Categorical Torelli theorems: results and open problems

We survey some recent results concerning the so called Categorical Torelli problem. This is to say how one can reconstruct a smooth projective variety up to isomorphism, by using the homological properties of special admissible subcategories of the bounded derived category of coherent sheaves of such a variety. The focus is on Enriques surfaces, prime Fano threefolds and cubic fourfolds.


Introduction
During the last decades, derived categories of coherent sheaves on smooth projective varieties have played a special role in algebraic geometry. In particular, their use in birational geometry and for the study of the geometry of moduli spaces has produced important and unexpected results.
One natural and related question is if a smooth projective variety can be reconstructed, up to isomorphism, from its derived category. Due to the seminal work by Bondal and Orlov [29] we know that this is indeed a theorem when the variety has canonical bundle which is either ample or anti-ample (meaning that its dual is ample). On the other hand, Mukai [114] showed that this is no longer the case when the canonical bundle is trivial.

3
Of course, one may start wondering how one can study the derived category of coherent sheaves and how one can extract geometric information from it. This is a fast growing research area where several fruitful ideas have come into the picture. Important results in this direction are due to the Russian school. The idea is to decompose the derived category in smaller pieces provided by nontrivial admissible subcategories which naturally generate the derived category and whose meaning is intrinsically connected to the geometry of the variety. This led to the notion of semiorthogonal decomposition which is certainly one of the main characters in this survey.
Semiorthogonal decompositions are not always available and when available they are not, in general, canonical. For example, again when the canonical bundle is trivial, the derived category is indecomposable. Nonetheless, when a semiorthogonal decomposition is given, then its components turn out to be extremely interesting. One special case, which is prominent in this paper, is when the derived category D b (X) of a smooth projective variety X contains a bunch of very simple objects, which are called exceptional, and a geometrically meaningful residual category, which we call Kuznetsov component.
In this paper we focus our attention on these components. Indeed, the problem we want to deal with can be now formulated in the following slightly vague form: Categorical Torelli problem. Let X 1 and X 2 be smooth projective varieties over a field, in the same deformation class and with Kuznetsov components Ku(X 1 ) and Ku(X 2 ) . Is it true that X 1 and X 2 are isomorphic if and only if their Kuznetsov components are equivalent?
As we mentioned above, semiorthogonal components are not in general canonical. Thus when such a problem has a positive answer, the corresponding Kuznetsov components have to emerge from very special geometric situations. This will be extensively explained in the examples of interest.
The Categorial Torelli theorems discussed in this paper are indeed the results that provide a positive answer to the Categorical Torelli problem above. As it turns out, we need to be more precise about the equivalence between the Kuznetsov component in the sense that, in some cases, it has to satisfy some additional property. We will discuss this along the paper and discover that some of these assumptions are probably removable once some related conjectures are proved. Others, unfortunately, cannot be avoided. Just to give a short summary, the Categorical Torelli theorems that we will review are the following geometric situations: (CT1) Enriques surfaces with an equivalence between the Kuznetsov components which is of Fourier-Mukai type (Theorem 4.5); (CT2) Cubic threefolds with no further assumptions on the equivalence between the Kuznetsov components (Theorem 6.4); (CT3) Several additional prime Fano threefolds (Section 6.5); (CT4) Cubic fourfolds with an equivalence between the Kuznetsov components which is compatible with the so called degree shift functor (Theorem 7.7).
One important feature of the above results is the variety of techniques that are used to prove them. Indeed, (CT1) is a consequence of a general statement (see Propsition 4.1) which allows us to extend a Fourier-Mukai equivalence between the Kuznetsov components of two Enriques surfaces to an equivalence of their bounded derived categories and then to apply what we call Derived Torelli theorem. This is somehow related to the possibility to enhance exact functors at the dg level. On the other hand, (CT2) and (CT3) use in an extensive way the notion of stability conditions which only recently were constructed on Kuznetsov components (see Section 6.1). The case of cubic fourfolds (CT4) can be handled either by using Hodge theoretic techniques as in [60] or, again, by using stability conditions [14]. Both approaches will be discussed in Section 7.2. The reader may actually view this survey as an occasion to review the most recent developments in so many different directions and to appreciate their power in combination with the theory of semiorthogonal decompositions.
In a negative direction, the Categorical Torelli theorem in the above formulation does not hold in the case of Fano threefolds of index 1. For instance, a consequence of the results in [90] provides the existence of non-isomorphic but birational Gushel-Mukai threefolds with equivalent Kuznetsov components. A refined version, which takes into account the preservation of special objects in the Kuznetsov components, has been recently proved in [67,69] (see Section 6.5 for more details). It becomes then natural to ask whether the existence of an isomorphism between the given varieties with equivalent Kuznetsov components in (CT1)-(CT4) is just a special instance of the following more general problem: Birational Categorical Torelli problem. Let X 1 and X 2 be smooth projective varieties over a field, in the same deformation class and with Kuznetsov components Ku(X 1 ) and Ku(X 2 ) . Is it true that if Ku(X 1 ) and Ku(X 2 ) are equivalent, then X 1 and X 2 are birational?
This will be carefully discussed in the paper but it is worth mentioning that a converse to the Birational Categorical Torelli problem should not hold true (see Remark 7.9).
As a related line of investigation, we recommend the reader to consult [94], where the authors study the problem of recovering the birational class of a smooth projective variety from its bounded derived category. More precisely, they conjecture that two smooth projective varieties are birational if there exists a (strongly) filtered exact equivalence between their bounded derived categories (see [94,Section 10] for some motivic foundations of this conjecture and more generally [73] for reconstruction problems). Another very interesting research topic which is tightly related to the discussion in this paper but which will not be covered in this survey is the so called infinitesimal version of the Categorical Torelli theorems for Fano threefolds (see [67,68]).
We conclude this presentation by pointing out that the paper is accompanied by a list of open problems in the form either of questions or of conjectures. Their relation to the existing results will be carefully explained, but we take the opportunity to stress that their role in this paper is as important as the one of the main results.
Plan of the paper The survey is organized as follows. In Section 2 we recollect some basic definitions and examples of semiorthogonal decompositions, and of Fourier-Mukai functors.
In Section 3 we focus on some special examples of smooth projective varieties of low dimension having a semiorthogonal decomposition with a nontrivial component, known as the Kunzetsov component. In particular, we consider Enriques surfaces, prime Fano threefolds of index 1 and 2 and cubic fourfolds, and we recall the properties of their Kuznetsov component.
Section 4 is devoted to the proof of the Categorical Torelli theorem for Enriques surfaces following [97,100]. The proof makes use of a general criterion, explained in Section 4.1, which allows us to extend Fourier-Mukai equivalences among admissible subcategories appearing in semiorthogonal decompositions under suitable assumptions. Then in Section 4.2 we characterize 3-spherical objects in the Kuznetsov component; this is used in the proof of the main theorem given in Section 4.3.
Section 5 provides a quick introduction to the notion of (weak) stability conditions, the definition of stability manifold together with the associated group actions, the construction via tilt-stability on surfaces and on some threefolds, and the conjectural approach through generalized Bogomolov inequalities. Section 6 is devoted to the case of cubic threefolds. In Section 6.1 we explain the first method to construct stability conditions on the Kuznetsov component of a cubic threefold and how to apply this result to prove the Categorical Torelli theorem; the main reference is [20]. In Section 6.2 we review the more recent method, introduced in [14], to induce stability conditions on admissible subcategories which are left orthogonal components to an exceptional collection in a triangulated category with a Serre functor. Then we introduce the notion of Serre-invariant stability conditions. This is applied to cubic threefolds in Section 6.3 to construct stability conditions on the associated Kuznetsov component which are Serre-invariant. In Section 6.4 we explain some applications of this result on Serreinvariant stability conditions to the study of the geometry of moduli spaces and to give an alternative proof of the Categorical Torelli theorem; the main references are [8,51,129]. In Section 6.5 we recall the state of art about these questions on Serre-invariant stability conditions and Categorical Torelli theorem for the Kuznetsov component of prime Fano threefolds of index 2 and 1.
Finally, in Section 7 we analyze the higher dimensional case of cubic fourfolds. We recall the construction of stability conditions on the Kuznetsov component with the method of [14]. Then we explain the two known ways to prove the Categorical Torelli theorem from [60] and [14], and how to deduce the Classical Torelli theorem from it. We end by discussing the analogous questions in the case of Gushel-Mukai fourfolds.

Notation/warning
All varieties and schemes appearing in this paper are defined over an algebraically closed field . All categories are linear over such a field as well. Depending on the results we are going to discuss, we will need to make further assumptions: either that char( ) is different from 2 or sufficiently large or equal to 0. We tried to point this out along the paper in the most careful way. But it is certainly true that the nonexpert reader might get confused at first sight. Such a reader is then encouraged to read the paper assuming that is algebraically closed of characteristic 0 and come back to the discussion about the possible more general statements only during a second read.

Semiorthogonal decompositions: general results
This section is devoted to a quick discussion about some basic facts concerning semiorthogonal decompositions (see Section 2.1) and Fourier-Mukai functors (see Section 2.3). We also recall some preliminary examples in Section 2.2.

The main definitions
Despite its relatively simple definition and even if it is one of the simplest examples of triangulated category, the bounded derived category D b (A) of an abelian category A has a very rich and often mysterious structure. This remains true when A is the category Coh(X) of coherent sheaves on a smooth projective variety X defined over a field .
In the latter case, there are several approaches to the study of the structure of D b (X) ∶= D b (Coh(X)): • We can look at the way D b (X) is generated; • Mimicking representation theory, we can look at the action of the autoequivalences group of D b (X) on a suitable vector space or lattice (e.g. the total cohomology of X); • We can decompose D b (X) into smaller pieces and try to understand how the geometry of X is encoded by those pieces.
The first strategy has a long history, initiated by the beautiful paper [31] (see also [132]). More recently, it was discovered that this is intimately related to the way D b (X) can be enhanced to higher categorical structures (see, for example, [28,34,39,105]). The second viewpoint has been widely adopted in the case of K3 surfaces, abelian varieties or, more generally, varieties with trivial canonical bundle (see, for example the seminal papers [116,121]). The reason being that, when the canonical bundle is trivial, D b (X) is indeed indecomposable by [23] and thus the third strategy cannot be pursued. On the contrary, the autoequivalences group is usually very rich and intimately related to the topology of the stability manifold which we will discuss later.
In this paper, we are mainly interested in the third approach which can be made precise by introducing the following definition. Let T be a triangulated category which, for simplicity, we assume from now on to be linear over a field . A semiorthogonal decomposition for T , denoted by is a sequence of full triangulated subcategories D 1 , … , D m of T such that: (1) Hom(F, G) = 0 , for all F ∈ D i , G ∈ D j and i > j; (2) For any F ∈ T , there is a sequence of morphisms We call the subcategories D i components of the decomposition.

Remark 2.1
It is a nice and relatively easy exercise to verify that (1) above yields that the factors i (F) in (2) are uniquely determined and functorial, for all F ∈ T and for all i = 1, … , m . Hence, in presence of a semiorthogonal decomposition, we get the i-th projec- Given a semiorthogonal decomposition for T as above, denote by i ∶ D i ↪ T the inclusion. We say that D i is admissible if i has left adjoint * i and right adjoint ! i . In presence of a semiorthogonal decomposition then 1 and 2 coincide with the left adjoint * 1 and the right adjoint ! 2 . Furthermore, if D is an admissible subcategory of T , we set to be the left orthogonal and right orthogonal subcategories of D , respectively. These triangulated subcategories yield semiorthogonal decompositions when D is admissible.
One special feature of the triangulated category T = D b (X) , for X a smooth projective variety, is that it has Serre functor. Recall that a Serre functor of a triangulated category T is an exact autoequivalence T ∶ T → T inducing, for all A and B in T , an isomorphism which is natural in both arguments. Such a functor is unique up to isomorphism of exact functors and thus we will refer to T as 'the' Serre functor of T .

