complex submanifolds in complex euclidean space

Question

Assume all manifolds are without boundaries. In Euclidean space $\mathbb{R}^n$, there are many submanifolds (Whitney Embedding Theorem).

In complex Euclidean space $\mathbb{C}^n$, are there any typical examples of complex submanifolds ?

My attempt: I only find the example:

(1). for any $m\leq n$, $\mathbb{C}^m$ is a complex submanifold of $\mathbb{C}^n$. And an open subset of $\mathbb{C}^m$ is a complex submanifold of $\mathbb{C}^n$.

(2). I find that any compact manifolds cannot be embedded as complex submanifolds of $\mathbb{C}^n$.

Are there any other nontrivial examples of complex submanifolds in $\mathbb{C}^n$?

Answer 1

You can take any projective manifold and take an open part of it that intersects with an affine chart $\{ (x_0 : \ldots : x_n) \mid x_n \neq 0 \}$ which is isomorphic to $\mathbb C^n$.

Complex manifolds that are closed submanifolds of $\mathbb C^n$ are known as Stein manifolds. Being Stein is the same as being acyclic, i.e. having all higher coherent sheaf cohomologies $H^i(M, \mathcal F)$ vanish, which makes Stein manifolds similar to affine manifolds in the algebraic world.

Answer 2

Consider a holomorphic map $f=(f_1,\dots,f_p):\mathbb C^n\to \mathbb C^p$ and its zero locus $X=\{x\in \mathbb C^n\vert f(x)=0\}$.
If the jacobian matrix $(\frac {\partial f_i}{\partial x_j}(x))$ has rank $p$ at every $x\in X$, then $X\subset \mathbb C^n$ is a complex submanifold.
The simplest example is a hypersurface i.e. the zero locus of a holomorphic function $f:\mathbb C^n\to \mathbb C $ whose gradient does not vanish on $X=f^{-1}(0)$.
These hypersurfaces are exactly the closed submanifolds of dimension $n-1$ of $\mathbb C^n$.

An explicit example
Take $f(x_1,\ldots,x_n)=x_1^k+\cdots+x^k_n-1\:(k\in \mathbb N^*)$ and obtain the Fermat hypersurface $x_1^k+\cdots+x^k_n=1$, a submanifold of dimension $n-1$ of $\mathbb C^n$.