2.1 Higgs bundles

As in the Introduction, throughout this paper $\Sigma$ will denote a closed Riemann surface of genus $g\geq 2$ , with structure sheaf $\mathcal{O}=\mathcal{O}_\Sigma$ and canonical bundle $K=K_\Sigma$ . Let $E\to \Sigma$ be a complex vector bundle. A Higgs bundle consists of a pair $(\mathcal{E},\varphi )$ , where $\mathcal{E}$ is a holomorphic bundle structure on $E$ and where $\varphi \in H^0 ({\rm End}(\mathcal{E})\otimes K)$ . If ${\rm rank}(E)=1$ , then a Higgs field is just an abelian differential $\omega$ . The pair $(\mathcal{E},\varphi )$ is called an ${\rm SL}(2,\mathbb{C})$ Higgs bundle if ${\rm rank}(E)=2$ , $\det (\mathcal{E})$ has a fixed isomorphism with the trivial bundle and if ${\rm Tr}(\varphi )=0$ . In this article we will focus mainly on ${\rm SL}(2,\mathbb{C})$ Higgs bundles, but the rank $1$ case will also be important.

Let $(\mathcal{E},\varphi )$ be an ${\rm SL}(2,\mathbb{C})$ Higgs bundle. A (proper) Higgs subbundle of $(\mathcal{E},\varphi )$ is a holomorphic line bundle $\mathcal{L}\subset \mathcal{E}$ that is $\varphi$ -invariant; that is, $\varphi :\mathcal{L}\to \mathcal{L}\otimes K$ . In this case the restriction $\varphi _{\mathcal{L}}:=\varphi \bigr |_{\mathcal L}$ makes $(\mathcal{L},\varphi _{\mathcal{L}})$ a rank $1$ Higgs bundle. Moreover, $\varphi$ induces a Higgs bundle structure on the quotient $\mathcal{E}/\mathcal{L}$ . We say that $(\mathcal{E},\varphi )$ is stable (resp. semistable) if for all Higgs subbundles $\mathcal{L}$ , $\deg \mathcal{L}\lt 0$ (resp. $\deg \mathcal{L}\leq 0$ ). We say that $(\mathcal{E},\varphi )$ is polystable if $(\mathcal{E},\varphi )\simeq (\mathcal{L},\omega )\oplus (\mathcal{L}^{-1},-\omega )$ , where $\mathcal{L}$ is a degree-zero holomorphic line bundle and $\omega \in H^0(K)$ .

If $(\mathcal{E},\varphi )$ is strictly semistable (that is, semistable but not polystable), the Seshadri filtration [Reference SeshadriSes67] gives the unique Higgs subbundle $0\subset (\mathcal{L},\omega )\subset (\mathcal{E},\varphi )$ , with $\deg (\mathcal{L})={1 \over 2}\deg (\mathcal{E})=0$ . If we write $(\mathcal{L}^{\prime},\omega ^{\prime}):=(\mathcal{E},\varphi )/(\mathcal{L},\omega )$ , then we have $\omega ^{\prime}=-\omega$ and $\mathcal{L}^{\prime}=\mathcal{L}^{-1}$ . The associated graded bundle ${\rm Gr}(\mathcal{E},\varphi )=(\mathcal{L},\omega )\oplus (\mathcal{L}^{-1},-\omega )$ of this filtration is a polystable ${\rm SL}(2,\mathbb{C})$ Higgs bundle. We say that $(\mathcal{E},\varphi )$ is S-equivalent to ${\rm Gr}(\mathcal{E},\varphi )$ .

Holomorphic bundles $\mathcal{E}$ , with underlying $C^\infty$ bundle $E$ , are in 1-to-1 correspondence, with $\bar \partial$ -operators $\bar \partial _E: \Omega^0(E)\to \Omega ^{0,1}(E)$ . We use the notation $\mathcal{E}:=(E,\bar{\partial }_E)$ . Let $\mathcal{C}$ denote the space of pairs $ (\bar \partial _E,\varphi )$ , $\bar \partial _E\varphi=0$ . Let $\mathcal{C}^s$ and $\mathcal{C}^{ss}$ denote the subspaces of $\mathcal{C}$ where the Higgs bundles are stable (resp. semistable). The complex gauge transformation group $\mathcal{G}_{\mathbb{C}}:={\rm Aut}(E)$ has a right-hand action on $\mathcal{C}$ by defining for $g\in \mathcal{G}_{\mathbb{C}}$ , $(\bar{\partial }_E,\varphi )g:=(g^{-1}\circ \bar{\partial }\circ g,g^{-1}\circ \varphi \circ g)$ .

There is a quasiprojective scheme $\mathcal{M}_{{\rm Dol}}$ whose closed points are in 1-to-1 correspondence with isomorphism classes of polystable ${\rm SL}(2,\mathbb{C})$ Higgs bundles constructed via (finite dimensional) Geometric Invariant Theory (see [Reference NitsureNit91, Reference SimpsonSim94]). In [Reference FanFan22b] it was shown that the infinite dimensional quotient $\mathcal{C}^{{\rm ss}}//\mathcal{G}_{\mathbb{C}}$ , where the double slash indicates that S-equivalent orbits are identified, admits the structure of a complex analytic space that is biholomorphic to the analytification $\mathcal{M}_{{\rm Dol}}^{\textrm{an}}$ of $\mathcal{M}_{{\rm Dol}}$ . Henceforth, we shall work in the complex analytic category; identify the algebro–geometric and gauge theoretic moduli spaces as complex analytic spaces; and simply denote them both by $\mathcal{M}_{{\rm Dol}}$ . We note that the set of stable Higgs bundles modulo gauge transformations, $\mathcal{M}_{{\rm Dol}}^s:=\mathcal{C}^{{\rm s}}/\mathcal{G}_{\mathbb{C}}$ , is a geometric quotient and an open subset of $\mathcal{M}_{{\rm Dol}}$ .

Finally, notice that the pair $(\mathcal{E},\varphi )$ is stable (resp. semistable) if and only if the same is true for $(\mathcal{E},\lambda \varphi )$ , $\lambda \in \mathbb{C}^\ast$ . Hence, $\mathcal{M}_{{\rm Dol}}$ admits an action of $\mathbb{C}^\ast$ that preserves $\mathcal{M}_{{\rm Dol}}^{s}$ . Although $\mathcal{M}_{{\rm Dol}}$ is only quasiprojective, the $\mathbb{C}^\ast$ action satisfies the Białynicki–Birula property:

Theorem 2.1. For any $[(\mathcal{E},\varphi )]\in \mathcal{M}_{{\rm Dol}}$ ,

\begin{align*} \lim _{\lambda \to 0} \lambda \cdot [(\mathcal {E},\varphi )]:=\lim _{\lambda \to 0}[(\mathcal {E},\lambda \varphi )] \end{align*}

exists in $\mathcal{M}_{{\rm Dol}}$ .

2.2 Spectral curves and the Hitchin fibration

The Hitchin map is defined as

\begin{align*} \mathcal {H}: \mathcal {M}_{\mathrm {Dol}} \longrightarrow H^0(K^2)\, \ [(\mathcal {E},\varphi )] \mapsto \det (\varphi )\, \end{align*}

where $H^0(K^2)=:\mathcal{B}$ is known as the Hitchin base. Hitchin [Reference HitchinHit87a, Reference HitchinHit87b] showed that $\mathcal{H}$ is a proper map and a fibration by abelian varieties over the open cone $\mathcal{B}^{\textrm{reg}}\subset \mathcal{B}$ consisting of nonzero quadratic differentials with only simple zeros. The discriminant locus $\mathcal{B}^{\textrm{sing}}:=\mathcal{B}\setminus \mathcal{B}^{\textrm{reg}}$ consists of quadratic differentials that either are identically zero or have at least one zero with multiplicity. For $q\in \mathcal{B}$ , let $\mathcal{M}_q:=\mathcal{H}^{-1}(q)$ . The ‘most singular fibre’ $\mathcal{M}_0$ is called the nilpotent cone.

Consider the total space ${\rm Tot}(K)$ of $K$ , along with its projection $\pi : {\rm Tot}(K) \to \Sigma$ . The pull-back bundle $\pi ^* K$ has a tautological section, which we denote by $\lambda \in H^0({\rm Tot}(K),\pi ^\ast K)$ . Given any $q\neq 0 \in H^0(K^2)$ , the spectral curve $S_q$ associated with $q$ is the zero scheme of the section $\lambda^2 - \pi ^\ast q \in H^0({\rm Tot}(K),\pi ^* K)$ . This is a reduced, but possibly reducible, projective algebraic curve. The restriction of $\pi$ to $S_q$ , also denoted by $\pi : S_q \to \Sigma$ , is a double covering branched along the zeros of $q$ .

The spectral curve $S_q$ is smooth if and only if $q$ has only simple zeros. It is reducible if and only if $q = -\omega \otimes \omega$ for some $\omega \in H^0(K)$ . In the latter case, we call such quadratic differentials reducible, and we otherwise refer to them as irreducible. There is a noteworthy observation regarding irreducible spectral curves.

Proof. Suppose $\mathcal{L}\subset \mathcal{E}$ is $\varphi$ -invariant, and let $\varphi _{\mathcal L}$ be the restriction. Then

\begin{align*} \det \varphi =-\frac {1}{2}\mathrm {Tr}(\varphi^2)=-(\varphi _{\mathcal L})^2\, \end{align*}

which contradicts the assumption.

Let us emphasise that being reducible is not the same as having only even zeros. To see this, suppose that ${\rm Div}(q)=2D$ . Then $K\simeq \mathcal{O}(D)\otimes \mathcal{I}$ , where $\mathcal{I}$ is a 2-torsion point in the Jacobian. The spectral curve $S_q$ is reducible if and only if $\mathcal{I}$ is trivial.

2.3 Rank 1 torsion-free sheaves and the BNR correspondence

In this subsection, we provide some background on rank 1 torsion-free sheaf theory over spectral curves in the context of the Hitchin and BNR correspondence, as developed in [Reference HitchinHit87b, Reference Narasimhan and RamananBNR89].

Let $S$ be a reduced and irreducible complex projective curve and $\mathcal{O}_S$ its structure sheaf. The moduli space of invertible sheaves on $S$ is denoted by ${\rm Pic}(S)$ . If $\mathcal F$ is a coherent analytic sheaf on $S$ , we can define its cohomology groups $H^i(S,\mathcal{F})$ . Since $\dim S=1$ , $H^i(S,\mathcal{F})=0$ for $i\geq 2$ . The Euler characteristic is defined as $\chi (\mathcal{F})=\dim H^0(S,\mathcal{F})-\dim H^1(S,\mathcal{F})$ . The degree of a torsion-free sheaf $\mathcal{F}$ is given by $\deg (\mathcal{F})=\chi (\mathcal{F})-{\rm rank}(\mathcal{F})\chi (\mathcal{O}_S)$ . If $\mathcal{F}$ is locally free, then $\deg (\mathcal{F})$ coincides with the degree of the invertible sheaf $\det (\mathcal{F})$ . We let ${\rm Pic}^d(S)\subset {\rm Pic}(S)$ denote the degree $d$ component.