Example 2.2
If X is a smooth projective variety, then the Serre functor X ∶= D b (X) takes the following explicit form The latter is a general fact: if T has Serre functor T , and it contains an admissible triangulated subcategory D , then D has a Serre functor D as well. The shape of D is exactly the one above with X replaced by T (see [27]).
Recall that T is -linear. An object E ∈ T is exceptional if Hom(E, E[p]) = 0 , for all integers p ≠ 0 , and Hom(E, E) ≅ . A set of objects {E 1 , … , E m } in T is an exceptional collection if E i is an exceptional object, for all i, and Hom(E i , E j [p]) = 0 , for all p and all i > j .

and for all integers p;
• full if the smallest full triangulated subcategory of T containing the exceptional collection is equal to T ; Finally, assume that T is a proper -linear triangulated category. This means that for all F and G in T . Its numerical Grothendieck group N (T) is defined as the quotient Here denotes the Euler form on K(T) defined by  [18] shows that the set of line bundles forms a full exceptional collection and so it yields a semiorthogonal decomposition It should be noted that the collection (2.3) is also full and strong.
Example 2.5 (Quadrics) Assume now that Q is an n-dimensional smooth quadric in ℙ n+1 defined by an equation {q = 0} . We assume char( ) ≠ 2 . According to [71], the category D b (Q) has a semiorthogonal decomposition by exceptional bundles whose explicit form depends on the parity of n. More precisely, if n = 2m + 1 is odd, where S is the spinor bundle on Q defined as coker( [124] for more details on spinor bundles.

Fourier-Mukai functors
We now briefly recall how to define special classes of exact functors between admissible subcategories. The reader can have a look to [36] for a survey or to [55] for an extensive treatment. Note that all functors are derived.
Let X 1 and X 2 be smooth projective varieties over a field with admissible embeddings for i = 1, 2.

Definition 2.6 An exact functor
Here the exact functor Φ E is given by where p i is the i-th natural projection.
By [79,Theorem 7.1], the projection functor onto an admissible subcategory D ↪ D b (X) is of Fourier-Mukai type. This motivates [85,Conjecture 3.7] which says that any exact equivalence ∶ D 1 → D 2 between admissible subcategories D i ↪ D b (X i ) , for X i smooth projective over , is of Fourier-Mukai type. Indeed, [79,Theorem 7.1] is a special case of this conjecture for = id.
Here we propose the following restatement: Question 2.7 Is any exact fully faithful functor ∶ D 1 → D 2 between admissible subcat- , for X i smooth projective over , of Fourier-Mukai type? This is motivated by Orlov's result that any fully faithful exact functor is of Fourier-Mukai type when X i is smooth projective over a field (see [121]). It should be noted that in [35,38] the assumptions on were weakened. In particular, assuming full is enough. More recently, [120] extended Orlov's result to the smooth and proper case. Note that Question 2.7 is related to [85,Conjecture 3.7].
Motivated by the recent work [39,105], we can actually state a weaker version of Question 2.7: Question 2.8 Let ∶ D 1 → D 2 be an exact equivalence between admissible subcatego- The problem above is mainly motivated by the recent developments about uniqueness of enhancements [105,Corollary 9.12] (see also [39,Section 7.2] and the recent improvements in [34]). The idea is that, given an equivalence ∶ (X 1 ) → (X 2 ) between the categories of perfect complexes on quasi-compact and quasi-separated schemes, one can replace with another equivalence which can be lifted to any dg model of (X i ) . Under additional suitable assumptions on X i (e.g. if X i is noetherian), the latter condition is equivalent to being of Fourier-Mukai type, due to [106,143].
We expect Question 2.7 and Question 2.8 to have negative answers in general also due to the results in [34,Section 6.4] (see, in particular, Corollary 6.12 and Remark 6.13 there). We will discuss later when one can positively answer them in the geometric settings we are interested in.

Semiorthogonal decompositions for special projective varieties
In this section we would like to go beyond Example 2.4 and Example 2.5 and consider semiorthogonal decompositions for D b (X) for special but very interesting smooth projective varieties. In most of the examples discussed in this section, the derived category will contain a small set of exceptional objects of geometric origin and a nontrivial admissible subcategory right orthogonal as in (2.1) to the exceptional objects and which we will call Kuznetsov component.

Smooth projective curves
It is well known that if C is a smooth complex projective curve, then D b (C) determines C up to isomorphism: Theorem 3.1 (Derived Torelli theorem for curves) Let C 1 and C 2 be smooth complex projective curves. Then As it is explained in the proof of [55,Corollary 5.46], the delicate case which requires using the cohomology of the curve and then the classical Torelli theorem is when the genus of the curves is 1. All the other cases follow from the following beautiful result by Bondal and Orlov [29]. Theorem 3.2 (Derived Torelli theorem for (anti)Fano manifolds) Let X 1 and X 2 be smooth projective varieties with ample or antiample canonical bundle (i.e. either X i or ∨ Apart from the case of ℙ 1 which is covered by Example 2.4, in genus greater than 0 the triangulated category D b (C) does not have nontrivial semiorthogonal decompositions by [119].

Remark 3.3
Note that Theorem 3.1 remains true over any algebraically closed field when the genus of the curves is not 1. This is because Theorem 3.2 holds in this more general setting. On the other hand, when the genus is 1, the assumption = ℂ cannot be removed as there are nonisomorphic smooth projective curves of genus 1 with equivalent derived categories (see [4,139]).

Enriques surfaces
Let be an algebraically closed field of characteristic different from 2. An Enriques surface is a smooth projective surface X defined over such that H 1 (X, O X ) = 0 and the dualizing line bundle X is nontrivial but 2-torsion. An Enriques surface X can be equivalently characterized as a quotient of a K3 surface by an involution acting without fixed points.
The derived category D b (X) of an Enriques surface determines the surface up to isomorphism in view of the following result which is a rewriting of [26, Proposition 6.1] and [54, Theorem 1.1]: Theorem 3.4 (Derived Torelli theorem for Enriques surfaces) Let X 1 and X 2 be smooth projective surfaces defined over an algebraically closed field of characteristic different from 2. If X 1 is an Enriques surface and there is an exact equivalence Clearly the situation becomes much more involved when has characteristic 2. In this case the definition has to be slightly modified: an Enriques surface is a minimal smooth projective surface whose canonical bundle is numerically trivial and such that the second Betti number is 10. If char( ) ≠ 2 , this definition coincides with the one above. But when char( ) = 2 , then one gets three families: • Classical Enriques surfaces: they are characterized by the fact that dim(H 1 (X, O X )) = 0 ; • Singular Enriques surfaces: in this case dim(H 1 (X, O X )) = 1 and such a cohomology group carries a nontrivial action of the Frobenius; • Supersingular Enriques surfaces: in this case dim(H 1 (X, O X )) = 1 and this cohomology group carries a trivial action of the Frobenius.
In the first case the canonical bundle is nontrivial and 2-torsion while, in the latter two cases, the canonical bundle is trivial. Singular Enriques surfaces are again realized as quotients of K3 surfaces. The reader can have a look at [44] and [48] for an extensive treatment of Enriques surfaces and [47] for a shorter but informative one. It is then natural to raise the following question: Question 3.5 Is Theorem 3.4 still true when char( ) = 2 for some/all of the three families above?
Remark 3. 6 It is clear that we should not expect an analogue of Theorem 3.4 to hold for all smooth projective surfaces. Indeed, already for abelian surfaces [114] and K3 surfaces [118,140], this is known to be false.
For the rest of this section we stick to the case where is algebraically close and char( ) ≠ 2 . We can further analyze D b (X) by means of the following result which is certainly well-known to experts.
We can illustrate the geometry attached to the collection L in some interesting cases.

Example 3.8 A generic
Enriques surface does not contain (−2)-curves. Thus the collection in Propsition 3.7 gets much simplified. In particular, with the above notation, we have n i = 1 for every i = 1, … , c , and we get 10 completely orthogonal blocks L i = ⟨L i ⟩ , where L i ∶= L i 1 is an exceptional line bundle, for all i = 1, … , 10. If = ℂ and X does not contain (−2)-curves, then [146] gives a very geometric interpretation of these orthogonal line bundles. Indeed, any ample polarization on X of degree 10 yields 10 elliptic pencils each containing 2 double fibers. Denote them by F + i and F − i . Then we can take L i ∶= O X (−F + i ) . One can prove that, for all i, we have the relation where K X is the canonical class. Using the Serre functor (see Example 2.2), one immediately sees that it is possible to change any L i to O X (−F − i ) and still get a completely orthogonal collection of 10 line bundles. In particular, D b (X) contains many distinct collections of exceptional objects, as we have at least 2 10 = 1024 possible choices of orthogonal exceptional collections of line bundles in D b (X) (see [97,Example 3.4] for a more detailed discussion).
Thus, if X is an Enriques surface and L is a collection of exceptional line bundles as in Propsition 3.7, then we get a semiorthogonal decomposition The admissible subcategory Ku(X, L ) ∶= L ⟂ is referred to as the Kuznetsov component of X. As the notation suggests, it is important to keep in mind that Ku (X, L ) depends on L and not just on X.
The Kuznetsov component is, at the moment, quite a mysterious subcategory. On one hand, it is certainly easy to show that it is nonzero. Indeed, its numerical Grothendieck group can be easily described: In Section 4.2 we will comment on the action of Ku (X,L ) on some special objects of Ku (X, L ) and we will mention why a positive answer to Question 3.9 (ii) may be interesting to provide yet another counterexample to the Jordan-Hölder conjecture. A positive answer to Question 3.9 (iii) (in either of the two forms) would yield a simpler statement for the Categorical Torelli theorem discussed later.

Prime Fano threefolds
We turn now to the case of Fano threefolds, i.e. smooth projective threefolds X defined over an algebraically closed field and such that ∨ X = O X (−K X ) is ample. We stick to the examples where the rank X of the Picard group Pic(X) of X is 1, which are called prime Fano threefolds. Under these assumptions, we denote by H X the primitive ample generator of Pic(X).
The classification of prime Fano threefolds was achieved in [65,117] in characteristic 0. These classification results have been extended to positive characteristic in [138]. These threefolds are classified by two numerical invariants. The first one is the index which is the positive integer i X such that The second one is the degree which is the positive integer d ∶= H 3 X . It turns out that i X ∈ {1, 2, 3, 4} . The cases i X = 3 and i X = 4 correspond to X = Q and X = ℙ 3 , respectively, where Q is a 3-dimensional quadric. In both cases, we know all about D b (X) in view of Example 2.4 and Example 2.5 (when char( ) ≠ 2 ), respectively. Thus we can stick to i X = 1, 2.
Prime Fano threefolds with index 1 are organized in 10 deformation types. Moreover, the degree of these Fano threefolds d X = 2g X − 2 is even and the deformation type is characterized by the choice of 2 ≤ g X ≤ 12 but g X ≠ 11 , where g X is called the genus of X. Now, for g X = 2, 3, 4, 5 we consider the semiorthogonal decomposition On the other hand, if g X is even and greater than 4, then we have where E 2 is a rank-2 stable vector bundle on X whose existence is claimed in [115] (see [14,Section 6] for a careful proof of this fact which is valid for algebraically closed fields of characteristic either 0 or sufficiently large). For odd genus g X = 7, 9 , we use again [115] which yields a rank-5 and rank-3 vector bundle E 5 and E 3 and semiorthogonal decompositions when X has genus 7 and 9, respectively. In all the above cases, the residual category Ku(X) is called Kuznetsov component. It is worth pointing out that Ku(X) can be better understood in some interesting cases. What is known is summarized in the following table, where the third column indicates the reference where the semiorthogonal decomposition in the second column is provided. As above, the base field has characteristic either zero or sufficiently large.
In the table, X d denotes a prime Fano threefold of index 1 and degree d = 2g − 2 , C g denotes a smooth curve of genus g while E i refer to vector bundles which are explicitly described in the references in the third column. It is worth to point out that in some cases the semiorthogonal decomposition of D b (X 6 ) can be refined. More precisely, note that X 6 is a complete intersection of a quadric hypersurface and a cubic hypersurface in ℙ 5 . When the quadric cutting X 6 is smooth, there is a semiorthogonal decomposition where S is the restriction to X of a spinor bundle on the quadric.
Prime Fano threefolds with index 2 are usually referred to as del Pezzo threefolds. They all have a canonical semiorthogonal decomposition where Ku(X) is, as usual, the Kuznetsov component. The degree d X is subject to the bounds 1 ≤ d X ≤ 5 and in some cases the Kuznetsov component can be further analyzed according to the following table: In the table, C 2 is a smooth curve of genus 2 and F 3 , F 2 are vector bundles of rank 3 and 2, respectively, described explicitely in the reference in the third column. In the above list, when i X = 2 and d X = 3 we get the celebrated case of cubic threefolds (i.e. smooth degree 3 hypersurfaces in ℙ 4 ) which are going to be important examples that we will analyze in full detail. The Kuznetsov components of cubic threefolds yield examples of the following special class of triangulated categories. The following result is going to be relevant later in the paper: Proof The admissible subcategory Ku(X) has Serre functor by Example 2.2. We can then simply apply [81,Corollary 4.4] (see also [81,Corollary 4.3] for a more general statement).
The following question is then natural. Question 3.13 Is Ku(X) indecomposable, when X is a cubic threefold? Remark 3.14 A list of other prime Fano threefolds with Kuznetsov component which is a fractional Calabi-Yau category can be deduced from [84] (see Section 2.4 therein) and [91]. One interesting example is provided by quartic threefolds, i.e. smooth hypersurfaces of degree 4 in ℙ 4 . According to the first table above ( = i = 1 and g = 3 ), if X is a quartic threefold, we have a semiorthogonal decomposition as in (3.3). In this case, Ku(X) is a fractional Calabi-Yau category of fractional dimension 10 4 . We will comment on this case later.
After discussing the Categorical Torelli theorem for cubic threefolds, we will explain that Question 2.8 has a positive answer for Ku(X) , when X is a cubic threefold (see Corollary 6.5).

