Fullness of the Kuznetsov–Polishchuk Exceptional Collection for the Spinor Tenfold

Abstract

Kuznetsov and Polishchuk provided a general algorithm to construct exceptional collections of maximal length for homogeneous varieties of type A, B, C, D. We consider the case of the spinor tenfold and we prove that the corresponding collection is full, i.e. it generates the whole derived category of coherent sheaves. We also verify strongness and purity of such collection. As a step of the proof, we construct some resolutions of homogeneous vector bundles which might be of independent interest.

Appendix A. Borel–Bott–Weil Computations

In this appendix, we will gather all cohomological computations done by means of the Borel–Bott–Weil theorem [29]. This well-known theorem provides a simple, algorithmic recipe to compute the cohomology of irreducible vector bundles.

Theorem 1.1

(Borel–Bott–Weil) Let ${\mathcal E}_\omega$ be a homogeneous, irreducible vector bundle on a rational homogeneous variety G/P, let us call $\rho$ the sum of all fundamental weights. Then one and only one of the following statements is true:

• 1. there exists a sequence $s_p$ of simple Weyl reflections of length p such that $s_p(\omega +\rho ) - \rho$ is dominant (i.e. all coefficients of its expansion in the fundamental weights are non-negative). Then $H^p(G/P, {\mathcal E}_\omega )\simeq V^G_{s_p(\omega +\rho ) - \rho }$ and all the other cohomology is trivial.

• 2. there is no such $s_p$ as in point (1). Then ${\mathcal E}_\omega$ has no cohomology.

While we chose to illustrate all computations in detail, the algorithm can be easily automatized. See, for example, the script [28].

1.1 A.2 Cohomology Computations on $D_5/P^4$

Here we follow the notation we fixed in Section 2.1.3.

Lemma 2.1

For every nonnegatgive $a,b,c,d\in {\mathbb Z}$, one has $H^\bullet (X, {\mathcal E}_{a\omega _1+b\omega _2+c\omega _3+d\omega _5}) = {\mathbb V}^{D_5}_{a\omega _1+b\omega _2+c\omega _3+d\omega _5}[0]$.

Proof

The weight $a\omega _1+b\omega _2+c\omega _3+d\omega _5$ is already dominant: by Theorem A.1 we immediately conclude.

Corollary 3.9

For every nonnegative integer r, one has $H^\bullet (X, {{\,\textrm{Sym}\,}}^r{\mathcal U}^\vee ) = {\mathbb V}^{D_5}_{r\omega _1}[0]$.

Proof

The weight associated to ${{\,\textrm{Sym}\,}}^r{\mathcal U}^\vee$ is $r\omega _1$ (see Eq. 2.2), hence we conclude by Lemma A.2.$\square$

Corollary A.3

There are the following isomorphisms:

$$\begin{aligned} {\text {Ext}}^\bullet ({\mathcal U}(2), {{\,\textrm{Sym}\,}}^2{\mathcal U}^\vee (2)) \simeq H^\bullet (X, {{\,\textrm{Sym}\,}}^3{\mathcal U}^\vee \oplus {\mathcal E}_{\omega _1+\omega _2})\simeq {\mathbb V}^{D_5}_{3\omega _1}[0]\oplus {\mathbb V}^{D_5}_{\omega _1+\omega _2}[0]. \end{aligned}$$

Proof

The first step is simply Lemma 2.1, then we conclude by Corollary A.3.$\square$

Lemma 2.2

For all integers $a\ge 0$, $b\ge 0$, $c>0$ and for $\epsilon \in \{1,2\}$ one has:

$$\begin{aligned} \begin{aligned} H^\bullet (X, {\mathcal E}_{a\omega _1+b\omega _2-\epsilon \omega _4})&= 0 \\ H^\bullet (X, {\mathcal E}_{a\omega _1+b\omega _2+ c\omega _3-2\omega _4})&= {\mathbb V}^{D_5}_{a\omega _1+b\omega _2+(c-1)\omega _3}[-1]. \end{aligned} \end{aligned}$$

Proof

An intuitive way to apply the Borel–Bott–Weil algorithm is to write the coefficients of the weight of the bundle as labels for the nodes of the Dynkin diagram, then add the sum $\rho$ of the fundamental weights, and then to apply the Weyl reflections associated to simple roots. By adding $\rho$ to the weight $a\omega _1+b\omega _2-\epsilon \omega _4$ we find:

Observe that for $\epsilon = 1$ we have a zero coefficient: hence, the weight is fixed by the action of $S_{\alpha _4}$ and there is no finite number of simple Weyl reflections which transforms it to a weight of the form $\rho + \lambda$ with $\lambda$ dominant. Thus, the associated bundle has no cohomology by Theorem A.1. On the other hand, if $\epsilon = 2$, one has

and the cohomology vanishes for the same reason as above. Let us now consider the second claim. By adding $\rho$ to $a\omega _1+b\omega _2+ c\omega _3-2\omega _4$ we obtain:

Then, applying $S_{\omega _4}$ yields:

Note that if we subtract $\rho$ to the resulting weight we obtain the dominant weight $a\omega _1+b\omega _2+(c-1)\omega _3$, and the result follows again by Theorem A.1.$\square$

Corollary A.4

One has $H^\bullet (X, \widehat{T}_X^\vee ) \simeq {\mathbb V}^{D_5}_{\omega _4}[0]$.

