# Serre-invariant stability conditions and Ulrich bundles on cubic threefolds

Soheyla Feyzbakhsh and Laura Pertusi

###### Abstract.

We prove a general criterion which ensures that a fractional Calabi–Yau
category of dimension $\leq 2$ admits a unique Serre-invariant stability
condition, up to the action of the universal cover of
$\text{GL}^{+}_{2}(\mathbb{R})$. We apply this result to the Kuznetsov
component $\operatorname{Ku}(X)$ of a cubic threefold $X$. In particular, we
show that all the known stability conditions on $\operatorname{Ku}(X)$ are
invariant with respect to the action of the Serre functor and thus lie in the
same orbit with respect to the action of the universal cover of
$\text{GL}^{+}_{2}(\mathbb{R})$. As an application, we show that the moduli
space of Ulrich bundles of rank $\geq 2$ on $X$ is irreducible, answering a
question asked by Lahoz, Macrì and Stellari in [LMS15].

###### Contents

  1. 1 Introduction
  2. 2 (Weak) Bridgeland stability conditions and tilting
  3. 3 Serre-invariant stability conditions
  4. 4 Kuznetsov component of cubic threefolds
  5. 5 Stability conditions on the Kuznetsov component via conic fibrations
  6. 6 Ulrich bundles

##  1\. Introduction

Let $X$ be a smooth cubic hypersurface in the complex projective space
$\mathbb{P}^{4}$. As first noted by Kuznetsov [Kuz04], the birational geometry
of $X$ is controlled by a certain full admissible subcategory
$\operatorname{Ku}(X)$ of the bounded derived category
$\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X),$ defined by the semiortogonal
decomposition

$\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)=\langle\operatorname{Ku}(X),\mathcal{O}_{X},\mathcal{O}_{X}(H)\rangle.$

We call $\operatorname{Ku}(X)$ the Kuznetsov component of $X$. It has been
shown that $\operatorname{Ku}(X)$ completely determines the isomorphism class
of the cubic threefold $X$ [BMMS12, BBF+20]. The proof involves the theory of
stability conditions for complexes in the derived category as introduced by
Bridgeland [Bri07], and the study of moduli spaces of stable objects in
$\operatorname{Ku}(X)$.

Up to now there are two different constructions of stability conditions on
$\operatorname{Ku}(X)$; one in [BMMS12] (see Section 5.2) and one in [BLMS17]
(see Section 4.2). One of the goals of this paper originates from the desire
to understand the connection between these two families of stability
conditions.

######  Main Results.

Let $\mathcal{T}$ be a $\mathbb{C}$-linear triangulated category with Serre
functor $S$. Recall that the space of stability conditions on $\mathcal{T}$
caries a right action of $\widetilde{\text{GL}}^{+}_{2}(\mathbb{R})$ which is
the universal cover of $\text{GL}^{+}_{2}(\mathbb{R})$, see Section 2.1. The
first result of this paper is a general criterion which implies the existence
of a unique orbit of stability conditions which are invariant with respect to
the action of the Serre functor (Definition 3.1).

######  Theorem 1.1 (Theorem 3.2).

Let $\sigma_{1}$ and $\sigma_{2}$ be Serre-invariant stability conditions on a
linear triangulated category $\mathcal{T}$ satisfying the conditions (C1),
(C2) and (C3) in Section 3. Then there exists
$\tilde{g}\in\widetilde{\emph{GL}}^{+}(2,\mathbb{R})$ such that
$\sigma_{1}=\sigma_{2}\cdot\tilde{g}$.

Theorem 1.1 applies for instance to the Kuznetsov component of a cubic
threefold $X$ to show that the known stability conditions on
$\operatorname{Ku}(X)$ are identified in the stability manifold up to the
action of $\widetilde{\text{GL}}^{+}_{2}(\mathbb{R})$, answering a question
asked by Macrì and Stellari (see [PY20, Remark 4.9]).

######  Theorem 1.2 (Corollary 5.7).

Let $X$ be a cubic threefold. The stability condition $\overline{\sigma}$ on
$\operatorname{Ku}(X)$ defined in [LMS15] is in the same orbit of the
stability conditions $\sigma(\alpha,\beta)$ introduced in [BLMS17] with
respect to the $\widetilde{\emph{GL}}^{+}_{2}(\mathbb{R})$ action.

Our second result provides a detailed description of the moduli spaces of
stable objects in $\operatorname{Ku}(X)$ with minimal dimension with respect
to any Serre-invariant stability condition. More precisely, let $M_{X}(v)$ and
$M_{X}(w)$ be the moduli of slope-stable sheaves on $X$ with Chern characters

$v=\left(1,0,-\frac{H^{2}}{3},0\right)\qquad\text{and}\qquad
w=\left(2,-H,-\frac{H^{2}}{6},\frac{H^{3}}{6}\right).$

Note that $v$ is the numerical class of the ideal sheaf $\mathcal{I}_{\ell}$
of a line $\ell$ in $X$ and in fact $M_{X}(v)$ is isomorphic to the Fano
variety of lines on $X$. We also denote by $M_{X}(v-w)$ the moduli of large
volume stable complexes on $X$ of class $v-w$, see Definition 4.10.

######  Theorem 1.3 (Theorem 4.5).

We have the isomorphisms

$\textstyle{M_{X}(v)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{L_{\mathcal{O}_{X}}(-\otimes\mathcal{O}(H))}$$\textstyle{M_{X}(w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{L_{\mathcal{O}_{X}}(-\otimes\mathcal{O}(H))}$$\textstyle{M_{X}(v-w).}$

Moreover, the above three moduli spaces are isomorphic to the moduli of
$\sigma$-stable objects of class $[\mathcal{I}_{\ell}]$ in
$\operatorname{Ku}(X)$ where $\sigma$ is any Serre-invariant stability
condition on $\operatorname{Ku}(X)$.

Finally, we apply Theorem 1.2 to study the moduli space of Ulrich bundles of
rank $\geq 2$ on $X$. Recall that an Ulrich bundle $E$ on $X$ is an
Arithmetically Cohen-Macauly vector bundle such that the graded module
$\oplus_{m\in\mathbb{Z}}H^{0}(X,E(mH))$ has $3\operatorname{rank}(E)$
generators in degree $1$ (see Section 6). Ulrich bundles all lie in
$\operatorname{Ku}(X)$. Thus the moduli spaces of stable objects in
$\operatorname{Ku}(X)$ become a useful tool in order to deduce properties of
certain classical moduli spaces of semistable sheaves on $X$, such as non-
emptyness and irreducibility. Applying these techniques, Lahoz, Macrì and
Stellari [LMS15, Theorem B] show that the moduli space $\mathfrak{M}^{sU}_{d}$
of stable Ulrich bundles of rank $d$ on $X$ is non-empty and smooth of the
expected dimension $d^{2}+1$. They leave as an open question its
irreducibility, which is our last result.

######  Theorem 1.4 (Theorem 6.1).

Let $X$ be a cubic threefold. The moduli space of Ulrich bundles of rank $\geq
2$ on $X$ is irreducible.

######  Related works and motivations.

The interest in the study of Serre-invariant stability conditions on
semiorthogonal components in the bounded derived category has grown recently,
mostly due to the applications to the study of moduli spaces (see for instance
[PY20], [LZ21]) and to understand more the Kuznetsov component. For Fano
threefolds of Picard rank $1$ and index $2$ the existence of Serre-invariant
stability conditions on their Kuznetsov component is proved in [PY20], making
use of the stability conditions constructed in [BLMS17]. In the upcoming paper
[PR22] the same result is proved for the Kuznetsov component of a Gushel–Mukai
threefold. On the other hand, in the recent paper [KP21] the authors show the
non-existence of Serre-invariant stability conditions on Kuznetsov components
of almost all Fano complete intersections of codimension $\geq 2$.

The assumptions in Theorem 1.1 require the category to be fractional
Calabi–Yau with numerical Grothendieck group of rank $2$ generated by objects
$E$ with small $\hom^{1}(E,E)$. The first condition allows to control the
phase of stable objects after the action of the Serre functor and the other
conditions make the category similar to the bounded derived category of a
curve. Since in the case of curves of genus $\geq 2$ there is a unique orbit
of stability conditions by Macrì’s result [Mac07], these are very natural
conditions in order to expect the category has a unique Serre-invariant
stability condition.

Theorem 1.1 applies to cubic threefolds and to other Fano threefolds (see
Remark 3.8). Note that in these cases the uniqueness result has been recently
proved independently by [JLLZ21]. We explain in Section 3 a related
application in the case of very general cubic fourfolds.

In Proposition 5.3 we apply the method in [BLMS17] to construct stability
conditions on $\operatorname{Ku}(X)$ via the embedding of
$\operatorname{Ku}(X)$ in
$\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathbb{P}^{2},\mathcal{B}_{0})$
[LMS15]. These stability conditions are Serre-invariant as shown in Section
5.3. Since the stability conditions constructed in [BLMS17] are also Serre-
invariant by [PY20, Corollary 5.5], we deduce that all the stability
conditions constructed on $\operatorname{Ku}(X)$ up to now are Serre-
invariant. Theorem 1.2 follows then from Theorem 1.1. An interesting question
would be to understand whether the property of Serre-invariance characterizes
the stability conditions on $\operatorname{Ku}(X)$. This fact together with
Theorem 1.1 would allow to show that there is a unique orbit of stability
conditions on $\operatorname{Ku}(X)$, in analogy to the case of curves of
genus $\geq 2$ [Mac07].

Another open question is to generalise Theorem 1.3 and Corollary 6.5 to
further study of moduli spaces of semistable objects in $\operatorname{Ku}(X)$
with respect to a Serre-invariant stability condition like their projectivity
or irreducibility. Some cases of small dimension have been studied in [PY20],
[APR19], [BBF+20], [Qin21], [LZ21].

The study of Ulrich bundles is a central theme in classical algebraic geometry
and commutative algebra (see for instance [ES03] and [Bea18] for a survey).
The existence of Ulrich bundles have been shown on cubic threefolds by [LMS15,
Theorem B]. In the rank-two case, namely the case of instanton sheaves of
minimal charge, we know a full description of the moduli space with class
$2[\mathcal{I}_{\ell}]$ for cubic threefolds [LMS15], and more generally for
Fano threefolds of Picard rank $1$, index $2$, degree $d\geq 3$ in [Qin21],
[LZ21]. For the case of cubic fourfolds we refer to [FK20].

Theorem 1.4 follows from an embedding of the moduli space of Ulrich bundles of
rank $d\geq 2$ on $X$ in a moduli space of semistable objects in
$\operatorname{Ku}(X)$ with class $d[\mathcal{I}_{\ell}]$, where
$\mathcal{I}_{\ell}$ is the ideal sheaf of a line in $X$. By Theorem 1.2, the
latter moduli space can be described via an irreducible moduli space
parameterising Gieseker semistable sheaves which are $\mathcal{B}_{0}$-modules
and it has the same dimension as $\mathfrak{M}^{sU}_{d}$.

######  Plan of the paper.

In Section 2 we recall the definitions and basic properties of (weak)
stability conditions, tilt stability, wall-crossing and a method to construct
stability conditions on Kuznetsov components. Section 3 is devoted to the
proof of Theorem 1.1. In Section 4 we review the construction of the stability
conditions $\sigma(\alpha,\beta)$ from [BLMS17] and then we prove Theorem 1.3.
In Section 5 we extend the construction of stability conditions on
$\operatorname{Ku}(X)$ from [BMMS12, LMS15] and we show they are Serre-
invariant. Finally, we deduce Theorem 1.2. In Section 6 we prove Theorem 1.4.

######  Acknowledgements.

We are very grateful to Chunyi Li, Emanuele Macrì, Paolo Stellari, Richard
Thomas, Xiaolei Zhao, Shizhuo Zhang for many useful conversations, and to
Arend Bayer for suggesting the strategy used in Theorem 1.1. We also thank an
anonymous referee for useful comments.

S.F. is supported by EPSRC postdoctoral fellowship EP/T018658/1. L.P. is
supported by the national research project PRIN 2017 Moduli and Lie Theory.

##  2\. (Weak) Bridgeland stability conditions and tilting

In this section we recall the definitions and properties of (weak) Bridgeland
stability conditions and wall-crossing. Our main reference is [BMS16]. We also
mention the stronger Bogomolov inequalities proved in [Li19] in the case of
cubic threefolds. Finally, we explain the method to construct stability
conditions on the orthogonal complement of an exceptional collection
introduced in [BLMS17].

###  2.1. Review on (weak) stability conditions

Let $\mathcal{T}$ be a $\mathbb{C}$-linear triangulated category and denote by
$K(\mathcal{T})$ its Grothendieck group. A (weak) stability condition on
$\mathcal{T}$ is defined by giving the heart of a bounded t-structure and a
(weak) stability function on it. Fix a finite rank lattice $\Lambda$ with a
surjective morphism $v\colon K(\mathcal{T})\twoheadrightarrow\Lambda$. We
denote by $\Re[-]$ ($\Im[-]$) the real (imaginary) part of a complex number.

######  Definition 2.1.

Let $\mathcal{A}$ be the heart of a bounded t-structure on $\mathcal{T}$. A
group homomorphism $Z\colon\Lambda\rightarrow\mathbb{C}$ is a _(weak)
stability function_ on $\mathcal{A}$ if for any $0\neq E\in\mathcal{A}$ we
have $\Im Z(v(E))\geq 0$, and in the case that $\Im Z(v(E))=0$, we have $\Re
Z(v(E))<(\leq)\,0$.

By abuse of notation, we will write $Z(E)$ for an object $E\in\mathcal{A}$
instead of $Z(v(E))$. The slope of $E\in\mathcal{A}$ with respect to $Z$ is
defined by

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

An object $F\in\mathcal{T}$ is called _(semi)stable_ with respect to the pair
$(\mathcal{A},Z)$ if some shift $F[k]$ lies in the heart $\mathcal{A}$ and for
every proper subobject $F^{\prime}\subset F[k]$ in $\mathcal{A}$ we have

$\mu_{Z}(F^{\prime})<(\leq)\ \mu_{Z}(F[k]/F^{\prime}).$

######  Definition 2.2.

A _(weak) stability condition_ (with respect to $\Lambda$) on $\mathcal{T}$ is
a pair $\sigma=(\mathcal{A},Z)$, where $\mathcal{A}$ is the heart of a bounded
t-structure on $\mathcal{T}$ and $Z$ is a (weak) stability function satisfying
the following properties:

  1. (a)

_HN property_ : Every object of $\mathcal{A}$ has a Harder–Narasimhan
filtration with $\sigma$-semistable factors.

  2. (b)

_Support property_ : There exists a quadratic form $Q$ on
$\Lambda\otimes\mathbb{R}$ such that the restriction of $Q$ to $\ker Z$ is
negative definite and $Q(E)\geq 0$ for all $\sigma$-semistable objects $E$ in
$\mathcal{A}$.

We denote by $\mathop{\mathrm{Stab}}\nolimits_{\Lambda}(\mathcal{T})$ the
space of stability conditions on $\mathcal{T}$ with respect to $\Lambda$ which
is actually a complex manifold [Bri07]. If $\Lambda$ is the numerical
Grothendieck group of $\mathcal{T}$, we call $\sigma$ a _numerical_ (weak)
stability condition. Any (weak) stability condition defines a slicing on
$\mathcal{T}$.

######  Definition 2.3.

The phase of a $\sigma$-semistable object $E\in\mathcal{A}$ is

$\phi(E)\coloneqq\frac{1}{\pi}\text{arg}(Z(E))\in(0,1].$

If $Z(E)=0$, then $\phi(E)=1$. For $F=E[n]$, we set

$\phi(E[n]):=\phi(E)+n.$

The slicing $\mathcal{P}_{\sigma}$ of $\mathcal{T}$ corresponding to $\sigma$
is a collection of full additive subcategories
$\mathcal{P}_{\sigma}(\phi)\subset\mathcal{T}$ for $\phi\in\mathbb{R}$, such
that:

  1. (i)

for $\phi\in(0,1]$, the subcategory $\mathcal{P}_{\sigma}(\phi)$ is given by
the zero object and all $\sigma$-semistable objects with phase $\phi$;

  2. (ii)

for $\phi+n$ with $\phi\in(0,1]$ and $n\in\mathbb{Z}$, we set
$\mathcal{P}_{\sigma}(\phi+n)\coloneqq\mathcal{P}_{\sigma}(\phi)[n]$.

The HN property of (weak) stability condition $\sigma$ implies that for $0\neq
E\in\mathcal{T}$ there is a unique finite sequence of real numbers

$\phi^{+}(E)\coloneqq\phi_{1}>\phi_{2}>\cdots>\phi_{n}\eqqcolon\phi^{-}(E)$

and a unique sequence of objects in $\mathcal{T}$

$0=E_{0}\xrightarrow{f_{1}}E_{1}\xrightarrow{f_{2}}E_{2}\xrightarrow{f_{3}}\cdots\xrightarrow{f_{n-1}}E_{n-1}\xrightarrow{f_{n}}E_{n}=E$

with cone$(f_{i})\in\mathcal{P}_{\sigma}(\phi_{i})$.

It is clear from the definition that $\mathcal{A}=\mathcal{P}_{\sigma}(0,1]$
where the latter is the full subcategory of $\mathcal{T}$ consisting of the
zero object together with those objects $0\neq E\in\mathcal{T}$ which satisfy
$0<\phi^{-}(E)\leq\phi^{+}(E)\leq 1$.

On the stability manifold
$\mathop{\mathrm{Stab}}\nolimits_{\Lambda}(\mathcal{T})$ we have:

  1. (a)

_a right action of the universal covering space
$\widetilde{\mathrm{GL}}^{+}_{2}(\mathbb{R})$ of
$\mathrm{GL}^{+}_{2}(\mathbb{R})$_: for a stability condition
$\sigma=(\mathcal{P},Z)\in\mathop{\mathrm{Stab}}\nolimits_{\Lambda}(\mathcal{T})$
and $\tilde{g}=(g,M)\in\widetilde{\mathrm{GL}}^{+}_{2}(\mathbb{R})$, where
$g:\mathbb{R}\to\mathbb{R}$ is an increasing function such that
$g(\phi+1)=g(\phi)+1$ and $M\in\mathrm{GL}^{+}_{2}(\mathbb{R})$, we define
$\sigma\cdot\tilde{g}$ to be the stability condition
$\sigma^{\prime}=(\mathcal{P}^{\prime},Z^{\prime})$ with
$Z^{\prime}=M^{-1}\circ Z$ and
$\mathcal{P}^{\prime}(\phi)=\mathcal{P}(g(\phi))$ (see [Bri09, Lemma 8.2]).

  2. (b)

_a left action of the group of linear exact autoequivalences
$\mathop{\mathrm{Aut}}\nolimits_{\Lambda}(\mathcal{T})$ of $\mathcal{T}$,
whose induced action on $K(\mathcal{T})$ is compatible with $v$_: for
$\Phi\in\mathop{\mathrm{Aut}}\nolimits_{\Lambda}(\mathcal{T})$ and
$\sigma\in\mathop{\mathrm{Stab}}\nolimits_{\Lambda}(\mathcal{T})$, we define
$\Phi\cdot\sigma=(\Phi(\mathcal{P}),Z\circ\Phi_{*}^{-1})$, where $\Phi_{*}$ is
the automorphism of $K(\mathcal{T})$ induced by $\Phi$.

###  2.2. Tilt stability

An important tool in the construction of weak stability conditions is the
procedure of tilting a heart. Let $\sigma=(\mathcal{A},Z)$ be a weak stability
condition on $\mathcal{T}$ and $\mu\in\mathbb{R}$. We consider the following
subcategories of $\mathcal{A}$:

$\displaystyle\mathcal{T}_{\sigma}^{\mu}\coloneqq\\{E\in\mathcal{A}|\text{
every Harder--Narasimhan factor $F$ of $E$ has $\mu_{Z}(F)>\mu$}\\};$
$\displaystyle\mathcal{F}_{\sigma}^{\mu}\coloneqq\\{E\in\mathcal{A}|\text{
every Harder--Narasimhan factor $F$ of $E$ has $\mu_{Z}(F)\leq\mu$}\\}.$

######  Theorem 2.4 ([HRS96]).

The category

$\mathcal{A}_{\sigma}^{\mu}\coloneqq\langle\mathcal{T}_{\sigma}^{\mu},\mathcal{F}_{\sigma}^{\mu}[1]\rangle_{\text{extension
closure}}$

is the heart of a bounded t-structure on $\mathcal{T}$.

We say that the heart $\mathcal{A}^{\mu}_{\sigma}$ is obtained by tilting
$\mathcal{A}$ with respect to the weak stability condition $\sigma$ at slope
$\mu$.

