2 Nakajima bundle varieties

The present section defines the main object of the current paper, Nakajima bundle representations. These are representations of double quivers in which vertices are represented by hermitian vector bundles, and edges by cohomology classes valued in Hom-bundles. There is a hermitian action of the group $\mathcal {G}=\prod _{v\in V}\text {Aut}(E_v)$ on the space of all such representations. Symplectic reduction produces a moduli space $\mathcal {M}^{r,d}_{\overline {Q}}(\tau )$ of Nakajima bundle representations.

A quiver Q consists of a tuple $(Q_0,Q_1,h,t)$ , where $Q_0=\{v_0,\ldots , v_n\}$ is the set of vertices, $Q_1=\{a:v_i\rightarrow v_j\}$ the set of edges, and $h,t:Q_1\rightarrow Q_0$ are functions specifying the head and tail of each edge.

Given any quiver Q, the double quiver $\overline {Q}$ is a quiver whose sets of vertices and edges satisfy $\overline {Q}_0 = Q_0$ and $\overline {Q}_1 = Q_1 \sqcup -Q_1$ . Here $-Q_1$ consists of a copy of each element of $Q_1$ but with opposite orientation.

In order to define representations of a quiver, it is necessary to equip it with a label. Given a quiver Q, a label of Q is a pair $(r,d)$ where $r=(r_v)_{v\in Q_0} \in \mathbb {N}^{\lvert Q_0\rvert }$ and $d=(d_v)_{v\in Q_0}\in \mathbb {Z}^{\lvert Q_0\rvert }$ . The data $(r,d)$ are referred to as the type of the representation.

Let $\mathcal {A}(E_v)$ denote the space of unitary connections on $E_v$ . The space $\text {Rep}(\overline {Q})$ of Nakajima bundle representations of $\overline {Q}$ is $\text {Rep}(\overline {Q}) = \text {Rep}(Q_0) \oplus \text {Rep}(Q_1)$ , where

and $\text {Rep}(Q_1)$ is the subset $\ker (\overline {\partial })$ of

$$ \begin{align*} \bigoplus_{a \in Q_1}(\Omega^0(X,\operatorname{\mathrm{Hom}}(E_{t(a)},E_{h(a)})) \; &\oplus \; \Omega^{0,1}(X,\operatorname{\mathrm{Hom}}(E_{h(a)},E_{t(a)})\otimes K)). \end{align*} $$

At each vertex, the group of unitary bundle automorphisms acts on $E_v$ , producing an action of $\mathcal {G}=\prod _{v\in Q_0} \mathcal {G}_v$ on $\text {Rep}(\overline {Q})$ :

$$ \begin{align*} g\cdot(A,\phi,x,y) = (g\cdot A, g_{v}\phi_v g^{-1}_{v}, g_{h(a)} x_ag^{-1}_{t(a)}, g_{t(a)} y_ag^{-1}_{h(a)}). \end{align*} $$

In [Reference HitchinHit87] it was shown that the action of $\mathcal {G}_v$ on $\mathcal {A}(E_v)\times \Omega ^{1,0}(X,\operatorname {End}(E_v))$ is Hamiltonian with real and complex moment maps given by Equation (1-1). This extends to the action of the product group $\mathcal {G}$ on $\text {Rep}(Q_0)$ whose real and complex moment maps are $\mu _{\mathbb {R}}^{\mathfrak {H}} =\bigoplus _{v\in Q_0}(\sqrt {-1}\Lambda F_{A_v}+[\phi _v,\phi _v^*])$ and $\mu _{\mathbb {C}}^{\mathfrak {H}} = \bigoplus _{v\in Q_0}\overline {\partial }_{E_v}\phi _v$ .

Consider the $\mathcal {G}$ -action on the space of edge representations $\text {Rep}(\overline {Q}_1)$ . For each ${a\in Q_1}$ let $y_a^{\mathrm {harm}}$ be the unique harmonic representative of the class $y_a$ . The representative $y_a^{\mathrm {harm}}$ may be viewed as a twisted morphism $E_{h(a)}\to E_{t(a)}\otimes \Omega ^{1,1}(X)$ . Using the metrics $h_{h(a)}$ and $h_{t(a)}$ on $E_{h(a)}$ and $E_{t(a)}$ respectively, there are adjoints ${x^{*h_a}:E_{h(a)}\to E_{t(a)}}$ and $(y_a^{\mathrm {harm}})^{*h_a}:E_{t(a)}\to E_{h(a)}\otimes \Omega ^{1,1}(X)$ . The proof of the following proposition is standard.

Proposition 2. The $\mathcal {G}$ -action is Hamiltonian with real and complex moment maps $\mu _{\mathbb {R}},\mu _{\mathbb {C}}$ whose components at the vertex v are given by

(2-1) $$ \begin{align} \mu_{\mathbb{R},v}^{\mathfrak{E}}(x,y) & = \sum_{a\in Q_1:h(a)=v} ( x_a\circ x_a^{*h_a} - \Lambda ((y_{a}^{\mathrm{harm}})^{*h_a}\circ y_{a}^{\mathrm{harm}}) ) \nonumber \\ & \quad + \sum_{a\in Q_1:t(a)=v} ( x_a^{*h_a} \circ x_a - \Lambda (y_{a}^{\mathrm{harm}}\circ (y_{a}^{\mathrm{harm}})^{*h_a}) ) \end{align} $$

and

(2-2) $$ \begin{align} \mu_{\mathbb{C},v}^{\mathfrak{E}}(x,y) = \Lambda\bigg(\sum_{a\in Q_1:h(a)=v} (x_a \otimes \operatorname{id}_{K}) y_a^{\mathrm{harm}} -\sum_{a\in Q_1:t(a)=v} y_a^{\mathrm{harm}} (x_a \otimes \operatorname{id}_K)\bigg). \end{align} $$

