Monads

Monad on Hirzebruch surfaces

Let \(\pi: \Sigma_{n}=\mathbf{P}(\mathcal{E}^{\vee})=\mathbf{P}(\mathcal{O}\oplus \mathcal{O}(-n))\rightarrow \mathbf{P}^1\) be the \(n\)-th Hirzebruch surfaces. Let \(H=c_{1}(\mathcal{O}_{\Sigma_{n}}(1))\), \(E\) be the class of \(\mathbf{P}(\mathcal{O}(-n))\), its self-intersection equals to \(-n\), \(F\) be a fiber class of \(\pi\). We have \(E=H-nF\). The Chow ring of \(\Sigma_{n}\) is given by \(A(\Sigma_{n})=A[\mathbf{P}^1][H]/(H^{2}-nFH)=\mathbf{Z}[F, H]/(F^2, H^{2}-nFH)\), naturally one has \(\operatorname{Pic}(\Sigma_{n})=\mathbf{Z} H\oplus \mathbf{Z} F\). From now on, we denote \(\mathcal{E}(p,q)=\mathcal{E}\otimes\mathcal{O}_{\Sigma_{n}}(pH+qF)\) for any sheaf of \(\mathcal{O}_{\Sigma_{n}}\)-modules. Recall that the relative Euler sequence computes the relative canonical sheaf

\[ 0\rightarrow \mathcal{O}_{\Sigma_{n}}(-1, 0)\rightarrow \pi^{*}(\mathcal{E}^{\vee})\rightarrow T_{\Sigma_{n}/\mathbf{P}^1}(-1,0)\rightarrow 0. \]

\[ \begin{aligned} \Omega_{\Sigma_{n}/\mathbf{P}^1}&=\mathcal{O}_{\Sigma_{n}}(-2, 0)\otimes \pi^{*}(\det \mathcal{E})=\mathcal{O}_{\Sigma_{n}}(-2,n)\\ T_{\Sigma_{n}/\mathbf{P}^1}&=\mathcal{O}_{\Sigma_{n}}(2,-n)\\ \omega_{\Sigma_{n}}&=\mathcal{O}_{\Sigma_{n}}(-2, n-2). \end{aligned} \]

For later computations, we need the following lemmata.

Lemma

The cohomology of any line bundle \(\mathcal{O}_{\Sigma_n}(p,q)\) on the Hirzebruch surface \(\Sigma_n\) is completely determined by the integers \(p\), \(q\), and \(n\) through the projection \(\pi:\Sigma_n\to\mathbf{P}^1\). Since \(\Sigma_n = \mathbf{P}(\mathcal{O}\oplus\mathcal{O}(-n))\), the higher direct images of \(\mathcal{O}_{\Sigma_n}(p)\) along fibers of \(\pi\) reduce the problem to cohomology on \(\mathbf{P}^1\).

Direct image computation. For \(p \geq 0\), the projection formula gives

\[ \pi_*\mathcal{O}_{\Sigma_n}(p,q) = \operatorname{Sym}^p(\mathcal{O}\oplus\mathcal{O}(-n))\otimes\mathcal{O}_{\mathbf{P}^1}(q) = \bigoplus_{j=0}^{p}\mathcal{O}_{\mathbf{P}^1}(q-jn), \]

while \(\pi_*\mathcal{O}_{\Sigma_n}(p,q) = 0\) for \(p < 0\) and \(R^1\pi_*\mathcal{O}_{\Sigma_n}(p,q) = 0\) for \(p \geq -1\).

