Factorization of proper holomorphic maps on irreducible bounded symmetric domains of rank \(\geqslant 2\)
Let \(\Omega\) be an irreducible bounded symmetric domain of rank \(\geq 2\) and \(H\) a finite subgroup of Aut \(\Omega\). The quotient \(\Omega/H\) carries a unique normal complex structure for which the canonical projection \(\pi:\Omega\rightarrow \Omega/H\) is holomorphic. The main result of the article is a factorization theorem for proper holomorphic maps \(F:\Omega\rightarrow Z\) from \(\Omega\) onto a normal complex space \(Z\): There exists a finite subgroup \(H\) of Aut \(\Omega\) and a biholomorphic map \(\nu:\Omega/H\rightarrow Z\) such that \(F=\nu\circ\pi\). This result has several interesting consequences. If \(Z\) is a complex manifold and \(\Omega\) is not isomorphic to a type-IV classical symmetric domain \(D_n^{IV}\) of dimension \(n\geq 3\), then \(F\) is biholomorphic. If in addition \(Z\) is a bounded convex domain, then \(Z\) is up to an affin-linear transformation the Harish-Chandra realization of \(\Omega\), cf. \textit{N. Mok} and \textit{I.-H. Tsai} [J. Reine Angew. Math. 431, 91--122 (1992; Zbl 0765.32017)]. If \(\Omega=D_n^{IV}, n\geq 3\), then \(F\) is either biholomorphic or a two-fold branched covering of \(Z\), ramified along a totally geodesic smooth complex hypersurface \(J\subset D_n^{IV}\), biholomorphic to \(D_{n-1}^{IV}\) and embedded into \(D_n^{IV}\) in the standard way. From the authors' abstract: ``The approach is along the line of the works of Mok and Tsai by considering radial limits of bounded holomorphic functions derived from \(F\) and proving that proper holomorphic maps between bounded symmetric domains preserve certain totally geodesic subdomains. In contrast to the previous works, in general we have to deal with multivalent holomorphic maps for which Fatou's theorem cannot be applied directly. We bypass the difficulty by devising a limiting process for taking radial limits of correspondences arising from proper holomorphic maps and by elementary estimates allowing us to define distinct univalent branches of the underlying multivalent map on certain subsets.''
scientific article; zbMATH DE number 5712119