Let $\overline{{\rm Pic}}^d(S)$ be the moduli space of degree $d$ , rank 1 torsion-free sheaves on $S$ , and let $\overline{{\rm Pic}}(S)=\prod _{d\in \mathbb{Z}}\overline{{\rm Pic}}^d(S)$ [D’S79]. Then, $\overline{{\rm Pic}}^d(S)$ is an irreducible projective scheme containing ${\rm Pic}^d(S)$ as an open subscheme. When $S$ is smooth, we have $\overline{{\rm Pic}}^d(S)={\rm Pic}^d(S)$ . The relationship to Higgs bundles is given by the following:

This correspondence gives the very useful exact sequence

(3) \begin{align} 0\to \mathcal{L}\otimes \mathcal{I}\to \pi ^{\ast }\mathcal{E}\xrightarrow{\pi ^{\ast }\varphi -\lambda }\pi ^{\ast }\mathcal{E}\otimes \pi ^{\ast }K\to \mathcal{L}\otimes \pi ^{\ast }K\to 0\, \end{align}

for some ideal sheaf $\mathcal{I}$ . If $S$ is smooth, then $\mathcal{I}=\mathcal{O}_S(-\Delta )$ , where $\Delta$ is the ramification divisor. The sequence (3) will be used in Section 6.

Let $q$ be a quadratic differential with only simple zeros, and to simplify notation, write $S=S_q$ . Let $\Lambda :={\rm Div}(\lambda )$ be the ramification divisor of the map $\pi :S\to \Sigma$ . By the Riemann–Hurwitz formula, the genus of $S$ is $g(S)=4g-3$ , where $g$ is the genus of $\Sigma$ . Furthermore, for any $\mathcal{L}\in {\rm Pic}(S)$ , Riemann–Roch gives $\deg (\pi _{\ast }\mathcal{L})=\deg (\mathcal{L})-(2g-2).$ The ${\rm SL}(2,\mathbb{C})$ Higgs bundles are characterized by

(4) \begin{align} \mathcal{T}:=\{\mathcal{L}\in {\rm Pic}^{2g-2}(S)\mid \det (\pi _{\ast }\mathcal{L})=\mathcal{O}_\Sigma \}. \end{align}

By the Hitchin–BNR correspondence (Theorem 2.3), the map $\chi _{{\rm BNR}}:\mathcal{T}\to \mathcal{M}_q$ is a bijection.

The branched double cover $\pi :S\to \Sigma$ is given by an involution $\sigma :S\to S$ . We have the norm map ${\rm Nm}_{S/\Sigma }:{\rm Jac}(S)\to {\rm Jac}(\Sigma )$ , where ${\rm Jac}(S)$ is the connected component of the trivial line bundle in ${\rm Pic}(S)$ and ${\rm Nm}_{S/\Sigma }(\mathcal{O}_{S}(D)):=\mathcal{O}_{\Sigma }(\pi (D))$ . The Prym variety is defined as

\begin{align*}\mathrm {Prym}(S/\Sigma ):=\ker (\mathrm {Nm}_{S/\Sigma })=\{\mathcal {L}\in \mathrm {Pic}(S)\mid \mathcal {L}\otimes \sigma ^{\ast }\mathcal {L}=\mathcal {O}_S\}.\end{align*}

Also, we have $\det (\pi _{\ast }\mathcal{L})\cong {\rm Nm}_{S/\Sigma }(\mathcal{L})\otimes K^{-1}$ . Thus, $\mathcal{T}$ can be expressed as

\begin{align*}\mathcal {T}=\{\mathcal {L}\in \mathrm {Pic}^{2g-2}(S)\mid \mathrm {Nm}_{S/\Sigma }(\mathcal {L})\cong K\}.\end{align*}

Hence, $\mathcal{T}$ is a torsor over ${\rm Prym}(S/\Sigma )$ . Explicitly, by choosing $\mathcal{L}_0\in \mathcal{T}$ , we obtain an isomorphism $\mathcal{T}\xrightarrow{\sim } {\rm Prym}(S/\Sigma )$ given by $\mathcal{L}\to \mathcal{L}\otimes \mathcal{L}_0^{-1}$ .

To summarize, we have the following:

If $q\neq 0$ is irreducible but nongeneric, the spectral curve $S$ is singular and irreducible. We may still define the set $\overline{\mathcal{T}}\subset \overline{{\rm Pic}}^{2g-2}(S)$ as follows:

\begin{align*} \overline {\mathcal {T}}:=\left\{\mathcal {L}\in \overline {\mathrm {Pic}}^{2g-2}(S)\mid \det (\pi _{\ast }\mathcal {L})\cong \mathcal {O}_\Sigma \right\}\ .\end{align*}

We also set $\mathcal{T}:=\overline{\mathcal{T}}\cap {\rm Pic}^{2g-2}$ . Then $\overline{\mathcal{T}}$ is the natural compactification of $\mathcal{T}$ induced by the inclusion ${\rm Pic}^{2g-2}(S)\subset \overline{{\rm Pic}}(S)$ . The BNR correspondence, as stated in Theorem 2.3, implies that $\chi _{{\rm BNR}}:\overline{\mathcal{T}}\to \mathcal{M}_q$ is an isomorphism.

2.4 The Hitchin moduli space and the nonabelian Hodge correspondence

We now recall the well-known nonabelian Hodge correspondence (NAH), which relates the space of flat ${\rm SL}(2,\mathbb{C})$ connections, Higgs bundles and solutions to the Hitchin equations. This result was developed in the work of Hitchin [Reference HitchinHit87a], Simpson [Reference SimpsonSim88], Corlette [Reference CorletteCor88] and Donaldson [Reference DonaldsonDon87].

As above, let $E$ be a trivial(ised), smooth, rank 2 vector bundle over the Riemann surface $\Sigma$ , and let $H_0$ be a fixed Hermitian metric on $E$ . We denote by $\mathfrak{sl}(E)$ (resp. $\mathfrak{su}(E)$ ) the bundle of traceless (resp. traceless skew-Hermitian) endomorphisms of $E$ . Let $A$ be a unitary (with respect to $H_0$ ) connection on $E$ that induces the trivial connection on $\det E$ , and let $\phi \in \Omega^1(i\mathfrak{su}(E))$ . We will sometimes also refer to $\phi$ as a Higgs field. The Hitchin equations for the pair $(A,\phi )$ are given by

(5) \begin{align} F_{A} + \phi \wedge \phi=0\quad d_A \phi = d_A^\ast \phi=0. \end{align}

If we split the Higgs field into type $\phi = \varphi + \varphi ^{\dagger }$ , with $\varphi \in \Omega ^{1,0}(\mathfrak{sl}(E))$ , then (5) is equivalent to

(6) \begin{align} F_{A} + [\varphi,\varphi ^{\dagger }]=0\quad \bar{\partial }_A \varphi=0. \end{align}

Notice that $(\bar \partial _E,\varphi )$ then defines an ${\rm SL}(2,\mathbb{C})$ Higgs bundle. The Hitchin moduli space, denoted by $\mathcal{M}_{{\rm Hit}}$ , is the moduli space of solutions to the Hitchin equations, which is given by

\begin{align*}\mathcal {M}_{\mathrm {Hit}} := \{(A,\phi ) \mid (A,\phi )\;\mathrm {satisfies}\;(5)\}/\mathcal {G},\end{align*}

where $\mathcal{G}$ is the gauge group of unitary automorphisms of $E$ . Recall that a flat connection $\mathcal{D}$ is called completely reducible if and only if it is a direct sum of irreducible flat connections. The NAH can be summarized as follows:

The nonabelian Hodge correspondence gives the Kobayashi–Hitchin homeomorphism (1), which, when restricted to the stable locus, is a diffeomorphism onto irreducible solutions of (5).

Finally, we note that there is an action of $S^1$ on $\mathcal{M}_{{\rm Hit}}$ defined by $(A,\phi ) \to (A,e^{i\theta }\cdot \phi )$ , where $e^{i\theta }\cdot \phi =e^{i\theta }\varphi +e^{-i\theta }\varphi ^\dagger$ . With respect to this and the $S^1\subset \mathbb{C}^\ast$ action on $\mathcal{M}_{{\rm Dol}}$ , the map $\Xi$ is $S^1$ -equivariant.

3.1 Filtered line bundles

Let $Z$ be a finite collection of distinct points on a closed Riemann surface $\Sigma$ , and let $\Sigma ^{\prime} = \Sigma \setminus Z$ . Viewing $\Sigma$ as a projective algebraic curve, an algebraic line bundle $L$ over the affine curve $\Sigma ^{\prime}$ is a line bundle defined by regular transition functions on Zariski open sets over $\Sigma ^{\prime}$ . The sheaf of sections of $L$ can be extended in infinitely many different ways over $Z$ to obtain (invertible) coherent analytic sheaves on $\Sigma$ . The sections of $L$ are then realised as meromorphic sections of any such extensions that are regular on $\Sigma ^{\prime}$ .

