# Ginzburg algebras and the Hitchin connection for parabolic $G$-bundles

Indranil Biswas  School of Mathematics, Tata Institute of Fundamental
Research, Mumbai-400005, India<EMAIL_ADDRESS>,  Swarnava
Mukhopadhyay  School of Mathematics, Tata Institute of Fundamental Research,
Mumbai-400005, India<EMAIL_ADDRESS>and  Richard Wentworth
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
<EMAIL_ADDRESS>
###### Abstract.

For a simple, simply connected, complex group $G$, we prove the existence of a
flat projective connection on the bundle of nonabelian theta functions on the
moduli space of semistable parabolic $G$-bundles over families of smooth
projective curves with marked points.

###### 2010 Mathematics Subject Classification:

Primary: 14H60, 32G34, 53D50; Secondary: 81T40, 14F08

I.B. was supported in part by a J.C. Bose fellowship and by DAE, India under
project no. 1303/3/2019/R&D/IIDAE/13820. S.M. was supported in part by was
partially supported by the DAE, India, under the same project no and also by
the Science and Engineering Research Board, India (SRG/2019/000513).R.W.
supported in part by NSF grant DMS-1906403.

###### Contents

  1. 1 Introduction
  2. 2 Quasi-Lie algebras and extensions of Atiyah algebras
    1. 2.1 Basic definitions
    2. 2.2 Principal bundles
  3. 3 Determinant of cohomology
    1. 3.1 Definitions
    2. 3.2 Relative Ginzburg complex
    3. 3.3 Associated bundles and pullback
  4. 4 Parabolic Atiyah algebras
    1. 4.1 Parabolic vector bundles
    2. 4.2 Parabolic Atiyah algebras and moduli of parabolic bundles
    3. 4.3 Parabolic $G$-bundles
  5. 5 Towards a parabolic Hitchin Symbol
    1. 5.1 Some canonical maps
    2. 5.2 Deformation of ${M}_{{G}}^{par,rs}$ via pointed curves
    3. 5.3 A fundamental commutative diagram
  6. 6 The parabolic Hitchin connection
    1. 6.1 Conditions for a flat connection
    2. 6.2 Definition of the symbol
    3. 6.3 Welters condition
  7. A Parabolic $G$-bundles
    1. A.1 Uniformization of quasi-parabolic bundles
    2. A.2 Parabolic bundles and associated constructions
    3. A.3 Line bundles on parabolic moduli spaces
  8. B $\Gamma$-equivariant $G$-bundles
    1. B.1 Uniformization of $(\Gamma,G)$-bundles of fixed local type
    2. B.2 Invariant direct image functor
    3. B.3 Determinant of cohomology for $\widehat{C}$ and invariant pushforward
  9. C The properness condition

##  1\. Introduction

In this paper, we prove the existence of a flat projective connection on
spaces of generalized theta functions on the moduli spaces of parabolic
$G$-bundles on a smooth projective curve with marked points. Before stating
the precise results, and since it is part of the larger and well-studied
program of geometric quantization, we first provide a brief historical context
to this subject.

Quantization as envisioned by Dirac, et al., can be thought of as a
deformation of a classical mechanical system depending on a parameter $\hbar$
that recovers the original classical system in the limit. Konstant-Kirillov-
Souriau developed and generalized this notion of “quantizing a function”, and
Auslander-Konstant [5] used it to construct unitary representations of a
connected Lie group (see also Kirillov [34]).

The starting point of the theory is a symplectic manifold $(M,\omega)$ where
$\omega$ is the curvature of a line bundle $\mathcal{L}$ with connection
$\nabla$. The quantum Hilbert space $\mathscr{H}$ is then the
$L^{2}$-completion of the space of global sections $\Gamma(M,\mathcal{L})$ of
this line bundle. The Lie algebra of functions on $M$ under the Poisson
bracket given by the form $\omega$ acts naturally on $\mathscr{H}$. This
process of assigning a function to this Lie algebra satisfying certain
commutativity constraint depending on $\hbar$ is known as quantization in the
present literature. However, it is not possible to achieve these commutativity
constraints in practice. To remedy this, Kostant [35] and Souriau [59] further
consider a compatible almost complex structure $I$ on $M$ such that
$(M,\omega,I)$ is a Kähler manifold. This induces a holomorphic structure on
the line bundle $\mathcal{L}$ and leads to the notion of geometric
quantization, where the Hilbert space $\mathscr{H}_{I}$ is reduced to the
space of holomorphic $L^{2}$-sections of $\mathcal{L}$. Because the
quantization process should arrive at a unique answer, it is natural to
investigate the dependence of the geometric quantization on the choice of
almost complex structure $I$ on $M$.

In [33], Hitchin analyzes this question in a very important setting (see also
[6, 26, 65, 1]). The Hilbert space in this case is canonically associated to
the choice of a simple, simply connected, complex Lie group $G$, a closed
orientable surface $\Sigma$, a conformal structure $I$ on $\Sigma$, and a
positive integer $k$ (known as the level). This is the vector space of
conformal blocks of the WZNW model at level $k$. It appears naturally in the
quantization of a $3d$-Chern-Simons theory to a $3d$-TQFT as considered by
Witten [68]. Here,

$M=\operatorname{Hom}^{irr}(\pi_{1}(\Sigma),K)/\operatorname{K}$

the moduli space of irreducible representations of the fundamental group
$\pi_{1}(\Sigma)$ to $K$, where $K\subset G$ is a maximal compact subgroup.
The group $K$ acts by conjugation on a representation $\rho:\pi_{1}(\Sigma)\to
K$, and we say $\rho$ is irreducible if the stabilizer under this action is
exactly the center of $K$. This space has a symplectic form defined by Atiyah-
Bott, Narasimhan, and Goldman [4, 45]. A choice of a complex structure $I$ on
$\Sigma$ endows $M$ with a Kähler structure, and via the Narasimhan-Seshadri-
Ramanathan theorem the complex manifold $M_{I}$ can be identified with the
space of regularly stable holomorphic principal $G$-bundles on $C:=(\Sigma,I)$
(see [52, Prop. 7.7 and Thm. 7.1]). The role of $\mathcal{L}$ is played by a
determinant of cohomology line bundle defined via some linear representation
of $G$, and $\mathscr{H}_{I}:=H^{0}(M_{I},\mathcal{L}^{\otimes k})$ is the
space of nonabelian theta functions of level $k$. The connection $\nabla$ is
the Chern connection of the Quillen metric. Hitchin found a flat projective
connection on the bundle of nonabelian theta functions over a family of curves
of fixed genus. His construction may be interpreted as a natural
identification between the spaces $\mathbb{P}(H^{0}(M_{I},\mathcal{L}^{\otimes
k})\cong\mathbb{P}(H^{0}(M_{I^{\prime}},\mathcal{L}^{\otimes k})$ via parallel
transport along a path connecting $I$ and $I^{\prime}$ in the Teichmüller
space.

As mentioned above, the vector spaces $\mathscr{H}_{I}$ that appear in
Hitchin’s geometric quantization also feature naturally in the WZNW-model of a
$2d$ rational conformal field theory constructed by Tsuchiya-Ueno-Yamada [64].
Let $\mathfrak{g}$ denote the Lie algebra of $G$. Given a positive integer $k$
and an $n$-tuple $\bm{\lambda}$ of dominant weights for $\mathfrak{g}$
satisfying a certain integrability condition depending on $k$, the paper [64]
constructs a vector bundle
$\mathbb{V}^{\dagger}_{\bm{\lambda}}(\mathfrak{g},k)$ on the Deligne-Mumford
compactification $\overline{\mathcal{M}}_{g,n}$ of stable $n$-pointed curves.
In the interior ${\mathcal{M}}_{g,n}$ parametrizing smooth curves,
$\mathbb{V}^{\dagger}_{\bm{\lambda}}(\mathfrak{g},k)$ admits a flat projective
connection. These vector bundles of conformal blocks satisfy the axioms of a
$2d$-rational conformal field theory. Moreover, due to work of Beauville-
Laszlo [12] and Kumar-Narasimhan-Ramanathan [38], in the case of a single
puncture with trivial weight, we get a canonical identification of
$\mathscr{H}_{I}$ with the fiber of
$\mathbb{V}^{\dagger}_{\bm{\lambda}}(\mathfrak{g},k)$ at the point
$C=(\Sigma,I)$ in $\mathcal{M}_{g,n}$. It is natural to ask whether the
connections of Hitchin [33] and Tsuchiya-Ueno-Yamada [64] coincide. That this
is indeed the case was proven by Laszlo [39].

A generalization of the identification of $\mathscr{H}_{I}$ with conformal
blocks also holds for smooth $C$ with an $n$-tuple of marked points $\bm{p}$.
Consider the moduli space
${M}_{G}^{par,rs}={M}_{G}^{par,rs}(C,{\bm{p}},{\bm{\lambda}})$ of regularly
stable parabolic $G$ bundles on a compact Riemann surface $C$ with $n$-marked
points $\bm{p}$ and parabolic structures $\bm{\lambda}$ at $\bm{p}$. Let
$\mathcal{L}_{\bm{\lambda},k}$ be a parabolic “determinant of cohomology” line
bundle on ${M}_{G}^{par,rs}$. Then there is a canonical isomorphism between
the finite dimensional vector space of holomorphic sections
$H^{0}({M}^{par,rs}_{G},\mathcal{L}_{\bm{\lambda},k})$ and the fiber of the
space of conformal blocks
$\mathbb{V}^{\dagger}_{\bm{\lambda}}(\mathfrak{g},k)\bigr{|}_{(C,\bm{p})}$
(see [50] and [40]). This identification between conformal blocks and
nonabelian theta functions is a mathematical analog of the Chern-Simons/WZNW
correspondence of Witten [68]. Since the vector bundle of conformal blocks is
endowed with a flat projective connection, it is very natural to ask the
following question:

Question. _Is there a flat projective connection on the family of spaces
$H^{0}({M}^{par,rs}_{G},\mathcal{L}_{\bm{\lambda},k})$ as the pointed Riemann
surface structure of $C$ varies?_

The motivation of the present paper is to give an affirmative answer to the
above question.

The curve $C$ and parabolic weights $\bm{\lambda}$ determine an orbifold curve
$\mathscr{C}$ (cf. Appendix C and Lemma C.3), with same stabilizers at each of
the marked points.

Main Theorem. _Let $\mathcal{C}\rightarrow S$ be a versal family of
$n$-pointed smooth projective curves of genus $g$. Furthermore, assume that
the genus of the orbifold curve $\mathscr{C}$ determined by the weights
$\bm{\lambda}$ is at least $2$. Let $G$ be a simple, simply connected complex
Lie group. Let $\pi:{M}^{par,rs}_{G}\rightarrow S$ be the relative moduli
space of regularly stable parabolic $G$ bundles on $\mathcal{C}$ for some
fixed parabolic weights $\bm{\lambda}$. Let $\mathcal{L}_{\phi}$ be the
determinant of cohomology line bundle on ${M}_{G}^{par,rs}$ determined by a
choice of representation $\phi:G\rightarrow\mathsf{SL}_{r}$. Then for any
$a\in\mathbb{Q}$, for which $\mathcal{L}_{\phi}^{\otimes a}$ defines a line
bundle on ${M}_{G}^{par,rs}$, the coherent sheaf
$\pi_{\ast}(\mathcal{L}_{\phi}^{\otimes a})$ has a flat projective connection.
_

Observe that for example, we can allow genus of $C$ to be zero or one in the
above theorem provided some inequalities are satisfied (cf. Example. C.2).

It is reasonable to expect that the TUY connection for conformal blocks and
the parabolic Hitchin connection constructed in the Main Theorem coincide
under the identification mentioned above. We postpone this question for a
future work.

Hitchin’s construction of a projective connection draws parallels with
Welters’ work on theta functions for abelian varieties [67]. The starting
point is the description of first order deformations of the triple
$(M_{I},\mathcal{L}^{\otimes k},s)$, where $s\in
H^{0}(M_{I},\mathcal{L}^{\otimes k})$, by an element of the first
hypercohomology group of the complex ${\rm At}(\mathcal{L}^{\otimes
k})\to\mathcal{L}^{\otimes k}$ defined by $s$. Here, ${\rm At}(\mathcal{L})$
denotes the Atiyah algebra of $\mathcal{L}$. Though Hitchin’s methods were
differential geometric in nature, in [65], van Geemen and de Jong
reinterpreted the construction in an algebraic manner closer to that of [67].
Using this framework, along with the fundamental results of Beilinson-
Schechtman [13] and Bloch-Esnault [20], Baier-Bolognesi-Martens-Pauly [7]
reproduced Hitchin’s connection for $G=\mathsf{SL}_{r}$. Moreover, their proof
works over fields of positive characteristic, with a few extra assumptions.
Other algebro-geometric constructions of the Hitchin connection have been
given in [26, 54, 51], and by Ginzburg in [27]. The approach in the present
paper is strongly motivated by [7] and [27].

The moduli space of $G$-bundles satisfies a “monotone” condition: the first
Chern class of the moduli space is a multiple of the first Chern class of the
prequantum line bundle. This property is an important technical point in
Hitchin’s construction of the connection (cf. [33, eqs. (2.8) and (3.9)]), and
it leads to a solution to the van Geemen-de Jong condition in Theorem 6.1 (i)
below. The main new feature in the case of parabolic bundles is the higher
rank of the Picard group of the moduli space, and because of this monotonicity
no longer holds, except in the special cases considered in [55]. A key
observation of this paper is that the fiducial symbol coming from the usual
construction can be inverted to obtain a modified symbol that now satisfies
the van Geemen-de Jong condition (see (70)). Another crucial ingredient is
Seshadri’s categorical equivalence of “$\pi$-bundles” on a ramified cover
$\widehat{C}\to C$ with parabolic bundles on $C$ ([57] and [16, 8, 58]). It
turns out that the line bundles on the moduli space of parabolic bundles
adapted to the parabolic weights correspond exactly to the restriction of the
determinants of cohomology to the locus of $\pi$-bundles (cf. [15, 21]).

In fact, one can work in the general setting of
$(\Gamma,\operatorname{Aut}(G))$-bundles. A moduli space of such pairs with a
fixed local type has been constructed by Balaji-Seshadri [13] (in the case of
($\Gamma,G$) bundles in characteristic zero) and by Heinloth [30] (in the more
general settings of Bruhat-Tits torsors in the sense of Pappas-Rapoport [49],
and over fields of arbitrary characteristic). We note that it has been not
verified whether the stability conditions of [9] and [30] coincide.
Nevertheless, the results in Section 5 generalize verbatim to moduli spaces of
$(\Gamma,\operatorname{Aut}(G))$-bundles. However, in order to produce a
Hitchin connection (as described in Section 6), the following additional
information would be required:

  * •

the base of the Hitchin map for the moduli of parahoric Higgs bundles for
$(\Gamma,\operatorname{Aut}(G))$ is an affine space, and the fibers of the
Hitchin map are connected;

  * •

the complement of the cotangent bundle of the moduli space of
$(\Gamma,\operatorname{Aut}(G))$-bundles in the parahoric Higgs bundles moduli
space has codimension at least 2.

There are some results in the direction of the first point by B. Wang [66],
who extends the result of Donagi-Pantev [22] to the set-up of parahoric
$(\Gamma,G)$-Higgs bundles. In full generality, however, the two items above
are not presently available in the literature, and we therefore restrict
ourselves here to the setting of parabolic bundles.

Regarding previous work on this problem: for parabolic vector bundles, a
Hitchin connection was constructed by Scheinhost-Schottenloher in [55] for the
special case of weights $\bm{\lambda}$ corresponding to powers of the
canonical bundle. In [19], Bjerre found a connection using the metaplectic
correction method from geometric quantization (see [2]). Ref. [61] uses the
results of [13] to extend Hitchin’s connection for logarithmic connections and
the moduli space of semistable torsion free sheaves on nodal curves.

The paper [7] cited above argues that it is of independent interest to
consider the Hitchin connection over field of positive characteristics from
the view point of the Grothendieck-Katz $p$-curvature conjecture and the
modular representations of the mapping class group. The constructions in this
paper follow those of [7], and will therefore also work over fields over
characteristic $p$, unless
$p\in\\{2,3,h^{\vee}(\mathfrak{g}),k,k+h^{\vee}\\}$. But even given these
constraints on $p$ it is not clear whether
$\pi_{*}\mathcal{L}_{\bm{\lambda},k}$ is locally free. For this, it would be
enough to show that $H^{1}(M^{par,ss}_{{G}},\mathcal{L}_{\bm{\lambda},k})$
vanishes. However, in the parabolic case the moduli spaces $M^{par,ss}_{{G}}$
are not Fano in general, even in characteristic zero. Moreover, there is no
analog of a Grauert-Riemenschneider type vanishing theorem in general. An
uniform approach to this vanishing result would follow if one can show that
$M^{par,ss}_{{G}}$ are Frobenius-split. There is some work in this direction
for ${G}=\mathsf{SL}_{2}$ by Mehta-Ramadas [42] and by Sun-Zhou [62], who show
that semistable parabolic bundles of rank $r$ and fixed determinant are
globally $F$-regular type. A general result on Frobenius splitting for moduli
of parabolic bundles is presently missing in the literature.

This paper is organized as follows: In Section 2, we describe Ginzburg’s
canonical construction of a differential graded Lie algebra from a quasi-Lie
algebra structure. The paradigmatic example is the Atiyah algebra of a
holomorphic bundle, where the dgla becomes what we call the _Ginzburg
complex_. We show in Section 2.2.4 that the Ginzburg complex is quasi-
isomorphic to the one introduced by Bloch-Esnault. Finally, we give an
explicit computation of the relative extension class of the exact sequence
obtained from the direct image of the Ginzburg complex and show that it is in
the class of the Atiyah-Bott symplectic form.

In Section 3, we introduce the determinant of cohomology line bundles on the
moduli space of $G$-bundles. These depend on choices of faithful linear
representations of $G$. We relate the Atiyah algebras of determinant line
bundles to the direct images of the Ginzburg and Bloch-Esnault complexes (cf.
Theorem 3.5).

Section 4 contains two parts, first for vector bundles and the second for
$G$-bundles. The goal in both parts is to define parabolic versions of the
Ginzburg and Bloch-Esnault complexes. The method is to view the parabolic
bundles as equivariant bundles on a ramified covering. This has the advantage
of embedding the parabolic weights into the structure of the equivariant push-
forward. We then relate the Atiyah algebras of determinant line bundles to the
direct images of these complexes (see Theorem 4.6 and Corollary 4.13).

In Section 5, we review the generalizations of Hitchin’s symbol and Kodaira-
Spencer maps in the parabolic bundle context. The important result here is
Theorem 5.3, which relates the fiducial Hitchin symbol to the relative
extension classes of the Atiyah algebras of the $G$-bundle and the determinant
of cohomology.

Finally, in Section 6 we prove that the modified Hitchin symbol satisfies the
constraint equations of van Geemen-de Jong. This leads to the proof of the
Main Theorem. The Appendix contains some definitions and technical results on
parabolic bundles, invariant push-forwards, and vanishing theorems, that are
used at various points in the paper. In particular, the determinant of
cohomology line bundles $\mathcal{L}_{\phi}$ associated to a linear
representation $\phi$ of $G$ are defined there. _Parabolic_ determinant of
cohomology line bundles are defined in A.10 and A.13. We also explain the
admissible values of $k$, how to realize the parabolic determinant of
cohomology bundles via the moduli space of $(\Gamma,G)$-bundles, and the
invariant push-forward functor construction.

_Acknowledgements_. The authors warmly thank Johan Martens for enlightening
discussions. R.W. also benefited from conversations with Florian Beck and
Gerard Freixas i Montplet.

##  2\. Quasi-Lie algebras and extensions of Atiyah algebras

###  2.1. Basic definitions

In this section, we recall a correspondence stated in Ginzburg [27] between
quasi-Lie algebras and their associated differential graded Lie algebras. We
also recall from Beilinson-Schechtman [13] a natural classes of Atiyah
algebras associated to a family of curves.

####  2.1.1. Quasi-Lie algebras

First we recall the definition of a quasi-Lie algebra. Let
$\widetilde{\mathfrak{g}}$ be a vector space equipped with a skew-symmetric
bilinear map $[\ ,\
]:\widetilde{\mathfrak{g}}\times\widetilde{\mathfrak{g}}\rightarrow\widetilde{\mathfrak{g}}$.
Let $Z\subset\widetilde{\mathfrak{g}}$ be a linear subspace.

