# Motive of Theta divisor I

Mohammad Reza Rahmati  Centro de Investigacion en Matematicas, CIMAT
<EMAIL_ADDRESS>
###### Abstract.

It is a well known fact that the Theta divisor on the Jacobian of a non-
singular curve is a determinantal variety, i.e. is defined by the zero set of
a determinant. A less known is for the generalized Jacobian of a singular
curve. We describe that the method of proof of the first statement can be
generalized to the case of singular curves. We continue to study the mixed
Hodge structure of the generalized Theta divisor using standard techniques in
Hodge theory.

###### Key words and phrases:

Mixed Hodge structure, Generalized Jacobian, Riemann-Roch Theorem, Motives,
Periods, Algebraic group

###### 1991 Mathematics Subject Classification:

14-xx

## Introduction

Long time, one probably looks for a valuable question in the theory of mixed
Hodge structures which involves a rich algebraic geometry. Despite of high
lighted questions such as Hodge conjecture or Standard conjectures which look
so hard to be solved, perhaps the question of studying the mixed Hodge
structure of Theta divisor on the Jacobian of curves is one of the very
complicated questions. I got acquainted with this question in a lecture by S.
Bloch in Madrid at ICMAT. His lectures on extensions of Hodge structures began
almost with this concept. The idea that the method can be generalized to the
singular curves reached to me at the same time. After searching a little on
the question I found, because of very singularities of Theta, the MHS or what
we call here the motive of Theta seems to be so complicated object to
describe.

In this note I have tried to interface the idea with my knowledge on algebraic
curves which I learnt from the book ’Algebraic groups and class fields’, by J.
P. Serre. We target to describe the MHS on the cohomologies of generalized
$\Theta$-divisors on (compactification of) the generalized Jacobians of
Singular curves. Our approach uses modulus on normal projective curves and the
theory of Riemann-Roch for singular curves. The aforementioned reference
provides an almost complete picture of the theory of generalized Jacobians
both over $\mathbb{C}$ and positive characteristic. We work only in char $0$
case, i.e. over the complex numbers. Theory of modulus on curves provides a
simple way to describe the generalized Jacobians and their relation with the
Jacobian in the smooth case as extensions of algebraic groups.

##  1\. singular Curves

The Riemann-Roch theorem in complex dimension 1, can be easily generalized to
singular projective curves, using theory of modulus. The proof is a
modification of that in smooth case, by careful study of linear series on
curves. In order to fix the idea, let $X$ be a complete, irreducible, non-
singular curve and let $\mathfrak{m}=\sum n_{p}.P$ be a modulus on $X$. let
$S$ be the support of $\mathfrak{m}$. Set $X^{\prime}=(X-S)\cup Q$, where $Q$
is one point, and put $\mathcal{O}^{\prime}_{Q}:=\cap_{P\to
Q}\mathcal{O}_{P}$. Define the sheaf $\mathcal{O}^{\prime}$ by setting it to
be equal to $\mathcal{O}$ for the points not in $S$. It is an easy exercise in
algebraic geometry to show that the sheaf $\mathcal{O}^{\prime}$ endows
$X^{\prime}$ with a structure of algebraic curve such that $X$ is its
normalization and its singular points are $Q$. One shows that any singular
curve is obtained in this way.

The basic example is $\mathfrak{m}=2P$ where one gets an ordinary cusp, and
another when $\mathfrak{m}=P_{1}+P_{2}$ where one obtains ordinary double
point, etc….

The Riemann-Roch theorem for singular curves is written as

$l^{\prime}(D)-i^{\prime}(D)=\deg(D)+1-\pi$

$l^{\prime}(D):=\dim H^{0}(X^{\prime},L^{\prime}(D)),\ i^{\prime}(D):=\dim
H^{1}(X^{\prime},L^{\prime}(D))$

where $\pi$ is the arithmetic genus. The sheaf $L^{\prime}(D)$ is the same as
$L(D)$ on $X-S$ and equals $\mathcal{O}_{Q}^{\prime}$ on $S$. We have the
short exact sequence

