Calabi-Yau fibrations, simple K-equivalence and mutations
aa r X i v : . [ m a t h . AG ] J un CALABI–YAU FIBRATIONS, SIMPLE K -EQUIVALENCE AND MUTATIONS MARCO RAMPAZZOAbstract. We consider families of homogeneous roofs of projective bundles over any smooth pro-jective variety, formulating a relative version of the duality of Calabi–Yau pairs of the type dis-cussed in [Kuz16, KR17]. Derived equivalence of such pairs lifts to Calabi–Yau fibrations, extend-ing a result of Bridgeland and Maciocia [BM02] to higher dimensional cases. We formulate a con-crete approach for proving that the DK -conjecture holds for a class of simple K -equivalent mapsarising from families of roofs. As an example, we propose a pair of eight dimensional Calabi–Yauvarieties fibered in dual Calabi–Yau threefolds, related by a GLSM phase transition, and we provederived equivalence with the methods above. Introduction
Dualities among Calabi–Yau varieties have been a popular subject of research in the courseof the last half century. In fact, from superstring theory and gauged linear sigma models tothe many longstanding conjectures on the derived and birational geometry of such varieties,Calabi–Yau pairs lie in the intersection of several diverse fields. In light of Bondal–Orlov’s re-construction theorem [BO01], Calabi–Yau varieties occupy a special place among algebraic va-rieties, namely it is possible to construct pairs of non isomorphic (or non birational) Calabi–Yauvarieties which are derived equivalent. A first example has been given in terms of the Pfaffian–Grassmannian pair [BC08]. Such example has a clear link with the idea of a phase transitionin a non-Abelian gauged linear sigma model [Rød00, ADS15]. This kind of constructions, incontrast with their Abelian counterpart, are quite rare, and proving derived equivalence forsuch pairs has very often relied on ad-hoc arguments. In fact, while some constructions like thePfaffian-Grassmannian above and the intersection of two translates of G (2 , [OR17, BCP17]can be now explained by the homological projective duality and categorical joins programs[Kuz07, KP19], there exists a class of conjecturally derived equivalent pairs of Calabi–Yau va-rieties [KR20, Conjecture 2.6] for which a general argument is missing. In the context of K -equivalence, the notion of roof of projective bundles has been introduced by Kanemitsu to de-fine special Fano manifolds which admit two projective bundle structures [Kan18]. It has beenshown that from the data of a hyperplane section of a roof of projective bundles one can definetwo equidimensional Calabi–Yau varieties [KR20]: several instances of this problem had beenpreviously investigated [Muk98, IMOU1606, Kuz16, KR17] but a working general approach toprove derived equivalence has yet to be found. Furthermore, while for constructions relatedto homological projective duality a link with the physics of gauged linear sigma models hasbeen provided [RS17], for the case of roofs of projective bundles a simple GLSM interpretationin terms non-Abelian phase transition is missing, even if the underlying equivalence of matrixfactorization categories [KR20] suggests its existence. In this paper we introduce the notion of roof bundles, which are families of roofs of projectivebundles with the structure of relative flag varieties on a smooth projective base. The main moti-vation for this construction arises in light of the DK -conjecture [BO02], [Kaw02]. In fact, Kane-mitsu showed that for a simple K -equivalent map, which is a birational morphism µ : X X between smooth projective varieties resolved by a single blowup X such that the pullbacks ofthe canonical bundles of X and X to X are isomorphic, the exceptional loci are both isomor-phic to a family of roofs [Kan18]. If we assume the exceptional locus to be a roof bundle, weconstruct fully faithful embeddings of D b coh( X ) and D b coh( X ) in the derived category of X and we prove derived equivalence of X and X for some classes of such birational pairs. Thisprovides evidence for the DK -conjecture in the form of an infinite list of examples.Furthermore, we formulate a relative version of the Calabi–Yau duality arising from a roof ofprojective bundles: by a hyperplane section of a roof bundle we define a pair of fibrations withCalabi–Yau fibers which are pairwise connected by the aforementioned duality. To address theproblem of derived equivalence, we construct semiorthogonal decompositions for the hyper-plane and develop a systematic approach based on mutations of exceptional objects, provingthat there exists a sequence of mutations defining a derived equivalence for the pair of fibra-tions if the associated problem of derived equivalence of the Calabi–Yau pair can be solved bymutations, under some mild hypotheses. As an example in Section 3 we propose a pair of eightdimensional Calabi–Yau fibrations over P such that for every point in P the fibers are nonbirational Calabi–Yau threefolds, extending a construction by Bridgeland and Maciocia [BM02]of derived equivalent elliptic and K fibrations to higher dimensional examples .Finally, we give a gauged linear sigma model describing the fibered Calabi–Yau eightfolds in-troduced in Section 3 as geometric phases. The model is strictly related to the constructiongiven in [KR17] and as the latter it admits a very simple description of the phase transition. Weexplain how the GLSM generalizes to an arbitrary smooth projective base, then we sketch ananalogous construction for higher dimensional fibers, though derived equivalence has yet to beproved.1.1. Organization of the paper.
In Section 2 we recall some definitions about roofs of projectivebundles and the associated Calabi–Yau pairs. Then we introduce roof bundles, fixing the nota-tion which will be used in the reminder of this paper. Furthermore, in Section 3, we discuss themain example of this construction: a pair of Calabi–Yau eightfolds fibered in Calabi–Yau three-folds over P . In Section 4 we review an approach based on semiorthogonal decompositionsand mutations for solving the problem of derived equivlence of a Calabi–Yau pair associatedto a given roof. Then we relativize the picture, providing a strategy for the problem of derivedequivalence of Calabi–Yau fibrations, based on derived equivalence of the fibers. In Section 5we establish a link between derived equivalence of a Calabi–Yau pair related to a given roofand derived equivalence of any simply K -equivalent pair of smooth projective varieties suchthat the exceptional locus of the associated blowup is a roof bundle whose fiber is isomorphicto the roof above. Finally, in Section 6, we give a GLSM interpretation of the fibered dualitydiscussed in Section 4, with particular attention to the example of Calabi–Yau eightfolds in-troduced in Section 3. We summarize all results about such pair of Calabi–Yau fibrations inTheorem 6.1. ALABI–YAU FIBRATIONS, SIMPLE K -EQUIVALENCE AND MUTATIONS 3 Acknowledgements.
I would like to thank my advisor Michał Kapustka for the constant sup-port and encouragement throughout this work. I am also very grateful to Alexander Kuznetsovfor reading an early draft of this paper and providing a detailed feedback, which led to impor-tant corrections. I am supported by the PhD program of the University of Stavanger2.
Construction
Homogeneous roofs of projective bundles.
Let us recall the definition of the followingclass of Mukai pairs [Muk88]:
Definition 2.1. [Kan18, Definition 0.1]
A simple Mukai pair ( Y, E ) is the data of a Fano variety Y ofPicard number one and an ample vector bundle E over Y such that: ◦ det( E ) ≃ ω ∨ Y ◦ There exists a vector bundle F over a variety Z satisfying rk( E ) = rk( F ) and P ( E ) ≃ P ( F ) . Definition 2.2. [Kan18, Definition 0.1]
A roof of rank r , or roof of P r − -bundles, is a Fano variety X which is isomorphic to the projectivization of a rank r vector bundle E over a Fano variety Y , where ( Y, E ) is a simple Mukai pair. The following picture emerges:(2.1) P ( E ) X P ( F ) Y Z h h Among roofs of projective bundles, nearly all known examples can be described in terms of G -homogeneous varieties of Picard number two where G is a semisimple Lie group, with theprojective bundle structures defined by the natural surjections to G -Grassmannians. This classof homogeneous roofs has remarkable properties: for example, as we will clarify below, a generalhyperplane section of a homogeneous roof defines a pair of Calabi–Yau varieties, which areconjectured to be derived equivalent [KR20, Conjecture 2.6]. In the present work we will entirelyfocus on homogeneous roofs. Definition 2.3.
A homogeneous roof of projective bundles is a roof which is isomorphic to a homogeneousvariety
G/P of Picard number two, where G is a semisimple Lie group and P is a parabolic subgroup. A complete list of homogeneous roofs has been given in [Kan18, Section 5.2.1]. Let us summa-rize its content in Table 1. We refer to the same nomenclature introduced by Kanemitsu, whichwill be adopted throughout the reminder of this work. Hereafter, given a semisimple Lie group G , G/P n ,...,n k denotes the quotient of G by its parabolic subgroup such that the Levi factor ofthe corresponding Lie algebra is the union of root spaces related to the simple roots n , . . . n k .The expressions G/P and G/P will denote the images of the two P r − -bundle structures h and h of the roof G/P . Where it is possible, we use the more standard notations for (orthogonaland symplectic) Grassmannians and flag varieties.
MARCO RAMPAZZO G roof G/P G/P G/P SL ( k + 1) × SL ( k + 1) A k × A k P k × P k P k P k SL ( k + 1) A Mk F (1 , k, k + 1) P k P k SL (2 k + 1) A G k F ( k, k + 1 , k + 1) G ( k, k + 1) G ( k + 1 , k + 1) Sp (3 k − ( k even) C k/ − IF ( k − , k, k − IG ( k − , k − IG ( k, k − Spin (2 k ) D k OG ( k − , k ) OG ( k, k ) + OG ( k, k ) − F F F /P , F /P F /P G G G /P , G /P G /P Table 1: Homogeneous roofsLet X be a roof, fix the notation of Diagram 2.1. Hence, by [Kan18, Proposition 1.5], there ex-ists a line bundle on X which restricts to O (1) on each P r − -fiber of both the projective bundlestructures. In the case of homogeneous roofs, the ample line bundle O (1 ,
1) := h ∗ O (1) ⊗ h ∗ O (1) satisfies such requirements.In the following, a Calabi–Yau variety is defined as an algebraic variety X such that ω X ≃ O X and H m ( X, O X ) = 0 for < m < dim( X ) . Moreover, we call Calabi–Yau fibration a fibration X −→ B such that the general fiber is a Calabi–Yau variety. Moreover, given a vector space V and k ∈ Z , we call V [ k ] the complex of vector spaces which is identically zero in every degreedifferent from k , where it is equal to V . Lemma 2.4.
