Inspired by the work of Engliš, we study the asymptotic behavior of the weighted Bergman kernel together with an application to the Lu Qi-Keng conjecture. Some comparison results between the weighted and the classical Bergman kernel are also obtained.

We study the limit behavior of weighted Bergman kernels on a sequence of domains in a complex space ${ℂ}^N$, and show that under some conditions on domains and weights, weighed Bergman kernels converge uniformly on compact sets. Then we give a weighted generalization of the theorem given by M. Skwarczyński (1980), highlighting some special property of the domains, on which the weighted Bergman kernels converge uniformly. Moreover, we show that convergence of weighted Bergman kernels implies this property,...

The integral representation for the multiplicity of an isolated zero of a holomorphic mapping $f:({ℂ}^n,0)\to ({ℂ}^n,0)$ by means of Weil’s formulae is obtained.