Строение производных категорий и геометрия многообразий Фано в грассманианах тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Гусева Ляля Андреевна

• Гусева Ляля Андреевна
• кандидат наук
• 2023

Оглавление диссертации кандидат наук Гусева Ляля Андреевна

Contents

Introduction

1 Preliminaries

1.1 Semiorthogonal decompositions and exceptional collections

1.2 The Borel-Bott-Weil theorem

1.2.1 Classical Grassmannian

1.2.2 Isotropic Grassmannian

1.3 The algebraic group G2

1.4 Conics on Grassmannians

2 The isotropic Grassmannian IGr(3, 8)

2.1 Some cohomology computations and the rectangular part of Db(IGr(3, 8))

2.1.1 The rectangular part

2.1.2 An extra bundle

2.1.3 Some semiorthogonalities

2.2 The bicomplex

2.2.1 Koszul and staircase complexes

2.2.2 Bicomplex

2.2.3 Some properties of the bicomplex

2.3 The non-rectangular part

2.3.1 Extra objects

2.3.2 Exceptionality

2.3.3 Exceptional collection

2.4 Fullness

2.4.1 Adding some objects

2.4.2 Fullness

2.5 Applications

2.5.1 Exceptional collections on flag varieties of Sp(8)

2.5.2 Fractional Calabi-Yau categories

2.5.3 Residual category

3 The Cayley Grassmannian

3.1 Quadric bundles constructions

3.1.1 Two general constructions

3.1.2 Quadrics containing the Veronese surface

3.1.3 Quadrics containing the Segre variety

3.1.4 Auxiliary lemma

3.2 G2-action

3.3 Self-dualities

3.3.1 Quadric bundles induced by the quadratic form

3.3.2 First self-duality

3.3.3 Quadric bundles induced by wedge product

3.3.4 Second self-duality

3.4 Fullness

3.4.1 Adding some objects

3.4.2 Fullness

3.5 Cohomology computations

3.5.1 Cohomology

3.5.2 Exceptionality

3.5.3 Residual category

3.6 Geometric constructions

3.6.1 The Cayley Grassmannian as the Hilbert scheme

3.6.2 The Hilbert scheme of lines on the Cayley Grassmannian

Conclusion

Bibliography

Введение диссертации (часть автореферата) на тему «Строение производных категорий и геометрия многообразий Фано в грассманианах»

Introduction

A fundamental question in algebraic geometry is to find invariants for a given class of geometric objects. In the case of algebraic varieties, natural invariants are the cohomology and the Chow groups. In looking for finer invariants one could think of the category of coherent sheaves; however, as it was shown by Gabriel in [16] two projective varieties are isomorphic if and only if the respective categories of coherent sheaves are equivalent. A straightforward extension of this idea is to look at the derived category of coherent sheaves. Derived categories were defined by Verdier in his thesis [45] back in the 1960s and were introduced as the right framework for the theory of duality developed by Grothendieck, see for example [17].

The origin of the theory as treated in the thesis goes back to celebrated Beilinson's paper [1], where he gave a simple characterization of the derived categories of projective spaces. Derived categories of coherent sheaves appeared again in the fundamental paper by Mukai [36], where geometrically motivated equivalences between derived categories of non-isomorphic varieties was constructed.

A radical change of perspective in connection with derived categories took place with a result of Bondal and Orlov, see [4, 5], according to which the derived category of a smooth projective variety Db(X) fully determines the variety X whenever the canonical bundle is either ample or anti-ample.

In general, the structure of Db(X) is quite difficult to describe. One of the approaches to this problem is to split Db(X) into smaller pieces, which leads to the notion of a semiorthogonal decomposition. Most interesting examples of semiorthogonal decompositions come from Fano varieties, i.e. varieties with the anti-ample canonical class. Fano varieties always possess an interesting semiorthogonal decomposition and, moreover, structure of components of this decomposition is closely related to the geometric propeties of a variety. In contrast, for varieties with trivial (or globally generated) canonical class there are no nontrivial decompositions of Db(X), see [6] and [23].

In the important case where Db(X) possesses a full exceptional collection (Ex,E2,... ,Em) the decomposition is as simple as possible: in this case every object of Db(X) admits a unique filtration with i-th subquotient being a direct sum of shifts of the objects Ej. Therefore, an exceptional collection can be considered as a kind of basis for the bounded derived category.

