You mean the Białynicki–Birula decomposition (usually abbreviated BB decomposition). It is one of the basic forms of “algebraic Morse theory”: a $\mathbf G_m$-action on a smooth projective variety decomposes the variety into attracting pieces governed entirely by the fixed locus and the weights on its normal bundle. The original foundational theorem is Białynicki-Birula’s 1973 Annals paper. (Annals of Mathematics)

I will first state the classical theorem precisely, then prove it in some detail, work out the isolated-fixed-point and flag-variety cases, and finally survey the major generalizations.

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1. The basic setup

Let $k$ be an algebraically closed field and let

$$\lambda:\mathbf G_m\times X\longrightarrow X$$

be an algebraic action on a smooth projective $k$-variety $X$. Write

$$F=X^{\mathbf G_m}$$

for the fixed-point scheme and decompose it into connected components

$$F=\coprod_{\alpha\in A}F_\alpha .$$

For $x\in X$, projectivity implies that the orbit map

$$\mathbf G_m\longrightarrow X,\qquad t\longmapsto t\cdot x$$

has limits at both $0$ and $\infty$. Define

$$X_\alpha^+ = \left\{ x\in X: \lim_{t\to0}t\cdot x\in F_\alpha \right\}$$

and similarly

$$X_\alpha^- = \left\{ x\in X: \lim_{t\to\infty}t\cdot x\in F_\alpha \right\}.$$

These are the attracting and repelling sets.

At a fixed point $p\in F_\alpha$, the tangent representation decomposes into weights:

$$T_pX= T_pX^-\oplus T_pX^0\oplus T_pX^+,$$

where

$$T_pX^\pm= \bigoplus_{\substack{m\in\mathbf Z\\ \pm m>0}} (T_pX)_m, \qquad T_pX^0=(T_pX)_0.$$

The key point is that the geometry of $X_\alpha^\pm$ is controlled by these subspaces.

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2. Classical Białynicki–Birula theorem

A convenient modern form is the following.

Theorem — smooth projective BB decomposition

Let $X$ be a smooth projective variety with an algebraic $\mathbf G_m$-action. Then:

$$X^{\mathbf G_m}=\coprod_\alpha F_\alpha$$

is smooth, and

$$X=\coprod_\alpha X_\alpha^+$$

is a decomposition into smooth locally closed subvarieties.

The limit map

$$p_\alpha^+:X_\alpha^+\longrightarrow F_\alpha, \qquad x\longmapsto\lim_{t\to0}t\cdot x,$$

is a Zariski-locally trivial affine-space fibration

$$\mathbf A^{d_\alpha}\longrightarrow X_\alpha^+ \xrightarrow{p_\alpha^+} F_\alpha ,$$

where

$$d_\alpha=\dim T_pX^+$$

for $p\in F_\alpha$. Moreover,

$$T_pF_\alpha=T_pX^0,$$

and

$$T_pX_\alpha^+ = T_pX^0\oplus T_pX^+.$$

Finally, the components can be ordered so that the decomposition is filtrable: there exists a filtration by closed subvarieties

$$\varnothing=Z_{-1} \subset Z_0 \subset\cdots\subset Z_N=X$$

such that

$$Z_i\setminus Z_{i-1}=X_{\alpha_i}^+.$$

The same statements hold with $+$ and $-$ interchanged. Modern formulations explicitly give the Zariski locally trivial affine fibrations and tangent-weight description. (Springer Nature Link)

A subtle point worth emphasizing is that $X_\alpha^+\to F_\alpha$ is naturally an affine-space bundle. It need not come with a canonical vector-bundle structure, even though its infinitesimal model along $F_\alpha$ is the positive-weight part of the normal bundle.

