K\"ahler surfaces with six-positive curvature operator of the second kind

TL;DR

This paper investigates the curvature operator of the second kind on Kähler surfaces, establishing that certain positivity conditions imply the surface is biholomorphic to complex projective space or a product of spheres.

Contribution

It classifies closed Kähler surfaces with six-positive or nonnegative curvature operator of the second kind, linking curvature conditions to complex geometric structures.

Findings

Kähler surface with six-positive curvature operator is biholomorphic to ^2.

Non-flat Kähler surface with six-nonnegative curvature operator is either ^2 or  imes .

Results connect curvature operator positivity to complex geometric classification.

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 $\mathbb{S}^2 \times \mathbb{S}^2$.