Hermitian-Yang-Mills connection on non-Kähler manifolds.

*(English)*Zbl 0664.53011
Mathematical aspects of string theory, Proc. Conf., San Diego/Calif. 1986, Adv. Ser. Math. Phys. 1, 560-573 (1987).

[For the entire collection see Zbl 0651.00012.]

The theorem of Donaldson and Uhlenbeck-Yau relating Hermitian-Yang-Mills connections to stable holomorphic bundles [S. K. Donaldson, Duke Math. J. 54, 231-247 (1987; Zbl 0627.53052); K. Uhlenbeck and S. T. Yau, Commun. Pure. Appl. Math. 39, S 257-S 293 (1986; Zbl 0615.58045)] is suspectible to generalizations in various directions. One needs on the one hand a gauge-theoretic differential equation of Yang-Mills type and on the other an algebro-geometric notion of stability. In this paper, the authors refine their original method to cover the case of non-Kähler Hermitian complex manifolds. The 2-dimensional case of this result, proved by a “direct” method was in fact done independently by N. P. Buchdahl [Math. Ann. 280, 625-648 (1988; Zbl 0617.32044)] and has already been used by Braam and Hurtubise to study instantons on Hopf surfaces. The authors have in mind a motivation arising from the demands of physicists studying string theory.

The theorem of Donaldson and Uhlenbeck-Yau relating Hermitian-Yang-Mills connections to stable holomorphic bundles [S. K. Donaldson, Duke Math. J. 54, 231-247 (1987; Zbl 0627.53052); K. Uhlenbeck and S. T. Yau, Commun. Pure. Appl. Math. 39, S 257-S 293 (1986; Zbl 0615.58045)] is suspectible to generalizations in various directions. One needs on the one hand a gauge-theoretic differential equation of Yang-Mills type and on the other an algebro-geometric notion of stability. In this paper, the authors refine their original method to cover the case of non-Kähler Hermitian complex manifolds. The 2-dimensional case of this result, proved by a “direct” method was in fact done independently by N. P. Buchdahl [Math. Ann. 280, 625-648 (1988; Zbl 0617.32044)] and has already been used by Braam and Hurtubise to study instantons on Hopf surfaces. The authors have in mind a motivation arising from the demands of physicists studying string theory.

Reviewer: N.Hitchin

##### MSC:

53C05 | Connections, general theory |

32L05 | Holomorphic bundles and generalizations |

35Q99 | Partial differential equations of mathematical physics and other areas of application |

81T08 | Constructive quantum field theory |