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Let $\pi \colon Z\to \Delta$ be a semistable degeneration with a simple normal crossings central fiber $\pi^{-1}(0)$. Here $Z$ is a smooth complex projective variety (or Kähler manifold), $\Delta$ is a small disk around origin in $\mathbb{C}$, generic fibers $X$ of $\pi$ are smooth, and the central fiber is a transverse union of smooth varieties $X_i$ along snc divisors $D_i\subset X_i$

Question: If fibers of $\pi$ are 2-dimensional (or more generally in higher dimensions) and $Z$ is minimal (or similar to exclude blowup components etc.), what is known about the relation between Kodaira dimension of smooth fibers $X$ and the logarithmic Kodaira dimensions of $(X_i,D_i)$?

For example (except for some trivial situations), if $X$ has Kodaira dimension $0$, then $(X_i,D_i)$ have logarithmic Kodaira dimension zero as well. (For basic degenerations with positive genus singular locus, this is even true in the symplectic category by a result of Michael Usher)

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    $\begingroup$ Greetings Mohammad! I think you need to add some minimality / ampleness condition. If you take an arbitrary $Z$ over $\Delta$, if you blow up a smooth point of the central fiber, then you produce a new irreducible component -- the exceptional divisor. With your definition, that pair is $(\mathbb{P}^n,\mathbb{P}^{n-1})$, and this does not have log Kodaira dimension $0$. $\endgroup$
    – Jason Starr
    Commented Dec 10, 2018 at 18:26
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    $\begingroup$ Definitely, I forgot to write that, thats what I meant by trivial situations in that example. Nice to hear from you Jason. $\endgroup$
    – Mohammad Farajzadeh-Tehrani
    Commented Dec 10, 2018 at 18:28

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