Special MMP for log canonical generalised pairs (with an appendix joint with Xiaowei Jiang)

Abstract

We show that minimal models of $$\mathbb {Q}$$ Q -factorial NQC log canonical generalised pairs exist, assuming the existence of minimal models of smooth varieties. More generally, we prove that on a $$\mathbb {Q}$$ Q -factorial NQC log canonical generalised pair $$ (X,B+M) $$ ( X , B + M ) we can run an MMP with scaling of an ample divisor which terminates, assuming that it admits an NQC weak Zariski decomposition or that $$ K_X+B+M$$ K X + B + M is not pseudoeffective. As a consequence, we establish several existence results for minimal models and Mori fibre spaces.

Semi-ampleness of generalized pairs

“…14E30,14C20.14E05. LT21,Has22,LX22]. These results almost complete the foundation of the minimal model program for glc g-pairs, or for g-pairs admitting an lc structure on the ambient variety.…”

The Nonvanishing Problem for varieties with nef anticanonical bundle

“…Finally, to treat the general case, we employ the recently developed Minimal Model Program for generalised pairs as in [28,61]. This is where the idea to use generalised pairs hinted at earlier becomes crucial.…”

“…Generalised pairs were introduced in [5,6] and they play a crucial role in this paper. Here we only recall their definition and refer for instance to [61] for a thorough discussion of their properties and most recent developments. Definition 3.8.…”

“…One defines singularities of g-pairs in a similar way as for usual pairs, see for instance [61,Definition 2.2]. With the same notation as above, an important property we need here is that if M is nef, then the g-pair .X; B C M / has log canonical (respectively klt, dlt) singularities if and only if the pair .X; B/ has log canonical (respectively klt, dlt) singularities.…”

“…The foundations of the MMP for Q-factorial NQC log canonical generalised pairs were established recently in [28,61]. We recall briefly below the results that we need in this paper.…”