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:

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 algebraic T-variety admits a T-stable cellular decomposition.

It is often described as algebraic Morse theory.[1]

See also

References

  • Altmann, Klaus; Ilten, Nathan Owen; Petersen, Lars; Süß, Hendrik; Vollmert, Robert (2012-08-15). "The Geometry of T-Varieties". arXiv:1102.5760. doi:10.4171/114. ISBN 978-3-03719-114-9. Cite journal requires |journal= (help)
  • 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.
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