Abstract Bergman kernel expansion and its applications (Q2796527)

This article about the existence of an asymptotic expansion for the Bergman kernel associated to high powers of a positive line bundle over a compact Kähler manifold extends the work of \textit{D. Catlin} [in: Analysis and geometry in several complex variables. Proceedings of the 40th Taniguchi symposium, Katata, Japan, 1997. Boston, MA: Birkhäuser. 1--23 (1999; Zbl 0941.32002)] and \textit{S. Zelditch} [Int. Math. Res. Not. 1998, No. 6, 317--331 (1998; Zbl 0922.58082)]. The authors give a proof based solely on techniques of complex geometry, including special coordinates termed \(K\)-coordinates and \(K\)-frames. Their abstract result provides improved control on the remainder term. In the real analytic case, the asymptotic expansion converges, and the remainder decays faster than every polynomial.