Kuznetsov's Fano threefold conjecture via K3 categories and enhanced group actions

We settle the last open case of Kuznetsov's conjecture on the derived categories of Fano threefolds. Contrary to the original conjecture, we prove the Kuznetsov components of quartic double solids and Gushel-Mukai threefolds are never equivalent, as recently shown independently by Zhang. On the other hand, we prove the modified conjecture asserting their deformation equivalence. Our proof of nonequivalence combines a categorical Enriques-K3 correspondence with the Hodge theory of categories. Along the way, we obtain a categorical description of the periods of Gushel-Mukai varieties, which we use to resolve a conjecture of Kuznetsov and the second author on the birational categorical Torelli problem, as well as to give a simple proof of a theorem of Debarre and Kuznetsov on the fibers of the period map. Our proof of deformation equivalence relies on results of independent interest about obstructions to enhancing group actions on categories.


Introduction
We work over the complex numbers.If V is a Fano 1 threefold of Picard number 1 with ample generator H ∈ Pic(V ), then the index of V is the integer i such that K V = −iH and the degree is the integer d = H 3 .The classification of Fano threefolds [IP99] shows that if i = 4 then V ∼ = P 3 , if i = 3 then V is a quadric, if i = 2 then 1 ≤ d ≤ 5, and if i = 1 then d is even, d = 20, and 2 ≤ d ≤ 22.Moreover, for any pair (i, d) satisfying these restrictions, there is a unique and explicitly described deformation class of Fano threefolds with these numerics.For instance: • Fano threefolds Y of Picard number 1, index 2, and degree 2 are quartic double solids, i.e.
double covers Y → P 3 branched along a quartic surface.• Fano threefolds X of Picard number 1, index 1, and degree 10 are Gushel-Mukai (GM) threefolds, i.e. either intersections X = Gr(2, 5) ∩ P 7 ∩ Q of the Grassmannian Gr(2, 5) with a codimension 2 linear subspace and a quadric in the Plücker embedding (in which case X is called ordinary), or double covers X → Gr(2, 5) ∩ P 6 of a codimension 3 linear section branched along a quadric section (in which case X is called special).
There are some curious classical "coincidences" between the families with numerical invariants (i = 2, d) and (i = 1, 4d + 2).For instance, the rationality of a (generic) Fano threefold in a given family is preserved under this correspondence (see [Bea16]).At the level of Hodge theory, the dimensions of the intermediate Jacobians on each side also match, except for d = 1 (see Remark 1.3 below).
1 Fano varieties are smooth by convention in this paper.
degeneration and resolution: there exists a smooth proper family of categories with generic fiber Ku(X) and special fiber a categorical resolution of Ku(Y ) for a nodal Y .
Bernardara and Tabuada [BT16] observed that Conjecture 1.1 also fails for d = 2, essentially for dimension reasons: the categories Ku(X) of GM threefolds vary in a 20-dimensional family, while M 2 2 is only 19-dimensional.Thus, if Z ⊂ M 2 2 × M 1 10 is a correspondence parameterizing Fano threefolds with equivalent Kuznetsov components, then Z does not dominate M 1 10 .This left open the question of whether there could be such a correspondence dominating M 2 2 , as suggested by the dimension count.
1.2.Main results.Our first main result says that, somewhat surprisingly, there does not even exist a nonempty correspondence Z ⊂ M 2 2 × M 1 10 parameterizing equivalent Kuznetsov components, and thus Conjecture 1.1 fails maximally for d = 2.
Theorem 1.4.Let Y be a quartic double solid, and let X be a GM threefold.Then Ku(Y ) and Ku(X) are not equivalent.
Remark 1.5.This result was also recently shown by Zhang [Zha21], via a completely different method, using uniqueness of (Serre-invariant) Bridgeland stability conditions and moduli spaces of stable objects.
In view of the failure of Conjecture 1.1 for d = 2, Kuznetsov suggested a weakening of the conjecture, which asserts that the categories Ku(Y ) and Ku(X) are "deformation equivalent".Theorem 1.4 can be thought of as a negative result in this direction, as the simplest way the modified conjecture could be true is if Ku(Y ) ≃ Ku(X) for some (Y, X) ∈ M 2 2 × M 1 10 .Nonetheless, our second main result confirms Kuznetsov's modified conjecture.
Theorem 1.6.There exists a smooth pointed curve (B, o) and a smooth proper B-linear category C such that: (1) The fiber C o is equivalent to Ku(Y ) for a quartic double solid Y .
(2) For b ∈ B \ {o}, the fiber C b is equivalent to Ku(X b ) for a GM threefold X b .
We expect this result to be a useful tool for relating Bridgeland moduli spaces of objects in Kuznetsov components of quartic double solids and GM threefolds.
Remark 1.7.See §2.1 for a summary on B-linear categories.In general, a B-linear category should be thought of as a "family of categories parameterized by B"; there is a well-behaved notion of base change for such categories, which in particular gives rise to a κ(b)-linear fiber category C b for any b ∈ B. Theorem 1.6 thus informally says that the Kuznetsov components of GM threefolds smoothly specialize to those of quartic double solids.
Our method of proof of Theorem 1.4 naturally leads to our third main result, concerning the "categorical Torelli problem".Namely, the intermediate Jacobian of a Fano threefold X is determined by its Kuznetsov component [Per20], and hence the association X → Ku(X) can be thought of as a categorical lift of the period map.The categorical Torelli problem then asks to what extent X is determined by Ku(X).Positive answers are known in many situations; see [PS22] for a recent survey and references.One particularly interesting open case is that of GM threefolds, for which the 3-dimensional case of a conjecture of Kuznetsov and the second author predicts the following.
We note that by the duality conjecture [KP18, Conjecture 3.7] proved in [KP19], there are indeed 2-dimensional families of birational GM threefolds with equivalent Kuznetsov component.More precisely, [DK18] introduces a notion of period partnership and duality for GM varieties (see §5.3 for definitions), shows that these relations imply birationality, and explicitly describes the locus of period partners and duals of a given GM variety in terms of an associated EPW sextic (the locus being 2-dimensional for a GM threefold), while the duality conjecture implies that the Kuznetsov components of period partners or duals are equivalent.We show that this is in fact the only way for GM threefolds to have equivalent Kuznetsov components, and therefore resolve Conjecture 1.8 while simultaneously computing the fiber of the "categorical period map": Theorem 1.9.Let X 1 and X 2 be GM threefolds.Then Ku(X 1 ) ≃ Ku(X 2 ) if and only if X 1 and X 2 are period partners or duals.In particular, if Ku(X 1 ) ≃ Ku(X 2 ), then X 1 and X 2 are birational.
One of the appeals of Theorem 1.9 is that the expected corresponding result for the ordinary period map, i.e. with Kuznetsov components replaced by intermediate Jacobians, is currently unknown.This illustrates the utility of the extra structure provided by working categorically.
Remark 1.10.Under a genericity assumption, the birationality of X 1 and X 2 above was recently shown in [JLLZ22], via a completely different method, and without the genericity assumption in the upcoming [JLZ22].
The proofs of Theorems 1.4, 1.6, and 1.9 involve ideas of independent interest, sketched below.
Categorical Enriques-K3 correspondence.The Kuznetsov components of quartic double solids and GM threefolds are Enriques categories, in the sense that their Serre functors are of the form τ • [2] where τ is a nontrivial involution generating a Z/2-action.To any Enriques category, there is an associated 2-Calabi-Yau (CY2) cover, defined as the invariant category for the Z/2-action.By [KP17], if Y → P 3 is a quartic double solid with branch locus a quartic K3 surface Y br ⊂ P 3 , then the CY2 cover of Ku(Y ) is D b (Y br ), while if X is a GM threefold, the CY2 cover of Ku(X) is the Kuznetsov component of the "opposite" GM variety X op (a Fano fourfold if X is ordinary or a K3 surface if X is special, see Definition 4.12); these CY2 categories are called K3 categories because their Hochschild homology agrees with that of a K3 surface.
As reviewed in §4, the CY2 cover of an Enriques category admits a residual Z/2-action, which should be thought of as an analogue of the covering involution of a K3 surface over an Enriques surface; in the case of D b (Y br ) and Ku(X op ) these actions can be described explicitly, see Theorem 4.15.One of our key observations is that two Enriques categories are equivalent if and only if their CY2 covers are Z/2-equivariantly equivalent (Lemma 4.9).This is useful as K3 categories are often easier to understand.
Outline of the proof of Theorem 1.4.In particular, Theorem 1.4 reduces to proving the nonexistence of a Z/2-equivariant equivalence D b (Y br ) ≃ Ku(X op ).To rule out such an equivalence, we study the induced Z/2-equivariant isometry H(Y br , Z) ∼ = H(Ku(X op ), Z) between their Mukai Hodge structures, whose definitions are reviewed in §5.1.This leads to a contradiction to constraints on the periods of GM fourfolds when X is ordinary (Theorem 7.1), and those of GM surfaces when X is special (Lemma 7.3).
Enhanced group actions.The above discussion elided a subtlety about Z/2-actions.In general, if G is a finite group, then there are several possible notions of an action of G on a category C. Naively, one might consider a homomorphism φ from G to the group of autoequivalences modulo isomorphisms of functors.However, more structure is needed to define a reasonable category C G of G-equivariant objects in C; namely, following Deligne [Del97], we need to specify suitably compatible isomorphisms of functors φ(g) • φ(g ′ ) ∼ = φ(g • g).This suffices if C is an ordinary category, but if C is triangulated then in general C G need not be (see [Ela15,Theorem 6.9] for a sufficient condition).
To correct this, we instead work with an enhanced triangulated category C -we use ∞categorical enhancements, see §1.6 -and consider ∞-categorical group actions on C. Then there is a well-behaved category C G of invariants, but the price we pay is that it is a priori much harder to specify a group action of G on C, as it requires an infinite hierarchy of data.At the first two levels, if hC denotes the triangulated homotopy category of C, then an ∞categorical action on C determines both a naive G-action on hC given by a homomorphism φ as above, as well as a G-action on hC in the sense of Deligne; we call the former a 1-categorical action on C, and the latter a 2-categorical action.We study obstructions to and uniqueness of lifts of 1-and 2-categorical actions to ∞-categorical ones.In particular, we show that if C satisfies a connectivity hypothesis on Hochschild cohomology (which holds for most categories of interest), then given a 1-categorical action there is a single obstruction to the existence of an ∞-categorical lift, and the set of lifts form a torsor over an explicit cohomology group (Corollary 3.4); moreover, this obstruction and torsor are the exact same as those controlling 2-categorical lifts.This in particular answers a question raised in [Ela15].In the special case where C is the derived category of a variety, the result is the following enhancement of [BO20, Theorem 2.1].
Theorem 1.11.Let X be a connected smooth proper variety over a field k.Let G be a finite group with a group homomorphism φ to the group of autoequivalences of D b (X).Then there is a canonical obstruction class ob(φ) ∈ H 3 (BG, k × ), where the G-action on k × is trivial, such that an ∞-categorical lift of φ exists if and only if ob(φ) = 0, in which case the set of equivalence classes of such lifts is an H 2 (BG, k × )-torsor.
The complete result in Corollary 3.4 applies in the relative setting where we consider categories C that are linear over a base scheme, instead of merely a field.An important technical result is that the vanishing of the obstruction mentioned above is an open condition in the étale topology of the base (Proposition 3.9).
Outline of the proof of Theorem 1.6.By our discussion above, if X is a special GM threefold and Y is a quartic double solid, then the CY2 covers of their Kuznetsov components are D b (X op ) and D b (Y br ), where X op is a GM K3 surface and Y br is a quartic K3 surface, and the Kuznetsov components can be recovered as the invariant categories for the residual Z/2actions.The idea of the proof of Theorem 1.6 is thus to find a specialization of X op to a quartic K3, with a Z/2-action that restricts on fibers to the residual Z/2-actions.To do so, we first construct such a specialization with a 1-categorical action on the family of derived categories of the K3 surfaces, and then use the general results discussed above to lift this to an ∞-categorical action.Passing to invariant categories gives the category C promised by Theorem 1.6, which is smooth and proper by a general result (Proposition 3.15) that we prove.
The role of derived algebraic geometry.We briefly explain the role played by derived algebraic geometry, more specifically stable ∞-categories, in this paper.We need the notion of a category linear over a base B, along with base change.Sometimes, one considers such categories as admissible B-linear subcategories of D perf (X) for a scheme X over B; this is e.g. the approach taken in [BLM + 21].However, we know of no such embedding of the category C in Theorem 1.6.
Instead, as explained above C is constructed as the invariant category for a Z/2-action on the derived category D perf (S) of the total space S of a family of K3 surfaces over B. A result of Elagin [Ela15, Corollary 6.10] allows to construct from a 2-categorical action on D perf (S) a triangulated structure on the invariant category, but it does not come with a natural B-linear structure that satisfies base change.Instead, our results in Section 3 show that the Z/2-action on the triangulated D perf (S) lifts to an action on its enhancement as a ∞-category; then the desired properties of the invariant category are automatic.
Categorical description of periods.As a byproduct of our proof of Theorem 1.4, we obtain a categorical description of the periods of even-dimensional GM varieties.GM varieties are generalizations of GM threefolds to dimensions 2 ≤ n ≤ 6, with similarly defined Kuznetsov components (see Definition 4.11 and (4.1)).If W is such a variety of dimension 4 or 6, then Ku(W ) is canonically the CY2 cover of an Enriques category (generalizing the discussion above for W = X op ), and hence carries a canonical residual Z/2-action.If n = dim(W ), the period map assigns to W the Hodge structure H n (W, Z) 0 given as the orthogonal to the sublattice H n (Gr(2, 5), Z) ⊂ H n (W, Z).Proposition 1.12.Let W be a GM variety of dimension n = 4 or 6.Let H(Ku(W ), Z) 0 denote the orthogonal to the invariant sublattice H(Ku(W ), Z) Z/2 ⊂ H(Ku(W ), Z) for the residual Z/2-action.Then there is an isometry of weight 2 Hodge structures , where ( n 2 − 1) on the right denotes a Tate twist.Our main application, to Theorem 1.9, is explained below.As another application, we give a simple proof of a recent result of Debarre and Kuznetsov [DK19], which identifies the periods of even-dimensional GM varieties that are "generalized partners or duals" (Theorem 5.11), and implies the period map factors through the moduli space of double EPW sextics.
Remark 1.13.In [Per20], canonical weight 0 and −1 Hodge structures K top 0 (A) and K top 1 (A) are constructed for any admissible subcategory A of the derived category of a smooth proper variety, which can be thought of as versions of even and odd degree cohomology of A. If A = Ku(W ) for an even-dimensional GM variety, then K top 0 (A) is up to Tate twist the Mukai Hodge structure.For many odd-dimensional Fano varieties (including GM varieties), if A is taken to be an appropriate Kuznetsov component, then K top 1 (A) recovers the middle Hodge structure of W on the nose.In this way, the categorical description of periods in even dimensions is more subtle than in odd dimensions.
Outline of the proof of Theorem 1.9.If X 1 and X 2 are GM threefolds with Ku(X 1 ) ≃ Ku(X 2 ), then passing to CY2 covers we obtain an equivalence Ku(X op 1 ) ≃ Ku(X op 2 ) equivariant for the residual Z/2-actions.By a trick one can reduce to the case where X 1 and X 2 are ordinary, so that X op 1 and X op 2 are GM fourfolds.Then Proposition 1.12 implies an isometry of Hodge structures H 4 (X op 1 , Z) 0 ∼ = H 4 (X op 2 , Z) 0 .Theorem 1.9 then follows by combining this with the factorization of the period map through the moduli space of double EPW sextics (mentioned above) and the injectivity of the period map for double EPW sextics (by Verbitsky's Torelli theorem).The moral of this argument is that passing to CY2 covers allows us to leverage Torelli theorems for hyperkähler fourfolds.
1.3.Further conjectures and questions.We highlight several further directions suggested by our work.
Birational geometry and intermediate Jacobians of the threefolds.Heuristic relations between derived categories and birational geometry [Kuz16] suggest that if Y were a quartic double solid which is birational to a GM threefold X, then Ku(Y ) ≃ Ku(X).Together with Theorem 1.4, this leads to: Conjecture 1.14.Let Y be a quartic double solid, and let X be a GM threefold.Then Y is not birational to X.

