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:

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

where

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

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

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

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

where

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

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

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

Note if ${\displaystyle \chi _{i}(t)=t}$ 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.^[1]

See also

• Sumihiro's theorem
• GKM variety
• Equivariant cohomology
• monomial ideal