发表机构
Institute of Science Tokyo（东京科学大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文定义正特征下的Hodge--Tate分裂，证明其与de Rham复形分解等价，建立相关准则并构造新簇例子，还将其应用于混合特征下的消失定理证明。
AI 中文摘要
我们引入正特征下概形的Hodge--Tate分裂概念。对于特征p>0的完美域k上的光滑簇X，若由绝对Frobenius诱导的自然态射$\udcFam{O}_X \to F_*\udcFam{\bullet}_{X/k}$在$\udcDf{\text{qcoh}}(X)$中存在一个分裂，则称X是Hodge--Tate分裂的。我们证明该条件等价于其de Rham复形存在分解$F_*\udcFam{\bullet}_{X/k} \backsimeq \bigoplus_{i=0}^{\text{dim}X}\udcFam{\bullet}_{X/k}[-i]$。因此，对于光滑射影簇，Hodge--Tate分裂蕴含Akizuki--Nakano消失定理及Hodge-to-de Rham谱序列的E₁退化。此外，我们建立了Hodge--Tate分裂的判定准则与稳定性性质，并利用它们构造了许多de Rham复形可分解的新簇例子，包括拟F-分裂簇沿简单正常相交除子层的胀开、环簇中的完全交，以及Hodge--Tate分裂簇的线性约化商。在这些例子中，我们得到了光滑射影簇，其Hodge-to-de Rham谱序列在E₁处退化，但Hochschild--Kostant--Rosenberg谱序列不退化。作为混合特征中的应用，我们证明了完美域的Witt环上的光滑射影全局+-正则簇的Akizuki--Nakano型消失定理。
英文摘要
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 $\mathcal{O}_X \to F_*Ω^\bullet_{X/k}$ induced by the absolute Frobenius admits a splitting in $\mathcal{D}_{\mathrm{qcoh}}(X)$. We prove that this condition is equivalent to the existence of a decomposition $F_*Ω^\bullet_{X/k} \simeq \bigoplus_{i=0}^{\dim X}Ω^i_{X/k}[-i]$ 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.
Comments76 pages; Comments are welcome