######  Definition 2.1.

A triple $(\widetilde{\mathfrak{g}},Z,[\ ,\ ])$ as above is called a quasi-Lie
algebra if the following holds:

  1. (i)

the subspace $Z$ is central with respect to $[\ ,\ ]$;

  2. (ii)

the bracket $[\ ,\ ]$ descends to give a Lie algebra structure on
$\widetilde{\mathfrak{g}}/Z$.

We will see that in the setting of Atiyah algebras, these quasi-Lie algebras
arise naturally. Now recall the notion of a differential graded Lie algebra
(dgla).

######  Definition 2.2.

A _differential graded Lie algebra_ (dgla) is a vector space
$\mathfrak{S}:=\oplus_{i}\mathfrak{S}^{i}$ together with a bilinear map
$\\{\mathfrak{S}^{i},\mathfrak{S}^{j}\\}\subset\mathfrak{S}^{i+j}$ and a
differential $d:\mathfrak{S}^{i}\rightarrow\mathfrak{S}^{i+1}$ satisfying the
following:

  * •

$\\{x,y\\}=(-1)^{|x||y|+1}\\{y,x\\}$, where $|z|=i$ for
$z\in\mathfrak{S}^{i}$,

  * •

$(-1)^{|x||z|}\\{x,\\{y,z\\}\\}+(-1)^{|y||x|}\\{y,\\{z,x\\}\\}+(-1)^{|y||z|}\\{z,\\{y,x\\}\\}=0$,

  * •

$d\\{x,y\\}=\\{dx,y\\}+(-1)^{|x|}\\{x,dy\\}$.

A morphism of dglas is a graded linear map
$\mathfrak{S}\rightarrow\mathfrak{S}^{\prime}$ that preserves the Lie bracket
and commutes with the differentials.

The following lemma of Ginzburg [27] gives a correspondence between quasi-Lie
algebras and a certain class of dglas.

######  Lemma 2.3.

Let $(\widetilde{\mathfrak{g}},Z,[\ ,\ ])$ be a quasi-Lie algebra equipped
with a symmetric $Z$-valued bilinear form $\langle\
,\rangle:\operatorname{Sym}^{2}\widetilde{\mathfrak{g}}\rightarrow Z$ such
that the following hold:

  * •

$\langle[x,y],z\rangle=0,$

  * •

$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=d(\langle[x,y],z\rangle)$.

Then there exists a dgla
$\mathfrak{S}=\mathfrak{S}^{-2}\oplus\mathfrak{S}^{-1}\oplus\mathfrak{S}^{0}$,
where $\mathfrak{S}^{-1}:=\widetilde{\mathfrak{g}}$, $\mathfrak{S}^{-2}:=Z$,
and with the bracket given by the formula

$\\{x,y\\}=\langle x,y\rangle;\ \mbox{and}\ \\{x,dy\\}=[x,y]+\langle
x,y\rangle\ \mbox{for $x,y\in\widetilde{\mathfrak{g}}$.}$

Conversely given a dgla as above there exists a quasi-Lie algebra along with a
symmetric $Z$-valued bilinear form.

####  2.1.2. Dglas associated to families of curves

Let $\pi:X\rightarrow T$ be a smooth morphism of relative dimension one
varieties parametrized by $T$; these varieties are not assumed to be proper.
The relative tangent bundle $\mathcal{T}_{X/T}$ fits in the exact sequence of
$\mathcal{O}_{X}$-modules

$0\longrightarrow\mathcal{T}_{X/T}\longrightarrow\mathcal{T}_{X}\longrightarrow\pi^{*}\mathcal{T}_{T}\longrightarrow
0.$

Let $\mathcal{T}_{X,\pi}$ denote the subsheaf
$d\pi^{-1}(\pi^{-1}\mathcal{T}_{T})$. Clearly $\mathcal{T}_{X,\pi}$ is a Lie
algebra with Lie bracket coming from that on $\mathcal{T}_{X}$ and there is an
exact sequence of Lie algebras

$0\longrightarrow\mathcal{T}_{X/T}\longrightarrow\mathcal{T}_{X,\pi}\stackrel{{\scriptstyle
d\pi}}{{\longrightarrow}}\pi^{-1}\mathcal{T}_{T}\longrightarrow 0.$

Consider the dgla given by
$\mathcal{T}_{\pi}^{\bullet}=\oplus_{i}\mathcal{T}^{i},$ where
$\mathcal{T}^{i}$ is zero for $i\neq\\{0,-1\\}$,
$\mathcal{T}^{-1}:=\mathcal{T}_{X/T}$ and
$\mathcal{T}^{0}:=\mathcal{T}_{X,\pi}$. The dgla $\mathcal{T}_{\pi}^{\bullet}$
carries a natural action of $\pi^{-1}\mathcal{O}_{T}$ and a map

(1) $\overline{\epsilon}:\mathcal{T}_{\pi}^{\bullet}\rightarrow
H^{0}(\mathcal{T}_{\pi}^{\bullet})=\pi^{-1}(\mathcal{T}_{T})$

given by $d\pi$.

The relative de Rham complex
$\Omega^{\bullet}_{X/T}=(\mathcal{O}_{X}\rightarrow\omega_{X/T})$ with
$V^{0}:=\mathcal{O}_{X}$ and $V^{1}:=\Omega_{X/T}$ is naturally a $dg$-module
$V^{\bullet}:=V^{0}\oplus V^{1}$ for $\mathcal{T}^{\bullet}_{\pi}$ which is
compatible with the $\pi^{-1}\mathcal{O}_{T}$ action on both sides.

####  2.1.3. Atiyah algebras as $R^{0}\pi_{*}$ of dglas

Let $\pi:X\rightarrow T$ be a family of curves as before. We discuss the
notion of $\pi$-algebras which are quasi-isomorphic to extensions of a complex
of Atiyah algebras by the de Rham complex $\Omega^{\bullet}_{X/T}$.

######  Definition 2.4.

An $\mathcal{O}_{S}$-Lie algebra $\mathcal{A}^{\bullet}$ on $X$ is a dgla
together with a $\pi^{-1}\mathcal{O}_{S}$-module structure and a morphism
$\epsilon_{\mathcal{A}}:\mathcal{A}^{\bullet}\rightarrow\mathcal{T}^{\bullet}_{\pi}$
that satisfies

$[a,fb]=\overline{\epsilon}_{\mathcal{A}}(a)(f)b+f[a,b],$

where
$\overline{\epsilon}_{\mathcal{A}}=\overline{\epsilon}\circ\epsilon_{\mathcal{A}}$
(see (1)). A $\pi$-algebra $\mathcal{A}^{\bullet}$ is an $\mathcal{O}_{S}$-Lie
algebra together with a three term filtration

$0=\mathcal{A}^{\bullet}_{-3}\subset\mathcal{A}^{\bullet}_{-2}\subset\mathcal{A}^{\bullet}_{-1}\subset\mathcal{A}^{\bullet}_{0}=\mathcal{A}^{\bullet}$

such that the following hold:

  1. (i)

$[\mathcal{A}^{\bullet}_{i},\mathcal{A}^{\bullet}_{j}]\subset\mathcal{A}^{\bullet}_{i+j}$,
$\mathcal{O}_{S}\cdot\mathcal{A}^{\bullet}_{i}\subset\mathcal{A}^{\bullet}_{i}$.

  2. (ii)

$\mathcal{A}^{\bullet}_{-2}\simeq\Omega_{X/S}^{\bullet}[2]$ as
$\mathcal{O}_{S}$-modules.

  3. (iii)

$\mathcal{A}^{\bullet}_{-1}/\mathcal{A}^{\bullet}_{-2}$ is acyclic.

  4. (iv)

$\epsilon_{\mathcal{A}}:\mathcal{A}^{\bullet}\rightarrow\mathcal{A}^{\bullet}/\mathcal{A}^{\bullet}_{-1}\simeq\mathcal{T}^{\bullet}_{\pi}$.

  5. (v)

The ad action of $\mathcal{A}^{\bullet}/\mathcal{A}^{\bullet}_{-1}$ on
$\mathcal{A}^{\bullet}_{-2}$ coincides with the
$\mathcal{T}^{\bullet}_{\pi}$-action on $\Omega^{\bullet}_{X/T}[2]$.

By the above definition of a $\pi$-algebra, we get an exact sequence

(2)
$0\longrightarrow\mathcal{A}^{\bullet}_{-2}\longrightarrow\mathcal{A}^{\bullet}\longrightarrow\mathcal{A}^{\bullet}/\mathcal{A}^{\bullet}_{-2}\longrightarrow
0.$

This will be called a $\Omega^{\bullet}_{X/S}$-extension of
$\mathcal{A}^{\bullet}/\mathcal{A}^{\bullet}_{-2}$. Assume that the map $\pi$
is proper. Suppose we are given any $\pi$-algebra $\mathcal{A}^{\bullet}$
fitting in an exact sequence of complexes

(3)
$0\longrightarrow\Omega_{X/T}^{\bullet}[2]\longrightarrow\mathcal{A}^{\bullet}\longrightarrow\mathcal{A}^{\bullet}/\mathcal{A}^{\bullet}_{-2}\longrightarrow
0.$

######  Proposition 2.5.

Taking $R^{0}\pi_{*}$ of (3), the following short exact sequence is obtained

$0\longrightarrow\mathcal{O}_{T}\longrightarrow
R^{0}\pi_{*}\mathcal{A}^{\bullet}\longrightarrow\mathcal{T}_{T}\longrightarrow
0.$

This defines an Atiyah algebra on $T$.

####  2.1.4. $\pi$-algebras associated to Atiyah algebras

Let $\pi:X\rightarrow T$ be a family of curve that are not necessarily
projective, and let $\mathcal{A}$ be an $R$-Atiyah algebra on $X$. There is a
natural $\pi$-algebra associated to $\mathcal{A}$. Consider the
$\mathcal{O}_{T}$-Lie algebra $\mathcal{A}^{\bullet}_{\pi}$ defined by the
following:

$\mathcal{A}_{\pi}^{-1}:=\epsilon_{\mathcal{A}}^{-1}\mathcal{T}_{X/T},\
\mbox{and}\
\mathcal{A}^{0}_{\pi}:=\epsilon_{\mathcal{A}}^{-1}\mathcal{T}_{X,\pi}.$

There is a canonical surjective map
$\epsilon_{\mathcal{A}}:\mathcal{A}_{\pi}^{\bullet}\rightarrow\mathcal{T}_{\pi}^{\bullet}$
whose kernel is $\operatorname{Cone}\operatorname{id}_{R}.$

######  Definition 2.6.

A $\Omega$-extension ${}^{\\#}\mathcal{A}^{\bullet}$ of $(\mathcal{A},R)$ is a
$\Omega$ extension of $\mathcal{A}_{\pi}^{\bullet}$ together with an
$\mathcal{O}_{X}$-module structure on ${}^{\\#}\mathcal{A}^{-1}$ such that

  * •

the $\mathcal{O}_{X}$-action is compatible with the action on
$\mathcal{A}_{\pi}^{-1}$, and

  * •

the component $[\
,]_{-1,-1}:{}^{\\#}\mathcal{A}^{-1}\otimes{}^{\\#}\mathcal{A}^{-1}\rightarrow{}^{\\#}\mathcal{A}^{-2}=\mathcal{O}_{X}$
is a differential operator along the fibers.

There is a commutative diagram

(4)
${0}$${0}$${\mathcal{O}_{X}}$${{}^{\\#}\mathcal{A}^{-2}}$${0}$${0}$${\Omega_{X/T}^{1}}$${{}^{\\#}\mathcal{A}^{-1}}$${\mathcal{A}_{\pi}^{-1}:=\mathcal{A}_{X/T}}$${0}$${0}$${{}^{\\#}\mathcal{A}^{0}}$${\mathcal{A}_{\pi}^{0}:=\mathcal{A}_{X,\pi}}$${0.}$

Observe that the $\Omega_{X/T}$-extension is, by definition, a $\pi$-algebra,
where the filtration ${}^{\\#}\mathcal{A}^{\bullet}_{-2}$ is given by
$\Omega_{X/T}^{\bullet}[2]$.

###  2.2. Principal bundles

####  2.2.1. The Ginzburg complex

We continue with the earlier notation. Let $G$ be a complex simple Lie group
with Lie algebra $\mathfrak{g}$. We will denote by $\kappa_{\mathfrak{g}}$ the
normalized Cartan-Killing form on $\mathfrak{g}$; let

(5) $\nu_{\mathfrak{g}}:\mathfrak{g}\xrightarrow{\simeq}\mathfrak{g}^{\vee}$

be the isomorphism given by it.

Let $\Pi:P\to X$ be a holomorphic principal $G$ bundle, where we assume $G$
acts on the right of $P$. Automorphisms $\operatorname{Aut}(P)|_{U}$ of $P$
over $U\subset X$ are by definition $G$-equivariant automorphisms of
$\Pi^{-1}(U)$, i.e. $F:P|_{U}\to P|_{U}$ satisfying $F(pg)=F(p)g$ for all
$g\in G$. The group of automorphisms $\operatorname{Aut}(P)|_{U}$ is generated
by the invariant vector fields:

$\operatorname{aut}_{U}(P):=\left\\{X\in\Gamma(\Pi^{-1}(U),TP)\mid(R_{g})_{\ast}X=X\
,\ \forall\ g\in G\right\\}.$

Then $\operatorname{aut}(P)$ defines a coherent sheaf of
$\mathcal{O}_{X}$-modules. We refer to the subsheaf ${\rm At}_{X/T}(P)$ of
vector fields that project by $\Pi_{\ast}$ to $\mathcal{T}_{X/T}$ as the
_relative Atiyah algebra of $P$_. We have an exact sequence

(6) $0\longrightarrow\operatorname{ad}(P)\longrightarrow{\rm
At}_{X/T}(P)\longrightarrow\mathcal{T}_{X/T}\longrightarrow 0.$

We explain the inclusion on the left. Recall that a section of
$\operatorname{ad}(P)$ is identified with a function $f:P\to\mathfrak{g}$
satisfying $f(pg)=\operatorname{Ad}_{g^{-1}}f(p)$. For $X\in\mathfrak{g}$, let
$X^{\sharp}$ denote the fundamental vector field on $P$ generated by $X$. Then
$X^{\sharp}(pg)=(R_{g})_{\ast}(\operatorname{Ad}_{g}X)^{\sharp}(p)$. The
inclusion $\operatorname{ad}(P)\to\operatorname{aut}(P)$ is $f\mapsto X$,
$X(p)=f(p)^{\sharp}$. With this definition,

$X(pg)=(\operatorname{Ad}_{g^{-1}}f(p))^{\sharp}(pg)=(R_{g})_{\ast}(f(p))^{\sharp}=X(p)$

so $X$ is invariant and lies in $\operatorname{aut}(P)$ as the kernel of
$\Pi_{\ast}$.

The following will be important when we discuss universal bundles:

######  Remark 2.7.

Let $Z(G)$ denote the (finite) center of $G$, and let $\overline{P}:=P/Z(G)$
the associated principal bundle for the adjoint group $\overline{G}:=G/Z(G)$.
Then there are canonical isomorphisms
$\operatorname{ad}(\overline{P})\simeq\operatorname{ad}(P)$ and ${\rm
At}_{X/T}(\overline{P})\simeq{\rm At}_{X/T}(P)$.

Dualizing (6) gives a quasi-Lie algebra structure on ${\rm
At}_{X/T}(P)^{\vee}$:

(7) $0\longrightarrow\Omega_{X/T}\longrightarrow{\rm
At}_{X/T}(P)^{\vee}\stackrel{{\scriptstyle
j}}{{\longrightarrow}}\operatorname{ad}(P)^{\vee}\longrightarrow 0.$

Identify $\operatorname{ad}(P)$ with $\operatorname{ad}(P)^{\vee}$ using
$\nu_{\mathfrak{g}}$ in (5), and denote ${\rm At}_{X/T}(P)^{\vee}$ by
$\widetilde{\mathfrak{g}}_{P}$. We have the following quasi-Lie algebra

(8) ${0}$${\Omega_{X/T}}$${{\rm
At}_{X/T}(P)^{\vee}}$${\operatorname{ad}(P)^{\vee}}$${0}$${0}$${\Omega_{X/T}}$${\widetilde{\mathfrak{g}}_{P}}$${\operatorname{ad}(P)}$${0.}$$\scriptstyle{\cong}$$\scriptstyle{\nu_{\mathfrak{g}}}$

As in Lemma 2.3, associated to $\widetilde{\mathfrak{g}}_{P}$ is a dgla,
$\mathfrak{S}^{i}(P)$, which we call the _Ginzburg complex_ for $P$.
Explicitly,

$\mathfrak{S}^{i}(P)=\begin{cases}0&i\neq-2,-1,0\\\ \mathcal{O}_{X}&i=-2\\\
\widetilde{\mathfrak{g}}_{P}&i=-1\\\ \mathfrak{g}_{P}&i=0.\end{cases}$

We will later consider a relative version of it.

####  2.2.2. The Bloch-Esnault complex

Let $\mathcal{E}\to X$ be a holomorphic vector bundle. Like the Ginzburg
complex, the Bloch-Esnault complex $\mathcal{B}^{\bullet}(\mathcal{E})$ [20]
is in nonzero in degrees $-2,-1$, and $0$, with
$\mathcal{B}^{-2}(\mathcal{E})=\mathcal{O}_{X}$. To define the other terms,
let $X\simeq\Delta\subset X\times_{T}X$ be the relative diagonal, and let
${\rm At}_{X/T}(\mathcal{E})$ denote the relative Atiyah algebra of $X\to T$.
Then $\mathcal{B}^{0}(\mathcal{E})=\operatorname{End}(\mathcal{E})$. Set
$\mathcal{E}^{\prime}:=\mathcal{E}\otimes\Omega_{X/T}$, and define

$\widetilde{\mathcal{B}}^{-1}(\mathcal{E})=\frac{\mathcal{E}\boxtimes\mathcal{E}^{\prime}(\Delta)}{\mathcal{E}\boxtimes\mathcal{E}^{\prime}(-\Delta)}$

on $X\times_{T}X$. Note that
$\mathcal{O}_{X\times_{T}X}(\Delta)\bigr{|}_{\Delta}=\mathcal{T}_{X/T}$. Then
$\mathcal{B}^{-1}(\mathcal{E})$ is defined by pushing out with the trace:

${0}$${\displaystyle\frac{\mathcal{E}\boxtimes\mathcal{E}^{\prime}}{\mathcal{E}\boxtimes\mathcal{E}^{\prime}(-\Delta)}}$${\displaystyle\frac{\mathcal{E}\boxtimes\mathcal{E}^{\prime}(\Delta)}{\mathcal{E}\boxtimes\mathcal{E}^{\prime}(-\Delta)}}$${\displaystyle\frac{\mathcal{E}\boxtimes\mathcal{E}^{\prime}(\Delta)}{\mathcal{E}\boxtimes\mathcal{E}^{\prime}}}$${0}$${0}$${\operatorname{End}(\mathcal{E})\otimes\Omega_{X/T}}$${\widetilde{\mathcal{B}}^{-1}(\mathcal{E})}$${\operatorname{End}(\mathcal{E})}$${0}$${0}$${\Omega_{X/T}}$${\mathcal{B}^{-1}(\mathcal{E})}$${\operatorname{End}(\mathcal{E})}$${0.}$$\scriptstyle{\operatorname{tr}}$

Here $\operatorname{tr}$ denote the trace of endomorphisms
$\operatorname{End}(\mathcal{E})\rightarrow\mathcal{O}_{X}$.

We will actually need the traceless version
$\mathcal{B}_{0}^{\bullet}(\mathcal{E})$:
$\mathcal{B}_{0}^{-2}(\mathcal{E})=\mathcal{B}^{-2}(\mathcal{E})$,
$\mathcal{B}_{0}^{0}(\mathcal{E})$ is defined by inclusion into the traceless
relative Atiyah algebra, and $\mathcal{B}_{0}^{-1}(\mathcal{E})$ is defined by
pulling back the extension $\mathcal{B}^{-1}(\mathcal{E})$ over the sheaf
$\operatorname{End}_{0}(\mathcal{E})$ of traceless endomorphisms.

####  2.2.3. Associated bundles

For any simple Lie algebra $\mathfrak{g}$, recall $\nu_{\mathfrak{g}}$ in (5).

