Calabi Yau and fractional Calabi Yau categories

J. reine angew. ath., Ahead of Print Journal für die reine und angewandte athematik DOI 10.1515/crelle-2017-0004 De Gruyter 2017 Calabi Yau and fractional Calabi Yau categories By Alexander Kuznetsov at oscow Abstract. We discuss Calabi Yau and fractional Calabi Yau semiorthogonal components of derived categories of coherent sheaves on smooth projective varieties. The main result is a general construction of a fractional Calabi Yau category from a rectangular Lefschetz decomposition and a spherical functor. We give many examples of applications of this construction and discuss some general properties of Calabi Yau categories. 1. Introduction Projective varieties with trivial canonical class (Calabi Yau varieties) form a very important class of varieties in algebraic geometry. Their importance is emphasized by the special role they play in irror Symmetry which associates with each Calabi Yau variety a mirror partner Y, such that the Hodge numbers of and Y are related by h p;q.y / D h q;n p./, where n D dim D dim Y. However, this relation shows that by considering only usual Calabi Yau varieties, we are missing some mirror partners. Indeed, if is a rigid Calabi Yau variety then h n 1;1./ D 0 and so one expects to have h 1;1.Y / D 0 for the mirror partner Y of. Thus Y cannot be projective. It is expected, however, that irror Symmetry extends to rigid Calabi Yau varieties, but their mirror partners are noncommutative Calabi Yau varieties. In other words, instead of an algebraic variety Y one expects to associate with a certain triangulated category T (thought of as the derived category of coherent sheaves on a noncommutative variety Y ). To express the Calabi Yau property of Y in terms of T it is natural to use the Serre functor S T. The Serre functor is one of the most important invariants of a triangulated category (see Section 2.3), which for derived categories of coherent sheaves on smooth projective varieties is the composition of the twist by the canonical bundle and the shift by the dimension. Thus derived categories of Calabi Yau varieties are characterized by the fact that their Serre functor is just a shift. This motivates the following definition. Definition 1.1. Let n 2 Z. A triangulated category T is an n-calabi Yau category if it has a Serre functor S T and, moreover, S T Š Œn. The integer n is called the CY-dimension of T. The author was partially supported by the Russian Academic Excellence Project 5-100, by RFBR grants 15-01-02164, 15-51-50045, and by the Simons foundation.

2 Kuznetsov, Calabi Yau and fractional Calabi Yau categories It is also natural to consider the following weakening of the Calabi Yau property. Definition 1.2. A triangulated category T is a fractional Calabi Yau category if it has a Serre functor S T and there are integers p and q 0 such that S q T Š Œp. The goal of this paper is to show that there are many examples of Fano varieties which have a semiorthogonal decomposition with one of the components being a fractional Calabi Yau category. The presence of a Calabi Yau component usually has a strong influence on the geometric properties of the Fano variety, which acquires some properties specific to Calabi Yau varieties (this was discussed from the Hodge-theoretic point of view in [11]). For example, if a variety has a semiorthogonal component which is a 2-Calabi Yau category then any moduli space of coherent sheaves on carries a closed 2-form, and some of them provide interesting examples of hyper-kähler varieties. This makes it interesting to find some general construction of Calabi Yau categories of geometric origin. The main result of this paper is such a construction. We start with a smooth projective variety with a rectangular Lefschetz decomposition (see Section 2.2 for a definition and Section 4.1 for examples of such varieties, the simplest example to have in mind is the projective space P n, or the Grassmannian Gr.k; n/ with coprime k and n). Further, consider a spherical functor ˆW D./! D. / between the bounded derived categories of coherent sheaves of another smooth projective variety and (see Section 2.4 for a definition and Section 3.1 for some examples, again the simplest example is the derived pushforward for a divisorial embedding,! ). Assuming some compatibility between the Lefschetz decomposition of D. / and the functor ˆ, we prove that D./ has a semiorthogonal decomposition, such that an appropriate power of the Serre functor of one of the components of this decomposition is isomorphic to a shift. The construction is explained in detail in Section 3 after a preparatory Section 2. In Section 4 we list some known varieties with a rectangular Lefschetz decomposition and some Calabi Yau categories arising from these. We pay special attention to K3 and 3-Calabi Yau categories coming from these examples. Finally, in Section 5 we discuss some general properties of Calabi Yau categories. We show that a connected Calabi Yau category is indecomposable, and prove an inequality between the CY-dimension of a Calabi Yau component of the derived category of a smooth projective variety and the dimension of the variety itself. We also discuss some interesting questions and conjectures related to Calabi Yau categories. Acknowledgement. I would like to thank Alex Perry for the suggestion to consider Example 3.3 of a spherical functor and for many valuable comments on the first draft of the paper. y thanks also go to Nick Addington and to the anonymous referee for their comments. 2. Preliminaries 2.1. Notations and conventions. All varieties considered in this paper are assumed to be smooth and projective over a field k, and all categories are k-linear. In the examples related to Grassmannians the field is assumed to be of zero (or sufficiently big positive) characteristic. For a variety we denote by D./ the bounded derived category of coherent sheaves on.

Kuznetsov, Calabi Yau and fractional Calabi Yau categories 3 All the pushforward, pullback, and tensor product functors are derived. All functors between triangulated categories are assumed to be triangulated. For a functor ˆW T 1! T 2 between triangulated categories T 1 and T 2 we denote by ˆ its left adjoint and by ˆŠ its right adjoint (if they exist). We denote the units and the co-units of the adjunctions by ˆ;ˆW id! ˆ ı ˆ and ˆ;ˆW ˆ ı ˆ! id; and if there is no risk of confusion we omit the lower indices. Recall that the compositions and ˆ ˆ;ˆıˆ! ˆ ı ˆ ı ˆ ˆıˆ;ˆ! ˆ ˆ ˆıˆ;ˆ! ˆ ı ˆ ı ˆ ˆ;ˆıˆ! ˆ are identity morphisms (in fact, this is one of the equivalent definitions of adjunction). Given an object E 2 D. Y / we can consider a functor D./! D.Y /; F 7! p Y.E p.f //; where p and p Y are the projections of Y to and Y, respectively. It is called the Fourier ukai functor with kernel E. A morphism of kernels induces a morphism of the corresponding Fourier ukai functors. Furthermore there is an operation of convolution of kernels, which corresponds to composition of functors. Finally, any Fourier ukai functor has both adjoints which are also Fourier ukai functors, and moreover, the unit and the counit of the adjunctions are induced by morphisms of kernels [2]. In what follows, to unburden notation we identify Fourier ukai functors with their kernels, and we consider only those morphisms of functors which are induced by morphisms of kernels. In particular, by a distinguished triangle of (Fourier ukai) functors we understand a distinguished triangle of kernels. Furthermore, an object F 2 D./ will be identified with the derived tensor product functor F, i.e. with the Fourier ukai functor whose kernel is the pushforward of F to under the diagonal embedding. In particular, given a line bundle L on, the same notation will be used for the tensor product L functor. Similarly, given an automorphism of we will write also for the autoequivalence of D./ it induces. 2.2. Semiorthogonal decompositions and mutation functors. For a review of semiorthogonal decompositions and their uses one can look into [20]. Definition 2.1. A semiorthogonal decomposition of a triangulated category T is a collection A 1 ; : : : ; A m of full triangulated subcategories in T such that for all i > j we have Hom.Ai ; A j / D 0; for any object T 2 T there is a filtration (i.e., a chain of morphisms) such that Cone.T i! T i 1 / 2 A i. 0 D T m! T m 1!! T 1! T 0 D T A semiorthogonal decomposition is denoted by T D ha 1 ; : : : ; A m i.

