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

TL;DR

This paper extends Abel's classical theorem and the Abel--Jacobi map to higher-dimensional manifolds and cycles, introducing Abel gerbes and establishing their relation to harmonic Deligne cohomology.

Contribution

It generalizes Abel's theorem and the Abel--Jacobi map to higher dimensions using Abel gerbes and harmonic Deligne cohomology, broadening classical results.

Findings

Defined Abel gerbes for higher-dimensional cycles.

Proved Abel's theorem holds in this generalized setting.

Established isomorphism between moduli space of Abel gerbes and harmonic Deligne cohomology.

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.