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Summer 2011 A generalization of Abel’s Theorem and the Abel–Jacobi map
Johan L. Dupont, Franz W. Kamber
Illinois J. Math. 55(2): 641-673 (Summer 2011). DOI: 10.1215/ijm/1359762406

Abstract

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.

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Johan L. Dupont. Franz W. Kamber. "A generalization of Abel’s Theorem and the Abel–Jacobi map." Illinois J. Math. 55 (2) 641 - 673, Summer 2011. https://doi.org/10.1215/ijm/1359762406

Information

Published: Summer 2011
First available in Project Euclid: 1 February 2013

zbMATH: 1271.53029
MathSciNet: MR3020700
Digital Object Identifier: 10.1215/ijm/1359762406

Subjects:
Primary: 55R20, 57R30
Secondary: 53C05, 53C12, 57R22

Rights: Copyright © 2011 University of Illinois at Urbana-Champaign

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Vol.55 • No. 2 • Summer 2011
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