Let X be a homogeneous roof of P r − -bundles with structure morphisms h i : X −→ Z i and let σ ∈ H ( F , O (1 , be a general section. Call E i := h i ∗ O (1 , . Then Y i = Z ( h i ∗ σ ) ⊂ Z i iseither empty or a Calabi–Yau variety of codimension r .Proof. Let us fix E i := h i ∗ O (1 , . Since O (1 , is an ample line bundle, E i is an ample vectorbundle. Let H = H ( X, O (1 , . By applying the derived pushforward functor to the surjection(2.2) H ⊗ O −→ O (1 , we conclude that E i is globally generated, thus Y i is of expected codimension by generality of σ . In fact, h i ∗ σ is general if σ is general. Since ( Z i , E i ) is a Mukai pair, Y i has vanishing firstChern class. By [Laz04, Example 7.1.5], since E i is ample, the restriction map H q ( Z i , Ω pZ i ) −→ H q ( Y i , Ω pY i ) are isomorphisms for p + q < dim( Y i ) , in particular H q ( Z i , O Z i ) ≃ H q ( Y i , O Y i ) for q < dim( Y i ) . But since Z i is homogeneous H • ( Z i , O Z i ) ≃ C [0] and this concludes the proof. (cid:3) Remark . Observe that in Lemma 2.4 the trivial case is represented only by roofs of type A k × A k . In fact, for these roofs, the projective bundle structures are given by projectivizations ALABI–YAU FIBRATIONS, SIMPLE K -EQUIVALENCE AND MUTATIONS 5 of vector bundles of rank k + 1 on P k , hence the zero loci of pushforwards of a general section σ ∈ H ( P k × P k , O (1 , are empty. In all other cases, the zero loci have nonnegative dimension. Definition 2.6. (cfr. [KR20, Definition 2.5])
Let X be a homogeneous roof, fix the notation of Lemma2.4. We say Y and Y are a Calabi–Yau pair associated to the roof X if Y and Y are nonempty and Y i ≃ Z ( h i ∗ σ ) for i ∈ {
1; 2 } , where σ ∈ H ( X, O (1 , is a general section. Homogeneous roof bundles.
While the problem of describing and classifying families ofroofs over a smooth projective variety has been addressed in [Kan18, ORS20], we focus on aspecial class of such families, which we call roof bundles . These objects provide a natural rela-tivization of homogeneous roofs of projective bundles, and retain many of the properties of thelatter objects in a relative setting.
Definition 2.7.
Fix a smooth projective variety B . Let G be a semisimple Lie group and P a parabolicsubgroup such that G/P is a homogeneous roof. We define a homogeneous roof bundle over B the variety V × G G/P , where V is a principal G -bundle over B .Remark . Note that
V × G G/P is a locally trivial fibration over B with fiber F b ≃ G/P , there-fore it is a G -flag bundle with respect to a given vector bundle W over B . More precisely, let W G be the fundamental representation space of G : there exists a vector bundle W = V × G W G with a fiberwise action of the fundamental representation of G such that V × G G/P is a relative G -homogeneous variety. Lemma 2.9.
Let G be a semisimple Lie group and P ⊂ G a parabolic subgroup such that G/P is ahomogeneous roof. Let
V −→ B be a principal G -bundle over a smooth projective variety B . Then thehomogeneous roof bundle F = V × G G/P admits two projective bundle structures p , p such that thefollowing diagram is commutative: (2.3) F F F B p p r r where r and r are smooth extremal contractions. Moreover, there exists a line bundle L on F such that L restricts to O (1) on the fibers of both p and p .Proof. Let F = V × G G/P be a homogeneous roof bundle over B . Let us call π : F −→ B themap induced by the structure map V −→ B . Then, for every b ∈ B we have π − ( b ) ≃ G/P . Since
G/P is a homogeneous variety of Picard number two, it admits two surjections
G/P −→ G/P and G/P −→ G/P to homogeneous varieties of Picard number one, the morphisms are defined MARCO RAMPAZZO by the natural inclusions of parabolic subgroups.(2.4)
G/PG/P G/P h h If we call F := V × G G/P and F := V × G G/P , we obtain the following diagram:(2.5) F F F B p p πr r where p and p , restricted to the preimage of a point b ∈ B , are the P r − -bundle structuresof the roof G/P , therefore they are P r − -fibrations over F and F . In particular, for each roofof the list [Kan18, Section 5.2.1], there exist homogeneous vector bundles E and E such that P ( E ) ≃ P ( E ) ≃ G/P . Hence, for i = 1 , , they have the form:(2.6) E i = G × P i V i for a given representation space V i . From the data of E i we can define vector bundles on F i with the following construction:(2.7) E i = V × G G × P i V i Note that for every b ∈ B , we have r − i ( b ) ≃ G/P i and E i | r − i ( b ) ≃ E i . Since G/P is a roof,this implies that ( r − i ( b ) , E i | r − i ( b ) ) is a simple Mukai pair. Observe that r is proper and everyfiber is isomorphic to a Fano variety of Picard number one G/P , which means that − K F is r -ample. Let us call ξ the generator of the Picard group of G/P and H the generator of ρ ( B ) .Fix K G/P = − rξ . Then, a Q -Cartier divisor D has class [ D ] = aH + bξ for a, b ∈ Q and for everytwo contracted curves C , C there exists q ∈ Q such that for every Cartier divisor D ⊂ F onehas C .D = qC .D . Therefore r is a smooth extremal contractions, and an identical argumentholds for r . Then, by [Kan18, Lemma 4.1], p and p are P r − -bundle structures, the diagramis commutative and there exists a line bundle L on F which restricts to O (1) on each P r − -fiber. (cid:3) Remark . Observe that, by restricting the relative Euler sequences of both the projectivebundle structures of F to π − ( b ) ≃ G/P for every b ∈ B , we obtain L| π − ( b ) ≃ O (1 , .Based on the existing classification of homogeneous roofs, [Kan18, Section 5.2.1], we can pro-duce a similar list for homogeneous roof bundles (Table 2). The notation is the same as inDiagram 2.5. For clarity, we also list the name of the associated roof according to Kanemitsu’snomenclature. We denote by F l , OF l , IF l respectively the linear, orthogonal and symplecticflag bundles. A similar notation will be adopted for linear, orthogonal and symplectic Grass-mann bundle. For the Grassmann and flag bundles of exceptional groups, the group will be ALABI–YAU FIBRATIONS, SIMPLE K -EQUIVALENCE AND MUTATIONS 7 indicated as a subscript. In each line, W is a vector bundle on B such that W b is the fundamen-tal representation space of G for b ∈ B (see Remark 2.8). In the following, we will often refer to roof bundles of type G/P to emphasize the homogeneous structure of the fibers. G roof F F F SL ( k + 1) × SL ( k + 1) A k × A k G r (1 , W ) × G r (1 , W ) G r (1 , W ) G r (1 , W ) SL ( k + 1) A Mk F l (1 , k, W ) G r (1 , W ) G r ( k, W ) SL (2 k + 1) A G k F l ( k, k + 1 , W ) G r ( k, W ) G r ( k + 1 , W ) Sp (3 k − C k/ − IF l ( k − , k, W ) IG r ( k − , W ) IG r ( k, W ) Spin (2 k ) D k OG r ( k − , W ) OG r ( k, W ) + OG r ( k, W ) − F F F l F (2 , , W ) G r F (2 , W ) G r F (3 , W ) G G F l G (1 , , W ) G r G (1 , W ) G r G (2 , W ) Table 2: Homogeneous roof bundles2.3.
Calabi–Yau fibrations.
Our main interest in Sections 3, 4 is to investigate the zero lociof pairs of sections of E and E which are pushforwards of a section Σ ∈ H ( F , L ) , hencerelativizing the setting of Definition 2.6. Let us make this clearer by the following lemma, thenotation is established in Diagrams 2.4 and 2.5. Lemma 2.11.
