Frobenius splitting

In algebraic geometry, a Frobenius splitting, introduced by Mehta and Ramanathan,^[1] is a splitting of the injective morphism ${\displaystyle {\mathcal {O}}_{X}\to F_{*}{\mathcal {O}}_{X}}$ from a structure sheaf ${\displaystyle {\mathcal {O}}_{X}}$ of a characteristic ${\displaystyle p>0}$ variety ${\displaystyle X}$ to its image ${\displaystyle F_{*}{\mathcal {O}}_{X}}$ under the Frobenius endomorphism ${\displaystyle F_{*}}$.

Brion and Kumar give a detailed discussion of Frobenius splittings.^[2]

A fundamental property of Frobenius-split projective schemes ${\displaystyle X}$ is that the higher cohomology ${\displaystyle H^{i}(X,L)}$ (${\displaystyle i>0}$) of ample line bundles ${\displaystyle L}$ vanishes.