# Torus actions, Morse homology, and the Hilbert scheme of points on affine
space

Burt Totaro  UCLA Mathematics Department, Box 951555, Los Angeles, CA
90095-1555<EMAIL_ADDRESS>
  *     * scAbstract. We formulate a conjecture on actions of the multiplicative group in motivic homotopy theory. In short, if the multiplicative group $\mathbf{G}_{m}$ acts on a quasi-projective scheme $U$ such that $U$ is attracted as $t$ approaches $0$ in $\mathbf{G}_{m}$ to a closed subset $Y$ in $U$, then the inclusion from $Y$ to $U$ should be an $\mathbf{A}^{1}$-homotopy equivalence. We prove several partial results. In particular, over the complex numbers, the inclusion is a homotopy equivalence on complex points. The proofs use an analog of Morse theory for singular varieties. Application: the Hilbert scheme of points on affine $n$-space is homotopy equivalent to the subspace consisting of schemes supported at the origin.

scKeywords. Torus action, Morse homology, Hilbert scheme of points, motivic
homotopy theory.

sc2020 Mathematics Subject Classification. 14L30, 14C05, 14F42, 55R80

sc[Français]

scActions du tore, homologie de Morse et schéma de Hilbert des points de
l’espace affine

scRésumé. Nous formulons une conjecture sur les actions du groupe
multiplicatif en théorie de l’homotopie motivique. En bref, si le groupe
multiplicatif $\mathbf{G}_{m}$ opère sur un schéma quasi-projectif $U$, de
sorte que lorsque $t$ tend vers $0$ dans $\mathbf{G}_{m}$, $U$ est contracté
sur un sous-ensemble fermé $Y$, alors l’inclusion de $Y$ dans $U$ devrait être
une $\mathbf{A}^{1}$-équivalence d’homotopie. Nous démontrons plusieurs
résultats partiels. En particulier, sur le corps des complexes, l’inclusion
est une équivalence d’homotopie sur les points complexes. Les preuves
utilisent des analogues de la théorie de Morse pour des variétés singulières.
Application : le schéma de Hilbert des points de l’espace affine de dimension
$n$ est homotopiquement équivalent au sous-espace dont les points sont les
schémas supportés à l’origine.

  *
cMay 25, 2021Received by the Editors on September 22, 2020.
Accepted on June 23, 2021.

UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555

sce-mail<EMAIL_ADDRESS>
This work was supported by NSF grant DMS-1701237. I spoke about it in the
Algebraic Geometry and Moduli Zoominar at ETH Zurich.

© by the author(s) This work is licensed under
http://creativecommons.org/licenses/by-sa/4.0/

###### Contents

  1. 1 Introduction
  2. 2 Main results, and a conjecture on $\mathbf{G}_{m}$-actions in motivic homotopy theory
  3. 3 $\mathbf{G}_{m}$-actions and broken trajectories
  4. 4 Openness
  5. 5 Proof of Theorem 2.2
  6. 6 The real case
  7. 7 $\mathbf{A}^{1}$-connectedness of the Hilbert scheme
  8. 8 Conservativity for the motivic stable homotopy category
  9. 9 $\mathbf{G}_{m}$-actions on smooth varieties and motivic homotopy theory

## 1\. Introduction

We formulate a conjecture on actions of the multiplicative group
$\mathbf{G}_{m}$ in algebraic geometry. (Over the complex numbers, this group
may be called $\text{\bf C}^{*}$.) In short, if $\mathbf{G}_{m}$ acts on a
quasi-projective scheme $U$ which is attracted as $t\to 0$ in $\mathbf{G}_{m}$
to a closed subset $Y$ in $U$, then the inclusion $Y\to U$ should be an
$\mathbf{A}^{1}$-homotopy equivalence (Conjecture 2.1). This is not obvious,
in that the action of $\mathbf{G}_{m}$ on $U$ usually does not extend to a
morphism $\mathbf{A}^{1}\times U\rightarrow U$; compare Figure 1. We show that
the inclusion $Y\to U$ over the complex numbers is at least a homotopy
equivalence in the classical topology (Theorem 2.2). This extends work of
Hausel and Rodriguez-Villegas on the case where $U$ is smooth [HR15, Corollary
1.3.6]. We prove several other results in the direction of the conjecture,
including a homotopy equivalence on real points (Theorem 6.1) and, when $U$ is
smooth, an $\mathbf{A}^{1}$-homotopy equivalence after a suitable suspension
(Theorem 9.1). The proofs use the ideas of Morse homology, translated into
algebraic geometry (Proposition 3.1).

YU Figure 1. Example of a $T$-action on $U\cong\text{\bf
P}^{1}\times\mathbf{A}^{1}$, $t([x_{0},x_{1}],y)=([x_{0},tx_{1}],ty)$, with
$Y=\text{\bf P}^{1}\times 0$ shown as the horizontal line
($T=\mathbf{G}_{m}$). The arrows point in the direction $t\to 0$. The fixed
point set $Y^{T}$ consists of two points.

We apply these results to the Hilbert scheme of points on affine space. The
Hilbert scheme of points on an algebraic surface is smooth, and its Betti
numbers were computed by Göttsche [Göt02]. The Hilbert scheme of points on a
higher-dimensional variety, even affine space $\mathbf{A}^{n}$, is more
mysterious. It has many irreducible components [Iar72], and for $n\geq 16$ its
singularities satisfy Murphy’s law up to retraction [Jel20]. Nonetheless,
progress was recently made toward understanding the homotopy type (and even
the $\mathbf{A}^{1}$-homotopy type) of
$\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n})$ for $n$ large compared to
$d$. In particular, in the limit where $n$ goes to infinity,
$\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{\infty})$ has the
$\mathbf{A}^{1}$-homotopy type of the infinite Grassmannian
$\mathrm{Gr}_{d-1}(\mathbf{A}^{\infty})\simeq\operatorname{BGL}(d-1)$
[HJNTY20]. There are also corresponding stability theorems. In particular,
over the complex numbers, the resulting homomorphism on integral cohomology,

$H^{*}(\operatorname{BGL}(d-1,\text{\bf C}),\text{\bf Z})=\text{\bf
Z}[c_{1},\ldots,c_{d-1}]\rightarrow
H^{*}(\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n}),\text{\bf Z}),$

is an isomorphism in degrees at most $2n-2d+2$ [HJNTY20].

This paper considers another homotopical property of the Hilbert scheme
$\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n})$ for finite $n$. Namely,
over the complex numbers, we show that
$\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n})$ (in the classical topology)
has the homotopy type of $\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n},0)$,
the (compact) subspace of schemes supported at the origin (Corollary 2.3).
This result is deduced from Theorem 2.2 on the topology of
$\mathbf{G}_{m}$-actions. For example, it follows that the weight filtration
on the rational cohomology
$H^{i}(\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n}),\text{\bf Q})$ is
concentrated in weights $\leq i$, since that holds for proper schemes over C
[Del75].

It remains open whether $\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n},0)$
is $\mathbf{A}^{1}$-homotopy equivalent to
$\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n})$, over the complex numbers
or any other field. This would follow from our general Conjecture 2.1. We can
say something about the unstable $\mathbf{A}^{1}$-homotopy type of these
spaces, namely that $\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n},0)$ and
$\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n})$ are
$\mathbf{A}^{1}$-connected (Theorems 7.1 and 7.3).

As a tool, we extend one of Bachmann’s conservativity theorems, relating the
motivic stable homotopy category to the derived category of motives along with
real realizations (Theorem 8.1).

