Shokurov's lower semicontinuity conjecture for minimal log discrepancies

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Let XX be a normal Q\mathbb{Q}-Gorenstein variety, and let Δ\Delta be an R\mathbb{R}-Weil divisor on XX such that KX+ΔK_X+\Delta is R\mathbb{R}-Cartier. For each dd, let X(d)X^{(d)} denote the set of points of dimension dd, and let

mld⁡(X,Δ):X(d)→  R∪{−∞}\operatorname{mld}_{(X,\Delta)}:X^{(d)}\xrightarrow{\ \ }\mathbb{R}\cup\{-\infty\}

be the minimal-log-discrepancy function. Shokurov's lower semicontinuity conjecture. For each dd, the function mld⁡(X,Δ)\operatorname{mld}_{(X,\Delta)} on X(d)X^{(d)} is lower semicontinuous.

This is the second conjecture in the paper's discussion of Shokurov's conjectures about minimal log discrepancies. The supplied text does not state whether it has been resolved.

References

Primary source

Christian Lehn and Gianluca Pacienza, “On the log minimal model program for irreducible symplectic varieties”, arXiv:1405.5649 (2014).

Additional references

2 papers in this index state this conjecture (2004–2014). The statement above is taken from the most recent of them; the others are arXiv:math/0409254.

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