# nLab Abel-Jacobi map

Contents

complex geometry

### Examples

#### Differential cohomology

differential cohomology

# Contents

## Idea

In algebraic/complex geometry The term Abel-Jacobi map refers to various group 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.

## Definition

### for curves

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

$\alpha \;\colon\; \Div^0(X) \longrightarrow J(X) ,\,$

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 (1981) is discussed in Esnault & Viehweg (1988), section 6, the refinement to Deligne cohomology in Esnault & Viehweg (1988), section 6. By the characterization of intermediate Jacobians as subgroups 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 (1988), 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:

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

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

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

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.

## References

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

Refinement of the Abel-Jacobi map to Hodge filtered differential MU-cobordism cohomology theory:

Introduction and survey:

Last revised on June 9, 2023 at 14:05:27. See the history of this page for a list of all contributions to it.