# BRIDGELAND STABILITY CONDITIONS ON THE CATEGORY OF HOLOMORPHIC TRIPLES OVER
CURVES

EVA MARTÍNEZ-ROMERO  Freie Univeristät Berlin, Arnimallee 3, 14195 Berlin,
Germany<EMAIL_ADDRESS>,  ALEJANDRA RINCÓN-HIDALGO  ICTP. Leonardo Da
Vinci Building, Strada Costiera 11, 34151 Trieste, Italy<EMAIL_ADDRESS>http://users.ictp.it/ arincon/ and  ARNE RÜFFER  Mary Immaculate College,
University of Limerick, Limerick, Ireland<EMAIL_ADDRESS>
###### Abstract.

We give a complete description of the Bridgeland stability manifold for the
bounded derived category of holomorphic triples over a smooth projective curve
of genus 1 as a connected, four dimensional complex manifold.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Review of t-structures and semiorthogonal decompositions.
    2. 2.2 Review of Bridgeland stability conditions
    3. 2.3 Bridgeland stability conditions on curves with $g>0$
  3. 3 Bridgeland stability conditions on $\mathcal{T}_{C}$
    1. 3.1 The triangulated category $\mathcal{T}_{C}$
    2. 3.2 Constructing pre-stability conditions
  4. 4 The stability manifold $\operatorname{Stab}(\mathcal{T}_{C})$
    1. 4.1 Pre-stability conditions in $\Theta_{12}$
    2. 4.2 Pre-stability conditions in $\Theta_{23}$ and $\Theta_{31}$
    3. 4.3 Non-gluing pre-stability conditions
    4. 4.4 Characterising CP-glued pre-stability conditions in $\Theta_{12}$
  5. 5 Support property for $\mathcal{T}_{C}$, with $g(C)\geq 1$
    1. 5.1 Support property for CP-glued pre-stability conditions
    2. 5.2 Duality and semistability on $\mathcal{T}_{C}$
    3. 5.3 Support property for non-gluing pre-stability conditions with negative discriminant and $g=1$
  6. 6 Topological description of $S_{12}$
  7. A Recollement

##  1\. Introduction

The concept of a holomorphic triple was introduced by Bradlow and García-Prada
in [8] and [9]. The abelian category of holomorphic triples
$\operatorname{TCoh}(C)$ consists of objects of the form $\varphi\colon
E_{1}\rightarrow E_{2}$, where $E_{1},E_{2}$ are coherent sheaves on a
nonsingular projective curve $C$ together with a morphism $\varphi$ between
them. It was shown in [8] that moduli spaces of semistable holomorphic triples
of vector bundles exist and are projective. This category has also played an
important role in the study of Higgs bundles [10]. Assuming $C$ to have genus
1, we will provide a complete description of the Bridgeland stability manifold
of the triangulated category $\mathcal{T}_{C}\coloneqq
D^{b}(\operatorname{TCoh}(C))$.

Stability conditions on triangulated categories have been introduced by
Bridgeland in [11] as a formalisation of Douglas’ work in [19] and [20]. The
main result in [11] asserts that the set of stability conditions has the
structure of a complex manifold (usually referred to as stability space).
Stability manifolds have several applications in algebraic geometry as for
example they serve as an important aid for the understanding of derived
categories or as a tool in birational geometry ([25], [4]). The description of
stability spaces is not an easy endeavour in geometric situations. For
nonsingular projective curves it is well understood ([11], [15], [33], [36])
and our strategy is to use this as a building stone in the case of holomorphic
triples.

Our first finding is that $D^{b}(\operatorname{Coh}(C))$ can be embedded as a
strictly full subcategory into $\mathcal{T}_{C}$ in three different ways.
These are

(1.1) $\displaystyle\mathcal{C}_{1}$ $\displaystyle\coloneqq\\{$
$\displaystyle X$ $\displaystyle\rightarrow$ $\displaystyle 0$
$\displaystyle\colon$ $\displaystyle X\in D^{b}(\operatorname{Coh}(C))\\}$
$\displaystyle\textnormal{ }\subset\mathcal{T}_{C},$
$\displaystyle\mathcal{C}_{2}$ $\displaystyle\coloneqq\\{$ $\displaystyle 0$
$\displaystyle\rightarrow$ $\displaystyle X$ $\displaystyle\colon$
$\displaystyle X\in D^{b}(\operatorname{Coh}(C))\\}$
$\displaystyle\textnormal{ }\subset\mathcal{T}_{C},$
$\displaystyle\mathcal{C}_{3}$ $\displaystyle\coloneqq\\{$ $\displaystyle X$
$\displaystyle\xrightarrow{\operatorname{id}}$ $\displaystyle X$
$\displaystyle\colon$ $\displaystyle X\in D^{b}(\operatorname{Coh}(C))\\}$
$\displaystyle\textnormal{ }\subset\mathcal{T}_{C}$

and subsequent pairing of two of the resulting subcategories, respectively,
leads to three semiorthogonal decompositions

(1.2)
$\displaystyle\mathcal{T}_{C}=\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle\textnormal{,
}\mathcal{T}_{C}=\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle\textnormal{ and
}\mathcal{T}_{C}=\langle\mathcal{C}_{3},\mathcal{C}_{1}\rangle.$

Following [7], we can prove the existence of the Serre functor on
$\mathcal{T}_{C}.$ Additionally, the reiterated application of the Serre
functor to any of the semiorthogonal decompositions (1.2) leads to permuting
through all three of them.

We construct discrete pre-stability conditions on $\mathcal{T}_{C}$ in two
different ways. The first approach is to construct pre-stability conditions on
$\mathcal{T}_{C}$ by using the semiorthogonal decompositions (1.2) to glue
stability conditions from $\operatorname{Stab}(D^{b}(\operatorname{Coh}(C))$
following [17]. Moreover, in the Appendix we compare this gluing procedure
with the well-known recollement from [6], which provides the same hearts when
the conditions for gluing as in [17] (we refer to them as CP-glued hearts) are
satisfied and we prove that in the case of $\mathcal{T}_{C}$ we cannot obtain
stability conditions from hearts constructed by recollement using hearts that
are not CP-glued hearts. The second approach is to tilt the standard heart in
$\mathcal{T}_{C}$ with respect to a certain torsion pair following [24].

The next step is then to study the structure of the stability manifold
$\operatorname{Stab}(\mathcal{T}_{C})$. We first prove that for each pre-
stability condition at least two of the three embeddings of $D^{b}(C)$ into
$\mathcal{T}_{C}$ have the property that they map all line bundles and all
skyscrapers to stable objects in Theorem 4.13. This finding is a crucial step
in the process of describing the entire stability space as a connected complex
manifold.

It turns out, however, that there are stability conditions that are not in a
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-orbit with a stability
condition obtained by CP-gluing. In Proposition 4.32 we obtain then, that up
to the $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action, the stability
space of $\mathcal{T}_{C}$ is given by the stability conditions obtained via
either CP-gluing via the semiorthogonal decompositions (1.2) or by tilting
with regard to the torsion pair from Lemma 3.43.

In order to prove our main result, we need to verify that tilting and CP-
gluing actually produces stability conditions in our situation. This includes
the verification of the support property. To prove that the support property
is fulfilled for CP-gluing pre-stability conditions we use a generalisation of
some of the equations of [8] to arbitrary stability conditions. For the non-
gluing case, under the condition $g(C)=1,$ we use the Euler form as a
Bogomolov-type inequality. We conjecture that the support property will hold
for genus $g(C)>1$ as well. We will now extend the Harder–Narashiman-property
using Bridgeland’s deformation result to the non-discrete case as in [12] and
[3]. Hence, we obtain the main theorem of this paper.

######  Theorem 1.1 (Theorem 6.13).

If

  1. (1)

$g(C)=1$ or

  2. (2)

$g(C)>1$ and all non-gluing pre-stability conditions with negative
discriminant satisfy the support property,

then $\operatorname{Stab}(\mathcal{T}_{C})$ is a connected 4-dimensional
complex manifold.

Finally, we conjecture that if $g(C)=1,$ then $\mathcal{T}_{C}$ is a
fractional Calabi–Yau of fractional dimension $\frac{4}{3}$.

As far as the organisation of this paper is concerned, we proceed as follows:
after providing the necessary framework in Section 2, we go on to introduce
and subsequently make use of the necessary techniques we require to construct
pre-stability conditions on $\mathcal{T}_{C}$ in Section 3. In Section 4 we
study the stability of skyscraper sheaves and line bundles under the
embeddings (1.1) and we prove that every stability condition on
$\mathcal{T}_{C}$ is obtained by the techniques introduced in Section 3. Next,
the support property is proved in Section 5, while a topological description
of the stability space is provided in Section 6. Finally, in the Appendix we
compare the two procedures to glue hearts from semiorthogonal decompositions
and justify why we restricted the first part of our construction to CP-glued
hearts.

#### Acknowledgments

We thank Bernd Kreussler and Alexander Schmitt for their invaluable advice on
the research necessary to produce this article. We also wish to thank Kotaro
Kawatani for pointing out a mistake in the proof of Lemma 4.1 in the
preliminary version of this paper. We thank the Freie Universität Berlin, the
Berlin Mathematical School, the IRTG GRK 1800 and the Mathematics department
of Mary Immaculate College, University of Limerick for the financial support
provided to the authors, as well as making possible to meet in order to
develop this work.

##  2\. Preliminaries

###  2.1. Review of t-structures and semiorthogonal decompositions.

Let $\mathcal{D}$ be a $\mathbb{C}$-linear triangulated category of finite
type. For the general theory of t-structures we suggest [6, Chapter 1].

######  Definition 2.1.

A _t-structure_ on a triangulated category $\mathcal{D}$ consists of a pair of
full additive subcategories $(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$,
with $\mathcal{D}^{\leq i}\coloneqq\mathcal{D}^{\leq 0}[-i]$ and
$\mathcal{D}^{\geq i}\coloneqq\mathcal{D}^{\geq 0}[-i]$ for $i\in\mathbb{Z}$,
such that:

  1. (1)

$\operatorname{Hom}_{\mathcal{D}}(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq
1})=0$.

  2. (2)

For all $E\in\mathcal{D}$, there is a distinguished triangle $T\rightarrow
E\rightarrow F\rightarrow T[1]$ with $T\in\mathcal{D}^{\leq 0}$ and
$F\in\mathcal{D}^{\geq 1}$.

  3. (3)

$\mathcal{D}^{\leq 0}\subset\mathcal{D}^{\leq 1}$ and $\mathcal{D}^{\geq
0}\supset\mathcal{D}^{\geq 1}$.

A t-structure is _bounded_ if every $E\in\mathcal{D}$ is contained in $D^{\leq
n}\cap D^{\geq-n}$ for some $n>0.$

######  Definition 2.2.

The _heart_ of a bounded t-structure $(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq
0})$ is defined as $\mathcal{A}\coloneqq\mathcal{D}^{\leq
0}\cap\mathcal{D}^{\geq 0}.$

######  Remark 2.3.

[34, Lemma 5.2] The heart of a bounded t-structure
$\mathcal{A}\subseteq\mathcal{D}$ is an abelian category whose short exact
sequences are precisely the exact triangles in $\mathcal{D}$ with objects in
$\mathcal{A}.$ A morphism $A\rightarrow B$ between two objects in
$\mathcal{A}$ is defined to be an inclusion if its cone is also in
$\mathcal{A},$ and it is defined to be a surjection if the cone is in
$\mathcal{A}[1].$

######  Example 2.4.

[6, Chapter 1]

  1. (1)

If $(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ is a t-structure, then the
pair $(\mathcal{D}^{\leq n},\mathcal{D}^{\geq n})$ is also a t-structure.

  2. (2)

Let $\mathcal{D}=D^{b}(\mathcal{A}),$ where $\mathcal{A}$ is an abelian
category. The standard bounded t-structure of $\mathcal{D}$ is given by
$\mathcal{D}^{\leq 0}\coloneqq\\{E\mid H^{i}(E)=0\textnormal{ , }i>0\\}$ and
$\mathcal{D}^{\geq 0}\coloneqq\\{E\mid H^{i}(E)=0\textnormal{ , }i<0\\}.$ Note
that $\mathcal{A}$ is the heart of the standard bounded t-structure.

######  Lemma 2.5.

[11, Lemma 3.2], [28, Rem. 1.16] Let $\mathcal{A}\subset\mathcal{D}$ be a full
additive subcategory of a triangulated category $\mathcal{D}$. Then
$\mathcal{A}$ is the heart of a bounded t-structure if and only if

  1. (1)

$\operatorname{Hom}_{\mathcal{D}}(\mathcal{A}[k_{1}],\mathcal{A}[k_{2}])=0$
for $k_{1}>k_{2}$.

  2. (2)

For every nonzero $E\in\mathcal{D}$ there exists a finite sequence of integers

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

and a collection of distinguished triangles

$\textstyle{0=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}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{m-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{m}=E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

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

######  Remark 2.6.

Let $\mathcal{A}\subseteq\mathcal{D}$ be the heart of a bounded t-structure.
For every object $E\in\mathcal{D},$ the objects $A_{j}\in\mathcal{A}[k_{j}]$
are its _cohomological objects_ with respect to $\mathcal{A}.$ They are
denoted by $A_{i}=H_{\mathcal{A}}^{-k_{i}}(E).$ Moreover, they induce a
cohomological functor i.e. they are functorial and induce a long exact
sequence of cohomology for any exact triangle. Note that a t-structure is
uniquely determined by its heart (see [28, Def. 1.13]).

Hearts of bounded t-structures play an important role in the definition of
Bridgeland stability conditions. Therefore, we will discuss different ways of
giving hearts of bounded t-structures.

#### Torsion pairs and tilting

######  Definition 2.7.

Let $\mathcal{A}$ be an abelian category. A _torsion pair_ for $\mathcal{A}$
consists of a pair $(\mathcal{T},\mathcal{F})$ of full subcategories such that

  1. (1)

$\operatorname{Hom}_{\mathcal{A}}(\mathcal{T},\mathcal{F})=0$.

  2. (2)

For all $E\in\mathcal{A}$, there is a short exact sequence $0\rightarrow
T\rightarrow E\rightarrow F\rightarrow 0$ with $T\in\mathcal{T}$ and
$F\in\mathcal{F}$.

######  Proposition 2.8.

[24, Prop. 2.1] and [34, Lemma 6.3] Let $\mathcal{A}$ be the heart of a
bounded t-structure on $\mathcal{D}.$ Given a torsion pair
$(\mathcal{T},\mathcal{F})$ in $\mathcal{A},$ then the full subcategory

$\mathcal{A}^{\sharp}=\\{E\in\mathcal{D}\mid
H_{\mathcal{A}}^{i}(E)=0\textnormal{ for }i\notin\\{-1,0\\}\textnormal{,
}H_{\mathcal{A}}^{-1}(E)\in\mathcal{F}\textnormal{ and
}H_{\mathcal{A}}^{0}(E)\in\mathcal{T}\\}$

is the heart of a bounded t-structure on $\mathcal{D}.$ We call
$\mathcal{A}^{\sharp}$ the _tilt_ of $\mathcal{A}$ with respect to
$(\mathcal{T},\mathcal{F}).$ Moreover, the torsion pair
$(\mathcal{T},\mathcal{F})$ gives rise to the torsion pair
$(\mathcal{F}[1],\mathcal{T})$ for the tilt $\mathcal{A}^{\sharp}.$

######  Remark 2.9.

Note that $\mathcal{A}^{\sharp}[-1]=(\mathcal{F},\mathcal{T}[-1])$ is also the
heart of a bounded t-structure. The heart
$\mathcal{A}^{\sharp}[-1]=(\mathcal{F},\mathcal{T}[-1])$ is called the _right
tilt_ of $\mathcal{A}$ with respect to $(\mathcal{T},\mathcal{F}).$

#### Admissible subcategories and semiorthogonal decompositions

In this section we introduce concepts to study a triangulated category
$\mathcal{D},$ namely semiorthogonal decompositions and Serre functors. We
refer to [27] and [7]. If $X$ is a smooth projective variety we refer to
$D^{b}(\operatorname{Coh}(X))$ just as $D^{b}(X).$

######  Definition 2.10.

Let $\mathcal{D}$ be a triangulated category. Let
$\mathcal{A}\subseteq\mathcal{D}$ be a strictly full triangulated subcategory.
The category $\mathcal{A}$ is _(left) right-admissible_ if the inclusion
functor $i\colon\mathcal{A}\rightarrow\mathcal{D}$ has a (left) right adjoint
($i^{*}\colon\mathcal{D}\rightarrow\mathcal{A}$)
$i^{!}\colon\mathcal{D}\rightarrow\mathcal{A}.$ It is called _admissible_ if
it is right and left admissible.

######  Definition 2.11.

Let $\mathcal{D}$ be a triangulated category. A semiorthogonal decomposition
of $\mathcal{D}$ consists of a collection
$\mathcal{A}_{1},\cdots,\mathcal{A}_{n}$ of full triangulated subcategories
such that

  1. (1)

$\operatorname{Hom}_{\mathcal{D}}(\mathcal{A}_{i},\mathcal{A}_{j})=0$ for
every $1\leq j<i\leq n$.

  2. (2)

$\mathcal{D}$ is generated by the $\mathcal{A}_{i}$.

We write $\mathcal{D}=\langle\mathcal{A}_{1},\cdots,\mathcal{A}_{n}\rangle$.

######  Lemma 2.12 ([7]).

Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{A}$ and
$\mathcal{B}$ be strictly full triangulated subcategories of $\mathcal{D}$.
Assume that $\operatorname{Hom}_{\mathcal{D}}(\mathcal{B},\mathcal{A})=0$.
Then, the following are equivalent:

  1. (1)

The category $\mathcal{D}$ is generated by $\mathcal{A}$ and $\mathcal{B}$
i.e. for each $X\in\mathcal{D}$, there exists a distinguished triangle
$B\rightarrow X\rightarrow A\rightarrow B[1]$ with $A\in\mathcal{A}$ and
$B\in\mathcal{B}$ i.e. $\mathcal{D}=\langle\mathcal{A},\mathcal{B}\rangle.$

  2. (2)

$\mathcal{B}=\prescript{\perp}{}{\mathcal{A}}\coloneqq\\{D\in\mathcal{D}\mid\operatorname{Hom}_{\mathcal{D}}(D,A)=0\textnormal{
for all }A\in\mathcal{A}\\}$ and there exists a functor
$i^{*}\colon\mathcal{D}\rightarrow\mathcal{A}$ which is left adjoint to the
inclusion $i\colon\mathcal{A}\hookrightarrow\mathcal{D}$ i.e. $\mathcal{A}$ is
left admissible.

  3. (3)

$\mathcal{A}=\mathcal{B}^{\perp}\coloneqq\\{D\in\mathcal{D}\mid\operatorname{Hom}_{\mathcal{D}}(B,D)=0\textnormal{
for all }B\in\mathcal{B}\\}$ and there exists a functor
$j^{!}\colon\mathcal{D}\rightarrow\mathcal{B}$ which is right adjoint to the
inclusion $j\colon\mathcal{B}\hookrightarrow\mathcal{D}$ i.e. $\mathcal{B}$ is
right admissible.

