Given a complex projective manifold X and a divisor D with normal crossings, we say that the logarithmic tangent bundle \(T_X(-\hbox {log}\, D)\) 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 \(D_{|F}\).

中文翻译：

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某些复曲面纤维束的数值表征

给定一个复杂的射影流形X和一个具有法线交叉的除数D，我们说对数正切束\（T_X（-\ hbox {log} \，D）\）如果回落到X中包含的任何有理曲线都是平凡向量束。此外，如果\（-（K_X + D）\）是nef，则对数正则除数\（K_X + D \）是扭转，并且最大合理链状连接的振动证明是平滑的局部琐碎纤维，典型纤维F为具有边界除数\（D_ {| F} \）的复曲面变体。