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Derived categories of Grassmannians: a survey A. V. Fonarev Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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    1. Derived categories and semiorthogonal decompositions1.1. Derived categoriesIn the middle of the 20th century, Alexander Grothendieck realized that it is not sufficient to work with classical derived functors if one wants to formulate what is now called coherent or Serre–Grothendieck–Verdier duality, a relative version of Serre duality. Together with his student Jean-Louis Verdier, he developed the notion of a derived category. As it often happens with Grothendieck’s ideas, the underlying insight is beautiful in its simplicity. By that time, it had been known for a while that homological constructions often operate with various kinds of resolutions constructed for modules, sheaves of modules, coherent sheaves, and so on. In modern terms, one could say that an abelian category $\mathcal{A}$ with enough projectives (or injectives) is equivalent to the homotopy category of projective (injective) resolutions. A resolution is, by definition, a complex with reasonable terms concentrated only in negative (or positive) degrees which has only one possibly non-trivial cohomology group in degree 0. What happens if we remove this restriction on cohomology? Meanwhile, when we deal with projective and/or injective resolutions, we treat them up to homotopy equivalence. We do this in order to identify resolutions of the same objects, that is, resolutions with isomorphic cohomology. Which equivalence do we put on arbitrary complexes? The answer that Grothendieck and Verdier give is simple and beautiful. Begin with an abelian category $\mathcal{A}$. Consider the category of complexes $C(\mathcal{A})$ (one can consider various versions of this category: bounded complexes, unbounded complexes, complexes bounded above or below, complexes with bounded cohomology, and so on). Now, invert formally all quasi-isomorphisms, that is, morphisms of complexes that induce isomorphisms on cohomology. That is precisely the derived category $D(\mathcal{A})$ of $\mathcal{A}$. While the idea is beautiful and simple, there is plenty of technical details to fill in. First, localization (the process of inverting a class of morphisms) is a rather delicate procedure. Fortunately, it can be done rather painlessly by first passing to the homotopy category $H(\mathcal{A})$: the category of complexes where morphisms are considered up to homotopy, just like we did it for resolutions. Since homotopic morphisms induce the same morphisms on homology, this first step does no harm. Now, the second step is to invert quasi-isomorphisms in the homotopy category. It turns out that the latter step can be done rather neatly since in the homotopy category quasi-isomorphisms satisfy the so-called Ore conditions that one can find in non-commutative algebra. Once the derived category is constructed, one should naturally ask questions about its structure. Say, we have started with a $\mathsf{k}$-linear abelian category over a field $\mathsf{k}$. While the derived category is naturally $\mathsf{k}$-linear and additive, it is no longer abelian. Grothendieck and Verdier came up with a beautiful notion of a triangulated category, which allows one to do much of the homological algebra just by considering exact triangles instead of short exact sequences. Given a smooth projective variety $X$ over a field $\mathsf{k}$, one defines its bounded derived category $D(X)$ as the derived category $D(\mathsf{Coh}(X))$ of the category of coherent sheaves $\mathsf{Coh}(X)$ on $X$. While its construction is simple, for many years derived categories remained mysterious black boxes with unfathomable internals. It all changed with the appearance of the paper [1] by Beilinson, who in 1978 gave a very explicit description of the bounded derived category of a projective space. That work was followed by papers by Kapranov, who gave a similar description of the derived categories of classical Grassmannians and quadrics. In both cases it was shown that the bounded derived categories of coherent sheaves on the corresponding varieties admit full exceptional collections. Since then, it has been conjectured that the same holds for the derived categories of all rational homogeneous varieties. In the present survey we discuss what we know about this conjecture. 1.2. Semiorthogonal decompositions and exceptional collectionsLet $\mathcal{T}$ be a triangulated category. We want to somehow split $\mathcal{T}$ into smaller pieces. One way of doing it is using the notion of a semiorthogonal decomposition. Definition 1.1. A semiorthogonal decomposition of a triangulated category $\mathcal{T}$ is a collection of full triangulated subcategories $\mathcal{A}_1,\mathcal{A}_2,\dots,\mathcal{A}_n\subset \mathcal{T}$ such that
A natural question is whether one can construct a semiorthogonal decomposition starting with a full triangulated subcategory $\mathcal{A}\subset \mathcal{T}$. Definition 1.2. A full triangulated subcategory $\mathcal{A}\subset \mathcal{T}$ is called admissible if the inclusion functor $\iota\colon\mathcal{A}\to \mathcal{T}$ admits both right and left adjoint functors $\iota^*,\iota^!\colon\mathcal{T}\to \mathcal{A}$. If $\mathcal{A}\subset \mathcal{T}$ is an admissible full triangulated subcategory, one can immediately construct two semiorthogonal decompositions of $\mathcal{T}$. First, define the right and left orthogonals to $\mathcal{A}$ as
$$
\begin{equation*}
\mathcal{A}^\perp = \langle X \in \mathcal{T} \mid \mathsf{Hom}(Y, X)=0\text{ for all }Y\in \mathcal{A} \rangle
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
{}^\perp{\mathcal{A}}= \langle X \in \mathcal{T} \mid \mathsf{Hom}(X, Y)=0\text{ for all }Y\in \mathcal{A} \rangle.
\end{equation*}
\notag
$$
Lemma 1.3 ([6]). If $\mathcal{A}\subset \mathcal{T}$ is an admissible full triangulated subcategory, then one has the semiorthogonal decompositions Due to fundamental results of Bondal and Kapranov (see [6]) all the subcategories that we consider in the present survey are admissible. From now on we assume that all our categories are $\mathsf{k}$-linear, where $\mathsf{k}$ is a field. The simplest example of an admissible subcategory is the one generated by an exceptional object. If $E\in \mathcal{T}$ is an exceptional object, the smallest full triangulated subcategory $\langle E \rangle$ generated by $E$ is simply equivalent to the derived category of finite-dimensional vector spaces over $\mathsf{k}$, and the latter is equivalent to the category of finite-dimensional $\mathbb{Z}$-graded vector spaces. Alternatively, $\langle E \rangle$ is equivalent to the bounded derived category $D(\mathsf{Spec}\mathsf{k})$ of a point. If a triangulated category admits a semiorthogonal decomposition such that every component is generated by an exceptional object, one says that the category admits a full exceptional collection. Here is a more common definition, where from now on for arbitrary objects in a triangulated category $\mathcal{T}$ by $\mathsf{Ext}^i(X,Y)$ we mean $\mathsf{Hom}_{\mathcal{T}}(X,Y[i])$. Definition 1.5. An exceptional collection in a triangulated category $\mathcal{T}$ is a sequence of exceptional objects $E_1,E_2,\dots,E_n$ such that for all $1\leqslant i < j\leqslant n$ one has $\mathsf{Ext}^\bullet(E_j,E_i)=0$. An exceptional collection is full if there is a semiorthogonal decomposition $\mathcal{T}=\langle \langle E_1\rangle, \langle E_2\rangle,\dots,\langle E_n\rangle\rangle$. One usually drops angle brackets for the subcategory generated by an exceptional object. For instance, one usually denotes by $\langle E_1,E_2,\dots,E_n\rangle$ the strictly full triangulated subcategory generated by an exceptional collection $E_1,E_2,\dots,E_n$. We are ready to give the first formulation of the pioneering result which launched a vast area of research in algebraic geometry. We warn the reader that the statement in [1] may look different, but this is only due to the fact that the terminology had not been developed at the time of publication. Theorem 1.6 ([1]). The bounded derived category $D(\mathbb{P}^n)$ of the projective space $\mathbb{P}^n$ over a field $\mathsf{k}$ admits a full exceptional collection: A similar result was soon proved by Kapranov for classical Grassmannians and quadrics. Before we present explicit formulations of these results, we can state the main folklore conjecture that came out of the work of Beilinson and Kapranov. Conjecture 1.7. Let $\mathbf{G}$ be a semisimple algebraic group over an algebraically closed field $\mathsf{k}$ of characteristic zero, and let $\mathbf{P}\subset \mathbf{G}$ be a parabolic subgroup. There is a full exceptional collection in the bounded derived category of coherent sheaves $D(\mathbf{G}/\mathbf{P})$. Let us make a few remarks on Conjecture 1.7. First, from the structure theory of semisimple algebraic groups, parabolic reduction, and some generalities on derived categories, it is rather easy to reduce the statement to the case when $\mathbf{G}$ is simple and $\mathbf{P}$ is maximal parabolic. Varieties of the form $\mathbf{G}/\mathbf{P}$ are often called generalized Grassmannians, and we restrict our attention to them. Next, one can impose additional conditions on exceptional objects. Since $\mathbf{G}/\mathbf{P}$ comes with a natural action of $\mathbf{G}$, it is natural to ask for the collection to consist of $\mathbf{G}$-equivariant objects. However, this is a harmless request since it was shown by Polishchuk in [34], Lemma 2.2, that any exceptional object in $D(\mathbf{G}/\mathbf{P})$ admits a $\mathbf{G}$-equivariant structure. Another restriction one can impose is to ask that all the objects are pure sheaves and not complexes (thus equivariant vector bundles). In the strongest form one could ask for extra relations between the exceptional objects. Definition 1.8. An exceptional collection $E_1,E_2,\dots,E_n$ is called strong if for all $1\leqslant i < j\leqslant n$ one has $\mathsf{Ext}^t(E_i, E_j)=0$ for all $t\ne 0$. The strongest version of Conjecture 1.7 for generalized Grassmannians could be the following. Conjecture 1.9. Let $\mathbf{G}$ be a simple algebraic group over an algebraically closed field $\mathsf{k}$ of characteristic zero, and let $\mathbf{P}\subset \mathbf{G}$ be a maximal parabolic subgroup. There is a full strong exceptional collection in the bounded derived category of coherent sheaves $D(\mathbf{G}/\mathbf{P})$ which consists of vector bundles. In the rest of the survey we discuss what is known about Conjecture 1.9. Before we begin, we need to discuss some other general notions from the theory of derived categories. 1.3. (Graded) dual exceptional collectionsThe notion of a dual exceptional collection could have appeared in the very same paper by Beilinson [1] and is analogous to the notion of a dual basis. Formally, it was studied by Bondal in [4]. However, since then several conventions related to shifts of objects have been used in literature. We will take a slightly alternative route and work with the definitions introduced in [16]. First, it will be convenient to work with exceptional collections indexed by a partially ordered set (poset). Many of our collections will naturally be indexed by some posets of Young diagrams. Definition 1.10. An exceptional collection indexed by a poset $(\mathcal{P},\preceq)$ is a collection of exceptional objects $\{E_x\}_{x\in\mathcal{P}}$ such that $\mathsf{Ext}^\bullet(E_x,E_y)=0$ unless $x\preceq y$. If $\mathcal{P}=\{1,2,\dots,n\}$ with its total ordering, then we obtain the usual notion of an exceptional collection. However, if $\mathcal{P}$ has no comparable elements, then an exceptional collection indexed by $\mathcal{P}$ consists of pairwise orthogonal objects. Let us now assume that $\mathcal{P}$ is finite and graded: there exists a grading function $|-|\colon\mathcal{P}\to \mathbb{Z}_{\geqslant 0}$ such that on each connected component of $\mathcal{P}$ it attains value 0, and for any pair $y,x\in\mathcal{P}$ such that $y$ covers $x$ one has $|y|=|x|+1$. Lemma 1.11 ([16], Lemma 2.5). Let $\langle E_x\mid x\in\mathcal{P} \rangle$ be an exceptional collection indexed by a finite graded poset $\mathcal{P}$. For any $y\in \mathcal{P}$ there exists a unique (up to isomorphism) object $E^\circ_y\in \langle E_x\mid x\in\mathcal{P} \rangle$ such that The objects $E^\circ_y$ form an exceptional collection with respect to the opposite poset $\mathcal{P}^\circ$. This collection is called the graded left dual, andWhen $\mathcal{P}$ is linearly ordered, the definitions of the graded left dual exceptional collection and the left dual exceptional collection from [4] agree. Remark 1.12. The usual definition of a graded poset simply requires the existence of a grading function. That is, a function $\nu\colon\mathcal{P}\to\mathbb{Z}$ such that $\nu(y)=\nu(x)$ whenever $y$ covers $x$. For instance, the opposite poset $\mathcal{P}^\circ$ is graded if and only if $\mathcal{P}^\circ$ is. If one wants to define the graded right dual to a graded exceptional collection, then one has to take a grading function different from $|-|$. Since graded right duals do not appear in the present text, we leave the details to the reader. 1.4. Lefschetz decompositionsIn many cases the collections that have been constructed in the derived categories of generalized Grassmannians are Lefschetz. We begin with the notion of a Lefschetz semiorthogonal decomposition, originally introduced by Kuznetsov in the context of Homological Projective Duality. For simplicity, fix a smooth projective variety $X$ with a very ample line bundle $\mathcal{O}(1)$. Definition 1.13 ([22]). A Lefschetz decomposition of $D(X)$ is a collection of full triangulated subcategories $D(X)\supset \mathcal{A}_0\supset \mathcal{A}_1\supset\cdots\supset\mathcal{A}_{m-1}$ such that there is a semiorthogonal decomposition where $\mathcal{B}(i)$ denotes the image of a subcategory $\mathcal{B}\subset D(X)$ under the autoequivalence given by the tensor product $-\bigotimes \mathcal{O}(i)$. If, in addition, $\mathcal{A}_0=\mathcal{A}_1=\cdots=\mathcal{A}_{m-1}$, then the decomposition is called rectangular.Lefschetz decompositions are particularly nice. Imagine, one has found a rectangular Lefschetz decomposition whose initial block $\mathcal{A}_0$ is generated by a full exceptional collection. Then it extends to a full exceptional collection in the whole derived category: one should simply add the appropriate twists of the exceptional objects. There are particularly interesting cases when one has a natural Lefschetz decomposition that is not rectangular, thus the following definition. Definition 1.14 ([11]). A Lefschetz basis is a pair consisting of an exceptional collection $(E_1,E_2,\dots, E_n)$ together with a function $o\colon\{1,2,\dots,n\}\to \mathbb{Z}_{> 0}$, called the support function, such that the collection of subcategories $\mathcal{A}_i=\langle E_j \mid i < o(j) \rangle$ forms a Lefschetz decomposition of $D(X)$. A Lefschetz exceptional collection is a Lefschetz basis whose support function is weakly decreasing. Note that the definition of a Lefschetz basis immediately extends to exceptional collections graded by posets. Assume further that the variety $X$ is such that its canonical bundle $\omega_X$ is isomorphic to $\mathcal{O}(-m)$ for some positive $m>0$. Then it immediately follows from Serre duality that $m$ is the maximum number of blocks in any Lefschetz decomposition of $D(X)$. Definition 1.15 ([25]). Assume that $\omega_X\simeq \mathcal{O}(-m)$. The residual category of a Lefschetz decomposition $D(X)=\langle \mathcal{A}_0, \mathcal{A}_1(1), \dots,\mathcal{A}_{m-1}(m-1)\rangle$ is defined as which is the left orthogonal to its rectangular part2. Projective spaces and classical Grassmannians2.1. Projective spacesThe problem of constructing a full exceptional collection consists essentially of two parts. First, one should find an exceptional collection of appropriate length. Indeed, if a triangulated category admits a full exceptional collection, then its Grothendieck group $K_0$ is finitely generated and free. Since the rank of $K_0$ can often be computed via geometric means, we know what the length of a full exceptional collection should be if there is one at all. Once the objects are found (which seems to be an art on its own), one needs to check that they are exceptional and verify semiorthogonality. The last two parts are essentially cohomological computations, and there are plenty of tools developed specifically for these purposes. Next, one needs to somehow show that the collection is full, and in each case this seems to be a much harder problem. Let us see how Beilinson dealt with the two tasks in [1]. To align our notation with the more general case of classical Grassmannians, let us fix a vector space $V$ over $\mathsf{k}$ of dimension $n$. We follow the anti-Grothendieck convention under which $\mathbb{P}(V)$ parameterizes one-dimensional subspaces in $V$. Then $\mathbb{P}(V)$ is of dimension $(n-1)$, and Theorem 1.6 states that the derived category $D(\mathbb{P}(V))$ admits a full exceptional collection
$$
\begin{equation}
\langle \mathcal{O},\mathcal{O}(1),\dots,\mathcal{O}(n-1)\rangle.
