Negatively curved metrics on Kodaira surfaces
We prove the following result. Let S be an algebraic surface admitting a Kodaira fibration over an algebraic curve C of \(genus>1\). We think of S as being a holomorphic family \(\{S_ p\}_{p\in C}\), where the \(S_ p's\) are the fibers of the fibration. To this family we can associate a holomorphic map f: \(C\to T\)- the Teichmüller space of the Riemann surface \(S_ p\). We show that S admits a Hermitian metric of negative holomorphic bisectional curvature if and only if the map f is an immersion. This implies a recent result that S has negative tangent bundle in the sense of Grauert [cf. \textit{M. Martin-Deschamps}, Ann. Inst. Fourier 39-64 (1984; Zbl 0547.14012), and \textit{M. Schneider} in Complex analysis and algebraic geometry, Proc. Conf., Göttingen 1985, Lect. Notes Math. 1194, 150-157 (1986; Zbl 0592.14016)].
scientific article; zbMATH DE number 4086474