# Higgs bundles twisted by a vector bundle

Guillermo Gallego  Facultad de Ciencias Matemáticas, UCM, Plaza de Ciencias 3,
Ciudad Universitaria, 28040 Madrid, SPAIN<EMAIL_ADDRESS>,  Oscar García-
Prada  Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Nicolás Cabrera,
13–15, 28049 Madrid, SPAIN<EMAIL_ADDRESS>and  M. S. Narasimhan
Department of Mathematics, Indian Institute of Science, Bangalore - 560012,
INDIA<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider a generalization of the theory of Higgs bundles
over a smooth complex projective curve in which the twisting of the Higgs
field by the canonical bundle of the curve is replaced by a rank $2$ vector
bundle. We define a Hitchin map and give a spectral correspondence. We also
state a Hitchin–Kobayashi correspondence for a generalization of the Hitchin’s
equations to this situation. In a certain sense, this theory lies halfway
between the theories of Higgs bundles on a curve and on a higher dimensional
variety.

###### 2020 Mathematics Subject Classification:

Primary 14H60; Secondary 14D23, 53C07

The first author’s research is supported by the UCM under the contract
CT63/19-CT64/19. The second author is partially supported by the Spanish
MINECO under the ICMAT Severo Ochoa grant No. SEV-2015-0554, and the Spanish
MICINN grant No. PID2019-109339GB-C31.

##  1\. Introduction

Higgs bundles over a smooth complex projective curve $X$ were introduced by
Hitchin in [hitchinselfduality] in his study of the self-duality equations,
referred now as Hitchin’s equations. A Higgs bundle over $X$ is a pair
$(E,\varphi)$ consisting of a vector bundle $E$ over $X$ and a twisted
endomorphism $\varphi:E\to E\otimes K_{X}$, where $K_{X}$ is the canonical
bundle of $X$. The moduli space of these objects has an extremely rich
geometric structure. In particular, the Hitchin map, given by the coefficients
of the characteristic polynomial of $\varphi$, defines a fibration which is an
algebraic completely integrable system — the Hitchin system. In
[hitchinsystem], Hitchin gives a description of the generic fibres in terms of
spectral curves. These are curves defined in the total space of the line
bundle $K_{X}$. This construction has been generalized in [bnr] for an
arbitrary twisting line bundle $L$ with $\deg L\geq\deg K_{X}$. The Hitchin
fibration plays a central role in the study of mirror symmetry and Langlands
duality [kapustinwitten, donagipantev], and has been a crucial ingredient in
the proof of the Fundamental Lemma by Ngô [ngolemme]. Higgs bundles on a
higher dimensional variety were introduced by Simpson [simpsonhodge]. In this
theory the role of the twisting bundle $K_{X}$ is taken by the cotangent
bundle of the variety, and there is the additional condition that
$\varphi\wedge\varphi=0$, which is trivally satisfied in dimension $1$.
Simpson [simpsonmoduliII] considered also the generalization of the Hitchin
map. Very recently, Chen and Ngô [chenngo] have studied this and obtained a
number of results in particular in the case of surfaces.

In this paper, we consider a problem that somehow is halfway between the case
of curves and the higher dimensional case. While we still work on a complex
projective curve $X$, the twisting line bundle $K_{X}$ is replaced by a higher
rank vector bundle $V$. As in higher dimensions, in this situation one
naturally considers the integrability condition $\varphi\wedge\varphi=0$. For
concreteness and to simplify computations, we limit ourselves here to the case
in which both the rank of $E$ and the rank of $V$ are two. This is the
simplest non-trivial situation where this problem can be considered.

After introducing the objects and defining stability, we give an algebraic
construction of our moduli space modeled on a construction of the moduli space
of Higgs bundles on a higher dimensional variety given by Simpson
[simpsonmoduliII]. We also study deformations generalizing the standard
approach followed in [biswasramanan]. We then define the Hitchin map in this
situation. For simplicity we restrict ourselves to the case in which the
determinant of $E$ is trivial and the trace of $\varphi$ vanishes. In Theorem
3.8 we give a spectral correspondence describing the fibres of the Hitchin
map. To do this, we follow a procedure inspired by Chen and Ngô’s
interpretation of the Hitchin morphism in terms of universal spectral data
[chenngo]. This spectral correspondence allows us to prove that the Hitchin
map is proper. We go on to study for what elements of the Hitchin base the
associated spectral curve has nice properties, like being integral or smooth.
The main results here are given by Propositions 3.10 and 3.14 and by Corollary
3.12. In future work, we will study what conditions can be imposed on the
twisting bundle $V$ in order to have generic smooth spectral curves. To
complete the circle of ideas relating to the original Higgs bundles introduced
by Hitchin, we consider natural gauge-theoretic equations in our situation,
generalizing the Hitchin equations and state a Hitchin–Kobayashi
correspondence. The existence of solutions follows from the general theorem
for twisted quiver bundles proved in [luisoscar].

The kind of pairs that we study in this paper were already mentioned by
Simpson [simpsonmoduliI]*p. 86 and have appeared in the literature
sporadically. See, for example the notes [pantev]*§2.2 and the papers of
Donagi with Markman [donagimarkman]*§9.3.4 and Gaitsgory
[donagigaitsgory]*§17.10. When the vector bundle is a direct sum
$V=L_{1}\oplus L_{2}$ and $L_{1}\otimes L_{2}=K_{X}$, these objects have been
considered by Xie and Yonekura in [xieyonekura], where they propose the
generalized Hitchin’s equations as a BPS equation for a certain class of
$\mathcal{N}=1$ supersymmetric gauge theories in four dimensions111The second
author thanks Greg Moore for pointing out this work to him.. Twisting by a
rank two vector bundle appears also in the work of Hitchin [hitchinnahm] on
the Nahm’s equations, regarded as a vector field on the moduli space of co-
Higgs bundles on the projective line, as well as in his study [hitchinloci] of
singular fibres of the ordinary Hitchin system.

When $\det V=K_{X}$, the total space of $V$ is a noncompact Calabi–Yau
threefold and our spectral description may be related to the study of
Donaldson–Thomas invariants [donaldsonthomas]. In fact, similar ideas have
already been considered in the study of ADHM sheaves by Diaconescu and others
[diaconescu, diaconescuwallcrossing] . This is an aspect that we would like to
explore in the future.

##  2\. $V$-twisted Higgs bundles

###  2.1. The main definitions

Let $X$ be a smooth projective curve over $\mathbb{C}$ and $V\rightarrow X$ a
rank $2$ vector bundle.

######  Definition 2.1.

A _$V$ -twisted Higgs bundle_ on $X$ is a pair $(E,\varphi)$, for
$E\rightarrow X$ a rank $2$ vector bundle and $\varphi:E\rightarrow E\otimes
V$ a twisted endomorphism satisfying the _commuting condition_

$\varphi\wedge\varphi=0$

as a homomorphism $E\rightarrow E\otimes\wedge^{2}V$. In the course of the
text sometimes we will refer to $V$-twisted Higgs bundles simply as _pairs_.

A _homomorphism_ of $V$-twisted Higgs bundles
$(E,\varphi)\rightarrow(E^{\prime},\varphi^{\prime})$ is a vector bundle
homomorphism $f:E\rightarrow E^{\prime}$ such that the following diagram
commutes

${E}$${E^{\prime}}$${E\otimes V}$${E^{\prime}\otimes
V.}$$\scriptstyle{f}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{f\otimes\operatorname{id}}$

######  Remark 2.2.

It is clear that with the obvious definitions all the constructions in this
section can be generalized to the case where $E$ and $V$ are vector bundles of
arbitrary rank.

######  Remark 2.3.

The commuting condition can be better understood in a local frame
$\\{v_{1},v_{2}\\}$ of $V$, so that $\varphi$ can be written as

$\varphi=\varphi_{1}\otimes v_{1}+\varphi_{2}\otimes v_{2},$

where the $\varphi_{i}$ are local sections of $\operatorname{End}E$. Now,

$\varphi\wedge\varphi=\varphi_{1}\circ\varphi_{2}\otimes v_{1}\wedge
v_{2}+\varphi_{2}\circ\varphi_{1}\otimes v_{2}\wedge
v_{1}=[\varphi_{1},\varphi_{2}]\otimes v_{1}\wedge v_{2}.$

Thus, locally we can regard $\varphi$ as a pair of commuting endomorphisms.

This commuting condition is the main aspect where the theory of $V$-twisted
Higgs bundles over a curve resembles the theory of usual Higgs bundles in
higher dimensions.

######  Definition 2.4.

We say that a $V$-twisted Higgs bundle $(E,\varphi)$ is _stable_ (resp.
_semistable_) if for every line subbundle $L\subset E$ such that
$\varphi(L)\subset L\otimes V$, we have

$\deg(L)<\tfrac{1}{2}\deg(E)\ \ \text{ (resp.
}\deg(L)\leq\tfrac{1}{2}\deg(E)).$

######  Remark 2.5.

Note that if a pair $(E,\varphi)$ is (semi)stable and $L$ is a line bundle,
then the associated pair $(E\otimes L,\varphi)$ (where we identify $\varphi$
with $\varphi\otimes\operatorname{id}_{L}$) is also (semi)stable.

The following result summarizes some important properties about stable pairs
and it can be seen as an analogue of the fact that stable vector bundles are
simple. The proof is essentially the same as the one for vector bundles
[seshadri]*p. 17, Prop. 6.

######  Proposition 2.6.

Let $(E_{1},\varphi_{1})$ and $(E_{2},\varphi_{2})$ be semistable $V$-twisted
Higgs bundles on $X$. Then

  * (a)

If $\deg(E_{1})>\deg(E_{2})$, then
$\operatorname{Hom}((E_{1},\varphi_{1}),(E_{2},\varphi_{2}))=0$.

  * (b)

If $E_{1}$ is stable and $\deg(E_{1})=\deg(E_{2})=d$, then
$\operatorname{Hom}((E_{1},\varphi_{1}),(E_{2},\varphi_{2}))=0$ or
$(E_{1},\varphi_{1})\cong(E_{2},\varphi_{2})$.

  * (c)

If $(E,\varphi)$ is stable, then
$\operatorname{End}((E,\varphi))=\mathbb{C}\cdot\operatorname{id}_{E}$.

###  2.2. The moduli space

We will construct a _moduli space_ $\mathcal{M}(d)$ that parametrizes
equivalence classes of semistable $V$-twisted Higgs bundles of degree $d$ on
$X$ based on Simpson’s “second construction” of the moduli space of (usual)
semistable Higgs bundles over projective varieties [simpsonmoduliII]. In
particular, there will be an open subvariety
$\mathcal{M}^{s}(d)\subset\mathcal{M}(d)$ parametrizing isomorphism classes of
stable $V$-twisted Higgs bundles of degree $d$.

####  2.2.1. The moduli functor

We will define a functor describing the moduli problem we have at hand. From
now on, it will be convenient to fix the usual notation
$T_{S}=T\times_{\mathbb{C}}S$, for $T$ and $S$ schemes over $\mathbb{C}$.
Moreover, if $s\in S$, we denote $T_{s}=T\times_{\mathbb{C}}\\{s\\}$.

If $S$ is a scheme of finite type over $\mathbb{C}$, a _family of semistable
$V$-twisted Higgs bundles of degree $d$ parametrized by $S$_ is a pair
$(E,\varphi)$, where $E$ is a vector bundle over $X_{S}$, flat over $S$, and
$\varphi:E\rightarrow E\otimes V_{S}$ is a morphism with
$\varphi\wedge\varphi=0$, such that, for each closed point $s\in S$,
$(E,\varphi)|_{X_{s}}$ is a semistable $V$-twisted Higgs bundle of degree $d$.
A _morphism of families_ $(E,\varphi)\rightarrow(E^{\prime},\varphi^{\prime})$
is a homomorphism $f:E\rightarrow E^{\prime}$ such that the following diagram
commutes

${E}$${E^{\prime}}$${E\otimes V_{S}}$${E^{\prime}\otimes
V_{S}.}$$\scriptstyle{f}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{f\otimes\operatorname{id}}$

Consider the functor
$\mathbf{Higgs}_{d}:\mathbf{Sch}/\mathbb{C}\rightarrow\mathbf{Set}$ sending a
$\mathbb{C}$-scheme $S$ to the set of isomorphism classes of families of
semistable $V$-twisted Higgs bundles on $X$ of degree $d$ parametrized by $S$.
We will construct a projective variety $\mathcal{M}(d)$ that _universally
corepresents_ the functor $\mathbf{Higgs}_{d}$, in the sense of Simpson
[simpsonmoduliI, simpsonmoduliII].

####  2.2.2. $V$-twisted Higgs bundles as sheaves on $V$

The key idea for the construction of the moduli space is to identify
$V$-twisted Higgs bundles with sheaves on $V$.

Let $Z$ denote a projective completion of $V$ and $D=Z\setminus V$ the divisor
at infinity. We can choose $Z$ so that the bundle projection $q:V\rightarrow
X$ extends to a map $q:Z\rightarrow X$.

######  Proposition 2.7.

There exists some $k\in\mathbb{N}$ such that the functor $q_{*}$ gives an
equivalence of categories between families $(E,\varphi)$ of $V$-twisted Higgs
bundles of degree $d$ parametrized by $S$ and coherent sheaves $\mathscr{E}$
on $Z_{S}$ of pure dimension $1$ with $\mathrm{supp}(\mathscr{E})\cap
D_{S}=\varnothing$ and Hilbert polynomial $p(\mathscr{E},n)=2kn+d$. In
particular, $(E,\varphi)$ is semistable if and only if $\mathscr{E}$ is.

