$\textstyle{{\widetilde{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{h}_{{\widetilde{C}}}}$$\scriptstyle{\pi}$$\scriptstyle{\mathsf{a}}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{h}_{\mathbb{P}^{1}}}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{h}_{D}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{h}_{C}}$$\textstyle{\mathbb{P}^{1}}$
(6)

where the maps $\mathsf{h}_{{\widetilde{C}}}$, $\mathsf{a}$, and $\pi$ have
covering involutions $\sigma$, $\rho$, and $\tau$ respectively.

Therefore the divisor $\tilde{t}_{1}+\tau(\tilde{t}_{2})$ is a fiber of the
map $\mathsf{h}_{{\widetilde{C}}}$ if and only if
$\tilde{t}_{1}=\sigma\circ\tau(\tilde{t}_{2})=\rho(\tilde{t}_{2})$. So in this
case the identity (5) becomes

$\mathcal{O}_{{\widetilde{C}}}(\tilde{t}_{1}-\rho(\tilde{t}_{1}))=\pi^{*}(A_{1}\otimes
A_{2}^{-1}).$ (7)

Note that the line bundle
$\mathcal{O}_{{\widetilde{C}}}(\tilde{t}_{1}-\rho(\tilde{t}_{1}))$ is a
$\rho$-antiinvariant line bundle on ${\widetilde{C}}$, i.e. as a point in
$\operatorname{Jac}^{0}({\widetilde{C}})$ it belongs to the subgroup

$\ker\left[\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
16.35837pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Jac}^{0}({\widetilde{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
18.8862pt\raise 5.54723pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.9639pt\hbox{$\scriptstyle{1+\sigma}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 40.35837pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
40.35837pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Jac}^{0}({\widetilde{C}})}$}}}}}}}\ignorespaces}}}}\ignorespaces\right]=\ker\left[\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
18.35837pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Jac}^{0}({\widetilde{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
20.8167pt\raise 5.82222pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.9611pt\hbox{$\scriptstyle{\mathsf{Nm}_{\mathsf{a}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 42.35837pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
42.35837pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Jac}^{0}(C)}$}}}}}}}\ignorespaces}}}}\ignorespaces\right]\
\subset\operatorname{Jac}^{0}({\widetilde{C}}).$

From the classical theory of Prym varieties [Mum74] of étale double covers, it
is known that $\ker(1+\rho)=\ker\mathsf{Nm}_{\mathsf{a}}$ is a disconnected
abelian subgroup with two connected components, and a connected component of
the identity equal to the (degree zero) Prym variety for the pair
$({\widetilde{C}},D)$:

$\mathsf{Prym}({\widetilde{C}},D)=\operatorname{im}\left[\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
16.35837pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Jac}^{0}({\widetilde{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
20.44176pt\raise 5.25555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{1-\sigma}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 40.35837pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
40.35837pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Jac}^{0}({\widetilde{C}})}$}}}}}}}\ignorespaces}}}}\ignorespaces\right].$

Next consider the Abel-Prym map

$\mathfrak{ap}_{({\widetilde{C}},D)}^{k}:\operatorname{Sym}^{k}{\widetilde{C}}\longrightarrow\ker\mathsf{Nm}_{\mathsf{a}}\subset\operatorname{Jac}^{0}({\widetilde{C}}),\
\mathfrak{d}\mapsto\mathcal{O}_{{\widetilde{C}}}(\mathfrak{d}-\rho(\mathfrak{d})).$

As explained in [Mum74] the image of this map lands in the identity component

$\mathsf{Prym}({\widetilde{C}},D)=\left(\ker\mathsf{Nm}_{\mathsf{a}}\right)_{o}\subset\ker\mathsf{Nm}_{\mathsf{a}}$

if and only if $k$ is even. Thus the image of Abel-Prym map

$\mathfrak{ap}_{({\widetilde{C}},D)}^{1}:{\widetilde{C}}\longrightarrow\ker\mathsf{Nm}_{\mathsf{a}}\subset\operatorname{Jac}^{0}({\widetilde{C}}),\
\tilde{t}\mapsto\mathcal{O}_{{\widetilde{C}}}(\tilde{t}-\rho(\tilde{t})),$

is contained in the non-neutral component of $\ker\mathsf{Nm}_{\mathsf{a}}$
and so the curve
$\mathfrak{ap}_{({\widetilde{C}},D)}^{1}({\widetilde{C}})\subset\operatorname{Jac}^{0}({\widetilde{C}})$
is disjoint444In our setting we can see this fact directly without appealing
to Mumford’s parity analysis [Mum74]. Indeed, by definition the curve $D$ is
the double cover of $\mathbb{P}^{1}$ branched at the $8$ points which are the
union of the $6$ branch points of $\mathsf{h}_{C}:C\to\mathbb{P}^{1}$ and the
two points $\boldsymbol{\mathsf{y}},\boldsymbol{\mathsf{z}}\in\mathbb{P}^{1}$.
Let $\mathsf{Y}$ denote the ramification point of
$\mathsf{h}_{D}:D\to\mathbb{P}^{1}$, sitting over
$\boldsymbol{\mathsf{y}}\in\mathbb{P}^{1}$. Then $\mathsf{a}^{-1}(\mathsf{Y})$
consists of two distinct points $\tilde{y},\rho(\tilde{y})$, which are both
ramification points of $\pi:{\widetilde{C}}\to C$. But in Case 2. above we
argued that the line bundle
$\mathcal{O}_{{\widetilde{C}}}(\tilde{y}-\rho(\tilde{y}))$ cannot be a
pullback from $\operatorname{Jac}^{0}(C)$. Thus the point
$\mathfrak{ap}_{({\widetilde{C}},D)}^{1}(\tilde{t})$ must belong to the non-
neutral component of $\ker\mathsf{Nm}_{\mathsf{a}}$. Since
$\mathfrak{ap}_{({\widetilde{C}},D)}^{1}({\widetilde{C}})$ is connected, this
implies that $\mathfrak{ap}_{({\widetilde{C}},D)}^{1}({\widetilde{C}})$ is
entirely contained in the non-neutral component.  from the two dimensional
abelian subvariety
$\mathsf{Prym}({\widetilde{C}},D)\subset\operatorname{Jac}^{0}({\widetilde{C}})$
inside $\operatorname{Jac}^{0}({\widetilde{C}})$. But the configuration (6) of
double covers implies that

$\mathsf{Prym}({\widetilde{C}},D)=\pi^{*}\operatorname{Jac}^{0}(C)\ \subset\
\operatorname{Jac}^{0}({\widetilde{C}}),$

and hence the equation (7) has no solution.

This completes the analysis of the third case and shows that the map (4) is
injective. To check that the map (4) also separates tangent directions,
consider the degree six version of the Prym for $({\widetilde{C}},C)$:

${\mathcal{P}}_{6}=\left\\{\left.M\in\operatorname{Jac}^{6}({\widetilde{C}})\,\right|\,\mathsf{Nm}_{\pi}(M)=\omega_{C}^{\otimes
3}\right\\}.$

We have a natural multiplication-by-2 map

$\mathsf{mult}_{2}:{\mathcal{P}}_{3}\to{\mathcal{P}}_{6},\quad M\mapsto
M^{\otimes 2},$

which fits in a commutative diagram

$\textstyle{{\widehat{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{{\mathsf{sq}}}}$$\textstyle{{\mathcal{P}}_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{mult}_{2}}$$\textstyle{{\widetilde{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{P}}_{6}}$
(8)

where the top horizontal map is the map (4), while the bottim horizontal map
${\widetilde{C}}\to{\mathcal{P}}_{6}$ is given by
$\tilde{t}\mapsto\pi^{*}(\omega_{C}(\boldsymbol{\mathsf{p}}))\otimes\mathcal{O}_{{\widetilde{C}}}(\tilde{t}-\tau(\tilde{t}))$.

Now observe that for any $\tilde{t}\in{\widetilde{C}}$ the divisor
$\tilde{t}+\sigma(\tilde{t})$ is in the hyperelliptic linear system on
${\widetilde{C}}$. By the same token
$\rho(\tilde{t})+\sigma\circ\rho(\tilde{t})=\rho(\tilde{t})+\tau(\tilde{t})$
is in the hyperelliptic linear system on ${\widetilde{C}}$. Thus we have a
linear equivalence

$\tilde{t}+\sigma(\tilde{t})\sim\rho(\tilde{t})+\tau(\tilde{t})$

and so

$\mathcal{O}_{{\widetilde{C}}}(\tilde{t}-\tau(\tilde{t}))\cong\mathcal{O}_{{\widetilde{C}}}(\rho(\tilde{t})-\sigma(\tilde{t}))=\mathcal{O}_{{\widetilde{C}}}(\rho(\tilde{t})-\tau(\rho(\tilde{t}))).$

Hence the map ${\widetilde{C}}\to{\mathcal{P}}_{6}$ factors through $D$. The
induced map $D\to\operatorname{Jac}^{0}({\widetilde{C}})$ is given by
$s\mapsto
a^{*}\mathcal{O}_{D}(s)\otimes\mathsf{h}_{{\widetilde{C}}}\mathcal{O}_{\mathbb{P}^{1}}(-1)$
for all $s\in D$. This shows that the map
${\widetilde{C}}\to{\mathcal{P}}_{6}$ factors as

$\textstyle{{\widetilde{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{a}}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{aj}}$$\textstyle{\operatorname{Jac}^{1}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{*}(-)\otimes\mathsf{h}_{{\widetilde{C}}}\mathcal{O}_{\mathbb{P}^{1}}(-1)\otimes\pi^{*}(\omega_{C}(\boldsymbol{\mathsf{p}}))}$$\textstyle{{\mathcal{P}}_{6}.}$

The map $\operatorname{Jac}^{1}(D)\to{\mathcal{P}}_{6}$ is an étale double
cover which is isomorphic to the quotient of $\operatorname{Jac}^{1}(D)$ by
the the translation action of the $2$-torsion line bundle defining the cover
${\widetilde{C}}\to D$ [Mum74]. From the diagram (6) we see that this
$2$-torsion line bundle is given explicitly as
$\mathcal{O}_{D}(\mathsf{Y}-\mathsf{Z})$, where $\mathsf{Y}$ and $\mathsf{Z}$
are the two Weierstrass points in $D$ that sit over
$\boldsymbol{\mathsf{y}},\boldsymbol{\mathsf{z}}\in\mathbb{P}^{1}$. Since the
Abel-Jacobi map $\mathfrak{aj}:D\to\operatorname{Jac}^{1}(D)$ is a closed
embedding, this shows that the map $D\to{\mathcal{P}}_{6}$ embeds
$D-\\{\mathsf{Y},\mathsf{Z}\\}$ and glues $\mathsf{Y}$ and $\mathsf{Z}$ into a
node. If we write $\underline{D}$ for the iimage of this nodal curve in
${\mathcal{P}}_{6}$ we get that the map
${\widetilde{C}}\to\underline{D}\subset{\mathcal{P}}_{6}$ is injective on
tangent spaces. But the map
$\widehat{{\mathsf{sq}}}:{\widehat{C}}\to{\widetilde{C}}$ is étale, and so the
composition
${\widehat{C}}\to{\widetilde{C}}\to\underline{D}\subset{\mathcal{P}}_{6}$ is
injective on tangent spaces. From the diagram (8) we see that this map also
factors as ${\widehat{C}}\to{\mathcal{P}}_{3}\to{\mathcal{P}}_{6}$ and so the
${\widehat{C}}\to{\mathcal{P}}_{3}$ is injective on tangent spaces. This
completes the proof of the lemma. ∎

###  3.3 The wobbly locus

We recall that Laumon defines the notion of very stable vector bundle as one
that does not admit a non-zero nilpotent Higgs field. Such bundles are
automatically stable [Lau88]. The first two authors therefore introduced the
complementary notion of wobbly as a semistable vector bundle that isn’t very
stable. The wobbly locus ${\rm Wob}\subset X$ thus consists of those
polystable vector bundles that admit a nonzero nilpotent Higgs field.

With indexation on the degree our notations for the wobbly loci are:

${\rm Wob}_{0}\subset X_{0},\;\;\;\;{\rm Wob}_{1}\subset X_{1}.$

These will be the supports for the parabolic structures and logarithmic poles
of the Higgs fields for the Higgs bundles (1).

In section 5 we show that the wobbly locus ${\rm Wob}_{0}$ is a divisor in the
$2$-theta space of $C$ comprising several familiar players in the classical
geometry of the quadric line complex. Specifically the divisor ${\rm Wob}_{0}$
in $X_{0}\cong\mathbb{P}^{3}$ has $17$ irreducible components: the quartic
Kummer surface $\mathsf{Kum}\subset\mathbb{P}^{3}$ and its $16$ trope planes
$\mathsf{Trope}_{\kappa}$, labeled by the theta characteristics of $C$. Thus

${\rm
Wob}_{0}=\mathsf{Kum}\cup\left[\cup_{\kappa\in\mathsf{Spin}(C)}\mathsf{Trope}_{\kappa}\right].$

From the classical $16_{6}$ configuration in the quadratic line complex [GH94]
we know that each trope plane $\mathsf{Trope}_{\kappa}$ is tangent to the
Kummer surface $\mathsf{Kum}$ along a trope conic $\mathfrak{C}_{\kappa}$.
Hence ${\rm Wob}_{0}$ fails to be normal crossing along the trope conics
$\left\\{\mathfrak{C}_{\kappa}\right\\}_{\kappa\in\mathsf{Spin}(C)}$.

It is also easy to characterize the components of the wobbly divisor from the
moduli point of view. The Kummer surface $\mathsf{Kum}$ parametrizes
$S$-equivalence classes of semistable bundles with polystable representatives
of the form $\mathfrak{a}\oplus\mathfrak{a}^{-1}$ for
$\mathfrak{a}\in\operatorname{Jac}^{0}(C)$. The trope plane
$\mathsf{Trope}_{\kappa}$ can be canonically identified with
$\mathbb{P}(H^{1}(C,\kappa^{-\otimes
2}))=\mathbb{P}(H^{1}(C,\omega_{C}^{-1}))$ and parametrizes all bundles that
can be realized as non-split extensions $0\to\kappa^{-1}\to F\to\kappa\to 0$.

Similarly, in section 4 we show that ${\rm Wob}_{1}\subset X_{1}$ is an
irreducible divisor. Specifically, as mentioned in the previous section, ${\rm
Wob}_{1}$ parametrizes all vector bundles $E\in X_{1}$ which arise as non-
split extensions $0\to A\to E\to A^{-1}(\boldsymbol{\mathsf{p}})\to 0$ for
some point $(A,t)\in{\overline{C}}$. In fact we will see that ${\rm Wob}_{1}$
is the tangent developable of the map $\mathsf{ql}:{\overline{C}}\to
X_{1}\subset\mathbb{P}^{5}$, i.e. the union of all projective tangent lines to
points in the curve $\mathsf{ql}({\overline{C}})$. This shows that ${\rm
Wob}_{1}$ is also a non normal crossings divisor in $X_{1}$. It has the curve
$\mathsf{ql}({\overline{C}})$ as its curve of cusps.

###  3.4 Hecke correspondences

The big Hecke correspondence ${\overline{\mathcal{H}}}\rightarrow X_{1}\times
X_{0}\times{\overline{C}}$ is the moduli of quadruples $(E,F,(A,t),\beta)$
such that $(E,F,(A,t))\in X_{1}\times X_{0}\times{\overline{C}}$ and
$\beta:F\otimes A^{-1}\to E$ is a map that fits in a short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F\otimes
A^{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{C}}_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

We will view ${\overline{\mathcal{H}}}$ either as a correspondence from
$X_{1}$ to $X_{0}\times{\overline{C}}$ or as a correspondence from $X_{0}$ to
$X_{1}\times{\overline{C}}$. We will label the respective projections as

$\textstyle{{\overline{\mathcal{H}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{X_{1}}$$\textstyle{X_{0}\times{\overline{C}}}$
$\textstyle{{\overline{\mathcal{H}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{b}$$\textstyle{X_{0}}$$\textstyle{X_{1}\times{\overline{C}}}$
(9)

where $p=\operatorname{pr}_{X_{1}}$, $d=\operatorname{pr}_{X_{0}}$,
$q=\left(\operatorname{pr}_{X_{0}},\operatorname{pr}_{{\overline{C}}}\right)$,
and
$b=\left(\operatorname{pr}_{X_{0}},\operatorname{pr}_{{\overline{C}}}\right)$.
In fact the moduli ${\overline{\mathcal{H}}}$ is a subvariety in the triple
product $X_{0}\times X_{1}\times{\overline{C}}$. To see this, note that when
$\beta$ exists for a given point $(E,F,(A,t))\in X_{0}\times
X_{1}\times{\overline{C}}$, then it is unique up to scale:
${\mathcal{O}}_{C}(t)$, and hence $t$, is determined as the ratio of
determinants, and then for this given $t$, the Hecke fiber is a line in one
direction, a conic in the other which excludes any self-intersections.

In some parts of the discussion we will look at Hecke transformations
supported at a single point. In those situations we will write
${\overline{\mathcal{H}}}(a)=\operatorname{pr}_{{\overline{C}}}^{-1}(\\{a\\})$
for the preimage of $a=(A,t)\in{\overline{C}}$ in ${\overline{\mathcal{H}}}$
and will view ${\overline{\mathcal{H}}}(a)$ as a correspondence between
$X_{0}$ and $X_{1}$.

The (big) Hecke correspondences appeared in Chapter 2, Sections 2.4 and 2.6,
from the synthetic viewpoint. We will review the comparison between the
synthetic and modular viewpoints in Section 6.3.

The big abelianized Hecke correspondence ${\widehat{\mathcal{H}}}^{\rm ab}$ is
the blow-up of $Y_{0}\times{\widehat{C}}$ along a copy of
${\widehat{C}}\times{\widehat{C}}$. As we will see in section 7.2 the map

$\textstyle{\boldsymbol{\imath}_{{\widehat{C}}\times{\widehat{C}}}:}$$\textstyle{{\widehat{C}}\times{\widehat{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{P}}_{2}\times{\widehat{C}}}$$\textstyle{((A_{1},\tilde{t}_{1}),(A_{2},\tilde{t}_{2}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(\pi^{*}\left(A_{2}\otimes
A_{1}^{\vee}(\boldsymbol{\mathsf{p}})\right)(\tilde{t}_{1}-\tilde{t}_{2}),(A_{2},\tilde{t}_{2})\right),}$

is a closed embedding. This embedding clearly preserves the second projections
to ${\widehat{C}}$ and the strict transform of
${\widehat{C}}\times{\widehat{C}}$ in $Y_{0}\times{\widehat{C}}$ is isomorphic
copy of ${\widehat{C}}\times{\widehat{C}}$ embedded as a subvariety of
$Y_{0}\times{\widehat{C}}$. The blow-up of $Y_{0}\times{\widehat{C}}$ along
this copy of ${\widehat{C}}\times{\widehat{C}}$ is the big abelianized Hecke
correspondence ${\widehat{\mathcal{H}}}^{\rm ab}$.

