Serre-invariant stability conditions and Ulrich bundles on cubic threefolds

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 $\text{Ku}(X)$ of a cubic threefold $X$. In particular, we show that all the known stability conditions on $\text{Ku}(X)$ 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, Macr\`i and Stellari.


Introduction
Let X be a smooth cubic hypersurface in the complex projective space P 4 .As first noted by Kuznetsov [Kuz04], the birational geometry of X is controlled by a certain full admissible subcategory Ku(X) of the bounded derived category D b (X), defined by the semiorthogonal decomposition We call Ku(X) the Kuznetsov component of X.It has been shown that Ku(X) completely determines the isomorphism class of the cubic threefold X; see [BMM + 12, 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 Ku(X).
Up to now there are two different constructions of stability conditions on Ku(X): one in [BMM + 12] (see Section 5.2) and one in [BLM + 17] (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 T be a C-linear triangulated category with Serre functor S. Recall that the space of stability conditions on T carries a right action of GL + 2 (R), which is the universal cover of GL + 2 (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 σ 1 and σ 2 be Serre-invariant stability conditions on a linear triangulated category T satisfying the conditions (C1), (C2) and (C3) in Section 3. Then there exists a g ∈ GL + (2, R) such that Theorem 1.1 applies for instance to the Kuznetsov component of a cubic threefold X to show that the known stability conditions on Ku(X) are identified in the stability manifold up to the action of GL + 2 (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 σ on Ku(X) defined in [LMS15] is in the same orbit as the stability conditions σ (α, β) introduced in [BLM + 17] with respect to the GL Our second result provides a detailed description of the moduli spaces of stable objects in 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 Note that v is the numerical class of the ideal sheaf I of a line 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.
Moreover, the above three moduli spaces are isomorphic to the moduli of σ -stable objects of class [I ] in Ku(X), where σ is any Serre-invariant stability condition on Ku(X).
Finally, we apply Theorem 1.2 to study the moduli space of Ulrich bundles of rank at least 2 on X. Recall that an Ulrich bundle E on X is an arithmetically Cohen-Macaulay vector bundle such that the graded module ⊕ m∈Z H 0 (X, E(mH)) has 3 rank(E) generators in degree 1 (see Section 6).Ulrich bundles all lie in Ku(X).Thus the moduli spaces of stable objects in Ku(X) become a useful tool 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 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 at least 2 on X is irreducible.

Related works and motivation
The interest in the study of Serre-invariant stability conditions on semiorthogonal components in the bounded derived category has grown recently, mostly due to applications to the study of moduli spaces (see for instance [PY20,LZ22]) and to the desire to better understand 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 [BLM + 17].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 at least 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 one 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 at least 2 there is a unique orbit of stability conditions by Macrì's result [Mac07], these are very natural conditions for one to expect the category to have 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 [JLL + 21].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 [BLM + 17] to construct stability conditions on Ku(X) via the embedding of Ku(X) in D b (P 2 , B 0 ); see [LMS15].These stability conditions are Serre invariant, as shown in Section 5.3.Since the stability conditions constructed in [BLM + 17] are also Serre invariant by [PY20, Corollary 5.5], we deduce that all the stability conditions constructed on Ku(X) up to now are Serre invariant.Theorem 1.2 then follows from Theorem 1.1.An interesting question would be to understand whether the property of Serre invariance characterises the stability conditions on Ku(X).This fact together with Theorem 1.1 would allow one to show that there is a unique orbit of stability conditions on Ku(X), in analogy to the case of curves of genus at least 2; see [Mac07].
Another open question is to generalise Theorem 1.3 and Corollary 6.5 to further study moduli spaces of semistable objects in 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, APR22, BBF + 20, Qin21, LZ22].
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 2 case, namely the case of instanton sheaves of minimal charge, we know a full description of the moduli space with class 2[I ] for cubic threefolds, see [LMS15], and more generally for Fano threefolds of Picard rank 1, index 2 and degree d ≥ 3; see [Qin21,LZ22].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 ≥ 2 on X in a moduli space of semistable objects in Ku(X) with class d[I ], where I 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 parametrising Gieseker-semistable sheaves which are B 0 -modules; it has the same dimension as 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 σ (α, β) from [BLM + 17], and then we prove Theorem 1.3.In Section 5 we extend the construction of stability conditions on Ku(X) from [BMM + 12, LMS15], and we show they are Serre invariant.Finally, we deduce Theorem 1.2.In Section 6 we prove Theorem 1.4.

