Białynicki-Birula decomposition for reductive groups

Abstract

We generalize the Białynicki-Birula decomposition from actions of G m on smooth varieties to actions of linearly reductive group G on finite type schemes and algebraic spaces. We also provide a relative version and briefly discuss the case of algebraic stacks.We define the Białynicki-Birula decomposition functorially: for a fixed G-scheme X and a monoid G which partially compactifies G, the BB decomposition parameterizes G-schemes over X for which the G-action extends to the G-action. The freedom of choice of G makes the theory richer than the G m -case.Corollary 1.2. Let X be a G-scheme locally of finite type over k. Then every point of X has an affine G-stable neighbourhood.Proof. Note that X = X + by Proposition 5.7. By Theorem 1.1 the projection π X is G-equivariant and affine. The required neighbourhood of x ∈ X is the preimage of any affine open neighbourhood of π X (x) ∈ X G . This is striking, since for an arbitrary, even normal, G-scheme X the existence of affine G-stable neighbourhoods is subtle, see Brion [Bri15]. As an example of non-normal scheme: the node of the nodal curve C = P 1 /(0 = ∞) with its natural G m -action does not admit an affine G m -stable neighbourhood; here C + is equal to A 1 . Now we describe X + for an affine X. Recall that every irreducible finite dimensional Grepresentation appears in H 0

Adjunction for Varieties With a ℂ* Action

“…See [3] for the original exposition. A vast generalization of this result, which is also valid for singular varieties, can be found in a recent paper [19] and references therein.…”

Elementary components of Hilbert schemes of points

“…Remark Representability of the functor follows also from [, Proposition 5.3] as the Hilbert scheme has a covering by ‐stable affine open subschemes [, Chapter 18]. However, the embedding is crucial for constructing obstruction theories for in Section 4.…”

“…Upon completion of this work, we learned that the Białynicki–Birula decomposition for algebraic spaces was constructed earlier by Drinfeld , by an entirely different method. His method was extended to actions of groups other than . While this paper was in review, a preprint appeared, which in particular claims to answer Question .…”

Motivic Chern classes and K‐theoretic stable envelopes

“…The Białynicki–Birula decomposition was studied recently by Drinfeld and Gaitsgory [23] with an application to derived categories. In their construction, the functorial properties of the stable and unstable sets for a –action play a crucial role, see also [35]. We need the functorial properties of the Białynicki–Birula decomposition to control the asymptotic behavior of the motivic Chern classes.…”