# The Fundamental Group and Extensions of Motives of Jacobians of Curves

Subham Sarkar and Ramesh Sreekantan

###### Abstract

In this paper we construct extensions of mixed Hodge structure coming from the
mixed Hodge structure on the graded quotients of the group ring of the
Fundamental group of a smooth projective pointed curve which correspond to the
regulators of certain motivic cohomology cycles on the Jacobian of the curve
essentially constructed by Bloch and Beilinson. This leads to a new iterated
integral expression for the regulator. This is a generalisation of a theorem
of Colombo [Col02] where she constructed the extension corresponding to
Collino’s cycles in the Jacobian of a hyperelliptic curve.

AMS Classification: 19F27, 11G55, 14C30, 14C35

###### Contents

  1. 1 Introduction
  2. 2 Iterated Integrals, Cycles, Extensions and Regulators
    1. 2.1 Iterated Integrals
    2. 2.2 Motivic Cohomology Cycles
    3. 2.3 The Cycle $Z_{QR,P}$ on on $J(C)$
    4. 2.4 Regulators
    5. 2.5 The Regulator of $Z_{QR,P}$
  3. 3 Extensions
    1. 3.1 Extensions of Mixed Hodge structures coming from the Fundamental Group
    2. 3.2 The Mixed Hodge structure on the Fundamental Group
    3. 3.3 The Extension $E^{3}_{Q,P}$
    4. 3.4 The Extensions $E^{4}_{Q,P}$ and $E^{4}_{R,P}$
    5. 3.5 The Baer sum
    6. 3.6 Rabi’s generalisation
    7. 3.7 The Extension $e^{4}_{QR,P}$
    8. 3.8 Statement of the Main Theorem
  4. 4 Carlson’s Representatives
    1. 4.1 Preliminaries
    2. 4.2 The Carlson Representative of $e^{4}_{QR,P}$

##  1 Introduction

A formula, usually called Beilinson’s formula — though independently due to
Deligne as well — describes the motivic cohomology group of a smooth
projective variety $X$ over a number field as the group of extensions in a
conjectured category of mixed motives, ${\mathcal{MM}}_{{\mathds{Q}}}$. If $i$
and $n$ are two integers then [Sch93],

$\operatorname{Ext}^{1}_{{\mathcal{MM}}_{{\mathds{Q}}}}({\mathds{Q}}(-n),h^{i}(X))=\begin{cases}CH^{n}_{\hom}(X)\otimes{\mathds{Q}}&\text{
if }i+1=2n\\\ H^{i+1}_{{\mathcal{M}}}(X,{\mathds{Q}}(n))&\text{ if }i+1\neq
2n\end{cases}$

Hence, if one had a way of constructing extensions in the category of mixed
motives by some other method, it would provide a way of constructing motivic
cycles.

One way of doing so is by considering the group ring of the fundamental group
of the algebraic variety ${\mathds{Z}}[\pi_{1}(X,P)]$. If $J_{P}$ is the
augmentation ideal — the kernel of the map from
${\mathds{Z}}[\pi_{1}(X,P)]\to{\mathds{Z}}$ — then the graded pieces
$J_{P}^{a}/J_{P}^{b}$ with $a<b$ are expected to have a motivic structure.
These give rise to natural extensions of motives — so one could hope that
these extensions could be used to construct natural motivic cycles.

Understanding the motivic structure on the fundamental group is complicated.
However, the Hodge structure on the fundamental group is well understood
[Hai87]. The regulator of a motivic cohomology cycle can be thought of as the
realisation of the corresponding extension of motives as an extension in the
category of mixed Hodge structures. So while we may not be able to construct
motivic cycles as extensions of motives coming from the fundamental group - we
can hope to construct their regulators as extensions of mixed Hodge structures
(MHS) coming from the fundamental group.

The aim of this paper is to describe this construction in the case of the
motivic cohomology group of the Jacobian of a curve. The first work in this
direction is due to Harris [Har83] and Pulte [Pul88], [Hai87]. They showed
that the Abel-Jacobi image of the modified diagonal cycle on the triple
product of a pointed curve $(C,P)$, or alternatively the Ceresa cycle in the
Jacobian $J(C)$ of the curve, is the same as an extension class coming from
$J_{P}/J_{P}^{3}$, where $J_{P}$ is the augmentation ideal in the group ring
of the fundamental group of $C$ based at $P$.

In [Col02], Colombo extended this theorem to show that the regulator of a
cycle in the motivic cohomology of a Jacobian of a hyperelliptic curve,
discovered by Collino [Col97], can be realised as an extension class coming
from $J_{P}/J_{P}^{4}$, where here $J_{P}$ is the augmentation ideal of a
related curve.

In this paper we extend Colombo’s result to more general curves. If $C$ is a
smooth projective curve of genus $g$ with a function $f$ with divisor
$\operatorname{div}(f)=NQ-NR$ for some points $Q$ and $R$ and some integer $N$
and such that $f(P)=1$ for some other point $P$, there is a motivic cohomology
cycle $Z_{QR,P}$ in $H^{2g-1}_{{\mathcal{M}}}(J(C),{\mathds{Z}}(g))$
discovered by Bloch [Blo00]. We show that the regulator of this cycle can be
expressed in terms of an extensions coming from $J_{P}/J_{P}^{4}$. When $C$ is
hyperelliptic and $Q$ and $R$ are ramification points of the canonical map to
${\mathds{P}}^{1}$, this is Colombo’s result.

A crucial step in Colombo’s work is to use the fact that the modified diagonal
cycle is torsion in the Chow group $CH^{2}_{\hom}(C^{3})$ when $C$ is a
hyperelliptic curve. This means the extension coming from $J_{P}/J_{P}^{3}$
splits and hence does not depend on the base point $P$. This allows her to
consider the extension for $J_{P}/J_{P}^{4}$. In general, that is not true —
in fact the known examples of non-torsion modified diagonal cycles come from
the curves we consider - namely modular and Fermat curves. Our main
contribution is to use an idea of Rabi [Rab01] to show that Colombo’s
arguments can be extended to work in our case as well. As a result we have a
more general situation — which has some arithmetical applications.

In Colombo’s paper there were errors in Propositions 3.2 and 3.3 which were
pointed out by a referee of an earlier version of this paper. Hence we had to
make some revisions. As it turned out the statement of the main result still
holds under some restricted conditions and much of the revisions we made were
to understand these conditions. Unfortunately this has made the paper a little
long.

We have the following theorem (Theorem (4.14)):

######  Theorem 1.1.

Let $C$ be a smooth projective curve of genus $g_{C}$ over ${\mathds{C}}$. Let
$P$, $Q$ and $R$ be three distinct points such that there is a function
$f_{QR}$ with $\operatorname{div}(f_{QR})=NQ-NR$ for some $N$ and
$f_{QR}(P)=1$. Let $Z_{QR}=Z_{QR,P}$ be the element of the motivic cohomology
group $H^{2g-1}_{{\mathcal{M}}}(J(C),{\mathds{Z}}(g))$ constructed below in
Section 2.3. There exists an extension class $\epsilon^{4}_{QR,P}$ in
$\operatorname{Ext}^{1}_{MHS}({\mathds{Z}}(-2),\wedge^{2}H^{1}(C))$
constructed from the mixed Hodge structures associated to the fundamental
groups $\pi_{1}(C\backslash Q,P)$ and $\pi_{1}(C\backslash R,P)$ such that

$\epsilon^{4}_{QR,P}=(2g_{C}+1)N\operatorname{reg}_{{\mathds{Z}}}(Z_{QR})$

in $\operatorname{Ext}^{1}_{MHS}({\mathds{Z}}(-2),\wedge^{2}H^{1}(C))$.

In other words our theorem states that the regulator of a natural cycle in the
motivic cohomology group of the Jacobian of curves, being thought of as an
extension class, is the same as the extension class of a natural extension of
mixed Hodge structures coming from the fundamental group of the curve.

The outline of the proof is as follows. In the second chapter we introduce
background material on iterated integrals, regulators, motivic cycles and
extensions. We then describe the cycle $Z_{QR,P}$ and give a formula for its
regulator. In the next chapter we construct an extension coming from the MHS
on the fundamental group that we expect to be related to the regulator of our
explicit cycle. Finally, we prove that that is indeed the case. This is the
place where there was an error in the paper of Colombo [Col02]. One has to
restrict to a certain subspace of $H_{1}(C\setminus\\{Q,R\\})$ which is
isomorphic to $H_{1}(C)$. There is no canonical way in which to choose this in
general, but in our case the function $f_{QR}$ determines such as space.

As a result of this we have a new iterated integral formula for the regulator
which is more amenable to computation. In a subsequent paper we apply this in
the case of Fermat curves to get an explicit expression for the regulator in
terms of hypergeometric functions — analogous to the works of Otsubo
[Ots11],[Ots12]. As the primary requirement is a nice basis for the homology
and the cohomology of the curve, we expect this will also work in the case of
modular curve, though that case is quite well understood.

Darmon-Rotger-Sols [DRS12] have used the modified diagonal cycle to construct
points on Jacobians of the curves and used the iterated integral approach to
find a formula for the Abel-Jacobi image of these points. Starting with Bloch
[Blo00] and later Collino [Col97] and Colombo [Col02] it has been known that
these null homologous cycles degenerate to higher Chow cycles on related
varieties. Recently Iyer and Müller-Stach [IM14] have shown that the modified
diagonal cycle degenerates to the kind of cycles we consider in some special
cases. This degeneration can be understood from the point of view of
extensions and in fact the iterated integral expression we have for the
regulator show that in a rather natural way in terms of the holomorphic forms
degenerating to logarithmic forms.

Acknowledgements: This work constitutes part of the PhD. thesis of the first
author. We would like to thank Najmuddin Fakhruddin, Noriyuki Otsubo, Satoshi
Kondo, Elisabetta Colombo, Jishnu Biswas, Manish Kumar, Ronnie Sebastian,
Ranier Kaenders, Harish Seshadri, Arvind Nair, Shreedhar Inamdar and Suresh
Nayak for their comments and suggestions. We would also like to thank the
referee of an earlier version of this manuscript for pointing out numerous
errors and specifically for pointing out the error in Colombo’s paper which we
had to rectify and the referee of this version for their comments and
suggestions.

Finally, it gives us great pleasure to thank the Indian Statistical Institute,
Bangalore for their support while this work was done. The first author would
also like to thank TIFR for their hospitality while these revisions were being
made.

##  2 Iterated Integrals, Cycles, Extensions and Regulators

###  2.1 Iterated Integrals

Let $X$ be a smooth projective variety over ${\mathds{C}}$. Let
$\alpha:[0,1]\to X$ be a path and $\omega_{1},\omega_{2},\dots,\omega_{n}$ be
$1$-forms on $X$. Suppose $\alpha^{*}(\omega_{i})=f_{i}(t)dt$. The iterated
integral of length n of ${\omega}_{1},{\omega}_{2}\dots\omega_{n}$ is defined
to be

$\int_{\alpha}\omega_{1}\omega_{2}\dots\omega_{n}:=\int_{0\leq t_{1}\leq
t_{2}\leq\dots\leq t_{n}\leq 1}f_{1}(t_{1})f_{2}(t_{2})\dots
f_{n}(t_{n})dt_{1}dt_{2}\dots dt_{n}.$

An iterated integral of length $\leq n$ is a linear combination of integrals
of the form above with lengths $\leq n$. It is said to be a homotopy
functional if it only depends on the homotopy class of the path $\alpha$. A
homotopy functional gives a functional on the group ring of the fundamental
group or path space.

Iterated integrals can be thought of as integrals on simplices and satisfy the
following basic properties. Here we have only stated the results for length
two iterated integrals, since that is the only type we will encounter in this
paper.

######  Lemma 2.1 (Basic Properties).

Let $\omega_{1}$ and $\omega_{2}$ be smooth $1$-forms on $X$ and $\alpha$ and
$\beta$ piecewise smooth paths on $X$ with $\alpha(1)=\beta(0)$. Then

  1. 1.

$\displaystyle{\int_{\alpha\cdot\beta}\omega_{1}\omega_{2}=\int_{\alpha}\omega_{1}\omega_{2}+\int_{\beta}\omega_{1}\omega_{2}+\int_{\alpha}\omega_{1}\int_{\beta}\omega_{2}}$

  2. 2.

$\displaystyle{\int_{\alpha}\omega_{1}\omega_{2}+\int_{\alpha}\omega_{2}\omega_{1}=\int_{\alpha}\omega_{1}\int_{\alpha}\omega_{2}}$

  3. 3.

$\displaystyle{\int_{\alpha}df\omega_{1}=\int_{\alpha}f\omega_{1}-f(\alpha(0))\int_{\alpha}\omega_{1}}$

  4. 4.

$\displaystyle{\int_{\alpha}\omega_{1}df=f(\alpha(1))\int_{\alpha}\omega_{1}-\int_{\alpha}f\omega_{1}}$

###### Proof.

This can be found in any article on iterated integrals, for instance Hain’s
excellent article [Hai87].∎

###  2.2 Motivic Cohomology Cycles

Let $X$ be a smooth projective algebraic variety of dimension $g$ defined over
${\mathds{C}}$. The motivic cohomology group
$H^{2g-1}_{{\mathcal{M}}}(X,{\mathds{Z}}(g))$ has the following presentation:
Generators are represented by finite sums

$Z=\sum_{i}(C_{i},f_{i})$

where $C_{i}$ are curves on $X$ and
$f_{i}:C_{i}\longrightarrow{\mathds{P}}^{1}$ are functions on them subject to
the co-cycle condition

$\sum_{i}\operatorname{div}(f_{i})=0.$

Relations in this group are defined as follows. If $Y$ is a surface on $X$ and
$f$ and $g$ are functions on $Y$, one has the Steinberg element $\\{f,g\\}$ in
$K_{2}({\mathds{C}}(Y))$, where ${\mathds{C}}(Y)$ is the function field of
$Y$. To such an element one can consider the sum, called the tame symbol of
$\\{f,g\\}$,

$\tau(\\{f,g\\})=\sum_{W\in
Y^{(1)}}(W,(-1)^{\operatorname{ord}_{W}(f)\operatorname{ord}_{W}(g)}\frac{f^{\operatorname{ord}_{W}(g)}}{g^{\operatorname{ord}_{W}(f)}}).$

where $Y^{(1)}$ is the collection of curves on $Y$. This is a finite sum and
satisfies the co-cycle condition, hence lies in the above group. An element is
said to be $0$ in $H^{2g-1}_{{\mathcal{M}}}(X,{\mathds{Z}}(g))$ if it lies in
the image of the free abelian group generated by the tame symbols of elements
of $K_{2}({\mathds{C}}(Y))$ for some surface $Y\subset X$. The group
$H^{2g-1}_{{\mathcal{M}}}(X,{\mathds{Z}}(g))\otimes{\mathds{Q}}$ is the same
as the higher Chow group $CH^{g}(X,1)\otimes{\mathds{Q}}$.

In the group $H^{2g-1}_{{\mathcal{M}}}(X,{\mathds{Z}}(g))$ there are certain
decomposable cycles coming from the product

$H^{2g-1}_{{\mathcal{M}}}(X,{\mathds{Z}}(g))_{dec}=\operatorname{Im}\left(H^{2g-2}_{{\mathcal{M}}}(X,{\mathds{Z}}(g-1)\otimes
H^{1}_{{\mathcal{M}}}(X,{\mathds{Z}}(1)))\longrightarrow
H^{2g-1}_{{\mathcal{M}}}(X,{\mathds{Z}}(g))\right).$

This is simply $CH^{1}(X)\times{\mathds{C}}^{*}$. The group of indecomposable
cycles is defined as the quotient

$H^{2g-1}_{{\mathcal{M}}}(X,{\mathds{Z}}(g))_{ind}=H^{2g-1}_{{\mathcal{M}}}(X,{\mathds{Z}}(g))/H^{2g-1}_{{\mathcal{M}}}(X,{\mathds{Z}}(g))_{dec}.$

In general it is not easy to find non trivial elements in this group.

###  2.3 The Cycle $Z_{QR,P}$ on on $J(C)$

In this section we construct a motivic cohomology cycle on $J(C)$, where $C$
is a smooth projective curve over ${\mathds{C}}$. This was first constructed
by Bloch [Blo00] in the case when $C$ is the modular curve $X_{0}(37)$. The
cycle is similar, in fact, generalises, the cycle constructed by Collino
[Col97]. This section generalises the work of Colombo [Col02] on constructing
the extension corresponding to the Collino cycle and hence many of the
arguments are adapted from her paper.

Let $C$ be a smooth projective curve defined over ${\mathds{C}}$. Let $Q$ and
$R$ be two distinct points on $C$ such that there is a function $f=f_{QR}$
with divisor

$\operatorname{div}(f_{QR})=NQ-NR$

for some $N\in{\mathds{N}}$. To determine the function precisely, we choose a
distinct third point $P$ and assume $f_{QR}(P)=1$.

There exist notable examples of curves where such functions can easily be
found. For instance, modular curves with $Q$ and $R$ being cusps, Fermat
curves with the two points being among the ‘trivial’ solutions of Fermat’s
Last Theorem, namely the points with one of the coordinates being $0$, and
hyperelliptic curves with the two points being Weierstrass points.

Let $C^{Q}$ denote the image of $C$ under the map $C\rightarrow J(C)$ given by
$x\rightarrow x-Q$. Similarly, let ${}_{R}C$ denote the image of $C$ under the
map $x\rightarrow R-x$ and let $f^{Q}$ and ${}_{R}f$ denote the function $f$
being considered as a function on $C^{Q}$ and ${}_{R}C$ respectively.