(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 [BLM + 17].

Review on (weak) stability conditions
Let T be a C-linear triangulated category, and denote by K(T ) its Grothendieck group.A (weak) stability condition on T is defined by giving the heart of a bounded t-structure and a (weak) stability function on it.Fix a finite-rank lattice Λ with a surjective morphism v : K(T ) Λ.We denote by [−] (resp. [−]) the real (resp.imaginary) part of a complex number.Definition 2.1.Let A be the heart of a bounded t-structure on T .A group homomorphism Z : Λ → C is a (weak) stability function on A if for any 0 E ∈ A, we have Z(v(E)) ≥ 0, and in the case that By abuse of notation, we will write Z(E) for an object E ∈ A instead of Z(v(E)).The slope of E ∈ A with respect to Z is defined by An object F ∈ T is called (semi )stable with respect to the pair (A, Z) if some shift F[k] lies in the heart A and for every proper subobject Definition 2.2.A (weak) stability condition (with respect to Λ) on T is a pair σ = (A, Z), where A is the heart of a bounded t-structure on T and Z is a (weak) stability function satisfying the following properties: (a) HN property: Every object of A has a Harder-Narasimhan (HN) filtration with σ -semistable factors.
(b) Support property: There exists a quadratic form Q on Λ ⊗ R such that the restriction of Q to ker Z is negative definite and Q(E) ≥ 0 for all σ -semistable objects E in A.
We denote by Stab Λ (T ) the space of stability conditions on T with respect to Λ.This space is actually a complex manifold; see [Bri07].
If Λ is the numerical Grothendieck group of T , we call σ a numerical (weak) stability condition.Any (weak) stability condition defines a slicing on T .
The slicing P σ of T corresponding to σ is a collection of full additive subcategories P σ (φ) ⊂ T for φ ∈ R, such that: (i) for φ ∈ (0, 1], the subcategory P σ (φ) is given by the zero object and all σ -semistable objects with phase φ; (ii) for φ + n with φ ∈ (0, 1] and n ∈ Z, we set The HN property of (weak) stability condition σ implies that for 0 E ∈ T , there exists a unique finite sequence of real numbers and a unique sequence of objects in T It is clear from the definition that A = P σ (0, 1], where the latter is the full subcategory of T consisting of the zero object together with those objects 0 E ∈ T which satisfy 0 < φ − (E) ≤ φ + (E) ≤ 1.

Tilt stability
An important tool in the construction of weak stability conditions is the procedure of tilting a heart.Let σ = (A, Z) be a weak stability condition on T and µ ∈ R. We consider the following subcategories of A:

Theorem 2.4 ([HRS96]). The category
is the heart of a bounded t-structure on T .
We say that the heart A µ σ is obtained by tilting A with respect to the weak stability condition σ at slope µ.The above construction applies to the case of the bounded derived category T = D b (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 := dim(X).The pair σ H = (Coh(X), Z H ), where Z H = − ch 1 H n−1 + ch 0 H n , is a weak stability condition on D b (X), known as slope stability, with respect to the rank 2 lattice generated by the elements of the form (H n ch 0 (E), H n−1 ch 1 (E)) for E ∈ D b (X); see [BLM + 17, Example 2.8].In particular, the slope of a coherent sheaf E on X is defined by Any µ H -semistable sheaf E satisfies the Bogomolov-Gieseker inequality We observe that if n = 1, then σ H is a stability condition.Now given β ∈ R, we denote by Coh β (X) the heart of a bounded t-structure obtained by tilting the weak stability condition σ H at the slope µ H = β (see Theorem 2.4).For E ∈ D b (X), the twisted Chern character is defined by ch β (E) := e −βH ch(E).Explicitly, the first three terms are Proposition 2.5 ([BLM + 17, Proposition 2.12]).There is a continuous family of weak stability conditions parametrised by R >0 × R, given by The weak stability condition σ α,β is called tilt stability, and if n = 2, then it defines a stability condition on D b (X) (see [Bri08,AB13]).There is a region in the upper half-plane where the notions of σ α,β -stability and µ H -stability are closely related.
We now recall the notions of walls and chambers for tilt stability.
Definition 2.7.Fix v ∈ Λ.A numerical wall for v is a subset of the upper half-plane of the form with respect to a vector w ∈ Λ.We say a point (α, β) ∈ W (v, w) is on an actual wall for v if and only if there is an object E ∈ Coh β (X) of class v(E) = v which is strictly σ α,β -semistable and unstable on 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 ∈ Λ are locally finite.Properties of walls are described by the following theorem, which was first proved in [Mac14] (see also [Sch20]).
(a) Numerical walls for v are either nested semicircles centred on the β-axis, or a vertical ray parallel to the α-axis.(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.(c) If ch 0 (v) 0, then there is a unique vertical numerical wall given by β = µ H (v). If ch 0 (v) = 0 and H n−1 ch 1 (v) 0, then there is no vertical wall.