Let F be a roof bundle of type G/P P n × P n over a smooth projective variety B andfix h i : G/P ≃ P ( E i ) −→ G/P i for i ∈ {
1; 2 } . Given a general section Σ ∈ H ( F , L ) , let us call X i := Z ( p i ∗ Σ) . Then there exist fibrations: (2.8) X X B f f such that for a general b ∈ B the varieties Y := f − ( b ) and Y := f − ( b ) are a Calabi–Yau pairassociated to the roof G/P in the sense of Definition . .Proof. We have p i ∗ L = E i , hence X i ⊂ F i is the zero locus of a section p i ∗ Σ of E i . Let us call f i := r i | X i . Since E i | r − i ( b ) ≃ E i and r − i ( b ) ≃ G/P i , it follows that ( r − i ( b ) , E i | r − i ( b ) ) is a Mukai MARCO RAMPAZZO pair. If b and Σ are general the varieties Y i = Z ( p i ∗ Σ | r − i ( b ) ) ⊂ r − i ( b ) are Calabi–Yau by Lemma2.4. Moreover, E i ≃ h i ∗ O (1 , and the varieties Y and Y are the zero loci of the pushforwardsof the same section Σ π − ( b ) , therefore they are a Calabi–Yau pair associated to the roof of type G/P as in Definition 2.6. (cid:3) A pair of Calabi–Yau eightfolds
Roof of type A G . We briefly recall a description of the roof of type A G and its relateddual Calabi–Yau threefolds. Let V be a vector space of dimension five. We call G (2 , V ) and G (3 , V ) the GL (5) -Grassmannians of respectively affine planes and affine 3-spaces in V . Oneach Grassmannian, there is a universal (tautological) short exact sequence:(3.1) −→ U −→ V ⊗ O −→ Q −→ where det U ∨ ≃ det Q ≃ O (1) . The flag variety F (2 , , V ) admits two projective bundle struc-tures, which define projections to the Grassmannians. These data define the roof of type A G ,illustrated by the following diagram:(3.2) P Q ∨ (2) F (2 , , V ) P U (2) G (2 , V ) G (3 , V ) h h There exists a line bundle O (1 , on F (2 , , V ) such that h ∗ O (1 , ≃ Q ∨ (2) and h ∗ O (1 , ≃U (2) . Sections of such bundles are Calabi–Yau threefolds. Moreover, for a general section S ∈ H ( F (2 , , V ) , O (1 , , the pushforwards h ∗ S and h ∗ S are a pair of non birational de-rived equivalent Calabi–Yau threefolds [KR17, Theorem 5.7]. The roof of type A G can be de-scribed by the following Dynkin diagrams:(3.3) . . Observe that, in the basis of fundamental weights { ω , . . . , ω } , O (1 , is the homogeneousline bundle whose highest weight is ω + ω , we write O (1 ,
1) = E ω + ω . Given a dominantweight ω , we denote V ω the associated representation space. By the Borel–Weil–Bott theoremwe have H ( F (2 , , V ) , O (1 , ≃ H ( G (2 , V ) , Q ∨ (2)) ≃ H ( G (3 , V ) , U (2)) ≃ V ω + ω whichis a 75-dimensional vector space.3.2. The roof bundle of type SL (5) /P , over P . Let us fix a vector space V ≃ C and thequotient bundle Q defined by the tautological sequence over G (1 , V ) :(3.4) −→ O ( − −→ V ⊗ O −→ Q −→ ALABI–YAU FIBRATIONS, SIMPLE K -EQUIVALENCE AND MUTATIONS 9 Hereafter we will define a roof bundle of type SL (5) /P , over P , with respect to the vectorbundle Q . In this setting, Diagram 2.5 becomes:(3.5) F l (2 , , Q ) G r (2 , Q ) G r (3 , Q ) P p p r r Note that G (1 , V ) ≃ P is an A -homogeneous variety and the whole construction can besketched in terms of crossed Dynkin diagrams: . . p p (3.6) . r r This picture is obtained extending the Dynkin diagrams of Diagram 3.3 with a new crossed rootfrom the left. The associated varieties are respectively F (1 , , , V ) , F (1 , , V ) , F (1 , , V ) and G (1 , V ) , hence Diagram 3.5 can be rewritten as:(3.7) F (1 , , , V ) F (1 , , V ) F (1 , , V ) G (1 , V ) p p πr r Here r and r are Grassmannian bundles, where the fibers are identified respectively with G (2 , V ) and G (3 , V ) . Note that there exist surjections ρ : F (1 , , V ) −→ G (3 , V ) and τ : F (1 , , V ) −→ G (4 , V ) .In the following, given a highest weight ω , we will call E ω the associated vector bundle. Given adominant weight ω , we will call V ω the associated representation space. On F (1 , , , V ) we de-fine a line bundle O (1 , ,
1) := p ∗ ρ ∗ O (1) ⊗ π ∗ O (1) ⊗ p ∗ τ ∗ O (1) . Fix a basis { ω , . . . , ω } of funda-mental weights for A . Observe that O (1 , ,
1) = E ω + ω + ω on F (1 , , , V ) has pushforwards to the Picard rank 2 flag varieties given by p ∗ O (1 , ,
1) = ρ ∗ Q ∨ (1 , and p ∗ O (1 , ,
1) = P (1 , where P is defined by the following short exact sequence on F (1 , , V ) :(3.8) −→ r ∗ U −→ τ ∗ U −→ P −→ . The line bundle O (1 , , is exactly the Grothendieck line bundle of the two projective bundlestructures of F (1 , , , V ) .3.3. Pairs of Calabi–Yau eightfolds.Lemma 3.1.
Let S ∈ H = H ( F (1 , , , V ) , O (1 , , be a general section. Then X = Z ( p ∗ O (1 , , and X = Z ( p ∗ O (1 , , are Calabi–Yau eightfolds of Picard number , and H ( X i , T X i ) ≃ H/ ( C ⊕ V ω + ω ) ≃ C .Proof. By adjunction formula, sections of E i := p i ∗ O (1 , , define eight dimensional varietieswith vanishing first Chern class for i ∈ {
1; 2 } . Since the Grothendieck line bundle of P ( E i ) is anample line bundle, E i is an ample vector bundle and we can use again [Laz04, Example 7.1.5]:the restriction maps H q ( F (1 , , V ) , Ω pF (1 , ,V ) ) −→ H q ( X , Ω pX ) H q ( F (1 , , V ) , Ω pF (1 , ,V ) ) −→ H q ( X , Ω pX ) (3.9)are isomorphisms for p + q < dim( X ) , and since F (1 , , V ) and F (1 , , V ) are homogeneousvarieties, their structure sheaves have cohomology of dimension one concentrated in degreezero. The Calabi–Yau condition follows from setting p = 0 in the isomorphism of Equation 3.9.In order to compute cohomology for the tangent bundle, let us first focus on X . We considerthe following two projections:(3.10) F (1 , , V ) G (1 , V ) G (3 , V ) r ρ and the following exact sequence(3.11) −→ O −→ ρ ∗ U (1 , − −→ T F (1 , ,V ) −→ ρ ∗ T G (3 ,V ) −→ which follows by the relative tangent bundle sequence of F (1 , , V ) −→ G (3 , V ) and the rela-tive Euler sequence of the projective bundle structure F (1 , , V ) ≃ P ( r ∗ U (1 , − .By the Borel–Weil–Bott theorem we get(3.12) H m ( X, T X ) ≃ V ω + ω + ω / ( C ⊕ V ω + ω ) m = 1 C m = 7 ALABI–YAU FIBRATIONS, SIMPLE K -EQUIVALENCE AND MUTATIONS 11 and this proves our claim. In fact, since Y is Calabi–Yau, by Serre duality we have:(3.13) H ( Y, T Y ) ≃ H ( Y, Ω Y ) = H (1 , ( Y ) and we conclude that the Picard number of Y is two by the long exact sequence of cohomologyof the exponential sequence. The case of X is identical: in fact, the sequence of Equation 3.11involves only bundles on F (1 , , V ) , and the weights of the bundles involved in the correspond-ing sequence on F (1 , , V ) are obtained by reversing the order of the fundamental weights onthe crossed Dynkin diagram of the flag variety. Therefore, the result is identical by the symme-try of the Dynkin diagram of type A . (cid:3) Derived equivalence of Calabi–Yau fibrations
Setup and general strategy.
Let
G/P be a homogeneous roof of rank r and M ⊂ G/P ageneral hyperplane. Let Y , Y be the associated Calabi–Yau pair, i.e. Y and Y are zero loci ofpushforwards of a section defining M along the projective bundle maps. One has the followingdiagram [KR20, Diagram 2.1]:(4.1) T M T G/PY G/P G/P Y h k l k ¯ h h h t t where T i are the preimages of Y i under of h i | M , and ¯ h i are the restrictions of h i | M to T i .There exist the following semiorthogonal decompositions of D b coh( M ) , which follow from anapplication of the Cayley trick [Orl03, Proposition 2.10]. D b coh( M ) ≃h h | ∗ M D b coh( G/P ) , . . . , h | ∗ M D b coh( G/P ) ⊗ O ( r − , r − , k ∗ ¯ h ∗ D b coh( Y ) i≃h h | ∗ M D b coh( G/P ) , . . . , h | ∗ M D b coh( G/P ) ⊗ O ( r − , r − , k ∗ ¯ h ∗ D b coh( Y ) i (4.2) Remark . Note that in the case of roofs of type A n × A n , we can proceed observing that the zerolocus M ⊂ P n × P n of a section of O (1 , is isomorphic to a flag variety F (1 , n, n + 1) . Hence,by Orlov’s formula for semiorthogonal decompositions of projective bundles [Orl92, Theorem4.3], we recover the same decomposition of Equation 4.2 except for the fact that D b coh( Y ) and D b coh( Y ) do not appear. This is of course not a surprise, since for roofs of type A n × A n thezero loci Y and Y are empty. Assume that D b coh( G/P ) and D b coh( G/P ) can be described by full exceptional collections ofhomogeneous vector bundles (see Remark 4.3 for a list of the cases where this is verified). Sup-pose there exists a sequence of mutations of exceptional objects realizing the following equiva-lence: D b coh( M ) ≃h h | ∗ M D b coh( G/P ) , . . . , h | ∗ M D b coh( G/P ) ⊗ O ( r − , r − , k ∗ ¯ h ∗ D b coh( Y ) i≃h h | ∗ M D b coh( G/P ) , . . . , h | ∗ M D b coh( G/P ) ⊗ O ( r − , r − , H ◦ k ∗ ¯ h ∗ D b coh( Y ) i (4.3)hence defining a Fourier–Mukai functor(4.4) D b coh( Y ) D b coh( Y ) where H is the action of the mutations on the Calabi–Yau component.The scope of this