# Invariant stability conditions of Local $\mathbb{P}^{1}\times\mathbb{P}^{1}$
(after Del Monte-Longhi)

Yirui Xiong  University of Sheffield, the Hicks building, Hounsfield Road,
Sheffield S3 7RH, UK<EMAIL_ADDRESS>
###### Abstract.

Let $X$ be the total space of canonical bundle of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, we study an invariant subspace of
stability conditions on $X$ under an autoequivalence of $D^{b}(X)$. We
describe the complete set of stable objects with respect to the invariant
stability conditions and characterize the space of invariant stability
conditions.

##  1\. Introduction

###  1.1. Background

Inspired by the Douglas’ work on $\Pi$-stability for D-branes, Bridgeland
introduced the notion of the stability condition on a triangulated category in
[12]. It was shown in [12] that to any triangulated category $\mathcal{D}$,
one can associate a complex manifold
$\operatorname{\mathrm{Stab}}(\mathcal{D})$ which parameterises stability
conditions on $\mathcal{D}$. Recall a stability condition on $\mathcal{D}$ is
a pair $\sigma=(Z,\mathcal{A})$, where $\mathcal{A}$ is a full subcategory of
$\mathcal{D}$ called the heart, and $Z$ is a group homomorphism called the
central charge from the Grothendieck group $K_{0}(\mathcal{A})$ to
$\mathbb{C}$ which satisfies the Harder-Narasimhan property[12] (see
Definition 2.15). Bridgeland showed that if
$\operatorname{\mathrm{Stab}}(\mathcal{D})$ is nonempty, then the forgetfull
map
$\operatorname{\mathrm{Stab}}(\mathcal{D})\rightarrow\operatorname{\mathrm{Hom}}(K_{0}(\mathcal{D}),\mathbb{C})$
which sends $(Z,\mathcal{P})$ to $Z$ is a local homeomorphism. Given the
triangulated category $\mathcal{D}$, one can ask the following three
questions:

  1. (1)

Can we find a stability condition on $\mathcal{D}$?

  2. (2)

What is $\operatorname{\mathrm{Stab}}(\mathcal{D})$ as a complex manifold?

  3. (3)

Given a stability condition $\sigma$, can we count the set of (semi)stable
objects in $\mathcal{D}$ for $\sigma$?

So far much progress towards the first two questions has been made for the
derived categories of projective and quasi-projective (local) varieties [13,
11, 14, 31, 24, 1].

The answer to the final question is usually very hard for both projective and
quasi-projective varieties. When $\mathcal{D}$ is a Calabi-Yau category of
dimension $3$, it is related to the Donaldson-Thomas invariants [26, 29].

We study the space of stability conditions for local
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ in this paper, i.e., the total space $X$
of canonical bundle over $\mathbb{P}^{1}\times\mathbb{P}^{1}$. The work is a
mathematical interpretation of the work of Del Monte-Longhi [21]. In their
paper, the physicists found that there were surprisingly complete answers to
all questions in the above when we restrict to an invariant subspace of
stability conditions under an autoequivalence of $D^{b}(X)$. The description
of all stable objects for local 3-Calabi-Yau variety seems to be the first in
mathematical literature.

###  1.2. Results

Denote by
$\pi:X=\operatorname{Tot}\omega_{\mathbb{P}^{1}\times\mathbb{P}^{1}}\rightarrow\mathbb{P}^{1}\times\mathbb{P}^{1}$
the bundle projection map and
$p_{i}:\mathbb{P}^{1}\times\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$, $i=1,\ 2$
the projection maps to each component. Write $\mathcal{O}(a,b)$ for the line
bundle $p_{1}^{*}\mathcal{O}(a)\otimes p_{2}^{*}\mathcal{O}(b)$. There is a
full and strong exceptional sequence of line bundles on
$\mathbb{P}^{1}\times\mathbb{P}^{1}$:

$\mathbb{E}=(E_{0},E_{1},E_{2},E_{3}):=\left(\mathcal{O}(0,0),\mathcal{O}(1,0),\mathcal{O}(1,1),\mathcal{O}(2,1)\right),$

which generates $D^{b}(\mathbb{P}^{1}\times\mathbb{P}^{1})$, and

$T=\bigoplus T_{i}:=\bigoplus_{i}\pi^{*}E_{i}$

is a tilting object in $D^{b}(X)$. Then by the derived Morita theory [27] $X$
is derived equivalent to a non-commutative algebra
$A=\operatorname{\mathrm{End}}(T)$ via the functor
$\mathbb{R}\mathrm{Hom}_{X}(T,-)$. Normally we present $A$ as the path algebra
of a quiver with relations, then $A=\oplus_{i\geq 0}A_{i}$ has a natural
grading by the length of paths. Denote by $D^{b}_{0}(X)$ the full subcategory
of $D^{b}(X)$ consisting of objects supported on
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, and $D^{b}_{0}(A)$ the full subcategory
of $D^{b}(A)$ consisting of objects whose cohomology modules are nilpotent,
here we say a right $A$-module $M$ is nilpotent if there exists $n>0$ such
that $MA_{n}=0$. Then $\mathbb{R}\mathrm{Hom}_{X}(T,-)$ restricts to an
equivalence between $\mathcal{C}$ and $D^{b}_{0}(A)$.

The presentation of the algebra $A$ as the path algebra of a quiver $Q$
subject to relations is as follows, the nodes of $Q$ correspond to the line
bundles $T_{i}$, and the number of arrows between two nodes can either be
calculated from the irreducible maps from $T_{i}$ to $T_{j}$, or the first
extension group of the pair of simple modules associated with the vertices.
The quiver of $A$ will be

(1.1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The symmetry of the shape of the quiver suggests that there should be an
autoequivalence of $D^{b}(X)$, denoted by $\Psi$, which cyclic permutes
cyclically the simple modules associated with the vertices. We will realize
$\Psi$ explicitly in Section 3.2 and let $\Phi=\Psi^{2}$. $\Phi$ restricts to
be an autoequivalence of $D^{b}_{0}(X)$. The space of stability conditions on
$D^{b}_{0}(X)$ is denoted by $\operatorname{\mathrm{Stab}}(X)$. Following Del
Monte-Longhi [21] we consider the space of stability conditions
$\operatorname{\mathrm{Stab}}(X)^{\Phi}$ which are invariant under $\Phi$,
which is called collimination chamber in their paper.

Let $\varphi$ be the automorphism of Grothendieck group $K_{0}(X)$ induced by
$\Phi$. Then $\operatorname{\mathrm{Stab}}(X)^{\Phi}$ is locally modelled on
$\operatorname{\mathrm{Hom}}(K_{0}(X),\mathbb{C})^{\varphi}$, the invariant
central charges under $\varphi$. Note that $K_{0}(X)\cong\mathbb{Z}^{4}$ has a
basis $\gamma_{0},\cdots,\gamma_{3}$ which corresponds to the vertices of $Q$.
Therefore
$\operatorname{\mathrm{Hom}}(K_{0}(X),\mathbb{C})^{\varphi}\cong\mathbb{C}^{2}$.

The construction of a stability condition for $D^{b}_{0}(X)$ is simple: denote
by $\mathcal{A}$ the heart in $D^{b}_{0}(X)$ by pulling back the standard
t-structure on $D^{b}_{0}(A)$, i.e. $\mathcal{A}$ is equivalent to
$\text{mod}_{0}\text{-}A$ via the functor $\mathbb{R}\mathrm{Hom}_{X}(T,-)$
where $\text{mod}_{0}\text{-}A$ is the category of nilpotent modules over $A$.
Since $\mathcal{A}$ is of finite-length and has a finite set of simple objects
$S_{i}$, let $Z:K_{0}(\mathcal{A})=K_{0}(X)\rightarrow\mathbb{C}$ assign the
class of each simple object $[S_{i}]$ to the semi-closed upper half plane

$H=\\{z=r\mathrm{exp}(i\pi\phi)|r>0,0<\phi\leq 1\\}\subset\mathbb{C},$

then $(Z,\mathcal{A})$ satisfies the Harder-Narasimhan property automatically
and is therefore a stability condition. Moreover we write
$\mathcal{U}(\mathcal{A})^{\Phi}$ for the subset of
$\operatorname{\mathrm{Stab}}(X)^{\Phi}$ consisting of $\Phi$-invariant
stability conditions with heart $\mathcal{A}$, since such stability condition
is uniquely determined by assigning each simple object to $H$, then
$\mathcal{U}(\mathcal{A})^{\Phi}$ is isomorphic to $H^{2}$. Such stability
conditions are called algebraic in [11][2].

First we characterize the stable objects for
$\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}$: note that the Kronecker quiver
$K_{2}$

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

can be embedded into $Q$ (1.1) in 4 different ways. Therefore
$\operatorname{\mathrm{rep}}(K_{2})$ embeds into $\mathcal{A}$ as full
subcategories. For a stability condition
$\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}$, $\sigma$ reduces to be a stability
function (Definition 2.14) $\bar{\sigma}$ on
$\operatorname{\mathrm{rep}}(K_{2})$. We are able to show that

######  Lemma 1.1 (=Lemma 5.10).

The stable objects in $\operatorname{\mathrm{rep}}(K_{2})$ with respect to
$\bar{\sigma}$ are stable in $\mathcal{A}$ with respect to $\sigma$.

The stable objects in $\operatorname{\mathrm{rep}}(K_{2})$ are well known to
be the indecomposable representations (with respect to certain stability
functions), their images in $\mathcal{A}$ are called the objects of special
Kronecker types I and II (Definition 5.6). The main result is that these are
in fact all the stable objects for $\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}$:

######  Theorem 1.2 (=Theorem 5.23).

Take $\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}$. Then the stable objects in
$D^{b}_{0}(X)$ for $\sigma$ and their classes in Grothendieck group (up to a
sign) are as follows:

  1. (1)

if $\arg Z(\gamma_{0})<\arg Z(\gamma_{1})$, then the classes of stable objects
are

$\displaystyle n\gamma_{0}+(n+1)\gamma_{1},$
$\displaystyle(n+1)\gamma_{0}+n\gamma_{1},$ $\displaystyle
n\gamma_{2}+(n+1)\gamma_{3},$ $\displaystyle(n+1)\gamma_{2}+n\gamma_{3},$
$\displaystyle\gamma_{0}+\gamma_{1},$ $\displaystyle\gamma_{2}+\gamma_{3}.$

each of the first 4 classes corresponds to a unique stable object (up to a
shift of degree), and each of the last two classes corresponds to a
$\mathbb{P}^{1}$-family of stable objects;

  2. (2)

if $\arg Z(\gamma_{1})<\arg Z(\gamma_{0})$, then the classes of stable objects
are

$\displaystyle n\gamma_{1}+(n+1)\gamma_{2},$
$\displaystyle(n+1)\gamma_{1}+n\gamma_{2},$ $\displaystyle
n\gamma_{3}+(n+1)\gamma_{0},$ $\displaystyle(n+1)\gamma_{3}+n\gamma_{0},$
$\displaystyle\gamma_{1}+\gamma_{2},$ $\displaystyle\gamma_{3}+\gamma_{0}.$

each of the first 4 classes corresponds to a unique stable object(up to a
shift of degree), and each of the last two classes corresponds to a
$\mathbb{P}^{1}$-family of stable objects;

  3. (3)

if $\arg Z(\gamma_{0})=\arg Z(\gamma_{1})$, then the classes of stable objects
are

$\gamma_{0},\ \gamma_{1},\ \gamma_{2},\ \gamma_{3}.$

each class corresponds to a unique stable object $S_{i}$.

This answers the question of counting the stable objects for the local
$\mathbb{P}^{1}\times\mathbb{P}^{1}$. We proceed to characterize a connected
component of $\operatorname{\mathrm{Stab}}(X)^{\Phi}$.

Let $K_{0}(X)^{-\varphi}$ be the subgroup of $K_{0}(X)$ whose elements are
antisymmetric under $\varphi$. We denote by
$\overline{K_{0}(X)}=K_{0}(X)/K_{0}(X)^{-\varphi}$ the quotient group. Note
that there is an isomorphism

$\operatorname{\mathrm{Hom}}_{\mathbb{Z}}(\overline{K_{0}(X)},\mathbb{C})\longrightarrow\operatorname{\mathrm{Hom}}_{\mathbb{Z}}(K_{0}(X),\mathbb{C})^{\varphi}.$

We write $\Delta\subset\overline{K_{0}(X)}$ for the image of the set of the
classes of the stable objects for $\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}$
in the quotient group. The connected component of
$\operatorname{\mathrm{Stab}}(X)^{\Phi}$ which contains
$\mathcal{U}(\mathcal{A})^{\Phi}$ is denoted by
$\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{0}$, we have

$\cdots$$\vdots$$\vdots$$\cdots$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{0})=Z(\gamma_{2})$
Figure 1. Real slice of $\mathcal{H}^{\operatorname{reg}}$

######  Theorem 1.3 (=Theorem 6.6 and 6.8).

The image of the forgetful map

$\mathcal{Z}:\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{0}\rightarrow\operatorname{\mathrm{Hom}}(\overline{K_{0}(X)},\mathbb{C})$

factors through

(1.2)
$\mathcal{Z}:\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{0}\rightarrow\mathcal{H}^{\operatorname{reg}}$

where

$\mathcal{H}^{\operatorname{reg}}:=\operatorname{\mathrm{Hom}}(\overline{K_{0}(X)},\mathbb{C})\setminus\bigcup_{\boldsymbol{v}\in\Delta}\boldsymbol{v}^{\perp},$

is the hyperplane complement of
$\boldsymbol{v}^{\perp}:=\\{Z\in\operatorname{\mathrm{Hom}}(\overline{K_{0}(X)},\mathbb{C})\mid
Z(\boldsymbol{v})=0\\}$ for $\boldsymbol{v}\in\Delta$. Moreover in
$(\ref{map:cover})$ $\mathcal{Z}$ is a covering map.

###  1.3. Relation with [21]

Finally we explain the relation between our work with [21]. We keep the
notations as above and introduce the notations for normalized stability
conditions
$\mathcal{U}^{n}(\mathcal{A})^{\Phi}\subset\mathcal{U}(\mathcal{A})^{\Phi}$ by

$\mathcal{U}^{n}(\mathcal{A})^{\Phi}=\\{(Z,\mathcal{P}):Z(\delta)=i\\},$

where $\delta$ is the class of a skyscrapper sheaf $\mathcal{O}_{x}$,
$x\in\mathbb{P}^{1}\times\mathbb{P}^{1}$. Note that
$Z(\gamma_{0})+Z(\gamma_{2})=Z(\gamma_{1})+Z(\gamma_{3})$ lies on the
imaginary axis. It was shown by Closset-Del Zotto [19, Appendix D] that there
is a unique stable object in each slicing $\mathcal{P}(\phi)$ where
$\phi\neq\frac{1}{2}+n$ ($n\in\mathbb{Z}$), and each such object corresponds
to a representation of the Kronecker quiver.

In this paper we analyze the special slicing $\mathcal{P}(\frac{1}{2})$ in
detail. For simplicity, we restrict to the subset
$\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{+}:=\bigl{\\{}(Z,\mathcal{P}):\arg
Z(\gamma_{0})<\arg Z(\gamma_{1})\bigr{\\}}$. We show that each stable object
in $\mathcal{P}(\frac{1}{2})$ is isomorphic to
$s_{*}\mathcal{O}_{\\{y\\}\times\mathbb{P}^{1}}$ or
$s_{*}\mathcal{O}_{\\{y\\}\times\mathbb{P}^{1}}(-1)[1]$ for
$y\in\mathbb{P}^{1}$ (Theorem 5.22). The case for
$\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{-}:=\bigl{\\{}(Z,\mathcal{P}):\arg
Z(\gamma_{1})<\arg Z(\gamma_{0})\bigr{\\}}$ is obtained by applying an
autoequivalence of $D^{b}(X)$. The key observation in the proof of the above
theorem was taken from [21], which is our Lemma 5.15: one can identify the
action of the autoequivalence $\mathcal{T}$ on the stability conditions in
$\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{+}$ with the action of $\widetilde{g}$
where
$\widetilde{g}\in\widetilde{\operatorname{\mathrm{GL}}}^{+}(2,\mathbb{R})$,
the universal covering space of
$\operatorname{\mathrm{GL}}^{+}(2,\mathbb{R})$. The former autoequivalence
$\mathcal{T}$ plays an important role in the tilting process (Theorem 4.1).

## Notation and Conventions

$\mathcal{D}$ |  Essentially small triangulated category.
---|---
$D^{b}(X)$ |  Bounded derived category of coherent sheaves on a noetherian and separated scheme $X$ over $\mathbb{C}$.
$D^{b}(A)$ |  Bounded derived category of right $A$-modules over a noetherian (possibly graded) $\mathbb{C}$-algebra $A$.
$\mathrm{mod}_{0}$-$A$ |  Category of nilpotent modules over a positively graded noetherian algebra over $\mathbb{C}$, where a module $M$ is said to be nilpotent, if there exists $n>0$ such that $M.A_{n}=0$.
$D^{b}_{0}(A)$ |  Full subcategory of $D^{b}(A)$ with complexes having nilpotent cohomology modules.
$K_{0}(\mathcal{D})$ (resp. $K_{0}(\mathcal{A})$). |  Grothendieck group of an triangulated category $\mathcal{D}$ (resp. an abelian category $\mathcal{A}$). In particular $K_{0}(X):=K_{0}(\operatorname{\mathrm{Coh}}X)$ for variety $X$.
$\operatorname{\mathrm{supp}}(F)$ |  Support of a complex of sheaves $F\in D^{b}(X)$.
$\operatorname{\mathrm{thick}}(T)$ |  Smallest thick subcategory of the object $T$ (or set of objects) in $\mathcal{D}$.
$\operatorname{\mathrm{rep}}Q$ |  Category of finite dimensional representations of a quiver $Q$.

Given a triangulated category $\mathcal{D}$, we write

$\operatorname{\mathrm{Hom}}_{\mathcal{D}}^{i}(A,B):=\operatorname{\mathrm{Hom}}_{\mathcal{D}}(A,B[i]),$

for $A,\ B\in\mathcal{D}$. We denote by
$\operatorname{\mathrm{Hom}}^{\bullet}_{\mathcal{D}}(A,B)=\bigoplus_{i\in\mathbb{Z}}\operatorname{\mathrm{Hom}}^{i}_{\mathcal{D}}(A,B[i])$
the total Hom-space.

Let $Q=(Q_{0},Q_{1})$ be a quiver specified by a set of vertices $Q_{0}$, a
set of arrows $Q_{1}$, and source and target maps $s,\ t:Q_{1}\rightarrow
Q_{0}$. We compose the arrows on the left, that is for $b,\ a\in Q_{1}$,
$ba=0$ unless $s(b)=t(a)$.

Denote by $\mathbb{C}Q$ the path algebra of $Q$, and given a two-sided
$I\subset\mathbb{C}Q$ generated by linear combinations of paths of length at
least $2$, let $A=A(Q,I)=\mathbb{C}Q/I$. We write
$\operatorname{\mathrm{rep}}(Q,I)=\mathrm{mod}_{\text{fd}}$-$A(Q,I)$ and
$\operatorname{\mathrm{rep}}_{\operatorname{\mathrm{nil}}}(Q,I)=\mathrm{mod}_{0}$-$A(Q,I)$.
For each vertex $i\in Q_{0}$ there is an associated one-dimensional simple
module $S_{i}\in\operatorname{\mathrm{rep}}(Q,I)$. Note that we have

$n_{ij}=\dim_{\mathbb{C}}\operatorname{\mathrm{Ext}}^{1}_{A}(S_{j},S_{i})$

where $n_{ij}$ is the number of arrows from vertex $i$ to $j$ in our
notations.

##  2\. Preliminaries

This section is a summary of the results about the tilting theory in the sense
of Happel-Reiten-Smalø[HRS], exceptional collections [8] and stability
conditions in [12, 13]. In this section let $\mathcal{D}$ be a
$\mathbb{C}$-linear triangulated category of finite type. The finite type
condition is the statement that for any two objects $A$, $B$ of $\mathcal{D}$
the vector space

$\operatorname{\mathrm{Hom}}^{\bullet}_{\mathcal{D}}(A,B)=\bigoplus_{i\in\mathbb{Z}}\operatorname{\mathrm{Hom}}^{i}_{\mathcal{D}}(A,B)$

is finite-dimensional.

