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 $\mathrm{rad}(G)$, 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 $\kappa$ 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 $\mathrm{rad}(G)$ 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, 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.