${\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)}$${\mathrm{Stab}^{L}\\!\left(\mathsf{C},\alpha(\mathsf{N})\right)}$${\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)}$${\mathrm{Stab}\\!\left(\mathsf{C}/\alpha(\mathsf{N})\right)}$$\scriptstyle{\alpha}$$\scriptstyle{\mu_{\mathsf{N}}}$$\scriptstyle{\mu_{\alpha(\mathsf{N})}}$$\scriptstyle{\alpha}$

commutes for each $\alpha\in\mathrm{Aut}_{\Lambda}(\mathsf{C})$. In particular
$\mu_{\mathsf{N}}$ is equivariant for the subgroup of
$\mathrm{Aut}_{\Lambda}(\mathsf{C})$ preserving $\mathsf{N}$.

###### Proof.

The actions of $\mathrm{Aut}_{\Lambda}(\mathsf{C})$ and $G$ extend to
continuous actions on
$\mathrm{Slice}(\mathsf{C})\times{\mathrm{Hom}}(\Lambda,\mathbb{C})$ which
preserve the subsets of lax and classical stability conditions. Hence they
preserve $\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$. The equivariance of
the charge map and the properties of the $\mu_{\mathsf{N}}$ are easy to
verify. ∎

###  5.4. Neighbourhoods of strata

Fix $0<\varepsilon<1/8$ and consider the open neighbourhood

$\mathrm{B}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)=\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)\cap\bigcup_{\mathclap{\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)}}B_{\varepsilon}(\sigma)$

of $\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ in
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$. Intuitively this is the subset
where objects of $\mathsf{N}$, and only those, are close to massless. These
neighbourhoods are $\mathbb{C}$-invariant and compatible with the action of
automorphisms in the sense that
$\alpha\cdot\mathrm{B}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)=\mathrm{B}^{L}_{\varepsilon}\\!\left(\mathsf{C},\alpha(\mathsf{N})\right)$.
Note that
$\mathrm{B}^{L}_{\varepsilon}\\!\left(\mathsf{C},0\right)=\mathrm{Stab}\\!\left(\mathsf{C}\right)$,
and that
$\mathrm{B}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{C}\right)=\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$
because the closure of the $\mathbb{C}$-orbit of any
$\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$ contains a lax
stability condition with massless subcategory $\mathsf{C}$.

For $\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$,
Corollary 5.8 shows that restriction
$(Q,W)\mapsto(Q\cap\mathsf{N},W|_{\Lambda_{\mathsf{N}}})$ gives a continuous
map $\rho_{\mathsf{N}}\colon
B_{\varepsilon}(\sigma)\cap\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)\to\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$.
It clearly extends to
$\mathrm{B}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$ so that
there is a commutative diagram

${\mathrm{B}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)}$${\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)}$${{\mathrm{Hom}}(\Lambda,\mathbb{C})}$${{\mathrm{Hom}}(\Lambda_{\mathsf{N}},\mathbb{C})}$$\scriptstyle{\rho_{\mathsf{N}}}$$\scriptstyle{\mathcal{Z}}$$\scriptstyle{\mathcal{Z}}$

of continuous maps which are equivariant for the right action of $\mathbb{C}$
and for the left action of the subgroup of automorphisms preserving
$\mathsf{N}$. More generally,
$\alpha\cdot\rho_{\mathsf{N}}(\sigma)=\rho_{\alpha(\mathsf{N})}(\alpha\cdot\sigma)$
as elements of $\mathrm{Stab}^{L}\\!\left(\alpha(\mathsf{N})\right)$ for any
$\alpha\in\mathrm{Aut}_{\Lambda}(\mathsf{C})$.

##  6\. Deforming lax stability conditions

The technical heart of the theory of stability conditions is Theorem 4.1 which
governs their deformation. We cannot expect such a simple result for lax
stability conditions, but it turns out that it is still possible to deform
them in a reasonable way. The heuristic is that the massive and massless parts
of a lax stability condition deform independently.

###  6.1. Tangential, normal and fibrewise deformations

For a lax stability condition
$\sigma=(P,Z)\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ with
massless subcategory $\mathsf{N}$, the base of the charge map
$\mathcal{Z}\colon\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)\to{\mathrm{Hom}}(\Lambda,\mathbb{C})$
decomposes as

${\mathrm{Hom}}(\Lambda_{\mathsf{N}},\mathbb{C})\oplus{\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C}),$

with $Z\in{\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$. Here, as
elsewhere, we consider ${\mathrm{Hom}}(\Lambda_{\mathsf{N}},\mathbb{C})$ as a
subspace of ${\mathrm{Hom}}(\Lambda,\mathbb{C})$ using the splitting arising
from the inner product on $\Lambda\otimes\mathbb{R}$ — see §2.3. It is
geometrically appealing to distinguish three (not mutually exclusive) cases of
deformation:

  1. (1)

A _tangential deformation_ of $\sigma$ is given by varying the charge in
${\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$. Such a deformation
fixes the massless subcategory and hence stays inside
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$.

  2. (2)

A _normal deformation_ of $\sigma$ is given by varying the charge in
${\mathrm{Hom}}(\Lambda_{\mathsf{N}},\mathbb{C})$. Such a deformation moves
out of $\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ into
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)$ for some thick
subcategory $\mathsf{M}$ of $\mathsf{N}$. We think of this as deforming in a
normal slice to the stratum.

  3. (3)

A _fibrewise deformation_ of $\sigma$ takes place when the charge function is
fixed, i.e. only the slicing is deformed. In the classical setting, the charge
map
$\mathrm{Stab}\\!\left(\mathsf{C}\right)\to{\mathrm{Hom}}(\Lambda,\mathbb{C})$
has discrete fibres, so there are no non-trivial fibrewise deformations.
However for the lax stability condition $\sigma$ we can potentially vary the
slicing on $\mathsf{N}$ in a continuous way, as this is not controlled by the
charge function.

We begin with a variant of Theorem 4.1 which shows that a lax stability
condition with massless subcategory $\mathsf{N}$ can be freely deformed in the
normal direction with respect to a nearby _stability_ condition on
$\mathsf{N}$.

######  Theorem 6.1.

Suppose $\sigma=(P,Z)$ is a lax pre-stability condition on $\mathsf{C}$ with
massless subcategory $\mathsf{N}$. Then there is some $0<\varepsilon_{0}<1/8$
such that for any $0<\varepsilon<\varepsilon_{0}$ and pre-stability condition
$\tau_{\mathsf{N}}=(Q_{\mathsf{N}},W_{\mathsf{N}})$ on $\mathsf{N}$ with

  * •

$||W_{\mathsf{N}}||_{\sigma}<\sin(\pi\varepsilon)$

  * •

and $d(P_{\mathsf{N}},Q_{\mathsf{N}})<\varepsilon$

there is a unique pre-stability condition $\tau=(Q,W)$ on $\mathsf{C}$ with

  * •

charge $W=Z+W_{\mathsf{N}}$,

  * •

restricted slicing $Q\cap\mathsf{N}=Q_{\mathsf{N}}$

  * •

and $d(P,Q)<\varepsilon$.

If $\sigma\in\mathrm{Stab}^{l}\\!\left(\mathsf{C}\right)$ and
$\tau_{\mathsf{N}}\in\mathrm{Stab}\\!\left(\mathsf{N}\right)$ then
$\tau\in\mathrm{Stab}\\!\left(\mathsf{C}\right)$.

######  Remark 6.2.

This is a normal deformation at least whenever $\tau_{\mathsf{N}}$ is a
stability condition — the charge $Z$ is changed by
$W_{\mathsf{N}}\in{\mathrm{Hom}}(\Lambda_{\mathsf{N}},\mathbb{C})$, and since
the deformation $\tau$ is in $\mathrm{Stab}\\!\left(\mathsf{C}\right)$ which
has discrete fibres over ${\mathrm{Hom}}(\Lambda,\mathsf{C})$, the slicing of
$\tau$ cannot be deformed in the fibre. The construction defines a continuous
map

(5)
$\delta_{\mathsf{N}}\colon\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\times_{\mathrm{Slice}(\mathsf{N})}\mathrm{Stab}\\!\left(\mathsf{N}\right)\dashrightarrow\mathrm{Stab}\\!\left(\mathsf{C}\right)$

where the fibre product denotes the set of pairs
$(\sigma,\tau_{\mathsf{N}})\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\times\mathrm{Stab}\\!\left(\mathsf{N}\right)$
whose slicings agree on $\mathsf{N}$ and the dashed arrow indicates that the
map is only defined on the open subset where the charge $W_{\mathsf{N}}$ of
$\tau_{\mathsf{N}}$ satisfies
$||W_{\mathsf{N}}||_{\sigma}<\sin(\pi\varepsilon)$ for some suitably small
$\varepsilon>0$.

The continuity of the charge $Z+W_{\mathsf{N}}$ of
$\delta_{\mathsf{N}}(\sigma,\tau_{\mathsf{N}})$ is evident; the continuity of
the slicing follows from the fact that
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ is locally homeomorphic to
${\mathrm{Hom}}(\Lambda,\mathbb{C})$ and Corollary 4.18.

###### Proof.

Choose $0<\varepsilon_{0}<1/8$ sufficiently small so that
$P(\varphi-4\varepsilon_{0},\varphi+4\varepsilon_{0})$ is a length category
for all $\varphi\in\mathbb{R}$ and fix $0<\varepsilon<\varepsilon_{0}$. Recall
that $P(s,t)$ is a _thin subcategory_ , see [6, Definition 7.2], if
$0<t-s<1-2\varepsilon$, and that this implies that it is quasi-abelian. The
charge $W=Z+W_{\mathsf{N}}$ defines a skewed stability function, see [6,
Definition 4.4], on any thin subcategory $P(s,t)$. That is, $W\colon
K(P(s,t))\to\mathbb{C}$ is a group homomorphism taking every non-zero object
into a rotated copy of the strict half-plane $\mathbb{H}\cup\mathbb{R}_{<0}$.
To see why, suppose that $c\in P(\varphi)$ for some $\varphi\in(s,t)$. Let $A$
be a finite multi-set of stable factors of $c$ in the quasi-abelian length
category $P(\varphi)$. If, on the one hand, $a\in A$ is a massless stable
object in $\mathsf{N}$ then $a\in
Q_{\mathsf{N}}(\varphi-\varepsilon,\varphi+\varepsilon)$ and so
$W(a)=(Z+W_{\mathsf{N}})(a)=W_{\mathsf{N}}(a)$ is non-zero and therefore one
can assign the phase $\frac{1}{\pi}\arg
W_{\mathsf{N}}(a)\in(s-\varepsilon,t+\varepsilon)$ to $a$. On the other hand,
if $a\not\in\mathsf{N}$ is a massive stable factor then

$|W(a)-Z(a)|<\sin(\pi\varepsilon)|Z(a)|$

because $||W-Z||_{\sigma}=||W_{\mathsf{N}}||_{\sigma}<\sin(\pi\varepsilon)$.
Therefore $W(a)\neq 0$ and differs in phase from $Z(a)$ by less than
$\varepsilon$ and again the phase of $a$ with respect to $W$ lies in
$(s-\varepsilon,t+\varepsilon)$. Since $W(c)=\sum_{a\in A}W(a)$ we conclude
that $W(c)\neq 0$ too and the phase of $c$, $\frac{1}{\pi}\arg
W(c)\in(s-\varepsilon,t+\varepsilon)$.

The remainder of the proof follows that of Theorem 4.1 in [6, §7] verbatim.
This is possible because, after the above initial step of showing that $W$
defines a skewed stability function on each thin subcategory, the charge $Z$
and the masses of objects with respect to $\sigma$ play no role in the proof,
one only uses the locally finite slicing $P$. Therefore the same argument goes
through even though $\sigma$ is lax, and we can construct a unique pre-
stability condition $\tau=(Q,W)$ with $d(P,Q)<\varepsilon$.

By Lemma 5.6 the slicing $Q$ restricts to a slicing $Q\cap\mathsf{N}$ on
$\mathsf{N}$ with $d(P_{\mathsf{N}},Q\cap\mathsf{N})<\varepsilon$. Since
$\sigma$ has massless subcategory $\mathsf{N}$ we know
$Z\in{\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$ so that
$W|_{\Lambda_{\mathsf{N}}}=W_{\mathsf{N}}$. Therefore $\tau=(Q,W)$ restricts
to a pre-stability condition on $\mathsf{N}$ with charge $W_{\mathsf{N}}$ and
slicing within distance $2\varepsilon$ of $Q_{\mathsf{N}}$. By Corollary 4.18
it follows that the restriction is $\tau_{\mathsf{N}}$, i.e. that
$Q\cap\mathsf{N}=Q_{\mathsf{N}}$ as claimed.

Finally, we must verify that $\tau$ satisfies the support property when
$\sigma$ and $\tau_{\mathsf{N}}$ do. We may assume that $\sigma$ and
$\tau_{\mathsf{N}}$ satisfy $K$-support for the same constant $K$. Suppose
$b\in Q(\varphi)$ and let $S$ be a multi-set of its $\sigma$-stable factors.
Since $d(P,Q)<\varepsilon$ these factors lie in
$P(\varphi-\varepsilon,\varphi+\varepsilon)\subset
Q(\varphi-2\varepsilon,\varphi+2\varepsilon)$. Therefore

$m_{\tau}(b)=|W(b)|\geq\cos(2\pi\varepsilon)\sum_{s\in S}|W(s)|.$

We consider massive factors in $S-\mathsf{N}$ and massless ones in
$S\cap\mathsf{N}$ separately. For a massive $\sigma$-stable factor $s$ the
triangle inequality in $\mathbb{C}$, the norm estimate
$||W-Z||_{\sigma}<\sin(\pi\varepsilon)$, and support for $\sigma$ imply that

$|W(s)|\geq|Z(s)|-|Z(s)-W(s)|>(1-\sin(\pi\varepsilon))|Z(s)|\geq\frac{1-\sin(\pi\varepsilon)}{K}||v(s)||.$

For a massless factor $s$ in $S\cap\mathsf{N}$ consider the set $T$ of its
$\tau$-semistable factors. Since $d(P,Q)<\varepsilon$ these lie in
$Q(\psi-\varepsilon,\psi+\varepsilon)$ where $s\in P(\psi)$. Moreover, since
the slicing $Q$ restricts to the slicing $Q_{\mathsf{N}}$ on $\mathsf{N}$ we
know that each $t\in T$ is a $\tau_{\mathsf{N}}$-semistable object in
$\mathsf{N}$, and in particular that
$W(t)=Z(t)+W_{\mathsf{N}}(t)=W_{\mathsf{N}}(t)$. Together with support for
$\tau_{\mathsf{N}}$ and the triangle inequality for the norm $||\cdot||$ these
observations yield the inequality

$|W(s)|\geq\cos(2\pi\varepsilon)\sum_{t\in
T}|W(t)|=\cos(2\pi\varepsilon)\sum_{t\in
T}|W_{\mathsf{N}}(t)|\geq\frac{\cos(2\pi\varepsilon)}{K}\sum_{t\in
T}||v(t)||\geq\frac{\cos(2\pi\varepsilon)}{K}||v(s)||.$

Combining these estimates for the massive and massless factors of $b$, and
using the triangle inequality for the norm again, gives

$m_{\tau}(b)\geq\frac{1}{L}\sum_{s\in S}||v(s)||\geq\frac{1}{L}||v(b)||$

where $L=K\max\\{1/(1-\sin(\pi\varepsilon)),1/\cos(\pi\varepsilon)\\}$.

Hence $\tau$ satisfies the support property and so is a stability condition in
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$. ∎

######  Example 6.3.

Let $\mathsf{C}=\mathsf{D}^{b}(\mathbb{P}^{1})$ and
$\Lambda=K(\mathbb{P}^{1})\cong\mathbb{Z}^{2}$ with basis
$[\mathcal{O}],[\mathcal{O}_{x}]$. The inner product is chosen so that this
basis is orthonormal. Let $\sigma=(P,Z)$ be the lax stability condition with
charge $Z(\mathcal{O})=0$, $Z(\mathcal{O}_{x})=-1$ and slicing $P=P_{b}$ from
Example 3.14, i.e.
$P(1)={\langle\,\mathcal{O}_{x},\mathcal{O}(n),\mathcal{O}(-n)[1]:x\in\mathbb{P}^{1},n\in\mathbb{N}_{>0}\,\rangle}$
and $P(1/2)={\langle\,\mathcal{O}\,\rangle}$. The massless subcategory
$\mathsf{N}=\mathsf{thick}(\mathcal{O})$ and the massive stable objects are,
up to shifts, the skyscrapers $\mathcal{O}_{x}$ for $x\in\mathbb{P}^{1}$, the
line bundle $\mathcal{O}(1)$ and the shifted line bundle $\mathcal{O}(-1)[1]$.
This lax stability condition can be deformed to a classical one using the
previous result. Let $\tau_{\mathsf{N}}=(Q_{\mathsf{N}},W_{\mathsf{N}})$ where
$Q_{\mathsf{N}}=P_{\mathsf{N}}$ is the restricted slicing with
$Q_{\mathsf{N}}(1/2)={\langle\mathcal{O}\rangle}$ and
$W_{\mathsf{N}}(\mathcal{O})=ri$ for some $r>0$. Considered as a charge in
${\mathrm{Hom}}(\Lambda,\mathbb{C})$ via the orthogonal splitting we also have
$W_{\mathsf{N}}(\mathcal{O}_{x})=0$. Therefore

$||W_{\mathsf{N}}||_{\sigma}=\sup\left\\{\frac{|W_{\mathsf{N}}(c)|}{|Z(c)|}:c\
\text{massive $\sigma$-stable}\right\\}=\sup\left\\{\frac{r}{n}:n\neq
0\right\\}=r$

and the conditions of Theorem 6.1 are satisfied. The deformed stability
condition $\tau=(Q,W)$ has charge $W(\mathcal{O}_{x})=-1$,
$W(\mathcal{O}(n))=-n+ri$ and heart
$Q(0,1]=\operatorname{\mathsf{coh}}(\mathbb{P}^{1})$. Note that
$d(P,Q)=\arctan(r)$ so that the slicing converges to $P$ as $r\to 0$.

