There's a connection (the Chern connection) on the Tangent Bundle of a Kahler Manifold which is compatible with both the hermitan metric, and the holomorphic structure. In general, I guess there's no reason for the connection to be holomorphic (i.e. there's no reason for holomorphic sections to go to holomorphic sections and one possible obstruction is that the tangent bundle needn't be flat?).
However, if my Kahler Manifold is actually a Hermitan symmetric domain, then that particular obstruction doesn't mess things up because the tangent bundle is trivial. My question is: is the Chern connection on a Hermitian symmetric domain holomorphic?