Consider the cycle in $J(C)$ given by

$Z_{QR,P}=(C^{Q},f^{Q})+(_{R}C,_{R}f)$

we have

$\operatorname{div}_{C^{Q}}(f^{Q})+\operatorname{div}_{{}_{R}C}(_{R}f)=N(0)-N(R-Q)+N(R-Q)-N(0)=0.$

Hence the cycle $Z_{QR,P}$ gives an element of
$H^{2g-1}_{{\mathcal{M}}}(J(C),{\mathds{Z}}(g))$.

###  2.4 Regulators

Let $X$ be a smooth projective variety of dimension $g$ over ${\mathds{C}}$.
The regulator map of Beilinson is a map from the motivic cohomology group to
the Deligne cohomology group.

$\operatorname{reg}_{{\mathds{Z}}}:H^{2g-1}_{{\mathcal{M}}}(X,{\mathds{Z}}(g))\rightarrow
H^{2g-1}_{{\mathcal{D}}}(X,{\mathds{Z}}(g))=\frac{(F^{1}H^{2}(X,{\mathds{C}}))^{*}}{H_{2}(X,{\mathds{Z}}(1))}.$

where $*$ denotes the ${\mathds{C}}$-linear dual and $F^{\bullet}$ denotes the
Hodge filtration. The group $H^{2g-1}_{{\mathcal{D}}}(X,{\mathds{Z}}(g))$ is a
generalised torus.

The map is defined as follows [Col97]: Let $Z=\sum_{i}(C_{i},f_{i})$ be a
cycle in $H^{2g-1}_{{\mathcal{M}}}(X,{\mathds{Z}}(g))$, so $C_{i}$ are curves
on $X$ and $f_{i}$ are functions on them satisfying the cocycle condition. Let
$[0,\infty]$ denote the positive real axis in ${\mathds{P}}^{1}$ and
$\gamma_{i}=f_{i}^{-1}([0,\infty])$. Then
$\sum_{i}\operatorname{div}(f_{i})=0$ implies that the $1$-chain
$\sum_{i}\gamma_{i}$ is closed and in fact torsion. If $H_{1}(X,{\mathds{Z}})$
has no torsion – as is in the case of the Jacobian of curves, it is exact.
Assuming that, we have

$\sum_{i}\gamma_{i}=\partial(D)$

for some $2$-chain $D$. For a closed $2$-form $\omega$ whose cohomology class
lies in $F^{1}H_{DR}^{2}(X,{\mathds{C}})$,

$\operatorname{reg}_{{\mathds{Z}}}(Z)(\omega):=\sum_{i}\int_{C_{i}\setminus\gamma_{i}}\log(f_{i})\omega+2\pi
i\int_{D}\omega.$ (1)

Here $C_{i}\setminus\gamma_{i}$ is the Riemann surface with boundary obtained
as follows. Let $n_{\epsilon}(\gamma_{i})$ be an open tubular neighbourhood of
$\gamma_{i}$ in $C_{i}$ which is homeomorphic to
$(-\epsilon,\epsilon)\times\gamma_{i}$. $C_{i}\backslash
n_{\epsilon}(\gamma_{i})$ is a closed subset of $C_{i}$ with the structure of
a manifold with boundary. The boundary $\partial(C_{i}\setminus
n_{\epsilon}(\gamma_{i}))$ is made up of two copies of $\gamma_{i}$ with
opposite orientation as well as $(-\epsilon,\epsilon)\times\gamma_{i}(0)$ and
$(-\epsilon,\epsilon)\times\gamma_{i}(1)$ with opposite orientations.
$C_{i}\setminus\gamma_{i}$ is the manifold obtained by letting
$\epsilon\rightarrow 0$. For a decomposable element $(C,a)$, where
$a\in{\mathds{C}}^{*}$, the regulator is particularly simple:

$\operatorname{reg}_{{\mathds{Z}}}((C,a))(\omega)=\int_{C}\log(a)\omega=\log(a)\int_{C}\omega.$

###  2.5 The Regulator of $Z_{QR,P}$

Let $Z_{QR,P}$ be the motivic cohomology cycle in
$H^{2g-1}_{{\mathcal{M}}}(J(C),{\mathds{Z}}(g))$. We now obtain a formula for
its regulator. The regulator is a current on forms in
$F^{1}(H^{2}(J(C)_{{\mathds{C}}}))$. Since $H^{2}(J(C))=\wedge^{2}H^{1}(C)$
elements are of the form $\phi\wedge\psi$ where $\phi$ and $\psi$ are closed
$1$-forms on $C$ and one of $\phi$ or $\psi$ is of type $(1,0)$. We have the
following theorem:

######  Theorem 2.2.

Let $Z_{QR,P}$ be the motivic cohomology cycle in
$H^{2g-1}_{{\mathcal{M}}}(J(C),{\mathds{Z}}(g))$, $\phi$ and $\psi$ two closed
harmonic $1$-forms in $H^{1}(J(C))=H^{1}(C)$ with $\psi$ holomorphic. Then

$\operatorname{reg}_{{\mathds{Z}}}(Z_{QR,P})(\phi\wedge\psi)=2\int_{C\setminus\gamma}\log(f)\phi\wedge\psi+2\pi
i\int_{\gamma}(\phi\psi-\psi\phi)$

###### Proof.

Recall that $f=f_{QR}$ is a function on $C$ with divisor $NQ-NR$ for some $N$.
Let $\omega=\phi\wedge\psi$ and $\gamma=f^{-1}([0,\infty])$. As $f$ is of
degree $N$, $\gamma$ is the union of $N$ paths each lying on a different sheet
with only the points $Q$ and $R$ in common. We will denote them by
$\gamma^{i}$, $1\leq i\leq N$. Each $\gamma^{i}$ is a path from $Q$ to $R$.
Let $\gamma^{Q}$ and ${}_{R}\gamma$ denote the path $\gamma$ on $C^{Q}$,
${}_{R}C$ respectively and similarly for the components $\gamma^{i}$. Then
from the co-cycle condition one has

$\gamma^{Q}\cdot_{R}\gamma=\partial(D)$

where $D$ is a $2$-chain on $J(C)$.

From equation (1) one has

$\operatorname{reg}_{{\mathds{Z}}}(Z_{QR,P})(\omega)=\int_{C^{Q}\setminus\gamma^{Q}}\log(f_{Q})\omega+\int_{{}_{R}C\setminus_{R}\gamma}\log(_{R}f)\omega+2\pi
i\int_{D}\omega.$ (2)

Our aim is to find a more explicit expression for
$\operatorname{reg}_{{\mathds{Z}}}(Z_{QR,P})$. For this we need an explicit
description of $D$. This was done by Colombo [Col02], Lemma 1.2.

######  Lemma 2.3.

Let

$a(s,t)=t\hskip 56.9055pt{\rm and}\hskip
56.9055ptb(s,t)=\frac{t(1-s)}{1-s(1-t)}.$

Define $F_{i}:[0,1]\times[0,1]\longrightarrow J(C)$ by

$F_{i}(s,t)=\gamma^{i}(a(s,t))-\gamma^{i}(b(s,t))$

for $1\leq i\leq N$ and let

$D_{i}=\operatorname{Im}(F_{i}).$

Then, orienting counterclockwise,

$\partial(D_{i})=\gamma^{i,Q}\cdot_{R}\gamma^{i}.$

In particular, if $D=\cup_{i=1}^{N}D_{i}$ then

$\partial(D)=\gamma^{Q}\cdot_{R}\gamma$

###### Proof.

The proof is essentially identical to Colombo’s Lemma 1.2 - the only change is
that she does it for $N=2$ \- so we do not repeat it here.

∎

We can compute the last integral as an iterated integral as follows.

######  Lemma 2.4.

Let $\phi$ and $\psi$ be closed harmonic $1$-forms on $C$ and let $D_{i}$ be a
disc as in the above lemma. Then

$\int_{D_{i}}\phi\wedge\psi=\int_{\gamma^{i,Q}}\phi\psi-\int_{{{}_{R}\gamma}^{i}}\psi\phi=\int_{\gamma}(\phi\psi-\psi\phi)$

###### Proof.

This again is a slightly modified version of Colombo [Col02], Lemma 13. ∎

This completes the proof of Theorem 2.2.

##  3 Extensions

As stated in the introduction, conjecturally, there is a canonical description
of the motivic cohomology group as an extension in the category of mixed
motives. From now on, $\operatorname{Ext}$ will denote
$\operatorname{Ext}^{1}$. Further, we will use $H^{*}(X)$ to denote the group
$H^{*}(X({\mathds{C}}),{\mathds{Z}})$, the singular (Betti) cohomology group
with integral coefficients and $H^{*}(X)_{A}$ to denote
$H^{*}(X)\otimes_{{\mathds{Z}}}A$, where $A$ is typically
${\mathds{Q}},{\mathds{R}}$ or ${\mathds{C}}$.

In our case, if one has a suitable category of mixed motives over
${\mathds{Q}}$, ${\mathcal{MM}}_{{\mathds{Q}}}$, one expects for a variety $X$
[Sch93] and $i,n$ non-negative numbers with $i<2n-1$,

$H^{i+1}_{{\mathcal{M}}}(X,{\mathds{Q}}(n))\simeq\operatorname{Ext}_{{\mathcal{MM}}_{{\mathds{Q}}}}({\mathds{Q}}(-n),h^{i}(X)).$
(3)

where ${\mathds{Q}}(-n)$ denotes the twist of the Tate motive and $h^{i}(X)$
denotes the motive whose Hodge realisation is $H^{i}(X)$.

One knows that the Deligne cohomology can be considered as an extension in the
category of integral mixed Hodge structures,

$H^{i+1}_{{\mathcal{D}}}(X,{\mathds{Z}}(n))\simeq\operatorname{Ext}_{MHS}({\mathds{Z}}(-n),H^{i}(X))$

Assuming (3) holds at the level of integer coefficients, the regulator map
above then has a canonical description as the map induced by the realisation
map from the category of mixed motives to the category of mixed Hodge
structures,

$\operatorname{Ext}_{{\mathcal{MM}}_{{\mathds{Q}}}}({\mathds{Z}}(-n),h^{i}(X))\stackrel{{\scriptstyle\operatorname{reg}_{{\mathds{Z}}}}}{{\longrightarrow}}\operatorname{Ext}_{MHS}({\mathds{Z}}(-n),H^{i}(X)).$

###  3.1 Extensions of Mixed Hodge structures coming from the Fundamental
Group

The key point of this paper is that, in some cases, one can also obtain
extensions of mixed Hodge structures in other ways. For instance, if $(X,P)$
is a pointed algebraic variety, it was shown by Hain [Hai87] that the graded
quotients $J_{P}^{a}/J_{P}^{b}$, with $a\leq b$, where $J_{P}$ is the
augmentation ideal of the group ring of the fundamental group
${\mathds{Z}}[\pi_{1}(X,P)]$, carry mixed Hodge structures. Hence natural
exact sequences involving them lead to extensions of mixed Hodge structures.

Our aim is to first construct some natural motivic cohomology cycles in the
case when $X=J(C)$, the Jacobian of a curve of genus $g=g_{C}$. Their
regulators will give rise to extensions of mixed Hodge structures. We will
show that there are natural extensions of mixed Hodge structures coming from
the Hodge structure on the graded pieces of ${\mathds{Z}}[\pi_{1}(C,P)]$ for
some suitable point $P$ which give the same extensions. In particular, since
the constructions can be carried out in at the level of mixed motives, if we
had a good category of mixed motives the cycle itself would be an extension in
the conjectured category of mixed motives coming from the fundamental group.

###  3.2 The Mixed Hodge structure on the Fundamental Group

Let $C$ be a smooth projective curve over ${\mathds{C}}$ and $P$, $Q$ and $R$
be three distinct points on $C$. Consider the open curve $C_{Q}=C-\\{Q\\}$.
Let ${\mathds{Z}}[\pi_{1}(C_{Q},P)]$ be the group ring of the fundamental
group of $C_{Q}$ based at $P$. Let $J_{Q,P}:=J_{C_{Q},P}$ denote the
augmentation ideal

$J_{Q,P}:=J_{C_{Q},P}=\operatorname{Ker}\\{{\mathds{Z}}[\pi_{1}(C_{Q},P)]\stackrel{{\scriptstyle\operatorname{deg}}}{{\longrightarrow}}{\mathds{Z}}\\}.$

Let $H^{0}({\mathcal{B}}_{r}(C_{Q};P))$ denote the $F$-vector space, where $F$
is ${\mathds{R}}$ or ${\mathds{C}}$, of homotopy invariant iterated integrals
of length $\leq r$. Chen[Che77] showed that

$H^{0}({\mathcal{B}}_{r}(C_{Q};P))\simeq\operatorname{Hom}_{{\mathds{Z}}}({\mathds{Z}}[\pi_{1}(C_{Q},P)]/J_{C_{Q}.P}^{r+1},F)$

under the map

$I\longrightarrow I(\gamma)=\int_{\gamma}I.$

Using this Hain [Hai87] was able to put a natural mixed Hodge structure on the
graded pieces $J_{Q,P}/J_{Q,P}^{r}$.

###  3.3 The Extension $E^{3}_{Q,P}$

From this point on, we will use the following notation. For an extension $E$
of mixed Hodge structures,

$E:0\longrightarrow B\longrightarrow H\longrightarrow A\longrightarrow 0$

we use $m$ to denote its class in $\operatorname{Ext}^{1}_{MHS}(A,B)$ and $H$
to denote the middle term. We will also use the notation $N\cdot E$ to denote
$N$ times the extension with respect to the Baer sum, use $N\cdot m$ to denote
its class of this extension in the $\operatorname{Ext}$ group and $N\cdot H$
to denote its middle term .

For $r\geq 3$, one can consider the extensions of mixed Hodge structures

$E^{r}_{Q,P}:0\longrightarrow(J_{Q,P}/J_{Q,P}^{r-1})^{*}\longrightarrow(J_{Q,P}/J_{Q,P}^{r})^{*}\longrightarrow(J_{Q,P}^{r-1}/J_{Q,P}^{r})^{*}\longrightarrow
0$

where for a module $M$, $M^{*}=\operatorname{Hom}(M,{\mathds{Z}})$.

The simplest non-trivial case is when $r=3$. In this case
$(J_{Q,P}/J_{Q,P}^{2})^{*}\simeq H^{1}(C_{Q})\simeq H^{1}(C)$ and
$(J_{Q,P}^{2}/J_{Q,P}^{3})^{*}\simeq\otimes^{2}H^{1}(C)$ and the exact
sequence becomes

$E^{3}_{Q,P}:0\longrightarrow
H^{1}(C)\longrightarrow(J_{Q,P}/J_{Q,P}^{3})^{*}\longrightarrow\otimes^{2}H^{1}(C)\longrightarrow
0.$

Hence $E^{3}_{Q,P}$ gives an element $m^{3}_{Q,P}$ in
$\operatorname{Ext}(\otimes^{2}H^{1}(C),H^{1}(C))$. A similar construction
with $R$ in the place of $Q$ gives us the extension $E^{3}_{R,P}$, which also
lies in the same $\operatorname{Ext}$ group.

There is a surjection $\cup:\otimes^{2}H^{1}(C)\longrightarrow
H^{2}(C)\simeq{\mathds{Z}}(-1)$ coming from the cup product. Let $K$ be the
kernel of this map. The exact sequence of Hodge structures

$0\longrightarrow
K\longrightarrow\otimes^{2}H^{1}(C)\stackrel{{\scriptstyle\cup}}{{\longrightarrow}}{\mathds{Z}}(-1)\longrightarrow
0$

splits over ${\mathds{Q}}$ but not over ${\mathds{Z}}$. This happens as
follows: There is a bilinear form [Kae01]

$b:\otimes^{2}H^{1}(C)\times\otimes^{2}H^{1}(C)\longrightarrow{\mathds{Z}}$

defined by

$b(x_{1}\otimes x_{2},y_{1}\otimes y_{2})=(x_{1}\cup y_{2})\cdot(x_{2}\cup
y_{1}).$

Let $S$ denote the orthogonal complement of $K$ in $\otimes^{2}H^{1}(C)$ with
respect to this bilinear form. Then, under the cup product $S$ projects to
$2g_{C}{\mathds{Z}}(-1)$ where $g_{C}$ is the genus of $C$ and

$\otimes^{2}H^{1}(C)_{{\mathds{Q}}}=K_{{\mathds{Q}}}\oplus S_{{\mathds{Q}}}.$

Let $\bar{m}^{3}_{Q,P}$ denote the class in
$\operatorname{Ext}_{MHS}(S,H^{1}(C))$ corresponding to the extension

$0\longrightarrow H^{1}(C)\longrightarrow\bar{E}^{3}_{Q,P}\longrightarrow
S\longrightarrow 0$

obtained by restricting $E^{3}_{Q,P}$ to the extension of $S$ by $H^{1}(C)$.
From Kaenders [Kae01] one knows there is a covering map of complex tori,

$\operatorname{Ext}(\otimes^{2}H^{1}(C),H^{1}(C))\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}\operatorname{Ext}(K\oplus
S,H^{1}(C))=\operatorname{Ext}(K,H^{1}(C))\times\operatorname{Ext}(S,H^{1}(C)).$

It is well known that
$\operatorname{Ext}(S,H^{1}(C))=\operatorname{Ext}({\mathds{Z}}(-1),H^{1}(C))\simeq\operatorname{Pic}^{0}(C)$.
To understand the other term, from the work of Hain [Hai87], Pulte [Pul88],
Kaenders [Kae01] and Rabi [Rab01] one has the following theorem

######  Theorem 3.1.

The image of the class $m^{3}_{Q,P}$ of $E^{3}_{Q,P}$ in
$\operatorname{Ext}(\otimes^{2}H^{1}(C),H^{1}(C))$ is given by

$\phi(m^{3}_{Q,P})=(m^{3}_{P},\bar{m}^{3}_{Q,P})$

where $m^{3}_{P}\in\operatorname{Ext}(K,H^{1}(C))$ depends only on $P$ and
$\bar{m}^{3}_{Q,P}$ is given by

$2g_{C}Q-2P-\kappa_{C}\in\operatorname{Pic}^{0}(C)$

where $\kappa_{C}$ is the canonical divisor of $C$ and $g_{C}$ is the genus of
$C$.

