A generalized pair in birational algebraic geometry consists of a normal variety 
, an effective divisor 
, and nef divisor data on a
higher birational model. For rational coefficients, choose a projective birational
morphism 
and a rational nef divisor 
on 
. Its image 
is required to make 
a rational Cartier divisor.
The nef data are pulled back on higher models and remain fixed under subsequent birational
changes (Birkar and Zhang 2016).
Define 
by
 |
The generalized log discrepancy of a prime divisor 
on a higher model is one minus its coefficient in the corresponding boundary. The
generalized pair is log canonical when these discrepancies are all nonnegative.
With 
,
the generalized log canonical condition reduces to that for an ordinary log
canonical pair.
A generalized canonical pair has log discrepancy at least 1 at every exceptional prime divisor on a higher model. Analytic nefness means that the first Chern class lies in the closure of the Kähler cone in real Bott-Chern cohomology.
