Hello, thanks for the reply. Perhaps I should try to explain what I meant by Kahler-Einstein metrics being "restrictive". Hermitian-Yang-Mills connection (also known as Hermite-Einstein metrics) can be thought of a generalization of Kahler-Einstein metrics with Levi-Civita connection. In that sense, Kahler-Einstein metrics are trivial. Note that $T^{1,0}S$ can admit an Hermite-Einstein metric without $S$ being Kahler-Einstein. This can be achieved by using a different hermitian structure on $T^{1,0}S$ rather than the natural one coming from $(S,J,g,\omega)$.