Motivated by this, we also propose:
Conjecture 1.15.Let Y be a quartic double solid, and let X be a GM threefold.Then the intermediate Jacobian J(Y ) is not isomorphic to J(X) as a principally polarized abelian variety.
We note that J(X) and J(Y ) are both 10-dimensional.One could hope to address Conjecture 1.15 by proving a description for the singular locus of the theta divisor of J(X) in terms of Bridgeland moduli spaces for Ku(X).Our main interest in Conjecture 1.15 is that it explains both Theorem 1.4 and Conjecture 1.14.
Conjecture 1.15 =⇒ Conjecture 1.14.For a Fano threefold W , the Clemens-Griffiths component J CG (W ) -defined as the product of the principally polarized factors of J(W ) that are not Jacobians of curves -is a birational invariant [CG72].It follows from [Voi88] that Theorem 1.17.For j = 1, 2, let C j be a smooth, proper, connected S j -linear category for a scheme S j .Then the locus In §2, we review the notion of connectedness for a linear category, and explain a useful criterion (Corollary 2.11) for checking it based on [Kuz15].The criterion applies to the Kuznetsov components of many Fano varieties, including those of Fano threefolds of Picard number 1 2 2 See [BLMS21, §6] for a (somewhat ad hoc) definition of the Kuznetsov component without restrictions on the degree.and of cubic fourfolds [Kuz10].By Theorem 1.17 the locus where connected Kuznetsov components become equivalent in the product of two moduli spaces is a countable union of locally closed subspaces.We believe such loci deserve further study.For instance: Question 1.18.When do equivalences between Kuznetsov components specialize, i.e. when are the locally closed subspaces above actually closed?
This specialization property holds in the setting of Conjecture 1.1: for d = 3, 4, 5 by the results of [Kuz09] combined with the categorical Torelli theorems for Fano threefolds Y of Picard number 1, index 2, and degree d = 3, 4, 5; for d = 2 by Theorem 1.4; and for d = 1 by Remark 1.3.For cubic fourfolds, the results of [BLM + 21] give a partial answer to Question 1.18: they imply that derived equivalences between Kuznetsov components of cubic fourfolds and K3 surfaces specialize, as they are given by 2-dimensional moduli spaces of stable objects.

Related work.
There are alternative approaches to some of our results: Theorem 1.4 was proved independently by Zhang in [Zha21], based on a study of Bridgeland moduli spaces, while Theorem 1.6 was proved independently by Kuznetsov and Shinder in work in preparation [KS22] (see also [Kuz21,§5.4]),based on a degeneration argument and a theory of "absorption of singularities".Our paper and these two use completely different methods, which we believe are interesting in their own right.
After posting the first version of this paper, we learned that Zoë Schroot has obtained results similar to ours in §3 on enhancing group actions on categories.
We also note that some of the ideas in this paper are used in the upcoming [PPZ21] to describe Bridgeland moduli spaces for Enriques categories, like the Kuznetsov components of GM threefolds or quartic double solids.1.5.Organization of the paper.In §2 we review the formalism of enhanced triangulated categories linear over a base scheme, as well as their Hochschild cohomology.In §3 we discuss ∞-categorical group actions on (linear) categories, and in particular study the obstruction to lifting an action on the homotopy category to the ∞-level.In §4 we explain the correspondence between Enriques categories and their CY2 covers.In §5 we review Mukai Hodge structures and prove Proposition 1.12.In §6 we prove Theorem 1.9, in §7 we prove Theorem 1.4, and finally in §8 we prove Theorem 1.6.1.6.Conventions.Schemes are tacitly assumed to be quasi-compact and quasi-separated.A variety over a field k is an integral scheme which is separated and of finite type over k.Fano varieties are smooth by convention.
For a scheme X, D perf (X) denotes the category of perfect complexes, D qc (X) denotes the unbounded derived category of quasi-coherent sheaves, and D b (X) denotes the bounded derived category of coherent sheaves.(In fact in all cases where we consider D b (X) in this paper, D b (X) = D perf (X), so it is just a matter of notation.)All functors are derived by convention.In particular, for a morphism f : X → Y of schemes we write f * and f * for the derived pushforward and pullback functors, and for E, F ∈ D perf (X) we write E ⊗ F for the derived tensor product.
For technical convenience all categories in the paper are considered as enhanced categories.More precisely, instead of k-linear triangulated categories we consider k-linear categories and functors between them in the sense of [Per19a], i.e. we consider small idempotent-complete stable ∞-categories equipped with a module structure over D b (Spec(k)) (or equivalently, when char(k) = 0, DG categories over k).In particular, for a variety X over k we regard D b (X) as such a category; its homotopy category hD b (X) is the classical triangulated derived category.We note that giving a semiorthogonal decomposition of D b (X) is equivalent to giving one of hD b (X), and by the results of [BZFN10] and [Kuz11] if C ⊂ D b (X) and D ⊂ D b (Y ) are semiorthogonal components of smooth proper varieties, then any functor C → D of k-linear categories is induced by a Fourier-Mukai kernel on X × Y .
If C is a k-linear category, we use the notation Hom k (E, F ) ∈ D qc (Spec(k)) for the mapping object between objects E, F ∈ C, see §2.1; in case C ⊂ D b (X) is a semiorthogonal component of the derived category of a variety, then Hom k (E, F ) coincides with the classical derived Hom complex RHom(E, F ).
In several places, we also need the general notion of categories linear over a base scheme, briefly reviewed in §2.1.
As stated in the introduction, our main results are over the complex numbers, and correspondingly in §5- §8 we work in this setting.However, in the foundational part of the paper, §2- §4, we work over more general bases, as explained there.1.7.Acknowledgements.We thank Sasha Kuznetsov, Laura Pertusi, and Xiaolei Zhao for helpful discussions about this work.We are especially grateful to Bhargav Bhatt for explaining to us the proof of Lemma 3.10 and suggesting Example 3.12, and to Sasha Kuznetsov for carefully reading a preliminary version of this paper.We also thank the referee for their careful reading.

