Author
Jihao Liu
Bio: Jihao Liu is an academic researcher. The author has contributed to research in topics: Mathematics & Combinatorics. The author has an hindex of 3, co-authored 12 publications receiving 73 citations.
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TL;DR: In this article, the authors prove the existence of complements for pairs with DCC coefficients and the ACC for minimal log discrepancies of exceptional singularities, and show its existence for pairs of real coefficients.
Abstract: We prove the existence of $n$-complements for pairs with DCC coefficients and the ACC for minimal log discrepancies of exceptional singularities. In order to prove these results, we develop the theory of complements for real coefficients. We introduce $(n,\Gamma_0)$-decomposable $\mathbb{R}$-complements, and show its existence for pairs with DCC coefficients.
53 citations
TL;DR: In this paper, the minimal log discrepancy of any Q -Gorenstein non-canonical three-fold is shown to be ≤ 12 13, which is the optimal bound.
15 citations
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TL;DR: In this article, it was shown that the minimal log discrepancy of any noncanonical Gorenstein non-canonical non-Gorewitzian is Θ( √ 12/13), where √ 13 is the optimal bound.
Abstract: We show that the minimal log discrepancy of any $\mathbb Q$-Gorenstein non-canonical threefold is $\leq\frac{12}{13}$, which is an optimal bound.
8 citations
TL;DR: In this paper, it was shown that any two birational Mori fiber spaces of a factorial gklt g-pairs are connected by a finite sequence of Sarkisov links.
Abstract: In this paper we show that any two birational Mori fiber spaces of $\mathbb{Q}$-factorial gklt g-pairs are connected by a finite sequence of Sarkisov links.
3 citations
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TL;DR: In this paper, it was shown that the singularity of $(\epsilon,\delta)$-lc is bounded up to a deformation, and that these singularities are not bounded in higher dimensions.
Abstract: In this paper we study $(\epsilon,\delta)$-lc singularites, i.e. $\epsilon$-lc singularities admitting a $\delta$-plt blow-up. We prove that $n$-dimensional $(\epsilon,\delta)$-lc singularities are bounded up to a deformation, and $2$-dimensional $(\epsilon,\delta)$-lc singularities form a bounded family. Furthermore, we give an example which shows that $(\epsilon,\delta)$-lc singularities are not bounded in higher dimensions, even in the analytic sense.
3 citations
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TL;DR: In this paper, the stability threshold is shown to be a constructible function on the fibers of log Fano pairs and any special test configuration arises from a log canonical place of a bounded complement.
Abstract: In this paper, we prove the openness of K-semistability in families of log Fano pairs by showing that the stability threshold is a constructible function on the fibers. We also prove that any special test configuration arises from a log canonical place of a bounded complement and establish properties of any minimizer of the stability threshold.
85 citations
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TL;DR: In this paper, it was shown that the minimal log discrepancies of a variety in its closed points define a lower semi-continuous function for toric varieties with dimension at most 3.
Abstract: An explanation to the boundness of minimal log discrepancies conjectured by V.V. Shokurov would be that the minimal log discrepancies of a variety in its closed points define a lower semi-continuous function. We check this lower semi-continuity behaviour for varieties of dimension at most 3 and for toric varieties of arbitrary dimension.
48 citations
Book•
01 Jan 2014
TL;DR: In this article, the authors present a set of variations of making singularities, including lifting up-covering singularities and hyperplane sections of singularities by cutting-hypersurface singularities.
Abstract: 0. Preliminaries: Variations of making singularities0.1. By cutting-hypersurface singularities, hyperplane section of singularities0.2. By taking quotients-quotient singularities, quotient of singularities0.3. By lifting up-covering singularities0.4. By contractions 1. Sheaves, algebraic varieties and analytic spaces1.1. Preliminaries on sheaves1.2. Sheaves on a topological space1.3. Analytic space and Algebraic variety1.4. Coherent sheaves2. Homological algebra and duality2.1. Injective resolutions2.2. i-th derived functors2.3. Ext2.4. Cohomologies with the coefficients on sheaves2.5. Derived functors and duality2.6. Spectral sequence3. Singularities, algebraization and resolutions of singularities3.1. Definition of a singularity3.2. Algebraization theorem3.3. Blowups and resolutions of the singularities3.4. Toric resolutions of the singularities4. Divisors on a variety and the corresponding sheaves4.1. Locally free sheaves, invertible sheaves and divisorial sheaves4.2. Divisors4.3. The canonical sheaves and a canonical divisor4.4. Intersections of divisors5. Differential forms around the singularities5.1. Ramification formula5.2. Canonical singularities, terminal singularities and rational singularities6. Two dimensional singularities6.1. Resolutions of two-dimensional singularities6.2. The fundamental cycle6.3. Rational singularities6.4. Quitient singularities6.5. Rational double points6.6. Elliptic singularities6.7. Two-dimensional Du Bois singularities6.8. Classification of two-dimensional singularities by kappa7. Higher dimensional singularities7.1. Mixed Hodge structures and Du Bois singularities7.2. Minimal model problem7.3. Higher dimensional canonical singularities and terminal singularities7.4. Higher dimensional 1-Gorenstein singularities8. Deformations of singularities8.1. Change of properties under deformations8.2. Versal deformationsAppendix: Recent resultsReferences
44 citations
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TL;DR: In this article, a general approach to prove K-stability of Fano varieties was developed, which is used to compute the stability thresholds for hypersurfaces at generalized Eckardt points and for cubic surfaces at all points.
Abstract: We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kahler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) to compute the stability thresholds for hypersurfaces at generalized Eckardt points and for cubic surfaces at all points, and (c) to provide a new algebraic proof of Tian's criterion for K-stability, amongst other applications.
43 citations