Log smooth deformation theory

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.