Under the assumption of the minimal model theory for projective klt pairs of dimension n, we establish the minimal model theory for lc pairs $(X/Z,\Delta )$ such that the log canonical divisor is relatively log abundant and its restriction to any lc center has relative numerical dimension at most n. We also give another detailed proof of results by the second author, and study termination of log MMP with scaling.

中文翻译：

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对数丰富LC对的最小模型理论

在n维射影klt对的最小模型理论的假设下，建立lc对的最小模型理论。$（X/ž，\Delta ）$使得对数正则除数相对对数丰富，并且其对任何lc中心的限制最多具有n的相对数值维。我们还提供了第二作者对结果的另一个详细证明，并研究了具有缩放功能的log MMP终止。