Germs of measure-preserving holomorphic maps from bounded symmetric domains to their Cartesian products
The main result of the article under review is the following general extension theorem: Let \(\Omega\) be an irreducible bounded symmetric domain in \(\mathbb C^n\), \(n\geq 2\), and \(d\mu_\Omega\) the volume form of the Bergman metric on \(\Omega\). Let \(\Omega_\alpha=\Omega\) denote the \(\alpha\)-th direct factor of \(\Omega^d\), \(1\leq\alpha\leq d\), and \(\pi_\alpha:\Omega^d\rightarrow\Omega_\alpha\) the natural surjection. Let \(d_1, d_2\in \mathbb N_+\) and \(f=(f_1,\dots,f_{d_2}):(\Omega,d_1d\mu_\Omega;0)\rightarrow (\Omega^{d_2},\pi_1^*d\mu_\Omega+\cdots+\pi_{d_2}^*d\mu_\Omega;0)\) the germ of a measure preserving holomorphic map such that each \(f_\alpha\), \(1\leq\alpha\leq d_2\), is of maximal rank at some point. Then \(d_1=d_2\) and \(f\) extends to a totally geodesic holomorphic embedding \(f:\Omega\rightarrow \Omega^{d_2}\).NEWLINENEWLINE The proof is based on methods of analytic continuation, in particular on a local characterization of automorphisms of \(\Omega\). When rank \(\Omega=1\) a theorem of \textit{T. Alexander} [Math. Ann. 209, 249--256 (1974; Zbl 0272.32006)] is applied which says that a nonconstant holomorphic map \(\varphi:U\rightarrow \mathbb C^n\) on a neighborhood \(U\) of \(b\in \partial B^n\) with \(\varphi(U\cap \partial B^n)\subset \partial B^n\) can be extended to an automorphism of the unit ball \(B^n\). For rank \(\Omega\geq 2\) the authors prove and apply an analogous theorem for \(\Omega\). Alternatively they deduce the total geodesy from a local characterization of automorphisms of \(\Omega\) by \textit{A. E. Tumanov} and \textit{G. M. Khenkin} [Sov. Math., Dokl. 26, 702--705 (1982); translation from Dokl. Akad. Nauk SSSR 267, 796--799 (1982; Zbl 0529.32014)].NEWLINENEWLINE The extension theorem cited at the beginning yields an affirmative answer to a question raised by \textit{L. Clozel} and \textit{E. Ullmo} [J. Reine Angew. Math. 558, 47--83 (2003; Zbl 1042.11027)] whether measure preserving algebraic correspondences on the quotient \(X=\Omega/\Gamma\) of \(\Omega\) by a torsion-free discrete subgroup \(\Gamma\) of Aut \(\Omega\) are totally geodesic. Suppose that \(Y\) is an algebraic correspondence on \(X\) in the sense that \(Y\) is a purely \(n\)-dimensional quasi-projective subvariety of \(X\times X\) and the natural projections pr\(_i:Y\rightarrow X_i=X\), \(i=1, 2\), to each of the two factors are surjective finite proper maps. If the general fiber of pr\(_2^{-1}(x)\) consists of \(d_2\) points for a general point \(x\in X\), then pr\(_1\)pr\(_2^{-1}(x) =\{x_1,\dots,x_{d_2}\}\) induces a germ of a holomorphic map \(f_0: (X,x)\rightarrow (X, x_1)\times\dots\times(X, x_{d_2})\). After lifting \(X\) locally to \(\Omega\) the above result can be applied. The case \(n=1\) was solved by L. Clozel and E. Ullmo.
scientific article; zbMATH DE number 6081358