| Sbornik: Mathematics |
|
|
|
|
|
This article is cited in 1 scientific paper (total in 1 paper) Secondary staircase complexes on isotropic Grassmannians A. A. Novikov Faculty of Mathematics, National Research University Higher School of Economics, Moscow, Russia
Bibliographic databases:
    § 1. IntroductionThe bounded derived category of coherent sheaves $\mathrm{D}^b(X)$ is an important invariant of an algebraic variety $X$, but its structure can be complicated. The simplest case is when $\mathrm{D}^b(X)$ has a full exceptional collection $(E_{1}, E_{2}, \dots, E_{m})$. Then each object of $\mathrm{D}^b(X)$ admits a unique filtration in which the $i$th subquotient is a direct sum of shifts of the objects $E_i$. Therefore, an exceptional collection serves as a kind of a noncommutative basis for $\mathrm{D}^b(X)$. A long-standing conjecture predicts the existence of full exceptional collections on all projective homogeneous varieties of reductive algebraic groups; see [8]. The first example of a full exceptional collection was constructed by Beilinson, who showed in [1] that the twists of the structure sheaf $\mathcal{O},\mathcal{O}(1), \dots, \mathcal{O}(n)$ form such a collection on the projective space $\mathbb{P}^n$. Afterwards, Kapranov [6] constructed full exceptional collections on the Grassmannians and flag varieties of the groups $\mathrm{GL}_n$ and on smooth quadrics. Subsequently, Fonarev constructed in [2] and [3] other exceptional collections on the Grassmannians and proposed a new proof of fullness, based on a consistent application of so-called staircase complexes. For symplectic isotropic Grassmannians $\operatorname{IGr}(k,2n)$ the progress has been slower. As the case $\operatorname{IGr}(1, 2n) = \mathbb{P}^{2n - 1}$ is covered by the result of Beilinson, the first new case is the isotropic Grassmannian of lines $\operatorname{IGr}(2, 2n)$; in this case a full exceptional collection was constructed in [7]. Subsequently, Kuznetsov and Polishchuk [8], by extending the ideas from [11] and [10], where the cases of $\operatorname{IGr}(3,6)$, $\operatorname{IGr}(4, 8)$ and $\operatorname{IGr}(5, 10)$ had been discussed, developed a general approach to constructing exceptional collections of expected length on all Grassmannians of classical groups. However, the fullness of these collections was proved only in the Lagrangian case $\operatorname{IGr}(n,2n)$ by Fonarev [4]. Finally, full exceptional collections on $\operatorname{IGr}(3,8)$ and $\operatorname{IGr}(3,10)$ were constructed in [5] and [9] by Guseva and this author, respectively. The above papers conveyed an important message: to construct sufficiently long exceptional collections we have to consider equivariant vector bundles which are not necessarily irreducible, and to prove the fullness of the collections constructed we have to find exact sequences relating these bundles, similarly to the staircase complexes that work so well in the case of $\mathrm{GL}_n$. Some complexes of this sort already appeared in the papers mentioned above. For instance, the key step in the proof in [7] of the fullness of the exceptional collection on $\operatorname{IGr}(2,2n)$ was a construction of a certain bicomplex in [7], Proposition 5.3, which can be considered as a complex consisting of the objects represented by its rows; one of our results is another construction of such a complex, which fixes an inaccuracy in [7]; see Remarks 5 and 6 and Theorem 3. Similarly, the proof of fullness of an exceptional collection on $\operatorname{IGr}(3,8)$ and $\operatorname{IGr}(3,10)$ uses bicomplexes described in [5], § 5.2, and [9], § 3.2, respectively. The main result of this paper is a considerable generalization of the above constructions. We describe a class of (nonirreducible) $\mathrm{Sp}_{2n}$-equivariant vector bundles $\mathcal{K}_{\operatorname{IGr}(k,2n)}^{\alpha_1,\alpha_2}$ on $\operatorname{IGr}(k,2n)$ defined as appropriate truncations of staircase complexes, show that they can be assembled into natural complexes and identify their cohomology sheaves. To state our results we need to introduce some notation. Let $V$ be a vector space of dimension $2n$, and consider the Grassmannian $\operatorname{Gr}(k,V) = \operatorname{Gr}(k,2n)$. We also fix a symplectic form $\omega$ on $V$ and, for $2 \leqslant k \leqslant n$, consider the isotropic Grassmannian $\operatorname{IGr}(k,V) = \operatorname{IGr}(k,2n)$. We denote by $\mathcal{U}_k$ the tautological subbundle of rank $k$ in $V \otimes \mathcal{O}$ on $\operatorname{Gr}(k,V)$, and for a dominant weight $(\alpha_1,\alpha_2,\dots,\alpha_k)$ of $\mathrm{GL}_k$ we denote by $\Sigma^{\alpha_1,\alpha_2,\dots,\alpha_k}\mathcal{U}_k^\vee$ the result of the application of the corresponding Schur functor to $\mathcal{U}_k^\vee$, the dual tautological bundle. In particular, $\Sigma^{a,0,\dots,0}\mathcal{U}_k^\vee \cong \mathrm{S}^a \mathcal{U}_k^\vee$. For arbitrary integers $2n-k\geqslant \alpha_1 \geqslant \alpha_2 \geqslant 0$ and $2 \leqslant k \leqslant n$ we define a $\mathrm{GL}_{2n}$-equivariant vector bundle $\mathcal{K}_{\operatorname{Gr}(k,V)}^{\alpha_1,\alpha_2}$ on $\operatorname{Gr}(k,V)$ as a special case of the duals to the bundles $\mathcal{E}^{\lambda,\mu}$ defined in [3], § 3:
$$
\begin{equation}
\mathcal{K}^{\alpha} :=(\mathcal{E}^{(\alpha_2),(1^{\alpha_1})})^\vee, \quad\text{where } (1^{\alpha_1})=(\underbrace{1, \dots, 1}_{\alpha_1}).
\end{equation}
\tag{1.1}
$$
Using the exact sequences from Theorem 4.3 in [3] and the isomorphisms $\mathcal{E}^{\lambda,0} \cong \Sigma^\lambda \mathcal{U}_k$, we obtain the following $GL_{2n}$-equivariant resolution (see (3.6)):
$$
\begin{equation}
\begin{aligned} \, \notag 0 &\to\mathcal{K}_{\operatorname{Gr}(k,V)}^{\alpha_1,\alpha_2} \\ \notag &\to \wedge^{\alpha_1+1} V^\vee \otimes \Sigma^{\alpha_2-1,0,\dots,0}\mathcal{U}_k^\vee \to \dotsb \to\wedge^{\alpha_1-\alpha_2+2} V^\vee \otimes \Sigma^{\alpha_2-1,\alpha_2-1,0,\dots,0}\mathcal{U}_k^\vee \\ &\to \wedge^{\alpha_1-\alpha_2} V^\vee \otimes \Sigma^{\alpha_2,\alpha_2,0,\dots,0}\mathcal{U}_k^\vee \to \dotsb \to\Sigma^{\alpha_1,\alpha_2,0,\dots,0}\mathcal{U}_k^\vee \to 0. \end{aligned}
\end{equation}
\tag{1.2}
$$
We furthermore define $\mathcal{K}_{\operatorname{IGr}(k,V)}^{\alpha_1,\alpha_2} := \mathcal{K}_{\operatorname{Gr}(k,V)}^{\alpha_1,\alpha_2}\vert_{\operatorname{IGr}(k,V)}$. We also consider the tautological bundle $\mathcal{U}_k^\perp$ of rank $2n-k$ defined by the exact sequence
$$
\begin{equation}
0 \to \mathcal{U}_k^\perp \to V^\vee \otimes \mathcal{O} \to \mathcal{U}_k^\vee \to 0.
