Introduction to complements

We recall the notion of a log canonical (lc) complement and
its natural version -- an lc n-complement. It is expected
a boundedness of $n$-complements in any fixed dimension.
A related conjecture with additional properties of complements
will be discussed in details. The conjecture was completely
established only in dimension $1$ and under certain restrictions
in any dimension (Birkar). Some important applications
of the conjecture will be presented.