Proof. Let $s$ be a holomorphic section of ${\rm ad} (E_G)$ defined over an open subset $U\, \subset \, X$ . Then $s$ defines a $G$ –invariant holomorphic vector field on $p^{-1}(U)\, \subset \, E_G$ which is vertical for the projection $p$ in (2.2); this vertical vector field on $p^{-1}(U)$ will be denoted by $\widetilde s$ . Take any $t\, \in \, H^0(U,\, TX(-D))$ . Let

\begin{align*}\widetilde {t}\, \,:=\,\, \widehat {\nabla }(t)\, \in \, H^0(p^{-1}(U),\, TE_G(-\log p^{-1}(D)))^G\end{align*}

be the horizontal lift of $t$ for the $D$ -twisted connection $\widehat \nabla$ in (2.12). Now consider the Lie bracket of vector fields

\begin{align*} [\widetilde {t},\, \widetilde {s}]\, \in \, H^0(p^{-1}(U),\, TE_G) . \end{align*}

Note that $[\widetilde {t},\, \widetilde {s}]$ is $G$ –invariant because both $\widetilde {s}$ and $\widetilde {t}$ are. Furthermore, $[\widetilde {t}, \, \widetilde {s}]$ is vertical for the projection $p$ , because $\widetilde {s}$ is vertical and $\widetilde {t}$ is $G$ –invariant. Indeed, for any holomorphic function $f$ on $U$ , evidently $\widetilde {s}(f\circ p)\,=\, 0$ (recall that $\widetilde s$ is vertical), and we also have that $\widetilde {t}(f\circ p)$ is $G$ -invariant, so $\widetilde {s}(\widetilde {t} (f\circ p))\, =\, 0$ . Consequently, $[\widetilde {t},\, \widetilde {s}]$ produces a holomorphic section of ${\rm ad}(E_G)$ over $U$ ; this section of ${\rm ad} (E_G)$ over $U$ will be denoted by $[t,\, s]'$ . Next, note that a holomorphic function $f$ on $U$ satisfies

\begin{align*} [\widetilde {f\cdot t},\, \widetilde {s}]\,=\, (f\circ p)\cdot [\widetilde {t},\, \widetilde {s}] - \widetilde {s} (f\circ p)\cdot \widetilde {t}\,=\, (f\circ p)\cdot [\widetilde {t},\, \widetilde {s}] \end{align*}

because $\widetilde {s} (f\circ p)\,=\, 0$ (recall that $\widetilde s$ is a vertical vector field). Consequently, there is a homomorphism

\begin{align*} \nabla \,\, :\,\, {\rm ad} (E_G) \longrightarrow {\rm ad} (E_G) \otimes K_X (D) \end{align*}

uniquely defined by the equation

\begin{align*} \langle \nabla (s),\, t\rangle \,=\, [t,\, s]'\, , \end{align*}

where $s$ and $t$ are locally defined holomorphic sections of ${\rm ad} (E_G)$ and $TX(-D)$ respectively, while $\langle -,\, -\rangle$ is the natural pairing $TX(-D)\otimes K_X(D)\, \longrightarrow \, {\mathcal O}_X$ .

Recall from Definition 2.3 that ${\rm res}_{x} ( \nabla )\, \in \, \mathcal{H}^{\perp }_{x}$ . Therefore, from the property of residues mentioned earlier, it follows immediately that $\widetilde {t}(x)\, \in \, {\mathcal H}^\perp _x$ for every $x\, \in \, D$ . Now if $s$ is a locally defined holomorphic section of ${\rm ad}_{\phi } (E_G)$ , then $\widetilde {s}(x)\, \in \, {\mathcal H}_x$ . Next, note that

(2.15) \begin{align} [{\mathcal H}^\perp _x,\, {\mathcal H}_x]\, \subset \, {\mathcal H}^\perp _x\, , \end{align}

because

\begin{align*} \widehat {\sigma }(x)([a,\, b]\otimes c)\,=\, \widehat {\sigma }(x)(a\otimes [b,\, c]) \end{align*}

for all $a,\, b,\, c\,\in \, {\rm ad}(E_G)_x$ (this is derived using the given condition on $\sigma$ that it is $G$ -invariant). As a consequence of (2.15), the homomorphism $\nabla$ maps the subsheaf ${\rm ad}_{\phi } (E_G)$ to ${\rm ad}^n_{\phi } (E_G) \otimes K_X (D)$ .

Proof. The space ${\rm Aut}(E,\,\phi ,\,\nabla )$ has the structure of a group scheme of finite type over $\mathbb{C}$ . We can see that the tangent space of ${\rm Aut}(E,\,\phi ,\,\nabla )$ at the identity element is isomorphic to

\begin{align*} {\rm ker}(H^0(X,\,\mathcal{E}nd_{\phi }(E) )\, \xrightarrow {\,\,\nabla \,\,}\, H^0( X,\, \mathcal{E}nd_{\phi }^n(E)\otimes K_X (D)) ), \end{align*}

which is zero because $(E,\,\phi ,\,\nabla )$ is simple. Consequently, ${\rm Aut}(E,\,\phi ,\,\nabla )$ is a finite group.

Proof. An automorphism $\sigma \,\in \,{\rm Aut}(E,\,\phi ,\,\nabla )$ is an automorphism of the vector bundle $E$ such that $\nabla \circ \sigma \,=\, \sigma \circ \nabla$ and $\phi _x\circ \sigma |_x\circ \phi _x^{-1}\,\in \, H_x$ for all $x \,\in \, D$ . Since $H_x\,=\,\{ e\}$ by the assumption, it follows that $\sigma |_x\,=\,{\rm Id}|_{E|_x}$ for all $x\,\in \, D$ . Now set

\begin{align*} \widetilde {\sigma } \,\,:=\,\, {\rm Id}_{E} - \sigma \,\colon \, E \,\longrightarrow \, E\, . \end{align*}

Then $\widetilde {\sigma }|_x\,=\,0$ for all $x\,\in \, D$ , and it is straightforward to check that $\nabla \circ \widetilde {\sigma } - \widetilde {\sigma } \circ \nabla \,=\,0$ , that is,

\begin{align*} \widetilde {\sigma } \,\in \, {\rm ker}(H^0(X,\,\mathcal{E}nd_{\phi }(E))\, \xrightarrow {\,\nabla \,}\, H^0(X,\, \mathcal{E}nd^n_{\phi }(E)\otimes K_X(D)))\, . \end{align*}

Since $(E,\,\phi ,\,\nabla )$ is simple, it follows that $\widetilde {\sigma }\,=\,0$ , and hence $\sigma \,=\, {\rm Id}_{E}$ .

Proof. The tangent space of ${\rm Aut}(E,\,\phi ,\,\nabla )/(\mathbb{C}^* \cdot \textrm { Id})$ is zero, because we have that $(E,\,\phi ,\,\nabla )$ is simple. Consequently, ${\rm Aut}(E,\,\phi ,\,\nabla )/(\mathbb{C}^* \cdot \textrm { Id})$ is a finite group.

Proof. Fix a very ample line bundle $\mathcal{O}_{X}(1)$ on the curve $X$ . Define a polynomial $\theta _d(m)$ in $m$ to be

\begin{align*}\theta _d(m)\,=\,r d_X m+d+r(1-g)\, ,\end{align*}

where $d_X\,\,:=\,\, \deg \mathcal{O}_X(1)$ and $g$ is the genus of $X$ . Let

(3.1) \begin{align} \Sigma _{m_0}^{d} \end{align}

denote the fibered category whose objects are simple framed ${\rm GL}(r,\mathbb{C})$ -connections $(E,\,\phi ,\,\nabla )$ on $X\times S$ such that:

1. (1) $H^1(X,\, E_s (m_0-1))\,=\,0$ for each $s\,\in \, S$ ;

2. (2) $\chi (E_s(m))\,= \,\theta _d(m)$ for each $s \,\in \, S$ and all $m \,\in \, \mathbb{Z}$ .

The fibered categories $\Sigma _{m_0}^{d}$ in (3.1) form an open covering of $\mathcal{M}^{H}_{\textrm {FC}} (d)$ . So we only have to prove that each $\Sigma _{m_0}^{d}$ is a Deligne–Mumford stack.

Let

\begin{align*}\mathcal{O}_{X\times Q^d_{m_0}}(-m_0)^{\oplus \theta _d(m_0)}\,\longrightarrow \, \mathcal{E}\end{align*}

be the universal quotient sheaf of the Quot-scheme ${\rm Quot}^{\theta _d}_{(\mathcal{O}_X(-m_0)^{\oplus \theta _d(m_0)}/X)}$ . Define the open subset $Q^d_{m_0}$ of ${\rm Quot}^{\theta _d}_{(\mathcal{O}_X(-m_0)^{\oplus \theta _d(m_0)}/X)}$ by

\begin{align*} Q^d_{m_0} \, \,:=\, \,\left \{ s \,\in \, {\rm Quot}^{\theta _d}_{(\mathcal{O}_X(-m_0)^{\oplus \theta _d(m_0)}/X)} \ \middle | \ \begin{array}{l} \rm (i)\ h^0(X,\,\ \mathcal{E}_s(m_0)\ )\,=\,\theta\ _d(m_0);\\ \rm (ii)\ h^i(X,\,\ \mathcal{E}_s(m_0-i)\ )\,=\,0\,\,\ for\ all\ i\,\gt\ \,0;\ and\\ \rm (iii)\ \mathcal{E}_s\ is\ locally\ free. \end{array} \right \}. \end{align*}

