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:
- [math]\displaystyle{ V = \bigoplus_{\chi} V_{\chi} }[/math]
where
- [math]\displaystyle{ \chi: T \to \mathbb{G}_m }[/math] is a group homomorphism, a character of T.
- [math]\displaystyle{ V_{\chi} = \{ v \in V | t \cdot v = \chi(t) v \} }[/math], T-invariant subspace called the weight subspace of weight [math]\displaystyle{ \chi }[/math].
The decomposition exists because the linear action determines (and is determined by) a linear representation [math]\displaystyle{ \pi: T \to \operatorname{GL}(V) }[/math] and then [math]\displaystyle{ \pi(T) }[/math] 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 ([math]\displaystyle{ \pi }[/math] is called rational; see below for an example). Alternatively, one uses functional analysis; for example, uses a Hilbert-space direct sum.
Example: Let [math]\displaystyle{ S = k[x_0, \dots, x_n] }[/math] be a polynomial ring over an infinite field k. Let [math]\displaystyle{ T = \mathbb{G}_m^r }[/math] act on it as algebra automorphisms by: for [math]\displaystyle{ t = (t_1, \dots, t_r) \in T }[/math]
- [math]\displaystyle{ t \cdot x_i = \chi_i(t) x_i }[/math]
where
- [math]\displaystyle{ \chi_i(t) = t_1^{\alpha_{i, 1}} \dots t_r^{\alpha_{i, r}}, }[/math] [math]\displaystyle{ \alpha_{i, j} }[/math] = integers.
Then each [math]\displaystyle{ x_i }[/math] is a T-weight vector and so a monomial [math]\displaystyle{ x_0^{m_0} \dots x_r^{m_r} }[/math] is a T-weight vector of weight [math]\displaystyle{ \sum m_i \chi_i }[/math]. Hence,
- [math]\displaystyle{ S = \bigoplus_{m_0, \dots m_n \ge 0} S_{m_0 \chi_0 + \dots + m_n \chi_n}. }[/math]
Note if [math]\displaystyle{ \chi_i(t) = t }[/math] 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. doi:10.4171/114. ISBN 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.
 |  |
