Hodge--Tate splitting and Akizuki--Nakano vanishing in positive characteristic
Abstract
We introduce the notion of Hodge--Tate splitting for schemes in positive characteristic. For a smooth variety X over a perfect field k of characteristic p > 0, we say that X is Hodge--Tate split if the natural morphism 𝒪X → F*Ω•X/k induced by the absolute Frobenius admits a splitting in 𝒟qcoh(X). We prove that this condition is equivalent to the existence of a decomposition F*Ω•X/k ≃ ⨁i=0dim XΩiX/k[−i] of its de Rham complex. Consequently, for smooth projective varieties, Hodge--Tate splitting implies Akizuki--Nakano vanishing and the E1-degeneration of the Hodge-to-de Rham spectral sequence. Furthermore, we establish criteria and permanence properties for Hodge--Tate splitting and use them to construct many new examples of varieties whose de Rham complexes decompose. These include blow-ups of quasi-F-split varieties along strata of simple normal crossings divisors, complete intersections in toric varieties, and linearly reductive quotients of Hodge--Tate split varieties. Among these examples, we obtain smooth projective varieties whose Hodge-to-de Rham spectral sequences degenerate at E1, whereas their Hochschild--Kostant--Rosenberg spectral sequences do not degenerate. As an application in mixed characteristic, we prove an Akizuki--Nakano-type vanishing theorem for smooth projective globally +-regular varieties over the Witt ring of a perfect field.
Related
- A Thullen-type Extension Theorem for Nakano Semi-Positive Holomorphic Vector Bundles
- Nakano-positive determinants outside the Hodge-Riemann cone
- Characteristic drops for high-order vanishing on the hypercube
- Gromov's Simplicial Volume Vanishing Conjecture for Positive Scalar Curvature
- Abundance for threefolds in positive characteristic
BibTeX
@misc{ishizuka2026hodgetate,
title = {{Hodge--Tate splitting and Akizuki--Nakano vanishing in positive characteristic}},
author = {Ryo Ishizuka and Shou Yoshikawa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00567},
eprint = {2609.00567},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00567},
}
RIS
TY - UNPB
TI - Hodge--Tate splitting and Akizuki--Nakano vanishing in positive characteristic
AU - Ryo Ishizuka
AU - Shou Yoshikawa
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00567
UR - https://arxiv.org/abs/2609.00567
AB - We introduce the notion of Hodge--Tate splitting for schemes in positive characteristic. For a smooth variety X over a perfect field k of characteristic p > 0, we say that X is Hodge--Tate split if the natural morphism 𝒪_X → F_*Ω^•_X/k induced by the absolute Frobenius admits a splitting in 𝒟_qcoh(X). We prove that this condition is equivalent to the existence of a decomposition F_*Ω^•_X/k ≃ ⨁_i=0^dim XΩ^i_X/k[−i] of its de Rham complex. Consequently, for smooth projective varieties, Hodge--Tate splitting implies Akizuki--Nakano vanishing and the E_1-degeneration of the Hodge-to-de Rham spectral sequence. Furthermore, we establish criteria and permanence properties for Hodge--Tate splitting and use them to construct many new examples of varieties whose de Rham complexes decompose. These include blow-ups of quasi-F-split varieties along strata of simple normal crossings divisors, complete intersections in toric varieties, and linearly reductive quotients of Hodge--Tate split varieties. Among these examples, we obtain smooth projective varieties whose Hodge-to-de Rham spectral sequences degenerate at E_1, whereas their Hochschild--Kostant--Rosenberg spectral sequences do not degenerate. As an application in mixed characteristic, we prove an Akizuki--Nakano-type vanishing theorem for smooth projective globally +-regular varieties over the Witt ring of a perfect field.
ER -