A Class Of Hartogs Domain Of The Bergman Kernel Function And Lu Qikeng

The Bergman kernel function, introduced by S. Bergman in 1921 and generalized in 1933 respectively, is a useful tool in the research of the branches of mathematics, such as Complex Analysis, Differential Geometry, Mathematical Physics and so on. However, it is difficult to calculate the Bergman kernel with explicit formula, even one focus only on the bounded domains in Cⁿ. Therefore, it has naturally become an important research field in several complex variables.The Lu Qi-Keng problem, which is actually on the zeros of the Bergman kernel, originated from the paper titled "On constant curvature Kahler manifolds" by Lu in 1966. Lu asked in that paper whether the Bergman kernel function of a simple connecteddomain in Cⁿ(n > 1) has no zeros. Many counterexamples occurred since the question was posed. Since the zeros set is an analytic invariant under the biholomorphictransformations, the research on Lu Qi-Keng problem can also be regarded as a powerful tool to tell that when two particular domains are biholomorphically inequivalent.In this thesis, we obtained explicitly the Bergman kernel function for the domainThe formula isFirstly, we compute the Bergman kernel function of the domainΩ(1,N₁,N₂; K, L) by it's holomorphic automorphism group and the complete orthonormal system, and then according to inflation principle we get the Bergman kernel function of the domainΩ(N₀,N₁,N₂;K,L).Secondly, we discuss the Lu Qi-Keng problem on the domainIn this part, we transform several variables into single variable problem with the holomorphicautomorphism ofΩ, get an inequality using Rouche theorem , and finally gain a series of Lu Qi-Keng domains, whose fibre's dimension N₀≥(?), when we fix the bottom spaces' dimensions N₁, N₂ and the parameters K, L. So we not only give lots of obverse examples but also offer an basis for finding geometric judgement of Lu Qi-Keng domain.