\end{equation}
\tag{2}
$$
It is easy to check that the collection (2) is exceptional. Indeed, a classical computation of the cohomology of line bundles on a projective space implies that
$$
\begin{equation}
\mathsf{Ext}^\bullet(\mathcal{O}(j), \mathcal{O}(i))= H^\bullet(\mathbb{P}(V), \mathcal{O}(i-j))=\begin{cases} S^{i-j}V^* & \text{if}\ 1\leqslant j\leqslant i\leqslant n-1, \\ 0 & \text{if}\ 1\leqslant i < j\leqslant n-1. \end{cases}
\end{equation}
\tag{3}
$$
We thus simultaneously check both exceptionality of the objects from (2) and semiorthogonality. Fullness was checked by Beilinson via a trick that is now called the resolution of diagonal argument. For an arbitrary smooth projective variety one may consider the diagram where $p$ and $q$ are the projections onto the first and second multiple, respectively, and $\Delta$ denotes the diagonal embedding. With the use of the projection formula, the identity functor from $D(X)$ to itself can cleverly be rewritten as the composition
$$
\begin{equation}
\operatorname{Id}_{D(X)}\simeq q_*\bigl(p^*(-)\otimes \Delta_*\mathcal{O}_X\bigr),
\end{equation}
\tag{5}
$$
where, unless mentioned otherwise, all the functors are derived. Now, if one finds a good enough resolution for $\Delta_*\mathcal{O}_X$ in $D(X\times X)$, then one can deduce statements about generation in $D(X)$. Let us return to the case $X=\mathbb{P}(V)$. The structure sheaf of the diagonal $\Delta_*\mathcal{O}$ has a Koszul resolution of the form
$$
\begin{equation}
\begin{aligned} \, \nonumber 0&\to p^*\bigl(\Omega^{n-1}(n-1)\bigr)\otimes q^*\bigl(\mathcal{O}(-n+1)\bigr) \to\cdots \to p^*\bigl(\Omega^{1}(1)\bigr)\otimes q^* \bigl(\mathcal{O}(-1)\bigr) \\ &\to \mathcal{O}\to \Delta_*\mathcal{O} \to 0, \end{aligned}
\end{equation}
\tag{6}
$$
where $\Omega^i=\Lambda^i\Omega^1$ is the locally free sheaf of differential $i$-forms. From the projection formula and (5) we conclude that every object $F\in D(\mathbb{P}(V))$ belongs to the subcategory generated by
$$
\begin{equation*}
\begin{aligned} \, q_* \bigl(p^*(F)\otimes p^*(\Omega^{i}(i))\otimes q^* \bigl(\mathcal{O}(-i)\bigr)\bigr) & \simeq q_* \bigl(p^* (F\otimes \Omega^{i}(i))\otimes q^* \bigl(\mathcal{O}(-i)\bigr)\bigr) \\ & \simeq q_*p^* \bigl(F\otimes \Omega^{i}(i)\bigr)\otimes \mathcal{O}(-i) \\ & \simeq R\Gamma\bigl(\mathbb{P}(V), F\otimes \Omega^{i}(i)\bigr) \otimes_{\mathsf{k}}\mathcal{O}(-i). \end{aligned}
\end{equation*}
\notag
$$
We thus see that the objects $(\mathcal{O}(-n+1),\dots,\mathcal{O}(-1),\mathcal{O})$ generate $D(\mathbb{P}(V))$. Using the (derived) duality anti-autoequivalence, we deduce that the objects from (2) generate $D(\mathbb{P}(V))$ as well. Let us make a few remarks. First, on the projective space there is a very special Euler short exact sequence where by $V^*$ we denote the trivial bundle with fibre $V^*$ (similarly, we will denote by $V$ the trivial bundle with fibre $V$). If we denote by $\mathcal{L}=\mathcal{O}(-1)$ the tautological line bundle on $\mathbb{P}(V)$, then we can rewrite (7) as
$$
\begin{equation}
0\to \mathcal{L}^\perp \to V^* \to \mathcal{L}^*\to 0,
\end{equation}
\tag{8}
$$
where $\mathcal{L}^\perp \simeq (V/\mathcal{L})^*$. The latter presentation will be of great importance for Grassmannians. Second, the objects
$$
\begin{equation}
\bigl\langle\Omega^{n-1}(n-1),\dots,\Omega^{2}(2),\Omega^{1}(1), \mathcal{O}\bigr\rangle=\langle\Lambda^{n-1}\mathcal{L}^\perp,\dots, \Lambda^2\mathcal{L}^\perp, \mathcal{L}^\perp, \mathcal{O}\rangle
\end{equation}
\tag{9}
$$
also form a full exceptional collection in $D(\mathbb{P}(V))$, which turns out to be left dual to the collection (2) (we will see this below in a more general setting of classical Grassmannians). It is not a coincidence that the terms of the resolution (6) are of the form $E^\circ_x\boxtimes (E_x)^*$, where $(-)^*$ denotes the usual (derived) dual. Unfortunately, the resolution of diagonal argument is rather hard to apply to cases other than projective spaces and classical Grassmannians (we will see how it works for the latter in a moment). Another way of showing the fullness of Beilinson’s collection relies on a simple lemma first proved by Orlov. Recall that a classical generator of a triangulated category $\mathcal{T}$ is an object $E\in\mathcal{T}$ such that $\mathcal{T}$ coincides with the smallest strictly full triangulated subcategory of $\mathcal{T}$ which contains $E$ and is closed under direct summands. For instance, an exceptional collection $\langle E_1, E_2,\dots,E_r\rangle$ is full if and only if $E=\bigoplus E_i$ is a classical generator of $\mathcal{T}$. For convenience, we formulate the lemma for smooth projective varieties. Theorem 2.1 ([32], Theorem 4). Let $X$ be a smooth projective variety of dimension $d$, and let $\mathcal{L}$ be a very ample line bundle on $X$. For any integer $t$ the object $\mathcal{E}=\bigoplus_{i=t}^{t+d}\limits\mathcal{L}^i$ is a classical generator of $D(X)$. Let us try to use Theorem 2.1 for Beilinson’s exceptional collection (2). Putting $t=0$, we see that $D(\mathbb{P}(V))$ is classically generated by $\mathcal{O}\oplus\mathcal{O}(1)\oplus\cdots\oplus\mathcal{O}(n)$, which is one twist more than the sum of the objects in (2): $\mathcal{O}\oplus\mathcal{O}(1)\oplus\cdots\oplus\mathcal{O}(n-1)$. Thus, to show that Beilinson’s collection is full it is enough to show that $\mathcal{O}(n)$ belongs to the subcategory generated by $\mathcal{O}, \mathcal{O}(1),\dots,\mathcal{O}(n-1)$. Consider the regular nowhere vanishing section $s$ of $V(1)$ which corresponds to the identity map in
$$
\begin{equation*}
\Gamma(\mathbb{P}(V), V(1))\simeq V\otimes V^* \simeq \mathsf{Hom}(V,V).
\end{equation*}
\notag
$$
Since it is nowhere vanishing, the Koszul resolution of its zero locus is simply an exact complex of the form
$$
\begin{equation}
0\to \Lambda^{n}V^*(-n)\to \cdots \to \Lambda^2V^*(-2)\to V^*(-1) \to \mathcal{O}\to 0.
\end{equation}
\tag{10}
$$
Twisting the complex (10) by $\mathcal{O}(n)$, we obtain a complex of the form
$$
\begin{equation}
0\to \Lambda^{n}V^*\to \cdots \to \Lambda^2V^*(n-2)\to V^*(n-1)\to \mathcal{O}(n)\to 0.
\end{equation}
\tag{11}
$$
Treating (11) as a resolution for it rightmost term $\mathcal{O}(n)$, we see that $\mathcal{O}(n)\in \langle \mathcal{O},\mathcal{O}(1),\dots, \mathcal{O}(n-1) \rangle$. Finally, $\mathcal{O}\oplus\mathcal{O}(1) \oplus\cdots\oplus\mathcal{O}(n)\in \langle \mathcal{O},\mathcal{O}(1),\dots,\mathcal{O}(n-1) \rangle$, so by Theorem 2.1 Beilinson’s collection is full. The argument presented above can be reformulated as the following proposition. Proposition 2.2. Let $X$ be a smooth projective variety, let $\mathcal{L}$ be a very (anti-)ample line bundle on $X$, and let $\langle E_1, E_2,\dots, E_r \rangle\subset D(X)$ be an exceptional collection such that $\mathcal{O}\in \langle E_1,E_2,\dots,E_r \rangle$. If for all $i=1,\dots,r$ the object $E_i\otimes \mathcal{L}$ belongs to $\langle E_1,E_2,\dots,E_r \rangle$, then the collection is full. Proof. By induction, the category $\mathcal{T}=\langle E_1,E_2,\dots,E_r\rangle$ contains the $\mathcal{L}^i$ for all $i\geqslant 0$. By Theorem 2.1, $\mathcal{T}$ contains the classical generator $\bigoplus\limits_{i=0}^{\dim X}\mathcal{L}^i$. Thus, $\mathcal{T}=D(X)$. The proposition is proved. To conclude our discussion of projective spaces, we turn to the relative case established by Orlov. Theorem 2.3 ([31]). Let $X$ be a smooth projective variety, and let $\mathcal{V}$ be a vector bundle on $X$ of rank $n$. Consider the projectivization $\pi\colon\mathbb{P}_X(\mathcal{V})\to X$, and denote by $\mathcal{O}_{\mathcal{V}}(1)$ the relative dual tautological line bundle. There is a semiorthogonal decomposition where $D(X)(i)$ denotes the subcategory $\pi^*D(X)\otimes \mathcal{O}_{\mathcal{V}}(i)$. In particular, if $D(X)$ admits a full exceptional collection, then so does $D(\mathbb{P}_X(\mathcal{V}))$.Since partial flag varieties of the form $\mathsf{Fl}(1,\dots,i;V)$ and $\mathsf{Fl}(j,\dots,n-1;V)$ are iterated projectivizations of vector bundles, we deduce the following. Corollary 2.4. Let $V$ be a vector space of dimension $n$. For all $1\leqslant i,j\leqslant n-1$ the derived categories of the flag varieties $\mathsf{Fl}(1,\dots,i;V)$ and $\mathsf{Fl}(j,\dots,n-1;V)$ admit full exceptional collections consisting of line bundles. Since more general flag varieties are iterated Grassmannian bundles over Grassmannians, we will need a little more to show that the derived categories of all of them admit full exceptional collections. To prove the latter will need the following result concerning fibrewise full exceptional collections. Theorem 2.5 ([37]). Let $\pi\colon X\to S$ be a flat proper morphism between smooth schemes $X$ and $S$ over $\mathsf{k}$. Assume that the objects $E_1,E_2,\dots,E_r\in D(X)$ are such that for any closed point $s\in S$ the restrictions of $E_i$ to the fibre $X_s$ form a full exceptional collection in $D(X_s)$. Then there is a semiorthogonal decomposition In particular, if $D(S)$ admits a full exceptional collection, then so does $D(X)$.Theorem 2.5 has some immediate consequences. The most obvious one is that, given smooth projective varieties $X$ and $Y$ whose derived categories admit full exceptional collections $D(X)=\langle E_1,E_2,\dots,E_r\rangle$ and $D(Y)=\langle F_1, F_2,\dots,F_s\rangle$, the bounded derived category of their product admits a full exceptional collection
$$
\begin{equation*}
D(X\times Y)=\langle E_i\boxtimes E_j\mid i=1,\dots,r;\ j=1,\dots,s\rangle.
\end{equation*}
\notag
$$
Observe that the latter collection is most naturally indexed by the product of partially ordered sets $\{1,\dots,r\}\times \{1,\dots,s\}$. 2.2. Classical GrassmanniansBeilinson’s theorem was generalized to classical Grassmannians by Kapranov in his 1984 paper [19]. We begin with a general setup. As before, let $V$ be a vector space of dimension $n$ over an algebraically closed field $\mathsf{k}$ of characteristic zero (for results concerning arbitrary characteristic we refer the reader to the works [7] and [9]). Fix an integer $1\leqslant k \leqslant n-1$ and consider the Grassmannian $\mathsf{Gr}(k,V)$ of $k$-dimensional subspaces in $V$. It comes with a rank $k$ tautological subbundle $\mathcal{U}$. One has a short exact sequence of vector bundles
$$
\begin{equation}
0\to \mathcal{U}^\perp \to V^* \to \mathcal{U}^*\to 0,
\end{equation}
\tag{12}
$$
which specializes to (8) in the case $k=1$. Denote by $\mathrm{Y}_{h,w}$ the set of Young diagrams of height at most $h$ and width at most $w$, which can be identified with the set of sequences
$$
\begin{equation*}
\mathrm{Y}_{h,w}=\{\lambda \in \mathbb{Z}^h \mid w \geqslant \lambda_1 \geqslant \lambda_2\geqslant \cdots \geqslant \lambda_h\geqslant 0\}.
