# Stability manifolds of Kuznetsov components of prime Fano threefolds

Changping Fan  College of Mathematics, Sichuan University, Chengdu, Sichuan
Province 610064, P. R. China<EMAIL_ADDRESS>,  Zhiyu Liu  Institute
for Advanced Study in Mathematics, Zhejiang University, Hangzhou, Zhejiang
Province 310030, P. R. China<EMAIL_ADDRESS>and  Songtao Ma
Department of Mathematics, University of Michigan - Ann Arbor, Ann Arbor,
Michigan 48104, United States<EMAIL_ADDRESS>
###### Abstract.

Let $X$ be a cubic threefold, quartic double solid or Gushel–Mukai threefold,
and $\mathcal{K}u(X)\subset\mathrm{D}^{b}(X)$ be its Kuznetsov component. We
show that a stability condition $\sigma$ on $\mathcal{K}u(X)$ is Serre-
invariant if and only if its homological dimension is at most $2$. As a
corollary, we prove that all Serre-invariant stability conditions on
$\mathcal{K}u(X)$ form a contractible connected component of the stability
manifold.

###### Key words and phrases:

Derived categories, Bridgeland stability conditions, Kuznetsov components,
Fano threefolds, Stability manifolds.

###### 2010 Mathematics Subject Classification:

Primary 14F05; secondary 14J45, 14D20, 14D23

###### Contents

  1. 1 Introduction
  2. 2 Kuznetsov components
  3. 3 Bridgeland stability conditions
  4. 4 Preliminary results
  5. 5 A criterion of Serre-invariant stability conditions

##  1\. Introduction

Motivated by the concept of $\Pi$-stability condition on string theory by
Douglas, the notion of a stability condition on a triangulated category
$\mathcal{D}$ was introduced by Bridgeland in [6]. It is proved in [6] that
the set $\operatorname{Stab}(\mathcal{D})$ of all stability conditions (with
respect to a fixed lattice) on $\mathcal{D}$ has a natural structure of the
complex manifold. In recent years, the geometry of
$\operatorname{Stab}(\mathcal{D})$ has played a central role in the theory of
homological mirror symmetry, representation theory, symplectic geometry, and
moduli of sheaves, see e.g. [16, 9, 1, 4, 3, 17, 10, 7].

One of the most fundamental examples is $\mathcal{D}=\mathrm{D}^{b}(C)$ where
$C$ is a smooth projective curve. Due to [29] and [27],
$\operatorname{Stab}(\mathrm{D}^{b}(C))$ is a contractible manifold and has a
complete description. More general, there is a folklore conjecture that if
$\operatorname{Stab}(\mathcal{D})$ is non-empty, then it is simply connected.
However, with current techniques, it still seems to be impossible to prove
some basic properties of $\operatorname{Stab}(\mathcal{D})$, even for
connectedness. Therefore, people found that it was better to study a
distinguished connected component of $\operatorname{Stab}(\mathcal{D})$. For
example, when $\mathcal{D}=\mathrm{D}^{b}(X)$ for a smooth projective variety
$X$, we say a component is distinguished if it contains a geometric stability
condition, with respect to which skyscraper sheaves of closed points are all
stable with the same phase. Results along this direction include [8, 5, 1, 15,
26, 36, 37, 12, 11, 38], etc.

The examples above mainly focus on the situation when $\mathcal{D}$ is the
bounded derived category of a certain smooth variety or the derived category
of representations of a well-described quiver or algebra. On the other hand,
by a series of work [21, 22, 23], we can associate a very interesting
triangulated subcategory $\mathcal{K}u(X)$ of $\mathrm{D}^{b}(X)$ to a prime
Fano threefold $X$, called _the Kuznetsov component_ of $X$ (cf. Section 2),
which are not of this form in many cases. They come from semiorthogonal
decompositions and have been widely studied in recent times. For instance,
when $X\subset\mathbb{P}^{4}$ is a smooth cubic threefold, $\mathcal{K}u(X)$
is defined as the right orthogonal of line bundles
$\operatorname{\mathcal{O}}_{X}$ and $\operatorname{\mathcal{O}}_{X}(1)$,
which is a $\frac{5}{3}$-Calabi–Yau category in the sense of [24], and is not
equivalent to $\mathrm{D}^{b}(X)$ or the derived category of a nice quiver or
algebra. And by a celebrated result of Orlov [30], $\mathcal{K}u(X)$ is also
equivalent to the category of graded matrix factorization of $X$, which plays
an important role in the theory of Landau–Ginzburg models.

Moreover, for many prime Fano threefolds $X$, the category $\mathcal{K}u(X)$
can be regarded as a non-commutative smooth projective curve for many reasons:
it is a non-commutative smooth projective variety in the sense of [31]; the
rank of its numerical Grothendieck group is the same as $\mathrm{D}^{b}(C)$;
its Serre dimension is smaller than $2$, etc. Therefore, it is natural to
study $\operatorname{Stab}(\mathcal{K}u(X))$ and expect its geometry is
similar to $\operatorname{Stab}(\mathrm{D}^{b}(C))$,

All prime Fano threefolds are classified in [32] by their index and degree
$d$. When $X$ is of index $\geq 3$, or index $2$ and degree $d>3$, or index
$1$ and degree $d>10$, $\mathcal{K}u(X)$ is either equivalent to
$\mathrm{D}^{b}(C)$ for a smooth projective curve $C$ or the derived category
of representations of a generalized Kronecker quiver, whose
$\operatorname{Stab}(\mathcal{K}u(X))$ is contractible and has been studied in
[27, 12]. When $X$ is of index $2$ and degree $d<2$ or index $1$ and degree
$d<10$, $\mathcal{K}u(X)$ behaves badly and it is not reasonable to regard
$\mathcal{K}u(X)$ as a non-commutative curve in these cases. Therefore, the
only remaining examples are cubic threefolds (index $2$ and degree $d=3$),
quartic double solids (index $2$ and degree $d=2$), and Gushel–Mukai
threefolds (index $1$ and degree $d=10$). In these cases, skyscraper sheaves
are not in $\mathcal{K}u(X)$, so there is no notion of geometric stability
conditions. However, we can consider the projection objects of skyscrapers
into $\mathcal{K}u(X)$, and these objects are stable with respect to _Serre-
invariant_ stability conditions on $\mathcal{K}u(X)$ (cf. [20, 13]). Then we
say a connected component of $\operatorname{Stab}(\mathcal{K}u(X))$ is
distinguished if it contains a Serre-invariant stability condition (see
Section 3.3 for the definition).

By Remark 3.11, $\mathcal{K}u(X)$ contains a unique distinguished component.
Our first theorem states that in these three cases, the distinguished
connected component of $\operatorname{Stab}(\mathcal{K}u(X))$ actually
consists of Serre-invariant stability conditions and is contractible.

######  Theorem 1.1 (Corollary 5.9).

Let $X$ be a cubic threefold, quartic double solid or Gushel–Mukai threefold.
Then the subspace of all Serre-invariant stability conditions is a
contractible connected component of $\operatorname{Stab}(\mathcal{K}u(X))$.

When $X$ is a quartic double solid, Theorem 1.1 follows from [34, Theorem 1.3]
and Theorem 3.10, in which a geometric description of the Serre functor of
$\mathcal{K}u(X)$ is used and can only work for quartic double solids. Our
method is different from [34, Theorem 1.3] and can be generalized to other
cases.

In our paper, we prove the following criterion of Serre-invariant stability
conditions:

######  Theorem 1.2 (Theorem 5.8).

Let $X$ be a cubic threefold, quartic double solid or Gushel–Mukai threefold,
and $\sigma$ be a stability condition on $\mathcal{K}u(X)$. Then the following
conditions are equivalent:

  1. (1)

$\mathrm{homdim}(\sigma)\leq 2$,

  2. (2)

$\mathrm{gldim}(\sigma)\leq 2$, and

  3. (3)

$\sigma$ is Serre-invariant.

Here, $\mathrm{homdim}(\sigma)$ is the _homological dimension_ of $\sigma$ and
$\mathrm{gldim}(\sigma)$ is the _global dimension_ of $\sigma$ (cf. Section
4.1).

The implications (3) $\Rightarrow$ (2) and (2) $\Rightarrow$ (1) in Theorem
1.2 are easy. However, it is quite surprising for the authors that (1) implies
(3), since (2) and (3) are preserved by
$\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$-action, while the condition (1) is
not in general.

Finally, we conjecture that Theorem 1.2 holds for any stability condition on
$\mathcal{K}u(X)$ (see Conjecture 5.10), which together with Theorem 1.1
implies
$\operatorname{Stab}(\mathcal{K}u(X))\cong\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})\cong\mathbb{C}\times\mathbb{H}$.

### Notation and conventions

  * •

We denote the phase and slope with respect to a stability condition $\sigma$
by $\phi_{\sigma}$ and $\mu_{\sigma}$, respectively. The maximal/minimal phase
of the Harder–Narasimhan factors of a given object will be denoted by
$\phi_{\sigma}^{+}$ and $\phi_{\sigma}^{-}$, respectively.

  * •

We use $\hom$ and $\operatorname{ext}^{i}$ to represent the dimension of the
vector spaces $\operatorname{Hom}$ and $\operatorname{Ext}^{i}$. We denote
$\operatorname{RHom}(-,-)=\bigoplus_{i\in\mathbb{Z}}\operatorname{Hom}(-,-[i])[-i]$.

  * •

All triangulated categories are assumed to be $\mathbb{C}$-linear of finite
type, i.e. for any two objects $E,F$ we have
$\sum_{i\in\mathbb{Z}}\operatorname{ext}^{i}(E,F)<+\infty$.

  * •

We denote the numerical class in the numerical Grothendieck group by $[E]$ for
any object $E$. In our setting, giving a numerical class is equivalent to
giving a Chern character.

###  1.1. Plan of the paper

In Section 2, we introduce Kuznetsov components $\mathcal{K}u(X)$ and
recollect some basic properties. In Section 3, we recall some basic
definitions and properties of Bridgeland stability conditions on
$\mathcal{K}u(X)$. Then we review the definition of global dimension and
homological dimension, and prove some general preliminary results in Section
4. Finally, we prove our main results Theorem 5.7, Theorem 5.8 and Corollary
5.9 in Section 5.