###  2.1. Simple tilts

The reader is assumed to be familiar with the concept of a t-structure [22].
We are only considering the bounded t-structures. Recall that a t-structure
$(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ is bounded in $\mathcal{D}$, if
for every object $E\in\mathcal{D}$, there exists an integer $n>0$ such that
$E[n]\in\mathcal{D}^{\leq 0}$ and $E[-n]\in\mathcal{D}^{\geq 0}$. The bounded
t-structure is determined by its heart:

######  Lemma 2.1 ([12, Lemma 3.2]).

Let $\mathcal{A}\subset\mathcal{D}$ be a full additive subcategory of
$\mathcal{D}$. Then $\mathcal{A}$ is the heart of a bounded t-structure
$(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ if and only if it satisfies the
following conditions:

  1. (1)

if $n_{1}>n_{2}$ then
$\operatorname{\mathrm{Hom}}_{\mathcal{D}}(A[n_{1}],B[n_{2}])=0$ for any $A,\
B\in\mathcal{A}$;

  2. (2)

for every nonzero object $E\in\mathcal{D}$ there are a finite sequence of
integers:

$k_{1}>k_{2}>\cdots>k_{n}$

and a collection of triangles

(2.1) |  |
---|---|---
$\textstyle{0_{\
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{\
}}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

with $A_{i}\in\mathcal{A}[k_{i}]$ for all $i$.

Given $\mathcal{A}$ the heart of a bounded t-structure and any nonzero object
$E$, we denote by $\operatorname{\mathrm{H}}^{i}_{\mathcal{A}}(E):=A_{i}$ the
$i$th-graded cohomology group with respect to $\mathcal{A}$, where $A_{i}$
appears in (2.1).

A heart of some t-structure will be called finite-length if it is artinian and
noetherian as an abelian category.

The following definition comes from Happel-Reiten-Smalø [23].

######  Definition 2.2 (Torsion pair).

Let $\mathcal{A}$ be a heart of some bounded t-structure in the triangulated
category $\mathcal{D}$. A pair of full subcategories
$(\mathcal{T},\mathcal{F})$ of $\mathcal{A}$ is called a torsion pair in
$\mathcal{A}$ if it satisfies the following conditions

  1. (1)

$\operatorname{\mathrm{Hom}}_{\mathcal{A}}(T,F)=0$ for $T\in\mathcal{T}$ and
$F\in\mathcal{F}$;

  2. (2)

for any object $A\in\mathcal{A}$, there exist $M\in\mathcal{T}$ and
$N\in\mathcal{F}$ such that they fit into a short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

The following theorem was proved in [23, Proposition 2.1].

######  Theorem 2.3 (Happel-Reiten-Smalø).

Let $(\mathcal{T},\mathcal{F})$ be a torsion pair in a heart $\mathcal{A}$.
Let

$\displaystyle\mathcal{A}^{\sharp}$ $\displaystyle:=$
$\displaystyle\left\\{E\in\mathcal{D}\ |\
\mathrm{H}^{1}_{\mathcal{A}}(E)\in\mathcal{T},\
\mathrm{H}^{0}_{\mathcal{A}}(E)\in\mathcal{F},\
\mathrm{H}^{i}_{\mathcal{A}}(E)=0\text{ for }i\neq 0,\ 1\right\\},$
$\displaystyle\mathcal{A}^{\flat}$ $\displaystyle:=$
$\displaystyle\left\\{E\in\mathcal{D}\ |\
\mathrm{H}^{-1}_{\mathcal{A}}(E)\in\mathcal{F},\
\mathrm{H}^{0}_{\mathcal{A}}(E)\in\mathcal{T},\
\mathrm{H}^{i}_{\mathcal{A}}(E)=0\text{ for }i\neq-1,\ 0\right\\},$

then $\mathcal{A}^{\sharp}$ and $\mathcal{A}^{\flat}$ are hearts of bounded
t-structures in $\mathcal{D}$.

A special case of the tilting construction will be particularly important [28,
Definition 3.7]. Suppose that $\mathcal{A}$ is a finite-length heart and
$S\in\mathcal{A}$ is a simple object. Let $\langle S\rangle$ be the full
subcategory consisting of objects $E\in\mathcal{A}$ all of whose simple
factors are isomorphic to $S$. Define the full subcategories

$S^{\perp}:=\\{E\in\mathcal{A}\ |\
\operatorname{\mathrm{Hom}}_{\mathcal{A}}(S,E)=0\\},\quad{}^{\perp}S:=\\{E\in\mathcal{A}\
|\ \operatorname{\mathrm{Hom}}_{\mathcal{A}}(E,S)=0\\}.$

Then we can either view $(\langle S\rangle,S^{\perp})$ or $(^{\perp}S,\langle
S\rangle)$ as a torsion pair. Then we can define new tilted hearts

(2.2) $L_{S}\mathcal{A}:=\langle S[1],^{\perp}S\rangle,\quad
R_{S}\mathcal{A}:=\langle S^{\perp},S[-1]\rangle,$

which we refer to as the left and right simple tilts of the heart
$\mathcal{A}$ at the simple object $S$.

######  Remark 2.4.

It is easy to see that $S[-1]$ is a simple object of $R_{S}\mathcal{A}$ and
that if the category is of finite-length, then
$L_{S[-1]}R_{S}\mathcal{A}=\mathcal{A}$. Similarly, if $L_{S}\mathcal{A}$ is
of finite-length then $R_{S[1]}L_{S}\mathcal{A}=\mathcal{A}$.

The following lemmas will be useful.

######  Lemma 2.5.

Let $(D^{\leq 0},D^{\geq 0})$ and $(\widetilde{D}^{\leq 0},\widetilde{D}^{\geq
0})$ be two bounded t-structures of $\mathcal{D}$, and we denote by
$\mathcal{A}$ and $\mathcal{A}^{\prime}$ their hearts respectively. If
$\mathcal{A}\subset\mathcal{A}^{\prime}$ then
$\mathcal{A}=\mathcal{A}^{\prime}$.

###### Proof.

Let $E\in\mathcal{A}^{\prime}$. Since for any object $F\in D^{\leq 0}$, $F$
has a finite filtration by objects in
$\mathcal{A}[k_{i}]\subset\mathcal{A}^{\prime}[k_{i}]$ for $k_{i}\geq 0$ by
Lemma 2.1, so we have

$\operatorname{\mathrm{Hom}}_{\mathcal{D}}(F,E[-1])=0.$

Therefore $E[-1]\in D^{>0}$.

Similarly for any object $G\in D^{>0}$, $G$ has a finite filtration by objects
in $\mathcal{A}[k_{i}]\subset\mathcal{A}^{\prime}[k_{i}]$ for $k_{i}<0$,
therefore

$\operatorname{\mathrm{Hom}}_{\mathcal{D}}(E,G)=0.$

Therefore $E\in D^{\leq 0}$. So we have $E\in D^{\leq 0}\cap
D^{>0}[1]=\mathcal{A}$. This proves the lemma. ∎

######  Lemma 2.6.

Take an autoequivalence $\Phi\in\operatorname{\mathrm{Aut}}(\mathcal{D})$. Let
$\mathcal{A}\subset\mathcal{D}$ be a heart of some bounded t-structure and of
finite-length, $S\in\mathcal{A}$ be a simple object. Then we have

$\Phi(L_{S}\mathcal{A})=L_{\Phi(S)}\Phi(\mathcal{A}),\quad\Phi(R_{S}\mathcal{A})=R_{\Phi(S)}\Phi(\mathcal{A}).$

###### Proof.

By the definition of simple tilts (2.2), we have
$R_{\Phi(S)}\Phi(\mathcal{A})=\left\langle\left(\Phi(S)\right)^{\perp},\Phi(S)[-1]\right\rangle$.
It is easy to check $\Phi(S^{\perp})=(\Phi(S))^{\perp}$, therefore
$R_{\Phi(S)}\Phi(\mathcal{A})\subset\Phi(R_{S}\mathcal{A})$ by definition. By
Lemma 2.5 we have $R_{\Phi(S)}\Phi(\mathcal{A})=\Phi(R_{S}\mathcal{A})$. The
proof of the left tilt case is similar. ∎

Given a heart of bounded t-structure $\mathcal{A}\subset\mathcal{D}$, we
denote by $\operatorname{\mathrm{Sim}}\mathcal{A}$ the set of all non-
isomorphic simple objects in $\mathcal{A}$. The following theorem
characterizes the new simple objects in the tilted hearts.

######  Proposition 2.7 ([28, Proposition 5.4]).

Assume $\operatorname{\mathrm{Sim}}\mathcal{A}$ is finite and $\mathcal{A}$ is
of finite-length. Let $S\in\mathcal{A}$ be such that
$\operatorname{\mathrm{Ext}}_{\mathcal{A}}^{1}(S,S)=0$. Then after taking a
left or right simple tilt, the new simple objects are:

(2.3) $\displaystyle\operatorname{\mathrm{Sim}}R_{S}\mathcal{A}$
$\displaystyle=$ $\displaystyle\\{S[-1]\\}\ \cup\
\\{\phi_{S}(X):X\in\operatorname{\mathrm{Sim}}\mathcal{A},\ X\neq S\\}$ (2.4)
$\displaystyle\operatorname{\mathrm{Sim}}L_{S}\mathcal{A}$ $\displaystyle=$
$\displaystyle\\{S[1]\\}\ \cup\
\\{\psi_{S}(X):X\in\operatorname{\mathrm{Sim}}\mathcal{A},\ X\neq S\\}$

where

$\displaystyle\phi_{S}(X)$ $\displaystyle=$
$\displaystyle\operatorname{\mathrm{Cone}}\
\left(S[-1]\otimes\operatorname{\mathrm{Ext}}^{1}(S,X)\longrightarrow
X\right),$ $\displaystyle\psi_{S}(X)$ $\displaystyle=$
$\displaystyle\operatorname{\mathrm{Cone}}\ \left(X\longrightarrow
S[1]\otimes\operatorname{\mathrm{Ext}}^{1}(X,S)^{*}\right)[-1].$

On the other hand, the concept of tilting objects gives another method to
construct new bounded t-structures. Recall that an object $T$ in $\mathcal{D}$
is called a tilting object if it satisfies the following conditions

  1. (1)

$\mathbb{R}\mathrm{Hom}_{\mathcal{D}}\left(T,T[n]\right)=0$, unless $n=0$;

  2. (2)

$T$ is a classical generator of $\mathcal{D}$, i.e., the smallest thick
subcategory containing $T$ which we denote by
$\operatorname{\mathrm{thick}}(T)$ is $\mathcal{D}$.

Suppose $X$ is a smooth quasi-projective variety and $\mathcal{D}=D^{b}(X)$.
Then $\mathbb{R}\mathrm{Hom}_{X}(T,-)$ induces a derived equivalence between
$X$ and noncommutative algebra $B=\operatorname{\mathrm{End}}_{X}(T)$ [33,
27]:

$\mathbb{R}\mathrm{Hom}_{X}(T,-):D^{b}(X)\longrightarrow D^{b}(B).$

Pulling back the standard t-structure on $D^{b}(B)$ via the equivalence gives
us a new t-structure on $D^{b}(X)$.

###  2.2. Exceptional collections

Usually tilting objects in $D^{b}(X)$ can break up into small pieces called
the exceptional objects.

######  Definition 2.8 (Exceptional collection).

An object $E$ in $\mathcal{D}$ is said to be exceptional if

$\operatorname{\mathrm{Hom}}^{k}_{\mathcal{D}}(E,E)=\left\\{\begin{array}[]{cc}\mathbb{C}&\
\text{if }k=0,\\\ 0&\ \text{otherwise}.\end{array}\right.$

An exceptional collection ${\mathbb{E}}\subset\mathcal{D}$ is a sequence of
exceptional objects

${\mathbb{E}}=(E_{0},\cdots,E_{n})$

such that for all $0\leq i<j\leq n$, we have
$\operatorname{\mathrm{Hom}}^{\bullet}_{\mathcal{D}}(E_{j},E_{i})=0$.

An exceptional collection $\mathbb{E}=(E_{0},\cdots,E_{n})$ is said to be
strong if for all $i,\ j$

$\operatorname{\mathrm{Hom}}^{k}_{\mathcal{D}}(E_{i},E_{j})=0,\ \ \text{unless
}k=0.$

We write $\operatorname{\mathrm{thick}}(\mathbb{E})\subset\mathcal{D}$ for the
smallest thick subcategory of $\mathcal{D}$ containing the elements of an
exceptional collection $\mathbb{E}\subset\mathcal{D}$. An exceptional
collection $\mathbb{E}$ is said to be full if
$\operatorname{\mathrm{thick}}(\mathbb{E})=\mathcal{D}$. From the definitions
above, we have that for a full and strong exceptional collection $\mathbb{E}$,
the object $\bigoplus_{i=0}^{n}E_{i}$ is a tilting object in $\mathcal{D}$.
The first full and strong exceptional collection was found in
$D^{b}(\mathbb{P}^{N})$ by Beilinson [4].

######  Example 2.9.

$D^{b}(\mathbb{P}^{N})$ admits a full and strong exceptional collection
$\bigl{(}\mathcal{O},\mathcal{O}(1),\cdots,\mathcal{O}(N)\bigr{)}$.
$\mathbb{P}^{N}$ is derived equivalent to the path algebra of quiver

$\cdots$$\vdots$$\vdots$$\cdots$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{0})=Z(\gamma_{2})$${0}$${1}$${\cdots}$${N-2}$${N-1}$${\vdots}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{N}}$${\vdots}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{N}}$${\vdots}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{N}}$${\vdots}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{N}}$

subject to the relations $f_{j+1}f_{j}=f_{j}f_{j+1}$.

Given an exceptional collection $\mathbb{E}$ in $\mathcal{D}$, the right
orthogonal subcategory to $\mathbb{E}$ is the full triangulated subcategory

$\mathbb{E}^{\bot}=\left\\{X\in\mathcal{D}:\
\operatorname{\mathrm{Hom}}^{\bullet}_{\mathcal{D}}(E,X)=0\ \text{for
}E\in\mathbb{E}\right\\}.$

Similarly, the left orthogonal subcategory to $\mathbb{E}$ is

${}^{\bot}\mathbb{E}=\left\\{X\in\mathcal{D}:\
\operatorname{\mathrm{Hom}}^{\bullet}_{\mathcal{D}}(X,E)=0\ \text{for
}E\in\mathbb{E}\right\\}.$

The subcategory $\langle\mathbb{E}\rangle$ is admissible due to [7, Theorem
3.2], i.e. the inclusion functor
$i:\langle\mathbb{E}\rangle\rightarrow\mathcal{D}$ has left and right adjoint
functors. Thus the fullness of $\mathbb{E}$ is equivalent to
$\mathbb{E}^{\bot}=0$ or ${}^{\bot}\mathbb{E}=0$.

We suppose $E\in\mathcal{D}$ to be exceptional. Given an object
$X\in\mathcal{D}$, the left mutation of $X$ through $E$ is the object
$L_{E}(X)$ defined up to isomorphism by the triangle

$\textstyle{L_{E}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{\mathrm{Hom}}^{\bullet}_{\mathcal{D}}(E,X)\otimes
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ev}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{E}(X)[1],}$

where $ev$ denotes the evaluation map. Similarly, given $X\in\mathcal{D}$, the
right mutation of $X$ through $E$ is the object $R_{E}X$ defined by the
triangle

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{coev}$$\textstyle{\operatorname{\mathrm{Hom}}^{\bullet}_{\mathcal{D}}(X,E)^{*}\otimes
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{E}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X[1],}$

where $coev$ denotes the coevaluation map. Moreover, consider the left and
right orthogonal subcategories of $E$, these two operations define mutually
inverse equivalences of categories (see [18, Appendix B])

(2.5)
$\textstyle{{}^{\bot}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{E}}$$\textstyle{E^{\bot}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R_{E}}$

######  Definition 2.10 (Standard mutation).

Given a full exceptional collection $\mathbb{E}=(E_{0},\cdots,E_{n})$, the
mutation operation $\sigma_{i}$ for each $0<i\leq n$ is defined by the rule

$\displaystyle\sigma_{i}(E_{0},\cdots,E_{i-2},E_{i-1},E_{i},E_{i+1},\cdots,E_{n})$
$\displaystyle=$
$\displaystyle(E_{0},\cdots,E_{i-2},L_{E_{i-1}}(E_{i}),E_{i-1},E_{i+1},\cdots,E_{n})$

This operation takes exceptional collections to exceptional collections [7,
Lemma 2.1]. And it takes full collections to full collections [7, Lemma 2.2].

The following definition is due to Bondal[7], and we refer our reader to [18,
Appendix B] for the proof of (2.6).

######  Definition 2.11 (Dual objects).

Let $\mathbb{E}=(E_{0},\cdots,E_{n})$ be a full exceptional collection and
define

$F_{j}=L_{E_{0}}L_{E_{1}}\cdots L_{E_{j-1}}(E_{j})[j],\ \ 0\leq j\leq n.$

Then $F_{j}$ is called the dual object to $E_{j}$ and satisfies

(2.6)
$\operatorname{\mathrm{Hom}}^{k}_{\mathcal{D}}(E_{i},F_{j})=\left\\{\begin{array}[]{cc}\mathbb{C}&\text{if}\
i=j\ \text{and}\ k=0,\\\ 0&\text{otherwise.}\end{array}\right.$

###  2.3. Stability conditions

We collect some properties and theorems on the space of stability conditions
introduced in [12].

######  Definition 2.12 (Slicing).

A slicing of $\mathcal{D}$ is a collection of full subcategories
$\mathcal{P}(\phi)$ indexed by $\phi\in\mathbb{R}$, satisfying the following
axioms:

  1. (1)

$\mathcal{P}(\phi+1)=\mathcal{P}(\phi)[1]$;

  2. (2)

$\operatorname{\mathrm{Hom}}_{D}\left(\mathcal{P}(\phi_{1}),\mathcal{P}(\phi_{2})\right)=0$
for $\phi_{1}>\phi_{2}$;

  3. (3)

for any nonzero object $E\in\mathcal{D}$, we have a collection of triangles

|  |
---|---|---
$\textstyle{0_{\
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{\
}}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

such that $A_{i}\in\mathcal{P}(\phi_{i})$, and

$\phi_{1}>\phi_{2}>\cdots>\phi_{n}.$

For any nonzero object $E$, we denote by $\phi^{+}(E)=\phi_{1}$ and
$\phi^{-}(E)=\phi_{n}$ where $\phi_{i}$ is defined as above. For any interval
$I\subset\mathbb{R}$, $\mathcal{P}(I)$ is defined to be the extension-closed
subcategory of $\mathcal{D}$ generated by objects $E\in\mathcal{P}(\phi)$ for
$\phi\in I$.

We denote by $\mathrm{Slice}(\mathcal{D})$ the set of all slicings on
$\mathcal{D}$. Bridgeland introduced a generalized metric in
$\mathrm{Slice}(\mathcal{D})$:

######  Definition 2.13 ([12, Section 6] ).

Let $\mathcal{P}_{1}$, $\mathcal{P}_{2}\in\mathrm{Slice}(\mathcal{D})$, then
the generalized metric
$d:\mathrm{Slice}(\mathcal{D})\times\mathrm{Slice}(\mathcal{D})\rightarrow[0,+\infty]$
is defined as

$d(\mathcal{P}_{1},\mathcal{P}_{2}):=\mathop{\mathrm{sup}}_{E\neq
0\in\mathcal{D}}\left\\{|\phi_{1}^{+}(E)-\phi_{2}^{+}(E)|,|\phi_{1}^{-}(E)-\phi_{2}^{-}(E)|\right\\}.$

Before recalling stability condition on $\mathcal{D}$, we first recall the
stability function on an abelian category $\mathcal{A}$ [34].

######  Definition 2.14.

A stability function on $\mathcal{A}$ is a group homomorphism
$Z:K_{0}(\mathcal{A})\rightarrow\mathbb{C}$ such that for any nonzero object
$A\in\mathcal{A}$, the complex number $Z(A)$ lies in the subset

$H=\\{z=r\mathrm{exp}(i\pi\phi)|r>0,0<\phi\leq 1\\}\subset\mathbb{C}.$

The phase of $A$ is defined to be
$\phi(A)=\frac{1}{\pi}\mathrm{arg}Z(A)\in(0,1]$. An object $E\in\mathcal{A}$
is said to be (semi)stable if for any subobject $A\subset E$ we have

$\phi(A)<(\leq)\phi(E).$

######  Definition 2.15 (Harder-Narasimhan property[12, Definition 2.3]).

Let $Z:K_{0}(\mathcal{A})\rightarrow\mathbb{C}$ be a stability function on the
abelian category $\mathcal{A}$. Then $Z$ is said to have Harder-Narasimhan
property if for any nonzero object $E\in\mathcal{A}$ there is a filtration

$0=E_{0}\subset E_{1}\subset E_{2}\subset\cdots E_{n-1}\subset E_{n}=E$

such that each $F_{i}=E_{i}/E_{i-1}$ is a semistable object of phase
$\phi_{i}$ and $\phi_{1}>\phi_{2}\cdots>\phi_{n-1}>\phi_{n}$.

######  Definition 2.16 (Stability condition).

A stability condition for $\mathcal{D}$ is a pair $\sigma=(Z,\mathcal{A})$
which consists of a heart of a bounded t-structure $\mathcal{A}$ in
$\mathcal{D}$, and a stability function (called the central charge of
$\sigma$) $Z:K_{0}(\mathcal{A})\rightarrow\mathbb{C}$ such that $Z$ satisfies
the Harder-Narasimhan property.