######  Remark 2.13.

Let $F\colon\mathcal{D}\rightarrow\mathcal{D}$ be an autoequivalence. If there
is a semiorthogonal decomposition
$\mathcal{D}=\langle\mathcal{A},\mathcal{B}\rangle,$ then $\mathcal{D}=\langle
F(\mathcal{A}),F(\mathcal{B})\rangle.$

######  Definition 2.14.

Let $\mathcal{D}$ be a $k$-linear triangulated category of finite type, where
$k$ is a field. We say that $\mathcal{D}$ is right (resp. left) saturated if
every contravariant (resp. covariant) cohomology functor
$\mathcal{D}\rightarrow\operatorname{Vect}_{k}$ of finite type is
representable.

######  Proposition 2.15 ([7, Proposition 2.6]).

Let $\mathcal{A}$ be right (resp. left) saturated. Suppose $\mathcal{A}$ is
embedded in a triangulated category $\mathcal{D}$ as a full triangulated
subcategory. Then $\mathcal{A}$ is right (resp. left) admissible.

######  Theorem 2.16 ([7, Theorem 2.14]).

Let $X$ be a smooth projective variety. Then, $D^{b}(X)$ is right and left
saturated.

#### Serre functor

Let $\mathcal{D}$ be a $\mathbb{C}$-linear triangulated category.

######  Definition 2.17.

A Serre functor on $\mathcal{D}$ is an exact autoequivalence
$S\colon\mathcal{D}\rightarrow\mathcal{D}$ such that for any
$E,F\in\mathcal{D}$, there is an isomorphism
$\eta_{E,F}\colon\operatorname{Hom}_{\mathcal{D}}(E,F)\rightarrow\operatorname{Hom}_{\mathcal{D}}(F,S(E))^{*}$
(of $\mathbb{C}$-vector spaces) which is functorial in $E$ and $F$.

######  Remark 2.18.

For $\mathcal{D}$ of finite type, a Serre functor, if it exists, is unique up
to isomorphism. Moreover, it commutes with equivalences, i.e. for
$F:\mathcal{D}\rightarrow\mathcal{D}^{\prime}$ an equivalence,
$S_{\mathcal{D}^{\prime}}\circ F\cong F\circ S_{\mathcal{D}}$. Furthermore,
given an admissible subcategory $\mathcal{X}\subset\mathcal{D},$ by Serre
duality $S_{\mathcal{D}}$ sends $\prescript{\perp}{}{\mathcal{X}}$ to
$\mathcal{X}^{\perp}.$

######  Example 2.19.

Let $X$ be a smooth projective variety defined over $\mathbb{C}.$ The
autoequivalence of $D^{b}(X)$ given by
$\mathcal{S}_{X}(E)=E\otimes\omega_{X}[\dim X],$ where $\omega_{X}$ is the
dualizing line bundle, is the Serre functor on $D^{b}(X)$. In particular, if
$X$ is a Calabi–Yau variety, its Serre functor is simply the shift functor
$S_{X}=[\dim X]$.

######  Definition 2.20.

Let $n\in\mathbb{Z}$. A triangulated category $\mathcal{D}$ is a $n$-Calabi-
Yau category if it has a Serre functor $S_{\mathcal{D}}$ and
$S_{\mathcal{D}}\cong[n]$. The integer $n$ is called the CY-dimension of
$\mathcal{D}$.

######  Definition 2.21.

A triangulated category $\mathcal{D}$ is a fractional Calabi–Yau category if
it has a Serre functor $S_{\mathcal{D}}$ and there are integers $p$ and $q\neq
0$ such that $S^{q}_{\mathcal{D}}\cong[p]$. In this case we say that
$\mathcal{D}$ has (CY-)fractional dimension $p/q$.

The following result plays an important role in proving the existence of the
Serre functor on $\mathcal{T}_{C}.$

######  Proposition 2.22 ([7, Proposition 3.8]).

Let $\mathcal{D}$ be a triangulated category and
$\mathcal{B}\subset\mathcal{D}$ an admissible full triangulated subcategory
with $\mathcal{C}\coloneqq\mathcal{B}^{\perp}$ admissible. If $\mathcal{B}$
and $\mathcal{C}$ have Serre functors, then there exists a Serre functor on
$\mathcal{D}$.

###  2.2. Review of Bridgeland stability conditions

In this section we define Bridgeland stability conditions on a
$\mathbb{C}$-linear triangulated category $\mathcal{D}$ of finite type. We
follow Bridgeland’s papers [11] and [12]. We also recommend the following
lecture notes on Bridgeland stability theory [34], [1], [28] and [3, Appendix
B.].

######  Definition 2.23.

The _Grothendieck group_ $K(\mathcal{D})$ of a triangulated category
$\mathcal{D}$ is the abelian group generated by the isomorphism classes of
objects in $\mathcal{D}$ subject to the relation $[A]=[C]+[B],$ where
$C\rightarrow A\rightarrow B\rightarrow C[1]$ is an exact triangle. We
consider the Euler bilinear form given by
$\chi(E,F)=\sum_{i}(-1)^{i}\operatorname{Hom}_{\mathcal{D}}(E,F[i]).$ We
define the _numerical Grothendieck group_ $\mathcal{N}(\mathcal{D})$ as the
quotient $K(\mathcal{D})/K(\mathcal{D})^{\perp},$ where
$K(\mathcal{D})^{\perp}$ denotes the right orthogonal with respect to the
Euler form. Moreover, if $\mathcal{N}(\mathcal{D})$ has finite rank then
$\mathcal{D}$ is called _numerically finite_.

######  Example 2.24.

  1. (1)

If $X$ is a smooth projective variety over $\mathbb{C}$, then $D^{b}(X)$ is
numerically finite.

  2. (2)

If $\mathcal{D}=\langle D_{1},\dots,D_{n}\rangle,$ then
$K(\mathcal{D})=\oplus_{i}K(D_{i}).$

Throughout this entire section, we assume that $\mathcal{D}$ is numerically
finite.

######  Definition 2.25.

If $\mathcal{A}$ is an abelian category, we define the Grothendieck group
$K(\mathcal{A})$ as the abelian group generated by isomorphism classes of
objects of $\mathcal{A}$ subject to the relation $[A]=[C]+[B],$ where
$0\rightarrow C\rightarrow A\rightarrow B\rightarrow 0$ is a short exact
sequence in $\mathcal{A}.$ If $\mathcal{A}\subseteq\mathcal{D}$ is a heart of
a bounded t-structure then $K(\mathcal{D})\cong K(\mathcal{A}).$

We now fix a finite rank $\mathbb{Z}$-lattice $\Lambda$ and a surjective
homomorphism $v\colon K(\mathcal{D})\twoheadrightarrow\Lambda.$ Since
$\mathcal{D}$ is numerically finite, then $\mathcal{N}(\mathcal{D})$ is a
finite rank $\mathbb{Z}$-lattice. We often choose
$\Lambda=\mathcal{N}(\mathcal{D})$ and $v$ as the natural projection.

The definition of a Bridgeland stability condition has two main components:
the heart of a bounded t-structure and a stability function.

######  Definition 2.26.

Let $\mathcal{A}$ be an abelian category. We say that a group homomorphism
$Z\colon K(\mathcal{A})\rightarrow\mathbb{C}$ is a _weak stability function_
on $\mathcal{A}$ if, for all $E\in\mathcal{A}$, we have $\Im(Z([E]))\geq 0$,
with $\Im(Z([E]))=0$ implying $\Re(Z([E]))\leq 0.$ If additionally, for $E\neq
0,$ we have that if $\Im(Z([E]))=0$ then $\Re(Z([E]))<0$, we say that $Z$ is a
_stability function_ on $\mathcal{A}$. Note that in this case, the image of
$Z$ is contained in the semi-closed upper half plane
$\overline{\mathbb{H}}=\\{\alpha\in\mathbb{C}\mid\Im(\alpha)\geq 0\textnormal{
and if }\Im(\alpha)=0\textnormal{, then }\Re(\alpha)<0\\}.$

We consider a group homomorphism $Z\colon\Lambda\rightarrow\mathbb{C},$ such
that $Z\circ v\colon K(\mathcal{A})\rightarrow\mathbb{C}$ is a stability
function on $\mathcal{A}$. We define the slope $\mu_{Z}\colon
K(\mathcal{A})\rightarrow\mathbb{R}\cup\infty$ by

$\mu_{Z}(E)=\begin{cases}-\frac{\Re(Z(E))}{\Im(Z(E))}&\textnormal{ if
}\Im(Z(E))\neq 0\\\ +\infty&\textnormal{otherwise,}\end{cases}$

where $Z(E)\coloneqq Z(v([E])).$ We say that $0\neq E\in\mathcal{A}$ is _Z-
semistable (stable)_ if for all proper subobjects $F\subseteq E,$ we have that
$\mu_{Z}(F)\leq\mu_{Z}(E)(\mu_{Z}(F)<\mu_{Z}(E)).$ We also define the phase of
$0\neq E$ as $\phi(E)=\arg(Z(E))\frac{1}{\pi}\in(0,1].$ Note that $E$ is
$Z$-semistable if and only if for all proper subobjects $F\subseteq E,$ we
have that $\phi(F)\leq\phi(E).$ We will constantly use the correspondence
between slope and phase given for the complex numbers in the semi-closed upper
half plane.

######  Definition 2.27.

A stability function $Z\colon K(\mathcal{A})\rightarrow\mathbb{C}$ satisfies
the _Harder–Narasimhan property_ (HN-property, for short) on $\mathcal{A}$ if
for every $0\neq E\in\mathcal{A},$ there is a filtration
$0=E_{0}\subseteq\cdots\subseteq E_{m-1}\subseteq E_{m}=E$ on $\mathcal{A},$
such that $E_{i}/E_{i-1}$ is $Z$-semistable for $i=1,\ldots,m$ and
$\phi(E_{1}/E_{0})>\cdots>\phi(E_{m}/E_{m-1}).$ Moreover, as the HN-filtration
is unique, we define the _HN-factors_ of $E$ as the quotients $E_{i}/E_{i-1}.$

######  Definition 2.28.

A _(weak) pre-stability condition_ on $\mathcal{D}$ is a pair
$\sigma=(Z,\mathcal{A}),$ where $\mathcal{A}\subseteq\mathcal{D}$ is the heart
of a bounded t-structure and $Z\colon\Lambda\rightarrow\mathbb{C}$ is a group
homomorphism such that $Z\circ v\colon
K(\mathcal{A})(=K(\mathcal{D}))\rightarrow\mathbb{C}$ is a (weak) stability
function on $\mathcal{A}$ satisfying the HN-property. The homomorphism $Z$ is
also called a central charge.

######  Remark 2.29.

[2, Proposition 2.9] Let $\sigma=(Z,\mathcal{A})$ be a weak pre-stability
condition, and let $\alpha\in\mathbb{R}.$ We form the following subcategories
of $\mathcal{A}$

$\mathcal{T}_{\sigma}^{\alpha}\coloneqq\\{E\in\mathcal{A}\mid\textnormal{The
HN-factors $F$ of $E$ satisfy }\mu_{Z}(F)>\alpha\\},$
$\mathcal{F}_{\sigma}^{\alpha}\coloneqq\\{E\in\mathcal{A}\mid\textnormal{The
HN-factors $F$ of $E$ satisfy }\mu_{Z}(F)\leq\alpha\\}.$

By the HN-property, we obtain that
$(\mathcal{T}_{\sigma}^{\alpha},\mathcal{F}_{\sigma}^{\alpha})$ is a torsion
pair on $\mathcal{A}.$ Therefore, by [24, Proposition 2.1], after tilting we
obtain a heart of a bounded t-structure denoted by
$\mathcal{A}_{\sigma}^{\alpha}$ with a torsion pair given by
$(\mathcal{F}_{\sigma}^{\alpha}[1],\mathcal{T}^{\alpha}_{\sigma}).$

We will now define a slicing. Intuitively, a heart of a bounded t-structure
$\mathcal{A}\subseteq\mathcal{D}$ breaks up every object in $\mathcal{D}$ in
terms of its cohomology indexed by $\mathbb{Z},$ a slicing further refines the
heart of a bounded t-structure, which allows us to break up each object into
pieces indexed by the real numbers.

######  Definition 2.30.

[11] A _slicing_ $\mathcal{P}$ on $\mathcal{D}$ is a collection of full
subcategories $\mathcal{P}(\phi)$ for all $\phi\in\mathbb{R}$ satisfying:

  1. (1)

$\mathcal{P}(\phi)[1]=\mathcal{P}(\phi+1)$, for all $\phi\in\mathbb{R}$.

  2. (2)

If $\phi_{1}>\phi_{2}$ and $E_{i}\in\mathcal{P}(\phi_{i})$, $i=1,2$, then
$\operatorname{Hom}_{\mathcal{D}}(E_{1},E_{2})=0$.

  3. (3)

For every nonzero object $E\in\mathcal{D}$ there exists a finite sequence of
maps

$0=E_{0}\xrightarrow{f_{0}}E_{1}\xrightarrow{f_{1}}\cdots\rightarrow
E_{m-1}\xrightarrow{f_{m-1}}E_{m}=E$

and of real numbers $\phi_{0}>\cdots>\phi_{m}$ such that the cone of $f_{j}$
is in $\mathcal{P}(\phi_{j})$ for $j=0,\cdots,{m-1}$.

For every interval $I\subseteq\mathbb{R}$ we define $\mathcal{P}(I)$ as the
extension-closed subcategory generated by the subcategories
$\mathcal{P}(\phi)$ with $\phi\in\mathbb{R}.$

######  Proposition 2.31 ([11, Proposition 5.3]).

To give a pre-stability condition $\sigma$ on $\mathcal{D}$ is equivalent to
giving a slicing $\mathcal{P}$ and a group homomorphism
$Z\colon\Lambda\rightarrow\mathbb{C}$ such that for every $0\neq
E\in\mathcal{P}(\phi),$ we have that $Z(E)\in\mathbb{R}_{>0}\cdot
e^{i\pi\phi}.$

######  Remark 2.32.

The key point of the proof of [11, Proposition 5.3] is to show that if
$\mathcal{P}$ is a slicing, then $\mathcal{A}=\mathcal{P}(0,1]$ is the heart
of a bounded t-structure satisfying the HN-property. If we have a pre-
stability condition $\sigma=(Z,\mathcal{A}),$ we define $\mathcal{P}(\phi),$
for $\phi\in(0,1]$ as the set of all $Z$-semistable objects in $\mathcal{A}.$

######  Remark 2.33.

Let $\sigma=(Z,\mathcal{P})$ be a pre-stability condition. The objects of
$\mathcal{P}(\phi)$ are called _$\sigma$ -semistable_ and the simple ones
_$\sigma$ -stable_ objects. By Definition 2.30, for every $E\in\mathcal{D},$
there is a filtration associated to $E$, that we also refer to as the
Harder–Narasimhan filtration. The semistable objects in the filtration are
called _Harder–Narasimhan factors_ (HN-factors, for short). Moreover, we write
$\phi^{+}(E),\phi^{-}(E)$ for the largest and the smallest phase appearing in
this filtration respectively. If $E$ is $\sigma$-semistable,
$\phi^{+}(E)=\phi^{-}(E)=\phi(E).$

######  Remark 2.34.

  1. (1)

Let $\sigma=(Z,\mathcal{A})$ be a pre-stability condition. By definition if
$E\in\mathcal{D}$ is $\sigma$-semistable, then there exists $n\in\mathbb{Z}$
such that $E[n]\in\mathcal{A}.$

  2. (2)

If $E,A\in\mathcal{D}$ and $\phi^{-}(E)>\phi^{+}(A),$ then
$\operatorname{Hom}_{\mathcal{D}}(E,A)=0.$

  3. (3)

Consider the last triangle $E_{m-1}\rightarrow E\rightarrow A_{m}\rightarrow
E_{m-1}[1]$ of the HN-filtration of $E\in\mathcal{D},$ where $A_{m}$ is the
cone of $f_{m-1}.$ We have that $\operatorname{Hom}_{\mathcal{D}}^{\leq
0}(E_{m-1},A_{m})=0.$

Let $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ be the universal covering
of $\rm{GL}^{+}(2,\mathbb{R}),$ whose elements are given by pairs $(T,f)$
where $T\in\rm{GL}^{+}(2,\mathbb{R})$ and
$f\colon\mathbb{R}\rightarrow\mathbb{R}$ is a continuous increasing function
that satisfies $f(x+1)=f(x)+1$ for all $x\in\mathbb{R}$ such that the induced
maps of $T$ and $f$ on
$S^{1}=\mathbb{R}/2\mathbb{Z}=(\mathbb{R}^{2}-\\{(0,0)\\})/\mathbb{R}_{>0}$
coincide. In the next section we study in detail
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ and its action on the set of
pre-stability conditions.

We define a right action of $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ on
the set of pre-stability conditions. If $\sigma=(Z,\mathcal{A})$ is a pre-
stability condition and
$g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}),$ then we define
$\sigma^{\prime}=\sigma g=(Z^{\prime},\mathcal{P}^{\prime})$ as $Z=T^{-1}\circ
Z$ and $\mathcal{P}^{\prime}(\phi)=\mathcal{P}(f(\phi)),$ where $\mathcal{P}$
and $\mathcal{P}^{\prime}$ are the slicings of $Z$ and $Z^{\prime}$
respectively. Note that the
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action preserves the
semistable objects, but relabels their phases.

Let us consider the group $\operatorname{Aut}_{\Lambda}(\mathcal{D})$ of
autoequivalences $\Phi$ on $\mathcal{D}$ whose induced automorphism $\phi_{*}$
of $K(\mathcal{D})$ is compatible with the map $v\colon
K(\mathcal{D})\rightarrow\Lambda.$

We define a left action of the group
$\operatorname{Aut}_{\Lambda}(\mathcal{D})$ on the set of pre-stability
conditions. For $\Phi\in\operatorname{Aut}_{\Lambda}(\mathcal{D})$ of
$\mathcal{D}.$ We define $\Phi(\sigma)=(Z^{\prime},\mathcal{P}^{\prime})$ as
$Z^{\prime}=Z\circ\phi^{-1}_{*}$ and
$\mathcal{P}^{\prime}(\phi)=\Phi(\mathcal{P}(\phi)).$ Note that if $E$ is a
$\sigma$-semistable object, then $\Phi(E)$ is $\Phi(\sigma)$-semistable.

######  Definition 2.35.

A pre-stability condition $\sigma$ is _locally finite_ if there is some
$\epsilon>0$ such that each category
$\mathcal{P}((\phi-\epsilon,\phi+\epsilon)),$ for $\phi\in\mathbb{R},$ is of
finite length.

######  Definition 2.36.

A pre-stability condition is _discrete_ if the image of $Z$ is a discrete
subgroup of $\mathbb{C}.$

######  Lemma 2.37 ([12, Lemma 4.5]).

Suppose that $\sigma=(Z,\mathcal{P})$ is a discrete pre-stability condition
and fix $0<\epsilon<\frac{1}{2}.$ Then for each $\phi\in\mathbb{R}$ the
category $\mathcal{P}((\phi-\epsilon,\phi+\epsilon))$ is of finite length. In
particular $\sigma$ is locally finite.