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3. Step 1 of the proof: existence of the limits

Take $x\in X$. Its orbit defines

$$\phi_x:\mathbf G_m\rightarrow X.$$

Since $X$ is proper, the valuative criterion applied at $0\in\mathbf A^1$ extends $\phi_x$ uniquely to

$$\overline\phi_x:\mathbf A^1\rightarrow X.$$

Set

$$x_0=\overline\phi_x(0).$$

Then $x_0$ is fixed. Indeed, for $s\in\mathbf G_m$,

$$s\cdot x_0 = s\cdot\lim_{t\to0}t\cdot x = \lim_{t\to0}st\cdot x = x_0.$$

Hence

$$x_0\in X^{\mathbf G_m}.$$

Thus every point belongs to exactly one attracting set:

$$X=\coprod_\alpha X_\alpha^+.$$

Applying the same argument after replacing $t$ by $t^{-1}$ produces the repelling decomposition.

This is the first place properness enters decisively. For a general quasi-projective variety the limit may simply leave the variety.

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4. Step 2: smoothness of the fixed locus

Let $p\in X^{\mathbf G_m}$. We examine $X$ formally at $p$.

Because $X$ is smooth,

$$\widehat{\mathcal O}_{X,p} \simeq k[[x_1,\ldots,x_n]].$$

The torus $\mathbf G_m$ is diagonalizable. Consequently its representation on

$$\mathfrak m_p/\mathfrak m_p^2$$

splits into weight spaces. Choose homogeneous coordinates $x_i$ lifting a weight basis. Say

$$t^*x_i=t^{w_i}x_i.$$

Separate them as

$$w_1,\ldots,w_a<0, \qquad w_{a+1},\ldots,w_{a+b}=0, \qquad w_{a+b+1},\ldots,w_n>0.$$

A point is fixed precisely when all nonzero-weight coordinates vanish. Thus formally

$$\widehat{X^{\mathbf G_m}}_p \simeq \operatorname{Spf} k[[x_{a+1},\ldots,x_{a+b}]].$$

So the fixed locus is formally smooth of dimension $b$, hence smooth near $p$.

Moreover,

$$T_p(X^{\mathbf G_m}) = (T_pX)_0.$$

This proves the first part of the theorem.

The same argument works over much more general fields because diagonalizable groups are linearly reductive in the relevant sense; arbitrary-field versions were developed after the original algebraically closed-field treatment. Modern references state the theorem for a smooth projective $k$-variety over an arbitrary field. (OUP Academic)

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5. Step 3: the local attractor calculation

The essential algebraic calculation is easiest on an invariant affine open subset.

Suppose

$$U=\operatorname{Spec}A$$

is $\mathbf G_m$-stable. The action gives a $\mathbf Z$-grading

$$A=\bigoplus_{m\in\mathbf Z} A_m.$$

An equivariant map

$$\mathbf A^1\longrightarrow U$$

with $\mathbf G_m$ acting on $\mathbf A^1$ by

$$t\cdot z=tz$$

cannot have a nonzero negative-weight coordinate: a degree $-r$ function would have to pull back to something proportional to $z^{-r}$, which is not regular at $0$.

Consequently the affine attractor is

$$U^+ = \operatorname{Spec} \left( A/ (A_{<0}) \right),$$

where $(A_{<0})$ denotes the ideal generated by all negative-weight elements.

After passing to

$$C=A/(A_{<0}),$$

only nonnegative weights survive:

$$C=\bigoplus_{m\ge0}C_m.$$

The fixed locus is

$$F_U = \operatorname{Spec} C_0,$$

and evaluation at $0$ gives the morphism

$$p:U^+\rightarrow F_U.$$

Thus the attractor carries a canonical contraction

$$\mathbf A^1\times U^+ \longrightarrow U^+$$

whose value at $0$ is $p$.

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6. Why the attractor is smooth

At a fixed point, the formal coordinate description gives

$$\widehat{U^+}_p \simeq \operatorname{Spf} k[[ x_{a+1},\ldots,x_n ]],$$

because the negative coordinates

$$x_1,\ldots,x_a$$

have been set equal to zero.