The above construction applies to the case of the bounded derived category
$\mathcal{T}=\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)$ of coherent sheaves
on a smooth projective variety $X$, to define tilt stability conditions. Fix
an ample divisor $H$ on $X$ and set $n:=\text{dim}(X)$. The pair
$\sigma_{H}=(\operatorname{Coh}(X),\
Z_{H}=-\operatorname{ch}_{1}H^{n-1}+\operatorname{ch}_{0}H^{n})$ is a weak
stability condition on $\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)$, known
as _slope stability_ , with respect to the rank-two lattice generated by the
elements of the form
$(H^{n}\operatorname{ch}_{0}(E),H^{n-1}\operatorname{ch}_{1}(E))$ for
$E\in\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)$ [BLMS17, Example 2.8]. In
particular, the slope of a coherent sheaf $E$ on $X$ is defined by

$\mu_{H}(E)=\begin{cases}\frac{H^{n-1}\operatorname{ch}_{1}(E)}{H^{n}\operatorname{ch}_{0}(E)}&\text{if
}\operatorname{ch}_{0}(E)>0\\\ +\infty&\text{otherwise}.\end{cases}$

Any $\mu_{H}$-semistable sheaf $E$ satifies the Bogomolov–Gieseker inequality

$\Delta_{H}(E):=(H^{n-1}\operatorname{ch}_{1}(E))^{2}-2H^{n}\operatorname{ch}_{0}(E)\cdot
H^{n-2}\operatorname{ch}_{2}(E)\geq 0.$

We remark that if $n=1$, then $\sigma_{H}$ is a stability condition.

Now given $\beta\in\mathbb{R}$, we denote by $\operatorname{Coh}^{\beta}(X)$
the heart of a bounded $t$-structure obtained by tilting the weak stability
condition $\sigma_{H}$ at the slope $\mu_{H}=\beta$ (see Theorem 2.4). For
$E\in\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)$, the twisted Chern
character is defined by $\operatorname{ch}^{\beta}(E):=e^{-\beta
H}\operatorname{ch}(E)$. Explicitly, the first three terms are

$\operatorname{ch}_{0}^{\beta}(E):=\operatorname{ch}_{0}(E),\quad\operatorname{ch}_{1}^{\beta}(E):=\operatorname{ch}_{1}(E)-\beta
H\operatorname{ch}_{0}(E)$

and

$\operatorname{ch}_{2}^{\beta}(E):=\operatorname{ch}_{2}(E)-\beta
H\operatorname{ch}_{1}(E)+\frac{\beta^{2}H^{2}}{2}\operatorname{ch}_{0}(E).$

######  Proposition 2.5.

[BLMS17, Proposition 2.12] There is a continuous family of weak stability
conditions parametrized by $\mathbb{R}_{>0}\times\mathbb{R}$, given by

$(\alpha,\beta)\in\mathbb{R}_{>0}\times\mathbb{R}\mapsto\sigma_{\alpha,\beta}=(\emph{Coh}^{\beta}(X),Z_{\alpha,\beta})$

with

$\displaystyle Z_{\alpha,\beta}(E)\colon\Lambda\cong\mathbb{Z}^{3}\ $
$\displaystyle\ \rightarrow\ \mathbb{C}$
$\displaystyle\left(H^{3}\operatorname{ch}_{0}(E),H^{2}\operatorname{ch}_{1}(E),H\operatorname{ch}_{2}(E)\right)\
$ $\displaystyle\ \mapsto\
\frac{1}{2}\alpha^{2}H^{n}\operatorname{ch}_{0}^{\beta}(E)-H^{n-2}\operatorname{ch}_{2}^{\beta}(E)+iH^{n-1}\operatorname{ch}_{1}^{\beta}(E).$

The weak stability condition $\sigma_{\alpha,\beta}$ is called _tilt
stability_ and if $n=2$, then it defines a stability condition on
$\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)$ (see [Bri08, AB13]). There is a
region in the upper-half plane where the notions of
$\sigma_{\alpha,\beta}$-stability and $\mu_{H}$-stability are closely related.

######  Lemma 2.6.

[BMS16, Lemma 2.7] Assume $E\in\operatorname{Coh}(X)$ is a $\mu_{H}$-stable
sheaf of positive rank. Then $E\in\operatorname{Coh}^{\beta}(X)$ for
$\beta<\mu_{H}(E)$ and is $\sigma_{\alpha,\beta}$-stable for $\alpha\gg 0$.

We now recall the notions of walls and chambers for tilt stability.

######  Definition 2.7.

Fix $v\in\Lambda$. A numerical wall for $v$ is a subset of the upper-half
plane of the form

$W(v,w)=\\{(\alpha,\beta)\in\mathbb{R}_{>0}\times\mathbb{R}\colon\mu_{\alpha,\beta}(v)=\mu_{\alpha,\beta}(w)\\}$

with respect to a vector $w\in\Lambda$. We say a point $(\alpha,\beta)\in
W(v,w)$ is on an actual wall for $v$ if and only if there is an object
$E\in\operatorname{Coh}^{\beta}(X)$ of class $v(E)=v$ which is strictly
$\sigma_{\alpha,\beta}$-semistable and unstable in one side of the wall.

A chamber for class $v$ is a connected component in the complement of the
union of walls in the upper-half plane.

Tilt stability satisfies well-behaved wall and chamber structure, in the sense
that walls with respect to a class $v\in\Lambda$ are locally finite.
Properties of walls are described by the following Theorem which was first
proved in [Mac14] (see also [Sch20]).

######  Theorem 2.8 (Bertram’s Nested Wall Theorem).

Fix $v\in\Lambda$.

  1. (a)

Numerical walls for $v$ are either nested semicircles centered on the
$\beta$-axis, or a vertical ray parallel to the $\alpha$-axis.

  2. (b)

If two numerical walls intersect, then they are the same. If a numerical wall
contains a point defining an actual wall, then the numerical wall is an actual
wall.

  3. (c)

If $\operatorname{ch}_{0}(v)\neq 0$, then there is a unique vertical numerical
wall given by $\beta=\mu_{H}(v)$. If $\operatorname{ch}_{0}(v)=0$ and
$H^{n-1}\operatorname{ch}_{1}(v)\neq 0$, then there is no vertical wall.

###  2.3. Cubic threefolds

We know any slope-semistable sheaf and more generally any
$\sigma_{\alpha,\beta}$-semistable object satisfies $\Delta_{H}(E)\geq 0$
[BMS16, Theorem 3.5]. In the case of a smooth complex cubic threefold $X$, we
have stronger Bogomolov inequalities as follows.

######  Proposition 2.9.

[Li19, Corollary 3.4] Let $E$ be a slope-semistable coherent sheaf of positive
rank on $X$. If $\left\lvert\mu_{H}(E)\right\rvert\leq\frac{1}{2}$, then
$\operatorname{ch}_{2}(E)\leq 0$, and if
$\frac{1}{2}<\left\lvert\mu_{H}(E)\right\rvert\leq 1$, then

$H\operatorname{ch}_{2}(E)\leq\left\lvert
H^{2}\operatorname{ch}_{1}(E)\right\rvert-\frac{1}{2}H^{3}\operatorname{ch}_{0}(E).$

######  Theorem 2.10.

For any $\sigma_{\alpha,\beta}$-semistable object
$E\in\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)$, we have

$0\leq
Q_{\alpha,\beta}(E)\coloneqq\left(\frac{\alpha^{2}}{2}+\frac{\beta^{2}}{2}\right)\left(C_{1}^{2}-2C_{0}C_{2}\right)+\beta\left(3C_{0}C_{3}-C_{1}C_{2}\right)+\left(2C_{2}^{2}-3C_{1}C_{3}\right)$

where $C_{i}:=\operatorname{ch}_{i}(E).H^{3-i}$.

###### Proof.

[BMS16, Conjecture 5.3] is proved for cubic threefolds in [Li19, Theorem 0.1].
Thus [BMS16, Theorem 5.4] implies that [BMS16, Conjecture 4.1] holds on cubic
threefold. One can easily show that its statement is equivalent to
$Q_{\alpha,\beta}(E)\geq 0$. ∎

###  2.4. Induced stability conditions

We finish this section by recalling induced stability conditions on an
admissible subcategory. Let $\mathcal{T}$ be a $\mathbb{C}$-linear
triangulated category with Serre functor $S$. Assume $\mathcal{T}$ has a
semiorthogonal decomposition of the form

$\mathcal{T}=\langle\mathcal{D},E_{1},\cdots,E_{m}\rangle,$

where $E_{1},\dots,E_{m}$ are an exceptional collection and
$\mathcal{D}:=\langle E_{1},\dots,E_{m}\rangle^{\perp}$. The next proposition
gives a criterion in order to construct a stability condition on $\mathcal{D}$
from a weak stability condition on $\mathcal{T}$.

######  Proposition 2.11 ([BLMS17, Proposition 5.1]).

Let $\sigma=(\mathcal{A},Z)$ be a weak stability condition on $\mathcal{T}$.
Assume that the exceptional collection $\\{E_{1},\cdots,E_{m}\\}$ satisfies
the following conditions:

  1. (a)

$E_{i}\in\mathcal{A}$;

  2. (b)

$S(E_{i})\in\mathcal{A}[1]$;

  3. (c)

$Z(E_{i})\neq 0$ for all $i=0,1,\cdots,m$.

If moreover there are no objects $0\neq
F\in\mathcal{A}^{\prime}:=\mathcal{A}\cap\mathcal{D}$ with $Z(F)=0$, then the
pair $\sigma^{\prime}=(\mathcal{A}^{\prime},Z|_{K(\mathcal{A})})$ is a
stability condition on $\mathcal{D}$.

##  3\. Serre-invariant stability conditions

Let $\mathcal{T}$ be a triangulated category which is linear of finite type
over a field $K$, i.e. $\oplus_{i}\operatorname{Hom}(E,F[i])$ is a finite-
dimensional vector space over $K$. Assume that $\mathcal{T}$ has Serre functor
$S$. Since $S$ is a linear exact autoequivalence, we can have the following
definition.

######  Definition 3.1.

A stability condition $\sigma$ on $\mathcal{T}$ is _Serre-invariant_ (or _$S$
-invariant_) if $S.\sigma=\sigma.\tilde{g}$ for some
$\tilde{g}\in\widetilde{\text{GL}}_{2}^{+}(\mathbb{R})$.

Assume $\mathcal{T}$ satisfies the following conditions:

  1. (C1)

The Serre functor $S$ of $\mathcal{T}$ satisfies $S^{r}=[k]$ when $0<k/r<2$ or
$r=2$ and $k=4$.

  2. (C2)

The numerical Grothendieck group $\mathcal{N}(\mathcal{T})$ is of rank 2 and
$\ell_{\mathcal{T}}\coloneqq\max\\{\chi(v,v)\colon 0\neq
v\in\mathcal{N}(\mathcal{T})\\}<0$.

  3. (C3)

There is an object $Q\in\mathcal{T}$ satisfying

(1) $-\ell_{\mathcal{T}}+1\leq\hom^{1}(Q,Q)<-2\ell_{\mathcal{T}}+2.$

If $k=2r=4$, there are two objects $Q_{1},Q_{2}\in\mathcal{T}$ satisfying (1)
such that $Q_{1}$ is not isomorphic to $Q_{2}$ or $Q_{2}[1]$,
$\hom(Q_{2},Q_{1})\neq 0$ and $\hom(Q_{1},Q_{2}[1])\neq 0$.

We can slightly relax (C3) in the case $k=2r=4$, see Lemma 3.7.

The goal of this section is to prove the following theorem.

######  Theorem 3.2.

Let $\sigma_{1}$ and $\sigma_{2}$ be $S$-invariant numerical stability
conditions on a triangulated category $\mathcal{T}$ satisfying the above
conditions (C1), (C2) and (C3). Then there exists
$\tilde{g}\in\widetilde{\emph{GL}}^{+}(2,\mathbb{R})$ such that
$\sigma_{1}=\sigma_{2}\cdot\tilde{g}$.

######  Remark 3.3.

It is easy to see that if the Serre functor satisfies $S=[2]$, then every
stability condition on $\mathcal{T}$ is Serre-invariant.

Before proving Theorem 3.2, we apply similar arguments as in [PY20] to
investigate some of the properties of $S$-invariant stability conditions.

######  Proposition 3.4.

Let $\mathcal{T}$ be a triangulated category satisfying conditions (C1) and
(C2) above, and let $\sigma=(Z,\mathcal{A})$ be an $S$-invariant stability
condition on $\mathcal{T}$.

  1. (a)

If $E$ is a $\sigma$-semistable object of phase $\phi(E)$, then
$\phi(S(E))\leq\phi(E)+2$. The inequality is strict if $k/r<2$.

  2. (b)

(Weak Mukai Lemma) Let $A\to E\to B$ be an exact triangle in $\mathcal{T}$
such that $\operatorname{Hom}(A,B)=0$ and the $\sigma$-semistable factors of
$A$ have phases greater (or equal in case $k/r<2$) than the phases of the
$\sigma$-semistable factors of $B$. Then

(2) $\hom^{1}(A,A)+\hom^{1}(B,B)\leq\hom^{1}(E,E).$

  3. (c)

Any non-zero object $E\in\mathcal{T}$ satisfies

$\hom^{1}(E,E)\geq-\ell_{\mathcal{T}}+1.$

  4. (d)

A $\sigma$-semistable object $E$ satisfies $\phi(E)+1\leq\phi(S(E))$. The
inequality is strict if $E$ is $\sigma$-stable.

  5. (e)

If $E\in\mathcal{T}$ satisfies
$-\ell_{\mathcal{T}}+1\leq\hom^{1}(E,E)<-2\ell_{\mathcal{T}}+2$, then it is
$\sigma$-stable.

###### Proof.

Since $\sigma$ is an $S$-invariant stability condition on $\mathcal{T}$, there
is $(g,M)\in\widetilde{\text{GL}}_{2}^{+}(\mathbb{R})$ where
$g\colon\mathbb{R}\rightarrow\mathbb{R}$ is an increasing map with
$g(\phi+1)=g(\phi)+1$ such that $S.\sigma=\sigma.\tilde{g}$. Thus
$\phi(S(F))=g(\phi(F))$ and for any $n>0$,

(3)
$\phi(S^{n}(F))=g(\phi(S^{n-1}(F)))=g^{2}(\phi(S^{n-2}(F)))=.....=g^{n}(\phi(F)).$

If $\phi(S(F))=g(\phi(F))\ (\geq)\ \phi(F)+2$, then since $g$ is increasing,

$\phi(F)+k=\phi(S^{r}(F))\overset{\eqref{equality}}{=}g^{r}(\phi(F))\ (\geq)\
g^{r-1}(\phi(F)+2)=g^{r-1}(\phi(F))+2\ (\geq)\ ...(\geq)\ \phi(F)+2r$

which gives $k/r\ (\geq)\ 2$, a contradiction. This completes the proof of
part (a).

To prove part (b), let $A\rightarrow E\rightarrow B$ be an exact triangle in
$\mathcal{T}$ with $\operatorname{Hom}(A,B)=0$. If $\phi^{+}(B)<\phi^{-}(A)$,
then part (a) implies $\phi^{+}(S(B))-2\leq\phi^{+}(B)$ which gives

(4) $\phi^{+}(S(B))<\phi^{-}(A[2]).$

If $k/r<2$, we may assume $\phi^{+}(B)\leq\phi^{-}(A)$ because by part (a)
again (4) holds. Then

(5) $\hom(B,A[2])=\hom(A[2],S(B))=0$

and the same argument as in [BB17, Lemma 2.4] implies the claim (2).

To prove part (c), first assume $E$ is $\sigma$-semistable. Up to shift we may
assume $E\in\mathcal{P}(0,1]$. Thus $\hom(E,E[i])=0$ for $i<0$. Also by part
(a) $\phi(S(E))\ \leq\ 3$, thus $\hom(E,E[i])=\hom(E[i],S(E))=0$ for $i\geq
3$. Hence

(6) $\hom(E,E[1])=-\chi(E,E)+\hom(E,E)+\hom(E,E[2])\geq-\ell_{\mathcal{T}}+1$

as claimed. If $E$ is not $\sigma$-semistable, then let $E_{1}\rightarrow
E\rightarrow E/E_{1}$ be the first piece in its HN filtration. Then applying
weak Mukai Lemma in part (b) to this sequence gives

$\hom^{1}(E_{1},E_{1})\leq\hom^{1}(E,E).$

Thus the claim in part (c) follows by the first part of the argument because
$E_{1}$ is $\sigma$-semistable.

Take a $\sigma$-semistable object $E$. Part (c) implies that
$\hom(E[1],S(E))=\hom(E,E[1])\neq 0$, thus $\phi(E)+1\leq\phi(S(E))$ as
claimed in part (d). If $E$ is $\sigma$-stable, then $S(E)$ is also
$\sigma$-stable. Since there is a non-zero map from $E$ to $S(E)[-1]$, they
cannot have the same phase. Note that $S(E)\neq E[1]$, otherwise
$\hom^{1}(E,E)=\hom(E,E)=1$ which is not possible by part (c).

Finally, to prove part (e), take an object $E\in\mathcal{T}$ satisfying the
minimality condition

$-\ell_{T}+1\leq\hom^{1}(E,E)<-2\ell_{T}+2$

on $\hom^{1}(E,E)$. Assume for a contradiction that $E$ is not
$\sigma$-semistable and let $A\rightarrow E\rightarrow B$ be the first piece
in the HN filtration of $E$, so $A$ is $\sigma$-semistable and

$\phi(A)=\phi^{+}(E)>\phi^{+}(B).$

Thus $\hom(A,B)=0$ and we can apply weak Mukai Lemma in part (b). Then part
(c) implies

$2(-\ell_{\mathcal{T}}+1)\leq\hom^{1}(A,A)+\hom^{1}(B,B)\leq\hom^{1}(E,E)$

which is not possible by our assumption.

Now assume $E$ is strictly $\sigma$-semistable. First suppose $k/r<2$. Let
$Q\hookrightarrow E$ be a $\sigma$-stable factor of $E$ with quotient
$E_{1}\in\mathcal{P}(\phi(E))$. If $\hom(Q,E_{1})\neq 0$, then
$\sigma$-stability of $Q$ implies that there is an embedding $Q\hookrightarrow
E_{1}$ with quotient $E_{2}\in\mathcal{P}(\phi(E))$. By continuing this
process, we get an exact triangle $A\rightarrow E\rightarrow B$ in
$\mathcal{P}(\phi(E))$ such that $A$ is $S$-equivalent to $\oplus_{k}Q$ for
some $k>0$ (i.e. all $\sigma$-stable factors of $A$ are isomorphic to $Q$) and
$\hom(A,B)=0$. If $B\neq 0$, then weak Mukai Lemma and part (c) lead to a
contradiction as above. Thus $[E]=k[Q]\in\mathcal{N}(\mathcal{T})$ and
$\chi(E,E)=k^{2}\chi(Q,Q)\leq k^{2}\ell_{\mathcal{T}}$. This gives

$\hom^{1}(E,E)=-\chi(A,A)+\hom(A,A)+\hom(A,A[2])\geq-k^{2}\ell_{\mathcal{T}}+1.$

Then our assumption implies
$-k^{2}\ell_{\mathcal{T}}+1<-2\ell_{\mathcal{T}}+2$ which is possible only if
$k=1$ because $\ell_{T}\leq-1$. Thus $E$ is $\sigma$-stable. This completes
the proof of (e) for the case $k/r<2$.

If $E$ is strictly $\sigma$-semistable and $k=2r=4$, we apply the next Lemma
3.5. In case (ii) of Lemma 3.5, the same argument as in weak Mukai Lemma in
part (b) implies that

$\hom^{1}(E_{1},E_{1})+\hom^{1}(E_{2},E_{2})\leq\hom^{1}(E,E)$

which is not possible by the minimality assumption on $\hom^{1}(E,E)$. Thus
case (i) of Lemma 3.5 happens, i.e. $E$ is $S$-equivalent to
$G^{\oplus{k_{1}}}\oplus S(G)^{\oplus{k_{2}}}[-2]$ for a stable object $G$.
Hence $\chi(E,E)=(k_{1}+k_{2})^{2}\chi(G,G)$ because

$\chi(G,G)=\chi(G,S(G))=\chi(S(G),S(G))=\chi(S(G),S^{2}(G))=\chi(S(G),G).$

Therefore (6) gives

$\hom^{1}(E,E)\geq-\chi(E,E)+1=-(k_{1}+k_{2})^{2}\chi(G,G)+1.$

If $k_{1}+k_{2}\geq 2$, then

$-2\ell_{T}+1>\hom^{1}(E,E)\geq-4\ell_{T}+1$

which implies $0>-2\ell_{T}$, a contradiction. Thus $k_{1}+k_{2}=1$, i.e. $E$
is isomorphic to either $G$ or $S(G)[-2]$, so $E$ is $\sigma$-stable as
claimed. ∎

######  Lemma 3.5.

Suppose $k=2r=4$ and $E$ is a strictly $\sigma$-semistable object, then either

  1. (i)

$E$ is $S$-equivalent to $G^{\oplus{k_{1}}}\oplus S(G)^{\oplus{k_{2}}}[-2]$
for a stable object $G$ which has same phase as $S(G)[-2]$ if $k_{2}\neq 0$,
or

  2. (ii)

there is a non-trivial triangle $E_{1}\rightarrow E\rightarrow E_{2}$ of
$\sigma$-semistable objects of the same phase such that $\hom(E_{1},E_{2})=0$
and $\hom(E_{2},E_{1}[2])=0$.

