Holomorphic sectional curvature and the Calabi operator on Kähler C-spaces

Abstract

We study holomorphic sectional curvature and the Calabi curvature operator on compact Kähler manifolds. First we show that every homogeneous Kähler metric on a generalized flag manifold has positive holomorphic sectional curvature. For the standard homogeneous Kähler-Einstein metric with Einstein constant 2, it is shown that the spectrum of the Calabi operator lies in $(-2,2]$. We determine its sharp upper bound in terms of the anticanonical degrees and characterize the 2-eigenspace. The spectral bounds imply nonnegative cross quadratic bisectional curvature and positive dual cross quadratic bisectional curvature, proving conjectures of Ni and Zheng. We also show that these conditions do not characterize homogeneity: we construct a nonhomogeneous Kähler-Einstein 13-fold whose Calabi spectrum lies strictly inside $(-2,2)$ after the same normalization, as well as nonhomogeneous examples in dimensions 4 and 5. The proofs involve an integral identity for holomorphic sectional curvature, projective embeddings, root-space estimates, and explicit metrics on projective bundles.