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).

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 The decomposition exists because the linear action determines (and is determined by) a linear representation

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.

be a polynomial ring over an infinite field k. Let

act on it as algebra automorphisms by: for

is a T-weight vector and so a monomial

for all i, then this is the usual decomposition of the polynomial ring into homogeneous components.

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.

This geometry-related article is a stub.