The variety ${\widehat{\mathcal{H}}}^{\rm ab}$ maps to $Y_{0}\times
Y_{1}\times{\widehat{C}}$ and thus gives correspondences

$\textstyle{{\widehat{\mathcal{H}}}^{\rm
ab}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\rm
ab}}$$\scriptstyle{q^{\rm
ab}}$$\textstyle{Y_{1}}$$\textstyle{Y_{0}\times{\widehat{C}}}$
$\textstyle{{\widehat{\mathcal{H}}}^{\rm
ab}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{\rm
ab}}$$\scriptstyle{b^{\rm
ab}}$$\textstyle{Y_{0}}$$\textstyle{Y_{1}\times{\widehat{C}}}$

There is also map

$g:{\widehat{\mathcal{H}}}^{\rm ab}\rightarrow{\overline{\mathcal{H}}},$

such that altogether we get a map of abelianized and usual Hecke
correspondences. The formulas for the maps $p^{\rm ab}$, $q^{\rm ab}$, $d^{\rm
ab}$, $b^{\rm ab}$, and $g$ are given in section 7.2.

Again, most of the time, we will work with a single fiber of the abelianized
Hecke correspondence. If $\tilde{a}=(A,\tilde{t})\in{\widehat{C}}$ is a fixed
point we will focus on the fiber ${\widehat{\mathcal{H}}}^{\rm
ab}(\tilde{a})=\operatorname{pr}_{{\widehat{C}}}^{-1}(\tilde{a})$.

###  3.5 Spectral line bundles

In the abelianization strategy to the Hecke eigensheaf problem we use spectral
data on $C$ to describe the eigenvalue Higgs bundle $(E,\theta)$ and spectral
data on $X_{0}$ and $X_{1}$ to describe the eigensheaf parabolic Higgs bundles
$(\mathscr{F}_{0,\bullet},\Phi_{0})$ and $(\mathscr{F}_{1,\bullet},\Phi_{1})$.
The spectral covers for these spectral data are $\pi:{\widetilde{C}}\to C$ and
$f_{0}:Y_{0}\to X_{0}$ and $f_{1}:Y_{1}\to X_{1}$ respectively.

Suppose $({\widetilde{C}}\subset T^{\vee}C,\boldsymbol{\mathit{N}})$ is the
spectral data for $(E,\theta)$, i.e.
$(E,\theta)=\left(\pi_{*}\boldsymbol{\mathit{N}},\pi_{*}\left(\lambda\otimes(-)\right)\right)$,
where $\lambda:{\widetilde{C}}\to T^{\vee}C$ is the embedding, and
$\boldsymbol{\mathit{N}}\in{\mathcal{P}}_{2}$ is the spectral line bundle. We
will use $\boldsymbol{\mathit{N}}$ to construct the spectral line bundles
${\mathscr{L}}_{0}$ on $Y_{0}$, and ${\mathscr{L}}_{1}$ on $Y_{1}$ which will
define our parabolic eigensheaf Higgs bundles.

An appropriately normalized Fourier-Mukai transform on the Jacobian of
${\widetilde{C}}$ (see section 7.2 for the precise details of the
normalization) sends the skyscraper sheaf
$\mathcal{O}_{\boldsymbol{\mathit{N}}}\in
D^{b}_{\operatorname{coh}}(\operatorname{Jac}^{2}({\widetilde{C}}))$ to a line
bundle on $\operatorname{Jac}^{2}({\widetilde{C}})$ with a vanishing first
Chern class. We denote the restriction of this line bundle to
${\mathcal{P}}_{2}\subset\operatorname{Jac}^{2}({\widetilde{C}})$ by
$\mathfrak{L}_{0}$ and define a spectral line bundle $\mathscr{L}_{0}$ on the
modular spectral cover $f_{0}:Y_{0}\to X_{0}$ by setting

$\mathscr{L}_{0}=\left(\boldsymbol{\varepsilon}_{0}^{*}\mathfrak{L}_{0}\right)(\boldsymbol{\mathit{E}}_{0})\otimes
f_{0}^{*}\mathcal{O}_{X_{0}}(2)\ \text{in}\ \operatorname{Pic}(Y_{0}).$

Using the modular spectral data $(Y_{0},\mathscr{L}_{0})$ we can now define a
meromorphic Higgs bundle on $X_{0}$ by

$\left({\mathscr{F}}_{0,0},\Phi_{0}\right):=\left(f_{0*}\mathscr{L}_{0},f_{0*}(\boldsymbol{\alpha}_{0}\otimes(-))\right),$

where $\boldsymbol{\alpha}_{0}:Y_{0}\to T^{\vee}_{X_{0}}(\log{\rm Wob}_{0})$
is the tautological map, defined away from the tacnodes of ${\rm Wob}_{0}$. We
will later provide this with a parabolic structure to build a parabolic rank
$8$ bundle ${\mathscr{F}}_{0,\bullet}$. Away from the non normal crossing
codimension two strata of ${\rm Wob}_{0}$, the parabolic structures are
trivial; the bundle needs to have a parabolic structure on a blown-up version
of $X_{0}$ at the tacnodes. The geometry of the blow-up and the construction
of the parabolic structure will be explained in section 5 and section 9.

In this analysis we will also see that the Higgs field of
$\Phi_{0}:{\mathscr{F}}_{0,\bullet}\to{\mathscr{F}}_{0,\bullet}\otimes\Omega^{1}_{X_{0}}(\log{\rm
Wob}_{0})$ has logarithmic poles on both the trope planes and the Kummer
surface. The residues are nilpotent and have Jordan blocks as follows: two
Jordan blocks of size $2$ and four of size $1$ over the tropes, and four
Jordan blocks of size $2$ over the Kummer.

To describe the spectral line bundle $\mathscr{L}_{1}$ on $Y_{1}$ it is
convenient to fix a point
$\widetilde{\boldsymbol{\mathsf{p}}}\in{\widetilde{C}}$ lying over
$\boldsymbol{\mathsf{p}}$. Recall the assumption that the spectral curve has
two branches over $\boldsymbol{\mathsf{p}}$.

The choice of $\widetilde{\boldsymbol{\mathsf{p}}}$ gives an isomorphism
$\boldsymbol{t}_{-\widetilde{\boldsymbol{\mathsf{p}}}}:{\mathcal{P}}_{3}\stackrel{{\scriptstyle\cong}}{{\rightarrow}}{\mathcal{P}}_{2}$
sending $L$ to $L(-\widetilde{\boldsymbol{\mathsf{p}}})$ and we can pullback
$\mathfrak{L}_{0}$ by this isomorphism to define a line bundle
$\mathfrak{L}_{1}=\boldsymbol{t}_{-\widetilde{\boldsymbol{\mathsf{p}}}}^{*}\mathfrak{L}_{0}$
on ${\mathcal{P}}_{3}$ with vanishing first Chern class. With this notation we
now set

$\mathscr{L}_{1}:=\boldsymbol{\varepsilon}_{1}^{*}\mathfrak{L}_{1}\otimes\mathcal{O}_{X_{1}}(1),$

and

$\left({\mathscr{F}}_{1,0},\Phi_{1}\right):=\left(f_{1*}\mathscr{L}_{1},f_{1*}\left(\boldsymbol{\alpha}_{1}\otimes(-)\right)\right).$

In section 4 this is going to be given a parabolic structure over ${\rm
Wob}_{1}$, with parabolic weights $0$ and $1/2$, such that the associated
graded bundle of grading $1/2$ has rank $2$. This means that

${\mathscr{F}}_{1,s}={\mathscr{F}}_{1}\;\;\;\mbox{for}\;\;\;s\in[0,1/2).$

###  3.6 Orbits of the ${\mathbb{C}}^{\times}$-action

An important aspect of the geometry of the Hitchin moduli space is the
${\mathbb{C}}^{\times}$-action. In this section, $\mathsf{Higgs}$ will denote
a moduli space of Higgs bundles of some degree and fixed determinant. In our
situation, it means $\mathsf{Higgs}=\mathsf{Higgs}_{0}$ or
$\mathsf{Higgs}_{1}$. Some of this discussion takes place in more general
settings so the notation is left non-specific when possible.

An element $z\in{\mathbb{C}}^{\times}$ sends $(E,\theta)\in\mathsf{Higgs}$ to
$(E,z\theta)$. The fixed-point locus $\mathsf{Higgs}^{{\mathbb{C}}^{\times}}$
decomposes

$\mathsf{Higgs}^{{\mathbb{C}}^{\times}}=\mathsf{Higgs}^{{\mathbb{C}}^{\times},\rm
u}\sqcup\mathsf{Higgs}^{{\mathbb{C}}^{\times},\rm nu}$

into the “unitary piece” that is just the moduli of bundles (= moduli of
bundles equipped with zero Higgs fields), i.e.
$\mathsf{Higgs}^{{\mathbb{C}}^{\times},\rm u}\cong X$, and the disjoint union
of remaining pieces that are moduli spaces of polystable Hodge bundles with
nonzero Higgs field.

We recall (see [Sim92]) that a Higgs bundle is called a Hodge bundle if
$E=\bigoplus E^{p}$ and $\theta:E^{p}\rightarrow E^{p-1}\otimes\omega_{C}$.
These correspond under the nonabelian Hodge correspondence to complex
variations of Hodge structure. In our case, we consider bundles of rank $2$,
so a non-unitary Hodge bundle is a direct sum of two line bundles

$E\cong L^{1}\oplus L^{0},\;\;\;\;\theta:L^{1}\rightarrow
L^{0}\otimes\omega_{C}.$

We remark that a Hodge bundle with a nonzero Higgs field that is semistable
but not stable could be $S$-equivalent to a polystable unitary Hodge bundle,
that is a to a Hodge bundle with a zero Higgs field. This will happen in our
case if $L^{p}$ are both line bundles of degree $0$. Those don’t count as
points in $\mathsf{Higgs}^{{\mathbb{C}}^{\times},\rm nu}$ since the moduli
space parametrizes $S$-equivalence classes and the polystable representative
would be the same bundle with trivial Higgs field.

######  Proposition 3.3.

For the case of rank $2$ Higgs bundles on a curve of genus $2$ considered in
this paper, the non-unitary fixed point loci in $\mathsf{Higgs}_{0}$ and
$\mathsf{Higgs}_{1}$ are described as follows:

  * (a)

$\mathsf{Higgs}_{0}^{{\mathbb{C}}^{\times},\rm nu}$ is a disjoint union of
$16$ points parametrizing the uniformizing Higgs bundles
$\\{(E_{\kappa},\theta_{\kappa})\\}_{\kappa\in\mathsf{Spin}(C)}$, where
$E_{\kappa}=\kappa\oplus\kappa^{-1}$, and
$\theta_{\kappa}:\kappa\to\kappa^{-1}\otimes\omega_{C}=\kappa$ is an
isomorphism;

  * (b)

$\mathsf{Higgs}_{1}^{{\mathbb{C}}^{\times},\rm nu}$ is connected, isomorphic
to the curve ${\overline{C}}$ that is a $16$-sheeted étale covering of $C$.

###### Proof.

Suppose $E\cong L^{1}\oplus L^{0}$ is a Hodge bundle. Polystability with
nontrivial Higgs field implies (since we are in the rank $2$ case) that the
Higgs bundle is in fact stable, and $(L^{0},0)$ is a sub-Higgs bundle, so
$\deg(L^{0})<(\deg E)/2$. If $E$ has degree $0$ with determinant
${\mathcal{O}}_{C}$ then $L^{0}=(L^{1})^{\vee}$, $\deg L^{0}<0$, and the only
possibility is that $\deg(L^{0})=-1$, the Higgs field $\theta:L^{1}\to
L^{0}\otimes\omega_{C}$ is an isomorphism, and hence $L^{1}=\kappa$ is one of
the $16$ square-roots of the canonical bundle. If $E$ has degree $1$ with
determinant ${\mathcal{O}}_{C}(\boldsymbol{\mathsf{p}})$ then $E\cong L\oplus
L^{-1}(\boldsymbol{\mathsf{p}})$, $\deg L\leq 0$, and
$\theta:L^{-1}(\boldsymbol{\mathsf{p}})\to L\otimes\omega_{C}$ is a non-zero
map. Therefore we must have $\deg L=0$ and so the Higgs field $\theta\in
H^{0}(C,L^{\otimes 2}(\boldsymbol{\mathsf{p}}))$ has a single zero at some
point $t\in C$, and we get
$L^{\otimes-2}\cong{\mathcal{O}}_{C}(\boldsymbol{\mathsf{p}}-t)$. This
corresponds to a point of the connected curve ${\overline{C}}$. ∎

The Hitchin fibration
$\boldsymbol{h}:\mathsf{Higgs}\rightarrow{\mathcal{B}}={\mathbb{A}}^{N}$ is
equivariant for the ${\mathbb{C}}^{\times}$-action, where the weights of the
action on the base are strictly positive, determined by the degrees of the
invariant polynomials on the Lie algebra or, equivalently, by the expression
in terms of sections of powers of $\omega_{C}$. In the present case, the
Hitchin base is simply $H^{0}(C,\omega_{C}^{\otimes 2})$ so there is a single
weight $2$.

Properness of the Hitchin fibration implies that if $y\in\mathsf{Higgs}$ is
any point, then the limit $\lim_{z\rightarrow 0}(zy)$ exists and is a
${\mathbb{C}}^{\times}$-fixed point. This fixed point is a unitary Higgs
bundle, i.e. a semistable vector bundle with zero Higgs field, if and only if
$y=(E,\theta)$ with semistable underlying vector bundle $E$, and in this case
$\lim_{z\rightarrow 0}(zy)=(E,0)$. Let $\mathsf{Higgs}^{\rm
su}\subset\mathsf{Higgs}^{\rm seu}\subset\mathsf{Higgs}$ denote the open
subsets of Higgs bundles whose underlying vector bundle is stable or
semistable respectively. The limiting construction provides a regular map
$\mathsf{Higgs}^{\rm seu}\rightarrow X$ to the moduli space of semistable
bundles.

Over the open subset of stable bundles, the map that we will write as
$\mathsf{Higgs}^{\rm su}\rightarrow X^{\rm s}$ may be identified with the
cotangent bundle

$\mathsf{Higgs}^{\rm su}\cong T^{\vee}(X^{\rm s}),$

compatibly with ${\mathbb{C}}^{\times}$ actions, the one on the right being
the scaling action on the total space of the cotangent bundle. This
identification preserves the symplectic structure [Hit87b], in particular the
fiber over any point of $X^{\rm s}$ is a Lagrangian subspace.

Let $X^{\rm vs}\subset X$ be the open subset of very stable bundles, and let
$\mathsf{Higgs}^{\rm vs}$ be the open subset of Higgs bundles whose underlying
bundle is very stable. We have $\mathsf{Higgs}^{\rm
vs}\subset\mathsf{Higgs}^{\rm su}$ and also $\mathsf{Higgs}^{\rm vs}\cong
T^{\vee}(X^{\rm vs})$.

The limiting map over the open subset $\mathsf{Higgs}^{\rm seu}$ provides a
rational map

$\mathsf{Higgs}\dashrightarrow X.$

One may think of this in terms of broken orbits. Those are defined as maximal
${\mathbb{C}}^{\times}$-invariant subsets of the form

$Z_{0}\cup Z_{1}\cup\cdots\cup Z_{k}\subset\mathsf{Higgs}$

such that $Z_{0}\cong{\mathbb{A}}^{1}$ and $Z_{i}\cong{\mathbb{P}}^{1}$ for
$i=1,\ldots,k$, provided with points $I_{i},O_{i}\in Z_{i}$ (except $I_{0}$)
such that $O_{i}=I_{i+1}$, where $O_{i}$ corresponds to the origin and $I_{i}$
to $\infty$ in $Z_{i}\cong{\mathbb{P}}^{1}$ (or just the origin for
$Z_{0}\cong{\mathbb{A}}^{1}$), such that everything is compatible with the
${\mathbb{C}}^{\times}$ action. The input point $y$ corresponding to
$1\in{\mathbb{A}}^{1}=Z_{0}$ is mapped by the correspondence of the rational
map, to the output point $O_{k}\in\mathsf{Higgs}^{{\mathbb{C}}^{\times},\rm
u}$. The intermediate points $O_{i}=I_{i+1}$ are non-unitary fixed points. The
data needed to determine such a broken orbit consists of fixing the downward
or outgoing direction (see below) of the next orbit $Z_{i+1}$ whenever the
limiting point $O_{i}$ is a non-unitary fixed point. The process stops when we
get to a unitary fixed point.

If we start at a general point $y_{\varepsilon}$ nearby to $y$, the orbit of
$y_{\varepsilon}$ limiting to a unitary fixed point, will be near to a broken
orbit starting at $y$.

Let $\mathfrak{Q}\subset\mathsf{Higgs}$ be the complement of
$\mathsf{Higgs}^{\rm seu}$. It is the set of points $y$ such that
$\lim_{z\rightarrow 0}(zy)\in\mathsf{Higgs}^{{\mathbb{C}}^{\times},\rm nu}$.
The limiting map provides a constructible map from $\mathfrak{Q}$ to the non-
unitary fixed point set. In general, this will not be a regular map. However,
in our case there are not very many non-unitary fixed points so broken orbits
have at most one break. Points of $\mathfrak{Q}$ are those having one break
which is a single well-defined point of
$\mathsf{Higgs}^{{\mathbb{C}}^{\times},\rm nu}$.

######  Lemma 3.4.

In our case, the map $y\mapsto\lim_{z\rightarrow 0}(zy)$ is a regular map from
$\mathfrak{Q}$ to the fixed-point set
$\mathsf{Higgs}^{{\mathbb{C}}^{\times},\rm nu}$.

###### Proof.

In the general situation the map can be non-regular if there is a broken orbit
with two non-unitary fixed points joined by an orbit. However, in view of the
description of Proposition 3.3, this does not happen in our case. ∎

In order to understand how broken orbits can work, or equivalently what their
nearby orbits could look like, we need to consider the local picture of
$\mathsf{Higgs}$ at a fixed point. Assume we are at a fixed point $y$ that is
a stable Higgs bundle, so it is a smooth point of $\mathsf{Higgs}$. The
general theory of Bialynicki-Birula [BB73] tells us that the fixed point set
is smooth at $y$. The ${\mathbb{C}}^{\times}$-action on $\mathsf{Higgs}$
determines an action of ${\mathbb{C}}^{\times}$ on its tangent space at the
fixed point $y$.

######  Lemma 3.5.

The tangent space decomposes into pieces according to weights of the
${\mathbb{C}}^{\times}$-action:

$T_{y}(\mathsf{Higgs})=\bigoplus T_{y}(\mathsf{Higgs})^{p}$

such that $T_{y}(\mathsf{Higgs})^{p}$ is Serre-dual to
$T_{y}(\mathsf{Higgs})^{1-p}$ and in particular they have the same dimension.
The tangent space of the fixed point set is $T_{y}(\mathsf{Higgs})^{0}$. In
our case, at a non-unitary fixed point, the weights that occur are $-1$ and
$2$ in the case of $\mathsf{Higgs}_{0}$, and $-1,0,1,2$ in the case of
$\mathsf{Higgs}_{1}$.

