$\Omega_{c}(3)=\omega^{5}\mathbb{Q}$. The previous corollary produces classes
in $H^{0}(\mathrm{SL}_{3}(\mathbb{Z});\mathbb{R})$, as well as
$H^{0}(\mathrm{SL}_{4}(\mathbb{Z});\mathbb{R})$,
$H^{3}(\mathrm{SL}_{4}(\mathbb{Z});\mathbb{R})$.

For $g=5$, we have forms of non-compact type
$\Omega_{nc}(5)=\mathbb{Q}\oplus\mathbb{Q}\omega^{5}$. They give rise, via the
previous corollary, to classes in
$H^{0}(\mathrm{SL}_{5}(\mathbb{Z});\mathbb{R})$ and
$H^{5}(\mathrm{SL}_{5}(\mathbb{Z});\mathbb{R})$, and also
$H^{0}(\mathrm{SL}_{6}(\mathbb{Z});\mathbb{R})$ and
$H^{5}(\mathrm{SL}_{6}(\mathbb{Z});\mathbb{R})$. The forms of compact type
$\Omega_{c}(5)=\omega^{9}\mathbb{Q}\oplus(\omega^{5}\wedge\omega^{9})\mathbb{Q}$
give rise to two further classes in
$H^{5}(\mathrm{SL}_{6}(\mathbb{Z});\mathbb{R})$ and
$H^{10}(\mathrm{SL}_{6}(\mathbb{Z});\mathbb{R})$.

######  Remark 14.11.

A natural way to phrase the corollary is to say that there is a complex

$\Omega^{n}_{nc}(g)\otimes_{\mathbb{Q}}\mathbb{R}\longrightarrow
H^{n}(\mathrm{SL}_{g+1}(\mathbb{Z});\mathbb{R})\longrightarrow\left(\Omega_{c}^{d_{g+1}-n-2}(g)\otimes_{\mathbb{Q}}\mathbb{R}\right)^{\vee}$

where the first map is injective, and the second map is surjective.

###  14.5. Stability

Let $g>1$ be odd. By Borel [Bor74], there exists a $\kappa(g)$ which grows
linearly in $g$ such that

(14.8)
$\Omega_{nc}^{n}(g)\otimes_{\mathbb{Q}}\mathbb{R}\overset{\sim}{\longrightarrow}H^{n}(\mathrm{GL}_{g}(\mathbb{Z});\mathbb{R})\quad\hbox{
for all }\,n<\kappa(g)\ .$

Borel showed that $\kappa(g)$ is of the order of $g/4$. Using the recent work
[KMP22, Theorem A], and (14.6), it may be improved to $\kappa(g)=g$. For $n$
in this range, $\Omega_{nc}^{n}(g)=\Omega^{n}(g)$.

######  Corollary 14.12.

Let $g>1$ be odd and let $\kappa(g)$ be such that (14.8) holds. Then

$\displaystyle H^{n}_{dR}(\left|L\mathcal{A}_{g}^{\mathrm{trop}}\right|)$
$\displaystyle\cong$
$\displaystyle\Omega^{n}_{c}(g)\otimes_{\mathbb{Q}}\mathbb{R}$ $\displaystyle
H^{n-1}_{dR}(\left|L\mathcal{A}_{g-1}^{\mathrm{trop}}\right|)$
$\displaystyle=$ $\displaystyle 0\ ,$

for all $n\geq d_{g}-\kappa(g)$.

###### Proof.

Let $g>1$ be odd. Since $L\mathcal{P}_{g}/\mathrm{GL}_{g}(\mathbb{Z})$ is
orientable, Poincaré duality implies that
$H_{c}^{n}(L\mathcal{P}_{g}/\mathrm{GL}_{g}(\mathbb{Z});\mathbb{R})\cong
H^{\ell_{g}-n}(L\mathcal{P}_{g}/\mathrm{GL}_{g}(\mathbb{Z});\mathbb{R})^{\vee}$.
By (14.8), (14.4) and duality, the cohomology of the compactly supported
cohomology is given in the stable range by:

$H_{c}^{n}(L\mathcal{P}_{g}/\mathrm{GL}_{g}(\mathbb{Z});\mathbb{R})\cong\Omega^{n}_{c}(g)\otimes_{\mathbb{Q}}\mathbb{R}\quad\hbox{
for }n>\ell_{g}-\kappa(g)$

In this range, proposition 14.6 becomes

$0\longrightarrow
H_{dR}^{n-1}(\left|L\mathcal{A}^{\mathrm{trop}}_{g-1}\right|)\longrightarrow\Omega^{n}_{c}(g)\overset{(*)}{\longrightarrow}H_{dR}^{n}(\left|L\mathcal{A}^{\mathrm{trop}}_{g}\right|)\longrightarrow
0$

By theorem 14.1, the map $\Omega^{n}_{c}(g)\rightarrow
H_{dR}^{n}(\left|L\mathcal{A}^{\mathrm{trop}}_{g}\right|)$ is injective, hence
$(*)$ is an isomorphism, which implies that its kernel vanishes. ∎

###  14.6. Canonical forms and the tropical Torelli map

The composition

$\Omega^{n}_{c}(g)\longrightarrow
H^{n}_{dR}\left(|L\mathcal{A}_{g}^{\mathrm{trop}}|\right)\overset{\lambda^{*}}{\longrightarrow}H^{n}_{dR}\left(|L\mathcal{M}_{g}^{\mathrm{trop}}|\right)$

sends the majority of forms to zero. Indeed, if $\eta\in\Omega^{d}_{c}(g)$ has
degree $d$ then

(14.9) $\lambda^{*}(\omega_{!}^{2g-1}\wedge\eta)=0\quad\hbox{ if }\quad d>g-2\
.$

This is because the left-hand side restricts to the form denoted by
$\omega^{2g-1}_{G}\wedge\eta_{G}$ in the notation of [Bro21] on the simplex
$\sigma_{G}$ associated to a graph $G$. By [Bro21, Corollary 6.19], it
vanishes if its degree is greater than $3g-3$. It might be possible to reduce
the degree $d$ in (14.9) by a more detailed analysis of vanishing of graph
forms [Bro21, §6.5]. Evidence suggests that $\lambda^{*}(\omega)$ is non-zero
if and only if $\omega$ is primitive, or more precisely, that the image of

$\lambda^{*}:H_{dR}^{n}(L\mathcal{P}_{g}/\mathrm{GL}_{g}(\mathbb{Z}))\longrightarrow
H^{n}_{dR}(\left|L\mathcal{M}_{g}^{\mathrm{trop}}\right|)$

is one-dimensional: $\mathrm{Im}(\lambda^{*})=\mathbb{Q}[\omega^{2g-1}]$, and
is dual to the class of the wheel with $g$ spokes.

Nevertheless, a conjecture in _loc. cit._ states that the free Lie algebra
embeds:

$\mathrm{Lie}\,(\Omega^{\mathrm{can}})\hookrightarrow\bigoplus_{g\geq
1}H^{\bullet}\left(|L\mathcal{M}_{g}^{\mathrm{trop}}|\right)\ .$

It is corroborated by all known evidence in low degrees, and suggests that all
canonical forms appear in the cohomology of
$H^{\bullet}_{dR}\left(|L\mathcal{M}_{g}^{\mathrm{trop}}|\right)$ but for a
higher than expected genus.

##  15\. Periods

We work in a suitable neutral Tannakian category $\mathcal{H}_{\mathbb{Q}}$ of
motivic realisations over $\mathbb{Q}$. There are many possible variants
[Del89]. We assume that it has two fiber functors

$M\mapsto M_{B},M_{dR}\ :\
\mathcal{H}_{\mathbb{Q}}\longrightarrow\mathrm{Vec}_{\mathbb{Q}}$

and that there is a canonical element
$\mathrm{comp}_{B,dR}\in\mathrm{Isom}_{\mathcal{H}_{\mathbb{Q}}}^{\otimes}(\omega_{dR},\omega_{B})(\mathbb{C})$.
A convenient option is to let $\mathcal{H}_{\mathbb{Q}}$ be the category of
mixed Hodge structures with additional $\mathbb{Q}$-de Rham structure studied
in [Bro17b]. The element $\mathrm{comp}_{B,dR}$ defines an isomorphism

$M_{dR}\otimes_{\mathbb{Q}}\mathbb{C}\overset{\sim}{\longrightarrow}M_{B}\otimes_{\mathbb{Q}}\mathbb{C}\
,$

which is natural in $M$, which may be interpreted as a period pairing
$M_{dR}\otimes_{\mathbb{Q}}M_{B}^{\vee}\rightarrow\mathbb{C}$.

Consider the affine group scheme:

$G_{dR}=\mathrm{Aut}^{\otimes}_{\mathcal{H}_{\mathbb{Q}}}(\omega_{dR})$

given by the automorphisms of the fiber functor $\omega_{dR}:M\mapsto M_{dR}$.
By the Tannaka theorem [SR72], the fiber functor $\omega_{dR}$ defines an
equivalence of categories:

$\omega_{dR}:\mathcal{H}_{\mathbb{Q}}\overset{\sim}{\longrightarrow}\mathrm{Rep}\,G_{dR}\
.$

###  15.1. ‘Motives’ of quadratic forms

######  Definition 15.1.

Let $Q$ be a positive definite quadratic form on $V$ such that $\sigma_{Q}>0$.
Recall that
$\mathsf{c}_{Q}^{\mathcal{B}}=(P^{\mathcal{B}_{Q}},L^{\mathcal{B}_{Q}},\sigma^{\mathcal{B}}_{Q})$.
Let $d=\dim\sigma_{Q}$. Define

(15.1)
$\mathrm{mot}_{Q}=H^{d}\left(P^{\mathcal{B}_{Q}}\backslash\widetilde{\mathrm{Det}}\
,\
L^{\mathcal{B}_{Q}}\backslash(L^{\mathcal{B}_{Q}}\cap\widetilde{\mathrm{Det}})\right)\
.$

By corollary 11.5, we have
$[\sigma^{\mathcal{B}}_{Q}]\in(\mathrm{mot}_{Q})_{B}^{\vee}$. Any canonical
form $\omega\in\Omega^{d}_{\mathrm{can}}$ defines a class
$[\omega]\in(\mathrm{mot}_{Q})_{dR}$, and so we may define the ‘motivic
period’:

(15.2)
$I^{\mathfrak{m}}_{Q}(\omega)=[\mathrm{mot}_{Q},[\sigma_{Q}],[\omega]]^{\mathfrak{m}}\quad\in\quad\mathcal{O}(\mathrm{Isom}^{\otimes}_{\mathcal{H}_{\mathbb{Q}}}(\omega_{dR},\omega_{B}))\
.$

Note that for cones $\sigma_{Q}$ which are contained in the determinant locus,
[MBW22, Lemma 4.9] implies that they are equivalent to the cone of a quadratic
form of smaller rank, so definition 15.1 in fact covers all cases. The period
of $I^{\mathfrak{m}}_{Q}(\omega)$ is the convergent integral

$\mathrm{per}\left(I^{\mathfrak{m}}_{Q}(\omega)\right)=\int_{\sigma_{Q}}\omega$

which is finite by corollary 13.12. Furthemore, the relation of theorem 13.13
is motivic: it holds verbatim for the objects $I^{\mathfrak{m}}_{Q}(\omega)$.
In the case when $Q=Q_{G}$ comes from a graph, the object
$I^{\mathfrak{m}}_{Q_{G}}(\omega)$ is equal to the ‘canonical motivic Feynman
integral’ $I^{\mathfrak{m}}_{G}(\omega)$ defined in [Bro21].

######  Remark 15.2.

Every isomorphism $[Q]\cong[Q^{\prime}]$ in the category
$\mathcal{D}^{\mathrm{perf},\mathcal{B}}$ gives rise to an isomorphism
$\mathrm{mot}_{Q}\cong\mathrm{mot}_{Q}^{\prime}$. Similarly, face diagrams
give rise to a diagram of morphisms in the category $\mathcal{H}_{\mathbb{Q}}$
(see [Bro21] for the case of graphs), but extraneous blow-ups are _not_
necessarily isomorphisms (although they do induce equivalences of motivic
periods (15.2)).

