I have a question on a remark from Daniel Hyubrechts' Complex Geometry Complex Geometry on page 107.
Definition 2.6.13 A holomorphic map $f: X \to Y$ is smooth at a point $x \in X$ if the induced map $\mathcal{T}_X(x) \to (f^*\mathcal{T}_Y)(x) = \mathcal{T}_Y(f(x))$ is surjective.
As an immediate consequence of Corollary 1.1.12 one finds:
Corollary 2.6.14 Let $f : X \to Y$ be a holomorphic map and $y \in Y$. Assume that $f$ is smooth in all points of the fibre $f^{-1}(y)$. Then the fibre $f^{-1}(y)$ is a smooth complex submanifold of $X$.
Corollary 1.1.12 tells me that:
Corollary 1.1.12 Let $U \subset \mathbb{C}^m$ be an open subset and let $f: U \to \mathbb{C}^n$ be a holomorphic map. Assume that $z_0 \in U$ such that $rank(J(f)(z_0))$ is maximal.
i) If $m>n$ then there exists a biholomorphic map $h : V \to U'$, where $U'$ is an open subset of $U$ containing $z_0$, such that $f(h(z_1,\dots, z_m)) = ( z_1 , \dots , z_n)$ for all $(z_1, \dots, z_m) \in V$.
ii) If $m < n$ then there exists a biholomorphic map $g : V \to V$, where $V$ is an open subset of $\mathbb{C}^n$ containing $f(z_o)$, such that $g(f(z)) = ( z_1 , \dots , z_m, 0 , \dots , 0 )$.
Q: I do not understand why Corollary 2.6.14 is a consequence of corollary 1.1.12 and how it is applied to our case. Essentially, 1.1.12 is nothing but the inverse functions theorem giving criterion on local invertibility in an open neighborhood. As $f$ smooth, we are in case i): $m >n$ as by assumption the differential map is surjective. The fibre $f^{-1}(z_0)$ is in general a closed set. I don't understand how the existence of the biholomorphic map as described above imply that the fibre is a submanifold of $X$. I would be very thankful if somebody could take some time to explain the argument presented in the book.
This terminology is quite nonstandard, such maps are usually called (holomorphic) submersions. Corollary 1.1.12 is stronger than the Inverse Mapping Theorem, it is usually called the "Constant Rank Theorem". Once you assume it, then Corollary 2.6.14 is immediate since (by a local biholomorphic change of variables in the domain and the range) the problem reduces to the case when $f: {\mathbb C}^m\to {\mathbb C}^n$ is a surjective linear map $(z_1,...,z_m)\mapsto (z_1,...,z_n)$. The preimage of ${\mathbf 0}$ under such a map is a linear subspace of dimension $m-n$, i.e. a smooth complex submanifold of the expected dimension.