Residual categories for (co)adjoint Grassmannians in classical types
aa r X i v : . [ m a t h . AG ] M a r RESIDUAL CATEGORIES FOR (CO)ADJOINT GRASSMANNIANSIN CLASSICAL TYPES
ALEXANDER KUZNETSOV AND MAXIM SMIRNOV
Abstract.
In our previous paper we suggested a conjecture relating the structure of thesmall quantum cohomology ring of a smooth Fano variety of Picard number 1 to the struc-ture of its derived category of coherent sheaves. Here we generalize this conjecture, make itmore precise, and support by the examples of (co)adjoint homogeneous varieties of simplealgebraic groups of Dynkin types A n and D n , i.e., flag varieties Fl(1 , n ; n + 1) and isotropicorthogonal Grassmannians OG(2 , n ); in particular we construct on each of those an excep-tional collection invariant with respect to the entire automorphism group.For OG(2 , n ) this is the first exceptional collection proved to be full. Introduction
This paper is devoted to the study of derived categories of coherent sheaves on homoge-neous varieties of semisimple algebraic groups and the relation to their quantum cohomology.Recall that Dubrovin’s conjecture (see [7]) predicts, that the existence of a full exceptionalcollection in the bounded derived category D b ( X ) of coherent sheaves on a smooth projec-tive variety X is equivalent to the generic semisimplicity of its big quantum cohomologyring BQH( X ) (for background on quantum cohomology we refer to [9, 20]). The big quan-tum cohomology ring is usually very hard to compute, in contrast to the small quantumcohomology QH( X ). Of course, if QH( X ) is generically semisimple, then so is BQH( X ).Thus, from Dubrovin’s conjecture we conclude that generic semisimplicity of QH( X ) shouldimply the existence of a full exceptional collection in D b ( X ). On the other hand, it wasobserved that the opposite implication is not true, see [6, 10, 13, 23].This observation suggests that on the one hand, some mildly non-simple factors of thegeneral fiber of QH( X ) should not obstruct the existence of a full exceptional collectionin D b ( X ), and on the other hand, generic semisimplicity of QH( X ) should have strongerimplications for D b ( X ) than just the existence of a full exceptional collection.In [19] we stated for varieties of Picard number 1 a conjecture saying that the structureof the general fiber of QH( X ) determines the structure of the exceptional collection on X .In this paper we suggest a more general and precise version of this conjecture and supportit by new examples.To state the conjecture we will need some notation. First, recall that the index of a smoothprojective variety X is the maximal integer m such that K X is divisible by m in Pic( X ); weusually assume that X is a Fano variety over an algebraically closed field of characteristiczero and write ω X ∼ = O X ( − m ) , where O X (1) is a primitive ample line bundle on X . A.K. was supported by the Russian Science Foundation under grant 19-11-00164. e will say that an exceptional collection E , . . . , E k in D b ( X ) extends to a rectangularLefschetz collection , if the collection E , E , . . . , E k ; E (1) , E (1) , . . . , E k (1); . . . ; E ( m − , E ( m − , . . . , E k ( m −
1) (1.1)is also exceptional. The orthogonal complement R = D E , . . . , E k ; . . . ; E ( m − , . . . , E k ( m − E ⊥ ⊂ D b ( X ) (1.2)is called the residual category of the above collection. In [19, Theorem 2.8] we checked that R is endowed with an autoequivalence τ R : R → R (called the induced polarization of R ) suchthat τ m R ∼ = S − R [dim X ] , where S R is the Serre functor of R . Thus, τ R plays in R the same role as the twist by O (1)plays in D b ( X ).Assume X is a smooth Fano variety and let r be the Picard rank of X , so that QH( X )is an algebra over the ring Q [ q , . . . , q r ] of functions on the affine space Pic( X ) ⊗ Q over Q .Let QH can ( X ) := QH( X ) ⊗ Q [ q ,...,q r ] C be the base change of QH( X ) to the point of Spec( Q [ q , . . . , q r ]) corresponding to the canon-ical class of X ; this is a C -algebra, whose underlying vector space is canonically isomorphicto H • ( X, C ). Assume that H odd ( X, C ) = 0 and let m be the index of X . Then QH can ( X ) iscommutative, so we consider the finite schemeQS X := Spec(QH can ( X ))and call it the (canonical) quantum spectrum of X . The natural grading of H • ( X, C ) inducesa Z /m -grading of QH can ( X ) such thatdeg(H i ( X, Q )) ≡ i (mod m ) , which gives rise to an action of the group µ m on the quantum spectrum QS X .Furthermore, let − K X ∈ H ( X, C ) ⊂ QH can ( X )be the anticanonical class. It defines a morphism of algebras C [ κ ] → QH can ( X ), κ
7→ − K X from a polynomial algebra in the variable κ , which geometrically can be understood as amorphism of schemes κ : QS X → A , (1.3)which is µ m -equivariant for the action on QS X defined above and the standard action on A . Remark . The notion of quantum spectrum is parallel to that of spectral cover in thetheory of Frobenius manifolds (see [11, 20]). The map (1.3) is analogous to the restriction ofa Landau–Ginzburg potential to its critical locus.
Example 1.2.
Let X = P n , so that m = n + 1. ThenQH( X ) ∼ = Q [ h, q ] / ( h n +1 − q ) , hence QH can ( X ) ∼ = C [ h ] / ( h n +1 − . The quantum spectrum QS P n is the reduced subscheme of A with points ζ i , 0 ≤ i ≤ n ,where ζ is a primitive ( n +1)-st root of unity. The Z / ( n +1)-grading is defined by deg( h ) = 1,it induces the natural action of µ n +1 on A under which QS P n is invariant. Finally, wehave − K X = ( n + 1) h , and the map κ up to rescaling is the natural inclusion QS P n ֒ → A . ur conjecture is based on an analogy between the µ m -action on QS X and the twistby O X (1) on Lefschetz exceptional collections in D b ( X ): in this analogy the parts of QS X supported over the complement of the origin and the origin of A (with respect to the map κ )QS × X := κ − ( A \ { } ) , QS ◦ X := QS X \ QS × X . correspond to the rectangular part and the residual category of the Lefschetz collection.Note that the µ m -action on QS × X is free (because it is free on A \ { } ), hence there exists afinite subscheme Z ⊂ QS × X such that the action map µ m × Z → QS × X is an isomorphism; thelength of Z is equal to the length of QS × X divided by m , and we consider it as an analogueof the subcollection E , . . . , E k in (1.1). Conjecture 1.3.
Let X be a Fano variety of index m over an algebraically closed fieldof characteristic zero with H odd ( X, C ) = 0 and assume that the big quantum cohomol-ogy BQH( X ) is generically semisimple. (1) There is an
Aut( X ) -invariant exceptional collection E , . . . , E k in D b ( X ) , where k is the length of QS × X divided by m ; this collection extends to a rectangular Lefschetzcollection (1.1) in D b ( X ) . (2) The residual category R of this collection ( defined by (1.2)) has a completely orthog-onal Aut( X ) -invariant decomposition R = M ξ ∈ QS ◦ X R ξ with components indexed by closed points ξ ∈ QS ◦ X ; moreover, the component R ξ of R is generated by an exceptional collection of length equal to the length of thelocalization (QS ◦ X ) ξ at ξ . (3) The induced polarization τ R permutes the components R ξ ; more precisely, for eachpoint ξ ∈ QS ◦ X it induces an equivalence τ R : R ξ ∼ −−→ R g ( ξ ) , where g is a generator of µ m . Note that any exceptional object on X is invariant with respect to any connected reductivegroup acting on X (see [24, Lemma 2.2]), however, the group Aut( X ) is not connected ingeneral. Thus, Aut( X )-invariance of the collection is an extra constraint (cf. the discussionsin § § X )-invariance statements, andfor the action of τ R on the residual category).If the Picard rank of X is 1, so that QH( X ) is a Q [ q ]-algebra, we haveQH( X ) ⊗ Q [ q ] Q ( q ) ∼ = (QH( X ) ⊗ Q [ q ] Q ) ⊗ Q Q ( q )because QH( X ) is graded with deg( q ) = m , hence semisimplicity of QH can ( X ) is equiva-lent to generic semisimplicity of QH( X ). Moreover, if QH( X ) is generically semisimple thelength of each localization (QS ◦ X ) ξ is 1, thus Conjecture 1.3 predicts the existence in D b ( X )of a rectangular Lefschetz collection whose residual category is generated by a completely or-thogonal exceptional collection, which is equivalent to the prediction of [19, Conjecture 1.12]. n particular, numerous examples listed in the introduction to [19] together with the mainresult of [19] support both conjectures.Let us also discuss a couple of simple examples of varieties with higher Picard rank. Example 1.4.