\end{equation}
\tag{1.3}
$$
In particular, if $\alpha_2 = 0$, then there is no second line in (1.2), and the resolution is
$$
\begin{equation*}
0 \to \mathcal{K}_{\operatorname{Gr}(k,V)}^{\alpha_1, 0} \to \wedge^{\alpha_1} V^\vee \otimes \mathcal{O} \to \wedge^{\alpha_1-1} V^\vee \otimes \mathcal{U}_k^\vee \to \dotsb \to \mathrm {S}^{\alpha_1}\mathcal{U}_k^\vee \to 0.
\end{equation*}
\notag
$$
So we obtain $\mathcal{K}_{\operatorname{Gr}(k,V)}^{\alpha_1, 0} \cong \Lambda^{\alpha_1} \mathcal{U}_k^{\perp}$. The symplectic form $\omega$ induces, after the restriction of $\mathcal{U}_k$ and $\mathcal{U}_k^\perp$ to $\operatorname{IGr}(k,V)$, the natural embedding $\mathcal{U}_k \hookrightarrow \mathcal{U}_k^\perp$, and we denote the quotient bundle by $\mathcal{S}_k$, so that we have an exact sequence
$$
\begin{equation}
0 \to \mathcal{U}_k \xrightarrow{\omega} \mathcal{U}_k^\perp \xrightarrow{\quad} \mathcal{S}_k \to 0.
\end{equation}
\tag{1.4}
$$
Finally, note that the symplectic form $\omega$ induces a symplectic structure on $\mathcal{S}_k$, which we denote by $\omega_\mathcal{S}$, and allows us to define for $0 \leqslant i \leqslant n-k=\frac12\operatorname{rank}(\mathcal{S}_k)$ the symplectic wedge powers
$$
\begin{equation}
\wedge^i_{\mathrm{Sp}}\mathcal{S}_k := \operatorname{Coker}(\wedge^{i-2}\mathcal{S}_k \xrightarrow{\omega_\mathcal{S}} \wedge^i\mathcal{S}_k).
\end{equation}
\tag{1.5}
$$
Theorem 1. For any $0 \leqslant t \leqslant 2n - k$ there exists an $\operatorname{Sp}(V)$-equivariant complex of vector bundles We are planning to use these results in our future work to study the fullness of the exceptional collection on $\operatorname{IGr}(k,2n)$ constructed in [8] for all $n$ and $2 \leqslant k \leqslant n$, and for minimal Lefschetz exceptional collections on $\operatorname{IGr}(3, 2n)$ for all $n$. To prove Theorem 1 we first consider in detail the case $k = 2$. In this case the bundles $\mathcal{K}_{\operatorname{Gr}(2,V)}^{\alpha_1,\alpha_2}$, up to duality and twist, coincide with the bundles $\mathcal{E}^{a,b}$ that can be defined as truncations of the Koszul complex (see (2.9)) or as extensions (2.12). These bundles first appeared in [8], Conjecture 9.8, and were extensively studied in [3]. The crucial computation with these objects is carried out in Proposition 1, where we combine the restrictions to $\operatorname{IGr}(2,V)$ of the bundles $\mathcal{E}^{a,b}$ into a complex and relate this complex to two Koszul complexes of (1.4). This allows us to compute the cohomology of these complexes (see Theorem 2) and, after dualization and twist, deduce Theorem 1 in the case $k = 2$. To deduce Theorem 1 for arbitrary $k$ we use the natural Fourier–Mukai functors
$$
\begin{equation*}
\widetilde\Phi_k \colon \mathrm{D}^b(\operatorname{Gr}(2,V)) \to \mathrm{D}^b(\operatorname{Gr}(k,V))\quad\text{and} \quad \Phi_k \colon \mathrm{D}^b(\operatorname{IGr}(2,V)) \to \mathrm{D}^b(\operatorname{IGr}(k,V))
\end{equation*}
\notag
$$
(see § 3.1). In Lemmas 2 and 3 we compute their actions on appropriate Schur functors applied to the dual tautological bundle $\mathcal{U}_2^\vee$ of $\operatorname{IGr}(2,V)$ and on symplectic wedge powers of the symplectic bundle $\mathcal{S}_2$ of $\operatorname{IGr}(2,V)$. We use the first to show that in some cases $\widetilde\Phi_k$ takes a staircase complex on $\operatorname{Gr}(2,V)$ to a staircase complex on $\operatorname{Gr}(k,V)$ (see Proposition 2) and then deduce from this that it takes the objects $\mathcal{K}_{\operatorname{Gr}(2,V)}^{\alpha_1,\alpha_2}$ to $\mathcal{K}_{\operatorname{Gr}(k,V)}^{\alpha_1,\alpha_2}$; see Lemma 4. After that Theorem 1 follows easily. Notation and conventionsWe work over an algebraically closed field $\Bbbk$ of characteristic zero. AcknowledgementsI would like to thank my scientific advisor Alexander Kuznetsov, without whom this work would not be possible, Lyalya Guseva for her attention to my work and an anonymous referee for their useful suggestions. § 2. Grassmannians of isotropic linesThroughout this section we use the notation $\mathcal{U} := \mathcal{U}_2$ for the tautological bundle on $\operatorname{Gr}(2,V)$ of rank 2, as well as for its restriction to $\operatorname{IGr}(2,V)$, and $\mathcal{S} := \mathcal{S}_2$ for the symplectic vector bundle (defined by (1.4)) of rank $2n - 4$ on $\operatorname{IGr}(2,V)$. The main result of this section is a proof of Theorem 1 for $\operatorname{IGr}(2,V)$. 2.1. Vector bundles $\mathcal{E}^{a,b}$We define vector bundles $\mathcal{E}^{a,b}$ on $\operatorname{Gr}(2,V)$ for arbitrary as a special case of the more general class of bundles $\mathcal{E}^{\lambda,\mu}$ from [3]:
$$
\begin{equation}
\mathcal{E}^{a,b} := \mathcal{E}^{(b),(1^{2n-2-a})}(1), \quad \text{where } (1^{2n-2-a})= (\underbrace{1, \dots, 1}_{2n-2-a}).
\end{equation}
\tag{2.2}
$$
In particular, the definition (1.1) of $\mathcal{K}_{\operatorname{Gr}(2,V)}$ implies that
$$
\begin{equation}
\mathcal{E}^{a,b} \cong (\mathcal{K}_{\operatorname{Gr}(2,V)}^{2n-2-a,b})^\vee(1).
\end{equation}
\tag{2.3}
$$
For our purposes it is, again, more convenient to interpret these bundles in terms of staircase complexes, rather than as pushforwards from a partial flag variety. Note that the combination of (2.3) with (1.2) already provides the left resolution:
$$
\begin{equation}
\begin{aligned} \, \notag 0 &\to\mathrm{S}^{c-b}\mathcal{U}(-1-b) \to V \otimes \mathrm{S}^{c-b-1}\mathcal{U}(-1-b) \to \dotsb \to \wedge^{c-b}V \otimes \mathcal{O}(-1-b) \\ \notag &\to \wedge^{c-b+2}V^\vee {\otimes}\, \mathcal{O}(-b) \to \wedge^{c-b+3}V^\vee {\otimes}\, \mathcal{U}(-b+1) \to \dotsb \to \wedge^{c+1}V^\vee \!\otimes \mathrm{S}^{-b+1}\mathcal{U}(-1) \\ &\to \mathcal{E}^{a,b} \to 0, \end{aligned}
\end{equation}
\tag{2.4}
$$
where $c=2n-2-a$. Next, we recall the definition of staircase complexes in the case of $\operatorname{IGr}(2, V)$. For any pair of integers $\alpha_1 \geqslant \alpha_2$ we consider the $\mathrm{GL}(V)$-equivariant vector bundle
$$
\begin{equation}
\Sigma^{\alpha_1,\alpha_2}\mathcal{U}^\vee \cong \mathrm{S}^{\alpha_1-\alpha_2}\mathcal{U}^\vee \otimes \mathcal{O}(\alpha_2) \cong \mathrm{S}^{\alpha_1-\alpha_2}\mathcal{U} \otimes \mathcal{O}(\alpha_1).