######  Lemma 2.8.

Let $\phi_{\ast}:\mathfrak{g}\rightarrow\mathfrak{s}$ be a nonzero
homomorphism of simple Lie algebras. Consider the linear map
$\underline{\psi}:\mathfrak{s}\rightarrow\mathfrak{g}$ given by the following
diagram:

(9)
${\mathfrak{s}^{\vee}}$${\mathfrak{g}^{\vee}}$${\mathfrak{g}}$${\mathfrak{s}}$${\mathfrak{g}.}$$\scriptstyle{\phi_{\ast}^{t}}$$\scriptstyle{\cong}$$\scriptstyle{\nu_{\mathfrak{g}}^{-1}}$$\scriptstyle{\phi_{\ast}}$$\scriptstyle{\underline{\psi}}$$\scriptstyle{\cong}$$\scriptstyle{\nu_{\mathfrak{s}}}$

Then
$\underline{\psi}\circ\phi_{\ast}=m_{\phi}\operatorname{Id}_{\mathfrak{g}}$,
where $m_{\phi}$ is the Dynkin index of the map $\phi_{\ast}$. Moreover,
replacing $\underline{\psi}$ by $\psi:=(m_{\phi})^{-1}\underline{\psi}$, we
get ${\psi}\circ\phi_{\ast}=\operatorname{Id}_{\mathfrak{g}}$.

###### Proof.

This follows form a direct calculation and the definition of Dynkin index ∎

The following is straightforward.

######  Lemma 2.9.

Let
${\psi}=\frac{1}{m_{\phi}}(\nu_{\mathfrak{g}}^{-1}\circ\phi_{*}^{t}\circ\nu_{\mathfrak{s}})$
be as in Lemma 2.8, and consider the map
$\varphi:\mathfrak{g}\rightarrow\mathfrak{s}$ defined by
$\nu_{\mathfrak{s}}^{-1}\circ\psi^{t}\circ\nu_{\mathfrak{g}}$:

(10)
${\mathfrak{g}}$${\mathfrak{g}^{\vee}}$${\mathfrak{s}^{\vee}}$${\mathfrak{s}.}$$\scriptstyle{\nu_{\mathfrak{g}}}$$\scriptstyle{\varphi}$$\scriptstyle{{\psi}^{t}}$$\scriptstyle{\nu_{\mathfrak{s}}^{-1}}$

Then $\varphi=\frac{1}{m_{\phi}}\phi_{\ast}$.

Let $\phi:G\to\mathsf{SL}(V)$ be a nontrivial holomorphic representation and
$\mathcal{E}_{\phi}=P\times_{G}V$ the vector bundle associated to a principal
$G$-bundle $\Pi:P\to X$. Sections of $\mathcal{E}_{\phi}$ are functions
$\sigma:P\to V$ satisfying $\sigma(pg)=\phi(g^{-1})\sigma(p)$. The adjoint
bundle $\operatorname{ad}(P)=P\times_{G}\mathfrak{g}$ maps to the traceless
endomorphism bundle $\operatorname{End}_{0}(\mathcal{E}_{\phi})$ using
$\phi_{\ast}:=d\phi:\mathfrak{g}\to\mathfrak{sl}(V)$, and we shall use the
same notation $\phi_{\ast}$ for this map
$\operatorname{ad}(P)\to\operatorname{End}_{0}(\mathcal{E}_{\phi})$.

Notice that in this case, the map $\psi$ defined in Lemma 2.8 is
$G$-equivariant, and hence defines a homomorphism
$\operatorname{End}_{0}(\mathcal{E}_{\phi})\to\operatorname{ad}(P)$. An
invariant vector field $X$ on $P$ defines a differential operator on sections
by $X(\sigma)$ ($=d\sigma(X)$). Invariance of $X$ guarantees that $X(\sigma)$
is again equivariant with respect to $\phi$, an so defines a section of
$\mathcal{E}_{\phi}$. It is clear that the symbol of this operator is
$\Pi_{\ast}X$. Therefore, denoting the relative Atiyah algebra of
$\mathcal{E}_{\phi}\to X$ by ${\rm At}_{X/T}(\mathcal{E}_{\phi})$, we have
constructed a map (cf. Atiyah [3])

(11) $\widetilde{\phi}:{\rm At}_{X/T}(P)\to{\rm
At}_{X/T}(\mathcal{E}_{\phi}).$

The following is a consequence of the above.

######  Proposition 2.10.

For a principal $G$-bundle $P\to X$, representation
$\phi:G\rightarrow\mathsf{SL}(V)$, and associated vector bundle
$\mathcal{E}_{\phi}\to X$, there is a natural map $\widetilde{\psi}:{\rm
At}_{X/T}(\mathcal{E}_{\phi})\to{\rm At}_{X/T}(P)$ that makes the following
diagram commutative:

(12) ${0}$${\operatorname{ad}(P)}$${{\rm
At}_{X/T}(P)}$${\mathcal{T}_{X/T}}$${0}$${0}$${\operatorname{End}_{0}(\mathcal{E}_{\phi})}$${{\rm
At}_{X/T}(\mathcal{E}_{\phi})}$${\mathcal{T}_{X/T}}$${0}$${0}$${\operatorname{ad}(P)}$${{\rm
At}_{X/T}(P)}$${\mathcal{T}_{X/T}}$${0.}$$\scriptstyle{\phi}$$\scriptstyle{\widetilde{\phi}}$$\scriptstyle{\psi}$$\scriptstyle{\widetilde{\psi}}$$\scriptstyle{\operatorname{Id}}$$\scriptstyle{\operatorname{Id}}$

Here
$\psi:\operatorname{End}_{0}(\mathcal{E}_{\phi})\rightarrow\operatorname{ad}(P)$
is induced from the map $\psi$ constructed in Lemma 2.8 and $\widetilde{\psi}$
is the map obtained by pushing out the exact sequence of
$At_{X/T}(\mathcal{E}_{\phi})$ via $\psi$.

####  2.2.4. Relating the Ginzburg and Bloch-Esnault complexes

We now compare the Ginzburg complex for a principal bundle $P$ with the Bloch-
Esnault complex for the bundle $\mathcal{E}_{\phi}$ associated to $P$ via a
nontrivial representation $\phi:G\to\mathsf{SL}(V)$.

Identifying the endomorphism bundle
$\operatorname{End}_{0}(\mathcal{E}_{\phi})$ with its dual using the trace
homomorphism, consider the map

$\phi_{\ast}^{t}:\operatorname{End}_{0}(\mathcal{E}_{\phi})\cong(\operatorname{End}_{0}(\mathcal{E}_{\phi}))^{\vee}\longrightarrow\operatorname{ad}(P)^{\vee}.$

Let $\mathcal{K}_{\phi}=\ker\phi_{\ast}^{t}$. The following is important for
us.

######  Proposition 2.11.

There is a lift of the inclusion:

(13)
${\mathcal{B}_{0}^{-1}(\mathcal{E}_{\phi})}$${\mathcal{K}_{\phi}}$${\operatorname{End}_{0}(\mathcal{E}_{\phi})}$${0.}$

###### Proof.

We begin by describing a general situation. Namely, for bundles $\mathcal{E}$,
$\mathcal{F}$, we find a local lifting $\Phi_{\alpha}$ of the map

$\frac{\mathcal{E}\boxtimes\mathcal{F}^{\prime}(\Delta)}{\mathcal{E}\boxtimes\mathcal{F}^{\prime}(-\Delta)}\longrightarrow\frac{\mathcal{E}\boxtimes\mathcal{F}^{\prime}(\Delta)}{\mathcal{E}\boxtimes\mathcal{F}^{\prime}}\simeq\operatorname{Hom}(\mathcal{F},\mathcal{E})\longrightarrow
0.$

Choose local trivializations of $X\to T$ and coordinate neighborhoods
$U_{\alpha},z_{\alpha}$ on a fixed fiber. We set
$\varphi_{\alpha\beta}=z_{\alpha}\circ z_{\beta}^{-1}$, so

$\varphi_{\alpha\beta}^{\prime}=\frac{dz_{\alpha}}{dz_{\beta}}.$

The lift is defined by choosing a (holomorphic) connection $\nabla_{\alpha}$
on $\mathcal{F}^{\ast}$. Given sections $u$ and $v$ of $\mathcal{E}$ and
$\mathcal{F}^{\ast}$ respectively on $U_{\alpha}$, define on $U_{\alpha}\times
U_{\alpha}$

(14) $\Phi_{\alpha}(u(q)\otimes v(q))=\frac{u(p)\boxtimes
v(q)dz_{\alpha}(q)}{z_{\alpha}(p)-z_{\alpha}(q)}+u(p)\boxtimes\nabla_{\alpha}v(q)\
\mod\ \mathcal{I}_{\Delta}.$

To show that this is well-defined, let $f(p,q)\in\mathcal{I}_{\Delta}$.
Multiplying on the right hand side, we have

(15)
$\left(\frac{f(p,q)dz_{\alpha}(q)}{z_{\alpha}(p)-z_{\alpha}(q)}+\frac{\partial
f}{\partial q}(p,q)\right)u(p)\boxtimes v(q)\ \mod\ \mathcal{I}_{\Delta}.$

Since

$f(p,q)=\frac{\partial f}{\partial q}(p,p)(z_{\alpha}(q)-z_{\alpha}(p))\ \mod\
\mathcal{I}_{\Delta}^{2},$

we see that (15) vanishes, and so (14) gives a well-defined lift.

Set

$\mathscr{A}_{\alpha\beta}=\nabla_{\alpha}-\nabla_{\beta}\,\,,\,\,\quad\Theta_{\alpha\beta}=\frac{\varphi^{\prime\prime}_{\alpha\beta}}{\varphi^{\prime}_{\alpha\beta}}dz_{\beta}.$

Then
$\mathscr{A}_{\alpha\beta}\in\operatorname{End}(\mathcal{F}^{\ast})\otimes\Omega_{X/T}$
is a 1-cocycle representing the Atiyah class of $\mathcal{F}^{\ast}$, and
$\Theta_{\alpha\beta}\in\Omega_{X/T}$ is a cocycle for the affine structure
(cf. [28, p. 164]).

Notice that

$\displaystyle z_{\alpha}(p)-z_{\alpha}(q)$
$\displaystyle=\varphi_{\alpha\beta}(z_{\beta}(p))-\varphi_{\alpha\beta}(z_{\beta}(q))$
$\displaystyle=\varphi^{\prime}_{\alpha\beta}(z_{\beta}(q))(z_{\beta}(p)-z_{\beta}(q))+\frac{1}{2}\varphi^{\prime\prime}_{\alpha\beta}(z_{\beta}(q))(z_{\beta}(p)-z_{\beta}(q))^{2}+\cdots$

from which we have

$\frac{dz_{\alpha}(q)}{z_{\alpha}(p)-z_{\alpha}(q)}=\frac{dz_{\beta}(q)}{z_{\beta}(p)-z_{\beta}(q)}-\frac{1}{2}\Theta_{\alpha\beta}\
\mod\ \mathcal{I}_{\Delta}.$

The cocycle

$\Phi_{\alpha\beta}=\Phi_{\alpha}-\Phi_{\beta}\in\operatorname{Hom}\left(\operatorname{Hom}(\mathcal{F},\mathcal{E}),\operatorname{Hom}(\mathcal{F},\mathcal{E})\otimes\Omega_{X/T}\right)$

is then given by

$\Phi_{\alpha\beta}(u\otimes
v)=u\otimes\mathscr{A}_{\alpha\beta}v-\frac{1}{2}u\otimes
v\otimes\Theta_{\alpha\beta}.$

In the case of $\mathcal{F}=\mathcal{E}$, we may write simply

$\Phi_{\alpha\beta}:\operatorname{End}(\mathcal{E})\longrightarrow\operatorname{End}(\mathcal{E})\otimes\Omega_{X/T}:\beta\mapsto\beta\left(\mathscr{A}_{\alpha\beta}-\frac{1}{2}\Theta_{\alpha\beta}\cdot{\bf
I}\right).$

The extension class for $\mathcal{B}_{0}^{-1}(\mathcal{E})$ is then given by
the trace of endomorphisms

(16)
$\operatorname{tr}\Phi_{\alpha\beta}:\operatorname{End}_{0}(\mathcal{E})\longrightarrow\Omega_{X/T}:\beta\mapsto\operatorname{tr}(\beta\mathscr{A}_{\alpha\beta}).$

since $\operatorname{tr}\beta=0$.

Finally, to prove the proposition we must show that if
$\mathcal{E}=\mathcal{E}_{\phi}$, then $\operatorname{tr}\Phi_{\alpha\beta}$
vanishes on $\mathcal{K}_{\phi}$. When $\mathcal{E}=\mathcal{E}_{\phi}$, we
may choose local holomorphic connections on $P$, so that
$\mathscr{A}_{\alpha\beta}$ is in the image of $\phi_{\ast}$. But
$\mathcal{K}_{\phi}$ consists precisely of endomorphisms that are orthogonal
to these under the trace pairing. Thus, the proposition follows from the
expression in (16). ∎

It is shown in [7, Thm. B.2.6] that the exact sequence for
$\mathcal{B}_{0}^{-1}(\mathcal{E})$ is dual to the (traceless) Atiyah algebra
sequence for $\mathcal{E}$ using the trace map. In the case of
$\mathcal{E}_{\phi}$, we have the natural map
$\widetilde{\phi}_{\ast}^{t}:{\rm At}_{X/T}(\mathcal{E}_{\phi})^{\vee}\to{\rm
At}_{X/T}(P)^{\vee}$. Then, by Proposition 2.11 we have the following:

######  Corollary 2.12.

Let $P\to X$ be a principal $G$-bundle, $\phi:G\to\mathsf{SL}(V)$ a
holomorphic representation, and $\mathcal{E}_{\phi}\to X$ the associated
vector bundle. Then the degree $-1$ part of the Bloch-Esnault complex
$\mathcal{B}^{-1}(\mathcal{E}_{\phi})$ is the pullback of the $-1$ part of the
Ginzburg complex in the commutative diagram

${0}$${\Omega_{X/T}}$${\mathcal{B}_{0}^{-1}(\mathcal{E}_{0})}$${\operatorname{End}_{0}(\mathcal{E}_{\phi})}$${0}$${0}$${\Omega_{X/T}}$${\mathfrak{S}^{-1}(P)}$${\operatorname{ad}(P)}$${0.}$$\scriptstyle{\widetilde{\phi}_{\ast}^{t}}$$\scriptstyle{\phi_{\ast}^{t}}$

Consider the map $\widetilde{\psi}:{\rm
At}_{X/T}(\mathcal{E}_{\phi})\rightarrow{\rm At}_{X/T}(P)$ obtained in
Proposition 2.10 along with its dual $\widetilde{\psi}^{*}:{\rm
At}_{X/T}(P)^{\vee}\rightarrow{\rm At}_{X/T}(\mathcal{E}_{\phi})$. We
summarize the discussion above in the following commutative diagram:

(17)
${0}$${\Omega_{X/T}}$${\mathfrak{S}^{-1}(P)}$${\operatorname{ad}(P)}$${0}$${0}$${\Omega_{X/T}}$${{\rm
At}_{X/T}(P)^{\vee}}$${\operatorname{ad}(P)^{\vee}}$${0}$${0}$${\Omega_{X/T}}$${{\rm
At}_{X/T}(\mathcal{E}_{\phi})^{\vee}}$${\operatorname{End}_{0}(\mathcal{E}_{\phi})^{\vee}}$${0}$${0}$${\Omega_{X/T}}$${\mathcal{B}_{0}^{-1}(\mathcal{E}_{\phi})}$${\operatorname{End}_{0}(\mathcal{E}_{\phi})}$${0.}$$\scriptstyle{m_{\phi}\operatorname{Id}}$$\scriptstyle{\cong}$$\scriptstyle{\widehat{\phi}}$$\scriptstyle{\nu_{\mathfrak{g}}}$$\scriptstyle{-\phi}$$\scriptstyle{m_{\phi}\operatorname{Id}}$$\scriptstyle{m_{\phi}\widetilde{\psi}^{*}}$$\scriptstyle{m_{\phi}\psi^{t}}$$\scriptstyle{\cong}$$\scriptstyle{-\nu^{-1}_{\mathfrak{sl}(r)}}$

In (17), $\widehat{\phi}$ is just the map between $\mathfrak{S}^{-1}(P)$ and
$\mathcal{B}_{0}^{-1}(\mathcal{E}_{\phi})$ obtained by composition in the
middle column of diagram (17). Observe that

(18) $\nu_{\mathfrak{sl}(r)}^{-1}\circ
m_{\phi}\psi^{*}\circ\nu_{\mathfrak{g}}=m_{\phi}(\nu_{\mathfrak{sl}(r)}^{-1}\circ\psi^{*}\circ\nu_{\mathfrak{g}})=m_{\phi}(\frac{1}{m_{\phi}}\phi)=\phi.$

Composing the maps that appear in the above diagram, we get the following.

######  Proposition 2.13.

The natural map
$\phi:\operatorname{ad}(P)\rightarrow\operatorname{End}_{0}(\mathcal{E}_{\phi})$
extends to an map
$\widehat{\phi}:\mathfrak{S}^{-1}(P)\rightarrow\mathcal{B}_{0}^{-1}(\mathcal{E}_{\phi})$
which restricts to multiplication by the Dynkin index $m_{\phi}$ on
$\Omega_{X/T}$. Taking push-forward by $R^{1}\pi$ yields the commutative
diagram

(19)
${0}$${\mathcal{O}_{T}}$${R^{1}\pi_{*}(\mathfrak{S}^{-1}(P))}$${R^{1}\pi_{*}(\operatorname{ad}(P))}$${0}$${0}$${\mathcal{O}_{T}}$${R^{1}\pi_{*}(\mathcal{B}_{0}^{-1}(\mathcal{E}_{\phi}))}$${R^{1}\pi_{*}(\operatorname{End}_{0}(\mathcal{E}_{\phi}))}$${0.}$$\scriptstyle{m_{\phi}\operatorname{Id}}$$\scriptstyle{\widehat{\phi}}$$\scriptstyle{-\phi}$

####  2.2.5. The relative extension class

In this section we consider the special case where $X\to T$ is simply a
product $X=C\times T$. Let $P\to X$ be a principal $G$-bundle such that the
restriction to every point $C\times\\{t\\}$ is regularly stable. We wish to
compute the extension class of the top (and hence also bottom) row of (19).
Since we assume the curve is fixed, for future reference we call this the
_relative extension class_.

To state the result, let $M_{G}^{rs}$ denote the moduli space of regularly
stable bundles on $C$. Since this a coarse moduli space there is a morphism
$\varphi:T\to M_{G}^{rs}$. By the deformation theory of $G$-bundles there is a
homomorphism:

$\mathcal{T}T\longrightarrow R^{1}\pi_{\ast}(\operatorname{ad}(P)).$

Via the Dolbeault isomorphism,

$H^{1}(T,(R^{1}\pi_{\ast}(\operatorname{ad}(P))^{\vee})\longrightarrow
H^{1}(T,(\mathcal{T}T)^{\vee})\simeq H^{1,1}_{\overline{\partial}}(T)$

and so the extension class of (19) defines a class in
$H^{1,1}_{\overline{\partial}}(T)$. On the other hand, there is another
natural class $\varphi^{\ast}[\Theta]\,\in\,H^{1,1}_{\overline{\partial}}(T)$
due to Atiyah-Bott, which we define below. The result is then:

######  Theorem 2.14.

Let $X=C\times T$ as above. The image of the extension class of (19) in
$H^{1,1}_{\overline{\partial}}(T)$ is exactly the pullback
$\varphi^{\ast}[\Theta]$ of the Atiyah-Bott form.

We first define $\Theta$. Under the assumptions,
$R^{1}\pi_{\ast}(\operatorname{ad}(P))\to T$ is locally free with fiber at
$t\in T$ isomorphic to $H^{1}(C,\operatorname{ad}(P_{t}))$, where
$P_{t}=P|_{C\times\\{t\\}}$. By Dolbeault isomorphism, this is isomorphic to
$H_{\overline{\partial}}^{0,1}(C,\operatorname{ad}(P_{t}))$.

Fix a maximal compact subgroup $K\subset G$. Let $\rho_{K}$ be the Cartan
involution of $\mathfrak{g}$ that fixes ${\rm Lie}(K)$ and acts on ${\rm
Lie}(K)^{\perp}$ as multiplication by $-1$. The Narasimhan-Seshadri-Ramanathan
theorem [48, 52] asserts the existence of a $C^{\infty}$ reduction of
structure group $P^{t}_{K}\subset P_{t}$ of $P_{t}$ to $K$, satisfying the
condition that the associated Chern connection on $P_{t}$ is flat. The Chern
connection defines harmonic representatives
$\mathcal{H}^{0,1}(C,\operatorname{ad}(P_{t}))$ of the Dolbeault group
$\mathcal{H}_{\overline{\partial}}^{0,1}(C,\operatorname{ad}(P_{t}))$.

