# A parabolic analog of a theorem of Beilinson and Schechtman

Indranil Biswas  Mathematics Department, Shiv Nadar University, NH91, Tehsil
Dadri, Greater Noida, Uttar Pradesh 201314, India<EMAIL_ADDRESS><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 an explicit
formula to compute the Atiyah class of parabolic determinant of cohomology
line bundle on the moduli space of parabolic $G$-bundles. This generalizes an
earlier result of Beilinson-Schechtman.

###### 2020 Mathematics Subject Classification:

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

I.B. is supported in part by a J.C. Bose fellowship and both I.B. and S.M. are
also partly supported by DAE, India under project no.
1303/3/2019/R&D/IIDAE/13820. S.M. received additional funding from the Science
and Engineering Research Board, India (SRG/2019/000513). R.W. is supported in
part by National Science Foundation grants DMS-1906403 and DMS-2204346.

##  1\. Introduction

Lie algebroids play an important role in the geometry of sheaves on manifolds.
For the case of Atiyah algebras associated to principal bundles, the Atiyah
exact sequence packages the information of connections, and more generally
twisted differential operators, on the bundle. Of particular interest are
Atiyah algebras associated to line bundles on moduli spaces of bundles on
curves. These are almost always constructed via descent from a bigger
parameter space. A fundamental question is therefore how to relate the
behavior of Atiyah algebras of these natural line bundles to infinitesimal
joint deformations of the moduli spaces under this correspondence.

This question was addressed in the fundamental work of Beilinson-Schechtman
[5]. In the context of relative moduli stacks of vector bundles over families
of smooth projective curves, the main result of [5] describes the Atiyah
algebra of the determinant of cohomology in terms of a direct image of a
_trace complex_ constructed from the Atiyah algebra of the universal bundle.
This construction is closely related to the “localization functor” [4, 7].

In [15], Ginzburg gave an alternative construction that is more amenable to
the case of principal bundles. This is based on a general correspondence
between quasi-Lie algebras and certain differential graded Lie algebras.
Applied to the moduli problem, this time for principal bundles, it can be seen
from work of Bloch-Esnault [12] that the direct image of the dgla constructed
by Ginzburg also computes the Atiyah algebra of the determinant of cohomology.

The main goal of this paper is to extend these constructions to the case of
moduli stacks of principal bundles with parabolic structures. In order to
state the result, let us introduce some notation. Let
$\mathcal{C}\,\longrightarrow\,S$ be a versal family of smooth projective
curves with $n$ marked points $p_{1},\,\cdots,\,p_{n}$. Fix a simple, simply
connected complex algebraic group $G$ with Lie algebra $\mathfrak{g}$. Choose
parabolic subgroups $P_{1},\,\cdots,\,P_{n}$ of $G$ and associated weights
$\bm{\alpha}\,=\,(\alpha_{1},\,\cdots,\,\alpha_{n})$. Let
$M_{G}^{\bm{\alpha},rs}\,=\,M_{G}^{\bm{\alpha},rs}(\mathcal{C}/S)\rightarrow
S$ be the relative moduli space (over $S$) of regularly stable parabolic
$G$-bundles. On $\mathcal{C}\times_{S}M_{G}^{\bm{\alpha},rs}$ there exists
local universal bundles $\mathcal{P}$. Let
${}^{par}\mathfrak{S}^{\bullet}_{\mathcal{C}\times_{S}M_{G}^{\bm{\alpha},rs},\pi,S}(\mathcal{P})$
denote the relative Ginzburg complex associated to $\mathcal{P}$. These local
complexes glue together to give a global complex on
$\mathcal{C}\times_{S}M_{G}^{\bm{\alpha},rs}$ (even though the global
$\mathcal{P}$ does not exist).

On the other hand, we consider the sheaf of strongly parabolic relative Atiyah
algebras
${}^{spar}\operatorname{At}_{\mathcal{C}\times_{S}M_{G}^{\bm{\alpha},rs}/M_{G}^{\bm{\alpha},rs}}(\mathcal{P})$
defined via the parabolic orbifold correspondence satisfying the short exact
sequence

$0\rightarrow\operatorname{SPar}(\mathcal{P})\rightarrow{}^{spar}\operatorname{At}_{\mathcal{C}\times_{S}M_{G}^{\bm{\alpha},rs}/M_{G}^{\bm{\alpha},rs}}(\mathcal{P})\rightarrow\mathcal{T}_{\mathcal{C}\times_{S}M_{G}^{\bm{\alpha},rs}/M_{G}^{\bm{\alpha},rs}}(-D)\rightarrow
0,$

where $\operatorname{SPar}(\mathcal{P})$ is the sheaf of strongly parabolic
endomorphisms and $D$ is the divisor of marked points. Now consider the sheaf
of quasi Lie algebras
${}^{spar}\widetilde{\operatorname{At}}_{\mathcal{C}\times_{S}M_{G}^{\bm{\alpha},rs}/M_{G}^{\bm{\alpha},rs}}(\mathcal{P})$
obtained as the pull back of the following sequence

$0\rightarrow{\Omega}_{\mathcal{C}\times_{S}M_{G}^{\bm{\alpha},rs}/M_{G}^{\bm{\alpha},rs}}\rightarrow({}^{spar}{\operatorname{At}}_{\mathcal{C}\times_{S}M_{G}^{\bm{\alpha},rs}/M_{G}^{\bm{\alpha},rs}}(\mathcal{P})(D))^{\vee}\rightarrow(\operatorname{SPar}(\mathcal{P})(D))^{\vee}\rightarrow
0,$

via the isomorphism of sheaf of parabolic endomorphisms
$\operatorname{Par}(\mathcal{P})\cong(\operatorname{SPar}(\mathcal{P})(D))^{\vee}$.
Moreover by construction
$R^{1}\pi_{*}{}^{spar}\widetilde{\operatorname{At}}_{\mathcal{C}\times_{S}M_{G}^{\bm{\alpha},rs}/M_{G}^{\bm{\alpha},rs}}(\mathcal{P})$
is a sheaf of relative Atiyah algebras on $M_{G}^{\bm{\alpha},rs}$, where
$\pi:\mathcal{C}\times_{S}M_{G}^{\bm{\alpha},rs}\rightarrow
M_{G}^{\bm{\alpha},rs}$ is the projection. We first show that

$R^{1}\pi_{*}({}^{par}\mathfrak{S}^{\bullet}_{\mathcal{C}\times_{S}M_{G}^{\bm{\alpha},rs},\pi,S}(\mathcal{P}))\simeq
R^{1}\pi_{\ast}({}^{par}\mathfrak{S}^{-1}_{\mathcal{C}\times_{S}M_{G}^{\bm{\alpha},rs}/M_{G}^{\bm{\alpha},rs}}(\mathcal{P}))\simeq
R^{1}\pi_{*}({}^{spar}\widetilde{\operatorname{At}}_{\mathcal{C}\times_{S}M_{G}^{\bm{\alpha},rs}/M_{G}^{\bm{\alpha},rs}}(\mathcal{P}))$

We refere the reader to Sections 3, 4 for more details. Finally we related it
to relative Atiyah algebras of parabolic determinant of cohomologies.

Given a nontrivial holomorphic embedding
$\phi\,:\,G\,\longrightarrow\,\mathsf{SL}_{r}$, there is an associated
determinant of cohomology line bundle
$\mathcal{L}_{\phi}\,\longrightarrow\,M_{G}^{\bm{\alpha},rs}$. Let ${\rm
At}_{M_{G}^{\bm{\alpha},rs}/S}(\mathcal{L}_{\phi})$ denote the relative Atiyah
algebra of $\mathcal{L}_{\phi}$. Then the main result of this paper is the
following.

######  Theorem 1.1.

On $M_{G}^{\bm{\alpha},rs}$ there is a natural isomorphism of Atiyah algebras

$\frac{1}{m_{\phi}}{\rm
At}_{M_{G}^{\bm{\alpha},rs}/S}(\mathcal{L}_{\phi})\,\simeq\,R^{1}\pi_{\ast}({}^{par}\mathfrak{S}^{-1}_{\mathcal{C}\times_{S}M_{G}^{\bm{\alpha},rs}/M_{G}^{\bm{\alpha},rs}}(\mathcal{P}))\simeq
R^{1}\pi_{*}({}^{spar}\widetilde{\operatorname{At}}_{\mathcal{C}\times_{S}M_{G}^{\bm{\alpha},rs}/M_{G}^{\bm{\alpha},rs}}(\mathcal{P})),$

where $m_{\phi}$ is the Dynkin index of the associated homomorphism
$\phi_{\ast}\,:\,\mathfrak{g}\,\to\,\mathfrak{sl}_{r}$ given by the ration of
the normalized Killing forms.

In a recent paper [2] the result of Beilinson-Schechtman was used in an
integral way to give an algebraic proof of the existence of a flat projective
connection (a _Hitchin connection_) on the bundle of generalized theta
functions for vector bundles on families of curves. One of the main
motivations of the present paper was to apply Theorem 1.1 in the same way to
obtain a Hitchin connection for theta functions associated to parabolic
$G$-bundles. This is carried out in [11].