######  Example 6.4.

In contrast there are lax pre-stability conditions which cannot be deformed to
classical ones. Again on $\mathsf{C}=\mathsf{D}^{b}(\mathbb{P}^{1})$, let
$\tau=(Q,W)$ be defined by the charge $W(\mathcal{O}_{x})=0$,
$W(\mathcal{O})=i$ and slicing $Q=P_{t}$ from Example 3.14, i.e.
$Q(1/2)={\langle\,\mathcal{O}(n):n\in\mathbb{Z}\,\rangle}$ and
$Q(1)={\langle\,\mathcal{O}_{x}:x\in\mathbb{P}^{1}\,\rangle}$. The massless
subcategory $\mathsf{N}=\mathsf{thick}(\mathcal{O}_{x}:x\in\mathbb{P}^{1})$
and the massive stable objects are, up to shifts, the line bundles
$\mathcal{O}(n)$ for $n\in\mathbb{Z}$. Let
$\tau_{\mathsf{N}}=(Q_{\mathsf{N}},W_{\mathsf{N}})$ where
$W_{\mathsf{N}}(\mathcal{O}_{x})=w$ for some $0\neq w\in\mathbb{C}$ and
$Q_{\mathsf{N}}$ is a compatible slicing. Considering $W_{\mathsf{N}}$ as a
charge in ${\mathrm{Hom}}(\Lambda,\mathbb{C})$ via the orthogonal splitting we
also have $W_{\mathsf{N}}(\mathcal{O})=0$. Therefore

$||W_{\mathsf{N}}||_{\tau}=\sup\left\\{\frac{|W_{\mathsf{N}}(c)|}{|W(c)|}:c\text{
massive $\tau$-stable}\right\\}=\sup\\{|n||w|:n\in\mathbb{Z}\\}=\infty.$

Thus the conditions of Theorem 6.1 are not satisfied. Indeed we have already
seen in Examples 4.12 and 4.17 that $\tau$ is not a lax stability condition,
so not in $\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$, because it does not
satisfy the support property as well as not being in the closure of
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$.

Theorem 6.1 leads to the following inductive criterion for recognising when a
lax stability condition is in $\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$.

######  Corollary 6.5.

Suppose $\sigma\in\mathrm{Stab}^{l}\\!\left(\mathsf{C}\right)$ is a lax
stability condition with massless subcategory $\mathsf{N}$. Then
$\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)\iff\rho_{\mathsf{N}}(\sigma)\in\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$.

###### Proof.

If $\sigma=(P,Z)\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$
then $\rho_{\mathsf{N}}(\sigma)\in\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$
by Lemma 5.6.

Conversely, if
$\rho_{\mathsf{N}}(\sigma)=(P_{\mathsf{N}},0)\in\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$
then we can choose a sequence of stability conditions
$(Q_{n},W_{n})\in\mathrm{Stab}\\!\left(\mathsf{N}\right)$ converging to
$(P_{\mathsf{N}},0)$ in the sense that $d(P_{\mathsf{N}},Q_{n})\to 0$ and
$W_{n}\to 0$ in the operator norm on
${\mathrm{Hom}}(\Lambda_{\mathsf{N}},\mathbb{C})$. It follows that $W_{n}\to
0$ in the operator norm on ${\mathrm{Hom}}(\Lambda,\mathbb{C})$, where as
usual we consider $W_{n}\in{\mathrm{Hom}}(\Lambda,\mathbb{C})$ via the fixed
splitting
${\mathrm{Hom}}(\Lambda_{\mathsf{N}},\mathbb{C})\hookrightarrow{\mathrm{Hom}}(\Lambda,\mathbb{C})$.
Since $\sigma$ satisfies the support property this implies
$||W_{n}||_{\sigma}\to 0$. Therefore we can apply Theorem 6.1 to lift this
sequence uniquely to a sequence of stability conditions
$(P_{n},Z+W_{n})\in\mathrm{Stab}\\!\left(\mathsf{C}\right)$ converging to
$(P,Z)=\sigma$. Hence $\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$
as claimed. ∎

The above criterion is tautological, and useless, when every object is
massless i.e. when $\mathsf{N}=\mathsf{C}$. In that case we have the following
result.

######  Lemma 6.6.

Suppose $P$ is a slicing. Then

$(P,0)\in\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)\iff
P\in\overline{\\{Q:(Q,W)\in\mathrm{Stab}\\!\left(\mathsf{C}\right)\\}}.$

###### Proof.

Suppose $(P,0)\in\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$. Then there is a
sequence $(P_{n},Z_{n})$ of stability conditions in
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ converging to it, in particular with
$P_{n}\to P$.

Conversely, suppose $P_{n}\to P$ and there exists a charge $Z_{n}$ such that
$(P_{n},Z_{n})\in\mathrm{Stab}\\!\left(\mathsf{C}\right)$ for each
$n\in\mathbb{N}$. Then $P$ is locally finite by Lemma 5.3. Moreover,
$(P_{n},Z_{n}/n||Z_{n}||)\in\mathrm{Stab}\\!\left(\mathsf{C}\right)$ and
converges to $(P,0)$ in
$\mathrm{Slice}(\mathsf{C})\times{\mathrm{Hom}}(\Lambda,\mathbb{C})$ as
$n\to\infty$. So $(P,0)\in\overline{\mathrm{Stab}\\!\left(\mathsf{C}\right)}$
and since the support property is automatic when all objects are massless in
fact $(P,0)\in\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$. ∎

######  Example 6.7.

Suppose $\mathsf{C}=\mathsf{D}^{b}({\mathbf{k}}A_{2})$, where $A_{2}$ is the
quiver $1\longrightarrow 2$. Define a sequence of stability conditions
$\tau_{n}=(Q_{n},W_{n})$ via $W_{n}(S_{1})=-1/n$ and $W_{n}(S_{2})=i/e^{n}$,
where $S_{1}$ and $S_{2}$ are the simple modules at $1$ and $2$ in the
standard heart, respectively. Note that the indecomposable projective module
$P_{1}$ in $\mathsf{mod}({\mathbf{k}}A_{2})$ is also $\tau_{n}$-semistable.
The limit slicing $Q=\lim_{n\to\infty}Q_{n}$ is given by
$Q(1/2)=\mathsf{add}(S_{2})$ and $Q(1)=\mathsf{add}(S_{1}\oplus P_{1})$. Note
that $Q$ is not a slicing for any (pre-)stability condition on
$\mathsf{D}^{b}({\mathbf{k}}A_{2})$ because the slice $Q(1)$ is not abelian.

We now refine Theorem 6.1 by considering deformations of a lax pre-stability
condition with massless subcategory $\mathsf{N}$ to one with massless
subcategory $\mathsf{M}\subset\mathsf{N}$. We saw in the proof of Proposition
5.10 that if $\sigma=(P,Z)$ is in the closure of
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)$ then $P$ is well-
adapted to $\mathsf{M}$. Therefore it is natural to impose this condition. The
result is weaker than Theorem 6.1 in that even if we start with lax stability
conditions on $\mathsf{C}$ and $\mathsf{N}$, we only show that the resulting
deformation is a lax pre-stability condition which satisfies support on the
quotient $\mathsf{C}/\mathsf{M}$ but not necessarily on $\mathsf{C}$; the set
of such is denoted $\mathrm{Stab}^{ls}\\!\left(\mathsf{C},\mathsf{M}\right)$
below. In the notation of Remark 6.2, we get a continuous map

(6)
$\delta_{\mathsf{N},\mathsf{M}}\colon\mathrm{Stab}^{l}\\!\left(\mathsf{C},\mathsf{N}\right)\times_{\mathrm{Slice}(\mathsf{N})}\mathrm{Stab}^{l}\\!\left(\mathsf{N},\mathsf{M}\right)\dashrightarrow\mathrm{Stab}^{ls}\\!\left(\mathsf{C},\mathsf{M}\right)$

where the domain is the set of pairs $(\sigma,\tau_{\mathsf{N}})$ whose
slicings agree on $\mathsf{N}$, and are well-adapted to the subcategory
$\mathsf{M}$. It is not clear that this is an open subset of the fibre
product. It is a natural question when the image of this map is within lax
stability conditions; this is discussed in Subsection 6.2.

######  Proposition 6.8.

Suppose $\sigma=(P,Z)$ is a lax pre-stability condition on $\mathsf{C}$ with
massless subcategory $\mathsf{N}$ such that $P$ is well-adapted to the thick
subcategory $\mathsf{M}$ of $\mathsf{N}$. Then there is some
$0<\varepsilon_{0}<1/8$ such that for any $0<\varepsilon<\varepsilon_{0}$ and
lax pre-stability condition
$\tau_{\mathsf{N}}=(Q_{\mathsf{N}},W_{\mathsf{N}})$ on $\mathsf{N}$ with

  * •

massless subcategory $\mathsf{M}$,

  * •

$||W_{\mathsf{N}}||_{\sigma}<\sin(\pi\varepsilon)$

  * •

and $d(P_{\mathsf{N}},Q_{\mathsf{N}})<\varepsilon$

there is a unique lax pre-stability condition $\tau=(Q,W)$ on $\mathsf{C}$
with

  * •

massless subcategory $\mathsf{M}$,

  * •

$W=Z+W_{\mathsf{N}}$,

  * •

restricted slicing $Q\cap\mathsf{N}=Q_{\mathsf{N}}$

  * •

and $d(P,Q)<\varepsilon$.

If in addition $\sigma\in\mathrm{Stab}^{l}\\!\left(\mathsf{C}\right)$ and
$\tau_{\mathsf{N}}\in\mathrm{Stab}^{l}\\!\left(\mathsf{N}\right)$ then
$\tau\in\mathrm{Stab}^{ls}\\!\left(\mathsf{C}\right)$.

######  Remark 6.9.

We do not know if a version of the last statement holds that includes the
closure property, i.e. whether
$\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$ and
$\tau_{\mathsf{N}}\in\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$ implies
$\tau\in\mathrm{Stab}^{LS}\\!\left(\mathsf{C}\right)$.

###### Proof.

For $\mathsf{M}=0$ this is Theorem 6.1. When $\mathsf{M}\neq 0$ the strategy
is to reduce to this case by taking the quotient by $\mathsf{M}$ and then
lifting back up to $\mathsf{C}$ using Proposition 3.10.

We first observe that $Q_{\mathsf{N}}$ is well-adapted to $\mathsf{M}$. It is
adapted to $\mathsf{M}$ by Proposition 4.4, so we just need to see that
$Q_{\mathsf{N}/\mathsf{M}}$ is locally finite. The restricted slicing
$P_{\mathsf{C}/\mathsf{M}}(\varphi)\cap\mathsf{N}/\mathsf{M}=P_{\mathsf{N}/\mathsf{M}}(\varphi)$
for each $\varphi\in\mathbb{R}$ because the proof of Proposition 3.2 shows
that each is the full subcategory of $\mathsf{N}/\mathsf{M}$ consisting of
those objects having an HNfiltration with one factor in
$P(\varphi)\cap\mathsf{N}$ and all others in $\mathsf{M}$. Hence, since
$d(P_{\mathsf{N}/\mathsf{M}},Q_{\mathsf{N}/\mathsf{M}})\leq
d(P_{\mathsf{N}},Q_{\mathsf{N}})<\varepsilon$ by Corollary 3.7, and
$P_{\mathsf{N}/\mathsf{M}}$ is a locally finite slicing, so is
$Q_{\mathsf{N}/\mathsf{M}}$. Now, by Proposition 4.13, $\tau_{\mathsf{N}}$
induces a pre-stability condition
$\mu_{\mathsf{M}}(\tau_{\mathsf{N}})=(Q_{\mathsf{N}/\mathsf{M}},W_{\mathsf{N}})$
on $\mathsf{N}/\mathsf{M}$. Moreover, since $P$ is well-adapted to
$\mathsf{M}$, Remark 4.14 shows there is a lax pre-stability condition
$\mu_{\mathsf{M}}(\sigma)=(P_{\mathsf{C}/\mathsf{M}},Z)$ on
$\mathsf{C}/\mathsf{M}$ with massless subcategory $\mathsf{N}/\mathsf{M}$. We
verify that these satisfy the conditions of Theorem 6.1.

The charge $W_{\mathsf{N}}$ is in
${\mathrm{Hom}}(\Lambda_{\mathsf{N}}/\Lambda_{\mathsf{M}},\mathbb{C})$ and the
splitting
${\mathrm{Hom}}(\Lambda_{\mathsf{N}},\mathbb{C})\hookrightarrow{\mathrm{Hom}}(\Lambda,\mathbb{C})$
restricts to one
${\mathrm{Hom}}(\Lambda_{\mathsf{N}}/\Lambda_{\mathsf{M}},\mathbb{C})\hookrightarrow{\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{M}},\mathbb{C})$.
We use this to consider $W_{\mathsf{N}}$ as an element of
${\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{M}},\mathbb{C})$. With this
identification
$||W_{\mathsf{N}}||_{\mu_{\mathsf{M}}(\sigma)}\leq||W_{\mathsf{N}}||_{\sigma}<\sin(\pi\varepsilon)$
by Lemma 4.15. Since the restricted slicing
$P_{\mathsf{C}/\mathsf{M}}(\varphi)\cap\mathsf{N}/\mathsf{M}=P_{\mathsf{N}/\mathsf{M}}(\varphi)$
for each $\varphi\in\mathbb{R}$ and
$d(P_{\mathsf{N}/\mathsf{M}},Q_{\mathsf{N}/\mathsf{M}})\leq
d(P_{\mathsf{N}},Q_{\mathsf{N}})<\varepsilon$ the conditions of Theorem 6.1
are satisfied. Applying that result we construct a pre-stability condition
$(Q_{\mathsf{C}/\mathsf{M}},W_{\mathsf{C}/\mathsf{M}})$ on
$\mathsf{C}/\mathsf{M}$ where $W_{\mathsf{C}/\mathsf{M}}=Z+W_{\mathsf{N}}$ and
$d(P_{\mathsf{C}/\mathsf{M}},Q_{\mathsf{C}/\mathsf{M}})<\varepsilon$.

Since $d(P_{\mathsf{M}},Q_{\mathsf{M}})\leq
d(P_{\mathsf{N}},Q_{\mathsf{N}})<\varepsilon$ we can use Proposition 3.10 to
glue $Q_{\mathsf{M}}$ and $Q_{\mathsf{C}/\mathsf{M}}$ to a locally finite
slicing $Q$ with $d(P,Q)<\varepsilon$ by Lemma 3.11. By Lemma 5.6 this slicing
$Q$ restricts to $\mathsf{N}$. It follows from the construction that the
restriction is the slicing glued from $Q_{\mathsf{M}}$ and
$Q_{\mathsf{N}/\mathsf{M}}$, which by uniqueness is $Q_{\mathsf{N}}$. Thus we
have constructed a lax pre-stability condition $\tau=(Q,W)$ with
$d(P,Q)<\varepsilon$, massless subcategory $\mathsf{M}$, charge
$W=Z+W_{\mathsf{N}}$ and slicing $Q$ restricting to $Q_{\mathsf{N}}$ on
$\mathsf{N}$. Corollary 4.18 implies that $\tau$ is unique with these
properties.

Finally, when $\sigma$ and $\tau_{\mathsf{N}}$ are lax stability conditions
then $\mu_{\mathsf{M}}(\sigma)$ is a lax stability condition and
$\mu_{\mathsf{M}}(\tau_{\mathsf{N}})$ a stability condition. Therefore by the
last part of Theorem 6.1 the pre-stability condition on the quotient
$\mu_{\mathsf{M}}(\tau)=(Q_{\mathsf{C}/\mathsf{M}},Z+W_{\mathsf{N}})$ is in
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{M}\right)$. Hence,
$\tau\in\mathrm{Stab}^{ls}\\!\left(\mathsf{C}\right)$ as claimed. ∎

The deformations described in this result are in general neither purely normal
nor fibrewise, but a mixture of the two: the charge $W$ is changed by
$W_{\mathsf{N}}\in{\mathrm{Hom}}(\Lambda_{\mathsf{N}},\mathbb{C})$, so for
$\mathsf{M}\neq\mathsf{N}$, a component of the deformation occurs in the
normal direction, but for $\mathsf{M}\neq 0$ the slicing on $\mathsf{N}$ is
not fully determined by $W_{\mathsf{N}}$, and the choice of $Q_{\mathsf{N}}$
determines the fibrewise deformation. When $\mathsf{M}=\mathsf{N}$ the
deformation is purely fibrewise since the charge, indeed the associated pre-
stability condition on the quotient, is fixed and only the massless slicing is
deformed.

######  Corollary 6.10.

Suppose $\sigma=(P,Z)$ is a lax pre-stability condition on $\mathsf{C}$ with
massless subcategory $\mathsf{N}$. Then there is some $\varepsilon_{0}>0$ such
that for any $0<\varepsilon<\varepsilon_{0}$ and
$Q_{\mathsf{N}}\in\mathrm{Slice}(\mathsf{N})$ with
$d(P_{\mathsf{N}},Q_{\mathsf{N}})<\varepsilon$ there is a unique lax pre-
stability condition $\tau=(Q,Z)$ on $\mathsf{C}$ with

  * •

massless subcategory $\mathsf{N}$

  * •

massless slicing $Q_{\mathsf{N}}$

  * •

and $d(P,Q)<\varepsilon$.

If $\sigma$ is a lax stability condition then we may choose
$\varepsilon_{0}=1/4$.

###### Proof.

This is the special case $\mathsf{M}=\mathsf{N}$ of Proposition 6.8, i.e.
$\tau_{\mathsf{N}}=(Q_{\mathsf{N}},0)$ except for the last statement. If
$\sigma$ is a lax stability condition then $P(I)$ is length for any interval
$I$ of length strictly less than one by Lemma 5.3. Therefore we may choose
$\varepsilon_{0}=1/4$. ∎

Finally, we consider tangential deformations where the charge is varied in
${\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$ and the massless
slicing remains fixed.

