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Bergman kernel function for Hartogs domains over bounded homogeneous domains. (English) Zbl 1376.32004

The aim of the paper is to give an explicit closed form of the Bergman kernel of the Hartogs-type domain \[ D_{m,s}=\big\{(z,\zeta)\in D\times\mathbb C^m:\|\zeta\|^2<K_D(z,z)^{-s}\big\},\quad m\in\mathbb N,\quad s\in\mathbb R, \] where \(K_D\) denotes the Bergman kernel of the bounded homogeneous domain \(D\), and to give sufficient conditions for the Bergman kernel \(K_{m,s}\) of \(D_{m,s}\) to be zero-free.
First, the authors obtain an explicit form of the weighted Bergman kernel \(K_s\) for the weighted Bergman space \(L^2_a(D,K_D(z,z)^{-s}\,dV(z))\). Next, using the virtual Bergman kernel, they express the Bergman kernel \(K_{m,s}\) as a rational function in the variable \(\|\zeta\|^2K_D(z,z)^s\). Finally, the authors apply their results to study the Lu Qi-Keng problem for the domains \(D_{m,s}\). They show that \(K_{m,s}\) is zero-free for large \(m\) (i.e., \(D_{m,s}\) is a Lu Qi-Keng domain). Moreover, they find the number \(m_0(D)\) such that \(D_{m,s}\) is a Lu Qi-Keng domain for all \(s\in\mathbb R\) and \(m\geq m_0(D)\), when \(D\) has dimension \(4\) or \(5\).

MSC:

32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32M10 Homogeneous complex manifolds
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