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 variety with a torus acting on it in such a way that there is a dense orbit is called a toric variety.

The Białynicki-Birula decomposition says that a smooth algebraic T-variety admits a T-stable celluar decomposition.

It is often described as algebraic Morse theory.

...
Wikipedia