complex algebraic morphisms as topological maps: every morphism is a topological fibration on a Zariski dense open subset? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T12:09:19Z http://mathoverflow.net/feeds/question/54092 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54092/complex-algebraic-morphisms-as-topological-maps-every-morphism-is-a-topological complex algebraic morphisms as topological maps: every morphism is a topological fibration on a Zariski dense open subset? mmm 2011-02-02T12:54:09Z 2011-02-03T00:17:42Z <p>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 $\Bbb C$). </p> <p>(i) Is every algebraic morphism of complex algebraic varieties necessarily a fibration in the (non-noetherian) complex topology on a <i>Zariski</i>-open <i>Zariski</i>-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 ?</p> <p>(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.</p> <p>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.)</p> <p>(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 ?</p> <p>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...</p> http://mathoverflow.net/questions/54092/complex-algebraic-morphisms-as-topological-maps-every-morphism-is-a-topological/54123#54123 Answer by Donu Arapura for complex algebraic morphisms as topological maps: every morphism is a topological fibration on a Zariski dense open subset? Donu Arapura 2011-02-02T18:44:49Z 2011-02-03T00:17:42Z <p>I have also wondered about question (i) in the past, and fortunately the answer is yes. Here is a reference:</p> <p>Verdier, Stratification de Whitney et theoreme de Bertini-Sard, Invent 1976, Cor 5.1</p> <p>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).</p> <p>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.]</p>