Vanishing theorems for Hodge numbers and the Calabi curvature operator

TL;DR

The paper advances vanishing and rigidity results for Hodge numbers on compact Kähler manifolds by introducing and exploiting the Calabi curvature operator $\mathcal{C}$. Through a novel calculus of curvature tensors and a weight principle that ties Calabi eigenvalues to the Lichnerowicz curvature term, it proves that $\frac{n}{2}$-positive $\mathcal{C}$ forces the manifold to have the rational cohomology of $\mathbf{P}^n$, with a sharp example given by the complex quadric in even dimensions. It further classifies manifolds with $\frac{n}{2}$-nonnegative $\mathcal{C}$, and, in the Kähler--Einstein case, refines vanishing results via the Kähler curvature operator restricted to $\mathfrak{su}(n)$, yielding precise conditions under which primitive harmonic forms are parallel or vanish. Collectively, these results unify curvature-driven vanishing phenomena, connect holonomy to topological rigidity, and extend Bochner-type techniques in Kähler geometry with explicit quantitative bounds.

Abstract

It is shown that a compact $n$-dimensional Kähler manifold with $\frac{n}{2}$-positive Calabi curvature operator has the rational cohomology of complex projective space. For even $n,$ this is sharp in the sense that the complex quadric with its symmetric metric has $\frac{n}{2}$-nonnegative Calabi curvature operator, yet $b_n =2.$ Furthermore, the compact Kähler manifolds with an $\frac{n}{2}$-nonnegative Calabi curvature operator are classified. In addition, the previously known results for the Kähler curvature operator are improved when the metric is Kähler--Einstein.

Theorems & Definitions (47)

• Theorem A
• Theorem B
• Theorem C
• Remark
• Theorem D
• Lemma 2.1
• proof
• Remark 2.2
• Remark 2.3
• Definition 2.4