A filtered line bundle $\mathcal{F}_{\ast }(L)$ is an algebraic line bundle $L\to \Sigma ^{\prime}$ , along with a collection $\{L_{\alpha }\}_{\alpha \in \mathbb R}$ of coherent extensions across the punctures $ Z$ , such that $L_{\alpha } \subset L_{\beta }$ for $\alpha \geq \beta$ for fixed, sufficiently small $\epsilon$ , $L_{\alpha -\epsilon }=L_{\alpha }$ and $L_{\alpha } = L_{\alpha+1} \otimes \mathcal{O}_\Sigma (Z)$ . Let ${\rm Gr}_{\alpha }= L_{\alpha }/L_{\alpha +\epsilon }$ denote the quotient (torsion) sheaf. A value $\alpha$ where ${\rm Gr}_{\alpha } \neq 0$ is called a jump. Since we are considering line bundles, for each $p$ in the support of ${\rm Gr}_{\alpha _p}$ , there is exactly one jump $\alpha _p$ in the interval $[0,1)$ . The collection of jumps $\alpha _p$ , $p\in Z$ , fully determines the filtered bundle structure. If we denote by $\mathcal{L}:=L_{0}$ , the degree of a filtered line bundle is defined as

\begin{align*}\deg (\mathcal {F}_{\ast }(L)) := \deg (\mathcal {L}) + \sum _{p\in Z}\alpha _p .\end{align*}

Alternatively, a weighted line bundle is a pair $(\mathcal{L},\chi )$ for which $\mathcal{L}\to \Sigma$ is a holomorphic line bundle and $\chi :Z\to \mathbb{R}$ is a weight function. The degree of a weighted bundle is defined as

\begin{align*}\deg (\mathcal {L},\chi ) := \deg (\mathcal {L}) + \sum _{p\in Z}\chi _p.\end{align*}

The notions of filtered and weighted line bundles are nearly equivalent: Namely, given a filtered line bundle $\mathcal{F}_{\ast }(L)$ , we define $\mathcal{L}:=L_{0}$ and $\chi _p=\alpha _p$ . Conversely, given a weighted line bundle $(\mathcal{L},\chi )$ , we let $\alpha _p=\chi _p+n_p$ , where $n_p\in \mathbb{Z}$ is the unique integer and $0\leq \chi _p+n_p\lt 1$ . A filtered bundle $\mathcal{F}_{\ast }(L)$ , $L:=\mathcal{L}\bigr |_{\Sigma ^{\prime}}$ , is then determined by setting $L_{0} = \mathcal{L}(-\sum _{p\in Z} n_p p)$ with jumps $\alpha _p$ . Clearly, $\deg (\mathcal{F}(L))=\deg (\mathcal{L},\chi )$ . We shall use the notation $\mathcal{F}_{\ast }(\mathcal{L},\chi )$ for the filtered bundle associated to a weighted bundle $(\mathcal{L},\chi )$ in this way.

Different weighted bundles can give rise to the same filtered bundle. The following is a fact that will be frequently used in this article. If $D=\sum _{x\in Z}d_xx$ is a divisor supported on $Z$ , let

\begin{align*} \chi _D(x) & :=\begin {cases}d_x & x\in Z\;\\ 0\, & x\in \Sigma \setminus Z. \end {cases} \end{align*}

Then for any weighted bundle $(\mathcal{L},\chi )$ , we have $\mathcal{F}_\ast (\mathcal{L}(D),\chi -\chi _{D})=\mathcal{F}_\ast (\mathcal{L},\chi )$ .

Let $(\mathcal{L}_1,\chi _1)$ and $(\mathcal{L}_2,\chi _2)$ be two weighted lines bundles. We define the tensor product

\begin{align*}(\mathcal {L}_1,\chi _1)\otimes (\mathcal {L}_2,\chi _2):=(\mathcal {L}_1\otimes \mathcal {L}_2,\chi _1+\chi _2)\ .\end{align*}

Then the degree is additive on tensor products. For filtered bundles, we define

(7) \begin{align} \mathcal{F}_\ast (\mathcal{L}_1,\chi _1)\otimes \mathcal{F}_\ast (\mathcal{L}_2,\chi _2):=\mathcal{F}_\ast (\mathcal{L}_1\otimes \mathcal{L}_2,\chi _1+\chi _2).\end{align}

The degree is again additive for the tensor product of filtered bundles. This agrees with the usual definition of tensor product for parabolic bundles.

3.2 Harmonic metrics for weighted line bundles

The metric obtained above is called the harmonic metric. For a weighted bundle $(\mathcal{L},\chi )$ , the holomorphic bundle $\mathcal{L}$ and the harmonic metric $h$ define a filtration as follows: Given $\alpha$ , define the sheaf

\begin{align*}L_{\alpha }(U):=\{s\in H^0(U,\mathcal {L}(\ast Z)) \mid |s|_h= O(r^{\alpha -\epsilon })\text { for all $\epsilon \gt 0$}\}\ \end{align*}

and any open set $U\subset \Sigma$ . Here, $r$ denotes the distance to $Z$ in any smooth, conformal metric on $\Sigma$ . It is straightforward to check that this defines a filtered bundle that matches $\mathcal{F}_\ast (\mathcal{L},\chi )$ under the correspondence given in the previous section.

Even though the harmonic metric is well-defined only up to a constant, the Chern connection $A=(\mathcal{L},h)$ is independent of this choice. The $(1,0)$ part of $A$ , denoted $\nabla _h$ , then defines logarithmic the connections $\nabla _h:L_{\alpha }\to L_{\alpha }\otimes K(Z).$

3.3 Convergence of weighted line bundles

In this subsection, we consider the convergence of weighted line bundles. The main result we prove here is a consequence of [Reference Mochizuki and SzabóMS23, Theorem 1.8]. For the reader’s convenience, we present a short proof in our situation.

Let $(\Sigma _0,g_0)$ be a metrised Riemann surface (that is, a Riemann surface $\Sigma _0$ with conformal metric $g_0$ ). We view $\Sigma _0$ as given by an underlying surface $C$ with almost complex structure $J_0$ . Consider a neighbourhood $U_1$ of $J_0$ in the moduli space of holomorphic structures and a neighbourhood $U_2$ of $g_0$ in the space of smooth metrics. We denote the product of these neighbourhoods by $U=U_1\times U_2$ . We can define the fibre bundle ${\rm Pic}_U \to U$ , where each fibre is the Picard group defined by the holomorphic structure. Let $(\Sigma _t=(C,J_t),g_t)$ be a family of metrised Riemann surfaces that converge smoothly to $(\Sigma _0,g_0)$ as $t\to 0$ . Let $Z_t\subset \Sigma _t$ be a collection of a finite number of points that converge to $Z_0$ in suitable symmetric products of $C$ . For each $p\in Z_0$ , we can write $Z_t=\cup _{p\in Z_0}Z_{t,p}$ such that all points in $Z_{t,p}$ converge to $p$ . We define the convergence of weighted line bundles as follows:

A sequence of filtered bundles $\mathcal{F}_{\ast }(\mathcal{L}_t)$ converges to $\mathcal{F}_{\ast }(\mathcal{L}_0)$ if the corresponding weighted bundles converge. The following theorem provides insight into the compactness of a sequence of weighted line bundles:

Proof. By the assumptions on weights, $\deg (\mathcal{L}_t)$ is a fixed, $t$ -independent constant. Let $\gamma _t=(J_t,g_t)$ be a path in $U$ . Then ${\rm Pic}_U|_{\gamma _t}$ is compact, and there exists an $\mathcal{L}_0\in {\rm Pic}(\Sigma _0)$ such that $\mathcal{L}_t$ converges to $\mathcal{L}_0$ . For $p\in Z_0$ , define $\chi _0(p)=\sum _{q\in Z_{t,p}}\chi _t(q)$ , after which you will obtain a weighted line bundle $(\mathcal{L}_0,\chi _0)$ . Choose a family of approximate harmonic metrics $h_t^{{\rm app}}$ such that $|z|^{-2\chi _p}h_t^{{\rm app}}$ extends to a smooth metric in a neighbourhood of $p$ , and $h_t^{{\rm app}}$ converges to $h_0^{{\rm app}}$ in $\mathcal{C}^{\infty }_{{\rm loc}}(\Sigma _0\setminus Z_0)$ . Moreover, write $h_t=h_t^{{\rm app}}e^{s_t}$ . After a suitable rescaling of $h_t$ , we can assume $\|s_t\|_{L^2}=1$ . Let $\rho _t:=\Delta _th_t^{{\rm app}}$ be the curvature defined by the metric $h^{{\rm app}}_t$ . Then $s_t$ satisfies the equation $\Delta _ts_t=\rho _t$ over $\Sigma$ . As $\rho _t$ converges to $\rho _0\in \mathcal{C}^{\infty }_{{\rm loc}}(\Sigma \setminus Z_0)$ and as $g_t$ is a family with bounded geometry, we obtain the estimate

\begin{align*} \|s_t\|_{\mathcal {C}^{k+2,\alpha }(\Sigma )}\leq C_{k,\alpha }(\|\rho _t\|_{\mathcal {C}^{k,\alpha }(\Sigma )}+1)\, \end{align*}

where $C_{k,\alpha }$ is a $t$ -independent constant. Therefore, passing to a subsequence, $s_t$ converges to $s_0$ in $\mathcal{C}^{\infty }(\Sigma )$ , which implies (i). The assertion (ii) follows from (i).

4.1 The algebraic compactification of the Dolbeault moduli space

In this subsection, we present the algebraic method for compactifying the Dolbeault moduli space. This technique is based on the $\mathbb{C}^{\ast }$ action on $\mathcal{M}_{{\rm Dol}}$ and was introduced in [Reference SimpsonSim97, Reference SchmittSch98, Reference HauselHau98, Reference de CataldodC21, Reference Katzarkov, Noll, Pandit and SimpsonKNPS15]. The gauge theoretic approach can be found in [Reference FanFan22a].

We apply this to the Dolbeault moduli space. The first step is to note that the possible isotropy subgroups are limited.

By this lemma, the space $(\mathcal{M}_{{\rm Dol}}\setminus \mathcal{H}^{-1}(0))/\mathbb{C}^{\ast }$ has an orbifold structure. In passing, we note that the fixed points of the $\mathbb{Z}/2$ action correspond to real representations under the nonabelian Hodge correspondence [Reference HitchinHit87a, Sec. 10].

By the properness of the Hitchin map $\mathcal{H}$ (see Theorem 2.1), it follows that $\displaystyle \lim _{t\to \infty }t\cdot \xi$ exists if and only if $\mathcal{H}(\xi )=0$ . Now define

(8) \begin{align} \overline{\mathcal{M}}_{{\rm Dol}} = \left \{(\mathcal{M}_{{\rm Dol}} \times \mathbb{C}^{\ast }) \coprod (\mathcal{M}_{{\rm Dol}} \setminus \mathcal{H}^{-1}(0))\right \}/\mathbb{C}^{\ast }. \end{align}

The analytic topology on the disjoint union is generated by open sets $U\times W_1$ and

\begin{align*}V\times (W_2\cap \mathbb {C}^{\ast })\amalg V\cap (\mathcal {M}_{\mathrm {Dol}} \setminus \mathcal {H}^{-1}(0))\,\end{align*}

where $U,V\subset \mathcal{M}_{{\rm Dol}}$ , $W_1,W_2\subset \mathbb{C}$ are open, and $0\not \in W_1$ , $0\in W_2$ . The topology on $\overline{\mathcal{M}}_{{\rm Dol}}$ is then the quotient topology, and it is straightforward to see that with this topology, it is compact.