###### Proof.

The weights can be understood by thinking of the Hodge bundle associated to
$y$ as being associated to a complex variation of Hodge structure $V$. The
hermitian structure of $V$ (given by a flat but indefinite hermitian form)
leads to a real structure on the VHS $\text{End}^{0}(V)$ of trace-free
endomorphisms. Thus [Zuc79] the first cohomology group has a real Hodge
structure of weight $1$:

$T_{y}(\mathsf{Higgs})\cong
H^{1}(C,\text{End}^{0}(V))=\bigoplus_{p+q=1}T^{p,q}$

with $T^{p,q}=\overline{T^{q,p}})$. This complex conjugation may be
identified, using the real structure, with Serre duality at the level of
Dolbeault cohomology. The ${\mathbb{C}}^{\times}$-action acts by weight $p$ on
the piece $T^{p,1-p}$.

More concretely if we write $E=\bigoplus E^{p}$ for the general case, then

$T_{y}(\mathsf{Higgs})^{p}=T^{p,1-p}={\mathbb{H}}^{1}\left[\bigoplus_{q}\text{Hom}(E^{q},E^{p+q})\stackrel{{\scriptstyle[-,\theta]}}{{\longrightarrow}}\bigoplus_{q}Hom(E^{q},E^{p+q-1})\otimes\omega_{C}\right]$

(where one should also include the trace-free condition but we didn’t put that
in so as not to complicate the notation). The complex is Serre-dual to itself
with a shift by $1$. The tangent space of the fixed point set is the set of
fixed points in the tangent space, as may be seen by the linearization result
to be mentioned shortly below. In our particular case, the VHS at a fixed
point has only two adjacent Hodge weights, so the VHS of trace-free
endomorphisms has Hodge weights $(-1,1)$, $(0,0)$ and $(1,-1)$. Thus, the
Hodge structure on $\mathbb{H}^{1}$ has weights obtained by adding $(1,0)$ and
$(0,1)$, namely $(-1,2)$, $(0,1)$, $(1,0)$ and $(2,-1)$. The
${\mathbb{C}}^{\times}$-weights are therefore $-1,0,1,2$. The piece of weight
$1$ is complex conjugate or Serre-dual to the piece of weight $0$, and this is
the tangent space of the fixed point set. Thus, in our case of
$\mathsf{Higgs}_{0}$ this vanishes whereas for $\mathsf{Higgs}_{1}$ it has
dimension $1$. We see that the piece of weight $-1$ has dimension $3$ for
$\mathsf{Higgs}_{0}$ and dimension $2$ for $\mathsf{Higgs}_{1}$. ∎

Define the incoming or upward directions to be the directions on which the
weight is $>0$, and the outgoing or downward directions to be those on which
the weight is $<0$. The general theory [BB73] of ${\mathbb{C}}^{\times}$
actions on complex manifolds tells us that there is an analytic open
neighborhood around a fixed point $z$ that is isomorphic, as a complex
manifold with ${\mathbb{C}}^{\times}$-action, to a neighborhood of the origin
in the tangent space $T_{y}(\mathsf{Higgs})$. In particular, the fixed points,
and incoming and outgoing manifolds are smooth and identified with the
corresponding subspaces of the tangent space. The incoming manifold is the
local subset of points whose limit as $z\rightarrow 0$ is equal to the fixed
point, whereas the outgoing manifold is the local subset of points whose
$z\rightarrow\infty$ is equal to the fixed point.

The terminologies “upward” and “downward” come from Hitchin’s picture [Hit87a]
of the energy, or $L^{2}$-norm of the Higgs field, as a Morse function on
$\mathsf{Higgs}$. We note that the orbit structure of the
${\mathbb{C}}^{\times}$-action has been identified, by work of Collier,
Wentworth, Wilkin and others, as being the same as the structure of flow lines
for the gradient of the Morse function [CW19, Wil20]. It is because of this
picture that we call the incoming directions “upward” (they are the directions
where one goes upwards by following the gradient of the energy function) and
the outgoing directions “downward” (they go downwards following the gradient
of the energy function). The flow stops at the “bottom” which is the unitary
fixed point set, the moduli space of bundles with zero Higgs field which is
obviously the minimum for the $L^{2}$ norm.

From Lemma 3.5 we see that the space of incoming directions (weights $p>0$) is
Serre dual to the tangent of the fixed points plus the space of outgoing
directions (weights $p\leq 0$). Thus, the space of incoming directions has
$1/2$ the dimension. In fact, each of the fibers of the projection over a
fixed point, is a Lagrangian subspace of the symplectic moduli space. The
space of outgoing directions, plus the fixed point directions, together form
the tangent space of the manifold of points that flow out of somewhere in that
fixed point set. This is locally (at a general point of the fixed point set)
one of the components of the nilpotent cone in $\mathsf{Higgs}$, also a
Lagrangian subspace.

In general, the space of directions that are outgoing from a given fixed point
will have dimension smaller than half, although it is also half the dimension
if the fixed point set is $0$-dimensional as is our case in
$\mathsf{Higgs}_{0}$. Similarly, the total space $\mathfrak{Q}$ of directions
incoming to the local piece of the fixed point set, will generally have
dimension greater than half, but again it is half if the fixed point set is
$0$-dimensional.

In general one will be considering a Prym variety
${\mathcal{P}}\subset\mathsf{Higgs}$ that is a general fiber of the Hitchin
map. The rational map $\mathsf{Higgs}\dashrightarrow X$ therefore provides a
rational map ${\mathcal{P}}\dashrightarrow X$, and to resolve it one should
blow up $\mathfrak{Q}\cap{\mathcal{P}}$ to get $Y\rightarrow{\mathcal{P}}$.
The dimension of $\mathfrak{Q}\cap{\mathcal{P}}$ is equal to the dimension of
the fixed point locus.

In our cases, for $\mathsf{Higgs}_{0}$ the dimension of $\mathfrak{Q}$ is $3$
and it is Lagrangian, while for $\mathsf{Higgs}_{1}$ the dimension of
$\mathfrak{Q}$ is $4$ and it fibers over ${\overline{C}}$ with fibers that are
Lagrangian. The locus $\mathfrak{Q}\cap{\mathcal{P}}$ to be blown up has
dimension $0$ for ${\mathcal{P}}\subset\mathsf{Higgs}_{0}$ and dimension $1$
for ${\mathcal{P}}\subset\mathsf{Higgs}_{1}$.

One can understand from the local description at the fixed point that a single
blowing-up will be sufficient in our case, yielding a morphism $Y\rightarrow
X$. We will denote the exceptional divisor by $\boldsymbol{\mathit{E}}\subset
Y$. In principle, one can use the description of the ${\mathbb{C}}^{\times}$
action to understand the resulting ramification of the map $Y\rightarrow X$
along $\boldsymbol{\mathit{E}}$. We have chosen instead to give more direct
proofs (Lemma 5.6, Proposition 5.12) that the map is simply ramified along the
exceptional locus.

In the general case when the ${\mathbb{C}}^{\times}$-action can have a wider
range of weights, and when there can be multiply broken flow lines,
understanding the birational transformation needed to resolve the rational
map, and understanding the resulting ramification, seem to be difficult
questions.

###  3.7 The logarithmic property

As before we will write $X$ for the coarse moduli space of semistable rank two
bundles on $C$ with fixed determinant $\mathbf{d}$ and $\mathsf{Higgs}$ for
the moduli space of semistable Higgs bundles with fixed determinant
$\mathbf{d}$. Recall that $\mathsf{Higgs}$ is equipped with an algebraic
$\mathbb{C}^{\times}$-action, scaling the Higgs fields, i.e.
$z\in\mathbb{C}^{\times}$ acts by $(E,\theta)\mapsto(E,z\theta)$. Note that
with this definition the Hitchin map
$\boldsymbol{h}:\mathsf{Higgs}\to{\mathcal{B}}$ becomes
$\mathbb{C}^{\times}$-equivariant once we equip
${\mathcal{B}}=H^{0}(\omega_{C}^{\otimes 2})$ with a scaling action of degree
$2$.

We will also keep using the notation
$\mathsf{Higgs}^{{\mathbb{C}}^{\times},\text{nu}}\subset\mathsf{Higgs}$ for
the union of components of the fixed point locus that are disjoint from $X$,
and let $\mathfrak{Q}\subset\mathsf{Higgs}$ denote the incoming variety to
$\mathsf{Higgs}^{{\mathbb{C}}^{\times},\text{nu}}$, that is

$\mathfrak{Q}=\left\\{(E,\theta)\in\mathsf{Higgs}\ \left|\ \lim_{z\rightarrow
0}(E,z\theta)\in\mathsf{Higgs}^{{\mathbb{C}}^{\times},\text{nu}}\right.\right\\}.$

In the degree $1$ case,
$\mathsf{Higgs}^{{\mathbb{C}}^{\times},\text{nu}}={\overline{C}}$ is an
irreducible curve that is a $16$-sheeted etale covering of $C$, whereas in the
degree $0$ case $\mathsf{Higgs}^{{\mathbb{C}}^{\times},\text{nu}}$ consists of
$16$ distinct points.

In our situation, the fixed point locus has only one higher level, that is to
say that the limit for $z\rightarrow 0$ along any outgoing direction at
$\mathsf{Higgs}^{{\mathbb{C}}^{\times},\text{nu}}$ lies in $X$. This will no
longer be the case for higher genus or higher rank. It allows for some
simplification.

######  Theorem 3.6.

The subvariety $\mathfrak{Q}$ is smooth and contained in the smooth locus of
$\mathsf{Higgs}$. It coincides with the locus of polystable Higgs bundles
whose underlying vector bundle is un(semi)stable.

  * (a)

Let $\widetilde{\mathsf{Higgs}}$ denote the blow-up of $\mathsf{Higgs}$ along
$\mathfrak{Q}$. Then this resolves the projection map to $X$, in other words
the rationally defined map $\mathsf{Higgs}\dashrightarrow X$ extends to a
morphism $\boldsymbol{\mathfrak{f}}:\widetilde{\mathsf{Higgs}}\rightarrow X$.

  * (b)

$\widetilde{\boldsymbol{h}}:\widetilde{\mathsf{Higgs}}\to{\mathcal{B}}$ denote
the composition of the blow-up morphism
$\widetilde{\mathsf{Higgs}}\to\mathsf{Higgs}$ with the Hitchin map. Let
$\mathsf{b}\in{\mathcal{B}}$ be a point corresponding to a smooth spectral
curve ${\widetilde{C}}$ and let
$Y=\widetilde{\boldsymbol{h}}^{-1}(\mathsf{b})\subset\widetilde{\mathsf{Higgs}}$
be the fiber of $\widetilde{\boldsymbol{h}}$. Then $Y$ is a smooth compact
threefold which is a blow-up of of the Hitchin fiber
${\mathcal{P}}=\boldsymbol{h}^{-1}(\mathsf{b})$ in a smooth center. The
induced map $f:=\boldsymbol{\mathfrak{f}}_{|Y}:Y\to X$ is finite and maps each
connected component of the exceptional divisor of the blow-up
$Y\to{\mathcal{P}}$ onto some irreducible component of the wobly divisor ${\rm
Wob}\subset X$.

###### Proof.

Note that $\mathsf{Higgs}$ is smooth in the degree $1$ case. In the degree $0$
case, the singular locus consists of the reducible Higgs bundles. A point of
$\mathsf{Higgs}^{{\mathbb{C}}^{\times},\text{nu}}$ is a Hodge bundle, for
which the underlying bundle is of the form $\kappa\oplus\kappa^{-1}$ with
$\kappa^{\otimes 2}=\omega_{C}$ and for which the Higgs field sends
$\kappa^{-1}$ to zero, and maps $\kappa$ isomorphically to
$\kappa^{-1}\otimes\omega_{C}$. Such Higgs bundles are stable. The stable
locus in $\mathsf{Higgs}$ is ${\mathbb{C}}^{\times}$-invariant and contains a
neighborhood of $\mathsf{Higgs}^{{\mathbb{C}}^{\times},\text{nu}}$ so it
contains $\mathfrak{Q}$. In the previous section we checked that
$\mathfrak{Q}$ itself is smooth as it is a rank three affine bundle over
${\overline{C}}$ in the degree one case and is the disjoint union of $16$
copies of a three dimensional affine space in the case of degree zero. This
proves the first statement.

For the second statement we will first deal with the degree zero case. For
this discussion we will write $\mathsf{Higgs}_{0}$, $\mathfrak{Q}_{0}$,
$X_{0}$, etc. By definition and by the stability comment in the previous
paragraph we have that a Higgs bundle $(E,\theta)$ belongs to
$\mathfrak{Q}_{o}$ if and only if $\lim_{z\to 0}(E,z\theta)$ exists as a
stable Higgs bundle and is isomoprphic to

$\left(\kappa\oplus\kappa^{-1},\begin{pmatrix}0&0\\\
\gamma&0\end{pmatrix}\right),\ \text{with}\ \lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
14.23608pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 4.60004pt\raise
0.0pt\hbox{$\textstyle{\gamma:\kappa\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.61108pt\raise 6.15pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\cong}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 26.23608pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
26.23608pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
4.60004pt\raise
0.0pt\hbox{$\textstyle{\kappa^{-1}\otimes\omega_{C}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

But if $(E,\theta)$ is a semistable Higgs bundle for which $E$ is not
semistable as bundle, then we can find a saturated line sub bundle
$\kappa\subset E$, such that $\deg\kappa>\deg E/2$, and
$\theta(\kappa)\not\subset\kappa\otimes\omega_{C}$. Since $E$ has trivial
determinant, this means that $E$ fits in a short exact sequence

$0\to\kappa\to E\to\kappa^{-1}\to 0,$ (10)

$\deg\kappa>0$, and the composite map

$\textstyle{\kappa\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{E\otimes\omega_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\kappa^{-1}\otimes\omega_{C}}$
(11)

is non-zero. Thus we must have $0<\deg\kappa\leq g(C)-1=1$, Since (11) is a
non-zero map between two line bundles of degree $1$, it must be an isomorphism
and $\kappa$ must be a theta characteristic on $C$. In particular this implies
that $\theta(\kappa)\otimes\omega_{C}^{-1}\subset E$ is a line sub bundle
which projects isomorphically onto $\kappa^{-1}$. Thus the short exact
sequence (10) is split and we have an isomorphism

$(E,\theta)\ \cong\
\left(\kappa\oplus\kappa^{-1},\begin{pmatrix}\alpha&\beta\\\
\gamma&-\alpha\end{pmatrix}\right),$ (12)

where $\alpha\in H^{0}(C,\omega_{C})$, $\beta\in H^{0}(C,\omega_{C}^{\otimes
2})$, and
$\textstyle{\gamma:\kappa\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\kappa^{-1}\otimes\omega_{C}}$.
The shape of the Higgs field in (12) can be rigidified somewhat by choosing
the identification $E\cong\kappa\oplus\kappa^{-1}$ more carefully. Indeed, the
automorphisms of the bundle $\kappa\oplus\kappa^{-1}$ that act trivially on
the determinant line bundle are given by matrices of the form

$\begin{pmatrix}a&b\\\ 0&a^{-1}\end{pmatrix},\quad a\in{\mathbb{C}}^{\times},\
b\in H^{0}(C,\omega_{C}).$

Conjugating by the authomorphism $\begin{pmatrix}u&-u^{-1}\alpha\\\
0&u^{-1}\end{pmatrix}$ with $u^{2}=\gamma$, we get an isomorphism

$(E,\theta)\ \cong\
\left(\kappa\oplus\kappa^{-1},\begin{pmatrix}0&-\mathsf{b}\\\
1&0\end{pmatrix}\right)$

where
$\mathsf{b}=-\alpha^{2}-\gamma\beta=\boldsymbol{h}(E,\theta)\in{\mathcal{B}}$
is the quadratic differential defining the spectral cover of $(E,\theta)$.
This shows that the locus
$\mathsf{Higgs}_{0}^{\text{unss}}\subset\mathsf{Higgs}_{0}$ of all trivial
determinant Higgs bundles whose underlying vector bundle is not semi-stable is
the disjoint union
$\mathsf{Higgs}_{0}^{\text{unss}}=\sqcup_{\kappa}\mathsf{Higgs}_{0,\kappa}^{\text{unss}}$
of the $16$ Hitchin sections

$\mathsf{Higgs}_{0,\kappa}^{\text{unss}}=\left\\{\left.\,\left(\kappa\oplus\kappa^{-1},\begin{pmatrix}0&-\mathsf{b}\\\
1&0\end{pmatrix}\right)\quad\right|\quad\mathsf{b}\in
H^{0}(C,\omega_{C}^{\otimes 2}\,\right\\}$

of $\boldsymbol{h}:\mathsf{Higgs}_{0}\to{\mathcal{B}}=\mathbb{A}^{3}$ labeled
by the theta characteristics $\kappa\in\mathsf{Spin}(C)$. Also, for any

$(E,\theta)\ \cong\
\left(\kappa\oplus\kappa^{-1},\begin{pmatrix}0&-\mathsf{b}\\\
1&0\end{pmatrix}\right)\in\mathsf{Higgs}_{0}^{\text{unss}}$

we have that whenever $z\neq 0$ the Higgs bundle $(E,z\theta)$ is isomorphic
to

$\left(\kappa\oplus\kappa^{-1},\begin{pmatrix}0&-z^{2}\mathsf{b}\\\
1&0\end{pmatrix}\right)$

where the isomorphism is given by conjugation by $\begin{pmatrix}u&0\\\
0&u^{-1}\end{pmatrix}$, with $u^{2}=z$. Thus

$\lim_{z\to 0}(E,z\theta)=\left(\kappa\oplus\kappa^{-1},\begin{pmatrix}0&0\\\
1&0\end{pmatrix}\right)$

and so $\mathsf{Higgs}^{\text{unss}}_{0}\subset\mathfrak{Q}_{0}$. Since both
the un(semi)stable locus and the incoming variety are conical Lagrangian
submanifolds which intersect at the $16$ non-unitary
${\mathbb{C}}^{\times}$-fixed points, we get that
$\mathsf{Higgs}^{\text{unss}}_{0}=\mathfrak{Q}_{0}$. To shorten the notation
we will write
$\mathfrak{Q}_{0,\kappa}=\mathsf{Higgs}^{\text{unss}}_{0,\kappa}$ for the
connected components of this locus.

