On Mixed Brieskorn Variety

2009/09/25 by Mutsuo Oka, Oka, Mutsuo · 1 citation
Engineering · Mathematics · #14J17 #58K05 #Algebraic Geometry (math.AG) #FOS: Mathematics #graph theory and CDMA systems #math.AG #msc:14J17 #msc:58K05

paper · pdf · doi:10.48550/arxiv.0909.4605

openalex publication_date 2009/09/25 · arxiv created 2009/09/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f_\bf a,\bf b(\bf z,\bf z)=z₁^a₁+b₁ z₁^b₁+...+z_n^a_n+b_n z_n^b_n be a polar weighted homogeneous mixed polynomial with a_j>0,b_j≥ 0, j=1,..., n and let f_{\bf a}(\bf z)=z₁^a₁+...+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)∩ S^2n-1 and K_{\bf a}=f_{\bf a}\inv(0)∩ S^2n-1. Ruas-Seade-Verjovsky \citeR-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^∞ equivalent and two links are diffeomorphic. We show the same assertion for f(\bf z,\bf z)=z₁^a₁+b₁ z₁^b₁z₂+...+z_n-1^a_n-1+b_n-1 z_n-1^b_n-1z_n+z_n^a_n+b_n z_n^b_n and its associated polynomial g(\bf z)=z₁^a₁z₂+...+ z_n-1^a_n-1z_n+z_n^a_n.

Cited by

• Non-degenerate mixed maps and contact structures

Related

• Bicomplex polar weighted homogeneous polynomials
• Topology of mixed hypersurfaces of cyclic type
• Mixed functions of strongly polar weighted homogeneous face type
• Topology of the Milnor fibrations of polar weighted homogeneous polynomials
• Topology of strongly polar weighted homogeneous links