2.1 Definitions

Let $X$ be a smooth, projective, geometrically connected curve of genus $g$ over a field $\Bbbk$ and let $D$ be a divisor on $X$ of degree $\ell$. Given $n\in \mathbb{N}$, let $Q=(I,H)$ be the quiver of type $A_{n-1}^{(1)}$, that is, let $I=\mathbb{Z}/n\mathbb{Z}$ be the set of vertices and $H=\{i\rightarrow i+1\mid i\in \mathbb{Z}/n\mathbb{Z}\}$ be the set of arrows. By definition, a $D$-twisted quiver sheaf (respectively, bundle) on $X$ is a tuple $\bar{E}=(E_{i},\unicode[STIX]{x1D703}_{i})_{i\in I}$ where $E_{i}$ is a coherent sheaf (respectively, vector bundle) on $X$ and $\unicode[STIX]{x1D703}_{i}\in \operatorname{Hom}(E_{i},E_{i+1}(D))$. As $D$ will be fixed throughout, we will often refer to such a data simply as a quiver sheaf (respectively, bundle).

To simplify notation, let ${\mathcal{A}}=\operatorname{Coh}(X)^{I}$ be the category of $I$-graded objects $E=(E_{i})_{i\in I}$ in $\operatorname{Coh}(X)$ and consider the shift functor

$$\begin{eqnarray}\displaystyle T:{\mathcal{A}}\rightarrow {\mathcal{A}},\quad E=(E_{i})_{i}\mapsto E[1]=(E_{i+1}(D))_{i}. & & \displaystyle \nonumber\end{eqnarray}$$

Then a quiver sheaf can be interpreted as a pair $\bar{E}=(E,\unicode[STIX]{x1D703})$, where $E\in {\mathcal{A}}$ and $\unicode[STIX]{x1D703}:E\rightarrow E[1]$ is a morphism in ${\mathcal{A}}$. We denote by ${\mathcal{A}}_{D}$ the category of quiver sheaves. It is an abelian category, with the obvious notion of morphism. Such categories have been studied by García-Prada, Gothen and collaborators; see, for example, [Reference García-Prada, Heinloth and SchmittGHS14, Reference Gothen and KingGK05]. Of particular importance are the Higgs case ($n=1$) in which one recovers the category of $D$-twisted (or meromorphic) Higgs sheaves, and the case $n=2$ which, for $\Bbbk =\mathbb{C}$ and $D=K$ the canonical divisor of $X$, yields a category equivalent to the (collection of) categories of Higgs bundles for the real groups $U(p,q)$; see [Reference GothenGot16]. Note also that, as any representation of a finite type $A$ quiver may trivially be regarded as a representation of a cyclic quiver, the categories of quiver sheaves considered here also contain the categories of quiver sheaves for finite type $A$ quivers (also known as ‘chains’; see [Reference García-Prada, Heinloth and SchmittGHS14]).

For $L$ a line bundle on $X$ and $E\in {\mathcal{A}}$, we define $E\otimes L=(E_{i}\otimes L)_{i}$. Similarly, for $\bar{E}=(E,\unicode[STIX]{x1D703})\in {\mathcal{A}}_{D}$, we define $\bar{E}\otimes L=(E\otimes L,\unicode[STIX]{x1D703}\otimes L)$ and we use a similar notation for the operation of twisting by a divisor. Similarly, we define $\bar{E}[1]=(E[1],\unicode[STIX]{x1D703}[1])$.

For a coherent sheaf $E\in \operatorname{Coh}(X)$, we define its class to be the pair $\operatorname{cl}E=(\operatorname{rk}E,\deg E)\in \mathbb{Z}^{2}$. The slope of a sheaf is

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}(E)=\frac{\deg E}{\operatorname{rk}E}\in \mathbb{Q}\cup \{\infty \}. & & \displaystyle \nonumber\end{eqnarray}$$

Similarly, for $E=(E_{i})_{i}\in {\mathcal{A}}$, we define

(5)$$\begin{eqnarray}\displaystyle & \operatorname{cl}E=(\operatorname{cl}E_{i})_{i}\in \unicode[STIX]{x1D6E4}=(\mathbb{Z}^{2})^{I}=\mathbb{Z}^{I}\oplus \mathbb{Z}^{I}, & \displaystyle\end{eqnarray}$$

(6)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D707}(E)=\frac{\deg E}{\operatorname{rk}E}\in \mathbb{Q}\cup \{\infty \},\quad \operatorname{rk}E=\sum \operatorname{rk}E_{i},\quad \deg E=\sum \deg E_{i}. & \displaystyle\end{eqnarray}$$

We extend this notation to quiver sheaves by setting $\operatorname{cl}\bar{E}=\operatorname{cl}E$ and $\unicode[STIX]{x1D707}(\bar{E})=\unicode[STIX]{x1D707}(E)$, for $\bar{E}=(E,\unicode[STIX]{x1D703})\in {\mathcal{A}}_{D}$. For any $\unicode[STIX]{x1D6FE}=(\bar{r},\bar{d})\in \unicode[STIX]{x1D6E4}$, define

(7)$$\begin{eqnarray}\unicode[STIX]{x1D6FE}[1]=(r_{i+1},d_{i+1}+\ell r_{i+1})_{i}\in \unicode[STIX]{x1D6E4}.\end{eqnarray}$$

Then $\operatorname{cl}(E)[1]=\operatorname{cl}(E[1])$ for any $E\in {\mathcal{A}}$.

For $E,F\in \operatorname{Coh}(X)$, we denote by $\unicode[STIX]{x1D712}(E,F)$ the Euler form on the category $\operatorname{Coh}(X)$, that is, we set

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}(E,F)=\dim \operatorname{Hom}(E,F)-\dim \operatorname{Ext}^{1}(E,F). & & \displaystyle \nonumber\end{eqnarray}$$

By the Riemann–Roch formula,

(8)$$\begin{eqnarray}\unicode[STIX]{x1D712}(E,F)=(1-g)\operatorname{rk}E\cdot \operatorname{rk}F+(\operatorname{rk}E\cdot \deg F-\operatorname{rk}F\cdot \deg E).\end{eqnarray}$$

Since $\unicode[STIX]{x1D712}(E,F)$ only depends on $\operatorname{cl}E$ and $\operatorname{cl}F$, we will sometimes denote this Euler form also by $\unicode[STIX]{x1D712}(\operatorname{cl}E,\operatorname{cl}F)$. The same notation is used for the Euler form on the category ${\mathcal{A}}$.

2.2 Homological properties

The categories $\operatorname{Coh}(X)$ and ${\mathcal{A}}$ are of homological dimension 1, while the category ${\mathcal{A}}_{D}$ is of homological dimension 2. More precisely, we have the following theorem.

Theorem 2.1 (cf. Gothen and King [Reference Gothen and KingGK05]).

Given $\bar{E}=(E,\unicode[STIX]{x1D703})$, $\bar{F}=(F,\unicode[STIX]{x1D703}^{\prime })$ in ${\mathcal{A}}_{D}$, there is a long exact sequence

$$\begin{eqnarray}\displaystyle 0 & \rightarrow & \displaystyle \operatorname{Hom}(\bar{E},\bar{F})\rightarrow \operatorname{Hom}(E,F)\rightarrow \operatorname{Hom}(E,F[1])\nonumber\\ \displaystyle & \rightarrow & \displaystyle \operatorname{Ext}^{1}(\bar{E},\bar{F})\rightarrow \operatorname{Ext}^{1}(E,F)\rightarrow \operatorname{Ext}^{1}(E,F[1])\rightarrow \operatorname{Ext}^{2}(\bar{E},\bar{F})\rightarrow 0\nonumber\end{eqnarray}$$

and the groups $\operatorname{Ext}^{i}(\bar{E},\bar{F})$ vanish for $i>2$.

Let us denote by

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}_{D}(\bar{E},\bar{F})=\dim \operatorname{Hom}(\bar{E},\bar{F})-\dim \operatorname{Ext}^{1}(\bar{E},\bar{F})+\dim \operatorname{Ext}^{2}(\bar{E},\bar{F}) & & \displaystyle \nonumber\end{eqnarray}$$

the Euler form in ${\mathcal{A}}_{D}$.

Observe that the Euler form $\unicode[STIX]{x1D712}$ on ${\mathcal{A}}_{D}$ is symmetric only in the case of Higgs sheaves (i.e. for $n=1$). Applying Serre duality for coherent sheaves, we obtain the following form of Serre duality for quiver sheaves.

Corollary 2.3. For any $\bar{E},\bar{F}\in {\mathcal{A}}_{D}$, we have

$$\begin{eqnarray}\displaystyle \operatorname{Ext}^{i}(\bar{E},\bar{F})\simeq \operatorname{Ext}^{2-i}(\bar{F},\bar{E}[-1](K))^{\ast }. & & \displaystyle \nonumber\end{eqnarray}$$

Recall that a coherent sheaf $E\in \operatorname{Coh}(X)$ is called semistable (respectively, stable) if for any proper subsheaf $F\subset E$ we have $\unicode[STIX]{x1D707}(F)\leqslant \unicode[STIX]{x1D707}(E)$ (respectively, $\unicode[STIX]{x1D707}(F)<\unicode[STIX]{x1D707}(E)$). The HN filtration of $E$ is the unique filtration

$$\begin{eqnarray}\displaystyle E=E_{1}\supset E_{2}\supset \cdots \supset E_{s}\supset E_{s+1}=0 & & \displaystyle \nonumber\end{eqnarray}$$

such that $E_{i}/E_{i+1}$ are semistable and

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}(E_{1}/E_{2})<\unicode[STIX]{x1D707}(E_{2}/E_{3})<\cdots <\unicode[STIX]{x1D707}(E_{s}). & & \displaystyle \nonumber\end{eqnarray}$$

We set

$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D70E}(E)=\{\unicode[STIX]{x1D707}(E_{1}/E_{2}),\unicode[STIX]{x1D707}(E_{2}/E_{3}),\ldots ,\unicode[STIX]{x1D707}(E_{s})\}\subset \mathbb{Q}\cup \{\infty \}, & \displaystyle \nonumber\\ \displaystyle & \unicode[STIX]{x1D707}_{\text{min}}(E)=\min \unicode[STIX]{x1D70E}(E)=\unicode[STIX]{x1D707}(E_{1}/E_{2}),\quad \unicode[STIX]{x1D707}_{\text{max}}(E)=\max \unicode[STIX]{x1D70E}(E)=\unicode[STIX]{x1D707}(E_{s}). & \displaystyle \nonumber\end{eqnarray}$$

In the same way we define semistable objects in ${\mathcal{A}}$ using the slope function (6). An object $E=(E_{i})_{i}\in {\mathcal{A}}$ is semistable if and only if all $E_{i}$ are semistable and have equal slope. We similarly define semistable objects in ${\mathcal{A}}_{D}$ using the slope function (6), that is, we say that $\bar{E}$ is semistable if for any quiver subsheaf $\bar{F}\subset \bar{E}$ we have $\unicode[STIX]{x1D707}(\bar{F})\leqslant \unicode[STIX]{x1D707}(\bar{E})$. We further say that $\bar{E}$ is stable if the inequality is strict for any proper quiver subsheaf $\bar{F}$. For $\unicode[STIX]{x1D708}\in \mathbb{Q}\cup \{\infty \}$ let us denote by ${\mathcal{A}}_{D}^{(\unicode[STIX]{x1D708})}$ the full subcategory of ${\mathcal{A}}_{D}$ whose objects are the semistable quiver sheaves of slope $\unicode[STIX]{x1D708}$.

We summarize the standard properties of ${\mathcal{A}}_{D}$ with respect to the above semistability notion in the following proposition, whose proof is left to the reader.

The following result will be crucial for our purposes.

Proof. By our assumption, $\deg (K-D)\leqslant 0$. Therefore

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}(\bar{E}[-1](K))\leqslant \unicode[STIX]{x1D707}(\bar{E})<\unicode[STIX]{x1D707}(\bar{F}). & & \displaystyle \nonumber\end{eqnarray}$$

By the semistability of $\bar{E}[-1](K)$ and $\bar{F}$, we conclude that

$$\begin{eqnarray}\displaystyle \operatorname{Hom}(\bar{F},\bar{E}[-1](K))=0. & & \displaystyle \nonumber\end{eqnarray}$$

By the Serre duality of Corollary 2.3, this implies that $\operatorname{Ext}^{2}(\bar{E},\bar{F})=0$.◻

2.3 Positive quiver sheaves

In this paragraph we introduce a suitable truncation ${\mathcal{A}}_{D}^{+}$ of ${\mathcal{A}}_{D}$ and prove that semistable objects of ${\mathcal{A}}_{D}$ having large slope are contained in ${\mathcal{A}}_{D}^{+}$.

We denote by $\operatorname{Coh}^{+}(X)$ the full subcategory of $\operatorname{Coh}(X)$ whose objects satisfy $\unicode[STIX]{x1D707}_{\text{min}}(E)\geqslant 0$. Equivalently, this means that all quotients of $E$ have nonnegative degree. The subcategory $\operatorname{Coh}^{+}(X)$ is closed under extensions and quotients, but not under taking subobjects. Similarly, we define ${\mathcal{A}}^{+}\subset {\mathcal{A}}$ to be the subcategory of ${\mathcal{A}}$ whose objects $E\in {\mathcal{A}}$ satisfy $\unicode[STIX]{x1D707}_{\text{min}}(E)\geqslant 0$. We define ${\mathcal{A}}_{D}^{+}$ as the full subcategory of ${\mathcal{A}}_{D}$ whose objects $\bar{E}=(E,\unicode[STIX]{x1D703})$ satisfy $E\in {\mathcal{A}}^{+}$. Obviously, if $\bar{E}\in {\mathcal{A}}_{D}^{+}$, then $\unicode[STIX]{x1D707}(\bar{E})\geqslant 0$.

