Derived categories of Gushel-Mukai varieties
aa r X i v : . [ m a t h . AG ] O c t DERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES
ALEXANDER KUZNETSOV AND ALEXANDER PERRY
Abstract.
We study the derived categories of coherent sheaves on Gushel–Mukai varieties.In the derived category of such a variety, we isolate a special semiorthogonal component,which is a K3 or Enriques category according to whether the dimension of the variety iseven or odd. We analyze the basic properties of this category using Hochschild homology,Hochschild cohomology, and the Grothendieck group.We study the K3 category of a Gushel–Mukai fourfold in more detail. Namely, we show thiscategory is equivalent to the derived category of a K3 surface for a certain codimension 1family of rational Gushel–Mukai fourfolds, and to the K3 category of a birational cubicfourfold for a certain codimension 3 family. The first of these results verifies a special caseof a duality conjecture which we formulate. We discuss our results in the context of therationality problem for Gushel–Mukai varieties, which was one of the main motivations forthis work.
Contents
1. Introduction 21.1. Background 21.2. GM categories 31.3. Conjectures on duality and rationality 31.4. Main results 41.5. Further directions 41.6. Organization of the paper 51.7. Notation and conventions 51.8. Acknowledgements 62. GM categories 62.1. GM varieties 62.2. Definition of GM categories 82.3. Serre functors of GM categories 92.4. Hochschild homology of GM categories 92.5. Hochschild cohomology of GM categories 102.6. Grothendieck groups of GM categories 132.7. Geometricity of GM categories 152.8. Self-duality of GM categories 163. Conjectures on duality and rationality 173.1. EPW sextics and moduli of GM varieties 17
Date : October 26, 2017.A.K. was partially supported by the Russian Academic Excellence Project “5-100”, by RFBR grant 15-01-02164, and by the Simons Foundation. A.P. was partially supported by NSF GRFP grant DGE1144152and NSF MSPRF grant DMS-1606460, and thanks the Laboratory of Algebraic Geometry NRU-HSE for itshospitality in December 2015, when part of this work was carried out.
Introduction
This paper studies the derived categories of coherent sheaves on smooth Gushel–Mukaivarieties, with a special focus on the relation to birational geometry and the case of fourfolds.1.1.
Background.
For the purpose of this paper, we use the following definition.
Definition 1.1. A Gushel–Mukai ( GM ) variety is a smooth n -dimensional intersection X = Cone(Gr(2 , ∩ P n +4 ∩ Q, ≤ n ≤ , where Cone(Gr(2 , ⊂ P is the cone over the Grassmannian Gr(2 , ⊂ P in its Pl¨uckerembedding, P n +4 ⊂ P is a linear subspace, and Q ⊂ P n +4 is a quadric hypersurface.We note that a more general definition of GM varieties, which includes singular varietiesand curves, is given in [10, Definition 2.1]. However, the definition there agrees with oursafter imposing the condition that a GM variety is smooth of dimension at least 2, see [10,Proposition 2.28]. The classification results of Gushel [15] and Mukai [41], generalized andsimplified in [10, Theorem 2.16], show that this class of varieties coincides with the classof all smooth Fano varieties of Picard number 1, coindex 3, and degree 10, together withBrill–Noether general polarized K3 surfaces of degree 10.In the Fano–Iskovskikh–Mori–Mukai classification of Fano threefolds, GM threefolds occupyan intermediate position between complete intersections in weighted projective spaces andlinear sections of homogeneous varieties, and possess a particularly rich birational geometry.The case of GM fourfolds is even more interesting, and was our original source of motivation.These fourfolds are similar to cubic fourfolds from several points of view: birational geometry,Hodge theory, and as we will see, derived categories.In terms of birational geometry, both types of fourfolds are unirational and rational exam-ples are known. On the other hand, a very general fourfold of either type is expected to beirrational, but to date irrationality has not been shown for a single example. ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 3
At the level of Hodge theory, a fourfold of either type has middle cohomology of K3 type,i.e. h , = 0 and h , = 1. Moreover, there is a classification of Noether–Lefschetz loci wherethe “non-special cohomology” is isomorphic to (a Tate twist of) the primitive cohomology ofa polarized K3 surface. This is due to Hassett for cubics [16], and Debarre–Iliev–Manivel [9]for GM fourfolds.Finally, the first author studied the derived categories of cubic fourfolds in [28]. For anycubic fourfold X ′ , a “K3 category” A X ′ is constructed as a semiorthogonal component of thederived category D b ( X ′ ), and it is shown for many rational X ′ that A X ′ is equivalent to thederived category of an actual K3 surface. Since their introduction, the categories A X ′ haveattracted a great deal of attention, see for instance [2], [17], [39], [7].1.2. GM categories.
We show in this paper that the parallel between GM and cubic four-folds persists at the level of derived categories. In fact, for any GM variety X — not necessarilyof dimension 4 — we define a semiorthogonal component A X of its derived category as theorthogonal to an exceptional sequence of vector bundles. Namely, projection from the ver-tex of Cone(Gr(2 , f : X → Gr(2 , U X on X . If n = dim X , we show in Proposition 2.3 that there is a semiorthogonaldecompositionD b ( X ) = h A X , O X , U ∨ X , O X (1) , U ∨ X (1) , . . . , O X ( n − , U ∨ X ( n − i . The
GM category A X is the main object of study of this paper. Its properties depend on theparity of the dimension n . For instance, we show that in terms of Serre functors, A X is a “K3category” or “Enriques category” according to whether n is even or odd (Proposition 2.6).We support the K3-Enriques analogy by showing that each GM category has a canonicalinvolution such that the corresponding equivariant category is equivalent to a GM categoryof opposite parity (Proposition 2.7).We also compute the Hochschild homology (Proposition 2.9), Hochschild cohomology (Corol-lary 2.11 and Proposition 2.12), and (in the very general case) the numerical Grothendieckgroup (Proposition 2.25 and Lemma 2.27) of GM categories. Our computation of Hochschildhomology and Grothendieck groups is based on their additivity, while for Hochschild coho-mology we rely on results on equivariant Hochschild cohomology from [52].We deduce from our computations structural properties of GM categories. Notably, weshow that for any GM variety of odd dimension or for a very general GM variety of evendimension greater than 2, the category A X is not equivalent to the derived category of anyvariety (Proposition 2.29).1.3. Conjectures on duality and rationality.
We formulate two conjectures about GMcategories. First we introduce a notion of “generalized duality” between GM varieties. Theprecise definition of this notion is somewhat involved (see § X is parameterized by the quotient of the projective space P by a finite group. We alsoformulate a similar notion of “generalized GM partners”, which reduces to [10, Definition 3.22]when the varieties have the same dimension. We conjecture that generalized dual GM varietiesand generalized GM partners have equivalent GM categories (Conjecture 3.7).Our second conjecture concerns the rationality of GM fourfolds, and is directly inspiredby an analogous conjecture for cubic fourfolds from [28]. Namely, we conjecture that the ALEXANDER KUZNETSOV AND ALEXANDER PERRY
GM category of a rational GM fourfold is equivalent to the derived category of a K3 surface(Conjecture 3.12). Together with Proposition 2.29, this conjecture implies that a very generalGM fourfold is not rational.1.4.
Main results.
Our first main result gives evidence for the above two conjectures. AGM variety as in Definition 1.1 is called ordinary if P n +4 does not intersect the vertexof Cone(Gr(2 , Theorem 1.2.
Let X be an ordinary GM fourfold containing a quintic del Pezzo surface.Then there is a K surface Y such that A X ≃ D b ( Y ) . For a more precise statement, see Theorem 4.1. The K3 surface Y is in fact a GM surfacewhich is generalized dual to X , and the GM fourfold X is rational (Lemma 4.7). Thus The-orem 1.2 verifies special cases of our duality and rationality conjectures. We note that GMfourfolds as in the theorem form a 23-dimensional (codimension 1 in moduli) family.By Theorem 1.2, the categories A X of GM fourfolds are deformations of the derived categoryof a K3 surface. Yet, as mentioned above, for very general X these categories are not equivalentto the derived category of a K3 surface. There even exist X such that A X is not equivalentto the twisted derived category of a K3 surface (see Remark 5.9). Families of categories withthese properties appear to be quite rare; this is the first example since [28].Our second main result shows that the K3 categories attached to GM and cubic fourfoldsare not only analogous, but in some cases even coincide. Theorem 1.3.
Let X be a generic ordinary GM fourfold containing a plane of type Gr(2 , .Then there is a cubic fourfold X ′ such that A X ≃ A X ′ . For a more precise statement, see Theorem 5.8. The cubic fourfold X ′ is given explicitlyby a construction of Debarre–Iliev–Manivel [9]. In fact, X ′ is birational to X and we use thestructure of this birational isomorphism to establish the result. We note that GM fourfoldsas in the theorem form a 21-dimensional (codimension 3 in moduli) family. Theorem 1.3 canbe considered as a step toward a 4-dimensional analogue of [26], which exhibits mysteriouscoincidences among the derived categories of Fano threefolds.1.5. Further directions.
The above results relate the K3 categories attached to three dif-ferent types of varieties: GM fourfolds, cubic fourfolds, and K3 surfaces (in the last case theK3 category is the whole derived category). We call two such varieties X and X derivedpartners if their K3 categories are equivalent. There is also a notion of X and X being Hodge-theoretic partners . Roughly, this means that there is an “extra” integral middle-degreeHodge class α i on X i , such that if K i ⊂ H dim( X i ) ( X i , Z ) denotes the lattice generated by α i and certain tautological algebraic cycles on X i , then the orthogonals K ⊥ and K ⊥ are iso-morphic as polarized Hodge structures (up to a Tate twist). This notion was studied in [16],[9], under the terminology that “ X is associated to X ”. Using lattice theoretic techniques,countably many families of GM fourfolds with Hodge-theoretic K3 and cubic fourfold partnersare produced in [9].We expect that a GM fourfold has a derived partner of a given type if and only if ithas a Hodge-theoretic partner of the same type. Theorems 1.2 and 1.3 can be thought of asevidence for this expectation, since by [9, § § ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 5 and Thomas [2] proved (generically) the analogous expectation for K3 partners of cubic four-folds. Their method is deformation theoretic, and requires as a starting point an analogue ofTheorem 1.2 for cubic fourfolds.Finally, we note that there are some other Fano varieties which fit into the above story,i.e. whose derived category contains a K3 category. One example is provided by a hyperplanesection of the Grassmannian Gr(3 , P × P × P × P in a K3 surface, and those of type (c7) are isomorphic to the blowupof a cubic fourfold in a Veronese surface. In particular, these fourfolds do indeed have a K3category in their derived category, but they reduce to known examples. Fourfolds of type (c5),however, conjecturally give rise to genuinely new K3 categories (see [34] for a discussion ofthe geometry of these fourfolds).1.6. Organization of the paper. In §
2, we define GM categories and study their basic prop-erties. After recalling some facts about GM varieties in § § § § § §
3, we formulate our conjectures about the duality and rationality of GM varieties. Thepreliminary § §
3] a description of the set of isomorphism classes of GMvarieties in terms of Lagrangian data. In § § § § § § § § § Notation and conventions.
We work over an algebraically closed field k of character-istic 0. A variety is an integral, separated scheme of finite type over k . A vector bundle on avariety X is a finite locally free O X -module. The projective bundle of a vector bundle E on avariety X is P ( E ) = Proj(Sym • ( E ∨ )) π / / X, with O P ( E ) (1) normalized so that π ∗ O P ( E ) (1) = E ∨ . We often commit the following convenientabuse of notation: given a divisor class D on a variety X , we denote still by D its pullbackto any variety mapping to X . Throughout the paper, we use V n to denote an n -dimensionalvector space. We denote by G = Gr(2 , V ) the Grassmannian of 2-dimensional subspaces of V . ALEXANDER KUZNETSOV AND ALEXANDER PERRY
In this paper, triangulated categories are k -linear and functors between them are k -linearand exact. For a variety X , by the derived category D b ( X ) we mean the bounded derivedcategory of coherent sheaves on X , regarded as a triangulated category. For a morphism ofvarieties f : X → Y , we write f ∗ : D b ( X ) → D b ( Y ) for the derived pushforward (provided f is proper), and f ∗ : D b ( Y ) → D b ( X ) for the derived pullback (provided f has finite Tor-dimension). For F , G ∈ D b ( X ), we write F ⊗ G for the derived tensor product.We write T = h A , . . . , A n i for a semiorthogonal decomposition of a triangulated category T with components A , . . . , A n . For an admissible subcategory A ⊂ T we write A ⊥ = { F ∈ T | Hom( G , F ) = 0 for all G ∈ A } , ⊥ A = { F ∈ T | Hom( F , G ) = 0 for all G ∈ A } , for its right and the left orthogonals, so that we have T = h A ⊥ , A i = h A , ⊥ A i .We regard graded vector spaces as complexes with trivial differential, so that any suchvector space can be written as W • = L n W n [ − n ], where W n denotes the degree n piece. Weoften suppress the degree 0 shift [0] from our notation.1.8. Acknowledgements.
We would like to thank Olivier Debarre, Joe Harris, and Johan deJong for many useful discussions. We are also grateful to Daniel Huybrechts, Richard Thomas,and Ravi Vakil for comments and questions. Finally, we thank the referee for suggestions aboutthe presentation of the paper. 2.
GM categories
In this section, we define GM categories and study their basic properties. We start with aquick review of the key features of GM varieties.2.1.
GM varieties.
Let V be a 5-dimensional vector space and G = Gr(2 , V ) the Grass-mannian of 2-dimensional subspaces. Consider the Pl¨ucker embedding G ֒ → P ( ∧ V ) and letCone( G ) ⊂ P ( k ⊕ ∧ V ) be the cone over G . Further, let W ⊂ k ⊕ ∧ V be a linear subspace of dimension n +5 with 2 ≤ n ≤
6, and Q ⊂ P ( W ) a quadric hypersurface.By Definition 1.1, if the intersection X = Cone( G ) ∩ Q (2.1)is smooth and transverse, then X is a GM variety of dimension n , and every GM variety canbe written in this form.There is a natural polarization H on a GM variety X , given by the restriction of thehyperplane class on P ( k ⊕ ∧ V ); we denote by O X (1) the corresponding line bundle on X .It is straightforward to check that H n = 10 and − K X = ( n − H. (2.2)Moreover, we haveif dim( X ) ≥ , then Pic( X ) = Z H , andif dim( X ) = 2 , then ( X, H ) is a Brill–Noether general K3 surface. (2.3)Conversely, by [10, Theorem 2.16] any smooth projective polarized variety of dimension ≥ ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 7
The intersection Cone( G ) ∩ Q does not contain the vertex of the cone, since X is smooth.Hence projection from the vertex defines a regular map f : X → G , called the Gushel map . Let U be the rank 2 tautological subbundle on G . Then U X = f ∗ U is a rank 2 vector bundle on X , called the Gushel bundle . By [10, § X , i.e. only depend on the abstract polarizedvariety ( X, H ) and not on the particular realization (2.1). In particular, so is the space V (being the dual of the space of sections of U ∨ X ), and we will sometimes write it as V ( X ) toemphasize this. The space W is also canonically associated to X , since its dual is the space ofglobal sections of O X (1). The quadric Q , however, is not canonically associated to X , see § M X = Cone( G ) ∩ P ( W )is called the Grassmannian hull of X . Note that X = M X ∩ Q is a quadric section of M X .Let W ′ be the projection of W to ∧ V . The intersection M ′ X = G ∩ P ( W ′ ) (2.4)is called the projected Grassmannian hull of X . Again by [10], both M X and M ′ X are canoni-cally associated to X .The Gushel map is either an embedding or a double covering of M ′ X , according to whetherthe projection map W → W ′ is an isomorphism or has 1-dimensional kernel. In the first case, W ∼ = W ′ and M X ∼ = M ′ X . Then considering Q as a subvariety of P ( W ′ ), we have X ∼ = M ′ X ∩ Q. (2.5)That is, X is a quadric section of a linear section of the Grassmannian G . A GM variety ofthis type is called ordinary .If the map W → W ′ has 1-dimensional kernel, then we have P ( W ) = Cone( P ( W ′ ))and M X = Cone( M ′ X ). As Q does not contain the vertex of the cone (by smoothness of X ),projection from the vertex gives a double cover X −−−→ M ′ X . (2.6)That is, X is a double cover of a linear section of the Grassmannian G . A GM variety of thistype is called special . Lemma 2.1 ([10, Proposition 2.22]) . Let X be a GM variety of dimension n . Then theintersection (2.4) defining M ′ X is dimensionally transverse. Moreover: (1) If n ≥ , or if n = 2 and X is special, then M ′ X is smooth. (2) If n = 2 and X is ordinary, then M ′ X has at worst rational double point singularities. By Lemma 2.1, if X is special then M ′ X is smooth. Further, by [10, § X ′ = G ∩ Q ′ , where Q ′ = Q ∩ P ( W ′ ) is aquadric hypersurface in P ( W ′ ). Hence, as long as n ≥
3, the branch divisor X ′ of (2.6) is anordinary GM variety of dimension n −
1. This gives rise to an operation taking a GM varietyof one type to the opposite type, by defining in this situation X op = X ′ and ( X ′ ) op = X. (2.7)Note that we have dim X op = dim X ±
1. The opposite GM variety is not defined for specialGM surfaces.
ALEXANDER KUZNETSOV AND ALEXANDER PERRY
Definition of GM categories.