######  Example 15.3.

We refer to [Bro21] for many examples of graphs for which $I_{G}(\omega)$ are
known. The most interesting was computed by Borinsky and Schnetz [BS21]:

(15.3)
$I_{K_{6}}(\omega^{5}\wedge\omega^{9})=3\,\frac{10!}{16}\left(12\,\zeta(3,5)-29\,\zeta(8)+23\,\zeta(3)\zeta(5)\right)\
.$

It was proven in [Bro21, lemmas 3.7, 6.8] that the canonical integrals are
stable under duality, which implies that the volume of the graphic cell
$\sigma_{K_{6}^{\vee}}$ where $K_{6}^{\vee}$ is the matroid dual to $K_{6}$,
is given by the same integral. It is the ‘principal cone’ [MBW22, Def. 2.13]
in the fundamental domain $\mathcal{F}_{5}$ for $\mathrm{GL}_{5}(\mathbb{Z})$.
By (14.2) the volume $\int_{\mathcal{F}_{5}}\omega^{5}\wedge\omega^{9}$ is a
proportional to a product $\zeta(3)\zeta(5)$. The appearance, therefore, of
the non-trivial multiple zeta value $\zeta(3,5)$ in (15.3) suggests that the
principal cone does not span the fundamental domain:
$\sigma_{K_{6}^{\vee}}\subsetneq\mathcal{F}_{5}$ and that the tropical Torelli
map is not surjective. Indeed, it is known (e.g. [EVGS13]) that the
fundamental domain involves two further cones. We deduce from (15.3) that
their volumes, which are not yet explicitly known, must involve non-trivial
multiple zeta values.

As noted in [Bro21, §10.3.1], the particular linear combination
$12\,\zeta(3,5)-29\,\zeta(8)$ is the same one which occurs in quantum field
theory as the Feynman residue of $K_{3,4}$. It is a very striking fact that
$\zeta(3,5)$ is only ever observed in this particular combination, and never
in isolation, amongst the vast array of periods in quantum field theory
[Sch10]. This surprising phenomenon is one particular consequence of the
‘Cosmic’ Galois group. The geometric interpretation (15.3) of this linear
combination points to an unexplored and deep connection between quantum fields
and the reduction theory of quadratic forms.

###  15.2. Locally finite homology ‘motive’ of $\mathrm{GL}_{g}(\mathbb{Z})$

Let $\mathcal{H}_{\mathbb{Q}}$, as above, be a Tannakian category of
realisations which contains every object $\mathrm{mot}_{Q}$ (15.1).

Consider the cohomology

$H_{c}^{d}(g)=H^{d}\left(\ker\left(\Omega^{\bullet}\left(L\mathcal{A}^{\mathrm{trop},\mathcal{B}}_{g}\
\backslash\
\widetilde{\mathrm{Det}}\right)\longrightarrow\Omega^{\bullet}\left(\partial
L\mathcal{A}^{\mathrm{trop},\mathcal{B}}_{g}\ \backslash\ (\partial
L\mathcal{A}^{\mathrm{trop},\mathcal{B}}_{g}\cap\widetilde{\mathrm{Det}})\right)\right)\right)$

of the complex of compatible systems (definition 2.8) of algebraic forms on
$L\mathcal{A}^{\mathrm{trop},\mathcal{B}}_{g}\backslash\widetilde{\mathrm{Det}}$
whose restriction to the boundary $\partial
L\mathcal{A}^{\mathrm{trop},\mathcal{B}}_{g}$ vanishes. There is a natural map

$H_{c}^{d}(g)\longrightarrow
H^{d}_{dR}\left(\left|L\mathcal{A}^{\mathrm{trop},\mathcal{B}}_{g}\right|\ ,\
\left|\partial L\mathcal{A}^{\mathrm{trop},\mathcal{B}}_{g}\right|\right)\cong
H^{d}_{c}(L\mathcal{P}_{g}/\mathrm{GL}_{g}(\mathbb{Z});\mathbb{R})$

via which elements of $H_{c}^{d}(g)$ may be interpreted as smooth compactly
supported differential forms on
$L\mathcal{P}_{g}/\mathrm{GL}_{g}(\mathbb{Z})$. There is a natural map
$\Omega^{d}_{c}(g)\rightarrow H_{c}^{d}(g)$ (which is injective).

######  Theorem 15.4.

For every $g>1$, and $d\geq 0$, there exists a minimal object
$\mathsf{M}^{d}_{g}$ of $\mathcal{H}_{\mathbb{Q}}$ which is equipped with a
pair of canonical linear maps

$\displaystyle H^{\mathrm{lf}}_{d}(\mathrm{GL}_{g}(\mathbb{Z});\mathbb{Q})$
$\displaystyle\longrightarrow$ $\displaystyle(\mathsf{M}^{d}_{g})_{B}^{\vee}$
$\displaystyle H^{d}_{c}(g)$ $\displaystyle\longrightarrow$
$\displaystyle(\mathsf{M}^{d}_{g})_{dR}$

such that the integration pairing

$H^{\mathrm{lf}}_{d}(\mathrm{GL}_{g}(\mathbb{Z});\mathbb{Q})\otimes_{\mathbb{Q}}H^{d}_{c}(g)\longrightarrow\mathbb{C}$

factors through the period pairing: $\langle\ ,\
\rangle:(\mathsf{M}^{d}_{g})_{B}^{\vee}\otimes_{\mathbb{Q}}(\mathsf{M}^{d}_{g})_{dR}\rightarrow\mathbb{C}$.

###### Proof.

First consider the object of $\mathcal{H}_{\mathbb{Q}}$ defined by:

$M^{d}_{g}=\bigoplus_{\dim\sigma_{Q}=d}\mathrm{mot}_{Q}^{\mathrm{Aut}(\sigma_{Q})}\
,$

where the sum is over isomorphism classes of cones $\sigma_{Q}$ of dimension
$d$. Consider the subspace
$Z_{g}^{d}\subset\Omega^{d}(L\mathcal{A}^{\mathrm{trop},\mathcal{B}}_{g}\backslash\widetilde{\mathrm{Det}})$
consisting of compatible systems of closed forms whose restriction to
$\partial L\mathcal{A}^{\mathrm{trop},\mathcal{B}}_{g}$ vanishes, and let
$B_{g}^{d}\subset Z^{d}_{g}$ be the subspace spanned by $d\omega$, where
$\omega$ are compatible systems of forms vanishing on $\partial
L\mathcal{A}^{\mathrm{trop},\mathcal{B}}_{g}$. There is a natural map
$Z_{g}^{d}\rightarrow(M^{d}_{g})_{dR}$. By the Tannaka theorem, there are
unique subobjects $\langle B_{g}^{d}\rangle\subset\langle
Z_{g}^{d}\rangle\subset M^{d}_{g}$ such that

$\langle Z_{g}^{d}\rangle_{dR}=G_{dR}\,Z_{g}^{d}\qquad\hbox{and}\qquad\langle
B_{g}^{d}\rangle_{dR}=G_{dR}\,B_{g}^{d}\ .$

Let $N_{g}^{d}=\langle Z_{g}^{d}\rangle/\langle B_{g}^{d}\rangle$. By
definition, there is a natural map
$H^{d}_{c}(g)\rightarrow\left(N^{d}_{g}\right)_{dR}$.

By theorem 11.3, inclusion defines a natural map

$H_{d}(\sigma_{Q},\partial\sigma_{Q})\longrightarrow(\mathrm{mot}_{Q})^{\vee}_{B}=H_{d}\left((P^{\mathcal{B}_{Q}}\backslash\widetilde{\mathrm{Det}})(\mathbb{C})\
,\
L^{\mathcal{B}_{Q}}\backslash(L^{\mathcal{B}_{Q}}\cap\widetilde{\mathrm{Det}})(\mathbb{C});\mathbb{Q}\right)$

which induces a map

$\bigoplus_{\dim\sigma_{Q}=d}H_{d}(\sigma_{Q},\partial\sigma_{Q})/\mathrm{Aut}(\sigma_{Q})\longrightarrow(M^{d}_{g})_{B}^{\vee}\
,$

where the direct sum is over all isomorphism classes of cones $\sigma_{Q}$ of
dimension $d$. Via the cellular interpretation of homology (theorem 3.2), we
deduce a map from closed cellular chains $\gamma$ in
$L\mathcal{A}_{g}^{\mathrm{trop},\mathcal{B}}$ to $(M_{g}^{d})^{\vee}_{B}$ and
thence to its quotient $\langle Z^{d}_{g}\rangle_{B}^{\vee}$. Since Stokes’
theorem (13.10) holds for motivic periods, the period pairing
$\langle\gamma,g\omega\rangle$ vanishes for all $g\in G^{dR}$ if $\omega\in
B_{g}^{d}$; and for all $g\in G^{dR}$ and $\omega\in Z_{g}^{d}$ if $\gamma$ is
supported on $\partial L\mathcal{A}_{g}^{\mathrm{trop},\mathcal{B}}$, or if
$\gamma=\partial\alpha$ is a boundary. This proves, at the same time, that the
image of a cellular chain $\gamma$ lands in the subspace $(\langle
Z^{d}_{g}\rangle/\langle B^{d}_{g}\rangle)_{B}^{\vee}$ of $\langle
Z^{d}_{g}\rangle_{B}^{\vee}$, and that it is zero if $\gamma$ is a boundary,
or supported on $\partial L\mathcal{A}_{g}^{\mathrm{trop},\mathcal{B}}$. Thus
by theorem 3.6, there is a well-defined map

$H_{d}\left(\left|L\mathcal{A}_{g}^{\mathrm{trop},\mathcal{B}}\right|\ ,\
\left|\partial
L\mathcal{A}_{g}^{\mathrm{trop},\mathcal{B}}\right|\right)\longrightarrow(N^{d}_{g})_{B}^{\vee}\
.$

The left-hand group is isomorphic to the locally finite homology
$H^{\mathrm{lf}}_{d}(\mathrm{GL}_{g}(\mathbb{Z});\mathbb{Q}),$ and so the
object $N^{d}_{g}$ satisfies the conditions of the theorem. This proves the
existence. For the minimality, one can repeat the argument of [Bro17b], to
show that any object $M$ of $\mathcal{H}_{\mathbb{Q}}$ equipped with maps
$V\hookrightarrow M_{B}^{\vee}$ and $W\hookrightarrow M_{dR}$, where $V,W$ are
$\mathbb{Q}$-vector spaces, has a unique smallest subquotient with the this
property (the case when $V$, and $W$ have dimension $1$ is proven in [Bro17b,
§2.4], but the general case is similar). ∎

It follows from (14.4) and the relative de Rham theorem 3.6 that the
composition

$\Omega_{c}^{d}(g)\longrightarrow
H^{d}_{c}(g)\longrightarrow(\mathsf{M}^{d}_{g})_{dR}$

is injective. Theorem 15.4 implies that any integral of a canonical form of
compact type over a locally finite homology class in
$H^{\mathrm{lf}}_{d}(\mathrm{GL}_{g}(\mathbb{Z}))$ is motivic. In particular,
we deduce a motivic interpretation of Minkowski’s volume integrals (14.2). A
motivic interpretation of the Borel regulator will be studied in a forthcoming
paper with Schnetz.

######  Remark 15.5.

An interesting question is whether the map
$H^{\mathrm{lf}}_{d}(\mathrm{GL}_{g}(\mathbb{Z});\mathbb{Q})\rightarrow(\mathsf{M}^{d}_{g})_{B}^{\vee}$
is injective or not: is it the case that the entire locally finite homology of
$\mathrm{GL}_{g}(\mathbb{Z})$ is motivic, or only a quotient of it? Note that
one can capture the entire cohomology by enlarging $H^{d}_{c}(g)$, for
instance by replacing the determinant locus with suitable linear subspaces,
since every cohomology class can be represented by a compatible family of
algebraic forms by remark 3.4. A deeper question is to describe the action of
$G_{dR}$ on $(\mathsf{M}^{d}_{g})_{dR}$.