### Acknowledgements

We would like to thank Chunyi Li and Alexander Perry for their careful reading
and many useful suggestions. The third author would also like to thank Dima
Arinkin and Andrei Caldararu for useful discussions. Part of the work was
finished when the second author visited the Hausdorff Research Institute for
Mathematics (HIM). He is grateful for the wonderful working environment and
hospitality.

##  2\. Kuznetsov components

In this paper, we consider a smooth projective threefold $X$ with $-K_{X}$
ample and Picard rank one, i.e. a prime Fano threefold. In the following, we
are mainly concerned about three types of prime Fano threefolds:

  * •

cubic threefolds: $X\subset\mathbb{P}^{4}$ is a smooth cubic hypersurface,

  * •

quartic double solids: there is a double cover $X\to\mathbb{P}^{3}$ branched
along a smooth quartic surface, and

  * •

Gushel–Mukai threefolds: a smooth intersection

$X=\mathrm{Cone}(\operatorname{Gr}(2,5))\cap Q,$

where $\mathrm{Cone}(\operatorname{Gr}(2,5))\subset\mathbb{P}^{10}$ is the
projective cone over the Plücker embedded Grassmannian
$\operatorname{Gr}(2,5)\subset\mathbb{P}^{9}$, and $Q\subset\mathbb{P}^{10}$
is a quadric hypersurface in a linear subspace
$\mathbb{P}^{7}\subset\mathbb{P}^{10}$.

When $X$ is a cubic threefold or quartic double solid, we have a
semiorthogonal decomposition

$\mathrm{D}^{b}(X)=\langle\mathcal{K}u(X),\operatorname{\mathcal{O}}_{X},\operatorname{\mathcal{O}}_{X}(1)\rangle,$

where $\operatorname{\mathcal{O}}_{X}(1)$ is the ample generator of
$\operatorname{Pic}(X)$.

When $X$ is a Gushel–Mukai threefold, there is also a semiorthogonal
decomposition given in [25]:

$\mathrm{D}^{b}(X)=\langle\mathcal{K}u(X),\operatorname{\mathcal{O}}_{X},\mathcal{E}^{\vee}_{X}\rangle,$

where $\mathcal{E}_{X}$ is the pullback of the tautological subbundle on
$\operatorname{Gr}(2,5)$ along $X\to\operatorname{Gr}(2,5)$.

######  Definition 2.1.

Let $X$ be a cubic threefold, quartic double solid, or Gushel–Mukai threefold.
The _Kuznetsov component_ of $X$ is the admissible triangulated subcategory
$\mathcal{K}u(X)\subset\mathrm{D}^{b}(X)$ constructed above.

Let $\mathrm{K}(\mathcal{K}u(X))$ denote the Grothendieck group of
$\mathcal{K}u(X)$. We have the bilinear Euler form on
$\mathrm{K}(\mathcal{K}u(X))$ defined by

$\chi([E],[F])=\sum_{i\in\mathbb{Z}}(-1)^{i}\operatorname{ext}^{i}(E,F)$

for $[E],[F]\in\mathrm{K}(\mathcal{K}u(X))$. The _numerical Grothendieck
group_ of $\mathcal{K}u(X)$ is defined as

$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X)):=\mathrm{K}(\mathcal{K}u(X))/\ker\chi.$

Therefore, the Euler form on $\mathrm{K}(\mathcal{K}u(X))$ can be defined on
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$ as well, which we also denote by
$\chi(-,-)$.

The Serre functor of a Kuznetsov component can be computed from the Serre
functor of $X$ and mutation functors. We define

$\mathbf{O}:=\bm{\mathrm{L}}_{\operatorname{\mathcal{O}}_{X}}\circ(-\otimes\operatorname{\mathcal{O}}_{X}(1)),$

where $\bm{\mathrm{L}}_{\operatorname{\mathcal{O}}_{X}}$ is the left mutation
functor along $\operatorname{\mathcal{O}}_{X}$. When $X$ is a cubic threefold
or a quartic double solid, it is an autoequivalence of $\mathcal{K}u(X)$.

  * •

When $X$ is a cubic threefold, we have
$S_{\mathcal{K}u(X)}\cong\mathbf{O}[1]$. In this case,
$S^{3}_{\mathcal{K}u(X)}\cong[5]$. The functor $S_{\mathcal{K}u(X)}$ acts non-
trivially on $\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$.

  * •

When $X$ is a quartic double solid or a Gushel–Mukai threefold, we have
$S^{2}_{\mathcal{K}u(X)}\cong[4]$. In this case we define
$\tau:=S_{\mathcal{K}u(X)}[-2]$. The functor $S_{\mathcal{K}u(X)}$ (and hence
$\tau$) acts trivially on $\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$.

Finally, we give a description of $\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
and its Euler form.

When $X$ is a cubic threefold or a quartic double solid, a computation using
[23, pp.6] shows that, the rank two lattice
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$ is generated by

$v=[\mathcal{I}_{l}],\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
w=[\mathbf{O}(\mathcal{I}_{l})[1]],$

where $\mathcal{I}_{l}$ is the ideal sheaf of a line $l\subset X$. The Euler
form $\chi(-,-)$ with respect to the basis $v$ and $w$ is given by the matrix

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

when $X$ is a cubic threefold, and

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

when $X$ is a quartic double solid.

For a Gushel–Mukai threefold $X$, $\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
is computed in [25] and [23, pp.5]. The numerical Grothendieck group
$\mathrm{K}_{\mathrm{num}}(\mathcal{A}_{X})$ is a rank 2 lattice with basis
vectors

$v=[\mathcal{I}_{C}],\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
w=[F],$

where $C\subset X$ is a conic and $F$ is a rank two slope-stable sheaf with
$c_{1}(F)=-1,c_{2}(F)=5$ and $c_{3}(F)=0$. The Euler form $\chi(-,-)$ with
respect to the basis is

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

##  3\. Bridgeland stability conditions

In this section, we recall basic definitions of Bridgeland stability
conditions on triangulated categories. Then we focus on the stability
conditions on Kuznetsov components. We follow [6] and [2].

###  3.1. Bridgeland stability conditions

Let $\mathcal{D}$ be a triangulated category and $\mathrm{K}(\mathcal{D})$ its
Grothendieck group.

######  Definition 3.1.

The _heart of a bounded t-structure_ on $\mathcal{D}$ is an abelian
subcategory $\mathcal{A}\subset\mathcal{D}$ such that the following conditions
are satisfied:

  1. (1)

for any $E,F\in\mathcal{A}$ and $n<0$, we have $\operatorname{Hom}(E,F[n])=0$;

  2. (2)

for any object $E\in\mathcal{D}$ there exist objects $E_{i}\in\mathcal{A}$ and
maps

$0=E_{0}\xrightarrow{\pi_{1}}E_{1}\xrightarrow{\pi_{2}}\cdots\xrightarrow{\pi_{m}}E_{m}=E$

such that $\operatorname{cone}(\pi_{i})=A_{i}[k_{i}]$ where
$A_{i}\in\mathcal{A}$ and the $k_{i}$ are integers such that
$k_{1}>k_{2}>\cdots>k_{m}$.

Fix a surjective morphism $v\colon\mathrm{K}(\mathcal{D})\rightarrow\Lambda$
to a finite rank lattice $\Lambda$.

######  Definition 3.2.

A _stability condition_ on $\mathcal{D}$ is a pair $\sigma=(\mathcal{A},Z)$
where $\mathcal{A}$ is the heart of a bounded t-structure on $\mathcal{D}$,
and $Z:\Lambda\rightarrow\mathbb{C}$ is a group homomorphism such that

  1. (1)

the composition $Z\circ
v:\mathrm{K}(\mathcal{A})=\mathrm{K}(\mathcal{D})\rightarrow\mathbb{C}$
satisfies: for any $E\neq 0\in\mathcal{D}$ we have
$\operatorname{Im}Z(v(E))\geq 0$ and if $\operatorname{Im}Z(v(E))=0$ then
$\operatorname{Re}Z(v(E))<0$. From now on, we write $Z(E)$ rather than
$Z(v(E))$.

We can define a _slope_ $\mu_{\sigma}$ for $\sigma$ using $Z$. For any
$E\in\mathcal{A}$, set

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

We say an object $0\neq E\in\mathcal{A}$ is $\sigma$-(semi)stable if
$\mu_{\sigma}(F)<\mu_{\sigma}(E)$ (respectively
$\mu_{\sigma}(F)\leq\mu_{\sigma}(E)$) for all proper subobjects $F\subset E$.

  1. (2)

Any object $E\in\mathcal{A}$ has a Harder–Narasimhan (HN) filtration in terms
of $\sigma$-semistability;

  2. (3)

There exists a quadratic form $Q$ on $\Lambda\otimes\mathbb{R}$ such that
$Q|_{\ker Z}$ is negative definite, and $Q(E)\geq 0$ for all
$\sigma$-semistable objects $E\in\mathcal{A}$.

######  Definition 3.3.

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

$\phi(E):=\frac{1}{\pi}\mathrm{arg}(Z(E))\in(0,1].$

Specially, if $Z(E)=0$ then $\phi(E)=1$. If $F=E[n]$, then we define

$\phi(F):=\phi(E)+n$

A _slicing_ $\mathcal{P}$ of $\mathcal{D}$ consists of full additive
subcategories $\mathcal{P}(\phi)\subset\mathcal{D}$ for each
$\phi\in\mathbb{R}$ satisfying

  1. (4)

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

  2. (5)

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

We will use both notations $\sigma=(\mathcal{A},Z)$ and
$\sigma=(\mathcal{P},Z)$ for a stability condition $\sigma$ with heart
$\mathcal{A}=\mathcal{P}((0,1])$ where $\mathcal{P}$ is the slicing of
$\sigma$.

In this paper, we let $\mathcal{D}$ be the Kuznetsov component and $\Lambda$
be the numerical Grothendieck group
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$, which is
$\mathrm{K}(\mathcal{K}u(X))$ modulo the kernel of the Euler form $\chi(-,-)$.

For a stability condition $\sigma=(\mathcal{A},Z)$ on $\mathcal{D}$, we define
the _homological dimension_ of $\mathcal{A}$ as the smallest integer
$\mathrm{homdim}(\mathcal{A})$ such that $\operatorname{Hom}(A,B[n])=0$ for
any $n>\mathrm{homdim}(\mathcal{A})$. The homological dimension of a stability
condition is defined as the homological dimension of its heart.

