1 Introduction↩︎

1.1 The geometric Eisenstein series↩︎

Eisenstein series and related functions appear naturally in the study of automorphic forms over global fields. In the function-field case, they have a geometric counterpart given by sheaf-theoretic functors between moduli stacks of bundles. In this paper we study the constant term of a particular Eisenstein series for the general linear group. The cohomology of the corresponding sheaf carries an action of a Hecke-type modification algebra. Remarkably, we show that it is isomorphic to an algebra of differential operators on a singular supervariety. Moreover, the constant term is encoded in the local cohomology of its resolution.

Let \(C\) be a smooth projective curve over \(\mathbb{F}_q\), and let \(G=GL_{n+1}\). We consider the mirabolic parabolic \(P\subset G\), whose Levi quotient is \(GL_n\). Laumon’s compactification of \(\operatorname{Bun}_P\) gives the geometric Eisenstein series correspondence [1]–[3] \[\operatorname{Bun}_{M} \xleftarrow{} \overline{\operatorname{Bun}}_{P} \xrightarrow{} \operatorname{Bun}_{G}.\] Fibers of the map to \(\mathrm{Bun}_{G}\) are also referred to as twisted quasimaps [4]–[6] from \(C\) to the projective space \(\mathbb{P}^{n}\). The constant term [7] along the Borel is obtained by a similar correspondence \[\operatorname{Bun}_{G} \xleftarrow{} \operatorname{Bun}_{B} \xrightarrow{} \operatorname{Bun}_{T}.\] The value of the constant term of the Eisenstein series is then given by the trace of Frobenius on the cohomology of a fiber of the map \[\overline{\operatorname{Bun}}_P \times_{\operatorname{Bun}_G} \operatorname{Bun}_B \rightarrow \mathrm{Bun}_{T}.\] The fiber over a \(T\)-bundle \[\vec{\mathcal{M}}=(\mathcal{M}_1,\ldots,\mathcal{M}_{n+1}) \in \mathrm{Bun}_{T}\] is the stack \(QM\vec{N}_{\vec{\mathcal{M}}}\) of triples \[(\mathcal{V}^\bullet,\mathcal{L},s),\] where \[0=\mathcal{V}^{(0)}\subset\mathcal{V}^{(1)}\subset\cdots \subset\mathcal{V}^{(n+1)}=\mathcal{V}\] is a flag of vector bundles with \[\mathcal{V}^{(k)}/\mathcal{V}^{(k-1)}\cong\mathcal{M}_k,\] and \[s:\mathcal{L}\hookrightarrow \mathcal{V}\] is an injective sheaf map. We regard \[H_c^*(QM\vec{N}_{\vec{\mathcal{M}}})\] as the categorified value of the constant term at \(\vec{\mathcal{M}}\). The stack \(QM\vec{N}_{\vec{\mathcal{M}}}\) carries a Hecke-type modification correspondence: one replaces the line bundle \(\mathcal{L}\) by \(\mathcal{L}(-x)\) at a point \(x\in C\). Pullback and pushforward along this correspondence give modification/monopole operators [8]–[12] \[e\langle \alpha\rangle,\qquad f\langle \beta\rangle, \qquad \alpha, \, \beta \in H^{*}_{c}(C)\] on \(H_c^*(QM\vec{N}_{\vec{\mathcal{M}}})\) which generate an algebra \(\mathcal{A}\). We show that the subalgebra of \(\mathcal{A}\) corresponding to \(H^{2}(C)\) is the algebra of regular functions on the \(A_n\)-surface: \[\mathcal{A}^{(2)} \cong \overline{\mathbb{Q}}_l[x,y,\eta]/(xy-\eta^{n+1}).\] The degree-one operators form a spin module over this surface, and the degree-zero operators act as first-order differential operators on it.

1.2 Main results↩︎

1.2.1 Symplectic dual description↩︎

Let \[X_0^\vee=\operatorname{Spec}\mathcal{A}^{(2)}\] and let \[\rho:X^\vee\to X_0^\vee\] be its minimal resolution. \(X_0^\vee\) is the Coulomb branch of a 3-dimensional \(\mathcal{N} = 4\) SUSY gauge theory [13]–[15], which is the symplectic/3d-mirror dual to the Higgs branch \(X = T^{*}\mathbb{P}^{n}\) [16], [17]; one can view the zero section \(\mathbb{P}^{n} \subset X\) as the target of the quasimaps. The surface \(X^{\vee}\) carries an action of a two-dimensional torus, coming from the cohomological and degree gradings on \(H^{*}_{c}(QM\vec{N})\). This action preserves a natural symplectic form on \(X^{\vee}\), so the attracting set \(L\subset X^{\vee}\) is Lagrangian.

A choice of Lagrangian subspace in \(H_{c}^{1}(C)\) gives rise to a supercommutative locally free \(\mathcal{O}_{X^\vee}\)-algebra \(\hat{\mathcal{O}}_{\mathrm{vir}}\). The operators in \(\mathcal{A}\) coming from \(H^{0}_{c}(C)\) can be identified with differential operators on the twisted sheaf \[\hat{\mathcal{O}}_{\mathrm{vir}} \otimes \mathcal{O}(\vec{\mathcal{M}});\] here \(\mathcal{O}(\vec{\mathcal{M}})\) is a certain equivariant line bundle depending on the tuple \(\vec{\mathcal{M}}\), whose restrictions to the exceptional curves have degree \(\deg \mathcal{M}_k-\deg \mathcal{M}_{k+1}\). We give an interpretation of the cohomology of quasimaps as the local cohomology of \(\hat{\mathcal{O}}_{\mathrm{vir}} \otimes \mathcal{O}(\vec{\mathcal{M}})\) with support in the Lagrangian subvariety \(L\). To make the following isomorphism torus-equivariant, we introduce a suitable shift 12 of the cohomology of quasimaps, denoted by \(\hat{H}^{*}_{c}(QM\vec{N})\).

As a consequence, the constant term can be expressed as the equivariant Euler characteristic of \(\hat{\mathcal{O}}_{\mathrm{vir}}\otimes\mathcal{O}(\vec{\mathcal{M}})\) on \(X^{\vee}\). This generalizes the analysis of [11], where the case of \(G = GL(2)\) is considered. The proof of Theorem 1 proceeds in two steps. First, we establish filtrations on both sides and explicitly compute the associated graded; we then compare the extensions. The filtration on the right-hand side comes from the decomposition of \(L\) into the attractors of the \(n+1\) fixed points. The filtration on \(\hat{H}^{*}_{c}(QM\vec{N})\) is provided by a Białynicki-Birula decomposition of \(QM\vec{N}\) into attractor orbits \(\mathrm{Attr}(k)\), \(k = 1, \ldots, n+1\). Importantly, to set up the filtration, we use the purity of the cohomology, which follows from the smoothness of the attractors. Note that the entire stack \(QM\vec{N}\) need not be smooth.

1.2.2 Nontrivial character↩︎

The construction generalizes from Eisenstein series to \(L\)-functions, the latter corresponding to cohomology of quasimaps with coefficients in local systems. We consider rank-one local systems \(\chi\) on \(QM\vec{N}_{\vec{\mathcal{M}}}\) pulled back from \(\mathrm{Pic}(C)\) by the natural map 11 . The local systems on \(\mathrm{Pic}(C)\) are prescribed by characters of \(\mathrm{Pic}(C)(\mathbb{F}_{q})\) and correspond, by geometric class field theory, to local systems \(\chi\) on the curve. On the dual side this gives an action of the geometric fundamental group \(\pi_{1}^{g}\) on the \(A_{n}\)-surface and its resolution. The scheme-theoretic fixed locus of the singular surface is the nonreduced thick point \[\operatorname{Spec}\overline{\mathbb{Q}}_l[\eta]/(\eta^{n+1}),\] whereas the fixed locus \((X^{\vee})^{\pi_{1}^{g}}\) of the resolved surface consists of \(n+1\) reduced points. Similarly to the trivial-character case, one can define a sheaf \(\hat{\mathcal{O}}_{\mathrm{vir}, \chi}\) on the fixed locus, a centered cohomology \(\hat{H}^{*}_{c}(QM\vec{N}, \chi)\), and an algebra \(\mathcal{A}_{\chi}\) of modification operators acting on the cohomology.

Let \(\mathcal{K}_{C}\) denote the canonical bundle of \(C\), and write \(\chi_C(\mathcal{F})= h^0(C,\mathcal{F})-h^1(C,\mathcal{F})\) for the Euler characteristic of a coherent sheaf \(\mathcal{F}\) on \(C\).

In any case, the associated graded of the Białynicki-Birula filtration on \(\hat{H}^{*}_{c}(QM\vec{N},\chi)\) matches \(H^{*}_{L}((X^{\vee})^{\pi_{1}^{g}}, \hat{\mathcal{O}}_{\mathrm{vir},\chi}\otimes \mathcal{O}(\vec{\mathcal{M}}))\); the conditions 1 are exactly the obstructions to splitting this filtration.

1.3 Organization↩︎

Section 2 defines the derived stack \(QM\vec{N}_{\vec{\mathcal{M}}}\) and computes its tangent complex. We then introduce the cohomology of \(QM\vec{N}\) with coefficients in a local system.

Section 3 constructs the modification correspondence and the operators \(e\langle\alpha\rangle\), \(f\langle\beta\rangle\). We prove the commutator and Casimir relations satisfied in the algebra \(\mathcal{A}_\chi\) for an arbitrary character \(\chi\).

Section 4 interprets \(\mathcal{A}\) geometrically: the degree-two part is the algebra of regular functions on the \(A_n\)-surface, the degree-one part is a Clifford algebra, and the degree-zero part is described by first-order differential operators.

Section 5 constructs the resolution \(X^\vee\) by gluing its affine charts. We describe the chart algebras in terms of the fixed components of the quasimap stack.

Section 6 computes the local cohomology of \(X^\vee\) with coefficients in \(\hat{\mathcal{O}}_{\mathrm{vir}}\otimes\mathcal{O}(\vec{\mathcal{M}})\).

Section 7 compares this local cohomology with \(H_c^*(QM\vec{N}_{\vec{\mathcal{M}}},\chi)\) for both the trivial and nontrivial cases. We first match the associated graded pieces and then analyze the extension problem.

Acknowledgements↩︎

I am grateful to Andrei Okounkov for suggesting this problem and for many illuminating discussions. I also thank Tommaso Botta, Ivan Danilenko, Andres Nunez, Che Shen, and Irina Sizova for exciting conversations.

2 The stack of quasimaps↩︎

In this section we introduce the moduli stack \(QM\vec{N}\) of quasimaps. We define its derived structure and compute the tangent complex, then introduce a local system and define a shift for its cohomology.

2.1 Notation and conventions↩︎

Let \(G = GL(n+1)\) and let \(T \subset G\) be the diagonal torus. Fix a geometrically smooth projective curve \(C\) over \(\mathbb{F}_{q}\) of genus \(g\), an \(\mathbb{F}_{q}\)-point \(p\in C\), and a tuple of line bundles \[\vec{\mathcal{M}} = (\mathcal{M}_{1}, \ldots, \mathcal{M}_{n+1}) \in \mathrm{Bun}_{T}(C), \qquad c_{1}(\mathcal{M}_{k}) = \mu_k, \qquad \deg(\mathcal{M}_{k}) = m_{k}.\] The point \(p\) gives an identification \(\mathrm{Pic}_{d}(C) \cong \mathrm{Pic}_{0}(C)\), used throughout. Fix an embedding \(\mathbb{F}_{q} \subset \overline{\mathbb{F}}_{q}\), and write \(C_{\overline{\mathbb{F}}_{q}}\) for the corresponding base change and \(\bar{p}\) for the induced geometric point. For a scheme or stack \(X\) over \(\mathbb{F}_{q}\) we write \(H^{*}(X)\) and \(H^{*}_{c}(X)\) (with coefficients in an understood local system) for the \(l\)-adic étale cohomology \(H^{*}_{\mathrm{et}}(X_{\overline{\mathbb{F}}_{q}})\) and compactly supported cohomology \(H^{*}_{\mathrm{et},c}(X_{\overline{\mathbb{F}}_{q}})\) respectively. We fix a square root of \(q\), \(q^{1/2} \in \overline{\mathbb{Q}}_{l}\), and use it to define the Tate twist \(\boldsymbol{(}\frac{1}{2}\boldsymbol{)}\) on Frobenius modules.

Let \(\mathbf{p} = p_{*}1 \in H^{2}(C)\) denote the class of the point \(p\). Fix a basis \[\begin{align} \gamma_{1}, \ldots, \gamma_{2g} \in H^{1}_{c}(C) = H^{1}_{c}(C, \overline{\mathbb{Q}}_{l}), \end{align}\] satisfying \[\begin{align} \gamma_{g + i} = \gamma_{i}^{\vee}, \qquad i = 1, \ldots, g, \end{align}\] where the dual basis \(\{ \gamma_{i}^{\vee} \}_{i = 1}^{2g}\) to \(\{ \gamma_{i} \}_{i = 1}^{2g}\) is defined by \[\begin{align} \gamma_{i} \cup \gamma_{j}^{\vee} = \delta_{ij}\mathbf{p}. \end{align}\] For a rank \(1\) local system \(\chi_{0}\) on \(C\) we also fix a basis \[\begin{align} \gamma_{1}^{\chi}, \ldots, \gamma_{2g-2}^{\chi} \in H^{1}_{c}(C, \chi^{n+1}_{0}) \end{align}\] and denote the dual basis by \[\begin{align} \gamma_{1}^{\chi^{-1}}, \ldots, \gamma_{2g-2}^{\chi^{-1}} \in H^{1}_{c}(C, \chi^{-n-1}_{0}). \end{align}\] We write \[\Theta = \sum_{i = 1}^{g} \gamma_{i}\, \gamma_{i}^{\vee} \in H^{2}(\mathrm{Pic}_{0}(C))\] for the theta divisor, and \[\begin{align} \tau = c_{1}(TC) \in H^{2}(C) \end{align}\] for the first Chern class of the tangent bundle of the curve.

Consider the projection to the Picard variety \[\label{piLmap} \pi_{\mathcal{L}}\colon QM\vec{N}_{d} \to \mathrm{Pic}_{-d}(C) \cong \mathrm{Pic}_{0}(C), \qquad (\mathcal{V}^{\bullet}, \mathcal{L}, s) \mapsto \mathcal{L}.\tag{2}\] With a slight abuse of notation, we use the same letter \(\mathcal{L}\) for the universal line bundle (the pullback of the universal line bundle from \(C \times \mathrm{Pic}(C)\) under \(\mathrm{id}_{C} \times \pi_{\mathcal{L}}\)), and \(\mathcal{V}\) for the universal vector bundle on \(C \times QM\vec{N}\).

2.2 The stack \(QM\vec{N}\)↩︎

We let \[QM\vec{N} = QM\vec{N}_{\vec{\mathcal{M}}}(C) = \{ (\mathcal{V}^{\bullet}, \mathcal{L}, s)\}\] denote the moduli stack of quasimaps. The degree of a quasimap \((\mathcal{V}^{\bullet}, \mathcal{L}, s)\) is \(d \overset{\mathrm{def}}{=} -\deg \mathcal{L}\); \(QM\vec{N}\) decomposes into connected components \(QM\vec{N}_{d}\), \(d \in \mathbb{Z}\). The stack \(QM\vec{N}\) is algebraic with unipotent stabilizers. Indeed, the projection onto the first two components maps \(QM\vec{N}\) to the scheme \(\{\mathcal{V}^{\bullet}\} \times \mathrm{Pic}(C)\), where \(\{\mathcal{V}^{\bullet}\}\) is the affine space of successive extensions. A fiber of this map over \((\mathcal{V}^{\bullet}, \mathcal{L})\), if nonempty, is the quotient of \(H^{0}(\mathcal{V} \otimes \mathcal{L}^{-1}) \setminus \{0\}\) by the automorphism group \(\mathrm{Aut}(\mathcal{V}^{\bullet}) \times \mathrm{Aut}(\mathcal{L})\), which acts on \(s\) by unipotent transformations and scalars, respectively. In other words, the fiber is the quotient of \(\mathbb{P}\left(H^{0}(\mathcal{V} \otimes \mathcal{L}^{-1})\right)\) by the unipotent group \(\mathrm{Aut}(\mathcal{V}^{\bullet})\), so any stabilizer, being a subgroup, is unipotent.

2.3 Derived structure↩︎

The algebraic stack \(QM\vec{N}\) admits a natural quasi-smooth derived enhancement. In this subsection we define the derived structure and compute the tangent complex.

For the quiver \(Q\) \[\label{quiver} \vcenter{\tikzset{vert/.style={draw, circle, minimum size=7.5mm, inner sep=0pt, font=}, midarrow/.style={postaction={decorate, decoration={markings, mark=at position 0.6 with {\arrow{>}}}}}}\begin{tikzcd}[column sep=small, row sep=small] |[vert]| 1 & |[vert]| 2 & {} & {} & |[vert]| n{+}1 \\ &&&& |[vert]| 1 \arrow[from=1-1, to=1-2, no head, midarrow] \arrow[from=1-2, to=1-3, no head, midarrow] \arrow[dotted, no head, from=1-3, to=1-4] \arrow[from=1-4, to=1-5, no head, midarrow] \arrow[from=2-5, to=1-5, no head, midarrow] \end{tikzcd}}\tag{4}\] with the specified dimension vector \(\mathbf{d}\), consider the moduli stack of \(Q\)-representations over \(\mathbb{F}_{q}\) \[\begin{align} \label{repmoduli} \mathfrak{M}_{\mathbf{d}} = \mathrm{Rep}_{Q}(\mathbf{d})/GL_{\mathbf{d}}, \end{align}\tag{5}\] where \[\begin{align} \mathrm{Rep}_{Q}(\mathbf{d}) = \mathrm{Hom}( \mathbb{A}^{1}_{\mathbb{F}_{q}} ,\mathbb{A}^{n+1}_{\mathbb{F}_{q}} ) \times \prod\limits_{k =1}^{n}\mathrm{Hom}(\mathbb{A}^{k}_{\mathbb{F}_{q}} , \mathbb{A}^{k+1}_{\mathbb{F}_{q}} ) \end{align}\] is the affine space of all representations and \[\begin{align} GL_{\mathbf{d}} = \left( GL_{1} \times \prod\limits_{k = 1}^{n+1} GL_{k} \right)/\mathbb{F}_{q} \end{align}\] is the gauge group. There is an open substack \[\begin{align} \mathfrak{M}_{\mathbf{d}}^{\zeta} \subset \mathfrak{M}_{\mathbf{d}} \end{align}\] defined by the stability condition \(\zeta\), which requires all maps in \(\mathrm{Rep}_{Q}(\mathbf{d})\) to be injective. The mapping stack \[\begin{align} \mathrm{Map}(C, \mathfrak{M}_{\mathbf{d}}) \end{align}\] parametrizes the data of vector bundles \((\mathcal{V}^{(1)}, ..., \mathcal{V}^{(n+1)}, \mathcal{L}), \; \mathrm{rk} \, \mathcal{V}^{(k)} = k,\;\; \mathrm{rk} \, \mathcal{L} = 1\) with sheaf maps \(\mathcal{V}^{(k)} \overset{i_{k}}{\rightarrow} \mathcal{V}^{(k+1)}, \; \mathcal{L} \overset{s}{\rightarrow} \mathcal{V}^{(n+1)} = \mathcal{V}\). Consider an open substack of the mapping stack \[\begin{align} \mathrm{Map}(C, \mathfrak{M}_{\mathbf{d}})^{\zeta} \subset \mathrm{Map}(C, \mathfrak{M}_{\mathbf{d}}) \end{align}\] which is defined by the condition that the generic point maps to \(\mathfrak{M}_{\mathbf{d}}^{\zeta}\). Equivalently, this condition means that the sheaf maps \(i_{k},\, k = 1, ..., n\) and \(s\) are injective. We can then consider a smaller open substack \[\begin{align} \mathrm{Map}(C, \mathfrak{M}_{\mathbf{d}})^{\zeta, \circ} \subset \mathrm{Map}(C, \mathfrak{M}_{\mathbf{d}})^{\zeta} \end{align}\] defined by the condition that the inclusions \(\mathcal{V}^{(k)} \rightarrow \mathcal{V}^{(k+1)}\) have torsion-free quotients. Taking these quotients gives a map \[\begin{align} \pi_{T} \colon \mathrm{Map}(C, \mathfrak{M}_{\mathbf{d}})^{\zeta, \circ} \rightarrow \mathrm{Pic}(C)^{n+1}. \end{align}\] Then the stack \(QM\vec{N}_{\vec{\mathcal{M}}}\) is the fiber of this map over the point \(\vec{\mathcal{M}} \in \mathrm{Pic}(C)^{n+1}\). Therefore, to define a derived structure on \(QM\vec{N}\) one can consider the natural derived structure on the mapping stack and on its open substack, and then take the derived fiber. Let us compute the corresponding tangent complex. With a slight abuse of notation, we use \(\mathcal{V}^{\bullet}, \mathcal{L}, s\) for the universal successive extensions, line bundle and sheaf map, respectively, on \[\begin{align} C \times \mathrm{Map}(C, \mathfrak{M}_{\mathbf{d}})^{\zeta, \circ}. \end{align}\] By the quotient representation 5 , the tangent complex of the moduli stack of \(Q\)-representations is \[\begin{align} &\mathbb{T}\mathfrak{M}_{\mathbf{d}} = \left[ \underline{\mathrm{Lie}(GL_{\mathbf{d}})} \rightarrow \underline{\mathrm{Rep}_{Q}(\mathbf{d})}\right][1]\\ &(u, u_1 , ..., u_{n+1}) \mapsto (u_{n+1} a - a u, u_2 a_1 - a_{1} u_1 , ..., u_{n+1}a_{n} - a_n u_{n}), \end{align}\] where \((a, a_{1}, ..., a_{n})\) is the universal section of the trivial bundle \(\underline{\mathrm{Rep}_{Q}(\mathbf{d})}\). Let \[\mathrm{ev} \colon C \times \mathrm{Map}(C, \mathfrak{M}_{\mathbf{d}})^{\zeta, \circ} \rightarrow \mathfrak{M}_{\mathbf{d}}\] be the evaluation map and let \(p\) be the projection onto the second factor. Then the tangent complex of the mapping stack is \[\begin{align} \mathbb{T} \mathrm{Map}(C, \mathfrak{M}_{\mathbf{d}})^{\zeta, \circ} &= p_{*}\mathrm{ev}^{*} \mathbb{T}\mathfrak{M}_{\mathbf{d}} \nonumber\\ &= p_{*}\left[\mathcal{E}\!{nd}(\mathcal{L}) \oplus \bigoplus_{k = 1}^{n+1} \mathcal{E}\!{nd}(\mathcal{V}^{(k)}) \rightarrow \mathcal{H}\!{om}(\mathcal{L}, \mathcal{V}) \oplus \bigoplus\limits_{k = 1}^{n} \mathcal{H}\!{om}(\mathcal{V}^{(k)}, \mathcal{V}^{(k+1)})\right][1] \end{align}\]

