Local vanishing for toric varieties

Let \(X\) be a toric variety. We establish vanishing (and non-vanishing) results for the sheaves \(R^if_*\Omega^p_{\tilde X}(\log E)\), where \(f: \tilde{X} \to X\) is a strong log resolution of singularities with reduced exceptional divisor \(E\). These extend the local vanishing theorem for toric varieties in [MOP20]. Our consideration of these sheaves is motivated by the notion of \(k\)-rational singularities introduced by Friedman and Laza [FL22b]. In particular, our results lead to criteria for toric varieties to have \(k\)-rational singularities, as defined in [SVV23].