There is a locally free $\mathcal{O}_{Q^d_{m_0}}$ –module $\mathscr{H}\,\,_D$ such that $\boldsymbol{V}^*(\mathscr{H}\,\,_D)\,\,:=\,\,{\rm Spec}({\rm Sym}^*\mathscr{H}\,\,_D)$ represents the functor

\begin{align*} S \,\,\longmapsto \,\,\bigoplus _{x \in D} {\rm Hom}_{x \times S} (\mathcal{O}^{\oplus r}_{x\times S},\, \, \mathcal{E}_{X\times S}|_{x\times S}) \,\, \in \,\, (\mathit{Sets}) \end{align*}

for any Noetherian schemes $S$ over $Q^d_{m_0}$ . There is a universal family

(3.2) \begin{align} \varphi ^{x}\, \colon \, \mathcal{O}^{\oplus r}_{x \times \boldsymbol{V}^*(\mathscr{H}\,\,_D)} \,\longrightarrow \, \mathcal{E}_{X \times \boldsymbol{V}^*(\mathscr{H}\,\,_D)} |_{x \times \boldsymbol{V}^*(\mathscr{H}\,\,_D)} \end{align}

for every $x \,\in \,D$ . Define $\widetilde {Q}^{d}_{m_0}$ as follows:

\begin{align*} \widetilde {Q}^{d}_{m_0}\,\,:=\,\, \left \{ s \,\in \, \boldsymbol{V}^*(\mathscr{H}\,\,_D) \ \middle | \ \rm \mathrm{coker}\ (\varphi\ ^{x}_s)\,=\,0 \, \right \}. \end{align*}

Consider the map $\widetilde {Q}^{d}_{m_0} \, \longrightarrow \, Q^d_{m_0}$ . For each Noetherian scheme $S$ over $Q^d_{m_0}$ , the natural transitive group action of $G \times S$ on

\begin{align*} {\rm Isom}_{x\times S} (\mathcal{O}^{\oplus r}_{x\times S},\,\, \mathcal{E}_{X\times S}|_{x\times S}) \end{align*}

induces an action on $\widetilde {Q}^{d}_{m_0}$ of the group scheme $\left (\prod _{x \in D}G\right )\times Q^d_{m_0}$ over $Q^d_{m_0}$ . The group scheme $\left (\prod _{x \in D}H_x\right )\times Q^d_{m_0}$ acts on $\widetilde {Q}^{d}_{m_0}$ by restricting this action of $(\prod _{x \in D}G)\times Q^d_{m_0}$ on $\widetilde {Q}^{d}_{m_0}$ . Set

\begin{align*} \widetilde {Q}^{d,H}_{m_0}\, \,:=\,\, \widetilde {Q}^{d}_{m_0}\Big / \Big ((\prod _{x \in D}H_x)\times Q^d_{m_0}\Big ). \end{align*}

Let $\widetilde {\mathcal{E}}$ be the pull-back of the family $\mathcal{E}$ under the map $X \times \widetilde {Q}^{d,H}_{m_0} \, \longrightarrow \, X \times Q^d_{m_0}$ . We have a family $\widetilde {\phi }_x$ of sections of

\begin{align*} {\rm Isom}_{\widetilde {Q}^{d,H}_{m_0}} (\mathcal{O}_{\widetilde {Q}^{d,H}_{m_0}}^{\oplus r}, \,\, \widetilde {\mathcal{E}}|_{x\times \widetilde {Q}^{d,H}_{m_0}} ) / (H_x \times \widetilde {Q}^{d,H}_{m_0}) \, \longrightarrow \, \widetilde {Q}^{d,H}_{m_0} \end{align*}

induced by $\varphi ^x$ . Put $\widetilde {\phi }\,\,:=\,\,\{\widetilde {\phi }_x \}_{x \in D}$ .

Let

(3.3) \begin{align} \pi \,\colon \, X \times \widetilde {Q}^{d,H}_{m_0} \,\longrightarrow \, \widetilde {Q}^{d,H}_{m_0} \end{align}

be the natural projection map. Let ${\rm At}(\widetilde {\mathcal{E}})$ be the Atiyah bundle for $\widetilde {\mathcal{E}}$ . Then there is a short exact sequence

\begin{align*} 0 \longrightarrow \mathcal{E}nd(\widetilde {\mathcal{E}})\longrightarrow {\rm At}(\widetilde {\mathcal{E}})\xrightarrow {{\rm symb}_1} T_{X\times \widetilde {Q}^{d,H}_{m_0}/\widetilde {Q}^{d,H}_{m_0}} \longrightarrow 0. \end{align*}

Set ${\rm At}_{D} (\widetilde {\mathcal{E}}) \,\,\,:=\,\,\, {\rm symb}_1^{-1}\left (T_{X\times \widetilde {Q}^{d,H}_{m_0} /\widetilde {Q}^{d,H}_{m_0}}(-D\times \widetilde {Q}^{d,H}_{m_0}) \right )$ . The natural short exact sequences of Atiyah bundles induces an exact sequence as follows.

In particular, there are two compositions of maps

(3.4) \begin{align} \begin{aligned} {\rm At}_{D} (\widetilde {\mathcal{E}}) \otimes K_X(D) \xrightarrow {\,\,\, p\,\,} \left ( {\rm At}_{D} (\widetilde {\mathcal{E}}) \otimes K_X(D) \right ) \Big /\left ( {\rm At}(\widetilde {\mathcal{E}}) \otimes K_{X} \right ) \\ \qquad \xrightarrow {\, q^{-1}\, } \mathcal{E}nd(\widetilde {\mathcal{E}}) \otimes K_{X}(D)|_{D\times \widetilde {Q}^{d,H}_{m_0}} \xrightarrow {\, {\rm res}_{D \times \widetilde {Q}^{d,H}_{m_0}} \,} {\rm End} (\widetilde {\mathcal{E}})|_{D\times \widetilde {Q}^{d,H}_{m_0}} \end{aligned} \end{align}

and

(3.5) \begin{align} \mathcal{E}nd(\widetilde {\mathcal{E}}) \otimes K_{X}(D) \xrightarrow {\, p'\, } \mathcal{E}nd(\widetilde {\mathcal{E}}) \otimes K_{X}(D)|_{D\times \widetilde {Q}^{d,H}_{m_0}} \xrightarrow {\, {\rm res}_{D \times \widetilde {Q}^{d,H}_{m_0}} \,} {\rm End} (\widetilde {\mathcal{E}})|_{D\times \widetilde {Q}^{d,H}_{m_0}}. \end{align}

Here, ${\rm res}_{D \times \widetilde {Q}^{d,H}_{m_0}}$ is the residue map

\begin{align*} {\rm res}_{D \times \widetilde {Q}^{d,H}_{m_0}} \,\,\colon \,\, {\rm End}(\widetilde {\mathcal{E}}) \otimes K_{X}(D)|_{D\times \widetilde {Q}^{d,H}_{m_0} } \longrightarrow {\rm End} (\widetilde {\mathcal{E}})|_{D\times \widetilde {Q}^{d,H}_{m_0}}. \end{align*}

Using the family $\widetilde {\phi }$ of framings and the Lie subgroups $H_x$ for each $x\,\in \,D$ , we may define a subsheaf $\mathcal{H}^{\perp }_{x \times \widetilde {Q}^{d,H}_{m_0}} \,\subset \, {\rm End} (\widetilde {\mathcal{E}})|_{D\times \widetilde {Q}^{d,H}_{m_0}}$ as in (2.10). Define subsheaves ${\rm At}^{\widetilde {\phi }}_{D} (\widetilde {\mathcal{E}}) \,\subset \, {\rm At}_{D} (\widetilde {\mathcal{E}})$ and $\mathcal{E}nd_{\widetilde {\phi }}^n(\widetilde {\mathcal{E}}) \,\subset \, \mathcal{E}nd(\widetilde {\mathcal{E}})$ such that ${\rm At}^{\widetilde {\phi }}_{D} (\widetilde {\mathcal{E}}) \otimes K_X(D)$ is the inverse image of $\mathcal{H}^{\perp }_{x \times \widetilde {Q}^{d,H}_{m_0}}$ under the composition of maps in (3.4) and $\mathcal{E}nd_{\widetilde {\phi }}^n(\widetilde {\mathcal{E}})\otimes K_X(D)$ is the inverse image of $\mathcal{H}^{\perp }_{x \times \widetilde {Q}^{d,H}_{m_0}}$ under the composition of maps in (3.5).

By [Reference GrothendieckGrot1, Theorem 7.7.6], there is a unique coherent sheaf $\mathscr{H}\,\,$ on $\widetilde {Q}^{d,H}_{m_0}$ (up to isomorphism) such that

(3.6) \begin{align} (\pi _{Q'})_*\left ( \left ( {\rm At}^{\widetilde {\phi }}_{D} (\widetilde {\mathcal{E}}) \otimes K_X(D)\right ) \otimes _{\mathcal{O}_{Q'}} M \right )\,\,\cong \,\, \mathcal{H}\textrm { om}_{\mathcal{O}_{Q'}} (\mathscr{H}\,\,_{Q'} , M) \end{align}

for any $\widetilde {Q}^{d,H}_{m_0}$ -scheme $Q'$ and any quasi-coherent sheaf $M$ on $Q'$ . Here, $\pi _{Q'}$ is the natural projection $X \times Q' \,\longrightarrow \, Q'$ . Set

\begin{align*}\boldsymbol{V} (\mathscr{H}\,\,) \,\,:=\,\, {\rm Spec}({\rm Sym}^*(\mathscr{H}\,\,))\, .\end{align*}

