Kobayashi’s problem; Fulton’s problem; Hartshorne–Schneider conjecture
For every smooth complex projective variety and every slope-stable ample holomorphic vector bundle , does there exist a smooth strongly pseudoconvex Finsler metric on with semipositive Kobayashi curvature?
References
Primary source
Additional references
- Stable ample vector bundles without semipositive Kobayashi curvature — arXiv — Kefeng Liu, Xueyuan Wan
Progress summary
An unrefereed preprint claims explicit counterexamples that refute several long-standing geometric conjectures, but the claim has not been independently checked.
The entry concerns Kobayashi’s problem, Fulton’s problem, and the Hartshorne–Schneider conjecture. A September 2026 preprint claims that one construction gives negative answers to all three questions.
September 2026 counterexamples
Kefeng Liu and Xueyuan Wan claim to construct stable ample vector bundles without semipositive Kobayashi curvature on every Hirzebruch surface, including one over a simply connected Fano base. If correct, the construction supplies counterexamples to the associated positivity and convexity conjectures; the preprint remains unrefereed and unverified.
Current status (as of September 2026): The preprint claims to settle the three conjectures negatively, but its counterexamples have not been independently verified.
Sources
Solutions 0
No solutions have been posted yet.