The above definition of stability condition is equivalent to the following
definition [12, Proposition 5.3]:

######  Definition 2.17.

A stability condition is a pair $\sigma=(Z,\mathcal{P})$ which consists of a
slicing $\mathcal{P}\in\mathrm{Slice}(\mathcal{D})$ and a group homormorphism
called the central charge $Z:K_{0}(D)\rightarrow\mathbb{C}$, such that it
satisfies the compatibility condition: if $0\neq E\in\mathcal{P}(\phi)$ for
some $\phi\in\mathbb{R}$, then

$Z(E)=r\mathrm{exp}(i\pi\phi),\ r>0.$

The objects in $\mathcal{P}(\phi)$ are called semistable of phase $\phi$, and
the simple objects in $\mathcal{P}(\phi)$ are called stable.

The following lemma will be useful later.

######  Lemma 2.18 ([12, Lemma 6.4]).

If the stability conditions $\sigma=(Z,\mathcal{P})$ and
$\tau=(Z,\mathcal{P}^{\prime})$ have the same central charge and
$d(\mathcal{P},\mathcal{P}^{\prime})<1$, then $\sigma=\tau$.

######  Definition 2.19 (Support property).

Let $\sigma=(Z,\mathcal{P})$ be a stability condition, by fixing a norm
$\lVert\ \cdot\ \rVert$ on
$K_{0}(\mathcal{D})_{\mathbb{R}}=K_{0}(\mathcal{D})\otimes_{\mathbb{Z}}\mathbb{R}$,
$\sigma$ is said to have support property if there exists a constant $C>0$
such that

$\lVert E\rVert\leq C|Z(E)|$

for any stable object $E$.

We denote by $\operatorname{\mathrm{Stab}}(\mathcal{D})$ the set of all
stability conditions with the support property. To define the topology on
$\operatorname{\mathrm{Stab}}(\mathcal{D})$, Bridgeland [12] introduced the
following definitions:

######  Definition 2.20.

Let $\sigma=(Z,\mathcal{P})\in\operatorname{\mathrm{Stab}}(\mathcal{D})$. The
function $\lVert\ \cdot\
\rVert_{\sigma}:\operatorname{\mathrm{Hom}}(K_{0}(\mathcal{D}),\mathbb{C})\rightarrow[0,+\infty]$
is defined as

$\lVert W\rVert_{\sigma}:=\sup\left\\{\frac{|W(E)|}{|Z(E)|}:\text{ E
semistable for }\sigma\right\\}.$

######  Lemma 2.21 ([12, Lemma 6.2]).

For $\sigma=(Z,\mathcal{P})\in\operatorname{\mathrm{Stab}}(\mathcal{D})$ and
$0<\epsilon<\frac{1}{4}$ let

$C_{\epsilon}(\sigma):=\left\\{\tau=(W,\mathcal{Q})\in\operatorname{\mathrm{Stab}}(\mathcal{D}):\lVert
W-Z\rVert_{\sigma}<\sin(\pi\epsilon),d(\mathcal{P},\mathcal{Q})<\epsilon\right\\}.$

Then by varying $\sigma$, we get a basis for the topology of
$\operatorname{\mathrm{Stab}}(\mathcal{D})$.

The following is the main result of [12]. The idea is that if
$\sigma=(Z,\mathbb{P})$ is a stability condition on $\mathcal{D}$ and one
deforms $Z$ to a new group homomorphism
$W:K_{0}(\mathcal{D})\rightarrow\mathbb{C}$ in such a way that the phase of
each semistable object in $\sigma$ changes in a uniformly bounded way, then it
is possible to define a new slicing $\mathcal{Q}(\psi)\subset\mathcal{D}$ so
that $(W,\mathcal{Q})$ is a stability condition on $\mathcal{D}$.

######  Theorem 2.22 (Deformation of stability conditions).

Let $\sigma=(Z,\mathcal{P})\in\operatorname{\mathrm{Stab}}(\mathcal{D})$. Then
there exists $0<\epsilon_{0}<\frac{1}{8}$ such that if
$0<\epsilon<\epsilon_{0}$ and for
$W\in\operatorname{\mathrm{Hom}}(K_{0}(\mathcal{D}),\mathbb{C})$ satisfying

$|W(E)-Z(E)|<\sin(\pi\epsilon)|Z(E)|$

for any semistable object $E\in\mathcal{D}$ with respect to $\sigma$. Then
there exists a unique stability condition $\tau=(W,\mathcal{P}^{\prime})$ such
that

$d(\mathcal{P},\mathcal{P}^{\prime})<\epsilon.$

######  Corollary 2.23 ([12, Theorem 1.2]).

Let $\mathcal{D}$ be a triangulated category. For each connected component
$\Sigma\subset\operatorname{\mathrm{Stab}}(\mathcal{D})$ there are a linear
subspace
$V(\Sigma)\subset\operatorname{\mathrm{Hom}}(K_{0}(\mathcal{D}),\mathbb{C})$,
with a well-defined linear topology, and a local homeomorphism
$\mathcal{Z}:\Sigma\rightarrow V$ which sends a stability condition to its
central charge $Z$.

Since $K_{0}(\mathcal{D})$ might have infinite rank, in practice we usually
assume there is a quotient group $\mathcal{N}$ of finite rank, and the
quotient map is denoted by $\mu:K_{0}(\mathcal{D})\rightarrow\mathcal{N}$.
Then let $\operatorname{\mathrm{Stab}}_{\mathcal{N}}(\mathcal{D})$ be the
subspace of $\operatorname{\mathrm{Stab}}(\mathcal{D})$ consisting of
stability conditions whose central charges
$Z:K_{0}(\mathcal{D})\rightarrow\mathbb{C}$ factor through $\mathcal{N}$. Then
the following result is an immediate consequence of Corollary 2.23.

######  Corollary 2.24 ([12, Corollary 1.3]).

For each connected component
$\Sigma\subset\operatorname{\mathrm{Stab}}_{\mathcal{N}}(\mathcal{D})$ there
are a linear subspace
$V(\Sigma)\subset\operatorname{\mathrm{Hom}}(\mathcal{N},\mathbb{C})$, and a
local homeomorphism $\mathcal{Z}:\Sigma\rightarrow V$ which sends a stability
condition to its central charge $Z$. In particular, $\Sigma$ is a finite-
dimensional complex manifold.

We recall some group actions on the space of stability conditions. Let
$\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$ be the universal covering of
$\mathrm{GL}^{+}(2,\mathbb{R})$. Note that an element in
$\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$ can be viewed as a pair $(g,f)$
where $g\in\mathrm{GL}^{+}(2,\mathbb{R})$ and
$f:\mathbb{R}\rightarrow\mathbb{R}$ is an increasing map with
$f(\phi+1)=f(\phi)+1$, such that $g$ and $f$ induce the same action on the
circle
$S^{1}=\\{e^{i\pi\phi}:\phi\in\mathbb{R}\\}=(\mathbb{R}^{2}\setminus\\{0\\})/\mathbb{R}_{>0}$.

######  Definition 2.25.

The space of stability conditions carries a right action by
$\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$. For
$\widetilde{g}=(g,f)\in\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$ and
$\sigma=(Z,\mathcal{P})\in\operatorname{\mathrm{Stab}}(\mathcal{D})$, then
$\sigma\cdot\widetilde{g}=(Z_{g},\mathcal{P}_{f})$ where for $[E]\in
K_{0}(\mathcal{D})$

$Z_{g}(E)=g^{-1}Z(E),\quad\mathcal{P}_{f}(\phi)=\mathcal{P}\left(f(\phi)\right).$

The space of stability conditions also carries a left action by
$\operatorname{\mathrm{Aut}}(\mathcal{D})$. For
$T\in\operatorname{\mathrm{Aut}}(D)$, denote by $t$ the automorphism of
$K_{0}(\mathcal{D})$ induced by $T$, then $T(\sigma)=(Z_{t},\mathcal{P}_{T})$
where for $[E]\in K_{0}(\mathcal{D})$

$Z_{t}(E)=Z(t^{-1}E),\quad\mathcal{P}_{T}(\phi)=T\left(\mathcal{P}(\phi)\right).$

######  Remark 2.26.

From the definition of $\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$-action,
$\sigma$ and $\sigma\cdot\widetilde{g}$ have the same set of semistable
objects, but the phases have been relabelled. In particular, note that the
additive group $\mathbb{C}$ acts on
$\operatorname{\mathrm{Stab}}(\mathcal{D})$, via the embedding
$\mathbb{C}\xhookrightarrow{}\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$: an
element $\lambda\in\mathbb{C}$ acts by

$\lambda:(Z,\mathcal{[}P])\mapsto(Z^{\prime},\mathcal{P}^{\prime}),\quad
Z^{\prime}(E)=e^{-i\pi\lambda}\cdot
Z(E),\quad\mathcal{P}^{\prime}(\phi)=\mathcal{P}\left(\phi+\mathrm{Re}(\lambda)\right).$

In the end of this subsection, we recall the following important lemma:

######  Lemma 2.27 ([17, Proposition 7.6]).

Fix $0\neq E\in D$. Then

  1. (1)

the set of stability conditions
$\sigma\in\operatorname{\mathrm{Stab}}(\mathcal{D})$ for which $E$ is
$\sigma$-stable is open;

  2. (2)

the set of stability conditions
$\sigma\in\operatorname{\mathrm{Stab}}(\mathcal{D})$ for which $E$ is
$\sigma$-semistable is closed.

######  Definition 2.28.

Let $\mathcal{A}$ be the heart of a bounded t-structure in $\mathcal{D}$ which
is of finite-length, then the subset of stability conditions
$\mathcal{U}(\mathcal{A})\subset\operatorname{\mathrm{Stab}}(\mathcal{D})$ is
defined to be

$\mathcal{U}(\mathcal{A})=\bigl{\\{}\sigma=(Z,\mathcal{P})|\mathcal{P}\bigl{(}(0,1]\bigr{)}=\mathcal{A}\bigr{\\}}.$

The relation between simple tilts and stability conditions is the following:

######  Lemma 2.29 ([10, Lemma 5.5]).

Suppose $\mathcal{A}$ is of finite-length. Let
$\sigma=(Z,\mathcal{P})\in\overline{\mathcal{U}}(\mathcal{A})$ the closure of
$\mathcal{U}(\mathcal{A})$. Suppose that $Z(S_{i})\in\mathbb{R}_{<0}$ for some
$i$, also $\mathrm{Im}Z(S_{j})>0$ for $j\neq i$, and $R_{S_{i}}\mathcal{A}$ is
finite length, then there is an open neighborhood $V$ of $\sigma$ such that
$V\subset\mathcal{U}(\mathcal{A})\cup\mathcal{U}(R_{S_{i}}\mathcal{A})$.
Similarly suppose $Z(S_{i})\in\mathbb{R}_{>0}$ for some $i$ and
$\mathrm{Im}Z(S_{j})>0$ for $j\neq i$, and $L_{S_{i}}\mathcal{A}$ is finite
length, then there is an open neighborhood $V^{\prime}$ of $\sigma$ such that
$V^{\prime}\subset\mathcal{U}(\mathcal{A})\cup\mathcal{U}(L_{S_{i}}\mathcal{A})$.

##  3\. Quiver symmetry and autoequivalence

This section we give explicit construction of the autoequivalence in the
introduction.

###  3.1. Quiver

There is a full and strong exceptional collection on $Z$:

$\mathbb{E}=\left(\mathcal{O}(0,0),\mathcal{O}(1,0),\mathcal{O}(1,1),\mathcal{O}(2,1)\right),$

which has the dual collection

$\mathbb{F}=\left(\mathcal{O}(0,0),\mathcal{O}(-1,0)[1],\mathcal{O}(1,-1)[1],\mathcal{O}(0,-1)[2]\right).$

As in the introduction, we denote by $\pi:X\rightarrow Z$ the bundle
projection map,
$p_{i}:Z=\mathbb{P}^{1}\times\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$, $i=1,\
2$ the projection maps to each component. We denote by $s:Z\hookrightarrow X$
the embedding map of the zero section.

######  Lemma 3.1.

The pull back $\mathcal{Q}=\bigoplus_{i}\pi^{*}E_{i}$ is a tilting object in
$D^{b}(X)$.

###### Proof.

We have

$\displaystyle\operatorname{\mathrm{Ext}}^{i}_{X}\left(\pi^{*}\mathcal{O}(a,b),\pi^{*}\mathcal{O}(c,d)\right)$
$\displaystyle=$
$\displaystyle\operatorname{\mathrm{Ext}}^{i}_{Z}\left(\mathcal{O}(a,b),\pi_{*}\pi^{*}\mathcal{O}(c,d)\right)$
$\displaystyle=$
$\displaystyle\bigoplus_{n=0}\operatorname{\mathrm{Ext}}^{i}_{Z}\left(\mathcal{O}(a,b),\mathcal{O}(c,d)\otimes(\omega^{*}_{\mathbb{P}^{1}\times\mathbb{P}^{1}})^{n}\right)$
$\displaystyle=$
$\displaystyle\bigoplus_{n=0}\mathrm{H}^{i}\left(\mathbb{P}^{1}\times\mathbb{P}^{1},\mathcal{O}(c-a+2n,d-b+2n)\right)$
$\displaystyle=$
$\displaystyle\bigoplus_{n=0}\bigoplus_{s+t=i}\mathrm{H}^{s}(\mathbb{P}^{1},\mathcal{O}(c-a+2n))\otimes\mathrm{H}^{t}(\mathbb{P}^{1},\mathcal{O}(d-b+2n)).$

For $i>0$,
$\mathrm{H}^{i}\left(\mathbb{P}^{1}\times\mathbb{P}^{1},\mathcal{O}(c-a+2n,d-b+2n)\right)=0$
unless

  1. (1)

$c-a+2n\leq-2,\ d-b+2n\geq 0$ or $d-b+2n\leq-2,\ c-a+2n\geq 0;$

  2. (2)

$c-a+2n\leq-2,\ d-b+2n\leq-2$.

Since $-1\leq d-b\leq 1$ in our case, so only the first part of case 1 is
possible. One can easily verify the bundles in our exceptional collection do
not belong to this case. Therefore
$\operatorname{\mathrm{Ext}}^{i}_{X}(\mathcal{Q},\mathcal{Q})=0$ for $i\neq
0$.

The proof of the generating property for $\mathcal{Q}$ is due to the general
result [30, Lemma 5.2.3]. We finished the proof. ∎

The endomorphism algebra $B=\operatorname{\mathrm{End}}_{X}(\mathcal{Q})$ is
noetherian [18, Theorem 3.6], therefore we can write it as the path algebra of
a quiver $Q$ subject to relations, and grade it by the length of paths. The
vertex is indexed by $i$, and the number of arrows from $i$ to $j$ is the
dimension of space of irreducible maps from $\pi^{*}E_{i}$ to $\pi^{*}E_{j}$,
i.e., the cokernel of the map

$\bigoplus_{k\neq i,\
j}\operatorname{\mathrm{Hom}}_{X}(\pi^{*}E_{i},\pi^{*}E_{k})\otimes\operatorname{\mathrm{Hom}}_{X}(\pi^{*}E_{k},\pi^{*}E_{j})\longrightarrow\operatorname{\mathrm{Hom}}_{X}(\pi^{*}E_{i},\pi^{*}E_{j}).$

The following corollary is from [9, Lemma 4.4].

######  Corollary 3.2.

There is a derived equivalence:

$\mathcal{R}_{\mathcal{Q}}:=\mathbb{R}\mathrm{Hom}(\mathcal{Q},-):D^{b}_{0}(X)=D^{b}_{0}(X)\longrightarrow
D^{b}_{0}(B).$

$D^{b}_{0}(B)$ inherits a t-structure from the standard t-structure on
$D^{b}(B)$, whose heart is $\mathrm{mod}_{0}$-$B$. From now on until the end
of the paper, we denote by $\mathcal{A}$ the heart of the t-structure by
pulling back the standard t-structure on $D^{b}_{0}(B)$, i.e., it is
equivalent to $\mathrm{mod}_{0}$-$B$.

######  Corollary 3.3.

The simple objects up to isomorphism in $\mathcal{A}$ are

$S_{0}=s_{*}\mathcal{O}(0,0),\quad S_{1}=s_{*}\mathcal{O}(-1,0)[1],\quad
S_{2}=s_{*}\mathcal{O}(1,-1)[1],\quad S_{3}=s_{*}\mathcal{O}(0,-1)[2].$

###### Proof.

We write $P_{i}$ the projective $B$-module and $C_{i}$ the simple $B$-module
associated with vertex $i$. Then $\mathrm{mod}_{0}$-$B$ is the extension-
closed subcategory of $\mathrm{mod}$-$B$ generated by $\\{C_{i}\\}_{i\in I}$
and $\\{C_{i}\\}_{i\in I}$ is the set of all simple $B$-modules in
$\mathrm{mod}_{0}$-$B$. By definition of $\mathcal{R}_{\mathcal{Q}}$,
$\pi^{*}E_{i}$ is sent to $P_{i}$. Then $s_{*}F_{i}$ is sent to $C_{i}$ which
follows from the definition of the dual collection:

$\operatorname{\mathrm{Hom}}^{\bullet}_{D^{b}(X)}(\pi^{*}E_{i},s_{*}F_{j})=\operatorname{\mathrm{Hom}}^{\bullet}_{D^{b}(Z)}(E_{i},F_{j})=\delta_{ij}\mathbb{C}.$

∎

######  Proposition 3.4.

The quiver $Q=(Q_{0},Q_{1})$ of the endomorphism algebra
$\mathrm{End}_{X}(\mathcal{Q})$ is

(3.1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x_{1}}$$\scriptstyle{y_{1}}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x_{2}}$$\scriptstyle{y_{2}}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x_{3}}$$\scriptstyle{y_{3}}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x_{4}}$$\scriptstyle{y_{4}}$

and the vertex $i$ corresponds to $S_{i}$.

###### Proof.

The arrows from $i$ to $j$ can be calculated by the dimension of the vector
space $\operatorname{\mathrm{Ext}}^{1}_{X}(S_{j},S_{i})$. By using the Koszul
resolution [25, Chapter 11] along the embedding map $s$, for any sheaf $F$ on
$Z$ we have

$s^{*}s_{*}F\cong F\oplus\left(F\otimes\omega_{Z}^{*}[1]\right)$

in $D^{b}(Z)$. Therefore

$\operatorname{\mathrm{Ext}}^{n}_{X}(s_{*}F_{i},s_{*}F_{j})=\operatorname{\mathrm{Ext}}^{n}_{Z}(F_{i},F_{j})\oplus\operatorname{\mathrm{Ext}}^{3-n}_{Z}(F_{j},F_{i})^{*}.$

For example

$\displaystyle\operatorname{\mathrm{Ext}}^{1}_{X}\left(s_{*}\mathcal{O}(1,-1)[1],s_{*}\mathcal{O}(-1,0)[1]\right)$
$\displaystyle=$
$\displaystyle\operatorname{\mathrm{Ext}}^{1}_{Z}\left(\mathcal{O}(1,-1),\mathcal{O}(-1,0)\right)\oplus\operatorname{\mathrm{Ext}}^{2}_{Z}\left(\mathcal{O}(-1,0),\mathcal{O}(1,-1)\right)^{*}$
$\displaystyle=$
$\displaystyle\mathrm{H}^{1}\left(\mathbb{P}^{1}\times\mathbb{P}^{1},\mathcal{O}(-2,1)\right)$
$\displaystyle=$ $\displaystyle\mathbb{C}^{2}.$

The other calculations are similar. Therefore we get the given quiver $Q$. ∎

We denote by $I$ the relations of paths in
$\operatorname{\mathrm{End}}_{X}(\mathcal{Q})$ and
$\operatorname{\mathrm{rep}}_{\mathrm{nil}}(Q,I)\cong\mathcal{A}$ the category
of nilpotent representations of quiver with relations.

