A Laplace integral, the T-Y-Z expansion and Berezin's transform on a Kaehler manifold

Let M be an n-dimensional complex manifold endowed with a C∞ Kähler metric g. We show that a certain Laplace-type integral Lm (x), when x varies in a sufficiently small open set U ⊂ M, has an asymptotic expansion Lm(x)= 1/mn ∑r≤0 m-rCr(f) (x), where Cr:C∞ (U) → C∞ (U) are smooth differential operators depending on the curvature of g and its covariant derivatives. As a consequence we furnish a different proof of Lu's theorem by computing the lower order terms of Tian-Yau-Zelditch expansion in terms of the operator Cj. Finally, we compute the differential operators Qj of the expansion Berm (f) = ∑r≤0 m-r Qr (f) of Berezin's transform in terms of the operators Cj.