section is to provide a method to extend such equivalence to zero loci ofpushforwards of general sections of L on a roof bundle. More precisely, let us consider a roofbundle F of type G/P over a smooth projective base B , with the locally trivial fibration π : F −→ B . Fix a general section Σ ∈ H ( F , L ) with zero locus M and the corresponding pairof Calabi–Yau fibrations X , X . We have the following diagram:(4.5) T M T F X F F X p m i m ¯ p p p u u Then, we prove that there exist fully faithful embeddings D b coh( X i ) ⊂ D b coh( M ) and a se-quence of mutations of exceptional objects providing a Fourier–Mukai functor D b coh X −→ D b coh X .4.2. Semiorthogonal decompositions for M . Let us first observe that, since M is a generalsection of H ( F , L ) and F is a P r − -bundle over both F and F , by Cayley trick we have thefollowing semiorthogonal decompositions: D b coh( M ) ≃h p | ∗M D b coh( F ) , . . . , p | ∗M D b coh( F ) ⊗ L ⊗ ( r − , φ D b coh( X ) i≃h p | ∗M D b coh( F ) , . . . , p | ∗M D b coh( F ) ⊗ L ⊗ ( r − , φ D b coh( X ) i (4.6)where φ i := m i ∗ ◦ ¯ p ∗ i .The next step is to construct semiorthogonal decompositions for F i . This is possible due tothe following theorem [Sam06, Thm 3.1]: Theorem 4.2 (Samokhin) . Let f : X −→ B be a flat proper morphism and {K , . . . , K N } ⊂ D b coh( X ) objects such that their restrictions {K | f − ( b ) , . . . , K N | f − ( b ) } ∈ D b coh( f − ( b )) are a fullexceptional collection for D b coh( f − ( b )) . Then there exist fully faithful embeddings φ i : D b coh( B ) −→ D b coh X E 7−→ f ∗ E ⊗ K i (4.7) ALABI–YAU FIBRATIONS, SIMPLE K -EQUIVALENCE AND MUTATIONS 13 and the following semiorthogonal decomposition of D b coh( X ) : (4.8) D b coh( X ) = h f ∗ D b coh( B ) ⊗ K , . . . f ∗ D b coh( B ) ⊗ K N i . Let us assume there exist objects {K , . . . , K N } ⊂ D b coh( F ) and { e K , . . . , e K N } ⊂ D b coh( F ) such that their restrictions to the fibers are full exceptional collections, the strength of this as-sumption will be discussed later. Then, applying Theorem 4.2 to Equation 4.6 we obtain thefollowing semiorthogonal decompositions:(4.9) D b coh( M ) ≃ h ¯ π ∗ B ⊗ p | ∗M K , . . . . . . . . . . . . . . . . . . , ¯ π ∗ B ⊗ p | ∗M K N ¯ π ∗ B ⊗ p | ∗M K ⊗ L , . . . . . . . . . . . . , ¯ π ∗ B ⊗ p | ∗M K N ⊗ L ... ....................................................................... ... ¯ π ∗ B ⊗ p | ∗M K ⊗ L ⊗ ( r − , . . . . . , ¯ π ∗ B ⊗ p | ∗M K N ⊗ L ⊗ ( r − , φ D b coh( X ) i≃ h ¯ π ∗ B ⊗ p | ∗M e K , . . . . . . . . . . . . . . . . . . , ¯ π ∗ B ⊗ p | ∗M e K N ¯ π ∗ B ⊗ p | ∗M e K ⊗ L , . . . . . . . . . . . . , ¯ π ∗ B ⊗ p | ∗M e K N ⊗ L ... ....................................................................... ... ¯ π ∗ B ⊗ p | ∗M e K ⊗ L ⊗ ( r − , . . . . . , ¯ π ∗ B ⊗ p | ∗M e K N ⊗ L ⊗ ( r − , φ D b coh( X ) i where B = D b coh( B ) and ¯ π = π ◦ i . Remark . In order to apply Theorem 4.2, it is required to have a full exceptional collection forevery fiber of r and r . The problem of finding full exceptional collections for homogeneousvarieties is still open, but there are many cases where a solution has been found. Let G/P be aroof with projective bundle structures h i : G/P −→ G/P i for i ∈ {
1; 2 } . Let us review the caseswhere a full exceptional collection is known for both G/P and G/P . ◦ Type A n × A n , A Mn and A G n : here G/P i is a SL ( V ) -Grassmannian for some vector space V . Full exceptional collections for these varieties have been constructed in [Kap88]. ◦ Type C n/ − : in this case G/P i is a symplectic Grassmannian. The only case where afull exceptional collection is known for both G/P and G/P is the roof of type C . Thecollections have been established in [Kuz08]. ◦ Type D n : the only two cases where both G/P i have a known full exceptional collectionare D and D In the former, by triality
G/P i are six dimensional quadrics, for whicha full exceptional collection has been found in [Kap88]. In the latter, the varieties G/P i are spinor tenfolds, a full exceptional collection for them is given in [Kuz06]. ◦ Type G : there are known full exceptional collections for both G/P and G/P [Kap88,Kuz06]. ◦ Type F : To the best of the author’s knowledge, no full exceptional collection is knownfor the homogeneous varieties F /P and F /P .Note that each of the collections listed above is given in terms of homogeneous vector bundles,hence, as in Equation 2.7, such bundles are restrictions of vector bundles on the associated roofbundle. Proposition 4.4.
Let
G/P be a roof and M j ֒ −−→ G/P the zero locus of a general section of O (1 , . Let F be a roof bundle of type G/P over a smooth projective base B , with structure map π : F −→ B .Call M l ֒ −−→ F a general section of L and fix ¯ π := π ◦ l . Consider two objects K , K ∈ D b coh( F ) and define K i := K i | π − ( b ) for i ∈ {
1; 2 } . Assume that the following conditions hold:(1) K and K are exceptional objects of D b coh( G/P ) and their restrictions to M are exceptionalobjects of D b coh( M ) .(2) Ext • G/P ( K , K ) = Ext • M ( K , K ) (3) Ext • G/P ( K , K ) = Ext • M ( K , K ) = 0 Then, the following is true for every b ∈ B and for every E ∈ D b coh( B ) : ( L h π ∗ D b coh( B ) ⊗K i K ⊗ π ∗ E ) π − ( b ) ≃ L K K ( R h π ∗ D b coh( B ) ⊗K i K ⊗ π ∗ E ) π − ( b ) ≃ R K K (4.10) Moreover, for general b ∈ B one has: ( L h ¯ π ∗ D b coh( B ) ⊗ l ∗ K i l ∗ K ⊗ ¯ π ∗ E ) ¯ π − ( b ) ≃ L j ∗ K j ∗ K ( R h ¯ π ∗ D b coh( B ) ⊗ l ∗ K i l ∗ K ⊗ ¯ π ∗ E ) ¯ π − ( b ) ≃ R j ∗ K j ∗ K (4.11) Proof.
We just need to check the claim for left mutations, since right mutations are just theirinverse functors. The main ingredient of this proof is the base change technique for kernelfunctors developed in [Kuz06]. We have the following expression for R K K :(4.12) R K K = Cone (cid:16) K −→ K ⊗ Ext • G/P ( K , K ) ∨ (cid:17) [ − and since Ext • G/P ( K , K ) = Ext • M ( K , K ) we conclude that j ∗ L K K ≃ L j ∗ K j ∗ K . Define X := F × F with projections pr and pr to its two factors. For every K ∈ D b coh( X ) weconsider the following kernel functors: Φ K : H −→ pr ∗ ( K ⊗ pr ∗ H )Φ ! K : H −→ pr ∗ E xt • X ( K , pr ∗ H ) (4.13)Note that Φ ! K is the right adjoint functor of Φ K .Since π is locally trivial, the following base change is faithful with respect to π for every b ∈ B :(4.14) F × B { b } F { b } B ρπ b πφ b ALABI–YAU FIBRATIONS, SIMPLE K -EQUIVALENCE AND MUTATIONS 15 Therefore, by [Kuz06, Lemma 2.42] the following identities hold for every b , where we defined F b := F × B { b } ≃ G/P : Φ K| Fb φ ∗ b = φ ∗ b Φ K i Φ K i φ b ∗ = φ b ∗ Φ K i | Fb Φ ! K i | Fb φ ∗ b = φ ∗ b Φ ! K i Φ ! K i φ b ∗ = φ b ∗ Φ ! K i | Fb (4.15)The mutation R h π ∗ D b coh( B ) ⊗K i K ⊗ π ∗ E can be described in terms of the following triangle in D b coh( F ) :(4.16) ΨΨ ! ( K ⊗ π ∗ E ) −→ K ⊗ π ∗ E −→ R h π ∗ D b coh( B ) ⊗K i K ⊗ π ∗ E where we define the functor Ψ as: Ψ : D b coh( B ) −−−→ D b coh( F ) E 7−→ π ∗ E ⊗ K (4.17)and we call Ψ ! its right adjoint functor. Once we note that Ψ = Φ pr ∗ K ◦ π ∗ , the claim(4.18) ( R h π ∗ D b coh( B ) ⊗K i K ⊗ π ∗ E ) π − ( b ) ≃ R K K follows from Equation 4.15 and the commutativity of Diagram 4.14.Let us now prove the last claim ( R h ¯ π ∗ D b coh( B ) ⊗ l ∗ K i l ∗ K ⊗ ¯ π ∗ E ) ¯ π − ( b ) ≃ R j ∗ K j ∗ K . We havethe following triangle:(4.19) Ψ M Ψ ! M ( l ∗ K ⊗ ¯ π ∗ E ) −→ l ∗ K ⊗ ¯ π ∗ E −→ R h ¯ π ∗ D b coh( B ) ⊗ l ∗ K i l ∗ K ⊗ ¯ π ∗ E where Ψ M is defined by: Ψ M : D b coh( B ) −−−→ D b coh( M ) E 7−→ ¯ π ∗ E ⊗ l ∗ K (4.20)Observe that the right adjoint of Ψ M is given for every H ∈ D b coh( M ) by(4.21) Ψ ! M : H −→ ¯ π ∗ E xt •M ( H , j ∗ K ) while the right adjoint of Ψ acts on G ∈ D b coh( F ) by:(4.22) Ψ ! : G −→ π ∗ E xt •F ( G , K ) Hence, we just need to prove that ¯ π ∗ E xt •M (¯ π ∗ E ⊗ l ∗ K , l ∗ K ) ≃ π ∗ E xt •F ( π ∗ E ⊗ K , K ) , sincefor general b and M one has that ¯ π − ( b ) is also the zero locus of a general section of O (1 , in G/P and this allows us to use the assumption
Ext • G/P ( K , K ) = Ext • M ( K , K ) .Recall that ¯ π ∗ = π ∗ l ∗ . It follows that:(4.23) ¯ π ∗ E xt •M (¯ π ∗ E ⊗ l ∗ K , l ∗ K ) ≃ π ∗ E xt •M ( π ∗ E ⊗ K , l ∗ l ∗ K ) . Since M is general, the following Koszul resolution is exact:(4.24) −→ L ∨ −→ O −→ l ∗ l ∗ O −→ hence we just need to prove that π ∗ E xt M ( π ∗ E ⊗ K , K ⊗ L ∨ ) has no cohomology. But this is aconsequence of the following, which holds for every b ∈ B : π ∗ E xt •F ( π ∗ E ⊗ K , K ⊗ L ∨ ) b ≃ H ( M, E xt •F ( π ∗ E ⊗ K , K ⊗ L ∨ )) ≃ Ext • G/P ( K , K ⊗ O ( − , − (4.25)where the last equality is due to the following exact Koszul resolution:(4.26) −→ O ( − , − −→ O −→ O M −→ and the fact that Ext • G/P ( K , K ) = Ext • M ( K , K ) . (cid:3) Theorem 4.5.
