ACC for local volumes and boundedness of singularities

Han, Jingjun; Liu, Yuchen; Qi, Lu

doi:10.1090/jag/799

Authors: Jingjun Han, Yuchen Liu and Lu Qi
Journal: J. Algebraic Geom. 32 (2023), 519-583
DOI: https://doi.org/10.1090/jag/799
Published electronically: November 1, 2022
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Abstract | References | Additional Information

Abstract: The ascending chain condition (ACC) conjecture for local volumes predicts that the set of local volumes of Kawamata log terminal (klt) singularities $x\in (X,\Delta )$ satisfies the ACC if the coefficients of $\Delta$ belong to a descending chain condition (DCC) set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of $\delta$-plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hold in dimension 2 as well as for 3-dimensional terminal singularities.

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Additional Information

Jingjun Han
Affiliation: Shanghai Center for Mathematical Sciences, Fudan University, Jiangwan Campus, Shanghai 200438, People’s Republic of China; Department of Mathematics, The University of Utah, Salt Lake City, Utah 84112; and Mathematical Sciences Research Institute, Berkeley, California 94720
MR Author ID: 1041405
ORCID: 0000-0001-9339-8502
Email: hanjingjun@fudan.edu.cn, jhan@math.utah.edu, jhan@msri.org

Yuchen Liu
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
MR Author ID: 1266987
ORCID: 0000-0002-4281-7057
Email: yuchenl@northwestern.edu

Lu Qi
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
MR Author ID: 1451607
Email: luq@princeton.edu

Received by editor(s): April 25, 2021
Published electronically: November 1, 2022
Additional Notes: The first author was supported by the National Key Research and Development Program of China (Grant No. 2020YFA0713200) and a grant from the Simons Foundation (Grant Number 814268, MSRI). The second author was partially supported by the NSF Grant No. DMS-2148266 (transferred from the NSF Grant No. DMS-2001317). Jingjun Han is the corresponding author.
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