Illinois Journal of Mathematics

A generalization of Abel’s Theorem and the Abel–Jacobi map

Johan L. Dupont and Franz W. Kamber

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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.

Article information

Source
Illinois J. Math., Volume 55, Number 2 (2011), 641-673.

Dates
First available in Project Euclid: 1 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1359762406

Digital Object Identifier
doi:10.1215/ijm/1359762406

Mathematical Reviews number (MathSciNet)
MR3020700

Zentralblatt MATH identifier
1271.53029

Subjects
Primary: 55R20: Spectral sequences and homology of fiber spaces [See also 55Txx] 57R30: Foliations; geometric theory
Secondary: 57R22: Topology of vector bundles and fiber bundles [See also 55Rxx] 53C05: Connections, general theory 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32]

Citation

Dupont, Johan L.; Kamber, Franz W. A generalization of Abel’s Theorem and the Abel–Jacobi map. Illinois J. Math. 55 (2011), no. 2, 641--673. doi:10.1215/ijm/1359762406. https://projecteuclid.org/euclid.ijm/1359762406


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