On some classes of hyperbolic complex spaces (Q5954622)

The authors give some characterization of hyperbolicity in the sense of Kobayashi for complex spaces. After studies of Hahn-Kim and Thai, they first prove that a complex space \(X\) is Kobayashi-hyperbolic if and only if \(X\) has the Landau property for any Finsler metric \(H\) on \(X\). Next they give the characterization of hyperbolicity by Gateaux holomorphic map. Let \(X\) be a compact complex space. Then they prove that \(X\) is hyperbolic if and only if every Gateaux holomorphic map from any open subset \(\Omega\) of any Banach space \(B\) into \(X\) is holomorphic. Furthermore, they construct the Kobayashi algebraic hyperbolic distance on an open subset of a Moishezon space and prove that this pseudodistance coincides with the Kobayashi pseudodistance. This result is an analogous assertion in the case of quasi-projective varieties which was proved by \textit{J.-P. Demailly, L. Lempert} and \textit{B. Shiffman} [Duke Math. J. 76, No.~2, 333-364 (1994; Zbl 0861.32006)].