Therefore $U^+$ is smooth along its fixed locus, and

$$T_pU^+ = T_pX^0\oplus T_pX^+.$$

Likewise the map

$$p:U^+\to F_U$$

is smooth along its zero section, with relative tangent space

$$T_pX^+.$$

Now use contraction.

The nonsmooth locus of $p$ is closed and $\mathbf G_m$-stable. Suppose it contained $x$. Since

$$\lim_{t\to0}t\cdot x=p(x),$$

closedness and invariance would imply that $p(x)$ also belonged to the nonsmooth locus.

But we have just proved that $p$ is smooth along the zero section.

Contradiction.

Hence

$$p:U^+\longrightarrow F_U$$

is smooth everywhere.

This little contraction argument is one of the conceptual cores of BB theory: a property holding near the fixed locus propagates to the entire attracting set because every orbit flows into the fixed locus.

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7. Why the fibers are affine spaces

Fix $q\in F_U$. The fiber has coordinate ring

$$C_q=C\otimes_{C_0}k(q).$$

It is a finitely generated positively graded algebra

$$C_q=k(q)\oplus (C_q)_1\oplus(C_q)_2\oplus\cdots$$

with a unique fixed point corresponding to its positive-degree ideal

$$\mathfrak m_+ = \bigoplus_{m>0}(C_q)_m.$$

The fiber is smooth. Let

$$d=\dim_{k(q)} \mathfrak m_+/\mathfrak m_+^2.$$

Choose homogeneous elements

$$y_1,\ldots,y_d$$

mapping to a basis of this cotangent space.

Because of positive grading, a graded Nakayama argument says that the $y_i$ generate $C_q$ as an algebra. Hence there is a surjection

$$k(q)[Y_1,\ldots,Y_d]\twoheadrightarrow C_q.$$

Both sides have dimension $d$: smoothness tells us that the fiber has dimension $d$, while the source is affine $d$-space.

The kernel therefore has height zero. Since the fiber is smooth and connected under the contracting action, the kernel is zero. Consequently

$$C_q\simeq k(q)[Y_1,\ldots,Y_d]$$

and therefore

$$p^{-1}(q)\simeq\mathbf A^d.$$

The integer $d$ is exactly

$$d=\dim T_qX^+.$$

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8. Why the fibration is Zariski locally trivial

The previous argument can be performed relatively.

Let

$$B=C_0, \qquad C_+=\bigoplus_{m>0}C_m.$$

Along the zero section,

$$C_+/C_+^2$$

is the conormal bundle. Since $p$ is smooth, this is a locally free $B$-module of rank $d$.

After restricting to a sufficiently small Zariski open subset

$$V\subset F_U,$$

choose a homogeneous basis

$$\overline y_1,\ldots,\overline y_d$$

and lift it to homogeneous elements $y_i\in C_+$.

The positive grading again shows that the $y_i$ generate $C$ over $B$, giving

$$B[Y_1,\ldots,Y_d]\twoheadrightarrow C.$$

On every fiber this morphism is an isomorphism by the preceding argument. Shrinking $V$ if necessary, the kernel therefore vanishes, and

$$p^{-1}(V) \simeq V\times\mathbf A^d.$$

Hence $p$ is a Zariski-locally trivial affine-space bundle.

This proves the local geometric heart of the BB theorem.

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9. Globalization

To globalize, one uses invariant affine neighborhoods.

For torus actions on normal varieties, Sumihiro-type local linearization gives sufficiently many invariant affine neighborhoods of fixed points. Since $X$ is smooth, it is normal.

Let $p\in F_\alpha$, and choose a $\mathbf G_m$-invariant affine open

$$U\ni p.$$

If $x\in X_\alpha^+$ and its limit lies in $U$, then for sufficiently small nonzero $t$,

$$t\cdot x\in U.$$

But $U$ is invariant, so

$$x=t^{-1}\cdot(t\cdot x)\in U.$$

Thus the affine computations above actually describe the entire portion of $X_\alpha^+$ lying over $F_\alpha\cap U$.

