Is the Riemann Sphere a submanifold of $\mathbb{C}^2$? This feels like a stupid question, but I am missing some imagination here.
Anyway, I know that the Riemann sphere $\mathbb{C}\cup\{\infty\}$ is a one-dimensional complex manifold and that I can also view it as a 2-sphere embedded as a real submanifold into $\mathbb{R}^3$.
However, is the Riemann sphere also an (immersed or embedded?) one-dimensional complex submanifold of $\mathbb{C}^2$?
Somehow, since $\mathbb{C}^2$ has one (real) dimension more than $\mathbb{R}^3$, I feel it should work. Does it?
What about a (one-dimensional complex) Torus?

There is no compact complex analytic submanifolds $M$ in $\mathbb C^n$ (except for points), since the coordinate functions $(z_1, \cdots z_n)$ restrict to complex analytic functions on it, which has to be constant as $M$ is compact.