The Cartan involution $\rho_{K}$ produces a conjugate linear involution
$\rho^{\prime}$ of $\operatorname{ad}(P_{t})\otimes T^{*}C\otimes{\mathbb{C}}$
that exchanges $\Omega^{1,0}_{C}(\operatorname{ad}(P_{t}))$ and
$\Omega^{0,1}_{C}(\operatorname{ad}(P_{t}))$ and preserves the harmonic forms;
this $\rho^{\prime}$ is also called the Hodge $\ast$ operator. Then a
hermitian inner product on $\mathcal{H}^{0,1}(C,\operatorname{ad}(P_{t}))$ is
given by

$\langle\alpha,\beta\rangle=i\int_{C}(\alpha\wedge\rho^{\prime}(\beta))_{\mathfrak{g}},$

where $\alpha$, $\beta$ are harmonic representatives. The almost complex
structure on $M_{G}^{rs}$ at $\varphi(t)$ is given by $\rho^{\prime}$ on
harmonic $1$-forms. The tangent space to the space of flat $K$-connections at
the point corresponding to the Narasimhan-Seshadri-Ramanathan connection on
$P^{t}_{K}$ coincides with
$H^{1}(C,\underline{\operatorname{ad}}(P^{t}_{K}))$, where
$\underline{\operatorname{ad}}(P^{t}_{K})$ is the local system. We note that
$H^{1}(C,\underline{\operatorname{ad}}(P^{t}_{K}))$ is identified with
$H^{0,1}_{\overline{\partial}}(C,\operatorname{ad}(P))$ by the map
$\alpha\,\longmapsto\,\alpha-\sqrt{-1}\rho(\alpha)$. The almost complex
structure on $H^{1}(C,\underline{\operatorname{ad}}(P^{t}_{K}))$ is given by
$\rho^{\prime}$. For $a,b\in
H^{1}(C,\underline{\operatorname{ad}}(P^{t}_{K}))$, the Riemannian metric is
given by

$\langle a,b\rangle=-\int_{C}(a\wedge\ast
b)_{\mathfrak{g}}=2\text{Re}\,\langle\alpha,\beta\rangle,$

where $\alpha,\beta\in H^{0,1}_{\overline{\partial}}(C,\operatorname{ad}(P))$
correspond to $a,b$ respectively. Finally, the _Atiyah-Bott-Narasimhan-
Goldman_ symplectic form is given by

$\Theta(a,b)=2\text{Im}\,\langle\alpha,\beta\rangle.$

It is closed and of type $(1,1)$, and so defines a class in
$H^{1,1}_{\overline{\partial}}(T)$.

###### Proof of Theorem 2.14.

First, we note the following:

  1. (i)

${\rm At}(P)^{\vee}$ is the sheaf of invariant $1$-forms on $P$, i.e.
$\varphi\in\Gamma(\pi^{-1}(U),T^{\ast}P)$ such that
$R_{g}^{\ast}\varphi=\varphi$ for all $g\in G$.

  2. (ii)

$\operatorname{ad}(P)^{\vee}$ is the sheaf of maps from
$P\to\mathfrak{g}^{\ast}$ that are equivariant with respect to the co-adjoint
action, i.e. $f(pg)=f(p)\circ\operatorname{Ad}_{g}$.

  3. (iii)

the map $j$ is defined explicitly by:

$j(\varphi)(p)(X)=\varphi(X^{\sharp}_{p})$

for $X\in\mathfrak{g}$. We have

$\displaystyle j(\varphi)(pg)(X)$ $\displaystyle=\varphi(pg)(X^{\sharp}_{pg})$
$\displaystyle=\varphi(pg)\left((R_{g})_{\ast}(\operatorname{Ad}_{g}X)^{\sharp}_{p}\right)$
$\displaystyle=R_{g}^{\ast}\varphi((\operatorname{Ad}_{g}X)^{\sharp}_{p})$
$\displaystyle=j(\varphi)\circ\operatorname{Ad}_{g}(X)$

consequently $j(\varphi)$ satisfies the correct invariance.

A relative holomorphic connection $\omega$ gives a holomorphic splitting of
(7). Explicitly, if $\varphi$ is a local section of
$\operatorname{ad}(P)^{\vee}$, then define $\omega(\varphi)$ in ${\rm
At}(P)^{\vee}$, by: $\omega(\varphi)(X)=\varphi(\omega(X))$. Notice that

$j(\omega(\varphi))(p)(X)=\omega(\varphi)(X^{\sharp}_{p})=\varphi(\omega(X^{\sharp}_{p}))=\varphi(X)$

so this map is a splitting of the sequence in (7).

Let $M^{dR}_{G}$ denote the moduli space of holomorphic $G$-connections on a
fixed curve $C$. Let $M^{dR,rs}_{G}$ denote the open subset where the
underlying bundle $P$ is regularly stable. Hence, there is the forgetful map
$p:M^{dR,rs}_{G}\to M_{G}^{rs}$, and this is a submersion. Hence, we can find
local holomorphic sections.

With this understood, let $\\{\alpha_{j}\\}$ be a local holomorphic frame for
the bundle $R^{1}\pi_{\ast}(\operatorname{ad}(P)^{\vee})$ over an open set
$U\subset T$, and let $\\{\alpha_{j}^{\ast}\\}$ be the dual frame. From the
previous paragraph, after shrinking $U$, we may find a holomorphic family of
relatively flat connections $\omega_{U}^{hol}$ for the restriction of $P$ to
$C\times U$. Then a lift of the identity endomorphism of
$R^{1}\pi_{\ast}(\operatorname{ad}(P)^{\vee})$ to $R^{1}\pi_{\ast}({\rm
At}(P)^{\vee})\otimes(R^{1}\pi_{\ast}(\operatorname{ad}(P)^{\vee}))^{\vee}$ is
given by

$s_{U}=\sum_{j}\omega_{U}^{hol}(\alpha_{j})\otimes\alpha_{j}^{\ast}.$

For sets $U$, $V$, the difference $\sigma_{UV}=s_{U}-s_{V}$ is valued in
$R^{1}\pi_{\ast}(\Omega_{C})\otimes(R^{1}\pi_{\ast}(\operatorname{ad}(P)^{\ast}))^{\ast}$,
and the $1$-cocycle $\\{\sigma_{UV}\\}$ represents the extension class.

We now shift to the Dolbeault picture. First, the identification
$R^{1}\pi_{\ast}(\Omega_{C})\simeq\mathbb{C}$ is given by integration along
the fiber $C$. Next, using the Killing form we identify
$\operatorname{ad}P\simeq(\operatorname{ad}P)^{\vee}$. Consider the map
$\mathcal{T}T\to R^{1}\pi_{\ast}(adP)$; then as discussed above the extension
class defines via the Dolbeault isomorphism a $\overline{\partial}$-closed
$(1,1)$-form on $T$.

Let $P_{K}\subset P$ be the reduction of structure group given by the
Narasimhan-Seshadri-Ramanathan theorem. The Chern connection on $P_{K}$
extends to a connection $\omega_{A}$ on $P$ that restricts to a flat
connection on each $P_{t}$, although $\omega_{A}$ does not vary
holomorphically in $t\in T$. We can write:
$\omega^{hol}_{U}=\omega_{A}+B_{U}$, where $B_{U}$ is an invariant
$\mathfrak{g}$-valued relative $1$-form on $P$ that vanishes on vertical
vector fields. Let

$b_{U}=\sum_{j}B_{U}^{hol}(\alpha_{j})\otimes\alpha_{j}^{\ast}.$

Then since $\omega_{A}$ is globally defined, we have
$s_{U}-s_{V}=b_{U}-b_{V}$. Hence, $\\{b_{U}\\}$ gives a $C^{\infty}$
trivialization of the 1-cocycle $\\{\sigma_{UV}\\}$. By definition, the
Dolbeault representative of the extension class is therefore given by the
global $(0,1)$-form $\\{\overline{\partial}b_{U}\\}$. Now, since
$\omega^{hol}_{U}$ is holomorphic, the extension class is represented by

$\Omega_{A}=-\sum_{j}\overline{\partial}\omega_{A}(\alpha_{j})\otimes\alpha_{j}^{\ast}.$

We calculate this form at a given point $t\in U$. Choose local holomorphic
coordinates $s_{1},\ldots,s_{N}$ centered at the point $[P_{t}]$ in
$M_{G}^{rs}$ corresponding to $t$. We may arrange that the holomorphic
sections of $\mathcal{T}M_{G}^{rs}$ in a neighborhood $[P_{t}]$,
$\alpha_{j}(s_{1},\ldots,s_{N})$, satisfy $\alpha_{j}(0)=\partial/\partial
s_{j}$. Thus, we have also $\alpha_{j}^{\ast}(0)=ds_{j}$. Each $\alpha_{j}(s)$
defines a Dolbeault class in $H^{0,1}(C,\operatorname{ad}\mathcal{P}_{s})$,
which we denote by the same notation. Let $\rho_{s}(\alpha_{j}(s))$ be the
hermitian conjugate with respect to $\rho_{s}$.

With this notation, the Chern connection is given by

$\omega_{A}(s)=\omega_{A}(0)+\sum
t_{j}(s)\alpha_{j}(s)+\sum\overline{t}_{j}(s)\rho_{s}(\alpha_{j}(s))\ \mod\
\mathcal{T}^{\vee}M_{G}^{rs},$

where $t_{j}(0)=0$, $\partial t_{j}/\partial s_{i}(0)=\delta_{ij}$, $\partial
t_{j}/\partial\overline{s}_{i}(0)=0$. We therefore have

$\displaystyle\omega_{A}(\alpha_{k})$
$\displaystyle=\omega_{A}(0)(\alpha_{k})+\sum\overline{t}_{j}(\rho(\alpha_{j}),\alpha_{k})$
$\displaystyle\overline{\partial}\omega_{A}(\alpha_{k})(0)$
$\displaystyle=\sum(\rho(\alpha_{j}),\alpha_{k})d\overline{s}_{j}$

Recalling that $H^{1}(C,K_{C})\simeq\mathbb{C}$ is gotten by integration over
$C$ we have that $\Omega_{A}$ is the pullback of the form

$\sum\int_{C}(\rho(\alpha_{j}),\alpha_{k})ds_{k}\wedge
d\overline{s}_{j}=i\sum\langle\alpha_{k},\alpha_{j}\rangle ds_{k}\wedge
d\overline{s}_{j}$

and this is precisely the Atiyah-Bott symplectic form. ∎

##  3\. Determinant of cohomology

In this section we reinterpret Theorem 2.14 in terms of the quasi-isomorphism
between the Bloch-Esnault complex, the trace complex of Beilinson-Schechtman,
and the Ginzburg complex.

###  3.1. Definitions

Recall the notation
${\mathcal{E}}^{\prime}:={\mathcal{E}}\otimes\Omega_{{X}/T}$. The sheaf
$\mathcal{D}^{\leq 1}(\mathcal{E})$ of first order differential operators on
$\mathcal{E}$ can be identified with
$\frac{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(2\Delta)}{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}}$.
There is an exact sequence of sheaves

(20)
$0\longrightarrow\frac{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(\Delta)}{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}}\longrightarrow\frac{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(2\Delta)}{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}}\longrightarrow\frac{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(2\Delta)}{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(\Delta)}\longrightarrow
0.$

Similarly, we use another short exact sequence from [13]:

(21)
$0\longrightarrow\frac{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}}{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(-\Delta)}\longrightarrow\frac{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(2\Delta)}{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(-\Delta)}\longrightarrow\frac{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(2\Delta)}{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}}\longrightarrow
0.$

Since we have ${\rm At}_{X/T}(\mathcal{E})\subset\mathcal{D}^{\leq
1}_{X/T}(\mathcal{E})$, pulling back (21) we get a quasi-Lie algebra that fits
into the following short exact sequence:

(22)
$0\longrightarrow\operatorname{End}(\mathcal{E})\otimes\Omega_{X/T}\longrightarrow{}^{tr}\widetilde{\mathcal{A}}(\mathcal{E})^{-1}\longrightarrow{\rm
At}_{X/T}(\mathcal{E})\longrightarrow 0.$

Pushing it forward via the trace
$\operatorname{tr}:\operatorname{End}(\mathcal{E})\rightarrow\mathcal{O}_{X}$
we get

(23)
$0\longrightarrow\Omega_{X/T}\longrightarrow{}^{tr}\mathcal{A}(\mathcal{E})^{-1}\longrightarrow{\rm
At}_{X/T}(\mathcal{E})\longrightarrow 0.$

The Beilinson-Schechtman trace complex
${}^{tr}\mathcal{A}(\mathcal{E})^{\bullet}$ is the $\Omega$-extension of the
dgla associated the quasi-Lie algebra
${}^{tr}\widetilde{\mathcal{A}}(\mathcal{E})^{-1}$, i.e.
${}^{tr}\mathcal{A}^{0}=\operatorname{At}_{X,\pi}(\mathcal{E})$,
${}^{tr}\mathcal{A}^{-2}=\mathcal{O}_{X}$ satisfying the diagram in (4).

Recall that
$\widetilde{\mathcal{B}}^{-1}(\mathcal{E}):=\frac{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(\Delta)}{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(-\Delta)}$
and define $\widetilde{\mathcal{B}}_{0}^{-1}(\mathcal{E})$ to be the trace
free version. Now consider the pushforwards of
$\widetilde{\mathcal{B}}^{-1}(\mathcal{E})$ and
$\widetilde{\mathcal{B}}_{0}^{-1}(\mathcal{E})$ via $\operatorname{tr}$, and
denote them by ${\mathcal{B}}^{-1}(\mathcal{E})$ and
$\mathcal{B}_{0}^{-1}(\mathcal{E})$, respectively. The natural inclusion

$\operatorname{End}(\mathcal{E})\cong\frac{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(\Delta)}{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}}\hookrightarrow\frac{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(2\Delta)}{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}}$

fits the above objects into the following commutative diagram

(24)
${0}$${\operatorname{End}(\mathcal{E})\otimes\Omega_{{X}/T}}$${{}^{tr}\widetilde{\mathcal{A}}(\widetilde{\mathcal{E}})^{-1}}$${\operatorname{At}_{{X}/T}(\mathcal{E})}$${0}$${0}$${\operatorname{End}(\mathcal{E})\otimes\Omega_{\widetilde{X}/T}}$${\widetilde{\mathcal{B}}^{-1}(\mathcal{E})}$${\operatorname{End}(\mathcal{E})}$${0}$${0}$${\operatorname{End}(\mathcal{E})\otimes\Omega_{\widetilde{X}/T}}$${\widetilde{\mathcal{B}}_{0}^{-1}(\mathcal{E})}$${\operatorname{End}_{0}(\mathcal{E})}$${0}$${0}$${\Omega_{{X}/T}}$${\mathcal{B}_{0}^{-1}(\mathcal{E})}$${\operatorname{End}_{0}(\mathcal{E})}$${0}$${0}$${\Omega_{{X}/T}}$${\mathcal{B}^{-1}(\mathcal{E})}$${\operatorname{End}(\mathcal{E})}$${0.}$$\scriptstyle{\operatorname{tr}}$$\scriptstyle{\operatorname{tr}_{*}}$$\scriptstyle{\operatorname{tr}}$$\scriptstyle{\operatorname{tr}_{*}}$

Now, consider the relative set-up. Therefore, we have smooth scheme $T$ over
$S$ with connected fibers, and $\pi:X\rightarrow T$ is a family of connected
smooth curves of genus $g$.

We have short the exact sequence:

$0\rightarrow\mathcal{T}_{X/T}\rightarrow\mathcal{T}_{X/S}\rightarrow\pi^{*}\mathcal{T}_{T/S}\rightarrow
0.$

Now as $\mathcal{O}_{T}$ modules,
$\pi^{-1}(\mathcal{T}_{T/S})\subset\pi^{\ast}(\mathcal{T}_{T/S})$. Define:
$\mathcal{T}_{X,\pi}=d\pi^{-1}(\pi^{-1}(\mathcal{T}_{T/S}))$. Then we have:

(25)
$0\longrightarrow\mathcal{T}_{X/T}\longrightarrow\mathcal{T}_{X,\pi}\stackrel{{\scriptstyle
d\pi}}{{\longrightarrow}}\pi^{-1}\mathcal{T}_{T/S}\longrightarrow 0.$

Through out the rest of this section we will have the assumption that there is
a splitting of the short exact sequence in (25). This condition holds for
example in the case of fiber products. We then use the splitting of (25) to
pull-back the Atiyah algebra $\operatorname{At}_{X,\pi}(\mathcal{E})$ further:

${0}$${\operatorname{End}{\mathcal{E}}}$${\operatorname{At}_{X,\pi}(\mathcal{E})}$${\mathcal{T}_{X,\pi}}$${0}$${0}$${\operatorname{End}{\mathcal{E}}}$${\mathcal{B}^{0}(\mathcal{E})}$${\pi^{-1}\mathcal{T}_{T/S}}$${0.}$

Let $\mathcal{B}_{0}^{0}(\mathcal{E})$ be the pushout of the exact sequence
defining $\mathcal{B}^{0}(\mathcal{E})$ via the quotient homomorphism
$\operatorname{End}(\mathcal{E})\rightarrow\operatorname{End}_{0}(\mathcal{E})$.

The Bloch-Esnault complex $\mathcal{B}^{\bullet}(\mathcal{E})$ is the three
term complex consisting of the locally free sheaves
$\mathcal{B}^{0}(\mathcal{E})$, $\mathcal{B}^{-1}(\mathcal{E})$ and
$\mathcal{B}^{-2}(\mathcal{E})\cong\mathcal{O}_{X}$ in degrees $0$, $-1$ and
$-2$ respectively and zero otherwise. We will denote by
$\mathcal{B}_{0}^{\bullet}$ the traceless Bloch-Esnault complex as considered
in [61] and [7].

There is a relative line bundle $\mathcal{L}$ over $T$ associated to the
family of vector bundle $\mathcal{E}$ on $X\rightarrow T$. We refer the reader
to Section A.3 for more details on determinant of cohomology $\mathcal{L}$ (in
our convention $\mathcal{L}$ is dual to the determinant $\det
R^{\bullet}\pi_{*}\mathcal{E}$). We recall the following result of Beilinson-
Schechtman [13] that connects the Atiyah algebra of
$\operatorname{At}(\mathcal{L})$ and the trace complex of $\mathcal{E}$ and a
result of Bloch-Esnault [20] connecting the trace complex with the Bloch-
Esnault complex.

######  Proposition 3.1.

The Atiyah algebra $\operatorname{At}(\mathcal{L}^{-1})$ is isomorphic to the
Atiyah algebra $R^{0}\pi_{*}({}^{tr}\mathcal{A}(\mathcal{E})^{\bullet})$. If
$\mathcal{B}^{\bullet}(\mathcal{E})$ is the relative Bloch-Esnault complex
associated to a family of curves $X\rightarrow T$, and (25) splits, then
$\mathcal{B}^{\bullet}(\mathcal{E})$ is quasi-isomorphic to the trace complex
${}^{tr}\mathcal{A}(\mathcal{E})^{\bullet}$.

We now define the analog of the Bloch-Esnault complex for principal
$G$-bundles.

###  3.2. Relative Ginzburg complex

Let $\mathcal{P}$ be a principal ${G}$-bundle on $X\to T$, and consider the
Atiyah algebra $\operatorname{At}_{X,\pi}(\mathcal{P})$ with the fundamental
exact sequence

$0\longrightarrow\mathfrak{g}\rightarrow\operatorname{At}_{X,\pi}(\mathcal{P})\longrightarrow\mathcal{T}_{X,\pi}\longrightarrow
0.$

We then use the splitting of the exact sequence in (25) to pull-back further
to define $\mathfrak{S}^{0}(\mathcal{P})$.

${0}$${\mathfrak{g}_{\mathcal{P}}}$${\operatorname{At}_{X,\pi}(\mathcal{P})}$${\mathcal{T}_{X,\pi}}$${0}$${0}$${\mathfrak{g}_{\mathcal{P}}}$${\mathfrak{S}^{0}(\mathcal{P})}$${\pi^{-1}\mathcal{T}_{T/S}}$${0.}$

We consider the following three term complex
$\mathfrak{S}^{\bullet}(\mathcal{P})$ which we refer to as the relative
Ginzburg complex:

$\mathfrak{S}^{\bullet}(\mathcal{P}):=\begin{cases}\mathfrak{S}^{-2}(\mathcal{P})=\mathcal{O}_{X}\
\mbox{if $i=-2$},\\\ \mathfrak{S}^{-1}(\mathcal{P})\ \mbox{if $i=-1$},\\\
\mathfrak{S}^{0}(\mathcal{P})\ \mbox{if $i=0$},\\\ 0,\
\mbox{otherwise}.\end{cases}$

Let $\phi:{G}\rightarrow\mathsf{SL}(V)$ be a representation of dimension $r$,
and let $\mathcal{E}_{\phi}$ be the associated bundle. Then one recovers the
Bloch-Esnault complex $\mathcal{B}_{0}^{\bullet}(\mathcal{E}_{\phi})$.

