Cambridge Journal of Mathematics

Volume 11 (2023)

Number 4

Existence of flips for generalized $\operatorname{lc}$ pairs

Pages: 795 – 828



Christopher D. Hacon (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)

Jihao Liu (Department of Mathematics, Northwestern University, Evanston, Illinois, U.S.A.)


$\def\lc{\operatorname{lc}} \def\Q{\mathbb{Q}}$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 conjecture of Han–Li–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 $\leq 3$ and pseudo-effective $\Q$-factorial NQC generalized $\lc$ pairs in dimension $4$.


generalized pairs, minimal model program, flips, cone theorem

2010 Mathematics Subject Classification

, 14J35. Primary 14C20, 14E30. Secondary 14E05, 14J17, 14J30.

The authors are partially supported by NSF research grants DMS-1801851 and DMS-1952522; and by a grant from the Simons Foundation, Award Number 256202.

Received 14 February 2022

Published 29 September 2023