Proof. Let $\unicode[STIX]{x1D703}\in \operatorname{Hom}(E,E[1])$ and $\bar{E}=(E,\unicode[STIX]{x1D703})$. There exists a (unique) short exact sequence

$$\begin{eqnarray}\displaystyle 0\rightarrow E^{\prime }\rightarrow E\rightarrow E^{\prime \prime }\rightarrow 0 & & \displaystyle \nonumber\end{eqnarray}$$

in ${\mathcal{A}}$ with $\unicode[STIX]{x1D707}_{\text{min}}(E^{\prime })\geqslant \unicode[STIX]{x1D708}_{k+1}$ and $\unicode[STIX]{x1D707}_{\text{max}}(E^{\prime \prime })\leqslant \unicode[STIX]{x1D708}_{k}$. Since $\unicode[STIX]{x1D707}_{\text{min}}(E^{\prime })>\unicode[STIX]{x1D707}_{\text{max}}(E^{\prime \prime })+\ell$ we deduce that $\operatorname{Hom}(E^{\prime },E^{\prime \prime }[1])=0$ and thus $\unicode[STIX]{x1D703}(E^{\prime })\subset E^{\prime }[1]$. But then $\bar{E}^{\prime }=(E^{\prime },\unicode[STIX]{x1D703})$ is a destabilizing subobject of $\bar{E}$.◻

Proof. If $\ell \leqslant 0$, then by Lemma 2.7 the object $E$ is semistable, hence $\bar{E}\in \mathfrak{M}_{D}^{+}$. Assume that $\ell >0$. Let us write $\unicode[STIX]{x1D70E}(E)=\{\unicode[STIX]{x1D708}_{1}<\unicode[STIX]{x1D708}_{2}<\cdots <\unicode[STIX]{x1D708}_{s}\}$ and let $r_{k},d_{k}$ be the rank and degree of the $k$th factor of the HN filtration of $E$, so that

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{k}d_{k}=d,\quad \mathop{\sum }_{k}r_{k}=r,\quad \unicode[STIX]{x1D708}_{k}=\frac{d_{k}}{r_{k}}. & & \displaystyle \nonumber\end{eqnarray}$$

By Lemma 2.7 we have $\unicode[STIX]{x1D708}_{k+1}-\unicode[STIX]{x1D708}_{k}\leqslant \ell$, hence $\unicode[STIX]{x1D708}_{k}\leqslant \unicode[STIX]{x1D708}_{1}+(k-1)\ell$, for all $k$, and

$$\begin{eqnarray}\displaystyle \ell \frac{(r-1)}{2}\leqslant \frac{d}{r}=\frac{\mathop{\sum }_{k}\unicode[STIX]{x1D708}_{k}r_{k}}{r}\leqslant \frac{\mathop{\sum }_{k}(\unicode[STIX]{x1D708}_{1}+(k-1)\ell )r_{k}}{r}\leqslant \frac{s}{r}\unicode[STIX]{x1D708}_{1}+\ell \frac{(r-1)}{2} & & \displaystyle \nonumber\end{eqnarray}$$

which implies that $\unicode[STIX]{x1D708}_{1}\geqslant 0$, hence $E\in {\mathcal{A}}^{+}$. We used here the fact that if $\sum _{k}r_{k}=r$ and $r_{k}\geqslant 1$ for all $k$ then $\sum _{k}(k-1)r_{k}\leqslant \binom{r}{2}$.◻

2.4 Notation for stacks

Let us denote by $\mathfrak{M}(r,d)$ the stack of coherent sheaves of rank $r$ and degree $d$ on $X$. This stack is locally of finite type and of finite volume; see, for example, [Reference SchiffmannSch16, Theorem 4.1.] Let $\mathfrak{M}^{+}(r,d)$ be the substack parametrizing positive coherent sheaves. This open substack is of finite type. Similarly, given $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}=\mathbb{Z}^{I}\oplus \mathbb{Z}^{I}$, let $\mathfrak{M}(\unicode[STIX]{x1D6FE})$ be the stack of objects in ${\mathcal{A}}$ having class $\unicode[STIX]{x1D6FE}$ and let $\mathfrak{M}^{+}(\unicode[STIX]{x1D6FE})$ be its substack parametrizing positive objects.

Let $\mathfrak{M}_{D}(\unicode[STIX]{x1D6FE})$ (respectively, $\mathfrak{M}_{D}^{\text{vec}}(\unicode[STIX]{x1D6FE})$) be the stack parametrizing quiver sheaves (respectively, bundles) on $X$ having class $\unicode[STIX]{x1D6FE}$. Let $\mathfrak{M}_{D}^{+}(\unicode[STIX]{x1D6FE})$ (respectively, $\mathfrak{M}_{D}^{+,\text{vec}}(\unicode[STIX]{x1D6FE})$) be the substack parametrizing positive quiver sheaves (respectively, bundles). Contrary to $\mathfrak{M}_{D}(\unicode[STIX]{x1D6FE})$, the stack $\mathfrak{M}_{D}^{+}(\unicode[STIX]{x1D6FE})$ is of finite type and hence of finite volume. Finally, let $\mathfrak{M}_{D}^{\text{ss}}(\unicode[STIX]{x1D6FE})\subset \mathfrak{M}_{D}(\unicode[STIX]{x1D6FE})$ be the substack of semistable quiver sheaves (they are automatically quiver bundles if the rank is positive).

3.1 Hall algebras and quantum torus

Let ${\mathcal{A}}$ be an abelian category, linear over a finite field $\Bbbk =\mathbb{F}_{q}$, of finite homological dimension and such that $\dim \operatorname{Ext}^{k}(M,N)<\infty$ for all objects $M,N$ and all $k\geqslant 0$. Let $\unicode[STIX]{x1D712}:K_{0}({\mathcal{A}})\otimes _{\mathbb{Z}}K_{0}({\mathcal{A}})\rightarrow \mathbb{Z}$ denote the Euler form. Let also $\unicode[STIX]{x1D6E4}$ be a lattice equipped with a skew-symmetric form $\langle -,-\rangle$ and with a group homomorphism $\operatorname{cl}:K_{0}({\mathcal{A}})\rightarrow \unicode[STIX]{x1D6E4}$ such that

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}(E,F)-\unicode[STIX]{x1D712}(F,E)=\langle \operatorname{cl}E,\operatorname{cl}F\rangle ,\quad E,F\in {\mathcal{A}}. & & \displaystyle \nonumber\end{eqnarray}$$

The algebra $\mathbb{T}=\mathbb{Q}(q^{1/2})[\unicode[STIX]{x1D6E4}]$ equipped with the product

(9)$$\begin{eqnarray}t^{\unicode[STIX]{x1D6FC}}\circ t^{\unicode[STIX]{x1D6FD}}=(-q^{1/2})^{\langle \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\rangle }t^{\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}},\quad \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6E4},\end{eqnarray}$$

is called the quantum (affine) torus. Let ${\mathcal{H}}$ be the Hall algebra of ${\mathcal{A}}$ (see, for example, [Reference SchiffmannSch12]). It is a vector space generated by isomorphism classes of objects in ${\mathcal{A}}$ equipped with an associative multiplication

$$\begin{eqnarray}\displaystyle (f\circ g)(X)=\mathop{\sum }_{U\subset X}f(U)g(X/U),\quad f,g\in {\mathcal{H}}. & & \displaystyle \nonumber\end{eqnarray}$$

Both ${\mathcal{H}}$ and $\mathbb{T}$ are graded by the lattice $\unicode[STIX]{x1D6E4}$. We will occasionally consider their completions, which we will still denote by ${\mathcal{H}}$ and $\mathbb{T}$ respectively for simplicity when there is no risk of confusion. One defines the integration map

(10)$$\begin{eqnarray}I:{\mathcal{H}}\rightarrow \mathbb{T},\quad [E]\mapsto (-q^{1/2})^{\unicode[STIX]{x1D712}(E,E)}\frac{t^{\operatorname{cl}E}}{\#\operatorname{Aut}E}.\end{eqnarray}$$

A crucial property of $I$ is that it is a ring homomorphism if ${\mathcal{A}}$ has homological dimension $1$ [Reference ReinekeRei06]. More generally, it satisfies

(11)$$\begin{eqnarray}I([E]\circ [F])=I(E)\circ I(F)\end{eqnarray}$$

if $\operatorname{Ext}^{k}(F,E)=0$ for $k\geqslant 2$. This explains the significance of Corollary 2.6.

3.2 Generating functions

We will denote the Hall algebra of ${\mathcal{A}}_{D}$ by ${\mathcal{H}}_{D}$. Set $\unicode[STIX]{x1D6E4}=(\mathbb{Z}^{2})^{I}=\mathbb{Z}^{I}\oplus \mathbb{Z}^{I}$, consider the map $\operatorname{cl}:K_{0}({\mathcal{A}}_{D})\rightarrow \unicode[STIX]{x1D6E4}$ defined in (5), and equip $\unicode[STIX]{x1D6E4}$ with bilinear forms (cf. Corollary 2.2)

(12)$$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D712}_{D}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE}^{\prime }) & = & \displaystyle \mathop{\sum }_{i\in I}\biggl((1-g)r_{i}r_{i}^{\prime }+(g-1-\ell )r_{i}r_{i+1}^{\prime }+\left|\begin{array}{@{}cc@{}}r_{i} & r_{i}^{\prime }-r_{i+1}^{\prime }\\ d_{i} & d_{i}^{\prime }-d_{i+1}^{\prime }\end{array}\right|\biggr),\end{eqnarray}$$

(13)$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE}^{\prime }\rangle & = & \displaystyle \unicode[STIX]{x1D712}_{D}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE}^{\prime })-\unicode[STIX]{x1D712}_{D}(\unicode[STIX]{x1D6FE}^{\prime },\unicode[STIX]{x1D6FE}),\end{eqnarray}$$

for $\unicode[STIX]{x1D6FE}=(\bar{r},\bar{d})\in \unicode[STIX]{x1D6E4}$ and $\unicode[STIX]{x1D6FE}^{\prime }=(\bar{r}^{\prime },\bar{d}^{\prime })\in \unicode[STIX]{x1D6E4}$. In the above equation, $|\cdot |$ stands for the determinant. Observe that when $n=1$ (i.e. in the Higgs case) the form $\langle -,-\rangle$ vanishes, hence the quantum torus $\mathbb{T}=\mathbb{Q}(q^{1/2})[\mathbb{Z}^{2}]$ is commutative.

Given $\unicode[STIX]{x1D6FE}=(\bar{r},\bar{d})\in \unicode[STIX]{x1D6E4}=\mathbb{Z}^{I}\oplus \mathbb{Z}^{I}$, we will use variables

$$\begin{eqnarray}\displaystyle z^{\bar{d}}=t^{(0,\bar{d})},\quad T^{\bar{r}}=t^{(\bar{r},0)} & & \displaystyle \nonumber\end{eqnarray}$$

in the quantum torus. Recall that $\mathfrak{M}_{D}^{\text{ss}}(\unicode[STIX]{x1D6FE})=\mathfrak{M}_{D}^{\text{ss}}(\bar{r},\bar{d})$ is the stack of semistable objects $\bar{E}=(E,\unicode[STIX]{x1D703})\in {\mathcal{A}}_{D}$ with $E$ having class $\unicode[STIX]{x1D6FE}$. Note that if $(E,\unicode[STIX]{x1D703})$ is semistable and has positive rank then $E=(E_{i})_{i}$ is an $I$-graded vector bundle. Define

(14)$$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D7D9}_{\unicode[STIX]{x1D6FE}}^{\text{ss}} & = & \displaystyle \mathop{\sum }_{\bar{E}\in \mathfrak{M}_{D}^{\text{ss}}(\unicode[STIX]{x1D6FE})(\Bbbk )}[\bar{E}]\in {\mathcal{H}}_{D},\end{eqnarray}$$

(15)$$\begin{eqnarray}\displaystyle \displaystyle \hat{\unicode[STIX]{x1D6FA}}_{D}(\unicode[STIX]{x1D6FE})t^{\unicode[STIX]{x1D6FE}} & = & \displaystyle I(\unicode[STIX]{x1D7D9}_{\unicode[STIX]{x1D6FE}}^{\text{ss}})=(-q^{1/2})^{\unicode[STIX]{x1D712}_{D}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE})}\mathop{\sum }_{\bar{E}\in \mathfrak{M}_{D}^{\text{ss}}(\unicode[STIX]{x1D6FE})(\Bbbk )}\frac{1}{\#\operatorname{Aut}\bar{E}}t^{\unicode[STIX]{x1D6FE}},\end{eqnarray}$$

(16)$$\begin{eqnarray}\displaystyle \displaystyle \hat{\unicode[STIX]{x1D6FA}}_{D,\unicode[STIX]{x1D70F}} & = & \displaystyle \mathop{\sum }_{\unicode[STIX]{x1D707}(\unicode[STIX]{x1D6FE})=\unicode[STIX]{x1D70F}}\hat{\unicode[STIX]{x1D6FA}}_{D}(\unicode[STIX]{x1D6FE})t^{\unicode[STIX]{x1D6FE}},\quad \unicode[STIX]{x1D70F}\in \mathbb{Q}.\end{eqnarray}$$

Tensoring by a line bundle preserves semistability. There always exists a line bundle of degree $1$ over a smooth, projective, geometrically connected curve defined over a finite field, hence we conclude that $\hat{\unicode[STIX]{x1D6FA}}_{D}(\bar{r},\bar{d})=\hat{\unicode[STIX]{x1D6FA}}_{D}(\bar{r},\bar{d}+\bar{r})$.

We likewise define the elements

(17)$$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D7D9}_{\unicode[STIX]{x1D6FE}}^{+} & = & \displaystyle \mathop{\sum }_{\bar{E}\in \mathfrak{M}_{D}^{+}(\unicode[STIX]{x1D6FE})(\Bbbk )}[\bar{E}]\in {\mathcal{H}}_{D},\end{eqnarray}$$

(18)$$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D6E9}_{D}^{+}(\unicode[STIX]{x1D6FE})t^{\unicode[STIX]{x1D6FE}} & = & \displaystyle I(\unicode[STIX]{x1D7D9}_{\unicode[STIX]{x1D6FE}}^{+})=(-q^{1/2})^{\unicode[STIX]{x1D712}_{D}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE})}\mathop{\sum }_{\bar{E}\in \mathfrak{M}_{D}^{+}(\unicode[STIX]{x1D6FE})(\Bbbk )}\frac{1}{\#\operatorname{Aut}\bar{E}}t^{\unicode[STIX]{x1D6FE}}.\end{eqnarray}$$

Observe that the stacks $\mathfrak{M}_{D}^{+}(\unicode[STIX]{x1D6FE})$ have finitely many objects up to isomorphisms so that the above sums are well defined. We can uniquely factorize

(19)$$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{D}^{+}(T,z)=\mathop{\sum }_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6E9}_{D}^{+}(\unicode[STIX]{x1D6FE})t^{\unicode[STIX]{x1D6FE}}=\mathop{\prod }_{\unicode[STIX]{x1D70F}\downarrow }\tilde{\unicode[STIX]{x1D6FA}}_{D,\unicode[STIX]{x1D70F}}^{+},\quad \tilde{\unicode[STIX]{x1D6FA}}_{D,\unicode[STIX]{x1D70F}}^{+}=\mathop{\sum }_{\unicode[STIX]{x1D707}(\unicode[STIX]{x1D6FE})=\unicode[STIX]{x1D70F}}\tilde{\unicode[STIX]{x1D6FA}}_{D}^{+}(\unicode[STIX]{x1D6FE})t^{\unicode[STIX]{x1D6FE}}.\end{eqnarray}$$

Note that unless $n=1$, the product is ordered as the quantum torus is not commutative.