Note that each of the Hitchin sections
$\mathsf{Higgs}^{\text{unss}}_{0,\kappa}$ has a trivial normal bundle inside
$\mathsf{Higgs}_{0}$. Indeed, the Hitchin sections intersect each Hitchin
fiber at smooth points, and by the Lagrangian property we have that the normal
bundle of $\mathsf{Higgs}^{\text{unss}}_{0,\kappa}$ inside
$\mathsf{Higgs}_{0}$ is isomorphic to the cotangent bundle of
$\mathsf{Higgs}^{\text{unss}}_{0,\kappa}$. Since
$\mathsf{Higgs}^{\text{unss}}_{0,\kappa}\cong{\mathcal{B}}=H^{0}(C,\omega_{C}^{\otimes
2})$ we conclude that
$N_{\mathsf{Higgs}^{\text{unss}}_{0,\kappa}/\mathsf{Higgs}_{0}}\cong\mathcal{O}_{\mathsf{Higgs}^{\text{unss}}_{0,\kappa}}\otimes
H^{0}(C,\omega_{C}^{\otimes 2})^{\vee}$. This implies that when we form the
blow-up
$\widetilde{\mathsf{Higgs}}_{0}=\mathsf{Bl}_{\mathfrak{Q}_{0}}\mathsf{Higgs}_{0}$
the exceptional divisor $\widetilde{\mathfrak{Q}}_{0}$ is a product

$\widetilde{\mathfrak{Q}}_{0}=\mathfrak{Q}_{0}\times\mathbb{P}\left(H^{0}(C,\omega_{C}^{\otimes
2})^{\vee}\right)\cong\bigsqcup_{\kappa\in\mathsf{Spin}(C)}\mathfrak{Q}_{0,\kappa}\times\mathbb{P}^{2}.$

The question of whether the rational map $\mathsf{Higgs}_{0}\dashrightarrow
X_{0}$ extends to a morphism $\widetilde{\mathsf{Higgs}}_{0}\to X_{0}$ is
local near $\mathfrak{Q}_{0}\subset\mathsf{Higgs}_{0}$. Therefore to check
that this happens it suffices to check that if $S$ is a scheme and
$({}^{S}E,{}^{S}\theta)$ is a relative semistable Higgs bundle on $S\times C$
that gives an étale map $S\to\mathsf{Higgs}_{0}$, then the composite rational
map $S\to\mathsf{Higgs}_{0}\dashrightarrow X_{0}$ extends to a morphism
$\widetilde{S}=S\times_{\mathsf{Higgs}_{0}}\widetilde{\mathsf{Higgs}}_{0}\to
X_{0}$. This is equivalent to checking that the vector bundle
${}^{S}E_{|(S-\mathfrak{Q}_{0})\times C}$ has a canonical (unique up to unique
isomorphism) extension to a vector bundle ${}^{\widetilde{S}}E$ on
$\widetilde{S}\times C$ which is semistable of trivial determinant on all
geometric fibers over $\widetilde{S}$.

As a first approximmation to ${}^{\widetilde{S}}E$ one can take the pullback

${}^{\widetilde{S}}E^{\text{naive}}:=(\widetilde{S}\times C\to S\times
C)^{*}\left({}^{S}E\right).$

This is a rank two vector bundle on $\widetilde{S}\times C$ which has trivial
determinant on all geometric fibers over $\widetilde{S}$ and is semistable on
all geometric fibers over points in
$\widetilde{S}-\widetilde{\mathfrak{Q}}_{0}=S-\mathfrak{Q}_{0}$. However by
construction the restriction of ${}^{\widetilde{S}}E^{\text{naive}}$ to
$\\{\tilde{y}\\}\times C$ for any closed point
$\tilde{y}\in\mathfrak{Q}_{0,\kappa}\times\mathbb{P}^{2}\subset\widetilde{\mathfrak{Q}}$
is isomorphic to $\kappa\oplus\kappa^{-1}$ which is unstable.

To construct the actual bundle ${}^{\widetilde{S}}E$ let
${}^{S}\mathfrak{Q}_{0}=S\times_{\mathsf{Higgs}_{0}}\mathfrak{Q}_{0}$ and let
${}^{\widetilde{S}}\mathfrak{Q}_{0}$ be the Cartier divisor
$\widetilde{S}\times_{S}({}^{S}\mathfrak{Q}_{0})$, i.e. the exceptional
divisor of the blow-up $\widetilde{S}\to S$. We will also write
${}^{S}\mathfrak{Q}_{0,\kappa}=\times_{\mathsf{Higgs}_{0}}\mathfrak{Q}_{0,\kappa}$
and
${}^{\widetilde{S}}\mathfrak{Q}_{0,\kappa}=\widetilde{S}\times_{S}({}^{S}\mathfrak{Q}_{0,\kappa})$
for the connected components of these loci.

Shrinking $S$ if necessary we get that the restriction of the bundle
${}^{\widetilde{S}}E^{\text{naive}}$ to
$({}^{\widetilde{S}}\mathfrak{Q}_{0,\kappa})\times C$ has a natural surjective
homomorphism to $\operatorname{pr}_{C}^{*}\kappa^{-1}$ which is unique up to
multiplication by an invertible function on ${}^{S}\mathfrak{Q}_{0,\kappa}$.
Define ${}^{\widetilde{S}}E$ to be the Hecke transform of
${}^{\widetilde{S}}E^{\text{naive}}$ centered at this homomorphism, i.e.

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{\widetilde{S}}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{\widetilde{S}}E^{\text{naive}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{\kappa\in\mathsf{Spin}(C)}\left({}^{\widetilde{S}}\mathfrak{Q}_{0,\kappa}\times
C\to\widetilde{S}\times
C\right)_{*}\operatorname{pr}_{C}^{*}\kappa^{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Note that scaling the last map in the sequence by an invertible function on
${}^{S}\mathfrak{Q}_{0}$ does not change the kernel sheaf
${}^{\widetilde{S}}E$ and so it is canonically defined as a subsheaf in
${}^{\widetilde{S}}E^{\text{naive}}$.

The fact that ${}^{\widetilde{S}}E$ is a relative semistable bundle on
$\widetilde{S}\times C$ now comes from the following

######  Lemma 3.7.

Suppose
$\tilde{y}=(y,\mathfrak{e})\in{}^{\widetilde{S}}\mathfrak{Q}_{0,\kappa}=({}^{S}\mathfrak{Q}_{0,\kappa})\times\mathbb{P}(H^{0}(\omega_{C}^{\otimes
2})^{\vee})$. Then the vector bundle
${}^{\widetilde{S}}E_{|\\{\tilde{y}\\}\times C}$ is isomorphic to the semi-
stable rank two bundle $E_{\mathfrak{e}}$ which is the extension

$0\to\kappa^{-1}\to E_{\mathfrak{e}}\to\kappa\to 0$

corresponding to the extension class
$\mathfrak{e}\in\mathbb{P}(H^{0}(\omega_{C}^{\otimes
2})^{\vee})=\mathbb{P}(H^{1}(\omega_{C}^{-1}))=\mathbb{P}(H^{1}(\kappa^{\otimes-2}))$.

###### Proof.

Since $\mathfrak{e}\in\mathbb{P}(H^{1}(\kappa^{\otimes-2}))$ the extension
$E_{\mathfrak{e}}$ is non-split. But if $L\subset E_{\mathfrak{e}}$ is a
destibilazing line sub bundle, then $\deg L>0$ and hence as subsheaves in
$E_{\mathfrak{e}}$ we must have $L\cap\kappa^{-1}=0$ this shows that $L$ maps
injectively into $\kappa$, and hence we will have $\deg L=1$ and
$L\cong\kappa$. This will split the extension which is contradiction. This
proves that $E_{\mathfrak{e}}$ is semi-stable.

The identification of ${}^{\widetilde{S}}E_{|\\{\tilde{y}\\}\times C}$ and
$E_{\mathfrak{e}}$ is a general fact in deformation theory which we explain
next. Suppose $S$ is the spectrum of a DVR with uniformizer
$\boldsymbol{\mathsf{x}}$ and closed point $o\in S$. Let $\mathcal{E}\to
S\times C$ be an algebraic vector bundle and let
$E=\mathcal{E}_{|\\{o\\}\times C}$. Assume that $E$ fits in a short exact
sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\imath}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\jmath}$$\textstyle{\mathsf{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(13)

of vector bundles on $C$. Consider the Hecke transform

$0\to\mathcal{E}^{\prime}\to\mathcal{E}\to\left(\\{o\\}\times C\to S\times
C\right)_{*}\mathsf{B}\to 0.$

Then the restriction $E^{\prime}=\mathcal{E}^{\prime}_{|\\{o\\}\times C}$ is a
vector bundle that fits in a short exact sequence

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{B}\otimes\mathcal{O}_{S\times
C}(-\\{o\\}\times C)_{|\\{o\\}\times
C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\mathsf{B}}$
(14)

Write

$\mathsf{ks}^{\mathcal{E}}:H^{0}(S,T_{S})\to
H^{1}(C,\operatorname{End}_{0}(E))\subset\operatorname{Hom}(E,E[1])$

for the Kodaira-Spencer map of $\mathcal{E}$. A direct calculation with either
square zero extenions or Čech cocycles gives now that the extension class of
the short exact sequence (14) is

$\jmath\circ\mathsf{ks}^{\mathcal{E}}(\partial_{\boldsymbol{\mathsf{x}}})\circ\imath\in\operatorname{Hom}(\mathsf{A},\mathsf{B}[1])=\operatorname{Ext}^{1}(\mathsf{A},\mathsf{B}).$

If in addition the sequence (13) is split, i.e.
$E=\mathsf{A}\oplus\mathsf{B}$, then we can write the Kodaira-Spencer class
$\mathsf{ks}^{\mathcal{E}}(\partial_{\boldsymbol{\mathsf{x}}})$ as a block
matrix

$\mathsf{ks}^{\mathcal{E}}(\partial_{\boldsymbol{\mathsf{x}}})=\begin{pmatrix}\eta_{\mathsf{A}\mathsf{A}}&\eta_{\mathsf{A}\mathsf{B}}\\\
\eta_{\mathsf{B}\mathsf{A}}&\eta_{\mathsf{B}\mathsf{B}}\end{pmatrix}$

where $\eta_{\mathsf{A}\mathsf{A}}:\mathsf{A}\to\mathsf{A}[1]$,
$\eta_{\mathsf{A}\mathsf{B}}:\mathsf{B}\to\mathsf{A}[1]$,
$\eta_{\mathsf{B}\mathsf{A}}:\mathsf{A}\to\mathsf{B}[1]$, and
$\eta_{\mathsf{B}\mathsf{B}}:\mathsf{B}\to\mathsf{B}[1]$. In particular, the
extension class of (14) will be
$\eta_{\mathsf{B}\mathsf{A}}:\mathsf{A}\to\mathsf{B}[1]$.

Suppose next that $S$ is the spectrum of a DVR which maps to
$\mathsf{Higgs}_{0}$ so that the closed point $o\in S$ maps to a closed point
$y\in\mathfrak{Q}_{0,\kappa}$ and the differential of the map
$S\to\mathsf{Higgs}_{0}$ maps $\partial_{\boldsymbol{\mathsf{x}}}$ to a
tangent vector in $T_{y}\mathsf{Higgs}_{0}$ which projects to the normal line
$\mathfrak{e}\subset H^{1}(\kappa^{\otimes-2})$. In particular the map
$S\to\mathsf{Higgs}_{0}$ is given by a relative Higgs bundle
$({}^{S}E,{}^{S}\theta)$ on $S\times C$ and the component
$\eta_{\kappa,\kappa^{-1}}$ of
$\mathsf{ks}^{({}^{S}E)}(\partial_{\boldsymbol{\mathsf{x}}})$ is just equal to
$\mathfrak{e}$. This concludes the proof of the lemma. ∎

These considerations prove assertion (a) in the statement of Theorem 3.6 for
moduli of Higgs bundles of determinant $\mathbf{d}=\mathcal{O}_{C}$. In fact,
the above discussion also proves assertion (b) in the trivial determinant
case. Indeed, when $\mathbf{d}={\mathcal{O}}_{C}$ the discussion above shows
that the connected component $\mathfrak{Q}_{0,\kappa}$ of the center of teh
blow-up $\widetilde{\mathsf{Higgs}}\to\mathsf{Higgs}$ intesects the Hitchin
fiber
$\boldsymbol{h}^{-1}(\mathsf{b})={\mathcal{P}}_{2}=\\{L\in\operatorname{Jac}^{2}({\widetilde{C}})\
|\ \mathsf{Nm}_{\pi}(L)\cong\omega_{C}\\}$ at a single point, namely the point
$\pi^{*}\kappa\in{\mathcal{P}}_{2}$. By the universal property of the blow-up
it follows that $Y$ is the blow-up of ${\mathcal{P}}_{2}$ at the $16$ distinct
points $\pi^{*}\kappa$, $\kappa\in\mathsf{Spin}(C)$. But the morphism
${\mathcal{P}}_{2}-\\{\pi^{*}\kappa\\}_{\kappa\in\mathsf{Spin}(C)}\to X$ is
quasi-finite, and also by the construction in Lemma 3.7 the map
$f=\boldsymbol{\mathfrak{f}}_{|Y}:Y\to X$ maps the exceptional divisor
$\boldsymbol{\mathit{E}}_{0,\kappa}\subset Y$ corresponding to the point
$\pi^{*}\kappa\in{\mathcal{P}}_{2}$ isomorphically to the plane in $X=X_{0}$
parametrizing all semistable bundles which are non-trivial extensions of
$\kappa$ by $\kappa^{-1}$. Thus $f:Y\to X$ is everywhere quasi-finite and
since it is proper, it is finite. Furthermore note that the plane
parametrizing non-trivial extensions of $\kappa$ by $\kappa^{-1}$ is precisely
the trope component $\mathsf{Trope}_{\kappa}$ of ${\rm Wob}_{0}$ which
completes the proof of assertion (b) in this case

Finally we analyze the degree one case. Again we will write
$\mathsf{Higgs}_{1}$, $\mathfrak{Q}_{1}$, $X_{1}$, etc.

######  Lemma 3.8.

Suppose $(E,\theta)\in\mathsf{Higgs}_{1}$ is a semistable Higgs bundle, such
that $E$ is not semistable as a bundle. Then

$(E,\theta)\ \cong\ \left(A^{-1}(\boldsymbol{\mathsf{p}})\oplus
A,\begin{pmatrix}\alpha&\beta\\\ \gamma&-\alpha\end{pmatrix}\right)$

where $A\in\operatorname{Jac}^{0}(C)$ satisfying $A^{\otimes
2}(\boldsymbol{\mathsf{p}})=\mathcal{O}_{C}(t)$ for some $t\in C$, $\alpha\in
H^{0}(C,\omega_{C})$, $\beta\in H^{0}(C,\omega_{C}(t^{\prime}))$, $\gamma\neq
0\in H^{0}(C,\mathcal{O}_{C}(t))$, and $t^{\prime}$ is the image of $t$ under
the hyperelliptic involution on $C$.

###### Proof.

Let $L\subset E$ be a saturated destabilizing line bundle, i.e. a saturated
line subbundle s.t. $\deg L\geq\deg E/2=1/2$. Since $(E,\theta)$ is semistable
as a Higgs bundle, $L$ can not be $\theta$-invariant. In other words, we must
have $\theta(L)\neq 0\subset E\otimes\omega_{C}$ which is equivalent to the
composite map $\theta:L\to E\otimes\omega_{C}\to(E/L)\otimes\omega_{C}$ being
non-zero. Thus we must have $1\leq\deg L\leq 1-\deg L+\deg\omega_{C}=3-\deg
L$, i.e. $\deg L=1$. Writing $L=A^{-1}(\boldsymbol{\mathsf{p}})$ with $\deg
A=0$ we get a non-zero (hence injective) map of locally free sheaves
$A^{-1}(\boldsymbol{\mathsf{p}})\to A(2\boldsymbol{\mathsf{p}})$ and so we
must have $A^{\otimes 2}(\boldsymbol{\mathsf{p}})=\mathcal{O}_{C}(t)$ for some
point $t\in C$, i.e. we must have that $(A,t)\in{\overline{C}}$.

Also $E$ fits in a short exact sequence

$0\to A^{-1}(\boldsymbol{\mathsf{p}})\to E\to A\to 0,$ (15)

and viewing $\theta$ as a map $E\otimes\omega_{C}^{-1}\to$ we get a subsheaf
$\theta\left(A^{-1}(\boldsymbol{\mathsf{p}})\otimes\omega_{C}^{-1}\right)\subset
E$ which maps injectively into $A$ via the map $E\to A$. This implies that the
extension (15) splits over the subsheaf
$\theta:A^{-1}(\boldsymbol{\mathsf{p}})\otimes\omega_{C}^{-1}\hookrightarrow
A$. Since $(A,t)\in{\overline{C}}$ we have that
$A^{-1}(\boldsymbol{\mathsf{p}})\otimes\omega_{C}^{-1}\cong A(-t)$ and under
this isomorphism the map $\theta$ gets identified with the natural inclusion
$A(-t)\subset A$.

By the same token we have

$\operatorname{Ext}^{1}(A,A^{-1}(\boldsymbol{\mathsf{p}}))=H^{1}(C,A^{\otimes-2})(\boldsymbol{\mathsf{p}}))=H^{1}(C,\mathcal{O}_{C}(2\boldsymbol{\mathsf{p}}-t))=H^{1}(C,\mathcal{O}_{C}(t^{\prime})),$

while

$\operatorname{Ext}^{1}(A(-t),A^{-1}(\boldsymbol{\mathsf{p}}))=H^{1}(C,A^{\otimes-2}(t+\boldsymbol{\mathsf{p}}))=H^{1}(C,\mathcal{O}_{C}(2\boldsymbol{\mathsf{p}}))=H^{1}(C,\mathcal{O}_{C}(t^{\prime}+t)),$

and under these identifications the map

$\operatorname{Ext}^{1}(A,A^{-1}(\boldsymbol{\mathsf{p}}))\to\operatorname{Ext}^{1}(A(-t),A^{-1}(\boldsymbol{\mathsf{p}}))$

gets identified with the map on cohomology

$H^{1}(C,\mathcal{O}_{C}(t^{\prime}))\to
H^{1}(C,\mathcal{O}_{C}(t^{\prime}+t))$ (16)

induced from the natural inclusion
$\mathcal{O}_{C}(t^{\prime})\subset\mathcal{O}_{C}(t^{\prime}+t)$.

From the long exact sequence in cohomology associated to the short exact
sequence of sheaves

$0\to\mathcal{O}_{C}(t^{\prime})\to\mathcal{O}_{C}(t^{\prime}+t)\to\mathcal{O}_{t}\to
0$

we see that the map (16) is surjective. on the other hand both
$H^{1}(C,\mathcal{O}_{C}(t^{\prime}))$ and
$H^{1}(C,\mathcal{O}_{C}(t^{\prime}+t))$ are one dimensional and so the map
(16) is an isomorphism. This implies that if an extension class in
$\operatorname{Ext}^{1}(A,A^{-1}(\boldsymbol{\mathsf{p}}))$ maps to zero in
$\operatorname{Ext}^{1}(A(-t),A^{-1}(\boldsymbol{\mathsf{p}}))$, then this
class is zero to begin with. Hence the extension (15) is split as claimed. The
statement about the matrix entries of $\theta$ under the decomposition $E\cong
A^{-1}(\boldsymbol{\mathsf{p}})\oplus A$ now follows tautologically. Note that
the entry $\gamma$ is precisely the composite map

$\textstyle{A^{-1}(\boldsymbol{\mathsf{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{E\otimes\omega_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes\omega_{C}=A(2\boldsymbol{\mathsf{p}})}$

and hence is non-zero. This finishes the proof of the lemmma. ∎

Again, the shape of the Higgs field in the pair $(E,\theta)$ can be simplified
by choosing the isomorphism $E\cong A^{-1}(\boldsymbol{\mathsf{p}})\oplus A$
more carefully. The global automorphisms of the bundle
$A^{-1}(\boldsymbol{\mathsf{p}})\oplus A$ are all of the form

$\begin{pmatrix}a&b\\\ 0&a^{-1}\end{pmatrix}$

where $a\in{\mathbb{C}}^{\times}$, and $b\in
H^{0}(C,\mathcal{O}_{C}(t^{\prime}))$. Conjugating
$z\cdot\begin{pmatrix}\alpha&\beta\\\ \gamma&-\alpha\end{pmatrix}$ by
$\begin{pmatrix}u&0\\\ 0&u^{-1}\end{pmatrix}$ with $u^{2}=z$ we get

$\begin{pmatrix}z\alpha&z^{2}\beta\\\ \gamma&-z\alpha\end{pmatrix}.$

Hence

$\lim_{z\to 0}(E,z\theta)=\left(\,A^{-1}(\boldsymbol{\mathsf{p}})\oplus
A,\begin{pmatrix}0&0\\\ \gamma&0\end{pmatrix}\,\right),$

and so the locus $\mathsf{Higgs}_{1}^{\text{unss}}$ parametrizing stable Higgs
bundles with unstable underlying vector bundles is contained in the incoming
variety $\mathfrak{Q}_{1}$. By the explicit description both of these are four
dimensional irreducible subvarieties in $\mathsf{Higgs}_{1}$ and therefore
$\mathsf{Higgs}_{1}^{\text{unss}}=\mathfrak{Q}_{1}$.