By the discussion in § X of dimen-sion n ≥ M ′ X of G by taking a quadric section ora branched double cover. To describe a natural semiorthogonal decomposition of D b ( X ), wefirst recall that G and its smooth linear sections of dimension at least 3 admit rectangularLefschetz decompositions (in the sense of [25, § O G , U ∨ form an exceptional pair in D b ( G ), where recall U denotes the tautologicalrank 2 bundle. Let B = h O G , U ∨ i ⊂ D b ( G ) (2.8)be the triangulated subcategory they generate. The following result holds by [24, § Lemma 2.2.
Let M be a smooth linear section of G ⊂ P ( ∧ V ) of dimension N ≥ . Let i : M ֒ → G be the inclusion. (1) The functor i ∗ : D b ( G ) → D b ( M ) is fully faithful on B ⊂ D b ( G ) . (2) Denoting the essential image of B by B M , there is a semiorthogonal decomposition D b ( M ) = h B M , B M (1) , . . . , B M ( N − i . (2.9)The next result gives a semiorthogonal decomposition of the derived category of a GMvariety. Proposition 2.3.
Let X be a GM variety of dimension n ≥ . Let f : X → G be the Gushelmap. (1) The functor f ∗ : D b ( G ) → D b ( X ) is fully faithful on B ⊂ D b ( G ) . (2) Denoting the essential image of B by B X , so that B X = h O X , U ∨ X i , there is a semiorthog-onal decomposition D b ( X ) = h A X , B X , B X (1) , . . . , B X ( n − i , (2.10) where A X is the right orthogonal category to h B X , . . . , B X ( n − i ⊂ D b ( X ) .Thus D b ( X ) has a semiorthogonal decomposition with the category A X and n − exceptionalobjects as components. Remark 2.4. If n = 2 we set A X = D b ( X ), so that (2.10) still holds. Proof.
The Gushel map factors through the map X → M ′ X to the projected Grassmannianhull M ′ X defined by (2.4). By Lemma 2.1, M ′ X is smooth and has dimension n + 1 if X is ordinary, or dimension n if X is special. In particular, D b ( M ′ X ) has a semiorthogonaldecomposition of the form (2.9). Further, X → M ′ X realizes X as a quadric section (2.5) if X is ordinary, or as a double cover (2.6) if X is special. Now applying [36, Lemmas 5.1 and 5.5]gives the result. (cid:3) Definition 2.5.
Let X be a GM variety. The GM category of X is the category A X definedby the semiorthogonal decomposition (2.10).More explicitly, using the definition (2.8) of B , the defining semiorthogonal decompositionof a GM category A X can be written asD b ( X ) = h A X , O X , U ∨ X , . . . , O X ( n − , U ∨ X ( n − i . (2.11)The GM category A X is the main object of study of this paper. As we will see below, itsproperties depend strongly on the parity of dim( X ). For this reason, we sometimes emphasizethe parity of dim( X ) by calling A X an even or odd GM category according to whether dim( X )is even or odd. ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 9
Serre functors of GM categories.
Recall from [6] that a
Serre functor for a triangu-lated category T is an autoequivalence S T : T → T with bifunctorial isomorphismsHom( F , S T ( G )) ∼ = Hom( G , F ) ∨ for F , G ∈ T . If a Serre functor exists, it is unique. If X is a smooth proper variety, then D b ( X )has a Serre functor given by the formulaS D b ( X ) ( F ) = F ⊗ ω X [dim X ] . (2.12)Moreover, given an admissible subcategory A ⊂ T , if T admits a Serre functor then so does A .Using [30], we can describe the Serre functor of a GM category. Proposition 2.6.
Let X be a GM variety of dimension n . (1) If n is even, the Serre functor of the GM category A X satisfies S A X ∼ = [2] . (2) If n is odd, the Serre functor of the GM category A X satisfies S A X ∼ = σ ◦ [2] for anontrivial involutive autoequivalence σ of A X . If in addition X is special, then σ isinduced by the involution of the double cover (2.6) .Proof. If n = 2, then A X = D b ( X ) and X is a K3 surface, so the result holds by (2.12).If n ≥
3, then as in the proof of Proposition 2.3 we may express X as a quadric section ordouble cover of the smooth variety M ′ X . It is easy to see the length m of the semiorthogonaldecomposition of D b ( M ′ X ) given by Lemma 2.2 satisfies K M ′ X = − mH , where H is therestriction of the ample generator of Pic( G ). Hence we may apply [30, Corollaries 3.7 and 3.8]to see that the Serre functors have the desired form.If σ were trivial, then the Hochschild homology HH − ( A X ) would be nontrivial (see Propo-sition 2.10), which contradicts the computation of Proposition 2.9 below. (cid:3) Proposition 2.6 shows that even GM categories can be regarded as “noncommutative K3surfaces”, and odd GM categories can be regarded as “noncommutative Enriques surfaces”.This analogy goes further than the relation between Serre functors. For instance, any Enriquessurface (in characteristic 0) is the quotient of a K3 surface by an involution. Similarly, theresults of [36] show that odd GM categories can be described as “quotients” of even GMcategories by involutions. To state this precisely, recall from § X is a special GMsurface, there is an associated GM variety X op of the opposite type and parity of dimension.The following result is proved in [36, § Proposition 2.7.
Let X be a GM variety which is not a special GM surface. Then there isa Z / -action on the GM category A X such that if A Z / X denotes the equivariant category, thenthere is an equivalence A Z / X ≃ A X op . If σ is the autoequivalence generating the Z / -action on A X , then σ is induced by the in-volution of the double covering X → M ′ X if X is special, and σ = S A X ◦ [ − if dim( X ) isodd. Hochschild homology of GM categories.
Given a suitably enhanced triangulatedcategory A , there is an invariant HH • ( A ) called its Hochschild homology , which is a graded k -vector space. We will exclusively be interested in admissible subcategories of the derived cat-egory of a smooth projective variety. For a definition of Hochschild homology in this context,see [27]. If A = D b ( X ), we write HH • ( X ) for HH • ( A ). The Hochschild–Kostant–Rosenberg (HKR)isomorphism gives the following explicit description of Hochschild homology in this case [40]:HH i ( X ) ∼ = M q − p = i H q ( X, Ω pX ) . (2.13)An important property of Hochschild homology is that it is additive under semiorthogonaldecompositions. Theorem 2.8 ([27, Theorem 7.3]) . Let X be a smooth projective variety. Given a semiorthog-onal decomposition D b ( X ) = h A , A , . . . , A m i , there is an isomorphism HH • ( X ) ∼ = m M i =1 HH • ( A i ) . By combining this additivity property with the HKR isomorphism for GM varieties, we cancompute the Hochschild homology of GM categories.
Proposition 2.9.
Let X be a GM variety of dimension n . Then HH • ( A X ) ∼ = (cid:26) k [2] ⊕ k ⊕ k [ − if n is even, k [1] ⊕ k ⊕ k [ − if n is odd.Proof. By (2.11) there is a semiorthogonal decomposition of D b ( X ) with A X and 2( n − k . Hence byadditivity, HH • ( X ) ∼ = HH • ( A X ) ⊕ k n − . By (2.13) the graded dimension of HH • ( X ) can be computed by summing the columns of theHodge diamond of X , which looks as follows (see [38], [20], [43], and [11]):dim( X ) = 2 dim( X ) = 3 dim( X ) = 4 dim( X ) = 5 dim( X ) = 6
10 01 20 10 01 10 00 1 00 10 10 00 1 00 01 10 00 1 00 0 0 00 1 22 1 00 0 0 00 1 00 01 10 00 1 00 0 0 00 0 2 0 00 0 10 10 0 00 0 2 0 00 0 0 00 1 00 01 10 00 1 00 0 0 00 0 2 0 00 0 0 0 0 00 0 1 22 1 0 00 0 0 0 0 00 0 2 0 00 0 0 00 1 00 01
Now the lemma follows by inspection. (cid:3)
Hochschild cohomology of GM categories.
Given a suitably enhanced triangulatedcategory A , there is also an invariant HH • ( A ) called its Hochschild cohomology , which has thestructure of a graded k -algebra. Again, for a definition in the case where A is an admissiblesubcategory of the derived category of a smooth projective variety, see [27].If A = D b ( X ), we write HH • ( X ) for HH • ( A ). There is the following version of the HKRisomorphism for Hochschild cohomology [40]:HH i ( X ) ∼ = M p + q = i H q ( X, ∧ p T X ) . (2.14) ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 11
Hochschild cohomology is not additive under semiorthogonal decompositions, and so itis generally much harder to compute than Hochschild homology. There is, however, a casewhen the computation simplifies considerably. Recall that a triangulated category A is called n -Calabi–Yau if the shift functor [ n ] is a Serre functor for A . Proposition 2.10 ([30, Proposition 5.2]) . Let A be an admissible subcategory of D b ( X ) fora smooth projective variety X . If A is an n -Calabi–Yau category, then for each i there is anisomorphism of vector spaces HH i ( A ) ∼ = HH i − n ( A ) . This immediately applies to even GM categories, as by Proposition 2.6 they are 2-Calabi–Yau.
Corollary 2.11.
Let X be a GM variety of even dimension. Then HH • ( A X ) ∼ = k ⊕ k [ − ⊕ k [ − . The Hochschild cohomology of odd GM categories is significantly harder to compute. Ourstrategy is to exploit the fact that there is a Z / homology of an odd GM category. Proposition 2.12.
Let X be a GM variety of odd dimension. Then HH • ( A X ) ∼ = k ⊕ k [ − ⊕ k [ − . Proof.
Recall that by Proposition 2.7 there is a Z / A X such that if σ : A X → A X denotes the corresponding involutive autoequivalence, then:(1) S A X = σ ◦ [2] is a Serre functor for A X .(2) A Z / X ≃ A X op , where X op is the opposite variety to X .As stated, these are results at the level of triangulated categories, but they also hold atthe enhanced level. Namely, in the terminology of [52], there is a k -linear stable ∞ -categoryD b ( X ) enh (denoted Perf( X ) in [52]) with homotopy category D b ( X ). The category D b ( X ) enh admits a semiorthogonal decomposition of the same form as (2.10), which defines a k -linearstable ∞ -category A enh X whose homotopy category is A X . If σ enh : A enh X → A enh X denotes thecorresponding involutive autoequivalence, then (1) and (2) above hold with A X , S A X , σ ,and A X op replaced by their enhanced versions, and (1) and (2) are recovered by passing tohomotopy categories. The Hochschild (co)homology of A X and A X op agree with the corre-sponding invariants of their enhancements. Hence [52, Corollary 1.3] givesHH • ( A X op ) ∼ = HH • ( A X ) ⊕ (HH • ( A X ) Z / [ − Z / • ( A X ) is induced by σ .Since X has odd dimension (and hence X op has even dimension), by Corollary 2.11 we haveHH • ( A X op ) ∼ = k ⊕ k [ − ⊕ k [ − , and by Proposition 2.9 we haveHH • ( A X ) ∼ = k [1] ⊕ k ⊕ k [ − . Combined with (2.15), this immediately shows HH • ( A X ) Z / is concentrated in degree 0, i.e.HH • ( A X ) Z / ∼ = k d for some 0 ≤ d ≤
2, andHH • ( A X ) ∼ = k ⊕ k − d [ − ⊕ k [ − . To prove d = 2, we apply [53, Corollary 3.11], which gives an equality X i ( − i dim HH i ( A X ) = X i ( − i Tr((S − A X ) ∗ : HH i ( A X ) → HH i ( A X )) . (2.16)Note that since S A X = σ ◦ [2], the map (S − A X ) ∗ : HH i ( A X ) → HH i ( A X ) induced by S − A X onHochschild homology coincides with the map induced by σ , and in particular squares to theidentity. It follows that the right side of (2.16) is bounded above by P i dim HH i ( A X ) = 22.But the left side of (2.16) equals 24 − d where 0 ≤ d ≤
2, so d = 2. (cid:3) Remark 2.13.
As a byproduct, the above proof shows that S A X acts on HH i ( A X ) by ( − i for any GM category A X . Remark 2.14.
It is possible to show d = 2 in the above proof without appealing to the equal-ity (2.16), as follows. Note that the statement is deformation invariant, since it is equivalentto the Euler characteristic P i ( − i dim HH i ( A X ) being 22. So we may assume X is special.Then the Z / A X is induced by the involution of the double cover X → M ′ X . Wewant to show that Z / ( A X ). But HH • ( A X ) is canonically a summandof HH • ( X ), and we claim that the involution of the double cover acts trivially on HH ( X ).Indeed, since X is odd-dimensional, pullback under X → M ′ X induces a surjection on even-degree cohomology and hence on HH . The claim follows. Remark 2.15.
Proposition 2.12 can also be deduced from Conjecture 3.7 stated below. In-deed, the conjecture implies that the GM category of any GM variety of odd dimension isequivalent to that of an ordinary GM threefold, whose Hochschild cohomology can be com-puted using [27, Theorem 8.8]. Yet another method for computing the Hochschild cohomologyof GM categories is via the normal Hochschild cohomology spectral sequence of [31], but thismethod becomes long and complicated for GM varieties of dimension bigger than 3.As an application, we discuss the indecomposability of GM categories. Recall that a trian-gulated category T is called indecomposable if it admits no nontrivial semiorthogonal decom-positions, i.e. if T = h A , A i implies either A ≃ A ≃
0. In general, there are very fewtechniques for proving indecomposability of a triangulated category. However, for Calabi–Yaucategories, we recall a simple criterion below.If A is an admissible subcategory of the derived category of a smooth projective variety,we say A is connected if HH ( A ) = k (see [30, § Proposition 2.16 ([30, Proposition 5.5]) . Let A be a connected admissible subcategory ofthe derived category of a smooth projective variety. Then A admits no nontrivial completelyorthogonal decompositions. If furthermore A is Calabi–Yau, then A is indecomposable. Corollary 2.17.
Let X be a GM variety of dimension n . (1) If n is even, then A X is indecomposable. (2) If n is odd, then A X admits no nontrivial completely orthogonal decompositions.Proof. This follows from Proposition 2.16, the connectivity of A X , and the fact that A X isCalabi–Yau if n is even. (cid:3) Remark 2.18.
It is plausible that A X is indecomposable if X is an odd-dimensional GMvariety, but we do not know how to prove this. ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 13
Grothendieck groups of GM categories.
The
Grothendieck group K ( T ) of a trian-gulated category T is the free group on isomorphism classes [ F ] of objects F ∈ T , modulo therelations [ G ] = [ F ] + [ H ] for every distinguished triangle F → G → H .Assume T is proper , i.e. that L i Hom( F , G [ i ]) is finite dimensional for all F , G ∈ T . Forinstance, this holds if T is admissible in the derived category of a smooth projective variety.Then for F , G ∈ T , we set χ ( F , G ) = X i ( − i dim Hom( F , G [ i ]) . This descends to a bilinear form χ : K ( T ) × K ( T ) → Z , called the Euler form . In generalthis form is neither symmetric nor antisymmetric. However, if T admits a Serre functor (e.g.if T is admissible in the derived category of a smooth projective variety), then the left andright kernels of the form χ agree, and we denote this common subgroup of K ( T ) by ker( χ ).In this situation, the numerical Grothendieck group is the quotientK ( T ) num = K ( T ) / ker( χ ) . Note that K ( T ) num is torsion free, since ker( χ ) is evidently saturated.If X is a smooth projective variety, we writeK ( X ) = K (D b ( X )) and K ( X ) num = K (D b ( X )) num . Further, let CH( X ) and CH( X ) num denote the Chow rings of cycles modulo rational andnumerical equivalence. The following well-known consequence of Hirzebruch–Riemann–Rochrelates the (numerical) Grothendieck group of X to its (numerical) Chow group. Lemma 2.19.
Let X be a smooth projective variety. Then there are isomorphisms K ( X ) ⊗ Q ∼ = CH( X ) ⊗ Q and K ( X ) num ⊗ Q ∼ = CH( X ) num ⊗ Q . Proof.
