holomorphic structure $\partial^{\nabla}$ on $\overline{\Sigma}$. Due to [7]
this assumption does not hold for all irreducible flat ${\rm
SL}(n,{\mathbb{C}})$-connections. Under the assumption, we obtain a section
$s\,=\,s^{\nabla}$ as follows. If $\overline{\partial}^{\nabla}$ is stable

$\lambda\,\longmapsto\,(\lambda,\,\overline{\partial}^{\nabla},\,\lambda\partial^{\nabla})$

is an irreducible section over ${\mathbb{C}}\,\subset\,{\mathbb{C}}P^{1}$. If
$\overline{\partial}^{\nabla}$ is unstable, we consider its destabilizing
subbundle $L\,\subset\,V$ of positive degree. The connection induces a
nilpotent Higgs field $\Phi$ on the holomorphic vector bundle
$L\oplus(V/L)\,=\,L\oplus L^{*}$ via

$\Phi\,=\,\pi^{V/L}\circ\nabla_{\mid L}.$

This is a special case of [45] and can be interpreted from a gauge theoretic
point of view (see also [4, $\S~{}4$] for details): Consider a complementary
bundle $\widetilde{L}\,\subset\,V$ of $L$, and the family of gauge-
transformations

$g(\lambda)\,=\,\begin{pmatrix}1&0\\\ 0&\lambda\end{pmatrix}.$

The family

$\lambda\,\longmapsto\,(\lambda,\,\overline{\partial}^{\nabla.g(\lambda)},\,\partial^{\nabla.g(\lambda)})$

extends to an irreducible (stable) Higgs pair at $\lambda\,=\,0$ which
identifies with $(L\oplus L^{*},\,\Phi)$

##  4\. Energy functional on sections of the Deligne–Hitchin moduli space

###  4.1. The energy as a moment map

It was proven in [4, Corollary 3.11] that the energy of an irreducible section
$s$ with lift $\widehat{s}$ as in (3.23) is given by

$\mathcal{E}(s)\,=\,\tfrac{1}{2\pi\mathsf{i}}\int_{\Sigma}\mathrm{tr}(\Phi\wedge\Psi).$
(4.1)

In particular, this integral is independent of the lift $\widehat{s}$. The
reader should be aware of the different prefactors in (4.1) and in (3.11). In
particular, if we think of $\mathcal{E}$ as the energy of a harmonic map, it
should be real-valued, while we want a moment map for the $S^{1}$-action to be
$\mathsf{i}{\mathbb{R}}$-valued. Working with the prefactor
$\tfrac{1}{2\pi\mathsf{i}}$ also has the advantage that we get fewer factors
of $2\pi\mathsf{i}$ in the statements of the results below.

######  Remark 4.1.

As pointed out in [6, Remark 2.3], the energy in the present example is
defined for all local sections around $\lambda\,=\,0$ which admit a lift as in
(3.23).

Let us write again
$\mathcal{S}^{\prime}\,=\,\mathcal{S}_{\mathcal{M}_{\mathrm{DH}}}^{\prime}$
for the space of _irreducible_ sections whose normal bundle is isomorphic to
$\mathcal{O}_{{\mathbb{C}}P^{1}}(1)^{\oplus 2d}$. Take any
$s\,\in\,\mathcal{S}^{\prime}$. In terms of lifts of sections, a tangent
vector $V\in T_{s}\mathcal{S}^{\prime}$ is expressed as follows. Let
$\widehat{s}$ be a lift of $s$ as in (3.23), and denote the curvature of the
connection $\partial+\overline{\partial}$ by
$F^{\partial+\overline{\partial}}\,=\,\overline{\partial}\partial+\partial\overline{\partial}$.
Expanding the integrability condition

$\overline{\partial}(\lambda)D(\lambda)+D(\lambda)\overline{\partial}(\lambda)\,=\,0$
(4.2)

in powers of $\lambda$, the zeroth and first order coefficients yield

$\displaystyle\overline{\partial}\Phi$ $\displaystyle\,=\,0$ (4.3)
$\displaystyle F^{\partial+\overline{\partial}}+[\Phi\wedge\Psi]$
$\displaystyle\,=\,0\,.$ (4.4)

Consider a family of sections
$(s_{t}\,\in\,{\mathcal{S}}_{\mathcal{M}_{\mathrm{DH}}})_{t}$ with
$s\,=\,s_{0}$ which represents $V\,\in\,T_{s}\mathcal{S}^{\prime}$. The
corresponding (lifted) infinitesimal variation
$\dot{\widehat{s}}\,=\,(\dot{\overline{\partial}}(\lambda),\,\dot{D}(\lambda),\,\lambda)$
satisfies the linearisation of (4.2), i.e.,

$\overline{\partial}(\lambda)(\dot{D}(\lambda))+D(\lambda)(\dot{\overline{\partial}}(\lambda))\,=\,0.$
(4.5)

Expanding $\dot{\widehat{s}}$ into a power series

$\dot{\widehat{s}}(\lambda)\,=\,\left(\sum_{k=0}^{\infty}\psi_{k}\lambda^{k},\,\sum_{k=0}^{\infty}\varphi_{k}\lambda^{k},\,\lambda\right)\,,$
(4.6)

for
$\varphi_{k}\,\in\,\Omega^{1,0}(\mathfrak{sl}(E)),\,\psi_{k}\,\in\,\Omega^{0,1}(\mathfrak{sl}(E))$,
the linearisation of (4.3) becomes

$\displaystyle\overline{\partial}\varphi_{0}+[\psi_{0}\wedge\Phi]$
$\displaystyle\,=\,0$ (4.7)
$\displaystyle\overline{\partial}\varphi_{1}+\partial\psi_{0}+[\varphi_{0}\wedge\Psi]+[\Phi\wedge\psi_{1}]$
$\displaystyle\,=\,0\,.$ (4.8)

Variations along the gauge orbit of $\widehat{s}$ are determined by
infinitesimal gauge transformations
${\mathbb{C}}\,\ni\,\lambda\,\longmapsto\,\xi(\lambda)\,\in\,\Gamma(\Sigma,\,\operatorname{\mathfrak{sl}}(E))$
and are of the form

$(\overline{\partial}(\lambda)\xi(\lambda),\,D(\lambda)\xi(\lambda),\,\lambda).$
(4.9)

By expanding $\xi(\lambda)\,=\,\sum_{k=0}^{\infty}\xi_{k}\lambda^{k}$, we get
with (4.9) and (3.23)

$\displaystyle\overline{\partial}(\lambda)\xi(\lambda)$
$\displaystyle\,=\,\overline{\partial}\xi_{0}+(\overline{\partial}\xi_{1}+[\Psi,\,\xi_{0}])\lambda+O(\lambda^{2})$
(4.10) $\displaystyle D(\lambda)\xi(\lambda)$
$\displaystyle\,=\,[\Phi,\,\xi_{0}]+(\partial\xi_{0}+[\Phi,\xi_{1}])\lambda+O(\lambda^{2})\,.$
(4.11)

Now let $s\,\in\,\mathcal{S}^{\prime}$ with lift $\widehat{s}$ over
${\mathbb{C}}$, and consider $V_{j}\,\in\,T_{s}\mathcal{S}^{\prime}$,
$j\,=\,1,\,2$, represented by

$\dot{\widehat{s}}_{j}\,=\,(\dot{\overline{\partial}}_{j}(\lambda),\,\dot{D}_{j}(\lambda),\,\lambda)\,=\,(\psi^{(j)}_{0}+\psi^{(j)}_{1}\lambda,\,\varphi_{0}^{(j)}+\varphi_{1}^{(j)}\lambda,\lambda)+O(\lambda^{2}).$
(4.12)

Then we define, recalling the definition of $\omega_{\lambda}$ given in
(3.22),

$\displaystyle\widehat{\Omega}_{\widehat{s}}(V_{1},\,V_{2})$
$\displaystyle\,=\,-\tfrac{\mathsf{i}}{2}\tfrac{\partial}{\partial\lambda}_{|\lambda=0}\omega_{\lambda}(V_{1}(\lambda),\,V_{2}(\lambda))$
(4.13)
$\displaystyle=-\tfrac{\mathsf{i}}{2}\tfrac{\partial}{\partial\lambda}_{|\lambda=0}2\mathsf{i}\int_{\Sigma}\mathrm{tr}\left(-\dot{D}_{1}(\lambda)\wedge\dot{\overline{\partial}}_{2}(\lambda)+\dot{D}_{2}(\lambda)\wedge\dot{\overline{\partial}}_{1}(\lambda)\right)$
(4.14)
$\displaystyle=\,\int_{\Sigma}\mathrm{tr}\left(-\varphi_{0}^{(1)}\wedge\psi_{1}^{(2)}+\varphi_{0}^{(2)}\wedge\psi_{1}^{(1)}-\varphi_{1}^{(1)}\wedge\psi_{0}^{(2)}+\varphi_{1}^{(2)}\wedge\psi_{0}^{(1)}\right).$
(4.15)

We view $\widehat{\Omega}$ as a two-form on the infinite-dimensional space of
germs of sections of $\varpi$ at $\lambda\,=\,0$. Note that the formula for
$\widehat{\Omega}$ is exactly (1.32) in the present context.