######  Proposition 6.11.

Suppose $\sigma=(P,Z)$ is a lax pre-stability condition on $\mathsf{C}$ with
massless subcategory $\mathsf{N}$. Then for any $0<\varepsilon<1/8$ and
$W\in{\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$ satisfying
$||W-Z||_{\sigma}<\sin(\pi\varepsilon)$ there is a unique lax pre-stability
condition $\tau=(Q,W)$ with $d(P,Q)<\varepsilon$ and massless slicing
$Q_{\mathsf{N}}=P_{\mathsf{N}}$. If in addition
$\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ and
$\tau\in\mathrm{Stab}^{l}\\!\left(\mathsf{C}\right)$ then
$\tau\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$.

###### Proof.

By Lemma 4.15,
$||W-Z||_{\mu_{\mathsf{N}}(\sigma)}=||W-Z||_{\sigma}<\sin(\pi\varepsilon)$
where $\mu_{\mathsf{N}}(\sigma)=(P_{\mathsf{C}/\mathsf{N}},Z)$ is the induced
stability condition in $\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)$.
Therefore, Theorem 4.1 allows us to construct a stability condition
$(Q_{\mathsf{C}/\mathsf{N}},W)$ in
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)$ with
$d(P_{\mathsf{C}/\mathsf{N}},Q_{\mathsf{C}/\mathsf{N}})<\varepsilon$.

By Proposition 3.10 the slicings $P_{\mathsf{N}}$ and
$Q_{\mathsf{C}/\mathsf{N}}$ can be glued to a locally finite slicing $Q$ on
$\mathsf{C}$. Since $Q(\varphi)\subseteq Q_{\mathsf{C}/\mathsf{N}}(\varphi)$
for all $\varphi\in\mathbb{R}$, we have thus constructed a lax pre-stability
condition $\tau=(Q,W)$ with massless subcategory $\mathsf{N}$, restricted
slicing $Q_{\mathsf{N}}=P_{\mathsf{N}}$, and such that $d(P,Q)<\varepsilon$.
Uniqueness follows from Corollary 4.18. Since $P_{\mathsf{N}}=Q_{\mathsf{N}}$
the last statement follows from Corollary 6.5. ∎

###  6.2. Support propagation

In the classical case support propagates in components of the stability space:
all nearby deformations of a stability condition are also stability
conditions, and not just pre-stability conditions. We now discuss the extent
to which this remains true for lax stability conditions.

######  Definition 6.12.

We say that _support propagates_ from

  * •

a lax stability condition
$\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ if there is
$\varepsilon>0$ such that any $\tau=(Q,W)\in B_{\varepsilon}(\sigma)$ with
$||Z-(W-W_{\mathsf{N}})||_{\sigma}<\sin(\pi\varepsilon)$ and
$\rho_{\mathsf{N}}(\tau)\in\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$ is in
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$. See Figure 1 for a schematic
illustration of the charge conditions.

  * •

a component $\Sigma$ of
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ if there is an
$\varepsilon>0$ such that support propagates from all $\sigma\in\Sigma$ with
respect to $\varepsilon$.

  * •

$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ if it propagates from
all components.

${\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$$Z$$||Z-(W-W_{\mathsf{N}})||_{\sigma}<\sin(\pi\varepsilon)$$||W-Z||_{\sigma}<\sin(\pi\varepsilon)$
Figure 1. The definition of support propagation from
$\sigma=(P,Z)\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$
imposes two conditions on the charge of $\tau=(Q,W)$. These are illustrated
schematically above. We require two conditions because $||\cdot||_{\sigma}$ is
only a _semi_ -norm. When $Z-W$ is in the subspace
${\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$ on which it
restricts to a norm the conditions coincide, and agree with the condition
$||W-Z||_{\mu_{\mathsf{N}}(\sigma)}<\sin(\pi\varepsilon)$ for the quotient
stability condition $\mu_{\mathsf{N}}(\sigma)$. In particular when
$\mathsf{N}=0$ they reduce to the usual condition in the classical case. At
the other extreme, when $\mathsf{N}=\mathsf{C}$, the conditions are vacuous
because $||\cdot||_{\sigma}=0$. Another special case which we use frequently
is when $Z-W\in{\mathrm{Hom}}(\Lambda_{\mathsf{N}},\mathbb{C})$ in which case
they again reduce to $||W-Z||_{\sigma}<\sin(\pi\varepsilon)$.

This condition on the lax stability condition $\sigma$ means that nearby lax
pre-stability conditions $\tau$ have the support property provided
$||Z-(W-W_{\mathsf{N}})||_{\sigma}<\sin(\pi\varepsilon)$ and
$\rho_{\mathsf{N}}(\tau)\in\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$. The
last condition is necessary because it is implied by $\tau\in
B_{\varepsilon}(\sigma)\cap\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$. We
illustrate this in the degenerate example below.

######  Example 6.13.

Consider the classical geometric stability condition
$\sigma_{g}=(P_{g},Z_{g})$ on $\mathsf{C}=\mathsf{D}^{b}(\mathbb{P}^{1})$ from
Examples 3.14 and 5.4. By sending all charges to zero uniformly, we obtain a
lax stability condition $\sigma=(P_{g},0)$ with massless subcategory
$\mathsf{N}=\mathsf{C}$. Consider the lax stability condition $\tau=(Q,0)$
given by $Q(\varphi)=P_{g}(\varphi)$ for $\varphi\notin\frac{1}{2}\mathbb{Z}$
and
$Q(\frac{1}{2}+\varepsilon)=P_{g}(\frac{1}{2})={\langle\mathcal{O}\rangle}$
for sufficiently small $\varepsilon>0$. Since we have not changed the phase of
any other slices, $Q$ cannot be compatible with any linear charge. Hence,
$\tau\in B_{\varepsilon}(\sigma)$ is a lax stability condition with the same
charge as $\sigma$ but
$\tau\notin\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)=\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$.

######  Example 6.14.

Support propagates in the following cases:

  1. (1)

For $\mathrm{Stab}\\!\left(\mathsf{C}\right)$, i.e. classical stability
conditions, by Theorem 4.1.

  2. (2)

For $\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{C}\right)$, i.e. for lax
stability conditions $(P,0)$ with massless subcategory
$\mathsf{N}=\mathsf{C}$. The propagation condition is tautological in this
case because $||\cdot||_{\sigma}=0$ and
$\rho_{\mathsf{N}}=\operatorname{{\mathsf{id}}}$.

  3. (3)

For $\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ where
$\mathrm{rk}\,(\Lambda_{\mathsf{N}})=1$. This result is a combination of
Theorem 6.1 (normal deformations) and Theorem 9.4 (tangential deformations).
The condition that
$\rho_{\mathsf{N}}(\tau)\in\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$ is
necessary in this case.

  4. (4)

For any component of $\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$
in the boundary of a finite type component
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$, i.e. a component in which every
heart is a length abelian category, by Corollary 10.7. In this case we take
$\Lambda=K(\mathsf{C})$.

We do not know of any examples in which support does not propagate. However,
it is entirely possible that such examples exist.

The next result is a refinement of Corollary 6.5.

######  Corollary 6.15.

Suppose support propagates from
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$. Let $\sigma$ be a
lax stability condition with massless subcategory $\mathsf{N}$ and
$\mathsf{M}\subset\mathsf{N}$ be a thick subcategory. Then

$\sigma\in\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)}\iff\rho_{\mathsf{N}}(\sigma)\in\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{N},\mathsf{M}\right)}.$

###### Proof.

If
$\sigma=(P,Z)\in\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)}$
then the continuity of $\rho_{\mathsf{N}}$ implies
$\rho_{\mathsf{N}}(\sigma)\in\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{N},\mathsf{M}\right)}$.

Now suppose that $\rho_{\mathsf{N}}(\sigma)=(P_{\mathsf{N}},0)$ is in the
closure of $\mathrm{Stab}^{L}\\!\left(\mathsf{N},\mathsf{M}\right)$. We always
have
$\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{N},\mathsf{M}\right)}\subseteq\mathrm{Stab}^{l}\\!\left(\mathsf{N}\right)\cap\overline{\mathrm{Stab}\\!\left(\mathsf{N}\right)}=\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$.
Thus $\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ by
Corollary 6.5. Next, $P_{\mathsf{N}}$ is well-adapted to $\mathsf{M}$ by the
proof of Proposition 5.10, i.e. $P_{\mathsf{N}/\mathsf{M}}$ is locally finite
and $P_{\mathsf{N}}$ is adapted to $\mathsf{M}$. Thus $P$, being well-adapted
to $\mathsf{N}$ by Proposition 4.13, is also adapted to $\mathsf{M}$ and
moreover $P_{\mathsf{C}/\mathsf{N}}$ is locally finite. As
$P_{\mathsf{C}/\mathsf{M}}$ is compatible with the pair
$(P_{\mathsf{N}/\mathsf{M}},P_{\mathsf{C}/\mathsf{N}})$, we conclude from
Lemma 3.3 that $P_{\mathsf{C}/\mathsf{M}}$ is locally finite. Hence $P$ is
actually well-adapted to $\mathsf{M}$.

Thus we can apply Proposition 6.8 to $\sigma$ and a sequence of lax stability
conditions in $\mathrm{Stab}^{L}\\!\left(\mathsf{N},\mathsf{M}\right)$
converging to $\rho_{\mathsf{N}}(\sigma)$ to construct a sequence of lax pre-
stability conditions with massless subcategory $\mathsf{M}$ which satisfy
support on the quotient $\mathsf{C}/\mathsf{M}$ and which converge to
$\sigma$. We conclude by noting that support propagation implies this is
eventually a sequence of lax stability conditions in
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)$. ∎

######  Remark 6.16.

If we do not assume support propagates then we can only conclude that $\sigma$
is in $\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ and is in the
limit in $\mathrm{Slice}(\mathsf{C})\times{\mathrm{Hom}}(\Lambda,\mathbb{C})$
of a sequence of lax pre-stability conditions with massless subcategory
$\mathsf{M}$ which satisfy support on the quotient $\mathsf{C}/\mathsf{M}$.

Definition • ‣ 6.12 asks for two properties of lax pre-stability conditions
occurring as small deformations of
$\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$: firstly the
support property and secondly the property of being in the closure of
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$. The next result shows that the
latter is automatic. Thus the terminology ‘support propagation’ is justified.

######  Theorem 6.17.

Suppose
$\sigma=(P,Z)\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$. Then
for any sufficiently small $\varepsilon>0$ the set of lax pre-stability
conditions $\tau=(Q,W)\in B_{\varepsilon}(\sigma)$ with
$||Z-(W-W_{\mathsf{N}})||_{\sigma}<\sin(\pi\varepsilon)$ and
$\rho_{\mathsf{N}}(\tau)\in\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$ is
contained in $\overline{\mathrm{Stab}\\!\left(\mathsf{C}\right)}$.

###### Proof.

Let $\sigma=(P,Z)\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ so
that $\rho_{\mathsf{N}}(\sigma)=(P_{\mathsf{N}},Z_{\mathsf{N}}=0)$. Since
$\sigma$ satisfies the support property there is some $K>0$ for which
$||v(c)||\leq K|Z(c)|$ whenever $c$ is a massive stable object. Recall from
Proposition 4.11 that this implies that $||\cdot||_{\sigma}\leq K||\cdot||$ on
${\mathrm{Hom}}(\Lambda,\mathbb{C})$.

Let $\tau=(Q,W)\in B_{\varepsilon}(\sigma)$ be a lax pre-stability condition
with $\rho_{\mathsf{N}}(\tau)\in\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$
and $||Z-(W-W_{\mathsf{N}})||_{\sigma}<\sin(\pi\varepsilon)$. Thus
$d(P_{\mathsf{N}},Q_{\mathsf{N}})\leq d(P,Q)<\varepsilon$ by Corollary 3.7 and

$||W_{\mathsf{N}}||_{\sigma}\leq||Z-W||_{\sigma}+||Z-(W-W_{\mathsf{N}})||_{\sigma}<2\sin(\pi\varepsilon)$

by the triangle inequality. To construct
$\tau^{\prime}\in\mathrm{Stab}\\!\left(\mathsf{C}\right)$ suitably close to
$\tau$ we proceed in a sequence of steps. We first deform $\sigma$ to
$\sigma^{\prime}\in\mathrm{Stab}\\!\left(\mathsf{C}\right)$ using Theorem 6.1.
We then apply Bridgeland’s deformation theorem, Theorem 4.1, to deform the
charge of $\sigma^{\prime}$ to obtain a stability condition $\tau^{\prime}$
whose charge is closer to that of $\tau$. Finally, we check that
$\tau^{\prime}$ is indeed suitably close to $\tau$.

Step 1: Deforming $\sigma$ to
$\sigma^{\prime}\in\mathrm{Stab}\\!\left(\mathsf{C}\right)$.

Since
$\rho_{\mathsf{N}}(\tau)\in\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)\subset\overline{\mathrm{Stab}\\!\left(\mathsf{N}\right)}$,
for any $\varepsilon>0$ there is
$\tau^{\prime}_{\mathsf{N}}=(Q^{\prime}_{\mathsf{N}},W_{\mathsf{N}}+W^{\prime}_{\mathsf{N}})\in\mathrm{Stab}\\!\left(\mathsf{N}\right)$
such that $d(Q_{\mathsf{N}},Q^{\prime}_{\mathsf{N}})<\varepsilon$ and
$||(W_{\mathsf{N}}+W^{\prime}_{\mathsf{N}})-W_{\mathsf{N}}||=||W^{\prime}_{\mathsf{N}}||<\sin(\pi\varepsilon)/K$.
The latter inequality implies
$||W_{\mathsf{N}}^{\prime}||_{\sigma}\leq\sin(\pi\varepsilon)$ and then the
triangle inequality gives

$d(P_{\mathsf{N}},Q^{\prime}_{\mathsf{N}})<2\varepsilon\quad\text{and}\quad||W_{\mathsf{N}}+W^{\prime}_{\mathsf{N}}||_{\sigma}<3\sin(\pi\varepsilon).$

Set $\delta\coloneqq\max\\{2\varepsilon,\arcsin(3\sin(\pi\varepsilon))/\pi\\}$
and note that $\delta\to 0$ as $\varepsilon\to 0$. In particular we can choose
$\varepsilon>0$ sufficiently small that $\delta<1/8$. By Theorem 6.1 there is
a unique stability condition $\sigma^{\prime}=(P^{\prime},Z^{\prime})\in
B_{\delta}(\sigma)$ with charge
$Z^{\prime}=Z+W_{\mathsf{N}}+W^{\prime}_{\mathsf{N}}$ and restricted slicing
$P^{\prime}\cap\mathsf{N}=Q^{\prime}_{\mathsf{N}}$.

Step 2: Deform the charge of $\sigma^{\prime}$ to get
$\tau^{\prime}\in\mathrm{Stab}\\!\left(\mathsf{C}\right)$.

Set $U\coloneqq W-Z-
W_{\mathsf{N}}\in{\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$.
Then by Lemma 5.11

$||U||_{\sigma^{\prime}}\leq\frac{||U||_{\sigma}}{C}\leq\frac{||W-Z||_{\sigma}+||W^{\prime}_{\mathsf{N}}||_{\sigma}}{C}\leq\frac{2\sin(\pi\varepsilon)}{C}<\sin(3\pi\varepsilon)$

where $C\coloneqq(1-\sin(\pi\delta))\cos(2\pi\delta)\to 1$ as $\varepsilon\to
0$ so that the final inequality holds for sufficiently small $\varepsilon>0$.
Possibly further shrinking $\varepsilon$ we can apply Theorem 4.1 to obtain a
stability condition $\tau^{\prime}=(Q^{\prime},W^{\prime})\in
B_{3\varepsilon}(\sigma^{\prime})$ with
$W^{\prime}=Z^{\prime}+U=W+W^{\prime}_{\mathsf{N}}$.

Step 3: $\tau^{\prime}$ is close to $\tau$.

By construction
$||W^{\prime}-W||=||W^{\prime}_{\mathsf{N}}||<\sin(\pi\varepsilon)$. Therefore
it suffices to show that $d(Q^{\prime},Q)<\varepsilon$. Consider a
$\tau^{\prime}$-semistable object $b\in Q^{\prime}(\varphi)$. Using
$d(P,Q)<\varepsilon$, $d(P,P^{\prime})<\delta$ and
$d(P^{\prime},Q^{\prime})<3\varepsilon$ gives
$d(Q,Q^{\prime})<\delta+4\varepsilon$ and hence

$Q^{\prime}(\varphi)\subset
Q(\varphi-\delta-4\varepsilon,\varphi+\delta+4\varepsilon)\subset
P(\varphi-\delta-5\varepsilon,\varphi+\delta+5\varepsilon)\eqqcolon\mathsf{A}.$

For sufficiently small $\varepsilon>0$ Lemma 5.3 ensures that $\mathsf{A}$ is
a quasi-abelian length category. The simple objects in $\mathsf{A}$ are the
$\sigma$-stable objects. The HNfiltration of $b$ with respect to $\tau$ is
obtained by grouping the simple factors in some composition series of $b$ in
$\mathsf{A}$ into a sequence of $\tau$-semistable objects of strictly
decreasing phase. Therefore it suffices to show that the phases of these
simple factors with respect to $\tau$ and $\tau^{\prime}$ differ by less than
$\varepsilon$.

If $a\in\mathsf{A}$ is a massless simple object then there is nothing to prove
because $d(Q^{\prime}_{\mathsf{N}},Q_{\mathsf{N}})<\varepsilon$. If
$a\in\mathsf{A}$ is massive then it suffices to show that
$|W_{\mathsf{N}}^{\prime}(a)|/|W(a)|$ can be made arbitrarily small, uniformly
for all such $a$, by choosing $||W_{\mathsf{N}}^{\prime}||$ sufficiently
small. We do so by making a series of estimates.