Similarly, the pullback of the tangent complex of \(\mathrm{Pic}(C)^{n+1}\) under \(\pi_{T}\) is given by \[\begin{align} \label{tpic} \pi_{T}^{*}\mathbb{T}\mathrm{Pic}(C)^{n+1} &= p_{*} \left[ \bigoplus_{k = 1}^{n+1}\mathcal{E}\!{nd}(\mathcal{V}^{(k)}/\mathcal{V}^{(k-1)}) \right][1] \nonumber \\ &\overset{\mathrm{qis}}{\cong} p_{*}\left[ \bigoplus_{k = 1}^{n+1} \mathcal{H}\!{om}(\mathcal{V}^{(k)}, \mathcal{V}^{(k)}/\mathcal{V}^{(k-1)}) \rightarrow \bigoplus_{k = 2}^{n+1} \mathcal{H}\!{om}(\mathcal{V}^{(k-1)}, \mathcal{V}^{(k)}/\mathcal{V}^{(k-1)})\right][1]. \end{align}\tag{6}\]

We have a distinguished triangle \[\mathbb{T}_{\pi_{T}} \rightarrow \mathbb{T} \mathrm{Map}(C, \mathfrak{M}_{\mathbf{d}})^{\zeta, \circ} \rightarrow \pi_{T}^{*}\mathbb{T}\mathrm{Pic}(C)^{n+1} \rightarrow \mathbb{T}_{\pi_{T}}[1],\] which in representation 6 is given by the natural projection. Since this projection is component-wise epimorphic, the tangent complex of \(\pi_{T}\) can be computed simply by taking kernels: \[\begin{align} \mathbb{T}_{\pi_{T}} = p_{*}\left[ \mathcal{E}\!{nd}(\mathcal{L}) \oplus \bigoplus_{k = 1}^{n+1} \mathcal{H}\!{om}(\mathcal{V}^{(k)}, \mathcal{V}^{(k-1)}) \rightarrow \mathcal{H}\!{om}(\mathcal{L}, \mathcal{V}) \oplus \bigoplus_{k = 1}^{n}\mathcal{H}\!{om}(\mathcal{V}^{(k)}, \mathcal{V}^{(k)}) \right] [1]. \end{align}\] Consider the sheaf \[\begin{align} \mathcal{E}\!{nd}(\mathcal{V}^{\bullet}) = \{\psi \in \mathcal{E}\!{nd}(\mathcal{V}): \psi(\mathcal{V}^{(k)}) \subset \mathcal{V}^{(k-1)}\}. \end{align}\] It is easy to see that it fits into the short exact sequence \[\begin{align} 0 \rightarrow \mathcal{E}\!{nd}(\mathcal{V}^{\bullet}) \rightarrow \bigoplus_{k = 1}^{n+1} \mathcal{H}\!{om}(\mathcal{V}^{(k)}, \mathcal{V}^{(k-1)}) &\rightarrow \bigoplus_{k = 1}^{n}\mathcal{H}\!{om}(\mathcal{V}^{(k)}, \mathcal{V}^{(k)})\rightarrow 0\\ (\psi_{k})_{k = 1}^{n+1} &\mapsto (\psi_{k+1}i_{k} - i_{k}\psi_{k})_{k=1}^{n+1}, \end{align}\] where the first map is given by restriction to the subbundles. Using it, we can rewrite the tangent complex of \(\pi_{T}\) as \[\label{eq:TpiT} \mathbb{T}_{\pi_{T}} = p_{*}\left[ \mathcal{E}\!{nd}(\mathcal{L}) \oplus \mathcal{E}\!{nd}(\mathcal{V}^{\bullet}) \rightarrow \mathcal{H}\!{om}(\mathcal{L}, \mathcal{V}) \right] [1].\tag{7}\] Then the tangent complex of \(QM\vec{N}\) is the restriction of 7 to the fiber over \(\vec{\mathcal{M}}\). We see that it is a perfect complex of Tor-amplitude \([-1, 1]\), hence \(QM\vec{N}\) is quasi-smooth [18] over \(\mathbb{F}_{q}\).

2.4 Virtual tangent space↩︎

It follows from 7 that for a point \((\mathcal{V}^{\bullet}, \mathcal{L}, s)\) in \(QM\vec{N}\), the tangent complex is given by the hypercohomology groups of the complex \[\begin{align} C^{\bullet} = \Big[ \overset{-1}{\mathcal{E}\!{nd}(\mathcal{L}) \oplus \mathcal{E}\!{nd}(\mathcal{V}^{\bullet})} &\overset{\Delta_{s}}{\longrightarrow} \overset{0}{\mathcal{H}\!{om}(\mathcal{L}, \mathcal{V})} \Big] \\ (\varphi, \psi) &\mapsto \psi \circ s - s \circ \varphi. \end{align}\] The hypercohomology groups \(\mathbb{H}^{-1}, \mathbb{H}^{0}, \mathbb{H}^{1}\) are referred to as infinitesimal automorphisms, deformations, and obstructions. One can easily compute the infinitesimal automorphisms \[\begin{align} \mathbb{H}^{-1}(C^{\bullet}) &= \mathrm{Ker}\left( \mathrm{End}(\mathcal{L}) \oplus \mathrm{End}(\mathcal{V}^{\bullet}) \overset{\Delta_{s}}{\longrightarrow} \mathrm{Hom}(\mathcal{L}, \mathcal{V})\right), \end{align}\] and, using Serre duality, the obstructions \[\begin{align} \mathbb{H}^{1}(C^{\bullet}) &= \mathrm{Coker}\left( H^{1}\left(\mathcal{E}\!{nd}(\mathcal{L})\right) \oplus H^{1}\left(\mathcal{E}\!{nd}(\mathcal{V}^{\bullet})\right) \overset{\Delta_{s}}{\longrightarrow} \mathrm{Ext}^{1}(\mathcal{L}, \mathcal{V})\right) \nonumber\\ &=\mathrm{Ker}\left( H^{0}(\mathcal{V}^{\vee} \otimes \mathcal{L} \otimes \mathcal{K}_{C}) \overset{\Delta_{s}^{\vee}}{\longrightarrow} H^{0}\left(\mathcal{E}\!{nd}(\mathcal{L})\otimes \mathcal{K}_{C}\right) \oplus H^{0}\left(\mathcal{E}\!{nd}(\mathcal{V}^{\bullet})^{\vee} \otimes \mathcal{K}_{C} \right)\right)^{\vee}, \label{obstructions} \end{align}\tag{8}\] where \[\begin{align} \label{Deltaadj} \Delta_{s}^{\vee}(\alpha) = (-\alpha \circ s,\; s \circ \alpha). \end{align}\tag{9}\] We will compute later that at the point \((\mathcal{V}^{\bullet}_{\mathrm{fix}}, \mathcal{M}_{k}(-D), s)\): \[\begin{align} &\mathbb{H}^{-1}(C^{\bullet}) = \bigoplus_{l < r \neq k} H^{0}(\mathcal{M}_{l} \otimes \mathcal{M}_{r}^{-1}),\\ &\mathbb{H}^{1}(C^{\bullet}) = \bigoplus_{l > k} H^{1}(\mathcal{M}_{l}\otimes \mathcal{M}_{k}^{-1}(D)). \end{align}\] The virtual tangent space at a point \((\mathcal{V}^{\bullet}, \mathcal{L}, s)\in QM\vec{N}\) is given by the \(K\)-theoretic difference \[\begin{align} T^{\mathrm{vir}}_{(\mathcal{V}^{\bullet}, \mathcal{L}, s)} QM\vec{N} &= \mathbb{H}^{0}(C^{\bullet}) - \mathbb{H}^{-1}(C^{\bullet}) - \mathbb{H}^{1}(C^{\bullet}) = H^{*}\left(\mathcal{H}\!{om}(\mathcal{L}, \mathcal{V})\right) - H^{*} \bigl( \mathcal{E}\!{nd}(\mathcal{L}) \oplus \mathcal{E}\!{nd}(\mathcal{V}^{\bullet}) \bigr)\nonumber \\ &=\sum\limits_{l = 1}^{n+1}\mathrm{Ext}^{\bullet}(\mathcal{L}, \mathcal{M}_{l})\otimes \mathcal{O}_{QM\vec{N}}(1) \big|_{(\mathcal{V}^{\bullet}, \mathcal{L}, s)} -\mathrm{Ext}^{\bullet}(\mathcal{L}, \mathcal{L}) - \sum\limits_{l < r}\mathrm{Ext}^{\bullet}(\mathcal{M}_{r}, \mathcal{M}_{l}). \label{TvirQMN} \end{align}\tag{10}\]

As a corollary, we obtain the virtual dimension of \(QM\vec{N}_{d}\): \[\begin{align} \mathrm{virdim}\, QM\vec{N}_{d} = - \sum\limits_{l < r}(m_{l} - m_{r}+1-g) + \sum\limits_{l = 1}^{n+1}(m_l + d + 1 - g) + g-1. \end{align}\]

2.5 The centered cohomology↩︎

Consider the map \[\begin{align} \label{Ns} \det\colon QM\vec{N} &\rightarrow \mathrm{Pic}(C),\\ (\mathcal{V}^{\bullet}, \mathcal{L}, s) &\mapsto \det(\mathcal{V}/\mathcal{L} \otimes \mathcal{L}^{-1}) = \mathcal{M}_{1} \otimes \cdots \otimes \mathcal{M}_{n+1} \otimes \mathcal{L}^{-n-1}. \nonumber \end{align}\tag{11}\] Let \(\chi_{0}\) be a one-dimensional local system on \(C\) with a fixed trivialisation \(\chi_{0}\big|_{\bar{p}} \cong \overline{\mathbb{Q}}_{l}\). The action of the geometric fundamental group \(\pi^{g}_{1} = \pi_{1, \mathrm{et}}(C_{\overline{\mathbb{F}}_{q}}, \bar{p})\) on the fiber of \(\chi_{0}\) over \(\bar{p}\) is given by a character \[\begin{align} \chi_{0}\colon \pi^{g}_{1} \rightarrow \overline{\mathbb{Q}}_{l}^{\times}. \end{align}\] Let \(A^{\vee} \overset{a}{\cong} \overline{\mathbb{Q}}_{l}^{\times}\) be a one-dimensional torus with coordinate \(a\). We denote by \(\chi\) the \(A^{\vee}\)-equivariant local system on \(C\) obtained from \(\chi_{0}\) by the fiberwise \(A^{\vee}\)-action given by multiplication by \(a^{-1}\). Accordingly, the group \(\pi_{1}^{g} \times A^{\vee}\) acts on the fiber of \(\chi\) over \(\bar{p}\) by the character \[\begin{align} \chi = \chi_{0}\,a^{-1}\colon \pi_{1}^{g} \times A^{\vee} \rightarrow \overline{\mathbb{Q}}_{l}^{\times}. \end{align}\]

The local system \(\chi_{0}\) on \(C\) also gives rise to a one-dimensional local system (still denoted \(\chi_{0}\)) on \(\mathrm{Pic}_{0}(C)\). Pulling it back under the map \[\begin{align} \mathrm{Pic}(C) &\rightarrow \mathrm{Pic}_{0}(C),\\ x &\mapsto x \otimes \mathcal{O}(-p)^{\otimes \deg x}, \nonumber \end{align}\] we obtain a local system on \(\mathrm{Pic}(C)\), which we still call \(\chi\). We make \(\chi\) on \(\mathrm{Pic}\) into an \(A^{\vee}\)-equivariant local system by declaring \(A^{\vee}\) to act on the fibers over \(\mathrm{Pic}_{d}(C)\) with multiplication by \(a^{-d}\). Pulling back under 11 yields a local system on \(QM\vec{N}\), which we also denote \(\chi\). The space \(QM\vec{N}\) and its cohomology carry a geometric Frobenius action; we denote by \(\mathrm{Fr} \cong \mathbb{Z}\) the group generated by this operator.

In order to make the isomorphism of Theorem 1 cohomologically graded and \(\mathrm{Fr}\times A^{\vee}\)-equivariant, we introduce a cohomological shift and \(\mathrm{Fr} \times A^{\vee}\)-twist \[\begin{align} \label{Hcent} \hat{H}^{*}_{c}(QM\vec{N}_{d}, \chi) = H^{*}_{c}(QM\vec{N}_{d}, \chi)[[\mathrm{virdim} \,QM\vec{N}_{d}]] \otimes a^{(n+1)(g-1)}, \end{align}\tag{12}\] where \[\begin{align} [[i]] = [i]\boldsymbol{(}i/2\boldsymbol{)} \end{align}\] denotes the pure combination of the cohomological shift and Tate twist, and \(a^{(n+1)(g-1)}\) stands for the one-dimensional \(A^{\vee}\)-representation given by the indicated character.

A quasimap of degree \(d=-\deg\mathcal{L}\) is sent by 11 to \(\det(\mathcal{V}/\mathcal{L} \otimes \mathcal{L}^{-1})\), which has degree \(\sum_{l = 1}^{n+1}(d + m_{l})\). We will show later that the cohomology \(H^{*}_{c}(QM\vec{N}, \chi)\) is pure, so the Frobenius eigenvalues on \(H^{i}_{c}\) are equal to \(\pm q^{i/2}\). Therefore, up to sign, \(\mathrm{Fr} \times A^{\vee}\) acts on the cohomology of quasimaps by \[\begin{align} \label{A-action} \mathrm{Fr} \times A^{\vee}\colon &\hat{H}^{i}_{c}(QM\vec{N}_{d}, \chi) \rightarrow \hat{H}^{i}_{c}(QM\vec{N}_{d}, \chi),\\ &\alpha \mapsto \# q^{i/2}(q^{1/2}a)^{ - \sum\limits_{l = 1}^{n+1} m_l + d + 1 - g}\, \alpha, \nonumber \end{align}\tag{13}\] where the constant \(\#\) is equal to \[\# = (q^{1/2})^{1 - g + \sum\limits_{l < r}(m_l - m_r + 1 - g)}.\] The factor \[\begin{align} q^{\displaystyle -\rho_{P}(\vec{\mathcal{M}}\otimes\mathcal{L}^{-1})}, \;\;\; \rho_{P}(\vec{\mathcal{M}}\otimes\mathcal{L}^{-1}) \mathrel{\vcenter{:}}= \frac{1}{2} \sum\limits_{l = 1}^{n+1} \chi_C(\mathcal{M}_{l} \otimes \mathcal{L}^{-1}) \end{align}\] in 13 corresponds to a shift by the half-sum of roots in the unipotent radical of \(P\). Such a shift is a general feature in the theory of Eisenstein series. The centering 12 can be viewed as a geometric way to introduce it.

3 Algebra of correspondences↩︎

The space \(\hat{H}_{c}^{*}(QM\vec{N}, \chi)\) carries a family of natural operators labelled by cohomology classes on \(C\). In this section we construct them from a single closed-immersion correspondence on \(QM\vec{N}\), compute the resulting pull–push identities in terms of tautological Chern classes, and derive relations among the corresponding operators on cohomology.

3.1 The modification correspondence↩︎

We introduce a correspondence \(\iota\) on \(C\times QM\vec{N}\), which is a special case of the action of a Coulomb branch [8], [13]–[15]. Then we compute relations in the algebra generated by pull-push operators along \(\iota\).

The map \(\iota\) is not in general a regular embedding, but the pushforward is still defined.

We have the \(A^{\vee}\)-equivariant pullback and pushforward maps (note the unusual cohomological shift due to 12 ): \[\begin{align} \iota^{*}\colon\;& \hat{H}^{i}_{c}(C \times QM\vec{N}_{d+1}, \chi^{-n-1} \boxtimes \chi) \to \hat{H}^{i + n + 1}_{c}(C \times QM\vec{N}_{d}, \overline{\mathbb{Q}}_{l} \boxtimes \chi),\\ &\hat{H}^{i}_{c}(C \times QM\vec{N}_{d+1}, \overline{\mathbb{Q}}_{l} \boxtimes \chi) \to \hat{H}^{i + n + 1}_{c}(C \times QM\vec{N}_{d}, \chi^{n+1}\boxtimes \chi),\\ \iota_{*}\colon\;& \hat{H}_{c}^{i}(C \times QM\vec{N}_{d}, \chi^{n+1} \boxtimes \chi) \to \hat{H}^{i+n+1}_{c}(C \times QM\vec{N}_{d+1}, \overline{\mathbb{Q}}_{l} \boxtimes \chi),\\ &\hat{H}_{c}^{i}(C \times QM\vec{N}_{d}, \overline{\mathbb{Q}}_{l} \boxtimes \chi) \to \hat{H}^{i+n+1}_{c}(C \times QM\vec{N}_{d+1}, \chi^{-n-1} \boxtimes \chi). \end{align}\]

For the following computations we need a formula for the Euler class of \(\mathcal{N}\) and its pullback under \(\iota\) in terms of tautological classes. Applying the splitting principle to the filtration 3 of \(\mathcal{V}\), using \(\mathcal{N} = \mathcal{V} \otimes \mathcal{L}^{-1} \otimes \mathcal{O}_{QM\vec{N}}(1)\) and 15 , we get \[\begin{align} \label{eq:chern-N} c_{n+1}(\mathcal{N}) &= \prod_{l=1}^{n+1}\bigl(\mu_{l}\otimes 1 - \lambda + 1 \otimes \eta\bigr),\\ c_{n+1}(\iota^{*}\mathcal{N}) &= \prod_{l=1}^{n+1}\bigl(\tau \otimes 1 + \mu_{l}\otimes 1 - \lambda + 1 \otimes \eta\bigr). \nonumber \end{align}\tag{16}\]

3.2 Commutator and Casimir relations↩︎

In the following we compute the commutator of \(e\) and \(f\) explicitly in terms of tautological classes.

Let \(\Delta_{C}\colon C \to C\times C\) be the diagonal map.

3.3 The algebra of correspondences↩︎

For any cohomology class on \(C\), the operators \(e, f\) specialise to honest endomorphisms of \(H^{*}_{c}(QM\vec{N}, \chi)\). We record the relations in the algebra generated by these operators.

We use the notation \([\;,\;]\) for the supercommutator \[[e\langle \alpha \rangle, f \langle \beta \rangle] = e\langle \alpha \rangle f \langle \beta \rangle - (-1)^{\deg \alpha \cdot \deg \beta} f \langle \beta \rangle e\langle \alpha \rangle.\] By Definition 5 and ?? , it is given by \[\label{compsup} [e\langle\alpha\rangle, f\langle\beta\rangle] = \int_{C}\psi\cup\alpha\cup\beta.\tag{19}\] We collect the supercommutator relations in Appendix 8.

4 The symplectic dual variety↩︎

The aim of this section is to construct the action of a certain Clifford algebra over an affine singular surface \(X_{0}^{\vee}\) on the cohomology of quasimaps. This is an example of the action of a Coulomb branch.

4.1 Action of the \(A_{n}\)-surface coordinate ring↩︎

Let \(R\) be the commutative ring \[\begin{align} R = \overline{\mathbb{Q}}_{l}[x, y, \eta]/(xy - \eta^{n+1}). \end{align}\] Here, with a slight abuse of notation, \(\eta\) denotes a formal variable. The spectrum \[X_{0}^{\vee} = \mathrm{Spec} \, R\] is a singular surface of \(A_{n}\)-type, which is the symplectic dual [16], [17] of \(X = T^{*}\mathbb{P}^{n}\). In this situation \(X\) is referred to as the Higgs branch, and \(X_{0}^{\vee}\) is the Coulomb branch. The following proposition is an example of a Coulomb branch action.

