Complete intersections in complex manifolds

Question

Let $X$ be a complex manifold of dimension $n$ and $Y\subset X$ a closed submanifold of codimension $k$.
a) Say that $Y$ is a complete intersection if the ideal $I(Y)\subset \mathcal O(X)$ of global holomorphic functions vanishing on $Y$ can be generated by $k$ functions $f_1,\cdots,f_k\in \mathcal O(X).$
b) Under what circumstances is this equivalent to the existence of a holomorphic mapping $F:X\to \mathbb C^k$ satisfying $Y=F^{-1}(0)$ and such that $F$ is a submersion at every $y\in Y$?
This seems quite plausible if $X$ is Stein.
Moreover the conditions on $F$ ensure that the normal bundle to $Y$ in $X$ is trivial.
Does this triviality play a role in the potential equivalence of a) and b) ?