Loading Web-Font TeX/Main/Regular
| Register
\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

4. Estimates for Bergman Kernel

    1. Problem 4.1.

      [Christ] Let M be a compact complex manifold without boundary and L be a holomorphic line bundle over M with positive curvature. Let B_k(z,w) be the Bergman kernel associated to L^k with respect to a fixed volume form. Suppose that for any \delta>0, there exists constants c_1 and c_2 (depending on \delta) such that |B_{k}(z,w)|\leq c_1e^{-c_2k} \text{ for all }z, w\in M \text{ with } \it{dist}(z,w)>\delta.
      Does it follow that the metric on L is real analytic?
        • Problem 4.2.

          [Catlin] Determine the boundary behavior of the Bergman kernel on Reinhardt domains. In particular, assume that the domain is smooth, complete and finite type. Start with the domains \left\{|z_1|^2+|z_2|^{2q}<1\right\} and \left\{|z_1|^4+|z_2|^{4}<1\right\}, where the kernels are known explicitly.

              Cite this as: AimPL: Cauchy-Riemann equations in several variables, available at http://aimpl.org/crscv.