Torus action explained

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:

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 (and is determined by) 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 (

is called rational; see below for an example). 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 = integers.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.^[1]

See also

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

References

• Book: Altmann. Klaus. Ilten. Nathan Owen. Petersen. Lars. Süß. Hendrik. Vollmert. Robert. 2012-08-15. The Geometry of T-Varieties. 1102.5760. 10.4171/114. 978-3-03719-114-9.
• A. Bialynicki-Birula, "Some Theorems on Actions of Algebraic Groups," Annals of Mathematics, Second Series, Vol. 98, No. 3 (Nov., 1973), pp. 480–497
• M. Brion, C. Procesi, Action d'un tore dans une variété projective, in Operator algebras, unitary representations, and invariant theory (Paris 1989), Prog. in Math. 92 (1990), 509–539.

Notes and References

1. Web site: Konrad Voelkel » Białynicki-Birula and Motivic Decompositions «.