Covering $F_\alpha$ by such opens proves globally that

$$X_\alpha^+$$

is smooth and locally closed and that

$$p_\alpha^+:X_\alpha^+\rightarrow F_\alpha$$

is an affine-space bundle.

Because the weights of an equivariant vector bundle are locally constant on a connected fixed component,

$$d_\alpha=\dim(T_pX)^+$$

does not depend on the chosen $p\in F_\alpha$.

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10. Filtrability: where projectivity matters

The preceding argument produces a decomposition into locally closed pieces. Projectivity gives more: they can be arranged into a filtration by closed subsets.

Take an equivariant projective embedding

$$X\hookrightarrow \mathbf P(V),$$

with

$$V=\bigoplus_{m\in\mathbf Z}V_m$$

the weight decomposition.

Write a point of $\mathbf P(V)$ as

$$[v],\qquad v=\sum_m v_m.$$

Then

$$t\cdot[v] = \left[\sum_m t^m v_m\right].$$

If

$$m_0=\min\{m:v_m\neq0\},$$

then

$$\lim_{t\to0}t\cdot[v] = [v_{m_0}].$$

Hence the attracting component is controlled by the smallest occurring weight.

For an integer $a$, put

$$Z_{\ge a} = X\cap \mathbf P \left( \bigoplus_{m\ge a}V_m \right).$$

This is closed in $X$.

Moreover,

$$Z_{\ge a}\setminus Z_{\ge a+1}$$

is exactly the union of the attracting strata whose limiting fixed points have weight $a$.

Within such a layer, the inverse images of distinct connected components of the fixed locus are simultaneously open and closed. They can therefore be inserted one at a time into the filtration.

This yields

$$\varnothing=Z_{-1} \subset Z_0 \subset\cdots\subset Z_N=X$$

with

$$Z_i-Z_{i-1}=X_{\alpha_i}^+.$$

This is the filtrable BB decomposition.

Projectivity should not casually be replaced by mere completeness here. The relation between BB cells and filtrability on complete nonprojective varieties is genuinely subtler; recent work continues to study precisely these structural issues. (arXiv)

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11. Classical special case: isolated fixed points

Suppose

$$X^{\mathbf G_m} = \{p_1,\ldots,p_N\}.$$

Then every base $F_\alpha$ is a point, so

$$X_i^+\simeq\mathbf A^{d_i}, \qquad d_i=\dim(T_{p_i}X)^+.$$

Thus

$$X= \coprod_{i=1}^N\mathbf A^{d_i}.$$

This is an actual affine paving.

This is probably the most famous form of the BB theorem: a smooth projective variety with a $\mathbf G_m$-action having isolated fixed points possesses a decomposition into affine spaces. (CiteSeerX)

Over $\mathbf C$, this immediately implies

$$H^{2r+1}(X,\mathbf Z)=0$$

in the usual cellular situations, and the number of $r$-dimensional BB cells records the corresponding even Betti number. In particular,

$$\chi(X)=\#X^{\mathbf G_m}.$$

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12. Example: projective space

Take

$$X=\mathbf P^n$$

and choose strictly increasing integers

$$a_0<a_1<\cdots<a_n.$$

Let

$$t\cdot[x_0:\cdots:x_n] = [t^{a_0}x_0:\cdots:t^{a_n}x_n].$$

The fixed points are

$$p_i=[0:\cdots:1:\cdots:0].$$

For a point $x$, let $i$ be the smallest index for which $x_i\neq0$. Then

$$\lim_{t\to0}t\cdot x=p_i.$$

Consequently

$$X_i^+ = \{ x_0=\cdots=x_{i-1}=0,\quad x_i\neq0 \}.$$

Normalize $x_i=1$. The remaining coordinates

$$x_{i+1},\ldots,x_n$$

are arbitrary, so

$$X_i^+\simeq\mathbf A^{n-i}.$$

Thus

$$\mathbf P^n = \mathbf A^n \sqcup \mathbf A^{n-1} \sqcup\cdots\sqcup \mathbf A^0.$$

At $p_i$, using $x_j/x_i$ as local coordinates, the tangent weights are

$$a_j-a_i.$$

The positive ones are exactly those with $j>i$, so

$$\dim(T_{p_i}\mathbf P^n)^+ = n-i,$$

exactly as predicted.