Since $(\mathcal{M}_{{\rm Dol}} \times \mathbb{C}^{\ast })/\mathbb{C}^{\ast }=\mathcal{M}_{{\rm Dol}}$ , there is the natural inclusion

\begin{align*} \iota : \mathcal {M}_{\mathrm {Dol}} \to \overline {\mathcal {M}}_{\mathrm {Dol}},\;\iota (\xi ) = [(\xi,1)] \end{align*}

where brackets denote the equivalence class under the $\mathbb{C}^{\ast }$ action. The boundary of $\overline{\mathcal{M}}_{{\rm Dol}}$ is

\begin{align*}\partial \overline {\mathcal {M}}_{\mathrm {Dol}} = \overline {\mathcal {M}}_{\mathrm {Dol}} \setminus \iota (\mathcal {M}_{\mathrm {Dol}})=(\mathcal {M}_{\mathrm {Dol}} \setminus \mathcal {H}^{-1}(0))/\mathbb {C}^{\ast }.\end{align*}

There is also the boundary map

\begin{align*} \iota _{\partial } : \mathcal {M}_{\mathrm {Dol}} \setminus \mathcal {H}^{-1}(0) \longrightarrow \partial \overline {\mathcal {M}}_{\mathrm {Dol}},\; \xi \mapsto [( \xi, 0)]\, \end{align*}

which is invariant under the $\mathbb{C}^{\ast }$ action; that is, $\iota _{\partial }(\lambda \xi ) = \iota _{\partial }(\xi )$ for $\lambda \in \mathbb{C}^{\ast }$ .

The $\mathbb{C}^{\ast }$ action on $\mathcal{M}_{{\rm Dol}}$ covers the square of the action on $\mathcal{B}$ . Hence, it is natural to compactify $\mathcal{B}$ by projectivising:

\begin{align*} \overline {\mathcal {B}} := \mathbb {P}(H^0(K^2)\oplus \mathbb {C}) . \end{align*}

The inclusion is given, as usual, by

\begin{align*} \iota _0:\mathcal {B}\to \overline {\mathcal {B}},\;\iota (q)=[q\times \{1\}]\, \end{align*}

where $q\times \{1\}\in H^0(K^2)\oplus \mathbb{C}$ . We also define $\partial \overline{\mathcal{B}}=\overline{\mathcal{B}}\setminus \iota _0(\mathcal{B})\simeq \mathbb{P}(H^0(K^2))$ , with boundary projection map

\begin{align*} \iota _{0,\partial }:\mathcal {B}\setminus \{0\}\to \partial \overline {\mathcal {B}}\,\ \iota _{0,\partial }(q)=[q\times \{0\}]. \end{align*}

The Hitchin map $\mathcal{H}:\mathcal{M}_{{\rm Dol}}\to \mathcal{B}$ extends to $\overline{\mathcal{H}}:\overline{\mathcal{M}}_{{\rm Dol}}\to \overline{\mathcal{B}}$ , where $\overline{\mathcal{H}}|_{\mathcal{M}_{{\rm Dol}}}:=\iota _0\circ \mathcal{H}$ , and for every $[(\mathcal{E},\varphi )]/\mathbb{C}^{\ast }\in \partial \overline{\mathcal{M}}_{{\rm Dol}}$ ,

\begin{align*} \overline {\mathcal {H}}([(\mathcal {E},\varphi )]/\mathbb {C}^{\ast }):=[(\mathcal {H}(\varphi ), 0)]\subset \overline {\mathcal {B}}. \end{align*}

This is well-defined, since $\det (\varphi ) \neq 0$ if $[(\mathcal{E},\varphi )]/\mathbb{C}^{\ast }\in \partial \overline{\mathcal{M}}_{{\rm Dol}}$ . Moreover,

commutes.

There is a good algebraic structure on this compactification.

The following characterisation of sequential convergence is useful: As $H^0(K^2)$ is a finite dimensional space, the $L^2$ norm on $q\in H^0(K_{\Sigma }^2)$ can be chosen arbitrarily, and we fix one such choice.

4.2 The analytic compactification of the Hitchin moduli space

We next describe the compactification of the Hitchin moduli space, as developed in [Reference Mazzeo, Swoboda, Weiß and WittMSWW14, Reference MochizukiMoc16, Reference TaubesTau13a].

4.2.1 Decoupled Hitchin equations

We begin by defining the decoupled Hitchin equations. Recalling the notation from Section 2.4, let $E$ be a trivial, smooth, rank 2 vector bundle over a Riemann surface $\Sigma$ , and let $H_0$ be a background Hermitian metric on $E$ . Let $Z$ be a finite set of distinct points in $\Sigma$ . For a smooth unitary connection $A$ on $E|_{\Sigma \setminus Z}$ and a smooth $\phi =\varphi +\varphi ^{\dagger }\in \Omega^1(i\mathfrak{su}(E))|_{\Sigma \setminus Z}$ , the decoupled Hitchin equations on $\Sigma \setminus Z$ are:

(9) \begin{align} F_A=0\,\ [\varphi,\varphi ^{\dagger }]=0\,\ \bar{\partial }_A\varphi=0. \end{align}

Solutions to (9) alone may be quite singular near $Z$ , so we make the following restriction:

Definition 4.5. A solution $(A,\phi )$ to (9) is called admissible if $\phi \neq 0$ and if $|\phi |$ extends to a continuous function on $\Sigma$ with $|\phi |^{-1}(0)=Z$ .

By a limiting configuration, we always mean an admissible solution to the decoupled Hitchin equations. Clearly, $Z$ is determined by $(A,\phi )$ . Admissibility guarantees that $\det (\varphi )$ extends to a holomorphic quadratic differential $q=\det (\varphi )$ on $\Sigma$ , with $Z=q^{-1}(0)$ being the zero locus. Hence, the spectral curve $S_q$ is well-defined. We emphasize that $Z$ may vary for different admissible solutions, but one always has $\# Z\leq 4g-4$ .

The equivalence relation on limiting configurations is that $(A_1,\phi _1)\sim (A_2,\phi _2)$ if $Z_1=Z_2$ and if $(A_1,\phi _1)g=(A_2,\phi _2)$ for a smooth unitary gauge transformation $g$ on $\Sigma \setminus Z_1$ . The moduli space of decoupled Hitchin equations is then

\begin{align*} \mathcal {M}_{\mathrm {Hit}}^{\mathrm {Lim}}=\{\text {admissible solutions to (9)}\}/\sim . \end{align*}

We denote by $\mathcal{M}_{{\rm Hit},q}^{{\rm Lim}}$ the elements in $\mathcal{M}_{{\rm Hit}}^{{\rm Lim}}$ , with the determinant of the Higgs field equal to a quadratic differential $q$ . In this case, the equivalence relation is induced by the action of the unitary gauge group over $\Sigma \setminus Z$ , $Z=q^{-1}(0)$ .

There is a natural $\mathbb{C}^{\ast }$ action on the moduli space $\mathcal{M}_{{\rm Hit}}^{{\rm Lim}}$ : Given $(A,\phi =\varphi +\varphi ^{\dagger })\in \mathcal{M}_{{\rm Hit}}^{{\rm Lim}}$ and $t\in \mathbb{C}^{\ast }$ , we set $t\cdot [(A,\phi )]=[(A,t\varphi +\bar{t}\varphi ^{\dagger })]$ , which is also a solution to (9).

4.2.3 The topology on the compactified space

We now carefully define the topology on the space $\mathcal{M}_{{\rm Hit}} \coprod \mathcal{M}_{{\rm Hit}}^{{\rm Lim}}/\mathbb{C}^{\ast }$ . Choose a metric in the conformal class on $\Sigma$ . Let $W^{k,2}$ denote the Sobolev spaces on $\Sigma$ of distributional sections with at least $k$ derivatives in $L^2$ . For a finite set of points $Z \subset \Sigma$ (or indeed any closed subset),

\begin{align*} W^{k,2}_{{\rm loc}}(\Sigma \setminus Z):=\{f\mid f\in W^{k,2}(K),\;K\subset \Sigma \setminus Z,\;K\;{\rm compact}\}. \end{align*}

These definitions extend easily to the space of connections and $\Omega^1(i \mathfrak{su}(E))$ for a Hermitian vector bundle $(E,H_0)$ over $\Sigma$ with a fixed, smooth background connection.

Let $\omega _n$ be a nested collection of open sets with $\omega _n\subset \overline{\omega _n} \subset \omega _{n+1}$ and with $\bigcup _{n}\omega _n= \Sigma \setminus Z$ . We then define the seminorms $\|f\|_n:=\|f\|_{W^{k,2}(\omega _n)}$ ; in terms of these, $W^{k,2}_{{\rm loc}}(\Sigma \setminus Z)$ is a Fréchet space.

For any $q\in H^0(K^2)\setminus \{0\}$ , set $Z_q:=q^{-1}(0)$ , and consider the moduli space

\begin{align*} \mathbb {M}_q=\Big\{[(A,\phi )]\in \mathcal {M}_{\mathrm {Hit},q^\ast }\cap W^{k,2}(\Sigma )\Big\}\cup \Big\{[(A,\phi )]\in \mathcal {M}_{\mathrm {Hit},q}^{\mathrm {Lim}}\cap W^{k,2}_{\mathrm {loc}}(\Sigma \setminus Z_q)\Big\}/\mathbb {C}^{\ast }. \end{align*}

Here we give a more precise explanation of the above notation. The space $\mathcal{M}_{{\rm Hit},q^*}$ consists of solutions $(A,\phi =\varphi +\varphi ^{\dagger })$ to the Hitchin equations such that $\det (\varphi )=tq$ for some nonzero complex number $t$ . Moreover, the notation $[(A,\phi )]\in \mathcal{M}_{{\rm Hit},q^\ast }\cap W^{k,2}(\Sigma )$ refers to the equivalence class of $(A,\phi )\in W^{k,2}(\Sigma )$ modulo unitary gauge transformations in $W^{k+1,2}(\Sigma )$ . Similarly, $[(A,\phi )]\in \mathcal{M}_{{\rm Hit},q}^{{\rm Lim}}\cap W^{{\rm loc}}$ consists of the equivalence class of $(A,\phi )\in W^{k,2}_{{\rm loc}}(\Sigma \setminus Z_q)$ modulo unitary gauge transformations in $W^{k+1,2}_{{\rm loc}}(\Sigma \setminus Z_q)$ , and the $\mathbb{C}^{\ast }$ action is given by $t\cdot [(A,\phi )]\to [(A,t\phi )].$ By classical bootstrapping of the gauge-theoretic elliptic equations, $\mathbb{M}_q$ is independent of $k\geq 2$ .

