If ${X}_{mer}$ has maximal degeneration at $t=0$ then

$$diam(X_{t},g_{X_{t}})=(\log|t|)^{-1}\exp(O(1))$$

and there is a limit $(B,g_{B})$ of $X_{t}^{new}$ in the Gromov-Hausdorff metric as $t\to 0$, such that:

a)

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

b)

$B^{sm}$ carries a ${\bf Z}$-affine structure.

c)

The metric $g_{B^{sm}}$ has a potential. This means that it is locally given in affine coordinates by a symmetric matrix $(g_{ij})=(\partial^{2}F/\partial x_{i}\partial x_{j})$, where $F$ is a smooth function (defined modulo adding an affine function).

d)

In affine coordinates the metric volume element is constant, i.e.

$$\det(g_{ij})=\det(\partial^{2}F/\partial x_{i}\partial x_{j})=const$$

(real Monge-Ampère equation).