###### Proof.

This is essentially the same as Lemma 6.8 and Corollary 6.9 in
[simpsonmoduliII]. ∎

####  2.2.3. Construction of the moduli space

Let $Z$ be the projective completion of $V$ and $k$ as in Proposition 2.7, and
consider the polynomial $P(n)=2kn+d$.

Simpson [simpsonmoduliI]*Thm. 1.21 showed that the functor
$\mathbf{Sh}_{Z,P}:\mathbf{Sch}/\mathbb{C}\rightarrow\mathbf{Set}$ sending a
$\mathbb{C}$-scheme $S$ to the set of isomorphism classes of families of
semistable sheaves over $Z$ with Hilbert polynomial $P$ parametrized by $S$ is
universally corepresented by a moduli space $M(Z,P)$. Simpson constructs this
moduli space by identifying the set of classes of semistable sheaves of pure
dimension with a subset $Q_{1}$ of a certain Quot scheme and considering a
subset $Q_{2}\subset Q_{1}$. The moduli space $M(Z,P)$ is obtained as a good
quotient of $Q_{2}$ by the action of a change of basis.

Let $Q_{3}\subset Q_{2}$ the open subset parametrizing quotient sheaves
$\mathscr{E}$ whose support does not meet $D$. This set $Q_{3}$ is invariant
by change of basis and the inverse image of a subset of $M(Z,P)$, so a good
quotient of $Q_{3}$ exists and it is equal to an open subset $M(V,P)\subset
M(Z,P)$. Finally, from the fact that $M(Z,P)$ universally corepresents
$\mathbf{Sh}_{Z,P}$ and from Proposition 2.7, we conclude that $M(V,P)$
universally corepresents the functor $\mathbf{Higgs}_{d}$, so we can define
the _moduli space of semistable $V$-twisted Higgs bundles of degree $d$ on
$X$_ as

$\mathcal{M}(d)=M(V,P).$

####  2.2.4. $S$-equivalence and polystability

We have constructed a moduli space $\mathcal{M}(d)$ universally corepresenting
$\mathbf{Higgs}_{d}$. However, we still have not said what is what the closed
points of $\mathcal{M}(d)$ parametrize. We answer now this question.

Suppose that $(E,\varphi)$ is a semistable $V$-twisted Higgs bundle that is
not stable. Then there exists a line subbundle $L\subset E$ with
$\varphi(L)\subset L\otimes V$ such that $\deg L=\tfrac{1}{2}\deg E$. Now,
consider the line bundle $E/L$ and the induced map
$\bar{\varphi}:E/L\rightarrow E/L\otimes V$. We define the _graded $V$-twisted
Higgs bundle associated to $(E,\varphi)$_ as the direct sum

$\operatorname{gr}(E,\varphi)=(L,\varphi|_{L})\oplus(E/L,\bar{\varphi}).$

This object is clearly well defined by $(E,\varphi)$ up to isomorphism.

When $(E,\varphi)$ is stable, we define
$\operatorname{gr}(E,\varphi)=(E,\varphi)$. We say that two semistable
$V$-twisted Higgs bundles are _$S$ -equivalent_ if their associated graded
objects are isomorphic. We denote by $S(d)$ the set of $S$-equivalence classes
of semistable $V$-twisted Higgs bundles of degree $d$ on $X$. Now, it is clear
that the subset $S^{s}(d)$ formed by stable objects is the set of isomorphism
classes of stable pairs of degree $d$.

We say that a $V$-twisted Higgs bundle $(E,\varphi)$ is _polystable_ if it is
stable or if it can be expressed as a direct sum
$(E,\varphi)=(L_{1},\varphi_{1})\oplus(L_{2},\varphi_{2})$, for $L_{i}$ line
bundles with $\deg L_{i}=\tfrac{1}{2}\deg E$ and $\varphi_{i}:L_{i}\rightarrow
L_{i}\otimes V$ with $\varphi_{i}\wedge\varphi_{i}=0$. Clearly, a polystable
pair is semistable, so the map
$[E,\varphi]\mapsto[\operatorname{gr}(E,\varphi)]$ defines a bijection between
$S(d)$ and the set of isomorphism classes of polystable pairs of degree $d$.

The natural transformation
$\tau:\mathbf{Higgs}_{d}\rightarrow\underline{\mathcal{M}(d)}$ corepresenting
$\mathbf{Higgs}_{d}$ induces a map
$\tau_{\mathbb{C}}:\mathbf{Higgs}_{d}(\mathbb{C})\rightarrow\underline{\mathcal{M}(d)}(\mathbb{C})$,
where $\mathbf{Higgs}_{d}(\mathbb{C})$ is the set of isomorphism classes of
semistable $V$-twisted Higgs bundles of degree $d$ and
$\underline{\mathcal{M}(d)}(\mathbb{C})$ is the set of closed points of
$\mathcal{M}(d)$.

######  Proposition 2.8.

The map $\tau_{\mathbb{C}}$ factors through the quotient map
$\mathbf{Higgs}_{d}(\mathbb{C})\rightarrow S(d)$ induced by the
$S$-equivalence relation. Moreover, the induced map
$S(d)\rightarrow\underline{\mathcal{M}(d)}(\mathbb{C})$ is a bijection.

###### Proof.

The proof is standard and identic to others in the literature. See
[simpsonmoduliI]*Thm. 1.21(3) and [nitsure]*Prop. 4.5. ∎

As a consequence of this result, we have that the set $S^{s}(d)$ of
isomorphism classes of stable pairs is in bijection with the set of closed
points of the open subvariety $\mathcal{M}^{s}(d)$ parametrizing stable
bundles. This shows that $\mathcal{M}^{s}(d)$ is a _coarse moduli space_ for
stable $V$-twisted Higgs bundles of degree $d$.

###  2.3. Infinitesimal deformations

To finish the section, we give a description of the space of infinitesimal
deformations of a $V$-twisted Higgs bundle. Our discussion is a generalization
of the results of Biswas and Ramanan [biswasramanan].

Consider the ring

$\mathbb{C}[\varepsilon]=\mathbb{C}[X]/(X^{2})=\left\\{a+b\varepsilon:a,b\in\mathbb{C}\right\\},$

which is an Artinian $\mathbb{C}$-algebra with maximal ideal $(\varepsilon)$.
Given a $\mathbb{C}$-scheme $S$, we denote
$S[\varepsilon]=S\times_{\mathbb{C}}\mathrm{Spec}(\mathbb{C}[\varepsilon])$
and identify $S$ with $S_{(\varepsilon)}$. Recall that the Zariski tangent
space of a variety $M$ at a point $p$ can be identified with the set of maps
$\mathrm{Spec}(\mathbb{C}[\varepsilon])\rightarrow M$ mapping $(\varepsilon)$
to the point $p$. Therefore, in order to study the “tangent” space of the
moduli space of $V$-twisted Higgs bundles at a point $(E,\varphi)$, it makes
sense to consider the space $T_{(E,\varphi)}$ of isomorphism classes of
families $(\tilde{E},\tilde{\varphi})$ of $V$-twisted Higgs bundles
parametrized by $\mathrm{Spec}(\mathbb{C}[\varepsilon])$ such that
$(\tilde{E},\tilde{\varphi})|_{X}\cong(E,\varphi)$. We call this
$T_{(E,\varphi)}$ the _space of infinitesimal deformations_ of $(E,\varphi)$.

######  Proposition 2.9.

Let $(E,\varphi)$ be a $V$-twisted Higgs bundle on $X$ and consider the chain
complex

${\mathscr{C}_{(E,\varphi)}:\operatorname{End}E}$${\operatorname{End}E\otimes
V}$${\operatorname{End}E\otimes\wedge^{2}V.}$$\scriptstyle{\varphi\wedge-}$$\scriptstyle{\varphi\wedge-}$

The space of infinitesimal deformations $T_{(E,\varphi)}$ of $(E,\varphi)$ is
canonically isomorphic to the hypercohomology space
$\mathbb{H}^{1}(\mathscr{C}_{(E,\varphi)})$.

######  Remark 2.10.

Notice the similarity between the complex $\mathscr{C}_{(E,\varphi)}$ and the
deformation complex of an usual Higgs bundle over a projective surface. In
particular, the space $\mathbb{H}^{3}(\mathscr{C}_{(E,\varphi)})$ will play an
important role in the deformation theory of $(E,\varphi)$.

###### Proof.

We start by computing $\mathbb{H}^{1}(\mathscr{C}_{(E,\varphi)})$. In order to
do this we take the Čech resolution of the complex
$\mathscr{C}_{(E,\varphi)}$, which yields the double complex

${\operatorname{End}E}$${\operatorname{End}E\otimes
V}$${\operatorname{End}E\otimes\wedge^{2}V\operatorname{End}E}$${K^{0,0}=\sum_{i}\tilde{M}_{i}}$${K^{0,1}=\sum_{i}\tilde{M}_{i}\otimes
H^{0}(U_{i},V)}$${K^{0,2}=\sum_{i}\tilde{M}_{i}\otimes
H^{0}(U_{i},\wedge^{2}V)}$${K^{1,0}=\sum_{ij}\tilde{M}_{ij}}$${K^{1,1}=\sum_{ij}\tilde{M}_{ij}\otimes
H^{0}(U_{ij},V)}$${K^{1,2}=\sum_{ij}\tilde{M}_{ij}\otimes
H^{0}(U_{ij},\wedge^{2}V).}$$\scriptstyle{\varphi\wedge-}$$\scriptstyle{\varphi\wedge-}$

This complex has been constructed by covering $X$ by affine open subsets
$U_{i}=\mathrm{Spec}(R_{i})$ with affine intersections
$U_{ij}=\mathrm{Spec}(R_{ij})$. The bundle $\operatorname{End}E|_{U_{i}}$
(respectively $\operatorname{End}E|_{U_{ij}}$) can be seen as the sheaf
associated to an $R_{i}$-module $M_{i}$ (respectively, an $R_{ij}$-module
$M_{ij}$). The hypercohomology group
$\mathbb{H}^{1}(\mathscr{C}_{(E,\varphi)})$ is now obtained as the cohomology
group $H^{1}(K_{\mathrm{tot}})$ for the total complex $K_{\mathrm{tot}}$
associated to the double complex $K$.

The total complex $K_{\mathrm{tot}}$ is obtained as follows

$\displaystyle K_{\mathrm{tot}}^{0}$
$\displaystyle=K^{0,0}=\sum_{i}\tilde{M}_{i},$ $\displaystyle
K_{\mathrm{tot}}^{1}$ $\displaystyle=K^{1,0}\oplus
K^{0,1}=\sum_{ij}\tilde{M}_{ij}\oplus\sum_{i}\tilde{M}_{i}\otimes
H^{0}(U_{i},V),$ $\displaystyle K_{\mathrm{tot}}^{2}$
$\displaystyle=K^{2,0}\oplus K^{1,1}\oplus K^{0,2}$
$\displaystyle=\sum_{ijk}\tilde{M}_{ijk}\oplus\sum_{ij}\tilde{M}_{ij}\otimes
H^{0}(U_{ij},V)\oplus\sum_{i}\tilde{M}_{i}\otimes H^{0}(U_{i},\wedge^{2}V),$
$\displaystyle K_{\mathrm{tot}}^{0}$ $\displaystyle\longrightarrow
K_{\mathrm{tot}}^{1}$ $\displaystyle s_{i}$
$\displaystyle\longmapsto(s_{i}-s_{j},\varphi\wedge s_{i})$ $\displaystyle
K_{\mathrm{tot}}^{1}$ $\displaystyle\longrightarrow K_{\mathrm{tot}}^{2}$
$\displaystyle(s_{ij},t_{i})$
$\displaystyle\longmapsto(s_{ij}+s_{jk}-s_{ik},\varphi\wedge
s_{ij}-(t_{i}-t_{j}),\varphi\wedge t_{i}).$

Therefore, $\mathbb{H}^{1}(\mathscr{C}_{(E,\varphi)})$ can be computed as the
quotient $Z/B$, where $Z$ consists of pairs $(s_{ij},t_{i})$, satisfying the
equations

$\begin{cases}s_{ij}+s_{jk}=s_{ij}\\\ t_{i}-t_{j}=\varphi\wedge s_{ij}\\\
\varphi\wedge t_{i}=0\end{cases}$

and $B$ is the set of pairs of the form $(s_{i}-s_{j},\varphi\wedge s_{i})$.
We will give now an isomorphism $Z/B\rightarrow T_{(E,\varphi)}$.

For every $i$ we can consider the bundle
$\tilde{E}_{i}=E|_{U_{i}}[\varepsilon]$, over $U_{i}[\varepsilon]$. Now,
consider an element $(s_{ij},t_{i})\in Z$. Then, the restriction
$\tilde{E}_{i}|_{U_{j}}$ can be identified with $\tilde{E}_{j}|_{U_{i}}$
through the isomorphism $\operatorname{id}+s_{ij}\varepsilon$. The first
equation above ensures that under these identifications the patches
$\tilde{E}_{i}$ can be glued to give a vector bundle $\tilde{E}$ on
$X[\varepsilon]$.