The isomorphisms are induced by the Chern character ch : K ( X ) → CH( X ) ⊗ Q . Forthe first, see [14, Example 15.2.16(b)]. The second then follows from the observation that,by Riemann–Roch, the kernel of the Euler form is precisely the preimage under the Cherncharacter of the subring of numerically trivial cycles. (cid:3) The following well-known lemma says that Grothendieck groups are additive.
Lemma 2.20.
Let X be a smooth projective variety. Given a semiorthogonal decomposition D b ( X ) = h A , A , . . . , A m i , there are isomorphisms K ( X ) ∼ = m M i =1 K ( A i ) and K ( X ) num ∼ = m M i =1 K ( A i ) num . Proof.
The embedding functors A i ֒ → D b ( X ) induce a map L i K ( A i ) → K ( X ), whoseinverse is the map induced by the projection functors D b ( X ) → A i . This isomorphism alsodescends to numerical Grothendieck groups. (cid:3) Now let X be a GM variety. If X is a surface then A X = D b ( X ), so the Grothendieck groupof A X coincides with that of X . Below we describe K ( A X ) num if X is odd dimensional, orif X is a fourfold or sixfold which is not “Hodge-theoretically special” in the following sense.First, we note that if n denotes the dimension of X , then by Lefschetz theorems (see [11,Proposition 3.4(b)]) the Gushel map f : X → G induces an injectionH n ( G , Z ) ֒ → H n ( X, Z ) . If n is odd, then H n ( G , Z ) simply vanishes. But if n = 4 or 6, then H n ( G , Z ) = Z is generatedby Schubert cycles, and the vanishing cohomology H n van ( X, Z ) is defined as the orthogonal toH n ( G , Z ) ⊂ H n ( X, Z ) with respect to the intersection form. Definition 2.21 ([9]) . Let X be a GM variety of dimension n = 4 or 6. Then X is Hodge-special if H n , n ( X ) ∩ H n van ( X, Q ) = 0 . Lemma 2.22 ([9]) . If X is a very general GM fourfold or sixfold, then X is not Hodge-special. Remark 2.23.
Very general here means that the moduli point [ X ] ∈ M n ( k ) lies in thecomplement of countably many proper closed substacks of M n , where n = dim( X ) and M n isthe moduli stack of n -dimensional GM varieties discussed in Appendix A. Proof.
In the fourfold case, this is [9, Corollary 4.6]. The main point of the proof is thecomputation that the local period map for GM fourfolds is a submersion. The sixfold casecan be proved by the same argument. (cid:3)
Remark 2.24.
Lemma 2.22 can also be proved by combining the the description of themoduli of GM varieties in terms of EPW sextics (see Remark 3.3) with [11, Theorem 5.1].
Proposition 2.25.
Let X be a GM variety of dimension n ≥ . If n is even assume alsothat X is not Hodge-special. Then K ( A X ) num ≃ Z .Proof. The proof is similar to that of Proposition 2.9. First, note that by Proposition 2.3there is a semiorthogonal decomposition of D b ( X ) with A X and 2( n −
2) exceptional objectsas components. Since the category generated by an exceptional object is equivalent to thederived category of a point, both its usual and numerical Grothendieck group is Z . Hence byadditivity, K ( X ) num ∼ = K ( A X ) num ⊕ Z n − . On the other hand, K ( X ) num ⊗ Q ∼ = CH( X ) num ⊗ Q . But under our assumptions on X ,the rational Hodge classes on X are spanned by the restrictions of Schubert cycles on G .In particular, the Hodge conjecture holds for X . So numerical equivalence coincides withhomological equivalence, and CH( X ) num ⊗ Q ∼ = L k H k,k ( X, Q )where H k,k ( X, Q ) = H k,k ( X ) ∩ H k ( X, Q ). Thus using the Hodge diamond of X (recorded inthe proof of Proposition 2.9) and the assumption that X is not Hodge-special if n is even, wefind dim(K ( X ) num ⊗ Q ) = 2 n − . Combined with the above, this shows the rank of K ( A X ) num is 2. Since K ( A X ) num is torsionfree, we conclude K ( A X ) num ∼ = Z . (cid:3) Remark 2.26.
Let X be a GM variety of dimension n = 4 or 6. The proof of the propositionshows that rank(K ( A X ) num ) = dim Q H n , n ( X, Q )if the Hodge conjecture holds for X . The Hodge conjecture holds for any uniruled smoothprojective fourfold [8], so for n = 4 the above equality is unconditional. If n = 6 the Hodgeconjecture can be proved using the correspondences studied in [11], but we do not discuss thedetails here. ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 15
Lemma 2.27.
Let X be a GM variety as in Proposition . Then in a suitable basis, theEuler form on K ( A X ) num = Z is given by (cid:18) − − (cid:19) if n = 3 , (cid:18) − − (cid:19) if n = 4 . Remark 2.28.
The duality conjecture (Conjecture 3.7) implies that if X is as in Propo-sition 2.25, then for n = 5 or 6 the lattice K ( A X ) num = Z is isomorphic to the latticedescribed in Lemma 2.27 for n = 3 or 4, respectively. Proof.
For n = 3, this is shown in the proof of [26, Proposition 3.9].For n = 4, we sketch the proof. First, note that any GM variety contains a line, since bytaking a hyperplane section we reduce to the case of dimension 3, where the result is well-known. Let P ∈ X be a point, L ⊂ X be a line, Σ be the zero locus of a regular section of U ∨ X , S be a complete intersection of two hyperplanes in X , and H be a hyperplane section of X .The key claim is that K ( X ) num = Z h [ O P ] , [ O L ] , [ O Σ ] , [ O S ] , [ O H ] , [ O X ] i , (2.17)i.e. the structure sheaves of these subvarieties give an integral basis of K ( X ) num . Once this isknown, as in the proof of [26, Proposition 3.9], the lemma reduces to a (tedious) computation,which we omit.Using [26, Remark 5.9] it is easy to see X is AK-compatible in the sense of [26, Defini-tion 5.1], hence to prove the claim it is enough to show thatCH( X ) num = Z h [ P ] , [ L ] , [Σ] , [ S ] , [ H ] , i . Clearly, this is equivalent to CH ( X ) num = Z h [Σ] , [ S ] i . But CH ( X ) num coincides with thegroup CH ( X ) hom ⊂ H ( X, Z ) of 2-cycles modulo homological equivalence (see the proof ofProposition 2.25), and Z h [Σ] , [ S ] i is the image of the inclusion H ( G , Z ) ֒ → CH ( X ) hom . Henceit suffices to show the cokernel of this inclusion is torsion free. Even better, the cokernel ofH ( G , Z ) ֒ → H ( X, Z )is torsion free. Indeed, we may assume X is ordinary, and then the statement holds by theproof of the Lefschetz hyperplane theorem, see [37, Example 3.1.18]. (cid:3) Geometricity of GM categories.
Now we consider the question of whether A X isequivalent to the derived category of a variety. The following two results show that in almost allcases, the answer is negative. In § Proposition 2.29.
Let X be a GM variety of dimension n . (1) If n is even and S is a variety such that A X ≃ D b ( S ) , then S is a K surface. (2) If n is odd, then A X is not equivalent to the derived category of any variety. (3) If n = 4 or n = 6 and X is not Hodge-special ( in particular, if X is very general ) ,then A X is not equivalent to the derived category of any variety.Proof. Suppose S is a variety such that A X ≃ D b ( S ). Then S is smooth by [24, Lemma D.22],and proper by [51, Proposition 3.30]. In particular, D b ( S ) has a Serre functor given byS D b ( S ) ( F ) = F ⊗ ω S [dim( S )] , which is unique up to isomorphism. Thus by Proposition 2.6, S is a surface with trivial (if n iseven) or 2-torsion (if n is odd) canonical class. Hence S is a K3, Enriques, abelian, or biellipticsurface. Using the HKR isomorphism and the Hodge diamonds of such surfaces, we findHH • ( S ) = k [2] ⊕ k ⊕ k [ −
2] if S is K3 , k if S is Enriques , k [2] ⊕ k [1] ⊕ k ⊕ k [ − ⊕ k [ −
2] if S is abelian , k [1] ⊕ k ⊕ k [ −
1] if S is bielliptic . Now parts (1) and (2) follow by comparing with HH • ( A X ) as given by Proposition 2.9. For (3)note that if A X ≃ D b ( S ), then K ( A X ) num ∼ = K ( S ) num . But on a projective surface powersof the hyperplane class give 3 independent elements in CH( S ) num ⊗ Q ∼ = K ( S ) num ⊗ Q . Henceby Proposition 2.25, X is Hodge-special. (cid:3) Self-duality of GM categories.
The derived category of a smooth proper variety X is self-dual : if D b ( X ) op denotes the opposite category of D b ( X ) (note that this has nothingto do with the opposite GM variety), there is an equivalence D b ( X ) ≃ D b ( X ) op given by thedualization functor F R H om ( F , O X ). In general, this self-duality property is not inheritedby semiorthogonal components of D b ( X ). Nonetheless, we show below that all GM categoriesare self-dual, which can be thought of as a weak geometricity property.For the proof, we recall some facts about mutation functors (see [5], [6] for more details).For any admissible subcategory A ⊂ T of a triangulated category, there are associated left and right mutation functors L A : T → T and R A : T → T . These functors annihilate A , andtheir restrictions L A | ⊥ A : ⊥ A → A ⊥ and R A | A ⊥ : A ⊥ → ⊥ A are mutually inverse equivalences[6, Lemma 1.9]. If T = h A , . . . , A n i is a semiorthogonal decomposition with admissible com-ponents, then for 1 ≤ i ≤ n − T = h A , . . . , A i − , L A i ( A i +1 ) , A i , A i +2 , . . . , A n i , T = h A , . . . , A i − , A i +1 , R A i +1 ( A i ) , A i +2 , . . . , A n i , and equivalences L A i ( A i +1 ) ≃ A i +1 and R A i +1 ( A i ) ≃ A i (2.18)induced by the mutation functors L A i : T → T and R A i +1 : T → T . When T admits a Serrefunctor S T , the effect of mutating A n or A to the opposite side of the semiorthogonal decom-position of T can be described as follows [6, Proposition 3.6]: T = h S T ( A n ) , A , . . . , A n − i and T = h A , . . . , A n , S − T ( A ) i . (2.19)That is, L h A ,..., A n − i ( A n ) = S T ( A n ) and R h A ,..., A n i ( A ) = S − T ( A ). Lemma 2.30.
For any GM variety X the corresponding GM category A X is self-dual, i.e. A X ≃ A op X . Proof.
If dim( X ) = 2 then A X = D b ( X ), so the result holds by self-duality of D b ( X ). Nowassume dim( X ) ≥
3. Applying the dualization functor F R H om ( F , O X ) to the semiorthog-onal decomposition (2.11), we obtain a new semiorthogonal decompositionD b ( X ) = h U X ( − ( n − , O X ( − ( n − , . . . , U X , O X , A ′ X i (2.20)and an equivalence A ′ X ≃ A op X . It remains to show A ′ X ≃ A X . (2.21) ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 17
We mutate the subcategory h U X ( − ( n − , O X ( − ( n − , . . . , U X i to the far right sideof (2.20). By (2.19), the formula (2.12) for the Serre functor of D b ( X ), and the formula (2.2)for − K X , the result isD b ( X ) = h O X , A ′ X , U X (1) , O X (1) , . . . , U X ( n − i . Using the isomorphism U X (1) ∼ = U ∨ X and comparing this decomposition with (2.11), we deducethat A X = L O X ( A ′ X ). Hence A X ≃ A ′ X by (2.18). (cid:3) Remark 2.31.
A similar argument shows that the K3 category associated to a cubic fourfold(as defined by (3.1) below) is self-dual.3.
Conjectures on duality and rationality
In this section, we propose two conjectures related to the variation of GM categories A X as X varies in moduli. We begin by briefly recalling a description of the moduli of GM varietiesin terms of EPW sextics from [10, §
3] (see Appendix A for some basic results about themoduli stack of GM varieties). Using this, we formulate a duality conjecture (Conjecture 3.7),which in particular implies that A X is constant in families of GM varieties with the sameassoicated EPW sextic. Next we discuss the rationality problem for GM varieties in termsof GM categories. This problem is most interesting for GM fourfolds, where by analogy withcubic fourfolds we conjecture that the GM category of a rational GM fourfold is equivalentto the derived category of a K3 surface (Conjecture 3.12).3.1. EPW sextics and moduli of GM varieties.
Let V be a 6-dimensional vector space.Its exterior power ∧ V has a natural det( V )-valued symplectic form, given by wedge product.For any Lagrangian subspace A ⊂ ∧ V , we consider the following stratification of P ( V ): Y ≥ k A = { v ∈ P ( V ) | dim( A ∩ ( v ∧ ( ∧ V ))) ≥ k } ⊂ P ( V ) . We write Y k A for the complement of Y ≥ k +1 A in Y ≥ k A , and Y A for Y ≥ A . The variety Y A is called an EPW sextic (for Eisenbud, Popescu, and Walter, who first defined it), and the sequence Y k A is called the EPW stratification .We say A has no decomposable vectors if P ( A ) does not intersect Gr(3 , V ) ⊂ P ( ∧ V ).O’Grady [44, 45, 46, 47, 48, 49] extensively investigated the geometry of EPW sextics, andproved in particular that (see also [10, Theorem B.2]) if A has no decomposable vectors, then: • Y A = Y ≥ A is a normal irreducible sextic hypersurface, smooth along Y A ; • Y ≥ A = Sing( Y A ) is a normal irreducible surface of degree 40, smooth along Y A ; • Y A = Sing( Y ≥ A ) is finite and reduced, and for general A is empty; • Y ≥ A = ∅ .For any Lagrangian subspace A ⊂ ∧ V , its orthogonal A ⊥ = ker( ∧ V ∨ → A ∨ ) ⊂ ∧ V ∨ isalso Lagrangian, and A has no decomposable vectors if and only if the same is true for A ⊥ . Inparticular, A ⊥ gives rise to an EPW sequence of subvarieties of P ( V ∨ ), which can be writtenin terms of A as follows: Y ≥ k A ⊥ = { V ∈ P ( V ∨ ) | dim( A ∩ ∧ V ) ≥ k } ⊂ P ( V ∨ ) . This stratification has the same properties as the stratification Y ≥ k A . By O’Grady’s work Y A ⊥ is projectively dual to Y A , and for this reason is called the dual EPW sextic to Y A . We notethat Y A ⊥ is not isomorphic to Y A for general A (see [45, Theorem 1.1]). One of the main results of [10] is the following description of the set of all isomorphismclasses of smooth ordinary GM varieties. If X ⊂ P ( W ) is a GM variety, then the space ofquadrics in P ( W ) containing X is 6-dimensional vector space [10, Theorem 2.3], which wedenote by V ( X ). The space of Pl¨ucker quadrics defining the Grassmannian G = Gr(2 , V ( X ))is canonically identified with V ( X ), so since X ⊂ Cone( G ) we have an embedding V ( X ) ⊂ V ( X ) . The hyperplane V ( X ) is called the Pl¨ucker hyperplane of X and the corresponding point p X ∈ P ( V ( X ) ∨ )is called the Pl¨ucker point of X . Furthermore, in [10, Theorem 3.10] it is shown that there isa natural Lagrangian subspace A ( X ) ⊂ ∧ V ( X )associated to X . If X op is the opposite variety of X as defined by (2.7), then A ( X op ) = A ( X )and p X op = p X . Theorem 3.1 ([10, Theorem 3.10]) . For any n ≥ the maps X → X op and X ( A ( X ) , p X ) define bijections between (1) the set of ordinary GM varieties X of dimension n ≥ whose Grassmannian hull M X is smooth, up to isomorphism, (2) the set of special GM varieties of dimension n + 1 ≥ , up to isomorphism, and (3) the set of pairs ( A , p ) , where A ⊂ ∧ V is a Lagrangian subspace with no decomposablevectors and p ∈ Y − n A ⊥ , up to the action of PGL( V ) . Note that by Lemma 2.1, M X is automatically smooth for ordinary GM varieties of dimen-sion n ≥ Remark 3.2.
To include all GM surfaces into the above bijection, we must allow a moregeneral class of Lagrangian subspaces in Theorem 3.1, namely those that contain finitelymany decomposable vectors, cf. [10, Theorem 3.16 and Remark 3.17].
Remark 3.3.