The operator norm estimate and the support property for $\sigma$ provide a
bound

$|W^{\prime}_{\mathsf{N}}(a)|\leq||W^{\prime}_{\mathsf{N}}||\cdot||v(a)||\leq
K||W^{\prime}_{\mathsf{N}}||\cdot|Z(a)|$

for some constant $K>0$. Moreover, since
$Z\in{\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$ and
$d(P,Q)<\varepsilon$ we have

$|Z(a)|\leq\sum_{t\in T}|Z(t)|\leq\sum_{t\in
T}||Z||_{\tau}|W(t)|\leq\frac{||Z||_{\tau}}{\cos(2\pi\varepsilon)}|W(a)|$

where $T$ is a set of $\tau$-stable factors of the object $a$. Finally,

$||Z||_{\tau}\leq\frac{||Z||_{\sigma}}{(1-\sin(\pi\varepsilon))\cos(2\pi\varepsilon)}=\frac{1}{(1-\sin(\pi\varepsilon))\cos(2\pi\varepsilon)}$

by Lemma 5.11 because
$Z\in{\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$. Combining these
estimates, we have the bound

$\frac{|W_{\mathsf{N}}^{\prime}(a)|}{|W(a)|}\leq\frac{K||W^{\prime}_{\mathsf{N}}||}{(1-\sin(\pi\varepsilon))\cos(2\pi\varepsilon)^{2}},$

uniformly in $a$. The result follows because we can make the right hand side
smaller than any given $\varepsilon>0$ by rescaling $W_{\mathsf{N}}^{\prime}$.
∎

######  Proposition 6.18.

Suppose support propagates from
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$. Then the map
$\delta_{\mathsf{N}}$ defined in Remark 6.2 extends to a continuous map

$\delta_{\mathsf{N}}\colon\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\times_{\mathrm{Slice}(\mathsf{N})}\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)\dashrightarrow\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$

defined on the open subset of pairs
$(\sigma,\tau_{\mathsf{N}})=\big{(}(P,Z),(P\cap\mathsf{N},W_{\mathsf{N}})\big{)}$
with $||W_{\mathsf{N}}||_{\sigma}<\sin(\pi\varepsilon)$ for some suitably
small $\varepsilon>0$.

###### Proof.

Let $\mathsf{M}\subset\mathsf{N}$ be the massless subcategory of
$\tau_{\mathsf{N}}$. Then $P_{\mathsf{N}}=P\cap\mathsf{N}$ is adapted to
$\mathsf{M}$ by Proposition 4.4. It follows that the $P$-semistable factors of
an object $c\in\mathsf{M}$ are the $P_{\mathsf{N}}$-semistable factors, thus
are in $\mathsf{M}$. Moreover, for any strict length one interval $I$ the
intersection $P(I)\cap\mathsf{M}=P_{\mathsf{N}}(I)\cap\mathsf{M}$ is a Serre
subcategory of $P_{\mathsf{N}}(I)$, which is in turn a Serre subcategory of
$P(I)$. Hence, $P$ is adapted to $\mathsf{M}$. Therefore, by Proposition 3.6,
there is an induced slicing $P_{\mathsf{C}/\mathsf{M}}$ on
$\mathsf{C}/\mathsf{M}$. This induced slicing is compatible with
$P_{\mathsf{C}/\mathsf{N}}$ and $P_{\mathsf{N}/\mathsf{M}}$, both of which are
locally finite because they are respectively the slicings of the classical
stability conditions $\mu_{\mathsf{N}}(\sigma)$ and
$\mu_{\mathsf{M}}(\tau_{\mathsf{N}})$. Hence $P_{\mathsf{C}/\mathsf{M}}$ is
locally finite by Lemma 3.3 so that $P$ is well-adapted to $\mathsf{M}$.

Thus Proposition 6.8 lets us define
$\delta_{\mathsf{N}}(\sigma,\tau_{\mathsf{N}})$ as the unique lax pre-
stability condition $(Q,W=Z+W_{\mathsf{N}})$ with $d(P,Q)<\varepsilon$ and
restricted slicing $Q\cap\mathsf{N}=P_{\mathsf{N}}$. The assumption that
support propagates implies
$\delta_{\mathsf{N}}(\sigma,\tau_{\mathsf{N}})\in\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$
so that $\delta_{\mathsf{N}}$ is well-defined as a map of sets.

The restriction of $\delta_{\mathsf{N}}$ to (an open subset of)
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\times_{\mathrm{Slice}(\mathsf{N})}\mathrm{Stab}\\!\left(\mathsf{N}\right)$
is continuous by Remark 6.2. The proof of Theorem 6.17 shows that the
extension is continuous on each normal slice
$\\{\sigma\\}\times_{\mathrm{Slice}(\mathsf{N})}\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$.
Together these facts imply the extension is continuous. ∎

##  7\. The topology of the space of lax stability conditions

We apply the deformation results of §6 to study the topology of the space of
lax stability conditions. Our starting point is the following consequence of
support propagation.

######  Proposition 7.1.

Suppose that support propagates from
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$. Then the following
maps are homeomorphisms onto unions of components:

  1. (1)

$\mu_{\mathsf{N}}\times\rho_{\mathsf{N}}\colon\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\to\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\times\mathrm{Stab}^{L}\\!\left(\mathsf{N},\mathsf{N}\right)$;

  2. (2)

$\mu_{\mathsf{N}}\times\operatorname{{\mathsf{id}}}\colon\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\times_{\mathrm{Slice}(\mathsf{N})}\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)\to\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\times\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$.

Here
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\times_{\mathrm{Slice}(\mathsf{N})}\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$
denotes the set of pairs $(P,Z)$ in
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ and
$(Q_{\mathsf{N}},W_{\mathsf{N}})$ in
$\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$ such that the restricted slicing
$P_{\mathsf{N}}=Q_{\mathsf{N}}$.

###### Proof.

(1) The quotient map
$\mu_{\mathsf{N}}\colon\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\to\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)$
is well-defined by Proposition 4.13 and continuous by Proposition 5.10. The
restriction map
$\rho_{\mathsf{N}}\colon\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\to\mathrm{Stab}^{L}\\!\left(\mathsf{N},\mathsf{N}\right)$
is continuous by Corollary 5.8. The product
$\mu_{\mathsf{N}}\times\rho_{\mathsf{N}}$ is injective: given
$\sigma=(P,Z)\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$, the
slicings $P_{\mathsf{N}}$ and $P_{\mathsf{C}/\mathsf{N}}$ determine $P$
uniquely by Propositions 3.2, 3.6 and 4.4; the charge $Z$ is determined by its
factorisation through $\Lambda/\Lambda_{\mathsf{N}}$.

By Corollary 6.10, $\sigma$ can be deformed along nearby slicings in
$\mathrm{Stab}^{L}\\!\left(\mathsf{N},\mathsf{N}\right)\subseteq\mathrm{Slice}(\mathsf{N})$
to a unique lax pre-stability condition which, by the assumption that support
propagates from $\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$, is
in fact in $\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$.
Proposition 6.11 is the analogous statement about tangential deformations,
i.e. deforming charges in
${\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$ or, equivalently,
deforming the stability condition on the quotient $\mathsf{C}/\mathsf{N}$.
Therefore, there is an open metric ball around
$(\mu_{\mathsf{N}}(\sigma),\rho_{\mathsf{N}}(\sigma))$ in the image of
$\mu_{\mathsf{N}}\times\rho_{\mathsf{N}}$. Thus the map in (1) is open. The
statement about surjectivity on connected components follows.

(2) As in the first part, the map
$\mu_{\mathsf{N}}\times\operatorname{{\mathsf{id}}}$ is injective because the
slicing of $\sigma$ in
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ can be reconstructed
uniquely from the slicing of $\mu_{\mathsf{N}}(\sigma)$ and the slicing on
$\mathsf{N}$. It is surjective because given
$\left(\sigma_{\mathsf{C}/\mathsf{N}},\sigma_{\mathsf{N}}\right)$ in a
component meeting the image of
$\mu_{\mathsf{N}}\times\operatorname{{\mathsf{id}}}$ we can apply the first
part to construct
$\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ with
$\mu_{\mathsf{N}}(\sigma)=\sigma_{\mathsf{C}/\mathsf{N}}$ and restricted
slicing that of $\sigma_{\mathsf{N}}$. Then $(\sigma,\sigma_{\mathsf{N}})$ is
the required lift of
$\left(\sigma_{\mathsf{C}/\mathsf{N}},\sigma_{\mathsf{N}}\right)$. The inverse
is continuous because the reconstruction of the slicing on $\mathsf{C}$ from
those on $\mathsf{C}/\mathsf{N}$ and $\mathsf{N}$ is continuous in the slicing
metric; see Lemma 3.11. ∎

We show in Corollary 7.4 that the product description of
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ extends to a
deformation retract neighbourhood, and we describe how pairs of strata fit
together in Proposition 7.8.

###  7.1. Neighbourhoods of strata

Our aim is to construct a deformation retract neighbourhood of
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$. The construction
depends on the splitting
${\mathrm{Hom}}(\Lambda_{\mathsf{N}},\mathbb{C})\hookrightarrow{\mathrm{Hom}}(\Lambda,\mathbb{C})$
induced from the inner product on $\Lambda\otimes\mathbb{R}$, and therefore we
need a neighbourhood which is adapted to this. We define one as follows. Fix
$0<\varepsilon<1/8$ and for
$\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ set

$V_{\varepsilon}(\sigma)=\\{\tau=(Q,W)\in
B_{\varepsilon}(\sigma):||W_{\mathsf{N}}||_{\sigma}<\sin(\pi\varepsilon)\\}$

where $W_{\mathsf{N}}$ is the restriction of $W$ to $\Lambda_{\mathsf{N}}$
considered as an element of ${\mathrm{Hom}}(\Lambda,\mathbb{C})$ via the
splitting. Thus $V_{\varepsilon}(\sigma)$ is open in
$\mathrm{Slice}(\mathsf{C})\times{\mathrm{Hom}}(\Lambda,\mathbb{C})$. Then
define an open subset of $\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$ by

$\mathrm{V}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)=\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)\cap\bigcup_{\mathclap{\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)}}V_{\varepsilon}(\sigma)$

These smaller neighbourhoods have the same good properties as the
$\mathrm{B}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$, namely
they are $\mathbb{C}$-invariant and satisfy
$\alpha\cdot\mathrm{V}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)=\mathrm{V}^{L}_{\varepsilon}\\!\left(\mathsf{C},\alpha(\mathsf{N})\right)$
for automorphisms $\alpha\in\mathrm{Aut}_{\Lambda}(\mathsf{C})$. Furthermore,
$\mathrm{V}^{L}_{\varepsilon}\\!\left(\mathsf{C},0\right)=\mathrm{Stab}\\!\left(\mathsf{C}\right)$
and
$\mathrm{V}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{C}\right)=\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$
because the closure of the $\mathbb{C}$-orbit of any
$\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$ contains a lax
stability condition with massless subcategory $\mathsf{C}$.

######  Lemma 7.2.

Assume that support propagates from
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$. Then, for
sufficiently small $\varepsilon>0$, the identity on
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ extends to a
continuous map

$\Phi_{\mathsf{N}}\colon\mathrm{V}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\longrightarrow\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$

mapping $(Q,W)$ to the unique element of
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ with charge
$W-W_{\mathsf{N}}$, massless slicing $Q_{\mathsf{N}}$ and slicing within
distance $5\varepsilon$ of $Q$. Moreover, $\Phi_{\mathsf{N}}$ is
$\mathbb{C}$-equivariant and satisfies
$\alpha\cdot\Phi_{\mathsf{N}}(\tau)=\Phi_{\alpha(\mathsf{N})}(\alpha\cdot\tau)$
for each automorphism $\alpha\in\mathrm{Aut}_{\Lambda}(\mathsf{C})$ and
$\tau\in\mathrm{V}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$.

######  Remark 7.3.

The map $\Phi_{\mathsf{N}}$ is induced by the projection $\pi_{\mathsf{N}}$ of
charges onto their $\Lambda/\Lambda_{\mathsf{N}}$-component,
$\pi_{\mathsf{N}}(W)=W-W_{\mathsf{N}}$:

${\mathrm{V}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)}$${\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)}$${{\mathrm{Hom}}(\Lambda,\mathbb{C})}$${{\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})}$${{\mathrm{Hom}}(\Lambda,\mathbb{C})}$$\scriptstyle{\Phi_{\mathsf{N}}}$$\scriptstyle{\mathcal{Z}}$$\scriptstyle{\mathcal{Z}}$$\scriptstyle{\pi_{\mathsf{N}}}$

###### Proof.

Given
$\tau=(Q,W)\in\mathrm{V}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$
choose $\sigma=(P,Z)\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$
such that $\tau\in B_{\varepsilon}(\sigma)$ and
$||W_{\mathsf{N}}||_{\sigma}<\sin(\pi\varepsilon)$ where
$\rho_{\mathsf{N}}(\tau)=(Q_{\mathsf{N}},W_{\mathsf{N}})$ and the slicing
restricts by Lemma 5.6. Then
$W-W_{\mathsf{N}}\in{\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$
and

$||Z-(W-W_{\mathsf{N}})||_{\sigma}\leq||Z-W||_{\sigma}+||W_{\mathsf{N}}||_{\sigma}<2\sin(\pi\varepsilon)\leq\sin(3\pi\varepsilon)$

where we assume $\varepsilon\leq 1/6$ which enables the final trigonometric
inequality.

Hence, for sufficiently small $\varepsilon>0$, we can apply Proposition 6.11
to construct a lax pre-stability condition
$\sigma^{\prime}=(P^{\prime},W-W_{\mathsf{N}})$ with restricted slicing
$P_{\mathsf{N}}^{\prime}=P_{\mathsf{N}}$ and $d(P,P^{\prime})<3\varepsilon$.
Therefore $\sigma^{\prime}\in B_{3\varepsilon}(\sigma)$, the restriction
$\rho_{\mathsf{N}}(\sigma^{\prime})=\rho_{\mathsf{N}}(\sigma)$, and
$||Z-(W-W_{\mathsf{N}})||_{\sigma}<\sin(3\pi\varepsilon)$. Thus we can apply
the support propagation assumption to conclude that $\sigma^{\prime}$ is in
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ when $\varepsilon$ is
sufficiently small.

Since $\tau\in B_{\varepsilon}(\sigma)$ we know that $d(P,Q)<\varepsilon$.
Hence $d(P_{\mathsf{N}},Q_{\mathsf{N}})<\varepsilon$ too. Therefore we can
deform the massless slicing of $\sigma^{\prime}$ using Corollary 6.10 to
obtain a lax pre-stability condition
$\Phi_{\mathsf{N}}(\tau)=(Q^{\prime},W-W_{\mathsf{N}})$ with massless slicing
$Q_{\mathsf{N}}^{\prime}=Q_{\mathsf{N}}$ and
$d(P^{\prime},Q^{\prime})<\varepsilon$, from which it follows that
$d(Q,Q^{\prime})<5\varepsilon$. It follows from Corollary 4.18 that
$\Phi_{\mathsf{N}}(\tau)$ is well-defined and independent of the particular
choice of $\sigma$ used to construct it. For sufficiently small
$\varepsilon>0$ the support propagation assumption implies that
$\Phi_{\mathsf{N}}(\tau)$ is in
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$.

By construction $\Phi_{\mathsf{N}}$ is continuous in the charge and in the
restricted slicing on $\mathsf{N}$. Since
$\rho_{\mathsf{N}}\Phi_{\mathsf{N}}(\tau)=(Q_{\mathsf{N}},0)$, in order to see
that
$\Phi_{\mathsf{N}}\colon\mathrm{V}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\to\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$
is continuous, it is sufficient to show that
$\mu_{\mathsf{N}}\Phi_{\mathsf{N}}(\tau)$ varies continuously in $\tau$. If
$\tau^{\prime}$ is sufficiently close to $\tau$ then, since the slicing
distance between $\tau$ and $\Phi_{\mathsf{N}}(\tau)$ is less than
$5\varepsilon$, the slicing distance between $\Phi_{\mathsf{N}}(\tau)$ and
$\Phi_{\mathsf{N}}(\tau^{\prime})$ can be made strictly less than $1$.
Therefore the slicing distance between
$\mu_{\mathsf{N}}\Phi_{\mathsf{N}}(\tau)$ and
$\mu_{\mathsf{N}}\Phi_{\mathsf{N}}(\tau^{\prime})$ is also strictly less than
$1$. Deformations of classical stability conditions with slicing within
distance $1$ are uniquely determined by the deformation of the charge, i.e. if
the charge deforms continuously then so does the stability condition. Since
$\mu_{\mathsf{N}}\Phi_{\mathsf{N}}(\tau)$ and
$\mu_{\mathsf{N}}\Phi_{\mathsf{N}}(\tau^{\prime})$ are classical stability
conditions whose slicings are within distance $1$ of each other whose charges
are also close, it follows that $\mu_{\mathsf{N}}\Phi_{\mathsf{N}}(\tau)$
varies continuously with $\tau$.

Moreover $\Phi_{\mathsf{N}}$ restricts to the identity on
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$. The characterisation
of $\Phi_{\mathsf{N}}(\tau)$ in the statement follows immediately from the
above construction, as does the compatibility with the actions of $\mathbb{C}$
and of $\mathrm{Aut}_{\Lambda}(\mathsf{C})$. ∎

Using the map $\Phi_{\mathsf{N}}$ we can, for sufficiently small
$\varepsilon>0$, define a smaller, and even better behaved, neighbourhood of
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ namely

$\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)=\left\\{\tau\in\mathrm{V}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\colon\tau\in
B_{\varepsilon}(\Phi_{\mathsf{N}}(\tau))\right\\}.$

The compatibility of $\Phi_{\mathsf{N}}$ with the actions of $\mathbb{C}$ and
$\mathrm{Aut}_{\Lambda}(\mathsf{C})$ ensures that these neighbourhoods are
$\mathbb{C}$-invariant and that
$\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$ is mapped
to $\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\alpha(\mathsf{N})\right)$
by $\alpha\in\mathrm{Aut}_{\Lambda}(\mathsf{C})$. In the two extreme cases of
$\mathsf{N}=0$ and $\mathsf{C}$ the extra condition is vacuous, and we have
equalities

$\displaystyle\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},0\right)$
$\displaystyle=\mathrm{V}^{L}_{\varepsilon}\\!\left(\mathsf{C},0\right)=\mathrm{B}^{L}_{\varepsilon}\\!\left(\mathsf{C},0\right)=\mathrm{Stab}\\!\left(\mathsf{C}\right),$
$\displaystyle\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{C}\right)$
$\displaystyle=\mathrm{V}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{C}\right)=\mathrm{B}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{C}\right)=\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right).$

######  Corollary 7.4.

Suppose support propagates from
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$. Then, for
sufficiently small $\varepsilon>0$ the map

$\Phi_{\mathsf{N}}\times\rho_{\mathsf{N}}\colon\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\longrightarrow\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\times_{\mathrm{Slice}(\mathsf{N})}\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$

is a homeomorphism onto the set of pairs $(\sigma,\tau_{\mathsf{N}})$ with
$||W_{\mathsf{N}}||_{\sigma}<\sin(\pi\varepsilon)$ where
$\tau_{\mathsf{N}}=(Q_{\mathsf{N}},W_{\mathsf{N}})$.

