Is an algebraic geometer's fibration also an algebraic topologist's fibration? 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!
 A: 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.
