Properties of the fibers of Albanese map?

Good afternoon,

I encounter the notion of Albanese map $alb$ from a compact Kahler manifold $X$ to its Albanese torus. I would like to know any properties of the fibers of this map, i.e. the set $alb^{-1}(y)$ with $y$ an element of the Albanese torus.

Any help is appreciated. Thanks in advances,

Duc Anh

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The generic fiber, hence most fibers, have trivial Albanese, which implies a bunch of things. Also +1 Michael (any relation?). –  Felipe Voloch Mar 9 '13 at 17:31
Thank you. But could you please give me some links/references? –  Đức Anh Mar 9 '13 at 17:40
@Felipe Voloch: I wonder about the singularity of the fibers. Are they non-singular varieties or not? Is there an Albanese torus for a singular projective variety? Do you know any information? –  Đức Anh Mar 9 '13 at 17:56
Since you didn't get an answer yet, let me say something here. There is no reason in general to expect the Albanese map to have smooth fibres. In fact here is a result in the opposite direction: it is a theorem (I think by Hacon--Kovacs and Popa--Schnell) that if $X$ is a smooth projective variety of general type, then $X$ does not have any smooth morphism $f:X \rightarrow A$ to an abelian variety of positive dimension. So in particular if $X$ is of general type, has positive irregularity and surjective Albanese morphism, then the Albanese morphism must have some singular fibres. –  Artie Prendergast-Smith Mar 11 '13 at 16:34
Thank you very much. I'll try to read these results. –  Đức Anh Mar 11 '13 at 18:13
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