###### Proof.

The map $\Phi_{\mathsf{N}}\times\rho_{\mathsf{N}}$ is continuous and, by
definition of
$\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$, has image
in the subset

$\left\\{\left(\sigma,\tau_{\mathsf{N}}=(Q_{\mathsf{N}},W_{\mathsf{N}})\right):||W_{\mathsf{N}}||_{\sigma}<\sin(\pi\varepsilon)\right\\}.$

The assumption that support propagates means that the map
$\delta_{\mathsf{N}}$ defined in Proposition 6.18 exists. We claim it is
inverse to $\Phi_{\mathsf{N}}\times\rho_{\mathsf{N}}$.

Let $\tau=\delta_{\mathsf{N}}(\sigma,\tau_{\mathsf{N}})$, where
$\sigma=(P,Z)$. By construction $\rho_{\mathsf{N}}(\tau)=\tau_{\mathsf{N}}$.
Moreover
$\tau\in\mathrm{V}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$ so
that $\Phi_{\mathsf{N}}(\tau)$ is well-defined. Then $\Phi_{\mathsf{N}}(\tau)$
has charge $Z$, massless subcategory $\mathsf{N}$ and massless slicing
$Q_{\mathsf{N}}$. The slicing of $\Phi_{\mathsf{N}}(\tau)$ is within distance
$5\varepsilon$ of $P$ so that $\Phi_{\mathsf{N}}(\tau)=\sigma$ by Corollary
4.18 provided that $5\varepsilon<1$. We conclude that $\tau\in
B_{\varepsilon}(\Phi_{\mathsf{N}}(\tau))$ so that
$\tau\in\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$ and
$(\Phi_{\mathsf{N}}\times\rho_{\mathsf{N}})\circ\delta_{\mathsf{N}}$ is the
identity.

Now set
$\tau^{\prime}=\delta_{\mathsf{N}}(\Phi_{\mathsf{N}}(\tau),\rho_{\mathsf{N}}(\tau))$.
This is a lax stability condition with the same charge, massless subcategory
and massless slicing as $\tau$. Moreover, its slicing is within distance
$5\varepsilon$ of that of $\tau$. Thus $\tau^{\prime}=\tau$ by Corollary 4.18
if again $5\varepsilon<1$ and
$\delta_{\mathsf{N}}\circ(\Phi_{\mathsf{N}}\times\rho_{\mathsf{N}})$ is the
identity too. ∎

Clearly
$\\{(\sigma,(Q_{\mathsf{N}},W_{\mathsf{N}})):||W_{\mathsf{N}}||_{\sigma}<\sin(\pi\varepsilon)\\}$
is an open neighbourhood of
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ in
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\times_{\mathrm{Slice}(\mathsf{N})}\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$,
and therefore
$\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$ is an open
neighbourhood of $\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$.
Indeed it is a deformation retract neighbourhood.

######  Definition 7.5.

Suppose support propagates from
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ so that
$\Phi_{\mathsf{N}}\times\rho_{\mathsf{N}}$ is a homeomorphism on
$\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$ for some
sufficiently small $\varepsilon>0$. For $t\in[0,1]$ let

$\Phi_{\mathsf{N},t}\colon\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\to\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$

be the map corresponding to
$(\sigma,(Q_{\mathsf{N}},W_{\mathsf{N}}))\mapsto(\sigma,(Q_{\mathsf{N}},tW_{\mathsf{N}}))$
under $\Phi_{\mathsf{N}}\times\rho_{\mathsf{N}}$. Note that
$\Phi_{\mathsf{N},0}=\Phi_{\mathsf{N}}$ and
$\Phi_{\mathsf{N},1}=\operatorname{{\mathsf{id}}}$.

When $\mathsf{N}=\mathsf{C}$ the retraction $\Phi_{\mathsf{N},t}$ is defined
on the entire space of lax stability conditions because
$\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{C}\right)=\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$.
In this case $\Phi_{\mathsf{N},t}(Q,W)=(Q,tW)=(Q,W)\cdot i\log(t)/\pi$ is
obtained by dilating the masses of objects using the right action of
$i\mathbb{R}\subset\mathbb{C}$ defined in §5.3. More generally

(7)
${\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)}$${\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)}$${\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)}$${\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)}$$\scriptstyle{\Phi_{\mathsf{N},t}}$$\scriptstyle{\rho_{\mathsf{N}}}$$\scriptstyle{\rho_{\mathsf{N}}}$$\scriptstyle{\Phi_{\mathsf{N},t}}$

commutes for each $t\in[0,1]$ where the bottom map is given by mass dilation
in $\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$. It is also clear that the
maps $\Phi_{\mathsf{N},t}$ preserve the intersections
$\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\cap\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)}$
for thick $\mathsf{M}\subset\mathsf{N}$.

###  7.2. Pairs of strata

We now consider how the subsets
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)$ and
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ fit together when
$\mathsf{M}\subset\mathsf{N}$ are thick subcategories. The first observation
is that the _frontier condition_ fails: that is in general

$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\cap\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)}\neq\varnothing\centernot\implies\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\subset\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)}.$

######  Example 7.6.

Let $\mathsf{C}=\mathsf{D}^{b}(\mathbb{P}^{1})$ and consider the pair
$\mathsf{M}=\mathsf{thick}(\mathcal{O})\subset\mathsf{D}^{b}(\mathbb{P}^{1})=\mathsf{N}$.
The above intersection contains the slicings $P$ occurring in
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ for which $\mathcal{O}$ is
semistable, so is non-empty. However, it is not the whole of
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{C}\right)$ because it does not
contain any slicings for which $\mathcal{O}$ is unstable, such as those in
which the only two semistable objects are $\mathcal{O}(1)$ and
$\mathcal{O}(2)$ and their shifts.

Nevertheless the situation is quite well-behaved. When we restrict to
$\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)}$ the maps
$\Phi_{\mathsf{N}}$ and $\rho_{\mathsf{N}}$ are compatible with passing to the
quotient via $\mu_{\mathsf{M}}$. Effectively then we can reduce to the case
$\mathsf{M}=0$.

######  Lemma 7.7.

Suppose $\mathsf{M}\subset\mathsf{N}$ are thick subcategories such that
support propagates from both
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ and
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}/\mathsf{M},\mathsf{N}/\mathsf{M}\right)$.
Then for sufficiently small $\varepsilon>0$ the map $\mu_{\mathsf{M}}$
restricts to a map

$\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\cap\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)}\longrightarrow\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C}/\mathsf{M},\mathsf{N}/\mathsf{M}\right).$

###### Proof.

The support propagation assumptions ensure that the maps $\Phi_{\mathsf{N},t}$
for $t\in[0,1]$ and $\Phi_{\mathsf{N}/\mathsf{M}}$ are defined.

Let
$\tau=(Q,W)\in\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\cap\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)}$
and $\rho_{\mathsf{N}}(\tau)=(Q_{\mathsf{N}},W_{\mathsf{N}})$. Then
$\Phi_{\mathsf{N}}(\tau)$ is also in
$\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)}$ because
$\Phi_{\mathsf{N},t}$ preserves the intersection
$\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\cap\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)}$,
and $\mu_{\mathsf{M}}(\Phi_{\mathsf{N}}(\tau))$ is well-defined and lies in
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}/\mathsf{M},\mathsf{N}/\mathsf{M}\right)$;
see Remark 4.14. Then

$||W_{\mathsf{N}}||_{\mu_{\mathsf{M}}(\Phi_{\mathsf{M}}(\tau))}\leq||W_{\mathsf{N}}||_{\Phi_{\mathsf{M}}(\tau)}<\sin(\pi\varepsilon)$

by Lemma 4.15 and, by Lemma 5.7, the distance between the slicings of
$\mu_{\mathsf{M}}(\Phi_{\mathsf{N}}(\tau))$ and $\mu_{\mathsf{M}}(\tau)$ is
less than that between those of $\Phi_{\mathsf{N}}(\tau)$ and $\tau$, which in
turn is less than $\varepsilon$ because $\tau\in
B_{\varepsilon}(\Phi_{\mathsf{N}}(\tau))$. Therefore, we also have
$\mu_{\mathsf{M}}(\tau)\in
B_{\varepsilon}(\mu_{\mathsf{M}}(\Phi_{\mathsf{M}}(\tau)))$. This shows that
$\mu_{\mathsf{M}}(\tau)$ is in
$\mathrm{V}^{L}_{\varepsilon}\\!\left(\mathsf{C}/\mathsf{M},\mathsf{N}/\mathsf{M}\right)$
so that $\Phi_{\mathsf{N}/\mathsf{M}}(\mu_{\mathsf{M}}(\tau))$ is well-
defined. Applying Corollary 4.18 then shows that

$\Phi_{\mathsf{N}/\mathsf{M}}(\mu_{\mathsf{M}}(\tau))=\mu_{\mathsf{M}}(\Phi_{\mathsf{N}}(\tau))$

because they have the same charge $W-W_{\mathsf{N}}$, the same massless
subcategory $\mathsf{N}/\mathsf{M}$ and massless slicing, and their slicings
are within distance one of each other for sufficiently small $\varepsilon>0$.
Therefore, $\mu_{\mathsf{M}}(\tau)\in
B_{\varepsilon}(\Phi_{\mathsf{N}/\mathsf{M}}(\mu_{\mathsf{M}}(\tau)))$ and so
is in
$\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C}/\mathsf{M},\mathsf{N}/\mathsf{M}\right)$
as claimed. ∎

######  Proposition 7.8.

Suppose $\mathsf{M}\subset\mathsf{N}$ are thick subcategories such that
support propagates from both
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ and
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}/\mathsf{M},\mathsf{N}/\mathsf{M}\right)$.
Then for sufficiently small $\varepsilon>0$ there is a commutative diagram

${\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)}$${\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\cap\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)}}$${\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\cap\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)}}$${\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{N},\mathsf{M}\right)}}$${\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)}$${\mathrm{Stab}^{L}\\!\left(\mathsf{C}/\mathsf{M},\mathsf{N}/\mathsf{M}\right)}$${\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C}/\mathsf{M},\mathsf{N}/\mathsf{M}\right)}$${\mathrm{Stab}^{L}\\!\left(\mathsf{N}/\mathsf{M}\right).}$$\scriptstyle{\mu_{\mathsf{M}}}$$\scriptstyle{\mu_{\mathsf{N}}}$$\scriptstyle{\Phi_{\mathsf{N}}}$$\scriptstyle{\rho_{\mathsf{N}}}$$\scriptstyle{\mu_{\mathsf{M}}}$$\scriptstyle{\mu_{\mathsf{M}}}$$\scriptstyle{\mu_{\mathsf{N}/\mathsf{M}}}$$\scriptstyle{\Phi_{\mathsf{N}/\mathsf{M}}}$$\scriptstyle{\rho_{\mathsf{N}/\mathsf{M}}}$

###### Proof.

The support propagation assumptions and the previous lemma ensure the maps in
the central square are well-defined, and that the square commutes. Consider
the left hand square and choose

$\sigma=(P,Z)\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\cap\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)}.$

Recall $\mu_{\mathsf{N}}(\sigma)=(P_{\mathsf{C}/\mathsf{N}},Z)$ from
Proposition 4.13 where $P_{\mathsf{C}/\mathsf{N}}(\varphi)$ is the isomorphism
closure of $P(\varphi)$ in $\mathsf{C}/\mathsf{N}$. Similarly,
$\mu_{\mathsf{M}}(\sigma)=(P_{\mathsf{C}/\mathsf{M}},Z)$ where
$P_{\mathsf{C}/\mathsf{M}}(\varphi)$ is the isomorphism closure of
$P(\varphi)$ in $\mathsf{C}/\mathsf{M}$. Applying
$\mu_{\mathsf{N}/\mathsf{M}}$ the charge remains the same, namely $Z$, and the
category of semistable objects of phase $\varphi$ becomes the isomorphism
closure of $P_{\mathsf{C}/\mathsf{M}}(\varphi)$ in
$(\mathsf{C}/\mathsf{M})/(\mathsf{N}/\mathsf{M})\simeq\mathsf{C}/\mathsf{N}$
which is, as before, the isomorphism closure of $P(\varphi)$. Hence
$\mu_{\mathsf{N}}=\mu_{\mathsf{N}/\mathsf{M}}\circ\mu_{\mathsf{M}}$.

Now consider the right hand square. Choose

$\sigma=(P,Z)\in\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\cap\overline{\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)}.$

Both $\mu_{\mathsf{M}}\circ\rho_{\mathsf{N}}(\sigma)$ and
$\rho_{\mathsf{\mathsf{N}/\mathsf{M}}}\circ\mu_{\mathsf{M}}(\sigma)$ have
charge $Z|_{\Lambda_{\mathsf{N}}}$. On the one hand, the category of
semistable objects of $\mu_{\mathsf{M}}\circ\rho_{\mathsf{N}}(\sigma)$ with
phase $\varphi$ is the isomorphism closure of $P(\varphi)\cap\mathsf{N}$ in
$\mathsf{N}/\mathsf{M}$. On the other hand, that of
$\rho_{\mathsf{\mathsf{N}/\mathsf{M}}}\circ\mu_{\mathsf{M}}(\sigma)$ is the
isomorphism closure of $P(\varphi)$ in $\mathsf{C}/\mathsf{M}$ intersected
with $\mathsf{N}/\mathsf{M}$. This clearly contains the former and hence is
the same as the former since nested slicings are equal. Therefore
$\mu_{\mathsf{M}}\circ\rho_{\mathsf{N}}=\rho_{\mathsf{\mathsf{N}/\mathsf{M}}}\circ\mu_{\mathsf{M}}$.
∎

##  8\. The space of quotient stability conditions

###  8.1. Definition

Define an equivalence relation on the points of
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$ by $\sigma\sim\tau$ if they have
the same charge and lie in the same connected component of the corresponding
fibre of
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)\to{\mathrm{Hom}}(\Lambda,\mathbb{C})$.
We refer to an equivalence class as a _quotient stability condition on
$\mathsf{C}$_ and denote the class of $\sigma$ by $[\sigma]$. If
$\sigma\sim\tau$ then by Corollary 5.14 they have the same massless
subcategory, $\mathsf{N}$ say, and induce the same stability condition
$\mu_{\mathsf{N}}(\sigma)=\mu_{\mathsf{N}}(\tau)$ in
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)$.

######  Remark 8.1.

If support propagates from
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ then

$\mu_{\mathsf{N}}\times\rho_{\mathsf{N}}\colon\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\to\mathrm{Stab}^{L}\\!\left(\mathsf{C}/\mathsf{N}\right)\times\mathrm{Stab}^{L}\\!\left(\mathsf{N},\mathsf{N}\right)$

is a homeomorphism onto a union of components by Proposition 7.1. In this case
$\sigma$ and $\tau$ are in the same component of the fibre of the charge map
precisely when $\rho_{\mathsf{N}}(\sigma)$ and $\rho_{\mathsf{N}}(\tau)$ are
in the same component of
$\mathrm{Stab}^{L}\\!\left(\mathsf{N},\mathsf{N}\right)$.

Conjecturally, non-empty stability spaces are contractible, in particular
connected. If this is the case, and support propagates from all strata
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$, then a quotient
stability condition is specified by a choice of massless subcategory
$\mathsf{N}$ and stability condition in
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)$. Which thick
subcategories $\mathsf{N}$ arise as massless categories remains a subtle
question.

The _space of quotient stability conditions_ is defined to be
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)=\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)/{\sim}$
equipped with the quotient topology. By definition the charge map factors
through the quotient, and we denote this factorisation also by $\mathcal{Z}$.

The theme of this section is that
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)$ is a stratified space in a
reasonable way. The first step is to specify a decomposition. Let
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)=\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)/{\sim}$
be the subspace of quotient stability conditions with massless subcategory
$\mathsf{N}$. Evidently

$\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)=\bigsqcup_{\mathrm{thick}~{}\mathsf{N}\subset\mathsf{C}}\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$

is a disjoint union of these subsets. The _strata_ are the connected
components of these pieces.

The maximal and minimal dimensional strata are easy to identify. The fibres of
the charge map on $\mathrm{Stab}\\!\left(\mathsf{C}\right)$ are discrete so
there are homeomorphisms

$\mathrm{Stab}\\!\left(\mathsf{C}\right)\cong\mathrm{Stab}^{L}\\!\left(\mathsf{C},0\right)\cong\mathrm{Stab}^{Q}\\!\left(\mathsf{C},0\right).$

Thus the usual space of stability conditions embeds continuously in
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C},0\right)$ as a union of strata. At the
other extreme
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{C}\right)=\pi_{0}\left(\mathcal{Z}^{-1}(0)\right)$
is the set of components of the fibre
$\mathcal{Z}^{-1}(0)=\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{C}\right)$,
with the discrete topology. These are the $0$-dimensional strata. If
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ is connected there is a unique such
stratum by Lemma 6.6.

###  8.2. The stratification of the space of quotient stability conditions

We show that, under suitable technical assumptions,
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)$ is a _stratified space_. By this
we mean that the strata
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$ are locally closed
subspaces satisfying the frontier condition, i.e. the closure of each such
stratum is a union of strata.

For $\sigma\in\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$ and
$c\in\mathsf{C}$ the mass $m_{\sigma}(c)$ is the mass of $c$ in the
corresponding stability condition on $\mathsf{C}/\mathsf{N}$. For massive
$c\in\mathsf{C}$ we define $\tilde{\varphi}^{\pm}_{\sigma}(c)$ to be the
minimal and maximal phases of the HNfactors of $c$ in $\mathsf{C}/\mathsf{N}$.