###### Proof.

Considering the JH filtration, we can write the short exact sequence
$A\rightarrow E\rightarrow G$, where $G$ is stable of the same phase as $E$.
If $\hom(A,G)\neq 0$, then we gain a surjection $A\twoheadrightarrow G$ with
kernel $A_{1}$:

$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G}$$\textstyle{G}$

Thus the cone of $A_{1}\to E$, denoted by $F_{1}$, is $S$-equivalent to
$G^{\oplus 2}$. If $\hom(A_{1},G)\neq 0$, we continue the above argument to
reach a triangle $A_{m}\rightarrow E\rightarrow F_{m}$ such that
$\hom(A_{m},G)=0$ and $F_{m}$ is $S$-equivalent to $G^{\oplus\,m+1}$. If
$A_{m}=0$, then $E$ is $S$-equivalent to $G^{\oplus\,m+1}$ as in case (i).

We know $S(G)$ is $\sigma$-stable and $\phi(S(G))\leq\phi(G)+2$ by Proposition
3.4, part (a). If $\phi(S(G))<\phi(G)+2$, then

$\hom(G,A_{m}[2])=\hom(A_{m}[2],S(G))=0.$

Hence if $\phi(S(G))<\phi(G)+2$ or $\hom(A_{m},S(G)[-2])=0$ , then
$\hom(F_{m},A_{m}[2])=0$, thus we get a triangle as in case (ii). But if
$\phi(S(G)[-2])=\phi(G)$ and $\hom(A_{m},S(G)[-2])\neq 0$, then there is a
commutative diagram

$\textstyle{S(G)[-2]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{m}}$$\textstyle{S(G)[-2]}$

where $E_{1}$ is $S$-equivalent to $G^{\oplus\,m+1}\oplus S(G)[-2]$. If
$\hom(B_{1},S(G)[-2])\neq 0$, we continue the above argument to get a triangle
$B_{n}\rightarrow E\rightarrow E_{n}$ such that $E_{n}$ is $S$-equivalent to
$G^{\oplus\,m+1}\oplus S(G)^{\oplus\,n}[-2]$ and $\hom(B_{n},S(G)[-2])=0$. If
$B_{n}=0$, then we are in case (i) and if $B_{n}\neq 0$, we continue the above
process to obtain an exact triangle $E_{1}\rightarrow E\rightarrow E_{2}$ such
that $E_{2}$ is $S$-equivalent to $G^{\oplus\,k_{1}}\oplus
S(G)^{\oplus\,k_{2}}[-2]$ and

$\hom(E_{1},G)=0=\hom(E_{1},S(G)[-2]).$

This implies that

$\hom(E_{1},E_{2})=0.$

Moreover, we have $\hom(E_{2},E_{1}[2])=\hom(E_{1},S(E_{2})[-2])$ and
$S(E_{2})[-2]$ is $S$-equivalent to $S(G)[-2]^{\oplus k_{1}}\oplus
S^{2}(G)^{\oplus k_{2}}[-4]=S(G)[-2]^{\oplus k_{1}}\oplus G^{\oplus k_{2}}$,
thus $\hom(E_{1},S(E_{2})[-2])=0$ as claimed in case (ii). ∎

Now assume our triangulated category $\mathcal{T}$ satisfies the condition
(C3). If $k/r<2$ fix an object $Q$ satisfying (1). By Proposition 3.4 part (e)
both $Q$ and $S(Q)[-2]$ are stable. Moreover, parts (a) and (d) imply

$\phi(Q)-1<\phi(S(Q)[-2])<\phi(Q).$

Hence, up to $\widetilde{\text{GL}}_{2}^{+}(\mathbb{R})$-action, we may assume
$Q$ is stable of phase one and $S(Q)[-2]$ is of phase $1/2$.

If $k=2r=4$, we fix two objects $Q_{1}$ and $Q_{2}$ described in condition
(C3). By part (e) of Proposition 3.4, they are $\sigma$-stable. Since
$\hom(Q_{2},Q_{1})\neq 0$, we gain $\phi(Q_{2})<\phi(Q_{1})$ and since
$\hom(Q_{1},Q_{2}[1])\neq 0$ we get $\phi(Q_{1})-1<\phi(Q_{2})$. Thus up to
$\widetilde{\text{GL}}_{2}^{+}(\mathbb{R})$-action, we may assume $Q_{1}$ is
of phase one and $Q_{2}$ is of phase $1/2$. From now on, in case $k/r<2$, we
define

$Q_{1}\coloneqq Q\qquad\text{and}\qquad Q_{2}\coloneqq S(Q)[-2].$

As a consequence, we have:

######  Lemma 3.6.

Let $\sigma=(Z,\mathcal{A})$ be an $S$-invariant stability condition on
$\mathcal{T}$. Up to $\widetilde{\emph{GL}}_{2}^{+}(\mathbb{R})$-action, we
may assume $Z(Q_{1})=-1$ and $Z(Q_{2})=i$.

###### Proof of Theorem 3.2.

Lemma 3.6 implies that after acting by
$\widetilde{\text{GL}}^{+}(2,\mathbb{R})$, we can assume $\sigma_{1}$ and
$\sigma_{2}$ have the same central charge $Z$ given by the bijective group
homomorphism

$Z\colon\mathcal{N}(\mathcal{T})\rightarrow\mathbb{C}\qquad Z([Q_{1}])=-1\
,\qquad Z([Q_{2}])=i\ .$

By Proposition 3.4, part (c), any $\sigma_{i}$-semistable object $E$ satisfies
$\hom^{1}(E,E)\geq-\ell_{\mathcal{T}}+1$. We show by induction on
$\hom^{1}(E,E)$ that if $E$ is a $\sigma_{1}$-(semi)stable object in
$P_{\sigma_{1}}(0,1]$ with phase $\phi_{\sigma_{1}}(E)$ then

  1. (i)

$E$ is $\sigma_{2}$-(semi)stable, and

  2. (ii)

$E$ has the same phase as with respect to $\sigma_{1}$, i.e.
$\phi_{\sigma_{2}}(E)=\phi_{\sigma_{1}}(E)$.

Step 1.(Base of the induction) If $\hom^{1}(E,E)$ is minimal, i.e.

$-\ell_{T}+1\leq\hom^{1}(E,E)<-2\ell_{T}+2,$

Proposition 3.4 part (e) implies that $E$ is $\sigma_{i}$-stable. Thus it
remains to show that if $E$ is in the heart $P_{\sigma_{1}}(0,1]$ with phase
$\phi_{\sigma_{1}}(E)$, then $E\in P_{\sigma_{2}}(0,1]$ with
$\phi_{\sigma_{2}}(E)=\phi_{\sigma_{1}}(E)$.

To do this, note that we only need to show $E\in P_{\sigma_{2}}(0,1]$, then
since $\sigma_{1}$ and $\sigma_{2}$ have the same central charge, $E$ must
have the same phase with respect to them. Assume otherwise, so there is $0\neq
m\in\mathbb{Z}$ such that $E[2m]\in P_{\sigma_{2}}(0,1]$. In fact, since
$\sigma_{1}$ and $\sigma_{2}$ have the same central charge, an even shift of
$E$ lies in the heart for $\sigma_{2}$. By Lemma 3.6, $E$ is not isomorphic to
$Q_{1}$ or $Q_{2}$, since $Q_{1}$ and $Q_{2}$ are in the heart of $\sigma_{1}$
and $\sigma_{2}$. We want to show that

(7) $\chi(Q_{1},E)=\chi(Q_{2},E)=0.$

Indeed, since $E[i]\in P_{\sigma_{1}}(i,i+1]$ and $E[2m]\in
P_{\sigma_{2}}(0,1]$, we gain

$\hom(Q_{1},E[i])=\hom(Q_{1},E[2m][i-2m])=0\qquad\text{when
$i\leq\max\\{0,2m\\}$}.$

Proposition 3.4 parts (a) and (d) give $S(Q_{1})[-2]\in P_{\sigma_{i}}(0,1]$,
thus

$\hom(Q_{1},E[i])=\hom(E[-2+i],S(Q_{1})[-2])=\hom(E[2m][-2+i-2m],S(Q)[-2])=0$

if $i>\min\\{2,2m+2\\}$. Hence, in total we gain $\chi(Q_{1},E)=0$. Similarly,
we have

$\hom(Q_{2},E[i])=\hom(Q_{2},E[2m][i-2m])=0\qquad\text{when
$i<\max\\{0,2m\\}$}.$

The above vanishing holds for $i=\max\\{0,2m\\}$ if
$\phi_{\sigma_{1}}(E)\in(0,1/2]$. Moreover, Proposition 3.4 parts (a) and (d)
imply that $S(Q_{2})[-2]\in P_{\sigma_{i}}(-1/2,1/2]$. Thus

$\hom(Q_{2},E[i])=\hom(E[i-2],S(Q_{2})[-2])=\hom(E[2m][i-2-2m],S(Q_{2})[-2])=0$

if $i>\min\\{2,2m+2\\}$. Also the above vanishing holds for
$i=\min\\{2,2m+2\\}$ when $\phi_{\sigma_{1}}(E)\in(1/2,1]$. Therefore
$\chi(Q_{2},E)=0$ as claimed in (7).

Since $Z(Q_{1})$ and $Z(Q_{2})$ are linearly independent, the classes
$[Q_{1}]$ and $[Q_{2}]$ are also linearly independent in
$\mathcal{N}(\mathcal{T})$. Hence (7) implies that $\chi([F],[E])=0$ for any
$[F]\in\mathcal{N}(\mathcal{T})$, thus $[E]=0$ by definition of
$\mathcal{N}(\mathcal{T})$, a contradiction. This completes the proof for the
case that $\hom^{1}(E,E)$ is minimal.

Step 2. Now assume $\hom^{1}(E,E)\geq-2\ell_{\mathcal{T}}+2$. There are 3
possibilities:

  1. (a)

$E$ is not $\sigma_{2}$-semistable.

  2. (b)

$E$ is strictly $\sigma_{2}$-semistable.

  3. (c)

$E$ is $\sigma_{2}$-stable.

First suppose case (a) happens, then we show that $E\in P_{\sigma_{2}}(0,1]$.
If not, consider its HN filtration and let $E_{1}$ be the first object of
maximum phase and $E_{n}$ be the last object of minimum phase with respect to
$\sigma_{2}$. Then the weak Mukai Lemma (2) applied to the triangle
$E_{1}\rightarrow E\rightarrow E/E_{1}$ and using part(c) of Proposition 3.4
imply $\hom^{1}(E_{1},E_{1})<\hom^{1}(E,E)$. Thus, by induction assumption,
$E_{1}$ is $\sigma_{1}$-semistable with
$\phi_{\sigma_{1}}(E_{1})=\phi_{\sigma_{2}}(E_{1})$. The existence of a non-
zero map $E_{1}\rightarrow E$ and the stability of $E$ with respect to
$\sigma_{1}$ imply that

(8)
$\phi_{\sigma_{2}}(E_{1})=\phi_{\sigma_{1}}(E_{1})\leq\phi_{\sigma_{1}}(E)\leq
1.$

Applying the same argument to the last piece in the HN filtration
$Q\rightarrow E\rightarrow E_{n}$ we get

(9)
$0<\phi_{\sigma_{1}}(E)\leq\phi_{\sigma_{1}}(E_{n})=\phi_{\sigma_{2}}(E_{n}).$

Therefore $E\in P_{\sigma_{2}}(0,1]$. Moreover, since the stability functions
for $\sigma_{1}$ and $\sigma_{2}$ are the same, we gain
$\phi_{\sigma_{1}}(E)=\phi_{\sigma_{2}}(E)$, thus (8) implies that
$\phi_{\sigma_{2}}(E_{1})\leq\phi_{\sigma_{2}}(E)$ which is not possible. Thus
$E$ is $\sigma_{2}$-semistable.

Now suppose case (b) happens and $E$ is strictly $\sigma_{2}$-semistable.
First assume $k/r<2$. Let $\\{E_{i}\\}_{i\in I}$ be the Jordan–Hölder
filtration of $E$ with respect to $\sigma_{2}$. By the Weak Mukai Lemma (2)
and induction assumption, it follows that the $E_{i}$’s are
$\sigma_{1}$-stable of the same phase as with respect to $\sigma_{2}$. Thus
$E$ is strictly $\sigma_{1}$-semistable of the same phase as with respect to
$\sigma_{2}$. If $k=2r=4$, we apply Lemma 3.5. If $E$ is $S$-equivalent to
$G^{\oplus k_{1}}\oplus S(G)[-2]^{\oplus k_{2}}$ for a $\sigma_{2}$-stable
object $G$ and $k_{1}+k_{2}>1$, then (6) gives

$\hom^{1}(E,E)\geq-\chi(E,E)+1=-(k_{1}+k_{2})^{2}\chi(G,G)+1>-\chi(G,G)+2\geq\hom^{1}(G,G).$

Thus $G$ and $S(G)$ are $\sigma_{1}$-stable of the same phase. Hence, $E$ is
strictly $\sigma_{1}$-semistable and
$\phi_{\sigma_{1}}(E)=\phi_{\sigma_{2}}(E)$. If the second case (ii) of Lemma
3.5 happens, then the same argument as in Weak Mukai Lemma implies that
$\hom^{1}(E_{i},E_{i})<\hom^{1}(E,E)$, thus again the claim follows by the
induction assumption.

Finally, assume (c) happens. If $E$ is strictly $\sigma_{1}$-semistable, then
the same argument as in case (b) implies that $E$ is also strictly
$\sigma_{2}$-semistable. Thus $E$ must be $\sigma_{1}$-stable and we only need
to show $E$ has the same phase with respect to both $\sigma_{1}$ and
$\sigma_{2}$.

We know $E\in P_{\sigma_{1}}(0,1]$. Since $E$ is $\sigma_{2}$-stable, there is
$m\in\mathbb{Z}$ such that $E[2m]\in P_{\sigma_{2}}(0,1]$. If $m\neq 0$, then
the same argument as in Step 1 implies that $\chi(Q_{j},E)=0$ for $j=1,2$, and
so $[E]=0$ in $\mathcal{N}(\mathcal{T})$ which is not possible. Thus $m=0$ and
$E$ has the same phase with respect to both $\sigma_{1}$ and $\sigma_{2}$. ∎

As suggested to us by Zhiyu Liu and Shizhuo Zhang, we can slightly relax the
condition (C3).

######  Lemma 3.7.

If $k=2r=4$, we can change the condition (C3) to the following: there are
three objects $Q_{1},Q_{2},Q_{2}^{\prime}\in\mathcal{T}$ satisfying (1) such
that

  1. (a)

$Q_{2}$ and $Q_{2}^{\prime}$ have the same class in
$\mathcal{N}(\mathcal{T})$,

  2. (b)

$Q_{1}$ is not isomorphic to $Q_{2}$, or $Q_{2}^{\prime}[1]$,

  3. (c)

$\hom(Q_{2},Q_{1})\neq 0$ and $\hom(Q_{1},Q_{2}^{\prime}[1])\neq 0$,

  4. (d)

$\hom(Q_{2}^{\prime},Q_{2}[3])=0$.

###### Proof.

Let $\sigma=(Z,\mathcal{A})$ be an $S$-invariant stability condition on
$\mathcal{T}$. By Proposition 3.4, part (e), we know all three objects
$Q_{1},Q_{2},Q_{2}^{\prime}$ are $\sigma$-stable. Hence, up to
$\widetilde{\text{GL}}_{2}^{+}(\mathbb{R})$-action, we may assume $Q_{1}$ is
of phase one and $Z(Q_{1})=-1$. If $\phi(Q_{2})\in(0,1]$, then we can assume
$Z(Q_{2})=i$ and proceed as before. So assume for a contradiction that
$Q_{2}\notin\mathcal{A}$. The assumptions (b) and (c) implies that

$\phi(Q_{2})<\phi(Q_{1})=1$

If $\phi(Q_{2})\in(-\infty,-1)$, then $\phi(S(Q_{2}))<1$ and so
$\hom(Q_{1},S(Q_{2}))=0$ which is not possible by condition (c). This implies
that

(10) $-1\leq\phi(Q_{2})\leq 0.$

On the other hand, conditions (b) and (c) give
$1=\phi(Q_{1})<\phi(Q_{2}^{\prime})+1$. This implies that
$1\leq\phi(Q_{2}^{\prime})$ because $Q_{2}$ and $Q_{2}^{\prime}$ have the same
class. Since $\hom(Q_{2}^{\prime}[1],S(Q_{1}))\neq 0$, we gain

$1\leq\phi(Q_{2}^{\prime})\leq 2.$

Thus $\phi(Q_{2}^{\prime})=\phi(Q_{2})+2$ which implies

$\hom(Q_{2}^{\prime},Q_{2}[i])=0\qquad\text{ for $i\leq 1$ and $i\geq 5$}.$

Therefore

$\displaystyle-\hom(Q_{2}^{\prime},Q_{2}[3])+\hom(Q_{2}^{\prime},Q_{2}[2])+\hom(Q_{2}^{\prime},Q_{2}[4])=\chi(Q_{2}^{\prime},Q_{2})=\chi(Q_{2},Q_{2})<0$

and so $\hom(Q_{2}^{\prime},Q_{2}[3])\neq 0$ which is in contradiction to the
assumption (d). ∎

######  Remark 3.8.

Theorem 3.2 can be applied to prove uniqueness of Serre-invariant stability
conditions on the Kuznetsov component of certain Fano threefolds of Picard
rank one and index $2$ (see [Kuz09, Theorem 2.3] for the classification in
terms of the degree). More precisely, as explained in Corollary 4.3 and Remark
4.4 it applies to the Kuznetsov component of a cubic threefold (degree $3$
case) and to the quartic double solid (degree $2$ case), respectively. Theorem
3.2 (via the relaxed condition in Lemma 3.7) also applies to the Kuznetsov
component of a Gushel–Mukai threefold, which is a Fano threefold of Picard
rank $1$, index $1$ and genus $6$, by [PR22, Corollary 4.5]. Recently, the
uniqueness of Serre-invariant stability conditions for the above examples has
been independently proved in [JLLZ21, Theorem 4.25] and [Zha21, Theorem 4.24].
But we believe that the general criterion given in Theorem 3.2 could be
potentially applied in other interesting geometric examples. For instance, in
the next Section we explain an application to the Kuznetsov component of a
very general cubic fourfolds.

### Aside: very general cubic fourfolds

If the Serre functor of $\mathcal{T}$ is $S=[2]$, i.e. $\mathcal{T}$ is a
$2$-Calabi-Yau category, then clearly any stability condition is Serre-
invariant. An example of such category is the Kuznetsov component of a cubic
fourfold. In this section we apply Theorem 3.2 to show that if $X$ is a very
general cubic fourfold, then there is a unique
$\widetilde{\text{GL}}^{+}(2,\mathbb{R})$-orbit of stability conditions on the
Kuznetsov component of $X$.

Recall that the bounded derived category of a cubic fourfold $X$ has a
semiorthogonal decomposition of the form

$\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)=\langle\operatorname{Ku}(X),\mathcal{O}_{X},\mathcal{O}_{X}(1),\mathcal{O}_{X}(2)\rangle$

by [Kuz10], where
$\operatorname{Ku}(X)=\langle\mathcal{O}_{X},\mathcal{O}_{X}(1),\mathcal{O}_{X}(2)\rangle^{\perp}$.
The Serre functor of $\operatorname{Ku}(X)$ satisfies $\mathcal{S}=[2]$. By
[AT14] the numerical Grothendieck group of $X$ contains two classes
$\lambda_{1}$ and $\lambda_{2}$ spanning a rank two $A_{2}$-lattice

$\langle\lambda_{1},\lambda_{2}\rangle\cong\begin{pmatrix}2&-1\\\
-1&2\end{pmatrix}.$

with respect to the intersection pairing $(-,-):=-\chi(-,-)$. We say that $X$
is very general if
$\mathcal{N}(\operatorname{Ku}(X))=\langle\lambda_{1},\lambda_{2}\rangle$.

######  Corollary 3.9.

Let $X$ be a very general cubic fourfold. If $\sigma_{1}$, $\sigma_{2}$ are
stability conditions on $\operatorname{Ku}(X)$, Then there exists
$\tilde{g}\in\widetilde{\emph{GL}}^{+}(2,\mathbb{R})$ such that
$\sigma_{1}=\sigma_{2}\cdot\tilde{g}$.

###### Proof.

The Serre functor of $\operatorname{Ku}(X)$ satisfies $\mathcal{S}^{2}=[4]$ as
required in (C1). Since $X$ is very general, we have

$\mathcal{N}(\operatorname{Ku}(X))\cong\langle\lambda_{1},\lambda_{2}\rangle\cong\begin{pmatrix}2&-1\\\
-1&2\end{pmatrix}.$

For every $v\in\mathcal{N}(\operatorname{Ku}(X))$ we have
$v=a\lambda_{1}+b\lambda_{2}$. Thus

$v^{2}=2a^{2}+2b^{2}-2ab\geq 2.$

It follows that $\chi(v,v)=-v^{2}\leq-2$, which implies
$\ell_{\operatorname{Ku}(X)}:=\text{max}\\{\chi(v,v):v\neq
0\in\mathcal{N}(\operatorname{Ku}(X))\\}=-2<0$ as required in (C2).

