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. Toric Varieties
  2. Bialynicki-Birula decomposition of a non-singular quasi-projective scheme.

References
  • Sumihiro, Hideyasu (1974), "Equivariant completion", J. Math. Kyoto Univ., 14: 1–28, doi:10.1215/kjm/1250523277.
  • Alper, Jarod; Hall, Jack; Rydh, David (2015). "A Luna étale slice theorem for algebraic stacks". arXiv:1504.06467 [math.AG].


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