\end{equation}
\tag{2.5}
$$
By [2], for any weight $(\alpha_1,\alpha_2)$ of $\mathrm{GL}_2$ such that
$$
\begin{equation}
\alpha_1 \geqslant \alpha_2 \geqslant \alpha_1-2n+2,
\end{equation}
\tag{2.6}
$$
there exists an exact sequence of vector bundles, called the staircase complex, that looks like
$$
\begin{equation}
\begin{aligned} \, \notag 0 &\to\wedge^{2n}V^\vee \otimes \Sigma^{\alpha_2-1,\alpha_1-2n+1}\mathcal{U}^\vee \to \dotsb \to\wedge^{\alpha_1-\alpha_2+2}V^\vee \otimes \Sigma^{\alpha_2-1,\alpha_2-1}\mathcal{U}^\vee \\ &\to\wedge^{\alpha_1-\alpha_2}V^\vee \otimes \Sigma^{\alpha_2,\alpha_2}\mathcal{U}^\vee \to \dotsb \to V^\vee \otimes \Sigma^{\alpha_1-1,\alpha_2}\mathcal{U}^\vee \to \Sigma^{\alpha_1,\alpha_2}\mathcal{U}^\vee \to0. \end{aligned}
\end{equation}
\tag{2.7}
$$
Here in the top row the second component of the weight increases from $\alpha_1 - 2n + 1$ to $\alpha_2-1$, while the exponent of the wedge power decreases from $2n$ to $\alpha_1 - \alpha_2 + 2$, and in the bottom row the first component of the weight increases from $\alpha_2$ to $\alpha_1$, while the exponent of the wedge power decreases from $\alpha_1 - \alpha_2$ to $0$. Remark 1. The staircase complex is $\mathrm{GL}(V)$-equivariant, and its differentials are the unique nonzero $\mathrm{GL}(V)$-equivariant maps between the corresponding terms. In particular, the dual of the staircase complex (2.7) with parameters $(\alpha_1,\alpha_2)$ is the staircase complex with parameters $(2n - 1 - \alpha_1, 1 - \alpha_2)$. It also follows that the bottom line of (2.7) is, up to twist, the Koszul complex We consider the special case of staircase complexes where $\alpha_1 \geqslant 0 \geqslant \alpha_2$, and for convenience we write $(\alpha_1,\alpha_2) = (a,-b)$. Then condition (2.6) translates into inequalities (2.1). Using the isomorphism $\wedge^iV \cong \wedge^{2n-i}V^\vee$ and the uniqueness of staircase complexes explained in Remark 1, we note that the resolution (2.4) is a truncation of the exact sequence (2.7). So we finally obtain our right resolution of the bundle $\mathcal{E}^{a,b}$:
$$
\begin{equation}
0 \to \mathcal{E}^{a,b} \to \wedge^{a}V^\vee \otimes \Sigma^{0,-b}\mathcal{U}^\vee \to \dotsb \to V^\vee \otimes \Sigma^{a-1,-b}\mathcal{U}^\vee \to \Sigma^{a,-b}\mathcal{U}^\vee \to 0.
\end{equation}
\tag{2.9}
$$
It is the truncation of the staircase complex (2.7) with $(\alpha_1,\alpha_2) = (a,-b)$ at the term with weight $(0,-b)$. Alternatively, $\mathcal{E}^{a,b}$ is a truncation of the Koszul complex (2.8) with $m = a + b$ twisted by $\mathcal{O}(-b)$. Note that $\Sigma^{0,-b}\mathcal{U}^\vee \cong \mathrm{S}^b\mathcal{U}$, so $\mathcal{E}^{a,b}$ is a $\mathrm{GL}(V)$-equivariant subbundle in $\wedge^{a} V^\vee \otimes \mathrm{S}^b\mathcal{U}$. In particular, $\mathcal{E}^{0,b} \cong \mathrm{S}^b\mathcal{U}$. Next, if $a \geqslant 1$, then by Theorem 4.4 in [3] there exists a $\mathrm{GL}(V)$-equivariant exact sequence:
$$
\begin{equation}
0 \to \mathcal{E}^{a,b} \to \wedge^{a} V^\vee \otimes \mathrm{S}^b\mathcal{U} \to \mathcal{E}^{a-1,b+1}(1) \to 0.
\end{equation}
\tag{2.10}
$$
Further, the tautological sequence (1.3) induces a filtration on the bundle ${\wedge^{a} V^\vee \otimes \mathrm{S}^b\mathcal{U}}$ with factors
$$
\begin{equation}
\wedge^{a} \mathcal{U}^\perp \otimes \mathrm{S}^b\mathcal{U}, \quad \wedge^{a-1} \mathcal{U}^\perp \otimes \mathrm{S}^{b-1}\mathcal{U}, \quad \wedge^{a-1} \mathcal{U}^\perp \otimes \mathrm{S}^{b+1}\mathcal{U}(1)\quad\text{and} \quad \wedge^{a-2} \mathcal{U}^\perp \otimes \mathrm{S}^b\mathcal{U}(1).
\end{equation}
\tag{2.11}
$$
Here, by convention, all wedge and symmetric powers $\wedge^i(-)$ and $\mathrm{S}^i(-)$ with negative $i$ are zero. Using recursion on $a \geqslant 0$ we obtain the following exact sequence for $\mathcal{E}^{a,b}$, since we already know it for $a = 0$:
$$
\begin{equation}
0 \to \wedge^{a} \mathcal{U}^\perp \otimes \mathrm{S}^b\mathcal{U} \to \mathcal{E}^{a,b} \to \wedge^{a-1} \mathcal{U}^\perp \otimes \mathrm{S}^{b-1}\mathcal{U} \to 0.
\end{equation}
\tag{2.12}
$$
Remark 2. By Proposition 3.5 in [3] the bundles $\mathcal{E}^{a,b}$ are exceptional. In particular, the extension (2.12) is nonsplit. 2.2. Secondary staircase complexes on $\operatorname{IGr}(2,V)$Now we consider the restrictions of the bundles $\mathcal{E}^{a,b}$ to $\operatorname{IGr}(2,V)$. By abusing notation we will still write $\mathcal{E}^{a,b}$ for these restrictions. In this subsection we construct $\operatorname{Sp}(V)$-equivariant complexes on $\operatorname{IGr}(2,V)$ from these bundles. Recall that $\omega$ denotes the symplectic form on $V$. First we consider two $\operatorname{Sp}(V)$-equivariant maps between the ambient bundles of $\mathcal{E}^{a,b}$ and $\mathcal{E}^{a+1,b-1}$:
$$
\begin{equation*}
\mathbf{d}_1,\mathbf{d}_2 \colon \wedge^a V^\vee \otimes \mathrm{S}^b\mathcal{U} \to \wedge^{a+1}V^\vee \otimes \mathrm{S}^{b-1}\mathcal{U}.
