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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, this is even true in the symplectic category by a result of Michael Usher)

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)

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. (This is even true in the symplectic category by a result of Michael Usher)

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, this is even true in the symplectic category by a result of Michael Usher)

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.

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. (This is even true in the symplectic category by a result of Michael Usher)