Deligne–Beilinson Cohomology and the Abel–Jacobi map
In this chapter, we define a refined invariant of an analytic cycle homologous to 0 on a compact Kähler manifold, namely its Abel–Jacobi invariant, which generalises the Abel–Jacobi invariant for the 0-cycles on curves (see Arbarello et al. 1985). In the last section, following Deligne, we will show that we can even construct a Deligne class, which determines the Hodge class, and which is equal to the Abel–Jacobi invariant for a cycle homologous to 0.
The intermediate Jacobians J2k−1(X) of such a manifold X are the complex tori defined by
When a cycle Z is homologous to 0, if we interpret ...