Higher dimensional Fano manifolds: cubic fourfolds
A cubic fourfold is a smooth cubic hypersurface in ℙ 5 . Here we assume that X is defined over an algebraically closed field such that char( ) ≠ 2.
The derived category of a cubic fourfold X has again a natural semiorthogonal decomposition where H is the class of a hyperplane section of X. The admissible subcategory Ku(X) is the Kuznetsov component of X.

Proposition 3.15 If X is a cubic fourfold, then the Kuznetsov component Ku(X) is an indecomposable 2-Calabi-Yau category for which Question 2.7 has a positive answer.
Proof The fact that Ku(X) is a 2-Calabi-Yau category is again a consequence of [81, Corollary 4.3]. Moreover, it is an indecomposable admissible subcategory due to the wellknown argument in [84, Section 2.6]. The fact that Question 2.7 has a positive answer is the content of [98] (based on [34,37]).
It was observed by Kuznetsov in [80] that, in many cases, one can realize the Kuznetsov component of a cubic fourfold as the derived category of a K3 surface. For example, this happens for Pfaffian cubic fourfolds. Subsequent recent work carried out in [2] and [13] completely classified all cubic fourfolds whose Kuznetsov component is equivalent to the derived category of a K3 surface. This body of work is very much related to the following very influential conjecture (see [80, Conjecture 1.1]) which we state even though it is not directly related to the rest of this paper.

Conjecture 3.16 (Kuznetsov) A cubic fourfold X is rational if and only if Ku(X) is equivalent to the bounded derived category of a K3 surface.
None of the two implications is clear but the conjecture perfectly matches the classical Hodge theoretic still conjectural characterization of rational cubic fourfolds due to Harris and Hassett. Some important results in this direction are contained in [133,134].

Remark 3.17
As it was first pointed out in [80], there are cubic fourfolds X such that Ku(X) is equivalent to the derived category D b (S, ) of twisted coherent sheaves, where S is a K3 surface and is an element in the Brauer group Br(S) ∶= H 2 (S, O * S ) tor of S (see [33, Chapter 1] for an extensive introduction to twisted coherent sheaves and their derived categories). The complete classification of all cubic fourfolds for which this is true was carried out in [13,57]. In view of Conjecture 3.16 it is certainly interesting to understand which geometric property of X corresponds to having an equivalence Ku(X) ≅ D b (S, ).
Because of the many similarities with the derived categories of K3 surfaces-only few of which have been discussed here-the Kuznetsov component Ku(X) is an example of so called noncommutative K3 surfaces.

Remark 3.18
For actual K3 surfaces, the description of the autoequivalences group of their bounded derived categories is a challenging problem (see, for example, [9,59,116,121]). One could of course try to describe the autoequivalences group of a noncommutative K3 surface as well. We will go back to this issue later in the paper. For later use, we content ourselves with the observation from [81] that, for a cubic fourfold X, the category Ku(X) has always an autoequivalence where * is the left adjoint of the inclusion ∶ Ku(X) ↪ D b (X) and H is a hyperplane class. Such an autoequivalence is called degree shift functor. It is not difficult to see that, by definition, X is of Fourier-Mukai type.
Assume now that = ℂ . It was observed in [2] (see also [111,Section 3.4]) that the Kuznetsov component Ku(X) of a cubic fourfold X is equipped with an even unimodular lattice H (Ku(X), ℤ) with a weight-2 Hodge structure induced by the Hodge decomposition of H 4 (X, ℂ) . Such a lattice is usually referred to as the Mukai lattice of X.
For the convenience of the reader let us spell out some details in the construction. First observe that the topological K-theory K top (X) of X comes equipped with the pairing where p ∶ X → pt . As a group H (Ku(X), ℤ) is defined as the set of classes in K top (X) which are orthogonal, with respect to (−, −) to the classes of the three line bundles O X (iH) , with i = 0, 1, 2 . This is nothing but the topological K-theory K top (Ku(X)) of the admissible subcategory Ku(X) . Then the restriction of (−, −) ∶= − (−, −) to K top (Ku(X)) defines a pairing on it, called the Mukai pairing.

Remark 3.19 It was proved in
is Hodge isometric to the Mukai lattice of the K3 surface. In particular, as a lattice, where U is the hyperbolic lattice and E 8 (−1) is the twist by −1 of the lattice corresponding to the root system E 8 .
The lattice H (Ku(X), ℤ) comes with a weight-2 Hodge structure defined as follows. Consider the natural map which is usually called Mukai vector. See [2, Section 2] for more details. Taking its complexification, we then set For later use, we are interested in describing special classes in H Hodge (Ku(X), ℤ) . To this extent, if we denote by * ∶ D b (X) → Ku(X) the projection functor as above and by any line in X, we can consider the classes in the numerical Grothendieck group N (Ku(X)) of Ku(X).

Remark 3.20
The integral Hodge conjecture for cubic fourfolds was originally proved in [144,Theorem 18] and then reproved in [13,Corollary 29.8] (see also [125] for a general treatment). As for the Kuznetsov component, one can prove that H Hodge (Ku(X), ℤ) is naturally isometric to the numerical Grothendieck group N (Ku(X)) of the Kuznetsov component and thus not only it contains interesting classes as observed above but it entirely consists of algebraic classes (see [13,Theorem 29.2]).
It is then clear that the classes 1 and 2 are in � H Hodge (Ku(X), ℤ) ⊆ � H(Ku(X), ℤ) as well. We set to be the primitive sublattice of H Hodge (Ku(X), ℤ) generated by the classes 1 and 2 . With the choice of these generators, A 2 is the free ℤ-module ℤ ⊕ ℤ with intersection form given by the matrix

Enriques surfaces
In this section we want to state and prove the Categorical Torelli theorem for Enriques surfaces as in [97,100]. As we will see, we need to consider the semiorthogonal decompositions in Section 3.2. The key idea is to prove and use an extension result for Fourier-Mukai equivalences between admissible subcategories. In particular, no stability conditions are needed here.H (3.10)

Extending Fourier-Mukai equivalences
In this section we illustrate a general criterion which was proved in [97]. It allows us to extend Fourier-Mukai equivalences between admissible subcategories under some assumptions on the nature of the semiorthogonal decompositions.
In general, such a criterion is not easy to apply. We will try to clarify this with a brief vague discussion. Assumption (b) is hard to verify in concrete examples and, in some cases, it might happen that (b) is not satisfied by a given Fourier-Mukai equivalence Φ E which must then be composed with some additional autoequivalence of D 2 (or D 1 ). Heuristically, we should expect to be able to apply such a result either when we have a good grip on the autoequivalence groups of D 1 and D 2 or when we have a good understanding of the object 1 ! 1 (E) . We will show in the next section that for Enriques surfaces we are in the second scenario.
Of course, the criterion can be iterated when the orthogonal complement of D i consists of more than one exceptional object. As we will see, this makes computations more complicated.

Remark 4.2
We conclude this section by pointing out that the above criterion is very much related to the gluing theory for dg categories and dg functors developed in [88]. The reader can have a look at [97, Section 2.3] for an extensive discussion.

Special objects and their classification
As we mentioned, the Categorical Torelli theorem for Enriques surfaces will be obtained as an application of Theorem 3.4 and Propsition 4.1. And, as we commented above, this is made possible by a complete understanding of the projection into the Kuznetsov component of the 10 exceptional line bundles in L .
Let us begin with a general discussion.

Definition 4.3 Let T be a triangulated category that is linear over a field and with Serre
Spherical objects were introduced and studied in [137]. They often appear in triangulated categories of Calabi-Yau type and, more specifically, in the derived category of smooth projective Calabi-Yau varieties. They naturally define special autoequivalences which are called spherical twists and which correspond, under Mirror Symmetry, to Dehn twists in the mirror Fukaya category.
The notions of spherical object and spherical twist have been widely extended and generalized. Actually, n-pseudoprojective objects are part of this more general picture. Indeed, the graded vector space of derived endomorphisms is, up to multiplying by 2 the degree, the same as the graded vector space of the total cohomology of an n-dimensional complex projective space. Hence, n-pseudoprojective objects are slight generalizations of the kind of objects studied in [61,76].
Let us go back to the geometric setting and let us assume that X is an Enriques surface. As we explained in Section 3.2, we have a semiorthogonal decomposition as in (3.1), where the 10 exceptional line bundles in L are as in Propsition 3.7. Without loss of generality, we can reorganize these exceptional objects to get a semiorthogonal decomposition into blocks such that, if c ≠ 10 , then there is a positive integer 1 ≤ d ≤ c such that L j consists of more than one object if 1 ≤ j ≤ d and of just one object if d < j ≤ c.
Consider now the corresponding objects S i ∈ Ku(X, L ) defined in (3.2). They provide a complete classification of 3-spherical and 3-pseudoprojective objects in Ku(X, L ) according to the following result.
Furthermore, all these 3-spherical and 3-pseudoprojective objects are not isomorphic.
As we pointed out in [100, Remark 2.9], this result (see also its easy consequence [100, Corollary 2.8]) together with a positive answer to Question 3.9 (ii) would provide another counterexample to the Jordan-Hölder property for semiorthogonal decompositions. Roughly, such a property predicts that if X is a smooth projective variety then the semiorthogonal decompositions of D b (X) are essentially unique, up to a reordering of the components and up to equivalence. The fact that this property does not hold in general is not new due to the counterexamples in [7,77].

The categorical Torelli theorem
We are now ready to state the main result of this section which is a combination of [ For simplicity, we illustrate the proof of Theorem 4.5 when N 1 and N 2 consist of 10 completely orthogonal line bundles. Hence, for i = 1, 2 . By Theorem 4.4 (1), the projections S i,j of L i,j into Ku(X i , N i ) are, up to shift and isomorphism, the only 3-spherical objects in Ku(X i , N i ) . The more general case where N i contains blocks with more then one object is dealt similarly by using Theorem 4.4 (2).
The idea is to extend the Fourier-Mukai equivalence ∶ Ku(X 1 , N 1 ) → Ku(X 2 , N 2 ) step by step by adding all the 10 exceptional objects in N i . It is not difficult to see that, since these objects are completely orthogonal, it is enough to show how to add one of them. Thus, let us consider L 1,1 and its projection S 1,1 . Since being 3-spherical is a property which is invariant under equivalence, the object (S 1,1 ) is 3-spherical as well. By Theorem 4.4 (1), up to shift, there is j ∈ {1, … , 10} and an isomorphism Again, by orthogonality, we can permute the exceptional objects L 2,j 's and assume, without loss of generality, that j = 1. Now, a direct application of Propsition 4.1 implies that the Fourier-Mukai equivalence extends to a Fourier-Mukai equivalence The same argument can be applied again for the other exceptional objects in N 1 and, in a finite number of steps, we get an equivalence Theorem 3.4 and deduce that X 1 ≅ X 2 . The careful reader might have noticed that, a priori, the argument gets more complicated when we add more exceptional objects: one should classify 3-spherical objects in categories which are larger that the original Kuznetsov component. But this is in reality much simpler because the line bundles L i,j are completely orthogonal and the projection of L i,j onto is the same as the projection onto Ku(X i , N i ) . Furthermore the situation gets even more involved when we deal with Enriques surfaces whose Kuznetsov component contains pseudoprojective objects. The detailed explanation about how these problems can be overcome are not suited for this survey and we refer to the original papers [97] and [100].