We now check (C3). Recall that given a line $\ell\subset X$, we can define the
objects $F_{\ell}$, $P_{\ell}\in\operatorname{Ku}(X)$ sitting in

$0\to F_{\ell}\to\mathcal{O}_{X}^{\oplus 4}\to\mathcal{I}_{\ell}(1)\to 0,$
$\mathcal{O}_{X}(-1)[1]\to P_{\ell}\to\mathcal{I}_{\ell}.$

Their classes in $\mathcal{N}(\operatorname{Ku}(X))$ are
$v(F_{\ell})=\lambda_{1}$, $v(P_{\ell})=\lambda_{1}+\lambda_{2}$ as computed
in [LLMS18, Section 6.3]. By [BLMS17, Lemma A.5] they are stable with respect
to every stability condition on $\operatorname{Ku}(X)$ and by [LLMS18,
(6.3.1)] their phases with respect to the stability conditions constructed in
[BLMS17] satisfy

(11) $\phi(F_{\ell})<\phi(P_{\ell})<\phi(F_{\ell})+1.$

Since $\chi(F_{\ell},F_{\ell})=-2=2-\hom^{1}(F_{\ell},F_{\ell})$ and similarly
for $P_{\ell}$, we have

$3\leq\hom^{1}(F_{\ell},F_{\ell})=\hom^{1}(P_{\ell},P_{\ell})=4<6.$

For two lines $\ell$ and $\ell^{\prime}$, we have

$\displaystyle\hom(F_{\ell},P_{\ell^{\prime}})$
$\displaystyle=\hom(\mathcal{I}_{\ell}(1),P_{\ell^{\prime}}[1])$
$\displaystyle=\hom(\mathcal{I}_{\ell}(1),\mathcal{I}_{\ell^{\prime}}[1])$
$\displaystyle=\hom(\mathcal{I}_{\ell}(1),\mathcal{O}_{\ell^{\prime}})$
$\displaystyle=\hom(\mathcal{O}_{\ell}(1),\mathcal{O}_{\ell^{\prime}}[1]).$

The latter is equal to $0$ if $\ell\cap\ell^{\prime}=\emptyset$. Assume $\ell$
and $\ell^{\prime}$ intersect in a point. Then by the local to global spectral
sequence, we only have to consider
$H^{0}(\mathcal{E}xt^{1}(\mathcal{O}_{\ell}(1),\mathcal{O}_{\ell^{\prime}}))$.
Since $\mathcal{E}xt^{1}(\mathcal{O}_{\ell}(1),\mathcal{O}_{\ell^{\prime}})$
is supported on the intersection point, we conclude that
$\hom(\mathcal{O}_{\ell}(1),\mathcal{O}_{\ell^{\prime}}[1])=1$ and thus
$\hom(F_{\ell},P_{\ell^{\prime}})=1\neq 0$ if $\ell$, $\ell^{\prime}$
intersect in a point.

We know
$\chi(P_{\ell^{\prime}},F_{\ell})=-(\lambda_{1}+\lambda_{2},\lambda_{1})=-1$.
By (11) and Serre duality $\hom(P_{\ell^{\prime}},F_{\ell}[i])=0$ for $i\neq
0,1,2$, hence

$-1=\chi(P_{\ell^{\prime}},F_{\ell})=\hom(P_{\ell^{\prime}},F_{\ell})-\hom^{1}(P_{\ell^{\prime}},F_{\ell})+\hom^{2}(P_{\ell^{\prime}},F_{\ell})$

which implies $\hom(P_{\ell^{\prime}},F_{\ell}[1])\geq 1$ for every pair of
lines $\ell$, $\ell^{\prime}$. Thus (C3) holds if we set
$Q_{1}=P_{\ell^{\prime}}$ and $Q_{2}=F_{\ell}$. ∎

##  4\. Kuznetsov component of cubic threefolds

From now on, we assume $X$ is a smooth complex cubic threefold, and
$\mathcal{O}_{X}(H)$ denotes the corresponding very ample line bundle.

###  4.1. Kuznetsov component

The Kuznetsov component $\operatorname{Ku}(X)$ is the right orthogonal
complement of the exceptional collection $\mathcal{O}_{X},\mathcal{O}_{X}(H)$
in $\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)$ sitting in the
semiorthogonal decomposition

$\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)=\langle\operatorname{Ku}(X),\mathcal{O}_{X},\mathcal{O}_{X}(H)\rangle$

(see [Kuz04]). By [BMMS12, Proposition 2.7] the numerical Grothendieck group
$\mathcal{N}(\operatorname{Ku}(X))$ of $\operatorname{Ku}(X)$ is a rank 2
lattice

$\mathcal{N}(\operatorname{Ku}(X))=\mathbb{Z}\,[\mathcal{I}_{\ell}]\oplus\mathbb{Z}\,[\mathcal{S}(\mathcal{I}_{\ell})]$

where $\mathcal{I}_{\ell}$ is the ideal sheaf of a line $\ell$ in $X$ and
$\mathcal{S}$ denotes the Serre functor of $\operatorname{Ku}(X)$. With
respect to this basis, the Euler form $\chi_{\operatorname{Ku}(X)}(-,-)$ on
$\mathcal{N}(\operatorname{Ku}(X))$ is represented by

(12) $\begin{pmatrix}-1&-1\\\ 0&-1\end{pmatrix}.$

Note that by [Kuz04, Lemma 4.1, Lemma 4.2], the functor

$\mathrm{O}\colon\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)\to\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X),\quad\mathrm{O}(-)=L_{\mathcal{O}_{X}}(-\otimes\mathcal{O}_{X}(H))[-1]$

satisfies $\mathrm{O}|_{\operatorname{Ku}(X)}^{2}=\mathcal{S}^{-1}[1]$ and
$\mathrm{O}|_{\operatorname{Ku}(X)}^{3}\cong[-1]$. Thus

$\mathcal{S}[-3]=\mathcal{S}^{-2}[2]=\mathrm{O}|_{\operatorname{Ku}(X)}^{4}=\mathrm{O}|_{\operatorname{Ku}(X)}[-1]$

which implies that

$\mathcal{S}=\mathrm{O}|_{\operatorname{Ku}(X)}[2]=L_{\mathcal{O}_{X}}(-\otimes\mathcal{O}_{X}(H))[1].$

###  4.2. Stability conditions on $\operatorname{Ku}(X)$

Stability conditions on $\operatorname{Ku}(X)$ have been first constructed in
[BMMS12] and more recently in [BLMS17]. We recall here the definition of the
latter, while the former will be reviewed in Section 5.2.

For $\beta\in\mathbb{R}$, denote by $\operatorname{Coh}^{\beta}(X)$ the heart
of a bounded t-structure obtained by tilting $\operatorname{Coh}(X)$ with
respect to slope stability at slope $\mu_{H}=\beta$. By [BLMS17, Proposition
2.12] for $\alpha\in\mathbb{R}_{>0}$ the pair
$\sigma_{\alpha,\beta}=(\operatorname{Coh}^{\beta}(X),Z_{\alpha,\beta})$
defines a weak stability condition on
$\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)$, known as tilt stability, with
respect to the lattice $\Lambda\cong\mathbb{Z}^{3}$ generated by the elements
of the form
$(H^{3}\operatorname{ch}_{0}(E),H^{2}\operatorname{ch}_{1}(E),H\operatorname{ch}_{2}(E))\in\mathbb{Q}^{3}$
for $E\in\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)$ (see Section 2.2). Here

$Z_{\alpha,\beta}\coloneqq\frac{1}{2}\alpha^{2}H^{3}\operatorname{ch}_{0}^{\beta}-H\operatorname{ch}_{2}^{\beta}+iH^{2}\operatorname{ch}_{1}^{\beta}$

with $\operatorname{ch}^{\beta}\coloneqq
e^{-\beta}\operatorname{ch}=(\operatorname{ch}_{0},\operatorname{ch}_{1}-\beta
H\operatorname{ch}_{0},\operatorname{ch}_{2}-\beta
H\operatorname{ch}_{1}+\frac{\beta^{2}H^{2}}{2}\operatorname{ch}_{0},\dots)$.
Its associated slope is

$\mu_{\alpha,\beta}(E)=-\frac{\Re Z_{\alpha,\beta}(E)}{\Im
Z_{\alpha,\beta}(E)}\quad\text{for }\Im Z_{\alpha,\beta}(E)\neq 0$

where $\Re[-]$ and $\Im[-]$ are the real and imaginary parts.

Now consider the tilted heart
$\operatorname{Coh}^{0}_{\alpha,\beta}(X)=\langle\mathcal{F}_{\alpha,\beta}[1],\mathcal{T}_{\alpha,\beta}\rangle$
where $\mathcal{F}_{\alpha,\beta}$ ($\mathcal{T}_{\alpha,\beta}$) is the
subcategory of objects in $\operatorname{Coh}^{\beta}(X)$ with
$\mu^{+}_{\alpha,\beta}\leq 0$ ( $\mu^{-}_{\alpha,\beta}>0$). By [BLMS17,
Proposition 2.14], the pair
$\sigma^{0}_{\alpha,\beta}=\left(\operatorname{Coh}^{0}_{\alpha,\beta}(X),-iZ_{\alpha,\beta}\right)$
is a weak stability condition on
$\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)$.

Sheaves supported in codimension 3 are the only objects in the heart
$\operatorname{Coh}^{\beta}(X)$ whose central charge $Z_{\alpha,\beta}$
vanishes. We denote by $\operatorname{Coh}_{0}(X)$ the category of sheaves on
$X$ of co-dimension 3.

######  Proposition 4.1.

Any $\sigma^{0}_{\alpha,\beta}$-(semi)stable object
$E\in\operatorname{Coh}^{0}_{\alpha,\beta}(X)$ is
$\sigma_{\alpha,\beta}$-(semi)stable if it does not lie in an exact triangle
of the form

$F[1]\rightarrow E\rightarrow T$

where $F\in\mathcal{F}_{\alpha,\beta}$ and $T\in\operatorname{Coh}_{0}(X)$.
Conversely, take a $\sigma_{\alpha,\beta}$-(semi)stable object $E$ such that
either

  1. (a)

$E\in\mathcal{T}_{\alpha,\beta}$ and
$\operatorname{Hom}(\operatorname{Coh}_{0}(X),E)=0$, or

  2. (b)

$E\in\mathcal{F}_{\alpha,\beta}$ and
$\operatorname{Hom}(\operatorname{Coh}_{0}(X),E[1])=0$.

Then $E$ is $\sigma^{0}_{\alpha,\beta}$-(semi)stable.

###### Proof.

First assume $E\in\operatorname{Coh}^{0}_{\alpha,\beta}$ is
$\sigma^{0}_{\alpha,\beta}$-(semi)stable, so by definition it lies in an exact
triangle

(13) $F[1]\rightarrow E\rightarrow T$

for some $F\in\mathcal{F}_{\alpha,\beta}$ and
$T\in\mathcal{T}_{\alpha,\beta}$. If $Z_{\alpha,\beta}(T)\neq 0$, i.e.
$T\notin\operatorname{Coh}_{0}(X)$, we have

$\Re[-iZ_{\alpha,\beta}(T)]\geq
0\qquad\text{and}\qquad\Re[-iZ_{\alpha,\beta}(F[1])]<0$

This shows that phase of $F[1]$ is bigger than that of $T$ with respect to
$\sigma^{0}_{\alpha,\beta}$. Thus the exact triangle (13) implies that one of
the following cases happens:

  1. (i)

$E=T\in\mathcal{T}_{\alpha,\beta}$,

  2. (ii)

$E=F[1]\in\mathcal{F}_{\alpha,\beta}[1]$, or

  3. (iii)

$T\in\operatorname{Coh}_{0}(X)$.

The third case (iii) is excluded in the statement, so we may assume the first
two cases happens. We only consider the case $E\in\mathcal{T}_{\alpha,\beta}$,
the other one can be shown by a similar argument. Suppose for a contradiction
that $E$ is not $\sigma_{\alpha,\beta}$-(semi)stable, and let

(14) $E_{1}\rightarrow E\rightarrow E_{2}$

be a destabilising sequence in $\operatorname{Coh}^{\beta}(X)$. We may assume
$E_{1}$ is $\sigma_{\alpha,\beta}$-semistable, thus

$\mu_{\alpha,\beta}^{-}(E_{1})=\mu_{\alpha,\beta}(E_{1})\geq\mu_{\alpha,\beta}(E_{2})\geq\mu_{\alpha,\beta}^{-}(E_{2})\geq\mu_{\alpha,\beta}^{-}(E)\geq
0.$

The right inequality follows from the definition of
$\mathcal{T}_{\alpha,\beta}$. Thus both $E_{1}$ and $E_{2}$ lie in
$\mathcal{T}_{\alpha,\beta}$, hence (14) is also a destabilising sequence for
$E$ in $\operatorname{Coh}^{0}_{\alpha,\beta}$ with respect to
$\sigma^{0}_{\alpha,\beta}$, a contradiction.

Now assume $E\in\mathcal{T}_{\alpha,\beta}$ is
$\sigma_{\alpha,\beta}$-(semi)stable and
$\operatorname{Hom}(\operatorname{Coh}_{0}(X),E)=0$. In the rest of the
argument, all $T_{i}$’s lie in $\mathcal{T}_{\alpha,\beta}$ and all $F_{i}$’s
lie in $\mathcal{F}_{\alpha,\beta}$. Suppose $E_{1}\rightarrow E\rightarrow
E_{2}$ is a destabilising sequence with respect to
$\sigma^{0}_{\alpha,\beta}$. We can assume $E_{2}$ is
$\sigma^{0}_{\alpha,\beta}$-semistable. Thus by the above argument, we have
either (i) $E_{2}\in\mathcal{F}_{\alpha,\beta}[1]$, or (ii)
$E_{2}\in\mathcal{T}_{\alpha,\beta}$, or (iii) $E_{2}$ lies in an exact
triangle $F_{2}[1]\rightarrow E_{2}\rightarrow T_{2}$ where
$T_{2}\in\operatorname{Coh}_{0}(X)$.

Since $E\in\mathcal{T}_{\alpha,\beta}$, the phase of $E$ with respect to
$\sigma^{0}_{\alpha,\beta}$ is less than or equal to $1/2$. But we know phase
of objects in $\mathcal{F}_{\alpha,\beta}[1]$ is bigger than $1/2$, thus the
first case (i) cannot happen. In third case (iii), we know phase of $E_{2}$ is
equal to the phase of $F_{2}[1]$, so again this case cannot happen. Hence, we
may assume $E_{2}\in\mathcal{T}_{\alpha,\beta}$.

By definition, we know $E_{1}$ lies in an exact triangle $F_{1}[1]\rightarrow
E_{1}\rightarrow T_{1}$. But the composition of the injections
$F_{1}[1]\rightarrow E_{1}\rightarrow E$ is zero because
$\hom(\mathcal{F}_{\alpha,\beta}[1],\mathcal{T}_{\alpha,\beta})=0$. Thus
$E_{1}\in\mathcal{T}_{\alpha,\beta}$ and so the destabilising sequence is an
exact sequence in the original heart $\operatorname{Coh}^{\beta}(X)$ which
(semi)destabilises $E$ with respect to $\sigma_{\alpha,\beta}$, a
contradiction.

Finally suppose $E\in\mathcal{F}_{\alpha,\beta}$ is
$\sigma_{\alpha,\beta}$-(semi)stable and
$\hom(\operatorname{Coh}_{0}(X),E[1])=0$. Let $E_{1}\rightarrow
E[1]\rightarrow E_{2}$ be a destabilising sequence with respect to
$\sigma^{0}_{\alpha,\beta}$. We may assume $E_{1}$ is
$\sigma^{0}_{\alpha,\beta}$-semistable. We know
$E_{1}\notin\operatorname{Coh}_{0}(X)$ and phase of objects in
$\mathcal{T}_{\alpha,\beta}\setminus\operatorname{Coh}_{0}(X)$ with respect to
$\sigma_{\alpha,\beta}^{0}$ is less than phase of objects in
$\mathcal{F}_{\alpha,\beta}[1]$. Thus the same argument as in case (a) implies
that $E_{1}$ lies in an exact triangle $F_{1}[1]\rightarrow E_{1}\rightarrow
T_{1}$ where $T_{1}\in\operatorname{Coh}_{0}(X)$ and $F_{1}\neq 0$. By
definition, $E_{2}$ lies in an exact triangle $F_{2}[1]\rightarrow
E_{2}\rightarrow T_{2}$. Since
$\hom(\mathcal{F}_{\alpha,\beta}[1],\mathcal{T}_{\alpha,\beta})=0$, the
composition of surjections $E[1]\rightarrow E_{2}\rightarrow T_{2}$ is zero.
Thus $T_{2}=0$ and $E_{2}=F_{2}[1]$. Taking cohomology from the destabilising
sequence with respect to the heart $\operatorname{Coh}^{\beta}(X)$ gives the
long exact sequence

$0\rightarrow F_{1}\rightarrow E\rightarrow F_{2}\rightarrow T_{1}\rightarrow
0.$

Thus $F_{1}$ is a subobject of $E$ in $\operatorname{Coh}^{\beta}(X)$. We know
phase of $E_{1}$ is equal to phase of $F_{1}[1]$ with respect to
$\sigma^{0}_{\alpha,\beta}$ and it is bigger than or equal to the phase of
$E[1]$. This implies that phase of $F_{1}$ is bigger than or equal to the
phase of $E$ with respect to $\sigma_{\alpha,\beta}$, a contradiction. ∎

By restricting weak stability conditions $\sigma^{0}_{\alpha,\beta}$ to the
Kuznetsov component $\operatorname{Ku}(X)$, we obtain a stability condition on
it.

######  Theorem 4.2.

[BLMS17, Theorem 6.8] [PY20, Theorem 3.3 and Corollary 5.5] For every
$(\alpha,\beta)$ in the set

(15)
$V\coloneqq\left\\{(\alpha,\beta)\in\mathbb{R}_{>0}\times\mathbb{R}:-\frac{1}{2}\leq\beta<0,\alpha<-\beta,\text{
or }-1<\beta<-\frac{1}{2},\alpha\leq 1+\beta\right\\},$

the pair $\sigma(\alpha,\beta)=(\mathcal{A}(\alpha,\beta),Z(\alpha,\beta))$ is
a $\mathcal{S}$-invariant stability condition on $\operatorname{Ku}(X)$ where

$\mathcal{A}(\alpha,\beta):=\operatorname{Coh}_{\alpha,\beta}^{0}(X)\cap\operatorname{Ku}(X)\qquad\text{and}\qquad
Z(\alpha,\beta)\coloneqq-iZ_{\alpha,\beta}|_{\operatorname{Ku}(X)}.$

As an application of Theorem 3.2, we get the following.

######  Corollary 4.3.

All $\mathcal{S}$-invariant stability conditions on $\operatorname{Ku}(X)$
such as $\sigma(\alpha,\beta)$ for $(\alpha,\beta)\in V$ lay in the same orbit
with respect to the action of $\widetilde{\emph{GL}}^{+}(2,\mathbb{R})$.

###### Proof.

We know $\mathcal{S}^{3}=[5]$ and the numerical Grothendieck group
$\mathcal{N}(\operatorname{Ku}(X))$ is a lattice of rank $2$ such that
$\chi_{\operatorname{Ku}(X)}(v,v)\leq-1$ for any $0\neq
v\in\mathcal{N}(\operatorname{Ku}(X))$. Thus the conditions (C1) and (C2) hold
for $\operatorname{Ku}(X)$. We also have
$\dim_{\mathbb{C}}\operatorname{Hom}^{1}(\mathcal{I}_{\ell},\mathcal{I}_{\ell})=2$
as the Hilbert scheme of lines in $X$ is a smooth surface [AK77, Section 1].
Thus the claim follows from Theorem 3.2. ∎

######  Remark 4.4.

Theorem 3.2 also applies to the Kuznetsov component of the quartic double
solid $Y$, which is a double cover $Y\to\mathbb{P}^{3}$ ramified in a quartic.
In fact, by [Kuz09] there is a semiorthogonal decomposition of the form
$\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(Y)=\langle\operatorname{Ku}(Y),\mathcal{O}_{Y},\mathcal{O}_{Y}(H)\rangle$,
where $H$ is an ample class, and the numerical Grothendieck group of
$\operatorname{Ku}(Y)$ is a rank two lattice represented by the matrix

$\begin{pmatrix}-1&-1\\\ -1&-2\end{pmatrix}.$

An easy computation shows that (C2) holds. Moreover, the Serre functor of
$\operatorname{Ku}(Y)$ satisfies $S_{\operatorname{Ku}(Y)}=\iota[2]$ where
$\iota$ is an involutive autoequivalence of $\operatorname{Ku}(Y)$ by [Kuz19,
Corollary 4.6]. Thus (C1) holds with $k=2r=4$. It remains to find two objects
as in (C3). Consider the ideal sheaf of a line $\mathcal{I}_{\ell}$ and its
derived dual
$\mathcal{J}_{\ell}:=R\operatorname{Hom}(\mathcal{I}_{\ell},\mathcal{O}(-H)[1])$.
By [PY20, Remark 3.8] for proper choices of lines, they provide the required
objects.

###  4.3. Stable objects in $\operatorname{Ku}(X)$ with minimal $\hom^{1}$

By (12) the classes in $\mathcal{N}(\operatorname{Ku}(X))$ with self-
intersection $-1$ are

$\pm[\mathcal{I}_{\ell}]\,,\quad\pm[\mathcal{S}(\mathcal{I}_{\ell})]\
\quad\text{and}\qquad\pm[\mathcal{S}^{2}(\mathcal{I}_{\ell})]=\pm\left([\mathcal{I}_{\ell}]-[\mathcal{S}(\mathcal{I}_{\ell})]\right).$

In this section, we investigate the moduli of semistable objects of the above
classes. We denote by $M^{\sigma}_{\operatorname{Ku}(X)}(v)$ the moduli space
of $\sigma$-semistable objects of class
$v\in\mathcal{N}(\operatorname{Ku}(X))$.

