Using the maximum modulus principle you can show that $\mathbb{C}^n$ doesn't have any compact complex submanifolds of positive dimension. It follows that lots of complex manifolds, such as complex grassmannians and projective spaces for example, do not embed into $\mathbb{C}^n$.Complex manifolds that admit embeddings into $\mathbb{C}^n$ are called Stein manifolds.
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Using the maximum modulus principle you can show that $\mathbb{C}^n$ doesn't have any compact complex submanifolds of positive dimension. It follows that lots of complex manifolds, such as complex grassmannians and projective spaces for example, do not embed into $\mathbb{C}^n$. Complex manifolds that admit embeddings into $\mathbb{C}^n$ are called Stein manifolds. |
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Using the maximum modulus principle you can show that a compact complex manifold has no non-constant global holomorphic functions. It follows that $\mathbb{C}^n$ has no doesn't have any compact complex submanifolds of positive dimension. (If $X \hookrightarrow \mathbb{C}^n$ is a compact complex submanifoldIt follows that lots of manifolds, then the holomorphic functions $X \hookrightarrow \mathbb{C}^n \stackrel{\pi_i}{\to} \mathbb{C}$, where the $\pi_i$ are the projectionssuch as complex grassmannians and projective spaces for example, are all constant; hence do not embed into $X$ must be a point.)\mathbb{C}^n$. Complex manifolds that admit embeddings into $\mathbb{C}^n$ are called Stein manifolds.(They will be necessarily non-compact.) |
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