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

## TL;DR

This paper introduces new Bochner-Kodaira formulas incorporating quadratic curvature terms on compact Kähler manifolds, leading to novel vanishing theorems and Hodge number estimates under weak curvature conditions.

## Contribution

It establishes new Weitzenböck formulas with quadratic curvature terms, extending classical results and enabling weaker curvature assumptions for vanishing theorems.

## Key findings

- Derived new vanishing theorems for Hodge numbers.
- Provided estimates for Hodge numbers under weak curvature conditions.
- Extended formulas to both Riemannian and Kähler manifolds.

## Abstract

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.