Cubic threefolds
We know any slope-semistable sheaf and, more generally, any σ α,β -semistable object satisfies ∆ H (E) ≥ 0; see [BMS16, Theorem 3.5].In the case of a smooth complex cubic threefold X, we have stronger Bogomolov inequalities as follows.

Induced stability conditions
We finish this section by recalling induced stability conditions on an admissible subcategory.Let T be a C-linear triangulated category with Serre functor S. Assume T has a semiorthogonal decomposition of the form where {E 1 , . . ., E m } is an exceptional collection and The next proposition gives a criterion to construct a stability condition on D from a weak stability condition on T .
Proposition 2.11 ([BLM + 17, Proposition 5.1]).Let σ = (A, Z) be a weak stability condition on T .Assume that the exceptional collection {E 1 , . . ., E m } satisfies the following conditions: If moreover there are no objects 0 F ∈ A := A ∩ D with Z(F) = 0, then the pair σ = (A , Z| K(A) ) is a stability condition on D.

Serre-invariant stability conditions
Let T be a triangulated category which is linear of finite type over a field K; i.e. ⊕ i Hom(E, F[i]) is a finite-dimensional vector space over K. Assume that T has Serre functor S. Since S is a linear exact autoequivalence, we can have the following definition.
Assume T satisfies the following conditions: (C1) The Serre functor S of T satisfies S r = [k] when 0 < k/r < 2 or r = 2 and k = 4. (C2) The numerical Grothendieck group N (T ) is of rank 2, and we have the inequality 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 σ 1 and σ 2 be S-invariant numerical stability conditions on a triangulated category T satisfying the above conditions (C1), (C2) and (C3).Then there exists a g ∈ GL Remark 3.3.It is easy to see that if the Serre functor satisfies S = [2], then every stability condition on 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 T be a triangulated category satisfying conditions (C1) and (C2) above, and let σ = (Z, A) be an S-invariant stability condition on T .
an exact triangle in T such that Hom(A, B) = 0 and the σ -semistable factors of A have phases greater than (or equal to in case k/r < 2) the phases of the σ -semistable factors of B. Then Proof.Since σ is an S-invariant stability condition on T , there is a (g, M) ∈ GL + 2 (R), where g : R → R is an increasing map with g(φ + 1) = g(φ) + 1, such that S • σ = σ • g.Thus φ(S(F)) = g(φ(F)) and for any n > 0, which gives k/r > (≥) 2, leading to a contradiction.This completes the proof of part (a).
To prove part (c), first assume E is σ -semistable.Up to shift, we may assume be the first piece in its HN filtration.Then applying weak Mukai lemma in part (b) to this sequence gives Thus the claim in part (c) follows by the first part of the argument because Finally, to prove part (e), take an object E ∈ T satisfying the minimality condition on hom 1 (E, E).Assume for a contradiction that E is not σ -semistable, and let A → E → B be the first piece in the HN filtration of E. So A is σ -semistable and Thus hom(A, B) = 0, and we can apply weak Mukai lemma in part (b).Then part (c) implies which is not possible by our assumption.
Now assume . By continuing this process, we get an exact triangle A → E → B in P (φ(E)) such that A is S-equivalent to ⊕ k Q for some k > 0 (i.e.all σ -stable factors of A are isomorphic to Q) and hom(A, B) = 0.If B 0, then the weak Mukai lemma and part (c) lead to a contradiction as above.
T .This gives Then our assumption implies This completes the proof of (e) for the case k/r < 2.
If E is strictly σ -semistable and k = 2r = 4, we apply Lemma 3.5 below.In Lemma 3.5(b), the same argument as in the weak Mukai lemma in part (b) implies that which is not possible by the minimality assumption on hom 1 (E, E).Thus Lemma 3.5(a) happens; i.e.E is Therefore, (3.6) gives Lemma 3.5.Suppose k = 2r = 4 and E is a strictly σ -semistable object.Then either Proof.Considering the Jordan-Hölder filtration, we can write the short exact sequence A → E → G, where G is stable of the same phase as E. If hom(A, G) 0, then we gain a surjection A G with kernel A 1 : Thus the cone of where If B n = 0, then we are in case (a) and if B n 0, we continue the above process to obtain an exact triangle This implies that Now assume our triangulated category T satisfies condition (C3).If k/r < 2, fix an object Q satisfying (3.1).By Proposition 3.4(e), both Q and S(Q)[−2] are stable.Moreover, parts (a) and (d) of the proposition imply Hence, up to GL + 2 (R)-action, we may assume Q is stable of phase 1 and S(Q)[−2] is of phase 1 2 .
To do this, note that we only need to show E ∈ P σ 2 (0, 1].Namely, in that case, since σ 1 and σ 2 have the same central charge, E must have the same phase with respect to them.Assume otherwise, so there is a non-zero m ∈ Z such that E[2m] ∈ P σ 2 (0, 1].In fact, since σ 1 and σ 2 have the same central charge, an even shift of E lies in the heart for σ 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 σ 1 and σ 2 .We want to show that

