Białynicki-Birula and Motivic Decompositions

This is about Białynicki-Birula's paper from '72 on actions of reductive linear algebraic groups on non-singular varieties, in particular Gm-operations on smooth projective varieties. I give a proof sketch of Theorem 4.1 therein and explain a little bit how Brosnan applied these results in 2005 to get decompositions of the Chow motive of smooth projective varieties with Gm-operation. Wendt used these methods in 2010 to lift such a decomposition on the homotopy-level, to prove that smooth projective spherical varieties admit stable motivic cell decompositions. Most of this blogpost consists of an outline of the B-B paper.

Białynicki-Birula's algebraic Morse theory

The paper is essentially about algebraic torus actions on varieties and relating the induced action on the tangent space of a fixed point to the variety itself. The most simple torus is just the multiplicative group $\mathbb{G}_m$ (think of $\mathbb{C}^\times$ or $\mathbb{R}^\times$). In classical Morse theory, one considers "Morse functions", which are a particular kind of function $X \to \mathbb{R}$, and their gradient flow, which is the flow associated to the gradient vector field. Such a flow is nothing but a $\mathbb{G}_m$-action! Where the Morse-theory people look at smooth manifolds and apply the exponential function from the tangent space (of a critical point of the Morse function, i.e. a fixed point of the flow) to the whole space $X$, an algebraic geometer has to do something else (as the exponential function is not algebraic). This something else is a gimmick invented by Białynicki-Birula. With this gimmick, a $\mathbb{G}_m$-action with isolated fixed points provides a cell decomposition, like the CW decomposition from classical Morse theory.

Proof outline

We work over an algebraically closed field $k$. Let $X$ be a quasi-affine algebraic scheme and $a \in X$ a nonsingular closed point. We denote by $G$ a reductive algebraic group, though in the end only the 1-dimensional torus $\mathbb{G}_m$ is relevant.

Given a $G$-action on a scheme $X$ with fixed point $a \in X$, the tangent space $T_a(X)$ gets a natural $G$-action. For any vector space $V$ with $\mathbb{G}_m$-action there is a decomposition $V = V^- \oplus V^0 \oplus V^+$ into the weight-graded pieces. I call the action definite if either the minus- or the plus-part vanishes, and fully definite if also the zero-part vanishes.

Theorem 2.1: Given a reductive group $G$ acting on $X$ with a closed irreducible $G$-invariant subscheme $X_0$ containing a closed fixed point $a$ nonsingular in $X_0$ and $X$, to any $G$-invariant subspace $U_1$ of the tangent space $T_a(X)$ that contains $T_a(X_0)$ one can find a closed irreducible $G$-invariant subscheme $X_1$ that contains $X_0$ and has the prescribed tangent space.
(This is what I consider a replacement for the exponential function).

Proof idea: The maximal ideal $\mathfrak{m} \leq k[X]$ corresponding to $a \in X^G$ maps $G$-equivariant surjective to $\mathfrak{m}_a/\mathfrak{m}_a^2 = T_a(X)^\vee$. Denote by $U_0 \subset T_a(X)$ the tangent space of $X_0$ and by $\mathfrak{n}_0 \leq \mathfrak{m}$ the ideal corresponding to $X_0$. Since $G$ is reductive, there exists a $G$-submodule $N_1 \subset \mathfrak{n}_0$ that maps isomorphically to $U_1^\perp \subset T_a(X)^\vee$. Then $\mathfrak{n}_1 := N_1k[X]$ is an ideal in $\mathfrak{n_0}$, so the corresponding closed subscheme of $X$ has an irreducible component $X_1$ containing $X_0$. By construction, $T_a(X_1) = U_1$.

Uniqueness of the subspace is also discussed in Theorem 2.2, in particular we have a Corollary to Theorem 2.2: Let $\mathbb{G}_m$ act on $X$ with fixed point $a$. If $U_1$ is either the positive, the negative, the non-negative, the non-positive or the zero-part of the graded vector space $T_a(X)$, then there exists exactly one closed, irreducible and reduced $\mathbb{G}_m$-invariant subscheme $X_1$ through $a$ such that $a \in X_1$ is non-singular and $T_a(X_1)=U_1$.

