On Mixed Brieskorn Variety

Abstract

Let $f_{{\bf a},\{bf b}}({\bf z},\bar{\bf z})=z_1^{a_1+b_1}\bar z_1^{b_1}+...+z_n^{a_n+b_n}\bar z_n^{b_n}$ be a polar weighted homogeneous mixed polynomial with $a_j>0,b_j\ge 0$, $j=1,..., n$ and let $f_{{\bf a}}({\bf z})=z_1^{a_1}+...+z_n^{a_n}$ be the associated weighted homogeneous polynomial. Consider the corresponding link variety $K_{{\bf a},{\bf b}}=f_{{\bf a},{\bf b}}\inv(0)\cap S^{2n-1}$ and $K_{{\bf a}}=f_{{\bf a}}\inv(0)\cap S^{2n-1}$. Ruas-Seade-Verjovsky \cite{R-S-V} proved that the Milnor fibrations of $f_{{\bf a},{\bf b}}$ and $f_{{\bf a}}$ are topologically equivalent and the mixed link $K_{{\bf a},{\bf b}}$ is homeomorphic to the complex link $K_{{\bf a}}$. We will prove that they are $C^\infty$ equivalent and two links are diffeomorphic. We show the same assertion for $ f({\bf z},\bar{\bf z})=z_1^{a_1+b_1}\bar z_1^{b_1}z_2+...+z_{n-1}^{a_{n-1}+b_{n-1}}\bar z_{n-1}^{b_{n-1}}z_n+z_n^{a_n+b_n}\bar z_n^{b_n}$ and its associated polynomial $ g({\bf z})=z_1^{a_1}z_2+...+ z_{n-1}^{a_{n-1}}z_n+z_n^{a_n}$.