Theorem 3.1 suggests there is a morphism from the moduli stack M n of n -dimensional GM varieties (see Appendix A) to the quotient stack LG( ∧ V ) / PGL( V )(where LG( ∧ V ) is the Lagrangian Grassmannian) given by X A ( X ) at the level of points,whose fiber over a point A is the union of two EPW strata Y − n A ⊥ ⊔ Y − n A ⊥ , modulo the actionof the stabilizer of A in PGL( V ). This morphism will be discussed in detail in [12]. Let ussimply note that it gives a geometric way to compute dim M n (cf. Proposition A.2). Namely,the quotient stack LG( ∧ V ) / PGL( V ) has dimension 20, and the fibers of the supposed mor-phism have dimension 5, 5, 4, or 2 for n = 6 , ,
4, or 3, respectively. Finally, for n = 2 themorphism is no longer dominant, as its image is the divisor of those A such that Y A ⊥ = ∅ ,and its fibers are finite.The above discussion shows the utility of the EPW stratification of P ( V ∨ ) from the pointof view of moduli. The following proposition gives a geometric interpretation of the EPWstratification of P ( V ), which will be essential later.As mentioned before, the quadric Q defining X in (2.1) is not unique; such quadrics areparameterized by the affine space P ( V ( X )) \ P ( V ( X )) of non-Pl¨ucker quadrics. In otherwords, a quadric Q defining X in (2.1) corresponds to a quadric point q ∈ P ( V ( X )) ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 19 such that ( q , p X ) does not lie on the incidence divisor in P ( V ( X )) × P ( V ( X ) ∨ ). Proposition 3.4 ([10, Proposition 3.13(b)]) . Let X be a GM variety. Under the identificationof the affine space P ( V ( X )) \ P ( V ( X )) with the space of non-Pl¨ucker quadrics containing X ,the stratum Y k A ( X ) ∩ ( P ( V ( X )) \ P ( V ( X ))) corresponds to the quadrics Q such that dim(ker( Q )) = k . The symmetry between the Pl¨ucker point p X and the quadric point q is the basis for theduality of GM varieties, discussed below.3.2. The duality conjecture.
The following definition extends [10, Definitions 3.22 and 3.26].
Definition 3.5.
Let X and X be GM varieties.(1) If there exists an isomorphism V ( X ) ∼ = V ( X ) identifying A ( X ) ⊂ ∧ V ( X )with A ( X ) ⊂ ∧ V ( X ), then we say: • X and X are period partners if dim( X ) = dim( X ), and • X and X are generalized partners if dim( X ) ≡ dim( X ) (mod 2).(2) If there exists an isomorphism V ( X ) ∼ = V ( X ) ∨ identifying A ( X ) ⊂ ∧ V ( X )with A ( X ) ⊥ ⊂ ∧ V ( X ) ∨ , then we say: • X and X are dual if dim( X ) = dim( X ), and • X and X are generalized dual if dim( X ) ≡ dim( X ) (mod 2). Remark 3.6. If X is a GM variety, then either A ( X ) does or does not contain decomposablevectors, and these two cases are preserved by generalized partnership and duality. The firstcase happens only when X is an ordinary surface with singular Grassmannian hull or X is aspecial surface, see [10, Theorem 3.16 and Remark 3.17]. In this paper, we focus on the casewhere A ( X ) does not contain decomposable vectors.One of the main results of [10, §
4] is that period partners or dual GM varieties of dimensionat least 3 are birational. Our motivation for defining generalized partners and duals is thefollowing conjecture.
Conjecture 3.7.
Let X and X be GM varieties such that the subspaces A ( X ) and A ( X ) do not contain decomposable vectors, and let A X and A X be their GM categories. (1) If X and X are generalized partners, there is an equivalence A X ≃ A X . (2) If X and X are generalized duals, there is an equivalence A X ≃ A X . By Proposition 2.6, GM varieties with equivalent GM categories must have dimensions ofthe same parity, which explains the parity condition in Definition 3.5. We note that part (1)of the conjecture follows from part (2), since by Definition 3.5 and Theorem 3.1 generalizedperiod partners have a common generalized dual GM variety. For this reason, we refer toConjecture 3.7 as the duality conjecture .As evidence for the duality conjecture, we prove in § X is anordinary GM fourfold and X is a (suitably generic) generalized dual GM surface. In fact,the approach of § Proposition 3.28] that duality of ordinary GM varieties can be interpreted in terms of projec-tive duality of quadrics (see also § Lemma 3.8.
Let X be an n -dimensional GM variety, and assume A ( X ) has no decomposablevectors. Then any quadric point q ∈ P ( V ( X )) corresponds to a generalized dual X ∨ q of X .If q lies in the stratum Y k A ( X ) for some k , we have: • If − k ≡ n (mod 2) , then X ∨ q is an ordinary GM variety of dimension − k . • If − k ≡ n (mod 2) , then X ∨ q is a special GM variety of dimension − k .Similarly, any point p ∈ P ( V ( X ) ∨ ) corresponds to a generalized partner X p of X .Conversely, any generalized dual of X arises as X ∨ q for some q ∈ P ( V ( X )) and anygeneralized partner of X arises as X p for some p ∈ P ( V ( X ) ∨ ) .Proof. The variety X ∨ q corresponding to a quadric point q ∈ P ( V ) is just the ordinaryGM variety of dimension 5 − k or the special GM variety of dimension 6 − k associated byTheorem 3.1 to the pair ( A ( X ) ⊥ , q ) (with V = V ( X ) ∨ ). It also follows from Theorem 3.1that any generalized dual of X arises in this way.The same argument also works for generalized partners. (cid:3) The argument of Lemma 3.8 shows that the set of isomorphism classes of generalizedduals of X can be identified with the quotient of P ( V ( X )) by the action of the stabilizerof A ( X ) in PGL( V ( X )). Analogously, the isomorphism classes of generalized partners of X are parameterized by a quotient of P ( V ( X ) ∨ ) by the same group.Let us list more explicitly the type of X ∨ q according to the stratum Y k A ( X ) of q and theparity of n : k X ∨ q for n even X ∨ q for n odd0 special sixfold ordinary fivefold1 ordinary fourfold special fivefold2 special fourfold ordinary threefold3 ordinary surface special threefoldRecall that the stratum Y k A ( X ) is always nonempty for k = 0 , ,
2, generically empty for k = 3,and always empty for k ≥ A ( X ) contains no decomposablevectors). In fact, the condition that Y A ( X ) is nonempty is divisorial in M n (see Remark 4.3).In the same way, one can describe the types of generalized partners X p of X depending onthe stratum Y k A ( X ) ⊥ of p and the parity of n .Conjecture 3.7 says there are equivalences A X ≃ A X p ≃ A X ∨ q for every p ∈ P ( V ( X ) ∨ ) and every q ∈ P ( V ( X )). In particular, it predicts that often GMcategories are equivalent to those of lower-dimensional GM varieties, namely that:(1) If X is a sixfold, then its GM category is equivalent to a fourfold’s GM category.(2) If X is a fivefold, then its GM category is equivalent to a threefold’s GM category. ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 21 (3) If X is a fourfold such that Y A ( X ) ⊥ = ∅ or Y A ( X ) = ∅ , then its GM category isequivalent to the derived category of a GM surface.As mentioned above, in § Y A ( X ) = ∅ and an additional genericityassumption holds, namely Y A ( X ) P ( V ( X )). Remark 3.9.
Using Theorem 3.1, it is easy to see that to prove the duality conjecture infull generality, it is enough to prove A X ≃ A X ∨ q for all X and q ∈ P ( V ( X )) \ P ( V ( X )). Asimilar reduction was used in [10, §
4] to prove birationality of period partners and of dualGM varieties.
Remark 3.10.
A GM variety X as in (1)–(3) above is rational (see the discussion belowand Lemma 4.7). It seems likely that for such an X there is a rationality construction thatinvolves a blowup of a generalized partner or dual variety of dimension 2 less, and gives riseto an equivalence of GM categories. Our approach to (3) in § Rationality conjectures.
Let us recall what is known about rationality of GM varieties.A general GM threefold is irrational by [3, Theorem 5.6], while every GM fivefold or sixfold isrational by [10, Proposition 4.2] (for a general
GM fivefold or sixfold this was already knownto Roth). The intermediate case of GM fourfolds is more mysterious, and closely parallels thesituation for cubic fourfolds: some rational examples are known [9], but while a very generalGM fourfold is expected to be irrational, it has not been proven that a single GM fourfold isirrational. Below, we analyze this state of affairs from the point of view of derived categories.Following [33, § • For a triangulated category A , the geometric dimension gdim( A ) is defined as the min-imal integer m such that there exists an m -dimensional connected smooth projectivevariety M and an admissible embedding A ֒ → D b ( M ). • If Y is a smooth projective variety and D b ( Y ) = h A , . . . , A m i is a maximal semiorthog-onal decomposition (i.e. the components are indecomposable), then A i is called a Griffiths component if gdim( A i ) ≥ dim Y − Y did not depend on the choice of maximal semiorthogonaldecomposition, then it would be a birational invariant [33, Lemma 3.10]; in particular, it wouldbe empty if Y is rational of dimension at least 2. Unfortunately, there are examples showingthis is not true (see [33, § § X ′ ⊂ P is a smooth cubic fourfold, there is a semiorthogonaldecomposition D b ( X ′ ) = h A X ′ , O X ′ , O X ′ (1) , O X ′ (2) i , (3.1)where A X ′ is a K3 category (see [23, Corollary 4.3] or [30, Corollary 4.1]). If A X ′ is equivalentto the derived category of a K3 surface, then gdim( A X ′ ) = 2 and hence (3.1) contains noGriffiths components. If A X ′ is not geometric (which holds for a very general cubic fourfoldby an argument similar to Proposition 2.29), then we expect A X ′ to be a Griffiths com-ponent, although this remains an interesting open problem, cf. [30, Conjecture 5.8]. Theseconsiderations motivated the following conjecture. Conjecture 3.11 ([28]) . If X ′ is a rational cubic fourfold, then A X ′ is equivalent to thederived category of a K3 surface. As evidence, this conjecture was proved in [28] for all rational X ′ known at the time.Since then, a nearly complete answer to when A X ′ is equivalent to the derived category ofa K3 surface has been given [2], and some new families of rational cubic fourfolds have beenproduced [1].The same philosophy can be applied to GM fourfolds. If the GM category A X of a GMfourfold X is geometric, then (2.11) contains no Griffiths components, and otherwise we ex-pect A X to be a Griffiths component. This suggests the following analogue of Conjecture 3.11. Conjecture 3.12. If X is a rational GM fourfold, then the GM category A X is equivalent tothe derived category of a K surface. One of the main results of this paper, Theorem 1.2 (or rather Theorem 4.1), verifies Con-jecture 3.12 for a certain family of rational GM fourfolds. Another result, Theorem 1.3 (orrather Theorem 5.8), builds a bridge between Conjectures 3.12 and 3.11. Finally, recall thatwe proved the GM category of a very general GM fourfold is not equivalent to the derivedcategory of a K3 surface (Proposition 2.29). Hence Conjecture 3.12 is consistent with theexpectation that a very general GM fourfold is irrational.Now we consider GM varieties of other dimensions from the perspective of derived cat-egories. The next result shows that for a GM threefold X , any maximal semiorthogonaldecomposition of D b ( X ) obtained by refining (2.11) contains a Griffiths component. We viewthis as evidence that any smooth GM threefold is irrational. Lemma 3.13 (cf. [33, Proposition 3.12]) . Let X be a GM threefold. Then A X does not admita semiorthogonal decomposition with all components of geometric dimension at most .Proof. It is easy to see that any category of geometric dimension 0 is equivalent to D b (Spec( k )).Further, by [50] any category of geometric dimension 1 is equivalent to the derived categoryof a curve. Note that HH • (Spec( k )) = k , and if C is a curve of genus g thenHH • ( C ) = k g [1] ⊕ k ⊕ k g [ − . Thus if A X has a semiorthogonal decomposition with all components of geometric dimensionat most 1, Proposition 2.9 and Theorem 2.8 imply A X ≃ D b ( C ) for a genus 10 curve C . Thiscannot happen by Proposition 2.29. (cid:3) If X is a GM fivefold or sixfold, then by the discussion in § X has a generalizeddual X ∨ with dim( X ∨ ) ≤ dim( X ) −
2. The duality conjecture (Conjecture 3.7(2)) predictsthat A X ≃ A X ∨ , and hence gdim( A X ) ≤ dim( X ) −
2. So assuming the duality conjecture, wesee that (2.11) has no Griffiths components, which is consistent with the rationality of X .4. Fourfold-to-surface duality
In this section we prove Conjecture 3.7 for ordinary fourfolds with a generalized dual surfacecorresponding to a non-Pl¨ucker quadric point.4.1.
Statement of the result.
Recall that for any GM fourfold X and a quadric point q ∈ P ( V ( X )), we associated in § X ∨ q , which is an ordinary GMsurface if q ∈ Y A ( X ) . ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 23
Theorem 4.1.
Let X be an ordinary GM fourfold such that Y A ( X ) ∩ ( P ( V ( X )) \ P ( V ( X ))) = ∅ . Then for any point q ∈ Y A ( X ) ∩ ( P ( V ( X )) \ P ( V ( X ))) , there is an equivalence A X ≃ D b ( X ∨ q ) . The proof of this theorem takes the rest of this section. We start by noting an immediateconsequence for period partners.
Corollary 4.2.
Assume X and q are as in Theorem , and let X p be a period partner of X such that ( q , p ) does not lie on the incidence divisor in P ( V ( X )) × P ( V ( X ) ∨ ) . Then thereis an equivalence of GM categories A X p ≃ A X .Proof. By Theorem 4.1 applied to X and X p we have a pair of equivalences A X ≃ D b ( X ∨ q )and A X p ≃ D b ( X ∨ q ). Combining them we obtain an equivalence A X p ≃ A X . (cid:3) A key ingredient in the proof of Theorem 4.1 is the theory of homological projective dual-ity [25]. Very roughly, this theory relates the derived categories of linear sections of an ambientvariety to those of orthogonal linear sections of a “dual” variety. As we explain below, thevarieties X and X ∨ q from Theorem 4.1 can be thought of as intersections of G ⊂ P ( ∧ V ) andits dual G ∨ = Gr(2 , V ∨ ) ⊂ P ( ∧ V ∨ ) with projectively dual quadric subvarieties. To proveTheorem 1.2, we thus establish a “quadratic” version of homological projective duality, in thecase where the ambient variety is G . Much of our argument is not special to G , however, andshould have interesting applications in other settings. Remark 4.3.
GM fourfolds X as in the theorem form a 23-dimensional (codimension 1in moduli) family. This can be seen using Theorem 3.1. Indeed, by [48, Proposition 2.2]Lagrangian subspaces A ⊂ ∧ V with no decomposable vectors such that Y A = ∅ form adivisor in the moduli space of all A , and hence form a 19-dimensional family. Having fixedsuch an A there are finitely many q ∈ Y A , and in order for q ∈ P ( V ( X )) \ P ( V ( X )) thePl¨ucker point p of X can be any point of Y A ⊥ such that ( q , p ) is not on the incidence divisor.In other words, p ∈ Y A ⊥ \ q ⊥ , so we have a 4-dimensional family of choices.Recall from § X is an ordinary GM fourfold, there is a (canonical) hyperplane W ⊂ ∧ V ( X ) and a (non-canonical) quadric Q ⊂ P ( W ) such that X = G ∩ Q . The fourfoldssatisfying the assumption of Theorem 4.1 admit several different characterizations. Lemma 4.4.
Let X be an ordinary GM fourfold. The following are equivalent: (1) Y A ( X ) ∩ ( P ( V ( X )) \ P ( V ( X ))) = ∅ . (2) There is a rank quadric Q ⊂ P ( W ) such that X = G ∩ Q . (3) X contains a quintic del Pezzo surface, i.e. a smooth codimension linear section ofthe Grassmannian G ⊂ P ( ∧ V ( X )) .Proof. The equivalence of (1) and (2) follows from Proposition 3.4 since dim W = 9. Notethat since Y A ( X ) = ∅ , the same proposition also shows that if X = G ∩ Q then rank( Q ) ≥ I ⊂ W for Q has dimension 6, so G ∩ P ( I ) is a quintic del Pezzo contained in X , providedthis intersection is transverse. By the argument of [10, Lemma 4.1] (or by Lemma 4.6 below),this is true for a general I . Conversely, assume (3) holds, i.e. assume there is a 6-dimensional subspace I ⊂ W suchthat Z = G ∩ P ( I ) ⊂ X is a quintic del Pezzo. The restriction map V ( X ) → H ( I Z/ P ( I ) (2))from quadrics in P ( W ) containing X to those in P ( I ) containing Z is surjective with one-dimensional kernel. If Q ⊂ P ( W ) is the quadric corresponding to this kernel, then X = G ∩ Q and P ( I ) ⊂ Q . It follows that rank( Q ) ≤
6. But as we noted above, the reverse inequalityalso holds. (cid:3)
For the rest of the section, we fix an ordinary GM fourfold X satisfying the equivalentconditions of Lemma 4.4 and a point q ∈ Y A ( X ) ∩ ( P ( V ( X )) \ P ( V ( X ))). Further, to easenotation, we denote the generalized dual of X corresponding to the quadric point q (seeLemma 3.8) by Y = X ∨ q . Note that Y is a GM surface.4.2. Setup and outline of the proof.