There is a natural morphism $\varphi \,\in \, \mathcal{H}\textrm {om}_{\mathcal{O}_{\boldsymbol{V} (\mathscr{H}\,\,)}} (\mathscr{H}\,\,_{\boldsymbol{V} (\mathscr{H}\,\,)} ,\, \mathcal{O}_{\boldsymbol{V} (\mathscr{H}\,\,)})$ by the definition of $\boldsymbol{V} (\mathscr{H}\,\,)$ . In view of the isomorphism in (3.6), there is an element $\varphi ' \,\in \, \pi _*( ( {\rm At}^{\widetilde {\phi }}_{D} (\widetilde {\mathcal{E}}) \otimes K_X(D) )_{\boldsymbol{V} (\mathscr{H}\,\,)})$ corresponding to $\varphi$ . The morphism ${\rm symb}_1^{D}$ induces a morphism

\begin{align*} (\pi _{\boldsymbol{V} (\mathscr{H}\,\,)})_* \left ( \left ( {\rm At}^{\widetilde {\phi }}_{D} (\widetilde {\mathcal{E}}) \otimes K_X(D) \right )_{\boldsymbol{V} (\mathscr{H}\,\,)}\right ) \, \longrightarrow \, (\pi _{\boldsymbol{V} (\mathscr{H}\,\,)})_*(\mathcal{O}_{X \times \boldsymbol{V} (\mathscr{H}\,\,)}) \cong \mathcal{O}_{\boldsymbol{V} (\mathscr{H}\,\,)}. \end{align*}

Using this morphism, there is a function $f_{{\rm symb}_1^{D}}$ on $\boldsymbol{V} (\mathscr{H}\,\,)$ corresponding to $\varphi '$ . Denote by $I_{{\rm symb}_1^{D}}$ the ideal sheaf of $\mathcal{O}_{\boldsymbol{V} (\mathscr{H}\,\,)}$ generated by $f_{{\rm symb}_1^{D}}-1$ . Put

\begin{align*}Z^{d,H}_{m_0} \,\,:=\,\, {\rm Spec}(\mathcal{O}_{\boldsymbol{V} (\mathscr{H}\,\,)}\, /\, I_{{\rm symb}_1^{D}})\, .\end{align*}

Also, denote by $\widetilde {\mathcal{E}}$ the pull-back of $\widetilde {\mathcal{E}}$ under the natural morphism $X \times Z^{d,H}_{m_0} \,\longrightarrow \, X\times \widetilde {Q}^{d,H}_{m_0,{\rm spl}}$ , and let

\begin{align*} \widetilde {\nabla }\,\, \colon \,\, \widetilde {\mathcal{E}} \longrightarrow \widetilde {\mathcal{E}} \otimes _{\mathcal{O}_{X \times \boldsymbol{V}^* (\mathscr{H}\,\,)_{\epsilon }}} K_{X}(D) \end{align*}

be a universal relative connection on $\widetilde {\mathcal{E}}$ , which is determined by $\varphi '$ . Define the open subset $(Z^{d,H}_{m_0})'$ of $Z^{d,H}_{m_0}$ by

\begin{align*} (Z^{d,H}_{m_0})'\, =\, \left \{ s\in Z^{d,H}_{m_0}\,\,\middle |\,\,\, (\widetilde {\mathcal E},\widetilde {\phi},\widetilde {\nabla})|_{X\times \{s\}}\ \rm is\ a\ simple\ framed\ connection \right \} \end{align*}

and denote by $\big (\widetilde {\mathcal{E}},\, \widetilde {\phi }\,=\, \big \{[\widetilde {\phi }_{x\times (Z^{d,H}_{m_0})'}] \big \}_{x\in D},\, \widetilde {\nabla }\big )$ a universal family of $m_0$ -regular simple framed $G$ -connections on $X\times (Z^{d,H}_{m_0})'$ . Here, $m_0$ -regular means $H^1(X,\, \widetilde {\mathcal{E}}_s (m_0-1))\,=\,0$ for each $s\,\in \, (Z^{d,H}_{m_0})'$ .

Now, consider the case where $\mathbb{C}^* \cdot \textrm { Id}\, \not \subset \, H_x$ for some $x\,\in \, D$ . There exists an action of ${\rm GL}(\theta _d(m_0),\mathbb{C})$ on $(Z^{d,H}_{m_0})'$ given by

\begin{align*} \left (\mathcal{O}_{X\times S}^{\oplus \theta _{d}(m)}\,\stackrel {q}{\longrightarrow }\, E,\, \phi ,\,\nabla \right ) \,\,\longmapsto \,\, \left (\mathcal{O}_{X\times S}^{\oplus \theta _{d}(m)}\,\xrightarrow {\, q \circ g \,} \, E,\, \phi , \, \nabla \right ) \end{align*}

on $S$ -points for $g \,\in \, {\rm GL}(\theta _d(m_0),\mathbb{C})_S$ . Consider the map

\begin{align*} \begin{aligned} (Z^{d,H}_{m_0})' \,\longrightarrow \, \Sigma _{m_0}^{d} \\ (\mathcal{O}_{X\times S}^{\oplus \theta _{d}(m)}\,\xrightarrow {\,\,q\,\,}\, E, \,\phi ,\, \nabla ) \,\longmapsto \, (S,\, E,\, \phi ,\, \nabla ). \end{aligned} \end{align*}

This map gives an isomorphism

\begin{align*} \Sigma _{m_0}^{d}\,\,\cong \,\, \big [ \, (Z^{d,H}_{m_0})' \, \big /\, {\rm GL}(\theta _d(m_0),\mathbb{C}) \, \big ] \,. \end{align*}

Here, $\big [ \, (Z^{d,H}_{m_0})' \, \big / \, {\rm GL}(\theta _d(m_0),\mathbb{C}) \, \big ]$ is a quotient stack. Consequently, $\Sigma _{m_0}^{d}$ is an algebraic stack. Using Proposition 3.2, it follows that $\Sigma _{m_0}^{d}$ is in fact a Deligne–Mumford stack (see [Reference OlssonOls, Corollary 8.4.2]).

Next, we consider the case where $\mathbb{C}^* \cdot \textrm { Id}\, \subset \, H_x$ for every $x\,\in \, D$ . The $\mathbb{C}^*$ -action on $(Z^{d,H}_{m_0})'$ is trivial, because $\mathbb{C}^* \cdot \textrm { Id}\, \subset \, H_x$ for all $x\,\in \, D$ . There exists a natural action of ${\rm PGL}(\theta _d(m_0),\,\mathbb{C})$ on $(Z^{d,H}_{m_0})'$ . Define a map

\begin{align*} \begin{aligned} (Z^{d,H}_{m_0})' \,\,&\longrightarrow \,\, \Sigma _{m_0}^{d}\\ (\mathcal{O}_{X\times S}^{\oplus \theta _{d}(m)}\xrightarrow {\,\,q\,\,} E,\, \phi ,\,\nabla ) \,&\longmapsto \, (S,\, E,\, \phi ,\, \nabla ). \end{aligned} \end{align*}

It is straightforward to check that this map gives an isomorphism

\begin{align*}\Sigma _{m_0}^{d}\, \,\cong \,\, \big [ \, (Z^{d,H}_{m_0})' \, \big / \, {\rm PGL}(\theta _d(m_0),\mathbb{C}) \, \big ].\end{align*}

Then $\Sigma _{m_0}^{d}$ is an algebraic stack. By Proposition 3.4, it follows that $\Sigma _{m_0}^{d}$ is in fact a Deligne–Mumford stack (see [Reference OlssonOls, Corollary 8.4.2]).

Proof. Fix a very ample line bundle $\mathcal{O}_{X}(1)$ on the curve $X$ . Let $\theta _d(m)$ be a polynomial in $m$ defined as in Proposition 3.5. Let $(\Sigma _{m_0}^{d})^{\circ }$ denote the substack of $\mathcal{M}^{H}_{\textrm {FC}}(d)^{\circ }$ whose objects are framed ${\rm GL}(r,\mathbb{C})$ -connections $(E,\,\phi ,\,\nabla )$ on $X$ such that:

1. (1) $(E,\,\phi )$ is simple;

2. (2) $H^1(X,\, E (m_0-1))\,=\,0$ ;

3. (3) $\chi (E(m))\,= \,\theta _d(m)$ for all $m \,\in \, \mathbb{Z}$ .

To prove the proposition it suffices to show that $(\Sigma _{m_0}^{d})^{\circ }$ is irreducible.

Let $V$ be a $\theta _d(m_0)$ -dimensional vector space so that the underlying vector bundle $E$ of any object of $(\Sigma _{m_0}^{d})^{\circ }$ is described as the following quotient:

\begin{align*} V \otimes \mathcal{O}_X (-m_0)\, \longrightarrow E. \end{align*}

Take a subspace $V_r \,\subset \, V$ such that $\dim (V_r) \,=\,r$ . Taking the dual of the above quotient, and tensoring with $\mathcal{O}_X (-m_0)$ , we have the following short exact sequence

\begin{align*} 0 \, \longrightarrow \, E^{\vee }(-m_0) \, \longrightarrow \, V_r^{\vee } \otimes \mathcal{O}_X \, \longrightarrow \, F \, \longrightarrow \, 0, \end{align*}

where $F$ is the quotient for the injective map $E^{\vee }(-m_0) \,\longrightarrow \, V_r^{\vee } \otimes \mathcal{O}_X$ . So for each object of $(\Sigma _{m_0}^{d})^{\circ }$ , there is a point of ${\rm Quot}^N_{(V_r^{\vee }\otimes \mathcal{O}_X)/X}$ which determines the underlying vector bundle of the object. Here $N$ is the length of $F$ . Note that $N$ remains constant for the underlying vector bundles. We will show that ${\rm Quot}^N_{(V_r^{\vee }\otimes \mathcal{O}_X)/X}$ is irreducible.