Over $U_{i}[\varepsilon]$ we have the section
$\tilde{\varphi}_{i}=\varphi+t_{i}\varepsilon$ of
$\operatorname{End}(\tilde{E}_{i})\otimes V[\varepsilon]$. Since the
transition functions of $\tilde{E}$ are given by
$\operatorname{id}+s_{ij}\varepsilon$, the associated transition functions on
$\operatorname{End}(\tilde{E})$ are
$\tilde{\varphi}_{i}\mapsto\tilde{\varphi}_{i}+s_{ij}\varepsilon\wedge\tilde{\varphi}_{i}$
. Therefore, the sections $\tilde{\varphi}_{i}$ will glue to give a global
section $\tilde{\varphi}$ of $\operatorname{End}(\tilde{E})\otimes
V[\varepsilon]$ if

$\tilde{\varphi}_{j}=\tilde{\varphi}_{i}+s_{ij}\varepsilon\wedge\tilde{\varphi}_{i}$

but this is true since

$\tilde{\varphi}_{i}+s_{ij}\varepsilon\wedge\tilde{\varphi}_{i}=\varphi+t_{i}\varepsilon+s_{ij}\wedge\varphi\varepsilon=\varphi+t_{i}\varepsilon-(t_{i}-t_{j})\varepsilon=\varphi+t_{j}\varepsilon=\tilde{\varphi}_{j}.$

Finally, since $\varphi\wedge t_{i}=0$, we have

$\tilde{\varphi}_{i}\wedge\tilde{\varphi}_{i}=(\varphi+t_{i}\varepsilon)\wedge(\varphi+t_{i}\varepsilon)=\varphi\wedge\varphi+(\varphi\wedge
t_{i}+t_{i}\wedge\varphi)\varepsilon=0.$

Therefore, the section $\tilde{\varphi}\in
H^{0}(X,\operatorname{End}(\tilde{E})\otimes V[\varepsilon])$ satisfies the
commuting condition $\tilde{\varphi}\wedge\tilde{\varphi}=0$.

To sum up, we have defined a map

$\displaystyle Z$ $\displaystyle\longrightarrow T_{(E,\varphi)}$
$\displaystyle(s_{ij},t_{i})$
$\displaystyle\longmapsto[(\tilde{E},\tilde{\varphi})].$

In order to see that this map descends to a map $Z/B\rightarrow
T_{(E,\varphi)}$ we have to check that an element of the form
$(s_{ij},t_{i})=(s_{i}-s_{j},\varphi\wedge s_{i})$, for
$(s_{i})\in\sum_{i}\tilde{M}_{i}$, induces a pair
$(\tilde{E},\tilde{\varphi})$ isomorphic to the “trivial” pair
$(E[\varepsilon],\varphi)$. Indeed, the identification
$\tilde{E}_{i}\rightarrow\tilde{E}_{j}$ on $U_{ij}$ is given by the
automorphism $1+(s_{i}-s_{j})\varepsilon$. Therefore, the following diagram
commutes

${\tilde{E}_{ij}}$${\tilde{E}_{ij}}$${\tilde{E}_{ij}}$${\tilde{E}_{ij}.}$$\scriptstyle{1+s_{i}\varepsilon}$$\scriptstyle{1+s_{ij}\varepsilon}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{1+s_{j}\varepsilon}$

This implies that the bundle $\tilde{E}$ is isomorphic to the trivial bundle
$E[\varepsilon]$. Moreover, the isomorphism $1+s_{i}\varepsilon$ sends
$\tilde{\varphi}_{i}$ to

$\tilde{\varphi}_{i}+s_{i}\varepsilon\wedge\tilde{\varphi}_{i}=\varphi+t_{i}\varepsilon+s_{i}\wedge\varphi\varepsilon=\varphi.$

We thus have defined a map
$\mathbb{H}^{1}(\mathscr{C}_{(E,\varphi)})\rightarrow T_{(E,\varphi)}$. We
define now an inverse map
$T_{(E,\varphi)}\rightarrow\mathbb{H}^{1}(\mathscr{C}_{(E,\varphi)})$. Let us
take then a family $(\tilde{E},\tilde{\varphi})$ of $V$-twisted Higgs bundles
parametrized by $\mathrm{Spec}(\mathbb{C}[\varepsilon])$, such that
$(\tilde{E},\tilde{\varphi})|_{X}\cong(E,\varphi)$. Since the $U_{i}$ are
affine, the restriction $\tilde{E}|_{U_{i}[\varepsilon]}$ must be of the form
$E_{i}[\varepsilon]$, with $E_{i}$ a vector bundle over $U_{i}$. Clearly,
$E_{i}\cong E|_{U_{i}}$. Therefore, $\tilde{E}$ is obtained by gluing the
restrictions of $E[\varepsilon]$ to $U_{i}[\varepsilon]$ via automorphisms
over $U_{ij}[\varepsilon]$. These have to be of the form
$1+s_{ij}\varepsilon$, with the $s_{ij}$ satisfying the cocycle condition:
$s_{ij}+s_{jk}=s_{ik}$ over the triple intersection $U_{ijk}$.

Moreover, the homomorphism $\tilde{\varphi}$ over each $U_{i}[\varepsilon]$
must be of the form $\tilde{\varphi}_{i}=\varphi+t_{i}\varepsilon$ for
sections $t_{i}\in H^{0}(U_{i},\operatorname{End}E\otimes V)$. The global
nature of $\tilde{\varphi}$ imposes the following compatibility condition

$\tilde{\varphi}_{i}+s_{ij}\varepsilon\wedge\tilde{\varphi}_{i}=\tilde{\varphi}_{j}.$

This implies that $\varphi\wedge s_{ij}=t_{i}-t_{j}$.

Finally, from the fact that $\tilde{\varphi}_{i}\wedge\tilde{\varphi}_{i}=0$
we deduce that $\varphi\wedge t_{i}=0$. Putting all this together, we see that
$(s_{ij},t_{i})$ is an element of $Z$. Hence we have given a map
$T_{(E,\varphi)}\rightarrow\mathbb{H}^{1}(\mathscr{C}_{(E,\varphi)})$. It is
easy to check that this and the map
$\mathbb{H}^{1}(\mathscr{C}_{(E,\varphi)})\rightarrow T_{(E,\varphi)}$
constructed above are inverse of each other. ∎

We can now study the complex $\mathscr{C}_{(E,\varphi)}$ in terms of a simpler
complex

${\mathscr{D}_{(E,\varphi)}:\operatorname{End}E}$${\operatorname{End}E\otimes
V.}$$\scriptstyle{\varphi\wedge-}$

This allows us to obtain the following.

######  Proposition 2.11.

If $(E,\varphi)$ is a $V$-twisted Higgs bundle on $X$, we have

$\dim\mathbb{H}^{1}(\mathscr{C}_{(E,\varphi)})-\dim\mathbb{H}^{2}(\mathscr{C}_{(E,\varphi)})+\dim\mathbb{H}^{3}(\mathscr{C}_{(E,\varphi)})=\dim\mathbb{H}^{0}(\mathscr{D}_{(E,\varphi)}).$

###### Proof.

There is an obvious short exact sequence of complexes

${0}$${\operatorname{End}E\otimes\wedge^{2}V[2]}$${\mathscr{C}_{(E,\varphi)}}$${\mathscr{D}_{(E,\varphi)}}$${0.}$

Here, we use the standard notation in derived categories where $[n]$ after a
chain complex stands for shifting the complex $n$ places. This induces a long
exact sequence in cohomology

$\displaystyle 0$
$\displaystyle\rightarrow\mathbb{H}^{1}(\mathscr{C}_{(E,\varphi)})\rightarrow\mathbb{H}^{1}(\mathscr{D}_{(E,\varphi)})\rightarrow
H^{0}(X,\operatorname{End}E\otimes\wedge^{2}V)$
$\displaystyle\rightarrow\mathbb{H}^{2}(\mathscr{C}_{(E,\varphi)})\rightarrow\mathbb{H}^{2}(\mathscr{D}_{(E,\varphi)})\rightarrow
H^{1}(X,\operatorname{End}E\otimes\wedge^{2}V)\rightarrow\mathbb{H}^{3}(\mathscr{C}_{(E,\varphi)})\rightarrow
0.$

Computing the Euler characteristic we get

$\displaystyle\dim\mathbb{H}^{1}(\mathscr{C}_{(E,\varphi)})$
$\displaystyle-\dim\mathbb{H}^{2}(\mathscr{C}_{(E,\varphi)})+\dim\mathbb{H}^{3}(\mathscr{C}_{(E,\varphi)})$
$\displaystyle=\dim\mathbb{H}^{1}(\mathscr{D}_{(E,\varphi)})-\dim\mathbb{H}^{2}(\mathscr{D}_{(E,\varphi)})-\chi(\operatorname{End}E\otimes\wedge^{2}V).$

On the other hand, we have the short exact sequence

${0}$${\operatorname{End}E\otimes
V[1]}$${\mathscr{D}_{(E,\varphi)}}$${\operatorname{End}E}$${0,}$

giving the long exact sequence

$\displaystyle 0\rightarrow\mathbb{H}^{0}(\mathscr{D}_{(E,\varphi)})$
$\displaystyle\rightarrow H^{0}(X,\operatorname{End}E)\rightarrow
H^{0}(X,\operatorname{End}E\otimes
V)\rightarrow\mathbb{H}^{1}(\mathscr{D}_{(E,\varphi)})$
$\displaystyle\rightarrow H^{1}(X,\operatorname{End}E)\rightarrow
H^{1}(X,\operatorname{End}E\otimes
V)\rightarrow\mathbb{H}^{2}(\mathscr{D}_{(E,\varphi)})\rightarrow 0.$

Again, computing the Euler characteristic we get

$\dim\mathbb{H}^{1}(\mathscr{D}_{(E,\varphi)})-\dim\mathbb{H}^{2}(\mathscr{D}_{(E,\varphi)})=\chi(\operatorname{End}E\otimes
V)-\chi(\operatorname{End}E)+\dim\mathbb{H}^{0}(\mathscr{D}_{(E,\varphi)}).$

Putting this together with the above we have

$\displaystyle\dim\mathbb{H}^{1}(\mathscr{C}_{(E,\varphi)})$
$\displaystyle-\dim\mathbb{H}^{2}(\mathscr{C}_{(E,\varphi)})+\dim\mathbb{H}^{3}(\mathscr{C}_{(E,\varphi)})$
$\displaystyle=\chi(\operatorname{End}E\otimes
V)-\chi(\operatorname{End}E)-\chi(\operatorname{End}E\otimes\wedge^{2}V)+\dim\mathbb{H}^{0}(\mathscr{D}_{(E,\varphi)}).$

Now, by Riemann–Roch it is easy to see that

$\displaystyle\chi(\operatorname{End}E\otimes
V)-\chi(\operatorname{End}E)-\chi(\operatorname{End}E\otimes\wedge^{2}V)=0.$

This concludes the proof. ∎

######  Remark 2.12.

Notice that the vanishing of the relation between the Euler characteristics is
a very special situation that only occurs when $V$ has rank $2$.

More can be said about the chain complex $\mathscr{C}_{(E,\varphi)}$ when
$(E,\varphi)$ is assumed to be stable.

######  Proposition 2.13.

If $(E,\varphi)$ is a stable $V$-twisted Higgs bundle on $X$, then

$\mathbb{H}^{0}(\mathscr{D}_{(E,\varphi)})\cong\mathbb{C}.$

###### Proof.

It is clear from the relations defining
$\mathbb{H}^{0}(\mathscr{D}_{(E,\varphi)})$ in the Čech resolution of
$\mathscr{D}_{(E,\varphi)}$ that

$\mathbb{H}^{0}(\mathscr{D}_{(E,\varphi)})=\left\\{\xi\in
H^{0}(X,\operatorname{End}E):\varphi\wedge\xi=0\right\\}=\operatorname{End}(E,\varphi).$

Since $(E,\varphi)$ is stable, by Proposition 2.6, elements
$\xi\in\mathbb{H}^{0}(\mathscr{D}_{(E,\varphi)})$ are of the form
$\xi=\lambda\operatorname{id}_{E}$. ∎

This proposition implies that, when $(E,\varphi)$ is stable, we have the
relation

$\dim\mathbb{H}^{1}(\mathscr{C}_{(E,\varphi)})-\dim\mathbb{H}^{2}(\mathscr{C}_{(E,\varphi)})+\dim\mathbb{H}^{3}(\mathscr{C}_{(E,\varphi)})=1.$

Finally, assuming some conditions on $V$ we obtain:

######  Corollary 2.14.

Let $K_{X}$ be the canonical line bundle of $X$ and denote $L=\wedge^{2}V$.
Assume that $\deg L\geq\deg K_{X}$. Then

$\dim\mathbb{H}^{1}(\mathscr{C}_{(E,\varphi)})=\begin{cases}\dim\mathbb{H}^{2}(\mathscr{C}_{(E,\varphi)}),&\text{if
}K_{X}\cong L,\\\
\dim\mathbb{H}^{2}(\mathscr{C}_{(E,\varphi)})+1,&\text{else}.\\\ \end{cases}$

Moreover, if $V$ is isomorphic to $V^{\vee}\otimes K_{X}$, then there exists a
perfect pairing

$\mathbb{H}^{1}(\mathscr{C}_{(E,\varphi)})\times\mathbb{H}^{2}(\mathscr{C}_{(E,\varphi)})\rightarrow\mathbb{C}.$