######  Remark 3.5.

Since $B=\operatorname{\mathrm{End}}_{X}(\mathcal{Q})$ is graded 3-Calabi-Yau
in the sense that the full subcategory consisting of objects with finite
dimensional cohomology modules $D^{b}_{\text{fin}}(B)$ has Serre duality:

$\operatorname{\mathrm{Ext}}^{i}_{B}(M,N)=\operatorname{\mathrm{Ext}}^{3-i}_{B}(N,M)^{*},\quad
M,\ N\in D^{b}_{\text{fin}}(B).$

Then work of Bocklandt [6] shows that the relations of
$\operatorname{\mathrm{End}}_{X}(\mathcal{Q})$ can be encoded in compact form
in a potential. Thus we can write

$B=B(Q,W)=\mathbb{C}Q/(\partial_{a}W:a\in Q_{1})$

for some non-uniquely defined element
$W\in\mathbb{C}Q/[\mathbb{C}Q,\mathbb{C}Q]$. In fact we can write down the
potential $W$ explicitly here: keep the notations as in (3.1). Then

$W=x_{4}x_{3}x_{2}x_{1}+y_{4}y_{3}y_{2}y_{1}-y_{4}x_{3}y_{2}x_{1}-x_{4}y_{3}x_{2}y_{1}.$

The corresponding relations are (for $j\in\mathbb{Z}_{4}$):

$\displaystyle\partial_{x_{j}}W=x_{j+3}x_{j+2}x_{j+1}-y_{j+3}x_{j+2}y_{j+1}=0,$
$\displaystyle\partial_{y_{j}}W=y_{j+3}y_{j+2}y_{j+1}-x_{j+3}y_{j+2}x_{j+1}=0.$

###  3.2. Autoequivalence and invariant stability conditions

The spherical object and spherical twist were introduced by Seidel and Thomas
[35]. We briefly recall the definition and property here: for our use we
simply consider $\mathcal{D}:=D^{b}(\mathcal{V})$ where $\mathcal{V}$ is a
local Calabi-Yau variety of dimension $n$.

######  Definition 3.6.

An object $S\in\mathcal{D}$ is called $n$-spherical if the following
conditions are satisfied:

  1. (1)

For any $F\in\mathcal{D}$,
$\operatorname{\mathrm{Hom}}^{\bullet}_{\mathcal{D}}(F,S)$ and
$\operatorname{\mathrm{Hom}}^{\bullet}_{\mathcal{D}}(S,F)$ have finite (total)
dimension over $\mathbb{C}$.

  2. (2)

We have

$\operatorname{\mathrm{Ext}}^{k}_{\mathcal{D}}(S,S)=\left\\{\begin{array}[]{cc}\mathbb{C}&k=0,\
n,\\\ 0&\text{otherwise.}\end{array}\right.$

Let $S$ be a spherical object in $\mathcal{D}$, then the spherical twist
$\operatorname{Tw}_{S}(E)$ of $E\in\mathcal{D}$ is defined to be the cone of
the canonical evaluation morphism:

$\textstyle{\operatorname{\mathrm{Hom}}^{\bullet}(S,E)\otimes
S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Tw}_{S}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[1]}$

The following important lemma is due to Seidel and Thomas.

######  Lemma 3.7 ([35]).

Let $S$ be a spherical object in $\mathcal{D}$. Then $\mathrm{Tw}_{S}$ is an
exact autoequivalence of $\mathcal{D}$.

######  Lemma 3.8.

If $E$ is an exceptional object on $\mathbb{P}^{1}\times\mathbb{P}^{1}$, then
$s_{*}E$ is a 3-spherical object in $D^{b}(X)$.

###### Proof.

By using the Koszul resolution as in Proposition 3.4, we have

$\displaystyle\operatorname{\mathrm{Ext}}^{n}_{X}(s_{*}E,s_{*}E)$
$\displaystyle\cong$
$\displaystyle\operatorname{\mathrm{Ext}}^{n}_{Z}(E,E)\oplus\operatorname{\mathrm{Ext}}_{Z}^{3-n}(E,E)$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{cc}\mathbb{C}&n=0,\
3,\\\ 0&\text{otherwise.}\end{array}\right.$

∎

Define
$\tau:\mathbb{P}^{1}\times\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}\times\mathbb{P}^{1}$,
$\tau(x,y):=(y,x)$, it has a natural extension to an automorphism of
$X=\operatorname{Tot}\mathcal{O}(-2,-2)$ which we also denote by $\tau$. We
consider the following functor

$\Psi:=\tau^{*}\circ\operatorname{Tw}_{S_{0}}\circ\left(-\otimes\pi^{*}\mathcal{O}(0,1)\right)$

which is an autoequivalence of $D^{b}(X)$ since it is a composition of
autoequivalences.

######  Lemma 3.9.

Let $\mathcal{A}$ be the heart of the bounded t-structure induced by
$\mathcal{Q}$, and $S_{i}$ be the simple objects in $\mathcal{A}$ defined in
Corollary 3.3. Then

$\Psi(S_{i})=S_{i+1},\quad i\in\mathbb{Z}_{4}.$

Therefore $\Psi$ reduces to be an autoequivalence of $\mathcal{A}$.

###### Proof.

By the projection formula, we have

$\displaystyle s_{*}\mathcal{O}(a,b)\otimes\pi^{*}\mathcal{O}(j,k)$
$\displaystyle=$ $\displaystyle s_{*}\left(\mathcal{O}(a,b)\otimes
s^{*}\pi^{*}\mathcal{O}(j,k)\right)$ $\displaystyle=$ $\displaystyle
s_{*}\left(\mathcal{O}(a,b)\otimes\mathcal{O}(j,k)\right)$ $\displaystyle=$
$\displaystyle s_{*}\mathcal{O}(a+j,b+k).$

  1. (1)

Recall $S_{3}=s_{*}\mathcal{O}(0,-1)[2]$. Thus
$S_{3}\otimes\pi^{*}\mathcal{O}(0,1)=s_{*}\mathcal{O}(0,0)[2]=S_{0}[2]$. Now

$\Psi(S_{3})=\tau^{*}\operatorname{Tw}_{S_{0}}(S_{0}[2])=\tau^{*}S_{0}=\tau_{*}s_{*}\mathcal{O}(0,0)=S_{0},$

where the second equality follows from the standard result, that if $S$ is an
$n$-spherical object, then

$\operatorname{Tw}_{S}(S)\cong S[1-n],$

and $n=3$ in our case.

  2. (2)

For $S_{0}=s_{*}\mathcal{O}(0,0)$,
$S_{0}\otimes\pi^{*}\mathcal{O}(0,1)=s_{*}\mathcal{O}(0,1)$. By using the
similar calculations in Proposition 3.4, we have

$\operatorname{\mathrm{Hom}}^{\bullet}\left(s_{*}\mathcal{O}(0,0),s_{*}\mathcal{O}(0,1)\right)=\mathbb{C}^{2},$

then $\operatorname{Tw}_{S_{0}}\left(s_{*}\mathcal{O}(0,1)\right)$ fits into
the triangle:

$\textstyle{s_{*}\mathcal{O}(0,0)^{\oplus
2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{s_{*}\mathcal{O}(0,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Tw}_{S_{0}}\left(s_{*}\mathcal{O}(0,1)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[1]}$

By applying the exact functor $s_{*}$ to the short exact sequence on $Z$:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(0,-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}^{\oplus
2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(0,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

and comparing with the above triangle, we have

$\operatorname{Tw}_{S_{0}}\left(s_{*}\mathcal{O}(0,1)\right)\cong
s_{*}\mathcal{O}(0,-1)[1].$

Thus
$\Psi(S_{0})=\tau^{*}s_{*}\mathcal{O}(0,-1)[1]=s_{*}\mathcal{O}(-1,0)[1]=S_{1}$.

  3. (3)

For $S_{1}=s_{*}\mathcal{O}(-1,0)[1]$,
$S_{1}\otimes\pi^{*}\mathcal{O}(0,1)=s_{*}\mathcal{O}(-1,1)[1]$. Since

$\operatorname{\mathrm{Hom}}^{\bullet}\left(s_{*}\mathcal{O}(0,0),s_{*}\mathcal{O}(-1,1)[1]\right)=0,$

then $\operatorname{Tw}_{S_{0}}\left(s_{*}\mathcal{O}(-1,1)[1]\right)\cong
s_{*}\mathcal{O}(-1,1)[1]$. So we have

$\Psi(S_{1})=\tau^{*}s_{*}\mathcal{O}(-1,1)[1]=s_{*}\mathcal{O}(1,-1)[1]=S_{2}.$

  4. (4)

For $S_{2}=s_{*}\mathcal{O}(1,-1)[1]$,
$S_{2}\otimes\pi^{*}\mathcal{O}(0,1)=s_{*}\mathcal{O}(1,0)[1]$. Since

$\operatorname{\mathrm{Hom}}^{\bullet}\left(s_{*}\mathcal{O}(0,0),s_{*}\mathcal{O}(1,0)[1]\right)=\mathbb{C}^{2}[1],$

then $\operatorname{Tw}_{S_{0}}\left(s_{*}\mathcal{O}(1,0)\right)$ fits into
the triangle:

$\textstyle{s_{*}\mathcal{O}(0,0)^{\oplus
2}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{s_{*}\mathcal{O}(1,0)[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Tw}_{S_{0}}\left(s_{*}\mathcal{O}(1,0)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[1]}$

by the same argument as above, we have

$\operatorname{Tw}_{S_{0}}\left(s_{*}\mathcal{O}(1,0)\right)\cong
s_{*}\mathcal{O}(-1,0)[2].$

So we have

$\Psi(S_{2})=\tau^{*}s_{*}\mathcal{O}(-1,0)[2]=s_{*}\mathcal{O}(0,-1)[2]=S_{3}.$

∎

######  Definition 3.10.

We define the autoequivalence of $D^{b}_{0}(X)$

(3.3) $\Phi=\Psi^{2}.$

We denote by $\varphi$ and $\psi$ the automorphisms of $K_{0}(X)$ induced by
$\Phi$ and $\Psi$ respectively.

Let $\operatorname{\mathrm{Stab}}(X)$ denote the space of stability conditions
satisfying the support property on $D^{b}_{0}(X)$.

######  Definition 3.11 ($\Phi$-invariant stability conditions).

The space of stability conditions which are invariant under $\Phi$ is denoted
by $\operatorname{\mathrm{Stab}}(X)^{\Phi}$. Let
$\mathcal{U}(\mathcal{A})^{\Phi}$ be the set of $\Phi$-invariant stability
conditions with the fixed heart $\mathcal{A}$. We denote the connected
component of $\operatorname{\mathrm{Stab}}(X)^{\Phi}$ which contains
$\mathcal{U}(\mathcal{A})^{\Phi}$ by

$\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{0}.$

From now on, we denote by $\gamma_{i}=[S_{i}]$ the class of $S_{i}$ in
$K_{0}(\mathcal{A})=K_{0}(X)$, $i=0,\cdots,3$.

The subgroup of $K_{0}(\mathcal{A})$ whose elements are antisymmetric under
$\varphi$ is generated by $\gamma_{0}-\gamma_{2}$ and $\gamma_{1}-\gamma_{3}$,
and is denoted by $K_{0}(\mathcal{A})^{-\varphi}$. The quotient group is
denoted by

$\overline{K_{0}(\mathcal{A})}:=K_{0}(\mathcal{A})/K_{0}(\mathcal{A})^{-\varphi}.$

The quotient map is denoted by
$\nu:K_{0}(\mathcal{A})\rightarrow\overline{K_{0}(\mathcal{A})}$. Note that
$\overline{K_{0}(\mathcal{A})}$ is free abelian of rank $2$ with basis
$\bar{\gamma_{0}},\bar{\gamma_{1}}$ (we will abuse notation and still denote
$\gamma_{i}$ in the quotient group). And there is a natural isomorphism

$\operatorname{\mathrm{Hom}}_{\mathbb{Z}}(\overline{K_{0}(\mathcal{A})},\mathbb{C})\longrightarrow\operatorname{\mathrm{Hom}}_{\mathbb{Z}}(K_{0}(\mathcal{A}),\mathbb{C})^{\varphi}.$

Therefore we can equally define the $\Phi$-invariant stability conditions to
be those whose central charges $Z:K_{0}(\mathcal{A})\rightarrow\mathbb{C}$
factor through $\overline{K_{0}(\mathcal{A})}$ and the slicings are invariant
under $\Phi$. For technical reason we will work with this definition.

By Corollary 2.24, the forgetful map
$\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{0}\rightarrow\operatorname{\mathrm{Hom}}_{\mathbb{Z}}(\overline{K_{0}(\mathcal{A})},\mathbb{C})\cong\mathbb{C}^{2}$
is a local homeomorphism.

At the end of this section, we recall the following definition from [16].

######  Definition 3.12.

Let $\operatorname{\mathrm{Aut}}(D^{b}_{0}(X))$ be the group of exact
$\mathbb{C}$-linear autoequivalences of the category $D^{b}_{0}(X)$, then
$\operatorname{\mathrm{Aut}}_{*}(D^{b}_{0}(X))$ is defined to be the
subquotient consisting of autoequivalences which preserve the connected
component $\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{0}$, modulo
those which acts trivially on it.

##  4\. Simple tilts and autoequivalence

In this section we use Proposition 2.7 to figure out the double simple tilts
of $\mathcal{A}$, that is we will calculate
$\mathcal{A}^{\prime}=L_{S_{i+2}}L_{S_{i}}\mathcal{A}$ and
$R_{S_{i+2}}R_{S_{i}}\mathcal{A}$ for $i\in\mathbb{Z}_{4}$.

We simply write $L_{i}:=L_{S_{i}}$ and $R_{i}:=R_{S_{i}}$. Recall that there
are $4$ simple objects in $\mathcal{A}$ up to isomorphism:

$S_{0}=s_{*}\mathcal{O}(0,0),\quad S_{1}=s_{*}\mathcal{O}(-1,0)[1],\quad
S_{2}=s_{*}\mathcal{O}(1,-1)[1],\quad S_{3}=s_{*}\mathcal{O}(0,-1)[2].$

Proposition 3.4 shows that there is no extension between $S_{i}$ and
$S_{i+2}$, for $i\in\mathbb{Z}_{4}$, therefore
$L_{i}L_{i+2}\mathcal{A}=L_{i+2}L_{i}\mathcal{A}$.

  1. $L_{0}\mathcal{A}$:

Since the only non-trivial extension to $S_{0}$ is
$\operatorname{\mathrm{Ext}}^{1}(S_{1},S_{0})$, thus the new simple objects
are

$S_{0}^{\prime}=S_{0}[1],\quad S_{1}^{\prime},\quad S_{2}^{\prime}=S_{2},\quad
S_{3}^{\prime}=S_{3},$

where $S_{1}^{\prime}$ fits into the triangle

$S_{0}^{\oplus 2}\rightarrow S_{1}^{\prime}\rightarrow S_{1}\rightarrow
S_{0}^{\oplus 2}[1].$

Thus $S_{1}^{\prime}$ fits into the short exact sequence

(4.1) $0\rightarrow s_{*}\mathcal{O}(-1,0)\rightarrow s_{*}\mathcal{O}^{\oplus
2}\rightarrow S_{1}^{\prime}\rightarrow 0.$

We already see the above short exact sequence in part 4 of the proof of Lemma
3.9, therefore $S_{1}^{\prime}=s_{*}\mathcal{O}(1,0)$.

  2. $L_{2}L_{0}\mathcal{A}$:

The new simple objects are

$\widetilde{S_{0}}=S_{0}^{\prime},\quad\widetilde{S_{1}}=S_{1}^{\prime},\quad\widetilde{S_{2}}=S_{2}^{\prime}[1],\quad\widetilde{S_{3}}$

where $\widetilde{S_{3}}$ fits into the triangle

$S_{2}^{\oplus 2}\rightarrow\widetilde{S_{3}}\rightarrow
S_{3}^{\prime}\rightarrow S_{2}^{\oplus 2}[1].$

Thus $\widetilde{S_{3}}[-1]$ fits into the short exact sequence

(4.2) $0\rightarrow s_{*}\mathcal{O}(0,-1)\rightarrow
s_{*}\mathcal{O}(1,-1)^{\oplus 2}\rightarrow\widetilde{S_{3}}[-1]\rightarrow
0.$

We obtain $\widetilde{S_{3}}=s_{*}\mathcal{O}(2,-1)[1]$.

Therefore in $L_{2}L_{0}\mathcal{A}$ we have the following simple objects up
to isomorphism:

$\widetilde{S_{0}}=s_{*}\mathcal{O}(0,0)[1],\quad\widetilde{S_{1}}=s_{*}\mathcal{O}(1,0),\quad\widetilde{S_{2}}=s_{*}\mathcal{O}(1,-1)[2],\quad\widetilde{S_{3}}=s_{*}\mathcal{O}(2,-1)[1].$

######  Theorem 4.1.

Let $\mathcal{T}=-\otimes\pi^{*}\mathcal{O}(1,0)$ and
$\mathcal{T}_{\Psi}=\Psi\circ\mathcal{T}\circ\Psi^{-1}$, then we have

$\displaystyle L_{2}L_{0}\mathcal{A}$ $\displaystyle=$
$\displaystyle\mathcal{T}\mathcal{A};$ $\displaystyle R_{3}R_{1}\mathcal{A}$
$\displaystyle=$ $\displaystyle\mathcal{T}^{-1}\mathcal{A};$ $\displaystyle
L_{3}L_{1}\mathcal{A}$ $\displaystyle=$
$\displaystyle\mathcal{T}_{\Psi}\mathcal{A};$ $\displaystyle
R_{2}R_{0}\mathcal{A}$ $\displaystyle=$
$\displaystyle\mathcal{T}_{\Psi}^{-1}\mathcal{A}.$

###### Proof.

$\mathcal{T}\mathcal{A}\subset L_{2}L_{0}\mathcal{A}$ follows directly from
the comparison of the simple objects after reordering them. By Lemma 2.5 we
have $\mathcal{T}\mathcal{A}=L_{2}L_{0}\mathcal{A}$. We also have

$\displaystyle\Psi\circ\mathcal{T}\circ\Psi^{-1}(\mathcal{A})$
$\displaystyle=$ $\displaystyle\Psi L_{2}L_{0}\circ\Psi^{-1}\mathcal{A}$
$\displaystyle=$
$\displaystyle\Psi\circ\Psi^{-1}L_{3}L_{1}\mathcal{A}\quad\text{by Lemma
\ref{aut_sim_tilt} and Lemma \ref{aut_quiver} }$ $\displaystyle=$
$\displaystyle L_{3}L_{1}\mathcal{A}.$

This proves the third identity. For the right mutation
$R_{3}R_{1}\mathcal{A}$, by using Remark 2.4 we have

$R_{S_{0}[1]}R_{S_{2}[1]}L_{S_{2}}L_{S_{0}}\mathcal{A}=\mathcal{A}.$

Note that
$R_{S_{0}[1]}R_{S_{2}[1]}L_{S_{2}}L_{S_{0}}\mathcal{A}=R_{S_{0}[1]}R_{S_{2}[1]}\mathcal{T}\mathcal{A}=\mathcal{T}R_{S_{1}}R_{S_{3}}\mathcal{A}$
by Lemma 2.6, and $\mathcal{T}(S_{1})=S_{0}[1]$,
$\mathcal{T}(S_{3})=S_{2}[1]$. Therefore combining with the above identities
we have

$R_{S_{1}}R_{S_{3}}\mathcal{A}=\mathcal{T}^{-1}\mathcal{A}.$

Finally for $R_{2}R_{0}\mathcal{A}$, the calculation is quite similar to that
of $L_{3}L_{1}\mathcal{A}$ and we leave it to the reader. ∎

$\mathcal{T}$ and $\mathcal{T}_{\Psi}$ induce automorphisms $t$ and $t_{\psi}$
of the Grothendieck group $K_{0}(\mathcal{A})$. The following results will be
useful later:

######  Lemma 4.2.

With respect to the basis $\\{\gamma_{i}\\}$ of $K_{0}(\mathcal{A})$, the
automorphisms $t$ and $t_{\psi}$ have the matrix forms:

(4.3) $t=\left[\begin{array}[]{c c c c}2&-1&0&0\\\ 1&0&0&0\\\ 0&0&2&-1\\\
0&0&1&0\end{array}\right]\quad t_{\psi}=\left[\begin{array}[]{c c c
c}0&0&0&1\\\ 0&2&-1&0\\\ 0&1&0&0\\\ -1&0&0&2\end{array}\right]\quad$

###### Proof.

Follows directly from the calculations above. ∎

It is easy to check that $t$ and $t_{\psi}$ preserve the subgroup
$K_{0}(\mathcal{A})^{-\varphi}$. Therefore $t$ and $t_{\psi}$ can be regarded
as the actions on $\overline{K_{0}(\mathcal{A})}$. In fact, when reducing to
$\overline{K_{0}(\mathcal{A})}$, $t$ and $t_{\psi}$ have the matrix forms with
respect to the basis $\\{\gamma_{0},\gamma_{1}\\}$:

(4.4) $t|_{\overline{K_{0}(\mathcal{A})}}=\left[\begin{array}[]{c c}2&-1\\\
1&0\end{array}\right]\quad
t_{\psi}|_{\overline{K_{0}(\mathcal{A})}}=\left[\begin{array}[]{c c}0&1\\\
-1&2\end{array}\right],$

we have

(4.5)
$\left(t|_{\overline{K_{0}(\mathcal{A})}}\right)^{-1}=t_{\psi}|_{\overline{K_{0}(\mathcal{A})}}.$

We have the following relation in $K_{0}(\mathcal{A})$.

