We prove that termination of lower dimensional flips for generalized $\operatorname{klt}$ pairs implies termination of flips for $\operatorname{log}$ canonical generalized pairs with a weak Zariski decomposition. Under the same hypothesis we prove that the existence of weak Zariski decompositions for pseudo-effective log canonical pairs implies the existence of weak Zariski decompositions for pseudo-effective generalized log canonical pairs. As an application, we prove the termination of any minimal model program for generalized log canonical pseudo-effective $4$‑folds.

中文翻译：

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关于弱Zariski分解和翻转终止

我们证明，终止$ \ operatorname {klt} $对的低维翻转意味着通过弱Zariski分解终止$ \ operatorname {log} $规范化广义对的翻转。在相同的假设下，我们证明了伪有效对数典范对的弱Zariski分解的存在暗示了伪有效对数典范对的弱Zariski分解的存在。作为一个应用程序，我们证明了对通用对数规范伪有效$ 4 $折的所有最小模型程序的终止。