In the case of collapsing K3 surfaces the corresponding intergal Monge-Ampère manifold has an explicit description.

Let $S$ be a complex surface endowed with a holomorphic non-vanishing volume form $\Omega_{S}$, and $\pi:S\to C$ be a holomorphic fibration over a complex curve $C$, such that fibers of $\pi$ are non-singular elliptic curves.

We define a metric $g_{C}$ on $C$ as the Kähler metric associated with the $(1,1)$-form $\pi_{\ast}(\Omega_{S}\wedge\overline{\Omega}_{S})$. Let us choose (locally on $C$) a basis $(\gamma_{1},\gamma_{2})$ in $H_{1}(\pi^{-1}(x),{{\bf Z}}),x\in C$. We define two closed 1-forms on $C$ by the formulas

$$\alpha_{i}=Re\left(\int_{\gamma_{i}}\Omega_{S}\right),\,\,\,i=1,2\,\,.$$

It follows that $\alpha_{i}=dx_{i}$ for some functions $x_{i},i=1,2$. We define a ${\bf Z}$-affine structure on $C$, and the corresponding connection $\nabla$, by saying that $(x_{1},x_{2})$ are ${\bf Z}$-affine coordinates (compare with 3.2.4). One can check directly that $(C,g_{C},\nabla)$ is a Monge-Ampère manifold. In a typical example of elliptic fibration of a K3 surface, one gets $C={{\bf C}P}^{1}\setminus\{x_{1},...,x_{24}\}$, where $\{x_{1},...,x_{24}\}$ is a set of distinct $24$ points in ${{\bf C}P}^{1}$. M. Gross and P. Wilson (see [GW]) proved that there exists a family of K3 surfaces with Calabi-Yau metrics collapsing to $S^{2}\simeq{\bf C}P^{1}$ with the intergal Monge-Ampère structure described above.