######  Lemma 4.3.

Let $x=([a:b],[c:d])$ be a closed point in
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, $\delta$ be the class of
$\mathcal{O}_{x}$ in $K_{0}(\mathcal{A})$. Then we have

$\delta=\sum_{i=0}^{3}\gamma_{i}.$

###### Proof.

We write $F_{1}=s_{*}{\mathcal{O}}_{[a:b]\times\mathbb{P}^{1}},\
F_{2}=s_{*}\mathcal{O}_{[a:b]\times\mathbb{P}^{1}}(-1)[1]$. First we have the
short exact sequences

$\displaystyle 0\rightarrow s_{*}\mathcal{O}(-1,0)\rightarrow
s_{*}\mathcal{O}\rightarrow F_{1}\rightarrow 0,$ $\displaystyle 0\rightarrow
s_{*}\mathcal{O}(0,-1)\rightarrow s_{*}\mathcal{O}(1,-1)\rightarrow
F_{2}[-1]\rightarrow 0.$

This gives $[F_{1}]=\gamma_{0}+\gamma_{1}$, $[F_{2}]=\gamma_{2}+\gamma_{3}$.
Then we consider the short exact sequence

(4.6) $0\rightarrow
s_{*}\mathcal{O}_{[a:b]\times\mathbb{P}^{1}}(-1)\rightarrow
s_{*}{\mathcal{O}}_{[a:b]\times\mathbb{P}^{1}}\rightarrow
s_{*}\mathcal{O}_{x}\rightarrow 0$

which gives $\delta=[F_{1}]+[F_{2}]=\sum_{i}\gamma_{i}$ as required. ∎

The above lemma shows that $[\mathcal{O}_{x}]$ does not depend on $x\in Z$.

##  5\. Semistable Objects

###  5.1. (Semi)stable objects

In this section we describe the set of stable objects for stability conditions
$\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}$ . The description relies on the
known properties of stability conditions for the Kronecker quiver.

Denote by $K_{2}$ the Kronecker quiver

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

and $\operatorname{\mathrm{rep}}(K_{2})$ the category of representations. We
denote by $C_{0}$ and $C_{1}$ the simple objects at vertices $0$ and $1$.
Recall the underlying quiver $Q$ of $\mathcal{A}$ (Proposition 3.4) is

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

######  Definition 5.1.

We define full subcategories of
$\mathcal{A}\cong\operatorname{\mathrm{rep}}_{\operatorname{\mathrm{nil}}}(Q,I)$
which can naturally be identified with $\operatorname{\mathrm{rep}}K_{2}$. The
objects of the full subcategories are

Subcategories | Objects | Dimension vectors | Class in $K_{0}(\mathcal{A})$
---|---|---|---
$\mathcal{K}^{I}_{1}$ | $\textstyle{\mathbb{C}^{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\mu_{2}}$$\textstyle{\mathbb{C}^{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ | $(p,q,0,0)$ | $p\gamma_{0}+q\gamma_{1}$
$\mathcal{K}^{I}_{2}$ | $\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}^{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}^{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\mu_{2}}$ | $(0,0,p,q)$ | $p\gamma_{2}+q\gamma_{3}$
$\mathcal{K}^{II}_{1}$ | $\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}^{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\mu_{2}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}^{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ | $(0,p,q,0)$ | $p\gamma_{1}+q\gamma_{2}$
$\mathcal{K}^{II}_{2}$ | $\textstyle{\mathbb{C}^{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}^{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\mu_{2}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ | $(q,0,0,p)$ | $p\gamma_{3}+q\gamma_{0}$

The objects in $\mathcal{K}^{I}_{i}$, $i=1,\ 2$ are called Kronecker type I,
and the objects in $\mathcal{K}^{II}_{i}$, $i=1,\ 2$ are called Kronecker type
II.

The following lemma is obvious

######  Lemma 5.2.

The full subcategories $\mathcal{K}^{I}_{i}$ and $\mathcal{K}^{II}_{i}$,
$i=1,\ 2$ are equivalent to $\operatorname{\mathrm{rep}}(K_{2})$. They are
Serre subcategories of
$\mathcal{A}\cong\operatorname{\mathrm{rep}}_{\operatorname{\mathrm{nil}}}(Q,I)$,
i.e., they are closed under taking quotients and subobjects.

We denote by $\Xi_{i}^{I}$ and $\Xi_{i}^{II}$ the corresponding embedding
functors from the full subcategories to $\mathcal{A}$.

Recall that for a finite acyclic quiver $Q$,
$K_{0}(\operatorname{\mathrm{rep}}Q)\cong\mathbb{Z}^{\oplus|Q_{0}|}$ is
generated by the simple modules $S_{i}$ at each vertex $i$. We denote by
$n_{ij}$ the number of arrows from vertex $i$ to $j$. Then the Euler form on
$K_{0}(\operatorname{\mathrm{rep}}Q)$ is defined by
$\chi([S_{i}],[S_{j}]):=\delta_{ij}-n_{ji}$. For
$\boldsymbol{\alpha}=(\alpha_{i})_{i\in Q_{0}}\in K_{0}(Q)$, the quadratic
form $q(-)$ is defined as
$q(\boldsymbol{\alpha}):=\chi(\boldsymbol{\alpha},\boldsymbol{\alpha})$. The
associated matrix of $q$ is a symmetrization of the associated matrix of
$\chi$.

When $Q$ is Dynkin or affine Dynkin (for example, the Kronecker quiver), it is
well-known that $q(-)$ is positive semi-definite. $\boldsymbol{\alpha}$ is
called a real root if $q(\boldsymbol{\alpha})=1$ and an imaginary root if
$q(\boldsymbol{\alpha})=0$. We need the following well-known result (for
example, see [5, Theorem 4.3.2]).

######  Theorem 5.3 (Indecomposable representations of Kronecker quiver).

We identify $K_{0}(\operatorname{\mathrm{rep}}K_{2})\cong\mathbb{Z}^{2}$ using
the basis $([C_{0}],[C_{1}])$. Then

  1. (1)

for each real root $(n,n+1)$ or $(n+1,n)$ ($n\geq 0$), there is a unique
indecomposable representation with this class in
$K_{0}(\operatorname{\mathrm{rep}}K_{2})$, up to isomorphism, which we will
denote by $E_{n,n+1}$ or $E_{n+1,n}$;

  2. (2)

for each imaginary root $(n,n)$ ($n\geq 1$), there is a family of
indecomposable representations indexed by $\mathbb{P}^{1}$ with this class in
$K_{0}(\operatorname{\mathrm{rep}}K_{2})$, which we denote by
$E^{\lambda}_{n}$ where $\lambda=[a:b]\in\mathbb{P}^{1}$.

The above are all the indecomposable representations in
$\operatorname{\mathrm{rep}}K_{2}$ up to isomorphism.

We have the following characterization of stability conditions for Kronecker
quivers by Okada [32].

######  Lemma 5.4.

Take a stability function
$Z:K_{0}(\operatorname{\mathrm{rep}}K_{2})\rightarrow\mathbb{C}$ and denote by
$\phi(E)$ the phase of a nonzero object
$E\in\operatorname{\mathrm{rep}}(K_{2})$

  1. (1)

if $\phi(C_{0})<\phi(C_{1})$, then every indecomposable representation of
$K_{2}$ is semistable, moreover, all indecomposable representations except for
$E_{m}^{\lambda}$ when $m>1$ are stable.

  2. (2)

If $\phi(C_{1})<\phi(C_{0})$, then the only stable objects are $C_{0}$ and
$C_{1}$. The semistable objects are $C_{0}^{\oplus k},\ C_{1}^{\oplus k}$ for
$k>1$.

  3. (3)

If $\phi(C_{0})=\phi(C_{1})$, then all objects are semistable, and only
$C_{0}$, $C_{1}$ are stable.

###### Proof.

Since $\operatorname{\mathrm{rep}}K_{2}$ is of finite-length, $Z$ satisfies
the Harder-Narasimhan property automatically, therefore $Z$ can be extended to
a stability condition for $D^{b}(K_{2})\cong D^{b}(\mathbb{P}^{1})$, and is
denoted by $(Z,\mathcal{P})$.

  1. (1)

Let $T=\mathcal{O}\oplus\mathcal{O}(1)$ be the tilting object in
$D^{b}(\mathbb{P}^{1})$, the functor $\mathbb{R}\mathrm{Hom}(T,-)$ sends
$\mathcal{O}$ and $\mathcal{O}(-1)[1]$ to $C_{0}$ and $C_{1}$ respectively. If
$\phi(C_{0})<\phi(C_{1})$, after rotating by
$\lambda=i\bigl{(}\pi-\phi(\mathcal{O}_{x})\bigr{)}$ where $x$ is a closed
point of $\mathbb{P}^{1}$, the resulting stability condition
$\lambda\cdot(Z,\mathcal{P})=(\overline{Z},\overline{\mathcal{P}})$ has heart
$\overline{\mathcal{P}}(0,1]=\mathrm{Coh}\ \mathbb{P}^{1}$ (see the following
figures).

$\cdots$$\vdots$$\vdots$$\cdots$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{0})=Z(\gamma_{2})$$\dots$$Z\bigl{(}\mathcal{O}(-1)\bigr{)}$$Z(\mathcal{O})$$Z\bigl{(}\mathcal{O}(1)\bigr{)}$$Z\bigl{(}\mathcal{O}(2)\bigr{)}$$Z(\mathcal{O}_{x})$$\ddots$$Z\bigl{(}\mathcal{O}(-1)[1]\bigr{)}$$Z(\mathcal{O}_{x})$$Z(\mathcal{O})$$Z\bigl{(}\mathcal{O}(1)\bigr{)}$$Z\bigl{(}\mathcal{O}(2)\bigr{)}$Central
charges of $\operatorname{\mathrm{rep}}K_{2}$Central charges after rotating

Therefore all line bundles and torsion sheaves are semistable, and in fact all
line bundles and skyscraper sheaves are stable. They correspond to the
indecomposable representations of $K_{2}$ by the functor
$\mathbb{R}\mathrm{Hom}(T,-)$.

  2. (2)

The second statement follows from the fact that $C_{0}$ is a simple subobject
of every indecomposable representation except $C_{1}$ and $C_{1}$ is a simple
factor object of every indecomposable representation except $C_{0}$.

  3. (3)

If $\phi(C_{0})=\phi(C_{1})$, then all nonzero objects in
$\operatorname{\mathrm{Rep}}(K_{2})$ have the same phase and therefore are
semistable.

∎

######  Remark 5.5.

Since $\mathcal{K}^{I}_{i}$ and $\mathcal{K}^{II}_{i}$ are equivalent to
$\operatorname{\mathrm{rep}}(K_{2})$ by Lemma 5.2, therefore by Theorem 5.3 we
can describe the indecomposable objects of Kronecker type I and II.

Since we will only be interested in the stable objects in
$\operatorname{\mathrm{rep}}(K_{2})$ by Lemma 5.4, by definition the stable
representations are bricks, that is,
$\operatorname{\mathrm{End}}(M)=\mathbb{C}$ if
$M\in\operatorname{\mathrm{rep}}(K_{2})$ is stable. Then the following
definition will be useful:

######  Definition 5.6 (Special Kronecker type).

We call the indecomposable object of Kronecker type I and II special if it is
a brick, or equivalently it is not isomorphic to the image of
$E^{\lambda}_{m}$ under $\Xi^{I}_{i}$ or $\Xi^{II}_{i}$ for $m>1$.

The following proposition follows gives us the geometric description of
objects of special Kronecker types. The calculations are direct and we leave
them to the reader.

######  Proposition 5.7.

Let $l\geq 0$. We have the following correspondences between objects in
$\mathcal{A}$ and $\operatorname{\mathrm{rep}}_{\mathrm{nil}}(Q,I)$ under the
equivalence
$\mathcal{R}_{\mathcal{Q}}:\mathcal{A}\rightarrow\operatorname{\mathrm{rep}}_{\mathrm{nil}}(Q,I)$
(we denote by $x$ a closed point of $\mathbb{P}^{1}$):

Objects of special | Classes in $K_{0}(\mathcal{A})$ | Objects of special | Classes in $K_{0}(\mathcal{A})$
---|---|---|---
Kronecker type I |  | Kronecker type II |
$s_{*}\mathcal{O}(l,0)$ | $(l+1)\gamma_{0}+l\gamma_{1}$ | $\Psi(s_{*}\mathcal{O}(l,0))$ | $(l+1)\gamma_{1}+l\gamma_{2}$
$s_{*}\mathcal{O}(l+1,-1)[1]$ | $(l+1)\gamma_{2}+l\gamma_{3}$ | $\Psi(s_{*}\mathcal{O}(l+1,-1)[1])$ | $(l+1)\gamma_{3}+l\gamma_{0}$
$s_{*}\mathcal{O}(-l-1,0)[1]$ | $l\gamma_{0}+(l+1)\gamma_{1}$ | $\Psi(s_{*}\mathcal{O}(-l-1,0)[1])$ | $l\gamma_{1}+(l+1)\gamma_{2}$
$s_{*}\mathcal{O}(-l,-1)[2]$ | $l\gamma_{2}+(l+1)\gamma_{3}$ | $\Psi(s_{*}\mathcal{O}(-l,-1)[2])$ | $l\gamma_{3}+(l+1)\gamma_{0}$
$F_{1}=s_{*}\mathcal{O}_{\\{x\\}\times\mathbb{P}^{1}}$ | $\gamma_{0}+\gamma_{1}$ | $\Psi(s_{*}\mathcal{O}_{\\{lx\\}\times\mathbb{P}^{1}})$ | $\gamma_{1}+\gamma_{2}$
$F_{2}=s_{*}\mathcal{O}_{\\{x\\}\times\mathbb{P}^{1}}(-1)[1]$ | $\gamma_{2}+\gamma_{3}$ | $\Psi(s_{*}\mathcal{O}_{\\{x\\}\times\mathbb{P}^{1}}(-1)[1])$ | $\gamma_{3}+\gamma_{0}$

From now on, we often identify objects in
$\operatorname{\mathrm{rep}}_{\mathrm{nil}}(Q,I)$ with the corresponding
objects in $\mathcal{A}$ without further comment.

######  Definition 5.8.

We introduce the open subsets of $\mathcal{U}(\mathcal{A})^{\Phi}$:

$\displaystyle\mathcal{U}(\mathcal{A})^{\Phi}_{+}$ $\displaystyle=$
$\displaystyle\\{\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}:\phi(S_{0})=\phi(S_{2})<\phi(S_{1})=\phi(S_{3})\\},$
$\displaystyle\mathcal{U}(\mathcal{A})^{\Phi}_{-}$ $\displaystyle=$
$\displaystyle\\{\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}:\phi(S_{1})=\phi(S_{3})<\phi(S_{0})=\phi(S_{1})\\}$

$\cdots$$\vdots$$\vdots$$\cdots$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{0})=Z(\gamma_{2})$$\dots$$Z\bigl{(}\mathcal{O}(-1)\bigr{)}$$Z(\mathcal{O})$$Z\bigl{(}\mathcal{O}(1)\bigr{)}$$Z\bigl{(}\mathcal{O}(2)\bigr{)}$$Z(\mathcal{O}_{x})$$\ddots$$Z\bigl{(}\mathcal{O}(-1)[1]\bigr{)}$$Z(\mathcal{O}_{x})$$Z(\mathcal{O})$$Z\bigl{(}\mathcal{O}(1)\bigr{)}$$Z\bigl{(}\mathcal{O}(2)\bigr{)}$Central
charges of $\operatorname{\mathrm{rep}}K_{2}$Central charges after
rotating$Z(\gamma_{0})=Z(\gamma_{2})$$Z(\gamma_{1})=Z(\gamma_{3})$ Figure 2.
Central charges of $\mathcal{U}(\mathcal{A})^{\Phi}_{+}$

$\cdots$$\vdots$$\vdots$$\cdots$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{0})=Z(\gamma_{2})$$\dots$$Z\bigl{(}\mathcal{O}(-1)\bigr{)}$$Z(\mathcal{O})$$Z\bigl{(}\mathcal{O}(1)\bigr{)}$$Z\bigl{(}\mathcal{O}(2)\bigr{)}$$Z(\mathcal{O}_{x})$$\ddots$$Z\bigl{(}\mathcal{O}(-1)[1]\bigr{)}$$Z(\mathcal{O}_{x})$$Z(\mathcal{O})$$Z\bigl{(}\mathcal{O}(1)\bigr{)}$$Z\bigl{(}\mathcal{O}(2)\bigr{)}$Central
charges of $\operatorname{\mathrm{rep}}K_{2}$Central charges after
rotating$Z(\gamma_{0})=Z(\gamma_{2})$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{0})=Z(\gamma_{2})$
Figure 3. Central charges of $\mathcal{U}(\mathcal{A})^{\Phi}_{-}$

######  Lemma 5.9.

The autoequivalence $\Psi$ induces a bijection between
$\mathcal{U}(\mathcal{A})^{\Phi}_{+}$ and
$\mathcal{U}(\mathcal{A})^{\Phi}_{-}$.

###### Proof.

Given $\sigma=(Z,\mathcal{A})\in\mathcal{U}(\mathcal{A})^{\Phi}_{+}$, we
denote by $\Psi(\sigma)=(Z_{\psi},\Psi(\mathcal{A})=\mathcal{A})$. By Lemma
3.9, $Z_{\psi}(\gamma_{i})=Z(\gamma_{i-1})$, therefore
$\Psi(\sigma)\in\mathcal{U}(\mathcal{A})^{\Phi}_{-}$. The statement follows
immediately. ∎

Let $\sigma=(Z,\mathbb{P})\in\mathcal{U}(\mathcal{A})^{\Phi}$. We define a
stability function $Z$ on $\operatorname{\mathrm{rep}}K_{2}$ by setting
$\overline{Z}(C_{0})=Z(\gamma_{j})$, $\overline{Z}(C_{1})=Z(\gamma_{j+1})$
where $C_{i}$ are the simple modules at the vertex $i$.

######  Lemma 5.10.

The stable objects in $\operatorname{\mathrm{rep}}K_{2}$ with respect to
$\overline{Z}$ are stable in $\mathcal{A}$ with respect to $\sigma$.

###### Proof.

By Lemma 5.2 $\operatorname{\mathrm{rep}}K_{2}$ is closed under taking
quotients and subobjects, and since $\Xi^{I}_{i}$ and $\Xi^{II}_{i}$ preserve
the ordering by phases, it follows that if
$E\in\operatorname{\mathrm{rep}}K_{2}$ is stable, then $\Xi^{I}_{i}(E)$ and
$\Xi^{II}_{i}(E)$ are stable in $\mathcal{A}$. ∎

We immediately have the following theorem

######  Theorem 5.11.

Let $\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}_{+}$. Then the objects of
special Kronecker type I are stable for $\sigma$. For
$\tau\in\mathcal{U}(\mathcal{A})^{\Phi}_{-}$, then the objects of special
Kronecker type II are stable for $\tau$.

######  Corollary 5.12.

Let $x\in\mathbb{P}^{1}\times\mathbb{P}^{1}$, then $\mathcal{O}_{x}$ is
semistable with respect to the stability condition
$\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}_{+}$. For
$\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}_{-}$, there is a semistable object
whose class in $K_{0}(\mathcal{A})$ is $\delta=[\mathcal{O}_{x}]$.

###### Proof.

Let $F_{1}=s_{*}\mathcal{O}_{\\{x_{1}\\}\times\mathbb{P}^{1}},\
F_{2}=s_{*}\mathcal{O}_{\\{x_{1}\\}\times\mathbb{P}^{1}}(-1)[1]$ where
$x_{1}=p_{1}(x)$. By (4.6), there is a short exact sequence in $\mathcal{A}$:

$0\rightarrow F_{1}\rightarrow\mathcal{O}_{x}\rightarrow F_{2}\rightarrow 0.$

By Proposition 5.7 and Theorem 5.11 $F_{1}$ and $F_{2}$ are stable for the
stability condition $\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}_{+}$. Moreover,
since $Z(F_{1})=Z(\gamma_{0})+Z(\gamma_{1})$ and
$Z(F_{2})=Z(\gamma_{2})+Z(\gamma_{3})$ therefore $\phi(F_{1})=\phi(F_{2})$. So
$\mathcal{O}_{x}$ is (strictly) semistable for $\sigma$ with the same phase as
$\phi(F_{i})$.

Suppose $\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}_{-}$. We take
$\Psi^{-1}(\sigma)$, by Lemma 5.9
$\Psi^{-1}(\sigma)\in\mathcal{U}(\mathcal{A})^{\Phi}_{+}$. Therefore
$\Psi(\mathcal{O}_{x})$ is semistable for $\sigma$. Since

$\psi(\delta)=\sum_{i=0}^{3}\psi(\gamma_{i})=\sum_{i=0}^{3}\gamma_{i}=\delta,$

the claim is proved. ∎

######  Remark 5.13.

For $\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}$, we can conclude that there is
a semistable object whose class in $K_{0}(\mathcal{A})$ is $\delta$. By the
above corollary, the remaining case we need to verify is that when
$\phi(S_{i})=\phi(\mathcal{O}_{x})$ for all $i$, however, each object in
$\mathcal{A}$ is semistable in this case.

