### Abstract

If (X,B) is a log canonical pair, it is natural to study the locus nklt(X,B) of points where the pair is not klt. In particular, this chapter proves Kawamata's adjunction formula: if W is an irreducible subvariety of nklt(X,B), then the restriction of K+B to W is expressed naturally in terms of the canonical class of W. This topic provides a simultaneous generalization of the classical adjunction formula, the formula for the canonical class of a smooth blow up, and Kodaira's formula for the canonical class of a relatively minimal elliptic surface. The ideas have many applications in higher dimensional algebraic geometry.

Original language | English (US) |
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Title of host publication | Flips for 3-folds and 4-folds |

Publisher | Oxford University Press |

ISBN (Electronic) | 9780191717703 |

ISBN (Print) | 9780198570615 |

DOIs | |

State | Published - Sep 1 2007 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- Codimension one adjunction formula
- Iitaka program
- Kawamata
- Lc centre
- Log canonical normalization
- Log canonical purity
- Non-klt locus
- Tie-breaking method

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## Cite this

Kollár, J. (2007). Kodaira's canonical bundle formula and adjunction. In

*Flips for 3-folds and 4-folds*Oxford University Press. https://doi.org/10.1093/acprof:oso/9780198570615.003.0008