We improve on the result of Hacon and Witaszek by showing that the MMP for semi-stable fourfolds in mixed characteristic terminates in several new situations. In particular, we show the validity of the MMP for strictly semi-stable fourfolds over excellent Dedekind schemes globally when the residue fields are perfect and have characteristics > 5.

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We establish a Kollár-type gluing theory for generalized log canonical pairs associated with crepant log structures and use it to prove semi-ampleness results of generalized pairs. As consequences, we prove the existence of flips for any generalized log canonical pair, and show that generalized log canonical singularities are Du Bois. Idea of the proof of Theorem 1.1 and some examplesThere are several natural approaches to prove the main results in this paper, however the nuances of glc pairs seem to pose some serious difficulties. Before

We study the behavior of generalized lc pairs with b-log abundant nef part, a meticulously designed structure on algebraic varieties. We show that this structure is preserved under the canonical bundle formula and sub-adjunction formulas, and is also compatible with the non-vanishing conjecture and the abundance conjecture in the classical minimal model program. * A canonical bundle formula for a projective surjective morphism f is given as the composition of a Fujino-Gongyo type canonical bundle formula [FG12] and a Kodaira type canonical bundle formula [Kod63] via the Stein factorization of f . We refer the reader to Subsection 2.4 formal definitions.

We provide a detailed proof of the validity of the Minimal Model Program for threefolds over excellent Dedekind separated schemes whose residue fields do not have characteristic 2 or 3.