### Acknowledgments

Thanks to Tom Bachmann, David Hemminger, Marc Hoyois, Joachim Jelisiejew,
Denis Nardin, Maria Yakerson, and the referee for their suggestions.

## 2\. Main results, and a conjecture on $\mathbf{G}_{m}$-actions in motivic
homotopy theory

In this section, we formulate a general conjecture about actions of the
multiplicative group $T=\mathbf{G}_{m}$ in motivic homotopy theory. (For
motivic homotopy theory as defined by Morel and Voevodsky, a reference is
[MV99] and an introduction is [AE17].) Roughly, if $T$ acts on a quasi-
projective scheme $U$ which is attracted as $t\to 0$ in $T$ to a closed subset
$Y$ in $U$, then the inclusion $Y\to U$ should be an $\mathbf{A}^{1}$-homotopy
equivalence (Conjecture 2.1). We show that (over the complex numbers) the
inclusion $Y\to U$ is at least a homotopy equivalence in the classical
topology (Theorem 2.2, proved in Section 5).

Let $\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n})$ be the quasi-projective
scheme of zero-dimensional degree-$d$ closed subschemes of affine space
$\mathbf{A}^{n}$ over a field $k$. When $k$ is the complex numbers, we deduce
from Theorem 2.2 that $\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n})$ has
the homotopy type of $\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n},0)$, the
(compact) subspace of schemes supported at the origin (Corollary 2.3).

Here is our general conjecture on actions of the multiplicative group. Let $X$
be a projective scheme over a field $k$ with an action of $T=\mathbf{G}_{m}$.
Suppose that there is a $T$-equivariant ample line bundle on $X$. Let $Y$ be a
$T$-invariant closed subset of $X$ such that every point $x$ in $X$ with
$\lim_{t\to\infty}(tx)\in Y$ is in $Y$. Suppose that the fixed point set
$Y^{T}$ is open in $X^{T}$. Let $U$ be the subset of points $x$ in $X$ such
that $\lim_{t\to 0}(tx)$ is in $Y$. We show in Lemma 4.1 that $Y$ is contained
in $U$ and $U$ is open in $X$.

######  Conjecture 2.1.

The inclusion $Y\to U$ is an $\mathbf{A}^{1}$-homotopy equivalence (that is,
an isomorphism in the $\mathbf{A}^{1}$-homotopy category $H(k)$).

The assumption that $X$ has a $T$-equivariant ample line bundle is automatic
if $X$ is normal [Sum75, Theorem 1.6].

Over the complex numbers, the proof of Theorem 2.2 shows that $Y$ is an
attracting set for the $T$-action on $X$, in the terminology of topological
dynamics, and $U$ is the basin of attraction for $Y$ [Mil85, section 1]. To
say that $Y$ is an attracting set means that there is a neighborhood $N_{1}$
of $Y$ (in the classical topology) for which the images $t(N_{1})$ converge to
$Y$, meaning that for every neighborhood $N_{2}$ of $Y$, there is an $r>0$
such that $t(N_{1})\subset N_{2}$ for all $t\in\text{\bf C}^{*}$ with $|t|<r$.

The conjecture would be useful for motivic homotopy theory, since
$\mathbf{G}_{m}$-actions occur everywhere. When $U$ is smooth, both $Y$ and
$U$ are unions of affine bundles over the connected components of $Y^{T}$, by
Białynicki-Birula [Bia73]. But even then, we only know how to prove that $Y\to
U$ is an $\mathbf{A}^{1}$-homotopy equivalence after a suitable suspension
(Theorem 9.1). Regardless of whether $U$ is smooth, the conjecture would be
clear if the $T$-action on $U$ extended to a morphism

$\mathbf{A}^{1}\times U\to U,$

since $0\times U$ would map into $Y$; but in general there is no such
morphism. Even the $T$-action on $Y$ need not extend to a morphism
$\mathbf{A}^{1}\times Y\to Y$: consider the case where $Y$ is $\text{\bf
P}^{1}$ with the standard action of $T$, where $\lim_{t\to 0}tx=0$ if
$x\neq\infty$ but $\lim_{t\to 0}t(\infty)=\infty$.

Another way to describe the same situation is Drinfeld’s analog of the
Białynicki-Birula decomposition for singular varieties, although we will not
use that explicitly in what follows. Namely, Drinfeld defines an algebraic
space $Y^{+}$, the “attractor” of $Y$, as the space of $T$-equivariant
morphisms $\mathbf{A}^{1}\to Y$; roughly speaking, a point of $Y^{+}$ is a
point $x$ of $Y$ together with a limit point $\lim_{t\to 0}tx$. Drinfeld shows
that $Y^{+}\to Y$ is bijective for $Y$ proper over $k$, although usually not
an isomorphism [Dri13, Proposition 1.4.11]. For example, if $Y=\text{\bf
P}^{1}$ with the standard $T$-action, then the space $Y^{+}$ is the disjoint
union of $\mathbf{A}^{1}$ and the point at infinity. In a sense, the
difficulty for Conjecture 2.1 is that the action of $T$ on $U$ does not extend
to an action of the multiplicative monoid $\mathbf{A}^{1}$. The action of $T$
on $U^{+}$ does extend to an action of $\mathbf{A}^{1}$; but that does not
obviously help, because the morphism $U^{+}\to U$ is usually not a homotopy
equivalence.

As evidence for Conjecture 2.1 in the singular case, we prove the following
weaker statement in section 5. Theorem 2.2 was proved in the case where $U$ is
smooth by Hausel and Rodriguez-Villegas [HR15, Corollary 1.3.6].

######  Theorem 2.2.

Under the assumptions of Conjecture 2.1 with base field C, the inclusion $Y\to
U$ is a homotopy equivalence $($in the classical topology$)$.

######  Corollary 2.3.

Over the complex numbers, the inclusion from
$\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n},0)$ to
$\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n})$ $($in the classical
topology$)$ is a homotopy equivalence.

###### Proof of Corollary 2.3.

Let $X=\operatorname{\mathrm{Hilb}}_{d}(\text{\bf P}^{n})$ and
$Y=\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n},0)$. The idea is to use the
action of the multiplicative group $T$ (that is, $\text{\bf C}^{*}$) on
$\operatorname{\mathrm{Hilb}}_{d}(\text{\bf P}^{n})$, coming from the action
of $T$ on $\text{\bf P}^{n}$ by

$t([x_{0},\ldots,x_{n}])=[x_{0},tx_{1},\ldots,tx_{n}].$

(We identify $\mathbf{A}^{n}$ with the open subset $x_{0}\neq 0$ in $\text{\bf
P}^{n}$.) Here $X$ has a $\operatorname{GL}(n+1)$-equivariant ample line
bundle by construction. (Namely, Grothendieck constructed the Hilbert scheme
as a closed subscheme of the Grassmannian of subspaces of the vector space of
homogeneous polynomials of sufficiently high degree, sending a closed
subscheme $S\subset\text{\bf P}^{n}$ to the linear subspace of polynomials
that vanish on $S$ [Kol95, Section I.1]. The standard ample line bundle $O(1)$
on the Grassmannian is $GL(n+1)$-equivariant.) In particular, $X$ has a
$T$-equivariant ample line bundle.

