## Illinois Journal of Mathematics

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

#### 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

https://projecteuclid.org/euclid.ijm/1359762406

Digital Object Identifier
doi:10.1215/ijm/1359762406

Mathematical Reviews number (MathSciNet)
MR3020700

Zentralblatt MATH identifier
1271.53029

#### 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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