Definition 3. Let $\mathcal {Z}(\mathfrak {g})$ denote the centre of $\mathfrak {g}=\text {Lie}(\mathcal {G})$ . For a fixed type $(r,d),$ and $\tau = (\tau _{\mathbb {R}},\tau _{\mathbb {C}}) \in \mathcal {Z}(\mathfrak {g}) \oplus (\mathcal {Z}(\mathfrak {g})\otimes \mathbb {C})$ the Nakajima bundle variety at level $\tau $ , $\mathcal {M}_{\overline {Q}}^{r,d}(\tau )$ is the moduli space of Nakajima quiver representations that are solutions of the equations

(2-3) $$ \begin{align} \mu_{\mathbb{R}}(A,\phi,x,y)_v &: \mu_{\mathbb{R},v}^{\mathfrak{H}}(A,\phi) +\mu_{\mathbb{R},v}^{\mathfrak{E}}(x,y) = \tau_{\mathbb{R}}, \end{align} $$

(2-4) $$ \begin{align} \mu_{\mathbb{C}}(A,\phi,x,y)_v &: \mu_{\mathbb{C},v}^{\mathfrak{H}}(A,\phi) +\mu_{\mathbb{C},v}^{\mathfrak{E}}(x,y) = \tau_{\mathbb{C}}. \end{align} $$

That is, it is the quotient

We close this section with a few remarks regarding the definition above.

3 $\overline {Q}$ -bundles

The main purpose of this section is to define an alternative notion of a quiver bundle for a double quiver $\overline {Q}$ , which we refer to as $\overline {Q}$ -bundles, such that stable $\overline {Q}$ -bundles provide solutions to Equations (2-3) and (2-4). One of the main differences in the current work from that of previous correspondences, such as [Reference Álvarez-Cónsul and García-PradaAaCGP03], lies in the fact that the data associated to a reverse edge are given by a cohomology class in degree $1$ rather than a true morphism of vector bundles. This is similar to the study of p-cohomology triples in [Reference Bradlow and García-PradaBGP95] and Higgs bundles on higher-dimensional varieties as in [Reference BottacinBot00].

A morphism of $\overline {Q}$ -bundles $f:(E,\phi ,x,y)\rightarrow (E',\phi ',x',y')$ , is a collection $f_v:E_v \rightarrow E^{\prime }_v$ of bundle morphisms such that the induced morphisms on cohomology satisfy $ f_{t(a)*}(y') = (f_{h(a)}\otimes 1_K)^*(y)$ , and the required squares commute for all v and a:

In order to identify $\overline {Q}$ -bundles with solutions to Equations (2-1) and (2-2), it is necessary to restrict to certain subsets of stable $\overline {Q}$ -bundles.

Definition 7. Let $\mathcal {E}$ be a $\overline {Q}$ -bundle on X. For a tuple of real numbers $\tau = (\tau _v)_{v\in Q_0}$ , define the degree and the $\tau $ -rank of $\mathcal {E}$ ,

The $\tau $ -slope of $\mathcal {E}$ , is then defined as

$$ \begin{align*} \mu_\tau(\mathcal{E}) = \frac{\operatorname{deg}(\mathcal{E})+ \operatorname{rk}_\tau(\mathcal{E})}{\operatorname{rk}_1(\mathcal{E})}. \end{align*} $$

A $\overline {Q}$ -bundle $\mathcal {E}$ is semi-stable if for all subbundles $\mathcal {E}'$ , the slope satisfies ${\mu _\tau (\mathcal {E}') \leq \mu _\tau (\mathcal {E})}$ . It is said to be stable if the inequality is strict, and polystable if $\mathcal {E}$ is a direct sum of stable $\overline {Q}$ -bundles all of which have the same slope.

Stable $\overline {Q}$ -bundles behave similarly to stable vector bundles. In particular, a standard proof as in [Reference KobayashiKob87, Corollary 5.7.12] shows the following proposition.

4 Stable $\overline {Q}$ -bundles and Nakajima Representations

The ideas behind the correspondence of Nakajima quiver bundles with $\overline {Q}$ -bundles are not new, having been used to prove many related correspondences (see, for example, [Reference Uhlenbeck and YauUY86, Reference HitchinHit87, Reference SimpsonSim92, Reference Álvarez-Cónsul and García-PradaAaCGP03]). However, greater care is required in order to account for the higher cohomology classes that appear in the current setting. In order to do this, we work with the unique harmonic representatives $y_a^{\mathrm {harm}}$ of the classes $y_a$ . This will reduce the problem to the correspondence for quiver bundles. The end result is the following correspondence relating stable $\overline {Q}$ -bundles with solutions of the moment map equations.

One direction of the correspondence is a straightforward generalization of the argument in [Reference NakajimaNak94, Reference Álvarez-Cónsul and García-PradaAaCGP03].

The remainder of this section is devoted to proving the second half of Proposition 10,

4.1 Preliminaries

Fix a compact Riemann surface X, and a $\overline {Q}$ -bundle $\mathcal {E}$ such that $\mu ^{\mathfrak {E}}_{\mathbb {C}}=0$ . For each $v\in Q_0$ , let $\mathrm {Met}_v$ denote the space of hermitian metrics on the bundle $E_v$ . Fix a $k_v\in \mathrm {Met}_v$ such that the metric on $\mathrm {det}(E_v)$ induced by $k_v$ satisfies $\sqrt {-1}F_{\text {det}(k_v)}=d_v$ . Given a metric $h_v\in \mathrm {Met}_v$ , denote the space of smooth $h_v$ -selfadjoint endomorphisms of $E_v$ by $S(h_v)$ . Any $h_v\in \mathrm {Met}_v$ is related to $k_v$ through a section $s_v\in S(k_v)$ by $h_v=k_v e^{s_v}$ . This notation extends to the direct sum $E=\oplus E_v$ by dropping the mention of the vertex.