We have the following commutative diagram in which the horizontal map is the
relative Ginzburg quasi-Lie algebra associated to a relative principal bundle
$\mathcal{P}$

(26)
${\mathcal{O}_{X}}$${\mathcal{O}_{X}}$${0}$${\Omega_{X/T}}$${\mathfrak{S}^{-1}(\mathcal{P})}$${\mathfrak{g}_{\mathcal{P}}}$${0}$${\mathfrak{S}^{0}(\mathcal{P})}$${\mathfrak{S}^{0}(\mathcal{P}).}$

The above diagram can be written as a short exact sequence of complexes

(27) $\displaystyle
0\rightarrow\Omega^{\bullet}_{X/T}[2]\rightarrow\mathfrak{S}^{\bullet}(\mathcal{P})\rightarrow\mathfrak{R}^{\bullet}(\mathcal{P}):=(\mathfrak{g}_{\mathcal{P}}\rightarrow\mathfrak{S}^{0}(\mathcal{P}))\rightarrow
0.$

The complex $\mathfrak{R}^{\bullet}(\mathcal{P})$ is quasi-isomorphic to
$\pi^{-1}\mathcal{T}_{T/S}$. Hence, there is a natural map of complexes

(28)
$\pi^{-1}\mathcal{T}_{T/S}\cong\mathfrak{R}^{\bullet}(\mathcal{P})\rightarrow\mathfrak{g}_{\mathcal{P}}[1].$

Taking pushforward with $\pi$, we get a map
$\mathcal{T}_{T/S}\cong\pi_{*}(\pi^{-1}\mathcal{T}_{T/S})\rightarrow
R^{1}\pi_{n*}\mathfrak{g}_{\mathcal{P}}$. We wish to compute
$R^{0}\pi_{\ast}\mathfrak{S}^{\bullet}(\mathcal{P})$.

To compute $R^{0}\pi_{\ast}\mathfrak{S}^{\bullet}(\mathcal{P})$, we need to
compute the zero-th hypercohomology. Choose fine resolutions (local on
$\mathcal{M}$),

${\mathfrak{G}^{-2}(\mathcal{P})}$${\mathfrak{G}^{-1}(\mathcal{P})}$${\mathfrak{G}^{0}(\mathcal{P})}$${\mathcal{L}_{0}(\mathcal{P})}$${\mathcal{M}_{0}(\mathcal{P})}$${\mathcal{N}_{0}(\mathcal{P})}$${\mathcal{L}_{1}(\mathcal{P})}$${\mathcal{M}_{1}(\mathcal{P})}$${\mathcal{N}_{1}(\mathcal{P})}$${\mathcal{L}_{2}(\mathcal{P})}$${\mathcal{M}_{2}(\mathcal{P})}$${\mathcal{N}_{2}(\mathcal{P})}$${\mathcal{L}_{3}(\mathcal{P})}$${\mathcal{M}_{3}(\mathcal{P})}$${\mathcal{N}_{3}(P)}$${\vdots}$${\vdots}$${\vdots}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{d}$$\scriptstyle{g}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{f}$$\scriptstyle{d}$$\scriptstyle{g}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{f}$$\scriptstyle{d}$$\scriptstyle{g}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{f}$$\scriptstyle{d}$$\scriptstyle{g}$$\scriptstyle{d}$$\scriptstyle{d}$

and compute the zero-th cohomology of the total complex

${\mathcal{C}_{-1}}$${\mathcal{C}_{0}}$${\mathcal{C}_{1}.}$$\scriptstyle{D_{-1}}$$\scriptstyle{D_{0}}$

Here,

$\mathcal{C}_{-1}=\mathcal{L}_{1}\oplus\mathcal{M}_{0}\ ,\
\mathcal{C}_{0}=\mathcal{L}_{2}\oplus\mathcal{M}_{1}\oplus\mathcal{N}_{0}\ ,\
\mathcal{C}_{1}=\mathcal{L}_{3}\oplus\mathcal{M}_{2}\oplus\mathcal{N}_{1}$

and

$\displaystyle
D_{-1}:\mathcal{C}_{-1}\to\mathcal{C}_{0}:(\ell_{1},m_{0})\mapsto(d\ell_{1},dm_{0}-f\ell_{1},gm_{0})$
$\displaystyle
D_{0}:\mathcal{C}_{0}\to\mathcal{C}_{1}:(\ell_{2},m_{1},n_{0})\mapsto(d\ell_{2},dm_{1}+f\ell_{2},dn_{0}-gm_{1}).$

We note three facts in the next lemma the proofs of which are immediate from
the assumptions:

######  Lemma 3.2.

Let $\pi:X\rightarrow T$ be as above. Then the following hold:

  1. (i)

$R^{2}\pi_{\ast}\mathcal{O}_{X}=\\{0\\}$ (since the relative dimension is
$1$).

  2. (ii)

$R^{0}\pi_{\ast}\mathfrak{g}_{\mathcal{P}}=\\{0\\}$ (at the regularly stable
points).

  3. (iii)

Since the the fibers connected, assume that the natural map

(29) $\pi_{\ast}(\pi^{-1}\mathcal{T}_{T/S})\cong\mathcal{T}_{T/S}\to
R^{1}\pi_{\ast}\mathfrak{g}_{P}$

is an isomorphism; then the map
$R^{1}\pi_{\ast}(\mathfrak{S}^{-1}(\mathcal{P}))\to
R^{1}\pi_{\ast}(\mathfrak{S}^{0}(\mathcal{P}))$ is zero as the map factors
through $R^{1}\pi_{\ast}\mathfrak{g}_{\mathcal{P}}$.

We prove the next proposition under the assumption of Condition (iii) in Lemma
3.2.

######  Proposition 3.3.

There is an isomorphism between
$R^{0}\pi_{*}\mathfrak{S}^{\bullet}(\mathcal{P})$ and
$R^{1}\pi_{*}\mathfrak{S}^{-1}(\mathcal{P})$ that is the identity on
$\mathcal{O}_{T}$ and $\mathcal{T}_{T/S}$.

###### Proof.

Let $(\ell_{2},m_{1},n_{0})\in\ker D_{0}$. By Lemma 3.2(i) we may take
$\ell_{2}=0$. Hence,

$(0,m_{1},n_{0})\mapsto(0,dm_{1},dn_{0}-gm_{1})=(0,0,0).$

So $m_{1}$ defines a class in
$R^{1}\pi_{\ast}(\mathfrak{S}^{-1}(\mathcal{P}))$. The second condition says
that $gm_{1}$ defines the zero class in
$R^{1}\pi_{\ast}(\pi^{-1}\mathcal{T}_{T/S})$. But by Part (iii) of Lemma 3.2
this is automatic. Now $n_{0}$ is such that $dn_{0}=gm_{1}$. By (ii) and (iii)
of Lemma 3.2, we have $R^{0}\pi_{\ast}\mathfrak{S}^{0}(\mathcal{P})=\\{0\\}$.
This means that $n_{0}$ is uniquely determined. Hence, the hypercohomology
gives $R^{1}\pi_{\ast}\mathfrak{S}^{-1}(\mathcal{P})$. ∎

Pick a representation $\phi:\mathfrak{g}\rightarrow\mathfrak{sl}_{r}$, and
consider the adjoint representation of
$\operatorname{ad}:\mathfrak{sl}_{r}\rightarrow\mathfrak{sl}(\mathfrak{sl}_{r})$.
This gives a map of the corresponding simply connected groups, and now the
corresponding associated construction first via $\phi$ gives a vector bundle
$\mathcal{E}_{\phi}$ and then taking the adjoint we have the bundle
$\operatorname{End}(\mathcal{E}_{\phi})$.

Assuming the condition stated in (iii) of Lemma 3.2, we have the following
proposition, which is a generalization of results in [7, Proposition 5.0.2]
and [61]:

######  Proposition 3.4.

There is a natural isomorphism between the direct image
$R^{0}\pi_{*}\mathcal{B}^{\bullet}(End_{0}(\mathcal{E}_{\phi}))$ of the Bloch-
Esnault complex and the direct image
$R^{0}\pi_{*}\mathfrak{S}^{\bullet}(\mathcal{P})$ of the relative Ginzburg
complex, along with the following diagram:

(30)
${0}$${R^{0}\pi_{*}\Omega^{\bullet}_{X/T}[2]\cong\mathcal{O}_{T}}$${R^{0}\pi_{*}\mathfrak{S}^{\bullet}(\mathcal{P})}$${\mathcal{T}_{T/S}}$${0}$${0}$${R^{0}\pi_{*}\Omega^{\bullet}_{X/T}[2]\cong\mathcal{O}_{T}}$${R^{0}\pi_{*}\mathcal{B}^{\bullet}(\operatorname{End}_{0}(\mathcal{E}_{\phi}))}$${\mathcal{T}_{T/S}}$${0.}$$\scriptstyle{m_{\phi}\cdot
2r}$$\scriptstyle{\cong}$$\scriptstyle{-\operatorname{id}}$

###### Proof.

The composition of homomorphisms
$\operatorname{ad}\circ\phi:\mathfrak{g}\rightarrow\mathfrak{sl}(\mathfrak{sl}_{r})$
gives a map

${\rm At}_{X/T}(\mathcal{P})\rightarrow{\rm
At}_{X/T}(\operatorname{End}_{0}(\mathcal{E}_{\phi}))$

(see (11) and ref. [3]). Moreover, the representation also gives the map

$\mathfrak{S}^{-1}(\mathcal{P})\rightarrow\mathcal{B}_{0}^{-1}(\operatorname{End}(\mathcal{E}_{\phi}))$

(see Proposition 2.13). These two together give a map between the complexes

(31)
${0}$${\Omega^{\bullet}_{X/T}[2]}$${\mathfrak{S}^{\bullet}(\mathcal{P})}$${\pi^{-1}\mathcal{T}_{T/S}}$${0}$${0}$${\Omega^{\bullet}_{X/T}[2]}$${\mathcal{B}^{\bullet}(\operatorname{End}(\mathcal{E}_{\phi}))}$${\pi^{-1}\mathcal{T}_{T/S}}$${0.}$$\scriptstyle{m_{\phi}\cdot
2r}$$\scriptstyle{-\operatorname{id}}$

Taking pushforward $R^{1}\pi_{*}$ of the above, we obtain the desired result.
∎

Now assume that $R^{1}\pi_{*}\mathfrak{g}_{\mathcal{P}}$ is isomorphic to
$\mathcal{T}_{T/S}$ under the natural map in (29). Consequently, combining
Propositions 3.3 and 3.4, we have the following diagram

(32)
${0}$${R^{1}\Omega_{X/T}\cong\mathcal{O}_{T}}$${R^{1}\pi_{*}\mathfrak{S}^{-1}(\mathcal{P})}$${\mathcal{T}_{T/S}}$${0}$${0}$${R^{0}\pi_{*}\Omega^{\bullet}_{X/T}[2]\cong\mathcal{O}_{T}}$${R^{0}\pi_{*}\mathfrak{S}^{\bullet}(\mathcal{P})}$${\mathcal{T}_{T/S}}$${0}$${0}$${R^{0}\pi_{*}\Omega^{\bullet}_{X/T}[2]\cong\mathcal{O}_{T}}$${R^{0}\pi_{*}\mathcal{B}^{\bullet}(\operatorname{End}_{0}(\mathcal{E}_{\phi}))}$${\mathcal{T}_{T/S}}$${0}$${0}$${\mathcal{O}_{T}}$${R^{0}\pi_{*}{}^{tr}\mathcal{At}^{\bullet}(\operatorname{End}_{0}(\mathcal{E}_{\phi}))}$${\mathcal{T}_{T/S}}$${0}$${0}$${\mathcal{O}_{T}}$${\operatorname{At}_{X/T}(\mathcal{L}^{-\otimes
2r}_{\phi})}$${\mathcal{T}_{T/S}}$${0.}$$\scriptstyle{\cong}$$\scriptstyle{m_{\phi}.2r}$$\scriptstyle{\cong}$$\scriptstyle{-\operatorname{id}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

The isomorphism between the second row and third row is due to Bloch-Esnault
[20] and the third row and the fourth row is due to Beilinson-Schechtman [13].
Here $\mathcal{L}_{\phi}$ is the determinant of cohomology associated to the
family $\mathcal{E}_{\phi}$.

We have the following theorem under the assumption that
$\mathcal{T}_{T/S}\cong R^{1}\pi_{*}\mathfrak{g}_{\mathcal{P}}$ for the map in
(29):

######  Theorem 3.5.

The relative Atiyah algebra $\frac{1}{m_{\phi}}{\rm At}(\mathcal{L}_{\phi})$
is isomorphic to the exact sequence

(33) $0\longrightarrow\mathcal{O}_{M^{rs}_{G}}\longrightarrow
R^{1}\pi_{n,\ast}(\mathfrak{S}^{-1}(\mathcal{P}))\longrightarrow\mathcal{T}_{T/S}\longrightarrow
0.$

This justifies the calculation of the relative extension class of the short
exact sequence via Dolbeault methods in Theorem 2.14.

###  3.3. Associated bundles and pullback

In this section we discuss the relation between the relative Ginzburg complex
for a group ${G}$ and the moduli space $M_{G}$ and the pull-back of Bloch-
Esnault complex of on $M_{\mathsf{SL}_{r}}$ associated to a representation
$\phi:G\hookrightarrow\mathsf{SL}_{r}$.

As previously, let $M^{rs}_{G}$ denote the moduli space parametrizing
regularly stable principal $G$ bundles, and denote by
$M^{s}_{\mathsf{SL}_{r}}$ the moduli space of stable rank $r$ vector bundles.
Let $\phi:G\hookrightarrow\mathsf{SL}_{r}$ be such that $G$ is not contained
in any proper-parabolic subgroup of $\mathsf{SL}_{r}$. With this condition, we
know from [17] that a stable ${G}$-bundle $P$ produces a stable
$\mathsf{SL}_{r}$ bundle.

Consider the map $\phi:M^{rs}_{G}\rightarrow M^{s}_{\mathsf{SL}_{r}}$ taking
$[P]$ to $[\mathcal{E}_{\phi}]$, where
$\mathcal{E}_{\phi}=P\times_{\phi}\mathbb{C}^{r}$ is the associated vector
bundle. Let $\mathscr{E}\to M^{s}_{\mathsf{SL}_{r}}\times_{S}\mathcal{C}$ and
$\mathscr{P}\to M^{rs}_{G}\times_{S}\mathcal{C}$ be the universal bundles
(which we can assume to exist without loss of generality). Then the associated
bundle $\mathscr{E}_{\phi}=\mathscr{P}\times_{\phi}\mathbb{C}^{r}$ is the
pull-back of $\mathscr{E}$ via $\phi$. We have the following diagram of maps:

(34)
${\mathscr{E}_{\phi}}$${\mathscr{E}}$${\mathfrak{X}_{G}:=M^{rs}_{G}\times_{S}\mathcal{C}}$${\mathfrak{X}_{\mathsf{SL}_{r}}:=M^{s}_{\mathsf{SL}_{r}}\times_{S}\mathcal{C}}$${M^{rs}_{G}}$${M^{s}_{\mathsf{SL}_{r}}}$${\mathcal{C}}$${\mathcal{C}}$${S}$${S.}$$\scriptstyle{f}$$\scriptstyle{\pi_{n,G}}$$\scriptstyle{\pi_{w,{G}}}$$\scriptstyle{\pi_{n,\mathsf{SL}_{r}}}$$\scriptstyle{\pi_{w,\mathsf{SL}_{r}}}$$\scriptstyle{\phi}$$\scriptstyle{\pi_{s,G}}$$\scriptstyle{\pi_{s,\mathsf{SL}_{r}}}$$\scriptstyle{\pi_{e,G}}$$\scriptstyle{\pi_{e,\mathsf{SL}_{r}}}$

Recall that we have the following commutative diagram on
$M^{s}_{\mathsf{SL}_{r}}$ that connects the Atiyah algebra of the determinant
bundle with the Atiyah algebra defined by Bloch-Esnault [20]

(35)
${0}$${\Omega_{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}}$${\operatorname{At}(\mathscr{E})^{\vee}}$${\operatorname{End}_{0}(\mathscr{E})^{\vee}}$${0}$${0}$${\Omega_{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}}$${\mathcal{B}_{0}^{-1}(\mathscr{E})}$${\operatorname{End}_{0}(\mathscr{E})}$${0.}$$\scriptstyle{\cong}$$\scriptstyle{-\nu_{\mathfrak{sl}(r)}}$$\scriptstyle{\cong}$

Taking pushforward and combining the results of Beilinson-Schechtman [13],
Bloch-Esnault [20], and Baier-Bolognesi-Martens-Pauly [7], we get

${0}$${R^{1}\pi_{n,\mathsf{SL}_{r}*}\Omega_{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}}$${R^{1}\pi_{n,\mathsf{SL}_{r}*}(\operatorname{At}(\mathscr{E})^{\vee})}$${R^{1}\pi_{n,\mathsf{SL}_{r}*}(\operatorname{End}_{0}(\mathscr{E})^{\vee})}$${0}$${0}$${R^{1}\pi_{n,\mathsf{SL}_{r}*}\Omega_{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}}$${R^{1}\pi_{n,\mathsf{SL}_{r}*}(\mathcal{B}_{0}^{-1}(\mathscr{E}))}$${R^{1}\pi_{n,\mathsf{SL}_{r}*}(\operatorname{End}_{0}(\mathscr{E}))}$${0}$${0}$${\mathcal{O}_{M^{s}_{\mathsf{SL}_{r}}}}$${\operatorname{At}_{M^{s}_{\mathsf{SL}_{r}}/S}(\mathcal{L}^{-1})}$${\mathcal{T}_{M^{s}_{\mathsf{SL}_{r}}/S}}$${0.}$$\scriptstyle{\cong}$$\scriptstyle{-\nu_{\mathfrak{sl}(r)}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

where $\mathcal{L}=\det(R\pi_{n*}\mathscr{E})$ is the determinant of
cohomology of the family $\mathscr{E}_{\phi}$. Pulling back the exact sequence
at the bottom of the diagram by the map $\phi:M^{rs}_{G}\rightarrow
M^{s}_{\mathsf{SL}_{r}}$, we get

$0\longrightarrow\mathcal{O}_{M^{rs}_{G}}\longrightarrow\operatorname{At}_{M^{rs}_{G}/S}(\phi^{*}(\mathcal{L}))\longrightarrow\mathcal{T}_{M^{rs}_{G}/S}\longrightarrow
0.$

We wish to connect the exact sequence

(36)
$0\longrightarrow\phi^{*}(R^{1}\pi_{n,\mathsf{SL}_{r}*}\Omega_{\mathfrak{X}_{\mathsf{SL}_{r}}/M_{\mathsf{SL}_{r}}})\longrightarrow\phi^{*}(R^{1}\pi_{n,\mathsf{SL}_{r}*}(\mathcal{B}_{0}^{-1}(\mathscr{E})))\longrightarrow\phi^{*}R^{1}\pi_{n,\mathsf{SL}_{r}*}(\operatorname{End}_{0}(\mathscr{E}))\longrightarrow
0,$

with

$0\longrightarrow\mathcal{O}_{M^{rs}_{G}}\cong
R^{1}\pi_{n*}\Omega_{\mathfrak{X}_{G}/M^{rs}_{G}}\longrightarrow
R^{1}\pi_{n*}(\mathfrak{S}^{-1}(\mathscr{P}))\longrightarrow\mathcal{T}_{M^{rs}_{G}/S}\longrightarrow
0.$

By Proposition 2.13, it is enough to relate (36) with

$0\longrightarrow\mathcal{O}_{M^{rs}_{G}}\longrightarrow
R^{1}\pi_{n,G\,\ast}(\mathcal{B}_{0}^{-1}(\mathscr{E}_{\phi}))\longrightarrow
R^{1}\pi_{n*}(\operatorname{End}_{0}(\mathscr{E}_{\phi}))\longrightarrow 0.$

in (19).

