Alexeev–Borisov boundedness conjecture for delta-log-canonical weak log Fano varieties

Let δ>0\delta>0 be a real number. A delta-log-canonical weak log Fano pair is a pair (X/pt.,B)(X/\operatorname{pt.},B) of dimension dd that is δ\delta-log canonical and weak log Fano, where BB is a boundary. Alexeev–Borisov boundedness conjecture. The varieties XX for which (X/pt.,B)(X/\operatorname{pt.},B) is a dd-dimensional δ\delta-log-canonical weak log Fano pair form a bounded family.

The paper attributes this conjecture to Alexeev, A. Borisov, and L. Borisov, and states that in characteristic zero its dimension-33 case implies termination of sequences of log flips starting with four-dimensional log canonical pairs of nonnegative Kodaira dimension. The supplied text does not state whether the conjecture had been resolved.

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Primary source

Caucher Birkar, “ACC for log canonical thresholds and termination of log flips”, arXiv:math/0502116 (2005).

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