Given a family of compact complex manifold $\mathcal{X} \rightarrow B$, what are the standard techniques to prove two distinct fibers $\mathcal{X}_a$ and $\mathcal{X}_b$ are not birational?
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2$\begingroup$ Are your manifolds Moishezon? If not, what do you mean by "birational"? Did you mean to write "bimeromorphic"? For Kaehler manifolds, the weight 1 Hodge structures are birational invariants. Thus, if you know that the classifying map from $B$ to the period domains of weight 1 Hodge structures is injective, then distinct fibers are not bimeromorphic. $\endgroup$– Jason StarrMar 18, 2015 at 2:56
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$\begingroup$ You can try to ensure the fibers are canonical representatives of their birational equivalence classes, for instance by showing the canonical bundle is ample, and then show that two fibers are not isomorphic. $\endgroup$– Will SawinMar 18, 2015 at 3:01
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