Discreteness of log discrepancies over log canonical triples on a fixed pair
type: Item
instance of: scholarly article
title: Discreteness of log discrepancies over log canonical triples on a fixed pair
zbMATH Open document ID: 1343.14014
DOI: 10.1090/S1056-3911-2014-00630-5
author: Masayuki Kawakita
published in: Journal of Algebraic Geometry
publication date: 2014-10-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1204.5248
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14F18
zbMATH DE Number: 6353751
MaRDI profile type: Publication
arXiv ID: 1204.5248
cites work: On minimal log discrepancies
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cites work: Log canonical thresholds on varieties with bounded singularities
cites work: Limits of log canonical thresholds
cites work: Inversion of adjunction for local complete intersection varieties
cites work: Towards boundedness of minimal log discrepancies by the Riemann-Roch theorem
cites work: Ideal-adic semi-continuity problem for minimal log discrepancies
cites work: 3-fold log models
cites work: Q3409139
cites work: Desingularization of quasi-excellent schemes in characteristic zero
OpenAlex ID: W2128706163
review text: Let \((X, \Delta)\) be a pair of data with a normal variety \(X\) and a boundary \(\Delta\) which is an effective \(\mathbb{R}\)-divisor such that \(K_X+ \Delta\) is an \(\mathbb{R}\)-Cartier \(\mathbb{R}\)-divisor. In the triple \((X, \Delta, \mathfrak{a})\), \(\mathfrak{a}\) is a formal product of finite many coherent ideal sheaves \({\mathfrak{a}}_j\) with real exponents \(r_j\in {\mathbb{R}}_{\geq 0}\): NEWLINE\[NEWLINE\mathfrak{a}=\prod_{j=1}^e {\mathfrak{a}}_j^{r_j}. NEWLINE\]NEWLINENEWLINENEWLINELet \(E\) be a prime divisor on a normal variety \(X'\) and \(\phi: X'\rightarrow X\) a proper birational morphism. The log discrepancy \(a_E(X, \Delta, \mathfrak{a})\) of \(E\) with respect to the triple \((X, \Delta, \mathfrak{a})\) is defined by NEWLINE\[NEWLINE a_E(X, \Delta, \mathfrak{a})=1+{\text{ord}_E}(K_{X'}-\phi^*(K_{X}+\Delta))- {\text{ord}_E}\mathfrak{a}, NEWLINE\]NEWLINE where \( {\text{ord}_E}\mathfrak{a}= \sum_{j=1}^e r_j {\text{ord}_E}\mathfrak{a}_j\).NEWLINENEWLINEThe main theorem of this paper isNEWLINENEWLINETheorem 1.1. In the triple \((X, \Delta, \mathfrak{a})\), the set NEWLINE\[NEWLINE \{ a_E(X, \Delta, \mathfrak{a}) | \mathfrak{a}_j \subset {\mathcal{O}}_X, (X, \Delta, \mathfrak{a})\text{ lc at }\eta_{\phi (E)} \} NEWLINE\]NEWLINE is discrete in \(\mathbb{R} \), where \(r_1,\dots, r_e\in {\mathbb{R}}_{\geq 0} \) and \(\eta_{\phi (E)}\) is the generic point of \(\phi (E)\).NEWLINENEWLINETheorem 1.1 is proved by developing the approach to the ACC for log canonical thresholds due to de Fernex, Ein, Mustat\(\breve{\text{a}}\) and Koll\(\acute{\text{a}}\)r.NEWLINENEWLINEThe author also discusses some applications of Theorem 1.1. For example, an immediate application of Theorem 1.1 is that the set NEWLINE\[NEWLINE\{ {\text{mld}_{\eta_Z}} (X, \Delta, \mathfrak{a}) | \mathfrak{a}_j \subset {\mathcal{O}}_X, Z\subset X\} NEWLINE\]NEWLINE is finite, where the minimal log discrepancy \({\text{mld}_{\eta_Z}}(X, \Delta, \mathfrak{a}) \) at \(\eta_Z\) is the …
reviewed by: Jing Zhang
description: scientific article; zbMATH DE number 6353751