Frobenius splitting

In mathematics, a Frobenius splitting, introduced by Mehta and Ramanthan (1985), is a splitting of the injective morphism O_X→F_*O_X from a structure sheaf O_X of a characteristic p > 0 variety X to its image F_*O_X under the Frobenius endomorphism F_*.

Brion & Kumar (2005) give a detailed discussion of Frobenius splittings.

A fundamental property of Frobenius-split projective schemes X is that the higher cohomology Hⁱ(X,L) (i > 0) of ample line bundles L vanishes.