Remark 4.6
(i) The Categorical Torelli theorem above has a trivial converse: if X 1 and X 2 are isomorphic Enriques surfaces and D b (X 1 ) has a semiorthogonal decomposition as in Section 3.2, then D b (X 2 ) has a semiorthogonal decomposition of the same type and there is a Fourier-Mukai equivalence Ku(X 1 , L 1 ) ≅ Ku(X 2 , L 2 ) , for appropriate L 1 and L2 . This is simply because any isomorphism X 1 ≅ X 2 induces a Fourier-Mukai equivalence between the whole derived categories which then trivially restricts to a Fourier-Mukai equivalence between the Kuznetsov components. (ii) The technique used in the proof is quite powerful and it was also used in [97]  We conclude this section by observing that our proof of Theorem 4.5 is almost completely characteristic free. The only point where we use that is not only algebraically closed but also such that char( ) ≠ 2 is when we invoke Theorem 3.4. Thus, if we can answer Question 3.5 in the positive, then we can extend Theorem 4.5 to Enriques surfaces over fields of characteristic 2. Furthermore, if we provide a positive answer to Question 3.9 (iii), we can avoid assuming that the equivalence between the Kuznetsov components is of Fourier-Mukai type.

A brief introduction to (weak) stability conditions
Stability conditions on triangulated categories have been defined for the first time by Bridgeland in [24], generalizing the notion of slope stability for sheaves on curves. Since then, the development of the theory has led to applications in classical algebraic geometry and in the study of moduli spaces of stable objects in admissible subcategories of the bounded derived category. In this section, we review the definition of (weak) stability conditions and the construction in the case of the bounded derived category of a smooth projective variety via tilt stability. Our main references are [11,13,24].

Definitions
Let be an algebraically closed field of arbitrary characteristic. Let X be a smooth projective variety over and D be a full admissible subcategory of D b (X) . A (weak) stability condition on D is essentially the data of the heart of a bounded t-structure and of a (weak) stability function detecting the semistable objects, satisfying certain compatibility conditions.
Note that the heart of a bounded t-structure A is not a triangulated category. By [19] we have that A is an abelian category. We denote by K(A ) the Grothendieck group of A . As an example, the abelian category Coh(X) of coherent sheaves on X is the heart of a bounded t-structure on D = D b (X).

Remark 5.2
Given E ∈ D , the objects A i in Definition 5.1 are uniquely determined and functorial (also the integers k i are unique). They are called the cohomology objects of E in the heart A .

Definition 5.3
Let A be the heart of a bounded t-structure on D . A group homomorphism Given a (weak) stability function Z, the slope of E ∈ A is and the phase of E is . Fix a finite rank free abelian group Λ and a surjective morphism ∶ K(D) ↠ Λ.

Definition 5.4 A weak stability condition (with respect to ) on D is a pair
where A is the heart of a bounded t-structure on D and Z ∶ Λ → ℂ is a group morphism called central charge, satisfying the following properties: We say that an object If Z• is a stability function, we say that is a Bridgeland stability condition on D (with respect to ).

Remark 5.5
(i) It is possible to verify that the filtration in Definition 5.4(b) is unique and functorial. Moreover, the Harder-Narasimhan property and Definition 5.1(ii) imply that every object in D has a filtration in -semistable ones, which are called HN factors. We denote by + (E) (resp. − (E) ) the largest (resp. smallest) phase of the HN factors of 0 ≠ E ∈ D. (ii) A (weak) stability condition = (A, Z) determines a slicing, i.e. a collection of full additive subcategories P( ) ⊂ D for ∈ ℝ , defined as follows: (1) for ∈ (0, 1] , the subcategory P( ) is the union of the zero object and all -semistable objects with phase ; (2) for + n with ∈ (0, 1] and n ∈ ℤ , set P( + n) ∶= P( )[n]. We will use the notation P(I) , where I ⊂ ℝ is an interval, to denote the extensionclosed subcategory of D generated by the subcategories P( ) with ∈ I . By Defini- The notion of weak stability condition is very useful for the construction of Bridgeland stability conditions, as we will explain in Section 5.3.
A Bridgeland stability condition is a stability condition in the sense of [24]. Note that it is not clear whether there exist moduli spaces parametrizing semistable objects with a fixed class in Λ , since they do not have a GIT description. Following the recent developments in [13] about the theory of families of stability conditions and [3] about the existence of good moduli spaces, we introduce the notion of stability condition with moduli spaces, which is a Bridgeland stability condition with "well-behaved" moduli functors.
Assume that the base field is of characteristic 0. Given a Bridgeland stability condition on D with respect to , fix v ∈ Λ and ∈ ℝ such that Z(v) ∈ ℝ >0 e i . Consider the functor from the category of schemes over to the category of groupoids, which associates to T ∈ Ob(Sch) the groupoid M (D, v)(T) of all perfect complexes E ∈ D(X × T) such that, for every point t ∈ T , the restriction E t of E to the fiber X × Spec(k(t)) belongs to D , is -semistable of phase and (E t ) = v.
If is a stability condition with moduli spaces on D with respect to , then by [13,Theorem 21.24 (3)], which makes use of [3], for every v ∈ Λ it follows that M (D, v) admits a good moduli space M (D, v) which is a proper algebraic space over .
A natural choice for the lattice Λ is the numerical Grothendieck group N(D) of D . In fact, the numerical Grothendieck group is a free abelian group of finite rank. This follows from the fact that D is an admissible subcategory of D b (X) , thus we have the semiorthogonal decomposition D b (X) = ⟨D, ⟂ D⟩ . Since N(−) is additive, we have that N(D) is a subgroup of the numerical Grothendieck group of X, which is a free abelian group of finite rank [52, 19.3.2]. This motivates the following definition.  If n = 1 , i.e. X is a curve, then H is a numerical stability condition on D b (X) , recovering the classical notion of slope stability.

Stability manifold and actions
Assume is an algebraically closed field of arbitrary characteristic 3 . We denote by Stab Λ (D) the set of stability conditions on D with respect to . We consider on Stab Λ (D) the coarsest topology such that the maps Assume that is a stability condition with moduli spaces. We observe that, by definition, moduli spaces with respect to ⋅ (M, g) and Φ ⋅ are isomorphic to moduli spaces with respect to . Λ ≅ H 0 (X, ℤ) ⊕ H 2 (X, ℤ) ≅ ℤ ⊕2 . 3 In this section we could work in more generality without assuming is algebraically closed, see [ where H is the slope stability defined in Example 5.8.
(ii) (Action of the Serre functor) The Serre functor D on D (see Example 2.2) defines an element in Aut N(D) (D) = Aut(D) , thus we can consider its action on numerical stability conditions. In Section 6.2 we will introduce the notion of Serre-invariant stability conditions, which are numerical stability conditions preserved by the Serre functor, up to the G L + 2 (ℝ)-action. In fact, this notion plays an important role in a proof of the Categorical Torelli theorem for cubic threefolds and in the study of the geometry of moduli spaces of stable objects in the Kuznetsov component, as we will see in Section 6.3.

Tilt stability
The construction of stability conditions is a difficult task, even in the case of D b (X) for a smooth projective variety X of dimension n ≥ 3 over an algebraically closed field . A conjectural approach is via the notion of tilt stability, whose definition is summarized in this section and which works perfectly for X of dimension n = 2 (see [1,25]).
Let (X, H) be a polarized smooth projective variety. We have seen in Example 5.8 that slope stability H = (Coh(X), Z H ) defines a stability condition on D b (X) when X is a curve. However, in higher dimension this is no longer true, as Z H vanishes on torsion sheaves supported in codimension ≥ 2 , and in higher dimensions slope stability is only a weak stability condition. Actually, the choice of Coh(X) as heart is not the correct one, since by [141,Lemma 2.7] it cannot be the heart of a numerical stability condition if dim(X) ≥ 2.
To overcome this problem, one can consider a new heart by tilting Coh(X) . More precisely, let = (A, Z) be a weak stability condition on a triangulated -linear category. Fix s ∈ ℝ and define the following subcategories of A : As proved in [53], the category is the heart of a bounded t-structure on T and is called the tilting of A with respect to at slope s. In this context ⟨−, −⟩ means the smallest full subcategory closed under extensions and containing the two additive subcategories T s and F s [1] . Note that if s > s ′ , then we have Indeed, consider F ∈ A and -semistable with Z (F) > s , which is an object in A s . Then [1] . By the definition of A s , we deduce the desired property.
In the case of slope stability, we consider the heart

3
obtained by tilting Coh(X) with respect to H at slope s. In analogy to the curve case, define where ch 2 (E) denotes the degree-2 part of the Chern character of E, and set Λ H ∶= Im( ) . For s, q ∈ ℝ , define Z s,q ∶ Λ H → ℂ by The slope associated to Z s,q is denoted by s,q . We can now state the main result of this section.
Theorem 5.11 ([1, 11, 12, 24]) Let X be a smooth projective variety of dimension n ≥ 2 . There is a continuous family of weak stability conditions with respect to , parametrized by defined as with a locally-finite wall and chamber structure.
If n = 2 , i.e. X is a surface, then { s,q } is a continuous family of stability conditions on X.

3
The proof of Theorem 5.11 is slightly involved and we refer to [112, Sections 6.2, 6.3] for a detailed summary. We only mention that a key ingredient is the classical Bogomolov Inequality, which implies that every H -semistable sheaf E satisfies Moreover, one can choose Δ H as the quadratic form fulfilling the support property for s,q . This also implies that s,q satisfies well-behaved wall-crossing: for fixed v ∈ Λ H , the boundary of the locus in Δ where an object of class v is stable is defined by a locally-finite set of submanifolds of real codimension one.
The weak stability conditions s,q defined in Theorem 5.11 are called tilt-stability conditions. A useful way to visualize tilt stability conditions has been introduced in [101, Section 1]. Note that given E ∈ D b (X) such that Z s,q (E) ≠ 0 , its truncated at degree 2 Chern character defines the point in a projective plane ℙ 2 ℝ . If ch 0 (E) ≠ 0 , we consider the affine coordinates By (5.4) and [11,Theorem 3.5], we have that s,q -semistable objects correspond to points below the parabola q = 1 2 s 2 and points above parametrize tilt stability conditions. Furthermore, the phase of a s,q -semistable object E ∈ Coh s (X) is the angle between the line connecting (s, q) and (s(E), q(E)) and the vertical half-ray from (s, q) to −∞ divided by (see Figure 1).
It is now natural to wonder whether the above procedure can be generalized to construct stability conditions when X has dimension > 2 . In this case, it is easy to see that Z s,q does not define a stability function, as it vanishes on objects supported in codimension ≥ 3.
Assume X is three-dimensional. In analogy to the surface case, in [12] the authors consider a new heart obtained by tilting Coh s (X) with respect to the tilt stability s,q and a weak stability function on it, involving the third Chern character of the objects. The key observation, made in [11,12], is that this new weak stability condition is a stability condition on D b (X) if and only if tilt semistable objects satisfy some quadratic inequality, called generalized Bogomolov inequality, involving the Chern characters till degree 3. Following this approach, stability conditions have been constructed on the bounded derived categories of Fano threefolds [12,21,95,110,135], abelian threefolds [11,107,108], some resolutions of finite quotients of abelian threefolds [11], and threefolds with nef tangent bundle [21,74].
A further progress in this direction has been made in [96], where a stronger Bogomolov inequality, namely an inequality involving the Chern characters up to degree 2 of slope semistable torsion free sheaves (see [96,Theorem 1.1] for the precise statement), has been proved in the case of quintic threefolds, giving the first highly nontrivial example of stability conditions on a strict (i.e. simply-connected) Calabi-Yau threefold. The involved strategy makes use (among other things) of a restriction lemma, first appeared in [50], which allows one to reduce to show a Clifford type bound for the dimensions of global sections of stable vector bundles on curves defined as complete intersections of two quadrics and a (5.4) Δ H (E) ∶= (H n−1 ch 1 (E)) 2 − 2(H n ch 0 (E))(H n−2 ch 2 (E)) ≥ 0.
quintic hypersurface in ℙ 4 . The existence of stability conditions is then obtained by using this stronger Bogomolov inequality to prove the following statement (see [11,Conjecture 4.1]): if E is s,q -semistable for a certain choice of (s, q) ∈ Δ , then E satisfies . This method has been recently generalized to the case of some three-dimensional weighted hypersurfaces in weighted projective spaces [75], and to three-dimensional complete intersections of quartic and quadric hypersurfaces in ℙ 5 [103]. An interesting problem would be trying to adapt this argument to treat other examples of Calabi-Yau complete intersections in ℙ N . For instance, the next case to address could be the following: Question 5.12 Let X be a complete intersection of a cubic and two quadric hypersurfaces in ℙ 6 . Is it possible to show a stronger Bogomolov inequality for slope semistable torsion free sheaves on X, similarly to [ We end this section with a sort of negative twist. Indeed, while the possibility to prove the generalized Bogomolov inequality (5.5) is still plausible for threefolds with trivial canonical bundle, we know that (5.5) is false in general. The first counterexample was provided in [136]. A replacement for this too optimistic guess was recently provided in [10]. In order to state the revised conjecture, we proceed as follows. Let (X, H) be a polarized projective variety of dimension n. We denote by CH • num (X) the Chow ring of X modulo numerical equivalence. Let B ∈ NS(X) ⊗ ℝ , where NS(X) = CH 1 num (X) , let Γ ∈ CH 2 num (X) ⊗ ℝ be such that Γ ⋅ H n−2 = 0 , and pick 3 , … n to be arbitrary classes with i ∈ CH i num (X) ⊗ ℝ . Let and set ch (−) ∶= ⋅ ch(−) . Using the class and the modified Chern character, one can repeat the discussion above, define -slope semistable objects and thus the notion of -tilt stability s,q for (s, q) ∈ Δ (see [ for all (s, q) ∈ Δ , such that q > f (s).
We do not explain the origin of this conjecture here. We only remark that it is proved for the above mentioned cases (prime Fano threefolds, abelian threefolds, quintic threefolds, complete intersections of quadratic and quartic hypersurfaces in ℙ 5 , the blow-up of ℙ 3 at a point, threefolds with nef tangent bundle) which justifies the way it is formulated. For an extensive and very interesting discussion, one can have a look at [10,Section 4]. The key point is that, if n = 3 , the conjecture implies the existence of stability conditions on D b (X) . Finally, it is clear that the original conjecture in [11,12] implies Conjecture 5.13 for some particular function f while the counterexamples mentioned above do not apply in this case.