There is a morphism-version of Theorem 2.1, which is slightly weaker. Roughly, to a $G$-isomorphism of some tangent spaces of two $G$-schemes, you get a third scheme with étale maps to the two others, and if you already have a $G$-isomorphism on subschemes, this is taken into account. The precise statement is

Theorem 2.4: Given for $i=1,2$ sequences $\{a_i\} \to Y_i \to X_i$ of closed immersions of $G$-invariant subschemes of quasi-affine algebraic $G$-schemes and a $G$-isomorphism $\alpha : (Y_1,a_1) \to (Y_2,a_2)$, such that the $G$-modules $T_{a_1}(X_1)$ and $T_{a_2}(X_2)$ are isomorphic, there exists such a sequence $\{a_0\} \to Y_0 \to X_0$ and étale $G$-morphisms $\beta_i : (X_0,Y_0,a_0) \to (X_i,Y_i,a_i)$ that map $Y_0$ onto an open subscheme of $Y_i$.

Proof idea: Inside $X_1\times X_2$ embed $Y_1$ as $Y_0' := Y_1 \times \alpha Y_1$ and apply Theorem 2.1 to get a subscheme $X' \subset X_1\times X_2$ that contains $a_0 = (a_1,a_2)$ and $Y_0'$, with $T_{a_0}(X')=\Delta$. The projections to the factors $X_i$ are étale at $a_0$, hence over a smaller subscheme $X''$ that still contains $a_0$. Denote by $Y'$ the union of the preimages of the $Y_i$ in $X''$, then $Y_0 := Y_0' \cap Y'$ and $X_0 := X'' \setminus (Y' \setminus Y_0')$ do the job.

The local structure of affine "cells" comes from

Theorem 2.5: For any torus $G$ acting on $X$ such that $a$ is a fixed point and the induced action on $T_a(X)$ is definite, there exists a $G$-invariant open neighborhood $U$ of $a$ which is $G$-isomorphic to $(U\cap X^G) \times V$, with $V$ a finite-dimensional fully definite $G$-module.

Proof idea: The $V$ arises as the complement of $T_a(X^G) \subset T_a(X)$.
WLOG (as one can show) $X$ is reduced, irreducible and $G=\mathbb{G}_m$ acts effectively. Apply Theorem 2.4 to $X_1 := X$, $X_2 := X^G \times V$, $Y_1 := X^G$, $Y_2 := X^G \times 0$, $a_1 := a$, $a_2 := (a,0)$, then the resulting $\beta_i : X_0 \to X_i$ are not only étale, but also birational (as one can show), hence open immersions. Then $X_0$ contains an open subscheme $X_0'$ which is $G$-isomorphic to $U_0 \times V$ for $U_0$ some open beighbourhood of $a$ in $X^G$ and $U := \beta_1(X_0')$ gives the statement of the theorem.

In the full proof of Theorem 2.5, the notion of a universal domain $\Omega$ is frequently used. This is a device to handle generic points without talking about prime ideals, which I explained in this blog posts about points.

Given a $G$-rep $\alpha : G \to GL(V)$ one defines the notion of an $\alpha$-fibration $X \to Y$, which carries a $G\times Y$-action on $X$ and Zariski-locally on $Y_i \subset Y$ looks like $V \times Y_i \to Y_i$, with $G \times Y_i$-action induced by $\alpha$. We call $\dim V$ the dimension of the $\alpha$-fibration.
One should remark that an $\alpha$-fibration needn't be a vector bundle, since there might be more $G$-equivariant automorphisms of $V$ than the linear ones.

The following gives us a uniqueness property for $\alpha$-fibration-structures on maps $X \to X^G$.

Corollary to Proposition 3.1: For any torus $G$ acting on a nonsingular $X$, two $G$-representations $\alpha_i : G \to GL(V_i)$ for $i=1,2$ and $\alpha_i$-fibrations $\gamma_i : X \to X^G$ (respectively), then $\alpha_1$ is equivalent to $\alpha_2$. If furthermore $a$ is a $G$-fixed point and $T_a(X)$ is definite, then $\gamma_1 = \gamma_2$.

Proof idea: For any closed point $a \in X^G$, as $G$-modules, $V_i \cong T_a(X)/T_a(X^G)$, so the $\alpha_i$ are equivalent. Note that the $U$ in $U \oplus T_a(X^G) = T_a(X)$ is uniquely determined, since there is no nonzero $G$-homomorphism $T_a(X^G) \to T_a(X)/T_a(X^G)$. By (the corollary to) Theorem 2.2 there exists exactly one $G$-invariant subscheme $X_a$ with $a \in X_a$ nonsingular and $T_a(X_a)=U$, but $\gamma_1^{-1}(a)$ and $\gamma_2^{-1}(a)$ both fulfill these conditions, hence $\gamma_1^{-1}(a) = \gamma_2^{-1}(a)$. This shows $\gamma_1 = \gamma_2$.