The \(\mathrm{Fr} \times A^{\vee}\)-action 13 on the cohomology of quasimaps induces an action on \(\mathcal{A}_{\chi}\). In order for the map ?? to be equivariant, \(\mathrm{Fr} \times A^{\vee}\) has to act on \(R\) by: \[\begin{align} x \mapsto (q^{1/2})^{n+1}a^{-n-1}x, \qquad y \mapsto (q^{1/2})^{n+1}a^{n+1}y, \qquad \eta &\mapsto \eta. \nonumber \end{align}\] The variety \(X_{0}^{\vee}\) has a Poisson structure given by an \(\mathrm{Fr}\times A^{\vee}\)-equivariant map \[\begin{align} \label{Poissonbraket} \{\cdot , \cdot \} : q^{-1} \otimes \bigwedge\nolimits^{2} \Omega_{R} \rightarrow R \qquad \{x, y \} = (n+1)\eta^{n}, \qquad \{x , \eta\} = x, \qquad \{y, \eta\} = -y, \end{align}\tag{20}\] where \(\Omega_{R}\) is the module of Kähler differentials of \(R\) over \(\overline{\mathbb{Q}}_{l}\), and \(q^{-1}\) denotes the one-dimensional \(\mathrm{Fr}\)-representation with the action by the corresponding character.

In the nontrivial local system case the action of \(R\) simplifies significantly.

It has an interpretation in terms of the action of \(\pi_{1}^{g}\) on \(X_{0}^{\vee}\).

4.2 Action of a Clifford algebra↩︎

Descendants of cohomology classes of degrees two and one generate a noncommutative algebra \(\mathcal{A}^{\geq 1}\). The homomorphism ?? makes it an \(R\)-algebra. In this subsection we consider a natural Clifford algebra over \(R\) and realise it inside \(\mathcal{A}^{\geq 1}\).

Consider the \(R\)-module \[\label{eq:RmoduleE} \mathcal{E} = q^{-1}\otimes \Omega_{R} \otimes H^{1}_{c}(C).\tag{22}\] Let us introduce the notation for the generators of \(\mathcal{E}\): \[\theta^{x}_{i} = d x \otimes \gamma_{i}, \qquad \theta^{y}_{i} = - d y \otimes \gamma_{i}, \qquad i = 1, ..., 2g,\] and use \(\gamma_i\) for the generator \(\gamma_{i} = d \eta \otimes \gamma_{i}\).

Consider a \(\mathrm{Fr}\times A^{\vee}\)-equivariant symmetric pairing \[\begin{align} \label{clifpair} \langle \cdot, \cdot \rangle \colon S^{2}\left( \mathcal{E} \right) \rightarrow R, \end{align}\tag{23}\] which is given by the tensor product of the Poisson bracket 20 with the symplectic intersection form on \(H^{1}_{c}(C)\). Let \(Cl\) be a Clifford algebra \[\begin{align} Cl = \mathrm{Cliff}_{R}(\mathcal{E}, \langle \cdot, \cdot \rangle ). \end{align}\] It acts on the cohomology of quasimaps.

In the nontrivial case \(\chi_{0}^{n+1} \neq 1\) we define an \(R_{\mathrm{fix}}\)-module \[\begin{align} \mathcal{E}_{\chi} = H^{1}\left(\pi_{1}^{g}, q^{-1}\Omega_{R}\otimes R_{\mathrm{fix}}\right), \end{align}\] where the \(\pi_{1}^{g}\times A^{\vee}\)-action on \(\Omega_{R}\) and its restriction to the fixed locus are induced from the action 21 on \(X_{0}^{\vee}\). Let us introduce the notation for the generators of \(\mathcal{E}_{\chi}\): \[\begin{align} \theta_{\chi, i}^{x} = dx \otimes \gamma_{i}^{\chi}, \qquad \theta_{\chi, i}^{y} = - dy \otimes \gamma_{i}^{\chi^{-1}}, \qquad i = 1, ..., 2g-2, \end{align}\] and again use \(\gamma_{i}\) for the generator \(\gamma_{i}\otimes \eta\). There is an analogous pairing \[\begin{align} \langle \cdot, \cdot \rangle \colon \mathrm{Sym}^{2}\left( \mathcal{E}_{\chi} \right) \rightarrow R_{\mathrm{fix}}, \end{align}\] which allows us to define a Clifford algebra \[\begin{align} Cl_{\chi} = \mathrm{Cliff}(\mathcal{E}_{\chi}, \langle \cdot, \cdot\rangle). \end{align}\] The algebra \(Cl_{\chi}\) plays the role of \(Cl\) in the case of a nontrivial local system.

In Appendix 9 we also identify degree-zero descendants with differential operators on \(X_{0}^{\vee}\).

5 The resolution of the singular surface↩︎

In this section we use the torus action on \(QM\vec{N}\) to produce a compatible family of maps \[\begin{align} \label{A2ktoA2} \mathcal{A}^{2} \rightarrow \mathcal{A}^{(k)2}, \;\;\; k = 1, \ldots, n+1, \end{align}\tag{24}\] to the polynomial algebras in two variables \(\mathcal{A}^{(k)2} \cong \overline{\mathbb{Q}}_{l}[x_k , y_k ]\). This allows us to prove that the map ?? is an isomorphism. The homomorphisms 24 then correspond to birational maps \[\mathrm{Spec}\, \mathcal{A}^{(k)2} \rightarrow X_{0}^{\vee}.\] Gluing them along the transition maps produces a smooth symplectic resolution \[\begin{align} X^{\vee} \rightarrow X^{\vee}_{0}. \end{align}\] Analogously to \(\mathcal{A}^2\), the algebras \(\mathcal{A}^{(k)2}\) have natural modules \(\mathcal{E}_{k}\) equipped with symmetric pairings. The Clifford algebras of \(\mathcal{E}_{k}\) glue to a locally free sheaf of Clifford algebras on \(X^{\vee}\), whose global sections contain \(Cl\) as a subalgebra.

5.1 Torus fixed loci and attracting strata↩︎

We begin by introducing a torus \(T\) acting on \(QM\vec{N}\) through rescaling of the graded pieces \(\mathcal{M}_{k}\), identifying its fixed loci, and computing the virtual normal bundles to the attracting strata. A Białynicki-Birula-type stratification then yields a filtration on \(\hat{H}^{*}_{c}(QM\vec{N})\) whose graded pieces are the cohomologies of the fixed loci.

The torus \(T \cong \mathrm{Aut}(\vec{\mathcal{M}}) \cong \mathrm{Aut}(\mathcal{M}_{1}) \times \cdots \times \mathrm{Aut}(\mathcal{M}_{n+1})\) acts on the space of filtrations \(\{\mathcal{V}^{\bullet}\}\) by \[\begin{align} t(\mathcal{V}^{(k-1)} \hookrightarrow \mathcal{V}^{(k)} \twoheadrightarrow \mathcal{M}_{k}) = (\mathcal{V}^{(k-1)} \hookrightarrow \mathcal{V}^{(k)} \twoheadrightarrow \mathcal{M}_{k} \overset{t_{k}}{\rightarrow} \mathcal{M}_{k}), \end{align}\] where \(t_{k}\) is the coordinate on \(\mathrm{Aut}(\mathcal{M}_{k})\). The associated natural isomorphism \[\begin{align} \rho_{t}\colon \mathcal{V} \rightarrow t\mathcal{V} \end{align}\] lifts the \(T\)-action to \(QM\vec{N}\) via \[\begin{align} \label{Taction} &T \ni t \colon QM\vec{N} \rightarrow QM\vec{N},\\ &(\mathcal{V}^{\bullet}, \mathcal{L}, s) \mapsto (t\mathcal{V}^{\bullet}, \mathcal{L}, \rho_{t} \circ s). \nonumber \end{align}\tag{25}\] The set of fixed points decomposes into \(n + 1\) components \[\begin{align} QM\vec{N}^{T} = \bigsqcup\limits_{k = 1}^{n+1}QM\vec{N}(k), \end{align}\] where \[\begin{align} QM\vec{N}(k) = \{\big( &\mathcal{V}^{\bullet}_{\mathrm{fix}} = (\mathcal{V}^{(l)} = \mathcal{M}_{1} \oplus \cdots \oplus \mathcal{M}_{l})_{l = 1}^{n+1}, \mathcal{L} = \mathcal{M}_{k}(- D), \\ &s\colon \mathcal{M}_{k}(- D) \rightarrow \mathcal{M}_{k} \subset \mathcal{V} \big) : D \in \mathrm{Div}_{+}(C) \cong S^{\bullet}C\}. \nonumber \end{align}\] Choose the one-dimensional subtorus \[\begin{align} \label{1torus} \mathbb{G}_{m} &\hookrightarrow T\\ &z \mapsto \left(t_{k} = z^{n+1 - k} \right)_{k = 1}^{n+1}. \nonumber \end{align}\tag{26}\] The set of fixed points for \(\mathbb{G}_{m}\) is the same as for \(T\), \[\begin{align} QM\vec{N}^{\mathbb{G}_{m}} \cong QM\vec{N}^{T}, \end{align}\] so we obtain a Białynicki-Birula-type decomposition \[\begin{align} \label{attrdecomp} QM\vec{N} = \bigsqcup\limits_{k = 1}^{n+1} \mathrm{Attr}(k) \end{align}\tag{27}\] with attractor cells \[\begin{align} \mathrm{Attr}(k) = \underset{z \rightarrow 0}{\mathrm{Attr}}(QM\vec{N}(k)) = \{(\mathcal{V}^{\bullet}, \mathcal{L}, s): s(\mathcal{L}) \subset \mathcal{V}^{(k)}\} \setminus \{(\mathcal{V}^{\bullet}, \mathcal{L}, s): s(\mathcal{L}) \subset \mathcal{V}^{(k-1)}\}. \end{align}\]

An important feature of the decomposition 27 is the smoothness of the attracting orbits \(\mathrm{Attr}(k)\)

It follows that \(\mathrm{Attr}(k)\) is a vector bundle stack over the smooth projective variety \(QM\vec{N}(k) \cong S^{\deg\,D}C\), and its cohomology is pure. In particular, this allows us to establish a filtration on the cohomology of the quasimaps.

5.2 Correspondences on fixed loci↩︎

The modification correspondence \(\iota\) preserves the fixed loci, hence acts on their cohomology. In this subsection, analogously to Section 3, we record pull-push identities, define the associated operators \(e^{(k)}, f^{(k)}\), and derive the commutator and Casimir relations they satisfy.

There are \(n+1\) fixed loci \(QM\vec{N}(k)\), \(k = 1, \ldots, n+1\), of the torus action 26 on \(QM\vec{N}\). The correspondence \(\iota\) preserves each of them, and we consider the closed immersion \[\begin{align} \iota^{(k)}\colon C \times QM\vec{N}_{d}(k) \rightarrow C \times QM\vec{N}_{d+1}(k), \end{align}\] the restriction of \(\iota\) to the corresponding fixed locus. Each component \(QM\vec{N}_{d}(k)\) is a symmetric product of the curve \(C\), and \(\iota^{(k)}\) “adds a point”.

It defines pullbacks and pushforwards (analogously to Lemma 2) on cohomology: \[\begin{align} & \iota^{(k)*}\colon \hat{H}^{i}_{c}(C \times QM\vec{N}_{d+1}(k), \chi^{-n-1}\boxtimes \chi) \rightarrow \hat{H}^{i + n + 1}_{c}(C \times QM\vec{N}_{d}(k), \overline{\mathbb{Q}}_{l}\boxtimes \chi) \\ & \iota^{(k)*}\colon \hat{H}^{i}_{c}(C \times QM\vec{N}_{d+1}(k), \overline{\mathbb{Q}}_{l}\boxtimes \chi) \rightarrow \hat{H}^{i + n + 1}_{c}(C \times QM\vec{N}_{d}(k), \chi^{n+1}\boxtimes \chi) \nonumber \end{align}\] \[\begin{align} &\iota^{(k)}_{*}\colon \hat{H}^{i}_{c}(C \times QM\vec{N}_{d}(k), \chi^{n+1}\boxtimes \chi) \rightarrow \hat{H}^{i-n+1}_{c}(C \times QM\vec{N}_{d+1}(k), \overline{\mathbb{Q}}_{l}\boxtimes \chi)\\ &\iota^{(k)}_{*}\colon \hat{H}^{i}_{c}(C \times QM\vec{N}_{d}(k), \overline{\mathbb{Q}}_{l}\boxtimes \chi) \rightarrow \hat{H}^{i-n+1}_{c}(C \times QM\vec{N}_{d+1}(k), \chi^{-n-1}\boxtimes \chi) \nonumber \end{align}\]

Consider a vector bundle \[\begin{align} \mathcal{N}^{(k)} = \mathcal{V}^{(k)}\otimes\mathcal{L}^{-1}\otimes \mathcal{O}_{QM\vec{N}}(1) \end{align}\] on \(C \times QM\vec{N}(k)\). The quotient bundle \[\begin{align} \mathcal{N}^{(k)}/\mathcal{N}^{(k-1)} \cong \mathcal{M}_{k} \otimes\mathcal{L}^{-1}\otimes \mathcal{O}_{QM\vec{N}}(1) \end{align}\] carries a natural section \(\bar{s}^{(k)}(x)\) obtained by restricting \(s(x)\) to \(C \times QM\vec{N}(k)\), and the image of \(\iota^{(k)}\) is cut out by the equation \(\bar{s}^{(k)}(x) = 0\). Hence \[\begin{align} \label{chernNi} \iota^{(k)}_{*}\iota^{(k)*} = c_{1}(\mathcal{N}^{(k)}/\mathcal{N}^{(k-1)}) \; \cup, \end{align}\tag{29}\] where \(c_{1}(\mathcal{N}^{(k)}/\mathcal{N}^{(k-1)}) \in H^{2}(QM\vec{N}(k))\) is the Euler class of the line bundle \(\mathcal{N}^{(k)}/\mathcal{N}^{(k-1)}\). Analogously to 14 , the virtual normal bundle \(N_{\iota^{(k)}}\) to the image of \(\iota^{(k)}\) is \[\begin{align} N_{\iota^{(k)}} = (\iota^{(k)*} - \mathrm{id})T^{\mathrm{vir}}(C \times QM\vec{N}(k)) = \iota^{(k)*}\mathcal{N}^{(k)}/\mathcal{N}^{(k-1)} \end{align}\] which implies \[\begin{align} \label{cherniNi} \iota^{(k)*}\iota^{(k)}_{*} = c_{1}(\iota^{(k)*}\mathcal{N}^{(k)}/\mathcal{N}^{(k-1)}) \; \cup = c_{1}(\mathcal{N}^{(k)}/\mathcal{N}^{(k-1)} \otimes TC ) \; \cup. \end{align}\tag{30}\] Using 3 , the top Chern class of \(\mathcal{N}^{(k)}/\mathcal{N}^{(k-1)}\) is \[\begin{align} c_{1}(\mathcal{N}^{(k)}/\mathcal{N}^{(k-1)}) = c_{1}(\mathcal{M}_{k} \otimes \mathcal{L}^{-1} \otimes \mathcal{O}_{QM\vec{N}}(1)) = \mu_{k}\otimes 1 - \lambda + 1 \otimes \eta, \end{align}\] and \[\begin{align} c_{1}(\iota^{*}\mathcal{N}^{(k)}/\mathcal{N}^{(k-1)}) = c_{1}(\mathcal{N}^{(k)}/\mathcal{N}^{(k-1)} \otimes TC) = \tau \otimes 1 + \mu_{k}\otimes 1 - \lambda + 1 \otimes \eta. \end{align}\]

The identities 29 and 30 yield the commutation relation \[\begin{align} e^{(k)}f_{2}^{(k)} - f^{(k)}e_{1}^{(k)} = \int\limits_{C} \Delta_{C}^{*}. \end{align}\] This implies the supercommutator identities of descendants, which we collect in Appendix 8.

From 29 and 30 we also obtain a Casimir identity \[\begin{align} \frac{1}{2}(e^{(k)}f_{2}^{(k)} + f^{(k)}e_{1}^{(k)})\Delta_{C*} = - \int\limits_{C} (1 \otimes \eta + \mathbf{p}\otimes 1 (- \deg\,\mathcal{L} + m_{k} + 1 - g) - \sum\limits_{i = 1}^{2g} \gamma_{i} \otimes \gamma_{i}^{\vee}). \end{align}\]

Analogously to Definition 6 we define an algebra of correspondences acting on the fixed loci.

Let us use the notation for the descendants \[\begin{align} x_{k} = e^{(k)}\langle \mathbf{p} \rangle, \;\;\; y_{k} = -f^{(k)}\langle \mathbf{p} \rangle, \;\;\; \end{align}\] \[\begin{align} \theta^{x_k}_{i} = e^{(k)}\langle \gamma_{i} \rangle, \;\;\; \theta^{y_k}_{i} = - f^{(k)}\langle \gamma_{i} \rangle, \;\;\; i = 1, \ldots, 2g, \end{align}\] \[\begin{align} \label{eq:thetaveek} \theta_{i}^{\vee x_k} = e^{(k)}\langle \gamma_{i}^{\vee} \rangle,\;\;\; \theta_{i}^{\vee y_k} = - f^{(k)}\langle \gamma_{i}^{\vee} \rangle, \;\;\; i = 1, \ldots, 2g, \end{align}\tag{32}\] \[\begin{align} \theta^{x_k}_{\chi, i} = e^{(k)} \langle \gamma_{i}^{\chi} \rangle, \qquad \theta^{y_k}_{\chi, i} = - f^{(k)} \langle \gamma_{i}^{\chi^{-1}} \rangle, \qquad i = 1, ..., 2g-2 \end{align}\]

5.3 Algebras of correspondences on the fixed loci↩︎

We now show that the algebra of descendants acting on \(H^{*}_{c}(QM\vec{N}(k), \chi)\) is a Weyl-Clifford superalgebra. Its commutative part is the coordinate ring of \(\pi_{1}^{g}\)-fixed points of the affine chart \(U_{k} \cong \mathbb{A}^{2}\), and the generators of degree one and two assemble into a Clifford algebra.

We briefly review the Weyl-Clifford algebra. Suppose \(V = V^{\mathrm{even}} \oplus V^{\mathrm{odd}}\) is a finite-dimensional super vector space over \(\overline{\mathbb{Q}}_{l}\), equipped with a nondegenerate bilinear form \(\langle \cdot, \cdot \rangle\), satisfying \[\langle u , v \rangle = (-1)^{\mathrm{deg}\,u \, \mathrm{deg}\,v}\langle v , u\rangle\] for homogeneous \(u, v \in V\). The actual example of interest for us is \[\label{V61HH} V = H^{*}_{c}(C, \chi^{n+1}) \oplus H^{*}_{c}(C, \chi^{n+1})^{*} \cong H^{*}_{c}(C, \chi^{n+1}) \oplus H^{*}_{c}(C, \chi^{-n-1}),\tag{33}\] where \(H_{c}^{*}(C, \chi^{-n-1})\) is identified with \(H_{c}^{*}(C, \chi^{n+1})^{*}\) by the Poincaré pairing. The \(\mathbb{Z}/2\mathbb{Z}\)-grading is provided by the cohomological degree, and the natural pairing has the form \[\begin{align} \langle \alpha_{1} \oplus \beta_{1}, \alpha_{2} \oplus \beta_{2} \rangle = \beta_{1}(\alpha_2 ) + (-1)^{\mathrm{deg}\,\alpha_1 \, \mathrm{deg}\,\beta_2}\beta_{2}(\alpha_1) \end{align}\] for homogeneous \(\alpha_{1} \oplus \beta_{1}\) and \(\alpha_{2} \oplus \beta_{2}\).

The supercommutator relations of Subsection 8.3 imply that the algebra \(WCl_{\overline{\mathbb{Q}}_{l}}(V)\) acts on the space \(\hat{H}_{c}^{*}(QM\vec{N}(k), \chi )[[-2\, \mathrm{rk}N(k)]]\) by \[\begin{align} \label{WeylCliffact} &WCl_{\overline{\mathbb{Q}}_{l}}\left(V\right) \rightarrow \mathcal{A}^{(k)}_{\chi} \\ &\alpha \mapsto e \langle \alpha \rangle ,\;\; \alpha \in H_{c}^{*}(C, \chi^{n+1}) \nonumber \\ &\beta \mapsto f \langle \beta \rangle, \;\; \beta \in H_{c}^{*}(C, \chi^{n+1})^{*}. \nonumber \end{align}\tag{34}\] We use the following result [11], [22], which is a one-dimensional analog of Nakajima’s action on the Hilbert schemes of points of surfaces [23].

