sequence $(w_{n})$ in $\mathbb{D}$ converging to $w\in\partial\mathbb{D}$ and
with $\sigma_{w_{n}}\to\sigma_{w}^{\psi}$ by controlling the limiting phase of
objects in $P(\varphi)$ appropriately. ∎

######  Lemma 11.3.

Suppose $w=e^{2\pi i\varphi}\in\partial\mathbb{D}$ and $P(\varphi)\neq 0$ and
moreover that the lax pre-stability condition $\sigma_{w}^{\psi}$ is in the
closure of the orbit $\sigma\cdot G$. Then it satisfies the support property
if, and only if, there is some $\varepsilon>0$ such that either

  1. (1)

$P(\varphi-\varepsilon,\varphi)=0$ and no simple object in
$P[\varphi,\varphi+1)$ lies in $P(\varphi,\varphi+\varepsilon)$; or,

  2. (2)

$P(\varphi,\varphi+\varepsilon)=0$ and no simple object in
$P(\varphi,\varphi+1]$ lies in $P(\varphi+1-\varepsilon,\varphi+1)$.

In particular, if $\varphi$ is isolated both conditions are satisfied and
$\sigma_{w}^{\psi}$ satisfies the support property.

###### Proof.

Under the assumption $P(\varphi-\varepsilon,\varphi)=0$ the heart
$P[\varphi,\varphi+1)$ is length so that it makes sense to speak of simple
objects, and similarly for the second case. We only need to check the support
property for massive stable objects. Each such is also $\sigma$ stable. If
$\varphi$ is an isolated point of $\Phi_{\sigma}$ then any massive
$\sigma_{w}^{\psi}$ stable object lies in a slice $P(\varphi^{\prime})$ whose
phase $\varphi^{\prime}$ is bounded away from $\varphi$. Again the support
property follows as in the proof of Lemma 11.1.

If $\varphi$ is an accumulation point of $\Phi_{\sigma}$ with
$P(\varphi-\varepsilon,\varphi)=0$ then $\psi=1$ by Lemma 11.2. It is enough
to consider massive stable objects in $P_{w}^{\psi}(1)=P[\varphi,\varphi+1)$.
These are the simple objects of this heart which are not in $P(\varphi)$.
Therefore each such lies in $P(\varphi^{\prime})$ for some
$\varphi^{\prime}\in(\varphi,\varphi+1)$. The result then follows as in the
proof of Lemma 11.1. The other case is similar. ∎

######  Remark 11.4.

If $\mathsf{C}$ has an algebraic heart then any length heart is algebraic,
i.e. has finitely many iso-classes of simple objects. In this situation the
conditions simplify to $P(\varphi-\varepsilon,\varphi)=0$ or
$P(\varphi,\varphi+\varepsilon)=0$, i.e. to $\varphi$ being isolated or at
worst a one-sided accumulation point.

######  Definition 11.5.

We define subsets of the closure of the Poincaré disk by

$\displaystyle\mathbb{D}_{\sigma}$ $\displaystyle=\mathbb{D}\cup\\{e^{2\pi
i\varphi}:\varphi\not\in\overline{\Phi_{\sigma}}\\}$ $\displaystyle\text{and}\
\mathbb{D}^{Q}_{\sigma}$ $\displaystyle=\mathbb{D}\cup\\{e^{2\pi
i\varphi}:\varphi\ \text{satisfies either (\ref{accumulation above}) or
(\ref{accumulation below}) of Lemma~{}\ref{lem: degenerate support} for some}\
\varepsilon>0\\}.$

Note that $\mathbb{D}_{\sigma}\subset\mathbb{D}^{Q}_{\sigma}$ because (1) and
(2) are satisfied when $\varphi\not\in\overline{\Phi_{\sigma}}$. For each
$w\in\mathbb{D}^{Q}_{\sigma}$ we have defined a quotient stability condition
$\overline{\sigma}_{w}\in\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)^{*}$,
namely $\overline{\sigma}_{w}=[\sigma_{w}]$ for $w\in\mathbb{D}_{\sigma}$ and
$\overline{\sigma}_{w}=[\sigma^{\psi}_{w}]$ for
$w\in\mathbb{D}^{Q}_{\sigma}-\mathbb{D}_{\sigma}$.

######  Proposition 11.6.

Let $\overline{\sigma\cdot G}/\mathbb{C}$ denote the closure of the orbit of
$\sigma$ in $\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$. Then

$\mathbb{D}^{Q}_{\sigma}\to\overline{\sigma\cdot G}/\mathbb{C}\quad\text{given
by}\quad w\mapsto\overline{\sigma}_{w}\cdot\mathbb{C}$

is a bijection, and restricts to a bijection between $\mathbb{D}_{\sigma}$ and
the closure of the orbit of $\sigma$ in
$\mathrm{Stab}(\mathsf{C})/\mathbb{C}$.

###### Proof.

By construction, and Lemma 11.2, $\overline{\sigma}_{w}$ is in
$\overline{\sigma\cdot G}$ so the map is well-defined. Since the image of the
charge $Z$ is the whole of $\mathbb{C}$ we can find
$\lambda,\mu\in\Lambda\otimes\mathbb{R}$ with $Z(\lambda)=i$ and $Z(\mu)=1$.
Then the assignment

$(Q,W)\mapsto f^{-1}\left(\frac{W(\lambda)}{W(\mu)}\right)$

descends to a map $\overline{\sigma\cdot
G}/\mathbb{C}\to\overline{\mathbb{D}}$ taking $\overline{\sigma}_{w}$ to $w$.
We claim this is the inverse. Certainly, if $w$ is the image of $(Q,W)$ then
$W$ is in the same $\mathbb{C}$ orbit as $Z_{w}$. It follows that the image of
this map is precisely $\mathbb{D}^{Q}_{\sigma}$, because we have shown that
there are no quotient stability conditions in $\overline{\sigma\cdot G}$ with
charge $Z_{w}$ for $w\not\in\mathbb{D}^{Q}_{\sigma}$. Moreover, there is a
unique quotient stability condition in $\overline{\sigma\cdot G}$ with charge
$Z_{w}$, which establishes the claim. ∎

######  Corollary 11.7.

The orbit $\sigma\cdot G$ is closed in
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ and in
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)^{*}$ if and only if the phase
diagram $\Phi_{\sigma}$ is dense.

###### Proof.

This follows immediately from the above result since
$\mathbb{D}_{\sigma}^{Q}=\mathbb{D}$ when $\Phi_{\sigma}$ is dense.
Alternatively, note that the Bridgeland metric induces the standard hyperbolic
metric on the quotient $(\sigma\cdot G)/\mathbb{C}\cong\mathbb{D}$ up to a
factor [34, Proposition 4.1]. Since the hyperbolic metric is complete, the
orbit is closed. ∎

Finally we describe the topology on $\mathbb{D}^{Q}_{\sigma}$ for which the
bijection in Proposition 11.6 is a homeomorphism. This is the topology in
which a sequence $(w_{n})$ converges to $w$ if and only if the charges
$Z_{w_{n}}$ converge in ${\mathrm{Hom}}(\Lambda,\mathbb{C})/\mathbb{R}_{>0}$
and the slicings $P_{w_{n}}$ converge in $\mathrm{Slice}(\mathsf{C})$. Since
$d(P_{w},P_{w^{\prime}})\leq 1$ for $w,w^{\prime}\in\mathbb{D}^{Q}_{\sigma}$
the convergence of charges implies uniform convergence of the phases of
massive objects. This occurs whenever $w_{n}\to w$ in the subspace topology
from $\mathbb{C}$. To ensure that the phase of the massless objects in
$P(\varphi)$ converges we have to refine this topology in a neighbourhood of
each $w\in\mathbb{D}_{\sigma}^{Q}-\mathbb{D}_{\sigma}$ so that convergence
also implies that

$\lim_{n\to\infty}\frac{w_{n}-w}{|w_{n}-w|}$

is a well-defined unit tangent vector in $T_{w}\mathbb{C}$. In other words,
the required topology arises from the real blow-up
$\beta\colon\widetilde{\mathbb{C}}\to\mathbb{C}$ at the points in
$\mathbb{D}^{Q}_{\sigma}-\mathbb{D}_{\sigma}$. More precisely, let
$\widetilde{\mathbb{D}}^{Q}_{\sigma}$ be the subspace of
$\beta^{-1}(\mathbb{D}^{Q}_{\sigma})$ consisting of
$\beta^{-1}(\mathbb{D}_{\sigma})$ and those points $(w=e^{2\pi i\varphi},v)$
in the exceptional divisors where the unit tangent vector $v\in
T_{w}\mathbb{C}$ is such that $\varphi$ and

$\psi=\frac{1}{\pi}\arg d_{w}f(v)$

satisfy one of the conditions (1), (2) or (3) in Lemma 11.2. The above
discussion leads to the following description of orbit closures.

######  Corollary 11.8.

Equip $\mathbb{D}^{Q}_{\sigma}$ with the quotient topology from
$\beta\colon\widetilde{\mathbb{D}}^{Q}_{\sigma}\to\mathbb{D}^{Q}_{\sigma}$.
Then

$\mathbb{D}^{Q}_{\sigma}\to\overline{\sigma\cdot G}/\mathbb{C}\colon
w\mapsto\overline{\sigma}_{w}\cdot\mathbb{C}$

is a homeomorphism, and restricts to a homeomorphism between
$\mathbb{D}_{\sigma}$ and the closure of the orbit of $\sigma$ in
$\mathrm{Stab}(\mathsf{C})/\mathbb{C}$.

######  Remark 11.9.

In fact, although we have not filled in all the details, the subspace
$\widetilde{\mathbb{D}}^{Q}_{\sigma}$ is homeomorphic to the closure of
$(\sigma\cdot G)/\mathbb{C}$ in
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)^{*}\\!/\mathbb{C}$.

##  12\. Two-dimensional stability spaces

We illustrate our results in the simplest non-trivial case in which the
stability space is a $2$-dimensional complex manifold. In this context there
is a very close relationship between the boundary strata we add and the wall-
and-chamber structure of the stability space.

###  12.1. Walls and chambers

Suppose that $\Lambda$ is a rank two lattice, and moreover that if
$\mathsf{C}$ contains a length heart then that heart has two iso-classes of
simple objects and $\Lambda=K(\mathsf{C})\cong\mathbb{Z}^{2}$. This assumption
is reasonable because otherwise the stability space of $\mathsf{C}$ is
naturally higher-dimensional. Since $\mathrm{rk}\,(\Lambda)=2$ the quotient
$\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ is a non-compact Riemann surface and

$\mathrm{Stab}\\!\left(\mathsf{C}\right)\cong\mathrm{Stab}(\mathsf{C})/\mathbb{C}\times\mathbb{C}$

as a complex manifold because all holomorphic bundles on a non-compact Riemann
surface are holomorphically trivial [16, Theorem 30.4]. Therefore it suffices
to describe $\mathrm{Stab}(\mathsf{C})/\mathbb{C}$.

The action of $\mathbb{C}$ preserves the set of semistable objects, so the
wall-and-chamber structure of the stability space descends to a partition of
$\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ into open chambers and real codimension
one walls between them. The charge map descends to a holomorphic map
$\mathrm{Stab}(\mathsf{C})/\mathbb{C}\to\mathbb{P}({\mathrm{Hom}}(\Lambda,\mathbb{C}))\cong\mathbb{C}\mathbb{P}^{1}$
which by abuse of notation we also denote $\mathcal{Z}$. The equatorial copy
of $\mathbb{R}\mathbb{P}^{1}$ consists of the charges with rank one image.

The action of the universal cover $G$ of $\mathrm{GL}_{2}^{+}(\mathbb{R})$
also preserves the sets of semistable objects, so each chamber and each wall
is a union of orbits. The image of a free orbit in
$\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ is a copy of
$G/\mathbb{C}\cong\mathbb{H}$ biholomorphic to its image under the charge map.
This image is either the Southern or Northern hemisphere in
$\mathbb{C}\mathbb{P}^{1}$. We refer to these free orbits as _cells_.

The image of a non-free orbit in $\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ is a
point. The heart of a stability condition in such an orbit is length, and so
by our assumption has two isoclasses of simple objects, say $s$ and $t$. There
is a one-parameter family consisting of images of orbits through stability
conditions for which $s$ and $t$ are semistable of the same phase and where
the ratio $m(s)/m(t)$ varies in $\mathbb{R}_{>0}$. This describes a real
analytic curve in $\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ which we refer to as
a _cell-wall_. Each cell-wall is isomorphic to its image in
$\mathbb{P}{\mathrm{Hom}}(\Lambda,\mathsf{C})$ which is an arc in the equator
from the point $Z(s)=0$ to $Z(t)=0$. The following lemma is immediate.

######  Lemma 12.1.

Suppose $\mathsf{C}$ admits an algebraic heart with two simple objects. Then
there is a bijection between the set of cell-walls and the set of algebraic
hearts (up to shift).

If there are non-split extensions between the simple objects $s$ and $t$ then
a cell-wall is a genuine wall in the stability space along which these
extensions are strictly semistable. All walls are of this form; in particular
no two walls intersect. If there are no non-split extensions, for instance if
the heart is semisimple, then the cell-wall lies in the same chamber as the
two neighbouring cells.

