Strict plurisubharmonicity of Bergman kernels on generalized annuli

Yanyan Wang¹

¹Department of Mathematics Tongji University 200092 Shanghai, China

Annales Polonici Mathematici, Tome 111 (2014) no. 3, pp. 237-243

Résumé

Let $A_\zeta=\varOmega-\overline{\rho(\zeta)\cdot\varOmega}$ be a family of generalized annuli over a domain $U$. We show that the logarithm of the Bergman kernel $K_{\zeta}(z)$ of $A_\zeta$ is plurisubharmonic provided $\rho\in {\rm PSH}(U)$. It is remarkable that $A_\zeta$ is non-pseudoconvex when the dimension of $A_\zeta$ is larger than one. For standard annuli in ${\mathbb C}$, we obtain an interesting formula for $\partial^2 \log K_{\zeta}/\partial \zeta\partial\bar{\zeta}$, as well as its boundary behavior.

Détail

DOI : 10.4064/ap111-3-2

Keywords: zeta varomega overline rho zeta cdot varomega family generalized annuli domain logarithm bergman kernel zeta zeta plurisubharmonic provided rho psh remarkable zeta non pseudoconvex dimension zeta larger standard annuli mathbb obtain interesting formula partial log zeta partial zeta partial bar zeta its boundary behavior