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

###  2.1. Basic definitions

In this section, we recall a correspondence stated in Ginzburg [15] between
quasi-Lie algebras and their associated differential graded Lie algebras. We
also recall from Beilinson-Schechtman [5] 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

$[\ ,\
]\,:\,\widetilde{\mathfrak{g}}\times\widetilde{\mathfrak{g}}\,\longrightarrow\,\widetilde{\mathfrak{g}}.$

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 $[\ ,\ ]$, and

  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
$\mathfrak{S}\,:=\,\bigoplus_{i}\mathfrak{S}^{i}$ together with a bilinear map
$\\{\mathfrak{S}^{i},\mathfrak{S}^{j}\\}\,\subset\,\mathfrak{S}^{i+j}$ and a
differential $d\,:\,\mathfrak{S}^{i}\,\longrightarrow\,\mathfrak{S}^{i+1}$
satisfying the following:

  * •

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

  * •

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

  * •

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

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

The following lemma of Ginzburg [15, Lemma 7.7] 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, $Z$-invariant bilinear form $\langle\ ,\
\rangle\,:\,\operatorname{Sym}^{2}\widetilde{\mathfrak{g}}\,\longrightarrow\,Z$
such that the following hold:

  * •

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

  * •

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

Then there exists a dgla
$\mathfrak{S}\,=\,\mathfrak{S}^{-2}\oplus\mathfrak{S}^{-1}\oplus\mathfrak{S}^{0}$,
where $\mathfrak{S}^{0}:=\mathfrak{g}$,
$\mathfrak{S}^{-1}\,:=\,\widetilde{\mathfrak{g}}$, $\mathfrak{S}^{-2}\,:=\,Z$,
the differential is given by inclusion and quotient, and with the bracket
given by the formula $\\{x,\,y\\}\,=\,\langle x,\,y\rangle$; and
$\\{x,\,dy\\}\,=\,[x,\,y]+\langle x,\,y\rangle$ for
$x,\,y\,\in\,\widetilde{\mathfrak{g}}$. Conversely given a dgla as above
satisfying $\\{\mathfrak{S}^{-2},d\mathfrak{S}^{-1}\\}\subset\ker d$, there
exists a quasi-Lie algebra along with a symmetric $Z$-valued, $Z$-invariant
bilinear form.

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

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

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

Let $\mathcal{T}_{X,\pi}\,\subset\,\mathcal{T}_{X}$ denote the subsheaf
$d\pi^{-1}(\pi^{-1}\mathcal{T}_{T})$. Clearly the sheaf $\mathcal{T}_{X,\pi}$
has the structure of 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
$\mathcal{T}_{\pi}^{\bullet}\,=\,\bigoplus_{i}\mathcal{T}^{i},$ where
$\mathcal{T}^{i}$ is zero for $i\,\neq\,\\{0,\,-1\\}$,
$\mathcal{T}^{-1}\,:=\,\mathcal{T}_{X/T}$ and
$\mathcal{T}^{0}\,:=\,\mathcal{T}_{X,\pi}$. This dgla
$\mathcal{T}_{\pi}^{\bullet}$ carries a natural action of
$\pi^{-1}\mathcal{O}_{T}$ and a map

(2.1)
$\overline{\epsilon}\,:\,\mathcal{T}_{\pi}^{\bullet}\,\longrightarrow\,H^{0}(\mathcal{T}_{\pi}^{\bullet})\,=\,\pi^{-1}(\mathcal{T}_{T})$

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 $\pi\,:\,X\,\longrightarrow\,T$ be a family of curves as before. We
discuss the notion of $\pi$-algebras following Beilinson-Scechtman [5] which
are quasi-isomorphic to extensions of a complex of Atiyah algebras by the de
Rham complex $\Omega^{\bullet}_{X/T}$.

######  Definition 2.4 ([6, § 1.2.1]).

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
$\epsilon_{\mathcal{A}}\,:\,\mathcal{A}^{\bullet}\,\longrightarrow\,\mathcal{T}^{\bullet}_{\pi}$
that satisfies the condition
$[a,\,fb]\,=\,\overline{\epsilon}_{\mathcal{A}}(a)(f)b+f[a,b]$, where
$\overline{\epsilon}_{\mathcal{A}}\,=\,\overline{\epsilon}\circ\epsilon_{\mathcal{A}}$
(see (2.1)). 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)

$\epsilon_{\mathcal{A}}\,:\,\mathcal{A}^{\bullet}\,\longrightarrow\,\mathcal{A}^{\bullet}/\mathcal{A}^{\bullet}_{-1}\,\simeq\,\mathcal{T}^{\bullet}_{\pi}$,
and

  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

(2.2)
$0\,\longrightarrow\,\mathcal{A}^{\bullet}_{-2}\,\longrightarrow\,\mathcal{A}^{\bullet}\,\longrightarrow\,\mathcal{A}^{\bullet}/\mathcal{A}^{\bullet}_{-2}\,\longrightarrow\,0.$

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

(2.3)
$0\,\longrightarrow\,\Omega_{X/T}^{\bullet}[2]\,\longrightarrow\,\mathcal{A}^{\bullet}\,\longrightarrow\,\mathcal{A}^{\bullet}/\mathcal{A}^{\bullet}_{-2}\,\longrightarrow\,0.$

######  Proposition 2.5 ([6, § 1.2.3]).

The short exact sequence

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

defines an Atiyah algebra on $T$.

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

Let $\pi\,:\,X\,\longrightarrow\,T$ be a family of curves 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:
$\mathcal{A}_{\pi}^{-1}\,:=\,\epsilon_{\mathcal{A}}^{-1}\mathcal{T}_{X/T}$ and
$\mathcal{A}^{0}_{\pi}\,:=\,\epsilon_{\mathcal{A}}^{-1}\mathcal{T}_{X,\pi}$.
There is a canonical surjective map
$\epsilon_{\mathcal{A}}\,:\,\mathcal{A}_{\pi}^{\bullet}\,\longrightarrow\,\mathcal{T}_{\pi}^{\bullet}$
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 $[\
,]_{-1,-1}\,:\,{}^{\\#}\mathcal{A}^{-1}\otimes{}^{\\#}\mathcal{A}^{-1}\,\longrightarrow\,{}^{\\#}\mathcal{A}^{-2}\,=\,\mathcal{O}_{X}$
is a differential operator along the fibers.

There is a commutative diagram

(2.4)
${0}$${0}$${\mathcal{O}_{X}}$${{}^{\\#}\mathcal{A}^{-2}}$${0}$${0}$${\Omega_{X/T}}$${{}^{\\#}\mathcal{A}^{-1}}$${\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
$\Omega_{X/T}^{\bullet}[2]$.

Later will also need to vary $T$ with respect to $S$ and consider
$\mathcal{T}_{\pi,S}^{\bullet}$ where degree $-1$ term is same as
$\mathcal{T}_{\pi}^{\bullet}$ and the degree zero term is
$\mathcal{T}_{X,\pi,S}\,:=\,d\pi^{-1}(\pi^{-1}(\mathcal{T}_{T/S}))$. In the
relative set-up one similarly defines $\mathcal{A}_{X,\pi,S}^{\bullet}$ by
modifying the zero-th term to be
$\mathcal{A}_{X,\pi,S}^{-0}:=\epsilon_{\mathcal{A}}^{-1}\mathcal{T}_{X,\pi,S}$.
The resulting pushforward $R^{0}\pi_{*}\mathcal{A}_{X,\pi,S}^{\bullet}$ is a
relative Atiyah algebra satisfying the fundamental exact sequence

$0\rightarrow\mathcal{O}_{T}\rightarrow
R^{0}\pi_{*}(\mathcal{A}_{X,\pi,S}^{\bullet})\rightarrow\mathcal{T}_{T/S}\rightarrow
0$

###  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 $\mathfrak{g}$ and consider the
corresponding isomorphism

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

Let $\Pi\,:\,P\,\longrightarrow\,X$ be a holomorphic principal $G$ bundle; we
use the convention that $G$ 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.
$F\,:\,P|_{U}\,\longrightarrow\,P|_{U}$ satisfying $F(pg)\,=\,F(p)g$ for all
$g\,\in\,G$; we do not assume that $\Pi\circ F\,=\,\Pi$. The group of
automorphisms $\operatorname{Aut}(P)|_{U}$ is generated by the invariant
vector fields

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

Then $\operatorname{aut}(P)$ defines a coherent sheaf of
$\mathcal{O}_{X}$-modules. We refer to the subsheaf ${\rm
At}_{X/T}(P)\,\subset\,\operatorname{aut}(P)$ that projects by $d\Pi$ to
$\mathcal{T}_{X/T}\,\subset\,TX$ as the _relative Atiyah algebra of $P$_. We
have an exact sequence

(2.6) $0\,\longrightarrow\,\operatorname{ad}(P)\,\longrightarrow\,{\rm
At}_{X/T}(P)\,\longrightarrow\,\mathcal{T}_{X/T}\,\longrightarrow\,0.$

We will explain the inclusion map on the left. Recall that a section of
$\operatorname{ad}(P)$ is identified with a function
$f\,:\,P\,\longrightarrow\,\mathfrak{g}$ satisfying
$f(pg)\,=\,\operatorname{Ad}_{g^{-1}}f(p)$. For $Y\,\in\,\mathfrak{g}$, let
$Y^{\sharp}$ denote the fundamental vector field on $P$ generated by $Y$. Then
$Y^{\sharp}(pg)\,=\,(R_{g})_{\ast}(\operatorname{Ad}_{g}Y)^{\sharp}(p)$. The
map $\operatorname{ad}(P)\,\longrightarrow\,\operatorname{aut}(P)$ in (2.6) is
$f\,\longmapsto\,Y$, where $Y(p)\,=\,f(p)^{\sharp}$. With this definition,

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

so $Y$ is invariant and lies in $\operatorname{aut}(P)$ as the kernel of
$d\Pi$. The following will be important when we investigate 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 (2.6) gives a quasi-Lie algebra structure on ${\rm
At}_{X/T}(P)^{\vee}$:

(2.7) $0\,\longrightarrow\,\Omega_{X/T}\,\longrightarrow\,{\rm
At}_{X/T}(P)^{\vee}\,\stackrel{{\scriptstyle
j}}{{\longrightarrow}}\,\operatorname{ad}(P)^{\vee}\,\longrightarrow\,0.$

Identify $\operatorname{ad}(P)$ with $\operatorname{ad}(P)^{\vee}$ using
$\nu_{\mathfrak{g}}$ in (2.5), and denote ${\rm At}_{X/T}(P)^{\vee}$ by
$\widetilde{\mathfrak{g}}_{P}$. We have the following quasi-Lie algebra

(2.8) ${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,
$\mathfrak{S}^{i}_{X/T}(P)$, which we call the _Ginzburg complex_ for $P$.
Explicitly,

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

We will later consider a relative version of it where $T$ varies.

