PL density invariant for type II degenerating K3 surfaces, Moduli compactification and hyperKahler metrics

A protagonist here is a new-type invariant for type II degenerations of K3 surfaces, which is explicit PL (piecewise linear) convex function from the interval with at most 18 non-linear points. Forgetting its actual function behaviour, it also classifies the type II degenerations into several combinatorial types, depending on the type of root lattices as appeared in classical examples. From differential geometric viewpoint, the function is obtained as the density function of the limit measure on the collapsing hyperKahler metrics to conjectural segments, as in the work of Honda-Sun-Zhang. On the way, we also reconstruct a moduli compactification of elliptic K3 surfaces in the works of Brunyate, Ascher-Bejleri, Alexeev-Brunyate-Engel in a more elementary manner, analyze the cusps more explicitly. We also interpret the glued hyperKahler fibration of Hein-Sun-Viaclovsky-Zhang as a special case from our viewpoint, discuss other cases, and possible relations with Landau-Ginzburg models in the mirror symmetry context.