Cubic threefolds and beyond
As we mentioned in Section 3.3, if X is a cubic threefold, we have a semiorthogonal decomposition where H is a hyperplane section. In this section we investigate the existence of stability conditions on Ku(X) and discuss several applications including, of course, the Categorical Torelli theorem for these hypersurfaces.

Cubic threefolds: first approach
In this section we want to recall the approach in [20] to the construction of stability conditions on the Kuznetsov component of a cubic threefold. Even though a much more modern technology is, at the moment, available (see Section 6.2 and Section 6.3 below), this circle of ideas have interesting applications to geometric problems and it is an important illustration of what will be proven later about cubic fourfolds.
Assume that X is defined over a field which is algebraically closed with char( ) ≠ 2 . The starting point is the following observation. Fix a line 0 in a cubic threefold X ↪ ℙ 4 and consider a plane ℙ 2 ⊆ ℙ 4 which is skew with respect to 0 . The rational map 0 ∶ X ⤏ ℙ 2 given by the projection from 0 can be resolved as follows. Consider the blow-up P 4 of ℙ 4 along 0 and the strict transform X of X inside P 4 . They all sit in the commutative diagram where q is the ℙ 2 -bundle induced by the projection from 0 and its restriction ̃ 0 to X is the corresponding conic fibration. We denote by h the pull-back to X of a hyperplane section of ℙ 2 , by H the pull-back to X of a hyperplane section of ℙ 4 and by e the exceptional divisor.
As a beautiful application of [83] (see also [6] for the case of arbitrary odd characteristic), the conic bundle ̃ 0 ∶X → ℙ 2 yields a sheaf B 0 on ℙ 2 of even part of Clifford

3
algebras. The explicit description of B 0 is not needed here but it is worth remembering that we have an isomorphism of O ℙ 2-modules. Moreover, B 0 is noncommutative and is an Azumaya algebra away from the discriminant of the conic bundle. One can further consider the abelian category Coh(ℙ 2 , B 0 ) of coherent B 0 -modules and the corresponding derived category D b (ℙ 2 , B 0 ).
It is clear from the discussion above that we can regard X both as a blow-up of X and as a conic fibration on ℙ 2 . In the first case, [123]  This is particularly rewarding when we want to construct stability conditions on Ku(X) . Indeed, as we explained in the previous sections, we have standard strategies to construct stability conditions on surfaces, the situation is less clear but, by now, manageable for threefolds and mostly obscure in higher dimension. We show now how this works for the Kuznetsov component of cubic threefolds. Then in Section 6.2 we will see that there is a more direct way to contruct stability conditions in the cubic threefold case, without relying on a dimension reduction trick. On the other hand, we will observe later that the dimension reduction trick is crucial while dealing with cubic fourfolds.
Let us now go back to X a cubic threefold. As we observed in the previous section, a standard way to construct (weak) stability conditions on surfaces is by using slope stability and tilt with respect to it.
In the case of the abelian category Coh(ℙ 2 , B 0 ) , the fact that B 0 is an O ℙ 2-algebra (see (6.1)), allows us to define a forgetful functor and thus the functions where N(ℙ 2 , B 0 ) stands for the numerical Grothendieck group of D b (ℙ 2 , B 0 ) . For F ∈ Coh(ℙ 2 , B 0 ) with rk(F) ≠ 0 , we define the slope As usual we can recast h in terms of a central charge Given F ∈ Coh(ℙ 2 , B 0 ) , when we say that F is either torsion-free or torsion of dimension d, we always mean that Forg(F) has this property. It is then not difficult to show (see [20,Section 2.3

]) that the pair
Note that if F ∈ Coh(ℙ 2 , B 0 ) has rank 0, then h (F) = +∞ . Given this, we can apply the tilting procedure described in the previous section and define the two full additive subcategories Tilting with respect to the pair (F 0 , T 0 ) yields the abelian category Coh − 5 4 (ℙ 2 , B 0 ) , which is the heart of a bounded t-structure. Set It turns out that A 0 is the heart of a bounded t-structure on Ku(X) (see [20,Lemma 3.4]).

Remark 6.1 A useful computation shows that if ⊆ X is any line, then the ideal sheaf I is contained in A 0 (see [20, Proposition 3.3]).
Consider now the function on the numerical Grothendieck group N(ℙ 2 , B 0 ) of D b (ℙ 2 , B 0 ) . The restriction Z 0 ∶= Z| N(Ku(X)) of Z to the numerical Grothendieck group of Ku(X) is a stability function. For this, see [20,Lemma 3.5]. This means that it satisfies the condition in Definition 5.3.
These preliminary observations allow us to conclude that we have stability conditions on Ku(X) . Assume that the base field has characteristic 0. Applying the recent results in [3, 13, Proposition 25.3] one can further show that 0 is a stability condition with moduli spaces. For later use, we are also interested in studying special moduli spaces of stable objects. The precise result is the following:
(2) Let be any line in X. The moduli space M 0 (Ku(X), [I ]) of 0 -stable objects in A 0 and with numerical class [I ] is isomorphic to the Fano surface of lines F 1 (X) in X.

Remark 6.3
The construction of (weak) stability conditions on D b (ℙ 2 , B 0 ) and on Ku(X) , for X a cubic threefold, was realized in [92] in a similar fashion as above, and it has been used to answer some very geometric questions. In particular, in [92, Theorem B] the authors reprove and generalize the main result of [40] about the nonemptiness of moduli spaces of Ulrich bundles of arbitrary rank on cubic threefolds. Let us recall that an Ulrich bundle E is an aCM (arithmetically Cohen-Macaulay) bundle whose graded module ⨁ m∈ℤ H 0 (X, E(m)) has 3 rk(E) generators in degree 1. On the other hand, a vector bundle E on a cubic threefold X is aCM if dim H i (X, E(j)) = 0 , for i = 1, 2 and all j ∈ ℤ . We will come back to related questions in Section 6.4 where the results mentioned above are stated in a precise form (see also [92,Remark 2.20] for comments on the used normalization in the definition of Ulrich bundle).
The result above is the key to prove the following which is indeed [20, Theorem 1.1]. Theorem 6.4 (Categorical Torelli theorem for cubic threefolds) Let X 1 and X 2 be cubic threefolds defined over an algebraically closed field with char( ) ≠ 2, 3 . Then there exists an exact equivalence Ku(X 1 ) ≅ Ku(X 2 ) if and only if X 1 ≅ X 2 .
Proof (Idea of proof) One implication is trivial. Indeed, if X 1 ≅ X 2 , then the isomorphism preserves the line bundles O X i and O X i (H X i ) . Thus we get an induced exact equivalence Ku(X 1 ) ≅ Ku(X 2 ).
Let us now start with an exact equivalence ∶ Ku(X 1 ) → Ku(X 2 ) . Let ⊆ X 1 be a line, seen as a point in F 1 (X 1 ) . By Theorem 6.2 (2), the object I is 0 -stable in Ku(X 1 ) . We would like to conclude that (I ) is again the ideal sheaf of a line in X 2 , but this is not true in general. Nevertheless, by the result of the discussion in [20, Section 5.1] based on a delicate argument in [20,Section 4], we can compose with a suitable power of the Serre functor Ku(X 2 ) and a shift, in order to get another exact equivalence � ∶ Ku(X 1 ) → Ku(X 2 ) such that: • The class [ � (I )] is in N (Ku(X 2 )) the class of the ideal of a line in X 2 ; • � (I ) is 0 -stable and contained in the heart A 0 in Ku(X 2 ).
Thus, Theorem 6.2 (2) implies that ′ induces a bijection between F 1 (X 1 ) and F 1 (X 2 ) . By [20, Section 5.2] such a bijection can be turned into an isomorphism f ∶ F 1 (X 1 ) → F 1 (X 2 ) . It was observed in [42] that F 1 (X i ) is a surface of general type whose canonical bundle coincides with the ample class induced by the Plücker embedding of the Grassmannian of lines in ℙ 4 . Thus, the isomorphism f must preserve the induced ample class given by the Plücker embedding. The trick in [41,Proposition 4] applies and thus f induces an isomorphism X 1 ≅ X 2 .
One interesting consequence is the following.

Corollary 6.5 Question 2.8 has a positive answer for cubic threefolds.
Proof Let X 1 and X 2 be cubic threefolds with an exact equivalence ∶ Ku(X 1 ) → Ku(X 2 ) . By Theorem 6.4, there is an isomorphism f ∶ X 1 → X 2 and, as observed in the proof of the result above, f yields an exact equivalence between the Kuznetsov components. For the same reason as in Remark 4.6 (i), such an equivalence is of Fourier-Mukai type.
Let us conclude this section with the following question: Question 6.6 Is it possible to prove Theorem 6.4 by using the extension techniques that we discussed in Section 4.1?

Inducing stability conditions
In this section we aim at explaining the methods introduced in [14], which allow us to induce stability conditions on the Kuznetsov component of prime Fano threefolds defined in Section 3.3, so in particular of cubic threefolds, and, with a dimension reduction trick as in Section 6.1, of cubic fourfolds. Then we introduce the notion of Serre-invariant stability conditions, whose existence in the context of cubic threefolds will have interesting consequences on the study of moduli spaces. Let T be a proper triangulated category which is linear over an algebraically closed field and with Serre functor T . Assume T has an exceptional collection E 1 , … , E m . Then we have a semiorthogonal decomposition of the form where D ∶= ⟨E 1 , … , E m ⟩ ⟂ . Note that rk(N(D)) = rk(N(T)) − m by additivility of the numerical Grothendieck group. This could vaguely suggest that D has "smaller dimension" than T , so it might be easier to construct Bridgeland stability conditions on D than on T .
In fact, the following key result provides a criterion which guarantees that a weak stability condition on T restricts to a Bridgeland stability condition on D.
Then the pair � = (A � , Z � ) is a Bridgeland stability condition on D.
Note that, if the weak stability condition in the statement is with respect to a lattice Λ , then ′ is defined over the sublattice Λ � of Λ determined by the image of K(A � ).
We point out that the conditions (1),(2) are used to show that the restriction A � = A ∩ D is the heart of a bounded t-structure on D , since they imply that the cohomology factors T = ⟨D, E 1 , … , E m ⟩ with respect to A of an object in D belong to D as well. By (4) the restriction Z ′ is a stability function on A ′ . For the proof of the Harder-Narasimhan property and the support property the authors make use of the notions of Harder-Narasimhan polygon and mass of an object in the heart. We suggest the interested reader to consult [14, Section 5] for more details.
Remark 6. 8 We should not expect Propsition 6.7 to be applicable in all situations where D b (X) has semiorthogonal decomposition with a Kuznetsov component which is residual to finitely many exceptional objects. One crucial example is when X is an Enriques surface with a semiorthogonal decomposition as in (3.1). Indeed, more should be true: if that X is very general, then Ku(X, L ) should not carry Bridgeland stability conditions at all. The heuristic reason is the following. The objects S i ∈ Ku(X, L ) in (3.2) are, at the same time, 3-spherical and numerically trivial. The latter condition implies that the stability function of any stability condition on Ku(X, L ) would map the class of S i to 0. On the other hand, motivated by the case of K3 surfaces, one would expect that, if a stability condition on Ku(X, L ) exists, then the fact that S i is spherical implies that it is also -stable. These two facts together would immediately lead to a contradiction and thus to the fact that such a cannot exist. Of course, it would be very interesting to make the previous argument rigorous and show that S i is stable in any stability condition on the Kuznetsov component.
An interesting consequence of Propsition 6.7 is the existence of Bridgeland stability conditions on the whole category T , using results in [43] about gluings of t-structures. When T = D b (X) , slope stability and tilt-stability, defined in Section 5.3, are weak stability conditions and we could ask whether they satisfy the conditions in Propsition 6.7. We will actually use the following tilting of s,q defined in Theorem 5.11. For ∈ ℝ , let u be the unit vector in the upper half plane with = − ℜ(u) ℑ(u) . We denote by the heart of a bounded t-structure obtained by tilting Coh s (X) with respect to s,q at s,q = . Set If we were working with Bridgeland stability conditions, the above process would correspond to consider u as an element of GL 1 (ℂ) ⊂ GL + 2 (ℝ) , lift it to ũ ∈ � GL + 2 (ℝ) and act on s,q by ũ . On the other hand, the G L + 2 (ℝ)-action is not a priori well-defined on weak stability conditions, because of the existence of objects with vanishing central charge. For instance, consider an extension E of the form where F is s,q -semistable in Coh s (X) with s,q (F) ≤ and T ∈ Coh(X) supported in codimension > 2 . Then E ∈ Coh s,q (X) , but E is not in Coh s (X) and thus cannot be s,q -semistable. This suggests that there could be objects which are semistable with respect to s,q , but not with respect to s,q (see [131,Lemma 5.4] for a concrete example). This prevents us to have a well-defined G L + 2 (ℝ)-action on these weak stability conditions. Nevertheless, we have the following result. We end this paragraph by introducing a notion which will be very useful in the case of prime Fano threefolds.