\end{equation*}
\notag
$$
We denote the composition of $V^\vee \twoheadrightarrow \mathcal{U}^\vee$ and the trace map $\mathcal{U}^\vee \otimes \mathcal{U} \to \mathcal{O}$ by
$$
\begin{equation*}
\operatorname{tr} \colon V^\vee \otimes \mathcal{U} \to \mathcal{O}.
\end{equation*}
\notag
$$
It induces the map
$$
\begin{equation*}
\operatorname{tr} \colon \wedge^a V^\vee \otimes \mathrm{S}^b\mathcal{U} \to \wedge^{a-1}V^\vee \otimes \mathrm{S}^{b-1}\mathcal{U}.
\end{equation*}
\notag
$$
The first morphism is defined as the composition of $\operatorname{tr}$ and $-\wedge \omega$:
$$
\begin{equation*}
\mathbf{d}_1 \colon \wedge^a V^\vee \otimes \mathrm{S}^b\mathcal{U} \xrightarrow{\operatorname{tr}} \wedge^{a-1}V^\vee \otimes \mathrm{S}^{b-1}\mathcal{U} \xrightarrow{- \wedge \omega} \wedge^{a+1}V^\vee \otimes \mathrm{S}^{b-1}\mathcal{U}.
\end{equation*}
\notag
$$
The second is also induced by the composition of multiplication by $\omega$ and $\operatorname{tr}$, but in the reverse order:
$$
\begin{equation*}
\begin{aligned} \, \mathbf{d}_2 \colon \wedge^a V^\vee \otimes \mathrm{S}^b \mathcal{U} &\xrightarrow{- \otimes \omega \otimes -} \wedge^a V^\vee \otimes (\wedge^2 V^\vee \otimes \mathrm{S}^{b}\mathcal{U}) \\ &\xrightarrow{- \otimes \operatorname{tr}} \wedge^{a} V^\vee \otimes V^\vee \otimes \mathrm{S}^{b-1}\mathcal{U} \to \wedge^{a+1}V^\vee \otimes \mathrm{S}^{b-1}\mathcal{U}, \end{aligned}
\end{equation*}
\notag
$$
where the last arrow is just a canonical surjection onto the direct summand. Remark 3. The abstract descriptions of $\mathbf{d}_1$ and $\mathbf{d}_2$ are not easy to work with. Specifically, it is difficult to compute their components between the subquotients of the filtrations (2.11) and (2.12), which is our main goal. On the other hand, the direct computation in the case $k=2$ is straightforward, unambiguous and turns out to be quite simple. Fix a point $[U] \in \operatorname{IGr}(2,V)$ and choose a basis $e_1$, $e_2$ of $U$. Then $\operatorname{tr}$ acts at this point by a convolution with $\sum e_i \otimes \partial/\partial e_i$. So the maps are given by the formulae:
$$
\begin{equation}
\mathbf{d}_1(\lambda \otimes P) := \lambda_1 \wedge \omega \otimes P_1+ \lambda_2 \wedge \omega \otimes P_2
\end{equation}
\tag{2.13}
$$
and
$$
\begin{equation}
\mathbf{d}_2(\lambda \otimes P) := \lambda \wedge (\omega_1 \otimes P_1+\omega_2 \otimes P_2),
\end{equation}
\tag{2.14}
$$
where $\lambda \in \wedge^aV^\vee$ and $P \in \mathrm{S}^b U$ is considered as a homogeneous polynomial of degree $b$ on $U^\vee$. Moreover, we write $\lambda_i$ and $\omega_i$ for the evaluation of the corresponding skew-form on $e_i$ and $P_i$ for the derivative of $P$ in the direction $e_i$. The maps $\mathbf{d}_1$ and $\mathbf{d}_2$ are obviously independent of the choice of a basis. Proposition 1. For any $(a,b)$ such that (2.1) holds the map Proof. We verify all claims at one point $[U] \in \operatorname{IGr}(2,V)$. Since all vector bundles and morphisms involved are $\operatorname{Sp}(V)$-equivariant, this is sufficient to prove the proposition.
We start by computing the map $\mathbf{d}$ on the first factor $\wedge^a U^\perp \otimes \mathrm{S}^b U$ of the filtration (2.12). Then $\lambda \in \wedge^aU^\perp$, hence $\lambda_1 = \lambda_2 = 0$, and therefore the map $\mathbf{d}_1$ vanishes on this factor. On the other hand, since $\omega_1,\omega_2 \in U^\perp$ (because $U$ is isotropic), we have $\lambda \wedge \omega_1, \lambda \wedge \omega_2 \in \wedge^{a+1}U^\perp$, hence $\mathbf{d}_2$ preserves the first factor. It is also clear, that $\mathbf{d}_2$ coincides on the first factor with the differential of the Koszul complex
$$
\begin{equation}
0 \to \mathrm{S}^t \mathcal{U} \xrightarrow{\mathbf{d}_2} \mathcal{U}^\perp \otimes \mathrm{S}^{t-1} \mathcal{U} \xrightarrow{\mathbf{d}_2} \dotsb \xrightarrow{\mathbf{d}_2} \wedge^{t -1} \mathcal{U}^\perp \otimes \mathcal{U} \xrightarrow{\mathbf{d}_2} \wedge^{t} \mathcal{U}^\perp \to \wedge^t\mathcal{S} \to 0
\end{equation}
\tag{2.16}
$$
of the short exact sequence (1.4).
Now consider the second factor of the filtration (2.12). To describe it explicitly, we consider the dual basis of $e_1,e_2 \in U$ and lift its elements to linear functions $e^1,e^2 \in V^\vee$. This allows us to lift an element $\mu \otimes Q$ from the second factor $\wedge^{a-1}U^\perp \otimes \mathrm{S}^{b-1}U$ of (2.12) to $\wedge^aV^\vee \otimes \mathrm{S}^bU$ as
$$
\begin{equation*}
\xi := (\mu \wedge e^1) \otimes (e_1 \cdot Q)+(\mu \wedge e^2) \otimes (e_2 \cdot Q),
\end{equation*}
\notag
$$
where $\mu \in \wedge^{a-1}U^\perp$ and $Q \in \mathrm{S}^{b - 1}U$ is a polynomial of degree $b - 1$ in $e_1$ and $e_2$. Then
$$
\begin{equation*}
\mathbf{d}_1(\xi)=\mu \wedge (\omega \otimes 2Q+\omega \otimes e_1 \cdot Q_1+\omega \otimes e_2 \cdot Q_2)=(b+1)\mu \wedge \omega \otimes Q,
\end{equation*}
\notag
$$
where $Q_1$ and $Q_2$ are the derivatives of $Q$ with respect to $e_1$ and $e_2$. Similarly,
$$
\begin{equation*}
\begin{aligned} \, \mathbf{d}_2(\xi) &=\mu \wedge (e^1 \wedge \omega_1 \otimes (Q+e_1 \cdot Q_1)+e^1 \wedge \omega_2 \otimes e_1 \cdot Q_2) \\ &\qquad +\mu \wedge (e^2 \wedge \omega_1 \otimes e_2 \cdot Q_1+e^2 \wedge \omega_2 \otimes (Q+e_2 \cdot Q_2)). \end{aligned}
\end{equation*}
\notag
$$
Now consider the skew-form
$$
\begin{equation}
\overline\omega := \omega+e^1 \wedge \omega_1+e^2 \wedge \omega_2 \in \wedge^2V^\vee.
