Comb smoothing and local triviality of homogeneous spaces over a relative curve

Let $R$ be a Henselian local ring, let $\kappa$ 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 $\mathrm{r}\mathrm{a}\mathrm{d}(G)$, and let $E\to C$ be a $G$-torsor. We show that, if either the kernel of the central isogeny $G^{\mathrm{s}\mathrm{c}}{\times }_C\mathrm{r}\mathrm{a}\mathrm{d}(G)\to G$ is étale over $C$ or $\kappa$ is large, the Zariski-local triviality of $E_{\kappa }\to C_{\kappa }$ implies the Zariski-local triviality of $E\to C$. We also prove an averaged form of this result, assuming only that $\mathrm{r}\mathrm{a}\mathrm{d}(G)$ 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, a Henselian version of a theorem of Drinfeld–Simpson, and an injectivity result for the Brauer–Azumaya group of $C$ not covered by earlier work of Colliot-Thélène–Ojanguren–Parimala. 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.