Recall that in the group $\operatorname{Ext}$, addition is given by the Baer
sum. We will denote this by $\oplus_{B}$ (or $\ominus_{B}$ if we are taking
differences). Let $m^{3}_{QR,P}$ denote the Baer difference
$m^{3}_{Q,P}\ominus_{B}m^{3}_{R,P}$.

######  Lemma 3.2.

Under the hypothesis that there is a function with divisor
$\operatorname{div}(f_{QR})=NQ-NR$ the extension class $m^{3}_{QR,P}$ is
torsion in $\operatorname{Ext}(H^{1}C)\otimes H^{1}(C),H^{1}(C))$. Precisely,

$N\cdot H^{3}_{QR,P}\simeq H^{1}(C)\bigoplus\otimes^{2}H^{1}(C)$

where by $N\cdot H^{3}_{QR,P}$ we mean the middle term of the exact sequence
obtained by adding the sequence $E^{3}_{QR,P}$ to itself $N$-times using the
Baer sum.

###### Proof.

This follows from [Kae01][Theorem 2.5] which states that the map

$\operatorname{Pic}^{0}(C)\longrightarrow\operatorname{Ext}(H^{1}(C)\otimes
H^{1}(C),H^{1}(C))$

given by

$Q-R\longrightarrow m^{3}_{Q,P}-m^{3}_{R,P}$

is well defined and injective. Hence, since $N(Q-R)=0$ in
$\operatorname{Pic}^{0}(C)$, $N(m^{3}_{Q,P}-m^{3}_{R,P})=N(m^{3}_{QR,P})=0$ in
$\operatorname{Ext}(H^{1}(C)\otimes H^{1}(C),H^{1}(C))$. ∎

A consequence of this is that there is a morphism of integral mixed Hodge
structures

$r_{3}:N\cdot H^{3}_{QR,P}\longrightarrow H^{1}(C)$

given by the projection.

######  Remark 3.3.

This extension represents the class $Q-R$, at least up to a integral multiple,
and is hence the first example of the theme of this paper - namely the Abel-
Jacobi image of a null-homologous cycle is described in terms of extensions
coming from the fundamental group.

∎

###  3.4 The Extensions $E^{4}_{Q,P}$ and $E^{4}_{R,P}$

From the work of Hain, Pulte, Harris and others one knows that the class
$m^{3}_{P}$ in $\operatorname{Ext}(K,H^{1}(C))$ corresponds to the extension
of mixed Hodge structures determined by the Ceresa cycle in $J(C)$, or
alternately, the modified diagonal cycle in $C^{3}$. We would like to
construct a similar class corresponding to the motivic cohomology cycle
$Z_{QR,P}$. To that end, we now consider, with $C,P,Q$ and $R$ as before, the
extension corresponding to $r=4$

$E_{Q,P}^{4}:0\longrightarrow(J_{Q,P}/J_{Q,P}^{3})^{*}\longrightarrow(J_{Q,P}/J_{Q,P}^{4})^{*}\longrightarrow(J_{Q,P}^{3}/J_{Q,P}^{4})^{*}\longrightarrow
0.$

We have that $(J_{Q,P}^{3}/J_{Q,P}^{4})^{*}\simeq\otimes^{3}H^{1}(C)$ and this
does not depend on $P,Q$ or $R$. However, from Theorem 3.1,
$(J_{Q.P}/J_{Q,P}^{3})^{*}$ depends on $Q$ and $P$. Similarly
$(J_{R.P}/J_{R,P}^{3})^{*}$ depends on $R$ and $P$. Hence we get classes in
$\operatorname{Ext}(\otimes^{3}H^{1}(C),(J_{Q.P}/J_{Q,P}^{3})^{*})$ and
$\operatorname{Ext}(\otimes^{3}H^{1}(C),(J_{R.P}/J_{R,P}^{3})^{*})$ – which
are different groups - hence we cannot take their difference.

When $C$ is hyperelliptic the extension classes $m^{3}_{Q,P}$ and
$m^{3}_{R,P}$ are 2-torsion in
$\operatorname{Ext}(\otimes^{2}H^{1}(C),H^{1}(C))$. Hence one gets two classes

$2m^{4}_{Q,P},2m^{4}_{R,P}\in\operatorname{Ext}(\otimes^{3}H^{1}(C),\otimes^{2}H^{1}(C)\oplus
H^{1}(C))$

and one can project to get two classes $e^{4}_{Q,P}$ and $e^{4}_{R,P}$ in
$\operatorname{Ext}(\otimes^{3}H^{1}(C),H^{1}(C))$. Colombo [Col02] shows that
the class

$e^{4}_{QR,P}=e^{4}_{Q,P}\ominus_{B}e^{4}_{R,P}\in\operatorname{Ext}(\otimes^{3}H^{1}(C),H^{1}(C))$

corresponds to the extension determined by the cycle $Z_{QR}$ — after pulling
back and pushing forward with some standard maps.

Unfortunately, in general the extension classes $m^{3}_{Q,P}$ and
$m^{3}_{R,P}$ are not torsion in the $\operatorname{Ext}$ group. They
correspond to the instances where the Ceresa cycle is non-torsion — which is
the generic case. In fact, the instances where it is known that the cycles are
non-torsion are precisely the cases we have in mind — modular curves and
Fermat curves [Har83],[Blo84]. Hence we cannot use this argument immediately.
However, since we know from Lemma 3.2 that their difference $m^{3}_{QR,P}$ is
torsion, we would like to get an extension of the form

$0\longrightarrow
H^{3}_{QR,P}\otimes{\mathds{Q}}\longrightarrow``H^{4}_{QR,P}"\otimes{\mathds{Q}}\longrightarrow\otimes^{3}H^{1}(C)\otimes{\mathds{Q}}\longrightarrow
0$

where $``H^{4}_{QR,P}"$ is the middle term of a sort of generalised Baer
difference of the two extensions $E^{4}_{Q,P}$ and $E^{4}_{R,P}$. We could
then push-forward this extension using the splitting to get a class in
$\operatorname{Ext}(\otimes^{3}H^{1}(C)_{{\mathds{Q}}},H^{1}(C)_{{\mathds{Q}}})$.
We cannot simply consider $E^{4}_{QR,P}=E^{4}_{Q,P}\ominus_{B}E^{4}_{R,P}$ as
the two extensions lie in different $\operatorname{Ext}$ groups. So we have to
consider a generalisation of Baer sums to not necessarily exact sequences
which we came across in a paper of Rabi [Rab01].

###  3.5 The Baer sum

This is well known but we recall it to fix notation in order to describe
Rabi’s work. Recall that if we have two exact sequences of modules

$\begin{CD}E_{j}:0@>{}>{}>A@>{f_{j}}>{}>B_{j}@>{p_{j}}>{}>C@>{}>{}>0\end{CD}$

for $j\in\\{1,2\\}$, the Baer difference $E_{1}\ominus_{B}E_{2}$ is
constructed as follows. We have

$\begin{CD}0@>{}>{}>A\oplus A@>{f_{1}\oplus f_{2}}>{}>B_{1}\oplus
B_{2}@>{p_{1}\oplus p_{2}}>{}>C\oplus<EMAIL_ADDRESS>
Let $\psi:B_{1}\oplus B_{2}\longrightarrow C$ be the map

$\psi(b_{1},b_{2})=p_{1}(b_{1})-p_{2}(b_{2})$

and let
$H=\operatorname{Ker}(\psi)=\\{(b_{1},b_{2})|\;p_{1}(b_{1})=p_{2}(b_{2})\\}$.
Let $D$ be the image of $\tilde{f}:A\longrightarrow A\oplus A\longrightarrow
H$

$\tilde{f}(a)=(f_{1}(a),f_{2}(a))$

Let $B=H/D$. The map $f:A\oplus A\longrightarrow B$ given by

$f(a_{1},a_{2})=(f_{1}(a_{1}),f_{2}(a_{2}))$

factors through $(A\oplus A)/A\simeq A$ and so one has a map
$\bar{f}:A\longrightarrow B$,

$a\longrightarrow(f_{1}(a),0)=(0,-f_{2}(a))$

and an exact sequence

$\begin{CD}0@>{}>{}>A@>{\bar{f}}>{}>B@>{p_{1}(or\;p_{2})}>{}>C@>{}>{}>0\end{CD}$

The class of this exact sequence in $\operatorname{Ext}(C,A)$ is the Baer
difference $E_{1}\ominus_{B}E_{2}$. The Baer sum $E_{1}\oplus_{B}E_{2}$ is the
sequence obtained when one of the maps $f_{2}$ or $p_{2}$ is replaced by its
negative. The Baer sum is essentially the push-out over $A$ in the category of
modules.

###  3.6 Rabi’s generalisation

Now suppose we have diagrams of the following type:

$\begin{CD}&&0\\\ &&@V{}V{}V\\\ &&A_{1}\\\ &&@V{}V{i_{j}}V\\\
0@>{}>{}>B_{1}^{j}@>{f_{j}}>{}>B_{2}^{j}@>{p_{j}}>{}>B_{3}@>{}>{}>0\\\
&&@V{}V{\pi_{j}}V\\\ &&C_{1}\\\ &&@V{}V{}V\\\ &&0\end{CD}$

where the vertical and horizontal sequences are exact for $j\in\\{1,2\\}$. Let
$E_{j}$ denote the horizontal exact sequences:

$\begin{CD}E_{j}:0@>{}>{}>B_{1}^{j}@>{f_{j}}>{}>B_{2}^{j}<EMAIL_ADDRESS>
We would like to take the Baer difference of the $E_{j}$ — but since they do
not lie in the same $\operatorname{Ext}$ group we cannot quite do that.
However, we can still salvage something.

One gets two types of extension classes in $\operatorname{Ext}$ groups which
do not depend on $j$. The vertical exact sequences give classes in
$\operatorname{Ext}(C_{1},A_{1})$. We can form their Baer difference to get an
exact sequence

$\begin{CD}0@>{}>{}>A_{1}@>{}>{}>{\mathbf{B}}_{1}<EMAIL_ADDRESS>
The horizontal exact sequences give extensions in
$\operatorname{Ext}(B_{3},B_{1}^{j})$. These depend on $j$ but their push
forward under $\pi_{j}$ give classes ${\mathbf{f}}_{B_{2}^{j}}$ in
$\operatorname{Ext}(B_{3},C_{1})$.

Define ${\mathbf{B}}_{2}$ as follows: Let $H_{2}=\operatorname{Ker}(\psi)$,
where $\psi$ is the ‘difference’ map

$\psi:B_{2}^{1}\oplus B_{2}^{2}\longrightarrow B_{3}$
$\psi((b_{2}^{1},b_{2}^{2}))=(p_{1}(b_{2}^{1})-p_{2}(b_{2}^{2}))$

Let $D_{2}$ be the image of the map

$A_{1}\longrightarrow B_{1}^{1}\oplus B_{1}^{2}\longrightarrow H_{2}$
$a\longrightarrow(f_{1}(i_{1}(a)),f_{2}(i_{2}(a)))$

Define ${\mathbf{B}}_{2}=H_{2}/D_{2}$. We call this the generalised Baer
difference of $E_{1}$ and $E_{2}$ and denote it by $\tilde{\ominus}_{B}$.
Observe that this is almost the Baer difference of $E_{1}$ and $E_{2}$ in the
sense that if $B_{1}=B_{1}^{1}=B_{1}^{2}$, then we could take the difference
in $\operatorname{Ext}(B_{3},B_{1})$. Since that is not the case, we do the
best we can — we take the difference of the inexact sequences

$\begin{CD}0@>{}>{}>A_{1}@>{}>{}>B_{2}^{j}<EMAIL_ADDRESS>
As a result of this one has a complex

$\begin{CD}0@>{}>{}>{\mathbf{B}}_{1}@>{f_{1}\oplus
f_{2}}>{}>{\mathbf{B}}_{2}@>{p_{1}(or\;p_{2})}>{}>B_{3}@>{}>{}>0\end{CD}$

However, this complex is not exact since $\operatorname{Ker}(p_{1})$ is larger
than $(f_{1}\oplus f_{2})({\mathbf{B}}_{1})$. The next lemma describes this
difference.

######  Lemma 3.4 (Rabi[Rab01]).

Let
${\mathbf{F}}={\mathbf{F}}_{B^{1}_{2}\tilde{\ominus}_{B}B^{2}_{2}}={\mathbf{B}}_{2}/{\mathbf{B}}_{1}$.
Then one has the following diagram, in which the horizontal and vertical
sequences are exact.

$\begin{CD}&&&&&&0\\\ &&&&&&@V{}V{}V\\\ &&&&&&C_{1}\\\ &&&&&&@V{}V{\phi}V\\\
0@>{}>{}>{\mathbf{B}}_{1}@>{f}>{}>{\mathbf{B}}_{2}@>{\eta}>{}>{\mathbf{F}}@>{}>{}>0\\\
&&&&&&@V{}V{\bar{p}}V\\\ &&&&&&B_{3}\\\ &&&&&&@V{}V{}V\\\ &&&&&&0\end{CD}$

###### Proof.

[Rab96], Appendix B. We repeat the proof here as that is unpublished. The
horizontal sequence is exact by definition. To show the vertical sequence is
exact we have to first describe be map $\phi$. It is defined as follows. One
has maps $\pi_{j}:B_{1}^{j}\longrightarrow C_{1}$. Consider the natural map

$\begin{CD}\tilde{\phi}:C_{1}\oplus C_{1}@>{}>{}>(B_{1}^{1}\oplus
B_{1}^{2})/\Delta_{A_{1}}@>{(f_{1},f_{2})}>{}>{\mathbf{B}}_{2}=H_{2}/D_{2}\end{CD}$
$\tilde{\phi}(c_{1},c_{2})\rightarrow(\pi_{1}^{-1}(c),\pi_{2}^{-1}(c))\rightarrow(f_{1}(\pi_{1}^{-1}(c_{1})),f_{2}(\pi_{2}^{-1}(c_{2})))$

where $\Delta_{A_{1}}=\\{(i_{1}(a),i_{2}(a))|a\in A_{1}\\}$. $\phi$ gives a
well defined map

$(C_{1}\oplus
C_{1})/\Delta_{C_{1}}\longrightarrow{\mathbf{B}}_{2}/\tilde{\phi}(\Delta_{C_{1}})$

where $\Delta_{C_{1}}=\\{(c,-c)|c\in C_{1}\\}$ is the anti-diagonal. This is
well defined as if $(b_{1},b_{2})$ and $(b^{\prime}_{1},b^{\prime}_{2})$ are
in $(\pi_{1}^{-1}(c_{1}),\pi_{2}^{-1}(c_{2}))$ we have to show

$(f_{1}(b_{1}),f_{2}(b_{2}))\equiv(f_{1}(b^{\prime}_{1}),f_{2}(b^{\prime}_{2}))\operatorname{mod}\tilde{\phi}(\Delta_{C_{1}})$

or

$(f_{1}(b_{1}-b^{\prime}_{1}),f_{2}(b_{2}-b^{\prime}_{2}))\in\tilde{\phi}(\Delta_{C_{1}}).$

From exactness, we have $b_{1}-b^{\prime}_{1}=i_{1}(a_{1})$ and
$b_{2}-b^{\prime}_{2}=i_{2}(a_{2})$ with $a_{i}\in A_{1}$. The image of
$\Delta_{C_{1}}$ under $(\pi_{1}^{-1},\pi_{2}^{-1})$ consists of
$(b,b^{\prime})$ such that $\pi_{1}(b)=\pi_{2}(b^{\prime})$.
$(i_{1}(a_{1}),i_{2}(a_{2}))$ lie in this image, hence

$(f_{1}(i_{1}(a_{1})),f_{2}(i_{2}(a_{2})))=(f_{1}(b_{1}-b^{\prime}_{1}),f_{2}(b_{2}-b^{\prime}_{2}))\in\tilde{\phi}(\Delta_{C_{1}}).$

Note that the pre-image $(\pi_{1}^{-1},\pi_{2}^{-1})(\Delta_{C_{1}})$ in
$B_{1}^{1}\oplus B_{1}^{2})/\Delta_{A_{1}}$ is the Baer difference
${\mathbf{B}}_{1}$. Further, $(C_{1}\oplus C_{1})/\Delta_{C_{1}})\simeq
C_{1}$. Hence one has a map
$\phi:C_{1}\rightarrow{\mathbf{F}}={\mathbf{B}}_{2}/{\mathbf{B}}_{1}$ and we
get a exact sequence

$0\longrightarrow
C_{1}\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}{\mathbf{F}}_{B^{1}_{2}\tilde{\ominus}_{B}B^{2}_{2}}\stackrel{{\scriptstyle\bar{p}}}{{\longrightarrow}}B_{3}\longrightarrow
0$

This sequence is exact as if $b=(b^{1}_{2},b^{2}_{2})$ is in
${\mathbf{F}}_{B^{1}_{2}\tilde{\ominus}_{B}B^{2}_{2}}$ and $\bar{p}(b)=0$,
then $p_{1}(b^{1}_{2})=p_{2}(b^{2}_{2})=0$. So $b^{1}_{2}$ and $b^{2}_{2}$ lie
in the image of $B_{1}^{1}\oplus B_{1}^{2}$ — say $b_{2}^{1}=f_{1}(b_{1}^{1})$
and $b_{2}^{2}=f_{2}(b_{1}^{2})$. Let $c_{i}=\pi_{1}(b_{1}^{1})$ and
$c_{2}=\pi_{2}(b_{1}^{2})$. Then

$b=\phi(c_{1},c_{2})$

so it lies in the image of $\phi$.