It will be convenient to consider the above invariants simultaneously for all finite field extensions of $\Bbbk =\mathbb{F}_{q}$. Hoping that this will not cause confusion, we define volumes (see Appendix A)

(20)$$\begin{eqnarray}\displaystyle \displaystyle \hat{\unicode[STIX]{x1D6FA}}_{D}(\unicode[STIX]{x1D6FE}) & = & \displaystyle (-q^{1/2})^{\unicode[STIX]{x1D712}_{D}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE})}[\mathfrak{M}_{D}^{\text{ss}}(\unicode[STIX]{x1D6FE})]\in {\mathcal{V}},\end{eqnarray}$$

(21)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E9}_{D}^{+}(\unicode[STIX]{x1D6FE}) & = & \displaystyle (-q^{1/2})^{\unicode[STIX]{x1D712}_{D}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE})}[\mathfrak{M}_{D}^{+}(\unicode[STIX]{x1D6FE})]\in {\mathcal{V}}.\end{eqnarray}$$

We can similarly define invariants $\unicode[STIX]{x1D6E9}_{D}^{+,\text{vec}}(\unicode[STIX]{x1D6FE})\in {\mathcal{V}}$ counting quiver bundles. The quantum torus $\mathbb{T}$ is now the algebra ${\mathcal{V}}[\unicode[STIX]{x1D6E4}]$ (or its completion) with the multiplication (9) using $\boldsymbol{q}=(q^{n})_{n\geqslant 1}\in {\mathcal{V}}$ instead of $q$ (which we will still denote by $q$ for convenience).

We will determine the series $\unicode[STIX]{x1D6E9}_{D}^{+}(T,z)$ for $\ell \geqslant 2g-2$ in the next sections. Then the series $\tilde{\unicode[STIX]{x1D6FA}}_{D,\unicode[STIX]{x1D70F}}^{+}$ (19) can be computed by the formula of Reineke [Reference ReinekeRei03] (cf. [Reference MozgovoyMoz10, Reference Mozgovoy and ReinekeMR14]). After that the following statement allows us to determine $\hat{\unicode[STIX]{x1D6FA}}_{D}(\unicode[STIX]{x1D6FE})$.

Proof. Choose $\unicode[STIX]{x1D70F}=\ell (r-1)/2$. Given any $\bar{E}\in \mathfrak{M}_{D}^{+}$ or rank ${\leqslant}r$, there is a unique exact sequence

$$\begin{eqnarray}\displaystyle 0\rightarrow \bar{E}^{\prime }\rightarrow \bar{E}\rightarrow \bar{E}^{\prime \prime }\rightarrow 0, & & \displaystyle \nonumber\end{eqnarray}$$

where $\bar{E}^{\prime }$ has HN factors (in ${\mathcal{A}}_{D}$) of slope ${\geqslant}\unicode[STIX]{x1D70F}$ and $\bar{E}^{\prime \prime }$ has HN factors of slope ${<}\unicode[STIX]{x1D70F}$. The object $\bar{E}^{\prime \prime }$ is contained in ${\mathcal{A}}_{D}^{+}$ as a quotient of $\bar{E}$.

Conversely, assume that we have an exact sequence as above, with $\operatorname{rk}(\bar{E})\leqslant r$, $\bar{E}^{\prime }$ having HN factors of slope ${\geqslant}\unicode[STIX]{x1D70F}$ and $\bar{E}^{\prime \prime }\in {\mathcal{A}}_{D}^{+}$ having HN factors of slope ${<}\unicode[STIX]{x1D70F}$. Given an HN factor $\bar{F}$ of $\bar{E}^{\prime }$ having rank $r^{\prime }$ and degree $d^{\prime }$, we have $d^{\prime }/r^{\prime }\geqslant \unicode[STIX]{x1D70F}=\ell (r-1)/2\geqslant \ell (r^{\prime }-1)/2$. Therefore $\bar{F}\in {\mathcal{A}}_{D}^{+}$ by Lemma 2.8. We conclude that $\bar{E}\in {\mathcal{A}}_{D}^{+}$.

This observation implies an identity in the Hall algebra which translates, by applying (11) and Corollary 2.6, to the identity in the quantum torus

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E9}_{D}^{+}(T,z)=\mathop{\prod }_{\unicode[STIX]{x1D708}\downarrow ,\unicode[STIX]{x1D708}\geqslant \unicode[STIX]{x1D70F}}\hat{\unicode[STIX]{x1D6FA}}_{D,\unicode[STIX]{x1D708}}\circ \mathop{\sum }_{\unicode[STIX]{x1D707}(\unicode[STIX]{x1D6FC})<\unicode[STIX]{x1D70F}}C_{\unicode[STIX]{x1D6FC}}t^{\unicode[STIX]{x1D6FC}} & & \displaystyle \nonumber\end{eqnarray}$$

modulo powers of $t$ having rank ${>}r$, where $C_{\unicode[STIX]{x1D6FC}}$ are some invariants. Comparing it with (19), we conclude that $\hat{\unicode[STIX]{x1D6FA}}_{D,\unicode[STIX]{x1D708}}=\tilde{\unicode[STIX]{x1D6FA}}_{D,\unicode[STIX]{x1D708}}^{+}$, for $\unicode[STIX]{x1D708}\geqslant \unicode[STIX]{x1D70F}$, modulo powers of $t$ having rank ${>}r$. This means that $\hat{\unicode[STIX]{x1D6FA}}_{D}(\unicode[STIX]{x1D6FC})=\tilde{\unicode[STIX]{x1D6FA}}_{D}^{+}(\unicode[STIX]{x1D6FC})$ for $\operatorname{rk}\unicode[STIX]{x1D6FC}\leqslant r$ and $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D6FC})\geqslant \unicode[STIX]{x1D70F}$. In particular, $\operatorname{rk}\unicode[STIX]{x1D6FE}=r$ and $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D6FE})=d/r\geqslant \unicode[STIX]{x1D70F}$, hence $\hat{\unicode[STIX]{x1D6FA}}_{D}(\unicode[STIX]{x1D6FE})=\tilde{\unicode[STIX]{x1D6FA}}_{D}^{+}(\unicode[STIX]{x1D6FE})$.◻

3.3 Donaldson–Thomas invariants

The special case $n=1$ is the most important as it corresponds to the moduli stacks of (meromorphic) Higgs bundles. The quantum torus is commutative in this case, as the form $\langle -,-\rangle$ on $\unicode[STIX]{x1D6E4}=\mathbb{Z}^{2}$ vanishes. We define the DT invariants $\unicode[STIX]{x1D6FA}_{D}(r,d)$ by the formula

(22)$$\begin{eqnarray}\mathop{\sum }_{d/r=\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D6FA}_{D}(r,d)}{q-1}T^{r}z^{d}=\operatorname{Log}\biggl(\mathop{\sum }_{d/r=\unicode[STIX]{x1D70F}}\hat{\unicode[STIX]{x1D6FA}}_{D}(r,d)T^{r}z^{d}\biggr),\quad \unicode[STIX]{x1D70F}\in \mathbb{Q},\end{eqnarray}$$

where $\operatorname{Log}$ is the plethystic logarithm. Comparing equations for $\unicode[STIX]{x1D70F}$ and $\unicode[STIX]{x1D70F}+1$, we obtain that $\unicode[STIX]{x1D6FA}_{D}(r,d)=\unicode[STIX]{x1D6FA}_{D}(r,d+r)$. Various tests justify the conjecture that if $\deg D\geqslant 2g-2$, then $\unicode[STIX]{x1D6FA}_{D}(r,d)$ are independent of $d$ (cf. [Reference Chuang, Diaconescu and PanCDP11, Conjecture 1.9]). Note that if $r,d$ are coprime then

$$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D6FA}}_{D}(r,d)=\frac{\unicode[STIX]{x1D6FA}_{D}(r,d)}{q-1}. & & \displaystyle \nonumber\end{eqnarray}$$

For $D=K$ and coprime $r,d$, independence of $\hat{\unicode[STIX]{x1D6FA}}_{D}(r,d)$ of $d$ was conjectured (in the closely related case of $\operatorname{SL}_{n}$-Higgs bundles) in [Reference HauselHau05, Conjecture 3.2].

We also consider the truncated version (cf. (19))

(23)$$\begin{eqnarray}\mathop{\sum }_{d/r=\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D6FA}_{D}^{+}(r,d)}{q-1}T^{r}z^{d}=\operatorname{Log}\biggl(\mathop{\sum }_{d/r=\unicode[STIX]{x1D70F}}\tilde{\unicode[STIX]{x1D6FA}}_{D}^{+}(r,d)T^{r}z^{d}\biggr),\quad \unicode[STIX]{x1D70F}\in \mathbb{Q}.\end{eqnarray}$$

As the quantum torus is commutative, we obtain from (19) that

(24)$$\begin{eqnarray}\mathop{\sum }_{r,d}\frac{\unicode[STIX]{x1D6FA}_{D}^{+}(r,d)}{q-1}T^{r}z^{d}=\operatorname{Log}\biggl(\mathop{\sum }_{r,d}\unicode[STIX]{x1D6E9}_{D}^{+}(r,d)T^{r}z^{d}\biggr).\end{eqnarray}$$

We can similarly define $\unicode[STIX]{x1D6FA}_{D}^{+,\text{vec}}(r,d)$ using invariants $\unicode[STIX]{x1D6E9}_{D}^{+,\text{vec}}(r,d)$, but we will actually see in Corollary 4.2 that $\unicode[STIX]{x1D6FA}_{D}^{+,\text{vec}}(r,d)=\unicode[STIX]{x1D6FA}_{D}^{+}(r,d)$ for $r\geqslant 1$.

In § 6 we will use our computation of $\unicode[STIX]{x1D6E9}_{D}^{+}(r,d)$ for negative $D$ (see § 5) to give a closed expression for the truncated DT invariants $\unicode[STIX]{x1D6FA}_{D}^{+}(r,d)$. Because $\unicode[STIX]{x1D6FA}_{D}(r,d)=\unicode[STIX]{x1D6FA}_{D}(r,d+r)$, this will be enough to fully determine the DT invariants $\unicode[STIX]{x1D6FA}_{D}(r,d)$.

4.1 Nilpotent quiver sheaves

We will say that a quiver sheaf $\bar{E}=(E,\unicode[STIX]{x1D703})$ is nilpotent if there exists $k>0$ such that the composition $\unicode[STIX]{x1D703}^{k}$,

(25)$$\begin{eqnarray}E\rightarrow E[1]\rightarrow E[2]\rightarrow \cdots \rightarrow E[k],\end{eqnarray}$$

vanishes. We call the minimal $k$ satisfying this property the nilpotency index of $\bar{E}$.

Let $\mathfrak{M}_{D,\text{nil}}^{+}(\unicode[STIX]{x1D6FE})$ denote the stack of nilpotent positive quiver sheaves having class $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}$ and let $\mathfrak{M}_{D,\text{nil}}^{+,\text{vec}}(\unicode[STIX]{x1D6FE})$ denote its open substack whose objects are quiver bundles. Observe that if $\ell <0$ then any quiver bundle is automatically nilpotent, hence

(26)$$\begin{eqnarray}\mathfrak{M}_{D}^{+,\text{vec}}(\unicode[STIX]{x1D6FE})=\mathfrak{M}_{D,\text{nil}}^{+,\text{vec}}(\unicode[STIX]{x1D6FE}).\end{eqnarray}$$

We define (cf. (21))

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E9}_{D,\text{nil}}^{+}(r,d)=(-q^{1/2})^{-\ell r^{2}}[\mathfrak{M}_{D,\text{nil}}^{+}(r,d)],\quad \unicode[STIX]{x1D6E9}_{D,\text{nil}}^{+,\text{vec}}(r,d)=(-q^{1/2})^{-\ell r^{2}}[\mathfrak{M}_{D,\text{nil}}^{+,\text{vec}}(r,d)] & & \displaystyle \nonumber\end{eqnarray}$$

and in the Higgs case we may also define $\unicode[STIX]{x1D6FA}_{D,\text{nil}}^{+}(\unicode[STIX]{x1D6FE})$ and $\unicode[STIX]{x1D6FA}_{D,\text{nil}}^{+,\text{vec}}(\unicode[STIX]{x1D6FE})$ as in (24).

Lemma 4.1. We have

(27)$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6E9}_{D}^{+}(\unicode[STIX]{x1D6FE})t^{\unicode[STIX]{x1D6FE}} & = & \displaystyle \mathop{\sum }_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6E9}_{D}^{+,\text{vec}}(\unicode[STIX]{x1D6FE})t^{\unicode[STIX]{x1D6FE}}\cdot \mathop{\sum }_{\bar{d}}\unicode[STIX]{x1D6E9}_{D}^{+}(0,\bar{d})t^{(0,\bar{d})},\end{eqnarray}$$

(28)$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6E9}_{D,\text{nil}}^{+}(\unicode[STIX]{x1D6FE})t^{\unicode[STIX]{x1D6FE}} & = & \displaystyle \mathop{\sum }_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6E9}_{D,\text{nil}}^{+,\text{vec}}(\unicode[STIX]{x1D6FE})t^{\unicode[STIX]{x1D6FE}}\cdot \mathop{\sum }_{\bar{d}}\unicode[STIX]{x1D6E9}_{D,\text{nil}}^{+}(0,\bar{d})t^{(0,\bar{d})}.\end{eqnarray}$$

Proof. Let us prove just the second statement; the proof of the first is similar. Let $E\in {\mathcal{A}}^{+}$, and let $E=F\oplus T$ be a decomposition as a direct sum of a torsion-free and a torsion part. Observe that $F$ and $T$ belong to ${\mathcal{A}}^{+}$. We have

$$\begin{eqnarray}\displaystyle \operatorname{Hom}(E,E[1])=\operatorname{Hom}(F,F[1])\oplus \operatorname{Hom}(T,T[1])\oplus \operatorname{Hom}(F,T[1]), & & \displaystyle \nonumber\end{eqnarray}$$

and $\unicode[STIX]{x1D703}\in \operatorname{Hom}(E,E[1])$ is nilpotent if and only if its projections to $\operatorname{Hom}(F,F[1])$ and $\operatorname{Hom}(T,T[1])$ are. On the other hand, there is a canonical exact sequence

$$\begin{eqnarray}\displaystyle 1\rightarrow \operatorname{Hom}(F,T)\rightarrow \operatorname{Aut}E\rightarrow \operatorname{Aut}F\times \operatorname{Aut}T\rightarrow 1. & & \displaystyle \nonumber\end{eqnarray}$$

We deduce that

$$\begin{eqnarray}\displaystyle \frac{\#\operatorname{Hom}^{\text{nil}}(E,E[1])}{\#\operatorname{Aut}E}=\frac{\#\operatorname{Hom}^{\text{nil}}(F,F[1])}{\#\operatorname{Aut}F}\cdot \frac{\#\operatorname{Hom}^{\text{nil}}(T,T[1])}{\#\operatorname{Aut}T}.\Box & & \displaystyle \nonumber\end{eqnarray}$$

In the Higgs case ($n=1$) this implies the following corollary.

