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A Levi-form criterion for Oka complements
Abstract
We prove that if M is a connected Stein manifold of complex dimension at least three with the density property, then M∖ K is a connected Oka manifold for every nonempty compact KSubset M with smooth boundary whose Levi form has at least two nonnegative eigenvalues at every boundary point for local defining functions negative on intK. For compact sets in ℂn with smooth boundary, this settles both implications between pseudoconvexity of ∂ K and the Oka property of ℂn∖ K: pseudoconvexity is sufficient but not necessary for n≥3, and necessary but not sufficient for n=2. In dimension two, we give an explicit compact set in ℂ2 with strongly pseudoconvex boundary and non-Oka complement.
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BibTeX
@misc{du2026leviform,
title = {{A Levi-form criterion for Oka complements}},
author = {Yun-Heng Du},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34426},
eprint = {2610.34426},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34426},
}
RIS
TY - UNPB
TI - A Levi-form criterion for Oka complements
AU - Yun-Heng Du
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34426
UR - https://arxiv.org/abs/2610.34426
AB - We prove that if M is a connected Stein manifold of complex dimension at least three with the density property, then M∖ K is a connected Oka manifold for every nonempty compact KSubset M with smooth boundary whose Levi form has at least two nonnegative eigenvalues at every boundary point for local defining functions negative on intK. For compact sets in ℂ^n with smooth boundary, this settles both implications between pseudoconvexity of ∂ K and the Oka property of ℂ^n∖ K: pseudoconvexity is sufficient but not necessary for n≥3, and necessary but not sufficient for n=2. In dimension two, we give an explicit compact set in ℂ^2 with strongly pseudoconvex boundary and non-Oka complement.
ER -