######  Lemma 12.2.

Each chamber of $\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ is a linear chain of
cells, separated by cell-walls. If it is a doubly-infinite chain then it is
biholomorphic to $\mathbb{C}$, otherwise it is biholomorphic to $\mathbb{H}$.

###### Proof.

Let $C$ be a cell. If it has no cell-walls in its closure then it is a chamber
and we are done. Let $W$ be a cell-wall in the closure, and let $s$ and $t$ be
simple objects in a corresponding algebraic heart. Their phases agree on $W$
and we may assume $\varphi(t)-1<\varphi(s)<\varphi(t)$ in $C$.

Suppose $W^{\prime}$ is another cell-wall in the closure of $C$ which is not
an actual wall. Let $s^{\prime}$ and $t^{\prime}$ be simple objects in a
corresponding algebraic heart. Since $s$ and $t$ are semistable on
$W^{\prime}$, some shifts $s[m]$ and $t[n]$ lie in this heart. Indeed, since
we assume $\varphi(s)<\varphi(t)$ in $C$, we may apply a shift so that the
heart contains $s[1]$ and $t$. Since these are indecomposable and there are no
non-split extensions between $s^{\prime}$ and $t^{\prime}$ we deduce, after
swapping $s^{\prime}$ and $t^{\prime}$ if necessary, that $s[1]$ and $t$ are
respectively self-extensions of $s^{\prime}$ and $t^{\prime}$. However, the
classes of $s[1]$ and $t$ are primitive in $\Lambda=K(\mathsf{C})$ so in fact
$s^{\prime}=s[1]$ and $t^{\prime}=t$. It follows that the closure of $C$ in
$\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ is precisely $W\cup C\cup W^{\prime}$.

Arguing inductively we conclude that the chamber is either a linear chain of
cells as claimed or a cycle. The possibility that it is a cycle is easily
excluded since the phase difference between $s$ and $t$ is well-defined and
monotonic as we move along a chain of cells.

For the final part note that the chamber is an open subset of the universal
cover $\mathbb{C}$ of
$\mathbb{P}{\mathrm{Hom}}(\Lambda,\mathsf{C})-\\{Z(s)=0,Z(t)=0\\}$. If the
chain of cells is doubly-infinite then it is the entirety of $\mathbb{C}$.
Otherwise it is a proper simply-connected open subset of $\mathbb{C}$ and is
therefore biholomorphic to $\mathbb{H}$ by the Riemann Mapping Theorem. ∎

###  12.2. The Speiser graph

The topology and geometry of $\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ is encoded
combinatorially in what we call the _Speiser graph_. This is the dual graph
$\mathrm{Sp}(\mathsf{C})$ to the cell and cell-wall decomposition, i.e. it has
a vertex for each cell and an edge between two vertices when they have a
common cell-wall in their closures. (More properly the Speiser graph is
defined when
$\mathrm{Stab}(\mathsf{C})/\mathbb{C}\to\mathbb{P}{\mathrm{Hom}}(\Lambda,\mathsf{C})$
has a finite set $S$ of singular values. It is the preimage in
$\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ of the dual graph in
$\mathbb{P}{\mathrm{Hom}}(\Lambda,\mathsf{C})-S$ to a Jordan curve passing
through the singular values in $S$ in cyclic order, see for example [12, §2].
However, this definition is too restrictive for our setting where there may be
infinitely many singular values.) Since the cells and cell-walls are
contractible the Speiser graph encodes the homotopy type of
$\mathrm{Stab}(\mathsf{C})/\mathbb{C}$. Indeed we can embed the Speiser graph
in $\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ so that each vertex is in the
corresponding cell and each edge is a smooth curve crossing the corresponding
cell-wall once transversely (and not crossing any other cell-wall). Then
$\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ deformation retracts onto the embedded
Speiser graph.

######  Conjecture 12.3.

Each component of the Speiser graph $\mathrm{Sp}(\mathsf{C})$ is a tree and
therefore each component of $\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ is
contractible.

When the Speiser graph is a tree the Uniformisation Theorem, see for example
[16, Theorem 27.9], tells us that $\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ is
biholomorphic to either $\mathbb{C}$ (parabolic type) or to the Poincaré disk
$\mathbb{D}$ (hyperbolic type). Either case may occur, for instance if
$\mathsf{C}$ is the bounded derived category of representations of the quiver
with two vertices and no arrows then
$\mathrm{Stab}(\mathsf{C})/\mathbb{C}\cong\mathbb{C}$ whereas if
$\mathsf{C}=\mathsf{D}^{b}(X)$ is the bounded derived category of coherent
sheaves on a smooth complex projective curve $X$ of strictly positive genus
then $\mathrm{Stab}(\mathsf{C})/\mathbb{C}\cong\mathbb{D}$ [6, 28]. The type
is parabolic if the Brownian motion on $\mathrm{Stab}(\mathsf{C})/\mathbb{C}$
is recurrent and hyperbolic if it is transient [21, 22]. The idea behind the
following conjecture is that the Brownian motion can be combinatorially
modelled by the random walk on the Speiser graph with suitable transition
probabilities on each edge (provided at least that we are not in the trivial
case in which the Speiser graph has a single vertex and no edges). See [12]
for a nice discussion.

######  Conjecture 12.4.

Assume the Speiser graph contains at least one edge. Equip the charge space
$\mathbb{P}{\mathrm{Hom}}(\Lambda,\mathbb{C})\cong\mathbb{C}\mathbb{P}^{1}$
with the constant curvature Riemannian metric in which the equator (consisting
of charges with rank one image) has length one. Assign each edge in the
Speiser graph a transition probability given by the length of the image in
charge space of the corresponding cell-wall. Then
$\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ is parabolic if the resulting random
walk is recurrent and hyperbolic if it is transient.

The following criterion, proved by comparing with the random walk on the
standard $2$-dimensional lattice, is useful for establishing recurrence and
shows that it does not depend delicately on the transition probabilities.

######  Theorem 12.5 ([13, §2.4]).

Suppose $\Gamma$ is an infinite connected graph, with a uniform upper bound on
the valency of its vertices. Assign transition probabilities in
$[\varepsilon,1-\varepsilon]$, where $\varepsilon>0$, to each edge. Then the
resulting random walk is recurrent if the vertices of $\Gamma$ can be embedded
in $\mathbb{R}^{2}$ with a uniform lower bound on the distance between any two
vertices, and a uniform upper bound on the length of each edge.

###  12.3. The space of quotient stability conditions

We remove the boundary strata of $\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$
where all objects are massless, and denote the resulting space by
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)^{*}$. Therefore
$\mathrm{rk}(\Lambda_{\mathsf{N}})=1$ when the massless subcategory
$\mathsf{N}\neq 0$. In this case
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ is a union of
components each homeomorphic to $\mathbb{C}\times\mathbb{R}$ by Theorem 9.4.
The action of $\mathbb{C}$ on
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)^{*}$ is free and the quotient is
a union of the open subset $\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ and a copy
of $\mathbb{R}$ for each boundary component in
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)^{*}$. Passing to
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)^{*}$ replaces each of these
copies of $\mathbb{R}$ by a point.

Recall, we abuse notation by using $\mathcal{Z}$ to denote the map
$\mathrm{Stab}^{L}(\mathsf{C})^{*}\\!/\mathbb{C}\to\mathbb{P}({\mathrm{Hom}}(\Lambda,\mathbb{C}))\cong\mathbb{C}\mathbb{P}^{1}$
induced from the charge map on the space of lax stability conditions. This
maps points of
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)\\!/\mathbb{C}$ where
$\mathsf{N}$ is massless to the point
${\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$. By Corollary 8.6
such a point has a punctured neighbourhood in
$\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$ isomorphic as a complex
manifold to the universal cover of a punctured disk centred at
${\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$. The partial
compactification $\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$ is obtained
locally by adding a point to the universal cover over the puncture. Such a
point is called a _logarithmic singularity_ of $\mathcal{Z}$. It follows that
$\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$ embeds in the _Mazurkiewicz
completion_ which is the union of $\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ with
all logarithmic singularities of $\mathcal{Z}$, see for example [15]. We do
not claim that this embedding is a homeomorphism, i.e. that every logarithmic
singularity occurs as a boundary point in
$\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$.

The space $\mathrm{Stab}^{L}(\mathsf{C})^{*}\\!/\mathbb{C}$ is recovered by
performing a real blow-up at each of the boundary points, resolving them to
copies of $\mathbb{R}$. Note that
$\mathrm{Stab}^{L}(\mathsf{C})^{*}\\!/\mathbb{C}$ is _not_ a Riemann surface
with boundary because each boundary stratum has a holomorphic function which
vanishes along it, namely the charge of (any) object which becomes massless on
that stratum.

The next result classifies the stable massless objects.

######  Proposition 12.6.

An object is massless and stable at some point in
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)^{*}$ precisely if it is simple in
some algebraic heart of $\mathsf{C}$.

###### Proof.

Suppose $s$ is simple in an algebraic heart $\mathsf{H}$. Let $t$ be the other
simple object. Define stability conditions $(P,Z_{n})$ for $n\in\mathbb{N}$
with slicing $P(1)=P(0,1]=\mathsf{H}$ and charge $Z_{n}(s)=-1/n$ and
$Z_{n}(t)=-1$. The limit $\sigma=\lim_{n\to\infty}(P,Z_{n})$ is a lax
stability condition in $\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)^{*}$ for
which $s$ is massless and stable. (By construction the limit is in the closure
of $\mathrm{Stab}\\!\left(\mathsf{C}\right)$ with locally-finite slicing $P$,
and the support property follows immediately because, up to shift, $t$ is the
only massive stable object.)

Now suppose $s$ is a massless stable object at some point in
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)^{*}$. Since
$\mathrm{rk}(\Lambda)=2$ each free $G$-orbit in
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ is open, and
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)^{*}$ is the union of their
closures. Hence $s$ is massless and stable at some point in the closure of a
free orbit $\sigma\cdot G$ where
$\sigma=(P,Z)\in\mathrm{Stab}\\!\left(\mathsf{C}\right)$. Then by Remark 11.4
and Proposition 11.6 the object $s$ is stable in some slice $P(\varphi)$ such
that $P(\varphi-\varepsilon,\varphi)=0$ or $P(\varphi,\varphi+\varepsilon)=0$
for sufficiently small $\varepsilon>0$ (or both). In the first case
$P[\varphi,\varphi+1)$ is a length heart in which $s$ is simple and in the
second $P(\varphi-1,\varphi]$. ∎

######  Corollary 12.7.

The boundary points of $\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$ are
in bijection with iso-classes of massless stable objects up to shift.

###### Proof.

We must show that there is a unique boundary point of
$\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$ at which a given stable
object is massless. Suppose
$\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ where
$\mathsf{N}=\mathsf{triang}_{\mathsf{C}}(s)$ is generated by the stable
massless object $s$. By Proposition 12.6, $s$ is simple in some algebraic
heart and our standing assumption $K(\mathsf{C})\cong\mathbb{Z}^{2}$ implies
$K(\mathsf{C}/\mathsf{N})\cong\mathbb{Z}$ and
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\cong\mathbb{C}$. Thus the
induced stability condition
$\sigma_{\mathsf{C}/\mathsf{N}}=(P_{\mathsf{C}/\mathsf{N}},Z_{\mathsf{C}/\mathsf{N}}=Z)$
on the quotient is fixed up to the action of $\mathbb{C}$. By Proposition 3.2
the slicing $P$ is uniquely determined by $P_{\mathsf{C}/\mathsf{N}}$ and
$P_{\mathsf{N}}$, i.e. by the slicing of the quotient and the phase of $s$.
However, by Theorem 9.4 the component of $\sigma$ in
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ is isomorphic to
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\times\mathbb{R}\cong\mathbb{C}\times\mathbb{R}$
so that all possible such $\sigma$ occur in the same component of
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$. Hence they all
determine the same point of $\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$
as claimed. ∎

Recall that cell-walls in $\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ correspond to
algebraic hearts (up to shift) and that the masses of the two simple objects
vanish at the respective ends of the cell-wall. Therefore the cell-wall starts
and ends at the boundary points where these simple objects become massless.

###  12.4. The exchange graph

The _exchange graph_ $\mathrm{EG}(\mathsf{C})$ has one vertex for each
algebraic heart of a stability condition in
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ and an edge whenever two hearts are
related by a simple HRS tilt; see [19, 33]. Each vertex of
$\mathrm{EG}(\mathsf{C})$ is $4$-valent because we can tilt left or right at
each of the two simple objects of the corresponding heart. The shift acts
freely on $\mathrm{EG}(\mathsf{C})$ and we denote the quotient, the
_projective exchange graph_ , by $\mathrm{EG}(\mathsf{C})/\mathbb{Z}$. This
quotient has one vertex for each algebraic heart up to shift, i.e. one vertex
for each cell-wall. If there is an edge between two vertices then they share a
common end point in the boundary of
$\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$. The massless stable object
at this boundary point is (up to shift) the common simple object of the hearts
corresponding to the two vertices.

The exchange graph can be embedded into
$\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$ by placing a vertex on each
cell-wall and embedding edges as smooth curves in the unique cell containing
the two cell-walls corresponding to its vertices in its closure.