####  2.2.2. The Bloch-Esnault complex

Let $\mathcal{E}\,\longrightarrow\,X$ be a holomorphic vector bundle. Like the
Ginzburg complex, the Bloch-Esnault complex
$\mathcal{B}_{X/T}^{\bullet}(\mathcal{E})$, [12], is nonzero in degrees
$-2,-1$, and $0$, with
$\mathcal{B}_{X/T}^{-2}(\mathcal{E})\,=\,\mathcal{O}_{X}$. 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 $X\,\longrightarrow\,T$. Then
$\mathcal{B}_{X/T}^{0}(\mathcal{E})\,=\,\operatorname{End}(\mathcal{E})$. Set
$\mathcal{E}^{\prime}\,:=\,\mathcal{E}\otimes\Omega_{X/T}$, and define

$\widetilde{\mathcal{B}}_{X/T}^{-1}(\mathcal{E})\,=\,\frac{\mathcal{E}\boxtimes\mathcal{E}^{\prime}(\Delta)}{\mathcal{E}\boxtimes\mathcal{E}^{\prime}(-\Delta)}$

on $X\times_{T}X$. Note that
$\mathcal{O}_{X\times_{T}X}(\Delta)\big{|}_{\Delta}\,=\,\mathcal{T}_{X/T}$.
Then $\mathcal{B}^{-1}_{X/T}(\mathcal{E})$ 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}}$${\widetilde{\mathcal{B}}^{-1}_{X/T}(\mathcal{E})}$${\operatorname{End}(\mathcal{E})}$${0}$${0}$${\Omega_{X/T}}$${\mathcal{B}^{-1}_{X/T}(\mathcal{E})}$${\operatorname{End}(\mathcal{E})}$${0.}$$\scriptstyle{\operatorname{tr}}$

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

We will actually need the traceless version
$\mathcal{B}_{0,X/T}^{\bullet}(\mathcal{E})$, where
$\mathcal{B}_{0,X/T}^{-2}(\mathcal{E})\,=\,\mathcal{B}^{-2}_{X/T}(\mathcal{E})$
while $\mathcal{B}_{0,X/T}^{0}(\mathcal{E})$ is defined by the inclusion map
into the traceless relative Atiyah algebra and
$\mathcal{B}_{0}^{-1}(\mathcal{E})$ is defined by pulling back the extension
$\mathcal{B}^{-1}_{X/T}(\mathcal{E})$ 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
(2.5).

######  Lemma 2.8.

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

(2.9)
${\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 as mentioned in the Introduction, $m_{\phi}$ is the Dynkin index of
$\phi_{\ast}$. Moreover, replacing $\underline{\psi}$ by
$\psi\,:=\,(m_{\phi})^{-1}\underline{\psi}$,

${\psi}\circ\phi_{\ast}\,=\,\operatorname{Id}_{\mathfrak{g}}.$

###### Proof.

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

The following lemma 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
$\varphi\,:\,\mathfrak{g}\,\longrightarrow\,\mathfrak{s}$ defined by
$\nu_{\mathfrak{s}}^{-1}\circ\psi^{t}\circ\nu_{\mathfrak{g}}$:

(2.10)
${\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 $\phi\,:\,G\,\longrightarrow\,\mathsf{SL}(V)$ be a nontrivial holomorphic
representation and $\mathcal{E}_{\phi}\,=\,P\times_{G}V$ the corresponding
vector bundle associated to a principal $G$-bundle
$\Pi\,:\,P\,\longrightarrow\,X$. Sections of $\mathcal{E}_{\phi}$ are
functions $\sigma\,:\,P\,\longrightarrow\,V$ satisfying the condition that
$\sigma(pg)\,=\,\phi(g^{-1})\sigma(p)$ for all $g\,\in\,G$. The adjoint bundle
$\operatorname{ad}(P)\,=\,P\times_{G}\mathfrak{g}$ maps to the traceless
endomorphism bundle $\operatorname{End}_{0}(\mathcal{E}_{\phi})$ using
$\phi_{\ast}\,:=\,d\phi\,:\,\mathfrak{g}\,\longrightarrow\,\mathfrak{sl}(V)$,
and we shall use the same notation $\phi_{\ast}$ for this map
$\operatorname{ad}(P)\,\longrightarrow\,\operatorname{End}_{0}(\mathcal{E}_{\phi})$.
Notice that in this case, the map $\psi$ defined in Lemma 2.8 is
$G$-equivariant, and hence defines a homomorphism
$\operatorname{End}_{0}(\mathcal{E}_{\phi})\,\longrightarrow\,\operatorname{ad}(P)$.
A $G$-invariant vector field $Y$ on $P$ defines a differential operator on
sections by $Y(\sigma)$ ($=\,d\sigma(Y)$). Invariance of $Y$ guarantees that
$Y(\sigma)$ is again equivariant with respect to $\phi$, and so defines a
section of $\mathcal{E}_{\phi}$. It is clear that the symbol of this operator
is $\Pi_{\ast}Y$. Therefore, denoting the relative Atiyah algebra of
$\mathcal{E}_{\phi}\,\longrightarrow\,X$ by ${\rm
At}_{X/T}(\mathcal{E}_{\phi})$, we have constructed a map (cf. Atiyah [1, Page
188])

(2.11) $\widetilde{\phi}\,\,:\,\,{\rm At}_{X/T}(P)\,\,\longrightarrow\,\,{\rm
At}_{X/T}(\mathcal{E}_{\phi}).$

The following is a consequence of the above.

######  Proposition 2.10.

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

(2.12) ${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
$\psi\,:\,\operatorname{End}_{0}(\mathcal{E}_{\phi})\,\longrightarrow\,\operatorname{ad}(P)$
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 $\phi\,:\,G\,\longrightarrow\,\mathsf{SL}(V)$.
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
$\mathcal{K}_{\phi}\,\longrightarrow\,\operatorname{End}_{0}(\mathcal{E}_{\phi})$:

(2.13)
${\mathcal{B}_{0,X/T}^{-1}(\mathcal{E}_{\phi})}$${\mathcal{K}_{\phi}}$${\operatorname{End}_{0}(\mathcal{E}_{\phi})}$

###### 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 $X\,\longrightarrow\,T$ and coordinate
neighborhoods $U_{\alpha},\,z_{\alpha}$ on a fixed fiber. We set
$\varphi_{\alpha\beta}\,=\,z_{\alpha}\circ z_{\beta}^{-1}$, so
$\varphi_{\alpha\beta}^{\prime}\,=\,dz_{\alpha}/dz_{\beta}$. 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 $U_{\alpha}\times U_{\alpha}$

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

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

(2.15)
$\left(\frac{f(p,q)dz_{\alpha}(q)}{z_{\alpha}(p)-z_{\alpha}(q)}+\frac{\partial
f}{\partial q}(p,q)\right)u(p)\boxtimes v(q)\ \mod\ \mathcal{I}_{\Delta}.$

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 (2.15) vanishes, and so (2.14) gives a well-defined lift. Set
$\mathscr{A}_{\alpha\beta}\,=\,\nabla_{\alpha}-\nabla_{\beta}$ and
$\Theta_{\alpha\beta}\,=\,(\varphi^{\prime\prime}_{\alpha\beta}/\varphi^{\prime}_{\alpha\beta})dz_{\beta}$.
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. [16, 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

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

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

(2.16)
$\operatorname{tr}\Phi_{\alpha\beta}\,:\,\operatorname{End}_{0}(\mathcal{E})\,\longrightarrow\,\Omega_{X/T},\
\ \beta\,\longmapsto\,\operatorname{tr}(\beta\mathscr{A}_{\alpha\beta}).$

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

Finally, to complete the proof 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 (2.16). ∎

It is shown in [2, Thm. B.2.6] that the exact sequence for
$\mathcal{B}_{0,X/T}^{-1}(\mathcal{E})$ 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
$\widetilde{\phi}_{\ast}^{t}\,:\,{\rm
At}_{X/T}(\mathcal{E}_{\phi})^{\vee}\,\longrightarrow\,{\rm
At}_{X/T}(P)^{\vee}$. Then, by Proposition 2.11 we have the following:

######  Corollary 2.12.

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

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

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

(2.17)
${0}$${\Omega_{X/T}}$${\mathfrak{S}^{-1}_{X/T}(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}}$${\mathcal{B}_{0,X/T}^{-1}(\mathcal{E}_{\phi})}$${\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 (2.17), $\widehat{\phi}$ is just the map obtained by composition in the
middle column of (2.17). Observe that

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

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

######  Proposition 2.13.

The map
$\phi\,:\,\operatorname{ad}(P)\,\longrightarrow\,\operatorname{End}_{0}(\mathcal{E}_{\phi})$
extends to
$\widehat{\phi}\,:\,\mathfrak{S}^{-1}_{X/T}(P)\,\longrightarrow\,\mathcal{B}_{0,X/T}^{-1}(\mathcal{E}_{\phi})$
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

(2.19)
${0}$${\mathcal{O}_{T}}$${R^{1}\pi_{*}(\mathfrak{S}^{-1}_{X/T}(P))}$${R^{1}\pi_{*}(\operatorname{ad}(P))}$${0}$${0}$${\mathcal{O}_{T}}$${R^{1}\pi_{*}(\mathcal{B}_{0,X/T}^{-1}(\mathcal{E}_{\phi}))}$${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 $X\,\longrightarrow\,T$ is
simply a product $X\,=\,C\times T$. Let $P\,\longrightarrow\,X$ be a principal
$G$-bundle such that the restriction of it to $C\times\\{t\\}$ is regularly
stable for every $t$. We wish to compute the extension class of the top (and
hence also the bottom) row of (2.19). 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
$\varphi\,:\,T\,\longrightarrow\,M_{G}^{rs}$. By the deformation theory of
principal $G$-bundles there is a homomorphism
$\mathcal{T}T\,\longrightarrow\,R^{1}\pi_{\ast}(\operatorname{ad}(P))$. Via
the Dolbeault isomorphism, we have

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

and so the extension class of (2.19) 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 (2.19) in
$H^{1,1}_{\overline{\partial}}(T)$ is exactly the pullback
$\varphi^{\ast}[\Theta]$ of the Atiyah-Bott-Narasimhan-Goldman form.

We first define $\Theta$. Under the assumptions, the direct image
$R^{1}\pi_{\ast}(\operatorname{ad}(P))\,\longrightarrow\,T$ is locally free
and its 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 ([17], [18]) 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

$\langle\alpha,\,\beta\rangle\,=\,\sqrt{-1}\int_{C}(\alpha\wedge\rho^{\prime}(\beta))_{\mathfrak{g}},$

where $\alpha$ and $\beta$ 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
$P\,\longrightarrow\,\mathfrak{g}^{\ast}$ 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

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

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

A relative holomorphic connection $\omega$ on $P$ gives a holomorphic
splitting of (2.7). Explicitly, if $\varphi$ is a local section of
$\operatorname{ad}(P)^{\vee}$, then define $\omega(\varphi)$ in ${\rm
At}(P)^{\vee}$, by $\omega(\varphi)(Y)\,=\,\varphi(\omega(Y))$. Notice that

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

so this map is a splitting of the sequence in (2.7). 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
$p\,:\,M^{dR,rs}_{G}\,\longrightarrow\,M_{G}^{rs}$, and this is a submersion.
Therefore, 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 open subsets $U$, $V$, 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
$\mathcal{T}T\,\longrightarrow\,R^{1}\pi_{\ast}(adP)$; 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 $s_{1},\,\cdots,\,s_{N}$ centered at the point $[P_{t}]$ in
$M_{G}^{rs}$ corresponding to $t$. We may so arrange that the holomorphic
sections $\alpha_{j}(s_{1},\cdots,s_{N})$ of $\mathcal{T}M_{G}^{rs}$ in a
neighborhood $[P_{t}]$ satisfy the condition
$\alpha_{j}(0)\,=\,\partial/\partial s_{j}$. Thus, we have also
$\alpha_{j}^{\ast}(0)\,=\,ds_{j}$. Each $\alpha_{j}(s)$ defines a Dolbeault
class in $H^{0,1}(C,\,\operatorname{ad}\mathcal{P}_{s})$, for which we use 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 $t_{j}(0)\,=\,0$, $\partial t_{j}/\partial s_{i}(0)\,=\,\delta_{ij}$ and
$\partial t_{j}/\partial\overline{s}_{i}(0)\,=\,0$. 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-Narasimhan-Goldman symplectic form. ∎

##  3\. Determinant of cohomology and Beilinson-Schechtman classes

In this section we reinterpret Theorem 2.14 in terms of the quasi-isomorphisms
between the Bloch-Esnault complex, the trace complex of Beilinson-Schechtman,
and the Ginzburg complex. Finally we related all of these to the Atiyah class
of the determinant of coholomology line bundle.

###  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

(3.1)
$0\,\longrightarrow\,\frac{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(\Delta)}{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}}\,\longrightarrow\,\frac{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(2\Delta)}{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}}\,\longrightarrow\,\frac{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(2\Delta)}{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(\Delta)}\,\longrightarrow\,0.$

Similarly, we use another short exact sequence from [5]:

(3.2)
$0\,\longrightarrow\,\frac{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}}{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(-\Delta)}\,\longrightarrow\,\frac{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(2\Delta)}{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(-\Delta)}\,\longrightarrow\,\frac{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}(2\Delta)}{{\mathcal{E}}\boxtimes{\mathcal{E}}^{\prime}}\,\longrightarrow\,0.$

Since we have ${\rm At}_{X/T}(\mathcal{E})\,\subset\,\mathcal{D}^{\leq
1}_{X/T}(\mathcal{E})$, pulling back (3.2) we get a quasi-Lie algebra that
fits into the following short exact sequence:

(3.3)
$0\,\longrightarrow\,\operatorname{End}(\mathcal{E})\otimes\Omega_{X/T}\,\longrightarrow\,{}^{tr}\widetilde{\mathcal{A}}_{X/T}(\mathcal{E})^{-1}\,\longrightarrow\,{\rm
At}_{X/T}(\mathcal{E})\,\longrightarrow\,0.$

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

(3.4)
$0\,\longrightarrow\,\Omega_{X/T}\,\longrightarrow\,{}^{tr}\mathcal{A}_{X/T}(\mathcal{E})^{-1}\,\longrightarrow\,{\rm
At}_{X/T}(\mathcal{E})\,\longrightarrow\,0.$

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

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

fits the above objects into the following commutative diagram

(3.5)
${0}$${\operatorname{End}(\mathcal{E})\otimes\Omega_{{X}/T}}$${{}^{tr}\widetilde{\mathcal{A}}_{X/T}(\mathcal{E})^{-1}}$${\operatorname{At}_{{X}/T}(\mathcal{E})}$${0}$${0}$${\operatorname{End}(\mathcal{E})\otimes\Omega_{{X}/T}}$${\widetilde{\mathcal{B}}_{X/T}^{-1}(\mathcal{E})}$${\operatorname{End}(\mathcal{E})}$${0}$${0}$${\operatorname{End}(\mathcal{E})\otimes\Omega_{{X}/T}}$${\widetilde{\mathcal{B}}_{0,X/T}^{-1}(\mathcal{E})}$${\operatorname{End}_{0}(\mathcal{E})}$${0}$${0}$${\Omega_{{X}/T}}$${\mathcal{B}_{0,X/T}^{-1}(\mathcal{E})}$${\operatorname{End}_{0}(\mathcal{E})}$${0}$${0}$${\Omega_{{X}/T}}$${\mathcal{B}_{X/T}^{-1}(\mathcal{E})}$${\operatorname{End}(\mathcal{E})}$${0.}$$\scriptstyle{\operatorname{tr}}$$\scriptstyle{\operatorname{tr}_{*}}$$\scriptstyle{\operatorname{tr}}$$\scriptstyle{\operatorname{tr}_{*}}$

###  3.2. Relative set-up

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

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

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

(3.6)
$0\,\longrightarrow\,\mathcal{T}_{X/T}\,\longrightarrow\,\mathcal{T}_{X,\pi,S}\,\stackrel{{\scriptstyle
d\pi}}{{\longrightarrow}}\,\pi^{-1}\mathcal{T}_{T/S}\,\longrightarrow\,0.$

Let $\operatorname{At}_{X,\pi,S}(\mathcal{E})$ denote the Atiyah algebra
satisfying the fundamental exact sequence

$0\longrightarrow\operatorname{End}(\mathcal{E})\rightarrow\operatorname{At}_{X,\pi,S}(\mathcal{E})\longrightarrow\mathcal{T}_{X,\pi,S}\longrightarrow
0$

Observe that
$\operatorname{At}_{X/T}(\mathcal{E})\subset\operatorname{At}_{X,\pi,S}(\mathcal{E})\subset\operatorname{At}_{X/S}(\mathcal{E})$.

######  Definition 3.1.

The Beilinson-Schechtman trace complex
${}^{tr}\mathcal{A}_{X,\pi,S}(\mathcal{E})^{\bullet}$ is the
$\Omega_{X/T}$-extension of the dgla associated the quasi-Lie algebra
${}^{tr}{\mathcal{A}}_{X/T}(\mathcal{E})^{-1}$, i.e.,
${}^{tr}\mathcal{A}_{X,\pi,S}(\mathcal{E})^{(0)}\,=\,\operatorname{At}_{X,\pi,S}(\mathcal{E})$,
${}^{tr}\mathcal{A}_{X,\pi,S}(\mathcal{E})^{-1}={}^{tr}{\mathcal{A}}_{X/T}(\mathcal{E})^{-1}$
and ${}^{tr}\mathcal{A}_{X,\pi,S}(\mathcal{E})^{(-2)}\,=\,\mathcal{O}_{X}$.

