complex algebraic morphisms as topological maps: every morphism is a topological fibration on a Zariski dense open subset?

2011-02-02T12:54:09Z

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...

Answer by Donu Arapura for complex algebraic morphisms as topological maps: every morphism is a topological fibration on a Zariski dense open subset?

2011-02-02T18:44:49Z

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.]

complex algebraic morphisms as topological maps: every morphism is a topological fibration on a Zariski dense open subset? - MathOverflow

complex algebraic morphisms as topological maps: every morphism is a topological fibration on a Zariski dense open subset?

Answer by Donu Arapura for complex algebraic morphisms as topological maps: every morphism is a topological fibration on a Zariski dense open subset?