Hochschild cohomology of linear categories
In this section we discuss categories linear over a base scheme and their Hochschild cohomology.In §2.1 we recall some of the basic formalism of linear categories, in §2.2 we define Hochschild cohomology and review some of its properties, and in §2.3 we define the notion of connectedness of a linear category (which appears as a hypothesis in Theorem 1.17) and explain a convenient method for checking it in practice.
2.1.Linear categories.Fix a base scheme S. We use the notion of S-linear categories as in [Per19a].Namely, the derived category D perf (S) is a commutative algebra object in the ∞-category of small idempotent-complete stable ∞-categories, and an S-linear category is a module object over D perf (S); in particular, an S-linear category C is equipped with an action functor D perf (S) × C → C.
There is a well-behaved base change operation along any morphism T → S which produces a T -linear category This construction is compatible with semiorthogonal decompositions in the following sense.We say a semiorthogonal decomposition C = C 1 , . . ., C n is S-linear if the D perf (S)-action preserves each of the components C i , in which case C i inherits the structure of an S-linear category.Then for any morphism T → S, there is an induced T -linear semiorthogonal decomposition For a point s ∈ S, we use the following terminology.The fiber C s of C over s is the base change along Spec(κ(s)) → S. Similarly, if F : C → D is a functor between S-linear categories, its fiber over s is the functor F s : C s → D s obtained by base change.
We also recall that for objects E, F ∈ C, there is a mapping object Hom S (E, F ) ∈ D qc (S) characterized by equivalences where Map(−, −) denotes the space of maps in an ∞-category.
Example 2.1.Let f : X → S be a morphism of schemes.Then D perf (X) is naturally an S-linear category.By [BZFN10], for T → S the base changed category D perf (X) T recovers D perf (X T ), where X T is the derived base change (which agrees with the classical base change if e.g. where Hom X (E, F ) ∈ D qc (X) denotes the derived sheaf Hom on X.For i ∈ Z we denote by HH i (C/S) the i-th cohomology sheaf of HH * (C/S), and by HH i (C/S) the i-th cohomology module of HH * (C/S).When C = D perf (X) for a morphism of schemes X → S, we use the simplified notation HH * (X/S) and HH * (X/S) for Hochschild cohomology.
Warning 2.3.Sometimes different notation is used for Hochschild cohomology; for instance, in [Per20] which we shall reference several times below, sheafy Hochschild cohomology is denoted by HH * (C/S).
Example 2.4.Let X → S be a morphism of schemes.Then there is an equivalence where O ∆ ∈ D qc (X × S X) is the structure sheaf of the diagonal ∆ ⊂ X × S X. Indeed, HH * (X/S) can be computed as the Hochschild cohomology of the presentable S-linear category D qc (X) = Ind(D perf (X)) (see [Per20, Remark 4.2]), and then the claim follows from the equivalence D qc (X × S X) ≃ Fun S (D qc (X), D qc (X)) of [BZFN10] which sends O ∆ to id Dqc(X) .
Remark 2.5.Sheafy Hochschild cohomology satisfies a base change formula: if C is an Slinear category and g : T → S is a morphism, then there is a natural equivalence (2.3)This morphism is constructed as follows.The base change formalism gives a functor which induces a morphism on mapping objects where the first equivalence is the Künneth formula for mapping objects [Per19a, Lemma 2.10].That this morphism is an equivalence is proved (in a more general context) in [Per20, Lemma 4.3].
Base change also induces maps on Hochschild cohomology.Namely, taking RΓ(−) of the adjoint of the map (2.3) gives a natural map HH * (C/S) → HH * (C T /T ) (which is usually not an equivalence).
We will be concerned with the case of smooth proper S-linear categories.We refer to [Per19a,§4] for background on this notion.In particular, we note that if C is an S-linear semiorthogonal component of D perf (X) where X → S is a smooth proper morphism, then C is a smooth proper S-linear category [Per19a, Lemma 4.9].Although not strictly necessary for our purposes, we observe that in the smooth proper case Hochschild cohomology satisfies the following finiteness property.
Lemma 2.6.Let C be a smooth proper S-linear category.Then HH * (C/S) ∈ D perf (S) is a perfect complex.
Proof.By definition, it suffices to show the functor category Fun S (C, C) is proper.In fact, it is smooth and proper.Indeed, an S-linear category is smooth and proper if and only if it is dualizable, in which case the dual is given by the opposite category C op ([Per19a, Lemma 4.8]).Thus there is an equivalence Fun S (C, C) ≃ C op ⊗ D perf (S) C, and it follows that Fun S (C, C) is dualizable, being the tensor product of such categories.Definition 2.7.Let C be an S-linear category.We say C is connected (over S) if for every morphism of schemes T → S, we have HH i (C T /T ) = 0 for i < 0 and O T → HH 0 (C T /T ) is an isomorphism.
Remark 2.8.By base change for sheafy Hochschild cohomology (Remark 2.5), it suffices to consider only affine schemes T in the definition of connectedness.
The source of this terminology is the following example.
Example 2.9.Let f : X → S be a morphism of schemes.It follows from Example 2.4 that HH i (X/S) = 0 for i < 0 and HH 0 (X/S) ≃ R 0 f * O X .If f : X → S is a flat proper surjective morphism with geometrically reduced and connected fibers, then the morphism The above example can sometimes be leveraged to deduce connectivity of a semiorthogonal component of D perf (X).Recall that if C → D perf (X) is the embedding of an S-linear semiorthogonal component, then there is a restriction morphism HH * (X/S) → HH * (C/S).
Kuznetsov [Kuz15] introduced a general method for controlling the cocone of this morphism, which is particularly effective when C is defined as the orthogonal to an exceptional collection.In [Kuz15] everything is done relative to a base field, but the arguments work similarly over a base ring, which is the case we will need.We recall the result below after introducing some notation.
Let f : X → S be a smooth proper morphism of schemes with S = Spec(A) affine.Let E 1 , . . ., E n ∈ D perf (X) be a relative exceptional collection, i.e.Ext • A (E i , E j ) = 0 for i > j and Ext Then there is an S-linear semiorthogonal decomposition (2.4) The pseudoheight of the collection is the inverse of the relative Serre functor.
Proposition 2.10 ([Kuz15]).Let f : X → S be a smooth proper morphism of schemes with S affine, let E 1 , . . ., E n ∈ D perf (X) be a relative exceptional collection, and let C be defined by the semiorthogonal decomposition (2.4).Then the restriction morphism HH i (X/S) Using this, we can give a simple criterion for connectedness of a semiorthogonal component.
Corollary 2.11.Let f : X → S be a smooth proper surjective morphism of schemes with geometrically connected fibers.Let E 1 , . . ., E n be finite locally free sheaves which form a relative exceptional collection in D perf (X).Let C be defined by the semiorthogonal decomposition (2.4).
If the relative dimension satisfies dim(X/S) ≥ n + 1, then C is a connected S-linear category.
Proof.All of our assumptions are preserved by base change along a morphism T → S, so we may assume S is affine and must prove HH i (C/S) = 0 for i < 0 and O S ∼ − → HH 0 (C/S) is an isomorphism.It follows from the definitions and the assumption that the E i are locally free sheaves that ph(E 1 , . . ., E n ) ≥ dim(X/S) − n + 1.Therefore, if dim(X/S) ≥ n + 1 then Proposition 2.10 shows the map HH i (X/S) → HH i (C/S) is an isomorphism for i ≤ 0. By Example 2.9 this finishes the proof.
Corollary 2.11 implies connectedness of many Kuznetsov components, including those of Fano threefolds of Picard number 1 or of cubic fourfolds.Let us spell out explicitly how this verifies the hypotheses of Theorem 1.17 in one example.
is smooth, proper, and connected over S, and satisfies Ku(Y) s ≃ Ku(Y s ) for every geometric point s ∈ S. Indeed, O Y , O Y (1) is a relative exceptional collection because this is so fiberwise.The S-linear category Ku(Y) is smooth and proper as Y → S is so, connected by Corollary 2.11, and by base change has as fibers the Kuznetsov components of the fibers of Y → S.

Group actions on categories
Throughout this section, G denotes a finite group.There are two notions of an action of G on a triangulated category C appearing in the literature.The first one, often considered in mirror symmetry, is simply a group homomorphism φ from G to the group of autoequivalences, considered up to natural transformations.A second, finer notion was originally introduced by Deligne in [Del97] and requires a choice of natural transformations compatible with triple compositions.This finer notion is necessary for the definition of Gequivariant objects; we refer to [BO20] for a recent account, including of the obstruction to lifting the former notion to the latter.
In this section, we consider actions on the homotopy category hD of an ∞-category D that lift to an action on D. In this context, the first and second notion above correspond to 1-categorical and 2-categorical group actions on D .
First in §3.1 we discuss some generalities on group actions, focusing on obstructions to lifting 1-categorical group actions to ∞-categorical actions.In §3.2 we specialize to the case of group actions on linear categories, and show that for connected linear categories over a base scheme there is a single obstruction, whose vanishing is an open condition in the étale topology of the base (Proposition 3.9).Finally, in §3.3 we study the category of (co)invariants for a group action on a linear category; this is the ∞-categorical analogue of Elagin's notion [Ela15] of the triangulated category of G-equivariant objects.In particular, we show that if the order of the group is invertible on the base scheme, then invariants commute with base change (Lemma 3.14) and preserve the property of being smooth and proper (Proposition 3.15).

3.1.
Group actions ∞-categorically.We freely use the language of ∞-categories, as developed in [Lur09].We often think of ∞-groupoids interchangeably as topological spaces, under the standard correspondence (given by passage to geometric realizations and singular simplicial sets).
We denote by BG the classifying space of G.When regarded as an ∞-groupoid, BG is the nerve of the ordinary category with a single object whose endomorphisms are G.We write * ∈ BG for the unique object.
Definition 3.1.Let D be an ∞-category, and let X ∈ D be an object.An action of G on X is a functor φ : Let us relate this ∞-categorical definition to some more classical notions.
3.1.1.1-categorical actions.Suppose that D is an 1-category.For such a category, we denote by N (D) its nerve, which is an ∞-category.Let (BG) 1 denote the 1-category with a single object * whose endomorphisms are G, so that N ((BG) 1 ) = BG by definition.Recall [Lur21, Tag 002Y] the nerve construction induces a bijection where the left side is the set of all functors of 1-categories (BG) 1 → D and the right is the set of all functors of ∞-categories BG → N (D).Note that a functor (BG) 1 → D taking * to an object X ∈ D is equivalent to the data of a homomorphism G → Aut X to the group of automorphisms of X; we call this a 1-categorical action of G on X.Thus under the nerve construction, Definition 3.1 recovers the notion of a 1-categorical action.where the left side denotes the set of all strictly unitary lax functors (BG) 1 → D (in the sense of [Lur21, Tag 008R]) and we regard (BG) 1 as a 2-category with only identity 2-morphisms.
For simplicity let us assume D is a strict (2, 1)-category; then concretely, a strictly unitary lax functor (BG) 1 → D taking * to X amounts to the following data, which we call a 2-categorical action of G on X: 3.1.3.Obstructions to ∞-actions.Now suppose D is an ∞-category.For an object X ∈ D, let Aut X denote the space of automorphisms and let B Aut X denote its classifying space.As the sub-∞-groupoid of D spanned by X is equivalent to B Aut X, a G-action on X is tantamount to the data of a functor ψ : BG → B Aut X.We say two G-actions φ, φ ′ : BG → D are equivalent if there is an equivalence φ ≃ φ ′ of functors.By the previous remark, equivalence classes of G-actions on an object X ∈ D are in bijection with the set of homotopy classes of maps from BG to B Aut X.This perspective is useful for building G-actions.Namely, consider the Postnikov tower (Here and below, we typically suppress basepoints when dealing with homotopy groups.)Note that there is an isomorphism in particular, it follows τ ≤1 (B Aut X) ≃ B π 0 (Aut X).Now suppose we are given a map BG → τ ≤1 (B Aut X).By the preceding observation, such a map corresponds via taking π 1 to a group homomorphism φ 1 : G → π 0 (Aut X).We call φ 1 a 1-categorical action of G on X, because when X is regarded as an object of the homotopy category hD then φ is precisely a 1-categorical action in the sense of §3.1.1.We say a G-action φ recovers φ 1 (upon taking π 1 ).It is also convenient to study an intermediate notion.Namely, we call a map φ n : BG → τ ≤n (B Aut X) an n-lift of φ 1 if the composition recovers φ 1 .We say two n-lifts are equivalent if they have the same homotopy class.
The n-lifts of φ 1 can be studied via obstruction theory; below we spell out the simplest case of 2-lifts.Note that π 0 (Aut X) acts on π i (Aut X) for i ≥ 1 via conjugation; under the identifications π 0 (Aut X) ∼ = π 1 (B Aut X) and π i (Aut X) ∼ = π i+1 (B Aut X), this is the usual action of the fundamental group on higher homotopy groups.In particular, a 1-categorical action φ 1 : G → π 0 (Aut X) also induces an action of G on π i (Aut X) for i ≥ 1.
Lemma 3.2.Let D be an ∞-category, let X ∈ D be an object, and let φ 1 : G → π 0 (Aut X) be a 1-categorical action of G on X. Regarding π 1 (Aut X) as a local system on BG via the action of G described above, then there is a canonical obstruction class such that a 2-lift of φ 1 exists if and only if ob(φ 1 ) = 0, and in this case the set of equivalence classes of 2-lifts is a H 2 (BG, π 1 (Aut X))-torsor.If π i (Aut X) = 0 for i ≥ 2, then the same conclusion holds for ∞-lifts.
Proof.The first claim holds by standard obstruction theory.If π i (Aut X) = 0 for i ≥ 2, then by (3.1) we have B Aut X ≃ τ ≤2 (B Aut X), so ∞-lifts are the same as 2-lifts.
3.2.Group actions on linear categories.Let S be a base scheme.Recall that the collection of all S-linear categories (with morphisms between them the exact S-linear functors) can be organized into an ∞-category Cat S [Per19a].Thus, using the formalism of §3.1 we can make sense of G-actions on S-linear categories.
3.2.1.Obstructions in terms of Hochschild cohomology.Our main observation is that when negative Hochschild cohomology vanishes, then it is easy to classify ∞-lifts of 1-categorical G-actions.For this, we need a preliminary lemma.If C is an S-linear category, to emphasize the dependence on the S-linear structure we write Aut(C/S) for the space of S-linear autoequivalences of C, i.e. the automorphism space of C as an object of Cat S .Note also that HH 0 (C/S) = H 0 (Hom S (id C , id C )) has a Γ(S, O S )-algebra structure, so we may consider the group of units HH 0 (C/S) × .Lemma 3.3.There are natural group isomorphisms Proof.Note that in general, if D is an ∞-groupoid and X ∈ D is an object, then there is an isomorphism π where the left side denotes the homotopy group of D (thought of as a topological space) based at the point X.Further, suppose that D ֒→ D ′ is the sub-∞-groupoid spanned by some collection of objects in an ∞-category D ′ .If Map D ′ (X, X) denotes the space of endomorphisms of X as an object of D ′ , then π 0 (Map D ′ (X, X)) is a monoid, whose group of invertible elements we denote by π 0 (Map D ′ (X, X)) × .Then there are isomorphisms Indeed, for i = 0 this holds by the definition of Aut X, while for i ≥ 1 this holds because k-morphisms in D ′ are invertible (and hence coincide with k-morphisms in D) for k ≥ 2. As Aut(C/S) is the sub-∞-groupoid spanned by the autoequivalences in the ∞-category Fun S (C, C), combining the above observations shows By the characterizing property of mapping objects (2.1) and the definition (2.2) of Hochschild cohomology, we have Taking homotopy groups, we conclude All together, this proves the claimed formula for π i (Aut(C/S)).
Corollary 3.4.Let C be an S-linear category, and let φ 1 : G → π 0 (Aut(C/S)) be a 1categorical action of G on C. Regarding HH 0 (C/S) × as a local system on BG via the action of G described above, then there is a canonical obstruction class ob(φ 1 ) ∈ H 3 (BG, HH 0 (C/S) × ) such that a 2-lift of φ 1 exists if and only if ob(φ 1 ) = 0, and in this case the set of equivalence classes of 2-lifts is a H 2 (BG, HH 0 (C/S) × )-torsor.If HH i (C/S) = 0 for i < 0, then the same conclusion holds for ∞-lifts.
Remark 3.5.Suppose C is an S-linear category such that the map Γ(S, O S ) → HH 0 (C/S) is an isomorphism; this holds if C is a connected S-linear category, such as C = D perf (X) for a morphism X → S with assumptions as in Example 2.9.Then π 0 (Aut(C/S)) acts trivially on HH 0 (C/S), because the conjugation action fixes id : id C → id C and the isomorphism Γ(S, O S ) → HH 0 (C/S) takes 1 to id C .Hence if this condition holds in the setup of Corollary 3.4, then the obstruction class lies in H 3 (BG, Γ(S, O S ) × ), where Γ(S, O S ) × is a constant local system on BG.
Remark 3.6.Let us make some comments about Aut(C/S) for an S-linear category C.
(1) Suppose C = D perf (X) where X → S is a smooth proper morphism of schemes.Then by [BZFN10] there is an equivalence D perf (X × S X) ≃ Fun S (D perf (X), D perf (X)) that takes K ∈ D perf (X × S X) to the Fourier-Mukai functor Φ K = pr 2 * (pr * 1 (−) ⊗ K).Therefore, Aut(D perf (X)/S) is equivalent to the sub-∞-groupoid of D perf (X × S X) spanned by objects K such that Φ K is an equivalence.
(2) Suppose S = Spec A is affine.The homotopy category hC is a triangulated category which is A-linear in the sense that it is enriched in A-modules.Let π 0 (Aut(hC/A)) be the group of A-linear exact autoequivalences of hC modulo isomorphisms of functors.
(Classically, this group may be denoted Aut(hC/A), but it is more consistent with our notation above to rather denote by Aut(hC/A) the groupoid whose objects are A-linear exact autoequivalences and whose morphisms are isomorphisms of functors.)There is a natural group homomorphism The question of when this map is an isomorphism is interesting and difficult.In the case where A = k is a field and C = D perf (X) for a smooth projective variety X over k, the answer is positive; indeed, this follows from (1) together with the existence and uniqueness of Fourier-Mukai kernels for triangulated equivalences [Orl97, Theorem 2.2].
Remark 3.7.Our arguments have similar consequences for group actions on 1-categories.
Let A be a ring.We use the term classical A-linear category to mean a 1-category which is enriched in A-modules.Let Cat cl A be the strict (2, 1)-category with objects the classical A-linear categories, 1-morphisms the A-linear functors, and 2-morphisms the isomorphisms of functors.For any C ∈ Cat cl A , we can consider the notion of a 2-categorical action on C (in the sense of §3.1.2).Any such action induces a homomorphism φ 1 : G → π 0 (Aut(C/A)), where π 0 (Aut(C/A)) denotes the group of A-linear autoequivalences modulo isomorphisms of functors.
Conversely, suppose we are given a homomorphism φ 1 : G → π 0 (Aut(C/A)), and we want to understand when it lifts to a 2-categorical action.By analogy with the case of linear categories, define the A-algebra HH 0 (C/A) := Hom(id C , id C ).Then there is a canonical obstruction class ob(φ 1 ) ∈ H 3 (BG, HH 0 (C/A) × ) such that a 2-categorical action lifting φ 1 exists if and only if ob(φ 1 ) = 0, and in this case the set of equivalence classes of ∞-lifts is a H 2 (BG, HH 0 (C/S) × )torsor.Indeed, the Duskin nerve N (Cat cl A ) is an ∞-category, C can be thought of as an object of N (Cat cl A ), and its corresponding automorphism space Aut(C/A) has π 0 as described above, π 1 (Aut(C/A)) ∼ = HH 0 (C/A) × , and vanishing higher homotopy groups; therefore, the claim follows from Lemma 3.2 and the correspondence between ∞-categorical actions on C ∈ N (Cat cl A ) and 2-categorical actions on C ∈ Cat cl A described in §3.1.2.In the case where HH 0 (C/A) ∼ = A and A = C is the field of complex numbers, this obstruction to 2-categorical actions was proved in [BO20, Theorem 2.1] by a hands-on cocycle argument; the advantage of our proof is that it is more conceptual and generalizes to ∞-categorical actions.We also refer to [BO20, §3.6] for some simple examples where this obstruction is nontrivial.By Remark 2.5, we also have a natural ring map HH 0 (C/S) → HH 0 (C T /T ), which is easily seen to be compatible with the actions of π 0 (Aut(C/S)) and π 0 (Aut(C T /T )) under the map π 0 (Aut(C/S)) → π 0 (Aut(C T /T )).
The next lemma follows by unwinding our construction of ob(φ 1 ) and using functoriality of all the constructions involved.
Lemma 3.8.Let C be an S-linear category, and let φ 1 : G → π 0 (Aut(C/S)) be a 1-categorical action of G on C. Then the obstruction class of Corollary 3.4 is functorial under base change in the sense that for any morphism T → S, the natural map takes ob(φ 1 ) to ob((φ 1 ) T ).Moreover, the map from the set of equivalence classes of 2-lifts of φ 1 to the set of those of (φ 1 ) T is compatible with the torsor structures under the map H 2 (BG, HH 0 (C/S) × ) → H 2 (BG, HH 0 (C T /T ) × ).

