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Let $X,Y$ be complex manifold, $Y$ Stein. It sounds quite reasonable to formulate the following claim: given $y_0\in Y$, for every $\epsilon>0$ and $M\subset X$ compact, there exists an holomorphic mapping $h\colon X\to Y$ with dense image (that is $\overline{h(X)}=Y$) such that $$ \|h-y_0\|_M<\epsilon $$ (the distance is taken with respect some Riemannian metric on $Y$).

This sounds like a very standard result; can somebody provide a reference? Or suggest some counter example, but I think this fact holds true. Thanks

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    $\begingroup$ Why don't you just take the constant map, equal to $y_0$? $\endgroup$
    – Moishe Kohan
    Commented Oct 12, 2021 at 1:09
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    $\begingroup$ Constant maps between complex analytic spaces are holomorphic. $\endgroup$
    – Jason Starr
    Commented Oct 12, 2021 at 1:10
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    $\begingroup$ Because I was tired and I forgot to add that I need $h$ to have dense image. Apologies. I'm going to edit $\endgroup$
    – Joe
    Commented Oct 12, 2021 at 5:21
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    $\begingroup$ This fails (with the modified question) for example for $X$ compact and $Y=\mathbb{C}$. To have dense image certainly restricts which $X$ and $Y$ for which it is possible. $\endgroup$
    – Richard Lärkäng
    Commented Oct 12, 2021 at 9:34
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    $\begingroup$ Even if both X, Y are Stein this fails. Maybe if X is a complex disk and Y is connected this holds. $\endgroup$
    – Moishe Kohan
    Commented Oct 12, 2021 at 23:40

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