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13. Example: Grassmannians and Schubert cells

Take

$$X=\operatorname{Gr}(r,n).$$

Let $\mathbf G_m$ act diagonally on $k^n$:

$$t\cdot e_i=t^{a_i}e_i, \qquad a_1<\cdots<a_n.$$

The fixed points of the induced action on the Grassmannian are the coordinate subspaces

$$E_I = \operatorname{span}(e_{i_1},\ldots,e_{i_r}), \qquad I=\{i_1<\cdots<i_r\}.$$

At $E_I$,

$$T_{E_I}\operatorname{Gr}(r,n) \simeq \operatorname{Hom}(E_I,k^n/E_I).$$

The elementary map

$$e_i\mapsto e_j$$

has weight

$$a_j-a_i.$$

Hence its weight is positive precisely when $j>i$.

Therefore

$$\dim X_I^+ = \#\{(i,j):i\in I,\ j\notin I,\ j>i\}.$$

Writing $I=\{i_1<\cdots<i_r\}$,

$$\dim X_I^+ = \sum_{p=1}^r (n-r+p-i_p).$$

These attracting cells are exactly one of the usual Schubert-cell decompositions, depending on the choice of sign convention for the one-parameter subgroup. Reversing

$$t\longmapsto t^{-1}$$

interchanges Schubert and opposite Schubert cells.

Thus the ordinary Schubert decomposition is a fundamental example of BB decomposition.

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14. Flag varieties and Bruhat decomposition

Let $G$ be reductive, $B\subset G$ a Borel subgroup and

$$X=G/B.$$

Choose a maximal torus $T\subset B$ and a regular one-parameter subgroup

$$\lambda:\mathbf G_m\rightarrow T.$$

Regularity means

$$\langle\alpha,\lambda\rangle\neq0$$

for every root $\alpha$.

The $\lambda(\mathbf G_m)$-fixed locus is then

$$(G/B)^{\lambda(\mathbf G_m)} = (G/B)^T \simeq W,$$

the Weyl group.

The BB attracting cells are the Bruhat cells for one choice of chamber, while the repelling cells are the opposite Bruhat cells. Replacing $\lambda$ by $-\lambda$ exchanges the two.

So the classical decomposition

$$G/B = \coprod_{w\in W} BwB/B$$

can be viewed as algebraic Morse theory.

This viewpoint is particularly powerful because the dimension of a cell is read off from tangent weights, reproducing Weyl-group length.

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15. Relation with Morse–Bott theory

Over $\mathbf C$, suppose $X$ is projective. Restrict

$$\mathbf C^\times$$

to its compact subgroup $S^1$. With an invariant Kähler form, the $S^1$-action has a moment map

$$\mu:X\rightarrow\mathbf R.$$

Its critical locus is

$$X^{S^1}=X^{\mathbf C^\times}.$$

The moment map is Morse–Bott, and the positive and negative normal directions at a critical component are precisely the positive and negative $\mathbf C^\times$-weight spaces.

The BB attracting manifolds are the algebraic analogues of stable manifolds of the gradient flow.

Schematically,

$$\boxed{ \text{Morse critical component} \longleftrightarrow F_\alpha }$$

and

$$\boxed{ \text{stable manifold} \longleftrightarrow X_\alpha^+ }.$$

This is why BB theory is frequently called algebraic Morse theory.

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16. Non-isolated fixed components

The version with positive-dimensional $F_\alpha$ is more important than the cell case in moduli problems.