4 Kuznetsov, Calabi Yau and fractional Calabi Yau categories If T D D./ is the derived category of a smooth projective variety, every component of a semiorthogonal decomposition is an admissible subcategory of D./ (see [6]), i.e. its embedding functor has both left and right adjoints. Conversely, if B D./ is an admissible subcategory with the embedding functor ˇ and its adjoints ˇ and ˇŠ, respectively, it extends in two ways to a semiorthogonal decomposition where D./ D hb? ; Bi; B? D F 2 D./ j Hom.B; F / D 0 D./ D hb;? Bi; and? B D F 2 D./ j Hom.F; B/ D 0 : oreover, the compositions ˇˇ, ˇˇŠW D./! D./ are Fourier ukai functors, whose kernels can be constructed as follows. First, using [19], one extends the above semiorthogonal decompositions to the square D. / D hb? D./; B D./i; D. / D hb D./;? B D./i: Then the decompositions of the structure sheaf of the diagonal with respect to these semiorthogonal decompositions are given by the distinguished triangles P R B! O! P L B? ; P R? B! O! P L B ; with P L B ; P R B 2 B D./, P L 2 B? D./, and P R? B? B 2? B D./, respectively. By [19, Theorem 7.1] the projection functors ˇˇ and ˇˇŠ are isomorphic to the Fourier ukai functors with kernels P L B and P R, respectively. In what follows we always implicitly make B this identification. The functors with kernels P L and P R? appearing in the above triangles are known B? B as the left and the right mutation functors through B. They are denoted by L B and R B, respectively. Using this notation, the above triangles can be rewritten as (2.1) ˇˇŠ! id! L B ; R B! id! ˇˇ: The following two results about mutations are straightforward, but quite useful. Lemma 2.2. (i) If B D hb 1 ; : : : ; B k i is a semiorthogonal decomposition of an admissible subcategory B D./ then L B D L B1 ı ı L Bk and R B D R Bk ı ı R B1 : (ii) If W D./! D./ is an autoequivalence then ı L B ı 1 D L.B/ and ı R B ı 1 D R.B/ : Assume is a smooth projective variety and L is a line bundle on. A Lefschetz decomposition of D. / is a semiorthogonal decomposition in which each component is embedded into the L twist of the previous component. The formal definition is: A Lefschetz decomposition of D. / is a semiorthogonal decom- Definition 2.3 ([15]). position of the form D. / D hb 0 ; B 1 L ; : : : ; B m 1 L m 1 i; where B 0 B 1 B m 1. A Lefschetz decomposition is rectangular if B 0 D B 1 D D B m 1.

Kuznetsov, Calabi Yau and fractional Calabi Yau categories 5 2.3. Serre functor. One of the main characteristics of a triangulated category is its Serre functor. Definition 2.4 ([6]). Let T be a triangulated category. A Serre functor in T is an autoequivalence S T W T! T with a bifunctorial isomorphism for all F; G 2 T. Hom.F; G/ _ Š Hom.G; S T.F // If a Serre functor exists then it is unique up to a canonical isomorphism. If T D D./ is the bounded derived category of a smooth projective variety then S.F / WD F! Œdim is a Serre functor for D./. The following properties of Serre functors are quite useful. Lemma 2.5. (i) Let T 1 and T 2 be triangulated categories with Serre functors S T1 and S T2, respectively. If ˆW T 1! T 2 is a functor then its left adjoint ˆ exists if and only if its right adjoint ˆŠ exists and ˆŠ ı S T2 D S T1 ı ˆ: (ii) The Serre functor of a triangulated category T commutes with all its autoequivalences. Another useful feature is a relation of the Serre functor of a triangulated category with Serre functors of components of any semiorthogonal decomposition. Lemma 2.6. Let T D ha; Bi be a semiorthogonal decomposition with admissible A and B, and assume that a Serre functor of T exists. Then Serre functors of A and B exist and S B Š.R A ı S T /j B and S 1 A D.L B ı S 1 T /j A: The following compatibility with rectangular Lefschetz decompositions will be useful later. Lemma 2.7. Let be a smooth projective variety and D. / D hb; B L ; : : : ; B L m 1 i a rectangular Lefschetz decomposition. Then one has S.B L i / D B Li i 2 Z. Proof. First, tensoring the decomposition by L i mc1, we deduce that B L i D? hb L i mc1 ; : : : ; B L i 1 i: From the definition of a Serre functor it then follows that S.B L i / D hb Li mc1 ; : : : ; B L i 1 Comparing this with the initial decomposition tensored by L i equality. m i? : m for each, we deduce the required

6 Kuznetsov, Calabi Yau and fractional Calabi Yau categories 2.4. Spherical functors. Spherical functors were introduced in [1], see also [3] for a more recent development. We suggest an alternative definition. Definition 2.8 (cf. [1]). A Fourier ukai functor ˆW D./! D.Y / is spherical if the following two maps are isomorphisms: (2.2) (2.3) ˆ ˆŠ ˆ ı ˆ ı ˆŠ ˆŠ ;ˆıˆCˆŠıˆ;ˆ! ˆŠ ı ˆ ı ˆ; ˆıˆ;ˆŠ Cˆ ;ˆıˆŠ! ˆ ˆŠ: One of the advantages of Definition 2.8 in comparison with the original definition is that it uses neither the triangulated structures nor enhancements of D./ and D.Y /, and can be used for arbitrary functors between additive categories. On the other hand, one can show Definition 2.8 to be equivalent to the original definition. We refrain from that here; instead we show that a spherical functor induces autoequivalences of both the source and the target categories. Proposition 2.9. If the two conditions of Definition 2.8 are satisfied by a functor ˆW D./! D.Y /, then the functors T and T 0 as well as the functors T Y and TY 0 defined by the distinguished triangles (2.4) (2.5) T Y! id ˆ;ˆ! ˆ ı ˆ; ˆ ı ˆ ˆ;ˆ! id! T ; (2.6) (2.7) ˆ ı ˆŠ ˆ;ˆŠ! id! T 0 Y ; T 0! id ˆŠ ;ˆ! ˆŠ ı ˆ: are mutually inverse autoequivalences of D./ and D.Y /. The idea behind the proof is very simple assuming equality abc D a C c, one can deduce from it.1 ab/.1 cb/ D 1 by multiplying the equality with b. The argument below is a categorical version of this, taking care of all the subtleties. Proof. Denote the connecting morphism ˆŠ ı ˆ! T 0 Œ1 in (2.7) by ı. Composing a rotation of (2.7) with ˆ on the right, we get a distinguished triangle ˆ ˆŠ;ˆıˆ! ˆŠ ı ˆ ı ˆ ııˆ! T 0 Œ1 ı ˆ: Using that (2.2) is an isomorphism, we conclude that the map.ı ı ˆ/ ı.ˆš ı ˆ;ˆ/W ˆŠ! ˆŠ ı ˆ ı ˆ! T 0 Œ1 ı ˆ is an isomorphism, hence the middle arrow in the diagram (2.8) below is an isomorphism.