Similarly to the degree zero case the identifictaion
$\mathsf{Higgs}_{1}^{\text{unss}}=\mathfrak{Q}_{1}$ is compatible both with
the Hitchin map $\boldsymbol{h}:\mathsf{Higgs}_{1}\to{\mathcal{B}}$ and with
the flow limit map
$\mathfrak{Q}_{1}\to\mathsf{Higgs}_{1}^{{\mathbb{C}}^{\times},\text{nu}}$.
Indeed,if

$\left(\,A^{-1}(\boldsymbol{\mathsf{p}})\oplus
A,\begin{pmatrix}\alpha&\beta\\\ \gamma&-\alpha\end{pmatrix}\,\right)\ \in\
\mathsf{Higgs}_{1}^{\text{unss}},$

then the determinant of the Higgs field is
$\mathsf{b}=-\alpha^{2}-\gamma\beta$. This shows that
$\mathsf{Higgs}_{1}^{\text{unss}}=\mathfrak{Q}_{1}$ surjects onto the Hitchin
base ${\mathcal{B}}$. In Section 3.2 we also checked that the fiber of
$\mathsf{Higgs}_{1}^{\text{unss}}$ over a general point
$\mathsf{b}\in{\mathcal{B}}$ of the Hitchin base is a copy of
${\widehat{C}}_{\mathsf{b}}={\overline{C}}\times_{C}{\widetilde{C}}_{\mathsf{b}}$
embedded in the corresponding Hitchin fiber
${\mathcal{P}}_{3,\mathsf{b}}=\boldsymbol{h}^{-1}(\mathsf{b})$. Thus we can
view the smooth variety $\mathfrak{Q}_{1}$ as a family of curves over
${\mathcal{B}}$.

At the same time, the above caluclation of $\lim_{z\to 0}zy$ for
$y\in\mathfrak{Q}_{1}$ identifies the limiting flow map
$\mathbf{fl}_{1}:\mathfrak{Q}_{1}\to{\overline{C}}$ with the natural
projection of “forgetting the Higgs field”, i.e. for every

$y=\left(A^{-1}(\boldsymbol{\mathsf{p}})\oplus
A,\,\begin{pmatrix}\alpha&\beta\\\
\gamma&-\alpha\end{pmatrix}\right)\in\mathfrak{Q}_{1}$

we have $\mathbf{fl}_{1}(y)=A\in{\overline{C}}$.

With this observation at hand we can now compute the normal bundle of smooth
subvariety $\mathfrak{Q}_{1}\subset\mathsf{Higgs}_{1}$ and the exceptional
divisor of the blow up
$\widetilde{\mathsf{Higgs}}_{1}=\operatorname{Bl}_{\mathfrak{Q}_{1}}\mathsf{Higgs}_{1}\to\mathsf{Higgs}_{1}$.
Specifically we have

######  Lemma 3.9.

  * (a)

The normal bundle of $\mathfrak{Q}_{1}\subset\mathsf{Higgs}_{1}$ is given by

$N_{\mathfrak{Q}_{1}/\mathsf{Higgs}_{1}}=H^{0}(C,\omega_{C})^{\vee}\otimes\mathbf{fl}_{1}^{*}{\mathsf{sq}}^{*}\omega_{C}.$

Thus the exceptional divisor $\widetilde{\mathfrak{Q}}_{1}$ of the blow up
$\widetilde{\mathsf{Higgs}}_{1}\to\mathsf{Higgs}_{1}$ is

$\widetilde{\mathfrak{Q}}_{1}=\mathbb{P}(N_{\mathfrak{Q}_{1}/\mathsf{Higgs}_{1}})=\mathfrak{Q}_{1}\times\mathbb{P}(H^{0}(C,\omega_{C})^{\vee})=\mathfrak{Q}_{1}\times\mathbb{P}^{1}.$

  * (b)

Suppose $\pi:{\widetilde{C}}\to C$ is a smooth spectral curve. Let
${\mathcal{P}}_{3}$ denote the corresponding Hitchin fiber, and let
${\widehat{C}}\subset{\mathcal{P}}_{3}$ be the correponding fiber of the map
$\boldsymbol{h}_{|\mathfrak{Q}_{1}}:\mathfrak{Q}_{1}\to{\mathcal{B}}$. Then

$N_{{\widehat{C}}/{\mathcal{P}}_{3}}=H^{0}(C,\omega_{C})^{\vee}\otimes\widehat{{\mathsf{sq}}}^{*}\pi^{*}\omega_{C},$

and the exceptional divisor $\boldsymbol{\mathit{E}}_{1}\subset Y_{1}$ of the
blow up $\boldsymbol{\varepsilon}_{1}:Y_{1}\to{\mathcal{P}}_{3}$ is given by

$\boldsymbol{\mathit{E}}_{1}={\widehat{C}}\times\mathbb{P}(H^{0}(C,\omega_{C})^{\vee})={\widehat{C}}\times\mathbb{P}^{1}.$

###### Proof.

To prove (a) note that since $\mathfrak{Q}_{1}$ is
${\mathbb{C}}^{\times}$-stable, we have that the normal bundle
$N_{\mathfrak{Q}_{1}/\mathsf{Higgs}_{1}}$ is
${\mathbb{C}}^{\times}$-equivariant. Furthermore, as we explained in the
previous section, in our situation there are no broken orbits in
$\mathfrak{Q}_{1}$ and therefore the normal bundle will be the pullback of the
bundle of outgoing directions to the non-unitary fix point set
$\mathsf{Higgs}_{1}^{{\mathbb{C}}^{\times},\text{nu}}={\overline{C}}$, i.e.

$N_{\mathfrak{Q}_{1}/\mathsf{Higgs}_{1}}=\mathbf{fl}_{1}^{*}\left(T_{\mathsf{Higgs}_{1}}\right)^{(-1)}_{|{\overline{C}}}$

is the piece in the tangent bundle to $\mathsf{Higgs}_{1}$ of
${\mathbb{C}}^{\times}$-weight $-1$ along the fixed point curve
${\overline{C}}$. Note that here ${\overline{C}}$ is embedded in
$\mathsf{Higgs}_{1}$ by the map sending $A\in{\overline{C}}$ to the
isomorphism class of the Higgs bundle

$(E_{A},\theta_{A}):=\left(A^{-1}(\boldsymbol{\mathsf{p}})\oplus
A,\,\begin{pmatrix}0&0\\\ \gamma&0\end{pmatrix}\right)$

with $\gamma:A^{-1}(\boldsymbol{\mathsf{p}})\to A\otimes\omega_{C}$ being the
unique (up to scale) non-zero map.

From the cohomological description of the tangent space of
$\mathsf{Higgs}_{1}$ at a point we see that the piece of
${\mathbb{C}}^{\times}$-weight $-1$ of the tangent space at a point
$A\in{\overline{C}}\subset\mathsf{Higgs}_{1}$ is precisely the image of the
composition map

$T_{\mathsf{Higgs}_{1},(E_{A},\theta_{A})}\to\mathbb{H}^{1}\left(C,\operatorname{End}_{0}(E_{A})\stackrel{{\scriptstyle\left[\theta_{A},-\right]}}{{\longrightarrow}}\operatorname{End}_{0}(E_{A})\otimes\omega_{C}\right)\to
H^{1}(C,\operatorname{End}_{0}(E_{A})).$

This image is precisely the matrix coefficient piece $H^{1}(C,A^{\otimes
2}(-\boldsymbol{\mathsf{p}}))\subset H^{1}(C,\operatorname{End}_{0}(E_{A}))$
which gives a canonical identification

$T^{(-1)}_{\mathsf{Higgs}_{1},(E_{A},\theta_{A})}=H^{1}(C,A^{\otimes
2}(-\boldsymbol{\mathsf{p}})).$

Next recall, that $A$ is a point in ${\overline{C}}$ precisely when
$A^{\otimes 2}\cong\mathcal{O}_{C}(t-\boldsymbol{\mathsf{p}})$ for some point
$t\in C$. Thus $A^{\otimes 2}(-\boldsymbol{\mathsf{p}})$ is isomorphic to
$\mathcal{O}_{C}(-t^{\prime})$ where $t^{\prime}$ is the hyperelliptic
conjugate of $t$. But the inclusion
$\mathcal{O}_{C}(-t^{\prime})\subset\mathcal{O}_{C}$ induces an isomorphism on
$H^{1}$’s and so we conclude that we get a natural, unique up to scale,
isomorphism

$T^{(-1)}_{\mathsf{Higgs}_{1},(E_{A},\theta_{A})}\cong
H^{1}(C,\mathcal{O}_{C}).$ (17)

More precisely, the isomorphism (17) is the map induced on $H^{1}$’s from the
map of locally free rank one sheaves $A^{\otimes
2}(-\boldsymbol{\mathsf{p}})\cong{\mathcal{O}}_{C}(-t^{\prime})\hookrightarrow{\mathcal{O}}_{C}$.
Since both the isomorphism $A^{\otimes
2}(-\boldsymbol{\mathsf{p}})\cong{\mathcal{O}}_{C}(-t^{\prime})$ and the
inclusion ${\mathcal{O}}_{C}(-t^{\prime})\hookrightarrow{\mathcal{O}}_{C}$ are
uniquely defined up to scale it follows that isomorphism (17) is also unique
up to scale.

This implies that we have a line bundle $L$ on ${\overline{C}}$ so that

$(T^{(-1)}_{\mathsf{Higgs}_{1}})_{|{\overline{C}}}\cong
H^{1}(C,\mathcal{O}_{C})\otimes L=H^{0}(C,\omega_{C})^{\vee}\otimes L.$

To finish the proof of (a) we need to compute the line bundle $L$. But note
that the restriction of the normal bundle of
$\mathfrak{Q}_{1}\subset\mathsf{Higgs}_{1}$ to a general fiber ${\widehat{C}}$
of $\boldsymbol{h}_{|\mathfrak{Q}_{1}}$ is the normal bundle of
${\widehat{C}}\subset{\mathcal{P}}_{3}$. Since the map
$\widehat{\pi}:{\widehat{C}}\to{\overline{C}}$ induces an injective map on
Picard varieties, we only need to compute $\widehat{\pi}^{*}L$. Thus (a) will
follow immediately once we prove (b).

To prove (b) fix a smooth spectral cover $\pi:{\widetilde{C}}\to C$ whose
branch locus does not include any Weirstrass point on $C$. Let
${\mathcal{P}}_{3}=\operatorname{Prym}^{3}({\widetilde{C}},C)\subset\mathsf{Higgs}_{1}$
denote the corresponding Hitchin fiber and let
${\widehat{C}}=\mathfrak{Q}_{1}\cap{\mathcal{P}}_{3}$ be the unstable locus in
${\mathcal{P}}^{3}$. Recall that
${\widehat{C}}={\overline{C}}\times_{C}{\widetilde{C}}$ and the embedding
${\widehat{C}}\hookrightarrow{\mathcal{P}}_{3}$ is given by
$(A,\tilde{t})\mapsto\pi^{*}(A^{-1}(\boldsymbol{\mathsf{p}}))(\tilde{t})$. The
tangent bundle to ${\mathcal{P}}_{3}$ is trivial with fiber identified with
the anti-invariants
$H^{1}({\widetilde{C}},\mathcal{O}_{{\widetilde{C}}})^{-}\subset
H^{1}({\widetilde{C}},\mathcal{O}_{{\widetilde{C}}})$ for the covering
involution of $\pi:{\widetilde{C}}\to C$. By Serre duality we can identify
$H^{1}({\widetilde{C}},\mathcal{O}_{{\widetilde{C}}})^{-}$ with the anti-
invariants in $H^{0}({\widetilde{C}},\omega_{{\widetilde{C}}})^{\vee}$. As
${\widetilde{C}}\to C$ is a spectral cover we have

$\displaystyle H^{0}({\widetilde{C}},\omega_{{\widetilde{C}}})$
$\displaystyle=H^{0}({\widetilde{C}},\pi^{*}\omega_{C}^{\otimes 2})$
$\displaystyle=H^{0}(C,\pi_{*}\pi^{*}\omega_{C}^{\otimes 2})$
$\displaystyle=H^{0}(C,\omega_{C}^{\otimes 2}\oplus\omega_{C})$
$\displaystyle=H^{0}(C,\omega_{C}^{\otimes 2})\oplus H^{0}(C,\omega_{C}),$

and so the anti-invariants in
$H^{0}({\widetilde{C}},\omega_{{\widetilde{C}}})^{\vee}$ are identified
intirinsically with

$H^{0}(C,\omega^{\otimes 2}_{C}).$

With this identification the normal sequence for the embedding
${\widehat{C}}\subset{\mathcal{P}}_{3}$ can be written as

$0\to T_{{\widehat{C}}}\to H^{0}(C,\omega^{\otimes
2}_{C})^{\vee}\otimes\mathcal{O}_{{\widehat{C}}}\to
N_{{\widehat{C}}/{\mathcal{P}}_{3}}\to 0,$

where the first map is the Kodaira-Spencer map for the varying family of line
bundles $\pi^{*}(A^{-1}(\boldsymbol{\mathsf{p}}))(\tilde{t})$ on
${\widetilde{C}}$. Dually, the conormal sequence reads

$0\to N_{{\widehat{C}}/{\mathcal{P}}_{3}}^{\vee}\to H^{0}(C,\omega^{\otimes
2}_{C})\otimes\mathcal{O}_{{\widehat{C}}}\to\omega_{{\widetilde{C}}}\to 0,$
(18)

where the last map is the composition

$\textstyle{H^{0}(C,\omega^{\otimes
2}_{C})\otimes\mathcal{O}_{{\widehat{C}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$$\textstyle{H^{0}({\widetilde{C}},\omega_{{\widetilde{C}}}))\otimes\mathcal{O}_{{\widetilde{C}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ev}}$$\textstyle{\omega_{{\widetilde{C}}}.}$

But $\omega_{{\widetilde{C}}}=\pi^{*}\omega_{C}^{\otimes 2}$ is a pull-back of
$\mathcal{O}(2)$ from the hyperelliptic $\mathbb{P}^{1}$ and
$H^{0}(C,\omega_{C}^{\otimes 2})$ is the pullback of
$H^{0}(\mathbb{P}^{1},\mathcal{O}(2))$. Thus the conormal sequence (18) is a
pullback of the canonical evaluation sequence of vector bundles on the
hyperelliptic $\mathbb{P}^{1}$

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\mathbb{P}^{1},\mathcal{O}(2))\otimes\mathcal{O}_{\mathbb{P}^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

But this is a standard Koszul sequence and so we have a canonical
identification

$\mathcal{K}=H^{0}(\mathbb{P}^{1},\mathcal{O}(1))\otimes\mathcal{O}(-1).$

Equivalently we can note that by definition the bundle $\mathcal{K}$ has no
cohomology and so is isomorphic to a two dimensional vector space tensored
with $\mathcal{O}(-1)$. Since the whole sequence is $SL(2,\mathbb{C})$
equivariant, this two dimensional vector space has to be the fundamental
representation as claimed.

Altogether we get that

$N^{\vee}_{{\widehat{C}}/{\mathcal{P}}_{3}}=H^{0}(\mathbb{P}^{1},\mathcal{O}(1))\otimes\widehat{{\mathsf{sq}}}^{*}\pi^{*}\mathsf{h}_{C}^{*}\mathcal{O}(-1)=H^{0}(C,\omega_{C})\otimes\widehat{{\mathsf{sq}}}^{*}\pi^{*}\omega_{C}^{-1}$

which finishes the proof of part (b) and the lemma. ∎

Again, the question of whether the rational map
$\mathsf{Higgs}_{1}\dashrightarrow X_{1}$ is resolved by blowing up
$\mathfrak{Q}_{1}\subset\mathsf{Higgs}_{1}$ is local near $\mathfrak{Q}_{1}$.
Similarly to the degree zero case it suffices to check that if $S$ is a scheme
and $({}^{S}E,{}^{S}\theta)$ is a relative semistable Higgs bundle on $S\times
C$, which corresponds to an étale map $S\to\mathsf{Higgs}_{1}$, then the
composite rational map $S\to\mathsf{Higgs}_{1}\dashrightarrow X_{1}$ extends
to a morphism
$\widetilde{S}=S\times_{\mathsf{Higgs}_{1}}\widetilde{\mathsf{Higgs}}_{1}\longrightarrow
X_{1}$. Thus we must show that the vector bundle
${}^{S}E_{|(S-\mathfrak{Q}_{1})\times C}$ has a canonical extension to a
vector bundle ${}^{\widetilde{S}}E$ on $\widetilde{S}\times C$ which is
semistable and of determinant $\mathcal{O}_{C}(\boldsymbol{\mathsf{p}})$ on
all geometric fibers over $\widetilde{S}$.

Let ${}^{S}\mathfrak{Q}_{1}=S\times_{\mathsf{Higgs}_{1}}\mathfrak{Q}_{1}$ and
let
${}^{\widetilde{S}}\mathfrak{Q}_{1}=\widetilde{S}\times_{\widetilde{\mathsf{Higgs}}_{1}}\mathfrak{Q}_{1}$
denote the exceptional divisor for the blow-up $\widetilde{S}\to S$. Shrinking
$S$ if necessary we can assume without loss of generality that the restriction
of ${}^{S}E$ to ${}^{S}\mathfrak{Q}_{1}\times C$ is a direct sum

${}^{S}E=\left({}^{S}\\!\\!\mathcal{A}\right)^{-1}(\operatorname{pr}_{C}^{*}\boldsymbol{\mathsf{p}})\oplus\left({}^{S}\\!\\!\mathcal{A}\right)$

where ${}^{S}\\!\\!\mathcal{A}$ denotes the pullback

${}^{S}\\!\\!\mathcal{A}=\left(\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
20.69849pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 4.60004pt\raise
0.0pt\hbox{$\textstyle{{}^{S}\mathfrak{Q}_{1}\times
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
50.69849pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
50.69849pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
4.60004pt\raise 0.0pt\hbox{$\textstyle{\mathfrak{Q}_{1}\times
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
91.65823pt\raise 6.075pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.78612pt\hbox{$\scriptstyle{\mathbf{fl}_{1}\times\operatorname{id}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 118.3388pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
118.3388pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
4.60004pt\raise 0.0pt\hbox{$\textstyle{{\overline{C}}\times
C}$}}}}}}}\ignorespaces}}}}\ignorespaces\right)^{*}\mathcal{A}$

of the normalized Poincaré line bundle $\mathcal{A}\to{\overline{C}}\times C$,
which is uniquely characterized by the conditions $\mathcal{A}_{\\{A\\}\times
C}\cong A$ for all $A\in{\overline{C}}$, and
$\mathcal{A}_{{\overline{C}}\times\\{\boldsymbol{\mathsf{p}}\\}}\cong\mathcal{O}_{{\overline{C}}}$.

