ACC for log canonical thresholds

By Christopher D. Hacon, J. McKernan. Annals of Mathematics, 2012

305 citations recorded by the source corpus.

We show that log canonical thresholds satisfy the ACC

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b93c2216f1de44c61efb650a0108852ed843d8ed

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In Computational Theory and Mathematics

1. M. Mustaţă
2. Qi’an Guan
3. Chi Li
4. De-Qi Zhang
5. Xiaowei Wang
6. Xiangyu Zhou
7. C. Birkar

In Geometry and Topology

1. Chi Li
2. Xiaowei Wang
3. Qi’an Guan
4. H. Blum
5. Xiangyu Zhou
6. Mattias Jonsson
7. Yuchen Liu
8. Robert J. Berman

Citations in the corpus

• Anti-pluricanonical systems on Fano varieties (2016, methodology): “The main tools used in this paper are the minimal model program [36][8], the theory of complements [40][39][42], creating families of non-klt centres using volumes [17][18][31, §6], and the theory of generalised polarised pairs [9].”
• Uniqueness of K-polystable degenerations of Fano varieties (2018, unknown): “= 0. Hence, if k≫ 0, Proposition 2.2 yields an extraction τk: ZW k → Zof Wk such that the pair (ZW k ,τk −1 ∗ (Γ+Z0)+(1− ak)Wk) is lc. Since lim k→∞ (1 − ak) = 1, the ACC for log canonical thresholds [HMX14] implies (ZW k ,τk −1 ∗ (Γ+Z0)+Wk) must be lc when k≫ 0. From now on, we fix a positive intege”
• Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs (2014, methodology): “1) and use the concept of potential birationality [10] to reduce to the special case of Proposition 3.”
• Algebraicity of the metric tangent cones and equivariant K-stability (2018, background): “to (X,D+(1−δ)ck ·Ik) is less than ǫ, and (X (2) 0 ,D (2) 0 +(1−δ)ck ·in(Ik)) is log canonical for a suitable choice of small δ (see (18)). The it follows from the ACC of log canonical thresholds (see [HMX14]) that (Y(2) k ,E (2) k +(µ (2))−1 ∗ D 2 +(Y(2) k )t) is log canonical for any t ∈C. The rela”
• Finiteness of algebraic fundamental groups (2012, methodology): “above by a constant C= C(M,n) which only depends on Mand n= dim(X). Thus deg(f) is bounded from above by C (−KX−∆)n . As we mentioned in the introduction, here we use the strong boundedness result in [HMX12]. We can argue more straightforwardly using Theorem 1 in the same dimension. Later we will se”
• Effectiveness of Demailly’s strong openness conjecture and related problems (2014, methodology): “In [10], the authors proved that the above conjecture is implied by the ACC conjecture (see [36] or [28]) whose proofs can referred to [12,24] Note that c log | f | is a plurisubharmonic function, then we [21] replaced c log | f | by general plurisubharmonic functions, and obtained the following low”
• Singularities of linear systems and boundedness of Fano varieties (2016, background): “f T on Xis a closed point x. Then there is a Q-divisor Λ ≥ 0 such that • nΛ is integral, • mA−Λ is ample, • (X,Λ) is lc near x, and • T is a lc place of (X,Λ). Proof. Step 1. By ACC for lc thresholds [11], there is ǫ′ ∈ (0,ǫ) depending only on dso that if (Y,(1−ǫ′)S) is a Q-factorial klt pair of dim”