######  Remark 2.38 ([11, Lemma 5.2]).

The categories $\mathcal{P}(\phi)$ with $\phi\in\mathbb{R}$ are abelian. If
$\sigma$ is locally finite, then $\mathcal{P}(\phi)$ has finite length.
Therefore, a $\sigma$-semistable object $E\in\mathcal{P}(\phi)$ admits a
finite Jordan-Hölder filtration, i.e. a finite filtration
$E_{0}\subset\ldots\subset E_{n}=E$ with stable quotient
$E_{i+1}/E_{i}\in\mathcal{P}(\phi),$ as the stable objects are the simple
objects in $\mathcal{P}(\phi).$

We will now recall the support property. It plays an important role in proving
good deformation properties and a well-behaved wall and chamber decomposition.
We suggest [5, App. A] to understand better the relation between the support
property and effective deformations of Bridgeland stability conditions.

######  Definition 2.39.

A pre-stability condition $\sigma=(Z,\mathcal{A})$ satisfies the _support
property_ if there is a symmetric bilinear form $Q$ on
$\Lambda\otimes\mathbb{R}\coloneqq\Lambda_{\mathbb{R}}$ which satisfies

  1. (1)

All $\sigma$-semistable objects $E\in\mathcal{A},$ satisfy $Q(v(E),v(E))\geq
0.$

  2. (2)

All nonzero vectors $v\in\Lambda_{\mathbb{R}}$ with $Z(v)=0$ satisfy
$Q(v,v)<0.$

######  Remark 2.40.

If $\operatorname{rk}(\Lambda)=2$ and $Z\colon\Lambda\rightarrow\mathbb{C}$ is
injective, then every pre-stability condition $\sigma=(Z,\mathcal{A})$
trivially satisfies the support property with respect to any positive
semidefinite quadratic form.

######  Definition 2.41.

A pre-stability condition $\sigma=(Z,\mathcal{A})$ that satisfies the support
property is called a _Bridgeland stability condition_. The set of Bridgeland
stability conditions with respect to $(\Lambda,v)$ is denoted by
$\operatorname{Stab}_{\Lambda}(\mathcal{D}).$ If
$\Lambda=\mathcal{N}(\mathcal{D})$ and $v$ the natural projection, then the
set of stability conditions is denoted by $\operatorname{Stab}(\mathcal{D}).$

######  Remark 2.42.

There is a generalized metric on the set of slicings
$\operatorname{Slice}(\mathcal{D}),$ i.e. a metric that does not need to be
finite: given two slicings $\mathcal{P}$ and $\mathcal{Q},$ we define

$d(\mathcal{P},\mathcal{Q})=\sup_{0\neq
E\in\mathcal{D}}\\{\lvert\phi_{\mathcal{P}}^{+}(E)-\phi_{\mathcal{Q}}^{+}(E)\rvert,\lvert\phi_{\mathcal{P}}^{-}(E)-\phi_{\mathcal{Q}}^{-}(E)\rvert\\}\in[0,\infty].$

In order to study $\operatorname{Stab}_{\Lambda}(\mathcal{D}),$ we study the
projection
$\mathcal{Z}\colon\operatorname{Stab}_{\Lambda}(\mathcal{D})\rightarrow\operatorname{Hom}_{\mathbb{Z}}(\Lambda,\mathbb{C}).$

######  Theorem 2.43 ([11, Theorem. 7.1]).

Let $\sigma=(Z,\mathcal{P})$ be a stability condition. If $1/8>\epsilon>0,$
then for any group homomorphism $W\colon K(\mathcal{D})\rightarrow\mathbb{C}$
with $||W-Z||_{\sigma}<\sin(\pi\epsilon),$ there exists a stability condition
$\tau=(W,\mathcal{Q})$ on $\mathcal{D}$ with
$d(\mathcal{P},\mathcal{Q})<\epsilon.$

######  Theorem 2.44 ([11]).

The map
$\mathcal{Z}\colon\operatorname{Stab}_{\Lambda}(\mathcal{D})\rightarrow\operatorname{Hom}(\Lambda,\mathbb{C})$
is a local homeomorphism. Particularly, this implies that
$\operatorname{Stab}_{\Lambda}(\mathcal{D})$ is a complex manifold of
dimension $\operatorname{rk}(\Lambda).$

For a complete proof see [1, Sec 5.5].

######  Lemma 2.45 ([12, Lemma 4.5], [3, Proposition B.4]).

Let $\sigma$ be a Bridgeland stability condition and fix
$0<\epsilon<\frac{1}{2}.$ Then, the quasi-abelian subcategory
$\mathcal{P}((\phi-\epsilon,\phi+\epsilon))$ is of finite length, for each
$\phi\in\mathbb{R}$. As a consequence, Bridgeland stability conditions are
locally-finite.

The main point of the proof of the last lemma is that for a Bridgeland
stability condition $\sigma$ there exist discrete stability conditions
arbitrarily close to $\sigma.$

######  Remark 2.46.

Let $\sigma\in\operatorname{Stab}_{\Lambda}(\mathcal{D})$ and
$g\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}),$ then $\sigma g$ is also a
Bridgeland stability condition. Indeed, if $\sigma$ satisfies the support
property with respect to $Q,$ then $\sigma g$ also satisfies the support
property with respect to $Q.$ If
$\Phi\in\operatorname{Aut}_{\Lambda}(\mathcal{D}),$ then $\Phi(\sigma)$
satisfies the support property with respect to $Q\circ\phi^{-1}_{*}.$ By [11,
Lemma 8.2], the right action of
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ and the left action of
$\operatorname{Aut}_{\Lambda}(\mathcal{D})$ on
$\operatorname{Stab}_{\Lambda}(\mathcal{D})$ commute.

###  2.3. Bridgeland stability conditions on curves with $g>0$

Let $C$ be a nonsingular projective curve of genus $g(C)>0.$ To describe
$\operatorname{Stab}(C)\coloneqq\operatorname{Stab}(D^{b}(C)),$ we follow
closely [11, Section 9] and [33]. Note that there is an isomorphism
$\mathcal{N}(D^{b}(C))\cong\mathbb{Z}^{2}$ given by
$(\deg(E),\operatorname{rk}(E))$ for $E\in D^{b}(C).$

To describe $\operatorname{Stab}(C),$ one studies the last triangle of their
HN-filtrations $E\rightarrow X\rightarrow G\rightarrow E[1],$ where $X$ is
either $\mathbb{C}(x)$ or $\mathcal{L}.$ Note that by Remark 2.34, we have
that $\operatorname{Hom}^{\leq 0}(E,G)=0.$ The next lemma is a strong
consequence of $D^{b}(C)$ being hereditary, i.e. it has homological dimension
$1$ and the fact that $g(C)>0.$

######  Lemma 2.47 ([23, Lemma 7.2]).

Given a distinguished triangle $E\rightarrow X\rightarrow G\rightarrow E[1]$
in $D^{b}(C)$, with $X\in\operatorname{Coh}(C)$ and $\operatorname{Hom}^{\leq
0}_{D^{b}(C)}(E,G)=0$, then $E,G\in\operatorname{Coh}(C)$.

######  Proposition 2.48 ([33]).

For any $\sigma\in\operatorname{Stab}(D^{b}(C))$, every line bundle
$\mathcal{L}$ and skyscraper sheaf $\mathbb{C}(x)$ of a point $x\in C$ are
$\sigma$-stable.

Moreover, there is a distinguished stability condition given by the standard
slope stability: $\sigma_{\mu}\coloneqq(Z_{\mu},\operatorname{Coh}(C))$ where
$Z_{\mu}(r,d)=-d+ir$ with $(r,d)\in\mathbb{Z}^{2}$ with
$r\coloneqq\operatorname{rk}(E)$ and $d\coloneqq\deg(E)$ for
$[E]\in\mathcal{N}(D^{b}(C)).$ The next theorem states that this is in fact
the only stability condition in $\operatorname{Stab}(D^{b}(C))$ up to the
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action.

######  Theorem 2.49 ([11, Theorem. 9.1], [33, Theorem 2.7]).

The action of $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ on
$\operatorname{Stab}(D^{b}(C))$ is free and transitive. In particular,

(2.1)
$\operatorname{Stab}(D^{b}(C))\cong\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}).$

######  Remark 2.50.

Let us consider $\sigma_{\mu}=(Z_{\mu},\operatorname{Coh}(C))$ and its
corresponding slicing $\mathcal{P}_{\mu}.$ Note that
$\operatorname{Coh}^{r}(C)\coloneqq\mathcal{P}_{\mu}(r,r+1]$ for
$r\in\mathbb{R}$ is a heart of a bounded t-structure. All the hearts appearing
in the stability conditions $\sigma=(Z,\mathcal{A})\in\operatorname{Stab}(C)$
are of this form. Indeed, if $\sigma=\sigma_{\mu}g$ with
$g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}),$ we obtain that
$\mathcal{A}=\mathcal{P}_{\mu}(f(0),f(1)=f(0)+1].$ If $f(0)=n+\theta,$ with
$n\in\mathbb{Z}$ and $\theta\in[0,1)$ then we also have
$\mathcal{A}=\operatorname{Coh}^{\theta}(C)[n].$

For our purpose it is important to understand the isomorphism (2.1). We define
the following matrices in $\operatorname{SL}(2,\mathbb{R})$

$K_{\phi}=\begin{bmatrix}\cos(\phi)&-\sin(\phi)\\\
\sin(\phi)&\cos(\phi),\end{bmatrix}\textnormal{, }A_{a}=\begin{bmatrix}a&0\\\
0&\frac{1}{a}\end{bmatrix}\textnormal{ and }N_{x}=\begin{bmatrix}1&x\\\
0&1\end{bmatrix},$

for $\phi\in[0,2\pi),$ $x,a\in\mathbb{R}$ and $a>0.$

######  Lemma 2.51 (Iwasawa decomposition [26, Section 16.3]).

For every $T\in\rm{GL}^{+}(2,\mathbb{R}),$ there are real numbers
$\phi\in[0,2\pi),$ $k,a\in\mathbb{R}_{>0}$ and $x\in\mathbb{R},$ such that
$T=kK_{\phi}A_{a}N_{x}.$ Moreover, this representation is unique.

Let us consider $g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$
where $T=kK_{\phi}A_{a}N_{x}$ and $f(0)=n+\theta,$ as above. We will now
relate $\phi$ and $\theta.$

######  Remark 2.52.

  1. (1)

Let us consider $g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$
where $T=kK_{\phi}A_{a}N_{x}$ as above, then there is $m\in\mathbb{Z},$ such
that $2m+\frac{\phi}{\pi}=f(0).$ Moreover, we have two cases: either

$n=2m\textnormal{ with }\theta=\frac{\phi}{\pi}\textnormal{ or
}n=2m+1\textnormal{ with }\theta=\frac{\phi}{\pi}-1.$

  2. (2)

Let $\sigma=(Z,\mathcal{A})\in\operatorname{Stab}(C).$ If
$\sigma=\sigma_{\mu}g$ with
$g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}),$ where
$T=kK_{\phi}A_{a}N_{x}$ then
$\mathcal{A}=\operatorname{Coh}^{\frac{\phi}{\pi}}(C)[2m].$

  3. (3)

Let $\sigma=(Z,\mathcal{A})\in\operatorname{Stab}(C).$ If
$\sigma^{\prime}=\sigma g$ with
$g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ where $T=\pm
kA_{a}N_{x},$ then there is $l\in\mathbb{Z}$ with
$\mathcal{A}^{\prime}=\mathcal{A}[l]$ where
$\sigma^{\prime}=(Z^{\prime},\mathcal{A}^{\prime}).$

  4. (4)

Let $\sigma=\sigma_{\mu}g=(Z,\mathcal{A})$ with
$g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ and
$T^{-1}=\begin{bmatrix}-A&B\\\ -D&C\end{bmatrix}$. We obtain that
$Z=T^{-1}Z_{\mu},$ i.e. $Z(d,r)=Ad+Br+i(Cr+Dd).$ Moreover, if
$T=kK_{\phi}A_{a}N_{x},$ then $\arg(C+Di)=\phi.$ Indeed, as
$C=\cos(\phi)\frac{a}{k}\textnormal{ and }D=\sin(\phi)\frac{a}{k},$ then

$\arg(C+Di)=\arg(\cos(\phi)+\sin(\phi)i)=\phi.$

  5. (5)

Let $\theta\in(0,1).$ First note, that by the HN-property, if
$\mathcal{T}_{\theta}=\mathcal{P}_{\mu}(\theta,1]$ and
$\mathcal{F}_{\theta}=\mathcal{P}_{\mu}(0,\theta],$ then
$(\mathcal{T}_{\theta},\mathcal{F}_{\theta})$ is a torsion pair of
$\operatorname{Coh}(C).$ By [24, Proposition 2.1] it induces a heart of a
bounded t-structure which is precisely $\operatorname{Coh}^{\theta}(C).$
Moreover, if we define $\alpha=-\cot(\pi\theta),$ then by Remark 2.29 we
obtain that
$\operatorname{Coh}^{\theta}(C)=\operatorname{Coh}(C)^{\alpha}_{{\sigma}_{\mu}}.$
Indeed, it is enough to notice that
$\mathcal{T}_{\theta}=\mathcal{T}_{\sigma_{\mu}}^{\alpha}$ and that
$\mathcal{F}_{\theta}=\mathcal{F}_{\sigma_{\mu}}^{\alpha},$ because
$\mu(E)>\alpha\textnormal{ if and only if }\phi(E)>\theta.$

Let $\sigma=\sigma_{\mu}g=(Z,\mathcal{A})$ with
$g\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ and $\mathcal{P}$ its
slicing. The objects $\mathbb{C}(x)$ and $\mathcal{L}$ are $\sigma$-stable. By
the definition of the $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action,
this implies that for every $\sigma\in\operatorname{Stab}(C)$ we obtain
$\phi_{0}\coloneqq\phi_{\sigma}(\mathbb{C}(x))$ for $x\in C.$

By definition of the $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action,
we obtain that
$\mathcal{P}(\phi_{0})=\mathcal{P}_{\mu}(f(\phi_{0}))=\mathcal{P}_{\mu}(1),$
i.e. $f(\phi_{0})=1.$ Analogously, if
$\phi_{1}=\phi_{\sigma}(\mathcal{O}_{C}),$ then $f(\phi_{1})=\frac{1}{2}.$

######  Remark 2.53.

  1. (1)

We have that $-n<\phi_{0}=f^{-1}(1)\leq-n+1\textnormal{ if and only if
}f(0)=n+\theta,$ with $n\in\mathbb{Z}$ and $\theta\in[0,1).$

  2. (2)

There is a homeomorphism

$\displaystyle\rho\colon\operatorname{Stab}(C)$ $\displaystyle\rightarrow$
$\displaystyle\\{(m_{0},m_{1},\phi_{0},\phi_{1})\in\mathbb{R}^{4}\mid\phi_{1}<\phi_{0}<\phi_{1}+1,\textnormal{
and }m_{0},m_{1}>0\\}$ $\displaystyle\sigma=(Z,\mathcal{A})$
$\displaystyle\mapsto$ $\displaystyle(m_{0},m_{1},\phi_{0},\phi_{1}),$

where $m_{0}=|Z([\mathbb{C}(x)])|,m_{1}=|Z([\mathcal{O}_{C}])|.$

##  3\. Bridgeland stability conditions on $\mathcal{T}_{C}$

Let $C$ denote a smooth projective curve of genus $g>0$.

######  Definition 3.1.

A holomorphic triple $E=(E_{1},E_{2},\varphi)$ (also denoted by
$E=E_{1}\overset{\varphi}{\rightarrow}E_{2}$) on $C$ consists of two coherent
sheaves $E_{1},E_{2}\in\operatorname{Coh}(C)$ and a sheaf morphism
$\varphi\colon E_{1}\rightarrow E_{2}$ between them. We denote by
$\operatorname{TCoh}(C)$ the category of holomorphic triples on $C.$

######  Remark 3.2.

The category $\operatorname{TCoh}(C)$ is abelian (see [29, Theorem 1]).

###  3.1. The triangulated category $\mathcal{T}_{C}$

To be studied in this subsection is the category $\mathcal{T}_{C}\coloneqq
D^{b}(\operatorname{TCoh}(C)).$ We construct different semiorthogonal
decompositions for $\mathcal{T}_{C}.$ We prove the existence of the Serre
functor.

We note that we have three different ways to embed $\operatorname{Coh}(C)$
into $\operatorname{TCoh}(C)$:

$\displaystyle i_{*}\colon\operatorname{Coh}(C)$
$\displaystyle\hookrightarrow$ $\displaystyle\operatorname{TCoh}(C)$
$\displaystyle X$ $\displaystyle\mapsto$ $\displaystyle X\rightarrow 0,$
$\displaystyle j_{*}\colon\operatorname{Coh}(C)$
$\displaystyle\hookrightarrow$ $\displaystyle\operatorname{TCoh}(C)$
$\displaystyle X$ $\displaystyle\mapsto$ $\displaystyle 0\rightarrow X,$
$\displaystyle l_{*}\colon\operatorname{Coh}(C)$
$\displaystyle\hookrightarrow$ $\displaystyle\operatorname{TCoh}(C)$
$\displaystyle X$ $\displaystyle\mapsto$ $\displaystyle
X\xrightarrow{\operatorname{id}}X.$

As these functors are exact, we take their corresponding derived functors and
by [14, Theorem 2.4], we obtain three different exact embeddings from
$D^{b}(C)$ into $\operatorname{TCoh}(C)$. We adapt the same notation $i_{*}$,
$j_{*}$, $l_{*}$. We denote the strictly full subcategories of
$\mathcal{T}_{C}$ obtained as images of $D^{b}(C)$ in $\mathcal{T}_{C}$ under
each embedding ($i_{*}$, $j_{*}$, $l_{*}$ respectively) by $\mathcal{C}_{i}$
for $i=1,2,3$.

Since the subcategories $\mathcal{C}_{i}$ of $\mathcal{T}_{C}$ are equivalent
to $D^{b}(C),$ they are also saturated.

