The Calabi conjecture for bi-polyhedral Kähler affine structures

Let $\Sigma$ be the underlying sphere or torus equipped with a discriminant locus $D$, and let $(\lambda,\nu)$ specify a class of bi-polyhedral Kähler affine structures on $\Sigma\backslash D$. A metric is Monge–Ampère when, in affine coordinates, its determinant satisfies a constant-volume condition.

The Calabi conjecture. There is a unique bi-polyhedral Kähler affine structure on $\Sigma\backslash D$ of type $(\lambda,\nu)$ such that the metric is Monge–Ampère:

$$\det g_{ij}=\bigl(\operatorname{Vol}\partial\Delta^\vee_\lambda\bigr)^{-1}\cdot\operatorname{Vol}\partial\Delta_\nu.$$

Its metric completion to $\Sigma$ is expected to be the limit of the Ricci-flat metrics on families of Calabi–Yau hypersurfaces. The conjecture asserts both uniqueness and the normalization of the Monge–Ampère metric; the supplied text does not state a resolution.