For an odd prime $p$, it is shown that if $G = AB$ is a finite $p$-group, for subgroups $A$ and $B$ such that $A$ is cyclic and $B$ is abelian of exponent at most $p^{k}$, then $\Omega_{k}(A)B \unlhd G$, where $\Omega_{k}(A) = \langle g \in A \mid g^{ p^{k}} = 1 \rangle$.

This note is a summary of our work [OO], which provides an explicit and global moduli-theoretic framework for the collapsing of Ricci-flat Kähler metrics and we use it to study especially the K3 surfaces case. For instance, it allows us to discuss their Gromov-Hausdorff limits along any sequences, which are even not necessarily “maximally degenerating”. Our results also give a proof of Kontsevich-Soibelman [KS06,Conjecture 1] (cf., [GW00, Conjecture 6.2]) in the case of K3 surfaces as a byproduct.