######  Remark 3.3 ([7]).

As a consequence of Proposition 2.15, we obtain that the strictly full
subcategories $\mathcal{C}_{i}$ are admissible in $\mathcal{T}_{C}$ for all
$i=1,2,3.$

######  Proposition 3.4.

The triangulated category $\mathcal{T}_{C}$ admits 3 semiorthogonal
decompositions

$\mathcal{T}_{C}=\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle\textnormal{,
}\mathcal{T}_{C}=\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle\textnormal{ and
}\mathcal{T}_{C}=\langle\mathcal{C}_{3},\mathcal{C}_{1}\rangle.$

###### Proof.

By Remark 3.3 we already know that the $\mathcal{C}_{i}$ are admissible in
$\mathcal{T}_{C}$ for all $i=1,2,3.$

We define the following functors

$\displaystyle l_{*}$
$\displaystyle\colon\mathcal{C}_{3}\rightarrow\mathcal{T}_{C}$ $\displaystyle
X$ $\displaystyle\mapsto X\overset{\operatorname{id}}{\rightarrow}X$
$\displaystyle l^{*}$
$\displaystyle\colon\mathcal{T}_{C}\rightarrow\mathcal{C}_{3}$ $\displaystyle
E_{1}\overset{\varphi}{\rightarrow}E_{2}$ $\displaystyle\mapsto E_{2}$
$\displaystyle l^{!}$
$\displaystyle\colon\mathcal{T}_{C}\rightarrow\mathcal{C}_{3}$ $\displaystyle
E_{1}\overset{\varphi}{\rightarrow}E_{2}$ $\displaystyle\mapsto E_{1}.$

Left adjointness, $(l^{*},l_{*})$ and right adjointness, $(l_{*},l^{!})$
follow directly from the definitions.

Additionally, we define the functors

$\displaystyle i_{*}$
$\displaystyle\colon\mathcal{C}_{1}\rightarrow\mathcal{T}_{C}$ $\displaystyle
X$ $\displaystyle\mapsto X\rightarrow 0$ $\displaystyle i^{*}$
$\displaystyle\colon\mathcal{T}_{C}\rightarrow\mathcal{C}_{1}$ $\displaystyle
E_{1}\overset{\varphi}{\rightarrow}E_{2}$ $\displaystyle\mapsto E_{1}.$

Left adjointness, $(i^{*},i_{*})$ follows directly from the definitions.

Finally, we have to prove that
$\mathcal{C}_{3}^{\perp}=\\{E\in\mathcal{T}_{C}\mid\operatorname{Hom}_{\mathcal{T}_{C}}(l_{*}\mathcal{C},E)=0\\}=\mathcal{C}_{2}.$
Indeed, by adjointness we have
$\operatorname{Hom}_{\mathcal{T}_{C}}(l_{*}(X),E)=\operatorname{Hom}_{\mathcal{C}}(X,l^{!}(E))$
for any $E=E_{1}\overset{\varphi}{\rightarrow}E_{2}\in\mathcal{T}_{C}$ and any
$X\in\mathcal{C}$. This means that
$\operatorname{Hom}_{\mathcal{C}}(X,E_{1})=0$ for all $X\in\mathcal{C}$. This
happens if and only if $E_{1}=0$.

With a similar argument, one can see that
$\prescript{\perp}{}{\mathcal{C}}_{3}=\mathcal{C}_{1}.$ and
$\prescript{\perp}{}{\mathcal{C}}_{1}=\mathcal{C}_{2}.$ ∎

######  Remark 3.5.

Let $E=E_{1}\xrightarrow{\varphi}E_{2}\in\mathcal{T}_{C}.$

  1. (1)

The semiorthogonal decomposition
$\mathcal{T}_{C}=\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle$ induces the
distinguished triangle $j_{*}(E_{2})\rightarrow E\rightarrow
i_{*}(E_{1})\rightarrow j_{*}(E_{2})[1].$

  2. (2)

The semiorthogonal decomposition
$\mathcal{T}_{C}=\langle\mathcal{C}_{3},\mathcal{C}_{1}\rangle$ induces the
distinguished triangle $i_{*}i^{!}(E)\rightarrow E\rightarrow
l_{*}(E_{2})\rightarrow i_{*}i^{!}(E)[1].$

  3. (3)

The semiorthogonal decomposition
$\mathcal{T}_{C}=\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle$ induces the
distinguished triangle $l_{*}(E_{1})\rightarrow E\rightarrow
j_{*}(j^{*}(E))\rightarrow l_{*}(E_{1})[1].$

By admissibility of the full subcategories $\mathcal{C}_{i}$ for $i=1,2,3$,
and the unicity of the triangles associated to the semiorthogonal
decompositions, we have that $l^{*}=j^{!}$, where $j^{!}$ satisfies right
adjunction $(j_{*},j^{!})$, similarly $l^{!}=i^{*}$.

In order to describe the precise distinguished triangles associated to the
semiorthogonal decompositions of $\mathcal{T}_{C}$ we need to understand the
functors $i^{!}$ and $j^{*}$ satisfying the adjunction relations
$(i_{*},i^{!})$ and $(j^{*},j_{*})$ respectively. We will now describe the
functors $i^{!}$ and $j^{*}$ at the level of objects and some features at the
level of morphisms.

######  Lemma 3.6.

Let $E=(E_{1},E_{2},\varphi_{E})$ and
$F=(F_{1},F_{2},\varphi_{F})\in\mathcal{T}_{C}.$ The functor $i^{!}$ is given
by

(3.1) $\displaystyle i^{!}\colon\mathcal{T}_{C}$ $\displaystyle\rightarrow$
$\displaystyle D^{b}(C)$ $\displaystyle E$ $\displaystyle\mapsto$
$\displaystyle C(\varphi_{E})[-1],$

where $C(-)$ is the cone, at the level of objects. If $\psi\colon E\rightarrow
F$ is a morphism of triples, then the following diagram commutes

(3.2)
$\textstyle{C(\varphi_{E})[-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{!}(\psi)}$$\scriptstyle{{\pi}_{E}}$$\textstyle{C(\varphi_{F})[-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\pi}_{F}}$$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}(\psi)}$$\textstyle{F_{1}.}$

###### Proof.

From the triangle in part (3) from Remark 3.5 and the fact that $i^{*}$ is an
exact functor, we obtain the triangle
$i^{!}(E)\xrightarrow{\pi_{E}}E_{1}\xrightarrow{\varphi}E_{2}\rightarrow
i^{!}(E),$ therefore $i^{!}(E)=C(\varphi_{E})[-1].$ We use the naturality of
the adjunction and we obtain $\psi\circ\pi_{E}=i_{*}i^{!}(\psi)\circ\pi_{F}$
and by taking $i^{*}$ in both sides we obtain the square (3.2). ∎

######  Lemma 3.7.

Let $E=(E_{1},E_{2},\varphi_{E})$ and
$F=(F_{1},F_{2},\varphi_{F})\in\mathcal{T}_{C}.$ The functor $j^{*}$ is given
by

(3.3) $\displaystyle j^{*}\colon\mathcal{T}_{C}$ $\displaystyle\rightarrow$
$\displaystyle D^{b}(C)$ $\displaystyle E$ $\displaystyle\mapsto$
$\displaystyle C(\varphi_{E})$

at the level of objects. If $\psi\colon E\rightarrow F$ is a morphism of
triples, then the following diagram commutes

(3.4)
$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{!}(\psi)}$$\scriptstyle{\tau_{E}}$$\textstyle{F_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{F}}$$\textstyle{C(\varphi_{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}(\psi)}$$\textstyle{C(\varphi_{F}).}$

###### Proof.

The proof goes along the lines of Lemma 3.6. ∎

######  Remark 3.8.

In many cases, we express a morphism
$\Phi\in\operatorname{Hom}_{\mathcal{T}_{\mathcal{C}}}(E,F)$ as two horizontal
arrows, but note that they just represent $i^{*}(\Phi)$ and $j^{!}(\Phi)$ and
they do not characterize the morphism $\Phi.$

We denote by $\mathcal{N}(C)=\mathcal{N}(D^{b}(C))$ the numerical Grothendieck
group of $\operatorname{Coh}(C).$

######  Remark 3.9.

Note that $K(\mathcal{T}_{C})=K(D^{b}(C))\oplus K(D^{b}(C)).$

Moreover, the Euler form
$\chi(E,E^{\prime})\coloneqq\sum\limits_{i}(-1)^{i}\dim_{\mathbb{C}}\operatorname{Hom}_{\mathcal{T}_{C}}(E,E^{\prime}[i])$
on $\operatorname{TCoh}(C)$ vanishes if and only if $r_{1}=r_{2}=0$ and
$d_{1}=d_{2}=0$, where $d_{i}\coloneqq\deg(E_{i})$,
$r_{i}\coloneqq\operatorname{rk}(E_{i})$ for $i=1,2$. Hence,
$\mathcal{N}(\mathcal{T}_{C})$ is isomorphic to $\mathbb{Z}^{4}$ by
identifying the class $[E]\in\mathcal{N}(\mathcal{T}_{C})$ of a holomorphic
triple $E=E_{1}\overset{\varphi}{\rightarrow}E_{2}$ with the point
$(r_{1},d_{1},r_{2},d_{2})\in\mathbb{Z}^{4}.$

#### Serre functor

By Proposition 2.22 and Proposition 3.4, we have the existence of the Serre
functor $S_{\mathcal{T}_{C}}$ on $\mathcal{T}_{C}$. The adjunction properties
of the Serre functor provide us with the following lemma.

######  Lemma 3.10 ([37, Lemma 2.42]).

Let $X\in D^{b}(C)$. The following equalities hold:

  1. (1)

$S_{\mathcal{T}_{C}}(i_{*}(X))=j_{*}(S_{C}(X))[1],$

  2. (2)

$S_{\mathcal{T}_{C}}(j_{*}(X))=l_{*}(S_{C}(X)),$

  3. (3)

$S_{\mathcal{T}_{C}}(l_{*}(X))=i_{*}(S_{C}(X)),$ where $S_{C}$ is as in
Example 2.19.

######  Remark 3.11.

Let $E=E_{1}\xrightarrow{\varphi}E_{2}$ be an object in
$\mathcal{T}_{\mathcal{C}}.$ As we have
$F^{!}S_{\mathcal{T}_{\mathcal{C}}}=S_{C}F^{*}$ for $F=i,j$ or $l,$ we
directly $i^{!}(S_{\mathcal{T}_{\mathcal{C}}}(E))=S_{C}(E_{1})\textnormal{ and
}j^{!}(S_{\mathcal{T}_{\mathcal{C}}}(E))=S_{C}(C(\varphi)).$

######  Remark 3.12 ([6, Proposition 1.3.3] and [6, Corollary 1.1.10]).

Let us consider the triangle $j_{*}j^{!}(E)\rightarrow E\rightarrow
i_{*}i^{*}(E)\xrightarrow{t_{E}}j_{*}j^{!}(E)[1].$ Note that the morphism
$t_{E}\colon i_{*}i^{*}(E)\rightarrow j_{*}j^{!}(E)[1]$ characterizes the
triangle, i.e. it is the unique morphism representing the isomorphism class of
the triangle induced by the semiorthogonal decomposition
$\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle$ for $E.$ Moreover, note that
$C(t_{E})[-1]$ is isomorphic to $E$ up to a nonunique isomorphism in
$\mathcal{T}_{C}.$

######  Remark 3.13.

Let us consider the triangle $j_{*}j^{!}(E)\rightarrow E\rightarrow
i_{*}i^{*}(E)\xrightarrow{t_{E}}j_{*}j^{!}(E)[1].$ After applying the Serre
functor we obtain a triangle

$S_{\mathcal{T}_{C}}(j_{*}j^{!}(E))\rightarrow
S_{\mathcal{T}_{C}}(E)\rightarrow
S_{\mathcal{T}_{C}}(i_{*}i^{*}(E))\xrightarrow{S_{\mathcal{T}_{C}}(t_{E})}S_{\mathcal{T}_{C}}(j_{*}j^{!}(E)[1].$

As $S_{\mathcal{T}_{C}}({{}^{\perp}}\mathcal{D})=\mathcal{D}^{\perp}$ for an
admissible triangulated subcategory $\mathcal{D}\subseteq\mathcal{T}_{C},$
then $S_{\mathcal{T}_{C}}(j_{*}j^{!}(E))\in\mathcal{C}_{3}\textnormal{ and
}S_{\mathcal{T}_{C}}(i_{*}i^{*}(E))\in\mathcal{C}_{2}.$ By the uniqueness of
the triangle induced by the semiorthogonal decomposition, we obtain precisely
the corresponding triangle of $S_{\mathcal{T}_{C}}(E)$ induced by the
semiorthogonal decomposition
$\mathcal{T}_{C}=\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle$ up to
isomorphism. Moreover $S_{\mathcal{T}_{C}}(t_{E})$ is the unique morphism that
characterizes the triangle as mentioned in Remark 3.12. See [32, Lemma 2.3]
and [6, Proposition 1.3.3].

The following remark connects the triple $E$ with the roof
$[\varphi_{E}]\coloneqq(\operatorname{id},\varphi_{E})$ in
$\operatorname{Mor}(D^{b}(C),D^{b}(C))$ and it plays a role in the explicit
construction of the Serre functor.

######  Remark 3.14 ([37, Lemma 2.49]).

$t_{E}=t_{l_{*}(E_{2})}\circ i_{*}([\varphi]).$

We will now describe the Serre functor at the level of objects.

######  Proposition 3.15 ([37, Lemma 2.51]).

Let $E=E_{1}\xrightarrow{\varphi}E_{2}$ be an object of $\mathcal{T}_{C}.$ If
$S_{C}(i_{E})$ is a morphism of complexes, where
$E_{2}\xrightarrow{i_{E}}C(\varphi)$ is the morphism of complexes given by the
injection, then $S_{\mathcal{T}_{C}}(E)$ is isomorphic to the triple
$S_{C}(E_{2})\xrightarrow{S_{C}(i_{E})}S_{C}(C(\varphi))$ in
$\mathcal{T}_{C}.$

On the other hand, if $S_{C}^{-1}(p_{E})$ is a morphism of complexes, where
$C(\varphi)[-1]\xrightarrow{p_{E}}E_{1}$ is the projection, then
$S^{-1}_{\mathcal{T}_{C}}(E)$ is isomorphic to
$S^{-1}_{C}(C(\varphi)(-1))\xrightarrow{S^{-1}_{C}(p_{E})}S^{-1}_{C}(E_{1}).$

######  Remark 3.16.

First of all, note that by Serre duality and the semiorthogonal decomposition
described above, it is clear that the homological dimension of
$\operatorname{TCoh}(C)$ is $\leq 2$ (see [35, Proposition 2.7.24]). Moreover,
given $E_{1}\in\operatorname{Coh}(C)$, if we take $i_{*}(E_{1})$ and
$j_{*}(E_{1})$, by Serre duality and Lemma 3.10 we get

$\displaystyle\operatorname{Hom}_{\mathcal{T}_{C}}(i_{*}(E_{1}),j_{*}(E_{1})[2])$
$\displaystyle=\operatorname{Hom}_{\mathcal{T}_{C}}(j_{*}(E_{1})[2],S_{\mathcal{T}_{C}}(i_{*}(E_{1})))^{*}$
$\displaystyle=\operatorname{Hom}_{D^{b}(C)}(E_{1},E_{1}\otimes\omega_{C})^{*}[2]\neq
0.$

This implies that the category $\operatorname{TCoh}(C)$ has homological
dimension 2.

If $g(C)=1$, we have the following conjecture.

######  Conjecture 3.17.

Let $C$ be an elliptic curve. Then $S_{\mathcal{T}_{C}}^{3}=[4]$, which
implies that $\mathcal{T}_{C}$ is a fractional Calabi–Yau category of
fractional dimension $4/3$.

###  3.2. Constructing pre-stability conditions

####  3.2.1. CP-gluing

In [17] (and Collins in [16]) Collins and Polishchuk introduced a way to
construct hearts from semiorthogonal decompositions of a triangulated category
$\mathcal{D}$ which later allow us to define stability conditions on
$\mathcal{D}$ in a natural way.

Let $\mathcal{D}$ be a triangulated category equipped with a semiorthogonal
decomposition $\mathcal{D}=\langle\mathcal{D}_{1},\mathcal{D}_{2}\rangle$. Let
$\rho_{2}\colon\mathcal{D}\rightarrow\mathcal{D}_{2}$ be the right adjoint
functor to the full embedding
$i_{2}\colon\mathcal{D}_{2}\rightarrow\mathcal{D}$ and let
$\lambda_{1}\colon\mathcal{D}\rightarrow\mathcal{D}_{1}$ be the left adjoint
functor to the full embedding
$i_{1}\colon\mathcal{D}_{1}\rightarrow\mathcal{D}$.

######  Proposition 3.18 ([17, Lemma 2.1]).

With the above notations, assume that we have t-structures
$(\mathcal{D}_{i}^{\leq 0},\mathcal{D}_{i}^{\geq 0})$ with hearts
$\mathcal{A}_{i}$ on $\mathcal{D}_{i}$, for $i=1,2$, such that

(3.5) $\operatorname{Hom}_{\mathcal{D}}^{\leq
0}(i_{1}\mathcal{A}_{1},i_{2}\mathcal{A}_{2})=0.$

Then there is a t-structure on $\mathcal{D}$ with the heart

(3.6)
$\operatorname{gl}(i_{1}\mathcal{A}_{1},i_{2}\mathcal{A}_{2})=\\{E\in\mathcal{D}\mid\rho_{2}E\in\mathcal{A}_{2},\lambda_{1}E\in\mathcal{A}_{1}\\}.$

Moreover,
$i_{k}\mathcal{A}_{k}\subset\mathcal{A}\coloneqq\operatorname{gl}(i_{1}\mathcal{A}_{1},i_{2}\mathcal{A}_{2})$
for $k=1,2$.

######  Definition 3.19.

We will refer to hearts of the form (3.6) as hearts obtained by CP-gluing.

######  Notation 3.20.

In the case of the semiorthogonal decompositions
$\mathcal{T}_{C}=\langle\mathcal{C}_{i},\mathcal{C}_{j}\rangle$, with
$ij\in\\{12,23,31\\}$, we will denote the heart obtained by CP-gluing of
hearts $\mathcal{A}_{i}$ on $\mathcal{C}_{i}$ and $\mathcal{A}_{j}$ on
$\mathcal{C}_{j}$ by
$\operatorname{gl}_{ij}(\mathcal{A}_{i},\mathcal{A}_{j})$. Recall that for
$r\in\mathbb{R},$ we define $\operatorname{Coh}^{r}(C)$ as in Remark 2.50.

######  Proposition 3.21.

We distinguish 3 cases according to the semiorthogonal decomposition of
$\mathcal{T}_{C}$.

  1. (1)

If $\mathcal{T}_{C}=\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle$, we have
$\operatorname{Hom}_{\mathcal{T}_{C}}^{\leq
0}(\operatorname{Coh}_{1}^{r_{1}}(C),\operatorname{Coh}_{2}^{r_{2}}(C))=0$ if
and only if $r_{1}\geq r_{2}$.

  2. (2)

If $\mathcal{T}_{C}=\langle\mathcal{C}_{3},\mathcal{C}_{1}\rangle$, we have
$\operatorname{Hom}_{\mathcal{T}_{C}}^{\leq
0}(\operatorname{Coh}_{3}^{r_{3}}(C),\operatorname{Coh}_{1}^{r_{1}}(C))=0$ if
and only if $r_{3}\geq r_{1}+1$.

  3. (3)

If $\mathcal{T}_{C}=\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle$, we have
$\operatorname{Hom}_{\mathcal{T}_{C}}^{\leq
0}(\operatorname{Coh}_{2}^{r_{2}}(C),\operatorname{Coh}_{3}^{r_{3}}(C))=0$ if
and only if $r_{2}\geq r_{3}+1$.

