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:

  1. $(X,B^+)$ is log canonical.
  2. $n(K_X+B^+) \sim 0$, and
  3. $B^+ \geq B$.

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).

  • $\begingroup$ I don't have any familiarity with this notion, but I have some vague thoughts. Condition (3) says that you are only adding effective divisors, and (1) says that doing so does not make the pair more singular. I would regard these as basically standard technical conditions in birational geometry. Condition (2) is then real key. First of all, a $1$-complement is just an element of the anticanonical system $|-K_X|$ satisfying (1) and (3). A cheap answer would then be that an $n$-complement is some kind of $n$-torsion version of a $1$-complement. OTOH, let's see... $\endgroup$ – Tabes Bridges May 31 at 4:58
  • $\begingroup$ ...$nB^+ \in |-nK_X|$. This means that rather than being an element of the $n$-anticanonical system (anti-($n$-canonical) system???), $B^+$ is roughly the support of a divisor in the system-that-has-no-concise-name. Perhaps this is all obvious to you? But I've never been convinced that many of the definitions in birational geometry are anything other than notational conveniences, rather than genuine geometric ideas. $\endgroup$ – Tabes Bridges May 31 at 4:58

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