A divisor complement in this birational-geometric setting replaces a boundary by an effective larger boundary so that a prescribed multiple of the log canonical
divisor becomes Cartier and linearly trivial. An index-
monotonic klt complement of 
near a base point is a rational divisor 
with 
Kawamata log terminal and
 |
where the multiple is integral Cartier on the inverse image of a neighborhood. Linear triviality means triviality of the associated line bundle and is stronger than numerical triviality.
A log canonical complement allows zero log discrepancies; a klt complement keeps all of them positive. Reducing the coefficients of an lc complement need not preserve its linear triviality. The bounded klt complement conjecture asks for an index depending only on specified dimension, discrepancy, and boundary-coefficient data.