Corollary 4.2. We have, for $r\geqslant 1$ and $d\geqslant 0$,

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA}_{D}^{+}(r,d)=\unicode[STIX]{x1D6FA}_{D}^{+,\text{vec}}(r,d),\quad \unicode[STIX]{x1D6FA}_{D,\text{nil}}^{+}(r,d)=\unicode[STIX]{x1D6FA}_{D,\text{nil}}^{+,\text{vec}}(r,d). & & \displaystyle \nonumber\end{eqnarray}$$

Lemma 4.3. We have

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{d\geqslant 0}\unicode[STIX]{x1D6E9}_{D,\text{nil}}^{+}(0,d)z^{d}=\operatorname{Exp}\biggl(\frac{[X]}{q-1}\mathop{\sum }_{d\geqslant 1}z^{d}\biggr)=\operatorname{Exp}\biggl(\frac{[X]}{q-1}\cdot \frac{z}{1-z}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$

Proof. This formula is proved as the second equality of Theorem 4.9. Observe that the number of absolutely indecomposable torsion sheaves of degree $d>0$ is $\#X(\Bbbk )$, hence

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{d\geqslant 0}\unicode[STIX]{x1D6E9}_{D,\text{nil}}^{+}(0,d)z^{d}=\operatorname{Exp}\biggl(\frac{[X]}{q-1}\mathop{\sum }_{d\geqslant 1}z^{d}\biggr).\Box & & \displaystyle \nonumber\end{eqnarray}$$

4.2 Consequences of Serre duality

Proof. We have, by definition,

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E9}_{D}^{+}(\unicode[STIX]{x1D6FE})=(-q^{1/2})^{\unicode[STIX]{x1D712}_{D}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE})}\mathop{\sum }_{\bar{E}\in \mathfrak{M}_{D}^{+}(\unicode[STIX]{x1D6FE})}\frac{1}{\#\operatorname{Aut}(\bar{E})}=(-q^{1/2})^{\unicode[STIX]{x1D712}_{D}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE})}\mathop{\sum }_{E\in \mathfrak{M}^{+}(\unicode[STIX]{x1D6FE})}\frac{q^{h^{0}(E,E[1])}}{\#\operatorname{Aut}(E)}, & & \displaystyle \nonumber\end{eqnarray}$$

where we have set

$$\begin{eqnarray}\displaystyle h^{k}(E,F)=\dim \operatorname{Ext}^{k}(E,F),\quad k=0,1;\,E,F\in {\mathcal{A}}. & & \displaystyle \nonumber\end{eqnarray}$$

Given $E=(E_{i})_{i}\in {\mathcal{A}}^{+}(\unicode[STIX]{x1D6FE})$, consider $F=(E_{-i})_{i}\in {\mathcal{A}}^{+}(\unicode[STIX]{x1D6FE}^{\ast })$. Then $(F,\unicode[STIX]{x1D711})\in \mathfrak{M}_{K-D}^{+}(\unicode[STIX]{x1D6FE}^{\ast })$ means that

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}\in \mathop{\prod }_{i}\operatorname{Hom}(E_{i},E_{i-1}(K-D))=\operatorname{Hom}(E,E[-1](K)). & & \displaystyle \nonumber\end{eqnarray}$$

Therefore, we have to prove that

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}_{D}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE})+2h^{0}(E,E[1])=\unicode[STIX]{x1D712}_{K-D}(\unicode[STIX]{x1D6FE}^{\ast },\unicode[STIX]{x1D6FE}^{\ast })+2h^{0}(E,E[-1](K)) & & \displaystyle \nonumber\end{eqnarray}$$

or, equivalently, by Serre duality, that

(29)$$\begin{eqnarray}\unicode[STIX]{x1D712}_{D}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE})+2\unicode[STIX]{x1D712}(E,E[1])=\unicode[STIX]{x1D712}_{K-D}(\unicode[STIX]{x1D6FE}^{\ast },\unicode[STIX]{x1D6FE}^{\ast }).\end{eqnarray}$$

By Corollary 2.2, we have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}_{D}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE}) & = & \displaystyle \unicode[STIX]{x1D712}(E,E)-\unicode[STIX]{x1D712}(E,E[1]),\nonumber\\ \displaystyle \unicode[STIX]{x1D712}_{K-D}(\unicode[STIX]{x1D6FE}^{\ast },\unicode[STIX]{x1D6FE}^{\ast }) & = & \displaystyle \unicode[STIX]{x1D712}(E,E)-\unicode[STIX]{x1D712}(E,E[-1](K)).\nonumber\end{eqnarray}$$

This and the fact that $\unicode[STIX]{x1D712}(E,F)=-\unicode[STIX]{x1D712}(F,E(K))$, for any $E,F\in {\mathcal{A}}$, imply (29). The statement concerning Higgs sheaves follows from the definition of the DT invariants (24).◻

From (26) and Proposition 4.4 we immediately deduce the following result.

Corollary 4.5. If $\ell >2g-2$ then, for any $\unicode[STIX]{x1D6FE}=(\bar{r},\bar{d})\in \unicode[STIX]{x1D6E4}$, we have $\unicode[STIX]{x1D6E9}_{D}^{+,\text{vec}}(\unicode[STIX]{x1D6FE})=\unicode[STIX]{x1D6E9}_{K-D,\text{nil}}^{+\text{vec}}(\unicode[STIX]{x1D6FE}^{\ast })$ or, equivalently (cf. (29) and (8)),

$$\begin{eqnarray}\displaystyle [\mathfrak{M}_{D}^{+,\text{vec}}(\unicode[STIX]{x1D6FE})] & = & \displaystyle q^{\unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE}[1])}[\mathfrak{M}_{K-D,\text{nil}}^{+,\text{vec}}(\unicode[STIX]{x1D6FE}^{\ast })],\nonumber\\ \displaystyle \unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE}[1]) & = & \displaystyle (1-g+\ell )\mathop{\sum }_{i}r_{i}r_{i+1}+\left|\begin{array}{@{}cc@{}}r_{i} & r_{i+1}\\ d_{i} & d_{i+1}\end{array}\right|.\nonumber\end{eqnarray}$$

In the Higgs case we have, for $r\geqslant 1$,

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA}_{D}^{+}(r,d)=\unicode[STIX]{x1D6FA}_{D}^{+,\text{vec}}(r,d)=\unicode[STIX]{x1D6FA}_{K-D,\text{nil}}^{+,\text{vec}}(r,d)=\unicode[STIX]{x1D6FA}_{K-D,\text{nil}}^{+}(r,d). & & \displaystyle \nonumber\end{eqnarray}$$

4.3 From Higgs sheaves to nilpotent Higgs sheaves

The aim of this section is to prove a result somewhat similar to Corollary 4.5 in the case $D=K$. We begin with the Higgs case, for which things can be made very explicit in terms of DT invariants and Kac polynomials of curves. Let $\mathsf{A}_{X,r,d}$ denote the number of absolutely indecomposable coherent sheaves on $X$ of rank $r$ and degree $d$. Similarly, let $\mathsf{A}_{X,r,d}^{+}$ denote the number of positive (i.e. contained in ${\mathcal{A}}^{+}$) absolutely indecomposable vector bundles of rank $r$ and degree $d$. Both of these numbers are the evaluation, at the collection of Weil numbers of $X$, of certain polynomials determined in [Reference SchiffmannSch16] which only depend on the genus of $X$. For simplicity, we will drop the index $X$ from the notation when the curve is understood. We will use the same notation for the corresponding volumes (see Appendix A), hoping it will not cause any confusion.

This is proved in [Reference SchiffmannSch16, Proposition 2.5]. We provide below a proof for the convenience of the reader.

Proof. Assume that there is a gap of length greater than $2g-2$, say $\unicode[STIX]{x1D708}_{k+1}-\unicode[STIX]{x1D708}_{k}>2g-2$. Then there exists an exact sequence

$$\begin{eqnarray}\displaystyle 0\rightarrow E^{\prime }\rightarrow E\rightarrow E^{\prime \prime }\rightarrow 0, & & \displaystyle \nonumber\end{eqnarray}$$

where $E^{\prime }\in {\mathcal{A}}^{{\geqslant}\unicode[STIX]{x1D708}_{k+1}}$ and $E^{\prime \prime }\in {\mathcal{A}}^{{\leqslant}\unicode[STIX]{x1D708}_{k}}$. This implies that

$$\begin{eqnarray}\displaystyle E^{\prime \prime }(K)\in {\mathcal{A}}^{{\leqslant}\unicode[STIX]{x1D708}_{k}+2g-2}\subset {\mathcal{A}}^{{<}\unicode[STIX]{x1D708}_{k+1}} & & \displaystyle \nonumber\end{eqnarray}$$

and therefore $\operatorname{Ext}^{1}(E^{\prime \prime },E^{\prime })\simeq \operatorname{Hom}(E^{\prime },E^{\prime \prime }(K))^{\ast }=0$. We conclude that the above sequence splits and $E$ is not indecomposable.◻

Proposition 4.6 follows.

The first formula of the next result was proved by the first author [Reference MozgovoyMoz11] in the case of quiver representations. The second formula was proved by the second author [Reference SchiffmannSch16].

Theorem 4.9. We have

(30)$$\begin{eqnarray}\displaystyle \mathop{\sum }_{r,d}\unicode[STIX]{x1D6E9}_{K}^{+}(r,d)T^{r}z^{d} & = & \displaystyle \mathop{\sum }_{r,d}\unicode[STIX]{x1D6E9}_{0}^{+}(r,d)T^{r}z^{d}=\operatorname{Exp}\biggl(\frac{q\mathop{\sum }_{r,d}\mathsf{A}_{r,d}^{+}\;T^{r}z^{d}}{q-1}\biggr),\end{eqnarray}$$

(31)$$\begin{eqnarray}\displaystyle \mathop{\sum }_{r,d}\unicode[STIX]{x1D6E9}_{0,\text{nil}}^{+}(r,d)T^{r}z^{d} & = & \displaystyle \operatorname{Exp}\biggl(\frac{\mathop{\sum }_{r,d}\mathsf{A}_{r,d}^{+}\;T^{r}z^{d}}{q-1}\biggr).\end{eqnarray}$$

Proof. Let $\operatorname{Ind}^{+}$ be the set of indecomposable positive coherent sheaves, up to isomorphism. For each $I\in \operatorname{Ind}^{+}$, let $\Bbbk _{I}=\operatorname{End}(I)/\text{rad}\operatorname{End}(I)$ be the residue field and $l_{I}=[\Bbbk _{I}:\Bbbk ]$ be its degree. The isomorphism classes of positive coherent sheaves are thus indexed by maps $\unicode[STIX]{x1D719}:\operatorname{Ind}^{+}\rightarrow \mathbb{N}$ with finite support and we set $E_{\unicode[STIX]{x1D719}}=\bigoplus _{I}I^{\oplus \unicode[STIX]{x1D719}(I)}$. Note that (see, for example, [Reference SchiffmannSch16, § 2.2])

$$\begin{eqnarray}\displaystyle \frac{\#\operatorname{End}(E_{\unicode[STIX]{x1D719}})}{\#\operatorname{Aut}(E_{\unicode[STIX]{x1D719}})}=\mathop{\prod }_{I}\frac{\#\mathfrak{g}\mathfrak{l}(\unicode[STIX]{x1D719}(I),\Bbbk _{I})}{\#\operatorname{GL}(\unicode[STIX]{x1D719}(I),\Bbbk _{I})}=\mathop{\prod }_{I}\frac{1}{(q^{-l_{I}})_{\unicode[STIX]{x1D719}(I)}}, & & \displaystyle \nonumber\end{eqnarray}$$

where $(q)_{n}=(1-q)\cdots (1-q^{n})$. Now the forgetful map

$$\begin{eqnarray}\displaystyle \mathfrak{M}_{K}^{+}(r,d)\rightarrow \mathfrak{M}^{+}(r,d) & & \displaystyle \nonumber\end{eqnarray}$$

has the fiber over $E\in \mathfrak{M}^{+}(r,d)$ that is equal to

$$\begin{eqnarray}\displaystyle \operatorname{Hom}(E,E\otimes K)\simeq \operatorname{Ext}^{1}(E,E)^{\ast }. & & \displaystyle \nonumber\end{eqnarray}$$

If $E\simeq E_{\unicode[STIX]{x1D719}}$ then the contribution of the fiber of $E$ to $[\mathfrak{M}_{K}^{+}(r,d)]$ is equal to

$$\begin{eqnarray}\displaystyle \frac{\#\operatorname{Hom}(E,E\otimes K)}{\#\operatorname{Aut}(E)}=\frac{\#\operatorname{Ext}^{1}(E,E)}{\#\operatorname{Aut}(E)}=\frac{q^{-\unicode[STIX]{x1D712}(E,E)}}{\mathop{\prod }_{I}(q^{-l_{I}})_{\unicode[STIX]{x1D719}(I)}}. & & \displaystyle \nonumber\end{eqnarray}$$

Note that $\unicode[STIX]{x1D712}(E,E)=r^{2}(1-g)$ and $\unicode[STIX]{x1D712}_{D}(E,E)=2r^{2}(1-g)$. We conclude from the proof of [Reference MozgovoyMoz11, Theorem 5.1] that (cf. (21))

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{r,d}\unicode[STIX]{x1D6E9}_{K}^{+}(r,d)T^{r}z^{d}=\mathop{\sum }_{r,d}q^{r^{2}(1-g)}[\mathfrak{M}_{K}^{+}(r,d)]T^{r}z^{d}=\mathop{\sum }_{\unicode[STIX]{x1D719}}\mathop{\prod }_{I}\frac{t^{\unicode[STIX]{x1D719}(I)\operatorname{cl}I}}{(q^{-l_{I}})_{\unicode[STIX]{x1D719}(I)}}=\mathop{\prod }_{I}\biggl(\mathop{\sum }_{n\geqslant 0}\frac{t^{n\operatorname{cl}I}}{(q^{-l_{I}})_{n}}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$

Using Heine’s formula and [Reference SchiffmannSch16, Lemma 2.6], we get

$$\begin{eqnarray}\displaystyle \mathop{\prod }_{I}\biggl(\mathop{\sum }_{n\geqslant 0}\frac{t^{n\operatorname{cl}I}}{(q^{-l_{I}})_{n}}\biggr)=\mathop{\prod }_{I}\exp \biggl(\mathop{\sum }_{k\geqslant 1}\frac{1}{k}\frac{t^{k\operatorname{cl}I}}{1-q^{-kl_{I}}}\biggr)=\operatorname{Exp}\biggl(\mathop{\sum }_{r,d}\frac{\mathsf{A}_{r,d}^{+}\;T^{r}z^{d}}{1-q^{-1}}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$

The proof of the second formula proceeds along the same lines. Consider the forgetful map

$$\begin{eqnarray}\displaystyle \mathfrak{M}_{0,\text{nil}}^{+}(r,d)\rightarrow \mathfrak{M}^{+}(r,d). & & \displaystyle \nonumber\end{eqnarray}$$

The contribution of the fiber of $E_{\unicode[STIX]{x1D719}}$ in $[\mathfrak{M}_{0,\text{nil}}^{+}(r,d)]$ is equal to [Reference SchiffmannSch16, Corollary 2.4]

$$\begin{eqnarray}\displaystyle \frac{\#\operatorname{Hom}^{\text{nil}}(E_{\unicode[STIX]{x1D719}},E_{\unicode[STIX]{x1D719}})}{\#\operatorname{Aut}E_{\unicode[STIX]{x1D719}}}=\mathop{\prod }_{I}\frac{q^{-\unicode[STIX]{x1D719}(I)l_{I}}}{(q^{-l_{I}})_{\unicode[STIX]{x1D719}(I)}}. & & \displaystyle \nonumber\end{eqnarray}$$

Applying the same argument as above, we conclude that

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{r,d}\unicode[STIX]{x1D6E9}_{0,\text{nil}}^{+}(r,d)T^{r}z^{d} & = & \displaystyle \mathop{\sum }_{r,d}[\mathfrak{M}_{0,\text{nil}}^{+}(r,d)]T^{r}z^{d}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{\unicode[STIX]{x1D719}}\mathop{\prod }_{I}\frac{q^{-\unicode[STIX]{x1D719}(I)l_{I}}t^{\unicode[STIX]{x1D719}(I)\operatorname{cl}I}}{(q^{-l_{I}})_{\unicode[STIX]{x1D719}(I)}}=\operatorname{Exp}\biggl(\mathop{\sum }_{r,d}\frac{q^{-1}\mathsf{A}_{r,d}^{+}\;T^{r}z^{d}}{1-q^{-1}}\biggr).\Box \nonumber\end{eqnarray}$$

Let us now turn to the case of quiver sheaves. We do not know of a formula similar to those of Theorem 4.9 expressing the volume of $\mathfrak{M}_{0}^{+}(\unicode[STIX]{x1D6FE})$ or $\mathfrak{M}_{0,\text{nil}}^{+}(\unicode[STIX]{x1D6FE})$ in terms of Kac polynomials $\mathsf{A}_{r,d}$. However, one still has the following relation between the volumes of $\mathfrak{M}_{0}^{+}(\unicode[STIX]{x1D6FE})$ and $\mathfrak{M}_{0,\text{nil}}^{+}(\unicode[STIX]{x1D6FE})$.

