On generalized minimal log discrepancy

Chen, Wei‐Chung; Gongyo, Yoshinori; Nakamura, Yusuke

doi:10.48550/arxiv.2112.09501

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2022

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“…The ACC conjecture for MLDs in dimension 2 was proved by Alexeev [1] and Shokurov [19] independently. We refer readers to [5,Lemma 4.5,Theorems B.1 and B.4] and [9, Theorem 1.5, Appendix A] for detailed proofs following Alexeev's and Shokurov's arguments, respectively, see also [6,Theorem 1.5]. The ACC conjecture for MLDs is still widely open in dimension 3 in general.…”

Section: Introductionmentioning

confidence: 99%

On Boundedness of Divisors Computing Minimal Log Discrepancies for Surfaces

Han

Luo²

2022

J. Inst. Math. Jussieu

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Let $\Gamma $ be a finite set, and $X\ni x$ a fixed kawamata log terminal germ. For any lc germ $(X\ni x,B:=\sum _{i} b_iB_i)$ , such that $b_i\in \Gamma $ , Nakamura’s conjecture, which is equivalent to the ascending chain condition conjecture for minimal log discrepancies for fixed germs, predicts that there always exists a prime divisor E over $X\ni x$ , such that $a(E,X,B)=\mathrm {mld}(X\ni x,B)$ , and $a(E,X,0)$ is bounded from above. We extend Nakamura’s conjecture to the setting that $X\ni x$ is not necessarily fixed and $\Gamma $ satisfies the descending chain condition, and show it holds for surfaces. We also find some sufficient conditions for the boundedness of $a(E,X,0)$ for any such E.

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Section: Introductionmentioning

confidence: 99%