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## 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.