Chinese Annals of Mathematics. Series B (Chinese Ann. Math. Ser. B) (20220101), 43, no.~2, 265-280. ISSN: 0252-9599 (print).eISSN: 1860-6261.
Subject
32 Several complex variables and analytic spaces -- 32A Holomorphic functions of several complex variables 32A25 Integral representations; canonical kernels
This paper studies zeros of the Bergman kernels of the first-type Cartan-Hartogs domains: $$ \Omega_{m,n,\mu}=\left\{ \left(\xi,Z\right)\in\Bbb{C}\times R_{I}\left(m,n\right):\left|\xi\right|^{2\mu}0$, $R_{I}(m,n)\coloneq \{ Z\in\Bbb{C}^{mn}:I-Z\overline{Z}^{t}>0\} $, and $N_{I}(Z,Z)\coloneq \det(I-Z\overline{Z}^{t})$. \par The Bergman kernel of $\Omega_{m,n,\mu}$ was studied earlier by W.~P. Yin [Chinese Sci. Bull. {\bf 44} (1999), no.~21, 1947--1951; MR1752411]. \par The work under review here shows that for any $m,n$, there exists $\mu_{0}>0$ such that the Bergman kernel of $\Omega_{m,n,\mu}$ has zeros when $\mu<\mu_{0}$ (Theorem 3.1). The authors also obtain various topological properties of the zero set of the Bergman kernels.