Let X = P n × P n , so that m = n + 1. By the quantum K¨unneth formula(see [12, 15]) we haveQH( X ) ∼ = QH( P n ) ⊗ Q QH( P n ) ∼ = Q [ h , h , q , q ] / ( h n +11 − q , h n +12 − q ) . The canonical class direction corresponds to q = q , so the canonical quantum cohomologyring can be computed by specializing both q and q to 1:QH can ( X ) ∼ = C [ h , h ] / ( h n +11 − , h n +12 − . Its spectrum QS P n × P n is a reduced scheme of length ( n + 1) with points ( ζ i , ζ j ), 0 ≤ i, j ≤ n ,where ζ is the primitive ( n + 1)-st root of unity. The function κ (up to rescaling) is givenby κ = h + h , so κ ( ζ i , ζ j ) = ζ i + ζ j andQS ◦ P n × P n = ( ∅ , if n = 2 k is even, { ( ζ i , ζ k +1+ i ) } , if n = 2 k + 1 is odd. (1.4)Furthermore, the generator of µ n +1 acts by ( ζ i , ζ j ) ( ζ i +1 , ζ j +1 ); in particular the actionof µ n +1 on QS ◦ P n × P n is transitive.The formula (1.4) exhibits a difference between the case of even and odd n ; it also appearson the level of derived category. If n = 2 k the collection of 2 k + 1 line bundles A = h O , O (1 , , O (0 , , . . . , O ( k, , O (0 , k ) i (1.5)extends to an Aut( X )-invariant rectangular Lefschetz collection D b ( X ) = h A , . . . , A ( n ) i oftotal length ( n + 1)(2 k + 1) = ( n + 1) whose residual category is zero (this can be easilyproved by the argument of Lemma 2.2).If n = 2 k +1 the collection (1.5) still extends to an Aut( X )-invariant rectangular Lefschetzcollection in D b ( X ) of length ( n + 1) n , and it can be checked (a similar computation ina more complicated situation can be found in §
2) that its residual category is generatedby n + 1 = 2 k + 2 completely orthogonal exceptional vector bundles F = O ( − , k ) ,F = τ R ( F ) ∼ = O ⊠ Ω k +1 ( k + 1) ,F = τ R ( F ) ∼ = O (1) ⊠ Ω k +2 ( k + 2) , ... F k +2 = τ R ( F k +1 ) ∼ = O ( k ) ⊠ Ω k +1 (2 k + 1) ∼ = O ( k, − ,F k +3 = τ R ( F k +2 ) ∼ = Ω k +1 ( k + 1) ⊠ O ,F k +4 = τ R ( F k +3 ) ∼ = Ω k +2 ( k + 2) ⊠ O (1) , ... F k +2 = τ R ( F k +1 ) ∼ = Ω k (2 k ) ⊠ O ( k − τ R on this exceptional collectionis transitive, analogously to the action of µ n +1 on QS ◦ P n × P n . xample 1.5. Let X = ( P ) n , so that m = 2. Applying again the quantum K¨unnethformula one can check that QS ( P ) n = { ( ± , ± , . . . , ± } ⊂ A n is a reduced scheme of length 2 n and the function κ is given by the sum of coordinates.Therefore, QS ◦ ( P ) n is empty when n is odd, while for even n = 2 k it contains exactly (cid:0) kk (cid:1) points, and the µ -action splits this set into (cid:0) kk (cid:1) free orbits.On the level of derived categories the same thing happens. If n is odd, a rectangu-lar Aut(( P ) n )-invariant Lefschetz collection in D b (( P ) n ) with zero residual category wasconstructed in [21, Theorem 4.1]. If n = 2 k is even, using [21, Theorem 4.1] it is easy to showthat the residual category is generated by (cid:0) kk (cid:1) exceptional line bundles and the τ R -actionswaps them (up to shift) pairwise.Using other results from [21] one can verify Conjecture 1.3 for some other products ( P n ) k .In all these examples, however, the ring QH can ( X ) is semisimple. Below we discuss moreintricate examples with non-semisimple ring QH can ( X ), provided by homogeneous varietiesof simple algebraic groups, where quite a lot is known both about quantum cohomology andderived categories.Perhaps, the most interesting case here is that of adjoint and coadjoint homogeneousvarieties. Recall that an adjoint (resp. coadjoint ) homogeneous variety of a simple algebraicgroup G is the highest weight vector orbit in the projectivization of the irreducible G-representation, whose highest weight is the highest long (resp. short ) root of G; in particular,if the group G is simply laced, the adjoint and coadjoint varieties coincide.For classical Dynkin types adjoint and coadjoint varieties are:Dynkin type group G adjoint variety coadjoint varietyA n SL( n + 1) Fl(1 , n ; n + 1) Fl(1 , n ; n + 1)B n Spin(2 n + 1) OG(2 , n + 1) Q n − C n Sp(2 n ) P n − IG(2 , n )D n Spin(2 n ) OG(2 , n ) OG(2 , n )Here Fl(1 , n ; n + 1) is the partial flag variety, Q k is a (smooth) k -dimensional quadric,while IG(2 , n ) and OG(2 , n ) are the symplectic and orthogonal isotropic Grassmanniansof 2-dimensional subspaces, respectively. Note that the Picard rank of a (co)adjoint varietyis 1, except for the A n -case, where it is 2.The small quantum cohomology ring of (co)adjoint varieties was computed in [3–5, 14] interms of generators and relations. Using these results, the fiber QS ◦ X of the map κ definedin (1.3) was computed by Nicolas Perrin and the second named author in [22]. To state theresults of [22] we will need some notation.Let T(G) be the Dynkin diagram of G, and let T short (G) be the subdiagram of T(G)with vertices corresponding to short roots. For reader’s convenience we collect the resultingDynkin types in a table: T A n B n C n D n E n F G T short A n A A n − D n E n A A The following theorem describes QS ◦ X for adjoint and coadjoint varieties. heorem 1.6 ([22]) . Let X ad and X coad be the adjoint and coadjoint varieties of a simplealgebraic group G , respectively. (1) If T(G) = A n , then QS ◦ X ad = QS ◦ X coad = ∅ . (2) If T(G) = A n , then QS ◦ X coad is a single non-reduced point and the localizationof QH can ( X ) at this point is isomorphic to the Jacobian ring of a simple hypersurfacesingularity of type T short (G) . (3) If T(G) is simply laced, then X ad = X coad , hence QS ◦ X ad = QS ◦ X coad . (4) If T(G) is not simply laced, then QS ◦ X ad = ∅ . A combination of Theorem 1.6 with Conjecture 1.3 allows to make predictions about thestructure of derived categories of adjoint and coadjoint varieties. Our expectation is statedin the following two conjectures.
Conjecture 1.7.
Let X be the adjoint variety of a simple algebraic group G over an alge-braically closed field of characteristic zero. If T(G) is not simply laced then D b ( X ) has afull Aut( X ) -invariant rectangular Lefschetz exceptional collection. Conjecture 1.8.
Let X be the coadjoint variety of a simple algebraic group G over analgebraically closed field of characteristic zero. Then D b ( X ) has an Aut( X ) -invariant rect-angular Lefschetz exceptional collection with residual category R and (1) if T(G) = A n and n is even, then R = 0 ; (2) otherwise, R is equivalent to the derived category of representations of a quiver ofDynkin type T short (G) . In non-simply laced Dynkin types B n , C n , and G these expectations agree with knownresults about D b ( X ). Indeed, for adjoint varieties full rectangular Lefschetz decomposi-tions were constructed in [17, Theorem 7.1] for type B n , [19, Example 1.4] for type C n ,and [16, § . For coadjoint varieties the residual categories were computedin [6, Theorem 9.6] for type C n and in [19, Example 1.6] for types B n and G . We plan toreturn to the remaining non-simply laced type F in the future (we expect that the restric-tion of the exceptional collection from [8] should give the required result for the coadjointvariety).The main result of this paper is the proof of Conjecture 1.8 for Dynkin types A n and D n .Since these Dynkin types are simply laced, the coadjoint and adjoint varieties coincide. Theorem 1.9.
Let X be the ( co ) adjoint variety of a simple algebraic group G of Dynkintype A n or D n over an algebraically closed field of characteristic zero. Then D b ( X ) hasan Aut( X ) -invariant rectangular Lefschetz exceptional collection with residual category R and (1) if T(G) = A n and n is even, then R = 0 ; (2) if T(G) = A n and n is odd, then R ∼ = D b (A n ) ; (3) if T(G) = D n , then R ∼ = D b (D n ) ;where D b (A n ) and D b (D n ) are the derived categories of representations of quivers of Dynkintypes A n and D n , respectively. More precise versions of these results can be found in Theorem 2.1 (for type A n ) andTheorem 3.1 (for type D n ) in the body of the paper. We leave the remaining exceptionaltypes E , E , E for future work. ote that a part of the statement of this theorem is a construction of a full exceptionalcollection in D b (OG(2 , n )), which was not known before (see [18] for a survey of resultsabout exceptional collections on homogeneous varieties). Remark . As one can see from Theorem 1.6 and Conjecture 1.8, the case of Dynkintype A n with even n is somewhat special. In this case, the Picard rank is equal to 2,and the canonical quantum cohomology ring QH can ( X ) is semisimple, so a singularity oftype T short (G) = A n does not show up. However, one can see this singularity in the skew-canonical quantum cohomology ring, i.e., the ring obtained from QH( X ) by base change tothe point of Pic( X ) ⊗ Q corresponding to the line bundle O (1 , − Acknowledgements.
We are indebted to Nicolas Perrin for sharing with us his results ofquantum cohomology computations and attracting our attention to the coadjoint varietiesof types A n and D n , that eventually led to this paper. We thank Giordano Cotti, AntonFonarev, Sergey Galkin, and Anton Mellit for useful discussions. Further, we are very gratefulto Pieter Belmans for his kind permission to reuse some parts of the code written for [1],which was instrumental for this paper, and for his comments on the first draft of this paper.Finally, M.S. would like to thank ICTP in Trieste, and MPIM in Bonn, where a part of thiswork was accomplished, for their hospitality.2. Type A n In this section we prove parts (1) and (2) of Theorem 1.9, restated in a more precise formin Theorem 2.1 below. In this section we work here over an arbitrary field k .2.1. Statement of the theorem.
Let V be a vector space of dimension n + 1. Throughoutthis section we put X = Fl(1 , n ; n + 1) = Fl(1 , n ; V ) ⊂ P ( V ) × P ( V ∨ );note that in this embedding X is a hypersurface of bidegree (1 , X is the semidirect productAut( X ) ∼ = PGL( V ) ⋊ Z / , where the factor Z / n ) that is induced by the morphism σ B : P ( V ) × P ( V ∨ ) B × B − −−−−−→ P ( V ∨ ) × P ( V )given by a choice of non-degenerate bilinear form B on V .It is elementary to construct a rectangular Lefschetz decomposition for D b ( X ) by usingthe P n − -fibration structure X → P ( V ) of X (see the proofs of Lemma 2.2 and Lemma 2.3).This Lefschetz decomposition is automatically PGL( V )-invariant and its residual categoryis trivial. However, it is not Aut( X )-invariant, since the outer automorphism takes it to adecomposition associated with the other P n − -fibration X → P ( V ∨ ), and so does not agreewith the original one.In this section we construct in D b ( X ) a rectangular Aut( X )-invariant Lefschetz collectionand compute its residual category. To state the result we need some notation. Let0 ֒ → U ֒ → U n ֒ → V ⊗ O e the tautological flag of rank-1 and rank- n subbundles in the trivial vector bundle. We set E := U n / U (2.1)for the intermediate quotient. Note that σ ∗ B E ∼ = E ∨ and det( E ) ∼ = O (1 , − O ( a , a ) the restriction to X of the line bundle O ( a ) ⊠ O ( a ) on P ( V ) × P ( V ∨ ).We prove the following Theorem 2.1.
