Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

A gap theorem for minimal log discrepancies of noncanonical singularities in dimension three


Author: Chen Jiang
Journal: J. Algebraic Geom. 30 (2021), 759-800
DOI: https://doi.org/10.1090/jag/759
Published electronically: June 9, 2021
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Abstract: We show that there exists a positive real number $\delta >0$ such that for any normal quasi-projective $\mathbb {Q}$-Gorenstein $3$-fold $X$, if $X$ has worse than canonical singularities, that is, the minimal log discrepancy of $X$ is less than $1$, then the minimal log discrepancy of $X$ is not greater than $1-\delta$. As applications, we show that the set of all noncanonical klt Calabi–Yau $3$-folds are bounded modulo flops, and the global indices of all klt Calabi–Yau $3$-folds are bounded from above.


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Chen Jiang
Affiliation: Shanghai Center for Mathematical Sciences, Fudan University, Jiangwan Campus, 2005 Songhu Road, Shanghai, 200438, People’s Republic of China
Email: chenjiang@fudan.edu.cn

Received by editor(s): August 26, 2019
Received by editor(s) in revised form: November 21, 2019
Published electronically: June 9, 2021
Additional Notes: Part of this work was supported by the National Science Foundation under Grant No. DMS-1440140.
Article copyright: © Copyright 2021 University Press, Inc.