The open subset $U\subset X$ of 0-dimensional schemes that converge as $t\in
T$ approaches 0 to a subscheme supported at $[1,0,\ldots,0]$ is exactly
$\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n})$. The action of $T$ on $U$
need not extend to a morphism $\mathbf{A}^{1}\times U\to U$ (or even
$\mathbf{A}^{1}\times Y\to Y$). Nonetheless, the desired homotopy equivalence
follows from Theorem 2.2. ∎

######  Remark 2.4.

The action of $T$ on $Y=\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n},0)$
does not extend to a morphism $\mathbf{A}^{1}\times Y\to Y$ in any case where
$Y\neq Y^{T}$. For example, for $Y=\operatorname{\mathrm{Hilb}}_{3}(A^{2},0)$,
the point $S_{a}:=\\{x=ay^{2},y^{3}=0\\}$ in $Y$ has $\lim_{t\to
0}t(S_{a})=Z:=\\{x^{2}=0,xy=0,y^{2}=0\\}$ for any $a\neq 0\in\text{\bf C}$,
whereas the point $S_{0}:=\\{x=0,y^{3}=0\\}$ in $Y$ is fixed by $T$ and hence
has $\lim_{t\to 0}t(S_{0})=S_{0}$.

## 3\. $\mathbf{G}_{m}$-actions and broken trajectories

We show here that for an action of the multiplicative group $T=\mathbf{G}_{m}$
on a projective scheme, every limit of $T$-orbits is a broken trajectory,
meaning a chain of $T$-orbits that connect a finite sequence of $T$-fixed
points. This is analogous to fundamental results in Morse homology. Namely,
given a smooth function on a closed Riemannian manifold satisfying some mild
conditions, every limit of gradient flow lines is a broken trajectory, meaning
a chain of gradient flow lines that connect a finite sequence of critical
points [BH10, Theorem 4.9, Definition 4.10]. For a $T$-action on a smooth
complex projective variety, one can in fact deduce the results here from those
in Morse homology, applied to a Hamiltonian function for the $T$-action.
Instead, we give a direct proof over any field. It turns out that smoothness
is irrelevant.

######  Proposition 3.1.

Let $X$ be a projective scheme over a field $k$ with an action of
$T=\mathbf{G}_{m}$. Suppose that there is a $T$-equivariant ample line bundle
on $X$. Then every limit of $T$-orbit closures in $X$ $($in the Chow variety
of effective 1-cycles on $X)$ is a broken trajectory, that is, a chain of
$T$-orbits $($with some positive multiplicities$)$ connecting some $T$-fixed
points.

In more detail: let $C$ be a smooth curve over $k$ with a morphism $f\colon
C\to X$, not mapping into $X^{T}$. Composing $f_{T}\colon T\times C\to T\times
X$ with the action of $T$ gives a morphism $T\times C\to X$, which extends to
a morphism $\text{\bf P}^{1}\times(C-Z)\to X$ for some 0-dimensional closed
subset $Z$ in $C$. This gives a morphism $e$ from $C-Z$ to the Chow variety of
1-cycles on $X$, which extends to all of $C$ by properness of Chow varieties.
Then for each $k$-point $c$ in $C$ $($possibly in $Z)$, $e(c)$ is a broken
trajectory over $k$, meaning the sum of $T$-orbits $($with some positive
multiplicities$)$ of points $y_{1},\ldots,y_{n}$ in $X(k)$ that connect
$T$-fixed points $x_{0},\ldots,x_{n}$ in $X(k)$. More precisely, $\lim_{t\to
0}t(y_{i})=x_{i-1}$ and $\lim_{t\to\infty}t(y_{i})=x_{i}$ for each $1\leq
i\leq n$.

If the image of $f\colon C\to X$ is contained in $X^{T}$, then the morphism
$e$ to the Chow variety of 1-cycles is constant (equal to zero as a 1-cycle).
The proposition would still be true in that case if suitably interpreted:
namely, any limit of $T$-fixed points in $X$ is a $T$-fixed point.

###### Proof.

There is a $T$-equivariant embedding of $X$ into the projective space $P(V)$
for some representation $V$ of $T$. Given that, we can assume that $X=P(V)$;
this greatly simplifies the situation. Then $T$ acts on $X=\text{\bf P}^{r}$
by $t([z_{0},\ldots,z_{r}])=[t^{a_{0}}z_{0},\ldots,t^{a_{r}}z_{r}]$ for some
integers $a_{i}$. We can assume that $a_{0}\leq\cdots\leq a_{r}$.

Composing $f\colon C\to X$ with the action of $T$ on $X$ gives a morphism
$T\times C\to X$, which can be viewed as a rational map $G\colon\text{\bf
P}^{1}\times C\dashrightarrow X$ over $k$. Since $X$ is proper over $k$, $G$
becomes a morphism $W\to X$, where $W$ is a surface obtained by blowing up
$\text{\bf P}^{1}\times C$ finitely many times at closed points. In
particular, $G$ restricts to a morphism $\text{\bf P}^{1}\times(C-Z)\to X$ for
some 0-dimensional closed subset $Z$ of $X$.

Because $C$ is normal and all fibers of $W\to C$ have dimension 1, the fibers
of $W\to C$ form a well-defined family of effective 1-cycles on $W$, and hence
they give a morphism from $C$ to the Chow variety of 1-cycles on $W$ [Kol95,
Theorem I.3.17]. Pushing cycles forward makes the Chow variety covariantly
functorial under arbitrary morphisms [Kol95, Theorem I.6.8]. Therefore, the
morphism $W\to X$ gives a morphism $e$ from $C$ to the Chow variety of
1-cycles on $X$.

For each $k$-point $c$ in $C$ (possibly in $Z$), $e(c)$ is an effective
1-cycle in $X$ whose support $S$ is the image under $G$ of the inverse image
of $c$ in $W$. Let us describe this image using power series. The completed
local ring of $C$ at $c$ is isomorphic to the power series ring $k[[u]]$. So
the curve $f\colon C\to X=\text{\bf P}^{r}$ near $c$ is given by some power
series $[z_{0}(u),\ldots,z_{r}(u)]$ with $z_{i}(u)\in k[[u]]$, not all zero.
Every point in the inverse image of $c$ in $W$ is contained in some curve in
$W$ that meets the open set $T\times(C-Z)$ in $W$. After completion, this
curve determines a finite extension $F$ of the field $k((u))$, along with an
$F$-point of $T\times(C-Z)$ over the given $k((u))$-point of $C$. Therefore,
the support $S$ of the limit 1-cycle $e(c)$, viewed as a subset of
$X(\overline{k})$, is the set of all $\overline{k}$-points in $X$ that can be
written as

$p=\lim_{u\to 0}[g(u)^{a_{0}}z_{0}(u),\ldots,g(u)^{a_{r}}z_{r}(u)]$

for some $g$ in the algebraic closure $\overline{k((u))}$, $g\neq 0$.

This limit point depends mainly on the rational number
$b:=\operatorname{ord}_{u}(g)$. The situation is described by the Newton
polygon of the pairs $(a_{i},\operatorname{ord}_{u}(z_{i}))$ in $\text{\bf
Z}\times(\text{\bf Z}\cup{\infty})$, as in Figure 2. (Here
$\operatorname{ord}_{u}(z_{i})=\infty$ if $z_{i}(u)$ is identically zero.)

$a_{i}$$\operatorname{ord}_{u}(z_{i})$ Figure 2. Newton polygon of the pairs
$(a_{i},\operatorname{ord}_{u}(z_{i}))$

Namely, let $I$ be the set of numbers $i\in\\{0,\ldots,r\\}$ such that
$ba_{i}+\operatorname{ord}_{u}(z_{i})$ reaches its minimum value. (That is,
let $l$ be the unique line of slope $-b$ that meets the Newton polygon but not
the region above it; then $I$ corresponds to the points
$(a_{i},\operatorname{ord}_{u}(z_{i}))$ that lie on $l$.) Then we compute that
the limit point $p$ defined above has all coordinates zero except the $i$th
coordinate for elements $i\in I$. Replacing $g$ by another function with the
same value of $b$ (that is, multiplying $g$ by a unit $h(u)$) just replaces
$p$ by $h(0)(p)$, another point in the same $T$-orbit as $p$.