Throughout the rest of this section we will have the assumption that there is
a splitting of the short exact sequence in (3.6).

This condition holds for example in the case of fiber products. We then use
the splitting of (3.6) to pull-back the Atiyah algebra
$\operatorname{At}_{X,\pi,S}(\mathcal{E})$ further via
$\pi^{-1}\mathcal{T}_{T/S}$ to obtain $\mathcal{B}_{X,\pi,S}$:

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

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

######  Definition 3.2.

The Bloch-Esnault complex $\mathcal{B}^{\bullet}_{X,\pi,S}(\mathcal{E})$ is
the three term complex consisting of the locally free sheaves
$\mathcal{B}^{0}_{X,\pi,S}(\mathcal{E})$,
$\mathcal{B}^{-1}_{X/T}(\mathcal{E})$ and
$\mathcal{B}^{-2}_{X,\pi,S}(\mathcal{E})\,\cong\,\mathcal{O}_{X}$ in degrees
$0$, $-1$ and $-2$ respectively, and zero otherwise.

We will denote by $\mathcal{B}_{0,X/T,S}^{\bullet}$ the traceless Bloch-
Esnault complex as considered in [20] and [2].

###  3.3. Determinant of cohomology

Let $\pi:X\rightarrow T$ be a family of smooth projective curves and let
$\mathcal{E}$ be a vector bundle $X$. Then the object $R\pi_{*}\mathcal{E}$ in
the bounded derived category of $T$ is represented by a two term complex
$\mathcal{E}_{0}\rightarrow\mathcal{E}_{1}$. The determinant of cohomology
$\mathcal{L}$ upto isomorphism is defined by

$\mathcal{L}:=\bigwedge^{top}\mathcal{E}_{1}\otimes\bigwedge^{top}\mathcal{E}_{0}^{\vee}$

We refer the reader to [11, § 6.3] for more details on determinant of
cohomology $\mathcal{L}$. We recall the following result of Beilinson-
Schechtman, [5], that connects the Atiyah algebra of
$\operatorname{At}_{X/T}(\mathcal{L})$ and the trace complex of $\mathcal{E}$
and also a result of Bloch-Esnault [12] connecting the trace complex with the
Bloch-Esnault complex.

######  Proposition 3.3.

The Atiyah algebra $\operatorname{At}_{T/S}(\mathcal{L}^{-1})$ is isomorphic
to the relative Atiyah algebra
$R^{0}\pi_{*}({}^{tr}\mathcal{A}_{X,\pi,S}(\mathcal{E})^{\bullet})$. If
$\mathcal{B}_{X,\pi,S}^{\bullet}(\mathcal{E})$ is the relative Bloch-Esnault
complex associated to a family of curves $X\,\longrightarrow\,T$ parametrized
by $T\rightarrow S$ such that the exact sequence in Equation (3.6) splits,
then $\mathcal{B}^{\bullet}_{X,\pi,S}(\mathcal{E})$ is quasi-isomorphic to the
trace complex ${}^{tr}\mathcal{A}_{X,\pi,S}(\mathcal{E})^{\bullet}$.

###  3.4. Relative Ginzburg complex

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

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

We then use the splitting of the exact sequence in (3.6) to pull-back further
to define $\mathfrak{S}^{0}(\mathcal{P})$:

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

We consider the following three term complex
$\mathfrak{S}^{\bullet}_{X},\pi,S(\mathcal{P})$ which will be referred to as
the relative Ginzburg complex:

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

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

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

(3.7)
${\mathcal{O}_{X}}$${\mathcal{O}_{X}}$${0}$${\Omega_{X/T}}$${\mathfrak{S}^{-1}_{X/T}(\mathcal{P})}$${\mathfrak{g}_{\mathcal{P}}}$${0}$${\mathfrak{S}^{0}_{X,\pi,S}(\mathcal{P})}$${\mathfrak{S}^{0}_{X,\pi,S}(\mathcal{P}).}$

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

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

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

To compute $R^{0}\pi_{\ast}\mathfrak{S}_{X,\pi,S}^{\bullet}(\mathcal{P})$, 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})}$${\vdots}$${\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}$

and compute the zero-th cohomology of the total complex
${\mathcal{C}_{-1}}$${\mathcal{C}_{0}}$${\mathcal{C}_{1},\mbox{where}}$$\scriptstyle{D_{-1}}$$\scriptstyle{D_{0}}$

$\mathcal{C}_{-1}\,=\,\mathcal{L}_{1}\oplus\mathcal{M}_{0}\ ,\
\mathcal{C}_{0}\,=\,\mathcal{L}_{2}\oplus\mathcal{M}_{1}\oplus\mathcal{N}_{0}\
,\
\mathcal{C}_{1}\,=\,\mathcal{L}_{3}\oplus\mathcal{M}_{2}\oplus\mathcal{N}_{1},\
\mbox{and the differentials are}$ $\displaystyle
D_{-1}\,:\,\mathcal{C}_{-1}\,\longrightarrow\,\mathcal{C}_{0},\,\
(\ell_{1},\,m_{0})\,\longmapsto\,(d\ell_{1},\,dm_{0}-f\ell_{1},\,gm_{0})$
$\displaystyle D_{0}\,:\,\mathcal{C}_{0}\,\longrightarrow\,\mathcal{C}_{1},\,\
(\ell_{2},\,m_{1},\,n_{0})\,\longmapsto\,(d\ell_{2},\,dm_{1}+f\ell_{2},\,dn_{0}-gm_{1}).$

We note three facts in the next lemma the proofs of which are immediate in
view of the assumptions.

######  Lemma 3.4.

Let $\pi\,:\,X\,\longrightarrow\,T$ 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)

$R^{0}\pi_{\ast}\mathfrak{g}_{\mathcal{P}}\,=\,\\{0\\}$ (assuming the stable
locus is non-empty).

  3. (iii)

Since the fibers are connected, assume that the natural map

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

is an isomorphism. Then the map
$R^{1}\pi_{\ast}(\mathfrak{S}_{X/T}^{-1}(\mathcal{P}))\,\longrightarrow\,R^{1}\pi_{\ast}(\mathfrak{S}_{X,\pi,S}^{0}(\mathcal{P}))$
is zero as it factors through $R^{1}\pi_{\ast}\mathfrak{g}_{\mathcal{P}}$.

We prove the next proposition under the assumption of Statement (iii) in Lemma
3.4.

######  Proposition 3.5.

There is an isomorphism
$R^{0}\pi_{*}\mathfrak{S}_{X,\pi,S}^{\bullet}(\mathcal{P})\simeq
R^{1}\pi_{*}\mathfrak{S}_{X/T}^{-1}(\mathcal{P})$ 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 3.4(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
$R^{1}\pi_{\ast}(\mathfrak{S}_{X/T}^{-1}(\mathcal{P}))$. 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 3.4
this is automatic. Now $n_{0}$ is such that $dn_{0}=gm_{1}$. By (ii) and (iii)
of Lemma 3.4, we have
$R^{0}\pi_{\ast}\mathfrak{S}_{X,\pi,S}^{0}(\mathcal{P})=\\{0\\}$. This means
that $n_{0}$ is uniquely determined. Hence, the hypercohomology gives
$R^{1}\pi_{\ast}\mathfrak{S}_{X/T}^{-1}(\mathcal{P})$. ∎

Pick a representation $\phi:\mathfrak{g}\rightarrow\mathfrak{sl}_{r}$, and
consider the adjoint
$\operatorname{ad}:\mathfrak{sl}_{r}\rightarrow\mathfrak{sl}(\mathfrak{sl}_{r})$.
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 3.4, we have the following proposition, which is a
generalization of results in [2, Proposition 5.0.2] and [20]:

######  Proposition 3.6.

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

(3.10)
${0}$${R^{0}\pi_{*}\Omega^{\bullet}_{X/T}[2]\cong\mathcal{O}_{T}}$${R^{0}\pi_{*}\mathfrak{S}^{\bullet}_{X,\pi,S}(\mathcal{P})}$${\mathcal{T}_{T/S}}$${0}$${0}$${R^{0}\pi_{*}\Omega^{\bullet}_{X/T}[2]\cong\mathcal{O}_{T}}$${R^{0}\pi_{*}\mathcal{B}^{\bullet}_{X,\pi,S}(\operatorname{End}_{0}(\mathcal{E}_{\phi}))}$${\mathcal{T}_{T/S}}$${0.}$$\scriptstyle{m_{\phi}\cdot
2r}$$\scriptstyle{\cong}$$\scriptstyle{-\operatorname{id}}$

###### Proof.

The composition of maps
$\operatorname{ad}\circ\phi\,:\,\mathfrak{g}\,\longrightarrow\,\mathfrak{sl}(\mathfrak{sl}_{r})$
gives a map ${\rm At}_{X/T}(\mathcal{P})\,\longrightarrow\,{\rm
At}_{X/T}(\operatorname{End}_{0}(\mathcal{E}_{\phi}))$ (see (2.11) and [1]).
Moreover, the representation also gives the map
$\mathfrak{S}_{X/T}^{-1}(\mathcal{P})\,\longrightarrow\,\mathcal{B}^{-1}_{0,X/T}(\operatorname{End}(\mathcal{E}_{\phi}))$
(see Proposition 2.13). These two together give a map between the complexes