The Quot-scheme ${\rm Quot}^N_{(V_r^{\vee }\otimes \mathcal{O}_X)/X}$ is smooth, because the obstructions to deformations of

\begin{align*}[\, q \,\colon \, V_r^{\vee } \otimes \mathcal{O}_X \,\longrightarrow \, F \,] \, \in \,{\rm Quot}^N_{(V_r^{\vee }\otimes \mathcal{O}_X)/X}\end{align*}

lie in

\begin{align*}{\rm Ext}^1 ({\rm Ker}\, q , \,F) \,\cong \, H^1 (({\rm Ker}\, q)^{\vee } \otimes F)\,=\,0\, .\end{align*}

Define the map

\begin{align*} \begin{aligned} f_N \,\,\colon \,\, {\rm Quot}^N_{(V_r^{\vee }\otimes \mathcal{O}_X)/X} \, \longrightarrow \, {\rm Hilb}^N_{X}\\ [\, q \colon V_r^{\vee } \otimes \mathcal{O}_X \longrightarrow F \,] \, \longmapsto \, {\rm Divisor} ( {\rm det} ({\rm Ker}\, q \,\longrightarrow \, V_r^{\vee } \otimes \mathcal{O}_X)). \end{aligned} \end{align*}

Let $H'$ be the Zariski open subset of ${\rm Hilb}^N_{X}$ which consists of distinct points on $X$ , or in other words, $H'$ parametrizes the reduced subschemes. This Zariski open subset $H'$ is in fact irreducible. The map

\begin{align*}f_N^{-1}(H') \ \longrightarrow \ H'\end{align*}

is a $(\mathbb{P}^{r-1}\times \cdots \times \mathbb{P}^{r-1})$ -bundle; here $\mathbb{P}^{r-1}\times \cdots \times \mathbb{P}^{r-1}$ is the product of $N$ -copies of $\mathbb{P}^{r-1}$ . Hence $f_N^{-1}(H')$ is irreducible. Take a point $x \,=\, N_1 z_1+ \cdots +N_l z_l$ on ${\rm Hilb}^N_{X}$ , where $\sum _{i=1}^l N_i\, =\,N$ and $z_1,\,\cdots ,\,z_l$ are distinct points on $X$ . A point on $f^{-1}_N(x)$ can be described as a collection $\left ( \, q_i \,\colon \, V_r^{\vee } \otimes \mathcal{O}_{z_i,X} \,\longrightarrow \, F_i\, \right )_{i=1}^l$ for which ${\rm length}(F_i)\,=\,N_i$ . Consider the map $({\rm Ker}\, q_i )_{z_i} \,\longrightarrow \, V_r^{\vee } \otimes \mathcal{O}_{z_i,X}$ corresponding to a point on $f^{-1}_N(x)$ . Note that $({\rm Ker}\, q_i )_{z_i}\,\cong \, \mathcal{O}_{z_i,X}^{\oplus r}$ . We have a matrix representation of $({\rm Ker}\, q_i )_{z_i} \,\longrightarrow \, V_r^{\vee } \otimes \mathcal{O}_{z_i,X}$ as follows:

\begin{align*} \left(\ \begin{matrix}1 &&&&& \\ &\ddots &&&& \\ &&1 &&& \\ &&& z^{l_{i,1}} && \\ &&&& \ddots & \\ &&&&&z^{l_{i,s_i}} \end{matrix}\right), \end{align*}

where the maximal ideal $\mathfrak{m}_{z_i}$ is $\{z=0 \}$ , and $1,\,z^{l_{i,1}},\,\ldots ,\,z^{l_{i,s_i}}$ are invariant factors of

\begin{align*}({\rm Ker}\, q_i )_{z_i} \,\longrightarrow \,V_r^{\vee } \otimes \mathcal{O}_{z_i,X}\, .\end{align*}

For any tuple of complex numbers $a_1^{(i)},\,\ldots ,\,a_{s_i}^{(i)}$ , there is a deformation of $({\rm Ker}\, q_i )_{z_i} \,\longrightarrow \, V_r^{\vee } \otimes \mathcal{O}_{z_i,X}$

\begin{align*} \left( \begin{matrix} 1&&&& \\& \ddots &&&&\\&& 1&&& \\ &&&z^{l_{i,1}}+t a_{1}^{(i)}&& \\&&&& \ddots& \\ &&&&&z^{l_{i,s_i}}+ t a_{s_i}^{(i)} \end{matrix}\right), \end{align*}

over $X \times {\rm Spec}\, \mathbb{C}[t]$ . When the complex numbers $a_1^{(i)},\,\cdots ,\,a_{s_i}^{(i)}$ are generic, we have a deformation moving from a point on $f^{-1}_N(x)$ , where $x \,=\, N_1 z_1+ \cdots +N_l z_l$ , to a point on $f_N^{-1}(H')$ . Therefore, ${\rm Quot}^N_{(V_r^{\vee }\otimes \mathcal{O}_X)/X}$ is irreducible.

Consider the open subset

\begin{align*} Q' \ \,:=\, \ \left \{ [\, q \,] \, \in \,{\rm Quot}^N_{(V_r^{\vee }\otimes \mathcal{O}_X)/X} \ \middle | \ \rm E_q\ satisfies\ H^1(E_q(m_0-1))\,=\,0 \right \} \, \subset \, {\rm Quot}^N_{(V_r^{\vee }\otimes \mathcal{O}_X)/X}. \end{align*}

Here, denote $E_q\,\,:=\,\,({\rm ker} q)^{\vee } (-m_0)$ for a quotient $q$ . By definition, $E_q$ is locally free and satisfies the condition $\chi (E_q (m)) = \theta _d(m)$ for all $m \, \in \, \mathbb{Z}$ . Let $\widetilde {Q}'$ be the scheme over $Q'$ which parametrizes quotients $q$ in $Q'$ and framings of $E_q$ , which is constructed as in the proof of Proposition 3.5. Now define $(\widetilde {Q}')^{\circ }$ as follows:

\begin{align*} (\widetilde {Q}')^{\circ }\,\,:=\,\, \left \{ s \,in \, \widetilde {Q}' \middle | \ \rm (\widetilde {\mathcal{E}}_s ,\, \widetilde {\phi}_s ) is\ simple \right \}. \end{align*}

Here, $(\widetilde {\mathcal{E}} ,\, \widetilde {\phi } )$ is the family of vector bundles $E_q$ and framings of $E_q$ in $\widetilde {Q}'$ induced by the universal family of $\widetilde {Q}'$ . Since ${\rm Quot}^N_{(V_r^{\vee }\otimes \mathcal{O}_X)/X}$ is irreducible, $(\widetilde {Q}')^{\circ }$ is also irreducible. Let $(Z')^{\circ }$ be the scheme over $(\widetilde {Q}')^{\circ }$ which parametrizes quotients $q$ with framings of $E_q$ in $(\widetilde {Q}')^{\circ }$ and connections on $E_q$ that are compatible with the framings. The scheme $(Z')^{\circ }$ is also constructed as in the proof of Proposition 3.5. It is straightforward to check that $(Z')^{\circ }$ is smooth and each fiber of $(Z')^{\circ } \,\longrightarrow \, (\widetilde {Q}')^{\circ }$ is an affine space which is isomorphic to $H^0(X,\, {\rm End}_{\phi }^n (E) \otimes K_X(D))$ . So $(Z')^{\circ }$ is irreducible. Since a natural map from $(Z')^{\circ }$ to $(\Sigma _{m_0}^{d})^{\circ }$ is induced and this map is surjective, we conclude that $(\Sigma _{m_0}^{d})^{\circ }$ is irreducible. This completes the proof of the proposition.

Proof. Let $v$ and $v'$ be tangent vectors of $T^*\mathcal{N}^e(d)$ at $(E,\,\phi ,\,\theta ) \,\in \, T^*\mathcal{N}^e(d)$ . Let

\begin{align*}D_v \,\,\colon \,\, \mathcal{O}_{T^*\mathcal{N}^e(d)} \,\longrightarrow \, \mathcal{O}_{T^*\mathcal{N}^e(d)}\end{align*}

be the derivative corresponding to $v$ . Take an affine open subset $U\,\subset \, T^*\mathcal{N}^e(d)$ such that $(E,\,\phi ,\,\theta ) \,\in \, U$ , and also take an affine open covering $\{ U_{\alpha } \}$ of $X\times U$ such that there is a trivialization

\begin{align*}g_{\alpha } \,\colon \,E|_{U_{\alpha }} \,\stackrel {\sim }{\longrightarrow } \, \mathcal{O}_{U_{\alpha }}^{\oplus r}\end{align*}

for each $U_{\alpha }$ . Set $g_{\alpha \beta }\,\,:=\,\,g_{\alpha } \circ g^{-1}_{\beta }$ and $\theta _{\alpha }\,\,:=\,\, g_{\alpha } \circ \theta |_{U_{\alpha }} \circ g_{\alpha }^{-1}$ . We may describe the tangent vector $v$ as

\begin{align*}v\,=\,[(\{ u_{\alpha \beta }\} ,\, \{ v_{\alpha } \} )]\, ,\end{align*}

where $u_{\alpha \beta }\,\,:=\,\, g_{\alpha }^{-1} \circ ( D_{v}(g_{\alpha \beta } )g_{\alpha \beta }^{-1} ) \circ g_{\alpha }$ and $v_{\alpha }\,\,:=\,\, g_{\alpha }^{-1} \circ ( D_v \theta _{\alpha }) \circ g_{\alpha }$ . The exterior derivative of $\phi _{\mathcal{N}^e(d) }$ is computed as follows:

\begin{align*} \begin{aligned} D_{v}\phi _{\mathcal{N}^e(d)} ( v') & - D_{v'}\phi _{\mathcal{N}^e(d)} (v) + \phi _{\mathcal{N}^e(d) }([v,\, v']) \, =\, D_{v} \left ( {\rm Tr}( \theta _{\alpha } D_{v'}(g_{\alpha \beta } )g_{\alpha \beta }^{-1} ) \right )\\ & - D_{v'} \left ( {\rm Tr} ( \theta _{\alpha } D_{v}(g_{\alpha \beta } )g_{\alpha \beta }^{-1} ) \right ) + \left ( {\rm Tr} ( \theta _{\alpha } (D_{v'} \circ D_{v} - D_{v} \circ D_{v'} )(g_{\alpha \beta } ) )g_{\alpha \beta }^{-1} \right ) \end{aligned} \end{align*}