For all but finitely many rational numbers $b$, the limit point $p$ above
belongs to a set $\\{x_{0},x_{1},\ldots,x_{n}\\}$ of $T$-fixed points, these
being indexed by the vertices of the Newton polygon. (For these values of $b$,
all nonzero coordinates $i$ in the set $I$ above have the same weight $a_{i}$,
which means that $p$ is a $T$-fixed point.) For the remaining $n$ values of
$b$, corresponding to the non-vertical edges of the Newton polygon, the limit
point can be anywhere in the $T$-orbit of a certain point $y_{i}$ in $X$ with
$\lim_{t\to 0}t(y_{i})=x_{i-1}$ and $\lim_{t\to\infty}t(y_{i})=x_{i}$. Here
the points $y_{1},\ldots,y_{n}$ (and hence the points
$x_{0},x_{1},\ldots,x_{n}$) can be taken to be $k$-points of $X$, by choosing
the function $g\in\overline{k((u))}$ with a given value of
$\operatorname{ord}_{u}(g)\in\text{\bf Q}$ to lie in a totally ramified
extension of $k((u))$, for example in $k((u^{1/e}))$ for a positive integer
$e$. ∎

## 4\. Openness

We now prove that the subset $U$ attracted in $Y$ in Conjecture 2.1 is open in
$X$.

######  Lemma 4.1.

As in Conjecture 2.1, let $X$ be a projective scheme over a field $k$ with an
action of $T=\mathbf{G}_{m}$. Suppose that there is a $T$-equivariant ample
line bundle on $X$. Let $Y$ be a $T$-invariant closed subset of $X$ such that
every point $x$ in $X$ with $\lim_{t\to\infty}(tx)\in Y$ is in $Y$. Suppose
that the fixed point set $Y^{T}$ is open in $X^{T}$. Let $U$ be the subset of
points $x$ in $X$ such that $\lim_{t\to 0}(tx)$ is in $Y$. Then $Y$ is
contained in $U$, and $U$ is open in $X$.

###### Proof.

Because $Y$ is a $T$-invariant closed subset of $X$, $Y$ is contained in $U$.

Clearly $U$ is a constructible subset of $X$; in Drinfeld’s notation from
section 2 above, $U$ is the image in $X$ of some connected components of
$X^{+}$, those whose limit as $t\to 0$ is in $Y^{T}$. (By assumption, $Y^{T}$
is a union of some connected components of $X^{T}$.) It suffices to prove that
$U$ is open in $X$ after replacing $k$ by its algebraic closure. If $U$ is not
open in $X$, then there is a morphism $f$ from a smooth curve $C$ to $X$ with
a $k$-point $c\in C$ such that $f(c)\in U$ and $f(d)\notin U$ for all $d\neq
c$ in $C$. Thus $\lim_{t\to 0}t(f(d))\in X^{T}-Y^{T}$ for $d\neq c$, whereas
$\lim_{t\to 0}t(f(c))\in Y^{T}$.

By Proposition 3.1, the limit of the $T$-orbit closures of the points $f(d)$
as $d$ approaches $c$ is a broken trajectory containing $f(c)$. By what we
have said about $f(d)$, this broken trajectory ends (in the $t\to 0$
direction) at a point in $X^{T}-Y^{T}$. But this broken trajectory also
contains $f(c)$ and hence the point $\lim_{t\to 0}t(f(c))$. Therefore, there
is a broken trajectory from $\lim_{t\to 0}t(f(c))\in Y^{T}$ down (in the $t\to
0$ direction) to a point in $X^{T}-Y^{T}$. This contradicts our assumption on
$Y$, namely that $Y$ is a $T$-invariant closed subset such that every point
$x$ in $X$ with $\lim_{t\to\infty}(tx)\in Y$ is in $Y$. We have shown that $U$
is open in $X$. ∎

## 5\. Proof of Theorem 2.2

###### Proof.

To recall the assumptions: we have a projective scheme $X$ over C with an
action of $T=\text{\bf C}^{*}$, and there is a $T$-equivariant ample line
bundle on $X$. We have a $T$-invariant closed subscheme $Y$ of $X$ such that
every point $x$ in $X$ with $\lim_{t\to\infty}(tx)\in Y$ is in $Y$, and the
fixed point set $Y^{T}$ is open in $X^{T}$. Let $U$ be the subset of points
$x$ in $X$ such that $\lim_{t\to 0}(tx)$ is in $Y$; then $U$ is Zariski open
in $X$, and $Y$ is contained in $U$, by Lemma 4.1. We want to show that the
inclusion $Y\to U$ is a homotopy equivalence in the classical topology.

######  Lemma 5.1.

Let $q_{1},q_{2},\ldots$ be a sequence of complex points in $X$ that converge
to a $T$-fixed point $w$. Let $t_{1},t_{2},\ldots$ be a sequence in $\text{\bf
C}^{*}$ that converges to zero in C. Then any limit point of the sequence
$t_{i}(q_{i})$ in $X$ lies in a broken trajectory “below $w$”. That is, such a
limit point belongs to the union of the $T$-orbits of some points
$y_{1},\ldots,y_{n}$ in $X$ and some $T$-fixed points $x_{0},\ldots,x_{n}=w$
such that $\lim_{t\to 0}t(y_{i})=x_{i-1}$ and
$\lim_{t\to\infty}t(y_{i})=x_{i}$ for each $1\leq i\leq n$.

###### Proof.

We largely follow the proof of Proposition 3.1. Choose a $T$-equivariant
embedding of $X$ into $P(V)$ for some representation $V$ of $T$. We can write
the action of $T$ on $\text{\bf P}^{r}=P(V)$ by
$t([z_{0},\ldots,z_{r}])=[t^{a_{0}}z_{0},\ldots,t^{a_{r}}z_{r}]$ with
$a_{0}\leq\cdots\leq a_{r}$. After passing to a subsequence, we can assume
that the points $q_{1},q_{2},\ldots$ all have the same lowest weight $a_{j}$
of a nonzero coefficient. On the locally closed subset $K$ in $X$ of points
with this lowest weight, the $T$-action $T\times K\to K$ extends to a morphism
$f\colon\mathbf{A}^{1}\times K\rightarrow\overline{K}$, by inspection. Here
$\overline{K}$ denotes the closure of $K$ in $X$.

By assumption, the points $(t_{i},q_{i})$ in $A_{1}\times K$ converge to the
point $(0,w)$ in $\mathbf{A}^{1}\times\overline{K}$. The rational map
$f\colon\mathbf{A}^{1}\times\overline{K}\dashrightarrow\overline{K}$ becomes a
morphism after some blow-up $M\to\mathbf{A}^{1}\times\overline{K}$ that is an
isomorphism over the complement of $0\times(\overline{K}-K)$. So any limit
point of the sequence $t_{i}(q_{i})$ in $X$ is equal to $f(m)$ for some point
$m$ in $M$ over $(0,w)\in\mathbf{A}^{1}\times\overline{K}$. In particular, we
can choose a smooth algebraic curve with a morphism to $M$ that goes through
$m$ and meets the open set $T\times K$.