###  3.2. The stability manifold

The set of stability conditions $\operatorname{Stab}(\mathcal{D})$ has a
natural topology induced by a generalized metric. Moreover, the following
theorem of Bridgeland states that the generalized metric space
$\operatorname{Stab}(\mathcal{D})$ is a complex manifold.

######  Theorem 3.4.

_(Bridgeland Deformation Theorem,[6])_ The continuous map

$\mathcal{Z}:\operatorname{Stab}(\mathcal{D})\rightarrow\operatorname{Hom}(\Lambda,\mathbb{C}),\quad(\mathcal{A},Z)\mapsto
Z$

is a local homeomorphism. In particular, the generalized metric space
$\operatorname{Stab}(\mathcal{D})$ has the structure of a complex manifold of
dimension $\operatorname{rk}(\Lambda)$.

Next, we recall two natural group actions on
$\operatorname{Stab}(\mathcal{D})$.

  1. (1)

An element $\tilde{g}=(g,G)$ in the universal covering
$\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$ of the group
$\mathrm{GL}^{+}(2,\mathbb{R})$ consists of an increasing function
$g:\mathbb{R}\rightarrow\mathbb{R}$ such that $g(\phi+1)=g(\phi)+1$ and matrix
$G\in\mathrm{GL}^{+}(2,\mathbb{R})$ with $\det(G)>0$. It acts on the right on
the stability manifold by $\sigma\cdot\tilde{g}:=(G^{-1}\circ
Z,\mathcal{P}(g(\phi))$ for any
$\sigma=(\mathcal{P},Z)\in\operatorname{Stab}(\mathcal{D})$ (see [6, Lemma
8.2]).

  2. (2)

Let $\operatorname{Aut}_{\Lambda}(\mathcal{D})$ be the group of exact
autoequivalences of $\mathcal{D}$, whose action $\Phi_{*}$ on
$\mathrm{K}(\mathcal{D})$ is compatible with
$v\colon\mathrm{K}(\mathcal{D})\to\Lambda$. For
$\Phi\in\operatorname{Aut}_{\Lambda}(\mathcal{D})$ and
$\sigma=(\mathcal{P},Z)\in\operatorname{Stab}(\mathcal{D})$, we define a left
action of the group of linear exact autoequivalences
$\operatorname{Aut}_{\Lambda}(\mathcal{D})$ by
$\Phi\cdot\sigma=(\Phi(\mathcal{P}),Z\circ\Phi_{*}^{-1})$, where $\Phi_{*}$ is
the automorphism of $\mathrm{K}(\mathcal{D})$ induced by $\Phi$.

###  3.3. Serre-invariant stability conditions on Kuznetsov components

######  Theorem 3.5 ([2, Proposition 6.8]).

Let $X$ be a cubic threefold, quartic double solid or Gushel–Mukai threefold.
Then there is a family of stability conditions on $\mathcal{K}u(X)$ with
respect to the rank two lattice $\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
and natural surjection
$\mathrm{K}(\mathcal{K}u(X))\twoheadrightarrow\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$.

######  Definition 3.6.

Let $\sigma$ be a stability condition on a triangulated category $\mathcal{D}$
with the Serre functor $S_{\mathcal{D}}$. It is called _Serre-invariant_ if
$S_{\mathcal{D}}\cdot\sigma=\sigma\cdot\tilde{g}$ for some
$\tilde{g}\in\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$.

By virtue of [34, Corollary 5.5] and [33, Theorem 1.1], all stability
conditions constructed in Theorem 3.5 are Serre-invariant.

We recall several properties of Serre-invariant stability conditions on
Kuznetsov components.

######  Proposition 3.7.

Let $X$ be a cubic threefold, quartic double solid or Gushel-Mukai threefold
and $\sigma$ be a Serre-invariant stability condition on $\mathcal{K}u(X)$.

  1. (1)

the homological dimension of the heart of $\sigma$ is two,

  2. (2)

For any $E$ and $F$ in $\mathcal{K}u(X)$ with phases
$\phi^{+}_{\sigma}(E)<\phi^{-}_{\sigma}(F)$, we have $\mathrm{Hom}(E,F[2])=0$.

  3. (3)

If $X$ is a cubic threefold and $E\in\mathcal{K}u(X)$ is $\sigma$-semistable,
then $\operatorname{Ext}^{2}(E,E)=0$.

  4. (4)

If $X$ is a quartic double solid or Gushel–Mukai threefold and
$E\in\mathcal{K}u(X)$ is $\sigma$-stable, then
$\operatorname{ext}^{2}(E,E)\leq 1$.

  5. (5)

$\mathrm{ext}^{1}(E,E)\geq 2$ for every non-zero object $E\in\mathcal{K}u(X)$.

  6. (6)

If $\operatorname{ext}^{1}(E,E)\leq 3$, then $E$ is $\sigma$-stable.

###### Proof.

See [34, Section 5], [19, Section 4.7] and [14, Proposition 3.4]. ∎

######  Lemma 3.8.

Let $X$ be a cubic threefold, quartic double solid or Gushel-Mukai threefold,
and $E\in\mathcal{K}u(X)$ be a non-zero object with
$\operatorname{ext}^{1}(E,E)\leq 3$. Then $E$ is stable with respect to any
Serre-invariant stability condition and
$[E]\in\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$ is primitive.

###### Proof.

The stability of $E$ follows from (6) of Proposition 3.7. Thus $\hom(E,E)=1$.
Moreover, by (3) and (4) of Proposition 3.7, we see $\chi(E,E)=-1$ when $X$ is
a cubic threefold, and $\chi(E,E)=-1$ or $-2$ otherwise. Then by a computation
using Euler form, we know that
$[E]\in\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$ is primitive. ∎

######  Lemma 3.9.

Let $X$ be a cubic threefold, quartic double solid or Gushel–Mukai threefold.
Suppose $\sigma$ is a Serre-invariant stability on $\mathcal{K}u(X)$ and $E$
is a $\sigma$-semistable object.

  1. (1)

When $X$ is a cubic threefold, we have

$\phi_{\sigma}(E)+1\leq\phi_{\sigma}(S_{\mathcal{K}u(X)}(E))<\phi_{\sigma}(E)+2.$

The first inequality is strict if $E$ is $\sigma$-stable.

  2. (2)

When $X$ is a quartic double solid or Gushel–Mukai threefold, we have

$\phi_{\sigma}(S_{\mathcal{K}u(X)}(E))=\phi_{\sigma}(E)+2.$

###### Proof.

(1) follows from [14, Proposition 3.3(a),(d)]. In the case of (2), we also
have
$\phi_{\sigma}(E)+1\leq\phi_{\sigma}(S_{\mathcal{K}u(X)}(E))\leq\phi_{\sigma}(E)+2$.
Since $[S_{\mathcal{K}u(X)}(E)]=[E]$, we have

$\phi_{\sigma}(E)-\phi_{\sigma}(S_{\mathcal{K}u(X)}(E))\in 2\mathbb{Z}.$

Thus the we get $\phi_{\sigma}(S_{\mathcal{K}u(X)}(E))=\phi_{\sigma}(E)+2$. ∎

Finally, we recall the following uniqueness result of Serre-invariant
stability conditions.

######  Theorem 3.10 ([19, Theorem 4.25], [14, Theorem 3.1]).

Let $X$ be a cubic threefold, quartic double solid or Gushel–Mukai threefold.
Then all Serre-invariant stability conditions on $\mathcal{K}u(X)$ are in the
same $\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$-orbit.

######  Remark 3.11.

Let $\mathsf{K}$ be the subset of all Serre-invariant stability conditions
with induced topology from $\operatorname{Stab}(\mathcal{K}u(X))$. Then
Theorem 3.10 implies
$\mathsf{K}\cong\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})\cong\mathbb{C}\times\mathbb{H}$.

##  4\. Preliminary results

In this section, we provide some preliminary results.

We start with a useful lemma. We say a non-zero element $a$ in a lattice
$\Lambda$ is _primitive_ if it can not be written as $a=nb$ for $b\in\Lambda$
and $n\in\mathbb{Z}_{>1}$.

######  Lemma 4.1.

Let $\sigma$ be a stability condition on $\mathcal{D}$ with respect to the
lattice $\Lambda$ and $v\colon K(\mathcal{D})\to\Lambda$. If a non-zero object
$E\in\mathcal{D}$ is not $\sigma$-stable and $v(E)\in\Lambda$ is primitive,
then we can find an exact triangle

$A\to E\to B$

such that $A$ is $\sigma$-semistable,
$\phi_{\sigma}(A)\geq\phi^{+}_{\sigma}(B)$ and $\operatorname{Hom}(A,B)=0$.
Moreover, we can assume all Jordan–Hölder factors of $A$ are isomorphic to
each other.

###### Proof.

If $E$ is strictly $\sigma$-semistable, since $v(E)$ is primitive,
Jordan–Hölder factors of $E$ can not be isomorphic to each other. Then the
existence of exact triangle $A\to E\to B$ follows from the existence of
Jordan–Hölder filtration of $A$.

If $E$ is not $\sigma$-semistable, then by the existence of Harder–Narasimhan
filtration, we can find an exact triangle

$A^{\prime}\to E\to B^{\prime}$

such that $A^{\prime}$ is $\sigma$-semistable and
$\phi_{\sigma}(A^{\prime})>\phi^{+}_{\sigma}(B^{\prime})$. In particular, we
see $\operatorname{Hom}(A^{\prime},B^{\prime})=0$. If Jordan–Hölder factors of
$A^{\prime}$ are isomorphic to each other, then we take $A:=A^{\prime}$ and
$B:=B^{\prime}$. If $A^{\prime}$ has at least two non-isomorphic Jordan–Hölder
factors, then as in the previous case, from the existence of Jordan–Hölder
filtration of $A^{\prime}$, we can find an exact triangle $A\to A^{\prime}\to
A^{\prime\prime}$ such that $A$ is $\sigma$-semistable with
$\operatorname{Hom}(A,A^{\prime\prime})=0$, and all Jordan–Hölder factors of
$A$ are isomorphic to each other. We take $B:=\operatorname{cone}(A\to E)$,
where the map $A\to E$ is the composition of $A\to A^{\prime}$ and
$A^{\prime}\to E$. Hence we have the following commutative diagram

${A}$${A}$${0}$${{A^{\prime}}}$${E}$${{B^{\prime}}}$${{A^{\prime\prime}}}$${B}$${{B^{\prime}}}$

with all rows and columns are exact triangles. Since
$\phi_{\sigma}(A)=\phi_{\sigma}(A^{\prime})>\phi_{\sigma}^{+}(B^{\prime})$, we
obtain $\operatorname{Hom}(A,B^{\prime})=0$. Hence from
$\operatorname{Hom}(A,A^{\prime\prime})=0$, we see
$\operatorname{Hom}(A,B)=0$. ∎

We have the following standard spectral sequence, see e.g. [35, Lemma 2.27].

