Weitzenb\"ock-Bochner-Kodaira formulas with quadratic curvature terms

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