∎

In general, for any ${\mathds{Z}}$-linear combination $m\cdot
B^{1}_{2}\tilde{\ominus}_{B}\;n\cdot B^{2}_{2}$ of $B_{2}^{1}$ and $B_{2}^{2}$
we get an extension class ${\mathbf{f}}_{m\cdot
B^{1}_{2}\tilde{\ominus}_{B}\;n\cdot B^{2}_{2}}$ in
$\operatorname{Ext}(B_{3},C_{1})$ corresponding to ${\mathbf{F}}_{m\cdot
B^{1}_{2}\tilde{\ominus}_{B}\;n\cdot B^{2}_{2}}$. The relation between this
and the extension classes constructed above is given as follows:

######  Corollary 3.5.

Let ${\mathbf{f}}_{B_{2}^{j}}$ and ${\mathbf{f}}_{m\cdot
B_{2}^{1}\tilde{\ominus}_{B}\;n\cdot B^{2}_{2}}$ be the extensions in
$\operatorname{Ext}(B_{3},C_{1})$ described above. Then,

${\mathbf{f}}_{m\cdot B_{2}^{1}\tilde{\ominus}_{B}\;n\cdot
B^{2}_{2}}=m\cdot{\mathbf{f}}_{B_{2}^{1}}\ominus_{B}\;n\cdot{\mathbf{f}}_{B_{2}^{2}}.$

###### Proof.

This follows from the construction of the map $\phi$. ∎

In the next section we apply these constructions in our particular case to get
the extension class we want.

###  3.7 The Extension $e^{4}_{QR,P}$

In this section we construct an extension $e^{4}_{QR,P}$ in
$\operatorname{Ext}(\otimes^{3}H^{1}(C),H^{1}(C))$ which generalises the
element $e^{4}_{Q,P}\ominus_{B}e^{4}_{R,P}$ constructed by Colombo. Recall
that we have an exact sequence

$E^{3}_{Q,P}:0\longrightarrow
H^{1}(C)\longrightarrow(J_{Q,P}/J_{Q,P}^{3})^{*}\longrightarrow\otimes^{2}H^{1}(C)\longrightarrow
0$

and a similar one $E^{3}_{R,P}$. Also, we have the exact sequence

$E_{Q,P}^{4}:0\longrightarrow(J_{Q,P}/J_{Q,P}^{3})^{*}\longrightarrow(J_{Q,P}/J_{Q,P}^{4})^{*}\longrightarrow(J_{Q,P}^{3}/J_{Q,P}^{4})^{*}\longrightarrow
0$

and a similar $E^{4}_{R,P}$. This gives us diagrams as in Lemma 3.4, with
$B_{1}^{1}=(J_{Q,P}/J_{Q,P}^{3})^{*}$, $B_{1}^{2}=(J_{R,P}/J_{R,P}^{3})^{*}$,
$B_{2}^{1}=(J_{Q,P}/J_{Q,P}^{4})^{*}$, $B_{2}^{2}=(J_{R,P}/J_{R,P}^{4})^{*}$
and $A_{1}=H^{1}(C)$, $B_{3}=\otimes^{2}H^{1}(C)$ and finally
$C_{1}=\otimes^{2}H^{1}(C)$.

Let ${\mathbf{f}}_{Q}$, ${\mathbf{f}}_{R}$ and ${\mathbf{f}}_{QR}$ denote the
classes in $\operatorname{Ext}(\otimes^{3}H^{1}(C),\otimes^{2}H^{1}(C))$ with
middle terms $H^{23}_{Q,P},H^{23}_{R,P}$ and $H^{23}_{QR,P}$ corresponding to
the diagrams for $Q$, $R$ and their generalised Baer difference.
${\mathbf{f}}_{Q}$ and ${\mathbf{f}}_{R}$ are the push-forwards of
$m^{4}_{Q,P}$ and $m^{4}_{R,P}$ respectively. From Corollary 3.5 one has

${\mathbf{f}}_{QR}={\mathbf{f}}_{Q}-{\mathbf{f}}_{R}.$

######  Lemma 3.6.

${\mathbf{f}}_{QR}$ is $N$-torsion in
$\operatorname{Ext}(\otimes^{3}H^{1}(C),\otimes^{2}H^{1}(C))$. Namely,

$N\cdot H^{23}_{QR,P}=\otimes^{2}H^{1}(C)\oplus\otimes^{3}H^{1}(C).$

###### Proof.

From Rabi [Rab01], Corollary 3.3, one has that the class $H^{23}_{Q,P}$ in
$\operatorname{Ext}(\otimes^{3}H^{1}(C),\otimes^{2}H^{1}(C))$ is given by

$H^{23}_{Q,P}=H^{1}(C)\otimes E^{3}_{QP}\oplus_{B}E^{3}_{QP}\otimes H^{1}(C).$

Similarly,

$H^{23}_{R,P}=H^{1}(C)\otimes E^{3}_{RP}\oplus_{B}E^{3}_{RP}\otimes H^{1}(C).$

Taking their difference gives

$H^{23}_{R,P}-H^{23}_{Q,P}=H^{1}(C)\otimes(E^{3}_{R,P}-E^{3}_{Q,P})\oplus_{B}(E^{3}_{RP}-E^{3}_{QP})\otimes
H^{1}(C)$

From Lemma 3.2 we have
$E^{3}_{R,P}-E^{3}_{Q,P}=(0,2g(Q-R))\in\operatorname{Ext}_{MHS}(\otimes^{2}H^{1}(C),H^{1}(C))$.
As $Q-R$ is $N$-torsion, we have

$N\cdot(H^{23}_{R,P}-H^{23}_{Q,P})\cong H^{1}(C)\oplus\otimes^{2}H^{1}(C).$

∎

We also know from Lemma 3.2 that $m^{3}_{QR,P}$ is $N$-torsion. Hence from
Lemma 3.4 we get an exact sequence

$\begin{CD}0@>{}>{}>\otimes^{2}H^{1}(C)\oplus H^{1}(C)@>{}>{}>N\cdot
H^{4}_{QR,P}<EMAIL_ADDRESS>
which gives a class in
$\operatorname{Ext}(\otimes^{3}H^{1}(C)\oplus\otimes^{2}H^{1}(C),\otimes^{2}H^{1}(C)\oplus
H^{1}(C))$. From the Künneth theorem,

$\operatorname{Ext}(\otimes^{3}H^{1}(C)\oplus\otimes^{2}H^{1}(C),\otimes^{2}H^{1}(C)\oplus
H^{1}(C))=\prod_{i\in\\{2,3\\},j\in\\{1,2\\}}\operatorname{Ext}(\otimes^{i}H^{1}(C),\otimes^{j}H^{1}(C)).$

Define

$e^{4}_{QR,P}\in\operatorname{Ext}(\otimes^{3}H^{1}(C),H^{1}(C))$ (4)

to be the projection onto that component. Note that if $C$ is hyperelliptic,
this class $e^{4}_{QR,P}$ is precisely the class
$e^{4}_{QR,P}=e^{4}_{Q,P}\ominus_{B}e^{4}_{R,P}$ constructed by Colombo.

###  3.8 Statement of the Main Theorem

Armed with the class
$e^{4}_{QR,P}\in\operatorname{Ext}(\otimes^{3}H^{1}(C),H^{1}(C))$ we can
proceed as in Colombo.

Let $\Omega$ denote the pullback of the polarisation on $J(C)$ in
$H^{2}(J(C))(1)$ to $\otimes^{2}H^{1}(C)(1)$. There is an injection obtained
by tensoring with $\Omega$,

$J_{\Omega}=\otimes\Omega:H^{1}(C)(-1)\longrightarrow\otimes^{3}H^{1}(C).$

We first pull back the class using the map $J_{\Omega}$ to get a class in

$J_{\Omega}^{*}(e^{4}_{QR,P})\in\operatorname{Ext}(H^{1}(C)(-1),H^{1}(C)).$

Tensoring with $H^{1}(C)$ we get a class

$J_{\Omega}^{*}(e^{4}_{QR,P})\otimes
H^{1}(C)\in\operatorname{Ext}(\otimes^{2}H^{1}(C)(-1),\otimes^{2}H^{1}(C)).$

Once again pulling back using the map
$\beta:{\mathds{Z}}(-1)\rightarrow\otimes^{2}H^{1}(C)$ gives us a class

$\epsilon_{QR,P}^{4}\in\operatorname{Ext}({\mathds{Z}}(-2),\otimes^{2}H^{1}(C))\subset\operatorname{Ext}({\mathds{Z}}(-2),H^{2}(C\times
C))$ (5)

Our main theorem is

######  Theorem 3.7.

Let $C$ be a smooth projective curve of genus $g_{C}$ and $P$, $Q$ and $R$ be
three distinct points. Let $Z_{QR,P}$ be the element of the motivic cohomology
group $H^{3}_{{\mathcal{M}}}(J(C),{\mathds{Z}}(2))$ constructed above. Let
$\epsilon^{4}_{QR,P}$ be the extension in
$\operatorname{Ext}_{MHS}({\mathds{Z}}(-2),\wedge^{2}H^{1}(C))$ constructed
above. Then

$\epsilon^{4}_{QR,P}=(2g_{C}+1)N\operatorname{reg}_{{\mathds{Z}}}(Z_{QR})$

in $\operatorname{Ext}_{MHS}({\mathds{Z}}(-2),\wedge^{2}H^{1}(C))$.

In other words our theorem states that the regulator of a natural cycle in the
motivic cohomology group of a product of curves, being thought of as an
extension class is the same as that as a natural extension of MHS coming from
the fundamental group of the curve. In fact, it is an extension of pure Hodge
structures.

######  Remark 3.8 (Dependence on $P$).

This is not so serious. If we do not normalise $f_{QR}$ with the condition
that $f_{QR}(P)=1$ then one has to add an expression of the form
$\log(f_{QR}(P))\int_{C}\cdot$ to the term — and this corresponds to adding a
decomposable element of the form $(\Delta_{C},\log(f_{QR}(P)))$ to our element
$Z_{QR}$.

##  4 Carlson’s Representatives

The proof of the main theorem will follow by showing that both the algebraic
cycle constructed earlier and the extension class constructed above induce the
same current. For that we have to understand the how an extension class
induces a current. This comes from understanding the Carlson representative.
In the section we once again follow Colombo [Col02] and adapt her arguments to
our situation.

If $V$ is a MHS all of whose weights are negative, then the Intermediate
Jacobian of $V$ is defined to be

$J_{0}(V)=\frac{V_{{\mathds{C}}}}{F^{0}V_{{\mathds{C}}}\oplus
V_{{\mathds{Z}}}}.$

This is a generalised torus - namely a group of the form
${\mathds{C}}^{a}/{\mathds{Z}}^{b}\simeq({\mathds{C}}^{*})^{b}\times({\mathds{C}})^{a-b}$
for some $a$ and $b$.

An extension of mixed Hodge structures

$0\longrightarrow
A\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}H\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}B\longrightarrow
0$

is called separated if the lowest non-zero weight of $B$ is greater than the
largest non-zero weight of $A$. This implies that
$\operatorname{Hom}_{MHS}(B,A)$ has negative weights. Carlson [Car80] showed
that

$\operatorname{Ext}_{MHS}(B,A)\simeq J_{0}(\operatorname{Hom}(B,A)).$

This is defined as follows. As an extension of Abelian groups, the extension
splits. So one has a map $r_{{\mathds{Z}}}:H\rightarrow A$ which is a
retraction, $r_{{\mathds{Z}}}\circ\iota=id$. Let $s_{F}$ be a section in
$\operatorname{Hom}(B_{{\mathds{C}}},H_{{\mathds{C}}})$ preserving the Hodge
filtration. Then the Carlson representative of an extension is defined to be
the class of

$r_{{\mathds{Z}}}\circ s_{F}\in J_{0}(Hom(B,A)).$

We now describe explicitly the Carlson representative of the extension
$\epsilon^{4}_{QR,P}$ constructed in the previous section. This is done in a
few steps, first we describe the representative for $e^{4}_{QR,P}$ and then
for its various pullbacks and push forwards to obtain that for
$\epsilon^{4}_{QR,P}$.

###  4.1 Preliminaries

As before, let $C$ be a smooth projective curve of genus $g=g_{C}$. We first
describe the Carlson representative of the extension

$e^{4}_{QR,P}\in\operatorname{Ext}_{MHS}(\otimes^{3}H^{1}(C),H^{1}(C)).$

From the above discussion this is an element of

$J_{0}(\operatorname{Hom}(\otimes^{3}H^{1}(C),H^{1}(C))=\frac{\operatorname{Hom}(\otimes^{3}H^{1}(C)_{{\mathds{C}}},H^{1}(C)_{{\mathds{C}}})}{F^{0}\operatorname{Hom}(\otimes^{3}H^{1}(C)_{{\mathds{C}}},H^{1}(C)_{{\mathds{C}}})\oplus\operatorname{Hom}(\otimes^{3}H^{1}(C),H^{1}(C))}$

so explicitly, given an element of $\otimes^{3}H^{1}(C)_{{\mathds{C}}}$ we get
an element of $H^{1}(C)_{{\mathds{C}}}$ which we can think of as a functional
on $H_{1}(C)_{{\mathds{C}}}$

Let $C_{QR}$ denote the open curve $C\setminus\\{Q,R\\}$. In fact we will
describe the functional as an iterated integral made up of forms in
$H^{1}(C_{QR})_{{\mathds{C}}}$ and will naturally be a functional on
$H_{1}(C_{QR})$. We have a natural inclusion

$i:C_{QR}\hookrightarrow C$

which induces $i_{*}$ on homology and $i^{*}$ on cohomology. In order to
consider the iterated integral as a functional on $H_{1}(C)$ we have to make a
choice of an embedding $H_{1}(C)\hookrightarrow H_{1}(C_{QR})$ which splits
the map $i_{*}$. There are many ways of doing this, but for our formula to
work, we need to make a particular choice. In this section we first construct
a ‘natural’ splitting of the map $i_{*}$ – namely a subgroup of
$H_{1}(C_{QR})$ which maps isomorphically to $H_{1}(C)$ under $i_{*}$.

Consider the group $\pi_{1}(C_{QR};P)$. This is a free group on $2g+1$
generators. The generators have the following description. The fundamental
polygon of $C$ is a $4g$ sided polygon with the edges $e_{i}$ and $e_{i+g}$
identified. The end points of the edges are identified and so they give $2g$
loops $\alpha^{\prime}_{i}$ in $C_{QR}$ which we consider as loops based at
$P$. Let $\beta_{Q}$ be a small simple loop around $Q$ based at $P$. Then
$\pi_{1}(C_{QR};P)=<\alpha^{\prime}_{1},\dots,\alpha^{\prime}_{2g},\beta_{Q}>$.

The map $f:C\rightarrow{\mathds{P}}^{1}$ restricts to give
$f:C_{QR}\rightarrow{\mathds{P}}^{1}-\\{0,\infty\\}$ and this induces

$f_{*}:\pi_{1}(C_{QR};P)\longrightarrow\pi_{1}({\mathds{P}}^{1}-\\{0,\infty\\};1)$

One knows $\pi_{1}({\mathds{P}}^{1}-\\{0,\infty\\})\simeq{\mathds{Z}}$. Let
$\beta_{0}$ denote the generator. Let $H=\operatorname{Ker}(f_{*})$.
$f_{*}(\pi_{1}(C_{QR};P))$ is a subgroup of ${\mathds{Z}}$. In a deleted
neighbourhood of $0$ the map looks like $z\rightarrow z^{N}$ where $N$ is the
degree. Hence loop $\beta_{Q}$ is taken to $\beta_{0}^{N}$. Let
$f_{*}(\alpha^{\prime}_{i})=\beta_{0}^{m_{i}}$ for some
$m_{i}\in{\mathds{Z}}$. Then
$\alpha_{i}={\alpha^{\prime}_{i}}^{N}\beta_{Q}^{-m_{i}}$ satisfies
$f_{*}(\alpha_{i})=0$. Let $G$ denote the subgroup of
$H=\operatorname{Ker}(f_{*})$ generated by the $\\{\alpha_{i}\\}$.

The inclusion map $i$ also induces $i_{*}$ on the fundamental groups. Since
$i_{*}(\beta_{Q})=0$, $i_{*}(\alpha_{i})=i_{*}(\alpha^{\prime}_{i})^{N}$. The
fundamental group of $C$ is
$\pi_{1}(C;P)=\\{<i_{*}(\alpha^{\prime}_{1}),\dots,i_{*}(\alpha^{\prime}_{2g})>/\prod[i_{*}(\alpha^{\prime}_{i}),i_{*}(\alpha^{\prime}_{i+g})]=0\\}$.
Hence one has a map $G\rightarrow\pi_{1}(C;P)$ whose image is the subgroup
generated by the $N^{th}$-powers of $\alpha^{\prime}_{i}$.

