Bergman kernel function for Hartogs domains over bounded homogeneous domains (Q2012974)

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 \(5\).