Vanishing of obstructions on étale neighborhoods. Recall that a ring A is called a
Grothendieck ring if it is noetherian and for every p ∈ Spec A the completion A p → A p of the local ring at p is a regular map of rings [Sta21, Tag 07GG].(This is often called a Gring, but we will not use that terminology to avoid confusion with the group G.) A scheme S is called a Grothendieck scheme if for every open affine U ⊂ S the ring O S (U ) is a Grothendieck ring.This is a very mild condition, which includes all excellent schemes.Proposition 3.9.Let S be a Grothendieck scheme and let C be a connected S-linear category.Let φ 1 : G → π 0 (Aut(C/S)) be a 1-categorical action of G on C. Let s ∈ S be a point such that the characteristic of the residue field κ(s) is prime to the order of G, and the obstruction ob((φ 1 ) s ) ∈ H 3 (BG, κ(s) × ) vanishes.Then there exists an étale neighborhood U → S of s such that the obstruction ob((φ 1 ) U ) ∈ H 3 (BG, Γ(U, O U ) × ) vanishes, and thus the set of equivalence classes of ∞-lifts of (φ 1 ) U is a nonempty H 2 (BG, Γ(U, O U ) × )-torsor.
Proof.As C is connected, the natural map Γ(T, O T ) → HH 0 (C T /T ) is an isomorphism for any T → S, and G acts trivially on HH 0 (C T /T ) × (Remark 3.5); in particular, the obstructions indeed lie in the stated groups.Further, as the claim is local on S, we may assume S = Spec A is affine.By functoriality of the obstruction under base change (Lemma 3.8), the result is then a consequence of the following lemma.
Lemma 3.10.Let A be a Grothendieck ring.Let α ∈ H n (BG, A × ) where n ≥ 1 and A × has the trivial G-action.Let p ∈ Spec A be a point such that the characteristic of κ(p) is prime to the order of G, and such that α maps to zero under H n (BG, A × ) → H n (BG, κ(p) × ).Then there exists an affine étale neighborhood Spec(B) → Spec(A) of p such that α maps to zero under Before proving the result in general, we handle the case of complete local rings: Lemma 3.11.Let A be a complete local ring with residue field κ of characteristic prime to the order of G. Then for n ≥ 1 the map H n (BG, A × ) → H n (BG, κ × ) is an isomorphism, where A × and κ × have the trivial G-actions.
Proof.First consider the case where A is artinian.Then the maximal ideal satisfies m i A = 0 for some i ≥ 1, so by considering the factorization we reduce to proving the following claim: if A → B is a surjection of artinian local rings whose kernel I is annihilated by m A , then for n ≥ 1 the map H n (BG, A × ) → H n (BG, B × ) is an isomorphism.In this case, we have an exact sequence and as an abelian group 1 + I is isomorphic to the κ-vector space I; as |G| is invertible in κ this implies H n (BG, 1 + I) = 0 for n ≥ 1, and hence the claim.Now consider the case of a general complete local ring A. Then A × = lim i A × i where By the artinian case handled above, it suffices to show that the first term in this sequence vanishes.But if n ≥ 2 then again by the artinian case the transition maps for the system H n−1 (BG, A × i ) are isomorphisms, while if n = 1 then the transition maps are surjective; in either case, the system is Mittag-Leffler and we conclude that their R 1 lim i vanishes.
Proof of Lemma 3.10.By assumption α dies in H n (BG, κ(p) × ), so by Lemma 3.11 it also dies in H n (BG, A p × ).As A is a Grothendieck ring, the composition A → A p → A p is a regular map of noetherian rings, so by Popescu's theorem [Sta21, Tag 07GC] we can write A p = colim A i as a filtered colimit of smooth ring maps and hence α must die in H n (BG, A × i ) for some i.As Spec(A i ) → Spec(A) is smooth and its image contains p, we can find étale neighborhood Spec(B) → Spec(A) of p over which Spec(A i ) → Spec(A) has a section.In other words, the map A → B factors through A → A i , and therefore α dies in H n (BG, B × ).
Example 3.12.In the conclusion of Lemma 3.10, "étale neighborhood" cannot be replaced by "Zariski neighborhood" in general.Indeed, let G = Z/m for m ≥ 2, and let However, it is easy to see that for any f ∈ A, the map to the localization A → A f induces a split injection A × → (A f ) × on groups of units, and hence H n (BG, A × ) → H n (BG, (A f ) × ) is also a split injection.It follows that any 0 = α ∈ H n (BG, A × ) dies after restriction to any closed point of Spec A, but has nonzero image in H n (BG, Γ(U, O U ) × ) for any Zariski open U ⊂ Spec A.
3.3.Invariant categories.The (co)invariants for a group action can be formulated in the ∞-categorical setting as follows.
Definition 3.13.Let D be an ∞-category, let X ∈ D be an object, and let φ : BG → D be an action of G on X.The G-invariants X G and G-coinvariants X G of the action φ are defined by X G = lim(φ) and X G = colim(φ), provided the displayed limit and colimit exist.
We will be interested in the case where D = Cat S is the ∞-category of S-linear categories over some base scheme S, and G acts on an S-linear category C ∈ Cat S .Note that Cat S has all limits and colimits; see for instance [Mat16, §2.1]where this result is explained for Cat st ∞ , the ∞-category of small stable ∞-categories, and it similarly holds for Cat S .The G-invariants C G and coinvariants C G thus always exist in this situation.A basic example to keep in mind is that for a scheme X with a G-action, D perf (X) G ≃ D perf ([X/G]) where [X/G] is the quotient stack.(2) There is a canonical norm functor Nm : C G → C G , which under our assumption on |G| is an equivalence; see [Per21,Proposition 3.4] where this result is stated for S a field, but the same proof works in general.Therefore, the claim reduces to (1) proved above.Proof.Properness of C G is the assertion that for any objects E, F ∈ C G their mapping object Hom S (E, F ) ∈ D qc (S) lies in D perf (S) (see e.g.[Per19a, Lemma 4.7]).This mapping object can be described by the formula

G-invariants
where Forg : C G → C is the forgetful functor, and the right side is the group invariants for the induced G-action on Hom S (Forg(E), Forg(F )) (see e.g.[Per21, §3.1]).The object Hom S (Forg(E), Forg(F )) is perfect by the properness of C. By our assumption on |G| the object Hom S (E, F ) is a summand of Hom S (Forg(E), Forg(F )), hence also perfect.Indeed, more generally if A ∈ D perf (S) is an object equipped with a G-action, then A G is a summand of A: if φ g denotes the automorphism of A corresponding to g ∈ G, then 1 |G| g∈G φ g : A → A G gives the splitting.
Smoothness of C G is the assertion that id Ind(C G ) ∈ Fun S (Ind(C G ), Ind(C G )) is a compact object (see [Per19a,§4] for background on ind completions and smoothness of categories).To prove this, we use some results from [Per21]; we note that while results there are stated for categories linear over a field, they also hold relative to a base scheme S by the same arguments.First we note that Ind(C G ) ≃ Ind(C) G ; indeed, by [Per21, Proposition 3.4 and Lemma 3.5], the assertion is equivalent to the analogous statement Ind(C G ) ≃ Ind(C) G for coinvariant categories, which holds by [Per21, Lemma 2.3(1)].By [Per21, Lemma 4.7] (or rather the corresponding result for presentable S-linear categories, see [Per21, Remark 4.6]), we have an equivalence Fun S (Ind(C) G , Ind(C) G ) ≃ Fun S (Ind(C), Ind(C)) G×G which sends id Ind(C) G to the functor g∈G Ind(φ g ) : Ind(C) → Ind(C), where φ g : C → C is the equivalence corresponding to the action of g ∈ G on C. By [Per21, Lemma 3.7] the functor g∈G Ind(φ g ) is compact as an object of Fun S (Ind(C), Ind(C)) G×G if and only if it is compact as an object of Fun S (Ind(C), Ind(C)).
The S-linear category C, being smooth and proper, is dualizable with dual the opposite category C op (see [Per19a, Lemma 4.8]), and the presentable S-linear category Ind(C) is dualizable with dual Ind(C op ) (see [Per19a,Lemma 4.3]).Thus we have equivalences where the second line holds by the definition of the tensor product of S-linear categories (see [Per19a,§2.3]).This shows that for a smooth proper S-linear category C, the compact objects of Fun S (Ind(C), Ind(C)) are precisely those in the image of Fun S (C, C) → Fun S (Ind(C), Ind(C)).
In particular, we see that the functor g∈G Ind(φ g ) from above is compact as an object of Fun S (Ind(C), Ind(C)), because it is the image of the object g∈G φ g ∈ Fun S (C, C).Hence C G is smooth.