######  Proposition 4.2.

The two-form $\widehat{\Omega}$ descends to a holomorphic two-form on the
space of irreducible sections, which on
$\mathcal{S}_{\mathcal{M}_{\mathrm{DH}}}^{\prime}$ coincides with the
holomorphic symplectic form $\Omega_{0}$ defined in (1.32).

###### Proof.

We will show that $\widehat{\Omega}_{\widehat{s}}$ is degenerate along the
gauge orbits. To this end, let $\widehat{s}$ be a germ of a section near
$\lambda\,=\,0$, and let
$\xi(\lambda)\,=\,\sum_{k=0}^{\infty}\xi_{k}\lambda^{k}$ be an infinitesimal
gauge transformation. The corresponding tangent vector $V_{1}$ is represented
by

$\dot{\widehat{s}}_{1}\,=\,(\overline{\partial}(\lambda)\xi(\lambda),\,D(\lambda)\xi(\lambda),\,\lambda).$

Then for an arbitrary tangent vector $V_{2}$ represented by
$\dot{\widehat{s}}_{2}\,=\,(\dot{\overline{\partial}}(\lambda),\,\dot{D}(\lambda),\,\lambda)$,
we find

$\displaystyle\widehat{\Omega}_{\widehat{s}}(V_{1},\,V_{2})$
$\displaystyle\,=\,\tfrac{\partial}{\partial\lambda}_{|\lambda=0}\int_{\Sigma}\mathrm{tr}\left(-\dot{D}(\lambda)\wedge\overline{\partial}(\lambda)\xi(\lambda)+D(\lambda)\xi(\lambda)\wedge\dot{\overline{\partial}}(\lambda)\right)$
(4.16) (Stokes)
$\displaystyle\,=\,\tfrac{\partial}{\partial\lambda}_{|\lambda=0}\int_{\Sigma}\mathrm{tr}\left(\overline{\partial}(\lambda)(\dot{D}(\lambda))+D(\lambda)(\dot{\overline{\partial}}(\lambda)))\xi(\lambda)\right)$
(4.17) $\displaystyle\,=\,0;$ (4.18)

we used (4.5). This shows that $\widehat{\Omega}$ descends to
$\mathcal{S}^{\prime}$. ∎

Theorem 2.3 thus allows us to make the following conclusion.

######  Corollary 4.3.

The restriction of
$2\pi\mathsf{i}\mathcal{E}\,:\,\mathcal{S}^{\prime}_{\mathcal{M}_{\mathrm{DH}}}\,\longrightarrow\,{\mathbb{C}}$
is a holomorphic moment map for the natural ${\mathbb{C}}^{*}$-action on
$\mathcal{S}^{\prime}_{\mathcal{M}_{\mathrm{DH}}}$ with respect to the
holomorphic symplectic form $\Omega_{0}$. In particular, the
${\mathbb{C}}^{*}$-orbits in
$\mathcal{S}^{\prime}_{\mathcal{M}_{\mathrm{DH}}}$ are exactly the critical
points of $\mathcal{E}|_{\mathcal{S}^{\prime}_{\mathcal{M}_{\mathrm{DH}}}}$.

###  4.2. Explicit description of some ${\mathbb{C}}^{*}$-fixed sections

Corollary 4.3 shows a close relationship between ${\mathbb{C}}^{*}$-orbits in
$\mathcal{S}_{\mathcal{M}_{\mathrm{DH}}}$ and the energy functional. We
therefore examine the ${\mathbb{C}}^{*}$-orbits more closely in this section.
Before explicitly determining the ${\mathbb{C}}^{*}$-fixed _irreducible_
sections, we first observe:

######  Lemma 4.4.

The set $\mathcal{S}_{\mathcal{M}_{\mathrm{DH}}}^{{\mathbb{C}}^{*}}$ of all
${\mathbb{C}}^{*}$-fixed sections is in a natural bijection with
$\mathcal{M}_{\mathrm{dR}}$, the moduli space of flat completely reducible
${\rm SL}(n,{\mathbb{C}})$-connections.

In particular, the critical points of
$\mathcal{E}\,\colon\,\mathcal{S}_{\mathcal{M}_{\mathrm{DH}}}^{\prime}\,\longrightarrow\,{\mathbb{C}}$
correspond to an open subset of $\mathcal{M}_{\mathrm{dR}}^{irr}$, the moduli
space of flat irreducible ${\rm SL}(n,{\mathbb{C}})$-connections.

###### Proof.

Let $\nabla\,\in\,\mathcal{M}_{\mathrm{dR}}$. As in Section 2.3, we obtain the
following ${\mathbb{C}}^{*}$-invariant section
$s_{\nabla}\,\colon\,{\mathbb{C}}^{*}\,\longrightarrow\,\mathcal{M}_{\mathrm{DH}}$:

$s_{\nabla}(\lambda)\,=\,[(\overline{\partial}^{\nabla},\,\lambda\partial^{\nabla},\,\lambda)],\quad\partial^{\nabla}\,=\,\nabla^{1,0},\quad\overline{\partial}^{\nabla}\,=\,\nabla^{0,1}.$
(4.19)

By a crucial result of Simpson ([44] for existence and [45] for a more
explicit approach), the limits of $s_{\nabla}(\lambda)$ for $\lambda\to 0$ and
$\lambda\infty$ _always_ exist in $\mathcal{M}_{\mathrm{Higgs}}(\Sigma,{\rm
SL}(n,{\mathbb{C}}))$ and $\mathcal{M}_{\mathrm{Higgs}}(\overline{\Sigma},{\rm
SL}(n,{\mathbb{C}}))$ respectively. The resulting section, also denoted by
$s_{\nabla}\,\in\,\mathcal{M}_{\mathrm{DH}}$, is ${\mathbb{C}}^{*}$-invariant
by continuity. Evaluation of sections
$s\,\colon\,{\mathbb{C}}P^{1}\,\longrightarrow\,\mathcal{M}_{\mathrm{DH}}$ at
$\lambda\,=\,1$ gives the inverse of the map
$\nabla\,\longmapsto\,s_{\nabla}$.

The last statement in the lemma is a direct consequence of Theorem 2.3 and
Corollary 4.3. ∎

We next determine explicitly the ${\mathbb{C}}^{*}$-fixed sections
$s\,\in\,\mathcal{S}_{\mathcal{M}_{\mathrm{DH}}}$ such that $s$ is irreducible
over ${\mathbb{C}}$, by using some results of [10]. In terms of Lemma 4.4,
these are precisely the sections $s_{\nabla}$ such that $s_{\nabla}(0)$ is
stable. Indeed, since irreducibility is an open condition,
$s_{\nabla}(\lambda)$ is an irreducible $\lambda$-connection for $\lambda$
close to $0$. Using the ${\mathbb{C}}^{*}$-invariance, we see that
$s(\lambda)$ is irreducible for every $\lambda\,\in\,{\mathbb{C}}$.