\begin{align*} \begin{aligned} & \quad=\,{\rm Tr} \left ( D_{v}(g_{\alpha \beta }^{-1} \theta _{\alpha } ) D_{v'}(g_{\alpha \beta } ) \right ) -{\rm Tr} \left ( D_{v'}( g_{\alpha \beta }^{-1}\theta _{\alpha })D_{v}(g_{\alpha \beta } ) \right ) \\ & \quad= \,\left ( {\rm Tr}( D_{v}(\theta _{\alpha } ) D_{v'}(g_{\alpha \beta } )g_{\alpha \beta }^{-1} ) - {\rm Tr} ( D_{v'}( \theta _{\alpha })D_{v}(g_{\alpha \beta } )g_{\alpha \beta }^{-1} ) \right ) \\ & \quad\qquad -\left ( {\rm Tr}(D_{v}(g_{\alpha \beta })g_{\alpha \beta }^{-1} \theta _{\alpha } D_{v'}(g_{\alpha \beta } )g_{\alpha \beta }^{-1} ) \right ) +\left ( {\rm Tr} ( D_{v'}( g_{\alpha \beta })g_{\alpha \beta }^{-1} \theta _{\alpha }D_{v}(g_{\alpha \beta } ) g_{\alpha \beta }^{-1} ) \right ) \\ & \quad= \,\left ( {\rm Tr} ( v_{\alpha } u'_{\alpha \beta } )- {\rm Tr} (v'_{\alpha }u_{\alpha \beta } ) \right ) + {\rm Tr} ( [u'_{\alpha \beta } ,\, \theta _{\alpha }]u_{\alpha \beta } ) \\ & \quad=\, {\rm Tr} ( v_{\alpha } u'_{\alpha \beta } )- {\rm Tr} (u_{\alpha \beta } v'_{\beta }). \end{aligned} \end{align*}

This completes the proof of the lemma.

Proof. Let $v\,=\, [(\{u_{\alpha \beta } \} ,\, \{ v_{\alpha } \})]$ be an element of $\mathbb{H}^1 (\mathcal{C}_{\bullet })$ . We compute $\Theta ^e(v,\,v)$ as follows:

\begin{align*} \begin{aligned}\Theta ^e(v,\,v) &=\, [(\{ {\rm Tr} (u_{\alpha \beta }u_{\beta \gamma }) \},\, - \{{\rm Tr}(u_{\alpha \beta }v_{\beta })-{\rm Tr}(v_{\alpha }u_{\alpha \beta }) \})] \\ &=\, [(\{ {\rm Tr} (u_{\alpha \beta }u_{\beta \gamma }) \},\, - \{{\rm Tr}(u_{\alpha \beta }(v_{\beta }-v_{\alpha })) \})] \\ &=\, [(\{ {\rm Tr} (u_{\alpha \beta }u_{\beta \gamma }) \},\, - \{{\rm Tr}(u_{\alpha \beta }(\nabla \circ u_{\alpha \beta } - u_{\alpha \beta } \circ \nabla ) ) \})] \\ &=\, \left [\left (\left \{ {\rm Tr} (u_{\alpha \beta }u_{\beta \gamma }) \right \},\, - \left \{ d \left ( \tfrac {1}{2}{\rm Tr}(u^2_{\alpha \beta } ) \right )\right \}\right )\right ]. \end{aligned} \end{align*}

On the other hand,

\begin{align*} \begin{aligned} \tfrac {1}{2}{\rm Tr}(u^2_{\alpha \beta } ) -\tfrac {1}{2}{\rm Tr}(u^2_{\alpha \gamma }) +\tfrac {1}{2}{\rm Tr}(u^2_{\beta \gamma }) &=\, \tfrac {1}{2}{\rm Tr}((u_{\alpha \beta } -u_{\alpha \gamma } )(u_{\alpha \beta } +u_{\alpha \gamma } )) +\tfrac {1}{2}{\rm Tr}(u^2_{\beta \gamma }) \\ &=\, \tfrac {1}{2}{\rm Tr}((u_{\beta \gamma } )(u_{\beta \gamma } -u_{\alpha \beta } -u_{\alpha \gamma } )) \\ &=\,- {\rm Tr}(u_{\alpha \beta }u_{\beta \gamma }). \end{aligned} \end{align*}

Combining these, it follows that $\Theta ^e(v,\,v)\,=\,0$ in $\mathbb{H}^2(\mathbb{K})$ .

Proof. Take a point $z\,=\,(E,\,\phi ,\,\nabla )$ of $p_1^{-1}(U)$ . Let $\nabla (E,\,\phi )$ be the connection associated to the point $s\circ p_1(z)$ . The image of $z$ under the map $P^{-1}_1$ in (4.6) is as follows:

\begin{align*} P_1^{-1}(z)\,=\,P^{-1}_1(E,\,\phi ,\,\nabla ) \,= \,(E,\,\phi ,\, \nabla - \nabla (E,\phi ))\, . \end{align*}

Let $[(\{u_{\alpha \beta } \} ,\,\{ v_{\beta } \})]$ be an element of $\mathbb{H}^1((\mathcal{C}_{\bullet })_z)$ , where $(\mathcal{C}_{\bullet })_z$ is the complex in (2.11) associated to $z\,=\,(E,\,\phi ,\, \nabla )$ . Recall from Lemma 2.4 that $\mathbb{H}^1((\mathcal{C}_{\bullet })_z)$ is the tangent space of $p_1^{-1}(U)$ at $z$ . Note that $u_{\alpha \beta }$ and $v_{\alpha }$ satisfy the equality

\begin{align*}v_{\beta }-v_{\alpha }\,=\, \nabla \circ u_{\alpha \beta }-u_{\alpha \beta } \circ \nabla .\end{align*}

Let $[(\{u_{\alpha \beta }\},\, \{ v^s_{\alpha } \})]$ be the element of $\mathbb{H}^1((\mathcal{C}_{\bullet })_{s\circ p_1(z)})$ such that

\begin{align*}(s \circ p_1)_{*} ([(\{u_{\alpha \beta } \} ,\,\{ v_{\beta } \})]) \,=\, [(\{u_{\alpha \beta } \} ,\, \{ v^s_{\beta } \})]\, . \end{align*}

Note that $u_{\alpha \beta }$ and $v^s_{\alpha }$ satisfy the equality

\begin{align*}v^s_{\beta }-v^s_{\alpha }\,=\, \nabla (E,\phi )\circ u_{\alpha \beta }-u_{\alpha \beta } \circ \nabla (E,\phi )\, .\end{align*}

Since

\begin{align*} (v_{\beta } - v^s_{\beta })- (v_{\alpha } - v^s_{\alpha }) \,=\, [ \nabla - \nabla (E,\phi ),\, u_{\alpha \beta }]\, , \end{align*}

it follows that $[(\{ u_{\alpha \beta } \} ,\,\{v_{\alpha }-v_{\alpha }^s \})]$ is an element of $\mathbb{H}^1 (\mathcal{C}^{{\rm Higgs}}_{\bullet })$ ; recall that $\mathbb{H}^1 (\mathcal{C}^{{\rm Higgs}}_{\bullet })$ is the tangent space of $T^*U$ at $P_1^{-1}(z)$ (see (4.6)). There is a map

\begin{align*} \begin{aligned} \mathbb{H}^1 ((\mathcal{C}_{\bullet })_z) &\longrightarrow \mathbb{H}^1 (\mathcal{C}^{{\rm Higgs}}_{\bullet }) \\ [(\{u_{\alpha \beta } \},\,\{ v_{\alpha } \})] &\longmapsto [(\{ u_{\alpha \beta } \} ,\,\{v_{\alpha }-v_{\alpha }^s \})] . \end{aligned} \end{align*}

This map coincides with $(P_1^{-1})_* \,\colon \,\mathbb{H}^1 ((\mathcal{C}_{\bullet })_z) \, \longrightarrow \,\mathbb{H}^1 (\mathcal{C}^{{\rm Higgs}}_{\bullet })$ .

Now we compute the map

\begin{align*} (\Theta ^e - (P_1^{-1})^* \Phi _U) \,\colon \, \mathbb{H}^1 ((\mathcal{C}_{\bullet })_z) \otimes \mathbb{H}^1 ((\mathcal{C}_{\bullet })_z) \,\longrightarrow \, \mathbb{H}^2(\mathbb{K}) \end{align*}

as follows:

\begin{align*} \begin{aligned} (\Theta ^e - (P_1^{-1})^* \Phi _U)(v,v') \,&=\, [(\{ {\rm Tr} (u_{\alpha \beta }u'_{\beta \gamma }) \},\, - \{ {\rm Tr}(u_{\alpha \beta }v'_{\beta })-{\rm Tr}(v_{\alpha }u'_{\alpha \beta }) \})] \\ &\qquad - [ \{ 0 \},\,- \{ {\rm Tr}(u_{\alpha \beta }(v'_{\beta }-(v_{\beta }^s)')) -{\rm Tr}((v_{\alpha }-v_{\alpha }^s) u'_{\alpha \beta }) \} ]\\ &=\,[(\{ {\rm Tr} (u_{\alpha \beta }u'_{\beta \gamma }) \},\, - \{{\rm Tr}(u_{\alpha \beta }(v^s_{\beta })')-{\rm Tr}(v^s_{\alpha }u'_{\alpha \beta }) \})] \,\in \,\mathbb{H}^2(\mathbb{K}). \end{aligned} \end{align*}

On the other hand, we compute

\begin{align*} p_1^* (s^* \Theta ^e) \,\colon \, \mathbb{H}^1 ((\mathcal{C}_{\bullet })_z) \otimes \mathbb{H}^1 ((\mathcal{C}_{\bullet })_z) \xrightarrow {(s\circ p_1)_* \otimes (s\circ p_1)_* } \mathbb{H}^1 ((\mathcal{C}_{\bullet })_{s\circ p_1(z)}) \otimes \mathbb{H}^1 ((\mathcal{C}_{\bullet })_{s\circ p_1(z)}) \xrightarrow {\ \Theta ^e \ } \mathbb{H}^1(\mathbb{K}) \end{align*}

as follows:

\begin{align*} \begin{aligned} p_1^* (s^* \Theta ^e)(v,\,v') = p_1^* (s^* \Theta ^e)([(\{u_{\alpha \beta }\},\, \{ v^s_{\alpha } \})] ,[(\{u'_{\alpha \beta }\}, \,\{ (v^s_{\alpha })' \})]) \\ =[(\{ {\rm Tr} (u_{\alpha \beta }u'_{\beta \gamma }) \},\, - \{{\rm Tr}(u_{\alpha \beta }(v^s_{\beta })')-{\rm Tr}(v^s_{\alpha }u'_{\alpha \beta }) \})] \,\in \, \mathbb{H}^1(\mathbb{K}). \end{aligned} \end{align*}

Therefore, we have the equality $\Theta ^e|_{p_1^{-1}(U)} - (P_1^{-1})^* \Phi _U \,=\, p_1^* (s^* \Theta ^e|_{p_1^{-1}(U)})$ .