The central charges of $\delta$ and other stable objects for
$\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}_{+}$ are depicted in the figure 4.

$\cdots$$\vdots$$\vdots$$\cdots$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{0})=Z(\gamma_{2})$$\dots$$Z\bigl{(}\mathcal{O}(-1)\bigr{)}$$Z(\mathcal{O})$$Z\bigl{(}\mathcal{O}(1)\bigr{)}$$Z\bigl{(}\mathcal{O}(2)\bigr{)}$$Z(\mathcal{O}_{x})$$\ddots$$Z\bigl{(}\mathcal{O}(-1)[1]\bigr{)}$$Z(\mathcal{O}_{x})$$Z(\mathcal{O})$$Z\bigl{(}\mathcal{O}(1)\bigr{)}$$Z\bigl{(}\mathcal{O}(2)\bigr{)}$Central
charges of $\operatorname{\mathrm{rep}}K_{2}$Central charges after
rotating$Z(\gamma_{0})=Z(\gamma_{2})$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{0})=Z(\gamma_{2})$$\vdots$$Z(\gamma_{0})=Z(\gamma_{2})$$Z(\gamma_{1})=Z(\gamma_{2})$$\vdots$$Z(\gamma_{0}+\gamma_{1})=Z(\gamma_{2}+\gamma_{3})$$Z(n\gamma_{0}+(n+1)\gamma_{1})=Z(n\gamma_{2}+(n+1)\gamma_{3})$$Z((n+1)\gamma_{0}+n\gamma_{1})=Z((n+1)\gamma_{2}+n\gamma_{3})$$Z(\delta)$
Figure 4. Central charges of stable objects and $\mathcal{O}_{x}$ for
$\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}_{+}$

Recall that $\mathcal{T}=-\otimes\pi^{*}\mathcal{O}(1,0)$ and
$\mathcal{T}_{\Psi}=\Psi\circ\mathcal{T}\circ\Psi^{-1}$, the simple objects in
$\mathcal{A}$ are $S_{0}=s_{*}\mathcal{O}(0,0)$,
$S_{1}=s_{*}\mathcal{O}(-1,0)[1]$, $s_{*}\mathcal{O}(1,-1)[1]$ and
$S_{3}=s_{*}\mathcal{O}(0,-1)[2]$. Finally we mention another description of
some objects of special Kronecker types I and II,

######  Lemma 5.14.

Let $n\geq 0$

  1. (1)

$\mathcal{T}^{n}(S_{0}),\ \mathcal{T}^{n}(S_{2}),\ \mathcal{T}^{-n}(S_{1}),\
\mathcal{T}^{-n}(S_{3})$ are objects of special Kronecker type I with classes
in $K_{0}(\mathcal{A})$:

$(n+1)\gamma_{0}+n\gamma_{1},\quad(n+1)\gamma_{2}+n\gamma_{3},\quad
n\gamma_{0}+(n+1)\gamma_{1},\quad n\gamma_{2}+(n+1)\gamma_{3}.$

  2. (2)

$\mathcal{T}^{n}_{\Psi}(S_{1}),\ \mathcal{T}^{n}_{\Psi}(S_{3}),\
\mathcal{T}^{-n}_{\Psi}(S_{0}),\ \mathcal{T}^{-n}_{\Psi}(S_{2})$ are objects
of special Kronecker type II with classes in $K_{0}(\mathcal{A})$:

$(n+1)\gamma_{1}+n\gamma_{2},\quad(n+1)\gamma_{3}+n\gamma_{0},\quad
n\gamma_{1}+(n+1)\gamma_{2},\quad n\gamma_{3}+(n+1)\gamma_{0}.$

###### Proof.

Note that
$\mathcal{T}^{n}\bigl{(}s_{*}\mathcal{O}(a,b)\bigr{)}=s_{*}\mathcal{O}(a+n,b)$
by the projection formula, the result follows directly from the table in
Proposition 5.7. ∎

For $\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}_{+}$, we can alternatively
illustrate the central charges of semistable objects in the complex plane:

$\cdots$$\vdots$$\vdots$$\cdots$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{0})=Z(\gamma_{2})$$\dots$$Z\bigl{(}\mathcal{O}(-1)\bigr{)}$$Z(\mathcal{O})$$Z\bigl{(}\mathcal{O}(1)\bigr{)}$$Z\bigl{(}\mathcal{O}(2)\bigr{)}$$Z(\mathcal{O}_{x})$$\ddots$$Z\bigl{(}\mathcal{O}(-1)[1]\bigr{)}$$Z(\mathcal{O}_{x})$$Z(\mathcal{O})$$Z\bigl{(}\mathcal{O}(1)\bigr{)}$$Z\bigl{(}\mathcal{O}(2)\bigr{)}$Central
charges of $\operatorname{\mathrm{rep}}K_{2}$Central charges after
rotating$Z(\gamma_{0})=Z(\gamma_{2})$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{0})=Z(\gamma_{2})$$\vdots$$Z(\gamma_{0})=Z(\gamma_{2})$$Z(\gamma_{1})=Z(\gamma_{2})$$\vdots$$Z(\gamma_{0}+\gamma_{1})=Z(\gamma_{2}+\gamma_{3})$$Z(n\gamma_{0}+(n+1)\gamma_{1})=Z(n\gamma_{2}+(n+1)\gamma_{3})$$Z((n+1)\gamma_{0}+n\gamma_{1})=Z((n+1)\gamma_{2}+n\gamma_{3})$$Z(\delta)$$\vdots$$Z(S_{0})=Z(S_{2})$$Z(S_{1})=Z(S_{3})$$\vdots$$Z(\mathcal{T}^{-n}(S_{1}))$$Z(\mathcal{T}^{n}(S_{0}))$$Z(\delta)$$Z(\mathcal{T}^{-1}(S_{1}))$$Z(\mathcal{T}(S_{0}))$

###  5.2. There are no other stable objects

This subsection is the main part of this paper. We prove that for
$\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}$, there are no other stable objects
other than the ones in Theorem 5.11.

For simplicity, we first restrict ourselves to the normalized stability
conditions

$\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{n}:=\\{\sigma=(Z,\mathcal{P}):Z(\delta)=i\\}\subset\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{0},$

where $\delta$ is the class of skyscrapper sheaf $\mathcal{O}_{x}$ in
$K_{0}(\mathcal{A})$. Note that
$\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{n}$ is a connected
submanifold of $\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{0}$.

Let
$\mathcal{U}^{n}(\mathcal{A})^{\Phi}\subset\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{n}$
which consists of normalized stability conditions $(Z,\mathcal{A})$ with the
fixed heart $\mathcal{A}$, and
$\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{\pm}=\\{(Z,\mathcal{A})\in\mathcal{U}(\mathcal{A})^{\Phi}_{\pm}:Z(\delta)=i\\}$.
For $\sigma\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{+}$, we have

(5.1)
$\phi(S_{0})=\phi(S_{2})<\phi(\mathcal{O}_{x})=\frac{1}{2}<\phi(S_{1})=\phi(S_{3}).$

Recall the group action on the stability conditions in Definition 2.25. Let
$\mathcal{T}$, $\mathcal{T}_{\Psi}$ be the autoequivalences defined in Theorem
4.1. We first make the following important observartion

######  Lemma 5.15.

  1. (1)

For $\sigma\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{+}$, then
$\mathcal{T}(\sigma)=\sigma\cdot\widetilde{g}$ where
$\widetilde{g}=(g,f)\in\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$ such that
$f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies
$f\left(\frac{1}{2}\right)=\frac{1}{2}$.

  2. (2)

For $\sigma\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{-}$, then
$\mathcal{T}_{\Psi}(\sigma)=\sigma\cdot\widetilde{g}$ where
$\widetilde{g}=(g,f)\in\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$ such that
$f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies
$f\left(\frac{1}{2}\right)=\frac{1}{2}$.

###### Proof.

Let $\sigma\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{+}$. We write
$\mathcal{T}(\sigma)=(Z_{t},\mathcal{P}_{\mathcal{T}})$. By viewing
$\mathbb{C}\cong\mathbb{R}^{2}$, we let

$\boldsymbol{e_{0}}=Z(\gamma_{0})=Z(\gamma_{2}),\quad\boldsymbol{e_{1}}=Z(\gamma_{1})=Z(\gamma_{3}).$

Then by (4.5), we have

$\displaystyle Z_{t}(\gamma_{0})$ $\displaystyle=$ $\displaystyle
Z\left(t^{-1}\gamma_{0}\right)=Z\left(-\gamma_{1}\right)=-\boldsymbol{e_{1}};$
$\displaystyle Z_{t}(\gamma_{1})$ $\displaystyle=$ $\displaystyle
Z\left(t^{-1}\gamma_{1}\right)=Z\left(\gamma_{0}+2\gamma_{1}\right)=\boldsymbol{e_{0}}+2\boldsymbol{e_{1}}.$

We define $g\in\mathrm{GL}^{+}(2,\mathbb{R})$ such that

$g(\boldsymbol{e_{0}})=2\boldsymbol{e_{0}}+\boldsymbol{e_{1}},\quad
g(\boldsymbol{e_{1}})=-\boldsymbol{e_{0}},$

Note that
$2(\boldsymbol{e_{0}}+\boldsymbol{e_{1}})=\sum_{i}Z(\gamma_{i})=Z(\delta)=i$
by Lemma 4.3, and
$g(\boldsymbol{e_{0}}+\boldsymbol{e_{1}})=\boldsymbol{e_{0}}+\boldsymbol{e_{1}}$,
therefore we see $g$ preserves the positive imaginary axis. We can take
$\widetilde{g}=(g,f)\in\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$ be the
unique lift of $g$ such that $f(\frac{1}{2})=\frac{1}{2}$. Let
$\sigma\cdot\widetilde{g}=(Z_{g},\mathcal{P}_{f})$. Then by definition
$Z_{g}(\gamma_{i})=Z_{t}(\gamma_{i})$ for $i=0,\ 1$. Therefore $Z_{g}=Z_{t}$.

We claim that the bounded hearts $\mathcal{P}_{\mathcal{T}}(0,1]$ and
$\mathcal{P}_{f}(0,1]$ are the same, thus finishing the proof of the first
case. By Theorem 4.1 we have

$\mathcal{P}_{\mathcal{T}}(0,1]=\mathcal{T}(\mathcal{A})=L_{S_{0}}L_{S_{2}}\mathcal{A}.$

Suppose $S_{1},S_{3}\in\mathcal{P}(\phi)$, $\phi\in(0,1]$. Since
$g(\boldsymbol{e_{1}})=-\boldsymbol{e_{0}}=-Z(\gamma_{0})=-Z(\gamma_{2})$,
therefore $f(\phi)=\phi(S_{0}[i])=\phi(S_{2}[i])$ for some odd number $i$.
Note that $\phi\in(1/2,1]\subset(1/2,3/2)$, so

$f(\phi)\in\left(f(\frac{1}{2}),f(\frac{3}{2})\right)=(1/2,3/2).$

By our assumption,

$\phi(S_{0}[1])=\phi(S_{2}[1])\in(1/2,3/2).$

Therefore, $i=1$ and $S_{0}[1],S_{2}[1]\in\mathcal{P}_{f}(\phi)$ which is
contained in $\mathcal{A}^{\prime}=\mathcal{P}_{f}(0,1]$.

Suppose $S_{0},S_{2}\in\mathcal{P}(\omega)$, $\omega\in(0,1]$. Then
$g(\boldsymbol{e_{0}})=2\boldsymbol{e_{0}}+\boldsymbol{e_{1}}=Z(\widetilde{S_{1}})=Z(\widetilde{S_{3}})$
where $\widetilde{S_{1}}=\mathcal{T}(S_{0})$ and
$\widetilde{S_{3}}=\mathcal{T}(S_{2})$. By Lemma 5.14 they are of special
Kronecker type I, therefore they are semistable for $\sigma$ by Theorem 5.11.
We have $f(\omega)=\phi(\widetilde{S_{1}}[n])=\phi(\widetilde{S_{3}}[n])$ for
some even number $n$. Since $\omega\in(0,1/2)\subset(-1/2,1/2)$, so
$f(\omega)\in(-1/2,1/2)$. By our assumption,
$\phi(\widetilde{S_{1}})=\phi(\widetilde{S_{2}})\in(-1/2,1/2)$. Therefore
$n=0$ and $\widetilde{S_{1}},\widetilde{S_{3}}\in\mathcal{A}^{\prime}$.

Recall the simple objects in $L_{S_{0}}L_{S_{1}}\mathcal{A}$ are exactly
$S_{0}[1]$, $S_{2}[1]$, $\widetilde{S_{1}}$ and $\widetilde{S_{3}}$ by our
computations in Section 4. We just proved
$L_{S_{0}}L_{S_{2}}\mathcal{A}=\langle
S_{0}[1],S_{2}[1],\widetilde{S_{1}},\widetilde{S_{3}}\rangle\subset\mathcal{A}^{\prime}$,
so by Lemma 2.5 the two hearts are equivalent. We finished the proof of the
first case.

For $\sigma\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{-}$, note that
$\Psi^{-1}(\sigma)\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{+}$ by Lemma 5.9,
therefore by the first part we have

$\displaystyle\mathcal{T}_{\Psi}(\sigma)$ $\displaystyle=$
$\displaystyle\Psi\circ\mathcal{T}\circ\Psi^{-1}(\sigma)$ $\displaystyle=$
$\displaystyle\Psi(\Psi^{-1}(\sigma)\cdot\widetilde{g})$ $\displaystyle=$
$\displaystyle\sigma\cdot\widetilde{g},$

where
$\widetilde{g}=(g,f)\in\widetilde{\operatorname{\mathrm{GL}}}^{+}(2,\mathbb{R})$
such that $f(1/2)=1/2$. This finishes the proof. ∎

######  Corollary 5.16.

If $\sigma=(Z,\mathcal{P})\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{+}$, then
$\mathcal{T}^{\pm
1}\left(\mathcal{P}(\frac{1}{2})\right)=\mathcal{P}(\frac{1}{2})$. Similarly,
if $\tau=(W,\mathcal{P}^{\prime})\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{-}$,
then $\mathcal{T}_{\Psi}^{\pm
1}\left(\mathcal{P}^{\prime}(\frac{1}{2})\right)=\mathcal{P}^{\prime}(\frac{1}{2})$.

###### Proof.

Let $\sigma=(Z,\mathcal{P})\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{+}$ and
$\mathcal{T}(\sigma)=(Z_{t},\mathcal{P}_{\mathcal{T}})$. By the above Lemma,
$\mathcal{T}(\sigma)=\sigma\cdot\widetilde{g}=(Z_{g},\mathcal{P}_{f})$ for
$\widetilde{g}=(g,f)\in\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$ such that
$f(1/2)=1/2$ , we have

$\mathcal{P}_{f}(1/2)=\mathcal{P}(1/2).$

The proof for the second statement is the same. ∎

Recall for any interval $I\subset\mathbb{R}$, $\mathcal{P}(I)$ is the
extension-closed subcategory of $D^{b}_{0}(X)$ generated by the subcategories
$\mathcal{P}(\phi)$ for $\phi\in I$. Recall the definition of
$\operatorname{\mathrm{Aut}}_{*}(D^{b}_{0}(X))$ in Definition 3.12. The
following proposition will be useful:

######  Proposition 5.17.

Let $W$ be an element of $\operatorname{\mathrm{Aut}}_{*}(D^{b}_{0}(X))$ such
that for a stability condition $\sigma=(Z,\mathcal{P})$ we have
$W(\sigma)=\sigma\cdot\widetilde{g}$ for some
$\widetilde{g}=(g,f)\in\widetilde{\operatorname{\mathrm{GL}}}^{+}(2,\mathbb{R})$.
Suppose that $E,\ F$ are two semistable objects with phases
$\phi_{E}<\phi_{F}$, then

$W\bigl{(}\mathcal{P}(\phi_{E},\phi_{F})\bigr{)}=\mathcal{P}\bigl{(}\phi_{W(E)},\phi_{W(F)}\bigr{)}.$

###### Proof.

We write $\sigma\cdot\widetilde{g}=(Z_{g},\mathcal{P}_{f})$, then
$\mathcal{P}_{f}(I)=\mathcal{P}\bigl{(}f(I)\bigr{)}$ for any interval
$I\subset\mathbb{R}$ by definition. Therefore by our assumption

$W\bigl{(}\mathcal{P}(\phi_{E},\phi_{F})\bigr{)}=\mathcal{P}\bigl{(}f(\phi_{E}),f(\phi_{F})\bigr{)}.$

Since $\sigma\cdot\widetilde{g}$ and $\sigma$ contain the same set of
semistable objects, therefore $W(E)$ and $W(F)$ are semistable for $\sigma$.
So we have $\phi_{W(E)}=f\bigl{(}\phi_{E}\bigr{)}$ and
$\phi_{W(F)}=f\bigl{(}\phi_{F}\bigr{)}$, this proves the result. ∎

######  Lemma 5.18.

Given a stability condition $\sigma=(Z,\mathcal{A})$ such that $\mathcal{A}$
is of finite-length with the finite set of simple objects (up to isomorphism)
$\\{S_{0},S_{1},\cdots,S_{n}\\}$. Define the linear cone in $\mathbb{C}$:

$C:=\\{z\in\mathbb{C}:z=\sum_{i=0}^{n}\lambda_{i}Z(S_{i}),\ \lambda_{i}\geq
0\\}\setminus\\{0\\}.$

Then for any non-zero semistable object $E\in\mathcal{P}(\psi)$,
$\psi\in\mathbb{R}$, we have

$Z(E)\in C\cup(-C).$

###### Proof.

After shifting $E$, we may assume that $E\in\mathcal{A}$. Since $\mathcal{A}$
is of finite-length, $E$ has a finite filtration by the simple objects
$S_{i}$. Then in $K_{0}(\mathcal{A})$ we have
$[E]=\sum_{i=0}^{n}\lambda_{i}[S_{i}],\ \lambda_{i}\geq 0$ (at least one
$\lambda_{j}\neq 0$). Therefore the result follows from the linearity of $Z$.
∎

The following important theorem characterizes the stable objects outside the
ray $\phi=\frac{1}{2}$ in the upper half complex plane:

######  Theorem 5.19.

Take $\sigma\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{+}$, there is no stable
object whose phase lies in the intervals $\bigl{(}0,\phi(S_{0})\bigr{)}$,
$\bigl{(}\phi(S_{1}),1\bigr{)}$, nor in the intervals

$\Bigl{(}\phi\left(\mathcal{T}^{m}(S_{0})\right),\phi\left(\mathcal{T}^{m+1}(S_{0})\right)\Bigr{)}\text{
or
}\Bigl{(}\phi\left(\mathcal{T}^{-m-1}(S_{1})\right),\phi\left(\mathcal{T}^{-m}(S_{1})\right)\Bigr{)},$

for any integer $m\geq 0$. Moreover, the stable objects of phases
$\phi\bigl{(}\mathcal{T}^{m}(S_{0})\bigr{)}$ and
$\phi\bigl{(}\mathcal{T}^{-m}(S_{1})\bigr{)}$ for $m\geq 0$ are of special
Kronecker type I.

###### Proof.

Given $\sigma=(Z,\mathcal{P})\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{+}$, it
is clear that there is no stable object of phase in the interval
$(0,\phi(S_{0}))\cup(\phi(S_{1}),1)$ by Lemma 5.18.

By Lemma 5.15 $\mathcal{T}(\sigma)=\sigma\cdot\widetilde{g}$ for some
$\widetilde{g}\in\widetilde{\operatorname{\mathrm{GL}}}^{+}(2,\mathbb{R})$,
therefore by Proposition 5.17 we only need to check that there is no stable
object of phase in the intervals
$\left(\phi(S_{0}),\phi(\mathcal{T}(S_{0}))\right)$ and
$\left(\phi(\mathcal{T}^{-1}(S_{1})),\phi(S_{1})\right)$, then apply
$\mathcal{T}^{\pm m}$ we see that there is no stable object of phase in other
open intervals.

