Sphere Bundles Transverse to Holomorphic Vector Fields

Abstract

We prove that for every pair of nonzero complex numbers λ ₁ and λ ₂ with $\frac {\lambda _{1}}{\lambda _{2}}\not \in \mathbb {R}$ there is an embedding $S^{2}\times S^{1}\rightarrow \mathbb {C}^{2}$ transverse to the linear holomorphic vector field $Z(x,y)=\lambda _{1}x\frac {\partial }{\partial x}+\lambda _{2} y\frac {\partial }{\partial y}$. This extends a previous result by Ito (1989).