Shokurov's minimal discrepancy conjecture

Let VV be a normal affine variety of complex dimension nn with an isolated singularity at 00 that is numerically Q\mathbb{Q}-Gorenstein. Write min.disc(V,0)\operatorname{min.disc}(V,0) for its minimal discrepancy. Shokurov's conjecture.

min.disc(V,0)n1\operatorname{min.disc}(V,0)\leq n-1

Moreover, the singularity is smooth if and only if min.disc(V,0)=n1\operatorname{min.disc}(V,0)=n-1. This conjecture relates minimal discrepancy to smoothness and is open for varieties of complex dimension greater than 33.

Sources & referencesView supporting material

Primary source

Shahnaz Shamim Shahul, “Topological Obstructions to Dynamical Convexity”, arXiv:2512.03893 (2026).

Additional references

9 papers in this index state this conjecture (1996–2025). The statement above is taken from the most recent of them; the others are arXiv:2409.10275, arXiv:2104.15072, arXiv:1712.03820, arXiv:1612.05349, arXiv:1404.1857, arXiv:1304.7012, arXiv:math/9903060, arXiv:alg-geom/9608013.

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