Writing ${}^{\widetilde{S}}\\!\\!\mathcal{A}$ for the pullback line bundle

${}^{\widetilde{S}}\\!\\!\mathcal{A}=\left({}^{\widetilde{S}}\mathfrak{Q}_{1}\times
C\to{}^{S}\mathfrak{Q}_{1}\times C\right)^{*}{}^{S}\\!\\!\mathcal{A}$

we can repeat the reasoning we used in the degree zero case we define
${}^{\widetilde{S}}E$ to be the Hecke transform

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{\widetilde{S}}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(\widetilde{S}\times
C\to S\times
C\right)^{*}\left({}^{S}E\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left({}^{\widetilde{S}}\mathfrak{Q}_{1}\times
C\to\widetilde{S}\times
C\right)_{*}\left({}^{\widetilde{S}}\\!\\!\mathcal{A}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

By the calculation in the previous lemma we see that the fiber of the normal
bundle to $\mathfrak{Q}_{1}$ at a point $(E,\theta)$ with
$E=A^{-1}(\boldsymbol{\mathsf{p}})\oplus A$ are precisely the parametrizing
the extension space

$\operatorname{Ext}^{1}(A^{-1}(\boldsymbol{\mathsf{p}}),A)=H^{1}(C,A^{\otimes
2}(-\boldsymbol{\mathsf{p}}))=H^{1}(C,\mathcal{O}_{C}(-t^{\prime}))=H^{1}(C,\mathcal{O}_{C}).$

Suppose now that $S$ is the spectrum of a DVR which maps to
$\mathsf{Higgs}_{1}$ so that the closed point $o\in S$ maps to
$y=(E,\theta)\in\mathfrak{Q}_{1}$ with
$E=A^{-1}(\boldsymbol{\mathsf{p}})\oplus A$, and such that the differential of
$S\to\mathsf{Higgs}_{1}$ maps the tangent vector
$\partial_{\boldsymbol{\mathsf{x}}}$ to a normal vector $\mathfrak{e}\neq 0\in
H^{1}(C,\mathcal{O}_{C}(-t^{\prime}))=H^{1}(C,\mathcal{O}_{C})$. Again the map
$S\to\mathsf{Higgs}_{1}$ is given by a relative Higgs bundle
$({}^{S}E,{}^{S}\theta)$ on $S\times S$ for which the lower left corner of the
matrix representing the Kodairra-Spencer class
$\mathsf{ks}^{\left({}^{S}E\right)}(\partial_{\boldsymbol{\mathsf{x}}})$ is
equal to $\mathfrak{e}$. The deformation theory calculation in Lemma 3.7 now
implies that the bundle ${}^{\widetilde{S}}E_{|\\{[\mathfrak{e}]\\}\times C}$
is the extension of $A^{-1}(\boldsymbol{\mathsf{p}})$ by $A$ given by the
class $\mathfrak{e}$. This showas that for every $\mathfrak{e}$ the bundle
${}^{\widetilde{S}}E_{|\\{[\mathfrak{e}]\\}\times C}$ is stable and completes
the proof of the Theorem 3.6(a) in the case of Higgs bundles with determinant
$\mathbf{d}={\mathcal{O}}_{C}(\boldsymbol{\mathsf{p}})$. We have also proven
Theorem 3.6(b) in this case. Indeed, in Lemma 3.9(b) we saw that $Y=Y_{1}$ is
the blow-up of ${\mathcal{P}}_{3}$ centered at a copy of
${\widehat{C}}\subset{\mathcal{P}}_{3}$. Thus $Y$ is a smooth projective
threefold. Furthermore, Lemma 3.9(b) and the discussion that follows
immediately the proof of Lemma 3.9 we see that the map
$f=\boldsymbol{\mathfrak{f}}_{|X}:Y\to X$ sends the exceptional divisor
$\boldsymbol{\mathit{E}}_{1}$ onto the wobbly divisor ${\rm Wob}_{1}\subset
X_{1}$ and the map $f_{1}:\boldsymbol{\mathit{E}}_{1}\to{\rm Wob}_{1}$ factors
as a composition
$\boldsymbol{\mathit{E}}_{1}={\widehat{C}}\times{\mathbb{P}}^{1}\to{\overline{C}}\times{\mathbb{P}}^{1}\to{\rm
Wob}_{1}$, where
${\widehat{C}}\times{\mathbb{P}}^{1}\to{\overline{C}}\times{\mathbb{P}}^{1}$
is the natural double cover, and ${\overline{C}}\times{\mathbb{P}}^{1}\to{\rm
Wob}_{1}$ is the normalization map. In other words, in this case
$\boldsymbol{\mathit{E}}_{1}$ and ${\rm Wob}_{1}$ are irreducible and
$f_{|\boldsymbol{\mathit{E}}_{1}}:\boldsymbol{\mathit{E}}_{1}\to{\rm Wob}_{1}$
is a double cover. Similarly to the degree zero case this implies that $f:Y\to
X$ is quasi-finite everywhere and hence is finite since it is a proper map.
The theorem is proven. ∎

###  3.8 The main construction

The description of the resolution in Theorem 3.6 shows in particular that the
exceptional divisor of the blowup maps onto the wobbly locus in the moduli of
bundles. This allows us to understand the relationship between the moduli of
Higgs bundles and the cotangent bundle of the moduli of vector bundles not
only over the very stable locus but also in codimension one in the stable
locus, i.e. over the generic point of the wobbly divisor.

Let $\widetilde{\mathfrak{Q}}\subset\widetilde{\mathsf{Higgs}}$ denote the
exceptional divisor, that is - the inverse image of $\mathfrak{Q}$. Recall
that we have a natural morphism

$\psi:T^{\vee}X\rightarrow\mathsf{Higgs}$

inducing an isomorphism on dense open subsets.

Let ${\rm Wob}^{\rm sing}$ denote the singular locus of the wobbly divisor
${\rm Wob}$, and set

$X^{\circ}:=X-{\rm Wob}^{\rm sing},\;\;\;\;{\rm Wob}^{\circ}:={\rm Wob}\cap
X^{\circ}={\rm Wob}-{\rm Wob}^{\rm sing}.$

We note that ${\rm Wob}^{\circ}\subset X^{\circ}$ is now a smooth divisor.

Let
$\widetilde{\mathsf{Higgs}^{\circ}}:=\boldsymbol{\mathfrak{f}}^{-1}(X^{\circ})\subset\widetilde{\mathsf{Higgs}}$
and let $\widetilde{\mathfrak{Q}}^{\circ}$ be the intersection of
$\widetilde{\mathfrak{Q}}$ with $\widetilde{\mathsf{Higgs}^{\circ}}$. With
this notation we now have

######  Proposition 3.10.

Let $T^{\vee}X^{\circ}(\log{\rm Wob}^{\circ})$ denote the total space of the
locally free sheaf $\Omega^{1}_{X^{\circ}}(\log{\rm Wob}^{\circ})$. There is a
commutative diagram

$\textstyle{T^{\vee}X^{\circ}(\log{\rm
Wob}^{\circ})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{\log}}$$\textstyle{\widetilde{\mathsf{Higgs}^{\circ}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\boldsymbol{\mathfrak{f}}}$$\textstyle{X^{\circ}}$

such that the map $\psi^{\rm log}$ coincides with $\psi$ over a dense open
subset, and such that $\psi^{\rm log}$ maps the zero-section of
$T^{\vee}X^{\circ}(\log{\rm Wob}^{\circ})$ isomorphically to
$\widetilde{\mathfrak{Q}}^{\circ}$.

###### Proof.

This follows tautologically from the Hecke description of the families of
Higgs bundles describing the map $\boldsymbol{\mathfrak{f}}$ over the
exceptional $\widetilde{\mathfrak{Q}}$ that we gave in the proof of Theorem
3.6. ∎

The construction: Again we write $\mathsf{Higgs}$ for the moduli space od
Higgs bundles of determinant
$\mathbf{d}={\mathcal{O}}_{C}(k\cdot\boldsymbol{\mathsf{p}})$ where $k$
isfixed to be either $0$ or $1$. Let
$\boldsymbol{h}:\mathsf{Higgs}\to{\mathcal{B}}$ denote the corresponding
Hitchin map and let $\mathsf{b}\in{\mathcal{B}}$ be a general point in the
Hitchin base. Concretely, choose $\mathsf{b}$ so that the correspondng
spectral cover $\pi:{\widetilde{C}}={\widetilde{C}}_{\mathsf{b}}\to C$ is
smooth and unramified above the Weierstrass points of $C$. Write
${\mathcal{P}}=\boldsymbol{h}^{-1}(\mathsf{b})$ for the corresponding fiber of
the Hitchin map. Then ${\mathcal{P}}$ is smooth, and isomorphic to the Prym
variety of the cover ${\widetilde{C}}\to C$. Let $Y$ denote the pullback of
${\mathcal{P}}$ in the blown up moduli space $\widetilde{\mathsf{Higgs}}$.
Then $Y$ is a smooth projective threefold, which by Theorem 3.6 is obtained by
blowing up ${\mathcal{P}}$ in the smooth subvariety
${\mathcal{P}}\cap\mathfrak{Q}$ which in the case of degree zero consists of
$16$ points - the $16$ line bundles
$\\{\pi^{*}\kappa\\}_{\kappa\in\mathsf{Spin}(C)}$ and in the case of degree
one is a copy of the curve ${\widehat{C}}$. Furthermore from the proof of
Theorem 3.6(b) we see that the restriction
$f:=\boldsymbol{\mathfrak{f}}_{|Y}:Y\to X$ is a finite morphism which maps the
exceptional divisor in $Y$ onto a union of components of the wobbly divisor in
$X$. In the degree zero case the exceptional divisor obtained by blowing up
the point $\pi^{*}\kappa\in{\mathcal{P}}$ maps birationally onto the trope
plane $\mathsf{Trope}_{\kappa}\subset{\rm Wob}_{0}\subset X_{0}$ labeled by
the theta characteristic $\kappa$. In the degree one case the exceptional
divisor obtained by blowing up ${\widehat{C}}\subset{\mathcal{P}}$ maps two-
to-one onto the wobbly divisor ${\rm Wob}_{1}\subset X_{1}$. Recall also that
as explained in the introduction, the finite morphism $f:Y\to X$ has degree
$2^{3g(C)-3}=8$.

######  Definition 3.11.

Let $\Omega^{1}_{X}(\log{\rm Wob})^{+}$ denote the unique reflexive sheaf on
$X$ whose restriction to $X^{\circ}$ is the vector bundle
$\Omega^{1}_{X^{\circ}}(\log{\rm Wob}^{\circ})$.

If we let $j^{\circ}:X^{\circ}\hookrightarrow X$, then we can set more
precisely

$\Omega^{1}_{X}(\log{\rm
Wob})^{+}:=\left(j^{\circ}_{*}\Omega^{1}_{X^{\circ}}(\log{\rm
Wob}^{\circ})\right)^{\vee\vee}.$

Note that $\Omega^{1}_{X}(\log{\rm Wob})^{+}$ is locally free over $X^{\circ}$
and there it is equal to $\Omega^{1}_{X^{\circ}}(\log{\rm Wob}^{\circ})$.

Let ${\mathcal{L}}$ be a line bundle on $Y$. Then set
${\mathcal{V}}:=f_{*}({\mathcal{L}})$. It is a rank $8$ vector bundle on $X$.
It has a meromorphic Higgs field coming from the fact that a dense open subset
of $Y$ may be identified with a subvariety of $T^{\vee}X$ via the inverse to
$\psi$.

######  Corollary 3.12.

Under the main construction ${\mathcal{V}}=f_{*}({\mathcal{L}})$ comes
equipped with a meromorphic Higgs field which comes from a morphism

$\Phi:{\mathcal{V}}\rightarrow{\mathcal{V}}\otimes\Omega^{1}_{X}(\log{\rm
Wob})^{+}.$

Viewed as a sheaf with operators, ${\mathcal{V}}$ has no nontrivial
$\Phi$-invariant subsheaves.

###### Proof.

Since $\Omega^{1}_{X}(\log{\rm Wob})^{+}$ is defined as a reflexive hull, it
suffices to see this over $X^{\circ}$, and there it comes from the commutative
diagram of Proposition 3.10.

The spectral variety of $\Phi$ is birational to the covering $f:Y\rightarrow
X$, and this is irreducible, so there can be no $\Phi$-invariant subsheaves of
${\mathcal{V}}$, indeed such a subsheaf would correspond to a nontrivial
decomposition of $Y$ as a union of closed subsets. ∎

###  3.9 Nonabelian Hodge outside codimension $2$

In Corollary 3.12, we are not able to say that we construct a “logarithmic
Higgs field” because the divisor ${\rm Wob}$ does not have normal crossings.
The objective of this section is to investigate how the data of a logarithmic
Higgs bundle defined outside of codimension $2$ determines a local system by
the nonabelian Hodge correspondence. We will also include parabolic
structures.

In this section we consider a general variety $X$, a divisor $W$, and an open
subset $X^{\circ}\subset X$ complement of a closed subset of codimension $\geq
2$, such that $W^{\circ}:=W\cap X^{\circ}$ is smooth.

Suppose we are given a reflexive sheaf ${\mathcal{V}}$ over $X$ that is
locally free over $X^{\circ}$, and a parabolic structure for ${\mathcal{V}}$
along $W^{\circ}$, denoted ${\mathcal{V}}_{\bullet}$. Suppose
$\Phi:{\mathcal{V}}\rightarrow{\mathcal{V}}\otimes\Omega^{1}_{X}(\log W)^{+}$
is an operator with values in the sheaf of differentials that is the reflexive
extension of $\Omega^{1}_{X^{\circ}}(\log W^{\circ})$ as defined previously.
We suppose that the residues of $\Phi$ along components of $W^{\circ}$, acting
on the parabolic associated graded pieces, are nilpotent.

Consider a projective morphism $g:Z\rightarrow X$ and let
$Z^{\circ}:=g^{-1}(X^{\circ})$. Suppose $D\subset Z$ is a simple normal
crossings divisor such that $D^{\circ}:=D\cap Z^{\circ}$ contains
$g^{-1}(W^{\circ})$. The following statement is a recasting of Mochizuki’s
extension theorem [Moc07b] to our setting.

######  Theorem 3.13.

Suppose that $g:(Z,D)\rightarrow(X,W)$ exists as above, and suppose that there
exists a semistable logarithmic parabolic Higgs sheaf ${\mathcal{F}}$ on
$(Z,D)$ whose restriction to $Z^{\circ}$ is isomorphic to
$g^{*}(({\mathcal{V}}_{\bullet},\Phi)|_{X^{\circ}})$. Suppose furthermore that
$c_{1}^{\rm par}({\mathcal{F}})=0$ and $c_{2}^{\rm par}({\mathcal{F}})=0$.
Then there is a mixed twistor ${\mathscr{D}}$-module ${\bf E}$ over $X$, with
singularities along $W$, such that the restriction of ${\bf E}$ to $X^{\circ}$
corresponds to $({\mathcal{V}}_{\bullet},\Phi)|_{X^{\circ}}$. Furthermore,
${\bf E}$ is unique up to isomorphism, and the pullback $g^{*}({\bf E})$
corresponds to ${\mathcal{F}}$.

If $S\subset X^{\circ}$ is any projective curve then the restriction of ${\bf
E}$ to $S$ is the mixed twistor ${\mathscr{D}}$-module that corresponds to the
logarithmic parabolic Higgs bundle on a curve
$({\mathcal{V}}_{\bullet},\Phi)|_{S}$.

The mixed twistor ${\mathscr{D}}$-module ${\bf E}$ has Betti realization that
restricts to a local system on $X-W$, and this local system has quasi-unitary
monodromy transformations around the components of $W^{\circ}$ such that the
arguments of eigenvalues of the monodromy are the parabolic weights and the
unipotent part of the monodromy for each eigenvalue has the same Jordan form
as the residue of $\Phi$ on the corresponding graded piece. This comes from
the restriction to curves $S\subset X^{\circ}$.

Stated more compactly, the conclusion of the above theorem is that in order to
construct such a local system on $X-W$ it suffices to know the parabolic
structure and logarithmic Higgs field over an open subset $X^{\circ}\subset X$
whose complement has codimension two, provided we can find some covering $Z/X$
where the divisor has normal crossings and where there exists an extension
with vanishing parabolic Chern classes.

This is what we will be doing in our concrete situations.

###  3.10 Parabolic structures

In this subsection we discuss some aspects of parabolic structures. We refer
to the numerous available references for the general theory of parabolic
bundles and their Chern classes, including for example [Bis97b, Bis97a, Bod91,
BH95, Bor07, BV12, Kon93, MY92, IS08, IS07, Tah10, Tah13]. The present
discussion will be tailored to our specific needs. Suppose $(X,D)$ is a pair
consisting of a smooth variety and a reduced divisor. Let $X^{\circ}\subset X$
be the complement of a subset of codimension $\geq 2$, and let
$D^{\circ}:=D\cap X^{\circ}$ (with this notation extended in a similar way to
other objects). We assume that $D^{\circ}$ is smooth. That is notably attained
by taking $X^{\circ}$ to be the complement of the non-smooth locus of $D$, but
one might want to throw out other subsets of codimension $\geq 2$ as well.

Define the notion of _quasi-parabolic bundle_ on $(X,D)$ to consist of a
reflexive sheaf ${\mathcal{V}}$ on $X$ together with a filtration by strict
subsheaves $0=F_{0}\subset F_{1}\subset\cdots\subset
F_{k}={\mathcal{V}}|_{D^{\circ}}$ of the restriction of ${\mathcal{V}}$ to
$D^{\circ}$. A _parabolic bundle_ is a quasi-parabolic bundle together with an
assignment of real parabolic weights to the subquotients of the filtration
over connected components of $D^{\circ}$.

If $D$ has simple normal crossings, we obtain by extension filtrations over
each smooth irreducible component of $D$. Assuming $D$ has simple normal
crossings, a _locally abelian parabolic bundle_ on $(X,D)$ is a parabolic
bundle such that ${\mathcal{V}}$ is locally free, the filtrations on divisor
components have locally free subquotients, and locally at any point there
exists a frame adapted to the filtrations of all divisor components passing
through that point. If, in addition, the parabolic weights are rational, then
these objects correspond to vector bundles on a root stack of $(X,D)$ [Bis97b,
Bod91, Bor07, BV12].

For the purposes of Theorem 3.13, we are interested in parabolic Chern numbers
of the form $c_{1}^{\rm par}\cdot[\omega]^{n-1}$ and $c_{2}^{\rm
par}\cdot[\omega]^{n-2}$, and for those it suffices to know the parabolic
structure outside of a subset of codimension $3$ in $X$. If $D$ has normal
crossings outside of codimension $3$, any parabolic structure satisfies the
locally abelian condition outside of codimension $3$.