###  12.5. Dense phase case

Suppose that $\mathrm{Stab}\\!\left(\mathsf{C}\right)$ contains a stability
condition $\sigma$ whose phase diagram $\Phi_{\sigma}$ is dense. Then the
$G$-orbit of $\sigma$ is free and by Corollary 11.7 closed. Since the orbit is
also open it is an entire connected component, consisting of a single chamber
with no walls. This is the case in which the Speiser graph is trivial. Every
stability condition in this component has dense phases and therefore no
stability condition in this component has an algebraic heart. Our theory adds
no boundary points to this component.

This situation occurs for the space $\mathrm{Stab}\\!\left(X\right)$ of
numerical stability conditions on a smooth complex projective curve $X$ of
genus $g>0$, see [6, 28] respectively for the elliptic curve and higher genus
cases. It also occurs for the space $\mathrm{Stab}\\!\left(Q\right)$ of
stability conditions on the bounded derived category of finite-dimensional
representations of a $2$-vertex quiver $Q$ with an oriented cycle, see [11,
Remark 3.33] for the existence of a dense-phase stability condition.

###  12.6. Non-dense phase case

Now suppose that there is at least one $\sigma$ in
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ with non-dense phases. By the above,
every stability condition in the component of $\sigma$ has non-dense phases,
and therefore lies in the $\mathbb{C}$ orbit of a stability condition with
length heart. In this situation we assume
$\Lambda=K(\mathsf{C})\cong\mathbb{Z}^{2}$.

Using the description of the closure of the orbit $(\sigma\cdot G)/\mathbb{C}$
in Proposition 11.6 and Corollary 11.8, we can describe the component of
$\sigma$ in $\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$. First we
determine the chamber containing $\sigma$ and its walls. The latter correspond
to non-trivial algebraic hearts whose simple objects are stable in the
chamber. Then we find the stable objects in the adjacent chambers and repeat.
In the examples we consider there are only finitely many chambers up to the
action of $\mathrm{Aut}_{\Lambda}(\mathsf{C})$ so this is an effective
strategy.

It is easy to find the cell decompositions of chambers (since each chamber is
a linear chain of cells) and thence construct the Speiser graph. In our
examples this is always a tree, so $\mathrm{Stab}\\!\left(\mathsf{C}\right)$
is contractible. The type of $\mathrm{Stab}(\mathsf{C})/\mathbb{C}$, either
parabolic or hyperbolic, is known from previous results and we confirm
Conjecture 12.4 in these cases by showing that the random walk on the Speiser
graph is respectively recurrent or transient. In the former case
$\mathrm{Aut}_{\Lambda}(\mathsf{C})$ acts by rotations and translations of
$\mathbb{C}$, and in the latter by rotations, translations and ideal rotations
of $\mathbb{D}$.

###  12.7. The $A_{2}$ quiver

Let $Q$ be the $A_{2}$ quiver with two vertices and one arrow. In this section
we consider several stability spaces associated with algebras associated with
the $A_{2}$ quiver, namely the classical $A_{2}$ path algebra, and the the
associated $2$-Calabi–Yau Ginzburg algebra.

####  12.7.1. Classic $A_{2}$

First we consider the bounded derived category $\mathsf{D}^{b}(A_{2})$ of
finite-dimensional representations of the classic $A_{2}$. Its stability space
$\mathrm{Stab}\\!\left(A_{2}\right)\cong\mathbb{C}^{2}$ was first described by
King [24], see [8] for further references.

The standard heart of $\mathsf{D}^{b}(A_{2})$ has two exceptional simple
objects $s$ and $t$ with one non-split extension $0\to s\to e\to t\to 0$
between them. It is easy to construct a stability condition in which $s$, $e$
and $t$ are the only stable objects up to shift. Its phase diagram has three
isolated phases. By Corollary 11.8 the cell containing this stability
condition has three cell-walls, and three boundary points where respectively
$s$, $e$ and $t$ become massless. The object $e$ destabilises as we cross the
cell-wall along which $s$ and $t$ have the same phase, and we enter a chamber
in which only $s$ and $t$ are stable. Since $e$ is the unique indecomposable
extension between any shifts of $s$ and $t$, this chamber is a chain of cells
indexed by $\mathbb{N}$. Similar considerations apply to the other two walls
of the initial cell. Thus the stability space has four chambers, one in which
$s$, $e$ and $t$ are all stable and three in which pairs of them are stable.
The Speiser graph has one central vertex with three infinite linear graphs
attached. The random walk on this is recurrent by Theorem 12.5 in agreement
with the fact that $\mathrm{Stab}(\mathsf{C})/\mathbb{C}\cong\mathbb{C}$ is
parabolic. The category $\mathsf{D}^{b}(A_{2})$ is fractional Calabi–Yau; the
Serre functor $S$ satisfies $S^{3}=[1]$. This acts by rotation on
$\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$ preserving the central
chamber and cyclically permuting the other three chambers, and also the three
boundary points. See Figure 12.7.2 for an illustration.

####  12.7.2. $2$-Calabi–Yau $A_{2}$

Now consider the $2$-Calabi–Yau category $\mathsf{D}^{b}(\Gamma_{\\!2}A_{2})$,
where $\Gamma_{\\!2}A_{2}$ is the Ginzburg dg algebra of the $A_{2}$ quiver;
see [23, §7.2], for example, for details of the construction.

The stability space
$\mathrm{Stab}(\Gamma_{\\!2}A_{2})/\mathbb{C}\cong\mathbb{D}$ is the universal
cover of the thrice punctured Riemann sphere and was first described in [32].
See [8] for a detailed discussion and further references, and also [25, 30]
for more general discussions of the stability spaces of the Ginzburg dg-
algebras associated to Dynkin quivers.

The standard heart of $\mathsf{D}^{b}(\Gamma_{\\!2}A_{2})$ has two
$2$-spherical simple objects $s$ and $t$, with one non-split extension $0\to
s\to e\to t\to 0$ and $0\to t\to f\to s\to 0$ in each direction. Each
$2$-spherical object in $\mathsf{D}^{b}(\Gamma_{\\!2}A_{2})$ generates a twist
automorphism. For example, applying the twist $\textsc{tw}_{s}$ about $s$ to
the triangle $s\to e\to t\to s[1]$ yields the triangle $s[-1]\to t\to f\to s$,
and then applying $\textsc{tw}_{t}$ yields the rotation $e[-1]\to t[-1]\to
s\to e$ of the original triangle. In particular $e$ and $f$ are also
$2$-spherical. The subgroup of
$\mathrm{Aut}(\mathsf{D}^{b}(\Gamma_{\\!2}A_{2}))$ generated by
$\textsc{tw}_{s}$ and $\textsc{tw}_{t}$ is isomorphic to the Artin–Tits braid
group $B_{3}$ of the $A_{2}$ quiver, i.e. the braid group on three strands.
The centre $Z(B_{3})$ is generated by a single automorphism which acts as the
Serre functor $S=[2]$. Let $\mathcal{S}$ be the set of equivalence classes of
spherical objects in $\mathsf{D}^{b}(\Gamma_{\\!2}A_{2})$ up to isomorphism
and shift. We abuse notation by using the same notation for spherical objects
and their classes in $\mathcal{S}$; this is harmless since the twists
$\textsc{tw}_{s}=\textsc{tw}_{s[1]}$ agree. The quotient
$B_{3}/Z(B_{3})\cong\mathrm{PSL}_{2}(\mathbb{Z})$ acts on $\mathcal{S}$ and
the stabiliser of $s$ is the infinite cyclic subgroup generated by
$\textsc{tw}_{s}$. From the above examples $t$ is in the orbit of $s$ (indeed
the action on $\mathcal{S}$ is transitive although we do not need this).

Now consider the stability space
$\mathrm{Stab}\\!\left(\Gamma_{\\!2}A_{2}\right)$. As in the
$\mathsf{D}^{b}(A_{2})$ case one can easily construct a stability condition in
which, up to shift, the stable objects are the two simple objects $s$ and $t$
of the standard heart and one, $e$ say, of the two extensions between them.
The phase diagram has three isolated phases so the corresponding cell in
$\mathrm{Stab}(\Gamma_{\\!2}A_{2})/\mathbb{C}$ has three cell-walls. As before
$e$ destabilises as we cross the wall where $s$ and $t$ have the same phase,
but now the other extension $f$ becomes stable on the far side of the wall.
Thus we enter the chamber obtained by applying $\textsc{tw}_{s}$ to the
initial one. Similar considerations apply to the other walls of the initial
chamber. Therefore $\mathrm{PSL}_{2}(\mathbb{Z})$ acts transitively on the
chambers in $\mathrm{Stab}(\Gamma_{\\!2}A_{2})/\mathbb{C}$, each of which is a
single cell bounded by three walls. There are three stable $2$-spherical
objects in each chamber whose respective masses vanish at the three boundary
points of the chamber. The action of $\mathrm{PSL}_{2}(\mathbb{Z})$ on
chambers is free because no pair of distinct spherical objects, a fortiori no
triple, is fixed. The action on walls is also free and it quickly follows from
the examples of twist actions that it is transitive.

In conformity with Corollary 10.4 and Corollary 12.7, the only massless stable
objects are the simple objects of the hearts of stability conditions (all of
which are algebraic; see e.g. [30]) and there is one boundary point in
$\mathrm{Stab}^{Q}(\Gamma_{\\!2}A_{2})^{*}\\!/\mathbb{C}$ for each (up to
shift and isomorphism). The Speiser graph is the Cayley graph of
$\mathrm{PSL}_{2}(\mathbb{Z})$ with respect to the generating set consisting
of the images of $\textsc{tw}_{s}$, $\textsc{tw}_{e}$ and $\textsc{tw}_{t}$.
It is an infinite trivalent tree as expected from Conjecture 12.3 and the
random walk on it is transient as expected from Conjecture 12.4. The twist
$\textsc{tw}_{s}$ acts by a hyperbolic isometry fixing the boundary point at
which $s$ is massless. Therefore, $\textsc{tw}_{s}$ acts either by an ideal
rotation about that point or a translation, since, locally at the fixed point
the action universally covers the action on
$\mathbb{P}{\mathrm{Hom}}(\Lambda,\mathbb{C})$. This is given by the matrix

$\begin{pmatrix}-1&0\\\ 1&1\end{pmatrix}$

with respect to the basis $\\{[s],[t]\\}$ of
$\Lambda=K(\mathsf{D}^{b}(\Gamma_{\\!2}A_{2}))$. Since the eigenvalues are
$\pm 1$ the twist acts by an ideal rotation. Up to isometry, the three
boundary points where $s$, $e$ and $t$ are massless can be chosen arbitrarily
on $\partial\mathbb{D}$ and this fixes the remaining boundary points of
$\mathrm{Stab}^{Q}({\Gamma_{\\!2}A_{2}})^{*}\\!/\mathbb{C}$ uniquely. They
form a dense subset of $\partial\mathbb{D}$. See Figure 12.7.2 for an
illustration.

&

$e$$s$$t$

$A_{2}$ quiver Ginzburg algebra $\Gamma_{2}A_{2}$

$s$$t_{0}$$t_{1}$$t_{2}$$t_{-1}$$\mathcal{O}_{x}$$\mathcal{O}(1)$$\mathcal{O}$

$\Lambda_{2,1,0}$ quiver Coherent sheaves on $\mathbb{P}^{1}$

Figure 2. Illustrations of $\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$.
In each case this is a contractible non-compact Riemann surface. We depict it
as a disk, which is holomorphically accurate in the hyperbolic case (Ginzburg
algebra $\Gamma_{2}A_{2}$, shown orange). In the parabolic cases (purple) it
is only topologically accurate, but has the advantage that we can more easily
visualise the partial compactification. This is obtained by adding logarithmic
singularities on the boundary circle at which certain objects become massless.
These points are labelled by the corresponding stable massless object.
Chambers are indicated by the shading. The set of stable objects in a chamber
(up to shift) is the set of massless stable objects at its boundary vertices.
Walls are shown as arcs of circles spanning a pair of points on the boundary.
The objects labelling these two boundary points have equal phase on the wall;
they are the two simple objects of the corresponding algebraic heart. The
subset of stable objects which become strictly semistable on a wall is
therefore the set of massless stable objects labelling the other vertices of
the chamber. The Speiser graph $\mathrm{Sp}(\mathsf{C})$ is shown in red and
the quotient $\mathrm{EG}(\mathsf{C})/\mathbb{Z}$ of the exchange graph by the
shift in blue. To avoid clutter cell-walls which are not walls are omitted;
these can be inferred from the Speiser and exchange graphs.

###  12.8. A discrete derived category

Let $Q=\Lambda_{2,1,0}$ be the bound quiver with two vertices, one arrow in
each direction, and the zero relation given by the composite of these arrows.
The (principal component of the) stability space
$\mathrm{Stab}\\!\left(\Lambda_{2,1,0}\right)\cong\mathbb{C}^{2}$ was first
described in [33], see also [9, 30] for proofs that the stability space is
connected and generalisations to other discrete derived categories.

