When some papers say"XXX fibration", I see it seems that it is just that the surjective map f: X ---> Y, such that the fiber is XXX,but it is really not "fibration",I didn't see it prove that it is a fibration. So could you tell me if I am wrong? Thanks!
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$\begingroup$ Please add some context, such as a paper where you have seen this, and perhaps an example of what you've seen for XXX. Also please say what algebraic topology you already know (do you know the difference between a Serre fibration and a Hurewicz fibration, for example?). $\endgroup$– Andrew StaceyCommented Jun 14, 2010 at 21:05
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$\begingroup$ In complex analytic or algebraic geometry, "fibration" sometimes simply means "proper, surjective morphism". For instance, a "Lefschetz fibration" is a fibration with non-degenerate critical points and distinct critical values. This usage has nothing to do with homotopical usage. The critical fibres of a Lefschetz fibration have different homotopy types to the regular fibres, so a LF with at least one critical point is not a Hurewicz fibration (nor a Serre fibration, for that matter). $\endgroup$– Tim PerutzCommented Jun 14, 2010 at 21:15
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$\begingroup$ @Tim: probably also should always include flatness to get a handle on decent properties, right? For example, blow-up at a point is presumably not a reasonable notion of "fibration" in any circumstance, whereas any proper surjective map between smooth connected schemes or analytic spaces of pure dimension is automatically flat (and hence falls into this notion of fibration). For example, such "fibrations" have sections over a finite flat cover locally on the base. Lefschetz fibrations are also flat. $\endgroup$– BoyarskyCommented Jun 14, 2010 at 22:00
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$\begingroup$ Boyarksy: probably so. I was careless and didn't specify source and target of the morphism (for a Lefschetz fibration, both are non-singular and the target is a curve). I was thinking of the convention in Barth et al., "Compact complex surfaces", p. 110, which seems to be that a fibration on a non-singular, connected complex surface is a proper, surjective, holomorphic map to a non-singular curve. $\endgroup$– Tim PerutzCommented Jun 14, 2010 at 22:26
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$\begingroup$ Thanks,Tim Perutz. I always assume it should be the definition of fibration in standard algebraic topology (the homotopical fibration as you mentioned) $\endgroup$– HYYYCommented Jun 14, 2010 at 23:00
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Now that the intent of the question has become clear, I'll attempt to take it out of limbo by transferring the content of the comments - my own (TP) and Boyarsky's - into a community wiki answer.
In algebraic or complex analytic geometry, a fibration is a map from a variety to a lower-dimensional variety having some reasonable properties (proper, surjective, flat). I'm not sure if there's a generally accepted, precise definition in this generality, but for instance, a Lefschetz fibration on a connected complex manifold $M$ is a proper, surjective holomorphic map $M\to C$ to a Riemann surface, with non-degenerate critical points and distinct critical values.
This usage is inconsistent with that of algebraic topologists.
A Lefschetz fibration, unless it happens to be a submersion (hence a smooth fibre bundle, by Ehresmann's theorem), is not a fibration in the senses of algebraic topology (Serre or Hurewicz). A singular fibre has the homotopy-type of a regular fibre with a middle-dimensional cell attached. So, the topological Euler characteristics of regular and singular fibres differ by 1.
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3$\begingroup$ How did algebraic geometers come to adopt "fibration" as an article of terminology? At some point someone must have been concerned about re-use of the same term. $\endgroup$ Commented Jun 15, 2010 at 0:47
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2$\begingroup$ Unwanted adjectives get dropped, things become ambiguous... So, long ago they played association football and Rugby football, and then (I suppose) the US version seemed close enough to Rugby football for American football to make sense as a name. But then two adjectives plus one noun get dropped, and we're left with two footballs plus one English public (that is, private) school... And it all made sense at the time. $\endgroup$ Commented Jun 15, 2010 at 2:02
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$\begingroup$ Grothendieck's descent theory allows one to treat faithfully flat quasi-compact (fpqc) maps "as if" they're topological fibrations. Lots of theorems say "the locus of points in the base when property P holds is open" for a proper fppf (faithfully flat, finitely presented) map, and fibral properties often imply relative properties (e.g., smooth morphisms) for fppf maps. In the algebro-geometric study of torsors for group schemes, fppf-local (or etale-local) sections are a good notion of "local triviality". So the word "fibration" for fppf map (especially in the proper case) is appropriate. $\endgroup$– BoyarskyCommented Jun 15, 2010 at 2:12
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$\begingroup$ I'm baffled by the statement: "Now that the intent of the question has become clear"! It's certainly not clear to me. The one addendum by the original questioner seems to imply that the question is about algebraic topology not geometry, so I don't understand either the retagging or the algebraic geometrical answer. $\endgroup$ Commented Jun 15, 2010 at 6:18
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$\begingroup$ Andrew, fair enough - I read HYYY's comment as confirming my interpretation, which was initially just a hunch. I could have misinterpreted the comment as well as the question. I don't propose to bump the question by editing my answer, but HYYY should edit his/her question! $\endgroup$ Commented Jun 15, 2010 at 13:11