######  Examples 15.6.

In the case $g=3,d=5$, the map

$\mathbb{Q}\cong
H_{5}^{\mathrm{lf}}(\mathrm{GL}_{3}(\mathbb{Z});\mathbb{Q})\longrightarrow(M^{5}_{3})^{\vee}_{B}$

is injective. In fact, $M_{5}^{3}$ is given by the graph ‘motive’ [BEK06] of
the wheel with 3 spokes $W_{3}$, which, one can show, is of rank two, and is a
non-trivial extension of $\mathbb{Q}(-3)$ by $\mathbb{Q}(0)$. The space
$\mathbb{Q}(-3)_{dR}$ is spanned by the class of the canonical form
$\omega^{5}$; the group
$H_{5}^{\mathrm{lf}}(\mathrm{GL}_{3}(\mathbb{Z});\mathbb{Q})$ is spanned by
the fundamental class which is the relative homology class of
$\sigma^{\mathcal{B}}_{W_{3}}$.

##  16\. An example

Let $G$ be the wheel with 3 spokes, with inner edges oriented outwards from
the center and outer edges oriented counter-clockwise.

Consider a basis $h_{1},h_{2},h_{3}$ for the homology given by the cycles
consisting of edges $\\{1,2,4,5\\}$, $\\{2,4,6\\}$, $\\{3,5,4\\}$.

$1$$3$$2$$5$$6$$4$

With respect to this basis and orientation, the graph Laplacian is represented
by the $3\times 3$ matrix

$\Lambda_{G}=\mathcal{H}^{T}_{G}\,D_{G}\,\mathcal{H}_{G}=\begin{pmatrix}x_{1}+x_{2}+x_{4}+x_{5}&-x_{2}-x_{4}&-x_{4}-x_{5}\\\
-x_{2}-x_{4}&x_{2}+x_{4}+x_{6}&-x_{4}\\\
-x_{4}-x_{5}&-x_{4}&x_{3}+x_{4}+x_{5}\end{pmatrix}\ .$

Consider the subgraph $\gamma$ spanned by the edges $1,2,4,5$. The quotient
graph $G/\gamma$ has a single vertex, and two edges $3,6$. Setting $x_{e}=0$
for $e\in E_{\gamma}$ one obtains

$\Lambda_{G}\big{|}_{\gamma}=\begin{pmatrix}0&0&0\\\ 0&x_{6}&0\\\
0&0&x_{3}\end{pmatrix}\ ,\qquad\hbox{ with Zariski closure}\qquad
V_{\sigma_{G/\gamma}}=\left\\{\begin{pmatrix}0&0&0\\\ 0&*&0\\\
0&0&*\end{pmatrix}\right\\}$

It defines a quadratic form on $V=\bigoplus_{i=1}^{3}h_{i}\mathbb{Q}\cong
H^{1}(G;\mathbb{Q})$ whose kernel is the one-dimensional subspace
$K=h_{1}\mathbb{Q}$. The space of quadratic forms $\mathcal{Q}(V/K)$ with null
space $K$ may be identified with the space of symmetric matrices of the form

$\mathcal{Q}(V/K)\cong\left\\{\begin{pmatrix}0&0&0\\\ 0&*&*\\\
0&*&*\end{pmatrix}\right\\}\ .$

The image of $\Lambda_{G}\big{|}_{\gamma}$ has codimension $1$ in it, and
therefore is not Zariski-dense. This example illustrates the difference
between $L\mathcal{M}_{g}^{\mathrm{trop},\mathcal{B}}$ and
$L\mathcal{A}_{g}^{\mathrm{trop},\mathcal{B}}$. The former involves blowing up
the subspace $L_{\gamma}=\\{x_{e}=0,e\in E_{\gamma}\\}$, corresponding to the
core subgraph $\gamma$, whose image under the Torelli map is the projective
space $\mathbb{P}(V_{\sigma_{G/\gamma}})$. However, in
$L\mathcal{A}_{g}^{\mathrm{trop},\mathcal{B}}$ it is the space
$\mathbb{P}(\mathcal{Q}(V/K))$ which is blown up, which strictly contains
$\mathbb{P}(V_{\sigma_{G/\gamma}})$. Note that
$\mathbb{P}(V_{\sigma})(\mathbb{R})$ and
$\mathbb{P}(\mathcal{Q}(V/K))(\mathbb{R})$ both meet $\sigma_{G}$ along the
same face $\sigma_{G/\gamma}$.

##  17\. Appendix: Blow-ups and Borel-Serre for $\mathrm{GL}_{n}$

###  17.1. Statements

Let $V$ be a vector space of dimension $n$ over $\mathbb{Q}$. Let
$\mathcal{Q}(V)$ denote the space of quadratic forms on $V$ and
$\mathbb{P}(\mathcal{Q}(V))$ the associated projective space. Let

(17.1) $X\quad\subset\quad\mathbb{P}(\mathcal{Q}(V))(\mathbb{R})$

denote the space of positive definite real forms. It is contained in the
complement of the determinant hypersurface
$\mathrm{Det}\subset\mathbb{P}(\mathcal{Q}(V))$. For any $0\neq K\subset V$,
the space $\mathbb{P}(\mathcal{Q}(V/K))\subset\mathrm{Det}$ is the subspace of
quadratic forms with null space $K$.

######  Definition 17.1.

Let $\pi_{\mathfrak{B}}:\mathfrak{B}\rightarrow\mathbb{P}(\mathcal{Q}(V))$
denote the space obtained by blowing up all linear subspaces of
$\mathbb{P}(\mathcal{Q}(V))$ of the form $\mathbb{P}(\mathcal{Q}(V/K))$ for
$0\neq K\subset V$, in increasing order of dimension. Let
$\widetilde{\mathrm{Det}}\subset\mathfrak{B}$ denote the strict transform of
$\mathrm{Det}$.

Let $X^{\mathrm{BS}}$ denote the Borel-Serre compactification of $X$. It has a
stratification

$X^{\mathrm{BS}}=X\cup\bigcup_{P}e(P)$

where $P$ ranges over rational parabolics (see §17.2). The exceptional divisor
$\mathcal{E}$ for $\pi_{\mathfrak{B}}$ defines a stratification on
$\mathfrak{B}$ by taking intersections of irreducible components. Since
$\pi_{\mathfrak{B}}(\mathcal{E})\subset\mathrm{Det}$, the inclusion (17.1)
induces an injective map:

(17.2) $X\ \longrightarrow\
\left(\mathbb{P}(\mathcal{Q}(V))\setminus\mathrm{Det}\right)(\mathbb{R})\overset{\pi_{\mathfrak{B}}^{-1}}{\longrightarrow}\left(\mathfrak{B}\setminus\mathcal{E}\right)(\mathbb{R})\
.$

######  Theorem 17.2.

There is a continuous injective map of stratified spaces

(17.3) $f:X^{\mathrm{BS}}\longrightarrow\left(\mathfrak{B}\ \setminus\
\widetilde{\mathrm{Det}}\right)(\mathbb{R})$

whose restriction to the big open stratum is the embedding (17.2). The image
of (17.3) is the closure of $X$, for the analytic topology, inside
$\mathfrak{B}(\mathbb{R})$. The map $f$ is equivariant with respect to the
natural actions of $\mathrm{GL}(V)$ on both $X^{\mathrm{BS}}$ and
$\mathfrak{B}$.

Let $P$ denote the rational parabolic associated to a nested sequence of
strict subspaces $0\subset V_{d}\subset V_{d-1}\subset\ldots\subset
V_{1}\subset V$. The map (17.3) restricts to a map

(17.4)
$f:e(P)\longrightarrow\mathcal{E}_{P}\backslash\left(\widetilde{\mathrm{Det}}\cap\mathcal{E}_{P}\right)\,(\mathbb{R})$

where $\mathcal{E}_{P}\subset\mathfrak{B}$ is the exceptional locus associated
to the iterated blow-up of

$\mathbb{P}(\mathcal{Q}(V/V_{1}))\subset\ldots\subset\mathbb{P}(\mathcal{Q}(V/V_{d}))\
$

in increasing order of dimension. The complement of the strict transform of
$\mathrm{Det}$ is canonically isomorphic to a product of hypersurface
complements:

$\mathcal{E}_{P}\backslash\left(\widetilde{\mathrm{Det}}\cap\mathcal{E}_{P}\right)\cong\prod_{k=0}^{d}\left(\mathbb{P}\left(\frac{\mathcal{Q}(V/V_{k+1})}{\mathcal{Q}(V/V_{k})}\right)\
\setminus\ \mathrm{Det}_{V_{k}/V_{k+1}}\right)$

where we write $V_{0}=V$ and $V_{d+1}=0$, and $\mathrm{Det}_{V_{k}/V_{k+1}}$
is the vanishing locus of the homogeneous polynomial given by the determinant
of the restriction of a quadratic form on $V/V_{k+1}$ to the subspace
$V_{k}/V_{k+1}$.

In short, the space $\mathfrak{B}$ is an algebraic incarnation of the Borel-
Serre compactification with identical combinatorial structure. The space
$X^{\mathrm{BS}}$ is identified with a semi-algebraic subset of its real
points $\mathfrak{B}(\mathbb{R})$. It is defined by infinitely many algebraic
inequalities of the form $u>0$, where $u$ is a homogeneous polynomial given by
the determinant of a matrix minor.

Note that although $\mathfrak{B}$ is defined by infinitely many blow-ups, and
is therefore not of finite type, the local structure of $X^{\mathrm{BS}}$ in
terms of spaces $X(P)$ (see §17.2) may be studied by embedding each $X(P)$
into a space obtained by performing only finitely many blow-ups of
$\mathbb{P}(\mathcal{Q}(V))$. This is how we shall prove theorem 17.2.

######  Corollary 17.3.

Let $|L\mathcal{A}_{g}|^{\mathrm{trop},\mathfrak{B}}$ be the topological space
defined in (11.4). Then the map (17.3) induces a canonical homeomorphism

$f:X_{g}^{\mathrm{BS}}/\mathrm{GL}_{g}(\mathbb{Z})\overset{\sim}{\longrightarrow}|L\mathcal{A}_{g}|^{\mathrm{trop},\mathfrak{B}}\
.$

###  17.2. Reminders on the Borel-Serre compactifcation for $\mathrm{GL}_{n}$

Let $V$ be a vector space of dimension $n$ over $\mathbb{R}$ equipped with an
inner product, or equivalently, an isomorphism $V\cong V^{\vee}$. The group of
automorphisms $\mathrm{GL}(V)$ is isomorphic to $\mathrm{GL}_{n}(\mathbb{R})$,
and the subgroup preserving the inner product defines a maximal compact
subgroup $K=O(V)$ of $\mathrm{GL}(V)$. It is isomorphic to
$O_{n}(\mathbb{R})$.

Given any filtration $F$ of $V$ of length $d$ by subspaces

(17.5) $F:\qquad 0=V_{d+1}\subset V_{d}\subset\ldots\subset V_{1}\subset
V_{0}=V\ ,$

where all inclusions are strict, denote by $P_{F}\leq\mathrm{GL}(V)$ the
parabolic subgroup of automorphisms of $V$ which preserve $F$. A filtration
$F^{\prime}$

$F^{\prime}:\qquad 0\subset V_{i_{k}}\subset\ldots\subset V_{i_{1}}\subset V\
$

where $1\leq i_{1}<\ldots<i_{k}\leq d$ is obtained by omitting elements from
the filtration $F$, is denoted by $F^{\prime}\leq F$. The set of filtrations
$F^{\prime}$ such that $F^{\prime}\leq F$ forms a finite poset. Since for
$F^{\prime}\leq F$, one has $P_{F}\leq P_{F^{\prime}}$, the set
$\\{P_{F^{\prime}}:F^{\prime}\leq F\\}$ with respect to inclusion of groups is
isomorphic to the opposite poset.