We have

$$X_\alpha^+ \rightarrow F_\alpha$$

with fiber

$$\mathbf A^{d_\alpha}.$$

Thus in the Grothendieck ring of varieties,

$$[X] = \sum_\alpha \mathbb L^{d_\alpha}[F_\alpha],$$

where

$$\mathbb L=[\mathbf A^1].$$

Over $\mathbf C$, the Hodge–Deligne polynomial satisfies

$$E(X;u,v) = \sum_\alpha (uv)^{d_\alpha} E(F_\alpha;u,v).$$

This is often far more useful than an affine paving, because complicated geometry of $X$ is reduced to typically much simpler fixed components.

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17. Motivic refinement

The BB decomposition is stronger than an equality of Grothendieck classes.

Brosnan used BB theory together with a result of Karpenko to obtain a decomposition of the Chow motive of a smooth projective $\mathbf G_m$-variety into Tate-twisted motives of its fixed components. In one common convention,

$$M(X) \simeq \bigoplus_\alpha M(F_\alpha)(d_\alpha).$$

The precise sign of the Tate twist depends on covariant versus contravariant conventions. Brosnan's work is a major reason BB decompositions became a standard tool in the theory of motives and projective homogeneous varieties. (ResearchGate)

At the crude Grothendieck-ring level, the formula

$$[X] = \sum_\alpha\mathbb L^{d_\alpha}[F_\alpha]$$

already follows immediately from the affine bundles; the motivic statement upgrades this additive identity to an actual direct-sum decomposition.

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18. Arbitrary ground fields

The original theorem was formulated under stronger hypotheses, but the standard modern theorem works over an arbitrary field $k$.

For a smooth projective $k$-variety with a $\mathbf G_m$-action,

$$X^{\mathbf G_m}$$

is a smooth closed $k$-subscheme, and after appropriately indexing its components there is a filtration with affine fibrations over the fixed components. This form is commonly attributed collectively to Białynicki-Birula, Hesselink, and Iversen. (OUP Academic)

The essential technical improvement is that the proof is made scheme-theoretic and descent-compatible rather than depending on the existence of sufficiently many algebraically closed points.

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19. Smooth quasi-projective and semiprojective varieties

Projectivity is stronger than is needed for the local geometry.

Suppose $X$ is smooth quasi-projective and satisfies:

$$\lim_{t\to0}t\cdot x$$

exists in $X$ for every $x$, and the fixed locus is proper. Such actions are commonly called semiprojective.

Then essentially the same theorem holds:

$$X=\coprod_\alpha X_\alpha^+$$

and

$$X_\alpha^+\to F_\alpha$$

is a Zariski locally trivial affine-space fibration, with the same tangent-weight formula. This is widely used for Higgs-bundle moduli spaces, quiver varieties, and related noncompact varieties. (Springer Nature Link)

What has changed is not the local attractor theorem but the global existence of limits. In projective geometry existence is automatic; in the semiprojective setting it is an extra hypothesis.

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20. Higher-dimensional torus actions

Let

$$T\simeq\mathbf G_m^r$$

act on $X$.

A torus has no preferred direction of flow, so choose a cocharacter

$$\lambda:\mathbf G_m\to T.$$

For a generic $\lambda$, one can arrange

$$X^{\lambda(\mathbf G_m)}=X^T.$$

The BB decomposition for the induced $\mathbf G_m$-action then gives

$$X= \coprod_\alpha X_{\alpha,\lambda}^+.$$

Different choices of $\lambda$ can give different decompositions.

The cocharacter space

$$X_*(T)\otimes_\mathbf Z\mathbf R$$

is divided by finitely many weight hyperplanes into chambers. Within a chamber, the signs of all normal weights are constant, and hence the BB decomposition is constant. Crossing a wall changes which normal directions are attracting and repelling.

This chamber structure is one conceptual bridge between BB theory, geometric invariant theory and variation of GIT quotients.

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21. What survives for singular varieties?