######  Theorem 4.5.

Let $\sigma$ be an $\mathcal{S}$-invariant stability condition on
$\operatorname{Ku}(X)$.

  1. (a)

The moduli space $M^{\sigma}_{\operatorname{Ku}(X)}([\mathcal{I}_{\ell}])$ is
isomorphic to the moduli space $M_{X}(v)$ of slope-stable sheaves on $X$ with
Chern character
$v\coloneqq\operatorname{ch}(\mathcal{I}_{\ell})=\left(1,0,-\frac{H^{2}}{3},0\right)$.

  2. (b)

The moduli space
$M^{\sigma}_{\operatorname{Ku}(X)}([\mathcal{S}(\mathcal{I}_{\ell})])$ is
isomorphic to the moduli space $M_{X}(w)$ of slope-stable sheaves on $X$ with
Chern character
$w\coloneqq\left(2,-H,-\frac{H^{2}}{6},\frac{H^{3}}{6}\right)$.

  3. (c)

The moduli space
$M^{\sigma}_{\operatorname{Ku}(X)}([\mathcal{S}^{2}(\mathcal{I}_{\ell})])$ is
isomorphic to the moduli space $M_{X}(v-w)$ of large volume limit stable
complexes of character $v-w$.

In particular, all the above three moduli spaces are isomorphic to the Fano
variety $\Sigma(X)$ of lines in $X$. This proves Theorem 1.3. The first part
(a) of Theorem 4.5 is proved in [PY20]; however we prove it here via a
slightly different argument.

We start with a slope-stable sheaf $E$ with Chern character $v$. So $E$ is
torsion-free with double dual $E^{\vee\vee}=\mathcal{O}_{X}$. Thus it lies in
the exact sequence

$E\hookrightarrow\mathcal{O}_{X}\twoheadrightarrow Q$

where $Q$ is a torsion sheaf of character $\left(0,0,\frac{H^{2}}{3},0\right)$
so is isomorphic to the structure sheaf $\mathcal{O}_{\ell}$ of a line $\ell$
in $X$. Thus $E$ is isomorphic to the ideal sheaf $\mathcal{I}_{\ell}$.

By Lemma 2.6 $\mathcal{I}_{\ell}$ is $\sigma_{\alpha,\beta}$-stable for
$\beta<0$ and $\alpha\gg 0$. Applying [Fey21, Lemma 3.5] to
$\mathcal{I}_{\ell}$ implies that there is no wall for it crossing the
vertical line $\beta=\beta_{0}=-\frac{1}{H^{3}}$. Thus $\mathcal{I}_{\ell}$ is
$\sigma_{\alpha,\beta_{0}}$-stable for any $\alpha>0$. Since
$\mu_{\alpha,\beta_{0}}(\mathcal{I}_{\ell})<0$, we have
$\mathcal{I}_{\ell}[1]\in\operatorname{Coh}_{\alpha,\beta_{0}}^{0}(X)$. For
any sheaf $Q\in\operatorname{Coh}_{0}(X)$, we know

$\hom(Q,\mathcal{O}_{\ell})=0=\hom(Q,\mathcal{O}_{X}[1]).$

Thus $\hom(Q,\mathcal{I}_{\ell}[1])=0$ and so Proposition 4.1 implies that
$\mathcal{I}_{\ell}[1]$ is $\sigma^{0}_{\alpha,\beta_{0}}$-stable. Since
$\mathcal{I}_{\ell}\in\operatorname{Ku}(X)$, we immediately get
$\mathcal{I}_{\ell}$ is $\sigma(\alpha,\beta_{0})$-stable. Thus by Corollary
4.3 we get the following.

######  Lemma 4.6.

The ideal sheaf $\mathcal{I}_{\ell}$ for any line $\ell$ in $X$ is stable with
respect to any $\mathcal{S}$-invariant stability condition on
$\operatorname{Ku}(X)$.

By definition $\mathcal{S}(\mathcal{I}_{\ell})$ is also stable with respect to
any $\mathcal{S}$-invariant stability condition on $\operatorname{Ku}(X)$. The
next step is to describe the complex $\mathcal{S}(\mathcal{I}_{\ell})$. We
know $\mathcal{I}_{\ell}(H)$ is of character $(1,H,H^{2}/6,-H^{3}/6)$ so
$\chi(\mathcal{O}_{X},\mathcal{I}_{\ell}(H))=3$. Consider the short exact
sequence

$\mathcal{I}_{\ell}(H)\hookrightarrow\mathcal{O}_{X}(H)\twoheadrightarrow\mathcal{O}_{\ell}(H).$

Since the map
$\operatorname{Hom}(\mathcal{O}_{X},\mathcal{O}_{X}(H))\rightarrow\operatorname{Hom}(\mathcal{O}_{X},\mathcal{O}_{\ell}(H))$
is surjective and $H^{i}(\mathcal{O}_{X}(H))=0$ for $i>0$ by Kodaira
vanishing, we get

$\hom(\mathcal{O}_{X},\mathcal{I}_{\ell}(H)[1])=\hom(\mathcal{O}_{X},\mathcal{I}_{\ell}(H)[2])=0.$

Thus we gain

$\mathcal{S}(\mathcal{I}_{\ell})[-1]=L_{\mathcal{O}_{X}}(\mathcal{I}_{\ell}(H))=cone(\mathcal{O}_{X}^{\oplus
3}\xrightarrow{ev}\mathcal{I}_{\ell}(H))$

Since Pic$(X)=\mathbb{Z}.H$, [Fey21, Lemma 3.5] implies that
$\mathcal{I}_{\ell}(H)$ is $\sigma_{\alpha,\beta=0}$-stable for any
$\alpha>0$. Thus $\mathcal{I}_{\ell}(H)$ is $\sigma_{\alpha,\beta}$-semistable
along the numerical wall $W(\mathcal{I}_{\ell}(H),\mathcal{O}_{X}[1])$ where
$\mathcal{I}_{\ell}(H)$ and $\mathcal{O}_{X}[1]$ have the same phase.
Therefore, their extension $L_{\mathcal{O}_{X}}(\mathcal{I}_{\ell}(H))$ is
also $\sigma_{\alpha,\beta}$-semistable along the wall. One can easily show
that $W(\mathcal{I}_{\ell}(H),\mathcal{O}_{X}[1])$ intersects the horizontal
line $\alpha=0$ at two points with $\beta$-values $0$ and $\frac{1}{3}$. Thus
by definition of the heart $\operatorname{Coh}^{\beta}(X)$,
$\mathcal{S}(\mathcal{I}_{\ell})$ is a two-term complex with cohomology in
degree -2 and -1, and

(16)
$\mu_{H}^{+}\left(\mathcal{H}^{-2}(\mathcal{S}(\mathcal{I}_{\ell}))\right)\leq
0.$

Also, since $\mathcal{H}^{-1}(\mathcal{S}(\mathcal{I}_{\ell}))$ is a quotient
sheaf of $\mathcal{I}_{\ell}(H)$, we get

$1\leq\mu_{H}\left(\mathcal{H}^{-1}(\mathcal{S}(\mathcal{I}_{\ell}))\right).$

We know
$\operatorname{ch}\left(\mathcal{S}(\mathcal{I}_{\ell})\right)=w=\left(2,-H,-\frac{H^{2}}{6},\frac{H^{3}}{6}\right)$.
The rank of $\mathcal{H}^{-1}(\mathcal{S}(\mathcal{I}_{\ell}))$ is lower than
$1=\operatorname{rank}(\mathcal{I}_{\ell}(H))$. If it is of rank one, then the
image im$(ev)$ in the category of coherent sheaves is of rank zero which is
not possible because $\mathcal{I}_{\ell}(H)$ is torsion-free. Thus
$\mathcal{H}^{-1}(\mathcal{S}(\mathcal{I}_{\ell}))$ is of rank zero, so

$(-1)^{i+1}\operatorname{ch}_{1}\left(\mathcal{H}^{i}(\mathcal{S}(\mathcal{I}_{\ell}))\right).H^{2}\geq
0$

for $i=-2,-1$. Thus
$\operatorname{ch}_{1}\left(\mathcal{H}^{-1}(\mathcal{S}(\mathcal{I}_{\ell}))\right)$
is either equal to zero or $H$. In the latter case, we will have
$\operatorname{ch}_{\leq
2}\left(\mathcal{H}^{-2}(\mathcal{S}(\mathcal{I}_{\ell}))\right)=(2,0,-\alpha)$
for some $\alpha\geq 0$ because of Gieseker-semistability of
$\mathcal{O}_{X}^{\oplus 3}$. Then im$(ev)$ is a subsheaf of
$\mathcal{I}_{\ell}(H)$ of rank one, so it is torsion-free of class
$\operatorname{ch}(\text{im}(ev))=(1,0,\alpha)$ and so $\alpha\leq 0$, thus in
total we gain $\alpha=0$.

Applying a similar argument also implies that
$\operatorname{ch}_{3}\left(\mathcal{H}^{-2}(\mathcal{S}(\mathcal{I}_{\ell}))\right)=0$,
thus
$\mathcal{H}^{-2}(\mathcal{S}(\mathcal{I}_{\ell}))=\mathcal{O}_{X}^{\oplus 2}$
which is not possible by the definition of the evaluation map $ev$. Therefore
$\mathcal{H}^{-1}(\mathcal{S}(\mathcal{I}_{\ell}))$ is a sheaf supported in
dim $\leq 1$.

We also claim $\mathcal{H}^{-2}(\mathcal{S}(\mathcal{I}_{\ell}))$ is slope-
stable. Otherwise, there is a subsheaf $F$ of bigger slope. By (16), it must
be of class $\operatorname{ch}_{\leq 1}(F)=(1,0)$. Semistability of
$\mathcal{S}(\mathcal{I}_{\ell})[-1]$ along the wall implies that
$\mathcal{H}^{-2}(\mathcal{S}(\mathcal{I}_{\ell}))$ is a reflexive sheaf. If
not, $\mathcal{H}^{-2}(\mathcal{S}(\mathcal{I}_{\ell}))[1]$ and so
$\mathcal{S}(\mathcal{I}_{\ell})[-1]$ has a subobject supported in dimension
$\leq 1$ of phase 1 which is not possible. Since
$\mathcal{H}^{-2}(\mathcal{S}(\mathcal{I}_{\ell}))$ is of rank 2, its quotient
sheaf $\mathcal{H}^{-2}(\mathcal{S}(\mathcal{I}_{\ell}))/F$ is stable and it
must be also reflexive. Taking double dual implies that $F^{\vee\vee}=F$, thus
$F$ is a line bundle and so $F\cong\mathcal{O}_{X}$ which is in contradiction
to the definition of the evaluation map $ev$. Therefore
$\mathcal{H}^{-2}(\mathcal{S}(\mathcal{I}_{\ell}))$ is slope-stable.

The next proposition implies that
$\mathcal{H}^{-1}(\mathcal{S}(\mathcal{I}_{\ell}))$ is zero and so the
evaluation map $ev$ is surjective and the kernel is a reflexive sheaf of Chern
character $w$.

######  Lemma 4.7.

Take an $H$-slope stable sheaf $E$ of Chern character
$(2,-H,\operatorname{ch}_{2},\operatorname{ch}_{3})$. Then
$\operatorname{ch}_{2}.H\leq-\frac{1}{6}H^{3}$, and if
$\operatorname{ch}_{2}.H=-\frac{1}{6}H^{3}$, then
$\operatorname{ch}_{3}\leq\frac{H^{3}}{6}$. This, in particular, implies that
any $H$-slope stable sheaf of Chern character $w$ is a reflexive sheaf.

###### Proof.

Proposition 2.9 implies that $\operatorname{ch}_{2}.H\leq 0$. Since
$c_{2}(E)=H^{2}/2-\operatorname{ch}_{2}$ is an integral class, we obtain
$\operatorname{ch}_{2}.H\leq-\frac{H^{3}}{6}$. Assume
$\operatorname{ch}_{2}.H=-\frac{H^{3}}{6}$. There is no wall for $E$ crossing
the vertical line $\beta=-1$ by [Fey21, Lemma 3.5], thus $E$ is
$\sigma_{\alpha,\beta=-1}$-stable for any $\alpha>0$. One can easily compute
that

$\lim_{\alpha\rightarrow
0}\mu_{\alpha,\,\beta=-1}(\mathcal{O}_{X}(-2H)[1])<\lim_{\alpha\rightarrow
0}\mu_{\alpha,\,\beta=-1}(E)$

which implies $\hom(E,\mathcal{O}_{X}(-2H)[1])=0$ because
$\mathcal{O}_{X}(-2H)[1]$ is $\sigma_{\alpha,\beta}$-stable for any $\alpha>0$
and $\beta\in\mathbb{R}$ [BMS16, Corollary 3.11]. Moreover, since

$\mu_{H}(E)=\frac{-1}{2}<\mu_{H}(\mathcal{O}_{X}),$

we get $\hom(\mathcal{O}_{X},E)=0$. Therefore
$\chi(\mathcal{O}_{X},E)=\operatorname{ch}_{3}(E)-\frac{H^{3}}{6}\leq 0$ as
claimed. ∎

Thus we gain

$\mathcal{S}(\mathcal{I}_{\ell})=L_{\mathcal{O}_{X}}(\mathcal{I}_{\ell}(H))[1]=K_{\ell}[2]$

for a slope-stable reflexive sheaf $K_{\ell}$ of Chern character $w$. The next
proposition shows that any slope-stable sheaf $K$ of class $w$ is of the form
$L_{\mathcal{O}_{X}}(\mathcal{I}_{\ell}(H))[-2]$ for a line $\ell$ on $X$.

######  Proposition 4.8.

Take a slope-stable sheaf $K$ of Chern character $w$. There exists a line
$\ell$ on $X$ such that $K$ lies in the short exact sequence

$0\rightarrow K\rightarrow\mathcal{O}_{X}^{\oplus
3}\rightarrow\mathcal{I}_{\ell}(H)\rightarrow 0.$

In other words $K=\mathcal{S}(\mathcal{I}_{\ell})[-2]$.

###### Proof.

Lemma 4.7 implies $K$ is a reflexive sheaf, thus $K[1]$ is
$\sigma_{\alpha,\beta}$-stable for $\beta\geq-\frac{1}{2}$ and $\alpha\gg 0$
[BBF+20, Proposition 4.17]. Since there is no wall for $K[1]$ crossing the
vertical line $\beta=0$, it is $\sigma_{\alpha,\beta}$-semistable along the
numerical wall $W(K[1],\mathcal{O}_{X}[1])$. We have

$\lim_{\alpha\rightarrow
0}\mu_{\alpha,\,\beta=0}(K[1])<\lim_{\alpha\rightarrow
0}\mu_{\alpha,\,\beta=0}(\mathcal{O}_{X}(2H)).$

Thus $\hom(K,\mathcal{O}_{X}[2])=\hom(\mathcal{O}_{X}(2H),K[1])=0$. Also
$\hom(K,\mathcal{O}_{X}[3])=\hom(\mathcal{O}_{X}(2H),K)=0$ because
$\mu_{H}(K)<0$. Therefore

$3=\chi(K,\mathcal{O}_{X})=\hom(K,\mathcal{O}_{X})-\hom(K,\mathcal{O}_{X}[1])$

which implies that there is a non-zero map
$K[1]\rightarrow\mathcal{O}_{X}[1]$. Therefore $K[1]$ is unstable below the
numerical wall $W(K[1],\mathcal{O}_{X}[1])$.

Choose three linearly independent sections of $K^{\vee}$:

$\psi\colon K[1]\rightarrow\mathcal{O}_{X}^{\oplus 3}[1]$

We know $K[1]$ and $\mathcal{O}_{X}[1]$ are $\sigma_{\alpha,\beta}$-semistable
of the same phase $\phi$ and $\mathcal{O}_{X}[1]$ is a simple object in
$\mathcal{P}(\phi)$. Therefore, the map $\psi$ is surjective in
$\mathcal{P}(\phi)$. The kernel $\ker\psi$ in $\mathcal{P}(\phi)$ is of Chern
character $\left(1,H,H^{2}/6,-H^{3}/6\right)$. Taking cohomology gives the
long exact sequence of coherent sheaves

$0\rightarrow\mathcal{H}^{-1}(\ker\psi)\rightarrow
K\rightarrow\mathcal{O}_{X}^{\oplus
3}\rightarrow\mathcal{H}^{0}(\ker\psi)\rightarrow 0.$

The wall $W(K[1],\mathcal{O}_{X}[1])$ intersects the horizontal line
$\alpha=0$ at two points with $\beta=0,\frac{1}{3}$. Thus moving along the
wall implies that

(17) $\frac{1}{6}\leq\mu_{H}^{-}\left(\mathcal{H}^{0}(\ker\psi)\right).$

Moreover, slope stability of $K$ implies that

(18) $\mu_{H}\left(\mathcal{H}^{-1}(\ker\psi)\right)\leq-\frac{1}{2}.$

Therefore
$0<(-1)^{i}\operatorname{ch}_{1}\left(\mathcal{H}^{i}(\ker\psi)\right).H$ and
$\operatorname{ch}_{1}\left(\mathcal{H}^{0}(\ker\psi)\right)-\operatorname{ch}_{1}\left(\mathcal{H}^{-1}(\ker\psi)\right)=H$.
Thus one of $\operatorname{ch}_{1}\left(\mathcal{H}^{i}(\ker\psi)\right)$ must
be zero and the other one is equal to $(-1)^{i}H$.

First assume $\mathcal{H}^{-1}(\ker\psi)$ is non-zero, so its rank is less
than $2=\operatorname{rank}(K)$. Note that it cannot be of rank 2, otherwise
the image of $\psi$ is torsion. Then (18) implies that
$\operatorname{ch}_{1}\left(\mathcal{H}^{-1}(\ker\psi)\right)H^{2}\leq-\frac{1}{2}H^{3}$.
Thus $\operatorname{ch}_{\leq
1}\left(\mathcal{H}^{-1}(\ker\psi)\right)=(1,-H)$ and so
$\operatorname{ch}_{\leq 1}\left(\mathcal{H}^{0}(\ker\psi)\right)$ is equal to
$(2,0)$ which is not possible by (17). Thus $\mathcal{H}^{-1}(\ker\psi)$ is
zero and $\ker\psi$ is a sheaf.

We finally claim that $\ker\psi$ is torsion-free. If not, by (17), its torsion
part is supported in dimension $\leq 1$. Since $\ker\psi$ is semistable along
the wall of phase lower than one, it cannot have a subobject of phase one.
Hence its torsion part is trivial, i.e. $\ker\psi$ is a torsion-free sheaf.
This immediately implies $\ker\psi=\mathcal{I}_{\ell}(H)$ for a line $\ell$ on
$X$. ∎

The next step is to compute $\mathcal{S}^{2}(\mathcal{I}_{\ell})$.

######  Lemma 4.9.

We have $h^{0}(K_{\ell}(H))=3$.

