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For smooth manifolds, we can define an embedded submanifold to be either (1) a subset locally cut out by "slice" charts, or (2) a subset that is a manifold in the subspace topology and admits a smooth structure such that the inclusion is smooth. These are equivalent (modulo details).

What about the complex case? By analogy, we have:

Definition 1. A subset $N$ of a complex manifold $M^m$ is called a complex submanifold of dimension $r$ if each point in $N$ has a neighborhood $U \subset M$ and a holomorphic chart $\phi: U \to \mathbb{C}^m$ such that $\phi(N \cap U)=\phi(U) \cap \mathbb{C}^r$.

$$\text{vs.}$$

Definition 2. A subset $N$ of a complex manifold $M$ is called a complex submanifold if it is a smooth submanifold of $M$ admitting a complex structure such that the inclusion is holomorphic.

Are these equivalent definitions of an embedded complex submanifold?

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  • $\begingroup$ I should note that Definition 2 encompasses Definition 1 (essentially because of the equivalence of the analogous smooth definitions). But I'm not sure about the other direction, since I don't know how to get a holomorphic "slice" chart from a smooth one. $\endgroup$ – Kyle Aug 21 '15 at 19:15
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If your interpretation of "smooth submanifold" is smooth embedded submanifold (meaning that it has the subspace topology), then these two definitions are equivalent. The proof of the existence of a holomorphic slice works just like the smooth case, but using the holomorphic version of the inverse function theorem instead of the smooth one. Here's a sketch of the proof.

Suppose $N\subseteq M$ is a smooth embedded submanifold with a complex structure, and let $\iota\colon N\to M$ denote the inclusion map. Given $p\in N$, we can choose holomorphic coordinates on a neighborhood $V$ of $p$ in $N$, and holomorphic coordinates on a neighborhood $W$ of $\iota(p)$ in $M$. Since $N$ has the subspace topology, after shrinking $V$ and $W$ if necessary we can assume that $V=W\cap \iota(N)$. Since the question is local, we might as well replace $M$ and $N$ by $V$ and $W$, identified with open subsets of $\mathbb C^r$ and $\mathbb C^n$, respectively. Then $d\iota_p$ is an injective complex-linear map from $\mathbb C^r$ to $\mathbb C^n$. After a complex-linear change of coordinates, we may assume that the image of $d\iota_p$ is the span of the first $r$ standard coordinate vectors. Let $Y$ be the span of the complementary $n-r$ coordinate vectors, and define $\Phi\colon \mathbb C^n = \mathbb C^r\times Y \to \mathbb C^n$ by $\Phi(x,y) =\iota(x)+y$. The holomorphic inverse function theorem shows that $\Phi$ is a biholomorphism onto some neighborhood $U$ of $(p,0)$, and $\Phi^{-1}$ is the required local holomorphic chart.

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  • $\begingroup$ Perfect, this really clears things up! $\endgroup$ – Kyle Aug 21 '15 at 20:31

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