Topology of mixed hypersurfaces of cyclic type

Abstract

Let $f_{II}({\boldsymbol{z}}, \bar{{\boldsymbol{z}}}) = z_{1}^{a_{1}+b_{1}}\bar{z}_{1}^{b_{1}}z_{2} + \cdots + z_{n-1}^{a_{n-1}+b_{n-1}}\bar{z}_{n-1}^{b_{n-1}}z_{n} + z_{n}^{a_{n}+b_{n}}\bar{z}_{n}^{b_{n}}z_{1}$ be a mixed weighted homogeneous polynomial of cyclic type and $g_{II}({\boldsymbol{z}}) = z_{1}^{a_{1}}z_{2} + \cdots + z_{n-1}^{a_{n-1}}z_{n} + z_{n}^{a_{n}}z_{1}$ be the associated weighted homogeneous polynomial where $a_{j} \geq 1$ and $b_{j} \geq 0$ for $j = 1, \dots, n$. We show that two links $S^{2n-1}_{\varepsilon} \cap f_{II}^{-1}(0)$ and $S^{2n-1}_{\varepsilon} \cap g_{II}^{-1}(0)$ are diffeomorphic and their Milnor fibrations are isomorphic.