Proposition 4.11. We have the following equality of formal series in $\mathbb{T}$:

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6E9}_{0}^{+}(\unicode[STIX]{x1D6FE})t^{\unicode[STIX]{x1D6FE}}=\biggl(\mathop{\sum }_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6E9}_{0,\text{nil}}^{+}(\unicode[STIX]{x1D6FE})t^{\unicode[STIX]{x1D6FE}}\biggr)\cdot \operatorname{Exp}\biggl(\mathop{\sum }_{r,d}\mathsf{A}_{r,d}^{+}\;T^{r\unicode[STIX]{x1D6FF}}z^{d\unicode[STIX]{x1D6FF}}\biggr), & & \displaystyle \nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}=(1,1,\ldots ,1)\in \mathbb{Z}^{I}$.

Proof. Note that the subalgebra $\bigoplus _{r,d}{\mathcal{V}}\cdot T^{r\unicode[STIX]{x1D6FF}}z^{d\unicode[STIX]{x1D6FF}}$ of $\mathbb{T}$ is commutative, hence the plethystic exponential is well defined. Let ${\mathcal{A}}_{0}^{+,\text{iso}}$ be the full subcategory of ${\mathcal{A}}_{0}^{+}$ consisting of quiver sheaves $\bar{E}=(E_{i},\unicode[STIX]{x1D703}_{i})_{i}$ for which $\unicode[STIX]{x1D703}_{i}:E_{i}\simeq E_{i+1}$ for all $i$. We claim that any object $\bar{E}\in {\mathcal{A}}_{0}^{+}$ has a unique subobject $\bar{E}^{\prime }$ satisfying

$$\begin{eqnarray}\displaystyle \bar{E}^{\prime }\in {\mathcal{A}}_{0}^{+,\text{iso}},\quad \bar{E}/\bar{E}^{\prime }\in {\mathcal{A}}_{0}^{+,\text{nil}}. & & \displaystyle \nonumber\end{eqnarray}$$

To see this, consider the decreasing filtration $\bar{E}\supset \unicode[STIX]{x1D703}(\bar{E})\supset \unicode[STIX]{x1D703}^{2}(\bar{E})\cdots \,$. Since $\operatorname{End}(\bigoplus _{i}E_{i})$ is finite-dimensional and hence finite, this filtration stabilizes and we let $\bar{E}^{\prime }$ denote its limit. By construction and because $\operatorname{Coh}^{+}(X)$ is stable under taking quotients, $\bar{E}^{\prime }\in {\mathcal{A}}_{0}^{+,\text{iso}}$ and $\bar{E}/\bar{E}^{\prime }\in {\mathcal{A}}_{0}^{+,\text{nil}}$. This shows the existence of a filtration of the desired form. Unicity comes from the easily checked fact that $\operatorname{Hom}(\bar{E}^{\prime },\bar{E}^{\prime \prime })=\{0\}$ whenever $\bar{E}^{\prime }\in {\mathcal{A}}_{0}^{\text{iso}}$ and $\bar{E}^{\prime \prime }\in {\mathcal{A}}_{0}^{\text{nil}}$. Setting $\bar{E}^{\prime \prime }=\operatorname{Ker}(\unicode[STIX]{x1D703}^{n})$ for $n\gg 0$ yields in fact a canonical splitting of the exact sequence $0\rightarrow \bar{E}^{\prime }\rightarrow \bar{E}\rightarrow \bar{E}/\bar{E}^{\prime }\rightarrow 0$ but we will not need this. Let

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E9}_{0,\text{iso}}^{+}(\unicode[STIX]{x1D6FE})=\mathop{\sum }_{\bar{E}\in {\mathcal{A}}_{0}^{+,\text{iso}}(\unicode[STIX]{x1D6FE})}(-q^{1/2})^{\unicode[STIX]{x1D712}_{0}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE})}\frac{1}{\#\operatorname{Aut}\bar{E}}. & & \displaystyle \nonumber\end{eqnarray}$$

From the uniqueness of the filtration $\bar{E}^{\prime }\subseteq \bar{E}$ above we have, by a standard argument in the Hall algebra,

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6E9}_{0}^{+}(\unicode[STIX]{x1D6FE})T^{\bar{r}}z^{\bar{d}}=\biggl(\mathop{\sum }_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6E9}_{0,\text{nil}}^{+}(\unicode[STIX]{x1D6FE})T^{\bar{r}}z^{\bar{d}}\biggr)\cdot \biggl(\mathop{\sum }_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6E9}_{0,\text{iso}}^{+}(\unicode[STIX]{x1D6FE})T^{\bar{r}}z^{\bar{d}}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$

Observe that $\unicode[STIX]{x1D6E9}_{0,\text{iso}}^{+}(\unicode[STIX]{x1D6FE})=0$ unless $\unicode[STIX]{x1D6FE}=(r\unicode[STIX]{x1D6FF},d\unicode[STIX]{x1D6FF})$ for some $(r,d)$. All that remains to prove is the following equality:

(32)$$\begin{eqnarray}\mathop{\sum }_{r,d}\unicode[STIX]{x1D6E9}_{0,\text{iso}}^{+}(r\unicode[STIX]{x1D6FF},d\unicode[STIX]{x1D6FF})T^{r\unicode[STIX]{x1D6FF}}z^{d\unicode[STIX]{x1D6FF}}=\operatorname{Exp}\biggl(\mathop{\sum }_{r,d}\mathsf{A}_{r,d}^{+}\;T^{r\unicode[STIX]{x1D6FF}}z^{d\unicode[STIX]{x1D6FF}}\biggr).\end{eqnarray}$$

The proof of that last statement is of a similar nature to that of Theorem 4.9. Let $\mathfrak{M}_{0,\text{iso}}^{+}(r\unicode[STIX]{x1D6FF},d\unicode[STIX]{x1D6FF})$ be the stack parametrizing objects in ${\mathcal{A}}_{0}^{+,\text{iso}}$ of class $(r\unicode[STIX]{x1D6FF},d\unicode[STIX]{x1D6FF})$. Note that $\unicode[STIX]{x1D712}_{0}((r\unicode[STIX]{x1D6FF},d\unicode[STIX]{x1D6FF}),(r\unicode[STIX]{x1D6FF},d\unicode[STIX]{x1D6FF}))=0$. Consider the forgetful map $\unicode[STIX]{x1D70B}:\mathfrak{M}_{0,\text{iso}}^{+}(r\unicode[STIX]{x1D6FF},d\unicode[STIX]{x1D6FF})\rightarrow {\mathcal{A}}^{+}(r,d)$. For any positive coherent sheaf $E\in {\mathcal{A}}^{+}(r,d)$, the fiber of $\unicode[STIX]{x1D70B}$ contributes a volume of $\prod _{i}\#\operatorname{Aut}E/\prod _{i}\#\operatorname{Aut}E=1$. It follows that

(33)$$\begin{eqnarray}\#\mathfrak{M}_{0,\text{iso}}^{+}(r\unicode[STIX]{x1D6FF},d\unicode[STIX]{x1D6FF})(\Bbbk )=\#\{E\in \operatorname{Coh}^{+}(X)\mid \operatorname{cl}E=(r,d)\}/\sim .\end{eqnarray}$$

Let us denote the right-hand side of (33) by $m_{r,d}$. Equality (32) is now a consequence of the next lemma (cf. [Reference MozgovoyMoz07, Lemma 5]).◻

Proof. The proof is close to that of [Reference SchiffmannSch16, Proposition 2.2]. For any $l\in \mathbb{N}$, let us denote by $\operatorname{Ind}_{(r,d),l}^{+}$ the set of isoclasses of indecomposable positive coherent sheaves $E$ on $X$ of class $(r,d)$ for which $E\otimes _{\Bbbk }\overline{\Bbbk }$ splits as a direct sum of $l$ geometrically indecomposable coherent sheaves. This means that $[\Bbbk _{E}:\Bbbk ]=l$, where $\Bbbk _{E}=\operatorname{End}(E)/\!\operatorname{rad}\operatorname{End}(E)$ (cf. the proof of Theorem 4.9). Note that $\operatorname{Ind}_{(r,d),l}^{+}$ is empty unless $l\mid \gcd (r,d)$; see [Reference SchiffmannSch16, Lemma 2.6]. Let $\Bbbk =\mathbb{F}_{q}$ and let $X_{n}=X\otimes _{\mathbb{F}_{q}}\mathbb{F}_{q^{n}}$, for $n\geqslant 1$. By [Reference SchiffmannSch16, (2.4), (2.5)] we have, for every $n,r,d$,

$$\begin{eqnarray}\displaystyle \mathsf{A}_{X_{n},r,d}^{+}=\mathop{\sum }_{l\mid n}l\#\operatorname{Ind}_{(lr,ld),l}^{+} & & \displaystyle \nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{n\geqslant 1}\mathop{\sum }_{r,d}\frac{1}{n}\mathsf{A}_{X_{n},r,d}^{+}\;T^{nr}z^{nd}=\mathop{\sum }_{l\geqslant 1}\mathop{\sum }_{r,d}\frac{1}{l}\#\operatorname{Ind}_{r,d}^{+}T^{lr}z^{ld}. & & \displaystyle \nonumber\end{eqnarray}$$

We deduce that

$$\begin{eqnarray}\displaystyle \operatorname{Exp}\biggl(\mathop{\sum }_{r,d}\mathsf{A}_{X,r,d}^{+}\;T^{r}z^{d}\biggr) & = & \displaystyle \exp \biggl(\mathop{\sum }_{n\geqslant 1}\mathop{\sum }_{r,d}\frac{1}{n}\mathsf{A}_{X_{n},r,d}\;T^{nr}z^{nd}\biggr)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\prod }_{r,d}\exp \biggl(\mathop{\sum }_{l\geqslant 1}\frac{1}{l}\#\operatorname{Ind}_{r,d}^{+}\;T^{lr}z^{ld}\biggr)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\prod }_{r,d}\frac{1}{(1-T^{r}z^{d})^{\#\operatorname{Ind}_{r,d}^{+}}}=\mathop{\sum }_{r,d}m_{r,d}T^{r}z^{d},\nonumber\end{eqnarray}$$

as desired. ◻

5.1 Jordan stratification

We do not assume that $\ell \leqslant 0$ here. Let $\bar{E}=(E,\unicode[STIX]{x1D703})\in {\mathcal{A}}_{D}$. For any $k\geqslant 0$, define $\unicode[STIX]{x1D703}^{k}$ to be the composition (cf. (25))

$$\begin{eqnarray}\displaystyle E\rightarrow E[1]\rightarrow \cdots \rightarrow E[k], & & \displaystyle \nonumber\end{eqnarray}$$

and set $F_{k}=\operatorname{Im}\unicode[STIX]{x1D703}^{k}[-k]\subset E$. Assume that $(E,\unicode[STIX]{x1D703})$ is a nilpotent quiver sheaf, of nilpotency index $s$. By construction, we have a chain of inclusions

$$\begin{eqnarray}\displaystyle E=F_{0}{\hookleftarrow}F_{1}{\hookleftarrow}F_{2}{\hookleftarrow}\cdots {\hookleftarrow}F_{s}=0 & & \displaystyle \nonumber\end{eqnarray}$$

and a chain of epimorphisms

$$\begin{eqnarray}\displaystyle E=F_{0}{\twoheadrightarrow}F_{1}[1]{\twoheadrightarrow}F_{2}[2]{\twoheadrightarrow}\cdots {\twoheadrightarrow}F_{s}[s]=0. & & \displaystyle \nonumber\end{eqnarray}$$

Let us set

$$\begin{eqnarray}\displaystyle F_{k}^{\prime }=\operatorname{Ker}(F_{k}\rightarrow F_{k+1}[1]),\quad F_{k}^{\prime \prime }=\operatorname{Coker}(F_{k+1}\rightarrow F_{k}). & & \displaystyle \nonumber\end{eqnarray}$$

Then we have a chain of inclusions

$$\begin{eqnarray}\displaystyle F_{0}^{\prime }{\hookleftarrow}F_{1}^{\prime }{\hookleftarrow}F_{2}^{\prime }{\hookleftarrow}\cdots {\hookleftarrow}F_{s}^{\prime }=0 & & \displaystyle \nonumber\end{eqnarray}$$

and a chain of epimorphisms

$$\begin{eqnarray}\displaystyle F_{0}^{\prime \prime }{\twoheadrightarrow}F_{1}^{\prime \prime }[1]{\twoheadrightarrow}F_{2}^{\prime \prime }[2]{\twoheadrightarrow}\cdots {\twoheadrightarrow}F_{s}^{\prime \prime }[s]=0. & & \displaystyle \nonumber\end{eqnarray}$$

Finally, let us set

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}_{k+1}=\operatorname{cl}F_{k}^{\prime \prime }[k]-\operatorname{cl}F_{k+1}^{\prime \prime }[k+1]\in \unicode[STIX]{x1D6E4},\quad k\geqslant 0, & & \displaystyle \nonumber\end{eqnarray}$$

and call the tuple $\bar{\unicode[STIX]{x1D6FC}}=(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{s})$ the Jordan type of $\bar{E}=(E,\unicode[STIX]{x1D703})$.

