Pure and Applied Mathematics Quarterly

Volume 11 (2015)

Number 1

Valuations and log canonical thresholds

Pages: 49 – 86

DOI: http://dx.doi.org/10.4310/PAMQ.2015.v11.n1.a3

Author

Zhengyu Hu (Center of Mathematical Sciences, Zhejiang University, Hangzhou, China)

Abstract

The goal of this paper is to continue the investigation of valuative quasi-plurisubharmonic functions (qpsh for short) on certain valuation spaces of a regular scheme, in line with the works [4], [5], [6] of Boucksom, Favre, Jonsson, and the works [31], [32] of Jonsson, Mustaţă. We divide this paper into two parts. In the first part we mainly discuss those valuations which compute the log canonical thresholds of qpsh functions. We expect them to be useful for the conjecture [[31], Conjecture B] raised by Jonsson and Mustaţă. In the second part we define the restriction of a valuative qpsh function to a regular subscheme and prove a number of expected results including the restriction theorem and the inversion of adjunction. We also treat some applications in complex algebraic geometry such as extensions of pluri-canonical forms on a dlt pair under an abundance assumption.

Keywords

pluri-canonical extensions, log canonical thresholds, multiplier ideals, valuations

2010 Mathematics Subject Classification

Primary 14F18. Secondary 12J20.

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