######  Lemma 4.2.

Let $X$ be a smooth projective variety. Suppose that there are two exact
triangles in $\mathrm{D}^{b}(X)$:

$A_{1}\to B_{1}\to C_{1},\quad A_{2}\to B_{2}\to C_{2}.$

There exist a spectral sequence which degenerates at $E_{3}$ and converges to
$\operatorname{Ext}^{*}(C_{1},C_{2})$, with $E_{1}$-page

$E^{p,q}_{1}=\left\\{\begin{aligned}
\mathrm{Ext}^{q}(B_{1},A_{2})&,\leavevmode\nobreak\ p=-1\\\
\mathrm{Ext}^{q}(A_{1},A_{2})\oplus\mathrm{Ext}^{q}(B_{1},B_{2})&,\leavevmode\nobreak\
p=0\\\ \mathrm{Ext}^{q}(A_{1},B_{2})&,\leavevmode\nobreak\ p=1\\\
0&,\leavevmode\nobreak\ p\notin[-1,1]\end{aligned}\right.$

Moreover, differentials $d_{1}^{p,q}\colon E^{p,q}_{1}\to E^{p+r,q-r+1}_{1}$
are given by compositions with morphisms $A_{1}\to B_{1}$ and $A_{2}\to
B_{2}$.

The following lemma is a generalization of [1, Lemma 2.5].

######  Lemma 4.3.

_(Weak Mukai Lemma)_ Let $X$ be a smooth projective variety and

$F\to E\to G$

be an exact triangle in $\mathrm{D}^{b}(X)$ such that
$\mathrm{Hom}(F,G)=\mathrm{Hom}(G,F[2])=0$. Then we have

$\operatorname{ext}^{1}(F,F)+\operatorname{ext}^{1}(G,G)\leq\operatorname{ext}^{1}(E,E)$

###### Proof.

Applying the standard spectral sequence in Lemma 4.2 to the exact triangle

$G[-1]\to F\to E,$

i.e. take $A_{1}=A_{2}=G[-1],B_{1}=B_{2}=F$ and $C_{1}=C_{2}=E$, we have
$E_{1}^{-1,1}=\operatorname{Hom}(F,G)$,
$E_{1}^{0,1}=\operatorname{Ext}^{1}(G,G)\oplus\operatorname{Ext}^{1}(F,F)$ and
$E_{1}^{1,1}=\operatorname{Hom}(G,F[2])$ and $E_{1}^{p,1}=0$ for
$p\notin\\{-1,0,1\\}$. Then by the assumption we get
$E_{1}^{0,1}=E_{\infty}^{0,1}$, which implies the result. ∎

Specializing in Kuznetsov components, we have the following lemma.

######  Lemma 4.4.

Let $X$ be a cubic threefold, quartic double solid or Gushel–Mukai threefold.
Assume that there is an exact triangle of $E\in\mathcal{K}u(X)$

$F\to E\to G$

such that $\mathrm{Hom}(F,G)=\mathrm{Hom}(G,F[2])=0$. Then we have

$\mathrm{ext}^{1}(F,F)<\mathrm{ext}^{1}(E,E),\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \mathrm{ext}^{1}(G,G)<\mathrm{ext}^{1}(E,E).$

###### Proof.

This follows from Lemma 4.3 and Proposition 3.7. ∎

The following spectral sequence is constructed in [28, Proposition 2.4].

######  Proposition 4.5.

Let $\mathcal{D}_{0}$ be the bounded derived category of an abelian category
with enough injective objects, and $\mathcal{D}\subset\mathcal{D}_{0}$ be a
full triangulated subcategory of finite type. Then for any heart $\mathcal{A}$
of $\mathcal{D}$ and objects $E,F\in\mathcal{D}$, there exists a spectral
sequence with

$E_{2}^{p,q}=\bigoplus_{i\in\mathbb{Z}}\operatorname{Ext}^{p}(\mathcal{H}^{i}_{\mathcal{A}}(E),\mathcal{H}^{i+q}_{\mathcal{A}}(F))$

and converges to $\operatorname{Ext}^{*}(E,F)$.

As a corollary, we see:

######  Lemma 4.6.

Let $\mathcal{D}_{0}$ be the bounded derived category of an abelian category
with enough injective objects, and $\mathcal{D}\subset\mathcal{D}_{0}$ be a
full triangulated subcategory of finite type. Then for any heart $\mathcal{A}$
of $\mathcal{D}$ with $\mathrm{homdim}(\mathcal{A})\leq 2$ and object
$E\in\mathcal{D}$, we have

$\sum_{i\in\mathbb{Z}}\operatorname{ext}^{1}(\mathcal{H}^{i}_{\mathcal{A}}(E),\mathcal{H}^{i}_{\mathcal{A}}(E))\leq\operatorname{ext}^{1}(E,E).$

###### Proof.

Since $\mathrm{homdim}(\mathcal{A})\leq 2$, we see $E_{2}^{p,q}=0$ for
$p>2,0>p$ and any $q$. Therefore, we see $E^{1,q}_{2}=E^{1,q}_{\infty}$. If we
take $q=0$, then we have $\dim E^{1,0}_{\infty}=\dim
E^{1,0}_{2}\leq\operatorname{ext}^{1}(E,E)$, which proves the lemma. ∎

In the rest of our paper, we always set
$\mathcal{D}_{0}=\mathrm{D}^{b}(\mathrm{QCoh}(X))$ and
$\mathcal{D}=\mathcal{K}u(X)$.

###  4.1. The global dimension function and homological dimension

First, we recall the definition of the homological dimension of a heart and
stability condition.

######  Definition 4.7.

Let $\mathcal{A}$ be the heart of a bounded t-structure of a triangulated
category $\mathcal{D}$. For an integer $n>0$, we say $\mathcal{A}$ has
_homological dimension at most $n$_ and denote by
$\mathrm{homdim}(\mathcal{A})\leq n$ if for any two non-zero objects
$E,F\in\mathcal{A}$, we have

$\operatorname{Hom}(E,F[k])=0,\forall k>n.$

We say a stability condition $\sigma$ on $\mathcal{D}$ has _homological
dimension at most $n$_ and denote by $\mathrm{homdim}(\sigma)\leq n$ if its
heart $\mathcal{A}$ satisfies $\mathrm{homdim}(\mathcal{A})\leq n$.

There is another notion of dimension for stability conditions, called the
global dimension, which is introduced in [18].

######  Definition 4.8.

Let $\sigma$ be a stability condition on a triangulated category $\mathcal{D}$
with the slicing $\mathcal{P}$, then the global dimension of $\sigma$ is
defined by

$\mathrm{gldim}(\sigma):=\sup\\{\phi_{2}-\phi_{1}|\leavevmode\nobreak\
\operatorname{Hom}(E_{1},E_{2})\neq 0,\leavevmode\nobreak\
E_{i}\in\mathcal{P}(\phi_{i}),\leavevmode\nobreak\ i=1\leavevmode\nobreak\
\text{or}\leavevmode\nobreak\ 2\\}$

Therefore, we have a function

$\mathrm{gldim}\colon\operatorname{Stab}(\mathcal{D})\to\mathbb{R}_{\geq
0},\quad\sigma\mapsto\mathrm{gldim}(\sigma),$

which is continuous by [18, Lemma 5.7].

In the following, we prove some basic properties of global dimension.

######  Lemma 4.9.

Let $\sigma$ be a stability condition on a triangulated category $\mathcal{D}$
and $n$ be a positive integer. Then the following are equivalent:

  1. (1)

For any non-zero objects $E$ and $F$ with
$\phi_{\sigma}^{+}(E)<\phi_{\sigma}^{-}(F)$, we have
$\operatorname{Hom}(E,F[n])=0$.

  2. (2)

For any $\sigma$-semistable objects $E$ and $F$ with
$\phi_{\sigma}(E)<\phi_{\sigma}(F)$, we have $\operatorname{Hom}(E,F[n])=0$.

  3. (3)

$\mathrm{gldim}(\sigma)\leq n$.

###### Proof.

(1) $\Rightarrow$ (2). It is clear by taking $E$ and $F$ to be semistable.

(2) $\Rightarrow$ (1). Since $\phi_{\sigma}^{+}(E)<\phi_{\sigma}^{-}(F)$, for
any Harder–Narasimhan factors $E_{i}$ and $F_{j}$ of $E$ and $F$,
respectively, we have $\phi_{\sigma}(E_{i})<\phi_{\sigma}(F_{j})$, and hence
$\operatorname{Hom}(E_{i},F_{j}[n])=0$. As there is no morphism between
Harder–Narasimhan factors of $E$ and $F[n]$, we also get
$\operatorname{Hom}(E,F[n])=0$.

(2) $\Rightarrow$ (3). Suppose $\mathrm{gldim}(\sigma)=n+\epsilon$ for some
$\epsilon>0$. This implies there exist $0<\delta\leq\epsilon$, together with
$\sigma$-semistable objects $E\in\mathcal{P}(\phi_{\sigma}(E))$ and
$F\in\mathcal{P}(\phi_{\sigma}(F))$, such that
$\phi_{\sigma}(F)-\phi_{\sigma}(E)>n+\delta$ and $\operatorname{Hom}(E,F)\neq
0$. But this implies $\operatorname{Hom}(E,(F[-n])[n])\neq 0$, while
$\phi_{\sigma}(F[-n])-\phi_{\sigma}(E)=\phi_{\sigma}(F)-\phi_{\sigma}(E)-n>\delta>0$,
which contradicts to our hypothesis.