The above vanishing holds for
Since Z(Q 1 ) and Z(Q 2 ) are linearly independent, the classes [Q 1 ] and [Q 2 ] are also linearly independent in N (T ).Hence (3.7) implies that χ([F], [E]) = 0 for any [F] ∈ N (T ); thus [E] = 0 by the definition of N (T ), leading to a contradiction.This completes the proof for the case that hom 1 (E, E) is minimal.
Step 2. Now assume hom 1 (E, E) ≥ −2 T + 2. There are three possibilities: First suppose case (i) happens; then we show that E ∈ P σ 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 σ 2 .Then the weak Mukai lemma (3.2) applied to the triangle E 1 → E → E/E 1 and Proposition 3.4(c) imply hom 1 (E 1 , E 1 ) < hom 1 (E, E).Thus, by the induction assumption, E 1 is σ 1 -semistable with φ σ 1 (E 1 ) = φ σ 2 (E 1 ).The existence of a non-zero map E 1 → E and the stability of E with respect to σ 1 imply that Applying the same argument to the last piece in the HN filtration Therefore, E ∈ P σ 2 (0, 1].Moreover, since the stability functions for σ 1 and σ 2 are the same, we obtain φ σ 1 (E) = φ σ 2 (E); thus (3.8) implies that φ σ 2 (E 1 ) ≤ φ σ 2 (E), which is not possible.Thus E is σ 2 -semistable.Now suppose case (ii) happens, so E is strictly σ 2 -semistable.First assume k/r < 2. Let {E i } i∈I be the Jordan-Hölder filtration of E with respect to σ 2 .By the weak Mukai lemma (3.2) and the induction assumption, it follows that the E i are σ 1 -stable of the same phase as with respect to σ 2 .Thus E is strictly σ 1 -semistable of the same phase as with respect to σ 2 .If k = 2r = 4, we apply Lemma 3.5.
Thus G and S(G) are σ 1 -stable of the same phase.Hence, E is strictly σ 1 -semistable and φ σ 1 (E) = φ σ 2 (E).If case (b) of Lemma 3.5 happens, then the same argument as in the 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 case (iii) happens.If E is strictly σ 1 -semistable, then the same argument as in case (ii) implies that E is also strictly σ 2 -semistable.Thus E must be σ 1 -stable, and we only need to show E has the same phase with respect to both σ 1 and σ 2 .
We know E ∈ P σ 1 (0, 1].Since E is σ 2 -stable, there is an m ∈ Z such that E[2m] ∈ P σ 2 (0, 1].If m 0, then the same argument as in Step 1 implies that χ(Q j , E) = 0 for j = 1, 2, and so [E] = 0 in N (T ), which is not possible.Thus m = 0, and E has the same phase with respect to both σ 1 and σ 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 Proof.Let σ = (Z, A) be an S-invariant stability condition on T .By Proposition 3.4(e), we know all three objects Q 1 , Q 2 , Q 2 are σ -stable.Hence, up to GL + 2 (R)-action, we may assume Q 1 is of phase 1 and Z(Q 1 ) = −1.If φ(Q 2 ) ∈ (0, 1], then we can assume Z(Q 2 ) = i and proceed as before.So assume for a contradiction that Q 2 A. The assumptions (b) and (c) imply that ) < 1, and so hom(Q 1 , S(Q 2 )) = 0, which is not possible by condition (c).This implies that (3.10) On the other hand, conditions (b) and (c) give Therefore, and so hom(Q 2 , Q 2 [3]) 0, which is in contradiction to assumption (d). Remark

Aside: Very general cubic fourfolds
If the Serre functor of T is S = [2], i.e.T is a 2-Calabi-Yau category, then clearly any stability condition is Serre invariant.An example of such a 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 GL + (2, R)-orbit of stability conditions on the Kuznetsov component of X.