######  Lemma 4.1.

The abelianization of $G$ is isomorphic to the subgroup of index $N^{2g}$ of
the abelianization of $\pi_{1}(C;P)$.

$G/[G,G]\simeq N\cdot\pi_{1}(C)/[\pi_{1}(C),\pi_{1}(C)]$

where $N\cdot$ denotes multiplication by $N$.

###### Proof.

Let $\alpha=\prod\alpha_{a_{i}}^{b_{i}}$ be a word in $G$. For a generator
$\alpha_{i}$ of $G$ define

$\operatorname{ord}_{\alpha_{i}}(\alpha)=\sum_{a_{i}=i}b_{i}$

namely, the number of times $\alpha_{i}$ appears in the word. Define

$\Psi:G\rightarrow{\mathds{Z}}^{2g}$
$\Psi(\alpha)=\left(\operatorname{ord}_{\alpha_{i}}(\alpha),\dots,\operatorname{ord}_{\alpha_{2g}}(\alpha)\right)$

Let $K=\ker(\Psi)$. Clearly $[G,G]\subset K$. Further, the map $\Psi$ factors
through $i_{*}$ and is surjective. We claim $K=[G,G]$. To see this, observe
that if $a,b\in G$

$ab\equiv ba\,\operatorname{mod}\,[G,G]$

Repeatedly applying this one can see that any word

$\alpha=\prod\alpha_{a_{i}}^{b_{i}}\equiv\prod_{i=1}^{2g}\alpha_{i}^{\operatorname{ord}_{\alpha_{i}}(\alpha)}\,\operatorname{mod}\,[G,G]$

In particular, if $\operatorname{ord}_{\alpha_{i}}(\alpha)=0$ for all $i$,
$\alpha\in[G,G]$. Hence $K=[G,G]$. Hence

$G/[G,G]\simeq{\mathds{Z}}^{2g}$

The map $i_{*}$ takes $\alpha_{i}$ to ${\alpha^{\prime}}_{i}^{N}$. One has a
similar map $\Psi^{\prime}:\pi_{1}(C;P)\rightarrow{\mathds{Z}}^{2g}$ using
$\alpha^{\prime}_{i}$ instead of $\alpha_{i}$ which shows that the
abelianization of $\pi_{1}(C;P)$ is ${\mathds{Z}}^{2g}$ as well. However,
under this map $\Psi^{\prime}(\alpha_{i})=N$ and hence $G/[G,G]$ is carried to
the subgroup $N\cdot{\mathds{Z}}^{2g}$. Multiplication by $N$ is an
isomorphism so the map

$i_{N}:=\frac{1}{N}\circ
i_{*}:G/[G,G]\longrightarrow\pi_{1}(C)/[\pi_{1}(C),\pi_{1}(C)]$

is an isomorphism between the two abelianizations.

∎

Let $V=G/[G,G]$. The abelianization of the fundamental group of $C_{QR}$ is
$H_{1}(C_{QR})$ and so $V$ is a subgroup of $H_{1}(C_{QR})$. The
abelianization of $\pi_{1}(C)$ is $H_{1}(C)$. Hence the map $i_{N}$ is an
isomorphism between $V$ and $H_{1}(C)$. Let $j_{N}:H_{1}(C)\longrightarrow V$
be the inverse isomorphism. This gives an embedding of $H_{1}(C)$ in
$H_{1}(C_{QR})$. As discussed above, the Carlson representative is a
functional on $H_{1}(C)$. However, we will obtain a functional on
$H_{1}(C_{QR})$ which will be the Carlson representative of the extension when
considered as a functional on $V$.

Let $[\alpha]$ denote the homology class of a loop $\alpha$. The collection
$\\{[\alpha_{i}^{\prime}]\\}$ has the property that their images
$\\{[i_{*}(\alpha^{\prime}_{i})]\\}$ in $H_{1}(C)$ form a symplectic basis.
Since $i_{*}([\beta_{Q}])=0$,
$i_{*}([\alpha_{i}])=Ni_{*}([\alpha^{\prime}_{i}])$. Hence under the
isomorphism, $i_{N}([\alpha_{i}])=[i_{*}(\alpha^{\prime}_{i})]$. Let
$\\{dx_{i}\\}$ be the dual basis of harmonic forms in $H^{1}(C,{\mathds{C}})$
satisfying

$\int_{[i_{*}(\alpha_{i}^{\prime})]}dx_{j}=\delta_{ij}$

where $\delta_{ij}$ is the Kronecker Delta function. With this choice of
$\\{[\alpha^{\prime}_{i}]\\}s$ and $\\{dx_{i}\\}s$, the volume form on
$H^{2}(C)$ can be expressed as follows. Let

$c(i)=\begin{cases}1&\text{ if }i\leq g_{C}\\\ -1&\text{ if
}i>g_{C}\end{cases}$

and $\sigma(i)=i+c(i)g_{C}$. The volume form is

$\sum_{i=1}^{2g_{C}}c(i)dx_{i}\wedge dx_{\sigma(i)}.$

From that one gets that a Poincaré dual of $c(i)dx_{\sigma(i)}$ is
$[i_{*}(\alpha_{i}^{\prime})]$.

The group $V$ is a subgroup of $H_{1}(C_{QR})$. Recall that for the non-
compact manifold $C_{QR}$, Poincaré duality states that

$H^{1}_{c}(C_{QR})\simeq H_{1}(C_{QR}),$

where $H^{1}_{c}(C_{QR})$ is the cohomology with compact support of $C_{QR}$.
This group has a mixed Hodge structure determined by identifying it with the
relative cohomology group $H^{1}(C_{QR},\\{Q,R\\})$. Unlike $H^{1}(C_{QR})$
which has nontrivial weight $1$ and weight $2$ pieces, cohomology with compact
support has weight $0$ and weight $1$ pieces and is covariant. However, we
have an isomorphism of the weight $1$ graded pieces,

$Gr^{W}_{1}H^{1}_{c}(C_{QR})_{{\mathds{Q}}}\simeq
Gr^{W}_{1}H^{1}(C_{QR})_{{\mathds{Q}}}\simeq H^{1}(C)_{{\mathds{Q}}}$

Here the first isomorphism is induced by $id_{*}$, the identity map and the
second by $i^{*}$.

The space $V$ determines a splitting of the Hodge structure on
$H_{1}(C_{QR})$. The space $V^{*}$ of Poincaré duals of element of $V$ is a
subspace of $H^{1}_{c}(C_{QR})$ which determines a splitting of the Hodge
structure on $H^{1}_{c}(C_{QR})$. Further, $V^{*}$ is isomorphic to
$H^{1}(C)$. Hence if $\eta$ is a form in $H^{1}(C)$ it is cohomologous in
$C_{QR}$ to a compactly supported form in $V^{*}\subset H^{1}_{c}(C_{QR})$.
One has

$H^{1}_{c}(V_{QR})_{{\mathds{Q}}}=V_{{\mathds{Q}}}\oplus{\mathds{Q}}\cdot\omega_{Q}$

where $\omega_{Q}$ is a Poincaré dual of $\beta_{Q}$.
${\mathds{Q}}\cdot\omega_{Q}\simeq{\mathds{Q}}(0)$. Note that

$\int_{[\alpha_{j}]}\omega_{Q}=\int_{C_{QR}}i^{*}(c(j)dx_{\sigma(j)})\wedge\omega_{Q}=$
$=-\int_{C_{QR}}\omega_{Q}\wedge
i^{*}(c(j)dx_{\sigma(j)})=-\int_{\beta_{Q}}i^{*}(c(j)dx_{\sigma(j)})=-\int_{i_{*}([\beta_{Q}])}c(j)dx_{\sigma(j)}=0$

since $i_{*}(\beta_{Q})=0$. Further

$\int_{[\alpha_{i}]}i^{*}(dx_{j})=\int_{i_{*}([\alpha_{i}])}dx_{j}=\int_{Ni_{*}([\alpha^{\prime}_{i}])}dx_{j}=N\delta_{ij}.$

Hence the dual of $[\alpha_{i}]$ is $\frac{i^{*}(dx_{i})}{N}$ and under the
dual map $dx_{i}$ is taken to $\frac{dx_{i}}{N}$ in
$V^{*}=\operatorname{Hom}(V,{\mathds{Z}})$. Further, note that

$\int_{[\alpha_{i}]}\frac{df}{f}=\int_{f_{*}(c(k)\alpha_{\sigma(k)})}\frac{dz}{z}=0$

since $[\alpha_{i}]\in\operatorname{Ker}(f_{*})$.

We now construct a cover of $C_{QR}$ which has the property that its first
homology group is $G/[G,G]$ and the form $\frac{df}{f}$ is exact. Further, the
loops $\alpha_{i}$ lift to loops on this cover. We do that as follows. Let
$u:X\rightarrow C_{QR}$ denote the universal cover of $C_{QR}$. The group $G$
acts on $X$ as a group of deck transformations. Let $\tilde{C}=X/G$ denote the
quotient and $q:\tilde{C}\rightarrow C_{QR}$ denote the covering map. This is
a cover

$q:(\tilde{C},\tilde{P})\longrightarrow(C_{QR},P)$ (6)

such that $\pi_{1}(\tilde{C};\tilde{P})=G$, where $\tilde{P}$ is a point in
$q^{-1}(P)$. Now by homotopy lifting ([Hat02], Proposition $1.31$), loops
based at $P$ whose homotopy class lie in $G\subset\pi_{1}(C_{QR})$ will lift
to loops in $\tilde{C}$ based at $\tilde{P}$. Thus $\alpha_{i}\in G$ will lift
to a unique, upto homotopy loop $\tilde{\alpha}_{i}$ based at $\tilde{P}$ such
that $q_{*}(\tilde{\alpha}_{i})=\alpha_{i}$.

######  Proposition 4.2.

$q^{*}(\frac{df}{f})=0$ in $H^{1}(\tilde{C})$. Hence there is a function,
which we call $\log(q^{*}(f))$, defined on $\tilde{C}$ such that
$d\log(q^{*}(f))=q^{*}(\frac{df}{f})$.

###### Proof.

From Lemma 4.1,

$H_{1}(\tilde{C})\simeq G/[G,G]\simeq V\simeq
N\cdot\pi_{1}(C)/[\pi_{1}(C;P),\pi_{1}(C;P)]\simeq N\cdot H_{1}(C)\simeq
H_{1}(C)$

By the de Rham isomorphism, $H^{1}(\tilde{C})\simeq H^{1}(C)$.

The maps $q^{*}$ and $q_{*}$ are adjoint with respect to the de Rham pairing –
namely if $\sigma\in H_{1}(\tilde{C})$ and $\omega\in
H^{1}(C_{QR},{\mathds{C}})$ then

$\int_{q_{*}(\sigma)}\omega=\int_{\sigma}q^{*}(\omega).$

Further, $q^{*}(\omega)$ is $0$ in $H^{1}(\tilde{C})$ if and only if
$\int_{q_{*}(\sigma)}\omega=0$ for all $\sigma\in H_{1}(\tilde{C})$. Applying
this to $\omega=\frac{df}{f}$ and using the fact that
$[\tilde{\alpha}_{i}],1\leq i\leq 2g$ give a basis for $H_{1}(\tilde{C})$, we
have

$q^{*}(\frac{df}{f})=0\in
H^{1}(\tilde{C})\Leftrightarrow\int_{[\tilde{\alpha}_{i}]}q^{*}(\frac{df}{f})=0\text{
for all }i\Leftrightarrow\int_{[\alpha_{i}]}\frac{df}{f}=0\text{ for all }i.$

The map $f$ induces

$f_{*}:H_{1}(C_{QR})\longrightarrow H_{1}({\mathds{P}}^{1}-\\{0,\infty\\}).$

The form $\frac{df}{f}=f^{*}(\frac{dz}{z})$. Hence, one has

$\int_{[\alpha_{i}]}\frac{df}{f}=\int_{[\alpha_{i}]}f^{*}(\frac{dz}{z})=\int_{f_{*}([\alpha_{i}])}\frac{dz}{z}.$

However, since $\alpha_{i}\in G$ and by choice
$G\subset\operatorname{Ker}(f_{*})$, we have $f_{*}(\alpha_{i})=0$ so
$f_{*}([\alpha_{i}])=0$ and finally

$\int_{f_{*}([\alpha_{i}])}\frac{dz}{z}=0$

Hence $q^{*}(\frac{df}{f})=0\in H^{1}(\tilde{C})$. Therefore integration of
$\frac{df}{f}$ is path independent and we have a well defined function

$\log(q^{*}(f))(x)=\int_{\tilde{P}}^{x}q^{*}(\frac{df}{f})$

on $\tilde{C}$. Note that $\log(q^{*}(f)(\tilde{P}))=0$.

∎

Hence the space $V$ can be understood as the homology of the space $\tilde{C}$
and the map $q_{*}$ gives a rational splitting of the map $i_{*}$. We also
have the following description of $V_{{\mathds{Q}}}$.

######  Lemma 4.3.

Let $f:C_{QR}\longrightarrow{\mathds{P}}^{1}-\\{0,\infty\\}$ be the map with
divisor $\operatorname{div}(f)=NQ-NR$ and $f(P)=1$ and $V=G/[G,G]$ as above.
Let
$W_{{\mathds{Q}}}=\operatorname{Ker}(f_{*}:H^{1}(C_{QR})_{{\mathds{Q}}}\longrightarrow
H^{1}({\mathds{P}}^{1}-\\{0,\infty\\}))$. Then
$V_{{\mathds{Q}}}=W_{{\mathds{Q}}}$

###### Proof.

Since $V\subset\operatorname{Ker}(f_{*})$, $V_{{\mathds{Q}}}\subset
W_{{\mathds{Q}}}$. However, both $V_{{\mathds{Q}}}$ and $W_{{\mathds{Q}}}$ are
subspaces of codimension $1$ in $H_{1}(C_{QR})_{{\mathds{Q}}}$. Hence they are
isomorphic. ∎

Note that it does not appear to be true that $V=\operatorname{Ker}(f_{*})$ as
${\mathds{Z}}$-modules. Intrinsically, the reason why there is such a $V$ is
the following. If $C$ and $C_{QR}$ are as above, there is an exact sequence of
mixed Hodge structures

$0\longrightarrow{\mathds{Z}}(1)\longrightarrow H_{1}(C_{QR})\longrightarrow
H_{1}(C)\longrightarrow 0$

induced by the inclusion map. Hence $H_{1}(C_{QR})$ determines a class in
$\operatorname{Ext}(H_{1}(C),{\mathds{Z}}(1))$. From the Carlson isomorphism
one knows

$\operatorname{Ext}_{MHS}(H_{1}(C),{\mathds{Z}}(1))\simeq\operatorname{Ext}_{MHS}({\mathds{Z}}(-1),H^{1}(C))\simeq
CH^{1}_{hom}(C)$

and the class determined by $H_{1}(C_{QR})$ is nothing but the class of $Q-R$
in $CH^{1}_{hom}(C)$. Since there exists a function $f$ with
$\operatorname{div}(f)=NQ-NR$ it implies that this sequence splits rationally.
Hence there is a map

$p:H_{1}(C_{QR})_{{\mathds{Q}}}\longrightarrow{\mathds{Q}}(1)$

which splits the exact sequence. This map can be seen to be

$p(\sigma)=\int_{\sigma}\frac{df}{f}=\int_{f_{*}(\sigma)}\frac{dz}{z}$

and if $V_{{\mathds{Q}}}$ is the kernel, then $V_{{\mathds{Q}}}\simeq
H_{1}(C)_{{\mathds{Q}}}$. Clearly
$\sigma\in\operatorname{Ker}(p)\Leftrightarrow\sigma\in\operatorname{Ker}(f_{*})$.
Hence $\operatorname{Ker}(f_{*})$ is isomorphic to $H_{1}(C)_{{\mathds{Q}}}$.
The $V$ defined above is only contained in $\operatorname{Ker}(f_{*})$ but is
a subgroup of the integral homology $H_{1}(C)$ – so has a little more
information.

###  4.2 The Carlson Representative of $e^{4}_{QR,P}$

The Carlson representative of $e^{4}_{QR,P}$ is given by

$p_{1}\circ r_{{\mathds{Z}}}\circ s_{F}\circ i_{3},$

where

  * •

$p_{1}$ is the projection of $N\cdot H^{3}_{QR,P}\simeq
H^{1}(C)\oplus\otimes^{2}H^{1}(C)\stackrel{{\scriptstyle
p_{1}}}{{\longrightarrow}}H^{1}(C)$.

  * •

$i_{3}$ is the inclusion map $\otimes^{3}H^{1}(C)\stackrel{{\scriptstyle
i_{3}}}{{\hookrightarrow}}\otimes^{3}H^{1}(C)\oplus\otimes^{2}H^{1}(C)$.

To describe $s_{F}$ we need a little more. Let $C_{\bullet}=C-\\{\bullet\\}$
for $\bullet\in\\{Q,R\\}$. The inclusion map

$i_{\bullet}:C_{\bullet}\hookrightarrow C$

induces isomorphisms on the first homology and cohomology groups – and in what
follows we will identify elements of $H_{1}(C_{\bullet})$ with their images in
$H_{1}(C)$ and similarly elements of $H^{1}(C)$ with their images in
$H^{1}(C_{\bullet})$.

