On Mixed Brieskorn Variety

Mutsuo Oka

paper

On Mixed Brieskorn Variety

arXiv:0909.4605

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 $a_{j} > 0, b_{j} \geq 0$ , $j = 1, ..., n$ and let $f_{a} (z) = z_{1}^{a_{1}} + ... + z_{n}^{a_{n}}$ 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 $f_{a, b}$ and $f_{a}$ are topologically equivalent and the mixed link $K_{a, b}$ is homeomorphic to the complex link $K_{a}$ . We will prove that they are $C^{\infty}$ equivalent and two links are diffeomorphic. We show the same assertion for $f (z, z ˉ) = z_{1}^{a_{1} + b_{1}} z ˉ_{1}^{b_{1}} z_{2} + ... + z_{n - 1}^{a_{n - 1} + b_{n - 1}} z ˉ_{n - 1}^{b_{n - 1}} z_{n} + z_{n}^{a_{n} + b_{n}} z ˉ_{n}^{b_{n}}$ and its associated polynomial $g (z) = z_{1}^{a_{1}} z_{2} + ... + z_{n - 1}^{a_{n - 1}} z_{n} + z_{n}^{a_{n}}$ .