Existence of log canonical closures

By Christopher D. Hacon. 2011

199 citations recorded by the source corpus.

Let f:X→U be a projective morphism of normal varieties and (X,Δ) a dlt pair. We prove that if there is an open set U0⊂U, such that (X,Δ)×UU0 has a good minimal model over U0 and the images of all the non-klt centers intersect U0, then (X,Δ) 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/85dd4b3b45bc9afa6d13fad72d9b054fdbd1e6cb

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Citations in the corpus

Stability of valuations and Kollár components (2016, unknown): “nt of Z 0 except X 0. We also note that ˚ 1 1 S0+ (1 )H+ (0) 1 D + H0+ (1 2) is horizontal over C. We run an MMP of (Z;˚ 1 1 S0+ (1 )H+ (0) 1 D + H0+ (1 2)) over Xwith the scaling of H0. By [HX13], we can get a relative minimal model f: Z99K W. Since K W+f ˚ 1(S0+(1 1)H+(0) 1D)+ 1H0+(1 2) ˘ Q f ”
ACC for log canonical thresholds (2012, methodology): “In this case [7] or by Theorem 1.6 of [16], we may replace Y by a log terminal model of ( Y, E + Γ − εT ) over X , gaining the fact that − T is nef over X , at the expense of temporarily losing the property that ( Y, Γ + E ) is divisorially log terminal, whilst preserving the condition that K Y + E…”
Jordan property for Cremona groups (2012, background): “Taking a log canonical closure ( ¯ V , ¯ D ) of the log pair ( V,D ) as in [14, Corollary 1.2], we see that Z is still a minimal center of non-klt singularities of the new pair ( ¯ V , ¯ D ) , and all other centers of non-klt singularities of ( ¯ V , ¯ D ) are disjoint from Z .”
The dual complex of Calabi–Yau pairs (2015, unknown): “Then, by [1,18], the (X, =1)-MMP terminates with a Fano-contraction q : Xr → Z .”
Semipositivity theorems for moduli problems (2012, background): “As the culmination of the works of several authors (see, for example, [Ale94], [Ale96], [AM04], [HMX14], [HX13], [KM97], [Kol08], [Kol16], [KSB88]), we have Corollary 1.4.”
On a generalized canonical bundle formula and generalized adjunction (2018, background): “Then, by [24], we may assume that Z is projective.”
The essential skeleton of a degeneration of algebraic varieties (2013, background): “dimension n= dim(XK). Then Sk(XK) is an n-dimensional closed pseudo-manifold. Proof. By running MMP for Y over C, we know that Xhas a good minimal dlt model X with reduced special ﬁber (see [Fu11] or [HX13]). Then one sees as in the proof of Theorem 4.1.4 that KX is trivial over C. Our assumption on”