It is important to stress that having a full exceptional collection is a very restrictive condition on a variety X. For example, the Grothendieck group of the variety must necessarily be free, and the Hodge numbers hp'q (X) should be zero whenever p = q.

In terms of exceptional collections the result of Beilinson [1] can be formulated as follows: the

collection of line bundles O, O(1),..., O(n) on P" is a full exceptional collection. In 1988 Kapranov [21] generalized this result and constructed full exceptional collections on Grassmannians and flag varieties of groups SLn and on smooth quadrics. It has been conjectured afterwards that:

Conjecture 0.0.1. If G is a semisimple algebraic group over an algebraically closed field of characteristic 0 and P C G is a parabolic subgroup then there is a full exceptional collection of vector bundles on G/P.

The conjecture easily reduces to the case when G is a simple group and P is its maximal parabolic subgroup, see [30, Section 1.2]. In the case of simple G and maximal P the conjecture is known to be true for the following series (we use the Bourbaki indexing of simple roots):

• G is of Dynkin type A: An/Pk = Gr(k, n + 1), see [21]

• G is of Dynkin type B:

- Bn/Pi = Q2n-1 is a quadric, see [21]

- Bn/P2 = OGr(2, 2n +1), see [25]

• G is of Dynkin type C:

- Cn/Pi = P2n-1, see [1]

- Cn/P2 = IGr(2, 2n), see [25]

- Cn/Pn = IGr(n, 2n), see [13]

• G is of Dynkin type D:

- Dn/Pi = Q2n-2, see [21];

- Dn/P2 = OGr(2, 2n), see [32];

and for several sporadic cases. Besides that, an exceptional collection of maximal possible length (equal to the rank of the Grothendieck group) has been constructed on G/P for all classical groups (i.e. for groups of Dynkin types ABCD) and all their maximal parabolic subgroups, see [30]; however the fullness of these collections is not yet known.

There is even stronger conjecture by Lunts that the bounded derived category of a cellular variery possesses a full exceptional collection, see [11, Conjecture 1.2]. While the general conjecture remains widely open, it is interesting to provide new examples of full exceptional collections that could shed some light upon the general case.

The aim of the thesis is to construct full exceptional collections in the bounded derived categories of coherent sheaves for two Fano varieties in Grassmannians.

The first variety corresponds to the case (C4,P3) in Conjecture 0.0.1, where the group is the symplectic group G = Sp(8) and the parabolic subgroup corresponds to the third simple root. The corresponding homogeneous space IGr(3, 8) is the Grassmannian of 3-dimensional isotropic subspaces in

an 8-dimensional symplectic vector space. This result was already generalized to IGr(3,10) by Novikov in [37].

The second variety is the so called Cayley Grassmannian CG, the subvariety of the Grassman-nian Gr(3, 7) parametrizing 3-dimensional subspaces that are annihilated by a general 4-form. The Cayley Grassmannian CG is a spherical variety with respect to an action of the exceptional simple Lie group G2. In fact, CG is a smooth projective symmetric G2-variety and it first appeared in [42], where such varieties with Picard number one were classified. Most of the varieties in this classification are homogeneous under their full automorphism group, or are just hyperplane sections of homogeneous spaces. One of the remarkable properties of CG is that it is not homogeneous, but still possesses a full exceptional collection consisting of vector bundles, which supports Lunts's conjecture in this case.

The geometry and cohomology of CG were studied in [34] and [2]. In particular, in [2] the semisim-plicity of the small quantum cohomology ring of CG was proved. Thus, the result confirms in this case Dubrovin's conjecture predicting that the semisimplicity of the quantum cohomology ring implies the existence of a full exceptional collection.

The exceptional collections that we construct are Lefschetz collections, [25, 27, 26]. Recall that a Lefschetz exceptional collection with respect to a line bundle L is just an exceptional collection which consists of several blocks, each of them is a sub-block of the previous one twisted by L, see Definition 1.1.6 for more details. If all blocks are the same, the collection is called rectangular.