Set k := ⌊ n/ ⌋ . The collection of k + 1 line bundles A := (cid:10) O (0 , , O (1 , , O (0 , , O (2 , , O (0 , , . . . , O ( k, , O (0 , k ) (cid:11) (2.2) in D b ( X ) is exceptional and extends to an Aut( X ) -invariant semiorthogonal decomposition D b ( X ) = h R , A , A ⊗ O (1 , , . . . , A ⊗ O ( n − , n − i (2.3) where R is the residual category. (1) If n = 2 k the residual category is zero. (2) If n = 2 k + 1 the residual category is generated by the Aut( X ) -invariant exceptionalcollection R = h O ( − , k ) , O ( k, − E ( − , k − , E ∨ ( k − , − . . . ; Λ k − E ( − , , Λ k − E ∨ (1 , − k E ( − , ∼ = Λ k E ∨ (0 , − i (2.4) of length n = 2 k + 1 and is equivalent to the derived category of the Dynkin quiver A n . See Remark 2.8 for the description of A and R in terms of weights of SL( n + 1).We construct the rectangular part of (2.3) in Lemma 2.2 (for even n ) and Lemma 2.5(for odd n ); in the construction we use the advantage of already knowing a full exceptionalcollection (thanks to the P n − -fibration mentioned above), so it is enough to rearrange itappropriately by a sequence of mutations. Part (1) of the theorem is proved in Lemma 2.2and part (2) in Lemma 2.7.We denote the natural projections of X by p : X → P ( V ) and p : X → P ( V ∨ ) . For any pair of coherent sheaves on P ( V ) and P ( V ∨ ) we set F ⊠ X F := p ∗ F ⊗ p ∗ F ∼ = ( F ⊠ F ) | X . Even n . In this section we prove Theorem 2.1 for even n . Lemma 2.2. If n = 2 k then D b ( X ) = h A , A ⊗ O (1 , , . . . , A ⊗ O (2 k − , k − i , where A is defined by (2.2) .Proof. The P k − -fibration p gives rise to the semiorthogonal decomposition D b ( X ) = (cid:10) p ∗ ( D b ( P ( V ))) , p ∗ ( D b ( P ( V ))) ⊗ O (0 , , . . . , p ∗ ( D b ( P ( V ))) ⊗ O (0 , k − (cid:11) . Choosing the exceptional collection D b ( P ( V )) = h O ( i − k ) , O ( i − k + 1) , . . . , O ( i + k + 1) i , n the i -th component, we obtain a full exceptional collection in D b ( X ) that takes the form D b ( X ) = D O ( − k, , O (1 − k, , . . . , O ( k, , O (1 − k, , O (2 − k, , . . . , O ( k + 1 , ,. . . O ( k − , k − , O ( k, k − , . . . , O (3 k − , k − E . (2.5)The collection is shown in Picture 1, the objects are represented by black dots. Picture 1.
Mutation of (2.5) to a rectangular Lefschetz collection for k = 3.Now we perform a mutation: we consider the subcollection formed by the first k termsof the first line, the first k − k -th line of (2.5): { O ( − k, , O (1 − k, , . . . , O ( − , O (1 − k, , . . . , O ( − , O ( − , k − } ; (2.6)(this subcollection is depicted by the triangle in the left part of Picture 1) and mutate itto the far right of the exceptional collection. It is easy to see (cf. Lemma 2.4 below) thatthe objects in (2.6) are right-orthogonal to the objects in the rest of (2.5). Therefore, theirmutation to the far right is realized by the anticanonical twist. As ω − X ∼ = O (2 k, k ), thisreplaces the black dots in the triangle by the white dots in the dashed triangle in Picture 1.It remains to note that the resulting full exceptional collection gives the required rectangularLefschetz collection; indeed, the blocks formed by the twists of the subcategory A correspondto the L-shaped figures on the picture. (cid:3) Odd n : rectangular part. From now on we set n = 2 k + 1. First, we use the trickof Lemma 2.2 to construct a full exceptional collection that includes the rectangular partof (2.3) as a subcollection. In this case this is slightly more complicated, so we split theconstruction in two steps. emma 2.3. The category D b ( X ) has the following full exceptional collection: D b ( X ) = D O ( − k, , O (1 − k, , . . . , O ( k + 1 , , O (1 − k, , O (2 − k, , . . . , O ( k + 2 , ,. . . O ( − , k − , O (0 , k − , . . . , O (2 k, k − , O ( − , k ) , O (0 , k ) , . . . , O (2 k, k ) , O (0 , k + 1) , O (1 , k + 1) , . . . , O (2 k + 1 , k + 1) ,. . . O ( k − , k ) , O ( k, k ) , . . . , O (3 k, k ) E . (2.7)A graphical representation for the collection in case k = 3 can be found in Picture 2.The objects are depicted by black and red dots form the collection (2.7); the rows of (2.7)correspond to rows in the picture (and the shifts of rows match up). The mutation ofLemma 2.5 will take the objects corresponding to the black dots in the left triangle to theobjects corresponding to white dots in the dashed triangle at the top. The L-shaped figurescorrespond to blocks of the rectangular Lefschetz collection (the first of them is (2.2)). Picture 2.
Mutation of (2.7) to a rectangular Lefschetz collection for k = 3. Proof.
The P k -fibration p gives rise to the semiorthogonal decomposition D b ( X ) = (cid:10) p ∗ ( D b ( P ( V ))) , p ∗ ( D b ( P ( V ))) ⊗ O (0 , , . . . , p ∗ ( D b ( P ( V ))) ⊗ O (0 , k ) (cid:11) . This time for the first k components (i.e., for 0 ≤ i ≤ k −
1) we choose the collection D b ( P ( V )) = h O ( i − k ) , O ( i − k + 1) , . . . , O ( i + k + 1) i , and for the last k + 1 components (i.e., for k ≤ i ≤ k ) we choose the collection D b ( P ( V )) = h O ( i − k − , O ( i − k ) , . . . , O ( i + k ) i . As the result, we obtain (2.7). (cid:3)
It is useful to understand the nontrivial Ext-spaces between the objects in this (and other)exceptional collections. These are characterized by the following emma 2.4. The cohomology H • ( X, O ( a, b )) is non-zero if and only if a, b ∈ ( −∞ , − k − ∪ [0 , + ∞ ) except for ( a, b ) = (0 , − k − and ( a, b ) = ( − k − , .Proof. Consider the standard exact sequence0 → O P ( V ) × P ( V ∨ ) ( a − , b − → O P ( V ) × P ( V ∨ ) ( a, b ) → O ( a, b ) → . Clearly, the cohomology H • ( P ( V ) × P ( V ∨ ) , O P ( V ) × P ( V ∨ ) ( a, b )) is non-zero if and only if a, b ∈ ( −∞ , − k − ∪ [0 , + ∞ )and H • ( P ( V ) × P ( V ∨ ) , O P ( V ) × P ( V ∨ ) ( a − , b − a, b ∈ ( −∞ , − k − ∪ [1 , + ∞ ) . The union of these two regions gives the set of all ( a, b ), where H • ( X, O ( a, b )) = 0. (cid:3) It follows from Lemma 2.4 that the only nontrivial Ext-spaces among the objects in Pic-ture 2 are those going in the upper-right direction and additionally, there is non-trivialExt-space Ext k ( O (2 k, k − , O ( − , k )) = k (2.8)from the rightmost red dot to the leftmost one.Now we consider the same subcollection (2.6) as in the proof of Lemma 2.2 (this subcol-lection is depicted by the triangle in the left part of Picture 2) and mutate it to the far rightof the exceptional collection. It follows from the description of Lemma 2.4 that the objectsin (2.6) are right-orthogonal to the objects in the rest of (2.7). Therefore, their mutation tothe far right is realized by the anticanonical twist. As ω − X ∼ = O (2 k + 1 , k + 1), we deducethe following Lemma 2.5.
The category D b ( X ) has the following full exceptional collection: D b ( X ) = D O (0 , , O (1 , , . . . , O ( k + 1 , , O (0 , , O (1 , , . . . , O ( k + 2 , ,. . . O (0 , k − , O (1 , k − , . . . , O (2 k, k − , O ( − , k ) , O (0 , k ) , . . . , O (2 k, k ) , O (0 , k + 1) , O (1 , k + 1) , . . . , O (2 k + 1 , k + 1) ,. . . O ( k − , k ) , O ( k, k ) , . . . , O (3 k, k ) , O ( k + 1 , k + 1) , O ( k + 2 , k + 1) , . . . , O (2 k, k + 1) , O ( k + 2 , k + 2) , . . . , O (2 k, k + 2) .. . . O (2 k, k ) E . (2.9) he resulting collection in the case k = 3 is shown in Picture 2: the objects correspondingto the black dots in the lower-left triangle are replaced by the white dots in the upper-righttriangle; the rows of (2.9) correspond to the rows in the picture (and the shifts of rowsmatch).2.4. Odd n : residual category. We note that the resulting exceptional collection (2.9) isalready quite close to what we need. For instance, it already contains the rectangular partof (2.3). Indeed, the blocks A ⊗ O ( i, i ) of (2.3) are generated by the objects correspondingto the dots in Picture 2 joined into L-shaped figures.However, the objects which are not in the rectangular part O ( k + 1 , , O ( k + 2 , , . . . , O (2 k, k − , O ( − , k ) , O (0 , k + 1) , O (1 , k + 2) , . . . , O ( k − , k )(they are marked with red dots on the picture), are not yet right-orthogonal to the rectangularpart (hence are not yet contained in the residual category). The next step is to mutate themaccordingly. Proposition 2.6. If R ⊂ D b ( X ) is the residual category of the rectangular collection definedby (2.3) , then R is generated by the following exceptional collection R = h Ω k +1 ( k + 1) ⊠ X O , Ω k +2 ( k + 2) ⊠ X O (1) , . . . , Ω k (2 k ) ⊠ X O ( k − , O ( − , k ) , O ⊠ X Ω k +1 ( k + 1) , O (1) ⊠ X Ω k +2 ( k + 2) , . . . , O ( k − ⊠ X Ω k (2 k ) i . (2.10) The only non-trivial
Ext -spaces between the objects of this exceptional collection are
Hom(Ω k + i − ( k + i − ⊠ X O ( i − , Ω k + i ( k + i ) ⊠ X O ( i − k , ≤ i ≤ k Hom(Ω k (2 k ) ⊠ X O ( k − , O ( − , k )) = k , Ext k ( O ( − , k ) , O ⊠ X Ω k +1 ( k + 1)) = k , Hom( O ( i − ⊠ X Ω k + i − ( k + i − , O ( i − ⊠ X Ω k + i ( k + i )) = k , ≤ i ≤ k. (2.11) In particular, R is equivalent to the derived category of the Dynkin quiver A k +1 .Proof. It follows from the description of Lemma 2.4 that the object O ( − , k ) (correspondingto the circled red dot in Picture 2) is already in the residual category, so it is enough tomutate the other 2 k objects.Consider the exact sequences0 → Ω k + i ( k + i ) ⊠ X O ( i − → Λ k + i V ∨ ⊗ O (0 , i − →· · · → V ∨ ⊗ O ( k + i − , i − → O ( k + i, i − → X of the truncated Koszul complexon P ( V ), and analogous exact sequences0 → O ( i − ⊠ X Ω k + i ( k + i ) → Λ k + i V ⊗ O ( i − , →· · · → V ⊗ O ( i − , k + i − → O ( i − , k + i ) → . (2.13)If we show that the objects Ω k + i ( k + i ) ⊠ X O ( i −
1) and O ( i − ⊠ X Ω k + i ( k + i ) belong to R , itwill follow that these exact sequences express mutations of O ( k + i, i −
