On the Lower Order Terms of the Asymptotic Expansion of Zelditch

A projective algebraic manifold M is a complex manifold in certain projective space CPm, m ≥ dimC M = n. The hyperplane line bundle of CPm restricts to an ample line bundle L onM . The bundle is a polarization onM . Suppose g is a Kähler metric on M . We say g is a polarized Kähler metric with respect to L, if the corresponding Kähler form represents the first Chern class c1(L) of L in H 2(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 h induces a Hermitian metric hm on Lm. Let {Sm 0 , · · · , Sm dm−1} be an orthonormal basis of the space H 0(M,Lm) of all holomorphic global sections of Lm. Such a basis {Sm 0 , · · · , Sm dm−1} induces a holomorphic embedding φm of M into CP dm−1 by assigning the point x of M to [Sm 0 (x), · · · , Sm dm−1(x)] in CP dm−1. Let gFS be the standard Fubini-Study metric on CP dm−1, i.e., ωFS = √ −1 2π ∂∂ log dm−1 i=0 |wi| for a homogeneous coordinate system [w0, · · · , wdm−1] of CP dm−1. The 1 m multiple of gFS on CP dm−1 restricts to a Kähler metric 1 mφ ∗ mgFS on M . This metric is a polarized Kähler metric on M and is called the Bergmann metric with respect to L. One of the main theorem in [12] is the following