Semi-stability for bi-holomorphic pairs over compact bi-Hermitian Gauduchon manifolds. (English) Zbl 1481.53034
Summary: In this paper, we investigate canonical metrics on bi-holomorphic bundles with a nontrivial global holomorphic section, and we prove that the \(I_{\pm}\)-holomorphic pair \((E,\bar{\partial}_+,\bar{\partial}_-,\phi)\) is \((\alpha,\tau)\)-semi-stable if and only if it admits an approximate \((\alpha ,\tau)\)-Hermitian-Einstein structure over the compact bi-Hermitian manifold.
MSC:
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 32Q20 | Kähler-Einstein manifolds |
| 58E15 | Variational problems concerning extremal problems in several variables; Yang-Mills functionals |
Keywords:
\(I_{\pm}\)-holomorphic pair; \((\alpha, \tau)\)-semi-stability; approximate \((\alpha, \tau)\)-Hermitian-Einstein structure; Gauduchon manifoldsReferences:
| [1] | Alvarez-Consul, L.; García-Prada, O., Dimensional reduction, SL (2, C)-equivariant bundles and stable holomorphic chains, Int. J. Math., 12, 2, 159-201 (2001) · Zbl 1110.32305 · doi:10.1142/S0129167X01000745 |
| [2] | Bando, S., Siu, Y.T.: Stable sheaves and Einstein-Hermitian metrics. In: Geometry And Analysis On Complex Manifolds: Festschrift for Professor S Kobayashi’s 60th Birthday, pp. 39-50. World Scientific (1994) · Zbl 0880.32004 |
| [3] | Biquard, O., Sur les fibrés paraboliques sur une surface complexe, J. Lond. Math. Soc., 53, 2, 302-316 (1996) · Zbl 0862.53025 · doi:10.1112/jlms/53.2.302 |
| [4] | Biswas, I.; Bradlow, SB; Jacob, A.; Stemmler, M., Approximate Hermitian-Einstein connections on principal bundles over a compact Riemann surface, Ann. Glob. Anal. Geom., 44, 3, 257-268 (2013) · Zbl 1280.53026 · doi:10.1007/s10455-013-9365-1 |
| [5] | Biswas, I., Bradlow, S.B., Jacob, A., Stemmler, M.: Automorphisms and connections on Higgs bundles over compact Kähler manifolds. Differ. Geom. Appl. 32, 139-152 (2014) · Zbl 1372.53024 |
| [6] | Bradlow, S.; Garcia-Prada, O., Stable triples, equivariant bundles and dimensional reduction, Math. Ann., 304, 2, 225-252 (1996) · Zbl 0852.32016 · doi:10.1007/BF01446292 |
| [7] | Bradlow, SB, Vortices in holomorphic line bundles over closed Kähler manifolds, Commun. Math. Phys., 135, 1, 1-17 (1990) · Zbl 0717.53075 · doi:10.1007/BF02097654 |
| [8] | Bradlow, SB, Special metrics and stability for holomorphic bundles with global sections, J. Differ. Geom., 33, 1, 169-213 (1991) · Zbl 0697.32014 · doi:10.4310/jdg/1214446034 |
| [9] | Bruzzo, U.; Otero, BG, Metrics on semistable and numerically effective Higgs bundles, Journal für die reine und angewandte Mathematik, 2007, 612, 59-79 (2007) · Zbl 1138.14024 · doi:10.1515/CRELLE.2007.084 |
| [10] | Buchdahl, NP, Hermitian-Einstein connections and stable vector bundles over compact complex surfaces, Math. Ann., 280, 4, 625-648 (1988) · Zbl 0617.32044 · doi:10.1007/BF01450081 |
| [11] | Cardona, S., Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. I: generalities and the one-dimensional case, Ann. Glob. Anal. Geom., 42, 3, 349-370 (2012) · Zbl 1260.58006 · doi:10.1007/s10455-012-9316-2 |
| [12] | Donaldson, SK, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc., 3, 1, 1-26 (1985) · Zbl 0529.53018 · doi:10.1112/plms/s3-50.1.1 |
| [13] | Donaldson, SK, A new proof of a theorem of Narasimhan and Seshadri, J. Differ. Geom., 18, 2, 269-277 (1983) · Zbl 0504.49027 |
| [14] | Donaldson, SK, Infinite determinants, stable bundles and curvature, Duke Math. J., 54, 1, 231-247 (1987) · Zbl 0627.53052 · doi:10.1215/S0012-7094-87-05414-7 |
| [15] | García-Prada, O., Dimensional reduction of stable bundles, vortices and stable pairs, Int. J. Math., 5, 1, 1-52 (1994) · Zbl 0799.32022 · doi:10.1142/S0129167X94000024 |
| [16] | Hitchin, NJ, The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc., 3, 1, 59-126 (1987) · Zbl 0634.53045 · doi:10.1112/plms/s3-55.1.59 |
| [17] | Hong, M-C, Heat flow for the Yang-Mills-Higgs field and the Hermitian Yang-Mills-Higgs metric, Ann. Glob. Anal. Geom., 20, 1, 23-46 (2001) · Zbl 0988.53009 · doi:10.1023/A:1010688223177 |
| [18] | Hong, M-C; Tian, G., Asymptotical behaviour of the Yang-Mills flow and singular Yang-Mills connections, Math. Ann., 330, 3, 441-472 (2004) · Zbl 1073.53084 · doi:10.1007/s00208-004-0539-9 |
