Equivalence of two fibrations for logarithmic foliations. (Q1918336)

A famous result of Milnor asserts the equivalence of two descriptions of the so-called Milnor fibration defined by an analytic function in a neighborhood of the origin in \(\mathbb{C}^n\). The author extends this result to foliations of the logarithmic Pfaffian forms \[ \omega= f_1f_2 \cdots f_m \sum^m_{k=1} \lambda_k {df_k\over f_k} \quad \text{and} \quad \Omega = \sum^m_{k=1} \biggl[\text{Im} \lambda_kd \bigl(\text{Log} |f_k|\bigr) + \text{Re} \lambda_k d(\text{Arg} f_k) \biggr], \] where \(f_1,f_2, \dots, f_m\) are germs of holomorphic functions at the origin in \(\mathbb{C}^n\) and \(\lambda_1, \lambda_2, \dots, \lambda_m\) are complex numbers that are not all zero. (As usual Im, Re and Arg denote imaginary and real parts and the argument of functions). In particular, he proves that, for suitable \(\varepsilon,\eta\), there is a \(C^\infty\) diffeomorphism of the tube \(T_1= [|f_1^{\lambda_1} f_2^{\lambda_2} \cdots f_m^{\lambda_m} |= \eta] \cap B^{2n}_\varepsilon\) onto \(T_2= S_\varepsilon^{2n-1} \setminus (S_\varepsilon^{2n-1}\cap \{f_1f_2 \cdots f_m=0\})\) transforming the foliation defined by \(\omega\) on \(T_1\) to the foliation defined by \(\Omega\) on \(T_2\). (Here \(B_\varepsilon\) and \(S_\varepsilon\) denote the ball and sphere of radius \(\varepsilon\) in \(\mathbb{C}^N)\).