| Russian Mathematical Surveys |
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Brief communications Minimal Lefschetz Collections on Isotropic Grassmannians A. A. Novikov HSE University
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  Let $\Bbbk$ be an algebraically closed field of characteristic zero, $V$ be a $2n$-dimensional vector space over $\Bbbk$ with a symplectic form $\omega$, and $\operatorname{Sp}(V)$ be the corresponding symplectic group. Consider the Grassmannian of isotropic three-dimensional subspaces $X = \operatorname{IGr}(3,2n)$. Let $\mathcal{O}(1)$ be the ample generator of its Picard group; then the canonical bundle of the variety $X$ is isomorphic to $\mathcal{O}(-w)$, where $w=2n-2$. We study $\mathrm{D}^{b}(X)$, the bounded derived category of coherent sheaves on $X$. We construct minimal Lefschetz collections on the Grassmannians $\operatorname{IGr}(3,2n)$ and prove their fullness. A Lefschetz collection on $X$ is an exceptional collection
$$
\begin{equation}
\mathcal{E}_1,\dots,\mathcal{E}_{a_1}, \ \mathcal{E}_1(1),\dots,\mathcal{E}_{a_2}(1), \ \dots, \ \mathcal{E}_1(w-1),\dots,\mathcal{E}_{a_{w}}(w-1),
\end{equation}
\tag{1}
$$
where $a_1 \geqslant a_2 \geqslant \dots \geqslant a_{w}$. The set $\{\mathcal{E}_1, \dots, \mathcal{E}_{a_1}\}$ is called the first block of the Lefschetz collection. The collection is called minimal if its length equals the rank of $K_0(X)$ and $a_1$ is minimal, and it is called rectangular if $a_1 = \dots = a_w$. It is easy to see that $a_1 \geqslant \operatorname{rk} K_0(X)/w$, and for full rectangular collections equality is achieved, so they are automatically minimal. Minimal Lefschetz collections were constructed on $\operatorname{IGr}(k,2n)$ for $k=1$ and $k=2$ in [1] and [2], respectively, for $k=3$ and $n$ equal to $3$, $4$, or $5$ in [3], [4], and [5], respectively, and for $k = n$ equal to $4$ or $5$ in [6].
In this note we construct full minimal Lefschetz collections on $\operatorname{IGr}(3,2n)$ for arbitrary $n$, thus generalizing the known results. We denote by $\mathcal{O}$, $\mathcal{U} \subset V \otimes \mathcal{O}$, and $\mathcal{U}^{\vee}$ the structure sheaf, tautological vector bundle on $X$, and its dual, respectively. Thus, $\operatorname{rk} \mathcal{U} = 3$ and $\wedge^{3} \mathcal{U}^{\vee} \simeq \mathcal{O}(1)$. Since the bundle $\mathcal{U}$ is isotropic at each point, $\mathcal{U}$ is embedded into its orthogonal $\mathcal{U}^{\perp}$ with respect to $\omega$. Moreover, since the form $\omega$ is symplectic, it induces a symplectic form on the quotient $\mathcal{S} := \mathcal{U}^{\perp}/\mathcal{U}$. In particular, $\operatorname{rk} \mathcal{S} = 2n - 6$, and for any integer $0 \leqslant s \leqslant n - 3$ the symplectic exterior power
$$
\begin{equation*}
\wedge_{\mathrm{Sp}}^{s}\mathcal{S}:=\operatorname{ker}(\wedge^{s}\mathcal{S} \xrightarrow{\omega_{\mathcal{S}}}\wedge^{s-2} \mathcal{S})
\end{equation*}
\notag
$$
is defined and $\operatorname{Sp}(V)$-equivariant. For a weight $\alpha = (\alpha_1, \alpha_2, \alpha_3) \in \mathbb{Z}^3$, $\alpha_1 \geqslant \alpha_2 \geqslant \alpha_3$, applying the Schur functor $\Sigma^{\alpha}$ to the bundle $\mathcal{U}^{\vee}$ we obtain an irreducible $\operatorname{Sp}(V)$-equivariant vector bundle on $X$, and we have
$$
\begin{equation*}
\Sigma^{\alpha} \mathcal{U}^{\vee} \simeq \Sigma^{(\alpha_{1}-\alpha_{3}, \alpha_{2}-\alpha_{3}, 0)} \mathcal{U}^{\vee}(\alpha_{3}) \quad \text{and} \quad \Sigma^{(\alpha_{1}, 0,0)} \mathcal{U}^{\vee} \simeq S^{\alpha_{1}} \mathcal{U}^{\vee}.
