Sumihiro's theorem

In algebraic geometry, Sumihiro's theorem, introduced by ( Sumihiro 1974 ), states that a normal algebraic variety with an action of a torus can be covered by torus-invariant affine open subsets.

The "normality" in the hypothesis cannot be relaxed. ^[1] The hypothesis that the group acting on the variety is a torus can also not be relaxed. ^[2]

Notes

1. Cox, David A.; Little, John B.; Schenck, Henry K. (2011). Toric Varieties. American Mathematical Soc. ISBN   978-0-8218-4819-7.
2. "Bialynicki-Birula decomposition of a non-singular quasi-projective scheme". MathOverflow. Retrieved 2023-03-10.

References

• Sumihiro, Hideyasu (1974), "Equivariant completion", J. Math. Kyoto Univ., 14: 1–28, doi: 10.1215/kjm/1250523277 .

External links

• Alper, Jarod; Hall, Jack; Rydh, David (2015). "A Luna étale slice theorem for algebraic stacks". Annals of Mathematics. 191 (3). arXiv: 1504.06467 . doi:10.4007/annals.2020.191.3.1.