Stein neighborhood bases for complex--totally real unions
Abstract
We study Stein neighborhood bases for unions of closed complex and totally real submanifolds, with particular attention to a linear configuration with noncompact intersection. We prove that the set X={w=0}∪{ℑ z=ℑ w=0}⊂C2 admits a Stein neighborhood basis. The proof starts from the explicit plurisubharmonic defining function ρ(z,w)=(ℑ w)2+sinh2(ℑ z)|w|2 and obtains the necessary non-uniform control at infinity by adding a locally uniformly convergent series of nonnegative plurisubharmonic terms constructed by Oka--Weil approximation. We also give an example of a closed complex submanifold and a closed totally real submanifold in C2 with compact intersection whose union has no Stein neighborhood basis, showing that the unrestricted union problem has a negative answer.
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BibTeX
@misc{preda2026stein,
title = {{Stein neighborhood bases for complex--totally real unions}},
author = {Ovidiu Preda},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10191},
eprint = {2609.10191},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10191},
}
RIS
TY - UNPB
TI - Stein neighborhood bases for complex--totally real unions
AU - Ovidiu Preda
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10191
UR - https://arxiv.org/abs/2609.10191
AB - We study Stein neighborhood bases for unions of closed complex and totally real submanifolds, with particular attention to a linear configuration with noncompact intersection. We prove that the set X={w=0}∪{ℑ z=ℑ w=0}⊂C^2 admits a Stein neighborhood basis. The proof starts from the explicit plurisubharmonic defining function ρ(z,w)=(ℑ w)^2+sinh^2(ℑ z)|w|^2 and obtains the necessary non-uniform control at infinity by adding a locally uniformly convergent series of nonnegative plurisubharmonic terms constructed by Oka--Weil approximation. We also give an example of a closed complex submanifold and a closed totally real submanifold in C^2 with compact intersection whose union has no Stein neighborhood basis, showing that the unrestricted union problem has a negative answer.
ER -