On the termination of flips for log canonical generalized pairs

Abstract

We prove the termination of flips for 4-dimensional pseudoeffective NQC log canonical generalized pairs. As main ingredients, we verify the termination of flips for 3-dimensional NQC log canonical generalized pairs, and show that the termination of flips for pseudoeffective NQC log canonical generalized pairs which admit NQC weak Zariski decompositions follows from the termination of flips in lower dimensions.2010 Mathematics Subject Classification: 14E30.

Special MMP for log canonical generalised pairs

“…Part (i) of Corollary 1.3 was already shown in [HL21, Theorem 1.5] as a consequence of [CT20]. Part (iii) of Corollary 1.3 is a special case of the following generalisation of [LT19, Theorems C and 4.3].…”

“…Moreover, flips for Q-factorial NQC log canonical g-pairs exist by [HL21, Theorem 1.2]. All this implies that one may run a (K X + B + M )-MMP over Z, whose termination is not known in general; however, the paper [CT20] establishes the termination of flips for Q-factorial NQC log canonical g-pairs of dimension 3 and for pseudoeffective Q-factorial NQC log canonical g-pairs of dimension 4, see also [Mor18,HM20]. Note that Q-factoriality is preserved in any MMP by [HL21, Corollaries 9.19 and 9.20 and Theorem 10.3].…”

Existence of flips for generalized lc pairs

“…It has recently become apparent that the minimal model program (MMP) for generalized pairs is closely related to the minimal model program for usual pairs and varieties. In particular, generalized pairs have been used to prove the termination of pseudo-effective fourfold flips [Mor18,HL18,HM20,CT20]. For this, and other reasons, it is important to study the minimal model program for generalized pairs.…”

“…[HL18, Example 3.15]), however this is a natural assumption and is contained in the original definition of generalized pairs in [BZ16]. Under the NQC assumption, the known results on the termination of flips are similar to the usual pair case (in particular, in full generality in dimension ≤ 3 [CT20] and in the pseudo-effective case in dimension 4 [HM20,CT20]). However, the cone and contraction theorems and the existence of flips seem to be far more challenging even in dimension 3, and we only know some partial results when M descends to X, i.e., M X is nef [LP20a,LP20b].…”

“…Finally, together with the existence of flips we just proved, we know that we can run MMP for any Q-factorial NQC glc g-pair (Theorem 1.4). In other words, for Q-factorial NQC glc g-pairs, whenever we know the termination of flips, we will have the complete MMP, and Theorem 1.5 follows from [HM20,CT20].…”