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Uniruledness and the sign of total scalar curvature
Abstract
For every integer n≥3, we construct a smooth projective manifold X of complex dimension n whose canonical bundle is not pseudoeffective, or equivalently, which is uniruled, but every Kähler metric has negative total scalar curvature. In particular, X admits no Kähler metric of positive scalar curvature, while it admits a Riemannian metric of positive scalar curvature. Thus the equivalence between uniruledness and the existence of a Kähler metric with positive scalar curvature holds in complex dimensions one and two, but fails in higher dimensions.
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BibTeX
@misc{sha2026uniruledness,
title = {{Uniruledness and the sign of total scalar curvature}},
author = {Zehao Sha and Jian Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26640},
eprint = {2609.26640},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26640},
}
RIS
TY - UNPB
TI - Uniruledness and the sign of total scalar curvature
AU - Zehao Sha
AU - Jian Wang
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26640
UR - https://arxiv.org/abs/2609.26640
AB - For every integer n≥3, we construct a smooth projective manifold X of complex dimension n whose canonical bundle is not pseudoeffective, or equivalently, which is uniruled, but every Kähler metric has negative total scalar curvature. In particular, X admits no Kähler metric of positive scalar curvature, while it admits a Riemannian metric of positive scalar curvature. Thus the equivalence between uniruledness and the existence of a Kähler metric with positive scalar curvature holds in complex dimensions one and two, but fails in higher dimensions.
ER -