######  Lemma 8.2.

For each $c\in\mathsf{C}$ the mass
$m_{\bullet}(c)\colon\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)\to\mathbb{R}_{\geq
0}$ is continuous. The phases $\tilde{\varphi}_{\bullet}^{\pm}(c)\colon
m_{\bullet}(c)^{-1}(\mathbb{R}_{>0})\to\mathbb{R}$ are well-defined and
$\tilde{\varphi}_{\bullet}^{-}(c)$ and $\tilde{\varphi}_{\bullet}^{+}(c)$ are
respectively upper and lower semi-continuous.

###### Proof.

Recall that the mass of $c$ is continuous on
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$ and is constant on equivalence
classes, since it depends only on the massive HNfactors. The continuity of
masses on $\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)$ follows immediately.
The statements for phases follow similarly using the fact that the minimal and
maximal phases are respectively bounded above and below by the minimal and
maximal phases of the massive factors (which are constant on equivalence
classes). ∎

It follows immediately from the continuity of masses that the subsets
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$ and hence also their
connected components, i.e. the strata, are locally closed. The following
consequence of Proposition 7.1 identifies the strata as components of spaces
of stability conditions on various quotient categories.

######  Corollary 8.3.

Suppose that support propagates from
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$. Then the
factorisation of
$\mu_{\mathsf{N}}\colon\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\to\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)$
through the quotient map induces a homeomorphism between the component of
$\sigma\in\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$ and the
component of
$\mu_{\mathsf{N}}(\sigma)\in\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)$.
Thus each stratum of $\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)$ can be
given the structure of a complex manifold in such a way that the restriction
of the charge map is a local homeomorphism to
${\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$.

We begin by describing the equivalence relation in a neighbourhood of
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$.

######  Lemma 8.4.

Suppose support propagates from both
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ and
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}/\mathsf{M},\mathsf{N}/\mathsf{M}\right)$
for each thick $\mathsf{M}\subset\mathsf{N}$. Let
$\sigma,\tau\in\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$
for some sufficiently small $\varepsilon>0$. Then $\sigma\sim\tau$ if and only
if
$\mu_{\mathsf{N}}(\Phi_{\mathsf{N}}(\sigma))=\mu_{\mathsf{N}}(\Phi_{\mathsf{N}}(\tau))$
and $\rho_{\mathsf{N}}(\sigma)\sim\rho_{\mathsf{N}}(\tau)$ as elements of
$\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$.

###### Proof.

Suppose $\sigma\sim\tau$. Then
$\rho_{\mathsf{N}}(\sigma)\sim\rho_{\mathsf{N}}(\tau)$ as elements of
$\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$. Moreover, $\sigma$ and $\tau$
have the same massless subcategory, $\mathsf{M}$ say, so that
$\sigma,\tau\in\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\cap\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{M}\right)$.
Since we also have $\mu_{\mathsf{M}}(\sigma)=\mu_{\mathsf{M}}(\tau)$
Proposition 7.8 implies that
$\mu_{\mathsf{N}}(\Phi_{\mathsf{N}}(\sigma))=\mu_{\mathsf{N}}(\Phi_{\mathsf{N}}(\tau))$.

Conversely, suppose
$\mu_{\mathsf{N}}(\Phi_{\mathsf{N}}(\sigma))=\mu_{\mathsf{N}}(\Phi_{\mathsf{N}}(\tau))$
and $\rho_{\mathsf{N}}(\sigma)\sim\rho_{\mathsf{N}}(\tau)$. Then $\sigma$ and
$\tau$ have the same massless subcategory, namely the common massless
subcategory $\mathsf{M}$ of $\rho_{\mathsf{N}}(\sigma)$ and
$\rho_{\mathsf{N}}(\tau)$. Because $\Phi_{\mathsf{N}}(\sigma)$ and
$\Phi_{\mathsf{N}}(\tau)$ have the same charge, as do
$\rho_{\mathsf{N}}(\sigma)$ and $\rho_{\mathsf{N}}(\tau)$, we get that
$\sigma$ and $\tau$ have the same charge. Next,
$\rho_{\mathsf{M}}(\rho_{\mathsf{N}}(\sigma))$ and
$\rho_{\mathsf{M}}(\rho_{\mathsf{N}}(\tau))$ are in the same component of
$\mathrm{Stab}^{L}\\!\left(\mathsf{M},\mathsf{M}\right)$, since
$\rho_{\mathsf{N}}(\sigma)\sim\rho_{\mathsf{N}}(\tau)$. Then $\sigma\sim\tau$
by Remark 8.1, as $\rho_{\mathsf{M}}\circ\rho_{\mathsf{N}}=\rho_{\mathsf{M}}$.
∎

######  Lemma 8.5.

Suppose support propagates from both
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ and
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}/\mathsf{M},\mathsf{N}/\mathsf{M}\right)$
for each thick $\mathsf{M}\subset\mathsf{N}$. Then, for sufficiently small
$\varepsilon>0$, the open neighbourhood
$\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$ is a union
of equivalence classes.

###### Proof.

Suppose
$\tau\in\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$. By
definition, the class $[\tau]$ is connected in the subspace topology from
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$. Its intersection with
$\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$ is an open
subset of $[\tau]$, so it suffices to show this intersection is also closed in
$[\tau]$. By Corollary 7.4,
$\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$ is
homeomorphic via $\Phi_{\mathsf{N}}\times\rho_{\mathsf{N}}$ to

$\\{\left(\sigma,\tau_{\mathsf{N}}=(Q_{\mathsf{N}},W_{\mathsf{N}})\right):||W_{\mathsf{N}}||_{\sigma}<\sin(\pi\varepsilon)\\}\subset\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\times_{\mathrm{Slice}(\mathsf{N})}\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right).$

And by the previous result
$\tau^{\prime}\in[\tau]\cap\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$
if and only if
$\mu_{\mathsf{N}}(\Phi_{\mathsf{N}}(\tau^{\prime}))=\mu_{\mathsf{N}}(\Phi_{\mathsf{N}}(\tau))$
and $\rho_{\mathsf{N}}(\tau^{\prime})\sim\rho_{\mathsf{N}}(\tau)$. The set of
such $\tau^{\prime}$ is closed in the above fibre product, therefore also in
$\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$ and in
$[\tau]$. ∎

These lemmas show that
$\mathrm{U}^{Q}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)=\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)/{\sim}$
is an open neighbourhood of
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$ in
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)$ and allow us to describe the
stratification within this neighbourhood.

######  Corollary 8.6.

Suppose that support propagates from
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$. Then for
sufficiently small $\varepsilon>0$ the map

$\mu_{\mathsf{N}}\circ\Phi_{\mathsf{N}}\times[\rho_{\mathsf{N}}]\colon\mathrm{U}^{Q}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\to\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\times\mathrm{Stab}^{Q}\\!\left(\mathsf{N}\right),\quad[\sigma]\mapsto(\mu_{\mathsf{N}}\left(\Phi_{\mathsf{N}}(\sigma)\right),[\rho_{\mathsf{N}}(\sigma)])$

is a homeomorphism onto an open subset. Moreover, if for each thick
$\mathsf{M}\subset\mathsf{N}$ support also propagates from
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}/\mathsf{M},\mathsf{N}/\mathsf{M}\right)$
then the homeomorphism above restricts to a biholomorphism from
$\mathrm{U}^{Q}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\cap\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{M}\right)$
onto (an open subset of) a union of components of
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\times\mathrm{Stab}^{Q}\\!\left(\mathsf{N},\mathsf{M}\right)$.
In particular, locally we have

$\overline{\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{M}\right)}\cong\overline{\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\times\mathrm{Stab}^{Q}\\!\left(\mathsf{N},\mathsf{M}\right)}\cong\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\times\overline{\mathrm{Stab}^{Q}\\!\left(\mathsf{N},\mathsf{M}\right)}.$

###### Proof.

The existence of the homeomorphism, and the fact that it is stratum-
preserving, follow from Corollary 7.4 and Lemma 8.4. It restricts to a
holomorphic isomorphism because it is compatible with the charge maps. ∎

######  Theorem 8.7.

Suppose that support propagates from both
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ and
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}/\mathsf{M},\mathsf{N}/\mathsf{M}\right)$
for each thick $\mathsf{M}\subset\mathsf{N}$. Then the space of quotient
stability conditions is stratified by complex manifolds. More precisely,

$\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)=\bigsqcup_{\mathrm{thick}~{}\mathsf{N}\subset\mathsf{C}}\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$

decomposes into a disjoint union of locally closed subsets. Each
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$ is a complex manifold
of dimension $\mathrm{rk}(\Lambda/\Lambda_{\mathsf{N}})$; we refer to its
connected components as strata. This decomposition satisfies the frontier
condition: the closure of each stratum is a union of strata.

###### Proof.

We have already seen that there is such a decomposition into locally closed
subsets $\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$ and, by
Corollary 8.3, $\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$ is a
complex manifold locally homeomorphic to
${\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$.

For the last part, consider
$[\sigma]\in\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$. Note that
$[\rho_{\mathsf{N}}(\sigma)]\in\mathrm{Stab}^{Q}\\!\left(\mathsf{N},\mathsf{N}\right)$
which is a discrete set of points. If
$[\rho_{\mathsf{N}}(\sigma)]\in\overline{\mathrm{Stab}^{Q}\\!\left(\mathsf{N},\mathsf{M}\right)}$
then
$[\sigma]\in\overline{\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{M}\right)}$
by the last part of Corollary 8.6. Since
$\mathrm{Stab}^{Q}\\!\left(\mathsf{N},\mathsf{N}\right)$ is discrete and we
have a local product description, an open neighbourhood of $[\sigma]$ in
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$ is contained in

$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\times\\{\,[\rho_{\mathsf{N}}(\sigma)]\,\\}\subset\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\times\overline{\mathrm{Stab}^{Q}\\!\left(\mathsf{N},\mathsf{M}\right)}.$

Therefore by the formula at the end of Corollary 8.6 the connected component
of $[\sigma]$ in $\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$ is
in the closure of $\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{M}\right)$. ∎

######  Corollary 8.8.

Suppose support propagates from
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$. Let
$\Phi_{\mathsf{N},t}$ be the deformation retraction of Definition 7.5. Then

$[\Phi_{\mathsf{N},t}]\colon\mathrm{U}^{Q}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\to\mathrm{U}^{Q}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right),\quad[\sigma]\mapsto[\Phi_{\mathsf{N},t}(\sigma)]$

is an almost-stratum-preserving deformation retraction of
$\mathrm{U}^{Q}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$ onto
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$. By this we mean that
the track of each point under the deformation retraction remains in the same
stratum until the last point, at which it may enter a higher codimension
stratum.

###### Proof.

Suppose
$\sigma,\tau\in\mathrm{U}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$
and $\sigma\sim\tau$. Then
$\Phi_{\mathsf{N},t}(\sigma)\sim\Phi_{\mathsf{N},t}(\tau)$ by Lemma 8.4,
because
$\Phi_{\mathsf{N}}(\Phi_{\mathsf{N},t}(\sigma))=\Phi_{\mathsf{N}}(\sigma)$ and
$\rho_{\mathsf{N}}(\Phi_{\mathsf{N},t}(\sigma))=\rho_{\mathsf{N}}(\sigma)\cdot
i\log(t)/\pi$. Thus $\Phi_{\mathsf{N},t}$ descends to the quotient. Since
$\Phi_{\mathsf{N},t}(\sigma)$ has the same massless subcategory as $\sigma$
for $t\in(0,1]$ the resulting deformation retraction is almost-stratum-
preserving, i.e. its tracks remain in the same stratum until the last moment,
when they move into the deeper stratum
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$. ∎

###  8.3. Group actions

To complete our description of the space of quotient stability conditions, we
note that the actions of $\mathrm{Aut}_{\Lambda}(\mathsf{C})$ and $G$ descend
to it, and that they respect the stratification and local models.

######  Corollary 8.9.

The actions of $\mathrm{Aut}(\mathsf{C})$ and of $G$ on
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ extend uniquely to continuous
actions on $\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)$. The maps in the
diagram

${\mathrm{Stab}\\!\left(\mathsf{C}\right)}$${\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)}$${\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)}$${{\mathrm{Hom}}(\Lambda,\mathbb{C})}$$\scriptstyle{\mathcal{Z}}$$\scriptstyle{\mathcal{Z}}$$\scriptstyle{\mathcal{Z}}$

are equivariant with respect to these actions.

The actions respect the stratification: elements of $G$ preserve
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$ whereas the action of
an automorphism $\alpha\in\mathrm{Aut}_{\Lambda}(\mathsf{C})$ takes
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$ to
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\alpha(\mathsf{N})\right)$. Moreover,
the neighbourhood
$\mathrm{U}^{Q}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$ is
$\mathbb{C}$-invariant and $\alpha\in\mathrm{Aut}_{\Lambda}(\mathsf{C})$ maps
it to
$\mathrm{U}^{Q}_{\varepsilon}\\!\left(\mathsf{C},\alpha(\mathsf{N})\right)$.

When support propagates from
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ the actions are
compatible with the local model in Corollary 8.6 in that

${\mathrm{U}^{Q}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)}$${\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\times\mathrm{Stab}^{Q}\\!\left(\mathsf{N}\right)}$${\mathrm{U}^{Q}_{\varepsilon}\\!\left(\mathsf{C},\alpha(\mathsf{N})\right)}$${\mathrm{Stab}\\!\left(\mathsf{C}/\alpha(\mathsf{N})\right)\times\mathrm{Stab}^{Q}\\!\left(\alpha(\mathsf{N})\right)}$$\scriptstyle{\alpha}$$\scriptstyle{\alpha\times\alpha}$

commutes, and that the horizontal maps are $\mathbb{C}$-equivariant with
respect to the evident diagonal action of $\mathbb{C}$ on the right hand
terms.

###### Proof.

It is easy to check that the actions preserve the equivalence relation and so
descend to the quotient. Lemma 5.15 implies they respect the stratification as
stated. The final part follows from the properties of the maps
$\mu_{\mathsf{N}}$, $\rho_{\mathsf{N}}$ and $\Phi_{\mathsf{N}}$ — see Lemma
5.15, Section 5.4, and Lemma 7.2 respectively. ∎

##  9\. Codimension one strata

We investigate the case in which the massless subcategory is non-zero but ‘as
small as possible’. More precisely, throughout this section we fix a thick
subcategory $\mathsf{N}$ for which the saturation $\Lambda_{\mathsf{N}}$ of
the image of $K(\mathsf{N})\to K(\mathsf{C})\to\Lambda$ is a rank one lattice.
We abuse terminology by saying that the massless subcategory $\mathsf{N}$ has
rank one. We assume that
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ is non-empty. For
simplicity of notation we also assume that it is connected; otherwise we
consider each component separately. The main result of this section, Theorem
9.4, is that $\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ is a
component of
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\times\mathbb{R}$. In
particular it is a real codimension one boundary stratum in
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$.

###  9.1. Objects and phases

When $\mathrm{rk}(\Lambda_{\mathsf{N}})=1$ the massless subcategory
$\mathsf{N}$ has a simple form. In particular, up to shift, all the massless
objects must have the same phase. This remains true even for nearby stability
conditions.

######  Lemma 9.1.

Suppose $\sigma\in\overline{\mathrm{Stab}\\!\left(\mathsf{C}\right)}$ is a lax
pre-stability condition with rank one massless subcategory $\mathsf{N}$. Then
$\mathsf{N}=\mathsf{triang}(S)$ is the triangulated closure of a set $S$ of
stable objects of the same phase.

###### Proof.

Write $\sigma=(P,Z)$. Suppose $0\neq c\in\mathsf{N}$ is a $\sigma$-semistable
object with phase $0<\varphi(c)\leq 1$. Given $0<\varepsilon<1/8$ we can
choose $\tau=(Q,W)\in\mathrm{Stab}\\!\left(\mathsf{C}\right)$ with
$d(P,Q)<\varepsilon$. By Lemma 5.6 all $\tau$-semistable factors of $c$ are in
$\mathsf{N}$. Since they have phases in
$(\varphi(c)-\varepsilon,\varphi(c)+\varepsilon)$ and $m_{\tau}(c)>0$ we
deduce that $W|_{\Lambda_{\mathsf{N}}}\neq 0$. Therefore we can choose a
generator $\lambda\in\Lambda_{\mathsf{N}}$ with
$W(\lambda)\in\mathbb{H}\cup\mathbb{R}_{<0}$. Let $\psi\in(0,1]$ be the phase
of $W(\lambda)$. Then $|\varphi(c)-\psi|<\varepsilon$ and if
$c^{\prime}\in\mathsf{N}$ is another $\sigma$-semistable object with phase
$0<\varphi(c^{\prime})\leq 1$ we have

$|\varphi(c)-\varphi(c^{\prime})|\leq|\varphi(c)-\psi|+|\psi-\varphi(c^{\prime})|<2\varepsilon.$

It follows that $\varphi(c)=\varphi(c^{\prime})=\varphi$, say, for all
$\sigma$-semistable objects in $\mathsf{N}$, i.e. $\mathsf{N}\cap
P(0,1]\subset P(\varphi)$.

Since $\sigma\in\overline{\mathrm{Stab}\\!\left(\mathsf{C}\right)}$ its
slicing $P$ is locally finite so that $P(\psi)$ is a quasi-abelian length
category. The full subcategory $\mathsf{N}\cap P(0,1]$ is closed under
extensions, strict subobjects and strict quotients. Therefore $\mathsf{N}\cap
P(0,1]={\langle S\rangle}$ is the extension-closure of a subset $S$ of simple
objects of $P(\varphi)$. Hence $\mathsf{N}=\mathsf{triang}(S)$. ∎

######  Corollary 9.2.

Fix $0<\varepsilon<1/8$. There is a set $S$ of objects with
$\mathsf{N}=\mathsf{triang}(S)$ and such that for each $\sigma$ in the open
neighbourhood
$\mathrm{B}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)$ of
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ all objects in $S$
are stable of the same phase.

