Holomorphic tangent bundle
Since the coordinate functions are holomorphic, so are any derivatives of them, and so the transition functions of the holomorphic tangent bundle are also holomorphic. Thus the holomorphic tangent bundle is a genuine holomorphic vector bundle. Similarly the holomorphic cotangent bundle is a genuine holomorphic vector bundle, with transition functions given by the inverse of the Jacobian matrix. Notice that the anti-holomorphic tangent and cotangent bundles do not have holomorphic transition functions, but anti-holomorphic ones.