Kontsevich–Soibelman conjecture on maximal degeneration limits

Let ${X}_{mer}$ be an algebraic $n$-dimensional Calabi–Yau manifold over the field of meromorphic germs, with a non-vanishing volume form, and let $(X_t,\Omega_t)$ be the associated family equipped with its Calabi–Yau metrics $g_{X_t}$. Assume that ${X}_{mer}$ has maximal degeneration at $t=0$, meaning that the exponent of $\log|t|$ in the asymptotic volume formula is $n$. Rescale the metrics to obtain the diameter-one manifolds $X_t^{new}$. Kontsevich–Soibelman conjecture. If ${X}_{mer}$ has maximal degeneration at $t=0$, then

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

and $X_t^{new}$ has a Gromov–Hausdorff limit $(B,g_B)$ satisfying: $B$ is compact and contains a dense open smooth oriented Riemannian $n$-manifold $(B^{sm},g_{B^{sm}})$; the singular set $B^{sing}=B\setminus B^{sm}$ has Hausdorff dimension at most $n-2$; $B^{sm}$ has a ${\bf Z}$-affine structure; the metric has a local potential $g_{ij}=\partial^2F/\partial x_i\partial x_j$ in affine coordinates; and

$$\det(g_{ij})=\det\left(\frac{\partial^2F}{\partial x_i\partial x_j}\right)=const.$$

This is the metric-collapse formulation of the Strominger–Yau–Zaslow picture. The conjecture is known for abelian varieties, and substantial results are available for K3 surfaces and in dimension three, but the general statement remains open.