We outline here the strategy for proving Theorem 4.1.The starting point is the following explicit geometric relation between X and Y . By Propo-sition 3.4, the point q corresponds to a rank 6 quadric Q cutting out X , and the Pl¨uckerpoint p X ∈ P ( V ( X ) ∨ ) ∼ = P ( V ( Y )) of X corresponds to a quadric Q ′ cutting out Y . Because X and Y are ordinary, we may regard Q as a subvariety of P ( ∧ V ( X )) and Q ′ as a subvarietyof P ( ∧ V ( Y )). Then [10, Proposition 3.28] (which is stated for dual varieties but works justas well for generalized duals) says that there is an isomorphism V ( X ) ∼ = V ( Y ) ∨ identifying Q ′ ⊂ P ( ∧ V ( Y )) with the projective dual to Q ⊂ P ( ∧ V ( X )). Hence, fixing V = V ( X ),our setup is as follows: there is a hyperplane W ⊂ ∧ V and a rank 6 quadric Q ⊂ P ( W ) suchthat X = G ∩ Q and Y = G ∨ ∩ Q ∨ , where Q ∨ ⊂ P ( ∧ V ∨ ) is the projectively dual quadric to Q ⊂ P ( ∧ V ), and G ∨ = Gr(2 , V ∨ ) ⊂ P ( ∧ V ∨ )is the dual Grassmannian.From this starting point, the main steps of the proof are as follows. First, by consideringfamilies of maximal linear subspaces of Q and Q ∨ , we find P -bundles b X → X and b Y → Y ,together with morphisms b X → P and b Y → P realizing b X and b Y as families of mutuallyorthogonal linear sections of G and G ∨ . This allows us to apply homological projective du-ality to obtain a semiorthogonal decomposition of D b ( b X ) with D b ( b Y ) as a component. Bycomparing this (via mutation functors) with the decomposition of D b ( b X ) coming from its P -bundle structure over X , we show D b ( b Y ) has a decomposition into two copies of A X . On theother hand, as b Y → Y is a P -bundle, D b ( b Y ) also decomposes into two copies of D b ( Y ). Weshow these two decompositions of D b ( b Y ) coincide, and hence A X ≃ D b ( Y ). Our proof givesan explicit functor inducing this equivalence, see (4.15).4.3. Maximal linear subspaces of the quadrics.
We start by discussing a geometricrelation between Q and Q ∨ . Let K ⊂ W be the kernel of Q , regarded as a symmetric linearmap W → W ∨ . Since dim W = 9 and rank( Q ) = 6, we have dim K = 3. The filtration0 ⊂ K ⊂ W ⊂ ∧ V induces a filtration 0 ⊂ W ⊥ ⊂ K ⊥ ⊂ ∧ V ∨ where K ⊥ and W ⊥ are the annihilators of K and W , so that dim K ⊥ = 7 and dim W ⊥ = 1.The pairing between the dual spaces ∧ V and ∧ V ∨ induces a nondegenerate pairing be-tween W/K and K ⊥ /W ⊥ , and hence an isomorphism K ⊥ /W ⊥ ∼ = ( W/K ) ∨ . The quadric Q induces a smooth quadric ¯ Q in the 5-dimensional projective space P ( W/K ).The quadric ¯ Q can be identified with the Grassmannian Gr(2 , W/K ∼ = ∧ S for a 4-dimensional vector space S , with an identification¯ Q = Gr(2 , S ) ⊂ P ( ∧ S ) . The projective dual of ¯ Q is then the dual Grassmannian¯ Q ∨ = Gr(2 , S ∨ ) ⊂ P ( ∧ S ∨ ) = P (( W/K ) ∨ ) = P ( K ⊥ /W ⊥ ) . It follows that the projective dual of Q = Cone P ( K ) ¯ Q ⊂ P ( ∧ V ) (4.1)is given by Q ∨ = Cone P ( W ⊥ ) ¯ Q ∨ ⊂ P ( ∧ V ∨ ) . (4.2)Projective 3-space P ( S ) is (a connected component of) the space of maximal linear sub-spaces of the quadric ¯ Q = Gr(2 , S ). The universal family is the flag variety Fl(1 , S ) → P ( S ),with fiber over a point s ∈ P ( S ) the plane P ( s ∧ S ) ⊂ P ( ∧ S ). Analogously, the same flagvariety Fl(2 , S ∨ ) ∼ = Fl(1 , S ) is (a connected component of) the space of maximal linearsubspaces of ¯ Q ∨ = Gr(2 , S ∨ ), this time with fiber over a point s ∈ P ( S ) being the plane P ( ∧ s ⊥ ) ⊂ P ( ∧ S ∨ ). Note that the fibers of these two correspondences over a point s ∈ P ( S )are mutually orthogonal with respect to the pairing between ∧ S and ∧ S ∨ . We summarizethis discussion by the diagramFl(1 , S ) p ¯ Q { { ✈✈✈✈✈✈✈✈✈✈ π ¯ Q % % ❑❑❑❑❑❑❑❑❑❑ Fl(2 , S ∨ ) π ¯ Q ∨ y y rrrrrrrrrr p ¯ Q ∨ % % ❏❏❏❏❏❏❏❏❏❏ P ( ∧ S ) ⊃ ¯ Q P ( S ) ¯ Q ∨ ⊂ P ( ∧ S ∨ ) (4.3)with the inner arrows being P -bundles with mutually orthogonal fibers (as linear subspacesof P ( ∧ S ) and P ( ∧ S ∨ )), and the outer arrows being P -bundles.By (4.1) every maximal isotropic subspace of ¯ Q gives a maximal isotropic subspace of Q by taking its preimage under the projection W → W/K = ∧ S . Analogously, by (4.2) everymaximal isotropic subspace of ¯ Q ∨ gives a maximal isotropic subspace of Q ∨ by taking itspreimage under the projection K ⊥ → K ⊥ /W ⊥ = ∧ S ∨ . Note that for the pairing between W and K ⊥ induced by the pairing between ∧ V and ∧ V ∨ , the subspace K ⊂ W is the leftkernel, and the subspace W ⊥ ⊂ K ⊥ is the right kernel. Hence any s ∈ P ( S ) gives mutuallyorthogonal maximal isotropic spaces I s and I ⊥ s of Q and Q ∨ respectively. These spaces form thefibers of vector bundles I and I ⊥ over P ( S ) of ranks 6 and 4, which are mutually orthogonal subbundles of ∧ V ⊗ O P ( S ) and ∧ V ∨ ⊗ O P ( S ) . We can summarize this discussion by thefollowing diagram P P ( S ) ( I ) p Q | | ①①①①①①①①① π Q $ $ ■■■■■■■■■ P P ( S ) ( I ⊥ ) π Q ∨ y y tttttttttt p Q ∨ $ $ ■■■■■■■■■ P ( ∧ V ) ⊃ Q P ( S ) Q ∨ ⊂ P ( ∧ V ∨ ) (4.4)Here the inner arrows are P - and P -bundles with mutually orthogonal fibers, and the outerarrows are P -bundles (induced by the P -bundles of diagram (4.3)) away from the vertices P ( K ) and P ( W ⊥ ) of the quadrics (over which the fibers are isomorphic to P ( S ) ∼ = P ).4.4. Families of linear sections of the Grassmannians.
Now define b X := G × P ( ∧ V ) P P ( S ) ( I ) and b Y := G ∨ × P ( ∧ V ∨ ) P P ( S ) ( I ⊥ ) (4.5)to be the induced families of linear sections of G and G ∨ . They fit into a diagram b X p X (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) π X ! ! ❈❈❈❈❈❈❈❈❈ b Y π Y } } ④④④④④④④④④ p Y (cid:30) (cid:30) ❂❂❂❂❂❂❂❂ X P ( S ) Y (4.6)with the maps induced by those in (4.4) (remember that X = G ∩ Q and Y = G ∨ ∩ Q ∨ ).We will denote by H, H ′ , and h the ample generators of Pic( G ), Pic( G ∨ ), and P ( S ). Lemma 4.5.
There are rank vector bundles S X and S Y on X and Y with c ( S X ) = − H and c ( S Y ) = − H ′ , and isomorphisms b X ∼ = P X ( S X ) and b Y ∼ = P Y ( S Y ) , such that O P X ( S X ) (1) = π ∗ X O P ( S ) ( h ) and O P Y ( S Y ) (1) = π ∗ Y O P ( S ) ( h ) . In particular, b X is asmooth fivefold, b Y is a smooth threefold, and K b X = − H − h and K b Y = H ′ − h. (4.7) Proof.
Since X and Y are smooth, they do not intersect the vertices P ( K ) and P ( W ⊥ ) ofthe quadrics Q and Q ∨ , hence the maps p X and p Y are P -fibrations induced by those indiagram (4.4). In other words, we have fiber product squares b X / / p X (cid:15) (cid:15) Fl(1 , S ) p ¯ Q (cid:15) (cid:15) X / / ¯ Q and b Y / / p Y (cid:15) (cid:15) Fl(2 , S ∨ ) p ¯ Q ∨ (cid:15) (cid:15) Y / / ¯ Q ∨ . The map p ¯ Q is the projectivization of the tautological subbundle of S ⊗ O on ¯ Q = Gr(2 , S ),and p ¯ Q ∨ is the projectivization of the annihilator of the tautological subbundle of S ∨ ⊗ O on ¯ Q ∨ = Gr(2 , S ∨ ). So we can take S X and S Y to be the pullbacks to X and Y of thesebundles.To compute the canonical classes, note that the determinant of the tautological bundle(and of its annihilator) on Gr(2 , S ) is O Gr(2 ,S ) ( − ( S X ) = − H and c ( S Y ) = − H ′ . ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 27
Now apply the standard formula for the canonical bundle of the projectivization of a vectorbundle, taking into account that K X = − H and K Y = 0 by (2.2). (cid:3) Lemma 4.6.
The map π X : b X → P ( S ) is flat with general fiber a smooth quintic del Pezzosurface. The map π Y : b Y → P ( S ) is generically finite of degree .Proof. The fiber of π X over a point s ∈ P ( S ) is the intersection G ∩ P ( I s ), where the subspace P ( I s ) ⊂ P ( ∧ V ) has codimension 4. Thus the dimension of π − X ( s ) is at least 2. If thedimension were greater than 2, this fiber would give a divisor in X of degree at most 5,but by (2.3) and (2.2) every divisor in X has degree divisible by 10. Thus every fiber is adimensionally transverse intersection, and flatness follows.Furthermore, since b X is smooth, the general fiber of π X is a smooth quintic del Pezzosurface. Then by [10, Proposition 2.24] the general fiber of π Y is a dimensionally transverseand smooth linear section of G ∨ of codimension 6, hence is just 5 reduced points. (cid:3) As a byproduct of the above, we obtain:
Lemma 4.7.
The variety X is rational.Proof. The same argument as in [10, Proposition 4.2] works. Let e X ⊂ b X be the preimageunder the map π X of a general hyperplane P ⊂ P ( S ). By Lemma 4.6, the general fiberof e X → P is a smooth quintic del Pezzo surface. Hence by a theorem of Enriques [55], e X isrational over P , and so over k as well. On the other hand, the map e X → X is birational (infact, it is a blowup of a quintic del Pezzo surface), so X is rational too. (cid:3) Homological projective duality.
Homological projective duality (HPD) is a key toolin the proof of Theorem 4.1. Very roughly, HPD relates the derived categories of linear sectionsof a given variety to those of orthogonal linear sections of an “HPD variety”. We refer to [25]for the details of this theory, and to [29] or [57] for an overview. For us, the relevant point isthat the dual Grassmannian G ∨ is HPD to G . We spell out the precise consequence of thisthat we need below.Recall that by Lemma 2.2 there is a semiorthogonal decompositionD b ( G ) = h B , B ( H ) , B (2 H ) , B (3 H ) , B (4 H ) i . Let i : H ( G , G ∨ ) ֒ → G × G ∨ ⊂ P ( ∧ V ) × P ( ∧ V ∨ )be the incidence divisor defined by the canonical section of O ( H + H ′ ). Recall that U denotesthe tautological rank 2 bundle on G , and let V denote the tautological rank 2 bundle on G ∨ .The following was proved in [24, § Theorem 4.8.
The Grassmannian G ∨ → P ( ∧ V ∨ ) is HPD to G → P ( ∧ V ) with respect tothe above semiorthogonal decomposition. Moreover, the duality is implemented by a sheaf E on H ( G , G ∨ ) whose pushforward to G × G ∨ fits into an exact sequence → O G ⊠ V → U ∨ ⊠ O G ∨ → i ∗ E → . In fact, we shall only need a consequence of HPD, which we formulate below as Corollary 4.9.Note that the natural map b X × P ( S ) b Y → X × Y → G × G ∨ factors through H ( G , G ∨ ). Indeed, the fiber of b X × P ( S ) b Y over any point s ∈ P ( S ) is( P ( I s ) × P ( I ⊥ s )) ∩ ( G × G ∨ ) ⊂ H ( G , G ∨ ) . Note also that dim( b X × P ( S ) b Y ) = 5 , (4.8)since the map b X × P ( S ) b Y → b Y is flat of relative dimension 2 by Lemma 4.6, and dim( b Y ) = 3by Lemma 4.5.Denote by b E the pullback of the HPD object E to b X × P ( S ) b Y and by b Φ : D b ( b Y ) → D b ( b X )the corresponding Fourier–Mukai functor. Note that b Φ is P ( S )-linear (since b E is supportedon the fiber product b X × P ( S ) b Y ), i.e. b Φ( F ⊗ π ∗ Y G ) ∼ = b Φ( F ) ⊗ π ∗ X G for all F ∈ D b ( b Y ) and G ∈ D b ( P ( S )). By Lemma 4.5 and (4.8), the families b X and b Y oflinear sections of G and G ∨ satisfy the dimension assumptions of [25, Theorem 6.27]. Hencewe obtain: Corollary 4.9.
The functor b Φ : D b ( b Y ) → D b ( b X ) is fully faithful, and there is a semiorthog-onal decomposition D b ( b X ) = h b Φ(D b ( b Y )) , B X ( H ) ⊠ D b ( P ( S )) i , (4.9) where B X ( H ) ⊠ D b ( P ( S )) denotes the triangulated subcategory generated by objects of theform p ∗ X ( F ) ⊗ π ∗ X ( G ) for F ∈ B X ( H ) and G ∈ D b ( P ( S )) . Mutations.
Since p X : b X → X is a P -bundle (Lemma 4.5), we also have a semiorthog-onal decomposition D b ( b X ) = h p ∗ X D b ( X ) , p ∗ X D b ( X )( h ) i . Inserting the decomposition (2.10) of D b ( X ), we obtainD b ( X ) = h A b X , B , B ( H ) , A b X ( h ) , B ( h ) , B ( H + h ) i , (4.10)where to ease notation we write A b X for p ∗ X A X and simply B for p ∗ X B X . We find a sequence ofmutations bringing this decomposition into the form of (4.9). In doing so we will use severaltimes K X = − H , which holds by (2.2), and K b X = − H − h , which holds by (4.7). For abrief review of mutation functors and references, see the discussion in § Step 1.
Mutate B ( H ) to the left of A b X in (4.10). Since this is a mutation in p ∗ X D b ( X ) and K X = − H , by (2.19) we getD b ( b X ) = h B ( − H ) , A b X , B , A b X ( h ) , B ( h ) , B ( H + h ) i . Step 2.
Mutate B ( H + h ) to the far left. Since K b X = − H − h , by (2.19) we getD b ( b X ) = h B ( − h ) , B ( − H ) , A b X , B , A b X ( h ) , B ( h ) i . Step 3.
Mutate B ( − H ) to the left of B ( − h ). Since these two subcategories are completelyorthogonal (see the lemma below), we getD b ( b X ) = h B ( − H ) , B ( − h ) , A b X , B , A b X ( h ) , B ( h ) i . Lemma 4.10.
The categories B ( − H ) and B ( − h ) in D b ( b X ) are completely orthogonal. ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 29
Proof.
By Step 2, the pair ( B ( − h ) , B ( − H )) is semiorthogonal. On the other hand, by Serre du-ality and (4.7), semiorthogonality of ( B ( − H ) , B ( − h )) is equivalent to that of ( B ( − h ) , B (2 h )),which follows from (4.9) as ( O ( − h ) , O (2 h )) is semiorthogonal in D b ( P ( S )). (cid:3) Step 4.
