Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains
The main result of this paper states that if \(\phi\) is a plurisubharmonic function in a pseudoconvex domain~\(D\) in the product space \(\mathbb{C}^{k}_{t} \times \mathbb{C}^{n}_{z}\), and \(K_{t}(z,w)\) is the Bergman kernel function for the slice domain \(\{ z: (t,z)\in D\}\) with respect to the weight function \(e^{-\phi(t,z)}\), then the function \(\log K_{t}(z,z)\) is plurisubharmonic in~\(D\) (or else identically equal to \(-\infty\)). \textit{F.~Maitani} and \textit{H.~Yamaguchi} [Math. Ann. 330, No. 3, 477--489 (2004; Zbl 1077.32006)] previously obtained the simplest case when \(\phi=0\) and \(n=1\). The author gives two proofs and some interesting applications.
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