In writing up a paper, we need references (and help) for the following facts which are probably well-known. They concern morphisms of complex algebraic varieties as continious maps in complex topology. (Here by 'complex topology' I mean the topology induced by the metric on $\Bbb C$).

(i) Is every algebraic morphism of complex algebraic varieties necessarily a fibration in the (non-noetherian) complex topology on a Zariski-open Zariski-dense subset ? That is, does there exists a Zariski open subset $Y^0$ of $Y$ such that $f_{\Bbb C}: f_{\Bbb C}^{-1}(Y^0(\Bbb C)) \longrightarrow Y^0(\Bbb C)$ is a fibration in the (non-noetherian) complex topology ?

(ii) If $f:W\rightarrow Y$ is a dominant rational map of irreducible complex varieties, with Y normal, then the index of the image of $\pi_1(W) \rightarrow \pi_1(Y)$ divides the number of irreducible components of a generic fibre.

Does something like this holds in prime characteristic ? (In char 0 this appears in [Janos Kollar, "Shaferevich maps and automorphic forms", Lemma 2.10.2]; is there a more standard reference as I find it hard to follow the proof there, not being an algebraic geometer.)

(iii) Stein factorisation. Every proper morphism $f:X\rightarrow Y$ of algebraic varieties decomposes as $X\rightarrow^{f_1} X' \rightarrow^{f_2} Y$ such that $f_1$ has connected fibres and is a fibration in complex topology over a Zariski open dense subset, and $f_2$ is finite and etale on an Zariski open dense subset ?

The last question only makes sense if (i) is not always true; without the bit about complex topology Stein factorisation appears in EGA. These questions came up when trying to define a noetherian "Zariski-type" topology on the universal covering space of a complex algebraic variety that is weaker than the complex analytic topology, sort of a model of etale topology...

  • $\begingroup$ For (i) does "generic smoothness" not suffice? This is in Hartshorne, the chapter on smooth morphisms. $\endgroup$ Feb 2, 2011 at 22:43
  • $\begingroup$ If the initial space is smooth, then yes, certainly. But it didn't seem that mmm was assuming this. $\endgroup$ Feb 3, 2011 at 0:21
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    $\begingroup$ The result in Hartshorne (Lemma 10.5) only assumes that you have a dominant morphism of integral schemes of finite type over an algebraically closed field of charactertistic zero. Indeed, in this case there is a open dense subset where your schemes are non-singular varieties. If the morphism is also proper then we can use Ehresmann's theorem to deduce that it is a fibration, but I guess you need more in the non-proper case. $\endgroup$ Feb 3, 2011 at 11:17

1 Answer 1


I have also wondered about question (i) in the past, and fortunately the answer is yes. Here is a reference:

Verdier, Stratification de Whitney et theoreme de Bertini-Sard, Invent 1976, Cor 5.1

The result is probably also contained in Thom, Bull AMS 75 (1969), but it may be harder to extract (at least it was for me).

Wouldn't (iii) follow from (i) + Stein factorization, or is there something that I'm missing? [In rereading your question, I realized you posed this only in the event that (i) failed.]

  • $\begingroup$ Thank you, we shall look this up! Could you also suggest another reference for (ii), and whether anything is known in prime characteristic ? It would seem that sucha basic result should be kind of everywhere... $\endgroup$
    – mmm
    Feb 4, 2011 at 17:36

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