(3) $\Rightarrow$ (2). Directly follows from the definition. ∎

######  Lemma 4.10.

Let $\sigma=(\mathcal{A},Z)$ be a stability condition on a triangulated
category $\mathcal{D}$ and $\mathrm{gldim}(\sigma)\leq n$ for a positive
integer $n$. Then the homological dimension of the heart $\mathcal{A}$ of
$\sigma$ is at most $n$, i.e. $\operatorname{Hom}(E,F[m])=0$ for any two
objects $E,F\in\mathcal{A}$ and $m\geq n+1$.

###### Proof.

Since $E,F\in\mathcal{A}$, we know that $\phi^{+}(E),\phi^{-}(F)\in(0,1]$.
Then the vanishing
$\operatorname{Hom}(E,F[m])=\operatorname{Hom}(E[-m+n],F[n])=0$ for $m\geq
n+1$ follows from

$\phi^{+}(E)-m+n=\phi^{+}(E[-m+n])\leq\phi^{+}(E)-1=\phi^{+}(E[-1])\leq
0<\phi^{-}(F)$

and Lemma 4.9. ∎

######  Lemma 4.11.

Let $\sigma$ be a stability condition on a triangulated category $\mathcal{D}$
with the slicing $\mathcal{P}$ and $\mathrm{gldim}(\sigma)\leq n$ for an
integer $n$. Then $\mathrm{gldim}(\sigma\cdot\tilde{g})\leq n$ for any
$\tilde{g}\in\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$.

###### Proof.

Let $E_{1}\in\mathcal{P}(\phi_{1})$ and $E_{2}\in\mathcal{P}(\phi_{2})$ such
that $\operatorname{Hom}(E_{1},E_{2})\neq 0$. Then
$n\geq\mathrm{gldim}(\sigma)\geq\phi_{2}-\phi_{1}$. Now
$\phi_{\sigma\cdot\tilde{g}}(E_{i})=g(\phi_{\sigma}(E_{i}))$, where
$g\colon\mathbb{R}\to\mathbb{R}$ be an increasing function and $g(x+n)=g(x)+n$
for any $n\in\mathbb{Z}$. Hence we get

$\phi_{\sigma\cdot\tilde{g}}(E_{2})=g(\phi_{2})\leq
g(\phi_{1}+n)=g(\phi_{1})+n=\phi_{\sigma\cdot\tilde{g}}(E_{1})+n.$

This implies
$\phi_{\sigma\cdot\tilde{g}}(E_{2})-\phi_{\sigma\cdot\tilde{g}}(E_{1})\leq n$,
which gives $\mathrm{gldim}(\sigma\cdot\tilde{g})\leq n$. ∎

##  5\. A criterion of Serre-invariant stability conditions

In this section, we are going to prove the criteria Theorem 5.8 and Theorem
5.7.

In the following, $X$ will be a cubic threefold, quartic double solid, or
Gushel–Mukai threefold. We begin with some lemmas.

######  Lemma 5.1.

Let $\sigma=(\mathcal{A},Z)$ be a stability condition on $\mathcal{K}u(X)$
with $\mathrm{homdim}(\mathcal{A})\leq 2$. Then for any non-zero object
$E\in\mathcal{K}u(X)$ with $\operatorname{ext}^{1}(E,E)\leq 3$, $E$ is in
$\mathcal{A}$ up to shift.

###### Proof.

Let $N\in\mathbb{N}$ be the number of non-zero cohomology objects of $E$ with
respect to $\mathcal{A}$. By (5) of Proposition 3.7 and Lemma 4.6, we get

$3\geq\operatorname{ext}^{1}(E,E)\geq\sum_{i}\operatorname{ext}^{1}(\mathcal{H}^{i}_{\mathcal{A}}(E),\mathcal{H}^{i}_{\mathcal{A}}(E))\geq
2N.$

which implies $N=1$ and the result follows.

∎

######  Lemma 5.2.

Let $\sigma=(\mathcal{A},Z)$ and
$\sigma^{\prime}=(\mathcal{A}^{\prime},Z^{\prime})$ be stability conditions on
$\mathcal{K}u(X)$ with homological dimension at most $2$. Let $E_{1}$ and
$E_{2}$ be two non-zero objects with
$[E_{1}]=[E_{2}]\in\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$ such that
$E_{1}[m_{1}]$ and $E_{2}[m_{2}]$ are both in $\mathcal{A}$. If
$E_{1}[m^{\prime}_{1}]$ and $E_{2}[m^{\prime}_{2}]$ are also both in
$\mathcal{A}^{\prime}$, then $m_{1}-m_{2}=m^{\prime}_{1}-m^{\prime}_{2}$.

###### Proof.

Since $[E_{1}]=[E_{2}]$, we know that
$m_{1}-m_{2},m_{1}^{\prime}-m_{2}^{\prime}\in 2\mathbb{Z}$. Then we see that

$\chi(E_{1},E_{2})=\chi(E_{1}[m_{1}],E_{2}[m_{2}])=\chi(E_{1}[m_{1}^{\prime}],E_{2}[m_{2}^{\prime}])<0.$

Therefore if $E_{1}[m_{1}]$ and $E_{2}[m_{2}]$ are both contained in
$\mathcal{A}$, we have

$\chi(E_{1}[m_{1}],E_{2}[m_{2}])=\hom(E_{1}[m_{1}],E_{2}[m_{2}])-\hom(E_{1}[m_{1}],E_{2}[m_{2}+1])+\hom(E_{1}[m_{1}],E_{2}[m_{2}+2])<0.$

This implies $m_{2}-m_{1}+1$ is the unique integer $N$ satisfies

$\hom(E_{1},E_{2}[N])>\hom(E_{1},E_{2}[n])$

for any $n\neq N$. On the other hand, the same argument also shows that
$m_{2}^{\prime}-m_{1}^{\prime}+1=N$, so we obtain
$m_{1}-m_{2}=m^{\prime}_{1}-m^{\prime}_{2}$. ∎

###  5.1. Proof of the main theorem

First, we prove the stability of objects with small $\operatorname{ext}^{1}$.

######  Theorem 5.3.

Let $\sigma=(\mathcal{A},Z)$ be a stability condition on $\mathcal{K}u(X)$
with $\mathrm{homdim}(\mathcal{A})\leq 2$. Then for any non-zero object
$E\in\mathcal{K}u(X)$ with $\operatorname{ext}^{1}(E,E)\leq 2$ or
$\operatorname{ext}^{1}(E,E)=3$ and $\operatorname{ext}^{2}(E,E)=0$, $E$ is
$\sigma$-stable.

###### Proof.

By Lemma 3.8, $E$ is stable with respect to any Serre-invariant stability
condition on $\mathcal{K}u(X)$ and $[E]$ is primitive. Moreover,
$\operatorname{RHom}(E,E)=\mathbb{C}\oplus\mathbb{C}^{2}[-1]$ when $X$ is a
cubic threefold, and
$\operatorname{RHom}(E,E)=\mathbb{C}\oplus\mathbb{C}^{2}[-1]$ or
$\mathbb{C}\oplus\mathbb{C}^{3}[-1]$ when $X$ is a quartic double solid or
Gushel–Mukai threefold.

By virtue of Lemma 5.1, we can assume that $E\in\mathcal{A}$ (up to shift). If
$E$ is not $\sigma$-stable, since $[E]$ is primitive, by Lemma 4.1 we can find
an exact sequence in $\mathcal{A}$

(4) $A\to E\to B$

such that $\operatorname{Hom}(A,B)=0$,
$\phi_{\sigma}(A)\geq\phi^{+}_{\sigma}(B)$ and $A$ is $\sigma$-semistable with
isomorphic Jordan–Hölder factors. From the exact triangle $B[-1]\to A\to E$
and Lemma 4.2, we have a spectral sequence which degenerates at $E_{3}$
converging to $\mathrm{Ext}^{*}(E,E)$ with $E_{1}$-page being

$E^{p,q}_{1}=\left\\{\begin{aligned}
\mathrm{Ext}^{q}(A,B[-1])=\mathrm{Ext}^{q-1}(A,B)&,\leavevmode\nobreak\
p=-1\\\ \mathrm{Ext}^{q}(A,A)\oplus\mathrm{Ext}^{q}(B,B)&,\leavevmode\nobreak\
p=0\\\ \mathrm{Ext}^{q}(B[-1],A)=\mathrm{Ext}^{q+1}(B,A)&,\leavevmode\nobreak\
p=1\\\ 0&,\leavevmode\nobreak\ p\notin[-1,1]\end{aligned}\right.$

Since the homological dimension of $\mathcal{A}$ is at most $2$,
$A,B\in\mathcal{A}$ and $\operatorname{Hom}(A,B)=0$, we have

$E^{p,q}_{1}=\begin{array}[]{cc|cc}0&0&0&0\\\
0&\operatorname{Ext}^{2}(A,B)&0&0\\\
0&\operatorname{Ext}^{1}(A,B)&\operatorname{Ext}^{2}(A,A)\oplus\operatorname{Ext}^{2}(B,B)&0\\\
0&0&\operatorname{Ext}^{1}(A,A)\oplus\operatorname{Ext}^{1}(B,B)&\operatorname{Ext}^{2}(B,A)\\\
0&0&\operatorname{Hom}(A,A)\oplus\operatorname{Hom}(B,B)&\operatorname{Ext}^{1}(B,A)\\\
\hline\cr 0&0&0&\operatorname{Hom}(B,A)\end{array}$

Note that the differential

$d\colon E_{1}^{0,0}=\operatorname{Hom}(A,A)\oplus\operatorname{Hom}(B,B)\to
E_{1}^{1,0}=\operatorname{Ext}^{1}(B,A)$

maps $(\operatorname{id}_{A},0)$ and $(0,\operatorname{id}_{B})$ to the
element $[E]\in\operatorname{Ext}^{1}(B,A)$ corresponds to the extension (4),
we see $\dim\ker(d)\geq 1$. Thus from $\hom(E,E)=1$, we have $\dim
E^{0,0}_{2}=\dim E^{0,0}_{\infty}=1$, hence
$E_{1}^{1,-1}=\operatorname{Hom}(B,A)=0$. Moreover, since in each case we have
$\operatorname{Ext}^{2}(E,E)=0$, we see

