Semistable Twisted Holomorphic Chains on Non-Compact Kähler Manifolds

Abstract

In this paper, the author proves a generalized Donaldson-Uhlenbeck-Yau theorem for twisted holomorphic chain on a non-compact Kähler manifold. As an application, the author obtains a Bogomolov type Chern numbers inequality for semistable twisted holomorphic chain.

Keywords

Hermitian metric / Twisted holomorphic chain / Chern numbers inequality

References

[1] Álvarez-Cónsul L, García-Prada O. Dimensional reduction. SL(2, C)-equivariant bundles and stable holomorphic chains, Int. J. Math., 2001, 2: 159-201

[2] Álvarez-Cónsul L, García-Prada O. Hitchin-Kobayashi correpondence, quivers, and vortices. Comm. Math. Phys., 2003, 238: 1-33

[3] Álvarez-Cónsul L, García-Prada O. Dimensional reduction and quiver bundles. J. Reine Angew. Math., 2003, 556: 1-46

[4] Bando S, Siu Y T. Stable sheaves and Einstein-Hermitian metrics, Geometry and Analysis on Complex Manifolds, 1994, River Edge: World Scientific Publishing 39-50

[5] Biquard O. On parabolic bundles over a complex surface. J. London Math. Soc., 1996, 53(2): 302-316

[6] Bradlow S B. Vortices in holomorphic line bundles over closed Kähler manifolds. Commun. Math. Phys., 1990, 135: 1-17

[7] Bradlow S B, García-Prada O. Stable triples, equivariant bundles and dimensional reduction. Math. Ann., 1996, 304: 225-252

[8] Buchdahl N P. Hermitian-Einstein connections and stable vector bundles over compact complex surfaces. Math. Ann., 1988, 280(4): 625-648

[9] Cardona S A H. Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles, I: generalities and the one-dimensional case. Ann. Global Anal. Geom., 2012, 42(3): 349-370

[10] Cardona S A H. Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles, II: Higgs sheaves and admissible structure. Ann. Global Anal. Geom., 2013, 44(4): 455-469

[11] De Bartolomeis P, Tian G. Stability of complex vector bundles. J. Diff. Geom., 1996, 43: 232-275

[12] 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

[13] Donaldson S K. Infinite determinats, stable bundles and curvature. Duke Math. J., 1987, 54: 231-247

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

[15] Hitchin N J. The self-duality equations on a Riemann surface. Proc. London Math. Soc., 1987, 55: 59-126

[16] Jacob A. Existence of approximate Hermitian-Einstein structures on semi-stable bundles. Asian J. Math., 2014, 18(5): 859-883

[17] Kobayashi S. Differential geometry of complex vector bundles. Publications of the Mathematical Society of Japan, 1987, Princeton, NJ: Princeton University Press 15

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

[19] Li J Y. Hermitian-Einstein metrics and Chern number inequalities on parabolic stable bundles over Kähler manifolds. Comm. Anal. Geom., 2000, 8(3): 445-475

[20] Li J Y, Narasimhan M S. Hermitian-Einstein metrics on parabolic stable bundles. Acta Math. Sin., (Engl. Ser.), 1999, 15(1): 93-114

[21] Li J Y, Zhang X. Existence of approximate Hermitian-Einstein structures on semi-stable Higgs bundles. Calc. Var. Partial Differential Equations, 2015, 52(3–4): 783-795

[22] Mundet i Riera I. A Hitchin-Kobayashi correspondence for Kähler fibrations. J. Reine Angew. Math., 2000, 528: 41-80

[23] Narasimhan M S, Seshadri C S. Stable and unitary vector bundles on compact Riemann surfaces. Ann. of Math., 1965, 82: 540-567

[24] Simpson C T. Constructing variations of Hodge structures using Yang-Mills connections and applications to uniformization. J. Amer. Math. Soc., 1988, 1: 867-918

[25] 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

[26] Wang Y, Zhang X. Twisted holomorphic chains and vortex equations over non-compact Kähler manifolds. J. Math. Anal. Appl., 2011, 373(1): 179-202