###### Proof.

Let $\sigma=(P,Z)\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$.
By the previous lemma there is $0<\varphi\leq 1$ and a subset $S$ of simple
objects in $P(\varphi)$ with $\mathsf{N}=\mathsf{triang}(S)$. Suppose
$\tau=(Q,W)\in
B_{\varepsilon}(\sigma)\cap\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$. Lemma
5.6 implies that

${\langle S\rangle}\subset
Q(\varphi-\varepsilon,\varphi+\varepsilon)\cap\mathsf{N}\subset
P(\varphi-2\varepsilon,\varphi+2\varepsilon)\cap\mathsf{N}.$

Since $P(0,1]\cap\mathsf{N}={\langle S\rangle}$ and $0<\varepsilon<1/8$ the
right-hand side above is ${\langle S\rangle}[k]$ for $k=0$ or $\pm 1$. Since
it contains ${\langle S\rangle}$ we must have $k=0$ so that the above
containments are equalities. Therefore all $\tau$-semistable factors of each
$s\in S$ are also in ${\langle S\rangle}$, and since each object $s\in S$ is
simple in ${\langle S\rangle}$ this implies that each $s\in S$ is
$\tau$-semistable. It follows as in the proof of the previous lemma that all
$s\in S$ have the same phase, so that in fact ${\langle
S\rangle}=Q(\psi)\cap\mathsf{N}$ for some
$\psi\in(\varphi-\varepsilon,\varphi+\varepsilon)$. Since each $s\in S$ is
simple, we deduce that each $s\in S$ is actually $\tau$-stable. The result now
follows from the definition

$\mathrm{B}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)=\bigcup_{\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)}B_{\varepsilon}(\sigma)\cap\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$

of the neighbourhood and the fact that we have assumed
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ is connected. ∎

###  9.2. The neighbourhood of a stratum

The neighbourhood of the stratum
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ has a simple form
when $\mathrm{rk}(\Lambda_{\mathsf{N}})=1$: it is a fibration over
$\mathbb{C}$.

Recall that $\mathrm{Stab}\\!\left(\mathsf{N}\right)$ is the space of
stability conditions on $\mathsf{N}$ whose charges factor through
$\Lambda_{\mathsf{N}}\cong\mathbb{Z}$. The space of non-zero charges is
${\mathrm{Hom}}(\Lambda_{\mathsf{N}},\mathbb{C})-\\{0\\}\cong\mathbb{C}^{*}$.
As $\mathbb{C}$ acts freely on $\mathrm{Stab}\\!\left(\mathsf{N}\right)$ we
deduce that each component is homeomorphic to the universal cover
$\mathbb{C}$, with $\mathbb{C}$ acting freely and transitively. The image of
the continuous restriction map

$\rho_{\mathsf{N}}\colon\mathrm{B}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\to\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right),\quad(P,Z)\mapsto(P\cap\mathsf{N},Z|_{\Lambda_{\mathsf{N}}})$

consists of lax stability conditions in which the simple objects of the
generating set $S$ are stable and have a common phase.

######  Lemma 9.3.

Fix $0<\varepsilon<1/8$. The map $\rho_{\mathsf{N}}$ restricts to a
holomorphic fibration from
$\mathrm{B}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\cap\mathrm{Stab}\\!\left(\mathsf{C}\right)$
onto the component of $\mathrm{Stab}\\!\left(\mathsf{N}\right)$ consisting of
stability conditions with heart a shift of ${\langle S\rangle}$.

###### Proof.

If
$\sigma=(P,Z)\in\mathrm{B}^{L}_{\varepsilon}\\!\left(\mathsf{C},\mathsf{N}\right)\cap\mathrm{Stab}\\!\left(\mathsf{C}\right)$
then clearly
$\rho_{\mathsf{N}}(\sigma)\in\mathrm{Stab}\\!\left(\mathsf{N}\right)$.
Moreover the objects of $S$ are stable with respect to
$\rho_{\mathsf{N}}(\sigma)$ and by Corollary 9.2 have a common phase. It
follows that the heart $P(0,1]\cap\mathsf{N}$ is a shift of the abelian length
category ${\langle S\rangle}$. The $\mathbb{C}$-equivariance of
$\rho_{\mathsf{N}}$ implies that the restriction is surjective onto this
component. Indeed it is a holomorphic submersion whose fibres are
biholomorphic to one another, and therefore it is a holomorphic fibration. ∎

The only massless subcategory appearing in the boundary of this component is
$\mathsf{N}$ itself. A lax stability condition is determined by a common phase
$\varphi\in\mathbb{R}$ for the objects in $S$. Thus we have a commuting
diagram

${\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)}$${\mathbb{C}\cup(-\infty+i\mathbb{R})}$${{\mathrm{Hom}}(\Lambda_{\mathsf{N}},\mathbb{C})}$${\mathbb{C}}$$\scriptstyle{\mathcal{Z}}$$\scriptstyle{\simeq}$$\scriptstyle{\exp}$$\scriptstyle{\simeq}$

where we define $\exp(-\infty+ir)=0$ for $r\in\mathbb{R}$. In the next section
we show that the restriction of $\rho_{\mathsf{N}}$ to
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ is a fibration over
$-\infty+i\mathbb{R}$ with fibre
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)$.

###  9.3. The boundary stratum

In this section we show that the boundary stratum
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ is (a component of)
the product
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\times\mathbb{R}$ of the
space of stability conditions on the quotient category $\mathsf{C}/\mathsf{N}$
and a factor $\mathbb{R}$ recording the common phase of the massless objects.
Recall that, for simplicity of notation only, we assume that
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ is connected.

We first observe that there are inclusions

$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)\subset\mathrm{Stab}^{LS}\\!\left(\mathsf{C},\mathsf{N}\right)\subset\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\times\mathbb{R},$

where we abuse notation by identifying
$\sigma\in\mathrm{Stab}^{LS}\\!\left(\mathsf{C},\mathsf{N}\right)$ with its
image under the continuous inclusion
$\mu_{\mathsf{N}}\times\rho_{\mathsf{N}}$, and where
$\mathrm{Stab}^{LS}\\!\left(\mathsf{C},\mathsf{N}\right)$ denotes the set of
lax pre-stability conditions $\sigma$ in the closure of
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ with massless subcategory
$\mathsf{N}$ and such that $\mu_{\mathsf{N}}(\sigma)$ is a stability condition
on $\mathsf{C}/\mathsf{N}$; see Definition 5.1.

######  Theorem 9.4.

Support propagates from
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$. Therefore,
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ is a component of
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\times\mathbb{R}$ and
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$ is a component of
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)$.

###### Proof.

Let $\sigma=(P,Z)\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$.
We show that support propagates from $\sigma$ in two steps:

  1. (1)

for $\tau$ sufficiently close to $\sigma$, we have
$\tau\in\mathrm{Stab}^{LS}\\!\left(\mathsf{C},\mathsf{N}\right)$; and,

  2. (2)

any $\tau\in\mathrm{Stab}^{ls}\\!\left(\mathsf{C}\right)$ with massless
subcategory $\mathsf{N}$ satisfies the support property.

Step 1: Choose $\varepsilon>0$ sufficiently small that we can apply Theorem
6.1 and Theorem 6.17. Let $\tau=(Q,W)\in B_{\varepsilon}(\sigma)$. If
$W|_{\Lambda_{\mathsf{N}}}\neq 0$ then $\tau$ is a (classical) pre-stability
condition in $B_{\varepsilon}(\sigma)$ and is therefore actually a stability
condition by the argument at the end of the proof of Theorem 6.1. Therefore we
may assume that $W\in{\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})$.
Then by Theorem 6.17

(8) $\\{\tau=(Q,W)\in
B_{\varepsilon}(\sigma):W\in{\mathrm{Hom}}(\Lambda/\Lambda_{\mathsf{N}},\mathbb{C})\\}\subset\mathrm{Stab}^{LS}\\!\left(\mathsf{C},\mathsf{N}\right)$

because $\mu_{\mathsf{N}}(\sigma)$ is a stability condition and so therefore
is any sufficiently small deformation of it. (The condition that
$\rho_{\mathsf{N}}(\tau)$ is in $\mathrm{Stab}^{L}\\!\left(\mathsf{N}\right)$
is automatic because
$\mathrm{Stab}^{L}\\!\left(\mathsf{N},\mathsf{N}\right)\cong\mathbb{R}$.)

Step 2: It remains to show that any
$\tau\in\mathrm{Stab}^{LS}\\!\left(\mathsf{C},\mathsf{N}\right)$ satisfies the
support property, for which we need the following lemma, whose proof we defer.

######  Lemma 9.5.

Suppose
$\sigma=(P,Z)\in\mathrm{Stab}^{LS}\\!\left(\mathsf{C},\mathsf{N}\right)$. Let
$S\subset P(\varphi)$ be a set of stable objects such that
$\mathsf{N}=\mathsf{triang}(S)$. Then, up to shifts, any sequence $(b_{n})$ of
massive stable objects with $m(b_{n})/||v(b_{n})||\to 0$ contains a
subsequence $(c_{n})$ whose phases converge and such that

$\lim_{n\to\infty}\left(\frac{v(c_{n})}{||v(c_{n})||}\right)=\frac{v(s)}{||v(s)||}$

for any $s\in S$. Moreover,
$\lim_{n\to\infty}\varphi(c_{n})\in\varphi+2\mathbb{Z}$ for any such
subsequence.

Let $S\subset Q(\varphi)$ be a set of $\tau$-stable objects such that
$\mathsf{N}=\mathsf{triang}(S)$. By Lemma 9.5 any sequence $(b_{n})$ of
massive stable objects with $m(b_{n})/||v(b_{n})||\to 0$ has, up to shifts, a
subsequence $(c_{n})$ with $\varphi(c_{n})\to\varphi$. Without loss of
generality we may assume $\varphi(c_{n})\leq\varphi$ for all $n\in\mathbb{N}$;
the other case is similar. Since $c_{n}$ is stable and not in $S$ this implies
that ${\mathrm{Hom}}(s,c_{n})=0$ for all $s\in S$.

Perturbing the common phase of the massless objects slightly using Corollary
6.10, we can find $\tau^{\prime}=(Q^{\prime},W)$ also in
$\mathrm{Stab}^{LS}\\!\left(\mathsf{C},\mathsf{N}\right)$ with
$\mu_{\mathsf{N}}(\tau^{\prime})=\mu_{\mathsf{N}}(\tau)$ and $S\subset
Q^{\prime}(\varphi^{\prime})$ where $\varphi^{\prime}>\varphi$. The $c_{n}$
remain massive and stable for $\tau^{\prime}$ by Proposition 3.2 because

$c_{n}\in
Q_{\mathsf{C}/\mathsf{N}}(\varphi(c_{n}))=Q^{\prime}_{\mathsf{C}/\mathsf{N}}(\varphi(c_{n}))$

where $\varphi(c_{n})<\varphi^{\prime}$ and ${\mathrm{Hom}}(s,c_{n})=0$ for
all $s\in S$. However now
$\varphi^{\prime}(c_{n})=\varphi(c_{n})\to\varphi<\varphi^{\prime}$
contradicting Lemma 9.5. We conclude that there is no sequence $(b_{n})$ of
massive stable objects with $m(b_{n})/||v(b_{n})||\to 0$. Therefore $\tau$
satisfies the support property and so is in
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$.

The final statement on the space of quotient stability conditions follows from
Proposition 7.1 and the definition of
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C},\mathsf{N}\right)$ because
$\mathrm{Stab}^{L}\\!\left(\mathsf{N},\mathsf{N}\right)\cong\mathbb{R}$. ∎

###### Proof of Lemma 9.5.

If $\sigma$ satisfies the support property then there is no sequence $(b_{n})$
with $m(b_{n})/||v(b_{n})||\to 0$ and the result is vacuously true. Therefore
we assume $\sigma$ does not satisfy the support property i.e. there is a
sequence $(b_{n})$ of massive stable objects with $m(b_{n})/||v(b_{n})||\to
0$.

Since $\Lambda_{\mathsf{N}}$ has rank one and by Corollary 9.2 the objects in
$S$ are stable with a common phase in nearby classical stability conditions,
there is $\lambda\in\Lambda_{\mathsf{N}}\otimes\mathbb{R}$ with
$||\lambda||=1$ such that $v(s)\in\mathbb{R}_{>0}\cdot\lambda$ for all $s\in
S$. Lemma 4.16 tells us that there is a subsequence $(c_{n})$ such that

$\lim_{n\to\infty}\left(\frac{v(c_{n})}{||v(c_{n})||}\right)\in\Lambda_{\mathsf{N}}\otimes\mathbb{R}.$

By replacing $c_{n}$ with $c_{n}[1]$ if necessary we can ensure that the limit
is $\lambda=v(s)/||v(s)||$. Further shifting by an even integer if necessary
we may assume that $\varphi(c_{n})\in[\varphi-1,\varphi+1]$ for each
$n\in\mathbb{N}$. Therefore, we can pass to a subsequence $(c_{n})$ whose
phases converge. It remains to show that the limit is the common phase
$\varphi$ of the objects $s\in S$.

Suppose that $(\sigma_{m})$ is a sequence in
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ tending to $\sigma$. By the argument
in the proof of Corollary 9.2, for sufficiently large $m\in\mathbb{N}$, the
objects in $S$ are $\sigma_{m}$-stable with common phase $\varphi_{m}(s)$ for
all $s\in S$. For similar reasons, $c_{n}$ is massive for each $\sigma_{m}$
and its charge has a well-defined phase $\varphi_{m}(c_{n})$ in the interval
$(\varphi(c_{n})-1/8,\varphi(c_{n})+1/8)$. Since $d(P_{m},P)\to 0$ and
$\varphi(c_{n})\to\varphi^{\prime}$, say, for any $\varepsilon>0$ we can find
$M,N\in\mathbb{N}$ such that

$|\varphi_{m}(c_{n})-\varphi^{\prime}|\leq|\varphi_{m}(c_{n})-\varphi(c_{n})|+|\varphi(c_{n})-\varphi^{\prime}|<\varepsilon$

whenever $m\geq M$ and $n\geq N$. Therefore we can switch the limits to
compute

$\varphi^{\prime}=\lim_{n\to\infty}\varphi(c_{n})=\lim_{n\to\infty}\lim_{m\to\infty}\varphi_{m}(c_{n})=\lim_{m\to\infty}\lim_{n\to\infty}\varphi_{m}(c_{n})=\lim_{m\to\infty}\varphi_{m}(s)=\varphi$

where the penultimate step follows because the phase $\varphi_{m}$ only
depends on the normalised vector in $\Lambda\otimes\mathbb{R}$ and
$v(c_{n})/||v(c_{n})||\to v(s)/||v(s)||$. ∎

######  Remark 9.6.

In Step 2 of the proof of Theorem 9.4 we established the support property for
all elements of $\mathrm{Stab}^{LS}\\!\left(\mathsf{C},\mathsf{N}\right)$.
This special situation in the codimension one case is not obvious in general:
Example 6.4 gives an example of a lax pre-stability condition which satisfies
support on the quotient of $\mathsf{C}/\mathsf{N}$ but does not satisfy the
stronger support property required to be a lax stability condition. However,
this is not quite a counter-example because it is not in the closure of
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ since the slicings do not converge
as we approach it.

##  10\. Finite type components

Let $\mathsf{C}$ be a triangulated category with
$K(\mathsf{C})\cong\mathbb{Z}^{n}$ and set
$v=\operatorname{{\mathsf{id}}}\colon K(\mathsf{C})\to\Lambda$. For simplicity
of notation we assume that $\mathrm{Stab}\\!\left(\mathsf{C}\right)$ is
connected; otherwise consider a component. Say that a stability condition
$\sigma=(P,Z)$ has _discrete phase distribution_ if
$\\{\varphi\in\mathbb{R}:P(\varphi)\neq 0\\}$ is a discrete subset of
$\mathbb{R}$, and that it has _algebraic heart_ if $P(0,1]$ is a finite length
abelian category with finitely many simple objects. Finally, we say that
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ has _finite type_ if each stability
condition has an algebraic heart admitting only finitely many torsion pairs
[30].

######  Lemma 10.1.

Assume that $K(\mathsf{C})\cong\mathbb{Z}^{n}$ is free abelian. The following
are equivalent:

  1. (1)

Each $\sigma\in\mathrm{Stab}\\!\left(\mathsf{C}\right)$ has algebraic heart.

  2. (2)

$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ has finite type.

  3. (3)

$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ has dimension $n$ and each stability
condition has discrete phase distribution.

###### Proof.

$(1)\implies(2)$: Suppose each
$\sigma\in\mathrm{Stab}\\!\left(\mathsf{C}\right)$ has algebraic heart. Then
the set of hearts of stability conditions in
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ is closed under HRS-tilting at
simple objects of these hearts (see, for example, [19, 33] for the
definition), because a stability condition with such a heart is given by
freely assigning a charge in the upper half plane to each simple object of the
heart.

Moreover, by [30, Corollary 3.27] the poset of strata is pure of length
$n=\mathrm{rk}\,K(\mathsf{C})$, i.e. each maximal chain in the poset has
length $n$. This implies that each heart has only finitely many torsion pairs
because the poset of torsion pairs in the heart has a uniform bound on the
valency of each element and on the length of each chain. Therefore
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ has finite type.

$(2)\implies(3)$: Now suppose $\mathrm{Stab}\\!\left(\mathsf{C}\right)$ has
finite type. The phase distribution of each
$\sigma\in\mathrm{Stab}\\!\left(\mathsf{C}\right)$ must be discrete, for
otherwise rotating phases would yield an infinite sequence of tilts, and thus
infinitely many torsion pairs in the heart of $\sigma$.

$(3)\implies(1)$: Finally, suppose each
$\sigma\in\mathrm{Stab}\\!\left(\mathsf{C}\right)$ has discrete phase
distribution. Then for each $\sigma$ there exists $\varepsilon>0$ such that
$P_{\sigma}(0,\varepsilon)=0$. Hence by [30, Lemma 3.1] the heart of $\sigma$
is algebraic. ∎

In general, we do not know which thick subcategories $\mathsf{N}$ of
$\mathsf{C}$ can appear as massless subcategories. If $\mathsf{N}$ occurs as a
massless subcategory then, by Corollary 7.4, both
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)$ and
$\mathrm{Stab}\\!\left(\mathsf{N}\right)$ are non-empty. However this is not
sufficient, as can be already seen for $\mathsf{D}^{b}(\mathbb{P}^{1})$.