Let $s$ be the simple representation at the vertex with no relation, and
$t=t_{0}$ the other simple representation. The object $s$ is $2$-spherical and
$t_{0}$ is exceptional. Since $\textsc{tw}_{s}(s)=s[-1]$ the twist
$\textsc{tw}_{s}$ generates an infinite cyclic subgroup of automorphisms. Set
$t_{n}=\textsc{tw}_{s}^{n}(t_{0})$. There are unique non-split extensions
$0\to s\to t_{-1}\to t_{0}\to 0$ and $0\to t_{0}\to t_{1}\to s\to 0$. In
particular there is a stability condition in which $s$, $t_{-1}$ and $t_{0}$
are the only stable objects up to shift. This lives in a chamber with three
walls, and three boundary points at which these objects are respectively
massless. Crossing the wall where $t_{-1}$ destabilises we enter a chamber in
which $t_{1}$ is stable. As in the previous example this chamber is the image
of the initial one under the action of the twist $\textsc{tw}_{s}$, and
similarly crossing the wall where $t_{0}$ destabilises we enter the image of
the initial chamber under $\textsc{tw}_{s}^{-1}$. However, if we cross the
wall where the spherical object $s$ destabilises then we enter a chamber in
which only $t_{0}$ and $t_{1}$ are stable. Unlike the previous chambers which
consist of a single cell, this is the union of a sequence of cells, and cell-
walls upon which the phases of $t_{0}$ and $t_{1}[n]$ agree for
$n\in\mathbb{N}$. In summary, there is one free orbit of chambers with three
stable objects (one spherical and two exceptional) and one free orbit of
chambers with two stable objects (both exceptional) under the action generated
by $\textsc{tw}_{s}$. As expected, there is one boundary point in
$\mathrm{Stab}^{Q}(\Lambda_{2,1,0})^{*}\\!/\mathbb{C}$ at which each of $s$
and $\\{t_{n}:n\in\mathbb{Z}\\}$ is massless. Clearly $\textsc{tw}_{s}$ fixes
the former and acts freely and transitively on the latter. Its square
$\textsc{tw}_{s}^{2}$ acts trivially on the Grothendieck group so the images
of the boundary points labelled by the $t_{n}$ map to one of two points in
charge space according to whether $n$ is even or odd.

The Speiser graph has vertices $\mathbb{Z}\times\mathbb{N}$ with edges $(m,0)$
to $(m+1,0)$ and $(m,n)$ to $(m,n+1)$ for $m\in\mathbb{Z}$ and
$n\in\mathbb{N}$. This is a tree as predicted by Conjecture 12.3. Moreover,
the random walk on the Speiser graph is recurrent by Theorem 12.5, as
predicted by Conjecture 12.4. The twist $\textsc{tw}_{s}$ acts by an isometry
without fixed points on
$\mathrm{Stab}(\Lambda_{2,1,0})/\mathbb{C}\cong\mathbb{C}$ and therefore acts
by a translation of $\mathbb{C}$. This is illustrated in Figure 12.7.2.

###  12.9. The projective line

The stability space $\mathrm{Stab}(\mathbb{P}^{1})/\mathbb{C}\cong\mathbb{C}$
was first described in [29], see also [28]. The ‘classical’ stability
condition has heart the coherent sheaves, with stable objects the skyscrapers
$\mathcal{O}_{x}$ for $x\in\mathbb{P}^{1}$ and the line bundles
$\mathcal{O}(k)$ for $k\in\mathbb{Z}$. The cell containing it has a sequence
of cell-walls indexed by $\mathbb{Z}$ separated by boundary points at which
$\mathcal{O}(k)$ is massless for $k\in\mathbb{Z}$. An example for a lax
stability condition at the boundary is given by the slicing $P_{b}$ of Example
3.14 with charge map $Z(\mathcal{O})=0$ and $Z(\mathcal{O}(1))=i$. These
boundary points accumulate on the boundary at a point where the charge of the
skyscrapers vanishes. However, Proposition 12.6 shows this point is not in
$\mathrm{Stab}^{Q}(\mathbb{P}^{1})^{*}\\!/\mathbb{C}$ because the skyscrapers
are not simple in any algebraic heart and so cannot become massless (see also
Example 4.17). We indicate this omitted point by a white dot in Figure 12.7.2
and label it by $\mathcal{O}_{x}$ to indicate that the skyscrapers are stable
in the adjacent chamber.

Crossing the wall spanning the points where $\mathcal{O}(k)$ and
$\mathcal{O}(k+1)$ are massless the skyscrapers and all other line bundles
destabilise and we enter a chamber in which the only stable objects are
$\mathcal{O}(k)$ and $\mathcal{O}(k+1)$. This chamber is the union of a
sequence of cells separated by walls on which the phases of $\mathcal{O}(k+1)$
and $\mathcal{O}(k)[n]$ for $n>0$ agree.

The Speiser graph is the union of $\mathbb{Z}$ copies of the graph with
vertices $\mathbb{N}$ and edges from $n$ to $n+1$ joined at the $0$ vertices.
This is a tree as expected from Conjecture 12.3. The central vertex has
infinite valence so Theorem 12.5 does not apply. Nevertheless the random walk
is recurrent as expected from Conjecture 12.4.

The infinite cyclic group generated by the automorphism
$-\otimes\mathcal{O}(1)$ preserves the chamber containing the ‘classical’
stability condition (but does not fix any stability condition in this chamber)
and acts freely and transitively on the chambers in which only two objects are
stable. It also acts freely and transitively on the boundary points. It
follows that it acts by a translation on
$\mathrm{Stab}(\mathbb{P}^{1})/\mathbb{C}\cong\mathbb{C}$. See Figure 12.7.2
for an illustration.

Superficially, this closely resembles the previous example. However, there are
several important (and inter-related) differences. In this case there is a
chamber bounded by a countably infinite family of walls; there are stable
objects whose mass does not vanish; the images of the boundary points
accumulate in charge space.

##  13\. Comparisons with other constructions

We compare our partial compactification of the stability space with two
alternative approaches, namely Bolognese’s ‘local compactification’ [5] and
Bapat, Deopurkar and Licata’s ‘Thurston compactification’ [1].

###  13.1. Bolognese’s ‘local compactification’

In [5] Bolognese constructs an alternative ‘local compactification’ of
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ using a metric completion of
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$. In order to do so she assumes that
$\mathcal{Z}\colon\mathrm{Stab}\\!\left(\mathsf{C}\right)\to{\mathrm{Hom}}(\Lambda,\mathbb{C})$
is a cover of the complement of a locally finite union
$\Delta\subset{\mathrm{Hom}}(\Lambda,\mathbb{C})$ of submanifolds. She fixes
an inner product on the underlying real space of
${\mathrm{Hom}}(\Lambda,\mathbb{C})$ and gives
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ the geodesic metric $d_{B}$ induced
from the pullback of the associated metric. Since
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ is locally homeomorphic to
${\mathrm{Hom}}(\Lambda,\mathbb{C})$ with its norm topology this metric
induces the usual topology on $\mathrm{Stab}\\!\left(\mathsf{C}\right)$. Her
local compactification $\widehat{\mathrm{Stab}}\\!\left(\mathsf{C}\right)$ is
the subspace of the metric completion consisting of equivalence classes of
Cauchy sequences satisfying the limiting support property below. As a
topological space $\widehat{\mathrm{Stab}}\\!\left(\mathsf{C}\right)$ is
independent of the choice of inner product [5, Lemma 3.6].

######  Definition 13.1 (Limiting support property [5, Definition 4.3]).

A Cauchy sequence $(\sigma_{n})$ in the metric $d_{B}$ on
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ has the _limiting support property_
if $\liminf_{n\to\infty}C_{n}=C>0$ where for each $n\in\mathbb{N}$ the
constant $C_{n}$ is the infimum of those $K>0$ such that
$|Z_{n}(c)|>K||v(c)||$ for every $c\in P_{n}(\varphi)$ with
$\lim_{n\to\infty}Z_{n}(c)\neq 0$. (The set of such constants $K$ is non-empty
because each $\sigma_{n}$ is in $\mathrm{Stab}\\!\left(\mathsf{C}\right)$ and
so satisfies the support property.) This property is well-defined on
equivalence classes of Cauchy sequences by [5, Lemma 4.4].

In fact Bolognese shows that one can construct
$\widehat{\mathrm{Stab}}\\!\left(\mathsf{C}\right)$ using only
$\mathcal{Z}$-local Cauchy sequences, that is Cauchy sequences $(\sigma_{n})$
for $d_{B}$ which eventually lie in an open subset
$U\subset\mathrm{Stab}\\!\left(\mathsf{C}\right)$ homeomorphic to its image
via $\mathcal{Z}$. More precisely, she shows that any Cauchy sequence is
equivalent to a $\mathcal{Z}$-local one, and that if two $\mathcal{Z}$-local
Cauchy sequences are equivalent then they are $\mathcal{Z}$-local with respect
to the same open $U$ [5, Theorem 3.7]. Moreover, each $\mathcal{Z}$-local
Cauchy sequence determines a thick subcategory of objects which become
massless in the limit, and a well-defined stability condition on the quotient
category [5, Propositions 4.2 and Theorem 6.1]. Finally, $\mathcal{Z}$-local
Cauchy sequences are equivalent precisely when they determine the same
massless subcategory and stability condition on the quotient [5, Theorem 6.2].

######  Lemma 13.2.

Suppose $(\sigma_{n})$ is a $\mathcal{Z}$-local Cauchy sequence for $d_{B}$ on
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$. Then $(\sigma_{n})$ converges to a
lax pre-stability condition $\sigma$ in the product metric on
$\mathrm{Slice}(\mathsf{C})\times{\mathrm{Hom}}(\Lambda,\mathbb{C})$.

###### Proof.

By construction the sequence $(Z_{n})$ of charges converges in norm in
${\mathrm{Hom}}(\Lambda,\mathbb{C})$, say $Z_{n}\to Z$ as $n\to\infty$. The
full subcategories
$P(\varphi)\coloneqq\\{c\in\mathsf{C}:\varphi_{\sigma_{n}}^{\pm}(c)\to\varphi\\}$
define a slicing [5, Proposition 5.3]. Bolognese notes in the proof of [5,
Proposition 5.2] that the sequence $(P_{n})$ of slicings is Cauchy in the
slicing metric on $\mathrm{Slice}(\mathsf{C})$. It follows that $P_{n}\to P$
in $\mathrm{Slice}(\mathsf{C})$. For, if this were not the case, then, after
passing to a subsequence, we may assume there is some $\varepsilon>0$ with
$d(P,P_{n})\geq\varepsilon$ for all $n\in\mathbb{N}$. In other words there is
a sequence $(\varphi_{n})$ of phases and objects $c_{n}\in P(\varphi_{n})$
such that $c_{n}\not\in
P_{n}(\varphi_{n}-\varepsilon,\varphi_{n}+\varepsilon)$. However, we know that
for each $n\in\mathbb{N}$ there is some $N$ with $c_{n}\in
P_{m}(\varphi_{n}-\varepsilon/2,\varphi_{n}+\varepsilon/2)$ whenever $m\geq
N$. Choosing $N$ sufficiently large that we also have
$d(P_{n},P_{m})<\varepsilon/2$ for all $m,n\geq N$ leads to a contradiction.

Therefore $(\sigma_{n})$ converges to $(P,Z)$ in the product metric on
$\mathrm{Slice}(\mathsf{C})\times{\mathrm{Hom}}(\Lambda,\mathbb{C})$. It is
easy to confirm that $\sigma=(P,Z)$ is a lax pre-stability condition, i.e.
that $Z(c)\in\mathbb{R}_{\geq 0}e^{\pi i\varphi}$ whenever $c\in P(\varphi)$.
∎

Let $\mathrm{Stab}^{B}\\!\left(\mathsf{C}\right)$ be the subset of the closure
of $\mathrm{Stab}\\!\left(\mathsf{C}\right)$ in
$\mathrm{Slice}(\mathsf{C})\times{\mathrm{Hom}}(\Lambda,\mathbb{C})$
consisting of the limits of $\mathcal{Z}$-local Cauchy sequences for $d_{B}$
which satisfy the limiting support property. In order to compare Bolognese’s
local compactification with our constructions, we need to be able to compare
this with $\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$. Unfortunately the
relationship is not obvious since our support property is phrased in terms of
massive stable objects in some $P(\varphi)$ and Bolognese’s limiting support
property is phrased in terms of semistable objects in some $P_{n}(\varphi)$
which remain massive in the limit. Since the HNfiltration of a
$\sigma_{n}$-semistable object with respect to $\sigma$ may contain massless
objects, and similarly the other way round, there is no direct argument
relating the two notions of support.

If
$\mathrm{Stab}^{B}\\!\left(\mathsf{C}\right)=\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$
then there should be a homeomorphism
$\widehat{\mathrm{Stab}}\\!\left(\mathsf{C}\right)\cong\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)$.
For instance, this is so for the example of the $A_{1}$ quiver computed in
Bolognese’s paper. More generally an inclusion in either direction should
extend to a map between $\widehat{\mathrm{Stab}}\\!\left(\mathsf{C}\right)$
and $\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)$ in the corresponding
direction.

###  13.2. Bapat, Deopurkar and Licata’s ‘Thurston compactification’

In [1], the authors take a different approach to compactifying the stability
space, more precisely the quotient $\mathrm{Stab}(\mathsf{C})/\mathbb{C}$, of
a ${\mathbf{k}}$-linear triangulated category $\mathsf{C}$. By analogy with
Thurston’s compactification of Teichmüller space, they consider the map

$m\colon\mathrm{Stab}(\mathsf{C})/\mathbb{C}\to\mathbb{P}(\mathbb{R}^{\mathsf{C}})\colon\sigma\cdot\mathbb{C}\mapsto[m_{\sigma}(c):0\neq
c\in\mathsf{C}].$

When $\mathbb{P}(\mathbb{R}^{\mathsf{C}})$ has the topology induced from the
product topology on $\mathbb{R}^{\mathsf{C}}$ this map is continuous, for
instance by Proposition 5.12. Automorphisms of $\mathsf{C}$ act on
$\mathbb{P}(\mathbb{R}^{\mathsf{C}})$ by pre-composing a real-valued function
on the objects of $\mathsf{C}$ with the inverse automorphism. The map $m$ is
equivariant for $\mathrm{Aut}_{\Lambda}(\mathsf{C})$ because
$m_{\alpha\cdot\sigma(c)}=m_{\sigma}(\alpha^{-1}c)$.

