On weak Zariski decompositions and termination of flips

Abstract

We prove that termination of lower dimensional flips for generalized klt pairs implies termination of flips for log canonical generalized pairs with a weak Zariski decomposition. Moreover, we prove that the existence of weak Zariski decompositions for pseudo-effective klt pairs implies the existence of minimal models for such pairs.

On the termination of flips for log canonical generalized pairs

“…Finally, we obtain Theorem 1.3 directly from Corollary 4.2 due to [LMT20, Theorem 1.2], which reduces the special termination for NQC lc g-pairs to lowerdimensional termination of flips. Note also that Theorem 1.3, together with [LT19, Theorem E], yield a different proof of [LT19, Corollary G], which does not depend on [HM18].…”

“…The majority of these developments are outlined in [Bir20], while for further applications of gpairs we refer to the papers [HLi20, HLiu20, LMT20, Che20, CX20], just to name a few. Not only have g-pairs been implemented successfully in a wide range of contexts, but also the recent papers [Mor18,HM18,HLi18,LT19] indicate that it is in fact essential to understand their birational geometry, even if one is only interested in studying the birational geometry of varieties and investigating the main open problems in the Minimal Model Program (MMP) for usual pairs. Consequently, it is very natural to formulate and address problems regarding usual pairs in the setting of g-pairs; in doing so, one usually works with NQC g-pairs, as they behave well in the MMP [HLi18].…”

“…We refer to [HM18] for the rational coefficients case of Theorem 1.3 with a substantially different approach, and to [HLi18] for the termination of the MMP with scaling for NQC dlt g-pairs which admit NQC weak Zariski decompositions, see also [LT19,Section 4] for some refinements.…”

Generalised pairs in birational geometry

“…Various geometric aspects of generalised pairs have been studied in recent years, for example, see [12][13][50] [28] for the minimal model program and termination, [13] for birational boundedness of linear systems and ACC for generalised lc thresholds, [6] for boundedness of complements, [43][42] for abundance, [20] for boundedness of generalised pairs of general type, [19] for the canonical bundle formula, [46] for the Sarkisov program, [3] for birational boundedness of linear systems and boundedness of polarised varieties, [45] for accumulation points of generalised lc thresholds, [23] for invariance of plurigenera and boundedness, [22][21] [15] for more on boundedness of complements.…”

On numerical nonvanishing for generalized log canonical pairs

“…Basically, Q-linear equivalence should be replaced with the weaker numerical equivalence. Note that numerical nonvanishing for pseudo-effective generalized log canonical pairs implies the existence of weak Zariski decomposition, which in turn guarantees the existence of minimal models ( [HM18,HL18]).…”