Next define $\mathbb{M}:=\mathcal{M}_{0}\cup \bigcup _{q\in H^0(K^2)\setminus \{0\}}\mathbb{M}_q$ , and based on the definition, we have $\mathbb{M}=\mathcal{M}_{{\rm Hit}}\cup \mathcal{M}_{{\rm Hit}}^{{\rm Lim}}/\mathbb{C}^{\ast }$ . Its topology is generated by two types of open sets. For interior points $\xi =[(A,\phi )]\in \mathcal{M}_{{\rm Hit}}\subset \mathbb{M}$ , we use the open sets

\begin{align*} V_{\xi,\epsilon }:=\Big\{[(A^{\prime},\phi ^{\prime})]\in \mathcal{M}_{{\rm Hit}}\mid \|A^{\prime}-A\|_{W^{k,2}(\Sigma )}+\|\phi ^{\prime}-\phi \|_{W^{k,2}(\Sigma )}\lt \epsilon \Big\} \end{align*}

from the topology of $\mathcal{M}_{{\rm Hit}}$ . For any boundary point $\xi _0\in \mathcal{M}_{{\rm Hit}}^{{\rm Lim}}/\mathbb{C}^{\ast }$ , choose a representative $(A_0,\phi _0)$ with $\|\phi _0\|_{L^2}=1$ . Let $q=\det (\phi _0)$ , and fix any open set $\omega \Subset \Sigma \setminus Z_q$ . Then, setting $\mathcal{M}_{{\rm Hit}}^{\ast }=\mathcal{M}_{{\rm Hit}}\setminus \mathcal{H}^{-1}(0)$ ,

\begin{align*} U_{\xi _0,\omega,\epsilon } & := \Big\{(A,\phi )\in \mathcal{M}_{{\rm Hit}}^{\ast }\mid \|A-A_0\|_{W^{k,2}(\omega )}+\inf _{\theta \in S^1}\|\|\phi \|_{L^2}^{-\frac 12}\phi -e^{i\theta }\phi _0\|_{W^{k,2}(\omega )}\lt \epsilon,\;\|\phi \|_{L^2}\gt {1 \over \epsilon }\Big\}\\& \cup \, \Big\{(A,\phi )\in \mathcal{M}_{{\rm Hit}}^{{\rm Lim}}| \|A-A_0\|_{W^{k,2}(\omega )}+\|\phi -\phi _0\|_{W^{k,2}(\omega )}\lt \epsilon \Big\} \end{align*}

defines an open set around $\xi _0$ . The sets $U_{\xi _0,\omega,\epsilon }$ and $V_{\xi,\epsilon }$ generate the topology on $\mathbb{M}$ .

Theorem 4.8. The space $\mathbb{M}$ is Hausdorff and compact.

Proof. The Hausdorff property follows from the definition of the topology. By Propositions 4.6 and 4.7, $\mathbb{M}$ is sequentially compact. Moreover, using this explicit base for the topology, $\mathbb{M}$ is first countable and, hence, compact.

We may now define the compactification of the Hitchin moduli space as the closure $\overline{\mathcal{M}}_{{\rm Hit}}\subset \mathbb{M}$ ; we write $\partial \overline{\mathcal{M}}_{{\rm Hit}}$ for the boundary of the closure and $\overline{\mathcal{M}}_{{\rm Hit},q^\ast }:=\overline{\mathcal{M}}_{{\rm Hit}}\cap \mathbb{M}_q$ for the subset of elements with a fixed quadratic differential.

The following result is described in [Reference Mazzeo, Swoboda, Weiss and WittMSWW16, Reference Ott, Swoboda, Wentworth and WolfOSWW20, Reference Mazzeo, Swoboda, Weiss and WittMSWW19].

Theorem 4.9. If $q$ has only simple zeros, then $\overline{\mathcal{M}}_{{\rm Hit},q^\ast }=\mathbb{M}_q$ .

In other words, the compactification of any slice where $q$ does not lie in the discriminant locus is ‘the obvious one’.

5.1 Normalization of the spectral curve

Let $q\neq 0$ be a quadratic differential with an irreducible, singular spectral curve $S=S_q$ . The zeros of $q$ define the divisor ${\rm Div}(q) = \sum _{i=1}^{r_1} m_i p_i + \sum _{j=1}^{r_2} n_j p^{\prime}_j$ , where the $m_i$ and $n_j$ are even and odd integers, respectively; and, hence, $r_1$ and $r_2$ are the numbers of even and odd zeros, respectively, counted without multiplicity. Write $Z_{{\rm even}} = \{p_1, \dots, p_{r_1}\}$ , $Z_{{\rm odd}} = \{p^{\prime}_1, \dots, p^{\prime}_{r_2}\}$ and $Z = Z_{{\rm even}} \cup Z_{{\rm odd}}$ ; so $\#Z=r = r_1 + r_2$ .

The map $\pi : S \to \Sigma$ is a double covering branched along $Z$ ; hence, we may view $p_i$ and $p^{\prime}_i$ as points in $S$ . For $x \in S$ , let $\mathcal{O}_x$ be the algebraic local ring, $\mathcal{O}_x^\ast$ its group of units and $R_x$ the completion. We say that $S$ has an $A_n$ singularity at $x$ if $R_x \cong \mathbb{C}[[r,s]]/(r^2-s^{n+1})$ , where $n \geq 1$ . If $S$ has an $A_1$ singularity at $x$ , we call it a nodal singularity, and if $S$ has an $A_2$ singularity at $x$ , we call it a cusp singularity.

Let $p:\widetilde{S} \to S$ be the normalisation of $S$ , and let $\tilde{\pi } := \pi \circ p$ :

(10)

For even zeros $p_i$ , we write $p^{-1}(p_i) = \{\tilde{p}_i^{+}, \tilde{p}_i^{-}\}$ , and for odd zeros $p^{\prime}_i$ we write $p^{-1}(p^{\prime}_i) = \tilde{p}^{\prime}_i$ . Since $\pi :S\to \Sigma$ is a branched double cover, the involution $\sigma$ on $S$ extends to an involution of $\widetilde{S}$ , which we also denote by $\sigma$ . Note that $\sigma (\tilde{p}^{\prime}_i) = \tilde{p}^{\prime}_i$ while $\sigma (\tilde{p}_i^\pm ) = \tilde{p}_i^\mp$ .

The ramification divisor $\Lambda ^{\prime}={1 \over 2}\sum _{i=1}^{r_1}m_ip_i+\frac 12\sum _{j=1}^{r_2}(n_j-1)p^{\prime}_j$ , is a (Weil) divisor on $S$ , and there is an exact sequence:

(11) \begin{align} 0\longrightarrow \mathcal{O}_{S}\longrightarrow p_{\ast }\mathcal{O}_{\widetilde{S}}\longrightarrow \sum _{x\in {\rm Supp}(\Lambda ^{\prime})}\widetilde{\mathcal{O}}_x/\mathcal{O}_x\longrightarrow 0. \end{align}

The genus of $\widetilde{S}$ is $g(\widetilde{S})=4g-3-\deg (\Lambda ^{\prime})=2g-1+r_2/2$ .

5.2 Jacobian under the pull-back to the normalization

We now recall some facts about the Jacobian under the pull-back to the normalization (compare to [Reference Gothen and OliveiraGO13]). Let $x \in Z \subset S$ be a singular point; that is, either $x\in Z_{{\rm even}}$ or $x=p^{\prime}_j$ with $n_j\geq 3$ . Let $\widetilde{\mathcal{O}}_{x}$ be the integral closure of $\mathcal{O}_x$ . Set $V:=\prod _{x\in Z}\widetilde{\mathcal{O}}_x^{\ast }/\mathcal{O}_x^{\ast }$ . Then we have the following well-known short exact sequence:

(12) \begin{align} 0\longrightarrow V\longrightarrow {\rm Jac}(S)\xrightarrow{p^{\ast }} {\rm Jac}(\widetilde{S})\longrightarrow 0. \end{align}

This will play an important role later on.

5.2.1 Hitchin fibre

We examine the locally free part $\mathcal{T}$ of the Hitchin fibre under the pull-back. Here, $\mathcal{T}$ is defined to be the set of $L\in {\rm Pic}^{2g-2}(S)$ such that $\det (\pi _{\ast }L)=\mathcal{O}_{\Sigma }$ [see (4)]. Although $\Lambda ^{\prime}$ is a divisor $S$ , it could also be considered to be a divisor on $\Sigma$ by the identification of $p_i,p^{\prime}_j$ and $\pi (p_i), \pi (p^{\prime}_j)$ . To shorten the notation, we write $\mathcal{O}_\Sigma (\Lambda ^{\prime})$ for the corresponding line bundle on $\Sigma$ . For any $L\in {\rm Pic}(S)$ , from (11) we see that $\det (\tilde{\pi }_{\ast }p^{\ast }L)\cong \det (\pi _{\ast }L)\otimes \mathcal{O}_\Sigma (\Lambda ^{\prime})$ . We define a new set, $\widetilde{\mathcal{T}}$ , as follows:

\begin{align*} \widetilde {\mathcal {T}}:=\{\widetilde {L}\in \mathrm {Pic}^{2g-2}(\widetilde {S})\mid \det (\tilde {\pi }_{\ast }L)\cong \mathcal {O}(\Lambda ^{\prime})\}. \end{align*}

Then $p^{\ast }$ maps $\mathcal{T}$ to $\widetilde{\mathcal{T}}$ . Furthermore, if $L_1,L_2\in {\rm Pic}(S)$ satisfy $p^{\ast }L_1\cong p^{\ast }L_2$ , then we have $\pi _{\ast }L_1\cong \pi _{\ast }L_2$ . This means that the fibre of $p^{\ast }:{\rm Jac}(S)\to {\rm Jac}(\widetilde{S})$ is the same as that of $p^{\ast }:\mathcal{T}\to \widetilde{\mathcal{T}}$ , resulting in the following fibration:

(13) \begin{align} V\longrightarrow \mathcal{T}\xrightarrow{\hspace{.4cm}p^{\ast }\ }\widetilde{\mathcal{T}}. \end{align}

5.3 Torsion-free sheaves

Now we present Cook’s parametrisation of rank 1 torsion-free sheaves on curves with Gorenstein singularities (see [Reference CookCoo98, p. 40] and [Reference CookCoo93, Reference RegoReg80]). An explicit computation of the invariants used in this article is provided in Appendix A. Let $x \in Z$ be a singular point of $S$ , and let $L\to S$ be a rank 1 torsion-free sheaf. We again let $\mathcal{O}_x$ denote the local ring at $x$ , $\widetilde{\mathcal{O}}_x$ its integral closure, and $\delta _x = \dim _{\mathbb{C}}(\widetilde{\mathcal{O}}_x/\mathcal{O}_x)$ . According to [Reference Greuel and PfisterGP93, Lemma 1.1], there exists a fractional ideal $I_x$ that is isomorphic to $L_x$ and uniquely defined up to multiplication by a unit of $\widetilde{\mathcal{O}}_x$ such that $\mathcal{O}_x \subset I_x \subset \widetilde{\mathcal{O}}_x$ . We define $\ell _x := \dim _{\mathbb{C}} (I_x/\mathcal{O}_x)$ and $b_x:=\dim _{\mathbb{C}}({T}(I_x\otimes _{\mathcal{O}_x}\widetilde{\mathcal{O}}_x))$ , where $T$ means the torsion subsheaf. Then $\ell _x$ and $b_x$ are invariants of $L$ .

