A Note on Bogomolov-Type Inequality for Semi-stable Parabolic Higgs Bundles

Abstract

In this paper, we prove a Chern number inequality for Higgs bundles over some Kähler manifolds. As an application, we get the Bogomolov inequality for semi-stable parabolic Higgs bundles over smooth projective varieties.

Keywords

Parabolic Higgs bundle / Semi-stable / Bogomolov inequality

References

[1] Bando, S., Siu, Y.T.: Stable Sheaves and Einstein–Hermitian Metrics. Geometry and Analysis on Complex Manifolds, pp. 39–50. World Sci. Publ., River Edge (1994)

[2] Biquard O. Fibrés paraboliques stables et connexions singuli$\grave{\text{e}}$res plates. Bull. Soc. Math. France. 1991, 119 2 231-257

[3] Biquard O. Sur les fibrés paraboliques sur une surface complexe. J. Lond. Math. Soc. (2). 1996, 53 2 302-316

[4] Biswas I. Parabolic bundles as orbifold bundles. Duke Math. J.. 1997, 88 2 305-325

[5] Bogomolov FA. Holomorphic tensors and vector bundles on projective varieties. Math. USSR-Izv.. 1979, 13 3 499-555

[6] Bradlow SB. Special metrics and stability for holomorphic bundles with global sections. J. Differ. Geom.. 1991, 33 1 169-213

[7] Donaldson SK. Anti-self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. (3). 1985, 50 1 1-26

[8] Donaldson SK. Infinite determinants, stable bundles and curvature. Duke Math. J.. 1987, 54 1 231-247

[9] Garcia-Prada O. Dimensional reduction of stable bundles, vortices and stable pairs. Int. J. Math.. 1994, 5 1 1-52

[10] Hitchin NJ. The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. (3). 1987, 55 1 59-126

[11] Kobayashi S. Differential Geometry of Complex Vector Bundles. 1987 Princeton, NJ: Princeton University Press

[12] Kobayashi S. Curvature and stability of vector bundles. Proc. Jpn. Acad. Ser. A Math. Sci.. 1982, 58 4 158-162

[13] Li JY, Narasimhan MS. Hermitian–Einstein metrics on parabolic stable bundles. Acta Math. Sin. (Engl. Ser.). 1999, 15 1 93-114

[14] Li JY. Hermitian–Einstein metrics and Chern number inequalities on parabolic stable bundles over Kähler manifolds. Commun. Anal. Geom.. 2000, 8 3 445-475

[15] Li JY, Zhang X. Existence of approximate Hermitian–Einstein structures on semi-stable Higgs bundles. Calc. Var. Partial Differ. Equ.. 2015, 52 3–4 783-795

[16] Li JY, Zhang CJ, Zhang X. Semi-stable Higgs sheaves and Bogomolov type inequality. Calc. Var. Partial Differ. Equ.. 2017, 56 3 33

[17] Li, J., Yau, S.T.: Hermitian–Yang–Mills Connection on non-Kähler Manifolds. Mathematical Aspects of String Theory (San Diego, Calif., 1986), 560–573, Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore (1987)

[18] Lübke M. Stability of Einstein–Hermitian vector bundles. Manuscr. Math.. 1983, 42 2–3 245-257

[19] Mochizuki, T.: Kobayashi-Hitchin correspondence for tame harmonic bundles and an application. Astérisque, 309. Soc. Math. France, Paris (2006)

[20] Mehta VB, Seshadri CS. Moduli of vector bundles on curves with parabolic structures. Math. Ann.. 1980, 248 3 205-239

[21] Maruyama M, Yokogawa K. Moduli of parabolic stable sheaves. Math. Ann.. 1992, 293 1 77-99

[22] Nie YC, Zhang X. Semistable Higgs bundles over compact Gauduchon manifolds. J. Geom. Anal.. 2018, 28 1 627-642

[23] Simpson C. Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization. J. Am. Math. Soc.. 1988, 1 4 867-918

[24] Simpson C. Harmonic bundles on noncompact curves. J. Am. Math. Soc.. 1990, 3 3 713-770

[25] Uhlenbeck K, Yau ST. On the existence of Hermitian–Yang–Mills connections in stable vector bundles. Commun. Pure Appl. Math.. 1986, 39 S257-S293

[26] Zhang CJ. Semi-stability for holomorphic bundles with global sections. J. Partial Differ. Equ.. 2016, 29 3 204-217

[27] Zhang, C., Zhang, P., Zhang, X.: Higgs bundles over non-compact Gauduchon manifolds. arXiv:1804.08994