Theorem 3 implies injectivity of 34 , and surjectivity follows from the Casimir relations in Subsection 8.4, so 34 is an isomorphism. In particular, \[\mathcal{A}^{(k)2} \cong \overline{\mathbb{Q}}_{l}[x_k , y_k ].\] As an \(\mathcal{A}^{(k)2}\)-module, \(WCl_{\overline{\mathbb{Q}}_{l}}\) is a product of symmetric and exterior algebras: \[\begin{align} WCl_{\overline{\mathbb{Q}}_{l}} \cong \mathrm{Sym}^{\bullet}_{\mathcal{A}^{(k)2}} \langle e\langle 1 \rangle , f\langle 1 \rangle \rangle \otimes \bigwedge\nolimits^{\bullet}_{\mathcal{A}^{(k)2}} \langle \theta_{i}^{x_k } , \theta_{i}^{y_k }, \theta_i^{\vee x_k }, \theta_i^{\vee y_k }, \; i = 1, ..., g \rangle. \end{align}\]

We denote the spectrum of \(\mathcal{A}^{(k)2}\) by \(U_{k}\): \[\begin{align} U_{k} = \mathrm{Spec}(\mathcal{A}^{(k)2}) \cong \mathbb{A}^{2}_{\overline{\mathbb{Q}}_{l}}. \end{align}\] The \(\mathrm{Fr}\times A^{\vee}\)-action on cohomology of the fixed loci induces the action on \(\mathcal{A}^{(k)2}\) by: \[\begin{align} x_{k} \mapsto q^{-\frac{n+1}{2}+k}a^{-n-1}x_{k}, \qquad y_{k} \mapsto q^{\frac{n+1}{2}+1-k}a^{n+1}y_{k}. \nonumber \end{align}\] The smooth variety \(U_{k}\) has a symplectic structure given by an \(\mathrm{Fr}\times A^{\vee}\)-equivariant map \[\begin{align} \label{Poissonbraketk} \{\cdot , \cdot \} : q^{-1} \otimes \bigwedge\nolimits^{2} \Omega_{\mathcal{A}^{(k)2}} \rightarrow \mathcal{A}^{(k)2} \qquad \{x_{k}, y_{k} \} = 1. \end{align}\tag{35}\]

When \(\chi_{0}^{n+1} \neq 1\), the isomorphism 34 implies \[\begin{align} \mathcal{A}^{(k)2}_{\chi} = \overline{\mathbb{Q}}_{l}[\eta]/(\eta) \cong \overline{\mathbb{Q}}_{l}. \end{align}\]

Analogously to 22 consider an \(\mathcal{A}^{(k)2}\)-module \[\begin{align} \mathcal{E}^{(k)} = q^{-1}\Omega_{\mathcal{A}^{(k)2}}\otimes H^{1}_{c}(C). \end{align}\] It is equipped with the symmetric pairing \(\langle \cdot, \cdot \rangle\), provided by anticommutators 111 . The isomorphism 34 implies that \(\mathcal{A}_{\chi}^{(k)\geq 1}\) is identified with \[\begin{align} \mathcal{A}^{(k)\geq 1} \cong \mathrm{Cliff}_{\mathcal{A}^{(k)2}}(\mathcal{E}^{(k)}, \langle \cdot , \cdot \rangle). \end{align}\]

In Appendix 9 we describe the entire \(\mathcal{A}^{(k)}\) as an algebra of differential operators.

In the nontrivial case \(\chi_{0}^{n+1} \neq 1\), we define an \(\mathcal{A}^{(k)2}_{\chi}\)-module \[\begin{align} \mathcal{E}_{\chi}^{(k)} = H^{1}\left(\pi_{1}^{g}, q^{-1}\Omega_{\mathcal{A}^{(k)2}} \otimes \mathcal{A}_{\chi}^{(k)2}\right), \end{align}\] where the \(\pi_{1}^{g}\times A^{\vee}\)-action on \(\Omega^{1}_{\mathcal{A}^{(k)2}}\) and its restriction to the fixed locus are induced from the action 36 on \(U_{k}\). There is an analogous pairing \[\begin{align} \label{eq:pairingEk} \langle \cdot, \cdot \rangle \colon S^{2}\left( \mathcal{E}_{\chi}^{(k)} \right) \rightarrow \mathcal{A}^{(k)2}_{\chi}. \end{align}\tag{38}\] The isomorphism 34 for \(\chi_{0}^{n+1} \neq 1\) implies \[\begin{align} \mathcal{A}^{(k)\geq 1}_{\chi} = \mathrm{Cliff}(\mathcal{E}^{(k)}_{\chi}, \langle \cdot , \cdot \rangle). \end{align}\]

5.4 Restricting to attractor strata↩︎

We now relate the correspondences on the full stack \(QM\vec{N}\) to those on the fixed loci. The excess intersection theorem gives explicit formulas for the action of the operators \(e\langle\alpha\rangle, f\langle\beta\rangle\) on \(\hat{H}^{*}_{c}(QM\vec{N}(k))\) in terms of their fixed-locus analogues.

Consider the closed immersions \[\begin{align} j^{(k)}\colon C \times \bigcup\limits_{l = 1}^{k}\mathrm{Attr}(l) \hookrightarrow C \times QM\vec{N}, \end{align}\] \[\begin{align} i^{(k)}\colon C \times QM\vec{N}(k) \hookrightarrow C \times \bigcup\limits_{l = 1}^{k}\mathrm{Attr}(l). \end{align}\] The correspondence \(\iota\) preserves \(C \times \bigcup_{l = 1}^{k}\mathrm{Attr}(l)\) and \(C \times QM\vec{N}(k)\), and its restrictions \[\begin{align} \bar{\iota}^{(k)}\colon C \times \bigcup\limits_{l = 1}^{k}\mathrm{Attr}(l) \rightarrow C \times \bigcup\limits_{l = 1}^{k}\mathrm{Attr}(l), \end{align}\] \[\begin{align} \iota^{(k)}\colon C \times QM\vec{N}(k) \rightarrow C \times QM\vec{N}(k) \end{align}\] are the base changes of \(\iota\) along \(j^{(k)}\) and \(j^{(k)} \circ i^{(k)}\), respectively.

Via the pullback map \[\begin{align} j^{(k)*}\colon \hat{H}^{*}_{c}(C \times QM\vec{N}, \psi) \rightarrow \hat{H}^{*}_{c}(C \times \bigcup\limits_{l = 1}^{k}\mathrm{Attr}(l), j^{(k)*}\psi) \end{align}\] the operators \(\iota_{*}, \iota^{*}\) induce operators on \(\hat{H}^{*}_{c}(C \times \bigcup_{l=1}^{k}\mathrm{Attr}(l), j^{(k)*}\psi)\), which we denote by the same symbols (here \(\psi\) is a local system on \(C \times QM\vec{N}\)). Since pullbacks commute, \[\begin{align} \label{pbk} \iota^{*} = \bar{\iota}^{(k)*}, \end{align}\tag{39}\] while for the pushforward, by [18], \[\begin{align} \label{iotaiotabark} \iota_{*} = \bar{\iota}^{(k)}_{*}c_{n+1-k}(j^{(k)*}N_{\iota} - N_{\bar{\iota}^{(k)}}). \end{align}\tag{40}\] Similarly, via the pushforward \[\begin{align} i^{(k)}_{*}\colon \hat{H}^{*}_{c}(C \times QM\vec{N}(k), \phi ) \rightarrow \hat{H}^{*}_{c}(C \times \bigcup\limits_{l = 1}^{k}\mathrm{Attr}(l), i_{!}^{(k)}\phi) \end{align}\] the operators \(\bar{\iota}^{(k)}_{*}, \bar{\iota}^{(k)*}\) induce operators on \(\hat{H}^{*}_{c}(C \times QM\vec{N}(k), \phi)\), which we again denote by the same symbols. Since pushforwards commute, \[\begin{align} \label{iotakiotabark} \bar{\iota}^{(k)}_{*} = \iota^{(k)}_{*}, \end{align}\tag{41}\] while the excess intersection formula (Fulton, Intersection Theory, Theorem 6.3; for the derived \(l\)-adic version see [18]) gives \[\begin{align} \bar{\iota}^{(k)*} = c_{k-1}(i^{(k)*}N_{\bar{\iota}^{(k)}} - N_{\iota^{(k)}}) \iota^{(k)*}. \end{align}\]

We next simplify the excess normal bundle: \[\begin{align} i^{(k)*}N_{\bar{\iota}^{(k)}} - N_{\iota^{(k)}} = i^{(k)*}&\left(\bar{\iota}^{(k)*}T^{\mathrm{vir}}\bigcup\limits_{l = 1}^{k}\mathrm{Attr}(l) -T^{\mathrm{vir}}\bigcup\limits_{l = 1}^{k}\mathrm{Attr}(l)\right) \\ &-\left(\iota^{(k)*}T^{\mathrm{vir}}QM\vec{N}(k) - T^{\mathrm{vir}}QM\vec{N}(k)\right). \nonumber \end{align}\] Using \[\begin{align} i^{(k)*}\bar{\iota}^{(k)*}T^{\mathrm{vir}}\bigcup\limits_{l = 1}^{k}\mathrm{Attr}(l) = \iota^{(k)*}i^{(k)*}T^{\mathrm{vir}}\bigcup\limits_{l = 1}^{k}\mathrm{Attr}(l), \end{align}\] this rearranges to \[\begin{align} i^{(k)*}N_{\bar{\iota}^{(k)}} - N_{\iota^{(k)}} &=\iota^{(k)*} \big(i^{(k)*}T^{\mathrm{vir}}\bigcup\limits_{l = 1}^{k}\mathrm{Attr}(l) - T^{\mathrm{vir}}QM\vec{N}(k)\big) -\\ &-\big( i^{(k)*}T^{\mathrm{vir}}\bigcup\limits_{l = 1}^{k}\mathrm{Attr}(l) - T^{\mathrm{vir}}QM\vec{N}(k) \big)= \nonumber\\ &=\iota^{(k)*} N_{\mathrm{Attr}(k)/QM\vec{N}(k)} - N_{\mathrm{Attr}(k)/QM\vec{N}(k)} = \iota^{(k)*} N(k) - N(k). \nonumber \end{align}\] By the same computation as in 14 , \[\begin{align} &\iota^{(k)*} N(k) - N(k)=(\iota^{(k)*} - \mathrm{id} )\left[H^{*}(C, \mathcal{V}^{(k-1)}\otimes \mathcal{L}^{-1})\otimes\mathcal{O}_{QM\vec{N}}(1)\right] \\ &=\iota^{(k)*}\mathcal{V}^{(k-1)}\otimes \mathcal{L}^{-1}\otimes\mathcal{O}_{QM\vec{N}}(1) = \iota^{(k)*} \mathcal{N}^{(k-1)} = \mathcal{N}^{(k-1)} \otimes TC, \nonumber \end{align}\] which finally yields \[\begin{align} \label{iotapr} \bar{\iota}^{(k)*} = c_{k-1}(\mathcal{N}^{(k-1)} \otimes TC) \iota^{(k)*}. \end{align}\tag{42}\]

Combining 40 and 41 , \[\begin{align} \label{pushiotainchart} \iota_{*} = \iota^{(k)}_{*}c_{n+1-k}((j^{(k)} \circ i^{(k)})^{*}N_{\iota} - N_{\iota^{(k)}}) = \iota^{(k)}_{*}c_{n + 1 -k}\left( (\mathcal{N}/\mathcal{N}^{(k)}) \otimes TC \right), \end{align}\tag{43}\] and combining 39 and 42 , \[\begin{align} \label{pulliotainchart} \iota^{*} = c_{k-1}(\mathcal{N}^{(k-1)} \otimes TC) \iota^{(k)*}. \end{align}\tag{44}\] From 43 and 44 we obtain the following formulas for the action of descendants on \(\hat{H}^{*}_{c}(QM\vec{N}(k))\). \[\begin{align} &f \langle \mathbf{p} \rangle = -\int\limits_{C} \iota^{*} \mathbf{p} \otimes \cdot = -\int\limits_{C} c_{k-1}\left( \mathcal{N}^{(k-1)}\otimes TC \right) \iota^{(k)*}\mathbf{p}\otimes\cdot = f^{(k)} \langle \mathbf{p} \rangle \eta^{k-1}, \nonumber \\ &f \langle \gamma_i \rangle = -\int\limits_{C} c_{k-1}\left( \mathcal{N}^{(k-1)}\otimes TC \right) \iota^{(k)*}\gamma_i \otimes\cdot = \eta^{k-1} f^{(k)}\langle \gamma_i \rangle + (k-1)f^{(k)} \langle \mathbf{p} \rangle \eta^{k-2} \gamma_i , \tag{45}\\ &e \langle \mathbf{p} \rangle = \int\limits_{C} \iota_{*} \mathbf{p} \otimes \cdot = \int\limits_{C} c_{n+1 -k}\left( \mathcal{N}/\mathcal{N}^{(k)} \right) \iota^{(k)}_{*}\mathbf{p}\otimes\cdot =e^{(k)} \langle \mathbf{p} \rangle \eta^{n+1-k}, \nonumber \\ &e \langle \gamma_i \rangle = \int\limits_{C} c_{n+1-k}\left( \mathcal{N}/\mathcal{N}^{(k)} \right) \iota^{(k)}_{*}\gamma_i \otimes\cdot = \eta^{n+1-k} e^{(k)}\langle \gamma_i \rangle + (n+1-k)e^{(k)} \langle \mathbf{p} \rangle \eta^{n-k} \gamma_i . \tag{46} \end{align}\] We also derive analogous formulas for the degree-zero generators in Appendix 9.

For the nontrivial character case \(\chi_{0}^{n+1} \neq 1\), the action of descendants on \(\hat{H}^{*}_{c}(QM\vec{N}(k), \chi)\) is given by \[\begin{align} f \langle \gamma_{i}^{\chi^{-1}} \rangle &= \eta^{k-1}f^{(k)}\langle \gamma_{i}^{\chi^{-1}} \rangle = \delta_{k, 1} f^{(k)}\langle \gamma_{i}^{\chi^{-1}} \rangle,\\ e \langle \gamma_{i}^{\chi} \rangle &= \eta^{n+1-k}e^{(k)}\langle \gamma_{i}^{\chi} \rangle = \delta_{k, n+1} e^{(k)}\langle \gamma_{i}^{\chi} \rangle. \nonumber \end{align}\]

We thus have compatible homomorphisms \[\label{eq:AtoAk} \varphi^{(k)}_{\chi}: \mathcal{A}_{\chi} \rightarrow \mathcal{A}_{\chi}^{(k)}, \qquad \varphi^{(k)2}_{\chi}: \mathcal{A}_{\chi}^{2} \rightarrow \mathcal{A}_{\chi}^{(k)2}, \qquad \varphi^{(k)\geq 1}_{\chi}: \mathcal{A}_{\chi}^{\geq 1} \rightarrow \mathcal{A}_{\chi}^{(k)\geq 1}, \qquad k = 1, ..., n+1.\tag{47}\] We omit the subscript \(\chi\) when \(\chi_{0}^{n+1} = 1\).

5.5 Symplectic resolution↩︎

In this subsection we use 47 to prove the isomorphism \(R \cong \mathcal{A}^{2}\) and to define a smooth symplectic resolution \(X^{\vee}\to X_{0}^{\vee}\). Then we prove the isomorphism \(Cl \cong \mathcal{A}^{\geq 1}\). The morphisms of algebras \(\mathcal{A}^{\geq 1} \rightarrow \mathcal{A}^{(k)\geq 1}\) will then correspond to a morphism of sheaves of Clifford algebras \(\mathcal{A}^{\geq 1} \rightarrow \mathrm{Cliff}\) on \(X^{\vee}\).

From now on we make no distinction between \(R\) and \(\mathcal{A}^{2}\). The homomorphisms \(\varphi^{(k)2}\) correspond to birational maps \[\begin{align} \label{projectk} \pi_{k}\colon\; U_{k} \rightarrow X^{\vee}_{0} \qquad x = x_{k}^{n+2 - k}y_{k}^{n+1-k}, \qquad y = x_{k}^{k-1}y_{k}^{k}, \end{align}\tag{48}\] which preserve the Poisson structure and the \(\pi_{1}^{g}\times \mathrm{Fr} \times A^{\vee}\)-action. In particular, for \(\chi_{0}^{n+1} \neq 1\) the map \(\pi_{k}\) restricts to a morphism of \(\pi_{1}^{g}\)-fixed loci \[\begin{align} \mathrm{Spec}\, \mathcal{A}^{(k)2}_{\chi} = (U_{k})^{\pi_1^{g}} &\rightarrow (X^{\vee}_{0})^{\pi_{1}^{g}} = \mathrm{Spec}\, R_{\mathrm{fix}}. \end{align}\] It is easy to see that the corresponding map of algebras is exactly \[\varphi^{(k)2}_{\chi} \circ \Phi^{2}_{\chi} \colon R_{\mathrm{fix}} \rightarrow \mathcal{A}^{(k)2}_{\chi} \qquad \eta \mapsto 0.\] Proposition 19 extends to the generators of degree \(\geq 1\).

From now on we make no distinction between \(Cl\) and \(\mathcal{A}^{\geq 1}\).

The map \(\varphi^{(k)}\) preserves the pairing \(\langle \cdot , \cdot \rangle\) and the \(\pi_{1}^{g} \times \mathrm{Fr} \times A^{\vee}\)-action. In particular, for \(\chi_{0}^{n+1} \neq 1\) it induces the \(\mathcal{A}^{2}_{\chi}\)-algebra homomorphism \[\begin{align} \mathrm{Cliff}\left( H^{1}\left(\pi_{1}^{g}, q^{-1}\Omega_{R}\otimes R_{\mathrm{fix}}\right) , \langle \cdot , \cdot \rangle \right) &\rightarrow \mathrm{Cliff}\left( H^{1}\left(\pi_{1}^{g}, q^{-1}\Omega_{\mathcal{A}^{(k)2}}\otimes \mathcal{A}^{(k)2}_{\chi} \right) , \langle \cdot , \cdot \rangle \right) . \end{align}\] It is easy to see that it coincides with \[\begin{align} \label{chiAgeq1toAkgeq1} \varphi_{\chi}^{(k)\geq 1} \circ \Phi_{\chi}^{\geq 1} \colon Cl_{\chi} \rightarrow \mathcal{A}^{(k) \geq 1}_{\chi} \qquad \theta^{x}_{\chi, i} \mapsto \delta_{k, n+1} \theta^{x_k}_{\chi, i}, \qquad \theta^{y}_{\chi, i} \mapsto \delta_{k, 1} \theta^{y_k}_{\chi, i}, \qquad \gamma_{i} &\mapsto 0. \end{align}\tag{51}\]

The \(\pi_{1}^{g} \times \mathrm{Fr} \times A^{\vee}\)-actions on the \(U_{k}\) are compatible, and hence induce a \(\pi_{1}^{g} \times \mathrm{Fr} \times A^{\vee}\)-action on \(X^{\vee}\). The \(\pi_{1}^{g}\)-fixed points of \(X^{\vee}\) are identified with the disjoint union of the chart-wise fixed loci, \[\begin{align} (X^{\vee})^{\pi_{1}^{g}} = \bigsqcup_{k = 1}^{n+1} \mathrm{Spec}\, \mathcal{A}_{\chi}^{(k)2}. \end{align}\]

The \(\mathcal{A}^{(k)2}\)-modules \(\mathcal{E}^{(k)}\) glue to the vector bundle \(q^{-1} \Omega^{1}_{X^{\vee}} \otimes H^{1}_{c}(C)\) on \(X^{\vee}\), which inherits the symmetric pairing \(\langle \cdot, \cdot \rangle\). The associated Clifford algebra is a locally free sheaf of \(\mathcal{O}_{X^{\vee}}\)-algebras \[\begin{align} \mathrm{Cliff} = \mathrm{Cliff}( q^{-1}\Omega_{X^{\vee}} \otimes H^{1}_{c}(C), \langle \cdot , \cdot \rangle ), \end{align}\] and the homomorphisms 50 glue to an inclusion of sheaves of algebras \[\begin{align} \rho^{*}Cl \rightarrow \mathrm{Cliff}. \end{align}\] Note that this inclusion is not surjective on global sections. Indeed, \(\Omega_{R}\) has three generators \(dx, dy, d\eta\), but global sections of \(\Omega_{X^{\vee}}\) are generated by \(dx, dy, d\eta\), and \(\eta(\frac{dx}{x} - \frac{dy}{y})\).

Analogously, for \(\chi_{0}^{n+1} \neq 1\) the sheaves \(\mathcal{E}^{(k)}_{\chi}\) glue to the locally free sheaf of \(\mathcal{O}_{(X^{\vee})^{\pi_{1}^{g}}}\)-modules \[\begin{align} H^{1}\left(\pi_{1}^{g}, q^{-1}\Omega_{X^{\vee}}\big|_{(X^{\vee})^{\pi_{1}^{g}}} \otimes H^{1}_{c}(C)\right) = \bigoplus\limits_{k = 1}^{n+1}\mathcal{E}_{\chi}^{(k)}, \end{align}\] which inherits \(\langle \cdot, \cdot \rangle\). The associated Clifford algebra is a locally free sheaf of \(\mathcal{O}_{(X^{\vee})^{\pi_{1}^{g}}}\)-algebras \[\begin{align} \mathrm{Cliff}_{\chi} = \mathrm{Cliff}\left(\bigoplus\limits_{k = 1}^{n+1}\mathcal{E}^{(k)}_{\chi}, \langle \cdot, \cdot \rangle \right) = \mathop{\mathlarger{\mathlarger{\times}}}\limits_{k = 1}^{n+1} \mathrm{Cliff}\left(\mathcal{E}^{(k)}_{\chi}, \langle \cdot, \cdot \rangle \right), \end{align}\] and the homomorphisms 51 glue to a morphism of sheaves of algebras \[\begin{align} \rho^{*}\mathcal{A}^{\geq1}_{\chi} \rightarrow \mathrm{Cliff}_{\chi}. \end{align}\]

6 Cohomology of the resolution↩︎

We want to describe the cohomology of quasimaps as the cohomology of a natural sheaf on the resolution \(X^{\vee}\). In this section we construct this sheaf as the twisted tensor product \(\hat{\mathcal{O}}_{\mathrm{vir}} \otimes \mathcal{O}(\vec{\mathcal{M}})\) of a natural spin bundle and a line bundle depending on \(\vec{\mathcal{M}} \in \mathrm{Pic}(C)^{\times (n+1)}\), and compute the local cohomology \(H^{*}_{L}\) with support in the attracting subvariety \(L \subset X^{\vee}\).