| [19] | Hu, S., Moraru, R., Seyyedali, R.: A Kobayashi-Hitchin correspondence for \(I \pm \)-holomorphic bundles. Adv. Math. 287, 519-566 (2016) · Zbl 1327.32041 |
| [20] | Hu, Z., Huang, P.: The Hitchin-Kobayashi Correspondence for Quiver Bundles over Generalized Kähler Manifolds. J. Geom. Anal. 30(4), 3641-3671 (2020) · Zbl 1461.14016 |
| [21] | Jacob, A., Existence of approximate Hermitian-Einstein structures on semi-stable bundles, Asian J. Math., 18, 5, 859-884 (2014) · Zbl 1315.53079 · doi:10.4310/AJM.2014.v18.n5.a5 |
| [22] | Jost, J., Yang, Y.-H., Zuo, K.: Cohomologies of unipotent harmonic bundles over noncompact curves (2006) · Zbl 1128.53016 |
| [23] | Kobayashi, S., Differential Geometry of Holomorphic Vector Bundles (1986), Berkeley: Center for Pure and Applied Mathematics, University of California, Berkeley |
| [24] | Li, J., Hermitian-Einstein metrics and Chern number inequalities on parabolic stable bundles over Kähler manifolds, Commun. Anal. Geom., 8, 3, 445-475 (2000) · Zbl 0978.32026 · doi:10.4310/CAG.2000.v8.n3.a1 |
| [25] | Li, J.; Narasimhan, M., Hermitian-Einstein metrics on parabolic stable bundles, Acta Math. Sinica, 15, 1, 93-114 (1999) · Zbl 0963.32016 · doi:10.1007/s10114-999-0062-8 |
| [26] | Li, J., Yau, S.T.: Hermitian-Yang-Mills connection on non-Kähler manifolds. In: Mathematical Aspects of String Theory, pp. 560-573. World Scientific (1987) · Zbl 0664.53011 |
| [27] | Li, J.; Zhang, C.; Zhang, X., Semi-stable Higgs sheaves and Bogomolov type inequality, Calc. Var. Partial. Differ. Equ., 56, 3, 81 (2017) · Zbl 1421.53031 · doi:10.1007/s00526-017-1174-0 |
| [28] | Li, J.; Zhang, X., Existence of approximate Hermitian-Einstein structures on semi-stable Higgs bundles, Calc. Var. Partial. Differ. Equ., 52, 3-4, 783-795 (2015) · Zbl 1311.53022 · doi:10.1007/s00526-014-0733-x |
| [29] | Liu, D.; Zhang, P., Hermitian-Einstein metrics for higgs bundles over complete Hermitian manifolds, Acta Mathematica Scientia, 40, 1, 211-225 (2020) · Zbl 1499.53127 · doi:10.1007/s10473-020-0114-z |
| [30] | Lubke, M.; Teleman, A., The Kobayashi-Hitchin Correspondence (1995), Singapore: World Scientific, Singapore · Zbl 0849.32020 · doi:10.1142/2660 |
| [31] | Narasimhan, M.S., Seshadri, C.S.: Stable and unitary vector bundles on a compact Riemann surface. Ann. Math. 540-567 (1965) · Zbl 0171.04803 |
| [32] | Nie, Y.; Zhang, X., Semistable Higgs bundles over compact Gauduchon manifolds, J. Geom. Anal., 28, 1, 627-642 (2018) · Zbl 1391.32026 · doi:10.1007/s12220-017-9835-y |
| [33] | Pan, C.: A note on bogomolov-type inequality for semi-stable parabolic higgs bundles. Commun. Math. Stat. 1-12 (2021) |
| [34] | Shen, Z.; Zhang, P., Canonical metrics on holomorphic filtrations over compact Hermitian manifolds, Commun. Math. Stat., 8, 2, 219-237 (2020) · Zbl 1442.53018 · doi:10.1007/s40304-019-00199-y |
| [35] | Sibley, B.C.: Asymptotics of the Yang-Mills Flow for Holomorphic Vector Bundles over Kahler Manifolds. Ph.D. thesis (2013) · Zbl 1329.58006 |
| [36] | Simpson, CT, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Am. Math. Soc., 1, 4, 867-918 (1988) · Zbl 0669.58008 · doi:10.1090/S0894-0347-1988-0944577-9 |
| [37] | Uhlenbeck, K.; Yau, S-T, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Commun. Pure Appl. Math., 39, S1, S257-S293 (1986) · Zbl 0615.58045 · doi:10.1002/cpa.3160390714 |
| [38] | Wang, R.; Zhang, C., Semi-stability for holomorphic pairs on compact Gauduchon manifolds, Commun. Math. Stat., 7, 2, 191-206 (2019) · Zbl 1420.53030 · doi:10.1007/s40304-018-0152-y |
| [39] | Zhang, C., Semi-stability for Holomorphic Bundles with Global Sections, J. Partial Differ. Equ., 29, 3, 204-217 (2016) · Zbl 1374.53046 · doi:10.4208/jpde.v29.n3.4 |
| [40] | Zhang, C., Zhang, P., Zhang, X.: Higgs bundles over non-compact Gauduchon manifolds. arXiv preprint arXiv:1804.08994 (2018). · Zbl 1391.32026 |
| [41] | Zhang, P., Canonical metrics on holomorphic bundles over compact bi-Hermitian manifolds, J. Geom. Phys., 144, 15-27 (2019) · Zbl 1425.53032 · doi:10.1016/j.geomphys.2019.05.010 |
| [42] | Zhang, P.: Semi-stable holomorphic vector bundles over generalized kähler manifolds. Complex Variables Elliptic Equ. 1-15 (2021) |
| [43] | Zhang, X., Hermitian Yang-Mills-Higgs metrics on complete Kähler manifolds, Can. J. Math., 57, 4, 871-896 (2005) · Zbl 1083.58014 · doi:10.4153/CJM-2005-034-3 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