Lefschetz collections are the necessary ingredient for the theory of homological projective duality, that was introduced by Kuznetsov in [26] as a homological extension of the classical projective duality. If algebraic varieties X and Y in dual projective spaces are homologically projectively dual, then the orthogonal linear sections of X and Y admit semiorthogonal decompositions with an equivalent nontrivial component. Another application of Lefschetz collections is categorical resolutions of singularities, see [28].

Notation and conventions

Let k be an algebraically closed field of characteristic zero. For any k-vector space W we denote by A the wedge product operation for skew forms and polyvectors, and by j the convolution operation

ApW 0 Aq Wv Ap-q W (if p > q),

induced by the natural pairing W 0 Wv ^ k.

If p = n = dim(W) and 0 = u e AnW, the convolution with u gives an isomorphism

AqWv ~ An-qW, f ^ fv := u j f.

This isomorphism is canonical up to rescaling (since u is unique up to rescaling). We say that fv is the dual of f.

We say that a q-form f e AqWv annihilates a k-dimensional subspace U C W, if k < q and f j AkU = 0. Analogously, we say that U C W is isotropic for f if q < k and AkU j f = 0, i.e. f|^ = 0.

We denote by Gr(k, W) the Grassmannian of k-dimensional vector subspaces in W. The tautological vector subbundle of rank k on Gr(k, W) is denoted by Uk C W 0 OGr(k,W). The quotient bundle is denoted simply by Qn-k, and for its dual we use the notation U_k := Q^_k. We often use the following tautological exact sequences on Gr(k, W):

0 ^ Uk ^ W 0 O ^ Qn_k ^ 0 (1)

0 ^ U_k ^ Wv 0 O ^ Uvk ^ 0. (2)

The point of the Grassmannian corresponding to a subspace Uk c W is denoted by [Uk], or even just Uk. We recall that the line bundle det Uk — det Qn_k is the ample generator of Pic(Gr(k, W)); we will denote it by O(Hk).

Since we will mostly work with Gr(3, W), we abbreviate its tautological bundle by U, unless this leads to a confusion. The line bundle O(H3) will be denoted by O(1). We will frequently use the following natural isomorphism on Gr(3, W):

A2U — Uv(-1). (3)

On Gr(3, 7) we will also use

A2U^" — A2Q4(-1). (4)

For both exceptional collections we will need the following vector bundle on Gr(3, W)

£2^Uv := (Uv 0 A2Uv)/A3Uv. (5)

The canonical line bundle of a variety X will be denoted by uX.

The isotropic Grassmannian IGr(3, 8)

Recall that U denote the tautological sub-bundle on IGr(3, 8). Denote by F and F' the following collections of vector bundles on IGr(3, 8):

F :=( O,Uv,S2Uv,A2Uv, S2^), (6)

F' := (S2,:Uv(-1), O, Uv,S2Uv, A2Uv ), (7)

where S2,1 Uv = (Uv 0 S2Uv)/S3Uv. We will denote by F(i) and F'(i) the collections consisting of the corresponding five vector bundles twisted by O(i), and in the same way the subcategories of Db(IGr(3, 8)) generated by these. We denote by L and R the left and right mutation functors, see the precise definition in Subsection 1.1. The second main result of this thesis is the following theorem.

Theorem 0.0.2. The objects

T := (Lf(S3,1Uv))[-3] and F := RS2,iuv(_i)(T) are equivariant vector bundles on IGr(3, 8).

The collections of 32 vector bundles on IGr(3, 8)

F, F , F(1), F (1), F (2), F (3), F (4), F (5), and

T, F', T(1), F'(1), F'(2), F'(3), F'(4), F'(5)

are full Lefschetz collections with respect to the line bundle O(1). The collections of 32 vector bundles on IGr(3, 8)

F, F , F (1), F (2), F(3), F (3), F (4), F (5), and

T, F', F'(1), F'(2), T(3), F'(3), F'(4), F'(5)

are full rectangular Lefschetz collections with respect to the line bundle O(3).

Descriptions of the vector bundles T and F can be found in Lemma 2.3.1 and Remark 2.3.9 respectively. In particular, the bundle F is isomorphic to a twist of the vector bundle £2>0>0;1 constructed in [30].

