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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The vortex equation on affine manifolds

Authors: Indranil Biswas, John Loftin and Matthias Stemmler
Journal: Trans. Amer. Math. Soc. 366 (2014), 3925-3941
MSC (2010): Primary 53C07, 57N16
Published electronically: March 20, 2014
MathSciNet review: 3192624
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Abstract: Let $M$ be a compact connected special affine manifold equipped with an affine Gauduchon metric. We show that a pair $(E, \phi )$, consisting of a flat vector bundle $E$ over $M$ and a flat nonzero section $\phi$ of $E$, admits a solution to the vortex equation if and only if it is polystable. To prove this, we adapt the dimensional reduction techniques for holomorphic pairs on Kähler manifolds to the situation of flat pairs on affine manifolds.

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Additional Information

Indranil Biswas
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
MR Author ID: 340073

John Loftin
Affiliation: Department of Mathematics and Computer Science, Rutgers University at Newark, Newark, New Jersey 07102

Matthias Stemmler
Affiliation: Fachbereich Mathematik und Informatik, Philipps–Universität Marburg, Hans–Meerwein–Straße, Lahnberge, 35032 Marburg, Germany

Keywords: Affine manifold, vortex equation, stability, dimensional reduction
Received by editor(s): November 19, 2012
Received by editor(s) in revised form: January 26, 2013
Published electronically: March 20, 2014
Article copyright: © Copyright 2014 American Mathematical Society