\end{equation}
\tag{2.17}
$$
Of course, it depends on the choice of the lifts $e^i$, but for any choice of these the form $\overline\omega$ is contained in $\wedge^2U^\perp \subset \wedge^2V^\vee$. Using this notation and the above computations, we can write
$$
\begin{equation*}
\mathbf{d}(\xi)=\mu \wedge \overline\omega \otimes Q - \mu \wedge \bigl(\omega_1 \wedge (e^1 \otimes e_1+e^2 \otimes e_2) \cdot Q_1 +\omega_2 \wedge (e^1 \otimes e_1+e^2 \otimes e_2) \cdot Q_2\bigr),
\end{equation*}
\notag
$$
and since $\overline\omega \in \wedge^2U^\perp$, we see that the first term is contained in the first factor of the filtration (2.12), while the second term is in its second factor. It follows that
$$
\begin{equation*}
\mathbf{d}(\mathcal{E}^{a,b}) \subset \mathcal{E}^{a+1,-b+1},
\end{equation*}
\notag
$$
and thus the first statement of the proposition is proved.
Moreover, it follows that $\mathbf{d}$ acts on $\mathcal{E}^{a,b}\vert_{[U]} = (\wedge^a U^\perp \otimes \mathrm{S}^b U) \oplus (\wedge^{a-1} U^\perp \otimes \mathrm{S}^{b-1} U)$ (where the direct sum decomposition is induced by the lifts $e^i$ chosen above) by means of the matrix
$$
\begin{equation*}
\begin{pmatrix} \mathbf{d}_2 & \overline\omega \\ 0 & -\mathbf{d}_2 \end{pmatrix}.
\end{equation*}
\notag
$$
Thus, to deduce the second and third statements it remains to note that the wedge product with $\overline\omega$ commutes with the Koszul differential $\mathbf{d}_2$, which is obvious from (2.14).
Proposition 1 is proved. Remark 4. Using Corollary 3.10 in [3] it is easy to check that there are unique maps of vector bundles from $\mathcal{E}^{a,b}$ to $\mathcal{E}^{a+1,b-1}$ and no higher morphisms. Analogously, $\mathcal{E}^{a,b}$ is semiorthogonal to $\mathcal{E}^{a+i,b-i}$ for $i \geqslant 2$. This allows us to construct the complex (2.15) abstractly. However, it is not clear whether it and a pair of complexes (2.16) form via (2.12) the short exact sequence of complexes from Proposition 1. 2.3. The cohomology of secondary staircase complexes on $\operatorname{IGr}(2,V)$Now we consider the secondary staircase complex (2.15) for $a=0$. Recall that ${\mathcal{S} := \mathcal{U}^\perp/\mathcal{U}}$ is the symplectic vector bundle of rank $2n - 4$ on $\operatorname{IGr}(2,V)$ (see (1.4)), $\omega_S \in H^0(\operatorname{IGr}(2,V), \wedge^2\mathcal{S})$ is its symplectic form, and we denote by $\wedge_\mathrm{Sp}^i\mathcal{S}$, $0 \leqslant i \leqslant n - 2$, denote its symplectic wedge powers (see (1.5)). Theorem 2. For $0 \leqslant t \leqslant 2n - 2$ consider the complex Proof. We consider the complex (2.15) in Proposition 1 for $a=0$ and $b=t$. Applying the snake lemma and taking (2.16) into account, we obtain the exact sequence
$$
\begin{equation}
0 \to \mathcal{H}^{-1}(\mathcal{E}_t^\bullet) \xrightarrow{\quad} \wedge^{t-2}\mathcal{S} \xrightarrow{\quad} \wedge^t\mathcal{S} \xrightarrow{\quad} \mathcal{H}^0(\mathcal{E}_t^\bullet) \to 0
\end{equation}
\tag{2.20}
$$
and $\mathcal{H}^i(\mathcal{E}_t^\bullet) = 0$ for $i \notin \{-1,0\}$. Furthermore, the argument of Proposition 1 shows that at the point $[U] \in \operatorname{IGr}(2,V)$ the middle map $\wedge^{t-2}\mathcal{S} \to \wedge^t\mathcal{S}$ in (2.20) is induced by the map
$$
\begin{equation*}
\overline\omega \colon \wedge^{t-2}U^\perp \to \wedge^tU^\perp, \qquad \lambda \mapsto \lambda \wedge \overline\omega,
\end{equation*}
\notag
$$
where $\overline\omega$ is defined in (2.17), and therefore it coincides with the map induced by the symplectic form $\omega_\mathcal{S}$ on $\mathcal{S}$. In particular, when $0 \leqslant t \leqslant n - 2$, the middle map in (2.20) is injective, so its kernel is zero, and by (1.5) its cokernel is $\wedge^t_\mathrm{Sp}\mathcal{S}$. Similarly, when $n \leqslant t \leqslant 2n - 2$ the middle map in (2.20) is surjective, so its cokernel is zero, and by duality and (1.5) its kernel is $\wedge^{2n-2-t}_\mathrm{Sp}\mathcal{S}$. Finally, if $t = n - 1$, this map is an isomorphism.
Theorem 2 is proved. Remark 5. The case where $t = n - 1$ is particularly interesting. In this case the complex $\mathcal{E}_{n-1}^\bullet$ is acyclic. On the other hand, its terms $\mathcal{E}^{n-1-b,b}$ have resolutions (2.9). Using Lemma 5.1 in [5] it is easy to check that the morphisms of $\mathcal{E}_{n-1}^\bullet$ extend in a unique way to morphisms between these resolutions, so we obtain a bicomplex whose terms are $\wedge^{n-1-b-c}V^\vee \otimes \Sigma^{c,-b}\mathcal{U}^\vee$ (see (4.1)). In fact, these unique extensions can be written very explicitly, we do this in Theorem 3. Note that the bicomplex (4.1) has precisely the same form as the bicomplex of Proposition 5.3 in [7] that was crucial for the proof of the fullness of the exceptional collection on $\operatorname{IGr}(2,2n)$. Thus, our computation shows that the formula for the maps in the bicomplex proposed in [7] is incorrect (see Remark 6). Therefore, Theorem 2 fills a gap in the proof of Proposition 5.3 in [7]. Now we can prove the first case of our main theorem. Proof of Theorem 1 for $k=2$. Recall that the bundle $\mathcal{E}^{a,b}$ satisfies (2.3). Therefore, dualizing (2.18) and twisting it by $\mathcal{O}(-1)$, we obtain the complex (1.6). Now the isomorphisms (1.7) follow from (2.19) because the symplectic bundle $\mathcal{S} = \mathcal{S}_2$ is self-dual.
