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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 *J*^{2k−1}(*X*) of such a manifold *X* are the complex tori defined by

When a cycle *Z* is homologous to 0, if we interpret ...

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