Recall that the adjoints of $\phi _v$ , $x_a$ and $y_a^{\mathrm {harm}}$ depend on the choice of metrics $h_v$ . This dependence will be denoted by $\phi ^{*h},x^{*h}$ and $(y_a^{\mathrm {harm}})^{*h}$ . If the operations are with respect to the fixed $k_v$ , the operations will be denoted by $x^*$ .

For each $h\in \mathrm {Met}$ , there is a metric $h_a$ on $\operatorname {\mathrm {Hom}}(E_{ta},E_{ha})$ for each $a\in \overline {Q}_1$ , given by $(\zeta _a,\zeta _a')_{h_a} = \mathrm {tr}(\zeta _a \circ \zeta _a^{*h_a})$ . This gives a metric h on $\bigoplus _a \operatorname {\mathrm {Hom}}(E_{ta},E_{ha})$ by ${(\zeta ,\zeta ') = \sum _a (\zeta _a,\zeta _a')_{h_a}}$ , and the corresponding $L^p$ -inner products and norms are defined similarly.

Fix an integer $p>2$ . Let $W^{p,2}S(k_v)$ denote the Sobolev space of $k_v$ -selfadjoint sections whose derivatives up to second order have finite $L^p$ -norm, and $\mathrm {Met}(W^{p,2}S(k_v)) = \{ k_v e^{s_v} : s_v \in W^{p,2}S(k_v)\}$ . Given $h =k e^s\in \mathrm {Met}(W^{p,2}S(k))$ , define the h-adjoint of a section $\xi $ by $\xi ^{*_h}= e^{-s}\xi ^* e^s$ . Finally, define the connection $A_{h_v}$ by

$$ \begin{align*} d_{h_v} = d_{k_v} + e^{-s_v} \partial_{k_v}(e^{s_v}) \end{align*} $$

with curvature $F_{h_v} = F_{k_v} + \overline {\partial }_{E_v}(e^{-s_v}\partial _{k_v}(e^{s_v}))$ .

For each vertex $v\in Q_0$ , let $s_v\in S(k_v)$ . We consider the spectral resolution $s_v= \sum _i \lambda _{v,i}P_{v,i}$ with selfadjoint projection $P_{v,i}$ . Let $\psi :\mathbb {R}\to \mathbb {R}$ and $\Psi :\mathbb {R}^2\to \mathbb {R}$ be $C^1$ -bounded functions. We define fibrewise, and basis-independently,

$$ \begin{align*} \psi(s_v) := \sum_i \psi(\lambda_{v,i})\, P_{v,i}, \quad \Psi(s_v)[T] := \sum_{i,j}\Psi(\lambda_{v,i},\lambda_{v,j})\, P_{v,i}\,T\, P_{v,j} \end{align*} $$

for any $T\in \operatorname {End}(E_v)$ . These definitions remain valid even under eigenvalue multiplicities.

We let $\Psi (s)$ act on the data as bundle maps by

$$ \begin{align*} (\Psi(s)\,\phi)_v &:= \Psi(s_v)[\phi_v] \in \Omega^{1,0}(X,\operatorname{End} E_v),\\ (\Psi(s)\,x)_a &:= \sum_{i,j}\Psi(\lambda_{h(a),i},\lambda_{t(a),j})\,P_{h(a),i}\,x_a\,P_{t(a),j} \in \Omega^0(X,\operatorname{\mathrm{Hom}}(E_{t(a)},E_{h(a)})),\\ (\Phi(s)\,y^{\mathrm{harm}})_a &:= \sum_{i,j}\Phi(\lambda_{t(a),j},\lambda_{h(a),i})\,P_{t(a),j}\,y^{\mathrm{harm}}_a\,P_{h(a),i}\in \Omega^{0,1}\!(X,\operatorname{\mathrm{Hom}}(E_{h(a)},E_{t(a)})\!\otimes\!K). \end{align*} $$

If $\Psi (p,q)=\psi _1(p)\psi _2(q)$ for functions $\psi _1,\psi _2:\mathbb {R}\rightarrow \mathbb {R}$ , then

$$ \begin{align*} (\Psi(s)x)_a = \psi_1(s_{h(a)})\,x_a\,\psi_2(s_{t(a)}), \quad (\Psi(s)_y^{\mathrm{harm}})_a = \psi_1(s_{t(a)})\, y^{\mathrm{harm}}_a\,\psi_2(s_{h(a)}). \end{align*} $$

For any real number b, let $W^{p,2}_bS(k) \subset W^{p,2}S(k)$ be the closed subset of sections s with $\lvert s\rvert \leq b$ almost everywhere in X.

Lemma 13 (Surface case)

Assume $\psi \in C^1_b(\mathbb {R})$ and $\Psi \in C^1_b(\mathbb {R}^2)$ as above. If ${s\in W^{1,2}_bS(k)}$ is selfadjoint, then for any $q<\infty $ the following assertions hold.

1. (1) $\psi (s)\in W^{1,2}S(k)$ with $\|\psi (s)\|_{W^{1,2}}\le C\,(1+\|s\|_{W^{1,2}})$ .