1) and O ( i − , k + i ) hrough the rectangular part of (2.3), and that together with the object O ( − , k ) theseobjects generate the residual category R .So, we need to check that Ext • ( O ( a, b ) , − ) = H • ( X, − ⊗ O ( − a, − b )) vanishes on theseobjects when O ( a, b ) run through the set of objects generating the rectangular part of (2.3),i.e., for 0 ≤ a ≤ b ≤ a + k ≤ k and 0 ≤ b ≤ a ≤ b + k ≤ k. (2.14)We have exact sequences on P ( V ) × P ( V ∨ )0 → Ω k + i ( k + i − ⊠ O ( i − → Ω k + i ( k + i ) ⊠ O ( i − → Ω k + i ( k + i ) ⊠ X O ( i − → . To compute H • ( X, Ω k + i ( k + i − a ) ⊠ X O ( i − − b )) we thus need to compute two tensorproducts H • ( P ( V ) , Ω k + i ( k + i − a )) ⊗ H • ( P ( V ∨ ) , O ( i − − b )) ,H • ( P ( V ) , Ω k + i ( k + i − − a )) ⊗ H • ( P ( V ∨ ) , O ( i − − b )) . (2.15)By Bott’s formula [2, Proposition 14.4] the first factors in these products are zero, exceptfor a ≤ − , or a = k + i, or a ≥ k + 2 ,a ≤ − , or a = k + i − , or a ≥ k + 1 (2.16)(the conditions in the first (resp. second) line of (2.16) corresponds to non-vanishing of thefirst factors in the first (resp. second) line in (2.15)). Similarly, the second factors vanishexcept for b ≤ i − , or b ≥ k + i + 1 ,b ≤ i − , or b ≥ k + i (2.17)(with the same convention about the role of the lines). Clearly, (2.14) is not compatible withthe conditions a ≤ − a ≤ − | a − b | ≤ k ,which contradicts all conditions in (2.16), except for those in the last columns. However, inthe latter case both a and b are strictly bigger than 2 k (recall that 1 ≤ i ≤ k ), which alsocontradicts (2.14).Thus, we conclude that for all ( a, b ) satisfying (2.14) either of the factors in the tensorproducts (2.15) vanishes, hence the objects Ω k + i ( k + i ) ⊠ X O ( i −
1) belong to the residualcategory R . By symmetry it also follows that the objects O ( i − ⊠ X Ω k + i ( k + i ) belong to R .Thus, all the objects in the right side of (2.10) are in R . Moreover, as we already pointedout, it also follows that (2.12) and (2.13) are mutation sequences. Therefore, together withthe line bundle O ( − , k ) they generate R .Similarly, from Lemma 2.4 and the mutation sequences, semiorthogonality between therows of (2.10) follows, so it remains to compute Ext-spaces between objects in each row aswell as from objects in the first row to O ( − , k ), and from O ( − , k ) to objects in the last row.For this we use (2.12) and (2.13) as resolutions for the source objects. Since almost allterms in these sequences are in the rectangular part of (2.3), it follows that it is enough tocompute Ext-spaces from the rightmost objects only (i.e., for the red dots from Picture 2).Thus, we need to describe the tensor products in (2.15) for ( a, b ) satisfying0 ≤ a ≤ k − , b = a + k + 1 or 0 ≤ b ≤ k − , a = b + k + 1 . learly, the assumption 0 ≤ a ≤ k − ≤ b ≤ k − a = b + k + 1, then a ≤ k , so the only compatible conditions are( a, b ) = ( k + i, i −
1) or ( a, b ) = ( k + i − , i − . Furthermore, in the first of these cases the first tensor product in (2.15) is 1-dimensional (andlives in the cohomological degree k + i ), and in the second of these cases the second tensorproduct is 1-dimensional (and lives in the same cohomological degree). This computationproves that the only Ext-spaces between the objects in the first and the last rows of (2.10)are those listed in the first and last lines of (2.11).Next, we compute Ext-spaces from objects in the first row of (2.10) to O ( − , k ). As beforewe use (2.12) as resolutions. The only object that appears in (2.12) which is not semiorthog-onal to O ( − , k ) is O (2 k, k −
1) (see Lemma 2.4), from which we have a 1-dimensional Ext-space in degree 2 k . Therefore, the only Ext-space from objects of the first row to O ( − , k )is given by the second line of (2.11).Finally, we compute Ext-spaces from O ( − , k ) to objects in the last row of (2.10). First,we consider the Koszul exact sequence0 → O ( − , k ) → Λ k +1 V ∨ ⊗ O (0 , k ) → · · · → V ∨ ⊗ O (2 k, k ) → O (2 k + 1 , k ) → . All its terms (except for the leftmost and rightmost) are in the rectangular part of (2.3),henceExt p ( O ( − , k ) , O ( i − ⊠ X Ω k + i ( k + i )) ∼ = Ext p +2 k +1 ( O (2 k + 1 , k ) , O ( i − ⊠ X Ω k + i ( k + i )) . Now we use (2.13) as a resolution for O ( i − ⊠ X Ω k + i ( k + i ). Clearly, the only nontrivialExt-space from O (2 k + 1 , k ) to its terms isExt k ( O (2 k + 1 , k ) , O (0 , k + 1)) ∼ = H k ( X, O ( − k − , ∼ = k (which holds for i = k + 1). Therefore, the only Ext-space from O ( − , k ) to the last rowof (2.3) is given by the third line of (2.11).It is clear from the above description of Ext-spaces that the category R is equivalent tothe derived category of the Dynkin quiver A k +1 (with the objects in (2.10) correspondingto simple representations of the quiver, up to shift). (cid:3) So far, the description of R we obtained does not look symmetric with respect to outerautomorphisms of X , and also looks different from the description in Theorem 2.1. In thenext statement we show that the two descriptions agree. Lemma 2.7.
The category R defined by (2.10) is generated by the exceptional collection (2.4) .Proof. The definition (2.1) of the object E can be rewritten as the exact sequence0 → E → T( − ⊠ X O → O (0 , → . (2.18)It follows that rk( E ) = 2 k and det( E ) ∼ = O (1 , − i E ∨ ∼ = Λ k − i E ⊗ O ( − , . (2.19)Dualizing (2.18), we obtain0 → O (0 , − → Ω(1) ⊠ X O → E ∨ → . Taking its ( k + i )-th exterior power and twisting by O (0 , i − → Λ k + i − E ∨ (0 , i − → Ω k + i ( k + i ) ⊠ X O ( i − → Λ k + i E ∨ (0 , i − → . sing (2.19), we can rewrite this as0 → Λ k − i +1 E ( − , i − → Ω k + i ( k + i ) ⊠ X O ( i − → Λ k − i E ( − , i ) → . (2.20)Now we interpret these sequences as mutations of (2.10): • For i = k , the second arrow in (2.20) is the unique (by (2.11)) morphismΩ k (2 k ) ⊠ X O ( k − → O ( − , k ) , therefore its kernel (up to shift) is the right mutation of Ω k (2 k ) ⊠ X O ( k − O ( − , k ), and so by (2.20) the result of the mutation is E ( − , k − • For i = k −
1, the second arrow in (2.20) is the unique morphismΩ k − (2 k − ⊠ X O ( k − → E ( − , k − , therefore its kernel (up to shift) is the right mutation of Ω k − (2 k − ⊠ X O ( k − E ( − , k − E ( − , k − k − (2 k − ⊠ X O ( k −
2) through O ( − , k ) istrivial, because these objects are completely orthogonal by (2.11).Continuing in the same manner, we conclude that the mutation of Ω k − j (2 k − j ) ⊠ X O ( k − j − h Ω k − j +1 (2 k − j +1) ⊠ X O ( k − j ) , . . . , Ω k (2 k ) ⊠ X O ( k − , O ( − , k ) i is the same as itsmutation through h O ( − , k ) , E ( − , k − , . . . , Λ j E ( − , k − j ) i , and is the same as its mutationthrough the last object Λ j E ( − , k − j ). Therefore, it is realized by the complex (2.20)with i = k − j , hence the result of the mutation is Λ j +1 E ( − , k − j − R is generated by the following exceptional collection R = h O ( − , k ) , E ( − , k − , . . . , Λ k − E ( − , , Λ k E ( − , , O ⊠ X Ω k +1 ( k + 1) , O (1) ⊠ X Ω k +2 ( k + 2) , . . . , O ( k − ⊠ X Ω k (2 k ) i , (2.21)and moreover, the only Ext-spaces between the objects in the first row and the second rowof (2.21), areExt • (Λ i E ( − , k − i ) , O ( j − ⊠ X Ω k + j ( k + j )) = ( k [ i − k ] , if j = 10 , if 2 ≤ j ≤ k . (2.22)Note also that by (2.19) the last term in the first line of (2.21) is isomorphic to Λ k E ∨ (0 , − X to (2.20) we obtain exact sequences0 → Λ k − i +1 E ∨ ( i − , − → O ( i − ⊠ X Ω k + i ( k + i ) → Λ k − i E ∨ ( i, − → . (2.23)Now we interpret these sequences as mutations: • For i = 1, the first arrow in (2.23) is the unique (by (2.22)) morphismΛ k E ∨ (0 , − → O ⊠ X Ω k +1 ( k + 1) , therefore its cokernel (up to shift) is the left mutation of the object O ⊠ X Ω k +1 ( k + 1)through Λ k E ∨ (0 , − k − E ∨ (1 , − • For i = 2, the first arrow in (2.23) is the unique morphismΛ k − E ∨ (1 , − → O (1) ⊠ X Ω k +2 ( k + 2) , therefore its cokernel (up to shift) is the left mutation of O (1) ⊠ X Ω k +2 ( k + 2)through Λ k − E ∨ (1 , − k − E ∨ (2 , − ere that the intermediate mutation of O (1) ⊠ X Ω k +2 ( k + 2) through Λ k E ∨ (0 , −
1) istrivial, because these objects are completely orthogonal by (2.22).Continuing in the same manner, we conclude that the mutation of O ( i − ⊠ X Ω k + i ( k + i )through h Λ k E ∨ (0 , − , O ⊠ X Ω k +1 ( k + 1) , . . . , O ( i − ⊠ X Ω k + i − ( k + i − i is the same asits mutation through h Λ k − i +1 E ∨ ( i − , − , . . . , Λ k − E ∨ (1 , − , Λ k E ∨ (0 , − i , and the sameas its mutation through the first object Λ k − i +1 E ∨ ( i − , − k − i +1 E ∨ ( i − , − R is generated by (2.4). Moreover, we see that the onlyExt-spaces between the objects in (2.4) areExt j − i (Λ i E (0 , k − i ) , Λ j E (0 , k − j )) = k , ≤ i ≤ j ≤ k, Ext j − i (Λ i E ∨ (0 , k − i ) , Λ j E ∨ (0 , k − j )) = k , ≤ i ≤ j ≤ k. In other words, the configuration of these exceptional objects is the following: O ( − , k )[ − k ] / / E ( − , k − − k ] / / . . . / / Λ k − E ( − , − , , ❨❨❨❨❨❨ Λ k E ( − , ∼ = Λ k E ∨ (0 , − O ( k, − − k ] / / E ∨ ( k − , − − k ] / / . . . / / Λ k − E ∨ (1 , − − ❡❡❡❡❡❡❡ So, this exceptional collection corresponds to shifts of projective modules in a quiver ofDynkin type A k +1 . (cid:3) Remark . The flag variety X is a homogeneous space for the action of the reductivegroup SL( n + 1) = SL( V ). Using the group-theoretic notation introduced in § ω i the highest weight of the fundamental representation Λ i V ∨ ∼ = Λ n +1 − i V , onecan rewrite the exceptional collection of Theorem 2.1 as follows. The first block of therectangular part A of D b ( X ) can be written as A = h O , U ω , U ω n , U ω , U ω n , . . . , U kω , U kω n i Furthermore, if n = 2 k + 1 the residual category R can be written as R = h U − ω + kω n , U kω − ω n ; U − ω + ω n − +( k − ω n , U ( k − ω + ω − ω n ; . . .. . . ; U − ω + ω n − i +( k − i − ω n , U ( k − i − ω + ω i +1 − ω n ; . . . U − ω + ω k +2 , U ω k − ω n ; U − ω + ω k +1 − ω k +1 i . Type D n In this section we prove part (3) of Theorem 1.9, restated in a more precise form inTheorem 3.1 below. In this section we work over an algebraically closed field k of charac-teristic zero. Let us point out again that, unlike in type A n , a full exceptional collectionin D b (OG(2 , n )) was not known before. Throughout this section we assume n ≥ Equivariant bundles on homogeneous varieties.