Lemma 5.1. We have $\operatorname{cl}F_{k}^{\prime \prime }=f_{k}^{\prime \prime }(\bar{\unicode[STIX]{x1D6FC}})$ and $\operatorname{cl}F_{k}^{\prime }=f_{k}^{\prime }(\bar{\unicode[STIX]{x1D6FC}})$, where

(34)$$\begin{eqnarray}f_{k}^{\prime \prime }(\bar{\unicode[STIX]{x1D6FC}})=\mathop{\sum }_{j\geqslant k}\unicode[STIX]{x1D6FC}_{j+1}[-k]\in \unicode[STIX]{x1D6E4},\quad f_{k}^{\prime }(\bar{\unicode[STIX]{x1D6FC}})=\mathop{\sum }_{j\geqslant k}\unicode[STIX]{x1D6FC}_{j+1}[-j]\in \unicode[STIX]{x1D6E4}.\end{eqnarray}$$

Proof. The first statement is immediate from the definition of $\unicode[STIX]{x1D6FC}_{k}$:

$$\begin{eqnarray}\displaystyle \operatorname{cl}F_{k}^{\prime \prime }=\unicode[STIX]{x1D6FC}_{k+1}[-k]+\operatorname{cl}F_{k+1}^{\prime \prime }[1]=\unicode[STIX]{x1D6FC}_{k+1}[-k]+\unicode[STIX]{x1D6FC}_{k+2}[-k]+\operatorname{cl}F_{k+2}^{\prime \prime }[2]=\cdots \,. & & \displaystyle \nonumber\end{eqnarray}$$

The second statement is then a consequence of the relations $\operatorname{cl}F_{k}=\sum _{j\geqslant k}\operatorname{cl}F_{j}^{\prime \prime }$ and $\operatorname{cl}F_{k}^{\prime }=\operatorname{cl}F_{k}-\operatorname{cl}F_{k+1}[1]$:

$$\begin{eqnarray}\displaystyle \!\!\operatorname{cl}F_{k}^{\prime }=\mathop{\sum }_{j\geqslant k}\operatorname{cl}F_{j}^{\prime \prime }-\mathop{\sum }_{j\geqslant k}\operatorname{cl}F_{j+1}^{\prime \prime }[1]=\mathop{\sum }_{i\geqslant j\geqslant k}\unicode[STIX]{x1D6FC}_{i+1}[-j]-\mathop{\sum }_{i>j\geqslant k}\unicode[STIX]{x1D6FC}_{i+1}[-j-1+1]=\mathop{\sum }_{j\geqslant k}\unicode[STIX]{x1D6FC}_{j+1}[-j].\Box & & \displaystyle \nonumber\end{eqnarray}$$

Define $|\bar{\unicode[STIX]{x1D6FC}}|=\sum _{k}f_{k}^{\prime \prime }(\bar{\unicode[STIX]{x1D6FC}})\in \unicode[STIX]{x1D6E4}$, so that $\operatorname{cl}\bar{E}=|\bar{\unicode[STIX]{x1D6FC}}|$. Given a partition $\unicode[STIX]{x1D706}$, define

(35)$$\begin{eqnarray}|\unicode[STIX]{x1D706}|=\mathop{\sum }_{i\geqslant 1}\unicode[STIX]{x1D706}_{i},\quad n(\unicode[STIX]{x1D706})=\mathop{\sum }_{i\geqslant 1}(i-1)\unicode[STIX]{x1D706}_{i}=\mathop{\sum }_{i\geqslant 1}\binom{\unicode[STIX]{x1D706}_{i}^{\prime }}{2},\quad \langle \unicode[STIX]{x1D706},\unicode[STIX]{x1D706}\rangle =\mathop{\sum }_{i\geqslant 1}(\unicode[STIX]{x1D706}_{i}^{\prime })^{2},\end{eqnarray}$$

where $\unicode[STIX]{x1D706}^{\prime }$ is the partition conjugate to $\unicode[STIX]{x1D706}$. Then $\langle \unicode[STIX]{x1D706},\unicode[STIX]{x1D706}\rangle =2n(\unicode[STIX]{x1D706})+|\unicode[STIX]{x1D706}|$.

Lemma 5.2. Let $(E,\unicode[STIX]{x1D703})$ have Jordan type $\bar{\unicode[STIX]{x1D6FC}}=(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{s})$ with $\unicode[STIX]{x1D6FC}_{i}=(r_{i},d_{i})$. Let $\unicode[STIX]{x1D706}=(1^{r_{1}}2^{r_{2}}\cdots \,)$ be the corresponding partition and

(36)$$\begin{eqnarray}\operatorname{rk}(\bar{\unicode[STIX]{x1D6FC}})=|\unicode[STIX]{x1D706}|=\mathop{\sum }_{i\geqslant 1}ir_{i},\quad \deg _{D}(\bar{\unicode[STIX]{x1D6FC}})=\mathop{\sum }_{i\geqslant 1}id_{i}-\ell \cdot n(\unicode[STIX]{x1D706}^{\prime }).\end{eqnarray}$$

Then $\operatorname{rk}E=\operatorname{rk}(\bar{\unicode[STIX]{x1D6FC}})$ and $\deg E=\deg _{D}(\bar{\unicode[STIX]{x1D6FC}})$.

Proof. We have

$$\begin{eqnarray}\displaystyle \operatorname{cl}E=\operatorname{cl}F_{0}=\mathop{\sum }_{j\geqslant 0}\operatorname{cl}F_{j}^{\prime \prime }=\mathop{\sum }_{i\geqslant j\geqslant 0}\unicode[STIX]{x1D6FC}_{i+1}[-j]=\mathop{\sum }_{i\geqslant 1}\mathop{\sum }_{j=0}^{i-1}(r_{i},d_{i}-j\ell r_{i})=\mathop{\sum }_{i\geqslant 1}\biggl(ir_{i},id_{i}-\binom{i}{2}\ell r_{i}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$

We note that $\sum _{i\geqslant 1}\binom{i}{2}r_{i}=\sum _{i\geqslant 1}\binom{\unicode[STIX]{x1D706}_{i}}{2}=n(\unicode[STIX]{x1D706}^{\prime })$.◻

For any fixed tuple $\bar{\unicode[STIX]{x1D6FC}}=(\unicode[STIX]{x1D6FC}_{k})_{k}$, we write $\mathfrak{M}_{D,\text{nil}}(\bar{\unicode[STIX]{x1D6FC}})$ for the locally closed substack of $\mathfrak{M}_{D,\text{nil}}(|\bar{\unicode[STIX]{x1D6FC}}|)$ whose objects are nilpotent quiver sheaves of Jordan type $\bar{\unicode[STIX]{x1D6FC}}$. Intersecting it with $\mathfrak{M}_{D,\text{nil}}^{+}(|\bar{\unicode[STIX]{x1D6FC}}|)$ yields an open substack $\mathfrak{M}_{D,\text{nil}}^{+}(\bar{\unicode[STIX]{x1D6FC}})=\mathfrak{M}_{D,\text{nil}}^{+}(\bar{\unicode[STIX]{x1D6FC}})$.

5.2 The forgetful map

For $\bar{\unicode[STIX]{x1D6FC}}=(\unicode[STIX]{x1D6FC}_{0},\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{s-1})$ a tuple of elements of $\unicode[STIX]{x1D6E4}$, we let $\mathbf{Flag}(\bar{\unicode[STIX]{x1D6FC}})$ denote the stack of chains of epimorphisms in ${\mathcal{A}}$,

$$\begin{eqnarray}\displaystyle E_{0}{\twoheadrightarrow}E_{1}{\twoheadrightarrow}\cdots {\twoheadrightarrow}E_{s}=0, & & \displaystyle \nonumber\end{eqnarray}$$

such that

$$\begin{eqnarray}\displaystyle \operatorname{cl}\operatorname{Ker}(E_{k}{\twoheadrightarrow}E_{k+1})=\unicode[STIX]{x1D6FC}_{k},\quad \forall k=0,\ldots ,s-1. & & \displaystyle \nonumber\end{eqnarray}$$

We denote by $\mathbf{Flag}^{+}(\bar{\unicode[STIX]{x1D6FC}})$ the open substack of $\mathbf{Flag}(\bar{\unicode[STIX]{x1D6FC}})$ consisting of chains $E_{0}{\twoheadrightarrow}E_{1}{\twoheadrightarrow}\cdots \,$ such that $E_{0}\in {\mathcal{A}}^{+}$.

Consider the map (see § 5.1 for notation)

(37)$$\begin{eqnarray}\unicode[STIX]{x1D71B}_{\bar{\unicode[STIX]{x1D6FC}}}:\mathfrak{M}_{D,\text{nil}}(\bar{\unicode[STIX]{x1D6FC}})\rightarrow \mathbf{Flag}(\bar{\unicode[STIX]{x1D6FC}}),\quad (E,\unicode[STIX]{x1D703})\mapsto (F_{0}^{\prime \prime }{\twoheadrightarrow}F_{1}^{\prime \prime }[1]{\twoheadrightarrow}\cdots {\twoheadrightarrow}F_{s}^{\prime \prime }[s]=0).\end{eqnarray}$$

From the fact that the category ${\mathcal{A}}^{+}$ is closed under taking quotients, it follows that $\unicode[STIX]{x1D71B}_{\bar{\unicode[STIX]{x1D6FC}}}$ restricts to a map $\mathfrak{M}_{D,\text{nil}}^{+}(\bar{\unicode[STIX]{x1D6FC}})\rightarrow \mathbf{Flag}^{+}(\bar{\unicode[STIX]{x1D6FC}})$.

Theorem 5.3. The volume of every fiber of $\unicode[STIX]{x1D71B}_{\bar{\unicode[STIX]{x1D6FC}}}$ is equal to $q^{\mathbf{d}_{D}(\bar{\unicode[STIX]{x1D6FC}})}$, where (see (34))

(38)$$\begin{eqnarray}\mathbf{d}_{D}(\bar{\unicode[STIX]{x1D6FC}})=-\mathop{\sum }_{k\geqslant 0}\unicode[STIX]{x1D712}(f_{k}^{\prime \prime }(\bar{\unicode[STIX]{x1D6FC}}),f_{k+1}^{\prime }(\bar{\unicode[STIX]{x1D6FC}}))=-\mathop{\sum }_{k\geqslant 0}\mathop{\sum }_{\substack{ i\geqslant k \\ j\geqslant k+1}}\unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FC}_{i+1}[-k],\unicode[STIX]{x1D6FC}_{j+1}[-j]).\end{eqnarray}$$

Proof. Let ${\mathcal{T}}$ be the category of triples $(F^{(1)},F^{(2)},\unicode[STIX]{x1D703})$, where $F^{(1)},F^{(2)}\in {\mathcal{A}}$ and $\unicode[STIX]{x1D703}:F^{(1)}\rightarrow F^{(2)}[1]$. Given a nilpotent quiver sheaf $(E,\unicode[STIX]{x1D703})$, we can define objects $\bar{F}_{k}=(F_{k},F_{k+1},\unicode[STIX]{x1D703})\in {\mathcal{T}}$, for $k\geqslant 0$, together with monomorphisms

$$\begin{eqnarray}\displaystyle \bar{F}_{0}{\hookleftarrow}\bar{F}_{1}{\hookleftarrow}\bar{F}_{2}{\hookleftarrow}\cdots \,. & & \displaystyle \nonumber\end{eqnarray}$$

By the discussion in the previous section, the category of nilpotent quiver sheaves of Jordan type $\bar{\unicode[STIX]{x1D6FC}}$ is equivalent to the category ${\mathcal{D}}$ consisting of tuples $(\bar{F}_{k}\in {\mathcal{T}})_{k=0,\ldots ,s}$ equipped with a chain of monomorphisms

$$\begin{eqnarray}\displaystyle \bar{F}_{0}{\hookleftarrow}\bar{F}_{1}{\hookleftarrow}\bar{F}_{2}{\hookleftarrow}\cdots \,, & & \displaystyle \nonumber\end{eqnarray}$$

isomorphisms $F_{k+1}^{(1)}=F_{k}^{(2)}$ for all $k$, and satisfying $\operatorname{cl}F_{k}^{(1)}=\sum _{j\geqslant k}f_{j}^{\prime \prime }(\bar{\unicode[STIX]{x1D6FC}})$ for all $k$. Under the equivalence $\mathfrak{M}_{D,\text{nil}}(\bar{\unicode[STIX]{x1D6FC}})(\Bbbk )\simeq \langle {\mathcal{D}}\rangle$ (groupoid of the category ${\mathcal{D}}$), the map $\unicode[STIX]{x1D71B}_{\bar{\unicode[STIX]{x1D6FC}}}$ is given by the functor

$$\begin{eqnarray}\displaystyle \langle {\mathcal{D}}\rangle \rightarrow \mathbf{Flag}(\bar{\unicode[STIX]{x1D6FC}})(\Bbbk ),\quad (\bar{F}_{k})_{k}\mapsto (\bar{F}_{0}^{(1)}/\bar{F}_{1}^{(1)}{\twoheadrightarrow}(\bar{F}_{1}^{(1)}/\bar{F}_{2}^{(1)})[1]{\twoheadrightarrow}\cdots {\twoheadrightarrow}\bar{F}_{s}^{(1)}[s]=0). & & \displaystyle \nonumber\end{eqnarray}$$

Let $\bar{H}=(H_{0}{\twoheadrightarrow}H_{1}{\twoheadrightarrow}\cdots \,)$ be an object of $\mathbf{Flag}(\bar{\unicode[STIX]{x1D6FC}})(\Bbbk )$. Let $F_{k}^{\prime \prime }=H_{k}[-k]$, so that $\operatorname{cl}F_{k}^{\prime \prime }=f_{k}^{\prime \prime }(\bar{\unicode[STIX]{x1D6FC}})$. Define

$$\begin{eqnarray}\displaystyle \bar{F}_{k}^{\prime \prime }=(F_{k}^{\prime \prime },F_{k+1}^{\prime \prime },\unicode[STIX]{x1D703})\in {\mathcal{T}}, & & \displaystyle \nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D703}:F_{k}^{\prime \prime }{\twoheadrightarrow}F_{k+1}^{\prime \prime }[1]$ is induced by the map $H_{k}[-k]{\twoheadrightarrow}H_{k+1}[-k]$. By construction, an object of the fiber of $\bar{H}$ corresponds to an iterated extension, in the category ${\mathcal{T}}$ of the objects $\bar{F}_{k}^{\prime \prime }$. More precisely, we may canonically reconstruct objects $\bar{E}$ of the fiber of $\bar{H}$ as follows: we inductively build exact sequences in ${\mathcal{T}}$,

(39)$$\begin{eqnarray}0\rightarrow \bar{F}_{k+1}\rightarrow \bar{F}_{k}\rightarrow \bar{F}_{k}^{\prime \prime }\rightarrow 0,\end{eqnarray}$$

together with identifications $F_{k}^{(2)}=F_{k+1}^{(1)}=:F_{k+1}$,

starting from $\bar{F}_{s}=\bar{F}_{s}^{\prime \prime }=0$ and letting $k=s-1,\ldots ,0$; we then set $\bar{E}=(F_{0}^{(1)},\unicode[STIX]{x1D703})$, where $\unicode[STIX]{x1D703}$ is the composition $F_{0}^{(1)}\rightarrow F_{0}^{(2)}[1]\simeq F_{1}^{(1)}[1]{\hookrightarrow}F_{0}^{(1)}[1]$.

