Torus action

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:

$V=\bigoplus _{\chi }V_{\chi }$

where

• $\chi :T\to \mathbb {G} _{m}$ is a group homomorphism, a character of T.
• $V_{\chi }=\{v\in V|t\cdot v=\chi (t)v\}$, T-invariant subspace called the weight subspace of weight $\chi$.

The decomposition exists because the linear action determines (and is determined by) a linear representation $\pi :T\to \operatorname {GL} (V)$ and then $\pi (T)$ 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 ($\pi$ is called rational; see below for an example). Alternatively, one uses functional analysis; for example, uses a Hilbert-space direct sum.

Example: Let $S=k[x_{0},\dots ,x_{n}]$ be a polynomial ring over an infinite field k. Let $T=\mathbb {G} _{m}^{r}$ act on it as algebra automorphisms by: for $t=(t_{1},\dots ,t_{r})\in T$

$t\cdot x_{i}=\chi _{i}(t)x_{i}$

where

$\chi _{i}(t)=t_{1}^{\alpha _{i,1}}\dots t_{r}^{\alpha _{i,r}},$ $\alpha _{i,j}$ = integers.

Then each $x_{i}$ is a T-weight vector and so a monomial $x_{0}^{m_{0}}\dots x_{r}^{m_{r}}$ is a T-weight vector of weight $\sum m_{i}\chi _{i}$. Hence,

$S=\bigoplus _{m_{0},\dots m_{n}\geq 0}S_{m_{0}\chi _{0}+\dots +m_{n}\chi _{n}}.$

Note if $\chi _{i}(t)=t$ 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.