Cartier Isomorphism for Toric Varieties
نویسنده
چکیده
is an isomorphism. Here F : X −→ X denotes the Frobenius morphism on X and H denotes the a cohomology sheaf of F∗Ω•X . If the variety is not smooth, not much is known about the properties of the Cartier operator and the poor behaviour of the deRham complex in this case makes its study difficult. If one substitutes the deRham complex with the Zariski-deRham complex the situation is better. For example, the Zariski differentials, though not locally free, are reflexive and there is a natural duality pairing between them. We show how to extend the Cartier operator in a natural way to the Zariski differentials. Using a description of the Zariski-deRham complex due to Danilov [Dan78] we show that this newly defined Cartier operator is an isomorphism for toric varieties. Moreover, it is induced by a split injection
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