We start with a brief reminderabout equivariant vector bundles on homogeneous varieties.Let G be a connected simply connected semisimple algebraic group. We fix a Borel sub-group B ⊂ G. Let P ⊂ G be a parabolic subgroup such that B ⊂ P. Recall that there is amonoidal equivalence of categories between the category of G-equivariant vector bundles on he homogeneous variety G / P and the category Rep(P) of representations of the parabolicsubgroup P.Let P ։ L be the Levi quotient. The image of the Borel B is a Borel subgroup B L in L. Weidentify the category Rep(L) of representations of L with the subcategory of Rep(P) of rep-resentations with the trivial action of the unipotent radical. This equivalence is compatiblewith the monoidal structure of the categories.We denote by P L = P G the weight lattices of L and G with their natural identification,and by P +G ⊂ P +L the cones of dominant weights with respect to B and B L , respectively. For each λ ∈ P +L wedenote by V λ L the corresponding irreducible representation of L and by U λ the equivariantvector bundle on G / P corresponding to it via the above equivalences. Similarly, for λ ∈ P +G we denote by V λ G the corresponding irreducible representation of G.We denote by W the Weyl group of G, by W L ⊂ W the Weyl group of L, and by w ∈ Wand w L0 ∈ W L the longest elements.3.2. Statement of the theorem.
Consider the group G = Spin(2 n ), the simply connected(double) covering Spin(2 n ) → SO(2 n ) (3.1)of the special orthogonal group, and its maximal parabolic subgroup P ⊂ G correspondingto the vertex 2 (marked with black) of the Dynkin diagram D n ❝ ❝ ❝s . . . ❝ ❝ ❝❝ ✏✏✏PPP n − n − n − n (3.2)In this case G / P ∼ = OG(2 , n )is the Grassmannian of 2-dimensional isotropic subspaces in a vector space of dimension 2 n endowed with a non-degenerate symmetric bilinear form, i.e., the (co)adjoint variety oftype D n . Note that dim(OG(2 , n )) = 4 n − , n ) is the semidirect productAut(OG(2 , n )) ∼ = ( PSO(2 n ) ⋊ S , if n = 4PSO(2 n ) ⋊ S , if n = 4where the factors S and S act by outer automorphisms (corresponding to the symmetryof the Dynkin diagram D n ).The weight lattice of SO(2 n ) is the lattice Z n with the standard basis { ǫ i } ≤ i ≤ n , and theweight lattice of Spin(2 n ) is its overlattice generated by the fundamental weights ω i = ǫ + · · · + ǫ i , ≤ i ≤ n − ,ω n − = ( ǫ + · · · + ǫ n − − ǫ n ) ,ω n = ( ǫ + · · · + ǫ n − + ǫ n ) . (3.3)The Levi group in SO(2 n ) corresponding to the second vertex of the Dynkin diagram isisomorphic to GL(2) × SO(2( n − n ) is thedouble covering L → GL(2) × SO(2( n − nduced by (3.1).The fundamental weight ω (associated with the parabolic P) corresponds to the amplegenerator of the Picard group Pic(G / P), so we write O (1) := U ω . (3.5)Note that the canonical line bundle can be written as ω OG(2 , n ) ∼ = O OG(2 , n ) (3 − n ) . (3.6)Furthermore, we use the following notation U := ( U ω ) ∨ , S − := U ω n − , S + := U ω n . (3.7)Thus U is the tautological rank-2 bundle of OG(2 , n ) while S ± are the spinor bundles (see [17, § D b (OG(2 , n )) A := h O , U ∨ , S U ∨ , . . . , S n − U ∨ , S − , S + i , B := h O , U ∨ , S U ∨ , . . . , S n − U ∨ , S n − U ∨ , S − , S + i . (3.8)Note that A ⊂ B and A is Aut(OG(2 , n ))-invariant. Indeed, every object in A is PSO(2 n )-invariant by [24, Lemma 2.2]. Moreover, if n = 4 the outer automorphisms S -actionon OG(2 , n ) swaps the spinor bundles S − and S + , while in case n = 4 the outer automor-phisms S -action permutes S − , S + , and U ∨ (and in this case A = h O , U ∨ , S − , S + i ). Theorem 3.1.
There exists a full exceptional collection D b (OG(2 , n )) = h U ω n − ( − , U ω n ( − , A , B (1) , . . . , B ( n − , A ( n − , . . . , A (2 n − i . (3.9) In particular, the subcategory A extends to an Aut(OG(2 , n )) -invariant rectangular Lef-schetz collection D b (OG(2 , n )) = h R , A , A (1) , . . . , A (2 n − i . (3.10) Moreover, the residual category R is equivalent to the derived category of representations ofthe Dynkin quiver D n .Remark . It is easy to see that the exceptional collection (3.9) is mutation equivalent tothe full Lefschetz collection with the starting block E i = S i − U ∨ , ≤ i ≤ n − ,E n = S − ,E n +1 = S + ,E n +2 = R h A ( − , B i ( U ω n − ( − ,E n +3 = R h A ( − , B i ( U ω n ( − σ = ( n + 3 , ( n + 1) n − , n n − ). emark . If n = 3 we have isomorphisms Spin(6) ∼ = SL(4) and OG(2 , ∼ = Fl(1 ,
3; 4).Furthermore, under these isomorphisms S − ∼ = O (1 ,
0) and S + ∼ = O (0 , A in (2.2) and (3.8) coincide, hence (2.3) coincides with (3.10).We prove Theorem 3.1 in Sections 3.4, 3.6 and 3.7 after some preparation.3.3. Computational tools.
To check that a collection of vector bundles is exceptional,we need some tools to dualize the bundles, take their tensor products, and compute thecohomology.We start by reminding the general machinery. Recall that the Levi group L is reductive,hence Rep(L) is a semisimple category; in particular V λ L ⊗ V µ L is a direct sum of irreduciblerepresentations of L. Since the functor V λ L U λ from the category Rep(L) to the categoryof equivariant vector bundles on G / P is monoidal, we have the following
Lemma 3.4 ([18, (8)]) . If V λ L ⊗ V µ L = L V ν L , then U λ ⊗ U µ = L U ν . Thus, knowing tensor products of L-representations we can control tensor products ofthe corresponding equivariant bundles. Similarly, we can control the dualization operation.Recall that w L0 denotes the longest element in W L , the Weyl group of L. Lemma 3.5 ([18, (8)]) . We have ( U λ ) ∨ ∼ = U − w L0 λ . A combination of the last two lemmas shows that ( U λ ) ∨ ⊗ U µ is a direct sum of some U ν .The summands that show up in this decomposition can be characterised by the following Lemma 3.6 ([18, Lemma 2.8 and Lemma 2.9]) . We have ( U λ ) ∨ ⊗ U µ = M ν ∈ P +L ∩ Conv( µ − wλ ) w ∈ WL ( U ν ) ⊕ m ( λ,µ,ν ) , where Conv( − ) stands for the convex hull and m ( λ, µ, ν ) ∈ Z ≥ is the multiplicity of V ν L in (V λ L ) ∨ ⊗ V µ L . In particular, if ν = 0 then m ( λ, µ, ν ) = 0 only when λ = µ and m ( λ, λ,
0) = 1 . The most efficient way to compute cohomology is provided by the Borel–Bott–Weil the-orem. Let ℓ : W → Z be the length function and let ρ ∈ P +G be the sum of fundamentalweights of G. Theorem 3.7 (Borel–Bott–Weil) . Let λ ∈ P +L . If the weight λ + ρ lies on a wall of a Weylchamber for the W -action, then H • (G / P , U λ ) = 0 . Otherwise, if w ∈ W is the unique element such that the weight w ( λ + ρ ) is dominant, then H • (G / P , U λ ) = V w ( λ + ρ ) − ρ G [ − ℓ ( w )] . The Weyl group of Spin(2 n ) is the semidirect productW = S n ⋉ ( Z / n − , where S n acts on Z n ⊂ P Spin(2 n ) by permutations and ( Z / n − by changes of signs of evennumber of coordinates. The walls of the Weyl chambers are given by the equalities λ i = ± λ j , ≤ i = j ≤ n. (3.11) he Weyl group of L is the subgroupW L = S × ( S n − ⋉ ( Z / n − ) , where S acts by transposition of the first two coordinates, S n − by permutations of thelast n −
2, and ( Z / n − by changes of signs of even number of the last n − w L0 ∈ W L acts by w L0 ( λ , . . . , λ n − , λ n ) = ( λ , λ , − λ , . . . , − λ n − , − ( − n λ n ) . (3.12)Finally, the sum ρ of the fundamental weights in the standard basis takes the form ρ = ( n − , n − , . . . , , . (3.13)Applying the general machinery in our situation we obtain the following corollaries. First,combining Lemma 3.5 with (3.3), (3.5), and (3.12), we obtain Corollary 3.8.
For a ≥ we have (cid:0) U aω (cid:1) ∨ = U aω ( − a ) . For a ≥ and even n we have (cid:0) U aω n − (cid:1) ∨ = U aω n − ( − a ) and (cid:0) U aω n (cid:1) ∨ = U aω n ( − a ) . For a ≥ and odd n we have (cid:0) U aω n − (cid:1) ∨ = U aω n ( − a ) and (cid:0) U aω n (cid:1) ∨ = U aω n − ( − a ) . Similarly, using Lemma 3.4 we deduce the following tensor product decompositions.
Lemma 3.9.
We have isomorphisms U kω ⊗ U lω ∼ = min { k,l } M i =0 U ( k + l − i ) ω ( i ) for k, l ≥ , (3.14) U kω ⊗ U λ ∼ = U kω + λ if λ = n X i =2 λ i ω i , (3.15)Λ n − ( U ω − ω ) ∼ = U ω n − ( − ⊕ U ω n ( − . (3.16) Proof.