A significant part of the proof of Theorem 0.0.2 is based on the study of a certain interesting bicomplex

0 -E3'2uv(-3) -V8 (g> E2'1Uv(-2) —A2V8 (g> uv(-1) ->■ A4V8 (g> o -)■ A2V8 ( A2uv -)■ V8 ( S2'1uv -)■ E3'1uv -)■ 0

0 -)■ E3'3uv(-4) -)■ V8 ( E2'2UV(-3) -)■ A2V8 ( A2UV(-2) —A3 V8 ( o(-1) —> A2V8 ( o -V8 ( Uv -S2UV -0,

of vector bundles on IGr(3, 8), where V8 is the tautological 8-dimensional representation of Sp(8). This bicomplex is Sp(8)-equivariant, its lines are exact and are obtained as the restrictions of the so-called staircase complexes (see [14]) from Gr(3, 8). The vector bundle T is identified with the cohomology of the truncation (2.10) or (2.11) of this bicomplex, and using the bicomplex we prove an isomorphism

lf'(i),f'(2) (T (3)) = T (1)[4],

which is crucial for the proof of completeness of the above exceptional collections.

To prove the fullness of the exceptional collections in Theorem 0.0.2 we first prove that some special objects lie in the subcategory D of Db(IGr(3, 8)) generated by each of these collections. After that we consider the isotropic flag variety IFl(2, 3; 8) with its two projections

IFl(2, 3; 8)

IGr(2, 8) IGr(3, 8)

The first arrow is a P3-fibration. Using a certain variant of the Lefschetz exceptional collection on IGr(2, 8) from [25] and Orlov's projective bundle formula we construct a very special full exceptional collection on IFl(2, 3;8). The main property of this exceptional collection is that the pushforwards along the second arrow (which is a P2-fibration) of almost all objects constituting it are contained in the subcategory D, and for the few objects that do not enjoy this property, the pushforwards are

contained in the subcategory F(6) C Db(IGr(3, 8)). It follows from this that every object of Db(IGr(3, 8)) contained in the orthogonal to the subcategory D, belongs to F(6). The trivial observation

n F(6) = 0

(that follows immediately from the Serre duality on IGr(3, 8)) then shows that fD = 0, and completes the proof of the fullness of the collections.

The Cayley Grassmannian

There are several description of the Cayley Grassmannian CG.

The first description explains the name. We consider the complexified nonassociative 8-dimensional Cayley algebra O. The Cayley Grassmannian CG can be defined as the set of 4-dimensional subalgebras of O; it is a closed subvariety in the Grassmannian Gr(4, O) — Gr(4, 8) of 4-dimensional vector subspaces in a 8-dimensional vector space O. All these subalgebras contain the unit element e of O, so we can instead define CG as a closed subvariety in Gr(3, O/(C ■ e)) — Gr(3, 7), that parametrizes the imaginary parts of the four-dimensional subalgebras of O.

The second description makes sense over any algebraically closed field k of characteristic 0. We consider the Grassmannian Gr(3, V7) of 3-dimensional vector subspaces in a 7-dimensional vector space V7. Let us fix a general global section of (1), that is a general skew-symmetric 4-form

A e A4V7v.

By definition, the Cayley Grassmannian CG is the zero locus of A e H0(Gr(3,V7),Uf(1)). Explicitly, CG parametrizes 3-dimensional vector subspaces of V7 annihilated by A, i.e., U C V7 such that A(u1,u2,u3, —) = 0 for any u1,u2,u3 e U. From this description we immediately deduce that the Cayley Grassmannian is a smooth Fano eightfold of index 4.

The equivalence of these two descriptions comes from the fact that the stabilizer of a general 4-form on V7 is isomorphic to the algebraic group G2, that is the automorphism group of the octonions, see [34] for more details.

Also the Cayley Grassmannian can be described as the Hilbert scheme of conics on the adjoint homogeneous variety G2d, see Subsection 3.6.1 for the details.

To describe a full Lefschetz collection on CG, we need to make some preparations. The exceptional collection on CG that we construct is a Lefschetz collection: it consists of four blocks with respect to the PKicker line bundle O(1) of Gr(3, V7) restricted to CG. The common part of these four blocks (the rectangular part of the Lefschetz collection, see Definition 1.1.6) consists of three vector bundles (O,Uv, A2Uv).

