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 Σd_iD_i where the D_i are the distinct irreducible components of D and all coefficients are rational numbers with 0≤d_i≤1.

A logarithmic pair, or 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 form d log(z) = dz/z along components of the divisor given locally by z=0.

References

• Iitaka, Shigeru (1976), "Logarithmic forms of algebraic varieties", Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics, 23 (3): 525–544, ISSN 0040-8980, MR 0429884 
• Matsuki, Kenji (2002), Introduction to the Mori program, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98465-0, MR 1875410