Kuznetsov, Calabi Yau and fractional Calabi Yau categories 7 Now consider the diagram ˆŠ ı ˆ ˆŠıˆ;ˆ ıˆ ˆŠ ı ˆ ı ˆ ı ˆ ııˆıˆ T 0 Œ1 ı ˆ ı ˆ ˆŠ ı ˆ ˆŠıˆıˆ ;ˆ ı T 0 Œ1: T 0 Œ1ıˆ;ˆ The square commutes since the vertical and the horizontal arrows in it act on different variables, and the diagonal dashed arrow is the identity by the standard characterization of adjunction (composed with ˆŠ on the left). This means that in the diagram (2.8) id ˆŠ ;ˆ ˆŠ ı ˆ ı T 0 Œ1 T 0 ı T.ııˆıˆ/ı.ˆŠıˆ;ˆ ıˆ/ T 0 Œ1 ı ˆ ı ˆ T 0 Œ1ıˆ;ˆ T 0 Œ1 where the top line is the distinguished triangle (2.7) and the bottom line is the distinguished triangle (2.5) composed with T 0 Œ1 on the left, the right square is commutative. Since the vertical arrows are isomorphisms, it follows that there is a dotted vertical arrow on the left, which is also an isomorphism. Thus T 0 ı T Š id. Analogously one proves that the other compositions are isomorphic to the identity. For TY 0 ı T Y we use that (2.2) is an isomorphism, while for T ı T 0 and for T Y ı TY 0 we use that (2.3) is an isomorphism. Remark 2.10. It may well be that it is enough to assume only one of the conditions of Definition 2.8. Indeed, assuming for example that (2.2) is an isomorphism, we can prove that the compositions T 0 ı T and TY 0 ı T Y are isomorphic to identity. On the other hand, it is easy to see that T 0 and T Y 0 are right adjoint to T and T Y, respectively. So, it follows that T and T Y are fully faithful endofunctors. It is very tempting to conjecture that any such endofunctor of the derived category of a smooth projective variety is an autoequivalence then it would follow that T 0 and T Y 0 are quasiinverse of T and T Y and so are also autoequivalences. Up to now it is not clear how this conjecture can be proved. However, it can be easily deduced from the following conjecture. Conjecture 2.11 (Noetherian property). Any decreasing chain D./ D A 0 A 1 A 2 of admissible subcategories stabilizes, i.e. for sufficiently large n one has A i all i n. D A ic1 for We will give examples of spherical functors in the next section (Examples 3.1, 3.2, and 3.3). Further on we will also use the following standard property. Corollary 2.12. If ˆ is a spherical functor and T and T Y are the autoequivalences of D./ and D.Y / defined by (2.5) and (2.4), respectively, then there are canonical isomorphisms ˆ ı T Š T Y ı ˆŒ2 and T ı ˆ Š ˆ ı T Y Œ2:

8 Kuznetsov, Calabi Yau and fractional Calabi Yau categories Proof. In Proposition 2.9 it was proved that ˆŠ Š T 0 Œ1 ı ˆ. An analogous argument, using (2.4), shows that ˆŒ 1 Š ˆŠ ı T Y. Combining these two isomorphisms, we conclude that ˆ ı TY 1 Œ 1 Š ˆŠ Š T 0 ı ˆŒ1: Composing with T on the left and with T Y Œ1 on the right, we deduce the second isomorphism. Furthermore, passing to the right adjoint functors (and shifting by 1), we deduce the first isomorphism. 3. A construction of fractional Calabi Yau categories 3.1. The setup. Assume we are given a spherical functor ˆW D./! D. / between derived categories of smooth projective varieties (or stacks). Further on we will consider a Lefschetz decomposition of D. / with respect to some line bundle L, and impose some compatibility conditions on them. Before doing that, however, we will discuss a number of model situations. In all these examples, in fact, the functor ˆ is the (derived) pushforward for a morphism f W!. Example 3.1. The map f W! is a divisorial embedding with the image f./ being a divisor in the linear system L d for some d 1. Example 3.2. The map f W! is a double covering branched in a divisor in the linear system L 2d, again for some d 1. The third example is very similar to the second, but has some special features. Example 3.3. Let fqw Q! Q be a double covering branched in a divisor in the linear system L 2d Q for some d 1. This morphism is 2-equivariant, where the group 2 D ¹ 1º acts on Q via the covering involution, and on Q trivially. Let D Œ= Q 2 and D Œ Q = 2 be the quotient stacks (thus is Q with the 2 -stacky structure along the branch divisor of f Q, while is Q with the 2 -stacky structure everywhere). The map f Q descends to a map! which we denote by f. In the next proposition we check that in all these cases ˆ D f W D./! D. / is spherical, compute the corresponding spherical twists T and T, and check some of their properties. In all cases we denote L WD f L, the pullback of the line bundle L to. Recall that according to our conventions we also denote by L and L the autoequivalences of D. / and D./ defined as tensor products with L and L, respectively. Proposition 3.4. Let f W! be a map from either of Examples 3.1, 3.2, or 3.3. Then the functor ˆ D f W D./! D. / is spherical. oreover, the spherical twist T commutes with L, and an appropriate power of the functor D T ı L d Š Ld ı T is a shift. Finally, if! D L m for some m 2 Z, then an appropriate power of the functor D S ı T ı L m is also a shift.

Kuznetsov, Calabi Yau and fractional Calabi Yau categories 9 Proof. First assume that f W! is as in Example 3.1. The relative canonical class is! = D L d, the relative dimension is 1, hence f Š.F / Š f.f / L d Œ 1. Therefore f Š.f.f.F /// Š f Š.F f O / Š f Š.F.L d '! O // Š f.f /.L d Œ1 O / L d Œ 1 Š f.f / f Š.F /: Here the first isomorphism is the projection formula, the second is the Koszul resolution for f O (with ' being the equation of in ), the third is the definition of f Š combined with the fact that ' j D 0, and the fourth is the definition of f Š again. Computing analogously the composition f ıf ıf Š, we see that Definition 2.8 holds, so f is a spherical functor. Finally, the standard distinguished triangles F L d! F! f f.f / and F L d Œ1! f f.f /! F show that in this case the spherical twists are (3.1) T D L d and T D L d Œ2: Clearly T commutes with L and (3.2) D L d Œ2 ı Ld D Œ2; (3.3) D L d m Œdim 1 ı L d Œ2 ı Lm D Œdim C 1; so with our assumptions both these functors and are shifts. Now assume that f W! is as in Example 3.2. Then the relative canonical class is! = D L d, but the relative dimension is 0, hence f Š.F / Š f.f / L d. Therefore f Š.f.f.F /// Š f Š.F f O / Š f Š.F.L d O // Š f.f /.L d O / L d Š f.f / f Š.F /: Here again, the first isomorphism is the projection formula, the second is the definition of the double covering, the third and the fourth is the definition of f Š. Computing analogously the composition f ıf ıf Š, we see that Definition 2.8 holds, so f is a spherical functor. Finally, the standard distinguished triangles F! f f.f /! F L d and F L d! f f.f /! F; where is the involution of the covering, show that in this case the spherical twists are (3.4) T D L d Œ 1 and T D ı L d Œ1: Since.L / Š L, it follows that the twist T commutes with L and (3.5) D ı L d Œ1 ı Ld D Œ1; (3.6) D L d m Œdim ı ı L d Œ1 ı Lm D Œdim C 1; so with our assumptions 2 and 2 are shifts.

