Vanishing for Hodge ideals on toric varieties

In this article we construct a Koszul-type resolution of the p-th exterior power of the sheaf of holomorphic differential forms on smooth toric varieties and use this to prove a Nadel-type vanishing theorem for Hodge ideals associated to effective $\mathbb{Q}$-divisors on smooth projective toric varieties. This extends earlier results of Musta\c{t}\u{a} and Popa.