Recall $\tilde{\ominus}_{B}$ denotes the generalised Baer difference. Let

$s_{F}\circ i_{3}:\otimes^{3}H^{1}(C)\longrightarrow N\cdot H^{4}_{QR,P}\simeq
N\cdot\left((J_{Q,P}/J_{Q,P}^{4})^{*}\tilde{\ominus}_{B}(J_{R,P}/J_{R,P}^{4})^{*}\right)$

be the section preserving the Hodge filtration given by

$s_{F}(dx_{i}\otimes dx_{j}\otimes dx_{k})=(I^{ijk}_{Q},I^{ijk}_{R}).$

Here $I^{ijk}_{\bullet}\in(J_{\bullet,P}/J_{\bullet,P}^{4})^{*}$ for
$\bullet\in\\{Q,R\\}$ are iterated integrals with

$I^{ijk}_{\bullet}=N\left(\int
dx_{i}dx_{j}dx_{k}+dx_{i}\mu_{jk,\bullet}+\mu_{ij,\bullet}dx_{k}+\mu_{ijk,\bullet}\right).$
(7)

where $\mu_{ij,\bullet}$, $\mu_{jk,\bullet}$ and $\mu_{ijk,\bullet}$ are
smooth, logarithmic $1$-forms on $C_{\bullet}$ such that

  1. 1.

$d\mu_{jk,\bullet}+dx_{j}\wedge dx_{k}=0$

  2. 2.

$d\mu_{ij,\bullet}+dx_{i}\wedge dx_{j}=0$

  3. 3.

$dx_{i}\wedge\mu_{jk,\bullet}+\mu_{ij,\bullet}\wedge
dx_{k}+d\mu_{ijk,\bullet}=0.$

There are inclusion maps of $C_{QR}$ into $C_{Q}$ and $C_{R}$ and we can pull
back the forms $dx_{i}$, $\mu_{ij,\bullet}$ and $\mu_{ijk,\bullet}$ to
$C_{QR}$ and consider all the forms as forms on $C_{QR}$. To compute the
element of
$\operatorname{Hom}(\otimes^{3}H^{1}(C)_{{\mathds{C}}},H^{1}(C)_{{\mathds{C}}})$
obtained as the projection under $p_{1}$, we describe it as an element of
$H_{1}(C)_{{\mathds{C}}}^{*}=\operatorname{Hom}(H_{1}(C),{\mathds{C}})=H^{1}(C,{\mathds{C}})$.
The integrands $I^{ijk}_{\bullet}$ are made up of forms on $C_{QR}$ and so to
compute it on elements of $H_{1}(C)$ we have to choose an embedding of
$H_{1}(C)$ in $H_{1}(C_{QR})$. This is precisely what the subgroup $V$ gives
us.

Hence from now on if $\alpha$ is a homology class in $H_{1}(C)$ we think of it
as an element of $H_{1}(C_{QR})$ by identifying it with its image in
$V=q_{*}(H_{1}(\tilde{C}))$. Let $V^{*}$ denote its dual in
$H^{1}_{c}(C_{QR})$. The Poincaré dual of a element of $V$ also lies in
$V^{*}$.

The map from

$H^{1}(C)\longrightarrow(H^{1}(C)\oplus H^{1}(C))/\Delta_{H^{1}(C)}$

is given by

$x\longrightarrow(x,-x).$

Further, if $\alpha$ is a loop based at $P$ on $C_{QR}$, the class in
$H_{1}(C)=J_{\bullet,P}/J_{\bullet,P}^{2}$ corresponding to it is $1-\alpha$.
So one has $p_{1}\circ r_{{\mathds{Z}}}\circ s_{F}\circ
i_{3}\in\operatorname{Hom}(\otimes^{3}H^{1}(C)_{{\mathds{C}}},H^{1}(C)_{{\mathds{C}}})$.
As an integral, it is

$p_{1}\circ r_{{\mathds{Z}}}\circ s_{F}\circ i_{3}(dx_{i}\otimes dx_{j}\otimes
dx_{k})(\alpha)=\int_{1-\alpha}I^{ijk}_{Q}-\int_{1-\alpha}I^{ijk}_{R}$

where the first $1-\alpha$ is the class in $H_{1}(C_{Q})$ and the second is
the class in $H_{1}(C_{R})$. They are both carried to the same class in $V$
under the isomorphism, so we can take the difference of the integrals when we
consider $\alpha$ as a loop in $C_{QR}$ whose corresponding homology class
lies in $V$. This resulting expression is

$\int_{1-\alpha}I^{ijk}_{Q}-\int_{1-\alpha}I^{ijk}_{R}=N\left(\int_{1-\alpha}dx_{i}\left(\mu_{jk,Q}-\mu_{jk,R}\right)+\left(\mu_{ij,Q}-\mu_{ij,R}\right)dx_{k}+\left(\mu_{ijk,Q}-\mu_{ijk,R}\right)\right).$

We can choose the logarithmic forms $\mu_{ij,\bullet}$ and
$\mu_{ijk,\bullet}$, for $\bullet\in\\{Q,R\\}$, satisfying the following

  * •

$\mu_{ij,\bullet}=-\mu_{ji,\bullet}$.

  * •

For $|i-j|\neq g_{C}$, $\mu_{ij,\bullet}$ is smooth on $C$, as
$d\mu_{ij,\bullet}=dx_{j}\wedge dx_{i}=0$. As $H^{2}(C_{QR},{\mathds{Z}})=0$
and $\mu_{ij,\bullet}$ is smooth, it is orthogonal to all closed forms, that
is, $\mu_{ij,\bullet}\wedge dx_{k}$ is exact.

  * •

$\mu_{i\sigma(i),\bullet}$ has a logarithmic singularity at $\bullet$ with
residue $c(i)$.

  * •

$\mu_{ij,Q}-\mu_{ij,R}=0$ if $|i-j|\neq g_{C}$ as forms on $C_{QR}$.

  * •

$\mu_{i\sigma(i),Q}-\mu_{i\sigma(i),R}=\frac{c(i)}{N}\frac{df}{f}$, where
$f=f_{QR}$ is a function such that $\operatorname{div}(f_{QR})=NQ-NR$. We can
normalise $f_{QR}$ once again by requiring that $f_{QR}(P)=1$.

In terms of our basis of forms of $H^{1}(C)$, $\Omega\in\otimes^{2}H^{1}(C)$
is

$\Omega=\sum_{i=1}^{g_{C}}dx_{i}\otimes dx_{(i+g_{C})}-dx_{(i+g_{C})}\otimes
dx_{i}=\sum_{i=1}^{2g_{C}}c(i)dx_{i}\otimes dx_{\sigma(i)}$

With these choices of $\mu_{ij,\bullet}$ and $\mu_{ijk,\bullet}$, we have the
following theorem:

######  Theorem 4.4.

Let ${\bf
G}_{QR,P}\in\operatorname{Hom}(H^{1}(C)(-1)_{{\mathds{C}}},H^{1}(C)_{{\mathds{C}}})$
be the Carlson representative corresponding to the extension class
$J_{\Omega}^{*}(e^{4}_{QR,P})$. It is given by

${\bf G}_{QR,P}(dx_{k})(\alpha_{j})=p_{1}\circ r_{{\mathds{Z}}}\circ
s_{F}\circ
i_{3}(dx_{k}\otimes\Omega)(\alpha_{j})=(2g_{C}+1)\int_{\alpha_{j}}\frac{df}{f}dx_{k}-N\int_{\alpha_{j}}W(dx_{k}).$

in $J(\operatorname{Hom}(H^{1}(C)(-1),H^{1}(C)))$, where

$W(dx_{k})=\sum_{i=1}^{2g_{C}}c(i)(\mu_{ki\sigma(i),Q}-\mu_{ki\sigma(i),R})$

is a $1$-form on $C_{QR}$ which satisfies

$dW(dx_{k})=(2g+1)\frac{dx_{k}}{N}\wedge\frac{df}{f}$

###### Proof.

Let $S_{F}$ denote the map $S_{F}=s_{F}\circ i_{3}\circ
J_{\Omega}:H^{1}(C)(-1)\rightarrow N\cdot H^{4}_{QR,P}$. This is given by

$S_{F}(dx_{k})=\sum_{i=1}^{2g_{C}}c(i)s_{F}(dx_{k}\otimes dx_{i}\otimes
dx_{\sigma(i)})$

From (7) one has

$S_{F}(dx_{k})=\left(\sum_{i=1}^{2g_{C}}c(i)\int
I_{Q}^{ki\sigma(i)},\sum_{i=1}^{2g_{C}}c(i)\int I_{R}^{ki\sigma(i)}\right)$

Evaluating on a loop $\alpha_{j}$ based at $P$ using the maps described above,
this is

$\sum_{i=1}^{2g_{C}}\int_{1-\alpha_{j}}c(i)\left(I_{Q}^{ki\sigma(i)}-I_{R}^{ki\sigma(i)}\right)$
$\sum_{i=1}^{2g_{C}}N\left(\int_{1-\alpha_{j}}c(i)dx_{k}(\mu_{i\sigma(i),Q}-\mu_{i,\sigma(i),R})+(\mu_{ki,Q}-\mu_{ki,R})dx_{\sigma(i)}+(\mu_{ki\sigma(i),Q}-\mu_{ki\sigma(i),R})\right)$

From the choice of the forms $\mu_{ij,\bullet}$ and $\mu_{ijk,\bullet}$ above,
the leading terms and several of the lower order terms cancel out and

$\mu_{ki,Q}-\mu_{ki,R}=c(k)\delta_{k\sigma(i)}\frac{1}{N}\frac{df}{f}$

and

$\mu_{i\sigma(i),Q}-\mu_{i\sigma(i),R}=c(i)\frac{1}{N}\frac{df}{f}.$

Since $c(i)^{2}=1$ what remains is

$\sum_{i=1}^{2g_{C}}\int_{1-\alpha_{j}}dx_{k}\frac{df}{f}-\int_{1-\alpha_{j}}\frac{df}{f}dx_{k}+N\sum_{i=1}^{2g_{C}}c(i)\int_{1-\alpha_{j}}\left(\mu_{ki\sigma(i),Q}-\mu_{ki\sigma(i),R}\right).$

Let

$W(dx_{k})=\sum_{i=1}^{2g_{C}}c(i)\left(\mu_{ki\sigma(i),Q}-\mu_{ki\sigma(i),R}\right).$

Since integration over a point, which corresponds to the constant loop 1, is
$0$ and $\int_{\alpha_{j}}\frac{df}{f}=0$ by choice of $\alpha_{j}$, using
Lemma 2.1 (2) the integral becomes

${\bf
G}_{QR,P}(dx_{k})(\alpha_{j})=2g_{C}\int_{1-\alpha_{j}}dx_{k}\frac{df}{f}-\int_{1-\alpha_{j}}\frac{df}{f}dx_{k}+N\int_{1-\alpha_{j}}W(dx_{k}).$
$=-(2g_{C}+1)\int_{\alpha_{j}}dx_{k}\frac{df}{f}-N\int_{\alpha_{j}}W(dx_{k}).$

Now consider

$dW(dx_{k})=\sum_{i=1}^{2g_{C}}c(i)d\left(\mu_{ki\sigma(i),Q}-\mu_{ki\sigma(i),R}\right).$

From the choice of $\mu_{ijk,\bullet}$, one has

$d\mu_{ijk,\bullet}=-dx_{i}\wedge\mu_{jk,\bullet}-\mu_{ij,\bullet}\wedge
dx_{k}.$

So the sum becomes

$dW(dx_{k})=\sum_{i=1}^{2g_{C}}-c(i)\left(\left(dx_{k}\wedge\mu_{i\sigma(i),Q}+\mu_{ki,Q}\wedge
dx_{\sigma(i)}\right)-\left(dx_{k}\wedge\mu_{i\sigma(i),R}+\mu_{ki,R}\wedge
dx_{\sigma(i)}\right)\right)$
$=\sum_{i=1}^{2g_{C}}-c(i)\left(dx_{k}\wedge(\mu_{i\sigma(i),Q}-\mu_{i\sigma(i),R})+(\mu_{ki,Q}-\mu_{ki,R})\wedge
dx_{\sigma(i)}\right).$

In the second sum, only one term survives and one has

$=-c(\sigma(k))(\mu_{k\sigma(k),Q}-\mu_{k\sigma(k),R})\wedge
dx_{k}+\sum_{i=1}^{2g_{C}}-c(i)\left(dx_{k}\wedge\frac{c(i)}{N}\frac{df}{f}\right)$
$=-c(\sigma(k))(\frac{c(\sigma(k))}{N}\frac{df}{f})\wedge
dx_{k}+\sum_{i=1}^{2g_{C}}-c(i)\left(dx_{k}\wedge\frac{c(i)}{N}\frac{df}{f}\right)$
$=-\frac{(2g_{C}+1)}{N}\frac{df}{f}\wedge
dx_{k}=\frac{(2g_{C}+1)}{N}dx_{k}\wedge\frac{df}{f}.$

∎

We have computed the Carlson representative $\bf{G}_{QR,P}$ of our class in
$\operatorname{Ext}(H^{1}(C)(-1),H^{1}(C))$. We now tensor with $H^{1}(C)$ and
pull back using the map
$\otimes\Omega:{\mathds{Z}}(-1)\longrightarrow\otimes^{2}H^{1}(C)$. This gives
us an element of $\operatorname{Ext}({\mathds{Z}}(-2),\otimes^{2}H^{1}(C))$.
We denote its Carlson representative by ${\bf F}_{QR,P}$.

######  Lemma 4.5.

The Carlson representative of the class in
$\operatorname{Ext}({\mathds{Z}}(-2),\otimes^{2}H^{1}(C))$ is given by

${\bf F}_{QR,P}=({\bf G}_{QR,P}\otimes Id)\circ\otimes\Omega$

in $(\otimes^{2}H^{1}(C)_{{\mathds{C}}})^{*}$. On an element
$\alpha_{j}\otimes\alpha_{k}$ it is given by

${\bf
F}_{QR,P}(\Omega)(\alpha_{j}\otimes\alpha_{k})=c(\sigma(k))N\left(\int_{\alpha_{j}}(2g_{C}+1)\frac{df}{f}dx_{\sigma(k)}-NW(dx_{\sigma(k)})\right)$
(8)

###### Proof.

Recall that

$\Omega=\sum_{1}^{2g_{C}}c(i)dx_{i}\otimes dx_{\sigma(i)}.$

From above we have

$({\bf G}_{QR,P}\otimes
Id)(\Omega)(\alpha_{j}\otimes\alpha_{k})=\sum_{1}^{2g_{C}}c(i){\bf
G}_{QR,P}(dx_{i})(\alpha_{j})\cdot Id(dx_{\sigma(i)})(\alpha_{k}).$

From the choice of $\alpha_{k}$ one has

$Id(dx_{\sigma(i)})(\alpha_{k})=N\delta_{k\sigma(i)}.$

Hence, in the sum above, precisely one term survives, at $i=\sigma(k)$.
Therefore

$({\bf G}_{QR,P}\otimes
Id)(\Omega)(\alpha_{j}\otimes\alpha_{k})=Nc(\sigma(k)){\bf
G}_{QR,P}(dx_{\sigma(k)})(\alpha_{j}).$

In particular

$\displaystyle{\bf F}_{QR,P}(\Omega)(\alpha_{j}\otimes\alpha_{k})=$
$\displaystyle Nc(\sigma(k)){\bf G}_{QR,P}(dx_{\sigma(k)})(\alpha_{j})$
$\displaystyle=$ $\displaystyle
Nc(\sigma(k))\left(\int_{\alpha_{j}}(2g_{C}+1)\frac{df}{f}dx_{\sigma(k)}-NW(dx_{\sigma(k)})\right)$

∎

We now use Proposition 4.2 to convert the iterated integral in to an ordinary
integral. The iterated integral term in (8) is

$Nc(\sigma(k))(2g_{C}+1)\int_{\alpha_{j}}\frac{df}{f}dx_{k}$

which we can evaluate using Lemma 2.1(3) if $\frac{df}{f}$ is exact. However,
$\frac{df}{f}$ is not exact on $C_{QR}$ but it is exact on $\tilde{C}$ using
Proposition 4.2. So we do the integration on $\tilde{C}$. Precisely, we do
that as follows.

Let $\alpha$ be a loop such that $[\alpha]\in q_{*}(H_{1}(\tilde{C}))$, where
$q:\tilde{C}\longrightarrow C_{QR}$ is the cover. Let
$\alpha=q_{*}(\tilde{\alpha})$, where $\tilde{\alpha}$ is a loop based at
$\tilde{P}$ lying over the basepoint $P$ of $\alpha$. Let $\psi$ be another
closed $1$-form on $C_{QR}$.

