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Open AccessOpen Access JUSTC Mathematics 22 March 2022

Curvature estimate of the Yang-Mills-Higgs flow on Kähler manifolds

  • School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, China

Cite this:
https://doi.org/10.52396/JUSTC-2021-0221
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  • Author Bio:

    Zhenghan Shen is currently a postdoctoral fellow at Nanjing University of Science and Technology. He received his PhD degree in Geometric Analysis under the tutelage of Prof. Xi Zhang from University of Science and Technology of China. His research interests focus on the Higgs bundle, Hermitian-Einstein metric and Hermitian-Yang-Mills flow

  • Corresponding author: E-mail: mathszh@njust.edu.cn
  • Received Date: 04 August 2020
  • Accepted Date: 21 December 2021
    • Available Online: 22 March 2022
    Abstract

  • Abstract

    The curvature estimate of the Yang-Mills-Higgs flow on Higgs bundles over compact Kähler manifolds is  studied. Under the assumptions that the Higgs bundle is non-semistable and the Harder-Narasimhan-Seshadri filtration has no singularities with length one, it is proved that the curvature of the evolved Hermitian metric is uniformly bounded.

    Graphical abstract

      Using the decomposition of Donaldson’s functional and the properties of Harder-Nasimhan-Seshadri filtration, we get the C0-bound of the rescaled metrics

    Abstract

    The curvature estimate of the Yang-Mills-Higgs flow on Higgs bundles over compact Kähler manifolds is  studied. Under the assumptions that the Higgs bundle is non-semistable and the Harder-Narasimhan-Seshadri filtration has no singularities with length one, it is proved that the curvature of the evolved Hermitian metric is uniformly bounded.

    • Public Summary

    • By vast calculation and analysis, we drive many evolution equations in Higgs bundles.
    • Generalized the decomposition of the Donaldson’s functional in Higgs bundle case.
    • It provides an analytic method for studying the algebraic singular set and analytic singular set in the future work.