6.1 The symmetrised virtual structure sheaf↩︎

We begin by constructing a locally free \(\mathrm{Cliff}\)-module \(\mathrm{Spin}\). Twisting it by a square root of the canonical bundle produces the “symmetrised virtual structure sheaf” \(\hat{\mathcal{O}}_{\mathrm{vir}}\).

Let \(\mathrm{Cliff}\) be a sheaf of Clifford algebras. A spin module \(\mathrm{Spin}\) is an \(\mathcal{O}_{X^{\vee}}\)-locally free \(\mathrm{Cliff}\)-module satisfying \[\begin{align} \label{HAHB} \mathrm{Cliff} &\cong \mathrm{Spin}^{\vee} \mathbin{\mathop{\otimes}\limits_{\;\mathcal{O}_{X^{\vee}}}} \mathrm{Spin},\\ \mathrm{Spin}^{\vee} &\cong \mathrm{Spin}. \end{align}\tag{52}\]

Let us give a construction of \(\mathrm{Spin}\) in the case of the Clifford algebra sheaf defined in Subsection 5.5. Consider a decomposition of \(H^{1}_{c}(C)\) into two Lagrangian subspaces \[\begin{align} H^{1}_{c}(C) = H_{A} \oplus H_{B}, \end{align}\] where \[\begin{align} H_{A} = \langle \gamma_{1}, \ldots, \gamma_{g} \rangle_{\overline{\mathbb{Q}}_{l}}, \;\;\; H_{B} = \langle \gamma_{g + 1}, \ldots, \gamma_{2g} \rangle_{\overline{\mathbb{Q}}_{l}}. \end{align}\] Recall that the \(\gamma_{i}\) were chosen to be \(\mathrm{Fr}\)-eigenvectors with eigenvalues \(\alpha_{i} = \pm q^{1/2}\); since \(\gamma_{g + i} = \gamma_{i}^{\vee}\) for \(i = 1, \ldots, g\), we have \[\begin{align} \alpha_{i+g} = q/\alpha_{i} = \alpha_i , \; i = 1, \ldots, g. \end{align}\]

As an \(\mathcal{O}_{X^{\vee}}\)-module the quotient is the exterior algebra \[\begin{align} \mathrm{Cliff}/(q^{-1}\Omega^{1}_{X^{\vee}} \otimes H_{B}) \cong \bigwedge q^{-1}\Omega^{1}_{X^{\vee}} \otimes H_{A}, \end{align}\] and the square root in 53 can be chosen as \[\begin{align} \left( \bigwedge\nolimits^{2g} q^{-1}\Omega^{1}_{X^{\vee}} \otimes H_{B} \right)^{1/2} = \prod\limits_{i = 1}^{g}\frac{q^{1/2}}{\alpha_{i}} \; \mathcal{O}_{X^{\vee}}, \end{align}\] the numerical factor representing the \(\mathrm{Fr}\)-twist on the fibers of the trivial bundle. \(\mathrm{Spin}\) can be viewed as the structure sheaf of the supervariety \(\mathcal{X}^{\vee} = H_{A} \otimes T X^{\vee}\), with \(H_{A}\) placed in odd degree.

We define a square root of the canonical bundle \[\begin{align} (\mathcal{K}_{X^{\vee}})^{1/2} = (\Omega^{2}_{X^{\vee}})^{1/2} = q^{1/2}\mathcal{O}_{X^{\vee}}. \end{align}\]

As an \(\mathcal{O}_{X^{\vee}}\)-module it is the twisted exterior algebra \[\begin{align} \hat{\mathcal{O}}_{\mathrm{vir}} \cong q^{1/2}\prod\limits_{i = 1}^{g}\frac{q^{1/2}}{\alpha_{i}} \left(\bigwedge q^{-1}\Omega^{1}_{X^{\vee}} \otimes H_{A}\right). \end{align}\]

In the nontrivial case \(\chi_{0}^{n+1} \neq 1\), set \[\begin{align} \label{spinchidef} \mathrm{Spin}_{\chi} = \bigoplus\limits_{k = 1}^{n+1} \left( \mathrm{Cliff}_{k}(\mathcal{E}^{(k)}_{\chi}, \langle \cdot, \cdot \rangle)/\left(V_{k}\right) \right)\otimes \left( \bigwedge\nolimits^{2g-2} V_{k} \right)^{1/2}, \end{align}\tag{54}\] where \[\begin{align} V_{k} = q^{-1}(T^{*}_{p_{k}}X^{\vee})^{f}\otimes H^{1}_{c}(C, \chi^{-n-1} ), \end{align}\] and \((T^{*}_{p_{k}}X^{\vee})^{f}\) is the subspace of \(T^{*}_{p_{k}}X^{\vee}\) on which \(A^{\vee}\) acts by a positive power of \(a\): \[\begin{align} q^{-1}(T^{*}_{p_{k}}X^{\vee})^{f}\otimes H^{1}_{c}(C, \chi^{-n-1} ) \cong \langle f^{(k)}\langle \gamma_{1}^{\chi^{-1}} \rangle , \ldots, f^{(k)}\langle \gamma_{2g-2}^{\chi^{-1}} \rangle \rangle_{\overline{\mathbb{Q}}_{l}}. \end{align}\] The square root in 54 can be chosen as \[\begin{align} \left( \bigwedge\nolimits^{2g-2} V_{k} \right)^{1/2} \cong q^{\frac{1}{2}(1-g) + (k-1)(1-g)}(q^{1/2}a)^{-(n+1)(1-g)}\overline{\mathbb{Q}}_{l}. \end{align}\]

As a \(\overline{\mathbb{Q}}_{l}\)-vector space it is isomorphic to a direct sum of exterior algebras: \[\begin{align} \hat{\mathcal{O}}_{\mathrm{vir}, \chi} \cong \bigoplus\limits_{k = 1}^{n+1} \left( \bigwedge\nolimits_{\overline{\mathbb{Q}}_{l}} \langle e^{(k)}\langle \gamma_{1}^{\chi} \rangle, \ldots, e^{(k)}\langle \gamma_{2g-2}^{\chi}\rangle \rangle_{\overline{\mathbb{Q}}_{l}} \right) \otimes \left( \bigwedge\nolimits^{2g-2} V_{k} \right)^{1/2}. \end{align}\]

6.2 The line bundle \(\mathcal{O}(\vec{\mathcal{M}})\)↩︎

The entire construction starting from \(QM\vec{N}\) depends on the tuple \(\vec{\mathcal{M}} \in \mathrm{Bun}_{T}(C)\). In this subsection we encode this dependence on the \(X^{\vee}\)-side as a \(\mathrm{Fr} \times A^{\vee}\)-equivariant line bundle \(\mathcal{O}(\vec{\mathcal{M}})\) on \(X^{\vee}\), which will ultimately twist \(\hat{\mathcal{O}}_{\mathrm{vir}}\) in the main comparison.

Let \(\vec{\mathcal{M}} = (\mathcal{M}_{1}, \ldots, \mathcal{M}_{n+1}) \in \mathrm{Bun}_{T}(C)\) with \(\deg(\mathcal{M}_{k}) = m_k\). To this tuple we associate a \(\mathrm{Fr} \times A^{\vee}\)-equivariant line bundle \(\mathcal{O}(\vec{\mathcal{M}})\) on \(X^{\vee}\), uniquely determined up to isomorphism by the following conditions: \[\begin{align} &\mathrm{wt}_{\mathrm{Fr}\times A^{\vee}} \left( \mathcal{O}(\vec{\mathcal{M}})\big|_{p_1 } \right) = q^{-\frac{1}{2}\sum\limits_{l < r}(m_{l} - m_{r}+1-g)} (q^{1/2}a)^{-\sum\limits_{l=1}^{n+1}m_{l} - m_{1}}, \label{linearization} \\ &\mathcal{O}(\vec{\mathcal{M}})\big|_{\bar{L}_{k}} \cong \mathcal{O}_{\mathbb{P}^{1}}(m_k - m_{k+1} ), \;\;\; k = 1, ..., n. \nonumber \end{align}\tag{55}\] See Appendix 9 for the natural origin of this bundle. It follows that the \(\mathrm{Fr}\times A^{\vee}\)-weights of the fibers of \(\mathcal{O}(\vec{\mathcal{M}})\) at the fixed points are \[\begin{align} \mathrm{wt}_{\mathrm{Fr}\times A^{\vee}} \left( \mathcal{O}(\vec{\mathcal{M}})\big|_{p_k } \right) = q^{-\frac{1}{2}\sum\limits_{l < r}(m_{l} - m_{r} + 1 - g) + \sum\limits_{l = 1}^{k-1}m_{l} - m_{k}} (q^{1/2}a)^{-\sum\limits_{l=1}^{n+1}m_{l} - m_{k}}. \end{align}\] Since \(\mathcal{O}(\vec{\mathcal{M}})\) is trivialized over each \(U_{k}\), \[\begin{align} \mathcal{O}(\vec{\mathcal{M}})\big|_{U_{k} } \cong q^{-\frac{1}{2}\sum\limits_{l < r}(m_{l} - m_{r} + 1 - g) + \sum\limits_{l = 1}^{k-1}m_{l} - m_{k}} (q^{1/2}a)^{-\sum\limits_{l=1}^{n+1}m_{l} - m_{k}} \mathcal{O}_{X^{\vee}}. \end{align}\] The assignment \[\begin{align} \mathrm{Bun}_{T}(C) &\rightarrow \mathrm{Pic}_{\mathrm{Fr}\times A^{\vee}}(X^{\vee}),\\ \vec{\mathcal{M}} &\mapsto \mathcal{O}(\vec{\mathcal{M}}) \nonumber \end{align}\] is evidently a group homomorphism.

6.3 Local cohomology of \(X^{\vee}\)↩︎

We introduce the local cohomology \(H^{*}_{L}\) of \(X^{\vee}\) with support in the attracting set \(L\) of the \(A^{\vee}\)-action. The stratification of \(L\) into attracting cells \(L_{k}\) induces a filtration on \(H^{*}_{L}\).

Each closure \(\bar{L}_{k}\) is a smooth Lagrangian subvariety, and \(\bar{L}_{k}\) meets \(\bar{L}_{k+1}\) transversely at \(p_{k+1}\), for \(k = 1, \ldots, n\). For \(k = 1, \ldots, n+1\) define \[\begin{align} L^{k} = \bigcup\limits_{l = 1}^{k} L_{l}, \end{align}\] so that \(L^{n+1} = L\), and observe the decomposition into closed and open subsets \[\begin{align} \label{Lkdec} L^{k} = L_{k} \bigsqcup L^{k-1}, \;\;\; k = 2, \ldots, n + 1. \end{align}\tag{56}\]

The functor \(\Gamma_{L}\) of sections with support in \(L\) is defined by \[\begin{align} \Gamma_{L}\colon \mathrm{Qcoh}_{X^{\vee}} &\longrightarrow \Gamma(X^{\vee}, \mathcal{O}_{X^{\vee}})\text{-}\mathrm{mod}\\ \mathcal{F} &\mapsto \{s \in \Gamma(X^{\vee}, \mathcal{F}): \mathcal{I}^{N}_{L}s = 0, N \gg 0\}, \nonumber \end{align}\] where \(\mathcal{I}_{L}\) is the ideal sheaf of \(L\) generated by \(y\): \[\begin{align} \mathcal{I}_{L} = y \mathcal{O}_{X^{\vee}} \subset \mathcal{O}_{X^{\vee}}. \end{align}\] The local cohomology groups are defined as the right derived functors \[\begin{align} H^{i}_{L}(X^{\vee}, \mathcal{F}) = R^{i}\Gamma_{L}(\mathcal{F}). \end{align}\] The decomposition 56 yields the long exact sequence of a pair \[\begin{align} \label{Hfilt} H^{*}_{L_{k}}(X^{\vee}, \mathcal{F}) \rightarrow H^{*}_{L^{k}}(X^{\vee}, \mathcal{F}) \rightarrow H^{*}_{L^{k-1}}(X^{\vee}, \mathcal{F}) \rightarrow H^{*}_{L_{k}}(X^{\vee}, \mathcal{F})[1], \;\; k = 2, \ldots, n + 1. \end{align}\tag{57}\] In the following we compute \(H^{*}_{L}(X^{\vee}, \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}}))\). We will see that only the first cohomology is nontrivial, so the long exact sequence 57 becomes short and equips \(H^{*}_{L}(X^{\vee}, \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}}))\) with a filtration whose associated graded pieces are the \(H^{*}_{L_{k}}(X^{\vee}, \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}}))\).

6.4 Cohomology with support in attracting subvarieties↩︎

Since \(L_{k} \subset U_{k}\), the local cohomology can be computed in the chart \(U_{k}\): \[\begin{align} H^{*}_{L_{k}} (X^{\vee}, \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}})) = H^{*}_{L_{k}} (U_{k},\hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}})). \end{align}\] The subvariety \(L_{k}\) is closed in \(U_{k}\), with ideal sheaf \(\mathcal{I}_{L_{k}} = y_k \mathcal{O}_{X^{\vee}}\). Applying the exact sequence of a pair to \(L_{k} \subset U_{k}\), we obtain the long exact sequence \[\begin{align} 0 &\rightarrow H^{0}_{L_{k}}(U_{k}, \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}})) \rightarrow H^{0}(U_{k}, \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}})) \rightarrow H^{0}(U_{k}\setminus L_{k}, \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}})) \rightarrow\\ &\rightarrow H^{1}_{L_{k}}(U_{k}, \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}})) \rightarrow H^{1}(U_{k}, \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}})) \rightarrow ... \nonumber \end{align}\] The group \(H^{0}_{L_{k}}(U_{k}, \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}}))\) vanishes because the sheaf \(\hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}})\) is locally free. Since \(U_{k}\) and \(U_{k} \setminus L_{k}\) are affine, \(\hat{H}^{i}(U_{k}, \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}})) = 0\) for \(i > 1\). Consequently \[\begin{align} H^{1}_{L_{k}}(U_k , \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}})) = \frac{H^{0}(U_{k}\setminus L_{k}, \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}}))}{H^{0}(U_{k}, \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}}))} = \frac{\overline{\mathbb{Q}}_{l}[y_{k}^{\pm 1}]}{\overline{\mathbb{Q}}_{l}[y_{k}]} \otimes_{\overline{\mathbb{Q}}_{l}[y_{k}]} H^{0}(U_{k}, \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}})) \end{align}\] as an \(\mathcal{A}^{(k)}\)-module, and this is the only nonzero local cohomology group. Expanding further, \[\begin{align} &\frac{\overline{\mathbb{Q}}_{l}[y_{k}^{\pm 1}]}{\overline{\mathbb{Q}}_{l}[y_{k}]} \otimes_{\overline{\mathbb{Q}}_{l}[y_{k}]}H^{0}(U_{k}, \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}})) =\\ &q^{-\frac{1}{2}\sum\limits_{l < r}(m_{l} - m_{r} + 1 - g) + \sum\limits_{l = 1}^{k-1}m_{l} - m_{k}} (q^{1/2}a)^{-\sum\limits_{l=1}^{n+1}m_{l} - m_{k}} q^{1/2}\left(\prod\limits_{i = 1}^{g}\frac{q^{1/2}}{\alpha_{i}} \right)\frac{\overline{\mathbb{Q}}_{l}[y_{k}^{\pm 1}]}{\overline{\mathbb{Q}}_{l}[y_{k}]} \otimes_{\overline{\mathbb{Q}}_{l}[y_{k}]}\bigwedge \left(q^{-1}\Omega^{1}_{U_{k}} \otimes V_{A}\right).\label{spinkaka} \end{align}\tag{58}\] Explicitly, in the coordinates \(x_k, y_k, \theta^{x_{k}}_{i}, \theta^{y_{k}}_{i}\), \[\begin{align} \frac{\overline{\mathbb{Q}}_{l}[y_{k}^{\pm 1}]}{\overline{\mathbb{Q}}_{l}[y_{k}]} \otimes_{\overline{\mathbb{Q}}_{l}[y_{k}]}\bigwedge \left(q^{-1}\Omega^{1}_{U_{k}} \otimes V_{A}\right) = y_{k}^{-1}\overline{\mathbb{Q}}_{l}[x_{k}, y_{k}^{-1}] \otimes \bigwedge\nolimits_{\overline{\mathbb{Q}}_{l}} \langle \theta^{x_{k}}_{1}, \ldots, \theta^{x_{k}}_{g}, \theta^{y_{k}}_{1}, \ldots, \theta^{y_{k}}_{g} \rangle_{\overline{\mathbb{Q}}_{l}}, \end{align}\] with the action of \(\mathcal{A}^{(k)}\) given by \[\begin{align} &x_k = x_k \cdot \;\;\; D_{x_k } = - \partial_{y_{k}}\\ &y_k = y_k \cdot \;\;\; D_{y_k } = \partial_{x_{k}} \nonumber\\ &\theta_{i}^{x_k} = \theta_{i}^{x_k} \cdot \;\;\; \theta_{i}^{y_k} = \theta_{i}^{y_k} \cdot, \;\;\; i = 1, \ldots, g \nonumber\\ &\theta^{\vee y_k}_{i} = -\partial_{\theta^{x_k}_{i}}, \;\;\; \theta^{\vee x_k}_{i} = \partial_{\theta^{y_k}_{i}}, \;\;\; i = 1, \ldots, g. \nonumber \end{align}\] We see that \(x_{k}, D_{x_{k}}, \theta^{x_{k}}_{i}\) for \(i = 1, \ldots, 2g\) freely generate this module from the cyclic vector \[\begin{align} y_{k}^{-1} \theta_{1}^{y_{k}} \wedge \cdots \wedge \theta^{y_{k}}_{g}, \end{align}\] which has \(\mathrm{Fr} \times A^{\vee}\)-weight \(q^{k-1}\bigl(\prod_{i=1}^{g} q^{-k}\alpha_{i}\bigr)(q^{1/2}a)^{-(n+1)(1-g)}\). Keeping track of the twists in 58 , we conclude that, as an \(\mathcal{A}^{(k)}\)-module, \(H^{1}_{L_{k}}(U_k , \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}}))\) is isomorphic to \[\label{eq:oldisohk} q^{\frac{1}{2}\kappa_{k}(\vec{m})} a^{\rho_{k}(\vec{m})} \otimes \overline{\mathbb{Q}}_{l}[x_{k}, D_{x_{k}}]\otimes \bigwedge\nolimits_{\overline{\mathbb{Q}}_{l}} \langle \theta^{x_{k}}_{1}, \ldots, \theta^{x_{k}}_{2g} \rangle_{\overline{\mathbb{Q}}_{l}},\tag{59}\] where \[\begin{align} &\kappa_{k}(\vec{m}) = 1 - g -\sum\limits_{l < r}(m_{l} - m_{r} + 1 - g) + \sum\limits_{l = 1}^{k-1}(m_{l} - m_{k} + 1 - g) -\sum\limits_{l = k}^{n+1}(m_{l} - m_{k} + 1 - g),\\ &\rho_{k}(\vec{m}) = -\sum\limits_{l=1}^{n+1}(m_{l} - m_{k} + 1 - g). \end{align}\] In the case \(\chi_{0}^{n+1} \neq 1\), \[\begin{align} H^{*}_{L}((X^{\vee})^{\pi_{1}^{g}}, \hat{\mathcal{O}}_{\mathrm{vir},\chi}\otimes\mathcal{O}(\vec{\mathcal{M}})) = \bigoplus_{k = 1}^{n+1}H_{L_{k}}^{0}((X^{\vee})^{\pi_{1}^{g}}, \hat{\mathcal{O}}_{\mathrm{vir},\chi}\otimes \mathcal{O}(\vec{\mathcal{M}})), \end{align}\] where \(H_{L_{k}}^{0}((X^{\vee})^{\pi_{1}^{g}}, \hat{\mathcal{O}}_{\mathrm{vir},\chi}\otimes \mathcal{O}(\vec{\mathcal{M}}))\) is isomorphic to \[\begin{align} \label{eq:chioldisohk} q^{\frac{1}{2}\kappa_{k}(\vec{m})} a^{\rho_{k}(\vec{m})} \otimes \bigwedge\nolimits_{\overline{\mathbb{Q}}_{l}} \langle e^{(k)}\langle \gamma_{1}^{\chi} \rangle, \ldots, e^{(k)}\langle \gamma_{2g-2}^{\chi}\rangle \rangle_{\overline{\mathbb{Q}}_{l}}. \end{align}\tag{60}\]

7 Comparison of the cohomologies↩︎

In this section we compare the two filtrations constructed in the previous sections: on the \(QM\vec{N}\)-side, the Białynicki-Birula filtration of Proposition 16; on the \(X^{\vee}\)-side, the filtration given by the long exact sequence 57 . We first compute the cohomology of each fixed locus \(QM\vec{N}(k)\) as an \(\mathcal{A}^{(k)}\)-module, match the graded pieces of the two filtrations, and then prove that the matched filtrations assemble into an isomorphism of \(\mathcal{A}\)-modules (in all but an exceptional configuration).

