The purpose of this article is to initiate the investigation of the curvature operator of the second kind on Kähler manifolds. The main result asserts that a closed Kähler surface with six-positive curvature operator of the second kind is biholomorphic to . It is also shown that a closed non-flat Kähler surface with six-nonnegative curvature operator of the second kind is either biholomorphic to or isometric to x .