Non-strict negativity of holomorphic bisectional curvature for compact relative K\"ahler fibrations

Abstract

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 K\"ahler 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 K\"ahler 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-Amp\`ere 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.