To describe the nonrectangular part of the exceptional collection on CG we need to define an additional vector bundle. Note that on CG we have an embedding of vector bundles

i\: A2U ^ A2Uf

given by A, see Lemma 3.3.11 for more details. So on CG we can define the quotient bundle A2Uf/A2U. For the exceptional collection we will need its dual

R := (A2Uf/A2U)V.

The first main result of the thesis is the following theorem.

Theorem 0.0.3. The collection of 15 vector bundles on CG

{0,UV, A2UV, R, £2>1UV; 0(1),UV(1), A2UV(1), R(1); 0(2),UV(2), A2UV(2); 0(3),UV(3), A2UV(3)} (8)

"--' v-V-"--"--'

block 1 block 2 block 3 block 4

is a full Lefschetz collection with respect to 0(1).

Let us sketch the idea of the proof of Theorem 0.0.3. First, it will be more convenient for us to prove the fullness of a slightly different collection (3.34). We cover CG with the family of sub-

varieties CG/ 4 CG defined as zero loci of sufficiently general global sections f E H0(CG,UV). It turns out that CG/ is isomorphic to the isotropic Grassmannian IGr(3, 6) and Db(CG/) posseses a full exceptional collection, so using standard arguments from [25] we reduce the problem to the checking that some objects lie in the subcategory D C Db(CG) generated by the collection (3.34): it is enough to show that S2UV(m) E D for m = 0,1, 2 and that £2>1UV, £2>1UV(1) e D. This check is the most interesting part of the proof, so let us describe it more precisely.

First, we present two general constructions with quadric bundles. Roughly speaking, the first construction shows that we can glue two quadric bundles with isomorphic cokernel sheaves into a quadric bundle that determines self-dual isomorphism, see Proposition 3.1.1 for the details; and the second one allows to construct a new quadric bundle from a quadric bundle with the cokernel sheaf supported on a Cartier divisor, see Proposition 3.1.2. Using these contructions we obtain several interesting G2-equivariant quadric bundles on CG. Using again Proposition 3.1.1 we glue the obtained quadric bundles into the following self-dual vector bundles on CG

Eio(1) ^ £Vo and 8ia(-1) ^ £Ve, (9)

where E10 is an extension of Uf by S2U and E16 is an extension of A2UV © 0(1) by Uf ® A2UV. Using (9) and some standard exact sequences we prove that S2UV(m) E D for m = 0,1,2 and that S2'1UV, S2'1UV(1) E D.

Заключение диссертации по теме «Другие cпециальности», Гусева Ляля Андреевна

Conclusion

This thesis is devoted to the study of full exceptional collections of vector bundles in the derived categories of two Fano varieties in Grassmannians.

The first variety IGr(3,8) is the isotropic Grassmannian of 3-dimensional isotropic subspaces in an 8-dimensional symplectic vector space. This is the rational homogeneous variety for the symplectic group Sp(8). The second variety is the Cayley Grassmannian CG, the subvariety of the Grassmannian Gr(3, 7) parametrizing 3-dimensional subspaces that are annihilated by a general 4-form. The Cayley Grassmannian CG is a spherical variety with respect to an action of the exceptional simple Lie group G2. The exceptional collections that we construct are Lefschetz collections.

In Chapter 2 we construct full exceptional Lefschetz collections on the isotropic Grassmannian IGr(3, 8). To construct these collections we construct a certain interesting Sp(8)-equivariant bi-complex on IGr(3,8) and investigate its properties. Also we compute the residual category of IGr(3, 8), and construct a pair of (fractional) Calabi-Yau categories related to a half-anticanonical section and anticanonical double covering of IGr(3, 8).

In Chapter 3 we construct full exceptional Lefschetz collection on the Cayley Grassmannian. To construct the collection we present general constructions with quadric bundles. The first construction shows that we can glue two quadric bundles with isomorphic cokernel sheaves into a non-degenerate quadric bundle; and the second one allows to construct a new quadric bundle from a quadric bundle with the cokernel sheaf supported on a Cartier divisor. We describe the residual category for the constructed collection on CG. Also we present several geometric constructions for the Cayley Grassmannian: we show that CG is isomorphic to the Hilbert scheme of conics on and describe the Hilbert scheme of lines on CG.

