The moduli spaces of abelian varieties are compactified by Faltings-Chai. Recently, Kajiwara-Kato-Nakayama realized this compactification as the moduli space of log abelian varieties which are degenerated objects in the world of logarithmic geometry. In the study of the mod p-fiber of this compactified moduli space, log p-divisible groups occur as degenerated objects of p-divisible groups. In this talk, we introduce our results on slope filtrations of log p-divisible groups, applying them to the geometry of the compactified moduli spaces.

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