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!



Is an algebraic geometer's fibration also an algebraic topologist's fibration?
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 *in*consistent 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.