Mutate B ( − H ) to the far right. Again by (2.19), we getD b ( b X ) = h B ( − h ) , A b X , B , A b X ( h ) , B ( h ) , B (2 h ) i . Step 5.
Mutate A b X and A b X ( h ) to the far left. We getD b ( b X ) = h L B ( − h ) ( A b X ) , L h B ( − h ) , B i ( A b X ( h )) , B ( − h ) , B , B ( h ) , B (2 h ) i = h L B ( − h ) ( A b X ) , L h B ( − h ) , B i ( A b X ( h )) , B X ⊠ D b ( P ( S )) i , where we used the standard decomposition D b ( P ( S )) = h O ( − h ) , O , O ( h ) , O (2 h ) i . Step 6.
Twist the decomposition by O ( H ). We getD b ( b X ) = h L B ( H − h ) ( A b X ( H )) , L h B ( H − h ) , B ( H ) i ( A b X ( H + h )) , B X ( H ) ⊠ D b ( P ( S )) i . (4.11)To rewrite the first two components here, we used the following general fact: If A ⊂ T isan admissible subcategory of a triangulated category and F is an autoequivalence of T (inour case F is the autoequivalence of D b ( b X ) given by tensoring with O ( H )), then there is anisomorphism of functors F ◦ L A ∼ = L F ( A ) ◦ F. (4.12)Finally, we obtain: Proposition 4.11.
The functor b Φ ∗ ◦ ( − ⊗ O ( H )) : D b ( b X ) → D b ( b Y ) induces an equivalence h A b X , A b X ( h ) i ≃ D b ( b Y ) , where b Φ ∗ : D b ( b X ) → D b ( b Y ) denotes the left adjoint of b Φ .Proof. Comparing the decompositions (4.11) and (4.9), we see that b Φ induces an equivalence b Φ : D b ( b Y ) ∼ −−→ h L B ( H − h ) ( A b X ( H )) , L h B ( H − h ) , B ( H ) i ( A b X ( H + h )) i . Therefore its left adjoint b Φ ∗ gives an inverse equivalence. On the other hand, by semiorthog-onality of (4.9) the functor b Φ ∗ vanishes on B ( H − h ) and B ( H ), hence its composition withthe mutation functors through these categories is isomorphic to b Φ ∗ . Thus b Φ ∗ induces anequivalence between h A b X ( H ) , A b X ( H + h ) i ⊂ D b ( b X ) and D b ( b Y ). This is equivalent to theclaim. (cid:3) Proof of the theorem.
Since p Y : b Y → Y is a P -bundle (Lemma 4.5), we haveD b ( b Y ) = h p ∗ Y D b ( Y ) , p ∗ Y D b ( Y )( h ) i . (4.13)We aim to prove that this semiorthogonal decomposition coincides with the one obtained byapplying the fully faithful functor ( − ⊗ O ( − h )) ◦ b Φ ∗ ◦ ( − ⊗ O ( H )) to h A b X , A b X ( h ) i . For this,we consider the composition of functorsF := p Y ∗ ◦ ( − ⊗ O ( − h )) ◦ b Φ ∗ ◦ ( − ⊗ O ( H )) ◦ p ∗ X : D b ( X ) → D b ( Y ) . (4.14) Proposition 4.12.
The functor F vanishes on the subcategory A X ⊂ D b ( X ) . Before proving the proposition, let us show how it implies the equivalence A X ≃ D b ( Y ). Proof of Theorem . We claim that p Y ∗ ◦ ( − ⊗ O ( − h )) ◦ b Φ ∗ ◦ ( − ⊗ O ( H )) ◦ p ∗ X : D b ( X ) → D b ( Y ) (4.15)induces an equivalence A X ≃ D b ( Y ). Note that the functor p ∗ X is fully faithful on A X . Soby Proposition 4.11 the functor ( − ⊗ O ( − h )) ◦ b Φ ∗ ◦ ( − ⊗ O ( H )) ◦ p ∗ X gives a fully faithfulembedding A X ֒ → D b ( b Y ), whose image A satisfiesD b ( Y ) = h A , A ( h ) i . (4.16)On the other hand, by Proposition 4.12 the functor p Y ∗ annihilates A ( − h ). But the kernel ofthe functor p Y ∗ is p ∗ Y D b ( Y )( − h ), so A ( − h ) ⊂ p ∗ Y D b ( Y )( − h ), and thus A ⊂ p ∗ Y D b ( Y ) and A ( h ) ⊂ p ∗ Y D b ( Y )( h ) . In view of the decompositions (4.16) and (4.13), we see that equality holds in the aboveinclusions. Since p Y ∗ induces an equivalence p ∗ Y D b ( Y ) ≃ D b ( Y ), this finishes the proof. (cid:3) Now we turn to the proof of Proposition 4.12, which takes the rest of the section. Let f X : X → G and f Y : Y → G ∨ be the Gushel maps, and let p XY : b X × P ( S ) b Y → X × Y bethe natural morphism. Recall from § b X × P ( S ) b Y p XY −−−−→ X × Y f X × f Y −−−−−→ G × G ∨ factors through the incidence divisor H ( G , G ∨ ). Hence there is a commutative diagram b X × P ( S ) b Y p / / b g & & ◆◆◆◆◆◆◆◆◆◆◆ H ( X, Y ) g (cid:15) (cid:15) j / / X × Y f X × f Y (cid:15) (cid:15) H ( G , G ∨ ) i / / G × G ∨ (4.17)where H ( X, Y ) is by definition the pullback of H ( G , G ∨ ) along f X × f Y , and p XY = j ◦ p .We will need the following two lemmas. Lemma 4.13.
There is an isomorphism p ∗ O b X × P ( S ) b Y ∼ = O H ( X,Y ) .Proof. We have a diagram b X × P ( S ) b Y b ∆ / / (cid:15) (cid:15) b X × b Y π X × π Y (cid:15) (cid:15) p X × p Y / / X × Y P ( S ) ∆ / / P ( S ) × P ( S )where the square is cartesian, and also Tor-independent as the fiber product has expecteddimension by (4.8). To prove the lemma, we must show ( p X × p Y ) ∗ ( b ∆ ∗ O b X × P ( S ) b Y ) ∼ = O H ( X,Y ) .By Tor-independence, we have an isomorphism b ∆ ∗ O b X × P ( S ) b Y ∼ = ( π X × π Y ) ∗ ∆ ∗ O P ( S ) . ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 31
Pulling back the standard resolution of the diagonal on P ( S ) × P ( S ), we obtain an exactsequence0 → π ∗ X O P ( S ) ( − h ) ⊠ π ∗ Y Ω P ( S ) (3 h ) → π ∗ X O P ( S ) ( − h ) ⊠ π ∗ Y Ω P ( S ) (2 h ) →→ π ∗ X O P ( S ) ( − h ) ⊠ π ∗ Y Ω P ( S ) ( h ) → O b X × b Y → b ∆ ∗ O b X × P ( S ) b Y → b X × b Y . Using the identifications p X : b X = P X ( S X ) → X and p Y : b Y = P Y ( S Y ) → Y ofLemma 4.5, it is easy to compute: p Y ∗ π ∗ Y Ω P ( S ) (3 h ) ∼ = p Y ∗ π ∗ Y O ( − h ) = 0 ,p X ∗ π ∗ X O P ( S ) ( − h ) ∼ = det( S X )[ − ∼ = O X ( − H )[ − ,p Y ∗ π ∗ Y Ω P ( S ) (2 h ) ∼ = det( S Y ) ∼ = O Y ( − H ′ ) ,p X ∗ π ∗ X O P ( S ) ( − h ) = 0 , ( p X × p Y ) ∗ ( O b X × b Y ) ∼ = O X × Y . It follows that in the spectral sequence computing ( p X × p Y ) ∗ ( b ∆ ∗ O b X × P ( S ) b Y ) from the aboveresolution, the only nontrivial terms areR ( p X × p Y ) ∗ ( π ∗ X O ( − h ) ⊠ π ∗ Y Ω P ( S ) (2 h )) ∼ = O X × Y ( − H − H ′ ) , R ( p X × p Y ) ∗ ( O b X × b Y ) ∼ = O X × Y , and we get an exact sequence0 → O X × Y ( − H − H ′ ) → O X × Y → ( p X × p Y ) ∗ ( b ∆ ∗ O b X × P ( S ) b Y ) → , which gives the required isomorphism ( p X × p Y ) ∗ ( b ∆ ∗ O b X × P ( S ) b Y ) ∼ = O H ( X,Y ) . (cid:3) Lemma 4.14.
The functor F[ − is given by a Fourier–Mukai kernel K ∈ D b ( X × Y ) , whichfits into a distinguished triangle U X ( − H ) ⊠ O Y ( − H ′ ) → O X ( − H ) ⊠ V ∨ Y ( − H ′ ) → K . Proof.
The main term in the definition (4.14) of F is the left adjoint b Φ ∗ of b Φ. By definition b Φis given by the Fourier–Mukai kernel b E ∈ D b ( b X × P ( S ) b Y ), so by Grothendieck duality we findthat b Φ ∗ is given by the kernel b E ∨ ⊗ ω b X × P ( S ) b Y / b Y [2] = b E ∨ (2 h − H )[2] ∈ D b ( b X × P ( S ) b Y ) , where b E ∨ = R H om ( b E , O ) is the derived dual of b E on b X × P ( S ) b Y . Using this, it follows easilyfrom the definition of F that F[ −
2] is given by the kernel K := p XY ∗ ( b E ∨ ) ∈ D b ( X × Y ) . Using the diagram (4.17) and the definition of b E , we can rewrite this as K ∼ = j ∗ p ∗ R H om ( p ∗ g ∗ E , O b X × P ( S ) b Y ) ∼ = j ∗ R H om ( g ∗ E , p ∗ O b X × P ( S ) b Y ) ∼ = j ∗ R H om ( g ∗ E , O H ( X,Y ) ) , where the second line holds by the local adjunction between p ∗ and p ∗ , and the third byLemma 4.13. Now Grothendieck duality for the inclusion j : H ( X, Y ) → X × Y of the incidencedivisor (which has class H + H ′ ) gives K ∼ = j ∗ R H om ( g ∗ E , j ! O X × Y ( − H − H ′ )[1]) ∼ = R H om ( j ∗ g ∗ E , O X × Y ( − H − H ′ )[1]) . On the other hand, the fiber square in diagram (4.17) is Tor-independent because H ( X, Y )has expected dimension. Hence we have an isomorphism j ∗ g ∗ E ∼ = ( f X × f Y ) ∗ i ∗ E , and so, by the explicit resolution of i ∗ E from Theorem 4.8, a distinguished triangle O X ⊠ V Y → U ∨ X ⊠ O Y → j ∗ g ∗ E . Dualizing, twisting by O X × Y ( − H − H ′ ), and rotating this triangle, we obtain a distinguishedtriangle U X ( − H ) ⊠ O Y ( − H ′ ) → O X ( − H ) ⊠ V ∨ Y ( − H ′ ) → R H om ( j ∗ g ∗ E , O X × Y ( − H − H ′ )[1]) , which combined with the above expression for K finishes the proof. (cid:3) Finally, we prove Proposition 4.12.
Proof of Proposition . By Lemma 4.14, it suffices to show the Fourier–Mukai functorswith kernels U X ( − H ) ⊠ O Y ( − H ′ ) and O X ( − H ) ⊠ V ∨ Y ( − H ′ )vanish on A X . This is equivalent to the vanishingH • ( X, U X ( − H ) ⊗ F ) = 0 and H • ( X, O X ( − H ) ⊗ F ) = 0for all F ∈ A X , which holds since A X is right orthogonal to B X ( H ) = h O X ( H ) , U ∨ X ( H ) i bydefinition (see (2.10) and (2.8)). (cid:3) Cubic fourfold derived partners
In this section, we show that the K3 categories attached to GM and cubic fourfolds notonly behave similarly, but sometimes even coincide. For this, we will consider ordinary GMfourfolds satisfying the following condition: there is a 3-dimensional subspace V ⊂ V ( X )such that Gr(2 , V ) ⊂ X. (5.1) Remark 5.1.
GM fourfolds that satisfy (5.1) for some V form a 21-dimensional (codimen-sion 3 in moduli) family. This can be seen using Theorem 3.1, as follows. Let V = V ( X ).Then by [11, Theorem 4.5(c)], for a 3-dimensional subspace V ⊂ V condition (5.1) holds ifand only if dim( A ∩ (( ∧ V ) ∧ V )) ≥ p X ∈ P ( V ⊥ ) ⊂ P ( V ∨ ) . (5.2)By [19, Lemma 3.6] Lagrangian subspaces A ⊂ ∧ V with no decomposable vectors such thatthe first part of (5.2) holds for some V ⊂ V form a nonempty divisor in the moduli space ofall A , and hence form a 19-dimensional family. Having fixed such an A there are finitely manypoints V ∈ Gr(3 , V ) such that the first part of (5.2) holds [21]. By Theorem 3.1, for sucha V , the ordinary GM fourfolds X such that the second part of (5.2) holds are parameterizedby Y A ⊥ ∩ P ( V ⊥ ). By [11, Lemma 2.3] we have P ( V ⊥ ) ⊂ Y A ⊥ . Further, since Y ≥ A ⊥ is an ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 33 irreducible surface of degree 40, we have P ( V ⊥ ) Y ≥ A ⊥ . Thus Y A ⊥ ∩ P ( V ⊥ ) is an open subsetof the projective plane P ( V ⊥ ).From now on we write V = V ( X ) and fix a 3-dimensional subspace V ⊂ V such that (5.1)holds. We associate to X a birational cubic fourfold X ′ following [9, § X ′ issmooth, and in this case we prove there is an equivalence A X ≃ A X ′ where A X ′ is the K3category of the cubic fourfold defined by (3.1) (Theorem 5.8). The cubic X ′ is simply theimage of the linear projection from the plane Gr(2 , V ) in X . We begin by studying thisprojection as a map from the entire Grassmannian G .5.1. A linear projection of the Grassmannian.
Set P = P ( ∧ V ) = Gr(2 , V ) ⊂ G . Choose a complement V to V in V , and set B = ∧ V / ∧ V = ∧ V ⊕ ( V ⊗ V ) . Then the linear projection from P gives a birational isomorphism from G to P ( B ). Its struc-ture can be described as follows. Lemma 5.2.
Let p : e G → G be the blowup with center in P . Then the linear projectionfrom P induces a regular map q : e G → P ( B ) which identifies G with the blowup of P ( B ) in P ( V ) × P ( V ) ⊂ P ( V ⊗ V ) ⊂ P ( B ) . In other words, we have a diagram E / / (cid:0) (cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) e G p (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) q ! ! ❉❉❉❉❉❉❉❉❉ E ′ o o ' ' ◆◆◆◆◆◆◆◆◆◆◆◆ P / / G P ( B ) P ( V ) × P ( V ) o o (5.3) where • E is the exceptional divisor of the blowup p , and is mapped birationally by q onto thehyperplane P ( V ⊗ V ) ⊂ P ( B ) . • E ′ is the exceptional divisor of the blowup q , and is mapped birationally by p onto theSchubert variety Σ = { U ∈ G | U ∩ V = 0 } ⊂ G Proof.
Straightforward. (cid:3)
We denote by H and H ′ the ample generators of Pic( G ) and Pic( P ( B )). Lemma 5.3. On e G we have the relations ( H ′ = H − E,E ′ = H − E, or equivalently ( H = 2 H ′ − E ′ ,E = H ′ − E ′ , (5.4) as divisors modulo linear equivalence. Moreover, we have K e G = − H + 3 E = − H ′ + 2 E ′ . (5.5) Proof.
The equalities (5.5) follow from the standard formula for the canonical class of ablowup, and the equality H ′ = H − E holds by definition of p . Using these, the other equalitiesin (5.4) follow directly (note that Pic( e G ) ∼ = Z is torsion free). (cid:3) Later in this section we will need an expression for the vector bundle p ∗ U ∨ on e G in termsof the blowup q . For this, we consider the composition φ : ( V ∨ ⊕ V ∨ ) ⊗ O P ( B ) ֒ → V ⊗ V ∨ ⊗ ( V ∨ ⊕ V ∨ ) ⊗ O P ( B ) →→ V ⊗ ( ∧ V ∨ ⊕ ( V ∨ ⊗ V ∨ )) ⊗ O P ( B ) → V ⊗ O P ( B ) ( H ′ ) , where the first morphism is induced by the map k → V ⊗ V ∨ corresponding to the identityof V , the second is induced by the map V ∨ ⊗ V ∨ → ∧ V ∨ , and the third is induced by thecomposition ( ∧ V ∨ ⊕ ( V ∨ ⊗ V ∨ )) ⊗ O P ( B ) = B ∨ ⊗ O P ( B ) → O P ( B ) ( H ′ ) . Lemma 5.4.
