This paper lays a foundation for log smooth deformation theory. We study the infinitesimal liftings of log smooth morphisms and show that the log smooth deformation functor has a representable hull. This deformation theory gives, for example, the following two types of deformations: (1) relative deformations of a certain kind of a pair of an algebraic variety and a divisor on it, and (2) global smoothings of normal crossing varieties. The former is a generalization of the relative deformation theory introduced by Makio and others, and the latter coincides with the logarithmic deformation theory introduced by Kawamata and Namikawa.
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