Proof. For any point $(E,\,\phi ,\,\nabla )\, \in \, \mathcal{M}^{e}_{\textrm {FC}}(d)^{\circ }$ and any tangent vectors $v,\, w\, \in \, T_{(E,\phi ,\nabla )} \mathcal{M}^{e}_{\textrm {FC}}(d)^{\circ }$ , we have

\begin{align*} p_1^* s^* \Theta ^e (E,\phi ,\nabla )(v,\, w)\,=\, 0 \end{align*}

when one of $v$ and $w$ is vertical for the projection $p_1$ in (4.5). So, if $w$ is vertical, from Lemma 4.4 it follows that

(4.7) \begin{align} \Theta ^e(E,\,\phi ,\,\nabla )(v,\, w) \,=\, (P_1^{-1})^* \Phi _U (E,\,\phi ,\,\nabla )(v,\, w)\, . \end{align}

Since $\Phi _U$ is a symplectic form, there is a tangent vector $v\, \in \, T_{(E,\phi ,\nabla )} \mathcal{M}^{e}_{\textrm {FC}}(d)^{\circ }$ such that

\begin{align*}(P_1^{-1})^* \Phi _U (E,\,\phi ,\,\nabla )(v,\, w)\, \not =\, 0.\end{align*}

Now from (4.7) it follows that $\Theta ^e(E,\,\phi ,\,\nabla )(v,\, w)\, \not =\, 0$ .

Since the vertical tangent spaces for the projection $T^*U\, \longrightarrow \, U$ are Lagrangian for the Liouville $2$ -form $\Phi _U$ , given any non-vertical tangent vector

\begin{align*} v\, \in \, T_{(E,\phi ,\nabla )} \mathcal{M}^{e}_{\textrm {FC}}(d)^{\circ } \end{align*}

for the projection $T^*U\, \longrightarrow \, U$ , there is a vertical tangent vector

\begin{align*} w\, \in \, T_{(E,\phi ,\nabla )} \mathcal{M}^{e}_{\textrm {FC}}(d)^{\circ } \end{align*}

for the projection $T^*U\, \longrightarrow \, U$ such that $(P_1^{-1})^* \Phi _U (E,\,\phi ,\,\nabla )(v,\, w)\,\not =\, 0$ . Now from (4.7) it follows that $\Theta ^e(E,\,\phi ,\,\nabla )(v,\, w)\, \not =\, 0$ . Consequently, the form $\Theta ^e\vert _{p^{-1}(U)}$ is nondegenerate.

Proof. The moduli space $\mathcal{N}^e(d)$ in (4.1) has the open subset $\mathcal{N}^e(d)^{\circ }$ defined by

\begin{align*} \mathcal{N}^e(d)^{\circ }\, \,:=\,\, \left \{ (E,\,\phi ) \,\in \, \mathcal{N}^e(d) \ \big \vert \ \rm E\ is\ a\ stable\ vector\ bundle\right \}. \end{align*}

Also, let $\mathcal{M}_{{\rm FC}}^{e}(d)^{\circ \circ }\, \subset \, \mathcal{M}_{{\rm FC}}^{e}(d)^{\circ }$ (see Definition 3.1) be the open subset

(4.8) \begin{align} \mathcal{M}_{{\rm FC}}^{e}(d)^{\circ \circ }\,\,:=\,\, \left \{ (E,\,\phi ,\,\nabla ) \,\in \, \mathcal{M}_{{\rm FC}}^{e}(d)^{\circ } \ \big \vert \ E\rm \ is\ a\ stable\ vector\ bundle\right \}. \end{align}

The openness of both $\mathcal{N}^e(d)^{\circ }$ and $\mathcal{M}_{{\rm FC}}^{e} (d)^{\circ \circ }$ follows from [Reference MaruyamaMaru, p. 635, Theorem 2.8(B)]. The moduli spaces $\mathcal{N}^e(d)^{\circ }$ and $\mathcal{M}_{{\rm FC}}^{e}(d)^{\circ \circ }$ are non-empty because $g\,\geqslant \, 2$ .

To prove that the form $\Theta ^e|_{\mathcal{M}^{e}_{{\rm FC}}(d)^{\circ }}$ on $\mathcal{M}_{{\rm FC}}^{e}(d)^{\circ }$ is closed, it suffices to show that the restriction of $\Theta ^e|_{\mathcal{M}^{e}_{{\rm FC}}(d)^{\circ }}$ to $\mathcal{M}_{{\rm FC}}^{e}(d)^{\circ \circ }$ is closed.

Let $p_{1,0}\, :\, \mathcal{M}^{e}_{\textrm {FC}}(d)^{\circ \circ } \,\longrightarrow \,\mathcal{N}^e(d)^{\circ }$ be the restriction of the forgetful map $p_1$ in (4.5). Take a sufficiently small analytic open subset $U \, \subset \, \mathcal{N}^e(d)^{\circ }$ such that there is a holomorphic section

\begin{align*}s\,\colon \, U \,\longrightarrow \, p_{1,0}^{-1}(U)\, ,\end{align*}

over $U$ , of $p_{1,0}$ . Now, Lemma 4.4 says that

\begin{align*} \Theta ^e - (P_1^{-1})^* \Phi _U \,= \,p^*_{1,0} (s^* \Theta ^e) \end{align*}

on $p_{1,0}^{-1}(U)$ . This implies that

(4.9) \begin{align} d \Theta ^e\,=\, p^*_{1,0} d (s^* \Theta ^e)\, \end{align}

on $p_{1,0}^{-1}(U)$ , because the Liouville $2$ -form is $d$ -closed.

In view of (4.9), to prove the theorem it suffices to show the existence of a local holomorphic section $s\,\colon \, U \,\longrightarrow \, p^{-1}_{1,0}(U)$ of the map $p^{-1}_{1,0}$ such that $d(s^*\Theta ^e)\,=\, 0$ .

We shall construct a holomorphic section $s\,\colon \, U \,\longrightarrow \, p^{-1}_{1,0}(U)$ such that

\begin{align*}d(s^*\Theta ^e)\,=\, 0\, .\end{align*}

For that, first define a moduli space

\begin{align*} \mathcal{M}^{e}_{\textrm {FC}}(d)^{\circ \circ }_{0}\,\,:=\, \,\left \{ (E,\phi ,\nabla ) \ \middle | \ \begin{array}{l} E\ is\ a\ stable\ vector\ bundle\ of\ degree\ d,\ and\\ (E,\ \phi\ ,\nabla\ )\ is\ a\ framed\ \mathrm{GL}(r,\mathbb{C})-connection\ such\ that \\ \rm \mathrm{res}_{x_i}(\nabla\ )\,=\,0\ for\ i=1,\ldots\ ,n-1\ and\ \mathrm{res}_{x_n}(\ \nabla\ )=-\frac{d}{r}e \end{array} \right \}{\Big /}\Large {\sim }, \end{align*}

where $e$ is the identity matrix. There is the natural inclusion map

(4.10) \begin{align} \iota \,\,\colon \,\, \mathcal{M}^{e}_{\textrm {FC}}(d)^{\circ \circ }_{0}\,\,\hookrightarrow \,\, \mathcal{M}^{e}_{\textrm {FC}}(d)^{\circ \circ }\, , \end{align}

where $\mathcal{M}^{e}_{\textrm {FC}}(d)^{\circ \circ }$ is defined in (4.8). Also, define two moduli spaces

\begin{align*} \mathcal{M}(d)_{0}^{\circ \circ }\,\,:=\, \,\left \{ (E,\nabla ) \ \middle | \ \begin{array}{l} \rm E\ is\ a\ stable\ vector\ bundle\ of\ rank\ r\ and\ degree\ d,\ and\\ \rm \nabla\ \,\colon\ \,\ E\ \,\longrightarrow\ \,E\otimes\ K_X(D)\ is\ a\ connection\ such\ that\ \\ \rm \mathrm{res}_{x_i}(\ \nabla\ )\,=\,0\ for\ i\,=\,1,\ldots\ ,n-1\ and\ \mathrm{res}_{x_n}(\nabla\ )\,=\,-\frac{d}{r}e \end{array} \right \}{\Big /}\Large {\sim } \end{align*}

and

\begin{align*} \mathcal{N}(d)^{\circ }\,=\, \{ E \mid \rm E\ is\ a\ stable\ vector\ bundle\ of\ rank\ r\ and\ degree\ d \}/\sim . \end{align*}

There are the forgetful maps

(4.11) \begin{align} q_1 \,\colon \, \mathcal{M}(d)^{\circ \circ }_{0}\,\longrightarrow \,\mathcal{N}(d)^{\circ },\quad q_2 \,\colon \, \mathcal{M}^{e}_{\textrm {FC}}(d)^{\circ \circ }_{0}\,\longrightarrow \, \mathcal{M}(d)^{\circ \circ }_{0}\quad {\rm and}\quad\ p_2 \,\colon \, \mathcal{N}^e(d)^{\circ } \,\longrightarrow \,\mathcal{N}(d)^{\circ }\, , \end{align}

where $q_2$ and $p_2$ forget the framing while $q_1$ forgets the connection.