$0\to\mathcal{O}^{\prime}\to\mathcal{O}\to\mathcal{O}/\mathcal{O}^{\prime}\to
0$

The sheaf $\mathcal{O}/\mathcal{O}^{\prime}$ is concentrated on $S$ and its
annihilator ideal is called the conductor of $\mathcal{O}$ into
$\mathcal{O}^{\prime}$. Thus,

$\chi(X^{\prime},\mathcal{O}/\mathcal{O}^{\prime})=\dim
H^{0}(X^{\prime},\mathcal{O}/\mathcal{O}^{\prime})=\sum_{Q\in
S}\dim\mathcal{O}_{Q}/\mathcal{O}^{\prime}_{Q}=:\delta$

One can prove that $\pi=g+\delta$ is equal to the arithmetic genus, and in
case the curve is defined by an equation of degree $d$ in a projective space
$\pi=1+d(d-3)/2,\ g=1-\delta+d(d-3)/2$, the classical Plucker formulas,[S].

##  2\. Generalized Jacobians

Assume $X$ is a projective irreducible algebraic curve. If $f:X\to\mathcal{J}$
be a rational map the the set of points where $f$ is not regular is a finite
set $S$. When $\mathcal{J}$ is abelian, then $S=\emptyset$, and $f(D)=0$ if
and only if $D$ is the divisor of a rational function. For $f$ regular away
from $S$, there can be find a modulus $\mathfrak{m}$ with support $S$, such
that $f(D)=0$ iff $D=(h)$ with $h\equiv 1\mod\mathfrak{m}$ (i.e.
$v_{P}(1-h)>n_{P}$, where $\mathfrak{m}=\sum n_{P}P$), [S], Chap. V.

######  Theorem 2.1.

([S] Chap. V, page 88) For every modulus $\mathfrak{m}$, there exists a
commutative algebraic group $J_{\mathfrak{m}}$ and a rational map
$\phi_{\mathfrak{m}}:X\to J_{\mathfrak{m}}$ which is universal for all the
rational $f:X\to G$ regular away from $S$. This means $f$ factors through
$\phi_{\mathfrak{m}}$ as $f=\theta\circ\phi_{\mathfrak{m}}$ for a unique
rational homomorphism $\theta:J_{\mathfrak{m}}\to G$.

The map $\phi_{\mathfrak{m}}$ defines by extensions to divisor classes a
bijection from $C_{\mathfrak{m}}^{0}$: the group of divisors of degree zero
which are prime to $S$ modulo those of the form $(f)$ where $f\equiv
1\mod\mathfrak{m}$ to $J_{\mathfrak{m}}$. One has the following exact sequence

$0\to R_{\mathfrak{m}}/\mathbb{G}_{m}\to C_{\mathfrak{m}}^{0}\to J\to 0$

where $R_{\mathfrak{m}}=\prod_{P\in S}U_{P}/U_{P}^{(n_{P})}$, and
$U_{P}^{(n)}:=\\{f;v_{P}(1-f)>n\\}$. Each of the groups $U/U^{(n)}$ in the
product are isomorphic to a product of $\mathbb{G}_{m}$ with a unipotent group
of upper triangular matrices, [S] page 94.

The algebraic group $J_{\mathfrak{m}}$ is called the generalized Jacobian of
the curve $X$, and the case $\mathfrak{m}=0$ corresponds to the usual Jacobian
$J$. By standard arguments one shows that for
$\mathfrak{m}\geq\mathfrak{m}^{\prime}$ there is a unique surjective separable
homomorphism $F:J_{\mathfrak{m}}\to J_{\mathfrak{m}}^{\prime}$, with connected
kernel such that $\phi_{\mathfrak{m}}^{\prime}=F\circ\phi_{\mathfrak{m}}$.