\(\mathrm{H}^0\) criterion. By the Leray spectral sequence, \(\mathrm{H}^0(\Sigma_n, \mathcal{O}(p,q)) = \mathrm{H}^0(\mathbf{P}^1, \pi_*\mathcal{O}(p,q))\). This is nonzero precisely when \(p \geq 0\) and at least one summand \(\mathcal{O}_{\mathbf{P}^1}(q - jn)\) has nonnegative degree for some \(0 \leq j \leq p\), which amounts to \(np + q \geq 0\). Hence

\[ \mathrm{H}^{0}(\mathcal{O}_{\Sigma_{n}}(p, q))\neq 0 \quad\text{if and only if}\quad p\geq 0 \;\text{ and }\; np+q\geq 0. \]

\(\mathrm{H}^2\) by Serre duality. Since \(\omega_{\Sigma_n} = \mathcal{O}(-2, n-2)\), Serre duality gives \(\mathrm{H}^2(\mathcal{O}(p,q)) = \mathrm{H}^0(\mathcal{O}(-p-2, n-2-q))^{\vee}\). Applying the \(\mathrm{H}^0\) criterion to the dual: \(-p-2 \geq 0\) and \(n(-p-2) + (n-2-q) \geq 0\), i.e.,

\[ \mathrm{H}^{2}(\mathcal{O}_{\Sigma_{n}}(p, q))\neq 0 \quad\text{if and only if}\quad p\leq -2 \;\text{ and }\; np+q\leq -(n+2). \]

\(\mathrm{H}^1\) by exclusion. The Euler characteristic from Riemann--Roch on \(\Sigma_n\) is \(\chi(\mathcal{O}(p,q)) = \tfrac{1}{2}(p+1)(np+2q+2)\). Since \(\mathrm{H}^1\) is determined by \(\chi - h^0 + h^2\), the region where \(\mathrm{H}^1 \neq 0\) consists of exactly two disjoint components:

\[ \mathrm{H}^{1}(\mathcal{O}_{\Sigma_{n}}(p, q))\neq 0 \quad\text{if and only if}\quad \begin{cases}p\geq 0,\; q\leq -2\end{cases} \;\text{ or }\; \begin{cases}p\leq -2,\; q\geq n.\end{cases} \]

The first region corresponds to contributions from \(R^0\pi_*\) summands of negative degree on \(\mathbf{P}^1\) (specifically \(\mathrm{H}^1(\mathbf{P}^1, \mathcal{O}(q-jn))\) for \(j\) with \(q - jn \leq -2\)). The second region comes from \(R^1\pi_*\) when \(p \leq -2\), which by relative Serre duality contributes \(\mathrm{H}^0\) of a bundle on \(\mathbf{P}^1\) of sufficiently positive degree.

The classical Beilinson spectral sequence is a way to describe torsion-free sheaves on \(\mathbf{P}^2\) as the cohomology of certain three-term complexes---the so-called monad. With the isomorphism between the Hilbert scheme of points on \(\mathbf{C}^2\) and the moduli space of rank \(1\) torsion-free sheaves on \(\mathbf{P}^2\) which are trivial over infinity, the Hilbert scheme can be realized as the quiver variety with one vertex and one loop, see [Nakajima, Theorem 2.1]. The essential part of the construction is a resolution of the diagonal in \(\mathbf{P}^2\times \mathbf{P}^2\). For Hirzebruch surfaces, the diagonal can also be resolved (for references to the details of the construction, we refer to [Buchdahl, 1987] or [Aprodu, 2009]). We first briefly recall the Beilinson-type spectral sequence on \(\Sigma_{n}\), following the approach in [Nakajima] we give a monad description of \(\operatorname{Hilb}(T^*\mathbf{P}^1)\) as a Nakajima quiver variety of type \(A_{1}^{(1)}\), and we show that the two tautological bundles corresponding to the two vertices are exactly \(\mathcal{O}_{X}(1)^{[n]}\) and \(\mathcal{O}_{X}^{[n]}\).

We denote \(p_{i}:\Sigma_{n}\times \Sigma_{n}\) be the projections to the two factors, \(p: X\times X\rightarrow \mathbf{P}^1\times \mathbf{P}^1\) be the product of the ruling \(\pi\). Let \(\Delta_{\mathbf{P}^1}\subset \mathbf{P}^1\times \mathbf{P}^1\) be the diagonal divisor on \(\mathbf{P}^1\times \mathbf{P}^1\) and \(\Delta\) be the diagonal divisor on \(\Sigma_{n}\times \Sigma_{n}\), \(L= \mathcal{O}_{\mathbf{P}^1\times \mathbf{P}^1}(\Delta_{\mathbf{P}^1})\). Consider the line bundle

\[ \mathcal{F} = p_{1}^{*}(T_{\Sigma_{n}/\mathbf{P}^1}(-1, 0))\otimes p_{2}^{*}(\mathcal{O}_{\Sigma_{n}}(1, 0))=\mathcal{O}_{\Sigma_{n}}(1,-n)\boxtimes \mathcal{O}_{\Sigma_{n}}(1, 0). \]