In order to keep track of these successive extensions, we will use the following result. Let $\bar{E}=(E^{(1)},E^{(2)},\unicode[STIX]{x1D719})$, $\bar{F}=(F^{(1)},F^{(2)},\unicode[STIX]{x1D713})$ be a pair of objects of ${\mathcal{T}}$. Consider the groupoid ${\mathcal{C}}$ whose objects are short exact sequences

(40)$$\begin{eqnarray}0\rightarrow \bar{F}\rightarrow \bar{G}\rightarrow \bar{E}\rightarrow 0\end{eqnarray}$$

in ${\mathcal{T}}$, and the groupoid ${\mathcal{C}}^{\prime }$ whose objects are short exact sequences

(41)$$\begin{eqnarray}0\rightarrow F^{(2)}\rightarrow G\rightarrow E^{(2)}\rightarrow 0.\end{eqnarray}$$

The set of isoclasses of objects in ${\mathcal{C}}$ is $\operatorname{Ext}_{{\mathcal{T}}}^{1}(\bar{E},\bar{F})$ and, for any $\unicode[STIX]{x1D702}\in \operatorname{Ext}_{{\mathcal{T}}}^{1}(\bar{E},\bar{F})$, we have $\operatorname{Aut}(\unicode[STIX]{x1D702})=\operatorname{Hom}_{{\mathcal{T}}}(\bar{E},\bar{F})$. Likewise, the set of isoclasses of objects in ${\mathcal{C}}^{\prime }$ is $\operatorname{Ext}_{{\mathcal{A}}}^{1}(E^{(2)},F^{(2)})$ and, for any $\unicode[STIX]{x1D6FE}\in \operatorname{Ext}_{{\mathcal{A}}}^{1}(E^{(2)},F^{(2)})$, we have $\operatorname{Aut}(\unicode[STIX]{x1D6FE})=\operatorname{Hom}_{{\mathcal{A}}}(E^{(2)},F^{(2)})$. There is an obvious forgetful functor $\unicode[STIX]{x1D6F7}:{\mathcal{C}}\rightarrow {\mathcal{C}}^{\prime }$.

Lemma 5.4. Assume that $\unicode[STIX]{x1D713}:F^{(1)}\rightarrow F^{(2)}[1]$ is an epimorphism. Then the orbifold volume of any fiber of $\unicode[STIX]{x1D6F7}:{\mathcal{C}}\rightarrow {\mathcal{C}}^{\prime }$ is equal to $q^{-\unicode[STIX]{x1D712}(\bar{E},\bar{F})+\unicode[STIX]{x1D712}(E^{(2)},F^{(2)})}$.

Proof. By [Reference Gothen and KingGK05] there is a long exact sequence

(42)$$\begin{eqnarray}\displaystyle 0 & \rightarrow & \displaystyle \operatorname{Hom}_{{\mathcal{T}}}(\bar{E},\bar{F})\rightarrow \operatorname{Hom}_{{\mathcal{A}}}(E^{(1)},F^{(1)})\oplus \operatorname{Hom}_{{\mathcal{A}}}(E^{(2)},F^{(2)})\rightarrow \operatorname{Hom}_{{\mathcal{A}}}(E^{(1)},F^{(2)}[1])\nonumber\\ \displaystyle & \rightarrow & \displaystyle \operatorname{Ext}_{{\mathcal{T}}}^{1}(\bar{E},\bar{F})\rightarrow \operatorname{Ext}_{{\mathcal{A}}}^{1}(E^{(1)},F^{(1)})\oplus \operatorname{Ext}_{{\mathcal{A}}}^{1}(E^{(2)},F^{(2)})\rightarrow \operatorname{Ext}_{{\mathcal{A}}}^{1}(E^{(1)},F^{(2)}[1])\nonumber\\ \displaystyle & \rightarrow & \displaystyle \operatorname{Ext}_{{\mathcal{T}}}^{2}(\bar{E},\bar{F})\rightarrow 0.\end{eqnarray}$$

Because $\unicode[STIX]{x1D713}$ is an epimorphism and the category ${\mathcal{A}}$ is hereditary, the map

$$\begin{eqnarray}\displaystyle \operatorname{Ext}^{1}(E^{(1)},F^{(1)})\rightarrow \operatorname{Ext}^{1}(E^{(1)},F^{(2)}[1]) & & \displaystyle \nonumber\end{eqnarray}$$

is onto. It follows that $\operatorname{Ext}^{2}(\bar{E},\bar{F})=0$ and that the composed map

$$\begin{eqnarray}\displaystyle \operatorname{Ext}^{1}(\bar{E},\bar{F})\rightarrow \operatorname{Ext}^{1}(E^{(1)},F^{(1)})\oplus \operatorname{Ext}^{1}(E^{(2)},F^{(2)})\rightarrow \operatorname{Ext}^{1}(E^{(2)},F^{(2)}) & & \displaystyle \nonumber\end{eqnarray}$$

is surjective. Therefore the functor $\unicode[STIX]{x1D6F7}$ is essentially surjective on objects and the set of isoclasses of objects $\unicode[STIX]{x1D6F7}^{-1}(\unicode[STIX]{x1D6FE})$ is of cardinality $q^{\dim \operatorname{Ext}^{1}(\bar{E},\bar{F})-\dim \operatorname{Ext}^{1}(E^{(2)},F^{(2)})}$. Taking into account the automorphisms of objects and using the fact that $\unicode[STIX]{x1D712}(\bar{E},\bar{F})=\dim \operatorname{Hom}(\bar{E},\bar{F})-\dim \operatorname{Ext}^{1}(\bar{E},\bar{F})$ yields the statement of the lemma.◻

We may now finish the proof of Theorem 5.3. Starting from $\bar{F}_{s}=\bar{F}_{s}^{\prime \prime }$, we inductively build objects $\bar{F}_{k}\in {\mathcal{T}}$ and exact sequences (39) in such a way that $F_{k}^{(2)}=F_{k+1}^{(1)}=:F_{k+1}$ for all $k$. We obtain inductively that the maps $F_{k}^{(1)}\rightarrow F_{k}^{(2)}[1]$ are epimorphisms. By Lemma 5.4, each step contributes a factor of $q^{-\unicode[STIX]{x1D712}(\bar{F}_{k}^{\prime \prime },\bar{F}_{k+1})+\unicode[STIX]{x1D712}(F_{k+1}^{\prime \prime },F_{k+2})}$ to the volume of the fiber. It remains to observe that because of the exact sequence (42), we have

$$\begin{eqnarray}\displaystyle -\unicode[STIX]{x1D712}(\bar{F}_{k}^{\prime \prime },\bar{F}_{k+1})+\unicode[STIX]{x1D712}(F_{k+1}^{\prime \prime },F_{k+1}^{(2)}) & = & \displaystyle -\unicode[STIX]{x1D712}(F_{k}^{\prime \prime },F_{k+1}^{(1)})+\unicode[STIX]{x1D712}(F_{k}^{\prime \prime },F_{k+1}^{(2)}[1])\nonumber\\ \displaystyle & = & \displaystyle -\unicode[STIX]{x1D712}(F_{k}^{\prime \prime },F_{k+1}^{}-F_{k+2}^{}[1])\nonumber\\ \displaystyle & = & \displaystyle -\unicode[STIX]{x1D712}(F_{k}^{\prime \prime },F_{k+1}^{\prime }).\Box \nonumber\end{eqnarray}$$

Lemma 5.5. Let $\bar{\unicode[STIX]{x1D6FC}}=(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{s})$ with $\unicode[STIX]{x1D6FC}_{i}=(r_{i},d_{i})$ and let $\unicode[STIX]{x1D706}=(1^{r_{1}}2^{r_{2}}\cdots \,)$ be the corresponding partition of weight $r=|\unicode[STIX]{x1D706}|=\sum _{i}ir_{i}$. Then

$$\begin{eqnarray}\displaystyle \mathbf{d}_{D}(\bar{\unicode[STIX]{x1D6FC}})=\mathbf{d}_{0}(\bar{\unicode[STIX]{x1D6FC}})+\frac{\ell }{2}r^{2}-\frac{\ell }{2}\langle \unicode[STIX]{x1D706},\unicode[STIX]{x1D706}\rangle . & & \displaystyle \nonumber\end{eqnarray}$$

Proof. For $\unicode[STIX]{x1D6FC}=(r,d)$ and $\unicode[STIX]{x1D6FD}=(r^{\prime },d^{\prime })$, we have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FC}[1],\unicode[STIX]{x1D6FD})=\unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})-\ell rr^{\prime },\quad \unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}[1])=\unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})+\ell rr^{\prime }. & & \displaystyle \nonumber\end{eqnarray}$$

Therefore

$$\begin{eqnarray}\displaystyle \mathbf{d}_{D}(\bar{\unicode[STIX]{x1D6FC}}) & = & \displaystyle -\mathop{\sum }_{k\geqslant 0}\mathop{\sum }_{i\geqslant k}\mathop{\sum }_{j\geqslant k+1}\unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FC}_{i+1}[-k],\unicode[STIX]{x1D6FC}_{j+1}[-j])\nonumber\\ \displaystyle & = & \displaystyle -\mathop{\sum }_{k\geqslant 1}\mathop{\sum }_{i\geqslant k}\mathop{\sum }_{j>k}\unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FC}_{i}[-k+1],\unicode[STIX]{x1D6FC}_{j}[-j+1])=\mathbf{d}_{0}(\bar{\unicode[STIX]{x1D6FC}})+\ell \mathop{\sum }_{k\geqslant 1}\mathop{\sum }_{i\geqslant k}\mathop{\sum }_{j\geqslant k}(j-k)r_{i}r_{j}.\nonumber\end{eqnarray}$$

We note that

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{k\geqslant 1}\mathop{\sum }_{i\geqslant k}\mathop{\sum }_{j\geqslant k}(j-k)r_{i}r_{j} & = & \displaystyle \mathop{\sum }_{i\geqslant j}\binom{j}{2}r_{i}r_{j}+\mathop{\sum }_{i<j}\biggl(i(j-1)-\binom{i}{2}\biggr)r_{i}r_{j}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{i}\binom{i}{2}r_{i}^{2}+\mathop{\sum }_{i<j}i(j-1)r_{i}r_{j}\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2}\mathop{\sum }_{i,j}ijr_{i}r_{j}-\frac{1}{2}\mathop{\sum }_{i,j}\min \{i,j\}r_{i}r_{j}=\frac{1}{2}r^{2}-\frac{1}{2}\langle \unicode[STIX]{x1D706},\unicode[STIX]{x1D706}\rangle .\Box \nonumber\end{eqnarray}$$

The map (cf. § 2.4)

(43)$$\begin{eqnarray}\mathbf{Flag}(\bar{\unicode[STIX]{x1D6FC}})\rightarrow \mathop{\prod }_{k}\mathfrak{M}(\unicode[STIX]{x1D6FC}_{k}),\quad (E_{0}{\twoheadrightarrow}E_{1}{\twoheadrightarrow}\cdots {\twoheadrightarrow}E_{s}=0)\mapsto (\operatorname{Ker}(E_{k}{\twoheadrightarrow}E_{k+1}))_{k}\end{eqnarray}$$

is a stack vector bundle of rank $-\!\sum _{j>k}\unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FC}_{j},\unicode[STIX]{x1D6FC}_{k})$ (see [Reference García-Prada, Heinloth and SchmittGHS14, § 3.1]), hence

(44)$$\begin{eqnarray}[\mathbf{Flag}(\bar{\unicode[STIX]{x1D6FC}})]=q^{-\!\mathop{\sum }_{j>k}\unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FC}_{j},\unicode[STIX]{x1D6FC}_{k})}\mathop{\prod }_{k}[\mathfrak{M}(\unicode[STIX]{x1D6FC}_{k})].\end{eqnarray}$$

We obtain from (44) and Theorem 5.3 that

(45)$$\begin{eqnarray}[\mathfrak{M}_{D,\text{nil}}(\bar{\unicode[STIX]{x1D6FC}})]=q^{\mathbf{d}_{D}(\bar{\unicode[STIX]{x1D6FC}})-\mathop{\sum }_{j>k}\unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FC}_{j},\unicode[STIX]{x1D6FC}_{k})}\mathop{\prod }_{k}[\mathfrak{M}(\unicode[STIX]{x1D6FC}_{k})].\end{eqnarray}$$

5.3 Volume of stacks of positive nilpotent quiver sheaves

We assume that $\ell \leqslant 0$. Fix $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6E4}=(\mathbb{Z}^{2})^{I}$ such that $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D6FC})\geqslant 0$. There are only finitely many $\bar{\unicode[STIX]{x1D6FC}}$ satisfying $|\bar{\unicode[STIX]{x1D6FC}}|=\unicode[STIX]{x1D6FC}$ for which $\mathfrak{M}_{D,\text{nil}}^{+}(\bar{\unicode[STIX]{x1D6FC}})$ is not empty; indeed, there are finitely many possible choices for $(f_{k}^{\prime \prime }\in \unicode[STIX]{x1D6E4})_{k\geqslant 0}$ satisfying $\sum _{k}f_{k}^{\prime \prime }=\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D707}(f_{k,i}^{\prime \prime })\geqslant 0$ for $i\in I$.

Proposition 5.6. Assume that $\ell \leqslant 0$. Then the following diagram is cartesian:

where the horizontal arrows stand for the open immersions.

As an immediate corollary of Theorem 5.3 and Proposition 5.6 we obtain the following formula.

Corollary 5.7. Assume that $\ell \leqslant 0$. Then the volume of the stack $\mathfrak{M}_{D,\text{nil}}^{+}(\bar{\unicode[STIX]{x1D6FC}})$ is equal to

$$\begin{eqnarray}\displaystyle [\mathfrak{M}_{D,\text{nil}}^{+}(\bar{\unicode[STIX]{x1D6FC}})]=q^{\mathbf{d}_{D}(\bar{\unicode[STIX]{x1D6FC}})}[\mathbf{Flag}^{+}(\bar{\unicode[STIX]{x1D6FC}})], & & \displaystyle \nonumber\end{eqnarray}$$

where $\mathbf{d}_{D}(\bar{\unicode[STIX]{x1D6FC}})$ is defined in (38).

The volumes of the stacks $\mathbf{Flag}^{+}(\bar{\unicode[STIX]{x1D6FC}})=\prod _{i\in I}\mathbf{Flag}^{+}(\bar{\unicode[STIX]{x1D6FC}}_{i})$ have been explicitly computed in [Reference SchiffmannSch16]. This yields a closed (albeit complicated) formula for the volumes of all the stacks $\mathfrak{M}_{D,\text{nil}}^{+}(\bar{\unicode[STIX]{x1D6FC}})$.

6.1 The case $D=0$

In this section we will recall the main computation of [Reference SchiffmannSch16]: the volumes of the nilpotent stacks $\mathfrak{M}_{D,\text{nil}}^{+}(\bar{\unicode[STIX]{x1D6FC}})$ for $D=0$.

