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Do you know of any explicit smooth complex projective threefold $X$ with an Abelian surface fibration over $\mathbb{P}^1$ such that $K_X = [-F]$ where $F$ is the fiber class divisor.

I am not looking for examples which are fiber product of two elliptic fibrations.

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  • $\begingroup$ Well, there goes my idea then. I was going to suggest a fibration $E \to S$ with $S$ a rational elliptic surface... $\endgroup$
    – Simon Rose
    Sep 24, 2014 at 10:47

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