论文标题
奇数闭合歧管的多项式凸的嵌入
Polynomially convex embeddings of odd-dimensional closed manifolds
论文作者
论文摘要
结果表明，尺寸$ 2K + 1 $，$ k \ geq 2 $的任何平滑封闭的定向歧管都承认将嵌入到$ \ mathbb c^{3k} $中的平滑多项式凸。在此类嵌入的可能的环境复合尺寸上，这将提高$ 1 $（$ k = 1 $时，这是锋利的）。进一步表明，产生的嵌入具有该特性，即图像上的所有连续函数都可以通过全体形态多项式统一近似。最后，对相同的技术进行了修改，以构造其图像具有不容易分析磁盘的非平凡船体的嵌入。这种维度设置的区别特征是非分离的CR斑点的出现，该cr含量不能仅使用局部分析方法来解决（如此类早期结果所做的那样），并且需要一种拓扑方法。
It is shown that any smooth closed orientable manifold of dimension $2k + 1$, $k \geq 2$, admits a smooth polynomially convex embedding into $\mathbb C^{3k}$. This improves by $1$ the previously known lower bound of $3k+1$ on the possible ambient complex dimension for such embeddings (which is sharp when $k=1$). It is further shown that the embeddings produced have the property that all continuous functions on the image can be uniformly approximated by holomorphic polynomials. Lastly, the same technique is modified to construct embeddings whose images have nontrivial hulls containing no nontrivial analytic disks. The distinguishing feature of this dimensional setting is the appearance of nonisolated CR-singularities, which cannot be tackled using only local analytic methods (as done in earlier results of this kind), and a topological approach is required.
