Collapsing K3 Surfaces, Tropical Geometry and Moduli Compactifications of Satake, Morgan-Shalen Type
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Collapsing K3 Surfaces, Tropical Geometry and Moduli Compactifications of Satake, Morgan-Shalen Type

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This research monograph mainly discusses a canonical and explicit compactification of the moduli spaces of abelian varieties, K3 surfaces and compact hyperKähler varieties. For that, we use two theories of compactification -- Satake compactifications for locally symmetric spaces in terms of the Lie theory, and Morgan-Shalen compactifications of complex varieties in terms of valuations. We show they coincide for Shimura varieties. The obtained compactifications are no longer varieties but we provide geometric meanings to them.We partially prove that the boundary parametrizes collapsed limits o...