Definition of complex submanifold

Solution 1:

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.