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The paper shows that the Kodaira vanishing theorem can fail on smooth Fano varieties of any characteristic *p* > 0. Taking cones over some of these varieties, we give the first examples of terminal singularities which are not Cohen-Macaulay. By a different method, we construct a terminal singularity of dimension 3 (the lowest possible) in characteristic 2 which is not Cohen-Macaulay.

*Image: Palmerston, Foreign Office cat, sneaks up on cheese; from his DiploMog twitter feed*

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