Torus action
# 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 algebraic T-variety admits a T-stable cellular decomposition.

It is often described as algebraic Morse theory.

