  This paper gives a foundation of 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 of it, and
(2) global smoothings of normal crossing varieties. The former is a
generalization of the relative deformation theory introduced by Makio, and the
latter coincides with the logarithmic deformation theory introduced by Kawamata
and Namikawa.
