A note on scaling asymptotics for Bohr-Sommerfeld Lagrangian submanifolds

This paper is involved with the asymptotics associated to Bohr-Sommerfeld Lagrangian submanifolds of a compact Hodge 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 applying the Szegö kernel of the quantizing line bundle to certain delta functions concentrated along the submanifold. Let M be a d-dimensional complex projective manifold, with complex structure J ; consider an ample line bundle A on it, and let h be an Hermitian metric on A such that the unique compatible connection has curvature Ω = −2iω, where ω is a Kähler form. Then the unit circle bundle X ⊆ A∗, endowed with the connection one-form α, is a contact manifold. A BohrSommerfeld Lagrangian submanifold of M (or, more precisely, of (M,A, h)) is then simply a Legendrian submanifold Λ ⊆ X, conceived as an immersed submanifold of M . In a standard manner, X inherits a Riemannian structure for which the projection π : X → M is a Riemannian fibration. In view of this, in the following at places we shall implicitly identify (generalized) functions, densities and half-densities on X. Referring to §2 of [DP] for a more complete description of the prelimiaries involved, we recall that if Λ ⊆ X is a compact Legendrian submanifold, and λ is a half-density on it, there is a naturally induced generalized half-density δΛ,λ on X supported on Λ; following [BPU], we can then define a sequence of CR functions uk =: Pk (