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

2008/11/06 by Johan L. Dupont, Dupont, Johan L., Franz W. Kamber +1 · 1 citation
Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #math-ph #math.DG #math.MP #msc:55R20 #msc:57R30

paper · pdf · doi:10.48550/arxiv.0811.0961

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

arxiv created 2008/12/02 · arxiv updated 2009/12/01

Abstract

We generalize Abel's classical theorem on linear equivalence of divisors on a Riemann surface. For every closed submanifold M^d ⊂ Xⁿ 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 Λ_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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