§ 3. General isotropic GrassmanniansIn this section we construct secondary staircase complexes and compute their cohomology on the isotropic Grassmannians $\operatorname{IGr}(k,V)$ with $3 \leqslant k \leqslant n$. The main tool is the natural Fourier–Mukai functor between the derived categories of Grassmannians and isotropic Grassmannians. 3.1. Fourier–Mukai transformsConsider the following commutative diagram of (isotropic) partial flag varieties and their natural embeddings and projections onto (isotropic) Grassmannians: Consider also the induced Fourier–Mukai functors
$$
\begin{equation}
\begin{aligned} \, \Phi_k & := (\pi_k)_* \circ \pi_2^* \colon \mathrm{D}^b(\operatorname{IGr}(2,V)) \to \mathrm{D}^b(\operatorname{IGr}(k,V)), \\ \widetilde\Phi_k & := (\widetilde\pi_k)_* \circ \widetilde\pi_2^* \colon \mathrm{D}^b(\operatorname{Gr}(2,V)) \to \mathrm{D}^b(\operatorname{Gr}(k,V)), \end{aligned}
\end{equation}
\tag{3.2}
$$
where the pullbacks and pushforwards are derived. We start with a simple observation. Proof. In fact, the right square in (3.1) is Cartesian with flat horizontal arrows, hence where the first isomorphism follows from the commutativity of the left-hand square, and the second is the base change formula for the right-hand square. The proof is complete. We apply these functors to the simplest irreducible equivariant vector bundles. In the computations we use the Borel–Bott–Weil theorem for the morphisms $\widetilde\pi_k$ and $\pi_k$. Note that these morphisms are fibrations with fibre $\operatorname{Gr}(2,k)$, so to compute the derived pushforward of an equivariant vector bundle associated with a weight $\beta$ of $\mathrm{GL}(V)$ or $\mathrm{Sp}(V)$, we consider the sum $\overline\beta$ of the first $k$-components of the weight $\beta$ with the special weight of the group $\mathrm{GL}_k$. If all the components of $\overline\beta$ are distinct, the derived pushforward is isomorphic to the equivariant bundle whose first $k$ components are given by the weight $\sigma(\overline\beta) - \rho_k$ (here $\sigma \in \mathfrak{S}_k$ is the minimal permutation such that the components of $\sigma(\overline\beta)$ are decreasing), and the remaining components coincide with those of the weight $\beta$; this equivariant bundle is shifted to the right by the length of $\sigma$ in the derived category. Otherwise (if some components of $\overline\beta$ coincide), the derived pushforward is zero.Now we do the computations: the first of these is very easy. Lemma 2. Assume that $\alpha_1 \geqslant \alpha_2$ and $\alpha_1 \geqslant -1$. Then Proof. The bundle $\Sigma^{\alpha_1,\alpha_2}\mathcal{U}_2^\vee$ is $\mathrm{GL}(V)$-equivariant and the corresponding weight is
$$
\begin{equation*}
\overline\beta=(\alpha_1,\alpha_2,0,\dots,0)+\rho_k=(k+\alpha_1,k-1+\alpha_2,k-2,\dots,1).
\end{equation*}
\notag
$$
If $\alpha_2 \geqslant 0$, then this weight is strictly dominant, so the first line of (3.3) follows. If $-1 \geqslant \alpha_2 \geqslant 2 - k$, then the second component of the weight is equal to one of the other components, hence the pushforward is zero, so the second line of (3.3) follows. Finally, if $1 - k \geqslant \alpha_2$, then to make the weight dominant we need to move its second component to the far right. The length of the corresponding permutation is $k - 2$, and since
$$
\begin{equation*}
(k+\alpha_1,k-2,\dots,1,k-1+\alpha_2)-\rho_k=(\alpha_1,-1,\dots,-1,k-2+\alpha_2),
\end{equation*}
\notag
$$
the third line of (3.3) also follows.
The lemma is proved. The second computation is more complicated. Proof. The flag $0 \subset \mathcal{U}_2 \subset \mathcal{U}_k \subset \mathcal{U}_k^\perp \subset \mathcal{U}_2^\perp$ on $\operatorname{IFl}(2,k;V)$ induces a filtration on $\pi_2^*\mathcal{S}_2$ with factors
$$
\begin{equation*}
\mathcal{U}_k / \mathcal{U}_2, \qquad \mathcal{U}_k^\perp/\mathcal{U}_k \cong \pi_k^*(\mathcal{S}_k) \quad\text{and}\quad \mathcal{U}_2^\perp / \mathcal{U}_k^\perp \cong (\mathcal{U}_k / \mathcal{U}_2)^\vee.
\end{equation*}
\notag
$$
By (1.5) we need to compute $\Phi_k(\wedge^m\mathcal{S}_2(-1))$ for $m = t - 2$ and $m = t$ and take the cone of the morphism induced by $\omega_S$ between them. Note that $\wedge^m\mathcal{S}_2$ has a filtration with terms
Using Pieri’s formula we rewrite this product as
$$
\begin{equation*}
\bigoplus_{s=\lceil(i+j+2-k)/2\rceil}^{\min(i,j)} \Sigma^{(1^{j-s},\,0^{k-2-i-j+2s},\,(-1)^{i-s})} (\mathcal{U}_k/\mathcal{U}_2)^\vee \otimes \pi_k^*(\wedge^{m-i-j} \mathcal{S}_k).
\end{equation*}
\notag
$$
To compute $\Phi_k(\wedge^m \mathcal{S}_2(-1))$ we twist this filtration by $\pi_2^*(\mathcal{O}(-1))$ and push forward along $\pi_k$. Using the projection formula, we reduce the computation to the objects
$$
\begin{equation*}
(\pi_k)_*\bigl(\Sigma^{(1^{j-s},\,0^{k-2-i-j+2s},\,(-1)^{i-s})} (\mathcal{U}_k/\mathcal{U}_2)^\vee \otimes \pi_2^*(\mathcal{O}(-1))\bigr).
\end{equation*}
\notag
$$
The corresponding weight is $(-1,-1,1^{j-s},\,0^{k-2-i-j+2s},\,(-1)^{i-s})$, and its sum with $\rho_k$ is
$$
\begin{equation}
\overline\beta= (k-1,k-2, \underbrace{k-1,\dots,k-j+s}_{\text{$j-s$ terms}}, \underbrace{k-j+s-2,\dots,i-s+1}_{\text{$k-2-i-j+2s$ terms}}, \underbrace{i-s-1,\dots,0}_{\text{$i-s$ terms}}).
\end{equation}
\tag{3.5}
$$
If $j - s > 0$, then the first and third coordinates are the same, hence the pushforward is zero. If $j - s = 0$ but $k-2-i-j+2s = k - 2 - i + s > 0$, then the second and third coordinates are the same, hence the pushforward is zero again. Thus, the only case where the pushforward is nonzero is the case where $s = j = i + 2 - k$. But since $i \leqslant k - 2$ and $j \geqslant 0$, we conclude that we must have $s = j = 0$ and $i = k - 2$. In this case the weight (3.5) equals $(k -1, k - 2, k - 3, \dots, 0)$, and the pushforward of the corresponding term of the filtration of $\pi_2^*(\wedge^m \mathcal{S}_2(-1))$ is
$$
\begin{equation*}
(\pi_k)_*\bigl(\Sigma^{(-1,-1,\dots,-1)} (\mathcal{U}_k/\mathcal{U}_2)^\vee \otimes \pi_2^*(\mathcal{O}(-1)) \otimes \pi_k^*(\wedge^{m-k+2} \mathcal{S}_k) \bigr) \cong \wedge^{m-k+2} \mathcal{S}_k(-1).
\end{equation*}
\notag
$$
In particular, the irreducible bundles have no higher pushforwards. Thus,
$$
\begin{equation*}
\Phi_k(\wedge^t_{\mathrm{Sp}} \mathcal{S}_2(-1))=\operatorname{Coker}(\wedge^{t-k} \mathcal{S}_k(-1) \overset{\omega_\mathcal{S}}\hookrightarrow \wedge^{t-k+2} \mathcal{S}_k(-1)).
\end{equation*}
\notag
$$
Finally, note that both terms vanish if $t \leqslant k - 3$, which gives us the first line of (3.4), and if $k - 2 \leqslant t \leqslant n - 2$, we obtain the second line.