###### Proof.

As in Remark 2.52, we write each $r_{i}=n_{i}+\theta_{i}$ for unique
$n_{i}\in\mathbb{Z}$ and $\theta_{i}\in[0,1)$. Firstly, we want to determine
under which conditions
$\operatorname{Hom}_{\mathcal{T}_{C}}(\operatorname{Coh}_{1}^{\theta_{1}}(C)[n_{1}],\operatorname{Coh}_{2}^{\theta_{2}}(C)[n_{2}+i])=0$
for every $i\leq 0$. Assume $n_{2}=0$, up to shifting by $-n_{2}$. For a fixed
$i$, take $0\rightarrow E_{2}\in\operatorname{Coh}_{2}^{\theta_{2}}(C)$ and
$E_{1}\rightarrow 0\in\operatorname{Coh}_{1}^{\theta_{1}}(C)$. By Serre
duality we have

(3.7) $\operatorname{Hom}_{\mathcal{T}_{C}}(E_{1}[n_{1}]\rightarrow
0,0\rightarrow E_{2}[i])=\operatorname{Hom}_{\mathcal{T}_{C}}(0\rightarrow
E_{2},S_{\mathcal{T}_{C}}(E_{1}\rightarrow 0)[n_{1}-i])^{*}.$

By Lemma 3.10, $S_{\mathcal{T}_{C}}(E_{1}\rightarrow 0)[n_{1}]\in
j_{*}(S_{C}(\operatorname{Coh}^{\theta_{1}}(C)[n_{1}]))[1]$, so (3.7) vanishes
for all $i\leq 0$ if and only if $n_{1}\geq 0$ and if $n_{1}=0$, we see that
we need that $\theta_{1}\geq\theta_{2}$. Indeed, if $n_{1}=0$, recall that
each heart $\operatorname{Coh}^{\theta_{i}}(C)$ was defined by tilting
$\operatorname{Coh}^{\theta_{i}}(C)=\langle
F_{\theta_{i}}[1],T_{\theta_{i}}\rangle$, for $i=1,2$. By the previous
argument with the Serre functor, the only restriction appears when
$E_{1}\rightarrow 0\in T_{\theta_{1}}$ and $0\rightarrow E_{2}[1]\in
F_{\theta_{2}}[1]$. Here, if $\theta_{1}\geq\theta_{2}$, we have
$T_{\theta_{1}}\subset T_{\theta_{2}}$. Use Serre duality for $D^{b}(C)$, such
that

$\operatorname{Hom}_{D^{b}(C)}(E_{2}[1],E_{1}\otimes\omega_{C}[2-i])=\operatorname{Hom}_{D^{b}(C)}(E_{1}[1-i],E_{2}[1])^{*}.$

This vanishes for all $i\leq 0$, since $(T_{\theta_{2}},F_{\theta_{2}})$ is a
torsion pair.

Case 2 (resp. 3) follows applying the Serre functor $S_{\mathcal{T}_{C}}$
(resp. $S^{-1}_{\mathcal{T}_{C}}$) to case 1. ∎

######  Remark 3.22.

By Theorem 2.49, the CP-gluing conditions in Proposition 3.21 can be re-stated
in terms of the corresponding elements
$(T_{i},f_{i})\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ for $i=1,2,3$
by considering $r_{i}=f_{i}(0)$.

Trivial examples of hearts in $\mathcal{T}_{C}$ defined by CP-gluing can be
given by considering $\operatorname{Coh}(C)$ embedded in each triangulated
subcategory $\mathcal{C}_{i}$ for $i=1,2,3$ as follows.

######  Definition 3.23 ([17]).

Let $\sigma_{i}=(Z_{i},\mathcal{A}_{i})$ be stability conditions on
$\mathcal{D}_{i}$ for $i=1,2$, such that the hearts $\mathcal{A}_{i}$ satisfy
(3.5). Then we say that a pair $\sigma=(Z,\mathcal{A})$ on
$\mathcal{D}=\langle\mathcal{D}_{1},\mathcal{D}_{2}\rangle$ is a CP-glued pair
(from $\sigma_{1}$ and $\sigma_{2}$) if the heart $\mathcal{A}$ is given by
(3.6) and $Z\colon K(\mathcal{A})\rightarrow\mathbb{C}$ is given by

(3.8) $Z=Z_{1}\circ\lambda_{1}+Z_{2}\circ\rho_{2}.$

######  Remark 3.24.

Note that this CP-glued pair is uniquely determined by $\sigma_{1}$ and
$\sigma_{2}$. It is a pre-stability condition if and only if the
Harder–Narasimhan property holds for the stability function $Z$ on the glued
heart $\mathcal{A}$. We will check this property separately later.

######  Notation 3.25.

In the case of the semiorthogonal decompositions
$\mathcal{T}_{C}=\langle\mathcal{C}_{i},\mathcal{C}_{j}\rangle$ for
$ij\in\\{12,23,31\\}$ we will denote by
$\operatorname{gl}_{ij}(\sigma_{i},\sigma_{j})$ the CP-glued pair obtained by
CP-gluing of stability conditions $\sigma_{i}$ on $\mathcal{C}_{i}$ and
$\sigma_{j}$ on $\mathcal{C}_{j}$.

######  Definition 3.26 (Standard hearts).

Consider $\mathcal{T}_{C}=\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle.$ By
definition we have that

$\operatorname{TCoh}(C)=\operatorname{gl}_{12}(\operatorname{Coh}_{1}(C),\operatorname{Coh}_{2}(C)).$

Consider $\mathcal{T}_{C}=\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle,$ then
we define

$\mathcal{H}_{23}\coloneqq\operatorname{gl}_{23}(\operatorname{Coh}_{2}(C)[1],\operatorname{Coh}_{3}(C)).$

Consider $\mathcal{T}_{C}=\langle\mathcal{C}_{3},\mathcal{C}_{1}\rangle,$ then
we define

$\mathcal{H}_{31}\coloneqq\operatorname{gl}_{31}(\operatorname{Coh}_{3}(C)[1],\operatorname{Coh}_{1}(C)).$

######  Lemma 3.27 ([17, Proposition 2.2]).

  1. (1)

A pre-stability condition $\sigma=(Z,\mathcal{A})$ on $\mathcal{D}$ is CP-
glued from $\sigma_{1}=(Z_{1},\mathcal{A}_{1})$ on $\mathcal{D}_{1}$ and
$\sigma_{2}=(Z_{2},\mathcal{A}_{2})$ on $\mathcal{D}_{2}$ if and only if
$Z_{i}=Z|_{\mathcal{D}_{i}}$ for $i=1,2$,
$\operatorname{Hom}_{\mathcal{D}}^{\leq 0}(\mathcal{A}_{1},\mathcal{A}_{2})=0$
and $\mathcal{A}_{i}\subset\mathcal{A}$ for $i=1,2$.

  2. (2)

Let $\sigma=(Z,\mathcal{A})$ be a pre-stability condition on $\mathcal{D}$.
Assume that the heart $\mathcal{A}$ is glued from hearts
$\mathcal{A}_{1}\subset\mathcal{D}_{1}$ and
$\mathcal{A}_{2}\subset\mathcal{D}_{2}$, with
$\operatorname{Hom}_{\mathcal{D}}^{\leq
0}(\mathcal{A}_{1},\mathcal{A}_{2})=0$, such that (3.6) holds. Then, there
exists a pre-stability condition
$\sigma_{i}=(Z_{i}=Z|_{\mathcal{D}_{i}},\mathcal{A}_{i})$ on
$\mathcal{D}_{i}$, for $i=1,2$, such that $\sigma$ is glued from $\sigma_{1}$
and $\sigma_{2}$.

  3. (3)

If $\sigma=(Z,\mathcal{P})$ is CP-glued from the pre-stability conditions
$\sigma_{1}=(Z_{1},\mathcal{P}_{1})$ and $\sigma_{2}=(Z_{2},\mathcal{P}_{2})$,
then $\mathcal{P}_{1}(\phi)\subset\mathcal{P}(\phi)$ and
$\mathcal{P}_{2}(\phi)\subset\mathcal{P}(\phi)$ for every $\phi\in\mathbb{R}$.

We will now explain the behaviour of the CP-glued pre-stability conditions
under the $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action.

######  Lemma 3.28.

Let $\sigma_{i}=(Z_{i},\mathcal{A}_{i})$ be pre-stability conditions on
$\mathcal{D}_{i}$ for $i=1,2$ and let $\sigma$ be a CP-glued pre-stability
condition from $\sigma_{1}$ and $\sigma_{2}$ on a triangulated category
$\mathcal{D}$ with respect to the semiorthogonal decomposition
$\mathcal{D}=\langle\mathcal{D}_{1},\mathcal{D}_{2}\rangle.$ Let
$g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}),$ then $\sigma
g=(W,\mathcal{B})$ satisfies the following conditions: If
$\sigma_{i}g=(W_{i},\mathcal{B}_{i}),$ for $i=1,2$ then

  1. (1)

$i_{1}\mathcal{B}_{1}\subseteq\mathcal{B}$ and
$i_{2}\mathcal{B}_{2}\subseteq\mathcal{B}$

  2. (2)

${\left.\kern-1.2ptW\vphantom{\big{|}}\right|_{\mathcal{D}_{i}}}=W_{i},$ for
i=1,2.

Moreover, if $\operatorname{Hom}^{\leq
0}_{\mathcal{D}}(i_{1}\mathcal{B}_{1},i_{2}\mathcal{B}_{2})$ then $\sigma
g=\operatorname{gl}_{12}(\sigma_{1}g,\sigma_{2}g)$.

###### Proof.

Let us consider the slicing $\mathcal{P}_{i}$ of $\sigma_{i},$ for $i=1,2,$
and $\mathcal{P}$ the slicing of $\sigma.$ By definition of the
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action we have
$\mathcal{B}_{i}=\mathcal{P}_{i}(f(0),f(1)]$ and
$\mathcal{B}=\mathcal{P}(f(0),f(1)].$ By the third part of Proposition 3.27 we
obtain directly that $i_{1}\mathcal{B}_{1}\subseteq\mathcal{B}$ and
$i_{2}\mathcal{B}_{2}\subseteq\mathcal{B}.$

Next, recall that by definition we have that $W=T^{-1}\circ Z.$ Let
$E_{i}\in\mathcal{D}_{i}$ for $i=1,2$. We note that $\lambda_{1}(E_{1})=E_{1}$
and $\rho_{2}(E_{1})=0$ (resp. $\lambda_{1}(E_{2})=0$ and
$\rho_{2}(E_{2})=E_{2}$) which implies $W([E_{i}])=T^{-1}\circ
Z_{i}([E_{i}])=W_{i}$. Finally, if we also assume that
$\operatorname{Hom}^{\leq
0}_{\mathcal{D}}(i_{1}\mathcal{B}_{1},i_{2}\mathcal{B}_{2})=0,$ by the first
part of Proposition 3.27 we obtain that $\sigma
g=\operatorname{gl}_{12}(\sigma_{1}g,\sigma_{2}g).$ ∎

It is important to remark that the gluing condition may not be preserved after
applying the $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action, as we can
see in the following example.

######  Example 3.29.

Consider the CP-glued pair
$\sigma=\operatorname{gl}_{12}(\sigma_{1},\sigma_{\mu})$ with
$\sigma_{1}=\sigma_{\mu}g$ and $g=(N_{1},f)$ in
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}).$ Note that $f(0)=0$ and that
$\mathcal{P}_{1}(t)=\mathcal{P}_{\mu}(f(t)).$

We have $1>t>f(t)$ for all $t\in(0,1)$. Indeed, let $E\in\mathcal{P}_{\mu}(t)$
be an object of phase $t\in(0,1)$. In particular, the rank $r$ of $E$ is
strictly positive. Now, since
$E\in\mathcal{P}_{\mu}(t)=\mathcal{P}_{1}(f^{-1}(t))$, together with the fact
that $\Im Z_{1}(E)=\Im Z_{\mu}(E)=r$ and $\Re Z_{1}(E)=-d+r>\Re Z_{\mu}(E)$,
we have $0<f(t)<t<1$. The inequalities follow because $f$ is strictly
increasing. Let us consider $g^{\prime}=(K_{\phi\pi},f_{\phi\pi})$ for
$\phi\in(0,1).$ We will now study $\sigma_{1}^{\prime}=\sigma_{1}g^{\prime}$
and $\sigma_{2}^{\prime}=\sigma_{\mu}g^{\prime},$ which are also given by
$(T^{\prime}_{i},f^{\prime}_{i})\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$
for $i=1,2$ respectively. Note that $f^{\prime}_{1}(0)=f\circ
g^{\prime}(0)=f(\phi)$ and $f^{\prime}_{2}(0)=g^{\prime}(0)=\phi.$ Therefore,
we obtain $f^{\prime}_{1}(0)<f^{\prime}_{2}(0)$ and from Proposition 3.21 it
follows that $\sigma^{\prime}_{1}$ and $\sigma^{\prime}_{2}$ do not satisfy
gluing conditions with respect to the semiorthogonal decomposition
$\mathcal{T}_{C}=\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle.$ See
Subsection 4.4 for the general picture.

Now that we have hearts in $\mathcal{T}_{C}$ with the corresponding stability
functions, we have to check that they satisfy the Harder–Narasimhan property
and the support property. We begin with the Harder–Narasimhan property,
arguing along the lines of [17].

######  Theorem 3.30 ([17, Theorem 3.6]).

Let $(\mathcal{D}_{1},\mathcal{D}_{2})$ be a semiorthogonal decomposition of a
triangulated category $\mathcal{D}$. Suppose
$\sigma_{i}=(Z_{i},\mathcal{P}_{i})$ is a stability condition on
$\mathcal{D}_{i}$ for $i=1,2$ and let $a\in(0,1)$ be a real number. Assume the
following conditions hold:

  1. (1)

$\operatorname{Hom}_{\mathcal{D}}^{\leq
0}(i_{1}\mathcal{P}_{1}(0,1],i_{2}\mathcal{P}_{2}(0,1])=0$

  2. (2)

$\operatorname{Hom}_{\mathcal{D}}^{\leq
0}(i_{1}\mathcal{P}_{1}(a,a+1],i_{2}\mathcal{P}_{2}(a,a+1])=0$.

Then, there exists a locally finite pre-stability condition $\sigma$ glued
from $\sigma_{1}$ and $\sigma_{2}$.

######  Definition 3.31.

For a real number $a\in(0,1)$, we define the subset $S(a)$ as the subset of
pairs of stability conditions
$(\sigma_{1},\sigma_{2})\in\operatorname{Stab}(\mathcal{D}_{1})\times\operatorname{Stab}(\mathcal{D}_{2})$
satisfying the conditions in Theorem 3.30.

######  Theorem 3.32 ([17, Theorem 4.3]).

Let $\textnormal{gl}\colon S(a)\rightarrow\operatorname{Stab}(\mathcal{D})$ be
the map associating to
$(\sigma_{1},\sigma_{2})\in\operatorname{Stab}(\mathcal{D}_{1})\times\operatorname{Stab}(\mathcal{D}_{2})$
the corresponding glued pre-stability condition $\sigma$ on $\mathcal{D}$
(defined by Theorem 3.30). Then, the map $\rm{gl}$ is continuous on $S(a)$.

For $a\in(0,1)$, we have a precise description of the sets $S(a)$ for the
semiorthogonal decomposition
$\mathcal{T}_{C}=\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle$.

######  Proposition 3.33.

Consider the semiorthogonal decomposition
$\mathcal{T}_{C}=\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle$. For
$a\in(0,1)$, we have
$S(a)\cong\left\\{((T_{1},f_{1}),(T_{2},f_{2}))\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})\times\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})\colon
f_{1}(0)\geq f_{2}(0)\textnormal{ and }f_{1}(a)\geq f_{2}(a)\right\\}.$

###### Proof.

Suppose $\sigma_{i}=(Z_{i},\operatorname{Coh}_{i}^{\theta_{i}}(C)[n_{i}])$ is
a stability condition on $\mathcal{C}_{i}$ with $\theta_{i}\in[0,1)$ and
$n_{i}\in\mathbb{Z}$, for $i=1,2$. Assume that these stability conditions
satisfy the gluing condition, i.e. $n_{1}+\theta_{1}\geq n_{2}+\theta_{2}$.
Let $(T_{i},f_{i})$ be the elements in
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ corresponding to $\sigma_{i}$
under the equivalence in Theorem 2.49 (for $i=1,2$). Note that
$f_{i}(0)=n_{i}+\theta_{i}$ (for $i=1,2$) so the condition 1 in Theorem 3.30
is equivalent to $f_{1}(0)\geq f_{2}(0)$. We end the proof by showing that
condition 2 is equivalent to $f_{1}(a)\geq f_{2}(a)$. Indeed, we will have
$\operatorname{Hom}_{\mathcal{D}}^{\leq
0}(i_{*}\mathcal{P}_{1}(a,a+1],j_{*}\mathcal{P}_{2}(a,a+1])=0$ if and only if
the stability condition $\sigma^{\prime}$ obtained from $\sigma$, acting by
rotation of angle $a$ satisfies the gluing property. Hence, if we denote
$\mathcal{P}(0,1]=\operatorname{Coh}(C)$ the standard heart associated to
slope-stability $Z_{\mu}$, then
$\mathcal{P}_{i}(a,a+1]=\mathcal{P}(f_{i}(a),f_{i}(a)+1]=\operatorname{Coh}^{f_{i}(a)}(C),$
for $i=1,2$ and they will satisfy the gluing condition if and only if
$f_{1}(a)\geq f_{2}(a)$. ∎

######  Example 3.34.

Consider the semiorthogonal decomposition
$\mathcal{T}_{C}=\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle$. Let
$\sigma_{i}=(Z_{i},\operatorname{Coh}_{i}^{r_{i}}(C))$ be a stability
conditions on $\mathcal{C}_{i}$ with $r_{i}\in\mathbb{R}$, for $i=1,2$.

  1. (1)

If $r_{1}>r_{2}$, then $\operatorname{gl}_{12}(\sigma_{1},\sigma_{2})$ is a
locally finite pre-stability condition on $\mathcal{T}_{C}$

  2. (2)

The CP-glued pair $\sigma=\operatorname{gl}_{12}(\sigma_{1},\sigma_{\mu})$
with $\sigma_{1}=\sigma_{\mu}g$ and
$g=(N_{1},f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$, that we
considered in Example 3.29, does not satisfy the conditions in Theorem 3.30.

See [35, Example 2.9.11] for more details on these and more examples.

For small hearts that fulfill the condition 1 in Theorem 3.30 but not 2, we
may need a different strategy.

######  Proposition 3.35 (HN-property for $\mathbb{Q}$).

Let

$\operatorname{gl}_{12}(\sigma_{1},\sigma_{2})=(Z_{12}=Z_{1}\circ
i^{*}+Z_{2}\circ
j^{!},\mathcal{A}_{12}=\operatorname{gl}_{12}(\mathcal{A}_{1},\mathcal{A}_{2}))$

be a CP-glued pair in $\mathcal{T}_{C}$ obtained by CP-gluing the stability
conditions $\sigma_{i}=(Z_{i},\mathcal{A}_{i})$ on $\mathcal{C}_{i}$ with
$\mathcal{A}_{i}=\operatorname{Coh}_{i}^{\theta_{i}}(C)$ for
$\theta_{i}\in[0,1)$ for $i=1,2$. If $\tan(\pi\theta_{i})\in\mathbb{Q}$, for
all $i$, then $Z_{12}$ has the HN-property.

