Previous |  Up |  Next


Kähler manifold; complex vector bundle; holomorphic connection; Yang-Mills bar gradient flow
In this note we introduce a Yang-Mills bar equation on complex vector bundles $E$ provided with a Hermitian metric over compact Hermitian manifolds. According to the Koszul-Malgrange criterion any holomorphic structure on $E$ can be seen as a solution to this equation. We show the existence of a non-trivial solution to this equation over compact Kähler manifolds as well as a short time existence of a related negative Yang-Mills bar gradient flow. We also show a rigidity of holomorphic connections among a class of Yang-Mills bar connections over compact Käahler manifolds of positive Ricci curvature.
[1] Bourguignon, J. P., Lawson, H. B.: Stability phenomena for Yang-Mills fields. Commun. Math. Phys. 79 (1981), 189–230. DOI 10.1007/BF01942061 | MR 0612248
[2] Donaldson, S. K., Kronheimer, P. B.: The geometry of 4-manifolds. Clarendon Press, Oxford, 1990. MR 1079726
[3] Griffiths, P., Harris, J.: Principles of algebraic geometry. 2nd ed., Wiley Classics Library, New York, 1994. MR 1288523 | Zbl 0836.14001
[4] Hamilton, R.: Three manifold with positive Ricci curvature. J. Differential Geom. 17 (2) (1982), 255–306. MR 0664497
[5] Kobayashi, S.: Differential geometry of complex vector bundles. Iwanami Shoten Publishers and Princeton University Press, 1987. MR 0909698 | Zbl 0708.53002
[6] Koszul, J. L., Malgrange, B.: Sur certaines structures fibres complexes. Arch. Math. (Basel) 9 (1958), 102–109. DOI 10.1007/BF02287068 | MR 0131882
Partner of
EuDML logo