AI 中文总结
研究\(\mathbb{C}^2\)中具退化CR奇点实曲面局部多项式凸性，通过全纯映射得标准形式，重点关注\(k\geq3\)的情况，证明了不同\(t\)值下\(M_t\)的局部多项式凸性及相关解析圆盘情况。
AI 中文摘要
本文研究了\(\mathbb{C}^2\)中在原点具有孤立CR奇点且退化阶数更高的某些光滑实曲面的局部多项式凸性。在曲面可通过从\(\mathbb{C}^2\)到\(\mathbb{C}^2\)的适当全纯映射拉回到有限多个两两横截的完全实曲面的并集的假设下，得到了此类曲面在原点附近的标准形式。重点关注退化阶数\(k\geq3\)的曲面，证明了若\(t>\csc(\frac{\pi}{k})\)，\(M_t\)在原点局部多项式凸；对于\(0<t<\frac{1}{k - 2}\)，\(M_t\)在原点非局部多项式凸，且在\(0<t<\min\{\sin(\frac{\pi}{k}),\frac{1}{k - 2}\}\)时存在一族\((2k - 3)\)参数的附着于\(M_t\)的解析圆盘。
英文摘要
In this paper, we study the local polynomial convexity of certain smooth real surfaces in \(\mathbb{C}^2\) with isolated CR singularity at the origin with higher-order of degeneracy. Under the assumption that the surface can be pulled back to a union of finitely many pairwise transverse totally real surfaces by a proper holomorphic map from $\mathbb{C}^2$ to $\mathbb{C}^2$, we obtain a normal form for such surfaces near the origin as $\{(z,w)\in\mathbb{C}^2: w= \overline{z}^k+o(|z|^{k})\}$ or $M_t := \left\{ (z,w)\in\mathbb{C}^2 : w=(z+t\overline{z})^k+o(|z|^k) \right\}$, for some \(t>0\), where the parameter $t$ is a local biholomorphic invariant. We focus on the surfaces with order of degeneracy $k\geq 3$. We prove that $M_t$ is locally polynomially convex at the origin if $t>cosec\left(\fracπ{k}\right)$. On the other hand, for $0<t<\frac{1}{k-2}$, we will also show that $M_t$ fails to be locally polynomially convex at the origin; and furthermore, a $(2k-3)$-parameter family of analytic discs attached to $M_t$ for $0<t<\min\left\{\sin\left(\fracπ{k}\right),\frac{1}{k-2}\right\}$.
Comments22 pages, comments are welcome