######  Proposition 3.6.

There is an isomorphism

$R^{1}\pi_{n,G\,\ast}(\mathcal{B}_{0}^{-1}(\mathscr{E}_{\phi}))\cong\phi^{*}(R^{1}\pi_{n,\mathsf{SL}_{r}*}(\mathcal{B}_{0}^{-1}(\mathscr{E}))).$

###### Proof.

Consider the following digram

${\mathscr{E}_{\phi}\simeq
f^{*}(\mathscr{E})}$${\mathscr{E}}$${M^{rs}_{G}\times_{S}C=:\mathfrak{X}_{G}}$${\mathfrak{X}_{\operatorname{\mathsf{SL}_{r}}}:=M^{s}_{\mathsf{SL}_{r}}\times_{S}C}$${M^{rs}_{G}}$${M^{s}_{\mathsf{SL}_{r}}.}$$\scriptstyle{f}$$\scriptstyle{\pi_{n,G}}$$\scriptstyle{\pi_{n,\mathsf{SL}_{r}}}$$\scriptstyle{\phi}$

This induces a map of the following exact sequences of Atiyah algebras as in
[3]:

${0}$${\operatorname{End}_{0}(\mathscr{E}_{\phi})}$${\operatorname{At}(\mathscr{E}_{\phi})}$${\mathcal{T}_{\mathfrak{X}_{G}/M^{rs}_{G}}}$${0}$${0}$${f^{*}\operatorname{End}_{0}(\mathscr{E})}$${f^{*}\operatorname{At}(\mathscr{E})}$${f^{*}\mathcal{T}_{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}}$${0.}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

Dualizing, we obtain

${0}$${f^{*}\Omega_{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}}$${f^{*}(\operatorname{At}(\mathscr{E})^{\vee})}$${f^{*}((\operatorname{End}_{0}(\mathscr{E}))^{\vee})}$${0}$${0}$${\Omega_{\mathfrak{X}_{G}/M^{rs}_{G}}}$${\operatorname{At}(\mathscr{E}_{\phi})^{\vee}}$${(\operatorname{End}_{0}(\mathscr{E}_{\phi}))^{\vee}}$${0}$${0}$${{\Omega}_{\mathfrak{X}_{G}/M^{rs}_{G}}}$${\mathcal{B}_{0}^{-1}(\mathscr{E}_{\phi})}$${\operatorname{End}_{0}(\mathscr{E}_{\phi}))}$${0}$${0}$${f^{*}{\Omega}_{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}}$${f^{*}(\mathcal{B}_{0}^{-1}(\mathscr{E}))}$${f^{*}\operatorname{End}_{0}(\mathscr{E})}$${0.}$$\scriptstyle{\cong}$$\scriptstyle{q_{\mathscr{E}_{\phi}}^{-1}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{-\nu_{\mathfrak{sl}(r)}^{-1}}$$\scriptstyle{q_{\mathscr{E}}}$$\scriptstyle{-\nu_{\mathfrak{sl}(r)}}$

Hence, by composing we get the following diagram

${0}$${{\Omega}_{\mathfrak{X}_{G}/M^{rs}_{G}}}$${\mathcal{B}_{0}^{-1}(\mathscr{E}_{\phi})}$${\operatorname{End}_{0}(\mathscr{E}_{\phi})}$${0}$${0}$${f^{*}{\Omega}_{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}}$${f^{*}(\mathcal{B}_{0}^{-1}(\mathscr{E}))}$${f^{*}\operatorname{End}_{0}(\mathscr{E})}$${0.}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

Applying $R^{1}\pi_{n,G\,\ast}$, we get that the following extensions are
isomorphic:

${0}$${R^{1}\pi_{n,G\,\ast}\Omega_{\mathfrak{X}_{G}/M^{rs}_{G}}}$${R^{1}\pi_{n,G\,\ast}(\mathcal{B}_{0}^{-1}(\mathscr{E}_{\phi}))}$${R^{1}\pi_{n,G\,\ast}(\operatorname{End}_{0}(\mathscr{E}_{\phi}))}$${0}$${0}$${R^{1}\pi_{n,G\,\ast}f^{*}\Omega_{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}}$${R^{1}\pi_{n,G\,\ast}(f^{*}(\mathcal{B}_{0}^{-1}(\mathscr{E})))}$${R^{1}\pi_{n,G\,\ast}f^{*}(\operatorname{End}_{0}(\mathscr{E}))}$${0.}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

Thus to finish the proof of the proposition we need to show that

$R^{1}\pi_{n,{G}*}(f^{*}(\mathcal{B}_{0}^{-1}(\mathscr{E})))\cong\phi^{*}(R^{1}\pi_{n,\mathsf{SL}_{r}*}(\mathcal{B}_{0}^{-1}(\mathscr{E}))).$

But since the fibers of the morphism $\pi_{n,\mathsf{SL}_{r}}$ are smooth
projective curves, we conclude that $R^{2}\pi_{n,\mathsf{SL}_{r}*}\mathcal{F}$
is zero for any coherent sheaf $\mathcal{F}$ which is flat over
$M_{\mathsf{SL}_{r}}$. Moreover,
$R^{1}\pi_{n,\mathsf{SL}_{r}*}(\mathcal{B}_{0}^{-1}(\mathscr{E}))$ is locally
free. Hence by cohomology and base change for flat sheaves, we get the
required isomorphism. This completes the proof. ∎

Consider the following diagram where $m_{\phi}$ be the Dynkin index of the
embedding $\mathfrak{g}\rightarrow\mathfrak{sl}(V)$:

${0}$${\mathcal{O}_{M_{G}}}$${\phi^{*}\operatorname{At}(\mathcal{L}^{-1})}$${\phi^{*}\mathcal{T}_{M^{s}_{\mathsf{SL}_{r}}/S}}$${0}$${0}$${\phi^{*}\left(R^{1}\pi_{n,\mathsf{SL}_{r}\,\ast}\Omega_{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}\right)}$${\phi^{*}\left(R^{1}\pi_{n,\mathsf{SL}_{r}*}(\mathcal{B}_{0}^{-1}(\mathscr{E})\right)}$${\phi^{*}\left(R^{1}\pi_{n,\mathsf{SL}_{r}*}(\operatorname{End}_{0}(\mathscr{E})\right)}$${0}$${0}$${R^{1}\pi_{n,G\,\ast}\Omega_{\mathfrak{X}_{G}/M^{rs}_{G}}}$${R^{1}\pi_{n,G\,\ast}(\mathcal{B}_{0}^{-1}(\mathscr{E}_{\phi}))}$${R^{1}\pi_{n,G\,\ast}(\operatorname{End}_{0}(\mathscr{E}_{\phi}))}$${0}$${0}$${R^{1}\pi_{n,G\,\ast}\Omega_{\mathfrak{X}_{G}/M^{rs}_{G}}}$${R^{1}\pi_{n,G\,\ast}(\mathfrak{S}^{-1}(\mathscr{P}))}$${\mathcal{T}_{M^{rs}_{G}/S}}$${0.}$$\scriptstyle{\cong}$$\scriptstyle{KS_{M_{\mathsf{SL}_{r}}^{s}/S}}$$\scriptstyle{\cong}$$\scriptstyle{m_{\phi}}$$\scriptstyle{-\phi_{*}}$

In this diagram, by Proposition 3.6 we get the isomorphism of the first two
rows. Finally, the map between the second and the third row follows from
Proposition 2.13 and Proposition 19.

Thus we get $R^{1}\pi_{n,{G}*}(\mathfrak{S}^{-1}(\mathscr{P}))$ is isomorphic
to the Atiyah algebra
$\frac{1}{m_{\phi}}\phi^{\bullet}\operatorname{At}(\mathcal{L})$, where
$\phi^{\bullet}$ denote the pull-back in the category of Atiyah algebras.

##  4\. Parabolic Atiyah algebras

We now extend the previous considerations to the case of parabolic bundles. In
order to analyze parabolic Atiyah algebras for families of parabolic bundles
on a curve $C$, we adopt Seshadri’s notion of $\Gamma$-linearized bundles on a
Galois cover $\widehat{C}\to C$ with Galois group $\Gamma$. See Appendix B for
more details.

###  4.1. Parabolic vector bundles

Let $\widetilde{\mathcal{E}}$ be a vector bundle of rank $r$ on a family of
$\Gamma$-cover of curves $\widetilde{\pi}:\widetilde{X}\rightarrow{T}$
ramified along $\widehat{D}$. This comes with a natural projection map
$p:\widetilde{X}\rightarrow X$ and the family $\pi:\widetilde{X}\rightarrow T$
parametrize a family of curves. Let $D:=p(\widehat{D})\subset X$ be the
divisor or marked points. Let $\widetilde{\mathcal{E}}$ be a family of vector
bundles on $\widetilde{X}$ which is $\Gamma$-linearized. Let $\mathcal{E}$ be
the vector bundle obtained by the invariant pushforward of the $\Gamma$-bundle
$\widetilde{{\mathcal{E}}}$. By the discussion in Appendix B, the vector
bundle $\mathcal{E}$ comes equipped with a parabolic structure with support on
$D$.

Recall the trace-zero relative Atiyah sequence

(37)
$0\longrightarrow\operatorname{End}_{0}(\widetilde{\mathcal{E}})\longrightarrow\operatorname{At}_{\widetilde{X}/T}(\widetilde{\mathcal{E}})\longrightarrow\mathcal{T}_{\widetilde{X}/T}\longrightarrow
0.$

By Seshadri [57], we can identify the sheaf of parabolic endomorphism
$\operatorname{Par}(\widetilde{\mathcal{E}})$ with
$p_{*}^{\Gamma}\operatorname{End}_{0}(\widetilde{\mathcal{E})}$. With this
set-up we can consider the parabolic Atiyah algebra with the following
fundamental exact sequence

$0\longrightarrow{\operatorname{Par}_{0}(\widetilde{\mathcal{E}})\atop\cong
p_{*}^{\Gamma}(\operatorname{End}_{0}(\widetilde{\mathcal{E}})}\longrightarrow{{}^{par}{\rm
At}_{X/T}(\mathcal{E})\atop:=p_{*}^{\Gamma}{\rm
At}_{\widetilde{X}/T}(\widetilde{\mathcal{E}})}\longrightarrow\mathcal{T}_{X/T}(-D)\longrightarrow
0$

As before, $\widehat{D}$ is the relative ramification divisor in
$\widetilde{X}$; consider the log-relative traceless Atiyah sequence obtained
from (37)

$0\longrightarrow\operatorname{End}_{0}(\widetilde{\mathcal{E}})(-\widehat{D})\longrightarrow(\operatorname{At}_{\widetilde{X}/T}(\widetilde{\mathcal{E}}))(-\widehat{D})\longrightarrow\mathcal{T}_{\widetilde{X}/T}(-\widehat{D})\longrightarrow
0.$

Since all the objects are naturally $\Gamma$-linearized, we can apply the
invariant push-forward functor $p_{*}^{\Gamma}$ to get the _strongly parabolic
Atiyah algebra_ ${}^{spar}{\rm At}_{X/T}(\mathcal{E})$ with the fundamental
exact sequence

$0\longrightarrow{\operatorname{SPar}_{0}(\mathcal{E})\atop\cong
p_{*}^{\Gamma}\left(\operatorname{End}_{0}(\widetilde{\mathcal{E}})(-\widehat{D})\right)}\longrightarrow
p_{*}^{\Gamma}\left((\operatorname{At}_{\widetilde{X}/T}(\widetilde{\mathcal{E}})(-\widehat{D})\right)\longrightarrow\mathcal{T}_{X/T}(-D)\longrightarrow
0$

Tensoring with $\mathcal{O}_{X}(D)$ we get,

$0\longrightarrow\operatorname{SPar}_{0}(\mathcal{E})(D)\longrightarrow
p_{*}^{\Gamma}\left((\operatorname{At}_{\widetilde{X}/T}(\widetilde{\mathcal{E}}))(-\widehat{D})\right)(D)\longrightarrow\mathcal{T}_{X/T}\longrightarrow
0.$

As in Section 2.1, consider the dual exact sequence:

(38)
$0\longrightarrow\Omega_{{X}/T}\longrightarrow\left(p_{*}^{\Gamma}\left((\operatorname{At}_{\widetilde{X}/T}(\widetilde{\mathcal{E}}))(-\widehat{D})\right)(D)\right)^{\vee}\longrightarrow\left(\operatorname{SPar}_{0}(\mathcal{E})(D)\right)^{\vee}\longrightarrow
0.$

The trace pairing
$\kappa_{\mathfrak{sl}(r)}:\operatorname{End}_{0}(\widetilde{\mathcal{E}})\otimes\operatorname{End}_{0}(\widetilde{\mathcal{E}})\rightarrow\mathcal{O}_{\widetilde{X}}$
gives an $\mathcal{O}_{\widetilde{X}}(-\widehat{D})$ valued pairing

$\kappa_{\mathfrak{sl}(r)}:\operatorname{End}_{0}(\widetilde{\mathcal{E}})\otimes\operatorname{End}_{0}(\widetilde{\mathcal{E}})(-\widehat{D})\longrightarrow\mathcal{O}_{\widetilde{X}}(-\widehat{D}).$

Taking $\Gamma$-invariant push forward $p_{*}^{\Gamma}$ of the above exact
sequence we get a map

$\kappa_{\mathfrak{sl}(r)}:\operatorname{Par}_{0}(\mathcal{E})\otimes\operatorname{SPar}_{0}(\mathcal{E})\longrightarrow\mathcal{O}_{{X}}(-D).$

Now by multiplying by $\mathcal{O}_{X}(D)$ on both sides we get the following:

######  Proposition 4.1.

The trace induces a nondegenerate pairing

$\kappa_{\mathfrak{sl}(r)}:\operatorname{Par}_{0}({\mathcal{E}})\otimes\operatorname{SPar}_{0}({\mathcal{E}})(D)\
\longrightarrow\mathcal{O}_{X}$

which identifies
$(\operatorname{SPar}_{0}({\mathcal{E}})(D))^{\vee}\cong\operatorname{Par}_{0}({\mathcal{E}})$.

We pull back then sequence in (38) via the map $\nu_{\mathfrak{sl}(r)}$ to get
the following quasi-Lie algebra which we denote by
$\widetilde{\operatorname{SPar}{\rm At}}_{X/T}(\mathcal{E})$

(39)
${0}$${\Omega_{X/T}}$${\left(p_{*}^{\Gamma}\left((\operatorname{At}_{\widetilde{X}/T}(\widetilde{\mathcal{E}}))(-\widehat{D})\right)(D)\right)^{\vee}}$${(\operatorname{SPar}_{0}(\mathcal{E})(D))^{\vee}}$${0}$${0}$${\Omega_{X/T}}$${\widetilde{\operatorname{SPar}{\rm
At}}_{X/T}(\mathcal{E})}$${\operatorname{Par}_{0}({\mathcal{E}})}$${0.}$$\scriptstyle{\cong}$$\scriptstyle{\nu_{\mathfrak{sl}(r)}}$

####  4.1.1. Parabolic Bloch-Esnault complex

By the construction of Beilinson-Schechtman [13], we get an exact sequence of
sheaves:

$0\longrightarrow\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T}\longrightarrow{}^{tr}\widetilde{\mathcal{A}}(\widetilde{\mathcal{E}})^{-1}\longrightarrow\operatorname{At}_{\widetilde{X}/T}(\widetilde{\mathcal{E}})\longrightarrow
0.$

Recall that since $\widetilde{X}\rightarrow T$ is a $\Gamma$-cover of curves,
and now assume that $\widetilde{\mathcal{E}}$ is $\Gamma$-equivariant. This
implies that all terms in the above exact sequence have a $\Gamma$-action.
Taking $\Gamma$-invariant pushforward we get the following:

(40) $0\longrightarrow
p_{*}^{\Gamma}(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T})\longrightarrow
p_{*}^{\Gamma}({}^{tr}\widetilde{\mathcal{A}}(\widetilde{\mathcal{E}})^{-1})\longrightarrow
p_{*}^{\Gamma}(\operatorname{At}_{\widetilde{X}/T}(\widetilde{\mathcal{E}}))\longrightarrow
0.$

Here ${\rm At}_{\widetilde{X}/T}(\widetilde{\mathcal{E}})$ is the relative
Atiyah algebra of the bundle of the vector bundle $\widetilde{\mathcal{E}}$.
The last term of the above is the _parabolic Atiyah algebra_
$\operatorname{{}^{\emph{par}}At}({\mathcal{E}})$. Pulling back the exact
sequence in (40) via the natural inclusion
$\operatorname{Par}_{0}({\mathcal{E}})\hookrightarrow\operatorname{{}^{\emph{par}}At}({\mathcal{E}})$,
we get an exact sequence:

(41) $0\longrightarrow
p_{*}^{\Gamma}(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T})\longrightarrow{}^{par}\widetilde{\mathcal{B}}_{0}^{-1}({\mathcal{E}})\longrightarrow\operatorname{Par}_{0}({\mathcal{E}})\longrightarrow
0.$

Taking invariant pushforward with respect to the trace of an endomorphism
$\operatorname{tr}:\mathcal{E}nd(\widetilde{\mathcal{E}})\rightarrow\mathcal{O}_{\widetilde{X}}$,
we get another map

$\operatorname{tr}:p_{*}^{\Gamma}\operatorname{End}(\widetilde{\mathcal{E}}\otimes\Omega_{\widetilde{X}/T})\longrightarrow
p_{*}^{\Gamma}\Omega_{\widetilde{X}/T}.$

Since $\Omega_{\widetilde{X}/T}\cong
p^{*}\Omega_{{X}/T}\otimes\mathcal{O}(\widehat{D})$, where $\widehat{D}$ is
the relative ramification $p:\widetilde{X}\rightarrow X$. We can identify
$p_{*}^{\Gamma}(\Omega_{\widetilde{X}/T})\cong\Omega_{X/T}$. This in turn
gives a map

(42)
$\operatorname{tr}:p_{*}^{\Gamma}(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T})\longrightarrow{\Omega}_{X/T}.$

Taking pushforward of the exact sequence in (40) via the parabolic trace in
(42), we get the following exact sequence:

(43)
$0\longrightarrow\Omega_{{X}/T}\longrightarrow{}^{par}\mathcal{B}_{0}^{-1}({\mathcal{E}})\longrightarrow\operatorname{Par}_{0}({\mathcal{E}})\longrightarrow
0.$

######  Definition 4.2.

We define the parabolic Bloch-Esnault complex as follows:
${}^{par}\mathcal{B}_{0}^{-2}(\mathcal{E})=\mathcal{O}_{X}$;
${}^{par}\mathcal{B}_{0}^{-1}(\mathcal{E})$ is defined in (43), and
${}^{par}\mathcal{B}_{0}^{0}(\mathcal{E})$ is the pull-back of ${}^{par}{\rm
At}_{0}(\mathcal{E})$ by the inclusion
$\pi^{-1}\mathcal{T}_{X/T}\hookrightarrow\mathcal{T}_{X/T}$.

We can summarize the above discussion in the following commutative diagram:

${0}$${p_{*}\left(\operatorname{End}(\widetilde{{\mathcal{E}}})\otimes\Omega_{\widetilde{X}/T}\right)}$${p_{*}({}^{tr}\widetilde{\mathcal{A}}(\widetilde{{\mathcal{E}}})^{-1})}$${p_{*}\left(\operatorname{At}_{\widetilde{X}/T}(\widetilde{{\mathcal{E}}})\right)}$${0}$${0}$${p_{*}^{\Gamma}(\operatorname{End}(\widetilde{{\mathcal{E}}})\otimes\Omega_{\widetilde{X}/T})}$${p_{*}^{\Gamma}({}^{tr}\widetilde{\mathcal{A}}(\widetilde{{\mathcal{E}}})^{-1})}$${\operatorname{{}^{\emph{par}}At}_{\widetilde{X}/T}({{\mathcal{E}}})}$${0}$${0}$${p_{*}^{\Gamma}(\operatorname{End}(\widetilde{{\mathcal{E}}})\otimes\Omega_{\widetilde{X}/T})}$${{}^{par}\widetilde{\mathcal{B}}^{-1}({\mathcal{E}})}$${\operatorname{Par}_{0}({\mathcal{E}})}$${0}$${0}$${\Omega_{{X}/T}}$${{}^{par}\mathcal{B}_{0}^{-1}({\mathcal{E}})}$${\operatorname{Par}_{0}({\mathcal{E}})}$${0.}$$\scriptstyle{\operatorname{tr}}$$\scriptstyle{\operatorname{tr}_{*}}$

Recall from Section 2.1.1.1 in Beilinson-Schechtman [13, Lemma (a)], that
there is a residue pairing

$\widetilde{\operatorname{Res}}:\Omega_{\widetilde{X}/T}\boxtimes\Omega_{\widetilde{X}/T}(3\Delta)\longrightarrow\mathcal{O}_{\widetilde{X}}.$

The following theorem connects the parabolic Ginzburg dgla defined above to
the quasi Lie algebra $\widetilde{\operatorname{SPar}{\rm
At}}_{X/T}(\mathcal{E})$ defined by (39).