\end{equation*}
\notag
$$
The set $\mathrm{Y}_{h,w}$ comes with a natural partial order given by containment: for $\lambda,\mu \in \mathrm{Y}_{h,w}$ put
$$
\begin{equation*}
\lambda \succeq \mu \quad \text{if}\ \ \lambda_i \geqslant \mu_i\ \ \text{for all}\ \ i=1,\dots, h.
\end{equation*}
\notag
$$
Simultaneously, there is a complete lexicographical order on $\mathrm{Y}_{h,w}$ induced by the corresponding order on $\mathbb{Z}^h$, which we denote by $\geqslant$ . For a Young diagram $\lambda$ we denote by $|\lambda|=\displaystyle\sum\lambda_i$ its size and by $\lambda^T$ its transpose. The width of $\lambda$ is defined by $\mathsf{w}(\lambda)=\lambda_1$, and the height of $\lambda$ is $\mathsf{h}(\lambda)=\mathsf{w}(\lambda^T)$. Given a vector bundle $\mathcal{E}$ and a Young diagram $\lambda$, we denote by $\Sigma^\lambda\mathcal{E}$ the result of the application of the corresponding Schur functor. We follow the convention under which $\Sigma^{(t)}=S^t$ is the symmetric power functor. We are ready to formulate Kapranov’ result. In the following theorem we identify the posets $\mathrm{Y}_{k,n-k}$ and $\mathrm{Y}_{n-k,k}$ via transposition of diagrams. Theorem 2.6 ([19]). The bounded derived category of $\mathsf{Gr}(k,V)$ admits a full exceptional collection indexed by the poset $\mathrm{Y}_{k,n-k}$:
$$
\begin{equation}
D(\mathsf{Gr}(k, V))=\langle \Sigma^\lambda\mathcal{U}^* \mid \lambda\in\mathrm{Y}_{k, n-k}\rangle.
\end{equation}
\tag{13}
$$
Its graded left dual is given by That is, for $\lambda\in\mathrm{Y}_{k,n-k}$ and $\mu\in\mathrm{Y}_{n-k,k}$ one has When $k=1$, Theorem 2.6 produces Beilinson’s exceptional collection on $\mathbb{P}(V)=\mathsf{Gr}(1,V)$. Indeed, one has
$$
\begin{equation*}
\mathcal{O}(1)=\mathcal{U}^*,\quad \Omega^1(1)\simeq \mathcal{U}^\perp,\quad\text{and}\quad \mathrm{Y}_{1,n-1}=\{(i) \mid i=0,\dots,n-1\}.
\end{equation*}
\notag
$$
For each $0\leqslant i\leqslant n-1$ there are isomorphisms
$$
\begin{equation*}
\Sigma^{(i)}\mathcal{O}(1)=S^i\mathcal{O}(1)\simeq \mathcal{O}(i)\quad\text{and}\quad \Sigma^{(i)^T}\mathcal{U}^\perp=\Lambda^i(\Omega^1(1))\simeq \Omega^i(i).
\end{equation*}
\notag
$$
Thus, Theorem 2.6 is a generalization of Theorem 1.6. The original proof of Theorem 2.6 is analogous to Beilinson’s proof of Theorem 1.6. First, one needs to check that the objects from (13) are exceptional and semiorthogonal. Since all of them are irreducible equivariant vector bundles, one can apply the celebrated Borel–Bott–Weil theorem in combination with the Littlewood–Richardson rule. In order to show fullness, Kapranov applies a resolution of diagonal argument. Consider the diagram The composition of natural morphisms vanishes precisely along the diagonal, which can be described as the vanishing locus of the corresponding section $s\in\Gamma\bigl(\mathsf{Gr}(k, V)\times \mathsf{Gr}(k, V),\, (V/\mathcal{U}) \boxtimes \mathcal{U}^*\bigr)$, where $\mathcal{E}\boxtimes\mathcal{F}= p^*\mathcal{E}\otimes q^*\mathcal{F}$ and $(V/\mathcal{U})\simeq (\mathcal{U}^\perp)^*$. The Koszul resolution of $\Delta_*\mathcal{O}$ has terms of the form
$$
\begin{equation*}
\Lambda^i(\mathcal{U}^\perp \mathbin{\boxtimes\mathcal{U}}) \simeq \bigoplus_{|\lambda|=i} \Sigma^{\lambda^T} \mathcal{U}^\perp \boxtimes\Sigma^\lambda\mathcal{U},
\end{equation*}
\notag
$$
where the isomorphism follows from [39], Corollary 2.3.3. Note that the product $\Sigma^{\lambda^T} \mathcal{U}^\perp\boxtimes\Sigma^\lambda\mathcal{U}$ is non-zero if and only if $\lambda\in\mathrm{Y}_{k, n-k}$ since $\mathsf{rk}(\mathcal{U})=k$ and $\mathsf{rk}(\,\mathcal{U}^\perp)=n-k$. The rest of the argument goes just as the one we presented for projective spaces. The exceptional collection (13) is defined in an obvious way in the relative setting. Applying Theorem 2.5, we deduce the following. Theorem 2.7. Let $\mathcal{V}$ be a vector bundle of rank $n$ on a smooth projective variety $X$, and let $1\leqslant k\leqslant n-1$ be an integer. Consider the relative Grassmannian bundle $\pi\colon\mathsf{Gr}_X(k,\mathcal{V})\to X$, and let $\mathcal{U}\subset \pi^*\mathcal{V}$ denote the relative tautological bundle. Then there is a (graded) semiorthogonal decomposition of the form In particular, if $D(X)$ has a full exceptional collection (consisting of vector bundles), then $D(\mathsf{Gr}_X(k,\mathcal{V}))$ also has a full exceptional collection (consisting of vector bundles).Corollary 2.8. Let $X$ be a smooth projective variety, which can be obtained as an iterated Grassmannian bundle over a smooth projective variety $S$ such that $D(S)$ admits a full exceptional collection. Then so does $D(X)$. Since for all integer sequences $1\leqslant i_1 < i_2 < \cdots < i_s \leqslant n-1$ the partial flag variety $\mathsf{Fl}(i_1,\dots,i_s;V)$ is an iterated Grassmannian bundle, say, over the Grassmannian $\mathsf{Gr}(i_1,V)$, we conclude the following. Corollary 2.9. All (partial) flag varieties over an algebraically closed field of characteristic zero admit full exceptional collections consisting of equivariant vector bundles. Corollary 2.9 completely establishes Conjecture 1.7 in type A. 2.3. Staircase complexesA class of long exact sequences of equivariant vector bundles was introduced in [11] in order to study Lefschetz exceptional collections (bases) in $D(\mathsf{Gr}(k,V))$. In the present subsection we describe these sequences, which are called staircase complexes, and give a construction which is independent of the existence of Kapranov’s exceptional collection (this construction appeared, unpublished, in the author’s Ph. D. thesis). For any positive integers $h,w>0$ such that $h+w=n$ the set $\mathrm{Y}_{h,w}$ carries a cyclic group action: fix a generator $g$ of the cyclic group $\mathbb{Z}/n\mathbb{Z}$ and put
$$
\begin{equation}
\lambda \mapsto g\cdot\lambda=\begin{cases} (\lambda_1+1, \lambda_2+1,\dots,\lambda_h+1) & \text{if}\ \mathsf{w}(\lambda) < w, \\ (\lambda_2,\dots,\lambda_h, 0) & \text{if}\ \mathsf{w}(\lambda)=w. \end{cases}
\end{equation}
\tag{15}
$$
For brevity we set $\lambda'=g\cdot \lambda$. The easiest way to see that (15) defines indeed a cyclic action is the following. Recall that there is a bijection between $\mathrm{Y}_{h,w}$ and the set of binary sequences $a_1a_2\ldots a_n\in \{0,1\}^n$ such that $\displaystyle\sum a_i=h$: each element of $\mathrm{Y}_{h,w}$ represents an integer path going from the lower left corner of an $h\times w$ rectangle to the top right one, moving one step right or up at a time. For each step to the right we write $0$, and for each step going up we write $1$. Under this bijection the cyclic action on $\mathrm{Y}_{h,w}$ translates to the usual cyclic action on binary sequences, $g\cdot (a_1a_2\ldots a_n)=a_na_1\ldots a_{n-1}$, which obviously preserves the sum of the elements. For any sequence $\lambda\in\mathbb{Z}^h$ and any integer $t$ put
$$
\begin{equation*}
\lambda(t)=(\lambda_1+t, \lambda_2+t,\dots,\lambda_h+t).
\end{equation*}
\notag
$$
If $\lambda$ is weakly decreasing, then it corresponds to a dominant weight of $\mathsf{GL}_h$, and for any rank $h$ bundle $\mathcal{E}$ one has $\Sigma^{\lambda(t)}\mathcal{E}\simeq \Sigma^{\lambda}\mathcal{E}\otimes (\det\mathcal{E})^{\otimes t}$. Put Then we can reformulate (15) as $\lambda'=\lambda(1)$ whenever $\lambda(1)\in\mathrm{Y}_{h,w}$ and $\lambda'=\bar\lambda$ if $\lambda(1)\notin\mathrm{Y}_{h,w}$. Let $\lambda\in\mathrm{Y}_{h,w}$ be a diagram of maximum width, that is, $\mathsf{w}(\lambda)=w$, and let $\mu=\lambda^T$. Observe that the condition on $\mathsf{w}(\lambda)$ implies that $\mu_w > 0$. For $i=1,\dots,w$ define $\lambda^{(i)}$ according to the following rule:
$$
\begin{equation}
\begin{aligned} \, \lambda^{(1)} &=(\mu_1, \mu_2,\dots,\mu_{w-1}, 0)^T, \\ \lambda^{(i)} &=(\mu_1,\dots,\mu_{w-i},\, \mu_{w-i+2}-1,\dots,\mu_w-1, 0)^T \quad \text{for}\ \ 1 < i < w, \\ \lambda^{(w)} &=(\mu_2-1,\dots,\mu_w-1, 0)^T. \end{aligned}
\end{equation}
\tag{16}
$$
Formulae (16) may look rather mysterious; however, they have a rather simple combinatorial explanation. Informally speaking, in order to find $\lambda^{(i)}$, one should take the inner band of unit width along the border of $\lambda$ (this band is nothing but the skew Young diagram $\lambda/\lambda^{(w)}$) and cut $i$ rightmost columns of the band out of $\lambda$. In terms of binary sequences, if $a_1a_2\ldots a_n$ is the sequence corresponding to $\lambda$, let $n>z_1>z_2>\cdots > z_w\geqslant 1$ be the indices for which $a_{z_i}=0$. Then $\lambda^{(i)}$ is the diagram corresponding to the binary sequence (recall that $\mathsf{w}(\lambda)=w$, so $a_n=1$). We thus obtain a sequence of subdiagrams
$$
\begin{equation}
\lambda \supset \lambda^{(1)}\supset \cdots \supset \lambda^{(w)}.
\end{equation}
\tag{17}
$$
Example 2.10. Consider $\lambda=(4,4,2)\in \mathrm{Y}_{3,4}$. Then the filtration (17) becomes where $\lambda^{(i)}$ is the white part of the corresponding diagram.Proposition 2.11 ([11], Proposition 5.3). Let $\lambda\in\mathrm{Y}_{k,n-k}$ be a diagram such that $\mathsf{w}(\lambda)=n-k$. There is a long exact sequence of vector bundles on $\mathsf{Gr}(k,V)$ of the form where $b_\lambda^{(i)}=|\lambda/\lambda^{(i)}|=|\lambda|-|\lambda^{(i)}|$.Remark 2.12. The proof of Proposition 2.11 given in [11] relies on the fullness of Kapranov’s exceptional collection. Indeed, the long exact sequence (18) explicitly shows how the object $\Sigma^{\bar\lambda}\mathcal{U}^*$ can be generated by objects from Kapranov’s collection. As promised in the beginning of the present section, below we give a fullness-independent construction of (18). Proof of Proposition 2.11. Consider the diagram Denote by $\mathcal{F}^\bullet$ the dual of the exact complex (11) twisted by $\mathcal{O}(-n+1)$, which is an exact complex on $\mathbb{P}(V)$ of the form where $\mathcal{F}^0=\mathcal{O}(1)$.
Denote by $\mathcal{L}\subset \mathcal{U}$ the universal flag on $\mathsf{Fl}(1,k;V)$. Consider the object
$$
\begin{equation*}
X^\bullet=p_*\bigl(q^*\mathcal{F}^\bullet \otimes \Sigma^{\bar\lambda}(\mathcal{U}/\mathcal{L})\otimes (\det(\mathcal{U}/\mathcal{L}))^{-1}\bigr),
\end{equation*}
\notag
$$
where $\Sigma^{\bar\lambda}(\mathcal{U}/\mathcal{L})$ is non-zero since $\mathsf{h}(\bar{\lambda}) \leqslant h-1= \mathsf{rk}(\mathcal{U}/\mathcal{L})$. Since $\mathcal{F}^\bullet$ is exact, $X^\bullet$ is zero in $D(\mathsf{Gr}(k, V))$. Meanwhile, one has the standard spectral sequence
$$
\begin{equation*}
E^{ab}_1=R^bp_*\bigl(q^*\mathcal{F}^a \otimes \Sigma^{\bar\lambda}(\mathcal{U}/\mathcal{L})\otimes (\det(\mathcal{U}/\mathcal{L}))^{-1}\bigr)\Rightarrow H^{a+b}(X^\bullet).
\end{equation*}
\notag
$$
By definition, for $a=-n,\dots,0$ we obtain
$$
\begin{equation*}
\begin{aligned} \, E^{ab}_1 &=R^bp_*\bigl(q^*(\Lambda^{n+a}V(a+1)) \otimes \Sigma^{\bar\lambda}(\mathcal{U}/\mathcal{L})\otimes (\det(\mathcal{U}/\mathcal{L}))^{-1}\bigr) \\ & \simeq R^bp_*\bigl(\Lambda^{n+a}V\otimes \mathcal{L}^{-a-1} \otimes \Sigma^{\bar\lambda}(\mathcal{U}/\mathcal{L})\otimes (\det(\mathcal{U}/\mathcal{L}))^{-1}\bigr) \\ & \simeq \Lambda^{n+a}V(1) \otimes R^bp_*\bigl(\mathcal{L}^{-a} \otimes \Sigma^{\bar\lambda}(\mathcal{U}/\mathcal{L})\bigr), \end{aligned}
\end{equation*}
\notag
$$
where in the last isomorphism we used the identity $\det\mathcal{U}=\det(\mathcal{U}/\mathcal{L})\otimes \mathcal{L}$.
