MIMS Technical Report No.00011 (200806091)

GRAPH-THEORETIC ALBANESE MAPS REVISITED
MOTOKO KOTANI AND TOSHIKAZU SUNADA

1. Introduction
In [6], [5], we introduced the notion of Albanese maps in the graphtheoretic context (see also [7], [8]). An Albanese map is a harmonic
map of a finite graph as a 1-dimensional singular space into a flat torus
which, together with the flat metric, is characterized by a minimizing
property for certain energy functional, and is related to asymptotic
behaviors of random walks on crystal lattices. On the other hand, thenotion of Abel-Jacobi maps was brought in graph theory by R. Bacher,
P. De La Harpe, and T. Nagnibeda [1] (see also [3]). A graph versionof Abel-Jacobi maps is a certain class of harmonic functions defined onvertices with values in finite abelian groups. The aim of this note is to
give a relationship between these notions.
2. Albanese maps
We first explain Albanese maps in a bit different way from the original one given in [8].
Let X = (V, E) be a finite graph with a set of vertices V and a setof oriented edges E. By o(e) (resp. t(e)) we denote the origin (resp.
terminus) of e ∈ E. The symbol e stands for the inverse edge of e.
Define the bilinear form on C1 (X, Z), the group of 1-chains on X, by(e = e )1(1)e, e  = −1 (e = e )0(otherwise),

where e, e ∈ E, oriented edges in X. This extends to an inner product on C1 (X, R), and is restricted to the homology group H1 (X, R) =Ker ∂, where ∂ : C1 (X) −→ C0 (X) is the boundary map. The Albanesetorus A(X) is defined to be the flat torus H1 (X, R)/H1 (X, Z) with theflat metric induced from this inner product.
The Albanese map Φal : X −→ A(X) is defined as follows. Let P :C1 (X, R) −→ H1 (X0 , R) be the orthogonal projection. Fix a reference
12

MOTOKO KOTANI AND TOSHIKAZU SUNADApoint x0 ∈ V , and let c = (e1 , · · · , en ) be a path with o(c) = x0 , t(c) =x. Then put Φal (x0 ) = 0 andΦal (x) = P (e1 + · · · + en ) = P (e1 ) + · · · + P (en ) (mod H1 (X, Z)).
If c0 = (e01 , · · · , e0m ) be another path joining x0 and x, thene1 + · · · + en − (e01 + · · · + e0m ) ∈ H1 (X, Z),
so thatP (e1 + · · · + en ) = P (e01 ) + · · · + P (e0m ) (mod H1 (X, Z)).
Hence Φal as a map from V into A(X) is well-defined. We extend Φal toedges as a piecewise linear maps. The map Φal : X −→ A(X) obtainedin this way is a harmonic map in the sense thatX£¤∆Φal (x) =Φal (te) − Φal (oe) = 0,e∈Ex

where Ex = {e ∈ E; o(e) = x}. In fact, for any closed path c =(e1 , · · · , en ) in X,Xhe, ci = 0e∈Exsince, if t(ei ) = o(ei+1 ) = x, then hei , ci + hei+1 , ci = 0. Hence
Xe ∈ H1 (X, R)⊥ ,and ∆Φal (x) = P¡Pe∈Ex¢e∈Ex e= 0.

3. Abel-Jacobi maps into finite abelian groups
There are several definitions of Abel-Jacobi maps. We take up adefiniton which resembles the classical one in algebraic geometry.
Define the group of divisors of degree zero by
XXDiv0 (X) = {ax x ∈ C0 (X, Z)|ax = 0}x∈Vx

and the group of principal divisors by¡¢Prin(X) = ∂∂ ∗ C0 (X, Z)
where ∂ ∗ is the adjoint of ∂ with respect to the inner products on
C0 (X, R)(1 (x = y)x·y =0 (x 6= y)GRAPH-THEORETIC ALBANESE MAPS REVISITED

