On Mixed Brieskorn variety

Let fa,b(z, z̄) = z a1+b1 1 z̄ b1 1 + · · ·+z an+bn n z̄ bn n be a polar weighted homogeneous polynomial with aj > 0, bj ≥ 0, j = 1, . . . , n and let fa(z) = z1 1 + · · ·+z an n be the associated weighted homogeneous polynomial. Consider the corresponding link variety Ka,b = f −1 a,b (0) ∩ S2n−1 and Ka = f−1 a (0) ∩ S2n−1. Ruas-Seade-Verjovsky [4] proved that the Milnor fibrations of fa,b and fa are topologically equivalent and the mixed link Ka,b is homeomorphic to the complex link Ka. We will prove that they are C∞ equivalent and two links are diffeomorphic. We show the same assertion for f(z, z̄) = z11 1 z̄ b1 1 z2 + · · ·+ zn−1n−1 n−1 z̄ bn−1 n−1 zn + z an+bn n z̄ bn n and its associated polynomial g(z) = z1 1 z2 + · · ·+ z an−1 n−1 zn + z an n .