Kähler surfaces with six-positive curvature operator of the second kind

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 ${\mathbb{C}\mathbb{P}}^2$. It is also shown that a closed non-flat Kähler surface with six-nonnegative curvature operator of the second kind is either biholomorphic to ${\mathbb{C}\mathbb{P}}^2$ or isometric to ${\mathbb{S}}^2\times {\mathbb{S}}^2$.