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

Abstract

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.