$E^{-1,3}_{1}=E^{-1,3}_{\infty}=\operatorname{Ext}^{2}(A,B)=0,$

which implies $\chi(A,B)=-\operatorname{ext}^{1}(A,B)\leq 0$. Then the first
page becomes

$E^{p,q}_{1}=\begin{array}[]{cc|cc}0&0&0&0\\\ 0&0&0&0\\\
0&\operatorname{Ext}^{1}(A,B)&\operatorname{Ext}^{2}(A,A)\oplus\operatorname{Ext}^{2}(B,B)&0\\\
0&0&\operatorname{Ext}^{1}(A,A)\oplus\operatorname{Ext}^{1}(B,B)&\operatorname{Ext}^{2}(B,A)\\\
0&0&\operatorname{Hom}(A,A)\oplus\operatorname{Hom}(B,B)&\operatorname{Ext}^{1}(B,A)\\\
\hline\cr 0&0&0&0\end{array}$

Case 1. First, we assume that
$\operatorname{RHom}(E,E)=\mathbb{C}\oplus\mathbb{C}^{2}[-1]$. When $X$ is a
cubic threefold, we can assume that $[E]=v,v-w$ or $2v-w$ up to sign. In each
case, a calculation using the Euler form and $\chi(A,B)\leq 0$ implies that

$[A]=[S^{-1}_{\mathcal{K}u(X)}(E)],\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ [B]=[S_{\mathcal{K}u(X)}(E)].$

Thus $\chi(A,B)=-\operatorname{ext}^{1}(A,B)=0$. Therefore, we also have

$E_{1}^{0,2}=E^{0,2}_{\infty}=\operatorname{Ext}^{2}(A,A)\oplus\operatorname{Ext}^{2}(B,B)=0.$

Since $[A]$ is primitive, we see that $A$ has only one Jordan–Hölder factor
with respect to $\sigma$ and is $\sigma$-stable. Hence $\hom(A,A)=1$. Now from
$\chi(A,A)=-1$ and $\operatorname{Ext}^{\geq 2}(A,A)=0$, we see
$\operatorname{ext}^{1}(A,A)=2$. By Lemma 3.8, $A$ is stable with respect to
any Serre-invariant stability condition. Using
$\operatorname{Hom}(A,E)=\operatorname{Hom}(E,S_{\mathcal{K}u(X)}(A))\neq 0$
and the fact that $[E]=[S_{\mathcal{K}u(X)}(A)]$ and
$E,S_{\mathcal{K}u(X)}(A)$ are stable with respect to any Serre-invariant
stability condition, we get $A\cong S^{-1}_{\mathcal{K}u(X)}(E)$. But this
makes a contradiction, since $A,E\in\mathcal{A}$ and

$\operatorname{Hom}(A,E[-1])=\operatorname{Hom}(S^{-1}_{\mathcal{K}u(X)}(E),E[-1])=\operatorname{Ext}^{1}(E,E)\neq
0.$

When $X$ is a quartic double solid or a Gushel–Mukai threefold, we can assume
that $[E]=v$ or $w$. Then a simple computation using $\chi(A,B)\leq 0$ shows
that $[A]=0$ or $[B]=0$, which implies $A=0$ or $B=0$ and makes a
contradiction.

Case 2. Now we assume that
$\operatorname{RHom}(E,E)=\mathbb{C}\oplus\mathbb{C}^{3}[-1]$. In this case,
$X$ is a quartic double solid or a Gushel–Mukai threefold and we can assume
that $[E]=v-w$ or $v+w$. Then a simple computation using $\chi(A,B)\leq 0$
shows that $\\{[A],[B]\\}=\\{v,w\\}$ when $[E]=v+w$, and
$\\{[A],[B]\\}=\\{v,-w\\}$ when $[E]=v-w$. Hence
$\chi(A,B)=\operatorname{ext}^{1}(A,B)=0$. Therefore, we also have

$E_{1}^{0,2}=E^{0,2}_{\infty}=\operatorname{Ext}^{2}(A,A)\oplus\operatorname{Ext}^{2}(B,B)=0.$

Since $[A]$ is primitive, we know that $A$ has only one Jordan–Hölder factor,
hence is $\sigma$-stable. Thus $\hom(A,A)=1$, and
$\operatorname{ext}^{1}(A,A)=2$ since $\chi(A,A)=-1$. By the previous case,
$\tau(A):=S_{\mathcal{K}u(X)}(A)[-2]$ is $\sigma$-stable as well. Moreover, by
Lemma 3.8 and Lemma 3.9, $\tau(A)$ is stable with respect to any Serre-
invariant stability condition with the same phase as $A$. Thus
$A,\tau(A)\in\mathcal{A}$ by Lemma 5.2. Hence
$\phi_{\sigma}(A)=\phi_{\sigma}(\tau(A))$.

If $\operatorname{Ext}^{2}(B,A)=0$, then by Lemma 4.3 and (5) of Proposition
3.7 we get a contradiction. Thus we can assume that
$\operatorname{Ext}^{2}(B,A)=\operatorname{Hom}(A,\tau(B))=\operatorname{Hom}(\tau(A),B)\neq
0$. By $\sigma$-stability of $\tau(A)$, we get
$\phi_{\sigma}(\tau(A))\leq\phi^{+}_{\sigma}(B)$. Then from
$\phi_{\sigma}(A)\geq\phi^{+}_{\sigma}(B)$ and
$\phi_{\sigma}(A)=\phi_{\sigma}(\tau(A))$, we see
$\phi_{\sigma}(A)=\phi_{\sigma}(\tau(A))=\phi^{+}_{\sigma}(B)$, hence from
$\operatorname{Hom}(\tau(A),B)\neq 0$ and the $\sigma$-stability of $\tau(A)$,
we have an injection $\tau(A)\hookrightarrow B$ in $\mathcal{A}$. Since $[B]$
is primitive, by looking at the Jordan–Hölder filtration of the first
Harder–Narasimhan factor of $B$ with respect to $\sigma$, we have an exact
sequence $A^{\prime}\to B\to B^{\prime}$ in $\mathcal{A}$ such that all
Jordan–Hölder factors of $A^{\prime}$ are isomorphic to $\tau(A)$ and
$\operatorname{Hom}(A^{\prime},B^{\prime})=0$ as in Lemma 4.1. Applying the
spectral sequence in Lemma 4.2 as above, since $\operatorname{Ext}^{\geq
2}(B,B)=0$, we have $\operatorname{Ext}^{2}(A^{\prime},B^{\prime})=0$, which
implies $\chi(A^{\prime},B^{\prime})\leq 0$. However, we know that
$[A^{\prime}]=n[\tau(A)]=n[A]$ for an integer $n\geq 1$, hence

$\chi(A^{\prime},B^{\prime})=\chi(n[A],[B]-n[A])=-n^{2}\chi([A],[A])=n^{2}>0$

and we get a contradiction. ∎

As a corollary, we have:

######  Corollary 5.4.

Let $\sigma=(\mathcal{A},Z)$ be a stability condition on $\mathcal{K}u(X)$
with homological dimension at most $2$. Then the image of the central charge
$Z$ is not contained in a line, and we can find a Serre-invariant stability
condition $\sigma^{\prime}=(\mathcal{A}^{\prime},Z^{\prime})$ on
$\mathcal{K}u(X)$ with $Z=Z^{\prime}$.

###### Proof.

If $X$ is a cubic threefold or quartic double solid, let $l\subset X$ be a
line. Then as $\operatorname{ext}^{1}(\mathcal{I}_{l},\mathcal{I}_{l})=2$, by
Theorem 5.3, $\mathcal{I}_{l},S^{-1}_{\mathcal{K}u(X)}(\mathcal{I}_{l})$ and
$S_{\mathcal{K}u}(\mathcal{I}_{l})$ are $\sigma$-stable. Then the result
follows from [34, Remark 4.8].

When $X$ is a Gushel–Mukai threefold, the result follows from Theorem 5.3 and
[33, Lemma 4.7]. ∎

We need two useful lemmas before proving our main theorem.

######  Lemma 5.5.

Let $X$ be a cubic threefold, $\sigma=(\mathcal{A},Z)$ be a stability
condition on $\mathcal{K}u(X)$ with $\mathrm{homdim}(\sigma)\leq 2$ and
$E\in\mathcal{K}u(X)$ be an object with $\operatorname{ext}^{1}(E,E)=2$. Then
we have

(5)
$\phi_{\sigma}(E)+1\leq\phi_{\sigma}(S_{\mathcal{K}u(X)}(E))<\phi_{\sigma}(E)+2.$

###### Proof.

Up to shift, we can assume that $E\in\mathcal{A}$. By Theorem 5.3,
$E,S_{\mathcal{K}u(X)}(E)$ and $S^{-1}_{\mathcal{K}u(X)}(E)$ are all
$\sigma$-stable. Since
$\operatorname{Hom}(E,E[1])=\operatorname{Hom}(E[1],S_{\mathcal{K}u(X)}(E))\neq
0$, we have

$\phi_{\sigma}(E)+1\leq\phi_{\sigma}(S_{\mathcal{K}u(X)}(E)).$

And from Corollary 5.4, we can take a Serre-invariant stability condition
$\sigma^{\prime}=(\mathcal{A}^{\prime},Z^{\prime})$ with $Z=Z^{\prime}$ and
$\phi_{\sigma}(E)=\phi_{\sigma^{\prime}}(E)$. Hence
$E\in\mathcal{A}\cap\mathcal{A}^{\prime}$. Assume that
$S_{\mathcal{K}u(X)}(E)[m]\in\mathcal{A}^{\prime}$. To prove the statement, by
Lemma 3.9 and $Z=Z^{\prime}$, we only need to show
$S_{\mathcal{K}u(X)}(E)[m]\in\mathcal{A}$. To this end, assume that
$S_{\mathcal{K}u(X)}(E)[n]\in\mathcal{A}$, then since $Z=Z^{\prime}$, we see
$n-m\in 2\mathbb{Z}$. From
$\operatorname{Hom}(E,S_{\mathcal{K}u(X)}(E))=\operatorname{Hom}(E,S_{\mathcal{K}u(X)}(E)[n][-n])\neq
0$, we see $0\leq-n\leq 2$. And from
$\operatorname{Hom}(E[1],S_{\mathcal{K}u(X)}(E))=\operatorname{Hom}(E,S_{\mathcal{K}u(X)}(E)[n][-1-n])\neq
0$, we see $0\leq-n-1\leq 2$. Therefore, we obtain $n=-1$ or $-2$. Since
$n-m\in 2\mathbb{Z}$, we have $n=m$ and the result follows. ∎

