4. Estimates for Bergman Kernel
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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.