The cokernel of φ is the sheaf O P ( V ) × P ( V ) (2 , .Proof. Write φ ′ : V ∨ ⊗ O P ( B ) → V ⊗ O P ( B ) ( H ′ ) ,φ ′′ : V ∨ ⊗ O P ( B ) → V ⊗ O P ( B ) ( H ′ ) , for the components of φ . The morphism φ ′ is an isomorphism away from the hyperplane P ( V ⊗ V ) ⊂ P ( B ), and zero on it. Hence coker( φ ′ ) = V ⊗ O P ( V ⊗ V ) ( H ′ ). It follows that thecokernel of φ coincides with the cokernel of the morphism φ ′′| P ( V ⊗ V ) : V ∨ ⊗ O P ( V ⊗ V ) → V ⊗ O P ( V ⊗ V ) ( H ′ ) . But the morphism φ ′′| P ( V ⊗ V ) is generically surjective with degeneracy locus the Segre subva-riety P ( V ) × P ( V ) ⊂ P ( V ⊗ V ), and its restriction to this locus factors as the composition V ∨ ⊗ O P ( V ) × P ( V ) ։ O P ( V ) × P ( V ) (0 , ֒ → V ⊗ O P ( V ) × P ( V ) (1 ,
1) = V ⊗ O P ( V ) × P ( V ) ( H ′ ) . It follows that the cokernel of φ ′′| P ( V ⊗ V ) is isomorphic to O P ( V ) × P ( V ) (2 , (cid:3) Let F denote the class of a fiber of the natural projection E ′ → P ( V ) × P ( V ) → P ( V ). Proposition 5.5. On e G there is an exact sequence → p ∗ U ∨ → V ⊗ O e G ( H ′ ) → O E ′ ( H ′ + F ) → . (5.6) Proof.
By Lemma 5.4, we have an exact sequence V ∨ ⊗ O P ( B ) φ −→ V ⊗ O P ( B ) ( H ′ ) → O P ( V ) × P ( V ) (2 , → . Pulling back to e G , we obtain an exact sequence V ∨ ⊗ O e G → V ⊗ O e G ( H ′ ) → O E ′ ( H ′ + F ) → , Since E ′ is a divisor on e G , the kernel K of the epimorphism V ⊗ O e G ( H ′ ) → O E ′ ( H ′ + F ) isa rank 2 vector bundle on e G , which by the above exact sequence is a quotient of the trivialbundle V ∨ ⊗ O e G . Hence K induces a morphism e G → G . This morphism can be checked toagree with the blowdown morphism p , so K ∼ = p ∗ U ∨ . (cid:3) ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 35
Setup and statement of the result.
Recall that X is an ordinary GM fourfold con-taining the plane P = Gr(2 , V ). The following proposition describes the structure of therational map from X to P given by projection from P . We slightly abuse notation by us-ing the same symbols for the exceptional divisors and blowup morphisms as in the abovediscussion of G . Proposition 5.6.
Let p : e X → X be the blowup with center in P . Then the linear projectionfrom P induces a regular map q : e X → X ′ to a cubic fourfold X ′ containing a smooth cubicsurface scroll T , and identifies e X as the blowup of X ′ in T . In other words, we have a diagram E i / / p E (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ e X p (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ q ❅❅❅❅❅❅❅❅ E ′ j o o q E ′ (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ P / / X X ′ T o o where p and q are blowups with exceptional divisors E and E ′ . Moreover, the relations (5.4) continue to hold on e X , and K e X = − H + E = − H ′ + E ′ . (5.7) Finally, if X does not contain planes of the form P ( V ∧ V ) where V ⊂ V ⊂ V ⊂ V ,then X ′ is smooth.Proof. By § P ( W ) ⊂ P ( ∧ V ) and a quadric hypersurface Q ⊂ P ( W )such that X = G ∩ Q and P ⊂ Q . Consider the subspace C = W/ ∧ V ⊂ ∧ V / ∧ V = B, so that P ( C ) ⊂ P ( B ) is a hyperplane. We claim that the corresponding hyperplane section T = ( P ( V ) × P ( V )) ∩ P ( C )of P ( V ) × P ( V ) ⊂ P ( B ) is a smooth cubic surface scroll. For this it is enough to showthat P ( C ) ∩ P ( V ⊗ V ) is a hyperplane in P ( V ⊗ V ) whose equation, considered as anelement in V ∨ ⊗ V ∨ ∼ = Hom( V , V ∨ ), has rank 2. Assume on the contrary that the rank ofthis equation is at most 1. Then its kernel is a subspace of V of dimension at least 2, which iscontained in the kernel of the skew form ω on V defining W . So the rank of ω is 2. But thenthe Grassmannian hull M X = G ∩ P ( W ) of X is singular along P = Gr(2 , ker( ω )), and X issingular along P ∩ Q . This contradiction proves the claim.The proper transform of the Grassmannian hull M = M X under p : e G → G coincides withthe proper transform of P ( C ) under q : e G → P ( B ). Thus if f M = Bl P ( M ) → M is the blowupin P , then projection from P gives an identification f M ∼ = Bl T ( P ( C )) → P ( C ). Further, theproper transform of X = M ∩ Q under f M → M is cut out by a section of the line bundle O f M (2 H − E ) = O f M (3 H ′ − E ′ ) , and therefore coincides with the proper transform under f M → P ( C ) of a cubic fourfold X ′ ⊂ P ( C ) containing T . This proves the first part of the lemma.The relations (5.4) clearly restrict to e X , and the equalities (5.7) follow from the standardformula for the canonical class of a blowup.It remains to show that X ′ is smooth if X does not contain planes of the form P ( V ∧ V )where V ⊂ V ⊂ V ⊂ V . For this, first note that the blowup of X ′ in T is smooth, since it coincides with the blowup of X in P . Therefore, X ′ is smooth away from T . On the other hand, T is also smooth, so it is enough to check that T ⊂ X ′ is a locally complete intersection, i.e.that its conormal sheaf is locally free. Since E ′ → T is the exceptional divisor of the blowupof X ′ in T , it is enough to check that the map E ′ → T is a P -bundle. Since E ′ is cut out inthe exceptional divisor of (5.3) by fiberwise linear conditions, it is enough to show that thereare no points in T ⊂ P ( V ) × P ( V ) over which the fiber of E ′ is isomorphic to P . But such apoint would correspond to a choice of a V ⊂ V (giving a point in P ( V )) and V ⊂ V (givinga point of P ( V /V ) = P ( V )), such that the plane P ( V ∧ V ) is in X . Since we assumed thereare no such planes in X , we conclude that X ′ is smooth. (cid:3) The condition guaranteeing smoothness of X ′ in the final statement of Proposition 5.6 holdsgenerically: Lemma 5.7. If X is a general ordinary GM fourfold containing P = Gr(2 , V ) for some V ⊂ V , then X does not contain planes of the form P ( V ∧ V ) where V ⊂ V ⊂ V ⊂ V .Proof. By Theorem 3.1, an ordinary GM fourfold X corresponds to a pair ( A , p ) such that A has no decomposable vectors and p ∈ Y A ⊥ . By Remark 5.1, X contains the plane Gr(2 , V ) ifand only if (5.2) holds. Similarly, by [11, Theorem 4.3(c)], X contains a plane P ( V ∧ V ) ifand only if Y A ∩ P ( V ) = ∅ .By [19, Lemma 3.6] Lagrangians A ⊂ ∧ V with no decomposable vectors such that thereis V ⊂ V for which the first part of (5.2) holds are parameterized by an open subset of adivisor Γ ⊂ LG(10 , ∧ V ), and by [19, Lemma 3.7] this divisor has no common componentswith the divisor ∆ ⊂ LG(10 , ∧ V ) parameterizing A such that Y A = ∅ . Choose any A withno decomposable vectors such that there is V ⊂ V for which the first part of (5.2) holds,but Y A = ∅ . Then as explained in Remark 5.1, there is a 2-dimensional family of ordinaryGM fourfolds containing Gr(2 , V ); none of these contain a plane of the form P ( V ∧ V )since Y A = ∅ . (cid:3) Our goal is to prove the following result.
Theorem 5.8.
Assume the cubic fourfold X ′ associated to X by Proposition is smooth.Then there is an equivalence A X ≃ A X ′ , where A X is the GM category defined by (2.11) and A X ′ is defined by (3.1) . Remark 5.9.
Theorem 5.8 is of an essentially different nature than Theorem 4.1, in thatit does not “come from” K3 surfaces. More precisely, for a very general GM fourfold X satisfying (5.1) for some V , the category A X is not equivalent to the derived category of a K3surface, or even a twisted K3 surface. Indeed, the construction of Proposition 5.6 dominatesthe locus of cubic fourfolds containing a smooth cubic surface scroll, so it suffices to provethat given a very general such cubic, its K3 category is not equivalent to the twisted derivedcategory of a K3 surface. Since cubic fourfolds containing a cubic scroll have discriminant 12by [16, Example 4.1.2], this follows from [17, Theorem 1.4].5.3. Strategy of the proof.
From now on, we assume the hypothesis of Theorem 5.8 issatisfied. The proof of this theorem occupies the rest of this section. Here is our strategy.By Orlov’s decomposition of the derived category of a blowup, we haveD b ( e X ) = h p ∗ D b ( X ) , i ∗ p ∗ E D b ( P ) i . ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 37
Inserting (2.11) and the standard decomposition of D b ( P ) into the above decomposition, weobtain D b ( e X ) = h p ∗ A X , O , U ∨ , O ( H ) , U ∨ ( H ) , O E , O E ( H ) , O E (2 H ) i . (5.8)Here and below, to ease notation we write U ∨ for p ∗ U ∨ X . This decomposition of D b ( e X ) consistsof a copy of A X and 7 exceptional objects.On the other hand, from the expression of e X as a blowup of X ′ , we haveD b ( e X ) = h q ∗ D b ( X ′ ) , j ∗ q ∗ E ′ D b ( T ) i . Inserting the decomposition (3.1) for D b ( X ′ ), we obtainD b ( e X ) = h q ∗ A X ′ , O , O ( H ′ ) , O (2 H ′ ) , j ∗ q ∗ E ′ D b ( T ) i . (5.9)Note that D b ( T ) has a decomposition consisting of 4 exceptional objects, hence the decom-position (5.9) consists of one copy of A X ′ and again 7 exceptional objects.To prove the equivalence A X ≃ A X ′ , we will find a sequence of mutations transformingthe exceptional objects of (5.8) into those of (5.9). In doing so, we will explicitly identify afunctor giving the desired equivalence, see (5.14).5.4. Mutations.
We perform a sequence of mutations, starting with (5.8). For a brief reviewof mutation functors and references, see the discussion in § Step 1.
Mutate U ∨ ( H ) to the far left in (5.8). Since this is a mutation in D b ( X ) and wehave K X = − H , by (2.19) the result isD b ( e X ) = h U ∨ ( − H ) , p ∗ A X , O , U ∨ , O ( H ) , O E , O E ( H ) , O E (2 H ) i . Step 2.
Mutate U ∨ ( − H ) to the far right. Again by (2.19) and (5.7), the result isD b ( e X ) = h p ∗ A X , O , U ∨ , O ( H ) , O E , O E ( H ) , O E (2 H ) , U ∨ ( H − E ) i . Step 3.
Left mutate O E through h O , U ∨ , O ( H ) i . We haveExt • ( O ( H ) , O E ) = H • ( P, O P ( − H )) = 0 , Ext • ( U ∨ , O E ) = H • ( P, U P ) = 0 , Ext • ( O , O E ) = H • ( P, O P ) = k , where in the second line U P is the tautological rank 2 bundle on P = Gr(2 , V ), i.e. therestriction of U from G to P . Hence by the definition of the mutation functorL h O , U ∨ , O ( H ) i ( O E ) = Cone( O → O E ) = O ( − E )[1] , and the resulting decomposition isD b ( e X ) = h p ∗ A X , O ( − E ) , O , U ∨ , O ( H ) , O E ( H ) , O E (2 H ) , U ∨ ( H − E ) i . Step 4.
Left mutate O E (2 H ) through h O , U ∨ , O ( H ) , O E ( H ) i . Lemma 5.10.
We have L h O , U ∨ , O ( H ) , O E ( H ) i ( O E (2 H )) ∼ = O E ′ ( E ′ − F )[2] . Proof.
There is an isomorphism of functorsL h O , U ∨ , O ( H ) , O E ( H ) i ∼ = L O ◦ L U ∨ ◦ L O ( H ) ◦ L O E ( H ) . Hence to prove the result we successively left mutate O E (2 H ) through O E ( H ) , O ( H ) , U ∨ , O .To compute L O E ( H ) ( O E (2 H )), we may compute L O P ( H ) ( O P (2 H )) and pull back the result.We have Ext • ( O P ( H ) , O P (2 H )) = H • ( P, O P ( H )) = V , soL O P ( H ) ( O P (2 H )) = Cone( O P ( H ) ⊗ V → O P (2 H )) . The morphism O P ( H ) ⊗ V → O P (2 H ) is the twist by H of the tautological morphism, henceit is surjective with kernel U P ( H ) ∼ = U ∨ P . Thus the above cone is U ∨ P [1], andL O E ( H ) ( O E (2 H )) = U ∨ E [1] . Next note Ext • ( O ( H ) , U ∨ E ) = H • ( P, U ∨ P ( − H )) = 0, henceL O ( H ) ( U ∨ E ) = U ∨ E . Further, we have Ext • ( U ∨ , U ∨ E ) = H • ( P, U P ⊗ U ∨ P ) = k , henceL U ∨ ( U ∨ E ) = Cone( U ∨ → U ∨ E ) = U ∨ ( − E )[1] . Now we are left with the last and most interesting step — the mutation of U ∨ ( − E ) through O .First, using the exact sequence 0 → O ( − E ) → O → O E → U ∨ , we findExt • ( O , U ∨ ( − E )) = H • ( e X, U ∨ ( − E )) = ker( V ∨ → V ∨ ) = V ∨ . (5.10)Thus we need to understand the cone of the natural morphism V ∨ ⊗ O → U ∨ ( − E ). Restrict-ing (5.6) to e X , dualizing, twisting by H ′ = H − E , and using the isomorphism U ( H ) ∼ = U ∨ ,we obtain a distinguished triangle V ∨ ⊗ O → U ∨ ( − E ) → O E ′ ( E ′ − F ) . Thus L O ( U ∨ ( − E )) = O E ′ ( E ′ − F ) , (5.11)which completes the proof of the lemma. (cid:3) By the lemma, the result of the above mutation isD b ( e X ) = h p ∗ A X , O ( − E ) , O E ′ ( E ′ − F ) , O , U ∨ , O ( H ) , O E ( H ) , U ∨ ( H − E ) i . Step 5.
Left mutate O E ( H ) through O ( H ). We haveL O ( H ) ( O E ( H )) = Cone( O ( H ) → O E ( H )) = O ( H − E )[1] = O ( H ′ )[1] , so the result isD b ( e X ) = h p ∗ A X , O ( − E ) , O E ′ ( E ′ − F ) , O , U ∨ , O ( H ′ ) , O ( H ) , U ∨ ( H − E ) i . Step 6.
Right mutate U ∨ through O ( H ′ ). We haveExt • ( U ∨ , O ( H ′ )) = Ext • ( O , U ( H − E )) = Ext • ( O , U ∨ ( − E )) = V ∨ , where the last equality holds by (5.10). HenceR O ( H ′ ) ( U ∨ ) = Cone( U ∨ → V ⊗ O ( H ′ ))[ − . ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 39
Now restricting (5.6) to e X shows R O ( H ′ ) ( U ∨ ) = O E ′ ( H ′ + F )[ − b ( e X ) = h p ∗ A X , O ( − E ) , O E ′ ( E ′ − F ) , O , O ( H ′ ) , O E ′ ( H ′ + F ) , O ( H ) , U ∨ ( H − E ) i . Step 7.
Left mutate U ∨ ( H − E ) through O ( H ). By (5.11) and (4.12) we haveL O ( H ) ( U ∨ ( H − E )) = O E ′ ( H + E ′ − F ) = O E ′ (2 H ′ − F ) , so the result isD b ( e X ) = h p ∗ A X , O ( − E ) , O E ′ ( E ′ − F ) , O , O ( H ′ ) , O E ′ ( H ′ + F ) , O E ′ (2 H ′ − F ) , O ( H ) i . Step 8.
