Let $A,B$ be complex analytic spaces. Suppose that $[A,B]$ satisfies the h-principle: i.e. every class of continuous function $f:A \to B$ up to homotopy, contains a holomorphic representative. Let $C \subset A$ and $D\subset B$ be submanifolds (not necessarily complex).

Let $[(A,C), (B,D)]$ be the class of continuous functions $f:A \to B$ with $f(C)=D$ up to homotopy (where homotopies are required to preserve the relative parts).

When does exist a h-principle for $[(A,C), (B,D)]$? Or more concretely, when is it true that $f:A \to B$ is homotopic to a holomorphic function $g:A \to B$ where the homotopy $H(x,s):A \times [0,1] \to B$ satifies that $H(C,s)=D$ for all $s \in [0,1]$?

Also, if the question is too broad and depends on the particular cases: I ask for a reference where this type of problem is worked out (at least under some restrictions).


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