# nLab Abel-Jacobi map

complex geometry

### Examples

#### Differential cohomology

differential cohomology

## Ingredients

• cohomology

• differential geometry

• ## Connections on bundles

• connection on a bundle

• curvature

• Chern-Weil theory

• ## Higher abelian differential cohomology

• differential function complex

• differential orientation

• ordinary differential cohomology

• differential K-theory

• differential elliptic cohomology

• differential cobordism cohomology

• ## Higher nonabelian differential cohomology

• Chern-Weil theory in Smooth∞Grpd

• ∞-Chern-Simons theory

• ## Fiber integration

• higher holonomy

• fiber integration in differential cohomology

• ## Application to gauge theory

• gauge theory

• gauge field

• quantum anomaly

• # Contents

## Idea

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.

## 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

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

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:

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

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

• Fouad El Zein and Steven Zucker, Extendability of normal functions associated to algebraic cycles, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), Ann. of Math. Stud., vol. 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 269–288. MR 756857

• Hélène Esnault, Eckart Viehweg, Deligne-Beilinson cohomology in Rapoport, Schappacher, Schneider (eds.) Beilinson’s Conjectures on Special Values of L-Functions . Perspectives in Math. 4, Academic Press (1988) 43 - 91 (pdf)

• Claire Voisin, section 12 of Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3

• Donu Arapura, Kyungho Oh. On the Abel-Jacobi map for non-compact varieties. Osaka Journal of Mathematics 34 (1997), no. 4, 769–781. Project Euclid.

• Matt Kerr, James Lewis, Stefan Müller-Stach, The Abel-Jacobi map for higher Chow groups, 2004, arXiv:0409116.

• Wikipedia, Abel-Jacobi map

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

Last revised on December 8, 2014 at 19:39:13. See the history of this page for a list of all contributions to it.