Enriques categories and their CY2 covers
In this section we work over an algebraically closed field k with char(k) = 2.We define Enriques and CY2 categories, explain the correspondence between them via residual Z/2actions, and describe the examples of interest for this paper.4.1.Definitions.Recall that a smooth proper k-linear category C admits a Serre functor, i.e. an autoequivalence S C such that there are natural isomorphisms is a semiorthogonal component of the derived category of a variety, then Hom k (E, F ) coincides with the classical derived Hom complex RHom(E, F ). Definition 4.1.Let C be a smooth proper k-linear category.
(1) C is an Enriques category if it is equipped with a Z/2-action whose generator τ is a nontrivial autoequivalence of Remark 4.2.Let C be a k-linear category, and suppose τ is an autoequivalence of C satisfying τ • τ ≃ id C .Then by Corollary 3.4 there is a class ob(τ ) ∈ H 3 (B(Z/2), HH 0 (C/k) × ), which vanishes if and only if τ is the generator for a Z/2-action on C, in which case the set of such actions is a torsor under H 2 (B(Z/2), HH 0 (C/k) × ).If C is connected, then HH 0 (C/k) = k and H 3 (B(Z/2), k × ) ∼ = Z/2 and H 2 (B(Z/2), k × ) = 0 (where the Z/2-action on k × is trivial), so a Z/2-action with τ as a generator is unique if it exists.This remark applies to all of the Enriques categories we consider in examples below, as they will all be connected.Remark 4.4.We use the term K3 category to mean a CY2 category whose Hochschild homology agrees with that of the derived category of a K3 surface.All of the explicit examples of CY2 categories considered in this paper will in fact be K3 categories.Note that a K3 category is automatically connected, because for a CY2 category C there is an isomorphism 4.2.Enriques-CY2 correspondence.An interesting feature of invariant categories is that they come equipped with a natural group action.In the Z/2-case, this leads to an involution on the category of k-linear categories equipped with a Z/2-action.
Lemma 4.5.Let C be a k-linear category with a Z/2-action, and let D = C Z/2 be the invariant category.Then there is a natural Z/2-action on D, called the residual Z/2-action, such that there is an equivalence Proof.This is a special case of [Ela15] (where the triangulated version of the result is proved, but a similar argument also works for k-linear categories).In particular, we note that the residual action on D = C Z/2 is given by tensoring with characters of Z/2.
Considering the invariant category of the Z/2-action on an Enriques category leads to a correspondence between Enriques and CY2 categories.Remark 4.7.There is a natural generalization of Lemma 4.6 in which C is assumed to be a smooth proper k-linear category whose Serre functor has the form S C = σ • [n] where σ is the generator of a Z/q-action.In this case, assuming the characteristic of k is coprime to q, we get a correspondence between such categories C and n-dimensional Calabi-Yau categories D equipped with a residual Z/q-action.
The source of the terminology "CY2 cover" in Lemma 4.6 is the following example.The following observation plays a key role in this paper, as it lets us translate the condition that two Enriques categories are equivalent to a statement about their CY2 covers.Lemma 4.9.Let C 1 and C 2 be connected Enriques categories, with CY2 covers D 1 and D 2 .Then the following are equivalent: (1) There is an equivalence (2) There is an equivalence C 1 ≃ C 2 which is equivariant for the Z/2-actions generated by the (−2)-shifted Serre functors.(3) There is an equivalence D 1 ≃ D 2 which is equivariant for the residual Z/2-actions.
Remark 4.10.Lemma 4.9 also admits an obvious generalization to the situation of Remark 4.7.

4.3.
Examples.One of the main sources of Enriques and CY2 categories in this paper are Kuznetsov components of GM varieties.Generalizing the definition of 3-dimensional GM varieties from the introduction, we have: Definition 4.11.An n-dimensional GM variety, 2 ≤ n ≤ 6, is either a smooth intersection W = Gr(2, 5) ∩ P n+4 ∩ Q of the Plücker embedded Grassmannian Gr(2, 5) ⊂ P 9 with a linear subspace P n+4 ⊂ P 9 and a quadric hypersurface Q ⊂ P 9 , or a smooth double cover W → Gr(2, 5) ∩ P n+3 branched along Gr(2, 5) ∩ P n+3 ∩ Q where P n+3 ⊂ P 9 is a linear subspace and Q ⊂ P 9 is a quadric hypersurface.We say W is ordinary in the first case, and special in the second.Note that if n = 6 then W is necessarily special.
There is a natural correspondence between GM varieties of ordinary and special types.
When discussing derived categories of GM varieties, we will always assume for simplicity that char(k) = 0, as this is done in the references cited below; in fact, for char(k) sufficiently large, all of the results still hold, but we leave the details to the interested reader.The Kuznetsov component of a GM variety is defined by the semiorthogonal decomposition where U W and O W (1) denote the pullbacks to W of the tautological rank 2 subbundle and Plücker line bundle on Gr(2, 5).These categories were extensively studied in [KP18].Note that if W is a GM threefold, this agrees with the definition from (1.2).Further, if W is a GM surface, then W is a K3 surface of degree 10 and Ku(W ) = D b (W ).
Remark 4.13.Instead of the Kuznetsov component as we have defined it, [KP18] studies a category A W defined by the slightly different semiorthogonal decomposition , from which the stated equivalence follows, cf.[KP18, Lemma 2.30].We chose to define Ku(W ) by the decomposition (4.1) for consistency with the definition (1.2) from [Kuz09] in the case of a GM threefold.Proof.By [Kuz19] and the Hochschild homology computation in [KP18, Proposition 2.9], the category Ku(W ) in ( 3) is a K3 category, while the Serre functors in (1) and ( 2) are of the form τ • [2] for an involution τ .By [KP18, Proposition 2.6] the involution τ is nontrivial for Ku(X), and the same argument applies to Ku(Y ).Thus to show that Ku(Y ) and Ku(X) are Enriques categories, it remains to show that there is a Z/2-action with generator τ .By Remark 4.2 there is a potential obstruction to the existence of such an action; we show it vanishes by relating τ to a geometric Z/2-action.More precisely, in case (1) or in case (2) if X is special, [Kuz19] shows that τ can be described as the pushforward along the covering involution of Y or X; in particular, it follows that τ is the generator of a Z/2-action.If X is ordinary, then τ ≃ Φ X [−1] where Φ X is the "rotation functor" defined in (4.2) below.Note that the Kuznetsov component Ku(X op ) of the opposite variety has a Z/2-action generated by the covering involution.By [KP17, §8.2] there is an equivalence Ku(X op ) Z/2 ≃ Ku(X), such that Φ X [−1] is the generator of the residual Z/2-action on Ku(X); in particular, τ also corresponds to a Z/2-action in this case.
Finally, the connectedness of the categories in (2) and (3) holds by the computation of their Hochschild cohomology in [KP18, Corollary 2.11 and Proposition 2.12], and an analogous argument applies to the categories in (1).
The CY2 covers of Kuznetsov components of quartic double solids and odd-dimensional GM varieties can be described explicitly as follows.(2) Let W be a GM variety and assume char(k) = 0. Let where L U W and L O W denote the left mutation functors through U W and O W if dim(W ) ≥ 3 (in which case these objects are exceptional), or the spherical spherical twists around U W and O W if dim(W ) = 2 (in which case these objects are spherical).Then Φ W [−1] is an involutive autoequivalence of Ku(W ), such that: (3) Let W be an odd-dimensional GM variety and assume char(k) = 0. Then the CY2 cover of Ku(W ) is equivalent to Ku(W op ).Under this equivalence the residual Z/2-action on Ku(W op ) is generated by the autoequivalence Proof.This follows from the main results of [KP17], as explained in [KP17, §8.1-8.2].
Remark 4.16.In the upcoming paper [PPZ21], we will use the description of the CY2 covers in these and other examples to describe moduli spaces of Bridgeland stable objects in Enriques categories.

Hodge theory via Kuznetsov components
In this section we work over the complex numbers.After reviewing some facts about Mukai Hodge structures in §5.1, we prove Proposition 1.12 on the categorical description of GM periods in §5.2, and give an application to the period map in §5.3.[Per20], to any C-linear category C occuring as a semiorthogonal component in the derived category of a smooth proper complex variety, one can canonically attach a lattice equipped with a Hodge structure, which is additive under semiorthogonal decompositions.If C is a CY2 category then we get a weight 2 Hodge structure H(C, Z), called the Mukai Hodge structure of C (see [Per20, Definition 6.4]), which generalizes a construction of Addington-Thomas [AT14] in the case of Kuznetsov components of cubic fourfolds.Below we explicitly describe this Hodge structure in the cases of interest for this paper.

Mukai Hodge structure. As explained in
Example 5.1.Let T be a complex K3 surface.Then H(D b (T ), Z) coincides with the classical Mukai Hodge structure, defined as where (−1) and (1) denote Tate twists, with pairing ((r, c, s), (r Example 5.2.Let W be a GM fourfold.In this case, H(Ku(W ), Z) was originally defined and studied in [Per19b], and admits the following explicit description.As an abelian group, where K top 0 (X) denotes the complex topological K-theory of the space of complex points X(C), and χ top denotes the topological Euler pairing.The group H(Ku(W ), Z) is regarded as a lattice with the symmetric pairing (−, −) = −χ top (−, −).Further, the Chern character induces an embedding By taking appropriate Tate twists, we regard H even (W, Q) = 4 k=0 H 2k (X, Q)(k − 1) as a weight 2 Hodge structure.The Hodge filtration on H(Ku(W ), Z) ⊗ C is then the intersection of the corresponding filtration on H even (W, C) along the above embedding.
Example 5.3.Let W be a GM sixfold.Then there is a similar description of the Mukai Hodge structure: as an abelian group , v) = 0 for i = 0, 1, 2, 3 , with Hodge structure induced by the one on H even (W, Q).
The above description shows that for a GM fourfold or sixfold, H(Ku(W ), Z) is rationally quite closely related to the usual middle-degree cohomology H dim(W ) (W, Z).We will need the following integral relation.
(2) Let H(Ku(W ), Z) 0 denote the orthogonal sublattice to A ⊕2 1 ⊂ H(Ku(W ), Z), and let H n (W, Z) 0 denote the orthogonal sublattice to H n (Gr(2, 5), Z) ֒→ H n (W, Z).Then the Chern class c n/2 : K top 0 (W ) → H n (W, Z) induces an isometry of weight 2 Hodge structures , where ( n 2 − 1) on the right denotes a Tate twist.Remark 5.5.Proposition 5.7 below implies that our notation H(Ku(W ), Z) 0 above is consistent with that of Proposition 1.12.For later use in §7, we record a lift of the isomorphism in Proposition 5.4(2) to the level of quotient groups.We only consider the 4-dimensional case, as that is the one we shall need, but a similar statement holds in dimension 6.