Recall that the short exact sequence ?? , \[\begin{align} 0 \rightarrow \hat{H}^{*}_{c}(QM\vec{N}(k))[[-2\mathrm{rk}\,N(k)]] \rightarrow \hat{H}^{*}_{c}\left( \bigcup\limits_{l = 1}^{k}\mathrm{Attr}(l)\right) \rightarrow \hat{H}^{*}_{c}\left( \bigcup\limits_{l = 1}^{k-1}\mathrm{Attr}(l)\right) \rightarrow 0, \;\;\; k = 2, \ldots, n+1, \end{align}\] induces a filtration on \(H^{*}_{c}(QM\vec{N})\) with graded pieces \(H^{*}_{c}(QM\vec{N}(k))[[-2\mathrm{rk}\,N(k)]]\). We first compute the associated graded module.

7.1 The associated graded module↩︎

We explicitly compute the action of \(\mathcal{A}^{(k)}\) on \(\hat{H}^{*}_{c}(QM\vec{N}(k), \chi)\) and identify it with the local cohomology \(H^{*}_{L_{k}}(X^{\vee}, \hat{\mathcal{O}}_{\mathrm{vir}}\otimes \mathcal{O}(\vec{\mathcal{M}}))\) computed in Section 6.

Theorem 3 identifies \(\mathcal{A}^{(k)}\) with a Weyl-Clifford algebra, acting on the Verma module \[\begin{align} H^{*}_{c}(QM\vec{N}(k)) \cong H^{*}_{c}(S^{\bullet}C) \cong S^{\bullet}H^{\text{even}}(C) \otimes \bigwedge H^{\text{odd}}(C) \cong \overline{\mathbb{Q}}_{l}[\mathbf{p}, \mathbf{u}]\otimes \bigwedge \langle \gamma_1, \ldots, \gamma_{2g} \rangle_{\overline{\mathbb{Q}}_{l}}. \end{align}\] Explicitly, the action of \(\mathcal{A}^{(k)}\) is given by \[\begin{align} \label{Akactcoord} &x_{k} = \mathbf{p}\cdot, \;\;\; \theta^{x_{k}}_{i} = \gamma_{i} \cdot, \;\;\; D_{x_k } = \mathbf{u} \cdot,\\ &y_k = \partial_{\mathbf{u}}, \;\;\; \theta^{\vee y_{k}}_{i} = -\partial_{\gamma_i }, \;\;\; D_{y_k } = \partial_{\mathbf{p}}, \nonumber \end{align}\tag{61}\] where \(\mathbf{u} \in H^{0}(C)\) is the unit. The operators \(x_k, D_{x_k}, \theta_{i}^{x_{k}}\), \(i = 1, \ldots, 2g\), freely generate this module from the cyclic vector \[\begin{align} 1 \in H_{c}^{0}(S^{0}C) \cong H^{0}_{c}(QM\vec{N}_{-m_{k}}(k)), \end{align}\] which has \(\mathrm{Fr} \times A^{\vee}\)-weight \(\prod_{l = 1}^{n+1} a^{-\deg\left(\mathcal{M}_{l} \otimes \mathcal{M}_{k}^{-1}\right)}\). Keeping track of all the shifts, we get \[\begin{align} \label{hattiso} &\hat{H}^{*}_{c}(QM\vec{N}(k))[[-2\mathrm{rk}N(k)]] \cong \overline{\mathbb{Q}}_{l}[x_{k}, e \langle 1 \rangle ]\otimes \bigwedge\nolimits_{\overline{\mathbb{Q}}_{l}} \langle \theta^{x_{k}}_{1}, \ldots, \theta^{x_{k}}_{2g} \rangle_{\overline{\mathbb{Q}}_{l}} [[-\kappa_{k}(\vec{m})]] a^{\rho_{k}(\vec{m})}. \end{align}\tag{62}\] Analogously, in the nontrivial local system case, Theorem 3 gives \[\begin{align} \label{hqmnkchi} H^{*}_{c}(QM\vec{N}(k), \chi) \cong \bigoplus\limits_{m = 0}^{\infty} H^{*}_{c}\left( S^{m}C, (\chi^{n+1})^{(m)}\right) \cong \bigwedge H^{1}(C, \chi^{n+1}) \cong \bigwedge \langle \gamma_1^{\chi}, \ldots, \gamma_{2g-2}^{\chi} \rangle_{\overline{\mathbb{Q}}_{l}}, \end{align}\tag{63}\] where \((\chi^{n+1})^{(m)}\) is the natural local system on \(S^{m}C\) whose stalk at \(q_{1} + \ldots + q_{m} \in S^{m}C\) is \(\chi^{n+1}_{q_{1}} \otimes \cdots \otimes \chi^{n+1}_{q_{m}}\). The first isomorphism in 63 follows from the definition of the local system \(\chi\) on \(QM\vec{N}\) as the pullback of \(\chi\) from \(\mathrm{Pic}(C)\) under 11 . The action of \(\mathcal{A}_{\chi}^{(k)}\) is given by \[\begin{align} e\langle \gamma_{i}^{\chi} \rangle &= \gamma_{i}^{\chi} \cdot,\\ f\langle \gamma_{i}^{\chi^{-1}} \rangle &= - \partial_{\gamma_{i}^{\chi}}. \nonumber \end{align}\] The operators \(e\langle \gamma_{i}^{\chi} \rangle\) for \(i = 1, \ldots, 2g-2\) freely generate this module from the one-dimensional cyclic vector \[\begin{align} 1 \in H_{c}^{0}(S^{0}C) \cong H^{0}_{c}(QM\vec{N}_{-m_{k}}(k), \chi). \end{align}\] Applying the shifts, we obtain \[\begin{align} \label{chihattiso} \hat{H}^{*}_{c}(QM\vec{N}(k), \chi)[[-2\mathrm{rk}N(k)]] \cong \bigwedge\nolimits_{\overline{\mathbb{Q}}_{l}} \langle e^{(k)}\langle \gamma_{1}^{\chi} \rangle, \ldots, e^{(k)}\langle \gamma_{2g-2}^{\chi} \rangle \rangle_{\overline{\mathbb{Q}}_{l}} [[-\kappa_{k}(\vec{m})]] a^{\rho_{k}}. \end{align}\tag{64}\]

The associated graded modules of the two filtrations coincide.

7.2 Intermediate algebras↩︎

We introduce an auxiliary family of algebras that will be useful to analyze the filtration on \(\hat{H}_{c}^{*}(QM\vec{N})\). Consider the restriction of \(\iota\) to \(C \times \bigcup_{l = k}^{s}\mathrm{Attr}(l)\): \[\begin{align} \label{iotaks} \iota^{(k, s)}\colon C \times \bigcup\limits_{l = k}^{s}\mathrm{Attr}(l) \rightarrow C \times \bigcup\limits_{l = k}^{s}\mathrm{Attr}(l). \end{align}\tag{65}\] In analogy with 31 , we define operators \(e^{(k, s)}\langle \gamma_{i}^{\chi} \rangle, f^{(k, s)}\langle \gamma_{i}^{\chi^{-1}} \rangle\) on \(H_{c}^{*}(\bigcup_{l=k}^{s}\mathrm{Attr}(l), \chi)\) using 65 .

For \(\chi_{0}^{n+1} \neq 1\), the generators satisfy the following relations: \[\begin{align} &[e^{(k, s)}\langle \gamma_{i}^{\chi} \rangle, f^{(k, s)}\langle \gamma_{j}^{\chi^{-1}} \rangle] = \delta_{ij}(s - k + 1)\eta^{s-k}, \label{commutefsk}\\ &[e^{(k, s)}\langle \gamma_{i}^{\chi} \rangle, \gamma_{j}] = [f^{(k, s)} \langle \gamma_{i}^{\chi} \rangle, \gamma_{j}] = 0, \;\;\; \eta^{s - k + 1} = 0, \;\;\; \eta^{s - k}\gamma_{i} = 0, \nonumber\\ &\sum\limits_{i = 1}^{2g-2} f^{(k, s)}\langle \gamma_{i}^{\chi^{-1}} \rangle e^{(k, s)}\langle \gamma_{i}^{\chi} \rangle + \eta^{s-k}\chi_C(\mathcal{V}^{(s)}/\mathcal{V}^{(k-1)} \otimes \mathcal{K}_{C}^{-1/2} \otimes \mathcal{L}^{-1}) - (s - k)(s - k + 1)\eta^{s - k - 1}\Theta = 0. \nonumber \end{align}\tag{66}\] From the identities \[\begin{align} \iota_{*} &= \iota_{*}^{(k, s)}c_{n+1-s}\left( (\mathcal{N}/\mathcal{N}^{(s)}) \otimes TC \right),\\ \iota^{*} &= c_{k-1} \left(\mathcal{N}^{(k-1)} \otimes TC \right) \iota^{(k, s)*}, \nonumber \end{align}\] we obtain a natural homomorphism \[\begin{align} \mathcal{A}_{\chi} &\rightarrow \mathcal{A}_{\chi}^{(k, s)},\\ e \langle \gamma_{i}^{\chi} \rangle &= \eta^{n+1-s} e^{(k,s)}\langle \gamma_{i}^{\chi} \rangle, \nonumber\\ f \langle \gamma_{i}^{\chi^{-1}} \rangle &= \eta^{k-1} f^{(k, s)}\langle \gamma_{i}^{\chi^{-1}} \rangle, \; i = 1, ..., 2g-2. \nonumber \end{align}\] The action of \(\mathcal{A}_{\chi}^{(k, s)}\) on \(\hat{H}_{c}^{*}(QM\vec{N}(l), \chi)[[-2\mathrm{rk}\,N(l)]]\) for \(k \leq l \leq s\) is given by \[\begin{align} \tag{67} \mathcal{A}^{(k, s)}_{\chi} &\rightarrow \mathcal{A}^{(l)}_{\chi},\\ e^{(k, s)} \langle \gamma_{i}^{\chi} \rangle &= \delta_{l, s} e^{(l)}\langle \gamma_{i}^{\chi} \rangle, \tag{68}\\ f^{(k, s)} \langle \gamma_{i}^{\chi^{-1}} \rangle & = \delta_{l, k}f^{(l)}\langle \gamma_{i}^{\chi^{-1}} \rangle, \; i = 1, ..., 2g-2. \tag{69} \end{align}\] More generally, for any \(\chi\) and \(k \leq k' \leq s' \leq s\) there is a similar map \[\label{eq:restscal} \mathcal{A}^{(k, s)}_{\chi} \rightarrow \mathcal{A}^{(k', s')}_{\chi}.\tag{70}\]

7.3 The trivial-character case↩︎

In this subsection we prove the isomorphism between the cohomology of the quasimap space and that of the symplectic dual variety for the trivial local system.

7.4 The nontrivial-character extension problem↩︎

When \(\chi_0^{n+1}\neq1\), the fixed locus of the resolved surface \(X^\vee\) is reduced: it is the disjoint union of the \(n+1\) fixed points. By contrast, the scheme-theoretic fixed locus of the singular affinization \(X_0^\vee\) is the fat point \[\operatorname{Spec}\overline{\mathbb{Q}}_l[\eta]/(\eta^{n+1}).\] The difference between these two models is exactly the possible \(\eta\)-extension appearing in the \(QM\vec{N}\)-filtration. Thus the associated graded comparison is naturally expressed on the resolved fixed locus, while the filtered comparison can remember the nonreduced structure. In this subsection we show that the filtration splits precisely when \(\eta = 0\), and establish the range of \(g\) and \(\vec{\mathcal{M}}\) in which \(\eta\) acts nontrivially.

We have the decreasing filtration on the \(\mathcal{A}\)-module \(M = \hat{H}_{c}^{*}(QM\vec{N}, \chi)\) defined by \[\begin{align} F^{k}M = H^{*}_{c}\left(QM\vec{N} \setminus \bigcup\limits_{l = 1}^{k}\mathrm{Attr}(l) \right), \;\; k = 1, \ldots, n+1. \end{align}\] By 72 , the associated graded is \[\begin{align} \label{grFM} \mathrm{gr}\, F^{\bullet}M = \bigoplus\limits_{k = 1}^{n+1} \hat{H}_{c}^{*}(QM\vec{N}(k), \chi)[[-2\mathrm{rk}\,N(k)]]. \end{align}\tag{73}\] Since \[\begin{align} H^{*}_{L}((X^{\vee})^{\pi_{1}^{g}}, \hat{\mathcal{O}}_{\mathrm{vir}, \chi}\otimes \mathcal{O}(\vec{\mathcal{M}})) \cong \bigoplus\limits_{k = 1}^{n+1} H^{*}_{L^{k}}(X^{\vee}, \hat{\mathcal{O}}_{\mathrm{vir}, \chi}\otimes \mathcal{O}(\vec{\mathcal{M}})), \end{align}\] the module \(M\) is isomorphic to \(H^{*}_{L}((X^{\vee})^{\pi_{1}^{g}}, \hat{\mathcal{O}}_{\mathrm{vir}, \chi}\otimes \mathcal{O}(\vec{\mathcal{M}}))\) if and only if \(M \cong \mathrm{gr}\, F^{\bullet}M\). Let us give a useful characterization of when the filtration on \(M\) splits.

For what follows, it is convenient to pass to the dual spaces, namely, Borel–Moore homology.

The dual to the space 76 is \[\begin{align} \label{dualwtgenk} &\hat{H}^{BM}_{2g-2}(QM\vec{N}(k)_{2g-2 - m_{k}}, \chi^{-1})[[-2\mathrm{rk}\,N_{2g-2 - m_{k}}(k)]] =\\ &=\hat{H}_{2g-2}^{BM}(S^{2g-2}C, (\chi^{-n-1})^{(m)})[[-2\mathrm{rk}\,N_{2g-2 - m_{k}}(k)]], \end{align}\tag{81}\] which is supported on the linear system \(\mathbb{P}(H^{0}(\mathcal{K}_{C})) \cong \mathbb{P}^{g-1} \overset{i}{\hookrightarrow} S^{2g-2}C\) of the divisor class of \(\mathcal{K}_{C}\), i.e. \[\begin{align} i_{*}: H_{2g-2}^{BM}(\mathbb{P}^{g-1}) \otimes \bigwedge\nolimits^{2g-2}H_{1}^{BM}(C, \chi^{-n-1}) \overset{\cong}{\rightarrow} H_{2g-2}^{BM}(S^{2g-2}C, (\chi^{-n-1})^{(m)}). \end{align}\] Hence the class in 81 is supported in the fiber of the projection 2 over \(\mathcal{L} = \mathcal{M}_{k} \otimes \mathcal{K}_{C}^{-1}\). Analogously, the class in \[\begin{align} \hat{H}^{BM}_{2g-2}(QM\vec{N}(s)_{2g-2 - m_{s}}, \chi^{-1})[[-2\mathrm{rk}\,N_{2g-2 - m_{s}}(s)]] \end{align}\] is supported in the fiber over \(\mathcal{L} = \mathcal{M}_{s} \otimes \mathcal{K}_{C}^{-1}\). Therefore \(\eta\) can only be nonzero if \[\begin{align} \label{mkmsiso} \mathcal{M}_{k} \cong \mathcal{M}_{s}, \end{align}\tag{82}\] which refines the condition 78 . We use the notation \[\begin{align} QM\vec{N}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}} = \pi^{-1}_{\mathcal{L}}(\mathcal{M}_{k} \otimes \mathcal{K}_{C}^{-1}). \end{align}\] Consider the projection to the space of iterated framed extensions \[\begin{align} \pi_{\mathcal{V}^{\bullet}}\colon QM\vec{N}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}} \rightarrow \{\mathcal{V}^{\bullet}\}. \end{align}\] A fiber of this map over some \(\mathcal{V}^{\bullet}\) is a quotient stack \[\begin{align} \mathbb{P}\left( H^{0}(\mathcal{V} \otimes \mathcal{L}^{-1}) \right) / \mathrm{End}(\mathcal{V}^{\bullet}). \end{align}\] Let us consider a scheme \(\widetilde{QM\vec{N}}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}}\), which is mapped to the affine space of iterated extensions \[\begin{align} \label{pifromrqmntoext} \pi_{\mathcal{V}^{\bullet}}\colon \widetilde{QM\vec{N}}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}} \rightarrow \{\mathcal{V}^{\bullet}\} \end{align}\tag{83}\] with fibers \(\mathbb{P}\left( H^{0}(\mathcal{V} \otimes \mathcal{L}^{-1}) \right)\). The scheme \(\widetilde{QM\vec{N}}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}}\) has a natural derived structure with virtual tangent bundle (compare with 10 ) \[\begin{align} T^{\mathrm{vir}} \widetilde{QM\vec{N}}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}} = \sum\limits_{l = 1}^{n+1} \mathrm{Ext}^{\bullet}(\mathcal{L}, \mathcal{M}_{l})\otimes \mathcal{O}_{QM\vec{N}}(1) -\mathrm{Hom}(\mathcal{L}, \mathcal{L}) + \sum\limits_{l < r}\mathrm{Ext}^{1}(\mathcal{M}_{r}, \mathcal{M}_{l}). \end{align}\]

The pullback under the natural projection \[\begin{align} \label{rig} \widetilde{QM\vec{N}}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}} \rightarrow QM\vec{N}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}} \end{align}\tag{84}\] identifies Borel–Moore homology with shift \([[2\sum\limits_{l < r} h^{0}(\mathcal{M}_{l}\otimes \mathcal{M}_{r})]]\), and the action of \(\eta\) lifts to the homology of the rigidification. Let \(\mathrm{Attr}'(l)\) denote the preimage under 84 of the intersection of \(\mathrm{Attr}(l)\) and \(\pi^{-1}_{\mathcal{L}}\left( \mathcal{M}_{k} \otimes \mathcal{K}_{C}^{-1}\right)\).

We will need the following lemma.

After dualizing 75 , passing to the rigidification, and reducing common shifts, we obtain a map \[\begin{align} \label{etaBMh} \eta\colon H^{BM}_{\mathrm{top}}(\mathrm{Attr}'(s)) = \overline{\mathbb{Q}}_{l}[\mathrm{Attr}'(s)] \rightarrow \overline{\mathbb{Q}}_{l}[\mathrm{Attr}'(k)] = H^{BM}_{\mathrm{top}}(\mathrm{Attr}'(k)). \end{align}\tag{86}\]

From Corollary 3 we conclude that the boundary map \[\begin{align} H^{0}(\mathcal{M}_{s} \otimes \mathcal{M}_{k}^{-1}\otimes\mathcal{K}_{C}) \rightarrow H^{1}(\mathcal{V}^{(s-1)}\otimes \mathcal{M}_{k}^{-1}\otimes\mathcal{K}_{C}) \end{align}\] is surjective for any \(\mathcal{V}^{\bullet}\) (so \(\pi_{\mathcal{V}^{\bullet}}(\mathrm{Attr}'(s)) = \{\mathcal{V}^{\bullet}\}\)) and generically has one-dimensional kernel, hence for generic \(\mathcal{V}^{\bullet}\) we have \[\begin{align} h^{0}(\mathcal{V}^{(s)} \otimes \mathcal{M}_{k}^{-1}\otimes\mathcal{K}_{C}) = h^{0}(\mathcal{V}^{(s-1)} \otimes \mathcal{M}_{k}^{-1}\otimes\mathcal{K}_{C}) + 1 = h^{0}(\mathcal{V}^{(k)} \otimes \mathcal{M}_{k}^{-1}\otimes\mathcal{K}_{C}) + 1. \end{align}\]

We can now prove the following.

We can now prove the main result, Theorem 2.