Let
G/P be a roof of type A Mk , A G k , C k/ − , D or G . Let M ⊂ G/P be the zero locusof a general section of O (1 , on G/P . Call F a roof bundle of type G/P over a smooth projective base B with projective bundle structures p i : F −→ F i . Given a general section Σ ∈ H ( F , L ) with zerolocus M , let us define X i := Z ( p i ∗ Σ) .Let { m α } α ≤ T be a sequence of mutations in D b coh( G/P ) for some T ∈ N acting on the exceptionalcollection Equation 4.6 for ⊥ D b coh( Y ) such that the following holds: (4.27) m T | M ◦ · · · ◦ m | M ( ⊥ D b coh Y ) = ⊥ D b coh Y where we call m | M the mutation defined by the restriction of the triangle in D b coh( G/P ) which defines m . Then X and X are derived equivalent.Proof. In the notation of Diagram 4.1 there exist the following semiorthogonal decompositions: D b coh( M ) ≃h h | ∗ M D b coh( G/P ) , . . . , h | ∗ M D b coh( G/P ) ⊗ O ( r − , r − , k ∗ ¯ h ∗ D b coh( Y ) i≃h h | ∗ M D b coh( G/P ) , . . . , h | ∗ M D b coh( G/P ) ⊗ O ( r − , r − , k ∗ ¯ h ∗ D b coh( Y ) i (4.28)As we discussed in Remark 4.3, for the roofs of types above, both G/P and G/P admit fullexceptional collections D b coh( G/P ) = h K , . . . , K N i D b coh( G/P ) = h e K , . . . , e K N i (4.29)such that K i and e K i are homogeneous vector bundles for ≤ i ≤ N . Plugging Equation 4.29into Equation 4.28 we obtain:(4.30) D b coh( M ) ≃ h h | ∗ M K , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , h | ∗ M K N h | ∗ M K ⊗ O (1 , , . . . . . . . . . . . . . . . . . . . . , h | ∗ M K N ⊗ O (1 , ... ................................................................... ... h | ∗ M K ⊗ O ( r − , r − , . . . . . , h | ∗ M K N ⊗ O ( r − , r − , φ D b coh( Y ) i≃ h h | ∗ M e K , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , h | ∗ M e K N h | ∗ M e K ⊗ O (1 , , . . . . . . . . . . . . . . . . . . . . , h | ∗ M e K N ⊗ O (1 , ... ................................................................... ... h | ∗ M e K ⊗ O ( r − , r − , . . . . . , h | ∗ M e K N ⊗ O ( r − , r − , φ D b coh( Y ) i ALABI–YAU FIBRATIONS, SIMPLE K -EQUIVALENCE AND MUTATIONS 17 Furthermore, there exist objects {K , . . . K N } ⊂ D b coh( F ) such that K m | G/P = K m and asimilar collection { e K , . . . e K N } ⊂ D b coh( F ) such that e K m | G/P = e K m . These objects can beconstructed exactly as in Equation 2.7 from the data of the bundles on the fibers. By Theorem4.2 we recover the following semiorthogonal decompositions (Equation 4.9):(4.31) D b coh( M ) ≃ h π ∗ B ⊗ p | ∗M K , . . . . . . . . . . . . . . . . . . , ¯ π ∗ B ⊗ p | ∗M K N ¯ π ∗ B ⊗ p | ∗M K ⊗ L , . . . . . . . . . . . . , ¯ π ∗ B ⊗ p | ∗M K N ⊗ L ... ................................................................... ... ¯ π ∗ B ⊗ p | ∗M K ⊗ L ⊗ ( r − , . . . . . , ¯ π ∗ B ⊗ p | ∗M K N ⊗ L ⊗ ( r − , φ D b coh( X ) i≃ h ¯ π ∗ B ⊗ p | ∗M e K , . . . . . . . . . . . . . . . . . . , ¯ π ∗ B ⊗ p | ∗M e K N ¯ π ∗ B ⊗ p | ∗M e K ⊗ L , . . . . . . . . . . . . , ¯ π ∗ B ⊗ p | ∗M e K N ⊗ L ... ................................................................... ... ¯ π ∗ B ⊗ p | ∗M e K ⊗ L ⊗ ( r − , . . . . . , ¯ π ∗ B ⊗ p | ∗M e K N ⊗ L ⊗ ( r − , φ D b coh( X ) i where B = D b coh( B ) . Observe that all objects of the form p i | ∗M K j in Equation 4.31 are re-strictions of homogeneous vector bundles p ∗ i K j on F and that that for every b ∈ B one has p ∗ i K j | π − ( b ) = h ∗ i K j . Moreover, the set of bundles { h ∗ i K j } , restricted to the zero locus of a gen-eral section of O (1 , give exactly the collection of Equation 4.30.By assumption, there exists a sequence of mutations { m α } on D b coh( G/P ) such that their re-strictions { m α | M } give a derived equivalence ⊥ D b coh( Y ) ≃ ⊥ D b coh( Y ) , where the semiorthog-onal complements are taken in the collections of Equation 4.29. By Proposition 4.4, mutations ofexceptional objects on F restrict to the fiber π − ( b ) ≃ G/P to mutations of the restrictions of thecorresponding objects. For every m i of Equation 4.27 there exists a mutation M i on D b coh( F ) which restricts to m i on every fiber of π . Furthermore, again by Proposition 4.4, the restrictionof M i to M is computed by the restriction to M of the corresponding triangle in D b coh( F ) .Hence, since the collection of Equation 4.31 restricts to the one of Equation 4.30 on generalfibers of ¯ π , if m T | M ◦ · · · ◦ m | M identifies the semiorthogonal complements of D b coh( Y ) and D b coh( Y ) , we conclude that(4.32) M T | M ◦ · · · ◦ M | M ( ⊥ D b coh X ) = ⊥ D b coh X and this completes the proof. (cid:3) Theorem 4.5 can be immediately applied to all cases of roofs where a sequence of mutationsrealizing a derived equivalence of a Calabi–Yau pair is known, which are A G and G . Beforedoing this, let us investigate the two additional cases C and A Mn , so we can extend our resultto these examples as well. Remark . In [BM02] Bridgeland and Maciocia constructed derived equivalent fibrations withgeneral fiber isomorphic to a K surface or an elliptic curve. Namely, from a fibration X −→ B with general fiber F , they constructed a fibration e X −→ B with fiber given by a moduli spaceof stable objects on F . Then, they proved that X and e X are derived equivalent by extendingthe Fourier–Moukai kernel on the fibers to the whole fibrations. In Theorem 4.5 we address a similar problem with a class of examples of higher dimensional Calabi–Yau fibration, and wepropose a method to extend a fiberwise derived equivalence to the total space of the fibration.4.3. Derived equivalence for the roof of type C . A roof of type C is given by the followingdiagram:(4.33) IF (1 , , V ) IG (1 , V ) IG (2 , V ) h h where IG and IF denote, respectively symplectic Grassmannians and flag varieties. Note that IG (1 , V ) ≃ P and IG (2 , V ) is a three dimensional quadric in P . Both h and h are P -fibrations. Let us choose a general section σ ∈ H ( IF (1 , , V ) , O (1 , and call M = Z ( σ ) .Then, by dimensional reasons and Lemma 2.4, the zero loci Y = Z ( h ∗ σ ) and Y = Z ( h ∗ σ ) are elliptic curves. Lemma 4.7.
Let V be a vector space of dimension four. Consider M = Z ( σ ) for a general section σ ∈ H ( IF (1 , , V ) , O (1 , . Fix Y i = Z ( h i ∗ σ ) for i ∈ {
1; 2 } . There exists a sequence of mutationsin D b coh( M ) realizing a derived equivalence D b coh( Y ) −→ D b coh( Y ) .Proof. Our approach follows [Mor19] closely. By Cayley trick we write the following semiorthog-onal decompositions: D b coh( M ) ≃hO ( − , , O ( − , , O , O (1 , , φ D b coh( Y ) i≃hO , U ∨ , O (0 , , O (0 , , φ D b coh( Y ) i (4.34)where φ i = k i ∗ ¯ h ∗ i in the notation of Diagram 4.1. Let us start from the first collection. We cansend the first bundle to the far right, then move φ D b coh( Y ) one step to the right, obtaining D b coh( M ) ≃hO ( − , , O , O (1 , , O ( − , , R O ( − , φ D b coh( Y ) i (4.35)We have the following short exact sequence on IF (1 , , V ) (and on M ):(4.36) −→ O ( − , −→ U ∨ −→ O (1 , −→ Given the following result, which can be computed by Borel–Weil–Bott’s theorem:(4.37)
Ext • IF (1 , ,V ) ( O (1 , , O ( − , • M ( O (1 , , O ( − , C [ − we can mutate O (1 , and get: D b coh( M ) ≃hO ( − , , O , O ( − , , U ∨ , R O ( − , φ D b coh( Y ) i (4.38)Again by a simple application of Borel–Weil–Bott’s theorem, we compute:(4.39) Ext • IF (1 , ,V ) ( O , O ( − , • M ( O , O ( − , ALABI–YAU FIBRATIONS, SIMPLE K -EQUIVALENCE AND MUTATIONS 19 hence we can exchange the second and the third bundles, then we can move the first two to theend and send R O ( − , φ D b coh( Y ) to the far right. We find: D b coh( M ) ≃hO , U ∨ , O (0 , , O (0 , , R hO ( − , , O (0 , , O (0 , i φ D b coh( Y ) i (4.40)In the first four bundles we recognise D b coh( IG (2 , V )) . Hence, comparing Equation 4.34 withEquation 4.40 we prove our claim. (cid:3) Remark . Note that the derived equivalence D b coh( Y ) ≃ D b coh( Y ) is a consequence ofthe derived equivalence of local Calabi–Yau fivefolds described in [Mor19, Section 2]: in fact,one can follow the approach of [Ued19, Section 5] based on matrix factorization categories.In general, given a roof of type G/P with P r − -bundle structures h i : G/P −→ G/P i , let uscall E i := h i ∗ O (1 , and Y i = Z ( h i ∗ σ ) , where σ is a general section of O (1 , . Then, one candefine by the data of a section of E i a superpotential w i such that the derived category of matrixfactorizations of the Landau–Ginzburg model ( E ∨ i , w i ) is equivalent to D b coh( Y i ) via Kn ¨orrerperiodicity [Shi12, Theorem 3.4] (for more details, see [KR20, Section 5]). Then, by [Ued19]if there exists a derived equivalence D b coh( E ∨ ) ≃ D b coh( E ∨ ) satisfying a C ∗ -equivariancycondition, it lifts to a derived equivalence of the matrix factorization categories of ( E ∨ i , w i ) , and D b coh( Y ) ≃ D b coh( Y ) follows from this last equivalence composed with Kn ¨orrer periodicity.This gives a derived equivalence for Calabi–Yau pairs of type A G , C [Mor19] and G [Ued19].4.4. Derived equivalence for roofs of type A Mn . A roof of type A Mn is given by the followingdiagram:(4.41) F (1 , n, V ) G (1 , V ) G ( n, V ) h h where V = V b . Call O (1 ,
1) = L| π − ( b ) and σ = Σ | π − ( b ) . Call M = Z ( σ ) and define theclosed immersion l : M = Z ( σ ) ֒ −−→ F (1 , n, V ) . Then the zero loci Y i = Z ( h i ∗ σ ) are zero-dimensional. Nonetheless we discuss their derived equivalence, since it will be necessary toprove further results in Section 5. By Cayley trick we recover the following semiorthogonaldecompositions: D b coh( M ) ≃h h | ∗ M D b coh G (1 , V ) , . . . , h | ∗ M D b coh G (1 , V ) ⊗ O ( n − , n − , φ D b coh( Y ) i≃h h | ∗ M D b coh G ( n, V ) , . . . , h | ∗ M D b coh G ( n, V ) ⊗ O ( n − , n − , φ D b coh( Y ) i (4.42) where φ i = h i | ∗ h − i ( Y i ) ◦ l ∗ . By Beilinson’s full exceptional collection for P n [Bei78] we write: D b coh( M ) ≃hO M (0 , , . . . . . . . . . . . . . . . , O M ( n, , ... ... O M ( n − , n − , . . . , O M (2 n − , n − , φ D b coh( Y ) i≃hO M (0 , , . . . . . . . . . . . . . . . , O M (0 , n ) , ... ... O M ( n − , n − , . . . , O M ( n − , n − , φ D b coh( Y ) i (4.43)By Theorem 4.5 it is sufficient to find a sequence of mutations such that φ D b coh( Y ) ≃ φ D b coh( Y ) as subcategories of D b coh( M ) to prove that X and X are derived equivalent. The remainderof this section is devoted to this. First, we need the following vanishing result: Lemma 4.9.
