Frobenius splitting

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

Brion and Kumar give a detailed discussion of Frobenius splittings.

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

• Brion, Michel; Kumar, Shrawan (2005). Frobenius Splitting Methods in Geometry and Representation Theory. Progress in Mathematics. Vol. 231. Boston: Birkhäuser. ISBN 978-0-8176-4191-7. MR 2107324.

• Mehta, V. B.; Ramanathan, A. (1985). "Frobenius splitting and cohomology vanishing for Schubert varieties". Annals of Mathematics. 122 (1): 27–40. JSTOR 1971368. MR 0799251.