Proof

Consider the dual of the sequence 2.9. The claim follows once we observe that by Lemma A.5 the bundle ${\mathcal U}^\vee (-1) = {\mathcal E}_{\omega _1-\omega _4}$ has no cohomology.$\square$

Lemma 2.3

The vector bundle ${\mathcal U}$ has no cohomology.

Proof

Since ${\mathcal U}\simeq \wedge ^4{\mathcal U}(-2) \simeq {\mathcal E}_{-\omega _4+\omega _5}$, the proof is immediate: one has

and this allows to conclude as we did in the proof of Lemma A.5.$\square$

Lemma 2.4

We have ${\text {Ext}}^\bullet ({\mathcal U}(1), {\mathcal U}^\vee ) = 0$.

Proof

We begin by decomposing the relevant bundle in a sum of irreducible, by Lemma 2.1:

$$\begin{aligned} \begin{aligned} {\text {Ext}}^\bullet ({\mathcal U}(1), {\mathcal U}^\vee )&= H^\bullet (X, {\mathcal U}^\vee \otimes {\mathcal U}^\vee (-1)) \\&= H^\bullet (X, {{\,\textrm{Sym}\,}}^2{\mathcal U}^\vee (-1)\oplus \wedge ^2{\mathcal U}^\vee (-1))\\&=H^\bullet (X, {\mathcal E}_{2\omega _1-\omega _4}\oplus {\mathcal E}_{\omega _2-\omega _4}). \end{aligned} \end{aligned}$$

Then, the proof follows by applying Lemma A.5.$\square$

Lemma 2.5

The following holds:

$$\begin{aligned} H^0(X, {\mathcal U}\otimes {\mathcal U}\otimes {{\,\textrm{Sym}\,}}^2{\mathcal U}^\vee ) \cong {\mathbb C}\end{aligned}$$

Proof

The proof is a simple, but tedious application of the Littlewood–Richardson formula, together with the Borel–Weil–Bott theorem. First, let us decompose the bundle in irreducible summands:

$$\begin{aligned} \begin{aligned} {\mathcal U}\otimes {\mathcal U}\otimes {{\,\textrm{Sym}\,}}^2{\mathcal U}^\vee&\simeq \wedge ^4{\mathcal U}^\vee \otimes \wedge ^4{\mathcal U}^\vee \otimes {{\,\textrm{Sym}\,}}^2{\mathcal U}^\vee (-4) \\&\simeq (\wedge ^3{\mathcal U}^\vee (2)\oplus {\mathcal E}_{2\omega _4\oplus 2\omega _5})\otimes {{\,\textrm{Sym}\,}}^2{\mathcal U}^\vee (-4) \\&\simeq {\mathcal E}_{2\omega _1 + \omega _3-2\omega _4}\oplus {\mathcal E}_{\omega _1-\omega _4+\omega _5} \oplus {\mathcal E}_{2\omega _1-2\omega _4+2\omega _5} \oplus {\mathcal E}_{\omega _1-\omega _4+\omega _5}\oplus {\mathcal O}, \end{aligned} \end{aligned}$$

then we conclude by the Borel–Weil–Bott theorem as in the previous results of this appendix.$\square$

1.2 A.3 Cohomology Computations on $B_4/Q^4$

The notation of this paragraph has been established in Section 2.1.4.

Lemma 3.1

One has:

$$\begin{aligned} \begin{aligned} H^\bullet (X, \wedge ^3{\mathcal R}^\vee (-2))&\simeq {\mathbb C}[-1]\\ H^\bullet (X, {\mathcal E}_{\nu _1+\nu _2-2\nu _4})&= 0 \end{aligned} \end{aligned}$$

Proof

By Eq. 2.4 we see that $\wedge ^3{\mathcal R}^\vee (-2) = {\mathcal E}_{\nu _3-2\nu _4}$. We proceed with the same algorithm as before but for $B_4$. By adding $\rho$ we find:

By Theorem A.1, we have cohomology if with a finite number of simple Weyl reflections we find a dominant weight: hence, we act with $S_{\beta _4}$ in order to get rid of the negative coefficient. We find:

At this point, it is clear that if we subtract $\rho$ from the resulting weight, we get a dominant weight (the trivial one). Hence we conclude that the cohomology of ${\mathcal E}_{\nu _3-2\nu _4}$ is one dimensional in degree given by the number of simple Weyl reflections we used, which is one.