2. (2) For each vertex $v\in Q_0$ , $\Psi (s_v)$ defines a bounded operator on $\Omega ^0(\operatorname {End} E_v)$ and on $\Omega ^{1,0}(\operatorname {End} E_v)$ ; likewise on $\Omega ^0(\operatorname {\mathrm {Hom}}(E_{t(a)},E_{h(a)}))$ and $\Omega ^{0,1}(\operatorname {\mathrm {Hom}}(E_{h(a)},E_{t(a)})\!\otimes \!K)$ .

3. (3) For every $q\leq 2$ , the actions in $(2)$ extend continuously to $\Psi (s):L^2\to L^q$ , with

   $$ \begin{align*} \|\Psi(s)[T]\|_{L^q} \leq C_{X,q} \|\psi_1\|_{C^1}\, \|\psi_2(s)\|_{C^1}( 1 + \|s\|_{W^{1,2}})^2 \|T\|_{L^2}. \end{align*} $$

4. (4) The actions in $(2)$ extend continuously to $\Psi (s):L^2\to L^2$ with

   $$ \begin{align*} \|\Psi(s)[T]\|_{L^2} \leq \|\psi_1(s)\|_{L^\infty}\|\psi_2(s)\|_{L^\infty}\|T\|_{L^2}. \end{align*} $$

Proof. Since $\psi \in C^1_b$ , $\|\psi (\lambda _i)\|_{W^{1,2}} \leq C_i +C_i' \|\nabla \lambda _i\|^2_{L^2}$ . Then

$$ \begin{align*} \|\psi(s)\|^2_{w^{1,2}} &\leq \sum_i ( C_i + C_i'\|\lambda_i\|^2_{W^{1,2}}) \\ & \leq C\sum_i(1+ \|\lambda_i\|^2_{W^{1,2}}) \\ & \leq C_{X,q}(1 + \|s\|^2_{W^{1,2}}). \end{align*} $$

From $\Psi \in C^1_b(\mathbb {R}^2)$ , $(2)$ is clear.

To see $(3)$ , when $1/q=1/2 +2/p$ ,

$$ \begin{align*} \|\Psi(s)[T]\|_{L^q} &= \| \psi_1(s)\,T\,\psi_2(s)\|_{L^q}\\ &\leq \|\psi_1(s)\|_{L^p}\, \|T\|_{L^2} \, \|\psi_2(s)\|_{L^p}. \end{align*} $$

Since $W^{1,2}$ embeds into $L^p$ for any $p<\infty $ , the result follows from $(1)$ .

Using the assumption that $s\in W^{1,2}_b$ , $\psi _1(s),\psi _2(s)\in W^{1,2}_b$ as well. Then $(4)$ follows from Hölder’s inequality.

At this point, thinking of the tuple $(E,\phi ,x,y^{\mathrm {harm}})$ as a quiver bundle for the tripled quiver of Q, the majority of the proof is the same as that of Sections 3.3–3.7 of [Reference Álvarez-Cónsul and García-PradaAaCGP03]. The interested reader is encouraged to refer there for full details. First, construct a functional $M_\tau $ on the space of metrics $\mathrm {Met}(W^{p,2}S(k))$ such that the metrics that minimize the functional provide solutions of the moment map equation. The existence of minimizing metrics is then shown to reduce to a particular inequality on $M_\tau $ . Finally, the regularity results of [Reference Uhlenbeck and YauUY86] are used to show that if this inequality is not satisfied, then there exists a filtration of the quiver bundle, which contradicts stability. To conclude it is necessary to relate the destabilizing quiver subbundles constructed in [Reference Álvarez-Cónsul and García-PradaAaCGP03] to subbundles of the $\bar {Q}$ -bundle. This is accomplished by the following lemma, the proof of which is clear.

Lemma 14. Suppose $(E,\phi ,x,y)$ is a $\overline {Q}$ -bundle. If $(E',\phi ',x',y^{'\mathrm {harm}})$ is a quiver subbundle of $(E,\phi ,x,y^{\mathrm {harm}})$ , then $(E',\phi ',x',y')$ is a $\overline {Q}$ -subbundle of $(E,\phi ,x,y)$ .

5 Infinitesimal deformations

Let $\mathcal {R}=(A_v,\phi _v,x_a,y_a)$ be a representation of $\overline {Q}$ . In order to study deformations of $\mathcal {R}$ , a complex $C^\bullet (\mathcal {R})$ is constructed such that $\mathbb {H}^1(C^\bullet (\mathcal {R}))$ corresponds to the infinitesimal deformations of $\mathcal {R}$ . The treatment of this section follows that of [Reference BottacinBot00] in the study of Higgs bundles on n-dimensional varieties. For the deformation theory of ordinary quiver bundles see [Reference Gothen and KingGK05]. Due to the growing number of indices in this section, in an effort to improve readability, for the remainder of this section the subscripts indicating vertices and edges are replaced with superscripts.

Let $\mathcal {U}=\{U_i\}_{i\in I}$ be a Čech cover of X. The cohomology classes $(\phi ^v,x^a,y^a)$ defining the representation $\mathcal {R}$ can be described in terms of Čech cocycles ( $\{\phi ^v_{i}\},\{x^a_i\},\{y^a_{ij}\})$ .