Our particular application is to the case where $D={\rm Wob}$ is the wobbly
divisor in $X=X_{1}$, and furthermore the parabolic weights are $-1/2$ and
$0$. We will be using a technique of pullback to a ramified covering to
compensate for the fact that ${\rm Wob}$ has non-normal crossings
singularities, namely cuspidal singularities as well as nodes up to
$X$-codimension $2$. The fact that there are only two distinct parabolic
weights means that the filtration involves a single subsheaf.

In this setting, the notion of parabolic structure can be simplified further.
Given $(X,D)$ a _crude quasi-parabolic structure_ consists of a reflexive
sheaf ${\mathcal{V}}$ and a subsheaf
${\mathcal{V}}^{\prime}\subset{\mathcal{V}}$ such that ${\mathcal{V}}$ is
locally free on $X^{\circ}$, ${\mathcal{V}}^{\prime}$ is also reflexive, and
${\mathcal{U}}:={\mathcal{V}}/{\mathcal{V}}^{\prime}$ is a locally free sheaf
over $W^{\circ}$. We let $r_{\mathcal{U}}:=\text{rank}({\mathcal{U}})$ be the
rank of this quotient.

A _crude parabolic structure_ consists of a crude quasi-parabolic structure
plus a _parabolic weight_ which is a single number $\alpha\in(0,1]$. To these
data we associate a parabolic vector bundle on $(X^{\circ},W^{\circ})$ whose
weights are $\alpha$ for the associated-graded piece ${\mathcal{U}}$, and $0$
for other the associated-graded piece
${\mathcal{U}}^{\prime}:={\mathcal{V}}^{\prime}/{\mathcal{V}}(-W)$.

The filtered sheaf ${\mathcal{E}}^{\circ}_{\cdot}$ on $X^{\circ}$ is given by

${\mathcal{E}}^{\circ}_{a}={\mathcal{V}}^{\prime},\;\;0\leq
a<\alpha,\;\;\;\;{\mathcal{E}}^{\circ}_{a}={\mathcal{V}},\;\;\alpha\leq a<1.$

Let $\Omega^{1}_{X}(\log D)^{+}$ be the reflexive extension to $X$ of
$\Omega^{1}_{X^{\circ}}(\log D^{\circ})$ which is well-defined since we are
assuming that $D^{\circ}$ is smooth. This isn’t really the correct “sheaf of
logarithmic differentials” as we will confront when dealing with the tacnodes
of the wobbly locus in the degree $0$ moduli space, but let us leave that
discussion for later.

Say that a map
$\Phi:{\mathcal{V}}\rightarrow{\mathcal{V}}\otimes\Omega^{1}_{X}(\log D)^{+}$
is a _pre-logarithmic Higgs field_ if it satisfies $\Phi\wedge\Phi=0$
generically. It is called _nil-compatible_ with the parabolic structure if
$\Phi({\mathcal{V}}^{\prime})\subset{\mathcal{V}}^{\prime}\otimes\Omega^{1}_{X}(\log
W)^{+}$ and if the residue of $\Phi$ along $D^{\circ}$ is nilpotent. This
nilpotence is equivalent to demanding that the two associated-graded pieces of
the residue, operating on ${\mathcal{U}}$ and ${\mathcal{U}}^{\prime}$, are
nilpotent.

Given such a logarithmic Higgs field, then on the parabolic structure
associated to $({\mathcal{V}},{\mathcal{V}}^{\prime},\alpha)$ we obtain a
parabolic logarithmic Higgs field with nilpotent residues.

The following construction shows how to get a crude logarithmic parabolic
structure from a spectral covering.

######  Theorem 3.14.

Suppose $f:Y\rightarrow X$ is a finite morphism, and suppose we are given a
factorization through an inclusion $\psi:Y^{\circ}\rightarrow
T^{\vee}X^{\circ}(\log D^{\circ})$ over an open subset $X^{\circ}$ complement
of a subset of codimension $\geq 2$. Set $Y^{\circ}:=f^{-1}(X^{\circ})$ and so
forth. Let $B\subset Y$ be a divisor such that $B^{\circ}=B\cap Y^{\circ}$ is
smooth and $f(B)\subset D$. Assume that $\psi^{\circ}$ maps the reduced
inverse image $f^{-1}(D^{\circ})^{\rm red}$ to the zero-section of
$T^{\vee}X^{\circ}(\log D^{\circ})$. Let ${\mathcal{L}}$ be a line bundle on
$Y$, and let ${\mathcal{L}}^{\prime}:={\mathcal{L}}(-B)$. Then
${\mathcal{V}}:=f_{*}({\mathcal{L}})$ has a subsheaf
${\mathcal{V}}^{\prime}:=f_{*}({\mathcal{L}})$. For any choice of parabolic
weight $\alpha$, and letting $\Phi$ be the pre-logarithmic Higgs field
determined by $\psi$, we obtain a crude parabolic pre-logarithmic Higgs bundle
$({\mathcal{V}},{\mathcal{V}}^{\prime},\alpha,\Phi)$ on $(X,D)$ with nilpotent
residues.

###### Proof.

Indeed, the quotient ${\mathcal{U}}:={\mathcal{V}}/{\mathcal{V}}^{\prime}$ is
isomorphic to $f_{*}({\mathcal{L}}|_{B})$ and over $D^{\circ}$ this is a
locally free quotient of ${\mathcal{V}}|_{D}$. The fact that $\psi$ maps
$Y^{\circ}$ to $T^{\vee}X^{\circ}(\log D^{\circ})$ implies that $\Phi$ is
logarithmic over $X^{\circ}$, for both ${\mathcal{V}}$ and its subsheaf
${\mathcal{V}}^{\prime}$. On the reflexive extensions this yields a crude
parabolic pre-logarithmic Higgs bundle. The condition on the zero-section
insures that the residues of $\Phi$ are nilpotent. ∎

Terminology: In the situation of the theorem, we will call
${\mathcal{L}}^{\prime}$ the _spectral line bundle_ on $Y$, because in terms
of the filtered sheaf we have

${\mathcal{E}}_{0}=f_{*}({\mathcal{L}}^{\prime}).$

There are two main difficulties in the theoretical situation, both occasioned
by the fact that the wobbly divisor $D={\rm Wob}$ generally does not have
normal-crossings singularities:

  * •

How to calculate effectively the parabolic $c_{2}$ or the parabolic Bogomolov
$\Delta$-invariant?

  * •

How to recognize if the Higgs field $\Phi$ will correspond to a genuinely
logarithmic Higgs field on a normal crossings resolution?

Over the degree $0$ moduli space $X_{0}$, the crude parabolic structure is
going to be trivial. The divisor ${\rm Wob}$ then represents the location of
singularities of the Higgs field. In that case, the strategy will be to follow
the prescription in Theorem 3.13 to make a resolution of singularities of the
divisor ${\rm Wob}$ into a normal crossings divisor, and look for a parabolic
extension whose parabolic Chern class vanishes. In this case, the parabolic
structures will have more than two jumps so one needs to consider filtrations
of bigger length.

For the degree $1$ moduli space $X_{1}$, we will be using a crude parabolic
structure with parabolic weight $\alpha=1/2$. This was found partly by
computation and partly by guessing. Our strategy for the degree $1$ case
consists of passing to a ramified covering in order to remove the
singularities of $D={\rm Wob}$ or transform them to normal crossings. Although
the pullback of a parabolic bundle with arbitrary weight under a ramified
covering may be difficult to compute, in the case $\alpha=1/2$ we can give a
description.

######  Proposition 3.15.

Suppose
$\underline{{\mathcal{V}}}=({\mathcal{V}},{\mathcal{V}}^{\prime},\alpha,\Phi)$
is a crude parabolic pre-logarithmic Higgs bundle on $(X,D)$ with parabolic
weight $\alpha=1/2$. Suppose $g:Z\rightarrow X$ is a Galois covering from a
smooth variety, such that $g$ is ramified of order $2$ along
$B^{1}=g^{-1}(D^{\circ})^{\rm red}$ over $D^{\circ}$. Then the pullback
$g^{*}(\underline{{\mathcal{V}}})$ has trivial parabolic structure, so it
corresponds to a reflexive sheaf ${\mathcal{V}}_{Z}$ with logarithmic Higgs
field that is described in the following way. Let
${\mathcal{U}}:={\mathcal{V}}/{\mathcal{V}}^{\prime}$, let
${\mathcal{U}}_{Z}:=g^{*}({\mathcal{U}})$, and use a superscript $()^{\circ}$
to denote the restriction over $X^{\circ}$. Let ${\mathcal{U}}_{B^{\circ}}$ be
the restriction of ${\mathcal{U}}^{\circ}$ to $B^{\circ}$ seen as a coherent
sheaf on $Z^{\circ}$, also equal to
${\mathcal{U}}_{Z}^{\circ}/\mathcal{I}_{B^{\circ}\subset
Z^{\circ}}{\mathcal{U}}_{Z}^{\circ}$. We have an exact sequence

$0\rightarrow{\mathcal{V}}^{\circ}_{Z}\rightarrow
g^{*}{\mathcal{V}}^{\circ}\rightarrow{\mathcal{U}}_{B^{\circ}}\rightarrow 0$

and ${\mathcal{V}}_{Z}$ is the reflexive extension of
${\mathcal{V}}^{\circ}_{Z}$ from $Z^{\circ}$ to $Z$. The logarithmic Higgs
field $\Phi$ is induced from that of $g^{*}({\mathcal{V}})$.

###### Proof.

Follows immediately from the definition of a pullback of parabolic structures.
∎

###  3.11 Pushforward statement

In order to calculate the Hecke transforms of the local systems that we are
going to construct, we need a Dolbeault method for calculating higher direct
images. This is provided by the theory of [DPS16], which is based in turn on
the theory of twistor ${\mathscr{D}}$-modules of Sabbah and Mochizuki [Moc07a,
Moc07b, Sab05].

We will need to extend the discussion of [DPS16] in order to apply it to the
specific situations encountered for the Hecke transform. Luckily, the fibers
of the Hecke correspondence are $1$-dimensional, since we are dealing with
moduli spaces of rank $2$ bundles in this paper. In order to identify the
higher direct image local systems, it suffices to restrict to curves in the
target space, and to make things easy we can even use lines. Thus, the direct
image calculations are for maps from a surface to a curve, the main setup of
[DPS16]. The present situation differs in that the horizontal divisor
typically has simple ramification points, whereas the hypothesis of [DPS16]
was to have an etale horizontal divisor. Thus, some work needs to be done in
the direction of looking at a ramified cover of the base.

The full discussion will be deferred to Chapter 12. In this subsection we
summarize the basic knowledge to come out of that, as needed to treat the
Hecke transforms. The setup we discuss here is therefore closely tailored to
the situations that are encountered in the applications.

Suppose $f:X\rightarrow S$ is a projective morphism from a smooth surface to a
smooth curve. Suppose $D\subset X$ is a reduced divisor, each of whose
components dominates $S$. Classify the points of $(X,D)$ among the following
kinds:

3.11.1. Types of points of $(X,D)$

  1. (a)

Points in $X-D$ where $f$ is smooth;

  2. (b)

Points in $X-D$ where the fiber of $f$ has a simple normal crossing;

  3. (c)

Points on $D$ where $D$ is etale over $S$;

  4. (d)

Points on $D$ where $f$ is smooth and $D$ is smooth with simple ramification
over $S$;

  5. (e)

Points on $D$ where $f$ is smooth and $D$ has a normal crossing such that both
branches are etale over $S$;

  6. (f)

Other points.

Suppose we are given a normal variety $\Sigma$ that we will call (under a
small abuse of notation) the _spectral variety_ , with a map
$\pi_{\Sigma}:\Sigma\rightarrow X$, together with a line bundle
${\mathcal{L}}$ that we will call the _spectral line bundle_ and a $1$-form
$\alpha$ on the smooth locus of $\Sigma$ that we will call the _tautological
form_.

Set $E_{0}:=\pi_{\Sigma*}({\mathcal{L}})$. We assume that $\alpha$ leads to a
map $\Sigma\rightarrow T^{\vee}(X,\log D)$, implying that the tautological
form provides $E_{0}$ with a Higgs field $\theta:E_{0}\rightarrow
E_{0}\otimes\Omega^{1}_{X}(\log D)$.

Notice here that the image of $\Sigma$ in the logarithmic cotangent bundle
might not be normal, but we look at the normalization $\Sigma$ and call that
the spectral variety, whence the abuse of notation mentioned above.

We will consider two cases: the _nilpotent case_ , where the parabolic
structure is trivial and the Higgs field has a nilpotent residue along $D$
coming from the ramification of $\Sigma$; and the _parabolic case_ where the
parabolic weights are $0,1/2$. In this case, we suppose
$E_{1/2}=\pi_{\Sigma*}({\mathcal{L}}(R))$ where $R\subset\Sigma$ is the part
of the ramification divisor of $\Sigma\to X$ that sits over $D$. Note that
$\Sigma$ will in general have other ramification that is not included in $R$,
the hypothesis is that all the ramification over $D$ contributes to the
parabolic structure.

The Higgs field is given by the spectral $1$-form $\alpha\in
H^{0}(\Sigma,\Omega^{1}_{\Sigma})$ We assume that $\alpha$ is nonzero on
normal vectors to $R$ at general points. This implies, in the nilpotent case,
that the nilpotent residues of the Higgs field are nonzero, giving a size $2$
Jordan block at each point of $R$ over a general point of $D$.

Define the _upper critical locus_

$\widetilde{{\rm Crit}}(X/S,E_{\bullet},\theta)\subset\Sigma$

to be the closure of the zero-locus of the projection of $\alpha$ to a section
of relative differentials on $X/S$, from the open subset where $\Sigma$ is
smooth. The _lower critical locus_ ${\rm Crit}(X/S,E_{\bullet},\theta)$ is its
image in $X$.

Let $G$ denote the normalization of $\widetilde{{\rm
Crit}}(X/S,E_{\bullet},\theta)$, and let $g:G\rightarrow S$ be the induced
morphism. Let $j:G\rightarrow X$ denote the normalization of the inclusion
map.

######  Hypothesis 3.16.

We make the following genericity hypotheses, in addition to the basic
assumptions above.

  * •

The lower critical locus ${\rm Crit}(X/S,E_{\bullet},\theta)$ does not contain
any points of type (3.11.1(f));

  * •

The spectral variety $\Sigma$ is smooth, except that each point above a point
of type (3.11.1(e)) is an ordinary double point;

  * •

If there are points of type (3.11.1(e)) then we are in the nilpotent case;

  * •

The upper critical locus is smooth except over points of type (3.11.1(e)), in
particular its normalization $G$ is a smooth curve.

  * •

The restriction of the spectral $1$-form $\alpha$ to the vertical direction in
$\Sigma$ over a point of type (3.11.1(d)) is nonzero.

  * •

At each ordinary double point over a point of type (3.11.1(e)), the curve $G$
has two smooth branches whose tangent vectors are distinct.

Let $Q:=(G\cap R)$ be the divisor in $G$ that is the intersection with $R$.
The hypothesis that $\alpha$ is nonzero in the vertical direction at points of
type (3.11.1(d)) implies that $G$ is transverse to $R$ at those points, so $Q$
is reduced at such points (see Lemma 12.9).

In practice, $G$ will only meet $R$ over points of type (3.11.1(d)) or
(3.11.1(e)): in the nilpotent case that comes from constancy of the Jordan
form of the residue of the Higgs field along $D$, while in the parabolic case
it is a property of our setup, see Lemma 9.6. We can however state the
pushforward property without including such an hypothesis.

The basic philosophy of the pushforward statement, whose proof will be given
in Chapter 12, is that the cohomology along the fibers is calculated by
localizing at the zeros of the relative Higgs field, namely the upper critical
locus. Its normalization $G$ is a smooth curve destined to be the spectral
curve of the higher direct image Higgs bundle.

The contribution at a local point will come from the spectral line bundle,
tensored with the relative differentials of $X/S$. A correction is needed at
the points where $D$ is ramified over the base, so the expressions for the
higher direct image spectral line bundle will include correction terms at the
divisor $Q$ in addition to ${\mathcal{L}}|_{G}\otimes j^{*}\omega_{X/S}$.

The points of type (3.11.1(e)) introduce a difficulty since the spectral
variety is not smooth and the upper critical locus has normal crossings. Our
statement identifying the higher direct image Higgs bundle will require, in
this case, a global degree calculation—that does work out in our cases of
interest.

######  Theorem 3.17.

Assume the basic hypotheses explained at the start, assume that we are either
in the parabolic or nilpotent cases as described above, and assume the
hypotheses 3.16. Then, the parabolic Higgs bundle $(F_{\bullet},\Phi)$ on $S$
corresponding to the middle direct image of the perverse sheaf associated to
$(E_{\bullet},\theta)$ has the following descriptions depending on the case.

—In the parabolic case, $F_{\bullet}$ has trivial parabolic structure and

$F_{0}=g_{*}\left({\mathcal{L}}|_{G}\otimes
j^{*}\omega_{X/S}\otimes{\mathcal{O}}_{G}(Q)\right).$

—In the nilpotent case, $F_{\bullet}$ has a parabolic structure with weights
$0,1/2$ and contains a parabolic subsheaf $F^{\prime}_{\bullet}\hookrightarrow
F_{\bullet}$ defined in the following way:

$F^{\prime}_{0}=g_{*}\left({\mathcal{L}}|_{G}\otimes j^{*}\omega_{X/S}\right)$

and

$F^{\prime}_{-1/2}=g_{*}\left({\mathcal{L}}|_{G}\otimes
j^{*}\omega_{X/S}\otimes{\mathcal{O}}_{G}(-Q)\right).$

We have $F^{\prime}_{\bullet}=F_{\bullet}$ away from the images of points of
type (3.11.1(e)), in particular if there are no such points then this gives
the expression for $F_{\bullet}$.

If the parabolic sheaf $F^{\prime}_{\bullet}$ defined as above has global
parabolic degree $0$ then $F^{\prime}_{\bullet}=F_{\bullet}$ and these give
expressions for $F_{\bullet}$.

In both the nilpotent and parabolic cases, the Higgs field on $F_{0}$ comes
from the differential form on $G$ obtained by restricting the tautological
differential $\alpha$ from $\Sigma$.

###  3.12 Chern classes

For reference below, we recall here the required facts about Chern classes.
We’ll almost always be interested in what happens up to codimension two, so by
convention—unless otherwise specified—our formulas will be truncated at
codimension two. Thus for example we write

${\rm ch}(E)=r(E)+c_{1}(E)+(c_{1}(E)^{2}/2-c_{2}(E).$

In the other direction,

$c_{2}=c_{1}^{2}/2-{\rm ch}_{2}.$

An important invariant is Bogomolov’s discriminant [Bog78, Bog94]

$\Delta=\frac{1}{2r}c_{1}^{2}-ch_{2}=c_{2}-\frac{r-1}{2r}c_{1}^{2},$

having the property that $\Delta(E\otimes L)=\Delta(E)$ for a line bundle $E$.

The same hold for parabolic Chern classes.