######  Theorem 2.2.

([S] page 101)

$J_{\mathfrak{m}}\cong\Omega(-\mathfrak{m})^{\vee}/H_{1}(X-S,\mathbb{Z})$

The Abel theorem on the Jacobian of non-singular curves asserts that
$Pic^{0}(X)\cong J(X)$, where $Pic^{0}(X)$ stands for group of zero divisors
modulo rational equivalence. This isomorphism is given by integration along
paths whose boundray is the given $0$-cycle. Because $J(X)$ is an abelian
group, the universal map $\phi:X\to J(X)$, provides a map $sym^{n}(X)\to J(X)$
whose fibers are projective spaces. For $n=g$ the genus of the curve $X$ this
map is surjective and for $n=g-1$ its image is the $\Theta$-divisor on $J(X)$.

We consider $X\times J$ with two projections $pr_{1},pr_{2}$. The universal
Poincare line bundle $\wp$ on $X\times J$ is a line bundle s.t.
$\wp\mid_{X\times j}\cong\mathcal{O}_{X}(D(j))$ where $D(j)$ is a divisor
class of degree $0$ via Abel theorem. Fix a line bundle $L_{0}$ a line bundle
of degree $g-1$ on $X$. Set $\mathcal{L}:=pr_{1}^{*}L_{0}\otimes\wp$. Let
$D=x_{1}+...+x_{g}$ be a divisor of degree $g$ on $X$. Then $D\times J$ is a
divisor on $X\times J$ and $\mathcal{O}(D\times
J)=pr_{1}^{*}\mathcal{O}_{X}(D)$. Moreover we have
$\mathcal{L}(D)=\mathcal{L}\otimes pr_{1}^{*}(\mathcal{O}_{X}(D))$. The
following theorem is well-known.

######  Theorem 2.3.

The $\Theta$-divisor is a determinantal subvariety of $J$, i.e is given by the
zero set of a determinant function.

###### Proof.

This can be obtained by applying the Leray cohomological fiber functor
$R^{\bullet}pr_{2,*}$ to the short exact sequence of sheaves on $X\times J$,

(1) $0\to\mathcal{L}\to\mathcal{L}(D)\to\mathcal{L}(D)/\mathcal{L}\to 0$

In the derived functor long exact sequence one obtains

(2) $0\to pr_{2*}\mathcal{L}\to pr_{2*}\mathcal{L}(D)\to
pr_{2*}(\mathcal{L}(D)/\mathcal{L})\to R^{1}pr_{2*}\mathcal{L}\to
R^{1}pr_{2*}\mathcal{L}(D)$

We have $pr_{2*}\mathcal{L}=R^{1}pr_{2*}\mathcal{L}(D)=0$ by Riemann-Roch.
Thus

(3) $0\to pr_{2*}\mathcal{L}(D)\to pr_{2*}(\mathcal{L}(D)/\mathcal{L})\to
R^{1}pr_{2*}\mathcal{L}\to 0$

where the last sheaf has support the $\Theta$-divisor. Therefore the
determinant of the first homomorphism defines $\Theta$. ∎

In case of singular curves the canonical map $\phi_{\mathfrak{m}}:X\to
J_{\mathfrak{m}}$ by extension over divisors defines a homomorphism from the
group of divisors prime to $S$ onto the group $J_{\mathfrak{m}}$. Its kernel
is formed by divisors $\mathfrak{m}$-equivalent to an integral multiple of a
fixed point $P_{0}$ in $X^{\prime}$.

Similar to the non-singular case the map $sym^{\pi}(X)\to J_{\mathfrak{m}}$ is
surjective and the image of $sym^{\pi-1}(X)\to J_{\mathfrak{m}}$ is the locus
of generalized $\Theta$-divisor. Thus we may apply the argument in the theorem
to $X\times J_{\mathfrak{m}}$ and work with divisors prime to $S$ to obtain

######  Theorem 2.4.

The generalized $\Theta_{\mathfrak{m}}$ is deteminantal on $J_{\mathfrak{m}}$.

##  3\. Motive of Theta divisor

The motive associated to the $\Theta_{\mathfrak{m}}$ is mainly described by
the mixed Hodge structure of
$H^{\pi}(J_{\mathfrak{m}}\setminus\Theta_{\mathfrak{m}})$. From now on set
$J:=J_{\mathfrak{m}},\ \Theta:=\Theta_{\mathfrak{m}}$. According to what was
said, The $\Theta$-divisor is the singular fiber in a degenerate family
defined by a determinant function. The mixed Hodge structure in such a family
is determined by J. Steenbrink which uses the method of resolution of
singularities.