Let $\mathcal{K}_x$ be the field of fractions of $\mathcal{O}_x$ . The conductor of $I_x\subset \widetilde{\mathcal{O}}_x$ is defined to be

\begin{align*}C(I_x)=\{u\in \mathcal {K}_x\mid u\cdot \widetilde {\mathcal {O}}_x\subset I_x\}.\end{align*}

The singularity is characterized by the following dimensions:

(14) \begin{align} \underbrace{C(\mathcal{O}_x)\subset \overbrace{C(I_x)\subset \overbrace{\mathcal{O}_x\subset \underbrace{I_x\subset \widetilde{\mathcal{O}}_x}_{\delta _x-\ell _x}}^{\delta _x}}^{2\delta _x-2\ell _x}}_{2\delta _x}. \end{align}

For $x = p_i \in Z_{{\rm even}}$ , we have $\delta _{p_i}=m_i/2$ , and there are two maximal ideals $\mathfrak{m}_{\pm }$ in $\widetilde{\mathcal{O}}_x$ corresponding to the points $\tilde{p}_i^{\pm }$ . We let $(\widetilde{\mathcal{O}}_{p_i}/C(I_{p_i}))_{\mathfrak{m}_{\pm }}$ be the localization by the ideals $\mathfrak{m}_{\pm }$ , and we define $a_{\tilde{p}_i^{\pm }}:=\dim _{\mathbb{C}}(\widetilde{\mathcal{O}}_{p_i}/C(I_{p_i}))_{\mathfrak{m}_{\pm }}$ . Moreover, we have $\dim _{\mathbb{C}}(\widetilde{\mathcal{O}}_{p_i}/C(\mathcal{O}_{p_i}))_{\mathfrak{m}_{\pm }}=m_i/2=\delta _{p_i}$ . By Appendix A, $a_{\tilde{p}_i^{\pm }}=(m_i/2)-\ell _{p_i}$ , and therefore $a_{\tilde{p}_i^+}+a_{\tilde{p}_i^-}=2\delta _{p_i}-2\ell _{p_i}$ and also $b_{p_i}=\ell _{p_i}$ . Define

\begin{align*} V(L_{p_i})=\{(c_i^+,c_i^-)\mid c_{i}^{\pm }\in \mathbb {Z}_{\ge 0}\,\ c_i^++c_i^-=\ell _{p_i}\}. \end{align*}

For $x=p^{\prime}_i \in Z_{{\rm odd}}$ , we have $\delta _{p^{\prime}_i}=(n_i-1)/2$ , and the maximal ideal $\mathfrak{m}$ of $\widetilde{\mathcal{O}}_x$ is unique. We define $a_{\tilde{p}^{\prime}_i}:=\dim _{\mathbb{C}}(\widetilde{\mathcal{O}}_{p^{\prime}_i}/C(I_{p^{\prime}_i}))_{\mathfrak{m}}$ . By Appendix A, we have $a_{\tilde{p}^{\prime}_i}=2\delta _{p^{\prime}_i}-2\ell _{p^{\prime}_i}$ and $b_{p^{\prime}_i}=\ell _{p^{\prime}_i}$ . Moreover, $\dim _{\mathbb{C}}(\widetilde{\mathcal{O}}_{p^{\prime}_i}/C(\mathcal{O}_{p^{\prime}_i}))_{\mathfrak{m}}=n_i-1=2\delta _{p^{\prime}_i}$ . In this case we set $V(L_{p^{\prime}_i})=\{\ell _{p^{\prime}_i}\}$ .

Let $\eta :\widetilde{\mathcal{O}}_x\to \widetilde{\mathcal{O}}_x/C(\mathcal{O}_x)$ be the quotient map. We define

\begin{align*} S(L_x):=\{\mathcal {O}_x\text {-submodules}\ F_x\subset \widetilde {\mathcal {O}}_x/C(\mathcal {O}_x)\mid \dim _{\mathbb C}(F_x)=\delta _x\,\ \eta ^{-1}(F_x)\cong L_x\}. \end{align*}

Hence, if $J_x=\eta ^{-1}(F_x)$ with $F_x\in S(L_x)$ , there exists an ideal $\mathfrak{t}_x$ in $\widetilde{\mathcal{O}}_x$ such that $J_x=\mathfrak{t}_x\cdot L_x$ . For $x=p_i\in Z_{{\rm even}}$ , we obtain two integers: $c_{i}^{\pm }=\dim _{\mathbb{C}}(\widetilde{\mathcal{O}}_x/(\mathfrak{t}_x\cdot \widetilde{\mathcal{O}}_x))_{\mathfrak{m}_{\pm }}$ . By [Reference CookCoo98, Lemma 6], $(c_i^+,c_i^-)\in V(L_{p_i})$ , for $x=p^{\prime}_i\in Z_{{\rm odd}}$ and $\dim _{\mathbb{C}}(\widetilde{\mathcal{O}}_x/(\mathfrak{t}_x\cdot \widetilde{\mathcal{O}}_x))=\ell _{p^{\prime}_i}\in V(L_{p^{\prime}_i})$ , and these only depend only on $F_x$ . Hence, there is a well-defined map:

\begin{align*} \kappa _x:S(L_x)\longrightarrow V(L_x)\ :\ \begin {cases} F_x\to (c_i^+,c_i^-) & \mathrm {when}\;x=p_i,\;\\[3pt] F_x\to \ell _{p^{\prime}_i} & \mathrm {when}\;x=p^{\prime}_i . \end {cases} \end{align*}

Set $V(L):=\prod _{x\in Z}V(L_x)$ and $S(L):=\prod _{x\in Z}S(L_x)$ . Write $N(L) := |V(L)|$ for the number of points in $V(L)$ . There is a map

\begin{align*} \kappa :=\prod _{x\in Z}\kappa _x: S(L)\longrightarrow V(L). \end{align*}

For any ${\bf c}\in V(L)$ , write ${\bf c}=(c_1^{\pm },\ldots,c_{r_1}^{\pm },\ell _{p^{\prime}_1},\ldots,\ell _{p^{\prime}_{r_2}})$ . Associate to ${\bf c}$ the divisor

\begin{align*} D_{\mathbf {c}}=\sum _{i=1}^{r_1}(c_i^+\tilde {p}_i^++c_i^-\tilde {p}_i^-)+\sum _{i=1}^{r_2}\ell _{p^{\prime}_i}\tilde {p}^{\prime}_i \end{align*}

on $\widetilde{S}$ . Composing $\kappa$ with the map above, we define

(15) \begin{align} \varkappa : S(L)\longrightarrow {\rm Div}(\widetilde{S})\ :\ \prod _{x\in Z}F_x \mapsto {\bf c}\mapsto D_{{\bf c}}. \end{align}

The following result is straightforward but important:

5.4 Parabolic modules

In this subsection, we define the notion of a parabolic module, following [Reference RegoReg80, Reference CookCoo93, Reference CookCoo98]. First note that $\dim _{\mathbb{C}}(\widetilde{\mathcal{O}}_x/C(\mathcal{O}_x))=2\delta _x$ (compare to (14)). Let ${\rm Gr}(\delta _x,\widetilde{\mathcal{O}}_x/C(\mathcal{O}_x))$ be the Grassmannian of $\delta _x$ dimensional subspaces of the vector space $\widetilde{\mathcal{O}}_x/C(\mathcal{O}_x)$ . Then $\widetilde{\mathcal{O}}^{\ast }_x$ acts on ${\rm Gr}(\delta _x,\widetilde{\mathcal{O}}_x/C(\mathcal{O}_x))$ by multiplication, with fixed points corresponding to $\delta _x$ -dimensional $\mathcal{O}_x$ submodules of $\widetilde{\mathcal{O}}_x/C(\mathcal{O}_x)$ . We write $\mathcal{P}(x)$ for the (reduced) variety of fixed points. This is a closed subvariety of ${\rm Gr}(\delta _x,\widetilde{\mathcal{O}}_x/C(\mathcal{O}_x))$ .

Suppose $x$ is an $A_n$ singularity. For notational convenience, we write $\mathcal{P}(A_n):=\mathcal{P}(x)$ . We have the following:

Define $\mathscr{P}(S)=\prod _{x\in Z}\mathcal{P}(x)$ . This depends only on the curve singularity of $S$ . Let $J\in {\rm Pic}(\widetilde{S})$ . As vector spaces,

\begin{align*} J_{\tilde {p}_i^+}^{\oplus \frac {m_i}{2}}\oplus J_{\tilde {p}_i^-}^{\oplus \frac {m_i}{2}}\cong \widetilde {\mathcal {O}}_{p_i}/C(\mathcal {O}_{p_i})\, \ J_{\tilde {p}^{\prime}_i}^{\oplus (n_i-1)}\cong \widetilde {\mathcal {O}}_{p^{\prime}_i}/C\big(\mathcal {O}_{p^{\prime}_i}\big). \end{align*}

By [Reference CookCoo98, p. 41], ${\rm PMod}(\widetilde{S})$ has a natural algebraic structure. Let ${\rm pr}:{\rm PMod}(\widetilde{S})\to {\rm Jac}(\widetilde{S})$ be the projection to the first component. Then ${\rm pr}$ defines a fibration of ${\rm PMod}(\widetilde{S})$ with fibre $\mathscr{P}(S)$ . Moreover, there is a finite morphism $\tau : {\rm PMod}(\widetilde{S})\to \overline{{\rm Jac}}(S)$ defined by sending $(J,v) \to L$ , where $L$ is given by:

\begin{align*} 0\longrightarrow L\longrightarrow p_{\ast }J\longrightarrow (J\otimes \mathcal {O}_{\Lambda })/v\longrightarrow 0. \end{align*}

There is a corresponding diagram:

The map $\tau$ may be regarded as the compactification of the pull-back normalization map $p^{\ast }$ in (12).