###### Proof.

It is easy to see that $0$ is an isolated point of $\Im
Z_{i}(\mathcal{A}_{i})\subset\mathbb{R}_{\geq 0}$ for $i=1,2$. Then, it
follows from [17, Proposition 3.5] that $Z_{12}$ has the HN-property on
$\mathcal{A}_{12}$. ∎

######  Corollary 3.36.

Let $\operatorname{gl}_{12}(\sigma_{1},\sigma_{2})=(Z_{12}=Z_{1}\circ
i^{*}+Z_{2}\circ
j^{!},\mathcal{A}_{12}=\operatorname{gl}_{12}(\mathcal{A}_{1},\mathcal{A}_{2}))$
be a CP-glued pair in $\mathcal{T}_{C}$ obtained by CP-gluing the stability
conditions $\sigma_{i}=(Z_{i},\mathcal{A}_{i})$ on $\mathcal{C}_{i}$ with
$\mathcal{A}_{i}=\operatorname{Coh}_{i}^{\theta_{i}}(C)$ for
$\theta_{i}\in[0,1)$ for $i=1,2$, such that $\theta_{1}\geq\theta_{2}$ but it
does not belong to $S(a)$ for any $a\in(0,1)$. If
$\tan(\pi\theta_{i})\in\mathbb{Q}$, for all $i$, then
$\operatorname{gl}_{12}(\sigma_{1},\sigma_{2})$ is a pre-stability condition.

######  Example 3.37 (classic $\alpha$-stability as CP-gluing).

Note that by [18, Theorem 5.3.11], pre-stability conditions on
$\mathcal{T}_{C}$ with heart $\operatorname{TCoh}(C)$ are obtained as CP-
gluing of two stability conditions on $D^{b}(C)$ with heart
$\operatorname{Coh}(C)$ and the stability function is given as

$Z(r_{1},d_{1},r_{2},d_{2})\coloneqq-
A_{1}d_{1}-A_{2}d_{2}+B_{1}r_{1}+B_{2}r_{2}+i(C_{1}r_{1}+C_{2}r_{2})$

where $A_{i},B_{i},C_{i}\in\mathbb{R}$ such that $A_{i},C_{i}>0$, for $i=1,2$.
In particular, we can recover the classic notion of $\alpha$-stability for
holomorphic triples of vector bundles of García-Prada et al. in [21] and [8]
by taking $\sigma_{\alpha}=(Z_{\alpha},\operatorname{TCoh}(C))$ where
$Z_{\alpha}(r_{1},d_{1},r_{2},d_{2})\coloneqq-d_{1}-d_{2}-\alpha
r_{1}+i(r_{1}+r_{2})$ with $\alpha\in\mathbb{R}.$

######  Remark 3.38.

There is a way to construct hearts from semiorthogonal decompositions which
agrees with CP-gluing when the gluing condition (3.5) is satisfied. This is
the well-known recollement introduced in [6]. See Appendix A. and Lemma A.5
for examples of hearts that do not accept a stability function.

####  3.2.2. Constructing pre-stability conditions via tilting

In this section we construct pre-stability conditions in $\mathcal{T}_{C}$
whose hearts are not given by Proposition 3.18. We follow the steps of [12,
Lemma 6.1], i.e. we use weak stability functions on $\operatorname{TCoh}(C)$
to obtain torsion pairs on $\operatorname{TCoh}(C)$ via truncation of the HN-
filtrations. After tilting in the sense of [24, Proposition 2.1], we obtain
hearts that admit Bridgeland stability functions.

######  Remark 3.39.

The intuition of this construction comes from Proposition 4.18. This
proposition gives us a description of torsion pairs of
$\operatorname{TCoh}(C),$ which after tilting will give us a heart of a pre-
stability condition.

We define the following homomorphism:

(3.9) $\displaystyle Z\colon\mathbb{Z}^{4}$ $\displaystyle\rightarrow$
$\displaystyle\mathbb{C}$ $\displaystyle(r_{1},d_{1},r_{2},d_{2})$
$\displaystyle\mapsto$ $\displaystyle
D_{1}d_{1}+(C_{1}-1)r_{1}+i(r_{1}+r_{2}),$

where $D_{1},C_{1}\in\mathbb{Q},$ and $D_{1}<0.$ For every
$E\in\operatorname{TCoh}(C)$ with $E\neq j_{*}(T),$ where $T$ is a torsion
sheaf, we define the phase of $E$ as $\lambda(E)=(1/\pi)\arg(Z([E]))\in(0,1].$

######  Remark 3.40.

Let $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ be a short exact
sequence where $A,B,C\in\operatorname{TCoh}(C)$ and $A,B,C\neq 0\rightarrow
T,$ where $T$ is a torsion sheaf, then

$\lambda(A)<\lambda(B)\iff\lambda(B)<\lambda(C)\textnormal{ and
}\lambda(A)>\lambda(B)\iff\lambda(B)>\lambda(C).$

For a triple $E=E_{1}\xrightarrow{\varphi}E_{2}\in\operatorname{TCoh}(C),$ let
$T(E_{i})$ be the torsion part and $F(E_{i})$ be the torsion-free part of of
$E_{i}$ for $i=1,2.$ By the functoriality of the torsion part, we obtain
$T(E)=T(E_{1})\rightarrow T(E_{2})$ and the following short exact sequence

(3.10) $0\rightarrow T(E)\rightarrow E\rightarrow F(E)\rightarrow 0,$

where $F(E)=F(E_{1})\rightarrow F(E_{2})=E/T(E).$

A triple $E=E_{1}\xrightarrow{\varphi}E_{2}\in\operatorname{TCoh}(C)$ is
called _torsion-free_ if $T(E_{i})=0,$ for $i=1,2.$ We define the _torsion-
free triple_ of $E=E_{1}\xrightarrow{\varphi}E_{2}\textnormal{ as
}F(E)=F(E_{1})\xrightarrow{f}F(E_{2}).$

######  Definition 3.41.

A torsion-free triple $E\in\operatorname{TCoh}(C)$ is called
$\lambda$-semistable if for all non-zero subobjects $F\subseteq E$ we have
$\lambda(F)\leq\lambda(E).$

We will now show that the $\lambda$-semistable objects admit HN-filtrations.
The proof goes along the lines of the classical proof for $\mu$-stability on
curves.

######  Lemma 3.42 (HN-filtration for $\lambda$-stability.).

Let $F=F_{1}\xrightarrow{\varphi}F_{2}\in\operatorname{TCoh}(C)$ be a torsion-
free object, then there is a unique Harder–Narasimhan filtration i.e. there is
an increasing filtration $0\subseteq E_{1}\cdots\subseteq E_{n-1}\subseteq
E_{n}=F$ where $G_{i}=E_{i}/E_{i-1}$ is $\lambda$-semistable for each
$i=0,\ldots,n$ and $\lambda(G_{1})>\cdots>\lambda(G_{n-1})>\lambda(G_{n}).$
Moreover, this filtration is unique.

######  Proof 1.

If $F$ is $\lambda$-semistable, there is nothing to prove. Let us consider the
subobjects $E$ of $F.$ Note that $E$ is also torsion-free. Take the object
$E_{1}$ with maximal $\lambda(E_{1})$ among all the subobjects of $F$ and with
maximal imaginary part among all the subobjects of $F$ with maximal
$\lambda$-phase. This object exists because the phase is bounded. Indeed, by
the correspondence between slope and phase, it follows from the fact that
$-D>0$ and by Riemann–Roch. As a consequence, the subobject $E_{1}$ is
necessarily $\lambda$-semistable and $F/E_{1}$ is torsion free. We also have
that for all $E$ with $0\neq E/E_{1}\subseteq F/E_{1},$ we get
$\lambda(E/E_{1})<\lambda(E_{1}).$ We now apply the same construction to
$F/E_{1}.$ We get the desired filtration.

######  Lemma 3.43.

Let $\phi=3/4.$ There is a torsion pair $(\mathcal{T},\mathcal{F})$ on the
category $\operatorname{TCoh}(C)$ defined as follows: $E\in\mathcal{T}$ if the
Harder–Narasimhan $\lambda$-semistable factors $A_{i}$ of $F(E)$ satisfy
$\lambda(A_{i})>\phi$ and $i^{!}(E)\in\operatorname{Coh}(C).$ We say that
$E\in\mathcal{F}$ if either $i^{*}(E)$ is torsion-free and the
Harder–Narasimhan factors $A_{i}$ of $F(E)$ satisfy $\lambda(A_{i})\leq\phi$
or $i^{*}(E)=0.$

######  Proof 2.

Note that if $E\in\mathcal{T},$ by our definition of $\mathcal{T}$ and the
correspondence between slope and phase, we have that $F(E)$ satisfies

(3.11)
$\frac{-D_{1}d_{1}-(C_{1}-1)r_{1}}{r_{1}+r_{2}}>-\cot(3\pi/4)\textnormal{
i.e.\ }-D_{1}d_{1}-C_{1}r_{1}-r_{2}>0.$

where here $d_{i}=\deg(F(E_{i}))$ and $r_{i}=\operatorname{rank}(F(E_{i})),$
for $i=1$, $2.$

We show that $(\mathcal{T},\mathcal{F})$ is a torsion pair of
$\operatorname{TCoh}(C).$ First, we prove that
$\operatorname{Hom}_{\mathcal{T}_{C}}(\mathcal{T},\mathcal{F})=0.$ By our
definition of stability we have that
$\operatorname{Hom}_{\mathcal{T}_{C}}(E,F)=0,$ for all objects
$E\in\mathcal{T}$ and $F\in\mathcal{F}$ that are torsion-free.

Let $E=E_{1}\xrightarrow{\varphi}E_{2}\in\mathcal{T}$ and $G=G_{1}\rightarrow
G_{2}\in\mathcal{F}.$ Let us consider the following short exact sequences as
in the triangle (3.10)

$0\rightarrow T(E)\rightarrow E\rightarrow F(E)\rightarrow 0\textnormal{ and
}0\rightarrow T(G)\rightarrow G\rightarrow F(G)\rightarrow 0.$

By definition of $\mathcal{F}$, we have that $F(G)\in\mathcal{F}$ and as
$i^{*}(G)$ is torsion-free, we get $i^{*}(T(G))=0$ and $T(G)\in\mathcal{F}.$
Then it is enough to show $\operatorname{Hom}_{\operatorname{TCoh}(C)}(E,G)=0$
for $G=0\rightarrow H,$ for any $H\in\operatorname{Coh}(C)$ and
$G=G_{1}\rightarrow G_{2}\in\mathcal{F}$ where $G_{1},G_{2}$ are torsion-free.

Case 1: $G\in\mathcal{F}$ a torsion-free triple.

By definition $F(E)\in\mathcal{T}$ and by stability we have
$\operatorname{Hom}_{\operatorname{TCoh}(C)}(F(E),G)=0.$ Also
$\operatorname{Hom}_{\operatorname{TCoh}(C)}(T(E),G)=0$ as $G$ is torsion-
free. Therefore, it follows that
$\operatorname{Hom}_{\operatorname{TCoh}(C)}(E,G)=0.$

Case 2: $G=0\rightarrow H.$ We have
$\operatorname{Hom}_{\mathcal{T}_{C}}(E,G)=\operatorname{Hom}_{\mathcal{T}_{C}}(E,j_{*}(j^{!}(G))),$
by adjointness

$\operatorname{Hom}_{\mathcal{T}_{C}}(E,j_{*}(j^{!}(G)))=\operatorname{Hom}_{D^{b}(C)}(j^{*}(E),j^{!}(G))=\operatorname{Hom}_{D^{b}(C)}(\operatorname{Ker}(\varphi)[1],H)=0,$

because
$j^{*}(E)[1]=i^{!}(E)=\operatorname{Ker}(\varphi)\oplus\operatorname{Coker}(\varphi)[1]$
is in $\operatorname{Coh}(C),$ which implies that
$\operatorname{Coker}(\varphi)=0.$

Note that for a torsion-free object $E,$ by Lemma 3.42, there is a short exact
sequence $0\rightarrow T\rightarrow E\rightarrow F\rightarrow 0,$ with $T$
torsion-free such that the HN-factors $A_{i}$ of $T$ satisfy that
$\lambda(A_{i})>\frac{3}{4}$ and
$F=F_{1}\xrightarrow{f^{\prime}}F_{2}\in\mathcal{F}$ also torsion-free. Note
that $T=T_{1}\xrightarrow{t^{\prime}}T_{2}$ is not necessarily surjective,
however we have the following claim.

######  Claim 3.44.

Either $\operatorname{Coker}(t^{\prime})=0$ or
$\operatorname{Coker}(t^{\prime})$ is a torsion sheaf.

######  Proof 3.

Because of Lemma 3.42, it is enough to show the statement for a
$\lambda$-semistable object $T$ satisfying $\lambda(T)>\frac{3}{4}.$ Let us
assume that $\operatorname{Coker}(t^{\prime})\neq 0.$ Note that
$t^{\prime}\neq 0.$ Then
$\operatorname{Img}(t^{\prime}),\operatorname{Coker}(t^{\prime})\neq 0,$ and
as a consequence, we have

(3.12)
$\textstyle{T_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\prime}}$$\textstyle{T_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{T_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Coker}(t^{\prime})}$
and
$\textstyle{T_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\prime}}$$\textstyle{T_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\prime}}$$\textstyle{\operatorname{Img}(t^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{2}.}$

If $\operatorname{rk}(\operatorname{Coker}(t^{\prime}))>0,$ then by
$\lambda$-semistability of $T,$ we obtain
$\lambda(T)=\frac{1}{2}<\frac{3}{4},$ which gives us a contradiction.

We obtain the following short exact sequence

(3.13)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\prime}}$$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{F_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Img}(t^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{2}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Note that $T^{\prime}\coloneqq
T_{1}\xrightarrow{t^{\prime}}\operatorname{Img}(t^{\prime})$ is in
$\mathcal{T}.$ Indeed, let us consider the last short exact sequence in its
HN-decomposition $0\rightarrow E\rightarrow T^{\prime}\twoheadrightarrow
A\rightarrow 0$ with $A=A_{1}\rightarrow A_{2}$ a $\lambda$-semistable torsion
free sheaf. We want to show that $\lambda(A)>\frac{3}{4}.$ Note that $E$ is
also a subobject of $T,$ as a consequence we consider the short exact sequence
$0\rightarrow E\rightarrow T\rightarrow T/E\rightarrow 0.$ We have that $T/E$
is a quotient of $T$ and therefore $\lambda(F(T/E))>\frac{3}{4}.$ We also have
that $\frac{3}{4}<\lambda(F(T/E))=\lambda(A).$ Moreover, we get that $F$ is
the torsion-free part of $F_{1}\xrightarrow{f^{\prime}}F_{2}^{\prime}$ and
this implies that $F_{1}\xrightarrow{f^{\prime}}F_{2}^{\prime}\in\mathcal{F}.$
Therefore, if $E_{1}\rightarrow E_{2}$ is torsion-free, the triangle (3.13)
gives us the decomposition of $E$ in $(\mathcal{T},\mathcal{F}).$

Let $E=E_{1}\xrightarrow{\varphi}E_{2}\in\operatorname{TCoh}(C).$ Let us,
again, consider the short exact sequence

(3.14)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T(E_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{F(E_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T(E_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F(E_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Since $F(E)$ is torsion-free, as mentioned before there is a short exact
sequence

$0\rightarrow T^{{}^{\prime}}\rightarrow F(E)\rightarrow F^{\prime}\rightarrow
0,$

with $T^{\prime}\in\mathcal{T}$ and $F^{{}^{\prime}}\in\mathcal{F}.$
Explicitly

(3.15)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T^{\prime}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l_{1}}$$\scriptstyle{t^{\prime}}$$\textstyle{F(E_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\scriptstyle{f}$$\textstyle{F^{\prime}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T^{\prime}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l_{2}}$$\textstyle{F(E_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}}$$\textstyle{F^{\prime}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

After choosing a splitting $E_{i}=F(E_{i})\oplus T(E_{i}),$ for $i=1,2,$ we
have a morphism

$l\colon F(E_{1})\hookrightarrow E_{1}\xrightarrow{\varphi}E_{2}\rightarrow
T(E_{2}).$

We now define the following morphism: $\beta_{1}\colon
T^{{}^{\prime}}_{1}\oplus T(E_{1})\xrightarrow{\begin{bmatrix}l_{1}&0\\\
0&\operatorname{id}\\\ \end{bmatrix}}F(E_{1})\oplus T(E_{1}).$

We obtain the following short exact sequence:

(3.16)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{1}^{\prime}\oplus
T(E_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{1}}$$\scriptstyle{\varphi\circ\beta_{1}}$$\textstyle{F(E_{1})\oplus
T(E_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(g_{1},0)}$$\scriptstyle{\varphi}$$\textstyle{F^{{}^{\prime}}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l^{\prime}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Img}(\varphi\circ\beta_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{F(E_{2})\oplus
T(E_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Note that $\varphi(x,y)=(f(x),l(x)+t(y)),$ for $(x,y)\in F(E_{1})\oplus
T(E_{1}).$

We claim that $T^{\prime}_{1}\oplus
T(E_{1})\xrightarrow{\varphi\circ\beta_{1}}\operatorname{Img}(\varphi\circ\beta_{1})\in\mathcal{T}\textnormal{
and }F_{1}^{\prime}\xrightarrow{l^{\prime}}G\in\mathcal{F}.$ Indeed, note that
we have the following decomposition:

(3.17)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T(E_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\textstyle{T^{\prime}_{1}\oplus
T(E_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi\circ\beta_{1}}$$\textstyle{T^{\prime}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\prime}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}(\pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Img}(\varphi\circ\beta_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{T^{\prime}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Note that, as seen from triangle (3.15), we have that
$\pi\colon\operatorname{Img}(\varphi\circ\beta_{1})\subseteq
T^{\prime}_{2}\oplus T(E_{2})\rightarrow T^{\prime}_{2}$ is just the
projection. Note also that we abused the notation by ignoring the inclusion
$l_{2}.$ As
$\operatorname{Ker}(\pi)=\operatorname{Img}(\varphi\circ\beta_{1}\mid_{\operatorname{Ker}{t^{\prime}}\oplus
T(E_{1})})$ is given by the points
$(0,x)\in\operatorname{Img}(\varphi\circ\beta_{1})\subseteq
T^{\prime}_{2}\oplus T(E_{2})$ and we can see it as a subset of $\subseteq
T(E)_{2}.$ Thus, $\operatorname{Ker}(\pi)$ is a torsion sheaf and by
definition $T^{\prime}_{2}$ is torsion-free. As a consequence, the torsion-
free part of $T^{\prime}_{1}\oplus
T(E_{1})\xrightarrow{\varphi\circ\beta_{1}}\operatorname{Img}(\varphi\circ\beta_{1})\in\mathcal{T}$
is $T^{\prime}_{1}\xrightarrow{t^{\prime}}T^{\prime}_{2}.$

Analogously, we have the following decomposition:

(3.18)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{\prime}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l^{\prime}}$$\textstyle{F^{\prime}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}([g_{2},0])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[g\circ
g_{2},0]}$$\textstyle{F^{\prime}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

We first check that $[g_{2},0]$ is well-defined. As $G=F(E_{2})\oplus
T(E_{2})/\operatorname{Img}(\varphi\circ\beta_{1}),$ then if $(x,y)$ in
$\operatorname{Img}(\varphi\circ\beta_{1}),$ there are
$(x^{\prime},y^{\prime})\in T^{\prime}_{1}\oplus T(E_{1}),$ which satisfy
$(f(x^{\prime}),l(x^{\prime})+t(y^{\prime}))=(x,y).$ Note that
$f(x^{\prime})\in T^{\prime}_{2},$ therefore $g_{2}(f(x^{\prime}))=0.$ Since
$g_{2}$ is surjective, we have that $[g_{2},0]$ is clearly surjective.

