In this paper, we study the minimal model theory for threefolds in mixed characteristic. As a generalization of a result of Kawamata, we show that the MMP holds for strictly semi-stable schemes over an excellent Dedekind scheme V of relative dimension two without any assumption on the residue characteristics of V . We also prove that we can run a pK X{V `∆q-MMP over Z, where π : X Ñ Z is a projective birational morphism of Q-factorial quasi-projective V -schemes and pX, ∆q is a threedimensional dlt pair with Excpπq Ă t∆u.V -scheme Z. Assume that pS N , p1 ´εqA S `BS q is globally T -regular for all 0 ă ε ă 1
In this paper, we introduce a new generalization of log canonical singularities for non-$\mathbb{Q}$-Gorenstein varieties. We prove that if a normal complex projective variety has a non-invertible polarized endomorphism, then it has log canonical singularities in our sense. As a corollary, we give an affirmative answer to a conjecture of Broustet and Höring [ 6].
We prove Kawaguchi-Silverman conjecture for all surjective endomorphisms on every smooth rationally connected variety admitting an int-amplified endomorphism.
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