which $f$ is non-smooth, and assume that the monodromy at each point of $S$ is non-trivial. Jost and Zuo \[JZ\] proved the Arakelov–Yau type inequality: $$\mbox{deg} f_* \omega_{X/{{\mathbb P}}^1}\leq \mbox{deg}\,
\Omega_{{{\mathbb P}}^1}^1 (\mbox{log} S).$$

Let $\Delta\subset X$ be the pull-back of $S$. Let $\omega_{X/{{{\mathbb P}}}^1}$ be the canonical sheaf. The Kodaira–Spencer maps $\theta^{2,0}$ and $\theta^{1,1}$ are maps of sheaves fitting into the following diagram: $$f_*\Omega^2_{X/{{{\mathbb P}}}^1}(\mbox{log}\,\Delta)
\overset{\theta^{2,0}}\to
R^1f_*\Omega^1_{X/{{{\mathbb P}}}^1}(\mbox{log}\,\Delta)
\otimes \Omega^1_{{{{\mathbb P}}}^1}(\mbox{log}\,S)
\overset{\theta^{1,1}}\to R^2\,
f_*{{\mathcal O}}_{X/{{{\mathbb P}}}^1}\otimes\Omega^1_{{{{\mathbb P}}}^1}
(\mbox{log}\,S)^{{\otimes}2}.$$ The iterated Kodaira–Spencer map of $f$ is defined to be the map $\theta^{1,1}\theta^{2,0}$.

It is known (see \[STZ\]) that when the (iterated) Kodaira–Spencer map is $0$, one actually has the stronger inequality $$\mbox{deg} f_* \omega_{X/{{\mathbb P}}^1}\leq
\frac{1}{2}\mbox{deg}\,
\Omega_{{{\mathbb P}}^1}^1 (\mbox{log} S).$$

Assume from now on that $X$ is a Calabi–Yau threefold. Then the triviality of the canonical bundle implies that $\mbox{deg}\, f_*
\omega_{X/{{\mathbb P}}^1}=2$ (see (\[deg\]) below).

Recently X. Sun, S-L. Tan and K. Zuo \[STZ\] considered Calabi–Yau threefolds for which the Arakelov–Yau inequality becomes equality. Thus $S$ consists of $6$ points when the Kodaira–Spencer map is $0$ and of $4$ points otherwise.

As a consequence of the main theorem of \[STZ\], the following results were obtained.

[**Theorem 1**]{} (Corollary 0.4 