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

Abstract: The purpose of this article is to initiate the investigation of the curvature operator of the second kind on K\"ahler manifolds. The main result asserts that a closed K\"ahler surface with six-positive curvature operator of the second kind is biholomorphic to $\mathbb{CP}^2$. It is also shown that a closed non-flat K\"ahler surface with six-nonnegative curvature operator of the second kind is either biholomorphic to $\mathbb{CP}^2$ or isometric to $\mathbb{S}^2 \times \mathbb{S}^2$.