Kuznetsov component of cubic threefolds
From now on, we assume X is a smooth complex cubic threefold, and O X (H) denotes the corresponding very ample line bundle.

Kuznetsov component
The Kuznetsov component Ku(X) is the right-orthogonal complement of the exceptional collection O X , O X (H) in D b (X) sitting in the semiorthogonal decomposition (see [Kuz04]).By [BMM + 12, Proposition 2.7], the numerical Grothendieck group N (Ku(X)) of Ku(X) is a rank 2 lattice where I is the ideal sheaf of a line in X and S denotes the Serre functor of Ku(X).With respect to this basis, the Euler form χ Ku(X) (−, −) on N (Ku(X)) is represented by (4.1) −1 −1 0 −1 .

Stability conditions on Ku(X)
Stability conditions on Ku(X) have been first constructed in [BMM + 12] and more recently in [BLM + 17].We recall here the definition of the latter, while the former will be reviewed in Section 5.2.
Proof.First assume E ∈ Coh 0 α,β is σ 0 α,β -(semi)stable, so by definition it lies in an exact triangle (4.2) for some F ∈ F α,β and T ∈ T α,β .If Z α,β (T ) 0, i.e.T Coh 0 (X), we have This shows that the phase of F[1] is bigger than that of T with respect to σ 0 α,β .Thus the exact triangle (4.2) implies that one of the following cases happens: Case (iii) is excluded in the statement, so we may assume the first two cases happen.We only consider the case E ∈ T α,β ; the other one can be shown by a similar argument.Suppose for a contradiction that E is not σ α,β -(semi)stable, and let be a destabilising sequence in Coh β (X).We may assume E 1 is σ α,β -semistable; thus The right inequality follows from the definition of T α,β .Thus both E 1 and E 2 lie in T α,β ; hence (4.3) is also a destabilising sequence for E in Coh 0 α,β with respect to σ 0 α,β , which leads to a contradiction.
Now assume E ∈ T α,β is σ α,β -(semi)stable and Hom(Coh 0 (X), E) = 0.In the rest of the argument, all T i lie in T α,β and all F i lie in F α,β .Suppose E 1 → E → E 2 is a destabilising sequence with respect to σ 0 α,β .We can assume E 2 is σ 0 α,β -semistable.Thus by the above argument, we have either (1) Since E ∈ T α,β , the phase of E with respect to σ 0 α,β is less than or equal to 1 2 .But we know the phase of objects in F α,β [1] is bigger than 1 2 ; thus case (i) cannot happen.In case (iii) we know the 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 ∈ T α,β .
By definition, we know E 1 lies in an exact triangle , and so the destabilising sequence is an exact sequence in the original heart Coh β (X) which (semi)destabilises E with respect to σ α,β , leading to a contradiction.
Finally, suppose E ∈ F α,β is σ α,β -(semi)stable and we have hom(Coh 0 (X), be a destabilising sequence with respect to σ 0 α,β .We may assume E 1 is σ 0 α,β -semistable.We know E 1 Coh 0 (X), and the phase of objects in T α,β \ Coh 0 (X) with respect to σ 0 α,β is less than the phase of objects in F α,β [1].Thus the same argument as in case (a) implies that E 1 lies in an exact triangle Taking cohomology from the destabilising sequence with respect to the heart Coh β (X) gives the long exact sequence Thus F 1 is a subobject of E in Coh β (X).We know the phase of E 1 is equal to the phase of F 1 [1] with respect to σ 0 α,β , and it is bigger than or equal to the phase of E[1].This implies that the phase of F 1 is bigger than or equal to the phase of E with respect to σ α,β , leading to a contradiction.By restricting weak stability conditions σ 0 α,β to the Kuznetsov component Ku(X), we obtain a stability condition on it.Theorem 4.2 ([BLM + 17, Theorem 6.8], [PY20, Theorem 3.3 and Corollary 5.5]).For every (α, β) in the set ) is a S-invariant stability condition on Ku(X), where As an application of Theorem 3.2, we get the following.
Proof.We know S 3 = [5] and that the numerical Grothendieck group N (Ku(X)) is a lattice of rank 2 such that χ Ku(X) (v, v) ≤ −1 for any 0 v ∈ N (Ku(X)).Thus conditions (C1) and (C2) hold for Ku(X).We also have dim C Hom 1 (I , I ) = 2 as the Hilbert scheme of lines in X is a smooth surface; see [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 → P 3 ramified in a quartic.In fact, by [Kuz09] there is a semiorthogonal decomposition of the form , where H is an ample class, and the numerical Grothendieck group of Ku(Y ) is a rank 2 lattice represented by the matrix An easy computation shows that (C2) holds.Moreover, the Serre functor of Ku(Y ) where ι is an involutive autoequivalence of 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 I and its derived dual . By [PY20, Remark 3.8], for proper choices of lines, they provide the required objects.In this section we investigate the moduli of semistable objects of the above classes.We denote by M σ Ku(X) (v) the moduli space of σ -semistable objects of class v ∈ N (Ku(X)).

