Bergman functions and the equivalence problem of singular domains (Q6185123)

An open subset \(V\) in the singular variety \(\widetilde V=\{(u_1,u_2,u_3,u_4)\in\mathbb C^4:u_1u_4=u_2u_3\}\) is called a bounded complete Reinhardt domain (resp.~a bounded Reinhardt domain) if there exists a bounded complete Reinhardt domain (resp.~a bounded Reinhardt domain) \(D\subset\mathbb C^4\) such that \(V=\widetilde V\cap D\). One of the main results of the paper is to characterize biholomorphic maps between two bounded complete Reinhardt domains (resp.~bounded Reinhardt domains) \(V_j\) in \(\widetilde V\), \(j=1,2\). The authors prove that any such biholomorphism is the restriction of a linear map \(\ell:\mathbb C^4\longrightarrow\mathbb C^4\) satisfying some technical conditions. As an application, they solve the biholomorphic equivalence problem for two four-parameter families of the form \[ V_{a,b,c,d}^k=\{(u_1,u_2,u_3,u_4)\in\mathbb C^4:u_1u_4=u_2u_3,\ a|u_1|^{2k}+b|u_2|^{2k}+c|u_3|^{2k}+d|u_4|^{2k}<\varepsilon\}, \] where \(a,b,c,d>0\), \(\varepsilon>0\) is fixed, and \(k\in\{1,2\}\). Namely, the authors prove that for \(k=1,2\) \[ V_{a_1,b_1,c_1,d_1}^k\simeq V_{a_2,b_2,c_2,d_2}^k\iff\frac{a_1d_1}{b_1c_1}=\frac{a_2d_2}{b_2c_2}\text{ or }\frac{a_1d_1}{b_1c_1}=\frac{b_2c_2}{a_2d_2}. \]