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For minimal surfaces admitting an elliptic fibration over a smooth curve, there is a famous analysis of possible singular fibers and a canonical bundle formula due to Kodaira.

There are two papers of Kenji Ueno, in which he tries to classify the singular fibers of an abelian surface fibration over a smooth curve. But I don't see any formula for the canonical bundle of such threefolds?

Does any body know of any formula for that?

(All the varieties I am talking about are projective)

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  • $\begingroup$ Koll\'ar wrote a chapter on canonical bundle formulas for fibrations of relative Kodaira dimension 0 in the book "Flips for 3-folds and 4-folds". It's quite technical, but maybe the results simplify in the case you're interested in... $\endgroup$
    – user5117
    Commented Sep 20, 2011 at 17:00
  • $\begingroup$ @Mohammad F.Tehrani: What's the formula for the canonical bundle in the case of elliptic fibrations $\endgroup$
    – jlk
    Commented Sep 21, 2011 at 0:47
  • $\begingroup$ @jlk: It is a pull back of some rational divisor from base. see Griffiths-Harris. $\endgroup$
    – Mohammad Farajzadeh-Tehrani
    Commented Sep 21, 2011 at 13:44

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