Let $(X,B)$ be a log canonical pair equipped with a projective morphism $X \to Z$. Following Birkar [B], define a strong $n$-complement of $K_X+B$ over a point $z\in Z$ to be of the form $K_X + B^+$ where over some neighborhood of $z$, we have the following three conditions are satisfied:
- $(X,B^+)$ is log canonical.
- $n(K_X+B^+) \sim 0$, and
- $B^+ \geq B$.
Prior to giving the definition, Birkar writes that the theory of complements is motivated by the study of the systems $|-n(K_X+B)|$ for $n \in \mathbb{N}$, in a relative sense over $Z$. Of course, this is only interesting when these systems are not all empty, e.g., in the Fano case.
I hate to ask vague questions, but I am struggling to intuitively grasp what strong $n$-complements, and more generally, $n$-complements, give us.
Vague, hopefully, sensical request: If anyone has any (preferably geometric) intuition for $n$-complements, or motivating results concerning them that would assist me in getting an understanding of these things, it would be very much appreciated.
[B] Birkar, C., Birational Geometry of Algebraic Varieties, arXiv: 1801.00013v1, (2017).