Smoothness is essential for the conclusion

$$X_\alpha^+\to F_\alpha \quad\text{is an affine-space bundle}.$$

For singular $X$, the attracting locus still makes sense, but its fibers can be singular, reducible or nonreduced. In particular, isolated fixed points do not in general imply that the attracting pieces are affine spaces, and naive cohomological conclusions can fail. Contemporary work gives explicit warnings and counterexamples in the singular setting. (ScienceDirect)

The correct modern replacement is to define the attractor functorially:

$$X^+ = \operatorname{Maps}^{\mathbf G_m} (\mathbf A^1,X).$$

An $S$-point of $X^+$ is a $\mathbf G_m$-equivariant morphism

$$\mathbf A^1\times S\to X.$$

There are natural maps

$$X^{\mathbf G_m} \xleftarrow{\,q^+\,} X^+ \xrightarrow{\,p^+\,} X,$$

given respectively by evaluation at $0$ and $1$.

Similarly one defines the repeller

$$X^-.$$

This functorial formulation continues to make sense without smoothness and is now fundamental in geometric representation theory. (ResearchGate)

In the smooth case it specializes back to the classical BB varieties.

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22. Braden's hyperbolic localization

The singular theory becomes especially powerful at the sheaf-theoretic level.

Given

$$X^+ \quad\text{and}\quad X^-,$$

one can form two seemingly different ways of restricting a constructible sheaf from $X$ to $X^{\mathbf G_m}$: one uses the attracting correspondence and the other the repelling correspondence.

Braden's hyperbolic localization theorem says, under suitable equivariance hypotheses, that these two operations are canonically related. For intersection cohomology, hyperbolic localization preserves a remarkable amount of semisimplicity: the hyperbolic localization of an intersection-cohomology complex decomposes into intersection-cohomology complexes on fixed loci. (arXiv)

Drinfeld and Gaitsgory subsequently gave a new geometric proof based on an interpolation that degenerates $X$ toward the attractor–repeller correspondence. (arXiv)

Conceptually,

$$\text{classical BB}$$

studies the geometry of stable manifolds, while

$$\text{hyperbolic localization}$$

studies what sheaves and categories do under those same flows.

This is the form of BB theory that appears in geometric Satake, category $\mathcal O$, character-sheaf theory and many localization constructions.

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23. General linearly reductive groups

The classical theory privileges

$$\mathbf G_m$$

because $\mathbf A^1$ compactifies it in one direction.

Jelisiejew and Sienkiewicz developed a substantially broader framework for a linearly reductive group $G$. Their idea is to choose a monoid

$$\overline G \supset G$$

playing the role of

$$\mathbf A^1\supset\mathbf G_m.$$

One then defines a generalized BB functor parametrizing maps or families for which the $G$-action extends to an action of $\overline G$.

Their theory works for finite-type schemes and algebraic spaces, admits relative forms, and has extensions toward algebraic stacks. (arXiv)

Thus the conceptual abstraction is

$$(\mathbf G_m\subset\mathbf A^1) \quad\rightsquigarrow\quad (G\subset\overline G).$$

A still more recent monoid-theoretic formulation proves affine-space-fibration statements for smooth schemes under suitable locally linearizable diagonalizable-group actions. (Springer Nature Link)

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24. The hierarchy of BB-type statements

It is useful to distinguish several levels that are sometimes all called “the BB decomposition.”

Level	Statement	Needed hypotheses
Attractor	$X^+=\operatorname{Maps}^{\mathbf G_m}(\mathbf A^1,X)$ exists	very general finite-type settings
Point decomposition	Points grouped according to $t\to0$ limits	existence of limits
Smooth strata	$X_\alpha^+$ smooth	smoothness of $X$
Affine fibers	$X_\alpha^+\to F_\alpha$ has fibers $\mathbf A^{d_\alpha}$	classical smooth setting
Affine bundles	Zariski locally $F_\alpha\times\mathbf A^{d_\alpha}$	smooth, locally linearizable action
Filtration	strata occur as successive differences of closed sets	especially natural in projective case
Cell decomposition	every $F_\alpha$ is a point	isolated fixed locus

Much confusion in applying BB theory comes from jumping, for example, from existence of an attractor in a singular scheme to the conclusion that the attractor is an affine cell. That implication is false.