(3.11)
${0}$${\Omega^{\bullet}_{X/T}[2]}$${\mathfrak{S}_{X,\pi,S}^{\bullet}(\mathcal{P})}$${\pi^{-1}\mathcal{T}_{T/S}}$${0}$${0}$${\Omega^{\bullet}_{X/T}[2]}$${\mathcal{B}^{\bullet}_{X,\pi,S}(\operatorname{End}(\mathcal{E}_{\phi}))}$${\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 (3.9). Consequently, combining
Propositions 3.5 and 3.6, we have the following diagram

(3.12)
${0}$${R^{1}\Omega_{X/T}\cong\mathcal{O}_{T}}$${R^{1}\pi_{*}\mathfrak{S}_{X/T}^{-1}(\mathcal{P})}$${\mathcal{T}_{T/S}}$${0}$${0}$${R^{0}\pi_{*}\Omega^{\bullet}_{X/T}[2]\cong\mathcal{O}_{T}}$${R^{0}\pi_{*}\mathfrak{S}_{X,\pi,S}^{\bullet}(\mathcal{P})}$${\mathcal{T}_{T/S}}$${0}$${0}$${R^{0}\pi_{*}\Omega^{\bullet}_{X/T}[2]\cong\mathcal{O}_{T}}$${R^{0}\pi_{*}\mathcal{B}_{X,\pi,S}^{\bullet}(\operatorname{End}_{0}(\mathcal{E}_{\phi}))}$${\mathcal{T}_{T/S}}$${0}$${0}$${\mathcal{O}_{T}}$${R^{0}\pi_{*}({}^{tr}\mathcal{A}_{X,\pi,S}(\operatorname{End}_{0}(\mathcal{E}_{\phi}))^{\bullet})}$${\mathcal{T}_{T/S}}$${0}$${0}$${\mathcal{O}_{T}}$${\operatorname{At}_{T/S}(\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 third and fourth rows is due to Bloch-Esnault
[12], and the isomorphism between the fourth and fifth rows is due to
Beilinson-Schechtman [5]. 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 (3.9).

######  Theorem 3.7.

The relative Atiyah sequence for $\frac{1}{m_{\phi}}{\rm
At}_{T/S}(\mathcal{L}_{\phi})$ is isomorphic to

(3.13)
$0\,\longrightarrow\,\mathcal{O}_{M^{rs}_{G}}\,\longrightarrow\,R^{1}\pi_{n,\ast}(\mathfrak{S}^{-1}_{X/T}(\mathcal{P}))\,\longrightarrow\,\mathcal{T}_{T/S}\,\longrightarrow\,0\
.$

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

###  3.5. Associated bundles and pullback

In this section we discuss the relation between the relative Ginzburg complex
for the moduli space $M_{G}$ and the pull-back of Bloch-Esnault complex of
$M_{\mathsf{SL}_{r}}$ associated to a representation
$\phi\,:\,G\,\hookrightarrow\,\mathsf{SL}_{r}$. As previously, let
$M^{rs}_{G}$ denote the moduli space parametrizing the regularly stable
principal $G$-bundles, and denote by $M^{s}_{\mathsf{SL}_{r}}$ the moduli
space of stable rank $r$ vector bundles of trivial determinant. 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 [10] that a stable principal ${G}$-bundle $P$ produces
a stable $\mathsf{SL}_{r}$ bundle.

Consider the map
$\phi\,:\,M^{rs}_{G}\,\longrightarrow\,M^{s}_{\mathsf{SL}_{r}}$ 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
(since they exist on a covering by étale open subsets, we can treat as if they
exist). Then the associated vector 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:

(3.14)
${\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 [12]

(3.15)
${0}$${\Omega_{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}}$${\operatorname{At}_{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}(\mathscr{E})^{\vee}}$${\operatorname{End}_{0}(\mathscr{E})^{\vee}}$${0}$${0}$${\Omega_{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}}$${\mathcal{B}_{{}_{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}}^{-1}(\mathscr{E})}$${\operatorname{End}_{0}(\mathscr{E})}$${0.}$$\scriptstyle{\cong}$$\scriptstyle{-\nu_{\mathfrak{sl}(r)}}$$\scriptstyle{\cong}$

Taking pushforward and combining the results of Beilinson-Schechtman [5],
Bloch-Esnault, [12], and Baier-Bolognesi-Martens-Pauly, [2], we get the
following commutative diagram of maps

${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}_{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}(\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}}}}$${R^{1}\pi_{n,\mathsf{SL}_{r}*}(\mathcal{B}_{0,{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}}^{-1}(\mathscr{E}))}$${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
$\phi\,:\,M^{rs}_{G}\,\longrightarrow\,M^{s}_{\mathsf{SL}_{r}}$, we get the
exact sequence

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

We wish to connect the following two exact sequences

(3.16)
$0\longrightarrow\phi^{*}(R^{1}\pi_{n,\mathsf{SL}_{r}*}\Omega_{\mathfrak{X}_{\mathsf{SL}_{r}}/M_{\mathsf{SL}_{r}}})\longrightarrow\phi^{*}(R^{1}\pi_{n,\mathsf{SL}_{r}*}(\mathcal{B}_{0,{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}}^{-1}(\mathscr{E})))\longrightarrow\phi^{*}R^{1}\pi_{n,\mathsf{SL}_{r}*}(\operatorname{End}_{0}(\mathscr{E}))\longrightarrow
0,$
$0\,\longrightarrow\,\mathcal{O}_{M^{rs}_{G}}\,\cong\,R^{1}\pi_{n*}\Omega_{\mathfrak{X}_{G}/M^{rs}_{G}}\,\longrightarrow\,R^{1}\pi_{n*}(\mathfrak{S}^{-1}_{\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}(\mathscr{P}))\,\longrightarrow\,\mathcal{T}_{M^{rs}_{G}/S}\,\longrightarrow\,0.$

By Proposition 2.13, it is enough to relate (3.16) with the following (see
(2.19)):

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

######  Proposition 3.8.

There is an isomorphism

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

###### 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
[1]:

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

Applying $R^{1}\pi_{n,G\,\ast}$, we conclude 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,\mathfrak{X}_{G}/M^{rs}_{G}}^{-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}}}}$${R^{1}\pi_{n,G\,\ast}(f^{*}(\mathcal{B}_{0,\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}^{-1}(\mathscr{E})))}$${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

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

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 base change of cohomologies for flat morphisms, we get the
required isomorphism. This completes the proof. ∎

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

${0}$${\mathcal{O}_{M_{G}}}$${\phi^{*}\operatorname{At}_{M_{\mathsf{SL}_{r}}^{s}/S}(\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)}$${\phi^{*}\left(R^{1}\pi_{n,\mathsf{SL}_{r}*}(\mathcal{B}_{0,\mathfrak{X}_{\mathsf{SL}_{r}}/M^{s}_{\mathsf{SL}_{r}}}^{-1}(\mathscr{E})\right)}$${\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,{\mathfrak{X}_{G}/M^{rs}_{G}}}^{-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}}}$${R^{1}\pi_{n,G\,\ast}(\mathfrak{S}^{-1}_{\mathfrak{X}_{G}/M^{rs}_{G}}(\mathscr{P}))}$${\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 3.8 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 2.19. Thus we get that
$R^{1}\pi_{n,{G}*}(\mathfrak{S}^{-1}_{\mathfrak{X}_{G}/M^{rs}_{G}}(\mathscr{P}))$
is isomorphic to the Atiyah algebra
$\frac{1}{m_{\phi}}\phi^{\bullet}\operatorname{At}_{M^{s}_{\mathsf{SL}_{r}}/S}(\mathcal{L})$,
where $\phi^{\bullet}$ denote the pull-back in the category of Atiyah
algebras.

##  4\. Parabolic analog of Beilinson-Schechtman construction

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 notion of $\Gamma$-linearized bundles on a Galois
cover $\widehat{C}\,\longrightarrow\,C$ with Galois group $\Gamma$. See [11, §
6] for more details.

