Existence of flips for generalized lc pairs

Abstract

We prove the existence of flips for Q-factorial NQC generalized lc pairs, and the cone and contraction theorems for NQC generalized lc pairs. This answers a question of C. Birkar. As an immediate application, we show that we can run the minimal model program for Q-factorial NQC generalized lc pairs. In particular, we complete the minimal model program for Q-factorial NQC generalized lc pairs in dimension ≤ 3 and pseudo-effective Q-factorial NQC generalized lc pairs in dimension 4.

The Nonvanishing problem for varieties with nef anticanonical bundle

“…As above, the argument is inductive, hence it works in principle also in higher dimensions. In this paper we reduce to the terminal case via the Minimal Model Program for generalised pairs [HL21,LT21]; this is done in Section 8.…”

“…Finally, to treat the general case, we employ the recently developed Minimal Model Program for generalised pairs as in [HL21,LT21]. This is where the idea to use generalised pairs hinted at on page 3 becomes crucial.…”

Special MMP for log canonical generalised pairs

“…The MMP for generalised pairs. In this paper we use -frequently without explicit mention -the foundations of the Minimal Model Program for Q-factorial NQC log canonical generalised pairs, as established very recently in [HL21]. We recall briefly the main results.…”

“…The main reason for this additional assumption was that, in its presence, one can reduce several foundational results in the geometry of generalised pairs, such as the existence of surgery operations in the MMP, to statements about usual pairs, see [BZ16,HL18]. These foundational results (the Cone and Contraction theorems, the existence of divisorial contractions and flips) were very recently established in [HL21] for Q-factorial NQC log canonical generalised pairs. Finally, we obtain the following generalisation of [Bir12b, Corollary 1.6].…”

On the existence of flips for threefolds in mixed characteristic $(0,5)$

“…One of the fundamental goals of algebraic geometry is to classify all algebraic varieties (up to birational equivalence), which, conjecturally, can be achieved by means of the Minimal Model Program (MMP). In characteristic zero, the program holds for varieties with dimension ≤ 3, and a major part of MMP is known for varieties of general type in higher dimensions by [BCHM10], where they also established the existence of klt flips (see [Bir12,HX13,HL21] for results in a more general setting). In positive characteristic, this theory is now known to hold for threefolds over perfect fields of characteristic p > 3 (see [HX15, CTX15, Bir16, BW17, GNT19, HW19b]) and in some special cases for fourfolds ( [HW20,XX21]).…”