A rank \(2\) locally free sheaf \(\mathcal{G}\) is defined by an extension

\[ 0\rightarrow \mathcal{F}\rightarrow \mathcal{G}\rightarrow p^{*}(L)\rightarrow 0. \]

Note that \(p^{*}(L)=\mathcal{O}_{\Sigma_{n}}(0,1)\boxtimes \mathcal{O}_{\Sigma_{n}}(0,1)\). Buchdahl proved that [Buchdahl, 1987] the diagonal \(\Delta\) can be realized as the zero locus of a global section \(s\) of \(\mathcal{G}\), then the Koszul complex of \(s\) gives us the resolution of the diagonal \(C^{\bullet}\twoheadrightarrow\mathcal{O}_{\Delta}\):

\[ 0\rightarrow \wedge^{2}\mathcal{G}^{\vee}\rightarrow \mathcal{G}^{\vee}\xrightarrow{s} \mathcal{O}_{\Sigma_{n}\times \Sigma_{n}}\rightarrow \mathcal{O}_{\Delta}\rightarrow 0. \]

Then the Beilinson spectral sequences comes from different ways of computing the image of \(p_{1}^{*}\mathcal{E}\) under the composite functor \(p_{2*}(\mathcal{O}_{\Delta}\otimes -)\) . On the one hand, we trivially have \(p_{2*}(\mathcal{O}_{\Delta}\otimes p_{1}^{*}\mathcal{E})=\mathcal{E}\). If we take the cohomology of \(C^{\bullet}\) first in the double complex for the hyper direct image \(\mathbf{R}^{\bullet}p_{2*}(C^{\bullet}\otimes p_{1}^{*}\mathcal{E})\), the trivial identity tells us exactly that the corresponding spectral sequence degenerate at the \(E_{2}\) page, namely

\[ E_{2}^{p,q}=R^{q}p_{2*}(\mathrm{H}^{p}(C^{\bullet}\otimes p_{1}^{*}\mathcal{E}))=\begin{cases}\mathcal{E} & (p,q)=(0,0)\\ 0 & \text{otherwise }\end{cases}. \]

On the other hand, we can also take the direct image first, then we get the so-called Beilinson spectral sequence for \(\Sigma_{n}\).

Theorem: Buchdahl

For any torsion-free sheaf \(\mathcal{E}\) on the Hirzebruch surface \(\Sigma_n\), the two spectral sequences of the double complex \(\mathbf{R}p_{2*}(C^\bullet \otimes p_1^*\mathcal{E})\) produce a Beilinson-type spectral sequence

\[ E_{1}^{p,q} = R^{q}p_{2*}(\wedge^{-p}\mathcal{G}^{\vee}\otimes p_{1}^{*}\mathcal{E}) \Rightarrow \begin{cases}\mathcal{E} & p+q=0\\ 0 & \text{otherwise}\end{cases}, \]