###  4.1. Parabolic vector bundles

Let $\widetilde{\mathcal{E}}$ be a vector bundle of rank $r$ on a family of
ramifed $\Gamma$-cover of curves
$\widetilde{\pi}\,:\,\widetilde{X}\,\longrightarrow\,{T}$ ramified along
$\widehat{D}$. In other words, there is a natural projection
$p\,:\,\widetilde{X}\,\longrightarrow\,X$ which is a ramifed $\Gamma$-covering
such that $\widetilde{\pi}\,=\,\pi\circ p$. 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 on $X$
defined by the invariant pushforward of the $\Gamma$-bundle
$\widetilde{{\mathcal{E}}}$. By the discussion in [11, § 7], the vector bundle
$\mathcal{E}$ comes equipped with a parabolic structure supported on $D$.
Recall the trace-zero relative Atiyah sequence

(4.1)
$0\longrightarrow\operatorname{End}_{0}(\widetilde{\mathcal{E}})\longrightarrow\operatorname{At}_{\widetilde{X}/T}(\widetilde{\mathcal{E}})\longrightarrow\mathcal{T}_{\widetilde{X}/T}\longrightarrow
0.$

By Seshadri [19], 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 (4.1)

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

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:

(4.2)
$0\longrightarrow\Omega_{{X}/T}\longrightarrow\left(p_{*}^{\Gamma}\left(\operatorname{At}_{\widetilde{X}/T}(\widetilde{\mathcal{E}})(-\widehat{D})\right)(D)\right)^{\vee}\longrightarrow\left(\operatorname{SPar}_{0}(\mathcal{E})(D)\right)^{\vee}\longrightarrow
0.$

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 the sequence in (4.2) via the map $\nu_{\mathfrak{sl}(r)}$ to get
the following quasi-Lie algebra which we denote by ${}^{spar}\widetilde{{\rm
At}}_{X/T}(\mathcal{E})$

(4.3)
${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}}$${{}^{spar}\widetilde{{\rm
At}}_{X/T}(\mathcal{E})}$${\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 [5], we get an exact sequence of
sheaves:

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

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:

(4.4) $0\longrightarrow
p_{*}^{\Gamma}(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T})\longrightarrow
p_{*}^{\Gamma}({}^{tr}\widetilde{\mathcal{A}}_{\widetilde{X}/T}(\widetilde{\mathcal{E}})^{-1})\longrightarrow
p_{*}^{\Gamma}(\operatorname{At}_{\widetilde{X}/T}(\widetilde{\mathcal{E}}))\longrightarrow
0.$

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_
$\operatorname{{}^{\emph{par}}At}_{X/T}({\mathcal{E}})$. Pulling back the
exact sequence in (4.4) via the natural inclusion
$\operatorname{Par}_{0}({\mathcal{E}})\hookrightarrow\operatorname{{}^{\emph{par}}At}({\mathcal{E}})$,
we get an exact sequence:

(4.5) $0\longrightarrow
p_{*}^{\Gamma}(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T})\longrightarrow{}^{par}\widetilde{\mathcal{B}}_{0,{X}/T}^{-1}({\mathcal{E}})\longrightarrow\operatorname{Par}_{0}({\mathcal{E}})\longrightarrow
0.$

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

(4.6)
$\operatorname{tr}:p_{*}^{\Gamma}(\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T})\longrightarrow{\Omega}_{X/T}.$

Taking pushforward of the exact sequence in (4.4) via the parabolic trace in
(4.6), we get the following exact sequence:

(4.7)
$0\longrightarrow\Omega_{{X}/T}\longrightarrow{}^{par}\mathcal{B}_{0,X/T}^{-1}({\mathcal{E}})\longrightarrow\operatorname{Par}_{0}({\mathcal{E}})\longrightarrow
0.$

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

Recall from Section 2.1.1.1 in Beilinson-Schechtman [5, 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 ${}^{spar}\widetilde{{\rm At}}_{X/T}(\mathcal{E})$
defined by (4.3).

######  Theorem 4.2.

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

${0}$${\Omega_{{X}/T}}$${{}^{spar}\widetilde{{\rm
At}}_{X/T}(\mathcal{E})}$${\operatorname{Par}_{0}({\mathcal{E}})}$${0}$${0}$${\Omega_{{X}/T}}$${{}^{par}\mathcal{B}_{0,X/T}^{-1}({\mathcal{E}})}$${\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 [5] 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:

(4.8)
$\langle\,,\,\rangle:\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\otimes\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(2\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\longrightarrow\mathcal{O}_{\widetilde{X}}.$

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

(4.9)
$\langle\,,\,\rangle:\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\otimes\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}\longrightarrow\mathcal{O}_{\widetilde{X}},\
\mbox{and}\
\langle\,,\,\rangle:\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\otimes\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\longrightarrow\mathcal{O}_{\widetilde{X}}.$

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

(4.10)
$\displaystyle\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}$
$\displaystyle\cong\operatorname{End}(\widetilde{\mathcal{E}})\otimes\mathcal{T}_{\widetilde{X}/T},\
\mbox{and}\
\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\cong\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T}.$

Lemma B.2.8 in [2] shows that the bilinear form $\langle\,,\,\rangle$ in (4.8)
and (4.9) can be identified as

(4.11)
$\displaystyle-\kappa_{\mathfrak{gl}(r)}:\operatorname{End}(\widetilde{\mathcal{E}})\otimes\operatorname{End}(\widetilde{\mathcal{E}})\longrightarrow\mathcal{O}_{\widetilde{X}}\
\mbox{and}\
\kappa_{\mathfrak{gl}(r)}:\operatorname{End}(\widetilde{\mathcal{E}})\otimes\mathcal{T}_{\widetilde{X}/T}\otimes\operatorname{End}(\widetilde{\mathcal{E}})\otimes\Omega_{\widetilde{X}/T}\longrightarrow\mathcal{O}_{\widetilde{X}},$

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

(4.12)
$\displaystyle\langle\,,\,\rangle:p_{*}^{\Gamma}\left(\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}\right)\otimes\left(p_{*}^{\Gamma}\left(\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}(-\widehat{D})\right)\right)(D)\longrightarrow\mathcal{O}_{X}.$
(4.13)
$\displaystyle\langle\,,\,\rangle:p_{*}^{\Gamma}\left(\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(-\Delta)}\right)\otimes\left(p_{*}^{\Gamma}\left(\frac{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}}{\widetilde{\mathcal{E}}\boxtimes\widetilde{\mathcal{E}}^{\prime}(\Delta)}(-\widehat{D})\right)\right)(D)\longrightarrow\mathcal{O}_{X}.$

The identification of parabolic and strongly parabolic endomorphism as
invariant pushforward and invariants pushforward of the identification in
(4.11) tell us that the form $\langle\,,\,\rangle$ in (4.12) 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 (4.13) 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 ${}^{par}\mathcal{B}_{0,X/T}^{-1}(\mathcal{E})$
with ${}^{spar}\widetilde{{\rm At}}_{X/T}(\mathcal{E})$ 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 4.3.

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

###### Proof.

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

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

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,\widetilde{X}/T}^{-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})}$${p_{*}^{\Gamma}(\widetilde{\mathcal{B}}_{0,\widetilde{X}/T}^{-1}(\widetilde{\mathcal{E}}))}$${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{X}/T}(\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 (3.5). Since both
${}^{par}\widetilde{\mathcal{B}}_{0,X/T}^{-1}(\mathcal{E})$ and
$p_{*}^{\Gamma}(\widetilde{\mathcal{B}}_{0,\widetilde{X}/T}^{-1}(\widetilde{\mathcal{E}}))$
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
$p_{*}^{\Gamma}(\widetilde{\mathcal{B}}_{0,\widetilde{X}/T}^{-1}(\widetilde{\mathcal{E}}))\cong{}^{par}\widetilde{\mathcal{B}}_{0,X/T}^{-1}(\mathcal{E}).$
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,\widetilde{X}/T}^{-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)}$${{}^{par}\widetilde{\mathcal{B}}_{0,\widetilde{X}/T}^{-1}(\mathcal{E})}$${\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})}$${p_{*}\widetilde{\mathcal{B}}_{0,\widetilde{X}/T}^{-1}(\widetilde{\mathcal{E}})}$${p_{*}(\operatorname{End}_{0}(\widetilde{\mathcal{E}}))}$${p_{*}^{\Gamma}(\operatorname{End}(\widetilde{\mathcal{E}}))\otimes\Omega_{\widetilde{X}/T})}$${{}^{par}\widetilde{\mathcal{B}}_{0,{X}/T}^{-1}(\mathcal{E})}$${\operatorname{Par}_{0}(\mathcal{E})}$${p_{*}\Omega_{\widetilde{X}/T}}$${p_{*}\mathcal{B}_{0,\widetilde{X}/T}^{-1}(\widetilde{\mathcal{E}})}$${p_{*}(\operatorname{End}_{0}(\widetilde{\mathcal{E}}))}$${\Omega_{X/T}}$${{}^{par}\mathcal{B}_{0,{X}/T}^{-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 4.3, and using the fact that $R^{1}\widetilde{\pi}=R^{1}\pi\circ
p_{*}$, we get the following proposition:

######  Proposition 4.4.

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,\widetilde{X}/T}^{-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}}$${R^{1}{\pi}_{*}({}^{par}\mathcal{B}_{0,{X}/T}^{-1}({\mathcal{E}}))}$${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 [11, §7.2], 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 [9], 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 two 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 [11, Lemma 8.3] 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
$Y_{\mathsf{SL}_{r}}^{par,s}\hookrightarrow M_{\mathsf{SL}_{r}}^{par,s}$.

