PreprintCC BY 4.0arXiv
Nef Hattori classes and products of projective lines
Abstract
Let X be a smooth connected complex projective variety with an effective algebraic ℂ×-action, a nonempty finite fixed locus whose weights are ±1. We show that if its Hattori line bundles are nef, then X is algebraically isomorphic to (ℙ1)dim X, and these bundles become the hyperplane bundles of the factors. The product statement follows from the nef-complexity classification of Enwright and collaborators, and we describe the isomorphism through global sections. To test the required nefness we specialise the orbit-graph criterion of Romano and Wiśniewski. The Hirzebruch surface 𝔽2 is an example why the topological hypotheses alone do not imply either the product conclusion or the Fano property.
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BibTeX
@misc{alvarez2026hattori,
title = {{Nef Hattori classes and products of projective lines}},
author = {Daniel Álvarez and Andrew Clarke and Eduardo González},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12683},
eprint = {2610.12683},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12683},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nef Hattori classes and products of projective lines
AU - Daniel Álvarez
AU - Andrew Clarke
AU - Eduardo González
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12683
UR - https://arxiv.org/abs/2610.12683
AB - Let X be a smooth connected complex projective variety with an effective algebraic ℂ^×-action, a nonempty finite fixed locus whose weights are ±1. We show that if its Hattori line bundles are nef, then X is algebraically isomorphic to (ℙ^1)^dim X, and these bundles become the hyperplane bundles of the factors. The product statement follows from the nef-complexity classification of Enwright and collaborators, and we describe the isomorphism through global sections. To test the required nefness we specialise the orbit-graph criterion of Romano and Wiśniewski. The Hirzebruch surface 𝔽_2 is an example why the topological hypotheses alone do not imply either the product conclusion or the Fano property.
ER -