Proposition 5.6 ([Per19b]
).Let W be a GM fourfold.There is an isomorphism of abelian groups H(Ku(W ), Z) induced by the second Chern class c 2 : H(Ku(W ), Z) → H 4 (W, Z).Moreover, under the resulting correspondence between sublattices (2) K is non-degenerate if and only if K ′ is nondegenerate, in which case K has signature (r, s) if and only if K ′ has signature (s + 2, r − 2) and disc(K) = (−1) rkK disc(K ′ ).
(3) K ⊂ H 4 (W, Z) consists of Hodge classes if and only if K ′ ⊂ H(Ku(W ), Z) consists of Hodge classes.
5.2.Proof of Proposition 1.12.Proposition 5.4 reduces Proposition 1.12 to the following result.Note that the construction of Hodge structures for categories from [Per20] is functorial; in particular, a Z/2-action on a CY2 category does indeed induce a Z/2-action on its Mukai Hodge structure.
Proposition 5.7.Let W be a GM variety of dimension 4 or 6.Then the invariant sublattice H(Ku(W ), Z) Z/2 ⊂ H(Ku(W ), Z) for the residual Z/2-action equals the canonical sublattice Remark 5.8.The residual Z/2-action on H(Ku(W ), Z) is by isometries, so its −1-eigenspace must be exactly the orthogonal complement H(Ku(W ), Z) 0 of A ⊕2 1 .In particular, the residual Z/2-action on Ku(W ) is antisymplectic, in the sense that it acts by −1 on H 2,0 (Ku(W )).This induces antisymplectic involutions of Bridgeland moduli spaces of objects in Ku(W ) with class in A ⊕2 1 , giving a categorical interpretation for the antisymplectic involutions from [PPZ19, Proposition 5.16] whose existence was guaranteed there lattice theoretically.The geometry of the fixed loci of these involutions will be described in the upcoming work [PPZ21].
We will prove Proposition 5.7 by showing the claim is deformation invariant, and then checking it for a specific W where the claim is easy.For this, we will need to consider families of GM varieties and their Kuznetsov components.By a family of GM varieties over a base S, we mean a smooth proper morphism π : W → S equipped with a line bundle O W (1) on W, such that for every point s ∈ S the pair (W s , O Ws (1)) is a GM variety with the Plücker polarization.
The results of [DK18] show that for any family of GM varieties, there is a canonical rank 5 vector bundle V 5 on S and a morphism W → Gr S (2, V 5 ) which on fibers recovers the usual map to Gr(2, 5).We denote by U W the pullback to W of the tautological rank 2 subbundle on Gr S (2, V 5 ).
Lemma 5.9.Let π : W → S be a family of n-dimensional GM varieties.
(1) There is an S-linear semiorthogonal decomposition such that the fiber of Ku(W) over any point s ∈ S satisfies Ku(W) s ≃ Ku(W s ), where the right side is defined by (4.1).
(2) Let where for E = U W or O W the functor L E/S is defined by the exact triangle for F ∈ D perf (W).Then Φ W is an autoequivalence of Ku(W), whose fiber over any s ∈ S recovers the autoequivalence Φ Ws from Theorem 4.15(2) under the identification Ku(W) s ≃ Ku(W s ).
Remark 5.10.Note that we do not claim Φ W [−1] is necessarily an involution, but it is fiberwise an involution by Theorem 4.15(2).
Proof.The semiorthogonal decomposition in (1) follows from the decomposition (4.1) on fibers, cf.[BLM + 21, Lemma 3.22 and 3.25] and [Sam07], and the statement about the fibers of Ku(W) is a consequence of the compatibility of base change with semiorthogonal decompositions.For E = U W or O W , the functor L E/S is nothing but the left mutation through the admissible subcategory π * (D perf (S))⊗ E if n ≥ 3, and the spherical twist around the spherical functor D perf (S) → D perf (W), F → π * (F ) ⊗ E if n = 2; it follows in particular that Φ W is indeed an autoequivalence of Ku(W).The final claim about the fibers of Φ W is immediate from the definitions.
Proof of Proposition 5.7.Suppose π : W → S is a family of GM varieties of dimension n = 4 or 6.By [Per20] the Mukai Hodge structures H(Ku(W s ), Z), s ∈ S(C), form the fibers of a local system H(Ku(W)/S, Z) on the analytification S an .By functoriality the autoequivalence Φ W [−1] induces an automorphism of the local system H(Ku(W)/S, Z), which is necessarily an involution because it is so on fibers (see Remark 5.10).Therefore, we have a Z/2-action on the local system H(Ku(W)/S, Z) which fiberwise recovers the residual Z/2-action on the Mukai Hodge structures.So it suffices to prove the proposition for any particular fiber of the family W → S.
We may thus assume W is a special GM variety.Then by Theorem 4.15 the residual Z/2action on Ku(W ) is induced by the covering involution of W .By construction H(Ku(W ), Z) is a summand of K top 0 (W ), and by the previous remark the inclusion H(Ku(W ), Z) ⊂ K top 0 (W ) is Z/2-equivariant, where Z/2 acts by the residual action on the left and the covering involution on the right.By the description of A ⊕2 1 ⊂ H(Ku(W ), Z) from Proposition 5.4(1), it follows that A ⊕2 1 ⊂ H(Ku(W ), Z) Z/2 .Moreover, as H 4 (Gr(2, 5), Z) ⊂ H 4 (W, Z) is primitive by [DIM15, §5.1], it follows from Proposition 5.6(1) that A ⊕2 1 ⊂ H(Ku(W ), Z) is also primitive.Therefore, to finish the proof it suffices to show the inclusion A ⊕2 1 ⊂ H(Ku(W ), Z) Z/2 is rationally an isomorphism.By applying the Chern character, it is enough to show H * (Gr(2, 5), Q) surjects onto the invariants of H * (W, Q).If dim(W ) = 6 this is true because W → Gr(2, 5) is a double cover.If dim(W ) = 4 then W → Gr(2, 5) ∩ P 8 is a double cover, so it is enough to observe that H * (Gr(2, 5), Q) surjects onto H * (Gr(2, 5) ∩ P 8 , Q); this follows, for instance, from the semiorthogonal decompositions

Application to periods. In [DK18]
Debarre and Kuznetsov classified GM varieties in terms of linear algebraic data, by constructing for any GM variety W a Lagrangian data set is a Lagrangian subspace with respect to the wedge product, and proving that W is completely determined by its dimension and this data.Surprisingly, many properties of W only depend on A(W ).Recall that GM varieties W 1 and W 2 with dim(W 1 ) ≡ dim(W 2 ) (mod 2) are called generalized partners if there exists an isomorphism , and generalized duals if there exists an isomorphism In the case where dim(W 1 ) = dim(W 2 ), these definitions specialize to the notions of period partners and duals originally introduced in [DK18].
The main result of [DK19] shows that A(W ) determines the period of W in dimensions 4 and 6, which implies the period map factors through the moduli space of EPW sextics studied by O'Grady [O'G06].More precisely: Theorem 5.11 ([DK19]).Let W 1 and W 2 be GM varieties of (possibly unequal) dimensions n 1 , n 2 ∈ {4, 6}.Assume W 1 and W 2 are generalized partners or duals.Then there is an isometry of Hodge structures 2 ).This is proved in [DK19] by intricate geometric arguments, but here we note a simple categorical proof.This relies on the following slight enhancement of [KP19, Theorem 1.6].
Theorem 5.12.Let W 1 and W 2 be GM varieties which are not special GM surfaces.If W 1 and W 2 are generalized partners or duals, then there is an equivalence Ku(W 1 ) ≃ Ku(W 2 ) which is equivariant for the canonical Z/2-actions described in Theorem 4.15(2).
Proof.By [KP19, Theorem 1.6] there exists an equivalence Ku(W 1 ) ≃ Ku(W 2 ), not a priori Z/2-equivariant.If W 1 and W 2 are odd-dimensional, then by Lemma 4.9 this equivalence is necessarily Z/2-equivariant.If W 1 and W 2 are even-dimensional, then by Theorem 4.15(3) combined with Lemma 4.9, it suffices to show there is an equivalence Ku(W op 1 ) ≃ Ku(W op 2 ).But by construction the Lagrangian data of a GM variety and its opposite GM variety are the same, so W op 1 and W op 2 are odd-dimensional generalized partners or duals, and the equivalence holds again by [KP19, Theorem 1.6].

Birational categorical Torelli
In this section we work over the complex numbers.After reviewing some facts about the period map for GM fourfolds, we prove Theorem 1.9 in §6.2.
6.1.The period morphism.Let N denote the moduli stack of GM fourfolds.This is a smooth irreducible 24-dimensional Deligne-Mumford stack of finite type over C (see [KP18, Proposition A.2] or [DK20]).We denote by p : N → D the period map, where the period domain D is the 20-dimensional quasi-projective variety classifying Hodge structures on the middle cohomology H 4 (W 0 , Z) of a fixed GM fourfold W 0 for which the canonical rank 2 sublattice H 4 (Gr(2, 5), Z) ⊂ H 4 (W 0 , Z) consists of Hodge classes (see [DIM15] for details).We note that D is equipped with a canonical involution, denoted r D (see the discussion preceding [DIM15, Corollary 6.3]).
We will also need to consider the related moduli space N EPW of EPW sextics [O'G16].Recall that, if we fix V 6 a 6-dimensional vector space, then N EPW is the GIT quotient by PGL(V 6 ) of the space of Lagrangians A ⊂ ∧ 3 V 6 containing no decomposable vectors.The space N EPW has a natural involution r EPW , induced by sending A ⊂ ∧ 3 V 6 to its orthogonal sending A to the period of the associated double EPW sextic [O'G15].
The results of [DK18,DK20] show that there is a surjective morphism sending a GM fourfold W to its Lagrangian A(W ) (as in §5.3).In particular, by definition GM fourfolds W 1 and W 2 are period partners if and only if π(W 1 ) = π(W 2 ), and duals if and only if π(W 1 ) = r EPW (π(W 2 )).We will need the following two ingredients in our proof of Theorem 1.9.
Proof.This follows from [DK19], cf.[DK20, Proposition 6.1] and Theorem 5.11 above.6.2.Proof of Theorem 1.9.By the discussion preceding the statement of Theorem 1.9, we only need to prove the forward implication.So let X 1 and X 2 be GM threefolds such that Ku(X 1 ) ≃ Ku(X 2 ).Note that if X is a special GM threefold, then we may find an ordinary GM threefold X ′ which is a period partner of X (see e.g.[KP18, Lemma 3.8]), and hence satisfies Ku(X) ≃ Ku(X ′ ) by [KP19,Theorem 1.6].Therefore, we may assume that X 1 and X 2 are both ordinary.By Proposition 4.14, Lemma 4.9, and Theorem 4.15, passing to CY2 covers gives an equivalence Ku(X op 1 ) ≃ Ku(X op 2 ) which is equivariant for the residual Z/2-actions.This induces a Z/2-equivariant Hodge isometry H(Ku(X op 1 ), Z) ∼ − → H(Ku(X op 2 ), Z), and hence by Proposition 1.12 a Hodge isometry H 4 (X op 1 , Z) 0 ∼ = H 4 (X op 2 , Z) 0 .By the definition of the period morphism, it follows that either p(X op 1 ) = p(X op 2 ) or p(X op 1 ) = r D (p(X op 2 )), cf.[DK19, Lemma 5.26].By Theorems 6.1 and 6.2, this means that either π(X op 1 ) = π(X op 2 ) or π(X op 1 ) = r EPW (π(X op 2 )), i.e. either X op 1 and X op 2 are period partners or duals.As these relations are preserved under passing to opposite GM varieties, we conclude the same is true of X 1 and X 2 .