Let $\widehat{\mathcal{E}}$ be the universal bundle (which exist in the étale
toplogy) on
$\widehat{\mathcal{C}}\times_{S}\widehat{M}^{ss}_{\mathsf{SL}_{r}}$ and
$\widetilde{\mathcal{E}}$ be its pull-back to
$\widehat{\mathcal{C}}\times_{S}M^{\bm{\tau},s}_{\mathsf{SL}_{r}}$. We denote
by $\mathcal{E}$ the universal parabolic bundle which we can assume to exist
without loss of generality (see Remark 2.7). As in the diagram let
$\widetilde{\pi}_{n}:\widehat{C}\times_{S}Y_{\mathsf{SL}_{r}}^{\bm{\tau},s}\longrightarrow
Y_{\mathsf{SL}_{r}}^{\bm{\tau},s}$ denote the projection and similarly
consider the projection
$\widehat{\pi}_{n}:\widehat{C}\times_{S}\widehat{M}_{\mathsf{SL}_{r}}^{s}\longrightarrow\widehat{M}_{\mathsf{SL}_{r}}^{s}$

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 [2], Beilinson-Schechtman [5], Bloch-Esnault
[12], and Sun-Tsai [20], we get an isomorphism of the Atiyah algebras ${\rm
At}_{\widehat{M}^{s}_{\mathsf{SL}_{r}}/S}(\mathcal{L}^{-1})$ with
$R^{1}\widehat{\pi}_{n*}(\mathcal{B}_{0,\widehat{\mathcal{C}}\times_{S}\widehat{M}_{\mathsf{SL}_{r}}^{s}/\widehat{M}^{s}_{\mathsf{SL}_{r}}}^{-1}(\widehat{\mathcal{E}}))$
which makes the diagram of fundamental sequences of Atiyah algebras commute:

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

Pulling back by $\phi$ we get an isomorphism of $\phi^{*}{\rm
At}_{\widehat{M}^{s}_{\mathsf{SL}_{r}}/S}(\mathcal{L})$ and
$\phi^{*}R^{1}\widehat{\pi}_{n*}(\mathcal{B}_{0,\widehat{\mathcal{C}}\times_{S}\widehat{M}_{\mathsf{SL}_{r}}^{s}/\widehat{M}^{s}_{\mathsf{SL}_{r}}}^{-1}(\widehat{\mathcal{E}}))$.
Moreover the base change theorems implies that the later is isomorphic to
$R^{1}\widetilde{\pi}_{n*}(\mathcal{B}_{0,\widehat{\mathcal{C}}\times_{S}Y^{\bm{\tau},s}_{\mathsf{SL}_{r}}/Y^{\bm{\tau},s}_{\mathsf{SL}_{r}}}^{-1}(\widehat{\mathcal{E}}))$.
(Observe that $\phi^{*}$ of an Atiyah algebra may not be an Atiyah algebra.)
We have the following result.

######  Theorem 4.5.

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

###### Proof.

Applying Proposition 4.4 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
$\operatorname{At}_{Y_{\mathsf{SL}_{r}}^{\bm{\tau},s}/S}(\phi^{*}\mathcal{L})$
and
$R^{1}\pi_{n*}{}^{par}\mathcal{B}_{{0,\mathcal{C}\times_{S}Y^{\bm{\tau},s}_{\mathsf{SL}_{r}}/Y^{\bm{\tau},s}_{\mathsf{SL}_{r}}}}^{-1}(\mathcal{E})$
over $Y^{par,s}_{\mathsf{SL}_{r}}$. Now both these sheaves
$\operatorname{At}_{Y_{\mathsf{SL}_{r}}^{\bm{\tau},s}/S}({\phi^{*}\mathcal{L}^{-1}})$
and
$R^{1}\pi_{n*}({}^{par}\mathcal{B}_{0,\mathcal{C}\times_{S}Y^{\bm{\tau},s}_{\mathsf{SL}_{r}}/Y^{\bm{\tau},s}_{\mathsf{SL}_{r}}}^{-1}(\mathcal{E}))$
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 4.6.

The proof of Theorem 2.14 applies in this parabolic setting as well. The fact
that the Atiyah-Bott-Narasimhan-Goldman symplectic form is in the class of
$\phi^{\ast}\mathcal{L}$ is one of the results of [13] (see also [8]).

###  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 $\widehat{v}$ 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 4.7.

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 4.8.

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 4.9.

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 discussion
above, 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 4.10.

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

(4.14)
$\displaystyle\begin{split}0\longrightarrow\operatorname{Par}(\mathcal{P})\longrightarrow\operatorname{{}^{\emph{par}}At}_{{X}/T}(\mathcal{P})\longrightarrow\mathcal{T}_{{X}/T}(-D)\longrightarrow
0,\\\
0\longrightarrow\operatorname{SPar}(\mathcal{P})\longrightarrow\operatorname{{}^{\emph{spar}}At}_{{X}/T}(\mathcal{P})\longrightarrow\mathcal{T}_{{X}/T}(-D)\longrightarrow
0,\end{split}$

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

(4.15)
$0\longrightarrow\Omega_{{X}/T}\longrightarrow(\operatorname{{}^{\emph{par}}At}_{\widetilde{X}/T}(\mathcal{P})(D))^{\vee}\longrightarrow(\operatorname{SPar}(\mathcal{P})(D))^{\vee}\longrightarrow
0.$

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

(4.16)
$\nu_{\mathfrak{g}}^{-1}:(\operatorname{SPar}(\mathcal{P})(D))^{\vee}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\sim$}}\,}\operatorname{Par}(\mathcal{P}).$

Pulling back the exact sequence in (4.15) by the above isomorphism, we get a
quasi-Lie algebra ${}^{spar}\widetilde{{\rm At}}_{X/T}(\mathcal{P})$ 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}}$${{}^{spar}\widetilde{{\rm
At}}_{X/T}(\mathcal{P})}$${\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 [15]. 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}}$${\mathfrak{S}^{-1}_{\widetilde{X}/T}(\widehat{\mathcal{P}})}$${\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
${}^{par}\mathfrak{S}^{-1}_{{X}/T}(P):=p_{*}^{\Gamma}\mathfrak{S}^{-1}_{\widetilde{X}/T}(\widehat{\mathcal{P}})$.

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

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

We now have the following proposition:

######  Proposition 4.11.

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

###### Proof.

Recall that there is a natural nondegenerate pairing
$\langle\,,\,\rangle:\mathfrak{S}^{-1}_{\widetilde{X}/T}(\widehat{\mathcal{P}})\times{\rm
At}_{\widetilde{X}/T}(\widehat{\mathcal{P}})\rightarrow\mathcal{O}_{\widetilde{X}}.$
Tensoring it with $\mathcal{O}_{\widetilde{X}}(-\widehat{D})$, we get a
$\mathcal{O}_{\widetilde{X}}(-\widehat{D})$-valued pairing

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

Taking invariant pushforward, we get the following nondegenerate pairing:

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

This produces a duality between
${}^{par}\mathfrak{S}^{-1}_{{X}/T}(\mathcal{P})$ and $p_{*}^{\Gamma}({\rm
At}_{\widetilde{X}/T}(\widehat{\mathcal{P}})(-\widehat{D}))(D)$. This
completes the proof of the proposition. ∎

###  4.4. The general set of parabolic $G$-bundles

Let $M^{\bm{\tau},ss}_{G}$ (respectively, $M^{\bm{\tau},rs}_{G}$) be the
moduli space of semistable (respectively, regularly stable) $(\Gamma,G)$
bundles on a curve $\widehat{C}$, and let $\phi:G\rightarrow\mathsf{SL}_{r}$
be a representation. We assume without loss of generality that
$\widehat{M}_{G}^{rs}$ is non-empty. Note that for a semistable $(\Gamma,G)$,
the underlying $G$-bundle is semistable ([9], [3]). We also use the same
notation for a relative family of $\Gamma$ covers
$\widehat{\mathcal{C}}\rightarrow S$. Consider the induced maps
${M}^{\bm{\tau},ss}_{G}\to\widehat{M}^{ss}_{G}\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}\widehat{M}^{ss}_{\mathsf{SL}_{r}}$.
Let $\widehat{M}^{rs}_{G}$ be the locus of regularly stable bundles on
$\widehat{C}$. Then by [14], we get that the complement of the regularly
stable locus is at least two provided $g\geq 2$ and $G$ is any simple group
different from $\mathsf{SL}_{2}$ and $g\geq 3$ if the Lie algebra of $G$ has a
$\mathfrak{sl}(2)$ factor. Now as in the $\mathsf{SL}_{r}$ case, let
$Y^{\bm{\tau},rs}_{G}$ be the inverse image of $\widehat{M}^{rs}_{G}$ in
$M_{G}^{\bm{\tau},ss}$. Moreover, the complement of $Y^{\bm{\tau},rs}_{G}$ has
codimension at least two provided the genus $g(\mathscr{C})$ of the orbifold
curve $\mathscr{C}=[\widehat{C}/\Gamma]$ (cf. [11, Lemma 8.3]) determined by
$\bm{\tau}$ is at least three, or if the Lie algebra of $G$ has no
$\mathfrak{sl}(2)$ or $\mathfrak{sp}(4)$ factor, $g(\mathscr{C})\geq 2$.

As before let $\mathcal{L}_{\phi}$ be the pull-back of the determinant of
cohomology $\mathcal{L}$ to $M^{\bm{\tau},ss}_{G}$. Then by applying
Proposition 4.11, and Theorem 3.7, we get the following.

######  Corollary 4.12.

There is a natural isomorphism of Atiyah algebras over the regularly stable
locus $Y_{G}^{\bm{\tau},rs}$:

$\frac{1}{m_{\phi}}{\rm
At}_{M_{G}^{\bm{\tau},rs}/S}(\mathcal{L}_{\phi})\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\sim$}}\,}R^{1}\pi_{n\ast}\left({}^{par}\mathfrak{S}^{-1}_{\mathcal{C}\times_{S}M_{G}^{\bm{\tau},rs}/M_{G}^{\bm{\tau},rs}}(\mathcal{P})\right).$

Since by assumption, the complement of $Y_{G}^{\bm{\tau},rs}$ in
$M_{G}^{\bm{\tau},rs}$ is at least two, the above isomorphism extends over the
entire space $M_{G}^{{\bm{\tau}},rs}$.

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