Existence of log canonical closures

Abstract

Let f : X → U be a projective morphism of normal varieties and (X, ∆) a dlt pair. We prove that if there is an open set U 0 ⊂ U , such that (X, ∆) × U U 0 has a good minimal model over U 0 and the images of all the non-klt centers intersect U 0 , then (X, ∆) has a good minimal model over U . As consequences we show the existence of log canonical compactifications for open log canonical pairs, and the fact that the moduli functor of stable schemes satisfies the valuative criterion for properness.

Remarks on special kinds of the relative log minimal model program

“…Theorem 1.2 (cf. [HX,Theorem 1.1]). Let π : X → Z be a projective morphism of normal quasi-projective varieties and let (X, ∆) be a log canonical pair with a boundary R-divisor ∆.…”

“…With Theorem 1.1 and Theorem 1.2, we can prove R-divisor versions of some results which are known in Q-boundary divisor case: Corollary 1.3 (existence of lc closures, cf. [HX,Corollary 1.2]). Let U 0 be an open subset of a normal quasi-projective variety U, f 0 : X 0 → U 0 be a projective morphism, and (X 0 , ∆ 0 ) be a log canonical pair.…”

Interaction Between Singularity Theory and the Minimal Model Program

“…Although part of the MMP, including the abundance conjecture, remains to be conjectural, all the MMP results we need in this note are already proved in [BCHM10] and its extensions, e.g. [HX13].…”

On a generalized canonical bundle formula and generalized adjunction

“…Up to replacing Z with a higher model and X with the normalization of the fiber product, by Chow's lemma, we may assume that Z is quasi-projective. Then, by [24], we may assume that Z is projective. Let X η be the geometric generic fiber.…”