For any ${\mathbb{C}}^{*}$-fixed sections $s_{\nabla}$, its values at $0$ and
$\infty$ are ${\mathbb{C}}^{*}$-fixed Higgs bundles on $\Sigma$ and
$\overline{\Sigma}$ respectively. These are called complex variations of Hodge
structures (VHS). Let $(\overline{\partial},\,\Phi)$ be any VHS on $\Sigma$.
The fact that $(\overline{\partial},\,\Phi)$ is a ${\mathbb{C}}^{*}$-fixed
point yields a splitting

$E\,=\,\bigoplus_{j=1}^{l}E_{j}$ (4.20)

into a direct sum of holomorphic bundles. With respect to this splitting,
$\overline{\partial}$ and $\Phi$ are given in the following block form

$\overline{\partial}=\begin{pmatrix}\overline{\partial}_{E_{1}}&0&\dots&\dots&0\\\
0&\overline{\partial}_{E_{2}}&\ddots&&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\vdots\\\ \vdots&&\ddots&\ddots&0\\\
0&\dots&\dots&0&\overline{\partial}_{E_{l}}\end{pmatrix},\qquad\Phi=\begin{pmatrix}0&\dots&\dots&\dots&0\\\
\Phi^{(1)}&\ddots&&&\vdots\\\ 0&\Phi^{(2)}&\ddots&&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\vdots\\\ 0&\dots&0&\Phi^{(l-1)}&0\end{pmatrix}.$
(4.21)

where $\Phi^{(j)}\,\in\,H^{0}(\Sigma,\,{\rm Hom}(E_{j},E_{j+1})\otimes
K_{\Sigma})$. The sheaf $\mathfrak{sl}(E)$ of trace-free holomorphic
endomorphisms of $E$ further decomposes into

$\displaystyle\mathfrak{sl}(E)=\bigoplus_{k\in\mathbb{Z}}\mathfrak{sl}(E)_{k},\quad\mathfrak{sl}(E)_{k}=\\{\psi\in\mathfrak{sl}(E)~{}|~{}\psi(E_{i})\subset
E_{i-k}\\}.$ (4.22)

By construction,
$\Phi\,\in\,H^{0}(\Sigma,\,K_{\Sigma}\otimes\mathfrak{sl}(E)_{-1})$. To define
the next notion, let

$N_{+}\,=\,\bigoplus_{k>0}\mathfrak{sl}(E)_{k},\qquad
N_{-}\,=\,\bigoplus_{k<0}\mathfrak{sl}(E)_{k},\qquad\mathbb{L}\,=\,\mathfrak{sl}(E)_{0}.$
(4.23)

Note that $N_{+}$ (respectively, $N_{-}$) is the subspace of
$\mathfrak{sl}(E)$ consisting of endomorphisms of $E$ that are strictly upper
(respectively, lower) block-triangular with respect to the splitting (4.20),
while $\mathbb{L}$ is the space of block-diagonal elements of
$\mathfrak{sl}(E)$.

Now let $(\overline{\partial},\,\Phi)\,\in\,\mathcal{M}_{Higgs}({\rm
SL}(n,{\mathbb{C}}))$ be a _stable_ complex variation of Hodge structures.
Then the BB-slice ([10, Definition 3.7]) through
$(\overline{\partial},\,\Phi)$ is defined by

$\displaystyle\mathcal{B}^{+}_{(\overline{\partial},\Phi)}\,=\,\\{(\beta,\,\phi)\,\in\,\Omega^{0,1}(N_{+})$
$\displaystyle\oplus\Omega^{1,0}(\mathbb{L}\oplus
N_{+})~{}\mid~{}D^{\prime\prime}(\beta,\,\phi)+[\beta\wedge\phi]\,=\,0,\quad
D^{\prime}(\beta,\,\phi)\,=\,0\\}.$ (4.24)

Here we denote by

$D\,:=\,\overline{\partial}+\partial^{h}+\Phi+\Phi^{*_{h}}$ (4.25)

the non-abelian Hodge connection attached to $(\overline{\partial},\,\Phi)$
with harmonic metric $h$, and

$D^{\prime\prime}\,:=\,\overline{\partial}+\Phi,\qquad
D^{\prime}\,:=\,\partial^{h}+\Phi^{*_{h}}.$

Hence the equations in (4.24) are explicitly given by

$D^{\prime\prime}(\beta,\,\phi)+[\beta\wedge\phi]\,=\,\overline{\partial}\phi+[(\Phi+\phi)\wedge\beta]\,=\,0,\qquad
D^{\prime}(\beta,\,\phi)\,=\,\partial^{h}\beta+[\Phi^{*_{h}}\wedge\phi]\,=\,0.$
(4.26)

Note that $\mathcal{B}^{+}_{(\overline{\partial},\Phi)}$ is a finite-
dimensional affine space. Then, [10, Theorem 1.4 (3)] states that the map

$p\,\colon\,\mathcal{B}^{+}_{(\overline{\partial},\Phi)}\times{\mathbb{C}}\,\longrightarrow\,\mathcal{M}_{\mathrm{Hod}},\quad((\beta,\,\phi),\,\lambda)\,\longmapsto\,[\lambda,\,\overline{\partial}+\lambda\Phi^{*}+\beta,\,\lambda\partial^{h}+\Phi+\phi]$
(4.27)

is a holomorphic embedding onto the “attracting set”

$W(\overline{\partial},\,\Phi)\,=\,\\{m\,\in\,\mathcal{M}_{\mathrm{Hod}}^{irr}~{}\mid~{}\lim_{\zeta\to
0}\zeta\cdot m\,=\,(\overline{\partial},\,\Phi)\\}$

and is compatible with the obvious projections to ${\mathbb{C}}$. In
particular, if $W^{\lambda}(\overline{\partial},\,\Phi)$ denotes the
intersection of $W(\overline{\partial},\,\Phi)$ with the fiber
$\varpi^{-1}(\lambda)$, then $W^{\lambda}(\overline{\partial},\Phi)$ is
biholomorphic to the affine space
$\mathcal{B}^{+}_{(\overline{\partial},\Phi)}$ via the map
$p_{\lambda}\,:=\,p(\bullet,\,\lambda)$. Thus,
$\mathcal{M}_{\mathrm{Hod}}^{irr}$ is stratified by affine spaces.

Given $(\beta,\,\phi)\,\in\,\mathcal{B}^{+}_{(\overline{\partial},\,\Phi)}$,
we can use Lemma 4.4 and (4.19) to define the ${\mathbb{C}}^{*}$-fixed section

$s_{(\beta,\phi)}\,:=\,s_{p_{1}(\beta,\phi)}\,\in\,\mathcal{S}_{\mathcal{M}_{\mathrm{DH}}}.$
(4.28)

As observed earlier, $s_{(\beta,\phi)}$ is an irreducible section over
${\mathbb{C}}\,\subset\,{\mathbb{C}}P^{1}$ but not necessarily over all of
${\mathbb{C}}P^{1}$.

######  Proposition 4.5.

Over ${\mathbb{C}}$, the ${\mathbb{C}}^{*}$-fixed section $s_{(\beta,\phi)}$
may be expressed as

$s_{(\beta,\phi)}(\lambda)\,=\,\left[\lambda,\overline{\partial}+\lambda(\Phi^{*_{h}}+\beta_{1})+\sum_{j=2}^{l}\lambda^{j}\beta_{j},\Phi+\lambda\partial^{h}+\sum_{j=0}^{l}\lambda^{j+1}\phi_{j}\right],$
(4.29)

where $\beta\,=\,\sum_{j=1}^{l}\beta_{j}$, with
$\beta_{j}\,\in\,\Omega^{0,1}(\mathfrak{sl}(E)_{j})$ and
$\phi\,=\,\sum_{j=0}\phi_{j}$ with
$\phi_{j}\,\in\,\Omega^{1,0}(\mathfrak{sl}(E)_{j})$.

###### Proof.

Let $\nabla\,=\,p_{1}(\beta,\,\phi)\,=\,[D+\beta+\phi]$ so that

$\overline{\partial}^{\nabla}\,=\,\overline{\partial}+\Phi^{*_{h}}+\beta,\quad\partial^{\nabla}\,=\,\partial^{h}+\Phi+\phi.$

Hence $s_{(\beta,\phi)}\,=\,s_{\nabla}$ is given by

$s_{(\beta,\phi)}(\lambda)\,=\,[\lambda,\,\overline{\partial}+(\Phi^{*_{h}}+\beta),\,\lambda\partial^{h}+\lambda\Phi+\lambda\phi]$
(4.30)

for $\lambda\,\in\,{\mathbb{C}}^{*}$ (see (4.19)). This does _not_ give a lift
of $s_{(\beta,\phi)}$ over all of ${\mathbb{C}}$, unless the holomorphic
bundle $(E,\,\overline{\partial})$ is stable, in which case we must have
$\beta\,=\,0$ and $\Phi\,=\,0$.

