| Résume |
This talk is partially based on a joint work with Yoshiki Oshima (e.g. Memoir MSJ, vol.40 and its subsequent works). We give a conjectural but totally explicit description of collapsed limits of K3 surfaces with the hyperKahler metrics. The actual framework we found is more general and at the level of moduli, i.e., the Gromov-Hausdorff compactification of their moduli and identifies it with certain Satake / Morgan-Shalen type compactification. We do also for other Kahler-Einstein manifolds, which turned out to follow the similar rules. Focusing on K3 case, the case of collapse to S2 is fully proven by using the Siegel/Satake type reduction theory to reduce to some extension of the work of Gross-Wilson/Gross-Tosatti-Zhang, leading to a proof of Kontsevich-Soibelman conjecture. In the case of collapsing to interval, the limit density function is a piecewise-linear convex function with at
most 18th bend points, which we explicitly describe. Many parts of these works have various algebraic geometry / non-archimedean geometry interpretation, which we also partially explain, with a reminiscence of Yau-Tian-Donaldson correspondence
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