Under appropriate conditions on $\mathsf{C}$, Bapat, Deopurkar and Licata
conjecture that $m$ is a homeomorphism onto its image $M(\mathsf{C})$, and
that the closure $\overline{M(\mathsf{C})}$ is a manifold with boundary and
interior $M(\mathsf{C})$. Moreover, motivated by the description of boundary
points of Thurston’s compactification as functionals given by unsigned
intersections with closed curves, they conjecture that there is a suitable
class $\mathcal{S}$ of objects such that the functionals

$\overline{\mathrm{hom}}(s)\coloneqq\bigg{[}\sum_{n\in\mathbb{Z}}\dim_{\mathbf{k}}{\mathrm{Hom}}_{\mathsf{C}}(s,c[n])\bigm{|}0\neq
c\in\mathsf{C}\bigg{]}\qquad(s\in\mathcal{S})$

form a dense subset of the boundary $\partial M(\mathsf{C})$. These
conjectures hold for the $2$-Calabi–Yau category associated to the $A_{2}$
quiver [1, §5]. More generally, when $\mathsf{C}$ is the $2$-Calabi–Yau
category associated to any connected quiver, they show $m$ is injective, that
the closure of the image is compact, and that the functionals of all
$2$-spherical objects lie in the boundary.

Let
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)^{*}=\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)-\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{C}\right)$
be the space of lax stability conditions with the stratum where all objects
are massless deleted, and define
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)^{*}$ similarly. We can extend $m$
to a map from $\mathrm{Stab}^{L}(\mathsf{C})^{*}\\!/\mathbb{C}$ defined in the
same way; the extension is also continuous by Proposition 5.12. This extension
factorises continuously through the quotient

${\mathrm{Stab}^{L}(\mathsf{C})^{*}\\!/\mathbb{C}}$${\mathbb{P}(\mathbb{R}^{\mathsf{C}})}$${\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}}$$\scriptstyle{q}$$\scriptstyle{m}$$\scriptstyle{m}$

because the masses depend only on the associated quotient stability condition.
(We abuse notation by denoting the extension and the factorisation by $m$.) By
construction the images of both
$\mathrm{Stab}^{L}(\mathsf{C})^{*}\\!/\mathbb{C}$ and
$\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$ are contained within
$\overline{M(\mathsf{C})}$. At least when $\mathsf{C}$ is the $2$-Calabi–Yau
category associated to a connected quiver, Bapat, Deopurkar and Licata expect
the boundary points in the image of
$\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$ to be dense in the boundary
[1, Remark 4.9]. We show that this is true for the $A_{2}$ quiver. Before
discussing that example we consider the non Calabi–Yau case. We slightly
modify the construction by considering the map

$\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}\to\mathbb{P}(\mathbb{R}^{\mathcal{S}})\colon\sigma\mapsto[m_{\sigma}(s):s\in\mathcal{S}]$

where $\mathcal{S}$ is a suitable class of objects with the property that the
masses of objects in $\mathcal{S}$ uniquely determine the masses of all
objects.

######  Example 13.3.

Recall the description of $\mathrm{Stab}(A_{2})/\mathbb{C}$ in §12.7. In each
stability condition in $\mathrm{Stab}\\!\left(A_{2}\right)$ the semistable
objects are, up to shift, a subset of at least two of $\\{s,e,t\\}$ where $s$
and $t$ are the two simple representations and $e$ the extension between them.
It follows that the masses of objects in $\mathcal{S}=\\{s,e,t\\}$ determine
the masses of all objects. This remains true for lax stability conditions in
$\mathrm{Stab}^{L}\\!\left(A_{2}\right)$. Therefore it suffices to consider
the map

$\mathrm{Stab}(A_{2})/\mathbb{C}\to\mathbb{P}(\mathbb{R}^{3})\colon\sigma\mapsto[m_{\sigma}(s):m_{\sigma}(e):m_{\sigma}(t)].$

The image is cut out by the inequalities $x_{0},x_{1},x_{2}>0$ (the masses are
strictly positive) together with the cyclic permutations of the inequality
$x_{0}-x_{1}+x_{2}\leq 0$ (the mass of an extension is bounded by the sum of
the masses of its factors). If we normalise so that $x_{0}+x_{1}+x_{2}=1$ then
the image can be viewed as the shaded triangle, with vertices omitted, in the
$2$-simplex in Figure 3. In particular we see that the map is not injective
because when, for example, $e$ is unstable the masses of $s$ and $t$ do not
suffice to determine their phases. The three boundary points in
$\mathrm{Stab}^{Q}(A_{2})^{*}\\!/\mathbb{C}$ where the masses of $s$, $e$ and
$t$ respectively vanish are mapped to the three omitted vertices. So in this
example, $\mathrm{Stab}^{Q}(A_{2})^{*}\\!/\mathbb{C}$ surjects onto Bapat,
Deopurkar and Licata’s compactification, and the boundaries coincide. The
boundary points correspond precisely to the functionals
$\overline{\mathrm{hom}}(c)$ for $c\in\mathcal{S}$.

$x_{2}=0$$x_{0}=0$$x_{1}=0$ Figure 3. The image of
$\mathrm{Stab}(A_{2})/\mathbb{C}\to\Delta^{2}\colon\sigma\mapsto(\lambda
m_{\sigma}(s),\lambda m_{\sigma}(e),\lambda m_{\sigma}(t))$ where
$\lambda=m_{\sigma}(s)+m_{\sigma}(e)+m_{\sigma}(t)$ is shaded red. The chamber
of the stability space in which $s$, $e$ and $t$ are stable is mapped
homeomorphically to the interior. The chamber in which only $s$ and $t$ are
stable, together with its bounding wall, are projected down onto the edge
$x_{0}-x_{1}+x_{2}=0$, and similarly for the other two chambers. The three
boundary points in $\mathrm{Stab}^{Q}(A_{2})^{*}\\!/\mathbb{C}$ where the
masses of $s$, $e$ and $t$ respectively vanish are mapped to the three black
vertices.

######  Example 13.4.

Let $\mathcal{S}$ be the set of equivalence classes of spherical objects in
$\mathsf{D}^{b}(\Gamma_{\\!2}A_{2})$ up to isomorphism and shift. Then by [1,
Proposition 5.7] the map

$m\colon\mathrm{Stab}(\Gamma_{\\!2}A_{2})/\mathbb{C}\to\mathbb{P}(\mathbb{R}^{\mathcal{S}})\colon\sigma\mapsto[m_{\sigma}(s):s\in\mathcal{S}]$

is a homeomorphism onto its image which we denote $M(\Gamma_{\\!2}A_{2})$.
Moreover, $M(\Gamma_{\\!2}A_{2})$ is homeomorphic to an open disk by [1,
Proposition 5.20]. After choosing an element $s\in\mathcal{S}$ to map to
$[1:0]$, the action of the Artin–Tits braids group induces a bijection
$\mathcal{S}\cong\mathbb{P}(\mathbb{Z}^{2})$. Using this identification the
map

$\mathcal{S}\to\mathbb{P}(\mathbb{R}^{\mathcal{S}})\colon
s\mapsto[\mathrm{hom}(s)(s^{\prime}):s^{\prime}\in\mathcal{S}]$

extends uniquely to a homeomorphism from $\mathbb{P}(\mathbb{R}^{2})$ onto the
boundary of $M(\Gamma_{\\!2}A_{2})$ in $\mathbb{P}(\mathbb{R}^{S})$ by [1,
Propositions 5.13 and 5.18]. The closure $\overline{M(\Gamma_{\\!2}A_{2})}$ is
homeomorphic to a closed disk. The functional $\overline{\mathrm{hom}}(s)$ is
the unique fixed point of the spherical twist $\textsc{tw}_{s}$.

Now consider the extension
$m\colon\mathrm{Stab}^{Q}(\Gamma_{\\!2}A_{2})^{*}\\!/\mathbb{C}\to\overline{M(\Gamma_{\\!2}A_{2})}$.
Recall from §12.7 and Figure 12.7.2 that the partial compactification
$\mathrm{Stab}^{Q}(\Gamma_{\\!2}A_{2})^{*}\\!/\mathbb{C}$ contains one
boundary point for each $s\in\mathcal{S}$ at which the objects in the class
$s$ become massless. This boundary point is fixed by $\textsc{tw}_{s}$ because
$\textsc{tw}_{s}$ acts by a shift on $s$. The equivariance of $m$ implies that
this point is mapped to $\overline{\mathrm{hom}}(s)$. (At first sight this
looks odd because $\sum_{n\in\mathbb{Z}}{\mathrm{Hom}}(s,s[n])=2$ is non-zero.
However we are working in an infinite-dimensional projective space and [1,
Proposition 4.5] states that

$\lim_{n\to\infty}\frac{m_{\sigma}\left(\textsc{tw}^{n}_{s}(s^{\prime})\right)}{n}=m_{\sigma}(s)\,\overline{\mathrm{hom}}(s)(s^{\prime})$

for any $s^{\prime}\in\mathcal{S}$ and stability condition $\sigma$ in which
$s$ is stable. Since $\textsc{tw}_{s}$ fixes $s$ up to a shift, this implies
that $\overline{\mathrm{hom}}(s)(s)=0$ as expected.) We conclude that

$m\colon\mathrm{Stab}^{Q}(\Gamma_{\\!2}A_{2})^{*}\\!/\mathbb{C}\to\overline{M(\Gamma_{\\!2}A_{2})}$

is a continuous embedding, restricting to a homeomorphism between the
interiors, and whose image is dense in the boundary. This accords with the
expectations of [1, Remark 4.9], and provides a modular interpretation of the
boundary points $\overline{\mathrm{hom}}(s)$ as quotient stability conditions.

##  14\. Open questions

We end with four open questions. (1) and (2) relate to the properties of
composition series in quasi-abelian categories. This technical issue plays a
role because the slices $P(\varphi)$ of a lax stability condition are quasi-
abelian but, in contrast to the situation for classical stability conditions,
need not be abelian. (3) and (4) relate to support properties for lax
stability conditions. There are various different notions in the literature,
and it would be good to understand which are equivalent, and to what extent
each has the crucial propagation property enjoyed by support for classical
stability conditions.

  1. (1)

Are there examples where the Jordan-Hölder property fails for the slices
$P(\varphi)$ of a lax stability condition? If there are then, whilst the
HNfiltrations of objects are unique, their refinements to filtrations with
stable factors would not be. Presumably this would have implications for the
wall-and-chamber structure.

  2. (2)

Let $P$ be a locally finite slicing on $\mathsf{C}$ and
$\mathsf{N}\subset\mathsf{C}$ a thick subcategory such that $P$ descends to
$\mathsf{C}/\mathsf{N}$. Is the slicing $P_{\mathsf{C}/\mathsf{N}}$ on the
quotient also locally finite? What if $P$ is the slicing of a lax pre-
stability condition with massless subcategory $\mathsf{N}$? (If $P$ is the
slicing of a lax stability condition then the support property guarantees that
$P_{\mathsf{C}/\mathsf{N}}$ is locally finite, see Proposition 4.13.)

  3. (3)

Are there examples where support propagation fails? See Definition • ‣ 6.12
and the discussion on page 6.14. This is crucial for the range of
applicability of our results.

  4. (4)

What is the relationship between the support property for a lax pre-stability
condition $\sigma$ we use and Bolognese’s notion of limiting support for a
sequence of stability conditions converging to $\sigma$? Understanding this is
key to clarifying the relationship between our partial compactification and
Bolognese’s, see §13.1.