Take an analytic open subset $U_0\,\subset \, \mathcal{N}(d)^{\circ }$ . Assume that $U_0$ is small enough and that the image of $U$ under the forgetful map $p_2 \,\colon \, \mathcal{N}^e (d)^{\circ }\,\longrightarrow \, \mathcal{N}(d)^{\circ }$ is contained in $U_0$ , by shrinking sufficiently the analytic open subset $U$ . Take a holomorphic section

\begin{align*} \begin{aligned} s_0 \,\colon \,\, U_0 \,&\longrightarrow \, q_1^{-1}(U_0) \\ E \,&\longmapsto \, (E,\,\nabla (E)) \end{aligned} \end{align*}

of the forgetful map $q_1 \,\colon \, \mathcal{M}(d)^{\circ \circ }_{0}\,\longrightarrow \, \mathcal{N}(d)^{\circ }$ . Since $H_x\,=\,\{ e\}$ for all $x\in D$ , we may define a section $\widetilde {s}$ on $U$

\begin{align*} \begin{aligned} \widetilde {s} \,\colon \,\, U \,&\longrightarrow \, \mathcal{M}^{e}_{\textrm {FC}}(d)^{\circ \circ }_{0} \\ (E , \,\phi ) \,&\longmapsto \, (E, \,\phi ,\, \nabla (E)) \end{aligned} \end{align*}

using the section $s_0$ . Define the section $s$ on $U$ of $p_{1,0} \,\colon \, p_1^{-1}(U)\,\longrightarrow \, U$ by

\begin{align*}s \,=\, \iota \circ \widetilde {s}\, .\end{align*}

Now we shall prove that

(4.12) \begin{align} d (s^* \Theta ^e)\,=\,0 \end{align}

for such a section.

To prove (4.12), first recall that the moduli space $\mathcal{M}(d)^{\circ \circ }_{0}$ is equipped with a natural symplectic structure. We briefly describe this symplectic structure on $\mathcal{M}(d)^{\circ \circ }_{0}$ . The tangent space to $\mathcal{M}(d)^{\circ \circ }_{0}$ at any point $(E,\,\nabla )$ is isomorphic to the first hypercohomology $\mathbb{H}^1(\mathcal{C}^0_{\bullet })$ , where

(4.13) \begin{align} \mathcal{C}^0_{\bullet }\, \,\colon \,\, \mathcal{C}^0_0\,=\, \mathcal{E}nd (E) \, \xrightarrow {\ \nabla \ }\, \mathcal{C}^0_1\,=\,\mathcal{E}nd(E)\otimes K_X\, . \end{align}

Define a nondegenerate $2$ -form $\Theta _0$ on $\mathcal{M}(d)^{\circ \circ }_{0}$

\begin{align*} \Theta _0(E,\,\nabla ) \,\colon \, \mathbb{H}^1(\mathcal{C}^0_{\bullet }) \otimes \mathbb{H}^1(\mathcal{C}^0_{\bullet }) \,\longrightarrow \,\mathbb{C} \end{align*}

exactly as done in (4.4). This $2$ -form $\Theta _0$ is $d$ -closed, which is proved in [Reference GoldmanGol].

Secondly, we show that

(4.14) \begin{align} \iota ^* \Theta ^e \,=\, q_2^* \Theta _0\, , \end{align}

where $q_2$ and $\iota$ are the maps in (4.11) and (4.10) respectively. To prove (4.14), note that the tangent space of $\mathcal{M}^{e}_{\textrm {FC}}(d)^{\circ \circ }_{0}$ at $(E,\,\phi ,\,\nabla )$ is isomorphic to the first hypercohomology $\mathbb{H}^1(\mathcal{C}'_{\bullet })$ of the complex

(4.15) \begin{align} \mathcal{C}'_{\bullet } \,\colon \, \mathcal{C}'_0\,=\,\mathcal{E}nd (E) (-D) \,\xrightarrow {\ \nabla \ }\, \mathcal{C}'_1\,=\,\mathcal{E}nd (E) \otimes K_X\, . \end{align}

For $[(\{u_{\alpha \beta }\} ,\{ v_{\alpha }\})]\,\in \, \mathbb{H}^1(\mathcal{C}'_{\bullet })$ , we have that

\begin{align*} \iota _*[(\{ u_{\alpha \beta }\} , \,\{ v_{\alpha }\} ) ] \,=\,[ (\{ u_{\alpha \beta }\} , \, \{ v_{\alpha } \} ) ]\,\in \, \mathbb{H}^1(\mathcal{C}_{\bullet }) \end{align*}

and

\begin{align*} (q_2)_*[ (\{ u_{\alpha \beta } \} ,\, \{ v_{\alpha }\} ) ] \,=\,[ (\{ u_{\alpha \beta }\} , \,\{ v_{\alpha }\} ) ] \,\in \, \mathbb{H}^1(\mathcal{C}^0_{\bullet }). \end{align*}

Therefore, $\iota ^* \Theta ^e$ and $q_2^* \Theta _0$ have the following identical description:

\begin{align*} \begin{aligned} \mathbb{H}^1(\mathcal{C}'_{\bullet }) \otimes \mathbb{H}^1(\mathcal{C}'_{\bullet }) \, &\longrightarrow \, \mathbb{H}^2(\mathbb{K}) \,\cong \, \mathbb{C} \\ [(\{ u_{\alpha \beta } \} ,\,\{ v_{\alpha }\}) ] \otimes [(\{ u_{\alpha \beta }' \} ,\,\{ v_{\alpha }'\}) ]\, &\longmapsto \, [(\{ {\rm Tr} (u_{\alpha \beta }u'_{\beta \gamma }) \},\, - \{{\rm Tr}(u_{\alpha \beta }v'_{\beta })-{\rm Tr}(v_{\alpha }u'_{\alpha \beta }) \})]. \end{aligned} \end{align*}

This proves (4.14).

Thirdly, by the equality $\iota ^* \Theta ^e \,=\, q_2^* \Theta _0$ in (4.14), we have

\begin{align*} s^* \Theta ^e \,=\, \widetilde {s}^* (q_2^* \Theta _0)\, . \end{align*}

Since $\Theta _0$ is $d$ -closed, it follows that $d(s^* \Theta ^e)\,=\,0$ , proving (4.12).

Finally, from the combination of (4.12) and the equality $d \Theta ^e\,=\, p^*_{1,0} d (s^* \Theta ^e)$ (see (4.9)), it follows that $d \Theta ^e\,=\,0$ on $p^{-1}_{1,0}(U)$ . This implies that the $2$ -form $\Theta ^e$ is $d$ -closed on $\mathcal{M}_{\textrm {FC}}^{e}(d)^{\circ }$ . As noted before, this proves the theorem.

Proof. Consider the forgetful map $p_1\,\colon \, \mathcal{M}^{e}_{\textrm {FC}}(d)^{\circ } \,\longrightarrow \, \mathcal{N}^e(d)$ in (4.5), and take a sufficiently small analytic open subset $U\, \subset \, \mathcal{N}^e(d)$ . For a holomorphic section $s\,\colon \, U \,\longrightarrow \, p_1^{-1}(U)$ of $p_1$ such that

\begin{align*} d(s^*\Theta ^e|_{\mathcal{M}^{e}_{{\rm FC}}(d)^{\circ }})\,=\,0, \end{align*}

we have $d \:\! \Theta ^e|_{\mathcal{M}^{e}_{{\rm FC}}(d)^{\circ }}\,=\,0$ by the same argument as in the proof of Theorem 4.7. We will now construct such a section $s$ .

Let $\mathcal{N}_{{\rm par}} (d)$ be the moduli space of simple $\boldsymbol{x}$ -quasi-parabolic bundles of rank $r$ and degree $d$ . For each $x \,\in \, D$ , set the complex Lie proper subgroup $H_x$ to be the subgroup of ${\rm GL}(r,\mathbb{C})$ consisting of the upper triangular matrices. It may be mentioned that an $\boldsymbol{x}$ -quasi-parabolic bundle is the same as a framed bundle with respect to $\{ H_x\}_{x \in D}$ . For a framed bundle $(E,\,\phi )$ , we can associate a quasi-parabolic bundle $(E,\,\boldsymbol{l})$ whose filtration $l^{(i)}_*$ on $E|_{x_i}$ is induced by the framing $\phi _{x_i}$ of $E|_{x_i}$ for each $1\,\leqslant \, i\,\leqslant \, n$ . Setting

\begin{align*} {\mathcal N}^e(d)^{\circ }\,\,:=\,\, \left \{(E,\phi )\,\,\middle |\,\, \rm The\ quasi-parabolic\ bundle\ (E,\,\boldsymbol{l})\ induced\ by\ the\ framing\ \phi\ \ is\ simple \right \}, \end{align*}

there is a natural morphism

(4.16) \begin{align} q_B\ \colon \ {\mathcal N}^e(d)^{\circ }\ \longrightarrow \ {\mathcal N}^{{\rm par}}(d)^{\circ }. \end{align}