Let V be a vector space of dimension n + 1 and M ⊂ F (1 , n, V ) a general hyperplane.Then: (4.44) Ext • M ( O ( m, , O (1 , • F (1 ,n,V ) ( O ( m, , O (1 , . for < m < n − .Proof. Twisting the Koszul resolution for M yields:(4.45) −→ O ( − m, −→ O (1 − m, −→ O M (1 − m, −→ Observe that for every x ∈ G ( n, V ) one has:(4.46) R • h ∗ O ( a, b ) x = H • ( h − ( x ) , O ( a ) | h − ( x ) ) This is identically zero for a = − m or a = 1 − m , hence O M (1 − m, has no cohomology andthis concludes the proof. (cid:3) Lemma 4.10.
Let σ ∈ H ( F (1 , n, V ) , O (1 , be a general section and M = Z ( σ ) . Fix Y i = Z ( h i ∗ σ ) for i ∈ {
1; 2 } , the notation is the one of Diagram . . Then Y and Y are derived equivalent, and thereexists a sequence of mutations of exceptional objects of D b coh( M ) realizing such equivalence.Proof. Let us switch to a more compact notation: hereafter O a,b := O ( a, b ) . Hence, Equation4.42 becomes: D b coh( M ) ≃ hO , , . . . . . . . . . . . . . . . , O n, , O , , . . . . . . . . . . . . . . . , O n +1 , , ... ... O n − ,n − , . . . . . . , O n − ,n − , φ D b coh( Y ) i (4.47) ALABI–YAU FIBRATIONS, SIMPLE K -EQUIVALENCE AND MUTATIONS 21 First, let us move O , to the end of the collection, then move φ D b coh( Y ) one step to the right.We get: D b coh( M ) ≃ hO , . . . . . . . . . . . . . . . , O n, , O , , . . . . . . . . . . . . . . . , O n +1 , , ... ... O n − ,n − , . . . . . . , O n − ,n − , O n − ,n − , ψ φ D b coh( Y ) i (4.48)where ψ := R O n − ,n − . By Lemma 4.9 we can move O , leftwards until it finds O , . We canrepeat the same step on each line, we get: D b coh( M ) ≃ hO , , O , , O , , . . . . . . . . . . . . . . . . . . , O n, , O , , O , , O , , . . . . . . . . . . . . . . . . . . , O n +1 , , ... ... O n − ,n − , O n,n − , O n +1 ,n − , . . . . . . , O n − ,n − , ψ φ D b coh( Y ) i (4.49)Now we move the first two bundles to the end of the collection, and we mutate ψ φ D b coh( Y ) two steps to the right. Then, on each line, using Lemma 4.9 we shift the last two bundles all theway to the right of the first bundle. We find: D b coh( M ) ≃ hO , , O , , O , , O , , . . . . . . . . . . . . . . . . . . , O n, , O , , O , , O , , O , , . . . . . . . . . . . . . . . . . . , O n +1 , , ... ... O n,n − , O n,n − , O n,n , O n +1 ,n − , . . . . . . , O n − ,n − , ψ φ D b coh( Y ) i (4.50)where ψ = R hO n,n − , O n,n i ◦ ψ . This process can be iterated moving the first three bundlesto the end, then on each row sending the last three bundles to the right of the first one, andrepeating these steps increasing by one the number of bundles we move. We stop once weget a semiorthogonal decomposition given by n − twists of hO , , . . . O ,n i and the image of φ D b coh( Y ) under a composition of mutations. This eventually happens after n steps. We getthe following collection: D b coh( M ) ≃ hO n, , O n, , . . . . . . . . . . . . . . . . . . . . . , O n,n , O n +1 , , O n +1 , , . . . . . . . . . . . . . . . , O n +1 ,n +1 , ... ... O n − ,n − , O n − ,n − , . . . . . . , O n − , n − , ψ n φ D b coh( Y ) i (4.51)If we twist the whole collection by O − n, we obtain: D b coh( M ) ≃ hO , , O , , . . . . . . . . . . . . . . . . . . , O ,n , O , , O , , . . . . . . . . . . . . . . . . . . , O ,n +1 , ... ... O n − ,n − , O n − ,n − , . . . . . . , O n − , n − , T − n, ◦ ψ n φ D b coh( Y ) i (4.52) where T − n, is the twist functor. By comparing Equation 4.43 with Equation 4.52 we concludethe proof. (cid:3) Lemma 4.11.
Derived equivalences of Calabi–Yau pairs associated to roofs of type A Mn , A n × A n , A G , C and G satisfy the assumptions of Theorem 4.5.Proof. We can prove that this claim holds by direct computation working case by case with theBorel–Weil–Bott theorem and the following Koszul resolution:(4.53) −→ O ( − , − −→ O −→ O M −→ Roof of type A Mn : this follows from Lemma 4.10. In fact, the only mutations that we use are theorthogonality conditions of Lemma 4.9, which hold on M and on G/P as well, as it is provedin Lemma 4.10.Roof of type A n × A n : the claim in this case follows from the fact that, as we discussed inRemark 4.1, given a general section σ ∈ H ( P n × P n , O (1 , , its zero locus M is isomorphicto the flag variety F (1 , n, n + 1) . Hence the semiorthogonal decompositions for D b coh( M ) areidentical to the ones for the zero locus of a section of O (1 , on the roof of type A Mk discussedabove, except for the fact that the categories of the zero loci of pushforwards of σ do not appear,but there is an additional twist of D b coh( P n ) . Since the canonical bundle of M has also an ad-ditional twist by O (1 , , the mutations we use are exactly the same of the ones we needed forthe previous case, i.e. the orthogonality conditions defined by Lemma 4.9.Roof of type A G : Let V be a vector space of dimension five. We call U i and Q i the pullbacks oftautological and quotient bundles of G ( i, V ) to F (2 , , V ) . We need the following cohomolog-ical results, which can be readily obtained by Borel–Weil–Bott’s theorem: ◦ Ext • M ( O (0 , a ) , O (1 , • G/P ( O (0 , a ) , O (1 , for a ∈ {
3; 4 } . ◦ Ext • M ( Q (0 , a ) , O (1 , • G/P ( Q (0 , a ) , O (1 , for a ∈ {
2; 3; 4 } . ◦ Ext • M ( U (0 , , O (1 , • G/P ( U (0 , , O (1 , ◦ Ext • M ( Q i , O ) = Ext • G/P ( Q i , O ) for i ∈ {
1; 2 } . ◦ Ext • M ( Q (0 , , U (1 , • G/P ( Q (0 , , U (1 , .Therefore, every mutation of [KR17, Proposition 5.6] can be applied in D b coh( G/P ) .Roof of type C : all the cohomological results that we need are covered by Lemma 4.9, wherewe proved that they hold on both M and G/P .Roof of type G : Here it is enough to note the following facts: ◦ The vanishings of [Kuz16, Corollary 2] of vector bundles on M hold identically on G/P . ◦ The short exact sequence of [Kuz16, Proposition 3] is a pullback from
G/P . (cid:3) ALABI–YAU FIBRATIONS, SIMPLE K -EQUIVALENCE AND MUTATIONS 23 Corollary 4.12.
Let
G/P be a roof of type A Mn , A G , G or C and let M = Z ( σ ) ⊂ G/P be a generalhypersurface. Call Y , Y the pair of Calabi–Yau varieties given by pushforwards of σ along the maps G/P −→ G/P i . There exists a pair of derived equivalent fibrations f i : X i −→ B such that for every b ∈ B one has f − i ( b ) ≃ Y i .Proof. Let us fix a general Σ ∈ H ( F , L ) . Then, by Lemma 2.4, for general b ∈ B we have aCalabi–Yau pair ( Y , Y ) , where Y i = Z ( p i ∗ L| π − ( b ) ) . We obtain a pair of Calabi–Yau fibrations f i : X i −→ B once we set f i = r i | X i , by Lemma 2.11. The derived equivalence follows byapplying Theorem 4.5 and Lemma 4.11 to the mutations described in [Kuz16], [KR17] for roofsof type A G , Lemma 4.10 for roofs of type A Mn and Lemma 4.7 for roofs of type C . (cid:3) Corollary 4.13.
Let F be a roof bundle of type G/P , where
G/P is a roof of type A k × A k and Σ ∈ H ( F , L ) a general section. Then, X = Z ( p ∗ Σ) and X = Z ( p ∗ Σ) are derived equivalent.Proof. The claim follows from Theorem 4.5, Lemma 4.11 and Remark 4.1. In fact, by Remark 4.1,we just need to compare two semiorthogonal decompositions of a general section of O (1 , on G/P ≃ P k × P k . The mutations we need to perform are described in the proof of Lemma 4.10and Lemma 4.9. (cid:3) In all the roof bundles where a proof of derived equivalence based on mutations of the associ-ated Calabi–Yau pair is known, the corresponding Calabi–Yau fibrations are derived equivalent(Corollaries 4.12 and 4.13). Therefore, in light of [KR20, Conjecture 2.6], we formulate the fol-lowing:
Conjecture 4.14.