Nonexistence of equivalences
In this section we work over the complex numbers.After reviewing some restrictions on the periods of GM fourfolds and surfaces, we prove Theorem 1.4 in §7.2.
7.1.Restrictions on periods.As in our discussion of the period map p : N → D for GM fourfolds in §6.1, let W 0 be a fixed GM fourfold.Let K ⊂ H 4 (W 0 , Z) be a rank 3 primitive positive definite sublattice which contains H 4 (Gr(2, 5), Z).We consider the locus in D parameterizing Hodge structures on H 4 (W 0 , Z) for which K ⊂ H 4 (W 0 , Z) consists of Hodge classes.By [DIM15] this locus only depends on the discriminant d > 0 of K, so we denote it by D d .Moreover, the locus D d is nonempty for d ≡ 0, 2, 4 (mod 8), an irreducible divisor for d ≡ 0, 4 (mod 8), and the union of two irreducible divisors for d ≡ 2 (mod 8).
The following restriction on the image of the period morphism plays a crucial role in our proof of Theorem 1.4 below.).Some partial progress on this problem was made in [DIM15], where it is shown that p is dominant and p −1 (D d ) is nonempty for d ≥ 10.
Next we give a similar but much easier result restricting the periods of ordinary GM surfaces.We will use this in one of two proofs given below for Theorem 1.4 in the case of special GM threefolds.
Lemma 7.3.Let W = Gr(2, 5) ∩ P 6 ∩ Q be an ordinary GM surface.Then Pic(W ) does not contain a rank two lattice with intersection form given by 10 1 1 0 or 10 3 3 0 , with the first basis vector corresponding to the restriction of the Plücker polarization.
Proof.This is a special case of [GLT15, Lemma 2.8]; since a direct proof is short and easy, we give one here.In either case, the second basis vector is effective, and thus the class of a genus 1 curve C. The Plücker polarization would restrict to a very ample divisor on C of degree 1 or 3, respectively.This is immediately a contradiction in the first case.In the second case, the Plücker polarization would embed C as a plane cubic curve.However, as the equations of Gr(2, 5) are quadratic, W would contain the entire P 2 spanned by C, a contradiction.
7.2.Proof of Theorem 1.4.Let Y be a quartic double solid and X a GM threefold.Assume for sake of contradiction that there is an equivalence Ku(Y ) ≃ Ku(X).Combining Proposition 4.14, Lemma 4.9, and Theorem 4.15, we obtain an equivalence D b (Y br ) ≃ Ku(X op ) which is equivariant for the residual Z/2-actions, where Y br ⊂ P 3 is the branch locus of Y → P 3 and X op is the opposite GM variety.This induces a Z/2-equivariant Hodge isometry To derive a contradiction, we will use a description of the Z/2-invariants on each lattice.The case where X is ordinary so that X op is a fourfold was already addressed in Proposition 5.7.Note that Y br , as well as X op when X is special, is a K3 surface, so its Mukai lattice up to sign is just the full integral cohomology (Example 5.1).
Lemma 7.4.The invariant sublattice H(Y br , Z) Z/2 , as well as H(X op , Z) Z/2 when X is a special GM threefold, are isomorphic to A ⊕2 1 and given as follows.
Proof.Recall that a spherical twist T E acts on cohomology by the reflection ρ v(E) , defined by while tensoring with a line bundle L acts by multiplication with ch(L).In case (1), this shows Hence the two vectors are preserved by the action of Φ 2 Y br [−1], i.e. by Theorem 4.15(1) they are Z/2-invariant.In case (2), we instead apply Theorem 4.15(3); using v(U X op ) = (2, −B, 3) it is a straightforward computation that both classes are invariant under Φ X op [−1].One also sees immediately that Z/2 acts by −1 on the orthogonal complements of the sublattices As Z/2 evidently does not act as the identity on these rank 3 sublattices, this leaves the claimed rank 2 lattices as the only possibility for the invariants.
Now we can finish the proof of Theorem 1.4.We break the proof into two cases, depending on whether X is ordinary or special.
Assume X is ordinary.The primitive sublattice L Y br ⊂ H(Y br , Z) from (7.4) has signature (2, 1), contains the Z/2-invariant sublattice, consists of Hodge classes, and has discriminant −4.Therefore, its image K ′ ⊂ H(Ku(X op ), Z) under the Z/2-equivariant Hodge isometry (7.1) has the same properties.By Proposition 5.7 the Z/2-invariants of H(Ku(X op ), Z) is the canonical A ⊕2 1 sublattice.So by Proposition 5.6, K ′ corresponds to a rank 3 primitive positive definite sublattice K ⊂ H 4 (X op , Z) which contains H 4 (Gr(2, 5), Z), consists of Hodge classes, and has discriminant 4.This means the period of X op lies in the divisor D 4 , which contradicts Theorem 7.1.
Assume X is special.We give two proofs.The first is shorter but relies on a hard result from [KP19] to reduce to the ordinary case, while the second uses only the easy Lemma 7.3 and is the starting point for our proof of Theorem 1.6 in §8.
Proof 1.The description of generalized partners from [KP18, Lemma 3.8] shows that there exists an ordinary GM threefold X ′ which is a generalized partner of X.Then by [KP19, Theorem 1.6] we have an equivalence Ku(X) ≃ Ku(X ′ ), so we are reduced to the ordinary case handled above.
Proof 2. The rank 3 lattices L Y br and L X op from (7.4) and (7.5) are given explicitly by where the first two basis vectors are the Z/2-invariant ones from Lemma 7.4, and the third is the class of a point.As the isometry θ in (7.1) is Z/2-equivariant, it sends the basis of the invariant lattice A ⊕2 1 ⊂ L Y br to the basis of A ⊕2 1 ⊂ L X op , up to sign and permutation.By (7.3), one can precompose with an appropriate power of Φ Y br to ensure it sends the first vector to the first vector.As θ respects the standard orientation of positive definite 2-planes by [HMS09,Corollary 4.10], it also sends the second basis vector to the second basis vector.
Thus, the rank 4 lattice in H(X op , Z) generated by θ(L Y br ) and L X op has intersection form for some r ∈ Z.The discriminant of this lattice, which has to be positive, is −4r 2 − 12r + 1, and hence the only possibilities are r = −3, −2, −1, 0. The fourth basis vector is a class of the form (r, D, s) with The methods used in the second proof for the case where X is special above also lead to a short direct proof of this categorical Torelli statement, which we sketch briefly.Given an equivalence Ku(Y 1 ) ≃ Ku(Y 2 ), one considers the associated Z/2-equivariant Hodge isometry If θ sends the class of a point to a class in the lattice L Y 2 ,br , then up to composition with the involution induced by Φ Y 2 ,br it sends it to the class of a point; in this case, an easy argument shows that θ induces a Hodge isometry H 2 (Y 1,br , Z) ∼ − → H 2 (Y 2,br , Z) preserving the polarizations.By the Torelli theorem for K3 surfaces there is a polarized isomorphism Y 1,br → Y 2,br , and thus an isomorphism Y 1 ∼ = Y 2 .Otherwise, one considers again the rank 4 lattice spanned by L Y 2 ,br and the image of the class of a point; a computation exactly as in the proof above shows that H 2 (Y 2,br , Z) contains a square zero class of degree 2, which cannot exist on a quartic K3.

Deformation equivalence
We continue to work over the complex numbers.In this section, we prove Theorem 1.6, which says that Kuznetsov components of quartic double solids and of GM threefolds are deformation equivalent.The idea is based on the second proof of Theorem 1.4 in the case where the GM threefold is special.Namely, in the case r = −1 we used the fact that there is no GM K3 surface with Picard lattice 10 3 3 0 .However, there is such a polarized K3 surface S -indeed, as we will see it is given by a very general quartic containing a line.This directly suggest our strategy: we extend the Z/2-action on GM K3 surfaces, given by Theorem 4.15(2), to one acting also on S, in such a way that it becomes conjugate with the Z/2-action on S as a quartic K3 surface given by Theorem 4.15(1).To construct this as an enhanced Z/2-action in a family, we apply the general results from §3.Taking the associated Z/2-invariant category then proves Theorem 1.6.8.1.Quartics containing a line.From now on, let S ⊂ P 3 be a quartic K3 surface that is very general among quartics containing a line L ⊂ S; in particular, S has Picard rank two.The projection from the line induces an elliptic fibration S → P 1 ; let E be the class of a fiber, a plane cubic.The Picard lattice of S is given by 4 3 3 0 , with respect to the basis given by D, the class of the hyperplane section, and E. Both E and the class L = D − E of the line are classes of curves that can be contracted, and so generate extremal rays of the Mori cone.
A simple computation then confirms the following result.
Lemma 8.1.The Mori cone of S is given by E, D − E , and the nef cone by E, 3D − E .
In particular, H := D + E is a polarization of degree 10.Moreover, with respect to the basis H, E, the intersection matrix becomes 10 3 3 0 , one of the possibilities we had to exclude for GM K3s in Lemma 7.3 for the second proof of Theorem 1.4.As the general degree 10 K3 surface is obtained as a GM surface [Muk02], S is a degeneration of GM K3 surfaces.
8.2.Stable rank 2 bundles on S. In our arguments, we will sometimes have to show that rank 2 bundles on S that we construct are stable.In all cases, this will follow from the following classical lemma due to Mukai.
Lemma 8.2 ([Muk87, Corollary 2.8]).Let 0 → F → E → G → 0 be a short exact sequence of sheaves on a K3 surface with Hom(F, G) = 0. Then Recall from Example 5.1 the Mukai Hodge structure H(S, Z) associated to the K3 surface S. The Mukai vector of an object E ∈ D b (S) is defined by The Mukai pairing is defined by (r, c, s), (r ′ , c ′ , s ′ ) = cc ′ − rs ′ − r ′ s and satisfies −χ(E, F) = (v(E), v(F)).Given a polarization A, we define µ A -(semi)stability via the slope function Proposition 8.3.Let S be a very general quartic K3 surface containing a line, and let A be a polarization on S.
(1) Let V be a rank 2 spherical vector bundle on S. Then V is µ A -stable.
(2) Let V be a rank 2 vector bundle on S such that Hom(V, V) = C and Ext If A.(D − 2E) = 0 (e.g. if A = D) and V is unstable, then more precisely the destabilizing object is O(B) where the sign is chosen such that µ A (O(B)) > µ A (c 1 (V)); moreover, in this case the spherical twist T O(B) V is a µ A -stable vector bundle of the same Mukai vector.
Proof.If V is not stable, then there is a short exact sequence where L 1 , L 2 are line bundles with µ A (L 1 ) ≥ µ A (L 2 ), and Z is a zero-dimensional subscheme of S. Since Hom(V, V) = C we have L 1 = L 2 , and hence Hom(L 1 , L 2 ) = 0. Mukai's Lemma, Lemma 8.2, shows that Z is empty in case (1), and that Z is either empty or a single point in case (2).We write (8.1) We write C 1 − C 2 = dD + eE, which gives (C 1 − C 2 ) 2 = 4d 2 + 6de.
Finally, if V is not stable, we have seen that its Harder-Narasimhan filtration is This shows that T O(B) V is a rank 2 vector bundle with Hom(O(B), T O(B) V) = 0. Since T O(B) is an equivalence, it is also a simple vector bundle with Ext Applying the previous results of the proposition shows that T O(B) V is stable as claimed.
Let U be the µ D -stable spherical vector bundle with v(U ) = (2, −D − E, 3), whose existence and uniqueness were first proved in [Kul90, Theorem 2.1] and [Muk87, Corollary 3.5], respectively.Then by Proposition 8.3, U is stable with respect to any polarization.Moreover: Lemma 8.4.Let U be the slope-stable spherical vector bundle with v(U ) = (2, −D − E, 3).Then the restriction of U to any fiber of the elliptic fibration induced by E is stable.
Proof.By the previous observation, U is stable with respect to the ample polarization E + ǫD for ǫ > 0, and thus at least semistable with respect to the nef polarization E. As E.c 1 (U ) = −3 is odd, this means it is stable with respect to E (in the sense that any saturated subsheaf of U has strictly smaller slope).Now we can follow the proof of [Lan04, Theorem 5.2].Assume that there is a curve C ⊂ S of class E such that U C is unstable.Then there exists a line bundle L C on C of degree d ≤ −2 and a surjection Remark 8.5.As pointed out by Kuznetsov, the bundle U can be described explicitly as follows.Note that Hom(O(D), O L (2)) = C 2 and the corresponding map O(D) ⊕2 → O L (2) is surjective.Let F be the vector bundle defined by the short exact sequence A computation shows that Hom(F, F) = C and v(F ∨ ) = (2, −D − E, 3).Therefore, F ∨ is a rank 2 spherical vector bundle, and hence by Proposition 8.3.1 it is µ D -stable.We conclude that U ∼ = F ∨ .8.3.Conjugate autoequivalences.Since we can consider S either as a quartic K3 surface, or as a degeneration of a GM K3 surface, there are two natural autoequivalences associated to it by Theorem 4.15: where T O and T U are the spherical twists around O and U .We will show that T 2 O(−D) Φ GM and Φ quartic are conjugate to each other.Since by Theorem 4.15(3) the latter generates the residual Z/2-action on D b (S) as the CY2 cover of the Kuznetsov component of the associated quartic double solid, it will follow that the former also generates a Z/2-action on D b (S).
The following lemma will allow us to prove our identity by computing the images of skyscraper sheaves of points.
Lemma 8.6.Let S be a smooth projective K3 surface, and let F 1 , F 2 be two autoequivalences of D b (S).Assume that: (1) F 1 and F 2 have the same action on H(S, Z).
(2) Applying F 1 and F 2 to skyscraper sheaves of points gives the same set of objects: Then F 1 and F 2 are isomorphic functors.
Proof.Assumption (2) implies that F −1 1 • F 2 sends skyscraper sheaves of points to skyscraper sheaves of points.By [Huy06, Corollary 5.23], it is the composition of the pushforward along an automorphism f of S and tensoring with a line bundle M .By (1), F −1 1 • F 2 acts trivially on cohomology.So F −1 1 • F 2 preserves the class v(O S ) and M is trivial.But then the action of f * on H 2 (S, Z) is trivial, and thus f is the identity by the Torelli theorem for K3 surfaces.
Remark 8.7.Condition (2) holds automatically when the F i (O s ) are, up to the same shift, slope-stable vector bundles for the same polarization.Indeed, both F 1 and F 2 induce an injective map from S to the moduli space of vector bundles of class F i (O s ); since this moduli space is two-dimensional and irreducible, both maps are bijections on closed points.
We first prove that the actions of Φ quartic and Φ GM on cohomology are conjugate.(8.2) By Lemma 8.8, the two sides have the same action on H(S, Z).By Lemma 8.6 and Remark 8.7, it will thus be enough to show that applying the LHS and the RHS to skyscraper sheaves of points yields slope-stable vector bundles.
We first consider the LHS of (8.2) applied to the skyscraper sheaf at s ∈ S(C).Up to possibly passing to a cover of C, there exists a rank 2 vector bundle U on S whose restrictions to fibers S c is the unique H-stable vector bundle of Mukai vector (2, −H c , 3); so it is the tautological subbundle on GM fibers, and the unique stable bundle of Φ GM , the autoequivalence considered in Lemma 8.8 and Proposition 8.9.In particular, for every c ∈ C, the autoequivalence Π c is an involution that generates a Z/2-action on D perf (S c ).
The functor Π • Π is the identity on every fiber, and thus sends every skyscraper sheaf O s for s ∈ S to itself.Since Π • Π is a Fourier-Mukai transform by construction, one can easily adapt the proof of [Huy06,Corollary 5.23] to show that it is given by tensor product with a line bundle; this line bundle is trivial on the fibers of π, i.e. it is pulled back from C. Shrinking C further if necessary, we may assume this line bundle to be trivial, and therefore that Π is an involution of D perf (S).
We therefore have a homomorphism φ : Z/2 → π 0 (Aut(D perf (S)/C)) as in Corollary 3.4, and want to show that the obstruction to the existence of an ∞-lift vanishes.The restriction of φ to S o can be lifted to a Z/2-action, as by Proposition 8.9 it is conjugate to the residual action on D perf (S o ) coming from its realization as the CY2 cover of the Kuznetsov component of the corresponding quartic double solid.By Proposition 3.9, we can replace C by an étale neighborhood B of o ∈ C such that the obstruction to lifting φ vanishes.
We have thus obtained a Z/2-action on D perf (S) over B whose generator acts by the involution Π.The associated invariant category C := D perf (S) Z/2 has the properties claimed in Theorem 1.
Remark 1.16.Conjecture 1.15, and hence Conjecture 1.14, are shown for a generic GM threefold X in [DIM12, Corollary 7.6].Loci of equivalent Kuznetsov components.Conjecture 1.1 motivates studying in general the locus where Kuznetsov components are equivalent in families of Fano varieties.We note the following consequence of work of Anel and Toën [AT09, Corollaire 3.3].

