Abstract
Fix a C ∞ principal G–bundle \({E^0_G}\) on a compact connected Riemann surface X, where G is a connected complex reductive linear algebraic group. We consider the gradient flow of the Yang–Mills–Higgs functional on the cotangent bundle of the space of all smooth connections on \({E^0_G}\). We prove that this flow preserves the subset of Higgs G–bundles, and, furthermore, the flow emanating from any point of this subset has a limit. Given a Higgs G–bundle, we identify the limit point of the integral curve passing through it. These generalize the results of the second named author on Higgs vector bundles.
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References
Anchouche B., Biswas I.: Einstein-Hermitian connections on polystable principal bundles over a compact Kähler manifold. Am. J. Math. 123, 207–228 (2001)
Atiyah M.F., Bott R.: The Yang–Mills equations over Riemann surfaces. Philos. Trans. Roy. Soc. London Ser. A 308, 523–615 (1983)
Biswas I., Gómez T.L.: Connections and Higgs fields on a principal bundle. Ann. Glob. Anal. Geom. 33, 19–46 (2008)
Biswas I., Holla Y.I.: Harder–Narasimhan reduction of a principal bundle. Nagoya Math. J. 174, 201–223 (2004)
Biswas I., Subramanian S.: Flat holomorphic connections on principal bundles over a projective manifold. Trans. Am. Math. Soc. 356, 3995–4018 (2004)
Biswas I., Schumacher G.: Yang–Mills equation for stable Higgs sheaves. Int. J. Math. 20, 541–556 (2009)
Daskalopoulos G.D.: The topology of the space of stable bundles on a compact Riemann surface. J. Diff. Geom. 36, 699–746 (1992)
Daskalopoulos, G.D., Weitsman, J., Wentworth, R.A., Wilkin, G.: Morse theory and hyperkähler Kirwan surjectivity for Higgs bundles, arXiv:math/0701560
Daskalopoulos G.D., Wentworth R.A.: Convergence properties of the Yang-Mills flow on Kähler surfaces. J. Reine Angew. Math. 575, 69–99 (2004)
Dey A., Parthasarathi R.: On Harder–Narasimhan reductions for Higgs principal bundles. Proc. Indian Acad. Sci. Math. Sci. 115, 127–146 (2005)
Earl R., Kirwan F.C.: Complete sets of relations in the cohomology rings of moduli spaces of holomorphic bundles and parabolic bundles over a Riemann surface. Proc. London Math. Soc. 89, 570–622 (2004)
Humphreys J.E.: Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 21. Springer, New York (1987)
Huybrechts D., Lehn M.: The Geometry of Moduli Spaces of Sheaves, Aspects of Mathematics, E31. Friedr. Vieweg & Sohn, Braunschweig (1997)
Jeffrey L.C., Kirwan F.C.: Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface. Ann. Math. 148, 109–196 (1998)
Kirwan F.C.: On the homology of compactifications of moduli spaces of vector bundles over a Riemann surface. Proc. London Math. Soc. 53, 237–266 (1986)
Kirwan F.C.: The cohomology rings of moduli spaces of bundles over Riemann surfaces. J. Am. Math. Soc. 5, 853–906 (1992)
Råde J.: On the Yang-Mills heat equation in two and three dimensions. J. Reine Angew. Math. 431, 123–163 (1992)
Simpson C.T.: Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. 75, 5–95 (1992)
Wilkin G.: Morse theory for the space of Higgs bundles. Comm. Anal. Geom. 16, 283–332 (2008)
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Biswas, I., Wilkin, G. Morse theory for the space of Higgs G–bundles. Geom Dedicata 149, 189–203 (2010). https://doi.org/10.1007/s10711-010-9476-9
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DOI: https://doi.org/10.1007/s10711-010-9476-9