Suppose that all of the zeros of the quadratic differential $q$ are odd. Then for $L\in \overline{{\rm Jac}}(S)$ , $N(L)=1$ , and we can deduce the following:

Corollary 5.6. If $q^{-1}(0)=\{p^{\prime}_1,\ldots, p^{\prime}_r\}$ and all zeroes have odd multiplicity, then $\tau :{\rm PMod}(\widetilde{S})\to \overline{{\rm Jac}}(S)$ is a bijection. Moreover, for $L\in \overline{{\rm Jac}}(S)$ with $\tau (J,v)=L$ , we have

\begin{align*} p^{\ast }L/{T}(p^{\ast }L)=J\Big(-\sum \ell _{p^{\prime}_i}\tilde {p}^{\prime}_i\Big). \end{align*}

For convenience, we recall the canonical example of a parabolic module.

Example 5.7. Suppose $q$ contains $4g-2$ simple zeros and one zero $x$ of order $2$ . Then the spectral curve $S$ has one nodal singularity at $x$ . Denote $p:\widetilde{S}\to S$ as the normalization, with $p^{-1}(x)=\{\tilde{x}_+,\tilde{x}_-\}$ . Then $\mathscr{P}(S)=\mathbb{C}P^1$ , and we obtain a fibration $\mathbb{C}P^1\to {\rm PMod}(\widetilde{S})\to {\rm Jac}(\widetilde{S})$ . Let $L\in \overline{{\rm Jac}}(S)\setminus {\rm Jac}(S)$ . If we write $\widetilde{L}:=p^{\ast }L/{T}(p^{\ast }L)$ , then

\begin{align*}\tau ^{-1}(L)=\{(\widetilde {L}\otimes \mathcal {O}(\tilde {x}_+),v_+),(\widetilde {L}\otimes \mathcal {O}(\tilde {x}_-),v_-)\}.\end{align*}

We can define two sections:

\begin{align*}s_{\pm }:\mathrm {Jac}(\widetilde S)\longrightarrow \mathrm {PMod}(\widetilde {S}) :J\mapsto (J,v_{\pm })\,\end{align*}

where $v_+=[1,0]$ , $v_-=[0,1]$ . Then $\overline{{\rm Jac}}(S)$ is the quotient of ${\rm PMod}(\widetilde{S})$ given by the identification

\begin{align*}\overline {\mathrm {Jac}}(S)\cong \mathrm {PMod}(\widetilde {S})/(s_+\sim \mathcal {O}(\tilde {x}_-\tilde {x}_+)s_-).\end{align*}

In particular, ${\rm PMod}(\widetilde{S})$ is not a fibration over $\overline{{\rm Jac}}(S)$ .

Let $\mathcal{P}:=\{L\in {\rm Jac}(S)\mid \det (\pi _{\ast }L)\cong K^{-1}\}$ and $\overline{\mathcal{P}}$ be the closure of $\mathcal{P}$ in $\overline{{\rm Pic}}(S)$ . Since we focus on ${\rm SL}(2,\mathbb{C})$ Higgs bundles, we must consider the parabolic module compactification of the fibration

\begin{align*} 0\longrightarrow V\longrightarrow \mathcal {P}\xrightarrow {\hspace {.3cm}p^{\ast }\ } \mathrm {Prym}(\widetilde {S}/\Sigma )\longrightarrow 0. \end{align*}

Setting $\widehat{{\rm PMod}}(\widetilde{S}):=\tau ^{-1}(\overline{\mathcal{P}})$ , there is a diagram from [Reference Gothen and OliveiraGO13, p. 17]

(16)

Theorem 5.5 proves that ${\rm pr}\circ \tau ^{-1}|_{\mathcal{P}}=p^{\ast }$ .

5.5 Stratifications of the BNR data

Recall that $\overline{\mathcal{T}}$ (resp. $\overline{\mathcal{P}}$ ) is the natural compactification of $\mathcal{T}$ (resp. $\mathcal{P}$ ) induced by the inclusion ${\rm Pic}(S)\subset \overline{{\rm Pic}}(S)$ . Parabolic modules define a stratification of $\overline{\mathcal{P}}$ and $\overline{\mathcal{T}}$ . In the following, $\pi :S\to \Sigma$ is a branched double cover, $\sigma$ is the associated involution on $S$ , and by $\sigma$ we also denote its extension to an involution on the normalization $\widetilde{S}$ of $S$ .

For a rank 1 torsion-free sheaf $L\in \overline{{\rm Pic}}(S)$ , consider the map

\begin{align*} p^{\star }_{\mathrm {tf}}:\overline {\mathrm {Pic}}(S)\longrightarrow \mathrm {Pic}(\widetilde {S})\,\ p^{\star }_{\mathrm {tf}}(L):=p^{\ast }L/{T}(p^{\ast }L)\; \end{align*}

that is, the torsion-free part of the pull-back to the normalization. By [Reference RabinowitzRab79], $p^{\star }_{{\rm tf}}(L)=p^{\ast }L$ at $x\in \widetilde{S}$ if and only if $L$ is locally free at $p(x)\in S$ .

Using the previous conventions, recall that we have the divisor

\begin{align*}\Lambda =\sum _{i=1}^{r_1}\frac {m_i}{2}(\tilde {p}_i^++\tilde {p}_i^-)+\sum _{j=1}^{r_2}n_j\tilde {p}^{\prime}_j\end{align*}

on $\widetilde{S}$ .

The $\sigma$ -divisors play an important role in describing the singular Hitchin fibres.

For a $\sigma$ -divisor $D$ , define

(17) \begin{align} \widetilde{\mathcal{T}}_D & =\{J\in {\rm Pic}(\widetilde{S})\mid J\otimes \sigma ^{\ast }J=\mathcal{O}(\Lambda -D)\}\ ; \nonumber\\[3pt] \widetilde{\mathcal{P}}_D & =\{J\in {\rm Pic}(\widetilde{S})\mid J\otimes \sigma ^{\ast }J=\mathcal{O}(-D)\}. \end{align}

By [Reference HornHor22a, Prop. 5.6], when the number of odd zeros $r_2\gt 0$ or $D\neq 0$ , $\widetilde{\mathcal{T}}_D$ and $\widetilde{\mathcal{P}}_D$ are abelian torsors over ${\rm Prym}(\widetilde{S}/\Sigma )$ , with dimension $g(\widetilde{S})-g=g-1+\frac 12r_2$ . When $r_2=0$ and $D=0$ , $\widetilde{\mathcal{P}}_{D}$ and $\widetilde{\mathcal{T}}_D$ are torsors over ${\rm Nm}^{-1}(\mathcal{O}_{\Sigma })\cup {\rm Nm}^{-1}(I)$ , where ${\rm Nm}$ is the norm map of the covering $\tilde{\pi }: \tilde{S}\to \Sigma$ , and $\mathcal{I}$ is the unique non-trivial line bundle that satisfies $\tilde{\pi }^*\mathcal{I}\cong \mathcal{O}_{\widetilde{S}}$ . In addition, we define

(18) \begin{align} \overline{\mathcal{T}}_D & =\{L\in \overline{\mathcal{T}}\mid p^{\star }_{{\rm tf}} L\in \widetilde{\mathcal{T}}_D\}\ ;\nonumber\\[3pt] \overline{\mathcal{P}}_D & =\{L\in \overline{\mathcal{P}}\mid p^{\star }_{{\rm tf}} L\in \widetilde{\mathcal{P}}_D\}. \end{align}

Then the partial order on divisors defines a stratification of $\overline{\mathcal{T}}$ (resp. $\overline{\mathcal{P}}$ ) by $\cup _{D^{\prime}\leq D}\overline{\mathcal{T}}_{D^{\prime}}$ (resp. $\cup _{D^{\prime}\leq D}\overline{\mathcal{P}}_{D^{\prime}}$ ). The top strata are $\overline{\mathcal{T}}_{D=0}$ (resp. $\overline{\mathcal{P}}_{D=0}$ ), and these consist of the locally free sheaves. From the definition, $\mathcal{T}=\overline{\mathcal{T}}_{D=0}$ and $\mathcal{P}=\overline{\mathcal{P}}_{D=0}$ .

Theorem 5.11. (i) Suppose $q$ contains at least one zero of odd order. For each stratum indexed by a $\sigma$ -divisor $D$ , if we let $n_{ss}$ be the number of $p$ such that $D|_p=\Lambda |_p$ , then there are holomorphic fibre bundles

(19) \begin{align} & (\mathbb{C}^{\ast })^{k_1}\times \mathbb{C}^{k_2}\longrightarrow \overline{\mathcal{T}}_D\xrightarrow{\hspace{.2cm}p^{\star }_{{\rm tf}}\ } \widetilde{\mathcal{T}}_D\ ;\nonumber\\[3pt]& (\mathbb{C}^{\ast })^{k_1}\times \mathbb{C}^{k_2}\longrightarrow \overline{\mathcal{P}}_D\xrightarrow{\hspace{.2cm}p^{\star }_{{\rm tf}}\ } \widetilde{\mathcal{P}}_D\, \end{align}

where $k_1=r_1-n_{ss}$ , $k_2=2g-2-\frac 12\deg (D)-r_1+n_{ss}-{r_2 \over 2}$ and $r_1,r_2$ are the number of even and odd zeros. Footnote ¹

(ii) Suppose $q$ is irreducible but all zeros are of even order. Then there exists an analytic space $\overline{\mathcal{T}}^{\prime}_D$ and a double branched covering $p:\overline{\mathcal{T}}_D\to \overline{\mathcal{T}}^{\prime}_D$ , with $\overline{\mathcal{T}}^{\prime}_D$ a holomorphic fibration

\begin{align*} (\mathbb {C}^{\ast })^{k_1}\times \mathbb {C}^{k_2}\longrightarrow \overline {\mathcal {T}}^{\prime}_D \xrightarrow {\hspace {.25cm}p^{\star }_{\mathrm {tf}}\ } \widetilde {\mathcal {T}}_D. \end{align*}

In particular, $\dim (\overline{\mathcal{P}}_D)=\dim (\overline{\mathcal{T}}_D)=3g-3-\frac 12\deg (D).$

As explained in [Reference HornHor22a], via the BNR correspondence the stratification above translates into a stratification of the Hitchin fibre. Let $\chi _{{\rm BNR}}:\overline{\mathcal{T}}\xrightarrow{\sim }\mathcal{M}_q$ be the bijection in Theorem 2.3. Let $D$ be a $\sigma$ -divisor. Define $\mathcal{M}_{q,D}:=\chi _{{\rm BNR}}(\overline{\mathcal{T}}_D)$ . Then the stratification of $\overline{\mathcal{T}}$ induces a stratification on $\mathcal{M}_q=\bigcup _D\mathcal{M}_{q,D}$ .