(4) $\begin{CD}J_{\infty}@>{}>{}>U@>{}>{}>J@<{}<{}<D\\\
@V{f_{\infty}}V{}V@V{}V{f}V@V{}V{f}V@V{}V{}V\\\
H@>{e}>{}>\Delta^{*}@>{}>{}>\Delta @<{}<{}<0\end{CD}$

as in 2.9, where $D=\bigcup_{i}^{m}D_{i}$ is a normal crossing divisor.

######  Lemma 3.1.

[JS3] The spectral sequence of the double complex

$A_{D}^{pq}:=(a_{q+1})_{*}\Omega_{D^{(q+1)}}^{p},\ \ D^{(q+1)}=\amalg
D_{i_{1}}\cap...\cap D_{i_{q}}$

with horizontal arrows to be the de Rham differentials, and vertical arrows
$\sum_{j}(-1)^{q+j}\delta_{j}$, induced by the inclusions $\amalg
D_{i_{1}}\cap...\cap D_{i_{q}}\hookrightarrow\amalg
D_{i_{1}}\cap..\hat{D_{i_{j}}}..\cap D_{i_{q}}$ obtained by possible omitting
the indices, on $A_{D}^{pq}$, degenerates at $E_{2}$, with

$E_{1}^{pq}=H^{q}(D^{(p+1)},\mathbb{C})\Rightarrow H^{p+q}(D,\mathbb{C})$

and computes the cohomologies of $D$. Moreover, the two filtrations

$F^{p}A_{D}^{..}=\bigoplus_{r\geq p}A_{D}^{r.},\qquad
W_{q}A_{D}^{..}=\bigoplus_{s\geq-q}A_{D}^{.s}$

induce the Hodge and the weight filtrations on $H^{k}(D,\mathbb{C})$ for each
$k$, to define a mixed Hodge structure.

######  Proposition 3.2.

[JS2] The spectral sequence of $B^{pq}:=A^{pq}/W_{q}$ degenerates at $E_{2}$
term with

$E_{1}^{-r,q+r}=\bigoplus_{k\geq
0,r}H^{q-r-2k}(D^{(2k+r+1)},\mathbb{C})(-r-k)\Rightarrow
H^{q}(X_{\infty},\mathbb{C})$

and equips $H^{q}(X_{\infty},\mathbb{C})$ with a mixed Hodge structure.

Thus we assume after suitable blow ups the family is given by
$f:J\to\mathbb{C}$, where afar from $0$ the fibration looks like fibers of
determinant, and $J_{0}=f^{-1}(0)$ is a normal crossing divisor with quasi-
unipotent monodromies. One can equip $H^{m}(J_{t}),\ H^{m}(J_{0}),\
H_{m}(J_{0})$ with mixed Hodge structures. The formation of the weight and
Hodge filtrations is standard in this case, and one may filter the associated
double complexes naively in vertical and horizontal directions. The MHS fit in
the Clemens-Schmid exact sequence ([SP] page 285, [ITW]) as mixed Hodge
structures,

$...\to
H_{2n+2-m}(J_{0})\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}H^{m}(J_{0})\stackrel{{\scriptstyle
i_{t}^{*}}}{{\rightarrow}}H^{m}(J_{t})\stackrel{{\scriptstyle
N}}{{\rightarrow}}H^{m}(J_{t})\stackrel{{\scriptstyle\beta}}{{\rightarrow}}H_{2n-2m}(J_{0})\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}H^{m-2}(J_{0})\to...$

where $\alpha$ is induced by Poincare duality followed by projection, and
$\beta$ is by inclusion followed by Poincare duality. The monodromy weight
filtration on $H^{m}(J_{0})$ can be described using the hypercover structure
obtained by intersections of NC divisors. Then the weight filtration of
$H^{m}(J_{t})$ can be computed via the induced filtration on
$\ker(N):=K_{t}^{m}$, and satisfies;

