Proper holomorphic polynomial maps between bounded symmetric domains of classical type (Q2789871)

Let \(\Omega _1 ,\Omega _2\) be domains in \(\mathbb{C}^n\), and let \(f,g:\Omega _1 \to \Omega _2\) be holomorphic maps. We say that \(f\) is proper if \(f^{ - 1} \left( K \right)\) is compact for every compact subset \(K \subseteq \Omega _2\). We say that \(f\) and \(g\) are equivalent if and only if \(f = A\circ g \circ B\) for some \(B \in \mathrm{Aut}(\Omega _1)\) and \(A \in\mathrm{Aut}(\Omega _2)\). For a domain \(\Omega\), denote the group of automorphisms fixing \(p \in \Omega\) by \(\mathrm{Isot}_p(\Omega)\). Suppose that for fixed \(p \in \Omega\), \(f(p) = g(p)\). Then we say that \(f\) and \(g\) are isotropically equivalent at \(p\) if there are \(U \in \mathrm{Isot}_p(\Omega_1)\) and \(V \in \mathrm{Isot}_{g(p)}(\Omega_2)\) such that \(f = V\circ g\circ U\). NEWLINENEWLINENEWLINENEWLINE Proper holomorphic maps between balls have been studied since \textit{H. Alexander} [Math. Ann. 209, 249--256 (1974; Zbl 0272.32006)] proved that every proper holomorphic self-map of the \(n\)-dimensional unit ball \(\mathbb{B}_n\) with \(n \geq 2\) is a holomorphic automorphism. \textit{J. P. D'Angelo} [Mich. Math. J. 35, No. 1, 83--90 (1988; Zbl 0651.32014)] showed that any two proper holomorphic polynomial maps from \(\mathbb{B}_n\) to \(\mathbb{B}_N\) preserving the origin are equivalent if and only if they are isotropically equivalent at the origin, and as a consequence, NEWLINENEWLINE\[NEWLINEf_\theta \left( z \right) = \left( {z_1 ,\dots,z_{n - 1} ,\cos \theta z_n ,\sin \theta z_1 z_n ,\dots,\sin \theta z_n z_n } \right) NEWLINE\]NEWLINE are inequivalent for all \(0 \leq \theta \leq \frac{\pi }{2}\). NEWLINENEWLINENEWLINENEWLINE Furthermore, \textit{H. Hamada} [Math. Ann. 331, No. 3, 693--711 (2005; Zbl 1076.32014)] showed that any proper rational map from \(\mathbb B^n\) to \(\mathbb B^{2n}\) with \(n\geq4\) is equivalent to \(f_\theta\) for some \(\theta\), \(0 \leq \theta \leq \frac{\pi }{2}\). NEWLINENEWLINENEWLINENEWLINE In the paper under the review, the authors generalize the results of D'Angelo, which deals with proper holomorphic polynomial maps between balls, to bounded symmetric domains of classical type, and prove the following two main theorems. {\parindent=6mm \begin{itemize}\item[(1)] Let \(\Omega _1 ,\Omega _2\) be bounded symmetric domains of classical type, and let \(f,g: \Omega _1 \to \Omega _2\) be proper holomorphic polynomial maps such that \(f(0) = g(0) = 0\). Then \(f\) and \(g\) are equivalent if and only if they are isotropically equivalent at \(0\). \item [(2)] There are uncountably many inequivalent proper holomorphic nonstandard maps from \(\Omega _{r,s}^I\) to \(\Omega _{2r - 1,2s}^I\) for \(r \geq 2, s \geq 2\).NEWLINENEWLINE\end{itemize}}