Right mutate p ∗ A X through h O ( − E ) , O E ′ ( E ′ − F ) i . The result isD b ( e X ) = h O ( − E ) , O E ′ ( E ′ − F ) , Ψ p ∗ A X , O , O ( H ′ ) , O E ′ ( H ′ + F ) , O E ′ (2 H ′ − F ) , O ( H ) i , where Ψ = R h O ( − E ) , O E ′ ( E ′ − F ) i . Step 9.
Mutate h O ( − E ) , O E ′ ( E ′ − F ) i to the far right. By (2.19), the result isD b ( e X ) = h Ψ p ∗ A X , O , O ( H ′ ) , O E ′ ( H ′ + F ) , O E ′ (2 H ′ − F ) , O ( H ) , O (2 H ′ ) , O E ′ (3 H ′ − F ) i . Step 10.
Right mutate O ( H ) through O (2 H ′ ). We haveExt • ( O ( H ) , O (2 H ′ )) = H • ( e X, O ( E ′ )) = k and hence R O (2 H ′ ) ( O ( H )) = Cone( O ( H ) → O (2 H ′ ))[ − . The morphism O ( H ) → O (2 H ′ ) is the twist by 2 H ′ of O ( − E ′ ) → O , henceR O (2 H ′ ) ( O ( H )) = O E ′ (2 H ′ )[ − . Thus the result of the mutation is a decompositionD b ( e X ) = h Ψ p ∗ A X , O , O ( H ′ ) , O E ′ ( H ′ + F ) , O E ′ (2 H ′ − F ) , O (2 H ′ ) , O E ′ (2 H ′ ) , O E ′ (3 H ′ − F ) i . Step 11.
Left mutate O (2 H ′ ) through h O E ′ ( H ′ + F ) , O E ′ (2 H ′ − F ) i . By the semiorthogonalityof q ∗ D b ( X ′ ) and j ∗ q ∗ E ′ D b ( T ) in D b ( e X ), this mutation is just a transposition. Thus the resultisD b ( e X ) = h Ψ p ∗ A X , O , O ( H ′ ) , O (2 H ′ ) , O E ′ ( H ′ + F ) , O E ′ (2 H ′ − F ) , O E ′ (2 H ′ ) , O E ′ (3 H ′ − F ) i . It is straightforward to check thatD b ( T ) = h O T ( H ′ + F ) , O T (2 H ′ − F ) , O T (2 H ′ ) , O T (3 H ′ − F ) i , so the above decomposition can be written asD b ( e X ) = h Ψ p ∗ A X , O , O ( H ′ ) , O (2 H ′ ) , j ∗ q ∗ E ′ D b ( T ) i . (5.12)This completes the proof of Theorem 5.8. Indeed, comparing the decompositions (5.12)and (5.9) shows q ∗ ◦ R h O e X ( − E ) , O E ′ ( E ′ − F ) i ◦ p ∗ : A X → A X ′ (5.13)is an equivalence. (cid:3) Remark 5.11.
The functor (5.13) is in fact isomorphic to q ∗ ◦ R O e X ( − E ) ◦ p ∗ : A X → A X ′ . (5.14)To see this, observe that q ∗ kills O E ′ ( E ′ − F ): if j : T ֒ → X ′ denotes the inclusion, then q ∗ ( O E ′ ( E ′ − F )) = j ∗ q E ′ ∗ ( O E ′ ( E ′ − F )) = j ∗ ( q E ′ ∗ ( O E ′ ( E ′ )) ⊗ O T ( F )) = 0since q E ′ ∗ ( O E ′ ( E ′ )) = 0. Thus q ∗ ◦ R O E ′ ( E ′ − F ) ∼ = q ∗ , and the claim follows since there is anisomorphism of functors R h O e X ( − E ) , O E ′ ( E ′ − F ) i ∼ = R O E ′ ( E ′ − F ) ◦ R O e X ( − E ) . Appendix A. Moduli of GM varieties
Let (Sch / k ) denote the category of k -schemes. Definition A.1.
For 2 ≤ n ≤
6, the moduli stack M n of smooth n -dimensional GM va-rieties is the fibered category over (Sch / k ) whose fiber over S ∈ (Sch / k ) is the groupoidof pairs ( π : X → S, L ), where π : X → S is a smooth proper morphism of schemes and L ∈ Pic
X/S ( S ), such that for every geometric point ¯ s ∈ S the pair ( X ¯ s , L ¯ s ) is isomorphicto a smooth n -dimensional GM variety with its natural polarization (equivalently, ( X ¯ s , L ¯ s )satisfies conditions (2.3) and (2.2) with H the divisor corresponding to L ¯ s ). A morphism from( π ′ : X ′ → S ′ , L ′ ) to ( π : X → S, L ) is a fiber product diagram X ′ π ′ (cid:15) (cid:15) g ′ / / X π (cid:15) (cid:15) S ′ g / / S such that ( g ′ ) ∗ ( L ) = L ′ ∈ Pic X ′ /S ′ ( S ′ ).The following result gives the basic properties of the moduli stack M n . An explicit descrip-tion of M n will be given in [12]. We follow [56] for our conventions on algebraic stacks. Proposition A.2.
The moduli stack M n is a smooth and irreducible Deligne–Mumford stackof finite type over k . Its dimension is given by dim M n = 25 − (6 − n )(5 − n ) / , i.e. dim M = 19 , dim M = 22 , dim M = 24 , dim M = 25 , dim M = 25 . We will use the following lemma.
Lemma A.3.
Let X be a smooth GM variety of dimension n ≥ . Then: (1) The automorphism group scheme
Aut k ( X ) is finite and reduced. (2) H i ( X, T X ) = 0 for i = 1 . (3) dim H ( X, T X ) = 25 − (6 − n )(5 − n ) / .Proof. As our base field k has characteristic 0, Aut k ( X ) is automatically reduced by a theoremof Cartier [42, Lecture 25], and it is finite by [10, Proposition 3.21(c)]. Hence H ( X, T X ), beingthe tangent space to Aut k ( X ) at the identity, vanishes. Further, T X ∼ = Ω n − X ( n −
2) by (2.2)and hence H i ( X, T X ) = 0 for i ≥ ( X, T X ) is straightforward to compute using Riemann–Roch. (cid:3) Proof of Proposition
A.2 . First consider the case n = 2. Then by (2.3), M is the Brill–Noether general locus (and hence Zariski open) in the moduli stack of polarized K3 surfacesof degree 10. It is well-known that all the properties in the proposition hold for the moduli ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 41 stack of primitively polarized K3 surfaces of a fixed degree (see [18, Chapter 5]), so they alsohold for M .From now on assume n ≥
3. A standard Hilbert scheme argument shows that M n is analgebraic stack of finite type over k , whose diagonal is affine and of finite type. To prove M n is Deligne–Mumford, by [56, Tag 06N3] it suffices to show its diagonal is unramified. Asa finite type morphism is unramified if and only if all of its geometric fibers are finite andreduced, we are done by Lemma A.3(1) (note that for a GM variety of dimension n ≥
3, allautomorphisms preserve the natural polarization).Next we check smoothness of M n . Let ( X, L ) be a point of M n , i.e. X is a GM n -fold and L ∈ Pic( X ) is the ample generator. Let A L be the Atiyah extension of L , i.e. the extension0 → O X → A L → T X → L . Further, recall that H ( X, A L ) classifies first order deformationsof the pair ( X, L ), and H ( X, A L ) is the obstruction space for such deformations (see [54, § i ( X, A L ) ∼ = H i ( X, T X ) for i ≥
1. In particular, H ( X, A L ) = 0 by Lemma A.3(2), so the formal deformation spaceof M n at ( X, L ) is smooth of dimension dim H ( X, A L ) = dim H ( X, T X ). This implies thesmoothness of M n and, using Lemma A.3(3), the formula for its dimension.It remains to show that M n is irreducible. This follows from the defining expression (2.1)of any GM variety. Indeed, let P n be the space of pairs ( W, Q ) where W ⊂ k ⊕ ∧ V isan ( n + 5)-dimensional linear subspace and Q ⊂ P ( W ) is a quadric hypersurface, and let U n ⊂ P n be the open subset where Cone( G ) ∩ Q is smooth of dimension n . The projection P n → Gr( n + 5 , k ⊕ ∧ V ) is a projective bundle, hence P n and U n are irreducible. On theother hand, by (2.1), U n maps surjectively onto M n . Hence M n is irreducible as well. (cid:3) References [1] Nicolas Addington, Brendan Hassett, Yuri Tschinkel, and Anthony V´arilly-Alvarado,
Cubic fourfoldsfibered in sextic del Pezzo surfaces , arXiv preprint arXiv:1606.05321 (2016).[2] Nicolas Addington and Richard Thomas,
Hodge theory and derived categories of cubic fourfolds , DukeMath. J. (2014), no. 10, 1885–1927.[3] Arnaud Beauville,
Vari´et´es de Prym et jacobiennes intermediaires , Ann. Sci. ´Ec. Norm. Sup´er. (4) (1977), no. 3, 309–391.[4] Christian B¨ohning, Hans-Christian Graf von Bothmer, and Pawel Sosna, On the Jordan-H¨older propertyfor geometric derived categories , Adv. Math. (2014), 479–492.[5] Alexei Bondal,
Representations of associative algebras and coherent sheaves , Izv. Akad. Nauk SSSR Ser.Mat. (1989), no. 1, 25–44.[6] Alexei Bondal and Mikhail Kapranov, Representable functors, Serre functors, and mutations , Mathematicsof the USSR-Izvestiya (1990), no. 3, 519.[7] John Calabrese and Richard Thomas, Derived equivalent Calabi–Yau threefolds from cubic fourfolds , Math.Ann. (2016), no. 1, 155–172.[8] A. Conte and J. P. Murre,
The Hodge conjecture for fourfolds admitting a covering by rational curves ,Math. Ann. (1978), no. 1, 79–88.[9] Olivier Debarre, Atanas Iliev, and Laurent Manivel,
Special prime Fano fourfolds of degree and index Gushel–Mukai varieties: classification and birationalities , toappear in Algebraic Geometry. arXiv preprint arXiv:1510.05448 (2016).[11] ,
Gushel–Mukai varieties: linear spaces and periods , arXiv preprint arXiv:1605.05648 (2016).[12] ,
Gushel–Mukai varieties: moduli stacks and coarse moduli spaces , in preparation, 2017. [13] Olivier Debarre and Claire Voisin,
Hyper-K¨ahler fourfolds and Grassmann geometry , J. Reine Angew.Math. (2010), 63–87.[14] William Fulton,
Intersection theory , second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3.Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series.A Series of Modern Surveys in Mathematics, vol. 2, Springer-Verlag, Berlin, 1998.[15] N. P. Gushel’,
On Fano varieties of genus 6 , Izv. Math. (1983), no. 3, 445–459.[16] Brendan Hassett, Special cubic fourfolds , Compositio Math. (2000), no. 1, 1–23.[17] Daniel Huybrechts,
The K category of a cubic fourfold , to appear in Compos. Math. arXiv preprintarXiv:1505.01775 (2016).[18] Daniel Huybrechts, Lectures on K surfaces , Cambridge University Press, 2016.[19] Atanas Iliev, Grzegorz Kapustka, Michal Kapustka, and Kristian Ranestad, EPW cubes , to appear in J.Reine Angew. Math. arXiv preprint arXiv:1505.02389 (2016).[20] Atanas Iliev and Laurent Manivel,
Fano manifolds of degree ten and EPW sextics , Ann. Sci. ´Ec. Norm.Sup´er. (4) (2011), no. 3, 393–426.[21] Grzegorz Kapustka and Michal Kapustka, private communication , (2016).[22] Oliver K¨uchle, On Fano -folds of index and homogeneous vector bundles over Grassmannians , Math.Z. (1995), no. 4, 563–575.[23] Alexander Kuznetsov, Derived Categories of Cubic and V threefolds , Trudy Matematicheskogo Institutaim. V.A. Steklova (2004), 183–207.[24] , Hyperplane sections and derived categories , Izv. Math. (2006), no. 3, 447–547.[25] , Homological projective duality , Publ. Math. Inst. Hautes ´Etudes Sci. (2007), no. 105, 157–220.[26] ,
Derived categories of Fano threefolds , Tr. Mat. Inst. Steklova (2009), no. MnogomernayaAlgebraicheskaya Geometriya, 116–128.[27] ,
Hochschild homology and semiorthogonal decompositions , arXiv preprint arXiv:0904.4330 (2009).[28] ,
Derived categories of cubic fourfolds , Cohomological and geometric approaches to rationalityproblems, Springer, 2010, pp. 219–243.[29] ,
Semiorthogonal decompositions in algebraic geometry , Proceedings of the International Congressof Mathematicians, Vol. II (Seoul, 2014), 2014, pp. 635–660.[30] ,
Calabi–Yau and fractional Calabi–Yau categories , to appear in J. Reine Angew. Math. arXivpreprint arXiv:1509.07657 (2015).[31] ,
Height of exceptional collections and Hochschild cohomology of quasiphantom categories , J. ReineAngew. Math. (2015), 213–243.[32] ,
On K¨uchle varieties with Picard number greater than 1. , Izv. Math. (2015), no. 4, 698–709.[33] , Derived categories view on rationality problems , pp. 67–104, Springer International Publishing,Cham, 2016.[34] ,
K¨uchle fivefolds of type c
5, Math. Z. (2016), no. 3-4, 1245–1278. MR 3563277[35] Alexander Kuznetsov and Alexander Perry,
Categorical joins , in preparation (2017).[36] ,
Derived categories of cyclic covers and their branch divisors , Selecta Math. (N.S.) (2017),no. 1, 389–423.[37] Robert Lazarsfeld, Positivity in algebraic geometry. I , Ergebnisse der Mathematik und ihrer Grenzgebiete.3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rdSeries. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004, Classicalsetting: line bundles and linear series.[38] Dmitry Logachev,
Fano threefolds of genus 6 , Asian J. Math. (2012), no. 3, 515–559.[39] Emanuele Macr`ı and Paolo Stellari, Fano varieties of cubic fourfolds containing a plane , Math. Ann. (2012), no. 3, 1147–1176.[40] Nikita Markarian,
The Atiyah class, Hochschild cohomology and the Riemann–Roch theorem. , J. Lond.Math. Soc., II. Ser. (2009), no. 1, 129–143.[41] Shigeru Mukai, Biregular classification of Fano -folds and Fano manifolds of coindex
3, Proc. Nat. Acad.Sci. U.S.A. (1989), no. 9, 3000–3002.[42] David Mumford, Lectures on curves on an algebraic surface , with a section by G. M. Bergman. Annals ofMathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966.[43] J. Nagel,
The generalized Hodge conjecture for the quadratic complex of lines in projective four-space ,Math. Ann. (1998), no. 2, 387–401.
ERIVED CATEGORIES OF GUSHEL–MUKAI VARIETIES 43 [44] Kieran G. O’Grady,
Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics , Duke Math. J. (2006), no. 1, 99–137.[45] ,
Dual double EPW-sextics and their periods , Pure Appl. Math. Q. (2008), no. 2, part 1, 427–468.[46] , Moduli of double EPW-sextics , arXiv preprint arXiv:1111.1395 (2011).[47] ,
EPW-sextics: taxonomy , Manuscripta Math. (2012), no. 1-2, 221–272.[48] ,
Double covers of EPW-sextics , Michigan Math. J. (2013), no. 1, 143–184.[49] , Periods of double EPW-sextics , Math. Z. (2015), no. 1-2, 485–524.[50] Shinnosuke Okawa,
Semi-orthogonal decomposability of the derived category of a curve , Adv. Math. (2011), no. 5, 2869–2873.[51] Dmitri Orlov,
Smooth and proper noncommutative schemes and gluing of DG categories , Adv. Math. (2016), 59–105. MR 3545926[52] Alexander Perry,
Hochschild cohomology and group actions , preprint (2017).[53] Alexander Polishchuk,
Lefschetz type formulas for dg-categories , Selecta Math. (N.S.) (2014), no. 3,885–928.[54] Edoardo Sernesi, Deformations of algebraic schemes , Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences], vol. 334, Springer-Verlag, Berlin, 2006.[55] N. I. Shepherd-Barron,
The rationality of quintic Del Pezzo surfaces—a short proof , Bull. London Math.Soc. (1992), no. 3, 249–250.[56] The Stacks Project Authors, Stacks project , http://stacks.math.columbia.edu , 2015.[57] Richard Thomas, Notes on HPD , arXiv preprint arXiv:1512.08985 (2015).
Steklov Mathematical Institute of Russian Academy of Sciences,8 Gubkin str., Moscow 119991 RussiaThe Poncelet Laboratory, Independent University of MoscowNational Research University Higher School of Economics, Russian Federation
E-mail address : [email protected] Department of Mathematics, Columbia University, New York, NY 10027, USA
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