Canonical metrics on holomorphic filtrations over compact Hermitian manifolds. (English) Zbl 1442.53018
Summary: The purpose of this paper is twofold. We first solve the Dirichlet problem for \(\tau\)-Hermitian-Einstein equations on holomorphic filtrations over compact Hermitian manifolds. Secondly, by using Uhlenbeck-Yau’s continuity method, we prove the existence of approximate \(\tau\)-Hermitian-Einstein structure on holomorphic filtrations over closed Gauduchon manifolds.
MSC:
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 32Q20 | Kähler-Einstein manifolds |
Keywords:
\(\tau\)-Hermitian-Einstein equation; approximate \(\tau\)-Hermitian-Einstein structure; semi-stability; holomorphic filtration; Gauduchon manifoldReferences:
| [1] | Álvarez-Cónsul, L.; García-Prada, L., Dimensional reduction, Sl \((2,{\mathbb{C}} )\)-equivariant bundles and stable holomorphic chains, Int. J. Math., 12, 159-201 (2001) · Zbl 1110.32305 · doi:10.1142/S0129167X01000745 |
| [2] | Álvarez-Cónsul, L.; García-Prada, L., Hitchin-Kobayashi correspondence, quivers, and vortices, Commun. Math. Phys., 238, 1-33 (2003) · Zbl 1051.53018 · doi:10.1007/s00220-003-0853-1 |
| [3] | Biquard, O., On parabolic bundles over a complex surface, J. Lond. Math. Soc., 53, 302-316 (1996) · Zbl 0862.53025 · doi:10.1112/jlms/53.2.302 |
| [4] | Biswas, I.; Schumacher, G., Yang-Mills equation for stable Higgs sheaves, Int. J. Math., 20, 541-556 (2009) · Zbl 1169.53017 · doi:10.1142/S0129167X09005406 |
| [5] | Biswas, I.; Bruzzo, U.; Graña Otero, B.; Giudice, AL, Yang-Mills-Higgs connections on Calabi-Yau manifolds, Asian J. Math., 20, 989-1000 (2016) · Zbl 1362.32014 · doi:10.4310/AJM.2016.v20.n5.a8 |
| [6] | Bradlow, SB, Vortices in holomorphic line bundles over closed Kähler manifolds, Commun. Math. Phys., 135, 1-17 (1990) · Zbl 0717.53075 · doi:10.1007/BF02097654 |
| [7] | Bruzzo, U.; Graña Otero, B., Metrics on semistable and numerically effective Higgs bundles, J. Reine Angew. Math., 612, 59-79 (2007) · Zbl 1138.14024 |
| [8] | Bruzzo, U.; Graña Otero, B., Semistable and numerically effective principal (Higgs) bundles, Adv. Math., 226, 3655-3676 (2011) · Zbl 1263.14019 · doi:10.1016/j.aim.2010.10.026 |
| [9] | Bruzzo, U.; Graña Otero, B., Approximate Hitchin-Kobayashi correspondence for Higgs G-bundles, Int. J. Geom. Methods Mod. Phys., 11, 1460015 (2014) · Zbl 1312.53042 · doi:10.1142/S0219887814600159 |
| [10] | Buchdahl, NP, Hermitian-Einstein connections and stable vector bundles over compact complex surfaces, Math. Ann., 280, 625-648 (1988) · Zbl 0617.32044 · doi:10.1007/BF01450081 |
| [11] | Cardona, SAH, Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. I: generalities and the one-dimensional case, Ann. Glob. Anal. Geom., 42, 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, 1-26 (1985) · Zbl 0529.53018 · doi:10.1112/plms/s3-50.1.1 |
| [13] | Donaldson, SK, Boundary value problems for Yang-Mills fields, J. Geom. Phys., 8, 89-122 (1992) · Zbl 0747.53022 · doi:10.1016/0393-0440(92)90044-2 |
| [14] | Gauduchon, P., La 1-forme de torsion d’une variété hermitienne compacte, Math. Ann., 267, 495-518 (1984) · Zbl 0523.53059 · doi:10.1007/BF01455968 |
| [15] | Hamilton, RS, Harmonic Maps of Manifolds with Boundary (2006), Berlin: Springer, Berlin |
| [16] | Hitchin, NJ, The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc., 3, 59-126 (1987) · Zbl 0634.53045 · doi:10.1112/plms/s3-55.1.59 |
| [17] | Hong, MC, Heat flow for the Yang-Mills-Higgs field and the Hermitian Yang-Mills-Higgs metric, Ann. Glob. Anal. Geom., 20, 23-46 (2001) · Zbl 0988.53009 · doi:10.1023/A:1010688223177 |
| [18] | Jacob, A., Existence of approximate Hermitian-Einstein structures on semi-stable bundles, Asian J. Math., 18, 859-883 (2014) · Zbl 1315.53079 · doi:10.4310/AJM.2014.v18.n5.a5 |
