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-vertical bar F.
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