Frobenius splitting

In algebraic geometry, a Frobenius splitting, introduced by Mehta and Ramanathan,Template:Sfnp is a splitting of the injective morphism ${𝒪}_X\to F_{\ast }{𝒪}_X$ from a structure sheaf ${𝒪}_X$ of a characteristic $p>0$ variety $X$ to its image $F_{\ast }{𝒪}_X$ under the Frobenius endomorphism $F_{\ast }$.

Brion and Kumar give a detailed discussion of Frobenius splittings.Template:Sfnp

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

References

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