The obtained results can be used to further investigation of the derived categories of the rational homogeneous varieties and spherical varieties. For example, the construction of Sp(8)-equivariant bicomplex on IGr(3, 8) was already generalized to all isotropic Grassmannians IGr(3, 2n).

Список литературы диссертационного исследования кандидат наук Гусева Ляля Андреевна, 2023 год

Bibliography

[1] Beilinson, Alexander. Coherent sheaves on Pn and problems in linear algebra. (Russian) Funkt-sional. Anal. i Prilozhen. 12 (1978), no. 3, 68-69.

[2] Benedetti, Vladimiro; Manivel, Laurent. The small quantum cohomology of the Cayley Grassman-nian. Internat. J. Math. 31 (2020), no. 3, 2050019, 23 pp.

[3] Bondal, Alexey. Representations of associative algebras and coherent sheaves. (Russian); translated from Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 25-44 Math. USSR-Izv. 34 (1990), no. 1, 23-42.

[4] Bondal, Alexei; Orlov, Dmitri. Derived categories of coherent sheaves. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 47-56, Higher Ed. Press, Beijing, 2002.

[5] Bondal, Alexei; Orlov, Dmitri. Reconstruction of a variety from the derived category and groups of autoequivalences. Compositio Math. 125 (2001), no. 3, 327-344.

[6] Bridgeland, Tom. Equivalences of triangulated categories and Fourier-Mukai transforms. Bull. London Math. Soc. 31 (1999), no. 1, 25-34.

[7] Chen, Dawei; Coskun, Izzet. Stable base locus decompositions of Kontsevich moduli spaces. Michigan Math. J. 59 (2010), no. 2, 435-466.

[8] Chung, Kiryong; Hong, Jaehyun; Kiem, Young-Hoon. Compactified moduli spaces of rational curves in projective homogeneous varieties. J. Math. Soc. Japan 64 (2012), no. 4, 1211-1248.

[9] Debarre, Olivier; Kuznetsov, Alexander. Gushel-Mukai varieties: classification and birationalities. Algebr. Geom. 5 (2018), no. 1, 15-76.

[10] Ein, Lawrence; Shepherd-Barron, Nicholas. Some special Cremona transformations. Amer. J. Math. 111 (1989), no. 5, 783-800.

[11] Elagin, Alexey; Lunts, Valery. On full exceptional collections of line bundles on del Pezzo surfaces. Mosc. Math. J. 16 (2016), no. 4, 691-709.

[12] Faenzi, Daniele; Manivel, Laurent. On the derived category of the Cayley plane II. Proc. Amer. Math. Soc. 143 (2015), no. 3, 1057-1074.

[13] Fonarev, Anton. Full exceptional collections on Lagrangian Grassmannians. Int. Math. Res. Not. IMRN 2022, no. 2, 1081-1122.

[14] Fonarev, Anton. Minimal Lefschetz decompositions of the derived categories for Grassmannians. (Russian); translated from Izv. Ross. Akad. Nauk Ser. Mat. 77 (2013), no. 5, 203-224 Izv. Math. 77 (2013), no. 5, 1044-1065.

[15] Fulton, William; Harris, Joe. Representation theory. A first course. Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991.

[16] Gabriel, Pierre. Des catégories abéliennes. (French) Bull. Soc. Math. France 90 (1962), 323-448.

[17] Hartshorne, Robin. Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20 Springer-Verlag, Berlin-New York 1966 vii+423 pp.

[18] Huybrechts, D. Fourier-Mukai transforms in algebraic geometry. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2006.

[19] Iliev, Atanas; Manivel, Laurent. Fano manifolds of degree ten and EPW sextics. Ann. Sci. Ec. Norm. Super. (4) 44 (2011), no. 3, 393-426.

[20] Kapranov, Mikhail. Derived category of coherent sheaves on Grassmann manifolds. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 1, 192-202.

[21] Kapranov, Mikhail. On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92 (1988), no. 3, 479-508.

[22] Kapustka, Michal; Ranestad, Kristian. Vector bundles on Fano varieties of genus ten. Math. Ann. 356 (2013), no. 2, 439-467.

