View cylinder as the Riemannian surface $\mathbb{C}/ \mathbb{Z}$, how does one embed it into affine space holomorphically?
It is well known by the maximum principle that there is no holomorphic embedding of compact complex into the affine space $\mathbb C^n$. I’am trying to find some noncompact complex manifold that also can not embed into the affine space.
the function $f(z)=e^z$ give the embedding.More presicely, we can assume the cylinder is just a region in complex plane with $Im(z)$ belong to$[0,2\pi)$,since f is periodic and its period is actually $2\pi$ in the imaginary part.it is easy to find that $logz$ give the inverse of f.
A famous theorem claim that every noncompact riemannian surface is stein, which means that every noncompact riemannian surface can be embed into the affine space.So if the manifold exists,it's complex dimension must be bigger than 1.