To construct a lift over all of ${\mathbb{C}}$ we use the
${\mathbb{C}}^{*}$-family of gauge transformations

$g(\lambda)\,=\,\lambda^{m}\begin{pmatrix}\lambda^{1-l}\mathrm{id}_{E_{1}}&0&\dots&\dots&0\\\
0&\lambda^{2-l}\mathrm{id}_{E_{2}}&\ddots&&\vdots\\\
\vdots&0&\ddots&\ddots&\vdots\\\ \vdots&&\ddots&\ddots&0\\\
0&\dots&\dots&0&\lambda^{0}\mathrm{id}_{E_{l}}\end{pmatrix},$ (4.31)

where $m\,=\,\frac{1}{n}\sum_{j=1}^{l}(l-j)\mathrm{rk}(E_{j})$ in order to
ensure $\det g(\lambda)\,=\,1$. Then any $\xi\,\in\,\mathfrak{sl}(E)_{j}$
satisfies

$g(\lambda)^{-1}\xi g(\lambda)\,=\,\lambda^{j}\xi.$

Let $\beta\,=\,\sum_{j=1}^{l}\beta_{j}$, with
$\beta_{j}\,\in\,\Omega^{0,1}(\mathfrak{sl}(E)_{j})$, and similarly
$\phi\,=\,\sum_{j=0}\phi_{j}$ with
$\phi_{j}\,\in\,\Omega^{1,0}(\mathfrak{sl}(E)_{j})$. Then using
$\Phi\,\in\,H^{0}(K\otimes\mathfrak{sl}(E)_{-1})$ and
$\Phi^{*_{h}}\,\in\,\Omega^{0,1}(K\otimes\mathfrak{sl}(E)_{1})$, we get that

$(\overline{\partial}+(\Phi^{*_{h}}+\beta),\,\lambda\partial^{h}+\lambda\Phi+\lambda\phi).g(\lambda)\,=\,\left(\overline{\partial}+\lambda(\Phi^{*_{h}}+\beta_{1})+\sum_{j=2}^{l}\lambda^{j}\beta_{j},\,\Phi+\lambda\partial^{h}+\sum_{j=0}^{l}\lambda^{j+1}\phi_{j}\right).$

The result follows. ∎

We next discuss the implications for the ${\mathbb{C}}^{*}$-fixed leaves of
the foliation $\mathcal{F}^{+}$ on
$\mathcal{S}^{\prime}\,=\,\mathcal{S}_{\mathcal{M}_{\mathrm{DH}}}^{\prime}$.
Recall that these leaves consist, in particular, of _irreducible_ sections (on
all of ${\mathbb{C}}P^{1}$) by definition. We denote by
$\mathcal{S}_{(\overline{\partial},\Phi)}^{\prime}$ all sections in
$\mathcal{S}^{\prime}$ which pass through the stable complex variation of
Hodge structure
$(\overline{\partial},\,\Phi)\,\in\,\mathcal{M}_{\mathrm{Higgs}}^{{\mathbb{C}}^{*}}$
at $\lambda\,=\,0$.

######  Proposition 4.6.

The ${\mathbb{C}}^{*}$-fixed point locus
$(\mathcal{S}^{\prime}_{(\overline{\partial},\Phi)})^{{\mathbb{C}}^{*}}$ is
isomorphic to an open and non-empty subset of the affine space
$\mathcal{B}^{+}_{(\overline{\partial},\Phi)}$.

###### Proof.

Consider the section
$s_{(\beta,\phi)}\,\colon\,{\mathbb{C}}P^{1}\,\longrightarrow\,\mathcal{M}_{\mathrm{DH}}$
for $(\beta,\,\phi)\,\in\,\mathcal{B}^{+}_{(\overline{\partial},\Phi)}$ which
is irreducible over ${\mathbb{C}}$. Since the complement of
$\mathcal{M}_{\mathrm{Higgs}}^{irr}(\overline{\Sigma},\,{\rm
SL}(n,{\mathbb{C}}))$ in
$\mathcal{M}_{\mathrm{Higgs}}(\overline{\Sigma},\,{\rm SL}(n,{\mathbb{C}}))$
is closed and of codimension at least two (cf. [17]), it follows that
$s_{(\beta,\phi)}$ is an irreducible section for
$(\beta,\,\phi)\,\in\,\mathcal{B}^{+}_{(\overline{\partial},\Phi)}$ in an open
and dense subset of $\mathcal{B}^{+}_{(\overline{\partial},\Phi)}$.

Note that $(\beta,\,\phi)\,=\,(0,\,0)$ corresponds to the twistor line
$s_{(\overline{\partial},\Phi)}$ through $(\overline{\partial},\,\Phi)$, which
lies in $\mathcal{S}^{\prime}$. Since $\mathcal{S}^{\prime}$ is open and non-
empty in the space of all irreducible sections, we therefore see that the
irreducible and ${\mathbb{C}}^{*}$-fixed section $s_{(\beta,\phi)}$ has the
desired normal bundle for $(\beta,\,\phi)$ in an open and non-empty subset
$U\,\subset\,\mathcal{B}^{+}_{(\overline{\partial},\Phi)}$. Altogether we
obtain the isomorphism
$p_{1}^{-1}\circ\mathrm{ev}_{1}\,\colon\,(\mathcal{S}_{(\overline{\partial},\Phi)}^{\prime})^{{\mathbb{C}}^{*}}\,\overset{\cong}{\longrightarrow}\,U$.
∎

From Theorem 2.3, we immediately obtain:

######  Corollary 4.7.

The locus of critical points $s\,\in\,\mathcal{S}^{\prime}$ of
$\mathcal{E}\,\colon\,\mathcal{S}^{\prime}\,\longrightarrow\,{\mathbb{C}}$ is
isomorphic to an open and non-empty subset in
$\mathcal{M}_{\mathrm{dR}}^{irr}$. It is foliated by leaves which are
isomorphic to open and non-empty subsets of affine spaces.

###### Proof.

The first statement follows, by a genericity argument, from Lemma 4.4. The
second one is a consequence of Proposition 4.6. ∎

######  Remark 4.8.

Let
$s\,:\,{\mathbb{C}}P^{1}\,\longrightarrow\,\mathcal{\mathcal{M}_{\mathrm{DH}}}$
be a ${\mathbb{C}}^{*}$-fixed section such that
$s(0)\,=\,(\overline{\partial},\,\Phi)$ and $s(\infty)\,=\,(\partial,\,\Psi)$
are _stable_ VHS on $\Sigma$ and $\overline{\Sigma}$ respectively, with
respective splittings of the underlying smooth bundle $E$ of the form

$E\,=\,\bigoplus_{j=1}^{l}E_{j},\qquad
E\,=\,\bigoplus_{j=1}^{l^{\prime}}E^{\prime}_{j}.$

With respect to these splittings the respective holomorphic structures are
diagonal and the Higgs fields $\Phi$ and $\Psi$ are lower triangular as in
(4.21). Then we have the BB-slices
$\mathcal{B}^{+}_{(\overline{\partial},\Phi)}(\Sigma)$ and
$\mathcal{B}^{+}_{(\partial,\Psi)}(\overline{\Sigma})$. By Proposition 4.6 and
its analog on $\overline{\Sigma}$ we see that, on the one hand, $s$
corresponds to
$(\beta,\,\phi)\,\in\,\mathcal{B}^{+}_{(\overline{\partial},\Phi)}(\Sigma)$,
and on the other hand to
$(\widetilde{\beta},\,\widetilde{\phi})\,\in\,\mathcal{B}^{+}_{(\partial,\Psi)}(\overline{\Sigma})$.
Therefore, we obtain two distinguished lifts of $s$ over ${\mathbb{C}}$ and
${\mathbb{C}}^{*}\cup\\{\infty\\}$ of the form

$s(\lambda)\,=\,[\lambda,\,\widehat{s}_{(\beta,\phi)}(\lambda)]_{\Sigma}\,=\,\left[\lambda,\,\overline{\partial}+\lambda(\Phi^{*_{h}}+\beta_{1})+\sum_{j=2}^{l}\lambda^{j}\beta_{j},\,\Phi+\lambda\partial^{h}+\sum_{j=0}^{l}\lambda^{j+1}\phi_{j}\right]_{\Sigma},$
$s(\lambda)\,=\,[\lambda^{-1},\,\widehat{s}_{(\widetilde{\beta},\widetilde{\phi})}(\lambda^{-1})]_{\overline{\Sigma}}\,=\,\left[\lambda^{-1},\,\partial+\lambda^{-1}(\Psi^{*_{\widetilde{h}}}+\widetilde{\beta}_{1})+\sum_{j=2}^{l^{\prime}}\lambda^{-j}\widetilde{\beta}_{j},\,\Psi+\lambda^{-1}\overline{\partial}^{\widetilde{h}}+\sum_{j=0}^{l^{\prime}}\lambda^{-(j+1)}\widetilde{\phi}_{j}\right]_{\overline{\Sigma}}.$

Let $g_{0}$ be a gauge transformation such that

$(\partial+\overline{\partial}^{\widetilde{h}}+\Psi+\Psi^{*_{\widetilde{h}}}+\widetilde{\beta}+\widetilde{\phi}).g_{0}\,=\,\overline{\partial}+\partial^{h}+\beta+\phi.$

Going through the proof of Proposition 4.5 and writing $g(\lambda)$ and
$\widetilde{g}(\lambda^{-1})$ for the respective ${\mathbb{C}}^{*}$-families
of gauge transformations we get that

$\widehat{s}_{(\beta,\phi)}(\lambda)\,=\,\widehat{s}_{(\widetilde{\beta},\widetilde{\phi})}(\lambda^{-1}).\widetilde{g}(\lambda^{-1})^{-1}g_{0}g(\lambda)$

for any $\lambda\,\in\,{\mathbb{C}}^{*}$.

