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?
Best Answer
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.