######  Theorem 4.3.

There is an isomorphism induced by invariant push-forward of the residue
pairing between the quasi-Lie algebras $\widetilde{\operatorname{SPar}}{\rm
At}_{X/T}(\mathcal{E})$ and ${}^{par}\mathcal{B}_{0}^{-1}(\mathcal{E})$ which
induces an isomorphism of exact sequences:

${0}$${\Omega_{{X}/T}}$${\widetilde{\operatorname{SPar}{\rm
At}}_{X/T}(\mathcal{E})}$${\operatorname{Par}_{0}({\mathcal{E}})}$${0}$${0}$${\Omega_{{X}/T}}$${{}^{par}\mathcal{B}_{0}^{-1}({\mathcal{E}})}$${\operatorname{Par}_{0}({\mathcal{E}})}$${0.}$$\scriptstyle{\cong}$$\scriptstyle{-\operatorname{id}}$$\scriptstyle{\cong}$

###### Proof.

Let
$\widetilde{\mathcal{E}}^{\prime}:=\widetilde{\mathcal{E}}\otimes\Omega_{\widetilde{X}/T}$,
as in Beilinson-Schechtman, the sheaf of first order differential operators
$\mathcal{D}^{\leq 1}(\widetilde{\mathcal{E}})$ can be identified with
$\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(2\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}$.
There is a natural exact sequence of sheaves

$0\longrightarrow\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\longrightarrow\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(2\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\longrightarrow\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(2\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}\longrightarrow
0.$

Moreover, all the objects of the above exact sequence are $\Gamma$-linearized.
In particular apply the invariant pushforward functor we get the following
commutative diagram:

$0\longrightarrow
p_{*}^{\Gamma}\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\longrightarrow
p_{*}^{\Gamma}\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(2\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\longrightarrow
p_{*}^{\Gamma}\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(2\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}\longrightarrow
0.$

Similarly we have another short exact sequence from [13] that is used in
constructing the trace complex.

$0\longrightarrow\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\longrightarrow\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(2\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\longrightarrow\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(2\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\longrightarrow
0.$

Pulling back the exact sequence above by the inclusion

$\operatorname{End}(\widetilde{\mathcal{E}})\cong\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\hookrightarrow\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(2\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}.$

we get a short exact sequence

$0\longrightarrow\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\longrightarrow\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\longrightarrow\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\longrightarrow
0.$

Taking invariant pushforward functor we get the following commutative diagram:

${0}$${p_{*}^{\Gamma}\left(\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\right)}$${p_{*}^{\Gamma}\left(\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(2\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\right)}$${p_{*}^{\Gamma}\left(\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(2\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\right)}$${0}$${0}$${p_{*}^{\Gamma}\left(\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\right)}$${p_{*}^{\Gamma}\left(\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\right)}$${p_{*}^{\Gamma}\left(\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\right)}$${0.}$

Now there is a natural nondegenerate bilinear form:

(44)
$\langle\,,\,\rangle:\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\otimes\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(2\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\longrightarrow\mathcal{O}_{\widetilde{X}}.$

The above form $\langle\,,\,\rangle$ vanishes identically restricted to
$\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\otimes\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}$
and hence it descends to a form

(45)
$\langle\,,\,\rangle:\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\otimes\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}\longrightarrow\mathcal{O}_{\widetilde{X}},\
\mbox{and}\
\langle\,,\,\rangle:\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\otimes\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\longrightarrow\mathcal{O}_{\widetilde{X}}.$

Under the canonical identifications of
$\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\cong\operatorname{End}(\widetilde{\mathcal{E}})$
we get

(46)
$\displaystyle\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}$
$\displaystyle\cong\operatorname{End}(\widetilde{\mathcal{E}})\otimes\mathcal{T}_{\widetilde{X}/T},\
\mbox{and}\
\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\cong\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T}.$

Lemma B.2.8 in [7] shows that the bilinear form $\langle\,,\,\rangle$ in (44)
and (45) can be identified as

(47)
$\displaystyle-\kappa_{\mathfrak{gl}(r)}:\operatorname{End}(\widetilde{\mathcal{E}})\otimes\operatorname{End}(\widetilde{\mathcal{E}})\longrightarrow\mathcal{O}_{\widetilde{X}}$
(48)
$\displaystyle\kappa_{\mathfrak{gl}(r)}:\operatorname{End}(\widetilde{\mathcal{E}})\otimes\mathcal{T}_{\widetilde{X}/T}\otimes\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T}\longrightarrow\mathcal{O}_{\widetilde{X}},$

where $\kappa_{\mathfrak{gl}(r)}$ is the trace of the product of the two
endomorphisms.

Now the invariant pushforward of $\langle\,,\,\rangle$ induces a nondegenerate
$\mathcal{O}_{X}(-D)$ valued form

$\langle\,,\,\rangle:p_{*}^{\Gamma}\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\otimes
p_{*}^{\Gamma}\left(\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(2\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}(-\widehat{D})\right)\longrightarrow
p_{*}^{\Gamma}\mathcal{O}_{\widetilde{X}}(-\widehat{D})\cong\mathcal{O}_{X}(-D)$

which gives the following nondegenerate form

$\langle\,,\,\rangle:p_{*}^{\Gamma}\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\otimes\left(p_{*}^{\Gamma}\left(\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(2\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}(-\widehat{D})\right)\right)(D)\longrightarrow\mathcal{O}_{X}$

that restricts to

(49)
$\displaystyle\langle\,,\,\rangle:p_{*}^{\Gamma}\left(\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\right)\otimes\left(p_{*}^{\Gamma}\left(\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}(-\widehat{D})\right)\right)(D)\longrightarrow\mathcal{O}_{X}.$
(50)
$\displaystyle\langle\,,\,\rangle:p_{*}^{\Gamma}\left(\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\right)\otimes\left(p_{*}^{\Gamma}\left(\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}(-\widehat{D})\right)\right)(D)\longrightarrow\mathcal{O}_{X}.$

The identification of parabolic and strongly parabolic endomorphism as
invariant pushforward and invariants pushforward of the identification in (47)
tell us that the form $\langle\,,\,\rangle$ in (49) can be identified with

$-\kappa_{\mathfrak{gl}(r)}:\operatorname{Par}(\widetilde{\mathcal{E}})\otimes\operatorname{SPar}(\widetilde{\mathcal{E}})(D)\longrightarrow\mathcal{O}_{X}.$

Similarly the form in (50) can be identified with

$\langle\,,\,\rangle:p_{*}^{\Gamma}\left(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T}\right)\otimes\left(p_{*}^{\Gamma}\left(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\mathcal{T}_{\widetilde{X}/T}(-\widehat{D})\right)\right)(D)\longrightarrow\mathcal{O}_{X}$

Now the identification of
$\left(p_{*}^{\Gamma}\left(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\mathcal{T}_{\widetilde{X}/T}(-\widehat{D})\right)\right)(D)\cong\mathcal{T}_{X/T}$
gives the following commutative

${\langle\,,\,\rangle:\left(p_{*}^{\Gamma}\left(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T}\right)\right)\otimes\left(p_{*}^{\Gamma}\left(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\mathcal{T}_{\widetilde{X}/T}(-\widehat{D})\right)\right)(D)}$${\mathcal{O}_{X}}$${\Omega_{{X}/T}\otimes\mathcal{T}_{X/T}}$${\mathcal{O}_{X}.}$$\scriptstyle{\cong}$

This induces an isomorphism of ${}^{par}\mathcal{B}_{0}^{-1}(\mathcal{E})$
with $\widetilde{\operatorname{SPar}{\rm At}}_{X/T}(\mathcal{E})$ that
restricts to $\operatorname{id}$ on $\Omega_{X/T}$ and $-\operatorname{id}$ on
$\operatorname{Par}_{0}(\mathcal{E})$. ∎

The following proposition connects $R^{1}\pi_{*}$ of the parabolic Ginzburg
dgla with $R^{1}\widetilde{\pi}_{*}$ of the dgla constructed by Bloch-Esnault.

######  Proposition 4.4.

There are inclusion maps $\Omega_{X/T}\rightarrow
p_{*}\Omega_{\widetilde{X}/T}$ and $\operatorname{Par}_{0}(\mathcal{E})\cong
p_{*}^{\Gamma}\operatorname{End}_{0}(\mathcal{E})\hookrightarrow
p_{*}\operatorname{End}_{0}(\widetilde{{\mathcal{E}}})$ that extend to a map
of the following exact sequences:

${0}$${p_{*}\Omega_{\widetilde{X}/T}}$${p_{*}\mathcal{B}_{0}^{-1}(\widetilde{{\mathcal{E}}})}$${p_{*}\operatorname{End}_{0}(\widetilde{{\mathcal{E}}})}$${0}$${0}$${\Omega_{X/T}}$${{}^{par}\mathcal{B}_{0}^{-1}(\mathcal{E})}$${\operatorname{Par}_{0}(\mathcal{E})}$${0.}$

###### Proof.

First we prove that
$p_{*}^{\Gamma}(\widetilde{\mathcal{B}}_{0}^{-1}(\widetilde{\mathcal{E}}))\cong{}^{par}\widetilde{\mathcal{B}}_{0}^{-1}(\mathcal{E})$.
Consider the short exact sequence

$0\longrightarrow
p_{*}(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T})\longrightarrow
p_{*}(\widetilde{\mathcal{B}}_{0}^{-1}(\widetilde{\mathcal{E}}))\longrightarrow
p_{*}(\operatorname{End}_{0}(\widetilde{\mathcal{E}}))\longrightarrow 0.$

Taking invariants with respect to $\Gamma$, we get

${0}$${p_{*}(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T})}$${p_{*}\widetilde{\mathcal{B}}_{0}^{-1}(\widetilde{\mathcal{E}})}$${p_{*}(\operatorname{End}_{0}(\widetilde{\mathcal{E}}))}$${0}$${0}$${p_{*}^{\Gamma}(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T})}$${p_{*}^{\Gamma}(\widetilde{\mathcal{B}}_{0}^{-1}(\widetilde{\mathcal{E}}))}$${p_{*}^{\Gamma}(\operatorname{End}_{0}(\widetilde{\mathcal{E}}))}$${0}$${0}$${p_{*}^{\Gamma}(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T})}$${p_{*}^{\Gamma}({}^{tr}\widetilde{\mathcal{A}}(\widetilde{\mathcal{E}})^{-1})}$${p_{*}^{\Gamma}(\operatorname{At}_{\widetilde{X}/T}(\widetilde{\mathcal{E}}))}$${0.}$

The inclusion of the second row into the third follows from the invariant
pushforward of the first two rows of the diagram (24).

Since both ${}^{par}\widetilde{\mathcal{B}}_{0}^{-1}(\mathcal{E})$ and
$p_{*}^{\Gamma}(\widetilde{\mathcal{B}}_{0}^{-1}(\widetilde{\mathcal{E}}))$
are extensions of
$p_{*}^{\Gamma}(\operatorname{End}_{0}(\widetilde{\mathcal{E}}))$ by
$p_{*}^{\Gamma}(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T})$
obtained as a sub extension of
$p_{*}^{\Gamma}(\operatorname{At}_{\widetilde{X}/T}(\widetilde{\mathcal{E}}))$
by
$p_{*}^{\Gamma}(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T})$
via the inclusion
$p_{*}^{\Gamma}(\operatorname{End}_{0}(\widetilde{\mathcal{E}}))\hookrightarrow
p_{*}^{\Gamma}(\operatorname{At}_{\widetilde{X}/T}(\widetilde{\mathcal{E}}))$,
it follows that
$p_{*}^{\Gamma}(\widetilde{\mathcal{B}}_{0}^{-1}(\widetilde{\mathcal{E}}))\cong{}^{par}\widetilde{\mathcal{B}}_{0}^{-1}(\mathcal{E}).$

Now the above gives the following commutative diagram:

${0}$${p_{*}(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T})}$${p_{*}\widetilde{\mathcal{B}}_{0}^{-1}(\widetilde{\mathcal{E}})}$${p_{*}(\operatorname{End}_{0}(\widetilde{\mathcal{E}}))}$${0}$${0}$${p_{*}^{\Gamma}\left(\operatorname{End}(\widetilde{\mathcal{E}}))\otimes\Omega_{\widetilde{X}/T}\right)}$${{}^{par}\widetilde{\mathcal{B}}_{0}^{-1}(\mathcal{E})}$${\operatorname{Par}_{0}(\mathcal{E})}$${0.}$

Pushing forward with respect to the trace of endomorphism $\operatorname{tr}$,
the above commutative diagram implies the existence of the following
commutative diagram all of whose rows are short exact sequences:

${p_{*}(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T})}$${p_{*}\widetilde{\mathcal{B}}_{0}^{-1}(\widetilde{\mathcal{E}})}$${p_{*}(\operatorname{End}_{0}(\widetilde{\mathcal{E}}))}$${p_{*}^{\Gamma}(\operatorname{End}(\widetilde{\mathcal{E}}))\otimes\Omega_{\widetilde{X}/T})}$${{}^{par}\widetilde{\mathcal{B}}_{0}^{-1}(\mathcal{E})}$${\operatorname{Par}_{0}(\mathcal{E})}$${p_{*}\Omega_{\widetilde{X}/T}}$${p_{*}\mathcal{B}_{0}^{-1}(\widetilde{\mathcal{E}})}$${p_{*}(\operatorname{End}_{0}(\widetilde{\mathcal{E}}))}$${\Omega_{X/T}}$${{}^{par}\mathcal{B}_{0}^{-1}(\mathcal{E})}$${\operatorname{Par}_{0}(\mathcal{E}).}$$\scriptstyle{p_{*}\operatorname{tr}}$$\scriptstyle{\operatorname{tr}_{*}}$$\scriptstyle{\operatorname{tr}}$$\scriptstyle{\operatorname{tr}_{*}}$

The bottom level of the above diagram gives the required result. ∎

Recall that we have a diagram relating the families of curves parametrized by
$T$:

${\widetilde{X}}$${X}$${T}$$\scriptstyle{p}$$\scriptstyle{\widetilde{\pi}}$$\scriptstyle{\pi}$

Taking $R^{1}\pi_{*}$ of all the terms of the commutative diagram in
Proposition 4.4, and using the fact that $R^{1}\widetilde{\pi}=R^{1}\pi\circ
p_{*}$, we get the following proposition:

######  Proposition 4.5.

The following diagram is commutative:

${0}$${R^{1}\widetilde{\pi}_{*}\Omega_{\widetilde{X}/T}\simeq\mathcal{O}_{T}}$${R^{1}\widetilde{\pi}_{*}(\mathcal{B}_{0}^{-1}(\widetilde{\mathcal{E}}))}$${R^{1}\widetilde{\pi}_{*}\operatorname{End}_{0}(\widetilde{{\mathcal{E}}})}$${0}$${0}$${R^{1}{\pi}_{*}\Omega_{{X}/T}\simeq\mathcal{O}_{T}}$${R^{1}{\pi}_{*}({}^{par}\mathcal{B}_{0}^{-1}({\mathcal{E}}))}$${R^{1}{\pi}_{*}\operatorname{Par}_{0}({{\mathcal{E}}})}$${0.}$

###  4.2. Parabolic Atiyah algebras and moduli of parabolic bundles

Let $\mathcal{C}\rightarrow S$ be a family of curves, and let
$\widehat{\mathcal{C}}\rightarrow S$ be a family of $\Gamma$-covers. Consider
the relative family $\widehat{M}^{s}_{\mathsf{SL}_{r}}$ parametrizing the
moduli space of stable $\mathsf{SL}_{r}$ bundles on
$\widehat{\mathcal{C}}\rightarrow S$. Let
$M^{\bm{\tau},s}_{\mathsf{SL}_{r},s}$ and ${M}^{par,s}_{\mathsf{SL}_{r}}$ be
the relative moduli spaces of stable $(\Gamma,\mathsf{SL}_{r})$ and stable
parabolic bundles on $\widehat{\mathcal{C}}\rightarrow S$ and
$\mathcal{C}\rightarrow S$ respectively. Without loss of generality assume
that the interior of $\widehat{M}_{\mathsf{SL}_{r}}^{ss}$ is non-empty,
otherwise our main theorem is trivially true.

Now by the discussion in Appendix B, the invariant pushforward functor induces
an isomorphism between $M^{\bm{\tau},s}_{\mathsf{SL}_{r}}$ (respectively, the
semi-stable moduli space $M^{\bm{\tau},ss}_{\mathsf{SL}_{r}}$) and
${M}^{par,s}_{\mathsf{SL}_{r}}$ (respectively,
${M}^{par,ss}_{\mathsf{SL}_{r}}$). The data ${\bm{\tau}}$ and the covering
family $\widehat{\mathcal{C}}$ depend on the data of the parabolic weights
that defines the parabolic semistability. By [16], we get a map
$\phi:M^{\bm{\tau},ss}_{\mathsf{SL}_{r}}\rightarrow\widehat{M}^{ss}_{\mathsf{SL}_{r}}$
which extends to a map
$\widehat{\mathcal{C}}\times_{S}M^{\bm{\tau},ss}_{\mathsf{SL}_{r}}\rightarrow\widehat{\mathcal{C}}\times_{S}\widehat{M}^{ss}_{\mathsf{SL}_{r}}$.
The map $\phi$ may not preserve stability, however $\phi$ being finite, the
complement of the inverse image
$Y^{\bm{\tau},s}_{\mathsf{SL}_{r}}:=\phi^{-1}\widehat{M}^{s}_{\mathsf{SL}_{r}}$
of the stable locus has codimension at least twoa in
$M_{\mathsf{SL}_{r}}^{\tau,s}$ provided genus of the orbifold curve
$\mathscr{C}=[\widehat{C}/\Gamma]$ is at least $2$ if $r\neq 2$ and at least
$3$ if $r=2$. We refer the reader to Lemma C.3 for more details.

We have the following diagrams that connect all the objects described above:

${\widehat{\mathcal{C}}\times_{S}\widehat{M}^{s}_{\mathsf{SL}_{r}}}$${\widehat{M}^{s}_{\mathsf{SL}_{r}}}$${{\widehat{\mathcal{C}}\times_{S}M^{\bm{\tau},s}_{\mathsf{SL}_{r}}\atop{\overset{\rotatebox{90.0}{$\subseteq$}}{\widehat{\mathcal{C}}\times_{S}Y^{\bm{\tau},s}_{\mathsf{SL}_{r}}}}}}$${{M^{\bm{\tau},s}_{\mathsf{SL}_{r}}\atop{\overset{\rotatebox{90.0}{$\subseteq$}}{Y^{\bm{\tau},s}_{\mathsf{SL}_{r}}}}}}$${{\mathcal{C}\times_{S}{M}^{par,s}_{\mathsf{SL}_{r}}\atop{\overset{\rotatebox{90.0}{$\subseteq$}}{{\mathcal{C}\times_{S}Y_{\mathsf{SL}_{r}}^{par,s}}}}}}$${{{M}^{par,s}_{\mathsf{SL}_{r}}\atop{\overset{\rotatebox{90.0}{$\subseteq$}}{{Y_{\mathsf{SL}_{r}}^{par,s}}}}}}$${\widehat{\mathcal{C}}}$${S}$${\widehat{\mathcal{C}}}$${S}$${\mathcal{C}}$${S}$$\scriptstyle{\widehat{\pi}_{w}}$$\scriptstyle{\widehat{\pi}_{n}}$$\scriptstyle{\widehat{\pi}_{e}}$$\scriptstyle{\operatorname{id}\times\phi}$$\scriptstyle{\operatorname{id}\times\phi}$$\scriptstyle{p\times\cong}$$\scriptstyle{\widetilde{\pi}_{n}}$$\scriptstyle{\cong}$$\scriptstyle{\phi}$$\scriptstyle{\phi}$$\scriptstyle{{\pi}_{n}}$$\scriptstyle{\widehat{\pi}_{s}}$$\scriptstyle{\widehat{\pi}_{e}}$$\scriptstyle{\widetilde{\pi}_{w}}$$\scriptstyle{\widetilde{\pi}_{s}}$$\scriptstyle{p}$$\scriptstyle{\widetilde{\pi}_{e}}$$\scriptstyle{{\pi}_{s}}$$\scriptstyle{{\pi}_{w}}$$\scriptstyle{{\pi}_{e}}$

The rational maps are regular over $Y_{\mathsf{SL}_{r}}^{\bm{\tau},s}$ which
will be also denoted by the same notation. The image of
$Y_{\mathsf{SL}_{r}}^{\bm{\tau},s}$ under the invariant pushforward
isomorphism of $M_{\mathsf{SL}_{r}}^{\bm{\tau},ss}\cong
M_{\mathsf{SL}_{r}}^{par,ss}$ will be denoted by
$Y_{\mathsf{SL}_{r}}^{par,s}$. By definition
$\phi:Y_{\mathsf{SL}_{r}}^{par,s}\rightarrow M_{\mathsf{SL}_{r}}^{par,s}$.

