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Stable ample vector bundles without semipositive Kobayashi curvature
Abstract
We construct slope-stable ample rank-two vector bundles on every Hirzebruch surface 𝔽e that admit no smooth strongly pseudoconvex Finsler metric with semipositive Kobayashi curvature. These examples give a negative answer to Kobayashi's problem. The same bundles give counterexamples to the smooth strongly pseudoconvex Finsler formulation of Fulton's problem and to the Hartshorne--Schneider conjecture on the convexity of complements. For e=1, these counterexamples have a simply connected Fano base that is not a product.
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BibTeX
@misc{liu2026stable,
title = {{Stable ample vector bundles without semipositive Kobayashi curvature}},
author = {Kefeng Liu and Xueyuan Wan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37309},
eprint = {2609.37309},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37309},
}
RIS
TY - UNPB
TI - Stable ample vector bundles without semipositive Kobayashi curvature
AU - Kefeng Liu
AU - Xueyuan Wan
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37309
UR - https://arxiv.org/abs/2609.37309
AB - We construct slope-stable ample rank-two vector bundles on every Hirzebruch surface 𝔽_e that admit no smooth strongly pseudoconvex Finsler metric with semipositive Kobayashi curvature. These examples give a negative answer to Kobayashi's problem. The same bundles give counterexamples to the smooth strongly pseudoconvex Finsler formulation of Fulton's problem and to the Hartshorne--Schneider conjecture on the convexity of complements. For e=1, these counterexamples have a simply connected Fano base that is not a product.
ER -