Yang-Mills bar connections over compact Kähler manifolds

Hông Vân Lê

Address: Mathematical Institute of ASCR Žitná 25, CZ-11567 Praha 1, Czech Republic

E-mail: hvle@math.cas.cz

Abstract: 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.

AMSclassification: primary 53C55; secondary 53C44, 58E99.

Keywords: Kähler manifold, complex vector bundle, holomorphic connection, Yang-Mills bar gradient flow.