8 Relations in the algebra of correspondences↩︎

8.1 Supercommutator relations in \(\mathcal{A}_{\chi}\)↩︎

Substituting \(\alpha, \beta \in \{\mathbf{p}, \gamma_{i}, 1\}\) in 19 and using the explicit formula for \(\psi\) of Proposition 3 gives the relations in \(\mathcal{A}_{\chi}\) for the trivial \(\chi_{0}^{n+1}\): \[\begin{align} &[e\langle \mathbf{p} \rangle, f \langle \mathbf{p} \rangle] = 0, \tag{91}\\ &[e\langle \mathbf{p} \rangle, f \langle \gamma_{i} \rangle] = 0,\\ &[e\langle \gamma_{i} \rangle, f \langle \mathbf{p} \rangle] = 0,\\ &[e\langle \gamma_{i} \rangle, f \langle \gamma_{j}^{\vee} \rangle] = \delta_{ij} (n+1)\eta^{n},\tag{92}\\ &[e\langle 1 \rangle, f \langle 1 \rangle] = n\,\eta^{n-1}\chi_C(\mathcal{V}\otimes \mathcal{L}^{-1}) - (n+1)n(n-1)\eta^{n-2}\Theta,\tag{93}\\ &[e\langle 1 \rangle, f \langle \mathbf{p} \rangle] = -[f\langle 1 \rangle, e \langle \mathbf{p} \rangle] = (n+1)\eta^{n},\tag{94}\\ &[e\langle 1 \rangle, f \langle \gamma_{i} \rangle] = -[f\langle 1 \rangle, e \langle \gamma_{i} \rangle] = n(n+1)\eta^{n-1}\gamma_{i}.\tag{95} \end{align}\]

By Lemma 4, the operators on \(\hat{H}^{*}_{c}(QM\vec{N})\) of multiplication by \(\eta, \gamma_{i} \in H^{*}(QM\vec{N})\) have the following commutators: \[\begin{align} [e\langle \mathbf{p} \rangle, \eta] &= 0, &[f\langle \mathbf{p} \rangle, \eta] &= 0,\tag{96}\\ [e\langle \gamma_{i} \rangle, \eta] &= 0, &[f\langle \gamma_{i} \rangle, \eta] &= 0,\tag{97}\\ [e\langle 1 \rangle, \eta] &= -e\langle \mathbf{p} \rangle, &[f\langle 1 \rangle, \eta] &= f\langle \mathbf{p} \rangle,\tag{98}\\ [e\langle 1 \rangle, \gamma_{i}] &= -e \langle \gamma_{i} \rangle, &[f\langle 1 \rangle, \gamma_{i}] &= f \langle \gamma_{i} \rangle,\tag{99}\\ [e\langle \gamma_{i} \rangle, \gamma_{j}^{\vee}] &= -\delta_{ij} e\langle \mathbf{p} \rangle, &[f\langle \gamma_{i} \rangle, \gamma_{j}^{\vee}] &= \delta_{ij} f \langle \mathbf{p} \rangle,\tag{100}\\ [e\langle \mathbf{p} \rangle, \gamma_{i}] &= 0, &[f\langle \mathbf{p} \rangle, \gamma_{i}] &= 0,\tag{101} \end{align}\]

Commutators with \(\chi_C(\mathcal{V}\otimes \mathcal{L}^{-1})\) are given by \[\begin{align} [e\langle \alpha \rangle, \chi_C(\mathcal{V}\otimes \mathcal{L}^{-1})] &= -(n+1)e\langle \alpha \rangle, &[f\langle \alpha \rangle, \chi_C(\mathcal{V}\otimes \mathcal{L}^{-1})] &= (n+1)f\langle \alpha \rangle, \;\;\; \alpha \in H^{1}_{c}(C, \chi). \label{[epdeg]} \end{align}\tag{102}\] Using the identity \[\begin{align} H_{c}^{*}(C, \overline{\mathbb{Q}}_{l}) &\rightarrow H_{c}^{*}(C \times C, \chi_{0}^{n+1} \boxtimes \chi_{0}^{-n-1}),\\ \Delta_{C*}(1) &= -\sum\limits_{i = 1}^{2g-2}\gamma_{i}^{\chi} \otimes \gamma_{i}^{\chi^{-1}}, \end{align}\] we analogously compute supercommutators in \(\mathcal{A}_\chi\) when \(\chi_{0}^{n+1} \neq 1\): \[\label{commutinAchin} [e\langle \gamma_{i}^{\chi} \rangle, f\langle \gamma_{j}^{\chi^{-1}} \rangle] = \delta_{ij}(n+1)\eta^{n},\qquad [e\langle \gamma_{i}^{\chi} \rangle, \eta] = [f\langle \gamma_{i}^{\chi^{-1}} \rangle, \eta] =[e\langle \gamma_{i}^{\chi} \rangle, \gamma_{j}] = [f\langle \gamma_{i}^{\chi^{-1}} \rangle, \gamma_{j}] = 0.\tag{103}\]

8.2 Casimir relations in \(\mathcal{A}_{\chi}\)↩︎

By Proposition 4, and the identities \[\label{deltapush} \begin{align} \Delta_{C*}(\mathbf{p}) &= \mathbf{p} \otimes \mathbf{p},\\ \Delta_{C*}(\gamma_{i}) &= \mathbf{p} \otimes \gamma_{i} + \gamma_{i} \otimes \mathbf{p},\\ \Delta_{C*}(1) &= \mathbf{p} \otimes 1 + 1 \otimes \mathbf{p} - \sum_{i=1}^{2g} \gamma_{i} \otimes \gamma_{i}^{\vee}, \end{align}\tag{104}\] the operators in \(\mathcal{A}_{\chi}\) for the trivial \(\chi_{0}^{n+1}\) satisfy \[\begin{align} &e\langle \mathbf{p} \rangle f \langle \mathbf{p} \rangle + \eta^{n+1} = 0, \tag{105}\\ &e\langle \mathbf{p} \rangle f\langle \gamma_{i} \rangle + f\langle \mathbf{p} \rangle e\langle \gamma_{i} \rangle + (n+1)\eta^{n}\gamma_{i} = 0,\tag{106}\\ &\begin{aligned} &f\langle \mathbf{p} \rangle e \langle 1 \rangle + e \langle \mathbf{p} \rangle f\langle 1 \rangle - \sum_{i=1}^{g}\bigl( e \langle \gamma_{i} \rangle f \langle \gamma_{i}^{\vee} \rangle + f \langle \gamma_{i} \rangle e \langle \gamma_{i}^{\vee} \rangle \bigr)\\ &\quad + \eta^{n}\chi_C(\mathcal{V} \otimes \mathcal{L}^{-1}) - n(n+1) \eta^{n-1} \Theta \end{aligned} = 0\tag{107}. \end{align}\] For the nontrivial \(\chi_{0}^{n+1}\) we have \[\begin{align} &\eta^{n+1} = 0, \qquad \eta^{n}\gamma_{i} = 0, \qquad \sum\limits_{i = 1}^{2g-2}f\langle \gamma_{i}^{\chi^{-1}}\rangle e\langle\gamma_{i}^{\chi}\rangle + \eta^{n}\chi_C(\mathcal{V}\otimes\mathcal{K}_{C}^{-1/2}\otimes\mathcal{L}^{-1})-n(n+1)\eta^{n-1}\Theta=0. \nonumber \end{align}\]

8.3 Supercommutator relations in \(\mathcal{A}^{(k)}_{\chi}\)↩︎

Analogously to Subsection 8.1 we get supercommutators for descendants in \(\mathcal{A}_{\chi}^{(k)}\) for the trivial \(\chi_{0}^{n+1}\): \[\begin{align} &[e^{(k)}\langle \mathbf{p} \rangle , f^{(k)}\langle \mathbf{p} \rangle ] = 0, \tag{108}\\ &[e^{(k)}\langle 1 \rangle , f^{(k)}\langle 1 \rangle ] = 0, \tag{109}\\ &[e^{(k)}\langle 1 \rangle , f^{(k)}\langle \mathbf{p} \rangle ] = -[f^{(k)}\langle 1 \rangle , e^{(k)}\langle \mathbf{p} \rangle ] = 1,\tag{110}\\ &[e^{(k)}\langle 1 \rangle , \eta ] = - e^{(k)}\langle \mathbf{p} \rangle, \nonumber \\ &[f^{(k)}\langle 1 \rangle , \eta ] = f^{(k)}\langle \mathbf{p} \rangle, \nonumber\\ &[e^{(k)}\langle 1 \rangle , \gamma_{i}] = - e^{(k)}\langle \gamma_i \rangle, \nonumber\\ &[f^{(k)}\langle 1 \rangle , \gamma_{i}] = f^{(k)}\langle \gamma_i \rangle, \nonumber \\ &[e^{(k)}\langle \gamma_{i} \rangle , f^{(k)}\langle \gamma_{j}^{\vee} \rangle ] = \delta_{ij}, \tag{111}\\ &[e^{(k)}\langle \gamma_{i} \rangle , \gamma_{j}^{\vee} ] = - \delta_{ij} e^{(k)}\langle \mathbf{p} \rangle, \nonumber \\ &[f^{(k)}\langle \gamma_{i} \rangle , \gamma_{j}^{\vee} ] = \delta_{ij} f^{(k)}\langle \mathbf{p} \rangle. \nonumber \end{align}\] For \(\chi_{0}^{n+1} \neq 1\), the only nontrivial supercommutator is \[\begin{align} [e^{(k)}\langle \gamma_{i}^{\chi} \rangle, f^{(k)}\langle \gamma_{j}^{\chi^{-1}} \rangle] = \delta_{ij}. \end{align}\]

8.4 Casimir relations in \(\mathcal{A}^{(k)}_{\chi}\)↩︎

Using 104 we compute the Casimir relations in \(\mathcal{A}_{\chi}^{(k)}\) for the trivial \(\chi_{0}^{n+1}\): \[\begin{align} &e^{(k)}\langle \mathbf{p} \rangle f^{(k)} \langle \mathbf{p} \rangle + \eta = 0 \tag{112}\\ &e^{(k)}\langle \mathbf{p} \rangle f^{(k)}\langle \gamma_{i} \rangle + f^{(k)}\langle \mathbf{p} \rangle e^{(k)}\langle \gamma_{i} \rangle + \gamma_{i} = 0 \tag{113} \\ &f^{(k)}\langle \mathbf{p} \rangle e^{(k)} \langle 1 \rangle + e^{(k)} \langle \mathbf{p} \rangle f^{(k)}\langle 1 \rangle - \sum\limits_{i = 1}^{g} \left( e^{(k)} \langle \gamma_{i} \rangle f^{(k)} \langle \gamma_{i}^{\vee} \rangle + f^{(k)} \langle \gamma_{i} \rangle e^{(k)} \langle \gamma_{i}^{\vee} \rangle \right) + \nonumber \\ &+\chi_C(\mathcal{M}_{k} \otimes \mathcal{L}^{-1}) = 0 \tag{114} \end{align}\] Analogously, for \(\chi_{0}^{n+1} \neq 1\) the operators in \(\mathcal{A}_{\chi}^{(k)}\) satisfy \[\begin{align} & \eta = 0, \qquad \gamma_{i} = 0, \qquad \sum_{i = 1}^{2g-2} f^{(k)}\langle \gamma_{i}^{\chi^{-1}} \rangle e^{(k)}\langle \gamma_{i}^{\chi} \rangle + \chi_C(\mathcal{M}_{k} \otimes \mathcal{K}_{C}^{-1/2} \otimes \mathcal{L}^{-1}) = 0. \label{eta0onk} \end{align}\tag{115}\]

9 Degree zero descendants as differential operators↩︎

In this section we describe the full algebra of correspondences. The degree-zero descendants will be identified with the global vector fields on the resolution \(X^{\vee}\) preserving the exceptional divisor. This will produce a natural module for the algebra \(\mathcal{A}\).

9.1 The degree-zero descendants in \(\mathcal{A}_{\chi}\).↩︎

Consider the decomposition 52 of \(H^{1}_{c}(C)\) into two Lagrangian subspaces \[\begin{align} H^{1}_{c}(C) = H_{A} \oplus H_{B}. \end{align}\] This induces a Lagrangian submodule of \(\mathcal{E}\), \[\begin{align} \mathcal{E}_{A} = q^{-1}\Omega_{R} \otimes H_{A}. \end{align}\]

Let us introduce a super commutative algebra \[\mathcal{R} = \bigwedge\nolimits^{\bullet}_{R} \mathcal{E}_{A},\] and denote the corresponding super variety \(\mathcal{X}_{0}^{\vee}\). It has a natural super Poisson structure, extending the Poisson structure 20 on \(R\): \[\begin{align} &\pi \colon q^{-1} \otimes \mathcal{R}\otimes \mathcal{R} \rightarrow \mathcal{R} \\ &\{ x , \theta_{i}^{y} \} = - \{ y , \theta_{i}^{x} \} = n(n+1)\eta^{n-1}\gamma_i , \qquad \{x , \gamma_i \} = \theta_{i}^{x} \qquad \{y , \gamma_{i} \} = -\theta_i^{y},\\ &\{ x , \theta_{i}^{x}\} = \{ y , \theta_{i}^{y}\} = \{ \theta_{i}^{x} , \theta_{j}^{x}\} = \{ \theta_{i}^{x} , \theta_{j}^{x}\} = \{ \theta_{i}^{y} , \theta_{j}^{y}\} = 0, \;\; i, j = 1, ..., g. \end{align}\] For \(r \in R\) let \[\begin{align} \xi_{r} = - \{r , \cdot \} \in \mathrm{Der}(R) \end{align}\] be the corresponding Poisson derivation of \(R\). Let us denote by \[\begin{align} \mathrm{Der}_{\pi}(R) \subset \mathrm{Der}(R) \end{align}\] the submodule of Poisson derivations. The Lie derivative extends \(\xi_{r}\) to an even derivation of \(\mathcal{R}\): \[\mathrm{Lie}_{\xi_{r}} \in \mathrm{Der}(\mathcal{R}).\] With the help of the pairing 23 , the elements \(r \in R\) and \(\gamma_{i}^{\vee} \in H_{B},\; i = 1, ..., g\) produce a map \[\iota_{\xi_{r}} \otimes \langle \gamma_{i}^{\vee}, \cdot \rangle \colon \mathcal{E}_{A} \rightarrow R,\] which gives rise to an odd derivation \[\theta_{i}^{\vee r} \in \mathrm{Der}(\mathcal{R}).\] The elements \(\mathrm{Lie}_{\xi_{r}}, \theta_{i}^{\vee r}, \;\; i = 1, ..., g, \;\; r = x, y, \eta\) generate a (proper) super Lie subalgebra \[\begin{align} \mathrm{Der}_{\pi}(\mathcal{R}) \subset \mathrm{Der}(\mathcal{R}). \end{align}\] Let \(\mathcal{D}(\mathcal{R})\) denote the algebra of differential operators on \(\mathcal{R}\), and let \[\begin{align} \mathcal{D}_{\pi}(\mathcal{R}) \subset \mathcal{D}(\mathcal{R}) \end{align}\] denote the subalgebra generated by \(\mathrm{Der}_{\pi}(\mathcal{R})\). A possible construction of \(\mathcal{D}_{\pi}\) goes as follows. Let \(\mathfrak{a}_{\pi}\) be a super Lie algebroid \[\begin{align} \label{Liealgebroid} \mathfrak{a}_{\pi} = \mathcal{R} \oplus \mathrm{Der}_{\pi}(\mathcal{R}), \end{align}\tag{116}\] where the bracket between the summands is given by the natural pairing, and is trivial when restricted to \(\mathcal{R}\). The natural projection and inclusion define the symbol map and the connection, respectively, \[\begin{align} \label{symcon} \sigma: \mathfrak{a}_{\pi} \rightarrow \mathrm{Der}_{\pi}(\mathcal{R}) \qquad \nabla: \mathrm{Der}_{\pi}(\mathcal{R}) \rightarrow \mathfrak{a}_{\pi}. \end{align}\tag{117}\]

The algebra \(\mathcal{D}_{\pi}(\mathcal{R})\) is defined as \[\label{eq:UR} \mathcal{D}_{\pi}(\mathcal{R}) = U_{\mathcal{R}}(\mathfrak{a}_{\pi}) \mathrel{\vcenter{:}}= U_{\overline{\mathbb{Q}}_{l}}(\mathfrak{a}_{\pi})/\left(1_{\mathcal{R}} - 1_{U_{\overline{\mathbb{Q}}_{l}}(\mathfrak{a}_{\pi})}, r \otimes D - r \cdot D, \;\; r \in \mathcal{R} \subset \mathfrak{a}_{\pi}, D \in \mathfrak{a}_{\pi} \right).\tag{118}\] Let \(\mathcal{A}^{\geq 1.5} \subset \mathcal{A}^{\geq1}\) be a subalgebra of \(\mathcal{A}\), generated by \(\mathcal{A}^{2}\) and \(\theta_{i}^{x}\), \(\theta_{i}^{y}, \gamma_i \;\; i = 1, ..., g\). It follows from Proposition 20 that \(\Phi\) maps \(\mathcal{R} \subset \mathcal{D}_{\pi}(\mathcal{R})\) isomorphically onto \(\mathcal{A}^{\geq 1.5}\), and in particular, identifies \(R \subset \mathcal{D}_{\pi}(\mathcal{R})\) with \(\mathcal{A}^{2}\). The relations from Appendix 8 in the algebra \(\mathcal{A}\) imply that the \(\mathcal{R}\)-algebra map \[\begin{align} \label{DtoA} \Phi \colon &\mathcal{D}_{\pi}(\mathcal{R}) \rightarrow \mathcal{A} \qquad \mathrm{Lie}_{\xi_{x}} \mapsto e \langle 1 \rangle, \qquad \mathrm{Lie}_{\xi_{y}} \mapsto - f \langle 1 \rangle \qquad \mathrm{Lie}_{\xi_{\eta}} \mapsto \frac{\chi_{C}(\mathcal{V}\otimes\mathcal{L}^{-1})}{n+1}\\ & \theta_{i}^{\vee x} \mapsto e \langle \gamma_i^{\vee} \rangle \qquad \theta_{i}^{\vee y} \mapsto - f \langle \gamma_i^{\vee} \rangle \qquad \theta_{i}^{\vee \eta} \mapsto \gamma_{i}^{\vee}, \;\; i = 1, ..., g. \nonumber \end{align}\tag{119}\] is a homomorphism of algebras.

Analogously, we define an \(\mathcal{A}^{(k)2}\)-module \[\mathcal{E}_{A}^{(k)} = q^{-1}\Omega^{1}_{\mathcal{A}^{(k)2}} \otimes H_{A}\] and consider a super commutative algebra \[\mathcal{R}^{(k)} = \bigwedge\nolimits^{\bullet} \mathcal{E}^{(k)}_{A},\] which we view as a structure algebra of a super variety \(\mathcal{U}_{k}\). Also there is a natural extension of the symplectic form 35 to a super Poisson structure \[\begin{align} &\{ \cdot, \cdot\} \colon q^{-1}\otimes \mathcal{R}^{(k)} \otimes \mathcal{R}^{(k)} \rightarrow \mathcal{R}^{(k)}, \end{align}\] for which \(\theta_{i}^{x_{k}}, \theta_{i}^{y_{k}}, \; i = 1, ..., g\) are Poisson-central elements. The maps of super algebras \(\mathcal{R}^{(k)} \rightarrow \mathcal{R}\) can be glued in a natural way to define a super variety with a map \[\mathcal{X}^{\vee} \rightarrow \mathcal{X}_{0}^{\vee}.\] By Remark 21 the structure sheaf of \(\mathcal{X}^{\vee}\) is given by \[\mathcal{O}_{\mathcal{X}^{\vee}} = \bigwedge\nolimits^{\bullet} \Omega_{X^{\vee}} \otimes H_{A}.\]

Using the pairing 38 , for elements \(r \in \mathcal{A}^{(k)2}\) and \(\gamma_i^{\vee}, \; i = 1, ..., g\), we define a map \[\iota_{\xi_{r}} \otimes \langle \gamma_{i}^{\vee}, \cdot \rangle \colon \mathcal{E}_{A}^{(k)} \rightarrow \mathcal{A}^{(k)2},\] which gives rise to an odd derivation. With a slight abuse of notation 32 we denote this derivation by \[\theta_{i}^{\vee r} \in \mathrm{Der}(\mathcal{R}^{(k)}).\] The operators \(\mathrm{Lie}_{\xi_{x_k}}, \mathrm{Lie}_{\xi_{y_k}}, \theta_{i}^{\vee x_k}, \theta_{i}^{\vee y_k}\) generate a super Lie algebra \(\mathrm{Der}(\mathcal{R}^{(k)})\). Analogously to 116 , we consider a Lie algebroid \[\mathrm{At}(\mathcal{R}^{(k)}) = \mathcal{R}^{(k)} \oplus \mathrm{Der}(\mathcal{R}^{(k)}),\] of first-order differential operators on \(\mathcal{R}^{(k)}\), also referred to as the Atiyah algebra of \(\mathcal{R}^{(k)}\). The algebra of differential operators on \(\mathcal{R}^{(k)}\) can be defined as \[\mathcal{D}(\mathcal{R}^{(k)}) = U_{\mathcal{R}^{(k)}}\left(\mathrm{At}(\mathcal{R}^{(k)})\right)/\left( 1_{\mathcal{R}^{(k)}} - 1_{U}\right),\] where the ‘universal enveloping over \(\mathcal{R}^{(k)}\)’ is defined similarly to 118 . This algebra is another description of the Weyl-Clifford algebra in Definition 9. Accordingly, Theorem 3 can be reformulated as the isomorphism of \(\mathcal{R}^{(k)}\)-algebras \[\begin{align} &\mathcal{D}(\mathcal{R}^{(k)}) \rightarrow \mathcal{A}^{(k)} \qquad \mathrm{Lie}_{\xi_{x_k}} \mapsto e^{(k)} \langle 1 \rangle , \qquad \mathrm{Lie}_{\xi_{y_k}} \mapsto - f^{(k)} \langle 1 \rangle, \qquad \mathrm{Lie}_{ \xi_{\eta} } \mapsto \chi_{C}(\mathcal{M}_{k} \otimes \mathcal{L}^{-1}),\\ &\theta_{i}^{\vee x_{k}} \mapsto e\langle \gamma_{i}^{\vee} \rangle, \qquad \theta_{i}^{\vee y_{k}} \mapsto - f\langle \gamma_{i}^{\vee} \rangle, \qquad \theta_{i}^{\vee \eta} \mapsto - x_{k} f\langle \gamma_{i}^{\vee} \rangle - y_{k}e\langle \gamma_{i}^{\vee} \rangle, \; i = 1, ..., g. \end{align}\] We will not make a distinction between these algebras.

