Singularities of Abel-Jacobi maps and geometry of dissolving vortices

Nuno M. Romão

Journal of Singularities
volume 6 (2012), 146-157
Proceedings of the Workshop on Singularities in Geometry and Applications, Będlewo, 5 – 21 May 2011

Received: 5 January 2012. Received in revised form: 30 April 2012.

DOI: 10.5427/jsing.2012.6l

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Gauged vortices are configurations of fields for certain gauge theories in fibre bundles over a surface Σ. Their moduli spaces support natural L^2-metrics, which are Kähler, and whose geodesic flow approximates vortex scattering at low speed. This paper focuses on vortices in line bundles, for which the moduli spaces are modeled on the spaces Σ^(k) of effective divisors on Σ with a fixed degree k; we describe the behaviour of the underlying L^2-metrics in a "dissolving limit" where the L^2-geometry simplifies. In such limit, the metrics degenerate precisely at the singular locus of the Abel-Jacobi map AJ of Σ at degree k, and their geometry can be understood in terms of the variety W_k = AJ(Σ^(k)) inside the Jacobian of Σ. Some intuition about the behaviour of the geodesic flow close to a singularity is provided through the study of the simplest example (resolution of a double point on a surface), corresponding to two dissolving vortices moving on a hyperelliptic curve of genus three.

Author(s) information:

Nuno M. Romão
Departament d'Àlgebra i Geometria
Universitat de Barcelona
Gran Via de les Corts Catalanes 585
08007 Barcelona, Spain