10 Kuznetsov, Calabi Yau and fractional Calabi Yau categories Finally, assume that fqw Q! Q and f W! are as in Example 3.3, so that D./ D D. / Q 2 and D. / D D. Q / 2 are the 2 -equivariant derived categories of Q and Q, respectively. The functors f, f, and f Š can be thought of as f Q, fq, and f QŠ with their natural equivariant structures (see [25] for details). Denote L Q WD f Q L Q and let be the nontrivial character of 2 (so that 2 D 1). Note that equivariantly we have f O D.L d / O and! = D L d : Therefore, analogously to the previous case we have f Š.f.f.F /// Š f Š.F f O / Š f Š.F.L d O // Š f.f /.L d O / L d Š f.f / f Š.F /: Computing analogously the composition f ı f ı f Š, we see that Definition 2.8 holds, so f is a spherical functor. Finally, the standard distinguished triangles F! f f.f /! F L d and F L d! f f.f /! F (note that acts trivially on any equivariant sheaf) show that in this case the spherical twists are (3.7) T D L d Œ 1 and T D L d Œ1: Clearly, T commutes with L and (3.8) D ı L d Œ1 ı Ld D Œ1; D L d m (3.9) ı Œdim ı ı L d Œ1 ı Lm so with our assumptions 2 and are shifts. This finishes the proof. D Œdim C 1; Now we return to the abstract situation of a spherical functor ˆW D./! D. / with the corresponding spherical twists T and T. We consider the following autoequivalences of D./: (3.10) WD T ı L d ; (3.11) WD S ı T ı L m : Theorem 3.5. Assume that and are smooth projective varieties (or stacks) with a spherical functor ˆW D./! D. / between their derived categories. Let T and T be the spherical twists. Assume that D. / has a rectangular Lefschetz decomposition with respect to a line bundle L : (3.12) D. / D hb; B L ; : : : ; B L m 1 i: Assume that there is some 1 d < m such that for all i 2 Z we have (3.13) T.B L i / D B Li d : Assume further that there is a line bundle L on such that ˆ intertwines between L and L twists: (3.14) L ı ˆ Š ˆ ı L :

Kuznetsov, Calabi Yau and fractional Calabi Yau categories 11 Finally, assume that the twist T commutes with L : (3.15) T ı L Š L ı T : Then the functor ˆW D. /! D./ is fully faithful on the component B of D. / and induces a semiorthogonal decomposition (3.16) D./ D ha ; B ; B L ; : : : ; B L m d 1 i; where B D ˆ.B/ and A is the orthogonal subcategory. oreover, if c D gcd.d; m/ then the.d=c/-th power of the Serre functor of the category A can be expressed as S d=c A Š. m=c ı d=c /j A : In particular, if some powers of and are shifts then A is a fractional Calabi Yau category. Remark 3.6. If d D m then the functor ˆ is not fully faithful on B, but still for A D D./ the result of the theorem holds. Indeed, we have S A D S D 1 ı by (3.10) and (3.11), which agrees with the formula in the theorem since in this case c D d D m. Note that Examples 3.1 3.3 satisfy the assumptions (3.13) (3.15) of the theorem (in the last example we need to assume additionally that B D B, i.e. that the Lefschetz decomposition is induced by a Lefschetz decomposition of D. Q /). Indeed, (3.14) is given by the projection formula, (3.13) and (3.15) follow from the description of the functors T and T in (3.1), (3.4), and (3.7). Note also that if! D L m then the functors and are shifts in the first example, and their squares are shifts in the second and the third examples as it was observed in the proof of Proposition 3.4. Thus, in all these cases the constructed category A is a fractional Calabi Yau category. Below we rewrite the conclusion of Theorem 3.5 in all three examples explicitly, assuming a Lefschetz decomposition (3.12) of D. / is given,! D L m, and substituting the expressions (3.2) (3.3), (3.5) (3.6), and (3.8) (3.9) into the general formula. Corollary 3.7. Corollary 3.8. Corollary 3.9. If f W! is as in Example 3.1, then S d=c A D Œ.dim C 1/d=c 2m=c: If f W! is as in Example 3.2, then S d=c A D.m d/=c Œ.dim C 1/d=c m=c: If f W! is as in Example 3.3, then S d=c A D m=c Œ.dim C 1/d=c m=c: We do not know whether there are other examples of spherical functors for which the assumptions of Theorem 3.5 are satisfied. Of course, it is tempting to replace the double cover example with a cyclic cover of arbitrary degree k, but the corresponding pushforward functor is not spherical, so the theorem does not apply in this case. However, as Alex Perry notes, the pushforward functor for a cyclic covering is a so-called P k 1 -functor, so it may well be that a generalization of our construction does something in this case as well.

12 Kuznetsov, Calabi Yau and fractional Calabi Yau categories 3.2. The induced semiorthogonal decomposition. We start with the first part of Theorem 3.5 (full faithfulness and a semiorthogonal decomposition). This result in fact is quite simple. oreover, for this to be true we do not need to know that the Lefschetz collection in D. / generates the whole category. So we state here a slightly more general result. Lemma 3.10. Assume that B D. / is an admissible subcategory, (3.17) hb; B L ; : : : ; B L m 1 i D. / is a rectangular Lefschetz collection, and ˆW D./! D. / is a spherical functor such that (3.13) and (3.14) hold. Then the functor ˆj B W B! D./ is fully faithful and, denoting B WD ˆ.B/, the sequence of subcategories B ; B L ; : : : ; B L m d 1 is semiorthogonal and gives a Lefschetz collection (3.18) hb ; B L ; : : : ; B L m d 1 i D./ that extends to the semiorthogonal decomposition (3.16) of D./. Proof. Denote the embedding functor B! D. / by ˇ. The category B is admissible, hence ˇ has a right adjoint which we denote by ˇŠ W D. /! B. Therefore the functor ˆ ı ˇ W B! D./ also has a right adjoint ˇŠ ı ˆ. We want to show that the composition ˇŠ ı ˆ ı ˆ ı ˇ is the identity. For this we compose (2.4) with ˇŠ on the left and with ˇ on the right: ˇŠ ı T ı ˇ! ˇŠ ı ˇ! ˇŠ ı ˆ ı ˆ ı ˇ : Note that the functor in the middle is the identity of B (since ˇ is fully faithful), so it is enough to check that the functor on the left is zero. As the kernel of ˇŠ is the orthogonal B?, it is enough to check that the image of T ıˇ is contained in this subcategory. But this image is T.B/ and by (3.13) it is in B L d B? by the twist hb L 1 m; : : : ; B L 1; Bi of (3.17) as 1 d m 1. For the semiorthogonality of (3.18) we have to check that the composition of functors ˇŠ ı ˆ ı L i ı ˆ ı ˇ is zero for each 1 i m d 1. For this we use the intertwining property (3.14) and rewrite this composition as ˇŠ ıl i ıˆıˆıˇ. Then we compose (2.4) with ˇŠ ı L i on the left and with ˇ on the right: Clearly, ˇŠ ı L i ı T ı ˇ! ˇŠ ı L i ı ˇ! ˇŠ ı L i ı ˆ ı ˆ ı ˇ : Im.L i ı ˇ / D B L i i and Im.L ı T ı ˇ / D B L i d : So, as both these categories are in B?, they are killed by ˇŠ, hence the first two terms of the triangle are zero. Hence so is the third. As we already have checked the embedding functor of B has a right adjoint, the subcategory is right admissible and thus gives the required semiorthogonal decomposition. We denote by A the orthogonal of the collection (3.18) that extends (3.18) to (3.16): (3.19) A WD hb ; B L ; : : : ; B L m d 1 i? D./: Sometimes the following alternative description of A is useful.