Since $\mathsf{Fl}(1,k;V)$ is nothing but the projectivization $\mathbb{P}_{\mathsf{Gr}(k,V)}(\mathcal{U})$, we can use the Borel–Bott–Weil theorem to compute the higher push-forwards. It turns out that the spectral sequence degenerates into an exact complex, which is dual to (18). The proposition is proved. Example 2.13. A well-known example of a staircase complex comes from $\lambda=(n-k)$:
$$
\begin{equation*}
\begin{aligned} \, 0\to \mathcal{O}(-1) \to \Lambda^{n-k}V^*\otimes \mathcal{O} &\to \cdots \to \Lambda^{n-k-i}V^*\otimes S^i\mathcal{U}^*\to \cdots\\ &\to V^*\otimes S^{n-k-1}\mathcal{U}^* \to S^{n-k}\mathcal{U}^*\to 0. \end{aligned}
\end{equation*}
\notag
$$
When $k=1$, the only staircase complex is a shifted and dualized Koszul complex, the complex dual to (19). An immediate application of staircase complexes is an alternative proof of the fullness of Kapranov’s exceptional collection. Indeed, let $\lambda\in\mathrm{Y}_{h, w}$. If $\mathsf{h}(\lambda)=h$, then $\lambda(-1)\in\mathrm{Y}_{h, w}$. Since
$$
\begin{equation*}
\Sigma^{\lambda(-1)}\mathcal{U}^*\simeq\Sigma^{\lambda}\mathcal{U}^* \otimes (\det\mathcal{U}^*)^{-1}\simeq \Sigma^\lambda\mathcal{U}^*(-1),
\end{equation*}
\notag
$$
we conclude that for all such $\lambda$ the object $\Sigma^\lambda\mathcal{U}^*(-1)$ trivially belongs to the subcategory generated by the objects from Kapranov’s collection. If $\mathsf{h}(\lambda) < h$, then $\lambda=\bar\mu$, where $\mu=(n-k, \lambda_1,\dots,\lambda_{h-1})$, and the staircase complex (18) written for $\mu$ and treated as a right resolution for $\Sigma^{\lambda}\mathcal{U}^*(-1)$ shows that $\lambda=\bar\mu$ also belongs to the subcategory generated by Kapranov’s collection. Applying Proposition 2.2 we see that Kapranov’s collection is full. While this proof gives nothing new for classical Grassmannians, this method will be generalized below to the case of Lagrangian Grassmannians, and we expect that it can be used in even greater generality. 2.4. Lefschetz decompositions for classical GrassmanniansGiven a smooth projective Fano variety, it is natural to look for a Lefschetz basis (decomposition) which is as small as possible. There are various ways to interpret ‘minimal’: one can ask for the number of objects in the basis to be minimum or the rectangular part to be as large as possible. Recall that $\mathsf{Gr}(k,V)$ is a Fano variety whose Picard group is isomorphic to $\mathbb{Z}$ and is generated by $\mathcal{O}(1)\simeq \det\,\mathcal{U}^*$. Its (Fano) index equals $n$: $\omega_{\mathsf{Gr}(k, V)}\simeq \mathcal{O}(-n)$. The first example of a minimal (in any sense) Lefschetz basis in $D(\mathsf{Gr}(k,V))$ was constructed for $k=2$ by Kuznetsov in [23]. Theorem 2.14 ([23]). If the dimension $n=2t+1$ of $V$ is odd, then there is a rectangular Lefschetz decomposition in $D(\mathsf{Gr}(2,V))$ with the basis If the dimension $n=2t$ of $V$ is even, then there is a Lefschetz decomposition in $D(\mathsf{Gr}(2,V))$ with the basisRecall that the number of objects in any full exceptional collection equals the rank of the Grothendieck group. Since $\mathsf{rk}\, K_0(\mathsf{Gr}(k, V))=\binom{n}{k}$, the minimum possible number of objects one can obtain in the first block of a Lefschetz decomposition is $\bigl\lceil\binom{n}{k}/n\bigr\rceil$. (Recall that by Serre duality one cannot have more blocks than the Fano index of the variety.) For $k=2$ we obtain the lower bound $\bigl\lceil\binom{n}{2}/2\bigr\rceil=\lceil(n-1)/{2}\rceil$, which is achieved in Theorem 2.14. We also see that, for divisibility reasons, when $n$ is odd, $D(\mathsf{Gr}(2, V))$ cannot have a minimal rectangular Lefschetz basis. Remark 2.15. A more common way to write Lefschetz exceptional collections is in matrix form, where objects are read bottom to top and left to right. For instance, when $n=2t+1$ is odd, one has a full exceptional collection in $D(\mathsf{Gr}(2,V))$ of the form
$$
\begin{equation}
\begin{pmatrix} S^{t-1}\mathcal{U}^* & S^{t-1}\mathcal{U}^*(1) & \dots & S^{t-1}\mathcal{U}^*(2t) \\ S^{t-2}\mathcal{U}^* & S^{t-2}\mathcal{U}^*(1) & \dots & S^{t-2}\mathcal{U}^*(2t) \\ \vdots & \vdots & & \vdots \\ \mathcal{U}^* & \mathcal{U}^*(1) & \dots & \mathcal{U}^*(2t) \\ \mathcal{O}^* & \mathcal{O}^*(1) & \dots & \mathcal{O}^*(2t) \end{pmatrix},
\end{equation}
\tag{20}
$$
and when $n=2t$ is even, one has a full exceptional collection in $D(\mathsf{Gr}(2,V))$ of the form
$$
\begin{equation*}
\begin{pmatrix} S^{t-1}\mathcal{U}^* & S^{t-1}\mathcal{U}^*(1) & \dots & S^{t-1}(t-1) & \\ S^{t-2}\mathcal{U}^* & S^{t-2}\mathcal{U}^*(1) & \dots & S^{t-2}(t-1) & \dots & S^{t-2}\mathcal{U}^*(2t-1) \\ \vdots & \vdots & & \vdots & & \vdots \\ \mathcal{U}^* & \mathcal{U}^*(1) & \dots & \mathcal{U}(t-1) & \dots & \mathcal{U}^*(2t-1) \\ \mathcal{O}^* & \mathcal{O}^*(1) & \dots & \mathcal{O}(t-1) & \dots & \mathcal{O}^*(2t-1) \end{pmatrix}.
\end{equation*}
\notag
$$
The argument used in [23] to show the fullness of the two collections relies on the following rather common technique. First, cover your variety by a family of simpler subvarieties. In this case one can consider the zero loci of linear forms $\phi\in \Gamma(\mathsf{Gr}(k,V),\mathcal{U}^*)\simeq V^*$, which are isomorphic to $\mathsf{Gr}(k, \operatorname{Ker}\phi)$. Observe that if all the restrictions of an object $X\in D(\mathsf{Gr}(k,V))$ to subvarieties vanish, then the object itself is zero. Finally, check that if an object is orthogonal to all the objects in the collection the fullness of which one wants to prove, then its restriction is zero for all choices of $\phi$. Unfortunately, this method is not always easy to apply. We are about to give a generalization of Theorem 2.14 to arbitrary $\mathsf{Gr}(k,V)$, where fullness will be shown with the use of staircase complexes. Actually, we are going to describe three Lefschetz bases in $D(\mathsf{Gr}(k,V))$. When $k$ and $n$ are coprime, the first two coincide and give minimal rectangular decompositions for $D(\mathsf{Gr}(k,V))$. When $k$ and $n$ are not coprime, the first basis is slightly larger. However, this is the one that we know to be full in all cases (the second one is conjecturally full for $k$ and $n$ not coprime). Finally, the third basis is of the same ‘shape’ as the second one. It is minimal in the sense that its rectangular part is maximal possible, and we know that it is full, though the proof has not appeared in the literature just yet and will be available in [17]. Our strategy of constructing a Lefschetz basis will always be the same: pick a subset of $\mathrm{Y}_{k,n-k}$ and use the corresponding objects from Kapranov’s collection as the first block of the decomposition by assigning appropriate values to the support function. Definition 2.16. A diagram $\lambda\in\mathrm{Y}_{h,w}$ is called upper-triangular if for all $1\leqslant i\leqslant h$ one has The subset of upper-triangular diagrams in $\mathrm{Y}_{h,w}$ is denoted by $\mathrm{Y}^{\mathrm{u}}_{h,w}\subset\mathrm{Y}_{h, w}$.For any upper-triangular diagram $\lambda\in\mathrm{Y}^{\mathrm{u}}_{w,h}$ let where $i$ is the smallest positive integer for which inequality (21) becomes equality (it is always the case for $i=h$). In other words, $o^{\rm u}(\lambda)$ is the length of the path going along $\lambda$ from the top right corner of the $h \times w$ rectangle until one meets the diagonal again. In terms of the cyclic group action, $o^{\rm u}(\lambda)$ is the smallest positive integer $i$ such that $g^i\cdot\lambda\in \mathrm{Y}^{\mathrm{u}}_{h,w}$, where $g$ is the chosen generator of $\mathbb{Z}/n\mathbb{Z}$.Theorem 2.19 ([11], Theorem 4.1). There is a Lefschetz decomposition of the derived category $D(\mathsf{Gr}(k,V))$ with a Lefschetz basis given by and support function $o^{\rm u}$.Example 2.20. Consider $\mathsf{Gr}(3, 6)$. Theorem 2.19 produces a Lefschetz decomposition with the basis
$$
\begin{equation*}
\bigl\langle \mathcal{O}, \mathcal{U}^*, \Lambda^2\mathcal{U}^*, S^2\mathcal{U}^*, \Sigma^{2, 1}\mathcal{U}^* \bigr\rangle.
\end{equation*}
\notag
$$
The corresponding (full) Lefschetz exceptional collection in $D(\mathsf{Gr}(3, 6))$ is of the form
$$
\begin{equation}
\begin{pmatrix} \Sigma^{2, 1}\mathcal{U}^* & \Sigma^{2, 1}\mathcal{U}^*(1) & & & & \\ S^2\mathcal{U}^* & S^2\mathcal{U}^*(1) & & & & \\ \Lambda^2\mathcal{U}^* & \Lambda^2\mathcal{U}^*(1) & \Lambda^2\mathcal{U}^*(2) & \Lambda^2\mathcal{U}^*(3) & & \\ \mathcal{U}^* & \mathcal{U}^*(1) & \mathcal{U}^*(2) & \mathcal{U}^*(3) & \mathcal{U}^*(4) & \mathcal{U}^*(5) \\ \mathcal{O} & \mathcal{O}(1) & \mathcal{O}(2) & \mathcal{O}(3) & \mathcal{O}(4) & \mathcal{O}(5) \end{pmatrix}.
