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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,Δ){(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.

On minimal model theory for log abundant lc pairs