$Gr_{k}^{W}H^{m}(J_{t})\cong\begin{cases}Gr_{k}K_{t}^{m}\oplus
Gr_{m-2}K_{t}^{m}\oplus...\oplus Gr_{k-2[k/2]}K_{t}^{m},\qquad k\leq
m\\\\[8.5359pt] Gr_{2m-k}H^{m}(J_{t}),\qquad k>m\end{cases}$

The relations between the weight filtrations can be explained by the
followings, cf. [ITW];

  * •

$i_{t}^{*}$ induces

$Gr_{k}H^{m}(J_{0})\cong Gr_{k}K_{t}^{m}$

  * •

The following sequence is exact

$0\to Gr_{m-2}K_{t}^{m-2}\to
Gr_{m-2n-2}H_{2n+2-m}(J_{0})\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}Gr_{m}H^{m}(J_{0})\to
Gr_{m}K_{t}^{m}\to 0$

Now assume $J\hookrightarrow\bar{J}$ is a projective compactification such
that $J$ is Zariski dense in $\bar{J}$. Then we have the long exact sequence

$...\to H^{m-1}(\bar{J}\setminus J_{0})\to H^{m}(\bar{J},\bar{J}\setminus
J_{0})\to H^{m}(\bar{J})\to H^{m}(\bar{J}\setminus J_{0})\to...$

where the morphisms are of MHS, and the isomorphism

$H^{2n+2-m}(\bar{J},\bar{J}\setminus J_{0})\cong H_{c}^{2n+2-m}(J)\cong
H^{m}(J)^{\vee}$

as MHS. In our case if we blow up $\bar{J_{\mathfrak{m}}}$ along
$\Theta_{\mathfrak{m}}$ then the above method computes the mixed Hodge
structure of $H^{\pi}(\bar{J}\setminus
D)=H^{\pi}(\bar{J}_{\mathfrak{m}}\setminus\Theta_{\mathfrak{m}})$.

######  Remark 3.3.

[ITW] For non-singular projective curves of genus $g$ the
$H^{g-1}(\Theta,\mathbb{Z})$ contains a natural sublattice

$P=\ker(H^{g-1}(\Theta,\mathbb{Z})\to H^{g+1}(J,\mathbb{Z}))$

where the map is the pushforward by inclusion. Then, as Hodge structures

$H^{g-1}(\Theta)=P\oplus\Theta.H^{g-3}(J)$

In this text we apply some of the consequences of the graph tree theorem to
the theory of the motives of graphs and try to relate it to the mixed Hodge
structures of configuration polynomials.

##  4\. Motives of graphs

Assume $\Gamma$ is a connected graph, with
$h_{1}(\Gamma):=\text{rank}H^{1}(\Gamma)$, and $\sharp E(\Gamma)=m$. Then we
have the homology sequence

$0\to
H_{1}(\Gamma,\mathbb{Z})\to\mathbb{Z}[E(\Gamma)]\to\mathbb{Z}[V(\Gamma)]\to
H_{0}(\Gamma,\mathbb{Z})\to 0$

$\mathbb{Z}[E(\Gamma)]$ is canonically self dual, by simply taking the dual
basis to the basis given by the edges $e\in E$. For each edge $e\in E$, the
functionals $e^{\vee}$, restrict to $H$ and thus,
$(e^{\vee})^{2}:H\to\mathbb{C}$ defines a rank one quadric on $H$.

######  Definition 4.1.

The graph polynomial is defined by;

$\Psi_{\Gamma}:=\Psi_{H}=\det(\sum_{e}x_{e}.(e^{\vee})^{2})$

The definition depends only on the configuration $H\subset\mathbb{Z}[E]$ and
not on the graph.

######  Proposition 4.2.

[B1] $\Psi_{\Gamma}:=\sum_{T}\prod_{e\notin T}x_{e}$, where $T$ runs over all
the spanning trees $T\subset\Gamma$.

######  Proposition 4.3.

[B1] $\Psi_{\Gamma}=\prod_{e}x_{e}L(\Gamma)(x_{e})$, where $L$ is the
Laplacian calculated in edge weight of $e$ to be $x_{e}$

The motive of the graph $\Gamma$ is

$M(\Gamma):=H^{m-1}(\mathbb{P}^{m-1}-X_{\Gamma},\mathbb{Q}(m-1))$

where $X_{\Gamma}:\Psi=0$ is the graph hypersurface.

