I was wondering about a generalization of the following property of surfaces of general type.

Let $X$ be a smooth projective surface of general type. Then there is no pencil of rational or elliptic curves on $X$, i.e., there are no pencils of varieties with non-maximal Kodaira dimension on $X$. (edit: see below what i mean by pencil)

Something like this will probably already fail in dimension three, but let me still be brave and ask.

Let $X$ be a smooth projective variety of general type satisfying some minimality condition, say it has ample canonical bundle. Then is it true that $X$ does not admit any pencil of varieties with non-maximal Kodaira dimension?

Slightly weaker, if $n$ is the dimension of $X$, is it true that $X$ does not admit any pencil of $(n-1)$-dimensional varieties with non-maximal Kodaira dimension?

If this fails, what extra conditions would make this work. Does it help if we assume the locally free sheaf of differentials to be ample as well?

The minimality condition is certainly necessary. If not you could just blow-up and your variety would contain a projective space. That would not be good.

Edit: When I say pencil on a variety, i mean the data of a variety $B$ and a non-isotrivial family of varieties $X\to B$. A pencil of abelian varieties on a variety $X$ is thus a base variety $B$ admitting a flat projective non-isotrivial family $X\to B$ of abelian varieties. If the base is a curve, this is a good definition. In general, this might not be a good definition. Note that the weaker version of this question is precisely about the case where the base variety is a curve.

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