Lemma 3 is proved. 3.2. General secondary staircase complexesIn this subsection we show that the functor $\widetilde\Phi_k$ takes some staircase complexes of $\operatorname{Gr}(2,V)$ to staircase complexes of $\operatorname{Gr}(k,V)$. As a consequence, we show that $\widetilde\Phi_k$ takes bundles $\mathcal{K}_{\operatorname{Gr}(2,V)}^{\alpha_1,\alpha_2}$ to bundles $\mathcal{K}_{\operatorname{Gr}(k,V)}^{\alpha_1,\alpha_2}$. The general definition of a staircase complex on $\operatorname{Gr}(k,V) = \operatorname{Gr}(k,2n)$ can be found in [2]. Such a complex is defined for any weight $\alpha$ of $\mathrm{GL}_k$ satisfying the condition
$$
\begin{equation*}
\alpha_1 \geqslant \alpha_2 \geqslant \alpha_3 \geqslant \dots \geqslant \alpha_k \geqslant \alpha_1-2n+k .
\end{equation*}
\notag
$$
For our purposes it is sufficient to consider the case where $\alpha_3 = \dots = \alpha_k = 0$, hence
$$
\begin{equation*}
2n-k \geqslant \alpha_1 \geqslant \alpha_2 \geqslant 0.
\end{equation*}
\notag
$$
Under this assumption the staircase complex takes the form
$$
\begin{equation}
\begin{aligned} \, \notag 0 &\to \wedge^{2n}V^\vee \otimes \Sigma^{\alpha_2,0,\dots,0,\alpha_1-2n+k}\mathcal{U}^\vee(-1) \to \dots \to \wedge^{\alpha_1+k+2} V^\vee \otimes \Sigma^{\alpha_2,0,0,\dots,0}\mathcal{U}^\vee(-1) \\ \notag &\to \wedge^{\alpha_1+1} V^\vee \otimes \Sigma^{\alpha_2-1,0,0,\dots,0}\mathcal{U}^\vee \to \dots \to \wedge^{\alpha_1-\alpha_2+2} V^\vee \otimes \Sigma^{\alpha_2-1,\alpha_2-1,0,\dots,0}\mathcal{U}^\vee \\ &\to \wedge^{\alpha_1-\alpha_2} V^\vee \otimes \Sigma^{\alpha_2,\alpha_2,0,\dots,0}\mathcal{U}^\vee \to \dots \to \Sigma^{\alpha_1,\alpha_2,0,\dots,0}\mathcal{U}^\vee \to 0. \end{aligned}
\end{equation}
\tag{3.6}
$$
Here in the top row the last component of the weight increases gradually, in the middle row the second component does the same, and in the bottom row so does the first component. In the special case where $\alpha_2 = 0$ the middle row disappears. Proposition 2. If $2n - k \geqslant \alpha_1 \geqslant \alpha_2 \geqslant 0$, the staircase complex (3.6) is the image under the functor $\widetilde\Phi_k$ of the staircase complex (2.7). Proof. First, assuming that $\alpha_2 > 0$ we rewrite (2.7) by breaking for convenience its top line at the term where the second component of the weight is zero:
$$
\begin{equation}
\begin{aligned} \, \notag 0 &\to \wedge^{2n}V^\vee \otimes \Sigma^{\alpha_2-1,\alpha_1+1-2n}\mathcal{U}_2^\vee \to \dots \to \wedge^{\alpha_1+2}V^\vee \otimes \Sigma^{\alpha_2-1,-1}\mathcal{U}_2^\vee \\ \notag &\to\wedge^{\alpha_1+1}V^\vee \otimes \Sigma^{\alpha_2-1,0}\mathcal{U}_2^\vee \to \dots \to \wedge^{\alpha_1-\alpha_2+2}V^\vee \otimes \Sigma^{\alpha_2-1,\alpha_2-1}\mathcal{U}_2^\vee \\ &\to \wedge^{\alpha_1-\alpha_2}V^\vee \otimes \Sigma^{\alpha_2,\alpha_2}\mathcal{U}_2^\vee \to \dots \to \Sigma^{\alpha_1,\alpha_2}\mathcal{U}_2^\vee \to 0. \end{aligned}
\end{equation}
\tag{3.7}
$$
Applying the functor $\widetilde\Phi_k$ and using Lemma 2 we see that each line of (3.7) gives the corresponding line of (3.6) (the last $k-2$ terms of the first line of (3.7) are taken to zero by $\widetilde\Phi_k$, but the previous terms are shifted by $k-2$ to the right). By the uniqueness property of the staircase complex, the differentials in the resulting complex are the same as in (3.6).
The case $\alpha_2 = 0$ is analogous: we only need to compare the last lines of (3.6) and (3.7). The proposition is proved. Recall the resolution (1.2) for the bundles $\mathcal{K}_{\operatorname{IGr}(k,V)}^{\alpha_1,\alpha_2}$. Lemma 4. If $2n - 2 \geqslant \alpha_1 \geqslant \alpha_2 \geqslant 0$, then there is an isomorphism Proof. First assume that $\alpha_1 \leqslant 2n - k$. Since $\mathcal{K}_{\operatorname{Gr}(2,V)}^{\alpha_1,\alpha_2}$ is defined as a truncation of (3.7) and $\mathcal{K}_{\operatorname{Gr}(k,V)}^{\alpha_1,\alpha_2}$ is defined as a truncation of its image (3.6) under $\widetilde\Phi_k$, and the truncations coincide, it follows that $\widetilde\Phi_k(\mathcal{K}_{\operatorname{Gr}(2,V)}^{\alpha_1,\alpha_2}) \cong \mathcal{K}_{\operatorname{Gr}(k,V)}^{\alpha_1,\alpha_2}$. Therefore, applying Lemma 1, we deduce the top line in (3.8).
Now assume that $2n - k + 1 \leqslant \alpha_1 \leqslant 2n - 2$. Then we apply Lemma 2 to the first row of (3.7) and conclude that $\widetilde\Phi_k(\mathcal{K}_{\operatorname{Gr}(2,V)}^{\alpha_1,\alpha_2})$ is zero, as required. Proof of Theorem 1 for $k \geqslant 3$. We define the complex (1.6) by applying the functor $\Phi_k$ to $\mathcal{K}_{t+k-2}^\bullet$ taken on $\operatorname{IGr}(2,V)$ and using Corollary 4 to identify the terms. In particular, the last $k-2$ terms vanish, so the image is $\mathcal{K}_{t}^\bullet$. It follows from the case $k = 2$ of Theorem 1 that the complex $\mathcal{K}_{t}^\bullet$ obtained is quasi-isomorphic to
$$
\begin{equation*}
\mathcal{K}_{t}^\bullet \cong \begin{cases} \Phi_k(\wedge_{\mathrm{Sp}}^{t+k-2}\mathcal{S}_2(-1)) & \text{if }0 \leqslant t \leqslant n-2, \\ \Phi_k(\wedge_{\mathrm{Sp}}^{2n-2-(t+k-2)}\mathcal{S}_2(-1))[-1] & \text{if }n \leqslant t \leqslant 2n-k, \\ 0 & \text{otherwise}. \end{cases}
\end{equation*}
\notag
$$
Now we apply Lemma 3 and obtain (1.7). Thus we have considered all possible cases in Theorem 1. § 4. BicomplexesIn this section, for each $0 \leqslant t \leqslant 2n-2$ we construct on $\operatorname{IGr}(2, V)$ a bicomplex whose columns are the right resolutions of $\mathcal{E}^{t-b,b}$ from Theorem 2, and the horizontal arrows are chosen in such a way that the induced differentials on these bundles coincide with the differentials of the complex $\mathcal{E}_{t}^\bullet$.Remark 6. In particular, if $t = n-1$, then the totalization of the bicomplex is acyclic. This is the case studied in [7], Proposition 5.3 (see Remark 5). It was claimed there that the horizontal maps are equal to $\mathbf{d}_2$. However, as we prove below, they are given by a nontrivial linear combination of $\mathbf{d}_1$ and $\mathbf{d}_2$. So, since $\mathbf{d}_1 \neq 0$ in general and these morphisms are unique, as we discussed in Remark 4, we obtain a contradiction. Recall the morphism
$$
\begin{equation*}
\mathbf{d}=\frac1{b+1}\mathbf{d}_1+\mathbf{d}_2 \colon \wedge^{a} V^\vee \otimes \mathrm{S}^{b} \mathcal{U} \to \wedge^{a+1} V^\vee \otimes \mathrm{S}^{b-1} \mathcal{U}
\end{equation*}
\notag
$$
defined in (2.13) and (2.14):
$$
\begin{equation*}
\mathbf{d}(\lambda \otimes P)= \frac1{b+1}(\lambda_1 \wedge \omega \otimes P_1+ \lambda_2 \wedge \omega \otimes P_2)+ \lambda \wedge (\omega_1 \otimes P_1+\omega_2 \otimes P_2).