######  Example 10.2.

Let $\mathsf{C}=\mathsf{D}^{b}(\mathbb{P}^{1})$ with
$\Lambda=K(\mathsf{C})\cong\mathbb{Z}^{2}$ and let
$\mathsf{N}=\mathsf{thick}(\mathcal{O}_{x}:x\in\mathbb{P}^{1})$. Note
$\Lambda_{\mathsf{N}}=\mathrm{im}(K(\mathsf{N})\to
K(\mathsf{C})\to\Lambda)\cong\mathbb{Z}$ even though
$K(\mathsf{N})\cong\mathbb{Z}^{\mathbb{P}^{1}}$. With these lattices,
$\mathrm{Stab}\\!\left(\mathsf{N}\right)\cong\mathbb{C}$ and
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\cong\mathbb{C}$ are non-
empty. However, it follows from Proposition 12.6 that $\mathsf{N}$ does not
occur as a massless subcategory because the skyscrapers $\mathcal{O}_{x}$ are
not simple objects in any algebraic heart.

For a finite type component of a stability space the situation is much
simpler: we can classify the massless subcategories completely, and show that
support propagates from each boundary stratum.

######  Corollary 10.3.

Suppose $\mathrm{Stab}\\!\left(\mathsf{C}\right)$ has finite type and
$\sigma=(P,Z)\in\overline{\mathrm{Stab}\\!\left(\mathsf{C}\right)}$. Then for
any $\varphi\in\mathbb{R}$ the full subcategory $P(\varphi,\varphi+1]$ is the
heart of some classical stability condition in
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$, and so, in particular, is
algebraic. Moreover, $\sigma$ has discrete phase distribution.

###### Proof.

Rotating phases by the $\mathbb{C}$ action it suffices to prove the first part
for $\varphi=0$. Choose $\tau=(Q,W)$ in
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ with $d(P,Q)<1/2$ so that the hearts
$P(0,1]$ and $Q(0,1]$ are both contained in $Q(-1/2,3/2]$. Then $P(0,1]$ is
obtained by first tilting from $Q(0,1]$ to $Q(-1/2,1/2]$ and then tilting from
that to $P(0,1]$. Since the set of hearts of stability conditions in the
finite-type component $\mathrm{Stab}\\!\left(\mathsf{C}\right)$ is closed
under tilting, $P(0,1]$ is the heart of some stability condition in
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$. Therefore $P(0,1]$ is algebraic.

Since $P(\varphi,\varphi+1]$ is algebraic there exists $\varepsilon>0$,
depending on $\varphi$, for which $P(\varphi,\varphi+\varepsilon)=0$. It
follows that $\sigma$ has discrete phase distribution. ∎

######  Corollary 10.4.

Suppose $\mathrm{Stab}\\!\left(\mathsf{C}\right)$ has finite type. Then
$\mathsf{N}$ is the massless subcategory of some
$\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$ if and only if
$\mathsf{N}$ is the triangulated closure of a finite subset of simple objects
in the heart of some classical stability condition
$\tau\in\mathrm{Stab}\\!\left(\mathsf{C}\right)$.

###### Proof.

Suppose $\mathsf{N}$ is generated by a finite subset $I$ of simple objects in
the heart $Q(0,1]$ of some classical
$\tau=(Q,W)\in\mathrm{Stab}\\!\left(\mathsf{C}\right)$. By deforming the
charge we may assume that

$W(s)=\begin{cases}-r&\iff s\in I,\\\ -1&\iff s\not\in I,\end{cases}$

for some $r\in\mathbb{R}_{>0}$. Clearly the charges converge as $r\to 0$.
Moreover, since $Q(0,1]=Q(1)$ for all $r$ the slicings also converge as $r\to
0$. In the limit as $r\to 0$ we obtain a lax stability condition $\sigma$ with
massless subcategory $\mathsf{N}$ and the same slicing $Q$. Therefore the only
stable objects of $\sigma$ are the simple objects of $Q(0,1]$. Since there are
finitely many of these, $\sigma$ satisfies the support property and therefore
lies in $\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$.

Now suppose $\sigma=(P,Z)\in\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$. By
Lemma 4.5 the massless subcategory $\mathsf{N}$ is the triangulated closure of
the set of stable massless objects in $P(0,1]$. Since $P(0,1]$ is algebraic
each stable massless object has a composition series whose factors are
massless simple objects, each of which is of course also stable. Therefore
$\mathsf{N}=\mathsf{triang}(S)$ is the triangulated closure of the set $S$ of
massless simple objects in $P(0,1]$. The result follows because $P(0,1]$ is
the heart of some classical stability condition in
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$ by Corollary 10.3. ∎

######  Remark 10.5.

Corollary 10.4 tells us that massless subcategories need not be admissible. An
example is the derived-discrete algebra $A=\Lambda(2,1,0)$; see [4] for
notation and [26, §8] for a detailed description of $\mathsf{D}^{b}(A)$. One
of the two simple modules $S$ is a $2$-spherical object. Therefore, its thick
hull $\mathsf{N}\coloneqq\mathsf{thick}(S)$ occurs as a massless subcategory.
On the other hand, the duality property
${\mathrm{Hom}}(S,-)={\mathrm{Hom}}(-,S[2])^{*}$ implies that an adjoint of
the inclusion $\mathsf{N}\hookrightarrow\mathsf{D}^{b}(A)$ would lead to a
splitting $\mathsf{D}^{b}(A)\cong\mathsf{N}\oplus\mathsf{M}$ but
$\mathsf{D}^{b}(A)$ is indecomposable.

######  Proposition 10.6.

If $\mathrm{Stab}\\!\left(\mathsf{C}\right)$ has finite type, then
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)=\overline{\mathrm{Stab}\\!\left(\mathsf{C}\right)}$.

###### Proof.

Suppose $\sigma=(P,Z)\in\overline{\mathrm{Stab}\\!\left(\mathsf{C}\right)}$
does not satisfy the support property. Then we can find a sequence $(c_{n})$
of massive stable objects with $m_{\sigma}(c_{n})/||v(c_{n})||\to 0$. Shifting
the objects if necessary we may assume they all have phase in $(0,1]$ and
hence, by passing to a subsequence, that the phases $\varphi(c_{n})\to\varphi$
as $n\to\infty$. Corollary 10.3 states that $\sigma$ has a discrete phase
distribution so, again passing to a subsequence if required, we may assume
that $\varphi(c_{n})=\varphi$ for all $n\in\mathbb{N}$. Thus the $c_{n}$ are
simple objects in $P(\varphi)$, and therefore also simple in
$P(\varphi-1,\varphi]$. There are only finitely many iso-classes of such
simple objects because $P(\varphi-1,\varphi]$ is algebraic by Corollary 10.3.
So, passing to a subsequence for a final time, we may assume that $(c_{n})$ is
a constant sequence. Since the $c_{n}$ are massive this contradicts the fact
that $m_{\sigma}(c_{n})/||v(c_{n})||\to 0$. We conclude that no such sequence
exists, i.e. that $\sigma$ satisfies support after all. ∎

######  Corollary 10.7.

If $\mathrm{Stab}\\!\left(\mathsf{C}\right)$ has finite type, then support
propagates from $\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$.

###### Proof.

This follows immediately from the fact that
$\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)=\overline{\mathrm{Stab}\\!\left(\mathsf{C}\right)}$
and Theorem 6.17 which show that deformations of an element of
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ remain in the closure
of $\mathrm{Stab}\\!\left(\mathsf{C}\right)$. ∎

######  Corollary 10.8.

Suppose $\mathrm{Stab}\\!\left(\mathsf{C}\right)$ has finite type and that
$\sigma\in\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$. Then
$\mu_{\mathsf{N}}(\sigma)$ is in a finite type component of
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)$.

###### Proof.

By Corollary 10.7, the entire component of $\mu_{\mathsf{N}}(\sigma)$ in
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)$ is in the image of
$\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ under
$\mu_{\mathsf{N}}$. The slicing of each lax stability condition $\sigma=(P,Z)$
in $\mathrm{Stab}^{L}\\!\left(\mathsf{C},\mathsf{N}\right)$ has discrete phase
distribution by Corollary 10.3. Since $P_{\mathsf{C}/\mathsf{N}}(\varphi)=0$
when $P(\varphi)=0$ the slicing $P_{\mathsf{C}/\mathsf{N}}$ of
$\mu_{\mathsf{N}}(\sigma)$ also has discrete phase distribution. Thus the
component of $\mu_{\mathsf{N}}(\sigma)$ has finite type by Lemma 10.1. ∎

######  Remark 10.9.

We have thus shown that the conditions for the structure results of §7 and §8
are satisfied. If $\mathrm{Stab}\\!\left(\mathsf{C}\right)$ is a finite type
component then the space $\mathrm{Stab}^{L}\\!\left(\mathsf{C}\right)$ of lax
stability conditions is a union of components of
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)\times\mathrm{Stab}^{L}\\!\left(\mathsf{N},\mathsf{N}\right)$,
where $\mathsf{N}$ is the triangulated closure of a finite set of simple
objects in the heart of a stability condition in
$\mathrm{Stab}\\!\left(\mathsf{C}\right)$. Each component of
$\mathrm{Stab}\\!\left(\mathsf{C}/\mathsf{N}\right)$ which appears has finite
type. The space $\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)$ of quotient
stability conditions is therefore stratified by these finite type components.

##  11\. Closures of $G$-orbits

Recall that the the universal cover $G$ of $\mathrm{GL}_{2}^{+}(\mathbb{R})$
acts on stability spaces and our partial compactifications. In this section we
describe the closures of $G$-orbits. The phase diagrams
$\Phi_{\sigma}=\\{\varphi+\mathbb{Z}:P(\varphi)\neq
0\\}\subset\mathbb{R}/\mathbb{Z}$ of stability conditions $\sigma=(P,Z)$ in
the same orbit are related by orientation-preserving diffeomorphisms of the
circle $\mathbb{R}/\mathbb{Z}$. The structure of the closure of the orbit in
both $\mathrm{Stab}\\!\left(\mathsf{C}\right)$ and in
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)^{*}$ can be described in terms of
the phase diagram. Here $\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)^{*}$ is
the space obtained by removing the strata where all objects are massless from
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)$. This is more convenient to
consider because the action of $\mathbb{C}$ on
$\mathrm{Stab}^{Q}\\!\left(\mathsf{C}\right)^{*}$ is free and in fact we will
describe the closure of $(\sigma\cdot G)/\mathbb{C}$ in
$\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ and in
$\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$.

Fix a stability condition $\sigma=(P,Z)$. If there is only one point in the
phase diagram $\Phi_{\sigma}$ then there is a non-trivial stabiliser and
$(\sigma\cdot G)/\mathbb{C}$ is a point. In this case the orbit is closed in
$\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ and also in
$\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$. Henceforth, we assume that
$\Phi_{\sigma}$ consists of at least two points. In particular the image of
the charge is the whole of $\mathbb{C}$ so that the $G$-orbit through $\sigma$
is free and $(\sigma\cdot G)/\mathbb{C}$ is biholomorphic to the Poincaré disk
$\mathbb{D}$ because $G\cong\mathbb{C}\times\mathbb{H}$. We show that the
closures of the orbit in $\mathrm{Stab}(\mathsf{C})/\mathbb{C}$ and in
$\mathrm{Stab}^{Q}(\mathsf{C})^{*}\\!/\mathbb{C}$ are homeomorphic to subsets
of the closed disk, with appropriate topologies. Our constructions require an
explicit identification of the Poincaré disk and the strict upper half-plane;
we choose $f\colon\mathbb{D}\to\mathbb{H}$ given by

$f(w)=i\frac{1+w}{1-w}.$

Note that $f$ extends to a homeomorphism from the closure of the disk to
$\mathbb{H}\cup\mathbb{R}\cup\\{\infty\\}$. For each
$w\in\overline{\mathbb{D}}$ we choose a charge $Z_{w}=M_{w}\circ Z$ where
$M_{w}\in{\mathrm{End}}_{\mathbb{R}}(\mathbb{C})$ satisfies

$M_{w}(1)=1\quad\text{and}\quad M_{w}(i)=f(w)\qquad(w\neq 1)$

and $M_{1}(1)=0$, $M_{1}(i)=-1$. Note that $M_{w}(i)\in\mathbb{H}$ when
$w\in\mathbb{D}$ so that there is a unique compatible slicing $P_{w}$ with
$P_{w}(0,1]=P(0,1]$. Mapping $w$ to the $\mathbb{C}$-orbit of
$\sigma_{w}=(P_{w},Z_{w})$ gives an explicit identification
$\mathbb{D}\cong(\sigma\cdot G)/\mathbb{C}$. The reason for this particular
choice is that

$\displaystyle P(\varphi)\subset\ker Z_{w}$ $\displaystyle\iff
M_{w}(e^{i\pi\varphi})=0$ $\displaystyle\iff
M_{w}(1)\cos(\pi\varphi)+f(w)\sin(\pi\varphi)=0$ $\displaystyle\iff\left(w=1\
\text{and}\ \varphi=0\right)\ \text{or}\ \left(w=e^{2\pi i\varphi}\
\text{and}\ \varphi\neq 0\right)$ $\displaystyle\iff w=e^{2\pi i\varphi}$

so that points on $\partial\mathbb{D}$ correspond to charges for which
semistable objects of a certain phase have vanishing mass in a natural way.
For $w=e^{2\pi i\varphi}\in\partial\mathbb{D}$ and $c\in P(\varphi^{\prime})$
the sine rule yields

(9)
$Z_{w}(c)=\begin{cases}|Z(c)|\dfrac{\sin\pi(\varphi-\varphi^{\prime})}{\sin\pi\varphi}&\text{
if }w\neq 1;\\\ -|Z(c)|\sin\pi\varphi^{\prime}&\text{ if }w=1.\end{cases}$

When $P(\varphi)=0$ there is a unique choice of slicing $P_{w}$ for $w=e^{2\pi
i\varphi}$ which is compatible with $Z_{w}$ and with $P(0,1]\subset
P_{w}[0,1]$, namely

$P_{w}(1)=P(\varphi,\varphi+1]=P[\varphi,\varphi+1)$

and all slices with phase in $(0,1)$ are zero. One can verify that
$\sigma_{w}=(P_{w},Z_{w})$ is a pre-stability condition.

######  Lemma 11.1.

Suppose $w=e^{2\pi i\varphi}\in\partial\mathbb{D}$ and $P(\varphi)=0$. Then
$\sigma_{w}=(P_{w},Z_{w})$ is a stability condition if and only if $\varphi$
is not an accumulation point of the phase diagram $\Phi_{\sigma}$.

###### Proof.

Suppose $\varphi\neq 0$ so that $M_{w}(1)=1$. If $\varphi$ is an accumulation
point of $\Phi_{\sigma}$ and $\varepsilon>0$ then using the first equation in
(9) shows that one can choose $\varphi^{\prime}\in\Phi_{\sigma}$ sufficiently
close to $\varphi$ that

$|Z_{w}(c)|\leq\frac{\varepsilon}{||Z||}|Z(c)|\leq\varepsilon||v(c)||$

for any $c\in P(\varphi^{\prime})$. Hence the support property fails since $c$
remains semistable for $\sigma_{w}$.

Conversely, if $\varphi$ is not an accumulation point then there is $L>0$ such
that

$|Z_{w}(c)|\geq L|Z(c)|\geq KL||v(c)||$

for all $c\in P(\varphi^{\prime})$ where $\varphi^{\prime}\in(0,1]$, and $K$
is a support constant for $\sigma$. Therefore the same inequality holds for
all $c\in P_{w}(1)=P(\varphi,\varphi+1]$ so that $\sigma_{w}$ satisfies the
support property as claimed.

The case $\varphi=0$ is similar but uses the second equation in (9). ∎

When $P(\varphi)\neq 0$ there is a one-parameter family of compatible slicings
$P_{w}^{\psi}$ for $w=e^{2\pi i\varphi}$ compatible with $Z_{w}$ and with
$P(0,1]\subset P_{w}[0,1]$. They differ by the choice of phase for the
massless objects in $P(\varphi)$. Namely, for each $\psi\in[0,1]$ there is a
unique such slicing $P_{w}^{\psi}$ with

$P_{w}^{\psi}(1)\supset P(\varphi,\varphi+1)\quad\text{and}\quad
P_{w}^{\psi}(\psi)\supset P(\varphi)$

and all other slices with phase in $(0,1]$ zero. One can verify that
$\sigma^{\psi}_{w}=(P^{\psi}_{w},Z_{w})$ is a lax pre-stability condition. We
now give criteria for when it is in the closure of the orbit, and when it
satisfies the support property.

######  Lemma 11.2.

Suppose $w=e^{2\pi i\varphi}\in\partial\mathbb{D}$ and $P(\varphi)\neq 0$. The
lax pre-stability condition $\sigma_{w}^{\psi}$ is in the closure of the orbit
$\sigma\cdot G$ in
$\mathrm{Slice}(\mathsf{C})\times{\mathrm{Hom}}(\Lambda,\mathbb{C})$ if and
only if one of the following conditions holds

  1. (1)

$\varphi$ is an isolated point of $\Phi_{\sigma}$ and $\psi\in[0,1]$;

  2. (2)

$\varphi$ is an accumulation point of $\Phi_{\sigma}$ such that
$P(\varphi-\varepsilon,\varphi)=0$ for some $\varepsilon>0$ and $\psi=1$;

  3. (3)

$\varphi$ is an accumulation point of $\Phi_{\sigma}$ such that
$P(\varphi,\varphi+\varepsilon)=0$ for some $\varepsilon>0$ and $\psi=0$.

In particular, the slicing $P_{w}^{\psi}$ is locally-finite in each of the
above cases.

###### Proof.

If there is a sequence in $\Phi_{\sigma}$ tending to $\varphi$ from below then
the slicings of a sequence in $\sigma\cdot G$ can only converge to
$P_{w}^{\psi}$ if $\psi=0$. Similarly, if there is a sequence in
$\Phi_{\sigma}$ tending to $\varphi$ from above then the slicings of a
sequence in $\sigma\cdot G$ can only converge to $P_{w}^{\psi}$ if $\psi=1$.
It follows that if $\sigma_{w}^{\psi}$ is in the closure then we are in one of
the three cases in the statement. In each of those cases one can construct a