On weak Zariski decompositions and termination of flips

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.

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The first author was partially supported by NSF research grants no: DMS-1300750, DMS-1265285 and by a grant from the Simons Foundation; Award Number: 256202. He would also like to thank the Mathematics Department and the Research Institute for Mathematical Sciences, located Kyoto University.

Received 14 September 2018

Accepted 15 October 2020

Published 12 January 2021