Take an element

\begin{align*}\boldsymbol{\nu }\,=\,(\nu _{j}^{(i)} )^{1\leqslant i \leqslant n}_{0\leqslant j \leqslant r-1} \,\in \, \mathbb{C}^{nr}\end{align*}

such that $\sum _{i,j} \nu _{j}^{(i)} \,=\, -d$ . Let $\mathcal{M}_{{\rm PC}} (\boldsymbol{\nu })^{\circ }$ be the moduli space defined by

\begin{align*} \mathcal{M}_{{\rm PC}} (\boldsymbol{\nu })^{\circ } \,\,:=\,\,\left \{ (E,\,\nabla ,\, \boldsymbol{l})\in \mathcal{M}_{{\rm PC}} (\boldsymbol{\nu }) \, \middle | \, \begin{array}{l} \rm (E,\,\ \boldsymbol{l})\ is\ a\ simple\ \boldsymbol{x}-quasi-parabolic\ bundle \end{array} \right \}{\Big /}\Large {\sim }. \end{align*}

Define the locally closed subspace $\mathcal{M}_{{\rm FC}}^{e} (\boldsymbol{\nu })^{\circ }$ of $\mathcal{M}_{{\rm FC}}^{e} (d)^{\circ }$ by

\begin{align*} &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \mathcal{M}_{{\rm FC}}^{e} (\boldsymbol{\nu})^{\circ } \, \,:=\, \\ &\left \{ (E,\, \phi ,\,\nabla ) \,\in \, \mathcal{M}_{{\rm FC}}^{e} (d)^{\circ } \ \middle | \ \begin{array}{l} \rm the\ framed\ bundle\ (E,\phi\ )\ belongs\ to\ {\mathcal N}^e(d)^{\circ\ } \\ \rm and\ (E,\,\boldsymbol{l})\,:=\,q_B(E,\,\phi)\ satisfies \\ \rm (\mathrm{res}_{x}(\nabla)-\nu^{(i)}_{j}\ \mathrm{Id}_{E|_{x_i}})\ (l^{(i)}_{j})\,\subset\,l_{j+1}^{(i)}\ for\ 0\leqslant\ j\leqslant\ r-1 \end{array} \right \}{\Big /}\Large {\sim }. \end{align*}

In the above definition we have $\phi \,=\,\{\phi _x\}_{x\in D}$ , where $\phi _x\,\colon \, \mathcal{O}_X^{\oplus r}|_x \,\longrightarrow \, E\vert _x$ are isomorphisms defining a framing of $E$ over $D$ . Since a framing defines a parabolic structure, there is a natural map

(4.17) \begin{align} q_1^{{\rm par}} \ \colon \ \mathcal{M}_{{\rm FC}}^{e} (\boldsymbol{\nu })^{\circ } \ \longrightarrow \ \mathcal{M}_{{\rm PC}} (\boldsymbol{\nu })^{\circ }\, . \end{align}

Notice that $\mathcal{M}_{{\rm PC}} (\boldsymbol{\nu })^{\circ }$ is non-empty by virtue of the assumption in the proposition, and so is $\mathcal{M}_{{\rm FC}}^{e} (\boldsymbol{\nu })^{\circ }$ . Consider the complex

\begin{align*} \mathcal{D}_{\bullet }^{{\rm par}} \ \colon \ {\rm ad}_{\phi } (E_G) \ \xrightarrow {\,\ [\, \nabla ,\, \cdot \, ] \,\, \ } \ {\rm ad}^n_{\phi } (E_G) \otimes K_X (D)\, \end{align*}

for $\{ H_x\}_{x \in D}$ . Here, ${\rm ad}_{\phi } (E_G)$ and ${\rm ad}^n_{\phi } (E_G) \otimes K_X (D)$ are defined as in (2.16). The tangent space of $\mathcal{M}_{{\rm PC}} (\boldsymbol{\nu })^{\circ }$ at $(E,\,\nabla , \,\boldsymbol{l})$ is $\mathbb{H}^1(\mathcal{D}^{{\rm par}}_{\bullet })$ . There is also a natural morphism

\begin{align*} p_0^{{\rm par}}\ \colon \ \mathcal{M}_{{\rm PC}} (\boldsymbol{\nu })^{\circ } \ \longrightarrow \ {\mathcal N}_{{\rm par}}(d)^{\circ } \end{align*}

which is étale locally an affine space bundle whose fiber is isomorphic to $H^0(X,\,{\rm ad}^n_{\phi }(E_G)\otimes K_X(D))$ . So there is a non-empty analytic open subset $U\,\subset \, {\mathcal N}^{{\rm par}}(d)^{\circ }$ with a local section $s^{{\rm par}}\,\colon \, U \,\longrightarrow \, (p_0^{{\rm par}})^{-1}(U)$ of $p_0^{{\rm par}}$ . Consider the commutative diagram

whose left square is Cartesian. The local section $s^{{\rm par}}$ of $p_0^{{\rm par}}$ produces a local section

\begin{align*}s_1\,\colon \, q_B^{-1}(U)\,\longrightarrow \, (q_B\circ p_1)^{-1}(U)\end{align*}

of $p_1|_{\mathcal{M}_{{\rm FC}}^{e} (\boldsymbol{\nu })^{\circ }}$ . Let $\Theta _{{\rm par}}$ be the symplectic structure on $\mathcal{M}_{{\rm PC}}^{\boldsymbol{\alpha }} (\boldsymbol{\nu })$ constructed in [Reference InabaIna]. This symplectic form $\Theta _{{\rm par}}$ is described as follows:

(4.18) \begin{align} \begin{aligned} \Theta _{{\rm par}} \,\colon \, \mathbb{H}^1 (\mathcal{D}^{{\rm par}}_{\bullet }) \otimes \mathbb{H}^1 (\mathcal{D}^{{\rm par}}_{\bullet }) \,&\longrightarrow \, \mathbb{H}^2(\mathbb{K}) \,\cong \, \mathbb{C} \\ [(\{u_{\alpha \beta } \} ,\, \{ v_{\alpha } \})]\otimes [(\{u'_{\alpha \beta } \} ,\, \{ v'_{\alpha } \})] \,&\longmapsto \, [(\{ {\rm Tr} (u_{\alpha \beta }u'_{\beta \gamma }) \},\, - \{{\rm Tr}(u_{\alpha \beta }v'_{\beta })-{\rm Tr}(v_{\alpha }u'_{\alpha \beta }) \})] \end{aligned} \end{align}

in terms of the Čech cohomology constructed using an affine open covering $\{ U_{\alpha }\}$ (see [Reference InabaIna, Proposition 7.2]). The symplectic form $\Theta _{{\rm par}}$ is $d$ -closed [Reference InabaIna, Proposition 7.3]. Since the images of $\Theta ^e$ and $\Theta _{{\rm par}}$ in $\mathbb{H}^2(\mathbb{K})$ have the same description in terms of Čech cohomology (see (4.4) and (4.18)), it follows that

\begin{align*} (q_1^{{\rm par}})^*\Theta _{{\rm par}} |_{\mathcal{M}_{{\rm PC}} (\boldsymbol{\nu })^{\circ }} \, =\, \iota ^* \Theta ^e|_{\mathcal{M}_{{\rm FC}}^{e} (d)^{\circ } } \, . \end{align*}

Since $\Theta ^{{\rm par}}$ is $d$ -closed, so is $\iota ^* \Theta ^e|_{\mathcal{M}_{{\rm FC}}^{e} (d)^{\circ } }$ . Set $s\,\,:=\,\,\iota \circ s_1\,\colon \, U\,\longrightarrow \, p_1^{-1}(U)$ , which is a local section of $U_1$ . Then the pullback $s^*(\Theta ^e)\,=\, s_1^*\iota ^*\Theta ^e|_{\mathcal{M}_{{\rm FC}}^{e} (d)^{\circ } }$ is $d$ -closed and so is $\Theta ^e|_{\mathcal{M}_{{\rm FC}}^{e} (d)^{\circ } }$ by the first remark in this proof.

Proof. Recall that the 2-form $\Theta ^e$ is defined in (4.4). Let $\xi _{\Theta ^H} \,\colon \, \mathbb{H}^1({\mathcal C}_{\bullet })\,\longrightarrow \, \mathbb{H}^1({\mathcal C}_{\bullet })^*$ be the homomorphism induced by $\Theta ^H$ . Set $\mathcal{C}_0 \,\,:=\, \,{\mathcal E}nd(E)(-D)$ and $\mathcal{C}_1 \,\,:=\, \,{\mathcal E}nd(E)\otimes K_X(D)$ . For the above defined map $\xi _{\Theta ^H}$ , we have the following commutative diagram whose rows are exact:

where $b_1,\,b_2,\,b_3,\,b_4$ are Serre duality isomorphisms. So, from the five lemma, it follows that $\xi _{\Theta ^H}$ is an isomorphism. In other words, the $2$ -form $\Theta ^H$ is nondegenerate.

Proof. Take a reduced effective divisor $\widetilde {D}$ such that $\widetilde {D} \,\supset \, D$ and $H^0(X, \,\mathcal{E}nd (E_0) (-\widetilde {D}) ) \,=\,0$ . Moreover, take an isomorphism $\widetilde {\phi }_0 \,\colon \, \mathcal{O}_{\widetilde {D}}^{\oplus r} \,\longrightarrow \, E_0|_{\widetilde {D}}$ such that $\widetilde {\phi }_0 |_D \,=\, \phi _0$ . We will show that $(E_0,\,\widetilde {\phi }_0)$ is simple. For that, let $\textbf {g}$ be an automorphism of $(E_0,\,\widetilde {\phi }_0)$ , that is, $\textbf {g}$ is an automorphism of $E_0$ such that the diagram

is commutative. So the restriction $\textbf {g}|_{\widetilde {D}}$ is the identity map. Therefore, we have

\begin{align*}\textbf {g}- {\rm Id}_{E_0} \,\in \, H^0(X,\, \mathcal{E}nd (E_0) (-\widetilde {D}) ). \end{align*}

Since $H^0(X, \,\mathcal{E}nd (E_0) (-\widetilde {D})) =\,0$ , it follows that $\textbf {g}\,=\, {\rm Id}_{E_0}$ . In other words, $(E_0,\,\widetilde {\phi }_0)$ is simple.