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Elliptic complements of cubic hypersurfaces
Abstract
Let D⊂ℙn, ngeqslant2, be an arbitrary cubic hypersurface, and let Dred denote its reduced support. We prove that ℙn∖ D is holomorphically elliptic, and hence Oka, unless Dred is the union of three distinct hyperplanes containing a common codimension-two linear subspace. In the exceptional case, ℙn∖ D≅(ℂ∖{0,1})×ℂn−1, so the complement is not Oka. As applications, we prove that, for every elliptic curve E, the space of degree-three holomorphic maps E→ℙ1, and the space of degree-three holomorphic self-maps of ℙ1, are both holomorphically elliptic, and hence Oka. The second application is connected with the classification through an irreducible cubic hypersurface in ℙ4.
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BibTeX
@misc{xie2026elliptic,
title = {{Elliptic complements of cubic hypersurfaces}},
author = {Song-Yan Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.02413},
eprint = {2608.02413},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02413},
}
RIS
TY - UNPB
TI - Elliptic complements of cubic hypersurfaces
AU - Song-Yan Xie
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2608.02413
UR - https://arxiv.org/abs/2608.02413
AB - Let D⊂ℙ^n, ngeqslant2, be an arbitrary cubic hypersurface, and let D_red denote its reduced support. We prove that ℙ^n∖ D is holomorphically elliptic, and hence Oka, unless D_red is the union of three distinct hyperplanes containing a common codimension-two linear subspace. In the exceptional case, ℙ^n∖ D≅(ℂ∖{0,1})×ℂ^n−1, so the complement is not Oka. As applications, we prove that, for every elliptic curve E, the space of degree-three holomorphic maps E→ℙ^1, and the space of degree-three holomorphic self-maps of ℙ^1, are both holomorphically elliptic, and hence Oka. The second application is connected with the classification through an irreducible cubic hypersurface in ℙ^4.
ER -