where \(\mathcal{G}\) is the rank-\(2\) bundle on \(\Sigma_n \times \Sigma_n\) constructed from Buchdahl's resolution of the diagonal, and \(p_1, p_2\) are the two projections. The argument is a direct application of the two filtrations of a double complex. Buchdahl's resolution provides a Koszul complex \(C^\bullet \to \mathcal{O}_\Delta\) on \(\Sigma_n \times \Sigma_n\), which when tensored with \(p_1^*\mathcal{E}\) (exact since the Koszul terms are locally free) gives a double complex \(C^\bullet \otimes p_1^*\mathcal{E}\). The first filtration --- taking cohomology of \(C^\bullet\) first --- sees only \(\mathcal{O}_\Delta \otimes p_1^*\mathcal{E}\) in degree \(0\), whose pushforward under \(p_2\) is \(\mathcal{E}\) itself, so this spectral sequence degenerates immediately at \(E_2\) with abutment \(\mathcal{E}\) in bidegree \((0,0)\). The second filtration --- taking \(Rp_{2*}\) first --- produces the \(E_1\)-page \(E_1^{p,q} = R^q p_{2*}(\wedge^{-p}\mathcal{G}^\vee \otimes p_1^*\mathcal{E})\), supported in the range \(-2 \leq p \leq 0\) and \(0 \leq q \leq 2\) (since \(\mathcal{G}\) has rank \(2\) and \(\Sigma_n\) is a surface). Both spectral sequences converge to the same abutment, establishing the theorem. The \(E_1\)-page has three columns: \(p = 0\) corresponds to \(\wedge^0 \mathcal{G}^\vee = \mathcal{O}\), \(p = -1\) to \(\mathcal{G}^\vee\), and \(p = -2\) to \(\wedge^2 \mathcal{G}^\vee = \det(\mathcal{G}^\vee)\).

Remark

Buchdahl originally stated the spectral sequence for locally free sheaves, but the same argument extends to torsion-free sheaves without modification: tensoring with \(p_1^*\mathcal{E}\) preserves exactness of the Koszul complex because its terms are locally free on \(\Sigma_n \times \Sigma_n\), so no flatness assumption on \(\mathcal{E}\) is needed. There is also an asymmetry in the construction. One may use either \(p_{1*}(p_2^*\mathcal{E} \otimes \mathcal{O}_\Delta)\) or \(p_{2*}(p_1^*\mathcal{E} \otimes \mathcal{O}_\Delta)\) to produce the spectral sequence, and these two choices generally yield different \(E_1\)-pages. The convention above (using \(p_{2*}\) and \(p_1^*\mathcal{E}\)) is preferred because the bundle \(\mathcal{G}\) involves \(T_{\Sigma_n/\mathbf{P}^1}(-1,0)\) pulled back from the first factor, which interacts favorably with the cohomology vanishing from the line bundle lemma to produce better vanishing on the \(E_1\)-page. To compute the \(E_1\)-terms explicitly, take the exterior powers of the dual of the defining sequence of \(\mathcal{G}\) and tensor with \(p_1^*\mathcal{E}\). Since \(\mathcal{E}\) is torsion-free, \(-\otimes p_1^*\mathcal{E}\) is exact, giving

\[ 0\rightarrow \wedge^{-p-1}\mathcal{F}^{\vee}\otimes p^{*}(L^{\vee})\otimes p_{1}^{*}\mathcal{E}\rightarrow \wedge^{-p}\mathcal{G}^{\vee}\otimes p_{1}^{*}\mathcal{E}\rightarrow \wedge^{-p}\mathcal{F}^{\vee}\otimes p_{1}^{*}\mathcal{E}\rightarrow 0. \]

For \(p = 0\), the first sheaf vanishes and \(\wedge^0\mathcal{G}^\vee \otimes p_1^*\mathcal{E} = p_1^*\mathcal{E}\). For \(p = -2\), the last sheaf vanishes and \(\wedge^2\mathcal{G}^\vee \otimes p_1^*\mathcal{E} = \mathcal{E}(-1,n-1) \boxtimes \mathcal{O}_{\Sigma_n}(-1,-1)\). Taking the associated long exact sequences in higher direct images yields

\[ \begin{aligned} &E_{1}^{0,q}= \mathrm{H}^{q}(\mathcal{E})\otimes \mathcal{O}_{\Sigma_{n}}(0,0)\\ \dots\rightarrow \mathrm{H}^{q}(\mathcal{E}(0, -1))\otimes \mathcal{O}_{\Sigma_{n}}(0, -1)\rightarrow &E_{1}^{-1, q}\rightarrow \mathrm{H}^{q}(\mathcal{E}(-1, n))\otimes \mathcal{O}_{\Sigma_{n}}(-1, 0)\rightarrow \dots \\ &E_{1}^{-2, q}= \mathrm{H}^{q}(\mathcal{E}(-1, n-1))\otimes \mathcal{O}_{\Sigma_{n}}(-1,-1). \end{aligned} \]