Let $\widehat{\mathcal{E}}$ be the universal bundle on
$\widehat{\mathcal{C}}\times_{S}\widehat{M}^{s}_{\mathsf{SL}_{r}}$ and
$\widetilde{\mathcal{E}}$ be its pull-back to
$\widehat{\mathcal{C}}\times_{S}Y^{\bm{\tau},s}_{\mathsf{SL}_{r}}$. We denote
by $\mathcal{E}$ the universal parabolic bundle which we can assume to exist
without loss of generality (see Remark 2.7).

Let $\mathcal{L}$ be the determinant of cohomology line bundle on
$\widehat{M}^{s}_{\mathsf{SL}_{r}}$. Now, as before, combining the results of
Baier-Bolognesi-Martens-Pauly [7], Beilinson-Schechtman [13], Bloch-Esnault
[20], and Sun-Tsai [61], we get an isomorphism of ${\rm At}(\mathcal{L})$ with
$R^{1}\widehat{\pi}_{n*}(\mathcal{B}_{0}^{-1}(\widehat{\mathcal{E}}))$ which
makes the diagram of fundamental sequences of Atiyah algebras commute:

${0}$${\mathcal{O}_{\widehat{M}^{s}_{\mathsf{SL}_{r}}}}$${{\rm
At}(\mathcal{L}^{-1})}$${\mathcal{T}_{\widehat{M}^{s}_{\mathsf{SL}_{r}}/S}}$${0}$${0}$${\mathcal{O}_{\widehat{M}^{s}_{\mathsf{SL}_{r}}}}$${R^{1}\widehat{\pi}_{n*}(\mathcal{B}_{0}^{-1}(\widehat{\mathcal{E}}))}$${\mathcal{T}_{\widehat{M}^{s}_{\mathsf{SL}_{r}}/S}}$${0.}$$\scriptstyle{\cong}$

Pulling back by $\phi$ we get an isomorphism of the sheaves $\phi^{*}{\rm
At}(\mathcal{L})$ and
$R^{1}\widetilde{\pi}_{n*}(\mathcal{B}_{0}^{-1}(\widetilde{\mathcal{E}}))$
which in turn is isomorphic to
$\phi^{*}R^{1}\widehat{\pi}_{n*}(\mathcal{B}_{0}^{-1}(\widehat{\mathcal{E}}))$.

For the map
$\phi:Y^{\bm{\tau},s}_{\mathsf{SL}_{r}}\rightarrow\widehat{M}^{s}_{\mathsf{SL}_{r}}$,
its differential
$d\phi:\mathcal{T}_{{Y}^{\bm{\tau},s}_{\mathsf{SL}_{r}}/S}\rightarrow\mathcal{T}_{{M}^{s}_{{\mathsf{SL}_{r}}/S}}$
is given by $R^{1}\pi_{n*}\operatorname{Par}_{0}(\mathcal{E})\rightarrow
R^{1}\widehat{\pi}_{n*}\operatorname{End}_{0}(\widehat{\mathcal{E}})$. Recall
if $\phi:Y\rightarrow M$ is a smooth map, the pull back in the category of
Atiyah algebras is defined by the formula

$\phi^{\bullet}\mathcal{A}:=\phi^{*}\mathcal{A}\times_{\mathcal{T}_{M/S}}\mathcal{T}_{Y/S},$

where $\mathcal{A}$ is a relative Atiyah algebra. We have the following
result.

######  Theorem 4.6.

There is an isomorphism of the Atiyah algebra ${\rm At}(\phi^{*}\mathcal{L})$
with $R^{1}\pi_{n*}({}^{par}\mathcal{B}_{0}^{-1}(\mathcal{E}))$ that restricts
to the identity map on $\Omega_{M^{par,s}_{\mathsf{SL}_{r}}}$.

###### Proof.

Applying Proposition 4.5 with
$\widetilde{X}=\widehat{\mathcal{C}}\times_{S}Y^{\bm{\tau},s}_{\mathsf{SL}_{r}}$,
$X=\mathcal{C}\times_{S}Y^{\bm{\tau},s}_{\mathsf{SL}_{r}}$, and
$T=Y^{\bm{\tau},s}_{\mathsf{SL}_{r}}$, we get an isomorphism between
$\operatorname{At}(\phi^{*}\mathcal{L})$ and
$R^{1}\pi_{n*}{}^{par}\mathcal{B}_{0}^{-1}(\mathcal{E})$ over
$Y^{par,s}_{\mathsf{SL}_{r}}$. Now both these sheaves
$\operatorname{At}({\phi^{*}\mathcal{L}})$ and
$R^{1}\pi_{n*}({}^{par}\mathcal{B}_{0}^{-1}(\mathcal{E}))$ are locally free
(hence reflexive) and extend over $M^{par,s}_{\mathsf{SL}_{r}}$. Since they
are isomorphic on an open subset whose complement has codimension at least
two, the isomorphism actually extends to all of $M_{\mathsf{SL}_{r}}^{par,s}$.
∎

######  Remark 4.7.

The proof of Theorem 2.14 applies in this parabolic setting as well. The fact
that the Atiyah-Bott symplectic form is in the class of
$\phi^{\ast}\mathcal{L}$ is one of the results of [21] (see also [15]).

###  4.3. Parabolic $G$-bundles

####  4.3.1. Parabolic bundles

Let $p:\widehat{C}\to C$ be a ramified covering with Galois group $\Gamma$, so
$C=\widehat{C}/\Gamma$. Let $D\subset C$ and $\widehat{D}\subset\widehat{C}$
denote the branching loci in $C$ and $\widehat{C}$ respectively. Let
$\widehat{\pi}:\widehat{P}\to\widehat{C}$ be a $\Gamma$-principal $G$-bundle,
i.e., $\widehat{P}$ is a principal $G$-bundle, and there is a representation
$\Gamma\to\operatorname{Aut}(\widehat{P}):\gamma\mapsto F_{\gamma}$ such that
the actions of $\Gamma$ and $G$ on $\widehat{P}$ commute. For an open set
$U\subset C$, define

$\operatorname{aut}_{\Gamma}(\widehat{P})(U)=\left\\{X\in\operatorname{aut}(\widehat{P})(p^{-1}(U))\mid
X(F_{\gamma}(p))=(F_{\gamma})_{\ast}(X(p))\ ,\ \forall\
\gamma\in\Gamma\right\\}.$

This presheaf defines an $\mathcal{O}_{C}$-coherent sheaf. Notice that for
$X\in\operatorname{aut}_{\pi}(\widehat{P})$, the vector field $\hat{v}$ on
$\widehat{C}$ is $\Gamma$-invariant, and so descends to a vector field on $C$
that vanishes along the ramification divisor $D\subset C$. Hence, we have a
map $\operatorname{aut}_{\Gamma}(\widehat{P})\to TC(-D)$.

Let $P$ be a parabolic ${G}$-bundle on $C$ and let $\widehat{P}$ be a
$(\Gamma,{G})$-bundle on $\widehat{C}$ such that $P$ and $\widehat{P}$ are
related by invariant pushforward.

######  Definition 4.8.

We define the parabolic Atiyah algebra ${}^{par}{\rm At}(P)$ of $P$ to be
$\operatorname{aut}_{\Gamma}(\widehat{P})$.

Suppose $\widehat{P}$ is the frame bundle of a vector bundle
$\widehat{\mathcal{E}}\to\widehat{C}$, and let $\mathcal{E}\to C$ denote the
sheaf of invariant sections of $\widehat{\mathcal{E}}$. The parabolic Atiyah
algebra $\operatorname{{}^{\emph{par}}At}(\mathcal{E})$ is
$\operatorname{aut}_{\Gamma}(\widehat{P})$.

We wish to describe the kernel $\operatorname{ad}_{\Gamma}(\widehat{P})$ as a
subsheaf of $\operatorname{aut}_{\Gamma}(\widehat{P})\to TC(-D)$. Let
$\widehat{U}\subset\widehat{C}$ be a neighborhood of $w_{0}\in\widehat{D}$,
and let $\Gamma_{0}\subset\Gamma$ be the isotropy group of $w_{0}$. We assume
that $\Gamma_{0}$ stabilizes $\widehat{U}$ and only $w_{0}\in\widehat{U}$ has
nontrivial isotropy. Choose a section $s$ of $\widehat{P}$ over $\widehat{U}$.
For each $\gamma\in\Gamma_{0}$ there is $\rho_{\gamma}:\widehat{U}\to G$
defined by: $F_{\gamma}(s(w))=s(\gamma w)\rho_{\gamma}(w)$.

######  Definition 4.9.

Define

$\operatorname{ad}_{\Gamma}(\widehat{P})(U)=\left\\{f\in\operatorname{ad}(\widehat{P})(\widehat{U})\mid
f(s(\gamma w))=\operatorname{Ad}_{\rho_{\gamma}(w)}f(s(w))\right\\}.$

The following is straightforward.

######  Proposition 4.10.

The above definition is independent of the choice of section $s$.

Extending the definition for neighborhoods at each of the branch points
defines $\operatorname{ad}_{\Gamma}(\widehat{P})$ globally.

We also note that the inclusion map
$\operatorname{ad}(\widehat{P})\to\operatorname{aut}(\widehat{P})$ restricts
to an inclusion map
$\operatorname{ad}_{\Gamma}(\widehat{P})\to\operatorname{aut}_{\Gamma}(\widehat{P})$.
Indeed, for $f\in\operatorname{ad}_{\Gamma}(\widehat{P})$, it suffices to
check the condition on the vector field $X=f^{\sharp}$ along the section $s$,
and this follows from the equivariance of $F_{\gamma}$:

$f(F_{\gamma}(s(w))=f(s(\gamma
w)\rho_{\gamma}(w))=\operatorname{Ad}_{\rho_{\gamma}(w)^{-1}}f(s(\gamma
w))=f(s(w)).$

Hence we have the following short exact sequence of sheaves of Lie algebras on
$C$:

$0\longrightarrow\operatorname{ad}_{\Gamma}(\widehat{P})\longrightarrow\operatorname{aut}_{\Gamma}(\widehat{P})\longrightarrow
TC(-D)\longrightarrow 0.$

Suppose that $\widehat{P}$ is the frame bundle of a vector bundle
$\widehat{\mathcal{E}}\to\widehat{C}$, and let $\mathcal{E}\to C$ denote the
sheaf of invariant sections of $\widehat{\mathcal{E}}$. The parabolic
endomorphism bundle of $\mathcal{E}$, namely
$\operatorname{Par}(\mathcal{E})$, is
$\operatorname{ad}_{\Gamma}(\widehat{P})$. We refer to the above sequence as
the fundamental sequence for parabolic Atiyah algebras.

From the above discussion, we have an exact sequence

$0\,\longrightarrow\,\operatorname{Par}(\mathcal{E})\,\longrightarrow\
\operatorname{{}^{\emph{par}}At}(\mathcal{E})\longrightarrow
TC(-D)\longrightarrow 0.$

We note that this is nothing but the $\Gamma$-invariant push-forward of the
Atiyah algebra exact sequence on $\widehat{C}$.

####  4.3.2. Strongly parabolic Atiyah algebras

Next, we define a “strongly parabolic” version of this construction. Set

$\displaystyle\operatorname{aut}(\widehat{P})(-\widehat{D})$
$\displaystyle:=\\{X\in\operatorname{aut}(\widehat{P})\mid X(p)=0\text{ for
}\widehat{\pi}(p)\in\widehat{D}\\}.$
$\displaystyle\operatorname{ad}(\widehat{P})(-\widehat{D})$
$\displaystyle:=\\{f\in\operatorname{ad}(\widehat{P})\mid f(p)=0\text{ for
}\widehat{\pi}(p)\in\widehat{D}\\}.$

Then we have the restricted short exact sequence

$0\longrightarrow\operatorname{ad}(\widehat{P})(-\widehat{D})\longrightarrow\operatorname{aut}(\widehat{P})(-\widehat{D})\longrightarrow
T\widehat{C}(-\widehat{D})\longrightarrow 0.$

Now if $\widehat{P}$ is the frame bundle of $\widetilde{E}$, then by
Seshadri’s correspondence (see Appendix B), the $\Gamma$-invariant part
$\operatorname{ad}_{\Gamma}(\widehat{P}(-\widehat{D}))$ is identified with the
_strongly_ parabolic endomorphisms $\operatorname{SPar}(P)$. Motivated by this
observation, we have:

######  Definition 4.11.

The strongly parabolic Atiyah algebra ${}^{spar}{\rm At}(P)$ of a
$\Gamma$-linearized principal bundle $\widehat{P}$ on $\widehat{C}$ is defined
to be the $\Gamma$-invariant part of
$\operatorname{aut}(\widehat{P})(-\widehat{D})$.

If $\widehat{P}$ is the frame bundle of a vector bundle
$\widehat{\mathcal{E}}$, then we denote it by
$\operatorname{{}^{\emph{spar}}At}(\mathcal{E}):=\operatorname{aut}_{\Gamma}(\widehat{P})$.
Noting that $p_{\ast}(T\widehat{C}(-\widehat{D}))^{\Gamma}=TC(-D)$, we have
the following short exact sequence on $C$

$0\longrightarrow\operatorname{SPar}(\mathcal{E})\longrightarrow\operatorname{{}^{\emph{spar}}At}(\mathcal{E})\longrightarrow
TC(-D)\longrightarrow 0.$

####  4.3.3. Determinant line bundle for parabolic $G$-bundles

Let $\widehat{\mathcal{P}}$ be a family of $(\Gamma,G)$-bundles parametrized
by $T$ as in the previous section, and let $\mathcal{P}$ be the family of
parabolic $G$-bundles obtained by applying the invariant pushforward functor.

Consider the relative parabolic Atiyah algebra
$\operatorname{{}^{\emph{par}}At}_{\widetilde{X}/T}(\mathcal{P}):=p_{*}^{\Gamma}({\rm
At}_{\widetilde{X}/T}(\widehat{\mathcal{P}})$ and the strongly parabolic
Atiyah algebra
$\operatorname{{}^{\emph{spar}}At}_{\widetilde{X}/T}({\mathcal{P}})=p_{*}^{\Gamma}({\rm
At}_{\widetilde{X}/T}(\widehat{\mathcal{P}})(-\widehat{D}))$. As in the case
of parabolic vector bundles, they fit in the following fundamental exact
sequences

(51)
$\displaystyle\begin{split}0\longrightarrow\operatorname{Par}(\mathcal{P})\longrightarrow\operatorname{{}^{\emph{par}}At}_{{X}/T}(\mathcal{P})\longrightarrow\mathcal{T}_{{X}/T}(-D)\longrightarrow
0,\\\
0\longrightarrow\operatorname{SPar}(\mathcal{P})\longrightarrow\operatorname{{}^{\emph{spar}}At}_{{X}/T}(\mathcal{P})\longrightarrow\mathcal{T}_{{X}/T}(-D)\longrightarrow
0,\end{split}$

where $\operatorname{Par}(\mathcal{P})$ (respectively,
$\operatorname{SPar}(\mathcal{P})$) denote the parabolic (respectively,
strongly parabolic) endomorphism bundle of $\mathcal{P}$. As in the case of
parabolic vector bundles we get the following quasi-Lie algebra

(52)
$0\longrightarrow\Omega_{\widetilde{X}/T}\longrightarrow(\operatorname{{}^{\emph{par}}At}_{\widetilde{X}/T}(\mathcal{P})(D))^{\vee}\longrightarrow(\operatorname{SPar}(\mathcal{P})(D))^{\vee}\longrightarrow
0.$

The Cartan-Killing form $\kappa_{\mathfrak{g}}$ gives an identification

(53)
$\nu_{\mathfrak{g}}^{-1}:(\operatorname{SPar}(\mathcal{P})(D))^{\vee}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\sim$}}\,}\operatorname{Par}(\mathcal{P}).$

Pulling back the exact sequence in (52) by the above isomorphism, we get a
quasi-Lie algebra $\widetilde{\operatorname{SPar}{\rm At}}_{X/T}(\mathcal{P})$
fitting into the following exact sequence:

${0}$${\Omega_{{X}/T}}$${(p_{*}^{\Gamma}({\rm
At}_{\widetilde{X}/T}(\widehat{\mathcal{P}})(-\widehat{D}))(D))^{\vee}}$${{(\operatorname{SPar}(\mathcal{P})(D))^{\vee}\cong(p_{*}^{\Gamma}(\operatorname{ad}(\widehat{\mathcal{P}})(-\widehat{D}))(D))^{\vee}}}$${0}$${0}$${\Omega_{{X}/T}}$${\widetilde{\operatorname{SPar}{\rm
At}}_{X/T}(\mathcal{P})}$${\operatorname{Par}(\mathcal{P})}$${0}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

which should be considered as a parabolic $G$-bundle analog of Ginzburg’s dgla
considered in [27]. Recall that we have the quasi-Lie algebra
$\mathfrak{S}^{-1}(\widehat{\mathcal{P}})$ associated to a family
$\widehat{\mathcal{P}}$ of principal $G$-bundles satisfying:

${0}$${\Omega_{\widetilde{X}/T}}$${{\rm
At}_{\widetilde{X}/T}(\widehat{\mathcal{P}})^{\vee}}$${\operatorname{ad}(\widehat{\mathcal{P}})^{\vee}}$${0}$${0}$${\Omega_{\widetilde{X}/T}}$${\mathfrak{S}^{-1}(\widehat{\mathcal{P}})}$${\operatorname{ad}(\widehat{\mathcal{P}})}$${0.}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

Now since $\widehat{\mathcal{P}}$ is $\Gamma$-linearized, we conclude that all
objects in the above exact sequence are $\Gamma$-linearized. We define
${}^{par}\mathfrak{S}^{-1}(P):=p_{*}^{\Gamma}\mathfrak{S}^{-1}(\widehat{\mathcal{P}})$.

Taking $\Gamma$-invariant pushforward of the bottom row, we get the following
extension

$0\longrightarrow\Omega_{X/T}\longrightarrow{}^{par}\mathfrak{S}^{-1}(\mathcal{P})\longrightarrow\operatorname{Par}(\mathcal{P})\longrightarrow
0.$

We now have the following proposition:

######  Proposition 4.12.

There is an isomorphism
${}^{par}\mathfrak{S}^{-1}(\mathcal{P})\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\sim$}}\,}\widetilde{\operatorname{SPar}{\rm
At}}_{X/T}(\mathcal{P})$ which induces identity maps on $\Omega_{X/T}$ and
$\operatorname{Par}(\mathcal{P})$.

###### Proof.

Recall that there is a natural nondegenerate pairing

$\langle\,,\,\rangle:\mathfrak{S}^{-1}(\widehat{\mathcal{P}})\times{\rm
At}_{\widetilde{X}/T}(\widehat{\mathcal{P}})\rightarrow\mathcal{O}_{\widetilde{X}}.$

Tensoring it with $\mathcal{O}_{\widetilde{X}}(-\widehat{D})$, we get a
$\mathcal{O}_{\widetilde{X}}(-\widehat{D})$-valued pairing

$\langle\,,\,\rangle:\mathfrak{S}^{-1}(\widehat{\mathcal{P}})\times{\rm
At}_{\widetilde{X}/T}(\widehat{\mathcal{P}})(-\widehat{D})\longrightarrow\mathcal{O}_{\widetilde{X}}(-\widehat{D}).$

Taking invariant pushforward, we get the following nondegenerate pairing:

$p_{*}^{\Gamma}\langle\,,\,\rangle:p_{*}^{\Gamma}\mathfrak{S}^{-1}(\widehat{\mathcal{P}})\times
p_{*}^{\Gamma}({\rm
At}_{\widetilde{X}/T}(\widehat{\mathcal{P}})(-\widehat{D}))\longrightarrow
p_{*}^{\Gamma}\mathcal{O}_{\widetilde{X}}(-\widehat{D}).$

This produces a duality between ${}^{par}\mathfrak{S}^{-1}(\mathcal{P})$ and
$p_{*}^{\Gamma}({\rm