\end{equation*}
\notag
$$
The structure of the exceptional collection depends on the remainder of $n$ modulo $3$. The simplest case is $n = 3m$ for $m \in \mathbb{Z}$, $m \geqslant 1$. Below we present the construction of an exceptional collection and the scheme of the proof of fullness in this case; the remaining cases are discussed in Remark 2. We define the following set of bundles:
$$
\begin{equation}
\begin{aligned} \, \nonumber \mathrm{B}&:=\{\Sigma^{(\alpha_1\kern-1pt,0,\alpha_3)} \mathcal{U}^{\vee} \mid 0 \leqslant\alpha_1-\alpha_3 \leqslant 4m-3,\ -2m+2 \leqslant \alpha_3 \leqslant 0\} \\ &\qquad\sqcup \{\wedge_{\mathrm{Sp}}^{s} \mathcal{S} \mid 2m-1\leqslant s \leqslant 3m-3\}. \end{aligned}
\end{equation}
\tag{2}
$$
On $\mathrm{B}$ we introduce an arbitrary total order consistent with the natural partial order on the set $\{\Sigma^\alpha\mathcal{U}\}$ and such that $\Sigma^\alpha\mathcal{U} < \wedge_{\mathrm{Sp}}^{s} \mathcal{S}$ for all $\alpha$ and $s$.
Theorem 1. The collection of equivariant vector bundles is a full rectangular Lefschetz collection on $\operatorname{IGr}(3,6m)$. As the main tool for verifying the exceptionality of the collection we use the symplectic version of the Borel–Bott–Weil theorem. For irreducible bundles the proof reduces to a direct computation of cohomology for the corresponding tensor products. The proof of fullness follows [5] and uses staircase complexes [7] and secondary staircase complexes [8], with the help of which new equivariant bundles are successively constructed in the subcategory $\langle \mathrm{C} \rangle \subset \mathrm{D}^b(X)$ generated by the collection $\mathrm{C}$. Thus, for the set
$$
\begin{equation*}
\mathrm{T}:=\{\Sigma^{(\alpha_1,0,\alpha_3)} \mathcal{U}^\vee \mid 0 \leqslant \alpha_1-\alpha_3 \leqslant 6m-3,\ -6m+3 \leqslant \alpha_3 \leqslant 0\}
\end{equation*}
\notag
$$
the inclusion $\mathrm{T}(l) \subset \langle \mathrm{C} \rangle$ is first proved for $0 \leqslant l \leqslant 6m - 3$ and then for all integers $l$. Fullness now follows from the fact that $\bigoplus_{l = 0}^{\dim(X)} \mathcal{O}(l) \cong \bigoplus_{l = 0}^{18m-12} \Sigma^{(0,0,0)}\mathcal{U}^\vee(l)$ is a generator in $\mathrm{D}^b(X)$.
Remark 2. If $n = 3m + e$ for $e \in \{1, 2\}$, then in addition to the block of irreducible bundles similar to the block (2), namely, The proof is similar to the proof of Theorem 1. The mutation defining the object $\mathcal{H}$ is explicitly written as a truncation of a certain secondary staircase complex. Its terms are truncations of staircase complexes, which are expressed in terms of bundles of the form $\Sigma^\alpha\mathcal{U}^\vee$, allowing the proof in these cases to be reduced to computations with irreducible bundles.
Citation in format AMSBIB
\Bibitem{Nov25} |
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