For cocycles $\{\alpha _{j_0,\ldots , j_p}^v\}_{v\in Q_0}\in \bigoplus _{v\in Q_0} C^p(\operatorname {End}(E^v))$ define $\{[\alpha ,x]_{j_0,\ldots , j_p}^a\}_{a\in Q_1} \in \bigoplus _{a\in Q_1} C^p(\operatorname {\mathrm {Hom}}(E^{t(a)},E^{h(a)}))$ by

$$ \begin{align*} [\alpha,x]^a_{j_0\cdots j_p} = \alpha^{h(a)}_{j_0,\cdots j_p}\circ x^a_{j_p} - x^a_{j_p} \circ \alpha^{t(a)}_{j_0\cdots j_p} \end{align*} $$

and similarly for $[\alpha ,y]^a$ and $[\alpha ,\phi ]^v.$ These define morphisms of Čech complexes

$$ \begin{align*} [\cdot,\phi] &: \bigoplus_{v\in Q_0} C^p(\operatorname{End}(E_v)) \longrightarrow \bigoplus_{v\in Q_0} C^p(\operatorname{End}(E^v)\otimes K), \\ [\cdot,x] &:\bigoplus_{v\in Q_0} C^p(\operatorname{End}(E^v)) \longrightarrow \bigoplus_{a\in Q_1} C^p(\operatorname{\mathrm{Hom}}(E^{t(a)},E^{h(a)})), \\ [\cdot,y] &:\bigoplus_{v\in Q_0} C^p(\operatorname{End}(E^v)) \longrightarrow \bigoplus_{a\in Q_1} C^{p+1}(\operatorname{\mathrm{Hom}}(E^{h(a)},E^{t(a)})\otimes K). \end{align*} $$

Putting these together, we define a map

by $\Psi (f) = ([f,\phi ],[f,x],[f,y])$ . Additionally, there exists a map $\Phi $ from the codomain of $\Psi $ to $\bigoplus _{v\in Q_0} C^{p+1}(\operatorname {End}(E^v)\otimes K)$ given by

$$ \begin{align*} \Phi(\beta,\alpha,\mu) = \delta \beta +[\alpha,y]+[\mu,x]. \end{align*} $$

The desired complex of sheaves $C^\bullet (\mathcal {R})$ can be defined as follows. Let $C^0(\mathcal {R}) = \bigoplus _{v\in Q_0} C^0(\operatorname {End}(E^v))$ , and for $p=1,2$ ,

$$ \begin{align*} C^p(\mathcal{R}) = &\bigoplus_{v\in Q_0} ( C^p(\operatorname{End}(E^v))\oplus C^{p-1}(\operatorname{End}(E^v)\otimes K)) \\ \oplus &\bigoplus_{a\in Q_1} (C^{p-1}(\operatorname{\mathrm{Hom}}(E^{t(a)},E^{h(a)})) \oplus C^{p}(\operatorname{\mathrm{Hom}}(E^{h(a)},E^{t(a)})\otimes K)), \end{align*} $$

with differentials $d^0:C^0(\mathcal {R})\longrightarrow C^1(\mathcal {R})$ given by $d^0(f) = \big [\!\begin {smallmatrix}\delta \\ \Psi \end {smallmatrix}\!\big ]$ and $d^1:C^1(\mathcal {R})\longrightarrow C^2(\mathcal {R})$ by

$$ \begin{align*} d^1 = \begin{bmatrix} \delta & 0 \\ \Psi & -\delta \\ 0 & \Phi \end{bmatrix}. \end{align*} $$

Above, $-\delta $ is understood as taking in a triple of cocycles and applying minus the appropriate Čech differential to each.

Proof. Without loss of generality we may assume Q consists of a pair of vertices connected by a single edge. Let $(E^1,E^2,\phi ^1,\phi ^2,x,y)$ be a $\overline {Q}$ -bundle and $y',y"$ two cocycles representing y. Write $y" = y' +\delta \nu $ for $\nu \in C^0(\operatorname {\mathrm {Hom}}(E^1,E^2)\otimes K)$ . Denote by $C^\bullet (\overline {Q},y')$ and $C^\bullet (\overline {Q},y")$ the corresponding complexes. It is straightforward to check that the map

$$ \begin{align*} \mathbb{H}^1(C^\bullet(\overline{Q},y"))\longrightarrow \mathbb{H}^1(C^\bullet(\overline{Q},y')) \end{align*} $$

given by sending $(\eta ^1,\eta ^2,\beta ^1,\beta ^2,\alpha ,\mu )$ to $(\eta ^1,\eta ^2,\beta ^1,\beta ^2,\alpha ,\mu +[\eta ,\nu ])$ is an isomorphism.

It is now not difficult to extend the proof of [Reference BottacinBot00, BR94] to arrive at the main result.

Proof. Given a vector bundle $E^v$ , a cohomology class $\{\eta ^v_{ij}\}\in H^1(\operatorname {End}(E^v))$ uniquely determines a vector bundle $E^v[\epsilon ]$ on . The affine sets form a Čech cover of $X_\epsilon $ .

Consider a deformation $x^a[\epsilon ]:E^{t(a)}[\epsilon ] \longrightarrow E^{h(a)}[\epsilon ]$ . This can be described by morphisms $x^a[\epsilon ]_i: U^{t(a)}[\epsilon ]_i \longrightarrow U^{h(a)}[\epsilon ]_i$ that restrict to $x^a$ modulo $\epsilon $ and are compatible with the gluing isomorphisms $1+\epsilon \eta ^v_{ij}$ . Thus, we may write $x^a[\epsilon ]_i = x^a +\epsilon \alpha _i^a$ for some $\alpha ^a_i :U^{t(a)}_{i}\longrightarrow U^{h(a)}_{i}$ , and for each pair $(i,j)$ , the following diagram commutes:

Replacing the maps $x^a[\epsilon ]_j$ with $x^a+\epsilon \alpha ^a_i$ , and using the fact that $(1+\epsilon \eta _{ij}^v)^{-1} = 1-\epsilon \eta _{ij}^v$ , the above diagram is equivalent to

$$ \begin{align*} (\alpha^a_j-\alpha^a_i) \lvert_{U_{ij}} = \eta_{ij}^{h(a)} x^a - x^a\eta_{ij}^{t(a)} = [\eta_{ij},x]^a. \end{align*} $$

A similar argument shows that

$$ \begin{align*} (\beta^v_j -\beta^v_i)\lvert_{U^v_{ij}} = (\eta_{ij}^v \otimes \text{id}_K)\phi^v - \phi^v\eta_{ij}^v =[\eta_{ij},\phi]^v. \end{align*} $$

Now consider a deformation of $y^a$ , which can be described in terms of cocycles as $y^a[\epsilon ]_{ij} = y^a + \epsilon \mu _{ij}^a$ . Compatibility gives the condition

$$ \begin{align*} (1-\epsilon\eta_{ij}^{t(a)})(y^a[\epsilon]_{ij}+y^a[\epsilon]_{jk})(1+\epsilon\eta_{ij}^{h(a)}) = y^a[\epsilon]_{ik} \end{align*} $$

from which it follows that

$$ \begin{align*} (\mu_{ij} + \mu_{jk} - \mu_{ik})\lvert_{U_{ijk}} = [\eta_{ij},y]^a. \end{align*} $$

Being a deformation of $(\phi ^v,x^a,y^a)$ , the data $(\phi ^v[\epsilon ],x^a[\epsilon ],y^a[\epsilon ])$ must solve the moment map equation. Explicitly, we must have that

$$ \begin{align*} \sum_{v\in Q_0} \overline{\partial}\phi^v[\epsilon] + \sum_{a:ha=v} (x^a \otimes \text{id}_K)\circ y^a - \sum_{a:ta=v} y^a(x^a\otimes \text{id}_K) = 0. \end{align*} $$

Expanding the cocycles, and using the fact that $(\phi ^v,x^a,y^a)$ satisfy the moment map equation, the result follows.

Now let $(\tilde {E}^{v}[\epsilon ],\tilde {\phi }^{v}[\epsilon ],\tilde {x}^{a}[\epsilon ], \tilde {y}^{a}[\epsilon ])$ be a deformation that is isomorphic to $(E^v[\epsilon ],\phi ^v[\epsilon ],x^a[\epsilon ], y^a[\epsilon ])$ , and corresponds to $(\{\tilde {\eta }_{ij}^{v}\},\{\tilde {\beta }_i^{v}\},\{\tilde {\alpha }_i^{a}\},\{\tilde {\mu }_{ij}^{a}\})$ . Then it follows that $\tilde {\eta }_{ij}^{v} = \eta _{ij}^v +(\lambda _j^v-\lambda _i^v)\lvert _{U_{ij}[\epsilon ]}$ , where $\lambda ^v[\epsilon ] = 1 +\epsilon \lambda ^v$ is the isomorphism $E^{v'}[\epsilon ]\cong E^v[\epsilon ]$ . Moreover, the following diagram commutes:

from which we determine that $\alpha _i^a - \tilde {\alpha }^{a}_i = [\lambda ,x]^a$ . Similarly, it can be shown that $\beta _i^v-\tilde {\beta }_i^{v} = [\lambda ,\phi ]^v$ and $\tilde {\mu }_{ij}^{a} =\mu _{ij}^a + [\lambda ,y]^a$ . Thus, deformations correspond to cohomology classes in $\mathbb {H}^1(C^\bullet (\mathcal {R}))$ .

6 $\overline {Q}$ -bundles of type $A_1$

We now consider framed quiver representations. Using the notation of [Reference GinzburgGin12], associated to any quiver Q, one can construct a new quiver $Q^\heartsuit $ , where $Q^\heartsuit _0 = Q_0\sqcup Q_0$ . The vertices in the second copy of $Q_0$ are called the framing vertices and are denoted by $w_i$ . The edge set $Q^\heartsuit _1$ consists of the edges of Q together with an edge $v_i\rightarrow w_i$ . A type vector for $Q^\heartsuit $ is vector $((v,d),(w,d_w))$ . Then a framed representation of $\overline {Q}$ is a representation of $\overline {Q^\heartsuit }$ such that the data associated to the framing vertices is fixed. Thus, choosing a framing is equivalent to choosing an additional vector bundle $W_v$ of rank $w_v$ for each $v\in Q_0$ .

Let $Q^\heartsuit $ be the framed $A_1$ -quiver, pictured below with framing shown as a square vertex, and label $((1,d),(n,0))$ for $d>0$ and $n>1$ :

In [Reference NakajimaNak94] it was shown that linear representations of $\overline {Q^\heartsuit }$ are isomorphic to cotangent spaces of Grassmannians.

