Kähler metrics of the negative holomorphic (bi)sectional curvature on a compact relative Kähler fibration

Abstract For a compact relative Kähler fibration over a compact Kähler manifold with negative holomorphic sectional curvature, if the relative Kähler form on each fiber also exhibits the negative holomorphic sectional curvature, we can construct Kähler metrics with the negative holomorphic sectional curvature on the total space. Additionally, if this form induces a Griffiths negative Hermitian metric on the relative tangent bundle, and the base admits a Kähler metric with the negative holomorphic bisectional curvature, we can also construct Kähler metrics with the negative holomorphic bisectional curvature on the total space. As an application, for a non‐trivial fibration where both the fibers and base have Kähler metrics with negative holomorphic bisectional curvature, and the fibers are one‐dimensional, we can explicitly construct Kähler metrics of the negative holomorphic bisectional curvature on the total space, thus resolving a question posed by To and Yeung for the case where the fibers have dimension one.