Non-strict negativity of holomorphic bisectional curvature for compact relative Kähler 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 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.
Related
- Holomorphic Bisectional Curvature on Total Spaces of Holomorphic Vector Bundles
- Hermitian manifolds with constant holomorphic sectional curvature or real bisectional curvature
- The cross quadratic bisectional curvature of Kähler C-spaces of classical type
- Positive bisectional curvature, Ricci mass, and the fundamental group
- Compact Vaisman and l.c.K. manifolds with nonnegative bisectional curvature
BibTeX
@misc{wan2026nonstrict,
title = {{Non-strict negativity of holomorphic bisectional curvature for compact relative Kähler fibrations}},
author = {Xueyuan Wan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11388},
eprint = {2609.11388},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11388},
}
RIS
TY - UNPB
TI - Non-strict negativity of holomorphic bisectional curvature for compact relative Kähler fibrations
AU - Xueyuan Wan
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11388
UR - https://arxiv.org/abs/2609.11388
AB - 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.
ER -