We have

$\int_{\alpha}\frac{df}{f}\psi=\int_{\tilde{\alpha}}q^{*}(\frac{df}{f})q^{*}(\psi)$

From Proposition 4.2, $q^{*}(\frac{df}{f})$ is exact on $\tilde{\alpha}$. In
other words $q^{*}(\frac{df}{f})=d\log(q^{*}(f))$. Choose a primitive
$\log(q^{*}(f))$ such that $\log(q^{*}(f)(\tilde{P}))=0$. Using Lemma 2.1(3)
and the fact that we have chosen $\log(q^{*}(f))$ with
$\log(q^{*}(f)(\tilde{P}))=0$,

$\int_{\tilde{\alpha}}q^{*}({\frac{df}{f}})q^{*}(\psi)=\int_{\tilde{\alpha}}\log(q^{*}(f))q^{*}(\psi)$

Hence we have

$\int_{\alpha}\frac{df}{f}\psi=\int_{\tilde{\alpha}}\log(q^{*}(f))q^{*}(\psi)$

Applying this to the case at hand we have

${\bf
F}_{QR,P}(\Omega)(\alpha_{j}\otimes\alpha_{k})=Nc(\sigma(k))\left(\int_{\tilde{\alpha}_{j}}(2g_{C}+1)\log(q^{*}(f))q^{*}(dx_{\sigma}(k))-Nq^{*}(W(dx_{\sigma(k)}))\right).$
(9)

We have made a choice of $\tilde{\alpha_{j}}$. If we chose a different
basepoint, the value of $\log(q^{*}(f))$ will change by $2\pi iM$ for some
$M\in{\mathds{Z}}$. This will change the integral by $2\pi
iM\int_{\alpha_{j}}dx_{\sigma(k)}$. This does not affect the class in the
intermediate Jacobian.

We would like to connect the expression above, which is the Carlson
representative of the extension class $\epsilon^{4}_{QR,P}$, to the regulator
of an explicit cycle on the Jacobian of the curve. This is done by the
following revised version of Colombo’s Proposition 3.3. Recall that by
$C\setminus\gamma$ we mean

$\lim_{\epsilon\rightarrow 0}C\setminus{n_{\epsilon}(\gamma)}$

where ${n_{\epsilon}(\gamma)}$ is an open tubular neighbourhood of $\gamma$
homeomorphic to $(-\epsilon,\epsilon)\times\gamma$.

######  Proposition 4.6 (Colombo Proposition 3.3).

Let $f=f_{QR}$ be as before and $\psi$ a closed $1$-form on $C_{QR}$. Let
$W(\psi)$ be a $1$-form such that $dW(\psi)=\psi\wedge\frac{df}{f}$. Hence
$\Theta=\log(f)\psi+W(\psi)$ is a closed $1$ form on $C_{QR}\setminus\gamma$.
Let $\alpha$ be a loop in $C_{QR}$ such that $[\alpha]\in V$ and let
$\eta_{\alpha}\in H^{1}_{c}(C_{QR})$ be the Poincaré dual of $[\alpha]$
constructed below. Then we have the iterated integral

$\int_{\alpha}\frac{df}{f}\psi+W(\psi)=\int_{C\setminus\gamma}\eta_{\alpha}\wedge\Theta+2\pi
i\int_{\gamma}\eta_{\alpha}\psi\,\left(\operatorname{mod}\,2\pi
i\int_{\alpha}\psi{\mathds{Z}}\right)$

###### Proof.

We first need the following useful lemma.

######  Lemma 4.7.

Let $\Theta$ and $\psi$ be as above. Then

$\int_{\gamma}\psi=0$

###### Proof.

Let $C\setminus{n_{\epsilon}(\gamma)}$ be the manifold with boundary as
before. This is a closed set so the form $\Theta$ is closed and compactly
supported on $C\setminus{n_{\epsilon}(\gamma)}$. The boundary

$\partial(C\setminus{n_{\epsilon}(\gamma)})=Q\times(-\epsilon,\epsilon)\cup\gamma_{1}\cup\gamma_{2}^{-1}\cup(\epsilon,-\epsilon)\times
R$

where $\gamma_{1}$ and $\gamma_{2}$ are copies of $\gamma$. Applying Stokes
theorem to $d\Theta$ we get

$0=\int_{C\setminus{n_{\epsilon}(\gamma)}}d\Theta=\int_{\partial(C\setminus{n_{\epsilon}(\gamma)})}\Theta=\int_{\gamma_{1}}\Theta-\int_{\gamma_{2}}\Theta+\int_{Q\times(-\epsilon,\epsilon)}\Theta-\int_{R\times(-\epsilon,\epsilon)}\Theta$

Recall $\Theta=\log(f)\psi+W(\psi)$. The function $\log(f)$ differs by $2\pi
i$ on $\gamma_{1}$ and $\gamma_{2}$. The form $W(\psi)$ is defined on all of
$C_{QR}$ so the integrals over $\gamma_{1}$ and $\gamma_{2}$ cancel. Keeping
track of orientations we have

$0=-2\pi
i\int_{\gamma_{1}}\psi+\int_{Q\times(-\epsilon,\epsilon)}\Theta-\int_{R\times(-\epsilon,\epsilon)}\Theta$

As $\epsilon\rightarrow 0$ the shows that $-2\pi i\int_{\gamma}\psi=0.$ ∎

The subgroup $V$ is generated by the classes of
$\alpha_{i}={\alpha^{\prime}}_{i}^{N}\beta_{Q}^{-m_{i}}$ where
$\alpha^{\prime}_{i}$ is one of the ‘standard’ generators of $\pi_{1}(C)$
coming from the edges of the fundamental polygon and $\beta_{Q}$ is a small
simple loop around $Q$. These loops satisfy $f_{*}(\alpha_{i})=0$.

It suffices to prove the theorem for the $\alpha_{i}$ and extend linearly, so
from this point on we let $\alpha=\alpha_{i}$,
$\alpha^{\prime}=\alpha^{\prime}_{i}$. Let $\eta_{\alpha}$ be the compactly
supported Poincaré dual of $[\alpha]$ constructed as in [FK92] as follows.
Suppose $\delta$ is a simple loop in $C_{QR}$. Let $D=D_{\delta}$ be a tubular
neighbourhood of $\delta$. We can write $D_{\delta}-\delta=D_{\delta}^{+}\cup
D_{\delta}^{-}$ with $D_{\delta}^{-}$ to the left and $D_{\delta}^{+}$ to the
right of $\delta$. Let $D_{0}$ be a sub-tubular neighbourhood of $\delta$ in
$D$ and $D_{0}^{\pm}=D_{0}\cap D_{\delta}^{\pm}$. Let $G_{\delta}$ be a
function such that is smooth on $C_{QR}-\delta$ and

$G_{\delta}\equiv\begin{cases}1&\text{ on }D_{0}^{-}\cup{\delta}\\\ 0&\text{
outside }D_{\delta}^{-}\end{cases}$

Define

$\eta_{\delta}=\begin{cases}dG_{\delta}&\text{ on $D_{\delta}-\delta$}\\\
0&\text{ elsewhere}\end{cases}$

so the support $\operatorname{Supp}(\eta_{\delta})\subset D_{\delta}^{-}$. One
can then see that if $\psi$ is a closed $1$-form on $C_{QR}$

$\int_{C_{QR}}\eta_{\delta}\wedge\psi=\int_{D_{\delta}^{-}}dG_{\delta}\wedge\psi=\int_{D_{\delta}^{-}}dG_{\delta}\psi=\int_{\partial(D_{\delta}^{-})}G_{\delta}\psi=\int_{[\delta]}\psi$

since $G_{\delta}\equiv 1$ on $\delta$ and with this choice of orientation
$\partial(D_{\delta}^{-})=\delta$.

In our case, in general
$\alpha=\alpha_{i}={\alpha^{\prime}}^{N}\beta_{Q}^{-m_{i}}$, is not a simple
loop. However, $\alpha^{\prime}$ and $\beta_{Q}$ are. Let
$D_{\alpha}^{-}=D_{\alpha^{\prime}}^{-}\cup D_{\beta_{Q}}^{-}$. Define

$\eta_{\alpha}=N\eta_{\alpha^{\prime}}-m_{i}\eta_{\beta_{Q}}.$

$\eta_{\alpha}$ is supported in $D_{\alpha}^{-}$ and is a Poincaré dual of
$[\alpha]$ as, for a $1$-form $\psi$,

$\int_{C_{QR}}\eta_{\alpha}\wedge\psi=N\int_{C_{QR}}\eta_{\alpha^{\prime}}\wedge\psi-
m_{i}\int_{C_{QR}}\eta_{\beta_{Q}}\wedge\psi=\int_{N[\alpha^{\prime}]-m_{i}[\beta_{Q}]}\psi=\int_{[\alpha]}\psi$

Let $\tilde{\Theta}=q^{*}(\Theta)=\log(q^{*}(f))q^{*}(\psi)+q^{*}(W(\psi))$.
From the discussion above

$\int_{\alpha}\frac{df}{f}\psi+W(\psi)=\int_{\tilde{\alpha}}\tilde{\Theta}.$

where $\tilde{\alpha}$ is a lifting of $\alpha$ to a loop in $\tilde{C}$ such
that it is based at $\tilde{P}$ and $\log(q^{*}(f))$ is chosen such that
$\log(q^{*}(f)(\tilde{P})=0.$ We would like to compute this integral.

Let ${n_{\epsilon}(\gamma)}$ be as above and
$C\setminus{n_{\epsilon}(\gamma)}$ be as before. Choosing $\epsilon$ and the
tubular neighbourhoods carefully, we can assume without loss of generality
that $\alpha^{\prime}$ and $\beta_{Q}$ do not pass through the points $Q$ and
$R$ and that the tubular neighbourhoods $D_{\alpha^{\prime}}$ and
$D_{\beta_{Q}}$ do not intersect $(-\epsilon,\epsilon)\times Q$ and
$(-\epsilon,\epsilon)\times R$.

We have $\alpha=\alpha^{\prime N}\beta_{Q}^{-m_{i}}$. Let
$\tilde{\alpha}^{\prime}$ be the lift of $\alpha^{\prime}$ and
$\tilde{\beta}_{Q}$ the lift of $\beta_{Q}$. The restriction
$\alpha^{\prime}|_{C_{QR}\setminus{n_{\epsilon}(\gamma)}}=\bigcup_{j}\alpha^{\prime
j}$ is a union of a finite number of paths $\alpha^{\prime j}$. The covering
map $q$ induces a homeomorphism from each $\alpha^{\prime j}$ to a path
$\tilde{\alpha}^{\prime j}$ such that $\bigcup\tilde{\alpha}^{\prime
j}=\tilde{\alpha}^{\prime}|_{\tilde{C}\setminus
q^{-1}({n_{\epsilon}(\gamma)})}$. Let $D_{\alpha^{\prime j}}$, and similarly
$D_{\alpha^{\prime j}}^{-}$ denote the restriction of the tubular
neighbourhood $D_{\alpha^{\prime}}$ of $\alpha^{\prime}$ to a tubular
neighbourhood of the path $\alpha^{\prime j}$. We have

$D_{\alpha^{\prime}}^{-}\setminus{n_{\epsilon}(\gamma)}=\bigcup_{j}D_{\alpha^{\prime
j}}^{-}$

Hence we have

$\int_{C_{QR}\setminus{n_{\epsilon}(\gamma)}}\eta_{\alpha^{\prime}}\wedge\Theta=\int_{D_{\alpha}^{\prime-}\setminus{n_{\epsilon}(\gamma)}}\eta_{\alpha^{\prime}}\wedge\Theta=\sum_{j}\int_{D_{\alpha^{\prime
j}}^{-}}\eta_{\alpha^{\prime}}\wedge\Theta$

The boundary of the tubular neighbourhood $D_{\alpha^{\prime j}}^{-}$ is

$\partial(D_{\alpha^{\prime j}}^{-})=\alpha^{\prime
j}\cup({\gamma_{1}}\cap\overline{D_{\alpha^{\prime
j}}^{-}})\cup(\gamma_{2}^{-1}\cap\overline{D_{\alpha^{\prime j}}^{-}})$

Applying Stokes’ Theorem we get

$\int_{D_{\alpha^{\prime
j}}^{-}}\eta_{\alpha^{\prime}}\wedge\Theta=\int_{D_{\alpha^{\prime
j}}^{-}}dG_{\alpha^{\prime}}\Theta=\int_{\partial(D_{\alpha^{\prime
j}}^{-})}G_{\alpha^{\prime}}\Theta$ $=\int_{\alpha^{\prime
j}}\Theta+\int_{\gamma_{1}\cap\overline{D_{\alpha^{\prime
j}}^{-}}}G_{\alpha^{\prime}}\Theta-\int_{\gamma_{2}\cap\overline{D_{\alpha^{\prime
j}}^{-}}}G_{\alpha^{\prime}}\Theta$

Summing up over $j$ we have

$\sum_{j}\left(\int_{\gamma_{1}\cap\overline{D_{\alpha^{\prime
j}}^{-}}}G_{\alpha^{\prime}}\Theta-\int_{\gamma_{2}\cap\overline{D_{\alpha^{\prime
j}}^{-}}}G_{\alpha^{\prime}}\Theta\right)=\int_{\gamma_{1}}G_{\alpha^{\prime}}\Theta-\int_{\gamma_{2}}G_{\alpha^{\prime}}\Theta$

as $G_{\alpha^{\prime}}$ is supported in $D_{\alpha^{\prime}}^{-}$. Recall
that the value of $\log(f)$ on $\gamma_{1}$ and $\gamma_{2}$ differ by $-2\pi
i$. The value of $G_{\alpha^{\prime}}W(\psi)$ is the same on both $\gamma_{1}$
and $\gamma_{2}$ hence
$\int_{\gamma_{1}}G_{\alpha^{\prime}}W(\psi)-\int_{\gamma_{2}}G_{\alpha^{\prime}}W(\psi)=0$.
Hence we get

$\int_{\gamma_{1}}G_{\alpha^{\prime}}\Theta-\int_{\gamma_{2}}G_{\alpha^{\prime}}\Theta=-2\pi
i\int_{\gamma}G_{\alpha^{\prime}}\psi$

Lemma 4.7 and Lemma 2.1 (3) shows that the integral simplifies to

$\int_{\gamma_{1}}G_{\alpha^{\prime}}\Theta-\int_{\gamma_{2}}G_{\alpha^{\prime}}\Theta=-2\pi
i\int_{\gamma}G_{\alpha^{\prime}}\psi=-2\pi
i\int_{\gamma}\eta_{\alpha^{\prime}}\psi$

So we get

$\int_{C_{QR}\setminus{n_{\epsilon}(\gamma)}}\eta_{\alpha^{\prime}}\wedge\Theta=\sum_{j}\int_{\alpha^{\prime
j}}\Theta-2\pi i\int_{\gamma}\eta_{\alpha^{\prime}}\psi$

We can make a similar argument for $\beta_{Q}$.

Combining the two we get

$N\sum_{j}\int_{\alpha^{\prime j}}\Theta-
m_{i}\sum_{s}\int_{\beta_{Q}^{s}}\Theta=\int_{C_{QR}\setminus{n_{\epsilon}(\gamma)}}\eta_{\alpha}\wedge\Theta+2\pi
i\int_{\gamma}\eta_{\alpha}\psi$

Finally, since $\eta_{\alpha}$ is compactly supported in $C_{QR}$, we can
replace $C_{QR}\setminus{n_{\epsilon}(\gamma)}$ with
$C\setminus{n_{\epsilon}(\gamma)}$ to get

$N\sum_{j}\int_{\alpha^{\prime j}}\Theta-
m_{i}\sum_{s}\int_{\beta_{Q}^{s}}\Theta=\int_{C\setminus\gamma}\eta_{\alpha}\wedge\Theta+2\pi
i\int_{\gamma}\eta_{\alpha}\psi$

To link this to the integral over $\tilde{\alpha}$ we observe the following.
The loop $\alpha^{\prime N}$ lifts to a path in $\widetilde{\alpha^{\prime
N}}$ in $\tilde{C}$ which is made up of copies of $\tilde{\alpha}^{\prime}$.
Let $\tilde{\alpha}^{\prime}_{k}$ denote the lift of the $k^{th}$ copy of
$\alpha^{\prime}$ so
$\tilde{\alpha}^{\prime}_{k}(1)=\tilde{\alpha}^{\prime}_{k+1}(0)$. We can
choose the homeomorphisms between $\alpha^{\prime j}$ and
$\tilde{\alpha}^{\prime j}$ such that the $k^{th}$ copy of $\alpha^{\prime j}$
is homeomorphic to a path $\tilde{\alpha}^{\prime j}_{k}$ in
$\tilde{\alpha}^{\prime}_{k}$. So we have homeomorphisms

$\bigcup_{k=1}^{N}\bigcup_{j}\alpha^{\prime
j}\simeq\bigcup_{k=1}^{N}\bigcup_{j}\tilde{\alpha}^{\prime
j}_{k}\simeq\widetilde{\alpha^{\prime N}}-\partial(\tilde{C}\setminus
q^{-1}({n_{\epsilon}(\gamma)}))$

A similar situation holds for $\beta_{Q}$. Via these homeomorphisms

$N\sum_{j}\int_{\alpha^{\prime j}}\Theta-
m_{i}\sum_{s}\int_{\beta_{Q}^{s}}\Theta=\sum_{k=1}^{N}\sum_{j}\int_{\tilde{\alpha}^{\prime
j}_{k}}\tilde{\Theta}-\sum_{r=1}^{m_{i}}\sum_{s}\int_{\tilde{\beta}_{Q,r}^{s}}\tilde{\Theta}$

which is

$\int_{\widetilde{\alpha^{\prime
N}\beta_{Q}^{-m_{i}}}-\partial(\tilde{C}\setminus
q^{-1}({n_{\epsilon}(\gamma)}))}\tilde{\Theta}=\int_{\tilde{\alpha}-\partial(\tilde{C}\setminus
q^{-1}({n_{\epsilon}(\gamma)}))}\tilde{\Theta}$

Finally, as $\epsilon\rightarrow 0$ the set
$\tilde{\alpha}\cap\partial(\tilde{C}\setminus
q^{-1}({n_{\epsilon}(\gamma)}))$ becomes a set of measure $0$ so we have

$\lim_{\epsilon\rightarrow 0}\int_{\tilde{\alpha}-\partial(\tilde{C}\setminus
q^{-1}({n_{\epsilon}(\gamma)}))}\tilde{\Theta}=\int_{\tilde{\alpha}}\tilde{\Theta}$

We have made a choice of a lifting $\tilde{\alpha}$ of $\alpha$. A different
choice of basepoint $\tilde{P}^{\prime}$ would change the value of
$\log(q^{*}(f))$ by $2\pi iM$ for some $M\in{\mathds{Z}}$. This would change
the integral by $2\pi iM\int_{\alpha}\psi$. Hence this equality holds only up
to $2\pi i\int_{\alpha}\psi{\mathds{Z}}$.