Stable objects in
Theorem 4.5.Let σ be an S-invariant stability condition on Ku(X).
) is isomorphic to the moduli space M X (v − w) of large volume limit stable complexes of character v − w.
In particular, all the three moduli spaces above are isomorphic to the Fano variety Σ(X) of lines in X.This proves Theorem 1.3.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 ∨∨ = O X .Thus it lies in the exact sequence where Q is a torsion sheaf of character 0, 0, 1 3 H 2 , 0 , which is therefore isomorphic to the structure sheaf O of a line in X.Thus E is isomorphic to the ideal sheaf I .
Lemma 4.6.The ideal sheaf I for any line in X is stable with respect to any S-invariant stability condition on Ku(X).
By definition, S(I ) is also stable with respect to any S-invariant stability condition on Ku(X).The next step is to describe the complex S(I ).We know Since Pic(X) = Z.H, Lemma 3.5 of [Fey22] implies that I (H) is σ α,β=0 -stable for any α > 0. Thus I (H) is σ α,β -semistable along the numerical wall W (I (H), O X [1]) where I (H) and O X [1] have the same phase.Therefore, their extension L O X (I (H)) is also σ α,β -semistable along the wall.One can easily show that W (I (H), O X [1]) intersects the horizontal line α = 0 at two points with β-values 0 and 1 3 .Thus by the definition of the heart Coh β (X), S(I ) is a two-term complex with cohomology in degree -2 and -1, and . If the rank is 1, then the image im(ev) in the category of coherent sheaves is of rank zero, which is not possible because I (H) is torsion-free.Thus H −1 (S(I )) is of rank zero, so ) is equal to either zero or H.In the latter case, we will have ch ≤2 H −2 (S(I )) = (2, 0, −α) for some α ≥ 0 because of the Gieseker semistability of O ⊕3 X .Then im(ev) is a subsheaf of I (H) of rank 1, so it is torsion-free of class ch(im(ev)) = (1, 0, α) and so α ≤ 0; thus in total we obtain α = 0.
Applying a similar argument also implies that ch 3 H −2 (S(I )) = 0; thus H −2 (S(I )) = O ⊕2 X , which is not possible by the definition of the evaluation map ev.Therefore, H −1 (S(I )) is a sheaf supported in dimension at most 1.
We also claim H −2 (S(I )) is slope stable.Otherwise, there is a subsheaf F of bigger slope.By (4.5), it must be of class ch ≤1 (F) = (1, 0).The semistability of S(I )[−1] along the wall implies that H −2 (S(I )) is a reflexive sheaf.If not, H −2 (S(I ))[1], and so S(I )[−1], has a subobject supported in dimension at most 1 of phase 1, which is not possible.Since H −2 (S(I )) is of rank 2, its quotient sheaf H −2 (S(I ))/F is stable, and it must also be reflexive.Taking the double dual shows that F ∨∨ = F; thus F is a line bundle, and so F O X , which is in contradiction to the definition of the evaluation map ev.Therefore, H −2 (S(I )) is slope stable.
The next proposition implies that H −1 (S(I )) is zero and so the evaluation map ev is surjective and the kernel is a reflexive sheaf of Chern character w.Proof.Lemma 4.7 implies that K is a reflexive sheaf; thus K[1] is σ α,β -stable for β ≥ − 1 2 and α 0; see [BBF + 20, Proposition 4.17].Since there is no wall for K[1] crossing the vertical line β = 0, it is σ α,β -semistable along the numerical wall Choose three linearly independent sections of K ∨ : are σ α,β -semistable of the same phase φ and O X [1] is a simple object in P (φ).Therefore, the map ψ is surjective in P (φ).The kernel ker ψ in P (φ) is of Chern character 1, H, 1 6 H 2 , − 1 6 H 3 .Taking cohomology gives the long exact sequence of coherent sheaves ) intersects the horizontal line α = 0 at two points with β = 0, 1 3 .Thus moving along the wall implies that (4.6) Moreover, the slope stability of K implies that (4.7) Therefore, 0 < (−1) i ch 1 H i (ker ψ) .H and ch 1 H 0 (ker ψ) − ch 1 H −1 (ker ψ) = H.Thus one of the ch 1 H i (ker ψ) must be zero, and the other one is equal to (−1) i H.
We finally claim that ker ψ is torsion-free.If not, by (4.6), its torsion part is supported in dimension at most 1.Since ker ψ is semistable along the wall of phase lower than 1, it cannot have a subobject of phase 1. Hence its torsion part is trivial; i.e. ker ψ is a torsion-free sheaf.This immediately implies that ker ψ = I (H) for a line on X.The next step is to compute S 2 (I ).Lemma 4.9.We have h 0 (K (H)) = 3.