Take $X=\mathbb {P}^1$ and $\mathcal {O}^n$ as framing bundle. A framed Nakajima bundle representation of $\overline {Q}$ consists of a line bundle L on $\mathbb {P}^1$ , together with a pair $x\in H^0(\mathbb {P}^1,\operatorname {\mathrm {Hom}}(\mathcal {O}^n,L))$ and $y\in H^1(\mathbb {P}^1,\operatorname {\mathrm {Hom}}(L,\mathcal {O}^n)\otimes K) \cong H^0(\mathbb {P}^1,\operatorname {\mathrm {Hom}}(\mathcal {O}^n,L))^*$ . For $\tau =(\tau _{\mathbb {R}},0)$ satisfying ${\tau _{\mathbb {R}}> d}$ , it follows from the real moment map that x must be surjective, hence it defines a rank $1$ quotient bundle F of $\mathcal {O}^n$ having degree d. Moreover, from the complex moment map $xy=0$ we see that $yx \in H^1(\mathbb {P}^1,\operatorname {\mathrm {Hom}}(F,\text {ker}(x))\otimes K)$ . As shown in [Reference SernesiSer06], the tangent space at a point F in the Grassmannian can be identified with $\operatorname {\mathrm {Hom}}_{\mathbb {P}^1}(F,\mathcal {O}^n/F)$ , so the map $yx$ lies in the cotangent space to the point defined by x. This describes a map from $\mathcal {M}_{\overline {Q}}^{r,d}(\tau )$ to $T^*\text {Gr}^{d}(1,\mathcal {O}^n)$ .

Conversely, assume that $(F,\Omega )$ is a point in $T^*\text {Gr}^{d}(1,\mathcal {O}^n)$ . So $q: \mathcal {O}^n \twoheadrightarrow F$ is a quotient bundle of $\mathcal {O}^n$ having degree d and $\Omega \in H^1(\mathbb {P}^1,\operatorname {\mathrm {Hom}}(F,\text {ker}(q))\otimes K)$ lies in the cotangent space at $[q:\mathcal {O}^n \twoheadrightarrow F]$ . Consider the exact sequence

We may apply the functor $\operatorname {\mathrm {Hom}}(F,-)\otimes K$ to obtain a new short exact sequence

with the associated long exact sequence in cohomology

Since we are working over $\mathbb {P}^1$ , the $H^0(\mathbb {P}^1,\operatorname {\mathrm {Hom}}(F,F)\otimes K)$ vanishes and so the map $H^1(\mathbb {P}^1,\operatorname {\mathrm {Hom}}(F,\text {ker}(q))\otimes K) \longrightarrow H^1(\mathbb {P}^1,\operatorname {\mathrm {Hom}}(F,\mathcal {O}^n)\otimes K) $ is injective. Therefore, we can push forward $\Omega $ to get an element $y\in H^1(\mathbb {P}^1,\operatorname {\mathrm {Hom}}(F,\mathcal {O}^n)\otimes K)$ . The complex moment map condition together with the above sequence implies that y is in the image of the map $H^1(\mathbb {P}^1,\operatorname {\mathrm {Hom}}(F,\text {ker}(q))\otimes K) \longrightarrow H^1(\mathbb {P}^1,\operatorname {\mathrm {Hom}}(F,\mathcal {O}^n)\otimes K) $ . This defines a map $T^*\text {Gr}^d(1,\mathcal {O}^n)$ to $\mathcal {M}_{\overline {Q}}^{r,d}(\tau )$ . This describes an isomorphism between $T^*\mathrm {Gr}^d(1,\mathcal {O}^n)$ and $\mathcal {M}_{\overline {Q}}^{r,d}(\tau )$ .

When the genus of X is $g\geq 1$ , there exist nonzero sections of $H^0(X,\operatorname {End}(L)\otimes K)$ and so a representation includes a twisted endomorphism $\phi $ . Thinking now in terms of stability, the $\overline {Q}$ -bundle $(L,\phi ,x,y)$ is stable if and only if $(L,\phi )$ is a stable Higgs bundle of rank $1$ , and x is surjective.

7 $\mathbb {C}^\times $ -action

In the presence of a suitable action of a torus $\mathbb {T}$ on a (possibly singular) variety X, the theory of Białynicki-Birula [Reference Białynicki-Birula.ByB73] establishes that the rational (co)homology of X is determined by the rational (co)homologies of the connected components of the fixed point set $X^{\mathbb {T}}$ . In this section we define a $\mathbb {C}^\times $ -action on Nakajima bundle varieties in an attempt to better understand the fixed points, and as a result the topology of $\mathcal {M}_{\overline {Q}}^{r,d}(\tau )$ . In general, even in the case of vector space representations, a full description of the fixed points is difficult. Therefore, we focus on the specific case of the framed $A_1$ quiver.

Recall the $\mathbb {C}^\times $ -actions on the moduli space of Higgs bundles and on Nakajima quiver varieties that scale the cotangent directions. In the same way there is an action of $\mathbb {C}^\times $ on $\mathcal {M}_{\overline {Q}}^{r,d}(\tau )$ that preserves the moment map conditions $\mu _{\mathbb {R}}=\tau _{\mathbb {R}},\; \mu _{\mathbb {C}} = 0$ . If $(E,\phi ,x,y)$ is a $\overline {Q}$ -bundle, the action of $\lambda \in \mathbb {C}^\times $ is given by $\lambda \cdot (E,\phi ,x,y) =(E,\lambda \phi ,x,\lambda y).$

For the remainder of this section, assume the genus of X is not $1$ . If $(E,\phi ,x,y)$ is a fixed point, there exists a one-parameter family of gauge transformations $g_\lambda $ such that

$$ \begin{align*} g_\lambda(E,\phi,x,y)g_\lambda^{-1} = (E, \lambda\phi, x, \lambda y). \end{align*} $$

For each $E_v\in Q_0$ let $E_v=\oplus _i E_v^{w_i}$ be the weight decomposition of $E_v$ , in which $\mathbb {C}^\times $ acts on $E_v^{w_i}$ with weight $w_i$ . Being a fixed point of the action implies

$$ \begin{align*} \phi_v &:E_v^{w_i} \longrightarrow E_v^{w_i-1}\otimes K, \\ x_a &: E_{t(a)}^{w_i} \longrightarrow E_{h(a)}^{w_i}, \\ y_a & \in H^1(X,\operatorname{\mathrm{Hom}}(E_{h(a)}^{w_i},E_{t(a)}^{w_i-1} \otimes K)). \end{align*} $$