###### Proof.

We know
$h^{0}(K_{\ell}(H))=\chi(K_{\ell}(H))+h^{1}(K_{\ell}(H))=3+h^{1}(K_{\ell}(H))$.
There is no wall for $K_{\ell}(H)$ crossing the vertical line $\beta=0$.
Therefore, $K_{\ell}(H)$ is semistable along the numerical wall
$W(K_{\ell}(H),\mathcal{O}_{X})$. Suppose for a contradiction that there are 4
linearly independent global sections:

$s\colon\mathcal{O}_{X}^{\oplus 4}\rightarrow K_{\ell}(H)$

Since $\mathcal{O}_{X}$ is stable along the wall and have the same phase as
$K_{\ell}(H)$, the map $s$ is injective and the cokernel of $s$ is also
semistable of the same phase. It is of Chern character
$\operatorname{ch}(\text{cok}\ s)=\left(-2,H,-H^{2}/6,-H^{3}/6\right)$. The
numerical wall $W(K_{\ell}(H),\mathcal{O}_{X})$ intersects the horizontal line
$\alpha=0$ at two points with $\beta_{1},\beta_{2}=0,-\frac{1}{3}$. Thus by
definition of the heart, we get

(19) $\mu_{H}^{+}\left(\mathcal{H}^{-1}(\text{cok}\
s)\right)\leq-\frac{1}{3}.$

Since $\mathcal{H}^{0}(\text{cok}\ s)$ is a quotient of $K_{\ell}(H)$, we gain

$\frac{1}{2}\leq\mu_{H}\left(\mathcal{H}^{0}(\text{cok}\ s)\right).$

Therefore $\operatorname{ch}_{\leq 1}(\mathcal{H}^{-1}(\text{cok}\ s))=(2,-H)$
and $\mathcal{H}^{0}(\text{cok}\ s)$ is supported in dimension $\leq 1$.
Moreover (19) implies $\mathcal{H}^{-1}(\text{cok}\ s)$ is slope-stable, then
Proposition 2.9 implies that

$\operatorname{ch}_{2}(\mathcal{H}^{-1}(\text{cok}\ s)).H\leq 0.$

Hence

$\operatorname{ch}_{2}(\text{cok}\
s).H=\operatorname{ch}_{2}(\mathcal{H}^{0}(\text{cok}\
s)).H-\operatorname{ch}_{2}(\mathcal{H}^{-1}(\text{cok}\ s)).H\geq 0$

a contradiction. ∎

Therefore $h^{1}(K_{\ell}(H))=0$ and

$L_{\mathcal{O}_{X}}(K_{\ell}(H))=cone(\mathcal{O}_{X}^{\oplus 3}\rightarrow
K_{\ell}(H))\eqqcolon J_{\ell}\
,\qquad\mathcal{S}(K_{\ell})=J_{\ell}[1]\quad\Rightarrow\quad\mathcal{S}^{2}(I_{\ell})=J_{\ell}[3].$

Moreover

$\operatorname{ch}(J_{\ell})=\left(-1,H,-\frac{H^{2}}{6},-\frac{H^{3}}{6}\right)=v-w.$

Consider the numerical wall $W(K_{\ell}(H),\mathcal{O}_{X})$. A similar
argument as in the proof of Lemma 4.9 implies that

$\mu_{H}^{+}\left(\mathcal{H}^{-1}(J_{\ell})\right)\leq-\frac{1}{3}\qquad\text{and}\qquad\frac{1}{2}\leq\mu_{H}\left(\mathcal{H}^{0}(J_{\ell})\right).$

Thus $\operatorname{ch}_{\leq
1}\left(\mathcal{H}^{-1}(J_{\ell}\right))=(1,-H)$ and
$\mathcal{H}^{0}(J_{\ell})$ is supported in dim $\leq 1$. We claim
$\mathcal{H}^{-1}(J_{\ell})$ is a line bundle, otherwise there is a sheaf $Q$
supported in dimension $\leq 1$ and we have injections
$Q\hookrightarrow\mathcal{H}^{-1}(J_{\ell})[1]\hookrightarrow J_{\ell}$ which
is not possible because of semistability of $J_{\ell}$ along the numerical
wall $W(\mathcal{O}_{X},K_{\ell}(H))$. Thus
$\mathcal{H}^{-1}(J_{\ell})=\mathcal{O}_{X}(-H)$ and
$\mathcal{H}^{0}(J_{\ell})=\mathcal{O}_{\ell}(-H)$. This implies that
$J_{\ell}$ is large volume limit stable.

######  Definition 4.10.

A two-term complex $E\in\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)$
supported in degree 0 and $-1$ is said to be large volume limit stable if
$\mathcal{H}^{-1}(E)$ is a line bundle, $\mathcal{H}^{0}(E)$ is a sheaf
supported in dim $\leq 1$, and $\operatorname{Hom}(\operatorname{Coh}_{\leq
1}(X),E)=0$.

By [Tod13, Lemma 3.12 and Lemma 3.13(ii)], a complex
$E\in\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)$ is large volume limit
stable if and only if $E$ lies in $\operatorname{Coh}^{\beta}(X)$ and is
$\sigma_{\alpha,\beta}$-stable for $\beta>\mu_{H}(E)$ and $\alpha\gg 0$.

###### Proof of Theorem 4.5.

Any slope-semistable sheaf $E$ of class $v$ is isomorphic to
$\mathcal{I}_{\ell}$ for a line $\ell$ in $X$. By Lemma 4.6,
$\mathcal{I}_{\ell}$ is $\sigma$-stable for any $\mathcal{S}$-invariant
stability condition $\sigma$ on $\operatorname{Ku}(X)$.

Conversely, let $E\in\operatorname{Ku}(X)$ be a $\sigma$-stable object of
class $[\mathcal{I}_{\ell}]$. By Corollary 4.3, we may assume
$\sigma=\sigma(\alpha,\beta)$ where $(\alpha,\beta)\in V\cap\Gamma$ where
$\Gamma$ is the hyperbola with equation $\alpha^{2}=\beta^{2}-2/3$. One can
easily compute that

(20) $Z(\alpha,\beta)(E)=-1,$

So $E$ is of maximum phase, thus $E$ is
$\sigma^{0}_{\alpha,\beta}$-semistable. Since $E$ lies in
$\operatorname{Ku}(X)$ (so we have $\chi(\mathcal{O}_{X}(kH),E)=0$ for
$k=0,1$) and is of class $[\mathcal{I}_{\ell}]$, one can easily show that $E$
is of Chern character $\pm v$. Proposition 4.1 implies that either (i) $E$ is
$\sigma_{\alpha,\beta}$-semistable, or (ii) $E$ lies in an exact triangle

$F[1]\rightarrow E\rightarrow T$

where $F\in\mathcal{F}_{\alpha,\beta}$ and $T\in\operatorname{Coh}_{0}(X)$. In
the latter case, $F[1]$ is of phase one with respect to
$\sigma^{0}_{\alpha,\beta}$, so it is $\sigma_{\alpha,\beta}$-semistable by
Proposition 4.1. We have

$\operatorname{ch}(F)=-\operatorname{ch}(E)+\operatorname{ch}(T)=\left(1,0,-\frac{1}{3}H^{2},\operatorname{ch}_{3}(T)\right).$

Let $(\alpha_{0},\beta_{0})$ be the intersection point of $\partial V$ with
$\Gamma$. Closedness of semistability implies that $E$ in case (i) (or $F$ in
case (ii)) is $\sigma_{\alpha_{0},\beta_{0}}$-semistable. We know the point
$(\alpha_{0},\beta_{0})$ lies on the numerical wall
$W(E,\mathcal{O}_{X}(-H)[1])$. In case (ii), applying Theorem 2.10 to $F$ at
the limiting point $(\alpha,\beta)\rightarrow(0,-1)$ implies that

$\operatorname{ch}_{3}(T)=\operatorname{ch}_{3}(F)\leq\frac{5}{9}$

which gives $T=0$, so $E=F[1]$ and we are reduced to case (i). In case (i),
the next Lemma 4.11 implies that $E$ is $\sigma_{\alpha,\beta}$-stable for
$\alpha\gg 0$, so is a slope-stable sheaf up to a shift by [BMS16, Lemma 2.7].
This completes the proof of part (a).

For part (b), take a slope-stable sheaf $E$ of class $w$. By Proposition 4.8,
there is a line $\ell$ on $X$ such that $E=K_{\ell}$. Thus $E$ is in
$\operatorname{Ku}(X)$ and $\sigma$-stable for any $\mathcal{S}$-invariant
stability condition $\sigma$. Conversely, let $E$ be a $\sigma$-stable object
in $\operatorname{Ku}(X)$ of class $[\mathcal{S}(\mathcal{I}_{\ell})]$. Then
$\mathcal{S}^{-1}(E)$ is $\sigma$-stable of class $[\mathcal{I}_{\ell}]$.
Hence by part (a), $\mathcal{S}^{-1}(E)$ is up to a shift isomorphic to the
ideal sheaf of a line $\ell$ on $X$. Thus $E$ is isomorphic to
$\mathcal{S}(\mathcal{I}_{\ell})$ and so is a slope-stable sheaf up to a
shift.

Finally we prove part (c). Take a large volume limit stable complex $J$ of
class $v-w$. We know there is no wall for $J$ above the numerical wall
$W(\mathcal{O}_{X},J)$. Thus $J$ is $\sigma_{\alpha,\beta}$-stable of phase
lower than one if $-1<\beta<-\frac{1}{3}$. Sheaves supported in dimension zero
are $\sigma_{\alpha,\beta}$-semistable of phase one, so
$\operatorname{Hom}(\operatorname{Coh}_{0}(X),J)=0$. One can easily check that
$J\in\mathcal{T}_{\alpha,\beta}$ when $\beta\rightarrow-1$, thus Proposition
4.1 implies $J$ is $\sigma^{0}_{\alpha,\beta}$-stable. We claim that $J$ lies
in the Kuznetsov component $\operatorname{Ku}(X)$. By definition, $J$ lies in
the exact triangle

$\mathcal{O}_{X}(-H)[1]\rightarrow J\rightarrow\mathcal{O}_{\ell}(-H).$

We know $\operatorname{Hom}^{i}(\mathcal{O}_{X}(H),\mathcal{O}_{X}(-H))$
vanishes for $i\neq 3$ and is isomorphic to $\mathbb{C}$ for $i=3$. Also
$\operatorname{Hom}^{i}(\mathcal{O}_{X}(H),\mathcal{O}_{\ell}(-H))=0$ vanishes
for $i\neq 1$ and is isomorphic to $\mathbb{C}$ for $i=1$. Thus
$\hom^{i}(\mathcal{O}_{X}(H),J)=0$ for every $i\in\mathbb{Z}$. We also have

$\operatorname{Hom}^{i}(\mathcal{O}_{X},\mathcal{O}_{X}(-H))=\operatorname{Hom}^{i}(\mathcal{O}_{X},\mathcal{O}_{\ell}(-H))=\operatorname{Hom}^{i}(\mathcal{O}_{X},J)=0$

for all $i\in\mathbb{Z}$. Therefore $J\in\operatorname{Ku}(X)$ and so $J$ is
$\sigma(\alpha,\beta)$-stable when $-1<\beta<-\frac{1}{3}$.

Conversely, take a $\sigma$-stable object $E\in\operatorname{Ku}(X)$ of class
$[\mathcal{S}^{2}(\mathcal{I}_{\ell})]$ for an $\mathcal{S}$-invariant
stability condition $\sigma$. Thus $\mathcal{S}^{-1}(E)$ is $\sigma$-stable of
class $[K_{\ell}]$, so by the second part, there is a line $\ell$ in $X$ such
that $\mathcal{S}^{-1}(E)=K_{\ell}$. Thus $E$ is isomorphic to $J_{\ell}$ up
to shift. This completes the proof. ∎

######  Lemma 4.11.

Let $E$ be a $\sigma_{\alpha,\beta}$-semistable object of Chern character $v$
for $(\alpha,\beta)$ along the numerical wall $W(E,\mathcal{O}_{X}(-H)[1])$.
Then $E$ is $\sigma_{\alpha,\beta}$-stable for $\alpha\gg 0$ and $\beta>0$.

###### Proof.

Suppose for a contradiction that $W(E,\mathcal{O}_{X}(-H)[1])$ is an actual
wall and $F_{1}\rightarrow E\rightarrow F_{2}$ is a destabilising sequence.
Let $\operatorname{ch}_{\leq 2}(F_{1})=\left(r,cH,sH^{2}\right)$. Since the
boundary of the wall intersects the vertical line $\beta=-1$, we must have
$\Im[Z_{\alpha=0,\beta=-1}(F_{i})]\geq 0$, thus

$0\leq c+r\qquad\text{and}\qquad 0\leq-c+1-r\,.$

Hence by relabeling $F_{1}$ and $F_{2}$, we may assume $c=-r$. Since $F_{i}$’s
have the same phase as $E$ along the wall, one can easily show that
$s=\frac{r}{2}$. Therefore $\operatorname{ch}_{\leq 2}(F_{2})=\left(1-r,\ rH,\
H^{2}(-\frac{1}{3}-\frac{r}{2})\right)$ and

$\Delta_{H}(F_{2})=(H^{3})^{2}\left(\frac{r}{3}+\frac{2}{3}\right).$

By [BMS16, Corollary 3.10], we must have
$0\leq\Delta_{H}(F_{2})<\Delta_{H}(E)=\frac{2}{3}(H^{3})^{2}$. Since
$\Delta_{H}\in\frac{1}{3}(H^{3})^{2}\mathbb{Z}$, we gain $r=-1$, or $r=-2$. In
both cases $F_{2}$ is of positive rank and $F_{1}$ is of negative rank, so
$F_{1}$ must be the subobject in the destabilising sequence to have phase
bigger than $E$ above the wall.

Since $\Delta_{H}(F_{1})=0$, $F_{1}$ is $\sigma_{\alpha,\beta}$-semistable for
any $\alpha>0$ and $\beta>-1$ [BMS16, Corollary 3.11(a)]. Thus
$\mathcal{H}^{-1}(F_{1})$ is a slope-stable reflexive sheaf and
$\mathcal{H}^{0}(F_{1})$ is supported in dimension $\leq 1$. Since
$\Delta_{H}(F_{1})=0$, there is no room for
$\operatorname{ch}_{2}(\mathcal{H}^{0}(F_{1}))$, so $\mathcal{H}^{0}(F_{1})$
is indeed supported in dimension zero. Applying Theorem 2.10 to $F_{1}(H)$
implies that for $\beta>0$ we have

(21) $0\leq
3r\beta\left(\operatorname{ch}_{3}\left(\mathcal{H}^{0}\left(F_{1}(H)\right)\right)-\operatorname{ch}_{3}\left(\mathcal{H}^{-1}\left(F_{1}(H)\right)\right)\right)$

Moreover, since $\mathcal{H}^{-1}(F_{1}(H))$ is a $\mu_{H}$-stable sheaf,
Lemma 2.6 implies that it is $\sigma_{\alpha^{\prime},\beta^{\prime}}$-stable
for $\beta^{\prime}<0$ and $\alpha^{\prime}\gg 0$. Applying Theorem 2.10 to
$\mathcal{H}^{-1}(F_{1})$ at this region implies that

$0\leq-3r\beta^{\prime}\operatorname{ch}_{3}\left(\mathcal{H}^{-1}\left(F_{1}(H)\right)\right)\qquad\Rightarrow\qquad
0\leq-\operatorname{ch}_{3}\left(\mathcal{H}^{-1}\left(F_{1}(H)\right)\right)$

Combing this with (21) implies that
$\operatorname{ch}_{3}\left(\mathcal{H}^{i}\left(F_{1}(H)\right)\right)=0$ for
$i=-1,0$. Therefore $\mathcal{H}^{0}(F_{1})=0$ and
$[F_{1}]=r[\mathcal{O}_{X}(-H)]$, thus
$F_{1}\cong\mathcal{O}_{X}(-H)^{\oplus{-r}}[1]$.

If $r=-1$ and $F_{2}$ is $\sigma_{\alpha,\beta}$-stable along the wall, then
$F_{2}$ will be $\sigma_{\alpha,\beta}$-stable for $\alpha\gg 0$ and
$\beta>-\frac{1}{2}$ because there is no wall for it crossing the vertical
line $\beta=-1$. Hence $F_{2}$ is a slope-stable sheaf of character
$\operatorname{ch}_{\leq 2}(F_{2})=(2,-H,\frac{1}{6}H^{2})$ which is not
possible by Proposition 2.9. Thus $F_{2}$ is strictly semistable along the
wall and the same argument as above implies that one of its destabilising
subobject is $\mathcal{O}_{X}(-H)[1]$. Hence, we may assume $r=-2$. Then
$\Delta_{H}(F_{2})=0$, thus by [BMS16, Corollary 3.11(a)] it is a slope-stable
sheaf which is not again possible by Proposition 2.9. ∎

##  5\. Stability conditions on the Kuznetsov component via conic fibrations

Let $X$ be a cubic threefold as before. Fix a line $\ell_{0}$ in $X$ and
consider the linear projection to a disjoint $\mathbb{P}^{2}$ in the
$\mathbb{P}^{4}$ containing $X$, which induces a conic fibration structure on
the blow up of $X$ along $\ell_{0}$. Let $\mathcal{B}_{0}$ (resp.
$\mathcal{B}_{1}$) be the sheaf of even (resp. odd) parts of the Clifford
algebra corresponding to the conic fibration as in [Kuz08, Section 3]. We
denote by $\operatorname{Coh}(\mathbb{P}^{2},\mathcal{B}_{0})$ the abelian
category of right coherent $\mathcal{B}_{0}$-modules and by
$\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathbb{P}^{2},\mathcal{B}_{0})$
its bounded derived category. By [BMMS12, Section 2.1] there is a
semiorthogonal decomposition

(22)
$\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathbb{P}^{2},\mathcal{B}_{0})=\langle\Xi(\operatorname{Ku}(X)),\mathcal{B}_{1}\rangle,$

where
$\Xi:\operatorname{Ku}(X)\to\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathbb{P}^{2},\mathcal{B}_{0})$
is a fully faithful embedding. We set

$\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0}):=\Xi(\operatorname{Ku}(X)),$

which is equivalent to $\operatorname{Ku}(X)$ by definition. In [BMMS12,
LMS15], the authors use the semiorthogonal decomposition (22) to construct
stability conditions on $\operatorname{Ku}(X)$. In this section, we first
summarize their construction and then show that all these stability conditions
on $\operatorname{Ku}(X)$ are $\mathcal{S}$-invariant and lie in the same
orbit with respect to $\widetilde{\text{GL}}^{+}_{2}$-action.

###  5.1. Stability conditions on Noncommutative $\mathbb{P}^{2}$

The Chern character of an object
$E\in\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathbb{P}^{2},\mathcal{B}_{0})$
is

$\operatorname{ch}(\text{Forg}(E))\in
K(\mathbb{P}^{2})\otimes\mathbb{Q}\cong\mathbb{Q}^{\oplus 3}$

where
$\text{Forg}\colon\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathbb{P}^{2},\mathcal{B}_{0})\to\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathbb{P}^{2})$
is the forgetful functor, and by abuse of notation we denote it
$\operatorname{ch}(E)$ for simplicity. Recall that by [BMMS12, Proposition
2.12], we have

(23)
$\mathcal{N}(\mathbb{P}^{2},\mathcal{B}_{0})=\mathbb{Z}[\mathcal{B}_{-1}]\oplus\mathbb{Z}[\mathcal{B}_{0}]\oplus\mathbb{Z}[\mathcal{B}_{1}]$

and the Chern characters

$\operatorname{ch}(\mathcal{B}_{-1})=\left(4,-7,\frac{15}{2}\right),\qquad\operatorname{ch}(\mathcal{B}_{0})=\left(4,-5,\frac{9}{2}\right),\qquad\operatorname{ch}(\mathcal{B}_{1})=\left(4,-3,\frac{5}{2}\right)$

are linearly independent. For objects in
$\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathbb{P}^{2},\mathcal{B}_{0})$,
we have the Euler Chracteristic

$\chi(-,-)=\sum_{i}(-1)^{i}\hom^{i}_{\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathbb{P}^{2},\mathcal{B}_{0})}(-,-).$

Given
$[F]=x[\mathcal{B}_{-1}]+y[\mathcal{B}_{-1}]+z[\mathcal{B}_{-1}]\in\mathcal{N}(\mathbb{P}^{2},\mathcal{B}_{0})$
with $\operatorname{ch}(F)=(r,c_{1},\operatorname{ch}_{2})$, [LMS15, Remark
2.2(iv)] gives

$\displaystyle\chi(F,F)=\ $ $\displaystyle\ x^{2}+y^{2}+z^{2}+3xy+3yz+6xz$
$\displaystyle=\ $ $\displaystyle\
-\frac{7}{64}r^{2}-\frac{1}{4}c_{1}^{2}+\frac{1}{2}r\operatorname{ch}_{2}$

Slope-stability for torsion-free sheaves in
$\operatorname{Coh}(\mathbb{P}^{2},\mathcal{B}_{0})$ is defined via

$\operatorname{rank}(F)\coloneqq\operatorname{rank}(\text{Forg}(F)),\qquad\deg(F)=c_{1}(\text{Forg}(F)).c_{1}(\mathcal{O}_{\mathbb{P}^{2}}(1)).$

######  Lemma 5.1 ([BLMS17], Theorem 8.3).

Any slope-semistable torsion-free sheaf $E$ in
$\operatorname{Coh}(\mathbb{P}^{2},\mathcal{B}_{0})$ of character
$(r,c_{1},\operatorname{ch}_{2})$ satisfies the quadratic inequality

(24) $Q(E)\coloneqq c_{1}^{2}-2r\operatorname{ch}_{2}+\frac{11}{16}r^{2}\geq
0$

###### Proof.

If $E$ is slope-stable, then [LMS15, Lemma 2.4] implies that $\chi(E,E)\leq
1$. Since $r=\operatorname{rank}(E)$ is a multiple of $4$ by [BMMS12, Lemma
2.13(i)], we gain

$\chi(E,E)=-\frac{7}{64}r^{2}-\frac{1}{4}c_{1}^{2}+\frac{1}{2}r\operatorname{ch}_{2}\leq\frac{r^{2}}{16}$

which implies the claim. If $E$ is strictly slope-semistable with stable
factors $\\{E_{i}\\}_{i\in I}$, we get

$\frac{\operatorname{ch}_{2}(E_{i})}{r(E_{i})}\leq\frac{1}{2}\left(\frac{c_{1}(E_{i})}{r(E_{i})}\right)^{2}+\frac{11}{32}\
=\ \frac{1}{2}\left(\frac{c_{1}}{r}\right)^{2}+\frac{11}{32}$

which shows (24) holds for $E$. ∎

Consider the open subset

$U\coloneqq\left\\{(b,w)\in\mathbb{R}^{2}\colon\
w>\frac{b^{2}}{2}+\frac{11}{32}\right\\}.$

By (24) we have the following result, which is well-known in the commutative
setting by [BMT14, BMS16] (see also [PPZ19, Lemma 4.4]).

