Abstract
In this paper, we study numerically flat holomorphic vector bundles over a compact non-Kähler manifold X which admits an Astheno–Kähler metric. We prove that numerically flatness is equivalent to approximate Hermitian flatness and the existence of a filtration by sub-bundles whose quotients are Hermitian flat. This gives an affirmative answer to the question proposed by Demailly, Peternell and Schneider.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12141104
Award Identifier / Grant number: 11721101
Award Identifier / Grant number: 11625106
Award Identifier / Grant number: 12001548
Award Identifier / Grant number: 11571332
Funding statement: The research was supported by the National Key R and D Program of China 2020YFA0713100. All authors were supported in part by NSF in China, No. 12141104, 11721101, 11625106, 12001548 and 11571332. The first proof of this paper was completed when the first author and the third author were at School of Mathematical Sciences, University of Science and Technology of China, and the second author was at School of Mathematical Sciences, Xiamen University.
References
[1] A. Aeppli, On the cohomology structure of Stein manifolds, Proceedings of the Conference on Complex Analysis, Springer, Berlin (1965), 58–70. 10.1007/978-3-642-48016-4_7Search in Google Scholar
[2] I. Biswas and V. P. Pingali, A characterization of finite vector bundles on Gauduchon astheno-Kähler manifolds, Épijournal Géom. Algébrique 2 (2018), Article ID 6. 10.46298/epiga.2018.volume2.4209Search in Google Scholar
[3] F. A. Bogomolov, Holomorphic tensors and vector bundles on projective manifolds, Math. USSR Izv. 13 (1978), no. 3, 499–555. 10.1070/IM1979v013n03ABEH002076Search in Google Scholar
[4] R. Bott and S. S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math. 114 (1965), 71–112. 10.1007/BF02391818Search in Google Scholar
[5] L. Bruasse, Harder–Narasimhan filtration on non Kähler manifolds, Internat. J. Math. 12 (2001), no. 5, 579–594. 10.1142/S0129167X01000897Search in Google Scholar
[6] J.-P. Demailly, Singular Hermitian metrics on positive line bundles, Complex algebraic varieties, Lecture Notes in Math. 1507, Springer, Berlin (1992), 87–104. 10.1007/BFb0094512Search in Google Scholar
[7] J.-P. Demailly, T. Peternell and M. Schneider, Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994), no. 2, 295–345. Search in Google Scholar
[8] S. Diverio, Segre forms and Kobayashi–Lübke inequality, Math. Z. 283 (2016), no. 3–4, 1033–1047. 10.1007/s00209-016-1632-ySearch in Google Scholar
[9] S. K. Donaldson, Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. (3) 50 (1985), no. 1, 1–26. 10.1112/plms/s3-50.1.1Search in Google Scholar
[10] S. K. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987), no. 1, 231–247. 10.1215/S0012-7094-87-05414-7Search in Google Scholar
[11] A. Fino, G. Grantcharov and L. Vezzoni, Astheno-Kähler and balanced structures on fibrations, Int. Math. Res. Not. IMRN 2019 (2019), no. 22, 7093–7117. 10.1093/imrn/rnx337Search in Google Scholar
[12] P. Gauduchon, La 1-forme de torsion d’une variété hermitienne compacte, Math. Ann. 267 (1984), no. 4, 495–518. 10.1007/BF01455968Search in Google Scholar
[13] D. Guler, On Segre forms of positive vector bundles, Canad. Math. Bull. 55 (2012), no. 1, 108–113. 10.4153/CMB-2011-100-6Search in Google Scholar
[14] A. Jacob, Existence of approximate Hermitian–Einstein structures on semi-stable bundles, Asian J. Math. 18 (2014), no. 5, 859–883. 10.4310/AJM.2014.v18.n5.a5Search in Google Scholar
[15] J. Jost and S.-T. Yau, A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry, Acta Math. 170 (1993), no. 2, 221–254. 10.1007/BF02392786Search in Google Scholar
[16] S. Kobayashi, Differential geometry of complex vector bundles, Publ. Math. Soc. Japan 15, Princeton University, Princeton 1987. 10.1515/9781400858682Search in Google Scholar
[17] A. Latorre and L. Ugarte, On non-Kähler compact complex manifolds with balanced and astheno-Kähler metrics, C. R. Math. Acad. Sci. Paris 355 (2017), no. 1, 90–93. 10.1016/j.crma.2016.11.004Search in Google Scholar
[18] J. Li and S.-T. Yau, Hermitian–Yang–Mills connection on non-Kähler manifolds, Mathematical aspects of string theory, Adv. Ser. Math. Phys. 1, World Scientific, Singapore (1987), 560–573. 10.1142/9789812798411_0027Search in Google Scholar
[19] J. Li, S.-T. Yau and F. Zheng, On projectively flat Hermitian manifolds, Comm. Anal. Geom. 2 (1994), no. 1, 103–109. 10.4310/CAG.1994.v2.n1.a6Search in Google Scholar
[20] J. Li and X. Zhang, Existence of approximate Hermitian–Einstein structures on semi-stable Higgs bundles, Calc. Var. Partial Differential Equations 52 (2015), no. 3–4, 783–795. 10.1007/s00526-014-0733-xSearch in Google Scholar
[21] P. Li, Geometric analysis, Cambridge Stud. Adv. Math.134, Cambridge University, Cambridge 2012. 10.1017/CBO9781139105798Search in Google Scholar
[22] M. Lübke, Chernklassen von Hermite–Einstein-Vektorbündeln, Math. Ann. 260 (1982), no. 1, 133–141. 10.1007/BF01475761Search in Google Scholar
[23] M. Lübke and A. Teleman, The Kobayashi–Hitchin correspondence, World Scientific, River Edge 1995. 10.1142/2660Search in Google Scholar
[24] J. McNamara and Y. Zhao, Limiting behavior of Donaldson’s heat flow on non-Kähler surfaces, preprint (2014), https://arxiv.org/pdf/1403.8037.pdf. Search in Google Scholar
[25] Y. Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977), 225–237. 10.1007/BF01389789Search in Google Scholar
[26] M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540–567. 10.2307/1970710Search in Google Scholar
[27] Y. Nie and X. Zhang, Semistable Higgs bundles over compact Gauduchon manifolds, J. Geom. Anal. 28 (2018), no. 1, 627–642. 10.1007/s12220-017-9835-ySearch in Google Scholar
[28] Y. Nie and X. Zhang, The limiting behaviour of the Hermitian–Yang–Mills flow over compact non-Kähler manifolds, Sci. China Math. 63 (2020), no. 7, 1369–1390. 10.1007/s11425-018-9411-7Search in Google Scholar
[29] K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian–Yang–Mills connections in stable vector bundles, Commun. Pure Appl. Math. 39 (1986), S257–S293. 10.1002/cpa.3160390714Search in Google Scholar
[30] S. T. Yau, Calabi’s conjecture and some new results in algebraic geometry, Proc. Natl. Acad. Sci. USA 74 (1977), no. 5, 1798–1799. 10.1073/pnas.74.5.1798Search in Google Scholar PubMed PubMed Central
[31] X. Zhang, Hermitian–Einstein metrics on holomorphic vector bundles over Hermitian manifolds, J. Geom. Phys. 53 (2005), no. 3, 315–335. 10.1016/j.geomphys.2004.07.002Search in Google Scholar
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