Kuznetsov, Calabi Yau and fractional Calabi Yau categories 13 Lemma 3.11. Assume (3.12) and let A D./ be the subcategory defined by (3.19). Then A D F 2 D./ j ˆ.F / 2 hb L d ; : : : ; B L 1 i D. / : Proof. By definition we have A D F 2 D./ j Hom.ˆ.B/; F / D D Hom.ˆ.B L m d 1 /; F / D 0 : By adjunction this can be rewritten as A D F 2 D./ j Hom.B; ˆ.F // D D Hom.B L m d 1 ; ˆ.F // D 0 : So, the result follows from the twist of (3.12). D. / D hb L d ; : : : ; B L 1 d 1 ; B; : : : ; B Lm i In what follows we denote by ˇW B! D./ and ˇŠ W D./! B the fully faithful embedding constructed in Lemma 3.10 and its right adjoint functor, so that (3.20) ˇ D ˆ ı ˇ ; ˇŠ D ˇŠ ı ˆ; and consider the constructed Lefschetz collection. Further we will need the following lemma. Lemma 3.12. For the functors and defined by (3.10) and (3.11) we have ı ˆ Š ˆ ı T ı L d Œ2 and ı ˆ Š ˆ ı L m ı S Œ1: In particular, all components of (3.16) are preserved by and. Proof. The first equality follows from the definition of, assumption (3.14) and Corollary 2.12. The second is checked similarly: ı ˆ D S ı T ı L m ı ˆ Š S ı ˆ ı T ı L m Œ2 Š S ı ˆ ı S 1 ı T ı L m ı S Œ2 Š ˆŠ ı T ı L m ı S Œ2 Š ˆ ı L m ı S Œ1: Here the equality is the definition of, the first isomorphism is (3.14) and Corollary 2.12, the second and third is Lemma 2.5, and the last is established in the proof of Corollary 2.12. It remains to note that by (3.13).B / D. ı ˆ/.B/ D.ˆ ı T ı L d /.B/ D ˆ.B/ D B ; so preserves B. Since commutes with L by (3.15), it also preserves all the other components of (3.18). An analogous argument (with Lemma 2.7 used instead of (3.13)) works for (note that commutes with L by (3.15) and Lemma 2.5). By (3.19), the category A is also preserved by and.

14 Kuznetsov, Calabi Yau and fractional Calabi Yau categories 3.3. Rotation functors. Now we already have proved the first part of Theorem 3.5, so it remains to compute the Serre functor. The main instruments for this are rotation functors. In general, a rotation functor can be defined in a presence of a rectangular Lefschetz collection hb; B L Y ; : : : ; B L s 1 Y i D.Y / on a smooth projective variety (or a stack) Y. It is defined as the composition of the twist and the left mutation functors: O B WD L B ı L Y : The following straightforward observation is quite useful. Lemma 3.13. If hb; B L Y ; : : : ; B LY s 1 i D.Y / is a rectangular Lefschetz collection and O B is the corresponding rotation functor, then for any 0 i s..o B / i D L hb;b LY ;:::;B L i Y 1i ı Li Y Proof. By Lemma 2.2 we have.o B / i D O B ı O B ı ı O B D.L B ı L Y / ı.l B ı L Y / ı ı.l B ı L Y / Š L B ı.l Y ı L B ı LY 1 / ı.l2 Y ı L B ı LY 2 / ı ı.li Y 1 ı L B ı L 1 Y i / ı Li Y Š L B ı L B LY ı L B L 2 ı ı L Y B L i 1 ı L i Y Y Š L hb;b LY ;:::;B L i Y 1i ı Li Y ; and we are done. In what follows we will consider two rectangular Lefschetz collections: the first is (3.12) generating D. /, and the second is (3.18) (which is not full). We denote the corresponding rotation functors by O and O. So, by definition (2.1) of mutation functors we have the following distinguished triangles: and (3.21) ˇ ˇŠ L! L! O (3.22) ˇˇŠ L! L! O : It is easy to see that the functor O is nilpotent. Corollary 3.14. For each 0 i m the i-th power O i of the rotation functor vanishes on the subcategory hb L i; : : : ; B L 1i D. /. In particular, Om D 0. Proof. Indeed, the twist by L i i takes the subcategory hb L ; : : : ; B L 1i to the subcategory hb; : : : ; B L i 1i, which is killed by the mutation functor L hb;:::;b L i 1i.

Kuznetsov, Calabi Yau and fractional Calabi Yau categories 15 Finally, for i D m the subcategory equals D. / by (3.12). hb L m ; : : : ; B L 1 i D. / It is also easy to see that the functor O commutes with and : Lemma 3.15. We have ı O Š O ı and ı O Š O ı. Proof. Indeed, O is the composition of L with L B. But L commutes with and by (3.15) and Lemma 2.5, and L B commutes with and by Lemma 3.12 and Lemma 2.2 (ii). 3.4. The fundamental relation. In contrast to the nilpotency of O, the functor O induces an autoequivalence of the subcategory A. oreover, its d-th power coincides on A with the autoequivalence. This follows from a careful investigation of the relation between the rotation functors O and O, and in the end leads to the proof of Theorem 3.5. Lemma 3.16. For any 0 i d 1 there is a morphism of functors ˆ ı O i i! O i ı ˆ inducing an isomorphism ˆ ı O i Š Oi ı ˆ on the subcategory hb L d i ; : : : ; B Ld 1 i? D hb L d m ; B LdC1 m ; : : : ; B L d i 1 i D. /: Proof. For i D 0 there is nothing to prove, so consider the case i D 1. Then we have the following diagram: ˆˇ ˇŠ L ˇ ;ˇŠ ˆL ˆO ˆ;ˆ ˇˇŠ L ˆ ˇ ;ˇŠ L ˆ O ˆ where the rows are obtained by composing (3.21) and (3.22) with ˆ, the isomorphism in the middle column is induced by (3.14), while the arrow in the left column is given by the isomorphisms (3.20) and (3.14), altogether giving an isomorphism ˇˇŠ L ˆ Š ˆˇ ˇŠ L ˆˆ; and the unit of the adjunction ˆ;ˆW id! ˆˆ. The left square clearly commutes, hence it extends to a morphism of triangles by the dotted arrow on the right which we denote by. It remains to show that is an isomorphism on the subcategory.b L d 1 /? D hb L d m ; B LdC1 m ; : : : ; B L d 2 i D. /:

16 Kuznetsov, Calabi Yau and fractional Calabi Yau categories By construction of the left arrow in the diagram, the first column extends to a triangle ˆˇ ˇŠ L T! ˆˇ ˇŠ L ˆ;ˆ! ˇˇŠ L ˆ: Note that the first functor here vanishes on the subcategory.b L d 1/? D. /. Indeed, by (3.13) the functor T takes it into.b L 1/? D. /, then L takes it to B? D. / which is killed by ˇŠ. It follows that the left arrow in the above diagram is an isomorphism on the subcategory.b L d 1/?, hence so is the right arrow. Now assume that i > 1. We define the map ˆ ı O i! Oi ı ˆ by an iteration of the map : ˆ ı O i! O ı ˆ ı O i 1!! O i 1 ı ˆ ı O! O i ı ˆ and denote it by i. It remains to prove that it induces an isomorphism on the specified subcategory. We prove this by induction in i, the case i D 1 proved above being the base of the induction. So, assume that we already have proved that i 1 induces an isomorphism ˆ ı O i 1 Š Oi 1 ı ˆ on hb L d ic1 ; : : : ; B L d 1i? D. /. Assume now that Then by the induction hypothesis we have F 2 hb L d i ; : : : ; B Ld 1 i? D. /: O i ı ˆ.F / D O.O i 1 ı ˆ.F // Š O.ˆ ı O i 1.F // D.O ı ˆ/.O i 1.F //: On the other hand, by Lemma 3.13 we have O i 1.F / D L 1 hb;:::;b L i 2 i.f Li /: It is easy to see that F L i 1 1 2.B Ld /? and hb; : : : ; B L i 2 1 i.b Ld well. It follows from the definition of mutations that.f / D L 1 1 hb;:::;b L i 2 i.f Li / 2.B Ld O i 1 and hence the base of induction applies and /? ;.O ı ˆ/.O i 1.F // Š.ˆ ı O /.O i 1.F // D.ˆ ı O i /.F /: This completes the proof of the lemma. Consider the composition of maps /? as ˆ ı O i ı ˆ i! O i ı ˆ ı ˆ ˆ;ˆ! O i : Proposition 3.17. For each 0 i d there is a distinguished triangle of functors ˆ ı O i ı ˆ ˆ;ˆı i! O i! T ı L i :

Kuznetsov, Calabi Yau and fractional Calabi Yau categories 17 Proof. We prove this by induction in i. The base of the induction, the case i D 0, is provided by the triangle (2.5). So, assume that i > 0. Consider the diagram ˆ ı O i 1 ı ˇ ˇŠ L ı ˆ ˇ ;ˇŠ ˆ ı O i 1 ı L ı ˆ ˆ ı O i 1 ı O ı ˆ O i 1 O i 1 i 1 ı ˆ ı ˇ ˇŠ L ı ˆ i 1 ˇ ;ˇŠ O i 1 ı ˆ ı L ı ˆ ˆ ;ˆ ˆ ;ˆıˇ ı ˆ ;ˇŠ ı ˇ ˇŠ ı ˆ ı L O i 1 ı L O i 1 i 1 ı ˆ ı O ı ˆ ˆ ;ˆı O i 1 ı O : Here the first row is obtained by composing the triangle (3.21) with ˆ ıo i 1 on the left and ˆ on the right, the second row is obtained by composing it with O i 1 ı ˆ on the left and ˆ on the right, and the last row is obtained by composing the triangle (3.22) with O i 1 on the left (taking into account (3.20)). So the rows are distinguished triangles and the vertical maps form morphisms of distinguished triangles (for the first this is evident, and for the second this follows from the definition of in Lemma 3.16). Composing the morphisms of these triangles, we get the following commutative diagram: ˆ ı O i 1 ı ˇ ˇŠ L ı ˆ ˇ ;ˇŠ ˆ ı O i 1 ı L ı ˆ ˆ ı O i 1 ı O ı ˆ O i 1 i 1 ı ˇˇŠ L ˇ ;ˇŠ O i 1 ˆ ;ˆı i 1 ı L O i 1 ˆ ;ˆı i ı O where we have rewritten the first term of the bottom row via (3.20). Note that i d implies d.i 1/ Im ˇ D B hb L ; : : : ; B L d 1 i? hence the left arrow is an isomorphism by Lemma 3.16. oreover, by induction hypothesis the middle vertical map extends to a distinguished triangle by T ı L i. Therefore, the octahedron axiom implies that the right vertical arrow extends to a distinguished triangle and thus proves the required claim. ˆ ı O i ı ˆ ˆ;ˆı i! O i! T ı L i ; Corollary 3.18. The restriction of O to the subcategory A D./ is an autoequivalence such that (3.23) O d ja Š ja ; where is defined by (3.10). Proof. Let us restrict the triangle of Proposition 3.17 with i D d to A. The first term of the triangle then vanishes by a combination of Lemma 3.11 and Corollary 3.14. Therefore, the functors given by the second and the third terms are isomorphic, so it remains to use the definition (3.10) of.

18 Kuznetsov, Calabi Yau and fractional Calabi Yau categories Note that both functors in (3.23) preserve A. For this is proved in Lemma 3.12, and for O d this follows from (3.23). In fact, even O itself preserves A, see [25, Lemma 7.6]. 3.5. Proof of Theorem 3.5. In this subsection we prove Theorem 3.5. The last thing we need is a relation between the Serre functor of A and the rotation functor. Lemma 3.19. The Serre functor of the category A is given by (3.24) S 1 A Š.O m d ı ı 1 /j A : Proof. By Lemma 2.6 we have and by definition (3.11) of we have S 1 S 1 S 1 A D L hb ;:::;B L m d 1 i ı S 1 ja D Lm ı T ı 1. Combining this, we obtain A Š L hb ;:::;B L m d 1 i ı Lm ı T ı 1 j A Š.L hb ;:::;B L m d 1 Š.O m d ı ı 1 /j A : Here the last isomorphism is Lemma 3.13. i ı Lm d / ı.l d ı T / ı 1 j A Proof of Theorem 3.5. Let us note that and commute. Indeed, both are combinations of T, L, and S, but T and L commute by (3.15), and S commutes with any autoequivalence by Lemma 2.5. oreover, both and commute with O by Lemma 3.15. Therefore, taking the.d=c/-th power of (3.24), where c D gcd.d; m/, we obtain S d=c d.m d/=c A Š.O ı d=c ı d=c /j A : But O d.m d/=c Š.m d/=c on the subcategory A by (3.23), hence This completes the proof of Theorem 3.5. S d=c A Š. m=c ı d=c /j A : Remark 3.20. Note that we could generalize the results of Theorem 3.5 as follows. First, we could replace L and L by arbitrary autoequivalences (not necessarily tensoring with a line bundle). Second, we could replace D./ and D. / by admissible subcategories (in other words, we could let and be noncommutative varieties). The same proof would apply in this larger generality. However, we do not know whether there are interesting examples of this more general situation. 4. Explicit examples 4.1. Varieties with a rectangular Lefschetz decomposition. In fact, any variety has a rectangular Lefschetz decomposition of length m D 1 with respect to the anticanonical line bundle. However, this decomposition does not produce an interesting Calabi Yau category as in this case d D m D 1 and A D D./ (see Remark 3.6).