Remark 6.12
Note that the property of T -invariance is stable under the G L + 2 (ℝ)-action.

Remark 6.13
One may wonder why we use the word invariant in the previous definition, which would be more suitable to identify a stability condition which is fixed by the Serre functor. However, recall that stability conditions in the same orbit with respect to the G L + 2 (ℝ)-action have the same set of semistable objects. In particular, (if they exist!) the corresponding moduli spaces are isomorphic. From this viewpoint, we are interested in distinguishing or identifying stability conditions, up to the G L + 2 (ℝ)-action. Example 6.14 If X is a curve of genus g(X), then the slope stability H (see Example 5.8) is a Serre-invariant stability condition on D b (X) . Actually, if g(X) ≥ 1 , then there is a unique G L + 2 (ℝ)-orbit of stability conditions on D b (X) , so the condition in Definition 6.11 is trivially satisfied. Anyway, the fact that slope stability is Serre-invariant can be checked directly as follows. The Serre functor of D b (X) is Tensoring by the line bundle X preserves the slope stability of a coherent sheaf and X (Coh(X)) = Coh(X) [1] . On the level of central charges, since (rk( X ), deg( X )) = (1, 2g(X) − 2) , we have .
One could first wonder whether an admissible subcategory D of D b (X) of a smooth projective variety X admits Serre-invariant stability conditions. This has been recently proved to be false in [91], using the notion of Serre dimension, in the case of the Kuznetsov component of almost all Fano complete intersections. Anyway, we could focus on prime Fano threefolds and consider the following less general question: We will see in the next section that Question 6.15 has a positive answer for the Kuznetsov components of cubic threefolds and some other prime Fano threefolds. In these cases, we will also see that there is a unique G L + 2 (ℝ)-orbit of Serre-invariant stability conditions. Motivated by this, we could also ask the following: Question 6.16 Assume that D is an admissible subcategory of D b (X) which admits Serreinvariant stability conditions. If N(D) has rank 2, is there a unique orbit of Serre-invariant stability conditions with respect to the G L + 2 (ℝ)-action?
The question above is very much related to several expectations concerning the topology of the stability manifold of D . In particular, one my wonder whether such a manifold is connected. A positive answer to Question 6.16 may give a positive indication in the direction of this expectation as well.

Cubic threefolds: a modern view
Let us now apply the methods introduced in Section 6.2 to the Kuznetsov component of a cubic threefold X defined over an algebraically closed field .
Consider first the slope stability H = (Coh(X), Z H ) . Unfortunately, it does not satisfy the assumptions in Propsition 6.7 using the exceptional collection O X , O X (H X ) . Indeed, we see for instance that Let us try with the weak stability conditions s,q = (Coh s (X), Z s,q ) on D b (X) for (s, q) in the set Δ = (s, q) ∈ ℝ 2 ∶ q > 1 2 s 2 (see Section 5.3). Note that O X , O X (H X ) are slope stable, as they are line bundles, with slope 0 and 1, respectively. Choosing s < 0 , we have O X , O X (H X ) ∈ Coh s (X) . On the other hand, we observe that if s < −1 , then [2] . For later use, we point out that we have shown: Even if we cannot apply directly Propsition 6.7, this computation suggests that if we tilt another time, we could find a suitable heart. In fact, consider the weak stability condition s,q defined in Propsition 6.10. Proof By (6.5) we have that O X , O X (H X ) , O X (−2H X ) [1] , O X (−H X ) [1] belong to Coh s (X) for −1 ≤ s < 0 . Since they define points on the parabola q = 1 2 s 2 , by [11, Corollary 3.11(a)], they are s,q -stable for every (s, q) ∈ Δ . Comparing the slopes with respect to s,q via a picture as in Figure 2 (or by a direct computation), we deduce that for (s, q) as in the statement. Thus if satisfies (6.6), then O X , O X (H X ) , O X (−2H X ) [2] , O X (−H X ) [2] belong to Coh s,q (X) . Note that an object E ∈ Coh s,q (X) with Z s,q (E) = 0 is a torsion sheaf supported in codimension 3 by [14, Lemma 2.16], which is not in Ku(X) since Hom(O X , E) ≠ 0 . We can then apply Propsition 6.7 which proves that s,q | Ku(X) is a Bridgeland stability condition on Ku(X) with respect to Then s,q | Ku(X) is a numerical stability condition on Ku(X) and [3,13] (see, in particular, Proposition 25.3 in the latter paper) implies that it is a stability condition with moduli spaces. We leave to the reader the check of the second part of the statement, using [14,Lemma 4.3] and considering the rotation between two sides of the sharp angle on Figure 2 which is provided by the segment connecting the point (s, q) to O X and the segment which is the continuation of the one connecting O X (−H X ) to (s, q).
We denote the induced stability conditions on Ku(X) by where As we explain below, these stability conditions satisfy the nice property of being Serreinvariant, answering positively Question 6.15 for cubic threefolds.
From now on, we assume = ℂ as in the papers we are referring to, although the same proof should apply in more generality over an algebraically closed field of characteristic 0. where the first arrow is given by adjunction.
The second step is to show that the heart O X (Φ(A (− 1 2 , q, − 1 2 ))) is a tilt of A (− 1 2 , q, − 1 2 ) . This is the hardest part in the proof. One key ingredient is [102, Lemma 3], which allows us to control the slope of the semistable factors with respect to s ′ ,q ′ of a s,q -semistable object in Coh s (X) , when deforming (s, q) to (s � , q � ).
Finally, it is not difficult to show that there exists M ∈ GL + 2 (ℝ) such that ) . Then we can use a similar argument to the one in the proof of the second part of Theorem 6.17, to get the statement.
In this case we can also give a positive answer to Question 6.16, as a consequence of the following general criterion, which provides sufficient conditions in order for a fractional Calabi-Yau category to admit a unique G L

Remark 6.20
Note that [51, Theorem 3.2] also deals with the case r = 2 and k = 4 . In that situation, we need the additional condition that there are two objects Q 1 , Q 2 ∈ T satisfying (6.9) such that Q 1 is not isomorphic to Q 2 or Q 2 [1] , Hom(Q 2 , Q 1 ) ≠ 0 and Hom(Q 1 , The assumptions in the statement could seem unnatural at a first glance, but actually they are not. Indeed, in (1) we require T to be fractional Calabi-Yau of dimension < 2 , while in (2) we require N (T ) to have rank 2 as a noncommutative curve (although the analogy with curves is not totally correct, as for instance in one case the numerical Grothendieck group is negative definite, while in the latter it is not, as pointed out by the referee). Condition (3) is also inspired by the case of curves: the objects Q and T (Q) for T play a similar role as the one of the skyscraper sheaves and line bundles on a curve in Macrì's proof [109], e.g. they are stable with controlled phase.
We now see how to apply the above criterion to Ku(X) of a cubic threefold. Proof By Propsition 3.12 we know that Ku(X) is fractional Calabi-Yau of dimension 5∕3 < 2 . By [20, Proposition 2.7] and [82], the numerical Grothendieck group N (Ku(X)) is a rank-2 lattice and a basis is given by

3
where I is the ideal sheaf of a line in X, with respect to which the Euler form is represented by An easy computation shows that (v, v) ≤ −1 for every 0 ≠ v ∈ N (Ku(X)) , thus Ku(X) = −1 . Since dim Ext 1 (I , I ) = 2 , we see that Thus Theorem 6.19 applies to Ku(X) and implies the statement.
In the next section, we will see how to apply these results on Serre-invariant stability conditions to study the properties of moduli spaces of stable objects in Ku(X) and to reprove the Categorical Torelli theorem.

Cubic threefolds: applications
Assume = ℂ . The first application of the results presented in the previous section concerns the relation between the stability conditions (s, q, ) , defined in (6.7), and the stability condition 0 , introduced previously in Theorem 6.2. Similarly to Theorem 6.18, we can show the following statement.
, the pair is a stability condition on Ku(X) which is in the G L + 2 (ℝ)-orbit of 0 (see Figure 3).
An interesting consequence of these results is that all the known stability conditions on Ku(X) are Serre-invariant, hence belong to the same G L + 2 (ℝ)-orbit. A positive answer to the following (hard) question would complete the analogy of Ku(X) with the curve case in [109].

Question 6.25
Let be a stability condition on Ku(X) of a cubic threefold X. Is in the same orbit of (s, q, ) ? Equivalently, is Serre-invariant?
The second application, which makes use of Corollary 6.23, concerns the study of Ulrich bundles on X, whose definition is recalled in Remark 6.3. Note that an Ulrich bundle E belongs to Ku(X) . Indeed, by [

