Hermitian metrics of negative holomorphic sectional curvature on some hyperbolic manifolds
It is well-known that if a compact complex manifold admits a Hermitian metric with negative holomorphic sectional curvature, then it is hyperbolic. The converse is still open. The purpose of this note is to show that for some types of compact hyperbolic manifolds, we can construct a Hermitian metric on them with everywhere negative holomorphic sectional curvature. In particular, Kodaira's surfaces of general type and the example of \textit{H. Grauert} and \textit{H. Reckziegel} [Math. Z. 89, 108-125 (1965; Zbl 0135.125)] are included as special cases, while the local result of the latter is extended to a global one. In the case of Kodaira surfaces, we will show that they can in fact be given a Kähler metric with negative holomorphic sectional curvature.
negative holomorphic sectional curvature
scientific article; zbMATH DE number 4058356