In general, starting only with the lift $\widehat{s}_{(\beta,\phi)}$ over
${\mathbb{C}}$ obtained above, it seems hard to determine explicitly the lift
$\widehat{s}_{(\widetilde{\beta},\widetilde{\phi})}(\lambda^{-1})$ over
${\mathbb{C}}P^{1}\setminus\\{0\\}$ or even the limiting VHS
$s_{(\beta,\phi)}(\infty)$. The next two examples discuss some situations in
which the limit can be computed.

######  Example 4.9.

Suppose the holomorphic structure $\partial^{h}+\Phi+\phi$ is stable on
$\overline{\Sigma}$. Then we can argue as follows. For
$\lambda\in{\mathbb{C}}^{*}$ we can write, using the Deligne gluing:

$s_{(\beta,\phi)}(\lambda)=[\lambda,\overline{\partial}+\Phi^{*_{h}}+\beta,\lambda(\partial^{h}+\Phi+\phi)]_{\Sigma}=[\lambda^{-1},\partial^{h}+\Phi+\phi,\lambda^{-1}(\overline{\partial}+\Phi^{*_{h}}+\beta)]_{\overline{\Sigma}}.$

Under our assumption that $\partial^{h}+\Phi+\phi$ is stable, this allows us
to conclude $s_{(\beta,\phi)}(\infty)=(\partial^{h}+\Phi+\phi,0)$. We will see
in the proof of Theorem 4.17 that this situation does in fact occur, at least
for rank $2$ bundles.

######  Example 4.10.

Consider the rank two case, $n=2$. If $s$ is the twistor line through a VHS
$(\overline{\partial},\Phi)$ on $\Sigma$, then we have $E=V\oplus V^{*}$,
where $V$ is a line bundle with $0<\deg V\leq g-1$ and $V^{*}=\ker\Phi$. Then
$s(\infty)=(\partial^{h},\Phi^{*_{h}})$ and the corresponding splitting is
$E=V^{*}\oplus V$. Note that, since $\overline{\Sigma}$ and $\Sigma$ come with
opposite orientations, we have $\deg V^{*}>0$, as a bundle on
$\overline{\Sigma}$. Then $\widetilde{g}(\lambda^{-1})=g(\lambda)$ in this
case, as the order is reversed. The associated lifts are thus just the lifts
of $s$ over ${\mathbb{C}}$ and ${\mathbb{C}}^{*}$ given by the harmonic
metric, i.e. the associated solution of the self-duality equations.

######  Example 4.11 (Grafting sections).

In [24] a special class of ${\mathbb{C}}^{*}$-invariant sections of
$\mathcal{M}_{\mathrm{DH}}(\Sigma,{\rm SL}(2,{\mathbb{C}}))$, called _grafting
sections_ , have been constructed by using grafting of projective structures
on $\Sigma$. We recover them from the previous proposition as follows.

Consider the ${\mathbb{C}}^{*}$-fixed stable Higgs bundle
$(\overline{\partial},\Phi)$ with

$E=K_{\Sigma}^{\frac{1}{2}}\oplus
K^{-\frac{1}{2}}_{\Sigma},\quad\Phi=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}$
(4.32)

where $K_{\Sigma}^{\frac{1}{2}}$ is a square root of the canonical bundle
$K_{\Sigma}$. To determine (4.23) in this example, we define
$E_{1}:=K_{\Sigma}^{\frac{1}{2}}$, $E_{2}:=K_{\Sigma}^{-\frac{1}{2}}$. Then we
see that

$N_{+}\cong K_{\Sigma},\quad N_{-}=K^{-1}_{\Sigma},\quad
L\cong\mathcal{O}_{\Sigma}$ (4.33)

By (4.26), $(0,\phi)\in\mathcal{B}_{(\overline{\partial},\Phi)}^{+}$ if and
only if $\overline{\partial}\phi=0$ and $[\phi\wedge\Phi^{*_{h}}]=0$. Hence
$\phi$ is of the form

$\phi=\begin{pmatrix}0&q\\\ 0&0\end{pmatrix},\quad q\in
H^{0}(\Sigma,K_{\Sigma}^{\otimes 2}),$ (4.34)

with respect to the splitting $E=E_{1}\oplus E_{2}$. For those $q$ such that
the monodromy of the corresponding flat connection at $\lambda=1$ is real, the
sections $s_{(0,\phi)}$ are precisely the grafting sections of [24, §2.1].
Since $\beta=0$ in this case, we see that the energy of a grafting section is
the same as the energy of the twistor line associated with the stable Higgs
pair $(\overline{\partial},\Phi)$. If the monodromy of the corresponding flat
connection is real, then [24] shows that the section $s_{(0,\phi)}$ is real
and defines an element of
$(\mathcal{S}^{\prime}_{\mathcal{M}_{\mathrm{DH}}})^{\tau}$, in particular it
has the correct normal bundle $\mathcal{O}_{{\mathbb{C}}P^{1}}(1)^{\oplus
2d}$. But the section $s_{(0,\phi)}$ is not admissible and thus cannot
correspond to a solution of the self-duality equations. This shows that
$\mathcal{M}_{\mathrm{SD}}(\Sigma,{\rm
SL}(2,{\mathbb{C}}))\subsetneq(\mathcal{S}_{\mathcal{M}_{\mathrm{DH}}}^{\prime})^{\tau}$.

###  4.3. The energy of a ${\mathbb{C}}^{*}$-fixed section

Proposition 4.5 gives concrete formulas for all ${\mathbb{C}}^{*}$-fixed
points $s\,\in\,\mathcal{S}_{\mathcal{M}_{\mathrm{DH}}}^{{\mathbb{C}}^{*}}$
such that $s(0)$ is a stable VHS. We next compute the energy of such sections.

######  Proposition 4.12.

Let $(\overline{\partial},\,\Phi)$ be a stable ${\mathbb{C}}^{*}$-fixed ${\rm
SL}(n,{\mathbb{C}})$-Higgs bundle, and let $s_{(\beta,\phi)}$ be the
${\mathbb{C}}^{*}$-fixed section corresponding to
$(\beta,\,\phi)\,\in\,\mathcal{B}^{+}_{(\overline{\partial},\Phi)}$. Its
energy is given by

$\mathcal{E}(s_{(\beta,\phi)})\,=\,\mathcal{E}(s_{0})\,=\,\sum_{k=2}^{l}(k-1)\deg(E_{k}),$

where $s_{0}$ is the twistor line through $(\overline{\partial},\,\Phi)$.

###### Proof.

Write $s_{(\beta,\phi)}$ in a form as in (4.29). Then the definition of
$\mathcal{E}$ immediately implies that

$\mathcal{E}(s_{(\beta,\phi)})\,=\,\mathcal{E}(s_{0})+\tfrac{1}{2\pi\mathsf{i}}\int_{\Sigma}\mathrm{tr}(\Phi\wedge\beta_{1}).$

Next we will show that $\int_{\Sigma}\mathrm{tr}(\Phi\wedge\beta_{1})\,=\,0$.
To this end, let us write

$\Phi\,=\,\sum_{k=1}^{l-1}\Phi^{(k)},\qquad\beta_{1}\,=\,\sum_{k=1}^{l-1}\beta^{(k)},$

where $\Phi^{(k)}\,\in\,\Omega^{1,0}(\mathrm{Hom}(E_{k},\,E_{k+1}),$
$\beta^{(k)}\,\in\,\Omega^{0,1}(\mathrm{Hom}(E_{k+1},\,E_{k})$; see the block
form in (4.21). It follows that

$\mathrm{tr}(\Phi\wedge\beta_{1})\,=\,\sum_{k=1}^{l}\mathrm{tr}_{E_{k}}(\Phi^{(k-1)}\wedge\beta^{(k-1)}).$

Note that each summand $\Phi^{(k-1)}\wedge\beta^{(k-1})$ belongs to
$\Omega^{1,1}(\mathrm{End}(E_{k}))$ and we have adopted the convention that
$\Phi^{(k)}\,=\,0\,=\,\beta^{(k)}$ if $k\,=\,0,\,l$.

