  Given a complex projective manifold $X$ and a divisor $D$ with normal
crossings, we say that the logarithmic tangent bundle $T_X(-\log D)$ is R-flat
if its pull-back to the normalization of any rational curve contained in $X$ is
the trivial vector bundle. If moreover $-(K_X+D)$ is nef, then the log
canonical divisor $K_X+D$ is torsion and the maximally rationally chain
connected fibration turns out to be a smooth locally trivial fibration with
typical fiber $F$ being a toric variety with boundary divisor $D_{|F}$.