Thus, by considering the completion of this curve at the point that maps to
$m$, we have power series $g(u)\neq 0\in\text{\bf C}[[u]]$ and $z(u)\in
X(\text{\bf C}((u)))$ such that $g(0)=0$, $\lim_{u\to 0}z(u)=w$, and
$\lim_{u\to 0}(g(u))(z(u))$ is the given limit point in $X$. The proof of
Proposition 3.1 showed that the limit of the closures of $T$-orbits of $z(u)$
as $u$ approaches 0 is a broken trajectory in $X$, which clearly contains $w$
as one of the $T$-fixed points $x_{0},\ldots,x_{n}$, say $w=x_{j}$. Moreover,
since $g(0)=0$ (so that $b:=\operatorname{ord}_{u}(g)>0$), the explicit
calculation of $\lim_{u\to 0}(g(u))(z(u))$ in $\text{\bf P}^{r}$ shows that
this limit point is “below $w$”, that is, in the union of
$x_{0},\ldots,x_{j}=w$ and the $T$-orbits that connect them. ∎

We continue the proof of Theorem 2.2. By the triangulation of real
semialgebraic sets, there is a triangulation of $X$ with $Y$ as a subcomplex
[Hir75, Section 1]. Therefore, $Y$ has arbitrarily small simplicial regular
neighborhoods $N$ in $X$, and for these the inclusion $Y\to N$ is a homotopy
equivalence [RS72, Chapter 3]. Let $N$ be a (compact) regular neighborhood of
$Y$ contained in $U$.

Consider the submonoid $(0,1]$ of $T=\text{\bf C}^{*}$. It would be convenient
to have $(0,1]\cdot N\subset N$, but it is not obvious that we can arrange
that. Instead, we argue as follows. I claim that each point $w\in Y^{T}$ has a
neighborhood $N_{1}$ in $U$ such that $t(N_{1})\subset N$ for all $t\in(0,1]$.
If not, then there would be a sequence $q_{i}$ in $U$ converging to $w$ such
that for each positive integer $j$, $(0,1]\cdot q_{j}$ is not contained in
$N$. So there is a sequence $t_{i}\in(0,1]$ such that $t_{i}(q_{i})$ is not in
$N$. The sequence $t_{i}$ must converge to zero; otherwise, a subsequence of
$t_{i}(q_{i})$ would converge to the $T$-fixed point $w$ in $Y$ (and hence
infinitely many of those points would be in $N$).

After passing to subsequences, we can assume that $t_{i}(q_{i})$ converges to
a point $v$ in $X-\operatorname{int}(N)$, hence not in $Y$. By Lemma 5.1, $v$
belongs to the union of some finite chain of $T$-orbits going “down” from $w$,
meaning the $T$-orbits of some points $y_{1},\ldots,y_{n}$ in $X$ and some
$T$-fixed points $x_{0},\ldots,x_{n}=w$ such that $\lim_{t\to
0}t(y_{i})=x_{i-1}$ and $\lim_{t\to\infty}t(y_{i})=x_{i}$. By our assumption
that all points $x$ in $X$ with $\lim_{t\to\infty}tx\in Y$ are in $Y$, it
follows that $v$ is in $Y$, a contradiction. Thus we have proved the claim
that each point $w\in Y^{T}$ has a neighborhood $N_{1}$ in $U$ such that
$t(N_{1})\subset N$ for all $t\in(0,1]$.

More generally, for each point $x\in U$ (not just in $Y^{T}$), there is a real
number $a\in(0,1]$ and a neighborhood $N_{1}$ of $x$ in $U$ such that
$t(N_{1})\subset N$ for all $t\in(0,a]$. That follows from the previous
statement applied to the point $y=\lim_{t\to 0}t(x)\in Y^{T}$.

Therefore, for every compact subset $K$ of $U$, there is a real number
$a\in(0,1]$ such that $t(K)\subset N$ for all $t\in(0,a]$. Equivalently,
$K\subset a^{-1}(N)$. In particular, there is a real number $c>1$ such that
the compact neighborhood $N$ of $Y$ is contained in the interior of $c(N)$. It
also follows that $U$ is the union of the subsets $c^{j}(N)$ over all $j\geq
0$.

Since the inclusion $Y\to N$ is a homotopy equivalence, so is the inclusion
$Y\to c^{j}(N)$ for each integer $j$. Therefore, each of the inclusions
$c^{j}(N)\to c^{j+1}(N)$ is also a homotopy equivalence. Since $c^{j}(N)$ is a
closed subset contained in the interior of $c^{j+1}(N)$, the union of these
subsets (namely, $U$) has the colimit topology. Since this is a filtered
colimit, the colimit $U$ is equivalent to the homotopy colimit, and so the
inclusion $N\to U$ is a homotopy equivalence. Since the inclusion $Y\to N$ is
also a homotopy equivalence, we conclude that $Y\to U$ is a homotopy
equivalence. ∎

## 6\. The real case

######  Theorem 6.1.

Under the assumptions of Conjecture 2.1 with base field R, the inclusion
$Y(\text{\bf R})\to U(\text{\bf R})$ is a homotopy equivalence.

###### Proof.

This is similar to the complex case (Theorem 2.2). In particular, Lemma 5.1
holds by the same proof over R in place of C, using that Proposition 3.1
expresses any limit of $T$-orbits of R-points as the union of a finite chain
of $T$-orbits of R-points. Given that, the proof of Theorem 2.2 applies
verbatim (using a regular neighborhood of $Y(\text{\bf R})$ inside
$U(\text{\bf R})$) to show that the inclusion $Y(\text{\bf R})\to U(\text{\bf
R})$ is a homotopy equivalence. ∎

## 7\. $\mathbf{A}^{1}$-connectedness of the Hilbert scheme

Hartshorne showed that the Hilbert scheme of projective space over a field $k$
(of subschemes with a given Hilbert polynomial) is connected [Har66]. In
particular, $\operatorname{\mathrm{Hilb}}_{d}(\text{\bf P}^{n})$ is connected
for every $n\geq 1$ and $d\geq 0$. The argument was sharpened by Reeves and
Pardue [Ree95, Par96]. Reeves and Pardue showed that for an infinite field
$k$, any two $k$-points of $\operatorname{\mathrm{Hilb}}_{d}(\text{\bf
P}^{n})$ can be connected by a chain of affine lines over $k$. By Morel’s
results (Lemma 7.2 below), it follows that
$\operatorname{\mathrm{Hilb}}_{d}(\text{\bf P}^{n}_{k})$ is
$\mathbf{A}^{1}$-connected for $k$ infinite.

We now show that $\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n})$ and
$\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n},0)$ are
$\mathbf{A}^{1}$-connected over an infinite field $k$. This seems to be harder
for $\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n},0)$, because (for $d>1$)
this space contains no smooth subschemes of $\mathbf{A}^{n}$. When $n\geq d$,
the $\mathbf{A}^{1}$-connectedness of these Hilbert schemes can be proved
using the ideas of [HJNTY20], but here we want the results for all $n$ and
$d$.

######  Theorem 7.1.

Let $k$ be an infinite field, $n\geq 1$, $d\geq 0$. Then
$\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n})$ is
$\mathbf{A}^{1}$-connected over $k$.

###### Proof.

We use the following result of Morel’s:

######  Lemma 7.2.

Let $X$ be a separated scheme of finite type over a field $k$ such that $X$
has a $k$-point. Suppose that for every separable finitely generated field
extension $F$ of $k$, any two $F$-points of $X$ can be connected by a chain of
affine lines $\mathbf{A}^{1}_{F}\to X_{F}$. Then $X$ is
$\mathbf{A}^{1}$-connected.