For any such $P=P_{F}$, as above, denote by $K_{P}=P\cap K$. The natural map

$P\longrightarrow\prod^{d}_{i=0}\mathrm{GL}(V_{i}/V_{i+1})$

induces an isomorphism $K_{P}\cong\prod_{i=0}^{d}O(V_{i}/V_{i+1})$, the
product of orthogonal groups with respect to the induced inner product on
$V_{i}\rightarrow V\cong V^{\vee}\rightarrow V_{i}^{\vee}$.

Let $Z_{P}\leq K_{P}$ denote the subgroup of central elements acting via
scalar multiplication by $\mathbb{R}^{\times}_{>0}$ on each quotient
$V_{i}/V_{i+1}$ for $0\leq i\leq d$. Thus
$Z_{P}\cong(\mathbb{R}^{\times}_{>0})^{d+1}$.

######  Example 17.4.

Let $e_{1},\ldots,e_{n}$ denote a basis of $V$ compatible with $F$. In this
basis, automorphisms of $V$ may be identified with $n\times n$ matries, and
the groups $P$ and $A_{P}$ are block-lower triangular, and block diagonal
respectively:

$P=\begin{pmatrix}P_{00}&&\\\ P_{01}&P_{11}\\\ \vdots&&\ddots\\\
P_{0k}&P_{1k}&\ldots&P_{kk}\end{pmatrix}\qquad,\qquad
Z_{P}=\begin{pmatrix}\mu_{0}I_{0}&&\\\ &\mu_{1}I_{1}\\\ &&\ddots\\\
&&&\mu_{k}I_{k}\end{pmatrix}$

where $k=d$, $I_{0},\ldots,I_{k}$ are block identity matrices and
$\mu_{0},\ldots,\mu_{k}>0$. The $P_{ij}$ are likewise block matrices. The
group $K_{P}$ is isomorphic to the group of block diagonal matrices where the
matrix $O_{i}$ in the $i^{\mathrm{th}}$ block is orthogonal:
$O_{i}^{T}O_{i}=I_{i}$.

The group $Z_{P}$ acts upon $P$ by multiplication on the left. Denote the
group of central elements by $H=Z_{P}\cap
Z(\mathrm{GL}(V))\cong\mathbb{R}^{\times}_{>0}$. In example 17.4, $H\leq
Z_{P}$ is the subgroup of diagonal matrices. Let $X$ denote the space of
projective equivalence classes of real positive definite quadratic forms on
$V$.

####  17.2.1. Geodesic action

For any $P=P_{F}$ as above there is a homeomorphism:

(17.6) $\displaystyle HK_{P}\setminus P$
$\displaystyle\overset{\sim}{\longrightarrow}$ $\displaystyle X$
$\displaystyle g$ $\displaystyle\mapsto$ $\displaystyle g^{T}g\ .$

The action of $Z_{P}$ on $P$ passes to a well-defined action on
$HK_{P}\setminus P$. It induces an action on $X$ via (17.6) which is called
the geodesic action. Borel and Serre consider the following space of
$Z_{P}$-orbits:

$e(P)=Z_{P}\setminus
X\overset{\eqref{XasquotientofP}}{\cong}Z_{P}K_{P}\setminus P\ .$

It is homeomorphic to a Euclidean space $\mathbb{R}^{\binom{n+1}{2}-d-1}$.

The compactification of $X$ is defined as follows. Consider the isomorphism

(17.7) $\displaystyle Z\setminus Z_{P}$ $\displaystyle\cong$ $\displaystyle
A_{P}\cong(\mathbb{R}^{\times}_{>0})^{d}$
$\displaystyle(\mu_{0},\mu_{1},\ldots,\mu_{d})$ $\displaystyle\mapsto$
$\displaystyle(\lambda_{1},\ldots,\lambda_{d})$

where $\lambda_{i}=\mu_{i}/\mu_{i-1}$ for $1\leq i\leq d$. Consider the
multiplicative monoid

$\overline{A}_{P}=(\mathbb{R}^{\times}_{\geq 0})^{d}$

with coordinates $\lambda_{i}\geq 0$. The subgroup $A_{P}\leq\overline{A}_{P}$
has coordinates $\lambda_{i}>0$. By (17.7), the natural map
$Z_{P}\rightarrow\overline{A}_{P}$ has kernel exactly $H$. Borel and Serre
define

$X(P)=A_{P}\setminus(X\times\overline{A}_{P})\ .$

The space $\overline{A}_{P}$ is stratified by the closed subspaces
$\lambda_{i}=0$ for $i=1,\ldots,d$. The corresponding open stratification is a
union of copies of $(\mathbb{R}_{>0}^{\times})^{a}$ for $a\leq d$, which may
be identified with the subgroups $A_{P_{F^{\prime}}}$, for all $F^{\prime}\leq
F$. Thus

$X(P)=X\sqcup\coprod_{P^{\prime}}A_{P^{\prime}}\setminus
X=X\sqcup\coprod_{P^{\prime}}e(P^{\prime})$

is a disjoint union of spaces $e(P^{\prime})$, where
$P^{\prime}=P_{F^{\prime}}$ for all $F^{\prime}\leq F$.

####  17.2.2. Borel-Serre compactification

Now suppose that $V$ has a $\mathbb{Q}$-structure and that its inner product
is defined over $\mathbb{Q}$. A parabolic $P=P_{F}$ is called rational if the
vector spaces $V_{i}$ in the corresponding sequence (17.5) are all defined
over $\mathbb{Q}$.

The Borel-Serre compactification is defined to be

(17.8) $X^{\mathrm{BS}}=\bigcup_{P}X(P)$

where the union is over all rational parabolics $P$.

######  Example 17.5.

Let $V=\mathbb{R}^{3}$, $d=2$, and $V_{i}\cong\mathbb{R}^{3-i}$, for $0\leq
i\leq 2$. Write

$P=\begin{pmatrix}p_{00}&&\\\ p_{01}&p_{11}&\\\
p_{02}&p_{12}&p_{22}\end{pmatrix}\quad\ ,\ \quad Z=\begin{pmatrix}\mu_{0}&&\\\
&\mu_{1}&\\\ &&\mu_{2}\end{pmatrix}$

where $p_{ij}\in\mathbb{R}$ and $\det(P)=p_{00}p_{11}p_{22}\neq 0$. We set

$X=P^{T}P=\begin{pmatrix}p_{00}^{2}+p_{01}^{2}+p_{02}^{2}&p_{02}p_{12}+p_{01}p_{11}&p_{02}p_{22}\\\
p_{02}p_{12}+p_{01}p_{11}&p_{12}^{2}+p_{11}^{2}&p_{12}p_{22}\\\
p_{02}p_{22}&p_{12}p_{22}&p_{22}^{2}\end{pmatrix}$

Let $X_{ij}=x_{ij}$. The geodesic action $X\mapsto Z\circ X$ where $Z\circ
X=P^{T}Z^{T}ZP$, may, as one can check, be written intrinsically as a function
of $X$ via

$Z\circ X=\mu_{0}^{2}\begin{pmatrix}\frac{\det(X)}{\det(X^{11})}&&\\\ &&\\\
&&\end{pmatrix}+\mu_{1}^{2}\begin{pmatrix}\frac{\det(X^{21})^{2}}{\det(X^{11})x_{33}}&\frac{\det(X^{21})}{x_{33}}&\\\
\frac{\det(X^{21})}{x_{33}}&\frac{\det(X^{11})}{x_{33}}&\\\
&&\end{pmatrix}+\mu_{2}^{2}\begin{pmatrix}\frac{x_{13}^{2}}{x_{33}}&\frac{x_{13}x_{23}}{x_{33}}&x_{13}\\\
\frac{x_{13}x_{23}}{x_{33}}&\frac{x_{23}^{2}}{x_{33}}&x_{23}\\\
x_{13}&x_{23}&x_{33}\end{pmatrix}$

where $\det(X^{11})=x_{22}x_{33}-x_{23}^{2}$ and
$\det(X^{21})=x_{12}x_{33}-x_{13}x_{23}$ are determinants of minors of $X$.
The action $X\mapsto Z\circ X$ is clearly algebraic in the entries of $X$ and
is well-defined on the locus where $\det(X)\det(X^{11})x_{33}\neq 0$. This
example serves to illustrate, perhaps, that the coordinates given by the
geodesic action are not ideally suited to studying the asymptotic behaviour of
canonical differential forms at infinity.

###  17.3. Linear monoidal actions and blow-ups

The Borel-Serre construction is topological, and relies on the geodesic action
which is only defined on the space of positive definite matrices, since it
makes crucial use of the surjectivity of (17.6). In order to reformulate it in
the language of algebraic geometry, we first need some preliminaries on
monoidal actions on vector space schemes and their quotients by multiplicative
groups. A possible relation between Borel-Serre’s construction and blow-ups is
hinted at in [BS73, pg 437, line 5].

Let $k$ be a field. All schemes will be over $k$.

####  17.3.1. Multiplicative monoid

Recall that the multiplicative group is defined by
$\mathbb{G}_{m}=\mathrm{Spec}\,k[t,t^{-1}]$, where
$\mathcal{O}(\mathbb{G}_{m})=k[t,t^{-1}]$ is the Hopf algebra whose coproduct
$\Delta:\mathcal{O}(\mathbb{G}_{m})\rightarrow\mathcal{O}(\mathbb{G}_{m})\otimes_{k}\mathcal{O}(\mathbb{G}_{m})$
satisfies $\Delta t=t\otimes t$. The multiplication on $\mathbb{G}_{m}$ is
dual to $\Delta$. We denote the multiplicative monoid scheme to be

$\overline{\mathbb{G}}_{m}=\mathrm{Spec}\,k[t]$

where $k[t]$ is the coalgebra equipped with the coproduct
$\Delta:k[t]\rightarrow k[t]\otimes_{k}k[t]$ satisfying the same formula
$\Delta t=t\otimes t$. Thus, as a scheme, $\overline{\mathbb{G}}_{m}$ is
simply the affine line $\mathbb{A}^{1}$, but has a multiplication morphism
$\overline{\mathbb{G}}_{m}\times\overline{\mathbb{G}}_{m}\rightarrow\overline{\mathbb{G}}_{m}$.
It has a distinguished point $0\in\overline{\mathbb{G}}_{m}$ such that
$\overline{\mathbb{G}}_{m}\backslash 0=\mathbb{G}_{m}$, and with the property
that the multiplication satisfies $0\times\overline{\mathbb{G}}_{m}\rightarrow
0$.

####  17.3.2. Linear monoidal action

A vector space of dimension $n$ over a field $k$ will be viewed as the
$k$-points of an affine scheme $\mathbb{A}^{n}$ in the usual manner. Consider
a finite-dimensional graded vector space $W$ over a field $k$

$W=W_{n}\oplus W_{n-1}\oplus\ldots\oplus W_{1}\oplus W_{0}\ $

where $\dim W_{i}\geq 1$ for all $i$. The multiplicative group
$\mathbb{G}_{m}\times W_{i}\rightarrow W_{i}$ acts by scalar multiplication on
each component, denoted by $(\lambda,w)\mapsto\lambda w$. It extends to an
action $\overline{\mathbb{G}}_{m}\times W_{i}\rightarrow W_{i}$ which sends
$(0,w)$ to the origin of $W_{i}$. Consider the monoid action

(17.9) $m:\overline{\mathbb{G}}_{m}^{n}\times W\longrightarrow W\ ,$

where the $i^{\mathrm{th}}$ component of $\overline{\mathbb{G}}_{m}$ acts via

$\displaystyle\overline{\mathbb{G}}_{m}\times W$
$\displaystyle\longrightarrow$ $\displaystyle W$
$\displaystyle\lambda_{i}.(w_{n},\ldots,w_{0})$ $\displaystyle=$
$\displaystyle(\lambda_{i}w_{n},\ldots,\lambda_{i}w_{i},w_{i-1},\ldots,w_{0})\
.$

The action (17.9) restricts to an action of $\mathbb{G}_{m}^{n}$ on $W$.
Furthermore, let $\mathbb{G}_{m}^{n}$ act on $\overline{\mathbb{G}}_{m}^{n}$
componentwise via the map
$\mathbb{G}_{m}\times\overline{\mathbb{G}}_{m}\rightarrow\overline{\mathbb{G}}_{m}$
given by $(\mu,\lambda)\mapsto\lambda\mu^{-1}$, where $\mu^{-1}$ is the
inverse in the group $\mathbb{G}_{m}$. We deduce that $m$ is equivariant for
the diagonal action of $\mathbb{G}_{m}^{n}$ on
$\overline{\mathbb{G}}_{m}^{n}\times W$, in other words, the following diagram
commutes

$\begin{array}[]{ccc}\mathbb{G}_{m}^{n}\times(\overline{\mathbb{G}}_{m}^{n}\times
W)&\longrightarrow&\overline{\mathbb{G}}_{m}^{n}\times W\\\
\downarrow&&\downarrow_{m}\\\ \overline{\mathbb{G}}_{m}^{n}\times
W&\overset{m}{\longrightarrow}&W\end{array}$

where the vertical map on the left is projection onto
$\overline{\mathbb{G}}_{m}^{n}\times W$, and the horizontal map is the
diagonal action of $\mathbb{G}_{m}^{n}$ on both
$\overline{\mathbb{G}}_{m}^{n}$ and $W$. In the case $n=1$ this diagram merely
expresses the trivial fact that $m(\mu.(\lambda,w))=m(\lambda\mu^{-1},\mu
w)=\lambda w=m(\lambda,w)$.