## References

  * [1] Asilata Bapat, Anand Deopurkar, and Anthony M. Licata.  A Thurston compactification of the space of stability conditions, 2020, arXiv:2011.07908.
  * [2] Arend Bayer and Emanuele Macrì.  The space of stability conditions on the local projective plane.  Duke Math. J., 160(2):263–322, 2011.
  * [3] Arend Bayer, Emanuele Macrì, and Paolo Stellari.  The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds.  Invent. Math., 206(3):869–933, 2016.
  * [4] Grzegorz Bobiński, Christof Geiß, and Andrzej Skowroński.  Classification of discrete derived categories.  Cent. Eur. J. Math., 2(1):19–49, 2004.
  * [5] Barbara Bolognese.  A local compactification of the Bridgeland stability manifold, 2020, 2006.04189.
  * [6] Tom Bridgeland.  Stability conditions on triangulated categories.  Ann. of Math. (2), 166(2):317–345, 2007.
  * [7] Tom Bridgeland.  Stability conditions on $K3$ surfaces.  Duke Math. J., 141(2):241–291, 2008.
  * [8] Tom Bridgeland, Yu Qiu, and Tom Sutherland.  Stability conditions and the $A_{2}$ quiver.  Adv. Math., 365:107049, 33, 2020.
  * [9] Nathan Broomhead, David Pauksztello, and David Ploog.  Discrete derived categories II: the silting pairs CW complex and the stability manifold.  J. Lond. Math. Soc. (2), 93(2):273–300, 2016.
  * [10] Thomas Brüstle, Souheila Hassoun, Denis Langford, and Sunny Roy.  Reduction of exact structures.  J. Pure Appl. Algebra, 224(4):106212, 29, 2020.
  * [11] George Dimitrov, Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich.  Dynamical systems and categories.  In The influence of Solomon Lefschetz in geometry and topology, volume 621 of Contemp. Math., pages 133–170. Amer. Math. Soc., Providence, RI, 2014.
  * [12] Peter G. Doyle.  Random walk on the Speiser graph of a Riemann surface.  Bull. Amer. Math. Soc. (N.S.), 11(2):371–377, 1984.
  * [13] Peter G. Doyle and J. Laurie Snell.  Random walks and electric networks, volume 22 of Carus Mathematical Monographs.  Mathematical Association of America, Washington, DC, 1984.
  * [14] Haruhisa Enomoto.  The Jordan-Hölder property and Grothendieck monoids of exact categories.  Advances in Mathematics, 396:108–167, 2022.
  * [15] Alexandre Eremenko.  Singularities of inverse functions, 2021, arXiv: 2110.06134.
  * [16] Otto Forster.  Lectures on Riemann surfaces, volume 81 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 1991.  Translated from the 1977 German original by Bruce Gilligan, Reprint of the 1981 English translation.
  * [17] Sergei I. Gelfand and Yuri I. Manin.  Methods of homological algebra.  Springer-Verlag, Berlin, 1996.
  * [18] Alexey L. Gorodentsev, Sergej A. Kuleshov, and Alexei N. Rudakov.  $t$-stabilities and $t$-structures on triangulated categories.  Izv. Ross. Akad. Nauk Ser. Mat., 68(4):117–150, 2004.
  * [19] Dieter Happel, Idun Reiten, and Sverre O. Smalø.  Tilting in abelian categories and quasitilted algebras.  Mem. Amer. Math. Soc., 120(575):viii+ 88, 1996.
  * [20] Robin Hartshorne.  Residues and duality.  Lecture Notes in Mathematics, No. 20. Springer-Verlag, Berlin-New York, 1966.
  * [21] Shizuo Kakutani.  Two-dimensional Brownian motion and the type problem of Riemann surfaces.  Proc. Japan Acad., 21:138–140 (1949), 1945.
  * [22] Shizuo Kakutani.  Random walk and the type problem of Riemann surfaces.  In Contributions to the theory of Riemann surfaces, Annals of Mathematics Studies, no. 30, pages 95–101. Princeton University Press, Princeton, N.J., 1953.
  * [23] Bernhard Keller.  Cluster algebras and derived categories.  In Derived categories in algebraic geometry, EMS Ser. Congr. Rep., pages 123–183. Eur. Math. Soc., Zürich, 2012.
  * [24] Alastair King.  The Elephant and the Mouse. Talk in Edinburgh.  Early 2000s.
  * [25] Alastair King and Yu Qiu.  Exchange graphs and Ext quivers.  Adv. Math., 285:1106–1154, 2015.
  * [26] Steffen Koenig and Dong Yang.  Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras.  Doc. Math., 19:403–438, 2014.
  * [27] Maxim Kontsevich and Yan Soibelman.  Motivic Donaldson-Thomas invariants: summary of results.  In Mirror symmetry and tropical geometry, volume 527 of Contemp. Math., pages 55–89. Amer. Math. Soc., Providence, RI, 2010.
  * [28] Emanuele Macrì.  Stability conditions on curves.  Math. Res. Lett., 14(4):657–672, 2007.
  * [29] So Okada.  Stability manifold of $\mathbb{P}^{1}$.  J. Algebraic Geom., 15(3):487–505, 2006.
  * [30] Yu Qiu and Jon Woolf.  Contractible stability spaces and faithful braid group actions.  Geom. Topol., 22(6):3701–3760, 2018.
  * [31] Jean-Pierre Schneiders.  Quasi-abelian categories and sheaves.  Mém. Soc. Math. Fr. (N.S.), (76):vi+134, 1999.
  * [32] Richard Thomas.  Stability conditions and the braid group.  Comm. Anal. Geom., 14(1):135–161, 2006.
  * [33] Jon Woolf.  Stability conditions, torsion theories and tilting.  J. Lond. Math. Soc. (2), 82(3):663–682, 2010.
  * [34] Jon Woolf.  Some metric properties of spaces of stability conditions.  Bull. Lond. Math. Soc., 44(6):1274–1284, 2012.

Contact: |<EMAIL_ADDRESS>|<EMAIL_ADDRESS>
---|---|---
|<EMAIL_ADDRESS>|<EMAIL_ADDRESS>


# Local Plücker formulas for orthogonal groups

D. O. Degtyarev  AGHA Laboratory, Moscow Institute of Physics and Technology,
Russia<EMAIL_ADDRESS>
###### Abstract.

Local Plücker formulas relate the vector of metrics on a holomorphic curve in
the projective space, induced by the Fubini-Study metrics on projetive spaces
via Plücker embeddings of associated curves, to the corresponding vector of
curvatures. Givental noted that the vector of curvatures is expressed in terms
of the vector of metrics via Cartan matrix of type $A_{n}$ and extended this
observation from type $A_{n}$ to all nonexceptional types. In this paper we
show how local Plücker formulas for special orthogonal groups, i.e. for Cartan
matrices of type $B_{n}$ and $D_{n}$, can be obtained by reduction to the
classical $A_{n}$ case.

##  1\. Introduction

###  1.1. Classical case

Consider a holomorphic curve $S$ and a nondegenerate map
$f=f_{1}\mathrel{\mathop{\mathchar 58\relax}}S\to\mathbb{P}^{n}$. Define a
local lift of $f_{1}$ in a neighborhood $U$ of any point $p\in U\subset S$ by
a holomorphic vector-valued function
$v(z)=(v_{0}(z),\dots,v_{n}(z))\mathrel{\mathop{\mathchar
58\relax}}U\rightarrow\mathbb{C}^{n+1}$ and define the $k$ -th ($1\leqslant
k\leqslant n$):

$f_{k}\mathrel{\mathop{\mathchar 58\relax}}U\to
Gr(k,n+1)\subset\mathbb{P}\left(\bigwedge^{k}\mathbb{C}^{n+1}\right)$

by

$f_{k}(z)=[v(z)\wedge v^{\prime}(z)\ldots v^{(k)}(z)].$

Fubini-Study metric ($FS$) on projective space
$\mathbb{P}(\bigwedge^{k+1}\mathbb{C}^{n+1})$ induces Hermitian metric on the
$k$-th associated curve and the corresponding curvature forms via the Plücker
embedding of $Gr(k,n+1)$. In both cases they are $(1,1)$ \- forms on the
associated curve. We will denote by $\varphi_{k}$ the associated $(1,1)$-form
of the metric on $S$, and by $\theta_{k}$ the curvature form corresponding to
the metric $\varphi_{k}$. The classical local Plücker formulas express the
vector of curvatures $\boldsymbol{\vartheta}\mathrel{\mathop{\mathchar
58\relax}}=(\vartheta_{k})$ in terms of the vector of metrics
$\boldsymbol{\varphi}\mathrel{\mathop{\mathchar 58\relax}}=(\varphi_{k})$
using the Cartan matrix $C$ of the $A_{n}$ series in the following form

(1) $\boldsymbol{\vartheta}=C\boldsymbol{\phi}.$

###  1.2. Local Plücker formulas according to Givental

One can also construct another vector of associated $(1,1)$-forms of the
metrics induced by pullback of the Fubini-Study metrics and the corresponding
curvatures via the following morphisms of the associated curves to projective
spaces. We will follow the treatment from [1], [4].

Let $f\mathrel{\mathop{\mathchar 58\relax}}S\to F$ be a holomorphic curve on
the flag variety $F\mathrel{\mathop{\mathchar 58\relax}}=G/B$ where $G$ is a
complex semisimple Lie group of rank $n$, and $B$ is a Borel subgroup. We
denote the tangent space to the point $B\in F$ by $T_{B}F$. Note that $T_{B}F$
is canonically isomorphic to $\mathfrak{g}/\mathfrak{b}$ where $\mathfrak{g}$,
$\mathfrak{b}$ are Lie algebras of the groups $G$ and $B$ respectively.
Consider the $n$-dimensional distribution $\mathcal{N}$ on the flag variety
$F$:

(2)
$\mathcal{N}(B)=\\{x\in\mathfrak{g}|[x,[\mathfrak{b},\mathfrak{b}]]\subset\mathfrak{b}\\}/\mathfrak{b}\subset
T_{B}F.$

Let us now begin by considering the collection of line bundles
$\mathscr{L}_{1},\dots,\mathscr{L}_{n}$ on $F$ such that each
$\mathscr{L}_{i}$ corresponds to the fundamental characters $\chi_{i}$ of the
group $B$. The space $\Gamma(F,\mathscr{L}_{i})^{\vee}$ is the fundamental
representation $V_{i}$ of the group $G$. With the line bundle
$\mathscr{L}_{i}$ we can associate the projective morphism $\pi_{i}$:

(3) $\pi_{i}\mathrel{\mathop{\mathchar 58\relax}}G/B\to\mathbb{P}(V_{i})$
$[g]\mapsto[g\cdot v_{i}],$

where $v_{i}$ is highest weight vector of the representation $V_{i}$ and $g\in
G$.

The representation $V_{i}$ is irreducible, and therefore it has a Hermitian
inner product invariant with respect to the action of the compact form $K$ of
the group $G$. Note that a Hermitian inner product is uniquely defined up to
multiplication by a constant scalar factor. The Hermitian form determines the
Fubini-Study metric $FS$ on the space $\mathbb{P}(V_{i})$. The induced metric
on the original curve $S$ is denoted by $\varphi_{i}=f^{*}\pi_{i}^{*}(FS)$ ,
and we write $\theta_{i}$ for the corresponding curvature form. Givental
conjectured in [1]:

######  Conjecture 1.1.

If the holomorphic curve $f\mathrel{\mathop{\mathchar 58\relax}}S\to F$ is an
integral curve of the distribution $\mathcal{N}$, then the vector
$\boldsymbol{\theta}$ of curvature forms is expressed in terms of the vector
of metrics $\boldsymbol{\varphi}$ using the Cartan matrix $C$ of the Lie
algebra $\mathfrak{g}$:

(4) $\boldsymbol{\theta}=C\boldsymbol{\varphi}.$

###  1.3. Connection with the classical case

Consider the case when $G=SL(n+1,\mathbb{C})$. The condition that the curve
$f\mathrel{\mathop{\mathchar 58\relax}}S\to F$ is an integral curve of the
distribution $\mathcal{N}$ means that the preimage under
$\pi_{1}\mathrel{\mathop{\mathchar 58\relax}}F\rightarrow\mathbb{P}(V)$ of the
point $x\in\pi_{1}(S)$ is the flag of osculating $i$-planes of the curve
$\pi_{1}(S)$ at the point $x$, i.e. the curve $\pi_{i}(S)$ is the $i$-th
associated of $\pi_{1}(S)$.

In this case the exterior power $\bigwedge^{i}V$ of the standard
representation of $G$ is the fundamental representation. The projective
morphism $\pi_{i}$ associated with the line bundle $\mathscr{L}_{i}$ provides
the Plücker embedding of the $i$-th associated curve on $Gr(i,n+1)$ in
$\mathbb{P}(V_{i})$.

However Plücker embeddings of associated curves on Grassmannians do not always
coincide with embeddings into the projectivization of fundamental
representations. In particular this can be seen in the example of the special
orthogonal groups.

Positselski proved the conjecture 1.1 in [4] using the general technique for
line bundles over flag varieties. At the same time in [1], in addition to the
general formulation of the conjecture, Givental proposed the proof of the
local Plücker formulas for symplectic group $Sp(2n)$ based on the reduction of
the problem to the classical formulas for the group $SU(2n)$.

However, the question of the geometric interpretation in the case of a special
orthogonal group has remained unanswered. In this paper we prove the 1.1 for
orthogonal groups, i.e. for Cartan matrices of the series $B_{n}$ and $D_{n}$.
Our proof is based on the reduction to the classical case.

### Acknowledgements

The reported study was funded by RFBR and CNRS, project number 21-51-15000. I
am grateful to Leonid Positselski who introduced me to the problem some time
ago.

##  2\. Local Plücker formulas for the group $SO(2n,\mathbb{C})$

Let us begin by considering the case when the group $G=SO(2n,\mathbb{C})$ with
the Lie algebra $\mathfrak{g}=\mathfrak{so}(2n,\mathbb{B})$. Given a
$2n$-dimensional vector space $V$ over $\mathbb{C}$ equipped with a
nondegenerate symmetric bilinear form $Q$ which is preserved by $G$. We will
choose a basis $e_{1},\dots,e_{n},f_{1},\dots,f_{n}$ for $V$ in terms of which
the quadratic form is given by

(5) $Q(e_{i},f_{i})=Q(f_{i},e_{i})=1,$ $Q(e_{i},e_{j})=Q(f_{i},f_{j})=0,$

and

$Q(e_{s},f_{k})=Q(f_{k},e_{s})=0,$

if $k\neq s$. The action of $G$ on the set of isotropic flags in $V$ with a
given set of numerical invariants (i.e., the sequence of dimensions of its
successive terms) is transitive except the case when maximal isotropic
subspace in the flag has dimension $n$. In the latter case there are exactly
two $G$-orbits with the given numerical invariants.

