复-全实并集的Stein邻域基
Stein neighborhood bases for complex--totally real unions
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- University of Bucharest, Faculty of Mathematics and Computer Science（布加勒斯特大学数学与计算机科学系）
- Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西米恩·斯托伊洛瓦数学研究所）
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中文总结 AI 辅助
本文研究闭复与全实子流形并集的Stein邻域基，证明特定线性构型具有Stein邻域基，并给出无Stein邻域基的反例。
中文摘要 AI 辅助
我们研究闭复子流形与闭全实子流形并集的Stein邻域基，特别关注具有非紧交集的线性构型。我们证明集合$X=\{w=0\}\cup\{\Im z=\Im w=0\}\subset\C^2$具有Stein邻域基。证明从显式多重次调和定义函数$\rho(z,w)=(\Im w)^2+\sinh^2(\Im z)|w|^2$出发，并通过添加由Oka-Weil逼近构造的非负多重次调和项的局部一致收敛级数，获得所需的无穷远处非一致控制。我们还给出了$\C^2$中一个闭复子流形与一个闭全实子流形具有紧交集但其并集没有Stein邻域基的例子，表明无限制的并集问题具有否定答案。
英文摘要
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 $X=\{w=0\}\cup\{\Im z=\Im w=0\}\subset\C^2$ admits a Stein neighborhood basis. The proof starts from the explicit plurisubharmonic defining function $ρ(z,w)=(\Im w)^2+\sinh^2(\Im z)|w|^2$ 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.