For each $\sigma$ -divisor $D$ , since $\sigma ^{\ast }D=D$ and for any $x\in {\rm Fix}(\sigma )$ , $D|_x=d_xx$ , where $d_x\equiv 0\mod 2$ , we can write $D^{\prime}:={1 \over 2}\tilde{\pi }(D)$ . Then $D^{\prime}$ is an effective divisor with ${\rm supp\;} D^{\prime}\subset Z$ . Moreover, for $x\in q^{-1}(0)$ , $D^{\prime}_x\leq \frac 12\lfloor {\rm ord}_x(q) \rfloor$ . Therefore, $\mathcal{M}_q$ may be regarded as also being stratified by divisors $D^{\prime}$ defined over $\Sigma$ .

5.6 The structure of the parabolic module projection

We now explain the relationship between the divisor $D_v$ in Theorem 5.5 and the $\sigma$ -divisor. Given $L \in \overline{\mathcal{P}}$ , define

(20) \begin{align} \mathscr{N}_L & :=\{(J,v)\in \widehat{{\rm PMod}}(\widetilde{S})\mid \tau (J,v)=L\}\ ;\nonumber\\ \mathscr{D}_L & :=\{D_v\mid (J,v)\in \mathscr{N}_L\}\, \end{align}

That is, $\mathscr{N}_L=\tau ^{-1}(L)$ , and $\mathscr{D}_L$ is the collection of divisors $D_v$ such that $J(-D_v) = p^{\star }_{{\rm tf}}(L)$ . By Theorem 5.5, if $\tau (J, v) = \tau (J^{\prime}, v)$ , then $J^{\prime} = J(D_{v^{\prime}} - D_v)$ . By (16), as $L\in \overline{\mathcal{P}}$ , we have $J,J^{\prime}\in {\rm Prym}(\widetilde{S}/\Sigma )$ , which implies $D_v=D_{v^{\prime}}$ . Therefore, for the cardinalities, we have $|\mathscr{N}_L|=|\mathscr{D}_L|$ . Furthermore, we define $N_L:=| \mathscr{N}_L|=|\mathscr{D}_L|$ .

The divisor $D_v$ satisfies the following symmetry proposition:

As a consequence, we have the following:

There are relationships among the integers appearing in the construction of the parabolic module.

Proof. By Lemma 5.14, $V(L)$ can be rewritten as

\begin{align*} V(L)=\{(c_1^{\pm },\ldots,c_{r_1}^{\pm },c^{\prime}_{1}=l_{p^{\prime}_1},\ldots,c^{\prime}_{r_2}=l_{p^{\prime}_{r_2}})\mid c_i^++c_i^-=d_i,c_i^{\pm }\in \mathbb{Z}_{\geq 0}\}. \end{align*}

If we define $n_L$ to be the number of $D_v\in \mathscr{D}_L$ such that $\sigma ^{\ast }D_v\neq D_v$ , then we have the following:

We should note that the integer $n_L$ depends only on the Higgs divisor $D$ , and in the rest of this article, we define $n_D:={n_L \over 2}.$

6.1 Abelianization of a Higgs bundle

Let $q$ be a fixed irreducible quadratic differential with spectral curve $S$ and with normalisation $p:\widetilde{S}\to S$ . We define $\widetilde{K}:=\widetilde{\pi }^{\ast }K$ (but note that $\widetilde{K}\neq K_{\widetilde S}$ ) and $\widetilde{q}:=\widetilde{\pi }^{\ast }q\in H^0(\widetilde{K}^2)$ , where $\widetilde{\pi }$ is as in (10). Choose a square root $\omega \in H^0(\widetilde{K})$ such that $\widetilde{q}=-\omega \otimes \omega$ (that is, $\omega =p^\ast \lambda$ ). Let $\Lambda :={\rm Div}(\omega )$ and $\widetilde{Z}:={\rm supp}(\Lambda )$ . We can then write

(21) \begin{align} \Lambda =\sum _{i=1}^{r_1}{m_i \over 2}(\tilde{p}_i^++\tilde{p}_i^-)+\sum _{j=1}^{r_2}n_j\tilde{p}^{\prime}_j. \end{align}

If $\sigma :\widetilde{S}\to \widetilde{S}$ denotes the involution, then $\sigma ^{\ast }\omega =-\omega$ .

Let $(\mathcal{E},\varphi )$ be a Higgs bundle on $\Sigma$ with $\det \varphi =q$ . Consider the pull-back $(\widetilde{\mathcal{E}},\tilde{\varphi }):=(\widetilde{\pi }^{\ast }\mathcal{E},\widetilde{\pi }^{\ast }\varphi )$ to $\widetilde S$ . We have $\tilde{\varphi }\in H^0({\rm End}(\widetilde{\mathcal{E}})\otimes \widetilde{K})$ and $\widetilde{q}=\det (\tilde{\varphi })$ . Since $\widetilde{q}=-\omega \otimes \omega$ , $\pm \omega$ are well-defined eigenvalues of $\tilde{\varphi }$ over $\widetilde{S}$ . Let $\tilde{\lambda }$ be the canonical section of the pull-back of $\widetilde K$ to the total space ${\rm Tot}(\widetilde{K})$ . The spectral curve for $(\widetilde{\mathcal{E}},\tilde{\varphi })$ is defined by the equation

\begin{align*} \widetilde {S}^{\prime}:=\{\tilde {\lambda }^2-\tilde {q}=0\}. \end{align*}

The set $\widetilde{S}^{\prime}={\rm Im}(\omega )\cup {\rm Im}(-\omega )\subset {\rm Tot}(\widetilde{K})$ decomposes into two irreducible pieces:

Having fixed a choice of $\omega$ , the eigenvalues of $\tilde{\varphi }$ are globally well-defined, and we can define the line bundle $\widetilde{L}_+\subset \widetilde{\mathcal{E}}$ as $\widetilde{L}_+:=\ker (\tilde{\varphi }-\omega )$ . Since $\sigma ^{\ast }\omega =-\omega$ , $\widetilde{L}_-=\sigma ^{\ast }\widetilde{L}_+=\ker (\tilde{\varphi }+\omega )$ , and there is an isomorphism $\widetilde{\mathcal{E}}|_{\widetilde{S}\setminus \widetilde{Z}}\cong \widetilde{L}_+\oplus \widetilde{L}_-|_{\widetilde{S}\setminus \widetilde{Z}}$ .

There is also a local description of $(\widetilde{\mathcal{E}},\tilde{\varphi })$ .

Lemma 6.1. Let $x\in \widetilde{Z}$ and write $\Lambda |_x=m_xx$ . Let $U$ be a holomorphic coordinate neighborhood of $x$ . Then there exists a frame $\mathfrak{e}\in H^0(U,\widetilde{K})$ such that, under a suitable trivialization of $\mathcal{E}|_U\cong U\times \mathbb{C}^2$ , we can write

(22) \begin{align} \tilde{\varphi }=z^{d_x}\left(\begin{array}{c@{\quad}c} 0 & 1\\[4pt] z^{2m_x-2d_x} & 0 \end{array}\right) \otimes \mathfrak{e}. \end{align}

Moreover, if we define $D:=\sum _{x\in \widetilde{Z}}d_xx$ , then $D$ is a $\sigma$ -divisor.

Lemma 6.2. For the $\widetilde{L}_{\pm }$ defined above, we have $\widetilde{L}_+\otimes \widetilde{L}_-=\mathcal{O}_{\widetilde S}(D-\Lambda )$ . Moreover, if we denote $\widetilde{L}_0:=\widetilde{L}_+(\Lambda -D)$ and $\widetilde{L}_1:=\sigma ^{\ast }\widetilde{L}_0$ , then $\widetilde{L}_+=\widetilde{\mathcal{E}}\cap \widetilde{L}_0$ , $\widetilde{L}_-=\widetilde{\mathcal{E}}\cap \widetilde{L}_1$ , and we have the exact sequences

\begin{align*} 0\longrightarrow \widetilde{L}_+\longrightarrow \widetilde{\mathcal{E}} \longrightarrow \widetilde{L}_1\longrightarrow 0\ ;\\ 0\longrightarrow \widetilde{L}_-\longrightarrow \widetilde{\mathcal{E}} \longrightarrow \widetilde{L}_0\longrightarrow 0. \end{align*}

Proof. The inclusion of $\widetilde{L}_{\pm }\to \widetilde{\mathcal{E}}$ defines an exact sequence:

\begin{align*} 0\longrightarrow \widetilde {L}_+\oplus \widetilde {L}_-\longrightarrow \widetilde {\mathcal {E}}\longrightarrow \mathcal {T}\longrightarrow 0\, \end{align*}

where $\mathcal{T}$ is a torsion sheaf with ${\rm supp\;} \mathcal{T}\subset \widetilde{Z}$ . From the local description in (22), in the same trivialization, $\widetilde{L}_{\pm }$ are spanned by the bases $s_{\pm }=\binom{1}{\pm z^{m_x-d_x}}.$ Therefore, as $\det (\mathcal{E})=\mathcal{O}_\Sigma$ , we obtain $\widetilde{L}_+\otimes \widetilde{L}_-=\mathcal{O}_{\widetilde S}(D-\Lambda )$ . Since $s_+,s_-$ are linear independent away from $z$ , $\widetilde{\mathcal{E}}/\widetilde{L}_+$ is locally generated by the section $z^{d_x-m_x}s_-$ . Therefore, $\widetilde{\mathcal{E}}/\widetilde{L}_+\cong \widetilde{L}_-(\Lambda -D)=\widetilde{L}_1$ . Using the involution, we obtain the other exact sequence.

Therefore, if $\widetilde{L}\otimes \sigma ^{\ast }\widetilde{L}=\mathcal{O}_{\widetilde S}(D-\Lambda )$ , we have $\widetilde{L}_0=\widetilde{L}(\Lambda -D)\in \widetilde{\mathcal{T}}_D$ . In summary, the construction above leads us to consider the composition of the following maps given by the composition

\begin{align*} \delta : \mathcal{M}_q\rightarrow \widetilde{\mathcal{T}}_D,\;\;(\mathcal{E},\varphi )\mapsto \widetilde{L}_+\mapsto \widetilde{L}_+(\Lambda -D)\, \end{align*}

where the first map is obtained by taking the kernel of $(\tilde{\pi }^{\ast }\varphi -\omega )|_{\tilde{\pi }^{\ast }\mathcal{E}}$ .

This procedure is directly related to the torsion-free pull-back. Recall that $\chi _{{\rm BNR}}:\overline{\mathcal{T}}\to \mathcal{M}_q$ is the BNR correspondence map and that $p^{\star }_{{\rm tf}}:\overline{{\rm Pic}}(S)\to {\rm Pic}(\widetilde{S})$ is the torsion-free pull-back. Then we have