A note on scaling asymptotics for Bohr-Sommerfeld Lagrangian submanifolds
aa r X i v : . [ m a t h . S G ] N ov A note on scaling asymptotics forBohr-Sommerfeld Lagrangian submanifolds
Roberto Paoletti ∗ The purpose of this note is to improve an expansion in [DP] for the asymp-totics associated to Bohr-Sommerfeld Lagrangian submanifolds of a compactHodge manifold, in the context of geometric quantization (see e.g. [BW],[BPU], [GS3], [W]). We adopt the general framework for quantizing Bohr-Sommerfeld Lagrangian submanifolds presented in [BPU], based on applyingthe Szeg¨o kernel of the quantizing line bundle to certain delta functions con-centrated along the submanifold.Let M be a d-dimensional complex projective manifold, with complexstructure J ; consider an ample line bundle A on it, and let h be an Hermitianmetric on A such that the unique compatible connection has curvature Ω = − iω , where ω is a K¨ahler form. Then the unit circle bundle X ⊆ A ∗ ,endowed with the connection one-form α , is a contact manifold. A Bohr-Sommerfeld Lagrangian submanifold of M (or, more precisely, of ( M, A, h ))is then simply a Legendrian submanifold Λ ⊆ X , conceived as an immersedsubmanifold of M .In a standard manner, X inherits a Riemannian structure for which theprojection π : X → M is a Riemannian fibration. In view of this, in the fol-lowing at places we shall implicitly identify (generalized) functions, densitiesand half-densities on X .Referring to § ⊆ X is a compact Legendrian submanifold, and λ is a half-density on it, there is a naturally induced generalized half-density δ Λ ,λ on X supported on Λ; following [BPU], we can then define a sequenceof CR functions u k =: P k (cid:0) δ Λ ,λ (cid:1) ∈ H ( X ) k , ∗ Address:
Dipartimento di Matematica e Applicazioni, Universit`a degli Studi di Mi-lano Bicocca, Via R. Cozzi 53, 20125 Milano, Italy; e-mail : [email protected] H ( X ) k is the k -th isotype of the Hardy space with respect to the S -action, and P k : L ( X ) → H ( X ) k is the orthogonal projector (extended to D ′ ( X ) → H ( X ) k ). In the present setting there are natural unitary structureson H ( X ) k and the space of global holomorphic sections H (cid:0) M, A ⊗ k (cid:1) , anda natural unitary isomorphism H ( X ) k ∼ = H (cid:0) M, A ⊗ k (cid:1) . One thinks of the u k ’s as representing the quantizations of (Λ , λ ) at Planck’s constant 1 /k . Itis easily seen that u k is rapidly decaying as k → + ∞ on the complement of S · Λ = π − (cid:16) π (Λ) (cid:17) ; here π : X → M is the projection. On the other hand,the asymptotic concentration of the u k ’s along S · Λ poses an interestingproblem, already considered in Theorem 3.12 of [BPU].This theme was revisited in [DP], in a somewhat different technical set-ting; in particular, Corollary 1.1 of [DP] shows that the scaling asymptoticsof u k (to be defined shortly) near any x ∈ S · Λ admit an asymptotic expan-sion, and explicitly computes the leading order term. We shall give presentlya more precise description of this expansion, as a function on the tangentspace of M at m = π ( x ). Namely, we shall show that this asymptotic expan-sion may be factored as an exponentially decaying term in the component w ⊥ of w ∈ T m M orthogonal to Λ, times an asymptotic expansion with poly-nomial coefficients in w (more precisely, the expansion is generally given bya finite sum of terms of