###### Proof.

For $m\geq 0$, Morel showed that an $\mathbf{A}^{1}$-local pointed simplicial
Nisnevich sheaf $X$ over $k$ is $m$-connected if and only if the fiber $X(F)$
is $m$-connected for every separable finitely generated field extension $F$ of
$k$ [Mor05, Lemma 6.1.3]. Also, for a simplicial sheaf $X$,
$\pi_{0}(X)\to\pi_{0}^{\mathbf{A}^{1}}(X)$ is a surjection of Nisnevich
sheaves [MV99, Section 2, Corollary 3.22]. In particular, for a separated
scheme $X$ of finite type over $k$, $X(F)\to\pi_{0}^{\mathbf{A}^{1}}(X)(F)$ is
surjective for every separable finitely generated field extension $F$ over
$k$. This implies the lemma. ∎

By Lemma 7.2, it suffices to show that for an infinite field $k$, any two
$k$-points of $U:=\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n})$ can be
connected by a chain of affine lines $\mathbf{A}^{1}_{k}\to U_{k}$. So let $S$
be any $k$-point of $U$. That is, $S$ is a closed subscheme of
$\mathbf{A}^{n}$ over $k$ of dimension zero and degree $d$. We use a “Gröbner
degeneration”, as follows. Let $c$ be a large positive integer, and consider
the action of $T:=\mathbf{G}_{m}$ on $\mathbf{A}^{n}$ by

$t(x_{1},\ldots,x_{n})=(t^{c}x_{1},t^{c^{2}}x_{2},\ldots,t^{c^{n}}x_{n}).$

Then $S_{2}:=\lim_{t\to 0}t(S)$ exists in $U$. It is a closed subscheme
supported at the origin in $\mathbf{A}^{n}$, and it is fixed by this
$T$-action. That is, the defining ideal $I$ of $S_{2}$ as a subscheme of
$\mathbf{A}^{n}_{k}$ is homogeneous with respect to the weights
$(c,c^{2},\ldots,c^{n})$ on $x_{1},\ldots,x_{n}$. Taking $c$ big enough
compared to $d$ and $n$, it follows that $I$ is generated by monomials. By
construction, we can connect $S$ to $S_{2}$ by an affine line over $k$.

Since $S_{2}$ has dimension 0 and is defined by monomials, it is smoothable,
using Hartshorne’s proof by distraction; a specific reference is [CEVV09,
Proposition 4.15]. We need the more precise information given by the proof, as
follows. Let $I=(x^{M_{1}},\ldots,x^{M_{r}})$ be the minimal set of monomial
generators for the ideal $I$. We use multi-index notation, so
$x^{M_{i}}=\prodop\displaylimits_{j=1}^{n}x_{j}^{M_{ij}}$. Consider the
following flat family of ideals in $k[x_{1},\ldots,x_{n}]$ parametrized by
affine space $A^{d}$: for a point $(a_{0},\ldots,a_{d-1})$ in $A^{d}$, take
the ideal $J_{a}$ in $k[x_{1},\ldots,x_{n}]$ generated by the elements

$f_{i}:=\prodop\displaylimits_{j=1}^{n}(x_{j}-a_{0})(x_{j}-a_{1})\cdots(x_{j}-a_{M_{ij}-1}).$

The initial ideal of $J_{a}$ (with respect to any monomial order compatible
with the grading, say the graded reverse lexicographic order) is $I$; so we
have a flat family. This defines a morphism
$A^{d}\to\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n})$ over $k$, with the
origin mapping to the given monomial scheme $S_{2}$. When
$a_{0},\ldots,a_{d-1}$ are distinct elements of $k$, the subscheme $Z_{a}$ of
$\mathbf{A}^{n}$ defined by $J_{a}$ contains $d$ distinct $k$-points: namely,
for each of the $d$ monomials $x^{L}$ not in $I$, $Z_{a}$ contains the
$k$-point $(a_{L_{1}},\ldots,a_{L_{n}})$. Since the scheme $Z_{a}$ has degree
$d$, it must be smooth over $k$, equal to those $d$ $k$-points in
$\mathbf{A}^{n}$.

Since $k$ is infinite, it follows that we can connect $S_{2}$ by an affine
line in $\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n})$ to a scheme $S_{3}$
which consists of $d$ distinct $k$-points in $\mathbf{A}^{n}$. If $n\geq 2$,
since the condition for two points to be equal in $\mathbf{A}^{n}$ has
codimension at least 2, it is easy to connect $S_{3}$ by a chain of affine
lines over $k$ to a fixed arrangement $S_{4}$ of $d$ distinct $k$-points in
$\mathbf{A}^{n}$. Thus $\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n})$ is
$\mathbf{A}^{1}$-connected when $n\geq 2$. It is also
$\mathbf{A}^{1}$-connected when $n=1$, since
$\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{1})\cong A^{d}$. ∎

######  Theorem 7.3.

Let $k$ be an infinite field, $n\geq 1$, $d\geq 0$. Then
$\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n},0)$ is
$\mathbf{A}^{1}$-connected over $k$.

###### Proof.

By Lemma 7.2, it suffices to show that for every infinite field $k$, any two
$k$-points of $Y:=\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n},0)$ can be
connected by a chain of affine lines over $k$. For lack of a direct proof, we
will reduce this to Theorem 7.1.

Let $X=\operatorname{\mathrm{Hilb}}_{d}(\text{\bf P}^{n})$,
$U=\operatorname{\mathrm{Hilb}}_{d}(\mathbf{A}^{n})$, and $T=\mathbf{G}_{m}$.
Consider the action of $T$ on $X$ coming from the action of $T$ by scaling on
$\mathbf{A}^{n}$. Then $Y$ is a $T$-invariant closed subset of $U$, and
$\lim_{t\to 0}t(x)$ exists in $Y$ for each point $x$ in $U$. Clearly we can
connect any $k$-point $x$ in $Y$ to this limit point by an affine line in $Y$,
and the limit point is fixed by $T$. So it suffices to show that any two
$k$-points $p,q$ in $Y^{T}$ can be connected by a chain of affine lines in
$Y$.

We know by the proof of Theorem 7.1 that $p$ and $q$ can be connected by a
chain of affine lines in $U$. So it suffices to show that for any morphism
$f\colon\mathbf{A}^{1}\to U$ over $k$, we can connect $\lim_{t\to 0}t(f(0))$
to $\lim_{t\to 0}t(f(1))$ by a chain of affine lines in $Y$.

Composing $f$ with the action of $T$ on $U$ gives a morphism
$T\times\mathbf{A}^{1}\to U$ over $k$, which can be viewed as a rational map
$\text{\bf P}^{1}\times\text{\bf P}^{1}\dashrightarrow X$ over $k$. Since $X$
is proper over $k$, this map becomes a morphism after blowing up the domain
finitely many times at closed points. It follows that $g(s):=\lim_{t\to
0}t(f(s))$ defines a morphism $g\colon\mathbf{A}^{1}-Z\to Y$ for some
0-dimensional closed subset $Z$ of $\mathbf{A}^{1}$. Since $Y$ is proper over
$k$, $g$ extends to a morphism $g\colon\mathbf{A}^{1}\to Y$. As a result, for
any two $k$-points $s_{1},s_{2}$ in $\mathbf{A}^{1}-Z$, $\lim_{t\to
0}t(f(s_{1}))$ and $\lim_{t\to 0}t(f(s_{2}))$ can be connected by an affine
line in $Y$.

There remains the case where $0$ or 1 is in $Z$. It suffices to show that for
any $k$-point $s_{0}$ in $Z$ (which will be 0 or 1 for us), the point
$z_{0}:=\lim_{t\to 0}t(f(s_{0}))$ can be connected by a chain of affine lines
in $Y$ to $g(s_{0})$.