We will therefore consider two connected components of the variety of maximal
isotropic flags in $V$. We denote these components by $F_{+}$ and $F_{-}$ and
assume that the Borel subgroup $B$ stabilizes the standard maximal isotropic
flag $\langle e_{1}\rangle\subset\langle
e_{1},e_{2}\rangle\subset...\subset\langle e_{1},...e_{n}\rangle$ from
$F_{+}$.

In contrast with the classical case the exterior powers of the tautological
representation $V_{i}\mathrel{\mathop{\mathchar 58\relax}}=\bigwedge^{i}V$ of
$G$ are irreducible for $i=1,\ldots,n-1$. Moreover, the exterior powers
provide us the fundamental representations of the group $G$ only for
$i=1,\ldots,n-2$.

Let us denote by $C(Q)^{even}$ the subalgebra of the Clifford algebra $C(Q)$
generated by elements of even degree with respect to a
$\mathbb{Z}/2\mathbb{Z}$ grading. The remaining two fundamental
representations $V_{i}$ for $i=n-1,n$ are realized as $C(Q)^{even}$-modules.
We briefly recall the relevant constructions here. The decomposition
$V=W\oplus W^{\prime}$, where $W=\langle e_{1},e_{2},\ldots,e_{n}\rangle$ and
$W^{\prime}=\langle f_{1},f_{2},\ldots,f_{n}\rangle$, determines an
isomorphism of algebras

(6) $C(Q)\cong\text{End}(\wedge^{\bullet}W),$

and $C(Q)^{even}$ respects the splitting
$\bigwedge^{\bullet}W=\bigwedge^{even}W\oplus\bigwedge^{odd}W$. It then
follows that

(7)
$C(Q)^{even}\cong\text{End}(\wedge^{even}W)\oplus\text{End}(\wedge^{odd}W).$

Moreover we have an embedding of Lie algebras

(8) $\mathfrak{so}(2n,\mathbb{C})\subset
C(Q)^{even}\cong\mathfrak{gl}(\wedge^{even}W)\oplus\mathfrak{gl}(\wedge^{odd}W).$

These two representations $\wedge^{even}W$ and $\wedge^{odd}W$ are usually
called half-spin representation (see [3]) .

Let us denote by $E_{i,i}$ the diagonal matrix that takes $e_{i}$ to itself
and kills $e_{j}$ for $j\neq i$. With the above-mentioned choice (5), we may
take as Cartan subalgebra $\mathfrak{h}$ the subalgebra of diagonal matrices
generated by the $2n\times 2n$ matrices $H_{i}\mathrel{\mathop{\mathchar
58\relax}}=E_{i,i}-E_{i+n,i+n}$. We will correspondingly take as basis for the
dual vector space $\mathfrak{h}^{*}$ the dual basis $L_{i}$ where
$L_{i}(H_{j})=\delta_{i,j}$. For concreteness assume that the representation
$S_{+}$ corresponds to the fundamental weight
$\omega_{n-1}=\omega_{-}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}(L_{1}+\ldots+L_{n-1}-L_{n})$ and the representation
$S_{-}$ corresponds to the fundamental weight
$\omega_{n}=\omega_{+}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}(L_{1}+\ldots+L_{n-1}+L_{n})$.

We also denote the morphisms $\pi_{n-1}$ and $\pi_{n}$ associated with the
line bundles $\mathscr{L}_{n-1}$ and $\mathscr{L}_{n}$ by $\pi_{-}$ and
$\pi_{+}$ respectively, i.e. suppose that $\pi_{\pm}\mathrel{\mathop{\mathchar
58\relax}}G/B\to\mathbb{P}(S_{\pm})$, $\pi_{\pm}\mathrel{\mathop{\mathchar
58\relax}}g\mapsto[gv_{\pm}]$ , where $v_{+}$ and $v_{-}$ are highest weight
vectors of the representations $S_{+}$ and $S_{-}$.

Let $i_{+}\mathrel{\mathop{\mathchar 58\relax}}G/B\to Fl(V)$ be the embedding
that corresponds to the orbit of the standard maximal isotropic flag, i.e.
$i_{+}\mathrel{\mathop{\mathchar 58\relax}}[b]\mapsto\langle
e_{1}\rangle\subset\langle e_{1},e_{2}\rangle\subset\ldots\subset\langle
e_{1},e_{2},\ldots e_{n-1},e_{n}\rangle$. Define the analogous embedding that
corresponds to the another orbit $i_{-}\mathrel{\mathop{\mathchar
58\relax}}G/B\to Fl(V)$, $[b]\mapsto\langle e_{1}\rangle\subset\langle
e_{1},e_{2}\rangle\subset\ldots\subset\langle e_{1},e_{2},\ldots
e_{n-1},f_{n}\rangle$. Denote the composition of $i_{\pm}$, projection to
$Gr(n,2n)$ and the Plücker embedding by $p_{n-1}$:

(9) $p_{n-1}\mathrel{\mathop{\mathchar 58\relax}}G/B\rightarrow
Fl(V)\rightarrow Gr(n-1,2n)\rightarrow\mathbb{P}(\wedge^{n-1}V)$

For the future reference we prove the following lemma.

######  Lemma 1.

Let $\sigma\mathrel{\mathop{\mathchar
58\relax}}\mathbb{P}(S_{+})\times\mathbb{P}(S_{-})\to\mathbb{P}(S_{+}\otimes
S_{-})$ be the Segre embedding. Then there is a natural linear embedding
$e\mathrel{\mathop{\mathchar
58\relax}}\mathbb{P}(\wedge^{n-1}V)\to\mathbb{P}(S_{+}\otimes S_{-})$ such
that the following diagram is commutative:

(10)
${\mathbb{P}(S_{+})\times\mathbb{P}(S_{-})}$${G/B}$${\mathbb{P}(S_{+}\otimes
S_{-})}$
${\mathbb{P}(\wedge^{n-1}V)}$$(\pi_{+},\pi_{-})\,\,\,$$p_{n-1}\,\,$$\sigma$$e$

###### Proof.

Note that $\wedge^{n-1}V$ is the irreducible representation of $\mathfrak{g}$
with the highest weight $\omega_{+}+\omega_{-}$. The complete decomposition
into irreducible representations of $S_{+}\otimes S_{-}$ looks like

(11) $S_{+}\otimes S_{-}=\wedge^{n-1}V\bigoplus\limits_{j>0}V_{n-2j-1},$

where $V_{i}$ is the irreducible representation of $\mathfrak{g}$ with the
highest weight $\omega_{i}=L_{1}+L_{2}+\ldots+L_{n-2j-1}$. Note that the
vector $v_{+}\otimes v_{-}$ has weight $\omega_{n}+\omega_{n-1}$ and therefore
it is the highest weight vector of the first summand in (11). This completes
the proof of the lemma.

∎

The projective space $\mathbb{P}(S_{+}\otimes S_{-})$ admits a $K$ -invariant
Fubini - Study metric induced by the $K$ -invariant Hermitian inner products
on the spaces $S_{+}$ and $S_{-}$. We denote the corresponding $(1,1)$ \- form
of the metric on the space $\mathbb{P}(S_{+}\otimes S_{-})$ by $FS$. The
following standard properties of $(1,1)$-forms of the metrics will be needed
later.

######  Lemma 2.

Let $N,M$ be complex manifolds, $f\mathrel{\mathop{\mathchar 58\relax}}N\to M$
a holomorphic map such that the induced map $f_{*}\mathrel{\mathop{\mathchar
58\relax}}T^{\prime}_{z}(N)\to T^{\prime}_{f(z)}(M)$ is injective for all
$z\in N$, then the associated (1,1) -form of the induced metric on $N$ is the
pullback of the associated (1,1) -form of the metric on $M$.

######  Lemma 3.

Let $\mathbb{P}^{n}$ and $\mathbb{P}^{m}$ be projective spaces with
homogeneous coordinates $[z_{0},\ldots,z_{n}]$ and $[w_{0},\ldots,w_{m}]$
respectively, equipped with the standard Fubini-Study metrics, $\sigma$ is the
Segre embedding $\sigma\mathrel{\mathop{\mathchar
58\relax}}\mathbb{P}^{n}\times\mathbb{P}^{m}\to\mathbb{P}^{N}$

(12)
$\sigma([z_{0},\ldots,z_{n}],[w_{0},\ldots,w_{m}]=[(z_{i}w_{j})_{0\leqslant
i\leqslant n,0\leqslant j\leqslant m}].$

Denote by $\alpha\mathrel{\mathop{\mathchar
58\relax}}\mathbb{P}^{n}\times\mathbb{P}^{m}\to\mathbb{P}^{n}$ and
$\beta\mathrel{\mathop{\mathchar
58\relax}}\mathbb{P}^{n}\times\mathbb{P}^{m}\to\mathbb{P}^{m}$ the projections
onto the factors, then $\sigma$ is a holomorphic isometry, that is

(13) $\sigma^{*}FS=\alpha^{*}FS+\beta^{*}FS$

We shall need the following fact.

######  Lemma 4.

Let $\varphi_{+}$ and $\varphi_{-}$ denote the pullback of the $FS$ metrics on
the spaces $\mathbb{P}(S_{+})$ and $\mathbb{P}(S_{-})$ via morphims
$\pi_{+}\circ f$ and $\pi_{-}\circ f$ respectively and let $\phi_{n-1}$ be the
metric on the curve $S$ induced by pullback of the metric $FS$ on the space
$\mathbb{P}(\wedge^{n-1}V)$ via $p_{n-1}\circ f$. Then the following equality
holds

(14) $\varphi_{+}+\varphi_{-}=\phi_{n-1}.$

###### Proof.

The commutative diagram (10) shows that on the one hand

(15) $(e\circ p_{n-1}\circ f)^{*}FS=(p_{n-1}\circ f)^{*}FS=\phi_{n-1}.$

On the other hand, it follows from the above lemmas that the following
equalities hold up

(16) $(\sigma\circ(\pi_{+},\pi_{-})\circ f)^{*}FS=((\pi_{+},\pi_{-})\circ
f)^{*}(\alpha^{*}FS+\beta^{*}FS)=$ $=(\pi_{+}\circ f)^{*}FS+(\pi_{-}\circ
f)^{*}FS=\varphi_{+}+\varphi_{-}.$

To prove the lemma it is sufficient to note that the left-hand sides of (15)
and (16) coincide. Therefore the lemma is proven. ∎

Consider the morphisms $p_{\pm}$:

(17) $p_{\pm}\mathrel{\mathop{\mathchar 58\relax}}G/B\to Fl(V)\to
Gr(n,2n)\subset\mathbb{P}(\wedge^{n}\mathbb{C}^{2n})$

which are compositions of the embeddings $i_{\pm}$, projections to $Gr(n,2n)$
and the Plücker embedding. We shall need the following properties of the
morphisms $p_{\pm}$.

######  Lemma 5.

Let $v_{\pm}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{P}(S_{\pm})\to\mathbb{P}(Sym^{2}S_{\pm})$ be the quadratic
Veronese embeddings, then there exist natural linear embeddings
$e_{\pm}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{P}(\wedge^{n}V_{\pm})\to\mathbb{P}(\wedge^{n}Sym^{2}S_{\pm})$
such that the following diagram is commutative

(18) ${\mathbb{P}(S_{\pm})}$${G/B}$${\mathbb{P}(\mathrm{Sym}^{2}S_{\pm})}$
${\mathbb{P}(\wedge^{n}V_{\pm})}$$\pi_{\pm}\,$$v_{\pm}\,\,$ $e_{\pm}$$p_{\pm}$

###### Proof.

The representation $\wedge^{n}V$ is no longer an irreducible representation of
$\mathfrak{g}$ as in the classical case. It decomposes into a direct sum of
the irreducible representations $\wedge^{n}V_{-}$ and $\wedge^{n}V_{+}$ with
highest weights $2\omega_{-}$ and $2\omega_{+}$ respectively.

Note also that there is the following direct sum decomposition of
$Sym^{2}S_{\pm}$ into irreducible representations of $\mathfrak{g}$:

(19)
$\mathrm{Sym}^{2}S_{\pm}=\wedge^{n}V_{\pm}\oplus\bigoplus\limits_{j>0}\wedge^{n-4j}V$

Since the vectors $v_{+}\otimes v_{+}$ and $v_{-}\otimes v_{-}$ are vectors of
highest weight $2\omega_{+}$ and $2\omega_{-}$ respectively, we get the
natural linear embeddings $e_{\pm}$. ∎

######  Remark 2.1.

The morphisms $\pi_{+}$ and $\pi_{-}$ decompose into a composition as follows

(20) $\pi_{\pm}\mathrel{\mathop{\mathchar 58\relax}}G/B\to
G/P_{\pm}\xrightarrow{spin}\mathbb{P}(S_{\pm}),$

Note that $\mathrm{Pic}(G/P_{\pm})\simeq\mathbb{Z}$ due to the fact that
$P_{\pm}$ is a maximal parabolic subgroup, and the positive generator of the
group corresponds to the very ample bundle $\mathcal{O}_{\mathbb{S}_{\pm}}(1)$
where $\mathbb{S}_{+}$ and $\mathbb{S}_{+}$ are spinor varieties. The
embeddings $spin_{\pm}$ are associated to these generators.