####  17.3.3. Quotients and blowups

For any finite dimensional vector space $W$ over $k$ let
$W^{\times}=W\backslash\\{0\\}$. It is an open subscheme of $W$ stable under
the action of $\mathbb{G}_{m}$, but not under the extended action of
$\overline{\mathbb{G}}_{m}$, since $\overline{\mathbb{G}}_{m}\times
W^{\times}\rightarrow W$. Let us denote by

$W^{\star}=W^{\times}_{n}\oplus W^{\times}_{n-1}\oplus\ldots\oplus
W^{\times}_{1}\oplus W_{0}\ .$

The morphism (17.9) restricts to a morphism
$m:\overline{\mathbb{G}}_{m}^{n}\times W^{\star}\rightarrow W$, which, by the
previous paragraph, is equivariant with respect to the action of
$\mathbb{G}_{m}^{n}$.

We wish to define its quotient by $\mathbb{G}_{m}^{n}$ in the category of
schemes.

######  Example 17.6.

Let $x_{i}^{1},\ldots,x_{i}^{n}$ denote coordinates on the $k$-vector space
$W_{i}\cong\mathbb{A}^{n}$, where $n=\dim W_{i}$, and write

$W_{i}^{\times}=\bigcup_{j=1}^{n}U_{i}^{j}$

where $U_{i}^{j}\subset\mathbb{A}^{n}$ is the open complement of
$x_{i}^{j}=0$. Let $U_{i}^{jk}=U_{i}^{j}\cap U_{i}^{k}$. The opens
$U_{i}^{j},U_{i}^{jk}$ are stable under the action of $\mathbb{G}_{m}$, and
the inclusions $U_{i}^{jk}\hookrightarrow U_{i}^{j}$ are
$\mathbb{G}_{m}$-equivariant. One may define the quotient by $\mathbb{G}_{m}$:

$\mathbb{G}_{m}\setminus\\!\setminus
W_{i}^{\times}:=\bigcup_{i=1}^{n}\mathbb{G}_{m}\setminus\\!\setminus
U_{i}^{j}$

to be the union of schemes $\mathbb{G}_{m}\setminus\\!\setminus U_{i}^{i}$
glued along the $\mathbb{G}_{m}\setminus\\!\setminus U_{i}^{jk}$, where
$\mathbb{G}_{m}\setminus\\!\setminus
U_{i}^{j}=\mathrm{Spec}\,(\mathcal{O}(U^{j}_{i})^{\mathbb{G}_{m}})$ (and
similarly $\mathbb{G}_{m}\setminus\\!\setminus U_{i}^{jk}$) is the affine
scheme whose coordinate ring consists of $\mathbb{G}_{m}$-invariant
polynomials. Since
$\mathcal{O}(U_{i}^{j})^{\mathbb{G}_{m}}=k[\frac{x^{1}_{i}}{x_{i}^{j}},\ldots,\frac{x_{i}^{n}}{x_{i}^{j}}]$,
we recover the standard affine covering of projective space and thus

$\mathbb{G}_{m}\setminus\\!\setminus W_{i}^{\times}\cong\mathbb{P}(W_{i})$

is canonically isomorphic to the projective space of $W_{i}$.

######  Definition 17.7.

Consider a $\mathbb{G}_{m}$-equivariant covering
$W^{\times}_{i}=\bigcup_{1\leq j\leq\dim W_{i}}U_{i}^{j}$ of $W_{i}$ for every
$i=1,\ldots,n$ as in the previous example, and define a scheme:

(17.10)
$\mathbb{G}_{m}^{n}\setminus\\!\setminus(\overline{\mathbb{G}}_{m}^{n}\times
W^{\star}):=\bigcup_{i_{1},\ldots,i_{n}}\mathbb{G}_{m}^{n}\setminus\\!\setminus\left(\overline{\mathbb{G}}_{m}^{n}\times
U_{n}^{i_{n}}\times\ldots\times U_{1}^{i_{1}}\times W_{0}\right)\ ,$

where each $1\leq i_{k}\leq\dim W_{k}$. It does not depend on the choice of
coordinates (since $\mathbb{G}_{m}^{n}$ is central in the product of general
linear groups $\mathrm{GL}(W_{i})$).

Since it is $\mathbb{G}_{m}^{n}$-equivariant, the map $m$ defines a morphism
of schemes

(17.11) $m\ :\
\mathbb{G}_{m}^{n}\setminus\\!\setminus(\overline{\mathbb{G}}_{m}^{n}\times
W^{\star})\longrightarrow W\ .$

######  Proposition 17.8.

The scheme (17.10) canonically embeds as an open subscheme of the iterated
blow-up $B\rightarrow W$ of the affine space $W$ along the linear subspaces

$W_{0}\ \subset\ W_{0}\oplus W_{1}\ \subset\ \ldots\ \subset\ W_{0}\oplus
W_{1}\oplus\ldots\oplus W_{n-1}$

in increasing order of dimension. It is Zariski-dense in $B$.

###### Proof.

The most direct proof is via explicit coordinates generalising example 17.6.
Consider the coordinate ring of a particular open in (17.10)

$\mathcal{O}\left(\overline{\mathbb{G}}_{m}^{n}\times
U_{n}^{i_{n}}\times\ldots\times U_{1}^{i_{1}}\times W_{0}\right)$

where $1\leq i_{k}\leq\dim W_{k}$. With the labelling of coordinates of
example 17.6, it is

$k\left[\lambda_{1},\ldots,\lambda_{n},\left(x^{j}_{i}\right)_{\begin{subarray}{c}0\leq
i\leq n\\\ 1\leq j\leq\dim
V_{i}\end{subarray}},\frac{1}{x_{1}^{i_{1}}},\ldots,\frac{1}{x_{n}^{i_{n}}}\right]$

where $\lambda_{i}$ is the coordinate on the $i$th component of
$\overline{\mathbb{G}}_{m}$ (and the $x^{j}_{0}$ are coordinates on $W_{0}$).
The coaction of the $i$th component of $\mathbb{G}_{m}$ is equivalent to the
action

$\lambda_{j}\mapsto\begin{cases}\mu_{i}^{-1}\lambda_{j}\quad\hbox{ if }i=j\
,\\\ \lambda_{j}\quad\qquad\hbox{ if }i\neq j\end{cases}\quad\hbox{ and }\quad
x_{k}^{j}\mapsto\begin{cases}\mu_{i}x_{k}^{j}\quad\hbox{ if }k\geq i\\\
x_{k}^{j}\qquad\hbox{ if }k>i\end{cases}$

for all $j\geq 1$. The $\mathbb{G}_{m}^{n}$-invariants of
$\mathcal{O}\left(\overline{\mathbb{G}}_{m}^{n}\times
U_{n}^{i_{n}}\times\ldots\times U_{1}^{i_{1}}\times W_{0}\right)$ are

$\mathcal{O}(W_{0})\Bigg{[}\left(\frac{\alpha_{1}^{j_{1}}}{\alpha_{1}^{i_{1}}}\right)_{1\leq
j_{1}\leq\dim
W_{1}},\alpha_{1}^{i_{1}},\left(\frac{\alpha_{2}^{j_{2}}}{\alpha_{2}^{i_{2}}}\right)_{1\leq
j_{2}\leq\dim
W_{2}},\frac{\alpha_{2}^{i_{2}}}{\alpha_{1}^{i_{1}}},\quad\ldots\qquad\qquad$
$\qquad\qquad\qquad\qquad\qquad\ldots,\left(\frac{\alpha_{n}^{j_{n}}}{\alpha_{n}^{i_{n}}}\right)_{1\leq
j_{n}\leq\dim
W_{n}},\frac{\alpha_{n}^{i_{n}}}{\alpha_{n-1}^{i_{n-1}}}\Bigg{]}$

upon writing
$\alpha_{j}^{i}=\lambda_{1}\lambda_{2}\ldots\lambda_{i}x^{i}_{j}$, which are
themselves $\mathbb{G}_{m}^{n}$-invariant. These are identical to a system of
local coordinates on the iterated blow-up described in §5.2 (after reversing
the indexation of the subscripts of the $\alpha$’s) along

$V(\alpha_{i}^{j}:i\geq 1)\quad\subset\quad V(\alpha_{i}^{j}:i\geq
2)\quad\subset\quad\ldots\quad\subset\quad V(\alpha_{i}^{j}:i\geq n)$

and therefore gives an canonical open immersion of this affine chart of
(17.10) into $B$. Since these coordinate charts are compatible they glue
together to form an explicit open immersion of schemes from (17.10) to $B$. ∎

Inspection of the proof shows that the scheme (17.10) is canonically
isomorphic to the complement in $B$ of strict transforms of some linear
subspaces of $\mathbb{P}(W)$ which may be written in terms of the $W_{i}$,
although we will not need such an explicit description here.

######  Remark 17.9.

The recursive structure of the iterated blow-up may be interpreted using the
monoidal action. The blow-down morphism (17.11) factors through

$\displaystyle\pi_{n}:\overline{\mathbb{G}}_{m}^{n}\times W^{\star}$
$\displaystyle\longrightarrow$ $\displaystyle
W_{n}\times\left(\overline{\mathbb{G}}_{m}^{n-1}\times(W_{n-1}^{\star}\oplus\ldots\oplus
W_{1}^{\star}\oplus W_{0})\right)$
$\displaystyle(\lambda_{1},\ldots,\lambda_{n}),(w_{n},\ldots,w_{0})$
$\displaystyle\mapsto$
$\displaystyle\lambda_{n}w_{n},(\lambda_{1},\ldots,\lambda_{n-1}),(w_{n-1},\ldots,w_{1},w_{0})\
.$

It is $\mathbb{G}_{m}^{n}$-equivariant and hence defines a morphism

$\mathbb{G}_{m}^{n}\setminus\\!\setminus\left(\overline{\mathbb{G}}_{m}^{n}\times
W^{\star}\right)\longrightarrow
W_{n}\times\mathbb{G}_{m}^{n-1}\setminus\\!\setminus\left(\overline{\mathbb{G}}_{m}^{n-1}\times(W_{n-1}^{\star}\oplus\ldots\oplus
W_{1}^{\star}\oplus W_{0})\right)\ .$

The proposition implies that this is the blow-up along the subscheme defined
by

$Z_{n}=0\times\mathbb{G}_{m}^{n-1}\setminus\\!\setminus\left(\overline{\mathbb{G}}_{m}^{n-1}\times(W_{n-1}^{\star}\oplus\ldots\oplus
W_{1}^{\star}\oplus W_{0})\right)$

where $0$ is the origin in $W_{n}$. To justify this, observe that $\pi_{n}$
restricts to an isomorphism from the open locus defined by $\lambda_{n}\neq 0$
and the complement of $Z_{n}$. This is because if $w_{n}\neq 0$, the equation
$\lambda_{n}w_{n}\neq 0$ is equivalent to $\lambda_{n}\neq 0$. Thus $\pi_{n}$
is indeed an isomorphism on the complement of the exceptional divisor.

