K\"ahler manifolds and the curvature operator of the second kind

TL;DR

This paper studies the curvature operator of the second kind on K"ahler manifolds, establishing conditions under which the manifold has constant holomorphic sectional curvature or is biholomorphic to complex projective space.

Contribution

It provides new algebraic and pointwise conditions linking the curvature operator of the second kind to geometric properties of K"ahler manifolds, including biholomorphic equivalence to complex projective space.

Findings

Manifolds with certain nonnegative curvature operator have constant holomorphic sectional curvature.

Positive curvature operator conditions imply the manifold is biholomorphic to P^m.

Specific positivity conditions lead to positive orthogonal Ricci curvature.

Abstract

This article aims to investigate the curvature operator of the second kind on K\"ahler manifolds. The first result states that an $m$-dimensional K\"ahler manifold with $\frac{3}{2}(m^2-1)$-nonnegative (respectively, $\frac{3}{2}(m^2-1)$-nonpositive) curvature operator of the second kind must have constant nonnegative (respectively, nonpositive) holomorphic sectional curvature. The second result asserts that a closed $m$-dimensional K\"ahler manifold with $\left(\frac{3m^3-m+2}{2m}\right)$-positive curvature operator of the second kind has positive orthogonal bisectional curvature, thus being biholomorphic to $\mathbb{CP}^m$. We also prove that $\left(\frac{3m^3+2m^2-3m-2}{2m}\right)$-positive curvature operator of the second kind implies positive orthogonal Ricci curvature. Our approach is pointwise and algebraic.