Proof.We know h 0 (K (H)) = χ(K (H)) + h 1 (K (H)) = 3 + h 1 (K (H)).There is no wall for K (H) crossing the vertical line β = 0. Therefore, K (H) is semistable along the numerical wall W (K (H), O X ).Suppose for a contradiction that there are four linearly independent global sections: Since O X is stable along the wall and has the same phase as K (H), the map s is injective and the cokernel of s is also semistable of the same phase.It is of Chern character ch(cok s) = −2, H, − 1 6 H 2 , − 1 6 H 3 .The numerical wall W (K (H), O X ) intersects the horizontal line α = 0 at two points with β 1 , β 2 = 0, − 1 3 .Thus by the definition of the heart, we get (4.8) Therefore, ch ≤1 (H −1 (cok s)) = (2, −H), and H 0 (cok s) is supported in dimension at most 1.Moreover, (4.8) implies that H −1 (cok s) is slope stable; Proposition 2.9 then implies that which gives a contradiction.
Therefore, h 1 (K (H)) = 0 and Consider the numerical wall W (K (H), O X ).A similar argument as in the proof of Lemma 4.9 implies that Thus ch ≤1 H −1 (J ) = (1, −H), and H 0 (J ) is supported in dimension at most 1.We claim H −1 (J ) is a line bundle; otherwise, there is a sheaf Q supported in dimension at most 1, and we have injections → J , which is not possible because of the semistability of J along the numerical wall W (O X , K (H)).Thus H −1 (J ) = O X (−H) and H 0 (J ) = O (−H).This implies that J is large volume limit stable.
Definition 4.10.A two-term complex E ∈ D b (X) supported in degree 0 and −1 is said to be large volume limit stable if H −1 (E) is a line bundle, H 0 (E) is a sheaf supported in dimension at most 1, and Hom(Coh ≤1 (X), E) = 0.
By [Tod13, Lemma 3.12 and Lemma 3.13(ii)], a complex E ∈ D b (X) is large volume limit stable if and only if E lies in Coh β (X) and is σ α,β -stable for β > µ H (E) and α 0.
Proof of Theorem 4.5.Any slope-semistable sheaf E of class v is isomorphic to I for a line in X.By Lemma 4.6, I is σ -stable for any S-invariant stability condition σ on Ku(X).
For part (b), take a slope-stable sheaf E of class w.By Proposition 4.8, there is a line on X such that E = K .Thus E is in Ku(X) and is σ -stable for any S-invariant stability condition σ .Conversely, let E be a σ -stable object in Ku(X) of class [S(I )].Then S −1 (E) is σ -stable of class [I ].Hence by part (a), S −1 (E) is, up to a shift, isomorphic to the ideal sheaf of a line on X.Thus E is isomorphic to S(I ) 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 (O X , J).Thus J is σ α,β -stable of phase lower than 1 if −1 < β < − 1 3 .Sheaves supported in dimension zero are σ α,β -semistable of phase 1, so Hom(Coh 0 (X), J) = 0.One can easily check that J ∈ T α,β when β → −1; thus Proposition 4.1 implies J is σ 0 α,β -stable.We claim that J lies in the Kuznetsov component Ku(X).By definition, J lies in the exact triangle We know Hom i (O X (H), O X (−H)) vanishes for i 3 and is isomorphic to C for i = 3.Also, Hom i (O X (H), O (−H)) = 0 vanishes for i 1 and is isomorphic to C for i = 1.Thus hom i (O X (H), J) = 0 for every i ∈ Z.We also have for all i ∈ Z.Therefore, J ∈ Ku(X), and so J is σ (α, β)-stable when −1 < β < − 1 3 .Conversely, take a σ -stable object E ∈ Ku(X) of class [S 2 (I )] for an S-invariant stability condition σ .Thus S −1 (E) is σ -stable of class [K ], so by the second part, there is a line in X such that S −1 (E) = K .Thus E is isomorphic to J up to shift.This completes the proof.Lemma 4.11.Let E be a σ α,β -semistable object of Chern character v for (α, β) along the numerical wall W (E, O X (−H)[1]).Then E is σ α,β -stable for α 0 and β > 0.
Proof.Suppose for a contradiction that W (E, O X (−H)[1]) is an actual wall and F 1 → E → F 2 is a destabilising sequence.Let ch ≤2 (F 1 ) = (r, cH, sH 2 ).Since the boundary of the wall intersects the vertical line Hence by relabelling F 1 and F 2 , we may assume c = −r.Since the F i have the same phase as E along the wall, one can easily show that s = 1 2 r.Therefore, ch ≤2 (F 2 ) = 1 − r, rH, H 2 − 1 3 − 1 2 r and By [BMS16, Corollary 3.10], we must have 0 ≤ ∆ H (F 2 ) < ∆ H (E) = 2 3 (H 3 ) 2 .Since ∆ H ∈ 1 3 (H 3 ) 2 Z, we obtain 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.