######  Proposition 5.2.

There is a continuous family of Bridgeland stability conditions on
$\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathbb{P}^{2},\mathcal{B}_{0})$
parameterized by $U$ given by

$(b,w)\in
U\mapsto\overline{\sigma}_{b,w}=\big{(}\operatorname{Coh}^{b}(\mathbb{P}^{2},\mathcal{B}_{0}),\
\overline{Z}_{b,w}=-\operatorname{ch}_{2}+w\operatorname{ch}_{0}+i(\operatorname{ch}_{1}-b\operatorname{ch}_{0})\big{)}.$

###  5.2. Induced stability conditions

By [BMMS12, Proposition 2.9(ii)] the Serre functor
$S_{\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathbb{P}^{2},\mathcal{B}_{0})}$
of $\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathbb{P}^{2},\mathcal{B}_{0})$
is $-\otimes_{\mathcal{B}_{0}}\mathcal{B}_{-1}[2]$. [BLMS17, Remark 9.4] shows
that $\mathcal{B}_{0}$ and $\mathcal{B}_{1}$ are slope stable
$\mathcal{B}_{0}$-modules with $\mu(\mathcal{B}_{0})=-\frac{5}{4}$ and
$\mu(\mathcal{B}_{1})=-\frac{3}{4}$, respectively. Then $\mathcal{B}_{1}$,
$\mathcal{B}_{0}[1]=S_{\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathbb{P}^{2},\mathcal{B}_{0})}(\mathcal{B}_{1})[-1]$
are in $\operatorname{Coh}^{b}(\mathbb{P}^{2},\mathcal{B}_{0})$ for
$-\frac{5}{4}\leq b<-\frac{3}{4}$. Since $\overline{\sigma}_{b,w}$ is a
stability condition on
$\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathbb{P}^{2},\mathcal{B}_{0})$,
we see that the conditions in [BLMS17, Proposition 5.1] are all satisfied, as
we wanted.

######  Proposition 5.3.

For any $(b,w)\in U$ with $-\frac{5}{4}\leq b<-\frac{3}{4}$, the pair

(25)
$\overline{\sigma}(b,w)\coloneqq\big{(}\mathcal{A}_{b}\coloneqq\operatorname{Coh}^{b}(\mathbb{P}^{2},\mathcal{B}_{0})\cap\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0}),\
\overline{Z}_{b,w}|_{\mathcal{N}(\mathcal{A}_{b})}\big{)}$

is a stability condition on
$\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0})$. With respect to
$\widetilde{\emph{GL}}^{+}_{2}$-action, they all lie in the same orbit as the
stability condition222The stability condition $\overline{\sigma}$ is first
constructed in [BMMS12, Section 3].

(26)
$\overline{\sigma}=\left(\mathcal{A}_{-\frac{5}{4}}\,,\,\overline{Z}=\operatorname{ch}_{0}+i\left(\operatorname{ch}_{1}+\frac{5}{4}\operatorname{ch}_{0}\right)\right).$

###### Proof.

The first claim follows form [BLMS17, Proposition 5.1] (see Proposition 2.11).
Arguing as in [LPZ20, Proposition and Definition 2.15] we have the relation

$\operatorname{ch}_{2}(F)=-\operatorname{ch}_{1}(F)-\frac{3}{8}\operatorname{rank}(F)$

for every $F\in\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0})$. Thus we can
rewrite

$\Re[\overline{Z}_{b,w}|_{\mathcal{N}(\mathcal{A}_{b})}]=\operatorname{ch}_{1}+\left(\frac{3}{8}+w\right)\operatorname{ch}_{0}.$

As a consequence,

$\displaystyle\overline{\mu}_{b,w}|_{\mathcal{N}(\mathcal{A}_{b})}=-\frac{\Re[\overline{Z}_{b,w}|_{\mathcal{N}(\mathcal{A}_{b})}]}{\Im[\overline{Z}_{b,w}|_{\mathcal{N}(\mathcal{A}_{b})}]}$
$\displaystyle=\frac{-\operatorname{ch}_{1}-\left(\frac{3}{8}+w\right)\operatorname{ch}_{0}}{\operatorname{ch}_{1}-b\operatorname{ch}_{0}}$
$\displaystyle=\frac{-\operatorname{ch}_{1}-\left(\frac{3}{8}+w\right)\operatorname{ch}_{0}\,+\,\operatorname{ch}_{1}-b\operatorname{ch}_{0}}{\operatorname{ch}_{1}-b\operatorname{ch}_{0}}-1$
$\displaystyle=-\frac{\left(\frac{3}{8}+w+b\right)\operatorname{ch}_{0}}{\operatorname{ch}_{1}-b\operatorname{ch}_{0}}-1$

Since $\frac{3}{8}+w+b>0$ for $(b,w)\in U$, the stability function
$\overline{Z}_{b,w}|_{\mathcal{N}(\mathcal{A}_{b})}$ induces the same ordering
on $\mathcal{A}_{b}$ as the function

$Z_{b}\coloneqq\operatorname{ch}_{0}+i(\operatorname{ch}_{1}-b\operatorname{ch}_{0}).$

Thus the stability condition
$\left(\mathcal{A}_{b},\overline{Z}_{b,w}|_{\mathcal{N}(\mathcal{A}_{b})}\right)$
lies in the same orbit as $\sigma_{b}=\left(\mathcal{A}_{b},Z_{b}\right)$ with
respect to $\widetilde{\text{GL}}^{+}_{2}$-action. We know
$Z_{b}=T_{b}^{-1}\circ Z_{-\frac{5}{4}}$ where

$T_{b}=\begin{pmatrix}1&0\\\ b+\frac{5}{4}&1\end{pmatrix}.$

Since $b+\frac{5}{4}\geq 0$, there exists a cover
$g_{b}\coloneqq(T_{b},f_{b})\in\widetilde{\text{GL}}^{+}_{2}(\mathbb{R})$ such
that

$\mathcal{P}_{\sigma_{-\frac{5}{4}}}\left(f_{b}(0,1]\right)\subset\langle\mathcal{A}_{-\frac{5}{4}},\mathcal{A}_{-\frac{5}{4}}[1]\rangle.$

Also an easy computation shows that
$\operatorname{Coh}^{b}(\mathbb{P}^{2},\mathcal{B}_{0})\subset\langle\operatorname{Coh}^{-\frac{5}{4}}(\mathbb{P}^{2},\mathcal{B}_{0}),\operatorname{Coh}^{-\frac{5}{4}}(\mathbb{P}^{2},\mathcal{B}_{0})[1]\rangle$
(see for instance [PY20, Lemma 3.7]). Restricting to the heart
$\mathcal{A}_{b}$, this implies

$\mathcal{A}_{b}\subset\langle\mathcal{A}_{-\frac{5}{4}},\mathcal{A}_{-\frac{5}{4}}[1]\rangle$

by [BLMS17, Lemma 4.3]. Thus the stability conditions $\sigma_{b}$ and
$\sigma_{-\frac{5}{4}}\,.\,g_{b}$ have the same central charge and satisfy the
conditions of [BMS16, Lemma 8.11], hence they are the same. ∎

###  5.3. Serre invariance

In this section we show the following result.

######  Theorem 5.4.

The stability condition
$\overline{\sigma}=\left(\mathcal{A}_{-\frac{5}{4}},\overline{Z}\right)$ (26)
on $\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0})$ is Serre-invariant.

Therefore Proposition 5.3 implies that all stability conditions
$\overline{\sigma}(b,w)$ (25) are Serre invariant. By [Bon89] (see also
[Kuz04, Proposition 3.8]) the Serre functor $\mathcal{S}_{\mathcal{B}_{0}}$ of
$\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0})$ satisfies

$\mathcal{S}^{-1}_{\mathcal{B}_{0}}\cong
L_{\mathcal{B}_{1}}(-\otimes\mathcal{B}_{1})[-2]\cong(-\otimes\mathcal{B}_{1})\circ
L_{\mathcal{B}_{0}}(-)[-2].$

We first describe the image of the heart after the action of
$\mathcal{S}_{\mathcal{B}_{0}}^{-1}$.

######  Proposition 5.5.

We have
$L_{\mathcal{B}_{0}}(\mathcal{A}_{-\frac{5}{4}})\otimes\mathcal{B}_{1}\subset\langle\mathcal{A}_{-\frac{5}{4}},\mathcal{A}_{-\frac{5}{4}}[1]\rangle$.

###### Proof.

Let
$E\in\mathcal{A}_{-\frac{5}{4}}=\operatorname{Coh}^{-\frac{5}{4}}(\mathbb{P}^{2},\mathcal{B}_{0})\cap\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0})$
and set $E^{\prime}:=L_{\mathcal{B}_{0}}(E)$. We first show that

(27)
$E^{\prime}\otimes\mathcal{B}_{1}=L_{\mathcal{B}_{0}}(E)\otimes\mathcal{B}_{1}\in\langle\operatorname{Coh}^{-\frac{5}{4}}(\mathbb{P}^{2},\mathcal{B}_{0}),\operatorname{Coh}^{-\frac{5}{4}}(\mathbb{P}^{2},\mathcal{B}_{0})[1]\rangle.$

By Serre duality and the fact that
$\mathcal{B}_{0}[1],\mathcal{B}_{-1}[1]\in\operatorname{Coh}^{-\frac{5}{4}}(\mathbb{P}^{2},\mathcal{B}_{0})$,
we have that $\operatorname{Hom}^{i}(\mathcal{B}_{0}[1],E)$ can be non zero
only for $i=0,1,2$. Thus $E^{\prime}$ is defined by the triangle

$\mathcal{B}_{0}[1]^{\oplus k_{1}}\oplus\mathcal{B}_{0}^{\oplus
k_{2}}\oplus\mathcal{B}_{0}[-1]^{\oplus k_{3}}\xrightarrow{\text{ev}}E\to
E^{\prime}.$

Since $\mathcal{B}_{0}[1]$ is stable of phase one with respect to
$\overline{\sigma}_{b=-\frac{5}{4},w}$, the map $f$ in the following
commutative diagram is injective in
$\operatorname{Coh}^{-\frac{5}{4}}(\mathbb{P}^{2},\mathcal{B}_{0})$:

$\textstyle{\mathcal{B}_{0}[1]^{\oplus
k_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$id$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}_{0}[1]^{\oplus
k_{1}}\oplus\mathcal{B}_{0}^{\oplus
k_{2}}\oplus\mathcal{B}_{0}[-1]^{k_{3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}_{0}[1]^{\oplus
k_{2}}\oplus\mathcal{B}_{0}[-1]^{k_{3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}_{0}[1]^{\oplus
k_{2}}\oplus\mathcal{B}_{0}^{k_{3}}.}$

Hence, we gain
$G\in\operatorname{Coh}^{-\frac{5}{4}}(\mathbb{P}^{2},\mathcal{B}_{0})$.
Tensoring the third column by $\mathcal{B}_{1}$ gives the exact triangle

$G\otimes\mathcal{B}_{1}\to
E^{\prime}\otimes\mathcal{B}_{1}\to\mathcal{B}_{1}^{\oplus
k_{2}}[1]\oplus\mathcal{B}_{1}^{\oplus k_{3}}.$

By the next Lemma 5.6, we have
$G\otimes\mathcal{B}_{1}\in\langle\operatorname{Coh}^{-\frac{5}{4}}(\mathbb{P}^{2},\mathcal{B}_{0}),\operatorname{Coh}^{-\frac{5}{4}}(\mathbb{P}^{2},\mathcal{B}_{0})[1]\rangle$,
thus the claim (27) follows.

Since
$L_{\mathcal{B}_{0}}(E)\otimes\mathcal{B}_{1}=L_{\mathcal{B}_{1}}(E\otimes\mathcal{B}_{1})$,
we know
$L_{\mathcal{B}_{0}}(E)\otimes\mathcal{B}_{1}\in\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0})$.
If we consider the HN filtration of
$L_{\mathcal{B}_{0}}(E)\otimes\mathcal{B}_{1}$ with respect to
$\overline{\sigma}_{b=-\frac{5}{4},w}$, (27) implies the maximum and minimum
phase satisfies

$0<\phi^{-}_{\overline{\sigma}_{b=-\frac{5}{4},w}}(L_{\mathcal{B}_{0}}(E)\otimes\mathcal{B}_{1})\leq\phi^{+}_{\overline{\sigma}_{b=-\frac{5}{4},w}}(L_{\mathcal{B}_{0}}(E)\otimes\mathcal{B}_{1})\leq
2.$

Then definition of stability condition $\overline{\sigma}$ implies that

$\begin{split}0<\phi^{-}_{\overline{\sigma}_{b=-\frac{5}{4},w}}(L_{\mathcal{B}_{0}}(E)\otimes\mathcal{B}_{1})\leq\phi^{-}_{\overline{\sigma}}(L_{\mathcal{B}_{0}}(E)\otimes\mathcal{B}_{1})&\leq\phi^{+}_{\overline{\sigma}}(L_{\mathcal{B}_{0}}(E)\otimes\mathcal{B}_{1})\\\
&\leq\phi^{+}_{\overline{\sigma}_{b=-\frac{5}{4},w}}(L_{\mathcal{B}_{0}}(E)\otimes\mathcal{B}_{1})\leq
2.\end{split}$

Therefore,

$L_{\mathcal{B}_{0}}(E)\otimes\mathcal{B}_{1}\in\langle\operatorname{Coh}^{-\frac{5}{4}}(\mathbb{P}^{2},\mathcal{B}_{0})\cap\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0}),\operatorname{Coh}^{-\frac{5}{4}}(\mathbb{P}^{2},\mathcal{B}_{0})\cap\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0})[1]\rangle$

as claimed. ∎

######  Lemma 5.6.

We have
$\operatorname{Coh}^{b}(\mathbb{P}^{2},\mathcal{B}_{0})\otimes\mathcal{B}_{1}\subset\langle\operatorname{Coh}^{b}(\mathbb{P}^{2},\mathcal{B}_{0}),\operatorname{Coh}^{b}(\mathbb{P}^{2},\mathcal{B}_{0})[1]\rangle$.

###### Proof.

Since $\mathcal{B}_{1}$ is a flat $\mathcal{B}_{0}$-module and by (23), for
$F\in\operatorname{Coh}(\mathbb{P}^{2},\mathcal{B}_{0})$ we have that

$\operatorname{ch}_{1}(F\otimes\mathcal{B}_{1})=\operatorname{ch}_{1}(F)+\frac{1}{2}\operatorname{rank}(F).$

If $F\in\operatorname{Coh}(\mathbb{P}^{2},\mathcal{B}_{0})$ is slope
semistable with $\mu(F)>b$, then $F\otimes\mathcal{B}_{1}$ is slope semistable
with slope $\mu(F)+\frac{1}{2}>b$. Then $F\otimes\mathcal{B}_{1}$ is in
$\operatorname{Coh}^{b}(\mathbb{P}^{2},\mathcal{B}_{0})$. Otherwise assume
$F\in\operatorname{Coh}(\mathbb{P}^{2},\mathcal{B}_{0})$ is slope semistable
with $\mu(F)\leq b$. If $\mu(F)+\frac{1}{2}\leq b$, then
$F\otimes\mathcal{B}_{1}[1]\in\operatorname{Coh}^{b}(\mathbb{P}^{2},\mathcal{B}_{0})$,
while if $\mu(F)+\frac{1}{2}>b$, then
$F\otimes\mathcal{B}_{1}\in\operatorname{Coh}^{b}(\mathbb{P}^{2},\mathcal{B}_{0})$
and thus
$F\otimes\mathcal{B}_{1}[1]\in\operatorname{Coh}^{b}(\mathbb{P}^{2},\mathcal{B}_{0})[1]$.
This implies the claim. ∎

###### Proof of Theorem 5.4.

Since
$\mathcal{S}_{\mathcal{B}_{0}}^{2}[-3]=\mathcal{S}_{\mathcal{B}_{0}}^{-1}[2]$,
we only need to find
$g\coloneqq(T,f)\in\widetilde{\text{GL}}^{+}_{2}(\mathbb{R})$ such that

$\mathcal{S}_{\mathcal{B}_{0}}^{-1}[2]\,.\,\overline{\sigma}=\overline{\sigma}\,.\,g$

We know
$\mathcal{S}_{\mathcal{B}_{0}}^{-1}[2]\,.\,\overline{\sigma}=\left(\overline{Z}\circ\mathcal{S}_{\mathcal{B}_{0}}[-2]\,,\,\mathcal{S}_{\mathcal{B}_{0}}^{-1}[2]\left(\mathcal{A}_{-\frac{5}{4}}\cap\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0})\right)\right)$.
By [LMS15, Remark 2.2, Proposition 2.12], the classes
$v_{1}\coloneqq[\Xi(\mathcal{I}_{\ell})]=[\mathcal{B}_{1}]-[\mathcal{B}_{0}]$
and
$v_{2}\coloneqq[\Xi(S_{\operatorname{Ku}(X)}(\mathcal{I}_{\ell}))]=2[\mathcal{B}_{0}]-[\mathcal{B}_{-1}]$
are the basis of
$\mathcal{N}(\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0}))$. Also [Huy06,
Lemma 1.30] implies
$\mathcal{S}_{\mathcal{B}_{0}}(\Xi(\mathcal{I}_{\ell}))=\Xi(\mathcal{S}_{\operatorname{Ku}(X)}(\mathcal{I}_{\ell}))$
and

$\left[\mathcal{S}_{\mathcal{B}_{0}}\left(\Xi\left(\mathcal{S}_{\operatorname{Ku}(X)}(\mathcal{I}_{\ell})\right)\right)\right]=\left[\Xi\left(\mathcal{S}^{2}_{\operatorname{Ku}(X)}(\mathcal{I}_{\ell})\right)\right]=\left[\Xi\left(\mathcal{S}_{\operatorname{Ku}(X)}(\mathcal{I}_{\ell})\right)\right]-\left[\Xi(\mathcal{I}_{\ell})\right].$

Therefore
$\overline{Z}\circ\mathcal{S}_{\mathcal{B}_{0}}=T^{-1}\circ\overline{Z}$ for a
linear invertible function $T\colon\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$
defined as

$T^{-1}\left[\overline{Z}(v_{1})\right]=\overline{Z}(v_{2})\qquad\text{and}\qquad
T^{-1}\left[\overline{Z}(v_{2})\right]=\overline{Z}(v_{2})-\overline{Z}(v_{1}).$

Thus

$T\left[\overline{Z}(v_{1})\right]=\overline{Z}(v_{1})-\overline{Z}(v_{2})\qquad\text{and}\qquad
T\left[\overline{Z}(v_{2})\right]=\overline{Z}(v_{1}).$

We know $\overline{Z}(v_{1})=2i$ and $\overline{Z}(v_{2})=4+2i$, so one can
easily check that the linear function $T$ with respect to the standard basis
can be represented as

$\begin{pmatrix}1&-2\\\ \frac{1}{2}&0\end{pmatrix}.$

Thus there exists a cover $(T,f)\in\widetilde{\text{GL}}^{+}_{2}(\mathbb{R})$
such that $f(0,1)\subset(0,2)$, so

$\mathcal{P}_{\overline{\sigma}}(f(0,1])\subset\langle\mathcal{A}_{-\frac{5}{4}},\mathcal{A}_{-\frac{5}{4}}[1]\rangle.$

Thus the stability conditions $\sigma\cdot g$ and
$((-\otimes\mathcal{B}_{1})\circ L_{\mathcal{B}_{0}})\cdot\sigma$ have the
same stability functions and the condition of [BMS16, Lemma 8.11] holds for
their hearts by Proposition 5.5. Hence they are the same stability conditions.
∎

######  Corollary 5.7.

The stability condition $\overline{\sigma}$ (26) and the stability conditions
$\overline{\sigma}(b,w)$ defined in (25) are in the same orbit of
$\sigma(\alpha,\beta)$ defined in Theorem 4.2 with respect to the
$\widetilde{\emph{GL}}^{+}_{2}(\mathbb{R})$ action.

###### Proof.

As noted in Corollary 4.3, the conditions of Theorem 3.2 are satisfied by
$\operatorname{Ku}(X)$. Then the statement follows from Theorem 5.4, Theorem
4.2 and the uniqueness implied by Theorem 3.2. ∎

######  Remark 5.8.

Note that all the known stability conditions on $\operatorname{Ku}(X)$ are
Serre-invariant. An interesting question would be to understand whether the
property of $S$-invariance holds for every stability condition on
$\operatorname{Ku}(X)$. A positive answer would imply that there is a unique
$\widetilde{\text{GL}}^{+}_{2}(\mathbb{R})$-orbit of stability conditions by
Theorem 3.2, giving a complete description of the stability manifold of
$\operatorname{Ku}(X)$, in analogy to the case of the bounded derived category
of a genus $\geq 2$ curve studied in [Mac07].

##  6\. Ulrich bundles

Let $X$ be a smooth cubic threefold as before. We denote by
$\mathfrak{M}^{(s)U}_{d}$ the moduli of (stable) Ulrich bundles. By [LMS15,
Theorem B] the moduli space $\mathfrak{M}^{sU}_{d}$ of stable Ulrich bundles
of rank $d$ on $X$ is non-empty and smooth of dimension $d^{2}+1$. In this
section we apply our result on Serre-invariant stability conditions to show
the following.

######  Theorem 6.1.

The moduli space $\mathfrak{M}^{U}_{d}$ of Ulrich bundles of rank $d$ is
irreducible.

