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Let $X^{n+m}$ and $B^m$ be two compact Kähler manifolds of respective (complex) dimension $n+m$ and $m$, for $m > 0$. Let $\pi : X \to B$ be a holomorphic submersion with connected Calabi$-$Yau fibers and suppose $c_1(B) < 0$.

The following is well-known: The relative pluricanonical bundle of $\pi$ is $$K_{X/B}^{\ell} = K_X^{\ell} \otimes (\pi^{\ast} K_B^{\ell})^{\ast},$$ where $\ell$ is any positive integer. By the projection formula, we know that $$\pi_{\ast}(K_X^{\ell}) = (\pi_{\ast} (K_{X/B}^{\ell})) \otimes K_B^{\ell},$$ and when restricted to any fibre $X_y = \pi^{-1}(y)$, $$K_{X/B}^{\ell} \vert_{X_y} \cong K_{X_y}^{\ell}.$$

Note that since $X_y$ is Calabi$-$Yau, there is a positive integer $\ell$ such that $K_{X_y}^{\ell}$ is trivial.

Question: Can $\ell$ be taken to be $\ell =1$? By Calabi$-$Yau, at least in the context of Kähler geometry, we always mean $c_1(X_y) = -c_1(K_{X_y}) =0$, but perhaps when considering Calabi$-$Yau fibres, only a sufficiently large power of the canonical divisor is trivial?

If we do indeed know only that a sufficiently large power of the canonical divisor is trivial, can someone provide an example of this situation where $K_{X_y}$ need not be trivial?

References:

Fong, F. T.-H., Zhang, Z., The collapsing rate of the Kähler--Ricci flow with regular infinite time singularity, J. reine angew. Math. 703 (2015), 95$-$113.

Tosatti, V., Weinkove, B., Yang, X., The Kähler--Ricci flow, Ricci-flat metrics and collapsing limits, arXiv: 1408.0161v2 (2017)

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    $\begingroup$ Enriques surfaces have $K^2=0\neq K$. $\endgroup$ – Oven Jan 26 at 22:56
  • $\begingroup$ @Oven Thanks, I missed such a simple example! $\endgroup$ – Kyle Broder Feb 2 at 2:59

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