Inversion of adjunction on log canonicity
TL;DR: In this paper, the authors prove inversion of adjunction on log canonicity, and prove that adjunction is invertible on log canonicity, but not on log-canonicity.
Abstract: We prove inversion of adjunction on log canonicity.
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TL;DR: In this paper, it was shown that pl-flips exist in dimension n − 1, assuming finite generation in dimension N − 1 and assuming that pl flips exist in all dimensions.
Abstract: Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.
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TL;DR: In this paper, it was shown that any singular Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Fano varieties equipped with their anti-canonical polarization.
Abstract: It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of $$\mathbb {Q}$$
-Fano varieties equipped with their anti-canonical polarization. The proof is based on a new formula expressing the Donaldson-Futaki invariants in terms of the slope of the Ding functional along a geodesic ray in the space of all bounded positively curved metrics on the anti-canonical line bundle of X. One consequence is that a toric Fano variety X is K-polystable iff it is K-polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope P associated to X. The results also extend to the logarithmic setting and in particular to the setting of Kahler-Einstein metrics with edge-cone singularities. Applications to geodesic stability, bounds on the Ricci potential and Perelman’s $$\lambda $$
-entropy functional on K-unstable Fano manifolds are also given.
210 citations
Cites methods or result from "Inversion of adjunction on log cano..."
...hat X0 is reduced and, by adjunction, that its normalization has log canonical singularities (see [?, 2.7]). Finally, the converse follows from “inversion of adjunction”, i.e. from the main result of [38], previously conjectured by Shokurov (the special case when X0 has log terminal singularities follows from a previous result of Kollar et al [39, Theorem 7.5]). Combining the last point in the previou...
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...ow how to express l0 explicitly in terms of a certain log canonical threshold of the central fiberX0 (Proposition 3.8). Finally, the vanishing properties of qare obtained using inversion of adjunction [38] (which can be seen as an algebro-geometric incarnation of the Ohsawa-Takegoshi extension theorem in complex analysis [19, 39]). It should be pointed out that the information about the vanishing prope...
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TL;DR: In this article, it was shown that log canonical singularities are Du Bois, as conjectured in [Kol92, 1.13] and [1.10].
Abstract: A recurring difficulty in the Minimal Model Program (MMP) is that while log terminal singularities are quite well behaved (for instance, they are rational), log canonical singularities are much more complicated; they need not even be CohenMacaulay. The aim of this paper is to prove that, as conjectured in [Kol92, 1.13], log canonical singularities are Du Bois. The concept of Du Bois singularities, abbreviated as DB, was introduced by Steenbrink in [Ste83] as a weakening of rationality. It is not clear how to define Du Bois singularities in positive characteristic, so we work over a field of characteristic 0 throughout the paper. The precise definition is rather involved, see (1.10), but our main applications rely only on the following consequence:
209 citations
Posted Content•
TL;DR: In this article, the authors give a canonical synthetic construction of the mirror family to a pair (Y,D) of a smooth projective surface with an anti-canonical cycle of rational curves, as the spectrum of an explicit algebra defined in terms of counts of rational curve on Y meeting D in a single point.
Abstract: We give a canonical synthetic construction of the mirror family to a pair (Y,D) of a smooth projective surface with an anti-canonical cycle of rational curves, as the spectrum of an explicit algebra defined in terms of counts of rational curves on Y meeting D in a single point. In the case D is contractible, the family gives a smoothing of the dual cusp, and thus a proof of Looijenga's 1981 cusp conjecture.
185 citations
Posted Content•
TL;DR: In this article, it was shown that any singular, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization.
Abstract: It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization. The proof exploits convexity properties of the Ding functional along weak geodesic rays in the space of all bounded positively curved metrics on the anti-canonical line bundle of X and also gives a new proof in the non-singular case. One consequence is that a toric Fano variety X is K-polystable iff it is K-polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope P associated to X. The results also extend to the logarithmic setting and in particular to the setting of Kahler-Einstein metrics with edge-cone singularities. Furthermore, applications to geodesic stability, bounds on the Ricci potential and Perelman's entropy functional on K-unstable Fano manifolds are given.
183 citations
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"Inversion of adjunction on log cano..." refers methods in this paper
...The existence of log resolutions is guaranteed by Hironaka’s resolution theorem [ 2 ]....
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Book•
01 Jan 1998TL;DR: In this paper, the authors introduce the minimal model program and the canonical class of rational curves, and present the singularities of the model program, as well as three dimensional flops.
Abstract: 1. Rational curves and the canonical class 2. Introduction to minimal model program 3. Cone theorems 4. Surface singularities 5. Singularities of the minimal model program 6. Three dimensional flops 7. Semi-stable minimal models.
1,754 citations
Journal Article•
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