Abel-Jacobi map


Complex geometry

Differential cohomology



The Abel-Jacobi map refers to various homomorphisms from certain groups of algebraic cycles to some sorts of Jacobians or generalized Jacobians. Such maps generalize the classical Abel-Jacobi map from points of a complex algebraic curve to its Jacobian, which answers the question of which divisors of degree zero arise from meromorphic functions.


for curves

Let XX be a smooth projective complex curve. Recall that a Weil divisor on XX is a formal linear combination of closed points. Classically, the Abel-Jacobi map

α:Div 0(X)J(X), \alpha : \Div^0(X) \longrightarrow J(X),

on the group of Weil divisors of degree zero, is defined by integration. According to Abel’s theorem, its kernel consists of the principal divisors, i.e. the ones coming from meromorphic functions.

on Deligne cohomology

The cycle map to de Rham cohomology due to (Zein-Zucker 81) is discussed in (Esnault-Viehweg 88, section 6). The refinement to Deligne cohomology in (Esnault-Viehweg 88, section 6). By the characterization of intermediate Jacobians as a subgroup of the Deligne complex (see intermediate Jacobian – characterization as Hodge-trivial Deligne cohomology this induces a map from cycles to intermediate Jacobians. This is the Abel-Jacobi map (Esnault-Viehweg 88, theorem 7.11).

on higher Chow groups

An Abel-Jacobi map on higher Chow groups is discussed in K-L-MS 04.

via extensions of mixed Hodge structures

An alternate construction of the Abel-Jacobi map, via Hodge theory, is due to Arapura-Oh. By a theorem of Carlson, the Jacobian is identified with the following group of extensions in the abelian category of mixed Hodge structures:

J(X)=Ext MHS 1(Z(1),H 1(X,Z)) J(X) = Ext^1_{MHS}(\mathbf{Z}(-1), H^1(X, \mathbf{Z}))

where Z(1)\mathbf{Z}(-1) is the Tate Hodge structure. Given a divisor DD of degree zero, one can associate to it a certain class in the above extension group. This gives a map

α:Div 0(X)J(X) \alpha : Div^0(X) \longrightarrow J(X)

which is called the Abel-Jacobi map. The Abel theorem says that its kernel is precisely the subgroup of principal divisors?, i.e. divisors which come from invertible rational functions. See (Arapura-Oh, 1997) for details of this construction.


Remarks on generalization to the more general context of anabelian geometry are in