######  Lemma 6.

Let $\phi_{+}$ and $\phi_{-}$ denote the pullback of $FS$ metrics from the
spaces $\mathbb{P}(\wedge^{n}V_{+})$ and $\mathbb{P}(\wedge^{n}V_{-})$ via
morphims $p_{+}\circ f$ and $p_{-}\circ f$ respectively and let $\phi_{n-1}$
be the metric on the curve $S$ induced by pullback of metrics $FS$ from the
spaces $\mathbb{P}(\wedge^{n-1}V_{+})$ and $\mathbb{P}(\wedge^{n-1}V_{-})$ via
morphims $p_{+}\circ f$ and $p_{-}\circ f$. Then the following equalities hold
$\phi_{+}=2\varphi_{+}$ and $\phi_{-}=2\varphi_{-}$.

###### Proof.

In order to prove the above statement we need to return to the commutative
diagram (18). The spaces $\mathbb{P}(Sym^{2}S_{+})$ and
$\mathbb{P}(Sym^{2}S_{-})$ have the Fubini-Study metric induced by the
Hermitian scalar product on the spaces $S_{+}\otimes S_{+}$ and $S_{-}\otimes
S_{-}$ respectively.

Note that, the Segre embedding $\sigma_{+}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{P}(S_{+})\to\mathbb{P}(S_{+}\otimes S_{+})$ factors through
$\mathbb{P}(S_{+})\times\mathbb{P}(S_{+})$:

(21) $v_{+}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{P}(S_{+})\xrightarrow{\Delta}\mathbb{P}(S_{+})\times\mathbb{P}(S_{+})\xrightarrow{\sigma}\mathbb{P}(S_{+}\otimes
S_{+}),$

where $\Delta$ is the diagonal morphism, $\sigma$ – Segre embedding. We
already know by lemma 3 that $v_{+}^{*}(FS)=2FS$. The similar relation holds
for the morphism $v_{-}$. Thus we get:

(22) $2\varphi_{\pm}=(v_{\pm}\circ\pi_{\pm}\circ f)^{*}FS=(e_{\pm}\circ
p_{\pm}\circ f)^{*}FS=\phi_{\pm}$

∎

Let $\theta(\mathscr{L})$ denote the curvature of the $K$ \- invariant metric
in an arbitrary bundle $\mathscr{L}$ with the action of $G$ on $G/B$. Let $l$
be a local section of $\mathscr{L}$, then the following formula holds:

(23) $\theta(\mathscr{L})=-\partial\overline{\partial}\mathrm{ln}\|l\|.$

We slightly abuse notation and take certainly a standard candidate for Cartan
subalgebra of $\mathfrak{sl}(2n,\mathbb{C})$, namely the subalgebra of
diagonal matrices with zero trace and write $L_{i}$ for the element of the
dual basis such that $L_{i}(E_{j,j})=\delta_{i,j}$.

######  Lemma 7.

Suppose $\mathcal{N}_{\pm}$ are line bundles on the projective variety $G/B$
corresponding to the simple roots $\lambda_{\pm}=L_{n-1}\pm L_{n}$ of
$\mathfrak{so}(2n,\mathbb{C})$, and $\mathcal{N}_{n-1}$ is the line bundle on
the full flag varietiy $Fl(V)$ corresponding to the simple root
$\lambda_{n-1}=L_{n-1}-L_{n}$ of $\mathfrak{sl}(2n,\mathbb{C})$. Then there
are following identities:

$i_{\pm}^{*}(\theta(\mathscr{N}_{n-1})=\theta(\mathscr{N}_{\mp})$

Classical Plücker formulas for the group $SL(2n,\mathbb{C})$ give the
following relation on the curve $S$:

(24) $\boldsymbol{\vartheta}=C\boldsymbol{\phi},$

where $C$ is a Cartan matrix of type $A_{2n-1}$. We notice that relation (24)
descends to the analogous relation on $Fl(V)$:

(25) $\widehat{\boldsymbol{\vartheta}}=C\widehat{\boldsymbol{\phi}}$

where $\widehat{\boldsymbol{\vartheta}}$ is the vector of $(1,1)$-forms
$\widehat{\vartheta}_{i}\mathrel{\mathop{\mathchar
58\relax}}=\theta(\mathcal{N}_{i})$ on $Fl(V)$, $\mathcal{N}_{i}$ is the line
bundle on $Fl(V)$ corresponding to the simple root $\lambda_{i}=L_{i}-L_{i+1}$
of $\mathfrak{sl}(2n,\mathbb{C})$, $\widehat{\boldsymbol{\phi}}$ – vector of
$(1,1)$-forms with $\widehat{\phi}_{i}\mathrel{\mathop{\mathchar
58\relax}}=\widehat{p}_{i}^{*}(FS)$ and
$\widehat{p}_{i}\mathrel{\mathop{\mathchar
58\relax}}Fl(V)\rightarrow\mathbb{P}(\wedge^{i}V)$ is the standard projection.

######  Theorem 2.2.

The local Plücker formulas for the group $SO(2n,\mathbb{C})$ are obtained by
taking pullback of the classical Plücker formulas on $Fl(V)$ via the maps
$i_{\pm}\circ f$ of the first $n-1$ components of the vector
$\widehat{\boldsymbol{\vartheta}}$.

###### Proof.

We need only deal with the first $n-1$ components of the vector
$\widehat{\boldsymbol{\vartheta}}$. The first $n-3$ components of the vector
$\widehat{\boldsymbol{\vartheta}}$ remain unchanged as the pullback is taken
with respect to morphisms $i_{\pm}$. Note that the result does not depend on
the choice of embedding $i_{+}$ or $i_{-}$.

Consider the $(n-2)$-th row of the matrix relation (25):

(26)
$\widehat{\vartheta}_{n-2}=-\widehat{\phi}_{n-3}+2\widehat{\phi}_{n-2}-\widehat{\phi}_{n-1}.$

Taking into the account formulas (14) and applying pullback to the right- and
left-hand sides we have

(27) $\theta_{n-2}=(i_{+}\circ f)^{*}\widehat{\vartheta}_{n-2}=-(i_{+}\circ
f)^{*}\widehat{\phi}_{n-3}+(i_{+}\circ f)^{*}2\widehat{\phi}_{n-2}-(i_{+}\circ
f)^{*}\widehat{\phi}_{n-1}=$
$=-\varphi_{n-3}+2\varphi_{n-2}-\varphi_{+}-\varphi_{-}.$

Next we consider the $(n-1)$-th component of the curvature vector
$\widehat{\boldsymbol{\vartheta}}$ and the corresponding row of the matrix
equality (25):

(28)
$\widehat{\vartheta}_{n-1}=-\widehat{\phi}_{n-2}+2\widehat{\phi}_{n-1}-\widehat{\phi}_{n}.$

In view of Lemma 4 and Lemma 6 we observe that pullback of the right- and
left-hand sides of the equality (28) via the morphism $i_{+}\circ f$ yields
the desiring equality for the one of the last two rows of Plücker relations
for $SO(2n,\mathbb{C})$:

(29) $\theta_{-}=(i_{+}\circ f)^{*}(\widehat{\vartheta}_{n-1})=-(i_{+}\circ
f)^{*}(\widehat{\phi}_{n-2})+(i_{+}\circ
f)^{*}(2\widehat{\phi}_{n-1})-(i_{+}\circ f)^{*}(\widehat{\phi}_{n})=$
$=-\varphi_{n-2}+2\varphi_{+}+2\varphi_{-}-2\varphi_{+}=-\varphi_{n-2}+2\varphi_{-}.$

Then the same reasoning as before shows that the remaining row of Plücker
formulas for $SO(2n,\mathbb{C})$ is given by:

(30) $\theta_{+}=(i\circ f)^{*}_{-}(\widehat{\vartheta}_{n-1})=-(i\circ
f)^{*}_{-}(\widehat{\phi}_{n-2})+(i\circ
f)^{*}_{-}(2\widehat{\phi}_{n-1})-(i\circ f)^{*}_{-}(\widehat{\phi}_{n})=$
$=-\omega_{n-2}+2\omega_{+}+2\omega_{-}-2\omega_{-}=-\omega_{n-2}+2\omega_{+}.$

With matrix notation we get the full collection of Plücker formulas on the
curve $S$ for $SO(2n,\mathbb{C})$:

(31) $\boldsymbol{\theta}=A\boldsymbol{\varphi},$

where $A$ is a Cartan matrix of type $D_{n}$. ∎

##  3\. Local Plücker formulas for the group $SO(2n+1,\mathbb{C})$

Consider the case $G=SO(2n+1,\mathbb{C})$. Let $V$ be a complex vector space
of dimension $2n+1$ with a nondegenerate symmetric bilinear form $Q$ which is
preserved by $G$. In this case the exterior power
$V_{i}\mathrel{\mathop{\mathchar 58\relax}}=\wedge^{i}V$ of the standard
representation $V$ of $\mathfrak{so}(2n+1,\mathbb{C})$ is the irreducible
representation with highest weight $L_{1}+\ldots+L_{i}$ for $i=1,\dots n$.
Exterior powers of standard representation provide us the fundamental
representations of the group $G$ only for $i=1,\dots,n-1$. The remaining
fundamental representations $V_{n}$ with highest weight
$\omega_{n}\mathrel{\mathop{\mathchar 58\relax}}=(L_{1}+\ldots+L_{n})/2$ is
called the spin representation and is realized on $S\mathrel{\mathop{\mathchar
58\relax}}=\wedge^{\bullet}W$ where $W\subset V$ is the $n$ -dimensional
isotropic subspace. As in the case of even orthogonal Lie algebras, we can
consider the projective morphism associated with line bundle $\mathscr{L}_{i}$
on $G/B$ corresponding to the fundamental weight $\omega_{i}$.

We immediately get the first $(n-2)$-th rows of Plücker formulas for
$SO(2n+1,\mathbb{C})$. We briefly show that two remaining formulas for
$i=n-1,n$ can be reconstructed from classical formulas.

Note that for the odd case a lemma similar to the lemma 5 is true.

######  Lemma 8.

Let $v\mathrel{\mathop{\mathchar
58\relax}}\mathbb{P}(V_{n})\to\mathbb{P}(V_{n}\otimes V_{n}))$ be the
quadratic Veronese embedding and let $p_{n}\mathrel{\mathop{\mathchar
58\relax}}G/B\rightarrow\mathbb{P}(\wedge^{n}V)$ be the the composition of the
standard projection to Grassmannian with Plücker embedding, then there exists
a natural linear embedding $e\mathrel{\mathop{\mathchar
58\relax}}\mathbb{P}(\wedge^{n}V)\to\mathbb{P}(V_{n}\otimes V_{n})$ such that
the following diagram is commutative

(32) ${\mathbb{P}(V_{n})}$${G/B}$${\mathbb{P}(V_{n}\otimes V_{n})}$
${\mathbb{P}(\wedge^{n}V)}$$\pi_{n}\,$$v\,\,$$e\,$$p_{n}$

###### Proof.

The representation $V_{n}\otimes V_{n}$ of $\mathfrak{so}(2n+1,\mathbb{C})$
decomposes into a direct sum

(33) $V_{n}\otimes
V_{n}=\bigwedge^{n}V\oplus\bigwedge^{n-1}V\oplus\ldots\oplus\bigwedge^{0}V.$

Let $v_{n}$ be the highest weight vector with weight $\omega_{n}$ of the
representation $V_{n}$. Next observe that $v_{n}\otimes v_{n}$ is the vector
with weight $L_{1}+\ldots+L_{n}$, i.e. $v_{n}\otimes v_{n}$ is the highest
weight vector of the irreducible representation $\bigwedge^{n}V$. The lemma is
an immediate consequence of this fact. ∎

The technical statements in the odd dimensional case can be worked out as for
an even orthogonal groups.

######  Lemma 9.

Let $\varphi_{n}$ denote the pullback of the $FS$ metric on the space
$\mathbb{P}(S)$ via morphims $\pi_{n}\circ f$ and let $\phi_{n-1}$ be the
metric on the curve $S$ induced by pullback of the metric $FS$ on the space
$\mathbb{P}(\wedge^{n}V)$ via $p_{n}\circ f$. Then the following equality
holds

(34) $2\varphi_{n}=\phi_{n}.$

Combining the previous two lemmas, we can now formulate the orthogonal version
of Plücker formulas for odd orthogonal groups.

######  Theorem 3.1.

The local Plücker formulas for the group $SO(2n+1,\mathbb{C})$ are obtained by
taking pullback of the classical Plücker formulas on $Fl(V)$ via the map
$i\circ f$ of the first $n$ components of the vector
$\widehat{\boldsymbol{\vartheta}}$.

## References

  * [1] A. B. Givental/, Plücker formulae and Cartan matrices, Uspekhi Mat. Nauk, 44:3(267) (1989), 155 – 156; Russian Math. Surveys, 44:3 (1989), 193 – 194
  * [2] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, 1978.
  * [3] William Fulton and Joe Harris. Representation theory, volume 129 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. A first course, Readings in Mathematics.
  * [4] L. E. Positselski, Local Plücker formulas for a semisimple Lie group, Funktsional. Anal. i Prilozhen., 25:4 (1991), 74 – 76; Funct. Anal. Appl., 25:4 (1991), 291 – 292