Recall that an Ulrich bundle $E$ of rank $d\geq 2$ on $X$ is a vector bundle
on $X$ satisfying $H^{i}(X,E(jH))=0$ for all $i=1,2$, $j\in\mathbb{Z}$ and
such that the graded module $\oplus_{m\in\mathbb{Z}}H^{0}(X,E(mH))$ has $3d$
generators in degree $1$. [CHGS12, Proposition 2.9] shows that any Ulrich
bundle $E$ is Gieseker semistable, and if it is Gieseker stable, then it is
$\mu$-stable. Moreover, if an Ulrich bundle $E$ is Gieseker strictly
semistable, then its stable factors are also Ulrich bundles. Thus [LMS15,
Lemma 2.19] implies that

$\operatorname{ch}(E)=(d,0,-\frac{d}{3}H^{2},0)=d\operatorname{ch}(I_{\ell}).$

Gieseker semistability of $E$ implies that
$\operatorname{Hom}(\mathcal{O}_{X}(kH),E)=0$ for $k\geq 0$. We also have
$\chi(\mathcal{O}_{X}(kH),E)=0$ for $k=0,1$, thus the definition of Ulrich
bundles implies that $\operatorname{Hom}(\mathcal{O}_{X}(kH),E[3])=0$.
Therefore $E\in\operatorname{Ku}(X)$. The first step to prove Theorem 6.1 is
to show that Ulrich bundles are semistable objects in the Kuznetsov component
with respect to the stability conditions $\sigma(\alpha,\beta)$ defined in
Theorem 4.2.

######  Proposition 6.2.

Every $E\in\mathfrak{M}^{U}_{d}$ is $\sigma(\alpha,\beta)$-semistable in
$\operatorname{Ku}(X)$ for any $(\alpha,\beta)\in V$(15).

###### Proof.

Since $\mu_{H}(E)=0$ and $E$ is Gieseker-semistable, we get
$E\in\operatorname{Coh}^{\beta}(X)$ for $\beta<0$ and by [BMS16, Lemma 2.7],
$E$ is $\sigma_{\alpha,\beta}$-semistable for $\alpha\gg 0$. [Fey21, Theorem
3.1] implies that $E$ is $\sigma_{\alpha,\beta_{0}}$-semistable for any
$\alpha>0$ and

$\beta_{0}=-\frac{1}{3d(d-1)}.$

Since $E$ is locally-free, Proposition 4.1 implies that $E$ is
$\sigma^{0}_{\alpha,\beta_{0}}$-semistable, thus by definition $E$ is
$\sigma(\alpha,\beta_{0})$-semistable for $\alpha$ small enough because
$E\in\operatorname{Ku}(X)$. Hence the claim follows from Corollary 4.3. ∎

We denote by $M^{\sigma(\alpha,\beta)}_{\operatorname{Ku}(X)}(d[I_{\ell}])$
the moduli space parametrising $\sigma(\alpha,\beta)$-semistable objects in
$\operatorname{Ku}(X)$ of class $d[I_{\ell}]$. Proposition 6.2 implies we have
an embedding

(28) $\mathfrak{M}^{U}_{d}\hookrightarrow
M^{\sigma(\alpha,\beta)}_{\operatorname{Ku}(X)}(d[I_{\ell}]).$

The next step is to show that (28) is an open embedding. By Corollary 4.3, we
may assume the pair $(\alpha,\beta)\in V$ (15) are on the curve
$\alpha^{2}=\beta^{2}-2/3$, so (up to a shift) any $E\in
M^{\sigma(\alpha,\beta)}_{\operatorname{Ku}(X)}(d[I_{\ell}])$ have maximum
phase $1$ in the heart $\mathcal{A}(\alpha,\beta)$, as in (20). Therefore
$E\in\operatorname{Coh}^{0}_{\alpha,\beta}(X)$ is
$\sigma^{0}_{\alpha,\beta}$-semistable of phase one. We claim $E$ is
$\sigma_{\alpha,\beta}$-semistable, otherwise Proposition 4.1 implies that $E$
lies in the exact triangle

(29) $F[1]\rightarrow E\rightarrow T$

where $F\in\mathcal{F}_{\alpha,\beta}$ and $T\in\operatorname{Coh}_{0}(X)$. By
our choice of $(\alpha,\beta)$, we know $\mu_{\alpha,\beta}(F)=0$. Since
$F\in\mathcal{F}_{\alpha,\beta}$, we know

$\mu^{+}_{\alpha,\beta}(F)\leq 0=\mu_{\alpha,\beta}(F).$

Thus $F$ is $\sigma_{\alpha,\beta}$-semistable of slope zero. But we know
$\mu_{\alpha,\beta}(\mathcal{O}_{X}(-2H)[1])<0$, thus

$\hom(\mathcal{O}_{X},F[2])=\hom(F,\mathcal{O}_{X}(-2H)[1])=0.$

Since $E\in\operatorname{Ku}(X)$, we know $\hom(\mathcal{O}_{X},E)=0$, thus
the exact triangle (29) implies that $\hom(\mathcal{O}_{X},T)=0$ which is not
possible as $T$ is a skyscraper sheaf, and so $E$ is
$\sigma_{\alpha,\beta}$-semistable.

Hence, up to a shift, we may assume any $E\in
M^{\sigma(\alpha,\beta)}_{\operatorname{Ku}(X)}(d[I_{\ell}])$ lies in the
heart $\operatorname{Coh}^{\beta}(X)$. Thus $\mathcal{H}^{i}(E)=0$ if $i\neq
0,-1$ and if $\mathcal{H}^{-1}(E)\neq 0$, then it is a torsion-free sheaf. So
if we consider the locus of objects $E\in
M^{\sigma(\alpha,\beta)}_{\operatorname{Ku}(X)}(d[I_{\ell}])$ with
rank$(\mathcal{H}^{-1}(E))=0$, then we get precisely the locus of sheaves
which we know is non-empty and by [Gro63, Theorem 7.7.5] is an open sub-locus.
Moreover, being Ulrich is an open property [CHGS12], so $\mathfrak{M}^{U}_{d}$
is an open subset of
$M^{\sigma(\alpha,\beta)}_{\operatorname{Ku}(X)}(d[I_{\ell}])$ as claimed.

By Theorem 4.2 we have

$M^{\sigma(\alpha,\beta)}_{\operatorname{Ku}(X)}(d[I_{\ell}])\cong
M^{\sigma(\alpha,\beta)}_{\operatorname{Ku}(X)}(d[\mathcal{S}(I_{\ell})])$

and Corollary 5.7 gives the isomorphism

(30)
$M^{\sigma(\alpha,\beta)}_{\operatorname{Ku}(X)}(d[\mathcal{S}(I_{\ell})])\cong
M^{\overline{\sigma}(b,w)}_{\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0})}(2d[\mathcal{B}_{0}]-d[\mathcal{B}_{-1}]),$

where the stability conditions $\overline{\sigma}(b,w)$ are defined in
Proposition 5.3.

######  Proposition 6.3.

Take an object
$E\in\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathbb{P}^{2},\mathcal{B}_{0})$
of class $2d[\mathcal{B}_{0}]-d[\mathcal{B}_{-1}]=d(4,-3,\frac{3}{2})$. Then
$E$ is a shift of a Gieseker-(semi)stable sheaf if and only if $E$ lies in
$\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0})$ and is
$\overline{\sigma}(b,w)$-(semi)stable for some $(b,w)\in U$ where
$-\frac{5}{4}\leq b<-\frac{3}{4}$.

###### Proof.

First assume $E$ is a Gieseker-(semi)stable sheaf. Then its Gieseker-slope is
less than that of $\mathcal{B}_{1}$, so $\hom(\mathcal{B}_{1},E)=0$. Moreover,

$\hom(\mathcal{B}_{1},E[2])=\hom(E,\mathcal{B}_{0})=0$

thus
$\hom(\mathcal{B}_{1},E[1])=-\chi(\mathcal{B}_{1},E)=-2d\chi(\mathcal{B}_{1},\mathcal{B}_{0})+d\chi(\mathcal{B}_{1},\mathcal{B}_{-1})=0$.
This shows that $E\in\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0})$. Since
$E$ is Gieseker-(semi)stable, it is $\overline{\sigma}_{b,w}$-(semi)stable for
$b<-\frac{3}{4}$ and $w\gg 0$. Then [BLMS17, Remark 5.12] implies that $E$ is
$\overline{\sigma}(b,w)$-semistable.

Conversely, take a $\overline{\sigma}(b,w)$-(semi)stable object $E$ in
$\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0})$. We may assume
$E\in\mathcal{A}_{b}$ and by the uniqueness of
$\mathcal{S}_{\mathcal{B}_{0}}$-invariant stability conditions, we can assume
$w\gg 0$. Thus [BLMS17, Remark 5.12] shows that
$E\in\operatorname{Coh}^{b}(\mathbb{P}^{2},\mathcal{B}_{0})$ and
$\overline{\sigma}_{b,w}$-(semi)stable. This immediately implies that $E$ is a
Gieseker (semi)stable sheaf. ∎

Thus we get the isomorphism

$M^{\overline{\sigma}(b,w)}_{\operatorname{Ku}(\mathbb{P}^{2},\mathcal{B}_{0})}(2d[\mathcal{B}_{0}]-d[\mathcal{B}_{-1}])\cong
M_{(\mathbb{P}^{2},\
\mathcal{B}_{0})}^{ss}(2d[\mathcal{B}_{0}]-d[\mathcal{B}_{1}])\eqqcolon\mathcal{M}_{d}.$

where the latter parametrises Gieseker-semistable sheaves in
$\operatorname{Coh}(\mathbb{P}^{2},\mathcal{B}_{0})$ of class
$2d[\mathcal{B}_{0}]-d[\mathcal{B}_{1}]$. The last step is to show the
following.

######  Proposition 6.4.

The moduli space $\mathcal{M}_{d}$ is irreducible.

###### Proof.

The argument is the same as Step 3 in the proof of [LMS15, Theorem 2.12]; we
add it for completeness.

The first claim is that $\mathcal{M}_{d}$ is connected. If $d=1$, then (30)
gives

$\mathcal{M}_{1}\cong
M^{\sigma(\alpha,\beta)}_{\operatorname{Ku}(X)}([I_{\ell}]),$

and the latter is isomorphic to the Fano surface of lines in $X$ by [PY20,
Theorem 1.1]. In particular, $\mathcal{M}_{1}$ is connected. For $d>1$, the
strictly semistable locus is covered by the images of the natural maps
$\varphi_{d_{1},d_{2}}\colon\mathcal{M}_{d_{1}}\times\mathcal{M}_{d_{2}}\to\mathcal{M}_{d}$
for $1\leq d_{1}\leq d_{2}\leq d$ and $d_{1}+d_{2}=d$ which send any pair
$(E_{1},E_{2})$ to $E_{1}\oplus E_{2}$. Thus by induction we deduce that the
semistable locus of $\mathcal{M}_{d}$ is connected as well. Arguing exactly as
in [KLS06, Lemma 4.1] we obtain the non-existence of a connected component of
$\mathcal{M}_{d}$ consisting of purely stable sheaves. We conclude that
$\mathcal{M}_{d}$ is connected.

Finally, note that for all $E\in\mathcal{M}_{d}$, we have
Ext${}^{2}(E,E)=\operatorname{Hom}(E,E\otimes_{\mathcal{B}_{0}}\mathcal{B}_{-1})^{\vee}=0$.
This implies that $\mathcal{M}_{d}$ is normal, combining well-known results in
deformation theory and Quot schemes (see [LMS15, Theorem 2.12]). ∎

Since $\mathfrak{M}^{U}_{d}$ is an open subset of the irreducible space
$\mathcal{M}_{d}$, it is irreducible. This ends the proof of Theorem 6.1.
We also remark the following property which is a consequence of the previous
computations.

######  Corollary 6.5.

The moduli space
$M^{\sigma(\alpha,\beta)}_{\operatorname{Ku}(X)}(d[I_{\ell}])$ is irreducible
and projective.

###### Proof.

By the above computation we have
$M^{\sigma(\alpha,\beta)}_{\operatorname{Ku}(X)}(d[I_{\ell}])\cong\mathcal{M}_{d}$.
Since the latter is a moduli space of Gieseker semistable sheaves, constructed
as a GIT quotient of an open subset of a Quot-scheme, it is projective. Hence
Proposition 6.4 implies the statement. ∎

In particular, by Theorem 3.2 we deduce that
$M^{\sigma}_{\operatorname{Ku}(X)}(d[I_{\ell}])$ is irreducible and projective
for every Serre-invariant stability condition $\sigma$ on
$\operatorname{Ku}(X)$.

## References

  * [AB13] Daniele Arcara and Aaron Bertram.  Bridgeland-stable moduli spaces for $K$-trivial surfaces.  J. Eur. Math. Soc. (JEMS), 15(1):1–38, 2013.  With an appendix by Max Lieblich.
  * [AK77] Allen B. Altman and Steven L. Kleiman.  Foundations of the theory of Fano schemes.  Compositio Math., 34(1):3–47, 1977.
  * [APR19] Matteo Altavilla, Marin Petkovic, and Franco Rota.  Moduli spaces on the Kuznetsov component of Fano threefolds of index $2$, arXiv:1908.10986, 2019.
  * [AT14] Nicolas Addington and Richard Thomas.  Hodge theory and derived categories of cubic fourfolds.  Duke Math. J., 163(10):1885–1927, 2014.
  * [BB17] A. Bayer and T. Bridgeland.  Derived automorphism groups of K3 surfaces of Picard rank 1.  Duke Math. J., 166(1):75–124, 2017.
  * [BBF+20] Arend Bayer, Sjoerd Beentjes, Soheyla Feyzbakhsh, Georg Hein, Diletta Martinelli, Fatemeh Rezaee, and Benjamin Schmidt.  The desingularization of the theta divisor of a cubic threefold as a moduli space, arXiv:2011.12240, 2020.
  * [Bea18] Arnaud Beauville.  An introduction to Ulrich bundles.  Eur. J. Math., 4(1):26–36, 2018.
  * [BLMS17] A. Bayer, M. Lahoz, E. Macrì, and P. Stellari.  Stability conditions on Kuznetsov components, to appear in ASENS, arXiv:1703.10839, 2017.
  * [BMMS12] M. Bernardara, E. Macrì, S. Mehrotra, and P. Stellari.  A categorical invariant for cubic threefolds.  Adv. Math., 229(2):770–803, 2012.
  * [BMS16] A. Bayer, E. Macrì, and P. Stellari.  The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds.  Invent. Math., 206(3):869–933, 2016.
  * [BMT14] A. Bayer, E. Macrì, and Y. Toda.  Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities.  J. Algebraic Geom., 23(1):117–163, 2014.
  * [Bon89] A. I. Bondal.  Representations of associative algebras and coherent sheaves.  Izv. Akad. Nauk SSSR Ser. Mat., 53(1):25–44, 1989.
  * [Bri07] T. Bridgeland.  Stability conditions on triangulated categories.  Ann. of Math. (2), 166(2):317–345, 2007.
  * [Bri08] T. Bridgeland.  Stability conditions on $K3$ surfaces.  Duke Math. J., 141(2):241–291, 2008.
  * [Bri09] T. Bridgeland.  Spaces of stability conditions.  In Algebraic geometry—Seattle 2005. Part 1, volume 80 of Proc. Sympos. Pure Math., pages 1–21. Amer. Math. Soc., Providence, RI, 2009.
  * [CHGS12] Marta Casanellas, Robin Hartshorne, Florian Geiss, and Frank-Olaf Schreyer.  Stable Ulrich bundles.  Internat. J. Math., 23(8):1250083, 50, 2012.
  * [ES03] David Eisenbud and Frank-Olaf Schreyer.  Resultants and Chow forms via exterior syzygies.  J. Amer. Math. Soc., 16(3):537–579, 2003.  With an appendix by Jerzy Weyman.
  * [Fey21] S. Feyzbakhsh.  An effective restriction theorem via wall-crossing and Mercat’s conjecture, arXiv:1608.07825v2, 2021.
  * [FK20] Daniele Faenzi and Yeongrak Kim.  Ulrich bundles on cubic fourfolds, arXiv:2011.12622, 2020.
  * [Gro63] A. Grothendieck.  Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. II.  Inst. Hautes Études Sci. Publ. Math., (17):91, 1963.
  * [HRS96] D. Happel, I. Reiten, and Sverre O. Smalø.  Tilting in abelian categories and quasitilted algebras.  Mem. Amer. Math. Soc., 120(575):viii+ 88, 1996.
  * [Huy06] D. Huybrechts.  Fourier-Mukai transforms in algebraic geometry.  Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2006.
  * [JLLZ21] Augustinas Jacovskis, Zhiyu Liu, Xun Lin, and Shizhuo Zhang.  Hochschild cohomology and categorical Torelli for Gushel-Mukai threefolds, arXiv:2108.02946, 2021.
  * [KLS06] D. Kaledin, M. Lehn, and Ch. Sorger.  Singular symplectic moduli spaces.  Invent. Math., 164(3):591–614, 2006.
  * [KP21] Alexander Kuznetsov and Alexander Perry.  Serre functors and dimensions of residual categories, arXiv:2109.02026, 2021.
  * [Kuz04] A. G. Kuznetsov.  Derived category of a cubic threefold and the variety $V_{14}$.  Tr. Mat. Inst. Steklova, 246(Algebr. Geom. Metody, Svyazi i Prilozh.):183–207, 2004.
  * [Kuz08] Alexander Kuznetsov.  Derived categories of quadric fibrations and intersections of quadrics.  Adv. Math., 218(5):1340–1369, 2008.
  * [Kuz09] A. G. Kuznetsov.  Derived categories of Fano threefolds.  Tr. Mat. Inst. Steklova, 264(Mnogomernaya Algebraicheskaya Geometriya):116–128, 2009.
  * [Kuz10] Alexander Kuznetsov.  Derived categories of cubic fourfolds.  In Cohomological and geometric approaches to rationality problems, volume 282 of Progr. Math., pages 219–243. Birkhäuser Boston, Boston, MA, 2010.
  * [Kuz19] Alexander Kuznetsov.  Calabi-Yau and fractional Calabi-Yau categories.  J. Reine Angew. Math., 753:239–267, 2019.
  * [Li19] Chunyi Li.  Stability conditions on Fano threefolds of Picard number 1.  J. Eur. Math. Soc. (JEMS), 21(3):709–726, 2019.
  * [LLMS18] Martí Lahoz, Manfred Lehn, Emanuele Macrì, and Paolo Stellari.  Generalized twisted cubics on a cubic fourfold as a moduli space of stable objects.  J. Math. Pures Appl. (9), 114:85–117, 2018.
  * [LMS15] Martí Lahoz, Emanuele Macrì, and Paolo Stellari.  Arithmetically Cohen-Macaulay bundles on cubic threefolds.  Algebr. Geom., 2(2):231–269, 2015.
  * [LPZ20] Chunyi Li, Laura Pertusi, and Xiaolei Zhao.  Elliptic quintics on cubic fourfolds, O’Grady 10, and Lagrangian fibrations.  arXiv:2007.14108, 2020.
  * [LZ21] Zhiyu Liu and Shizhuo Zhang.  A note on Bridgeland moduli spaces and moduli spaces of sheaves on $x_{14}$ and $y_{3}$, arXiv:2106.01961, 2021.
  * [Mac07] E. Macrì.  Stability conditions on curves.  Math. Res. Lett., 14(4):657–672, 2007.
  * [Mac14] A. Maciocia.  Computing the walls associated to Bridgeland stability conditions on projective surfaces.  Asian J. Math., 18(2):263–279, 2014.
  * [PPZ19] Alexander Perry, Laura Pertusi, and Xiaolei Zhao.  Stability conditions and moduli spaces for Kuznetsov components of Gushel–Mukai varieties.  To appear in Geometry and Topology, arXiv:1912.06935, 2019.
  * [PR22] Laura Pertusi and Ethan Robinett.  Stability conditions on Kuznetsov components of Gushel–Mukai threefolds and Serre functor, arXiv:2112.04769, 2022.
  * [PY20] L. Pertusi and S. Yang.  Some remarks on Fano threefolds of index two and stability conditions, to appear in IMRN, arXiv:2004.02798v3, 2020.
  * [Qin21] Xuqiang Qin.  Bridgeland stability of minimal instanton bundles on Fano threefolds, arXiv:2105.14617, 2021.
  * [Sch20] Benjamin Schmidt.  Bridgeland stability on threefolds: some wall crossings.  J. Algebraic Geom., 29(2):247–283, 2020.
  * [Tod13] Yukinobu Toda.  Bogomolov–Gieseker-type inequality and counting invariants.  J. Topol., 6(1):217–250, 2013.
  * [Zha21] Shizhuo Zhang.  Bridgeland moduli spaces and Kuznetsov’s Fano threefold conjecture.  arXiv:2012.12193v2, 2021.

S. Feyzbakhsh, Department of Mathematics, Imperial College, London, SW7 2AZ,
United Kingdom.

E-mail address<EMAIL_ADDRESS>
URL: https://www.imperial.ac.uk/people/s.feyzbakhsh.

L. Pertusi, Dipartimento di Matematica F. Enriques, Università degli studi di
Milano, Via Cesare Saldini 50, 20133 Milano, Italy.

E-mail address<EMAIL_ADDRESS>
URL: http://www.mat.unimi.it/users/pertusi/.