The exceptional divisor $\pi_{n}^{-1}(Z_{n})\cong V(\lambda_{n})$ is
isomorphic to

$\left(\mathbb{G}_{m}\setminus\\!\setminus
W^{\times}_{n}\right)\times\mathbb{G}_{m}^{n-1}\setminus\\!\setminus\left(\overline{\mathbb{G}}_{m}^{n-1}\times(W_{n-1}^{\star}\oplus\ldots\oplus
W_{1}^{\star}\oplus W_{0})\right)\ .$

By example 17.6 this is isomorphic to $\mathbb{P}(W_{n})\times Z_{n}$ and is
isomorphic to the projectivised normal bundle of $Z_{n}$.

####  17.3.4. Variant: projective version

With notation as above, let

$W^{\circ}=W^{\times}_{n}\oplus W^{\times}_{n-1}\oplus\ldots\oplus
W^{\times}_{1}\oplus W_{0}^{\times}$

with the action of
$\mathbb{G}_{m}^{n+1}=\mathbb{G}_{m}^{n}\times\mathbb{G}_{m}$ where
$\mathbb{G}_{m}^{n}$ acts as above, but with an additional copy of
$\mathbb{G}_{m}$ (with coordinate $\lambda_{0}$) acting by scalar
multiplication on $W^{\circ}$. Then we may define the quotient
$\mathbb{G}_{m}^{n+1}\setminus\\!\setminus(\overline{\mathbb{G}}_{m}^{n}\times
W^{\circ})$ as before. The morphism (17.11) becomes:

$\mathbb{G}_{m}^{n+1}\setminus\\!\setminus(\overline{\mathbb{G}}_{m}^{n}\times
W^{\circ})\longrightarrow\mathbb{G}_{m}\setminus\\!\setminus(W\backslash\\{0\\})=\mathbb{P}(W)\
.$

Note that its image is the open complement of
$\mathbb{P}(W_{n}\oplus\ldots\oplus W_{1}\oplus\\{0\\})$ in $\mathbb{P}(W)$.
The projective version of proposition 17.8 is:

######  Proposition 17.10.

The scheme
$\mathbb{G}_{m}^{n+1}\setminus\\!\setminus(\overline{\mathbb{G}}_{m}^{n}\times
W^{\circ})$, relative to the morphism:

$\mathbb{G}_{m}^{n+1}\setminus\\!\setminus(\overline{\mathbb{G}}_{m}^{n}\times
W^{\circ})\longrightarrow\mathbb{P}(W)$

canonically embeds into the iterated blow-up $B_{W}\rightarrow\mathbb{P}(W)$
along the linear subspaces

$\mathbb{P}(W_{0})\ \subset\ \mathbb{P}(W_{0}\oplus W_{1})\ \subset\ \ldots\
\subset\ \mathbb{P}(W_{0}\oplus W_{1}\oplus\ldots\oplus W_{n-1})$

in increasing order of dimension. It is Zariski-dense in $B_{W}$.

####  17.3.5. Exceptional divisors

The exceptional divisors of
$\mathbb{G}_{m}^{n+1}\setminus\\!\setminus(\overline{\mathbb{G}}_{m}^{n}\times
W^{\circ})$ are in one-to-one correspondence with the divisors
$V(\lambda_{i})$ defined by $\lambda_{i}=0$ for $i=1,\ldots,n$. Furthemore,
the divisor $V(\lambda_{i})$ is canonically isomorphic to a product

$V(\lambda_{i})\ \cong\
\mathbb{G}_{m}^{n-i+1}\setminus\\!\setminus(\overline{\mathbb{G}}_{m}^{n-i}\times
W_{n}^{\times}\oplus\ldots\oplus W_{i}^{\times})\ \times\
\mathbb{G}_{m}^{i}\setminus\\!\setminus(\overline{\mathbb{G}}_{m}^{i-1}\times
W_{i-1}^{\times}\oplus\ldots\oplus W_{0}^{\times})$

which, if we write $W^{\prime}=W_{i-1}\oplus\ldots\oplus W_{0}$, embeds into
the product

$V(\lambda_{i})\ \hookrightarrow\
B_{W/W^{\prime}}\,\mathbb{P}(W/W^{\prime})\times
B_{W^{\prime}}\,\mathbb{P}(W^{\prime})$

where the blow-ups are with respect to the flags of subspaces by the
filtration $F_{i}=\bigoplus_{k=0}^{i}W_{k}$ on $W$ and the filtrations it
induces on $W^{\prime}$ and $W/W^{\prime}$.

The exceptional divisor $\mathcal{S}=\bigcup\mathcal{S}_{i}$ defines a
stratification on the blow-up $B_{W}$ of $\mathbb{P}(W)$. Consider the
corresponding quasi-projective stratification

(17.12) $B_{W}=\bigcup_{I}\mathcal{S}^{\circ}_{I}$

where the union is over all subsets $I\subset\\{1,\ldots,n\\}$, and where we
denote as is customary, $\mathcal{S}_{I}=\bigcap_{i\in I}\mathcal{S}_{i}$ ,
and $\mathcal{S}_{\emptyset}=\mathbb{P}(W)$, and define

$\mathcal{S}^{\circ}_{I}=\mathcal{S}_{I}\setminus\left(\bigcup_{j\notin
I}\mathcal{S}_{I}\cap\mathcal{S}_{j}\right)\ .$

It follows from the description above that each stratum of (17.12), and hence
each stratum of
$\mathbb{G}_{m}^{n+1}\setminus\\!\setminus(\overline{\mathbb{G}}_{m}^{n}\times
W^{\circ})$, is embedded in a product of projective spaces:

$\mathcal{S}^{\circ}_{I}\quad\subset\quad\prod^{|I|}_{k=1}\mathbb{P}\left(\bigoplus_{i_{k-1}\leq
j\leq
i_{k}}W_{j}\right)=\prod^{|I|}_{k=1}\mathbb{P}\left(\mathrm{gr}^{F^{\prime}}_{k}W\right)\
,$

where $F^{\prime}W$ is the filtration on $W$ whose $k$th graded is
$\bigoplus_{i_{k-1}\leq j\leq i_{k}}W_{j}$. The strata of (17.12) are in one-
to-one correspondence with filtrations $F^{\prime}\leq F$.

###  17.4. Spaces of parabolic and symmetric matrices

Using the previous construction, we may define an algebraic version of the
Borel-Serre construction $A_{P}\setminus(\overline{A}_{P}\times P)$ associated
to a rational parabolic $P$ and embed it in an iterated blow-up. We then
provide an algebraic interpretation of the space $X(P)$ by embedding it into
the real points of a scheme of finite type. We shall denote algebraic groups
with caligraphic letters to distinguish them from the topological groups
considered in §17.2.

###  17.5. Spaces of parabolic matrices

Let $V$ be a finite-dimensional vector space over $\mathbb{Q}$ and consider a
flag of (rational) vector subspaces

$F\ :\ 0\subset V_{d}\subset V_{d-1}\subset\ldots\subset V_{1}\subset V_{0}=V$

which is split: $V\cong\bigoplus_{i\geq 0}V_{i}/V_{i+1}$. Let

$\mathcal{P}_{F}\subset\mathrm{End}_{F}(V)$

denote the affine variety over $\mathbb{Q}$ of endomorphisms of $V$ which
preserve $F$. Using the splitting of $V$ we may write

$\mathrm{End}_{F}(V)=\mathrm{Hom}_{F}(V\ ,\
\bigoplus_{i}V_{i}/V_{i+1})=\bigoplus_{i}\mathrm{Hom}_{F}(V,V_{i}/V_{i+1})$

which consequently gives a decomposition

(17.13)
$\mathcal{P}_{F}=\mathcal{P}_{d}\oplus\mathcal{P}_{d-1}\oplus\ldots\oplus\mathcal{P}_{0}\
.$

Define a subscheme $\mathcal{P}_{F}^{\circ}$ as in §17.3.4. Let

$\mathcal{Z}_{P}=\mathbb{G}_{m}^{d+1}\ \hookrightarrow\
\mathrm{GL}(\bigoplus_{i}V_{i}/V_{i+1})=\prod_{i}\mathrm{GL}(V_{i}/V_{i+1})$

be the product of multiplicative groups whose $k^{\mathrm{th}}$ component,
where $0\leq k\leq d$, acts by scalar multiplication on $V_{k}/V_{k+1}$. Let
$\overline{\mathcal{A}}_{P}=\overline{\mathbb{G}}_{m}^{d}$ denote the
multiplicative monoid whose $k^{\mathrm{th}}$ component, with coordinate
$\lambda_{k}$ where $1\leq k\leq d$, acts by scalar multiplication on $V_{k}$
as in §17.3, i.e., for $v_{i}\in V_{i}/V_{i+1}$, for $0\leq i\leq d$ we have:

$(v_{d},\ldots,v_{1},v_{0})\mapsto(\lambda_{d}\ldots\lambda_{0}v_{d},\ldots,\lambda_{1}\lambda_{0}v_{1},\lambda_{0}v_{0})$

The morphism of monoids $\mathcal{Z}_{P}\rightarrow\overline{\mathcal{A}}_{P}$
is given in coordinates by:

(17.14) $\displaystyle\mathcal{O}(\overline{\mathcal{A}}_{P})\cong
k[\lambda_{1},\ldots,\lambda_{d}]$ $\displaystyle\rightarrow$
$\displaystyle\mathcal{O}(\mathcal{Z}_{P})\cong
k[\mu_{0}^{\pm},\ldots,\mu_{d}^{\pm}]$ $\displaystyle\lambda_{i}$
$\displaystyle\mapsto$ $\displaystyle\mu_{i-1}^{-1}\mu_{i}$

and has kernel $\mathbb{G}_{m}$. Denote by
$\mathcal{A}_{P}\leq\overline{\mathcal{A}}_{P}$ the subgroup $\lambda_{i}\neq
0$. Thus
$\mathcal{Z}_{P}/\mathbb{G}_{m}\cong\mathcal{A}_{P}\subset\overline{\mathcal{A}}_{P}$.

######  Proposition 17.11.

There is a canonical Zariski-dense embedding:

(17.15)
$\mathcal{A}_{P}\setminus\left(\overline{\mathcal{A}}_{P}\times\mathcal{P}^{\circ}_{F}\right)\longrightarrow
B_{F}\,\mathbb{P}(\mathcal{P}_{F})$

where $B_{F}\mathbb{P}(\mathcal{P}_{F})$ is the iterated blow-up of
$\mathbb{P}(\mathcal{P}_{F})$ along the subspaces

$\mathbb{P}(\mathcal{P}_{0})\ \subset\
\mathbb{P}(\mathcal{P}_{0}\oplus\mathcal{P}_{1})\ \subset\ \ldots\ \subset\
\mathbb{P}(\mathcal{P}_{0}\oplus\ldots\oplus\mathcal{P}_{d-1})\ .$

Let $I=\\{i_{1},\ldots,i_{k}\\}$ and $1\leq i_{1}<\ldots<i_{k}\leq d$. The
image of the locus $V(\lambda_{i},i\in I)$ under the morphism (17.15) is
contained in the codimension $k$ stratum $\mathcal{S}_{I}$ of the exceptional
divisor of $B_{F}\mathbb{P}(\mathcal{P}_{F})$ which corresponds to the flag
$F^{\prime}\leq F$:

(17.16) $F^{\prime}\ :\ 0=V_{i_{k+1}}\subset V_{i_{k}}\subset
V_{i_{k-1}}\subset\ldots\subset V_{i_{1}}\subset V_{i_{0}}=V\ .$

Both $V(\lambda_{i},i\in I)$ and its Zariski-closure $\mathcal{S}_{I}$ are
canonically isomorphic to products, and the morphism (17.15) restricts to the
embedding:

(17.17) $V(\lambda_{i},i\in
I)\cong\prod_{m=0}^{k}\left(\mathcal{A}_{P_{F_{m}}}\setminus\\!\setminus\left(\overline{\mathcal{A}}_{P_{F_{m}}}\times\mathcal{P}^{\circ}_{F_{m}}\right)\right)\longrightarrow\mathcal{S}_{I}\cong\prod_{m=0}^{k}B_{F_{m}}\mathbb{P}(P_{F_{m}})$

where $\mathcal{P}_{F_{m}}=\bigoplus_{i_{m}\leq j<i_{m+1}}\mathcal{P}_{j}$,
$\mathcal{P}^{\circ}_{F_{m}}=\bigoplus_{i_{m}\leq
j<i_{m+1}}\mathcal{P}^{\times}_{j}$ and $F_{m}$ is the filtration induced by
$F$ on the quotients $V_{i_{m}}/V_{i_{m+1}}$.

