Non-strict negativity of holomorphic bisectional curvature for compact relative Kähler fibrations

Strict negativity of holomorphic bisectional curvature need not pass from the base and fibers of a compact holomorphic fibration to its total space, even when the Kodaira-Spencer map is everywhere injective. We construct a compact relative Kähler fibration over a genus-two curve, with a smooth projective threefold as total space and an everywhere injective Kodaira–Spencer map, whose induced fiber metrics have strictly negative holomorphic bisectional curvature, whereas the total space admits no Kähler metric with this curvature property. This gives a negative answer to an open problem posed by To and Yeung. We also construct compact, effectively parametrized Monge-Ampère fibrations from universal quaternionic abelian surfaces over Shimura curves. For products of these families, the generalized Weil-Petersson metric has nonpositive holomorphic bisectional curvature and strictly negative holomorphic sectional curvature, but its mixed bisectional curvatures vanish. Thus strict bisectional negativity can fail both for the existence of a metric on the total space and for the natural metric on the parameter space, despite effective variation at every point.