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In characteristic zero, if a quotient variety has a crepant resolution, theEuler characteristic of the crepant resolution is equal to the number ofconjugacy classes of the acting group, by Batyrev's theorem. This is one of theMcKay correspondence. It is natural to consider the analogue statement in thepositive characteristic. In this paper, we present sequences of crepantresolutions of quotient varieties in the positive characteristic and show thatone of the sequences gives a counterexample to the analogue statement ofBatyrev's theorem.

Crepant resolutions of quotient varieties in positive characteristics and their Euler characteristics