The restriction of (17.17) to the open locus $\lambda_{j}\neq 0$ for all
$j\notin I$ gives an embedding

(17.18)
$\mathcal{A}_{P_{F^{\prime}}}\setminus\\!\setminus\mathcal{P}_{F}^{\circ}\quad\hookrightarrow\quad\mathcal{S}^{\circ}_{I}\subseteq\prod_{m=0}^{k}\mathbb{P}(P_{F_{m}})\
.$

###### Proof.

Apply proposition 17.10 to $\mathcal{P}_{F}$. The second part follows by
iterating the description of a codimension one exceptional divisor given in
§17.3.5. The only novelty is (17.18). It follows from the fact that the locus
$\lambda_{j}\neq 0$ for all $j\notin J$ is isomorphic to

$\left(\prod_{m=0}^{k}\mathcal{A}_{P_{F_{m}}}\setminus\\!\setminus\left(\overline{\mathcal{A}}_{P_{F_{m}}}\times\mathcal{P}^{\circ}_{F_{m}}\right)\right)\setminus\bigcup_{j\notin
J}V(\lambda_{j})\cong\mathcal{A}_{P_{F^{\prime}}}\setminus\\!\setminus\mathcal{P}_{F}^{\circ}$

This follows from
$\mathcal{P}_{F}^{\circ}=\bigoplus_{m=0}^{k}\mathcal{P}^{\circ}_{F_{m}}$, and
the fact that the natural morphism of multiplicative monoids
$\mathcal{A}_{P}\rightarrow\overline{\mathcal{A}}_{P}$ given by (17.14) has
kernel $\mathbb{G}_{m}$. ∎

Since the determinant is a homogeneous polynomial in matrix entries, we may
define the determinant hypersurface
$\mathrm{Det}\subset\mathbb{P}(\mathcal{P}_{F})$ to be its vanishing locus.
Let

$\widetilde{\mathrm{Det}}\quad\subset\quad B_{F}\mathbb{P}(\mathcal{P}_{F})$

denote its strict transform. Let
$P=\mathrm{GL}_{F}(V\otimes_{\mathbb{Q}}\mathbb{R})$ denote the (topological)
group of automorphisms considered in §17.2, and likewise $A_{P}\leq P$. Then
we may interpret

$P\ \cong\ \left(\mathcal{P}_{F}\backslash\mathrm{Det}\right)(\mathbb{R})$

and it follows from proposition 17.11 that there are injections:

(17.19) $A_{P}\setminus\left(\overline{A}_{P}\times
P\right)\longrightarrow\left(\mathcal{A}_{P}\setminus\left(\overline{\mathcal{A}}_{P}\times
P^{\circ}_{F}\right)\backslash\mathrm{Det}\right)(\mathbb{R})\longrightarrow\left(B_{F}\mathbb{P}(\mathcal{P}_{F})\setminus\widetilde{\mathrm{Det}}\right)(\mathbb{R})\
.$

######  Example 17.12.

Consider the case when $d=2$ and $F$ is $0\subset V_{2}\subset V_{1}\subset
V$. By choosing a basis of $V$ adapted to this filtration, the points of
$\mathcal{P}_{F}$ are given by block lower triangular matrices, and
$\overline{\mathcal{A}}_{P}\cong\overline{\mathbb{G}}_{m}^{2}$ acts by

$\begin{pmatrix}P_{0}&&\\\ P_{01}&P_{1}&\\\
P_{02}&P_{12}&P_{2}\end{pmatrix}\mapsto\begin{pmatrix}P_{0}&&\\\
\lambda_{1}P_{01}&\lambda_{1}P_{1}&\\\
\lambda_{1}\lambda_{2}P_{02}&\lambda_{1}\lambda_{2}P_{12}&\lambda_{1}\lambda_{2}P_{2}\end{pmatrix}$

where $\lambda_{1},\lambda_{2}$ are points on $\overline{\mathbb{G}}_{m}^{2}$.
The points of $\mathcal{P}_{i}$, for $i=0,1,2$ are given by the subsets of
matrices which vanish everywhere except in row $i+1$, and the subspaces
$\mathcal{P}_{0}\subset\mathcal{P}_{0}\oplus\mathcal{P}_{1}$ which are to be
blown up are the spaces of matrices of the form:

$\begin{pmatrix}P_{0}&0&0\\\ 0&0&0\\\ 0&0&0\end{pmatrix}\ \subseteq\
\begin{pmatrix}P_{0}&0&0\\\ P_{01}&P_{1}&0\\\ 0&0&0\end{pmatrix}\ .$

The points of the subspace $\mathcal{P}^{\circ}_{F}$ satisfy $P_{0}\neq 0$,
$P_{1}\neq 0$ and $P_{2}\neq 0$. This strictly contains the complement of the
locus $\mathrm{Det}$ defined by $\det(P_{0})\det(P_{1})\det(P_{2})=0$.

The stratification of
$\mathbb{G}_{m}\setminus\\!\setminus(\overline{\mathbb{G}}_{m}^{2}\times\mathcal{P}^{\circ}_{F})$
is generated by the equations $\lambda_{1}=0$ and $\lambda_{2}=0$ (below
left). The strata themselves are described as quotients (below, middle), which
embed into the following products of blow-ups (below, right):

$\begin{array}[]{rccc}V(\lambda_{1})\quad:&\left(\mathbb{G}_{m}\setminus\\!\setminus\mathcal{P}^{\circ}_{0}\right)\times\mathbb{G}_{m}^{2}\setminus\\!\setminus\left(\overline{\mathbb{G}}_{m}\times\mathcal{P}^{\circ}_{1}\oplus\mathcal{P}^{\circ}_{2}\right)&\subseteq&\mathbb{P}(\mathcal{P}_{0})\times
B_{\,\mathbb{P}(\mathcal{P}_{1})}\mathbb{P}(\mathcal{P}_{2}\oplus\mathcal{P}_{1})\\\
V(\lambda_{2})\quad:&\mathbb{G}_{m}^{2}\setminus\\!\setminus\left(\overline{\mathbb{G}}_{m}\times\mathcal{P}^{\circ}_{1}\oplus\mathcal{P}^{\circ}_{0}\right)\times\left(\mathbb{G}_{m}\setminus\\!\setminus\mathcal{P}^{\circ}_{2}\right)&\subseteq&B_{\,\mathbb{P}(\mathcal{P}_{0})}\mathbb{P}(\mathcal{P}_{0}\oplus\mathcal{P}_{1})\times\mathbb{P}(\mathcal{P}_{2})\\\
V(\lambda_{1},\lambda_{2})\quad:&\left(\mathbb{G}_{m}\setminus\\!\setminus\mathcal{P}^{\circ}_{0}\right)\times\left(\mathbb{G}_{m}\setminus\\!\setminus\mathcal{P}^{\circ}_{1}\right)\times\left(\mathbb{G}_{m}\setminus\\!\setminus\mathcal{P}^{\circ}_{2}\right)&\subseteq&\mathbb{P}(\mathcal{P}_{0})\times\mathbb{P}(\mathcal{P}_{1})\times\mathbb{P}(\mathcal{P}_{2})\
.\end{array}\ .$

These strata, respectively, corresponding to the flags: $0\subset V_{1}\subset
V$ for $\lambda_{1}=0$; $0\subset V_{2}\subset V$ for $\lambda_{2}=0$; and the
full flag $0\subset V_{2}\subset V_{1}\subset V$ for
$\lambda_{1}=\lambda_{2}=0$.

###  17.6. Spaces of quadratic forms, blow-ups and $X(P)$

Let $F$, $P$ be as in §17.5. There is a natural morphism of algebraic
varieties

(17.20) $\displaystyle\mathbb{P}(\mathcal{P}_{F})$
$\displaystyle\longrightarrow$ $\displaystyle\mathbb{P}(\mathcal{Q}(V))$
$\displaystyle M$ $\displaystyle\mapsto$ $\displaystyle M^{T}M$

which sends the subspace
$\mathbb{P}(\mathcal{P}_{0}\oplus\ldots\oplus\mathcal{P}_{i})$ to
$\mathbb{P}(\mathcal{Q}(V/V_{i+1}))\subset\mathbb{P}(\mathcal{Q}(V/V_{i+1}))$,
the projective space of the subspace of quadratic forms which vanish on
$V_{i+1}$.

Consequently, by the universal property of blow-ups, (17.20) induces a
morphism

(17.21) $B_{F}\mathbb{P}(\mathcal{P}_{F})\longrightarrow
B_{F}\mathbb{P}(\mathcal{Q}(V))$

where $B_{F}\mathbb{P}(\mathcal{Q}(V))$ is the iterated blow-up of
$\mathbb{P}(\mathcal{Q}(V))$ along the linear subspaces

$\mathbb{P}\left(\mathcal{Q}(V/V_{d})\right)\ \subset\
\mathbb{P}\left(\mathcal{Q}(V/V_{d-1})\right)\ \subset\ \ldots\ \subset\
\mathbb{P}\left(\mathcal{Q}(V/V_{1})\right)\ .$

associated to the flag $F$. Combining with proposition 17.11 we deduce a
natural map:

(17.22)
$\mathcal{A}_{P}\setminus\\!\setminus\left(\overline{\mathcal{A}}_{P}\times\mathcal{P}^{\circ}_{F}\right)\longrightarrow
B_{F}\mathbb{P}(\mathcal{P}_{F})\longrightarrow
B_{F}\mathbb{P}(\mathcal{Q}(V))$

which is an embedding. Recall from §9.5 that the exceptional divisor
$\mathcal{E}$ of $B_{F}\mathbb{P}(\mathcal{Q}(V))$ defines a stratification
whose strata $\mathcal{E}_{F^{\prime}}$ are in one-to-one correspondence with
flags $F^{\prime}\leq F$ (17.16), and are canonically isomorphic to a product
of iterated blow-ups

$\mathcal{E}_{F^{\prime}}\cong\prod_{m=0}^{k}B_{F_{i}}\mathbb{P}\left(\frac{\mathcal{Q}(V/V_{i_{m+1}})}{\mathcal{Q}(V/V_{i_{m}})}\right)\
,\ $

where $F_{i}$ is the induced filtration on the succesive quotients
$V_{i}/V_{i+1}$ of (17.16). Let
$\mathrm{Det}\subset\mathbb{P}(\mathcal{Q}(V))$ denote the vanishing locus of
the determinant, and $\widetilde{\mathrm{Det}}$ its strict transform in
$B_{F}\mathbb{P}(\mathcal{Q}(V))$. It was proven in proposition 11.4 that

(17.23)
$\mathcal{E}_{F^{\prime}}\setminus\left(\mathcal{E}_{F^{\prime}}\cap\widetilde{\mathrm{Det}}\right)\cong\prod_{m=0}^{k}\left(\mathbb{P}\left(\frac{\mathcal{Q}(V/V_{i_{m+1}})}{\mathcal{Q}(V/V_{i_{m}})}\right)\
\setminus\ \mathrm{Det}_{V_{i_{m}}/V_{i_{m+1}}}\right)\ $

where $\mathrm{Det}_{V_{i_{m}}/V_{i_{m+1}}}$ is the zero locus of the
homogeneous map

$\frac{\mathcal{Q}(V/V_{i_{m+1}})}{\mathcal{Q}(V/V_{i_{m}})}\longrightarrow\mathcal{Q}(V_{i_{m}}/V_{i_{m+1}})\overset{\mathrm{det}}{\longrightarrow}\mathbb{Q},$

where the first map is restriction of quadratic forms.