We call a morphism $X \to Y$ with $G \times Y$-action on $X$ a $G$-fibration if it is Zariski-locally over $Y_i \subset Y$ an $\alpha_i$-fibration $X \times_Y Y_i \to Y_i$ for some $G$-reps $\alpha_i$. If the dimensions of the $\alpha_i$ all coincide, we call that number the dimension of the $G$-fibration.

Now, let $G=\mathbb{G}_m$ and $X$ any non-singular reduced algebraic $G$-scheme that can be covered by $G$-invariant quasi-affine open subschemes (for example any smooth projective $X$ will do, maybe normal quasiprojective suffices, by Sumihiro's equivariant compactification).

Theorem 4.1: Let $X^G = \bigcup (X^G)_i$ be the decomposition into connected components. For any $i$ there exists a unique locally closed $G$-invariant subscheme $X_i^+ \subset X$ and a unique morphism $\gamma_i^+ : X_i^+ \to (X^G)_i$ such that

1. $\gamma_i^+$ is a retraction, i.e. $(X^G)_i$ is a closed subscheme of $X_i^+$ and $\gamma_i^+|_{(X^G)_i}$ is the identity,
2. $\gamma_i^+$ is a $G$-fibration,
3. for any closed fixed point $a \in (X^G)_i$, the tangent space is $T_a(X_i^+) = T_a(X)^0 \oplus T_a(X)^+$ and the dimension of the $G$-fibration $\gamma_i^+$ is $dim T_a(X)^+$.

Proof idea: Let $a \in X^G$. By Theorem 2.1 there exists a closed $G$-invariant irreducible subscheme $Y_a' \subset X$ with $a \in Y_a'$ nonsingular and $T_a(Y_a')=T_a(X)^0 \oplus T_a(X)^+$. By Theorem 2.5, there is an open $G$-stable nonsingular subscheme $Y_a \subset Y_a'$ that still contains $a$ and $\gamma_a : Y_a \to Y_a \cap X^G$ is a trivial $G$-fibration. Using the Corollary to Theorem 2.2 and the Corollary to Proposition 3.1 (the uniqueness statements) we know for $a,b \in X^G$ that $\gamma_a|_{Y_a \cap Y_b} = \gamma_b|_{Y_a \cap Y_b}$ and for any third fixed point $c \in Y_a \cap Y_b \cap X^G$ we have $Y_a \cap Y_b \supset \gamma_c^{-1}(Y_a \cap Y_b \cap Y_c \cap X^G)$.
Since every $(X^G)_i$ is noetherian, we find $\{a_1,\dots,a_n\} \subset (X^G)_i$ such that $(X^G)_i = \bigcup (Y_{a_i} \cap (X^G)_i)$, so $X_i^+ := \bigcup Y_{a_i}$ is a $G$-invariant, locally closed subscheme of $X$ and a $G$-fibration $\gamma_i^+ : X_i^+ \to (X^G)_i$ can be uniquely glued together from the $\gamma_{a_i}$.

Actually, there is also a minus-decomposition, where you use $T_a(X)^-$ instead. The interplay of these two decompositions for the same $\mathbb{G}_m$-action is explained in Theorem 4.2: Let $G=\mathbb{G}_m$ act on a quasi-affine $X$. For a rational point $t \in X(k)$, the orbit closure $\overline{G(k)t}$ intersects a connected component $(X^G)_i$ in a non-empty set iff $t \in X_i^+$ or $t \in X_i^-$. Moreover, $X_i^+ \cap X_i^- = (X^G)_i$ for all connected components.

Proof idea: The direction $t \in X_i^+ \cup X_i^- \Rightarrow \overline{G(k)t} \cap (X^G)_i \neq \emptyset$ is clear. For the other direction apply Theorem 2.4 to $X_1 := X$, $X_2 := (X^G)_i \times (T_a(X)^+ \oplus T_a(X)^-)$, $Y_1 := (X^G)_i$, $Y_2 := (X^G)_i \times 0$, $a_1 := a$, $a_2 := (a,0)$. For $\beta_1^{-1}(t) = \{t_1,\dots,t_s\}$ we have $\beta_1^{-1}(\overline{G(k)t}) = \bigcup \overline{G(k)t_i}$. Only one $\overline{G(k)t_i}$ contains $a_0$ and one can show (using again Theorem 2.1 and 2.2) that actually $a_0 \in G(k)t_i$ and $t \in X_i^+ \cup X_i^-$.
Moreover, from $((X^G)_i \times T_a(X)^+) \cap ((X^G)_i \times T_a(X)^-) = (X^G)_i \times 0$ we get $X_i^+ \cap X_i^- = (X^G)_i$.

