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Hodge Theory and Complex Algebraic Geometry I by Claire Voisin, Leila Schneps

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12

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 ...

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