On the anti-self-duality of the Yang-Mills connection over higher dimensional Kaehlerian manifold (Q756712)

Let M be a compact Kaehler manifold of complex dimension \(n\geq 2\), with a Kaehler form \(\Phi\). Using the operator \(\#=*^{-1}L^{(n-2)}/(n-2)!,\) where L denotes the multiplication by \(\Phi\) introduced by \textit{H. J. Kim} [Curvatures and holomorphic vector bundle (Ph. D. thesis Berkeley)] the author defines the self-duality and anti-self-duality of a connection A on a principal fibre bundle over M. The following two main theorems are proved: A. Let M be a complex n-dimensional compact Kaehler manifold for which the sum of any two distinct eigenvalues of the Ricci tensor is positive. Let A be an irreducible Yang-Mills connection. If \([F^{2,0}\wedge F^{0,2}]=0,\) then A is anti-self-dual. B. Let M be a compact Kaehler manifold with the same condition as in Theorem A. If \([F^{2,0}\wedge F^{1,1}]=0\) and \([F^ 0\wedge F^{2,0}]=0,\) then A is anti-self-dual.