First, we note that the bundles U kω and U lω correspond to representations of Lpulled back from representations V k, and V l, via the map (3.4), so (3.14) follows fromthe Clebsch–Gordan ruleV k ,k GL ⊗ V l ,l GL = min { k − k ,l − l } M j =0 V k + l − j,k + l + j GL . For (3.15) and (3.16) we need a slightly more detailed understanding of the Levi group L.Let f GL(2) → GL(2) be the unique connected double covering. Then there is a doublecovering f GL(2) × Spin(2( n − → L . (3.17)To understand tensor product of representations of L it is enough to understand tensorproduct of their pullbacks via the map (3.17). e will denote by ω ′ , ω ′ the fundamental weights of f GL(2) and by ω ′ i , 3 ≤ i ≤ n , thefundamental weights of Spin(2( n − n ). It is easy to see that they can be expressed in thebasis of fundamental weights ω i as follows: ω ′ = ω , ω ′ = ω , ω ′ i = ( ω i − ω , if 3 ≤ i ≤ n − ω i − ω , if i ∈ { n − , n } . (3.18)By (3.18) the representation V kω L pullbacks to V k, f GL , while V λ L with λ = P ni =2 λ i ω i pullbacksto V c ( λ ) ,c ( λ ) f GL(2) ⊗ V λ ′ Spin(2( n − , where λ ′ = P ni =3 λ i ω ′ i and c ( λ ) = 2 P n − i =2 λ i + λ n − + λ n . Clearly,V k, f GL ⊗ (cid:16) V c ( λ ) ,c ( λ ) f GL(2) ⊗ V λ ′ Spin(2( n − (cid:17) ∼ = V k + c ( λ ) ,c ( λ ) f GL(2) ⊗ V λ ′ Spin(2( n − , and the right side corresponds to V kω + λ L . This proves (3.15).Similarly, V ω − ω L corresponds to V ω ′ Spin(2( n − (which, according to our conventions, is thefirst fundamental representation of Spin(2( n − ω ′ n and 2 ω ′ n − . By (3.18) these weights correspond to the weights 2 ω n − ω and 2 ω n − − ω , andthe claim follows. This proves (3.16). (cid:3) Finally, we deduce from Borel–Bott–Weil a vanishing lemma, on which most of semiorthog-onality results for OG(2 , n ) in the next section rely. Lemma 3.10.
Let λ = ( λ , . . . , λ n ) be a dominant weight of L . Assume that one of thefollowing two conditions holds | λ i + n − i | < n − for all i, (3.19) or | λ i + n − i | < n − for all but one i. (3.20) Then U λ is acyclic.Proof. These are special cases of [18, Lemma 5.2]; we provide a proof for completeness.In view of (3.13), if (3.19) holds, then all coordinates of λ + ρ have absolute values lessthan n −
1. Therefore, at least two of them have the same absolute values. Similarly, if (3.20)holds, then all but one coordinates of λ + ρ have absolute values less than n −
2. Therefore,again at least two of them have the same absolute values. In both cases we see that λ + ρ lies on one of the walls (3.11), and by Borel–Bott–Weil theorem we conclude that U λ isacyclic. (cid:3) Exceptional collection.
In this section we prove that (3.9) is an exceptional collection.Recall from (3.7) that U denotes the dual bundle of U ω . Consequently, S k U ∨ ∼ = U kω .First, we discuss the “tautological” part of the collection. Recall that an exceptionalcollection E , . . . , E k of vector bundles is strong if Ext > ( E i , E j ) = 0 for all 1 ≤ i, j ≤ k . emma 3.11. The collection of vector bundles O , U ∨ , . . . , S n − U ∨ is a strong exceptionalcollection. Moreover, for ≤ k, l ≤ n − and ≤ t ≤ n − we have Ext • ( S k U ∨ ( t ) , S l U ∨ ) = k [4 − n ] , if k = l = n − , t = n − , k [3 − n ] , if k = l = n − , t = n − , , otherwise.Proof. We will show that Ext • ( S k U ∨ ( t ) , S l U ∨ ) is zero for 0 ≤ k, l ≤ n − ≤ t ≤ n − t = 0 and k ≤ l and t ∈ { n − , n − } , k = l = n −
2. By (3.6) and Serre dualitywe can assume 0 ≤ t ≤ n − S k U ∨ ( t )) ∨ ⊗ S l U ∨ ∼ = S k U ∨ ( − k − t ) ⊗ S l U ∨ ∼ = min { k,l } M j =0 S k + l − j U ∨ ( j − k − t ) = min { k,l } M j =0 U ( l − j − t,j − k − t, ,..., . So, it is enough to compute the cohomology of bundles U λ with λ = ( l − j − t, j − k − t, , . . . , − n ≤ λ ≤ n − , − n ≤ λ ≤ , λ = · · · = λ n = 0 . It is convenient to distinguish three cases:
Case 1:
If 4 − n < λ <
0, then | λ i + n − i | < n − i ≥
2. Hence, (3.20) holdsand U λ is acyclic by Lemma 3.10. Case 2: If λ = 0, then we necessarily have t = 0 and j = k . Further, since j ≤ l , weobtain k ≤ l . In particular, the weight λ is dominant, hence H > (OG(2 , n ) , U λ ) = 0.Moreover, if k = l , then λ = 0 and in this case the cohomology of U λ is isomorphicto k [0].A combination of Case 1 and Case 2 proves that O , U ∨ , . . . , S n − U ∨ is a strong exceptionalcollection. Case 3: If λ = 4 − n , then we necessarily have k = t = n − j = 0. In this casewe have 2 − n ≤ λ ≤ λ <
0, then | λ i + n − i | < n − i . Hence (3.19) holds and U λ is acyclicby Lemma 3.10.If λ = 0, then l = n −
2, and we have U λ = U (0 , − n, ,..., . In this case the Borel–Bott–Weil theorem implies that the cohomology of U λ is isomorphic to k [4 − n ].This completes the proof of the lemma. (cid:3) Next, we discuss the “spinor” part of the collection.
Lemma 3.12.
The collection of sheaves U ω n − , U ω n , U ω n − , U ω n , { U ω n − ( t ) , U ω n ( t ) } ≤ t ≤ n − is exceptional.Proof. We will compute Ext • ( U aω i ( t ) , U bω j ) where i, j ∈ { n − , n } , and • either a = 1, b ∈ { , } , and 1 ≤ t ≤ n − • or a, b ∈ { , } , a ≥ b , and t = 0. y (3.6) and Serre duality we can assume 0 ≤ t ≤ n − λ = aω i + tω = ( t + a , t + a , a , . . . , a , ± a ) and µ = bω j = ( b , b , b , . . . , b , ± b ).Since the W L -orbit of λ consists of points ( t + a , t + a , ± a , . . . , ± a , ± a ), by Lemma 3.6 wehave ( U λ ) ∨ ⊗ U µ = M ν ( U ν ) ⊕ m ( λ,µ,ν ) , (3.21)where ν runs over the integral or half-integral points satisfying ν = ν = b − a − t, ν , . . . , ν n − = b ± a ∈ [ − , , ν n = ± b ± a ∈ [ − , . It is convenient to distinguish three cases:
Case 1:
If 1 ≤ t ≤ n − a = 1, then ν = ν ∈ [2 − n, − ] , ν , . . . , ν n − ∈ [0 , ] , ν n ∈ [ − , ] . For any such point we have | ν i + n − i | < n − i , hence (3.19) holds for ν and,therefore, all summands in (3.21) are acyclic by Lemma 3.10. Case 2: If t = 0 and a > b , then ν = ν = − , ν , . . . , ν n − ∈ [ − , ] , ν n ∈ [ − , ] . For any such point we still have | ν i + n − i | < n − i , hence again all summandsin (3.21) are acyclic. Case 3: If t = 0 and a = b , then ν = ν = 0 , ν , . . . , ν n − ∈ [0 , , ν n ∈ [ − , . It is easy to see that the only non-acyclic bundle among U ν is the trivial bundle. ByLemma 3.6 it is a summand of ( U λ ) ∨ ⊗ U µ only when λ = µ and then its multiplicityis equal to 1. Hence, in this case we have Ext • ( U λ , U µ ) = k [0].This completes the proof of the lemma. (cid:3) In the next two lemmas we show that the “tautological” part of the collection can bemerged with the “spinor” part.
Lemma 3.13.
We have
Ext • ( U ω j ( t ) , S k U ∨ ) = 0 for all ≤ k ≤ n − , j ∈ { n − , n } ,and ≤ t ≤ n − .Proof. By Corollary 3.8 and (3.15) we have( U ω j ( t )) ∨ ⊗ S k U ∨ ∼ = U ω j ′ − ( t +1) ω ⊗ U kω ∼ = U kω − ( t +1) ω + ω j ′ , where j ′ = j if n is even and j ′ = 2 n − − j if n is odd. So, it is enough to compute thecohomology of bundles U λ with λ = (cid:0) k − t − , − t − , , . . . , , ± (cid:1) . If 0 ≤ t ≤ n −
5, then | λ i + n − i | < n − i ≥
2. Hence (3.20) holds and the bundle U λ is acyclic by Lemma 3.10. Similarly, if t = 2 n −
4, then we have | λ i + n − i | < n − i . Hence (3.19) holds and the bundle U λ is acyclic by Lemma 3.10. (cid:3) emma 3.14. If ≤ k ≤ n − , j ∈ { n − , n } , and ≤ t ≤ n − , then Ext • ( S k U ∨ ( t ) , U ω j ) = k [5 − n ] , if k = n − and t = n , k [2 − n ] , if k = n − and t = 1 , , otherwise.Proof. As before, we have( S k U ∨ ( t )) ∨ ⊗ U ω j ∼ = S k U ∨ ( − k − t ) ⊗ U ω j ∼ = U kω − ( k + t ) ω +2 ω j So, it is enough to compute the cohomology of the bundle U λ with λ = (1 − t, − k − t, , . . . , , ± . For this we directly apply the Borel–Bott–Weil theorem. We have λ + ρ = ( n − t, n − − k − t, n − , . . . , , ± . It is convenient to distinguish several cases:
Case 1:
If 2 ≤ t ≤ n − t = n , then the absolute value of the first coordinateof λ + ρ is equal to the absolute value of one of the last n − U λ is acyclic. Case 2: If t = n and 0 ≤ k ≤ n −
3, then n − − k − t = − k −
1, and the absolutevalue of the second coordinate of λ + ρ is equal to the absolute value of one of thelast n − U λ is acyclic. Case 3: If t = n and k = n −
2, then λ + ρ = (0 , − n, n − , . . . , , ± U λ equals k [5 − n ]. Case 4: If t = 1 and 0 ≤ k ≤ n −
3, then n − − k − t = n − − k , and the absolutevalue of the second coordinate of λ + ρ is equal to the absolute value of one of thelast n − U λ is acyclic. Case 5: If t = 1 and k = n −
2, then λ + ρ = ( n − , , n − , . . . , , ± U λ equals k [2 − n ].This completes the proof of the lemma. (cid:3) Now we deduce
Proposition 3.15.
