AI 中文总结
本文证明$\funnyA^1$丛或向量丛映射诱导平凡局部纤维化，据此得到第一类Korus-Russel三维簇的$\funnyunderline{Sing}(X)$为$\funnyA^1$局部，且任意光滑仿射复曲面的$\funnyA^1$连通层具同伦不变性。
AI 中文摘要
本文证明，$\funnyA^1$丛映射或向量丛映射$p: X \to Y$诱导平凡局部纤维化$\funnyunderline{Sing}(X) \to \funnyunderline{Sing}(Y)$。利用该结论，首先证明第一类Korus-Russel三维簇$X$的空间$\funnyunderline{Sing}(X)$是$\funnyA^1$局部的；随后证明任意光滑仿射复曲面$X$的$\funnyA^1$连通层是同伦不变的。
英文摘要
In this article we show that an $\mathbb{A}^1$ bundle map or a vector bundle map $p: X \to Y$ induces trivial local fibration $\underline{Sing}(X) \to \underline{Sing}(Y)$. Using this, we first show that for Korus Russel threefolds of first kind $X$ the space $\underline{Sing}(X)$ is $\mathbb{A}^1$ local. Then we show that for any smooth affine complex surface $X$, the $\mathbb{A}^1$- connected component sheaf is homotopy invariant.