Let
G/P be a homogeneous roof, and F a roof bundle of type G/P with projectivebundle structures p i : F −→ F i for i ∈ {
1; 2 } . Given a general section Σ ∈ H ( F , L ) , the Calabi–Yaufibrations X i := Z ( p i ∗ Σ) are derived equivalent. Simple K -equivalence and roof bundles Let X , X be smooth projective varieties. A K -equivalence is a birational morphism(5.1) µ : X X such that there exists a smooth projective variety X and the following diagram:(5.2) X X X g g µ where g and g are birational maps fulfilling g ∗ K X ≃ g ∗ K X . By the DK -conjecture [BO02,Kaw02], two K -equivalent varieties are expected to be derived equivalent. We can providesome evidence to this conjecture, and establish a method to verify it for the class of simple K -equivalent maps, under some assumption on the resolution X .A simple K -equivalent map, following the notation of Diagram 5.2, is a K -equivalence µ suchthat g and g are smooth blowups. Then, by the structure theorem for simple K -equivalence [Kan18, Thm. 0.2], the common exceptional locus of both the blowups is a family of roofs of pro-jective bundles over a smooth projective variety B . Let us focus our attention to the followingsetting: Definition 5.1.
We say that a simple K -equivalence µ is of type G/P if the exceptional locus of theblowup which resolves µ is isomorphic to a roof bundle of type G/P over a smooth projective variety B . Therefore, for every K -equivalence µ of type G/P there exists the following diagram:(5.3) F X F X X F AAAAB fp p g g r µ r which a simple adaptation of [Kan18, Diagram 0.2.1] to our setting.By constructing semiorthogonal decompositions for X in terms of the derived categories of X i and F i , we observe again a striking similarity with the pattern appearing in the two semiorthog-onal decompositions 4.2 for the zero locus m of a general section of O (1 , on G/P . Proposition 5.2.
Let
G/P be a roof of P r − -bundles with structure maps h i : G/P −→ G/P i for i =1 , . Let µ : X X be a simple K -equivalent map of type G/P and let M = Z ( σ ) ⊂ G/P , for a gen-eral section σ ∈ H ( G/P, O (1 , . Then, if there exists a sequence of mutations in D b coh( G/P ) suchthat their pullback to D b coh( M ) defines an equivalence of categories D b coh( h ∗ σ ) ≃ D b coh( h ∗ σ ) , X and X are derived equivalent. ALABI–YAU FIBRATIONS, SIMPLE K -EQUIVALENCE AND MUTATIONS 25 Proof.
Let us fix some notation. The relation between the family of roofs of type
G/P and the K -equivalent map is described by the following diagram:(5.4) T M T F X X F X X F X AAAAB m ¯ p i m ¯ p fp p g g r µ r while the restriction to b ∈ B gives rise to the picture:(5.5) T M T G/PY G/P G/P Y h k l k ¯ h h h t t Let us consider a general section Σ ∈ H ( F , L ) . and its zero locus M = Z (Σ) ⊂ F . Then, bythe discussion of Section 4, for general b ∈ B there exist the following semiorthogonal decom-positions for M ≃ π − ( b ) , where π = r ◦ p = r ◦ p : D b coh( M ) ≃h h | ∗ M D b coh( G/P ) , . . . , h | ∗ M D b coh( G/P ) ⊗ O ( r − , r − , k ∗ ¯ h ∗ D b coh( Y ) i≃h h | ∗ M D b coh( G/P ) , . . . , h | ∗ M D b coh( G/P ) ⊗ O ( r − , r − , k ∗ ¯ h ∗ D b coh( Y ) i (5.6)This expression is formally identical, up to overall twist, to the following one, which is obtainedby applying Orlov’s blowup formula [Orl92, Theorem 4.3] to Diagram 5.4: D b coh( X ) ≃ h f ∗ p ∗ D b coh( F ) ⊗ L ⊗ (1 − r ) , . . . , f ∗ p ∗ D b coh( F ) ⊗ L ⊗ ( − , g ∗ D b coh( X ) i≃ h f ∗ p ∗ D b coh( F ) ⊗ L ⊗ (1 − r ) , . . . , f ∗ p ∗ D b coh( F ) ⊗ L ⊗ ( − , g ∗ D b coh( X ) i (5.7)Due to Proposition 4.4, the proof reduces to show that mutations commute with the push-forward f ∗ . Given two objects K , K ∈ D b coh( F ) , let us consider the following triangle on D b coh( X ) :(5.8) ΞΞ ! ( f ∗ π ∗ E ⊗ f ∗ K ) −→ f ∗ π ∗ E ⊗ f ∗ K −→ R h f ∗ π ∗ D b coh( B ) ⊗ f ∗ K i f ∗ π ∗ E ⊗ f ∗ K where the fully faithful embedding Ξ i is given by: Ξ : D b coh B −−−→ D b coh F E 7−→ f ∗ π ∗ E ⊗ f ∗ K (5.9)Note that, in the notation of Equation 4.16, we have Ξ = f ∗ Ψ . Since f is a closed immersion itfollows that: Ξ ! ( f ∗ G ) = π ∗ f ∗ E xt •X ( f ∗ G , f ∗ K ) ≃ π ∗ E xt •F ( f ∗ f ∗ G , f ∗ f ∗ K ) ≃ π ∗ E xt •F ( G , K ) ≃ Ψ ! ( G ) (5.10)and this allows us to we deduce that mutations commute with f ∗ .Summing all up, there exist mutations { M α } on D b coh( F ) which, fiberwise, restrict to { m α } for every b ∈ B , and such mutations, restricted to M , are the ones which define the derivedequivalence D b coh( Y ) ≃ D b coh( Y ) . Moreover, as we showed above, the mutations { M α } induce corresponding mutations on D b coh( X ) by exactness of f ∗ , hence they can be appliedin Equation 5.7 providing an equivalence ⊥ D b coh( X ) ≃ ⊥ D b coh( X ) and this completes theproof. (cid:3) The following theorem is an extension of the results of [BO95, Kaw02, Nam03] on derived equiv-alence for varieties related by K -equivalence of type A n × A n and A Mn which are respectivelystandard flops and Mukai flops. Theorem 5.3.
Let µ : X X be a simple K -equivalent map of type G/P , where
G/P is a roof oftype A Mn , A n × A n , A G , C or G . Then X and X are derived equivalent.Proof. In all cases above there exist sequences of mutations proving derived equivalence for theassociated Calabi–Yau pairs: by Proposition 5.2 we just need to verify that such mutations arerestrictions of mutations on the roof
G/P . More precisely, let
G/P be one of the roofs listedabove, and M ⊂ G/P the zero locus of a general section of O (1 , . We are interested in muta-tions of pairs, hence, for two exceptional objects G and H in D b coh( G/P ) we have: L G H = Cone {G ⊗ Ext • ( G , H ) −→ H} R H G = Cone {G −→ H ⊗ Ext • ( G , H ) ∨ } [ − . (5.11)These mutations restrict to M if Ext • G/P ( G , H ) ≃ Ext • M ( G , H ) . This condition is fulfilled in eachone of the cases above by Lemma 4.11. (cid:3) Gauged linear sigma model and Calabi–Yau fibrations
Let us fix a roof bundle F of type G/P = F (2 , , V ) . Herefter we present a GLSM describ-ing the zero loci X and X as critical loci of a superpotential w related by a phase transition.Such physical phenomenon is described by means of a variation of GIT (VGIT). We will mainly ALABI–YAU FIBRATIONS, SIMPLE K -EQUIVALENCE AND MUTATIONS 27 focus our attention to the Calabi–Yau pair of Section 3.3, therefore we fix B = P and conse-quently F g = F (1 , , , . Further we will describe how the VGIT construction can be ex-tended. Namely, we can generalize the picture in the following directions: ◦ Replace P with a general smooth projective B , not necessarily homogeneous ◦ Substitute A with a bigger special linear algebra A k All these constructions yield a VGIT, but we are mainly interested in Calabi–Yau zero loci em-bedded in homogeneous varieties, therefore the case of the family of A M -roofs over P willoccupy a central place in the discussion below.6.1. Notation.
The geometry for B = P has been established in Section 3.2. Let us considerthe following GIT description of F (1 , , V ) :(6.1) F (1 , , V ) ≃ Hom( C , V ) \ ZG Here Z is the locus of rank smaller than four and(6.2) G = (cid:26)(cid:18) λ × h (cid:19)(cid:27) ⊂ GL (4) , λ ∈ C ∗ , h ∈ GL (3) . where × denotes the entries corresponding to a nilpotent subgroup, on which we have no con-ditions.The G -action defines an equivalence relation C ∼ Cg − which we use to take the quotient.Given a three dimensional vector space V , we can describe P (1 , as a G -equivariant vectorbundle over F (1 , , V ) in the following way:(6.3) P (1 ,
2) =
Hom( C ,V ) \ Z ⊕ V G F (1 , , V ) where the equivalence relation on Hom( C , V ) \ Z ⊕ V is ( C, x ) ∼ ( Cg − , λ − det h − hx ) . Infact, since O (1 ,
0) = t ∗ U ∨ and O (0 ,
1) = u ∗ det U ∨ , the weight of O (0 , under its associatedone dimensional representation is det g − = λ − det h − .A section s of such bundle is defined by an equivariant map ˆ s : Hom( C , V ) −→ C fulfill-ing the equivariancy condition s ([ C ]) = [ C, ˆ s ( C )] . Therefore it must satisfy(6.4) ˆ s ( Cg − ) = λ − det g − h ˆ s ( C ) . We can characterize this section by its image under the dashed arrow below:(6.5) V ⊗ O (1 , P (1 , t ∗ Q (1 , ff − ◦ ii In order to do that, let us rename v the first column of C and call B the rest of the matrix. Weuse the notation ( v | B ) for juxtaposition. Then, observe that the function ( v | B ) −→ B ˆ s (( v | B )) transforms like the fiber of V ⊗ O (1 , under the G -action. Moreover, since its image lies in the image of B , by the maximal rank condition on ( v | B ) it must lie in V / Span ( v ) , which is thefiber of t ∗ Q over v , where we identify v with t ( v, B ) ∈ G (1 , V ) . Note that, fixing v , we recoverexactly the description of the section of U G (3 ,V ) (2) of [KR17].6.2. The superpotential.