Kuznetsov, Calabi Yau and fractional Calabi Yau categories 19 So, one can get something interesting only from a rectangular Lefschetz decomposition of length greater than 1. In the following list we give a number of such decompositions. In most of these the line bundle L is the ample generator of the Picard group, so we always assume this is the case unless something else is specified. oreover, in all these cases! D L m. 4.1.1. A projective space P n has a rectangular Lefschetz decomposition of length m D n C 1 (Beilinson s collection). D.P n / D ho P n; O P n.1/; : : : ; O P n.n/i 4.1.2. A weighted projective space P.w 0 ; w 1 ; : : : ; w n / considered as a smooth toric stack has a rectangular Lefschetz decomposition D.P.w 0 ; w 1 ; : : : ; w n // D ho P.w0 ;w 1 ;:::;w n /; O P.w0 ;w 1 ;:::;w n /.1/; : : : ; O P.w0 ;w 1 ;:::;w n /.m 1/i of length m D w WD w 0 C w 1 C C w n, see [4, Proposition 4.1.1]. 4.1.3. A smooth quadric of dimension n D 4s C 2 has a rectangular Lefschetz decomposition D.Q 4sC2 / D hb; B.2s C 1/i of length m D 2 with respect to the line bundle O.2s C 1/, where B D ho; O.1/; : : : ; O.2s/; S.2s/i with S being one of the two spinor bundles (it can be obtained from Kapranov s collection [12] by a simple sequence of mutations). 4.1.4. A Grassmannian Gr.k; n/ with.k; n/ coprime has a rectangular Lefschetz decomposition D.Gr.k; n// D hb; B.1/; : : : ; B.n 1/i of length m D n, with the category B generated by the exceptional collection formed by the Schur functors U _, where U is the tautological rank k subbundle and runs through the set of all Young diagrams with at most k 1 rows and with p-th row of length at most.n k/.k p/=k: B D h U _ j 1 <.n k/.k 1/=k; 2 <.n k/.k 2/=k; : : : ; k 1 <.n k/=ki; see [9]. 4.1.5. An orthogonal Grassmannian OGr.2; 2n C 1/ has a rectangular Lefschetz decomposition D.OGr.2; 2n C 1// D hb; B.1/; : : : ; B.2n 3/i of length m D 2n 2, with the category B generated by the exceptional collection formed by symmetric powers of the dual tautological bundle and the spinor bundle: see [16]. B D ho; U _ ; : : : ; S n 2 U _ ; Si;

20 Kuznetsov, Calabi Yau and fractional Calabi Yau categories 4.1.6. Some other homogeneous spaces: some symplectic Grassmannians, e.g. D.SGr.3; 6// D hb; B.1/; B.2/; B.3/i; where B D ho; U _ ii some (connected components of) orthogonal Grassmannians, e.g. D.OGr C.5; 10// D hb; B.1/; : : : ; B.7/i; where B D ho; U _ ii the Grassmannian of the simple group of type G 2 (the highest weight orbit in the projectivization of the adjoint representation): D.G 2 Gr/ D hb; B.1/; B.2/i; where B D ho; U _ i; and with U being the restriction of the tautological bundle under the natural embedding G 2 Gr,! Gr.2; 7/, see [14]. 4.1.7. Some quasihomogeneous spaces, e.g. a hyperplane section IGr.2; 2n C 1/ of Gr.2; 2n C 1/: D.IGr.2; 2n C 1// D hb; B.1/; : : : ; B.2n 1/i; where B D ho; U _ ; : : : ; S n 1 U _ i; see [16]. One can also consider relative versions of the above decompositions. For example, if E is a vector bundle on a scheme S then its projectivization P S.E/ has a rectangular Lefschetz decomposition of length equal to the rank of E with the components equivalent to D.S/. In general, given a minimal homogeneous space D G=P (i.e. with semisimple G and maximal parabolic P ) it is expected that D. / has a rectangular Lefschetz decomposition as soon as the Euler characteristic of (which is equal to the rank of the Grothendieck group of D. / and which can be computed as the index of the Weyl group of P in the Weyl group of G) is divisible by the index of. For instance, it should exist on SGr.3; 6n/ and SGr.3; 6n C 4/ for any n, and many others. In some cases, when the rank of the Grothendieck group of such is not divisible by the index i, but they have a nontrivial common divisor m, it may be that there is a rectangular Lefschetz decomposition of length m with respect to O.i =m/. For instance, for an even dimensional quadric Q 2k the rank of the Grothendieck group is 2kC2, while the index is 2k, so the only nontrivial common divisor is 2. And indeed, if k is odd, D.Q 2k / admits a rectangular Lefschetz decomposition of length 2 with respect to O.k/ (see Section 4.1.3). However for even k it seems that there is no analogue for this decomposition. Another example of this sort is Gr.2; 6/, when the rank of the Grothendieck group is 15 and the index is 6, so one can take m D 3, and indeed there is a rectangular Lefschetz decomposition D.Gr.2; 6// D hb; B.2/; B.4/i; where B D ho; U _ ; S 2 U _ ; O.1/; U _.1/i. 4.2. Hypersurfaces. In this section we give explicit statements of Theorem 3.5 for hypersurfaces in some varieties with rectangular Lefschetz decompositions. The first result in fact can be found in [13].

