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Characteristic-free characterizations of projective space for smooth toric varieties
Abstract
We prove that a smooth projective toric variety over an algebraically closed field of arbitrary characteristic is isomorphic to a projective space if its tangent bundle contains an ample locally free subsheaf of positive rank. No torus-equivariant structure on this subsheaf is assumed. The proof uses primitive relations, the toric Euler sequence, and characteristic-free cohomological lifting. As an application of the same method, we establish the toric form of Beauville's cohomological characterization of projective spaces and quadrics, including the polarization. Without the toric hypothesis, both characterizations fail in characteristic two.
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BibTeX
@misc{fujino2026characteristicfree,
title = {{Characteristic-free characterizations of projective space for smooth toric varieties}},
author = {Osamu Fujino and Hiroshi Sato},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03282},
eprint = {2609.03282},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03282},
}
RIS
TY - UNPB
TI - Characteristic-free characterizations of projective space for smooth toric varieties
AU - Osamu Fujino
AU - Hiroshi Sato
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.03282
UR - https://arxiv.org/abs/2609.03282
AB - We prove that a smooth projective toric variety over an algebraically closed field of arbitrary characteristic is isomorphic to a projective space if its tangent bundle contains an ample locally free subsheaf of positive rank. No torus-equivariant structure on this subsheaf is assumed. The proof uses primitive relations, the toric Euler sequence, and characteristic-free cohomological lifting. As an application of the same method, we establish the toric form of Beauville's cohomological characterization of projective spaces and quadrics, including the polarization. Without the toric hypothesis, both characterizations fail in characteristic two.
ER -