######  Theorem 17.13.

The map (17.22) induces a continuous map

(17.24) $X(P)\longrightarrow\left(B_{F}\,\mathbb{P}(Q(V))\ \setminus\
\widetilde{\mathrm{Det}}\right)(\mathbb{R})\ .$

For every flag $F^{\prime}$ of the form (17.16) the map (17.24) restricts to a
map

(17.25)
$e(P^{\prime})\longrightarrow\mathcal{E}_{F^{\prime}}\setminus\left(\mathcal{E}_{F^{\prime}}\cap\widetilde{\mathrm{Det}}\right)(\mathbb{R})$

to the corresponding stratum in the exceptional locus of
$B_{F}\,\mathbb{P}(Q(V))$. The image of the map (17.24) is an isomorphism onto
the closure, in the analytic topology, in
$B_{F}\mathbb{P}(\mathcal{Q}(V))(\mathbb{R})$ of the locus of positive
definite quadratic forms inside $\mathbb{P}(\mathcal{Q}(V))(\mathbb{R})$.

###### Proof.

The equation $\det(M^{T}M)=\det(M)^{2}$ implies that (17.20) preserves the
determinant loci $\mathrm{Det}$ in $\mathbb{P}(\mathcal{P}_{F})$ and
$\mathbb{P}(\mathcal{Q}(V))$, and so the morphism (17.21) preserves their
strict transforms. It follows that (17.22) defines a natural map

(17.26)
$\mathcal{A}_{P}\setminus\\!\setminus\left(\overline{\mathcal{A}}_{P}\times\mathcal{P}^{\circ}_{F}\right)\longrightarrow
B_{F}\mathbb{P}(\mathcal{Q}(V))\setminus\ \widetilde{\mathrm{Det}}$

whose restriction to the open locus $\\{\lambda_{i}\neq 0,\hbox{ for }1\leq
i\leq d\\}$ is the morphism

$\mathbb{G}_{m}\setminus\\!\setminus\mathcal{P}^{\circ}_{F}\longrightarrow\mathbb{P}(\mathcal{Q}(V))\setminus\mathrm{Det}$

which, in coordinates given by a basis of $V$, sends a matrix $M$ to the
projective class of $M^{T}M$. This map is invariant by left-multiplication by
the subscheme

$\mathcal{K}_{P_{F}}\leq\mathcal{P}^{\circ}_{F}$

whose points are matrices $O$ such that $O^{T}O=1$, since one has
$(OM)^{T}OM=M$. Since this equation is algebraic, it remains true on the
Zariski closure, and hence (17.26) is also invariant under left multiplication
by $\mathcal{K}_{P_{F}}$. We deduce that the map

$A_{P}\setminus\left(\overline{A}_{P}\times
P\right)\longrightarrow\left(B_{F}\mathbb{P}(\mathcal{Q}(V))\setminus\
\widetilde{\mathrm{Det}}\right)(\mathbb{R})$

which is induced by taking real points of the map $\eqref{inproof: APtoBF}$
and composing with the first map in (17.19), is
$K_{P}\leq\mathcal{K}_{P_{F}}(\mathbb{R})$ invariant and so passes to the
quotient

$X(P)=A_{P}K_{P}\setminus\left(\overline{A}_{P}\times
P\right)\longrightarrow\left(B_{F}\mathbb{P}(\mathcal{Q}(V))\setminus\
\widetilde{\mathrm{Det}}\right)(\mathbb{R})\ .$

This gives a well-defined continuous map (17.24) such that

$\begin{array}[]{ccc}A_{P}\setminus\left(\overline{A}_{P}\times
P\right)&\longrightarrow&\left(B_{F}\mathbb{P}(P_{F})\setminus\
\widetilde{\mathrm{Det}}\right)(\mathbb{R})\\\ \downarrow&&\downarrow\\\
X(P)=A_{P}\setminus\left(\overline{A}_{P}\times
X\right)&\longrightarrow&\left(B_{F}\mathbb{P}(\mathcal{Q}(V))\setminus\
\widetilde{\mathrm{Det}}\right)(\mathbb{R})\end{array}$

commutes, where the horizontal map along the top is the morphism (17.15) and
the vertical maps are $M\mapsto M^{T}M$ on the left, and (17.21) on the right.

Now compute the restriction of the map (17.24) on each stratum indexed by a
flag $F^{\prime}$ of the form (17.16). The map (17.26) restricts to a map on
exceptional divisors

$\mathcal{A}_{P_{F^{\prime}}}\setminus\\!\setminus\mathcal{P}^{\circ}_{F}\
\overset{\eqref{OpenStratumAndAPquotient}}{\longrightarrow}\
\mathcal{S}^{\circ}_{F^{\prime}}\overset{\eqref{blowupMTM}}{\longrightarrow}\mathcal{E}_{F^{\prime}}\
.$

Since it is left-invariant by $\mathcal{K}_{P_{F}}$, on taking real points it
descends to the quotient

$e(P^{\prime})=A^{\prime}_{P}\setminus X\cong A^{\prime}_{P}K_{P}\setminus
P\longrightarrow\mathcal{E}_{F^{\prime}}\backslash\left(\mathcal{E}_{F^{\prime}}\setminus\widetilde{\mathrm{Det}}\right)(\mathbb{R})\
.$

This morphism is injective on each stratum $e(P^{\prime})$ since
$M^{T}M=N^{T}N$ for two invertible real matrices if and only if $M=ON$ for $O$
an orthogonal matrix. The final statement follows from the fact that (17.24)
is homeomorphic onto its image and identifies the open $X\subset X(P)$ with
the subspace of projective classes of positive-definite matrices inside
$\mathbb{P}(\mathcal{Q}(V))(\mathbb{R})$. ∎

######  Examples 17.14.

As in example 17.12, use the splitting $V\cong V_{2}\oplus V_{2}/V_{1}\oplus
V/V_{2}$ to write endomorphisms of $V$ in block matrix form. Consider the
deepest stratum which is associated to the full flag $F$, which corresponds to
$\mathcal{E}_{12}.$ The morphism (17.25)

$e(P)\longrightarrow\mathbb{P}(\mathcal{Q}(V/V_{1}))(\mathbb{R})\times\mathbb{P}\left(\frac{\mathcal{Q}(V/V_{2})}{\mathcal{Q}(V/V_{1})}\right)(\mathbb{R})\times\mathbb{P}\left(\frac{\mathcal{Q}(V)}{\mathcal{Q}(V/V_{2})}\right)(\mathbb{R})$

is given explicitly on matrix representatives in $P$ of classes in
$e(P)=A_{P}K_{P}\setminus P$ by:

$\begin{pmatrix}P_{0}&&\\\ P_{01}&P_{1}&\\\
P_{02}&P_{12}&P_{2}\end{pmatrix}\mapsto\begin{pmatrix}P_{0}^{T}P_{0}&0&0\\\
0&0&0\\\ 0&0&0\end{pmatrix}\ ,\ \begin{pmatrix}0&P_{01}^{T}P_{1}&0\\\
P_{1}^{T}P_{01}&P_{1}^{T}P_{1}&0\\\ 0&0&0\end{pmatrix}\ ,\
\begin{pmatrix}0&0&P_{02}^{T}P_{2}\\\ 0&0&P_{12}^{T}P_{2}\\\
P_{2}^{T}P_{02}&P_{2}^{T}P_{12}&P_{2}^{T}P_{2}\end{pmatrix}$

Now consider the stratum associated to the flag $F^{\prime}:0\subset
V_{1}\subset V$, which corresponds to the exceptional divisor
$\mathcal{E}_{1}$. The morphism (17.25) is given on matrix representatives in
$P$ of classes in $e(P^{\prime})=A_{P^{\prime}}K_{P}\setminus P$ by:

$\displaystyle e(P^{\prime})$ $\displaystyle\longrightarrow$
$\displaystyle\mathbb{P}(\mathcal{Q}(V/V_{1}))(\mathbb{R})\times\mathbb{P}\left(\frac{\mathcal{Q}(V)}{\mathcal{Q}(V/V_{1})}\right)(\mathbb{R})$
$\displaystyle\begin{pmatrix}A&0\\\ U&B\end{pmatrix}$ $\displaystyle\mapsto$
$\displaystyle\begin{pmatrix}A^{T}A&0\\\
0&0\end{pmatrix},\begin{pmatrix}0&B^{T}U\\\ B^{T}B&U^{T}B\end{pmatrix}$

where

$A=P_{0}\ ,\ U=\begin{pmatrix}P_{01}\\\ P_{02}\end{pmatrix}\ ,\
B=\begin{pmatrix}P_{1}&0\\\ P_{12}&P_{2}\end{pmatrix}\ .$

###  17.7. Proof of the statements in §17.1

####  17.7.1. Proof of theorem 17.2

Theorem 17.2 follows from theorem 17.13 together with the description (17.8)
of $X^{\mathrm{BS}}$ as a union of $X(P)$, for $P$ rational parabolics. It
follows from theorem 11.3 that the closure of $X$ in
$\mathfrak{B}(\mathbb{R})$ is equal to its closure in
$(\mathfrak{B}\backslash\widetilde{\mathrm{Det}})(\mathbb{R})$, since the
analytic closure of any polyhedron $\sigma$ in an admissible decomposition of
$X$ does not meet the strict transform of the determinant locus. The final
part of the theorem is simply (17.23).

Finally, the group of rational points $\mathrm{GL}(V)(\mathbb{Q})$ acts on $X$
via $P\mapsto Pg$. Correspondingly, the algebraic group $\mathrm{GL}(V)$ acts
upon $\mathbb{P}(\mathcal{Q}(V))$ via $(g,M)\mapsto g^{T}Mg$. Since it
permutes the set of flags (17.5) and hence the centers of all the blow-ups
which define $\mathfrak{B}$, it extends to a natural action of
$\mathrm{GL}(V)$ upon $\mathfrak{B}$. Its group of $\mathbb{Q}$-points
$\mathrm{GL}(V)(\mathbb{Q})$ acts algebraically upon $\mathfrak{B}$. This
action is by compatible, via $P\mapsto P^{T}P$, with the action of
$\mathrm{GL}(V)(\mathbb{Q})$ upon $X$.

####  17.7.2. Proof of corollary 17.3

Consider an $\mathrm{GL}_{g}(\mathbb{Z})$-equivariant admissible decomposition
$X_{g}=\bigcup_{\sigma}\sigma$. Consider the image of a polyhedron $\sigma$
under the map (17.3)

$f:X^{\mathrm{BS}}{\longrightarrow}\mathfrak{B}(\mathbb{R})\ .$

Denote its closure in $X\subset X^{\mathrm{BS}}$ by $\overline{\sigma}$. By
invariance under extraneous blow-ups (proposition 5.15), the closure of
$f(\sigma)$ in $\mathfrak{B}(\mathbb{R})$ (or equivalently, in
$(\mathfrak{B}\setminus\widetilde{\mathrm{Det}})(\mathbb{R})$) is canonically
homeomorphic to the polyhedron
$\sigma^{\mathcal{B}}=\sigma^{\mathcal{B}^{\min,\sigma}}$ in the minimal blow-
up of $\sigma$. By continuity, $f$ defines a homeomorphism
$f:\overline{\sigma}\overset{\sim}{\rightarrow}\sigma^{\mathcal{B}}$. Conclude
using the fact that, by definition (11.4),
$\left|L\mathcal{A}_{g}^{\mathrm{trop},\mathfrak{B}}\right|$ is the quotient
of the union of all $\sigma^{\mathcal{B}}$ by the action of
$\mathrm{GL}_{g}(\mathbb{Z})$, since the gluing relations in the diagram
category $\mathcal{D}_{g}^{\mathrm{trop},\mathcal{B}}$ are those induced by
$\mathrm{GL}_{g}(\mathbb{Z})$.

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