Stability conditions on the Kuznetsov component via conic fibrations
Let X be a cubic threefold as before.Fix a line 0 in X, and consider the linear projection to a disjoint P 2 in the P 4 containing X, which induces a conic fibration structure on the blowup of X along 0 .Let B 0 (resp.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 Coh(P 2 , B 0 ) the abelian category of right coherent B 0 -modules and by D b (P 2 , B 0 ) its bounded derived category.By [BMM + 12, Section 2.1], there is a semiorthogonal decomposition which is equivalent to Ku(X) by definition.In [BMM + 12, LMS15], the authors use the semiorthogonal decomposition (5.1) to construct stability conditions on Ku(X).In this section, we first summarise their construction and then show that all these stability conditions on Ku(X) are S-invariant and lie in the same orbit with respect to GL + 2 -action.

Stability conditions on non-commutative P 2
The Chern character of an object E ∈ D b (P 2 , B 0 ) is where Forg : D b (P 2 , B 0 ) → D b (P 2 ) is the forgetful functor; by abuse of notation, we denote it by ch(E) for simplicity.Recall that by [BMM + 12, Proposition 2.12], we have Proposition 6.2.Every E ∈ M U d is σ (α, β)-semistable in Ku(X) for any (α, β) ∈ V , with V as in (4.4).
Hence, up to a shift, we may assume any E ∈ M σ (α,β) Ku(X) (d[I ]) lies in the heart Coh β (X).Thus H i (E) = 0 if i 0, −1, and if H −1 (E) 0, then it is a torsion-free sheaf.So if we consider the locus of objects

The latter is equal to 0
if ∩ = ∅.Assume and intersect in a point.Then by the local-to-global spectral sequence, we only have to consider H 0 (Ext 1 (O (1), O )).Since Ext 1 (O (1), O ) is supported on the intersection point, we conclude that hom(O (1), O [1]) = 1 and thus hom(F , P ) = 1 0 if and intersect in a point.
as claimed.Thus we obtain S(I ) = L O X (I (H))[1] = K [2] for a slope-stable reflexive sheaf K of Chern character w.The next proposition shows that any slope-stable sheaf K of class w is of the form L O X (I (H))[−2] for a line on X. Proposition 4.8.Take a slope-stable sheaf K of Chern character w.There exists a line on X such that K lies in the short exact sequence 0 → K → O ⊕3 X → I (H) → 0. In other words, K = S(I )[−2].