the definition of Ulrich bundles implies that
By using the embedding of Ku(X) in D b (ℙ 2 , B 0 ) , the following existence result has been proved. Theorem 6.26 ([92], Theorem B) The moduli space of stable Ulrich bundles of rank d ≥ 2 on X is non-empty and smooth of dimension d 2 + 1.
The property of Serre-invariance of 0 and (s, q, ) and the uniqueness result allows us to prove the irreducibility of the moduli space of Ulrich bundles. where the latter is the moduli space of (s, q, )-semistable objects in Ku(X) with numerical class d[I ] (see Section 5.1). Now using that (s, q, ) is Serre-invariant and that 0 is in the same G L + 2 (ℝ)-orbit of (s, q, ) , we have the isomorphisms between moduli spaces The last step is to show that M 0 (Ku(X), dv) is identified to the moduli space d of Gieseker semistable sheaves in Coh(ℙ 2 , B 0 ) with class dv. A standard argument (see [92,Theorem 2.12], [70,Lemma 4.1]) whose key point is the vanishing of Ext 2 for sheaves in this moduli space, allows us to show that d is irreducible, and thus U d is such (see [51,Proposition 6.4] for more details). Proof Let E be a -stable object in Ku(X) . Up to shifting, we can assume that E is in the heart of . Since is Serre-invariant, we have that Ku(X) (E) is -stable. By using the relation 3 Ku(X) = [5] , it is not difficult to show (see [129,Lemma 5.9]) that the phases with respect to satisfy As a consequence, by Serre duality we have is identified with the tangent space to the moduli space at the point corresponding to E and the obstruction is given by Ext 2 (E, E) (see [93,125,Lemma 4.8]) which is vanishing, we conclude that the moduli space M (Ku(X), [E]) , where [E] ∈ N (Ku(X)) , is smooth of dimension 1 − (E, E) at the point corresponding to E.
Last but not least, we focus on the Categorical Torelli theorem. Using the same strategy as in Theorem 6.4, it is possible to reprove this theorem by using any Serre-invariant stability condition and by applying the modular description of the Fano surface of lines by Theorem 6.28 (see [129,Theorem 5.17]). We end this section by explaining an alternative beautiful proof, given in [8], which is based on a description of the desingularization of the theta divisor of the intermediate Jacobian of X as a moduli space of stable sheaves on X, and then of semistable objects in Ku(X).
Recall that the intermediate Jacobian of X is the complex torus It has the structure of a principally polarized abelian variety of dimension 5 and plays a key role in the seminal paper [42] for the proof of the (classical) Torelli Theorem and the nonrationality of X. If T denotes the closure in the Hilbert scheme of the subvariety parametrizing twisted cubic curves in X, then we have the Abel-Jacobi map ∶ T → J(X) defined by (t) = t − H 2 X (which is an element of J(X) through the cycle class map). Then by [16,Proposition 4.2] the image of in J(X) is a theta divisor Θ ⊂ J(X) and its generic fiber is isomorphic to ℙ 2 . In fact, the linear span of a twisted cubic C is ⟨C⟩ ≅ ℙ 3 , so C is contained in the cubic surface S = ⟨C⟩ ∩ X for C general. Then the generic fiber of is the ℙ 2 of twisted cubic curves which are linearly equivalent to C on S.
Let M X (v � ) be the moduli space of Gieseker stable sheaves on X with Chern character 3 − H − the projection K x in Ku(X) of the skyscraper sheaf of a point x ∈ X has numerical class v ′ and it turns out to be -stable with respect to any Serre-invariant stability condition on Ku(X) , which motivates the choice of this particular moduli space to recover the isomorphism class of X. Consider the Abel-Jacobi morphism ∶ M X (v � ) → J(X) defined by (E) =c 2 (E) − H 2 X , where c 2 (E) corresponds to the second Chern class c 2 (E) of E via the cycle map. The key ingredient is the following result. The proof of the above theorem makes use of classical results on the properties of Θ together with techniques of wall-crossing to describe the objects in M X (v � ) . In particular, the embedding of X in M X (v � ) is obtained by sending a point x ∈ X to the projection K x in Ku(X) of the corresponding skyscraper sheaf. The objects K x sweep a divisor in this moduli space. Also note that the Abel-Jacobi map factors through .
The second key ingredient is the identification of M X (v � ) with a moduli space of semistable objects in Ku(X). ) has been fully described in [5,Theorem 1.2] (see also [8,Proposition 8.5]). The proof then follows by computing the image via Ku(X) of the objects.
As a consequence, the authors get a new proof of the Categorical Torelli Theorem 6.4 (see [8,Theorem 8

.1]):
Proof (New proof of Theorem 6.4) As usual, there is only one nontrivial implication to prove. Assume that there is an exact equivalence Φ ∶ Ku(X 1 ) ∼ � � � � � � � → Ku(X 2 ) (not necessarily of Fourier-Mukai type). Up to composing with a power of the Serre functor and shifting, we can assume Φ * (v � ) = v � , namely that Φ maps the numerical class of the projection K x 1 of the skyscraper sheaf of a point x 1 ∈ X 1 to the numerical class of K x 2 for a point x 2 ∈ X 2 , using that the Serre functor acts transitively on all classes that have the same square as v ′ with respect to (see [8,Lemma 8.3]). Since is Serre invariant and the Serre functor commutes with equivalences, we have that Φ ⋅ is Serre invariant. Theorem 6.31 applied two times gives the isomorphisms between moduli spaces By [8,Lemma 7.5] we have that X 1 and X 2 are the unique rationally connected divisors in M X 1 (v � ) and M X 2 (v � ) , respectively, which get contracted by any morphism to a complex abelian variety. Thus the above chain of isomorphisms implies X 1 ≅ X 2 .

More prime Fano threefolds
After the success of the above techniques in the case of cubic threefolds, it is natural to wonder whether we can get similar results for other prime Fano threefolds of index 1 and 2 (see Section 3.3 for the classification and properties). The goal of this section is to summarize the state of art in these cases concerning the existence and uniqueness of Serreinvariant stability conditions and the Categorical Torelli theorem. We assume the base field is = ℂ for simplicity.
Let us start by considering the index 2 case. We denote by Y d a prime Fano threefold of index 2 and degree 1 ≤ d ≤ 5 5 and take its Kuznetsov component The strategy of [14], reviewed in Section 6.3 for cubic threefolds, works more generally for Y d and allows one to construct stability conditions on Ku(Y d ) , which we denote by (s, q, ) . We have the following results: (1) By [129,Proposition 5.7] the stability conditions (s, q, ) on Ku(Y d ) are Serre-invariant for every 1 ≤ d ≤ 5.
The case d = 1 remains mysterious. Recall that Y 1 is a hypersurface of degree 6 in the weighted projective space ℙ(1, 1, 1, 2, 3) , known as the Veronese double cone. The Serre functor of Ku(Y 1 ) satisfies 3 Ku(Y 1 ) = [7] by [78,Corollary 4.2]. As a consequence, we cannot apply the criterion of Theorem 6.19 and the homological dimension of the heart of a Serreinvariant stability condition is 3. This makes this case quite different from the others. A remarkable difference is that this time the Fano surfaces of lines is an irreducible component of a moduli space of stable objects in Ku(Y 1 ) and in [131] the authors classify all the objects in this moduli space. We can then formulate the following questions: We now focus on the index 1 case and denote by X d such a prime Fano threefold of degree d = 2g − 2 , where 2 ≤ g ≤ 12 , g ≠ 11 . We first consider g ≥ 6 , so that the Kuznetsov component is defined by Ku(X d ) ∶= ⟨E r , O X d ⟩ ⟂ , where E r is an exceptional vector bundle of rank r on X d (see Section 3.3). Again by [14], there are stability conditions (s, q, ) on Ku(X d ) defined following the same procedure explained in Section 6.3 for cubic threefolds. In these cases we have the following results: (1) By [130,Theorem 3.18] [130,Corollary 4.5]) this uniqueness result holds also for Ku(X 10 ) . In the remaining cases, i.e. d = 12, 16 , we have Ku(X d ) is equivalent to the bounded derived category of a curve of genus 7 and 3, respectively. In particular, by (5.1) the stability manifold of Ku(X d ) is identified with the G L + 2 (ℝ)-orbit of the slope stability on the curve. Thus there is a unique orbit of Serre-invariant stability conditions and actually every stability condition is Serre-invariant, using that the action of any autoequivalence commutes with the action of G L + 2 (ℝ) . We conclude that for every d ≥ 10 , there exists a non-empty and unique orbit of Serre-invariant stability conditions.
(3) The Categorical Torelli theorem does not hold for X d in the form stated above. For instance, by [90] it is known that there are Gushel-Mukai threefolds with equivalent Kuznetsov components but which are not isomorphic (only birational). Nevertheless, a refined version of the Categorical Torelli theorem has been proved in [67] for Gushel-Mukai threefolds and more generally in [69] for every X d with d ≥ 10 , making use (among all) of the existence and uniqueness of Serre-invariant stability conditions. The precise statement is the following. Consider X d and X ′ d two prime Fano threefolds of index 1 and same degree d ≥ 10 . Denote by i ! and i � ! the right adjoints of the embedding functors In other words, (the other direction is trivial) the knowledge of i ! (E r ) is necessary and sufficient to reconstruct X d from Ku(X d ) for d ∈ {10, 12, 14, 16, 18, 22}. As noted in the introduction to [67], the Categorical Torelli theorem for cubic threefolds and the Kuznetsov's correspondence, discussed in item (2) above, imply that the birational class of X 14 is determined by its Kuznetsov component. Inspired by [90,Conjecture 1.7], we can formulate the following question, which is nothing but the Birational Categorical Torelli theorem for Gushel-Mukai threefolds: Question 6.34 ([67], Question 1.1(1)) Let X 1 and X 2 be prime Fano threefolds of index 1 and degree 10. Assume that there is an equivalence Φ ∶ Ku(X 1 ) ∼ � � � � � � � → Ku(X 2 ) (of Fourier-Mukai type). Is it true that X 1 and X 2 are birational?
As noted by the referee X 12 , X 16 , X 18 , X 22 are rational (over an algebraically closed field), so the above question is void in these cases. Furthermore, this question has been positively answered in [67, Theorem 1.5] for general ordinary Gushel-Mukai threefolds.
Let us consider the remaining Fano threefolds X d with d = 2, 4, 6, 8 , where Ku(X d ) ∶= ⟨O X d ⟩ ⟂ . Even if the construction in [14] still applies and there are stability conditions (s, q, ) on Ku(X d ) , the situation is far less understood. Note that N (Ku(X d )) has rank 3 in this case, so the stability manifold of Ku(X d ) has a bigger dimension than in the previous cases and Ku(X d ) does not look like a noncommutative curve. Up to now, we know the following results: (1) By [ , then X 4 is either a quartic threefold (see Remark 3.14) or a double cover of a quadric hypersurface Q in ℙ 4 ramified in the intersection of Q with a quartic. Assume X 4 belongs to the second class. Using the recent results in [91], we can compute explicitly the Serre functor of Ku(X 4 ) and show that the induced stability conditions (s, q, ) cannot be Serre-invariant (see [63]).
We can focus on the case of quartic threefolds and ask the following questions: In these notes we have seen many methods to show the Categorical Torelli theorem for the Kuznetsov components of Enriques surfaces and cubic threefolds. The first tentative to approach Question 6.36 could be trying to adapt one of these techniques to the case of quartic threefolds. This is the line of investigation we are following in the work in progress [63].

Cubic fourfolds
Let us now move back to the case when X is a cubic fourfold defined over a field which is algebraically closed with char( ) ≠ 2 . Let us recall from Section 3.4 that we have a semiorthogonal decomposition where H is a hyperplane class. The aim of this section is to show that Ku(X) carries stability conditions which we will use to prove both a categorical and a classical Torelli theorem for these Fano fourfolds. We conclude this section with a brief discussion on Gushel-Mukai fourfolds (see Section 7.3) which, quite surprisingly, are at the same time very close in spirit to cubic fourfolds but very different for some key features.

Stability conditions on the Kuznetsov component
In this section we want to prove that Ku(X) carries stability conditions. Ideally, we would like to apply the techniques discussed in Section 6.2 but what prevents us from a successful output is the fact that X has dimension 4 while the inducing strategy works perfectly fine for threefolds.
Thus the idea is to embed Ku(X) into the derived category of a new threefold whose construction is intimately related to the geometry of X as in Section 6.1.
More precisely, we follow [14,Section 7]. Now, as in the case of cubic threefolds, we pick a line ⊆ X which is not contained in a plane in X and we consider the projection ∶ X ⤏ ℙ 3 onto a skew 3-dimensional projective plane. We further consider the blow-up X ∶= Bl (X) of X in which makes the rational map into an actual morphism whose fibers are conics.
On the categorical side, the conic fibration ̃ yields a sheaf B 0 of even parts of Clifford algebras which is analogous to the one considered in Section 6.1 (in particular, it is noncommutative and generically Azumaya). As in Section 6.1 the fact that we can either view X as a conic fibration or as a blow-up, gives two fully faithful embeddings A direct comparison between the two inclusions shows that the embedding of the Kuznetsov component can be made compatible by mutations with the one of D b (ℙ 3 , B 0 ) . Hence, according to [14,Proposition 7.7], we get a semiorthogonal decomposition where E i is an exceptional B 0 -coherent sheaf. As in (6.4), we omit the explicit description of the embedding Ku(X) ↪ D b (ℙ 3 , B 0 ) which is indeed relevant for computations but useless for the purposes of this paper.

Remark 7.1
Despite the analogy between (6.4) for cubic threefolds and (7.1) for cubic fourfolds, the complexity of the twisted projective space (ℙ 3 , B 0 ) and of its derived category has secretly increased a lot. Indeed, while the numerical Grothendieck group of D b (ℙ 2 , B 0 ) has always rank 3 for all cubic threefolds (see [20,Proposition 2.12]), the rank of the numerical Grothendieck group of D b (ℙ 3 , B 0 ) varies when the cubic fourfold varies.
This is, indeed, not surprising because the Kuznetsov component behaves like a noncommutative curve (with a rather simple cohomology) for cubic threefolds while it behaves like a noncommutative K3 surface (hence with a rich cohomology) for cubic fourfolds.
In order to construct stability conditions on Ku(X) we are now in a good position to apply the dimension reduction trick described in Section 6.1: indeed, Ku(X) is now an admissible subcategory of the derived category of a (twisted) threefold with residue category generated by three exceptional objects.
To make this precise, we proceed as in the cubic threefold case and take the forgetful The twisted Chern character is then defined as where L is the class of a line in ℙ 3 . We denote by ch B 0 ,i the degree i component of ch B 0 . Since this is a cohomology class of ℙ 3 , it gets naturally identified with a rational number. 11 32 is needed to provide a Bogomolov inequality for the twisted derived category D b (ℙ 3 , B 0 ) . The role of such an inequality is to give the correct quadratic inequality in the support property for the Kuznetsov component. This is similar to (5.4). For the purposes of this paper the numerical correction above can be ignored.

