Abstract
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 \(f: \tilde{X} \rightarrow X\) is a strong log resolution of singularities with reduced exceptional divisor E. These extend the local vanishing theorem for toric varieties in Mustaţă et al. (J. Inst. Math. Jussieu 19(3):801-819, 2020). Our consideration of these sheaves is motivated by the notion of k-rational singularities introduced by Friedman and Laza (Higher Du Bois and higher rational singularities, 2001). In particular, our results lead to criteria for toric varieties to have k-rational singularities, as defined in Shen et al. (On k-Du Bois and k-rational singularities, 2023).
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Acknowledgements
We would like to express our sincere gratitude to Mircea Mustaţă and Mihnea Popa for their constant support during the preparation of this paper.
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S.V. was partially supported by NSF grant DMS-2001132.
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