Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures

Abstract: For any degenerating Calabi-Yau family, we introduce new limit space which we call galaxy, whose dense subspace is the disjoint union of countably infinite open Calabi-Yau varieties, parametrized by the rational points of the Kontsevich-Soibelman's essential skeleton, while dominated by the Huber adification over the Puiseux series field. Other topics include: projective limits of toroidal compactifications, locally modelled on what we call the limit toric varieties, the way to attach tropicalized family to given Calabi-Yau family, which are weakly related to each other.