\end{equation}
\tag{23}
$$
The proof of Theorem 2.19 relies on some intricate homology computations using the Borel–Bott–Weil theorem and the Littlewood–Richardson rule for semiorthogonality and staircase complexes for fullness. Namely, using staircase complexes and induction, it was shown in [11] that the objects of Kapranov’s collection are generated by the objects given by the Lefschetz basis (22). When $k=2$, Theorem 2.19 recovers Theorem 2.14. When $k$ and $n$ are coprime, Theorem 2.19 produces a rectangular Lefschetz decomposition which is optimal due to numerical reasons: the number of objects in the basis equals $\mathsf{rk}\,K_0(\mathsf{Gr}(k,V))/ n$. However, when $k$ and $n$ are not coprime, one can find a smaller Lefschetz decomposition in $D(\mathsf{Gr}(k,V))$. This decomposition agrees with the one presented in Theorem 2.19 when $(k,n)=1$ and is expected to be full for arbitrary $k$ and $n$. Consider again the cyclic action of $\mathbb{Z}/n\mathbb{Z}$ on $\mathrm{Y}_{w,h}$. The following observation is a very pleasant and easy exercise. Lemma 2.21 ([11], Lemma 3.2). Every orbit of the cyclic action on $\mathrm{Y}_{w,h}$ contains an upper-triangular diagram. For each $\lambda$ denote by $o(\lambda)$ the length of the cyclic orbit containing $\lambda$. Our strategy for constructing a Lefschetz basis will be as follows: we want elements of the basis to be represented by elements of the orbits of the action, while the support function will be the length of the corresponding orbit. One way to pick an element in each orbit is to go for upper-triangular diagrams. However, by looking at the orbits for $h=3$ and $w=3$ we see that such representatives are not unique as soon as $h$ and $w$ are not coprime and $h, w > 2$. Indeed, the corresponding orbits are and we see that both $(2,0,0)$ and $(1,1,0)$ are upper-triangular elements lying in the same orbit. A rather straightforward idea is to pick in each orbit the upper-triangular element which is lexicographically minimal. Denote by $\mathrm{Y}^{\mathrm{mu}}_{h,w}$ the set of such diagrams.Theorem 2.22 ([11], Theorem 4.3). Let $\mathcal{A}_i=\langle \Sigma^\lambda \mathcal{U}^* \mid \lambda \in \mathrm{Y}^{\mathrm{mu}}_{k, n-k},\ i < o(\lambda) \rangle$. There is a semiorthogonal decomposition where $\mathcal{R}^{\rm mu}$ is expected to be zero.Conjecture 2.23 ([11], Conjecture 4.4). The orthogonal $\mathcal{R}^{\rm mu}$ in Theorem 2.22 vanishes. In other words, $\langle \Sigma^\lambda \mathcal{U}^* \mid \lambda \in \mathrm{Y}^{\mathrm{mu}}_{k, n-k}\rangle$ is a Lefschetz basis with the support function $o$. When $(k, n)=1$, one has $\mathrm{Y}^{\mathrm{mu}}_{k, n-k}=\mathrm{Y}^{\mathrm{u}}_{k, n-k}$, so we deduce from Theorem 2.19 that Conjecture 2.23 holds: the two bases coincide. The same holds true when $k=2$ or $n-k=2$. In the next simplest case, when $k=3$ and $n=6$, Conjecture 2.23 was established in [11], Proposition 5.7. In particular, one obtains a full Lefschetz exceptional collection
$$
\begin{equation}
\begin{pmatrix} \Sigma^{2, 1}\mathcal{U}^* & \Sigma^{2, 1}\mathcal{U}^*(1) & & & & \\ \Lambda^2\mathcal{U}^* & \Lambda^2\mathcal{U}^*(1) & \Lambda^2\mathcal{U}^*(2) & \Lambda^2\mathcal{U}^*(3) & \Lambda^2\mathcal{U}^*(4) & \Lambda^2\mathcal{U}^*(5) \\ \mathcal{U}^* & \mathcal{U}^*(1) & \mathcal{U}^*(2) & \mathcal{U}^*(3) & \mathcal{U}^*(4) & \mathcal{U}^*(5) \\ \mathcal{O} & \mathcal{O}(1) & \mathcal{O}(2) & \mathcal{O}(3) & \mathcal{O}(4) & \mathcal{O}(5) \end{pmatrix}
\end{equation}
\tag{25}
$$
(cf. (23)). In [25] Kuznetsov and Smirnov used decomposition (24) in order to study a certain generalization of Dubrovin’s conjecture and established Conjecture 2.23 for $k=3$ (see [25], Proposition A.1). It remains open in the other cases. Finally, we present our third Lefschetz decomposition. It has the same shape as the conjecturally full one constructed in Theorem 2.22, and one can prove that it is full, although the proof has not appeared in the literature just yet. Once again, we want to choose a representative in each orbit of the cyclic action on $\mathrm{Y}_{k,n-k}$ and use the length of the orbit $o$ as the support function (thus the same shape of the basis). Let us drop the upper-triangular condition and simply pick the lexicographically minimal element in each orbit. Denote the set of such elements by $\mathrm{Y}^{\mathrm{m}}_{k,n-k}$. Theorem 2.24 ([17]). There is a Lefschetz decomposition of $D(\mathsf{Gr}(k,V))$ with a Lefschetz basis given by and the support function $o$.Note that the sets $\mathrm{Y}^{\mathrm{m}}_{w,h}$ and $\mathrm{Y}^{\mathrm{mu}}_{w,h}$ are different even when $h$ and $w$ are coprime. Indeed, the diagram $(1,1,1,0)\in \mathrm{Y}^{\mathrm{m}}_{4,3}$ is minimal, but the only upper-triangular element in its orbit is $(2, 0, 0, 0)\in \mathrm{Y}^{\mathrm{mu}}_{4, 3}$. 3. Kuznetsov–Polishchuk construction3.1. Equivariant vector bundlesLet $\mathbf{G}$ be a simple algebraic group, and let $\mathbf{P}\subset \mathbf{G}$ be a maximal parabolic subgroup. Such subgroups are in a one-to-one correspondence with the simple roots of $\mathbf{G}$. We are interested in $D(X)$, where $X=\mathbf{G}/\mathbf{P}$. Since we know that all exceptional objects in $D(X)$ admit an equivariant structure, we can begin with studying the equivariant derived category $D^\mathbf{G}(X)$. It is well known that the category of $\mathbf{G}$-equivariant sheaves $\mathsf{Coh}^\mathbf{G}(X)$ is equivalent to the category of representations:
$$
\begin{equation*}
\mathsf{Coh}^\mathbf{G}(X) \cong \mathsf{Rep}(\mathbf{P})
\end{equation*}
\notag
$$
(see [5]). Under this equivalence irreducible equivariant vector bundles correspond to irreducible representations of $\mathbf{P}$, which is not a reductive group. Denote by $\mathbf{U}\subset \mathbf{P}$ the unipotent radical and by $\mathbf{L}=\mathbf{P}/\mathbf{U}$ the semisimple Levi quotient. The semisimple representations of $\mathbf{P}$ are precisely those restricted from $\mathbf{L}$. Denote by $P^+_{\mathbf{L}}$ the dominant weight lattice of $\mathbf{L}$ and by $\mathcal{U}^\lambda$ the $\mathbf{G}$-equivariant vector bundle on $X$ corresponding to $\lambda\in P^+_{\mathbf{L}}$. It is rather easy to construct an infinite exceptional collection in $D^\mathbf{G}(X)$. Theorem 3.1 ([24], Theorem 3.4). Let $\xi$ denote the fundamental weight corresponding to the simple root associated with the maximal parabolic $\mathbf{P}\subset\mathbf{G}$. Consider the partial order on $P^+_{\mathbf{L}}$ which is opposite to the so-called $\xi$-ordering given by the formula The irreducible equivariant bundles $\{\mathcal{U}^\lambda \mid \lambda\in \mathbf{P}\subset\mathbf{G}\}$ form an infinite full exceptional collection in $D^\mathbf{G}(X)$.There is an obvious forgetful functor The idea of Kuznetsov and Polishchuk was to find a condition under which a subcollection of the collection from Theorem 3.1 is mapped to an exceptional collection in the non-equivariant derived category. They found one and called it the exceptional block condition.3.2. Exceptional blocksSince the group $\mathbf{G}$ is reductive, the functor of taking $\mathbf{G}$-invariants is exact. Thus, for any equivariant objects $F, G\in D^\mathbf{G}(X)$ one has $\mathsf{Ext}_{\mathbf{G}}^i(F, G)= (\mathsf{Ext}_{D(X)}^i(F, G))^\mathbf{G}$, and there is little chance that even exceptional objects are sent to exceptional objects via the forgetful functor. For any $\lambda, \mu\in P^+_{\mathbf{L}}$ the forgetful functor induces a linear map
$$
\begin{equation*}
\mathsf{Fg}\colon \mathsf{Ext}_{\mathbf{G}}^i(\mathcal{U}^\lambda, \mathcal{U}^\mu) \to \mathsf{Ext}_{D(X)}^i (\mathcal{U}^\lambda, \mathcal{U}^\mu),
\end{equation*}
\notag
$$
and for any triple $\lambda, \mu, \nu \in P^+_{\mathbf{L}}$ one obtains a bilinear composition map
$$
\begin{equation*}
\mathsf{Hom}(\mathcal{U}^\nu, \mathcal{U}^\mu)\otimes_{\mathsf{k}} \mathsf{Ext}_{\mathbf{G}}^i(\mathcal{U}^\lambda, \mathcal{U}^\nu)\to \mathsf{Ext}_{D(X)}^i(\mathcal{U}^\lambda, \mathcal{U}^\mu).
\end{equation*}
\notag
$$
Definition 3.2 ([24], Definition 3.1). A subset $B\subset P^+_{\mathbf{L}}$ of dominant weights is called a (right) exceptional block if the natural morphism is an isomorphism for all $\lambda, \mu\in B$.The definition of a right exceptional block says that any non-equivariant extension between two irreducible equivariant bundles associated with two elements of the block can be decomposed in a unique way as a sum of compositions of equivariant extensions from the target into equivariant bundles associated with (other) elements of the block with non-equivariant homomorphisms into the target. The adjective right does not appear in [24]; however, it turns out that it is often convenient to consider left exceptional blocks. Definition 3.3. A subset $B\subset P^+_{\mathbf{L}}$ of dominant weights is called a left exceptional block if the natural morphism is an isomorphism for all $\lambda, \mu\in B$.Here we ask for each non-equivariant extension to decompose into a sum of non-equivariant homomorphisms followed by equivariant extensions. Let $B\subset P^+_{\mathbf{L}}$ be a right exceptional block. According to Theorem 3.1, the bundles $\{\mathcal{U}^\lambda \mid \lambda \in B\}$ form a graded exceptional collection in $D^\mathbf{G}(X)$. Let $\{\mathcal{E}^\lambda_B \mid \lambda \in B\}$ denote the graded right dual exceptional collection. Proposition 3.4 ([24], Proposition 3.9). Let $B\subset P^+_{\mathbf{L}}$ be a right exceptional block. The objects form a graded exceptional collection in $D(X)$ with respect to the $\xi$-ordering.There is a similar result for left exceptional blocks: for a left exceptional block $B\subset P^+_{\mathbf{L}}$ let $\{\mathcal{F}^\lambda_B \mid \lambda \in B\}$ denote the graded left dual exceptional collection to $\{\mathcal{U}^\lambda \mid \lambda \in B\}$. Proposition 3.5. Let $B\subset P^+_{\mathbf{L}}$ be a left exceptional block. The objects form a graded exceptional collection in $D(X)$ with respect to the $\xi$-ordering.The proof of Proposition 3.4 is quite easy. The idea of the exceptional block condition is beautiful and very powerful. Most of [24] is dedicated to constructing sufficiently many orthogonal exceptional blocks for isotropic Grassmannians (quotients by maximal parabolics of the simple groups of types $B$, $C$, and $D$) to produce exceptional collections of maximum possible length (equal to the rank of $K_0$). These results are covered below. The authors also formulated a certain curious conjecture for classical Grassmannians, which we discuss in the following two subsections. 3.3. Generalized staircase complexesWe have seen that the bundles $\Sigma^\lambda\mathcal{U}^*$ (dually, $\Sigma^\lambda\mathcal{U}$) are exceptional on $\mathsf{Gr}(k,V)$ for $\lambda\in\mathrm{Y}_{k,n-k}$, and so are the $\Sigma^\mu\mathcal{U}^\perp$ (dually, $\Sigma^\mu(V/\mathcal{U})$) for $\mu\in\mathrm{Y}_{n-k, k}$. It turns out, there is a way to ‘interpolate’ between the collections $\langle \Sigma^\lambda\mathcal{U}^* \mid \lambda\in \mathrm{Y}_{k, n-k} \rangle$ and $\langle\Sigma^\lambda(V/\mathcal{U})\mid \mu\in \mathrm{Y}_{n-k,k}\rangle$. In the process some interesting exceptional bundles appear naturally. In this subsection we give a geometric description of these bundles and present a generalization of staircase complexes. With a pair of diagrams $\lambda\in \mathrm{Y}_{k, n-k}$ and $\mu\in \mathrm{Y}_{n-k, k}$ such that
$$
\begin{equation}
\mathsf{w}(\lambda)+\mathsf{h}(\mu) \leqslant n-k \quad \text{and} \quad \mathsf{h}(\lambda)+\mathsf{w}(\mu) \leqslant k
\end{equation}
\tag{27}
$$
we associate an equivariant exceptional vector bundle $\mathcal{F}^{\lambda,\mu}$. Its dual is denoted by $\mathcal{E}^{\lambda,\mu}=(\mathcal{F}^{\lambda,\mu})^*$. Let $h=\mathsf{h}(\lambda)$, and assume that $0<h<k$. Consider the diagram We denote by $\mathcal{U}$ and $\mathcal{W}$ the tautological bundles on $\mathsf{Gr}(k, V)$ and $\mathsf{Gr}(k-h,V)$, respectively, as well as their pullbacks under $p$ and $q$. Thus, the universal flag on $\mathsf{Fl}(k-h,k;V)$ is $\mathcal{W}\subset\mathcal{U}\subset V$.Proposition 3.6 ([12]). If $\lambda$ and $\mu$ satisfy (27) and $h=\mathsf{h}(\lambda)$ satisfies $0 < h < k$, then and is an exceptional vector bundle on $\mathsf{Gr}(k,V)$.Since we introduced the extra condition on $h$, we extend the definition of $\mathcal{F}^{\lambda,\mu}$ by putting
$$
\begin{equation*}
\mathcal{F}^{\lambda,0}=\Sigma^\lambda\mathcal{U} \quad\text{and}\quad \mathcal{F}^{0,\mu}=\Sigma^\mu\mathcal{U}^\perp.
\end{equation*}
\notag
$$
Thus, the bundles $\mathcal{F}^{\lambda,\mu}$ become a generalization of the bundles $\Sigma^\lambda\mathcal{U}$ and $\Sigma^\mu\mathcal{U}^\perp$. There is an alternative geometric construction of $\mathcal{F}^{\lambda,\mu}$, which can be obtained via the usual isomorphism $\mathsf{Gr}(k, V)\simeq \mathsf{Gr}(n-k,V)$. Assume that $w=\mathsf{w}(\lambda)$ is such that $0 < w < n-k$. Consider the diagram Denote by $\mathcal{U}$ and $\mathcal{K}$ the tautological bundles on $\mathsf{Gr}(k,V)$ and $\mathsf{Gr}(n-w,V)$, respectively, as well as their pullbacks under $f$ and $g$. Thus, the universal flag on $\mathsf{Fl}(k, n-w;V)$ is $\mathcal{U}\subset\mathcal{K}\subset V$.Proposition 3.7 ([12]). If $\lambda$ and $\mu$ satisfy (27) and $w=\mathsf{h}(\lambda)$ satisfies $0 < w < n-k$, then andExample 3.8. The first non-trivial example of a diagram of the form $\mathcal{F}^{\lambda,\mu}$ is given by the universal extension the dual of which isIn what follows it will be convenient for us to work with the dual bundles $\mathcal{E}^{\lambda,\mu}$. Let us list some of their properties. First, as we have already seen, if $\mu$ is empty, then $\mathcal{E}^{\lambda,\mu}=\mathcal{E}^{\lambda,0}\simeq \Sigma^\lambda\mathcal{U}^*$, and if $\lambda$ is empty, then $\mathcal{E}^{0, \mu}=\Sigma^\mu(V/\mathcal{U})$. Second,
$$
\begin{equation*}
\mathcal{E}^{\lambda, \mu} \in \langle \Sigma^\alpha\mathcal{U}^*\otimes \Sigma^\beta (V/\mathcal{U})\mid \alpha \subset \lambda,\ \beta\subset \mu \rangle,
\end{equation*}
\notag
$$
and there is actually an epimorphism $\mathcal{E}^{\lambda, \mu} \to \Sigma^\lambda\mathcal{U}^*\otimes \Sigma^\mu (V/\mathcal{U})$ whose kernel belongs to the subcategory $\langle \Sigma^\alpha\mathcal{U}^*\otimes \Sigma^\beta (V/\mathcal{U}) \mid \alpha \subsetneq \lambda,\ \beta\subsetneq \mu \rangle$. Third, in Proposition 3.6 one could choose any integer $h$ as long as $\mathsf{h}(\lambda)\leqslant h$ and $\mathsf{w}(\mu) \leqslant k-h$. Similarly, in Proposition 3.7 one could choose any integer $w$ as long as $\mathsf{w}(\lambda)\leqslant w$ and $\mathsf{h}(\mu)\leqslant n-k-w$. Let $\lambda$ and $\mu$ be as above, and let $w=\mathsf{w}(\lambda)>0$. Recall that $\bar\lambda$ denotes the diagram obtained from $\lambda$ by removing the first row. Denote by $\mu(1)$ the diagram obtained from $\mu$ by adding a full column of height $n-k-w$. In what follows $\lambda^{(i)}$ are precisely the diagrams introduced in (16). Proposition 3.9. Let $w=\mathsf{w}(\lambda)$. There is an exact sequence of vector bundles on $\mathsf{Gr}(k, V)$ of the form where $b_\lambda^{(i)}=|\lambda/\lambda^{(i)}|=|\lambda|-|\lambda^{i}|$.Using the isomorphism between $\mathsf{Gr}(k,V)$ and $\mathsf{Gr}(n-k,V^*)$, one obtains the following dual version of Proposition 3.9. Proposition 3.10. Let $w'=\mathsf{w}(\mu)$. There is an exact sequence of vector bundles on $\mathsf{Gr}(k,V)$ of the form where $b_\mu^{(i)}=|\lambda/\lambda^{(i)}|=|\lambda|-|\lambda^{i}|$, and $\lambda(1)$ is obtained from $\lambda$ by adding a full column of height $k-w'$.Remark 3.12. The proofs of Propositions 3.9 and 3.10 appeared in [13] (see Theorems 4.3 and 4.4 there, respectively); however, what we denote by $\mathcal{F}^{\lambda,\mu}$ in the present survey is denoted by $\mathcal{E}^{\lambda,\mu}$ there. We present our apology to the reader for the change of notation, which we did it to make it in line with the definitions of right and left exceptional blocks. Long exact sequences (34) and (35) are obvious generalizations of (18), and are interesting even when $\lambda$ or $\mu$ is the zero diagram. For instance, when $\lambda=(w)$ for some $0 < w < n-k$ and $\mu$ is empty, one has $\mathcal{E}^{\lambda, \mu}\simeq S^w\mathcal{U}^*$. In this case $\bar\lambda$ is empty and $\mu(1)=(n-k-w)^T$, so $\mathcal{E}^{\bar\lambda, \mu(1)}(1)\simeq \Lambda^{n-k-w}(V/\mathcal{U})(1) \simeq \Lambda^w\mathcal{U}^\perp$, and the sequence (34) is the well-known long exact sequence
$$
\begin{equation*}
0\to \Lambda^w\mathcal{U}^\perp \to \Lambda^wV^*\otimes\mathcal{O} \to \Lambda^{w-1}V^*\otimes\mathcal{U}^* \to \cdots \to V^*\otimes S^{w-1}\mathcal{U}^*\to S^w\mathcal{U}^*\to 0
\end{equation*}
\notag
$$
obtained from the dual tautological short exact sequence $0\to \mathcal{U}^\perp\to V^*\to \mathcal{U}^*\to 0$. 3.4. Kuznetsov–Polishchuk collections for classical GrassmanniansThe first application of generalized staircase complexes that we encounter is related to the following story, which was conjectured in [24], § 9.6. The proof of the statements was originally announced in [12] and presented in full in [13]. Let $p=(w,h)\in\mathbb{R}^2$ be a point in our favorite rectangle of width $n-k$ and height $k$; that is, we assume that $0\leqslant w\leqslant n-k$ and $0\leqslant h\leqslant k$. With such a point we want to associate a collection of weights of the maximal parabolic $\mathrm{P}\subset \mathsf{GL}(V)$. Since we do not want to introduce the necessary notation, we simply describe this collection via the corresponding irreducible equivariant vector bundles: these are of the form $\Sigma^\lambda\mathcal{U}^*\otimes\Sigma^\mu(V/\mathcal{U})$ for $\lambda\in\mathrm{Y}_{n-k-w, h}$ and $\mu\in\mathrm{Y}_{k-h, w}$ (where for arbitrary $a, b\in \mathbb{R}$ we put $\mathrm{Y}_{a,b}=\mathrm{Y}_{\lfloor a\rfloor,\lfloor b\rfloor}$). Notice that such a pair $(\lambda,\mu)$ satisfies conditions (27). We denote this set of weights by $B_p$ and denote by $\mathcal{B}_p$ the subcategory generated by the corresponding bundles,
$$
\begin{equation*}
\mathcal{B}_p=\bigl\langle \Sigma^\lambda\mathcal{U}^*\otimes \Sigma^\mu(V/\mathcal{U})\mid (\lambda,\mu)\in \mathrm{Y}_{n-k-w,h} \times\mathrm{Y}_{k-h,w}\bigr\rangle.
