nLab Abel-Jacobi map

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

• Minhyong Kim, Galois Theory and Diophantine geometry, 2009 (pdf)