The collection (3.9) is exceptional.Proof.
A combination of Lemmas 3.11, 3.12, 3.13, 3.14, and Serre duality proves the propo-sition. (cid:3)
Some useful complexes.
In this section we construct some exact sequences and com-plexes that will be used to check that the collection (3.9) is full.Let V be a vector space of dimension 2 n endowed with a non-degenerate symmetric bilinearform and let OG(2 , V ) = OG(2 , n ) be the corresponding even orthogonal Grassmannian.We denote V = V ⊗ O the trivial vector bundle. Note that V and V are canonically self-dual (via the chosen symmetric bilinear form on V ). We have the tautological short exactsequence on OG(2 , n ) 0 → U → V → V / U → → U ⊥ → V → U ∨ → . aking exterior powers of the above sequences, we obtain the long exact sequences0 → S m U → V ⊗ S m − U → Λ V ⊗ S m − U → . . . · · · → Λ m − V ⊗ U → Λ m V ⊗ O → Λ m ( V / U ) → → Λ m U ⊥ → Λ m V ⊗ O → Λ m − V ⊗ U ∨ → . . . · · · → Λ V ⊗ S m − U ∨ → V ⊗ S m − U ∨ → S m U ∨ → S − := H (OG(2 , V ) , S − ) = V ω n − Spin(2 n ) , S + := H (OG(2 , V ) , S + ) = V ω n Spin(2 n ) (the half-spinor representations of Spin(2 n )). We set W i = ⌊ i/ ⌋ M s =0 Λ i − s V. (3.24)We will combine (3.22) and (3.23) to prove the following Proposition 3.16. On OG(2 , n ) there exists an exact sequence → S n − U → W ⊗ S n − U → W ⊗ S n − U → . . . · · · → W n − ⊗ O → S + ⊗ S + ⊕ S − ⊗ S − → W n − ⊗ O (1) → . . . · · · → W ⊗ S n − U ∨ (1) → W ⊗ S n − U ∨ (1) → S n − U ∨ (1) → . (3.25) Proof. If n is even, we apply [17, Proposition 6.7] and conclude that there is a filtrationon S + ⊗ S ∨ + ⊕ S − ⊗ S ∨− with factors of the form Λ s U ⊥ , where 0 ≤ s ≤ n −
1. Tensoringby O (1) and taking into account isomorphisms S ∨ + (1) ∼ = S + , S ∨− (1) ∼ = S − (Corollary 3.8) we obtain a filtration on S + ⊗ S + ⊕ S − ⊗ S − with factors of the form Λ s U ⊥ (1),where 0 ≤ s ≤ n − n is odd, the argument of [17, Proposition 6.7] proves that there is a filtrationon S + ⊗ S ∨− ⊕ S − ⊗ S ∨ + with factors of the form Λ s +1 U ⊥ , where 0 ≤ s ≤ n −
2. Tensoringby O (1) and taking into account isomorphisms S ∨ + (1) ∼ = S − , S ∨− (1) ∼ = S + , we finally obtain a filtration on S + ⊗ S + ⊕ S − ⊗ S − with factors of the form Λ s +1 U ⊥ (1),where 0 ≤ s ≤ n − S + ⊗ S + ⊕ S − ⊗ S − with factorsΛ n − − r U ⊥ (1) , Λ n − − r U ⊥ (1) , . . . , Λ n U ⊥ (1) , Λ n − U ⊥ (1) , . . . , Λ r U ⊥ (1) , Λ r U ⊥ (1) , where r = 0 if n is even and r = 1 if n is odd. Note that the total number of factors n − r is even in both cases. This means that there is an exact sequence of vector bundles0 → F − → S + ⊗ S + ⊕ S − ⊗ S − → F + → , where F − has a filtration with factors Λ n − − r U ⊥ (1) , Λ n − − r U ⊥ (1) , . . . , Λ n U ⊥ (1) and F + has a filtration with factors Λ n − U ⊥ (1) , . . . , Λ r U ⊥ (1) , Λ r U ⊥ (1). So, it is enough to show hat F − has a left resolution given by the first half of (3.25), and that F + has a rightresolution given by the second half of (3.25).Indeed, each factor of the filtration of F − can be rewritten asΛ n +2 k U ⊥ (1) ∼ = Λ n − − k ( V / U ) , where 0 ≤ k ≤ ( n − − r ) /
2, hence has the resolution (3.22) with m = n − − k .Since by Lemma 3.11 the exceptional collection { S i U } n − i =0 is strong , the extensions of thefiltration factors Λ n +2 k U ⊥ (1) in F − can be realized by a bicomplex with rows given by (3.22)with m = n − − k shifted by − k . The totalization of this bicomplex can be written as S n − U → W ⊗ S n − U → W ⊗ S n − U → · · · → W n − ⊗ O and thus provides a left resolution for F − .Similarly, each factor Λ n − − k U ⊥ (1) of the filtration of F + has resolution (3.23) twistedby O (1) (where m = n − − k ), and the above argument shows that F + has the rightresolution W n − ⊗ O (1) → · · · → W ⊗ S n − U ∨ (1) → W ⊗ S n − U ∨ (1) → S n − U ∨ (1) . This completes the proof of the proposition. (cid:3)
Note that the composition of morphisms U → V → U ∨ is zero, hence U ⊂ U ⊥ . Moreover, U ⊥ / U ∼ = U ω − ω . (3.26)We will need the following lemma. Lemma 3.17.
For any m ≤ n − on OG(2 , n ) there exists a double complex Λ m V ⊗ O / / Λ m − V ⊗ U ∨ / / Λ m − V ⊗ S U ∨ / / . . . / / V ⊗ S m − U ∨ / / S m U ∨ Λ m − V ⊗ U / / O O Λ m − V ⊗ U ⊗ U ∨ / / O O Λ m − V ⊗ U ⊗ S U ∨ / / O O . . . / / U ⊗ S m − U ∨ O O . . . O O . . . O O . . . O O ... Λ V ⊗ S m − U / / O O V ⊗ S m − U ⊗ U ∨ / / O O S m − U ⊗ S U ∨ O O V ⊗ S m − U / / O O S m − U ⊗ U ∨ O O S m U O O whose total complex has only one non-trivial cohomology in the middle term isomorphicto Λ m ( U ⊥ / U ) .Proof. The bicomplex is just the m -th exterior power of the complex U → V → U ∨ , which is quasiisomorphic to U ⊥ / U , hence the claim (cf. the argument of [17, Lemma 7.3]). (cid:3) Finally, we will need the following lemma from [17]. emma 3.18 ([17], Proposition 6.3) . There exists a complex → S − → S − ⊗ O (1) → S − (1) → , whose only non-trivial cohomology group is in the middle term and is isomorphic to S + ⊗ U ∨ . Fullness.
The proof of the fullness of the collection (3.9) goes via restriction to oddGrassmannians OG(2 , n − A (2 − n ) , . . . , A ( − , U ω n − ( − , U ω n ( − , A , B (1) , . . . , B ( n − . (3.27)By Serre duality, exceptionality of (3.9) implies exceptionality of (3.27). Conversely, fullnessof (3.27) implies fullness of (3.9).We will deduce fullness of (3.27) from the following key lemma. Lemma 3.19.
Let E be the set of bundles appearing in (3.27) , and set E ′ := { S k U ∨ ( t ) , S + ( t ) | ≤ k ≤ n − , − n ≤ t ≤ n − } ⊂ E . (3.28) Then for any object E ∈ E ′ we have E ⊗ O (1) ∈ h E i , (3.29) E ⊗ U ∨ ∈ h E i . (3.30) Proof.
The inclusion (3.29) follows immediately from comparison of (3.28), (3.8) and (3.27).Thus, we concentrate here on the inclusion (3.30).We split objects E ∈ E ′ into two classes: • S + ( t ) with t ∈ [2 − n, n −
3] (spinor bundles), and • S k U ∨ ( t ) with k ∈ [0 , n −
3] and t ∈ [2 − n, n −
3] (tautological bundles).and treat these separately.The class of spinor bundles is easy: from Lemma 3.18 we conclude S + ( i ) ⊗ U ∨ ∈ h S − ( i ) , O ( i + 1) , S − ( i + 1) i . and the inclusion (3.30) for S + ( i ) follows.The class of tautological bundles is a bit more tedious. First, note that by (3.14) we have S k U ∨ ⊗ U ∨ = S k +1 U ∨ ⊕ S k − U ∨ (1) . Thus, the inclusion of (3.30) would follow from the inclusions S j U ∨ ( t ) ∈ h E i , ≤ j ≤ n − , − n ≤ t ≤ n − . (3.31)So, proving these inclusions, we will prove the lemma.For j ≤ n − j = n −
2, 1 ≤ t ≤ n − O ( n − t ) and isomorphisms S l U ≃ S l U ∨ ( − l ) , we deduce (3.31) for j = n − − n ≤ t ≤ − S n − U ∨ ∈ h E i . or this we consider the double complex of Lemma 3.17 with m = n −
2. Using the directsum decompositions S k U ∨ ⊗ S l U ∼ = min { k,l } M j =0 S k + l − j U ∨ ( j − l )and the cases of the inclusion (3.31) proved above, we see that all the terms of the doublecomplex with a possible exception for the rightmost term S n − U ∨ are contained in thesubcategory generated by E . By (3.16) and (3.26) the unique cohomology sheaf U ω n − ( − ⊕ U ω n ( − h E i , therefore the last term S n − U ∨ is in the subcate-gory generated by E . (cid:3) Let v ∈ H (OG(2 , n ) , U ∨ ) = V ∨ ∼ = V be any non-zero global section and let v ⊥ ⊂ V be the orthogonal complement of v . Therestriction of the bilinear form to v ⊥ is nondegenerate if and only if v is non-isotropic. Inthis case the zero locus of v (considered as a section of U ∨ ) is the odd orthogonal Grassman-nian OG(2 , v ⊥ ) ∼ = OG(2 , n −
1) and we have the natural closed embedding i v : OG(2 , v ⊥ ) → OG(2 , V )and the Koszul resolution0 → O ( − → U → O → i v ∗ O OG(2 ,v ⊥ ) → . (3.32)Since n ≥
3, the union of OG(2 , v ⊥ ) for non-isotropic v ∈ V sweeps OG(2 , V ), hence wehave the following Lemma 3.20 ([17, Lemma 4.5]) . If for an object F ∈ D b (OG(2 , n )) the restrictions i ∗ v F vanish for all non-isotropic v ∈ V , then F = 0 . Now we are finally ready to prove the fullness of (3.27).