2. 2 .
Hochschild cohomology.Recall that given two S-linear categories C and D, there is a natural S-linear category Fun S (C, D) whose objects are the S-linear functors C → D. Definition 2.2.Let C be an S-linear category.The sheafy Hochschild cohomology over S of C is HH * (C/S) := Hom S (id C , id C ) ∈ D qc (S), (2.2) i.e. the endomorphism object of id C regarded as an object of Fun S (C, C).The Hochschild cohomology over S of C is the derived global sections HH * (C/S) := RΓ(HH * (C/S)) ∈ D(Mod Γ(S,O S ) )

2. 3 .
Connected linear categories.Note that for any S-linear category C, by the definition of HH * (C/S) there is a canonical morphism O S → HH 0 (C/S).
Example 2.12.Let Y → S be a smooth family of Fano threefolds of Picard number 1 and index 2, equipped with a line bundle O Y (1) whose restriction to each geometric fiber Y s is the ample generator of Pic(Y s ).Then O Y , O Y (1) is a relative exceptional collection (in the sense of [BLM + 21, §3.3], cf.[Sam07]), and the S-linear category Ku(Y) ⊂ D perf (Y) defined by

3.2. 2 .
Base change of G-actions.Let C be an S-linear category.As discussed in §2.1, for any morphism of schemes T → S we can form the base change category C T which is linear over T .Formation of base change is functorial, i.e. gives a functor Cat S → Cat T ; therefore, any G-action φ : BG → Cat S on C induces a base changed G-action φ T : BG → Cat T on C T by composition with this functor.Similarly, any 1-categorical action φ 1 : G → π 0 (Aut(C/S)) of G on C induces a base changed 1-categorical action (φ 1 ) T : G → π 0 (Aut(C T /T )) on C T , by composition with π 0 of the map Aut(C/S) → Aut(C T /T ).Note also that the base change map for group actions takes n-lifts of φ 1 (in the sense of §3.1.3)to n-lifts of (φ 1 ) T .

3.3. 1 .
Base change of G-(co)invariants.The operations of taking G-coinvariants commutes with base change, while the same is true for G-invariants if the order of G is invertible on the base scheme: Lemma 3.14.Let C be an S-linear category with a G-action, and let T → S be a morphism of schemes.(1)There is an equivalence(C G ) T ≃ (C T ) G of T -linear categories, where (C T ) G denotes the G-coinvariants for the induced G-action on the base change C T .(2) If |G| is invertible on S, then there is an equivalence (C G ) T ≃ (C T ) G of T -linear categories.Proof.(1) The base change functor Cat S → Cat T , C → C T has a right adjoint, given by the functor which regards a T -linear category as an S-linear category via restriction along D perf (S) → D perf (T ), and therefore commutes with colimits.
of smooth proper categories.Passage to G-invariants preserves smooth and properness of a category, as long as |G| is invertible on the base scheme: Proposition 3.15.Let C be a smooth proper S-linear category, where |G| is invertible on S. Then the G-invariant category C G is also smooth and proper.
Example 4.3.(1) If S is an Enriques surface, then D b (S) is an Enriques category with Z/2-action generated by tensoring by ω S , cf.Example 4.8.(2) If T is a smooth proper surface with K T = 0, i.e.T is a K3 or abelian surface, then D b (T ) is a CY2 category.

Lemma 4. 6 .
Let C be an Enriques category, and let D = C Z/2 be the invariant category for the Z/2-action.Then D is a CY2 category, called the CY2 cover of C. Proof.The category D is smooth and proper by Proposition 3.15 and has Serre functor [2] by [Per21, Lemma 6.5].
Example 4.8.If S is an Enriques surface, then its canonical bundle satisfies ω ⊗2 S ∼ = O S .The corresponding étale double cover T → S is a K3 surface.For the Z/2-action generated by the involution of T over S, we have D b (T ) Z/2 ≃ D b (S).Under this equivalence, the residual Z/2-action on D b (S) is generated by tensoring by ω S , and we have D b (S) Z/2 ≃ D b (T ).
where L O W is the left mutation functor through O W . Indeed, if we tensor the defining semiorthogonal decomposition (4.1) of Ku(W ) by O W (1) and mutate the object O W (dim(W ) − 2) to the far left, we get D b Recall from (1.1) the definition of the Kuznetsov component of a quartic double solid.Proposition 4.14.(1) If Y is a quartic double solid, then Ku(Y ) is a connected Enriques category.(2) If char(k) = 0 and X is an odd-dimensional GM variety, then Ku(X) is a connected Enriques category.(3) If char(k) = 0 and W is an even-dimensional GM variety, then Ku(W ) is a connected K3 category.
Theorem 4.15 ([KP17]).(1) Let Y → P 3 be a quartic double solid with branch locus Y br .Then the CY2 cover of Ku(Y ) is equivalent to D b (Y br ).Under this equivalence the residual Z/2-action on D b (Y br ) is generated by the autoequivalence Φ 2 Y br [−1], where Φ Y br = T O Y br • (− ⊗ O Y br (1)) and T O Y br is the spherical twist around O Y br .

Theorem 6. 2 .
The morphism p EPW : N EPW → D is an open embedding and commutes with the natural involutions, i.e. p EPW • r EPW = r D • p EPW .Proof.That p EPW is an open embedding follows from Verbitsky's Torelli theorem; see [O'G15, Theorem 1.3] for a more precise statement.That p EPW commutes with the involutions is proved in [O'G08].
We note that T O (O s ⊗ O(D)) = T O (O s ) = I s [1].Since O(D) is very ample, the sheaf I s (D) has three sections, no higher cohomology, and is globally generated; thus the image of O s is F s [2], where F s is defined by the short exact sequence 0 → F s → O ⊗ H 0 (I s (D)) → I s (D) → 0.It is a simple rank 2 bundle with Ext1 (F s , F s ) = Ext 1 (O s , O s ) = C 2 and c 1 (F s ) = −D.By Proposition 8.3, F s is µ D -slope stable unless it is destabilized by O −D − (D − 2E) 2 = O(−L),where we recall that L = D − E is the class of the line on S. Since the natural morphismHom(O(−L), O ⊗ H 0 (I s (D)) = H 0 (I s (D)) → Hom(O(−L), I s (D)) = H 0 (I s (D + L)),induced by multiplying with the defining section of O(L) is injective, Hom(O(−L), F s ) = 0, and thus F s is µ D -stable.Now we consider the RHS of (8.2), applied to O s .After the first three steps we reachT O(−D) (O s ⊗ O(−E)) ⊗ O(D + E) = I s (E)[1].Since O(E) is globally generated and has two sections (inducing the elliptic fibration), there is a unique section of I s (E) vanishing at the elliptic fiber E s containing s; thusT O (I s (E)[1]) = I s/Es [1],whereIs/Es denotes the image of the composition I s → O S → O Es .By Lemma 8.4, U | Es is a stable vector bundle on the elliptic curve E s of rank two and degree −3.By Serre duality, Ext 1 S (U , I s/Es ) = Ext 1 Es (U | Es , I s/Es ) = Hom Es (I s/Es , U | Es ) ∨ = 0, and therefore by Riemann-Roch Hom(U , I s/Es ) = C.Using stability of U | Es once more, we see that this map must be surjective.Therefore, T U I s/Es [1]) = V s [2] where the vector bundle V s with v(V s ) = (2, −D−2E, 4) is defined by the short exact sequence 0 → V s → U → I s/Es → 0. (8.3)By Proposition 8.3, V s is µ D -stable unless it is destabilized by O −D − 2E − (D − 2E) 2 = O(−D).We claim that Hom(O(−D), V s [i]) = 0 for all i if s / ∈ L, and Hom(O(−D), V s ) = 0 if s ∈ L. To prove the claim, first note that by stability Ext 2 (O(−D), U ) = Hom(U , O(−D)) ∨ = 0.Moreover, Ext 1 (O(−D), U ) = Ext 1 (U , O(−D)) ∨ = 0; otherwise, the corresponding extension would define µ D -stable vector bundle of Mukai vector (3, −2D − E, 6), whose square is −8, a contradiction.Therefore, Hom • (O(−D), U ) = C 2 [0] = Hom • (O(−D), I s/Es ).To prove the claim, we need to show that the long exact sequence obtained by applying Hom(O(−D), −) to the map U → I s/Es in (8.3) induces an isomorphism in degree 0 if and only if s / ∈ L. Every nonzero morphism O(−D) → U fits into short exact sequence 0 → O(−D) → U → I s ′ (−E) → 0 An easy computation shows that s ′ ∈ L, as otherwise Ext 1 (I s ′ (−E), O(−D)) = H 1 (I s ′ (L)) ∨ = 0. Conversely, if s ′ ∈ L, the extension exists, and thus arises from a morphism O(−D) → U .Now observe that the composition O(−D) → U → I s/Es is nonzero unless there is a morphism I s ′ (−E) → I s/Es , which exists if and only if s = s ′ .This proves the claim.Thus T O(−D) V s = V s if s / ∈ L, and Proposition 8.3 shows that T O(−D) V s is µ D -stable for all s ∈ S. Thus both the LHS and the RHS of (8.2) send O s to the shift by [2] of a µ D -stable vector bundle.By Lemma 8.6 and Remark 8.7, this completes the proof of the proposition.8.4.Z/2-action in families.Proof of Theorem 1.6.Consider the quasi-projective moduli space F 6 of degree 10 (genus 6) polarized K3 surfaces.It contains as a Noether-Lefschetz divisor the locus of K3 surfaces lattice polarizable with the lattice 4 3 3 0 spanned by D and E where D + E corresponds to the given polarization of degree 10.Let C ⊂ F 6 be a smooth curve intersecting this divisor transversely at a single point o ∈ C, such that o corresponds to a quartic K3 of Picard rank two containing a line.By base change to a finite cover if necessary, we can assume that there exists a family of K3 surfaces π : S → C with polarization H.The very general point of C necessarily corresponds to a K3 surface of Picard rank one, which is a GM surface by [Muk02] and [GLT15, Lemma 2.8]; shrinking C if necessary, we can assume C \ {o} parameterizes only GM surfaces.
Mukai vector (2, −D − E, 3) appearing in Lemma 8.4 on S o .Let i o : S o → S be the inclusion of the special fiber.Then the object i o * O(−D) is spherical; indeed, by our choice of the curve C ⊂ F 6 the object O(−D) does not deform in the family π : S → C, so the claim holds by [HT06, Proposition 1.4].Now consider the following autoequivalence of D perf (S): Π = T io * O(−D) • T U/C • T O/C • − ⊗ O S (H) [−1], where T io * O(−D) is the spherical twist around i o * O(−D), and T U/C and T O/C are the spherical twists associated to the spherical functors D perf (C) → D perf (S) given by F → π * F ⊗ U and F → π * F .Each of these three spherical twists are associated to C-linear spherical functors: in the case of T io * O(−D) for the functor D perf (o) → D perf (S), V → i o * O(−D) ⊗ V .Hence the spherical twists, and thus Π, are also C-linear.The autoequivalence Π induces an autoequivalence Π c on D perf (S c ) for every fiber by base change.For c = 0, it is the residual action Φ GM of Theorem 4.15 on GM K3s as the CY2 covers of Kuznetsov components of corresponding special GM threefolds.For c = o, by [HT06, Proposition 2.7 and Proposition 2.9] we get Π o = T 2 O(−D) • T U • T O • − ⊗ O(D + E) [−1] = T 2 O(−D) 6. Indeed, by Proposition 3.15 the category C is smooth and proper over B, and by Lemma 3.14 the fiber C b over b ∈ B is given by the invariant category D b (S b ) Z/2 for the induced Z/2-action φ b .Let Y → P 3 be the quartic double solid branched along S o , and for b = 0 let X b be the GM threefold opposite (in the sense of Definition 4.12) to the GM surface S b .By construction, for b = o the action of φ o on D b (S o ) is conjugate to the residual action on D b (S o ) from Theorem 4.15(1), and thus its invariant category is equivalent to Ku(Y ) (Lemma 4.5).Similarly, by construction and Remark 4.2, for b = o the action φ b is equivalent to the residual action on D b (S b ) from Theorem 4.15(3), so its invariant category is equivalent to Ku(X b ).