this form, one from each branch of Λ projecting to m ); the exponential term also contains a symplectic pairing between w ⊥ andthe component of w along Λ, w k . Furthermore, we shall give some relevantremainder estimates not mentioned in [DP].Before stating the results of this paper, let us recall that for any x ∈ X we can find a Heisenberg local chart for X centered at x , ρ : B ( ǫ ) × ( − π, π ) → X, ( p, q, θ ) r e iθ (cid:16) e (cid:0) ̺ ( p, q ) (cid:1)(cid:17) ;here B ( ǫ ) ⊆ R is a ball of radius ǫ centered at the origin, ̺ : B ( ǫ ) → M is a preferred local chart for M centered at m =: π ( x ), meaning that it triv-ializes the holomorphic and symplectic structures at m , and e is a unitarylocal frame of A ∗ , given by the unitarization of a preferred local frame (com-plete definitions are in [SZ]). Finally, r e iθ : X → X is the diffeomorphisminduced by the circle action. It is in this kind of local coordinates that thescaling limits of Szeg¨o kernels exhibit their universal nature (Theorem 3.1of [SZ]). If ρ is a system of Heisenberg local coordinates centered at x , and p, q ∈ R d , w = p + iq , one poses x + w =: ρ (cid:0) ( p, q ) , (cid:1) . For any θ , we have u k (cid:0) ρ ( p, q, θ ) (cid:1) = e ikθ u k (cid:0) ρ ( p, q, (cid:1) = e ikθ u k ( x + w ) . T m M and C d , with this understanding we can also consider theexpression x + w with w ∈ T m M .If x ∈ S · Λ, there are only finitely many elements h , . . . , h N x ∈ S such that x j =: r h j ( x ) ∈ Λ. Since Λ is Legendrian, hence horizontal for theconnection, for any j we may naturally identify the tangent space T x j Λ ⊆ T x j X with a subspace of T π ( x ) M . With this in mind, if w ∈ T π ( x ) M we canwrite w = w k j + w ⊥ j for unique w k j ∈ T x j Λ and w ⊥ j ∈ T x j Λ ⊥ ; the latter denotesthe orthocomplement of T x j Λ in T π ( x ) M in the Riemannian metric of M .Finally, let dens (1 / be the Riemannian half-density on Λ (for the inducedmetric); thus if λ is a C ∞ half-density on Λ we can write λ = F λ · dens (1 / for a unique F λ ∈ C ∞ (Λ). Theorem 1.
Let Λ ⊆ X be a compact Legendrian submanifold, and suppose λ is a smooth half-weight on it. For every k = 1 , , . . . , let u k =: P k (cid:0) δ Λ ,λ (cid:1) .Suppose x ∈ S · Λ , and choose a system of Heisenberg local coordinates for X centered at x . Let h , . . . , h N x ∈ S be the finitely many elements suchthat r h j ( x ) ∈ Λ . Then:1. Suppose a > . Uniformly for min j (cid:8) k w ⊥ j k (cid:9) & k a , we have u k (cid:18) x + w √ k (cid:19) = O (cid:0) k −∞ (cid:1) .
2. There exists polynomials a lj on C d such that the following holds: for w ∈ T π ( x ) M and k, ℓ = 1 , , . . . , let us define R k,ℓ ( x, w ) =: u k (cid:18) x + w √ k (cid:19) − (cid:18) kπ (cid:19) d / N x X j =1 h − kj e −k w ⊥ j k − iω π ( x ) ( w ⊥ j ,w k j ) F λ ( x j ) · ℓ X l =1 k − l/ a lj ( w ) ! . Then uniformly for k w k . k / we have (cid:12)(cid:12) R k,ℓ ( x, w ) (cid:12)(cid:12) ≤ C ℓ k (d − ℓ − / N x X j =1 e − − ǫ k w ⊥ j k . (1) Corollary 1. ∀ w ∈ T π ( x ) M , the following asymptotic expansion holds as k → + ∞ : u k (cid:18) x + w √ k (cid:19) ∼ (cid:18) kπ (cid:19) d / N x X j =1 h − kj e −k w ⊥ j k − iω π ( x ) ( w ⊥ j ,w k j ) F λ ( x j ) · X l ≥ k − l/ a lj ( w ) ! . Let us first prove 2. Thus, we want to investigate the asymptotics of u k (cid:16) x + w √ k (cid:17) as k → + ∞ , assuming that w ∈ T π ( x ) M , k w k ≤ C k / for some fixed C > k ∈ C ∞ ( X × X ) be the Schartz kernel of P k ; explicitly, if n s ( k ) r o isan orthonormal basis of H ( X ) k , thenΠ k ( y, y ′ ) = X r s ( k ) r ( y ) · s ( k ) r ( y ′ ) ( y, y ′ ∈ X ) . Let dens X and dens Λ denote, respectively, the Riemannian density on X andΛ. Then, in standard distributional short-hand, by definition of δ Λ ,λ for any x ′ ∈ X we have u k ( x ′ ) = Z X Π k ( x ′ , y ) δ Λ ,λ ( y ) dens X ( y )= h δ Λ ,λ , Π k ( x ′ , · ) i = Z Λ Π k ( x ′ , y ) F λ ( y ) dens Λ ( y ) . (2)Let us write dist M for the Riemannian distance function on M , pulled-back to a smooth function on X × X by the projection π × π . Let us set: V k =: (cid:8) x ′ ∈ X : dist M (cid:0) x, x ′ ) < C k − / (cid:9) ,V ′ k =: (cid:8) x ′ ∈ X : dist M (cid:0) x, x ′ ) > C k − / (cid:9) . If y ∈ V ′ k and k w k ≤ C k / , then dist M (cid:16) x + w √ k , y (cid:17) ≥ C k − / for k ≫
0; bythe off-diagonal estimates on Szeg¨o kernels in [C], therefore, Π k (cid:16) x + w √ k , y (cid:17) = O ( k −∞ ) uniformly for y ∈ V ′ k . 4or k ≫
0, Λ ∩ V k has N x connected components:Λ ∩ V k = N x [ j =1 Λ kj , where Λ kj is the connected component containing x j . Let { s k , s ′ k } be an S -invariant partition of unity on X , subordinate to the open cover { V k , V ′ k } . Inview of (2) and the previous discussion, we obtain u k (cid:18) x + w √ k (cid:19) = Z Λ Π k (cid:18) x + w √ k , y (cid:19) F λ ( y ) dens Λ ( y ) ∼ N x X j =1 Z Λ kj Π k (cid:18) x + w √ k , y (cid:19) F λ ( y ) s k ( y ) dens Λ ( y ) , (3)where ∼ means that the two terms have the same asymptotics. Let us nowevaluate the asymptotics of the j -th summand in (3).To this end, recall that the Heisenberg local chart ρ centered at x dependson the choice of the preferred local chart ̺ at π ( x ), and of the local frame e of A ∗ . We obtain a Heisenberg local chart ρ ′ j centered at x j by setting ρ ′ j ( p, q, θ ) =: r h j (cid:16) ρ ( p, q, θ ) (cid:17) . By the discussion in § ρ ′ j with a suitable transformation in ( p, q ) (thatis, a change of preferred local chart for M ) so as to obtain a Heisenberg localchart ρ j ( p, q, θ ) centered at x j with the following property: Λ is locally definednear x j by the conditions θ = f j ( q ) and p = 0, where f j vanishes to thirdorder at the origin. By construction, we have ρ j ( p, q, θ ) = r h j (cid:16) ρ ( p ′ , q ′ , θ ) (cid:17) for a certain local diffeomorphism ( p, q ) ( p ′ , q ′ ).Thus Λ is locally parametrized, near x j and in the chart ρ j , by the imag-inary vectors iq ; viewing the q ’s as local coordinates on Λ near x j , locally wehave dens Λ = D λ · | dq | , for a unique locally defined smooth function D Λ . Byconstruction of Heisenberg local coordinates, D λ (0) = 1.Applying a rescaling by k − / , we obtain Z Λ kj Π k (cid:18) x + w √ k , y (cid:19) F λ ( y ) s k ( y ) dens Λ ( y ) (4)= k − d / Z R d Π k (cid:18) x + w √ k , r e ifj ( q/ √ k ) (cid:18) x j + iq √ k (cid:19)(cid:19) F λ (cid:18) q √ k (cid:19) s k (cid:18) iq √ k (cid:19) D λ (cid:18) q √ k (cid:19) dq = k − d / Z R d e − ikf j “ q √ k ” Π k (cid:18) x + w √ k , x j + iq √ k (cid:19) F λ (cid:18) q √ k (cid:19) s k (cid:18) iq √ k (cid:19) D λ (cid:18) q √ k (cid:19) dq. x + w √ k = ρ (cid:16) ℜ ( w ) √ k , ℑ ( w ) √ k , (cid:17) (we use the Heisenberg chart to unitarilyidentify T m M with C d ), and x j + iq √ k = ρ j (cid:16) , q √ k , (cid:17) . Notice that s k (cid:16) iq √ k (cid:17) = 1for k q k . k / , s k (cid:16) iq √ k (cid:17) = 0 for k q k & k / . In particular, integration takesplace over a ball of radius ∼ k / . Also, Taylor expanding F λ and f j at theorigin we have asymptotic expansions F λ (cid:18) q √ k (cid:19) ∼ F λ ( x j ) + X r ≥ k − r/ b r ( q ) , D λ (cid:18) q √ k (cid:19) ∼ X r ≥ k − r/ c r ( q ) , and, since f j vanishes to third order at the origin, f j (cid:18) q √ k (cid:19) ∼ X r ≥ k − (3+ r ) / d r ( q ) , e − ikf j “ q √ k ” ∼ X r ≥ k − r/ e r ( q ) , for suitable polynomials b r , c r , d r , e r .Let w j ∈ C d correspond to w in the Heisenberg local coordinates ρ j . Bythe above, Taylor expanding the transformation ( p, q ) ( p ′ , q ′ ), we obtain x + w √ k = r h − j (cid:16) x j + w j √ k + H ( w, k ) (cid:17) , where H ( w, k ) ∼ P f ≥ k − f/ h f ( w ).Without affecting the leading order term of the resulting asymptotic expan-sion, we may pretend for simplicity that x + w √ k = r h − j (cid:16) x j + w j √ k (cid:17) .Write w j = p j + iq j , with p j , q j ∈ R d . Thus w ⊥ j = p j , w k j = iq j . In viewof Theorem 3.1 of [SZ], we haveΠ k (cid:18) x + w √ k , x j + iq √ k (cid:19) (5)= Π k (cid:18) r h − j (cid:18) x j + w j √ k (cid:19) , x j + iq √ k (cid:19) = h − kj Π k (cid:18) x j + w j √ k , x j + iq √ k (cid:19) ∼ h − kj (cid:18) kπ (cid:19) d e − ip j · q − k p j k − k q j − q k · X r ≥ k − r/ R j ( w, q ) ! , for certain polynomials R j in w and q . Furthermore, by the large ball estimateon the remainder discussed in § ≤ r ≤ R is bounded by C R k d − ( R +1) e − − ǫ ( k p j k + k q − q j k ) . (6)It follows that the product of these asymptotic expansions can be integratedterm by term; given this, we only lose a contribution which is O ( k −∞ ) bysetting s k = 1 and integrating over all of R d .6e have Z R d e − ip j · q − k q j − q k dq = e − ip j · q j Z R d e − ip j · s − k s k ds = (2 π ) d / e − ip j · q j − k p j k . Given (5), this implies that (4) is given by an asymptotic expansion, withleading order term h − kj (cid:18) kπ (cid:19) d / e −k w ⊥ j k − iω π ( x ) ( w ⊥ j ,w k j ) F λ ( x j ) . To determine the general term of the expansion, on the other hand, we areled to computing integrals of the form Z R d q β e − ip j · q − k q j − q k dq = e − ip j · q j Z R d ( s + q j ) β e − ip j · s − k s k ds. where q β is some monomial. Thus we led to a sum of terms of the form e − ip j · q j C γ ( q j ) Z R d s γ e − ip j · s − k s k ds, and the integral is the evaluation at p j of the Fourier transform of s γ e − k s k .Up to a scalar factor, the latter is an iterated derivative to e − k s k ; thereforewe are left with a summand of the form e − ip j · q j T ( q j , q j ) e − k p j k ds , where T is a polynomial in p j , q j . Given (5), this implies that the general term of theasymptotic expansion for (4) has the form h − kj (cid:18) kπ (cid:19) d / k − l/ e −k w ⊥ j k − iω π ( x ) ( w ⊥ j ,w k j ) F λ ( x j ) · a lj ( w )for an appropriate polynomial a lj ( w ). Finally, (1) (at x j ) follows by integrat-ing (6).To complete the proof of 2., we need only sum over j .Turning to the proof of 1., by definition of preferred local coordinates, if w ⊥ j ≥ C k a , say, then dist M (cid:18) x + v √ k , Λ kj (cid:19) ≥ C k a − , for all k ≫
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