[23] Kawatani, Kotaro; Okawa, Shinnosuke Nonexistence of semiorthogonal decompositions and sections of the canonical bundle. Preprint arXiv:1508.00682, 2018.

[24] Kuznetsov, Alexander. Calabi-Yau and fractional Calabi-Yau categories. J. Reine Angew. Math. 753 (2019), 239-267.

[25] Kuznetsov, Alexander. Exceptional collections for Grassmannians of isotropic lines. Proc. Lond. Math. Soc. (3) 97 (2008), no. 1, 155-182.

[26] Kuznetsov, Alexander. Homological projective duality. Publ. Math. Inst. Hautes Etudes Sci. No. 105 (2007), 157-220.

[27] Kuznetsov, Alexander. Hyperplane sections and derived categories. (Russian); translated from Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006), no. 3, 23-128 Izv. Math. 70 (2006), no. 3, 447-547.

[28] Kuznetsov, Alexander. Lefschetz decompositions and categorical resolutions of singularities. Selecta Math. (N.S.) 13 (2008), no. 4, 661-696.

[29] Kuznetsov, Alexander. Kuchle fivefolds of type c5. Math. Z. 284 (2016), no. 3-4, 1245-1278.

[30] Kuznetsov, Alexander; Polishchuk, Alexander. Exceptional collections on isotropic Grassmannians. J. Eur. Math. Soc. (JEMS) 18 (2016), no. 3, 507-574.

[31] Kuznetsov, Alexander; Smirnov, Maxim. On residual categories for Grassmannians. Proc. Lond. Math. Soc. (3) 120 (2020), no. 5, 617-641.

[32] Kuznetsov, Alexander; Smirnov, Maxim. Residual categories for (co)adjoint Grassmannians in classical types. Compos. Math. 157 (2021), no. 6, 1172-1206.

[33] Landsberg, J. M.; Manivel, L. The sextonions and E71. Adv. Math. 201 (2006), no. 1, 143-179.

[34] Manivel, Laurent. The Cayley Grassmannian. J. Algebra 503 (2018), 277-298.

[35] Mukai, Shigeru. Biregular classification of Fano 3-folds and Fano manifolds of coindex 3. Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 9, 3000-3002.

[36] Mukai, Shigeru. Duality between D(X) and D(X) with its application to Picard sheaves. Nagoya Math. J. 81 (1981), 153-175.

[37] Novikov, Aleksandr. Lefschetz Exceptional Collections on Isotropic Grassmannians. www.hse.ru/en/edu/vkr/369850047

[38] Onishchik, A. L.; Vinberg, E. B. Lie groups and algebraic groups. Translated from the Russian and with a preface by D. A. Leites. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1990.

[39] Orlov, Dmitri. Projective bundles, monoidal transformations, and derived categories of coherent sheaves. (Russian); translated from Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 852-862 Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 133-141.

[40] Pasquier, Boris. On some smooth projective two-orbit varieties with Picard number 1. Math. Ann. 344 (2009), no. 4, 963-987.

[41] Polishchuk, Alexander; Samokhin, Alexander. Full exceptional collections on the Lagrangian Grassmannians LG(4, 8) and LG(5,10). J. Geom. Phys. 61 (2011), no. 10, 1996-2014.

[42] Ruzzi, Alessandro. Geometrical description of smooth projective symmetric varieties with Picard number one. Transform. Groups 15 (2010), no. 1, 201-226.

[43] Samokhin, Alexander. On the derived category of coherent sheaves on a 5-dimensional Fano variety. C. R. Math. Acad. Sci. Paris 340 (2005), no. 12, 889-893.

[44] Samokhin, Alexander. The derived category of coherent sheaves on . (Russian); translated from Uspekhi Mat. Nauk 56 (2001), no. 3(339), 177-178 Russian Math. Surveys 56 (2001), no. 3, 592-594.

[45] Verdier, Jean-Louis. Catégories dérivées: quelques résultats (état 0). (French) (Institut des hautes études scientifiques, Paris, 1965)

[46] Weyman, Jerzy. Cohomology of vector bundles and syzygies. Cambridge Tracts in Mathematics, 149. Cambridge University Press, Cambridge, 2003.