\end{equation*}
\notag
$$
Proposition 3.13 ([13], Proposition 5.1). The subcategory $\mathcal{B}_p$ is generated by a full exceptional collection Remark 3.14. It was originally conjectured in [24] that the blocks $B_p$ are (right) exceptional blocks. Though it was stated in [12] that this is indeed the case, the proof of Proposition 3.13 does not rely on this fact. What we do obtain though is that the objects $\mathcal{E}^{\lambda,\mu}$ form the right dual exceptional collection to $\Sigma^\lambda\mathcal{U}^*\otimes\Sigma^\mu(V/\mathcal{U})$ in the equivariant derived category. Consider a monotone path $\gamma$ in $\mathbb{R}^2$ going from $(0,0)$ to $(n-k,k)$. Denote by $p_0,p_1,p_2,\dots,p_l$ the points of intersection of $\gamma$ with the grid $\mathbb{Z}\times\mathbb{R} \cup \mathbb{R}\times\mathbb{Z}$, listed from left to right. In particular, one has $p_0=(0, 0)$ and $p_l=(n-k, k)$. The following theorem was stated originally as a conjecture in [24] (see Conjecture 9.8). Theorem 3.15 ([13]). For any $\gamma$ as above, there is a semiorthogonal decomposition Generalized staircase complexes were specifically constructed to prove the fullness of the decompositions (36). Namely, via a combinatorial argument one can reduce to the case when $\gamma$ is of very specific form (goes very closely along the left border of the rectangle, then follows the top border), and it is not hard to see that the corresponding decompositions comes from Kapranov’s exceptional collection. Remark 3.16. When $n=2k$, one can choose $\gamma$ to be diagonal in the rectangle $[0,k]\times[0,k]$. The corresponding decomposition has some very nice symmetry. 4. Symplectic Grassmannians4.1. Symplectic GrassmanniansWe are ready to discuss isotropic Grassmannians associated with the simple group $\mathbf{G}=\mathsf{Sp}(V)$ of type $C_n$, where $V$ is a $2n$-dimensional vector spaces equipped with a non-degenerate skew-symmetric bilinear form $\omega\in\Lambda^2V^*$. There are precisely $n$ maximal parabolic subgroups $\mathbf{P}_i\subset\mathbf{G}$. We follow the Bourbaki numbering of simple roots, which is displayed in Fig. 1. The quotient $\mathbf{G}/\mathbf{P}_k \simeq \mathsf{IGr}(k,V)$ can be identified with the Grassmannian of subspaces $U\subset V$ isotropic with respect to $\omega$. In particular, $\mathsf{IGr}(k,V)$ embeds into $\mathsf{Gr}(k,V)$ and can actually be described as the zero locus of the regular section of $\Lambda^2\mathcal{U}^*$ on $\mathsf{Gr}(k,V)$ which corresponds to $\omega$ (recall that $\Gamma(\mathsf{Gr}(k, V), \Lambda^2\mathcal{U}^*)\simeq \Lambda^2V^*$). The isotropic Grassmannian $\mathsf{IGr}(k,V)$ has Fano index $2n-k+1$, and $\mathsf{rk}\,K_0(\mathsf{IGr}(k, V))=\binom{n}{k}2^n$. We denote by $\mathcal{U}$ the universal bundle (restricted from $\mathsf{Gr}(k,V)$) on $\mathsf{IGr}(k,V)$. The form $\omega$ induces an isomorphism $V\to V^*$, and one obtains a commutative diagram The quotient $\mathcal{U}^\perp /\mathcal{U}$ is naturally a symplectic vector bundle. By looking at the Levi subgroup of $\mathbf{P}_i$, we see that all irreducible equivariant vector bundles on $\mathsf{IGr}(k,V)$ are of the form
$$
\begin{equation*}
\Sigma^\lambda\mathcal{U}^*\otimes (\mathcal{U}^\perp /\mathcal{U})^{\langle \mu \rangle},
\end{equation*}
\notag
$$
where $\lambda$ is a dominant weight of $\mathsf{GL}_k$, $\mu$ is a dominant weight of $\mathsf{Sp}_{n-k}$, and $(-)^{\langle\mu\rangle}$ denotes the symplectic Schur functor. 4.2. Symplectic Grassmannians of planesThe first isotropic symplectic Grassmannian $\mathsf{Sp}(V)/\mathbf{P}_1$ is not very interesting. Indeed, $\mathsf{IGr}(1, V)\simeq \mathbb{P}(V)$, and this case was covered by Beilinson. Thus, we turn to the case $k=2$. Recall that $\mathsf{IGr}(k,V)$ is the zero locus of a regular section of $\Lambda^2\mathcal{U}^*$ on $\mathsf{Gr}(k, V)$. Thus, $\mathsf{IGr}(2, V)$ is just a smooth hyperplane section of $\mathsf{Gr}(2,V)$ with respect to $\mathcal{O}(1)\simeq \Lambda^2\mathcal{U}^*$. A full exceptional collection in $D(\mathsf{IGr}(2, V))$ was constructed by Kuznetsov in [23] using the following simple observation, which lies in the core of his theory of Homological Projective Duality. Proposition 4.1. Let $X$ be a smooth projective variety with a very ample line bundle $\mathcal{O}(1)$, and let $Y$ be a smooth hyperplane section of $X$ with respect to $\mathcal{O}(1)$. Denote by $\iota\colon Y \to X$ the embedding morphism. Assume that $D(X)$ is equipped with a Lefschetz decomposition Then for all $1\leqslant i < m$ the pullback functor induces a fully faithful embedding and there is a semiorthogonal decomposition of the formIn other words, if one has a Lefschetz decomposition with a rather small initial block, then one obtains quite a lot of information about the derived categories of its hyperplane sections: the subcategory generated by the remaining blocks embeds fully and faithfully into them. The best possible scenario happens when $\mathcal{A}_Y$ in (37) vanishes. It turns out that this is the case for $\mathsf{IGr}(2,V)$. Theorem 4.2 ([23], Theorem 5.1). There is a (full) Lefschetz decomposition of $D(\mathsf{IGr}(2,V))$ with the basis Remark 4.3. The exceptional collection produced by Theorem 4.2 is, up to a twist by $\mathcal{O}(-1)$, precisely the collection given in Remark 2.15 with the first column dropped. Namely, we obtain an exceptional collection of the form
$$
\begin{equation*}
\begin{pmatrix} S^{n-1}\mathcal{U}^* & S^{n-1}\mathcal{U}^*(1) & \dots & S^{n-1}(n-2) & \\ S^{n-2}\mathcal{U}^* & S^{n-2}\mathcal{U}^*(1) & \dots & S^{n-2}(n-2) & \dots & S^{n-2}\mathcal{U}^*(2n-2) \\ \vdots & \vdots & & \vdots & & \vdots \\ \mathcal{U}^* & \mathcal{U}^*(1) & \dots & \mathcal{U}(n-1) & \dots & \mathcal{U}^*(2n-2) \\ \mathcal{O}^* & \mathcal{O}^*(1) & \dots & \mathcal{O}(n-1) & \dots & \mathcal{O}^*(2n-2) \end{pmatrix}.
\end{equation*}
\notag
$$
4.3. Kuznetsov–Polishchuk collectionsIn [24] Kuznetsov and Polishchuk constructed exceptional blocks that produce exceptional collections of maximum length in $\mathsf{IGr}(k,V)$ for all $k$. The following results were formulated in [24], Theorem 9.2. We will also say that a collection of irreducible equivariant vector bundles forms an exceptional block if the corresponding dominant weights do. Proposition 4.4. Let $0\leqslant t \leqslant k-1$. Then the bundles form an exceptional block. We denote by $\mathcal{A}_t$ the corresponding admissible subcategory in $D(\mathsf{IGr}(k,V))$.Proposition 4.5. Let $k \leqslant t \leqslant 2n-k$. Then the bundles form an exceptional block. We denote by $\mathcal{A}_t$ the corresponding admissible subcategory in $D(\mathsf{IGr}(k,V))$.Theorem 4.6. The subcategories $\mathcal{A}_0,\mathcal{A}_1(1),\dots,\mathcal{A}_{2n-k}(2n-k)$ are semiorthogonal in $D(\mathsf{IGr}(k, V))$. Since each of them is generated by an exceptional collection, there is an exceptional collection in $D(\mathsf{IGr}(k,V))$ of length $2^k\binom{n}{k}$, which equals $\mathsf{rk} K_0(\mathsf{IGr}(k,V))$. Remark 4.7. Let us say a few words about Theorem 4.6.
Conjecture 4.8 ([24], Conjecture 1.6). The exceptional collections described in Theorem 4.6 are full. So far, Conjecture 4.8 has been proved only for Lagrangian Grassmannians in [15]. We discuss this result below. 4.4. Lagrangian Grassmannians and Lagrangian staircase complexesConsider the Lagrangian Grassmannian $\mathsf{LGr}(V)=\mathsf{IGr}(n,V)$. Theorem 4.6 produces a collection of subcategories
$$
\begin{equation*}
\mathcal{A}_t=\langle \mathcal{E}^\lambda_t \mid \lambda \subset \mathrm{Y}_{t, k-t} \rangle,
\end{equation*}
\notag
$$
where the objects $\mathcal{E}^\lambda_t$ form the graded right dual exceptional collection to $\langle \Sigma^\lambda\mathcal{U}^* \mid \lambda \subset \mathrm{Y}_{t,k-t}\rangle$ in the equivariant derived category. We first observe that the subscript $t$ can be dropped from $\mathcal{E}^\lambda_t$ since the corresponding object does not depend on the block. Indeed, for any $\lambda$ such that $\mathsf{h}(\lambda)+\mathsf{w}(\lambda)\leqslant n+1$ we are going to construct an exceptional equivariant vector bundle on $\mathsf{LGr}(V)$, which we denote by $\mathcal{E}^\lambda$. This bundle will coincide with $\mathcal{E}^\lambda_t$ if $\lambda \in \mathrm{Y}_{t,k-t}$. Remark 4.9. We actually construct more exceptional bundles than what Theorem 4.6 gives us: our condition is that $h+w=n+1$, while Theorem 4.6 is concerned with diagrams $\lambda$ such that $\mathsf{h}(\lambda)+\mathsf{w}(\lambda)\leqslant n$. There are two constructions of $\mathcal{E}^\lambda$ presented in [15]. In both cases we are concerned with the dual objects, which we denote by $\mathcal{F}^\lambda$ (a similar thing happened in subsection 3.4). Let $h,w\geqslant 1$ be such that $h+w=n+1$. Consider the diagram and denote by $\mathcal{W}\subset\mathcal{U}$ the universal flag on $\mathsf{IFl}(w,n;V)$.Proposition 4.10 ([15], Proposition 3.1). Let $\lambda\in\mathrm{Y}_{h-1,w}$. The object is an exceptional vector bundle on $\mathsf{LGr}(V)$ which depends only on $\lambda$ and not on $w$.If $w>1$, we also consider the diagram and denote by $\mathcal{H}\subset\mathcal{U}$ the universal flag on $\mathsf{IFl}(w-1,n;V)$.Proposition 4.11 ([15], Proposition 3.2). Assume $w\geqslant 2$. Let $\lambda\in\mathrm{Y}_{h,w}$ be such that $\lambda_h>0$. The object
$$
\begin{equation*}
\mathcal{F}^\lambda=\tilde{f}_*\bigl(\det(\mathcal{U}/\mathcal{H})\otimes \tilde{g}^* (\mathcal{H}^\perp / \mathcal{H})^{\langle \lambda(-1)\rangle}\bigr),
\end{equation*}
\notag
$$
where $\lambda(-1)=(\lambda_1-1,\dots,\lambda_h-1)$, is an exceptional vector bundle on $\mathsf{LGr}(V)$. If $\lambda \in \mathrm{Y}_{h'-1,w'}$ for some $h'+w'=n+1$, then $\mathcal{F}^\lambda$ coincides with the one defined in Proposition 4.10. As one can see, Propositions 4.10 and 4.11 cover all $\lambda$ with $\mathsf{h}(\lambda)+\mathsf{w}(\lambda)\leqslant n+1$, but not in a single take. The second construction does not have this disadvantage. We consider diagram (39) once again. First observe the following. Lemma 4.12 ([15], Lemma 3.4). There is an exceptional collection in $D(\mathsf{IGr}(w,V))$ of the form Denote the left graded dual exceptional collection to (40) by
$$
\begin{equation*}
\langle \mathcal{G}^\lambda \mid \lambda \in \mathrm{Y}_{h,w}\rangle.
\end{equation*}
\notag
$$
The defining property of this collection is the following:
$$
\begin{equation*}
\mathcal{G}^\lambda \in \langle \Sigma^\mu \mathcal{U}^* \mid \mu\in \mathrm{Y}_{w, h}\rangle,\qquad \mathsf{Ext}^\bullet(\Sigma^\mu\mathcal{U}^*,\mathcal{G}^\lambda)= \begin{cases} \mathsf{k}[-|\lambda|] & \text{if}\ \lambda=\mu^T, \\ 0 & \text{otherwise}. \end{cases}
\end{equation*}
\notag
$$
Proposition 4.13 ([15], Proposition 3.6). There is an isomorphism The descriptions of $\mathcal{F}^\lambda$ (thus, of $\mathcal{E}^\lambda$, which are simply the duals) given by Propositions 4.10, 4.11, and 4.13 allow one to construct what we call Lagrangian staircase complexes. Recall that the set $\mathrm{Y}_{h, w}$ for $w+h=n+1$ can be identified with the set of binary sequences of length $n+1$ with exactly $w$ terms equal to $0$. Thus, the disjoint union $\bigsqcup\limits_{w+w=n+1}\mathrm{Y}_{h, w}$ is in bijection with the set of all binary sequences of length $n+1$, which is $\{0,1\}^{n+1}$. Fix a generator $g\in \mathbb{Z}/(2n+2)\mathbb{Z}$ and consider the following action of the group $\mathbb{Z}/(2n+2)\mathbb{Z}$ on $\{0,1\}^{n+1}$:
$$
\begin{equation*}
g\colon a_0a_1\ldots a_{n} \mapsto (a_n-1)a_0a_1\ldots a_{n-1}.
\end{equation*}
\notag
$$
The action on the diagrams can be described as follows. Let $\lambda=(\lambda_1,\dots,\lambda_h)\in\mathrm{Y}_{w, h}$. Then
$$
\begin{equation*}
g\colon \lambda \mapsto \lambda'=\begin{cases} (\lambda_1,\dots,\lambda_h, 0) & \text{if}\ \lambda_1 < w, \\ (\lambda_2+1,\dots,\lambda_h+1) & \text{if}\ \lambda_1=w. \end{cases}
\end{equation*}
\notag
$$
In particular, $\lambda'\in \mathrm{Y}_{w-1, h+1}$ if $\lambda_1 < w$, and $\lambda'\in \mathrm{Y}_{w+1, h-1}$ if $\lambda_1=w$. We keep the same notation $\lambda'$ as in § 2.3, in order to show a direct analogy between the classical and Lagrangian cases. We strongly suggest that the reader compares the following proposition with Proposition 2.11. Proposition 4.15 ([15], Proposition 4.3). Let $w, h>0$ be such that $w+h=n+1$, and let $\lambda\in\mathrm{Y}_{h,w}$ be such that $\lambda_1=w$. There is an exact sequence of vector bundles on $\mathsf{LGr}(V)$ of the form
$$
\begin{equation}
0\to \mathcal{E}^{\lambda'}(-1) \to V^{[b_\lambda^{(w)}]}\otimes \mathcal{E}^{\lambda^{(w)}}\to \cdots \to V^{[b_\lambda^{(1)}]}\otimes \mathcal{E}^{\lambda^{(1)}} \to\mathcal{E}^\lambda \to 0,
\end{equation}
\tag{41}
$$
where $\lambda^{(i)}$ and $b_\lambda^{(i)}$ are the same as in Proposition 2.11, and $V^{[j]}$ denotes the $j$ th fundamental representation of $\mathsf{Sp}(V)$. Complexes of the form (41) are called Lagrangian staircase complexes. Lagrangian staircase complexes were introduced precisely to prove Conjecture 4.8 for Lagrangian Grassmannians. Theorem 4.16 ([15], Theorem 4.6). There is a semiorthogonal decomposition where for each $0\leqslant h\leqslant n$ the subcategory $\mathcal{A}_h$ is generated by an exceptional collection:Remark 4.17. For some strange reason we had to consider more exceptional objects than it is used in Theorem 4.16 (the condition $h+w=n$ was extended to $h+w=n+1$). This strange doubling phenomenon is quite common when dealing with Lagrangian Grassmannians. Using Theorem 2.5 and the existence of full exceptional collections on flag varieties of type $A$, one immediately deduces the following. Corollary 4.18. Let $1\leqslant i_1 < i_2 < \cdots < i_t < n$. Then the bounded derived category of $\mathsf{IFl}(i_1,\dots,i_t, n;\, V)$ admits a full exceptional collection consisting of vector bundles. Proof. Apply Theorem 2.5, the existence of full exceptional collections on flag varieties of type $A$, and the fact that the projection $\mathsf{IFl}(i_1,\dots,i_t, n;\, V)\to \mathsf{LGr}(V)$ is isomorphic to the relative flag bundle $\mathsf{Fl}_{\mathsf{LGr}(V)}(i_1,\dots,i_t;\, \mathcal{U})$. There is an interesting duality result for collections (42). Theorem 4.19 ([16], Theorem 3.1). Let $0\leqslant h \leqslant n$. The exceptional collection $\langle \mathcal{F}^\mu \mid \mu\in\mathrm{Y}_{n-h, h} \rangle$ is the graded left dual exceptional collection to $\langle \mathcal{E}^\lambda \mid \lambda \in \mathrm{Y}_{h, n-h}\rangle$. That is, for $\mu\in\in\mathrm{Y}_{n-h, h}$ one has
$$
\begin{equation*}
\mathcal{F}^\mu \in \langle \mathcal{E}^\lambda \mid \lambda\in \mathrm{Y}_{h, n-h}\rangle,\qquad \mathsf{Ext}^\bullet(\mathcal{F}^\mu, \mathcal{E}^\lambda)=\begin{cases} \mathsf{k}[-|\lambda|] & \textit{if}\ \lambda=\mu^T, \\ 0 & \textit{otherwise}. \end{cases}
\end{equation*}
\notag
$$
Finally, we note that the full exceptional collections given by Theorem 4.16 are in no reasonable way Lefschetz. We end this section with minimal Lefschetz exceptional collections for small $n$. Proposition 4.20 ([36], [35], [15]). For $n=2,\dots,5$ there are minimal Lefschetz exceptional collections on $\mathsf{LGr}(n)$ of the following form.
4.5. Sporadic casesThere are two more symplectic Grassmannians for which full exceptional collections have been constructed: on $\mathsf{IGr}(3,8)$ and $\mathsf{IGr}(3,10)$ by Guseva and Novikov, respectively. Theorem 4.21 ([18]). There is a Lefschetz decomposition of $\mathsf{IGr}(3,8)$ with Lefschetz basis given by where $F$ is a certain explicit exceptional vector bundle. In particular, $D(\mathsf{IGr}(3,8))$ admits a full exceptional collection.Theorem 4.22 ([30]). There is a rectangular Lefschetz semiorthogonal decomposition
$$
\begin{equation*}
D(\mathsf{IGr}(3,10))= \langle\mathcal{B},\mathcal{B}(1),\dots,\mathcal{B}(7)\rangle,
\end{equation*}
\notag
$$
where $\mathcal{B}$ is generated by an exceptional collection:
$$
\begin{equation*}
\mathcal{B}=\langle H,S^2\mathcal{U},\Sigma^{3,1}\mathcal{U}(1),\mathcal{U}, \Sigma^{2,1}\mathcal{U}(1),\Sigma^{3,1}\mathcal{U}^*(-1),\mathcal{O}, \mathcal{U}^*,S^2\mathcal{U}^*,S^3\mathcal{U}^* \rangle
\end{equation*}
\notag
$$
and $H$ is an explicit exceptional vector bundle on $\operatorname{\mathsf{IGr}}(3, 10)$. In particular, $D(\mathsf{IGr}(3,10))$ admits a full exceptional collection. 4.6. Odd symplectic GrassmanniansIt is very plausible that another class of varieties which are very close to being homogeneous admits full exceptional collections. Consider a vector space $V$ of dimension $2n+1$ equipped with a skew-symmetric form $\omega$ of maximum rank (which equals $2n$). As in the even case, one can define isotropic Grassmannians $\mathsf{IGr}(k,V)$ for all $k=1,\dots,n+1$ as the zero loci of the corresponding regular sections of $\Lambda^2\mathcal{U}^*$ on $\mathsf{Gr}(k,V)$. A good reference for the geometry of these varieties is [28]. Similarly, one can define odd symplectic flag varieties. Conjecture 4.23. Derived categories of odd symplectic flag varieties admit full exceptional collections. When $k=1$, one has $\mathsf{IGr}(k,V)\simeq \mathbb{P}(V)$. When $k=n+1$, it is not hard to check that $\mathsf{IGr}(n+1,V)\simeq \mathsf{LGr}(\bar{V})$, where $\bar{V}=V/\operatorname{Ker}\omega$. Thus, the first non-trivial case is given by $\mathsf{IGr}(2,V)$. Theorem 4.24 ([23], Remark 5.6). There is a rectangular Lefschetz exceptional collection in the derived category $\mathsf{IGr}(2,2n+1)$ which comes from the collection (20):
$$
\begin{equation*}
D(\mathsf{IGr}(2, 2n+1))=\begin{pmatrix} S^{n-1}\mathcal{U}^* & S^{n-1}\mathcal{U}^*(1) & \dots & S^{n-1}\mathcal{U}^*(2n-1) \\ S^{n-2}\mathcal{U}^* & S^{n-2}\mathcal{U}^*(1) & \dots & S^{n-2}\mathcal{U}^*(2n-1) \\ \vdots & \vdots & & \vdots \\ \mathcal{U}^* & \mathcal{U}^*(1) & \dots & \mathcal{U}^*(2n-1) \\ \mathcal{O}^* & \mathcal{O}^*(1) & \dots & \mathcal{O}^*(2n-1) \end{pmatrix}.