By Proposition 3.1, the $T$-orbits of the points $f(s)$ (for
$s\in\mathbf{A}^{1}-Z)$ converge as $s$ approaches $s_{0}$ to a “broken
trajectory” containing $f(s_{0})$. This means the union of $T$-orbits of
points $y_{1},\ldots,y_{n}$ in $X(k)$ that connect $T$-fixed points
$x_{0},\ldots,x_{n}$ in $X(k)$, in the sense that $\lim_{t\to
0}t(y_{i})=x_{i-1}$ and $\lim_{t\to\infty}t(y_{i})=x_{i}$.

Both the $k$-point $z_{0}=\lim_{t\to 0}t(f(s_{0}))$ and the $k$-point
$g(s_{0})$ lie in this union of $T$-orbit closures in $X$, and both are in the
closed subset $Y$. We know that every point $x$ in $X$ with
$\lim_{t\to\infty}(tx)\in Y$ is in $Y$. Therefore, all the orbit closures that
connect $z_{0}$ to $g(s_{0})$ are in $Y$. So these two points can be connected
by affine lines over $k$ in $Y$, as we want. ∎

## 8\. Conservativity for the motivic stable homotopy category

Extending one of Tom Bachmann’s results, we prove the following conservativity
theorem, relating the motivic stable homotopy category with the derived
category of motives along with real realizations. Thanks for Bachmann for his
suggestions. This result will be used in the proof of Theorem 9.1.

######  Theorem 8.1.

Let $k$ be a finitely generated field of characteristic zero $($that is, a
finitely generated extension field of $\text{\bf Q})$. Let $A$ be a compact
object in $SH(k)$ such that $M(A)=0$ in $DM(k)$ and for every embedding of $k$
into R, $H_{*}(A(\text{\bf R}),\text{\bf Z}[1/2])=0$. Then $A=0$.

If $k$ has no real embedding, Theorem 8.1 just says that $M(A)=0$ in $DM(k)$
implies $A=0$ in $SH(k)$.

###### Proof.

Bachmann showed (in particular) that if $A$ is a compact object in $SH(k)$
such that $M(A)=0$ in $DM(k)$ and for every $\sigma$ in the space
$\operatorname{Sper}(k)$ of orderings of $k$, $M_{\sigma}[1/2](A)=0$ in
$D(\text{\bf Z}[1/2])$, then $A=0$ [Bac18, Theorem 33]. When $\sigma$ comes
from an embedding of $k$ into R, $M_{\sigma}[1/2](A)$ is the complex that
computes the singular homology of the corresponding real realization of $A$,
$H_{*}(A(\text{\bf R}),\text{\bf Z}[1/2])$ [Bac18, Remark 1]. It remains to
show that we only need to consider orderings that come from real embeddings of
$k$.

We use the following property of the space $\operatorname{Sper}(k)$ of
orderings of $k$ [FHV94, Lemma 1.6]. The topology on $\operatorname{Sper}(k)$
is defined by taking the sets $\\{\sigma:a>_{\sigma}0\\}$ for $a\in k$ as a
sub-basis for the topology. This makes $\operatorname{Sper}(k)$ into a compact
Hausdorff totally disconnected space.

######  Lemma 8.2.

Let $k$ be a finitely generated field of characteristic zero. Then the set of
archimedean orderings of $k$ is dense in the topological space
$\operatorname{Sper}(k)$ of orderings of $k$, and every archimedean ordering
comes from an embedding of $k$ into R.

Given that, we are done if we can show that the support in
$X:=\operatorname{Sper}(k)$ of a compact object in $SH(k)$ is open as well as
closed. This is related to the general fact that for a tensor triangulated
category $K$, the support of an object of $K$ in the Balmer spectrum
$\operatorname{Spc}(K)$ is closed and its complement is quasi-compact [Bal05,
Proposition 2.14]. However, we will argue more directly.

We use that the functors $M_{\sigma}$ come from a functor from $SH(k)$ to the
derived category of sheaves $D(X,\text{\bf Z}[1/2])$, which takes compact
objects to compact objects. Indeed, by [Bac18, Lemma 21], the functor from
$SH(k)$ to Witt motives $DM_{W}(k,\text{\bf Z}[1/2])$ is monoidal; so it takes
rigid objects to rigid objects, and the rigid objects coincide with the
compact objects in these categories. (Some people say “strongly dualizable”
rather than “rigid”.) Furthermore, $DM_{W}(k,\text{\bf Z}[1/2])$ is equivalent
to $D(X,\text{\bf Z}[1/2])$ [Bac18, Lemma 26 and proof of Theorem 30].

A compact object in $D(X,\text{\bf Z}[1/2])$ is a perfect complex; that is, it
is locally isomorphic to a bounded complex of finitely generated projective
$\text{\bf Z}[1/2]$-modules. (Indeed, since $X$ is compact, Hausdorff, and
totally disconnected, every open subset of $X$ is a union of clopen subsets
(or equivalently, quasi-compact open subsets). It follows that every compact
object in $D(X,\text{\bf Z}[1/2])$ is a summand in $D(X,\text{\bf Z}[1/2])$ of
a bounded complex of sheaves which are finite direct sums of sheaves of the
form $j_{!}(\text{\bf Z}[1/2]_{U})$, with $j\colon U\hookrightarrow X$ the
inclusion of a quasi-compact open subset [Stacks, Lemma 094C]. Clearly such a
summand is a perfect complex of $\text{\bf Z}[1/2]$-modules on $X$.)

Because sections of the sheaf $\text{\bf Z}[1/2]$ on
$X=\operatorname{Sper}(k)$ are locally constant, the support of a perfect
complex on $X$ is open as well as closed. ∎

## 9\. $\mathbf{G}_{m}$-actions on smooth varieties and motivic homotopy
theory

We now consider Conjecture 2.1 in the special case where $U$ is smooth. (One
example where this applies is the inclusion from
$\operatorname{\mathrm{Hilb}}_{d}(A^{2},0)$ to
$\operatorname{\mathrm{Hilb}}_{d}(A^{2})$.) For $U$ smooth, we show that the
inclusion $Y\to U$ becomes an $\mathbf{A}^{1}$-homotopy equivalence after
suspending by $\mathbf{S}^{3,1}=\mathbf{S}^{2}\wedge\mathbf{G}_{m}$. It
follows that $Y$ and $U$ have many invariants in common, such as motivic
homology and cohomology, $l$-adic cohomology, and so on. On the other hand, it
remains open whether the Nisnevich sheaf $\pi_{0}^{\mathbf{A}^{1}}$ is the
same for $Y$ and $U$, and likewise for $\pi_{1}^{\mathbf{A}^{1}}$. At least
for $\pi_{0}^{\mathbf{A}^{1}}$, one might hope to imitate the proof of Theorem
7.3.

######  Theorem 9.1.

Under the assumptions of Conjecture 2.1 with base field $k$ of characteristic
zero, and assuming that $U$ is smooth over $k$, the inclusion $Y\to U$ becomes
an $\mathbf{A}^{1}$-homotopy equivalence after suspending by
$\mathbf{S}^{3,1}=\mathbf{S}^{2}\wedge\mathbf{G}_{m}$.

###### Proof.

We can assume that $U$ is connected, by arguing separately for each connected
component of $U$. Next, by equivariant resolution of singularities (using that
$k$ has characteristic zero), we can assume that $X$ (as well as $U$) is
smooth over $k$, while still having a $T$-action [Kol07, Proposition 3.9.1].