Now, equation (4.26) implies that

$\overline{\partial}\phi_{0}+[\Phi\wedge\beta_{1}]\,=\,0$

and we can write

$[\Phi\wedge\beta_{1}]\,=\,\sum_{k=1}^{l-1}\Phi^{(k-1)}\wedge\beta^{(k-1)}+\beta^{(k)}\wedge\Phi^{(k)}.$
(4.35)

Thus, for each $k\,=\,1,\,\cdots,\,l$,

$\overline{\partial}\phi_{0}^{(k)}+\Phi^{(k-1)}\wedge\beta^{(k-1)}+\beta^{(k)}\wedge\Phi^{(k)}\,=\,0.$

Consider the case of $k\,=\,l$:

$\overline{\partial}\phi_{0}^{(l)}+\Phi^{(l-1)}\wedge\beta^{(l-1)}\,=\,0.$

Taking the trace of this equation and integrating over $\Sigma$, we find,
using Stokes’ theorem, that

$\int_{\Sigma}\mathrm{tr}_{E_{l}}(\Phi^{(l-1)}\wedge\beta^{(l-1)})\,=\,0.$

Now assume that
$\int_{\Sigma}\mathrm{tr}_{E_{k+1}}(\Phi^{(k)}\wedge\beta^{(k)})\,=\,0$ for
all $k\,\geq\,k_{0}$. Then we have

$\overline{\partial}\phi_{0}^{(k_{0})}+\Phi^{(k_{0}-1)}\wedge\beta^{(k_{0}-1)}+\beta^{(k_{0})}\wedge\Phi^{(k_{0})}\,=\,0.$

Taking the trace and integrating yields

$\displaystyle 0$
$\displaystyle=\int_{\Sigma}\mathrm{tr}_{E_{k_{0}}}(\Phi^{(k_{0}-1)}\wedge\beta^{(k_{0}-1)}+\beta^{(k_{0})}\wedge\Phi^{(k_{0})})$
(4.36)
$\displaystyle=\int_{\Sigma}\mathrm{tr}_{E_{k_{0}}}(\Phi^{(k_{0}-1)}\wedge\beta^{(k_{0}-1)}-\int_{\Sigma}\mathrm{tr}_{E_{k_{0}+1}}(\Phi^{(k_{0})}\wedge\beta^{(k_{0})})$
(4.37)
$\displaystyle=\int_{\Sigma}\mathrm{tr}_{E_{k_{0}}}(\Phi^{(k_{0}-1)}\wedge\beta^{(k_{0}-1)}).$
(4.38)

It follows inductively that
$\int_{\Sigma}\mathrm{tr}(\Phi\wedge\beta_{1})\,=\,0$.

It remains to compute the energy of the twistor line $s_{0}$. To this end, we
observe that

$\mathcal{E}(s_{0})\,=\,\tfrac{1}{2\pi\mathsf{i}}\int_{\Sigma}\mathrm{tr}(\Phi\wedge\Phi^{*_{h}})\,=\,\tfrac{1}{2\pi\mathsf{i}}\int_{\Sigma}\sum_{k=2}^{l}\mathrm{tr}_{E_{k}}(\Phi^{(k-1)}\wedge(\Phi^{k-1})^{*_{h}})\,=\,\sum_{k=2}^{l}\mathcal{E}_{k}(s_{0}),$

where we put
$\mathcal{E}_{k}(s_{0})\,=\,\tfrac{1}{2\pi\mathsf{i}}\int_{\Sigma}\mathrm{tr}_{E_{k}}(\Phi^{(k-1)}\wedge(\Phi^{k-1})^{*_{h}})$
for $k\,\geq\,2$. The equation $F^{\nabla^{h}}+[\Phi\wedge\Phi^{*_{h}}]\,=\,0$
is block-diagonal with respect to the splitting
$E\,=\,\bigoplus_{k=1}^{l}E_{k}$, with components

$F^{\nabla^{h}_{E_{k}}}+\Phi^{(k-1)}\wedge(\Phi^{(k-1)})^{*_{h}}+(\Phi^{(k)})^{*_{h}}\wedge\Phi^{(k)}=0.$

This gives the following recursive relations:

$\displaystyle\mathcal{E}_{k}(s_{0})$
$\displaystyle=\tfrac{1}{2\pi\mathsf{i}}\int_{\Sigma}\mathrm{tr}_{E_{k}}(\Phi^{(k-1)}\wedge(\Phi^{(k-1)})^{*_{h}})$
(4.39)
$\displaystyle=\tfrac{\mathsf{i}}{2\pi}\int_{\Sigma}\mathrm{tr}_{E_{k}}(F^{\nabla^{h}_{E_{k}}})+\tfrac{1}{2\pi\mathsf{i}}\int_{\Sigma}\mathrm{tr}_{E_{k+1}}(\Phi^{(k)}\wedge(\Phi^{(k)})^{*_{h}})$
(4.40) $\displaystyle=\deg(E_{k})+\mathcal{E}_{k+1}(s_{0}).$ (4.41)

Thus, if $k\,=\,l$, we find that

$\mathcal{E}_{l}(s_{0})\,=\,\deg(E_{l})\,,$

and for general $k$ we get that

$\mathcal{E}_{k}(s_{0})\,=\,\sum_{j=k}^{l-1}\deg(E_{j})+\mathcal{E}_{l}(s_{0})\,=\,\sum_{j=k}^{l}\deg(E_{j}).$

Therefore,

$\mathcal{E}(s_{0})\,=\,\sum_{k=2}^{l}\mathcal{E}_{k}(s_{0})=\sum_{k=2}^{l}\sum_{j=k}^{l}\deg(E_{j})\,=\,\sum_{k=2}^{l}(k-1)\deg(E_{k}),$

and this completes the proof. ∎

###  4.4. The second variation of the Energy at a ${\mathbb{C}}^{*}$-fixed
section

Next we study the second variation of the energy functional $\mathcal{E}$ at a
${\mathbb{C}}^{*}$–fixed point.

Examining the proof of Proposition 4.5, we can check explicitly that the
sections $s_{(\beta,\phi)}$ satisfy for any $\zeta\in{\mathbb{C}}^{*}$ the
relation

$\zeta.\widehat{s}_{(\beta,\phi)}=\widehat{s}_{(\beta,\phi)}.g(\zeta)^{-1}.$

Moreover, if we use the notation of equation (4.31) and put

$\xi=\begin{pmatrix}(m+1-l)\mathrm{id}_{E_{1}}&0&\dots&\dots&0\\\
0&(m+2-l)\mathrm{id}_{E_{2}}&\ddots&&\vdots\\\
\vdots&0&\ddots&\ddots&\vdots\\\ \vdots&&\ddots&\ddots&0\\\
0&\dots&\dots&0&m\mathrm{id}_{E_{l}}\end{pmatrix},$ (4.42)

then $[\xi,\cdot]$ acts as multiplication by $k$ on $\mathfrak{sl}(E)_{k}$ and
we see that

$-\mathsf{i}\lambda\frac{d}{d\lambda}\overline{\partial}(\lambda)=\overline{\partial}(\lambda)\xi(\lambda),\qquad\mathsf{i}D(\lambda)-\mathsf{i}\lambda\frac{d}{d\lambda}D(\lambda)=D(\lambda)\xi(\lambda)$
(4.43)

with

$(\overline{\partial}(\lambda),D(\lambda))=\left(\overline{\partial}+\lambda(\Phi^{*_{h}}+\beta_{1})+\sum_{j=2}^{l}\lambda^{j}\beta_{j},\Phi+\lambda\partial^{h}+\sum_{j=0}^{l}\lambda^{j+1}\phi_{j}\right).$

For $\xi(\lambda)=\sum_{k=0}^{\infty}\xi_{k}\lambda^{k}$ we deduce from (4.43)
the following equations

$\displaystyle 0$ $\displaystyle=\overline{\partial}\xi_{0}$
$\displaystyle\Phi$ $\displaystyle=[\Phi,\xi_{0}]$ (4.44) $\displaystyle-\Psi$
$\displaystyle=\overline{\partial}\xi_{1}+[\Psi,\xi_{0}]$ $\displaystyle 0$
$\displaystyle=[\Phi,\xi_{1}]+\partial\xi_{0}.$ (4.45)

We can now compute the second variation of $\mathcal{E}$ at such fixed points.