Suppose $\pi:Y\rightarrow X$ is a map. Then the Grothendieck-Riemann-Roch
formula says that for a coherent sheaf $L$ on $Y$ we have

${\rm ch}(\pi_{*}(L))=\pi_{*}({\rm td}(Y/X)\cdot{\rm ch}(L)),$

where the relative Todd class is

${\rm td}(Y/X)={\rm td}(TY)\pi^{*}{\rm td}(TX)^{-1}$

with

${\rm td}(TY)=1+\frac{c_{1}(TY)}{2}+\frac{c_{1}(TY)^{2}+c_{2}(TY)}{12}$

and similarly for ${\rm td}(TX)$.

We will specialize to the main cases that arise in our study: projective
space, the intersection of two quadrics in ${\mathbb{P}}^{5}$, and blow-ups of
abelian varieties.

Projective spaces: Let us consider the case of projective space. Let $H$
denote the hyperplane class on ${\mathbb{P}}^{n}$, and as stated above by
convention we truncate to codimension $2$. The Euler exact sequence

$0\rightarrow{\mathcal{O}}_{{\mathbb{P}}^{n}}\rightarrow{\mathcal{O}}_{{\mathbb{P}}^{n}}(1)^{n+1}\rightarrow
T{\mathbb{P}}^{n}\rightarrow 0$

gives

${\rm ch}(T{\mathbb{P}}^{n})=n+(n+1)H+\frac{n+1}{2}H^{2}$

so

$c_{1}(T{\mathbb{P}}^{n})=(n+1)H,\;\;\;\;c_{2}(T{\mathbb{P}}^{n})=\frac{n^{2}+n}{2}H^{2}.$

Thus

$\displaystyle{\rm td}(T{\mathbb{P}}^{n})$
$\displaystyle=1+c_{1}/2+(c_{1}^{2}+c_{2})/12$
$\displaystyle=1+\frac{n+1}{2}H+\frac{3n^{2}+5n+2}{24}H^{2}.$

For $X={\mathbb{P}}^{3}$ this gives

${\rm td}(TX)=1+2H+\frac{11}{6}H^{2}.$

Intersection of quadrics: Suppose $X\subset{\mathbb{P}}^{5}$ a complete
intersection of two quadrics. Let $H$ also denote the restriction of the
hyperplane class of ${\mathbb{P}}^{5}$ to $X$. The normal bundle of $X$ is
$N_{X/{\mathbb{P}}^{5}}\cong{\mathcal{O}}_{X}(2)^{2}$ so the exact sequence

$0\rightarrow TX\rightarrow T{\mathbb{P}}^{5}|_{X}\rightarrow
N_{X/{\mathbb{P}}^{5}}\rightarrow 0$

gives

$\displaystyle{\rm ch}(TX)$ $\displaystyle={\rm ch}(T{\mathbb{P}}^{5})-2{\rm
exp}(2H)$ $\displaystyle=5+6H+3H^{2}-2-4H-4H^{2}$ $\displaystyle=3+2H-H^{2}.$

We get

$c_{1}(TX)=2H,\;\;\;\;c_{2}(TX)=3H^{2}.$

Thus,

$\displaystyle{\rm td}(TX)$ $\displaystyle=1+c_{1}/2+(c_{1}^{2}+c_{2})/12$
$\displaystyle=1+H+7H^{2}/12.$

Blow-ups of Pryms: Suppose ${\mathcal{P}}$ is a $3$-dimensional abelian
variety, so its tangent sheaf has trivial Chern classes, and we let
$\boldsymbol{\varepsilon}:Y\rightarrow{\mathcal{P}}$ be the blow-up along a
smooth subvariety $A\subset{\mathcal{P}}$. Let $E\subset Y$ denote the
exceptional divisor. There are two cases of interest.

Suppose first $A$ is a disjoint collection of $a$ points. Then $E$ is a
disjoint collection of $a$ planes ${\mathbb{P}}^{2}$. A point $y\in E$
corresponds to a tangent direction at the corresponding image point
$\boldsymbol{\varepsilon}(y)\in{\mathcal{P}}$, and the tangent vectors on $Y$
at $y$ are vectors that are constrained to map into this tangent direction in
$T_{\boldsymbol{\varepsilon}(y)}{\mathcal{P}}$. Over $E$ there is the
universal subbundle ${\mathcal{O}}_{E}(-1)={\mathcal{O}}_{E}(E)$, fitting into
an exact sequence

$0\rightarrow{\mathcal{O}}_{E}(E)\rightarrow\boldsymbol{\varepsilon}^{*}T{\mathcal{P}}|_{E}\rightarrow
R\rightarrow 0$

where the middle is a trivial bundle. Then, letting $R_{Y}$ be $R$ viewed as a
coherent sheaf on $Y$ supported on $E$, we then have an exact sequence

$0\rightarrow
TY\rightarrow\boldsymbol{\varepsilon}^{*}T{\mathcal{P}}\rightarrow
R_{Y}\rightarrow 0.$

We have

${\rm ch}(R_{Y})={\rm ch}(\boldsymbol{\varepsilon}^{*}T{\mathcal{P}}){\rm
ch}({\mathcal{O}}_{E})-{\rm ch}({\mathcal{O}}_{Y}(E)){\rm
ch}({\mathcal{O}}_{E}).$

Now ${\rm ch}(\boldsymbol{\varepsilon}^{*}T{\mathcal{P}})=3$ since the tangent
sheaf of ${\mathcal{P}}$ is trivial of rank $3$. The exact sequence

$0\rightarrow{\mathcal{O}}(-E)\rightarrow{\mathcal{O}}\rightarrow{\mathcal{O}}_{E}\rightarrow
0$

gives

${\rm ch}({\mathcal{O}}_{E})=E-E^{2}/2,$

so we get

$\displaystyle{\rm ch}(R_{Y})$ $\displaystyle=(3-(1+E+E^{2}/2))(E-E^{2}/2)$
$\displaystyle=2E-2E^{2}.$

Putting this into the other exact sequence we get

${\rm ch}(TY)=3-2E+2E^{2}.$

We note that the shape of this formula is independent of the number $a$ of
points that were blown up (but of course the full exceptional divisor $E$
contains that information).

Turn now to the other case: when $A$ is a smooth irreducible curve. The normal
bundle $N_{A/{\mathcal{P}}}$ is a rank $2$ vector bundle over $A$. Let
$\delta$ denote the degree of the normal bundle, i.e. the degree of its
determinant line bundle. A point $y\in E$ corresponds to a normal direction to
$A$ at $b(y)$, so as before the tangent vectors to $Y$ at $y$ are constrained
to have image in $T_{b(y)}{\mathcal{P}}$ that maps into this normal direction
under the map $T{\mathcal{P}}|_{A}\rightarrow N_{A/{\mathcal{P}}}$. Let
${\mathcal{O}}_{E}(-1)$ denote the tautological sub-bundle over $E$. It fits
into an exact sequence

$0\rightarrow{\mathcal{O}}_{E}(-1)\rightarrow\boldsymbol{\varepsilon}^{*}N_{A/{\mathcal{P}}}\rightarrow
R\rightarrow 0$

and as before, if $R_{Y}$ denotes the corresponding coherent sheaf on $Y$
supported over $E$, then we have the exact sequence

$0\rightarrow
TY\rightarrow\boldsymbol{\varepsilon}^{*}T{\mathcal{P}}\rightarrow
R_{Y}\rightarrow 0.$

The tautological subbundle is also the normal bundle to $E$ in $Y$, thus
${\mathcal{O}}_{E}(-1)={\mathcal{O}}_{E}(E)$ as in the previous case.

The rank $2$ bundle $N_{A/{\mathcal{P}}}$ over the curve $A$ is determined, up
to rational numerical equivalence, by its degree that we’ll call $\delta$.

Suppose $G$ is an ample divisor class on ${\mathcal{P}}$, then $G\cdot A$ is
the degree of $G$ restricted to $A$. Working numerically, we can write

${\rm det}N_{A/{\mathcal{P}}}\sim\frac{\delta}{G\cdot A}G|_{A}$

so $\boldsymbol{\varepsilon}^{*}N_{A/{\mathcal{P}}}$, considered as a coherent
sheaf on $Y$, is rationally numerically equivalent to the expression (with a
fractional divisor)
$(\boldsymbol{\varepsilon}^{*}{\mathcal{O}}_{Y}(\frac{\delta}{G\cdot
A}G)\oplus{\mathcal{O}}_{Y})\otimes{\mathcal{O}}_{E}$. This gives

${\rm
ch}(\boldsymbol{\varepsilon}^{*}N_{A/{\mathcal{P}}})=\left(2+\frac{\delta}{G\cdot
A}G\right){\rm ch}({\mathcal{O}}_{E}).$

Here we can truncate the expression on the left at codimension $1$ because it
is multiplied by ${\rm ch}({\mathcal{O}}_{E})$ of rank $0$.

Recall as above that ${\rm ch}({\mathcal{O}}_{E})=E-E^{2}/2$. Similarly ${\rm
ch}({\mathcal{O}}_{E}(E))=E+E^{2}/2$ and from the exact sequence,

$\displaystyle{\rm ch}(R_{Y})$ $\displaystyle=\left(2+\frac{\delta}{G\cdot
A}G\right)\cdot(E-E^{2}/2)-E-E^{2}/2$
$\displaystyle=E-\frac{3}{2}E^{2}+\frac{\delta}{G\cdot A}G\cdot E.$

From the other exact sequence and recalling that
$\boldsymbol{\varepsilon}^{*}T{\mathcal{P}}$ is trivial of rank $3$ we get

${\rm ch}(TY)=3-E+\frac{3}{2}E^{2}-\frac{\delta}{G\cdot A}G\cdot E.$

The last term could perhaps more easily be understood as a sum of fibers: let
$\mathsf{fib}$ denote the numerical class of a fiber of the projection
$E\rightarrow A$, viewed as a codimension $2$ class on $Y$. We have (writing
also $G$ for its pullback to $Y$)

$\mathsf{fib}=\frac{1}{G\cdot A}G\cdot E$

so we can also write

${\rm ch}(TY)=3-E+\frac{3}{2}E^{2}-\delta\cdot\mathsf{fib}.$

we get

$\displaystyle c_{1}(TY)$ $\displaystyle=-E,$ $\displaystyle c_{2}(TY)$
$\displaystyle=\delta\cdot\mathsf{fib}-E^{2},$

and

$\displaystyle{\rm td}(TY)$ $\displaystyle=1+c_{1}/2+(c_{1}^{2}+c_{2})/12$
$\displaystyle=1-E/2+(\delta/12)\cdot{\rm fib}.$

##  4 The degree one moduli space

For odd degree we have fixed the line bundle
${\mathcal{O}}_{C}(\boldsymbol{\mathsf{p}})$ to be the determinant, for the
chosen Weierstrass point $\boldsymbol{\mathsf{p}}\in C$. For purposes of the
present section, $X:=X_{1}$ is the moduli space of stable bundles $E$ with
${\rm det}(E)\cong{\mathcal{O}}_{C}(\boldsymbol{\mathsf{p}})$. Similarly,
${\rm Wob}:={\rm Wob}_{1}$ denotes the wobbly locus. Recall that
$\mathsf{Higgs}_{1}$ denotes the moduli space of Higgs bundles with
determinant ${\mathcal{O}}_{C}(\boldsymbol{\mathsf{p}})$ and trace of the
Higgs field equal to zero.

Recall from Proposition 3.1 that the $2:1$ spectral covering
$\pi:{\widetilde{C}}\rightarrow C$ has four branch points
$x,x^{\prime},y,y^{\prime}\in C$ such that $x$ and $x^{\prime}$ are opposite
under the hyperelliptic involution, and $y$ and $y^{\prime}$ are opposite
under the hyperelliptic involution. For a general point in the Hitchin base,
the resulting two points of the hyperelliptic ${\mathbb{P}}^{1}$ are general
with respect to the choice of $C$.

In this chapter, various notations such as $X,Y,{\rm Wob}$ etc. are used
without the subscript $1$ since we are talking about the degree $1$ moduli
space. For insertion into the rest of the paper one should read
$X_{1},Y_{1},{\rm Wob}_{1}$ and so on.

###  4.1 Geometry of the wobbly locus in degree one

We use the notations of subsections 3.1 and 3.7. In particular
$\mathsf{Higgs}_{1}^{{\mathbb{C}}^{\times},{\rm nu}}$ denotes the fixed point
locus of the ${\mathbb{C}}^{\times}$-action on $\mathsf{Higgs}_{1}$ and
${\mathcal{P}}^{\rm unss}$ denotes the unstable locus in ${\mathcal{P}}$ which
by Theorem 3.6 coincides with the intersection of
$\mathfrak{Q}\cap{\mathcal{P}}$ of ${\mathcal{P}}$ with the incoming variety
$\mathfrak{Q}$ defined in subsection 3.7. The geometric description of these
loci was already given in subsections 3.1 and 3.7 but we record it again in
the following Lemma for ease of reference.

######  Lemma 4.1.

  * •

$\mathsf{Higgs}_{1}^{{\mathbb{C}}^{\times},{\rm nu}}$ is equal to the curve
${\overline{C}}\subset\mathsf{Higgs}_{1}$, the $16$-sheeted etale covering of
$C$ defined in subsection 3.1.

  * •

If ${\mathcal{P}}$ is a generic Hitchin fiber corresponding to spectral curve
${\widetilde{C}}$, then ${\mathcal{P}}^{\rm
unss}=\mathfrak{Q}\cap{\mathcal{P}}\subset{\mathcal{P}}$ is a smooth curve
${\widehat{C}}$ that can be expressed as a fiber product

${\widehat{C}}={\widetilde{C}}\times_{C}{\overline{C}}.$

In particular, ${\widehat{C}}$ is a $16$-sheeted étale covering of
${\widetilde{C}}$.

These curves have genera respectively

$g_{C}=2,\;\;\;g_{{\widetilde{C}}}=5,\;\;\;g_{{\overline{C}}}=17,\;\;\;g_{{\widehat{C}}}=65.$

###### Proof.

The spectral curve ${\widetilde{C}}$ has genus $5$ and is a smooth double
cover ${\widetilde{C}}\rightarrow C$ ramified over $4$ points in two conjugate
pairs. This follows from our running genericity assumption requiring that
${\widetilde{C}}$ is not ramified over any of the Weierstrass points of $C$.
The Hitchin fiber ${\mathcal{P}}\subset\mathsf{Higgs}_{1}$ corresponding to
such ${\widetilde{C}}$ is the Prym variety ${\mathcal{P}}={\mathcal{P}}_{3}$
of line bundles on ${\widetilde{C}}$ whose direct image down to $C$ is a rank
$2$ bundle of determinant ${\mathcal{O}}_{C}(\boldsymbol{\mathsf{p}})$.

The statement that $\mathsf{Higgs}_{1}^{{\mathbb{C}}^{\times},{\rm
nu}}={\overline{C}}$ was already proven in Proposition 3.3 but we recall the
argument here for completeness. A fixed point that is in
$\mathsf{Higgs}_{1}^{{\mathbb{C}}^{\times},{\rm nu}}$ (i.e. such that the
underlying vector bundle is not stable) is a Higgs bundle of the form
$\left(A\oplus A^{\vee}(\boldsymbol{\mathsf{p}}),\theta\right)$, such that
$\theta:A^{\vee}(\boldsymbol{\mathsf{p}})\rightarrow A\otimes\omega_{C}$.
Stability of this Higgs bundle means $\deg(A)\leq 0$ and existence of the map
implies that $\deg(A)=0$. As
$\omega_{C}={\mathcal{O}}_{C}(2\boldsymbol{\mathsf{p}})$ this means that
$A^{\otimes 2}\cong{\mathcal{O}}_{C}(t-\boldsymbol{\mathsf{p}})$ for the point
$t\in C$ where $\theta$ vanishes. This description provides an isomorphism
$\mathsf{Higgs}_{1}^{{\mathbb{C}}^{\times},{\rm nu}}\cong{\overline{C}}$.

The incoming variety $\mathfrak{Q}\subset\mathsf{Higgs}_{1}$ consists of Higgs
bundles $(E,\theta)$ such that $\lim_{t\rightarrow
0}(E,t\theta)\in\mathsf{Higgs}_{1}^{{\mathbb{C}}^{\times},{\rm
nu}}\cong{\overline{C}}$. In subsection 3.6 we saw that the limiting map
$\mathfrak{Q}\rightarrow{\overline{C}}$ is well defined and has fibers
isomorphic to ${\mathbb{A}}^{3}$. We claim that

${\mathcal{P}}^{\rm unss}=\mathfrak{Q}\cap{\mathcal{P}}$

has the structure

${\widehat{C}}={\widetilde{C}}\times_{C}{\overline{C}}.$

In particular, it is seen to be a genus $65$ curve.

For a general Hitchin fiber the intersection ${\mathcal{P}}^{\rm unss}$ is
smooth of dimension $1$. A point of ${\mathcal{P}}^{\rm unss}$ is by
definition a line bundle $U$ of degree $3$ on ${\widetilde{C}}$ such that
$\pi_{*}U$ has determinant ${\mathcal{O}}_{C}(\boldsymbol{\mathsf{p}})$ but is
unstable. Instability means that there is a sub-line bundle of degree $1$ that
we will denote $A^{\vee}(\boldsymbol{\mathsf{p}})\hookrightarrow\pi_{*}U$. By
adjunction we have $\pi^{*}(A^{\vee}(\boldsymbol{\mathsf{p}}))\hookrightarrow
U$, and $\pi^{*}(A^{\vee}(\boldsymbol{\mathsf{p}}))$ has degree $2$. Thus,
there is a unique point $\tilde{t}\in{\widetilde{C}}$ such that
$U=\pi^{*}(A^{\vee}(\boldsymbol{\mathsf{p}}))(\tilde{t})$. If $t$ denotes the
image of $\tilde{t}$ in $C$, the determinant condition says that

$A^{\otimes-2}(2\boldsymbol{\mathsf{p}}+t)\otimes\omega_{C}^{-1}\cong{\mathcal{O}}_{C}(\boldsymbol{\mathsf{p}}).$

Thus, $A$ solves $A^{\otimes
2}\cong{\mathcal{O}}_{C}(t-\boldsymbol{\mathsf{p}})$ and $(A,t)$ is the
corresponding point of ${\overline{C}}$. This identifies ${\mathcal{P}}^{\rm
unss}$ with the image of
${\widehat{C}}={\widetilde{C}}\times_{C}{\overline{C}}$ under the map (4). But
by Lemma 3.2 this map is a closed embedding, and so we have
${\mathcal{P}}^{\rm unss}={\widehat{C}}$. ∎

######  Remark 4.2.

One can notice that the genera of the curves involved here are of the form
$4^{k}+1$. We do not know if that is significant or not.

######  Lemma 4.3.

Let $D$ denote the ${\mathbb{P}}^{1}$-bundle over ${\overline{C}}$
projectivization of the bundle of outgoing directions. Then the map sending a
point on ${\overline{C}}$ together with an outgoing direction, to the limit in
$X$ of the resulting ${\mathbb{C}}^{\times}$-orbit, provides a map
$D\rightarrow{\rm Wob}$ that is finite and generically injective. Thus, $D$ is
the normalization of ${\rm Wob}$.

###### Proof.

Given a ${\mathbb{C}}^{\times}$-fixed Higgs bundle $(A\oplus
A^{\vee}(\boldsymbol{\mathsf{p}}),\theta)$ corresponding to