We first show that the inclusion $Y\to U$ induces an isomorphism in the
derived category of motives $DM(k)$, $M(Y)\to M(U)$. Namely, since $X$ is
smooth over $k$, we have the Białynicki-Birula decomposition, as follows. The
fixed point set $X^{T}$ is smooth over $k$. Write $Z_{1},\ldots,Z_{m}$ for the
connected components of $X^{T}$. For each $i$, let $Z_{i}^{+}=\\{x\in
X:\lim_{t\to 0}tx\in Z_{i}\\}$ and $Z_{i}^{-}=\\{x\in X:\lim_{t\to\infty}tx\in
Z_{i}\\}$ be the stable and unstable manifolds of $Z_{i}$. Then the action of
$T$ gives morphisms $Z_{i}^{+}\to Z_{i}$ and $Z_{i}^{-}\to Z_{i}$ which are
affine-space bundles [Bia73].

Karpenko showed that this geometric decomposition gives a direct-sum
decomposition of Chow motives over $k$ [Kar00, Theorem 6.5], [Bro05, Theorem
3.5]:

$M(X)\cong\oplus_{i=1}^{m}M(Z_{i})\\{a_{i}\\},$

where $a_{i}:=\dim(Z_{i}^{+})-\dim(Z_{i})$. (Here $\text{\bf Z}\\{1\\}$
denotes the Lefschetz motive, with $M(\text{\bf P}^{1})=\text{\bf
Z}\\{0\\}\oplus\text{\bf Z}\\{1\\}$.) This implies another decomposition
$M(X)\cong\oplus_{i=1}^{m}M(Z_{i})\\{b_{i}\\}$, where
$b_{i}:=\dim(Z_{i}^{-})-\dim(Z_{i})$, by inverting the $T$-action on $X$. Here

$a_{i}+\dim(Z_{i})+b_{i}=n,$

by considering the action on $T$ on the tangent space to $X$ at a point of
$Z_{i}$.

The category of Chow motives is a full subcategory of the derived category of
motives, $DM(k)$: the thick subcategory generated by smooth projective schemes
over $k$ tensored with $\text{\bf Z}\\{a\\}=\text{\bf Z}(a)[2a]$ for integers
$a$ [Voe00]. Every scheme $X$ of finite type over $k$ has a motive $M(X)$ and
a compactly supported motive $M^{c}(X)$ in $DM(k)$. We can assume that
$Z_{1},\ldots,Z_{m}$ are ordered in such a way that the closure of $Z_{i}^{-}$
is contained in $X_{i}:=\cup_{j\leq i}Z_{j}^{-}$. Karpenko’s argument shows
that the exact triangle

$M^{c}(X_{i-1})\to M^{c}(X_{i})\to M^{c}(Z_{i}^{-})\cong M(Z_{i})\\{b_{i}\\}$

in $DM(k)$ is split [Kar00, Theorem 6.5, part (a)]. (Indeed, his splitting on
Chow groups is defined by an element of $CH_{\dim(Z_{i})+b_{i}}(Z_{i}\times
X_{i})$, and that is precisely
$\operatorname{Hom}(M(Z_{i})\\{b_{i}\\},M^{c}(X_{i}))$ since $Z_{i}$ is smooth
and proper over $k$.) In particular, it follows that

$M(X_{j})\cong\oplus_{i=1}^{j}M(Z_{i})\\{b_{i}\\}$

for each $1\leq j\leq m$, and its open complement $X-X_{j}$ satisfies

$M^{c}(X-X_{j})\cong\oplus_{i=j+1}^{m}M(Z_{i})\\{b_{i}\\}$

(Here $X_{j}$ need not be smooth, but it is proper over $k$, and so its motive
$M(X_{j})$ is the same as its compactly supported motive $M^{c}(X_{j})$.)

In the notation of Conjecture 2.1, we can assume that the closed subset $Y$ of
$X$ is equal to $X_{r}$ for some $r\leq m$. So
$M(Y)\cong\oplus_{i=1}^{r}M(Z_{i})\\{b_{i}\\}$. Likewise, the open subset $U$
is the union of the subsets $Z_{i}^{+}$ with $i\leq r$. By the splitting in
$DM(k)$ above, applied to the inverse action of $T$ on $X$, we have

$M^{c}(U)\cong\oplus_{i=1}^{r}M(Z_{i})\\{a_{i}\\}.$

Since $U$ is smooth of dimension $n$ over $k$, it follows that

$\displaystyle M(U)$ $\displaystyle\cong M^{c}(U)^{*}\\{n\\}$
$\displaystyle\cong\oplus_{i=1}^{r}M(Z_{i})^{*}\\{n-a_{i}\\}$
$\displaystyle\cong\oplus_{i=1}^{r}M(Z_{i})\\{n-a_{i}-\dim(Z_{i})\\}$
$\displaystyle\cong\oplus_{i=1}^{r}M(Z_{i})\\{b_{i}\\}.$

Thus $M(Y)$ is isomorphic to $M(U)$ in $DM(k)$. More precisely, the inclusion
$Y\to U$ induces an isomorphism $M(Y)\to M(U)$. To see this, one checks from
Karpenko’s construction of the splittings that for $i,j\in\\{1,\ldots,r\\}$,
the composition $M(Z_{i})\\{b_{i}\\}\to M(Y)\to M(U)\to M(Z_{j})\\{b_{j}\\}$
is the identity for $i=j$ and zero if $i<j$.

The schemes $Y,U,X$ with $T$-action are defined over some finitely generated
subfield of $k$. So we can assume that the field $k$ is finitely generated
over Q. Apply Theorem 8.1 to the cofiber $A={}^{\infty}(U/Y)$ in $SH(k)$. We
showed above that the motive of $A$ in $DM(k)$ is zero. Also, for every real
embedding of $k$, the real realization of $A$ is zero in the stable homotopy
category, by Theorem 6.1. (It may be that $k$ has no real embedding.)
Therefore, $A=0$ in $SH(k)$. That is, the inclusion $Y\to U$ induces an
isomorphism in $SH(k)$.

Again using that $k$ has characteristic zero, Bachmann showed that the
$\text{\bf P}^{1}$-suspension functor

$Q={}^{\infty}_{\text{\bf P}^{1}}\colon H(k)_{*}\to SH(k)$

is conservative on $\mathbf{A}^{1}$-simply connected spaces which can be
written as homotopy colimits of spaces $X_{+}\wedge\mathbf{G}_{m}$ with
$X\in\mathrm{Sm}_{k}$ [Bac21, Theorem 1.3].

The $\mathbf{S}^{2}$-suspension of every space in $H(k)_{*}$ is
$\mathbf{A}^{1}$-simply connected. The map
$\mathbf{S}^{2}\wedge\mathbf{G}_{m}\wedge
Y_{+}\to\mathbf{S}^{2}\wedge\mathbf{G}_{m}\wedge U_{+}$ is therefore a pointed
$\mathbf{A}^{1}$-homotopy equivalence. ∎

Theorem 9.1 can be slightly strengthened if in addition $Y$ and $U$ are
$\mathbf{A}^{1}$-connected. In that case, their $\mathbf{S}^{1}$-suspensions
are $\mathbf{A}^{1}$-simply connected, and so $\text{\bf P}^{1}\wedge
Y_{+}\to\text{\bf P}^{1}\wedge U_{+}$ is a pointed $\mathbf{A}^{1}$-homotopy
equivalence, using that $\text{\bf
P}^{1}=\mathbf{S}^{2,1}=\mathbf{S}^{1}\wedge\mathbf{G}_{m}$.

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