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A protagonist here is a new-type invariant for type II degenerations of K3surfaces, which is explicit PL (piecewise linear) convex function from theinterval with at most 18 non-linear points. Forgetting its actual functionbehaviour, it also classifies the type II degenerations into severalcombinatorial types, depending on the type of root lattices as appeared inclassical examples.  From differential geometric viewpoint, the function is obtained as thedensity function of the limit measure on the collapsing hyperKahler metrics toconjectural segments, as in the work of Honda-Sun-Zhang. On the way, we alsoreconstruct a moduli compactification of elliptic K3 surfaces in the works ofBrunyate, Ascher-Bejleri, Alexeev-Brunyate-Engel in a more elementary manner,analyze the cusps more explicitly.  We also interpret the glued hyperKahler fibration ofHein-Sun-Viaclovsky-Zhang as a special case from our viewpoint, discuss othercases, and possible relations with Landau-Ginzburg models in the mirrorsymmetry context.

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