######  Proposition 4.13.

The second variation of $\mathcal{E}$ at a ${\mathbb{C}}^{*}$–fixed point $s$
with lift $\widehat{s}$ as in (3.23) is given by

$d^{2}\mathcal{E}(\dot{s})=\,\frac{1}{2\pi\mathsf{i}}\int_{\Sigma}\mathrm{tr}(\psi_{0}\wedge[\varphi_{1},\xi]+\varphi_{1}\wedge[\psi_{0},\xi]+\psi_{1}\wedge[\varphi_{0},\xi])+\varphi_{0}\wedge[\psi_{1},\xi]+2\varphi_{0}\wedge\psi_{1})\,.$

###### Proof.

Let $(s_{t})$ be a family of sections with $s_{0}=s$. We compute, using the
notation for $\widehat{s}$ and $\dot{s}$ as in Section 4

$2\pi\mathsf{i}\frac{d^{2}}{dt^{2}}|_{t=0}\mathcal{E}(s_{t})=\int_{\Sigma}\mathrm{tr}(\Phi\wedge\dot{\psi}_{1}+\dot{\varphi}_{0}\wedge\Psi+2\varphi_{0}\wedge\psi_{1}).$
(4.46)

Since $s_{0}\,=\,s$ is fixed by the action of ${\mathbb{C}}^{*}$, we can use
(4.44) (with $\xi=\xi_{0},\xi_{1}=0$) to write

$\displaystyle 2\pi\mathsf{i}\frac{d^{2}}{dt^{2}}|_{t=0}\mathcal{E}(s_{t})$
$\displaystyle=\int_{\Sigma}\mathrm{tr}([\Phi,\xi]\wedge\dot{\psi}-\dot{\phi}\wedge[\Psi,\xi]+2\phi\wedge\psi)$
(4.47)
$\displaystyle=\int_{\Sigma}\mathrm{tr}(-\xi([\Phi\wedge\dot{\psi}]+[\dot{\phi}\wedge\Psi])+2\phi\wedge\psi)$
(4.48)
$\displaystyle=\int_{\Sigma}\mathrm{tr}(\xi(\overline{\partial}\dot{\varphi}_{1}+\partial\dot{\psi}_{0}+2[\psi_{0}\wedge\varphi_{1}]+2[\varphi_{0}\wedge\psi_{1}])+2\varphi_{0}\wedge\psi_{1})$
(4.49) $\displaystyle({\rm using}~{}\overline{\partial}\xi=0=\partial\xi)$
$\displaystyle=\int_{\Sigma}\mathrm{tr}(\xi(2[\psi_{0}\wedge\varphi_{1}]+2[\varphi_{0}\wedge\psi_{1}])+2\varphi_{0}\wedge\psi_{1})$
(4.50)
$\displaystyle=\int_{\Sigma}\mathrm{tr}(\psi_{0}\wedge[\varphi_{1},\xi]+\varphi_{1}\wedge[\psi_{0}\wedge\xi]+\psi_{1}\wedge[\varphi_{0},\xi])+\varphi_{0}\wedge[\psi_{1}\wedge\xi]+2\varphi_{0}\wedge\psi_{1})$
(4.51)

In the third equation from above we made use of the second linearisation of
(4.3). ∎

Proposition 4.13 shows that the second variation is closely related to the
infinitesimal ${\mathbb{C}}^{*}$-action on the tangent space. The following
proposition is obtained.

######  Proposition 4.14.

Let
$\dot{s}(\lambda)\,=\,(\dot{\overline{\partial}}(\lambda),\,\dot{D}(\lambda),\lambda)\,=\,(\sum_{k=0}^{\infty}\psi_{k}\lambda^{k},\,\sum_{k=0}^{\infty}\varphi_{k}\lambda^{k},\,\lambda)$
be an infinitesimal deformation of the critical point $s\,\in\,\mathcal{S}$.
Suppose that $\dot{s}$ satisfies

$[\psi_{0},\xi]=n_{0}\psi_{0},\qquad[\psi_{1},\xi]=n_{1}\psi_{1},\qquad[\varphi_{0},\xi]=m_{0}\varphi_{0},\qquad[\varphi_{1},\xi]=m_{1}\varphi_{1}$

for some $m_{i},\,n_{i}\,\in\,\mathbb{Z}$. Then

$d^{2}\mathcal{E}(\dot{s})=\frac{1}{2\pi\mathsf{i}}\int_{\Sigma}\mathrm{tr}((m_{1}+n_{0})\psi_{0}\wedge\varphi_{1}+(m_{0}+n_{1}+2)\psi_{1}\wedge\varphi_{0})\,.$

######  Remark 4.15.

Note that this resembles the discussion surrounding Eq. (8.10) in [28]. In
fact, it does reproduce Hitchin’s result in the case that $s$ is the twistor
line corresponding to a ${\mathbb{C}}^{*}$-fixed point in
$\mathcal{\mathcal{M}_{\mathrm{Higgs}}}$ and the deformation $\dot{s}$ is
real, so that $\psi_{1}=\varphi_{0}^{*},\psi_{0}=-\varphi_{1}^{*}$.

###  4.5. Sections and the degree of the hyperholomorphic line bundle

Our previous results together with the energy can be used to show that the
space of irreducible sections is not connected. We begin with the following

######  Proposition 4.16.

Let $(\overline{\partial},\Phi)$ be a stable ${\mathbb{C}}^{*}$-fixed Higgs
bundle and let $s_{(\beta,\phi)}$ be a ${\mathbb{C}}^{*}$-fixed section
corresponding to
$(\beta,\phi)\in\mathcal{B}^{+}_{(\overline{\partial},\Phi)}$. If
$s_{\beta,\phi}(\infty)$ is given by a VHS on $\overline{\Sigma}$ with
underlying holomorphic bundle $E=\bigoplus_{k=1}^{l^{\prime}}E^{\prime}_{k}$,
then we have

$\deg(s_{(\beta,\phi)}^{*}L_{Z})\,=\,\sum_{k=1}^{l}(k-1)\deg(E_{k})+\sum_{k=1}^{l^{\prime}}(k-1)\deg(E_{k}^{\prime}).$

###### Proof.

Proposition 4.12 allows us to compute $\mathcal{E}_{0}(s)$ and
$\mathcal{E}_{\infty}(s)$. The assertion now follows from the formula

$\deg(s_{(\beta,\phi)}^{*}L_{Z})=\mathcal{E}(s_{(\beta,\phi)})+\mathcal{E}_{\infty}(s_{(\beta,\phi)})\,.$

∎

######  Theorem 4.17.

There exist irreducible sections $s$ of
$\varpi\,:\,\mathcal{M}_{\mathrm{DH}}(\Sigma,{\rm
SL}(2,{\mathbb{C}}))\,\longrightarrow\,{\mathbb{C}}P^{1}$ such that the
pullback $s^{*}L_{Z}$ of the holomorphic line bundle
$L_{Z}\,\longrightarrow\,\mathcal{M}_{\mathrm{DH}}(\Sigma,{\rm
SL}(2,{\mathbb{C}}))$ has non-zero degree. In particular, the space of
irreducible sections is not connected.

###### Proof.

Let $K_{\Sigma}^{\frac{1}{2}}$ be a square-root of the canonical line bundle
$K_{\Sigma}$. Consider the uniformization (Fuchsian) flat connection

$\nabla^{Fuchs}=\begin{pmatrix}\nabla^{K_{\Sigma}^{\frac{1}{2}}}&1^{*}\\\
1&\nabla^{K_{\Sigma}^{-\frac{1}{2}}}\end{pmatrix}$

on the rank two bundle $K_{\Sigma}^{\frac{1}{2}}\oplus
K_{\Sigma}^{-\frac{1}{2}}.$ For generic holomorphic quadratic differential
$q\,\in\,H^{0}(\Sigma,\,K_{\Sigma}^{2})$, the anti-holomorphic structure

$\begin{pmatrix}\partial_{K_{\Sigma}^{\frac{1}{2}}}&q\\\
1&\partial_{K_{\Sigma}^{-\frac{1}{2}}}\end{pmatrix}$

is stable (i.e., it defines a stable holomorphic bundle on
$\overline{\Sigma}$). Then,

$\nabla:=\nabla^{Fuchs}+\begin{pmatrix}0&q\\\ 0&0\end{pmatrix}$

gives a ${\mathbb{C}}^{*}$-invariant section
$s_{\nabla}\in\mathcal{S}_{\mathcal{M}_{\mathrm{DH}}}$ by the construction of
Lemma 4.4. In view of Proposition 4.12, the energy at $\lambda=0$ is given by

$\deg{K_{\Sigma}^{-\frac{1}{2}}}=1-g\neq 0.$

By assumption, $\partial^{\nabla}$ is stable, so the anti-Higgs field of $s$
at $\lambda=\infty$ vanishes, and the energy at $\lambda=\infty$ is given by
$\mathcal{E}_{\infty}=0.$ Finally, we have

$\deg(s^{*}L_{Z})=\mathcal{E}(s)+\mathcal{E}_{\infty}(s)\neq 0$

by the residue formula for the pull-back under $s$ of the meromorphic
connection to ${\mathbb{C}}P^{1}$ (see Section 3 of [4]). ∎

Given a(n irreducible) section $s\in\mathcal{S}_{\mathcal{M}_{\mathrm{DH}}}$,
it is in general very difficult to compute its normal bundle $N_{s}$. However,
by using the methods of [24], it can be shown that the
${\mathbb{C}}^{*}$-fixed points considered in the proof of Theorem 4.17 do not
have normal bundles of generic type, i.e., their normal bundles admit
holomorphic sections with double zeros.