Note that $\operatorname{Ker}([g\circ g_{2},0])=T_{2}\oplus
T(E_{2})/\operatorname{Img}(\varphi\circ\beta_{1}).$ As a consequence,
$\operatorname{Ker}([g_{2}])$ is a torsion sheaf. We now obtain that the
torsion-free part of $F^{\prime}_{1}\xrightarrow{l^{\prime}}G$ is the same as
the one of $F^{\prime}_{1}\rightarrow F^{\prime}_{2}$ and this implies that
$F^{\prime}_{1}\xrightarrow{l^{\prime}}G\in\mathcal{F}.$

After tilting $\operatorname{TCoh}(C)$ with respect to the torsion pair of
Lemma 3.43 as in [24, Proposition 2.1], we obtain the heart

$\mathcal{A}_{r}=\\{E\in\mathcal{T}_{C}\mid H^{i}(E)=0\textnormal{ for }i\neq
1,0,H^{1}(E)\in\mathcal{T}\textnormal{ and }H^{0}(E)\in\mathcal{F}\\},$

where $r=\frac{\arg(C_{1}+D_{1}i)}{\pi}\in(-1,0].$ It has a corresponding
torsion pair $\mathcal{A}_{r}=(\mathcal{F},\mathcal{T}[-1]).$

We will now define a stability function $Z_{r}$ on $\mathcal{A}_{r}$ such that
the pair $(Z_{r},\mathcal{A}_{r})$ is a Bridgeland stability condition.

######  Remark 3.45.

If we have $\sigma=\operatorname{gl}_{12}(\sigma_{1},\sigma_{2})$ a pre-
stability condition and we act by $\sigma_{2}^{-1}=g_{2}$ in
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ and obtain a non CP-glued pre-
stability condition, then the heart of $\sigma g_{2}$ is going to be given by
$\mathcal{A}_{r}$ (see Subsection 4.1).

We now define

(3.19) $\displaystyle\noindent Z_{r}\colon\mathbb{Z}^{4}$
$\displaystyle\rightarrow$ $\displaystyle\mathbb{C}$
$\displaystyle(r_{1},d_{1},r_{2},d_{2})$ $\displaystyle\mapsto$ $\displaystyle
A_{1}d_{1}+B_{1}r_{1}-d_{2}+i(D_{1}d_{1}+r_{1}C_{1}+r_{2}),$

where $M=\begin{bmatrix}-A_{1}&B_{1}\\\ -D_{1}&C_{1}\\\ \end{bmatrix}$ with
$\det(M)>0,$ $\det(M+I)>0,$ $A_{1},B_{1}\in\mathbb{R},$
$C_{1},D_{1}\in\mathbb{Q}$ and $D_{1}<0.$

######  Remark 3.46.

Note that if $E\in\operatorname{TCoh}(C)$ is a $\lambda$-semistable torsion-
free triple, then $\lambda(E)>\frac{3}{4}$ if and only if $\Im(Z_{r}(E))<0.$
Moreover, $\lambda(E)\leq\frac{3}{4}$ if and only if $\Im(Z_{r}(E))\geq 0.$

If we consider the same $C_{1},D_{1}$ as above for the construction of
$\mathcal{A}_{r},$ we obtain the following lemma.

######  Lemma 3.47.

The group homomorphism $Z_{r}$ is a stability function on $\mathcal{A}_{r}$

######  Proof 4.

First of all, we show that the image of $E\in\mathcal{A}_{r}$ under $Z_{r}$
lies in $\mathbb{H}\cup\mathbb{R}_{<0}.$ Let $E\in\mathcal{T},$ then we
consider the short exact sequence $0\rightarrow T(E)\rightarrow E\rightarrow
F(E)\rightarrow 0.$ Note that by definition $F(E)\in\mathcal{T}$ and the right
exactness of $\operatorname{Coker}(-)$ we obtain that
$i^{!}(F(E))\in\operatorname{Coh}(C).$

We prove now that $Z_{r}(E[-1])\in\mathbb{H}\cup\mathbb{R}_{<0}.$ We first
show that $Z_{r}(F(E)[-1])\in\mathbb{H}\cup\mathbb{R}_{<0}.$ It is enough to
assume that $F(E)$ is $\lambda$-semistable and this follows directly from
Remark 3.46.

As $\operatorname{rk}(T(E_{1}))=\operatorname{rk}(T(E_{2}))=0,$ if
$T(E_{1})\neq 0,$ then $\deg(T(E_{1}))\geq 0$ and $\Im(Z_{r}((T(E)[-1]))$ is
precisely $-\deg(T(E_{1}))D_{1}>0.$ If $\deg(T(E_{1}))=0,$ then $T(E_{1})=0,$
and $F(E_{1})\cong E_{1}.$ This implies $\Im(Z_{r}(F(E)[-1]))=\Im
Z_{r}(E[-1])>0.$ Since $Z$ is additive with respect to short exact sequences,
we obtain that $Z(E[-1])\in\mathbb{H}\cup\mathbb{R}_{<0}.$

We will now show that if $E\in\mathcal{F},$ then
$Z_{r}(E)\in\mathbb{H}\cup\mathbb{R}_{<0}.$ Once again we consider the short
exact sequence $0\rightarrow T(E)\rightarrow E\rightarrow F(E)\rightarrow 0.$
As $i^{*}(E)$ is a torsion-free sheaf $T(E)=0\rightarrow T(E_{2}),$ where
$T(E_{2})$ is a torsion sheaf. Note that by definition $T(E)\in\mathcal{F}$
and $F(E)\in\mathcal{F}.$ Clearly $Z_{r}(T(E))=-\deg(T(E_{2}))<0,$ as
$T(E_{2})$ is a torsion-sheaf.

Let $F(E)\in\mathcal{F}$ be a torsion-free $\lambda-$semistable object. If
$F(E)$ satisfies $\lambda(F(E))<3/4,$ then $-D_{1}d_{1}-C_{1}r_{1}-r_{2}<0$
and $Z(E)$ lies in the upper-half plane.

We now assume that $F(E)=F(E_{1})\xrightarrow{f}F(E_{2})$ in $\mathcal{F}$ is
a torsion-free object with $D_{1}d_{1}+r_{1}C_{1}+r_{2}=0.$ It suffices to
show our statement for $F(E)$ a $\lambda$-semistable object. We will now prove
that $A_{1}d_{1}+B_{1}r_{1}-d_{2}<0.$

First note that if $F(E_{1})=0,$ then $0=D_{1}d_{1}+r_{1}C_{1}+r_{2}=r_{2}$
and this implies $F(E)=0.$ If $F(E_{2})=0,$ then $D_{1}d_{1}+r_{1}C_{1}=0,$
and $A_{1}d_{1}+B_{1}r_{1}-d_{2}=A_{1}d_{1}+B_{1}r_{1}<0,$ because
$\det(M)>0.$ Therefore, we assume $F(E_{1})$ and $F(E_{2})\neq 0.$ Moreover,
note that $\operatorname{rk}(\operatorname{Coker}(f))=0.$ Indeed, otherwise
$j_{*}(\operatorname{Coker}(f))$ would be a quotient of $F(E)$ with slope
$\frac{1}{2},$ which gives us a contradiction.

Note that $\lambda(F(E_{1})\rightarrow\operatorname{Img}(f))=\lambda(F(E)),$
and moreover
$[F(E)]=[F(E_{1})\rightarrow\operatorname{Img}(f)]+(0,0,0,d_{2}^{\prime\prime}),$
where $d_{2}^{\prime\prime}=\deg(\operatorname{Coker}(f))>0,$
$d^{\prime\prime}_{1}=\deg(\operatorname{Img}(f))$ and $d_{2}=\deg(F(E_{2})).$
As

$A_{1}d_{1}+B_{1}r_{1}-d_{2}=A_{1}d_{1}+B_{1}r_{1}-d^{\prime\prime}_{1}-d_{2}^{\prime\prime}<A_{1}d_{1}+B_{1}r_{1}-d^{\prime\prime}_{1}$

and $F(E_{1})\rightarrow\operatorname{Img}(f)\in\mathcal{F},$ since it is a
subobject of $F(E),$ it is enough to show our statement for torsion-free
objects $F(E_{1})\xrightarrow{f}F(E_{2})\in\mathcal{F}$ with
$\operatorname{Coker}(f)=0.$ As a consequence, we assume
$r_{1}=\operatorname{rk}(F(E_{1}))\geq\operatorname{rk}(F(E_{2}))=r_{2}.$

Since $K=i_{*}(i^{!}(F(E)))=\operatorname{Ker}(f)\rightarrow 0$ is a subobject
of $F(E)$ in $\operatorname{TCoh}(C)$ and $\mathcal{F}$ is closed under
subobjects, we obtain that $K\in\mathcal{F}.$ Since
$[i_{*}(i^{!}(E))]=[K]=(r_{1}-r_{2},d_{1}-d_{2},0,0),$ where
$d_{i}=\deg(F(E_{i}))$ for $i=1,2,$ it follows that
$-D_{1}(d_{1}-d_{2})-C_{1}(r_{1}-r_{2})\leq 0.$ By hypothesis
$D_{1}d_{1}+r_{1}C_{1}+r_{2}=0.$ Therefore we obtain
$D_{1}d_{2}\leq-(C_{1}+1)r_{2}.$

Now we want to show that $A_{1}d_{1}+B_{1}r_{1}-d_{2}<0.$ First note that
$d_{1}=\frac{r_{2}+C_{1}r_{1}}{-D_{1}},$ we obtain

$\displaystyle A_{1}d_{1}+B_{1}r_{1}-d_{2}$ $\displaystyle=$
$\displaystyle(\frac{1}{-D_{1}})(A_{1}r_{2}+r_{1}A_{1}C_{1}-r_{1}B_{1}D_{1}+D_{1}d_{2}).$

Since $-D_{1}>0,$ it is enough to show that
$A_{1}r_{2}+r_{1}A_{1}C_{1}-r_{1}B_{1}D_{1}+D_{1}d_{2}<0.$ We have

(3.20) $\displaystyle A_{1}r_{2}+r_{1}A_{1}C_{1}-r_{1}B_{1}D_{1}+D_{1}d_{2}$
$\displaystyle{\leq}$ $\displaystyle r_{2}(A_{1}-C_{1}-1)-r_{1}(\det(M))$
$\displaystyle\leq$ $\displaystyle r_{2}(A_{1}-C_{1}-1)-r_{2}(\det(M))$
$\displaystyle=$ $\displaystyle r_{2}(-\det(M+I))<0,$

as $-r_{1}\leq-r_{2}$ and also $\det(M)>0$ and $\det(M+I)>0.$ Since $Z_{r}$ is
additive with respect to short exact sequences we obtain that
$Z_{r}(E)\in\mathbb{H}\cup\mathbb{R}_{<0}$ for $E\in\mathcal{F}.$

######  Lemma 3.48.

If $A_{1},B_{1},C_{1},D_{1}\in\mathbb{Q},$ the pair $(Z_{r},\mathcal{A}_{r})$
is a pre-stability condition.

######  Proof 5.

We follow the steps of [12, Proposition 7.1]. First note that by [3,
Proposition B.2], it is enough to show that if $E\in\mathcal{A}_{r}$ and
$0\subset L_{1}\subset L_{2}\ldots\subset L_{i}\subset\ldots\subset E,$ where
$L_{i}$ belongs to the full subcategory $\mathcal{P}^{\prime}(1)$ of objects
with phase one, the sequence stabilizes. As
$L_{i}\in\mathcal{A}_{r}=(\mathcal{F},\mathcal{T}[-1]),$ we consider the short
exact sequence $0\rightarrow F_{i}\rightarrow L_{i}\rightarrow
T_{i}[-1]\rightarrow 0,$ where $F_{i}\in\mathcal{F}$ and
$T_{i}\in\mathcal{T}.$ As $\Im Z_{r}(L_{i})=0,$ we obtain that $\Im
Z_{r}(T_{i}[-1])=0.$ Note that this implies that $T_{i}=0.$ Therefore, we
obtain that
$\mathcal{P}^{\prime}(1)\subseteq\mathcal{F}\subseteq\operatorname{TCoh}(C),$
as $\operatorname{TCoh}(C)$ is noetherian, our result follows.

######  Remark 3.49.

  1. (1)

By Lemma 2.37 the pre-stability condition $\sigma_{r}=(Z_{r},\mathcal{A}_{r})$
is locally finite.

  2. (2)

Let $\sigma=(Z_{r},\mathcal{A}_{r})$ be a pre-stability condition constructed
in Lemma 3.48. Note that $i_{*}(\mathbb{C}(x))[-1],$
$l_{*}(\mathbb{C}(x))[-1]$ and $j_{*}(\mathbb{C}(x))$ are in $\mathcal{A}_{r}$
and $j_{*}(\mathbb{C}(x))$ is stable of phase one.

  3. (3)

We consider the torsion pair
$(\mathcal{T}_{1},\mathcal{F}_{1})=\operatorname{Coh}(C),$ which is given by
truncating the Harder–Narasimhan filtration with respect to
$\sigma_{1}=(Z(r,d)=D_{1}d+(C_{1}-1)r+ir,\operatorname{Coh}(C))$ at
$\frac{3}{4}$ in $\operatorname{Stab}(C).$

######  Lemma 3.50.

We have that $\operatorname{Coh}_{2}(C)\subseteq\mathcal{A}_{r}\textnormal{
and }\operatorname{Coh}_{1}^{r}(C)\subseteq\mathcal{A}_{r}\textnormal{ and
}\operatorname{Coh}_{3}^{r_{3}}(C)\subseteq\mathcal{A}_{r},$ where
$r_{3}=\frac{\textnormal{acot}(\frac{C_{1}+1}{D_{1}})}{\pi}.$

######  Proof 6.

Note that
$\operatorname{Coh}_{2}(C)\subseteq\mathcal{F}\subseteq\mathcal{A}_{r}.$ We
consider the torsion pair
$\operatorname{Coh}(C)=(\mathcal{T}_{1},\mathcal{F}_{1})$ of Remark 3.49,
which also induces the heart $\operatorname{Coh}^{r}_{1}(C)$ after a right
tilt, i.e.
$\operatorname{Coh}^{r}_{1}(C)=(\mathcal{F}_{1},\mathcal{T}_{1}[-1]).$ Let
$E\in\operatorname{Coh}_{1}(C)$ be a $\mu$-semistable object. If
$E=i_{*}(\mathbb{C}(x)),$ then by Remark 3.49, we have
$E[-1]\in\mathcal{A}_{r}.$ We assume that $E$ is torsion free. As the only
possible subobjects or quotients of $E$ are in $\operatorname{Coh}_{1}(C),$ we
have that $E$ is $\mu$-semistable if and only if it is $\lambda$-semistable.
It follows directly from the definition of $\mathcal{T}$ and $\mathcal{F}$
that $i_{*}(\mathcal{T}_{1})\subseteq\mathcal{T}\textnormal{ and
}i_{*}(\mathcal{F}_{1})\subseteq\mathcal{F}.$

We also consider a torsion pair
$\operatorname{Coh}(C)=(\mathcal{T}_{3},\mathcal{F}_{3}),$ as in Remark 2.52,
such that after taking the right tilt,
$\operatorname{Coh}^{r_{3}}(C)=(\mathcal{F}_{3},\mathcal{T}_{3}[-1]).$ Let
$E\in\operatorname{Coh}_{3}(C)$ be a $\mu$-semistable object. We have that
$E=l_{*}(\mathbb{C}(x))\in\mathcal{A}_{r}$ by Remark 3.49. It is easy to see
by computing the slope of the Harder–Narasimhan factors that
$l_{*}(\mathcal{T}_{3})\subseteq\mathcal{T}$ and
$l_{*}(\mathcal{F}_{3})\subseteq\mathcal{F}.$

##  4\. The stability manifold $\operatorname{Stab}(\mathcal{T}_{C})$

Lemma 2.47 is the main tool to prove that $\mathbb{C}(x)$ is a $\sigma$-stable
object for every $\sigma\in\operatorname{Stab}(C)$ and every $x\in C.$ In this
section, we prove the analogous statement for $\mathcal{T}_{C},$ to give a
characterization of every $\sigma\in\operatorname{Stab}(\mathcal{T}_{C})$ in
terms of the skyscraper sheaves. In Subsection 4.1, we follow closely the
steps of [12, Lemma 10.1] to describe the possible hearts appearing in pre-
stability conditions on $\mathcal{T}_{C}.$ We finally prove that every pre-
stability condition on $\mathcal{T}_{C}$ has to be given by one of the already
constructed pairs in Lemma 3.48 or Subsection 3.2.2 i.e. either by CP-gluing
or by tilting.

######  Remark 4.1.

Due to the assumption that $g(C)\geq 1,$ if
$\operatorname{Hom}_{\operatorname{Coh}(C)}(E,A)\neq 0,$ then
$\operatorname{Hom}_{\operatorname{Coh}(C)}(E,A\otimes\omega_{C})\neq 0.$ If
$f$ is a non-zero morphism in
$\operatorname{Hom}_{\operatorname{Coh}(C)}(E,A),$ we denote by $f_{C}$ the
non-zero morphism in
$\operatorname{Hom}_{\operatorname{Coh}(C)}(E,A\otimes\omega_{C})$ associated
to $f.$

######  Lemma 4.2.

If there is a distinguished triangle in $\mathcal{T}_{C}$ of the form

(4.1)
$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{E}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{A}}$$\textstyle{E_{1}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{E}[1]}$$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l}$$\textstyle{E_{2}[1]}$

with $X\in\operatorname{Coh}(C)$ and
$\operatorname{Hom}_{\mathcal{T}_{C}}^{\leq 0}(E,A)=0,$ then
$E_{1},A_{1}\in\operatorname{Coh}(C)$.

