Abstract
We study the limiting behaviour of solutions to abelian vortex equations when the volume of the underlying Riemann surface grows to infinity. We prove that the solutions converge smoothly away from finitely many points. The proof relies on a priori estimates for functions satisfying generalised Kazdan–Warner equations. We relate our results to the work of Hong, Jost, and Struwe on classical vortices, and that of Haydys and Walpuski on the Seiberg–Witten equations with multiple spinors.
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Notes
By slight abuse of notation, we will denote by the same symbol a divisor and the underlying set of points.
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Acknowledgements
The work presented in this article is part of my doctoral thesis at Stony Brook University. I am grateful to my advisor Simon Donaldson for his guidance and support. Thanks to Andriy Haydys and Thomas Walpuski for their encouragement and many helpful discussions, and to Gonçalo Oliveira, Oscar Garcia–Prada, Song Sun, Alex Waldron, and the anonymous referee for valuable comments on the previous versions of this paper. I am supported by the Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics.
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Communicated by J. Jost.
