A log canonical pair is a normal variety 
with an effective rational boundary divisor 
such that 
is a rational Cartier divisor
and all log discrepancies are nonnegative.
Here 
is a canonical divisor and a boundary has coefficients
between zero and one. For a normal birational model 
, define 
by
 |
The log discrepancy of a prime divisor 
on 
is 
.
Requiring these quantities to be nonnegative on every
birational model gives the definition (Kollár and Mori 1998). A smooth
variety with 
is log canonical.
Log canonical pairs allow controlled singularities in the minimal model program. A generalized pair adds nef divisor data on a higher birational model.
