﻿ Logarithmic pair : definition of Logarithmic pair and synonyms of Logarithmic pair (English)

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# definition - Logarithmic pair

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# Logarithmic pair

In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities. They were studied by Iitaka (1976).

## Definition

A boundary Q-divisor on a variety is a Q-divisor D of the form ΣdiDi where the Di are the distinct irreducible components of D and all coefficients are rational numbers with 0≤di≤1.

A logarithmic pair, lor log pair for short, is a pair (X,D) consisting of a normal variety X and a boundary Q-divisor D.

The log canonical divisor of a log pair (X,D) is K+D where K is the canonical divisor of X.

A logarithmic 1-form on a log pair (X,D) is allowed to have logarithmic singularities of the formd log(z) = dz/z along components of the divisor given locally by z=0.

## References

• Iitaka, Shigeru (1976), [Expression error: Missing operand for > "Logarithmic forms of algebraic varieties"], Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics 23 (3): 525–544, MR0429884, ISSN 0040-8980
• Matsuki, Kenji (2002), [Expression error: Missing operand for > Introduction to the Mori program], Universitext, Berlin, New York: Springer-Verlag, MR1875410, ISBN 978-0-387-98465-0

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