A generalization of Abel’s Theorem and the Abel–Jacobi map

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We generalize Abel's classical theorem on linear\break 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.