Proposition 3.21. If F ∈ D b (OG(2 , V )) is right orthogonal to all the vector bundles E inthe collection (3.27) , i.e., Ext • ( E, F ) = 0 , then F = 0 .Proof. The assumption of the proposition can be rewritten asExt • ( E, F ) = H • (OG(2 , V ) , E ∨ ⊗ F ) = 0 ∀ E ∈ E . (3.33)Now take any non-isotropic vector v and any bundle E ∈ E ′ , and tensor (3.32) by E ∨ ⊗ F .We obtain an exact sequence0 → O ( − ⊗ E ∨ ⊗ F → U ⊗ E ∨ ⊗ F → E ∨ ⊗ F → i v ∗ i ∗ v ( E ∨ ⊗ F ) → i v ∗ i ∗ v ( E ∨ ⊗ F ) has to vanish, and weconclude that H • (OG(2 , V ) , i v ∗ i ∗ v ( E ∨ ⊗ F )) = H • (OG(2 , v ⊥ ) , i ∗ v ( E ∨ ⊗ F )) = Ext • ( i ∗ v E, i ∗ v F ) = 0 (3.35)for any E ∈ E ′ . In other words, the object i ∗ v F is orthogonal to all objects i ∗ v E for E ∈ E ′ .Set C := h O , i ∗ v ( U ∨ ) , . . . , i ∗ v ( S n − U ∨ ) , i ∗ v ( S + ) i ⊂ D b (OG(2 , v ⊥ )) . ince i ∗ v ( U ∨ ) is the dual tautological bundle and i ∗ v ( S + ) is the spinor bundle on the oddorthogonal Grassmannian OG(2 , v ⊥ ) ∼ = OG(2 , n − D b (OG(2 , n − h C , C (1) , . . . , C (2 n − i . (3.36)Since ω OG(2 , n − ∼ = O (4 − n ), we also have a semiorthogonal decomposition D b (OG(2 , n − h C (2 − n ) , . . . , C ( − , C , C (1) , . . . , C ( n − i . (3.37)Thus the definition of the set E ′ implies that the objects i ∗ v E for E ∈ E ′ generate thecategory D b (OG(2 , v ⊥ )), hence (3.35) implies that i ∗ v F = 0. Since this holds for any non-isotropic v , it follows from Lemma 3.20 that F = 0. (cid:3) Residual category.
For 1 ≤ i ≤ n − F i to be the stupid right truncation ofcomplex (3.25) after n − − i terms (counting from the left). In other words, the objects F i are defined by the following exact sequences:0 → S n − U ∨ (2 − n ) → F n − → → S n − U ∨ (2 − n ) → W ⊗ S n − U ∨ (3 − n ) → F n − → → S n − U ∨ (2 − n ) → W ⊗ S n − U ∨ (3 − n ) → · · · → W n − ⊗ U ∨ ( − → F → . (3.38)Since (3.25) is exact, the objects F i are also quasiisomorphic to the stupid left truncationsof (3.25); in other words, we also have exact sequences0 → F n − → W ⊗ S n − U ∨ (3 − n ) → · · · → W ⊗ S n − U ∨ (1) → S n − U ∨ (1) → → F n − → W ⊗ S n − U ∨ (4 − n ) → · · · → W ⊗ S n − U ∨ (1) → S n − U ∨ (1) → → F → W n − ⊗ O → · · · → W ⊗ S n − U ∨ (1) → S n − U ∨ (1) → . (3.39)We will check that the objects F , . . . , F n − together with the objects U ω n − ( − U ω n ( − Lemma 3.22.
We have L h A , A (1) ,..., A ( i ) i (cid:0) S n − U ∨ ( i ) (cid:1) ∼ = F i ( i − n − i ] . Proof.
To prove the claim it is enough to show the following two facts: • the object F i ( i − n − i ] is contained in the orthogonal h A , A (1) , . . . , A ( i ) i ⊥ ; • there exists a morphism S n − U ∨ ( i ) → F i ( i − n − i ], whose cone lies in thecategory h A , A (1) , . . . , A ( i ) i .Twisting (3.38) by O ( i −
1) and using semiorthogonality of (3.9), we deduce the first ofthese facts. On the other hand, considering (3.39) twisted by O ( i −
1) as a Yoneda extensionof S n − U ∨ ( i ) by F i ( i −
1) of length n − i , i.e., as a morphism S n − U ∨ ( i ) → F i ( i − n − i ],we conclude that its cone is quasiisomorphic to the subcomplex of middle terms, hencebelongs to the subcategory h A , A (1) , . . . , A ( i ) i . This proves the second fact. (cid:3) hus, we have the following description for the residual category R = D U ω n − ( − , U ω n ( − , F , F (1) , . . . , F n − ( n − E . (3.40) Remark . For n = 4 the exceptional collection in (3.40) is Aut(OG(2 , n ))-invariant.For n = 4 it takes form S S − ( − , S S + ( − , ( V ⊗ U ) /S U , S U ∨ ( − , R = h S S − ( − , S S + ( − , S U ∨ ( − , ˜ T ( − i , where the bundle ˜ T is defined by the exact sequence 0 → O → ˜ T → T → T is thetangent bundle.It remains to compute the Ext-spaces between the objects of (3.40). Lemma 3.24.
We have
Ext • ( F i ( i − , F j ( j − ( k , if j = i + 1 or j = i , , otherwise.Furthermore, we have Ext • ( U ω n − ( − , F j ( j − • ( U ω n ( − , F j ( j − ( k [ − , if j = 1 , , otherwise.Proof. We start with the first claim. Since we already know that (3.40) is an exceptionalcollection, we may assume i < j . From (3.39) twisted by O ( i −
1) we deduce F i ( i − ∈ h A , A (1) , . . . , A ( i − , B ( i ) i and from (3.38) twisted by O ( j −
1) and with i replaced by j we deduce F j ( j − ∈ h B ( j − n + 1) , A ( j − n + 2) , . . . , A ( − i . (3.41)Using appropriate twist of (3.9) one easily sees that non-trivial Ext’s can only come fromExt • ( S n − U ∨ ( i ) , S n − U ∨ ( j − n + 1)) . By Lemma 3.11 this space is nonzero (and is isomorphic to k [4 − n ]) only if j = i + 1 andthe claim follows.For the second claim, using the inclusion (3.41), we see that non-trivial Ext’s can onlycome from Ext • ( U ω ε ( − , S n − U ∨ ( i − n + 1)) , where ε ∈ { n − , n } . By Lemma 3.14 and Serre duality this space is nonzero (and isisomorphic to k [2 − n ]) only if i = 1, and the claim follows. (cid:3) References [1] P. Belmans and M. Smirnov,
The Hochschild cohomology of generalised Grassmannians , available at arXiv:1911.09414 .[2] Raoul Bott,
Homogeneous vector bundles , Ann. of Math. (2) (1957), 203–248.[3] Anders Skovsted Buch, Andrew Kresch, and Harry Tamvakis, Quantum Pieri rules for isotropic Grass-mannians , Invent. Math. (2009), no. 2, 345–405.[4] P. E. Chaput and N. Perrin,
On the quantum cohomology of adjoint varieties , Proc. Lond. Math. Soc.(3) (2011), no. 2, 294–330.
5] Ionut¸ Ciocan-Fontanine,
On quantum cohomology rings of partial flag varieties , Duke Math. J. (1999),no. 3, 485–524.[6] J.A. Cruz Morales, A. Kuznetsov, A. Mellit, N. Perrin, and M. Smirnov, On quantum cohomology ofGrassmannians of isotropic lines, unfoldings of A n -singularities, and Lefschetz exceptional collections ,Annales de l’Institut Fourier (2019), no. 3, 955–991.[7] Boris Dubrovin, Geometry and analytic theory of Frobenius manifolds , Proceedings of the InternationalCongress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 315–326.[8] Daniele Faenzi and Laurent Manivel,
On the derived category of the Cayley plane II , Proc. Amer. Math.Soc. (2015), no. 3, 1057–1074.[9] W. Fulton and R. Pandharipande,
Notes on stable maps and quantum cohomology , Algebraic geometry—Santa Cruz 1995, 1997, pp. 45–96.[10] S. Galkin, A. Mellit, and M. Smirnov,
Dubrovin’s conjecture for IG (2 , (2015), no. 18, 8847–8859.[11] Claus Hertling, Frobenius manifolds and moduli spaces for singularities , Cambridge Tracts in Mathe-matics, vol. 151, Cambridge University Press, Cambridge, 2002.[12] Ralph Kaufmann,
The intersection form in H ∗ ( M n ) and the explicit K¨unneth formula in quantumcohomology , Internat. Math. Res. Notices (1996), 929–952.[13] Hua-Zhong Ke, On semisimplicity of quantum cohomology of P -orbifolds , J. Geom. Phys. (2019),1–14.[14] Bumsig Kim, Quantum cohomology of partial flag manifolds and a residue formula for their intersectionpairings , Internat. Math. Res. Notices (1995), 1–15.[15] M. Kontsevich and Yu. Manin, Quantum cohomology of a product , Invent. Math. (1996), no. 1-3,313–339. With an appendix by R. Kaufmann.[16] A. G. Kuznetsov,
Hyperplane sections and derived categories , Izv. Ross. Akad. Nauk Ser. Mat. (2006),no. 3, 23–128.[17] Alexander Kuznetsov, Exceptional collections for Grassmannians of isotropic lines , Proceedings of theLondon Mathematical Society. Third Series (2008), no. 1, 155–182.[18] Alexander Kuznetsov and Alexander Polishchuk, Exceptional collections on isotropic Grassmannians ,J. Eur. Math. Soc. (JEMS) (2016), no. 3, 507–574.[19] Alexander Kuznetsov and Maxim Smirnov, On residual categories for Grassmannians , Proceedings ofthe London Mathematical Society , no. 5, 617–641.[20] Yuri I. Manin,
Frobenius manifolds, quantum cohomology, and moduli spaces , American MathematicalSociety Colloquium Publications, vol. 47, American Mathematical Society, Providence, RI, 1999.[21] Mikhail Mironov,
Lefschetz exceptional collections in S k -equivariant categories of ( P n ) k , available at arXiv:1807.01534 .[22] N. Perrin and M. Smirnov, On the big quantum cohomology of (co)adjoint varieties , In preparation.[23] Nicolas Perrin,
Semisimple quantum cohomology of some Fano varieties , available at arXiv:1405.5914v1 .[24] A. Polishchuk, K -theoretic exceptional collections at roots of unity , J. K-Theory (2011), no. 1, 169–201.[25] `E. B. Vinberg and A. L. Onishchik, Seminar po gruppam Li i algebraicheskim gruppam , Second, URSS,Moscow, 1995.
Algebraic Geometry Section, Steklov Mathematical Institute of Russian Academy ofSciences, 8 Gubkin str., Moscow 119991 Russia
E-mail address : [email protected] Universit¨at Augsburg, Institut f¨ur Mathematik, Universit¨atsstr. 14, 86159 Augsburg,Germany
E-mail address : [email protected]@math.uni-augsburg.de