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This question has not the ambition of being very precise, instead it is more philosophical.

The question is the following: does the Stein factorization of a morphism have some kind of universal property? i.e. all morphism of some type factor through it, like the situation you have in the case of fibered products

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3 Answers 3

There is a kind of "universal property" of the Stein factorization of the following type.

Theorem (Stability of the factorization under deformation). Let $f \colon X \to Y$ be a surjective holomorphic map between complex projective varieties, and let $$X \stackrel{\alpha}{\longrightarrow} Z \stackrel{\beta}{\longrightarrow} Y$$ be the corresponding Stein factorization. Then any deformation $$f' \colon X \longrightarrow Y$$ of $f$ factors through $\beta$.

For a proof, see this paper by Kebekus and Peternell.

There is another kind of "universal property" which holds not only for the Stein factorization, but for any morphism with connected fibres. It is the following

Rigidity Lemma. Let $\alpha \colon X \to Z$ and $f \colon X \to Y$ be proper morphisms of complex projective varieties. Assume that $\alpha_* \mathcal{O}_X =\mathcal{O}_Z$.

If $f$ contracts all fibres of $\alpha$, then it factors through $\alpha$.

For a proof, see [Debarre, Higher-Dimensional Algebraic Geometry, Lemma 1.15 p. 12].

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For a proper morphism of schemes $f:X\to Y$, the Stein factorization may be defined simply as $Z=\mathrm{Spec} f_*\mathcal{O}_X$. This construction makes sense for any quasicompact and quasiseparated morphism (to ensure that $f_*\mathcal{O}_X$ is quasicoherent), and then $Z$ is the "affine hull" of $f$, universal for factorizations $X\to T\to Y$ where $T\to Y$ is affine. See EGA1, (9.1.21) (Springer edition).

This universal property is easy to prove; in the proper case, the property stated by Georges, related to the connected components of fibers, is deeper.

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Anton has called this "relative affinification" here mathoverflow.net/questions/404/… –  Martin Brandenburg Jul 13 '11 at 9:21

Dear Yusky, consider a proper holomorphic map of complex spaces $f:X\to Y$. (Please note that there is absolutely no condition on these complex spaces)

You can consider the set of connected components of the fibres of $f$: call that set $|Y'|$.
Stein factorization tells you that you can endow that set with the structure of a complex space $Y'=(|Y'|,\mathcal O_Y)$ so that there is a canonical factorization $f=\pi\circ f'$, where $\pi:Y'\to Y$ is a finite morphism and $f':Y \to Y'$ is holomorphic, proper, surjective with connected fibers and satisfies $f'_{\ast}\mathcal O_X=\mathcal O_Y'$.
The universal property you are looking for is the following :

Universal property If $g: X\to Z$ is a morphism of complex spaces constant on the connected components of the fibers of $f$, then there is is a unique holomorphic map $g':Y'\to Z$ with $g=g'\circ f'$.

This is due to Stein, Remmert and Cartan, and uses some powerful tool: either Remmert's proper mapping theorem, according to which the proper image of an analytic set is analytic or Grauert's celebrated (even stronger) direct image theorem, according to which the direct image of a coherent sheaf under a proper holomorphic map is coherent.

Bibliography What could be a better reference than a book written by the two giants of twentieth century complex analysis mentioned above? Look at page 219 of

Grauert, H. and Remmert, R.: Coherent analytic sheaves. Grundl. Math. Wiss. 265, Springer- Verlag (1984).

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Dear Georges, this looks like a version of Rigidity Lemma for arbitrary complex spaces, right? Can you suggest any reference? –  Francesco Polizzi Jul 13 '11 at 6:56
    
Dear Francesco, you are right: a reference is in order. I have added one –  Georges Elencwajg Jul 13 '11 at 9:10
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Dear Georges, thank you. It's really a pity that such a awesome book is currently out of print. I have looked several times at amazon.com, abebooks.com and similar websites, but it seems not possible to buy a used copy for less than 200 € :-( –  Francesco Polizzi Jul 13 '11 at 10:36

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