ) diam d i π − ( x ) < ε holds for any i ≥ i r,ε . If { ( X i , ¯ d i , ¯ ν i , ¯ p i ) } i converges to ( X, ¯ d, ¯ ν, ¯ p ) with re-spect to the pointed measured Gromov-Hausdorff topology, then { ( P i , d i , ν i , p i ) } s converges to ( X, ¯ d, ¯ ν, ¯ p ) in the sense of the pointed G -equivariant measuredGromov-Hausdorff topology. Here, the G -action on X is the trivial action.Proof. Let ¯ φ i : ( B X i (¯ p i , r ) , ¯ p i ) → ( X, ¯ p ) be ε -approximations given by thepointed Gromov-Hausdorff convergence of ( X i , ¯ p i ). Then one can see that φ := ¯ φ i ◦ π i : ( π − i ( B (¯ p i , r )) , p i ) → ( X, ¯ p ) are G -equivariant 2 ε -approximations.Using these maps one can show the assertion. (cid:3) Proof of Theorem 1.2.
Assume ♠ and that there is κ ∈ R such that Ric g Js ≥ κg J s . Let u ∈ S | µ − ( y ) and assume that µ − ( y ) is not a Bohr-Sommerfeldfiber of L m for any m >
0. On the neighborhood U of µ − ( y ), we may write g J s | U = g A for some A = A ( s, x, θ ). Here we consider the pointed measured Gromov-Hausdorff limit of ( X, g J s , µ gJs K √ s n , p ) as s → p ∈ µ − ( y ) and K > g A ′ ,where A ′ ( s, x, θ ) = sA (0 , θ ) and ¯ Q = Im( A )(0 , θ ) is independent of θ .Notice that we already had P ij (0 , θ ) = ¯ P ij + ∂ H ∂θ i ∂θ j in Subsection 7.3 and F ∗ s g A ′ = t (cid:18)p s ¯Θ dθ − p s ¯Θ − ¯ P ¯ Q − dx (cid:19) · (cid:18)p s ¯Θ dθ − p s ¯Θ − ¯ P ¯ Q − dx (cid:19) + s − · t dx · ¯Θ − · dx holds by (8), where F s ( x, θ ) = (cid:18) x + s ∂ H ∂θ , . . . , x n + s ∂ H ∂θ n , θ (cid:19) , ¯Θ = ¯ Q + ¯ P ¯ Q − ¯ P .
Then by the transformation y = √ s ¯Θ − x and τ = √ s ¯Θ θ , we have F ∗ s g A ′ = t (cid:0) dτ − ¯ P ¯ Q − dy (cid:1) · (cid:0) dτ − ¯ P ¯ Q − dy (cid:1) + t dy · dy. The above expression implies that ( y, τ ) y is the Riemannian submersionto the Euclidean space. Since the diameters of the fibers of the submersionconverge to 0 as s →
0, we have proved that (
X, F ∗ s g A ′ , p ) pointed Gromov-Hausdorff converges to ( R n , t dy · dy, (cid:3) As a consequence of 1.2 we obtain the latter half of Theorem 1.3, sincethe S -action on R n in Theorem 1.2 is trivial and ( C ∞ ( R n ) ⊗ C ) ρ k = { } for any k > Acknowledgment.
The author would like to express his gratitude to Pro-fessors Hajime Fujita, Hiroshi Konno and Takahiko Yoshida for their severaluseful comments and advices.
References [1] Jørgen Ellegaard Andersen. Geometric quantization of symplectic manifolds withrespect to reducible non-negative polarizations.
Communications in mathematicalphysics , 183(2):401–421, 1997.[2] V. I. Arnold and A. Avez.
Probl`emes ergodiques de la m´ecanique classique . Monogra-phies Internationales de Math´ematiques Modernes, No. 9. Gauthier-Villars, ´Editeur,Paris, 1967.[3] Thomas Baier, Carlos Florentino, Jos´e M. Mour˜ao, and Jo˜ao P. Nunes. Toric K¨ahlermetrics seen from infinity, quantization and compact tropical amoebas.