###### Proof.

Consider the dual complex of $\mathscr{C}_{(E,\varphi)}$,

${\mathscr{C}_{(E,\varphi)}^{*}:(\operatorname{End}E)^{\vee}\otimes
L^{-1}}$${(\operatorname{End}E)^{\vee}\otimes
V^{\vee}}$${(\operatorname{End}E)^{\vee}.}$$\scriptstyle{(\varphi\wedge-)^{\vee}}$$\scriptstyle{(\varphi\wedge-)^{\vee}}$

By the pairing induced by the Killing form of $\mathfrak{gl}(2,\mathbb{C})$,
we can identify this complex with the complex

${\mathscr{C}_{(E,\varphi)}^{*}:\operatorname{End}E\otimes
L^{-1}}$${\operatorname{End}E\otimes
V^{\vee}}$${\operatorname{End}E.}$$\scriptstyle{-(\varphi\wedge-)}$$\scriptstyle{-(\varphi\wedge-)}$

By Grothendieck–Serre duality for hypercohomology (see [dalakov]*§3.3 for a
comprehensive overview in a similar context), we have

$(\mathbb{H}^{3}(\mathscr{C}_{(E,\varphi)}))^{*}\cong\mathbb{H}^{0}(\mathscr{C}_{(E,\varphi)}^{*}\otimes
K_{X}),$

where $\mathscr{C}_{(E,\varphi)}^{*}\otimes K_{X}$ is the chain complex

${\mathscr{C}_{(E,\varphi)}^{*}\otimes K_{X}:\operatorname{End}E\otimes
L^{-1}K_{X}}$${\operatorname{End}E\otimes V^{\vee}\otimes
K_{X}}$${\operatorname{End}E\otimes
K_{X}.}$$\scriptstyle{-(\varphi\wedge-)}$$\scriptstyle{-(\varphi\wedge-)}$

Now,

$\displaystyle\mathbb{H}^{0}(\mathscr{C}_{(E,\varphi)}^{*}\otimes K_{X})$
$\displaystyle=\left\\{\xi\in H^{0}(X,\operatorname{End}E\otimes
L^{-1}K_{X}):\varphi\wedge\xi=0\right\\}$
$\displaystyle=\operatorname{Hom}((E,\varphi),(E\otimes
L^{-1}K_{X},\varphi)).$

From Remark 2.5 we have that, since $(E,\varphi)$ is stable, so will be
$(E\otimes L^{-1}K_{X},\varphi)$. Now, we assumed that
$\deg(K_{X})=2g-2\leq\deg(L)$, so $\deg(E\otimes L^{-1}K_{X})\leq\deg(E)$.

If $L$ is not isomorphic to $K_{X}$, Proposition 2.6 implies that

$\mathbb{H}^{0}(\mathscr{C}_{(E,\varphi)}^{*}\otimes
K_{X})=\operatorname{Hom}((E,\varphi),(E\otimes L^{-1}K_{X},\varphi))=0.$

On the other hand, if $L\cong K_{X}$ that same proposition asserts that

$\dim\mathbb{H}^{0}(\mathscr{C}_{(E,\varphi)}^{*}\otimes K_{X})=1.$

Finally, Grothendieck–Serre duality gives a perfect pairing

$\mathbb{H}^{1}(\mathscr{C}_{(E,\varphi)})\times\mathbb{H}^{2}(\mathscr{C}_{(E,\varphi)}^{*}\otimes
K_{X})\rightarrow\mathbb{C}.$

Now, notice that if $V\cong V^{\vee}\otimes K_{X}$, we have that
$\mathscr{C}_{(E,\varphi)}^{*}\otimes K_{X}=\mathscr{C}_{(E,\varphi)}$, and
thus we obtain the desired pairing. ∎

##  3\. The Hitchin morphism for $V$-twisted $\mathrm{SL}(2,\mathbb{C})$-Higgs
bundles

###  3.1. The Hitchin map

In this section we introduce the Hitchin morphism and study its fibres by
giving a _spectral correspondence_. For simplicity, we restrict to the case
where the “structure group” of the pair is $\mathrm{SL}(2,\mathbb{C})$.

######  Definition 3.1.

A _$V$ -twisted $\mathrm{SL}(2,\mathbb{C})$-Higgs bundle_ is a $V$-twisted
Higgs bundle $(E,\varphi)$ such that $\det E=\mathscr{O}_{X}$ and
$\operatorname{tr}\varphi=0$.

The _moduli space $\mathcal{M}_{0}$ of semistable $V$-twisted
$\mathrm{SL}(2,\mathbb{C})$-Higgs bundles_ can be obtained from the moduli
space of semistable $V$-twisted Higgs bundles of degree $0$ as the fibre of
the element $(\mathscr{O}_{X},0)$ by the map

$\displaystyle\mathcal{M}(0)$
$\displaystyle\longrightarrow\mathrm{Jac}(X)\times H^{0}(X,V)$
$\displaystyle(E,\varphi)$ $\displaystyle\longmapsto(\det
E,\operatorname{tr}(\varphi)).$

Analogously, we can obtain the moduli space $\mathcal{M}_{0}^{s}$ of stable
$V$-twisted $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles.

The _Hitchin morphism_ is defined as the map

$\displaystyle h:\mathcal{M}_{0}$ $\displaystyle\longrightarrow
H^{0}(X,\mathrm{Sym}^{2}V)$ $\displaystyle(E,\varphi)$
$\displaystyle\longmapsto\operatorname{tr}(\varphi^{2}).$

This map can be interpreted as some sort of “characteristic polynomial” since,
at least formally

$\det(\varphi-\operatorname{id}_{E}\otimes
T)=T^{2}-\operatorname{tr}(\varphi^{2}).$

This is why the elements in the image of $h$ are referred as _spectral data_.
We are interested in studying the fibres of this morphism. Note that a good
description of these fibres will give a way of constructing semistable
$V$-twisted Higgs bundles.

One of the main differences with the well-known case where the twisting bundle
is a line bundle is that in our situation the Hitchin morphism is not
surjective. In order to see this, suppose that $V$ splits as a direct sum
$V=L_{1}\oplus L_{2}$ (which can always be assumed locally). In that case, we
can write $\varphi=(\varphi_{1},\varphi_{2})$, with $\varphi_{i}:E\rightarrow
E\otimes L_{i}$. The commuting condition means now that
$[\varphi_{1},\varphi_{2}]=0$. The symmetric product $\mathrm{Sym}^{2}V$
splits as

$\mathrm{Sym}^{2}V=L_{1}^{2}\oplus L^{2}_{2}\oplus L_{1}L_{2}$

and $h(\varphi)$ can be written as

$h(\varphi)=\operatorname{tr}(\varphi^{2})=(\operatorname{tr}(\varphi_{1}^{2}),\operatorname{tr}(\varphi_{2}^{2}),\operatorname{tr}(\varphi_{1}\varphi_{2})).$

This implies that an element $b=(b_{1},b_{2},b_{3})\in
H^{0}(X,\mathrm{Sym}^{2}V)$ in the image of $h$ must satisfy the relation

$b_{3}^{2}=b_{1}b_{2}.$

We define the _Hitchin base_ as the space

$\mathcal{B}=\left\\{b\in H^{0}(X,\mathrm{Sym}^{2}V)\text{ such that locally
}b=(b_{1},b_{2},b_{3})\text{ with }b_{3}^{2}=b_{1}b_{2}\right\\}.$

As we just explained, the Hitchin morphism $h$ factors through the inclusion
$\mathcal{B}\hookrightarrow H^{0}(X,\mathrm{Sym}^{2}V)$. In the spirit of
[chenngo], we conjecture that the image of $h$ is precisely the Hitchin base
$\mathcal{B}$.

###  3.2. Universal spectral data

We follow a procedure inspired by Chen and Ngô’s [chenngo] interpretation of
the Hitchin morphism in terms of _universal spectral data_.

The intuition behind this construction is to consider the “models over a
point” of each of the elements involved in the Hitchin morphism. For example,
over a point a $V$-twisted Higgs bundle is simply a pair
$(\varphi_{1},\varphi_{2})$ of elements of $\mathfrak{sl}(2,\mathbb{C})$ with
$[\varphi_{1},\varphi_{2}]=0$. We define the _commuting variety_ as

$\mathfrak{C}=\left\\{(\varphi_{1},\varphi_{2})\in\mathfrak{sl}(2,\mathbb{C}):[\varphi_{1},\varphi_{2}]=0\right\\}.$

The point model of the Hitchin morphism is the map

$\displaystyle h:\mathfrak{C}$
$\displaystyle\longrightarrow\mathbb{C}^{3}\cong\mathrm{Sym}^{2}\mathbb{C}^{2}$
$\displaystyle(\varphi_{1},\varphi_{2})$
$\displaystyle\longmapsto(\operatorname{tr}(\varphi_{1})^{2},\operatorname{tr}(\varphi_{2})^{2},\operatorname{tr}(\varphi_{1}\varphi_{2})).$

The image of this map is the cone

$B=\left\\{(x,y,z)\in\mathbb{C}^{3}:z^{2}=xy\right\\}.$

Now, since an element of $\mathfrak{C}$ consists of a pair of traceless
commuting endomorphisms $\varphi_{1}$ and $\varphi_{2}$ of $\mathbb{C}^{2}$,
there exists a common eigenvector $v\in\mathbb{C}^{2}$ of $\varphi_{1}$ and
$\varphi_{2}$. If $\lambda_{i}$ denotes the eigenvalue of $\varphi_{i}$
associated to $v$, we can define the map

$\displaystyle\mathfrak{C}$ $\displaystyle\longrightarrow\mathbb{C}^{2}$
$\displaystyle(\varphi_{1},\varphi_{2})$
$\displaystyle\longmapsto(\lambda_{1},\lambda_{2}).$

In order to remove the dependence on the choice of the eigenvector $v$, we can
consider the quotient $\mathbb{C}^{2}/\mathbb{Z}_{2}$ by the action
$(\lambda_{1},\lambda_{2})\mapsto(-\lambda_{1},-\lambda_{2})$. Composing this
quotient with the previous map, we get the _universal spectral data morphism_

$\displaystyle\mathrm{sd}:\mathfrak{C}$
$\displaystyle\longrightarrow\mathbb{C}^{2}/\mathbb{Z}_{2}$
$\displaystyle(\varphi_{1},\varphi_{2})$
$\displaystyle\longmapsto[(\lambda_{1},\lambda_{2})]$

Moreover, note that $\mathbb{C}^{2}/\mathbb{Z}_{2}$ is in fact isomorphic to
the cone $B$ via the map

$\displaystyle\iota:\mathbb{C}^{2}/\mathbb{Z}_{2}$
$\displaystyle\longrightarrow\mathbb{C}^{3}$
$\displaystyle[(\lambda_{1},\lambda_{2})]$
$\displaystyle\longmapsto(\lambda_{1}^{2},\lambda_{2}^{2},\lambda_{1}\lambda_{2}).$

Thus, we get a factorization of $h$ as $\iota\circ\mathrm{sd}$. We conclude
that in this point model of the Hitchin morphism, spectral data will be given
by elements of the cone $B$.

######  Definition 3.2.

We define the _universal characteristic polynomial_ as the map

$\displaystyle\chi:\mathbb{C}^{2}\times B$
$\displaystyle\longrightarrow\mathbb{C}^{3}$
$\displaystyle\left((x,y),(b_{1},b_{2},b_{3})\right)$
$\displaystyle\longmapsto(x^{2}-b_{1},y^{2}-b_{2},xy-b_{3}).$

The _universal spectral cover_ is defined as the fibre $\chi^{-1}(0)$ with the
natural projection map $p:\chi^{-1}(0)\rightarrow B$.

######  Proposition 3.3.

The universal spectral cover $p:\chi^{-1}(0)\rightarrow B$ is generically
étale of degree $2$ over the “multiplicity-free” locus
$B\setminus\left\\{0\right\\}$, and it is not flat in general.

###### Proof.

If $b\in B\setminus\\{0\\}$, the corresponding points $(x,y)\in\mathbb{C}^{2}$
lying above $b$ are $\pm(\sqrt{b_{1}},\sqrt{b_{2}})$. Therefore, the
analytification $p^{\mathrm{an}}$ defines a local biholomorphism, which
implies that $p$ is étale of degree $2$ in $b$.

On the other hand, the fibre over $0$ is the spectrum of the
$\mathbb{C}$-algebra

$\mathbb{C}[X,Y]/(X^{2},Y^{2},XY),$

which is a $\mathbb{C}$-vector space of rank $3$. Therefore, $p$ cannot be
flat over $0$. ∎

Now, we can give the following generalization of the theorem of
Cayley–Hamilton.

######  Proposition 3.4.

(Universal Cayley–Hamilton theorem) For any
$(\varphi_{1},\varphi_{2})\in\mathfrak{C}$, the $\mathbb{C}[X,Y]$-module
induced by the map

$\displaystyle\Phi:\mathbb{C}[X,Y]$
$\displaystyle\longrightarrow\operatorname{End}\mathbb{C}^{2}$ $\displaystyle
f(X,Y)$ $\displaystyle\longmapsto f(\varphi_{1},\varphi_{2}),$

is supported in the fibre $p^{-1}(h(\varphi_{1},\varphi_{2}))$.