Lemma

Let \(\mathcal{E}\) be a torsion-free sheaf on \(\Sigma_n\) that is trivial at infinity, meaning \(\mathcal{E}|_E \cong \mathcal{O}_E^{\oplus r}\) where \(E\) is the negative section. The triviality condition controls the fiberwise splitting type and forces cohomological vanishing in two complementary ranges:

\[ \mathrm{H}^{0}(\mathcal{E}(p,q))=0 \quad \text{for } np+q\leq -1, \]

\[ \mathrm{H}^{2}(\mathcal{E}(p,q))=0 \quad \text{for } np+q\geq -(n+1). \]

For the \(\mathrm{H}^0\) vanishing, restrict to a general fiber \(F \cong \mathbf{P}^1\) of \(\pi\). On each fiber, the restriction \(\mathcal{E}(p,q)|_F\) has degree \(np + q\) (since \(H \cdot F = 1\) and \(F \cdot F = 0\)). Since \(\mathcal{E}\) is trivial at infinity, the splitting type on each fiber is nonnegative, so \(\mathcal{E}|_F \cong \bigoplus \mathcal{O}_{\mathbf{P}^1}(a_i)\) with each \(a_i \geq 0\). When \(np + q \leq -1\), the total degree is negative, so every summand \(\mathcal{O}_{\mathbf{P}^1}(a_i + np + q)\) has \(a_i + np + q \leq a_i - 1\), and for generic fibers \(a_i = 0\), giving degree \(\leq -1\). Thus any global section restricts to zero on every fiber, and \(\pi_*\mathcal{E}(p,q) = 0\) forces \(\mathrm{H}^0(\mathcal{E}(p,q)) = 0\). For the \(\mathrm{H}^2\) vanishing, Serre duality gives \(\mathrm{H}^2(\mathcal{E}(p,q)) = \mathrm{H}^0(\mathcal{E}^\vee(-p-2, n-2-q))^\vee\) using \(\omega_{\Sigma_n} = \mathcal{O}(-2,n-2)\). The dual \(\mathcal{E}^\vee\) is also trivial at infinity, so the \(\mathrm{H}^0\) vanishing applies to \(\mathcal{E}^\vee\): we need \(n(-p-2) + (n-2-q) \leq -1\), which rearranges to \(np + q \geq -(n+1)\).

Proposition

A torsion-free sheaf \(\mathcal{E}\) on \(\Sigma_n\) that is trivial at infinity admits a monad description: it is the cohomology of a three-term complex

\[ 0\rightarrow \mathrm{H}^{1}(\mathcal{E}(-2,n-1))\otimes \mathcal{O}_{\Sigma_{n}}(0, -1)\rightarrow E_{1}^{-1, 1}\otimes\mathcal{O}_{\Sigma_{n}}(1, 0)\rightarrow \mathrm{H}^{1}(\mathcal{E}(-1, 0))\otimes \mathcal{O}_{\Sigma_{n}}(1, 0)\rightarrow 0, \]

where the middle term fits in a split short exact sequence

\[ 0 \rightarrow \mathrm{H}^{1}(\mathcal{E}(-1, -1))\otimes\mathcal{O}_{\Sigma_n}(1, -1)\rightarrow E_{1}^{-1, 1}\otimes \mathcal{O}_{\Sigma_{n}}(1, 0)\rightarrow \mathrm{H}^{1}(\mathcal{E}(-2, n))\otimes \mathcal{O}_{\Sigma_{n}}(0, 0)\rightarrow 0. \]

