If ${\cal X}_{mer}$ has maximal degeneration at $q=0$ then there is a limit $(\overline{Y},g_{\overline{Y}})$ of $X_{q}^{new}$ in the Gromov-Hausdorff metric, such that:

a) $(\overline{Y},g_{\overline{Y}})$ is a compact metric space, which contains a smooth oriented Riemannian manifold $(Y,g_{Y})$ of dimension $n$ as a dense open metric subspace. The Hausdorff dimension of $Y^{sing}=\overline{Y}\setminus Y$ is less or equal than $n-2$.

b) $Y$ carries an integral affine structure. This means that it carries a torsion-free flat connection $\nabla$ with the holonomy contained in $SL(n,{{\bf Z}})$.

c) The metric $g_{Y}$ has a potential. This means that it is locally given in affine coordinates by a symmetric matrix $(g_{ij})=(\partial^{2}K/\partial x_{i}\partial x_{j})$, where $K$ is a smooth function (defined modulo adding an affine function, i.e. the sum of a linear function and a constant).

d) In affine coordinates the metric volume element is constant, $det(g_{ij})=det(\partial^{2}K/\partial x_{i}\partial x_{j})=const$ (real Monge-Ampère equation).