Remark 7.2 The mysterious numerical correction
Next, we define and consider the lattice Λ B 0 ∶= Im( � ) . Finally, we denote by Λ B 0 ,Ku(X) the image of the composition of the natural maps K(Ku(X)) → K(D b (ℙ 3 , B 0 )) → Λ B 0 . The main result is then the following. It is worth pointing out that the result is actually more precise. Following Bridgeland's notation, a full numerical stability condition on Ku(X) is a stability condition on Ku(X) with respect to the lattice H Hodge (Ku(X), ℤ) ≅ N (Ku(X)) whose definition is recalled in Section 3.4 (see Remark 3.20). For later use, a Mukai vector is an element in the image of the restriction ∶ K(Ku(X)) →H Hodge (Ku(X), ℤ) of ′ . We denote by Stab m (Ku(X)) the set of full numerical stability conditions with moduli spaces on Ku(X) . Note that the twisted Chern character induces a natural map u ∶H Hodge (Ku(X), ℤ) → Λ B 0 ,Ku(X) . [14,Section 9], that if = (A, Z) is a stability condition constructed in Theorem 7.3, then the pair Ku(X) ∶= (A, Z•u) is in Stab m (Ku(X)) which is then nonempty. Theorem 7.3 and Propsition 6.9 yield the following Corollary 7.5 Let X be a cubic fourfold. Then D b (X) has a stability condition. Here A 2 is the lattice in (3.10). Let P + 0 be the connected component of P 0 which contains ( Ku(X) ) , for Ku(X) as in Example 7.4. In addition, let Stab † (Ku(X)) be the connected component of Stab m (Ku(X)) which contains Ku(X) .

Example 7.4 It is explained in
The complete result is then the following.

Categorical Torelli theorem
In this section we want to discuss the following result which is one of the main results of [60].
Theorem 7.7 (Categorical Torelli theorem for cubic fourfolds) Let X 1 and X 2 be cubic fourfolds. Then X 1 ≅ X 2 if and only if there is an equivalence Φ ∶ Ku(X 1 ) Here X i is the degree shift autoequivalence of Ku(X i ) described in Remark 3.18. Note that we do not require Φ being of Fourier-Mukai type. The assumption concerning the compatibility between the equivalence Φ and the degree shift functors X i is crucial. Indeed, it was proved in [127, Theorem 1.1], that given any positive integer N one can find N non-isomorphic cubic fourfolds with equivalent Kuznetsov components. This is another striking similarity with the case of K3 surfaces (see, for example, [140]). It is then natural to raise the following question (see [111,Question 3.25] and the discussion therein): Question 7.8 (Huybrechts) Does a Birational Categorical Torelli theorem for cubic fourfolds hold? Namely, let X 1 and X 2 be cubic fourfolds. Is it true that the existence of an equivalence Ku(X 1 ) ∼ � � � � � � � → Ku(X 2 ) implies that X 1 and X 2 are birational? ∶ Stab m (Ku(X)) →H Hodge (Ku(X), ℂ) This comes with the ring structure induced by the composition (see [60, Section 3.1]). Indeed, applying the convolution with P i , we have a natural map Hom(P 0 , P j ) → Hom(P i , P i+j ) and thus requiring that the multiplication is trivial if i + j > N . Standard arguments show that this endows L(Y) with the structure of a graded ring. We denote by L j (Y) the graded piece of degree j. We set HH * (Ku(Y), Y ) to be the graded subalgebra of L(Y) generated by L 1 (Y) 7 .

Idea of proof 2 (stability conditions)
We now outline the strategy of proof via stability conditions which is pursued in the appendix to [14] and, as explained later, allows one to get another proof of the classical Torelli theorem as well.
In this second proof, we can assume that is an algebraically closed field with char( ) ≠ 2 . As above, there is only one implication in the statement that needs to be proved.
Hom(P 0 , P i ) × Hom(P 0 , P j ) → Hom(P 0 , P i ) × Hom(P i , P i+j ) → Hom(P 0 , P i+j ) The strategy is very close in spirit to the one in Section 6.1 and Section 6.3 where we proved the Categorical Torelli theorem for cubic threefolds. Indeed, let X be a cubic fourfold and let F 1 (X) be the Fano varieties of lines in X. By [17], when = ℂ , the variety F 1 (X) is a 4-dimensional smooth and projective irreducible symplectic manifold (i.e. a simply-connected manifold such that H 0 (F 1 (X), Ω 2 F 1 (X) ) is generated by an everywhere nondegenerate holomorphic 2-form). In general, as F 1 (X) is embedded in the Grassmannian of lines in ℙ 5 , it carries a privileged ample polarization which is the restriction of the Plüker polarization. To shorten the notation we will refer to such a polarization on F 1 (X) as the Plücker polarization.
The key point is that one can interpret F 1 (X) as a moduli space of stable objects in the Kuznetsov component Ku(X) . This approach was initiated in [113] and pursued in [14,99]. Let X be a cubic fourfold and fix a stability condition ∈ Stab † (Ku(X)) such that ( ) ∈ (A 2 ⊗ ℂ) ∩ P ⊆ P + 0 . As we observed in (7.2) any stability condition constructed in the proof of Theorem 7.3 would work. By the general theory of moduli spaces of (semi)stable objects in the Kuznetsov component Ku(X) which we discussed in Section 5.1, we can take the moduli space M (Ku(X), v) for every Mukai vector v. By the results in [13], any such moduli space M (Ku(X), v) carries a natural ample polarization .

Remark 7.13
In [99, Theorem 1.1] the previous result has been generalized by showing that the Fano variety of lines of every cubic fourfold is isomorphic to a moduli space of -stable objects for as in Section 7.1. Now, let X 1 and X 2 be cubic fourfolds with an equivalence Φ ∶ Ku(X 1 ) ∼ � � � � � � � → Ku(X 2 ) commuting with the rotation functors. By Propsition 3.15, Φ is of Fourier-Mukai type. Then Φ induces a Hodge isometry between the Mukai lattices. It is a simple exercise, using our assumption X 2 •Φ = Φ• X 1 and [57, Proposition 3.12], to show that Φ H sends the A 2 -lattice of X 1 to the corresponding one of X 2 . Remark 7.14 It is clear that, for this argument, one can weaken the assumption Analogously, d preserved the property of not having (−2)-classes. We thus conclude that Def(X 1 ) ≅ Def(X 2 ) restricts to an isomorphism D 1 ≅ D 2 . Thus we can set D ∶= D 1 ≅ D 2 . As explained in the appendix to [14], the set D is dense. Moreover, for all d ∈ D , the Hodge isometry d can be lifted to an equivalence by the derived Torelli theorem for twisted K3 surfaces [62, Theorem 0.1]. By construction, the isometry Φ H d commutes with the action of the degree shifts functors in cohomology. Now, for d ∈ D , pick 1 ∈ Stab † (Ku(X 1,d )) such that ( 1 ) ∈ (A 2 ⊗ ℂ) ∩ P ⊆ P + 0 and set 2 ∶= Φ d ( 1 ) . By [58, Theorem 1] the stability manifold Stab † (Ku(X i,d )) has a unique connected component of maximal dimension. Since the action of Φ d on the stability manifolds exchanges components of the same dimension, it follows that the image 2 of 1 belongs to Stab † (Ku(X 2,d )) . Moreover, by definition, ( 2 ) ∈ (A 2 ⊗ ℂ) ∩ P as well. Thus we can apply Theorem 7.12 twice and obtain a string of isomorphisms The isomorphism in the middle is induced by Φ d and, as explained in [14, Appendix A], it sends the polarization 1 to 2 . Thus, by Theorem 7.12, the whole sequence of isomorphisms sends the Plücker polarization on F 1 (X 1,d ) to the Plücker polarization on F 1 (X 2,d ) . The proof then continues as in the one of Theorem 6.4 and it consists in applying Chow's trick (see [41,Proposition 4]) in order to conclude that X 1,d ≅ X 2,d for all d ∈ D . Since D is dense, separatedness of the moduli space of cubic fourfolds implies that X 1 and X 2 have to be isomorphic (since any open neighborhood of the point corresponding to X 1 in Def(X 1 ) intersects any open neighborhood of the point corresponding to X 2 in Def(X 2 ) , precisely in the points which belong to D). This ends the proof.

Gushel-Mukai fourfolds
We could wonder whether the techniques explained in the previous sections may be adapted to other classes of Fano fourfolds. This turns out to be true in the case of Gushel-Mukai fourfolds (more generally for Gushel-Mukai varieties of even dimension).
Recall that a general complex Gushel-Mukai (GM) fourfold X is a smooth four-dimensional intersection of the form where Gr(2, 5) is the Plücker embedded Grassmannian and Q is a quadric hypersurface in a hyperplane section of ℙ 9 . GM fourfolds share many similarities with cubic fourfolds. For instance, from a geometric viewpoint, there are known examples of rational GM fourfolds, but it is still unknown whether the very general one is irrational or rational. From the point of view of derived categories, Kuznetsov and Perry proved in [89] that the bounded derived category of X has a semiorthogonal decomposition of the form where U X is the restriction of the tautological bundle of rank 2 on the Grassmannian Gr (2,5) and Ku(X) ∶= ⟨O X , U ∨ X , O X (1), U ∨ X (1)⟩ ⟂ . The residual component Ku(X) is the Kuznetsov component of X and is a noncommutative K3 surface.
Full numerical stability conditions on Ku(X) have been constructed in [126], using a dimension reduction trick. More precisely, the authors show that X is birational to a conic fibration over a quadric threefold Y and then provide an embedding of Ku(X) in the bounded derived category D b (Y, B 0 ) , where (Y, B 0 ) is a twisted quadric threefold.
On the other hand, it is known that the Torelli Theorem does not hold for GM fourfolds. In fact, in this case the period map has four-dimensional fibers by [45]. Nevertheless, we can still wonder whether a (refined) Categorical Torelli theorem holds for GM fourfolds.
More precisely, note that GM fourfolds in the same fiber of the period map have equivalent Kuznetsov components by [90,Theorem 1.6]. On the other hand, by [32,Theorem 1.3] there are examples of GM fourfolds with equivalent Kuznetsov components, but defining different period points. These considerations suggest that we have to impose some additional conditions to an equivalence between the Kuznetsov components to recover the period point or the isomorphism class of a GM fourfold.
Recall that the degree shift functor of Ku(X) is defined by where ⟨O X ,U ∨ X ⟩ is the left mutation in ⟨O X , U ∨ X ⟩ (see, for example, [78,Section 2] for the precise definition). We can formulate the following question: Question 7.16 Let X 1 and X 2 be Gushel-Mukai fourfolds. Assume that there is an equivalence Φ ∶ Ku(X 1 ) ∼ � � � � � � � → Ku(X 2 ) which commutes with the degree shift functors of X 1 and X 2 . Then under which assumptions on Φ we have that X 1 and X 2 are isomorphic?
To address this question, it may be helpful to use the stability conditions defined on Ku(X) and the associated moduli spaces having the structure of hyperkähler manifolds [126], as done for cubic fourfolds in the last part of Section 7.2.
Note that if X 1 and X 2 are very general, then the above statement holds for every exact equivalence Φ ∶ Ku(X 1 ) ∼ � � � � � � � → Ku(X 2 ) . This can be proved exactly as in [57,Corollary 3.6]. Indeed, by [98] every equivalence as above is of Fourier-Mukai type. Then by [57, Corollary 3.5], it induces a Hodge isometry Φ H between the Mukai lattices of Ku(X 1 ) and Ku(X 2 ) . Note that the condition of being very general means that the algebraic part of the Mukai lattice of Ku(X i ) is generated by two classes 1 , 2 (see [89] and [128,Lemma 2.4] for the precise definitions). As a consequence, the Hodge isometry Φ H restricts to a Hodge isometry H 4 van (X 1 , ℤ) ≅ H 4 van (X 2 , ℤ) . Of course, we cannot say X 1 and X 2 are isomorphic (and in general they are not), since there is no Torelli theorem.
We finally recall the following conjecture about Birational Categorical Torelli theorem for Gushel-Mukai varieties. The same comment in Remark 7.9 applies to this setting.