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    • References(26)
  • [1]
    Hitchin N J. The self-duality equations on a Riemann surface. Proc. London Math. Soc., 1987, (55): 59–126. doi: 10.1112/plms/s3-55.1.59
    [2]
    Simpson C T. Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformazation. J. Amer. Math. Soc., 1988, 1: 867–918. doi: 10.1090/S0894-0347-1988-0944577-9
    [3]
    Simpson C T. Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math., 1992, 75: 5–95. doi: 10.1007/BF02699491
    [4]
    Bando S, Siu Y T. Stable sheaves and Einstein-Hermitian metrics. In: Geometry and Analysis on Complex Manifolds. Singapore: World Scientific, 1994: 39-50.
    [5]
    Bartolomeis P D, Tian G. Stability of complex vector bundles. J. Differ. Geom., 1996, 43: 232–275. doi: 10.4310/jdg/1214458107
    [6]
    Biquard O. On parabolic bundles over a complex surface. J. London Math. Soc., 1996, 53 (2): 302–316. doi: 10.1112/jlms/53.2.302
    [7]
    Bradlow S B. Vortices in holomorphic line bundles over closed Kähler manifolds. Commun. Math. Phys., 1990, 135: 1–17. doi: 10.1007/BF02097654
    [8]
    Bradlow S B, Garcia-Prada O. Stable triples equivariant bundles and dimensional reduction. Math. Ann., 1996, 135: 225–252. doi: 10.1007/BF01446292
    [9]
    Álvarez-Cónsul L, Garcís-Prada O. Dimensional reduction, $SL(2, \mathbb{C})$ -equivariant bundles and stable holomorphic chains. Int. J. Math., 2001, 2: 159–201. doi: 10.1142/S0129167X01000745
    [10]
    Shen Z, Zhang P. Canonical metrics on holomorphic filtrations over compact Hermitian manifolds. Commun. Math. Stat., 2020, 8: 219–237. doi: 10.1007/s40304-019-00199-y
    [11]
    Garcia-Prada O. Dimensional reduction of stable bundles, vortices and stable pairs. Int. J. Math., 1994, 5: 1–52. doi: 10.1142/S0129167X94000024
    [12]
    Huybrechts D, Lehn M. Stable pairs on curves and surfaces. J. Algebr. Geom., 1995, 4 (1): 67–104.
    [13]
    Atiyah M, Bott R. The Yang-Mills equations over Riemann surfaces. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1982, 308: 524–615. doi: 10.1098/rsta.1983.0017
    [14]
    Li J Y, Zhang X. The gradient flow of Higgs pairs. J. Eur. Math. Soc., 2011, 13 (5): 1373–1422. doi: 10.4171/JEMS/284
    [15]
    Uhlenbeck K K. A priori estimates for Yang-Mills fields, unpublished manuscript. http://www.math.uwaterloo.ca/~karigian/ uhlenbeck-preprint.pdf.
    [16]
    Uhlenbeck K K. Connections with L p bounds on curvature. Commun. Math. Phys., 1982, 83 (1): 31–42. doi: 10.1007/BF01947069
    [17]
    Jacob A. The Yang-Mills flow and the Atiyah-Bott formula on compact Kähler manifolds. Amer. J. Math., 2016, 138 (2): 329–365. doi: 10.1353/ajm.2016.0011
    [18]
    Sibley B. Asymtotics of the Yang-Mills flow for holomorphic vector bundles over Kähler manifolds: The canonical structure of the limit. J. Reine Agnew. Math., 2015, 706: 123–191. doi: 10.1515/crelle-2013-0063
    [19]
    Daskalopoulos G, Wentworth R. On the blow-up set of the Yang-Mills flow on Kähler surfaces. Math. Z., 2007, 256: 301–310. doi: 10.1007/s00209-006-0075-2
    [20]
    Sibley B, Wentworth R. Analytic cycles, Bott-Chern forms, and singular sets for the Yang-Mills flow on Kähler manifolds. Adv. Math., 2015, 279: 501–531. doi: 10.1016/j.aim.2015.04.009
    [21]
    Li J Y, Zhang C J, Zhang X. A note on curvature estimate of the Hermitian-Yang-Mills flow. Commun. Math. Stat., 2018, 6: 319–358. doi: 10.1007/s40304-018-0135-z
    [22]
    Uhlenbeck K K, Yau S T. On existence of Hermitian-Yang-Mills connection in stable vector bundles. Comm. Pure Appl. Math., 1986, 39S: 257–293. doi: 10.1002/cpa.3160390714
    [23]
    Donaldson S K. Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc., 1985, 50: 1–26. doi: 10.1112/plms/s3-50.1.1
    [24]
    Cheng S Y, Li P. Heat kernel estimates and lower bound of eigenvalues. Comment. Math. Helv., 1981, 56: 327–338. doi: 10.1007/BF02566216
    [25]
    Bando S, Mabuchi T. Uniquessness of Einstein Kähler metrics modulo connected group actions. Advanced Studies in Pure Mathematics, 1987, 10: 11–40. doi: 10.2969/ASPM/01010011
    [26]
    Li J Y, Zhang C J, Zhang X. Semi-stable Higgs sheaves and Bogomolov type inequality. Calc. Var. Partial Differ. Equ., 2017, 56 (3): 81. doi: 10.1007/s00526-017-1174-0