Vanishing on the \(E_1\)-page. The key step is to apply the Beilinson spectral sequence not to \(\mathcal{E}\) but to the twist \(\mathcal{E}(-1,0)\). The \(E_1\)-page for \(\mathcal{E}(-1,0)\) is a \(3 \times 3\) grid (\(q\) rows, \(p\) columns):

\[ \begin{array}{c|ccc} q \backslash p & p=-2 & p=-1 & p=0 \\ \hline q=2 & \mathrm{H}^{2}(\mathcal{E}(-2,n\!-\!1))\otimes\mathcal{O}(-1,-1) & E_{1}^{-1,2} & \mathrm{H}^{2}(\mathcal{E}(-1,0))\otimes\mathcal{O}_{\Sigma_{n}} \\ q=1 & \mathrm{H}^{1}(\mathcal{E}(-2,n\!-\!1))\otimes\mathcal{O}_{\Sigma_{n}}(-1,-1) & E_{1}^{-1,1} & \mathrm{H}^{1}(\mathcal{E}(-1,0))\otimes\mathcal{O}_{\Sigma_{n}} \\ q=0 & \mathrm{H}^{0}(\mathcal{E}(-2,n\!-\!1))\otimes\mathcal{O}_{\Sigma_{n}}(-1,-1) & E_{1}^{-1,0} & \mathrm{H}^{0}(\mathcal{E}(-1,0))\otimes\mathcal{O}_{\Sigma_{n}} \end{array} \]

We use the vanishing lemma to kill the four corner entries. For \(\mathrm{H}^0\): the fiberwise degree of \(\mathcal{E}(-2,n-1)\) is \(n(-2) + (n-1) = -(n+1) \leq -1\), so \(\mathrm{H}^0 = 0\); for \(\mathcal{E}(-1,0)\), the fiberwise degree is \(-n \leq -1\), so \(\mathrm{H}^0 = 0\). For \(\mathrm{H}^2\): the fiberwise degree of \(\mathcal{E}(-2,n-1)\) is \(-(n+1) \geq -(n+1)\), so \(\mathrm{H}^2 = 0\); for \(\mathcal{E}(-1,0)\), the fiberwise degree is \(-n \geq -(n+1)\), so \(\mathrm{H}^2 = 0\). Degeneration to a monad. With all four corner entries vanishing, convergence of the spectral sequence forces \(E_1^{-1,2} = 0\) and \(E_1^{-1,0} = 0\) as well, since these would contribute to total degrees \(1\) and \(-1\) respectively with no surviving terms to cancel them. Only the \(q = 1\) row survives, and the spectral sequence degenerates at \(E_1\), expressing \(\mathcal{E}(-1,0)\) as the cohomology of

\[ 0\rightarrow \mathrm{H}^{1}(\mathcal{E}(-2,n-1))\otimes \mathcal{O}_{\Sigma_{n}}(-1, -1)\rightarrow E_{1}^{-1, 1}\rightarrow \mathrm{H}^{1}(\mathcal{E}(-1, 0))\otimes \mathcal{O}_{\Sigma_{n}}\rightarrow 0. \]

Tensoring with \(\mathcal{O}_{\Sigma_n}(1,0)\) yields the first statement. Splitting of the middle term. The vanishing lemma also gives \(\mathrm{H}^0(\mathcal{E}(-1,-1)) = 0\) (fiberwise degree \(-(n+1) \leq -1\)) and \(\mathrm{H}^2(\mathcal{E}(-1,-1)) = 0\) (fiberwise degree \(-(n+1)\), boundary). The long exact sequence from the exterior power filtration then degenerates to the claimed short exact sequence for \(E_1^{-1,1} \otimes \mathcal{O}_{\Sigma_n}(1,0)\). This sequence splits because the relevant Ext group vanishes: \(\operatorname{Ext}^1(\mathcal{O}_{\Sigma_n}, \mathcal{O}_{\Sigma_n}(1,-1)) = \mathrm{H}^1(\mathcal{O}_{\Sigma_n}(1,-1))\), and by the line bundle cohomology lemma, \(\mathrm{H}^1(\mathcal{O}_{\Sigma_n}(1,-1))\) requires either \(\{p \geq 0, q \leq -2\}\) or \(\{p \leq -2, q \geq n\}\). With \(p = 1\) and \(q = -1\), neither condition holds, so the Ext group vanishes and the sequence splits. The essential point is that the twist by \(\mathcal{O}(-1,0)\) is precisely what makes all four corner entries of the \(E_1\)-page vanish simultaneously, forcing the spectral sequence to collapse to a single row and producing the monad.