The bounded klt complement conjecture predicts a uniformly bounded index for monotonic klt complements of Fano type contractions
with a fixed positive log-discrepancy bound and a finite
set of rational boundary coefficients. Fix total dimension 
,
,
and finite ![I subset [0,1] intersection Q](/images/equations/BoundedKltComplementConjecture/Inline3.svg)
. For an 
-log-canonical pair 
with coefficients in 
and a Fano type contraction 
with 
relatively nef, the predicted index depends only on 
.
Bounded Klt Complement Conjecture
See also
Divisor Complement, Fano Contraction, Fano Type Variety, Kawamata Log Terminal Pair, Log Canonical Threshold, Log DiscrepancyExplore with Wolfram|Alpha
References
Chen, B. "Boundedness of klt Complements on Fano Fibrations over Surfaces." 19 Sep 2025. https://arxiv.org/abs/2403.01154v5.Cite this as:
Weisstein, Eric W. "Bounded Klt Complement Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BoundedKltComplementConjecture.html