Sphere Bundles Transverse to Holomorphic Vector Fields

We prove that for every pair of nonzero complex numbers λ1 and λ2 with λ1λ2∉ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac {\lambda _{1}}{\lambda _{2}}\not \in \mathbb {R}$\end{document} there is an embedding S2×S1→ℂ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S^{2}\times S^{1}\rightarrow \mathbb {C}^{2}$\end{document} transverse to the linear holomorphic vector field Z(x,y)=λ1x∂∂x+λ2y∂∂y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Z(x,y)=\lambda _{1}x\frac {\partial }{\partial x}+\lambda _{2} y\frac {\partial }{\partial y}$\end{document}. This extends a previous result by Ito (1989).