| [19] | Jacob, A.; Walpuski, T., Hermitian Yang-Mills metrics on reflexive sheaves over asymptotically cylindrical Kähler manifolds, Commun. PDE, 101, 1566-1598 (2018) · Zbl 1516.53029 · doi:10.1080/03605302.2018.1517792 |
| [20] | Kobayashi, S., Curvature and stability of vector bundles, Proc. Jpn. Acad. Ser. A, 58, 158-162 (1982) · Zbl 0538.32021 · doi:10.3792/pjaa.58.158 |
| [21] | Kobayashi, S., Differential Geometry of Complex Vector Bundles (1987), Princeton: Princeton University Press, Princeton · Zbl 0708.53002 |
| [22] | Li, J., Yau, S.T.: Hermitian-Yang-Mills connection on non-Kähler manifolds. In: Mathematical Aspects of String Theory, pp. 560-573. World Scientific, New York (1987) · Zbl 0664.53011 |
| [23] | Li, JY; Zhang, X., Existence of approximate Hermitian-Einstein structures on semi-stable Higgs bundles, Calc. Var., 52, 783-795 (2015) · Zbl 1311.53022 · doi:10.1007/s00526-014-0733-x |
| [24] | Li, JY; Zhang, CJ; Zhang, X., Semi-stable Higgs sheaves and Bogomolov type inequality, Calc. Var., 56, 1-33 (2017) · Zbl 1379.35074 · doi:10.1007/s00526-016-1094-4 |
| [25] | Li, JY; Zhang, CJ; Zhang, X., The limit of the Hermitian-Yang-Mills flow on reflexive sheaves, Adv. Math., 325, 165-214 (2018) · Zbl 1381.53046 · doi:10.1016/j.aim.2017.11.029 |
| [26] | Li, JY; Zhang, CJ; Zhang, X., A note on curvature estimate of the Hermitian-Yang-Mills flow, Commun. Math. Stat., 6, 319-358 (2018) · Zbl 1428.53031 · doi:10.1007/s40304-018-0135-z |
| [27] | Li, Z.; Zhang, X., Dirichlet problem for Hermitian Yang-Mills-Higgs equations over Hermitian manifolds, J. Math. Anal. Appl., 310, 68-80 (2005) · Zbl 1082.58022 · doi:10.1016/j.jmaa.2005.01.033 |
| [28] | Lübke, M., Stability of Einstein-Hermitian vector bundles, Manuscr. Math., 42, 245-257 (1983) · Zbl 0558.53037 · doi:10.1007/BF01169586 |
| [29] | Lübke, M.; Teleman, A., The Kobayashi-Hitchin correspondence (1995), Singapore: World Scientific Publishing, Singapore · Zbl 0849.32020 |
| [30] | Lübke, M.; Teleman, A., The Universal Kobayashi-Hitchin Correspondence on Hermitian Manifolds (2006), Providence: Mem. AMS, Providence · Zbl 1103.53014 |
| [31] | Mochizuki, T.: Kobayashi-Hitchin correspondence for tame harmonic bundles and an application. Astérisque Soc. Math. France 309, (2006) · Zbl 1119.14001 |
| [32] | Narasimhan, MS; Seshadri, CS, Stable and unitary vector bundles on a compact Riemann surface, Ann. Math., 82, 540-567 (1965) · Zbl 0171.04803 · doi:10.2307/1970710 |
| [33] | Nie, Y.; Zhang, X., Semistable Higgs bundles over compact Gauduchon manifolds, J. Geom. Anal., 28, 627-642 (2018) · Zbl 1391.32026 · doi:10.1007/s12220-017-9835-y |
| [34] | Simpson, CT, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Am. Math. Soc., 1, 867-918 (1988) · Zbl 0669.58008 · doi:10.1090/S0894-0347-1988-0944577-9 |
| [35] | Taylor, ME, Partial Differential Equations I (Applied Mathematical Sciences) (2011), New York: Springer, New York · Zbl 1206.35004 |
| [36] | Uhlenbeck, KK; Yau, ST, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Commun. Pure Appl. Math., 39S, S257-S293 (1986) · Zbl 0615.58045 · doi:10.1002/cpa.3160390714 |
| [37] | Zhang, C., Zhang, P., Zhang, X.: Higgs bundles over non-compact Gauduchon manifolds. arXiv:1804.08994 (2018) · Zbl 1391.32026 |
| [38] | Zhang, X., Hermitian-Einstein metrics on holomorphic vector bundles over Hermitian manifolds, J. Geom. Phys., 53, 315-335 (2005) · Zbl 1075.58010 · doi:10.1016/j.geomphys.2004.07.002 |
| [39] | Zhang, X., Hermitian Yang-Mills-Higgs metrics on complete Kähler manifolds, Can. J. Math., 57, 871-896 (2005) · Zbl 1083.58014 · doi:10.4153/CJM-2005-034-3 |
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