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A counterexample to Liu's conjecture on Fano indices in dimension four

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove that Liu's conjecture on Fano indices fails in dimension four: $1389$ is the index of a terminal Calabi--Yau fourfold and of a four-dimensional log canonical singularity, but it is not the Fano index of any canonical weighted projective space of dimension four.

A mix of human-written and AI-generated textHuman understanding: some partsCalabi–Yau varietyFano indexindexlog canonical singularityweighted projective space

Oct 22, 2026PDF ↗View work →

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A counterexample to Liu's conjecture on Fano indices in dimension four

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