Theorem 4.3: Let $G=\mathbb{G}_m$ act on a complete nonsingular algebraic $k$-scheme $X$, with $X^G = \bigcup (X^G)_i$ the decomposition of the fixed points into connected components. Then there exists a unique locally closed $G$-invariant decomposition $X = \bigcup X_i$ and $G$-fibrations $\gamma_i : X_i \to (X^G)_i$ such that $(X_i)^G = (X^G)_i$ and for any closed fixed point $a \in (X^G)_i$, $T_a(X_i) = T_a((X^G)_i) \oplus T_a(X)^+$.

Proof idea: Take the plus-decomposition from Theorem 4.1, then what's missing for the statement ($X_i \cap X_j = \emptyset$ for $i \neq j$ and $(X_i)^G = (X^G)_i$) follows from analyzing orbit closures (that is actually Theorem 4.2 together with Lemma 4.1 which I didn't include in this summary).

From this follows Theorem 4.4: If the $\mathbb{G}_m$-action in Theorem 4.3 has isolated fixed points, then any $X_i$ is isomorphic to an affine space $\mathbb{A}^{n_i}_k$.

(This looks like a cell structure!)

The proofs and results have been improved a little bit (Hesselink removed the assumption that $k$ is algebraically closed), so that the current level of generality provides the following
Theorem:
Let $X$ be a smooth projective $\mathbb{G}_m$-variety over a field $k$. Then
1) $X^{\mathbb{G}_m}$ is a smooth closed subscheme of $X$ (Iversen),
2) Given the connected components $X^{\mathbb{G}_m} = \bigcup_{i=1}^n Z_i$, there is a filtration $X = X_n \supset X_{n-1} \supset \cdots \supset X_0 \supset X_{-1} = \emptyset$ and affine fibrations $\phi_i : X_i \setminus X_{i-1} \to Z_i$,
3) The relative dimension of $\phi_i$ is the dimension of $T_a(X)^+$ for any $a \in Z_i$.

I want to remark that any generalization of this theorem to quasiprojective or singular situations would be a very impressive result.

The only generalizations I know of are papers of Skowera and Choudhury on Deligne-Mumford stacks and papers of Carrell and Sommese on the Kähler analogue.

Karpenko's, Chernousov-Gille-Merkurjev's and Brosnan's motivic decompositions

Theorem (Karpenko):
Let $X$ be a smooth projective variety over a field $k$, equipped with a filtration $X = X_n \supset X_{n-1} \supset \cdots \supset X_0 \supset X_{-1} = \emptyset$ where the $X_i$ are closed subvarieties, and affine fibrations $\phi_i : X_i \setminus X_{i-1} \to Z_i$ of relative dimension $n_i$. Then the Chow motive decomposes $h(X) = \bigoplus_{i=0}^n h(Z_i)(n_i)$.

Corollary (Brosnan):
Let $X$ be a smooth projective $\mathbb{G}_m$-variety over a field $k$. Then $h(X) = \bigoplus_{i=0}^n h(Z_i)(n_i)$, where the $Z_i$ are the connected components of the fixed point locus $X^{\mathbb{G}_m}$.

From this, Brosnan re-proved
Theorem (Chernousov-Gille-Merkurjev):
For $X$ a projective homogeneous variety (for a reductive group) over a field $k$, the kernel of the map $End(M(X)) \to End(M(X_{\overline{k}}))$ consists only of nilpotent elements.

Brosnan proved more, in particular how the motive of $G/P$ decomposes, using this method: $M(G/P) = \bigoplus_{w \in E} M(Z_w)(\ell(w))$, where $\ell(w)$ is length and $E$ is the set of minimal length coset representatives of $W_I\backslash W/W_J$, with $J$ the set of roots corresponding to $P$ and $I$ the set of roots that are killed by a non-central cocharacter of the maximal torus (taking care of the possible non-splitness of the maximal torus).

Wendt's cellular decomposition of the stable motivic homotopy type

Using the BB-decomposition, one gets a decomposition of the motive. Actually, one gets a bit more, namely a decomposition in the stable A¹-homotopy category. This is even more analogous to CW decompositions coming from Morse theory.

What you need for a cellular decomposition (in the homotopy-theoretic sense), but what's missing in a direct sum decomposition of the motive, are the gluing maps. One has to extract these gluing maps from the BB-decomposition. This was done by Wendt, who used this approach to show stable cellularity of connected split reductive groups and their classifying spaces, as well as stable cellularity of smooth projective spherical varieties under connected split reductive groups.
As this post is already too long, I might explain the motivic cell structures in another post. Actually, you can just take a look at the preprint.

It remains to see how these cell structures look like explicitly!