A generalization of Abel's Theorem and the Abel--Jacobi map

Details A generalization of Abel's Theorem and the Abel--Jacobi map A generalization of Abel's Theorem and the Abel--Jacobi map

2008-11-06

arxiv.org/abs/0811.0961v2

Mathematics

Differential Geometry

27 pages Added references; minor changes in text; corrected typos

Scientific paper

We generalize Abel's classical theorem on linear equivalence of divisors on a Riemann surface. For every closed submanifold $M^d \subset X^n$ in a compact oriented Riemannian $n$--manifold, or more generally for any $d$--cycle $Z$ relative to a triangulation of $X$, we define a (simplicial) $(n-d-1)$--gerbe $\Lambda_{Z}$, the Abel gerbe determined by $Z$, whose vanishing as a Deligne cohomology class generalizes the notion of `linear equivalence to zero'. In this setting, Abel's theorem remains valid. Moreover we generalize the classical Inversion Theorem for the Abel--Jacobi map, thereby proving that the moduli space of Abel gerbes is isomorphic to the harmonic Deligne cohomology; that is, gerbes with harmonic curvature.

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