Let us call V the vector space(6.6) V = Hom( C , V ) ⊕ Hom( C , V ) ⊕ V ∨ endowed with the following G -action:(6.7) G × V Vg, ( v, B, x ) ( vλ − , Bh − , λ det h xh − ) where g decomposes as in Equation 6.3. Given a smooth section s ∈ H ( F, , , V ) , P (1 , weconstruct a G -invariant function called superpotential :(6.8) V C ( v, B, x ) x · ˆ s ( v, B ) w where the dot is the usual contraction V ∨ × V −→ C .We define a family of characters(6.9) ρ τ : G −→ C ∗ g λ − τ det h − τ and we consider the associated variation of GIT related to the chambers τ > and τ < . Moreprecisely, fixed one of the two chambers, we investigate the locus Z ± ∈ V of triples ( v, B, x ) such that there exists a sequence { g n } ⊂ G with ρ − ± ( g n ) −→ ∞ and { g n ( v, B, x ) } converges.Then, the corresponding semistable locus is V ss ± = V \ Z ± .Let us fix a sequence of diagonal elements in G depending on four parameters k , . . . , k whoseelements are(6.10) g n = n k n k n k n k The chamber τ > . Here the condition ρ − ( g n ) −→ ∞ translates to P i k i < . Then ( v, B, x ) ∈ Z + if and only if (up to change of basis) there exist a quadruple k , . . . k satisfyinga set of inequalities:(6.11) P i k i < − a i ≤ k + k + 2 k + 2 k ≤ k + 2 k + k + 2 k ≤ k + 2 k + 2 k + k ≤ ALABI–YAU FIBRATIONS, SIMPLE K -EQUIVALENCE AND MUTATIONS 29 Solving these inequalities provides the following information:(6.12) V ss + = { ( v, B, x ) ∈ V | rk v = 1 , rk B = 3 } . Therefore, since(6.13)
V // + G = V ss + /G = P ∨ ( − , − . we conclude that Crit( w ) // + G ≃ X .6.2.2. The chamber τ < . Here the condition ρ − − ( g n ) −→ ∞ gives the inequality to P i k i > .The other inequalities are unchanged, but the solution is radically different:(6.14) V ss − = { ( v, B, x ) ∈ V | rk v = 1 , rk x = 1 , ker B ∩ ker x = 0 } . Acting with G we can reduce to the situation where x = (1 , , . Then the stabilizer has theform(6.15) G S = g ∈ G : g = λ z z z δ z m m z m m We observe that the action of the stabilizer on B preserves linear combinations of the secondand third columns, while the first one transforms like the image of the fiber of t ∗ Q ( − , − .Hence, the GIT quotient is(6.16) V // − G = V ss + /G = r ∗ Q ∨ ( − , − . The phase transition.
In order to prove that the critical locus in the second phase is iso-morphic to X , we need to describe the section s more explicitly. First let us describe S ∈ H ( F (1 , , , V ) , O (1 , , . In analogy with Equation 6.1, the flag variety F (1 , , , V ) is givenby the following GIT description:(6.17) F (1 , , , V ) ≃ Hom( C , V ) \ ZH where(6.18) H = λ × × h × δ ⊂ GL (4) , λ, δ ∈ C ∗ , h ∈ GL (2) . and the action is C ≃ Cg − for every g ∈ H . Let us write C = ( v | A | u ) ∈ Hom( C , V ) where v, u ∈ Hom( C , V ) and A ∈ Hom( C , V ) . Then, a section of O (1 , , acts in the followingway:(6.19) ( v | A | u ) −−−−−→ S (( v | A | u )) = S ijklmnpq v i ψ jkl ( v | A ) ψ mnpq ( v | A | u ) where ψ k ,...k r is the totally skew-symmetric tensor of minors obtained choosing the lines k , . . . k r ,hence it defines a Pl ¨ucker embedding. To unclutter the notation, we used Einstein’s summationconvention, which omits sums over repeated high and low indices. We observe that(6.20) S ( g. ( v | A | u )) = λ − det h − δ − S (( v | A | u )) which is the correct equivariancy condition since O (1 , , ≃ O (1) ⊠ O (1) ⊠ O (1) . Then, thepushforwards of this section to F (1 , , V ) and F (1 , , V ) are defined by the following equivari-ant functions:(6.21) ( v | A ) −−−−−→ ˆ σ r (( v | A | u )) = S ijklmnpq v i ψ jkl ( v | A ) δ r [ q ψ mnp ] ( v | A ) (6.22) ( v | B ) −−−−−→ ˆ s r (( v | B )) = S ijklmnpq v i ∂∂B rt [ ψ jklt ( v | B )] ψ mnpq ( v | B ) where square brackets around a set of indices means totally skew-symmetric. What is left toprove is that the quotient of ther critical locus of w restricted to V ss − by G is isomorphic to X .Let us write the superpotential explicitly: by Equations 6.8 and 6.22 we have(6.23) ( v, B, x ) −−−−−→ x r S ijklmnpq v i ∂∂B rt [ ψ jklt ( v | B )] ψ mnpq ( v | B ) As we showed before, for every G S -orbit in V ss − there exist a unique point such that x = x :=(1 , , . Let us work on such points. Define:(6.24) e V = { ( v, B ) : rk v = 1 , B r = 0 ∀ r ≤ } . We are interested in the locus(6.25) dw ∩ e V = { ( v, B, x ) : x = x , ( v, B ) ∈ e V , ˆ s ( v, B, x ) = 0 , x · ds ( v, B, x ) = 0 } . If ( v, B ) ∈ e V the first equation is automatically satisfied, since ψ ( v | B ) is identically zero forlower rank matrices, and the first column of B is zero. Let us now focus on the second equationdefining the critical locus. By Equation 6.23, restricted to ( e V , x ) it becomes (up to sign): x · ds ( v, B, x ) | ( v,B ) ∈ e V = S ijklmnpq v i ∂∂B t [ ψ jklt ( v | B )] ∂∂B z [ ψ mnpq ( v | B )] (cid:12)(cid:12)(cid:12)(cid:12) ( v,B ) ∈ e V = S ijklmnpq v i ψ jkl ( v | e A ) δ z [ q ψ q ] mnp ( v | e A ) (6.26)where e A ∈ Hom( C , V ) is the matrix resulting by removing the first (vanishing) column from B . This last equation coincides with 6.21, hence it describes a section of r ∗ Q ∨ (1 , on F (1 , , V ) .Summing all up, the critical locus of w on V ss − is a bundle over the zero locus of the six equations x · dw The last step is to observe that the action of the stabilizer G S described by Equation 6.7 is tran-sitive and free on { x = ( x , x , x ) } . Hence, quotienting by G S , we obtain the Calabi–Yaueightfold X . Theorem 6.1.
There exist a pair of derived equivalent Calabi–Yau eightfolds X , X of Picard numbertwo, and fibrations f : X −→ P and f : X −→ P such that for every b ∈ B Y := f − ( b ) and Y := f − ( b ) are non birational, derived equivalent Calabi–Yau threefolds. Moreover, X and X areisomorphic to the critical loci of two phases of a non Abelian gauged linear sigma model.Proof. Let us consider the roof bundle of type A M over P . By the discussion of Section 3.3, X and X are Calabi–Yau eightfolds. In particular, by Lemma 3.1, they have Picard number two.Derived equivalence follows from Corollary 4.12. By the above, X and X are isomorphic tothe critical loci of w in the two stability chambers τ < and τ > . Finally, the fibers Y := ALABI–YAU FIBRATIONS, SIMPLE K -EQUIVALENCE AND MUTATIONS 31 f − ( b ) and Y := f − ( b ) are a Calabi–Yau pair of type A M , hence, for a general M , they are nonbirational and derived equivalent by [KR17]. (cid:3) GLSM fibrations over a smooth projective base.
If we substitute P with a smooth pro-jective (and not necessarily homogeneous) base B , we obtain a relative version of the gaugedlinear sigma model described in [KR17] over B .More precisely, the model can be described by the following data: ◦ A vector bundle V of rank with a GL(5)-action given by the fundamental representa-tion:(6.27) g, ( b, v ) ( b, gv ) ◦ A three dimensional vector space W with a GL(3)-action given by the fundamental rep-resentation. This defines the flag bundle as(6.28) F l (3 , V ) ≃ W ∨ ⊗ V \ Z /GL ( W ) where Z is the subbundle of smaller rank morphisms from W ⊗ O to V . ◦ A GL ( W ) -equivariant morphism of vector bundles ˆ s defined as(6.29) W ∨ ⊗ V W ⊗ ( ∧ W ∨ ) ⊗ ⊗ O B ˆ s ( B ) w where the equivariancy condition is explictily described as(6.30) ˆ s ( Bg − ) = g det g − ˆ s ( B ) ◦ A GL ( W ) -invariant function w called superpotential:(6.31) W ∨ ⊗ V ⊕ W ∨ ⊗ ( ∧ W ) ⊗ ⊗ O C AAAAAAAAAAAAAAAAAA ( b, B, x ) x · ˆ s ( B ) w ◦ A character ρ τ : GL ( W ) −→ C ∗ defined by(6.32) g det g − τ . Since both the superpotential w and the behaviour of the semistable loci for the two chambers τ > and τ < do not depend on the choice of b ∈ B , we conclude that for every b there ex-ists a gauged linear sigma model describing a phase transiton between two three dimensionalCalabi–Yau phases Y and Y .In particular, if B = F ( k , . . . , k r , V k +5 ) with k r ≤ k + 1 one can directly generalize the ex-plicit GLSM formulation over P . Gauged linear sigma model for g = A M k . All the models above can be extended to describepairs of Calabi–Yau fibrations associated to roofs of type A M k . In particular, one would getCalabi–Yau fibers of dimension k − which are sections of respectively Q ∨ (2) on G ( k, k + 1) and U (2) on G ( k + 1 , k + 1) . However, up to the author’s knowledge, derived equivalence ofthe fibers is still not known. References [ADS15] Nicolas Addington, Will Donovan, Ed Segal.
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