$\cdots$$\vdots$$\vdots$$\cdots$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{0})=Z(\gamma_{2})$$\dots$$Z\bigl{(}\mathcal{O}(-1)\bigr{)}$$Z(\mathcal{O})$$Z\bigl{(}\mathcal{O}(1)\bigr{)}$$Z\bigl{(}\mathcal{O}(2)\bigr{)}$$Z(\mathcal{O}_{x})$$\ddots$$Z\bigl{(}\mathcal{O}(-1)[1]\bigr{)}$$Z(\mathcal{O}_{x})$$Z(\mathcal{O})$$Z\bigl{(}\mathcal{O}(1)\bigr{)}$$Z\bigl{(}\mathcal{O}(2)\bigr{)}$Central
charges of $\operatorname{\mathrm{rep}}K_{2}$Central charges after
rotating$Z(\gamma_{0})=Z(\gamma_{2})$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{0})=Z(\gamma_{2})$$\vdots$$Z(\gamma_{0})=Z(\gamma_{2})$$Z(\gamma_{1})=Z(\gamma_{2})$$\vdots$$Z(\gamma_{0}+\gamma_{1})=Z(\gamma_{2}+\gamma_{3})$$Z(n\gamma_{0}+(n+1)\gamma_{1})=Z(n\gamma_{2}+(n+1)\gamma_{3})$$Z((n+1)\gamma_{0}+n\gamma_{1})=Z((n+1)\gamma_{2}+n\gamma_{3})$$Z(\delta)$$\vdots$$Z(S_{0})=Z(S_{2})$$Z(S_{1})=Z(S_{3})$$\vdots$$Z(\mathcal{T}^{-n}(S_{1}))$$Z(\mathcal{T}^{n}(S_{0}))$$Z(\delta)$$Z(\mathcal{T}^{-1}(S_{1}))$$Z(\mathcal{T}(S_{0}))$$\vdots$$Z(S_{0})=Z(S_{2})$$Z(S_{1})=Z(S_{3})$$\vdots$$Z(\mathcal{T}^{-n}(S_{1}))$$Z(\mathcal{T}^{n}(S_{0}))$$Z(\delta)$$Z(\mathcal{T}^{-1}(S_{1}))$$Z(\mathcal{T}(S_{0}))$

Suppose $E\in\mathcal{P}\left(\phi(S_{0})+\epsilon\right)$ for
$0<\epsilon<\phi(\mathcal{T}(S_{0}))-\phi(S_{0})$. We will take a
$\mathbb{C}$-action on $\sigma$ and reduce to the case in the beginning: we
choose $0<\epsilon^{\prime}\ll\epsilon$ and let
$\lambda=-(\phi(S_{0})+\epsilon^{\prime})$. Then let
$\sigma^{\prime}:=\sigma\cdot\lambda=(Z^{\prime},\mathcal{P}^{\prime})$. The
phase of the semistable object for $\sigma^{\prime}$ is denoted by
$\phi^{\prime}(-)$. By definition of the $\mathbb{C}$-action (see Remark 2.26)
we have

$\displaystyle\phi^{\prime}(S_{0}[1])=\phi^{\prime}(S_{2}[1])$
$\displaystyle=$ $\displaystyle\phi(S_{0}[1])-\phi(S_{0})-\epsilon^{\prime}$
$\displaystyle=$ $\displaystyle 1-\epsilon^{\prime}\in(1/2,1),$
$\displaystyle\phi^{\prime}\left(\mathcal{T}(S_{0})\right)=\phi^{\prime}\left(\mathcal{T}(S_{2})\right)$
$\displaystyle=$
$\displaystyle\phi(\mathcal{T}(S_{0}))-\phi(S_{0})-\epsilon^{\prime}\in(0,1/2).$

We see that the heart $\mathcal{A}^{\prime}=\mathcal{P}^{\prime}(0,1]$
contains the simple objects
$\\{S_{0}[1],S_{2}[1],\mathcal{T}(S_{0}),\mathcal{T}(S_{2})\\}$ which generate
$L_{S_{0}}L_{S_{2}}\mathcal{A}$, we have
$L_{S_{0}}L_{S_{2}}(\mathcal{A})\subset\mathcal{A}^{\prime}$ therefore
$\mathcal{A}^{\prime}=L_{S_{0}}L_{S_{2}}(\mathcal{A})=\mathcal{T}\mathcal{A}$
by Theorem 4.1 (see figure 5).

$\cdots$$\vdots$$\vdots$$\cdots$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{0})=Z(\gamma_{2})$$\dots$$Z\bigl{(}\mathcal{O}(-1)\bigr{)}$$Z(\mathcal{O})$$Z\bigl{(}\mathcal{O}(1)\bigr{)}$$Z\bigl{(}\mathcal{O}(2)\bigr{)}$$Z(\mathcal{O}_{x})$$\ddots$$Z\bigl{(}\mathcal{O}(-1)[1]\bigr{)}$$Z(\mathcal{O}_{x})$$Z(\mathcal{O})$$Z\bigl{(}\mathcal{O}(1)\bigr{)}$$Z\bigl{(}\mathcal{O}(2)\bigr{)}$Central
charges of $\operatorname{\mathrm{rep}}K_{2}$Central charges after
rotating$Z(\gamma_{0})=Z(\gamma_{2})$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{1})=Z(\gamma_{3})$$Z(\gamma_{0})=Z(\gamma_{2})$$\vdots$$Z(\gamma_{0})=Z(\gamma_{2})$$Z(\gamma_{1})=Z(\gamma_{2})$$\vdots$$Z(\gamma_{0}+\gamma_{1})=Z(\gamma_{2}+\gamma_{3})$$Z(n\gamma_{0}+(n+1)\gamma_{1})=Z(n\gamma_{2}+(n+1)\gamma_{3})$$Z((n+1)\gamma_{0}+n\gamma_{1})=Z((n+1)\gamma_{2}+n\gamma_{3})$$Z(\delta)$$\vdots$$Z(S_{0})=Z(S_{2})$$Z(S_{1})=Z(S_{3})$$\vdots$$Z(\mathcal{T}^{-n}(S_{1}))$$Z(\mathcal{T}^{n}(S_{0}))$$Z(\delta)$$Z(\mathcal{T}^{-1}(S_{1}))$$Z(\mathcal{T}(S_{0}))$$\vdots$$Z(S_{0})=Z(S_{2})$$Z(S_{1})=Z(S_{3})$$\vdots$$Z(\mathcal{T}^{-n}(S_{1}))$$Z(\mathcal{T}^{n}(S_{0}))$$Z(\delta)$$Z(\mathcal{T}^{-1}(S_{1}))$$Z(\mathcal{T}(S_{0}))$$Z^{\prime}(E)$$Z^{\prime}(S_{0}[1])=Z^{\prime}(S_{2}[1])$$Z^{\prime}(\mathcal{T}(S_{0}))=Z^{\prime}(\mathcal{T}(S_{2}))$$Z^{\prime}(\delta)$
Figure 5. Central charges of simple objects and $E$, $\delta$ for
$\sigma^{\prime}$

Since $\mathcal{A}^{\prime}$ is of finite-length, and the phase of $E$ for
$\sigma^{\prime}$ is
$\phi(S_{0})+\epsilon-\phi(S_{0})-\epsilon^{\prime}=\epsilon-\epsilon^{\prime}\in\Bigl{(}0,\phi\bigl{(}\mathcal{T}(S_{0})\bigr{)}\Bigr{)}$,
therefore $E$ is not semistable for $\sigma^{\prime}$ by Lemma 5.18. Since
$\sigma^{\prime}$ and $\sigma$ contain the same set of semistable objects, so
$E$ is not semistable for $\sigma$ either.

Similarly, suppose $E\in\mathcal{P}\left(\phi(S_{1})-\epsilon\right)$, where
$\epsilon\in\left(0,\phi(S_{1})-\phi(\mathcal{T}^{-1}(S_{1}))\right)$, we
choose $0<\epsilon^{\prime}\ll\epsilon$. Let
$t=1+\epsilon^{\prime}-\phi(S_{1})$ and $\tau:=\sigma\cdot
t=(Z^{\prime},\mathcal{P}^{\prime})$. For $\tau$ we have the phases of

$\displaystyle\phi^{\prime}(S_{1}[-1])=\phi^{\prime}(S_{3}[-1])$
$\displaystyle=$ $\displaystyle\epsilon^{\prime}\in(0,1/2),$
$\displaystyle\phi^{\prime}\left(\mathcal{T}^{-1}(S_{1})\right)=\phi^{\prime}\left(\mathcal{T}^{-1}(S_{3})\right)$
$\displaystyle=$
$\displaystyle\phi\left(\mathcal{T}^{-1}(S_{1})\right)-\phi(S_{1})+1+\epsilon^{\prime}\in(1/2,1).$

The heart $\mathcal{A}^{\prime}=\mathcal{P}^{\prime}(0,1]$ contains the simple
objects
$\\{S_{1}[-1],S_{3}[-1],\mathcal{T}^{-1}(S_{1}),\mathcal{T}^{-1}(S_{3})\\}$
which generate $R_{S_{1}}R_{S_{3}}\mathcal{A}$,
$R_{S_{1}}R_{S_{3}}\mathcal{A}\subset\mathcal{A}^{\prime}$ therefore
$\mathcal{A}^{\prime}=R_{S_{1}}R_{S_{3}}\mathcal{A}=\mathcal{T}^{-1}\mathcal{A}$.
The phase of $E$ for $\tau$ is
$\phi(S_{1})-\epsilon+1+\epsilon^{\prime}-\phi(S_{1})=1+\epsilon^{\prime}-\epsilon\in(\mathcal{T}^{-1}(S_{1}),1]$,
therefore $E$ cannot be semistable for $\tau$ again by Lemma 5.18, and is also
not semistable for $\sigma$.

For the second statement, let $E\in\mathcal{P}\left(\phi(S_{0})\right)$ be a
stable object, we take the Jordan-Hölder filtration of $E$:

$0\subset E_{n}\subset E_{n-1}\subset\cdots E_{1}\subset E_{0}=E,$

such that $E_{i}/E_{i-1}=S_{j}$ for $j\in\\{0,\cdots 3\\}$. Since
$\phi(S_{0})=\phi(S_{2})\neq\phi(S_{1})=\phi(S_{3})$, by the linearity of $Z$
the only graded factors appear in the filtrations are $S_{0}$ and $S_{2}$.
Therefore $S_{0}$ or $S_{2}$ is a subobject of $E$, thus must be isomorphic to
$E$. Similarly if $E\in\mathcal{P}\left(\phi(S_{1})\right)$ is stable, we
prove that $E$ is one of $S_{1}$ and $S_{3}$ exactly in the same way. Now we
apply $\mathcal{T}^{\pm m}$ on $\sigma$ for $m\geq 0$. Using Lemma 5.15 again,
we see $\mathcal{T}^{\pm m}(\sigma)$ and $\sigma$ contain the same set of
stable objects. Therefore the stable objects of phase
$\phi\bigl{(}\mathcal{T}^{m}(S_{0})\bigr{)}$ are $\mathcal{T}^{m}(S_{0})$ and
$\mathcal{T}^{m}(S_{2})$, and the stable objects of phase
$\phi\bigl{(}\mathcal{T}^{-m}(S_{1})\bigr{)}$ are $\mathcal{T}^{-m}(S_{1})$
and $\mathcal{T}^{-m}(S_{3})$. By Lemma 5.14 they are of special Kronecker
type I. ∎

Let $\sigma\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{-}$, we take
$\Psi(\sigma)\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{+}$, then by the above
Lemma the stable objects for $\sigma$ outside the ray $\phi=\frac{1}{2}$ are
of special Kronecker type II.

The rest of this section is devoted to characterizing the stable objects on
the ray $\phi=\frac{1}{2}$.

######  Lemma 5.20.

Let $\sigma\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{+}$ and
$E\in\mathcal{P}(\frac{1}{2})$, then $\pi_{*}E$ is (set theoretically)
supported on $S\times\mathbb{P}^{1}$ where $S$ is a finite set of closed
points in $\mathbb{P}^{1}$.

###### Proof.

According to Corollary 5.16,
$\mathcal{T}^{n}\left(\mathcal{P}(\frac{1}{2})\right)=\mathcal{P}(\frac{1}{2})\subset\mathcal{A}$
for any integer $n$. Therefore we have the vanishing of cohomology groups
$\mathrm{Ext}^{k}(\mathcal{Q},E)=0$ for $k\neq 0$, in particular:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\operatorname{\mathrm{Ext}}^{k}_{X}\left(\pi^{*}\mathcal{O}\oplus\pi^{*}\mathcal{O}(1,1),E\otimes\pi^{*}\mathcal{O}(n,0)\right)$
$\displaystyle=$
$\displaystyle\operatorname{\mathrm{Ext}}^{k}_{Z}\left(\mathcal{O}\oplus\mathcal{O}(1,1),\pi_{*}E\otimes\mathcal{O}(n,0)\right)\quad(\text{projection
formula})$ $\displaystyle=$
$\displaystyle\mathbb{H}^{k}\bigl{(}\mathbb{P}^{1},(p_{1}\circ\pi)_{*}E\otimes\mathcal{O}(n)\bigr{)}\oplus\mathbb{H}^{k}\Bigl{(}\mathbb{P}^{1},(p_{1}\circ\pi)_{*}\bigl{(}E(0,-1)\bigr{)}\otimes\mathcal{O}(n-1)\Bigr{)}$

where $\mathbb{H}$ means the hypercohomology of complexes and $k\neq 0$. In
general for a complex $F^{\star}\in D^{b}(X)$, we have a spectral sequence
[25, p.74]

$E_{2}^{p,q}=\mathrm{H}^{q}\left(X,H^{p}(F^{\star})\right)\Rightarrow\mathbb{H}^{p+q}(X,F^{\star}).$

We write $E^{\prime}_{0}=(p_{1}\circ\pi)_{*}E$ and
$E^{\prime}_{1}=(p_{1}\circ\pi)_{*}\bigl{(}E(0,-1)\bigr{)}$. By taking $n\gg
0$, then
$\mathrm{H}^{i}\left(\mathbb{P}^{1},H^{j}(E^{\prime}_{m})\otimes\mathcal{O}(n)\right)=0$
for $i\neq 0$, therefore the spectral sequence degenerates, we have

(5.2)
$\mathbb{H}^{k}(\mathbb{P}^{1},E^{\prime}_{m}\otimes\mathcal{O}(n))=\bigoplus_{j}\mathrm{H}^{k-j}\left(\mathbb{P}^{1},H^{j}(E^{\prime}_{m})\otimes\mathcal{O}(n)\right)=0\quad\text{for
}k\neq 0.$

Fix $j\neq 0$. Then
$\mathrm{H}^{0}\left(\mathbb{P}^{1},H^{j}(E^{\prime}_{m})\otimes\mathcal{O}(n)\right)=0$
where $n\gg 0$. This implies that $H^{j}(E^{\prime}_{m})=0$. Therefore
$E^{\prime}_{m}$ is concentrated in degree $0$ and is indeed a sheaf.

For $m=0,1$, now we have

(5.3)
$\mathrm{H}^{k}(\mathbb{P}^{1},E^{\prime}_{m}\otimes\mathcal{O}(n))=0,\quad
k=1,\quad n\in\mathbb{Z}.$

By taking $n\ll 0$, and using the fact that every coherent sheaf on
$\mathbb{P}^{1}$ splits into line bundles and torsion sheaves [20], we have
$\mathrm{dim}(\mathrm{supp}\ E^{\prime}_{m})=0$.

We denote
$S:=\operatorname{\mathrm{supp}}E^{\prime}_{0}\cup\operatorname{\mathrm{supp}}E^{\prime}_{1}$.
Suppose $s\notin S$, we consider the following fibre product diagram with
naturally-defined morphisms:

(5.4)
$\textstyle{p_{1}^{-1}(s)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{pt}$$\textstyle{\mathbb{P}^{1}\times\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\textstyle{\\{s\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{\mathbb{P}^{1}}$

We apply the flat base change theorem [25, Chapter 3.3] to $\pi_{*}E,\
\pi_{*}\bigl{(}E(0,-1)\bigr{)}\in D^{b}(Z)$, for any integer $k$

$\displaystyle m=0:$
$\displaystyle\mathrm{H}^{k}(\mathbb{P}^{1},i^{*}\pi_{*}E)=j^{*}E^{\prime}_{0}=0$
$\displaystyle m=1:$
$\displaystyle\mathrm{H}^{k}\left(\mathbb{P}^{1},i^{*}\pi_{*}(E(0,-1))\right)=\mathrm{H}^{k}\left(\mathbb{P}^{1},(i^{*}\pi_{*}E)\otimes\mathcal{O}(-1)\right)=j^{*}E^{\prime}_{1}=0.$

These vanishings imply that $i^{*}\pi_{*}E=0$ (one can again use the structure
theorem of coherent sheaf on $\mathbb{P}^{1}$). Therefore $\pi_{*}E$ is
supported on $S\times\mathbb{P}^{1}$. ∎

######  Lemma 5.21.

Suppose $E\in\mathcal{A}$ is isomorphic to the shift of a sheaf, and
$\operatorname{\mathrm{End}}_{\mathcal{A}}(E)\cong\mathbb{C}$ then it is the
pushforward $E=s_{*}F[i]$ for some $F\in\operatorname{\mathrm{Coh}}Z$.

###### Proof.

We follow the idea in the proof of [3, Lemma 3.1]: let $Y$ be the scheme-
theoretic support of $E$. By definition, $\mathrm{H}^{0}(\mathcal{O}_{Y})$
acts faithfully on $E$, and
$\operatorname{\mathrm{End}}_{\mathcal{A}}(E)=\mathbb{C}.\mathrm{Id}$,
therefore $\mathrm{H}^{0}(\mathcal{O}_{Y})\cong\mathbb{C}$. Take the
composition of the embedding of $Y$ with the contraction

$f:Y\hookrightarrow
X\twoheadrightarrow\bar{X}=\operatorname{\mathrm{Spec}}\mathrm{H}^{0}(\mathcal{O}_{X}),$

as
$\mathrm{H}^{0}(f_{*}\mathcal{O}_{Y})=\mathrm{H}^{0}(\mathcal{O}_{Y})=\mathbb{C}$,
so the scheme-theoretic image of $Y$ under $f$ is a point of $\bar{X}$. By
definition of $f$, the point will be the origin (singular point), thus $Y$ is
contained scheme-theoretically in the fiber of the contraction map. Since the
scheme-theoretic fiber of the origin is exactly
$Z=\mathbb{P}^{1}\times\mathbb{P}^{1}$, so $E=s_{*}F$ for some
$F\in\operatorname{\mathrm{Coh}}Z$. ∎

Now we are able to characterize the stable objects on the ray
$\phi=\frac{1}{2}$.

######  Theorem 5.22.

If $\sigma\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{+}$,  let $E$ be a
$\sigma$-stable object in $\mathcal{P}(\frac{1}{2})$, then there exists a
point $x\in\mathbb{P}^{1}$ such that either
$E=F_{1}(x)=s_{*}\mathcal{O}_{\\{x\\}\times\mathbb{P}^{1}}$ or
$E=F_{2}(x)=s_{*}\mathcal{O}_{\\{x\\}\times\mathbb{P}^{1}}(-1)[1]$.

If $\tau\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{-}$, let $E$ be a
$\tau$-stable object in $\mathcal{P}(\frac{1}{2})$, then there exists a point
$x\in\mathbb{P}^{1}$ such that either $E=\Psi\left(F_{1}(x)\right)$ or
$E=\Psi\left(F_{2}(x)\right)$.

###### Proof.

We prove the first statement, the second statement follows since $\Psi$
exchanges the stability conditions in
$\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{+}$ and
$\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{-}$, as also exchanges the objects of
special Kronecker type I and II.

Suppose $E\in\mathcal{P}\left(\frac{1}{2}\right)$ is stable and not isomorphic
to $F_{1}(x_{1})$ and $F_{2}(x_{1})$ for any $x_{1}\in\mathbb{P}^{1}$. We will
show that there is a vector bundle $F\in\operatorname{\mathrm{Coh}}Z$ such
that $E\cong s_{*}F[1]$. We follow the idea of proof in [3, Lemma 3.2]: since
$F_{i}$ are stable, therefore

$\operatorname{\mathrm{Hom}}_{X}(E,F_{n}(x_{1}))=\operatorname{\mathrm{Hom}}_{X}(F_{n}(x_{1}),E)=0,\
n=1,\ 2.$

Note that we have the short exact sequence in $\mathcal{A}$:

$0\rightarrow F_{1}(x_{1})\rightarrow\mathcal{O}_{x}\rightarrow
F_{2}(x_{1})\rightarrow 0,$

where $x\in Z$ such that $p_{1}(x)=x_{1}$, therefore there cannot be any
nonzero map $E\rightarrow\mathcal{O}_{x}$ or $\mathcal{O}_{x}\rightarrow E$.
Since $\mathcal{O}_{x}$ is semistable of phase $1/2$, then
$\operatorname{\mathrm{Hom}}^{i}_{X}(E,\mathcal{O}_{x})=0$ for $i\leq 0$, and
Serre duality gives
$\operatorname{\mathrm{Hom}}_{X}(\mathcal{O}_{x}[i],E)=\operatorname{\mathrm{Hom}}_{X}(E,\mathcal{O}_{x}[i+3])=0$
for $i\geq 0$ and $x\in Z$. Since $E$ is supported on $Z$, there will be no
homomorphisms with shifts of skyscraper sheaves outside the zero-section.
Therefore we can apply [15, Proposition 5.4] and deduce that $E$ is isomorphic
to a two-term complex of locally-free sheaves

$E^{-2}\xrightarrow{d^{-2}}E^{-1}.$

Hence $H^{-2}(E)\subset E^{-2}$ is torsion free on $X$. However, since
$H^{-2}(E)$ is supported on $Z$, therefore it must vanish. The map $d^{-2}$ is
injective, so that $E$ is isomorphic to the shift of a sheaf $F^{\prime}[1]$.
Since $E$ is stable,
$\operatorname{\mathrm{End}}(F^{\prime})\cong\mathbb{C}.\mathrm{Id}$,
therefore we apply Lemma 5.21 and show that $F^{\prime}=s_{*}F$ where
$F\in\operatorname{\mathrm{Coh}}Z$. Since
$\operatorname{\mathrm{Hom}}_{X}(s_{*}\mathcal{O}_{x},s_{*}F[1])\cong\operatorname{\mathrm{Hom}}_{Z}(\mathcal{O}_{x},F[1])\oplus\operatorname{\mathrm{Hom}}_{Z}(\mathcal{O}_{x},F)=0$,
therefore $F$ has depth $2$ and by Auslander-Buchsbaum formula, $F$ is
actually locally free.