Kuznetsov, Calabi Yau and fractional Calabi Yau categories 21 Corollary 4.1. Let P n be a smooth hypersurface of degree d n C 1 and c D gcd.d; n C 1/. The derived category of has a semiorthogonal decomposition D./ D ha ; O ; : : : ; O.n d/i and the Serre functor of A has the property S d=c A D Œ.n C 1/.d 2/=c. In particular, if d divides n C 1 then A is a Calabi Yau category of dimension.n C 1/.d 2/=d. The most famous of these cases is that of a cubic fourfold (see [18]), when the category A can be thought of as a noncommutative K3 surface. The case of a cubic hypersurface of dimension 7 (when A is a 3-Calabi Yau category) was discussed in [11]. Corollary 4.2. Let P.w 0 ; w 1 ; : : : ; w n / be a smooth hypersurface of degree d w WD P w i in a weighted projective space (considered as a smooth toric stack) and c D gcd.d; w/. The derived category of has a semiorthogonal decomposition D./ D ha ; O ; : : : ; O.w d 1/i and the Serre functor of A has the property S d=c A D Œ..n C 1/d 2w/=c. In particular, if d divides w then A is a Calabi Yau category of dimension n C 1 2w=d. Corollary 4.3. Let Q 4sC2 be a hypersurface of degree 2s C 1 (thus is a complete intersection of type.2; 2s C 1/ in P 4sC3 ). The derived category of has a semiorthogonal decomposition D./ D ha ; O ; O.1/; : : : ; O.2s/; S.2s/ j i and A is a Calabi Yau category of dimension 4s 1. The case s D 1 appeared in [11]. Corollary 4.4. Assume gcd.k; n/ D 1 and let Gr.k; n/ be a hypersurface of degree d n and c D gcd.d; n/. The derived category of has a semiorthogonal decomposition D./ D ha ; B ; B.1/; : : : ; B.n d 1/i; where the category B is described in Section 4.1.4. The Serre functor of the category A has the property S d=c A D Œ.k.n k/ C 1/d=c 2n=c. In particular, if d divides n then A is a Calabi Yau category of dimension k.n k/ C 1 2n=d. Corollary 4.5. Let OGr.2; 2n C 1/ be a hypersurface of degree d 2n 2 and c D gcd.d; 2n 2/. The derived category of has a semiorthogonal decomposition D./ D ha ; B ; B.1/; : : : ; B.2n 3 d/i; where the category B is described in Section 4.1.5. The Serre functor of the category A has the property S d=c A D Œ4.n 1/.d 1/=c. In particular, if d divides 2n 2 then A is a Calabi Yau category of dimension 4.n 1/.d 1/=d. We leave to the reader to formulate analogous results in other cases.

22 Kuznetsov, Calabi Yau and fractional Calabi Yau categories 4.3. Double coverings. Here we restrict to stating what happens for double covers of projective spaces and Grassmannians. The reader is welcome to formulate the other results. Corollary 4.6. Let! P n be a double covering ramified in a smooth hypersurface of degree 2d with d n C 1 and let c D gcd.d; n C 1/. Let be the involution of the double covering. The derived category of has a semiorthogonal decomposition D./ D ha ; O ; : : : ; O.n d/i and the Serre functor of A has the property S d=c A D.nC1 d/=c Œ.n C 1/.d 1/=c. In particular, if d divides nc1 and.nc1/=d is odd then A is a Calabi Yau category of dimension.n C 1/.d 1/=d. The case n D 5, d D 2 appeared in [11]. Corollary 4.7. Assume that gcd.k; n/ D 1 and let! Gr.k; n/ be a double covering ramified in a smooth hypersurface of degree 2d with d n and let c D gcd.d; n/. Let be the involution of the double covering. The derived category of has a semiorthogonal decomposition D./ D ha ; B ; : : : ; B.n d 1/i; where the category B is described in Section 4.1.4, and the Serre functor of the category A has the property S d=c A D.n d/=c Œ.k.n k/ C 1/d=c n=c. In particular, if d divides n and n=d is odd then A is a Calabi Yau category of dimension k.n k/ C 1 n=d. One of the interesting cases here is formed by double covers of Gr.2; 5/ (i.e. k D 2, n D 5, d D 1), known as Gushel ukai 6-folds. See [24, 25] for more details. 4.4. K3 categories. Let us list the cases when the category A is a 2-Calabi Yau category: cubic fourfolds 3 P 5 ; hyperplane sections 1 Gr.3; 10/ (Debarre Voisin varieties, see [8]); double covers 2! Gr.2; 5/ ramified in a quadratic section (Gushel ukai varieties, see [24, 25]). In all these cases, using additivity of Hochschild homology (see [17, Corollary 7.5]), one can check that the category A has the same Hochschild homology as the derived category of a K3 surface. oreover, for special cubic fourfolds the category A 3 is equivalent to D.S/ for a K3 surface S (see [18]) and the same is expected to be true for some Gushel ukai sixfolds (see [24]). It is also expected that the same is true for special Debarre Voisin varieties. Thus, it is natural to consider all these categories as noncommutative K3 surfaces (or as K3 categories). Remark 4.8. In the last example one can replace Gr.2; 5/ by its linear section of codimension k 3 and then for odd k take to be a quadric section of and for even k take to be a double covering of ramified in a quadric. In all these cases A is a K3 category (see [24]).

Kuznetsov, Calabi Yau and fractional Calabi Yau categories 23 One of the interesting properties K3 surfaces have, is that moduli spaces of sheaves on them carry a symplectic structure, and so when smooth and compact they are hyper-kähler varieties. One can use K3 categories in the same way. In fact, it was shown in [23] that any moduli space of sheaves on a cubic fourfold 3 carries a closed 2-form, and if all the sheaves parameterized by this moduli space are objects of the category A 3, then the 2-form is nondegenerate. The same argument can be applied to any K3 category to show that a moduli space of objects in it carries a symplectic form. This allows constructing new examples of hyper-kähler varieties. In case of 3 this gives the classical Beauville Donagi fourfold [5] or a more recent eightfold [26]. Applied to 2 this gives a double EPW sextic [10] and for 1 presumably one can get the Debarre Voisin fourfold [8]. Other moduli spaces and other examples of K3 categories may give new hyper-kähler varieties. However, finding other examples of noncommutative K3 categories seems to be a difficult problem. For instance, one can obtain a long list of hypersurfaces in weighted projective spaces with A being a K3 category. But it looks like most of them are equivalent to derived categories of K3 surfaces, or reduce to one of the three above examples. For instance, one can take a degree 4 hypersurface 4 P.1; 1; 1; 1; 1; 3/. But the equation of 4 after an appropriate change of coordinates necessarily takes the form x 5 x 4 C f 4.x 0 ; : : : ; x 4 / D 0: Then 4 can be obtained from P 4 by the blowup of the K3 surface S D ¹x 4 D f 4.x 0 ; : : : ; x 4 / D 0º followed by the contraction of the proper preimage of the hyperplane ¹x 4 D 0º. This allows to show that A 4 Š D.S/. A potentially new example of a noncommutative K3 category is expected to be found inside the derived category of a Küchle fourfold of type.c5/, see [21, 22]. 4.5. 3-Calabi Yau categories. As Calabi Yau threefolds are of special interest for physics, let us also list some examples of varieties, containing a 3-Calabi Yau (3CY for short) category: a cubic 7-fold 3 P 8 ; an intersection of a quadric and a cubic 2;3 P 7 ; an intersection of Gr.2; 6/ and a quadric 2 Gr.2; 6/; a hyperplane section 1 Gr.3; 11/; a hyperplane section 0 1 Gr.4; 9/; an intersection of SGr.3; 6/ with a quadric 0 2 SGr.3; 6/; an intersection of OGrC.5; 10/ with a quadric 00 2 OGr C.5; 10/ P 15 ; an intersection of P 3 P 3 P 15 with a quadric 000 2 P 3 P 3 ; a double covering 0000 2! P 5 ramified in a quartic; a double covering 00000 2! G 2 Gr ramified in a quadric.