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25. Why the theorem is so useful

There are essentially three reasons.

First, it converts a global variety into fixed-point data:

$$X \rightsquigarrow \{F_\alpha,\;N_\alpha^+,\;N_\alpha^-\}.$$

Second, fixed loci are often dramatically simpler than the original variety. A complicated moduli space may have fixed components which are products of symmetric powers of curves, smaller moduli spaces, flag varieties, or discrete combinatorial objects.

Third, many invariants are additive for affine fibrations. Thus

$$[X],\quad E(X),\quad \chi(X),\quad CH_*(X),\quad M(X)$$

can often be reconstructed from the $F_\alpha$.

This explains the extraordinary range of applications: Schubert calculus, toric geometry, moduli spaces, Hilbert and Quot schemes, Higgs bundles, quiver varieties, motives, intersection cohomology and geometric representation theory.

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26. A useful conceptual summary of the proof

The entire classical proof can be compressed into the chain

$$\boxed{\text{properness}} \Longrightarrow \boxed{\text{every orbit has a limit}}$$

followed by

$$\boxed{\mathbf G_m\text{ is diagonalizable}} \Longrightarrow \boxed{ T_pX=T_pX^-\oplus T_pX^0\oplus T_pX^+ }$$

followed locally by

$$\boxed{ \text{kill negative-weight coordinates} } \Longrightarrow \boxed{ X_\alpha^+\text{ is smooth} }$$

and then

$$\boxed{\text{contraction to }F_\alpha} \Longrightarrow \boxed{ X_\alpha^+\to F_\alpha \text{ is an }\mathbf A^{d_\alpha}\text{-bundle} }$$

and finally

$$\boxed{\text{equivariant projective embedding}} \Longrightarrow \boxed{\text{closed filtration}}.$$

For isolated fixed points the base disappears:

$$F_\alpha=\{p_\alpha\}$$

and hence

$$X_\alpha^+=\mathbf A^{d_\alpha}.$$

That is the classical BB cell decomposition.

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27. Selected references

• A. Białynicki-Birula, “Some theorems on actions of algebraic groups,” Annals of Mathematics 98 (1973), 480–497 — the foundational paper. (Annals of Mathematics)
• A. Białynicki-Birula, “Some properties of the decompositions of algebraic varieties determined by actions of a torus,” 1976 — develops structural and filtrability properties. (ZBMath)
• P. Brosnan, “On motivic decompositions arising from the method of Białynicki-Birula,” Inventiones Mathematicae 161 (2005), 91–111 — motivic refinement and arbitrary-field applications. (ResearchGate)
• T. Braden, “Hyperbolic localization of intersection cohomology,” introducing the sheaf-theoretic extension to singular spaces. (arXiv)
• V. Drinfeld and D. Gaitsgory, “On a theorem of Braden,” giving the modern interpolation proof of hyperbolic localization. (arXiv)
• J. Jelisiejew and Ł. Sienkiewicz, “Białynicki-Birula decomposition for reductive groups,” J. Math. Pures Appl. 131 (2019), 290–325 — functorial generalization from $\mathbf G_m$ to linearly reductive groups, schemes and algebraic spaces. (ScienceDirect)

The essential slogan is therefore:

$$\boxed{ \text{BB theory identifies an algebraic variety with the stable manifolds of its torus-fixed locus.} }$$

Smoothness makes those stable manifolds affine bundles; isolated fixed points turn them into affine cells; projectivity orders them into a filtration; and the modern theory replaces cells by attractor/repeller correspondences that continue to work for singular schemes, sheaves, motives and much more.