## References

  * [1] D. V. Alekseevsky, V. Cortés and T. Mohaupt, Conification of Kähler and hyper-Kähler manifolds, Comm. Math. Phys. 324 no. 2, (2013), 637 – 655
  * [2] M. F. Atiyah and R. S. Ward, Instantons and algebraic geometry, Comm. Math. Phys. 55 no. 2, (1977), 117 – 124.
  * [3] R. J. Baston and M. G. Eastwood, The Penrose transform: its interaction with representation theory, Oxford Mathematical Monographs, Clarendon Press, Oxford (1989).
  * [4] F. Beck, S. Heller and M. Roeser, Energy of sections of the Deligne–Hitchin twistor space, Math. Ann. (2020), https://doi.org/10.1007/s00208-020-02042-0.
  * [5] I. Biswas and S. Heller, On the Automorphisms of a Rank One Deligne–Hitchin Moduli Space, SIGMA 13 (2017), https://doi.org/10.3842/SIGMA.2017.072.
  * [6] I. Biswas, S. Heller and M. Röser, Real holomorphic sections of the Deligne–Hitchin twistor space, Comm. Math. Phys. 366 (2019), 1099–1133.
  * [7] I. Biswas and S. Dumitrescu, and S. Heller, Irreducible flat $\mathrm{SL}(2,{\mathbb{R}})$-connections on the trivial holomorphic bundle, Jour. Math. Pures Appl. (to appear), arXiv:2003.06997.
  * [8] I. Biswas and N. Raghavendra, Line bundles over a moduli space of logarithmic connections on a Riemann surface, Geom. Funct. Anal. 15 (2005), 780–808.
  * [9] N. Buchdahl, On the relative de Rham sequence Proc. AMS 87, No. 2 (1983), 363–366
  * [10] B. Collier and R. Wentworth, Conformal limits and the Bialynicki-Birula stratification of the space of $\lambda$-connections, Adv. Math. 350 (2019), 1193–1225.
  * [11] K. Corlette, Flat $G$-bundles with canonical metrics, J. Diff. Geom. 28 (1988), 361 – 382
  * [12] S. K. Donaldson, Twisted Harmonic Maps and the self-duality equations, , Proc. London Math. Soc. (3) 55, no. 1, (1987), 127 – 131.
  * [13] J. Dorfmeister, F. Pedit and H. Wu, Weierstrass type representation of harmonic maps into symmetric spaces, Comm. Anal. Geom. 6 (1998), no. 4, 633–668.
  * [14] J.-M. Drézet and M. S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), 53–94.
  * [15] O. Dumitrescu, L. Fredrickson, G. Kydonakis, R. Mazzeo, M. Mulase and A. Neitzke, From the Hitchin section to opers through Nonabelian Hodge, Journal of Differential Geometry, 117, No. 2, (2021), 223 – 253
  * [16] O. Dumitrescu and M. Mulase, Interplay between opers, quantum curves, WKB analysis, and Higgs bundles, preprint arXiv:1702.00511v2 [math.AG]
  * [17] G. Faltings, Stable $G$-bundles and projective connections, Journal of Algebraic Geometry 2 (1993), No.3, 507–568
  * [18] B. Feix, Hyperkähler metrics on cotangent bundles, J. Reine Angew. Math. 532 (2001), 33–46.
  * [19] B. Feix, Hypercomplex manifolds and hyperholomorphic bundles, Math. Proc. Cam. Philos. Soc. 133, (2002), 443–457.
  * [20] B. Feix, Twistor spaces of hyperkähler manifolds with $S^{1}$-actions, Differential Geom. Appl. 19 (2003), 15–28.
  * [21] H. Grauert and R. Remmert, _Coherent analytic sheaves_ , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 265, Springer-Verlag, Berlin, 1984.
  * [22] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, (1977).
  * [23] A. Haydys, HyperKähler and quaternionic Kähler manifolds with $S^{1}$-symmetries, J. Geom. Phys. 58 (2008), 293–306.
  * [24] S. Heller, Real projective structures on Riemann surfaces and new hyper-kähler manifolds, arXiv:1906.10350, (2019)
  * [25] L. Heller and S. Heller, Higer solutions of Hitchin’s self-duality equations, Journal of Integrable Systems, Volume 5, Issue 1, (2020)
  * [26] N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55, no. 1, (1987) 59 –126.
  * [27] N. J. Hitchin, Harmonic maps from a 2-torus to the 3-sphere, J. Differential Geom. 31, no. 3, (1990), 627 – 710.
  * [28] N. J. Hitchin, Lie groups and Teichmüller space, Topology 31 (1992), 449 – 473
  * [29] N. J. Hitchin, On the hyperkähler/quaternion Kähler correspondence, Comm. Math. Phys. 324 (2013), 77 – 106.
  * [30] N. J. Hitchin, The hyperholomorphic line bundle, in Algebraic and Complex Geometry. In honour of Klaus Hulek’s 60th birthday, Springer Publishing (2014)
  * [31] N. J. Hitchin, A. Karlhede, U. Lindström and M. Roček, Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), 535–589.
  * [32] Z. Hu and P. Huang, Flat $\lambda$-Connections, Mochizuki Correspondence and Twistor Spaces, arXiv:1905.10765.
  * [33] P. Huang, Non-Abelian hodge theory and related topics, SIGMA 16 (2020), https://doi.org/10.3842/SIGMA.2020.029.
  * [34] S. A. Huggett and S. A. Merkulov, Twistor Transform of vector bundles, Math. Scand. Vol. 85, No. 2 (1999), pp. 219 – 244
  * [35] M. Jardim and M. Verbitsky, Trihyperkähler reduction and instanton bundles on ${\mathbb{C}}P^{3}$, Compositio Math. 150 (2014), 1836–1868.
  * [36] M. Jardim and M. Verbitsky, Moduli spaces of framed instanton bundles on ${\mathbb{C}}P^{3}$ and twistor sections of moduli spaces of instantons on ${\mathbb{C}}^{2}$ Adv. Math., 227 (2011), 1526–1538.
  * [37] C. LeBrun, Quaternionic-Kähler manifolds and conformal geometry. Math. Ann. 284, 353–376 (1989).
  * [38] M. Maruyama, Openness of a family of torsion free sheaves, Jour. Math. Kyoto Univ. 16 (1976), 627–637.
  * [39] M. Mayrand, Hyperkahler metrics near Lagrangian submanifolds and symplectic groupoids, arXiv:2011.09282.
  * [40] M. Namba, On maximal families of compact complex submanifolds of complex fiber spaces, Tohoku Math. J. (2) Volume 25, Number 2 (1973), 237–262.
  * [41] M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540–567.
  * [42] K. Pohlmeyer, Integrable Hamiltonian systems and interactions through quadratic constraints, Comm. Math. Phys. 46, no. 3, 1976.
  * [43] D. G. Quillen, Determinants of Cauchy–Riemann operators over a Riemann surface, Funct. Anal. Appl. 19 (1985), 31–34.
  * [44] C. Simpson, The Hodge filtration on nonabelian cohomology, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 217–281.
  * [45] C. Simpson, Iterated destabilizing modifications for vector bundles with connection, Contemporary Math. 522, 2010 (Proceedings of the Ramanan Conference, Madrid, 2008)., 2008.
  * [46] C. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95.
  * [47] K. Uhlenbeck, Harmonic maps into Lie groups (classical solutions of the chiral model),J. Diff. Geom., Vol. 30, pages 1–50, 1989.