One of the classical results in graph theory is the matrix-tree theorem which
asserts that the determinant of the cofactor of the combinatorial Laplacian is
equal to the number of spanning trees in the graph. The usual notion of
Laplacian for graphs considers edge weights. It is not obvious how to define
the Laplacian with vertex weight.

Let $G$ be a graph with vertex set $V=\\{v_{1},...,v_{n}\\}$ and edge set $E$.
The Laplacian of $G$ is defined by

$L(u,v)=\begin{cases}d_{v},\qquad u=v\\\ -1\qquad\ \text{if}\ u\sim v\
\text{adjacent}\\\ 0\end{cases}$

where $d_{v}$ denotes the degree of $v$. In order to generalize the definition
of the Laplacian to graphs with vertex weight $\alpha_{v}$. Define the matrix

$\mathbb{L}(u,v)=\begin{cases}\sum_{z\sim v}\alpha_{v},\qquad u=v\\\
-\alpha_{v}\qquad\ \text{if}\ u\sim v\ \text{adjacent}\\\ 0\end{cases}$

The matrix $\mathbb{L}$ is not symmetric. To make it symmetric define the
following matrix which is equivalent to $\mathbb{L}$;

$\mathcal{L}(u,v)=\begin{cases}\sum_{z\sim v}\alpha_{v},\qquad u=v\\\
-\sqrt{\alpha_{u}.\alpha_{v}}\qquad\text{if}\ u\sim v\ \text{adjacent}\\\
0\end{cases},\qquad\mathbb{L}=W^{-1/2}\mathcal{L}W^{1/2}$

where $W$ is the $n\times n$-diagonal matrix with $(v,v)$-entry $\alpha_{v}$.
Then $\mathcal{L}$ is symmetric. In fact if $B$ is the incidence matrix with
rows indexed by vertices and columns indexed by edges defined by

$B(v,e)=\begin{cases}\sqrt{\alpha_{v}},\qquad e={u,v},\ u=u_{i},v=v_{j},i<j\\\
-\sqrt{\alpha_{v}}\qquad\text{if}\ e={u,v},\ u=u_{i},v=v_{j},i>j\\\
0\end{cases}\Rightarrow\qquad\mathcal{L}=BB^{*}$

where $B^{*}$ is the transpose of $B$. Suppose Now the graph $G$ also has the
edge weight $w_{e}=w_{u,v}\geq 0$, and let $T$ be the diagonal matrix with
$(e,e)$-entry $w_{e}$. Then a result proved in [CL] is

(5) $\mathcal{L}=BTB^{*}$

This equation can be regarded as a generalization for the Laplacian when
vertex and edge weights.

Let $G$ be our graph with $v$ having weight $\alpha_{v}$ and $T$ a tree in
$G$. For $v\in T$ define the rooted directed tree $T_{v}$, by orienting every
edge of $T$ toward the root $v$. Define the edge set of $T_{v}$ by

$E(T_{v})=\\{\ (x,Y)\|\ d_{T}(v,x)>d_{T}(v,x)\ \\}$

where $d_{T}(v,x)$ is the distance in $T$ between $v$ and $x$. Define

$w(t_{v})=\prod_{(x,y)\in E(T_{v})}\alpha_{v}\qquad\text{weight of}T_{v}$
$k_{v}(G)=\sum_{T}w(T_{v}),\qquad k(G)=\sum_{v}k_{v}(G)$

If $\alpha_{v}=1$ then $k(G)$ is the number of rooted directed spanning trees
in $G$. We use this definition to extend the definition of the motive defined
by a graph when having weights. According to [CL], the determinant of any
cofactor of $\mathcal{L}$ obtained by deleting the $u$-th row and $v$-th
collumn is