∎

We have the following useful corollary to the above proposition, which says
that in fact, we can replace $\eta_{\alpha}$ by any form on $C$ which is
cohomologous to $i_{*}([\eta_{\alpha}])$.

######  Corollary 4.8.

Let $f$, $\psi$, and $W(\psi)$ be as above and $\phi_{\alpha}$ a closed $1$
form on $C$ which is cohomologous in $H^{1}(C)$ to $i_{*}([\eta_{\alpha}])$
for some $\alpha$ in $V$. Then

$\int_{\alpha}\frac{df}{f}\psi+W(\psi)=\int_{C\setminus\gamma}\phi_{\alpha}\wedge\Theta+2\pi
i\int_{\gamma}\phi_{\alpha}\psi\,\left(\operatorname{mod}\,2\pi
i\int_{\alpha}\psi{\mathds{Z}}\right)$

###### Proof.

Let $\phi_{\alpha}$ denote such a form. Both $\phi_{\alpha}$ and
$\eta_{\alpha}$ are Poincaré duals of the same homology class, so
$\phi_{\alpha}-\eta_{\alpha}=dg$. Let ${n_{\epsilon}(\gamma)}$ be as before.
One has

$\int_{C\setminus{n_{\epsilon}(\gamma)}}\phi_{\alpha}\wedge\Theta=\int_{C\setminus{n_{\epsilon}(\gamma)}}dg\wedge\Theta+\int_{C\setminus{n_{\epsilon}(\gamma)}}\eta_{\alpha}\wedge\Theta$

So it suffices to compute the two terms separately.

Any closed form $\phi$ on $C$ is compactly supported on the manifold with
boundary $C\setminus{n_{\epsilon}(\gamma)}$ as
$C\setminus{n_{\epsilon}(\gamma)}$ is a closed subset of $C$ and $C$ is
compact. Further, by choice of $V$ it will be cohomologous to
$[i_{*}(\eta_{\alpha})]$ for some $\alpha\in V$. In particular we know that
$[c(i)\frac{dx_{\sigma(i)}}{N}]$ is cohomologous to a Poincaré dual of
$[\alpha_{i}]$.

Since both $\phi_{\alpha}$ and $\eta_{\alpha}$ are compactly supported on
$C\setminus{n_{\epsilon}(\gamma)}$, so is $dg$ and hence $g$ and
$dg\wedge\Theta$ are compactly supported as well. From Stokes’ Theorem we get

$\int_{C\setminus{n_{\epsilon}(\gamma)}}dg\wedge\Theta=\int_{C\setminus{n_{\epsilon}(\gamma)}}d(g\Theta)=\int_{\partial(C\setminus{n_{\epsilon}(\gamma)})}g\Theta$

If we choose a different function $h$ such that $dh=dg$, then $h-g=c$ for some
constant $c$. Hence, from Lemma 4.7 we see

$\int_{\partial(C\setminus{n_{\epsilon}(\gamma)})}g\Theta-\int_{\partial(C\setminus{n_{\epsilon}(\gamma)})}h\Theta=c\int_{\partial(C\setminus{n_{\epsilon}(\gamma)})}\Theta=\int_{C\setminus{n_{\epsilon}(\gamma)}}d\Theta=0$

so it does not depend on the choice of primitive.

An argument similar to Lemma 4.7 with $g\Theta$ in the place of $\Theta$ shows
that

$\int_{\partial(C\setminus{n_{\epsilon}(\gamma)})}g\Theta=-2\pi
i\int_{\gamma}g\psi$

Lemma 4.7 along with Lemma 2.1 (3) shows further that

$\int_{C\setminus{n_{\epsilon}(\gamma)}}dg\wedge\Theta=\int_{\partial(C\setminus{n_{\epsilon}(\gamma)})}g\Theta=-2\pi
i\int_{\gamma}g\psi=-2\pi i\int_{\gamma}dg\psi.$

Taking the limit as $\epsilon\rightarrow 0$ one gets

$\int_{C\setminus\gamma}dg\wedge\Theta=-2\pi i\int_{\gamma}dg\psi.$

From Proposition 4.6 we have

$\int_{\alpha}\frac{df}{f}\psi+W(\psi)=\int_{C\setminus\gamma}\eta_{\alpha}\wedge\Theta+2\pi
i\int_{\gamma}\eta_{\alpha}\psi\,\left(\operatorname{mod}\,2\pi
i\int_{\alpha}\psi{\mathds{Z}}\right)$

Since the integral of $dg\wedge\Theta$ cancels the iterated integral term, we
have

$\int_{C\setminus\gamma}\phi_{\alpha}\wedge\Theta+2\pi
i\int_{\gamma}\phi_{\alpha}\psi=\int_{C\setminus\gamma}\left(\eta_{\alpha}\wedge\Theta+dg\wedge\Theta\right)+2\pi
i\int_{\gamma}\left(\eta_{\alpha}\psi+dg\psi\right)$
$=\int_{C\setminus\gamma}\eta_{\alpha}\wedge\Theta+2\pi
i\int_{\gamma}\eta_{\alpha}\psi=\int_{\alpha}\frac{df}{f}\psi+W(\psi)\left(\operatorname{mod}\,2\pi
i\int_{\alpha}\psi{\mathds{Z}}\right)$

∎

######  Remark 4.9.

We will apply this to compute the Carlson representative of an extension
class. This is an element of the intermediate Jacobian associated to the
extension, hence the two expressions, while they possibly differ by and
element of $(2\pi i\int_{\alpha}\psi){\mathds{Z}}$, will have the same class
in $\operatorname{Hom}_{{\mathds{Z}}}({\mathds{Z}}(-2),\otimes^{2}H^{1}(C))$
keeping in mind isomorphism of $H_{1}(C)$ with $q_{*}(H_{1}(\tilde{C}))\subset
H_{1}(C_{QR})$. We will use $\equiv$ to keep in mind the fact that all
equalities hold only in the intermediate Jacobian.

We now apply this in the case of interest to us.

######  Corollary 4.10.

Let $\alpha_{j}$ be as above. Then

$\displaystyle{\bf F}_{QR,P}(\Omega)(\alpha_{j}\otimes\alpha_{k})\equiv$
$\displaystyle
N(2g_{C}+1)c(j)c(\sigma(k))\left(\int_{C\setminus\gamma}dx_{\sigma(j)}\wedge\left(\log(f)dx_{\sigma(k)}-\frac{N}{(2g_{C}+1)}W(dx_{\sigma(k)})\right)\right.$
$\displaystyle+\left.2\pi i\int_{\gamma}dx_{\sigma(j)}dx_{\sigma(k)}\right)$

###### Proof.

This is a straightforward application of Corollary 4.8 to the expression in
Lemma 4.5.

∎

${\bf F}_{QR,P}(\Omega)$ determines an element of the intermediate Jacobian of
$(\otimes^{2}V_{{\mathds{C}}})^{*}$

$J(\otimes^{2}V_{{\mathds{C}}}^{*})\simeq\frac{F^{1}(\otimes^{2}V_{{\mathds{C}}}^{*})}{(\otimes^{2}V)^{*}}$

so to determine ${\bf F}_{QR,P}(\Omega)$ it suffices to evaluate it on
elements of $F^{1}(\otimes^{2}V_{{\mathds{C}}})^{*}$. From the above
Corollary, we can compute it on $F^{1}(\otimes^{2}H^{1}(C))$.

Let $\\{dz_{i}\\}$ be a basis of the holomorphic $1$-forms on $C$ such that

$\int_{i_{*}(\alpha_{i})}dz_{j}=N\delta_{ij}\hskip 72.26999pt1\leq i\leq g$

where $\\{\alpha_{i}\\}$ is the basis $V$.

$dz_{j}=dx_{j}+\sum_{i=1}^{g}A_{ji}dx_{i+g}\hskip 72.26999pt\text{ where
}A_{ji}=\frac{1}{N}\int_{i_{*}(\alpha_{i+g})}dz_{j}$

coming from the fact that $c(j)\frac{dx_{\sigma(j)}}{N}$ is dual to
$\alpha_{j}$. This expression continues to hold when we consider them as forms
in $C_{QR}$.

Let $\zeta_{j}=c(\sigma(j))\alpha_{\sigma(j)}+\sum_{1\leq i\leq
g}A_{ji}c(i)\alpha_{i},$ where $j\leq g$. Then $dz_{j}$ is cohomologous in
$H^{1}(C_{QR})$ to a Poincaré dual of $\zeta_{j}$. We have the following
proposition

######  Proposition 4.11.

The map ${\bf F}_{QR,P}(\Omega)$ evaluated on elements of the form
$\zeta_{i}\otimes\alpha_{j}$ is

${\bf F}_{QR,P}(\Omega)(\zeta_{i}\otimes
c(\sigma(j))\alpha_{\sigma(j)})\equiv(2g_{C}+1)N\left(\int_{C\setminus\gamma}\log(f)dz_{i}\wedge
dx_{j}+2\pi i\int_{\gamma}dz_{i}dx_{j}\right)$

In other words

$dz_{i}\wedge W(dx_{j})=0.$

###### Proof.

This is Colombo, [Col02]. Proposition 3.4. The point is that $W(dx_{j})$ and
$dz_{i}$ are both of type $(1,0)$ hence $dz_{i}\wedge W(dx_{j})=0$

∎

In fact, the theorem holds for the other term as well.

######  Proposition 4.12.

For a suitable choice of $\mu_{ijk,Q}$ and $\mu_{ijk,R}$ one has

$\displaystyle W(dz_{i}):=$ $\displaystyle
W(dx_{i})+\sum_{k}A_{ki}W(dx_{i+g})=0$

###### Proof.

[Col02] Lemma $3.1$. ∎

Hence we have

######  Proposition 4.13.

${\bf
F}_{QR,P}(\Omega)(c(\sigma(j))\alpha_{\sigma(j)}\otimes\zeta_{i})\equiv(2g_{C}+1)N\left(\int_{C\setminus\gamma}\log(f)dx_{j}\wedge
dz_{i}+2\pi i\int_{\gamma}dx_{j}dz_{i}\right).$

Comparing this with the regulator term in Theorem 2.2 we get

######  Theorem 4.14.

Let $Z_{QR}$ be the motivic cohomology cycle constructed above and
$\epsilon_{QR,P}^{4}$ the extension in
$\operatorname{Ext}_{MHS}({\mathds{Z}}(-2),\wedge^{2}H^{1}(C))$. We use
$\epsilon^{4}_{QR,P}$ to denote its Carlson representative as well. Then one
has

$\epsilon^{4}_{QR,P}(\omega)\equiv(2g_{C}+1)N\operatorname{reg}_{{\mathds{Z}}}(Z_{QR})(\omega)$

where $\omega\in F^{1}\wedge^{2}H^{1}(C)$.

###### Proof.

It suffices to check this on $dz_{i}\wedge dx_{j}=dz_{i}\otimes
dx_{j}-dx_{j}\otimes dz_{i}$. The result then follows by comparing the formula
for the Carlson representative ${\bf F}_{QR,P}$ in Lemma 4.5 with the
expression for the regulator in Theorem 2.2 using Proposition 4.11

From Theorem 4.13 and Lemma 4.5 we have

$\displaystyle{\bf
F}_{QR,P}(\Omega)(c(\sigma(j))\alpha_{\sigma(j)}\otimes\zeta_{i})\equiv$
$\displaystyle(2g_{C}+1)N\left(\int_{C\setminus\gamma}\log(f)dx_{j}\wedge
dz_{i}+2\pi i\int_{\gamma}dx_{j}dz_{j}\right)$

On the other hand, from Propostion 4.13 one has

$\displaystyle{\bf F}_{QR,P}(\Omega)(\zeta_{i}\otimes
c(\sigma(j))\alpha_{\sigma(j)})$
$\displaystyle=(2g_{C}+1)N\left(\int_{C\setminus\gamma}\log(f)dz_{i}\wedge
dx_{j}+2\pi i\int_{\gamma}dz_{i}dx_{j}\right)$
$\displaystyle=(2g_{C}+1)N\left(-\int_{C\setminus\gamma}\log(f)dx_{j}\wedge
dz_{i}+2\pi i\int_{\gamma}dz_{i}dx_{j}\right)$

Therefore we get

$\displaystyle{\bf
F}_{QR,P}(\Omega)(c(\sigma(j))\alpha_{\sigma(j)}\wedge\zeta_{i})\equiv$
$\displaystyle{\bf
F}_{QR,P}(\Omega)(c(\sigma(j))\alpha_{\sigma(j)}\otimes\zeta_{i})-{\bf
F}_{QR,P}(\Omega)(\zeta_{i}\otimes c(\sigma(j))\alpha_{\alpha_{\sigma(j)}})$
$\displaystyle\equiv$
$\displaystyle(2g_{C}+1)N\left(2\int_{C\setminus\gamma}\log(f)dx_{j}\wedge
dz_{i}+2\pi i\int_{\gamma}(dx_{j}dz_{i}-dz_{i}dx_{j})\right)$

On the other hand, from Theorem 2.2

$(2g_{C}+1)N\operatorname{reg}_{{\mathds{Z}}}(Z_{QR})(dx_{j}\wedge
dz_{i})=(2g_{C}+1)N\left(2\int_{C\setminus\gamma}\log(f)dx_{j}\wedge
dz_{i}+2\pi i\int_{\gamma}(dx_{j}dz_{i}-dz_{i}dx_{j})\right)$

∎

Recall that we have assumed in both cases that $f_{QR}(P)=1$. If we do not
make that assumption, then one has a term corresponding to a decomposable
element that one has to account for. However, if we work modulo the
decomposable cycles we can ignore that term.

As a result of this theorem, we get the following expression of the regulator
as an iterated integral over a loop - which is more amenable to computation.

######  Corollary 4.15.

Let $Z_{QR,P}$ be the element of
$H^{2g-1}_{{\mathcal{M}}}({J}(C),{\mathds{Z}}(g))$ and let $\eta$ and $\omega$
be two closed $1$ forms on $C$ with $\omega$ holomorphic. Let $\alpha$ be loop
in $C_{QR}$ based at $P$ such that $[\alpha]\in V$ and a Poincaré dual of
$[\alpha]$ is homologous to $\eta$ in $H^{1}(C_{QR})$. Then

$\operatorname{reg}_{{\mathds{Z}}}(Z_{QR,P})(\eta\wedge\omega)\equiv
2(2g_{C}+1)N\int_{\alpha}\frac{df}{f}\omega.$

###### Proof.

It suffices to check this for $\eta=dx_{j}$ and $\omega=dz_{i}$. From Theorem
4.14, one has

${\bf
F}_{QR,P}(\Omega)(c(\sigma(j))\alpha_{\sigma(j)}\wedge\zeta_{i})\equiv(2g_{C}+1)N\left(2\int_{C\setminus\gamma}\log(f)dx_{j}\wedge
dz_{i}+2\pi i\int_{\gamma}(dx_{j}dz_{i}-dz_{i}dx_{j})\right).$

From Lemma 2.1 we have

$\int_{\gamma}dx_{j}dz_{i}+\int_{\gamma}dz_{i}dx_{j}=\int_{\gamma}dx_{j}\int_{\gamma}dz_{i}$

From Lemma 4.7 one has $\int_{\gamma}dz_{i}=0$. Hence

$\int_{\gamma}dx_{j}dz_{i}=-\int_{\gamma}dz_{i}dx_{j}$

Therefore, we can simplify the regulator expression to get

$\operatorname{reg}_{{\mathds{Z}}}(Z_{QR,P})(dx_{j}\wedge
dz_{i})=2(2g_{C}+1)N\left(\int_{C\setminus\gamma}\log(f)dx_{j}\wedge
dz_{i}+2\pi i\int_{\gamma}dx_{j}dz_{i}\right)$
$=2(2g_{C}+1)N\left(\int_{C\setminus\gamma}dx_{j}\wedge\log(f)dz_{i}+2\pi
i\int_{\gamma}dx_{j}dx_{i}\right)$

Using Corollary 4.8 this becomes

$\operatorname{reg}_{{\mathds{Z}}}(Z_{QR,P})(dx_{j}\wedge
dz_{i})=2(2g_{C}+1)c(j)N\int_{\alpha}\frac{df}{f}dz_{i}.$

∎

######  Remark 4.16.

Darmol-Rotger-Sols [DRS12] and Otsubo [Ots12] have a similar formula for the
regulator of the modified diagonal cycle. Their expression is an iterated
integral of two holomorphic forms over the dual of a third forn - but none of
the forms are logarithmic. Similarly, the regulator of an element of $K_{2}$
can be expressed as the iterated integral of two logarithmic forms over the
dual of a third holomorphic form.

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Ramesh Sreekantan | Subham Sarkar
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Indian Statistical Institute | Tata Institute of Fundamental Research
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