Let ${\mathcal{J}}$ be the set of Jordan types $\bar{\unicode[STIX]{x1D6FC}}=(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{s})$ with $\unicode[STIX]{x1D6FC}_{s}\neq 0$ and let ${\mathcal{P}}$ be the set of partitions. Define the map

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70B}:{\mathcal{J}}\rightarrow {\mathcal{P}},\quad \bar{\unicode[STIX]{x1D6FC}}=(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{s})\mapsto (1^{r_{1}}2^{r_{2}}\cdots \,),\quad \unicode[STIX]{x1D6FC}_{i}=(r_{i},d_{i}). & & \displaystyle \nonumber\end{eqnarray}$$

Theorem 6.1 (see [Reference SchiffmannSch16, § 5.6]).

If $D=0$, then

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\bar{\unicode[STIX]{x1D6FC}}\in \unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D706})}[\mathfrak{M}_{D,\text{nil}}^{+}(\bar{\unicode[STIX]{x1D6FC}})]z^{\mathop{\sum }_{i}i\deg \unicode[STIX]{x1D6FC}_{i}} & = & \displaystyle \mathop{\sum }_{\bar{\unicode[STIX]{x1D6FC}}\in \unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D706})}q^{\mathbf{d}_{0}(\bar{\unicode[STIX]{x1D6FC}})}[\mathbf{Flag}^{+}(\bar{\unicode[STIX]{x1D6FC}})]z^{\mathop{\sum }_{i}i\deg \unicode[STIX]{x1D6FC}_{i}}\nonumber\\ \displaystyle & = & \displaystyle q^{(g-1)\langle \unicode[STIX]{x1D706},\unicode[STIX]{x1D706}\rangle }J_{\unicode[STIX]{x1D706}}(z)H_{\unicode[STIX]{x1D706}}(z)\cdot \operatorname{Exp}\biggl(\frac{[X]}{q-1}\cdot \frac{z}{1-z}\biggr),\nonumber\end{eqnarray}$$

where $\langle \unicode[STIX]{x1D706},\unicode[STIX]{x1D706}\rangle =\sum _{i\geqslant 1}(\unicode[STIX]{x1D706}_{i}^{\prime })^{2}$, $\unicode[STIX]{x1D706}^{\prime }$ is the partition conjugate to $\unicode[STIX]{x1D706}$, and the functions $J_{\unicode[STIX]{x1D706}}(z),H_{\unicode[STIX]{x1D706}}(z)$ are introduced in Appendix B.

6.2 The general case

Assume that $\ell =\deg D\leqslant 0$. Recall that

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E9}_{D,\text{nil}}^{+}(r,d)=(-q^{1/2})^{-\ell r^{2}}[\mathfrak{M}_{D,\text{nil}}^{+}(r,d)]. & & \displaystyle \nonumber\end{eqnarray}$$

Theorem 6.2. We have

$$\begin{eqnarray}\mathop{\sum }_{r,d}\unicode[STIX]{x1D6E9}_{D,\text{nil}}^{+}(r,d)T^{r}z^{d}=\mathop{\sum }_{\unicode[STIX]{x1D706}}z^{-\ell n(\unicode[STIX]{x1D706}^{\prime })}(-q^{1/2})^{(2g-2-\ell )\langle \unicode[STIX]{x1D706},\unicode[STIX]{x1D706}\rangle }J_{\unicode[STIX]{x1D706}}(z)H_{\unicode[STIX]{x1D706}}(z)T^{|\unicode[STIX]{x1D706}|}\cdot \operatorname{Exp}\biggl(\frac{[X]}{q-1}\cdot \frac{z}{1-z}\biggr).\end{eqnarray}$$

Proof. Given a Jordan type $\bar{\unicode[STIX]{x1D6FC}}=(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{s})$ with $\unicode[STIX]{x1D6FC}_{i}=(r_{i},d_{i})$, define

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E9}_{D,\text{nil}}^{+}(\bar{\unicode[STIX]{x1D6FC}})=(-q^{1/2})^{-\ell r^{2}}[\mathfrak{M}_{D,\text{nil}}^{+}(\bar{\unicode[STIX]{x1D6FC}})],\quad r=\sum ir_{i}. & & \displaystyle \nonumber\end{eqnarray}$$

By Lemma 5.2, we have

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{r,d}\unicode[STIX]{x1D6E9}_{D,\text{nil}}^{+}(r,d)T^{r}z^{d} & = & \displaystyle \mathop{\sum }_{\bar{\unicode[STIX]{x1D6FC}}\in {\mathcal{J}}}\unicode[STIX]{x1D6E9}_{D,\text{nil}}^{+}(\bar{\unicode[STIX]{x1D6FC}})T^{\operatorname{rk}\bar{\unicode[STIX]{x1D6FC}}}z^{\deg _{D}\bar{\unicode[STIX]{x1D6FC}}}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{\unicode[STIX]{x1D706}\in {\mathcal{P}}}T^{|\unicode[STIX]{x1D706}|}z^{-\ell n(\unicode[STIX]{x1D706}^{\prime })}\mathop{\sum }_{\bar{\unicode[STIX]{x1D6FC}}\in \unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D706})}\unicode[STIX]{x1D6E9}_{D,\text{nil}}^{+}(\bar{\unicode[STIX]{x1D6FC}})z^{\mathop{\sum }_{i}i\deg \unicode[STIX]{x1D6FC}_{i}}.\nonumber\end{eqnarray}$$

Given $\bar{\unicode[STIX]{x1D6FC}}\in {\mathcal{J}}$ with $\unicode[STIX]{x1D70B}(\bar{\unicode[STIX]{x1D6FC}})=\unicode[STIX]{x1D706}$ and $\operatorname{rk}\bar{\unicode[STIX]{x1D6FC}}=|\unicode[STIX]{x1D706}|=r$, we have, by Corollary 5.7 and Lemma 5.5,

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E9}_{D,\text{nil}}^{+}(\bar{\unicode[STIX]{x1D6FC}})=(-q^{1/2})^{-\ell r^{2}+2\mathbf{d}_{D}(\bar{\unicode[STIX]{x1D6FC}})}[\mathbf{Flag}^{+}(\bar{\unicode[STIX]{x1D6FC}})]=(-q^{1/2})^{2\mathbf{d}_{0}(\bar{\unicode[STIX]{x1D6FC}})-\ell \langle \unicode[STIX]{x1D706},\unicode[STIX]{x1D706}\rangle }[\mathbf{Flag}^{+}(\bar{\unicode[STIX]{x1D6FC}})]. & & \displaystyle \nonumber\end{eqnarray}$$

Finally, applying Theorem 6.1, we obtain

$$\begin{eqnarray}\displaystyle \!\!\!\mathop{\sum }_{\bar{\unicode[STIX]{x1D6FC}}\in \unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D706})}\unicode[STIX]{x1D6E9}_{D,\text{nil}}^{+}(\bar{\unicode[STIX]{x1D6FC}})z^{\mathop{\sum }_{i}i\deg \unicode[STIX]{x1D6FC}_{i}}=(-q^{1/2})^{-\ell \langle \unicode[STIX]{x1D706},\unicode[STIX]{x1D706}\rangle }q^{(g-1)\langle \unicode[STIX]{x1D706},\unicode[STIX]{x1D706}\rangle }J_{\unicode[STIX]{x1D706}}(z)H_{\unicode[STIX]{x1D706}}(z)\cdot \operatorname{Exp}\biggl(\frac{[X]}{q-1}\cdot \frac{z}{1-z}\biggr).\Box & & \displaystyle \nonumber\end{eqnarray}$$

Theorem 6.3 (cf. Theorem 1.1).

Define invariants $\unicode[STIX]{x1D6FA}_{D}^{+}(r,d)$ by the formula

(46)$$\begin{eqnarray}\mathop{\sum }_{r,d}\unicode[STIX]{x1D6FA}_{r,d}^{+}T^{r}z^{d}=(q-1)\operatorname{Log}\biggl(\mathop{\sum }_{\unicode[STIX]{x1D706}}z^{(\ell -2g+2)n(\unicode[STIX]{x1D706}^{\prime })}(-q^{1/2})^{\ell \langle \unicode[STIX]{x1D706},\unicode[STIX]{x1D706}\rangle }J_{\unicode[STIX]{x1D706}}(z)H_{\unicode[STIX]{x1D706}}(z)T^{|\unicode[STIX]{x1D706}|}\biggr).\end{eqnarray}$$

Then for all $r\geqslant 1$ and $d\geqslant \ell \binom{r}{2}$:

1. (i) if $\deg D>2g-2$, then $\unicode[STIX]{x1D6FA}_{D}(r,d)=\unicode[STIX]{x1D6FA}_{r,d}^{+}$;

2. (ii) if $D=K$, then $\unicode[STIX]{x1D6FA}_{D}(r,d)=q\unicode[STIX]{x1D6FA}_{r,d}^{+}$.

Proof. We obtain from Theorem 6.2 that $\unicode[STIX]{x1D6FA}_{r,d}^{+}=\unicode[STIX]{x1D6FA}_{K-D}^{+,\text{nil}}(r,d)$ for $r\geqslant 1$. If $\ell >2g-2$, then, by Lemma 3.2 and Corollary 4.5,

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA}_{D}(r,d)=\unicode[STIX]{x1D6FA}_{D}^{+}(r,d)=\unicode[STIX]{x1D6FA}_{K-D}^{+,\text{nil}}(r,d)=\unicode[STIX]{x1D6FA}_{r,d}^{+}. & & \displaystyle \nonumber\end{eqnarray}$$

If $D=K$, then, by Lemma 3.2 and Corollary 4.10, we have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA}_{D}(r,d)=\unicode[STIX]{x1D6FA}_{D}^{+}(r,d)=q\unicode[STIX]{x1D6FA}_{K-D}^{+,\text{nil}}(r,d)=q\unicode[STIX]{x1D6FA}_{r,d}^{+}.\Box & & \displaystyle \nonumber\end{eqnarray}$$

Theorem 6.4. Define the series $\unicode[STIX]{x1D6FA}_{r}^{+}(z)$ by the formula

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{r}\unicode[STIX]{x1D6FA}_{r}^{+}(z)T^{r}=(q-1)\operatorname{Log}\biggl(\mathop{\sum }_{\unicode[STIX]{x1D706}}z^{(\ell -2g+2)n(\unicode[STIX]{x1D706}^{\prime })}(-q^{1/2})^{l\langle \unicode[STIX]{x1D706},\unicode[STIX]{x1D706}\rangle }J_{\unicode[STIX]{x1D706}}(z)H_{\unicode[STIX]{x1D706}}(z)T^{|\unicode[STIX]{x1D706}|}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$

Then the DT invariants $\unicode[STIX]{x1D6FA}_{D}(r,d)$, for $r\geqslant 1$, are given by

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA}_{D}(r,d)=\left\{\begin{array}{@{}ll@{}}\displaystyle -\mathop{\sum }_{\unicode[STIX]{x1D709}\in \unicode[STIX]{x1D707}_{r}}\unicode[STIX]{x1D709}^{-d}\operatorname{Res}_{z=\unicode[STIX]{x1D709}}\unicode[STIX]{x1D6FA}_{r}^{+}(z)\frac{dz}{z},\quad & \deg D>2g-2,\\ \displaystyle -q\mathop{\sum }_{\unicode[STIX]{x1D709}\in \unicode[STIX]{x1D707}_{r}}\unicode[STIX]{x1D709}^{-d}\operatorname{Res}_{z=\unicode[STIX]{x1D709}}\unicode[STIX]{x1D6FA}_{r}^{+}(z)\frac{dz}{z},\quad & D=K,\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D707}_{r}$ stands for the set of $r$th roots of unity.

Proof. Let $\unicode[STIX]{x1D6FA}_{r}^{+}(z)=\sum _{d}\unicode[STIX]{x1D6FA}_{r,d}^{+}z^{d}$. Then, by the previous theorem, $\unicode[STIX]{x1D6FA}_{D}(r,d)=\unicode[STIX]{x1D6FA}_{r,d}^{+}$ if $\ell >2g-2$ or $\unicode[STIX]{x1D6FA}_{D}(r,d)=q\unicode[STIX]{x1D6FA}_{r,d}^{+}$ if $D=K$, for $d\gg 0$. Therefore $\unicode[STIX]{x1D6FA}_{r,d}^{+}$ is $r$-periodic for $d\gg 0$. This implies that the rational function $\unicode[STIX]{x1D6FA}_{r}^{+}(z)$ is regular outside of $r$th roots of unity and has at most simple poles. In addition, for large enough $N$,

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FA}_{r,d+Nr}^{+}=-\mathop{\sum }_{\unicode[STIX]{x1D709}\in \unicode[STIX]{x1D707}_{r}}\unicode[STIX]{x1D709}^{-d}\operatorname{Res}_{z=\unicode[STIX]{x1D709}}\unicode[STIX]{x1D6FA}_{r}^{+}(z)\frac{dz}{z}.\Box & & \displaystyle \nonumber\end{eqnarray}$$

Corollary 6.5. If $\ell =\deg D\leqslant 0$, then

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{r,d}\unicode[STIX]{x1D6E9}_{D,\text{nil}}^{+,\text{vec}}(r,d)T^{r}z^{d}=\mathop{\sum }_{\unicode[STIX]{x1D706}}z^{-\ell n(\unicode[STIX]{x1D706}^{\prime })}(-q^{1/2})^{(2g-2-\ell )\langle \unicode[STIX]{x1D706},\unicode[STIX]{x1D706}\rangle }J_{\unicode[STIX]{x1D706}}(z)H_{\unicode[STIX]{x1D706}}(z)T^{|\unicode[STIX]{x1D706}|}. & & \displaystyle \nonumber\end{eqnarray}$$

Corollary 6.6. If $\ell =\deg (D)>2g-2$, then

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{r,d}\unicode[STIX]{x1D6E9}_{D}^{+,\text{vec}}(r,d)T^{r}z^{d}=\mathop{\sum }_{\unicode[STIX]{x1D706}}z^{(\ell -2g+2)n(\unicode[STIX]{x1D706}^{\prime })}(-q^{1/2})^{\ell \langle \unicode[STIX]{x1D706},\unicode[STIX]{x1D706}\rangle }J_{\unicode[STIX]{x1D706}}(z)H_{\unicode[STIX]{x1D706}}(z)T^{|\unicode[STIX]{x1D706}|}. & & \displaystyle \nonumber\end{eqnarray}$$

Proof. By Corollary 4.5,

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{r,d}\unicode[STIX]{x1D6E9}_{D}^{+,\text{vec}}(r,d)T^{r}z^{d}=\mathop{\sum }_{r,d}\unicode[STIX]{x1D6E9}_{K-D,\text{nil}}^{+,\text{vec}}(r,d)T^{r}z^{d}. & & \displaystyle \nonumber\end{eqnarray}$$

We note that $\deg (K-D)=2g-2-\ell <0$ and apply the previous statement.◻