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Existence of log canonical closures. (English) Zbl 1282.14027
In this paper, the ground field is the field of complex numbers. The main result is Theorem 1.1: Let $$f:X\rightarrow U$$ be a projective morphism of normal varieties, $$\triangle$$ a $$\mathbb{Q}$$-divisor such that $$(X, \triangle)$$ is a dlt pair and $$S= \llcorner \triangle \lrcorner$$ the non-klt locus. Assume that there exists an open subsets $$U^0\subset U$$ such that $$(X^0, \triangle^0)= (X, \triangle)\times_U U^0$$ has a good minimal model over $$U^0$$, and that any stratum of $$S$$ intersects $$X^0$$. Then $$(X, \triangle)$$ has a good minimal model over $$U$$.
The proof of Theorem 1.1 is by induction on the dimension and is divided into three main steps. First, by Kollár’s gluing theory, the authors prove an analogous statement for the non-klt locus $$S= \llcorner \triangle \lrcorner$$ or for the sdlt pair $$(S, \triangle_S)$$ where $$K_S+ \triangle_S=(K_X+ \triangle)|_S$$ is defined by adjunction. Combining the first step with the theorems of Kawamata and Fujino, they show that any minimal model of $$(X, \triangle)$$ is a good model. So the proof of Theorem 1.1 is reduced to proving the existence of a minimal model for the dlt pair $$(X, \triangle)$$. This is the most technical part of the proof. Since the pair is not of log general type, they use Iitaka fibration and Kawamata’s canonical bundle formula. They do not directly show the termination of flips but show that it suffices to find a neutral model which contracts the right components. To construct such a neutral model, they reduce the problem to a special termination question by Shokurov’s idea so that they can apply induction on the dimension.
The authors also prove the following important applications of the main theorem.
1. Existence of log canonical closure
Let $$U^0$$ be an open subset of a normal quasi-projective variety $$U$$, $$f^0: X^0\rightarrow U^0$$ a projective morphism, and $$(X^0, \triangle^0)$$ a log canonical pair. Then there exists a projective morphism $$f:X\rightarrow U$$ and a log canonical pair $$(X, \triangle)$$ such that $$X^0= X\times _U U^0$$ is an open subset and $$\triangle^0 =\triangle|_{X^0}$$.
2. Existence of compactifications of log canonical morphisms (Kollár-Kovács conjecture).
Let $$Y$$ be a normal variety, $$g: Y\rightarrow U$$ a dominant morphism to a smooth curve $$U$$ and $$\triangle$$ an effective $$\mathbb{Q}$$-divisor such that $$K_Y+ \triangle$$ is $$\mathbb{Q}-$$Cartier. If $$(Y, \triangle +Y_p)$$ is lc for all closed points $$p\in U$$, then $$g$$ is called a log canonical morphism, or an lc morphism, where $$Y_p$$ is the fiber over $$p$$.
Now let $$U$$ be a smooth curve and $$f^0: X^0\rightarrow U$$ be an affine finite type lc morphism. Then there exists a finite dominating base change morphism $$\theta: \tilde{U} \rightarrow U$$ and a projective lc morphism $$f: X\rightarrow \tilde{U}$$ such that $$X^0\times _U\tilde{U}\subset X$$ and $$f|_{X^0\times _U \tilde{U}}= f^0\times_U \theta$$.
3. The properness of the moduli functor of stable schemes.
Let $$f^0: X^0\rightarrow U^0$$ be a projective morphism, $$(X^0, \triangle^0)$$ a log canonical pair, $$U$$ the germ of a smooth curve, $$p\in U$$ a closed point and $$U^0=U\backslash \{p\}$$. If $$K_{X^0}+ \triangle^0$$ is $$f^0-$$ample, then there is a finite dominating base change $$\theta: \tilde{U}\rightarrow U$$, a log canonical pair $$(X, \triangle)$$ and a projective lc morphism $$(X, \triangle) \rightarrow \tilde{U}$$ such that $$K_X+ \triangle$$ is ample over $$\tilde{U}$$ and the restriction of $$(X, \triangle)$$ to the pre-image $$\theta^{-1}(U^0)$$ is isomorphic to $$(X^0, \triangle^0)\times_U \tilde{U}$$.
4. Koll$$\acute{\text{a}}$$r’s conjecture and existence of log canonical flips.
Let $$f:X\rightarrow U$$ be a projective morphism of normal varieties, $$\triangle'$$ and $$\triangle''$$ effective $$\mathbb{Q}$$-divisors on $$X$$ such that $$(X, \triangle'+ \triangle'')$$ is a $$\mathbb{Q}$$-factorial lc pair, $$(X, \triangle'')$$ is dlt and $$K_X+ \triangle'+ \triangle''\sim_{{\mathbb{Q}}, U} 0$$. Then the $$(K_X+ \triangle'')$$-MMP with scaling over $$U$$ terminates with either a Mori fibration or a $${\mathbb{Q}}$$-factorial good minimal model. It implies the existence of log canonical flips: Let $$f: X\rightarrow Z$$ be a flipping contraction for a log canonical pair $$(X, \triangle)$$. Then the flip of $$f$$ exists.

##### MSC:
 1.4e+31 Minimal model program (Mori theory, extremal rays) 1.4e+06 Rational and birational maps
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