Termination of flips for projective log canonical fourfolds with rational boundary

Theorem 1.1: for a projective lc pair $(X,B)$ of dimension four with rational boundary over an algebraically closed field of characteristic zero, every negative extremal ray at every stage has its contraction and, when birational, a positive model; every permitted program has finitely many birational steps for every sequence of choices; and every maximal program ends at a nef adjoint or a Mori fibre space. Proved over $\mathbb C$ by a weighted difficulty with a topological branch estimate, then transferred to all such fields. Not shown: real boundaries, generalized pairs in the projective setting, any dimension above four, or semiampleness of the nef endpoint without the release's log abundance result. The Kahler companions prove termination for generalized lc flips with projective small diagrams on globally Weil Q-factorial compact Kahler fourfolds, a four-dimensional rational form of Hacon-Xie's Conjecture 1.15.

Theorem 1.1: for a projective lc pair $(X,B)$ of dimension four with rational boundary over an algebraically closed field of characteristic zero, every negative extremal ray at every stage has its contraction and, when birational, a positive model; every permitted program has finitely many birational steps for every sequence of choices; and every maximal program ends at a nef adjoint or a Mori fibre space. Proved over $\mathbb C$ by a weighted difficulty with a topological branch estimate, then transferred to all such fields. Not shown: real boundaries, generalized pairs in the projective setting, any dimension above four, or semiampleness of the nef endpoint without the release's log abundance result. The Kahler companions prove termination for generalized lc flips with projective small diagrams on globally Weil Q-factorial compact Kahler fourfolds, a four-dimensional rational form of Hacon-Xie's Conjecture 1.15.