###### Proof.

The suport of the module is $\mathrm{Spec}(\mathbb{C}[X,Y]/\ker\Phi)$. Let
$[(\lambda_{1},\lambda_{2})]$ be the spectral data of
$(\varphi_{1},\varphi_{2})$ and consider the ideal

$I=(X-\lambda_{1},Y-\lambda_{2})\cdot(X+\lambda_{1},Y+\lambda_{2})=(X^{2}-\lambda_{1}^{2},Y^{2}-\lambda_{2}^{2},XY-\lambda_{1}\lambda_{2}).$

Clearly $I\subset\ker\Phi$. Therefore,
$\mathrm{Spec}(\mathbb{C}[X,Y]/\ker\Phi)$ is naturally a subscheme of
$\mathrm{Spec}(\mathbb{C}[X,Y]/I)=p^{-1}(h(\varphi_{1},\varphi_{2}))$. ∎

######  Remark 3.5.

Notice that if $\lambda_{1},\lambda_{2}\neq 0$, then in fact $I=\ker\Phi$.
This is clear since there exists some basis of $\mathbb{C}^{2}$ where the
associated matrices to $\varphi_{1}$ and $\varphi_{2}$ are
$\operatorname{diag}(\lambda_{1},-\lambda_{1})$ and
$\operatorname{diag}(\lambda_{2},-\lambda_{2})$. Therefore, if
$f(\varphi_{1},\varphi_{2})=0$, in particular we have
$f(\lambda_{1},\lambda_{2})=0$ and $f(-\lambda_{1},-\lambda_{2})=0$, and thus
$f\in I$.

This is also true when just one of the eigenvalues (for example $\lambda_{2}$)
vanishes. Indeed, in that case we must have $\varphi_{2}=0$, but we also have
$I=(X^{2}-\lambda_{1}^{2},Y,XY)$ (notice that $\lambda_{1}^{2}Y=X(XY)$).

However, when $[(\lambda_{1},\lambda_{2})]=[(0,0)]$, in general we can at most
hope for a basis where

$\varphi_{1}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\ \text{ and }\
\varphi_{2}=\begin{pmatrix}0&a\\\ 0&0\end{pmatrix},$

for $a\in\mathbb{C}$. In this case $f(X,Y)=aX-Y\in\ker\Phi$, but $f\not\in
I=(X^{2},Y^{2},XY)$.

###  3.3. The spectral correspondence

We will obtain a spectral curve by twisting all the universal objects
constructed in the previous section by the $\mathrm{GL}(2,\mathbb{C})$-torsor
attached to the vector bundle $V\rightarrow X$. More precisely we take the
frame bundle $P$ of $V$, which is a $\mathrm{GL}(2,\mathbb{C})$-principal
bundle and consider the functor sending any $\mathbb{C}$-scheme $F$ with a
left action of $\mathrm{GL}(2,\mathbb{C})$ to the associated fibre bundle
$V(F)=P\times_{\mathrm{GL}(2,\mathbb{C})}F$ and equivariant morphisms to the
associated maps between the associated bundles.

The natural action of $\mathrm{GL}(2,\mathbb{C})$ on $\mathbb{C}^{2}$ induces
an action on the cone $\mathbb{C}^{2}/\mathbb{Z}_{2}$ and thus on $B$. The
sections of the associated fibre bundle $V(B)$ form precisely the Hitchin base

$\mathcal{B}=\left\\{b\in H^{0}(X,\mathrm{Sym}^{2}V)\text{ such that locally
}b=(b_{1},b_{2},b_{3})\text{ with }b_{3}^{2}=b_{1}b_{2}\right\\}.$

Moreover, the universal spectral cover is clearly equivariant by this action,
so it induces a fibre bundle morphism $V(p):V(\chi^{-1}(0))\rightarrow V(B)$.

######  Definition 3.6.

Given $b\in\mathcal{B}$, we define the _spectral cover associated to $b$_ as
the morphism $\pi:Y_{b}\rightarrow X$ given by the Cartesian diagram

${Y_{b}}$${V(\chi^{-1}(0))}$${X}$${V(B).}$$\scriptstyle{\pi}$$\scriptstyle{V(p)}$$\scriptstyle{b}$

The $1$-dimensional scheme $Y_{b}$ is called the _spectral curve associated to
$b$_.

It is clear from this construction that there is a closed embedding
$Y_{b}\hookrightarrow V(\mathbb{C}^{2})=V$. In fact, $\pi$ is obtained as
$q|_{Y_{b}}$, where $q:V\rightarrow X$ is the bundle projection. The
finiteness of $p$ implies that the spectral cover $\pi:Y_{b}\rightarrow X$ is
a finite morphism. Moreover, we can consider the subset
$\mathcal{B}^{\prime}\subset\mathcal{B}$ consisting of sections mapping the
generic point of $X$ to the multiplicity free locus
$B\setminus\left\\{0\right\\}$. If $b\in\mathcal{B}^{\prime}$, then $\pi$ is
generically étale of degree $2$.

The spectral cover $\pi$ will not be flat in general. However, we can make it
flat by the following process of _removing torsion_. Consider the sheaf
$\pi_{*}\mathscr{O}_{Y_{b}}$, which is a coherent sheaf on $X$. Since $X$ is a
smooth curve, the sheaf $\pi_{*}\mathscr{O}_{Y_{b}}$ splits as a direct sum
$F\oplus T$, where $T$ is a torsion sheaf and $F$ is a locally free sheaf of
rank $2$ over $X$. We define the subscheme $\tilde{Y}_{b}\subset Y_{b}$ as the
relative spectrum $\tilde{Y}_{b}=\underline{\mathrm{Spec}}(F)$ and the
morphism $\tilde{\pi}=\pi|_{\tilde{Y}_{b}}$. This morphism
$\tilde{\pi}:\tilde{Y}_{b}\rightarrow X$ is finite locally free, and thus
flat.

######  Remark 3.7.

It is clear that torsion removing does not affect the irreducible components,
so $\tilde{Y}_{b}$ will be irreducible if and only if $Y_{b}$ is. Moreover,
since $\tilde{\pi}:\tilde{Y}_{b}\rightarrow X$ is flat, $X$ is irreducible and
the generic fibre is reduced, we conclude that the curve $\tilde{Y}_{b}$ is
reduced. Therefore, we have shown that if $Y_{b}$ is irreducible, then
$\tilde{Y}_{b}$ is integral.

######  Theorem 3.8 (The spectral correspondence).

For any $b\in\mathcal{B}^{\prime}$ the functor $\pi_{*}$ gives an equivalence
of categories between the category of coherent sheaves $\mathscr{L}$ on
$Y_{b}$ of generic rank $1$ with $\pi_{*}\mathscr{L}$ locally free and
$\det\pi_{*}\mathscr{L}=\mathscr{O}_{X}$ and the category of $V$-twisted Higgs
bundles $(E,\varphi)$ with $\operatorname{tr}(\varphi^{2})=b$. In particular,
this category is nonempty.

Moreover, if $Y_{b}$ is irreducible, $\pi_{*}$ gives a bijection between
$h^{-1}(b)$ and the set of equivalence classes of torsion free sheaves
$\mathscr{L}$ on $\tilde{Y}_{b}$ of generic rank $1$ with
$\det\pi_{*}\mathscr{L}=\mathscr{O}_{X}$.

Finally, if $\tilde{Y}_{b}$ is a smooth curve of genus $g^{\prime}$, the fibre
$h^{-1}(b)$ is an abelian variety of dimension $g^{\prime}-g$, where $g$ is
the genus of $X$.

###### Proof.

We saw in Proposition 2.7 that a $V$-twisted Higgs bundle $(E,\varphi)$ on $X$
is the same that a coherent sheaf $\mathscr{E}$ on $V$ with
$E=q_{*}\mathscr{E}$ locally free of rank $2$ and $\det E=\mathscr{O}_{X}$.

The idea now is to apply the universal Cayley–Hamilton theorem. What this
result tells us in this situation is that a $V$-twisted Higgs bundle
$(E,\varphi)$ is supported as an $\mathscr{O}_{V}$-module on the spectral
curve $Y_{b}$, for $b=\operatorname{tr}(\varphi^{2})$. Therefore, for
$b\in\mathcal{B}^{\prime}$, there is a bijective correspondence between
$V$-twisted Higgs bundles with $\operatorname{tr}(\varphi^{2})=b$ and
isomorphism classes of quasi-coherent sheaves $\mathscr{L}$ on $Y_{b}$ with
$\pi_{*}\mathscr{L}$ locally free of rank $2$. Now, since $\pi$ is finite and
generically étale of degree $2$, asking that $\pi_{*}\mathscr{L}$ has rank $2$
is the same as asking that $\mathscr{L}$ has generic rank $1$.

Using the functoriality of taking pushforward and the equivalence of
categories given by the closed embedding $\tilde{Y}_{b}\hookrightarrow Y_{b}$
we have that, if $(E,\varphi)$ is a $V$-twisted Higgs bundle with
$\operatorname{tr}(\varphi^{2})=b$, then there is an equivalence of categories
between quasi-coherent sheaves $\mathscr{L}$ on $Y_{b}$ with
$\pi_{*}\mathscr{L}=E$ and quasi-coherent sheaves $\mathscr{L}$ on
$\tilde{Y}_{b}$ with $\tilde{\pi}_{*}\mathscr{L}=E$. It suffices then to show
that if a sheaf $\mathscr{L}$ on $Y_{b}$ satisfies $\pi_{*}\mathscr{L}=E$, it
is supported on $\tilde{Y}_{b}$. This is true since the morphism

$\pi_{*}\mathscr{O}_{Y_{b}}\longrightarrow\operatorname{End}E$

giving the structure of $\pi_{*}\mathscr{O}_{Y_{b}}$-module on $E$ must factor
through something which is torsion free, since $\operatorname{End}E$ is
torsion free. In particular, it must factor through
$\pi_{*}\mathscr{O}_{Y_{b}}/T=F=\tilde{\pi}_{*}\mathscr{O}_{\tilde{Y}_{b}}$.
Therefore, $\mathscr{L}$ is supported on $Y_{b}$.

Since $\tilde{\pi}:\tilde{Y}_{b}\rightarrow X$ is a flat morphism, for any
line bundle $L$ on $\tilde{Y}_{b}$, $\tilde{\pi}_{*}L$ will be locally free
and have the structure of a $V$-twisted Higgs bundle. The determinant of
$\tilde{\pi}_{*}L$ is determined by the formula [hartshorne]*Ex. IV.2.6(a)

$\det\tilde{\pi}_{*}L\cong\det\tilde{\pi}_{*}\mathscr{O}_{\tilde{Y}_{b}}\otimes\mathrm{Nm}_{\tilde{\pi}}L,$

where $\mathrm{Nm}_{\tilde{\pi}}$ is the norm map of $\tilde{\pi}$. Thus it
suffices to pick $L$ such that
$\mathrm{Nm}_{\tilde{\pi}}L\cong\det(\tilde{\pi}_{*}\mathscr{O}_{Y_{b}})^{-1}$
in order to have $\det\tilde{\pi}_{*}L\cong\mathscr{O}_{X}$.

Now, if $(E,\varphi)$ is a $V$-twisted Higgs bundle obtained as
$\pi_{*}\mathscr{L}$ and $L\subset E$ is a $\varphi$-invariant line subbundle,
then $\operatorname{tr}(\varphi|_{L}^{2})$ must divide
$\operatorname{tr}(\varphi^{2})$, and $Y_{b}$ is not irreducible. Therefore,
if $Y_{b}$ is irreducible $(E,\varphi)$ is stable.

Moreover, it is clear that if $\tilde{Y}_{b}$ is integral and
$\pi_{*}\mathscr{L}$ is locally free then $\mathscr{L}$ must be torsion-free.
The same goes for $\tilde{Y}_{b}$ smooth and $\mathscr{L}$ locally free.
Therefore, if $\tilde{Y}_{b}$ is smooth, the fibre $h^{-1}(b)$ consists on
those line bundles $L$ on $\tilde{Y}_{b}$ with
$\det\tilde{\pi}_{*}L\cong\mathscr{O}_{X}$ but, as we saw above, the set of
these bundles is the preimage of
$(\det\tilde{\pi}_{*}\mathscr{O}_{\tilde{Y}_{b}})^{-1}$ under the norm map.
This is an abelian variety of dimension $g^{\prime}-g$ (in fact, it can be
identified with the _Prym variety_ of $\pi$). ∎

One of the main consequences of the spectral correspondence is that it allows
us to prove the following.

######  Proposition 3.9.

The Hitchin morphism $h:\mathcal{M}_{0}\rightarrow\mathcal{B}$ is proper.

###### Proof.

We show this by using the valuative criterion. Thus, take
$\Delta=\mathrm{Spec}(R)$ the spectrum of a discrete valuation ring, the
closed point $p\in\Delta$ and the generic point
$\Delta^{\prime}=\Delta\setminus\\{p\\}=\mathrm{Spec}(K)$, where $K$ is the
quotient field of $R$. Consider a map
$g^{\prime}:\Delta^{\prime}\rightarrow\mathcal{M}_{0}$ such that $h\circ g$
extends to a map $f:\Delta\rightarrow\mathcal{B}$. We want to find a map
$g:\Delta\rightarrow\mathcal{M}_{0}$ extending $g^{\prime}$.

Recall from the construction of $\mathcal{M}_{0}$ that this space sits inside
the moduli space of sheaves $M(Z,P)$, for $Z$ a projective completion of $V$
and $P(n)=2kn$. Composing with this inclusion, $g^{\prime}$ defines a map
$\Delta^{\prime}\rightarrow M(Z,P)$ and, since $M(Z,P)$ is a projective
variety, this extends to a map $g:\Delta\rightarrow M(Z,P)$. It suffices to
show that $g(p)\in\mathcal{M}_{0}$.