Let us consider the second claim. Here, applying $\rho$ to the weight $\nu _1+\nu _2-2\nu _4$ we find:

Let us apply $S_{\beta _4}$:

As above, we found a zero coefficient: Therefore, by Theorem A.1, ${\mathcal E}_{\nu _1+\nu _2-2\nu _4}$ has no cohomology.$\square$

Lemma 3.2

We have ${\text {Ext}}_G^\bullet (\wedge ^2{\mathcal R}^\vee , {\mathcal R}^\vee ) = {\mathbb C}[-1]$

Proof

The proof follows immediately by the decomposition $\wedge ^2{\mathcal R}\otimes {\mathcal R}^\vee \simeq \wedge ^2{\mathcal R}^\vee \otimes \wedge ^2{\mathcal R}^\vee (-2)\simeq {\mathcal E}_{\nu _1+\nu _2-2\nu _4}\oplus {\mathcal E}_{\nu _3-2\nu _4}$ together with Lemma A.10.

Corollary A.6

One has ${\text {Ext}}^\bullet ({\mathcal O}, {\mathcal R}) = {\mathbb C}[-1]$.

Proof

Since ${\text {Ext}}^\bullet ({\mathcal O}, {\mathcal R}) \simeq H^\bullet (X, {\mathcal R})$, the proof follows simply by the isomorphism ${\mathcal R}\simeq \wedge ^3{\mathcal R}^\vee (-2)$ (Lemma 2.1) and by Lemma A.10.$\square$

Lemma 3.4

The vanishing $H^0(X, \wedge ^r{\mathcal R}(-c)) = 0$ holds for any $c\ge -1$ and $1\le r\le 3$.

Proof

The proof follows the same identical reasoning of the one of Lemma A.10, once we write $\wedge ^r{\mathcal R}(-c) = {\mathcal E}_{\nu _r-c\nu _4}$.

Lemma 3.5

One has $H^\bullet (X, {{\,\textrm{Sym}\,}}^2{\mathcal R}) = 0$.

Proof

A quick way to prove this claim is to resolve ${{\,\textrm{Sym}\,}}^2{\mathcal R}$ in terms of $D_5$-homogeneous bundles and then apply some results of the previous subsection. By the appropriate Schur power of the sequence 2.6 we obtain:

$$\begin{aligned} 0\longrightarrow {{\,\textrm{Sym}\,}}^2{\mathcal R}\longrightarrow {{\,\textrm{Sym}\,}}^2{\mathcal U}\longrightarrow {\mathcal U}\longrightarrow 0. \end{aligned}$$

(1.1)

The last bundle has no cohomology by Lemma A.7, while the middle term can be resolved by a Schur power of the sequence 2.3:

$$\begin{aligned} 0\longrightarrow {{\,\textrm{Sym}\,}}^2{\mathcal U}\longrightarrow {\mathbb V}^{D_5}_{\omega _1}\otimes {\mathcal U}\longrightarrow \wedge ^2 {\mathbb V}^{D_5}_{\omega _1}\otimes {\mathcal O}\longrightarrow \wedge ^2{\mathcal U}^\vee \longrightarrow 0. \end{aligned}$$

(2.1)

Note that the last arrow is an isomorphism at the level of cohomology (it is the evaluation map on sections and $\wedge ^2{\mathcal U}^\vee$ is globally generated). Then, we conclude by Lemma A.7 again.$\square$

Lemma A.2

We have $H^\bullet (X, {{\,\textrm{Sym}\,}}^2{\mathcal R}\otimes {\mathcal U}^\vee )^G = {\mathbb C}[-1]$.

Proof

By the tensor power of the sequence A.1 with ${\mathcal U}^\vee$ we find:

$$\begin{aligned} 0\longrightarrow {{\,\textrm{Sym}\,}}^2{\mathcal R}\otimes {\mathcal U}^\vee \longrightarrow {{\,\textrm{Sym}\,}}^2{\mathcal U}\otimes {\mathcal U}^\vee \longrightarrow {\mathcal U}\otimes {\mathcal U}^\vee \longrightarrow 0. \end{aligned}$$

(2.2)

The second term is resolved again by the tensor product of ${\mathcal U}^\vee$ with a Schur power of the sequence A.2:

$$\begin{aligned} 0\longrightarrow {{\,\textrm{Sym}\,}}^2{\mathcal U}\otimes {\mathcal U}^\vee \longrightarrow {{\,\textrm{Sym}\,}}^2{\mathbb V}^{D_5}_{\omega _1}\otimes {\mathcal U}^\vee \longrightarrow {\mathbb V}^{D_5}_{\omega _1}\otimes {\mathcal U}^\vee \otimes {\mathcal U}^\vee \longrightarrow \wedge ^2{\mathcal U}^\vee \otimes {\mathcal U}^\vee \longrightarrow 0. \end{aligned}$$

In this last sequence, the cohomology of all terms but the first can be computed by the Littlewood–Richardson formula together with Lemma A.2. We find $H^\bullet (X, {{\,\textrm{Sym}\,}}^2{\mathcal U}\otimes {\mathcal U}^\vee )\simeq {\mathbb V}^{D_5}_{\omega _1}[-1]$. On the other hand, the last term of the sequence A.3 contributes with ${\mathbb C}[-1]$ (${\mathcal U}$ is an exceptional vector bundle in $D^b(X)$). We conclude by taking the space of invariants.$\square$