$\sqrt{\alpha_{u}.\alpha_{v}}(\sum_{z}\alpha_{z})^{-1}k(G)$

A forest in a graph is a subgraph containing no loops. Let $S\subset V(G)$
with $|S|=s$, and $X\subset E$ with $|X|=n-s$ be subsets. If the subgraph with
vertex set $V(G)$ and edge set $X$ is a spanning forest and each of its
subtree contains exactly one vertex in $S$, Define the rooted directed
spanning forest $X_{S}$, consisting of all edges of $X$ oriented toward $S$.
Set

$w(X_{S})=\prod_{(x,y)\in E(X_{S})}\alpha_{y},\qquad\text{weight of}\ X_{S}$

$k_{S}(G)=\sum_{X_{S}}w(X_{S}),\qquad k_{S}(G)=\sum_{S,\ |S|=s}k_{S}(G),\
1\leq s\leq n$

Then we have the following well known theorem

######  Theorem 4.4.

[CL] The $s$-th coefficient of the characteristic polynomial of $\mathcal{L}$
is the sum of weights of all the rooted directed spanning forests with
$s$-roots.

Along these discussions we can generalize the definition of the motive by
graphs to weighted one. That is to define

$\Psi_{\Gamma}:=\prod_{e}x_{e}\mathcal{L}(\alpha_{v},x_{e})$

where $\mathcal{L}(\alpha_{v},x_{e})$ means the Laplacian is computed with
vertex weight $\alpha_{v}$ and edge weight $x_{e}$.

##  5\. Appendix: Extensions of groups

Let $A,B,C$ be three algebraic groups. A sequence of algebraic homomorphisms
$0\to B\to C\to A\to 0$ is called strictly exact If it is exact in the usual
sense, and if all the homomorphisms are separable, In other words if the
algebraic structure of $B$, and $A$ is induced by that of $C$. This would be
equivalent to

$0\to\mathfrak{t}_{B}\to\mathfrak{t}_{C}\to\mathfrak{t}_{A}\to 0$

is exact. A strict exact sequence would be called an extension of $A$ by $B$.
The set of isomorphism classes of all extensions of $A$ by $B$ is denoted
$Ext(A,B)$, which is endowed with a composition law as follows. For $f:B\to
B^{\prime}$ and $C\in Ext(A,B)$ then $f_{*}C\in Ext(A,B^{\prime})$ may be
defined as the quotient of $C\times B^{\prime}$ by the subgroup formed by
$(-b,f(b))$. The element $f_{*}C$ is the unique extension such that there
exists a homomorphism $F:C\to C^{\prime}$ making a commutative diagram

(6)
$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>B@>{}>{}>C@>{}>{}>A@>{}>{}>0\\\
\Big{\|}@V{}V{}V@V{}V{}V@V{}V{}V\Big{\|}\\\
0@>{}>{}>B^{\prime}@>{}>{}>C^{\prime}<EMAIL_ADDRESS>
Similarly one defines the pull back $g^{*}C$ for $g:A^{\prime}A$ as the
subgroup of $A^{\prime}\times C$ formed by pairs having the same image in $A$.
If $C,C^{\prime}$ are two elements in $Ext(A,B)$, then $C\times C^{\prime}$
can be considered as an element of $Ext(A\times A,B\times B)$, denoting by
$d:A\to A\times A$ and $s:B\times B\to B$ the diagonal of $A$ and composition
law of $B$, then $C+C^{\prime}=d^{*}s_{*}(C\times C^{\prime})$ in $Ext(A,B)$
and , makes it into an additive bi-functor, [S] Chap VII.

######  Remark 5.1.

([S] page 166) The category $\mathcal{C}_{k}$ of commutative algebraic groups
over the field $k$ is an additive category. It is an Abelian category, when
the characteristic of $k$ is $0$ and only in this case. However,
$\mathcal{C}_{k}$ contains neither enough injectives nor enough projectives.

## References

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  * [CL] F. Chung, R. Langlands, A combinatorial Laplacian with vertex weights, J. of Combinatorial theory, (A), 75,(1996), 316-327
  * [ITW] E. Izadi, CS. Tama, J. Wang, The primitive cohomology of the Theta divisor of an abelian 5-fold, Arxiv e-print 1311.6212,
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