\end{equation*}
\notag
$$
We checked in Proposition 1 that it induces a complex on the subbundles $\mathcal{E}^{a,b} \subset \wedge^{a} V^\vee \otimes \mathrm{S}^{b} \mathcal{U}$. Now we check that it also induces a complex of the ambient bundles. Lemma 5. The composition Proof. The composition of these arrows is the sum of four maps, namely, $b(b+1)\mathbf{d}_2^2$, $\mathbf{d}_1^2$, $b\,\mathbf{d}_2 \circ \mathbf{d}_1$ and $(b+1)\mathbf{d}_1 \circ \mathbf{d}_2$. These compositions take $\lambda \otimes P$ to
$$
\begin{equation*}
\begin{aligned} \, &b(b+1)\lambda \wedge (\omega_1 \wedge \omega_2 \otimes P_{12}+\omega_2 \wedge \omega_1 \otimes P_{12}), \\ &\lambda_1 \wedge \omega_1 \wedge \omega \otimes P_{11} +\lambda_2 \wedge \omega_1 \wedge \omega \otimes P_{12} \\ &\qquad+\lambda_1 \wedge \omega_2 \wedge \omega \otimes P_{12} +\lambda_2 \wedge \omega_2 \wedge \omega \otimes P_{22}, \\ &b\lambda_1 \wedge \omega \wedge (\omega_1 \otimes P_{11}+\omega_2 \otimes P_{12}) \\ &\qquad +b\lambda_2 \wedge \omega \wedge (\omega_1 \otimes P_{12}+\omega_2 \otimes P_{22}) \end{aligned}
\end{equation*}
\notag
$$
and where, as usual, we choose a basis $e_1$, $e_2$ in $U$ and write $\lambda_i$ and $\omega_i$ for the evaluation of the corresponding skew-forms on $e_i$ and $P_{ij}$ for the second derivatives of $P$ in the directions $e_i$ and $e_j$. It is easy to see that the first summand is zero and the last three summands annihilate.
The lemma is proved. Consider also the unique $\mathrm{GL}(V)$-equivariant morphism
$$
\begin{equation*}
\mathbf{d}_0 \colon \wedge^{a} V^\vee \otimes \mathrm{S}^{b} \mathcal{U} \to \wedge^{a-1} V^\vee \otimes \mathrm{S}^{b+1} \mathcal{U}(1).
\end{equation*}
\notag
$$
It is, up to twist, the differential of Koszul complex (2.8), and it is given by the following composition of canonical morphisms:
$$
\begin{equation*}
\begin{aligned} \, \wedge^{a} V^\vee \otimes \mathrm{S}^{b} \mathcal{U} &\to \wedge^{a-1} V^\vee \otimes V^\vee \otimes \mathrm{S}^{b} \mathcal{U} \to \wedge^{a-1} V^\vee \otimes \mathcal{U}^\vee \otimes \mathrm{S}^{b} \mathcal{U} \\ &\to \wedge^{a-1} V^\vee \otimes \Sigma^{b,-1} \mathcal{U} \cong \wedge^{a-1} V^\vee \otimes \mathrm{S}^{b+1} \mathcal{U}(1). \end{aligned}
\end{equation*}
\notag
$$
At the point $[U]$ with a basis $e_1, e_2$ of $U$ it acts by
$$
\begin{equation}
\mathbf{d}_0(\lambda \otimes P)=(\lambda_1 \otimes e_2 \cdot P-\lambda_2 \otimes e_1 \cdot P) \cdot \det U^\vee,
\end{equation}
\tag{4.2}
$$
since the composition of the first three maps is $\sum \partial/\partial e_i \otimes e_i$ and the last isomorphism is $e_i \mapsto e_j \cdot \det U^\vee$ for $\{i,j\}=\{1,2\}$. Proof. Using formulae (2.13), (2.14) and (4.2) it is straightforward to check that the composition $(\mathbf{d}_1 + (b + 2)\mathbf{d}_2) \circ \mathbf{d}_0$ takes $\lambda \otimes P$ to
$$
\begin{equation*}
\begin{aligned} \, & \bigl(\lambda_{12} \wedge \omega \otimes (P+e_1 \cdot P_1) +\lambda_{12} \wedge \omega \otimes (P+e_2 \cdot P_2) \\ &\qquad\qquad +(b+2) \lambda_1 \wedge (\omega_1 \otimes e_2 \cdot P_1+\omega_2 \otimes (P+e_2 \cdot P_2)) \\ &\qquad\qquad -(b+2) \lambda_2 \wedge (\omega_1 \otimes (P+e_1 \cdot P_1)+\omega_2 \otimes e_1 \cdot P_2)\bigr) \cdot \det U^\vee, \end{aligned}
\end{equation*}
\notag
$$
while the composition $\mathbf{d}_0 \circ (\mathbf{d}_1+(b+1)\mathbf{d}_2)$ takes $\lambda \otimes P$ to
$$
\begin{equation*}
\begin{aligned} \, & \bigl(\lambda_1 \wedge \omega_1 \otimes e_2 \cdot P_1-\lambda_{12} \wedge \omega \otimes e_2 \cdot P_2+\lambda_2 \wedge \omega_1 \otimes e_2 \cdot P_2 \\ &\qquad\qquad - \lambda_{12} \wedge \omega \otimes e_1 \cdot P_1-\lambda_1 \wedge \omega_2 \otimes e_1 \cdot P_1-\lambda_2 \wedge \omega_2 \otimes e_1 \cdot P_2 \\ &\qquad\qquad - (b+1) \lambda_1 \wedge (\omega_1 \otimes e_2 \cdot P_1+\omega_2 \otimes e_2 \cdot P_2) \\ &\qquad\qquad + (b+1) \lambda_2 \wedge (\omega_1 \otimes e_1 \cdot P_1+\omega_2 \otimes e_1 \cdot P_2)\bigr) \cdot \det U^\vee. \end{aligned}
\end{equation*}
\notag
$$
It remains to note that the sum of the first expression multiplied by $b$ and the second expression multiplied by $b + 2$ is zero.
The proposition is proved. As a corollary to the proposition, we obtain the following result. Theorem 3. The total complex of the diagram (4.1), where the vertical arrows are given by $\mathbf{d}_0$ and the horizontal arrow 
Citation in format AMSBIB
\Bibitem{Nov25} |
|
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|