The good quotient $Q_{2}\rightarrow M(Z,P)$ is a surjective map of finite
type, so we can obtain an integral extension $R\subset\tilde{R}$, where
$\tilde{R}$ is another discrete valuation ring over $\mathbb{C}$ with quotient
field $\tilde{K}$ so that, if we denote
$\tilde{\Delta}=\mathrm{Spec}(\tilde{R})$,
$\tilde{\Delta}^{\prime}=\mathrm{Spec}(\tilde{K})$ and
$\alpha:\tilde{\Delta}\rightarrow\Delta$ the map induced by the extension, the
following diagram commutes

${\tilde{\Delta}^{\prime}}$${\tilde{\Delta}}$${Q_{2}}$${\Delta^{\prime}}$${\Delta}$${M(Z,P).}$$\scriptstyle{\alpha}$$\scriptstyle{g}$

Since $Q_{2}$ represents a certain functor, the map $\tilde{\Delta}\rightarrow
Q_{2}$ can be identified with a sheaf $\mathscr{E}$ on $Z_{\tilde{\Delta}}$
and, for $s\in\tilde{\Delta}^{\prime}$, we have that
$g\circ\alpha(s)\in\mathcal{M}_{0}$. Thus,
$\mathscr{E}|_{\tilde{\Delta}^{\prime}}$ is supported inside of
$V_{\tilde{\Delta}^{\prime}}$ and it corresponds to a family
$(E^{\prime},\varphi^{\prime})$ of semistable $V$-twisted
$\mathrm{SL}(2,\mathbb{C})$-Higgs bundles on $X$ parametrized by
$\tilde{\Delta}^{\prime}$ with

$(E^{\prime}_{s},\varphi^{\prime}_{s})=g\circ\alpha(s).$

Therefore, the map

$\displaystyle\tilde{\Delta}^{\prime}$
$\displaystyle\longrightarrow\mathcal{B}$ $\displaystyle s$
$\displaystyle\longmapsto h(E^{\prime}_{s},\varphi^{\prime}_{s})$

extends to a map $b=f\circ\alpha:\tilde{\Delta}\rightarrow\mathcal{B}$.

From the spectral correspondence we have that $\mathscr{E}$ is supported
inside of the closed subscheme

$W=\coprod_{s\in\tilde{\Delta}}Y_{b(s)}\subset V_{\tilde{\Delta}}.$

Flatness of $\mathscr{E}$ over $\tilde{\Delta}$ implies that $\mathscr{E}$ is
supported in the closure of $W$, which is again $W$. Hence $\mathscr{E}$
corresponds to a family $(E,\varphi)$ of semistable $V$-twisted Higgs bundles
parametrized by $\tilde{\Delta}$, with
$(E,\varphi)|_{\tilde{\Delta}^{\prime}}=(E^{\prime},\varphi^{\prime})$. In
other words, we obtain a map $\tilde{\Delta}\rightarrow\mathcal{M}_{0}$
extending $g^{\prime}\circ\alpha:\tilde{\Delta}^{\prime}\rightarrow M(Z,P)$.
But since $M(Z,P)$ is separated, this map coincides with
$g\circ\alpha:\tilde{\Delta}\rightarrow M(Z,P)$. This implies that, if $s_{0}$
is the closed point of $\tilde{\Delta}$, we have
$g(p)=g(\alpha(s_{0}))\in\mathcal{M}_{0}$, as we wanted to show. ∎

###  3.4. Properties of the spectral curve

In the previous section we have obtained a general correspondence between
Higgs bundles with spectral data $b$ and sheaves on the corresponding spectral
curve $Y_{b}$, or on its modification $\tilde{Y}_{b}$. However, we still need
to know under what conditions we can assume that the spectral curve $Y_{b}$
has nice properties, like being integral or smooth.

First of all, it will be useful to write down explicitly the local equations
describing $Y_{b}$. Given a point $x\in X$, we can take an open neighbourhood
$U$ of $x$ so that $V|_{U}\cong\mathscr{O}_{X}^{2}|_{U}$ and $b|_{U}$ can be
seen as a section $b=(b_{1},b_{2},b_{3})$ of $\mathscr{O}_{X}|_{U}^{3}$, with
$b_{3}^{2}=b_{1}b_{2}$. Recall now that the spectral curve is obtained as a
relative spectrum $Y_{b}=\underline{\mathrm{Spec}}(\mathscr{F})$, for
$\mathscr{F}=\pi_{*}\mathscr{O}_{Y_{b}}$ a sheaf such that

(1)
$\mathscr{F}|_{U}=\frac{\mathscr{O}_{X}|_{U}[X,Y]}{(X^{2}-b_{1},Y^{2}-b_{2},XY-
b_{3})}.$

####  3.4.1. Integrality

From this description it is clear that $Y_{b}$ will be reducible if and only
if there exists a section $a\in H^{0}(X,V)$ such that $b=a^{2}$. Locally, this
means that $a=(a_{1},a_{2})$ and
$(b_{1},b_{2},b_{3})=(a_{1}^{2},a_{2}^{2},a_{1}a_{2})$, so

$(X^{2}-b_{1},Y^{2}-b_{2},XY-b_{3})=(X-a_{1},Y-a_{2})\cdot(X+a_{1},Y+a_{2}).$

In that case, the spectral curve $Y_{b}$ can be written as
$\operatorname{im}a\cup\operatorname{im}(-a)$.

The image of the natural map $H^{0}(X,V)\rightarrow\mathcal{B}$ taking $a$ to
$a^{2}$ is closed. Indeed, we can consider the variety
$\mathbb{P}(\mathcal{B})$ as the quotient
$\mathcal{B}\setminus\left\\{0\right\\}$ by the $\mathbb{C}^{*}$ action
$b\mapsto\lambda b$. This quotient is a variety since it is the subvariety of
$\mathbb{P}(H^{0}(X,\mathrm{Sym}^{2}V))$ corresponding to $\mathcal{B}$ (which
is locally defined by homogeneous polynomials). Now we can consider the map

$\displaystyle\mathbb{P}(H^{0}(X,V))$
$\displaystyle\longrightarrow\mathbb{P}(\mathcal{B})$ $\displaystyle[a]$
$\displaystyle\longmapsto[a^{2}],$

which is well defined since $(\lambda a)^{2}=\lambda^{2}a^{2}$. This is a
morphism between projective varieties and as such it has closed image $Z$.
Now, the image of the corresponding map $H^{0}(X,V)\rightarrow\mathcal{B}$ is
the union of $\left\\{0\right\\}$ with the preimage of $Z$ by the natural
projection
$\mathcal{B}\setminus\left\\{0\right\\}\rightarrow\mathbb{P}(\mathcal{B})$,
which is also closed, since this projection is continuous.

We conclude that $Y_{b}$ is either reducible for every $b\in\mathcal{B}$, or
irreducible for a generic element $b\in\mathcal{B}$. Moreover, recall that if
$Y_{b}$ is irreducible, then $\tilde{Y}_{b}$ is integral. Therefore, we have
proved:

######  Proposition 3.10.

If $V$ is such that the map $H^{0}(X,V)\rightarrow\mathcal{B}$ given by
$a\mapsto a^{2}$ is not surjective, then for a generic $b\in\mathcal{B}$, the
fibre of the Hitchin morphism $h^{-1}(b)$ is in bijection with the set of
equivalence classes of torsion free sheaves $\mathscr{L}$ on $Y_{b}$ of
generic rank $1$ with $\det\pi_{*}\mathscr{L}=\mathscr{O}_{X}$.

####  3.4.2. Étale spectral curves

We consider now the case where the spectral curves are best behaved, which is
when the section $b$ does not vanish.

######  Proposition 3.11.

If $x\in X$ is such that $b(x)\neq 0$, the induced map
$\pi^{\mathrm{an}}:Y_{b}^{\mathrm{an}}\rightarrow X^{\mathrm{an}}$ in the
analytifications is a local biholomorphism near any point $y$ with $\pi(y)=x$.

###### Proof.

This is clear from (1), since $y$ will be of the form
$\pm(\sqrt{b_{1}(x)},\sqrt{b_{2}(x)})$, and we can define the local inverse
$x^{\prime}\mapsto\pm(\sqrt{b_{1}(x^{\prime})},\sqrt{b_{2}(x^{\prime})})$
(making the choice of sign compatible with the value of $y$). ∎

We say that a section $b\in\mathcal{B}$ is _nonvanishing_ if $b(x)\neq 0$ for
every $x\in X$.

######  Corollary 3.12.

If $b\in\mathcal{B}$ is a nonvanishing section, the spectral cover
$\pi:Y_{b}\rightarrow X$ is étale. In particular, if $Y_{b}$ is irreducible,
the spectral curve $Y_{b}$ is a smooth projective curve of genus $2g-1$, where
$g$ is the genus of $X$.

Moreover, in that case the fibre of the Hitchin morphism $h^{-1}(b)$ will be
an abelian variety of dimension $g-1$.

###### Proof.

Since $b$ does not vanish, $\pi^{\mathrm{an}}$ is a local biholomorphism,
which is equivalent to say that $\pi$ is étale. If $\pi$ is étale and $X$ is
smooth, $Y_{b}$ must also be smooth. If this $Y_{b}$ is irreducible, its genus
$g^{\prime}$ is given by the Riemann–Hurwitz formula

$2g^{\prime}-2=\deg\pi(2g-2),$

so

$g^{\prime}=2g-1,$

since $\deg\pi=2$. ∎

######  Remark 3.13.

Of course, the existence of nonvanishing sections in $\mathcal{B}$ is not
guaranteed a priori. We expect that if $V$ satisfies certain conditions,
nonvanishing sections can be found generically. We will study this problem in
future work.

####  3.4.3. The general situation

We explore now the properties of the spectral curve $Y_{b}$ and of the
modification $\tilde{Y}_{b}$ in the more general case where $b$ vanishes at
some points of $X$. If we assume that $b$ is in the open subset
$\mathcal{B}^{\prime}\subset\mathcal{B}$ of sections that do not vanish at the
generic point, the set $b^{-1}(0)$ of points where $b$ vanishes will be a
finite subset of $X$.

From the above discussion, it is clear that over a point $x$ outside of
$b^{-1}(0)$ the spectral curve $Y_{b}$ is smooth and thus the spectral cover
$\pi$ is flat. Therefore, we are reduced to study the case where $b(x)=0$.

From (1), it is clear that the stalk of
$\mathscr{F}=\pi_{*}\mathscr{O}_{Y_{b}}$ at a point $x$ is

$\mathscr{F}_{x}=\frac{\mathscr{O}_{X,x}[X,Y]}{(X^{2}-b_{1},Y^{2}-b_{2},XY-
b_{3})},$

where the $b_{i}$ are actually their equivalence classes in
$\mathscr{O}_{X,x}$. This can in principle be decomposed as

$\mathscr{F}_{x}=\mathscr{O}_{X,x}\oplus X\cdot\mathscr{O}_{X,x}\oplus
Y\cdot\mathscr{O}_{X,x}.$

When $x\in b^{-1}(0)$, this module $\mathscr{F}_{x}$ will not be free, so it
will have a decomposition

$\mathscr{F}_{x}=F_{x}\oplus T_{x},$

where $F_{x}$ is a free module and $T_{x}$ is a torsion module. Let us find
the explicit form of this decomposition.

Take a small neighbourhood $U$ of $x$ and a local coordinate
$z:U\rightarrow\mathbb{C}$, with $z(x)=0$ and inverse $\phi:\Delta\rightarrow
U$, for $\Delta$ a disk centered at $0$, and define the polynomial functions
$f_{i}=b_{i}\circ\phi:\Delta\rightarrow\mathbb{C}$. Since $b(x)=0$, we have
that $f_{i}(0)=0$ for $i=1,2,3$. Therefore, we can write
$f_{1}(Z)=Z^{n}g_{1}(Z)$ and $f_{3}(Z)=Z^{m}g_{3}(Z)$, for $g_{1}$ and $g_{3}$
not vanishing at $0$. Moreover, by switching $f_{1}$ and $f_{2}$, we can
assume without loss of generality that $n\geq m$. If we define
$a_{i}=g_{i}\circ z$, we have

$\displaystyle b_{1}$ $\displaystyle=z^{n}a_{1}$ $\displaystyle b_{3}$
$\displaystyle=z^{m}a_{3}.$

We have the relation

$b_{3}X-b_{1}Y=(XY)X-X^{2}Y=0$

in the module $\mathscr{F}_{x}$, so

$z^{m}(a_{3}X-z^{n-m}a_{1}Y)=0.$

This implies that $a_{3}X-z^{n-m}a_{1}Y$ is a torsion element of
$\mathscr{F}_{x}$. Now, the determinant

$\begin{vmatrix}a_{3}&-z^{n-m}a_{1}\\\
z^{n-m}a_{1}&a_{3}\end{vmatrix}_{x}=[a_{3}^{2}+z^{2(n-m)}a_{1}^{2}]_{x}=a_{3}(0)^{2}\neq
0,$

so there is an isomorphism

$X\cdot\mathscr{O}_{X,x}\oplus
Y\cdot\mathscr{O}_{X,x}\cong[z^{n-m}a_{1}X+a_{3}Y]\cdot\mathscr{O}_{X,x}\oplus[a_{3}X-z^{n-m}a_{1}Y]\cdot\mathscr{O}_{X,x}.$

This gives a decomposition $\mathscr{F}_{x}=F_{x}\oplus T_{x}$, as a direct
sum of a free $\mathscr{O}_{X,x}$-module

$F_{x}=\mathscr{O}_{X,x}\oplus[z^{n-m}a_{1}X+a_{3}Y]\cdot\mathscr{O}_{X,x}$

and a torsion module

$T_{x}=[a_{3}X-z^{n-m}a_{1}Y]\cdot\mathscr{O}_{X,x}.$

Globally, we wrote $\mathscr{F}=F\oplus T$, with
$F=\tilde{\pi}_{*}\mathscr{O}_{\tilde{Y}_{b}}$ a locally free sheaf of rank
$2$ and $T$ the torsion part of $\mathscr{F}$. Now we have given an explicit
description of $T$ as a sum of skyscraper sheaves

$T=\bigoplus_{x\in b^{-1}(0)}i_{x,*}T_{x},$

with $i_{x}:\mathrm{Spec}(\mathscr{O}_{X,x})\rightarrow X$ the natural map
induced by localization.

