Weitzenböck-Bochner-Kodaira formulas with quadratic curvature terms

Abstract

In this paper we establish new Bochner-Kodaira formulas with quadratic curvature terms on compact Kähler manifolds: for any $\eta \in {\Omega }^{p,q}(M)$, $$⟨{\Delta }_{\overline{\partial }}\eta ,\eta ⟩=⟨{\Delta }_{{\overline{\partial }}_F}\eta ,\eta ⟩+\frac{1}{4}⟨\left(\mathcal{R}\otimes {\mathrm{Id}}_{{\Lambda }^{p+1,q-1}{T}^{\ast }M}\right)({\mathbb{T}}_{\eta }),{\mathbb{T}}_{\eta }⟩.$$ This linearized curvature term yields new vanishing theorems and provides estimates for Hodge numbers under exceptionally weak curvature conditions. Furthermore, we derive Weitzenböck formulas with quadratic curvature terms on both Riemannian and Kähler manifolds.