J. DifferentialGeom. , 89(3):411–454, 2011.[4] Thomas Baier, Jos´e M. Mour˜ao, and Jo˜ao P. Nunes. Quantization of abelian varieties:distributional sections and the transition from K¨ahler to real polarizations.
J. Funct.Anal. , 258(10):3388–3412, 2010.[5] Jeff Cheeger and Tobias H. Colding. On the structure of spaces with Ricci curvaturebounded below. III.
J. Differential Geom. , 54(1):37–74, 2000.[6] Shiing-shen Chern.
Complex manifolds without potential theory (with an appendixon the geometry of characteristic classes) . Universitext. Springer-Verlag, New York,second edition, 1995.[7] J. J. Duistermaat. On global action-angle coordinates.
Comm. Pure Appl. Math. ,33(6):687–706, 1980.
HE GEOMETRIC QUANTIZATIONS 37 [8] Hajime Fujita, Mikio Furuta, and Takahiko Yoshida. Torus fibrations and localizationof index i-polarization and acyclic fibrations.
J. Math. Sci. Univ. Tokyo , 17(1):1–26,2010.[9] Kenji Fukaya. Collapsing of Riemannian manifolds and eigenvalues of Laplace oper-ator.
Invent. Math. , 87(3):517–547, 1987.[10] Kenji Fukaya and Takao Yamaguchi. Isometry groups of singular spaces.
Math. Z. ,216(1):31–44, 1994.[11] Mark D. Hamilton and Hiroshi Konno. Convergence of K¨ahler to real polarizationson flag manifolds via toric degenerations.
J. Symplectic Geom. , 12(3):473–509, 2014.[12] A. J. Hoffman and H. W. Wielandt. The variation of the spectrum of a normal matrix.
Duke Math. J. , 20:37–39, 1953.[13] Atsushi Kasue. Spectral convergence of Riemannian vector bundles.
Sci. Rep.Kanazawa Univ. , 55:25–49, 2011.[14] Yosuke Kubota. The joint spectral flow and localization of the indices of ellipticoperators.
Ann. K-Theory , 1(1):43–83, 2016.[15] Kazuhiro Kuwae and Takashi Shioya. Convergence of spectral structures: a func-tional analytic theory and its applications to spectral geometry.
Comm. Anal. Geom. ,11(4):599–673, 2003.[16] John Lott. Collapsing and Dirac-type operators. In
Proceedings of the Euroconferenceon Partial Differential Equations and their Applications to Geometry and Physics(Castelvecchio Pascoli, 2000) , volume 91, pages 175–196, 2002.[17] John Lott. Collapsing and the differential form Laplacian: the case of a smooth limitspace.
Duke Math. J. , 114(2):267–306, 2002.[18] L. Markus and K. R. Meyer.
Generic Hamiltonian dynamical systems are neither inte-grable nor ergodic . American Mathematical Society, Providence, R.I., 1974. Memoirsof the American Mathematical Society, No. 144.[19] N. A. Tyurin. Letter to the editors: “Dynamic correspondence in algebraic Lagrangiangeometry [Izv. Ross. Akad. Nauk Ser. Mat. (2002), no. 3, 175–196; mr1921813]. Izv. Ross. Akad. Nauk Ser. Mat. , 68(3):219–220, 2004.[20] Nikolai A. Tyurin. Geometric quantization and algebraic Lagrangian geometry. In
Surveys in geometry and number theory: reports on contemporary Russian mathemat-ics , volume 338 of
London Math. Soc. Lecture Note Ser. , pages 279–318. CambridgeUniv. Press, Cambridge, 2007.[21] N. M. J. Woodhouse.
Geometric quantization . Oxford Mathematical Monographs.The Clarendon Press, Oxford University Press, New York, second edition, 1992. Ox-ford Science Publications.[22] Takahiko Yoshida. Adiabatic limits, theta functions, and geometric quantization. arXiv preprint arXiv:1904.04076 , 2019.
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