AI 中文总结
研究Hensel局部环上光滑射影曲线的约化群丛，证明在核条件或剩余域充分大时，特殊纤维的Zariski局部平凡性可提升到整体，并推广到射影齐性空间。
AI 中文摘要
设$R$为Hensel局部环，$\kappa$为$R$的剩余域，$C$为$R$上具有几何连通纤维的光滑射影曲线，$G$为$C$上具有等平凡根环面$\mathrm{rad}(G)$的约化群，$E\to C$为$G$-丛。我们证明，若中心同源$G^{\mathrm{sc}}\times_C \mathrm{rad}(G)\to G$的核在$C$上平展或$\kappa$充分大，则$E_\kappa\to C_\kappa$的Zariski局部平凡性蕴含$E\to C$的Zariski局部平凡性。我们还证明了该结果的平均形式（仅假设$\mathrm{rad}(G)$等平凡），以及对射影齐性空间在无$G$限制下的变体。作为推论，我们得到了Hensel离散赋值环上曲线函数域上丛的局部-整体原理（加强了Gille--Parimala--Suresh的工作）、Drinfeld--Simpson定理的Hensel版本，以及$C$的Brauer--Azumaya群的单射性结果（未涵盖于Colliot-Thélène--Ojanguren--Parimala的早期工作）。我们的证明是几何的，依赖于丛的紧化以及梳子光滑化技术的相对和算术版本，我们基于Kollár和Graber--Harris--Starr的工作详细发展了该技术。
英文摘要
Let $R$ be a Henselian local ring, let $κ$ be the residue field of $R$, let $C$ be a smooth projective curve over $R$ with geometrically connected fibers, let $G$ be a reductive $C$-group with isotrivial radical torus $\mathrm{rad}(G)$, and let $E\to C$ be a $G$-torsor. We show that, if either the kernel of the central isogeny $G^{\mathrm{sc}}\times_C \mathrm{rad}(G)\to G$ is étale over $C$ or $κ$ is large, the Zariski-local triviality of $E_κ\to C_κ$ implies the Zariski-local triviality of $E\to C$. We also prove an averaged form of this result, assuming only that $\mathrm{rad}(G)$ is isotrivial, as well as a variant for projective homogeneous spaces under no restrictions on $G$. As consequences, we obtain a local-global principle for torsors over function fields of curves over Henselian discrete valuation rings, strengthening work of Gille-Parimala-Suresh and a Henselian version of a theorem of Drinfeld-Simpson. Our proofs are geometric and rely on compactifications of torsors and on a relative and arithmetic version of the comb smoothing technique, which we develop in detail, building on work of Kollár and Graber-Harris-Starr.
CommentsMinor changes. 26 pages