To sum up, we have obtained that near a point $x\in b^{-1}(0)$, the flat
spectral curve $\tilde{Y}_{b}$ is described by the equations

$\begin{cases}X^{2}-f_{1}(Z)=0\\\ Y^{2}-f_{2}(Z)=0\\\ XY-f_{3}(Z)=0\\\
g_{3}(Z)X-Z^{n-m}g_{1}(Z)Y=0.\end{cases}$

This allows us to give conditions for smoothness over $x$. We just have to
apply the Jacobian criterion: the Jacobian at the point
$y=\tilde{\pi}^{-1}(x)$ has rank greater or equal than $2$, the codimension of
$\tilde{Y}_{b}$, whenever one of the $f_{i}^{\prime}(0)$ is nonzero; that is,
if $n$ or $m$ equals $1$.

We have proven:

######  Proposition 3.14.

If the flat spectral curve $\tilde{Y}_{b}$ is irreducible, then it is smooth
if and only if $b$ does not have a multiple zero.

##  4\. Gauge theoretic equations and Hitchin–Kobayashi correspondence

###  4.1. The generalized Hitchin equations

Higgs bundles twisted by a vector bundle appear as solutions to gauge
theoretic equations that are a generalization of the Hitchin equations. These
are defined over a compact Riemann surface that we also denote by $X$.

We fix once and for all a volume form $\omega_{X}\in\Omega^{1,1}(X)$ with
$\int_{X}\omega_{X}=1$. Consider $V$ a holomorphic vector bundle of rank $2$
over $X$ and fix $h$ a Hermitian metric on $V$. If the vector bundle $V$ is
stable, we can choose the solution to the Hermitian Yang–Mills equations given
by the theorem of Narasimhan–Seshadri.

Suppose that $(E,\varphi)$ a $V$-twisted Higgs bundle on $X$ and denote

$\lambda_{E}=-i\pi\deg
E\operatorname{id}_{E}\in\Omega^{0}(X,\operatorname{End}E).$

If $H$ is a Hermitian metric on $E$, the twisted endomorphism $\varphi$ has a
smooth adjoint morphism $\varphi^{\dagger}:E\otimes V\rightarrow E$ with
respect to the metric induced by $H$ and $h$, so it makes sense to consider
the composition
$\varphi^{\dagger}\circ\varphi\in\Omega^{0}(X,\operatorname{End}E)$. Moreover,
regarding $\varphi^{\dagger}$ as a smooth map $E\rightarrow E\otimes
V^{\vee}$, we can also consider the composition
$\varphi\circ\varphi^{\dagger}$ and thus the commutator
$[\varphi,\varphi^{\dagger}]\in\Omega^{0}(X,\operatorname{End}E)$.

######  Definition 4.1.

Let $(E,\varphi)$ be a $V$-twisted Higgs bundle. We say that a Hermitian
metric $H$ on $E$ is a _solution to the generalized Hitchin equations_ if

$\Lambda F_{H}+[\varphi,\varphi^{\dagger}]=\lambda_{E},$

where $F_{H}$ is the curvature of the Chern connection associated to the
metric $H$ on $E$ and $\Lambda$ is the adjoint operator of
$\omega_{X}\wedge-$.

We have the following _Hitchin–Kobayashi correspondence_ :

######  Proposition 4.2.

A $V$-twisted Higgs bundle $(E,\varphi)$ is polystable if and only if it
admits a solution to the generalized Hitchin equations.

###### Proof.

The generalized Hitchin equations are a special case of the _twisted quiver
vortex equations_ , indroduced in [luisoscar]. More precisely, we can regard
$(E,\varphi)$ as a holomorphic $V$-twisted $Q$-bundle on $X$, where $Q$ is the
quiver

${Q=\bullet}$.

The Hitchin–Kobayashi correspondence for twisted quiver bundles
[luisoscar]*Thm. 3.1 guarantees the existence of the solution $H$, given that
$(E,\varphi)$ is polystable, and viceversa. ∎

######  Remark 4.3.

Although the generalized Hitchin equations are a particular case of the more
general gauge equations studied in [luisoscar], in that paper the commuting
condition is not considered. However, this extra condition does not affect the
Hitchin–Kobayashi correspondence.

###  4.2. The moment map

Understanding the Hitchin–Kobayashi correspondence as an infinite-dimensional
analogue of the Kempf–Ness theorem, $V$-twisted Higgs bundles can be studied
in terms of a moment map. As above, we fix a rank $2$ holomorphic vector
bundle $V$ on $X$ and a Hermitian metric $h$ on $V$.

We consider $\mathbb{E}\rightarrow X$ a rank $2$ smooth complex vector bundle
and fix a Hermitian metric $H$ on $\mathbb{E}$. We denote by $\mathcal{A}_{H}$
the space of $H$-unitary connections on $\mathbb{E}$. Of course, given a
connection $A\in\mathcal{A}_{H}$, the associated operator $\bar{\partial}_{A}$
endows $\mathbb{E}$ with the structure of a holomorphic vector bundle.
Finally, we denote

$\mathcal{H}=\mathcal{A}_{H}\times\Omega^{0}(X,\operatorname{End}\mathbb{E}\otimes
V).$

The tangent space of $\mathcal{H}$ at any point $(A,\varphi)$ is the vector
space
$\Omega^{1}(X,\mathfrak{u}_{H}(\mathbb{E}))\times\Omega^{0}(X,\operatorname{End}\mathbb{E}\otimes
V)$, where $\mathfrak{u}_{H}(\mathbb{E})$ is the bundle of skew-Hermitian
endomorphisms of $\mathbb{E}$ for the metric $H$. We can define a symplectic
on $\mathcal{H}$ given by

$\Omega\left((B_{1},\psi_{1}),(B_{2},\psi_{2})\right)=-\int_{X}\operatorname{tr}(B_{1}\wedge
B_{2})+2i\mathrm{Im}(\psi_{1}^{\dagger}\psi_{2})\omega_{X}.$

The (real) _gauge group_ of $\mathbb{E}$ is
$\mathcal{G}=\Omega^{0}(X,\mathrm{U}_{H}(\mathbb{E}))$ the group of sections
of the bundle $\mathrm{U}_{H}(\mathbb{E})$ given by $H$-unitary endomorphisms
of $\mathbb{E}$. The group $\mathcal{G}$ acts symplectically by by conjugation
on $\mathcal{H}$. The Lie algebra of $\mathcal{G}$ is
$\Omega^{0}(X,\mathfrak{u}_{H}(\mathbb{E}))$. Therefore, the moment map for
the action of $\mathcal{G}$ can be regarded as a map

$\displaystyle\mu:\mathcal{H}$
$\displaystyle\longrightarrow\Omega^{0}(X,\mathfrak{u}_{H}(\mathbb{E})).$

The same arguments that those given in [hitchinselfduality]*§4 show that this
map is precisely

$\mu(A,\varphi)=\Lambda F_{A}+[\varphi,\varphi^{\dagger}].$

Now, we can consider the associated symplectic quotient
$\mu^{-1}(\lambda_{\mathbb{E}})/\mathcal{G}$. The space we are interested in
is the subvariety

$\mathcal{M}_{\mathrm{gauge}}(\mathbb{E})=\left\\{[(A,\varphi)]\in\mu^{-1}(\lambda_{\mathbb{E}})/\mathcal{G}:\bar{\partial}_{A,V}\varphi=0\text{
and }\varphi\wedge\varphi=0\right\\},$

where
$\bar{\partial}_{A,V}=\bar{\partial}_{A}\otimes\operatorname{id}_{V}+\operatorname{id}_{\operatorname{End}\mathbb{E}}\otimes\bar{\partial}_{V}$
is the Dolbeaut operator induced on $\operatorname{End}\mathbb{E}\otimes V$ by
the connection $A$ and the holomorphic structure of $V$.

The decomposition
$\operatorname{End}(\mathbb{E})=\mathfrak{u}_{H}(\mathbb{E})\oplus
i\mathfrak{u}_{H}(\mathbb{E})$ induces an isomorphism between
$\Omega^{1}(X,\mathfrak{u}_{H}(\mathbb{E}))$ and
$\Omega^{0,1}(X,\operatorname{End}\mathbb{E})$. This allows us to define a
complex structure on $\mathcal{A}_{H}$ by $J(\alpha)=i\alpha$. Together with
the natural complex structure on
$\Omega^{0}(X,\operatorname{End}\mathbb{E}\otimes V)$, this gives a complex
structure on $\mathcal{H}$. Moreover, this complex structure is compatible
with the symplectic structure $\Omega$ and thus they give a Kähler structure.
Since the action of the gauge group $\mathcal{G}$ preserves these structures,
the symplectic quotient is in fact a Kähler quotient. Therefore, in general
$\mu^{-1}(\lambda_{\mathbb{E}})/\mathcal{G}$ is a Kähler orbifold. Since the
conditions defining $\mathcal{M}_{\mathrm{gauge}}(\mathbb{E})$ are analytic,
the space $\mathcal{M}_{\mathrm{gauge}}(\mathbb{E})$ is a complex analytic
variety.

The Hitchin–Kobayashi correspondence given by the above Proposition 4.2
implies that there is a canonical bijection between the set $S(d)$ of
isomorphism classes of polystable $V$-twisted Higgs bundles on $X$ and the
space $\mathcal{M}_{\mathrm{gauge}}(\mathbb{E})$.

Moreover, suppose that $X$ is a smooth complex projective curve,
$X^{\mathrm{an}}$ denotes the Riemann surface obtained by analytification of
$X$ and $\mathcal{M}(d)^{\mathrm{an}}$ is the analytification of the moduli
space $\mathcal{M}(d)$ of semistable $V$-twisted Higgs bundles of degree $d$,
where $d=\deg\mathbb{E}$. We expect that the Hitchin–Kobayashi correspondence
gives an analytic isomorphism between $\mathcal{M}(d)^{\mathrm{an}}$ and
$\mathcal{M}_{\mathrm{gauge}}(\mathbb{E})$. This is in fact well-known
folklore in the field, and similar results are obtained by applying the
_Kuranishi slice method_. The reader may refer to [fan] for a detailed
exposition of these techniques for the moduli space of (usual) Higgs bundles.

## Acknowledgements

The first and second authors thank the Indian Institute of Science (Bangalore)
and the International Centre for Theoretical Sciences (Bangalore) for
hospitality during the visits that led to this collaboration, and during the
programme _Moduli of bundles and related structures_ (Code: ICTS/mbrs2020/02).

We thank Nigel Hitchin and Tony Pantev for fruitful discussions. We also thank
Peter Dalakov for useful comments.

## References


#  ProCoT: Stimulating Critical Thinking and Writing of Students through
Engagement with Large Language Models (LLM s)

Tosin Adewumi*, Lama Alkhaled, Claudia Buck, Sergio Hernández, Saga Brilioth,
Mkpe Kekung, Yelvin Ragimov, and Elisa Barney.  Luleå University of
Technology, Sweden;
*Corresponding author<EMAIL_ADDRESS>

###### Abstract

We introduce a novel writing method called Probing Chain-of-Thought (ProCoT),
which prevents students from cheating using a Large Language Model (LLM), such
as ChatGPT, while enhancing their active learning through such models. LLM s
have disrupted education and many other feilds. For fear of students cheating,
many educationists have resorted to banning their use, as their outputs can be
human-like and hard to detect in some cases. These LLM s are also known for
hallucinations (i.e. fake “facts”). We conduct studies with ProCoT in two
different courses with a combined total of about 66 students. The students in
each course were asked to prompt an LLM of their choice with one question from
a set of four and required to affirm or refute statements in the LLM output by
using peer-reviewed references. The results show two things: (1) ProCoT
stimulates creative/critical thinking and writing of students through
engagement with LLM s when we compare the LLM-solely output to ProCoT output
and (2) ProCoT can prevent cheating because of clear limitations in existing
LLM s when we compare students’ ProCoT output to LLM ProCoT output. We also
discover that most students prefer to give answers in fewer words than LLM s,
which are typically verbose. The average word counts for students, ChatGPT
(v3.5) and Phind (v8) are 208, 391 and 383, respectively.