By 43 and 44 the map 47 from \(\mathcal{A}\) to \(\mathcal{A}^{(k)}\) is defined on degree zero generators by \[\begin{align} &e \langle 1 \rangle \mapsto \int\limits_{C} c_{n+1-k} (\mathcal{N}/\mathcal{N}^{(k)} ) \iota^{(k)}_{*} 1 \otimes \cdot = \nonumber \\ &\eta^{n+1-k} e^{(k)}\langle 1 \rangle + x_{k} \eta^{n-k}\chi_C(\mathcal{V}/\mathcal{V}^{(k)}\otimes \mathcal{L}^{-1}) - x_{k} (n+1-k)(n-k)\eta^{n-k-1}\Theta \nonumber \\ &-(n+1-k)\eta^{n-k} \sum\limits_{i = 1}^{g} \left( e^{(k)}\langle \gamma_i \rangle \gamma_{i}^{\vee} + \gamma_{i} e^{(k)}\langle \gamma_i^{\vee} \rangle \right) \tag{120}\\ &f \langle 1 \rangle \mapsto -\int\limits_{C} c_{k-1} (\mathcal{N}^{(k-1)} \otimes TC) \iota^{(k)*} 1 \otimes \cdot = \nonumber \\ &\eta^{k-1}f^{(k)}\langle 1 \rangle - y_{k} \eta^{k-2}\chi_C(\mathcal{V}^{(k-1)}\otimes \mathcal{L}^{-1}) + y_{k} (k-1)(k-2)\eta^{k-3}\Theta \nonumber \\ &-(k-1)\eta^{k-2} \sum\limits_{i = 1}^{g} \left( f^{(k)}\langle \gamma_{i} \rangle \gamma_{i}^{\vee} + \gamma_{i} f^{(k)}\langle \gamma_{i}^{\vee} \rangle \right), \nonumber \\ &\chi_{C}(\mathcal{V}\otimes \mathcal{L}^{-1}) \mapsto (n+1)\chi_{C}(\mathcal{M}_{k}\otimes\mathcal{L}^{-1}) + \sum\limits_{l = 1}^{n+1}(m_l - m_k ) \tag{121} \end{align}\]

This homomorphism allows us to prove the following.

From now on, we make no distinction between \(\mathcal{D}_{\pi}(\mathcal{R})\) and \(\mathcal{A}\). With the help of the formulas 45 , 46 , 120 , and 121 , we can explicitly compute the \(\mathcal{R}\)-algebra maps \[\begin{align} \label{phikmaps} \varphi^{(k)} \colon \mathcal{A} &\rightarrow \mathcal{A}^{(k)} \\ \theta_{i}^{\vee x} &\mapsto (n+2-k)(x_k y_k )^{n+1-k}\theta^{\vee x_{k}}_{i} + (n+1-k)x_{k}^{n+2-k}y_k^{n-k}\theta^{\vee y_k}_{i}, \; i = 1, ..., g \nonumber \\ \theta_{i}^{\vee y} &\mapsto k(x_k y_k )^{k-1}\theta^{\vee y_{k}}_{i} + (k-1)x_k^{k-2}y_{k}^{k}\theta^{\vee x_k}_{i}, \;i = 1, ... g \nonumber \\ \theta_{i}^{\vee \eta} &\mapsto \gamma_{i}^{\vee}, \; i = 1, ..., g \nonumber \\ \mathrm{Lie}_{\xi_{x}} &\mapsto \mathrm{Lie}_{\xi_{x(x_k , y_k )}} + x_k^{n+1-k}y_k^{n-k} \sum\limits_{l > k}(m_l - m_k) \nonumber \\ \mathrm{Lie}_{\xi_{y}} &\mapsto \mathrm{Lie}_{\xi_{y(x_k , y_k)}} + x_{k}^{k - 2}y_{k}^{k-1}\sum\limits_{l < k}(m_{l} - m_{k}) \nonumber \\ \mathrm{Lie}_{\xi_{\eta}} &\mapsto \mathrm{Lie}_{\xi_{\eta(x_k , y_k)}} + \frac{1}{n+1} \sum\limits_{l = 1}^{n+1}(m_l - m_k) \nonumber \end{align}\tag{122}\] Here \(\mathrm{Lie}_{\xi_{x(x_k , y_k )}}\), \(\mathrm{Lie}_{\xi_{y(x_k , y_k )}}\), and \(\mathrm{Lie}_{\xi_{\eta(x_k , y_k)}}\) are the transformations under the change of variables 48 of Lie derivatives acting on \(\mathcal{O}_{X^{\vee}}\). Explicitly, they are given by \[\begin{align} &\mathrm{Lie}_{\xi_{x(x_k , y_k )}} = (n+2-k)(x_k y_k )^{n+1-k} \mathrm{Lie}_{\xi_{x_k}} + (n+1-k)x_{k}^{2}(x_k y_k )^{n-k}\mathrm{Lie}_{\xi_{y_k}} \nonumber \\ &- (n+2 - k)(n+1-k)x_k^{n+1-k}y_k^{n-k}\sum\limits_{i = 1}^{g}(\theta_{i}^{y_{k}} \theta_{i}^{\vee x_k } + \theta_{i}^{x_k}\theta_{i}^{\vee y_k }) \tag{123}\\ &- (n+2-k)(n+1-k)y_{k}(x_{k}y_{k})^{n-k}\sum\limits_{i = 1}^{g}\theta_{i}^{x_k}\theta_{i}^{\vee x_k } - (n+1-k)(n-k)x_{k}^{n+2-k}y_{k}^{n-1-k}\sum\limits_{i = 1}^{g}\theta_{i}^{y_{k}}\theta_{i}^{\vee y_k } \nonumber \\ &\mathrm{Lie}_{\xi_{y(x_k , y_k )}} = (k-1)x_k^{k-2} y_k^{k} \mathrm{Lie}_{\xi_{x_k}} + k (x_k y_k )^{k-1}\mathrm{Lie}_{\xi_{y_k}} \nonumber \\ &-k(k-1)x_k^{k-2}y_k^{k-1}\sum\limits_{i = 1}^{g}(\theta_{i}^{x_k}\theta_{i}^{\vee y_k} + \theta_{i}^{y_k}\theta_{i}^{\vee x_{k}}) \nonumber \\ &-(k-1)(k-2)x_{k}^{k-1}y_{k}^{k} \sum\limits_{i = 1}^{g} \theta_{i}^{x_k}\theta_{i}^{\vee x_k} - k(k-1)x_k (x_{k}y_{k})^{k-2}\sum\limits_{i = 1}^{g} \theta_{i}^{y_k}\theta_{i}^{\vee y_k} \tag{124}\\ &\mathrm{Lie}_{\xi_{\eta(x_k , y_k)}} = y_k \mathrm{Lie}_{\xi_{x(x_k , y_k)}} + x_k \mathrm{Lie}_{\xi_{y(x_k , y_k)}} - \sum\limits_{i = 1}^{g}(\theta_{i}^{x_k}\theta_{i}^{\vee y_k} + \theta_{i}^{y_k} \theta_{i}^{\vee x_k}) \tag{125} \end{align}\] The algebra \(\mathcal{A}^{(k)}\) can be viewed as an algebra of differential operators on the supervariety \(\mathcal{U}_{k}\). The homomorphisms 122 uniquely define the transition maps \[\begin{align} &\mathrm{At}(\mathcal{R}^{(k)})\big|_{\mathcal{U}_{k} \cap \mathcal{U}_{s}} \rightarrow \mathrm{At}(\mathcal{R}^{(s)})\big|_{\mathcal{U}_{s} \cap \mathcal{U}_{k}}, \qquad k < s\\ & \left( r , \;\mathrm{Lie}_{\xi_{x_{k}}} \right) \mapsto \left( r - \frac{1}{y_{k}}\sum\limits_{l = k+1}^{s - 1} (m_{l} - m_{s}) , \; \mathrm{Lie}_{\xi_{x_{k}(x_s , y_s)}} \right)\\ &\left( r , \; \mathrm{Lie}_{\xi_{y_{k}}} \right) \mapsto \left( r + \frac{1}{x_{k}} \sum\limits_{l = k}^{s - 1} (m_{l} - m_{s}) , \; \mathrm{Lie}_{\xi_{y_{k}(x_{s}, y_{s})}} \right) , \;\; r \in \mathcal{R}^{(k)}. \end{align}\]

Using the commutator relations 110 , we can identify \(\xi_{x_{k}}\) and \(\xi_{y_{k}}\) with the coordinate basis vectors \[\begin{align} \xi_{x_{k}} = -\partial_{y_{k}} \qquad \xi_{y_{k}} = \partial_{x_{k}}. \end{align}\] Thus, we can rewrite the transition maps in the case \(s = k+1\) in the form \[\begin{align} \label{transkk431} &\mathrm{At}(\mathcal{R}^{(k)})\big|_{\mathcal{U}_{k} \cap \mathcal{U}_{k+1}} \rightarrow \mathrm{At}(\mathcal{R}^{(s)})\big|_{\mathcal{U}_{k+1} \cap \mathcal{U}_{k}} \\ & \left( r ,\; \mathrm{Lie}_{\partial_{y_{k}}} \right) \mapsto \left( r , \;\mathrm{Lie}_{\partial_{y_{k}}} \right) \nonumber \\ &\left( r , \mathrm{Lie}_{\partial_{x_{k}}} \right) \mapsto \left( r + \frac{m_k - m_{k+1}}{x_{k}} , \; \mathrm{Lie}_{\partial_{x_{k}}} \right) , \;\; r \in \mathcal{R}^{(k)} \nonumber \end{align}\tag{126}\] Let \(\mathcal{O}_{X^{\vee}}(\vec{\mathcal{M}})\) be a \(\mathrm{Fr}\times A^{\vee}\)-equivariant invertible sheaf of \(\mathcal{O}_{X^{\vee}}\)-modules, satisfying \[\label{eq:OM} \mathcal{O}_{X^{\vee}}(\vec{\mathcal{M}})\big|_{\bar{L}_{k}} \cong \mathcal{O}_{\mathbb{P}^{1}}(m_k - m_{k+1} ), \;\;\; k = 1, ..., n.\tag{127}\] This condition defines \(\mathcal{O}_{X^{\vee}}(\vec{\mathcal{M}})\) uniquely up to linearization (i.e. a choice of \(\mathrm{Fr}\times A^{\vee}\)-weight of a fiber over some fixed point), which we specify in 55 . Then we define an invertible sheaf \[\mathcal{O}_{\mathcal{X}^{\vee}}(\vec{\mathcal{M}}) = \mathcal{O}_{\mathcal{X}^{\vee}}\underset{\mathcal{O}_{X^{\vee}}}{\otimes} \mathcal{O}_{X^{\vee}}(\vec{\mathcal{M}})\] of \(\mathcal{O}_{\mathcal{X}^{\vee}}(\vec{\mathcal{M}})\)-modules. The transition maps 126 glue the algebras \(\mathrm{At}(\mathcal{R}^{(k)})\) to a sheaf \(\mathrm{At}_{\mathcal{O}_{\mathcal{X}^{\vee}}(\vec{\mathcal{M}})}\) of first-order differential operators on \(\mathcal{O}_{\mathcal{X}^{\vee}}(\vec{\mathcal{M}})\). Indeed, the transition maps for \(\mathrm{At}_{\mathcal{O}_{\mathcal{X}^{\vee}}(\vec{\mathcal{M}})}\) have the form \[\begin{align} &\mathrm{At}_{\mathcal{O}_{\mathcal{X}^{\vee}}(\vec{\mathcal{M}})}\big|_{\mathcal{U}_{k}} \rightarrow \mathrm{At}_{\mathcal{O}_{\mathcal{X}^{\vee}}(\vec{\mathcal{M}})}\big|_{\mathcal{U}_{k+1}} \\ &\left( r , \; D \right) \mapsto \left( r + D \log g_{k , k+1}, \; D \right), \;\; r \in \mathcal{R}^{(k)}, \; D \in \mathrm{Der}(\mathcal{R}^{(k)}), \end{align}\] where \(g_{k, k+1}\) is the clutching function \[\mathcal{O}_{\mathcal{X}^{\vee}}(\vec{\mathcal{M}})\big|_{\mathcal{U}_{k}}\rightarrow \mathcal{O}_{\mathcal{X}^{\vee}}(\vec{\mathcal{M}})\big|_{\mathcal{U}_{k+1}} \qquad s_{k} \mapsto g_{k, k+1}s_{k}.\] The condition 127 forces it to be equal to \(g_{k , k+1} = x_{k}^{m_{k} - m_{k+1}}\) up to a multiplicative constant, so we recover 126 .

From the Atiyah algebra sheaf \(\mathrm{At}_{\mathcal{O}_{\mathcal{X}^{\vee}}(\vec{\mathcal{M}})}\) we reconstruct the sheaf of differential operators on \(\mathcal{O}_{\mathcal{X}^{\vee}}(\vec{\mathcal{M}})\) \[\mathcal{D}_{\mathcal{O}_{\mathcal{X}^{\vee}}(\vec{\mathcal{M}})} = U_{\mathcal{O}_{\mathcal{X}^{\vee}}}\left(\mathrm{At}_{\mathcal{O}_{\mathcal{X}^{\vee}}(\vec{\mathcal{M}})}\right),\] where the ‘universal enveloping over \(\mathcal{O}_{\mathcal{X}^{\vee}}\)’ is defined similarly to 118 . The homomorphisms 122 define an inclusion of \(\mathcal{A}\) into a space of global differential operators \[\label{eq:AtoGamma} \mathcal{A} \rightarrow \Gamma\left(\mathcal{X}^{\vee}, \mathcal{D}_{\mathcal{O}_{\mathcal{X}^{\vee}}(\vec{\mathcal{M}})}\right).\tag{128}\]

As an \(\mathcal{R}\)-algebra, \(\mathcal{A} \cong \mathcal{D}_{\pi}(\mathcal{R})\) is generated by \(\mathrm{Lie}_{\xi_{r}}, \; \theta_{i}^{\vee r}, \;\; i = 1, ..., g, \;\; r = x, y, \eta\). One can characterize the image of 128 by noting that \(\xi_{x}, \xi_{y}, \xi_{\eta}\) generate a Lie subalgebra \(TX^{\vee}(-\log E)\) of the global vector fields on \(X^{\vee}\) preserving the exceptional divisor \[E = \pi^{-1}(0) \subset X^{\vee}.\] It follows that the connection 117 gives rise to a flat logarithmic connection \[\begin{align} \nabla \colon \mathcal{T}\mathcal{X}^{\vee}(-\log (E)) \rightarrow \mathrm{At}_{\mathcal{O}_{\mathcal{X}^{\vee}}(\vec{\mathcal{M}})}. \end{align}\]

10 The action on attractor classes via equivariant localization↩︎

The following calculation is the only place where the exceptional conditions are used. The \(T\)-action 25 lifts to the action on \(\widetilde{QM\vec{N}}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}}\). We consider a one-dimensional subtorus of \(T\), slightly different from 26 : \[\begin{align} \label{newgm} &\mathbb{G}_{m} \hookrightarrow T\\ &t_{l} = \begin{cases} z^{n+1 - s}, \;k < l \leq s\\ z^{n+1 - l}, \;\text{otherwise.} \end{cases} \end{align}\tag{129}\] Since \(h^{0}(\mathcal{M}_{l} \otimes \mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}) = 0\) for \(k < l < s\), \(\mathrm{Attr}'(l) = \varnothing\) for this range of \(l\)’s, and the \(\mathbb{G}_{m}\)-action 129 contracts the space \(\{\mathcal{V}^{\bullet}\}\) to the single point \(\mathcal{V}_{\mathrm{fix}}^{\bullet}\), namely \(\mathcal{V}_{\mathrm{fix}} = \mathcal{M}_{1} \oplus \cdots \oplus \mathcal{M}_{n+1}\) with the natural filtration. The fiber \(F\) over \(\mathcal{V}^{\bullet}_{\mathrm{fix}}\) is a projective space \[\begin{align} \label{Finclusion} F = \mathbb{P}\left( H^{0}(\mathcal{V}_{\mathrm{fix}}\otimes \mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C})\right) \overset{i_{F}}{\hookrightarrow} \widetilde{QM\vec{N}}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}}. \end{align}\tag{130}\] Because of the vanishing \(h^{1}(\mathcal{M}_{l} \otimes \mathcal{M}_{k}^{-1}\otimes \mathcal{K}_{C})\), the attractor \(\mathrm{Attr}'(k)\) maps surjectively onto the space \(\{\mathcal{V}^{\bullet}\}\) of iterated extensions. The map 130 is quasi-smooth with normal bundle \[\begin{align} N_{i_{F}} = \sum\limits_{l < r}\mathrm{Ext}^{1}(\mathcal{M}_{r}, \mathcal{M}_{l}). \end{align}\] The fiber is quasi-smooth as well, with virtual tangent bundle \[\begin{align} T^{\mathrm{vir}}F = T^{\mathrm{vir}} \widetilde{QM\vec{N}}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}} - N_{i_{F}} = \sum\limits_{l = 1}^{n+1} \mathrm{Ext}^{\bullet}(\mathcal{L}, \mathcal{M}_{l})\otimes \mathcal{O}_{QM\vec{N}}(1) -H^{0}(\mathcal{O}_{C}). \end{align}\] The fiber \(F\) has a chain of subspaces \[\begin{align} F^{(r)} = \mathrm{Attr}'(r) \cap F = \mathbb{P}\left( H^{0}(\mathcal{V}^{(r)}\otimes \mathcal{M}_{k}^{-1}\otimes \mathcal{K}_{C})\right),\; r = 1, \ldots, n+1, \end{align}\] which carry a natural derived structure with virtual tangent bundle \[\begin{align} \label{TvirFr} T^{\mathrm{vir}}F^{(r)} = T^{\mathrm{vir}} \widetilde{QM\vec{N}}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}} - N_{i_{F}} = \sum\limits_{l = 1}^{r} \mathrm{Ext}^{\bullet}(\mathcal{L}, \mathcal{M}_{l})\otimes \mathcal{O}_{QM\vec{N}}(1) -H^{0}(\mathcal{O}_{C}). \end{align}\tag{131}\] Since the action is \(\pi_{\mathcal{V}^{\bullet}}\)-equivariant, the fixed locus \[\begin{align} \mathrm{Fix} \overset{i_{\mathrm{Fix}}}{\hookrightarrow} \widetilde{QM\vec{N}}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}} \end{align}\] of the \(\mathbb{G}_{m}\)-action is contained in \(F\). Since \(F\) is proper, attractors to fixed points cover the space \(\widetilde{QM\vec{N}}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}}\), hence the Gysin pullback in \(\mathbb{G}_{m}\)-equivariant Borel–Moore homology [19] \[\begin{align} i_{\mathrm{Fix}}^{*}\colon H^{BM, \mathbb{G}_{m}}_{*}(\widetilde{QM\vec{N}}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}}) \rightarrow H^{BM, \mathbb{G}_{m}}_{*}(\mathrm{Fix}) \end{align}\] is injective. Moreover, this map factorizes through the Gysin pullback on the fiber: \[\begin{align} i_{F}^{*}\colon H^{BM, \mathbb{G}_{m}}_{*}(\widetilde{QM\vec{N}}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}}) \rightarrow H^{BM, \mathbb{G}_{m}}_{*}(F), \end{align}\] which implies the injectivity of \(i_{F}^{*}\).

Let \[\begin{align} (u_{l} = d_{e}t_{l})_{l = 1}^{n+1}, \;\;\; u = d_{e}z \end{align}\] be coordinate systems on \(\mathrm{Lie}(T)\) and \(\mathrm{Lie}(\mathbb{G}_{m})\), respectively. From 131 we compute \[\begin{align} i_{F}^{*} [\mathrm{Attr}'(s)]^{\mathrm{equiv}} &= [F^{(s)}]^{\mathrm{equiv, vir}}=\\ &=\prod\limits_{l \leq s}(\eta + u_{l} - u_{k})^{h^{1}(\mathcal{M}_{l}\otimes \mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C})} \prod\limits_{l > s}(\eta + u_{l} - u_{k})^{h^{0}(\mathcal{M}_{l}\otimes \mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C})} \cap [F]=\\ &= \eta (\eta + (s-k)u)^{g-1} \prod\limits_{l > s}(\eta + u_{l} - u_{k})^{h^{0}(\mathcal{M}_{l}\otimes \mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C})} \cap [F], \end{align}\] \[\begin{align} i_{F}^{*} [\mathrm{Attr}'(k)]^{\mathrm{equiv}} &= [F^{(k)}]^{\mathrm{equiv, vir}}=\\ &= \prod\limits_{l \leq k}(\eta + u_{l} - u_{k})^{h^{1}(\mathcal{M}_{l}\otimes \mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C})} \prod\limits_{l > k}(\eta + u_{l} - u_{k})^{h^{0}(\mathcal{M}_{l}\otimes \mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C})} \cap [F]=\\ &= \eta (\eta + (s-k)u)^{g} \prod\limits_{l > s}(\eta + u_{l} - u_{k})^{h^{0}(\mathcal{M}_{l}\otimes \mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C})} \cap [F]. \end{align}\] We see that \[\begin{align} i^{*}_{F}\left( (\eta + (s-k)u ) \cap [\mathrm{Attr}'(s)]^{\mathrm{equiv}} - [\mathrm{Attr}'(k)]^{\mathrm{equiv}}\right)= 0. \end{align}\] Injectivity of \(i^{*}_{F}\) implies \[\begin{align} (\eta + (s-k)u ) \cap [\mathrm{Attr}'(s)]^{\mathrm{equiv}} = [\mathrm{Attr}'(k)]^{\mathrm{equiv}} \end{align}\] in \(\mathbb{G}_{m}\)-equivariant Borel–Moore homology of \(\widetilde{QM\vec{N}}_{\mathcal{M}_{k}^{-1} \otimes \mathcal{K}_{C}}\). The required statement follows by taking \(u = 0\).

References↩︎