# Trivial canonical divisor of the Calabi--Yau fibres of a holomorphic submersion between compact Kähler manifolds with connected fibres

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)

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