###### keywords:

LLM, ChatGPT, ProCoT, Cheating, Education, Pedagogy

††articletype: ARTICLE

##  1 Introduction

A chain of thought is a sequence of intermediate reasoning steps (Wei .,
2022). Chain-of-thought (CoT) prompting allows a sufficiently Large Language
Model (LLM)to solve a problem as a clear sequence of steps before a final
answer. An LLM, which is typically a deep neural probabilistic model (Brown .,
2020), such as ChatGPT111openai.com, has the ability to generate convincing
outputs when given inputs, called prompts. The use of such Artificial
Intelligence (AI)tools, particularly ChatGPT, has gained immense popularity in
many fields in the last one year (Allam ., 2023). They have also gained
notoriety for suspected cheating, particularly in the Education field (Cotton
., 2023; Ventayen, 2023). This is besides the hallucinations (or ungrounded
addition) problem with LLM s (Rawte ., 2023), which occurs partly because they
are typically trained on static data, with no updates through time. Humans, on
the other hand, can source and use real-world context to verify the veracity
of claims (Maslej ., 2023).

Topic-based writing has always been a common way of assessing students (Todd
., 2004). However, the capability of LLM s to completely or partially generate
content as desired by a user appears to be putting a threat to students
writing own their own. The research question we address in this work is: “Is
there a way to both prevent cheating in writing by students through LLM s and
also engage them in the use of such AI tools?” The generative power of these
LLM s makes it imperative to ask similar questions regarding coding,
mathematics and other similar tasks.

We introduce Probing Chain-of-Thought (ProCoT), a method aimed at stimulating
creative/critical thinking and writing of students through engagement with LLM
s, given that these models are here to stay. We demonstrate this method in two
courses (an Erasmus+ blended intensive program (BIP)(Commission, 2022) and a
full course) for two sets of students. The students were free to use the LLM
of their choice and 95% used ChatGPT222openai.com while 5% used
Phind333phind.com. We believe ProCoT will empower educators to no longer fear
LLM s but see them as technological tools and students will be better critical
thinkers. Also, it may be applied to any field that uses writing.

###  1.1 Our Contributions

  1. 1.

ProCoT prevents cheating through LLM s by students.

  2. 2.

ProCoT stimulates creative/critical thinking and writing of students through
engagement with LLM s

  3. 3.

ProCoT will provide valuable data to further train LLM s to be better models
without privacy issues.

The rest of this paper is organized as follows. Section 2 discusses the
background and literature review, including LLM s. Section 3 describes in
details the methods we follow, including the cases. Section 4 discusses the
outcome in results that also present ststistical analysis. Section 5 briefly
gives concluding remarks.

##  2 Background and Literature Review

In essay writing assessments, students are asked to demonstrate a point of
view (Todd ., 2004) and this may be supported by evidence (e.g. through peer-
reviewed references). This allows students to show what they have learnt and
build their writing skills.

###  2.1 Essay Writing Evaluation

The pedagogy of essay writing evaluation requires a comprehensive approach
that focuses on student learning and development. Central to this is the use
of rubrics, which are detailed guides laying out the expected criteria for an
essay (Lipnevich ., 2023). They may cover aspects like the strength of an
argument, the use of evidence (or references), and the organization of ideas
(Todd ., 2004). A formative approach to evaluation, based on quality feedback,
helps students identify their strengths and weaknesses and fosters a growth
mindset (Spector ., 2016; Porter, 2022). These methods also provide
transparency and ensure alignment with intended learning outcomes (ILOs)(Le .,
2023; Ramesh Sanampudi, 2022).

Peer review and self-assessment are also important in pedagogy. By engaging in
peer review, students develop critical thinking skills (Cheong ., 2023) while
self-assessment encourages students to reflect on their own writing, fostering
a sense of ownership and responsibility (Culver, 2023). Encouraging reflection
is vital in the pedagogical evaluation of any writing. It helps students to be
more independent and confident writers (Pang, 2022). The pedagogical
evaluation of writing is a comprehensive yet subjective process (Todd .,
2004). It is a crucial component of the educational journey that plays a
significant role in the overall development of students, both as critical
thinkers and skilled writers.

###  2.2 Large Language Models (LLM s)

An LLM aims to mimic human language patterns, responses, and general
linguistic structures after training. It is a deep probabilistic model that is
trained on big data to generate probabilities over a set of tokens (or words)
for predicting the next token in a sequence and there are many types with
different sizes (Brown ., 2020; Touvron ., 2023). These models are usually
based on the Transformer, which is a deep neural networks architecture
(Vaswani ., 2017). They are trained to perform Natural Language Processing
(NLP)tasks, like reading comprehension, summarization, and question answering
(Radford ., 2019; Touvron ., 2023).

By far, the most popular example of an LLM is OpenAI’s ChatGPT2, though there
are several others, like Gemini444deepmind.google/technologies/gemini by
Google DeepMind and LLAMA-2 (Touvron ., 2023) by Meta. ChatGPT, now in vesion
4 (Natalie, 2023), is an advanced LLM. It is the latest in the family of
Generative Pre-trained Transformer (GPT) models, which began with GPT-1 while
version 2 (Radford ., 2019) in 2019 is recognized as possibly the first LLM
because of its size of 1.5B parameters (Maslej ., 2023). ChatGPT (v3.5) is
available for free to users through a web interface or for a fee after a
certain period through an Application Programming Interface (API).

LLM s offer a myriad of benefits, including assistance in learning,
improvement in language skills, access to extensive information, and the
provision of customized learning experiences. Particularly, the capabilities
of ChatGPT to generate coherent and contextually relevant text based on
prompts have made it popular. Similarly to other machine learning (ML)models
(Yamamoto ., 2018), it may be used to assist in evaluating student essays and
providing feedback. The use of LLM s in education is not without challenges,
however (Sullivan ., 2023; Sok Heng, 2023; Meyer ., 2023). Over-reliance on
these models can potentially lead to diminished critical thinking abilities in
students. Another critical aspect to consider is the inherent biases in LLM s
(Meyer ., 2023). Since these models are trained on existing data, they may
unintentionally reflect and reinforce any biases that exist in the data they
were trained on, which could affect the quality and fairness of information
provided to students. Ensuring responsible AI use is paramount to maintaining
a safe and productive learning environment (Vinuesa ., 2020).

###  2.3 LLM Fact-Checking Methods

In recent years, LLM fact-checking methods have emerged for combating
misinformation. Retrieval Augmented Generation (RAG)models (Lewis ., 2020),
which combine information retrieval and generation, and special-purpose
knowledge bases or datasets are useful in this regard. A few datasets that are
used for fact-checking in LLM s are listed below. The availability of these
methods have not made LLM s error-free, indicating that more work still needs
to be done by the research community.

  1. 1.

FEVER (Thorne ., 2018) \- Fact Extraction and VERification (FEVER) is a widely
used dataset that consists of claims and evidence, allowing LLM s to learn to
validate or debunk statements based on the supporting information. The
statements from the dataset are from Wikipedia and claims are classified as
Supported, Refuted or NotEnoughInfo.

  2. 2.

ClaimBuster (Arslan ., 2020) \- ClaimBuster is a dataset of 23,533 statements
that is designed to identify claims which are worth fact-checking. The
statements are extracted from U.S. general election presidential debates.
There are three possible categories in the data: non-factual, unimportant
factual statement, and check-worthy factual statement.

##  3 Methodology

Probing the chain-of-thought (output) of an LLM involves students scrutinizing
the output by using a reference-based platform to provide time-relevant fact-
checked feedback on the output. The method is entrenched in the “Self-
Regulation” method (Zimmerman, 2013), which is the self-directive process by
which learners transform their mental abilities into task-related skills. It
involves the following components in an iterative process: forethought,
planning, monitoring, failure detection and correction. It leads to faster
independence and self-awareness of students. ProCoT may even be applied to
other outputs, besides those of LLM s.

We compare the quality of students’ writing, as evaluated by a human subject
expert and using statistical analysis, to establish the effectiveness of
ProCoT. We compare 1) LLM-only outputs, 2) students’ ProCoT outputs, and 3)
LLM ProCoT outputs. LLM-only output is the result of asking the model any of
the the original questions among the lists in the sub-sections below.
Meanwhile, a student’s ProCoT output is the feedback provided (with peer-
reviewed references) for the LLM-only output while an LLM ProCoT output is the
LLM feedback to their original (LLM-only) output. We give the same instruction
that is given to the students to the LLM when generating the LLM ProCoT
output. This is to determine if the students can cheat by generating ProCoT
feedback successfully with the LLM and pass them off as their own. These
evaluations are carried out in two cases involving two different courses and
two different sets of students.

###  3.1 Case 1

The first case involves 26 students, where about 88% of Masters students, 9%
Bachelor’s, and 3% PhD at a BIP. A BIP is a short programme that combines
physical mobility with a virtual part. The 6-day BIP was titled
“DigiHealth‐AI: Practice, Research, Ethics, and Regulation” and organized by
Deggendorf Institute of Technology (DIT, Germany) in collaboration with
partners from four European countries. These include Luleå University of
Technology (LTU, Sweden), AI Sweden, University of Agder (UiA, Norway), Czech
Technical University (CTU, Czechia), and Technical University of Catalonia
(UPC, Spain).

The students were asked to pose one out of the four questions below to any
LLM. The questions were randomly selected from two blogs.555wtamu.edu/
cbaird/sq/category/health; magazine.utoronto.ca/research-ideas Thereafter, the
ProCoT instruction was “Write 1 page to affirm or refute assertions/statements
made by ChatGPT/LLM in the response by using references from peer-reviewed
articles”. The students were under supervision for a 30-minute period while
providing answers.

  1. 1.

Did cancer exist before man-made chemicals were around to create it?

  2. 2.

Who will benefit from AI in healthcare?

  3. 3.

How long do you have to exercise for it to count?

  4. 4.

How will we avoid machine bias?

###  3.2 Case 2

This case had fourty students who were all Masters students. The course is
titled “Text Mining” and is a regular semester course at Luleå University of
Technology (LTU). Again, each student was asked to pose one out of the four
questions below to any LLM. The questions were randomly selected from a list
of 50 exam questions, which are dedicated to the course. Thereafter, the
ProCoT instruction was “Write, at least, 300 words (excluding the reference
section) to affirm or refute statements made by ChatGPT/LLM by using (at least
5) references from peer-reviewed articles or books”. The students were allowed
to go with the task and submit within a week as the deadline.

  1. 1.

What is a large language model (LLM)?

  2. 2.

What is the vanishing gradient problem and how is it tackled?

  3. 3.

Describe the two paradigms of using a pre-trained model in transfer learning:
feature extraction and fine-tuning.

  4. 4.

Name two similarity measurement metrics for vectorized semantic
representation. Which one may be preferred and why?

##  4 Results and Discussion

Two out of twenty-six submissions from students in case 1 were rejected for
lack of compliance to the ProCoT instructions. Results from the valid
students’ ProCoT answers show that cheating can be prevented by framing
questions as we have done with ProCoT. Comparing the LLM (ProCoT) feedback on
its answers to the original questions in both cases, ChatGPT (v3.5) expressly
says “I cannot provide direct references to peer-reviewed articles” on the one
hand while saying “numerous studies in peer-reviewed literature support” its
answer. Phind (v8), meanwhile, appears to be better at providing peer-reviewed
references, though some references are blog posts. However, it appears to lift
the same content supplied to it as if regurgitating (in what may be considered
plagiarism) more than paraphrasing the content or generating original response
for engaging communication. Therefore, the quality of students’ ProCoT answers
(on the basis of grounding by references) is better than those by ChatGPT,
though ChatGPT usually gives more comprehensive answers that are typically
verbose but not necessarily always factual. They are also better than those by
Phind on the basis of originality, though Phind may also give more verbose
answers.

(a) Box and Whisker plot of ProCoT number of words

(b) Pareto plot of ProCoT number of words

Figure 1: Quantitative plots of ProCoT number of words in student answers for
case 1.

The minimum, median, maximum, and average word counts for the students’ ProCoT
answers (in case 1) are 31, 189, 594, and 208, respectively, as shown in the
Box plot of Figure 1 (a). Figure 1 (b) shows the Pareto plot for the words in
their answers, where most students used word counts in the range 31 and 201.
This implies most students prefer using fewer words to express their answers.
ChatGPT gave an average word count of 391 in its ProCoT answers while Phind
gave 383. These indicators of the average word counts by students and LLM s
may be useful as cues for possible cheating, besides LLM artefacts that may
result from copying an LLM output unedited. It is important to emphasize that
this is not necessarily the case.

Observations from the results and ongoing research into LLM s imply these
models will improve over time, especially as new data from methods like ProCoT
can be used to retrain these models to be better. Hence, new methods may be
required to evaluate students’ writing performance and learning in the future.
We believe this is not a reason for educators to fear or bar students from
using these tools but to see the opportunity to stimulate critical thinking
through the engagement of students with these tools. This will be possible
through evolving, active and innovative methods.

##  5 Conclusion

ProCoT shows that educators need not fear LLM s but embrace them as powerful
tools to enrich the experiences of students in learning. Weaknesses currently
exist in the most popular free LLM\- ChatGPT, and this makes it useful for
teaching students to be better writers by grounding their writing in peer-
reviewed references. We anticipate LLM s to be better grounded in their
knowledge in the future, hence new innovative methods will be required for
preventing students from cheating. Multimodal evaluation may be the solution
in the future or a method that is grounded in interview/oral assessment.

## Funding

Part-funding for the writing of this publication was provided by the Computer
Science, Electrical and Space Engineering Department’s (SRT) Strategic
Investmenet fund of Luleå University of Technology (LTU).

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