\end{equation*}
\notag
$$
The only other two known cases are covered by the following results. Theorem 4.25 ([14]). There is a rectangular Lefschetz exceptional collection in the derived category of $\mathsf{IGr}(3, 7)$ given by
$$
\begin{equation*}
D(\mathsf{IGr}(3, 7))=\begin{pmatrix} \Lambda^2\mathcal{Q} & \Lambda^2\mathcal{Q}(1) & \Lambda^2\mathcal{Q}(2) & \Lambda^2\mathcal{Q}(3) & \Lambda^2\mathcal{Q}(4) \\ \mathcal{U}^* & \mathcal{U}^*(1) & \mathcal{U}^*(2) & \mathcal{U}^*(3) & \mathcal{U}^*(4) \\ \mathcal{O} & \mathcal{O}(1) & \mathcal{O}(2) & \mathcal{O}(3) & \mathcal{O}(4) \\ \mathcal{U} & \mathcal{U}(1) & \mathcal{U}(2) & \mathcal{U}(3) & \mathcal{U}(4) \end{pmatrix},
\end{equation*}
\notag
$$
where $\mathcal{Q}=V/\mathcal{U}$ is the universal quotient bundle. Theorem 4.26 ([8]). There is an explicit exceptional vector bundle $\mathcal{H}$ on $\mathsf{IGr}(3,9)$ and a rectangular Lefschetz decomposition where $\mathcal{B}$ is generated by an exceptional collection5. Orthogonal Grassmannians5.1. Orthogonal GrassmanniansWe consider Grassmannians for groups of type $B_n$ and $D_n$. First let $\mathbf{G}$ be of type $B_n$. We number the simple roots in accordance with Fig. 2. Then $\mathbf{G}/\mathbf{P}_k$ is isomorphic to the orthogonal Grassmannian $\mathsf{OGr}(k,V)$, where $V$ is a $(2n+1)$-dimensional vector space equipped with a non-degenerate symmetric form $q\in S^2V^*$. Recall that there is an isomorphism $S^2V^*\simeq \Gamma(\mathsf{Gr}(2, V), S^2\mathcal{U}^*)$. Similarly to the symplectic case, $\mathsf{OGr}(k, V)$ is the zero locus of the regular section of $S^2\mathcal{U}^*$ corresponding to $q$. Let $\mathbf{G}$ be of type $D_n$. We number the simple roots in accordance with Fig. 3. For $k\leqslant n-2$ the Grassmannian $\mathbf{G}/\mathbf{P}_k$ is isomorphic to the orthogonal Grassmannian $\mathsf{OGr}(k,V)$, where $V$ is a $2n$-dimensional vector space equipped with a non-degenerate symmetric form $q\in S^2V^*$. Again, $\mathsf{OGr}(k,V)$ is the zero locus of the regular section of $S^2\mathcal{U}^*$ corresponding to $q$. When $k=n-1$ or $k=n$, the Grassmannians $\mathbf{G}/\mathbf{P}_{n-1}$ and $\mathbf{G}/\mathbf{P}_{n}$ are isomorphic to the two (isomorphic) connected components of $\mathsf{OGr}(n, V)$, which are traditionally denoted by $\mathsf{OGr}_+(n, V)$ and $\mathsf{OGr}_-(n, V)$. Finally, there is a classical isomorphism of varieties 5.2. Quadrics and orthogonal Grassmannians of planesThe simplest orthogonal Grassmannian are quadrics. Consider an $m$-dimensional quadric $Q$. If $m$ is even, then $Q$ carries two very important vector bundles, called spinor bundles, which we denote by $\mathcal{S}_+$ and $\mathcal{S}_-$. If $m$ is odd, there is only one spinor bundle on $Q$, which we denote by $\mathcal{S}$. We refer the reader to [33] for the case of quadrics. For spinor bundles on general orthogonal Grassmannians, we refer the reader to [23; § 6]. It was shown by Kapranov in [20] that bounded derived categories of quadrics admit full exceptional collections. Theorem 5.1 ([20]). Let $Q$ be an $m$-dimensional quadric. There are full exceptional collections Remark 5.2. The decomposition (44) is obviously minimal Lefschetz, while (43) can be rewritten as which is minimal Lefschetz.The next two series of cases are $\mathsf{OGr}(2,V)$. The reader will immediately recognize that the following collections are closely related to minimal Lefschetz exceptional collections on classical Grassmannians of planes (see Theorem 2.14). We first consider $\mathsf{IGr}(2,2n+1)$. Full exceptional collections of vector bundles on these varieties were constructed by Kuznetsov in [23]. Theorem 5.3. Let $n\geqslant 2$. There is a full Lefschetz exceptional collection on $\mathsf{IGr}(2,2n+1)$ of the form
$$
\begin{equation}
D(\mathsf{OGr}(2, 2n+1))=\begin{pmatrix} \mathcal{S} & \mathcal{S}(1) & \dots & \mathcal{S}(2n-3) \\ S^{n-2}\mathcal{U}^* & S^{n-2}\mathcal{U}^*(1) & \dots & S^{n-2}\mathcal{U}^*(2n-3) \\ \vdots & \vdots & & \vdots \\ \mathcal{U}^* & \mathcal{U}^*(1) & \dots & \mathcal{U}^*(2n-3) \\ \mathcal{O}^* & \mathcal{O}^*(1) & \dots & \mathcal{O}^*(2n-3) \end{pmatrix},
\end{equation}
\tag{45}
$$
where $\mathcal{S}$ denotes the spinor bundle. For $\mathsf{IGr}(2,2n)$ an almost full exceptional collection was constructed in [23]. However, it was one object too short. The problem of constructing a full exceptional collection was solved by Kuznetsov and Smirnov in [26]. In its most elegant form it can be stated as follows. Theorem 5.4 ([26]). Consider the subcategory where $\mathcal{S}_+$ and $\mathcal{S}_-$ are the two spinor bundles. There is a semiorthogonal decomposition where $\mathcal{R}$ is equivalent to the derived category of representations of the Dynkin quiver $D_n$. In particular, $\mathcal{R}$ and, as a consequence, $D(\mathsf{OGr}(2,2n))$ admits a full exceptional collection.5.3. Kuznetsov–Polishchuk collectionsIn [24] the authors constructed sufficiently many exceptional blocks for orthogonal Grassmannians to obtain exceptional collections of maximum possible length. Accurate formulation of these results requires some setup from representation theory, so we refer the reader to [24], Theorems 9.1 and 9.3, for details. Theorem 5.5. Consider $\mathsf{OGr}(k,V)$, where $\dim V=2n+1$ and $k\leqslant n-1$. For each $0\leqslant t \leqslant k-1$ the sets of weights
$$
\begin{equation*}
\begin{aligned} \, B_t=\Biggl\{\lambda\in \mathbb{Z}^n \Biggm| 2n-1-k \geqslant \lambda_1 &\geqslant \cdots \geqslant \lambda_t \geqslant t=\lambda_{t+1}=\cdots=\lambda_k, \\ &\qquad\qquad \frac{k-t}{2} \geqslant \lambda_{k+1} \geqslant \cdots \geqslant \lambda_n \geqslant 0 \Biggr\} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, B_{t+1/2}=\!\biggl\{\lambda\,{\in}\,\biggl(\frac{1}{2}+\mathbb{Z}\biggr)^n \!\biggm|\!{}&2n-\frac{1}{2}-k \geqslant \lambda_1 \geqslant \cdots \geqslant \lambda_t \geqslant t+\frac{1}{2}=\lambda_{t+1}=\cdots=\lambda_k,\\ &\quad\qquad\qquad\qquad\qquad\,\,\frac{k-t}{2} \geqslant \lambda_{k+1} \geqslant \cdots \geqslant \lambda_n \geqslant 0 \biggr\} \end{aligned}
\end{equation*}
\notag
$$
are (right) exceptional blocks. For each $k\leqslant t \leqslant 2n-k-1$ the set of weights
$$
\begin{equation*}
B_t=\bigl\{\lambda\in \mathbb{Z}^n \bigm| 2n-1-k \geqslant \lambda_1 \geqslant \cdots \geqslant \lambda_{k-1} \geqslant \lambda_k=t, \ \lambda_{k+1}=\cdots=\lambda_n=0\bigr\}
\end{equation*}
\notag
$$
is a (right) exceptional block. Denote by $\mathcal{A}_t$ and $\mathcal{A}_{t+1/2}$ the categories generated by the exceptional collections $\langle \mathcal{E}^\lambda \mid \lambda \in B_t \rangle$ and $\langle \mathcal{E}^\lambda \mid \lambda \in B_{t+1/2} \rangle$, respectively. The subcategories are semiorthogonal.Theorem 5.6. Consider $\mathsf{OGr}(k,V)$, where $\dim V=2n$ and $k\leqslant n-2$. For each $0\leqslant t \leqslant k-1$ the sets of weights
$$
\begin{equation*}
\begin{aligned} \, B_t=\biggl\{\lambda\in \mathbb{Z}^n \biggm| 2n-2-k \geqslant \lambda_1 &\geqslant \cdots \geqslant \lambda_t \geqslant t=\lambda_{t+1}=\cdots=\lambda_k, \\ &\qquad \frac{k-t}{2} \geqslant \lambda_{k+1} \geqslant \cdots \geqslant \lambda_{n-1} \geqslant |\lambda_n| \biggr\} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, B_{t+1/2}=\biggl\{\lambda\,{\in}\,\biggl(\frac{1}{2}+\mathbb{Z}\biggr)^n \!\biggm|\! 2n-\frac{3}{2}-k \geqslant & \lambda_1 \geqslant \cdots \geqslant \lambda_t \geqslant t+\frac{1}{2}=\lambda_{t+1}=\cdots=\lambda_k, \\ &\frac{k-t}{2} \geqslant \lambda_{k+1} \geqslant \cdots \geqslant \lambda_{n-1} \geqslant |\lambda_n| \biggr\} \end{aligned}
\end{equation*}
\notag
$$
are (right) exceptional blocks. For each $k\leqslant t \leqslant 2n-k-2$ the set of weights
$$
\begin{equation*}
B_t=\bigl\{\lambda\in \mathbb{Z}^n \bigm| 2n-2-k \geqslant \lambda_1 \geqslant \cdots \geqslant \lambda_{k-1} \geqslant \lambda_k=t, \ \lambda_{k+1}=\cdots=\lambda_n=0 \bigr\}
\end{equation*}
\notag
$$
is a (right) exceptional block. Denote by $\mathcal{A}_t$ and $\mathcal{A}_{t+1/2}$ the categories generated by the exceptional collections $\langle \mathcal{E}^\lambda \mid \lambda \in B_t \rangle$ and $\langle \mathcal{E}^\lambda \mid \lambda \in B_{t+1/2} \rangle$, respectively. The subcategories are semiorthogonal.Theorem 5.7. Consider $\mathsf{OGr}(n, V)$, where $\dim V=2n+1$. For all $0\leqslant t \leqslant n-1$ the sets of weights
$$
\begin{equation*}
B_{2t}=\{ \lambda\in\mathbb{Z}^n \mid n-1 \geqslant \lambda_1 \geqslant \cdots \geqslant \lambda_t \geqslant t=\lambda_{t+1}=\cdots=\lambda_n \}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
B_{2t+1}=\biggl\{\lambda\in\biggl(\frac{1}{2}+\mathbb{Z}\biggr)^n \biggm| n-\frac{1}{2} \geqslant \lambda_1 \geqslant \cdots \geqslant \lambda_t \geqslant t+\frac{1}{2}=\lambda_{t+1}=\cdots=\lambda_n\biggr\}
\end{equation*}
\notag
$$
are (right) exceptional blocks. Denote by $\mathcal{A}_t$ the subcategory in $D(\mathsf{OGr}(n, V))$ generated by the exceptional collection $\langle \mathcal{E}^\lambda \mid \lambda \in B_t \rangle$. The subcategories $ \mathcal{A}_0,\mathcal{A}_1,\dots,\mathcal{A}_{2n-1} $ are semiorthogonal. 5.4. Sporadic casesWe have the following not very interesting cases of orthogonal Grassmannians when $n$ is small.
The derived category of $\mathsf{OGr}_+(5,10)$ was first studied by Kuznetsov. Theorem 5.9 ([21]). Let $X=\mathsf{OGr}_+(5, 10)$, which is also isomorphic to $\mathsf{OGr}(4,9)$. The bounded derived category of $X$ admits a full exceptional collection. Remark 5.10. In [29] Moschetti and Rampazzo showed that the collection given by the Kuznetsov–Polishchuk construction for $\mathsf{OGr}_+(5,10)$ is full. The next theorem by Benedetti, Faenzi, and Smirnov covers $\mathsf{OGr}_+(6,12)$, which is isomorphic to $\mathsf{OGr}(5,11)$. Theorem 5.11 ([3]). Let $X=\mathsf{OGr}_+(6, 12)$. There are unique $\mathsf{Spin}_{12}$-equivariant extensions and a Lefschetz decomposition of $D(X)$ with Lefschetz basis In particular, $D(\mathsf{OGr}_+(6,12))\simeq D(\mathsf{OGr}(5,11))$ admits a full exceptional collection of vector bundles.6. Grassmannians for exceptional groups6.1. Grassmannians for $E_6$, $E_7$, and $E_8$The only case when something is known is when $\mathbf{G}$ is of type $E_6$ and the parabolic group is either $\mathbf{P}_1$ or $\mathbf{P}_6$, where we use the numbering of roots shown in Fig. 4. The diagram has a symmetry, so $\mathbf{G}/\mathbf{P}_1$ and $\mathbf{G}/\mathbf{P}_6$ are isomorphic. This is a $16$-dimensional variety of Fano index $12$, which is also known as the Cayley plane $\mathbb{OP}^2$. Its derived category was originally studied by Manivel in [27]. The following result is due to Faenzi and Manivel. Theorem 6.1 ([10]). There is a Lefschetz decomposition of $D(\mathbb{CO}^2)$ with Lefschetz basis given by
$$
\begin{equation*}
\langle \mathcal{U}^{2\omega_6-\omega_1},\mathcal{U}^{\omega_6-\omega_1}, \mathcal{O} \rangle,\qquad o=(3, 12, 12),
\end{equation*}
\notag
$$
where $\mathcal{U}^{\omega}$ is the vector bundle associated with the $\mathbf{L}$-dominant weight $\omega$, and the $\omega_i$ are the dominant weights of $\mathbf{G}$ with respect to the numbering given in Fig. 4. In particular, $D(\mathbb{CO}^2)$ admits a full exceptional collection consisting on vector bundles. The only other known case is when $\mathbf{G}$ is of type $E_7$: in [3] Benedetti, Faenzi, and Smirnov constructed an exceptional collection of maximum length in $D(\mathbf{G}/\mathbf{P}_7)$. 6.2. Grassmannians for $F_4$Let $\mathbf{G}$ be of type $F_4$. We use the numbering of weights given in Fig. 5. The two cases when something is known correspond to the first and fourth roots. The variety $\mathbf{G}/\mathbf{P}_1$ is known as the adjoint Grassmannian in type $F_4$. It is a variety of dimension $15$, and its Fano index equals $8$. The following was shown by Smirnov. Theorem 6.2 ([38]). Let $X=\mathbf{G}/\mathbf{P}_1$ be the adjoint Grassmannian. There is a unique equivariant extension
$$
\begin{equation*}
0 \to \mathcal{O} \to \tilde{\mathcal{T}} \to \mathcal{T}_{X} \to 0
\end{equation*}
\notag
$$
of its tangent bundle $\mathcal{T}_X$ and a rectangular Lefschetz exceptional collection
$$
\begin{equation*}
D(X)=\begin{pmatrix}[r] \tilde{\mathcal{T}} & \tilde{\mathcal{T}}(1) & \tilde{\mathcal{T}}(2) & \tilde{\mathcal{T}}(3) & \tilde{\mathcal{T}}(4) & \tilde{\mathcal{T}}(5) & \tilde{\mathcal{T}}(6) & \tilde{\mathcal{T}}(7) \\ \mathcal{U}^{\omega_4} & \mathcal{U}^{\omega_4}(1) & \mathcal{U}^{\omega_4}(2) & \mathcal{U}^{\omega_4}(3) & \mathcal{U}^{\omega_4}(4) & \mathcal{U}^{\omega_4}(5) & \mathcal{U}^{\omega_4}(6) & \mathcal{U}^{\omega_4}(7) \\ \mathcal{O} & \mathcal{O} (1) & \mathcal{O} (2) & \mathcal{O} (3) & \mathcal{O} (4) & \mathcal{O} (5) & \mathcal{O} (6) & \mathcal{O} (7) \end{pmatrix},
\end{equation*}
\notag
$$
where $\mathcal{U}^{\omega_4}$ is the vector bundle associated with the $\mathbf{L}$-dominant fundamental weight $\omega_4$ of $\mathbf{G}$. The second case is the variety $\mathbf{G}/\mathbf{P}_4$, which is known as the coadjoint Grassmannian in type $F_4$. It is a variety of dimension $15$, and its Fano index equals $11$. Moreover, it can be realized as a general hyperplane section of the Cayley plane $\mathbb{CO}^2$. In [2] Belmans, Kuznetsov, and Smirnov showed that the collection constructed by Manivel and Faenzi (see Theorem 6.1) restricts to a full rectangular exceptional collection in $D(\mathbf{G}/\mathbf{P}_4)$. Theorem 6.3 ([2]). The coadjoint Grassmannian $\mathbf{G}/\mathbf{P}_4$ admits a full Lefschetz exceptional collection consisting of vector bundles with the support function 6.3. Grassmannians for $G_2$Finally, we consider the case when $\mathbf{G}$ is of type $G_2$. The root numbering is shown in Fig. 6. There are only two Grassmannians. The quotient $\mathbf{G}/\mathbf{P}_2$ is isomorphic to $Q_5$, a $5$-dimensional quadric. This case is covered by Kapranov’s work. The remaining case is $\mathbf{G}/\mathbf{P}_1$, which is a $5$-dimensional variety of Fano index $3$. Kuznetsov showed the following. Theorem 6.4 ([21]). The bounded derived category of $\mathbf{G}/\mathbf{P}_1$ admits a full exceptional collection.
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