A log Calabi-Yau pair is a pair 
whose log canonical divisor 
is rationally linearly trivial.
In the log canonical or semi-log canonical settings, 
is an effective rational boundary and the adjoint is rational
Cartier. Rational linear triviality means that some positive Cartier multiple has
trivial associated line bundle. The index is the least
such positive integer. For example, a smooth elliptic
curve with zero boundary is a log Calabi-Yau pair of index one.
Log Calabi-Yau Pair
See also
Calabi-Yau Manifold, Canonical Divisor, Iitaka Fibration, Kawamata Log Terminal Pair, Log Canonical Pair, Semi-Log Canonical PairExplore with Wolfram|Alpha
References
Birkar, C. "Boundedness of Fano Type Fibrations." Ann. Sci. École Norm. Sup. 57, 787-840, 2024. https://doi.org/10.24033/asens.2582.Fujino, O. "Fundamental Theorems for Semi Log Canonical Pairs." Algebr. Geom. 1, 194-228, 2014. https://doi.org/10.14231/AG-2014-011.Cite this as:
Weisstein, Eric W. "Log Calabi-Yau Pair." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LogCalabi-YauPair.html