In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraic torus on the variety. A variety equipped with an action of a torus T is called a T-variety. In differential geometry, one considers an action of a real or complex torus on a manifold (or an orbifold). A normal algebraic variety with a torus acting on it in such a way that there is a dense orbit is called a toric variety (for example, orbit closures that are normal are toric varieties).
Linear action of a torus A linear action of a torus can be simultaneously diagonalized, after extending the base field if necessary: if a torus T is acting on a finite-dimensional vector space V, then there is a direct sum decomposition:
V = ⨁ χ V χ {\displaystyle V=\bigoplus _{\chi }V_{\chi }}
where
χ : T → G m {\displaystyle \chi :T\to \mathbb {G} _{m}} is a group homomorphism, a character of T.
V χ = { v ∈ V | t ⋅ v = χ ( t ) v } {\displaystyle V_{\chi }=\{v\in V|t\cdot v=\chi (t)v\}} , T-invariant subspace called the weight subspace of weight χ {\displaystyle \chi } . The decomposition exists because the linear action determines (and is determined by) a linear representation π : T → GL ( V ) {\displaystyle \pi :T\to \operatorname {GL} (V)} and then π ( T ) {\displaystyle \pi (T)} consists of commuting diagonalizable linear transformations, upon extending the base field. If V does not have finite dimension, the existence of such a decomposition is tricky but one easy case when decomposition is possible is when V is a union of finite-dimensional representations ( π {\displaystyle \pi } is called rational; see below for an example). Alternatively, one uses functional analysis; for example, uses a Hilbert-space direct sum. Example: Let S = k [ x 0 , … , x n ] {\displaystyle S=k[x_{0},\dots ,x_{n}]} be a polynomial ring over an infinite field k. Let T = G m r {\displaystyle T=\mathbb {G} _{m}^{r}} act on it as algebra automorphisms by: for t = ( t 1 , … , t r ) ∈ T {\displaystyle t=(t_{1},\dots ,t_{r})\in T}
t ⋅ x i = χ i ( t ) x i {\displaystyle t\cdot x_{i}=\chi _{i}(t)x_{i}}
where
χ i ( t ) = t 1 α i , 1 … t r α i , r , {\displaystyle \chi _{i}(t)=t_{1}^{\alpha _{i,1}}\dots t_{r}^{\alpha _{i,r}},} α i , j {\displaystyle \alpha _{i,j}} = integers. Then each x i {\displaystyle x_{i}} is a T-weight vector and so a monomial x 0 m 0 … x r m r {\displaystyle x_{0}^{m_{0}}\dots x_{r}^{m_{r}}} is a T-weight vector of weight ∑ m i χ i {\displaystyle \sum m_{i}\chi _{i}} . Hence,
S = ⨁ m 0 , … m n ≥ 0 S m 0 χ 0 + ⋯ + m n χ n . {\displaystyle S=\bigoplus _{m_{0},\dots m_{n}\geq 0}S_{m_{0}\chi _{0}+\dots +m_{n}\chi _{n}}.}
Note if χ i ( t ) = t {\displaystyle \chi _{i}(t)=t} for all i, then this is the usual decomposition of the polynomial ring into homogeneous components.
Białynicki-Birula decomposition The Białynicki-Birula decomposition says that a smooth projective algebraic T-variety admits a T-stable cellular decomposition. It is often described as algebraic Morse theory.
See also Sumihiro's theorem GKM variety Equivariant cohomology monomial ideal
References
Altmann, Klaus; Ilten, Nathan Owen; Petersen, Lars; Süß, Hendrik; Vollmert, Robert (2012-08-15). The Geometry of T-Varieties. arXiv:1102.5760. doi:10.4171/114. ISBN 978-3-03719-114-9. A. Bialynicki-Birula, "Some Theorems on Actions of Algebraic Groups," Annals of Mathematics, Second Series, Vol. 98, No. 3 (Nov., 1973), pp. 480–497 M. Brion, C. Procesi, Action d'un tore dans une variété projective, in Operator algebras, unitary representations, and invariant theory (Paris 1989), Prog. in Math. 92 (1990), 509–539.