3and the one on C1 (X, R) defined in the previous section. The Picard
group is defined as
Pic(X) = Div0 (X)/Prin(X).
The order |Pic(X)| coincides with κ(X), the number of spanning treesin X. The discrete Abel-Jacobi map Φaj : V −→ Pic(X) is defined byΦaj (x) = [x − x0 ].
4. Discrete Albanese tori and Abel’s theorem
Let us now establish a relationship between Albanese maps and discrete Abel-Jacobi maps.
The homology group H1 (X, Z) with coefficients in Z is an integral
lattice in H1 (X, R) with respect to the inner product (1). Denote by
H1 (X, Z)# the dual lattice of H1 (X, Z) in H1 (X, R). Since the lattice
H1 (X, Z) is integral, we have H1 (X, Z) ⊂ H1 (X, Z)# . The discrete
Albanese torus A(X) is defined to be H1 (X, Z)# /H1 (X, Z) which is
identified with a finite subgroup of A(X).
For any e ∈ E and α ∈ H1 (X, Z), we find hP (e), αi = he, P (α)i =he, αi ∈ Z, and hence P (e) ∈ H1 (X, Z)# . Thus we haveLemma 4.1. Let Φal be the Albanese map of X into A(X). ThenΦ(V ) ⊂ A(X).
We shall call Φal |V : V −→ A(X) the discrete Albanese map.
In order to prove that Φab (V ) generates A(X), take a spanning tree
T of X, and let e1 , . . . , eb (b = rank H1 (X, Z)) be all edges not in T .
Then P (e1 ), . . . , P (eb ) constitute a Z-basis of H1 (X, Z)# since, if wetake circuits c1 , . . . , cb in X such that ci contains ei , then {c1 , . . . , cb }is a Z-basis of H1 (X, Z), and hci , P (ej )i = hP (ci ), ej i = hci , ej i = δij .
Theorem 4.1. (A discrete version of Abel’s theorem) The correspondence x ∈ V 7→ Φal (x) ∈ A(X) induces an isomorphism ϕ of Pic(X)onto A(X) such that ϕ ◦ Φaj = Φal .
Proof. This is actually a consequence of the universality of AbelJacobi maps (cf. [2]). For the completeness, we will give a proof.
Define the homomorphism ϕ : Div0 (X) −→ A(X) by setting ϕ(x −x0 ) = Φal (x) (note that {x − x0 ; x 6= x0 ∈ V } is a Z-basis of Div0 (X)).
On the other hand, an easy computation leads us to³X
´X X¡
¢ax x = −axt(e) − o(e) ,∂∂ ∗x∈Vx∈Ve∈Ex4

MOTOKO KOTANI AND TOSHIKAZU SUNADAand hence³³X´´X X¡¢ϕ ∂∂ ∗ax x = −ax
Φal (t(e)) − Φal (o(e)) = 0.
x∈Vx∈Ve∈Ex

which implies that ϕ induces a homomorphism ϕ : Pic(X) −→ A(X).
From what we have seen above, ϕ is surjective.
To check that ϕ is an isomorphism, it is enough to see that |A(X)| =κ(X). For this, we take a look at the following exact sequence
0 → A(X) → A(X) → H1 (X.R)/H1 (X, Z)# → 0.
We therefore have the following identity for the order of A(X).
|A(X)| = vol(A(X))/vol(H1 (X.R)/H1 (X, Z)# ).
We also havevol(H1 (X.R)/H1 (X, Z)# ) = vol(A(X))−1 ,and hence we obtain|A(X)| = vol(A(X))2 .
It is known ([5]) that vol(A(X))2 coincides with κ(X), and hence|A(X)| = κ(X).
A non-degenerate symmetric bilinear form on A(X) with values in
Q/Z is induced from the inner product on H1 (X, R). Thinking of
this form as an analogue of “polarization”, one may ask whether theTorelli type theorem holds in the discrete realm. More specifically, oneasks whether two regular graphs X1 and X2 with the same degree areisomorphic when there exists a group isomorphism between A(X1 ) andA(X2 ) preserving polarizations1.
References[1] R. Bacher, P. De La Harpe, and T. Nagnibeda, The lattice of integral flows andthe lattice of integral cuts on a finite graph, Bull. Soc. Math. France, 125(1997),167-198.
[2] M. Baker and S. Norine, Riemann-Roch and Abel-Jacobi theory on a finitegraph, Adv. in Math., 215(2007), 766-788.
[3] N. L. Biggs, Algebraic potential theory on graphs, Bull. London Math. Soc.,
29(1997), 641-682.
[4] P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley &Sons, 1978.
[5] M. Kotani and T. Sunada, Jacobian tori associated with a finite graph and its
abelian covering graphs, Advances in Apply. Math. 24(2000), 89-110.
1This problem was suggested by Kenichi Yoshikawa.

If we would remove theconditions on degree and polarizations, there are many examples of X1 , X2 withA(X1 ) ∼= A(X2 ) ([1]).

GRAPH-THEORETIC ALBANESE MAPS REVISITED

5[6] M. Kotani and T. Sunada, Standard realizations of crystal lattices via harmonicmaps, Trans. A.M.S. 353(2000), 1-20.
[7] M. Kotani and T. Sunada, Albanese maps and an off diagonal long time asymptotic for the heat kernel, Comm. Math. Phys. 209(2000), 633-670.
[8] M. Kotani and T. Sunada, Spectral geometry of crystal lattices, Contemporary.
Math. 338 (2003), 271-305.
[9] T. Nagnibeda, The Jacobian of a finite graph, Contemporary Math.,
206(1997), 149-151.
[10] T. Nagano and B. Smith, Minimal varieties and harmonic maps in tori,
Comm.Math.Helv. 50(1975), 249-265.