Białynicki-Birula Decomposition

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:
:$V = \bigoplus_ V_$
where
*$\chi: T \to \mathbb_m$ is a group homomorphism, a character of ''T''.
*$V_ = \$, ''T''-invariant subspace called the weight subspace of weight $\chi$.

The decomposition exists because the linear action determines (and is determined by) a linear representation $\pi: T \to \operatorname(V)$ and then $\pi(T)$ 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 ($\pi$ is called rational; see below for an example). Alternatively, one uses functional analysis; for example, uses a Hilbert-space direct sum.

Example: Let S = k _0, \dots, x_n/math> be a polynomial ring over an infinite field ''k''. Let $T = \mathbb_m^r$ act on it as algebra automorphisms by: for $t = (t_1, \dots, t_r) \in T$
:$t \cdot x_i = \chi_i(t) x_i$
where
:$\chi_i(t) = t_1^ \dots t_r^,$ $\alpha_$ = integers.
Then each $x_i$ is a ''T''-weight vector and so a monomial $x_0^ \dots x_r^$ is a ''T''-weight vector of weight $\sum m_i \chi_i$. Hence,
:$S = \bigoplus_ S_.$
Note if $\chi_i(t) = t$ 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.

See also

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

References

*
*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.

Algebraic geometry
Algebraic groups
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