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    [1]
    Hitchin N J. The self-duality equations on a Riemann surface. Proc. London Math. Soc., 1987, (55): 59–126. doi: 10.1112/plms/s3-55.1.59
    [2]
    Simpson C T. Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformazation. J. Amer. Math. Soc., 1988, 1: 867–918. doi: 10.1090/S0894-0347-1988-0944577-9
    [3]
    Simpson C T. Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math., 1992, 75: 5–95. doi: 10.1007/BF02699491
    [4]
    Bando S, Siu Y T. Stable sheaves and Einstein-Hermitian metrics. In: Geometry and Analysis on Complex Manifolds. Singapore: World Scientific, 1994: 39-50.
    [5]
    Bartolomeis P D, Tian G. Stability of complex vector bundles. J. Differ. Geom., 1996, 43: 232–275. doi: 10.4310/jdg/1214458107
    [6]
    Biquard O. On parabolic bundles over a complex surface. J. London Math. Soc., 1996, 53 (2): 302–316. doi: 10.1112/jlms/53.2.302
    [7]
    Bradlow S B. Vortices in holomorphic line bundles over closed Kähler manifolds. Commun. Math. Phys., 1990, 135: 1–17. doi: 10.1007/BF02097654
    [8]
    Bradlow S B, Garcia-Prada O. Stable triples equivariant bundles and dimensional reduction. Math. Ann., 1996, 135: 225–252. doi: 10.1007/BF01446292
    [9]
    Álvarez-Cónsul L, Garcís-Prada O. Dimensional reduction, $SL(2, \mathbb{C})$ -equivariant bundles and stable holomorphic chains. Int. J. Math., 2001, 2: 159–201. doi: 10.1142/S0129167X01000745
    [10]
    Shen Z, Zhang P. Canonical metrics on holomorphic filtrations over compact Hermitian manifolds. Commun. Math. Stat., 2020, 8: 219–237. doi: 10.1007/s40304-019-00199-y
    [11]
    Garcia-Prada O. Dimensional reduction of stable bundles, vortices and stable pairs. Int. J. Math., 1994, 5: 1–52. doi: 10.1142/S0129167X94000024
    [12]
    Huybrechts D, Lehn M. Stable pairs on curves and surfaces. J. Algebr. Geom., 1995, 4 (1): 67–104.
    [13]
    Atiyah M, Bott R. The Yang-Mills equations over Riemann surfaces. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1982, 308: 524–615. doi: 10.1098/rsta.1983.0017
    [14]
    Li J Y, Zhang X. The gradient flow of Higgs pairs. J. Eur. Math. Soc., 2011, 13 (5): 1373–1422. doi: 10.4171/JEMS/284
    [15]
    Uhlenbeck K K. A priori estimates for Yang-Mills fields, unpublished manuscript. http://www.math.uwaterloo.ca/~karigian/ uhlenbeck-preprint.pdf.
    [16]
    Uhlenbeck K K. Connections with L p bounds on curvature. Commun. Math. Phys., 1982, 83 (1): 31–42. doi: 10.1007/BF01947069
    [17]
    Jacob A. The Yang-Mills flow and the Atiyah-Bott formula on compact Kähler manifolds. Amer. J. Math., 2016, 138 (2): 329–365. doi: 10.1353/ajm.2016.0011
    [18]
    Sibley B. Asymtotics of the Yang-Mills flow for holomorphic vector bundles over Kähler manifolds: The canonical structure of the limit. J. Reine Agnew. Math., 2015, 706: 123–191. doi: 10.1515/crelle-2013-0063
    [19]
    Daskalopoulos G, Wentworth R. On the blow-up set of the Yang-Mills flow on Kähler surfaces. Math. Z., 2007, 256: 301–310. doi: 10.1007/s00209-006-0075-2
    [20]
    Sibley B, Wentworth R. Analytic cycles, Bott-Chern forms, and singular sets for the Yang-Mills flow on Kähler manifolds. Adv. Math., 2015, 279: 501–531. doi: 10.1016/j.aim.2015.04.009
    [21]
    Li J Y, Zhang C J, Zhang X. A note on curvature estimate of the Hermitian-Yang-Mills flow. Commun. Math. Stat., 2018, 6: 319–358. doi: 10.1007/s40304-018-0135-z
    [22]
    Uhlenbeck K K, Yau S T. On existence of Hermitian-Yang-Mills connection in stable vector bundles. Comm. Pure Appl. Math., 1986, 39S: 257–293. doi: 10.1002/cpa.3160390714
    [23]
    Donaldson S K. Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc., 1985, 50: 1–26. doi: 10.1112/plms/s3-50.1.1
    [24]
    Cheng S Y, Li P. Heat kernel estimates and lower bound of eigenvalues. Comment. Math. Helv., 1981, 56: 327–338. doi: 10.1007/BF02566216
    [25]
    Bando S, Mabuchi T. Uniquessness of Einstein Kähler metrics modulo connected group actions. Advanced Studies in Pure Mathematics, 1987, 10: 11–40. doi: 10.2969/ASPM/01010011
    [26]
    Li J Y, Zhang C J, Zhang X. Semi-stable Higgs sheaves and Bogomolov type inequality. Calc. Var. Partial Differ. Equ., 2017, 56 (3): 81. doi: 10.1007/s00526-017-1174-0
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