Abstract
We study the vortex equations on a line bundle over a compact Kähler manifold. These are a generalization of the classical vortex equations over ℝ2. We first prove an invariant version of the theorem of Donaldson, Uhlenbeck and Yau relating the existence of a Hermitian-Yang-Mills metric on a holomorphic bundle to the stability of such a bundle. We then show that the vortex equations are a dimensional reduction of the Hermitian-Yang-Mills equation. Using this fact and the theorem above we give a new existence proof for the vortex equations and describe the moduli space of solutions.
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Bradlow, S.B.: Vortices in Holomorphic Line Bundles over Closed Kähler Manifolds. Commun. Math. Phys.135, 1–17 (1990)
Bradlow, S.B.: Special Metrics and Stability for Holomorphic Bundles with Global Sections. J. Diff. Geom.33, 169–214 (1991)
Donaldson, S.K.: Anti-self-dual Yang-Mills Connections on a Complex Algebraic Surface and Stable Vector Bundles. Proc. Lond. Math. Soc.3, 1–26 (1985)
Donaldson, S.K.: Infinite Determinants, Stable Bundles and Curvature. Duke Math. J.54, 231–247 (1987)
Donaldson, S.K., Kronheimer, P.B.: The Geometry of Four Manifolds. Oxford Mathematical Monographs. Oxford: Oxford University Press, 1990
García-Prada, O.: The Geometry of the Vortex Equation. D. Phil. Thesis, University of Oxford 1991
García-Prada, O.: A Direct Existence Proof for the Vortex Equations over a Compact Riemann Surface. Bull. London Math. Soc., to appear
García-Prada, O.: Dimensional Reduction of Stable Bundles, Vortices and Stable Pairs, Int. J. Math., to appear
Ginsburg, V.L., Landau, L.D.: Zh. Eksp. Theor. Fiz.20, 1064 (1950)
Griffiths, P., Harris, J.: Principles of Algebraic Geometry. New York: Wiley 1978
Hitchin, N.J.: Metrics on Moduli Spaces. Contemporary Mathematics58, 157–178 (1986)
Hitchin, N.J.: The Self-Duality Equations on a Riemann Surface. Proc. Lond. Math. Soc.55, 59–126 (1987)
Jaffe, A., Taubes, C.H.: Vortices and Monopoles, Progress in Physics2, Boston: Birkhäuser 1980
Kobayashi, S.: Curvature and Stability of Vector Bundles. Proceedings of the Japan Academy58, A4, 158–162 (1982)
Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Princeton, NJ: Princeton University Press 1987
Lübke, M.: Stability of Einstein-Hermitian Vector Bundles. Manuscripta Mathematica42, 245–257 (1983)
Okonek, C., Schneider M., Spindler, H.: Vector Bundles on Complex Projective Spaces. Progress in Mathematics 3, Boston: Birkhäuser 1980
Simpson, C.T.: Constructing Variations of Hodge Structure Using Yang-Mills Theory, with Applications to Uniformisation. J. Am. Math. Soc.1, 867–918 (1989)
Taubes, C.H.: ArbitraryN-Vortex Solutions to the First Order Ginzburg-Landau Equations. Commun. Math. Phys.72, 277–292 (1980)
Taubes, C.H.: On the Equivalence of the First and Second Order Equations for Gauge Theories. Commun. Math. Phys.75, 207–227 (1980)
Uhlenbeck, K.K., Yau, S.-T.: On the Existence of Hermitian-Yang-Mills Connections on Stable Bundles Over Compact Kähler Manifolds. Commun. Pure and Appl. Math.39-S, 257–293 (1986)
Witten, E.: Some Exact Multipseudoparticle Solutions of Classical Yang-Mills Theory. Phys. Rev. Lett.38, 121 (1977)
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Communicated by A. Jaffe
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García-Prada, O. Invariant connections and vortices. Commun.Math. Phys. 156, 527–546 (1993). https://doi.org/10.1007/BF02096862
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DOI: https://doi.org/10.1007/BF02096862