######  Lemma 5.6.

Let $X$ be a cubic threefold, quartic double solid or Gushel–Mukai threefold.
Then there exist two objects $D_{1},D_{2}\in\mathcal{K}u(X)$ with $[D_{1}]=v$
and $[D_{2}]=w$, such that for any stability condition $\sigma$ on
$\mathcal{K}u(X)$ with $\mathrm{homdim}(\sigma)\leq 2$, $D_{1}$ and $D_{2}$
are $\sigma$-stable with phases

(6) $\phi_{\sigma}(D_{1})-1<\phi_{\sigma}(D_{2})<\phi_{\sigma}(D_{1}).$

###### Proof.

When $X$ is a cubic threefold or a quartic double solid, we set
$D_{1}=\mathcal{I}_{l}$ and $D_{2}=\mathbf{O}(\mathcal{I}_{l})[-1]$, where
$l\subset Y$ is a line in $Y$ and $\mathbf{O}$ is the rotation functor defined
in Section 2. Therefore, by [34, Lemma 5.16], we see

$\operatorname{RHom}(D_{i},D_{i})=\mathbb{C}\oplus\mathbb{C}^{2}[-1],\leavevmode\nobreak\
\forall i\in\\{1,2\\}.$

Hence $[D_{1}]=v$ and $[D_{2}]=w$, and are both $\sigma$-stable from Theorem
5.3. Thus (6) follows from [34, Remark 4.8], in which
$\mathcal{J}_{l}:=\mathbf{O}^{-1}(\mathcal{I}_{l})[1]$.

Now we assume that $X$ is a Gushel–Mukai threefold. We set
$D_{1}=\mathcal{I}_{C}$ and $D_{2}=F$, where $C\subset X$ is a general smooth
conic and $F$ is a general rank $2$ non-locally free slope-stable sheaf on $X$
with $c_{1}(F)=-1,c_{2}(F)=5$ and $c_{3}(F)=0$. By the smoothness of $C$, we
get $\operatorname{RHom}(D_{1},D_{1})=\mathbb{C}\oplus\mathbb{C}^{2}[-1]$.
Since $C$ is general in the Fano surface of conics, by [19, Proposition 7.1]
we have $D_{1}=\mathcal{I}_{C}\in\mathcal{K}u(X)$. Hence from Theorem 5.3,
$D_{1}$ is $\sigma$-stable. And by virtue of [19, Theorem 8.1], $F$ fits into
an exact sequence $0\to
F\to\mathcal{E}_{X}\to\operatorname{\mathcal{O}}_{l}(-1)\to 0$, where
$l\subset X$ is a line. By [19, Proposition 8.2], we see
$D_{2}=F\in\mathcal{K}u(X)$. Moreover, we can take $l$ to be general in the
Hilbert scheme of lines on $X$ so that
$\operatorname{RHom}(D_{2},D_{2})=\mathbb{C}\oplus\mathbb{C}^{2}[-1]$ (cf.
[19, Section 8]). Hence $D_{2}$ is $\sigma$-stable as well.

It remains to verify (6). By [19, Lemma 6.2],
$\operatorname{Hom}(\mathcal{E}_{X},D_{1})\neq 0$. Hence from the exact
sequence above defining $F$, we see
$\operatorname{Hom}(D_{2},D_{1})=\operatorname{Hom}(S^{-1}_{\mathcal{K}u(X)}(D_{1}),D_{2})\neq
0$. Thus we get
$\phi_{\sigma}(S^{-1}_{\mathcal{K}u(X)}(D_{1}))<\phi_{\sigma}(D_{2})<\phi_{\sigma}(D_{1})$.
Note that $S^{-1}_{\mathcal{K}u(X)}(D_{1})$ is $\sigma$-stable as well, and by
Lemma 3.8 and Lemma 3.9 $D_{1}$ and $S^{-1}_{\mathcal{K}u(X)}(D_{1})[2]$ are
stable with respect to any Serre-invariant stability conditions with the same
phase. Thus from $[D_{1}]=[S^{-1}_{\mathcal{K}u(X)}(D_{1})[2]]$ and Lemma 5.2,
we see
$\phi_{\sigma}(D_{1})=\phi_{\sigma}(S^{-1}_{\mathcal{K}u(X)}(D_{1})[2])$. Then
we obtain $\phi_{\sigma}(D_{1})-2<\phi_{\sigma}(D_{2})<\phi_{\sigma}(D_{1})$.

Now by Corollary 5.4, we can find a Serre-invariant stability condition
$\sigma^{\prime}=(\mathcal{A}^{\prime},Z^{\prime})$ such that $Z=Z^{\prime}$
and $\phi_{\sigma}(D_{1})=\phi_{\sigma^{\prime}}(D_{1})$. Then we get

$\phi_{\sigma^{\prime}}(D_{1})-2<\phi_{\sigma}(D_{2})<\phi_{\sigma^{\prime}}(D_{1}).$

However, from $Z=Z^{\prime}$, we see
$\phi_{\sigma}(D_{2})-\phi_{\sigma^{\prime}}(D_{2})\in 2\mathbb{Z}$, which
forces $\phi_{\sigma}(D_{2})=\phi_{\sigma^{\prime}}(D_{2})$. Then it remains
to check
$\phi_{\sigma^{\prime}}(D_{1})-1<\phi_{\sigma^{\prime}}(D_{2})<\phi_{\sigma^{\prime}}(D_{1})$.
By Theorem 3.10, up to $\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$-action, we
can take $\sigma^{\prime}$ to be the one constructed in [2], then the result
follows from a direct computation of tilt-stability. ∎

Now we are ready to prove our main theorems. We first prove that a stability
condition $\sigma$ on $\mathcal{K}u(X)$ with $\mathrm{homdim}(\sigma)\leq 2$
is determined by its central charge.

######  Theorem 5.7.

Let $X$ be a cubic threefold, quartic double solid or Gushel–Mukai threefold,
and $\sigma_{1}=(\mathcal{A}_{1},Z_{1}),\sigma_{2}=(\mathcal{A}_{2},Z_{2})$ be
a pair of stability conditions on $\mathcal{K}u(X)$. If $Z_{1}=Z_{2}$ and
$\mathrm{homdim}(\sigma_{i})\leq 2$ for any $i\in\\{1,2\\}$, then

$\sigma_{1}=[2m]\cdot\sigma_{2}$

for an integer $m\in\mathbb{Z}$.

###### Proof.

We take two objects $D_{1}$ and $D_{2}$ as in Lemma 5.6. Note that $[D_{1}]=v$
and $[D_{2}]=w$ and are both $\sigma_{1}$-stable and $\sigma_{2}$-stable.
Moreover, we have

(7)
$\phi_{\sigma_{k}}(D_{1})-1<\phi_{\sigma_{k}}(D_{2})<\phi_{\sigma_{k}}(D_{1})$

for any $k\in\\{1,2\\}.$ Up to shift, we can furthermore assume that
$D_{1}\in\mathcal{A}_{1}\cap\mathcal{A}_{2}$. Thus from (7), either
$D_{2}\in\mathcal{A}_{1}\cap\mathcal{A}_{2}$ or
$D_{2}[1]\in\mathcal{A}_{1}\cap\mathcal{A}_{2}$. Moreover, since
$Z_{1}=Z_{2}$, we get $\phi_{\sigma_{1}}(D_{i})=\phi_{\sigma_{2}}(D_{i})$ for
$i\in\\{1,2\\}$.

Claim 1. _If a non-zero object $E\in\mathcal{A}_{i}$ satisfies
$E[n]\in\mathcal{A}_{j}$, then $n=0$, where $\\{i,j\\}=\\{1,2\\}$._ Note that
from $Z_{1}=Z_{2}$, we have $n\in 2\mathbb{Z}$.

Since the claim and assumptions are symmetry between $\sigma_{1}$ and
$\sigma_{2}$, in the following we assume that $i=1$ and $j=2$. Then we have

(8) $n<\phi^{-}_{\sigma_{2}}(E),\quad\phi^{+}_{\sigma_{2}}(E)\leq n+1.$

Let $[E]=av+bw$ for $a,b\in\mathbb{Z}$. First, we assume that $X$ is a cubic
threefold. Assume $F\in\mathcal{A}_{1}\cap\mathcal{A}_{2}$ be an object with
$\operatorname{ext}^{1}(F,F)=2$. Then $F,S_{\mathcal{K}u(X)}(F)$ and
$S^{-1}_{\mathcal{K}u(X)}(F)$ are $\sigma_{1}$-stable and $\sigma_{2}$-stable.
Note that if $\chi(F,E)<0$ or $\chi(E,F)<0$, then from
$\mathrm{homdim}(\sigma_{1})\leq 2$, we get
$\operatorname{Hom}(F,E[1])=\operatorname{Hom}(E,S_{\mathcal{K}u(X)}(F)[-1])\neq
0$ or
$\operatorname{Hom}(E,F[1])=\operatorname{Hom}(S^{-1}_{\mathcal{K}u(X)}(F)[1],E)\neq
0$, which implies

$\phi_{\sigma_{2}}(F)-1\leq\phi^{+}_{\sigma_{2}}(E),\quad\phi_{\sigma_{2}}^{-}(E)\leq\phi_{\sigma_{2}}(S_{\mathcal{K}u(X)}(F))-1$

or

$\phi_{\sigma_{2}}(S^{-1}_{\mathcal{K}u(X)}(F))+1\leq\phi^{+}_{\sigma_{2}}(E),\quad\phi_{\sigma_{2}}^{-}(E)\leq\phi_{\sigma_{2}}(F)+1.$

But by (8) and (5), we always have $-1<n+1$ and $n<2$, which implies $n=0$
since $n\in 2\mathbb{Z}$. Therefore, in the following, we only need to find an
object $F\in\mathcal{A}_{1}\cap\mathcal{A}_{2}$ with
$\operatorname{ext}^{1}(F,F)=2$ and $\chi(F,E)<0$ or $\chi(E,F)<0$.