######  Proof 7.

First of all note that $l_{2}\coloneqq j^{!}(l)$ is an isomorphism in
$D^{b}(C).$ We now consider the functor
$F\colon\mathcal{T}_{C}\rightarrow\operatorname{Mor}(D^{b}(C))$ defined as
$F(E_{1}\xrightarrow{\varphi_{E}}E_{2})=E_{1}\xrightarrow{[\varphi_{E}]}E_{2}$
at the level of objects. For $\Phi$ in
$\operatorname{Hom}_{\mathcal{T}_{C}}(E,F),$ by 3.6, we get the commutative
square

(4.2)
$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}(\Phi)}$$\scriptstyle{[\varphi_{E}]}$$\textstyle{F_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\varphi_{F}]}$$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{!}(\Phi)}$$\textstyle{F_{2}}$

in $D^{b}(C),$ which is in fact a morphism in $\operatorname{Mor}(D^{b}(C)).$

######  Claim 4.3.

The functor $F\colon\mathcal{T}_{C}\rightarrow\operatorname{Mor}(D^{b}(C))$ is
full.

###### Proof.

Let
$(\varphi_{1},\varphi_{2})\in\operatorname{Hom}_{\operatorname{Mor}(D^{b}(C))}(F(E),F(G)),$
with $E,G\in\mathcal{T}_{C}.$ By definition we have the commutative diagram

(4.3)
$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{[\varphi_{E}]}$$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\varphi_{G}]}$$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{2}}$$\textstyle{G_{2}.}$

Note that we obtain a commutative square in $\mathcal{T}_{C}$ given by

(4.4)
$\textstyle{i_{*}(E_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t_{E}}$$\scriptstyle{\varphi_{1}}$$\textstyle{j_{*}(E_{2})[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{2}}$$\textstyle{i_{*}(G_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t_{F}}$$\textstyle{j_{*}(G_{2})[1].}$

By taking cones of the horizontal arrows and using Remark 3.12, we obtain that
there is a morphism $\Phi\in\operatorname{Hom}_{\mathcal{T}_{C}}(E,F)$ with
$F(\Phi)=(\varphi_{1},\varphi_{2}).$ ∎

By [27, Corollary 3.15], for every $G\in D^{b}(C),$ we have that
$G=\oplus_{i\in\mathbb{Z}}G_{i}[-i]$ with
$G_{i}=H^{i}(G)\in\operatorname{Coh}(C).$ We also have the canonical morphisms
(which come from the genuine chain maps)

$G^{\alpha}[-\alpha]\xrightarrow{i_{\alpha}}G\xrightarrow{\pi_{\beta}}G^{\beta}[-\beta],$

for $\alpha,\beta\in\mathbb{Z}.$ Moreover, note that if we have a morphism
$\psi\colon G_{1}\rightarrow G_{2}$ in $D^{b}(C)$, then
$H^{i}(\psi)[-i]=\pi^{2}_{i}\circ\psi\circ i^{1}_{i}\colon
G^{i}_{1}[-i]\rightarrow G_{2}^{i}[-i].$

For $G=G_{1}\xrightarrow{\varphi_{G}}G_{2}\in\mathcal{T}_{C}$ we get that
$F(G)\in\operatorname{Hom}_{D^{b}(C)}(\oplus_{i}G_{1}^{i}[-i],\oplus_{i}G_{2}^{i}[-i]),$
with $G_{j}^{i}\in\operatorname{Coh}(C),$ for $j=1,2$ and $i\in\mathbb{Z}.$
Due to the fact that $\operatorname{Coh}(C)$ has homological dimension one,
there is a non-zero morphism ${t_{G}}^{i}$ from $G^{i}_{1}[-i]\rightarrow
G_{2}^{i}[-i]\oplus G_{2}^{i-1}[-i+1]$ to
$G_{1}\xrightarrow{\varphi_{G}}G_{2}$ in $\operatorname{Mor}(D^{b}(C)).$ We
construct the morphism

$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{i}\circ[\varphi_{A}]}$$\scriptstyle{[\varphi_{A}]}$$\textstyle{A_{2}^{i}[-i]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{C}[-i]}$$\scriptstyle{\operatorname{id}}$$\textstyle{A^{i}_{2}[-i]\otimes{\omega_{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l[-i]\otimes\omega_{C}}$$\scriptstyle{l[-i]\otimes\omega_{C}}$$\textstyle{E^{i+1}_{2}[-i]\otimes\omega_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{i}_{\mathcal{S}_{\mathcal{T}_{C}}(E)}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
E_{2}\otimes\omega_{C}[1]}$$\scriptstyle{[i_{E}]}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{i}}$$\textstyle{A_{2}^{i}[-i]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{i+1}_{2}[-i]\otimes\omega_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}(\varphi_{E})^{i+1}[-i]\oplus\mathcal{C}(\varphi_{E})^{i}[-i+1]\otimes\omega_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C(\varphi_{E})[1]}$

in $\operatorname{Mor}(D^{b}(C)).$ Note that the morphism
$\operatorname{id}_{C}$ is given by Remark 4.1.

We obtain a morphishm $A\rightarrow\mathcal{S}_{\mathcal{T}_{C}}(E)$ in
$\operatorname{Mor}(D^{b}(C)).$ Since $F$ is full, we have a morphism in
$\operatorname{Hom}_{\mathcal{T}_{C}}(A,\mathcal{S}_{\mathcal{T}_{C}}(E)).$ By
Serre duality, we obtain

$\operatorname{Hom}_{\mathcal{T}_{C}}(A,\mathcal{S}_{\mathcal{T}_{C}}(E))\cong\operatorname{Hom}_{\mathcal{T}_{C}}(E,A)^{*}.$

By hypothesis, this morphism has to be zero. This directly implies that
$\operatorname{id}_{C}[-i]\circ(\pi_{i}\circ[\varphi_{A}])=0,$ because
$l\otimes\omega_{C}$ is an isomorphism. Since the construction does not depend
on the $i\in\mathbb{Z},$ we obtain that the induced morphism
$A^{i}_{1}[-i]\xrightarrow{\operatorname{id}_{C}[-i]\circ\pi_{i}\circ[\varphi_{A}]\circ
i_{i}}A^{i}_{2}[-i]\otimes\omega_{C}$ is zero. Due to the fact that it is
precisely the morphism $(\pi_{i}\circ[\varphi_{A}]\circ i_{i})_{C}[-i]$ in
$\operatorname{Coh}(C)[-i],$ as in Remark 4.1, we obtain that
$A^{i}_{1}\xrightarrow{\pi_{i}\circ[\varphi_{A}]\circ i_{1}}A^{i}_{2}$ is also
zero. The triangle (4.1) in $\mathcal{T}_{C}$ induces a long exact sequence in
cohomology and we get

$\textstyle{A_{1}^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\pi_{i}\circ[\varphi_{A}]\circ
i_{i}}$$\textstyle{E_{1}^{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{i+1}\circ[\varphi_{E}]\circ
i_{i+1}}$$\textstyle{A_{2}^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{E_{2}^{i+1},}$

as the morphism in the component $i$ in $\operatorname{Mor}(D^{b}(C))$ is
precisely the one given by the cohomology. Therefore
$\pi_{i}\circ[\varphi_{E}]\circ i_{1}=0,$ for all $i\neq-1,0.$

This implies that there is a morphism from $E^{i}_{1}[-i]\rightarrow
E^{i-1}_{2}[-i+1]$ to $E_{1}\xrightarrow{\varphi_{E}}E_{2},$ which, moreover,
is a split monomorphism for $i\neq 0,1.$ In addition, it provides a split
monomorphism from $A^{i}_{1}[-i]\rightarrow A^{i-1}_{2}[-i+1]$ to
$A_{1}\xrightarrow{\varphi_{A}}A_{2}$ for $i\neq-1,0.$

We get the isomorphism

$\textstyle{A^{i}_{1}[-i]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{A}}$$\textstyle{E_{1}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{E}[1]}$$\textstyle{E^{i+1}_{1}[-i]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{i-1}_{2}[-i+1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l}$$\textstyle{E_{2}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{i}_{2}[-i+1]}$

in $\operatorname{Mor}(D^{b}(C))$, were the morphism of the first row is the
isomorphism induced by the long exact sequence of cohomology of the triangle
(4.1).

From the isomorphism above, we construct the morphism

$\textstyle{E_{1}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\varphi_{E}]}$$\textstyle{E^{i+1}_{1}[-i]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{A^{i}_{1}[-i]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\varphi_{A}]}$$\textstyle{E_{2}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{i}_{2}[-i+1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{A^{i-1}_{2}[-i+1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}}$

in $\operatorname{Mor}(D^{b}(C)).$ Once again, since the functor $F$ is full,
this needs to be the zero morphism. As a consequence, we obtain that
$A^{i}_{1}[-i],A^{i}_{2}[-i]$ are zero for all $i\neq 0,-1.$ We now study the
remaining object in the long exact sequence of cohomology

(4.5)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{-1}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t_{1}}$$\scriptstyle{0}$$\textstyle{E^{0}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{0}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{t_{2}}$$\textstyle{E^{1}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{-1}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{E^{0}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{0}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{E^{1}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

We will now show that $t_{i}=0,$ for $i=1,2.$ If $t_{1}\neq 0,$ by Remark 4.1,
we obtain a non-zero morphism ${t_{1}}_{C}\colon A^{-1}_{1}\rightarrow
E^{0}_{1}\otimes\omega_{C}.$ By Serre duality

$\operatorname{Hom}_{D^{b}(C)}(A_{1}^{-1},E^{0}_{1}\otimes\omega_{C})\cong\operatorname{Hom}_{D^{b}(C)}(E^{0}_{1}[-1],A_{1}^{-1})$

and there is a non-zero morphism $t^{\prime}_{1}\colon
E^{0}_{1}[-1]\rightarrow A^{-1}_{1}.$ As there is a split monomorphism from
$E_{1}^{0}\rightarrow E^{0}_{2}$ to $E,$ we get the non-zero morphism

$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{E}}$$\textstyle{E^{0}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\prime}_{1}[1]}$$\textstyle{A_{1}^{-1}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{A}}$$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{0}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{A_{2}^{-1}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}}$

in $\operatorname{Mor}(D^{b}(C)).$ As above, this implies that
$t^{\prime}_{1}=0$ and therefore $t_{1}=0.$ We will now prove that $t_{2}=0$.

The triangle $A_{1}\rightarrow A_{2}\rightarrow C(\varphi_{A})$ induces a long
exact sequence of cohomology. It follows that $H^{i}(C(\varphi_{A}))=0$ unless
$i=-1,$ i.e. $C(\varphi_{A})\in\operatorname{Coh}(C)[1].$

Note that $A_{2}[-1]\rightarrow C(\varphi_{A})[-1]\rightarrow A_{1}$ is a
short exact sequence in $\operatorname{Coh}(C),$ then the morphism
$C(\varphi_{A})[-1]\xrightarrow{\pi_{A}}A_{1}$ is surjective in
$\operatorname{Coh}(C).$

Let us assume that $t_{2}\neq 0.$ By Remark 4.1, there is a non-zero morphism
$t^{\prime}_{2}\colon A_{1}\rightarrow E^{1}_{1}\otimes\omega_{C}.$ We now
consider $t_{3}\coloneqq t^{\prime}_{2}\circ\pi_{A}$ in
$\operatorname{Coh}(C).$ Since $\pi_{A}$ is surjective and $t^{\prime}_{2}\neq
0,$ then $t_{3}\neq 0.$

This induces the morphism $\Phi\colon
A\rightarrow\mathcal{S}_{\mathcal{T}_{C}}(E)$ given by

$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\prime}_{2}}$$\scriptstyle{\varphi_{A}}$$\textstyle{E^{1}_{1}\otimes\omega_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{E_{1}[1]\otimes\omega_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{E}\otimes\operatorname{id}}$$\textstyle{E_{2}[1]\otimes\omega_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}(\varphi_{E})[1]\otimes\omega_{C}}$

in $\operatorname{Hom}_{\mathcal{T}_{C}}(A,\mathcal{S}_{\mathcal{T}_{C}}(E)).$
Note that $i$ is a split monomorphism. By Serre duality,

$\operatorname{Hom}_{\mathcal{T}_{C}}(A,\mathcal{S}_{\mathcal{T}_{C}}(E))\cong\operatorname{Hom}_{\mathcal{T}_{C}}(E,A)^{*}=0.$

Therefore, we have that $\Phi=0.$ After applying $i^{!},$ we obtain

$i^{!}(\Phi)=\mathcal{C}(\varphi_{A})[-1]\xrightarrow{t_{3}}E^{1}_{1}\otimes\omega_{C}\xrightarrow{i}E_{1}[1]\otimes\omega_{C},$

which has to be zero. As consequence $t_{3}=0$ and we have a contradiction.
Hence, we get $A^{-1}_{1}=E^{1}_{1}=0,$ which implies that
$A_{1},E_{1}\in\operatorname{Coh}(C).$

######  Remark 4.4.

From now on we assume that all pre-stability conditions on $\mathcal{T}_{C}$
that we are considering are locally finite.

######  Proposition 4.5.

Let $X$ be either a skyscraper sheaf $\mathbb{C}(x)$ or a line bundle
$\mathcal{L}$ on $C$. For any pre-stability condition $\sigma$ on
$\mathcal{T}_{C}$, if $i_{*}(X)$ is not $\sigma$-semistable, then $j_{*}(X)$
and $l_{*}(X)$ are $\sigma$-stable.

######  Proof 8.

We assume that $i_{*}(X)$ is not $\sigma$-semistable. Therefore, we consider
the last triangle of its Harder–Narasimhan filtration $E\rightarrow
i_{*}(X)\rightarrow A\rightarrow E[1]$ with $\operatorname{Hom}^{\leq
0}_{\mathcal{T}_{\mathcal{C}}}(E,A)=0$ and $A\in\mathcal{T}_{C}$ semistable.
By Lemma 4.2, we have $E_{1},A_{1}\in\operatorname{Coh}(C),$ thus we obtain a
short exact sequence $0\rightarrow E_{1}\rightarrow X\rightarrow
A_{1}\rightarrow 0,$ in $\operatorname{Coh}(C)$ which is not possible. Hence,
either $E_{1}=0$ or $A_{1}=0.$ If $E_{1}=0,$ we obtain a contradiction. As a
consequence, $A_{1}=0$ and $E_{1}\cong X.$ By adjointness we have

$\operatorname{Hom}^{\leq 0}_{\mathcal{T}_{C}}(E,A)=\operatorname{Hom}^{\leq
0}_{\mathcal{T}_{C}}(E,j_{*}(A_{2}))=\operatorname{Hom}^{\leq
0}_{D^{b}(C)}(C(\varphi_{E}),A_{2})=0.$

Exactness of the functor $j^{*}$ yields the triangle

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
16.3193pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{C(\varphi_{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
40.3193pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
40.3193pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
89.9443pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
89.9443pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
130.24432pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
130.24432pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{C(\varphi_{E})[1]}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Since $\operatorname{Hom}^{\leq 0}(C(\varphi_{E}),A_{2})=0,$ due to Lemma
2.47, the classical GKR lemma for curves, we get that
$C(\varphi_{E})[-1],A_{2}[-1]\in\operatorname{Coh}(C)$ which once again is not
possible. As $A_{2}$ cannot be zero, we get $C(\varphi_{E})=0$ and $A_{2}\cong
X[1].$ This implies that $A\cong j_{*}(X)[1]$ and that $E\cong
l_{*}(X)\in\mathcal{C}_{3}.$ As a consequence, $j_{*}(X)$ is
$\sigma$-semistable.

We will now show that $l_{*}(X)$ is $\sigma$-semistable. We proceed by
contradiction. If $l_{*}(X)$ is not $\sigma$-semistable, we examine the last
triangle of its HN-filtration and we apply the Serre functor and the
autoequivalence $G_{\omega^{*}_{C}}.$ Arguing as above, we obtain $B\cong
i_{*}(X)$ and $i_{*}(X)$ is $\sigma$-semistable, which contradicts our
assumption. Therefore $l_{*}(X)$ is $\sigma$-semistable.

We prove now that $l_{*}(X)$ and $j_{*}(X)$ are $\sigma$-stable. We start by
proving by contradiction that $l_{*}(X)$ is stable. If $l_{*}(X)$ is not
$\sigma$-stable, we consider its Jordan-Hölder filtration. Note that all its
$\sigma$-stable factors $A_{i}$ have the same phase $\phi.$ We assume that
$\operatorname{Hom}_{\mathcal{T}_{C}}(A_{i_{0}},l_{*}(X))\neq 0$ for a
$\sigma$-stable factor $A_{i_{0}}.$ Therefore by [28, Example 1.6], we have
that all the stable factors of $l_{*}(X)$ are isomorphic to $A_{i_{0}}.$
Hence, $[l_{*}(X)]=n[A_{i_{0}}]$, where $n$ is the number of stable factors.
Since $[l_{*}(\mathbb{C}(x))]=(0,1,0,1)$ and
$[\mathcal{L}]=(1,\deg(\mathcal{L}),1,\deg(\mathcal{L}))$, we must have $n=1$,
i.e. $l_{*}(\mathbb{C}(x))$ and $l_{*}(\mathcal{L})$ are stable.

An analogous proof works for the stability of $j_{*}(X).$ Instead of using the
Serre functor, we use $\mathcal{S}^{-1}.$ Consequently, we obtain that
$j_{*}(\mathbb{C}(x))$ and $j_{*}(\mathcal{L})$ are stable.

######  Remark 4.6.

If $X$ is either $\mathbb{C}(x)$ or $\mathcal{L}$ as above, we use Proposition
4.5 to prove that if $j_{*}(X)$ ($l_{*}(X)$) is not $\sigma$-semistable, then
$i_{*}(X)$ and $l_{*}(X)$ ($j_{*}(X)$ and $i_{*}(X)$) are $\sigma$-stable.
Meaning that if one of the objects $i_{*}(X),j_{*}(X),l_{*}(X)$ is not
$\sigma$-semistable then the other two have to be $\sigma$-stable.

######  Remark 4.7.

If $i_{*}(X)$ is not $\sigma$-semistable, where $X$ is either $\mathbb{C}(x)$
or $\mathcal{L},$ then by Proposition 4.5, we obtain the HN-filtration for
$i_{*}(X).$ It is given precisely by

$l_{*}(X)\rightarrow i_{*}(X)\rightarrow j_{*}(X)[1]\rightarrow l_{*}(X)[1].$

After applying the Serre functor, we obtain the corresponding HN-filtration
for $j_{*}(X)$ and $l_{*}(X).$

Moreover, we define

$\phi^{0}_{x}\coloneqq\phi_{\sigma}(i_{*}(\mathbb{C}(x)))\textnormal{ and
}\phi^{1}_{\mathcal{L}}\coloneqq\phi_{\sigma}(i_{*}(\mathcal{L})),$
$\phi^{2}_{x}\coloneqq\phi_{\sigma}(j_{*}(\mathbb{C}(x)))\textnormal{ and
}\phi^{3}_{\mathcal{L}}\coloneqq\phi_{\sigma}(j_{*}(\mathcal{L})),$
$\phi^{4}_{x}\coloneqq\phi_{\sigma}(l_{*}(\mathbb{C}(x)))\textnormal{ and