We work with the specific example of the $\mathbb {C}^\times $ -action on $\mathcal {M}_{\overline {Q}}^{r,d}(\tau )$ , where Q is the framed $A_1$ quiver. From the fixed-point equations, a point $(\phi ,x,y)$ is fixed if there exists a one-parameter subgroup $g_\lambda $ such that for all $\lambda $ we have

$$ \begin{align*} \lambda\phi = g_\lambda \phi g^{-1}_\lambda , \quad x = g_\lambda x, \quad \lambda y = y g^{-1}_\lambda. \end{align*} $$

By stability, x is surjective, and therefore the only fixed points are of the form $(0,x,0)$ and the fixed-point set coincides with the $\mathrm {Gr}^d(1,\mathcal {O}^n)$ .

Replace the framing vertex with an unframed vertex. We are interested in the action

$$ \begin{align*} (E_1,E_2,\phi_1,\phi_2,x,y)\mapsto (E_1,E_2,\lambda \phi_1,\lambda\phi_2,x,\lambda y). \end{align*} $$

Using the weight space decomposition of $E_1,E_2$ , it is straightforward to verify that the fixed points can be described as pairs of holomorphic chains given by the $\phi _i:E_i^{w_i}\to E_i^{w_i-1}\otimes K$ , intertwined by the $x^k$ and $y^k$ data. Such a fixed point is shown in Figure 1. A more refined description, such as the length of the chains involved, will depend on stability and the topological type of the bundles $E_1,E_2$ .

Figure 1

A fixed point for the $\mathbb {C}^\times $ -action on the unframed $A_1$ quiver.

If $\lvert Q_0\rvert =n$ , the action above can be generalized to an action of the product $(\mathbb {C}^\times )^n$ on $\mathcal {M}_{\overline {Q}}^{r,d}(\tau )$ . For $\lambda = (\lambda _1,\ldots , \lambda _n)\in (\mathbb {C}^\times )^n$ , the action scales $\phi $ by $\lambda _v$ and y by $\lambda _{t(a)}$ .

A representation $(A,\phi ,x,y)$ is now a fixed point if there exists a homomorphism $g:(\mathbb {C}^\times )^n\rightarrow \mathcal {G}$ such that

$$ \begin{align*} \lambda_v\phi_v = g(\lambda)\phi_v g(\lambda)^{-1}, \quad x_a =g(\lambda)_{h(a)}x_a g(\lambda)_{t(a)}^{-1}, \quad \lambda_{t(a)}y_a = g(\lambda)_{t(a)}y_a g(\lambda)_{h(a)}^{-1}. \end{align*} $$

From here, the same argument as for Higgs bundles shows that in order for $\mathcal {E}= (E,\phi ,x,y)$ to be a fixed point, $\phi _v$ must be nilpotent for all v.

However, unlike in the case of ordinary Higgs bundles, the $\phi _v$ being nilpotent does not provide a sufficient condition for $\mathcal {E}$ to be fixed. As example, let ${\mathcal {E}=(E_1,E_2,\phi _1,\phi _2,x,y)}$ be a $\overline {Q}$ -bundle on the unframed $A_1$ quiver. Suppose ${E_1\cong E_2\cong \mathcal {O}_X\oplus \mathcal {O}_X}$ and

$$ \begin{align*} \phi_1 = \begin{bmatrix}0 & 0 \\ \alpha & 0 \end{bmatrix}, \quad \phi_2 = \begin{bmatrix}0 & 0 \\ \beta & 0 \end{bmatrix}, \quad x = \begin{bmatrix} 0 & x_{12} \\ x_{21} & 0 \end{bmatrix}, \quad y = \begin{bmatrix} 0 & 0 \\ y_{21} & 0 \end{bmatrix}. \end{align*} $$

Defining the nilpotent cone inside of $\mathcal {M}_{\overline {Q}}(\tau )$ to be those Nakajima representations with all Higgs fields $\phi _v$ nilpotent, all fixed points for the torus action are constrained to live inside the nilpotent cone.

As the underlying quiver becomes more complicated and as the ranks and degrees involved grow larger, the analysis of the fixed points becomes markedly more difficult. In the absence of a systematic approach, the understanding of the fixed points of the action relies on a case-by-case approach. However, once this is accomplished, the Białynicki-Birula decomposition of $\mathcal {M}_{\overline {Q}}^{r,d}(\tau )$ in terms of the fixed-point sets should allow for a more complete understanding of the topology of Nakajima bundle varieties.

On a more speculative note, the fixed points of the $\mathbb {C}^\times $ -action on the moduli space of Higgs bundles play an important role in understanding both the Hitchin system and Nakajima quiver varieties. The Hitchin system leads to a proper morphism ${\mathcal {H}:\mathcal {M}^{\text {Higgs}}(X)\rightarrow \mathcal {B}}$ to an affine space $\mathcal {B}$ , sending a Higgs field to the coefficients of its characteristic polynomial. The fixed points of the $\mathbb {C}^\times $ -action all reside in the nilpotent cone $\mathcal {H}^{-1}(0)$ of Higgs bundles $(E,\phi )$ with nilpotent $\phi $ . Given the presence of integrable systems on Nakajima quiver varieties [Reference Fisher and RayanFR16, Reference Rayan and SchaposnikRS21], as well as the similarities between Nakajima bundle varieties and Higgs moduli, the fixed points might facilitate a deeper understanding of the relationship between these systems.