Johan Dupont: A generalization of Abel's theorem and the Abel-Jacobi map

We generalize Abel's classical theorem on linear equivalence of divisors on a Riemann surface. For every closed submanifold M^d in a compact oriented Riemannian n-manifold X^n, 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 is isomorphic to the harmonic Deligne cohomology; that is, gerbes with harmonic curvature.