  * •

If $b=0$, then $[E]=a[D_{1}]$ and $a>0$. In this case, we have

$\chi(D_{1},E)=\chi(E,D_{1})=-a<0.$

  * •

Assume $D_{2}\in\mathcal{A}_{1}\cap\mathcal{A}_{2}$. If $a=0$, then $b>0$. In
this case, we have

$\chi(E,D_{2})=-a-b<0.$

  * •

Assume $D_{2}\in\mathcal{A}_{1}\cap\mathcal{A}_{2}$. If $a\neq 0$, $b\neq 0$,
then from

$\operatorname{Im}(Z_{1}(E))=a\cdot\operatorname{Im}(Z_{1}(D_{1}))+b\cdot\operatorname{Im}(Z_{1}(D_{2}))\geq
0,$

either $a<0<b$ or $0<a$. Then we have either $\chi(D_{2},E)=-b<0$ or
$\chi(E,D_{1})=-a<0$.

  * •

Assume $D_{2}[1]\in\mathcal{A}_{1}\cap\mathcal{A}_{2}$. If $a=0$, then $b<0$.
In this case, we have

$\chi(E,D_{2}[1])=a+b<0.$

  * •

Assume $D_{2}[1]\in\mathcal{A}_{1}\cap\mathcal{A}_{2}$. If $a\neq 0$, $b\neq
0$, then from

$\operatorname{Im}(Z_{1}(E))=a\cdot\operatorname{Im}(Z_{1}(D_{1}))-b\cdot\operatorname{Im}(Z_{1}(D_{2}[1]))\geq
0,$

either $0<a$ or $b<0,a<0$. Then we have either $\chi(E,D_{1})=-a<0$ or
$\chi(E,D_{2}[1])=a+b<0$.

Now we assume that $X$ is a quartic double solid or a Gushel–Mukai threefold.

  * •

If $b=0$, then $a>0$. In this case, we have

$\chi(D_{1},E)=\chi(E,D_{1})=-a<0,$

hence from $D_{1},E\in\mathcal{A}_{1}$ we get
$\operatorname{Hom}(D_{1},E[1])\neq 0$ and $\operatorname{Hom}(E,D_{1}[1])\neq
0$. Then we have

$\phi_{\sigma_{2}}(D_{1})-1\leq\phi^{+}_{\sigma_{2}}(E),\quad\phi_{\sigma_{2}}^{-}(E)\leq\phi_{\sigma_{2}}(D_{1})+1.$

But by (8), we get $-1<n+1$ and $n<2$, which implies $n=0$ since $n\in
2\mathbb{Z}$.

  * •

Assume $D_{2}\in\mathcal{A}_{1}\cap\mathcal{A}_{2}$. If $b\neq 0$, then $b>0$.
In this case, we have

$\chi(D_{2},E)=\chi(E,D_{2})=-b<0.$

Then by the same argument above, we get $\operatorname{Hom}(D_{2},E[1])\neq 0$
and $\operatorname{Hom}(E,D_{2}[1])\neq 0$. Then we have

$\phi_{\sigma_{2}}(D_{2})-1\leq\phi^{+}_{\sigma_{2}}(E),\quad\phi_{\sigma_{2}}^{-}(E)\leq\phi_{\sigma_{2}}(D_{2})+1.$

But by (8), we get $-1<n+1$ and $n<2$, which implies $n=0$ since $n\in
2\mathbb{Z}$.

  * •

Assume $D_{2}[1]\in\mathcal{A}_{1}\cap\mathcal{A}_{2}$. If $b\neq 0$, then
$b<0$. In this case, we have

$\chi(D_{2}[1],E)=\chi(E,D_{2}[1])=b<0.$

Then by the same argument above, we get $\operatorname{Hom}(D_{2}[1],E[1])\neq
0$ and $\operatorname{Hom}(E,D_{2}[2])\neq 0$. Then we have

$\phi_{\sigma_{2}}(D_{2}[1])-1\leq\phi^{+}_{\sigma_{2}}(E),\quad\phi_{\sigma_{2}}^{-}(E)\leq\phi_{\sigma_{2}}(D_{2}[1])+1.$

But by (8), we get $-1<n+1$ and $n<2$, which implies $n=0$ since $n\in
2\mathbb{Z}$.

Claim 2. _A non-zero object $E\in\mathcal{K}u(X)$ is in $\mathcal{A}_{1}$ if
and only if $E$ is in $\mathcal{A}_{2}$._

We prove this claim by induction on $\operatorname{ext}^{1}(E,E)$. When
$\operatorname{ext}^{1}(E,E)\leq 2$, we have $\operatorname{ext}^{1}(E,E)=2$
by Proposition 3.7 (5), hence $E$ is $\sigma_{i}$-stable by Theorem 5.3. then
the result follows from our assumption
$\phi_{\sigma_{1}}(E)=\phi_{\sigma_{2}}(E)$. Now we assume that the claim
holds for any object $E$ with $\operatorname{ext}^{1}(E,E)<N$ for an integer
$N>2$. We are going to prove the claim for $\operatorname{ext}^{1}(E,E)=N$. To
this end, we first assume that $E\in\mathcal{A}_{1}$ with
$\operatorname{ext}^{1}(E,E)=N$. If $E$ has at least two cohomology objects
with respect to $\mathcal{A}_{2}$, we denote $a,b\in\mathbb{Z}$ by unique
integers satisfy

$\mathcal{H}^{-a}_{\mathcal{A}_{2}}(E)\neq
0,\mathcal{H}^{k}_{\mathcal{A}_{2}}(E)=0,\forall k<-a$

and

$\mathcal{H}^{-b}_{\mathcal{A}_{2}}(E)\neq
0,\mathcal{H}^{k}_{\mathcal{A}_{2}}(E)=0,\forall k>-b.$

Therefore, we have two non-zero maps
$\mathcal{H}^{-a}_{\mathcal{A}_{2}}(E)[a]\to E$ and
$E\to\mathcal{H}^{-b}_{\mathcal{A}_{2}}(E)[b]$. Using (5) of Proposition 3.7
and Lemma 4.6, we see

$\operatorname{ext}^{1}(\mathcal{H}^{-a}_{\mathcal{A}_{2}}(E),\mathcal{H}^{-a}_{\mathcal{A}_{2}}(E))<N,\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\
\operatorname{ext}^{1}(\mathcal{H}^{-b}_{\mathcal{A}_{2}}(E),\mathcal{H}^{-b}_{\mathcal{A}_{2}}(E))<N.$

Thus by the induction hypothesis, we get
$\mathcal{H}^{-a}_{\mathcal{A}_{2}}(E),\mathcal{H}^{-b}_{\mathcal{A}_{2}}(E)\in\mathcal{A}_{1}$.
But since $E\in\mathcal{A}_{1}$, we have $a\leq 0$ and $b\geq 0$, which
contradicts $a>b$. This implies $E$ is in $\mathcal{A}_{2}$ up to a shift, but
Claim 1 shows $E\in\mathcal{A}_{2}$. When $E\in\mathcal{A}_{2}$ with
$\operatorname{ext}^{1}(E,E)=N$, the same argument as above also shows that
$E\in\mathcal{A}_{1}$. This completes our induction argument and proves Claim
2.

Therefore, by the Claim 2 above, we have $\mathcal{A}_{1}=\mathcal{A}_{2}$,
which together with $Z_{1}=Z_{2}$ implies $\sigma_{1}=\sigma_{2}$. ∎

Since we know that the homological dimension of any Serre invariant stability
condition on $\mathcal{K}u(X)$ is at most $2$, we get the following criterion.

######  Theorem 5.8.

Let $X$ be a cubic threefold, quartic double solid or Gushel–Mukai threefold,
and $\sigma$ be a stability condition on $\mathcal{K}u(X)$. Then the following
conditions are equivalent:

  1. (1)

$\mathrm{homdim}(\sigma)\leq 2$,

  2. (2)

$\mathrm{gldim}(\sigma)\leq 2$, and

  3. (3)

$\sigma$ is Serre-invariant.

###### Proof.

By Proposition 3.7, we see (3) implies (2), and by Lemma 4.10 we have (2)
implies (1).

By Theorem 5.3 and Corollary 5.4, if $\mathrm{homdim}(\sigma)\leq 2$, then we
can find a Serre-invariant stability condition $\sigma^{\prime}$ such that
$Z=Z^{\prime}$. Then (1) implies (3) follows from Theorem 5.7. ∎

###  5.2. The Serre-invariant component

Let $\mathsf{K}\subset\operatorname{Stab}(\mathcal{K}u(X))$ be the subspace of
all Serre-invariant stability conditions on $\mathcal{K}u(X)$. By Theorem
3.10, we have $\mathsf{K}=\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$. It is
clear that $\mathsf{K}$ is contractible since
$\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$ is.

######  Corollary 5.9.

Let $X$ be a cubic threefold, quartic double solid or Gushel–Mukai threefold.
Then $\mathsf{K}$ is a contractible connected component of
$\operatorname{Stab}(\mathcal{K}u(X))$.

###### Proof.

By [34, Remark 3.10], $\mathsf{K}$ is an open subset of a connected component
of $\operatorname{Stab}(\mathcal{K}u(X))$. Since
$\mathrm{gldim}\colon\operatorname{Stab}(\mathcal{K}u(X))\to\mathbb{R}$ is
continuous by [18, Lemma 5.7], the subspace
$\mathrm{gldim}^{-1}([0,2])\subset\operatorname{Stab}(\mathcal{K}u(X))$ is
closed. Moreover, by Theorem 5.8, $\mathsf{K}=\mathrm{gldim}^{-1}([0,2])$.
Since
$\mathsf{K}=\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})\cong\mathbb{C}\times\mathbb{H}$
is connected, open and closed, it is a contractible connected component. ∎

For an arbitrary stability condition $\sigma$ on $\mathcal{K}u(X)$,
Proposition 3.7 (5) implies $\mathrm{homdim}(\sigma)\geq 1$. So it is natural
to make the following conjecture, which together with Theorem 5.8 could
describe all stability conditions on $\mathcal{K}u(X)$:

######  Conjecture 5.10.

Let $X$ be a cubic threefold, quartic double solid or Gushel–Mukai threefold.
Then for any stability condition $\sigma$ on $\mathcal{K}u(X)$, we have
$\mathrm{homdim}(\sigma)\leq 2.$

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