Generalized Asymptotic Expansions of Tian-yau-zelditch (announcements)

Let M be an n-dimensional projective algebraic manifold in certain projective space CP . The hyperplane line bundle of CP restricts to an ample line bundle L on M , which is called a polarization of M . A Kähler metric g is called a polarized metric, if the corresponding Kähler form represents the first Chern class c1(L) of L in H (M,Z). Given any polarized Kähler metric g, there is a Hermitian metric h on L whose Ricci form is equal to ωg. For each positive integer m > 0, the Hermitian metric hL induces the Hermitian metric h m L on L . Let (E, hE) be a Hermitian vector bundle of rank r with a Hermitian metric hE. Consider the space Γ(M,L m ⊗ E) of all holomorphic sections for large m. For U, V ∈ Γ(M,L ⊗ E), the pointwise and the L inner products are defined as