However, this contradicts with Lemma 5.20 which says that $\pi_{*}E$ is
supported on a
$S\times\mathbb{P}^{1}\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$, where $S$ is
a finite set of points. Therefore we conclude that $E$ is isomorphic to either
$F_{1}(x)$ or $F_{2}(x)$ for some $x\in\mathbb{P}^{1}$. ∎

In summary, we have completed the description of the stable objects for
$\sigma\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}$:

######  Theorem 5.23.

  1. (1)

If $\sigma\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{+}$, then the stable objects
(up to a shift) are of special Kronecker type I, and the classes of stable
objects (up to a sign) in $K_{0}(\mathcal{A})$ are ($n\in\mathbb{N}$)

$\displaystyle n\gamma_{0}+(n+1)\gamma_{1},$
$\displaystyle(n+1)\gamma_{0}+n\gamma_{1},$ $\displaystyle
n\gamma_{2}+(n+1)\gamma_{3},$ $\displaystyle(n+1)\gamma_{2}+n\gamma_{3},$
$\displaystyle\gamma_{0}+\gamma_{1},$ $\displaystyle\gamma_{2}+\gamma_{3}.$

  2. (2)

if $\sigma\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}_{-}$, then the stable objects
(up to a shift) are of special Kronecker type II, and the classes of stable
objects (up to a sign) in $K_{0}(\mathcal{A})$ are ($n\in\mathbb{N}$)

$\displaystyle n\gamma_{1}+(n+1)\gamma_{2},$
$\displaystyle(n+1)\gamma_{1}+n\gamma_{2},$ $\displaystyle
n\gamma_{3}+(n+1)\gamma_{0},$ $\displaystyle(n+1)\gamma_{3}+n\gamma_{0},$
$\displaystyle\gamma_{1}+\gamma_{2},$ $\displaystyle\gamma_{3}+\gamma_{0}.$

  3. (3)

if $\sigma\in\mathcal{U}^{n}(\mathcal{A})^{\Phi}$ and
$\phi(S_{i})=\frac{1}{2}$ for each $i$, then the stable objects (up to a
shift) are only $\\{S_{i}\\}_{i}$, and the classes of stable objects (up to a
sign) in $K_{0}(\mathcal{A})$ are

$\gamma_{0},\ \gamma_{1},\ \gamma_{2},\ \gamma_{3}.$

###### Proof.

We have proved the first two cases in the above. For the last case, we do
induction on the length $l(E)$ of object $E\in\mathcal{A}$. When $l(E)=1$, it
is obvious. Then for $l(E)=n+1$, by taking the Jordan-Hölder filtration of
$E$, we have short exact sequence

$0\rightarrow E^{\prime}\rightarrow E\rightarrow S_{i}^{\oplus
n_{i}}\rightarrow 0.$

Then $E^{\prime}$ is semistable by our induction hypothesis, note that
$E^{\prime}$ has the same phase as $S_{i}$, we see that $E$ is also semistable
of phase of $S_{i}$. Therefore we proved that there are no other stable
objects other that $S_{i}$, $i=0,\cdots,3$. ∎

By applying $\mathbb{C}$-action we obtain the same description of stable
objects for general stability conditions in $\mathcal{U}(\mathcal{A})^{\Phi}$.

##  6\. Space of invariant stability conditions

Recall from the introduction the subset
$\Delta\subset\overline{K_{0}(\mathcal{A})}$ is defined to be the set of
classes of stable objects for $\sigma\in\mathcal{U}(\mathcal{A})^{\Phi}$ in
the quotient group
$\overline{K_{0}(\mathcal{A})}=K_{0}(\mathcal{A})/K_{0}(\mathcal{A})^{-\varphi}$.
By Theorem 5.23 $\Delta$ consists of the following elements:

$\Delta=\bigl{\\{}n\in\mathbb{Z}:n\gamma_{0}+(n+1)\gamma_{1},\
(n+1)\gamma_{0}+n\gamma_{1},\pm(\gamma_{0}+\gamma_{1})\bigr{\\}}.$

Recall also that

$\mathcal{H}^{\operatorname{reg}}:=\operatorname{\mathrm{Hom}}(\overline{K_{0}(\mathcal{A})},\mathbb{C})\setminus\bigcup_{\boldsymbol{v}\in\Delta}\boldsymbol{v}^{\perp},$

where
$\boldsymbol{v}^{\perp}:=\\{Z\in\operatorname{\mathrm{Hom}}(\overline{K_{0}(\mathcal{A})},\mathbb{C})|Z(\boldsymbol{v})=0\\}$
is the hyperplane complement.

Note that $\mathcal{H}^{\operatorname{reg}}$ is the complement of a family of
hyperplanes in
$\operatorname{\mathrm{Hom}}(\overline{K_{0}(\mathcal{A})},\mathbb{C})\cong\mathbb{C}^{2}$:

$\displaystyle nZ(\gamma_{0})+(n+1)Z(\gamma_{1})$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle(n+1)Z(\gamma_{0})+nZ(\gamma_{1})$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle Z(\gamma_{0})+Z(\gamma_{1})$
$\displaystyle=$ $\displaystyle 0$

for $n\geq 0$ (see Figure 1).

In the final section we prove Theorem 1.3 in the introduction: the forgetful
map

$\mathcal{Z}:\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{0}\rightarrow\operatorname{\mathrm{Hom}}(\overline{K_{0}(X)},\mathbb{C})$

factors through

$\mathcal{Z}:\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{0}\rightarrow\mathcal{H}^{\operatorname{reg}}.$

Moreover, the above is a covering map.

Recall that a continuous map $f:A\rightarrow B$ between topological spaces is
called a covering map, if every point $b\in B$ has an open neighborhood
$V\subset B$ such that the restriction of $f$ to each connected component of
$f^{-1}(V)$ is a homeomorphism onto $V$.

We first analyze the boundary of $\mathcal{U}^{n}(\mathcal{A})^{\Phi}$. Recall
$\\{S_{i}\\}_{i\in\mathbb{Z}_{4}}$ (Corollary 3.3) are the simple objects
which generate $\mathcal{A}$ and $\gamma_{i}=[S_{i}]$ are their classes. By
definition $\partial\ \mathcal{U}^{n}(\mathcal{A})^{\Phi}$ has four components
of codimension-one submanifolds (real lines), which are

$W^{+}_{i}:=\\{Z(\gamma_{i})=Z(\gamma_{i+2})\in\mathbb{R}_{>0}\\},\quad
W^{-}_{i}:=\\{Z(\gamma_{i})=Z(\gamma_{i+2})\in\mathbb{R}_{<0}\\}$

$i=0,\ 1$. Though we cannot apply Lemma 2.29 directly, however, since we are
deforming $\sigma$ while preserving the condition
$Z(\gamma_{i})=Z(\gamma_{i+2})$, the statement and proof are exactly the same
as there.

######  Lemma 6.1.

  1. (1)

For any stability condition on $W^{+}_{i}$ ($i=0,\ 1$) there exists an open
neighborhood $V$ such that
$V\subset\mathcal{U}^{n}(\mathcal{A})^{\Phi}\cup\mathcal{U}^{n}(L_{S_{i}}L_{S_{i+2}}\mathcal{A})^{\Phi}$.
Similarly, for any stability condition on $W^{-}_{i}$ ($i=0,\ 1$) there exists
an open neighborhood $V$ such that
$V\subset\mathcal{U}^{n}(\mathcal{A})^{\Phi}\cup\mathcal{U}^{n}(R_{S_{i}}R_{S_{i+2}}\mathcal{A})^{\Phi}$.

  2. (2)

We have
$W^{+}_{i}=\overline{\mathcal{U}^{n}}(\mathcal{A})^{\Phi}\cap\overline{\mathcal{U}^{n}}(\tau\mathcal{A})^{\Phi}$,
where $\tau=\mathcal{T}$ when $i=0$ and $\tau=\mathcal{T}_{\Psi}$ when $i=1$.
Similarly,
$W^{-}_{i}=\overline{\mathcal{U}^{n}}(\mathcal{A})^{\Phi}\cap\overline{\mathcal{U}^{n}}(\tau\mathcal{A})^{\Phi}$,
where $\tau=\mathcal{T}_{\Psi}^{-1}$ when $i=0$ and $\tau=\mathcal{T}^{-1}$
when $i=1$.

###### Proof.

In the following proof we will repeatedly use Lemma 2.5 that if $\mathcal{A},\
\mathcal{A}^{\prime}\subset\mathcal{D}$ are hearts of bounded t-structures and
$\mathcal{A}\subset\mathcal{A}^{\prime}$, then
$\mathcal{A}=\mathcal{A}^{\prime}$.

First we suppose $\sigma\in W^{+}_{0}$, that is
$Z(\gamma_{0})=Z(\gamma_{2})\in\mathbb{R}_{>0}$. The objects
$\widetilde{S_{1}}=\mathcal{T}(S_{0})$ and
$\widetilde{S_{3}}=\mathcal{T}(S_{2})$ lie in $\mathcal{A}$, and are in the
short exact sequences by the computations in Section 4:

(6.1) $0\rightarrow S_{i-1}^{\oplus 2}\rightarrow\widetilde{S_{i}}\rightarrow
S_{i}\rightarrow 0,\quad i=1,\ 3$

where $2=\dim_{\mathbb{C}}\operatorname{\mathrm{Ext}}^{1}(S_{i},S_{i-1})^{*}$.
Since
$\operatorname{\mathrm{Hom}}(\widetilde{S_{1}},S_{0})=\operatorname{\mathrm{Hom}}(\widetilde{S_{3}},S_{2})=0$
the objects $\widetilde{S_{i}}$ lie in $\mathcal{P}(0,1)$, and by choosing a
small enough open neighborhood $V$ of $\sigma$ we can assume this is the case
for all stability conditions $(Z,\mathcal{P})$ of $V$. We can split $V$ into
two pieces

$V_{+}=\\{\mathrm{Im}Z(S_{0})=\mathrm{Im}Z(S_{2})>0\\},\quad
V_{-}=\\{\mathrm{Im}\ Z(S_{0})=\mathrm{Im}Z(S_{2})\leq 0\\}.$

For $\sigma\in V_{+}$, we can shrink $V$ if necessarily such that
$S_{i}\in\mathcal{P}(0,1)$ for all $i$. This shows that
$\mathcal{A}\subset\mathcal{P}(0,1]$ for all stability conditions in $V_{+}$,
therefore $\mathcal{P}(0,1]=\mathcal{A}$ and so
$V_{+}\subset\mathcal{U}^{n}(\mathcal{A})^{\Phi}$. On the other hand, for any
stability condition $(Z,\mathcal{P})\in V_{-}$ the objects $S_{0}$ and $S_{2}$
are in $\mathcal{P}(-1/2,0]$, thus the heart $\mathcal{P}(0,1]$ contains the
objects $S_{0}[1],\ S_{2}[1],\ \widetilde{S_{1}}$ and $\widetilde{S_{3}}$.
Since these are the simple objects of the finite length category
$L_{S_{0}}L_{S_{2}}\mathcal{A}$, therefore
$\mathcal{P}(0,1]=L_{S_{0}}L_{S_{2}}\mathcal{A}$ and so
$V_{-}\subset\mathcal{U}^{n}(L_{S_{0}}L_{S_{2}}\mathcal{A})^{\Phi}$. Therefore
$V\subset\mathcal{U}^{n}(\mathcal{A})^{\Phi}\cup\mathcal{U}^{n}(L_{S_{0}}L_{S_{2}}\mathcal{A})^{\Phi}$.

By applying $\Psi$ on $\sigma$ then $\Psi(\sigma)\in W^{+}_{1}$, that is
$Z(\gamma_{1})=Z(\gamma_{3})\in\mathbb{R}_{>0}$, then there exists an open
neighborhood $V$ of $\Psi(\sigma)$ such that
$V\subset\mathcal{U}^{n}(\mathcal{A})^{\Phi}\cup\mathcal{U}^{n}(L_{S_{1}}L_{S_{3}}\mathcal{A})^{\Phi}$.

The proof for $\sigma\in W^{-}_{i}$ is essentially the same by replacing the
left double tilt with the right double tilt.

For the second statement, by the first part we have

$W^{+}_{i}=\overline{\mathcal{U}^{n}}(\mathcal{A})^{\Phi}\cap\overline{\mathcal{U}^{n}}(L_{S_{i}}L_{S_{i+2}}\mathcal{A})^{\Phi},\quad
W^{-}_{i}=\overline{\mathcal{U}^{n}}(\mathcal{A})^{\Phi}\cap\overline{\mathcal{U}^{n}}(R_{S_{i}}R_{S_{i+2}}\mathcal{A})^{\Phi}.$

By Theorem 4.1 we obtain the results. ∎

We denote by $H$ the subgroup in
$\operatorname{\mathrm{Aut}}_{*}(D^{b}_{0}(X))$ (see Definition 3.12)
generated by $\mathcal{T}$ and $\mathcal{T}_{\Psi}$.

######  Proposition 6.2.

(6.2) $\bigcup_{g\in
H}\overline{\mathcal{U}^{n}}(g\mathcal{A})^{\Phi}=\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{n}.$

###### Proof.

Since $t(\delta)=t_{\psi}(\delta)=\delta$ due to Lemma 4.2, therefore
$\overline{\mathcal{U}^{n}}(g\mathcal{A})^{\Phi}\subset\bigl{(}\operatorname{\mathrm{Stab}}(D^{b}_{0}(X))^{\Phi}\bigr{)}_{n}$
for any $g\in H$.

Now we show that the left side is open and closed, hence the inclusion is in
fact an equality.

First we prove the openess. For any
$\sigma\in\overline{\mathcal{U}^{n}}(g\mathcal{A})^{\Phi}$, considering the
preimage of $\sigma$ under the autoequivalence $g$, $\sigma=(Z,\mathcal{P})$
lies in $\overline{\mathcal{U}^{n}}(\mathcal{A})^{\Phi}$. Suppose first that
$\mathrm{Im}Z(\gamma_{i})>0$ for each $i$, then we can choose an open
neighborhood $U$ of $\sigma$ such that each simple object $S_{i}$ has phase
$(0,1)$ for all stability conditions $(Z,\mathcal{P})$ of $U$. Since
$\mathcal{A}$ is the smallest extension-closed subcategory of $\mathcal{D}$
containing $S_{i}$ it follows that $\mathcal{A}\subset\mathcal{P}(0,1]$ of all
stability conditions in $U$. Therefore $\mathcal{P}(0,1]=\mathcal{A}$ by Lemma
2.5 and so $U$ is contained in $\mathcal{U}^{n}(\mathcal{A})^{\Phi}$.

Now suppose $\sigma$ lies on the boundary of
$\mathcal{U}^{n}(\mathcal{A})^{\Phi}$, according to Lemma 6.1, there is an
open neighborhood $V$ of $\sigma$ such that
$V\subset\mathcal{U}^{n}(\mathcal{A})^{\Phi}\cup\mathcal{U}^{n}(\tau\mathcal{A})^{\Phi}$
where $\tau$ is one of the autoequivalences $\mathcal{T}^{\pm 1}$ and
$\mathcal{T}_{\Psi}^{\pm 1}$. This finishes the proof of openess.

To check the left side of (6.2) is closed, we only need to show the collection
of closed sets is locally finite. Suppose

$\sigma\in\bigcap_{g\in H^{\prime}\subset
H}\overline{\mathcal{U}^{n}}(g\mathcal{A})^{\Phi}.$

It is obvious that
$\mathcal{U}^{n}(g\mathcal{A})^{\Phi}\cap\mathcal{U}^{n}(g^{\prime}\mathcal{A})^{\Phi}=\emptyset$
if $g\mathcal{A}\neq g^{\prime}\mathcal{A}$. Taking the preimage under some
autoequivalence $g$, we suppose $\sigma$ lies on the boundary of
$\mathcal{U}^{n}(\mathcal{A})^{\Phi}$. Then this intersection is finite since
by Lemma 6.1, each boundary component corresponds to exactly one of the
autoequivalences $\mathcal{T}^{\pm 1}$ and $\mathcal{T}_{\Psi}^{\pm 1}$.

This finishes the proof of the equality (6.2). ∎

######  Lemma 6.3.

For any special stability condition
$\sigma=(Z,\mathcal{P})\in\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{0}$,
we have $Z(\delta)\neq 0$ where $\delta=[\mathcal{O}_{x}]$ for $x\in Z$.
Moreover, there is an exact equivalence $W\in H$ and $\lambda\in\mathbb{C}$
such that $\lambda W(\sigma)\in\overline{\mathcal{U}}(\mathcal{A})^{\Phi}$,
the closure of $\mathcal{U}(\mathcal{A})^{\Phi}$.

###### Proof.

Suppose there exists
$\tau=(Z_{1},\mathcal{P}_{1})\in\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{0}$
such that $Z_{1}(\delta)=0$. In particular there is no semistable object in
$\tau$ whose class in $K_{0}(\mathcal{A})$ is $\delta$. By Lemma 2.27 this is
true in an open neighborhood of $\tau$ in
$\bigl{(}\operatorname{\mathrm{Stab}}(X)^{\Phi}\bigr{)}_{0}$. Consider the
stability condition $\sigma=(Z_{2},\mathcal{P}_{2})$ in such neighborhood such
that $Z_{2}(\delta)\neq 0$. First we act by some $\lambda\in\mathbb{C}$ so
that $\lambda\cdot Z_{2}(\delta)=i$. Then by (6.2) there exists some $W\in H$
such that $\sigma^{\prime}=\lambda
W(\sigma)\in\overline{\mathcal{U}^{n}}(\mathcal{A})^{\Phi}$. By Remark 5.13
and Lemma 2.27 there does exist a semistable object whose class is $\delta$
for $\sigma^{\prime}$. Since $\delta$ is preserved by element of $H$,
therefore we conclude that there exists a semistable object $E$ for $\sigma$
whose class in $K_{0}(\mathcal{A})$ is $\delta$. This gives a contradiction.

Therefore for any stability condition $\sigma=(Z,\mathcal{P})$, we have
$Z(\delta)\neq 0$. Then by choosing $\lambda\in\mathbb{C}$ such that
$\lambda\cdot Z(\delta)=i$, and by (6.2) we can find $W\in H$ such that
$\lambda
W(\sigma)\in\overline{\mathcal{U}^{n}}(\mathcal{A})^{\Phi}\subset\overline{\mathcal{U}}(\mathcal{A})^{\Phi}$.
This finishes the proof. ∎

We let $t$ and $t_{\psi}$ be the automorphisms of
$\overline{K_{0}(\mathcal{A})}$ induced by $\mathcal{T}$ and
$\mathcal{T}_{\Psi}$, the following lemma is an easy consequence of Lemma 4.2:

######  Lemma 6.4.

The automorphisms $t^{\pm 1}$ and $t_{\psi}^{\pm 1}$ preserve $\Delta$.

###### Proof.

There are the following equivalent classes (up to a sign) in
$\Delta\subset\overline{K_{0}(\mathcal{A})}$:

(6.3)
$n\gamma_{0}+(n+1)\gamma_{1},\quad(n+1)\gamma_{0}+n\gamma_{1},\quad\gamma_{0}+\gamma_{1}.$

We apply Lemma 4.2 to the following calculations, for $t$, we have

$\displaystyle t\left(n\gamma_{0}+(n+1)\gamma_{1}\right)$ $\displaystyle=$
$\displaystyle(n-1)\gamma_{0}+n\gamma_{1};$ $\displaystyle
t\left((n+1)\gamma_{0}+n\gamma_{1}\right)$ $\displaystyle=$
$\displaystyle(n+2)\gamma_{0}+(n+1)\gamma_{1};$ $\displaystyle
t\left(\gamma_{0}+\gamma_{1}\right)$ $\displaystyle=$
$\displaystyle\gamma_{0}+\gamma_{1}.$

Therefore $t\Delta\subset\Delta$. Similarly, for $t_{\psi}$ we have