To–Yeung open problem on negatively curved Kähler fibrations
Let be a nontrivial compact holomorphic fibration, with and every fiber admitting Kähler metrics of strictly negative holomorphic bisectional curvature. Must admit a Kähler metric with strictly negative holomorphic bisectional curvature?
References
Primary source
Additional references
Progress summary
A September 2026 preprint claims to disprove the rule that negatively curved bases and fibers always give a negatively curved total space, but this has not been independently confirmed.
To and Yeung asked whether a nontrivial fibration whose base and fibers admit Kähler metrics of negative holomorphic bisectional curvature must have a total space admitting such a metric. The question is identified with their 2011 remark.
Known results
- Xueyuan Wan, 2026: under additional Griffiths-negativity assumptions on , constructs metrics with negative holomorphic bisectional curvature on ; the paper claims the one-dimensional-fiber case.
September 10, 2026 claimed disproof
A report linked to Xueyuan Wan's preprint Non-strict negativity of holomorphic bisectional curvature for compact relative Kähler fibrations says the proposed inheritance of strict negativity is false. If confirmed, this would settle the problem negatively; the claim currently rests on a preprint and is unverified.
Current status (as of September 2026): The inheritance conjecture is claimed false by a preprint, but the disproof is not independently verified; Wan's earlier positive result under extra hypotheses remains recorded.
Solutions 0
No solutions have been posted yet.