The log Calabi–Yau varieties have Frobenius split type conjecture

Let XX be a log Calabi–Yau variety of characteristic zero, meaning that XX admits an effective Q\mathbb Q-divisor Δ\Delta such that (X,Δ)(X,\Delta) is log canonical and KX+ΔK_X+\Delta is Q\mathbb Q-linearly equivalent to the trivial divisor. A variety is Frobenius split type if it has Frobenius-split reductions modulo a Zariski-dense set of primes.

Frobenius split type conjecture. If XX is a log Calabi–Yau variety of characteristic zero, then XX has Frobenius split type.

This is presented as the characteristic-zero analogue of the result that normal Frobenius-split projective varieties are log Calabi–Yau. The conjecture remains open in the source.

Sources & referencesView supporting material

Primary source

Karen E. Smith and Wenliang Zhang, “Frobenius Splitting in Commutative Algebra”, arXiv:1409.1169 (2014).

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