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.
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.

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:
where

is a group homomorphism, a character of T.
, T-invariant subspace called the weight subspace of weight.

The decomposition exists because the linear action determines a linear representation and then 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. Alternatively, one uses functional analysis; for example, uses a Hilbert-space [direct sum].
Example: Let be a polynomial ring over an infinite field k. Let act on it as algebra automorphisms by: for
where
Then each is a T-weight vector and so a monomial is a T-weight vector of weight. Hence,
Note if 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.