Collapsing K3 Surfaces, Tropical Geometry And Moduli Compactifications Of Satake, Morgan-shalen Type

This research monograph mainly discusses a canonical and explicit compactification of the moduli spaces of abelian varieties, K3 surfaces and compact hyperKaehler 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 of the Ricci-flat Kaehler metrics. Such limits also coincide with a posteriori defined 'tropicalized version' or equivalently the dual graphs of degenerations of original varieties. From differential geometric perspective, this work provides a moduli-theoretic framework for the limiting behavior of Ricci-flat Kaehler metrics. From Lie theoretic perspective, this work provides a geometric meaning to the Satake compactification associated to adjoint representations, which are not the same as the Baily-Borel compactifications. Applying our theory to the case of one parameter maximal degeneration of K3 surfaces, we obtain proofs of conjectures of Gross-Wilson and Kontsevich-Soibelman.We formulate general conjectures on the limits of Ricci-flat Kaehler metrics in the above framework and partially prove them, but they largely remain open.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets