Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 2

Metrics on twisted pluricanonical bundles and finite generation of twisted canonical rings

Pages: 451 – 485

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n2.a2


Bojie He (School of Mathematical Sciences, Peking University, Beijing, China)

Xiangyu Zhou (Institute of Mathematics, Academy of Mathematics and Systems Science, C.A.S., Beijing, China)


In this paper, we first introduce the notion of admissible Bergman metrics. Then we establish a connection between singularities of admissible Bergman metrics and finite generation of twisted pluricanonical rings with $m$-multiplier ideal sheaves on smooth projective pairs. It involves an analytic approach to Boucksom’s result about asymptotic multiplier ideal of a graded system of ideals. In the end, we give a few applications of our main theorem.


multiplier ideal sheaf, canonical ring, Siu-type metric, admissible Bergman metric, pluricanonical bundle

2010 Mathematics Subject Classification

14C30, 14F18, 32J25, 32L10, 32U05

The second author is partially supported by the National Natural Science Foundation of China (grant no. 11688101 and no. 12288201).

Received 6 July 2022

Accepted 13 July 2022

Published 7 April 2023